LMFDB - Embedded newform 6422.2.a.l.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6422,2,Mod(1,6422)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6422, 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("6422.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6422 = 2 \cdot 13^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6422.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:	$51.2799281781$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 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			1.2
Root			$0.445042$ of defining polynomial
Character	$\chi$	$=$	6422.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -2.35690 q^{3} +1.00000 q^{4} +2.69202 q^{5} +2.35690 q^{6} -2.49396 q^{7} -1.00000 q^{8} +2.55496 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.35690 q^{3} +1.00000 q^{4} +2.69202 q^{5} +2.35690 q^{6} -2.49396 q^{7} -1.00000 q^{8} +2.55496 q^{9} -2.69202 q^{10} -3.60388 q^{11} -2.35690 q^{12} +2.49396 q^{14} -6.34481 q^{15} +1.00000 q^{16} -4.54288 q^{17} -2.55496 q^{18} +1.00000 q^{19} +2.69202 q^{20} +5.87800 q^{21} +3.60388 q^{22} -2.21983 q^{23} +2.35690 q^{24} +2.24698 q^{25} +1.04892 q^{27} -2.49396 q^{28} -6.98792 q^{29} +6.34481 q^{30} -6.45473 q^{31} -1.00000 q^{32} +8.49396 q^{33} +4.54288 q^{34} -6.71379 q^{35} +2.55496 q^{36} +8.71379 q^{37} -1.00000 q^{38} -2.69202 q^{40} -9.70171 q^{41} -5.87800 q^{42} -4.09783 q^{43} -3.60388 q^{44} +6.87800 q^{45} +2.21983 q^{46} -2.35690 q^{48} -0.780167 q^{49} -2.24698 q^{50} +10.7071 q^{51} -0.176292 q^{53} -1.04892 q^{54} -9.70171 q^{55} +2.49396 q^{56} -2.35690 q^{57} +6.98792 q^{58} -7.28382 q^{59} -6.34481 q^{60} +8.51573 q^{61} +6.45473 q^{62} -6.37196 q^{63} +1.00000 q^{64} -8.49396 q^{66} -8.05861 q^{67} -4.54288 q^{68} +5.23191 q^{69} +6.71379 q^{70} -1.28083 q^{71} -2.55496 q^{72} +13.9705 q^{73} -8.71379 q^{74} -5.29590 q^{75} +1.00000 q^{76} +8.98792 q^{77} +2.17092 q^{79} +2.69202 q^{80} -10.1371 q^{81} +9.70171 q^{82} -15.7017 q^{83} +5.87800 q^{84} -12.2295 q^{85} +4.09783 q^{86} +16.4698 q^{87} +3.60388 q^{88} +12.1957 q^{89} -6.87800 q^{90} -2.21983 q^{92} +15.2131 q^{93} +2.69202 q^{95} +2.35690 q^{96} +6.89008 q^{97} +0.780167 q^{98} -9.20775 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{5} + 3 q^{6} + 2 q^{7} - 3 q^{8} + 8 q^{9}+O(q^{10}) $ 3 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 + 3 * q^5 + 3 * q^6 + 2 * q^7 - 3 * q^8 + 8 * q^9	$ 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{5} + 3 q^{6} + 2 q^{7} - 3 q^{8} + 8 q^{9} - 3 q^{10} - 2 q^{11} - 3 q^{12} - 2 q^{14} + 4 q^{15} + 3 q^{16} + 5 q^{17} - 8 q^{18} + 3 q^{19} + 3 q^{20} - 2 q^{21} + 2 q^{22} - 8 q^{23} + 3 q^{24} + 2 q^{25} - 6 q^{27} + 2 q^{28} - 2 q^{29} - 4 q^{30} + 3 q^{31} - 3 q^{32} + 16 q^{33} - 5 q^{34} - 12 q^{35} + 8 q^{36} + 18 q^{37} - 3 q^{38} - 3 q^{40} - 2 q^{41} + 2 q^{42} + 6 q^{43} - 2 q^{44} + q^{45} + 8 q^{46} - 3 q^{48} - q^{49} - 2 q^{50} + 2 q^{51} - 8 q^{53} + 6 q^{54} - 2 q^{55} - 2 q^{56} - 3 q^{57} + 2 q^{58} + 11 q^{59} + 4 q^{60} + 13 q^{61} - 3 q^{62} + 10 q^{63} + 3 q^{64} - 16 q^{66} + 7 q^{67} + 5 q^{68} + 36 q^{69} + 12 q^{70} - 15 q^{71} - 8 q^{72} + 7 q^{73} - 18 q^{74} - 2 q^{75} + 3 q^{76} + 8 q^{77} + 17 q^{79} + 3 q^{80} - 25 q^{81} + 2 q^{82} - 20 q^{83} - 2 q^{84} - 16 q^{85} - 6 q^{86} + 2 q^{87} + 2 q^{88} - q^{90} - 8 q^{92} + 25 q^{93} + 3 q^{95} + 3 q^{96} + 20 q^{97} + q^{98} - 10 q^{99}+O(q^{100}) $ 3 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 + 3 * q^5 + 3 * q^6 + 2 * q^7 - 3 * q^8 + 8 * q^9 - 3 * q^10 - 2 * q^11 - 3 * q^12 - 2 * q^14 + 4 * q^15 + 3 * q^16 + 5 * q^17 - 8 * q^18 + 3 * q^19 + 3 * q^20 - 2 * q^21 + 2 * q^22 - 8 * q^23 + 3 * q^24 + 2 * q^25 - 6 * q^27 + 2 * q^28 - 2 * q^29 - 4 * q^30 + 3 * q^31 - 3 * q^32 + 16 * q^33 - 5 * q^34 - 12 * q^35 + 8 * q^36 + 18 * q^37 - 3 * q^38 - 3 * q^40 - 2 * q^41 + 2 * q^42 + 6 * q^43 - 2 * q^44 + q^45 + 8 * q^46 - 3 * q^48 - q^49 - 2 * q^50 + 2 * q^51 - 8 * q^53 + 6 * q^54 - 2 * q^55 - 2 * q^56 - 3 * q^57 + 2 * q^58 + 11 * q^59 + 4 * q^60 + 13 * q^61 - 3 * q^62 + 10 * q^63 + 3 * q^64 - 16 * q^66 + 7 * q^67 + 5 * q^68 + 36 * q^69 + 12 * q^70 - 15 * q^71 - 8 * q^72 + 7 * q^73 - 18 * q^74 - 2 * q^75 + 3 * q^76 + 8 * q^77 + 17 * q^79 + 3 * q^80 - 25 * q^81 + 2 * q^82 - 20 * q^83 - 2 * q^84 - 16 * q^85 - 6 * q^86 + 2 * q^87 + 2 * q^88 - q^90 - 8 * q^92 + 25 * q^93 + 3 * q^95 + 3 * q^96 + 20 * q^97 + q^98 - 10 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−2.35690	−1.36075	−0.680377	−	0.732862i	$-0.738184\pi$
$3$	−2.35690	−1.36075	−0.680377	+	0.732862i	$0.738184\pi$
$4$	1.00000	0.500000
$5$	2.69202	1.20391	0.601954	−	0.798531i	$-0.294389\pi$
$5$	2.69202	1.20391	0.601954	+	0.798531i	$0.294389\pi$
$6$	2.35690	0.962199
$7$	−2.49396	−0.942628	−0.471314	−	0.881965i	$-0.656220\pi$
$7$	−2.49396	−0.942628	−0.471314	+	0.881965i	$0.656220\pi$
$8$	−1.00000	−0.353553
$9$	2.55496	0.851653
$10$	−2.69202	−0.851292
$11$	−3.60388	−1.08661	−0.543305	−	0.839536i	$-0.682827\pi$
$11$	−3.60388	−1.08661	−0.543305	+	0.839536i	$0.682827\pi$
$12$	−2.35690	−0.680377
$13$	0	0
$13$	0	0
$14$	2.49396	0.666539
$15$	−6.34481	−1.63822
$16$	1.00000	0.250000
$17$	−4.54288	−1.10181	−0.550905	−	0.834568i	$-0.685717\pi$
$17$	−4.54288	−1.10181	−0.550905	+	0.834568i	$0.685717\pi$
$18$	−2.55496	−0.602209
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	2.69202	0.601954
$21$	5.87800	1.28269
$22$	3.60388	0.768349
$23$	−2.21983	−0.462867	−0.231434	−	0.972851i	$-0.574342\pi$
$23$	−2.21983	−0.462867	−0.231434	+	0.972851i	$0.574342\pi$
$24$	2.35690	0.481099
$25$	2.24698	0.449396
$26$	0	0
$27$	1.04892	0.201864
$28$	−2.49396	−0.471314
$29$	−6.98792	−1.29762	−0.648812	−	0.760949i	$-0.724734\pi$
$29$	−6.98792	−1.29762	−0.648812	+	0.760949i	$0.724734\pi$
$30$	6.34481	1.15840
$31$	−6.45473	−1.15930	−0.579652	−	0.814864i	$-0.696811\pi$
$31$	−6.45473	−1.15930	−0.579652	+	0.814864i	$0.696811\pi$
$32$	−1.00000	−0.176777
$33$	8.49396	1.47861
$34$	4.54288	0.779097
$35$	−6.71379	−1.13484
$36$	2.55496	0.425826
$37$	8.71379	1.43254	0.716269	−	0.697824i	$-0.245848\pi$
$37$	8.71379	1.43254	0.716269	+	0.697824i	$0.245848\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	−2.69202	−0.425646
$41$	−9.70171	−1.51515	−0.757576	−	0.652747i	$-0.773617\pi$
$41$	−9.70171	−1.51515	−0.757576	+	0.652747i	$0.773617\pi$
$42$	−5.87800	−0.906995
$43$	−4.09783	−0.624914	−0.312457	−	0.949932i	$-0.601152\pi$
$43$	−4.09783	−0.624914	−0.312457	+	0.949932i	$0.601152\pi$
$44$	−3.60388	−0.543305
$45$	6.87800	1.02531
$46$	2.21983	0.327296
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	−2.35690	−0.340189
$49$	−0.780167	−0.111452
$50$	−2.24698	−0.317771
$51$	10.7071	1.49929
$52$	0	0
$53$	−0.176292	−0.0242156	−0.0121078	−	0.999927i	$-0.503854\pi$
$53$	−0.176292	−0.0242156	−0.0121078	+	0.999927i	$0.503854\pi$
$54$	−1.04892	−0.142740
$55$	−9.70171	−1.30818
$56$	2.49396	0.333269
$57$	−2.35690	−0.312178
$58$	6.98792	0.917559
$59$	−7.28382	−0.948272	−0.474136	−	0.880452i	$-0.657239\pi$
$59$	−7.28382	−0.948272	−0.474136	+	0.880452i	$0.657239\pi$
$60$	−6.34481	−0.819112
$61$	8.51573	1.09033	0.545164	−	0.838330i	$-0.316467\pi$
$61$	8.51573	1.09033	0.545164	+	0.838330i	$0.316467\pi$
$62$	6.45473	0.819752
$63$	−6.37196	−0.802792
$64$	1.00000	0.125000
$65$	0	0
$66$	−8.49396	−1.04553
$67$	−8.05861	−0.984515	−0.492258	−	0.870450i	$-0.663828\pi$
$67$	−8.05861	−0.984515	−0.492258	+	0.870450i	$0.663828\pi$
$68$	−4.54288	−0.550905
$69$	5.23191	0.629848
$70$	6.71379	0.802452
$71$	−1.28083	−0.152007	−0.0760034	−	0.997108i	$-0.524216\pi$
$71$	−1.28083	−0.152007	−0.0760034	+	0.997108i	$0.524216\pi$
$72$	−2.55496	−0.301105
$73$	13.9705	1.63512	0.817559	−	0.575844i	$-0.195327\pi$
$73$	13.9705	1.63512	0.817559	+	0.575844i	$0.195327\pi$
$74$	−8.71379	−1.01296
$75$	−5.29590	−0.611518
$76$	1.00000	0.114708
$77$	8.98792	1.02427
$78$	0	0
$79$	2.17092	0.244247	0.122124	−	0.992515i	$-0.461030\pi$
$79$	2.17092	0.244247	0.122124	+	0.992515i	$0.461030\pi$
$80$	2.69202	0.300977
$81$	−10.1371	−1.12634
$82$	9.70171	1.07137
$83$	−15.7017	−1.72349	−0.861743	−	0.507345i	$-0.830627\pi$
$83$	−15.7017	−1.72349	−0.861743	+	0.507345i	$0.830627\pi$
$84$	5.87800	0.641343
$85$	−12.2295	−1.32648
$86$	4.09783	0.441881
$87$	16.4698	1.76575
$88$	3.60388	0.384174
$89$	12.1957	1.29274	0.646369	−	0.763025i	$-0.276287\pi$
$89$	12.1957	1.29274	0.646369	+	0.763025i	$0.276287\pi$
$90$	−6.87800	−0.725005
$91$	0	0
$92$	−2.21983	−0.231434
$93$	15.2131	1.57753
$94$	0	0
$95$	2.69202	0.276196
$96$	2.35690	0.240550
$97$	6.89008	0.699582	0.349791	−	0.936828i	$-0.386253\pi$
$97$	6.89008	0.699582	0.349791	+	0.936828i	$0.386253\pi$
$98$	0.780167	0.0788088
$99$	−9.20775	−0.925414
$100$	2.24698	0.224698
$101$	5.46011	0.543301	0.271650	−	0.962396i	$-0.412431\pi$
$101$	5.46011	0.543301	0.271650	+	0.962396i	$0.412431\pi$
$102$	−10.7071	−1.06016
$103$	8.17390	0.805398	0.402699	−	0.915332i	$-0.368072\pi$
$103$	8.17390	0.805398	0.402699	+	0.915332i	$0.368072\pi$
$104$	0	0
$105$	15.8237	1.54424
$106$	0.176292	0.0171230
$107$	5.25667	0.508181	0.254091	−	0.967180i	$-0.418224\pi$
$107$	5.25667	0.508181	0.254091	+	0.967180i	$0.418224\pi$
$108$	1.04892	0.100932
$109$	7.70171	0.737690	0.368845	−	0.929491i	$-0.379753\pi$
$109$	7.70171	0.737690	0.368845	+	0.929491i	$0.379753\pi$
$110$	9.70171	0.925022
$111$	−20.5375	−1.94933
$112$	−2.49396	−0.235657
$113$	−6.09783	−0.573636	−0.286818	−	0.957985i	$-0.592598\pi$
$113$	−6.09783	−0.573636	−0.286818	+	0.957985i	$0.592598\pi$
$114$	2.35690	0.220744
$115$	−5.97584	−0.557250
$116$	−6.98792	−0.648812
$117$	0	0
$118$	7.28382	0.670530
$119$	11.3297	1.03860
$120$	6.34481	0.579200
$121$	1.98792	0.180720
$122$	−8.51573	−0.770978
$123$	22.8659	2.06175
$124$	−6.45473	−0.579652
$125$	−7.41119	−0.662877
$126$	6.37196	0.567659
$127$	−21.6625	−1.92223	−0.961117	−	0.276141i	$-0.910944\pi$
$127$	−21.6625	−1.92223	−0.961117	+	0.276141i	$0.910944\pi$
$128$	−1.00000	−0.0883883
$129$	9.65817	0.850354
$130$	0	0
$131$	3.72587	0.325531	0.162766	−	0.986665i	$-0.447959\pi$
$131$	3.72587	0.325531	0.162766	+	0.986665i	$0.447959\pi$
$132$	8.49396	0.739304
$133$	−2.49396	−0.216254
$134$	8.05861	0.696158
$135$	2.82371	0.243026
$136$	4.54288	0.389548
$137$	10.7463	0.918120	0.459060	−	0.888405i	$-0.348186\pi$
$137$	10.7463	0.918120	0.459060	+	0.888405i	$0.348186\pi$
$138$	−5.23191	−0.445370
$139$	8.68963	0.737045	0.368522	−	0.929619i	$-0.379864\pi$
$139$	8.68963	0.737045	0.368522	+	0.929619i	$0.379864\pi$
$140$	−6.71379	−0.567419
$141$	0	0
$142$	1.28083	0.107485
$143$	0	0
$144$	2.55496	0.212913
$145$	−18.8116	−1.56222
$146$	−13.9705	−1.15620
$147$	1.83877	0.151659
$148$	8.71379	0.716269
$149$	−6.64071	−0.544028	−0.272014	−	0.962293i	$-0.587690\pi$
$149$	−6.64071	−0.544028	−0.272014	+	0.962293i	$0.587690\pi$
$150$	5.29590	0.432408
$151$	20.2446	1.64748	0.823741	−	0.566967i	$-0.191883\pi$
$151$	20.2446	1.64748	0.823741	+	0.566967i	$0.191883\pi$
$152$	−1.00000	−0.0811107
$153$	−11.6069	−0.938359
$154$	−8.98792	−0.724267
$155$	−17.3763	−1.39570
$156$	0	0
$157$	−19.5308	−1.55873	−0.779364	−	0.626572i	$-0.784457\pi$
$157$	−19.5308	−1.55873	−0.779364	+	0.626572i	$0.784457\pi$
$158$	−2.17092	−0.172709
$159$	0.415502	0.0329514
$160$	−2.69202	−0.212823
$161$	5.53617	0.436311
$162$	10.1371	0.796443
$163$	−0.944378	−0.0739694	−0.0369847	−	0.999316i	$-0.511775\pi$
$163$	−0.944378	−0.0739694	−0.0369847	+	0.999316i	$0.511775\pi$
$164$	−9.70171	−0.757576
$165$	22.8659	1.78011
$166$	15.7017	1.21869
$167$	9.54527	0.738635	0.369318	−	0.929303i	$-0.379591\pi$
$167$	9.54527	0.738635	0.369318	+	0.929303i	$0.379591\pi$
$168$	−5.87800	−0.453498
$169$	0	0
$170$	12.2295	0.937961
$171$	2.55496	0.195383
$172$	−4.09783	−0.312457
$173$	16.7681	1.27485	0.637427	−	0.770511i	$-0.279999\pi$
$173$	16.7681	1.27485	0.637427	+	0.770511i	$0.279999\pi$
$174$	−16.4698	−1.24857
$175$	−5.60388	−0.423613
$176$	−3.60388	−0.271652
$177$	17.1672	1.29037
$178$	−12.1957	−0.914104
$179$	−7.37867	−0.551507	−0.275754	−	0.961228i	$-0.588927\pi$
$179$	−7.37867	−0.551507	−0.275754	+	0.961228i	$0.588927\pi$
$180$	6.87800	0.512656
$181$	−22.7875	−1.69378	−0.846889	−	0.531769i	$-0.821527\pi$
$181$	−22.7875	−1.69378	−0.846889	+	0.531769i	$0.821527\pi$
$182$	0	0
$183$	−20.0707	−1.48367
$184$	2.21983	0.163648
$185$	23.4577	1.72465
$186$	−15.2131	−1.11548
$187$	16.3720	1.19724
$188$	0	0
$189$	−2.61596	−0.190283
$190$	−2.69202	−0.195300
$191$	−3.54958	−0.256839	−0.128419	−	0.991720i	$-0.540990\pi$
$191$	−3.54958	−0.256839	−0.128419	+	0.991720i	$0.540990\pi$
$192$	−2.35690	−0.170094
$193$	26.6896	1.92116	0.960581	−	0.278001i	$-0.0896719\pi$
$193$	26.6896	1.92116	0.960581	+	0.278001i	$0.0896719\pi$
$194$	−6.89008	−0.494679
$195$	0	0
$196$	−0.780167	−0.0557262
$197$	−10.1032	−0.719824	−0.359912	−	0.932986i	$-0.617193\pi$
$197$	−10.1032	−0.719824	−0.359912	+	0.932986i	$0.617193\pi$
$198$	9.20775	0.654366
$199$	−4.39612	−0.311633	−0.155817	−	0.987786i	$-0.549801\pi$
$199$	−4.39612	−0.311633	−0.155817	+	0.987786i	$0.549801\pi$
$200$	−2.24698	−0.158885
$201$	18.9933	1.33968
$202$	−5.46011	−0.384172
$203$	17.4276	1.22318
$204$	10.7071	0.749646
$205$	−26.1172	−1.82411
$206$	−8.17390	−0.569503
$207$	−5.67158	−0.394202
$208$	0	0
$209$	−3.60388	−0.249285
$210$	−15.8237	−1.09194
$211$	−9.12498	−0.628190	−0.314095	−	0.949392i	$-0.601701\pi$
$211$	−9.12498	−0.628190	−0.314095	+	0.949392i	$0.601701\pi$
$212$	−0.176292	−0.0121078
$213$	3.01879	0.206844
$214$	−5.25667	−0.359338
$215$	−11.0315	−0.752339
$216$	−1.04892	−0.0713698
$217$	16.0978	1.09279
$218$	−7.70171	−0.521626
$219$	−32.9269	−2.22500
$220$	−9.70171	−0.654089
$221$	0	0
$222$	20.5375	1.37839
$223$	7.70709	0.516105	0.258052	−	0.966131i	$-0.416919\pi$
$223$	7.70709	0.516105	0.258052	+	0.966131i	$0.416919\pi$
$224$	2.49396	0.166635
$225$	5.74094	0.382729
$226$	6.09783	0.405622
$227$	−3.86294	−0.256392	−0.128196	−	0.991749i	$-0.540919\pi$
$227$	−3.86294	−0.256392	−0.128196	+	0.991749i	$0.540919\pi$
$228$	−2.35690	−0.156089
$229$	−12.1196	−0.800886	−0.400443	−	0.916322i	$-0.631144\pi$
$229$	−12.1196	−0.800886	−0.400443	+	0.916322i	$0.631144\pi$
$230$	5.97584	0.394035
$231$	−21.1836	−1.39378
$232$	6.98792	0.458779
$233$	−17.8019	−1.16624	−0.583122	−	0.812385i	$-0.698169\pi$
$233$	−17.8019	−1.16624	−0.583122	+	0.812385i	$0.698169\pi$
$234$	0	0
$235$	0	0
$236$	−7.28382	−0.474136
$237$	−5.11662	−0.332360
$238$	−11.3297	−0.734399
$239$	−6.78746	−0.439044	−0.219522	−	0.975607i	$-0.570450\pi$
$239$	−6.78746	−0.439044	−0.219522	+	0.975607i	$0.570450\pi$
$240$	−6.34481	−0.409556
$241$	−10.8116	−0.696438	−0.348219	−	0.937413i	$-0.613213\pi$
$241$	−10.8116	−0.696438	−0.348219	+	0.937413i	$0.613213\pi$
$242$	−1.98792	−0.127788
$243$	20.7453	1.33081
$244$	8.51573	0.545164
$245$	−2.10023	−0.134179
$246$	−22.8659	−1.45788
$247$	0	0
$248$	6.45473	0.409876
$249$	37.0073	2.34524
$250$	7.41119	0.468725
$251$	−5.50604	−0.347538	−0.173769	−	0.984786i	$-0.555595\pi$
$251$	−5.50604	−0.347538	−0.173769	+	0.984786i	$0.555595\pi$
$252$	−6.37196	−0.401396
$253$	8.00000	0.502956
$254$	21.6625	1.35922
$255$	28.8237	1.80501
$256$	1.00000	0.0625000
$257$	14.9879	0.934921	0.467460	−	0.884014i	$-0.345169\pi$
$257$	14.9879	0.934921	0.467460	+	0.884014i	$0.345169\pi$
$258$	−9.65817	−0.601291
$259$	−21.7318	−1.35035
$260$	0	0
$261$	−17.8538	−1.10512
$262$	−3.72587	−0.230185
$263$	11.4276	0.704655	0.352327	−	0.935877i	$-0.385390\pi$
$263$	11.4276	0.704655	0.352327	+	0.935877i	$0.385390\pi$
$264$	−8.49396	−0.522767
$265$	−0.474582	−0.0291533
$266$	2.49396	0.152914
$267$	−28.7439	−1.75910
$268$	−8.05861	−0.492258
$269$	−22.4155	−1.36670	−0.683349	−	0.730092i	$-0.739477\pi$
$269$	−22.4155	−1.36670	−0.683349	+	0.730092i	$0.739477\pi$
$270$	−2.82371	−0.171845
$271$	−4.39612	−0.267046	−0.133523	−	0.991046i	$-0.542629\pi$
$271$	−4.39612	−0.267046	−0.133523	+	0.991046i	$0.542629\pi$
$272$	−4.54288	−0.275452
$273$	0	0
$274$	−10.7463	−0.649209
$275$	−8.09783	−0.488318
$276$	5.23191	0.314924
$277$	−8.71140	−0.523417	−0.261709	−	0.965147i	$-0.584286\pi$
$277$	−8.71140	−0.523417	−0.261709	+	0.965147i	$0.584286\pi$
$278$	−8.68963	−0.521169
$279$	−16.4916	−0.987324
$280$	6.71379	0.401226
$281$	22.3913	1.33576	0.667878	−	0.744271i	$-0.267203\pi$
$281$	22.3913	1.33576	0.667878	+	0.744271i	$0.267203\pi$
$282$	0	0
$283$	−16.7332	−0.994684	−0.497342	−	0.867555i	$-0.665691\pi$
$283$	−16.7332	−0.994684	−0.497342	+	0.867555i	$0.665691\pi$
$284$	−1.28083	−0.0760034
$285$	−6.34481	−0.375834
$286$	0	0
$287$	24.1957	1.42823
$288$	−2.55496	−0.150552
$289$	3.63773	0.213984
$290$	18.8116	1.10466
$291$	−16.2392	−0.951959
$292$	13.9705	0.817559
$293$	1.84787	0.107954	0.0539769	−	0.998542i	$-0.482810\pi$
$293$	1.84787	0.107954	0.0539769	+	0.998542i	$0.482810\pi$
$294$	−1.83877	−0.107239
$295$	−19.6082	−1.14163
$296$	−8.71379	−0.506479
$297$	−3.78017	−0.219348
$298$	6.64071	0.384686
$299$	0	0
$300$	−5.29590	−0.305759
$301$	10.2198	0.589061
$302$	−20.2446	−1.16495
$303$	−12.8689	−0.739299
$304$	1.00000	0.0573539
$305$	22.9245	1.31265
$306$	11.6069	0.663520
$307$	32.4198	1.85030	0.925148	−	0.379606i	$-0.123940\pi$
$307$	32.4198	1.85030	0.925148	+	0.379606i	$0.123940\pi$
$308$	8.98792	0.512134
$309$	−19.2650	−1.09595
$310$	17.3763	0.986906
$311$	1.37329	0.0778721	0.0389360	−	0.999242i	$-0.487603\pi$
$311$	1.37329	0.0778721	0.0389360	+	0.999242i	$0.487603\pi$
$312$	0	0
$313$	−31.2295	−1.76520	−0.882598	−	0.470128i	$-0.844208\pi$
$313$	−31.2295	−1.76520	−0.882598	+	0.470128i	$0.844208\pi$
$314$	19.5308	1.10219
$315$	−17.1535	−0.966488
$316$	2.17092	0.122124
$317$	12.6353	0.709671	0.354836	−	0.934929i	$-0.384537\pi$
$317$	12.6353	0.709671	0.354836	+	0.934929i	$0.384537\pi$
$318$	−0.415502	−0.0233002
$319$	25.1836	1.41001
$320$	2.69202	0.150489
$321$	−12.3894	−0.691510
$322$	−5.53617	−0.308519
$323$	−4.54288	−0.252772
$324$	−10.1371	−0.563170
$325$	0	0
$326$	0.944378	0.0523043
$327$	−18.1521	−1.00382
$328$	9.70171	0.535687
$329$	0	0
$330$	−22.8659	−1.25873
$331$	−25.6819	−1.41160	−0.705801	−	0.708410i	$-0.749413\pi$
$331$	−25.6819	−1.41160	−0.705801	+	0.708410i	$0.749413\pi$
$332$	−15.7017	−0.861743
$333$	22.2634	1.22003
$334$	−9.54527	−0.522294
$335$	−21.6939	−1.18527
$336$	5.87800	0.320671
$337$	20.6703	1.12598	0.562990	−	0.826464i	$-0.309651\pi$
$337$	20.6703	1.12598	0.562990	+	0.826464i	$0.309651\pi$
$338$	0	0
$339$	14.3720	0.780578
$340$	−12.2295	−0.663239
$341$	23.2620	1.25971
$342$	−2.55496	−0.138156
$343$	19.4034	1.04769
$344$	4.09783	0.220940
$345$	14.0844	0.758280
$346$	−16.7681	−0.901458
$347$	5.75600	0.308999	0.154499	−	0.987993i	$-0.450624\pi$
$347$	5.75600	0.308999	0.154499	+	0.987993i	$0.450624\pi$
$348$	16.4698	0.882874
$349$	25.0224	1.33942	0.669708	−	0.742624i	$-0.266419\pi$
$349$	25.0224	1.33942	0.669708	+	0.742624i	$0.266419\pi$
$350$	5.60388	0.299540
$351$	0	0
$352$	3.60388	0.192087
$353$	27.3099	1.45356	0.726780	−	0.686871i	$-0.241016\pi$
$353$	27.3099	1.45356	0.726780	+	0.686871i	$0.241016\pi$
$354$	−17.1672	−0.912426
$355$	−3.44803	−0.183002
$356$	12.1957	0.646369
$357$	−26.7030	−1.41327
$358$	7.37867	0.389975
$359$	−22.7922	−1.20293	−0.601464	−	0.798900i	$-0.705416\pi$
$359$	−22.7922	−1.20293	−0.601464	+	0.798900i	$0.705416\pi$
$360$	−6.87800	−0.362503
$361$	1.00000	0.0526316
$362$	22.7875	1.19768
$363$	−4.68532	−0.245915
$364$	0	0
$365$	37.6088	1.96853
$366$	20.0707	1.04911
$367$	−14.8358	−0.774422	−0.387211	−	0.921991i	$-0.626561\pi$
$367$	−14.8358	−0.774422	−0.387211	+	0.921991i	$0.626561\pi$
$368$	−2.21983	−0.115717
$369$	−24.7875	−1.29038
$370$	−23.4577	−1.21951
$371$	0.439665	0.0228263
$372$	15.2131	0.788764
$373$	22.0978	1.14418	0.572091	−	0.820190i	$-0.306132\pi$
$373$	22.0978	1.14418	0.572091	+	0.820190i	$0.306132\pi$
$374$	−16.3720	−0.846574
$375$	17.4674	0.902013
$376$	0	0
$377$	0	0
$378$	2.61596	0.134550
$379$	27.2078	1.39757	0.698784	−	0.715333i	$-0.253725\pi$
$379$	27.2078	1.39757	0.698784	+	0.715333i	$0.253725\pi$
$380$	2.69202	0.138098
$381$	51.0562	2.61569
$382$	3.54958	0.181612
$383$	10.0411	0.513079	0.256539	−	0.966534i	$-0.417418\pi$
$383$	10.0411	0.513079	0.256539	+	0.966534i	$0.417418\pi$
$384$	2.35690	0.120275
$385$	24.1957	1.23313
$386$	−26.6896	−1.35847
$387$	−10.4698	−0.532210
$388$	6.89008	0.349791
$389$	20.3672	1.03266	0.516328	−	0.856391i	$-0.327298\pi$
$389$	20.3672	1.03266	0.516328	+	0.856391i	$0.327298\pi$
$390$	0	0
$391$	10.0844	0.509991
$392$	0.780167	0.0394044
$393$	−8.78150	−0.442968
$394$	10.1032	0.508992
$395$	5.84415	0.294051
$396$	−9.20775	−0.462707
$397$	38.8442	1.94953	0.974766	−	0.223229i	$-0.0716598\pi$
$397$	38.8442	1.94953	0.974766	+	0.223229i	$0.0716598\pi$
$398$	4.39612	0.220358
$399$	5.87800	0.294268
$400$	2.24698	0.112349
$401$	26.8552	1.34108	0.670542	−	0.741872i	$-0.266062\pi$
$401$	26.8552	1.34108	0.670542	+	0.741872i	$0.266062\pi$
$402$	−18.9933	−0.947299
$403$	0	0
$404$	5.46011	0.271650
$405$	−27.2892	−1.35601
$406$	−17.4276	−0.864916
$407$	−31.4034	−1.55661
$408$	−10.7071	−0.530080
$409$	−38.1909	−1.88842	−0.944209	−	0.329347i	$-0.893171\pi$
$409$	−38.1909	−1.88842	−0.944209	+	0.329347i	$0.893171\pi$
$410$	26.1172	1.28984
$411$	−25.3279	−1.24934
$412$	8.17390	0.402699
$413$	18.1655	0.893868
$414$	5.67158	0.278743
$415$	−42.2693	−2.07492
$416$	0	0
$417$	−20.4805	−1.00294
$418$	3.60388	0.176271
$419$	−0.968541	−0.0473163	−0.0236582	−	0.999720i	$-0.507531\pi$
$419$	−0.968541	−0.0473163	−0.0236582	+	0.999720i	$0.507531\pi$
$420$	15.8237	0.772118
$421$	5.59312	0.272592	0.136296	−	0.990668i	$-0.456480\pi$
$421$	5.59312	0.272592	0.136296	+	0.990668i	$0.456480\pi$
$422$	9.12498	0.444197
$423$	0	0
$424$	0.176292	0.00856150
$425$	−10.2078	−0.495149
$426$	−3.01879	−0.146261
$427$	−21.2379	−1.02777
$428$	5.25667	0.254091
$429$	0	0
$430$	11.0315	0.531984
$431$	−18.7289	−0.902137	−0.451069	−	0.892489i	$-0.648957\pi$
$431$	−18.7289	−0.902137	−0.451069	+	0.892489i	$0.648957\pi$
$432$	1.04892	0.0504661
$433$	−4.18705	−0.201217	−0.100608	−	0.994926i	$-0.532079\pi$
$433$	−4.18705	−0.201217	−0.100608	+	0.994926i	$0.532079\pi$
$434$	−16.0978	−0.772721
$435$	44.3370	2.12580
$436$	7.70171	0.368845
$437$	−2.21983	−0.106189
$438$	32.9269	1.57331
$439$	40.9487	1.95437	0.977187	−	0.212380i	$-0.0681214\pi$
$439$	40.9487	1.95437	0.977187	+	0.212380i	$0.0681214\pi$
$440$	9.70171	0.462511
$441$	−1.99330	−0.0949188
$442$	0	0
$443$	−35.2707	−1.67576	−0.837880	−	0.545854i	$-0.816205\pi$
$443$	−35.2707	−1.67576	−0.837880	+	0.545854i	$0.816205\pi$
$444$	−20.5375	−0.974667
$445$	32.8310	1.55634
$446$	−7.70709	−0.364941
$447$	15.6515	0.740289
$448$	−2.49396	−0.117828
$449$	17.1099	0.807467	0.403733	−	0.914877i	$-0.367712\pi$
$449$	17.1099	0.807467	0.403733	+	0.914877i	$0.367712\pi$
$450$	−5.74094	−0.270630
$451$	34.9638	1.64638
$452$	−6.09783	−0.286818
$453$	−47.7144	−2.24182
$454$	3.86294	0.181297
$455$	0	0
$456$	2.35690	0.110372
$457$	18.7899	0.878952	0.439476	−	0.898254i	$-0.355164\pi$
$457$	18.7899	0.878952	0.439476	+	0.898254i	$0.355164\pi$
$458$	12.1196	0.566312
$459$	−4.76510	−0.222416
$460$	−5.97584	−0.278625
$461$	−40.1769	−1.87122	−0.935612	−	0.353030i	$-0.885151\pi$
$461$	−40.1769	−1.87122	−0.935612	+	0.353030i	$0.885151\pi$
$462$	21.1836	0.985550
$463$	−6.13275	−0.285013	−0.142507	−	0.989794i	$-0.545516\pi$
$463$	−6.13275	−0.285013	−0.142507	+	0.989794i	$0.545516\pi$
$464$	−6.98792	−0.324406
$465$	40.9541	1.89920
$466$	17.8019	0.824659
$467$	−3.35391	−0.155201	−0.0776003	−	0.996985i	$-0.524726\pi$
$467$	−3.35391	−0.155201	−0.0776003	+	0.996985i	$0.524726\pi$
$468$	0	0
$469$	20.0978	0.928032
$470$	0	0
$471$	46.0320	2.12104
$472$	7.28382	0.335265
$473$	14.7681	0.679037
$474$	5.11662	0.235014
$475$	2.24698	0.103098
$476$	11.3297	0.519298
$477$	−0.450419	−0.0206233
$478$	6.78746	0.310451
$479$	38.7767	1.77175	0.885877	−	0.463921i	$-0.153558\pi$
$479$	38.7767	1.77175	0.885877	+	0.463921i	$0.153558\pi$
$480$	6.34481	0.289600
$481$	0	0
$482$	10.8116	0.492456
$483$	−13.0482	−0.593713
$484$	1.98792	0.0903599
$485$	18.5483	0.842233
$486$	−20.7453	−0.941024
$487$	4.12631	0.186981	0.0934905	−	0.995620i	$-0.470198\pi$
$487$	4.12631	0.186981	0.0934905	+	0.995620i	$0.470198\pi$
$488$	−8.51573	−0.385489
$489$	2.22580	0.100654
$490$	2.10023	0.0948786
$491$	3.31037	0.149395	0.0746975	−	0.997206i	$-0.476201\pi$
$491$	3.31037	0.149395	0.0746975	+	0.997206i	$0.476201\pi$
$492$	22.8659	1.03088
$493$	31.7453	1.42973
$494$	0	0
$495$	−24.7875	−1.11411
$496$	−6.45473	−0.289826
$497$	3.19434	0.143286
$498$	−37.0073	−1.65834
$499$	−4.85517	−0.217347	−0.108674	−	0.994077i	$-0.534660\pi$
$499$	−4.85517	−0.217347	−0.108674	+	0.994077i	$0.534660\pi$
$500$	−7.41119	−0.331438
$501$	−22.4972	−1.00510
$502$	5.50604	0.245747
$503$	15.0121	0.669356	0.334678	−	0.942333i	$-0.391373\pi$
$503$	15.0121	0.669356	0.334678	+	0.942333i	$0.391373\pi$
$504$	6.37196	0.283830
$505$	14.6987	0.654085
$506$	−8.00000	−0.355643
$507$	0	0
$508$	−21.6625	−0.961117
$509$	19.6775	0.872192	0.436096	−	0.899900i	$-0.356361\pi$
$509$	19.6775	0.872192	0.436096	+	0.899900i	$0.356361\pi$
$510$	−28.8237	−1.27634
$511$	−34.8418	−1.54131
$512$	−1.00000	−0.0441942
$513$	1.04892	0.0463108
$514$	−14.9879	−0.661089
$515$	22.0043	0.969626
$516$	9.65817	0.425177
$517$	0	0
$518$	21.7318	0.954842
$519$	−39.5206	−1.73476
$520$	0	0
$521$	1.24267	0.0544423	0.0272211	−	0.999629i	$-0.491334\pi$
$521$	1.24267	0.0544423	0.0272211	+	0.999629i	$0.491334\pi$
$522$	17.8538	0.781441
$523$	24.0000	1.04945	0.524723	−	0.851273i	$-0.324169\pi$
$523$	24.0000	1.04945	0.524723	+	0.851273i	$0.324169\pi$
$524$	3.72587	0.162766
$525$	13.2078	0.576434
$526$	−11.4276	−0.498266
$527$	29.3230	1.27733
$528$	8.49396	0.369652
$529$	−18.0723	−0.785754
$530$	0.474582	0.0206145
$531$	−18.6098	−0.807598
$532$	−2.49396	−0.108127
$533$	0	0
$534$	28.7439	1.24387
$535$	14.1511	0.611804
$536$	8.05861	0.348079
$537$	17.3907	0.750466
$538$	22.4155	0.966401
$539$	2.81163	0.121105
$540$	2.82371	0.121513
$541$	−22.2543	−0.956786	−0.478393	−	0.878146i	$-0.658781\pi$
$541$	−22.2543	−0.956786	−0.478393	+	0.878146i	$0.658781\pi$
$542$	4.39612	0.188830
$543$	53.7077	2.30482
$544$	4.54288	0.194774
$545$	20.7332	0.888111
$546$	0	0
$547$	29.5502	1.26347	0.631737	−	0.775183i	$-0.282342\pi$
$547$	29.5502	1.26347	0.631737	+	0.775183i	$0.282342\pi$
$548$	10.7463	0.459060
$549$	21.7573	0.928580
$550$	8.09783	0.345293
$551$	−6.98792	−0.297695
$552$	−5.23191	−0.222685
$553$	−5.41417	−0.230234
$554$	8.71140	0.370112
$555$	−55.2874	−2.34682
$556$	8.68963	0.368522
$557$	41.8950	1.77515	0.887574	−	0.460666i	$-0.152389\pi$
$557$	41.8950	1.77515	0.887574	+	0.460666i	$0.152389\pi$
$558$	16.4916	0.698144
$559$	0	0
$560$	−6.71379	−0.283709
$561$	−38.5870	−1.62914
$562$	−22.3913	−0.944522
$563$	27.6974	1.16731	0.583653	−	0.812003i	$-0.301623\pi$
$563$	27.6974	1.16731	0.583653	+	0.812003i	$0.301623\pi$
$564$	0	0
$565$	−16.4155	−0.690605
$566$	16.7332	0.703348
$567$	25.2814	1.06172
$568$	1.28083	0.0537425
$569$	−9.50604	−0.398514	−0.199257	−	0.979947i	$-0.563853\pi$
$569$	−9.50604	−0.398514	−0.199257	+	0.979947i	$0.563853\pi$
$570$	6.34481	0.265755
$571$	−26.6025	−1.11328	−0.556641	−	0.830753i	$-0.687910\pi$
$571$	−26.6025	−1.11328	−0.556641	+	0.830753i	$0.687910\pi$
$572$	0	0
$573$	8.36599	0.349494
$574$	−24.1957	−1.00991
$575$	−4.98792	−0.208011
$576$	2.55496	0.106457
$577$	−12.5633	−0.523018	−0.261509	−	0.965201i	$-0.584220\pi$
$577$	−12.5633	−0.523018	−0.261509	+	0.965201i	$0.584220\pi$
$578$	−3.63773	−0.151310
$579$	−62.9047	−2.61423
$580$	−18.8116	−0.781110
$581$	39.1594	1.62461
$582$	16.2392	0.673137
$583$	0.635334	0.0263129
$584$	−13.9705	−0.578102
$585$	0	0
$586$	−1.84787	−0.0763349
$587$	46.4215	1.91602	0.958010	−	0.286736i	$-0.0925703\pi$
$587$	46.4215	1.91602	0.958010	+	0.286736i	$0.0925703\pi$
$588$	1.83877	0.0758297
$589$	−6.45473	−0.265963
$590$	19.6082	0.807256
$591$	23.8122	0.979504
$592$	8.71379	0.358135
$593$	−31.1836	−1.28056	−0.640278	−	0.768143i	$-0.721181\pi$
$593$	−31.1836	−1.28056	−0.640278	+	0.768143i	$0.721181\pi$
$594$	3.78017	0.155102
$595$	30.4999	1.25038
$596$	−6.64071	−0.272014
$597$	10.3612	0.424056
$598$	0	0
$599$	−16.8702	−0.689299	−0.344650	−	0.938731i	$-0.612002\pi$
$599$	−16.8702	−0.689299	−0.344650	+	0.938731i	$0.612002\pi$
$600$	5.29590	0.216204
$601$	15.2185	0.620776	0.310388	−	0.950610i	$-0.399541\pi$
$601$	15.2185	0.620776	0.310388	+	0.950610i	$0.399541\pi$
$602$	−10.2198	−0.416529
$603$	−20.5894	−0.838465
$604$	20.2446	0.823741
$605$	5.35152	0.217570
$606$	12.8689	0.522764
$607$	43.5123	1.76611	0.883054	−	0.469271i	$-0.155483\pi$
$607$	43.5123	1.76611	0.883054	+	0.469271i	$0.155483\pi$
$608$	−1.00000	−0.0405554
$609$	−41.0750	−1.66444
$610$	−22.9245	−0.928187
$611$	0	0
$612$	−11.6069	−0.469179
$613$	22.4112	0.905179	0.452590	−	0.891719i	$-0.350500\pi$
$613$	22.4112	0.905179	0.452590	+	0.891719i	$0.350500\pi$
$614$	−32.4198	−1.30836
$615$	61.5555	2.48216
$616$	−8.98792	−0.362134
$617$	−6.13813	−0.247112	−0.123556	−	0.992338i	$-0.539430\pi$
$617$	−6.13813	−0.247112	−0.123556	+	0.992338i	$0.539430\pi$
$618$	19.2650	0.774953
$619$	48.6848	1.95681	0.978405	−	0.206699i	$-0.0662722\pi$
$619$	48.6848	1.95681	0.978405	+	0.206699i	$0.0662722\pi$
$620$	−17.3763	−0.697848
$621$	−2.32842	−0.0934363
$622$	−1.37329	−0.0550639
$623$	−30.4155	−1.21857
$624$	0	0
$625$	−31.1860	−1.24744
$626$	31.2295	1.24818
$627$	8.49396	0.339216
$628$	−19.5308	−0.779364
$629$	−39.5857	−1.57838
$630$	17.1535	0.683410
$631$	−9.66679	−0.384829	−0.192414	−	0.981314i	$-0.561632\pi$
$631$	−9.66679	−0.384829	−0.192414	+	0.981314i	$0.561632\pi$
$632$	−2.17092	−0.0863544
$633$	21.5066	0.854812
$634$	−12.6353	−0.501813
$635$	−58.3159	−2.31419
$636$	0.415502	0.0164757
$637$	0	0
$638$	−25.1836	−0.997028
$639$	−3.27247	−0.129457
$640$	−2.69202	−0.106411
$641$	0.733169	0.0289584	0.0144792	−	0.999895i	$-0.495391\pi$
$641$	0.733169	0.0289584	0.0144792	+	0.999895i	$0.495391\pi$
$642$	12.3894	0.488971
$643$	39.8840	1.57287	0.786435	−	0.617673i	$-0.211924\pi$
$643$	39.8840	1.57287	0.786435	+	0.617673i	$0.211924\pi$
$644$	5.53617	0.218156
$645$	26.0000	1.02375
$646$	4.54288	0.178737
$647$	−29.9275	−1.17657	−0.588286	−	0.808653i	$-0.700197\pi$
$647$	−29.9275	−1.17657	−0.588286	+	0.808653i	$0.700197\pi$
$648$	10.1371	0.398221
$649$	26.2500	1.03040
$650$	0	0
$651$	−37.9409	−1.48702
$652$	−0.944378	−0.0369847
$653$	0.469205	0.0183614	0.00918071	−	0.999958i	$-0.497078\pi$
$653$	0.469205	0.0183614	0.00918071	+	0.999958i	$0.497078\pi$
$654$	18.1521	0.709804
$655$	10.0301	0.391910
$656$	−9.70171	−0.378788
$657$	35.6939	1.39255
$658$	0	0
$659$	−26.2911	−1.02416	−0.512078	−	0.858939i	$-0.671124\pi$
$659$	−26.2911	−1.02416	−0.512078	+	0.858939i	$0.671124\pi$
$660$	22.8659	0.890055
$661$	−17.2707	−0.671751	−0.335876	−	0.941906i	$-0.609032\pi$
$661$	−17.2707	−0.671751	−0.335876	+	0.941906i	$0.609032\pi$
$662$	25.6819	0.998154
$663$	0	0
$664$	15.7017	0.609345
$665$	−6.71379	−0.260350
$666$	−22.2634	−0.862688
$667$	15.5120	0.600627
$668$	9.54527	0.369318
$669$	−18.1648	−0.702292
$670$	21.6939	0.838110
$671$	−30.6896	−1.18476
$672$	−5.87800	−0.226749
$673$	13.7345	0.529426	0.264713	−	0.964327i	$-0.414723\pi$
$673$	13.7345	0.529426	0.264713	+	0.964327i	$0.414723\pi$
$674$	−20.6703	−0.796188
$675$	2.35690	0.0907170
$676$	0	0
$677$	29.9323	1.15039	0.575196	−	0.818016i	$-0.304926\pi$
$677$	29.9323	1.15039	0.575196	+	0.818016i	$0.304926\pi$
$678$	−14.3720	−0.551952
$679$	−17.1836	−0.659446
$680$	12.2295	0.468981
$681$	9.10454	0.348887
$682$	−23.2620	−0.890750
$683$	−43.5445	−1.66619	−0.833093	−	0.553134i	$-0.813432\pi$
$683$	−43.5445	−1.66619	−0.833093	+	0.553134i	$0.813432\pi$
$684$	2.55496	0.0976913
$685$	28.9293	1.10533
$686$	−19.4034	−0.740826
$687$	28.5646	1.08981
$688$	−4.09783	−0.156228
$689$	0	0
$690$	−14.0844	−0.536185
$691$	−7.48666	−0.284806	−0.142403	−	0.989809i	$-0.545483\pi$
$691$	−7.48666	−0.284806	−0.142403	+	0.989809i	$0.545483\pi$
$692$	16.7681	0.637427
$693$	22.9638	0.872321
$694$	−5.75600	−0.218495
$695$	23.3927	0.887334
$696$	−16.4698	−0.624286
$697$	44.0737	1.66941
$698$	−25.0224	−0.947110
$699$	41.9573	1.58697
$700$	−5.60388	−0.211807
$701$	50.4198	1.90433	0.952165	−	0.305584i	$-0.0988518\pi$
$701$	50.4198	1.90433	0.952165	+	0.305584i	$0.0988518\pi$
$702$	0	0
$703$	8.71379	0.328647
$704$	−3.60388	−0.135826
$705$	0	0
$706$	−27.3099	−1.02782
$707$	−13.6173	−0.512131
$708$	17.1672	0.645183
$709$	6.69979	0.251616	0.125808	−	0.992055i	$-0.459848\pi$
$709$	6.69979	0.251616	0.125808	+	0.992055i	$0.459848\pi$
$710$	3.44803	0.129402
$711$	5.54660	0.208014
$712$	−12.1957	−0.457052
$713$	14.3284	0.536604
$714$	26.7030	0.999336
$715$	0	0
$716$	−7.37867	−0.275754
$717$	15.9973	0.597432
$718$	22.7922	0.850599
$719$	−14.2306	−0.530711	−0.265356	−	0.964151i	$-0.585489\pi$
$719$	−14.2306	−0.530711	−0.265356	+	0.964151i	$0.585489\pi$
$720$	6.87800	0.256328
$721$	−20.3854	−0.759191
$722$	−1.00000	−0.0372161
$723$	25.4819	0.947681
$724$	−22.7875	−0.846889
$725$	−15.7017	−0.583147
$726$	4.68532	0.173888
$727$	−24.8552	−0.921827	−0.460914	−	0.887445i	$-0.652478\pi$
$727$	−24.8552	−0.921827	−0.460914	+	0.887445i	$0.652478\pi$
$728$	0	0
$729$	−18.4832	−0.684563
$730$	−37.6088	−1.39196
$731$	18.6160	0.688536
$732$	−20.0707	−0.741834
$733$	−11.8834	−0.438923	−0.219461	−	0.975621i	$-0.570430\pi$
$733$	−11.8834	−0.438923	−0.219461	+	0.975621i	$0.570430\pi$
$734$	14.8358	0.547599
$735$	4.95002	0.182584
$736$	2.21983	0.0818241
$737$	29.0422	1.06978
$738$	24.7875	0.912439
$739$	17.8974	0.658366	0.329183	−	0.944266i	$-0.393227\pi$
$739$	17.8974	0.658366	0.329183	+	0.944266i	$0.393227\pi$
$740$	23.4577	0.862323
$741$	0	0
$742$	−0.439665	−0.0161406
$743$	−33.5453	−1.23066	−0.615328	−	0.788271i	$-0.710977\pi$
$743$	−33.5453	−1.23066	−0.615328	+	0.788271i	$0.710977\pi$
$744$	−15.2131	−0.557740
$745$	−17.8769	−0.654960
$746$	−22.0978	−0.809059
$747$	−40.1172	−1.46781
$748$	16.3720	0.598618
$749$	−13.1099	−0.479026
$750$	−17.4674	−0.637819
$751$	1.88876	0.0689217	0.0344608	−	0.999406i	$-0.489029\pi$
$751$	1.88876	0.0689217	0.0344608	+	0.999406i	$0.489029\pi$
$752$	0	0
$753$	12.9772	0.472914
$754$	0	0
$755$	54.4989	1.98342
$756$	−2.61596	−0.0951414
$757$	−4.86353	−0.176768	−0.0883840	−	0.996086i	$-0.528170\pi$
$757$	−4.86353	−0.176768	−0.0883840	+	0.996086i	$0.528170\pi$
$758$	−27.2078	−0.988230
$759$	−18.8552	−0.684399
$760$	−2.69202	−0.0976499
$761$	7.75063	0.280960	0.140480	−	0.990084i	$-0.455135\pi$
$761$	7.75063	0.280960	0.140480	+	0.990084i	$0.455135\pi$
$762$	−51.0562	−1.84957
$763$	−19.2078	−0.695367
$764$	−3.54958	−0.128419
$765$	−31.2459	−1.12970
$766$	−10.0411	−0.362801
$767$	0	0
$768$	−2.35690	−0.0850472
$769$	−0.369568	−0.0133270	−0.00666349	−	0.999978i	$-0.502121\pi$
$769$	−0.369568	−0.0133270	−0.00666349	+	0.999978i	$0.502121\pi$
$770$	−24.1957	−0.871951
$771$	−35.3250	−1.27220
$772$	26.6896	0.960581
$773$	38.9724	1.40174	0.700870	−	0.713290i	$-0.252796\pi$
$773$	38.9724	1.40174	0.700870	+	0.713290i	$0.252796\pi$
$774$	10.4698	0.376329
$775$	−14.5036	−0.520986
$776$	−6.89008	−0.247340
$777$	51.2197	1.83750
$778$	−20.3672	−0.730199
$779$	−9.70171	−0.347600
$780$	0	0
$781$	4.61596	0.165172
$782$	−10.0844	−0.360618
$783$	−7.32975	−0.261944
$784$	−0.780167	−0.0278631
$785$	−52.5773	−1.87656
$786$	8.78150	0.313226
$787$	−37.1075	−1.32274	−0.661370	−	0.750060i	$-0.730025\pi$
$787$	−37.1075	−1.32274	−0.661370	+	0.750060i	$0.730025\pi$
$788$	−10.1032	−0.359912
$789$	−26.9336	−0.958862
$790$	−5.84415	−0.207926
$791$	15.2078	0.540725
$792$	9.20775	0.327183
$793$	0	0
$794$	−38.8442	−1.37853
$795$	1.11854	0.0396705
$796$	−4.39612	−0.155817
$797$	38.2150	1.35365	0.676823	−	0.736146i	$-0.263356\pi$
$797$	38.2150	1.35365	0.676823	+	0.736146i	$0.263356\pi$
$798$	−5.87800	−0.208079
$799$	0	0
$800$	−2.24698	−0.0794427
$801$	31.1594	1.10096
$802$	−26.8552	−0.948289
$803$	−50.3478	−1.77674
$804$	18.9933	0.669842
$805$	14.9035	0.525279
$806$	0	0
$807$	52.8310	1.85974
$808$	−5.46011	−0.192086
$809$	11.5539	0.406213	0.203107	−	0.979157i	$-0.434896\pi$
$809$	11.5539	0.406213	0.203107	+	0.979157i	$0.434896\pi$
$810$	27.2892	0.958844
$811$	1.61224	0.0566133	0.0283066	−	0.999599i	$-0.490989\pi$
$811$	1.61224	0.0566133	0.0283066	+	0.999599i	$0.490989\pi$
$812$	17.4276	0.611588
$813$	10.3612	0.363383
$814$	31.4034	1.10069
$815$	−2.54229	−0.0890524
$816$	10.7071	0.374823
$817$	−4.09783	−0.143365
$818$	38.1909	1.33531
$819$	0	0
$820$	−26.1172	−0.912053
$821$	24.8200	0.866224	0.433112	−	0.901340i	$-0.357416\pi$
$821$	24.8200	0.866224	0.433112	+	0.901340i	$0.357416\pi$
$822$	25.3279	0.883414
$823$	−2.52888	−0.0881511	−0.0440755	−	0.999028i	$-0.514034\pi$
$823$	−2.52888	−0.0881511	−0.0440755	+	0.999028i	$0.514034\pi$
$824$	−8.17390	−0.284751
$825$	19.0858	0.664481
$826$	−18.1655	−0.632060
$827$	−21.0127	−0.730682	−0.365341	−	0.930874i	$-0.619048\pi$
$827$	−21.0127	−0.730682	−0.365341	+	0.930874i	$0.619048\pi$
$828$	−5.67158	−0.197101
$829$	−41.7271	−1.44924	−0.724620	−	0.689148i	$-0.757985\pi$
$829$	−41.7271	−1.44924	−0.724620	+	0.689148i	$0.757985\pi$
$830$	42.2693	1.46719
$831$	20.5319	0.712242
$832$	0	0
$833$	3.54420	0.122799
$834$	20.4805	0.709183
$835$	25.6961	0.889249
$836$	−3.60388	−0.124643
$837$	−6.77048	−0.234022
$838$	0.968541	0.0334577
$839$	18.8549	0.650944	0.325472	−	0.945552i	$-0.394477\pi$
$839$	18.8549	0.650944	0.325472	+	0.945552i	$0.394477\pi$
$840$	−15.8237	−0.545970
$841$	19.8310	0.683828
$842$	−5.59312	−0.192752
$843$	−52.7741	−1.81763
$844$	−9.12498	−0.314095
$845$	0	0
$846$	0	0
$847$	−4.95779	−0.170352
$848$	−0.176292	−0.00605389
$849$	39.4383	1.35352
$850$	10.2078	0.350123
$851$	−19.3432	−0.663075
$852$	3.01879	0.103422
$853$	−23.2838	−0.797223	−0.398611	−	0.917120i	$-0.630508\pi$
$853$	−23.2838	−0.797223	−0.398611	+	0.917120i	$0.630508\pi$
$854$	21.2379	0.726745
$855$	6.87800	0.235223
$856$	−5.25667	−0.179669
$857$	36.6025	1.25032	0.625160	−	0.780497i	$-0.285034\pi$
$857$	36.6025	1.25032	0.625160	+	0.780497i	$0.285034\pi$
$858$	0	0
$859$	40.7741	1.39119	0.695596	−	0.718433i	$-0.255140\pi$
$859$	40.7741	1.39119	0.695596	+	0.718433i	$0.255140\pi$
$860$	−11.0315	−0.376170
$861$	−57.0267	−1.94346
$862$	18.7289	0.637907
$863$	53.7318	1.82905	0.914527	−	0.404526i	$-0.132563\pi$
$863$	53.7318	1.82905	0.914527	+	0.404526i	$0.132563\pi$
$864$	−1.04892	−0.0356849
$865$	45.1400	1.53481
$866$	4.18705	0.142282
$867$	−8.57374	−0.291180
$868$	16.0978	0.546396
$869$	−7.82371	−0.265401
$870$	−44.3370	−1.50317
$871$	0	0
$872$	−7.70171	−0.260813
$873$	17.6039	0.595801
$874$	2.21983	0.0750870
$875$	18.4832	0.624846
$876$	−32.9269	−1.11250
$877$	9.19700	0.310561	0.155280	−	0.987870i	$-0.450372\pi$
$877$	9.19700	0.310561	0.155280	+	0.987870i	$0.450372\pi$
$878$	−40.9487	−1.38195
$879$	−4.35524	−0.146899
$880$	−9.70171	−0.327045
$881$	−52.7590	−1.77750	−0.888748	−	0.458397i	$-0.848424\pi$
$881$	−52.7590	−1.77750	−0.888748	+	0.458397i	$0.848424\pi$
$882$	1.99330	0.0671177
$883$	53.0810	1.78632	0.893158	−	0.449742i	$-0.148484\pi$
$883$	53.0810	1.78632	0.893158	+	0.449742i	$0.148484\pi$
$884$	0	0
$885$	46.2145	1.55348
$886$	35.2707	1.18494
$887$	17.5754	0.590124	0.295062	−	0.955478i	$-0.404660\pi$
$887$	17.5754	0.590124	0.295062	+	0.955478i	$0.404660\pi$
$888$	20.5375	0.689193
$889$	54.0253	1.81195
$890$	−32.8310	−1.10050
$891$	36.5327	1.22389
$892$	7.70709	0.258052
$893$	0	0
$894$	−15.6515	−0.523463
$895$	−19.8635	−0.663965
$896$	2.49396	0.0833173
$897$	0	0
$898$	−17.1099	−0.570965
$899$	45.1051	1.50434
$900$	5.74094	0.191365
$901$	0.800873	0.0266809
$902$	−34.9638	−1.16417
$903$	−24.0871	−0.801568
$904$	6.09783	0.202811
$905$	−61.3443	−2.03915
$906$	47.7144	1.58520
$907$	30.3672	1.00833	0.504163	−	0.863609i	$-0.331801\pi$
$907$	30.3672	1.00833	0.504163	+	0.863609i	$0.331801\pi$
$908$	−3.86294	−0.128196
$909$	13.9503	0.462704
$910$	0	0
$911$	−26.3067	−0.871578	−0.435789	−	0.900049i	$-0.643531\pi$
$911$	−26.3067	−0.871578	−0.435789	+	0.900049i	$0.643531\pi$
$912$	−2.35690	−0.0780446
$913$	56.5870	1.87276
$914$	−18.7899	−0.621513
$915$	−54.0307	−1.78620
$916$	−12.1196	−0.400443
$917$	−9.29218	−0.306855
$918$	4.76510	0.157272
$919$	−13.8780	−0.457793	−0.228897	−	0.973451i	$-0.573512\pi$
$919$	−13.8780	−0.457793	−0.228897	+	0.973451i	$0.573512\pi$
$920$	5.97584	0.197018
$921$	−76.4101	−2.51780
$922$	40.1769	1.32316
$923$	0	0
$924$	−21.1836	−0.696889
$925$	19.5797	0.643777
$926$	6.13275	0.201535
$927$	20.8840	0.685920
$928$	6.98792	0.229390
$929$	−3.66248	−0.120162	−0.0600811	−	0.998194i	$-0.519136\pi$
$929$	−3.66248	−0.120162	−0.0600811	+	0.998194i	$0.519136\pi$
$930$	−40.9541	−1.34294
$931$	−0.780167	−0.0255690
$932$	−17.8019	−0.583122
$933$	−3.23670	−0.105965
$934$	3.35391	0.109743
$935$	44.0737	1.44136
$936$	0	0
$937$	32.5332	1.06281	0.531406	−	0.847117i	$-0.321664\pi$
$937$	32.5332	1.06281	0.531406	+	0.847117i	$0.321664\pi$
$938$	−20.0978	−0.656218
$939$	73.6047	2.40200
$940$	0	0
$941$	−19.5931	−0.638718	−0.319359	−	0.947634i	$-0.603467\pi$
$941$	−19.5931	−0.638718	−0.319359	+	0.947634i	$0.603467\pi$
$942$	−46.0320	−1.49981
$943$	21.5362	0.701314
$944$	−7.28382	−0.237068
$945$	−7.04221	−0.229083
$946$	−14.7681	−0.480152
$947$	−31.8189	−1.03398	−0.516988	−	0.855993i	$-0.672947\pi$
$947$	−31.8189	−1.03398	−0.516988	+	0.855993i	$0.672947\pi$
$948$	−5.11662	−0.166180
$949$	0	0
$950$	−2.24698	−0.0729016
$951$	−29.7802	−0.965688
$952$	−11.3297	−0.367199
$953$	44.6762	1.44720	0.723602	−	0.690217i	$-0.242485\pi$
$953$	44.6762	1.44720	0.723602	+	0.690217i	$0.242485\pi$
$954$	0.450419	0.0145828
$955$	−9.55555	−0.309210
$956$	−6.78746	−0.219522
$957$	−59.3551	−1.91868
$958$	−38.7767	−1.25282
$959$	−26.8009	−0.865445
$960$	−6.34481	−0.204778
$961$	10.6635	0.343985
$962$	0	0
$963$	13.4306	0.432794
$964$	−10.8116	−0.348219
$965$	71.8491	2.31290
$966$	13.0482	0.419818
$967$	24.1026	0.775088	0.387544	−	0.921851i	$-0.373324\pi$
$967$	24.1026	0.775088	0.387544	+	0.921851i	$0.373324\pi$
$968$	−1.98792	−0.0638941
$969$	10.7071	0.343961
$970$	−18.5483	−0.595549
$971$	−31.4233	−1.00842	−0.504210	−	0.863581i	$-0.668216\pi$
$971$	−31.4233	−1.00842	−0.504210	+	0.863581i	$0.668216\pi$
$972$	20.7453	0.665404
$973$	−21.6716	−0.694759
$974$	−4.12631	−0.132215
$975$	0	0
$976$	8.51573	0.272582
$977$	5.51083	0.176307	0.0881535	−	0.996107i	$-0.471903\pi$
$977$	5.51083	0.176307	0.0881535	+	0.996107i	$0.471903\pi$
$978$	−2.22580	−0.0711732
$979$	−43.9517	−1.40470
$980$	−2.10023	−0.0670893
$981$	19.6775	0.628256
$982$	−3.31037	−0.105638
$983$	−29.9269	−0.954520	−0.477260	−	0.878762i	$-0.658370\pi$
$983$	−29.9269	−0.954520	−0.477260	+	0.878762i	$0.658370\pi$
$984$	−22.8659	−0.728939
$985$	−27.1981	−0.866602
$986$	−31.7453	−1.01097
$987$	0	0
$988$	0	0
$989$	9.09651	0.289252
$990$	24.7875	0.787797
$991$	−46.4975	−1.47704	−0.738521	−	0.674230i	$-0.764476\pi$
$991$	−46.4975	−1.47704	−0.738521	+	0.674230i	$0.764476\pi$
$992$	6.45473	0.204938
$993$	60.5295	1.92084
$994$	−3.19434	−0.101318
$995$	−11.8345	−0.375178
$996$	37.0073	1.17262
$997$	18.3811	0.582134	0.291067	−	0.956703i	$-0.405990\pi$
$997$	18.3811	0.582134	0.291067	+	0.956703i	$0.405990\pi$
$998$	4.85517	0.153688
$999$	9.14005	0.289178

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.l.1.2	✓	3
13.12	even	2		6422.2.a.r.1.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6422.2.a.l.1.2	✓	3	1.1	even	1	trivial
6422.2.a.r.1.2	yes	3	13.12	even	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:	\( N \)	\(=\)	\( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6422.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.35690 q^{3} +1.00000 q^{4} +2.69202 q^{5} +2.35690 q^{6} -2.49396 q^{7} -1.00000 q^{8} +2.55496 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.35690 q^{3} +1.00000 q^{4} +2.69202 q^{5} +2.35690 q^{6} -2.49396 q^{7} -1.00000 q^{8} +2.55496 q^{9} -2.69202 q^{10} -3.60388 q^{11} -2.35690 q^{12} +2.49396 q^{14} -6.34481 q^{15} +1.00000 q^{16} -4.54288 q^{17} -2.55496 q^{18} +1.00000 q^{19} +2.69202 q^{20} +5.87800 q^{21} +3.60388 q^{22} -2.21983 q^{23} +2.35690 q^{24} +2.24698 q^{25} +1.04892 q^{27} -2.49396 q^{28} -6.98792 q^{29} +6.34481 q^{30} -6.45473 q^{31} -1.00000 q^{32} +8.49396 q^{33} +4.54288 q^{34} -6.71379 q^{35} +2.55496 q^{36} +8.71379 q^{37} -1.00000 q^{38} -2.69202 q^{40} -9.70171 q^{41} -5.87800 q^{42} -4.09783 q^{43} -3.60388 q^{44} +6.87800 q^{45} +2.21983 q^{46} -2.35690 q^{48} -0.780167 q^{49} -2.24698 q^{50} +10.7071 q^{51} -0.176292 q^{53} -1.04892 q^{54} -9.70171 q^{55} +2.49396 q^{56} -2.35690 q^{57} +6.98792 q^{58} -7.28382 q^{59} -6.34481 q^{60} +8.51573 q^{61} +6.45473 q^{62} -6.37196 q^{63} +1.00000 q^{64} -8.49396 q^{66} -8.05861 q^{67} -4.54288 q^{68} +5.23191 q^{69} +6.71379 q^{70} -1.28083 q^{71} -2.55496 q^{72} +13.9705 q^{73} -8.71379 q^{74} -5.29590 q^{75} +1.00000 q^{76} +8.98792 q^{77} +2.17092 q^{79} +2.69202 q^{80} -10.1371 q^{81} +9.70171 q^{82} -15.7017 q^{83} +5.87800 q^{84} -12.2295 q^{85} +4.09783 q^{86} +16.4698 q^{87} +3.60388 q^{88} +12.1957 q^{89} -6.87800 q^{90} -2.21983 q^{92} +15.2131 q^{93} +2.69202 q^{95} +2.35690 q^{96} +6.89008 q^{97} +0.780167 q^{98} -9.20775 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{5} + 3 q^{6} + 2 q^{7} - 3 q^{8} + 8 q^{9}+O(q^{10}) \) 3 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 + 3 * q^5 + 3 * q^6 + 2 * q^7 - 3 * q^8 + 8 * q^9	\( 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{5} + 3 q^{6} + 2 q^{7} - 3 q^{8} + 8 q^{9} - 3 q^{10} - 2 q^{11} - 3 q^{12} - 2 q^{14} + 4 q^{15} + 3 q^{16} + 5 q^{17} - 8 q^{18} + 3 q^{19} + 3 q^{20} - 2 q^{21} + 2 q^{22} - 8 q^{23} + 3 q^{24} + 2 q^{25} - 6 q^{27} + 2 q^{28} - 2 q^{29} - 4 q^{30} + 3 q^{31} - 3 q^{32} + 16 q^{33} - 5 q^{34} - 12 q^{35} + 8 q^{36} + 18 q^{37} - 3 q^{38} - 3 q^{40} - 2 q^{41} + 2 q^{42} + 6 q^{43} - 2 q^{44} + q^{45} + 8 q^{46} - 3 q^{48} - q^{49} - 2 q^{50} + 2 q^{51} - 8 q^{53} + 6 q^{54} - 2 q^{55} - 2 q^{56} - 3 q^{57} + 2 q^{58} + 11 q^{59} + 4 q^{60} + 13 q^{61} - 3 q^{62} + 10 q^{63} + 3 q^{64} - 16 q^{66} + 7 q^{67} + 5 q^{68} + 36 q^{69} + 12 q^{70} - 15 q^{71} - 8 q^{72} + 7 q^{73} - 18 q^{74} - 2 q^{75} + 3 q^{76} + 8 q^{77} + 17 q^{79} + 3 q^{80} - 25 q^{81} + 2 q^{82} - 20 q^{83} - 2 q^{84} - 16 q^{85} - 6 q^{86} + 2 q^{87} + 2 q^{88} - q^{90} - 8 q^{92} + 25 q^{93} + 3 q^{95} + 3 q^{96} + 20 q^{97} + q^{98} - 10 q^{99}+O(q^{100}) \) 3 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 + 3 * q^5 + 3 * q^6 + 2 * q^7 - 3 * q^8 + 8 * q^9 - 3 * q^10 - 2 * q^11 - 3 * q^12 - 2 * q^14 + 4 * q^15 + 3 * q^16 + 5 * q^17 - 8 * q^18 + 3 * q^19 + 3 * q^20 - 2 * q^21 + 2 * q^22 - 8 * q^23 + 3 * q^24 + 2 * q^25 - 6 * q^27 + 2 * q^28 - 2 * q^29 - 4 * q^30 + 3 * q^31 - 3 * q^32 + 16 * q^33 - 5 * q^34 - 12 * q^35 + 8 * q^36 + 18 * q^37 - 3 * q^38 - 3 * q^40 - 2 * q^41 + 2 * q^42 + 6 * q^43 - 2 * q^44 + q^45 + 8 * q^46 - 3 * q^48 - q^49 - 2 * q^50 + 2 * q^51 - 8 * q^53 + 6 * q^54 - 2 * q^55 - 2 * q^56 - 3 * q^57 + 2 * q^58 + 11 * q^59 + 4 * q^60 + 13 * q^61 - 3 * q^62 + 10 * q^63 + 3 * q^64 - 16 * q^66 + 7 * q^67 + 5 * q^68 + 36 * q^69 + 12 * q^70 - 15 * q^71 - 8 * q^72 + 7 * q^73 - 18 * q^74 - 2 * q^75 + 3 * q^76 + 8 * q^77 + 17 * q^79 + 3 * q^80 - 25 * q^81 + 2 * q^82 - 20 * q^83 - 2 * q^84 - 16 * q^85 - 6 * q^86 + 2 * q^87 + 2 * q^88 - q^90 - 8 * q^92 + 25 * q^93 + 3 * q^95 + 3 * q^96 + 20 * q^97 + q^98 - 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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