LMFDB - Embedded newform 1849.2.a.q.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1849,2,Mod(1,1849)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1849, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1849.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1849 = 43^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1849.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:	$14.7643393337$
Analytic rank:	$1$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 27 x^{18} + 300 x^{16} - 1775 x^{14} + 6037 x^{12} - 11859 x^{10} + 12809 x^{8} - 6823 x^{6} + \cdots + 1 $ x^20 - 27x^18 + 300x^16 - 1775x^14 + 6037x^12 - 11859x^10 + 12809x^8 - 6823x^6 + 1500x^4 - 75*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-0.805985$ of defining polynomial
Character	$\chi$	$=$	1849.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.805985 q^{2} +1.88727 q^{3} -1.35039 q^{4} -1.04963 q^{5} -1.52111 q^{6} -0.595112 q^{7} +2.70036 q^{8} +0.561775 q^{9} +O(q^{10})$	\(q-0.805985 q^{2} +1.88727 q^{3} -1.35039 q^{4} -1.04963 q^{5} -1.52111 q^{6} -0.595112 q^{7} +2.70036 q^{8} +0.561775 q^{9} +0.845990 q^{10} +3.37853 q^{11} -2.54854 q^{12} -5.24147 q^{13} +0.479651 q^{14} -1.98094 q^{15} +0.524325 q^{16} +5.72667 q^{17} -0.452782 q^{18} -5.32309 q^{19} +1.41741 q^{20} -1.12313 q^{21} -2.72305 q^{22} -0.0573439 q^{23} +5.09630 q^{24} -3.89827 q^{25} +4.22455 q^{26} -4.60158 q^{27} +0.803632 q^{28} +8.57253 q^{29} +1.59661 q^{30} -8.23414 q^{31} -5.82332 q^{32} +6.37619 q^{33} -4.61561 q^{34} +0.624650 q^{35} -0.758615 q^{36} -0.373092 q^{37} +4.29033 q^{38} -9.89205 q^{39} -2.83439 q^{40} +5.92295 q^{41} +0.905230 q^{42} -4.56233 q^{44} -0.589659 q^{45} +0.0462183 q^{46} -5.75673 q^{47} +0.989541 q^{48} -6.64584 q^{49} +3.14194 q^{50} +10.8077 q^{51} +7.07802 q^{52} -0.292329 q^{53} +3.70880 q^{54} -3.54622 q^{55} -1.60702 q^{56} -10.0461 q^{57} -6.90933 q^{58} -1.78040 q^{59} +2.67504 q^{60} -0.570869 q^{61} +6.63659 q^{62} -0.334319 q^{63} +3.64486 q^{64} +5.50163 q^{65} -5.13911 q^{66} -13.1397 q^{67} -7.73322 q^{68} -0.108223 q^{69} -0.503459 q^{70} -3.55996 q^{71} +1.51700 q^{72} +12.7597 q^{73} +0.300706 q^{74} -7.35707 q^{75} +7.18823 q^{76} -2.01060 q^{77} +7.97285 q^{78} -8.49561 q^{79} -0.550350 q^{80} -10.3697 q^{81} -4.77381 q^{82} -0.846914 q^{83} +1.51667 q^{84} -6.01091 q^{85} +16.1787 q^{87} +9.12326 q^{88} -9.39121 q^{89} +0.475256 q^{90} +3.11926 q^{91} +0.0774366 q^{92} -15.5400 q^{93} +4.63984 q^{94} +5.58730 q^{95} -10.9902 q^{96} +8.30305 q^{97} +5.35645 q^{98} +1.89798 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 14 q^{4} - 32 q^{6} - 6 q^{9}+O(q^{10}) $ 20 * q + 14 * q^4 - 32 * q^6 - 6 * q^9	$ 20 q + 14 q^{4} - 32 q^{6} - 6 q^{9} + 12 q^{10} - 20 q^{11} + 6 q^{13} - 34 q^{14} - 34 q^{15} + 14 q^{16} - 8 q^{17} + 12 q^{21} - 46 q^{23} + 2 q^{24} + 20 q^{25} - 38 q^{31} - 76 q^{35} - 46 q^{36} - 68 q^{38} - 64 q^{40} - 56 q^{41} - 58 q^{44} - 24 q^{47} - 20 q^{49} - 20 q^{52} - 42 q^{53} + 66 q^{54} - 46 q^{56} + 2 q^{57} + 12 q^{58} - 90 q^{59} + 18 q^{60} - 28 q^{64} + 76 q^{66} - 62 q^{67} - 54 q^{68} + 24 q^{74} - 36 q^{78} - 10 q^{79} - 48 q^{81} - 68 q^{83} + 70 q^{84} - 12 q^{87} + 58 q^{90} - 8 q^{92} - 42 q^{95} - 22 q^{96} - 44 q^{97} - 38 q^{99}+O(q^{100}) $ 20 * q + 14 * q^4 - 32 * q^6 - 6 * q^9 + 12 * q^10 - 20 * q^11 + 6 * q^13 - 34 * q^14 - 34 * q^15 + 14 * q^16 - 8 * q^17 + 12 * q^21 - 46 * q^23 + 2 * q^24 + 20 * q^25 - 38 * q^31 - 76 * q^35 - 46 * q^36 - 68 * q^38 - 64 * q^40 - 56 * q^41 - 58 * q^44 - 24 * q^47 - 20 * q^49 - 20 * q^52 - 42 * q^53 + 66 * q^54 - 46 * q^56 + 2 * q^57 + 12 * q^58 - 90 * q^59 + 18 * q^60 - 28 * q^64 + 76 * q^66 - 62 * q^67 - 54 * q^68 + 24 * q^74 - 36 * q^78 - 10 * q^79 - 48 * q^81 - 68 * q^83 + 70 * q^84 - 12 * q^87 + 58 * q^90 - 8 * q^92 - 42 * q^95 - 22 * q^96 - 44 * q^97 - 38 * 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 $

$2$	−0.805985	−0.569917	−0.284959	−	0.958540i	$-0.591980\pi$
$2$	−0.805985	−0.569917	−0.284959	+	0.958540i	$0.591980\pi$
$3$	1.88727	1.08961	0.544807	−	0.838562i	$-0.316603\pi$
$3$	1.88727	1.08961	0.544807	+	0.838562i	$0.316603\pi$
$4$	−1.35039	−0.675194
$5$	−1.04963	−0.469411	−0.234706	−	0.972067i	$-0.575413\pi$
$5$	−1.04963	−0.469411	−0.234706	+	0.972067i	$0.575413\pi$
$6$	−1.52111	−0.620990
$7$	−0.595112	−0.224931	−0.112466	−	0.993656i	$-0.535875\pi$
$7$	−0.595112	−0.224931	−0.112466	+	0.993656i	$0.535875\pi$
$8$	2.70036	0.954722
$9$	0.561775	0.187258
$10$	0.845990	0.267526
$11$	3.37853	1.01867	0.509333	−	0.860570i	$-0.329892\pi$
$11$	3.37853	1.01867	0.509333	+	0.860570i	$0.329892\pi$
$12$	−2.54854	−0.735701
$13$	−5.24147	−1.45372	−0.726861	−	0.686784i	$-0.759022\pi$
$13$	−5.24147	−1.45372	−0.726861	+	0.686784i	$0.759022\pi$
$14$	0.479651	0.128192
$15$	−1.98094	−0.511477
$16$	0.524325	0.131081
$17$	5.72667	1.38892	0.694460	−	0.719531i	$-0.255643\pi$
$17$	5.72667	1.38892	0.694460	+	0.719531i	$0.255643\pi$
$18$	−0.452782	−0.106722
$19$	−5.32309	−1.22120	−0.610600	−	0.791939i	$-0.709072\pi$
$19$	−5.32309	−1.22120	−0.610600	+	0.791939i	$0.709072\pi$
$20$	1.41741	0.316944
$21$	−1.12313	−0.245088
$22$	−2.72305	−0.580555
$23$	−0.0573439	−0.0119570	−0.00597852	−	0.999982i	$-0.501903\pi$
$23$	−0.0573439	−0.0119570	−0.00597852	+	0.999982i	$0.501903\pi$
$24$	5.09630	1.04028
$25$	−3.89827	−0.779653
$26$	4.22455	0.828502
$27$	−4.60158	−0.885575
$28$	0.803632	0.151872
$29$	8.57253	1.59188	0.795939	−	0.605376i	$-0.206977\pi$
$29$	8.57253	1.59188	0.795939	+	0.605376i	$0.206977\pi$
$30$	1.59661	0.291500
$31$	−8.23414	−1.47890	−0.739448	−	0.673214i	$-0.764913\pi$
$31$	−8.23414	−1.47890	−0.739448	+	0.673214i	$0.764913\pi$
$32$	−5.82332	−1.02943
$33$	6.37619	1.10995
$34$	−4.61561	−0.791570
$35$	0.624650	0.105585
$36$	−0.758615	−0.126436
$37$	−0.373092	−0.0613359	−0.0306679	−	0.999530i	$-0.509763\pi$
$37$	−0.373092	−0.0613359	−0.0306679	+	0.999530i	$0.509763\pi$
$38$	4.29033	0.695983
$39$	−9.89205	−1.58400
$40$	−2.83439	−0.448157
$41$	5.92295	0.925010	0.462505	−	0.886617i	$-0.346951\pi$
$41$	5.92295	0.925010	0.462505	+	0.886617i	$0.346951\pi$
$42$	0.905230	0.139680
$43$	0	0
$43$	0	0
$44$	−4.56233	−0.687797
$45$	−0.589659	−0.0879012
$46$	0.0462183	0.00681452
$47$	−5.75673	−0.839705	−0.419853	−	0.907592i	$-0.637918\pi$
$47$	−5.75673	−0.839705	−0.419853	+	0.907592i	$0.637918\pi$
$48$	0.989541	0.142828
$49$	−6.64584	−0.949406
$50$	3.14194	0.444338
$51$	10.8077	1.51339
$52$	7.07802	0.981545
$53$	−0.292329	−0.0401544	−0.0200772	−	0.999798i	$-0.506391\pi$
$53$	−0.292329	−0.0401544	−0.0200772	+	0.999798i	$0.506391\pi$
$54$	3.70880	0.504704
$55$	−3.54622	−0.478173
$56$	−1.60702	−0.214747
$57$	−10.0461	−1.33064
$58$	−6.90933	−0.907240
$59$	−1.78040	−0.231788	−0.115894	−	0.993262i	$-0.536973\pi$
$59$	−1.78040	−0.231788	−0.115894	+	0.993262i	$0.536973\pi$
$60$	2.67504	0.345346
$61$	−0.570869	−0.0730922	−0.0365461	−	0.999332i	$-0.511636\pi$
$61$	−0.570869	−0.0730922	−0.0365461	+	0.999332i	$0.511636\pi$
$62$	6.63659	0.842848
$63$	−0.334319	−0.0421202
$64$	3.64486	0.455608
$65$	5.50163	0.682393
$66$	−5.13911	−0.632581
$67$	−13.1397	−1.60527	−0.802637	−	0.596468i	$-0.796570\pi$
$67$	−13.1397	−1.60527	−0.802637	+	0.596468i	$0.796570\pi$
$68$	−7.73322	−0.937791
$69$	−0.108223	−0.0130286
$70$	−0.503459	−0.0601748
$71$	−3.55996	−0.422490	−0.211245	−	0.977433i	$-0.567752\pi$
$71$	−3.55996	−0.422490	−0.211245	+	0.977433i	$0.567752\pi$
$72$	1.51700	0.178780
$73$	12.7597	1.49341	0.746704	−	0.665156i	$-0.231635\pi$
$73$	12.7597	1.49341	0.746704	+	0.665156i	$0.231635\pi$
$74$	0.300706	0.0349564
$75$	−7.35707	−0.849521
$76$	7.18823	0.824547
$77$	−2.01060	−0.229130
$78$	7.97285	0.902747
$79$	−8.49561	−0.955831	−0.477915	−	0.878406i	$-0.658607\pi$
$79$	−8.49561	−0.955831	−0.477915	+	0.878406i	$0.658607\pi$
$80$	−0.550350	−0.0615309
$81$	−10.3697	−1.15219
$82$	−4.77381	−0.527179
$83$	−0.846914	−0.0929609	−0.0464805	−	0.998919i	$-0.514801\pi$
$83$	−0.846914	−0.0929609	−0.0464805	+	0.998919i	$0.514801\pi$
$84$	1.51667	0.165482
$85$	−6.01091	−0.651975
$86$	0	0
$87$	16.1787	1.73453
$88$	9.12326	0.972543
$89$	−9.39121	−0.995466	−0.497733	−	0.867330i	$-0.665834\pi$
$89$	−9.39121	−0.995466	−0.497733	+	0.867330i	$0.665834\pi$
$90$	0.475256	0.0500964
$91$	3.11926	0.326987
$92$	0.0774366	0.00807332
$93$	−15.5400	−1.61142
$94$	4.63984	0.478563
$95$	5.58730	0.573245
$96$	−10.9902	−1.12168
$97$	8.30305	0.843047	0.421523	−	0.906818i	$-0.361496\pi$
$97$	8.30305	0.843047	0.421523	+	0.906818i	$0.361496\pi$
$98$	5.35645	0.541083
$99$	1.89798	0.190754
$100$	5.26417	0.526417
$101$	−14.8471	−1.47734	−0.738672	−	0.674065i	$-0.764547\pi$
$101$	−14.8471	−1.47734	−0.738672	+	0.674065i	$0.764547\pi$
$102$	−8.71088	−0.862506
$103$	−14.0712	−1.38647	−0.693237	−	0.720709i	$-0.743816\pi$
$103$	−14.0712	−1.38647	−0.693237	+	0.720709i	$0.743816\pi$
$104$	−14.1539	−1.38790
$105$	1.17888	0.115047
$106$	0.235613	0.0228847
$107$	−9.17979	−0.887444	−0.443722	−	0.896164i	$-0.646342\pi$
$107$	−9.17979	−0.887444	−0.443722	+	0.896164i	$0.646342\pi$
$108$	6.21392	0.597935
$109$	−12.0410	−1.15332	−0.576660	−	0.816984i	$-0.695644\pi$
$109$	−12.0410	−1.15332	−0.576660	+	0.816984i	$0.695644\pi$
$110$	2.85820	0.272519
$111$	−0.704123	−0.0668324
$112$	−0.312032	−0.0294842
$113$	−11.9319	−1.12246	−0.561230	−	0.827660i	$-0.689672\pi$
$113$	−11.9319	−1.12246	−0.561230	+	0.827660i	$0.689672\pi$
$114$	8.09699	0.758353
$115$	0.0601902	0.00561276
$116$	−11.5762	−1.07483
$117$	−2.94453	−0.272222
$118$	1.43497	0.132100
$119$	−3.40801	−0.312411
$120$	−5.34926	−0.488318
$121$	0.414471	0.0376792
$122$	0.460112	0.0416565
$123$	11.1782	1.00790
$124$	11.1193	0.998541
$125$	9.33993	0.835389
$126$	0.269456	0.0240051
$127$	−3.88031	−0.344322	−0.172161	−	0.985069i	$-0.555075\pi$
$127$	−3.88031	−0.344322	−0.172161	+	0.985069i	$0.555075\pi$
$128$	8.70894	0.769769
$129$	0	0
$130$	−4.43423	−0.388908
$131$	11.4076	0.996685	0.498343	−	0.866980i	$-0.333942\pi$
$131$	11.4076	0.996685	0.498343	+	0.866980i	$0.333942\pi$
$132$	−8.61033	−0.749433
$133$	3.16783	0.274686
$134$	10.5904	0.914874
$135$	4.82998	0.415698
$136$	15.4641	1.32603
$137$	−4.03930	−0.345101	−0.172550	−	0.985001i	$-0.555201\pi$
$137$	−4.03930	−0.345101	−0.172550	+	0.985001i	$0.555201\pi$
$138$	0.0872263	0.00742520
$139$	4.53005	0.384234	0.192117	−	0.981372i	$-0.438465\pi$
$139$	4.53005	0.384234	0.192117	+	0.981372i	$0.438465\pi$
$140$	−0.843520	−0.0712905
$141$	−10.8645	−0.914955
$142$	2.86927	0.240784
$143$	−17.7085	−1.48086
$144$	0.294553	0.0245461
$145$	−8.99803	−0.747246
$146$	−10.2841	−0.851120
$147$	−12.5425	−1.03449
$148$	0.503819	0.0414136
$149$	−3.86650	−0.316756	−0.158378	−	0.987379i	$-0.550626\pi$
$149$	−3.86650	−0.316756	−0.158378	+	0.987379i	$0.550626\pi$
$150$	5.92969	0.484157
$151$	19.2268	1.56466	0.782328	−	0.622867i	$-0.214032\pi$
$151$	19.2268	1.56466	0.782328	+	0.622867i	$0.214032\pi$
$152$	−14.3743	−1.16591
$153$	3.21710	0.260087
$154$	1.62052	0.130585
$155$	8.64284	0.694210
$156$	13.3581	1.06950
$157$	−8.56187	−0.683312	−0.341656	−	0.939825i	$-0.610988\pi$
$157$	−8.56187	−0.683312	−0.341656	+	0.939825i	$0.610988\pi$
$158$	6.84733	0.544745
$159$	−0.551702	−0.0437528
$160$	6.11236	0.483225
$161$	0.0341261	0.00268951
$162$	8.35785	0.656655
$163$	−1.77090	−0.138708	−0.0693539	−	0.997592i	$-0.522094\pi$
$163$	−1.77090	−0.138708	−0.0693539	+	0.997592i	$0.522094\pi$
$164$	−7.99828	−0.624561
$165$	−6.69267	−0.521024
$166$	0.682600	0.0529801
$167$	12.2878	0.950862	0.475431	−	0.879753i	$-0.342292\pi$
$167$	12.2878	0.950862	0.475431	+	0.879753i	$0.342292\pi$
$168$	−3.03287	−0.233991
$169$	14.4730	1.11331
$170$	4.84470	0.371572
$171$	−2.99038	−0.228680
$172$	0	0
$173$	11.8890	0.903901	0.451951	−	0.892043i	$-0.350728\pi$
$173$	11.8890	0.903901	0.451951	+	0.892043i	$0.350728\pi$
$174$	−13.0397	−0.988541
$175$	2.31990	0.175368
$176$	1.77145	0.133528
$177$	−3.36008	−0.252559
$178$	7.56918	0.567334
$179$	−14.0097	−1.04713	−0.523567	−	0.851984i	$-0.675399\pi$
$179$	−14.0097	−1.04713	−0.523567	+	0.851984i	$0.675399\pi$
$180$	0.796269	0.0593504
$181$	−9.61479	−0.714662	−0.357331	−	0.933978i	$-0.616313\pi$
$181$	−9.61479	−0.714662	−0.357331	+	0.933978i	$0.616313\pi$
$182$	−2.51408	−0.186356
$183$	−1.07738	−0.0796423
$184$	−0.154849	−0.0114157
$185$	0.391610	0.0287917
$186$	12.5250	0.918379
$187$	19.3477	1.41485
$188$	7.77382	0.566964
$189$	2.73845	0.199193
$190$	−4.50328	−0.326702
$191$	10.0769	0.729136	0.364568	−	0.931177i	$-0.381217\pi$
$191$	10.0769	0.729136	0.364568	+	0.931177i	$0.381217\pi$
$192$	6.87883	0.496436
$193$	0.637492	0.0458877	0.0229438	−	0.999737i	$-0.492696\pi$
$193$	0.637492	0.0458877	0.0229438	+	0.999737i	$0.492696\pi$
$194$	−6.69213	−0.480467
$195$	10.3830	0.743545
$196$	8.97447	0.641033
$197$	10.0785	0.718062	0.359031	−	0.933326i	$-0.383107\pi$
$197$	10.0785	0.718062	0.359031	+	0.933326i	$0.383107\pi$
$198$	−1.52974	−0.108714
$199$	−0.467094	−0.0331114	−0.0165557	−	0.999863i	$-0.505270\pi$
$199$	−0.467094	−0.0331114	−0.0165557	+	0.999863i	$0.505270\pi$
$200$	−10.5267	−0.744352
$201$	−24.7982	−1.74913
$202$	11.9666	0.841964
$203$	−5.10161	−0.358063
$204$	−14.5947	−1.02183
$205$	−6.21694	−0.434210
$206$	11.3412	0.790176
$207$	−0.0322144	−0.00223906
$208$	−2.74823	−0.190556
$209$	−17.9842	−1.24399
$210$	−0.950161	−0.0655673
$211$	9.71529	0.668828	0.334414	−	0.942426i	$-0.391462\pi$
$211$	9.71529	0.668828	0.334414	+	0.942426i	$0.391462\pi$
$212$	0.394757	0.0271120
$213$	−6.71859	−0.460350
$214$	7.39878	0.505770
$215$	0	0
$216$	−12.4259	−0.845478
$217$	4.90023	0.332650
$218$	9.70488	0.657298
$219$	24.0809	1.62724
$220$	4.78878	0.322859
$221$	−30.0162	−2.01911
$222$	0.567513	0.0380890
$223$	1.69217	0.113316	0.0566581	−	0.998394i	$-0.481955\pi$
$223$	1.69217	0.113316	0.0566581	+	0.998394i	$0.481955\pi$
$224$	3.46553	0.231550
$225$	−2.18995	−0.145997
$226$	9.61694	0.639709
$227$	16.7912	1.11447	0.557237	−	0.830354i	$-0.311861\pi$
$227$	16.7912	1.11447	0.557237	+	0.830354i	$0.311861\pi$
$228$	13.5661	0.898438
$229$	4.74962	0.313864	0.156932	−	0.987609i	$-0.449840\pi$
$229$	4.74962	0.313864	0.156932	+	0.987609i	$0.449840\pi$
$230$	−0.0485124	−0.00319881
$231$	−3.79455	−0.249663
$232$	23.1489	1.51980
$233$	23.2365	1.52227	0.761135	−	0.648593i	$-0.224642\pi$
$233$	23.2365	1.52227	0.761135	+	0.648593i	$0.224642\pi$
$234$	2.37325	0.155144
$235$	6.04246	0.394167
$236$	2.40423	0.156502
$237$	−16.0335	−1.04149
$238$	2.74680	0.178049
$239$	−18.3691	−1.18820	−0.594099	−	0.804392i	$-0.702491\pi$
$239$	−18.3691	−1.18820	−0.594099	+	0.804392i	$0.702491\pi$
$240$	−1.03866	−0.0670450
$241$	1.64184	0.105760	0.0528801	−	0.998601i	$-0.483160\pi$
$241$	1.64184	0.105760	0.0528801	+	0.998601i	$0.483160\pi$
$242$	−0.334058	−0.0214740
$243$	−5.76571	−0.369871
$244$	0.770894	0.0493515
$245$	6.97571	0.445662
$246$	−9.00945	−0.574422
$247$	27.9008	1.77529
$248$	−22.2352	−1.41193
$249$	−1.59835	−0.101292
$250$	−7.52784	−0.476103
$251$	27.5599	1.73957	0.869784	−	0.493433i	$-0.164258\pi$
$251$	27.5599	1.73957	0.869784	+	0.493433i	$0.164258\pi$
$252$	0.451461	0.0284393
$253$	−0.193738	−0.0121802
$254$	3.12748	0.196235
$255$	−11.3442	−0.710401
$256$	−14.3090	−0.894312
$257$	−12.4147	−0.774408	−0.387204	−	0.921994i	$-0.626559\pi$
$257$	−12.4147	−0.774408	−0.387204	+	0.921994i	$0.626559\pi$
$258$	0	0
$259$	0.222031	0.0137964
$260$	−7.42934	−0.460748
$261$	4.81584	0.298093
$262$	−9.19434	−0.568028
$263$	−27.5020	−1.69585	−0.847924	−	0.530117i	$-0.822148\pi$
$263$	−27.5020	−1.69585	−0.847924	+	0.530117i	$0.822148\pi$
$264$	17.2180	1.05970
$265$	0.306838	0.0188489
$266$	−2.55322	−0.156548
$267$	−17.7237	−1.08467
$268$	17.7437	1.08387
$269$	−12.4612	−0.759773	−0.379886	−	0.925033i	$-0.624037\pi$
$269$	−12.4612	−0.759773	−0.379886	+	0.925033i	$0.624037\pi$
$270$	−3.89289	−0.236914
$271$	−19.3300	−1.17421	−0.587107	−	0.809510i	$-0.699733\pi$
$271$	−19.3300	−1.17421	−0.587107	+	0.809510i	$0.699733\pi$
$272$	3.00263	0.182061
$273$	5.88688	0.356290
$274$	3.25562	0.196679
$275$	−13.1704	−0.794206
$276$	0.146143	0.00879680
$277$	−27.9341	−1.67840	−0.839200	−	0.543823i	$-0.816976\pi$
$277$	−27.9341	−1.67840	−0.839200	+	0.543823i	$0.816976\pi$
$278$	−3.65115	−0.218982
$279$	−4.62574	−0.276936
$280$	1.68678	0.100804
$281$	30.7018	1.83151	0.915757	−	0.401734i	$-0.131592\pi$
$281$	30.7018	1.83151	0.915757	+	0.401734i	$0.131592\pi$
$282$	8.75661	0.521449
$283$	−7.09606	−0.421817	−0.210908	−	0.977506i	$-0.567642\pi$
$283$	−7.09606	−0.421817	−0.210908	+	0.977506i	$0.567642\pi$
$284$	4.80733	0.285262
$285$	10.5447	0.624615
$286$	14.2728	0.843966
$287$	−3.52482	−0.208063
$288$	−3.27140	−0.192769
$289$	15.7947	0.929100
$290$	7.25228	0.425868
$291$	15.6701	0.918595
$292$	−17.2305	−1.00834
$293$	−6.27015	−0.366306	−0.183153	−	0.983084i	$-0.558630\pi$
$293$	−6.27015	−0.366306	−0.183153	+	0.983084i	$0.558630\pi$
$294$	10.1090	0.589572
$295$	1.86877	0.108804
$296$	−1.00748	−0.0585587
$297$	−15.5466	−0.902104
$298$	3.11634	0.180525
$299$	0.300567	0.0173822
$300$	9.93490	0.573592
$301$	0	0
$302$	−15.4965	−0.891724
$303$	−28.0205	−1.60973
$304$	−2.79103	−0.160076
$305$	0.599204	0.0343103
$306$	−2.59293	−0.148228
$307$	1.95203	0.111408	0.0557041	−	0.998447i	$-0.482260\pi$
$307$	1.95203	0.111408	0.0557041	+	0.998447i	$0.482260\pi$
$308$	2.71510	0.154707
$309$	−26.5561	−1.51072
$310$	−6.96600	−0.395642
$311$	2.77911	0.157589	0.0787944	−	0.996891i	$-0.474893\pi$
$311$	2.77911	0.157589	0.0787944	+	0.996891i	$0.474893\pi$
$312$	−26.7121	−1.51228
$313$	−14.3867	−0.813183	−0.406592	−	0.913610i	$-0.633283\pi$
$313$	−14.3867	−0.813183	−0.406592	+	0.913610i	$0.633283\pi$
$314$	6.90074	0.389431
$315$	0.350913	0.0197717
$316$	11.4724	0.645371
$317$	10.5665	0.593472	0.296736	−	0.954960i	$-0.404102\pi$
$317$	10.5665	0.593472	0.296736	+	0.954960i	$0.404102\pi$
$318$	0.444664	0.0249355
$319$	28.9626	1.62159
$320$	−3.82577	−0.213867
$321$	−17.3247	−0.966972
$322$	−0.0275051	−0.00153280
$323$	−30.4835	−1.69615
$324$	14.0032	0.777954
$325$	20.4327	1.13340
$326$	1.42732	0.0790520
$327$	−22.7246	−1.25667
$328$	15.9941	0.883127
$329$	3.42590	0.188876
$330$	5.39419	0.296940
$331$	−33.7181	−1.85331	−0.926657	−	0.375908i	$-0.877331\pi$
$331$	−33.7181	−1.85331	−0.926657	+	0.375908i	$0.877331\pi$
$332$	1.14366	0.0627667
$333$	−0.209594	−0.0114857
$334$	−9.90382	−0.541913
$335$	13.7919	0.753533
$336$	−0.588887	−0.0321264
$337$	12.8457	0.699750	0.349875	−	0.936796i	$-0.386224\pi$
$337$	12.8457	0.699750	0.349875	+	0.936796i	$0.386224\pi$
$338$	−11.6650	−0.634494
$339$	−22.5187	−1.22305
$340$	8.11706	0.440209
$341$	−27.8193	−1.50650
$342$	2.41020	0.130329
$343$	8.12080	0.438482
$344$	0	0
$345$	0.113595	0.00611575
$346$	−9.58232	−0.515149
$347$	7.89069	0.423594	0.211797	−	0.977314i	$-0.432068\pi$
$347$	7.89069	0.423594	0.211797	+	0.977314i	$0.432068\pi$
$348$	−21.8475	−1.17115
$349$	12.2378	0.655072	0.327536	−	0.944839i	$-0.393782\pi$
$349$	12.2378	0.655072	0.327536	+	0.944839i	$0.393782\pi$
$350$	−1.86981	−0.0999454
$351$	24.1191	1.28738
$352$	−19.6743	−1.04864
$353$	15.6654	0.833784	0.416892	−	0.908956i	$-0.363119\pi$
$353$	15.6654	0.833784	0.416892	+	0.908956i	$0.363119\pi$
$354$	2.70818	0.143938
$355$	3.73666	0.198321
$356$	12.6818	0.672133
$357$	−6.43182	−0.340408
$358$	11.2916	0.596780
$359$	−3.39665	−0.179268	−0.0896342	−	0.995975i	$-0.528570\pi$
$359$	−3.39665	−0.179268	−0.0896342	+	0.995975i	$0.528570\pi$
$360$	−1.59229	−0.0839212
$361$	9.33525	0.491329
$362$	7.74938	0.407298
$363$	0.782218	0.0410558
$364$	−4.21221	−0.220780
$365$	−13.3930	−0.701023
$366$	0.868353	0.0453896
$367$	−10.0821	−0.526281	−0.263140	−	0.964758i	$-0.584758\pi$
$367$	−10.0821	−0.526281	−0.263140	+	0.964758i	$0.584758\pi$
$368$	−0.0300668	−0.00156734
$369$	3.32737	0.173216
$370$	−0.315632	−0.0164089
$371$	0.173968	0.00903198
$372$	20.9851	1.08802
$373$	−11.5741	−0.599284	−0.299642	−	0.954052i	$-0.596867\pi$
$373$	−11.5741	−0.599284	−0.299642	+	0.954052i	$0.596867\pi$
$374$	−15.5940	−0.806345
$375$	17.6269	0.910251
$376$	−15.5453	−0.801685
$377$	−44.9327	−2.31415
$378$	−2.20715	−0.113524
$379$	15.6110	0.801883	0.400941	−	0.916104i	$-0.368683\pi$
$379$	15.6110	0.801883	0.400941	+	0.916104i	$0.368683\pi$
$380$	−7.54502	−0.387051
$381$	−7.32319	−0.375178
$382$	−8.12180	−0.415547
$383$	14.0266	0.716724	0.358362	−	0.933583i	$-0.383335\pi$
$383$	14.0266	0.716724	0.358362	+	0.933583i	$0.383335\pi$
$384$	16.4361	0.838751
$385$	2.11040	0.107556
$386$	−0.513809	−0.0261522
$387$	0	0
$388$	−11.2123	−0.569220
$389$	17.4632	0.885418	0.442709	−	0.896665i	$-0.354017\pi$
$389$	17.4632	0.885418	0.442709	+	0.896665i	$0.354017\pi$
$390$	−8.36858	−0.423759
$391$	−0.328390	−0.0166074
$392$	−17.9462	−0.906419
$393$	21.5292	1.08600
$394$	−8.12310	−0.409236
$395$	8.91729	0.448678
$396$	−2.56300	−0.128796
$397$	4.63227	0.232487	0.116244	−	0.993221i	$-0.462915\pi$
$397$	4.63227	0.232487	0.116244	+	0.993221i	$0.462915\pi$
$398$	0.376471	0.0188708
$399$	5.97854	0.299302
$400$	−2.04396	−0.102198
$401$	24.1879	1.20789	0.603944	−	0.797027i	$-0.293595\pi$
$401$	24.1879	1.20789	0.603944	+	0.797027i	$0.293595\pi$
$402$	19.9870	0.996859
$403$	43.1590	2.14990
$404$	20.0494	0.997494
$405$	10.8844	0.540852
$406$	4.11182	0.204066
$407$	−1.26050	−0.0624808
$408$	29.1848	1.44486
$409$	−17.8451	−0.882385	−0.441192	−	0.897413i	$-0.645444\pi$
$409$	−17.8451	−0.882385	−0.441192	+	0.897413i	$0.645444\pi$
$410$	5.01076	0.247464
$411$	−7.62324	−0.376027
$412$	19.0016	0.936139
$413$	1.05953	0.0521363
$414$	0.0259643	0.00127608
$415$	0.888951	0.0436369
$416$	30.5228	1.49650
$417$	8.54941	0.418667
$418$	14.4950	0.708974
$419$	32.3299	1.57942	0.789711	−	0.613480i	$-0.210231\pi$
$419$	32.3299	1.57942	0.789711	+	0.613480i	$0.210231\pi$
$420$	−1.59195	−0.0776791
$421$	36.2683	1.76761	0.883804	−	0.467856i	$-0.154974\pi$
$421$	36.2683	1.76761	0.883804	+	0.467856i	$0.154974\pi$
$422$	−7.83038	−0.381177
$423$	−3.23399	−0.157242
$424$	−0.789394	−0.0383363
$425$	−22.3241	−1.08288
$426$	5.41509	0.262362
$427$	0.339731	0.0164407
$428$	12.3963	0.599197
$429$	−33.4206	−1.61356
$430$	0	0
$431$	−6.89989	−0.332356	−0.166178	−	0.986096i	$-0.553143\pi$
$431$	−6.89989	−0.332356	−0.166178	+	0.986096i	$0.553143\pi$
$432$	−2.41272	−0.116082
$433$	25.3029	1.21598	0.607990	−	0.793945i	$-0.291976\pi$
$433$	25.3029	1.21598	0.607990	+	0.793945i	$0.291976\pi$
$434$	−3.94952	−0.189583
$435$	−16.9817	−0.814209
$436$	16.2601	0.778715
$437$	0.305247	0.0146019
$438$	−19.4089	−0.927392
$439$	−26.4938	−1.26448	−0.632239	−	0.774773i	$-0.717864\pi$
$439$	−26.4938	−1.26448	−0.632239	+	0.774773i	$0.717864\pi$
$440$	−9.57609	−0.456522
$441$	−3.73347	−0.177784
$442$	24.1926	1.15072
$443$	31.3997	1.49184	0.745922	−	0.666033i	$-0.232009\pi$
$443$	31.3997	1.49184	0.745922	+	0.666033i	$0.232009\pi$
$444$	0.950840	0.0451249
$445$	9.85734	0.467283
$446$	−1.36387	−0.0645809
$447$	−7.29712	−0.345142
$448$	−2.16910	−0.102480
$449$	26.8375	1.26654	0.633270	−	0.773931i	$-0.281712\pi$
$449$	26.8375	1.26654	0.633270	+	0.773931i	$0.281712\pi$
$450$	1.76507	0.0832060
$451$	20.0109	0.942275
$452$	16.1127	0.757878
$453$	36.2861	1.70487
$454$	−13.5335	−0.635158
$455$	−3.27409	−0.153492
$456$	−27.1281	−1.27039
$457$	−1.28399	−0.0600625	−0.0300312	−	0.999549i	$-0.509561\pi$
$457$	−1.28399	−0.0600625	−0.0300312	+	0.999549i	$0.509561\pi$
$458$	−3.82813	−0.178877
$459$	−26.3517	−1.22999
$460$	−0.0812801	−0.00378971
$461$	7.52374	0.350415	0.175208	−	0.984532i	$-0.443940\pi$
$461$	7.52374	0.350415	0.175208	+	0.984532i	$0.443940\pi$
$462$	3.05835	0.142287
$463$	34.3541	1.59657	0.798285	−	0.602280i	$-0.205741\pi$
$463$	34.3541	1.59657	0.798285	+	0.602280i	$0.205741\pi$
$464$	4.49479	0.208665
$465$	16.3113	0.756421
$466$	−18.7282	−0.867568
$467$	32.9370	1.52414	0.762071	−	0.647494i	$-0.224183\pi$
$467$	32.9370	1.52414	0.762071	+	0.647494i	$0.224183\pi$
$468$	3.97626	0.183803
$469$	7.81961	0.361076
$470$	−4.87014	−0.224643
$471$	−16.1585	−0.744546
$472$	−4.80771	−0.221293
$473$	0	0
$474$	12.9227	0.593561
$475$	20.7508	0.952113
$476$	4.60213	0.210938
$477$	−0.164223	−0.00751926
$478$	14.8052	0.677174
$479$	−5.89165	−0.269196	−0.134598	−	0.990900i	$-0.542974\pi$
$479$	−5.89165	−0.269196	−0.134598	+	0.990900i	$0.542974\pi$
$480$	11.5357	0.526528
$481$	1.95555	0.0891654
$482$	−1.32330	−0.0602746
$483$	0.0644050	0.00293053
$484$	−0.559697	−0.0254408
$485$	−8.71517	−0.395735
$486$	4.64708	0.210796
$487$	−26.4401	−1.19812	−0.599059	−	0.800705i	$-0.704458\pi$
$487$	−26.4401	−1.19812	−0.599059	+	0.800705i	$0.704458\pi$
$488$	−1.54155	−0.0697828
$489$	−3.34217	−0.151138
$490$	−5.62232	−0.253990
$491$	8.26315	0.372911	0.186455	−	0.982463i	$-0.440300\pi$
$491$	8.26315	0.372911	0.186455	+	0.982463i	$0.440300\pi$
$492$	−15.0949	−0.680530
$493$	49.0920	2.21099
$494$	−22.4876	−1.01177
$495$	−1.99218	−0.0895419
$496$	−4.31736	−0.193855
$497$	2.11857	0.0950310
$498$	1.28825	0.0577278
$499$	−27.7100	−1.24047	−0.620235	−	0.784416i	$-0.712963\pi$
$499$	−27.7100	−1.24047	−0.620235	+	0.784416i	$0.712963\pi$
$500$	−12.6125	−0.564050
$501$	23.1904	1.03607
$502$	−22.2129	−0.991410
$503$	−20.0472	−0.893862	−0.446931	−	0.894569i	$-0.647483\pi$
$503$	−20.0472	−0.893862	−0.446931	+	0.894569i	$0.647483\pi$
$504$	−0.902783	−0.0402131
$505$	15.5841	0.693482
$506$	0.156150	0.00694172
$507$	27.3145	1.21308
$508$	5.23993	0.232484
$509$	−12.8306	−0.568707	−0.284353	−	0.958720i	$-0.591779\pi$
$509$	−12.8306	−0.568707	−0.284353	+	0.958720i	$0.591779\pi$
$510$	9.14324	0.404870
$511$	−7.59344	−0.335914
$512$	−5.88505	−0.260085
$513$	24.4946	1.08146
$514$	10.0061	0.441349
$515$	14.7696	0.650826
$516$	0	0
$517$	−19.4493	−0.855379
$518$	−0.178954	−0.00786278
$519$	22.4376	0.984903
$520$	14.8564	0.651496
$521$	16.5700	0.725945	0.362972	−	0.931800i	$-0.381762\pi$
$521$	16.5700	0.725945	0.362972	+	0.931800i	$0.381762\pi$
$522$	−3.88149	−0.169888
$523$	7.99883	0.349764	0.174882	−	0.984589i	$-0.444046\pi$
$523$	7.99883	0.349764	0.174882	+	0.984589i	$0.444046\pi$
$524$	−15.4047	−0.672956
$525$	4.37828	0.191084
$526$	22.1662	0.966494
$527$	−47.1542	−2.05407
$528$	3.34319	0.145494
$529$	−22.9967	−0.999857
$530$	−0.247307	−0.0107423
$531$	−1.00018	−0.0434042
$532$	−4.27780	−0.185466
$533$	−31.0450	−1.34471
$534$	14.2851	0.618175
$535$	9.63543	0.416576
$536$	−35.4821	−1.53259
$537$	−26.4401	−1.14097
$538$	10.0435	0.433008
$539$	−22.4532	−0.967127
$540$	−6.52235	−0.280677
$541$	43.9049	1.88762	0.943809	−	0.330490i	$-0.107214\pi$
$541$	43.9049	1.88762	0.943809	+	0.330490i	$0.107214\pi$
$542$	15.5797	0.669205
$543$	−18.1457	−0.778706
$544$	−33.3482	−1.42979
$545$	12.6387	0.541382
$546$	−4.74474	−0.203056
$547$	−3.43587	−0.146907	−0.0734537	−	0.997299i	$-0.523402\pi$
$547$	−3.43587	−0.146907	−0.0734537	+	0.997299i	$0.523402\pi$
$548$	5.45462	0.233010
$549$	−0.320700	−0.0136871
$550$	10.6152	0.452632
$551$	−45.6323	−1.94400
$552$	−0.292242	−0.0124387
$553$	5.05584	0.214996
$554$	22.5145	0.956549
$555$	0.739073	0.0313719
$556$	−6.11733	−0.259432
$557$	25.2303	1.06904	0.534521	−	0.845155i	$-0.320492\pi$
$557$	25.2303	1.06904	0.534521	+	0.845155i	$0.320492\pi$
$558$	3.72827	0.157830
$559$	0	0
$560$	0.327520	0.0138402
$561$	36.5143	1.54163
$562$	−24.7452	−1.04381
$563$	−9.44716	−0.398150	−0.199075	−	0.979984i	$-0.563794\pi$
$563$	−9.44716	−0.398150	−0.199075	+	0.979984i	$0.563794\pi$
$564$	14.6713	0.617772
$565$	12.5241	0.526895
$566$	5.71931	0.240401
$567$	6.17115	0.259164
$568$	−9.61318	−0.403360
$569$	−32.0424	−1.34329	−0.671644	−	0.740874i	$-0.734411\pi$
$569$	−32.0424	−1.34329	−0.671644	+	0.740874i	$0.734411\pi$
$570$	−8.49889	−0.355979
$571$	−2.98592	−0.124957	−0.0624784	−	0.998046i	$-0.519900\pi$
$571$	−2.98592	−0.124957	−0.0624784	+	0.998046i	$0.519900\pi$
$572$	23.9133	0.999866
$573$	19.0177	0.794477
$574$	2.84095	0.118579
$575$	0.223542	0.00932234
$576$	2.04759	0.0853164
$577$	−4.02655	−0.167628	−0.0838138	−	0.996481i	$-0.526710\pi$
$577$	−4.02655	−0.167628	−0.0838138	+	0.996481i	$0.526710\pi$
$578$	−12.7303	−0.529510
$579$	1.20312	0.0499999
$580$	12.1508	0.504536
$581$	0.504009	0.0209098
$582$	−12.6298	−0.523523
$583$	−0.987642	−0.0409039
$584$	34.4558	1.42579
$585$	3.09068	0.127784
$586$	5.05365	0.208764
$587$	−11.7631	−0.485516	−0.242758	−	0.970087i	$-0.578052\pi$
$587$	−11.7631	−0.485516	−0.242758	+	0.970087i	$0.578052\pi$
$588$	16.9372	0.698479
$589$	43.8310	1.80603
$590$	−1.50620	−0.0620092
$591$	19.0208	0.782410
$592$	−0.195621	−0.00803998
$593$	−25.8787	−1.06271	−0.531354	−	0.847150i	$-0.678317\pi$
$593$	−25.8787	−1.06271	−0.531354	+	0.847150i	$0.678317\pi$
$594$	12.5303	0.514125
$595$	3.57716	0.146649
$596$	5.22128	0.213872
$597$	−0.881530	−0.0360786
$598$	−0.242252	−0.00990643
$599$	2.22118	0.0907551	0.0453776	−	0.998970i	$-0.485551\pi$
$599$	2.22118	0.0907551	0.0453776	+	0.998970i	$0.485551\pi$
$600$	−19.8668	−0.811057
$601$	−5.23331	−0.213471	−0.106736	−	0.994287i	$-0.534040\pi$
$601$	−5.23331	−0.213471	−0.106736	+	0.994287i	$0.534040\pi$
$602$	0	0
$603$	−7.38158	−0.300601
$604$	−25.9637	−1.05645
$605$	−0.435044	−0.0176870
$606$	22.5841	0.917416
$607$	31.8465	1.29261	0.646306	−	0.763079i	$-0.276313\pi$
$607$	31.8465	1.29261	0.646306	+	0.763079i	$0.276313\pi$
$608$	30.9981	1.25714
$609$	−9.62811	−0.390151
$610$	−0.482949	−0.0195540
$611$	30.1737	1.22070
$612$	−4.34433	−0.175609
$613$	−10.8433	−0.437958	−0.218979	−	0.975730i	$-0.570273\pi$
$613$	−10.8433	−0.437958	−0.218979	+	0.975730i	$0.570273\pi$
$614$	−1.57331	−0.0634935
$615$	−11.7330	−0.473121
$616$	−5.42936	−0.218755
$617$	−26.6014	−1.07093	−0.535466	−	0.844557i	$-0.679864\pi$
$617$	−26.6014	−1.07093	−0.535466	+	0.844557i	$0.679864\pi$
$618$	21.4038	0.860987
$619$	−16.3076	−0.655458	−0.327729	−	0.944772i	$-0.606283\pi$
$619$	−16.3076	−0.655458	−0.327729	+	0.944772i	$0.606283\pi$
$620$	−11.6712	−0.468726
$621$	0.263873	0.0105888
$622$	−2.23992	−0.0898127
$623$	5.58882	0.223911
$624$	−5.18665	−0.207632
$625$	9.68781	0.387513
$626$	11.5954	0.463447
$627$	−33.9410	−1.35547
$628$	11.5619	0.461368
$629$	−2.13657	−0.0851907
$630$	−0.282831	−0.0112682
$631$	−3.04228	−0.121111	−0.0605556	−	0.998165i	$-0.519287\pi$
$631$	−3.04228	−0.121111	−0.0605556	+	0.998165i	$0.519287\pi$
$632$	−22.9412	−0.912553
$633$	18.3353	0.728764
$634$	−8.51642	−0.338230
$635$	4.07291	0.161629
$636$	0.745012	0.0295417
$637$	34.8340	1.38017
$638$	−23.3434	−0.924174
$639$	−1.99990	−0.0791147
$640$	−9.14121	−0.361338
$641$	46.6520	1.84264	0.921321	−	0.388802i	$-0.127111\pi$
$641$	46.6520	1.84264	0.921321	+	0.388802i	$0.127111\pi$
$642$	13.9635	0.551094
$643$	−29.9485	−1.18105	−0.590526	−	0.807018i	$-0.701080\pi$
$643$	−29.9485	−1.18105	−0.590526	+	0.807018i	$0.701080\pi$
$644$	−0.0460834	−0.00181594
$645$	0	0
$646$	24.5693	0.966665
$647$	−30.5352	−1.20046	−0.600232	−	0.799826i	$-0.704925\pi$
$647$	−30.5352	−1.20046	−0.600232	+	0.799826i	$0.704925\pi$
$648$	−28.0020	−1.10002
$649$	−6.01512	−0.236114
$650$	−16.4684	−0.645944
$651$	9.24805	0.362460
$652$	2.39141	0.0936547
$653$	5.91977	0.231658	0.115829	−	0.993269i	$-0.463047\pi$
$653$	5.91977	0.231658	0.115829	+	0.993269i	$0.463047\pi$
$654$	18.3157	0.716201
$655$	−11.9738	−0.467855
$656$	3.10555	0.121251
$657$	7.16808	0.279653
$658$	−2.76122	−0.107644
$659$	30.9493	1.20561	0.602806	−	0.797888i	$-0.294049\pi$
$659$	30.9493	1.20561	0.602806	+	0.797888i	$0.294049\pi$
$660$	9.03770	0.351792
$661$	−40.3394	−1.56902	−0.784509	−	0.620117i	$-0.787085\pi$
$661$	−40.3394	−1.56902	−0.784509	+	0.620117i	$0.787085\pi$
$662$	27.1763	1.05624
$663$	−56.6485	−2.20004
$664$	−2.28698	−0.0887519
$665$	−3.32507	−0.128941
$666$	0.168929	0.00654588
$667$	−0.491583	−0.0190342
$668$	−16.5934	−0.642016
$669$	3.19358	0.123471
$670$	−11.1161	−0.429452
$671$	−1.92870	−0.0744565
$672$	6.54038	0.252300
$673$	−5.32093	−0.205107	−0.102553	−	0.994728i	$-0.532701\pi$
$673$	−5.32093	−0.205107	−0.102553	+	0.994728i	$0.532701\pi$
$674$	−10.3535	−0.398800
$675$	17.9382	0.690441
$676$	−19.5442	−0.751700
$677$	−40.4628	−1.55511	−0.777557	−	0.628813i	$-0.783541\pi$
$677$	−40.4628	−1.55511	−0.777557	+	0.628813i	$0.783541\pi$
$678$	18.1497	0.697036
$679$	−4.94124	−0.189627
$680$	−16.2316	−0.622455
$681$	31.6896	1.21435
$682$	22.4219	0.858580
$683$	26.5479	1.01583	0.507914	−	0.861408i	$-0.330417\pi$
$683$	26.5479	1.01583	0.507914	+	0.861408i	$0.330417\pi$
$684$	4.03817	0.154403
$685$	4.23979	0.161994
$686$	−6.54524	−0.249899
$687$	8.96381	0.341991
$688$	0	0
$689$	1.53223	0.0583734
$690$	−0.0915558	−0.00348547
$691$	−7.06641	−0.268819	−0.134409	−	0.990926i	$-0.542914\pi$
$691$	−7.06641	−0.268819	−0.134409	+	0.990926i	$0.542914\pi$
$692$	−16.0547	−0.610309
$693$	−1.12951	−0.0429064
$694$	−6.35978	−0.241414
$695$	−4.75490	−0.180364
$696$	43.6882	1.65600
$697$	33.9188	1.28476
$698$	−9.86345	−0.373337
$699$	43.8534	1.65869
$700$	−3.13277	−0.118408
$701$	34.4221	1.30011	0.650053	−	0.759889i	$-0.274747\pi$
$701$	34.4221	1.30011	0.650053	+	0.759889i	$0.274747\pi$
$702$	−19.4396	−0.733700
$703$	1.98600	0.0749034
$704$	12.3143	0.464112
$705$	11.4037	0.429490
$706$	−12.6261	−0.475188
$707$	8.83570	0.332301
$708$	4.53741	0.170526
$709$	−1.65029	−0.0619780	−0.0309890	−	0.999520i	$-0.509866\pi$
$709$	−1.65029	−0.0619780	−0.0309890	+	0.999520i	$0.509866\pi$
$710$	−3.01169	−0.113027
$711$	−4.77262	−0.178987
$712$	−25.3597	−0.950394
$713$	0.472178	0.0176832
$714$	5.18395	0.194004
$715$	18.5874	0.695131
$716$	18.9185	0.707019
$717$	−34.6674	−1.29468
$718$	2.73765	0.102168
$719$	−24.2131	−0.902996	−0.451498	−	0.892272i	$-0.649110\pi$
$719$	−24.2131	−0.902996	−0.451498	+	0.892272i	$0.649110\pi$
$720$	−0.309173	−0.0115222
$721$	8.37392	0.311861
$722$	−7.52407	−0.280017
$723$	3.09859	0.115238
$724$	12.9837	0.482536
$725$	−33.4180	−1.24111
$726$	−0.630456	−0.0233984
$727$	−19.0134	−0.705168	−0.352584	−	0.935780i	$-0.614697\pi$
$727$	−19.0134	−0.705168	−0.352584	+	0.935780i	$0.614697\pi$
$728$	8.42314	0.312182
$729$	20.2278	0.749176
$730$	10.7946	0.399525
$731$	0	0
$732$	1.45488	0.0537740
$733$	−15.4640	−0.571177	−0.285588	−	0.958352i	$-0.592189\pi$
$733$	−15.4640	−0.571177	−0.285588	+	0.958352i	$0.592189\pi$
$734$	8.12601	0.299937
$735$	13.1650	0.485599
$736$	0.333932	0.0123089
$737$	−44.3930	−1.63524
$738$	−2.68181	−0.0987187
$739$	19.9135	0.732531	0.366265	−	0.930510i	$-0.380636\pi$
$739$	19.9135	0.732531	0.366265	+	0.930510i	$0.380636\pi$
$740$	−0.528826	−0.0194400
$741$	52.6563	1.93438
$742$	−0.140216	−0.00514748
$743$	30.5882	1.12217	0.561086	−	0.827758i	$-0.310384\pi$
$743$	30.5882	1.12217	0.561086	+	0.827758i	$0.310384\pi$
$744$	−41.9637	−1.53846
$745$	4.05842	0.148689
$746$	9.32855	0.341542
$747$	−0.475776	−0.0174077
$748$	−26.1269	−0.955295
$749$	5.46300	0.199614
$750$	−14.2070	−0.518768
$751$	−30.3277	−1.10667	−0.553337	−	0.832957i	$-0.686646\pi$
$751$	−30.3277	−1.10667	−0.553337	+	0.832957i	$0.686646\pi$
$752$	−3.01840	−0.110070
$753$	52.0129	1.89546
$754$	36.2151	1.31887
$755$	−20.1811	−0.734466
$756$	−3.69798	−0.134494
$757$	−3.23614	−0.117619	−0.0588097	−	0.998269i	$-0.518731\pi$
$757$	−3.23614	−0.117619	−0.0588097	+	0.998269i	$0.518731\pi$
$758$	−12.5822	−0.457007
$759$	−0.365636	−0.0132717
$760$	15.0877	0.547290
$761$	−0.785975	−0.0284916	−0.0142458	−	0.999899i	$-0.504535\pi$
$761$	−0.785975	−0.0284916	−0.0142458	+	0.999899i	$0.504535\pi$
$762$	5.90238	0.213821
$763$	7.16576	0.259418
$764$	−13.6077	−0.492308
$765$	−3.37678	−0.122088
$766$	−11.3052	−0.408474
$767$	9.33189	0.336955
$768$	−27.0049	−0.974455
$769$	9.15194	0.330027	0.165014	−	0.986291i	$-0.447233\pi$
$769$	9.15194	0.330027	0.165014	+	0.986291i	$0.447233\pi$
$770$	−1.70095	−0.0612980
$771$	−23.4299	−0.843806
$772$	−0.860862	−0.0309831
$773$	11.1851	0.402299	0.201150	−	0.979561i	$-0.435532\pi$
$773$	11.1851	0.402299	0.201150	+	0.979561i	$0.435532\pi$
$774$	0	0
$775$	32.0989	1.15303
$776$	22.4212	0.804875
$777$	0.419032	0.0150327
$778$	−14.0751	−0.504615
$779$	−31.5284	−1.12962
$780$	−14.0211	−0.502037
$781$	−12.0274	−0.430375
$782$	0.264677	0.00946483
$783$	−39.4472	−1.40973
$784$	−3.48458	−0.124449
$785$	8.98684	0.320754
$786$	−17.3522	−0.618932
$787$	−15.7855	−0.562691	−0.281346	−	0.959607i	$-0.590781\pi$
$787$	−15.7855	−0.562691	−0.281346	+	0.959607i	$0.590781\pi$
$788$	−13.6099	−0.484831
$789$	−51.9037	−1.84782
$790$	−7.18720	−0.255709
$791$	7.10082	0.252476
$792$	5.12522	0.182117
$793$	2.99219	0.106256
$794$	−3.73354	−0.132498
$795$	0.579086	0.0205381
$796$	0.630758	0.0223566
$797$	−42.8003	−1.51606	−0.758032	−	0.652218i	$-0.773839\pi$
$797$	−42.8003	−1.51606	−0.758032	+	0.652218i	$0.773839\pi$
$798$	−4.81862	−0.170577
$799$	−32.9669	−1.16628
$800$	22.7009	0.802597
$801$	−5.27575	−0.186409
$802$	−19.4951	−0.688396
$803$	43.1090	1.52128
$804$	33.4872	1.18100
$805$	−0.0358199	−0.00126249
$806$	−34.7855	−1.22527
$807$	−23.5176	−0.827859
$808$	−40.0926	−1.41045
$809$	−24.4282	−0.858849	−0.429424	−	0.903103i	$-0.641284\pi$
$809$	−24.4282	−0.858849	−0.429424	+	0.903103i	$0.641284\pi$
$810$	−8.77269	−0.308241
$811$	−0.476723	−0.0167400	−0.00837001	−	0.999965i	$-0.502664\pi$
$811$	−0.476723	−0.0167400	−0.00837001	+	0.999965i	$0.502664\pi$
$812$	6.88916	0.241762
$813$	−36.4808	−1.27944
$814$	1.01595	0.0356089
$815$	1.85880	0.0651110
$816$	5.66677	0.198377
$817$	0	0
$818$	14.3829	0.502886
$819$	1.75232	0.0612312
$820$	8.39528	0.293176
$821$	7.65375	0.267118	0.133559	−	0.991041i	$-0.457359\pi$
$821$	7.65375	0.267118	0.133559	+	0.991041i	$0.457359\pi$
$822$	6.14421	0.214304
$823$	−23.3941	−0.815466	−0.407733	−	0.913101i	$-0.633681\pi$
$823$	−23.3941	−0.815466	−0.407733	+	0.913101i	$0.633681\pi$
$824$	−37.9973	−1.32370
$825$	−24.8561	−0.865378
$826$	−0.853969	−0.0297134
$827$	−40.8226	−1.41954	−0.709770	−	0.704434i	$-0.751201\pi$
$827$	−40.8226	−1.41954	−0.709770	+	0.704434i	$0.751201\pi$
$828$	0.0435020	0.00151180
$829$	−56.1364	−1.94970	−0.974849	−	0.222866i	$-0.928459\pi$
$829$	−56.1364	−1.94970	−0.974849	+	0.222866i	$0.928459\pi$
$830$	−0.716481	−0.0248694
$831$	−52.7192	−1.82881
$832$	−19.1044	−0.662327
$833$	−38.0585	−1.31865
$834$	−6.89070	−0.238605
$835$	−12.8978	−0.446345
$836$	24.2857	0.839937
$837$	37.8901	1.30967
$838$	−26.0574	−0.900140
$839$	−32.9025	−1.13592	−0.567960	−	0.823056i	$-0.692267\pi$
$839$	−32.9025	−1.13592	−0.567960	+	0.823056i	$0.692267\pi$
$840$	3.18341	0.109838
$841$	44.4883	1.53408
$842$	−29.2317	−1.00739
$843$	57.9424	1.99564
$844$	−13.1194	−0.451589
$845$	−15.1914	−0.522600
$846$	2.60655	0.0896149
$847$	−0.246657	−0.00847523
$848$	−0.153275	−0.00526349
$849$	−13.3922	−0.459617
$850$	17.9929	0.617150
$851$	0.0213945	0.000733395 0
$852$	9.07271	0.310826
$853$	20.8833	0.715030	0.357515	−	0.933907i	$-0.383624\pi$
$853$	20.8833	0.715030	0.357515	+	0.933907i	$0.383624\pi$
$854$	−0.273818	−0.00936985
$855$	3.13881	0.107345
$856$	−24.7888	−0.847263
$857$	−21.9576	−0.750056	−0.375028	−	0.927014i	$-0.622367\pi$
$857$	−21.9576	−0.750056	−0.375028	+	0.927014i	$0.622367\pi$
$858$	26.9365	0.919597
$859$	−0.561550	−0.0191598	−0.00957992	−	0.999954i	$-0.503049\pi$
$859$	−0.561550	−0.0191598	−0.00957992	+	0.999954i	$0.503049\pi$
$860$	0	0
$861$	−6.65227	−0.226709
$862$	5.56121	0.189416
$863$	14.4982	0.493524	0.246762	−	0.969076i	$-0.420633\pi$
$863$	14.4982	0.493524	0.246762	+	0.969076i	$0.420633\pi$
$864$	26.7965	0.911635
$865$	−12.4791	−0.424301
$866$	−20.3938	−0.693008
$867$	29.8088	1.01236
$868$	−6.61722	−0.224603
$869$	−28.7027	−0.973672
$870$	13.6870	0.464032
$871$	68.8715	2.33362
$872$	−32.5151	−1.10110
$873$	4.66445	0.157868
$874$	−0.246024	−0.00832190
$875$	−5.55830	−0.187905
$876$	−32.5186	−1.09870
$877$	30.0645	1.01521	0.507603	−	0.861591i	$-0.330532\pi$
$877$	30.0645	1.01521	0.507603	+	0.861591i	$0.330532\pi$
$878$	21.3536	0.720649
$879$	−11.8334	−0.399132
$880$	−1.85937	−0.0626794
$881$	19.0426	0.641560	0.320780	−	0.947154i	$-0.396055\pi$
$881$	19.0426	0.641560	0.320780	+	0.947154i	$0.396055\pi$
$882$	3.00912	0.101322
$883$	16.1318	0.542878	0.271439	−	0.962456i	$-0.412501\pi$
$883$	16.1318	0.542878	0.271439	+	0.962456i	$0.412501\pi$
$884$	40.5335	1.36329
$885$	3.52686	0.118554
$886$	−25.3077	−0.850228
$887$	15.2772	0.512957	0.256478	−	0.966550i	$-0.417438\pi$
$887$	15.2772	0.512957	0.256478	+	0.966550i	$0.417438\pi$
$888$	−1.90139	−0.0638064
$889$	2.30922	0.0774488
$890$	−7.94487	−0.266313
$891$	−35.0345	−1.17370
$892$	−2.28509	−0.0765104
$893$	30.6436	1.02545
$894$	5.88137	0.196702
$895$	14.7051	0.491537
$896$	−5.18279	−0.173145
$897$	0.567249	0.0189399
$898$	−21.6306	−0.721823
$899$	−70.5874	−2.35422
$900$	2.95728	0.0985761
$901$	−1.67407	−0.0557713
$902$	−16.1285	−0.537019
$903$	0	0
$904$	−32.2205	−1.07164
$905$	10.0920	0.335470
$906$	−29.2461	−0.971635
$907$	32.3002	1.07251	0.536256	−	0.844056i	$-0.319838\pi$
$907$	32.3002	1.07251	0.536256	+	0.844056i	$0.319838\pi$
$908$	−22.6747	−0.752486
$909$	−8.34075	−0.276645
$910$	2.63886	0.0874775
$911$	42.7184	1.41532	0.707662	−	0.706551i	$-0.249750\pi$
$911$	42.7184	1.41532	0.707662	+	0.706551i	$0.249750\pi$
$912$	−5.26741	−0.174421
$913$	−2.86133	−0.0946961
$914$	1.03488	0.0342306
$915$	1.13086	0.0373850
$916$	−6.41384	−0.211919
$917$	−6.78879	−0.224186
$918$	21.2391	0.700994
$919$	0.389948	0.0128632	0.00643160	−	0.999979i	$-0.497953\pi$
$919$	0.389948	0.0128632	0.00643160	+	0.999979i	$0.497953\pi$
$920$	0.162535	0.00535863
$921$	3.68400	0.121392
$922$	−6.06402	−0.199708
$923$	18.6594	0.614183
$924$	5.12411	0.168571
$925$	1.45441	0.0478207
$926$	−27.6889	−0.909913
$927$	−7.90484	−0.259629
$928$	−49.9206	−1.63872
$929$	27.9881	0.918261	0.459130	−	0.888369i	$-0.348161\pi$
$929$	27.9881	0.918261	0.459130	+	0.888369i	$0.348161\pi$
$930$	−13.1467	−0.431097
$931$	35.3764	1.15941
$932$	−31.3782	−1.02783
$933$	5.24492	0.171711
$934$	−26.5467	−0.868635
$935$	−20.3080	−0.664144
$936$	−7.95130	−0.259896
$937$	−43.6947	−1.42744	−0.713721	−	0.700430i	$-0.752992\pi$
$937$	−43.6947	−1.42744	−0.713721	+	0.700430i	$0.752992\pi$
$938$	−6.30249	−0.205784
$939$	−27.1515	−0.886056
$940$	−8.15967	−0.266139
$941$	3.60886	0.117646	0.0588228	−	0.998268i	$-0.481265\pi$
$941$	3.60886	0.117646	0.0588228	+	0.998268i	$0.481265\pi$
$942$	13.0235	0.424330
$943$	−0.339645	−0.0110604
$944$	−0.933506	−0.0303830
$945$	−2.87438	−0.0935035
$946$	0	0
$947$	4.70534	0.152903	0.0764514	−	0.997073i	$-0.475641\pi$
$947$	4.70534	0.152903	0.0764514	+	0.997073i	$0.475641\pi$
$948$	21.6514	0.703206
$949$	−66.8796	−2.17100
$950$	−16.7248	−0.542626
$951$	19.9418	0.646656
$952$	−9.20285	−0.298266
$953$	−50.1740	−1.62530	−0.812648	−	0.582755i	$-0.801975\pi$
$953$	−50.1740	−1.62530	−0.812648	+	0.582755i	$0.801975\pi$
$954$	0.132361	0.00428536
$955$	−10.5770	−0.342265
$956$	24.8054	0.802264
$957$	54.6601	1.76691
$958$	4.74858	0.153420
$959$	2.40384	0.0776239
$960$	−7.22026	−0.233033
$961$	36.8011	1.18713
$962$	−1.57614	−0.0508169
$963$	−5.15698	−0.166181
$964$	−2.21712	−0.0714087
$965$	−0.669134	−0.0215402
$966$	−0.0519094	−0.00167016
$967$	18.1088	0.582340	0.291170	−	0.956671i	$-0.405956\pi$
$967$	18.1088	0.582340	0.291170	+	0.956671i	$0.405956\pi$
$968$	1.11922	0.0359732
$969$	−57.5306	−1.84815
$970$	7.02429	0.225536
$971$	−17.6031	−0.564911	−0.282455	−	0.959280i	$-0.591149\pi$
$971$	−17.6031	−0.564911	−0.282455	+	0.959280i	$0.591149\pi$
$972$	7.78595	0.249735
$973$	−2.69589	−0.0864261
$974$	21.3104	0.682828
$975$	38.5619	1.23497
$976$	−0.299321	−0.00958102
$977$	−32.2522	−1.03184	−0.515920	−	0.856637i	$-0.672550\pi$
$977$	−32.2522	−1.03184	−0.515920	+	0.856637i	$0.672550\pi$
$978$	2.69373	0.0861361
$979$	−31.7285	−1.01405
$980$	−9.41991	−0.300908
$981$	−6.76435	−0.215969
$982$	−6.65997	−0.212528
$983$	−13.9780	−0.445827	−0.222914	−	0.974838i	$-0.571557\pi$
$983$	−13.9780	−0.445827	−0.222914	+	0.974838i	$0.571557\pi$
$984$	30.1852	0.962268
$985$	−10.5787	−0.337066
$986$	−39.5674	−1.26008
$987$	6.46558	0.205802
$988$	−37.6769	−1.19866
$989$	0	0
$990$	1.60567	0.0510315
$991$	−22.8877	−0.727051	−0.363526	−	0.931584i	$-0.618427\pi$
$991$	−22.8877	−0.727051	−0.363526	+	0.931584i	$0.618427\pi$
$992$	47.9501	1.52242
$993$	−63.6350	−2.01940
$994$	−1.70754	−0.0541599
$995$	0.490278	0.0155429
$996$	2.15840	0.0683914
$997$	7.03336	0.222749	0.111374	−	0.993779i	$-0.464475\pi$
$997$	7.03336	0.222749	0.111374	+	0.993779i	$0.464475\pi$
$998$	22.3338	0.706965
$999$	1.71681	0.0543175

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.2.a.q.1.7	✓	20
43.42	odd	2	inner	1849.2.a.q.1.14	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1849.2.a.q.1.7	✓	20	1.1	even	1	trivial
1849.2.a.q.1.14	yes	20	43.42	odd	2	inner

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:	\( N \)	\(=\)	\( 1849 = 43^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1849.a (trivial)

\(f(q)\)	\(=\)	\(q-0.805985 q^{2} +1.88727 q^{3} -1.35039 q^{4} -1.04963 q^{5} -1.52111 q^{6} -0.595112 q^{7} +2.70036 q^{8} +0.561775 q^{9} +O(q^{10})\)	\(q-0.805985 q^{2} +1.88727 q^{3} -1.35039 q^{4} -1.04963 q^{5} -1.52111 q^{6} -0.595112 q^{7} +2.70036 q^{8} +0.561775 q^{9} +0.845990 q^{10} +3.37853 q^{11} -2.54854 q^{12} -5.24147 q^{13} +0.479651 q^{14} -1.98094 q^{15} +0.524325 q^{16} +5.72667 q^{17} -0.452782 q^{18} -5.32309 q^{19} +1.41741 q^{20} -1.12313 q^{21} -2.72305 q^{22} -0.0573439 q^{23} +5.09630 q^{24} -3.89827 q^{25} +4.22455 q^{26} -4.60158 q^{27} +0.803632 q^{28} +8.57253 q^{29} +1.59661 q^{30} -8.23414 q^{31} -5.82332 q^{32} +6.37619 q^{33} -4.61561 q^{34} +0.624650 q^{35} -0.758615 q^{36} -0.373092 q^{37} +4.29033 q^{38} -9.89205 q^{39} -2.83439 q^{40} +5.92295 q^{41} +0.905230 q^{42} -4.56233 q^{44} -0.589659 q^{45} +0.0462183 q^{46} -5.75673 q^{47} +0.989541 q^{48} -6.64584 q^{49} +3.14194 q^{50} +10.8077 q^{51} +7.07802 q^{52} -0.292329 q^{53} +3.70880 q^{54} -3.54622 q^{55} -1.60702 q^{56} -10.0461 q^{57} -6.90933 q^{58} -1.78040 q^{59} +2.67504 q^{60} -0.570869 q^{61} +6.63659 q^{62} -0.334319 q^{63} +3.64486 q^{64} +5.50163 q^{65} -5.13911 q^{66} -13.1397 q^{67} -7.73322 q^{68} -0.108223 q^{69} -0.503459 q^{70} -3.55996 q^{71} +1.51700 q^{72} +12.7597 q^{73} +0.300706 q^{74} -7.35707 q^{75} +7.18823 q^{76} -2.01060 q^{77} +7.97285 q^{78} -8.49561 q^{79} -0.550350 q^{80} -10.3697 q^{81} -4.77381 q^{82} -0.846914 q^{83} +1.51667 q^{84} -6.01091 q^{85} +16.1787 q^{87} +9.12326 q^{88} -9.39121 q^{89} +0.475256 q^{90} +3.11926 q^{91} +0.0774366 q^{92} -15.5400 q^{93} +4.63984 q^{94} +5.58730 q^{95} -10.9902 q^{96} +8.30305 q^{97} +5.35645 q^{98} +1.89798 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 14 q^{4} - 32 q^{6} - 6 q^{9}+O(q^{10}) \) 20 * q + 14 * q^4 - 32 * q^6 - 6 * q^9	\( 20 q + 14 q^{4} - 32 q^{6} - 6 q^{9} + 12 q^{10} - 20 q^{11} + 6 q^{13} - 34 q^{14} - 34 q^{15} + 14 q^{16} - 8 q^{17} + 12 q^{21} - 46 q^{23} + 2 q^{24} + 20 q^{25} - 38 q^{31} - 76 q^{35} - 46 q^{36} - 68 q^{38} - 64 q^{40} - 56 q^{41} - 58 q^{44} - 24 q^{47} - 20 q^{49} - 20 q^{52} - 42 q^{53} + 66 q^{54} - 46 q^{56} + 2 q^{57} + 12 q^{58} - 90 q^{59} + 18 q^{60} - 28 q^{64} + 76 q^{66} - 62 q^{67} - 54 q^{68} + 24 q^{74} - 36 q^{78} - 10 q^{79} - 48 q^{81} - 68 q^{83} + 70 q^{84} - 12 q^{87} + 58 q^{90} - 8 q^{92} - 42 q^{95} - 22 q^{96} - 44 q^{97} - 38 q^{99}+O(q^{100}) \) 20 * q + 14 * q^4 - 32 * q^6 - 6 * q^9 + 12 * q^10 - 20 * q^11 + 6 * q^13 - 34 * q^14 - 34 * q^15 + 14 * q^16 - 8 * q^17 + 12 * q^21 - 46 * q^23 + 2 * q^24 + 20 * q^25 - 38 * q^31 - 76 * q^35 - 46 * q^36 - 68 * q^38 - 64 * q^40 - 56 * q^41 - 58 * q^44 - 24 * q^47 - 20 * q^49 - 20 * q^52 - 42 * q^53 + 66 * q^54 - 46 * q^56 + 2 * q^57 + 12 * q^58 - 90 * q^59 + 18 * q^60 - 28 * q^64 + 76 * q^66 - 62 * q^67 - 54 * q^68 + 24 * q^74 - 36 * q^78 - 10 * q^79 - 48 * q^81 - 68 * q^83 + 70 * q^84 - 12 * q^87 + 58 * q^90 - 8 * q^92 - 42 * q^95 - 22 * q^96 - 44 * q^97 - 38 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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