LMFDB - Embedded newform 75.12.a.i.1.3

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:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 6154x^{2} - 41770x + 5647125 $ x^4 - 2x^3 - 6154x^2 - 41770*x + 5647125
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 3\cdot 5^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$28.7345$ 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)

$f(q)$	$=$	$q+39.7345 q^{2} +243.000 q^{3} -469.173 q^{4} +9655.47 q^{6} -8306.83 q^{7} -100019. q^{8} +59049.0 q^{9} +O(q^{10})$	\(q+39.7345 q^{2} +243.000 q^{3} -469.173 q^{4} +9655.47 q^{6} -8306.83 q^{7} -100019. q^{8} +59049.0 q^{9} +337742. q^{11} -114009. q^{12} -332445. q^{13} -330067. q^{14} -3.01331e6 q^{16} +4.09893e6 q^{17} +2.34628e6 q^{18} +9.11010e6 q^{19} -2.01856e6 q^{21} +1.34200e7 q^{22} +2.88443e7 q^{23} -2.43045e7 q^{24} -1.32095e7 q^{26} +1.43489e7 q^{27} +3.89734e6 q^{28} -4.37488e7 q^{29} +9.03196e7 q^{31} +8.51055e7 q^{32} +8.20712e7 q^{33} +1.62869e8 q^{34} -2.77042e7 q^{36} +7.41610e8 q^{37} +3.61985e8 q^{38} -8.07841e7 q^{39} +6.88364e8 q^{41} -8.02063e7 q^{42} -2.14585e8 q^{43} -1.58459e8 q^{44} +1.14611e9 q^{46} +1.50451e9 q^{47} -7.32235e8 q^{48} -1.90832e9 q^{49} +9.96041e8 q^{51} +1.55974e8 q^{52} -2.39637e9 q^{53} +5.70146e8 q^{54} +8.30836e8 q^{56} +2.21376e9 q^{57} -1.73833e9 q^{58} +6.31089e9 q^{59} -2.31809e9 q^{61} +3.58880e9 q^{62} -4.90510e8 q^{63} +9.55289e9 q^{64} +3.26106e9 q^{66} +2.51799e9 q^{67} -1.92311e9 q^{68} +7.00917e9 q^{69} +2.10654e9 q^{71} -5.90599e9 q^{72} -2.23379e10 q^{73} +2.94675e10 q^{74} -4.27422e9 q^{76} -2.80556e9 q^{77} -3.20991e9 q^{78} +4.01526e10 q^{79} +3.48678e9 q^{81} +2.73518e10 q^{82} -9.34317e9 q^{83} +9.47054e8 q^{84} -8.52641e9 q^{86} -1.06309e10 q^{87} -3.37804e10 q^{88} +5.40157e10 q^{89} +2.76156e9 q^{91} -1.35330e10 q^{92} +2.19477e10 q^{93} +5.97807e10 q^{94} +2.06806e10 q^{96} +1.27556e11 q^{97} -7.58262e10 q^{98} +1.99433e10 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 46 q^{2} + 972 q^{3} + 4648 q^{4} + 11178 q^{6} + 68372 q^{7} + 386172 q^{8} + 236196 q^{9}+O(q^{10}) $ 4 * q + 46 * q^2 + 972 * q^3 + 4648 * q^4 + 11178 * q^6 + 68372 * q^7 + 386172 * q^8 + 236196 * q^9	\( 4 q + 46 q^{2} + 972 q^{3} + 4648 q^{4} + 11178 q^{6} + 68372 q^{7} + 386172 q^{8} + 236196 q^{9} - 99944 q^{11} + 1129464 q^{12} + 2306276 q^{13} + 3107022 q^{14} + 7622200 q^{16} - 3443816 q^{17} + 2716254 q^{18} - 4214548 q^{19} + 16614396 q^{21} - 34106532 q^{22} + 52691304 q^{23} + 93839796 q^{24} - 8766722 q^{26} + 57395628 q^{27} + 472439576 q^{28} + 217393304 q^{29} - 326317036 q^{31} + 788672072 q^{32} - 24286392 q^{33} - 564836244 q^{34} + 274459752 q^{36} + 273252872 q^{37} - 1762793006 q^{38} + 560425068 q^{39} - 22069456 q^{41} + 755006346 q^{42} - 705091900 q^{43} - 992474176 q^{44} + 2697853524 q^{46} + 187768360 q^{47} + 1852194600 q^{48} + 5213315400 q^{49} - 836847288 q^{51} + 7619620904 q^{52} - 6392224256 q^{53} + 660049722 q^{54} + 20338292700 q^{56} - 1024135164 q^{57} - 16176655332 q^{58} + 36710008 q^{59} + 11538870620 q^{61} - 4825203906 q^{62} + 4037298228 q^{63} + 55398609952 q^{64} - 8287887276 q^{66} + 37721158484 q^{67} - 45682615072 q^{68} + 12803986872 q^{69} + 8211316688 q^{71} + 22803070428 q^{72} + 5713413224 q^{73} + 136155026252 q^{74} - 78094966168 q^{76} + 72854549304 q^{77} - 2130313446 q^{78} + 45026381600 q^{79} + 13947137604 q^{81} + 55157549184 q^{82} - 104211315528 q^{83} + 114802816968 q^{84} + 77092671046 q^{86} + 52826572872 q^{87} - 175877029608 q^{88} + 111829609152 q^{89} + 225968327284 q^{91} + 88286181792 q^{92} - 79295039748 q^{93} + 193370717388 q^{94} + 191647313496 q^{96} + 77104304804 q^{97} - 139912937716 q^{98} - 5901593256 q^{99}+O(q^{100}) \) 4 * q + 46 * q^2 + 972 * q^3 + 4648 * q^4 + 11178 * q^6 + 68372 * q^7 + 386172 * q^8 + 236196 * q^9 - 99944 * q^11 + 1129464 * q^12 + 2306276 * q^13 + 3107022 * q^14 + 7622200 * q^16 - 3443816 * q^17 + 2716254 * q^18 - 4214548 * q^19 + 16614396 * q^21 - 34106532 * q^22 + 52691304 * q^23 + 93839796 * q^24 - 8766722 * q^26 + 57395628 * q^27 + 472439576 * q^28 + 217393304 * q^29 - 326317036 * q^31 + 788672072 * q^32 - 24286392 * q^33 - 564836244 * q^34 + 274459752 * q^36 + 273252872 * q^37 - 1762793006 * q^38 + 560425068 * q^39 - 22069456 * q^41 + 755006346 * q^42 - 705091900 * q^43 - 992474176 * q^44 + 2697853524 * q^46 + 187768360 * q^47 + 1852194600 * q^48 + 5213315400 * q^49 - 836847288 * q^51 + 7619620904 * q^52 - 6392224256 * q^53 + 660049722 * q^54 + 20338292700 * q^56 - 1024135164 * q^57 - 16176655332 * q^58 + 36710008 * q^59 + 11538870620 * q^61 - 4825203906 * q^62 + 4037298228 * q^63 + 55398609952 * q^64 - 8287887276 * q^66 + 37721158484 * q^67 - 45682615072 * q^68 + 12803986872 * q^69 + 8211316688 * q^71 + 22803070428 * q^72 + 5713413224 * q^73 + 136155026252 * q^74 - 78094966168 * q^76 + 72854549304 * q^77 - 2130313446 * q^78 + 45026381600 * q^79 + 13947137604 * q^81 + 55157549184 * q^82 - 104211315528 * q^83 + 114802816968 * q^84 + 77092671046 * q^86 + 52826572872 * q^87 - 175877029608 * q^88 + 111829609152 * q^89 + 225968327284 * q^91 + 88286181792 * q^92 - 79295039748 * q^93 + 193370717388 * q^94 + 191647313496 * q^96 + 77104304804 * q^97 - 139912937716 * q^98 - 5901593256 * 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 $

$2$	39.7345	0.878016	0.439008	−	0.898483i	$-0.355330\pi$
$2$	39.7345	0.878016	0.439008	+	0.898483i	$0.355330\pi$
$3$	243.000	0.577350
$3$	243.000	0.577350
$4$	−469.173	−0.229089
$5$	0	0
$5$	0	0
$6$	9655.47	0.506923
$7$	−8306.83	−0.186808	−0.0934041	−	0.995628i	$-0.529775\pi$
$7$	−8306.83	−0.186808	−0.0934041	+	0.995628i	$0.529775\pi$
$8$	−100019.	−1.07916
$9$	59049.0	0.333333
$10$	0	0
$11$	337742.	0.632303	0.316151	−	0.948709i	$-0.397609\pi$
$11$	337742.	0.632303	0.316151	+	0.948709i	$0.397609\pi$
$12$	−114009.	−0.132264
$13$	−332445.	−0.248331	−0.124166	−	0.992262i	$-0.539625\pi$
$13$	−332445.	−0.248331	−0.124166	+	0.992262i	$0.539625\pi$
$14$	−330067.	−0.164021
$15$	0	0
$16$	−3.01331e6	−0.718430
$17$	4.09893e6	0.700167	0.350084	−	0.936718i	$-0.386153\pi$
$17$	4.09893e6	0.700167	0.350084	+	0.936718i	$0.386153\pi$
$18$	2.34628e6	0.292672
$19$	9.11010e6	0.844070	0.422035	−	0.906579i	$-0.361316\pi$
$19$	9.11010e6	0.844070	0.422035	+	0.906579i	$0.361316\pi$
$20$	0	0
$21$	−2.01856e6	−0.107854
$22$	1.34200e7	0.555172
$23$	2.88443e7	0.934452	0.467226	−	0.884138i	$-0.345253\pi$
$23$	2.88443e7	0.934452	0.467226	+	0.884138i	$0.345253\pi$
$24$	−2.43045e7	−0.623053
$25$	0	0
$26$	−1.32095e7	−0.218039
$27$	1.43489e7	0.192450
$28$	3.89734e6	0.0427956
$29$	−4.37488e7	−0.396074	−0.198037	−	0.980195i	$-0.563457\pi$
$29$	−4.37488e7	−0.396074	−0.198037	+	0.980195i	$0.563457\pi$
$30$	0	0
$31$	9.03196e7	0.566621	0.283310	−	0.959028i	$-0.408567\pi$
$31$	9.03196e7	0.566621	0.283310	+	0.959028i	$0.408567\pi$
$32$	8.51055e7	0.448366
$33$	8.20712e7	0.365060
$34$	1.62869e8	0.614758
$35$	0	0
$36$	−2.77042e7	−0.0763628
$37$	7.41610e8	1.75819	0.879096	−	0.476645i	$-0.158147\pi$
$37$	7.41610e8	1.75819	0.879096	+	0.476645i	$0.158147\pi$
$38$	3.61985e8	0.741107
$39$	−8.07841e7	−0.143374
$40$	0	0
$41$	6.88364e8	0.927912	0.463956	−	0.885858i	$-0.346430\pi$
$41$	6.88364e8	0.927912	0.463956	+	0.885858i	$0.346430\pi$
$42$	−8.02063e7	−0.0946973
$43$	−2.14585e8	−0.222599	−0.111299	−	0.993787i	$-0.535501\pi$
$43$	−2.14585e8	−0.222599	−0.111299	+	0.993787i	$0.535501\pi$
$44$	−1.58459e8	−0.144853
$45$	0	0
$46$	1.14611e9	0.820464
$47$	1.50451e9	0.956876	0.478438	−	0.878121i	$-0.341203\pi$
$47$	1.50451e9	0.956876	0.478438	+	0.878121i	$0.341203\pi$
$48$	−7.32235e8	−0.414786
$49$	−1.90832e9	−0.965103
$50$	0	0
$51$	9.96041e8	0.404242
$52$	1.55974e8	0.0568898
$53$	−2.39637e9	−0.787111	−0.393556	−	0.919301i	$-0.628755\pi$
$53$	−2.39637e9	−0.787111	−0.393556	+	0.919301i	$0.628755\pi$
$54$	5.70146e8	0.168974
$55$	0	0
$56$	8.30836e8	0.201596
$57$	2.21376e9	0.487324
$58$	−1.73833e9	−0.347759
$59$	6.31089e9	1.14922	0.574612	−	0.818426i	$-0.305153\pi$
$59$	6.31089e9	1.14922	0.574612	+	0.818426i	$0.305153\pi$
$60$	0	0
$61$	−2.31809e9	−0.351412	−0.175706	−	0.984443i	$-0.556221\pi$
$61$	−2.31809e9	−0.351412	−0.175706	+	0.984443i	$0.556221\pi$
$62$	3.58880e9	0.497502
$63$	−4.90510e8	−0.0622694
$64$	9.55289e9	1.11210
$65$	0	0
$66$	3.26106e9	0.320528
$67$	2.51799e9	0.227847	0.113923	−	0.993490i	$-0.463658\pi$
$67$	2.51799e9	0.227847	0.113923	+	0.993490i	$0.463658\pi$
$68$	−1.92311e9	−0.160400
$69$	7.00917e9	0.539506
$70$	0	0
$71$	2.10654e9	0.138563	0.0692816	−	0.997597i	$-0.477929\pi$
$71$	2.10654e9	0.138563	0.0692816	+	0.997597i	$0.477929\pi$
$72$	−5.90599e9	−0.359720
$73$	−2.23379e10	−1.26115	−0.630575	−	0.776128i	$-0.717181\pi$
$73$	−2.23379e10	−1.26115	−0.630575	+	0.776128i	$0.717181\pi$
$74$	2.94675e10	1.54372
$75$	0	0
$76$	−4.27422e9	−0.193367
$77$	−2.80556e9	−0.118119
$78$	−3.20991e9	−0.125885
$79$	4.01526e10	1.46813	0.734065	−	0.679079i	$-0.237621\pi$
$79$	4.01526e10	1.46813	0.734065	+	0.679079i	$0.237621\pi$
$80$	0	0
$81$	3.48678e9	0.111111
$82$	2.73518e10	0.814721
$83$	−9.34317e9	−0.260354	−0.130177	−	0.991491i	$-0.541555\pi$
$83$	−9.34317e9	−0.260354	−0.130177	+	0.991491i	$0.541555\pi$
$84$	9.47054e8	0.0247081
$85$	0	0
$86$	−8.52641e9	−0.195445
$87$	−1.06309e10	−0.228674
$88$	−3.37804e10	−0.682355
$89$	5.40157e10	1.02536	0.512678	−	0.858581i	$-0.328653\pi$
$89$	5.40157e10	1.02536	0.512678	+	0.858581i	$0.328653\pi$
$90$	0	0
$91$	2.76156e9	0.0463903
$92$	−1.35330e10	−0.214072
$93$	2.19477e10	0.327139
$94$	5.97807e10	0.840152
$95$	0	0
$96$	2.06806e10	0.258864
$97$	1.27556e11	1.50819	0.754097	−	0.656763i	$-0.228075\pi$
$97$	1.27556e11	1.50819	0.754097	+	0.656763i	$0.228075\pi$
$98$	−7.58262e10	−0.847375
$99$	1.99433e10	0.210768
$100$	0	0
$101$	1.89892e11	1.79779	0.898895	−	0.438164i	$-0.144371\pi$
$101$	1.89892e11	1.79779	0.898895	+	0.438164i	$0.144371\pi$
$102$	3.95771e10	0.354931
$103$	−1.78532e11	−1.51744	−0.758719	−	0.651418i	$-0.774175\pi$
$103$	−1.78532e11	−1.51744	−0.758719	+	0.651418i	$0.774175\pi$
$104$	3.32506e10	0.267989
$105$	0	0
$106$	−9.52184e10	−0.691096
$107$	1.16210e11	0.801004	0.400502	−	0.916296i	$-0.368836\pi$
$107$	1.16210e11	0.801004	0.400502	+	0.916296i	$0.368836\pi$
$108$	−6.73212e9	−0.0440881
$109$	−2.32240e11	−1.44574	−0.722871	−	0.690983i	$-0.757178\pi$
$109$	−2.32240e11	−1.44574	−0.722871	+	0.690983i	$0.757178\pi$
$110$	0	0
$111$	1.80211e11	1.01509
$112$	2.50311e10	0.134209
$113$	8.11780e10	0.414483	0.207241	−	0.978290i	$-0.433551\pi$
$113$	8.11780e10	0.414483	0.207241	+	0.978290i	$0.433551\pi$
$114$	8.79624e10	0.427878
$115$	0	0
$116$	2.05257e10	0.0907361
$117$	−1.96305e10	−0.0827770
$118$	2.50760e11	1.00904
$119$	−3.40491e10	−0.130797
$120$	0	0
$121$	−1.71242e11	−0.600193
$122$	−9.21081e10	−0.308545
$123$	1.67272e11	0.535730
$124$	−4.23755e10	−0.129806
$125$	0	0
$126$	−1.94901e10	−0.0546735
$127$	3.75838e11	1.00944	0.504719	−	0.863284i	$-0.331596\pi$
$127$	3.75838e11	1.00944	0.504719	+	0.863284i	$0.331596\pi$
$128$	2.05283e11	0.528077
$129$	−5.21441e10	−0.128517
$130$	0	0
$131$	5.46002e10	0.123652	0.0618261	−	0.998087i	$-0.480308\pi$
$131$	5.46002e10	0.123652	0.0618261	+	0.998087i	$0.480308\pi$
$132$	−3.85056e10	−0.0836311
$133$	−7.56761e10	−0.157679
$134$	1.00051e11	0.200053
$135$	0	0
$136$	−4.09969e11	−0.755592
$137$	−9.76073e11	−1.72790	−0.863951	−	0.503576i	$-0.832017\pi$
$137$	−9.76073e11	−1.72790	−0.863951	+	0.503576i	$0.832017\pi$
$138$	2.78505e11	0.473695
$139$	−6.48305e11	−1.05974	−0.529868	−	0.848080i	$-0.677759\pi$
$139$	−6.48305e11	−1.05974	−0.529868	+	0.848080i	$0.677759\pi$
$140$	0	0
$141$	3.65595e11	0.552452
$142$	8.37020e10	0.121661
$143$	−1.12280e11	−0.157020
$144$	−1.77933e11	−0.239477
$145$	0	0
$146$	−8.87585e11	−1.10731
$147$	−4.63723e11	−0.557202
$148$	−3.47944e11	−0.402781
$149$	−3.99487e11	−0.445634	−0.222817	−	0.974860i	$-0.571525\pi$
$149$	−3.99487e11	−0.445634	−0.222817	+	0.974860i	$0.571525\pi$
$150$	0	0
$151$	−9.67557e11	−1.00301	−0.501503	−	0.865156i	$-0.667219\pi$
$151$	−9.67557e11	−1.00301	−0.501503	+	0.865156i	$0.667219\pi$
$152$	−9.11179e11	−0.910886
$153$	2.42038e11	0.233389
$154$	−1.11477e11	−0.103711
$155$	0	0
$156$	3.79017e10	0.0328453
$157$	−9.95877e11	−0.833216	−0.416608	−	0.909086i	$-0.636781\pi$
$157$	−9.95877e11	−0.833216	−0.416608	+	0.909086i	$0.636781\pi$
$158$	1.59544e12	1.28904
$159$	−5.82317e11	−0.454439
$160$	0	0
$161$	−2.39605e11	−0.174563
$162$	1.38545e11	0.0975573
$163$	−4.79168e11	−0.326179	−0.163089	−	0.986611i	$-0.552146\pi$
$163$	−4.79168e11	−0.326179	−0.163089	+	0.986611i	$0.552146\pi$
$164$	−3.22962e11	−0.212574
$165$	0	0
$166$	−3.71246e11	−0.228595
$167$	−2.66692e12	−1.58880	−0.794399	−	0.607396i	$-0.792214\pi$
$167$	−2.66692e12	−1.58880	−0.794399	+	0.607396i	$0.792214\pi$
$168$	2.01893e11	0.116391
$169$	−1.68164e12	−0.938332
$170$	0	0
$171$	5.37943e11	0.281357
$172$	1.00677e11	0.0509948
$173$	−1.90096e12	−0.932651	−0.466326	−	0.884613i	$-0.654423\pi$
$173$	−1.90096e12	−0.932651	−0.466326	+	0.884613i	$0.654423\pi$
$174$	−4.22415e11	−0.200779
$175$	0	0
$176$	−1.01772e12	−0.454265
$177$	1.53355e12	0.663505
$178$	2.14628e12	0.900279
$179$	6.26039e11	0.254630	0.127315	−	0.991862i	$-0.459364\pi$
$179$	6.26039e11	0.254630	0.127315	+	0.991862i	$0.459364\pi$
$180$	0	0
$181$	2.37886e12	0.910198	0.455099	−	0.890441i	$-0.349604\pi$
$181$	2.37886e12	0.910198	0.455099	+	0.890441i	$0.349604\pi$
$182$	1.09729e11	0.0407314
$183$	−5.63296e11	−0.202888
$184$	−2.88496e12	−1.00842
$185$	0	0
$186$	8.72078e11	0.287233
$187$	1.38438e12	0.442718
$188$	−7.05874e11	−0.219209
$189$	−1.19194e11	−0.0359513
$190$	0	0
$191$	2.67088e11	0.0760276	0.0380138	−	0.999277i	$-0.487897\pi$
$191$	2.67088e11	0.0760276	0.0380138	+	0.999277i	$0.487897\pi$
$192$	2.32135e12	0.642073
$193$	2.78544e12	0.748736	0.374368	−	0.927280i	$-0.377860\pi$
$193$	2.78544e12	0.748736	0.374368	+	0.927280i	$0.377860\pi$
$194$	5.06838e12	1.32422
$195$	0	0
$196$	8.95334e11	0.221094
$197$	3.06625e11	0.0736281	0.0368140	−	0.999322i	$-0.488279\pi$
$197$	3.06625e11	0.0736281	0.0368140	+	0.999322i	$0.488279\pi$
$198$	7.92437e11	0.185057
$199$	1.60906e12	0.365495	0.182747	−	0.983160i	$-0.441501\pi$
$199$	1.60906e12	0.365495	0.182747	+	0.983160i	$0.441501\pi$
$200$	0	0
$201$	6.11872e11	0.131547
$202$	7.54525e12	1.57849
$203$	3.63413e11	0.0739899
$204$	−4.67316e11	−0.0926072
$205$	0	0
$206$	−7.09387e12	−1.33233
$207$	1.70323e12	0.311484
$208$	1.00176e12	0.178408
$209$	3.07686e12	0.533708
$210$	0	0
$211$	1.11046e13	1.82789	0.913944	−	0.405841i	$-0.133021\pi$
$211$	1.11046e13	1.82789	0.913944	+	0.405841i	$0.133021\pi$
$212$	1.12431e12	0.180318
$213$	5.11888e11	0.0799995
$214$	4.61756e12	0.703294
$215$	0	0
$216$	−1.43516e12	−0.207684
$217$	−7.50269e11	−0.105849
$218$	−9.22791e12	−1.26938
$219$	−5.42812e12	−0.728126
$220$	0	0
$221$	−1.36267e12	−0.173873
$222$	7.16060e12	0.891267
$223$	−3.50005e12	−0.425009	−0.212504	−	0.977160i	$-0.568162\pi$
$223$	−3.50005e12	−0.425009	−0.212504	+	0.977160i	$0.568162\pi$
$224$	−7.06957e11	−0.0837585
$225$	0	0
$226$	3.22556e12	0.363923
$227$	−1.28426e13	−1.41420	−0.707100	−	0.707114i	$-0.749997\pi$
$227$	−1.28426e13	−1.41420	−0.707100	+	0.707114i	$0.749997\pi$
$228$	−1.03863e12	−0.111640
$229$	1.41749e13	1.48738	0.743692	−	0.668522i	$-0.233073\pi$
$229$	1.41749e13	1.48738	0.743692	+	0.668522i	$0.233073\pi$
$230$	0	0
$231$	−6.81752e11	−0.0681962
$232$	4.37568e12	0.427427
$233$	−1.67924e13	−1.60197	−0.800984	−	0.598685i	$-0.795690\pi$
$233$	−1.67924e13	−1.60197	−0.800984	+	0.598685i	$0.795690\pi$
$234$	−7.80008e11	−0.0726795
$235$	0	0
$236$	−2.96090e12	−0.263274
$237$	9.75708e12	0.847626
$238$	−1.35292e12	−0.114842
$239$	2.33977e13	1.94082	0.970408	−	0.241471i	$-0.0776298\pi$
$239$	2.33977e13	1.94082	0.970408	+	0.241471i	$0.0776298\pi$
$240$	0	0
$241$	2.09226e13	1.65776	0.828880	−	0.559426i	$-0.188978\pi$
$241$	2.09226e13	1.65776	0.828880	+	0.559426i	$0.188978\pi$
$242$	−6.80421e12	−0.526979
$243$	8.47289e11	0.0641500
$244$	1.08759e12	0.0805045
$245$	0	0
$246$	6.64648e12	0.470380
$247$	−3.02861e12	−0.209609
$248$	−9.03363e12	−0.611474
$249$	−2.27039e12	−0.150316
$250$	0	0
$251$	−2.52862e13	−1.60205	−0.801027	−	0.598628i	$-0.795713\pi$
$251$	−2.52862e13	−1.60205	−0.801027	+	0.598628i	$0.795713\pi$
$252$	2.30134e11	0.0142652
$253$	9.74193e12	0.590857
$254$	1.49337e13	0.886302
$255$	0	0
$256$	−1.14075e13	−0.648442
$257$	−2.58576e13	−1.43865	−0.719327	−	0.694671i	$-0.755550\pi$
$257$	−2.58576e13	−1.43865	−0.719327	+	0.694671i	$0.755550\pi$
$258$	−2.07192e12	−0.112840
$259$	−6.16043e12	−0.328445
$260$	0	0
$261$	−2.58332e12	−0.132025
$262$	2.16951e12	0.108569
$263$	7.31654e12	0.358549	0.179275	−	0.983799i	$-0.442625\pi$
$263$	7.31654e12	0.358549	0.179275	+	0.983799i	$0.442625\pi$
$264$	−8.20864e12	−0.393958
$265$	0	0
$266$	−3.00695e12	−0.138445
$267$	1.31258e13	0.591990
$268$	−1.18137e12	−0.0521971
$269$	1.59590e12	0.0690825	0.0345413	−	0.999403i	$-0.489003\pi$
$269$	1.59590e12	0.0690825	0.0345413	+	0.999403i	$0.489003\pi$
$270$	0	0
$271$	1.19634e13	0.497190	0.248595	−	0.968607i	$-0.420031\pi$
$271$	1.19634e13	0.497190	0.248595	+	0.968607i	$0.420031\pi$
$272$	−1.23514e13	−0.503021
$273$	6.71059e11	0.0267834
$274$	−3.87837e13	−1.51712
$275$	0	0
$276$	−3.28851e12	−0.123595
$277$	6.80599e12	0.250757	0.125378	−	0.992109i	$-0.459986\pi$
$277$	6.80599e12	0.250757	0.125378	+	0.992109i	$0.459986\pi$
$278$	−2.57600e13	−0.930465
$279$	5.33328e12	0.188874
$280$	0	0
$281$	−3.46056e11	−0.0117832	−0.00589158	−	0.999983i	$-0.501875\pi$
$281$	−3.46056e11	−0.0117832	−0.00589158	+	0.999983i	$0.501875\pi$
$282$	1.45267e13	0.485062
$283$	5.40349e13	1.76949	0.884746	−	0.466073i	$-0.154331\pi$
$283$	5.40349e13	1.76949	0.884746	+	0.466073i	$0.154331\pi$
$284$	−9.88330e11	−0.0317432
$285$	0	0
$286$	−4.46140e12	−0.137866
$287$	−5.71812e12	−0.173342
$288$	5.02540e12	0.149455
$289$	−1.74706e13	−0.509766
$290$	0	0
$291$	3.09962e13	0.870756
$292$	1.04804e13	0.288915
$293$	5.69268e13	1.54008	0.770042	−	0.637993i	$-0.220235\pi$
$293$	5.69268e13	1.54008	0.770042	+	0.637993i	$0.220235\pi$
$294$	−1.84258e13	−0.489232
$295$	0	0
$296$	−7.41747e13	−1.89737
$297$	4.84623e12	0.121687
$298$	−1.58734e13	−0.391273
$299$	−9.58914e12	−0.232054
$300$	0	0
$301$	1.78252e12	0.0415832
$302$	−3.84453e13	−0.880654
$303$	4.61438e13	1.03795
$304$	−2.74516e13	−0.606405
$305$	0	0
$306$	9.61725e12	0.204919
$307$	5.47860e13	1.14659	0.573296	−	0.819348i	$-0.305665\pi$
$307$	5.47860e13	1.14659	0.573296	+	0.819348i	$0.305665\pi$
$308$	1.31629e12	0.0270598
$309$	−4.33833e13	−0.876094
$310$	0	0
$311$	5.94582e13	1.15886	0.579428	−	0.815023i	$-0.303276\pi$
$311$	5.94582e13	1.15886	0.579428	+	0.815023i	$0.303276\pi$
$312$	8.07990e12	0.154723
$313$	−5.89791e12	−0.110970	−0.0554848	−	0.998460i	$-0.517670\pi$
$313$	−5.89791e12	−0.110970	−0.0554848	+	0.998460i	$0.517670\pi$
$314$	−3.95706e13	−0.731577
$315$	0	0
$316$	−1.88385e13	−0.336332
$317$	−6.77326e13	−1.18843	−0.594213	−	0.804308i	$-0.702536\pi$
$317$	−6.77326e13	−1.18843	−0.594213	+	0.804308i	$0.702536\pi$
$318$	−2.31381e13	−0.399005
$319$	−1.47758e13	−0.250439
$320$	0	0
$321$	2.82391e13	0.462460
$322$	−9.52056e12	−0.153269
$323$	3.73417e13	0.590990
$324$	−1.63591e12	−0.0254543
$325$	0	0
$326$	−1.90395e13	−0.286390
$327$	−5.64342e13	−0.834699
$328$	−6.88491e13	−1.00136
$329$	−1.24977e13	−0.178752
$330$	0	0
$331$	−2.52659e13	−0.349527	−0.174763	−	0.984610i	$-0.555916\pi$
$331$	−2.52659e13	−0.349527	−0.174763	+	0.984610i	$0.555916\pi$
$332$	4.38357e12	0.0596442
$333$	4.37913e13	0.586064
$334$	−1.05968e14	−1.39499
$335$	0	0
$336$	6.08255e12	0.0774854
$337$	−1.78886e13	−0.224187	−0.112094	−	0.993698i	$-0.535756\pi$
$337$	−1.78886e13	−0.224187	−0.112094	+	0.993698i	$0.535756\pi$
$338$	−6.68191e13	−0.823870
$339$	1.97262e13	0.239302
$340$	0	0
$341$	3.05047e13	0.358276
$342$	2.13749e13	0.247036
$343$	3.22774e13	0.367097
$344$	2.14625e13	0.240219
$345$	0	0
$346$	−7.55336e13	−0.818882
$347$	−1.13535e14	−1.21148	−0.605740	−	0.795663i	$-0.707123\pi$
$347$	−1.13535e14	−1.21148	−0.605740	+	0.795663i	$0.707123\pi$
$348$	4.98776e12	0.0523865
$349$	8.49062e13	0.877808	0.438904	−	0.898534i	$-0.355367\pi$
$349$	8.49062e13	0.877808	0.438904	+	0.898534i	$0.355367\pi$
$350$	0	0
$351$	−4.77022e12	−0.0477913
$352$	2.87437e13	0.283503
$353$	3.67232e13	0.356598	0.178299	−	0.983976i	$-0.442941\pi$
$353$	3.67232e13	0.356598	0.178299	+	0.983976i	$0.442941\pi$
$354$	6.09346e13	0.582568
$355$	0	0
$356$	−2.53427e13	−0.234897
$357$	−8.27394e12	−0.0755157
$358$	2.48753e13	0.223569
$359$	1.54790e14	1.37001	0.685003	−	0.728540i	$-0.259801\pi$
$359$	1.54790e14	1.37001	0.685003	+	0.728540i	$0.259801\pi$
$360$	0	0
$361$	−3.34962e13	−0.287545
$362$	9.45225e13	0.799168
$363$	−4.16118e13	−0.346522
$364$	−1.29565e12	−0.0106275
$365$	0	0
$366$	−2.23823e13	−0.178139
$367$	2.02722e13	0.158942	0.0794709	−	0.996837i	$-0.474677\pi$
$367$	2.02722e13	0.158942	0.0794709	+	0.996837i	$0.474677\pi$
$368$	−8.69170e13	−0.671338
$369$	4.06472e13	0.309304
$370$	0	0
$371$	1.99062e13	0.147039
$372$	−1.02973e13	−0.0749437
$373$	−1.37431e14	−0.985565	−0.492782	−	0.870153i	$-0.664020\pi$
$373$	−1.37431e14	−0.985565	−0.492782	+	0.870153i	$0.664020\pi$
$374$	5.50076e13	0.388713
$375$	0	0
$376$	−1.50478e14	−1.03262
$377$	1.45440e13	0.0983576
$378$	−4.73610e12	−0.0315658
$379$	1.76730e14	1.16090	0.580449	−	0.814296i	$-0.302877\pi$
$379$	1.76730e14	1.16090	0.580449	+	0.814296i	$0.302877\pi$
$380$	0	0
$381$	9.13285e13	0.582799
$382$	1.06126e13	0.0667535
$383$	1.40640e13	0.0871998	0.0435999	−	0.999049i	$-0.486117\pi$
$383$	1.40640e13	0.0871998	0.0435999	+	0.999049i	$0.486117\pi$
$384$	4.98837e13	0.304885
$385$	0	0
$386$	1.10678e14	0.657402
$387$	−1.26710e13	−0.0741995
$388$	−5.98460e13	−0.345510
$389$	2.57438e13	0.146538	0.0732688	−	0.997312i	$-0.476657\pi$
$389$	2.57438e13	0.146538	0.0732688	+	0.997312i	$0.476657\pi$
$390$	0	0
$391$	1.18231e14	0.654273
$392$	1.90868e14	1.04150
$393$	1.32679e13	0.0713907
$394$	1.21836e13	0.0646466
$395$	0	0
$396$	−9.35687e12	−0.0482844
$397$	−5.99711e13	−0.305207	−0.152603	−	0.988288i	$-0.548766\pi$
$397$	−5.99711e13	−0.305207	−0.152603	+	0.988288i	$0.548766\pi$
$398$	6.39352e13	0.320910
$399$	−1.83893e13	−0.0910361
$400$	0	0
$401$	2.76568e14	1.33201	0.666007	−	0.745946i	$-0.268002\pi$
$401$	2.76568e14	1.33201	0.666007	+	0.745946i	$0.268002\pi$
$402$	2.43124e13	0.115501
$403$	−3.00263e13	−0.140710
$404$	−8.90923e13	−0.411853
$405$	0	0
$406$	1.44400e13	0.0649643
$407$	2.50473e14	1.11171
$408$	−9.96225e13	−0.436241
$409$	1.76679e14	0.763319	0.381659	−	0.924303i	$-0.375353\pi$
$409$	1.76679e14	0.763319	0.381659	+	0.924303i	$0.375353\pi$
$410$	0	0
$411$	−2.37186e14	−0.997605
$412$	8.37624e13	0.347628
$413$	−5.24235e13	−0.214684
$414$	6.76768e13	0.273488
$415$	0	0
$416$	−2.82929e13	−0.111343
$417$	−1.57538e14	−0.611839
$418$	1.22257e14	0.468604
$419$	−2.87711e14	−1.08838	−0.544189	−	0.838963i	$-0.683163\pi$
$419$	−2.87711e14	−1.08838	−0.544189	+	0.838963i	$0.683163\pi$
$420$	0	0
$421$	3.95759e14	1.45841	0.729204	−	0.684296i	$-0.239890\pi$
$421$	3.95759e14	1.45841	0.729204	+	0.684296i	$0.239890\pi$
$422$	4.41235e14	1.60491
$423$	8.88395e13	0.318959
$424$	2.39681e14	0.849418
$425$	0	0
$426$	2.03396e13	0.0702408
$427$	1.92560e13	0.0656466
$428$	−5.45229e13	−0.183501
$429$	−2.72842e13	−0.0906558
$430$	0	0
$431$	1.13334e14	0.367060	0.183530	−	0.983014i	$-0.441248\pi$
$431$	1.13334e14	0.367060	0.183530	+	0.983014i	$0.441248\pi$
$432$	−4.32378e13	−0.138262
$433$	−2.49811e14	−0.788729	−0.394365	−	0.918954i	$-0.629035\pi$
$433$	−2.49811e14	−0.788729	−0.394365	+	0.918954i	$0.629035\pi$
$434$	−2.98115e13	−0.0929374
$435$	0	0
$436$	1.08961e14	0.331203
$437$	2.62775e14	0.788743
$438$	−2.15683e14	−0.639306
$439$	1.73432e14	0.507662	0.253831	−	0.967249i	$-0.418309\pi$
$439$	1.73432e14	0.507662	0.253831	+	0.967249i	$0.418309\pi$
$440$	0	0
$441$	−1.12685e14	−0.321701
$442$	−5.41449e13	−0.152664
$443$	3.23012e13	0.0899493	0.0449746	−	0.998988i	$-0.485679\pi$
$443$	3.23012e13	0.0899493	0.0449746	+	0.998988i	$0.485679\pi$
$444$	−8.45503e13	−0.232546
$445$	0	0
$446$	−1.39073e14	−0.373164
$447$	−9.70753e13	−0.257287
$448$	−7.93542e13	−0.207750
$449$	−3.89694e14	−1.00779	−0.503893	−	0.863766i	$-0.668099\pi$
$449$	−3.89694e14	−1.00779	−0.503893	+	0.863766i	$0.668099\pi$
$450$	0	0
$451$	2.32489e14	0.586721
$452$	−3.80865e13	−0.0949533
$453$	−2.35116e14	−0.579085
$454$	−5.10293e14	−1.24169
$455$	0	0
$456$	−2.21417e14	−0.525900
$457$	−5.73678e14	−1.34626	−0.673131	−	0.739524i	$-0.735051\pi$
$457$	−5.73678e14	−1.34626	−0.673131	+	0.739524i	$0.735051\pi$
$458$	5.63230e14	1.30595
$459$	5.88152e13	0.134747
$460$	0	0
$461$	1.24832e14	0.279236	0.139618	−	0.990205i	$-0.455412\pi$
$461$	1.24832e14	0.279236	0.139618	+	0.990205i	$0.455412\pi$
$462$	−2.70890e13	−0.0598773
$463$	3.90762e14	0.853526	0.426763	−	0.904363i	$-0.359654\pi$
$463$	3.90762e14	0.853526	0.426763	+	0.904363i	$0.359654\pi$
$464$	1.31829e14	0.284552
$465$	0	0
$466$	−6.67235e14	−1.40655
$467$	−6.44667e13	−0.134305	−0.0671526	−	0.997743i	$-0.521391\pi$
$467$	−6.44667e13	−0.134305	−0.0671526	+	0.997743i	$0.521391\pi$
$468$	9.21012e12	0.0189633
$469$	−2.09165e13	−0.0425636
$470$	0	0
$471$	−2.41998e14	−0.481058
$472$	−6.31206e14	−1.24020
$473$	−7.24743e13	−0.140750
$474$	3.87692e14	0.744229
$475$	0	0
$476$	1.59749e13	0.0299641
$477$	−1.41503e14	−0.262370
$478$	9.29694e14	1.70407
$479$	−7.25709e14	−1.31497	−0.657487	−	0.753466i	$-0.728381\pi$
$479$	−7.25709e14	−1.31497	−0.657487	+	0.753466i	$0.728381\pi$
$480$	0	0
$481$	−2.46544e14	−0.436614
$482$	8.31348e14	1.45554
$483$	−5.82239e13	−0.100784
$484$	8.03423e13	0.137497
$485$	0	0
$486$	3.36665e13	0.0563247
$487$	−7.77844e14	−1.28672	−0.643358	−	0.765565i	$-0.722459\pi$
$487$	−7.77844e14	−1.28672	−0.643358	+	0.765565i	$0.722459\pi$
$488$	2.31852e14	0.379229
$489$	−1.16438e14	−0.188319
$490$	0	0
$491$	−5.33222e14	−0.843257	−0.421629	−	0.906769i	$-0.638541\pi$
$491$	−5.33222e14	−0.843257	−0.421629	+	0.906769i	$0.638541\pi$
$492$	−7.84798e13	−0.122730
$493$	−1.79323e14	−0.277318
$494$	−1.20340e14	−0.184040
$495$	0	0
$496$	−2.72161e14	−0.407077
$497$	−1.74986e13	−0.0258847
$498$	−9.02127e13	−0.131979
$499$	7.21762e14	1.04434	0.522169	−	0.852842i	$-0.325123\pi$
$499$	7.21762e14	1.04434	0.522169	+	0.852842i	$0.325123\pi$
$500$	0	0
$501$	−6.48061e14	−0.917293
$502$	−1.00473e15	−1.40663
$503$	4.77237e14	0.660861	0.330431	−	0.943830i	$-0.392806\pi$
$503$	4.77237e14	0.660861	0.330431	+	0.943830i	$0.392806\pi$
$504$	4.90601e13	0.0671986
$505$	0	0
$506$	3.87090e14	0.518781
$507$	−4.08639e14	−0.541746
$508$	−1.76333e14	−0.231251
$509$	−6.45040e14	−0.836833	−0.418416	−	0.908255i	$-0.637415\pi$
$509$	−6.45040e14	−0.836833	−0.418416	+	0.908255i	$0.637415\pi$
$510$	0	0
$511$	1.85557e14	0.235593
$512$	−8.73690e14	−1.09742
$513$	1.30720e14	0.162441
$514$	−1.02744e15	−1.26316
$515$	0	0
$516$	2.44646e13	0.0294419
$517$	5.08134e14	0.605035
$518$	−2.44781e14	−0.288379
$519$	−4.61933e14	−0.538467
$520$	0	0
$521$	6.27179e14	0.715787	0.357894	−	0.933762i	$-0.383495\pi$
$521$	6.27179e14	0.715787	0.357894	+	0.933762i	$0.383495\pi$
$522$	−1.02647e14	−0.115920
$523$	−4.84276e14	−0.541170	−0.270585	−	0.962696i	$-0.587217\pi$
$523$	−4.84276e14	−0.541170	−0.270585	+	0.962696i	$0.587217\pi$
$524$	−2.56170e13	−0.0283273
$525$	0	0
$526$	2.90719e14	0.314812
$527$	3.70214e14	0.396729
$528$	−2.47306e14	−0.262270
$529$	−1.20815e14	−0.126799
$530$	0	0
$531$	3.72652e14	0.383075
$532$	3.55052e13	0.0361225
$533$	−2.28843e14	−0.230429
$534$	5.21547e14	0.519776
$535$	0	0
$536$	−2.51846e14	−0.245883
$537$	1.52128e14	0.147011
$538$	6.34122e13	0.0606555
$539$	−6.44521e14	−0.610237
$540$	0	0
$541$	−3.49238e14	−0.323994	−0.161997	−	0.986791i	$-0.551794\pi$
$541$	−3.49238e14	−0.323994	−0.161997	+	0.986791i	$0.551794\pi$
$542$	4.75358e14	0.436541
$543$	5.78062e14	0.525503
$544$	3.48842e14	0.313931
$545$	0	0
$546$	2.66642e13	0.0235163
$547$	2.09315e15	1.82755	0.913777	−	0.406216i	$-0.133152\pi$
$547$	2.09315e15	1.82755	0.913777	+	0.406216i	$0.133152\pi$
$548$	4.57947e14	0.395842
$549$	−1.36881e14	−0.117137
$550$	0	0
$551$	−3.98556e14	−0.334315
$552$	−7.01046e14	−0.582213
$553$	−3.33541e14	−0.274259
$554$	2.70432e14	0.220168
$555$	0	0
$556$	3.04167e14	0.242773
$557$	8.90039e13	0.0703405	0.0351702	−	0.999381i	$-0.488803\pi$
$557$	8.90039e13	0.0703405	0.0351702	+	0.999381i	$0.488803\pi$
$558$	2.11915e14	0.165834
$559$	7.13376e13	0.0552782
$560$	0	0
$561$	3.36405e14	0.255603
$562$	−1.37504e13	−0.0103458
$563$	1.46644e14	0.109262	0.0546309	−	0.998507i	$-0.482602\pi$
$563$	1.46644e14	0.109262	0.0546309	+	0.998507i	$0.482602\pi$
$564$	−1.71527e14	−0.126560
$565$	0	0
$566$	2.14705e15	1.55364
$567$	−2.89641e13	−0.0207565
$568$	−2.10693e14	−0.149532
$569$	−9.54948e14	−0.671216	−0.335608	−	0.942002i	$-0.608942\pi$
$569$	−9.54948e14	−0.671216	−0.335608	+	0.942002i	$0.608942\pi$
$570$	0	0
$571$	−1.66072e15	−1.14498	−0.572490	−	0.819911i	$-0.694023\pi$
$571$	−1.66072e15	−1.14498	−0.572490	+	0.819911i	$0.694023\pi$
$572$	5.26790e13	0.0359716
$573$	6.49025e13	0.0438946
$574$	−2.27206e14	−0.152197
$575$	0	0
$576$	5.64088e14	0.370701
$577$	2.80671e15	1.82697	0.913485	−	0.406873i	$-0.133381\pi$
$577$	2.80671e15	1.82697	0.913485	+	0.406873i	$0.133381\pi$
$578$	−6.94186e14	−0.447582
$579$	6.76862e14	0.432283
$580$	0	0
$581$	7.76121e13	0.0486363
$582$	1.23162e15	0.764538
$583$	−8.09354e14	−0.497693
$584$	2.23421e15	1.36098
$585$	0	0
$586$	2.26195e15	1.35222
$587$	7.25934e14	0.429920	0.214960	−	0.976623i	$-0.431038\pi$
$587$	7.25934e14	0.429920	0.214960	+	0.976623i	$0.431038\pi$
$588$	2.17566e14	0.127649
$589$	8.22821e14	0.478268
$590$	0	0
$591$	7.45099e13	0.0425092
$592$	−2.23470e15	−1.26314
$593$	−9.72930e14	−0.544855	−0.272427	−	0.962176i	$-0.587826\pi$
$593$	−9.72930e14	−0.544855	−0.272427	+	0.962176i	$0.587826\pi$
$594$	1.92562e14	0.106843
$595$	0	0
$596$	1.87429e14	0.102090
$597$	3.91002e14	0.211019
$598$	−3.81019e14	−0.203747
$599$	3.35341e15	1.77680	0.888401	−	0.459068i	$-0.151817\pi$
$599$	3.35341e15	1.77680	0.888401	+	0.459068i	$0.151817\pi$
$600$	0	0
$601$	−2.28228e15	−1.18730	−0.593649	−	0.804724i	$-0.702313\pi$
$601$	−2.28228e15	−1.18730	−0.593649	+	0.804724i	$0.702313\pi$
$602$	7.08274e13	0.0365107
$603$	1.48685e14	0.0759489
$604$	4.53952e14	0.229777
$605$	0	0
$606$	1.83350e15	0.911341
$607$	1.05542e15	0.519861	0.259931	−	0.965627i	$-0.416300\pi$
$607$	1.05542e15	0.519861	0.259931	+	0.965627i	$0.416300\pi$
$608$	7.75320e14	0.378453
$609$	8.83094e13	0.0427181
$610$	0	0
$611$	−5.00165e14	−0.237622
$612$	−1.13558e14	−0.0534668
$613$	−7.28605e14	−0.339985	−0.169992	−	0.985445i	$-0.554374\pi$
$613$	−7.28605e14	−0.339985	−0.169992	+	0.985445i	$0.554374\pi$
$614$	2.17689e15	1.00673
$615$	0	0
$616$	2.80608e14	0.127470
$617$	−1.02070e15	−0.459546	−0.229773	−	0.973244i	$-0.573798\pi$
$617$	−1.02070e15	−0.459546	−0.229773	+	0.973244i	$0.573798\pi$
$618$	−1.72381e15	−0.769224
$619$	−9.45033e13	−0.0417973	−0.0208987	−	0.999782i	$-0.506653\pi$
$619$	−9.45033e13	−0.0417973	−0.0208987	+	0.999782i	$0.506653\pi$
$620$	0	0
$621$	4.13884e14	0.179835
$622$	2.36254e15	1.01749
$623$	−4.48699e14	−0.191545
$624$	2.43428e14	0.103004
$625$	0	0
$626$	−2.34350e14	−0.0974330
$627$	7.47678e14	0.308136
$628$	4.67239e14	0.190880
$629$	3.03981e15	1.23103
$630$	0	0
$631$	−2.88945e15	−1.14988	−0.574941	−	0.818195i	$-0.694975\pi$
$631$	−2.88945e15	−1.14988	−0.574941	+	0.818195i	$0.694975\pi$
$632$	−4.01600e15	−1.58435
$633$	2.69842e15	1.05533
$634$	−2.69132e15	−1.04346
$635$	0	0
$636$	2.73208e14	0.104107
$637$	6.34412e14	0.239665
$638$	−5.87108e14	−0.219889
$639$	1.24389e14	0.0461877
$640$	0	0
$641$	−2.72923e15	−0.996140	−0.498070	−	0.867137i	$-0.665958\pi$
$641$	−2.72923e15	−0.996140	−0.498070	+	0.867137i	$0.665958\pi$
$642$	1.12207e15	0.406047
$643$	−1.69179e15	−0.606998	−0.303499	−	0.952832i	$-0.598155\pi$
$643$	−1.69179e15	−0.606998	−0.303499	+	0.952832i	$0.598155\pi$
$644$	1.12416e14	0.0399905
$645$	0	0
$646$	1.48375e15	0.518899
$647$	−3.42569e15	−1.18788	−0.593942	−	0.804508i	$-0.702429\pi$
$647$	−3.42569e15	−1.18788	−0.593942	+	0.804508i	$0.702429\pi$
$648$	−3.48743e14	−0.119907
$649$	2.13145e15	0.726657
$650$	0	0
$651$	−1.82315e14	−0.0611122
$652$	2.24813e14	0.0747238
$653$	−3.99603e15	−1.31706	−0.658530	−	0.752554i	$-0.728822\pi$
$653$	−3.99603e15	−1.31706	−0.658530	+	0.752554i	$0.728822\pi$
$654$	−2.24238e15	−0.732879
$655$	0	0
$656$	−2.07426e15	−0.666640
$657$	−1.31903e15	−0.420384
$658$	−4.96588e14	−0.156947
$659$	−3.16155e15	−0.990899	−0.495450	−	0.868637i	$-0.664997\pi$
$659$	−3.16155e15	−0.990899	−0.495450	+	0.868637i	$0.664997\pi$
$660$	0	0
$661$	4.23425e15	1.30517	0.652586	−	0.757714i	$-0.273684\pi$
$661$	4.23425e15	1.30517	0.652586	+	0.757714i	$0.273684\pi$
$662$	−1.00393e15	−0.306890
$663$	−3.31129e14	−0.100386
$664$	9.34490e14	0.280964
$665$	0	0
$666$	1.74002e15	0.514573
$667$	−1.26190e15	−0.370112
$668$	1.25125e15	0.363976
$669$	−8.50513e14	−0.245379
$670$	0	0
$671$	−7.82917e14	−0.222199
$672$	−1.71791e14	−0.0483580
$673$	3.41491e14	0.0953448	0.0476724	−	0.998863i	$-0.484820\pi$
$673$	3.41491e14	0.0953448	0.0476724	+	0.998863i	$0.484820\pi$
$674$	−7.10792e14	−0.196840
$675$	0	0
$676$	7.88981e14	0.214961
$677$	−3.72275e15	−1.00606	−0.503032	−	0.864268i	$-0.667782\pi$
$677$	−3.72275e15	−1.00606	−0.503032	+	0.864268i	$0.667782\pi$
$678$	7.83811e14	0.210111
$679$	−1.05959e15	−0.281743
$680$	0	0
$681$	−3.12075e15	−0.816488
$682$	1.21209e15	0.314572
$683$	−1.69711e15	−0.436915	−0.218458	−	0.975846i	$-0.570103\pi$
$683$	−1.69711e15	−0.436915	−0.218458	+	0.975846i	$0.570103\pi$
$684$	−2.52388e14	−0.0644556
$685$	0	0
$686$	1.28253e15	0.322317
$687$	3.44449e15	0.858742
$688$	6.46611e14	0.159922
$689$	7.96660e14	0.195464
$690$	0	0
$691$	7.05386e14	0.170332	0.0851662	−	0.996367i	$-0.472858\pi$
$691$	7.05386e14	0.170332	0.0851662	+	0.996367i	$0.472858\pi$
$692$	8.91879e14	0.213660
$693$	−1.65666e14	−0.0393731
$694$	−4.51124e15	−1.06370
$695$	0	0
$696$	1.06329e15	0.246775
$697$	2.82156e15	0.649694
$698$	3.37370e15	0.770729
$699$	−4.08054e15	−0.924897
$700$	0	0
$701$	−7.74751e15	−1.72867	−0.864337	−	0.502913i	$-0.832261\pi$
$701$	−7.74751e15	−1.72867	−0.864337	+	0.502913i	$0.832261\pi$
$702$	−1.89542e14	−0.0419615
$703$	6.75615e15	1.48404
$704$	3.22641e15	0.703185
$705$	0	0
$706$	1.45918e15	0.313099
$707$	−1.57740e15	−0.335842
$708$	−7.19499e14	−0.152001
$709$	−2.07265e15	−0.434481	−0.217241	−	0.976118i	$-0.569706\pi$
$709$	−2.07265e15	−0.434481	−0.217241	+	0.976118i	$0.569706\pi$
$710$	0	0
$711$	2.37097e15	0.489377
$712$	−5.40257e15	−1.10652
$713$	2.60521e15	0.529480
$714$	−3.28760e14	−0.0663040
$715$	0	0
$716$	−2.93721e14	−0.0583328
$717$	5.68564e15	1.12053
$718$	6.15049e15	1.20289
$719$	−9.04462e15	−1.75542	−0.877711	−	0.479190i	$-0.840930\pi$
$719$	−9.04462e15	−1.75542	−0.877711	+	0.479190i	$0.840930\pi$
$720$	0	0
$721$	1.48303e15	0.283470
$722$	−1.33096e15	−0.252469
$723$	5.08419e15	0.957108
$724$	−1.11610e15	−0.208516
$725$	0	0
$726$	−1.65342e15	−0.304252
$727$	1.75147e14	0.0319862	0.0159931	−	0.999872i	$-0.494909\pi$
$727$	1.75147e14	0.0319862	0.0159931	+	0.999872i	$0.494909\pi$
$728$	−2.76207e14	−0.0500625
$729$	2.05891e14	0.0370370
$730$	0	0
$731$	−8.79569e14	−0.155856
$732$	2.64284e14	0.0464793
$733$	5.21010e15	0.909441	0.454721	−	0.890634i	$-0.349739\pi$
$733$	5.21010e15	0.909441	0.454721	+	0.890634i	$0.349739\pi$
$734$	8.05506e14	0.139553
$735$	0	0
$736$	2.45481e15	0.418977
$737$	8.50431e14	0.144068
$738$	1.61509e15	0.271574
$739$	−6.31664e15	−1.05425	−0.527123	−	0.849789i	$-0.676729\pi$
$739$	−6.31664e15	−1.05425	−0.527123	+	0.849789i	$0.676729\pi$
$740$	0	0
$741$	−7.35951e14	−0.121018
$742$	7.90962e14	0.129102
$743$	−8.49495e15	−1.37633	−0.688165	−	0.725555i	$-0.741583\pi$
$743$	−8.49495e15	−1.37633	−0.688165	+	0.725555i	$0.741583\pi$
$744$	−2.19517e15	−0.353035
$745$	0	0
$746$	−5.46073e15	−0.865341
$747$	−5.51705e14	−0.0867848
$748$	−6.49515e14	−0.101422
$749$	−9.65340e14	−0.149634
$750$	0	0
$751$	6.40712e14	0.0978686	0.0489343	−	0.998802i	$-0.484418\pi$
$751$	6.40712e14	0.0978686	0.0489343	+	0.998802i	$0.484418\pi$
$752$	−4.53355e15	−0.687448
$753$	−6.14454e15	−0.924947
$754$	5.77900e14	0.0863595
$755$	0	0
$756$	5.59226e13	0.00823602
$757$	−1.66077e15	−0.242819	−0.121410	−	0.992602i	$-0.538741\pi$
$757$	−1.66077e15	−0.242819	−0.121410	+	0.992602i	$0.538741\pi$
$758$	7.02226e15	1.01929
$759$	2.36729e15	0.341131
$760$	0	0
$761$	1.19452e16	1.69659	0.848295	−	0.529524i	$-0.177629\pi$
$761$	1.19452e16	1.69659	0.848295	+	0.529524i	$0.177629\pi$
$762$	3.62889e15	0.511707
$763$	1.92917e15	0.270076
$764$	−1.25311e14	−0.0174171
$765$	0	0
$766$	5.58825e14	0.0765628
$767$	−2.09802e15	−0.285388
$768$	−2.77203e15	−0.374378
$769$	−1.80111e15	−0.241516	−0.120758	−	0.992682i	$-0.538533\pi$
$769$	−1.80111e15	−0.241516	−0.120758	+	0.992682i	$0.538533\pi$
$770$	0	0
$771$	−6.28341e15	−0.830608
$772$	−1.30685e15	−0.171527
$773$	−5.34907e15	−0.697094	−0.348547	−	0.937291i	$-0.613325\pi$
$773$	−5.34907e15	−0.697094	−0.348547	+	0.937291i	$0.613325\pi$
$774$	−5.03476e14	−0.0651484
$775$	0	0
$776$	−1.27580e16	−1.62758
$777$	−1.49698e15	−0.189628
$778$	1.02291e15	0.128662
$779$	6.27107e15	0.783223
$780$	0	0
$781$	7.11465e14	0.0876139
$782$	4.69784e15	0.574462
$783$	−6.27747e14	−0.0762245
$784$	5.75038e15	0.693359
$785$	0	0
$786$	5.27191e14	0.0626821
$787$	−9.60045e15	−1.13352	−0.566762	−	0.823882i	$-0.691804\pi$
$787$	−9.60045e15	−1.13352	−0.566762	+	0.823882i	$0.691804\pi$
$788$	−1.43860e14	−0.0168673
$789$	1.77792e15	0.207009
$790$	0	0
$791$	−6.74331e14	−0.0774288
$792$	−1.99470e15	−0.227452
$793$	7.70638e14	0.0872665
$794$	−2.38292e15	−0.267976
$795$	0	0
$796$	−7.54929e14	−0.0837307
$797$	2.59705e15	0.286062	0.143031	−	0.989718i	$-0.454315\pi$
$797$	2.59705e15	0.286062	0.143031	+	0.989718i	$0.454315\pi$
$798$	−7.30688e14	−0.0799312
$799$	6.16687e15	0.669973
$800$	0	0
$801$	3.18957e15	0.341786
$802$	1.09893e16	1.16953
$803$	−7.54445e15	−0.797429
$804$	−2.87074e14	−0.0301360
$805$	0	0
$806$	−1.19308e15	−0.123545
$807$	3.87804e14	0.0398848
$808$	−1.89927e16	−1.94010
$809$	1.39740e15	0.141777	0.0708883	−	0.997484i	$-0.477417\pi$
$809$	1.39740e15	0.141777	0.0708883	+	0.997484i	$0.477417\pi$
$810$	0	0
$811$	9.98117e15	0.999003	0.499502	−	0.866313i	$-0.333516\pi$
$811$	9.98117e15	0.999003	0.499502	+	0.866313i	$0.333516\pi$
$812$	−1.70504e14	−0.0169502
$813$	2.90710e15	0.287053
$814$	9.95240e15	0.976098
$815$	0	0
$816$	−3.00138e15	−0.290419
$817$	−1.95489e15	−0.187889
$818$	7.02023e15	0.670206
$819$	1.63067e14	0.0154634
$820$	0	0
$821$	1.15847e16	1.08392	0.541962	−	0.840403i	$-0.317681\pi$
$821$	1.15847e16	1.08392	0.541962	+	0.840403i	$0.317681\pi$
$822$	−9.42444e15	−0.875912
$823$	−5.28588e15	−0.487998	−0.243999	−	0.969775i	$-0.578459\pi$
$823$	−5.28588e15	−0.487998	−0.243999	+	0.969775i	$0.578459\pi$
$824$	1.78565e16	1.63756
$825$	0	0
$826$	−2.08302e15	−0.188496
$827$	−8.34089e15	−0.749777	−0.374888	−	0.927070i	$-0.622319\pi$
$827$	−8.34089e15	−0.749777	−0.374888	+	0.927070i	$0.622319\pi$
$828$	−7.99109e14	−0.0713574
$829$	1.46959e16	1.30360	0.651802	−	0.758389i	$-0.274013\pi$
$829$	1.46959e16	1.30360	0.651802	+	0.758389i	$0.274013\pi$
$830$	0	0
$831$	1.65385e15	0.144774
$832$	−3.17581e15	−0.276170
$833$	−7.82209e15	−0.675733
$834$	−6.25969e15	−0.537204
$835$	0	0
$836$	−1.44358e15	−0.122266
$837$	1.29599e15	0.109046
$838$	−1.14320e16	−0.955613
$839$	−1.76906e16	−1.46910	−0.734550	−	0.678555i	$-0.762607\pi$
$839$	−1.76906e16	−1.46910	−0.734550	+	0.678555i	$0.762607\pi$
$840$	0	0
$841$	−1.02866e16	−0.843125
$842$	1.57253e16	1.28051
$843$	−8.40916e13	−0.00680301
$844$	−5.20998e15	−0.418748
$845$	0	0
$846$	3.52999e15	0.280051
$847$	1.42248e15	0.112121
$848$	7.22101e15	0.565484
$849$	1.31305e16	1.02162
$850$	0	0
$851$	2.13912e16	1.64295
$852$	−2.40164e14	−0.0183270
$853$	3.83297e15	0.290614	0.145307	−	0.989387i	$-0.453583\pi$
$853$	3.83297e15	0.290614	0.145307	+	0.989387i	$0.453583\pi$
$854$	7.65126e14	0.0576388
$855$	0	0
$856$	−1.16232e16	−0.864410
$857$	7.73461e15	0.571536	0.285768	−	0.958299i	$-0.407751\pi$
$857$	7.73461e15	0.571536	0.285768	+	0.958299i	$0.407751\pi$
$858$	−1.08412e15	−0.0795972
$859$	6.21339e14	0.0453279	0.0226640	−	0.999743i	$-0.492785\pi$
$859$	6.21339e14	0.0453279	0.0226640	+	0.999743i	$0.492785\pi$
$860$	0	0
$861$	−1.38950e15	−0.100079
$862$	4.50328e15	0.322284
$863$	2.04952e15	0.145745	0.0728724	−	0.997341i	$-0.476783\pi$
$863$	2.04952e15	0.145745	0.0728724	+	0.997341i	$0.476783\pi$
$864$	1.22117e15	0.0862881
$865$	0	0
$866$	−9.92610e15	−0.692517
$867$	−4.24536e15	−0.294313
$868$	3.52006e14	0.0242489
$869$	1.35612e16	0.928303
$870$	0	0
$871$	−8.37093e14	−0.0565814
$872$	2.32283e16	1.56018
$873$	7.53207e15	0.502731
$874$	1.04412e16	0.692529
$875$	0	0
$876$	2.54673e15	0.166805
$877$	1.06115e16	0.690683	0.345341	−	0.938477i	$-0.387763\pi$
$877$	1.06115e16	0.690683	0.345341	+	0.938477i	$0.387763\pi$
$878$	6.89123e15	0.445735
$879$	1.38332e16	0.889168
$880$	0	0
$881$	1.76636e16	1.12127	0.560637	−	0.828061i	$-0.310556\pi$
$881$	1.76636e16	1.12127	0.560637	+	0.828061i	$0.310556\pi$
$882$	−4.47746e15	−0.282458
$883$	9.09839e15	0.570402	0.285201	−	0.958468i	$-0.407940\pi$
$883$	9.09839e15	0.570402	0.285201	+	0.958468i	$0.407940\pi$
$884$	6.39328e14	0.0398324
$885$	0	0
$886$	1.28347e15	0.0789769
$887$	2.02538e16	1.23859	0.619294	−	0.785160i	$-0.287419\pi$
$887$	2.02538e16	1.23859	0.619294	+	0.785160i	$0.287419\pi$
$888$	−1.80245e16	−1.09545
$889$	−3.12202e15	−0.188571
$890$	0	0
$891$	1.17763e15	0.0702558
$892$	1.64213e15	0.0973646
$893$	1.37062e16	0.807670
$894$	−3.85724e15	−0.225902
$895$	0	0
$896$	−1.70525e15	−0.0986491
$897$	−2.33016e15	−0.133976
$898$	−1.54843e16	−0.884852
$899$	−3.95137e15	−0.224424
$900$	0	0
$901$	−9.82256e15	−0.551110
$902$	9.23783e15	0.515151
$903$	4.33152e14	0.0240081
$904$	−8.11930e15	−0.447293
$905$	0	0
$906$	−9.34222e15	−0.508446
$907$	−2.92582e15	−0.158273	−0.0791367	−	0.996864i	$-0.525216\pi$
$907$	−2.92582e15	−0.158273	−0.0791367	+	0.996864i	$0.525216\pi$
$908$	6.02540e15	0.323977
$909$	1.12129e16	0.599264
$910$	0	0
$911$	2.70247e16	1.42695	0.713476	−	0.700679i	$-0.247120\pi$
$911$	2.70247e16	1.42695	0.713476	+	0.700679i	$0.247120\pi$
$912$	−6.67074e15	−0.350108
$913$	−3.15558e15	−0.164623
$914$	−2.27948e16	−1.18204
$915$	0	0
$916$	−6.65046e15	−0.340743
$917$	−4.53554e14	−0.0230993
$918$	2.33699e15	0.118310
$919$	−3.08657e16	−1.55325	−0.776624	−	0.629964i	$-0.783070\pi$
$919$	−3.08657e16	−1.55325	−0.776624	+	0.629964i	$0.783070\pi$
$920$	0	0
$921$	1.33130e16	0.661985
$922$	4.96015e15	0.245174
$923$	−7.00307e14	−0.0344095
$924$	3.19860e14	0.0156230
$925$	0	0
$926$	1.55267e16	0.749409
$927$	−1.05421e16	−0.505813
$928$	−3.72326e15	−0.177586
$929$	−1.90121e16	−0.901452	−0.450726	−	0.892662i	$-0.648835\pi$
$929$	−1.90121e16	−0.901452	−0.450726	+	0.892662i	$0.648835\pi$
$930$	0	0
$931$	−1.73850e16	−0.814614
$932$	7.87853e15	0.366993
$933$	1.44483e16	0.669066
$934$	−2.56155e15	−0.117922
$935$	0	0
$936$	1.96342e15	0.0893296
$937$	−1.38120e16	−0.624727	−0.312363	−	0.949963i	$-0.601121\pi$
$937$	−1.38120e16	−0.624727	−0.312363	+	0.949963i	$0.601121\pi$
$938$	−8.31106e14	−0.0373715
$939$	−1.43319e15	−0.0640683
$940$	0	0
$941$	1.48358e15	0.0655494	0.0327747	−	0.999463i	$-0.489566\pi$
$941$	1.48358e15	0.0655494	0.0327747	+	0.999463i	$0.489566\pi$
$942$	−9.61567e15	−0.422376
$943$	1.98554e16	0.867090
$944$	−1.90167e16	−0.825637
$945$	0	0
$946$	−2.87973e15	−0.123580
$947$	−3.53452e16	−1.50801	−0.754007	−	0.656866i	$-0.771882\pi$
$947$	−3.53452e16	−1.50801	−0.754007	+	0.656866i	$0.771882\pi$
$948$	−4.57776e15	−0.194181
$949$	7.42613e15	0.313183
$950$	0	0
$951$	−1.64590e16	−0.686138
$952$	3.40554e15	0.141151
$953$	−6.17191e15	−0.254336	−0.127168	−	0.991881i	$-0.540589\pi$
$953$	−6.17191e15	−0.254336	−0.127168	+	0.991881i	$0.540589\pi$
$954$	−5.62255e15	−0.230365
$955$	0	0
$956$	−1.09776e16	−0.444619
$957$	−3.59051e15	−0.144591
$958$	−2.88357e16	−1.15457
$959$	8.10807e15	0.322786
$960$	0	0
$961$	−1.72509e16	−0.678941
$962$	−9.79631e15	−0.383354
$963$	6.86211e15	0.267001
$964$	−9.81632e15	−0.379774
$965$	0	0
$966$	−2.31350e15	−0.0884901
$967$	−3.66178e16	−1.39266	−0.696332	−	0.717720i	$-0.745186\pi$
$967$	−3.66178e16	−1.39266	−0.696332	+	0.717720i	$0.745186\pi$
$968$	1.71274e16	0.647704
$969$	9.07404e15	0.341209
$970$	0	0
$971$	1.17235e16	0.435864	0.217932	−	0.975964i	$-0.430069\pi$
$971$	1.17235e16	0.435864	0.217932	+	0.975964i	$0.430069\pi$
$972$	−3.97525e14	−0.0146960
$973$	5.38535e15	0.197967
$974$	−3.09072e16	−1.12976
$975$	0	0
$976$	6.98514e15	0.252465
$977$	2.58955e16	0.930687	0.465343	−	0.885130i	$-0.345931\pi$
$977$	2.58955e16	0.930687	0.465343	+	0.885130i	$0.345931\pi$
$978$	−4.62659e15	−0.165347
$979$	1.82434e16	0.648336
$980$	0	0
$981$	−1.37135e16	−0.481914
$982$	−2.11873e16	−0.740393
$983$	3.17428e16	1.10307	0.551533	−	0.834153i	$-0.314043\pi$
$983$	3.17428e16	1.10307	0.551533	+	0.834153i	$0.314043\pi$
$984$	−1.67303e16	−0.578138
$985$	0	0
$986$	−7.12531e15	−0.243490
$987$	−3.03693e15	−0.103203
$988$	1.42094e15	0.0480190
$989$	−6.18955e15	−0.208008
$990$	0	0
$991$	−3.65070e16	−1.21331	−0.606654	−	0.794966i	$-0.707489\pi$
$991$	−3.65070e16	−1.21331	−0.606654	+	0.794966i	$0.707489\pi$
$992$	7.68669e15	0.254054
$993$	−6.13961e15	−0.201800
$994$	−6.95298e14	−0.0227272
$995$	0	0
$996$	1.06521e15	0.0344356
$997$	−1.86602e16	−0.599918	−0.299959	−	0.953952i	$-0.596973\pi$
$997$	−1.86602e16	−0.599918	−0.299959	+	0.953952i	$0.596973\pi$
$998$	2.86788e16	0.916945
$999$	1.06413e16	0.338364

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.12.a.i.1.3	yes	4
3.2	odd	2		225.12.a.q.1.2		4
5.2	odd	4		75.12.b.g.49.6		8
5.3	odd	4		75.12.b.g.49.3		8
5.4	even	2		75.12.a.h.1.2	✓	4
15.2	even	4		225.12.b.o.199.3		8
15.8	even	4		225.12.b.o.199.6		8
15.14	odd	2		225.12.a.s.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.12.a.h.1.2	✓	4	5.4	even	2
75.12.a.i.1.3	yes	4	1.1	even	1	trivial
75.12.b.g.49.3		8	5.3	odd	4
75.12.b.g.49.6		8	5.2	odd	4
225.12.a.q.1.2		4	3.2	odd	2
225.12.a.s.1.3		4	15.14	odd	2
225.12.b.o.199.3		8	15.2	even	4
225.12.b.o.199.6		8	15.8	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)

\(f(q)\)	\(=\)	\(q+39.7345 q^{2} +243.000 q^{3} -469.173 q^{4} +9655.47 q^{6} -8306.83 q^{7} -100019. q^{8} +59049.0 q^{9} +O(q^{10})\)	\(q+39.7345 q^{2} +243.000 q^{3} -469.173 q^{4} +9655.47 q^{6} -8306.83 q^{7} -100019. q^{8} +59049.0 q^{9} +337742. q^{11} -114009. q^{12} -332445. q^{13} -330067. q^{14} -3.01331e6 q^{16} +4.09893e6 q^{17} +2.34628e6 q^{18} +9.11010e6 q^{19} -2.01856e6 q^{21} +1.34200e7 q^{22} +2.88443e7 q^{23} -2.43045e7 q^{24} -1.32095e7 q^{26} +1.43489e7 q^{27} +3.89734e6 q^{28} -4.37488e7 q^{29} +9.03196e7 q^{31} +8.51055e7 q^{32} +8.20712e7 q^{33} +1.62869e8 q^{34} -2.77042e7 q^{36} +7.41610e8 q^{37} +3.61985e8 q^{38} -8.07841e7 q^{39} +6.88364e8 q^{41} -8.02063e7 q^{42} -2.14585e8 q^{43} -1.58459e8 q^{44} +1.14611e9 q^{46} +1.50451e9 q^{47} -7.32235e8 q^{48} -1.90832e9 q^{49} +9.96041e8 q^{51} +1.55974e8 q^{52} -2.39637e9 q^{53} +5.70146e8 q^{54} +8.30836e8 q^{56} +2.21376e9 q^{57} -1.73833e9 q^{58} +6.31089e9 q^{59} -2.31809e9 q^{61} +3.58880e9 q^{62} -4.90510e8 q^{63} +9.55289e9 q^{64} +3.26106e9 q^{66} +2.51799e9 q^{67} -1.92311e9 q^{68} +7.00917e9 q^{69} +2.10654e9 q^{71} -5.90599e9 q^{72} -2.23379e10 q^{73} +2.94675e10 q^{74} -4.27422e9 q^{76} -2.80556e9 q^{77} -3.20991e9 q^{78} +4.01526e10 q^{79} +3.48678e9 q^{81} +2.73518e10 q^{82} -9.34317e9 q^{83} +9.47054e8 q^{84} -8.52641e9 q^{86} -1.06309e10 q^{87} -3.37804e10 q^{88} +5.40157e10 q^{89} +2.76156e9 q^{91} -1.35330e10 q^{92} +2.19477e10 q^{93} +5.97807e10 q^{94} +2.06806e10 q^{96} +1.27556e11 q^{97} -7.58262e10 q^{98} +1.99433e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 46 q^{2} + 972 q^{3} + 4648 q^{4} + 11178 q^{6} + 68372 q^{7} + 386172 q^{8} + 236196 q^{9}+O(q^{10}) \) 4 * q + 46 * q^2 + 972 * q^3 + 4648 * q^4 + 11178 * q^6 + 68372 * q^7 + 386172 * q^8 + 236196 * q^9	\( 4 q + 46 q^{2} + 972 q^{3} + 4648 q^{4} + 11178 q^{6} + 68372 q^{7} + 386172 q^{8} + 236196 q^{9} - 99944 q^{11} + 1129464 q^{12} + 2306276 q^{13} + 3107022 q^{14} + 7622200 q^{16} - 3443816 q^{17} + 2716254 q^{18} - 4214548 q^{19} + 16614396 q^{21} - 34106532 q^{22} + 52691304 q^{23} + 93839796 q^{24} - 8766722 q^{26} + 57395628 q^{27} + 472439576 q^{28} + 217393304 q^{29} - 326317036 q^{31} + 788672072 q^{32} - 24286392 q^{33} - 564836244 q^{34} + 274459752 q^{36} + 273252872 q^{37} - 1762793006 q^{38} + 560425068 q^{39} - 22069456 q^{41} + 755006346 q^{42} - 705091900 q^{43} - 992474176 q^{44} + 2697853524 q^{46} + 187768360 q^{47} + 1852194600 q^{48} + 5213315400 q^{49} - 836847288 q^{51} + 7619620904 q^{52} - 6392224256 q^{53} + 660049722 q^{54} + 20338292700 q^{56} - 1024135164 q^{57} - 16176655332 q^{58} + 36710008 q^{59} + 11538870620 q^{61} - 4825203906 q^{62} + 4037298228 q^{63} + 55398609952 q^{64} - 8287887276 q^{66} + 37721158484 q^{67} - 45682615072 q^{68} + 12803986872 q^{69} + 8211316688 q^{71} + 22803070428 q^{72} + 5713413224 q^{73} + 136155026252 q^{74} - 78094966168 q^{76} + 72854549304 q^{77} - 2130313446 q^{78} + 45026381600 q^{79} + 13947137604 q^{81} + 55157549184 q^{82} - 104211315528 q^{83} + 114802816968 q^{84} + 77092671046 q^{86} + 52826572872 q^{87} - 175877029608 q^{88} + 111829609152 q^{89} + 225968327284 q^{91} + 88286181792 q^{92} - 79295039748 q^{93} + 193370717388 q^{94} + 191647313496 q^{96} + 77104304804 q^{97} - 139912937716 q^{98} - 5901593256 q^{99}+O(q^{100}) \) 4 * q + 46 * q^2 + 972 * q^3 + 4648 * q^4 + 11178 * q^6 + 68372 * q^7 + 386172 * q^8 + 236196 * q^9 - 99944 * q^11 + 1129464 * q^12 + 2306276 * q^13 + 3107022 * q^14 + 7622200 * q^16 - 3443816 * q^17 + 2716254 * q^18 - 4214548 * q^19 + 16614396 * q^21 - 34106532 * q^22 + 52691304 * q^23 + 93839796 * q^24 - 8766722 * q^26 + 57395628 * q^27 + 472439576 * q^28 + 217393304 * q^29 - 326317036 * q^31 + 788672072 * q^32 - 24286392 * q^33 - 564836244 * q^34 + 274459752 * q^36 + 273252872 * q^37 - 1762793006 * q^38 + 560425068 * q^39 - 22069456 * q^41 + 755006346 * q^42 - 705091900 * q^43 - 992474176 * q^44 + 2697853524 * q^46 + 187768360 * q^47 + 1852194600 * q^48 + 5213315400 * q^49 - 836847288 * q^51 + 7619620904 * q^52 - 6392224256 * q^53 + 660049722 * q^54 + 20338292700 * q^56 - 1024135164 * q^57 - 16176655332 * q^58 + 36710008 * q^59 + 11538870620 * q^61 - 4825203906 * q^62 + 4037298228 * q^63 + 55398609952 * q^64 - 8287887276 * q^66 + 37721158484 * q^67 - 45682615072 * q^68 + 12803986872 * q^69 + 8211316688 * q^71 + 22803070428 * q^72 + 5713413224 * q^73 + 136155026252 * q^74 - 78094966168 * q^76 + 72854549304 * q^77 - 2130313446 * q^78 + 45026381600 * q^79 + 13947137604 * q^81 + 55157549184 * q^82 - 104211315528 * q^83 + 114802816968 * q^84 + 77092671046 * q^86 + 52826572872 * q^87 - 175877029608 * q^88 + 111829609152 * q^89 + 225968327284 * q^91 + 88286181792 * q^92 - 79295039748 * q^93 + 193370717388 * q^94 + 191647313496 * q^96 + 77104304804 * q^97 - 139912937716 * q^98 - 5901593256 * q^99

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