LMFDB - Embedded newform 75.12.a.j.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,12,Mod(1,75)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 12, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("75.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	75.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$57.6257385420$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 10212x^{4} - 7696x^{3} + 23447760x^{2} - 17883600x - 4555384000 $ x^6 - 3x^5 - 10212x^4 - 7696x^3 + 23447760x^2 - 17883600*x - 4555384000
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{4}\cdot 5^{6} $
Twist minimal:	no (minimal twist has level 15)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$87.0120$ of defining polynomial
Character	$\chi$	$=$	75.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+85.0120 q^{2} -243.000 q^{3} +5179.04 q^{4} -20657.9 q^{6} -38732.4 q^{7} +266176. q^{8} +59049.0 q^{9} +O(q^{10})$	\(q+85.0120 q^{2} -243.000 q^{3} +5179.04 q^{4} -20657.9 q^{6} -38732.4 q^{7} +266176. q^{8} +59049.0 q^{9} -978590. q^{11} -1.25851e6 q^{12} -1.02859e6 q^{13} -3.29272e6 q^{14} +1.20215e7 q^{16} -4.83388e6 q^{17} +5.01987e6 q^{18} +6.37171e6 q^{19} +9.41198e6 q^{21} -8.31919e7 q^{22} -1.55756e7 q^{23} -6.46808e7 q^{24} -8.74422e7 q^{26} -1.43489e7 q^{27} -2.00597e8 q^{28} +3.13486e7 q^{29} -7.65267e7 q^{31} +4.76841e8 q^{32} +2.37797e8 q^{33} -4.10938e8 q^{34} +3.05817e8 q^{36} -2.61555e7 q^{37} +5.41672e8 q^{38} +2.49947e8 q^{39} -6.83829e8 q^{41} +8.00131e8 q^{42} +6.31194e8 q^{43} -5.06816e9 q^{44} -1.32411e9 q^{46} -5.74182e8 q^{47} -2.92122e9 q^{48} -4.77125e8 q^{49} +1.17463e9 q^{51} -5.32709e9 q^{52} -5.41573e9 q^{53} -1.21983e9 q^{54} -1.03096e10 q^{56} -1.54833e9 q^{57} +2.66501e9 q^{58} +3.70415e9 q^{59} +8.47598e9 q^{61} -6.50569e9 q^{62} -2.28711e9 q^{63} +1.59173e10 q^{64} +2.02156e10 q^{66} +9.26660e8 q^{67} -2.50348e10 q^{68} +3.78487e9 q^{69} -2.77181e10 q^{71} +1.57174e10 q^{72} -7.87844e9 q^{73} -2.22353e9 q^{74} +3.29994e10 q^{76} +3.79032e10 q^{77} +2.12485e10 q^{78} +1.41828e10 q^{79} +3.48678e9 q^{81} -5.81337e10 q^{82} +5.50909e10 q^{83} +4.87450e10 q^{84} +5.36591e10 q^{86} -7.61772e9 q^{87} -2.60477e11 q^{88} +8.53995e10 q^{89} +3.98397e10 q^{91} -8.06666e10 q^{92} +1.85960e10 q^{93} -4.88123e10 q^{94} -1.15872e11 q^{96} -9.23102e10 q^{97} -4.05614e10 q^{98} -5.77848e10 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 9 q^{2} - 1458 q^{3} + 8157 q^{4} + 2187 q^{6} - 64368 q^{7} + 29277 q^{8} + 354294 q^{9}+O(q^{10}) $ 6 * q - 9 * q^2 - 1458 * q^3 + 8157 * q^4 + 2187 * q^6 - 64368 * q^7 + 29277 * q^8 + 354294 * q^9	\( 6 q - 9 q^{2} - 1458 q^{3} + 8157 q^{4} + 2187 q^{6} - 64368 q^{7} + 29277 q^{8} + 354294 q^{9} - 160776 q^{11} - 1982151 q^{12} - 1777464 q^{13} - 2318022 q^{14} + 14360865 q^{16} - 7789536 q^{17} - 531441 q^{18} + 14306328 q^{19} + 15641424 q^{21} - 99540522 q^{22} - 56578896 q^{23} - 7114311 q^{24} + 156248982 q^{26} - 86093442 q^{27} - 29296674 q^{28} + 229450716 q^{29} + 419037696 q^{31} + 496142397 q^{32} + 39068568 q^{33} + 272636394 q^{34} + 481662693 q^{36} + 441738792 q^{37} - 99095256 q^{38} + 431923752 q^{39} - 1880639244 q^{41} + 563279346 q^{42} - 1081616760 q^{43} - 3162393414 q^{44} - 939739884 q^{46} - 1868657040 q^{47} - 3489690195 q^{48} + 511947750 q^{49} + 1892857248 q^{51} - 9215549766 q^{52} - 10349483256 q^{53} + 129140163 q^{54} - 21606208290 q^{56} - 3476437704 q^{57} - 25240036752 q^{58} - 707714568 q^{59} + 8985508380 q^{61} - 25995031416 q^{62} - 3800866032 q^{63} + 4944042873 q^{64} + 24188346846 q^{66} - 13687306776 q^{67} - 94791079962 q^{68} + 13748671728 q^{69} - 8562874608 q^{71} + 1728777573 q^{72} - 63672451056 q^{73} - 44262217422 q^{74} + 69695921208 q^{76} + 10135567584 q^{77} - 37968502626 q^{78} + 41427333360 q^{79} + 20920706406 q^{81} - 34598023506 q^{82} - 124949795688 q^{83} + 7119091782 q^{84} - 18826696116 q^{86} - 55756523988 q^{87} - 430278178518 q^{88} + 90885005748 q^{89} - 48271166064 q^{91} - 310396941828 q^{92} - 101826160128 q^{93} - 139084157808 q^{94} - 120562602471 q^{96} - 257134073616 q^{97} - 405310493781 q^{98} - 9493662024 q^{99}+O(q^{100}) \) 6 * q - 9 * q^2 - 1458 * q^3 + 8157 * q^4 + 2187 * q^6 - 64368 * q^7 + 29277 * q^8 + 354294 * q^9 - 160776 * q^11 - 1982151 * q^12 - 1777464 * q^13 - 2318022 * q^14 + 14360865 * q^16 - 7789536 * q^17 - 531441 * q^18 + 14306328 * q^19 + 15641424 * q^21 - 99540522 * q^22 - 56578896 * q^23 - 7114311 * q^24 + 156248982 * q^26 - 86093442 * q^27 - 29296674 * q^28 + 229450716 * q^29 + 419037696 * q^31 + 496142397 * q^32 + 39068568 * q^33 + 272636394 * q^34 + 481662693 * q^36 + 441738792 * q^37 - 99095256 * q^38 + 431923752 * q^39 - 1880639244 * q^41 + 563279346 * q^42 - 1081616760 * q^43 - 3162393414 * q^44 - 939739884 * q^46 - 1868657040 * q^47 - 3489690195 * q^48 + 511947750 * q^49 + 1892857248 * q^51 - 9215549766 * q^52 - 10349483256 * q^53 + 129140163 * q^54 - 21606208290 * q^56 - 3476437704 * q^57 - 25240036752 * q^58 - 707714568 * q^59 + 8985508380 * q^61 - 25995031416 * q^62 - 3800866032 * q^63 + 4944042873 * q^64 + 24188346846 * q^66 - 13687306776 * q^67 - 94791079962 * q^68 + 13748671728 * q^69 - 8562874608 * q^71 + 1728777573 * q^72 - 63672451056 * q^73 - 44262217422 * q^74 + 69695921208 * q^76 + 10135567584 * q^77 - 37968502626 * q^78 + 41427333360 * q^79 + 20920706406 * q^81 - 34598023506 * q^82 - 124949795688 * q^83 + 7119091782 * q^84 - 18826696116 * q^86 - 55756523988 * q^87 - 430278178518 * q^88 + 90885005748 * q^89 - 48271166064 * q^91 - 310396941828 * q^92 - 101826160128 * q^93 - 139084157808 * q^94 - 120562602471 * q^96 - 257134073616 * q^97 - 405310493781 * q^98 - 9493662024 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	85.0120	1.87852	0.939259	−	0.343210i	$-0.111514\pi$
$2$	85.0120	1.87852	0.939259	+	0.343210i	$0.111514\pi$
$3$	−243.000	−0.577350
$3$	−243.000	−0.577350
$4$	5179.04	2.52883
$5$	0	0
$5$	0	0
$6$	−20657.9	−1.08456
$7$	−38732.4	−0.871035	−0.435517	−	0.900180i	$-0.643435\pi$
$7$	−38732.4	−0.871035	−0.435517	+	0.900180i	$0.643435\pi$
$8$	266176.	2.87193
$9$	59049.0	0.333333
$10$	0	0
$11$	−978590.	−1.83207	−0.916033	−	0.401103i	$-0.868627\pi$
$11$	−978590.	−1.83207	−0.916033	+	0.401103i	$0.868627\pi$
$12$	−1.25851e6	−1.46002
$13$	−1.02859e6	−0.768339	−0.384169	−	0.923263i	$-0.625512\pi$
$13$	−1.02859e6	−0.768339	−0.384169	+	0.923263i	$0.625512\pi$
$14$	−3.29272e6	−1.63625
$15$	0	0
$16$	1.20215e7	2.86614
$17$	−4.83388e6	−0.825708	−0.412854	−	0.910797i	$-0.635468\pi$
$17$	−4.83388e6	−0.825708	−0.412854	+	0.910797i	$0.635468\pi$
$18$	5.01987e6	0.626173
$19$	6.37171e6	0.590353	0.295176	−	0.955443i	$-0.404622\pi$
$19$	6.37171e6	0.590353	0.295176	+	0.955443i	$0.404622\pi$
$20$	0	0
$21$	9.41198e6	0.502892
$22$	−8.31919e7	−3.44157
$23$	−1.55756e7	−0.504593	−0.252297	−	0.967650i	$-0.581186\pi$
$23$	−1.55756e7	−0.504593	−0.252297	+	0.967650i	$0.581186\pi$
$24$	−6.46808e7	−1.65811
$25$	0	0
$26$	−8.74422e7	−1.44334
$27$	−1.43489e7	−0.192450
$28$	−2.00597e8	−2.20270
$29$	3.13486e7	0.283811	0.141906	−	0.989880i	$-0.454677\pi$
$29$	3.13486e7	0.283811	0.141906	+	0.989880i	$0.454677\pi$
$30$	0	0
$31$	−7.65267e7	−0.480091	−0.240046	−	0.970762i	$-0.577162\pi$
$31$	−7.65267e7	−0.480091	−0.240046	+	0.970762i	$0.577162\pi$
$32$	4.76841e8	2.51217
$33$	2.37797e8	1.05774
$34$	−4.10938e8	−1.55111
$35$	0	0
$36$	3.05817e8	0.842943
$37$	−2.61555e7	−0.0620087	−0.0310044	−	0.999519i	$-0.509871\pi$
$37$	−2.61555e7	−0.0620087	−0.0310044	+	0.999519i	$0.509871\pi$
$38$	5.41672e8	1.10899
$39$	2.49947e8	0.443600
$40$	0	0
$41$	−6.83829e8	−0.921800	−0.460900	−	0.887452i	$-0.652473\pi$
$41$	−6.83829e8	−0.921800	−0.460900	+	0.887452i	$0.652473\pi$
$42$	8.00131e8	0.944692
$43$	6.31194e8	0.654766	0.327383	−	0.944892i	$-0.393833\pi$
$43$	6.31194e8	0.654766	0.327383	+	0.944892i	$0.393833\pi$
$44$	−5.06816e9	−4.63298
$45$	0	0
$46$	−1.32411e9	−0.947887
$47$	−5.74182e8	−0.365184	−0.182592	−	0.983189i	$-0.558449\pi$
$47$	−5.74182e8	−0.365184	−0.182592	+	0.983189i	$0.558449\pi$
$48$	−2.92122e9	−1.65477
$49$	−4.77125e8	−0.241298
$50$	0	0
$51$	1.17463e9	0.476723
$52$	−5.32709e9	−1.94300
$53$	−5.41573e9	−1.77885	−0.889425	−	0.457081i	$-0.848895\pi$
$53$	−5.41573e9	−1.77885	−0.889425	+	0.457081i	$0.848895\pi$
$54$	−1.21983e9	−0.361521
$55$	0	0
$56$	−1.03096e10	−2.50155
$57$	−1.54833e9	−0.340840
$58$	2.66501e9	0.533144
$59$	3.70415e9	0.674532	0.337266	−	0.941409i	$-0.390498\pi$
$59$	3.70415e9	0.674532	0.337266	+	0.941409i	$0.390498\pi$
$60$	0	0
$61$	8.47598e9	1.28492	0.642460	−	0.766319i	$-0.277914\pi$
$61$	8.47598e9	1.28492	0.642460	+	0.766319i	$0.277914\pi$
$62$	−6.50569e9	−0.901860
$63$	−2.28711e9	−0.290345
$64$	1.59173e10	1.85301
$65$	0	0
$66$	2.02156e10	1.98699
$67$	9.26660e8	0.0838512	0.0419256	−	0.999121i	$-0.486651\pi$
$67$	9.26660e8	0.0838512	0.0419256	+	0.999121i	$0.486651\pi$
$68$	−2.50348e10	−2.08807
$69$	3.78487e9	0.291327
$70$	0	0
$71$	−2.77181e10	−1.82323	−0.911617	−	0.411040i	$-0.865166\pi$
$71$	−2.77181e10	−1.82323	−0.911617	+	0.411040i	$0.865166\pi$
$72$	1.57174e10	0.957310
$73$	−7.87844e9	−0.444800	−0.222400	−	0.974956i	$-0.571389\pi$
$73$	−7.87844e9	−0.444800	−0.222400	+	0.974956i	$0.571389\pi$
$74$	−2.22353e9	−0.116485
$75$	0	0
$76$	3.29994e10	1.49290
$77$	3.79032e10	1.59579
$78$	2.12485e10	0.833311
$79$	1.41828e10	0.518578	0.259289	−	0.965800i	$-0.416512\pi$
$79$	1.41828e10	0.518578	0.259289	+	0.965800i	$0.416512\pi$
$80$	0	0
$81$	3.48678e9	0.111111
$82$	−5.81337e10	−1.73162
$83$	5.50909e10	1.53515	0.767574	−	0.640960i	$-0.221464\pi$
$83$	5.50909e10	1.53515	0.767574	+	0.640960i	$0.221464\pi$
$84$	4.87450e10	1.27173
$85$	0	0
$86$	5.36591e10	1.22999
$87$	−7.61772e9	−0.163858
$88$	−2.60477e11	−5.26157
$89$	8.53995e10	1.62110	0.810550	−	0.585669i	$-0.199168\pi$
$89$	8.53995e10	1.62110	0.810550	+	0.585669i	$0.199168\pi$
$90$	0	0
$91$	3.98397e10	0.669250
$92$	−8.06666e10	−1.27603
$93$	1.85960e10	0.277181
$94$	−4.88123e10	−0.686004
$95$	0	0
$96$	−1.15872e11	−1.45040
$97$	−9.23102e10	−1.09145	−0.545727	−	0.837963i	$-0.683746\pi$
$97$	−9.23102e10	−1.09145	−0.545727	+	0.837963i	$0.683746\pi$
$98$	−4.05614e10	−0.453283
$99$	−5.77848e10	−0.610689
$100$	0	0
$101$	6.21666e9	0.0588558	0.0294279	−	0.999567i	$-0.490631\pi$
$101$	6.21666e9	0.0588558	0.0294279	+	0.999567i	$0.490631\pi$
$102$	9.98579e10	0.895532
$103$	4.46918e10	0.379859	0.189930	−	0.981798i	$-0.439174\pi$
$103$	4.46918e10	0.379859	0.189930	+	0.981798i	$0.439174\pi$
$104$	−2.73785e11	−2.20661
$105$	0	0
$106$	−4.60402e11	−3.34160
$107$	1.13216e11	0.780367	0.390183	−	0.920737i	$-0.372412\pi$
$107$	1.13216e11	0.780367	0.390183	+	0.920737i	$0.372412\pi$
$108$	−7.43136e10	−0.486673
$109$	−1.08321e11	−0.674324	−0.337162	−	0.941447i	$-0.609467\pi$
$109$	−1.08321e11	−0.674324	−0.337162	+	0.941447i	$0.609467\pi$
$110$	0	0
$111$	6.35578e9	0.0358008
$112$	−4.65621e11	−2.49651
$113$	−2.41207e11	−1.23157	−0.615783	−	0.787916i	$-0.711160\pi$
$113$	−2.41207e11	−1.23157	−0.615783	+	0.787916i	$0.711160\pi$
$114$	−1.31626e11	−0.640274
$115$	0	0
$116$	1.62356e11	0.717710
$117$	−6.07370e10	−0.256113
$118$	3.14897e11	1.26712
$119$	1.87228e11	0.719221
$120$	0	0
$121$	6.72327e11	2.35647
$122$	7.20560e11	2.41374
$123$	1.66171e11	0.532201
$124$	−3.96335e11	−1.21407
$125$	0	0
$126$	−1.94432e11	−0.545418
$127$	1.57718e11	0.423604	0.211802	−	0.977313i	$-0.432067\pi$
$127$	1.57718e11	0.423604	0.211802	+	0.977313i	$0.432067\pi$
$128$	3.76587e11	0.968746
$129$	−1.53380e11	−0.378029
$130$	0	0
$131$	6.25501e11	1.41656	0.708282	−	0.705930i	$-0.249471\pi$
$131$	6.25501e11	1.41656	0.708282	+	0.705930i	$0.249471\pi$
$132$	1.23156e12	2.67485
$133$	−2.46792e11	−0.514218
$134$	7.87772e10	0.157516
$135$	0	0
$136$	−1.28666e12	−2.37138
$137$	−3.56082e11	−0.630358	−0.315179	−	0.949032i	$-0.602065\pi$
$137$	−3.56082e11	−0.630358	−0.315179	+	0.949032i	$0.602065\pi$
$138$	3.21759e11	0.547263
$139$	6.92439e11	1.13188	0.565940	−	0.824447i	$-0.308514\pi$
$139$	6.92439e11	1.13188	0.565940	+	0.824447i	$0.308514\pi$
$140$	0	0
$141$	1.39526e11	0.210839
$142$	−2.35637e12	−3.42498
$143$	1.00657e12	1.40765
$144$	7.09856e11	0.955381
$145$	0	0
$146$	−6.69762e11	−0.835564
$147$	1.15941e11	0.139314
$148$	−1.35460e11	−0.156809
$149$	−1.90289e10	−0.0212271	−0.0106135	−	0.999944i	$-0.503378\pi$
$149$	−1.90289e10	−0.0212271	−0.0106135	+	0.999944i	$0.503378\pi$
$150$	0	0
$151$	3.93414e11	0.407827	0.203914	−	0.978989i	$-0.434634\pi$
$151$	3.93414e11	0.407827	0.203914	+	0.978989i	$0.434634\pi$
$152$	1.69600e12	1.69545
$153$	−2.85436e11	−0.275236
$154$	3.22223e12	2.99773
$155$	0	0
$156$	1.29448e12	1.12179
$157$	−1.06892e12	−0.894325	−0.447162	−	0.894453i	$-0.647565\pi$
$157$	−1.06892e12	−0.894325	−0.447162	+	0.894453i	$0.647565\pi$
$158$	1.20571e12	0.974158
$159$	1.31602e12	1.02702
$160$	0	0
$161$	6.03281e11	0.439518
$162$	2.96419e11	0.208724
$163$	−2.17586e12	−1.48115	−0.740575	−	0.671973i	$-0.765447\pi$
$163$	−2.17586e12	−1.48115	−0.740575	+	0.671973i	$0.765447\pi$
$164$	−3.54158e12	−2.33107
$165$	0	0
$166$	4.68339e12	2.88380
$167$	−1.15608e12	−0.688727	−0.344363	−	0.938836i	$-0.611905\pi$
$167$	−1.15608e12	−0.688727	−0.344363	+	0.938836i	$0.611905\pi$
$168$	2.50524e12	1.44427
$169$	−7.34169e11	−0.409656
$170$	0	0
$171$	3.76243e11	0.196784
$172$	3.26898e12	1.65579
$173$	6.65869e11	0.326689	0.163345	−	0.986569i	$-0.447772\pi$
$173$	6.65869e11	0.326689	0.163345	+	0.986569i	$0.447772\pi$
$174$	−6.47597e11	−0.307811
$175$	0	0
$176$	−1.17641e13	−5.25096
$177$	−9.00109e11	−0.389442
$178$	7.25998e12	3.04527
$179$	2.79176e12	1.13550	0.567750	−	0.823201i	$-0.307814\pi$
$179$	2.79176e12	1.13550	0.567750	+	0.823201i	$0.307814\pi$
$180$	0	0
$181$	−1.09883e11	−0.0420433	−0.0210217	−	0.999779i	$-0.506692\pi$
$181$	−1.09883e11	−0.0420433	−0.0210217	+	0.999779i	$0.506692\pi$
$182$	3.38685e12	1.25720
$183$	−2.05966e12	−0.741849
$184$	−4.14585e12	−1.44916
$185$	0	0
$186$	1.58088e12	0.520689
$187$	4.73039e12	1.51275
$188$	−2.97371e12	−0.923486
$189$	5.55768e11	0.167631
$190$	0	0
$191$	4.26641e12	1.21445	0.607224	−	0.794531i	$-0.292283\pi$
$191$	4.26641e12	1.21445	0.607224	+	0.794531i	$0.292283\pi$
$192$	−3.86789e12	−1.06984
$193$	3.30742e11	0.0889046	0.0444523	−	0.999012i	$-0.485846\pi$
$193$	3.30742e11	0.0889046	0.0444523	+	0.999012i	$0.485846\pi$
$194$	−7.84748e12	−2.05032
$195$	0	0
$196$	−2.47105e12	−0.610202
$197$	4.93597e11	0.118525	0.0592623	−	0.998242i	$-0.481125\pi$
$197$	4.93597e11	0.118525	0.0592623	+	0.998242i	$0.481125\pi$
$198$	−4.91240e12	−1.14719
$199$	−2.00982e12	−0.456525	−0.228263	−	0.973600i	$-0.573304\pi$
$199$	−2.00982e12	−0.456525	−0.228263	+	0.973600i	$0.573304\pi$
$200$	0	0
$201$	−2.25178e11	−0.0484115
$202$	5.28490e11	0.110562
$203$	−1.21421e12	−0.247209
$204$	6.08347e12	1.20555
$205$	0	0
$206$	3.79934e12	0.713573
$207$	−9.19723e11	−0.168198
$208$	−1.23651e13	−2.20217
$209$	−6.23530e12	−1.08157
$210$	0	0
$211$	2.17542e12	0.358087	0.179044	−	0.983841i	$-0.442700\pi$
$211$	2.17542e12	0.358087	0.179044	+	0.983841i	$0.442700\pi$
$212$	−2.80483e13	−4.49841
$213$	6.73550e12	1.05264
$214$	9.62476e12	1.46593
$215$	0	0
$216$	−3.81933e12	−0.552703
$217$	2.96407e12	0.418176
$218$	−9.20862e12	−1.26673
$219$	1.91446e12	0.256805
$220$	0	0
$221$	4.97206e12	0.634424
$222$	5.40317e11	0.0672524
$223$	−4.65701e12	−0.565497	−0.282748	−	0.959194i	$-0.591246\pi$
$223$	−4.65701e12	−0.565497	−0.282748	+	0.959194i	$0.591246\pi$
$224$	−1.84692e13	−2.18819
$225$	0	0
$226$	−2.05055e13	−2.31352
$227$	−8.07743e12	−0.889470	−0.444735	−	0.895662i	$-0.646702\pi$
$227$	−8.07743e12	−0.889470	−0.444735	+	0.895662i	$0.646702\pi$
$228$	−8.01885e12	−0.861926
$229$	−4.40239e12	−0.461948	−0.230974	−	0.972960i	$-0.574191\pi$
$229$	−4.40239e12	−0.461948	−0.230974	+	0.972960i	$0.574191\pi$
$230$	0	0
$231$	−9.21047e12	−0.921332
$232$	8.34425e12	0.815086
$233$	6.58150e12	0.627867	0.313933	−	0.949445i	$-0.398353\pi$
$233$	6.58150e12	0.627867	0.313933	+	0.949445i	$0.398353\pi$
$234$	−5.16338e12	−0.481112
$235$	0	0
$236$	1.91840e13	1.70578
$237$	−3.44643e12	−0.299401
$238$	1.59166e13	1.35107
$239$	2.19078e13	1.81723	0.908616	−	0.417633i	$-0.137140\pi$
$239$	2.19078e13	1.81723	0.908616	+	0.417633i	$0.137140\pi$
$240$	0	0
$241$	2.21030e13	1.75129	0.875643	−	0.482959i	$-0.160438\pi$
$241$	2.21030e13	1.75129	0.875643	+	0.482959i	$0.160438\pi$
$242$	5.71559e13	4.42666
$243$	−8.47289e11	−0.0641500
$244$	4.38974e13	3.24934
$245$	0	0
$246$	1.41265e13	0.999750
$247$	−6.55386e12	−0.453591
$248$	−2.03696e13	−1.37879
$249$	−1.33871e13	−0.886319
$250$	0	0
$251$	−1.52439e13	−0.965809	−0.482905	−	0.875673i	$-0.660418\pi$
$251$	−1.52439e13	−0.965809	−0.482905	+	0.875673i	$0.660418\pi$
$252$	−1.18450e13	−0.734232
$253$	1.52421e13	0.924448
$254$	1.34079e13	0.795747
$255$	0	0
$256$	−5.84161e11	−0.0332057
$257$	−2.75696e13	−1.53390	−0.766952	−	0.641705i	$-0.778227\pi$
$257$	−2.75696e13	−1.53390	−0.766952	+	0.641705i	$0.778227\pi$
$258$	−1.30392e13	−0.710135
$259$	1.01306e12	0.0540118
$260$	0	0
$261$	1.85111e12	0.0946037
$262$	5.31751e13	2.66104
$263$	−2.59591e13	−1.27213	−0.636067	−	0.771634i	$-0.719440\pi$
$263$	−2.59591e13	−1.27213	−0.636067	+	0.771634i	$0.719440\pi$
$264$	6.32960e13	3.03777
$265$	0	0
$266$	−2.09803e13	−0.965967
$267$	−2.07521e13	−0.935943
$268$	4.79921e12	0.212045
$269$	−1.82205e13	−0.788722	−0.394361	−	0.918956i	$-0.629034\pi$
$269$	−1.82205e13	−0.788722	−0.394361	+	0.918956i	$0.629034\pi$
$270$	0	0
$271$	−2.21132e13	−0.919010	−0.459505	−	0.888175i	$-0.651973\pi$
$271$	−2.21132e13	−0.919010	−0.459505	+	0.888175i	$0.651973\pi$
$272$	−5.81104e13	−2.36660
$273$	−9.68104e12	−0.386391
$274$	−3.02713e13	−1.18414
$275$	0	0
$276$	1.96020e13	0.736716
$277$	−2.30014e13	−0.847454	−0.423727	−	0.905790i	$-0.639278\pi$
$277$	−2.30014e13	−0.847454	−0.423727	+	0.905790i	$0.639278\pi$
$278$	5.88656e13	2.12625
$279$	−4.51883e12	−0.160030
$280$	0	0
$281$	−3.89798e13	−1.32726	−0.663628	−	0.748063i	$-0.730984\pi$
$281$	−3.89798e13	−1.32726	−0.663628	+	0.748063i	$0.730984\pi$
$282$	1.18614e13	0.396064
$283$	7.83689e11	0.0256636	0.0128318	−	0.999918i	$-0.495915\pi$
$283$	7.83689e11	0.0256636	0.0128318	+	0.999918i	$0.495915\pi$
$284$	−1.43553e14	−4.61065
$285$	0	0
$286$	8.55701e13	2.64429
$287$	2.64864e13	0.802920
$288$	2.81570e13	0.837390
$289$	−1.09055e13	−0.318206
$290$	0	0
$291$	2.24314e13	0.630151
$292$	−4.08028e13	−1.12482
$293$	−4.44939e13	−1.20373	−0.601864	−	0.798599i	$-0.705575\pi$
$293$	−4.44939e13	−1.20373	−0.601864	+	0.798599i	$0.705575\pi$
$294$	9.85642e12	0.261703
$295$	0	0
$296$	−6.96196e12	−0.178085
$297$	1.40417e13	0.352581
$298$	−1.61769e12	−0.0398754
$299$	1.60209e13	0.387698
$300$	0	0
$301$	−2.44477e13	−0.570324
$302$	3.34449e13	0.766111
$303$	−1.51065e12	−0.0339804
$304$	7.65974e13	1.69204
$305$	0	0
$306$	−2.42655e13	−0.517036
$307$	2.14907e13	0.449769	0.224884	−	0.974385i	$-0.427800\pi$
$307$	2.14907e13	0.449769	0.224884	+	0.974385i	$0.427800\pi$
$308$	1.96302e14	4.03549
$309$	−1.08601e13	−0.219312
$310$	0	0
$311$	−5.50134e13	−1.07223	−0.536113	−	0.844146i	$-0.680108\pi$
$311$	−5.50134e13	−1.07223	−0.536113	+	0.844146i	$0.680108\pi$
$312$	6.65298e13	1.27399
$313$	−6.96518e11	−0.0131050	−0.00655252	−	0.999979i	$-0.502086\pi$
$313$	−6.96518e11	−0.0131050	−0.00655252	+	0.999979i	$0.502086\pi$
$314$	−9.08706e13	−1.68000
$315$	0	0
$316$	7.34535e13	1.31139
$317$	−3.10336e13	−0.544511	−0.272256	−	0.962225i	$-0.587770\pi$
$317$	−3.10336e13	−0.544511	−0.272256	+	0.962225i	$0.587770\pi$
$318$	1.11878e14	1.92927
$319$	−3.06775e13	−0.519961
$320$	0	0
$321$	−2.75116e13	−0.450545
$322$	5.12861e13	0.825643
$323$	−3.08001e13	−0.487459
$324$	1.80582e13	0.280981
$325$	0	0
$326$	−1.84974e14	−2.78237
$327$	2.63221e13	0.389321
$328$	−1.82019e14	−2.64734
$329$	2.22395e13	0.318088
$330$	0	0
$331$	−1.22335e14	−1.69238	−0.846189	−	0.532883i	$-0.821109\pi$
$331$	−1.22335e14	−1.69238	−0.846189	+	0.532883i	$0.821109\pi$
$332$	2.85318e14	3.88213
$333$	−1.54445e12	−0.0206696
$334$	−9.82806e13	−1.29379
$335$	0	0
$336$	1.13146e14	1.44136
$337$	−1.13534e14	−1.42286	−0.711428	−	0.702759i	$-0.751951\pi$
$337$	−1.13534e14	−1.42286	−0.711428	+	0.702759i	$0.751951\pi$
$338$	−6.24132e13	−0.769546
$339$	5.86132e13	0.711045
$340$	0	0
$341$	7.48883e13	0.879559
$342$	3.19852e13	0.369663
$343$	9.50669e13	1.08121
$344$	1.68009e14	1.88044
$345$	0	0
$346$	5.66068e13	0.613692
$347$	−9.94275e13	−1.06095	−0.530474	−	0.847701i	$-0.677986\pi$
$347$	−9.94275e13	−1.06095	−0.530474	+	0.847701i	$0.677986\pi$
$348$	−3.94525e13	−0.414370
$349$	1.20158e14	1.24226	0.621132	−	0.783706i	$-0.286673\pi$
$349$	1.20158e14	1.24226	0.621132	+	0.783706i	$0.286673\pi$
$350$	0	0
$351$	1.47591e13	0.147867
$352$	−4.66632e14	−4.60246
$353$	6.37335e12	0.0618881	0.0309440	−	0.999521i	$-0.490149\pi$
$353$	6.37335e12	0.0618881	0.0309440	+	0.999521i	$0.490149\pi$
$354$	−7.65201e13	−0.731573
$355$	0	0
$356$	4.42287e14	4.09949
$357$	−4.54964e13	−0.415242
$358$	2.37334e14	2.13306
$359$	−3.63893e12	−0.0322073	−0.0161036	−	0.999870i	$-0.505126\pi$
$359$	−3.63893e12	−0.0322073	−0.0161036	+	0.999870i	$0.505126\pi$
$360$	0	0
$361$	−7.58915e13	−0.651484
$362$	−9.34134e12	−0.0789791
$363$	−1.63376e14	−1.36051
$364$	2.06331e14	1.69242
$365$	0	0
$366$	−1.75096e14	−1.39358
$367$	−8.60250e13	−0.674468	−0.337234	−	0.941421i	$-0.609491\pi$
$367$	−8.60250e13	−0.674468	−0.337234	+	0.941421i	$0.609491\pi$
$368$	−1.87242e14	−1.44624
$369$	−4.03794e13	−0.307267
$370$	0	0
$371$	2.09764e14	1.54944
$372$	9.63094e13	0.700943
$373$	6.78611e12	0.0486657	0.0243328	−	0.999704i	$-0.492254\pi$
$373$	6.78611e12	0.0486657	0.0243328	+	0.999704i	$0.492254\pi$
$374$	4.02140e14	2.84173
$375$	0	0
$376$	−1.52833e14	−1.04878
$377$	−3.22448e13	−0.218063
$378$	4.72470e13	0.314897
$379$	−1.95850e14	−1.28650	−0.643248	−	0.765658i	$-0.722414\pi$
$379$	−1.95850e14	−1.28650	−0.643248	+	0.765658i	$0.722414\pi$
$380$	0	0
$381$	−3.83254e13	−0.244568
$382$	3.62696e14	2.28136
$383$	9.79877e13	0.607545	0.303772	−	0.952745i	$-0.401754\pi$
$383$	9.79877e13	0.607545	0.303772	+	0.952745i	$0.401754\pi$
$384$	−9.15106e13	−0.559306
$385$	0	0
$386$	2.81170e13	0.167009
$387$	3.72714e13	0.218255
$388$	−4.78078e14	−2.76010
$389$	−3.44461e14	−1.96073	−0.980363	−	0.197203i	$-0.936814\pi$
$389$	−3.44461e14	−1.96073	−0.980363	+	0.197203i	$0.936814\pi$
$390$	0	0
$391$	7.52905e13	0.416647
$392$	−1.26999e14	−0.692992
$393$	−1.51997e14	−0.817853
$394$	4.19617e13	0.222650
$395$	0	0
$396$	−2.99270e14	−1.54433
$397$	8.54847e13	0.435052	0.217526	−	0.976055i	$-0.430201\pi$
$397$	8.54847e13	0.435052	0.217526	+	0.976055i	$0.430201\pi$
$398$	−1.70859e14	−0.857591
$399$	5.99705e13	0.296884
$400$	0	0
$401$	−8.15227e13	−0.392631	−0.196315	−	0.980541i	$-0.562898\pi$
$401$	−8.15227e13	−0.392631	−0.196315	+	0.980541i	$0.562898\pi$
$402$	−1.91429e13	−0.0909418
$403$	7.87144e13	0.368873
$404$	3.21963e13	0.148836
$405$	0	0
$406$	−1.03222e14	−0.464387
$407$	2.55955e13	0.113604
$408$	3.12659e14	1.36912
$409$	2.85262e14	1.23244	0.616221	−	0.787574i	$-0.288663\pi$
$409$	2.85262e14	1.23244	0.616221	+	0.787574i	$0.288663\pi$
$410$	0	0
$411$	8.65280e13	0.363937
$412$	2.31461e14	0.960599
$413$	−1.43471e14	−0.587541
$414$	−7.81875e13	−0.315962
$415$	0	0
$416$	−4.90473e14	−1.93020
$417$	−1.68263e14	−0.653491
$418$	−5.30075e14	−2.03174
$419$	−1.07931e14	−0.408289	−0.204145	−	0.978941i	$-0.565441\pi$
$419$	−1.07931e14	−0.408289	−0.204145	+	0.978941i	$0.565441\pi$
$420$	0	0
$421$	2.77891e14	1.02406	0.512028	−	0.858969i	$-0.328895\pi$
$421$	2.77891e14	1.02406	0.512028	+	0.858969i	$0.328895\pi$
$422$	1.84936e14	0.672673
$423$	−3.39049e13	−0.121728
$424$	−1.44154e15	−5.10873
$425$	0	0
$426$	5.72598e14	1.97741
$427$	−3.28295e14	−1.11921
$428$	5.86352e14	1.97341
$429$	−2.44595e14	−0.812705
$430$	0	0
$431$	5.91140e13	0.191454	0.0957272	−	0.995408i	$-0.469482\pi$
$431$	5.91140e13	0.191454	0.0957272	+	0.995408i	$0.469482\pi$
$432$	−1.72495e14	−0.551590
$433$	6.07883e14	1.91927	0.959637	−	0.281242i	$-0.0907464\pi$
$433$	6.07883e14	1.91927	0.959637	+	0.281242i	$0.0907464\pi$
$434$	2.51981e14	0.785552
$435$	0	0
$436$	−5.61001e14	−1.70525
$437$	−9.92432e13	−0.297888
$438$	1.62752e14	0.482413
$439$	2.45207e14	0.717757	0.358878	−	0.933384i	$-0.383159\pi$
$439$	2.45207e14	0.717757	0.358878	+	0.933384i	$0.383159\pi$
$440$	0	0
$441$	−2.81738e13	−0.0804327
$442$	4.22685e14	1.19178
$443$	−3.88769e14	−1.08261	−0.541303	−	0.840827i	$-0.682069\pi$
$443$	−3.88769e14	−1.08261	−0.541303	+	0.840827i	$0.682069\pi$
$444$	3.29168e13	0.0905340
$445$	0	0
$446$	−3.95901e14	−1.06230
$447$	4.62403e12	0.0122554
$448$	−6.16514e14	−1.61404
$449$	5.26454e14	1.36146	0.680731	−	0.732534i	$-0.261662\pi$
$449$	5.26454e14	1.36146	0.680731	+	0.732534i	$0.261662\pi$
$450$	0	0
$451$	6.69189e14	1.68880
$452$	−1.24922e15	−3.11442
$453$	−9.55995e13	−0.235459
$454$	−6.86679e14	−1.67088
$455$	0	0
$456$	−4.12127e14	−0.978869
$457$	5.40988e14	1.26955	0.634773	−	0.772699i	$-0.281094\pi$
$457$	5.40988e14	1.26955	0.634773	+	0.772699i	$0.281094\pi$
$458$	−3.74256e14	−0.867777
$459$	6.93609e13	0.158908
$460$	0	0
$461$	−7.21391e14	−1.61367	−0.806837	−	0.590775i	$-0.798822\pi$
$461$	−7.21391e14	−1.61367	−0.806837	+	0.590775i	$0.798822\pi$
$462$	−7.83001e14	−1.73074
$463$	6.41008e14	1.40013	0.700064	−	0.714080i	$-0.253155\pi$
$463$	6.41008e14	1.40013	0.700064	+	0.714080i	$0.253155\pi$
$464$	3.76857e14	0.813444
$465$	0	0
$466$	5.59507e14	1.17946
$467$	−8.27804e14	−1.72458	−0.862292	−	0.506411i	$-0.830972\pi$
$467$	−8.27804e14	−1.72458	−0.862292	+	0.506411i	$0.830972\pi$
$468$	−3.14560e14	−0.647665
$469$	−3.58918e13	−0.0730373
$470$	0	0
$471$	2.59746e14	0.516339
$472$	9.85957e14	1.93721
$473$	−6.17680e14	−1.19958
$474$	−2.92988e14	−0.562430
$475$	0	0
$476$	9.69661e14	1.81879
$477$	−3.19793e14	−0.592950
$478$	1.86243e15	3.41370
$479$	2.18669e14	0.396226	0.198113	−	0.980179i	$-0.436519\pi$
$479$	2.18669e14	0.396226	0.198113	+	0.980179i	$0.436519\pi$
$480$	0	0
$481$	2.69032e13	0.0476437
$482$	1.87902e15	3.28982
$483$	−1.46597e14	−0.253756
$484$	3.48201e15	5.95910
$485$	0	0
$486$	−7.20297e13	−0.120507
$487$	9.72882e14	1.60935	0.804675	−	0.593715i	$-0.202339\pi$
$487$	9.72882e14	1.60935	0.804675	+	0.593715i	$0.202339\pi$
$488$	2.25610e15	3.69020
$489$	5.28734e14	0.855143
$490$	0	0
$491$	5.52095e14	0.873104	0.436552	−	0.899679i	$-0.356200\pi$
$491$	5.52095e14	0.873104	0.436552	+	0.899679i	$0.356200\pi$
$492$	8.60604e14	1.34585
$493$	−1.51535e14	−0.234345
$494$	−5.57157e14	−0.852078
$495$	0	0
$496$	−9.19964e14	−1.37601
$497$	1.07359e15	1.58810
$498$	−1.13806e15	−1.66496
$499$	−2.40097e13	−0.0347403	−0.0173702	−	0.999849i	$-0.505529\pi$
$499$	−2.40097e13	−0.0347403	−0.0173702	+	0.999849i	$0.505529\pi$
$500$	0	0
$501$	2.80927e14	0.397637
$502$	−1.29592e15	−1.81429
$503$	−6.55110e14	−0.907173	−0.453587	−	0.891212i	$-0.649856\pi$
$503$	−6.55110e14	−0.907173	−0.453587	+	0.891212i	$0.649856\pi$
$504$	−6.08774e14	−0.833851
$505$	0	0
$506$	1.29576e15	1.73659
$507$	1.78403e14	0.236515
$508$	8.16826e14	1.07122
$509$	1.38537e14	0.179729	0.0898646	−	0.995954i	$-0.471357\pi$
$509$	1.38537e14	0.179729	0.0898646	+	0.995954i	$0.471357\pi$
$510$	0	0
$511$	3.05151e14	0.387436
$512$	−8.20910e14	−1.03112
$513$	−9.14271e13	−0.113613
$514$	−2.34375e15	−2.88146
$515$	0	0
$516$	−7.94362e14	−0.955972
$517$	5.61889e14	0.669040
$518$	8.61227e13	0.101462
$519$	−1.61806e14	−0.188614
$520$	0	0
$521$	2.67933e14	0.305787	0.152893	−	0.988243i	$-0.451141\pi$
$521$	2.67933e14	0.305787	0.152893	+	0.988243i	$0.451141\pi$
$522$	1.57366e14	0.177715
$523$	3.44295e14	0.384744	0.192372	−	0.981322i	$-0.438382\pi$
$523$	3.44295e14	0.384744	0.192372	+	0.981322i	$0.438382\pi$
$524$	3.23950e15	3.58225
$525$	0	0
$526$	−2.20683e15	−2.38973
$527$	3.69921e14	0.396415
$528$	2.85868e15	3.03165
$529$	−7.10211e14	−0.745386
$530$	0	0
$531$	2.18727e14	0.224844
$532$	−1.27815e15	−1.30037
$533$	7.03378e14	0.708254
$534$	−1.76417e15	−1.75819
$535$	0	0
$536$	2.46655e14	0.240815
$537$	−6.78399e14	−0.655581
$538$	−1.54896e15	−1.48163
$539$	4.66910e14	0.442074
$540$	0	0
$541$	−1.05651e15	−0.980144	−0.490072	−	0.871682i	$-0.663030\pi$
$541$	−1.05651e15	−0.980144	−0.490072	+	0.871682i	$0.663030\pi$
$542$	−1.87989e15	−1.72638
$543$	2.67015e13	0.0242737
$544$	−2.30499e15	−2.07432
$545$	0	0
$546$	−8.23005e14	−0.725843
$547$	−6.74135e14	−0.588595	−0.294297	−	0.955714i	$-0.595086\pi$
$547$	−6.74135e14	−0.588595	−0.294297	+	0.955714i	$0.595086\pi$
$548$	−1.84416e15	−1.59407
$549$	5.00498e14	0.428306
$550$	0	0
$551$	1.99744e14	0.167549
$552$	1.00744e15	0.836671
$553$	−5.49336e14	−0.451699
$554$	−1.95540e15	−1.59196
$555$	0	0
$556$	3.58617e15	2.86233
$557$	−6.26291e14	−0.494963	−0.247481	−	0.968893i	$-0.579603\pi$
$557$	−6.26291e14	−0.494963	−0.247481	+	0.968893i	$0.579603\pi$
$558$	−3.84155e14	−0.300620
$559$	−6.49238e14	−0.503082
$560$	0	0
$561$	−1.14948e15	−0.873388
$562$	−3.31375e15	−2.49327
$563$	6.52805e14	0.486393	0.243196	−	0.969977i	$-0.421804\pi$
$563$	6.52805e14	0.486393	0.243196	+	0.969977i	$0.421804\pi$
$564$	7.22612e14	0.533175
$565$	0	0
$566$	6.66229e13	0.0482096
$567$	−1.35052e14	−0.0967817
$568$	−7.37789e15	−5.23620
$569$	−1.02763e15	−0.722300	−0.361150	−	0.932508i	$-0.617616\pi$
$569$	−1.02763e15	−0.722300	−0.361150	+	0.932508i	$0.617616\pi$
$570$	0	0
$571$	−1.03969e15	−0.716814	−0.358407	−	0.933565i	$-0.616680\pi$
$571$	−1.03969e15	−0.716814	−0.358407	+	0.933565i	$0.616680\pi$
$572$	5.21304e15	3.55970
$573$	−1.03674e15	−0.701162
$574$	2.25166e15	1.50830
$575$	0	0
$576$	9.39898e14	0.617671
$577$	8.28819e14	0.539502	0.269751	−	0.962930i	$-0.413059\pi$
$577$	8.28819e14	0.539502	0.269751	+	0.962930i	$0.413059\pi$
$578$	−9.27100e14	−0.597755
$579$	−8.03703e13	−0.0513291
$580$	0	0
$581$	−2.13381e15	−1.33717
$582$	1.90694e15	1.18375
$583$	5.29978e15	3.25897
$584$	−2.09705e15	−1.27743
$585$	0	0
$586$	−3.78251e15	−2.26122
$587$	−1.61993e15	−0.959370	−0.479685	−	0.877441i	$-0.659249\pi$
$587$	−1.61993e15	−0.959370	−0.479685	+	0.877441i	$0.659249\pi$
$588$	6.00466e14	0.352300
$589$	−4.87607e14	−0.283423
$590$	0	0
$591$	−1.19944e14	−0.0684302
$592$	−3.14427e14	−0.177726
$593$	7.31876e14	0.409861	0.204930	−	0.978777i	$-0.434303\pi$
$593$	7.31876e14	0.409861	0.204930	+	0.978777i	$0.434303\pi$
$594$	1.19371e15	0.662330
$595$	0	0
$596$	−9.85516e13	−0.0536796
$597$	4.88386e14	0.263575
$598$	1.36196e15	0.728298
$599$	−2.87751e15	−1.52465	−0.762324	−	0.647196i	$-0.775941\pi$
$599$	−2.87751e15	−1.52465	−0.762324	+	0.647196i	$0.775941\pi$
$600$	0	0
$601$	9.01633e14	0.469051	0.234526	−	0.972110i	$-0.424646\pi$
$601$	9.01633e14	0.469051	0.234526	+	0.972110i	$0.424646\pi$
$602$	−2.07835e15	−1.07136
$603$	5.47184e13	0.0279504
$604$	2.03751e15	1.03133
$605$	0	0
$606$	−1.28423e14	−0.0638328
$607$	3.64912e15	1.79742	0.898712	−	0.438540i	$-0.144504\pi$
$607$	3.64912e15	1.79742	0.898712	+	0.438540i	$0.144504\pi$
$608$	3.03830e15	1.48307
$609$	2.95053e14	0.142726
$610$	0	0
$611$	5.90596e14	0.280585
$612$	−1.47828e15	−0.696025
$613$	−1.41357e15	−0.659607	−0.329803	−	0.944050i	$-0.606982\pi$
$613$	−1.41357e15	−0.659607	−0.329803	+	0.944050i	$0.606982\pi$
$614$	1.82697e15	0.844898
$615$	0	0
$616$	1.00889e16	4.58301
$617$	−9.09097e13	−0.0409300	−0.0204650	−	0.999791i	$-0.506515\pi$
$617$	−9.09097e13	−0.0409300	−0.0204650	+	0.999791i	$0.506515\pi$
$618$	−9.23239e14	−0.411981
$619$	3.83817e15	1.69756	0.848782	−	0.528744i	$-0.177337\pi$
$619$	3.83817e15	1.69756	0.848782	+	0.528744i	$0.177337\pi$
$620$	0	0
$621$	2.23493e14	0.0971090
$622$	−4.67680e15	−2.01420
$623$	−3.30773e15	−1.41204
$624$	3.00473e15	1.27142
$625$	0	0
$626$	−5.92124e13	−0.0246181
$627$	1.51518e15	0.624442
$628$	−5.53596e15	−2.26159
$629$	1.26432e14	0.0512011
$630$	0	0
$631$	−7.54761e14	−0.300364	−0.150182	−	0.988658i	$-0.547986\pi$
$631$	−7.54761e14	−0.300364	−0.150182	+	0.988658i	$0.547986\pi$
$632$	3.77513e15	1.48932
$633$	−5.28626e14	−0.206742
$634$	−2.63823e15	−1.02287
$635$	0	0
$636$	6.81573e15	2.59716
$637$	4.90765e14	0.185399
$638$	−2.60795e15	−0.976756
$639$	−1.63673e15	−0.607745
$640$	0	0
$641$	−2.51894e15	−0.919387	−0.459693	−	0.888078i	$-0.652041\pi$
$641$	−2.51894e15	−0.919387	−0.459693	+	0.888078i	$0.652041\pi$
$642$	−2.33882e15	−0.846356
$643$	6.31572e14	0.226601	0.113301	−	0.993561i	$-0.463858\pi$
$643$	6.31572e14	0.226601	0.113301	+	0.993561i	$0.463858\pi$
$644$	3.12441e15	1.11147
$645$	0	0
$646$	−2.61838e15	−0.915700
$647$	4.73004e15	1.64018	0.820089	−	0.572236i	$-0.193924\pi$
$647$	4.73004e15	1.64018	0.820089	+	0.572236i	$0.193924\pi$
$648$	9.28098e14	0.319103
$649$	−3.62485e15	−1.23579
$650$	0	0
$651$	−7.20268e14	−0.241434
$652$	−1.12689e16	−3.74558
$653$	2.27574e15	0.750066	0.375033	−	0.927012i	$-0.377631\pi$
$653$	2.27574e15	0.750066	0.375033	+	0.927012i	$0.377631\pi$
$654$	2.23770e15	0.731347
$655$	0	0
$656$	−8.22064e15	−2.64201
$657$	−4.65214e14	−0.148267
$658$	1.89062e15	0.597533
$659$	−2.00034e15	−0.626951	−0.313475	−	0.949596i	$-0.601493\pi$
$659$	−2.00034e15	−0.626951	−0.313475	+	0.949596i	$0.601493\pi$
$660$	0	0
$661$	1.34866e15	0.415714	0.207857	−	0.978159i	$-0.433351\pi$
$661$	1.34866e15	0.415714	0.207857	+	0.978159i	$0.433351\pi$
$662$	−1.04000e16	−3.17916
$663$	−1.20821e15	−0.366285
$664$	1.46639e16	4.40884
$665$	0	0
$666$	−1.31297e14	−0.0388282
$667$	−4.88273e14	−0.143209
$668$	−5.98738e15	−1.74167
$669$	1.13165e15	0.326490
$670$	0	0
$671$	−8.29451e15	−2.35406
$672$	4.48802e15	1.26335
$673$	−6.07049e14	−0.169489	−0.0847443	−	0.996403i	$-0.527007\pi$
$673$	−6.07049e14	−0.169489	−0.0847443	+	0.996403i	$0.527007\pi$
$674$	−9.65175e15	−2.67286
$675$	0	0
$676$	−3.80229e15	−1.03595
$677$	3.49109e15	0.943459	0.471730	−	0.881743i	$-0.343630\pi$
$677$	3.49109e15	0.943459	0.471730	+	0.881743i	$0.343630\pi$
$678$	4.98283e15	1.33571
$679$	3.57540e15	0.950694
$680$	0	0
$681$	1.96282e15	0.513535
$682$	6.36641e15	1.65227
$683$	1.41897e15	0.365309	0.182655	−	0.983177i	$-0.441531\pi$
$683$	1.41897e15	0.365309	0.182655	+	0.983177i	$0.441531\pi$
$684$	1.94858e15	0.497633
$685$	0	0
$686$	8.08183e15	2.03108
$687$	1.06978e15	0.266706
$688$	7.58788e15	1.87665
$689$	5.57055e15	1.36676
$690$	0	0
$691$	1.20019e15	0.289816	0.144908	−	0.989445i	$-0.453711\pi$
$691$	1.20019e15	0.289816	0.144908	+	0.989445i	$0.453711\pi$
$692$	3.44856e15	0.826141
$693$	2.23815e15	0.531931
$694$	−8.45253e15	−1.99301
$695$	0	0
$696$	−2.02765e15	−0.470590
$697$	3.30555e15	0.761138
$698$	1.02149e16	2.33361
$699$	−1.59931e15	−0.362499
$700$	0	0
$701$	5.86487e15	1.30861	0.654304	−	0.756232i	$-0.272962\pi$
$701$	5.86487e15	1.30861	0.654304	+	0.756232i	$0.272962\pi$
$702$	1.25470e15	0.277770
$703$	−1.66655e14	−0.0366070
$704$	−1.55765e16	−3.39484
$705$	0	0
$706$	5.41811e14	0.116258
$707$	−2.40786e14	−0.0512655
$708$	−4.66170e15	−0.984831
$709$	−6.43443e15	−1.34882	−0.674412	−	0.738355i	$-0.735603\pi$
$709$	−6.43443e15	−1.34882	−0.674412	+	0.738355i	$0.735603\pi$
$710$	0	0
$711$	8.37482e14	0.172859
$712$	2.27313e16	4.65569
$713$	1.19195e15	0.242251
$714$	−3.86774e15	−0.780040
$715$	0	0
$716$	1.44587e16	2.87148
$717$	−5.32359e15	−1.04918
$718$	−3.09352e14	−0.0605019
$719$	4.32118e15	0.838674	0.419337	−	0.907831i	$-0.362263\pi$
$719$	4.32118e15	0.838674	0.419337	+	0.907831i	$0.362263\pi$
$720$	0	0
$721$	−1.73102e15	−0.330871
$722$	−6.45169e15	−1.22382
$723$	−5.37102e15	−1.01111
$724$	−5.69086e14	−0.106320
$725$	0	0
$726$	−1.38889e16	−2.55574
$727$	−7.82636e15	−1.42929	−0.714646	−	0.699487i	$-0.753412\pi$
$727$	−7.82636e15	−1.42929	−0.714646	+	0.699487i	$0.753412\pi$
$728$	1.06044e16	1.92204
$729$	2.05891e14	0.0370370
$730$	0	0
$731$	−3.05111e15	−0.540646
$732$	−1.06671e16	−1.87601
$733$	3.85053e15	0.672122	0.336061	−	0.941840i	$-0.390905\pi$
$733$	3.85053e15	0.672122	0.336061	+	0.941840i	$0.390905\pi$
$734$	−7.31316e15	−1.26700
$735$	0	0
$736$	−7.42709e15	−1.26762
$737$	−9.06821e14	−0.153621
$738$	−3.43274e15	−0.577206
$739$	1.89136e15	0.315667	0.157833	−	0.987466i	$-0.449549\pi$
$739$	1.89136e15	0.315667	0.157833	+	0.987466i	$0.449549\pi$
$740$	0	0
$741$	1.59259e15	0.261881
$742$	1.78325e16	2.91065
$743$	−6.95777e15	−1.12728	−0.563640	−	0.826021i	$-0.690599\pi$
$743$	−6.95777e15	−1.12728	−0.563640	+	0.826021i	$0.690599\pi$
$744$	4.94981e15	0.796044
$745$	0	0
$746$	5.76901e14	0.0914193
$747$	3.25306e15	0.511716
$748$	2.44989e16	3.82549
$749$	−4.38515e15	−0.679727
$750$	0	0
$751$	−1.05538e16	−1.61209	−0.806043	−	0.591857i	$-0.798395\pi$
$751$	−1.05538e16	−1.61209	−0.806043	+	0.591857i	$0.798395\pi$
$752$	−6.90251e15	−1.04667
$753$	3.70427e15	0.557610
$754$	−2.74119e15	−0.409635
$755$	0	0
$756$	2.87835e15	0.423909
$757$	−9.99905e15	−1.46195	−0.730973	−	0.682406i	$-0.760934\pi$
$757$	−9.99905e15	−1.46195	−0.730973	+	0.682406i	$0.760934\pi$
$758$	−1.66496e16	−2.41671
$759$	−3.70384e15	−0.533730
$760$	0	0
$761$	3.24768e15	0.461273	0.230636	−	0.973040i	$-0.425919\pi$
$761$	3.24768e15	0.461273	0.230636	+	0.973040i	$0.425919\pi$
$762$	−3.25812e15	−0.459425
$763$	4.19555e15	0.587360
$764$	2.20959e16	3.07113
$765$	0	0
$766$	8.33013e15	1.14128
$767$	−3.81004e15	−0.518269
$768$	1.41951e14	0.0191713
$769$	5.22452e15	0.700570	0.350285	−	0.936643i	$-0.386085\pi$
$769$	5.22452e15	0.700570	0.350285	+	0.936643i	$0.386085\pi$
$770$	0	0
$771$	6.69941e15	0.885599
$772$	1.71293e15	0.224824
$773$	−5.46690e15	−0.712449	−0.356225	−	0.934400i	$-0.615936\pi$
$773$	−5.46690e15	−0.712449	−0.356225	+	0.934400i	$0.615936\pi$
$774$	3.16851e15	0.409997
$775$	0	0
$776$	−2.45708e16	−3.13458
$777$	−2.46175e14	−0.0311837
$778$	−2.92833e16	−3.68326
$779$	−4.35717e15	−0.544187
$780$	0	0
$781$	2.71247e16	3.34029
$782$	6.40060e15	0.782678
$783$	−4.49819e14	−0.0546195
$784$	−5.73575e15	−0.691595
$785$	0	0
$786$	−1.29216e16	−1.53635
$787$	1.06026e16	1.25184	0.625921	−	0.779886i	$-0.284723\pi$
$787$	1.06026e16	1.25184	0.625921	+	0.779886i	$0.284723\pi$
$788$	2.55636e15	0.299728
$789$	6.30806e15	0.734467
$790$	0	0
$791$	9.34252e15	1.07274
$792$	−1.53809e16	−1.75386
$793$	−8.71828e15	−0.987253
$794$	7.26723e15	0.817252
$795$	0	0
$796$	−1.04089e16	−1.15447
$797$	−4.38617e15	−0.483130	−0.241565	−	0.970385i	$-0.577661\pi$
$797$	−4.38617e15	−0.483130	−0.241565	+	0.970385i	$0.577661\pi$
$798$	5.09821e15	0.557701
$799$	2.77552e15	0.301535
$800$	0	0
$801$	5.04275e15	0.540367
$802$	−6.93041e15	−0.737564
$803$	7.70977e15	0.814903
$804$	−1.16621e15	−0.122424
$805$	0	0
$806$	6.69167e15	0.692934
$807$	4.42759e15	0.455369
$808$	1.65472e15	0.169030
$809$	1.14268e16	1.15933	0.579665	−	0.814855i	$-0.303183\pi$
$809$	1.14268e16	1.15933	0.579665	+	0.814855i	$0.303183\pi$
$810$	0	0
$811$	6.93611e15	0.694227	0.347113	−	0.937823i	$-0.387162\pi$
$811$	6.93611e15	0.694227	0.347113	+	0.937823i	$0.387162\pi$
$812$	−6.28843e15	−0.625150
$813$	5.37350e15	0.530590
$814$	2.17592e15	0.213407
$815$	0	0
$816$	1.41208e16	1.36636
$817$	4.02179e15	0.386543
$818$	2.42507e16	2.31516
$819$	2.35249e15	0.223083
$820$	0	0
$821$	8.91632e14	0.0834254	0.0417127	−	0.999130i	$-0.486719\pi$
$821$	8.91632e14	0.0834254	0.0417127	+	0.999130i	$0.486719\pi$
$822$	7.35591e15	0.683662
$823$	−1.95753e15	−0.180722	−0.0903608	−	0.995909i	$-0.528802\pi$
$823$	−1.95753e15	−0.180722	−0.0903608	+	0.995909i	$0.528802\pi$
$824$	1.18959e16	1.09093
$825$	0	0
$826$	−1.21967e16	−1.10371
$827$	1.66771e16	1.49913	0.749565	−	0.661931i	$-0.230263\pi$
$827$	1.66771e16	1.49913	0.749565	+	0.661931i	$0.230263\pi$
$828$	−4.76328e15	−0.425343
$829$	3.12919e14	0.0277576	0.0138788	−	0.999904i	$-0.495582\pi$
$829$	3.12919e14	0.0277576	0.0138788	+	0.999904i	$0.495582\pi$
$830$	0	0
$831$	5.58935e15	0.489278
$832$	−1.63723e16	−1.42374
$833$	2.30637e15	0.199242
$834$	−1.43043e16	−1.22759
$835$	0	0
$836$	−3.22929e16	−2.73509
$837$	1.09808e15	0.0923936
$838$	−9.17541e15	−0.766979
$839$	−1.29785e16	−1.07779	−0.538896	−	0.842372i	$-0.681158\pi$
$839$	−1.29785e16	−1.07779	−0.538896	+	0.842372i	$0.681158\pi$
$840$	0	0
$841$	−1.12178e16	−0.919451
$842$	2.36241e16	1.92371
$843$	9.47208e15	0.766291
$844$	1.12666e16	0.905541
$845$	0	0
$846$	−2.88232e15	−0.228668
$847$	−2.60409e16	−2.05256
$848$	−6.51050e16	−5.09844
$849$	−1.90436e14	−0.0148169
$850$	0	0
$851$	4.07387e14	0.0312892
$852$	3.48834e16	2.66196
$853$	−1.77152e16	−1.34316	−0.671579	−	0.740933i	$-0.734384\pi$
$853$	−1.77152e16	−1.34316	−0.671579	+	0.740933i	$0.734384\pi$
$854$	−2.79090e16	−2.10245
$855$	0	0
$856$	3.01355e16	2.24116
$857$	−1.92138e16	−1.41977	−0.709887	−	0.704315i	$-0.751254\pi$
$857$	−1.92138e16	−1.41977	−0.709887	+	0.704315i	$0.751254\pi$
$858$	−2.07935e16	−1.52668
$859$	2.41763e16	1.76371	0.881855	−	0.471521i	$-0.156295\pi$
$859$	2.41763e16	1.76371	0.881855	+	0.471521i	$0.156295\pi$
$860$	0	0
$861$	−6.43619e15	−0.463566
$862$	5.02540e15	0.359651
$863$	1.75739e16	1.24971	0.624856	−	0.780740i	$-0.285158\pi$
$863$	1.75739e16	1.24971	0.624856	+	0.780740i	$0.285158\pi$
$864$	−6.84215e15	−0.483467
$865$	0	0
$866$	5.16774e16	3.60539
$867$	2.65004e15	0.183716
$868$	1.53510e16	1.05750
$869$	−1.38792e16	−0.950069
$870$	0	0
$871$	−9.53151e14	−0.0644261
$872$	−2.88326e16	−1.93661
$873$	−5.45083e15	−0.363818
$874$	−8.43686e15	−0.559588
$875$	0	0
$876$	9.91507e15	0.649416
$877$	7.55842e15	0.491964	0.245982	−	0.969274i	$-0.420890\pi$
$877$	7.55842e15	0.491964	0.245982	+	0.969274i	$0.420890\pi$
$878$	2.08455e16	1.34832
$879$	1.08120e16	0.694973
$880$	0	0
$881$	−2.73238e16	−1.73450	−0.867249	−	0.497875i	$-0.834114\pi$
$881$	−2.73238e16	−1.73450	−0.867249	+	0.497875i	$0.834114\pi$
$882$	−2.39511e15	−0.151094
$883$	−5.04860e15	−0.316510	−0.158255	−	0.987398i	$-0.550587\pi$
$883$	−5.04860e15	−0.316510	−0.158255	+	0.987398i	$0.550587\pi$
$884$	2.57505e16	1.60435
$885$	0	0
$886$	−3.30500e16	−2.03370
$887$	−5.85946e15	−0.358325	−0.179163	−	0.983819i	$-0.557339\pi$
$887$	−5.85946e15	−0.358325	−0.179163	+	0.983819i	$0.557339\pi$
$888$	1.69176e15	0.102817
$889$	−6.10879e15	−0.368974
$890$	0	0
$891$	−3.41213e15	−0.203563
$892$	−2.41188e16	−1.43004
$893$	−3.65852e15	−0.215587
$894$	3.93098e14	0.0230221
$895$	0	0
$896$	−1.45861e16	−0.843812
$897$	−3.89307e15	−0.223838
$898$	4.47549e16	2.55753
$899$	−2.39901e15	−0.136255
$900$	0	0
$901$	2.61790e16	1.46881
$902$	5.68891e16	3.17244
$903$	5.94078e15	0.329277
$904$	−6.42034e16	−3.53697
$905$	0	0
$906$	−8.12711e15	−0.442314
$907$	−3.06892e16	−1.66014	−0.830072	−	0.557656i	$-0.811701\pi$
$907$	−3.06892e16	−1.66014	−0.830072	+	0.557656i	$0.811701\pi$
$908$	−4.18333e16	−2.24932
$909$	3.67087e14	0.0196186
$910$	0	0
$911$	−3.36424e16	−1.77638	−0.888191	−	0.459475i	$-0.848038\pi$
$911$	−3.36424e16	−1.77638	−0.888191	+	0.459475i	$0.848038\pi$
$912$	−1.86132e16	−0.976897
$913$	−5.39114e16	−2.81249
$914$	4.59905e16	2.38486
$915$	0	0
$916$	−2.28001e16	−1.16819
$917$	−2.42272e16	−1.23388
$918$	5.89651e15	0.298511
$919$	−6.66548e15	−0.335425	−0.167713	−	0.985836i	$-0.553638\pi$
$919$	−6.66548e15	−0.335425	−0.167713	+	0.985836i	$0.553638\pi$
$920$	0	0
$921$	−5.22224e15	−0.259674
$922$	−6.13269e16	−3.03131
$923$	2.85105e16	1.40086
$924$	−4.77014e16	−2.32989
$925$	0	0
$926$	5.44933e16	2.63016
$927$	2.63901e15	0.126620
$928$	1.49483e16	0.712982
$929$	−1.94476e16	−0.922102	−0.461051	−	0.887374i	$-0.652528\pi$
$929$	−1.94476e16	−0.922102	−0.461051	+	0.887374i	$0.652528\pi$
$930$	0	0
$931$	−3.04011e15	−0.142451
$932$	3.40859e16	1.58777
$933$	1.33683e16	0.619050
$934$	−7.03733e16	−3.23966
$935$	0	0
$936$	−1.61667e16	−0.735538
$937$	3.46632e16	1.56784	0.783919	−	0.620863i	$-0.213217\pi$
$937$	3.46632e16	1.56784	0.783919	+	0.620863i	$0.213217\pi$
$938$	−3.05123e15	−0.137202
$939$	1.69254e14	0.00756620
$940$	0	0
$941$	−1.70221e16	−0.752089	−0.376045	−	0.926602i	$-0.622716\pi$
$941$	−1.70221e16	−0.752089	−0.376045	+	0.926602i	$0.622716\pi$
$942$	2.20816e16	0.969951
$943$	1.06510e16	0.465134
$944$	4.45294e16	1.93331
$945$	0	0
$946$	−5.25102e16	−2.25342
$947$	2.28346e16	0.974247	0.487124	−	0.873333i	$-0.338046\pi$
$947$	2.28346e16	0.974247	0.487124	+	0.873333i	$0.338046\pi$
$948$	−1.78492e16	−0.757134
$949$	8.10366e15	0.341757
$950$	0	0
$951$	7.54117e15	0.314374
$952$	4.98356e16	2.06555
$953$	−3.56877e16	−1.47064	−0.735321	−	0.677719i	$-0.762969\pi$
$953$	−3.56877e16	−1.47064	−0.735321	+	0.677719i	$0.762969\pi$
$954$	−2.71863e16	−1.11387
$955$	0	0
$956$	1.13461e17	4.59547
$957$	7.45462e15	0.300200
$958$	1.85895e16	0.744317
$959$	1.37919e16	0.549064
$960$	0	0
$961$	−1.95521e16	−0.769512
$962$	2.28709e15	0.0894995
$963$	6.68532e15	0.260122
$964$	1.14472e17	4.42870
$965$	0	0
$966$	−1.24625e16	−0.476685
$967$	−7.20157e15	−0.273893	−0.136947	−	0.990578i	$-0.543729\pi$
$967$	−7.20157e15	−0.273893	−0.136947	+	0.990578i	$0.543729\pi$
$968$	1.78957e17	6.76761
$969$	7.48442e15	0.281435
$970$	0	0
$971$	8.34042e15	0.310086	0.155043	−	0.987908i	$-0.450448\pi$
$971$	8.34042e15	0.310086	0.155043	+	0.987908i	$0.450448\pi$
$972$	−4.38814e15	−0.162224
$973$	−2.68198e16	−0.985906
$974$	8.27066e16	3.02319
$975$	0	0
$976$	1.01894e17	3.68276
$977$	1.82165e16	0.654704	0.327352	−	0.944902i	$-0.393844\pi$
$977$	1.82165e16	0.654704	0.327352	+	0.944902i	$0.393844\pi$
$978$	4.49487e16	1.60640
$979$	−8.35711e16	−2.96996
$980$	0	0
$981$	−6.39627e15	−0.224775
$982$	4.69347e16	1.64014
$983$	−2.17924e16	−0.757287	−0.378643	−	0.925543i	$-0.623609\pi$
$983$	−2.17924e16	−0.757287	−0.378643	+	0.925543i	$0.623609\pi$
$984$	4.42306e16	1.52845
$985$	0	0
$986$	−1.28823e16	−0.440222
$987$	−5.40419e15	−0.183648
$988$	−3.39427e16	−1.14705
$989$	−9.83122e15	−0.330391
$990$	0	0
$991$	3.70744e15	0.123216	0.0616082	−	0.998100i	$-0.480377\pi$
$991$	3.70744e15	0.123216	0.0616082	+	0.998100i	$0.480377\pi$
$992$	−3.64911e16	−1.20607
$993$	2.97275e16	0.977095
$994$	9.12680e16	2.98328
$995$	0	0
$996$	−6.93323e16	−2.24135
$997$	−5.02074e16	−1.61415	−0.807075	−	0.590449i	$-0.798951\pi$
$997$	−5.02074e16	−1.61415	−0.807075	+	0.590449i	$0.798951\pi$
$998$	−2.04111e15	−0.0652603
$999$	3.75302e14	0.0119336

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.12.a.j.1.6		6
3.2	odd	2		225.12.a.y.1.1		6
5.2	odd	4		15.12.b.a.4.12	yes	12
5.3	odd	4		15.12.b.a.4.1	✓	12
5.4	even	2		75.12.a.k.1.1		6
15.2	even	4		45.12.b.d.19.1		12
15.8	even	4		45.12.b.d.19.12		12
15.14	odd	2		225.12.a.v.1.6		6
20.3	even	4		240.12.f.d.49.11		12
20.7	even	4		240.12.f.d.49.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.12.b.a.4.1	✓	12	5.3	odd	4
15.12.b.a.4.12	yes	12	5.2	odd	4
45.12.b.d.19.1		12	15.2	even	4
45.12.b.d.19.12		12	15.8	even	4
75.12.a.j.1.6		6	1.1	even	1	trivial
75.12.a.k.1.1		6	5.4	even	2
225.12.a.v.1.6		6	15.14	odd	2
225.12.a.y.1.1		6	3.2	odd	2
240.12.f.d.49.5		12	20.7	even	4
240.12.f.d.49.11		12	20.3	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	75.a (trivial)

\(f(q)\)	\(=\)	\(q+85.0120 q^{2} -243.000 q^{3} +5179.04 q^{4} -20657.9 q^{6} -38732.4 q^{7} +266176. q^{8} +59049.0 q^{9} +O(q^{10})\)	\(q+85.0120 q^{2} -243.000 q^{3} +5179.04 q^{4} -20657.9 q^{6} -38732.4 q^{7} +266176. q^{8} +59049.0 q^{9} -978590. q^{11} -1.25851e6 q^{12} -1.02859e6 q^{13} -3.29272e6 q^{14} +1.20215e7 q^{16} -4.83388e6 q^{17} +5.01987e6 q^{18} +6.37171e6 q^{19} +9.41198e6 q^{21} -8.31919e7 q^{22} -1.55756e7 q^{23} -6.46808e7 q^{24} -8.74422e7 q^{26} -1.43489e7 q^{27} -2.00597e8 q^{28} +3.13486e7 q^{29} -7.65267e7 q^{31} +4.76841e8 q^{32} +2.37797e8 q^{33} -4.10938e8 q^{34} +3.05817e8 q^{36} -2.61555e7 q^{37} +5.41672e8 q^{38} +2.49947e8 q^{39} -6.83829e8 q^{41} +8.00131e8 q^{42} +6.31194e8 q^{43} -5.06816e9 q^{44} -1.32411e9 q^{46} -5.74182e8 q^{47} -2.92122e9 q^{48} -4.77125e8 q^{49} +1.17463e9 q^{51} -5.32709e9 q^{52} -5.41573e9 q^{53} -1.21983e9 q^{54} -1.03096e10 q^{56} -1.54833e9 q^{57} +2.66501e9 q^{58} +3.70415e9 q^{59} +8.47598e9 q^{61} -6.50569e9 q^{62} -2.28711e9 q^{63} +1.59173e10 q^{64} +2.02156e10 q^{66} +9.26660e8 q^{67} -2.50348e10 q^{68} +3.78487e9 q^{69} -2.77181e10 q^{71} +1.57174e10 q^{72} -7.87844e9 q^{73} -2.22353e9 q^{74} +3.29994e10 q^{76} +3.79032e10 q^{77} +2.12485e10 q^{78} +1.41828e10 q^{79} +3.48678e9 q^{81} -5.81337e10 q^{82} +5.50909e10 q^{83} +4.87450e10 q^{84} +5.36591e10 q^{86} -7.61772e9 q^{87} -2.60477e11 q^{88} +8.53995e10 q^{89} +3.98397e10 q^{91} -8.06666e10 q^{92} +1.85960e10 q^{93} -4.88123e10 q^{94} -1.15872e11 q^{96} -9.23102e10 q^{97} -4.05614e10 q^{98} -5.77848e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 9 q^{2} - 1458 q^{3} + 8157 q^{4} + 2187 q^{6} - 64368 q^{7} + 29277 q^{8} + 354294 q^{9}+O(q^{10}) \) 6 * q - 9 * q^2 - 1458 * q^3 + 8157 * q^4 + 2187 * q^6 - 64368 * q^7 + 29277 * q^8 + 354294 * q^9	\( 6 q - 9 q^{2} - 1458 q^{3} + 8157 q^{4} + 2187 q^{6} - 64368 q^{7} + 29277 q^{8} + 354294 q^{9} - 160776 q^{11} - 1982151 q^{12} - 1777464 q^{13} - 2318022 q^{14} + 14360865 q^{16} - 7789536 q^{17} - 531441 q^{18} + 14306328 q^{19} + 15641424 q^{21} - 99540522 q^{22} - 56578896 q^{23} - 7114311 q^{24} + 156248982 q^{26} - 86093442 q^{27} - 29296674 q^{28} + 229450716 q^{29} + 419037696 q^{31} + 496142397 q^{32} + 39068568 q^{33} + 272636394 q^{34} + 481662693 q^{36} + 441738792 q^{37} - 99095256 q^{38} + 431923752 q^{39} - 1880639244 q^{41} + 563279346 q^{42} - 1081616760 q^{43} - 3162393414 q^{44} - 939739884 q^{46} - 1868657040 q^{47} - 3489690195 q^{48} + 511947750 q^{49} + 1892857248 q^{51} - 9215549766 q^{52} - 10349483256 q^{53} + 129140163 q^{54} - 21606208290 q^{56} - 3476437704 q^{57} - 25240036752 q^{58} - 707714568 q^{59} + 8985508380 q^{61} - 25995031416 q^{62} - 3800866032 q^{63} + 4944042873 q^{64} + 24188346846 q^{66} - 13687306776 q^{67} - 94791079962 q^{68} + 13748671728 q^{69} - 8562874608 q^{71} + 1728777573 q^{72} - 63672451056 q^{73} - 44262217422 q^{74} + 69695921208 q^{76} + 10135567584 q^{77} - 37968502626 q^{78} + 41427333360 q^{79} + 20920706406 q^{81} - 34598023506 q^{82} - 124949795688 q^{83} + 7119091782 q^{84} - 18826696116 q^{86} - 55756523988 q^{87} - 430278178518 q^{88} + 90885005748 q^{89} - 48271166064 q^{91} - 310396941828 q^{92} - 101826160128 q^{93} - 139084157808 q^{94} - 120562602471 q^{96} - 257134073616 q^{97} - 405310493781 q^{98} - 9493662024 q^{99}+O(q^{100}) \) 6 * q - 9 * q^2 - 1458 * q^3 + 8157 * q^4 + 2187 * q^6 - 64368 * q^7 + 29277 * q^8 + 354294 * q^9 - 160776 * q^11 - 1982151 * q^12 - 1777464 * q^13 - 2318022 * q^14 + 14360865 * q^16 - 7789536 * q^17 - 531441 * q^18 + 14306328 * q^19 + 15641424 * q^21 - 99540522 * q^22 - 56578896 * q^23 - 7114311 * q^24 + 156248982 * q^26 - 86093442 * q^27 - 29296674 * q^28 + 229450716 * q^29 + 419037696 * q^31 + 496142397 * q^32 + 39068568 * q^33 + 272636394 * q^34 + 481662693 * q^36 + 441738792 * q^37 - 99095256 * q^38 + 431923752 * q^39 - 1880639244 * q^41 + 563279346 * q^42 - 1081616760 * q^43 - 3162393414 * q^44 - 939739884 * q^46 - 1868657040 * q^47 - 3489690195 * q^48 + 511947750 * q^49 + 1892857248 * q^51 - 9215549766 * q^52 - 10349483256 * q^53 + 129140163 * q^54 - 21606208290 * q^56 - 3476437704 * q^57 - 25240036752 * q^58 - 707714568 * q^59 + 8985508380 * q^61 - 25995031416 * q^62 - 3800866032 * q^63 + 4944042873 * q^64 + 24188346846 * q^66 - 13687306776 * q^67 - 94791079962 * q^68 + 13748671728 * q^69 - 8562874608 * q^71 + 1728777573 * q^72 - 63672451056 * q^73 - 44262217422 * q^74 + 69695921208 * q^76 + 10135567584 * q^77 - 37968502626 * q^78 + 41427333360 * q^79 + 20920706406 * q^81 - 34598023506 * q^82 - 124949795688 * q^83 + 7119091782 * q^84 - 18826696116 * q^86 - 55756523988 * q^87 - 430278178518 * q^88 + 90885005748 * q^89 - 48271166064 * q^91 - 310396941828 * q^92 - 101826160128 * q^93 - 139084157808 * q^94 - 120562602471 * q^96 - 257134073616 * q^97 - 405310493781 * q^98 - 9493662024 * q^99

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