LMFDB - Embedded newform 1875.2.a.e.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1875,2,Mod(1,1875)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1875, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1875.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1875 = 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1875.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:	$14.9719503790$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.5125.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 6x^{2} + 7x + 11 $ x^4 - 2x^3 - 6x^2 + 7*x + 11
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 75)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.12233$ of defining polynomial
Character	$\chi$	$=$	1875.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.12233 q^{2} -1.00000 q^{3} -0.740367 q^{4} -1.12233 q^{6} -1.11373 q^{7} -3.07561 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.12233 q^{2} -1.00000 q^{3} -0.740367 q^{4} -1.12233 q^{6} -1.11373 q^{7} -3.07561 q^{8} +1.00000 q^{9} +3.67008 q^{11} +0.740367 q^{12} +4.05204 q^{13} -1.24998 q^{14} -1.97112 q^{16} -2.12233 q^{17} +1.12233 q^{18} -4.06064 q^{19} +1.11373 q^{21} +4.11905 q^{22} -6.17438 q^{23} +3.07561 q^{24} +4.54774 q^{26} -1.00000 q^{27} +0.824573 q^{28} -2.25963 q^{29} +10.0520 q^{31} +3.93896 q^{32} -3.67008 q^{33} -2.38197 q^{34} -0.740367 q^{36} -7.37232 q^{37} -4.55739 q^{38} -4.05204 q^{39} -7.47214 q^{41} +1.24998 q^{42} -9.24998 q^{43} -2.71720 q^{44} -6.92971 q^{46} +3.12765 q^{47} +1.97112 q^{48} -5.75960 q^{49} +2.12233 q^{51} -3.00000 q^{52} +3.50961 q^{53} -1.12233 q^{54} +3.42541 q^{56} +4.06064 q^{57} -2.53606 q^{58} -6.59382 q^{59} -9.10408 q^{61} +11.2817 q^{62} -1.11373 q^{63} +8.36307 q^{64} -4.11905 q^{66} -2.62663 q^{67} +1.57131 q^{68} +6.17438 q^{69} +0.660827 q^{71} -3.07561 q^{72} -7.47542 q^{73} -8.27420 q^{74} +3.00637 q^{76} -4.08749 q^{77} -4.54774 q^{78} +8.53711 q^{79} +1.00000 q^{81} -8.38623 q^{82} -12.2639 q^{83} -0.824573 q^{84} -10.3816 q^{86} +2.25963 q^{87} -11.2877 q^{88} -15.2881 q^{89} -4.51290 q^{91} +4.57131 q^{92} -10.0520 q^{93} +3.51026 q^{94} -3.93896 q^{96} -13.5000 q^{97} -6.46419 q^{98} +3.67008 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} - 4 q^{3} + 8 q^{4} + 2 q^{6} - 2 q^{7} - 15 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q - 2 * q^2 - 4 * q^3 + 8 * q^4 + 2 * q^6 - 2 * q^7 - 15 * q^8 + 4 * q^9	$ 4 q - 2 q^{2} - 4 q^{3} + 8 q^{4} + 2 q^{6} - 2 q^{7} - 15 q^{8} + 4 q^{9} - 7 q^{11} - 8 q^{12} - q^{13} + 16 q^{14} + 4 q^{16} - 2 q^{17} - 2 q^{18} + 5 q^{19} + 2 q^{21} + 6 q^{22} - q^{23} + 15 q^{24} + 3 q^{26} - 4 q^{27} - 9 q^{28} - 20 q^{29} + 23 q^{31} - 12 q^{32} + 7 q^{33} - 14 q^{34} + 8 q^{36} - 2 q^{37} - 35 q^{38} + q^{39} - 12 q^{41} - 16 q^{42} - 16 q^{43} - 29 q^{44} - 17 q^{46} - 2 q^{47} - 4 q^{48} + 8 q^{49} + 2 q^{51} - 12 q^{52} + 4 q^{53} + 2 q^{54} + 5 q^{56} - 5 q^{57} + 25 q^{58} - 15 q^{59} - 2 q^{61} - 9 q^{62} - 2 q^{63} + 23 q^{64} - 6 q^{66} - 2 q^{67} + 11 q^{68} + q^{69} - 2 q^{71} - 15 q^{72} - 16 q^{73} - 19 q^{74} + 40 q^{76} - 19 q^{77} - 3 q^{78} + 35 q^{79} + 4 q^{81} + 6 q^{82} - 16 q^{83} + 9 q^{84} + 3 q^{86} + 20 q^{87} + 30 q^{88} - 35 q^{89} - 12 q^{91} + 23 q^{92} - 23 q^{93} - 9 q^{94} + 12 q^{96} - 12 q^{97} + q^{98} - 7 q^{99}+O(q^{100}) $ 4 * q - 2 * q^2 - 4 * q^3 + 8 * q^4 + 2 * q^6 - 2 * q^7 - 15 * q^8 + 4 * q^9 - 7 * q^11 - 8 * q^12 - q^13 + 16 * q^14 + 4 * q^16 - 2 * q^17 - 2 * q^18 + 5 * q^19 + 2 * q^21 + 6 * q^22 - q^23 + 15 * q^24 + 3 * q^26 - 4 * q^27 - 9 * q^28 - 20 * q^29 + 23 * q^31 - 12 * q^32 + 7 * q^33 - 14 * q^34 + 8 * q^36 - 2 * q^37 - 35 * q^38 + q^39 - 12 * q^41 - 16 * q^42 - 16 * q^43 - 29 * q^44 - 17 * q^46 - 2 * q^47 - 4 * q^48 + 8 * q^49 + 2 * q^51 - 12 * q^52 + 4 * q^53 + 2 * q^54 + 5 * q^56 - 5 * q^57 + 25 * q^58 - 15 * q^59 - 2 * q^61 - 9 * q^62 - 2 * q^63 + 23 * q^64 - 6 * q^66 - 2 * q^67 + 11 * q^68 + q^69 - 2 * q^71 - 15 * q^72 - 16 * q^73 - 19 * q^74 + 40 * q^76 - 19 * q^77 - 3 * q^78 + 35 * q^79 + 4 * q^81 + 6 * q^82 - 16 * q^83 + 9 * q^84 + 3 * q^86 + 20 * q^87 + 30 * q^88 - 35 * q^89 - 12 * q^91 + 23 * q^92 - 23 * q^93 - 9 * q^94 + 12 * q^96 - 12 * q^97 + q^98 - 7 * 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 $

$2$	1.12233	0.793610	0.396805	−	0.917903i	$-0.370119\pi$
$2$	1.12233	0.793610	0.396805	+	0.917903i	$0.370119\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−0.740367	−0.370184
$5$	0	0
$5$	0	0
$6$	−1.12233	−0.458191
$7$	−1.11373	−0.420952	−0.210476	−	0.977599i	$-0.567501\pi$
$7$	−1.11373	−0.420952	−0.210476	+	0.977599i	$0.567501\pi$
$8$	−3.07561	−1.08739
$9$	1.00000	0.333333
$10$	0	0
$11$	3.67008	1.10657	0.553285	−	0.832992i	$-0.313374\pi$
$11$	3.67008	1.10657	0.553285	+	0.832992i	$0.313374\pi$
$12$	0.740367	0.213726
$13$	4.05204	1.12383	0.561917	−	0.827194i	$-0.310064\pi$
$13$	4.05204	1.12383	0.561917	+	0.827194i	$0.310064\pi$
$14$	−1.24998	−0.334072
$15$	0	0
$16$	−1.97112	−0.492780
$17$	−2.12233	−0.514741	−0.257371	−	0.966313i	$-0.582856\pi$
$17$	−2.12233	−0.514741	−0.257371	+	0.966313i	$0.582856\pi$
$18$	1.12233	0.264537
$19$	−4.06064	−0.931575	−0.465787	−	0.884897i	$-0.654229\pi$
$19$	−4.06064	−0.931575	−0.465787	+	0.884897i	$0.654229\pi$
$20$	0	0
$21$	1.11373	0.243037
$22$	4.11905	0.878184
$23$	−6.17438	−1.28745	−0.643723	−	0.765258i	$-0.722611\pi$
$23$	−6.17438	−1.28745	−0.643723	+	0.765258i	$0.722611\pi$
$24$	3.07561	0.627806
$25$	0	0
$26$	4.54774	0.891886
$27$	−1.00000	−0.192450
$28$	0.824573	0.155830
$29$	−2.25963	−0.419603	−0.209802	−	0.977744i	$-0.567282\pi$
$29$	−2.25963	−0.419603	−0.209802	+	0.977744i	$0.567282\pi$
$30$	0	0
$31$	10.0520	1.80540	0.902700	−	0.430271i	$-0.141582\pi$
$31$	10.0520	1.80540	0.902700	+	0.430271i	$0.141582\pi$
$32$	3.93896	0.696316
$33$	−3.67008	−0.638878
$34$	−2.38197	−0.408504
$35$	0	0
$36$	−0.740367	−0.123395
$37$	−7.37232	−1.21200	−0.606001	−	0.795464i	$-0.707227\pi$
$37$	−7.37232	−1.21200	−0.606001	+	0.795464i	$0.707227\pi$
$38$	−4.55739	−0.739307
$39$	−4.05204	−0.648846
$40$	0	0
$41$	−7.47214	−1.16695	−0.583476	−	0.812131i	$-0.698308\pi$
$41$	−7.47214	−1.16695	−0.583476	+	0.812131i	$0.698308\pi$
$42$	1.24998	0.192876
$43$	−9.24998	−1.41061	−0.705304	−	0.708905i	$-0.749190\pi$
$43$	−9.24998	−1.41061	−0.705304	+	0.708905i	$0.749190\pi$
$44$	−2.71720	−0.409634
$45$	0	0
$46$	−6.92971	−1.02173
$47$	3.12765	0.456214	0.228107	−	0.973636i	$-0.426746\pi$
$47$	3.12765	0.456214	0.228107	+	0.973636i	$0.426746\pi$
$48$	1.97112	0.284507
$49$	−5.75960	−0.822799
$50$	0	0
$51$	2.12233	0.297186
$52$	−3.00000	−0.416025
$53$	3.50961	0.482083	0.241041	−	0.970515i	$-0.422511\pi$
$53$	3.50961	0.482083	0.241041	+	0.970515i	$0.422511\pi$
$54$	−1.12233	−0.152730
$55$	0	0
$56$	3.42541	0.457739
$57$	4.06064	0.537845
$58$	−2.53606	−0.333001
$59$	−6.59382	−0.858442	−0.429221	−	0.903199i	$-0.641212\pi$
$59$	−6.59382	−0.858442	−0.429221	+	0.903199i	$0.641212\pi$
$60$	0	0
$61$	−9.10408	−1.16566	−0.582829	−	0.812595i	$-0.698054\pi$
$61$	−9.10408	−1.16566	−0.582829	+	0.812595i	$0.698054\pi$
$62$	11.2817	1.43278
$63$	−1.11373	−0.140317
$64$	8.36307	1.04538
$65$	0	0
$66$	−4.11905	−0.507020
$67$	−2.62663	−0.320894	−0.160447	−	0.987044i	$-0.551294\pi$
$67$	−2.62663	−0.320894	−0.160447	+	0.987044i	$0.551294\pi$
$68$	1.57131	0.190549
$69$	6.17438	0.743307
$70$	0	0
$71$	0.660827	0.0784257	0.0392128	−	0.999231i	$-0.487515\pi$
$71$	0.660827	0.0784257	0.0392128	+	0.999231i	$0.487515\pi$
$72$	−3.07561	−0.362464
$73$	−7.47542	−0.874932	−0.437466	−	0.899235i	$-0.644124\pi$
$73$	−7.47542	−0.874932	−0.437466	+	0.899235i	$0.644124\pi$
$74$	−8.27420	−0.961856
$75$	0	0
$76$	3.00637	0.344854
$77$	−4.08749	−0.465813
$78$	−4.54774	−0.514930
$79$	8.53711	0.960500	0.480250	−	0.877132i	$-0.340546\pi$
$79$	8.53711	0.960500	0.480250	+	0.877132i	$0.340546\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−8.38623	−0.926104
$83$	−12.2639	−1.34614	−0.673069	−	0.739580i	$-0.735024\pi$
$83$	−12.2639	−1.34614	−0.673069	+	0.739580i	$0.735024\pi$
$84$	−0.824573	−0.0899683
$85$	0	0
$86$	−10.3816	−1.11947
$87$	2.25963	0.242258
$88$	−11.2877	−1.20327
$89$	−15.2881	−1.62054	−0.810268	−	0.586059i	$-0.800679\pi$
$89$	−15.2881	−1.62054	−0.810268	+	0.586059i	$0.800679\pi$
$90$	0	0
$91$	−4.51290	−0.473080
$92$	4.57131	0.476592
$93$	−10.0520	−1.04235
$94$	3.51026	0.362056
$95$	0	0
$96$	−3.93896	−0.402018
$97$	−13.5000	−1.37071	−0.685357	−	0.728207i	$-0.740354\pi$
$97$	−13.5000	−1.37071	−0.685357	+	0.728207i	$0.740354\pi$
$98$	−6.46419	−0.652982
$99$	3.67008	0.368856
$100$	0	0
$101$	7.22642	0.719055	0.359528	−	0.933134i	$-0.382938\pi$
$101$	7.22642	0.719055	0.359528	+	0.933134i	$0.382938\pi$
$102$	2.38197	0.235850
$103$	−5.27748	−0.520006	−0.260003	−	0.965608i	$-0.583723\pi$
$103$	−5.27748	−0.520006	−0.260003	+	0.965608i	$0.583723\pi$
$104$	−12.4625	−1.22205
$105$	0	0
$106$	3.93896	0.382585
$107$	−5.46682	−0.528498	−0.264249	−	0.964455i	$-0.585124\pi$
$107$	−5.46682	−0.528498	−0.264249	+	0.964455i	$0.585124\pi$
$108$	0.740367	0.0712419
$109$	11.3395	1.08613	0.543064	−	0.839692i	$-0.317264\pi$
$109$	11.3395	1.08613	0.543064	+	0.839692i	$0.317264\pi$
$110$	0	0
$111$	7.37232	0.699749
$112$	2.19531	0.207437
$113$	−4.83520	−0.454858	−0.227429	−	0.973795i	$-0.573032\pi$
$113$	−4.83520	−0.454858	−0.227429	+	0.973795i	$0.573032\pi$
$114$	4.55739	0.426839
$115$	0	0
$116$	1.67296	0.155330
$117$	4.05204	0.374611
$118$	−7.40046	−0.681268
$119$	2.36372	0.216681
$120$	0	0
$121$	2.46946	0.224496
$122$	−10.2178	−0.925078
$123$	7.47214	0.673740
$124$	−7.44220	−0.668330
$125$	0	0
$126$	−1.24998	−0.111357
$127$	−0.759596	−0.0674032	−0.0337016	−	0.999432i	$-0.510730\pi$
$127$	−0.759596	−0.0674032	−0.0337016	+	0.999432i	$0.510730\pi$
$128$	1.50823	0.133310
$129$	9.24998	0.814415
$130$	0	0
$131$	14.6551	1.28042	0.640211	−	0.768199i	$-0.278847\pi$
$131$	14.6551	1.28042	0.640211	+	0.768199i	$0.278847\pi$
$132$	2.71720	0.236502
$133$	4.52248	0.392148
$134$	−2.94796	−0.254665
$135$	0	0
$136$	6.52746	0.559725
$137$	−12.2914	−1.05012	−0.525062	−	0.851064i	$-0.675958\pi$
$137$	−12.2914	−1.05012	−0.525062	+	0.851064i	$0.675958\pi$
$138$	6.92971	0.589896
$139$	14.7340	1.24972	0.624861	−	0.780736i	$-0.285156\pi$
$139$	14.7340	1.24972	0.624861	+	0.780736i	$0.285156\pi$
$140$	0	0
$141$	−3.12765	−0.263395
$142$	0.741668	0.0622394
$143$	14.8713	1.24360
$144$	−1.97112	−0.164260
$145$	0	0
$146$	−8.38991	−0.694354
$147$	5.75960	0.475043
$148$	5.45822	0.448663
$149$	15.6498	1.28208	0.641041	−	0.767507i	$-0.278503\pi$
$149$	15.6498	1.28208	0.641041	+	0.767507i	$0.278503\pi$
$150$	0	0
$151$	3.95819	0.322113	0.161056	−	0.986945i	$-0.448510\pi$
$151$	3.95819	0.322113	0.161056	+	0.986945i	$0.448510\pi$
$152$	12.4889	1.01299
$153$	−2.12233	−0.171580
$154$	−4.58753	−0.369673
$155$	0	0
$156$	3.00000	0.240192
$157$	4.50061	0.359188	0.179594	−	0.983741i	$-0.442522\pi$
$157$	4.50061	0.359188	0.179594	+	0.983741i	$0.442522\pi$
$158$	9.58149	0.762262
$159$	−3.50961	−0.278330
$160$	0	0
$161$	6.87661	0.541953
$162$	1.12233	0.0881788
$163$	2.45389	0.192203	0.0961016	−	0.995372i	$-0.469363\pi$
$163$	2.45389	0.192203	0.0961016	+	0.995372i	$0.469363\pi$
$164$	5.53213	0.431987
$165$	0	0
$166$	−13.7642	−1.06831
$167$	3.06328	0.237043	0.118522	−	0.992951i	$-0.462184\pi$
$167$	3.06328	0.237043	0.118522	+	0.992951i	$0.462184\pi$
$168$	−3.42541	−0.264276
$169$	3.41904	0.263003
$170$	0	0
$171$	−4.06064	−0.310525
$172$	6.84839	0.522185
$173$	−11.2596	−0.856054	−0.428027	−	0.903766i	$-0.640791\pi$
$173$	−11.2596	−0.856054	−0.428027	+	0.903766i	$0.640791\pi$
$174$	2.53606	0.192258
$175$	0	0
$176$	−7.23416	−0.545296
$177$	6.59382	0.495622
$178$	−17.1584	−1.28607
$179$	10.0574	0.751722	0.375861	−	0.926676i	$-0.377347\pi$
$179$	10.0574	0.751722	0.375861	+	0.926676i	$0.377347\pi$
$180$	0	0
$181$	−19.4684	−1.44708	−0.723539	−	0.690283i	$-0.757486\pi$
$181$	−19.4684	−1.44708	−0.723539	+	0.690283i	$0.757486\pi$
$182$	−5.06498	−0.375441
$183$	9.10408	0.672993
$184$	18.9899	1.39996
$185$	0	0
$186$	−11.2817	−0.827218
$187$	−7.78912	−0.569597
$188$	−2.31561	−0.168883
$189$	1.11373	0.0810123
$190$	0	0
$191$	−11.6090	−0.840000	−0.420000	−	0.907524i	$-0.637970\pi$
$191$	−11.6090	−0.840000	−0.420000	+	0.907524i	$0.637970\pi$
$192$	−8.36307	−0.603552
$193$	3.38156	0.243410	0.121705	−	0.992566i	$-0.461164\pi$
$193$	3.38156	0.243410	0.121705	+	0.992566i	$0.461164\pi$
$194$	−15.1515	−1.08781
$195$	0	0
$196$	4.26422	0.304587
$197$	17.5881	1.25310	0.626550	−	0.779381i	$-0.284466\pi$
$197$	17.5881	1.25310	0.626550	+	0.779381i	$0.284466\pi$
$198$	4.11905	0.292728
$199$	20.0102	1.41849	0.709244	−	0.704963i	$-0.249037\pi$
$199$	20.0102	1.41849	0.709244	+	0.704963i	$0.249037\pi$
$200$	0	0
$201$	2.62663	0.185268
$202$	8.11045	0.570649
$203$	2.51663	0.176633
$204$	−1.57131	−0.110013
$205$	0	0
$206$	−5.92309	−0.412681
$207$	−6.17438	−0.429149
$208$	−7.98706	−0.553803
$209$	−14.9029	−1.03085
$210$	0	0
$211$	2.44261	0.168156	0.0840780	−	0.996459i	$-0.473206\pi$
$211$	2.44261	0.168156	0.0840780	+	0.996459i	$0.473206\pi$
$212$	−2.59840	−0.178459
$213$	−0.660827	−0.0452791
$214$	−6.13560	−0.419421
$215$	0	0
$216$	3.07561	0.209269
$217$	−11.1953	−0.759987
$218$	12.7267	0.861961
$219$	7.47542	0.505142
$220$	0	0
$221$	−8.59978	−0.578484
$222$	8.27420	0.555328
$223$	25.6369	1.71677	0.858386	−	0.513005i	$-0.171468\pi$
$223$	25.6369	1.71677	0.858386	+	0.513005i	$0.171468\pi$
$224$	−4.38695	−0.293116
$225$	0	0
$226$	−5.42671	−0.360979
$227$	−6.56336	−0.435625	−0.217813	−	0.975991i	$-0.569892\pi$
$227$	−6.56336	−0.435625	−0.217813	+	0.975991i	$0.569892\pi$
$228$	−3.00637	−0.199101
$229$	−7.96390	−0.526269	−0.263135	−	0.964759i	$-0.584756\pi$
$229$	−7.96390	−0.526269	−0.263135	+	0.964759i	$0.584756\pi$
$230$	0	0
$231$	4.08749	0.268937
$232$	6.94974	0.456273
$233$	9.42107	0.617195	0.308597	−	0.951193i	$-0.400140\pi$
$233$	9.42107	0.617195	0.308597	+	0.951193i	$0.400140\pi$
$234$	4.54774	0.297295
$235$	0	0
$236$	4.88185	0.317781
$237$	−8.53711	−0.554545
$238$	2.65288	0.171961
$239$	−11.2231	−0.725964	−0.362982	−	0.931796i	$-0.618241\pi$
$239$	−11.2231	−0.725964	−0.362982	+	0.931796i	$0.618241\pi$
$240$	0	0
$241$	22.6809	1.46101	0.730503	−	0.682910i	$-0.239286\pi$
$241$	22.6809	1.46101	0.730503	+	0.682910i	$0.239286\pi$
$242$	2.77155	0.178162
$243$	−1.00000	−0.0641500
$244$	6.74037	0.431508
$245$	0	0
$246$	8.38623	0.534686
$247$	−16.4539	−1.04694
$248$	−30.9161	−1.96318
$249$	12.2639	0.777193
$250$	0	0
$251$	−6.76819	−0.427205	−0.213602	−	0.976921i	$-0.568520\pi$
$251$	−6.76819	−0.427205	−0.213602	+	0.976921i	$0.568520\pi$
$252$	0.824573	0.0519432
$253$	−22.6604	−1.42465
$254$	−0.852520	−0.0534918
$255$	0	0
$256$	−15.0334	−0.939587
$257$	14.7934	0.922786	0.461393	−	0.887196i	$-0.347350\pi$
$257$	14.7934	0.922786	0.461393	+	0.887196i	$0.347350\pi$
$258$	10.3816	0.646328
$259$	8.21080	0.510194
$260$	0	0
$261$	−2.25963	−0.139868
$262$	16.4479	1.01616
$263$	−19.9688	−1.23133	−0.615665	−	0.788008i	$-0.711113\pi$
$263$	−19.9688	−1.23133	−0.615665	+	0.788008i	$0.711113\pi$
$264$	11.2877	0.694710
$265$	0	0
$266$	5.07573	0.311213
$267$	15.2881	0.935617
$268$	1.94467	0.118790
$269$	−10.2381	−0.624228	−0.312114	−	0.950045i	$-0.601037\pi$
$269$	−10.2381	−0.624228	−0.312114	+	0.950045i	$0.601037\pi$
$270$	0	0
$271$	12.3075	0.747630	0.373815	−	0.927503i	$-0.378050\pi$
$271$	12.3075	0.747630	0.373815	+	0.927503i	$0.378050\pi$
$272$	4.18338	0.253654
$273$	4.51290	0.273133
$274$	−13.7950	−0.833389
$275$	0	0
$276$	−4.57131	−0.275160
$277$	−31.5520	−1.89578	−0.947888	−	0.318603i	$-0.896786\pi$
$277$	−31.5520	−1.89578	−0.947888	+	0.318603i	$0.896786\pi$
$278$	16.5365	0.991791
$279$	10.0520	0.601800
$280$	0	0
$281$	−21.3119	−1.27136	−0.635681	−	0.771952i	$-0.719281\pi$
$281$	−21.3119	−1.27136	−0.635681	+	0.771952i	$0.719281\pi$
$282$	−3.51026	−0.209033
$283$	0.864403	0.0513834	0.0256917	−	0.999670i	$-0.491821\pi$
$283$	0.864403	0.0513834	0.0256917	+	0.999670i	$0.491821\pi$
$284$	−0.489254	−0.0290319
$285$	0	0
$286$	16.6906	0.986933
$287$	8.32198	0.491231
$288$	3.93896	0.232105
$289$	−12.4957	−0.735041
$290$	0	0
$291$	13.5000	0.791382
$292$	5.53456	0.323886
$293$	−3.17701	−0.185603	−0.0928014	−	0.995685i	$-0.529582\pi$
$293$	−3.17701	−0.185603	−0.0928014	+	0.995685i	$0.529582\pi$
$294$	6.46419	0.376999
$295$	0	0
$296$	22.6743	1.31792
$297$	−3.67008	−0.212959
$298$	17.5643	1.01747
$299$	−25.0188	−1.44688
$300$	0	0
$301$	10.3020	0.593799
$302$	4.44240	0.255632
$303$	−7.22642	−0.415147
$304$	8.00401	0.459062
$305$	0	0
$306$	−2.38197	−0.136168
$307$	0.507986	0.0289923	0.0144961	−	0.999895i	$-0.495386\pi$
$307$	0.507986	0.0289923	0.0144961	+	0.999895i	$0.495386\pi$
$308$	3.02624	0.172436
$309$	5.27748	0.300225
$310$	0	0
$311$	−14.6438	−0.830375	−0.415188	−	0.909736i	$-0.636284\pi$
$311$	−14.6438	−0.830375	−0.415188	+	0.909736i	$0.636284\pi$
$312$	12.4625	0.705549
$313$	14.9562	0.845372	0.422686	−	0.906276i	$-0.361087\pi$
$313$	14.9562	0.845372	0.422686	+	0.906276i	$0.361087\pi$
$314$	5.05119	0.285055
$315$	0	0
$316$	−6.32060	−0.355562
$317$	−18.1190	−1.01767	−0.508834	−	0.860865i	$-0.669923\pi$
$317$	−18.1190	−1.01767	−0.508834	+	0.860865i	$0.669923\pi$
$318$	−3.93896	−0.220886
$319$	−8.29302	−0.464320
$320$	0	0
$321$	5.46682	0.305128
$322$	7.71785	0.430099
$323$	8.61803	0.479520
$324$	−0.740367	−0.0411315
$325$	0	0
$326$	2.75408	0.152534
$327$	−11.3395	−0.627076
$328$	22.9813	1.26893
$329$	−3.48337	−0.192044
$330$	0	0
$331$	26.3514	1.44841	0.724203	−	0.689587i	$-0.242208\pi$
$331$	26.3514	1.44841	0.724203	+	0.689587i	$0.242208\pi$
$332$	9.07979	0.498318
$333$	−7.37232	−0.404000
$334$	3.43802	0.188120
$335$	0	0
$336$	−2.19531	−0.119764
$337$	10.6948	0.582584	0.291292	−	0.956634i	$-0.405915\pi$
$337$	10.6948	0.582584	0.291292	+	0.956634i	$0.405915\pi$
$338$	3.83731	0.208722
$339$	4.83520	0.262612
$340$	0	0
$341$	36.8918	1.99780
$342$	−4.55739	−0.246436
$343$	14.2108	0.767311
$344$	28.4493	1.53388
$345$	0	0
$346$	−12.6371	−0.679373
$347$	−16.1740	−0.868264	−0.434132	−	0.900849i	$-0.642945\pi$
$347$	−16.1740	−0.868264	−0.434132	+	0.900849i	$0.642945\pi$
$348$	−1.67296	−0.0896800
$349$	15.2383	0.815688	0.407844	−	0.913052i	$-0.366281\pi$
$349$	15.2383	0.815688	0.407844	+	0.913052i	$0.366281\pi$
$350$	0	0
$351$	−4.05204	−0.216282
$352$	14.4563	0.770522
$353$	29.5383	1.57217	0.786084	−	0.618120i	$-0.212106\pi$
$353$	29.5383	1.57217	0.786084	+	0.618120i	$0.212106\pi$
$354$	7.40046	0.393330
$355$	0	0
$356$	11.3188	0.599896
$357$	−2.36372	−0.125101
$358$	11.2877	0.596574
$359$	−13.1668	−0.694915	−0.347457	−	0.937696i	$-0.612955\pi$
$359$	−13.1668	−0.694915	−0.347457	+	0.937696i	$0.612955\pi$
$360$	0	0
$361$	−2.51120	−0.132168
$362$	−21.8501	−1.14842
$363$	−2.46946	−0.129613
$364$	3.34120	0.175127
$365$	0	0
$366$	10.2178	0.534094
$367$	3.88895	0.203001	0.101501	−	0.994835i	$-0.467636\pi$
$367$	3.88895	0.203001	0.101501	+	0.994835i	$0.467636\pi$
$368$	12.1704	0.634428
$369$	−7.47214	−0.388984
$370$	0	0
$371$	−3.90878	−0.202934
$372$	7.44220	0.385860
$373$	8.14326	0.421642	0.210821	−	0.977525i	$-0.432386\pi$
$373$	8.14326	0.421642	0.210821	+	0.977525i	$0.432386\pi$
$374$	−8.74200	−0.452038
$375$	0	0
$376$	−9.61941	−0.496083
$377$	−9.15613	−0.471564
$378$	1.24998	0.0642921
$379$	9.90720	0.508898	0.254449	−	0.967086i	$-0.418106\pi$
$379$	9.90720	0.508898	0.254449	+	0.967086i	$0.418106\pi$
$380$	0	0
$381$	0.759596	0.0389153
$382$	−13.0292	−0.666632
$383$	5.22215	0.266840	0.133420	−	0.991060i	$-0.457404\pi$
$383$	5.22215	0.266840	0.133420	+	0.991060i	$0.457404\pi$
$384$	−1.50823	−0.0769668
$385$	0	0
$386$	3.79524	0.193173
$387$	−9.24998	−0.470203
$388$	9.99493	0.507416
$389$	−3.72974	−0.189105	−0.0945526	−	0.995520i	$-0.530142\pi$
$389$	−3.72974	−0.189105	−0.0945526	+	0.995520i	$0.530142\pi$
$390$	0	0
$391$	13.1041	0.662702
$392$	17.7142	0.894705
$393$	−14.6551	−0.739253
$394$	19.7397	0.994473
$395$	0	0
$396$	−2.71720	−0.136545
$397$	−4.01562	−0.201538	−0.100769	−	0.994910i	$-0.532130\pi$
$397$	−4.01562	−0.201538	−0.100769	+	0.994910i	$0.532130\pi$
$398$	22.4581	1.12573
$399$	−4.52248	−0.226407
$400$	0	0
$401$	−24.9890	−1.24789	−0.623945	−	0.781468i	$-0.714471\pi$
$401$	−24.9890	−1.24789	−0.623945	+	0.781468i	$0.714471\pi$
$402$	2.94796	0.147031
$403$	40.7313	2.02897
$404$	−5.35020	−0.266183
$405$	0	0
$406$	2.82450	0.140178
$407$	−27.0570	−1.34116
$408$	−6.52746	−0.323158
$409$	−25.3768	−1.25480	−0.627401	−	0.778697i	$-0.715881\pi$
$409$	−25.3768	−1.25480	−0.627401	+	0.778697i	$0.715881\pi$
$410$	0	0
$411$	12.2914	0.606290
$412$	3.90728	0.192498
$413$	7.34376	0.361363
$414$	−6.92971	−0.340577
$415$	0	0
$416$	15.9608	0.782544
$417$	−14.7340	−0.721527
$418$	−16.7260	−0.818094
$419$	−30.8219	−1.50575	−0.752873	−	0.658165i	$-0.771333\pi$
$419$	−30.8219	−1.50575	−0.752873	+	0.658165i	$0.771333\pi$
$420$	0	0
$421$	8.54649	0.416530	0.208265	−	0.978072i	$-0.433218\pi$
$421$	8.54649	0.416530	0.208265	+	0.978072i	$0.433218\pi$
$422$	2.74142	0.133450
$423$	3.12765	0.152071
$424$	−10.7942	−0.524212
$425$	0	0
$426$	−0.741668	−0.0359339
$427$	10.1395	0.490686
$428$	4.04746	0.195641
$429$	−14.8713	−0.717993
$430$	0	0
$431$	−26.3815	−1.27075	−0.635376	−	0.772203i	$-0.719155\pi$
$431$	−26.3815	−1.27075	−0.635376	+	0.772203i	$0.719155\pi$
$432$	1.97112	0.0948356
$433$	9.37272	0.450424	0.225212	−	0.974310i	$-0.427693\pi$
$433$	9.37272	0.450424	0.225212	+	0.974310i	$0.427693\pi$
$434$	−12.5649	−0.603133
$435$	0	0
$436$	−8.39540	−0.402067
$437$	25.0719	1.19935
$438$	8.38991	0.400886
$439$	−0.515980	−0.0246264	−0.0123132	−	0.999924i	$-0.503920\pi$
$439$	−0.515980	−0.0246264	−0.0123132	+	0.999924i	$0.503920\pi$
$440$	0	0
$441$	−5.75960	−0.274266
$442$	−9.65183	−0.459091
$443$	17.8348	0.847357	0.423678	−	0.905813i	$-0.360739\pi$
$443$	17.8348	0.847357	0.423678	+	0.905813i	$0.360739\pi$
$444$	−5.45822	−0.259036
$445$	0	0
$446$	28.7731	1.36245
$447$	−15.6498	−0.740210
$448$	−9.31423	−0.440056
$449$	−4.16533	−0.196574	−0.0982870	−	0.995158i	$-0.531336\pi$
$449$	−4.16533	−0.196574	−0.0982870	+	0.995158i	$0.531336\pi$
$450$	0	0
$451$	−27.4233	−1.29131
$452$	3.57983	0.168381
$453$	−3.95819	−0.185972
$454$	−7.36628	−0.345716
$455$	0	0
$456$	−12.4889	−0.584848
$457$	17.6734	0.826725	0.413362	−	0.910567i	$-0.364354\pi$
$457$	17.6734	0.826725	0.413362	+	0.910567i	$0.364354\pi$
$458$	−8.93815	−0.417652
$459$	2.12233	0.0990620
$460$	0	0
$461$	−0.204956	−0.00954577	−0.00477288	−	0.999989i	$-0.501519\pi$
$461$	−0.204956	−0.00954577	−0.00477288	+	0.999989i	$0.501519\pi$
$462$	4.58753	0.213431
$463$	12.8176	0.595682	0.297841	−	0.954615i	$-0.403733\pi$
$463$	12.8176	0.595682	0.297841	+	0.954615i	$0.403733\pi$
$464$	4.45401	0.206772
$465$	0	0
$466$	10.5736	0.489812
$467$	−1.11373	−0.0515375	−0.0257687	−	0.999668i	$-0.508203\pi$
$467$	−1.11373	−0.0515375	−0.0257687	+	0.999668i	$0.508203\pi$
$468$	−3.00000	−0.138675
$469$	2.92537	0.135081
$470$	0	0
$471$	−4.50061	−0.207377
$472$	20.2800	0.933462
$473$	−33.9481	−1.56094
$474$	−9.58149	−0.440092
$475$	0	0
$476$	−1.75002	−0.0802120
$477$	3.50961	0.160694
$478$	−12.5961	−0.576132
$479$	−25.1750	−1.15027	−0.575136	−	0.818057i	$-0.695051\pi$
$479$	−25.1750	−1.15027	−0.575136	+	0.818057i	$0.695051\pi$
$480$	0	0
$481$	−29.8729	−1.36209
$482$	25.4555	1.15947
$483$	−6.87661	−0.312897
$484$	−1.82830	−0.0831048
$485$	0	0
$486$	−1.12233	−0.0509101
$487$	−10.3598	−0.469447	−0.234723	−	0.972062i	$-0.575418\pi$
$487$	−10.3598	−0.469447	−0.234723	+	0.972062i	$0.575418\pi$
$488$	28.0006	1.26753
$489$	−2.45389	−0.110969
$490$	0	0
$491$	−15.6571	−0.706595	−0.353297	−	0.935511i	$-0.614940\pi$
$491$	−15.6571	−0.706595	−0.353297	+	0.935511i	$0.614940\pi$
$492$	−5.53213	−0.249408
$493$	4.79569	0.215987
$494$	−18.4667	−0.830858
$495$	0	0
$496$	−19.8138	−0.889665
$497$	−0.735985	−0.0330135
$498$	13.7642	0.616788
$499$	35.7864	1.60202	0.801010	−	0.598651i	$-0.204296\pi$
$499$	35.7864	1.60202	0.801010	+	0.598651i	$0.204296\pi$
$500$	0	0
$501$	−3.06328	−0.136857
$502$	−7.59617	−0.339034
$503$	11.7791	0.525203	0.262601	−	0.964904i	$-0.415420\pi$
$503$	11.7791	0.525203	0.262601	+	0.964904i	$0.415420\pi$
$504$	3.42541	0.152580
$505$	0	0
$506$	−25.4326	−1.13061
$507$	−3.41904	−0.151845
$508$	0.562380	0.0249516
$509$	33.6507	1.49154	0.745771	−	0.666203i	$-0.232082\pi$
$509$	33.6507	1.49154	0.745771	+	0.666203i	$0.232082\pi$
$510$	0	0
$511$	8.32563	0.368304
$512$	−19.8889	−0.878976
$513$	4.06064	0.179282
$514$	16.6031	0.732332
$515$	0	0
$516$	−6.84839	−0.301483
$517$	11.4787	0.504833
$518$	9.21526	0.404895
$519$	11.2596	0.494243
$520$	0	0
$521$	−11.8448	−0.518929	−0.259465	−	0.965753i	$-0.583546\pi$
$521$	−11.8448	−0.518929	−0.259465	+	0.965753i	$0.583546\pi$
$522$	−2.53606	−0.111000
$523$	−4.09694	−0.179147	−0.0895733	−	0.995980i	$-0.528550\pi$
$523$	−4.09694	−0.179147	−0.0895733	+	0.995980i	$0.528550\pi$
$524$	−10.8502	−0.473992
$525$	0	0
$526$	−22.4117	−0.977195
$527$	−21.3338	−0.929314
$528$	7.23416	0.314827
$529$	15.1229	0.657518
$530$	0	0
$531$	−6.59382	−0.286147
$532$	−3.34829	−0.145167
$533$	−30.2774	−1.31146
$534$	17.1584	0.742515
$535$	0	0
$536$	8.07849	0.348938
$537$	−10.0574	−0.434007
$538$	−11.4906	−0.495393
$539$	−21.1382	−0.910485
$540$	0	0
$541$	13.9582	0.600109	0.300055	−	0.953922i	$-0.402995\pi$
$541$	13.9582	0.600109	0.300055	+	0.953922i	$0.402995\pi$
$542$	13.8132	0.593326
$543$	19.4684	0.835471
$544$	−8.35978	−0.358423
$545$	0	0
$546$	5.06498	0.216761
$547$	26.5045	1.13325	0.566625	−	0.823976i	$-0.308249\pi$
$547$	26.5045	1.13325	0.566625	+	0.823976i	$0.308249\pi$
$548$	9.10015	0.388739
$549$	−9.10408	−0.388553
$550$	0	0
$551$	9.17556	0.390892
$552$	−18.9899	−0.808266
$553$	−9.50808	−0.404325
$554$	−35.4119	−1.50451
$555$	0	0
$556$	−10.9086	−0.462627
$557$	−10.6860	−0.452781	−0.226391	−	0.974037i	$-0.572693\pi$
$557$	−10.6860	−0.452781	−0.226391	+	0.974037i	$0.572693\pi$
$558$	11.2817	0.477594
$559$	−37.4813	−1.58529
$560$	0	0
$561$	7.78912	0.328857
$562$	−23.9191	−1.00897
$563$	25.5750	1.07786	0.538928	−	0.842352i	$-0.318829\pi$
$563$	25.5750	1.07786	0.538928	+	0.842352i	$0.318829\pi$
$564$	2.31561	0.0975047
$565$	0	0
$566$	0.970149	0.0407784
$567$	−1.11373	−0.0467725
$568$	−2.03244	−0.0852794
$569$	6.10210	0.255813	0.127907	−	0.991786i	$-0.459174\pi$
$569$	6.10210	0.255813	0.127907	+	0.991786i	$0.459174\pi$
$570$	0	0
$571$	23.7396	0.993473	0.496737	−	0.867901i	$-0.334532\pi$
$571$	23.7396	0.993473	0.496737	+	0.867901i	$0.334532\pi$
$572$	−11.0102	−0.460361
$573$	11.6090	0.484974
$574$	9.34003	0.389845
$575$	0	0
$576$	8.36307	0.348461
$577$	44.1639	1.83857	0.919284	−	0.393595i	$-0.128769\pi$
$577$	44.1639	1.83857	0.919284	+	0.393595i	$0.128769\pi$
$578$	−14.0243	−0.583336
$579$	−3.38156	−0.140533
$580$	0	0
$581$	13.6587	0.566659
$582$	15.1515	0.628048
$583$	12.8805	0.533458
$584$	22.9915	0.951393
$585$	0	0
$586$	−3.56566	−0.147296
$587$	−0.511966	−0.0211311	−0.0105656	−	0.999944i	$-0.503363\pi$
$587$	−0.511966	−0.0211311	−0.0105656	+	0.999944i	$0.503363\pi$
$588$	−4.26422	−0.175853
$589$	−40.8177	−1.68187
$590$	0	0
$591$	−17.5881	−0.723478
$592$	14.5317	0.597250
$593$	18.8405	0.773687	0.386844	−	0.922145i	$-0.373565\pi$
$593$	18.8405	0.773687	0.386844	+	0.922145i	$0.373565\pi$
$594$	−4.11905	−0.169007
$595$	0	0
$596$	−11.5866	−0.474606
$597$	−20.0102	−0.818964
$598$	−28.0795	−1.14825
$599$	6.20712	0.253616	0.126808	−	0.991927i	$-0.459527\pi$
$599$	6.20712	0.253616	0.126808	+	0.991927i	$0.459527\pi$
$600$	0	0
$601$	34.0303	1.38813	0.694063	−	0.719915i	$-0.255819\pi$
$601$	34.0303	1.38813	0.694063	+	0.719915i	$0.255819\pi$
$602$	11.5623	0.471244
$603$	−2.62663	−0.106965
$604$	−2.93051	−0.119241
$605$	0	0
$606$	−8.11045	−0.329465
$607$	−16.2488	−0.659518	−0.329759	−	0.944065i	$-0.606968\pi$
$607$	−16.2488	−0.659518	−0.329759	+	0.944065i	$0.606968\pi$
$608$	−15.9947	−0.648670
$609$	−2.51663	−0.101979
$610$	0	0
$611$	12.6734	0.512709
$612$	1.57131	0.0635163
$613$	0.0679852	0.00274590	0.00137295	−	0.999999i	$-0.499563\pi$
$613$	0.0679852	0.00274590	0.00137295	+	0.999999i	$0.499563\pi$
$614$	0.570130	0.0230086
$615$	0	0
$616$	12.5715	0.506521
$617$	−32.8108	−1.32091	−0.660456	−	0.750864i	$-0.729637\pi$
$617$	−32.8108	−1.32091	−0.660456	+	0.750864i	$0.729637\pi$
$618$	5.92309	0.238262
$619$	−6.77712	−0.272395	−0.136198	−	0.990682i	$-0.543488\pi$
$619$	−6.77712	−0.272395	−0.136198	+	0.990682i	$0.543488\pi$
$620$	0	0
$621$	6.17438	0.247769
$622$	−16.4353	−0.658994
$623$	17.0269	0.682168
$624$	7.98706	0.319738
$625$	0	0
$626$	16.7858	0.670895
$627$	14.9029	0.595163
$628$	−3.33211	−0.132966
$629$	15.6465	0.623867
$630$	0	0
$631$	−7.66797	−0.305257	−0.152629	−	0.988284i	$-0.548774\pi$
$631$	−7.66797	−0.305257	−0.152629	+	0.988284i	$0.548774\pi$
$632$	−26.2568	−1.04444
$633$	−2.44261	−0.0970849
$634$	−20.3356	−0.807630
$635$	0	0
$636$	2.59840	0.103033
$637$	−23.3381	−0.924690
$638$	−9.30754	−0.368489
$639$	0.660827	0.0261419
$640$	0	0
$641$	−23.6911	−0.935744	−0.467872	−	0.883796i	$-0.654979\pi$
$641$	−23.6911	−0.935744	−0.467872	+	0.883796i	$0.654979\pi$
$642$	6.13560	0.242153
$643$	−2.16861	−0.0855218	−0.0427609	−	0.999085i	$-0.513615\pi$
$643$	−2.16861	−0.0855218	−0.0427609	+	0.999085i	$0.513615\pi$
$644$	−5.09122	−0.200622
$645$	0	0
$646$	9.67231	0.380552
$647$	−2.55541	−0.100463	−0.0502317	−	0.998738i	$-0.515996\pi$
$647$	−2.55541	−0.100463	−0.0502317	+	0.998738i	$0.515996\pi$
$648$	−3.07561	−0.120821
$649$	−24.1998	−0.949926
$650$	0	0
$651$	11.1953	0.438779
$652$	−1.81678	−0.0711505
$653$	−32.4137	−1.26845	−0.634224	−	0.773150i	$-0.718680\pi$
$653$	−32.4137	−1.26845	−0.634224	+	0.773150i	$0.718680\pi$
$654$	−12.7267	−0.497653
$655$	0	0
$656$	14.7285	0.575051
$657$	−7.47542	−0.291644
$658$	−3.90950	−0.152408
$659$	0.449951	0.0175276	0.00876381	−	0.999962i	$-0.497210\pi$
$659$	0.449951	0.0175276	0.00876381	+	0.999962i	$0.497210\pi$
$660$	0	0
$661$	−26.9827	−1.04951	−0.524753	−	0.851254i	$-0.675842\pi$
$661$	−26.9827	−1.04951	−0.524753	+	0.851254i	$0.675842\pi$
$662$	29.5751	1.14947
$663$	8.59978	0.333988
$664$	37.7189	1.46378
$665$	0	0
$666$	−8.27420	−0.320619
$667$	13.9518	0.540217
$668$	−2.26795	−0.0877496
$669$	−25.6369	−0.991178
$670$	0	0
$671$	−33.4127	−1.28988
$672$	4.38695	0.169230
$673$	−17.6224	−0.679292	−0.339646	−	0.940553i	$-0.610307\pi$
$673$	−17.6224	−0.679292	−0.339646	+	0.940553i	$0.610307\pi$
$674$	12.0032	0.462344
$675$	0	0
$676$	−2.53135	−0.0973595
$677$	19.3484	0.743621	0.371810	−	0.928309i	$-0.378737\pi$
$677$	19.3484	0.743621	0.371810	+	0.928309i	$0.378737\pi$
$678$	5.42671	0.208412
$679$	15.0354	0.577005
$680$	0	0
$681$	6.56336	0.251508
$682$	41.4049	1.58547
$683$	29.9460	1.14585	0.572926	−	0.819607i	$-0.305808\pi$
$683$	29.9460	1.14585	0.572926	+	0.819607i	$0.305808\pi$
$684$	3.00637	0.114951
$685$	0	0
$686$	15.9493	0.608946
$687$	7.96390	0.303842
$688$	18.2328	0.695120
$689$	14.2211	0.541781
$690$	0	0
$691$	−19.9349	−0.758361	−0.379181	−	0.925323i	$-0.623794\pi$
$691$	−19.9349	−0.758361	−0.379181	+	0.925323i	$0.623794\pi$
$692$	8.33627	0.316897
$693$	−4.08749	−0.155271
$694$	−18.1526	−0.689063
$695$	0	0
$696$	−6.94974	−0.263429
$697$	15.8584	0.600678
$698$	17.1025	0.647337
$699$	−9.42107	−0.356338
$700$	0	0
$701$	−49.0150	−1.85127	−0.925636	−	0.378415i	$-0.876469\pi$
$701$	−49.0150	−1.85127	−0.925636	+	0.378415i	$0.876469\pi$
$702$	−4.54774	−0.171643
$703$	29.9363	1.12907
$704$	30.6931	1.15679
$705$	0	0
$706$	33.1519	1.24769
$707$	−8.04831	−0.302688
$708$	−4.88185	−0.183471
$709$	−4.81347	−0.180774	−0.0903868	−	0.995907i	$-0.528810\pi$
$709$	−4.81347	−0.180774	−0.0903868	+	0.995907i	$0.528810\pi$
$710$	0	0
$711$	8.53711	0.320167
$712$	47.0202	1.76216
$713$	−62.0651	−2.32436
$714$	−2.65288	−0.0992815
$715$	0	0
$716$	−7.44614	−0.278275
$717$	11.2231	0.419136
$718$	−14.7775	−0.551491
$719$	32.2512	1.20277	0.601383	−	0.798961i	$-0.294617\pi$
$719$	32.2512	1.20277	0.601383	+	0.798961i	$0.294617\pi$
$720$	0	0
$721$	5.87771	0.218897
$722$	−2.81840	−0.104890
$723$	−22.6809	−0.843512
$724$	14.4138	0.535685
$725$	0	0
$726$	−2.77155	−0.102862
$727$	29.1747	1.08203	0.541015	−	0.841013i	$-0.318040\pi$
$727$	29.1747	1.08203	0.541015	+	0.841013i	$0.318040\pi$
$728$	13.8799	0.514423
$729$	1.00000	0.0370370
$730$	0	0
$731$	19.6315	0.726099
$732$	−6.74037	−0.249131
$733$	−5.20102	−0.192104	−0.0960521	−	0.995376i	$-0.530622\pi$
$733$	−5.20102	−0.192104	−0.0960521	+	0.995376i	$0.530622\pi$
$734$	4.36469	0.161104
$735$	0	0
$736$	−24.3206	−0.896469
$737$	−9.63994	−0.355092
$738$	−8.38623	−0.308701
$739$	−16.3206	−0.600363	−0.300182	−	0.953882i	$-0.597047\pi$
$739$	−16.3206	−0.600363	−0.300182	+	0.953882i	$0.597047\pi$
$740$	0	0
$741$	16.4539	0.604449
$742$	−4.38695	−0.161050
$743$	−53.4487	−1.96084	−0.980421	−	0.196914i	$-0.936908\pi$
$743$	−53.4487	−1.96084	−0.980421	+	0.196914i	$0.936908\pi$
$744$	30.9161	1.13344
$745$	0	0
$746$	9.13946	0.334619
$747$	−12.2639	−0.448712
$748$	5.76681	0.210856
$749$	6.08859	0.222472
$750$	0	0
$751$	−0.566970	−0.0206890	−0.0103445	−	0.999946i	$-0.503293\pi$
$751$	−0.566970	−0.0206890	−0.0103445	+	0.999946i	$0.503293\pi$
$752$	−6.16497	−0.224813
$753$	6.76819	0.246647
$754$	−10.2762	−0.374238
$755$	0	0
$756$	−0.824573	−0.0299894
$757$	−53.0708	−1.92889	−0.964445	−	0.264282i	$-0.914865\pi$
$757$	−53.0708	−1.92889	−0.964445	+	0.264282i	$0.914865\pi$
$758$	11.1192	0.403867
$759$	22.6604	0.822521
$760$	0	0
$761$	−27.5056	−0.997077	−0.498539	−	0.866867i	$-0.666130\pi$
$761$	−27.5056	−0.997077	−0.498539	+	0.866867i	$0.666130\pi$
$762$	0.852520	0.0308835
$763$	−12.6292	−0.457208
$764$	8.59495	0.310954
$765$	0	0
$766$	5.86100	0.211766
$767$	−26.7184	−0.964747
$768$	15.0334	0.542471
$769$	14.9716	0.539889	0.269944	−	0.962876i	$-0.412995\pi$
$769$	14.9716	0.539889	0.269944	+	0.962876i	$0.412995\pi$
$770$	0	0
$771$	−14.7934	−0.532771
$772$	−2.50360	−0.0901065
$773$	2.05427	0.0738871	0.0369436	−	0.999317i	$-0.488238\pi$
$773$	2.05427	0.0738871	0.0369436	+	0.999317i	$0.488238\pi$
$774$	−10.3816	−0.373158
$775$	0	0
$776$	41.5206	1.49050
$777$	−8.21080	−0.294561
$778$	−4.18601	−0.150076
$779$	30.3417	1.08710
$780$	0	0
$781$	2.42528	0.0867835
$782$	14.7072	0.525927
$783$	2.25963	0.0807527
$784$	11.3529	0.405459
$785$	0	0
$786$	−16.4479	−0.586678
$787$	16.8269	0.599815	0.299908	−	0.953968i	$-0.403044\pi$
$787$	16.8269	0.599815	0.299908	+	0.953968i	$0.403044\pi$
$788$	−13.0217	−0.463877
$789$	19.9688	0.710909
$790$	0	0
$791$	5.38513	0.191473
$792$	−11.2877	−0.401091
$793$	−36.8901	−1.31001
$794$	−4.50686	−0.159942
$795$	0	0
$796$	−14.8149	−0.525101
$797$	0.208891	0.00739930	0.00369965	−	0.999993i	$-0.498822\pi$
$797$	0.208891	0.00739930	0.00369965	+	0.999993i	$0.498822\pi$
$798$	−5.07573	−0.179679
$799$	−6.63791	−0.234832
$800$	0	0
$801$	−15.2881	−0.540179
$802$	−28.0460	−0.990337
$803$	−27.4354	−0.968173
$804$	−1.94467	−0.0685834
$805$	0	0
$806$	45.7141	1.61021
$807$	10.2381	0.360398
$808$	−22.2256	−0.781894
$809$	−16.9655	−0.596476	−0.298238	−	0.954491i	$-0.596399\pi$
$809$	−16.9655	−0.596476	−0.298238	+	0.954491i	$0.596399\pi$
$810$	0	0
$811$	−20.4684	−0.718742	−0.359371	−	0.933195i	$-0.617009\pi$
$811$	−20.4684	−0.718742	−0.359371	+	0.933195i	$0.617009\pi$
$812$	−1.86323	−0.0653866
$813$	−12.3075	−0.431644
$814$	−30.3669	−1.06436
$815$	0	0
$816$	−4.18338	−0.146447
$817$	37.5609	1.31409
$818$	−28.4812	−0.995822
$819$	−4.51290	−0.157693
$820$	0	0
$821$	38.3636	1.33890	0.669449	−	0.742858i	$-0.266530\pi$
$821$	38.3636	1.33890	0.669449	+	0.742858i	$0.266530\pi$
$822$	13.7950	0.481157
$823$	29.0098	1.01122	0.505609	−	0.862763i	$-0.331268\pi$
$823$	29.0098	1.01122	0.505609	+	0.862763i	$0.331268\pi$
$824$	16.2315	0.565449
$825$	0	0
$826$	8.24215	0.286781
$827$	−14.3567	−0.499233	−0.249616	−	0.968345i	$-0.580305\pi$
$827$	−14.3567	−0.499233	−0.249616	+	0.968345i	$0.580305\pi$
$828$	4.57131	0.158864
$829$	4.63563	0.161002	0.0805011	−	0.996755i	$-0.474348\pi$
$829$	4.63563	0.161002	0.0805011	+	0.996755i	$0.474348\pi$
$830$	0	0
$831$	31.5520	1.09453
$832$	33.8875	1.17484
$833$	12.2238	0.423529
$834$	−16.5365	−0.572611
$835$	0	0
$836$	11.0336	0.381605
$837$	−10.0520	−0.347449
$838$	−34.5924	−1.19498
$839$	4.92642	0.170079	0.0850395	−	0.996378i	$-0.472898\pi$
$839$	4.92642	0.170079	0.0850395	+	0.996378i	$0.472898\pi$
$840$	0	0
$841$	−23.8941	−0.823933
$842$	9.59201	0.330562
$843$	21.3119	0.734022
$844$	−1.80843	−0.0622486
$845$	0	0
$846$	3.51026	0.120685
$847$	−2.75032	−0.0945020
$848$	−6.91787	−0.237561
$849$	−0.864403	−0.0296662
$850$	0	0
$851$	45.5194	1.56039
$852$	0.489254	0.0167616
$853$	−53.6476	−1.83686	−0.918430	−	0.395584i	$-0.870542\pi$
$853$	−53.6476	−1.83686	−0.918430	+	0.395584i	$0.870542\pi$
$854$	11.3799	0.389413
$855$	0	0
$856$	16.8138	0.574684
$857$	27.1144	0.926210	0.463105	−	0.886303i	$-0.346735\pi$
$857$	27.1144	0.926210	0.463105	+	0.886303i	$0.346735\pi$
$858$	−16.6906	−0.569806
$859$	10.4190	0.355493	0.177747	−	0.984076i	$-0.443119\pi$
$859$	10.4190	0.355493	0.177747	+	0.984076i	$0.443119\pi$
$860$	0	0
$861$	−8.32198	−0.283612
$862$	−29.6088	−1.00848
$863$	43.6432	1.48563	0.742816	−	0.669495i	$-0.233490\pi$
$863$	43.6432	1.48563	0.742816	+	0.669495i	$0.233490\pi$
$864$	−3.93896	−0.134006
$865$	0	0
$866$	10.5193	0.357461
$867$	12.4957	0.424376
$868$	8.28864	0.281335
$869$	31.3319	1.06286
$870$	0	0
$871$	−10.6432	−0.360632
$872$	−34.8758	−1.18105
$873$	−13.5000	−0.456905
$874$	28.1391	0.951818
$875$	0	0
$876$	−5.53456	−0.186995
$877$	−53.6435	−1.81141	−0.905706	−	0.423907i	$-0.860658\pi$
$877$	−53.6435	−1.81141	−0.905706	+	0.423907i	$0.860658\pi$
$878$	−0.579102	−0.0195438
$879$	3.17701	0.107158
$880$	0	0
$881$	−54.8950	−1.84946	−0.924730	−	0.380623i	$-0.875710\pi$
$881$	−54.8950	−1.84946	−0.924730	+	0.380623i	$0.875710\pi$
$882$	−6.46419	−0.217661
$883$	19.0952	0.642603	0.321302	−	0.946977i	$-0.395880\pi$
$883$	19.0952	0.642603	0.321302	+	0.946977i	$0.395880\pi$
$884$	6.36700	0.214145
$885$	0	0
$886$	20.0166	0.672471
$887$	11.2162	0.376602	0.188301	−	0.982111i	$-0.439702\pi$
$887$	11.2162	0.376602	0.188301	+	0.982111i	$0.439702\pi$
$888$	−22.6743	−0.760901
$889$	0.845988	0.0283735
$890$	0	0
$891$	3.67008	0.122952
$892$	−18.9807	−0.635521
$893$	−12.7003	−0.424998
$894$	−17.5643	−0.587438
$895$	0	0
$896$	−1.67977	−0.0561173
$897$	25.0188	0.835354
$898$	−4.67489	−0.156003
$899$	−22.7139	−0.757552
$900$	0	0
$901$	−7.44857	−0.248148
$902$	−30.7781	−1.02480
$903$	−10.3020	−0.342830
$904$	14.8712	0.494608
$905$	0	0
$906$	−4.44240	−0.147589
$907$	36.4513	1.21034	0.605172	−	0.796095i	$-0.293104\pi$
$907$	36.4513	1.21034	0.605172	+	0.796095i	$0.293104\pi$
$908$	4.85930	0.161261
$909$	7.22642	0.239685
$910$	0	0
$911$	1.16533	0.0386091	0.0193045	−	0.999814i	$-0.493855\pi$
$911$	1.16533	0.0386091	0.0193045	+	0.999814i	$0.493855\pi$
$912$	−8.00401	−0.265039
$913$	−45.0094	−1.48959
$914$	19.8354	0.656097
$915$	0	0
$916$	5.89621	0.194816
$917$	−16.3219	−0.538997
$918$	2.38197	0.0786166
$919$	43.9475	1.44969	0.724847	−	0.688910i	$-0.241911\pi$
$919$	43.9475	1.44969	0.724847	+	0.688910i	$0.241911\pi$
$920$	0	0
$921$	−0.507986	−0.0167387
$922$	−0.230029	−0.00757561
$923$	2.67770	0.0881375
$924$	−3.02624	−0.0995561
$925$	0	0
$926$	14.3856	0.472739
$927$	−5.27748	−0.173335
$928$	−8.90060	−0.292176
$929$	31.2121	1.02404	0.512019	−	0.858974i	$-0.328898\pi$
$929$	31.2121	1.02404	0.512019	+	0.858974i	$0.328898\pi$
$930$	0	0
$931$	23.3876	0.766499
$932$	−6.97506	−0.228476
$933$	14.6438	0.479418
$934$	−1.24998	−0.0409006
$935$	0	0
$936$	−12.4625	−0.407349
$937$	45.1060	1.47355	0.736774	−	0.676139i	$-0.236348\pi$
$937$	45.1060	1.47355	0.736774	+	0.676139i	$0.236348\pi$
$938$	3.28324	0.107202
$939$	−14.9562	−0.488076
$940$	0	0
$941$	−39.1341	−1.27573	−0.637867	−	0.770146i	$-0.720183\pi$
$941$	−39.1341	−1.27573	−0.637867	+	0.770146i	$0.720183\pi$
$942$	−5.05119	−0.164577
$943$	46.1358	1.50239
$944$	12.9972	0.423023
$945$	0	0
$946$	−38.1011	−1.23877
$947$	48.8537	1.58753	0.793766	−	0.608223i	$-0.208117\pi$
$947$	48.8537	1.58753	0.793766	+	0.608223i	$0.208117\pi$
$948$	6.32060	0.205284
$949$	−30.2907	−0.983278
$950$	0	0
$951$	18.1190	0.587550
$952$	−7.26986	−0.235618
$953$	−22.7824	−0.737995	−0.368998	−	0.929430i	$-0.620299\pi$
$953$	−22.7824	−0.737995	−0.368998	+	0.929430i	$0.620299\pi$
$954$	3.93896	0.127528
$955$	0	0
$956$	8.30924	0.268740
$957$	8.29302	0.268075
$958$	−28.2547	−0.912868
$959$	13.6893	0.442052
$960$	0	0
$961$	70.0435	2.25947
$962$	−33.5274	−1.08097
$963$	−5.46682	−0.176166
$964$	−16.7922	−0.540841
$965$	0	0
$966$	−7.71785	−0.248318
$967$	−43.1927	−1.38898	−0.694492	−	0.719500i	$-0.744371\pi$
$967$	−43.1927	−1.38898	−0.694492	+	0.719500i	$0.744371\pi$
$968$	−7.59507	−0.244115
$969$	−8.61803	−0.276851
$970$	0	0
$971$	−32.0252	−1.02774	−0.513869	−	0.857869i	$-0.671788\pi$
$971$	−32.0252	−1.02774	−0.513869	+	0.857869i	$0.671788\pi$
$972$	0.740367	0.0237473
$973$	−16.4098	−0.526073
$974$	−11.6271	−0.372557
$975$	0	0
$976$	17.9452	0.574413
$977$	7.27011	0.232591	0.116296	−	0.993215i	$-0.462898\pi$
$977$	7.27011	0.232591	0.116296	+	0.993215i	$0.462898\pi$
$978$	−2.75408	−0.0880657
$979$	−56.1085	−1.79324
$980$	0	0
$981$	11.3395	0.362042
$982$	−17.5725	−0.560760
$983$	5.09155	0.162395	0.0811976	−	0.996698i	$-0.474126\pi$
$983$	5.09155	0.162395	0.0811976	+	0.996698i	$0.474126\pi$
$984$	−22.9813	−0.732619
$985$	0	0
$986$	5.38237	0.171410
$987$	3.48337	0.110877
$988$	12.1819	0.387559
$989$	57.1129	1.81608
$990$	0	0
$991$	−62.0762	−1.97192	−0.985958	−	0.166992i	$-0.946594\pi$
$991$	−62.0762	−1.97192	−0.985958	+	0.166992i	$0.946594\pi$
$992$	39.5946	1.25713
$993$	−26.3514	−0.836237
$994$	−0.826021	−0.0261998
$995$	0	0
$996$	−9.07979	−0.287704
$997$	−2.56895	−0.0813594	−0.0406797	−	0.999172i	$-0.512952\pi$
$997$	−2.56895	−0.0813594	−0.0406797	+	0.999172i	$0.512952\pi$
$998$	40.1643	1.27138
$999$	7.37232	0.233250

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1875.2.a.e.1.3		4
3.2	odd	2		5625.2.a.n.1.2		4
5.2	odd	4		1875.2.b.c.1249.5		8
5.3	odd	4		1875.2.b.c.1249.4		8
5.4	even	2		1875.2.a.h.1.2		4
15.14	odd	2		5625.2.a.i.1.3		4
25.2	odd	20		375.2.i.b.274.3		16
25.9	even	10		75.2.g.b.31.1	✓	8
25.11	even	5		375.2.g.b.226.2		8
25.12	odd	20		375.2.i.b.349.2		16
25.13	odd	20		375.2.i.b.349.3		16
25.14	even	10		75.2.g.b.46.1	yes	8
25.16	even	5		375.2.g.b.151.2		8
25.23	odd	20		375.2.i.b.274.2		16
75.14	odd	10		225.2.h.c.46.2		8
75.59	odd	10		225.2.h.c.181.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.2.g.b.31.1	✓	8	25.9	even	10
75.2.g.b.46.1	yes	8	25.14	even	10
225.2.h.c.46.2		8	75.14	odd	10
225.2.h.c.181.2		8	75.59	odd	10
375.2.g.b.151.2		8	25.16	even	5
375.2.g.b.226.2		8	25.11	even	5
375.2.i.b.274.2		16	25.23	odd	20
375.2.i.b.274.3		16	25.2	odd	20
375.2.i.b.349.2		16	25.12	odd	20
375.2.i.b.349.3		16	25.13	odd	20
1875.2.a.e.1.3		4	1.1	even	1	trivial
1875.2.a.h.1.2		4	5.4	even	2
1875.2.b.c.1249.4		8	5.3	odd	4
1875.2.b.c.1249.5		8	5.2	odd	4
5625.2.a.i.1.3		4	15.14	odd	2
5625.2.a.n.1.2		4	3.2	odd	2

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:	\( N \)	\(=\)	\( 1875 = 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1875.a (trivial)

\(f(q)\)	\(=\)	\(q+1.12233 q^{2} -1.00000 q^{3} -0.740367 q^{4} -1.12233 q^{6} -1.11373 q^{7} -3.07561 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.12233 q^{2} -1.00000 q^{3} -0.740367 q^{4} -1.12233 q^{6} -1.11373 q^{7} -3.07561 q^{8} +1.00000 q^{9} +3.67008 q^{11} +0.740367 q^{12} +4.05204 q^{13} -1.24998 q^{14} -1.97112 q^{16} -2.12233 q^{17} +1.12233 q^{18} -4.06064 q^{19} +1.11373 q^{21} +4.11905 q^{22} -6.17438 q^{23} +3.07561 q^{24} +4.54774 q^{26} -1.00000 q^{27} +0.824573 q^{28} -2.25963 q^{29} +10.0520 q^{31} +3.93896 q^{32} -3.67008 q^{33} -2.38197 q^{34} -0.740367 q^{36} -7.37232 q^{37} -4.55739 q^{38} -4.05204 q^{39} -7.47214 q^{41} +1.24998 q^{42} -9.24998 q^{43} -2.71720 q^{44} -6.92971 q^{46} +3.12765 q^{47} +1.97112 q^{48} -5.75960 q^{49} +2.12233 q^{51} -3.00000 q^{52} +3.50961 q^{53} -1.12233 q^{54} +3.42541 q^{56} +4.06064 q^{57} -2.53606 q^{58} -6.59382 q^{59} -9.10408 q^{61} +11.2817 q^{62} -1.11373 q^{63} +8.36307 q^{64} -4.11905 q^{66} -2.62663 q^{67} +1.57131 q^{68} +6.17438 q^{69} +0.660827 q^{71} -3.07561 q^{72} -7.47542 q^{73} -8.27420 q^{74} +3.00637 q^{76} -4.08749 q^{77} -4.54774 q^{78} +8.53711 q^{79} +1.00000 q^{81} -8.38623 q^{82} -12.2639 q^{83} -0.824573 q^{84} -10.3816 q^{86} +2.25963 q^{87} -11.2877 q^{88} -15.2881 q^{89} -4.51290 q^{91} +4.57131 q^{92} -10.0520 q^{93} +3.51026 q^{94} -3.93896 q^{96} -13.5000 q^{97} -6.46419 q^{98} +3.67008 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} - 4 q^{3} + 8 q^{4} + 2 q^{6} - 2 q^{7} - 15 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q - 2 * q^2 - 4 * q^3 + 8 * q^4 + 2 * q^6 - 2 * q^7 - 15 * q^8 + 4 * q^9	\( 4 q - 2 q^{2} - 4 q^{3} + 8 q^{4} + 2 q^{6} - 2 q^{7} - 15 q^{8} + 4 q^{9} - 7 q^{11} - 8 q^{12} - q^{13} + 16 q^{14} + 4 q^{16} - 2 q^{17} - 2 q^{18} + 5 q^{19} + 2 q^{21} + 6 q^{22} - q^{23} + 15 q^{24} + 3 q^{26} - 4 q^{27} - 9 q^{28} - 20 q^{29} + 23 q^{31} - 12 q^{32} + 7 q^{33} - 14 q^{34} + 8 q^{36} - 2 q^{37} - 35 q^{38} + q^{39} - 12 q^{41} - 16 q^{42} - 16 q^{43} - 29 q^{44} - 17 q^{46} - 2 q^{47} - 4 q^{48} + 8 q^{49} + 2 q^{51} - 12 q^{52} + 4 q^{53} + 2 q^{54} + 5 q^{56} - 5 q^{57} + 25 q^{58} - 15 q^{59} - 2 q^{61} - 9 q^{62} - 2 q^{63} + 23 q^{64} - 6 q^{66} - 2 q^{67} + 11 q^{68} + q^{69} - 2 q^{71} - 15 q^{72} - 16 q^{73} - 19 q^{74} + 40 q^{76} - 19 q^{77} - 3 q^{78} + 35 q^{79} + 4 q^{81} + 6 q^{82} - 16 q^{83} + 9 q^{84} + 3 q^{86} + 20 q^{87} + 30 q^{88} - 35 q^{89} - 12 q^{91} + 23 q^{92} - 23 q^{93} - 9 q^{94} + 12 q^{96} - 12 q^{97} + q^{98} - 7 q^{99}+O(q^{100}) \) 4 * q - 2 * q^2 - 4 * q^3 + 8 * q^4 + 2 * q^6 - 2 * q^7 - 15 * q^8 + 4 * q^9 - 7 * q^11 - 8 * q^12 - q^13 + 16 * q^14 + 4 * q^16 - 2 * q^17 - 2 * q^18 + 5 * q^19 + 2 * q^21 + 6 * q^22 - q^23 + 15 * q^24 + 3 * q^26 - 4 * q^27 - 9 * q^28 - 20 * q^29 + 23 * q^31 - 12 * q^32 + 7 * q^33 - 14 * q^34 + 8 * q^36 - 2 * q^37 - 35 * q^38 + q^39 - 12 * q^41 - 16 * q^42 - 16 * q^43 - 29 * q^44 - 17 * q^46 - 2 * q^47 - 4 * q^48 + 8 * q^49 + 2 * q^51 - 12 * q^52 + 4 * q^53 + 2 * q^54 + 5 * q^56 - 5 * q^57 + 25 * q^58 - 15 * q^59 - 2 * q^61 - 9 * q^62 - 2 * q^63 + 23 * q^64 - 6 * q^66 - 2 * q^67 + 11 * q^68 + q^69 - 2 * q^71 - 15 * q^72 - 16 * q^73 - 19 * q^74 + 40 * q^76 - 19 * q^77 - 3 * q^78 + 35 * q^79 + 4 * q^81 + 6 * q^82 - 16 * q^83 + 9 * q^84 + 3 * q^86 + 20 * q^87 + 30 * q^88 - 35 * q^89 - 12 * q^91 + 23 * q^92 - 23 * q^93 - 9 * q^94 + 12 * q^96 - 12 * q^97 + q^98 - 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more