LMFDB - Embedded newform 1075.6.a.j.1.24

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1075,6,Mod(1,1075)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1075, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("1075.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1075 = 5^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1075.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:	$172.412606299$
Analytic rank:	$0$
Dimension:	$37$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.24
Character	$\chi$	$=$	1075.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.95362 q^{2} -3.98544 q^{3} -16.3689 q^{4} -15.7569 q^{6} +105.027 q^{7} -191.232 q^{8} -227.116 q^{9} +O(q^{10})$	\(q+3.95362 q^{2} -3.98544 q^{3} -16.3689 q^{4} -15.7569 q^{6} +105.027 q^{7} -191.232 q^{8} -227.116 q^{9} -502.589 q^{11} +65.2371 q^{12} -941.796 q^{13} +415.237 q^{14} -232.256 q^{16} -1444.07 q^{17} -897.932 q^{18} -1557.68 q^{19} -418.578 q^{21} -1987.05 q^{22} -1620.94 q^{23} +762.144 q^{24} -3723.51 q^{26} +1873.62 q^{27} -1719.17 q^{28} +4587.72 q^{29} -7282.53 q^{31} +5201.18 q^{32} +2003.03 q^{33} -5709.32 q^{34} +3717.64 q^{36} -4976.02 q^{37} -6158.46 q^{38} +3753.47 q^{39} -17240.0 q^{41} -1654.90 q^{42} +1849.00 q^{43} +8226.81 q^{44} -6408.57 q^{46} +21855.2 q^{47} +925.643 q^{48} -5776.36 q^{49} +5755.26 q^{51} +15416.1 q^{52} +14576.8 q^{53} +7407.58 q^{54} -20084.5 q^{56} +6208.02 q^{57} +18138.1 q^{58} -12923.7 q^{59} -15018.8 q^{61} -28792.4 q^{62} -23853.3 q^{63} +27995.7 q^{64} +7919.24 q^{66} -35748.9 q^{67} +23637.8 q^{68} +6460.14 q^{69} -5700.21 q^{71} +43432.0 q^{72} +50592.8 q^{73} -19673.3 q^{74} +25497.4 q^{76} -52785.3 q^{77} +14839.8 q^{78} +40352.0 q^{79} +47722.1 q^{81} -68160.3 q^{82} -118184. q^{83} +6851.65 q^{84} +7310.25 q^{86} -18284.1 q^{87} +96111.1 q^{88} -137345. q^{89} -98913.9 q^{91} +26532.9 q^{92} +29024.1 q^{93} +86407.1 q^{94} -20729.0 q^{96} +62096.6 q^{97} -22837.5 q^{98} +114146. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 37 q + 630 q^{4} + 291 q^{6} + 213 q^{8} + 3535 q^{9}+O(q^{10}) $ 37 * q + 630 * q^4 + 291 * q^6 + 213 * q^8 + 3535 * q^9	\( 37 q + 630 q^{4} + 291 q^{6} + 213 q^{8} + 3535 q^{9} + 675 q^{11} - 4446 q^{12} + 1241 q^{13} + 2375 q^{14} + 10518 q^{16} + 1153 q^{17} - 6680 q^{18} + 4065 q^{19} + 9953 q^{21} + 9283 q^{22} - 360 q^{23} + 2265 q^{24} + 23695 q^{26} + 1323 q^{27} - 30375 q^{28} + 19290 q^{29} + 23291 q^{31} + 8166 q^{32} - 10388 q^{33} - 13153 q^{34} + 148705 q^{36} + 13501 q^{37} - 8127 q^{38} - 1327 q^{39} + 38345 q^{41} - 21835 q^{42} + 68413 q^{43} + 47768 q^{44} + 48755 q^{46} + 84859 q^{47} - 208720 q^{48} + 107255 q^{49} + 62027 q^{51} + 128320 q^{52} - 53559 q^{53} + 44158 q^{54} + 107538 q^{56} + 104239 q^{57} - 85186 q^{58} + 48186 q^{59} + 82364 q^{61} + 206506 q^{62} - 75269 q^{63} + 161467 q^{64} + 91969 q^{66} + 38168 q^{67} + 95991 q^{68} + 287103 q^{69} + 155302 q^{71} + 9979 q^{72} + 31927 q^{73} + 59946 q^{74} + 225407 q^{76} - 80007 q^{77} - 67815 q^{78} + 150174 q^{79} + 417489 q^{81} + 60603 q^{82} + 266568 q^{83} + 586273 q^{84} - 57554 q^{87} + 323054 q^{88} + 334356 q^{89} + 51747 q^{91} - 258529 q^{92} - 285287 q^{93} + 302744 q^{94} + 287282 q^{96} - 78640 q^{97} - 397117 q^{98} + 362152 q^{99}+O(q^{100}) \) 37 * q + 630 * q^4 + 291 * q^6 + 213 * q^8 + 3535 * q^9 + 675 * q^11 - 4446 * q^12 + 1241 * q^13 + 2375 * q^14 + 10518 * q^16 + 1153 * q^17 - 6680 * q^18 + 4065 * q^19 + 9953 * q^21 + 9283 * q^22 - 360 * q^23 + 2265 * q^24 + 23695 * q^26 + 1323 * q^27 - 30375 * q^28 + 19290 * q^29 + 23291 * q^31 + 8166 * q^32 - 10388 * q^33 - 13153 * q^34 + 148705 * q^36 + 13501 * q^37 - 8127 * q^38 - 1327 * q^39 + 38345 * q^41 - 21835 * q^42 + 68413 * q^43 + 47768 * q^44 + 48755 * q^46 + 84859 * q^47 - 208720 * q^48 + 107255 * q^49 + 62027 * q^51 + 128320 * q^52 - 53559 * q^53 + 44158 * q^54 + 107538 * q^56 + 104239 * q^57 - 85186 * q^58 + 48186 * q^59 + 82364 * q^61 + 206506 * q^62 - 75269 * q^63 + 161467 * q^64 + 91969 * q^66 + 38168 * q^67 + 95991 * q^68 + 287103 * q^69 + 155302 * q^71 + 9979 * q^72 + 31927 * q^73 + 59946 * q^74 + 225407 * q^76 - 80007 * q^77 - 67815 * q^78 + 150174 * q^79 + 417489 * q^81 + 60603 * q^82 + 266568 * q^83 + 586273 * q^84 - 57554 * q^87 + 323054 * q^88 + 334356 * q^89 + 51747 * q^91 - 258529 * q^92 - 285287 * q^93 + 302744 * q^94 + 287282 * q^96 - 78640 * q^97 - 397117 * q^98 + 362152 * 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 $

$2$	3.95362	0.698908	0.349454	−	0.936953i	$-0.386367\pi$
$2$	3.95362	0.698908	0.349454	+	0.936953i	$0.386367\pi$
$3$	−3.98544	−0.255666	−0.127833	−	0.991796i	$-0.540802\pi$
$3$	−3.98544	−0.255666	−0.127833	+	0.991796i	$0.540802\pi$
$4$	−16.3689	−0.511527
$5$	0	0
$5$	0	0
$6$	−15.7569	−0.178687
$7$	105.027	0.810131	0.405066	−	0.914288i	$-0.367249\pi$
$7$	105.027	0.810131	0.405066	+	0.914288i	$0.367249\pi$
$8$	−191.232	−1.05642
$9$	−227.116	−0.934635
$10$	0	0
$11$	−502.589	−1.25236	−0.626182	−	0.779677i	$-0.715383\pi$
$11$	−502.589	−1.25236	−0.626182	+	0.779677i	$0.715383\pi$
$12$	65.2371	0.130780
$13$	−941.796	−1.54560	−0.772802	−	0.634647i	$-0.781146\pi$
$13$	−941.796	−1.54560	−0.772802	+	0.634647i	$0.781146\pi$
$14$	415.237	0.566207
$15$	0	0
$16$	−232.256	−0.226813
$17$	−1444.07	−1.21190	−0.605950	−	0.795503i	$-0.707207\pi$
$17$	−1444.07	−1.21190	−0.605950	+	0.795503i	$0.707207\pi$
$18$	−897.932	−0.653224
$19$	−1557.68	−0.989904	−0.494952	−	0.868920i	$-0.664814\pi$
$19$	−1557.68	−0.989904	−0.494952	+	0.868920i	$0.664814\pi$
$20$	0	0
$21$	−418.578	−0.207123
$22$	−1987.05	−0.875288
$23$	−1620.94	−0.638920	−0.319460	−	0.947600i	$-0.603502\pi$
$23$	−1620.94	−0.638920	−0.319460	+	0.947600i	$0.603502\pi$
$24$	762.144	0.270090
$25$	0	0
$26$	−3723.51	−1.08024
$27$	1873.62	0.494620
$28$	−1719.17	−0.414404
$29$	4587.72	1.01298	0.506491	−	0.862245i	$-0.330942\pi$
$29$	4587.72	1.01298	0.506491	+	0.862245i	$0.330942\pi$
$30$	0	0
$31$	−7282.53	−1.36106	−0.680531	−	0.732719i	$-0.738251\pi$
$31$	−7282.53	−1.36106	−0.680531	+	0.732719i	$0.738251\pi$
$32$	5201.18	0.897897
$33$	2003.03	0.320187
$34$	−5709.32	−0.847007
$35$	0	0
$36$	3717.64	0.478091
$37$	−4976.02	−0.597555	−0.298778	−	0.954323i	$-0.596579\pi$
$37$	−4976.02	−0.597555	−0.298778	+	0.954323i	$0.596579\pi$
$38$	−6158.46	−0.691852
$39$	3753.47	0.395158
$40$	0	0
$41$	−17240.0	−1.60168	−0.800842	−	0.598876i	$-0.795614\pi$
$41$	−17240.0	−1.60168	−0.800842	+	0.598876i	$0.795614\pi$
$42$	−1654.90	−0.144760
$43$	1849.00	0.152499
$43$	1849.00	0.152499
$44$	8226.81	0.640619
$45$	0	0
$46$	−6408.57	−0.446547
$47$	21855.2	1.44314	0.721572	−	0.692340i	$-0.243420\pi$
$47$	21855.2	1.44314	0.721572	+	0.692340i	$0.243420\pi$
$48$	925.643	0.0579883
$49$	−5776.36	−0.343688
$50$	0	0
$51$	5755.26	0.309841
$52$	15416.1	0.790619
$53$	14576.8	0.712808	0.356404	−	0.934332i	$-0.384003\pi$
$53$	14576.8	0.712808	0.356404	+	0.934332i	$0.384003\pi$
$54$	7407.58	0.345694
$55$	0	0
$56$	−20084.5	−0.855838
$57$	6208.02	0.253085
$58$	18138.1	0.707982
$59$	−12923.7	−0.483345	−0.241673	−	0.970358i	$-0.577696\pi$
$59$	−12923.7	−0.483345	−0.241673	+	0.970358i	$0.577696\pi$
$60$	0	0
$61$	−15018.8	−0.516786	−0.258393	−	0.966040i	$-0.583193\pi$
$61$	−15018.8	−0.516786	−0.258393	+	0.966040i	$0.583193\pi$
$62$	−28792.4	−0.951258
$63$	−23853.3	−0.757177
$64$	27995.7	0.854361
$65$	0	0
$66$	7919.24	0.223781
$67$	−35748.9	−0.972917	−0.486458	−	0.873704i	$-0.661711\pi$
$67$	−35748.9	−0.972917	−0.486458	+	0.873704i	$0.661711\pi$
$68$	23637.8	0.619920
$69$	6460.14	0.163350
$70$	0	0
$71$	−5700.21	−0.134198	−0.0670989	−	0.997746i	$-0.521374\pi$
$71$	−5700.21	−0.134198	−0.0670989	+	0.997746i	$0.521374\pi$
$72$	43432.0	0.987366
$73$	50592.8	1.11117	0.555586	−	0.831459i	$-0.312494\pi$
$73$	50592.8	1.11117	0.555586	+	0.831459i	$0.312494\pi$
$74$	−19673.3	−0.417637
$75$	0	0
$76$	25497.4	0.506363
$77$	−52785.3	−1.01458
$78$	14839.8	0.276179
$79$	40352.0	0.727440	0.363720	−	0.931508i	$-0.381507\pi$
$79$	40352.0	0.727440	0.363720	+	0.931508i	$0.381507\pi$
$80$	0	0
$81$	47722.1	0.808178
$82$	−68160.3	−1.11943
$83$	−118184.	−1.88306	−0.941530	−	0.336929i	$-0.890612\pi$
$83$	−118184.	−1.88306	−0.941530	+	0.336929i	$0.890612\pi$
$84$	6851.65	0.105949
$85$	0	0
$86$	7310.25	0.106583
$87$	−18284.1	−0.258985
$88$	96111.1	1.32302
$89$	−137345.	−1.83797	−0.918986	−	0.394291i	$-0.870990\pi$
$89$	−137345.	−1.83797	−0.918986	+	0.394291i	$0.870990\pi$
$90$	0	0
$91$	−98913.9	−1.25214
$92$	26532.9	0.326825
$93$	29024.1	0.347977
$94$	86407.1	1.00863
$95$	0	0
$96$	−20729.0	−0.229562
$97$	62096.6	0.670098	0.335049	−	0.942201i	$-0.391247\pi$
$97$	62096.6	0.670098	0.335049	+	0.942201i	$0.391247\pi$
$98$	−22837.5	−0.240206
$99$	114146.	1.17050
$100$	0	0
$101$	75517.8	0.736624	0.368312	−	0.929702i	$-0.379936\pi$
$101$	75517.8	0.736624	0.368312	+	0.929702i	$0.379936\pi$
$102$	22754.1	0.216551
$103$	99507.3	0.924192	0.462096	−	0.886830i	$-0.347098\pi$
$103$	99507.3	0.924192	0.462096	+	0.886830i	$0.347098\pi$
$104$	180102.	1.63281
$105$	0	0
$106$	57631.2	0.498187
$107$	202915.	1.71339	0.856693	−	0.515827i	$-0.172515\pi$
$107$	202915.	1.71339	0.856693	+	0.515827i	$0.172515\pi$
$108$	−30669.0	−0.253012
$109$	−142027.	−1.14500	−0.572500	−	0.819904i	$-0.694027\pi$
$109$	−142027.	−1.14500	−0.572500	+	0.819904i	$0.694027\pi$
$110$	0	0
$111$	19831.6	0.152775
$112$	−24393.2	−0.183748
$113$	−204161.	−1.50410	−0.752048	−	0.659108i	$-0.770934\pi$
$113$	−204161.	−1.50410	−0.752048	+	0.659108i	$0.770934\pi$
$114$	24544.2	0.176883
$115$	0	0
$116$	−75095.8	−0.518168
$117$	213897.	1.44458
$118$	−51095.5	−0.337814
$119$	−151666.	−0.981798
$120$	0	0
$121$	91544.2	0.568418
$122$	−59378.7	−0.361186
$123$	68708.8	0.409496
$124$	119207.	0.696220
$125$	0	0
$126$	−94307.0	−0.529197
$127$	−240799.	−1.32478	−0.662391	−	0.749158i	$-0.730458\pi$
$127$	−240799.	−1.32478	−0.662391	+	0.749158i	$0.730458\pi$
$128$	−55753.3	−0.300777
$129$	−7369.07	−0.0389887
$130$	0	0
$131$	−366083.	−1.86381	−0.931905	−	0.362702i	$-0.881854\pi$
$131$	−366083.	−1.86381	−0.931905	+	0.362702i	$0.881854\pi$
$132$	−32787.4	−0.163784
$133$	−163598.	−0.801952
$134$	−141338.	−0.679980
$135$	0	0
$136$	276153.	1.28027
$137$	−250062.	−1.13827	−0.569137	−	0.822243i	$-0.692722\pi$
$137$	−250062.	−1.13827	−0.569137	+	0.822243i	$0.692722\pi$
$138$	25541.0	0.114167
$139$	283210.	1.24329	0.621643	−	0.783300i	$-0.286465\pi$
$139$	283210.	1.24329	0.621643	+	0.783300i	$0.286465\pi$
$140$	0	0
$141$	−87102.4	−0.368963
$142$	−22536.5	−0.0937919
$143$	473336.	1.93566
$144$	52749.2	0.211987
$145$	0	0
$146$	200025.	0.776608
$147$	23021.3	0.0878692
$148$	81451.9	0.305666
$149$	138439.	0.510851	0.255425	−	0.966829i	$-0.417785\pi$
$149$	138439.	0.510851	0.255425	+	0.966829i	$0.417785\pi$
$150$	0	0
$151$	−431706.	−1.54080	−0.770400	−	0.637561i	$-0.779943\pi$
$151$	−431706.	−1.54080	−0.770400	+	0.637561i	$0.779943\pi$
$152$	297878.	1.04575
$153$	327973.	1.13268
$154$	−208693.	−0.709098
$155$	0	0
$156$	−61440.0	−0.202134
$157$	−288043.	−0.932627	−0.466313	−	0.884620i	$-0.654418\pi$
$157$	−288043.	−0.932627	−0.466313	+	0.884620i	$0.654418\pi$
$158$	159536.	0.508414
$159$	−58094.9	−0.182241
$160$	0	0
$161$	−170242.	−0.517609
$162$	188675.	0.564842
$163$	211668.	0.624001	0.312001	−	0.950082i	$-0.399001\pi$
$163$	211668.	0.624001	0.312001	+	0.950082i	$0.399001\pi$
$164$	282199.	0.819305
$165$	0	0
$166$	−467256.	−1.31609
$167$	−19555.3	−0.0542591	−0.0271295	−	0.999632i	$-0.508637\pi$
$167$	−19555.3	−0.0542591	−0.0271295	+	0.999632i	$0.508637\pi$
$168$	80045.6	0.218809
$169$	515686.	1.38889
$170$	0	0
$171$	353774.	0.925199
$172$	−30266.0	−0.0780072
$173$	−63111.8	−0.160323	−0.0801614	−	0.996782i	$-0.525544\pi$
$173$	−63111.8	−0.160323	−0.0801614	+	0.996782i	$0.525544\pi$
$174$	−72288.3	−0.181007
$175$	0	0
$176$	116729.	0.284052
$177$	51506.6	0.123575
$178$	−543011.	−1.28457
$179$	24053.0	0.0561096	0.0280548	−	0.999606i	$-0.491069\pi$
$179$	24053.0	0.0561096	0.0280548	+	0.999606i	$0.491069\pi$
$180$	0	0
$181$	−39133.6	−0.0887879	−0.0443939	−	0.999014i	$-0.514136\pi$
$181$	−39133.6	−0.0887879	−0.0443939	+	0.999014i	$0.514136\pi$
$182$	−391068.	−0.875133
$183$	59856.5	0.132125
$184$	309975.	0.674967
$185$	0	0
$186$	114750.	0.243204
$187$	725775.	1.51774
$188$	−357744.	−0.738207
$189$	196780.	0.400707
$190$	0	0
$191$	522040.	1.03543	0.517715	−	0.855553i	$-0.326783\pi$
$191$	522040.	1.03543	0.517715	+	0.855553i	$0.326783\pi$
$192$	−111575.	−0.218431
$193$	−167917.	−0.324490	−0.162245	−	0.986751i	$-0.551873\pi$
$193$	−167917.	−0.324490	−0.162245	+	0.986751i	$0.551873\pi$
$194$	245507.	0.468337
$195$	0	0
$196$	94552.4	0.175806
$197$	267864.	0.491756	0.245878	−	0.969301i	$-0.420924\pi$
$197$	267864.	0.491756	0.245878	+	0.969301i	$0.420924\pi$
$198$	451290.	0.818075
$199$	563189.	1.00814	0.504071	−	0.863662i	$-0.331835\pi$
$199$	563189.	1.00814	0.504071	+	0.863662i	$0.331835\pi$
$200$	0	0
$201$	142475.	0.248742
$202$	298569.	0.514833
$203$	481834.	0.820649
$204$	−94207.1	−0.158492
$205$	0	0
$206$	393414.	0.645925
$207$	368141.	0.597157
$208$	218738.	0.350563
$209$	782870.	1.23972
$210$	0	0
$211$	−774536.	−1.19767	−0.598833	−	0.800874i	$-0.704369\pi$
$211$	−774536.	−1.19767	−0.598833	+	0.800874i	$0.704369\pi$
$212$	−238606.	−0.364621
$213$	22717.8	0.0343098
$214$	802250.	1.19750
$215$	0	0
$216$	−358296.	−0.522526
$217$	−764861.	−1.10264
$218$	−561523.	−0.800251
$219$	−201634.	−0.284089
$220$	0	0
$221$	1.36002e6	1.87312
$222$	78406.8	0.106775
$223$	665145.	0.895682	0.447841	−	0.894113i	$-0.352193\pi$
$223$	665145.	0.895682	0.447841	+	0.894113i	$0.352193\pi$
$224$	546263.	0.727415
$225$	0	0
$226$	−807174.	−1.05123
$227$	648850.	0.835756	0.417878	−	0.908503i	$-0.362774\pi$
$227$	648850.	0.835756	0.417878	+	0.908503i	$0.362774\pi$
$228$	−101618.	−0.129460
$229$	1.00874e6	1.27114	0.635569	−	0.772044i	$-0.280766\pi$
$229$	1.00874e6	1.27114	0.635569	+	0.772044i	$0.280766\pi$
$230$	0	0
$231$	210372.	0.259393
$232$	−877320.	−1.07013
$233$	364314.	0.439629	0.219815	−	0.975542i	$-0.429455\pi$
$233$	364314.	0.439629	0.219815	+	0.975542i	$0.429455\pi$
$234$	845669.	1.00963
$235$	0	0
$236$	211547.	0.247244
$237$	−160820.	−0.185982
$238$	−599632.	−0.686187
$239$	−449627.	−0.509164	−0.254582	−	0.967051i	$-0.581938\pi$
$239$	−449627.	−0.509164	−0.254582	+	0.967051i	$0.581938\pi$
$240$	0	0
$241$	1.07380e6	1.19092	0.595459	−	0.803386i	$-0.296970\pi$
$241$	1.07380e6	1.19092	0.595459	+	0.803386i	$0.296970\pi$
$242$	361931.	0.397272
$243$	−645483.	−0.701243
$244$	245841.	0.264350
$245$	0	0
$246$	271649.	0.286200
$247$	1.46701e6	1.53000
$248$	1.39265e6	1.43785
$249$	471016.	0.481434
$250$	0	0
$251$	562819.	0.563877	0.281939	−	0.959432i	$-0.409023\pi$
$251$	562819.	0.563877	0.281939	+	0.959432i	$0.409023\pi$
$252$	390452.	0.387317
$253$	814664.	0.800161
$254$	−952027.	−0.925902
$255$	0	0
$256$	−1.11629e6	−1.06458
$257$	752739.	0.710905	0.355452	−	0.934694i	$-0.384327\pi$
$257$	752739.	0.710905	0.355452	+	0.934694i	$0.384327\pi$
$258$	−29134.5	−0.0272495
$259$	−522616.	−0.484098
$260$	0	0
$261$	−1.04195e6	−0.946769
$262$	−1.44736e6	−1.30263
$263$	−646430.	−0.576278	−0.288139	−	0.957589i	$-0.593037\pi$
$263$	−646430.	−0.576278	−0.288139	+	0.957589i	$0.593037\pi$
$264$	−383045.	−0.338252
$265$	0	0
$266$	−646804.	−0.560491
$267$	547381.	0.469906
$268$	585169.	0.497673
$269$	−1.44976e6	−1.22156	−0.610779	−	0.791801i	$-0.709144\pi$
$269$	−1.44976e6	−1.22156	−0.610779	+	0.791801i	$0.709144\pi$
$270$	0	0
$271$	1.15042e6	0.951549	0.475775	−	0.879567i	$-0.342168\pi$
$271$	1.15042e6	0.951549	0.475775	+	0.879567i	$0.342168\pi$
$272$	335395.	0.274875
$273$	394215.	0.320130
$274$	−988652.	−0.795549
$275$	0	0
$276$	−105745.	−0.0835580
$277$	−181312.	−0.141980	−0.0709899	−	0.997477i	$-0.522616\pi$
$277$	−181312.	−0.141980	−0.0709899	+	0.997477i	$0.522616\pi$
$278$	1.11970e6	0.868944
$279$	1.65398e6	1.27210
$280$	0	0
$281$	1.35054e6	1.02034	0.510168	−	0.860075i	$-0.329583\pi$
$281$	1.35054e6	1.02034	0.510168	+	0.860075i	$0.329583\pi$
$282$	−344370.	−0.257871
$283$	787277.	0.584334	0.292167	−	0.956367i	$-0.405624\pi$
$283$	787277.	0.584334	0.292167	+	0.956367i	$0.405624\pi$
$284$	93306.1	0.0686458
$285$	0	0
$286$	1.87139e6	1.35285
$287$	−1.81066e6	−1.29757
$288$	−1.18127e6	−0.839206
$289$	665490.	0.468702
$290$	0	0
$291$	−247482.	−0.171321
$292$	−828147.	−0.568395
$293$	385504.	0.262337	0.131168	−	0.991360i	$-0.458127\pi$
$293$	385504.	0.262337	0.131168	+	0.991360i	$0.458127\pi$
$294$	91017.5	0.0614125
$295$	0	0
$296$	951576.	0.631269
$297$	−941659.	−0.619445
$298$	547337.	0.357038
$299$	1.52659e6	0.987518
$300$	0	0
$301$	194195.	0.123544
$302$	−1.70680e6	−1.07688
$303$	−300972.	−0.188330
$304$	361780.	0.224523
$305$	0	0
$306$	1.29668e6	0.791643
$307$	−1.25942e6	−0.762649	−0.381325	−	0.924441i	$-0.624532\pi$
$307$	−1.25942e6	−0.762649	−0.381325	+	0.924441i	$0.624532\pi$
$308$	864036.	0.518985
$309$	−396580.	−0.236284
$310$	0	0
$311$	−3.13453e6	−1.83769	−0.918843	−	0.394624i	$-0.870875\pi$
$311$	−3.13453e6	−1.83769	−0.918843	+	0.394624i	$0.870875\pi$
$312$	−717784.	−0.417453
$313$	1.14416e6	0.660123	0.330062	−	0.943959i	$-0.392930\pi$
$313$	1.14416e6	0.660123	0.330062	+	0.943959i	$0.392930\pi$
$314$	−1.13881e6	−0.651821
$315$	0	0
$316$	−660516.	−0.372105
$317$	−1.37693e6	−0.769598	−0.384799	−	0.923000i	$-0.625729\pi$
$317$	−1.37693e6	−0.769598	−0.384799	+	0.923000i	$0.625729\pi$
$318$	−229685.	−0.127369
$319$	−2.30574e6	−1.26862
$320$	0	0
$321$	−808706.	−0.438054
$322$	−673072.	−0.361761
$323$	2.24940e6	1.19966
$324$	−781156.	−0.413405
$325$	0	0
$326$	836854.	0.436120
$327$	566041.	0.292738
$328$	3.29684e6	1.69205
$329$	2.29538e6	1.16914
$330$	0	0
$331$	1.23247e6	0.618308	0.309154	−	0.951012i	$-0.399954\pi$
$331$	1.23247e6	0.618308	0.309154	+	0.951012i	$0.399954\pi$
$332$	1.93454e6	0.963236
$333$	1.13014e6	0.558496
$334$	−77314.1	−0.0379221
$335$	0	0
$336$	97217.4	0.0469781
$337$	−2.95257e6	−1.41620	−0.708101	−	0.706111i	$-0.750448\pi$
$337$	−2.95257e6	−1.41620	−0.708101	+	0.706111i	$0.750448\pi$
$338$	2.03883e6	0.970709
$339$	813669.	0.384546
$340$	0	0
$341$	3.66012e6	1.70455
$342$	1.39869e6	0.646629
$343$	−2.37186e6	−1.08856
$344$	−353588.	−0.161102
$345$	0	0
$346$	−249520.	−0.112051
$347$	−2.28600e6	−1.01919	−0.509593	−	0.860416i	$-0.670204\pi$
$347$	−2.28600e6	−1.01919	−0.509593	+	0.860416i	$0.670204\pi$
$348$	299290.	0.132478
$349$	1.54820e6	0.680399	0.340199	−	0.940353i	$-0.389505\pi$
$349$	1.54820e6	0.680399	0.340199	+	0.940353i	$0.389505\pi$
$350$	0	0
$351$	−1.76457e6	−0.764487
$352$	−2.61405e6	−1.12450
$353$	−4.06971e6	−1.73831	−0.869154	−	0.494541i	$-0.835336\pi$
$353$	−4.06971e6	−1.73831	−0.869154	+	0.494541i	$0.835336\pi$
$354$	203638.	0.0863675
$355$	0	0
$356$	2.24819e6	0.940172
$357$	604457.	0.251012
$358$	95096.5	0.0392154
$359$	1.53591e6	0.628971	0.314486	−	0.949262i	$-0.398168\pi$
$359$	1.53591e6	0.628971	0.314486	+	0.949262i	$0.398168\pi$
$360$	0	0
$361$	−49745.5	−0.0200903
$362$	−154720.	−0.0620546
$363$	−364844.	−0.145325
$364$	1.61911e6	0.640505
$365$	0	0
$366$	236650.	0.0923430
$367$	−2.63161e6	−1.01990	−0.509949	−	0.860205i	$-0.670336\pi$
$367$	−2.63161e6	−1.01990	−0.509949	+	0.860205i	$0.670336\pi$
$368$	376473.	0.144915
$369$	3.91548e6	1.49699
$370$	0	0
$371$	1.53096e6	0.577468
$372$	−475091.	−0.178000
$373$	−3.82365e6	−1.42300	−0.711501	−	0.702685i	$-0.751984\pi$
$373$	−3.82365e6	−1.42300	−0.711501	+	0.702685i	$0.751984\pi$
$374$	2.86944e6	1.06076
$375$	0	0
$376$	−4.17941e6	−1.52456
$377$	−4.32070e6	−1.56567
$378$	777995.	0.280058
$379$	−2.00058e6	−0.715415	−0.357707	−	0.933834i	$-0.616441\pi$
$379$	−2.00058e6	−0.715415	−0.357707	+	0.933834i	$0.616441\pi$
$380$	0	0
$381$	959687.	0.338702
$382$	2.06395e6	0.723670
$383$	−454929.	−0.158470	−0.0792349	−	0.996856i	$-0.525248\pi$
$383$	−454929.	−0.158470	−0.0792349	+	0.996856i	$0.525248\pi$
$384$	222201.	0.0768985
$385$	0	0
$386$	−663880.	−0.226789
$387$	−419938.	−0.142530
$388$	−1.01645e6	−0.342773
$389$	3.13023e6	1.04882	0.524412	−	0.851465i	$-0.324285\pi$
$389$	3.13023e6	1.04882	0.524412	+	0.851465i	$0.324285\pi$
$390$	0	0
$391$	2.34075e6	0.774307
$392$	1.10463e6	0.363078
$393$	1.45900e6	0.476513
$394$	1.05903e6	0.343692
$395$	0	0
$396$	−1.86844e6	−0.598745
$397$	−4.88254e6	−1.55478	−0.777391	−	0.629017i	$-0.783457\pi$
$397$	−4.88254e6	−1.55478	−0.777391	+	0.629017i	$0.783457\pi$
$398$	2.22664e6	0.704599
$399$	652009.	0.205032
$400$	0	0
$401$	−2.96868e6	−0.921941	−0.460970	−	0.887416i	$-0.652499\pi$
$401$	−2.96868e6	−0.921941	−0.460970	+	0.887416i	$0.652499\pi$
$402$	563292.	0.173848
$403$	6.85865e6	2.10366
$404$	−1.23614e6	−0.376803
$405$	0	0
$406$	1.90499e6	0.573558
$407$	2.50089e6	0.748357
$408$	−1.10059e6	−0.327322
$409$	5.78151e6	1.70896	0.854482	−	0.519481i	$-0.173875\pi$
$409$	5.78151e6	1.70896	0.854482	+	0.519481i	$0.173875\pi$
$410$	0	0
$411$	996607.	0.291018
$412$	−1.62882e6	−0.472749
$413$	−1.35734e6	−0.391573
$414$	1.45549e6	0.417358
$415$	0	0
$416$	−4.89845e6	−1.38779
$417$	−1.12871e6	−0.317866
$418$	3.09517e6	0.866451
$419$	5.44973e6	1.51649	0.758246	−	0.651969i	$-0.226057\pi$
$419$	5.44973e6	1.51649	0.758246	+	0.651969i	$0.226057\pi$
$420$	0	0
$421$	−3.41147e6	−0.938073	−0.469037	−	0.883179i	$-0.655399\pi$
$421$	−3.41147e6	−0.938073	−0.469037	+	0.883179i	$0.655399\pi$
$422$	−3.06222e6	−0.837059
$423$	−4.96367e6	−1.34881
$424$	−2.78755e6	−0.753024
$425$	0	0
$426$	89817.8	0.0239794
$427$	−1.57738e6	−0.418665
$428$	−3.32149e6	−0.876443
$429$	−1.88645e6	−0.494882
$430$	0	0
$431$	1.24118e6	0.321841	0.160921	−	0.986967i	$-0.448554\pi$
$431$	1.24118e6	0.321841	0.160921	+	0.986967i	$0.448554\pi$
$432$	−435160.	−0.112186
$433$	−5.56458e6	−1.42631	−0.713153	−	0.701008i	$-0.752734\pi$
$433$	−5.56458e6	−1.42631	−0.713153	+	0.701008i	$0.752734\pi$
$434$	−3.02397e6	−0.770643
$435$	0	0
$436$	2.32483e6	0.585699
$437$	2.52489e6	0.632470
$438$	−797186.	−0.198552
$439$	2.04664e6	0.506851	0.253426	−	0.967355i	$-0.418443\pi$
$439$	2.04664e6	0.506851	0.253426	+	0.967355i	$0.418443\pi$
$440$	0	0
$441$	1.31190e6	0.321222
$442$	5.37701e6	1.30914
$443$	3.46778e6	0.839542	0.419771	−	0.907630i	$-0.362110\pi$
$443$	3.46778e6	0.839542	0.419771	+	0.907630i	$0.362110\pi$
$444$	−324621.	−0.0781483
$445$	0	0
$446$	2.62973e6	0.626000
$447$	−551741.	−0.130607
$448$	2.94030e6	0.692144
$449$	−6.65111e6	−1.55696	−0.778481	−	0.627668i	$-0.784010\pi$
$449$	−6.65111e6	−1.55696	−0.778481	+	0.627668i	$0.784010\pi$
$450$	0	0
$451$	8.66461e6	2.00589
$452$	3.34188e6	0.769386
$453$	1.72054e6	0.393930
$454$	2.56531e6	0.584117
$455$	0	0
$456$	−1.18717e6	−0.267363
$457$	−6.04354e6	−1.35363	−0.676816	−	0.736152i	$-0.736641\pi$
$457$	−6.04354e6	−1.35363	−0.676816	+	0.736152i	$0.736641\pi$
$458$	3.98820e6	0.888409
$459$	−2.70564e6	−0.599430
$460$	0	0
$461$	−6.23444e6	−1.36630	−0.683148	−	0.730280i	$-0.739390\pi$
$461$	−6.23444e6	−1.36630	−0.683148	+	0.730280i	$0.739390\pi$
$462$	831733.	0.181292
$463$	17584.1	0.00381212	0.00190606	−	0.999998i	$-0.499393\pi$
$463$	17584.1	0.00381212	0.00190606	+	0.999998i	$0.499393\pi$
$464$	−1.06553e6	−0.229758
$465$	0	0
$466$	1.44036e6	0.307260
$467$	−3.73636e6	−0.792786	−0.396393	−	0.918081i	$-0.629738\pi$
$467$	−3.73636e6	−0.792786	−0.396393	+	0.918081i	$0.629738\pi$
$468$	−3.50125e6	−0.738940
$469$	−3.75459e6	−0.788190
$470$	0	0
$471$	1.14798e6	0.238441
$472$	2.47143e6	0.510615
$473$	−929286.	−0.190984
$474$	−635823.	−0.129984
$475$	0	0
$476$	2.48261e6	0.502216
$477$	−3.31063e6	−0.666215
$478$	−1.77766e6	−0.355859
$479$	−543266.	−0.108187	−0.0540933	−	0.998536i	$-0.517227\pi$
$479$	−543266.	−0.108187	−0.0540933	+	0.998536i	$0.517227\pi$
$480$	0	0
$481$	4.68640e6	0.923584
$482$	4.24541e6	0.832342
$483$	678488.	0.132335
$484$	−1.49848e6	−0.290761
$485$	0	0
$486$	−2.55199e6	−0.490105
$487$	9.35342e6	1.78710	0.893548	−	0.448968i	$-0.148208\pi$
$487$	9.35342e6	1.78710	0.893548	+	0.448968i	$0.148208\pi$
$488$	2.87208e6	0.545943
$489$	−843588.	−0.159536
$490$	0	0
$491$	1.40210e6	0.262468	0.131234	−	0.991351i	$-0.458106\pi$
$491$	1.40210e6	0.262468	0.131234	+	0.991351i	$0.458106\pi$
$492$	−1.12468e6	−0.209468
$493$	−6.62501e6	−1.22763
$494$	5.80001e6	1.06933
$495$	0	0
$496$	1.69141e6	0.308706
$497$	−598676.	−0.108718
$498$	1.86222e6	0.336478
$499$	−4.97133e6	−0.893760	−0.446880	−	0.894594i	$-0.647465\pi$
$499$	−4.97133e6	−0.893760	−0.446880	+	0.894594i	$0.647465\pi$
$500$	0	0
$501$	77936.2	0.0138722
$502$	2.22517e6	0.394099
$503$	−1.92414e6	−0.339092	−0.169546	−	0.985522i	$-0.554230\pi$
$503$	−1.92414e6	−0.339092	−0.169546	+	0.985522i	$0.554230\pi$
$504$	4.56152e6	0.799896
$505$	0	0
$506$	3.22088e6	0.559239
$507$	−2.05524e6	−0.355093
$508$	3.94160e6	0.677662
$509$	−6.92095e6	−1.18405	−0.592027	−	0.805918i	$-0.701672\pi$
$509$	−6.92095e6	−1.18405	−0.592027	+	0.805918i	$0.701672\pi$
$510$	0	0
$511$	5.31360e6	0.900195
$512$	−2.62928e6	−0.443264
$513$	−2.91849e6	−0.489626
$514$	2.97605e6	0.496857
$515$	0	0
$516$	120623.	0.0199438
$517$	−1.09842e7	−1.80734
$518$	−2.06623e6	−0.338340
$519$	251528.	0.0409890
$520$	0	0
$521$	1.04313e7	1.68362	0.841808	−	0.539777i	$-0.181491\pi$
$521$	1.04313e7	1.68362	0.841808	+	0.539777i	$0.181491\pi$
$522$	−4.11946e6	−0.661705
$523$	2.81981e6	0.450781	0.225390	−	0.974269i	$-0.427634\pi$
$523$	2.81981e6	0.450781	0.225390	+	0.974269i	$0.427634\pi$
$524$	5.99237e6	0.953390
$525$	0	0
$526$	−2.55574e6	−0.402766
$527$	1.05165e7	1.64947
$528$	−465218.	−0.0726225
$529$	−3.80891e6	−0.591781
$530$	0	0
$531$	2.93519e6	0.451751
$532$	2.67791e6	0.410220
$533$	1.62365e7	2.47557
$534$	2.16414e6	0.328422
$535$	0	0
$536$	6.83634e6	1.02781
$537$	−95861.7	−0.0143453
$538$	−5.73179e6	−0.853757
$539$	2.90313e6	0.430422
$540$	0	0
$541$	−7.00568e6	−1.02910	−0.514550	−	0.857461i	$-0.672041\pi$
$541$	−7.00568e6	−1.02910	−0.514550	+	0.857461i	$0.672041\pi$
$542$	4.54831e6	0.665046
$543$	155965.	0.0227000
$544$	−7.51088e6	−1.08816
$545$	0	0
$546$	1.55858e6	0.223742
$547$	−4.07099e6	−0.581744	−0.290872	−	0.956762i	$-0.593945\pi$
$547$	−4.07099e6	−0.581744	−0.290872	+	0.956762i	$0.593945\pi$
$548$	4.09323e6	0.582258
$549$	3.41102e6	0.483006
$550$	0	0
$551$	−7.14618e6	−1.00276
$552$	−1.23539e6	−0.172566
$553$	4.23804e6	0.589322
$554$	−716838.	−0.0992308
$555$	0	0
$556$	−4.63582e6	−0.635975
$557$	−3.68580e6	−0.503377	−0.251689	−	0.967808i	$-0.580986\pi$
$557$	−3.68580e6	−0.503377	−0.251689	+	0.967808i	$0.580986\pi$
$558$	6.53922e6	0.889079
$559$	−1.74138e6	−0.235702
$560$	0	0
$561$	−2.89253e6	−0.388035
$562$	5.33954e6	0.713121
$563$	−1.24931e7	−1.66112	−0.830559	−	0.556931i	$-0.811979\pi$
$563$	−1.24931e7	−1.66112	−0.830559	+	0.556931i	$0.811979\pi$
$564$	1.42577e6	0.188734
$565$	0	0
$566$	3.11260e6	0.408396
$567$	5.01210e6	0.654730
$568$	1.09006e6	0.141769
$569$	9.35048e6	1.21075	0.605374	−	0.795941i	$-0.293024\pi$
$569$	9.35048e6	1.21075	0.605374	+	0.795941i	$0.293024\pi$
$570$	0	0
$571$	−1.24948e6	−0.160375	−0.0801877	−	0.996780i	$-0.525552\pi$
$571$	−1.24948e6	−0.160375	−0.0801877	+	0.996780i	$0.525552\pi$
$572$	−7.74797e6	−0.990143
$573$	−2.08056e6	−0.264724
$574$	−7.15866e6	−0.906885
$575$	0	0
$576$	−6.35828e6	−0.798516
$577$	−1.51309e7	−1.89202	−0.946011	−	0.324134i	$-0.894927\pi$
$577$	−1.51309e7	−1.89202	−0.946011	+	0.324134i	$0.894927\pi$
$578$	2.63110e6	0.327580
$579$	669222.	0.0829609
$580$	0	0
$581$	−1.24125e7	−1.52553
$582$	−978451.	−0.119738
$583$	−7.32613e6	−0.892695
$584$	−9.67497e6	−1.17386
$585$	0	0
$586$	1.52414e6	0.183349
$587$	1.50652e7	1.80460	0.902299	−	0.431111i	$-0.141878\pi$
$587$	1.50652e7	1.80460	0.902299	+	0.431111i	$0.141878\pi$
$588$	−376833.	−0.0449475
$589$	1.13438e7	1.34732
$590$	0	0
$591$	−1.06756e6	−0.125725
$592$	1.15571e6	0.135533
$593$	−2.49472e6	−0.291330	−0.145665	−	0.989334i	$-0.546532\pi$
$593$	−2.49472e6	−0.291330	−0.145665	+	0.989334i	$0.546532\pi$
$594$	−3.72297e6	−0.432935
$595$	0	0
$596$	−2.26610e6	−0.261314
$597$	−2.24455e6	−0.257747
$598$	6.03557e6	0.690184
$599$	1.27142e7	1.44784	0.723920	−	0.689884i	$-0.242339\pi$
$599$	1.27142e7	1.44784	0.723920	+	0.689884i	$0.242339\pi$
$600$	0	0
$601$	−4.06846e6	−0.459456	−0.229728	−	0.973255i	$-0.573784\pi$
$601$	−4.06846e6	−0.459456	−0.229728	+	0.973255i	$0.573784\pi$
$602$	767772.	0.0863458
$603$	8.11916e6	0.909322
$604$	7.06654e6	0.788161
$605$	0	0
$606$	−1.18993e6	−0.131625
$607$	7.00912e6	0.772132	0.386066	−	0.922471i	$-0.373834\pi$
$607$	7.00912e6	0.772132	0.386066	+	0.922471i	$0.373834\pi$
$608$	−8.10175e6	−0.888832
$609$	−1.92032e6	−0.209812
$610$	0	0
$611$	−2.05831e7	−2.23053
$612$	−5.36854e6	−0.579399
$613$	−1.15860e7	−1.24533	−0.622664	−	0.782489i	$-0.713950\pi$
$613$	−1.15860e7	−1.24533	−0.622664	+	0.782489i	$0.713950\pi$
$614$	−4.97927e6	−0.533022
$615$	0	0
$616$	1.00943e7	1.07182
$617$	1.17982e7	1.24768	0.623842	−	0.781551i	$-0.285571\pi$
$617$	1.17982e7	1.24768	0.623842	+	0.781551i	$0.285571\pi$
$618$	−1.56793e6	−0.165141
$619$	−1.01497e7	−1.06470	−0.532351	−	0.846524i	$-0.678691\pi$
$619$	−1.01497e7	−1.06470	−0.532351	+	0.846524i	$0.678691\pi$
$620$	0	0
$621$	−3.03702e6	−0.316023
$622$	−1.23927e7	−1.28437
$623$	−1.44249e7	−1.48900
$624$	−871767.	−0.0896270
$625$	0	0
$626$	4.52357e6	0.461366
$627$	−3.12008e6	−0.316954
$628$	4.71493e6	0.477064
$629$	7.18574e6	0.724178
$630$	0	0
$631$	1.58235e7	1.58209	0.791043	−	0.611760i	$-0.209538\pi$
$631$	1.58235e7	1.58209	0.791043	+	0.611760i	$0.209538\pi$
$632$	−7.71660e6	−0.768481
$633$	3.08687e6	0.306202
$634$	−5.44387e6	−0.537879
$635$	0	0
$636$	950948.	0.0932210
$637$	5.44015e6	0.531205
$638$	−9.11601e6	−0.886652
$639$	1.29461e6	0.125426
$640$	0	0
$641$	−1.34533e7	−1.29325	−0.646626	−	0.762807i	$-0.723821\pi$
$641$	−1.34533e7	−1.29325	−0.646626	+	0.762807i	$0.723821\pi$
$642$	−3.19732e6	−0.306160
$643$	−5.84658e6	−0.557667	−0.278833	−	0.960340i	$-0.589948\pi$
$643$	−5.84658e6	−0.557667	−0.278833	+	0.960340i	$0.589948\pi$
$644$	2.78667e6	0.264771
$645$	0	0
$646$	8.89327e6	0.838456
$647$	1.04566e7	0.982037	0.491019	−	0.871149i	$-0.336625\pi$
$647$	1.04566e7	0.982037	0.491019	+	0.871149i	$0.336625\pi$
$648$	−9.12600e6	−0.853774
$649$	6.49531e6	0.605324
$650$	0	0
$651$	3.04831e6	0.281907
$652$	−3.46476e6	−0.319193
$653$	−2.16577e6	−0.198761	−0.0993803	−	0.995050i	$-0.531686\pi$
$653$	−2.16577e6	−0.198761	−0.0993803	+	0.995050i	$0.531686\pi$
$654$	2.23791e6	0.204597
$655$	0	0
$656$	4.00409e6	0.363282
$657$	−1.14904e7	−1.03854
$658$	9.07507e6	0.817119
$659$	−7.08898e6	−0.635872	−0.317936	−	0.948112i	$-0.602990\pi$
$659$	−7.08898e6	−0.635872	−0.317936	+	0.948112i	$0.602990\pi$
$660$	0	0
$661$	−2.93805e6	−0.261551	−0.130775	−	0.991412i	$-0.541747\pi$
$661$	−2.93805e6	−0.261551	−0.130775	+	0.991412i	$0.541747\pi$
$662$	4.87271e6	0.432141
$663$	−5.42028e6	−0.478892
$664$	2.26006e7	1.98930
$665$	0	0
$666$	4.46813e6	0.390338
$667$	−7.43641e6	−0.647215
$668$	320097.	0.0277550
$669$	−2.65089e6	−0.228995
$670$	0	0
$671$	7.54828e6	0.647205
$672$	−2.17710e6	−0.185975
$673$	−3.90020e6	−0.331932	−0.165966	−	0.986131i	$-0.553074\pi$
$673$	−3.90020e6	−0.331932	−0.165966	+	0.986131i	$0.553074\pi$
$674$	−1.16733e7	−0.989796
$675$	0	0
$676$	−8.44120e6	−0.710457
$677$	1.90810e7	1.60003	0.800017	−	0.599977i	$-0.204824\pi$
$677$	1.90810e7	1.60003	0.800017	+	0.599977i	$0.204824\pi$
$678$	3.21694e6	0.268762
$679$	6.52181e6	0.542868
$680$	0	0
$681$	−2.58595e6	−0.213674
$682$	1.44707e7	1.19132
$683$	−7.59368e6	−0.622875	−0.311437	−	0.950267i	$-0.600810\pi$
$683$	−7.59368e6	−0.622875	−0.311437	+	0.950267i	$0.600810\pi$
$684$	−5.79087e6	−0.473264
$685$	0	0
$686$	−9.37744e6	−0.760806
$687$	−4.02029e6	−0.324986
$688$	−429442.	−0.0345886
$689$	−1.37284e7	−1.10172
$690$	0	0
$691$	−8.45792e6	−0.673858	−0.336929	−	0.941530i	$-0.609388\pi$
$691$	−8.45792e6	−0.673858	−0.336929	+	0.941530i	$0.609388\pi$
$692$	1.03307e6	0.0820094
$693$	1.19884e7	0.948262
$694$	−9.03800e6	−0.712317
$695$	0	0
$696$	3.49650e6	0.273597
$697$	2.48958e7	1.94108
$698$	6.12100e6	0.475536
$699$	−1.45195e6	−0.112398
$700$	0	0
$701$	9.97360e6	0.766579	0.383290	−	0.923628i	$-0.374791\pi$
$701$	9.97360e6	0.766579	0.383290	+	0.923628i	$0.374791\pi$
$702$	−6.97643e6	−0.534306
$703$	7.75103e6	0.591522
$704$	−1.40703e7	−1.06997
$705$	0	0
$706$	−1.60901e7	−1.21492
$707$	7.93140e6	0.596762
$708$	−843105.	−0.0632119
$709$	−1.46549e7	−1.09488	−0.547440	−	0.836845i	$-0.684397\pi$
$709$	−1.46549e7	−1.09488	−0.547440	+	0.836845i	$0.684397\pi$
$710$	0	0
$711$	−9.16459e6	−0.679891
$712$	2.62648e7	1.94167
$713$	1.18045e7	0.869610
$714$	2.38980e6	0.175435
$715$	0	0
$716$	−393721.	−0.0287016
$717$	1.79196e6	0.130176
$718$	6.07242e6	0.439593
$719$	−1.54112e7	−1.11177	−0.555885	−	0.831259i	$-0.687620\pi$
$719$	−1.54112e7	−1.11177	−0.555885	+	0.831259i	$0.687620\pi$
$720$	0	0
$721$	1.04509e7	0.748716
$722$	−196675.	−0.0140413
$723$	−4.27957e6	−0.304477
$724$	640573.	0.0454174
$725$	0	0
$726$	−1.44245e6	−0.101569
$727$	7.74678e6	0.543607	0.271803	−	0.962353i	$-0.412380\pi$
$727$	7.74678e6	0.543607	0.271803	+	0.962353i	$0.412380\pi$
$728$	1.89155e7	1.32279
$729$	−9.02393e6	−0.628893
$730$	0	0
$731$	−2.67009e6	−0.184813
$732$	−979783.	−0.0675853
$733$	−1.65681e7	−1.13897	−0.569485	−	0.822002i	$-0.692857\pi$
$733$	−1.65681e7	−1.13897	−0.569485	+	0.822002i	$0.692857\pi$
$734$	−1.04044e7	−0.712815
$735$	0	0
$736$	−8.43078e6	−0.573685
$737$	1.79670e7	1.21845
$738$	1.54803e7	1.04626
$739$	−7.85461e6	−0.529070	−0.264535	−	0.964376i	$-0.585219\pi$
$739$	−7.85461e6	−0.529070	−0.264535	+	0.964376i	$0.585219\pi$
$740$	0	0
$741$	−5.84668e6	−0.391169
$742$	6.05282e6	0.403597
$743$	1.84189e7	1.22403	0.612013	−	0.790848i	$-0.290360\pi$
$743$	1.84189e7	1.22403	0.612013	+	0.790848i	$0.290360\pi$
$744$	−5.55033e6	−0.367610
$745$	0	0
$746$	−1.51173e7	−0.994548
$747$	2.68416e7	1.75997
$748$	−1.18801e7	−0.776366
$749$	2.13115e7	1.38807
$750$	0	0
$751$	−1.06056e7	−0.686179	−0.343089	−	0.939303i	$-0.611473\pi$
$751$	−1.06056e7	−0.686179	−0.343089	+	0.939303i	$0.611473\pi$
$752$	−5.07600e6	−0.327324
$753$	−2.24308e6	−0.144164
$754$	−1.70824e7	−1.09426
$755$	0	0
$756$	−3.22107e6	−0.204973
$757$	2.40404e7	1.52476	0.762381	−	0.647128i	$-0.224030\pi$
$757$	2.40404e7	1.52476	0.762381	+	0.647128i	$0.224030\pi$
$758$	−7.90954e6	−0.500009
$759$	−3.24679e6	−0.204574
$760$	0	0
$761$	5.80826e6	0.363567	0.181783	−	0.983339i	$-0.441813\pi$
$761$	5.80826e6	0.363567	0.181783	+	0.983339i	$0.441813\pi$
$762$	3.79424e6	0.236721
$763$	−1.49167e7	−0.927601
$764$	−8.54521e6	−0.529650
$765$	0	0
$766$	−1.79862e6	−0.110756
$767$	1.21715e7	0.747060
$768$	4.44890e6	0.272176
$769$	2.91283e7	1.77623	0.888114	−	0.459623i	$-0.152015\pi$
$769$	2.91283e7	1.77623	0.888114	+	0.459623i	$0.152015\pi$
$770$	0	0
$771$	−2.99999e6	−0.181754
$772$	2.74861e6	0.165985
$773$	−1.29224e7	−0.777849	−0.388925	−	0.921270i	$-0.627153\pi$
$773$	−1.29224e7	−0.777849	−0.388925	+	0.921270i	$0.627153\pi$
$774$	−1.66028e6	−0.0996158
$775$	0	0
$776$	−1.18749e7	−0.707905
$777$	2.08285e6	0.123767
$778$	1.23758e7	0.733031
$779$	2.68543e7	1.58551
$780$	0	0
$781$	2.86486e6	0.168065
$782$	9.25445e6	0.541170
$783$	8.59564e6	0.501042
$784$	1.34160e6	0.0779528
$785$	0	0
$786$	5.76834e6	0.333039
$787$	1.20860e7	0.695579	0.347789	−	0.937573i	$-0.386932\pi$
$787$	1.20860e7	0.695579	0.347789	+	0.937573i	$0.386932\pi$
$788$	−4.38464e6	−0.251546
$789$	2.57631e6	0.147335
$790$	0	0
$791$	−2.14423e7	−1.21852
$792$	−2.18284e7	−1.23654
$793$	1.41447e7	0.798747
$794$	−1.93037e7	−1.08665
$795$	0	0
$796$	−9.21877e6	−0.515692
$797$	−1.32106e7	−0.736677	−0.368339	−	0.929692i	$-0.620073\pi$
$797$	−1.32106e7	−0.736677	−0.368339	+	0.929692i	$0.620073\pi$
$798$	2.57780e6	0.143298
$799$	−3.15605e7	−1.74895
$800$	0	0
$801$	3.11933e7	1.71783
$802$	−1.17371e7	−0.644352
$803$	−2.54274e7	−1.39159
$804$	−2.33215e6	−0.127238
$805$	0	0
$806$	2.71165e7	1.47027
$807$	5.77791e6	0.312311
$808$	−1.44414e7	−0.778184
$809$	−4.35025e6	−0.233691	−0.116846	−	0.993150i	$-0.537278\pi$
$809$	−4.35025e6	−0.233691	−0.116846	+	0.993150i	$0.537278\pi$
$810$	0	0
$811$	3.12236e7	1.66698	0.833491	−	0.552533i	$-0.186339\pi$
$811$	3.12236e7	1.66698	0.833491	+	0.552533i	$0.186339\pi$
$812$	−7.88708e6	−0.419784
$813$	−4.58491e6	−0.243279
$814$	9.88759e6	0.523033
$815$	0	0
$816$	−1.33670e6	−0.0702760
$817$	−2.88014e6	−0.150959
$818$	2.28579e7	1.19441
$819$	2.24650e7	1.17030
$820$	0	0
$821$	−2.79299e7	−1.44614	−0.723072	−	0.690773i	$-0.757271\pi$
$821$	−2.79299e7	−1.44614	−0.723072	+	0.690773i	$0.757271\pi$
$822$	3.94021e6	0.203395
$823$	−1.21219e7	−0.623836	−0.311918	−	0.950109i	$-0.600971\pi$
$823$	−1.21219e7	−0.623836	−0.311918	+	0.950109i	$0.600971\pi$
$824$	−1.90290e7	−0.976333
$825$	0	0
$826$	−5.36640e6	−0.273674
$827$	−2.80935e7	−1.42838	−0.714188	−	0.699954i	$-0.753204\pi$
$827$	−2.80935e7	−1.42838	−0.714188	+	0.699954i	$0.753204\pi$
$828$	−6.02606e6	−0.305462
$829$	4.47138e6	0.225972	0.112986	−	0.993597i	$-0.463958\pi$
$829$	4.47138e6	0.225972	0.112986	+	0.993597i	$0.463958\pi$
$830$	0	0
$831$	722606.	0.0362994
$832$	−2.63662e7	−1.32050
$833$	8.34148e6	0.416515
$834$	−4.46251e6	−0.222159
$835$	0	0
$836$	−1.28147e7	−0.634151
$837$	−1.36447e7	−0.673209
$838$	2.15462e7	1.05989
$839$	−4.59792e6	−0.225505	−0.112753	−	0.993623i	$-0.535967\pi$
$839$	−4.59792e6	−0.225505	−0.112753	+	0.993623i	$0.535967\pi$
$840$	0	0
$841$	536041.	0.0261341
$842$	−1.34877e7	−0.655627
$843$	−5.38251e6	−0.260865
$844$	1.26783e7	0.612639
$845$	0	0
$846$	−1.96245e7	−0.942696
$847$	9.61460e6	0.460493
$848$	−3.38555e6	−0.161674
$849$	−3.13764e6	−0.149394
$850$	0	0
$851$	8.06582e6	0.381790
$852$	−371865.	−0.0175504
$853$	1.12123e7	0.527622	0.263811	−	0.964574i	$-0.415021\pi$
$853$	1.12123e7	0.527622	0.263811	+	0.964574i	$0.415021\pi$
$854$	−6.23636e6	−0.292608
$855$	0	0
$856$	−3.88039e7	−1.81005
$857$	−1.75530e7	−0.816393	−0.408197	−	0.912894i	$-0.633842\pi$
$857$	−1.75530e7	−0.816393	−0.408197	+	0.912894i	$0.633842\pi$
$858$	−7.45831e6	−0.345877
$859$	−5.17739e6	−0.239402	−0.119701	−	0.992810i	$-0.538194\pi$
$859$	−5.17739e6	−0.239402	−0.119701	+	0.992810i	$0.538194\pi$
$860$	0	0
$861$	7.21627e6	0.331745
$862$	4.90716e6	0.224937
$863$	−3.58979e7	−1.64075	−0.820374	−	0.571827i	$-0.806235\pi$
$863$	−3.58979e7	−1.64075	−0.820374	+	0.571827i	$0.806235\pi$
$864$	9.74502e6	0.444118
$865$	0	0
$866$	−2.20003e7	−0.996857
$867$	−2.65227e6	−0.119831
$868$	1.25199e7	0.564030
$869$	−2.02804e7	−0.911020
$870$	0	0
$871$	3.36682e7	1.50374
$872$	2.71602e7	1.20960
$873$	−1.41032e7	−0.626297
$874$	9.98248e6	0.442038
$875$	0	0
$876$	3.30053e6	0.145319
$877$	2.49898e7	1.09715	0.548573	−	0.836103i	$-0.315171\pi$
$877$	2.49898e7	1.09715	0.548573	+	0.836103i	$0.315171\pi$
$878$	8.09165e6	0.354242
$879$	−1.53640e6	−0.0670706
$880$	0	0
$881$	1.15913e7	0.503144	0.251572	−	0.967839i	$-0.419052\pi$
$881$	1.15913e7	0.503144	0.251572	+	0.967839i	$0.419052\pi$
$882$	5.18678e6	0.224505
$883$	−3.76711e7	−1.62595	−0.812973	−	0.582301i	$-0.802153\pi$
$883$	−3.76711e7	−1.62595	−0.812973	+	0.582301i	$0.802153\pi$
$884$	−2.22620e7	−0.958151
$885$	0	0
$886$	1.37103e7	0.586763
$887$	−1.24799e7	−0.532602	−0.266301	−	0.963890i	$-0.585801\pi$
$887$	−1.24799e7	−0.532602	−0.266301	+	0.963890i	$0.585801\pi$
$888$	−3.79245e6	−0.161394
$889$	−2.52903e7	−1.07325
$890$	0	0
$891$	−2.39846e7	−1.01213
$892$	−1.08877e7	−0.458166
$893$	−3.40433e7	−1.42857
$894$	−2.18138e6	−0.0912824
$895$	0	0
$896$	−5.85559e6	−0.243669
$897$	−6.08413e6	−0.252475
$898$	−2.62960e7	−1.08817
$899$	−3.34102e7	−1.37873
$900$	0	0
$901$	−2.10500e7	−0.863852
$902$	3.42566e7	1.40193
$903$	−773951.	−0.0315859
$904$	3.90421e7	1.58896
$905$	0	0
$906$	6.80236e6	0.275321
$907$	2.26764e7	0.915285	0.457643	−	0.889136i	$-0.348694\pi$
$907$	2.26764e7	0.915285	0.457643	+	0.889136i	$0.348694\pi$
$908$	−1.06209e7	−0.427512
$909$	−1.71513e7	−0.688475
$910$	0	0
$911$	9.59967e6	0.383231	0.191615	−	0.981470i	$-0.438627\pi$
$911$	9.59967e6	0.383231	0.191615	+	0.981470i	$0.438627\pi$
$912$	−1.44185e6	−0.0574029
$913$	5.93980e7	2.35828
$914$	−2.38939e7	−0.946065
$915$	0	0
$916$	−1.65120e7	−0.650221
$917$	−3.84486e7	−1.50993
$918$	−1.06971e7	−0.418947
$919$	2.82812e7	1.10461	0.552306	−	0.833642i	$-0.313748\pi$
$919$	2.82812e7	1.10461	0.552306	+	0.833642i	$0.313748\pi$
$920$	0	0
$921$	5.01934e6	0.194983
$922$	−2.46486e7	−0.954916
$923$	5.36844e6	0.207417
$924$	−3.44356e6	−0.132687
$925$	0	0
$926$	69520.7	0.00266432
$927$	−2.25997e7	−0.863782
$928$	2.38616e7	0.909555
$929$	−1.00811e7	−0.383237	−0.191618	−	0.981469i	$-0.561374\pi$
$929$	−1.00811e7	−0.383237	−0.191618	+	0.981469i	$0.561374\pi$
$930$	0	0
$931$	8.99769e6	0.340218
$932$	−5.96341e6	−0.224882
$933$	1.24925e7	0.469833
$934$	−1.47721e7	−0.554085
$935$	0	0
$936$	−4.09040e7	−1.52608
$937$	2.36037e7	0.878276	0.439138	−	0.898420i	$-0.355284\pi$
$937$	2.36037e7	0.878276	0.439138	+	0.898420i	$0.355284\pi$
$938$	−1.48443e7	−0.550873
$939$	−4.55997e6	−0.168771
$940$	0	0
$941$	4.49646e7	1.65538	0.827689	−	0.561188i	$-0.189655\pi$
$941$	4.49646e7	1.65538	0.827689	+	0.561188i	$0.189655\pi$
$942$	4.53867e6	0.166648
$943$	2.79449e7	1.02335
$944$	3.00162e6	0.109629
$945$	0	0
$946$	−3.67405e6	−0.133480
$947$	1.33413e7	0.483420	0.241710	−	0.970349i	$-0.422292\pi$
$947$	1.33413e7	0.483420	0.241710	+	0.970349i	$0.422292\pi$
$948$	2.63245e6	0.0951346
$949$	−4.76481e7	−1.71743
$950$	0	0
$951$	5.48767e6	0.196760
$952$	2.90035e7	1.03719
$953$	−4.89363e7	−1.74541	−0.872707	−	0.488243i	$-0.837638\pi$
$953$	−4.89363e7	−1.74541	−0.872707	+	0.488243i	$0.837638\pi$
$954$	−1.30890e7	−0.465623
$955$	0	0
$956$	7.35988e6	0.260451
$957$	9.18936e6	0.324344
$958$	−2.14787e6	−0.0756126
$959$	−2.62632e7	−0.922151
$960$	0	0
$961$	2.44061e7	0.852490
$962$	1.85283e7	0.645501
$963$	−4.60854e7	−1.60139
$964$	−1.75769e7	−0.609187
$965$	0	0
$966$	2.68249e6	0.0924900
$967$	4.92217e6	0.169274	0.0846371	−	0.996412i	$-0.473027\pi$
$967$	4.92217e6	0.169274	0.0846371	+	0.996412i	$0.473027\pi$
$968$	−1.75062e7	−0.600487
$969$	−8.96483e6	−0.306713
$970$	0	0
$971$	−1.27105e7	−0.432628	−0.216314	−	0.976324i	$-0.569403\pi$
$971$	−1.27105e7	−0.432628	−0.216314	+	0.976324i	$0.569403\pi$
$972$	1.05658e7	0.358705
$973$	2.97446e7	1.00723
$974$	3.69799e7	1.24902
$975$	0	0
$976$	3.48821e6	0.117214
$977$	−3.26784e7	−1.09528	−0.547639	−	0.836715i	$-0.684473\pi$
$977$	−3.26784e7	−1.09528	−0.547639	+	0.836715i	$0.684473\pi$
$978$	−3.33523e6	−0.111501
$979$	6.90282e7	2.30181
$980$	0	0
$981$	3.22567e7	1.07016
$982$	5.54339e6	0.183441
$983$	1.49358e6	0.0492998	0.0246499	−	0.999696i	$-0.492153\pi$
$983$	1.49358e6	0.0492998	0.0246499	+	0.999696i	$0.492153\pi$
$984$	−1.31393e7	−0.432599
$985$	0	0
$986$	−2.61928e7	−0.858004
$987$	−9.14809e6	−0.298908
$988$	−2.40133e7	−0.782636
$989$	−2.99711e6	−0.0974344
$990$	0	0
$991$	−1.79508e7	−0.580629	−0.290314	−	0.956931i	$-0.593760\pi$
$991$	−1.79508e7	−0.580629	−0.290314	+	0.956931i	$0.593760\pi$
$992$	−3.78777e7	−1.22209
$993$	−4.91191e6	−0.158080
$994$	−2.36694e6	−0.0759838
$995$	0	0
$996$	−7.70999e6	−0.246267
$997$	3.29472e7	1.04974	0.524869	−	0.851183i	$-0.324114\pi$
$997$	3.29472e7	1.04974	0.524869	+	0.851183i	$0.324114\pi$
$998$	−1.96547e7	−0.624656
$999$	−9.32317e6	−0.295563

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1075.6.a.j.1.24	yes	37
5.4	even	2		1075.6.a.i.1.14	✓	37

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1075.6.a.i.1.14	✓	37	5.4	even	2
1075.6.a.j.1.24	yes	37	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1075 = 5^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1075.a (trivial)

\(f(q)\)	\(=\)	\(q+3.95362 q^{2} -3.98544 q^{3} -16.3689 q^{4} -15.7569 q^{6} +105.027 q^{7} -191.232 q^{8} -227.116 q^{9} +O(q^{10})\)	\(q+3.95362 q^{2} -3.98544 q^{3} -16.3689 q^{4} -15.7569 q^{6} +105.027 q^{7} -191.232 q^{8} -227.116 q^{9} -502.589 q^{11} +65.2371 q^{12} -941.796 q^{13} +415.237 q^{14} -232.256 q^{16} -1444.07 q^{17} -897.932 q^{18} -1557.68 q^{19} -418.578 q^{21} -1987.05 q^{22} -1620.94 q^{23} +762.144 q^{24} -3723.51 q^{26} +1873.62 q^{27} -1719.17 q^{28} +4587.72 q^{29} -7282.53 q^{31} +5201.18 q^{32} +2003.03 q^{33} -5709.32 q^{34} +3717.64 q^{36} -4976.02 q^{37} -6158.46 q^{38} +3753.47 q^{39} -17240.0 q^{41} -1654.90 q^{42} +1849.00 q^{43} +8226.81 q^{44} -6408.57 q^{46} +21855.2 q^{47} +925.643 q^{48} -5776.36 q^{49} +5755.26 q^{51} +15416.1 q^{52} +14576.8 q^{53} +7407.58 q^{54} -20084.5 q^{56} +6208.02 q^{57} +18138.1 q^{58} -12923.7 q^{59} -15018.8 q^{61} -28792.4 q^{62} -23853.3 q^{63} +27995.7 q^{64} +7919.24 q^{66} -35748.9 q^{67} +23637.8 q^{68} +6460.14 q^{69} -5700.21 q^{71} +43432.0 q^{72} +50592.8 q^{73} -19673.3 q^{74} +25497.4 q^{76} -52785.3 q^{77} +14839.8 q^{78} +40352.0 q^{79} +47722.1 q^{81} -68160.3 q^{82} -118184. q^{83} +6851.65 q^{84} +7310.25 q^{86} -18284.1 q^{87} +96111.1 q^{88} -137345. q^{89} -98913.9 q^{91} +26532.9 q^{92} +29024.1 q^{93} +86407.1 q^{94} -20729.0 q^{96} +62096.6 q^{97} -22837.5 q^{98} +114146. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 37 q + 630 q^{4} + 291 q^{6} + 213 q^{8} + 3535 q^{9}+O(q^{10}) \) 37 * q + 630 * q^4 + 291 * q^6 + 213 * q^8 + 3535 * q^9	\( 37 q + 630 q^{4} + 291 q^{6} + 213 q^{8} + 3535 q^{9} + 675 q^{11} - 4446 q^{12} + 1241 q^{13} + 2375 q^{14} + 10518 q^{16} + 1153 q^{17} - 6680 q^{18} + 4065 q^{19} + 9953 q^{21} + 9283 q^{22} - 360 q^{23} + 2265 q^{24} + 23695 q^{26} + 1323 q^{27} - 30375 q^{28} + 19290 q^{29} + 23291 q^{31} + 8166 q^{32} - 10388 q^{33} - 13153 q^{34} + 148705 q^{36} + 13501 q^{37} - 8127 q^{38} - 1327 q^{39} + 38345 q^{41} - 21835 q^{42} + 68413 q^{43} + 47768 q^{44} + 48755 q^{46} + 84859 q^{47} - 208720 q^{48} + 107255 q^{49} + 62027 q^{51} + 128320 q^{52} - 53559 q^{53} + 44158 q^{54} + 107538 q^{56} + 104239 q^{57} - 85186 q^{58} + 48186 q^{59} + 82364 q^{61} + 206506 q^{62} - 75269 q^{63} + 161467 q^{64} + 91969 q^{66} + 38168 q^{67} + 95991 q^{68} + 287103 q^{69} + 155302 q^{71} + 9979 q^{72} + 31927 q^{73} + 59946 q^{74} + 225407 q^{76} - 80007 q^{77} - 67815 q^{78} + 150174 q^{79} + 417489 q^{81} + 60603 q^{82} + 266568 q^{83} + 586273 q^{84} - 57554 q^{87} + 323054 q^{88} + 334356 q^{89} + 51747 q^{91} - 258529 q^{92} - 285287 q^{93} + 302744 q^{94} + 287282 q^{96} - 78640 q^{97} - 397117 q^{98} + 362152 q^{99}+O(q^{100}) \) 37 * q + 630 * q^4 + 291 * q^6 + 213 * q^8 + 3535 * q^9 + 675 * q^11 - 4446 * q^12 + 1241 * q^13 + 2375 * q^14 + 10518 * q^16 + 1153 * q^17 - 6680 * q^18 + 4065 * q^19 + 9953 * q^21 + 9283 * q^22 - 360 * q^23 + 2265 * q^24 + 23695 * q^26 + 1323 * q^27 - 30375 * q^28 + 19290 * q^29 + 23291 * q^31 + 8166 * q^32 - 10388 * q^33 - 13153 * q^34 + 148705 * q^36 + 13501 * q^37 - 8127 * q^38 - 1327 * q^39 + 38345 * q^41 - 21835 * q^42 + 68413 * q^43 + 47768 * q^44 + 48755 * q^46 + 84859 * q^47 - 208720 * q^48 + 107255 * q^49 + 62027 * q^51 + 128320 * q^52 - 53559 * q^53 + 44158 * q^54 + 107538 * q^56 + 104239 * q^57 - 85186 * q^58 + 48186 * q^59 + 82364 * q^61 + 206506 * q^62 - 75269 * q^63 + 161467 * q^64 + 91969 * q^66 + 38168 * q^67 + 95991 * q^68 + 287103 * q^69 + 155302 * q^71 + 9979 * q^72 + 31927 * q^73 + 59946 * q^74 + 225407 * q^76 - 80007 * q^77 - 67815 * q^78 + 150174 * q^79 + 417489 * q^81 + 60603 * q^82 + 266568 * q^83 + 586273 * q^84 - 57554 * q^87 + 323054 * q^88 + 334356 * q^89 + 51747 * q^91 - 258529 * q^92 - 285287 * q^93 + 302744 * q^94 + 287282 * q^96 - 78640 * q^97 - 397117 * q^98 + 362152 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more