LMFDB - Embedded newform 1849.2.a.q.1.13

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			1.13
Root			$0.666460$ 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)

$f(q)$	$=$	$q+0.666460 q^{2} -2.64610 q^{3} -1.55583 q^{4} +3.91294 q^{5} -1.76352 q^{6} -0.421793 q^{7} -2.36982 q^{8} +4.00186 q^{9} +O(q^{10})$	\(q+0.666460 q^{2} -2.64610 q^{3} -1.55583 q^{4} +3.91294 q^{5} -1.76352 q^{6} -0.421793 q^{7} -2.36982 q^{8} +4.00186 q^{9} +2.60781 q^{10} +0.269572 q^{11} +4.11689 q^{12} -1.45607 q^{13} -0.281108 q^{14} -10.3540 q^{15} +1.53227 q^{16} +0.538717 q^{17} +2.66708 q^{18} -7.09731 q^{19} -6.08787 q^{20} +1.11611 q^{21} +0.179659 q^{22} -4.89199 q^{23} +6.27078 q^{24} +10.3111 q^{25} -0.970415 q^{26} -2.65102 q^{27} +0.656239 q^{28} +8.01489 q^{29} -6.90054 q^{30} -1.74156 q^{31} +5.76084 q^{32} -0.713316 q^{33} +0.359034 q^{34} -1.65045 q^{35} -6.22622 q^{36} +4.65176 q^{37} -4.73007 q^{38} +3.85292 q^{39} -9.27295 q^{40} -3.69690 q^{41} +0.743841 q^{42} -0.419409 q^{44} +15.6590 q^{45} -3.26032 q^{46} +2.39131 q^{47} -4.05455 q^{48} -6.82209 q^{49} +6.87191 q^{50} -1.42550 q^{51} +2.26541 q^{52} -8.33224 q^{53} -1.76680 q^{54} +1.05482 q^{55} +0.999573 q^{56} +18.7802 q^{57} +5.34160 q^{58} -6.65141 q^{59} +16.1091 q^{60} -9.65277 q^{61} -1.16068 q^{62} -1.68796 q^{63} +0.774818 q^{64} -5.69752 q^{65} -0.475397 q^{66} -9.35033 q^{67} -0.838153 q^{68} +12.9447 q^{69} -1.09996 q^{70} +4.97702 q^{71} -9.48368 q^{72} +1.25397 q^{73} +3.10021 q^{74} -27.2841 q^{75} +11.0422 q^{76} -0.113704 q^{77} +2.56782 q^{78} -1.95181 q^{79} +5.99569 q^{80} -4.99070 q^{81} -2.46384 q^{82} -2.96999 q^{83} -1.73648 q^{84} +2.10797 q^{85} -21.2082 q^{87} -0.638838 q^{88} -6.58503 q^{89} +10.4361 q^{90} +0.614162 q^{91} +7.61111 q^{92} +4.60836 q^{93} +1.59371 q^{94} -27.7713 q^{95} -15.2438 q^{96} -8.22265 q^{97} -4.54665 q^{98} +1.07879 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.666460	0.471258	0.235629	−	0.971843i	$-0.424285\pi$
$2$	0.666460	0.471258	0.235629	+	0.971843i	$0.424285\pi$
$3$	−2.64610	−1.52773	−0.763864	−	0.645377i	$-0.776700\pi$
$3$	−2.64610	−1.52773	−0.763864	+	0.645377i	$0.776700\pi$
$4$	−1.55583	−0.777916
$5$	3.91294	1.74992	0.874959	−	0.484197i	$-0.160888\pi$
$5$	3.91294	1.74992	0.874959	+	0.484197i	$0.160888\pi$
$6$	−1.76352	−0.719954
$7$	−0.421793	−0.159423	−0.0797114	−	0.996818i	$-0.525400\pi$
$7$	−0.421793	−0.159423	−0.0797114	+	0.996818i	$0.525400\pi$
$8$	−2.36982	−0.837857
$9$	4.00186	1.33395
$10$	2.60781	0.824663
$11$	0.269572	0.0812791	0.0406396	−	0.999174i	$-0.487060\pi$
$11$	0.269572	0.0812791	0.0406396	+	0.999174i	$0.487060\pi$
$12$	4.11689	1.18844
$13$	−1.45607	−0.403842	−0.201921	−	0.979402i	$-0.564718\pi$
$13$	−1.45607	−0.403842	−0.201921	+	0.979402i	$0.564718\pi$
$14$	−0.281108	−0.0751293
$15$	−10.3540	−2.67340
$16$	1.53227	0.383068
$17$	0.538717	0.130658	0.0653291	−	0.997864i	$-0.479190\pi$
$17$	0.538717	0.130658	0.0653291	+	0.997864i	$0.479190\pi$
$18$	2.66708	0.628636
$19$	−7.09731	−1.62823	−0.814117	−	0.580700i	$-0.802779\pi$
$19$	−7.09731	−1.62823	−0.814117	+	0.580700i	$0.802779\pi$
$20$	−6.08787	−1.36129
$21$	1.11611	0.243555
$22$	0.179659	0.0383035
$23$	−4.89199	−1.02005	−0.510025	−	0.860159i	$-0.670364\pi$
$23$	−4.89199	−1.02005	−0.510025	+	0.860159i	$0.670364\pi$
$24$	6.27078	1.28002
$25$	10.3111	2.06221
$26$	−0.970415	−0.190314
$27$	−2.65102	−0.510190
$28$	0.656239	0.124018
$29$	8.01489	1.48833	0.744164	−	0.667997i	$-0.232848\pi$
$29$	8.01489	1.48833	0.744164	+	0.667997i	$0.232848\pi$
$30$	−6.90054	−1.25986
$31$	−1.74156	−0.312794	−0.156397	−	0.987694i	$-0.549988\pi$
$31$	−1.74156	−0.312794	−0.156397	+	0.987694i	$0.549988\pi$
$32$	5.76084	1.01838
$33$	−0.713316	−0.124172
$34$	0.359034	0.0615737
$35$	−1.65045	−0.278977
$36$	−6.22622	−1.03770
$37$	4.65176	0.764745	0.382372	−	0.924008i	$-0.375107\pi$
$37$	4.65176	0.764745	0.382372	+	0.924008i	$0.375107\pi$
$38$	−4.73007	−0.767319
$39$	3.85292	0.616961
$40$	−9.27295	−1.46618
$41$	−3.69690	−0.577359	−0.288680	−	0.957426i	$-0.593216\pi$
$41$	−3.69690	−0.577359	−0.288680	+	0.957426i	$0.593216\pi$
$42$	0.743841	0.114777
$43$	0	0
$43$	0	0
$44$	−0.419409	−0.0632283
$45$	15.6590	2.33431
$46$	−3.26032	−0.480707
$47$	2.39131	0.348808	0.174404	−	0.984674i	$-0.444200\pi$
$47$	2.39131	0.348808	0.174404	+	0.984674i	$0.444200\pi$
$48$	−4.05455	−0.585224
$49$	−6.82209	−0.974584
$50$	6.87191	0.971835
$51$	−1.42550	−0.199610
$52$	2.26541	0.314155
$53$	−8.33224	−1.14452	−0.572260	−	0.820072i	$-0.693933\pi$
$53$	−8.33224	−1.14452	−0.572260	+	0.820072i	$0.693933\pi$
$54$	−1.76680	−0.240431
$55$	1.05482	0.142232
$56$	0.999573	0.133574
$57$	18.7802	2.48750
$58$	5.34160	0.701387
$59$	−6.65141	−0.865940	−0.432970	−	0.901408i	$-0.642534\pi$
$59$	−6.65141	−0.865940	−0.432970	+	0.901408i	$0.642534\pi$
$60$	16.1091	2.07968
$61$	−9.65277	−1.23591	−0.617956	−	0.786213i	$-0.712039\pi$
$61$	−9.65277	−1.23591	−0.617956	+	0.786213i	$0.712039\pi$
$62$	−1.16068	−0.147407
$63$	−1.68796	−0.212663
$64$	0.774818	0.0968523
$65$	−5.69752	−0.706691
$66$	−0.475397	−0.0585173
$67$	−9.35033	−1.14232	−0.571162	−	0.820837i	$-0.693507\pi$
$67$	−9.35033	−1.14232	−0.571162	+	0.820837i	$0.693507\pi$
$68$	−0.838153	−0.101641
$69$	12.9447	1.55836
$70$	−1.09996	−0.131470
$71$	4.97702	0.590664	0.295332	−	0.955395i	$-0.404570\pi$
$71$	4.97702	0.590664	0.295332	+	0.955395i	$0.404570\pi$
$72$	−9.48368	−1.11766
$73$	1.25397	0.146766	0.0733829	−	0.997304i	$-0.476620\pi$
$73$	1.25397	0.146766	0.0733829	+	0.997304i	$0.476620\pi$
$74$	3.10021	0.360392
$75$	−27.2841	−3.15050
$76$	11.0422	1.26663
$77$	−0.113704	−0.0129577
$78$	2.56782	0.290748
$79$	−1.95181	−0.219595	−0.109798	−	0.993954i	$-0.535020\pi$
$79$	−1.95181	−0.219595	−0.109798	+	0.993954i	$0.535020\pi$
$80$	5.99569	0.670338
$81$	−4.99070	−0.554522
$82$	−2.46384	−0.272085
$83$	−2.96999	−0.325998	−0.162999	−	0.986626i	$-0.552117\pi$
$83$	−2.96999	−0.325998	−0.162999	+	0.986626i	$0.552117\pi$
$84$	−1.73648	−0.189465
$85$	2.10797	0.228641
$86$	0	0
$87$	−21.2082	−2.27376
$88$	−0.638838	−0.0681003
$89$	−6.58503	−0.698012	−0.349006	−	0.937121i	$-0.613481\pi$
$89$	−6.58503	−0.698012	−0.349006	+	0.937121i	$0.613481\pi$
$90$	10.4361	1.10006
$91$	0.614162	0.0643817
$92$	7.61111	0.793513
$93$	4.60836	0.477864
$94$	1.59371	0.164379
$95$	−27.7713	−2.84928
$96$	−15.2438	−1.55581
$97$	−8.22265	−0.834884	−0.417442	−	0.908704i	$-0.637073\pi$
$97$	−8.22265	−0.834884	−0.417442	+	0.908704i	$0.637073\pi$
$98$	−4.54665	−0.459281
$99$	1.07879	0.108423
$100$	−16.0423	−1.60423
$101$	−0.701637	−0.0698155	−0.0349078	−	0.999391i	$-0.511114\pi$
$101$	−0.701637	−0.0698155	−0.0349078	+	0.999391i	$0.511114\pi$
$102$	−0.950040	−0.0940679
$103$	−15.4880	−1.52608	−0.763039	−	0.646353i	$-0.776294\pi$
$103$	−15.4880	−1.52608	−0.763039	+	0.646353i	$0.776294\pi$
$104$	3.45063	0.338362
$105$	4.36726	0.426201
$106$	−5.55310	−0.539365
$107$	6.91997	0.668978	0.334489	−	0.942400i	$-0.391436\pi$
$107$	6.91997	0.668978	0.334489	+	0.942400i	$0.391436\pi$
$108$	4.12454	0.396885
$109$	−13.1024	−1.25498	−0.627492	−	0.778623i	$-0.715919\pi$
$109$	−13.1024	−1.25498	−0.627492	+	0.778623i	$0.715919\pi$
$110$	0.702995	0.0670279
$111$	−12.3090	−1.16832
$112$	−0.646303	−0.0610698
$113$	−4.55965	−0.428936	−0.214468	−	0.976731i	$-0.568802\pi$
$113$	−4.55965	−0.428936	−0.214468	+	0.976731i	$0.568802\pi$
$114$	12.5163	1.17225
$115$	−19.1420	−1.78500
$116$	−12.4698	−1.15779
$117$	−5.82700	−0.538707
$118$	−4.43290	−0.408081
$119$	−0.227227	−0.0208299
$120$	24.5372	2.23993
$121$	−10.9273	−0.993394
$122$	−6.43319	−0.582433
$123$	9.78238	0.882048
$124$	2.70958	0.243327
$125$	20.7818	1.85878
$126$	−1.12496	−0.100219
$127$	17.7059	1.57114	0.785571	−	0.618771i	$-0.212369\pi$
$127$	17.7059	1.57114	0.785571	+	0.618771i	$0.212369\pi$
$128$	−11.0053	−0.972739
$129$	0	0
$130$	−3.79717	−0.333034
$131$	−7.13074	−0.623016	−0.311508	−	0.950244i	$-0.600834\pi$
$131$	−7.13074	−0.623016	−0.311508	+	0.950244i	$0.600834\pi$
$132$	1.10980	0.0965957
$133$	2.99360	0.259578
$134$	−6.23162	−0.538330
$135$	−10.3733	−0.892790
$136$	−1.27666	−0.109473
$137$	−5.45921	−0.466411	−0.233206	−	0.972427i	$-0.574922\pi$
$137$	−5.45921	−0.466411	−0.233206	+	0.972427i	$0.574922\pi$
$138$	8.62713	0.734390
$139$	−11.2699	−0.955896	−0.477948	−	0.878388i	$-0.658619\pi$
$139$	−11.2699	−0.955896	−0.477948	+	0.878388i	$0.658619\pi$
$140$	2.56782	0.217020
$141$	−6.32765	−0.532884
$142$	3.31699	0.278355
$143$	−0.392517	−0.0328239
$144$	6.13194	0.510995
$145$	31.3618	2.60445
$146$	0.835719	0.0691646
$147$	18.0520	1.48890
$148$	−7.23735	−0.594907
$149$	19.2989	1.58103	0.790514	−	0.612444i	$-0.209814\pi$
$149$	19.2989	1.58103	0.790514	+	0.612444i	$0.209814\pi$
$150$	−18.1838	−1.48470
$151$	4.43147	0.360628	0.180314	−	0.983609i	$-0.442289\pi$
$151$	4.43147	0.360628	0.180314	+	0.983609i	$0.442289\pi$
$152$	16.8193	1.36423
$153$	2.15587	0.174292
$154$	−0.0757790	−0.00610645
$155$	−6.81463	−0.547364
$156$	−5.99450	−0.479944
$157$	−9.21028	−0.735061	−0.367530	−	0.930012i	$-0.619797\pi$
$157$	−9.21028	−0.735061	−0.367530	+	0.930012i	$0.619797\pi$
$158$	−1.30080	−0.103486
$159$	22.0480	1.74852
$160$	22.5418	1.78208
$161$	2.06341	0.162619
$162$	−3.32610	−0.261323
$163$	−18.0702	−1.41537	−0.707685	−	0.706528i	$-0.750260\pi$
$163$	−18.0702	−1.41537	−0.707685	+	0.706528i	$0.750260\pi$
$164$	5.75176	0.449137
$165$	−2.79116	−0.217292
$166$	−1.97938	−0.153629
$167$	1.81299	0.140294	0.0701468	−	0.997537i	$-0.477653\pi$
$167$	1.81299	0.140294	0.0701468	+	0.997537i	$0.477653\pi$
$168$	−2.64497	−0.204064
$169$	−10.8798	−0.836911
$170$	1.40487	0.107749
$171$	−28.4024	−2.17199
$172$	0	0
$173$	−0.622184	−0.0473038	−0.0236519	−	0.999720i	$-0.507529\pi$
$173$	−0.622184	−0.0473038	−0.0236519	+	0.999720i	$0.507529\pi$
$174$	−14.1344	−1.07153
$175$	−4.34914	−0.328764
$176$	0.413059	0.0311355
$177$	17.6003	1.32292
$178$	−4.38866	−0.328944
$179$	9.05172	0.676558	0.338279	−	0.941046i	$-0.390155\pi$
$179$	9.05172	0.676558	0.338279	+	0.941046i	$0.390155\pi$
$180$	−24.3628	−1.81590
$181$	14.8044	1.10040	0.550202	−	0.835032i	$-0.314551\pi$
$181$	14.8044	1.10040	0.550202	+	0.835032i	$0.314551\pi$
$182$	0.409314	0.0303404
$183$	25.5422	1.88814
$184$	11.5931	0.854657
$185$	18.2020	1.33824
$186$	3.07128	0.225197
$187$	0.145223	0.0106198
$188$	−3.72047	−0.271343
$189$	1.11818	0.0813359
$190$	−18.5085	−1.34275
$191$	15.4601	1.11865	0.559327	−	0.828947i	$-0.311060\pi$
$191$	15.4601	1.11865	0.559327	+	0.828947i	$0.311060\pi$
$192$	−2.05025	−0.147964
$193$	6.68680	0.481327	0.240663	−	0.970609i	$-0.422635\pi$
$193$	6.68680	0.481327	0.240663	+	0.970609i	$0.422635\pi$
$194$	−5.48007	−0.393446
$195$	15.0762	1.07963
$196$	10.6140	0.758144
$197$	14.5253	1.03489	0.517443	−	0.855718i	$-0.326884\pi$
$197$	14.5253	1.03489	0.517443	+	0.855718i	$0.326884\pi$
$198$	0.718971	0.0510950
$199$	−21.1724	−1.50087	−0.750436	−	0.660943i	$-0.770157\pi$
$199$	−21.1724	−1.50087	−0.750436	+	0.660943i	$0.770157\pi$
$200$	−24.4353	−1.72784
$201$	24.7419	1.74516
$202$	−0.467613	−0.0329011
$203$	−3.38063	−0.237273
$204$	2.21784	0.155280
$205$	−14.4657	−1.01033
$206$	−10.3221	−0.719177
$207$	−19.5771	−1.36070
$208$	−2.23110	−0.154699
$209$	−1.91324	−0.132341
$210$	2.91060	0.200851
$211$	−24.2599	−1.67012	−0.835059	−	0.550160i	$-0.814567\pi$
$211$	−24.2599	−1.67012	−0.835059	+	0.550160i	$0.814567\pi$
$212$	12.9636	0.890341
$213$	−13.1697	−0.902374
$214$	4.61188	0.315262
$215$	0	0
$216$	6.28244	0.427466
$217$	0.734580	0.0498665
$218$	−8.73224	−0.591422
$219$	−3.31813	−0.224218
$220$	−1.64112	−0.110644
$221$	−0.784412	−0.0527653
$222$	−8.20348	−0.550581
$223$	16.2980	1.09140	0.545698	−	0.837982i	$-0.316265\pi$
$223$	16.2980	1.09140	0.545698	+	0.837982i	$0.316265\pi$
$224$	−2.42988	−0.162353
$225$	41.2634	2.75089
$226$	−3.03882	−0.202140
$227$	−10.8644	−0.721095	−0.360548	−	0.932741i	$-0.617410\pi$
$227$	−10.8644	−0.721095	−0.360548	+	0.932741i	$0.617410\pi$
$228$	−29.2188	−1.93507
$229$	−7.92199	−0.523500	−0.261750	−	0.965136i	$-0.584300\pi$
$229$	−7.92199	−0.523500	−0.261750	+	0.965136i	$0.584300\pi$
$230$	−12.7574	−0.841198
$231$	0.300872	0.0197959
$232$	−18.9938	−1.24701
$233$	23.7333	1.55482	0.777411	−	0.628993i	$-0.216533\pi$
$233$	23.7333	1.55482	0.777411	+	0.628993i	$0.216533\pi$
$234$	−3.88346	−0.253870
$235$	9.35703	0.610386
$236$	10.3485	0.673628
$237$	5.16468	0.335482
$238$	−0.151438	−0.00981626
$239$	−1.18150	−0.0764249	−0.0382124	−	0.999270i	$-0.512166\pi$
$239$	−1.18150	−0.0764249	−0.0382124	+	0.999270i	$0.512166\pi$
$240$	−15.8652	−1.02409
$241$	−18.9831	−1.22281	−0.611405	−	0.791318i	$-0.709395\pi$
$241$	−18.9831	−1.22281	−0.611405	+	0.791318i	$0.709395\pi$
$242$	−7.28263	−0.468145
$243$	21.1590	1.35735
$244$	15.0181	0.961435
$245$	−26.6944	−1.70544
$246$	6.51956	0.415672
$247$	10.3342	0.657550
$248$	4.12719	0.262077
$249$	7.85889	0.498037
$250$	13.8503	0.875967
$251$	19.6713	1.24164	0.620820	−	0.783953i	$-0.286800\pi$
$251$	19.6713	1.24164	0.620820	+	0.783953i	$0.286800\pi$
$252$	2.62618	0.165434
$253$	−1.31875	−0.0829088
$254$	11.8003	0.740414
$255$	−5.57790	−0.349301
$256$	−8.88422	−0.555264
$257$	20.2404	1.26256	0.631280	−	0.775555i	$-0.282530\pi$
$257$	20.2404	1.26256	0.631280	+	0.775555i	$0.282530\pi$
$258$	0	0
$259$	−1.96208	−0.121918
$260$	8.86439	0.549746
$261$	32.0745	1.98536
$262$	−4.75235	−0.293601
$263$	−4.98793	−0.307569	−0.153785	−	0.988104i	$-0.549146\pi$
$263$	−4.98793	−0.307569	−0.153785	+	0.988104i	$0.549146\pi$
$264$	1.69043	0.104039
$265$	−32.6035	−2.00282
$266$	1.99511	0.122328
$267$	17.4247	1.06637
$268$	14.5475	0.888632
$269$	−18.5503	−1.13103	−0.565517	−	0.824737i	$-0.691323\pi$
$269$	−18.5503	−1.13103	−0.565517	+	0.824737i	$0.691323\pi$
$270$	−6.91338	−0.420735
$271$	−2.68991	−0.163400	−0.0817001	−	0.996657i	$-0.526035\pi$
$271$	−2.68991	−0.163400	−0.0817001	+	0.996657i	$0.526035\pi$
$272$	0.825463	0.0500510
$273$	−1.62514	−0.0983577
$274$	−3.63834	−0.219800
$275$	2.77958	0.167615
$276$	−20.1398	−1.21227
$277$	−13.8763	−0.833743	−0.416872	−	0.908965i	$-0.636874\pi$
$277$	−13.8763	−0.833743	−0.416872	+	0.908965i	$0.636874\pi$
$278$	−7.51090	−0.450474
$279$	−6.96949	−0.417253
$280$	3.91127	0.233743
$281$	−27.2430	−1.62518	−0.812591	−	0.582835i	$-0.801943\pi$
$281$	−27.2430	−1.62518	−0.812591	+	0.582835i	$0.801943\pi$
$282$	−4.21712	−0.251126
$283$	12.4141	0.737940	0.368970	−	0.929441i	$-0.379710\pi$
$283$	12.4141	0.737940	0.368970	+	0.929441i	$0.379710\pi$
$284$	−7.74341	−0.459487
$285$	73.4858	4.35292
$286$	−0.261597	−0.0154686
$287$	1.55933	0.0920442
$288$	23.0541	1.35847
$289$	−16.7098	−0.982928
$290$	20.9013	1.22737
$291$	21.7580	1.27548
$292$	−1.95096	−0.114171
$293$	29.7013	1.73517	0.867584	−	0.497291i	$-0.165672\pi$
$293$	29.7013	1.73517	0.867584	+	0.497291i	$0.165672\pi$
$294$	12.0309	0.701656
$295$	−26.0265	−1.51532
$296$	−11.0238	−0.640747
$297$	−0.714643	−0.0414678
$298$	12.8619	0.745072
$299$	7.12310	0.411940
$300$	42.4495	2.45082
$301$	0	0
$302$	2.95340	0.169949
$303$	1.85660	0.106659
$304$	−10.8750	−0.623725
$305$	−37.7707	−2.16274
$306$	1.43680	0.0821365
$307$	20.1439	1.14967	0.574837	−	0.818268i	$-0.305065\pi$
$307$	20.1439	1.14967	0.574837	+	0.818268i	$0.305065\pi$
$308$	0.176904	0.0100800
$309$	40.9828	2.33143
$310$	−4.54167	−0.257950
$311$	−6.35107	−0.360136	−0.180068	−	0.983654i	$-0.557632\pi$
$311$	−6.35107	−0.360136	−0.180068	+	0.983654i	$0.557632\pi$
$312$	−9.13073	−0.516926
$313$	21.0404	1.18928	0.594638	−	0.803994i	$-0.297296\pi$
$313$	21.0404	1.18928	0.594638	+	0.803994i	$0.297296\pi$
$314$	−6.13828	−0.346403
$315$	−6.60487	−0.372142
$316$	3.03668	0.170827
$317$	−28.2430	−1.58629	−0.793144	−	0.609035i	$-0.791557\pi$
$317$	−28.2430	−1.58629	−0.793144	+	0.609035i	$0.791557\pi$
$318$	14.6941	0.824003
$319$	2.16059	0.120970
$320$	3.03181	0.169484
$321$	−18.3109	−1.02202
$322$	1.37518	0.0766357
$323$	−3.82344	−0.212742
$324$	7.76469	0.431371
$325$	−15.0137	−0.832809
$326$	−12.0431	−0.667005
$327$	34.6703	1.91727
$328$	8.76099	0.483745
$329$	−1.00864	−0.0556080
$330$	−1.86020	−0.102400
$331$	−16.0230	−0.880705	−0.440353	−	0.897825i	$-0.645147\pi$
$331$	−16.0230	−0.880705	−0.440353	+	0.897825i	$0.645147\pi$
$332$	4.62080	0.253599
$333$	18.6157	1.02013
$334$	1.20829	0.0661145
$335$	−36.5872	−1.99897
$336$	1.71018	0.0932981
$337$	−24.1488	−1.31547	−0.657735	−	0.753249i	$-0.728485\pi$
$337$	−24.1488	−1.31547	−0.657735	+	0.753249i	$0.728485\pi$
$338$	−7.25098	−0.394401
$339$	12.0653	0.655297
$340$	−3.27964	−0.177863
$341$	−0.469477	−0.0254236
$342$	−18.9291	−1.02357
$343$	5.83006	0.314794
$344$	0	0
$345$	50.6518	2.72700
$346$	−0.414661	−0.0222923
$347$	30.3396	1.62872	0.814358	−	0.580363i	$-0.197089\pi$
$347$	30.3396	1.62872	0.814358	+	0.580363i	$0.197089\pi$
$348$	32.9964	1.76879
$349$	13.9362	0.745989	0.372994	−	0.927834i	$-0.378331\pi$
$349$	13.9362	0.745989	0.372994	+	0.927834i	$0.378331\pi$
$350$	−2.89852	−0.154933
$351$	3.86009	0.206036
$352$	1.55296	0.0827732
$353$	−14.4651	−0.769900	−0.384950	−	0.922938i	$-0.625781\pi$
$353$	−14.4651	−0.769900	−0.384950	+	0.922938i	$0.625781\pi$
$354$	11.7299	0.623438
$355$	19.4748	1.03361
$356$	10.2452	0.542994
$357$	0.601267	0.0318224
$358$	6.03261	0.318833
$359$	28.1541	1.48591	0.742957	−	0.669339i	$-0.233423\pi$
$359$	28.1541	1.48591	0.742957	+	0.669339i	$0.233423\pi$
$360$	−37.1090	−1.95582
$361$	31.3718	1.65115
$362$	9.86655	0.518575
$363$	28.9148	1.51764
$364$	−0.955533	−0.0500835
$365$	4.90669	0.256828
$366$	17.0229	0.889800
$367$	26.7425	1.39595	0.697974	−	0.716123i	$-0.254085\pi$
$367$	26.7425	1.39595	0.697974	+	0.716123i	$0.254085\pi$
$368$	−7.49587	−0.390749
$369$	−14.7945	−0.770170
$370$	12.1309	0.630657
$371$	3.51448	0.182463
$372$	−7.16982	−0.371738
$373$	0.401160	0.0207713	0.0103856	−	0.999946i	$-0.496694\pi$
$373$	0.401160	0.0207713	0.0103856	+	0.999946i	$0.496694\pi$
$374$	0.0967855	0.00500466
$375$	−54.9909	−2.83972
$376$	−5.66697	−0.292251
$377$	−11.6703	−0.601050
$378$	0.745224	0.0383302
$379$	16.3998	0.842403	0.421202	−	0.906967i	$-0.361608\pi$
$379$	16.3998	0.842403	0.421202	+	0.906967i	$0.361608\pi$
$380$	43.2075	2.21650
$381$	−46.8516	−2.40028
$382$	10.3035	0.527175
$383$	−26.4332	−1.35067	−0.675337	−	0.737510i	$-0.736002\pi$
$383$	−26.4332	−1.35067	−0.675337	+	0.737510i	$0.736002\pi$
$384$	29.1211	1.48608
$385$	−0.444915	−0.0226750
$386$	4.45649	0.226829
$387$	0	0
$388$	12.7931	0.649469
$389$	14.3581	0.727983	0.363992	−	0.931402i	$-0.381414\pi$
$389$	14.3581	0.727983	0.363992	+	0.931402i	$0.381414\pi$
$390$	10.0477	0.508785
$391$	−2.63540	−0.133278
$392$	16.1671	0.816563
$393$	18.8687	0.951799
$394$	9.68054	0.487699
$395$	−7.63729	−0.384274
$396$	−1.67842	−0.0843436
$397$	−5.79544	−0.290865	−0.145432	−	0.989368i	$-0.546457\pi$
$397$	−5.79544	−0.290865	−0.145432	+	0.989368i	$0.546457\pi$
$398$	−14.1106	−0.707299
$399$	−7.92136	−0.396564
$400$	15.7994	0.789968
$401$	2.49915	0.124802	0.0624009	−	0.998051i	$-0.480124\pi$
$401$	2.49915	0.124802	0.0624009	+	0.998051i	$0.480124\pi$
$402$	16.4895	0.822421
$403$	2.53585	0.126319
$404$	1.09163	0.0543106
$405$	−19.5283	−0.970368
$406$	−2.25305	−0.111817
$407$	1.25399	0.0621578
$408$	3.37818	0.167245
$409$	−23.6485	−1.16934	−0.584671	−	0.811270i	$-0.698777\pi$
$409$	−23.6485	−1.16934	−0.584671	+	0.811270i	$0.698777\pi$
$410$	−9.64083	−0.476127
$411$	14.4456	0.712550
$412$	24.0967	1.18716
$413$	2.80552	0.138051
$414$	−13.0473	−0.641241
$415$	−11.6214	−0.570470
$416$	−8.38820	−0.411266
$417$	29.8212	1.46035
$418$	−1.27510	−0.0623670
$419$	29.6381	1.44791	0.723957	−	0.689845i	$-0.242321\pi$
$419$	29.6381	1.44791	0.723957	+	0.689845i	$0.242321\pi$
$420$	−6.79472	−0.331548
$421$	16.4026	0.799415	0.399707	−	0.916643i	$-0.369112\pi$
$421$	16.4026	0.799415	0.399707	+	0.916643i	$0.369112\pi$
$422$	−16.1682	−0.787057
$423$	9.56968	0.465294
$424$	19.7459	0.958945
$425$	5.55475	0.269445
$426$	−8.77708	−0.425251
$427$	4.07147	0.197032
$428$	−10.7663	−0.520409
$429$	1.03864	0.0501461
$430$	0	0
$431$	15.7160	0.757013	0.378506	−	0.925599i	$-0.376438\pi$
$431$	15.7160	0.757013	0.378506	+	0.925599i	$0.376438\pi$
$432$	−4.06209	−0.195438
$433$	1.57684	0.0757783	0.0378891	−	0.999282i	$-0.487937\pi$
$433$	1.57684	0.0757783	0.0378891	+	0.999282i	$0.487937\pi$
$434$	0.489568	0.0235000
$435$	−82.9864	−3.97889
$436$	20.3852	0.976272
$437$	34.7200	1.66088
$438$	−2.21140	−0.105665
$439$	16.3140	0.778626	0.389313	−	0.921106i	$-0.372713\pi$
$439$	16.3140	0.778626	0.389313	+	0.921106i	$0.372713\pi$
$440$	−2.49973	−0.119170
$441$	−27.3010	−1.30005
$442$	−0.522779	−0.0248661
$443$	−15.8227	−0.751758	−0.375879	−	0.926669i	$-0.622659\pi$
$443$	−15.8227	−0.751758	−0.375879	+	0.926669i	$0.622659\pi$
$444$	19.1508	0.908856
$445$	−25.7668	−1.22146
$446$	10.8620	0.514329
$447$	−51.0669	−2.41538
$448$	−0.326813	−0.0154405
$449$	3.56502	0.168244	0.0841218	−	0.996455i	$-0.473192\pi$
$449$	3.56502	0.168244	0.0841218	+	0.996455i	$0.473192\pi$
$450$	27.5004	1.29638
$451$	−0.996583	−0.0469272
$452$	7.09405	0.333676
$453$	−11.7261	−0.550942
$454$	−7.24068	−0.339822
$455$	2.40318	0.112663
$456$	−44.5057	−2.08417
$457$	21.1849	0.990987	0.495493	−	0.868612i	$-0.334987\pi$
$457$	21.1849	0.990987	0.495493	+	0.868612i	$0.334987\pi$
$458$	−5.27969	−0.246704
$459$	−1.42815	−0.0666604
$460$	29.7818	1.38858
$461$	28.4343	1.32432	0.662158	−	0.749364i	$-0.269641\pi$
$461$	28.4343	1.32432	0.662158	+	0.749364i	$0.269641\pi$
$462$	0.200519	0.00932899
$463$	−39.8312	−1.85111	−0.925557	−	0.378608i	$-0.876403\pi$
$463$	−39.8312	−1.85111	−0.925557	+	0.378608i	$0.876403\pi$
$464$	12.2810	0.570131
$465$	18.0322	0.836223
$466$	15.8173	0.732723
$467$	15.5142	0.717910	0.358955	−	0.933355i	$-0.383133\pi$
$467$	15.5142	0.717910	0.358955	+	0.933355i	$0.383133\pi$
$468$	9.06584	0.419068
$469$	3.94390	0.182113
$470$	6.23609	0.287649
$471$	24.3714	1.12297
$472$	15.7626	0.725534
$473$	0	0
$474$	3.44205	0.158099
$475$	−73.1808	−3.35777
$476$	0.353527	0.0162039
$477$	−33.3444	−1.52674
$478$	−0.787422	−0.0360158
$479$	−3.81547	−0.174333	−0.0871666	−	0.996194i	$-0.527781\pi$
$479$	−3.81547	−0.174333	−0.0871666	+	0.996194i	$0.527781\pi$
$480$	−59.6479	−2.72254
$481$	−6.77331	−0.308836
$482$	−12.6515	−0.576259
$483$	−5.45999	−0.248438
$484$	17.0011	0.772777
$485$	−32.1747	−1.46098
$486$	14.1016	0.639662
$487$	−14.7061	−0.666398	−0.333199	−	0.942856i	$-0.608128\pi$
$487$	−14.7061	−0.666398	−0.333199	+	0.942856i	$0.608128\pi$
$488$	22.8753	1.03552
$489$	47.8157	2.16230
$490$	−17.7907	−0.803704
$491$	−29.9469	−1.35148	−0.675742	−	0.737138i	$-0.736177\pi$
$491$	−29.9469	−1.35148	−0.675742	+	0.737138i	$0.736177\pi$
$492$	−15.2197	−0.686159
$493$	4.31776	0.194462
$494$	6.88734	0.309876
$495$	4.22124	0.189731
$496$	−2.66855	−0.119822
$497$	−2.09927	−0.0941653
$498$	5.23763	0.234704
$499$	−14.4277	−0.645874	−0.322937	−	0.946420i	$-0.604670\pi$
$499$	−14.4277	−0.645874	−0.322937	+	0.946420i	$0.604670\pi$
$500$	−32.3330	−1.44598
$501$	−4.79736	−0.214330
$502$	13.1101	0.585133
$503$	3.43184	0.153018	0.0765092	−	0.997069i	$-0.475623\pi$
$503$	3.43184	0.153018	0.0765092	+	0.997069i	$0.475623\pi$
$504$	4.00015	0.178181
$505$	−2.74546	−0.122171
$506$	−0.878891	−0.0390715
$507$	28.7892	1.27857
$508$	−27.5474	−1.22222
$509$	−20.3043	−0.899971	−0.449985	−	0.893036i	$-0.648571\pi$
$509$	−20.3043	−0.899971	−0.449985	+	0.893036i	$0.648571\pi$
$510$	−3.71744	−0.164611
$511$	−0.528915	−0.0233978
$512$	16.0896	0.711067
$513$	18.8151	0.830709
$514$	13.4894	0.594992
$515$	−60.6035	−2.67051
$516$	0	0
$517$	0.644631	0.0283508
$518$	−1.30765	−0.0574547
$519$	1.64636	0.0722674
$520$	13.5021	0.592106
$521$	9.55685	0.418693	0.209347	−	0.977841i	$-0.432866\pi$
$521$	9.55685	0.418693	0.209347	+	0.977841i	$0.432866\pi$
$522$	21.3763	0.935617
$523$	4.41173	0.192911	0.0964557	−	0.995337i	$-0.469249\pi$
$523$	4.41173	0.192911	0.0964557	+	0.995337i	$0.469249\pi$
$524$	11.0942	0.484654
$525$	11.5083	0.502262
$526$	−3.32426	−0.144944
$527$	−0.938211	−0.0408691
$528$	−1.09300	−0.0475665
$529$	0.931571	0.0405031
$530$	−21.7289	−0.943844
$531$	−26.6180	−1.15512
$532$	−4.65753	−0.201930
$533$	5.38296	0.233162
$534$	11.6128	0.502537
$535$	27.0774	1.17066
$536$	22.1586	0.957105
$537$	−23.9518	−1.03360
$538$	−12.3630	−0.533009
$539$	−1.83905	−0.0792134
$540$	16.1391	0.694515
$541$	−44.1148	−1.89664	−0.948322	−	0.317310i	$-0.897221\pi$
$541$	−44.1148	−1.89664	−0.948322	+	0.317310i	$0.897221\pi$
$542$	−1.79271	−0.0770037
$543$	−39.1740	−1.68112
$544$	3.10346	0.133060
$545$	−51.2689	−2.19612
$546$	−1.08309	−0.0463519
$547$	−0.982873	−0.0420246	−0.0210123	−	0.999779i	$-0.506689\pi$
$547$	−0.982873	−0.0420246	−0.0210123	+	0.999779i	$0.506689\pi$
$548$	8.49360	0.362829
$549$	−38.6290	−1.64865
$550$	1.85248	0.0789899
$551$	−56.8842	−2.42335
$552$	−30.6766	−1.30568
$553$	0.823258	0.0350085
$554$	−9.24797	−0.392908
$555$	−48.1645	−2.04447
$556$	17.5340	0.743607
$557$	−13.1879	−0.558791	−0.279395	−	0.960176i	$-0.590134\pi$
$557$	−13.1879	−0.558791	−0.279395	+	0.960176i	$0.590134\pi$
$558$	−4.64489	−0.196634
$559$	0	0
$560$	−2.52894	−0.106867
$561$	−0.384276	−0.0162241
$562$	−18.1564	−0.765880
$563$	−29.6127	−1.24803	−0.624014	−	0.781413i	$-0.714499\pi$
$563$	−29.6127	−1.24803	−0.624014	+	0.781413i	$0.714499\pi$
$564$	9.84475	0.414539
$565$	−17.8416	−0.750602
$566$	8.27349	0.347761
$567$	2.10504	0.0884035
$568$	−11.7946	−0.494892
$569$	30.7503	1.28912	0.644559	−	0.764555i	$-0.277041\pi$
$569$	30.7503	1.28912	0.644559	+	0.764555i	$0.277041\pi$
$570$	48.9753	2.05135
$571$	−2.59158	−0.108454	−0.0542271	−	0.998529i	$-0.517269\pi$
$571$	−2.59158	−0.108454	−0.0542271	+	0.998529i	$0.517269\pi$
$572$	0.610691	0.0255343
$573$	−40.9090	−1.70900
$574$	1.03923	0.0433766
$575$	−50.4416	−2.10356
$576$	3.10071	0.129196
$577$	20.1867	0.840382	0.420191	−	0.907436i	$-0.361963\pi$
$577$	20.1867	0.840382	0.420191	+	0.907436i	$0.361963\pi$
$578$	−11.1364	−0.463213
$579$	−17.6940	−0.735336
$580$	−48.7936	−2.02604
$581$	1.25272	0.0519716
$582$	14.5008	0.601078
$583$	−2.24614	−0.0930256
$584$	−2.97167	−0.122969
$585$	−22.8007	−0.942692
$586$	19.7947	0.817712
$587$	−6.23038	−0.257155	−0.128578	−	0.991699i	$-0.541041\pi$
$587$	−6.23038	−0.257155	−0.128578	+	0.991699i	$0.541041\pi$
$588$	−28.0858	−1.15824
$589$	12.3604	0.509302
$590$	−17.3456	−0.714109
$591$	−38.4355	−1.58103
$592$	7.12777	0.292950
$593$	−18.3729	−0.754485	−0.377243	−	0.926114i	$-0.623128\pi$
$593$	−18.3729	−0.754485	−0.377243	+	0.926114i	$0.623128\pi$
$594$	−0.476281	−0.0195420
$595$	−0.889126	−0.0364506
$596$	−30.0258	−1.22991
$597$	56.0244	2.29293
$598$	4.74726	0.194130
$599$	23.3909	0.955726	0.477863	−	0.878434i	$-0.341412\pi$
$599$	23.3909	0.955726	0.477863	+	0.878434i	$0.341412\pi$
$600$	64.6584	2.63967
$601$	−33.0222	−1.34701	−0.673503	−	0.739185i	$-0.735211\pi$
$601$	−33.0222	−1.34701	−0.673503	+	0.739185i	$0.735211\pi$
$602$	0	0
$603$	−37.4187	−1.52381
$604$	−6.89462	−0.280538
$605$	−42.7579	−1.73836
$606$	1.23735	0.0502640
$607$	19.2404	0.780944	0.390472	−	0.920615i	$-0.372312\pi$
$607$	19.2404	0.780944	0.390472	+	0.920615i	$0.372312\pi$
$608$	−40.8864	−1.65816
$609$	8.94548	0.362489
$610$	−25.1726	−1.01921
$611$	−3.48192	−0.140863
$612$	−3.35417	−0.135584
$613$	36.4676	1.47291	0.736456	−	0.676485i	$-0.236498\pi$
$613$	36.4676	1.47291	0.736456	+	0.676485i	$0.236498\pi$
$614$	13.4251	0.541794
$615$	38.2778	1.54351
$616$	0.269457	0.0108567
$617$	36.6238	1.47442	0.737210	−	0.675664i	$-0.236143\pi$
$617$	36.6238	1.47442	0.737210	+	0.675664i	$0.236143\pi$
$618$	27.3134	1.09871
$619$	15.0924	0.606613	0.303307	−	0.952893i	$-0.401909\pi$
$619$	15.0924	0.606613	0.303307	+	0.952893i	$0.401909\pi$
$620$	10.6024	0.425803
$621$	12.9688	0.520419
$622$	−4.23273	−0.169717
$623$	2.77752	0.111279
$624$	5.90373	0.236338
$625$	29.7627	1.19051
$626$	14.0226	0.560456
$627$	5.06263	0.202182
$628$	14.3296	0.571815
$629$	2.50598	0.0999201
$630$	−4.40188	−0.175375
$631$	30.4898	1.21378	0.606890	−	0.794786i	$-0.292417\pi$
$631$	30.4898	1.21378	0.606890	+	0.794786i	$0.292417\pi$
$632$	4.62542	0.183990
$633$	64.1941	2.55149
$634$	−18.8229	−0.747551
$635$	69.2819	2.74937
$636$	−34.3029	−1.36020
$637$	9.93347	0.393578
$638$	1.43995	0.0570081
$639$	19.9173	0.787918
$640$	−43.0630	−1.70221
$641$	12.7086	0.501961	0.250980	−	0.967992i	$-0.419247\pi$
$641$	12.7086	0.501961	0.250980	+	0.967992i	$0.419247\pi$
$642$	−12.2035	−0.481634
$643$	17.5888	0.693633	0.346816	−	0.937933i	$-0.387263\pi$
$643$	17.5888	0.693633	0.346816	+	0.937933i	$0.387263\pi$
$644$	−3.21031	−0.126504
$645$	0	0
$646$	−2.54817	−0.100257
$647$	−15.8275	−0.622242	−0.311121	−	0.950370i	$-0.600704\pi$
$647$	−15.8275	−0.622242	−0.311121	+	0.950370i	$0.600704\pi$
$648$	11.8271	0.464610
$649$	−1.79304	−0.0703829
$650$	−10.0060	−0.392468
$651$	−1.94377	−0.0761825
$652$	28.1142	1.10104
$653$	0.0522286	0.00204386	0.00102193	−	0.999999i	$-0.499675\pi$
$653$	0.0522286	0.00204386	0.00102193	+	0.999999i	$0.499675\pi$
$654$	23.1064	0.903532
$655$	−27.9021	−1.09023
$656$	−5.66467	−0.221168
$657$	5.01820	0.195779
$658$	−0.672216	−0.0262057
$659$	−4.36477	−0.170027	−0.0850136	−	0.996380i	$-0.527093\pi$
$659$	−4.36477	−0.170027	−0.0850136	+	0.996380i	$0.527093\pi$
$660$	4.34257	0.169034
$661$	18.0436	0.701815	0.350907	−	0.936410i	$-0.385873\pi$
$661$	18.0436	0.701815	0.350907	+	0.936410i	$0.385873\pi$
$662$	−10.6787	−0.415040
$663$	2.07564	0.0806110
$664$	7.03833	0.273140
$665$	11.7138	0.454240
$666$	12.4066	0.480746
$667$	−39.2088	−1.51817
$668$	−2.82071	−0.109137
$669$	−43.1262	−1.66735
$670$	−24.3839	−0.942033
$671$	−2.60212	−0.100454
$672$	6.42972	0.248032
$673$	41.6584	1.60581	0.802906	−	0.596106i	$-0.203286\pi$
$673$	41.6584	1.60581	0.802906	+	0.596106i	$0.203286\pi$
$674$	−16.0942	−0.619926
$675$	−27.3349	−1.05212
$676$	16.9272	0.651046
$677$	−13.3731	−0.513970	−0.256985	−	0.966415i	$-0.582729\pi$
$677$	−13.3731	−0.513970	−0.256985	+	0.966415i	$0.582729\pi$
$678$	8.04104	0.308814
$679$	3.46826	0.133100
$680$	−4.99550	−0.191569
$681$	28.7483	1.10164
$682$	−0.312888	−0.0119811
$683$	8.52135	0.326060	0.163030	−	0.986621i	$-0.447873\pi$
$683$	8.52135	0.326060	0.163030	+	0.986621i	$0.447873\pi$
$684$	44.1894	1.68962
$685$	−21.3615	−0.816182
$686$	3.88550	0.148349
$687$	20.9624	0.799765
$688$	0	0
$689$	12.1324	0.462206
$690$	33.7574	1.28512
$691$	3.16724	0.120488	0.0602438	−	0.998184i	$-0.480812\pi$
$691$	3.16724	0.120488	0.0602438	+	0.998184i	$0.480812\pi$
$692$	0.968014	0.0367984
$693$	−0.455027	−0.0172850
$694$	20.2201	0.767546
$695$	−44.0982	−1.67274
$696$	50.2596	1.90509
$697$	−1.99159	−0.0754367
$698$	9.28793	0.351553
$699$	−62.8008	−2.37535
$700$	6.76652	0.255750
$701$	−2.56813	−0.0969969	−0.0484985	−	0.998823i	$-0.515444\pi$
$701$	−2.56813	−0.0969969	−0.0484985	+	0.998823i	$0.515444\pi$
$702$	2.57259	0.0970962
$703$	−33.0150	−1.24518
$704$	0.208870	0.00787207
$705$	−24.7597	−0.932503
$706$	−9.64041	−0.362822
$707$	0.295946	0.0111302
$708$	−27.3831	−1.02912
$709$	−7.59840	−0.285364	−0.142682	−	0.989769i	$-0.545573\pi$
$709$	−7.59840	−0.285364	−0.142682	+	0.989769i	$0.545573\pi$
$710$	12.9792	0.487099
$711$	−7.81085	−0.292930
$712$	15.6053	0.584834
$713$	8.51971	0.319066
$714$	0.400720	0.0149966
$715$	−1.53589	−0.0574392
$716$	−14.0830	−0.526305
$717$	3.12637	0.116756
$718$	18.7635	0.700249
$719$	−22.3021	−0.831728	−0.415864	−	0.909427i	$-0.636521\pi$
$719$	−22.3021	−0.831728	−0.415864	+	0.909427i	$0.636521\pi$
$720$	23.9939	0.894200
$721$	6.53273	0.243292
$722$	20.9081	0.778117
$723$	50.2312	1.86812
$724$	−23.0332	−0.856022
$725$	82.6420	3.06925
$726$	19.2706	0.715198
$727$	18.9626	0.703283	0.351641	−	0.936135i	$-0.385624\pi$
$727$	18.9626	0.703283	0.351641	+	0.936135i	$0.385624\pi$
$728$	−1.45545	−0.0539427
$729$	−41.0167	−1.51914
$730$	3.27011	0.121032
$731$	0	0
$732$	−39.7394	−1.46881
$733$	35.2298	1.30124	0.650622	−	0.759402i	$-0.274508\pi$
$733$	35.2298	1.30124	0.650622	+	0.759402i	$0.274508\pi$
$734$	17.8228	0.657852
$735$	70.6361	2.60545
$736$	−28.1820	−1.03880
$737$	−2.52059	−0.0928471
$738$	−9.85993	−0.362949
$739$	−15.3335	−0.564052	−0.282026	−	0.959407i	$-0.591006\pi$
$739$	−15.3335	−0.564052	−0.282026	+	0.959407i	$0.591006\pi$
$740$	−28.3193	−1.04104
$741$	−27.3454	−1.00456
$742$	2.34226	0.0859871
$743$	−52.5033	−1.92616	−0.963079	−	0.269219i	$-0.913235\pi$
$743$	−52.5033	−1.92616	−0.963079	+	0.269219i	$0.913235\pi$
$744$	−10.9210	−0.400382
$745$	75.5153	2.76667
$746$	0.267357	0.00978863
$747$	−11.8855	−0.434866
$748$	−0.225943	−0.00826129
$749$	−2.91879	−0.106650
$750$	−36.6492	−1.33824
$751$	−14.6250	−0.533675	−0.266837	−	0.963742i	$-0.585979\pi$
$751$	−14.6250	−0.533675	−0.266837	+	0.963742i	$0.585979\pi$
$752$	3.66414	0.133617
$753$	−52.0522	−1.89689
$754$	−7.77777	−0.283250
$755$	17.3401	0.631070
$756$	−1.73970	−0.0632724
$757$	−11.8289	−0.429928	−0.214964	−	0.976622i	$-0.568963\pi$
$757$	−11.8289	−0.429928	−0.214964	+	0.976622i	$0.568963\pi$
$758$	10.9298	0.396989
$759$	3.48954	0.126662
$760$	65.8130	2.38729
$761$	−49.9435	−1.81045	−0.905226	−	0.424930i	$-0.860299\pi$
$761$	−49.9435	−1.81045	−0.905226	+	0.424930i	$0.860299\pi$
$762$	−31.2247	−1.13115
$763$	5.52651	0.200073
$764$	−24.0533	−0.870218
$765$	8.43579	0.304996
$766$	−17.6167	−0.636516
$767$	9.68495	0.349703
$768$	23.5086	0.848292
$769$	−2.67040	−0.0962973	−0.0481486	−	0.998840i	$-0.515332\pi$
$769$	−2.67040	−0.0962973	−0.0481486	+	0.998840i	$0.515332\pi$
$770$	−0.296518	−0.0106858
$771$	−53.5581	−1.92885
$772$	−10.4035	−0.374432
$773$	20.0637	0.721643	0.360821	−	0.932635i	$-0.382496\pi$
$773$	20.0637	0.721643	0.360821	+	0.932635i	$0.382496\pi$
$774$	0	0
$775$	−17.9574	−0.645048
$776$	19.4862	0.699514
$777$	5.19187	0.186257
$778$	9.56908	0.343068
$779$	26.2381	0.940076
$780$	−23.4561	−0.839862
$781$	1.34167	0.0480086
$782$	−1.75639	−0.0628083
$783$	−21.2477	−0.759329
$784$	−10.4533	−0.373332
$785$	−36.0392	−1.28630
$786$	12.5752	0.448543
$787$	−22.5537	−0.803954	−0.401977	−	0.915650i	$-0.631677\pi$
$787$	−22.5537	−0.803954	−0.401977	+	0.915650i	$0.631677\pi$
$788$	−22.5990	−0.805054
$789$	13.1986	0.469882
$790$	−5.08995	−0.181092
$791$	1.92323	0.0683822
$792$	−2.55654	−0.0908426
$793$	14.0552	0.499113
$794$	−3.86243	−0.137072
$795$	86.2722	3.05976
$796$	32.9407	1.16755
$797$	−14.3884	−0.509663	−0.254832	−	0.966985i	$-0.582020\pi$
$797$	−14.3884	−0.509663	−0.254832	+	0.966985i	$0.582020\pi$
$798$	−5.27927	−0.186884
$799$	1.28824	0.0455746
$800$	59.4003	2.10012
$801$	−26.3524	−0.931115
$802$	1.66559	0.0588139
$803$	0.338035	0.0119290
$804$	−38.4943	−1.35759
$805$	8.07398	0.284570
$806$	1.69004	0.0595291
$807$	49.0861	1.72791
$808$	1.66275	0.0584954
$809$	10.0909	0.354777	0.177389	−	0.984141i	$-0.443235\pi$
$809$	10.0909	0.354777	0.177389	+	0.984141i	$0.443235\pi$
$810$	−13.0148	−0.457294
$811$	−42.0752	−1.47746	−0.738730	−	0.674002i	$-0.764574\pi$
$811$	−42.0752	−1.47746	−0.738730	+	0.674002i	$0.764574\pi$
$812$	5.25968	0.184579
$813$	7.11777	0.249631
$814$	0.835731	0.0292924
$815$	−70.7077	−2.47678
$816$	−2.18426	−0.0764643
$817$	0	0
$818$	−15.7608	−0.551062
$819$	2.45779	0.0858821
$820$	22.5062	0.785952
$821$	28.6620	1.00031	0.500155	−	0.865936i	$-0.333276\pi$
$821$	28.6620	1.00031	0.500155	+	0.865936i	$0.333276\pi$
$822$	9.62743	0.335795
$823$	20.9468	0.730159	0.365079	−	0.930976i	$-0.381042\pi$
$823$	20.9468	0.730159	0.365079	+	0.930976i	$0.381042\pi$
$824$	36.7037	1.27864
$825$	−7.35505	−0.256070
$826$	1.86977	0.0650575
$827$	50.7198	1.76370	0.881851	−	0.471528i	$-0.156297\pi$
$827$	50.7198	1.76370	0.881851	+	0.471528i	$0.156297\pi$
$828$	30.4586	1.05851
$829$	−21.5277	−0.747688	−0.373844	−	0.927492i	$-0.621960\pi$
$829$	−21.5277	−0.747688	−0.373844	+	0.927492i	$0.621960\pi$
$830$	−7.74517	−0.268839
$831$	36.7180	1.27373
$832$	−1.12819	−0.0391131
$833$	−3.67518	−0.127337
$834$	19.8746	0.688202
$835$	7.09412	0.245502
$836$	2.97668	0.102951
$837$	4.61693	0.159584
$838$	19.7526	0.682342
$839$	46.2066	1.59523	0.797614	−	0.603168i	$-0.206095\pi$
$839$	46.2066	1.59523	0.797614	+	0.603168i	$0.206095\pi$
$840$	−10.3496	−0.357095
$841$	35.2385	1.21512
$842$	10.9317	0.376731
$843$	72.0878	2.48284
$844$	37.7443	1.29921
$845$	−42.5721	−1.46453
$846$	6.37781	0.219273
$847$	4.60907	0.158370
$848$	−12.7673	−0.438430
$849$	−32.8489	−1.12737
$850$	3.70202	0.126978
$851$	−22.7564	−0.780078
$852$	20.4899	0.701971
$853$	26.5345	0.908523	0.454261	−	0.890868i	$-0.349903\pi$
$853$	26.5345	0.908523	0.454261	+	0.890868i	$0.349903\pi$
$854$	2.71347	0.0928532
$855$	−111.137	−3.80080
$856$	−16.3991	−0.560509
$857$	−57.9900	−1.98090	−0.990450	−	0.137876i	$-0.955973\pi$
$857$	−57.9900	−1.98090	−0.990450	+	0.137876i	$0.955973\pi$
$858$	0.692213	0.0236317
$859$	18.6327	0.635740	0.317870	−	0.948134i	$-0.397032\pi$
$859$	18.6327	0.635740	0.317870	+	0.948134i	$0.397032\pi$
$860$	0	0
$861$	−4.12614	−0.140619
$862$	10.4741	0.356749
$863$	−24.3579	−0.829153	−0.414576	−	0.910015i	$-0.636070\pi$
$863$	−24.3579	−0.829153	−0.414576	+	0.910015i	$0.636070\pi$
$864$	−15.2721	−0.519568
$865$	−2.43457	−0.0827778
$866$	1.05090	0.0357111
$867$	44.2158	1.50165
$868$	−1.14288	−0.0387919
$869$	−0.526153	−0.0178485
$870$	−55.3071	−1.87509
$871$	13.6148	0.461319
$872$	31.0504	1.05150
$873$	−32.9059	−1.11370
$874$	23.1395	0.782704
$875$	−8.76564	−0.296333
$876$	5.16244	0.174423
$877$	47.1759	1.59302	0.796508	−	0.604628i	$-0.206678\pi$
$877$	47.1759	1.59302	0.796508	+	0.604628i	$0.206678\pi$
$878$	10.8726	0.366934
$879$	−78.5927	−2.65086
$880$	1.61627	0.0544845
$881$	−33.5969	−1.13191	−0.565954	−	0.824437i	$-0.691492\pi$
$881$	−33.5969	−1.13191	−0.565954	+	0.824437i	$0.691492\pi$
$882$	−18.1951	−0.612659
$883$	24.8491	0.836238	0.418119	−	0.908392i	$-0.362689\pi$
$883$	24.8491	0.836238	0.418119	+	0.908392i	$0.362689\pi$
$884$	1.22041	0.0410469
$885$	68.8689	2.31500
$886$	−10.5452	−0.354272
$887$	−38.6646	−1.29823	−0.649116	−	0.760690i	$-0.724861\pi$
$887$	−38.6646	−1.29823	−0.649116	+	0.760690i	$0.724861\pi$
$888$	29.1702	0.978887
$889$	−7.46822	−0.250476
$890$	−17.1725	−0.575624
$891$	−1.34535	−0.0450711
$892$	−25.3569	−0.849013
$893$	−16.9719	−0.567941
$894$	−34.0340	−1.13827
$895$	35.4188	1.18392
$896$	4.64195	0.155077
$897$	−18.8485	−0.629332
$898$	2.37594	0.0792862
$899$	−13.9584	−0.465540
$900$	−64.1989	−2.13996
$901$	−4.48872	−0.149541
$902$	−0.664182	−0.0221148
$903$	0	0
$904$	10.8055	0.359387
$905$	57.9287	1.92562
$906$	−7.81500	−0.259636
$907$	−25.6470	−0.851594	−0.425797	−	0.904819i	$-0.640006\pi$
$907$	−25.6470	−0.851594	−0.425797	+	0.904819i	$0.640006\pi$
$908$	16.9032	0.560951
$909$	−2.80785	−0.0931306
$910$	1.60162	0.0530932
$911$	−19.3513	−0.641136	−0.320568	−	0.947226i	$-0.603874\pi$
$911$	−19.3513	−0.641136	−0.320568	+	0.947226i	$0.603874\pi$
$912$	28.7764	0.952883
$913$	−0.800626	−0.0264969
$914$	14.1189	0.467011
$915$	99.9451	3.30408
$916$	12.3253	0.407239
$917$	3.00770	0.0993229
$918$	−0.951806	−0.0314143
$919$	−8.83762	−0.291526	−0.145763	−	0.989320i	$-0.546564\pi$
$919$	−8.83762	−0.291526	−0.145763	+	0.989320i	$0.546564\pi$
$920$	45.3632	1.49558
$921$	−53.3029	−1.75639
$922$	18.9503	0.624095
$923$	−7.24691	−0.238535
$924$	−0.468106	−0.0153996
$925$	47.9646	1.57707
$926$	−26.5459	−0.872353
$927$	−61.9808	−2.03572
$928$	46.1725	1.51569
$929$	28.8957	0.948038	0.474019	−	0.880515i	$-0.342803\pi$
$929$	28.8957	0.948038	0.474019	+	0.880515i	$0.342803\pi$
$930$	12.0177	0.394077
$931$	48.4185	1.58685
$932$	−36.9251	−1.20952
$933$	16.8056	0.550190
$934$	10.3396	0.338321
$935$	0.568249	0.0185837
$936$	13.8089	0.451359
$937$	40.4460	1.32131	0.660656	−	0.750688i	$-0.270278\pi$
$937$	40.4460	1.32131	0.660656	+	0.750688i	$0.270278\pi$
$938$	2.62845	0.0858220
$939$	−55.6751	−1.81689
$940$	−14.5580	−0.474828
$941$	−13.9033	−0.453234	−0.226617	−	0.973984i	$-0.572767\pi$
$941$	−13.9033	−0.453234	−0.226617	+	0.973984i	$0.572767\pi$
$942$	16.2425	0.529210
$943$	18.0852	0.588935
$944$	−10.1918	−0.331714
$945$	4.37538	0.142331
$946$	0	0
$947$	−1.47398	−0.0478980	−0.0239490	−	0.999713i	$-0.507624\pi$
$947$	−1.47398	−0.0478980	−0.0239490	+	0.999713i	$0.507624\pi$
$948$	−8.03537	−0.260977
$949$	−1.82587	−0.0592702
$950$	−48.7721	−1.58237
$951$	74.7340	2.42342
$952$	0.538488	0.0174525
$953$	−14.6115	−0.473312	−0.236656	−	0.971593i	$-0.576051\pi$
$953$	−14.6115	−0.473312	−0.236656	+	0.971593i	$0.576051\pi$
$954$	−22.2227	−0.719487
$955$	60.4944	1.95755
$956$	1.83821	0.0594521
$957$	−5.71715	−0.184809
$958$	−2.54286	−0.0821559
$959$	2.30266	0.0743566
$960$	−8.02249	−0.258925
$961$	−27.9670	−0.902160
$962$	−4.51414	−0.145542
$963$	27.6927	0.892386
$964$	29.5345	0.951242
$965$	26.1650	0.842282
$966$	−3.63886	−0.117079
$967$	14.7740	0.475101	0.237550	−	0.971375i	$-0.423656\pi$
$967$	14.7740	0.475101	0.237550	+	0.971375i	$0.423656\pi$
$968$	25.8958	0.832322
$969$	10.1172	0.325012
$970$	−21.4431	−0.688498
$971$	−41.7465	−1.33971	−0.669855	−	0.742492i	$-0.733644\pi$
$971$	−41.7465	−1.33971	−0.669855	+	0.742492i	$0.733644\pi$
$972$	−32.9198	−1.05590
$973$	4.75355	0.152392
$974$	−9.80105	−0.314046
$975$	39.7277	1.27231
$976$	−14.7907	−0.473439
$977$	−11.7020	−0.374380	−0.187190	−	0.982324i	$-0.559938\pi$
$977$	−11.7020	−0.374380	−0.187190	+	0.982324i	$0.559938\pi$
$978$	31.8672	1.01900
$979$	−1.77514	−0.0567338
$980$	41.5320	1.32669
$981$	−52.4340	−1.67409
$982$	−19.9584	−0.636898
$983$	−34.5520	−1.10204	−0.551018	−	0.834493i	$-0.685760\pi$
$983$	−34.5520	−1.10204	−0.551018	+	0.834493i	$0.685760\pi$
$984$	−23.1825	−0.739030
$985$	56.8367	1.81097
$986$	2.87761	0.0916419
$987$	2.66896	0.0849539
$988$	−16.0783	−0.511518
$989$	0	0
$990$	2.81329	0.0894121
$991$	36.1118	1.14713	0.573564	−	0.819161i	$-0.305560\pi$
$991$	36.1118	1.14713	0.573564	+	0.819161i	$0.305560\pi$
$992$	−10.0329	−0.318544
$993$	42.3986	1.34548
$994$	−1.39908	−0.0443762
$995$	−82.8463	−2.62640
$996$	−12.2271	−0.387431
$997$	−19.5507	−0.619176	−0.309588	−	0.950871i	$-0.600191\pi$
$997$	−19.5507	−0.619176	−0.309588	+	0.950871i	$0.600191\pi$
$998$	−9.61550	−0.304373
$999$	−12.3319	−0.390165

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.2.a.q.1.13	yes	20
43.42	odd	2	inner	1849.2.a.q.1.8	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1849.2.a.q.1.8	✓	20	43.42	odd	2	inner
1849.2.a.q.1.13	yes	20	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:	\( N \)	\(=\)	\( 1849 = 43^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1849.a (trivial)

\(f(q)\)	\(=\)	\(q+0.666460 q^{2} -2.64610 q^{3} -1.55583 q^{4} +3.91294 q^{5} -1.76352 q^{6} -0.421793 q^{7} -2.36982 q^{8} +4.00186 q^{9} +O(q^{10})\)	\(q+0.666460 q^{2} -2.64610 q^{3} -1.55583 q^{4} +3.91294 q^{5} -1.76352 q^{6} -0.421793 q^{7} -2.36982 q^{8} +4.00186 q^{9} +2.60781 q^{10} +0.269572 q^{11} +4.11689 q^{12} -1.45607 q^{13} -0.281108 q^{14} -10.3540 q^{15} +1.53227 q^{16} +0.538717 q^{17} +2.66708 q^{18} -7.09731 q^{19} -6.08787 q^{20} +1.11611 q^{21} +0.179659 q^{22} -4.89199 q^{23} +6.27078 q^{24} +10.3111 q^{25} -0.970415 q^{26} -2.65102 q^{27} +0.656239 q^{28} +8.01489 q^{29} -6.90054 q^{30} -1.74156 q^{31} +5.76084 q^{32} -0.713316 q^{33} +0.359034 q^{34} -1.65045 q^{35} -6.22622 q^{36} +4.65176 q^{37} -4.73007 q^{38} +3.85292 q^{39} -9.27295 q^{40} -3.69690 q^{41} +0.743841 q^{42} -0.419409 q^{44} +15.6590 q^{45} -3.26032 q^{46} +2.39131 q^{47} -4.05455 q^{48} -6.82209 q^{49} +6.87191 q^{50} -1.42550 q^{51} +2.26541 q^{52} -8.33224 q^{53} -1.76680 q^{54} +1.05482 q^{55} +0.999573 q^{56} +18.7802 q^{57} +5.34160 q^{58} -6.65141 q^{59} +16.1091 q^{60} -9.65277 q^{61} -1.16068 q^{62} -1.68796 q^{63} +0.774818 q^{64} -5.69752 q^{65} -0.475397 q^{66} -9.35033 q^{67} -0.838153 q^{68} +12.9447 q^{69} -1.09996 q^{70} +4.97702 q^{71} -9.48368 q^{72} +1.25397 q^{73} +3.10021 q^{74} -27.2841 q^{75} +11.0422 q^{76} -0.113704 q^{77} +2.56782 q^{78} -1.95181 q^{79} +5.99569 q^{80} -4.99070 q^{81} -2.46384 q^{82} -2.96999 q^{83} -1.73648 q^{84} +2.10797 q^{85} -21.2082 q^{87} -0.638838 q^{88} -6.58503 q^{89} +10.4361 q^{90} +0.614162 q^{91} +7.61111 q^{92} +4.60836 q^{93} +1.59371 q^{94} -27.7713 q^{95} -15.2438 q^{96} -8.22265 q^{97} -4.54665 q^{98} +1.07879 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 14 q^{4} - 32 q^{6} - 6 q^{9}+O(q^{10}) \) 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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