LMFDB - Embedded newform 6422.2.a.bk.1.4

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:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 2x^{8} - 12x^{7} + 14x^{6} + 54x^{5} - 11x^{4} - 84x^{3} - 48x^{2} - 4x + 1 $ x^9 - 2x^8 - 12x^7 + 14x^6 + 54x^5 - 11x^4 - 84x^3 - 48x^2 - 4x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.88062$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -1.60929 q^{3} +1.00000 q^{4} -4.22571 q^{5} -1.60929 q^{6} +0.325660 q^{7} +1.00000 q^{8} -0.410201 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.60929 q^{3} +1.00000 q^{4} -4.22571 q^{5} -1.60929 q^{6} +0.325660 q^{7} +1.00000 q^{8} -0.410201 q^{9} -4.22571 q^{10} -3.83154 q^{11} -1.60929 q^{12} +0.325660 q^{14} +6.80037 q^{15} +1.00000 q^{16} +2.40765 q^{17} -0.410201 q^{18} +1.00000 q^{19} -4.22571 q^{20} -0.524079 q^{21} -3.83154 q^{22} +2.11166 q^{23} -1.60929 q^{24} +12.8566 q^{25} +5.48799 q^{27} +0.325660 q^{28} +7.46895 q^{29} +6.80037 q^{30} -4.53149 q^{31} +1.00000 q^{32} +6.16605 q^{33} +2.40765 q^{34} -1.37614 q^{35} -0.410201 q^{36} +2.51608 q^{37} +1.00000 q^{38} -4.22571 q^{40} +8.17536 q^{41} -0.524079 q^{42} -8.05774 q^{43} -3.83154 q^{44} +1.73339 q^{45} +2.11166 q^{46} -1.42557 q^{47} -1.60929 q^{48} -6.89395 q^{49} +12.8566 q^{50} -3.87460 q^{51} +8.33758 q^{53} +5.48799 q^{54} +16.1910 q^{55} +0.325660 q^{56} -1.60929 q^{57} +7.46895 q^{58} +6.88289 q^{59} +6.80037 q^{60} +9.16884 q^{61} -4.53149 q^{62} -0.133586 q^{63} +1.00000 q^{64} +6.16605 q^{66} -1.83754 q^{67} +2.40765 q^{68} -3.39826 q^{69} -1.37614 q^{70} -11.6905 q^{71} -0.410201 q^{72} -12.3174 q^{73} +2.51608 q^{74} -20.6900 q^{75} +1.00000 q^{76} -1.24778 q^{77} -14.2281 q^{79} -4.22571 q^{80} -7.60113 q^{81} +8.17536 q^{82} -6.05693 q^{83} -0.524079 q^{84} -10.1740 q^{85} -8.05774 q^{86} -12.0197 q^{87} -3.83154 q^{88} +0.553738 q^{89} +1.73339 q^{90} +2.11166 q^{92} +7.29246 q^{93} -1.42557 q^{94} -4.22571 q^{95} -1.60929 q^{96} -2.52157 q^{97} -6.89395 q^{98} +1.57170 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{2} - 5 q^{3} + 9 q^{4} + q^{5} - 5 q^{6} - 13 q^{7} + 9 q^{8} + 10 q^{9}+O(q^{10}) $ 9 * q + 9 * q^2 - 5 * q^3 + 9 * q^4 + q^5 - 5 * q^6 - 13 * q^7 + 9 * q^8 + 10 * q^9	\( 9 q + 9 q^{2} - 5 q^{3} + 9 q^{4} + q^{5} - 5 q^{6} - 13 q^{7} + 9 q^{8} + 10 q^{9} + q^{10} - 13 q^{11} - 5 q^{12} - 13 q^{14} - q^{15} + 9 q^{16} - 12 q^{17} + 10 q^{18} + 9 q^{19} + q^{20} + 18 q^{21} - 13 q^{22} - 22 q^{23} - 5 q^{24} + 4 q^{25} - 26 q^{27} - 13 q^{28} + 12 q^{29} - q^{30} + q^{31} + 9 q^{32} - 28 q^{33} - 12 q^{34} - 18 q^{35} + 10 q^{36} - 25 q^{37} + 9 q^{38} + q^{40} + 11 q^{41} + 18 q^{42} - 10 q^{43} - 13 q^{44} - q^{45} - 22 q^{46} - 12 q^{47} - 5 q^{48} + 2 q^{49} + 4 q^{50} + 35 q^{51} + 9 q^{53} - 26 q^{54} + 18 q^{55} - 13 q^{56} - 5 q^{57} + 12 q^{58} + 10 q^{59} - q^{60} + 32 q^{61} + q^{62} - 63 q^{63} + 9 q^{64} - 28 q^{66} - 73 q^{67} - 12 q^{68} + 2 q^{69} - 18 q^{70} - 51 q^{71} + 10 q^{72} - 14 q^{73} - 25 q^{74} - 49 q^{75} + 9 q^{76} + 18 q^{77} - 28 q^{79} + q^{80} + 29 q^{81} + 11 q^{82} - 22 q^{83} + 18 q^{84} - 51 q^{85} - 10 q^{86} - 20 q^{87} - 13 q^{88} + 3 q^{89} - q^{90} - 22 q^{92} - 59 q^{93} - 12 q^{94} + q^{95} - 5 q^{96} + 2 q^{98} + 25 q^{99}+O(q^{100}) \) 9 * q + 9 * q^2 - 5 * q^3 + 9 * q^4 + q^5 - 5 * q^6 - 13 * q^7 + 9 * q^8 + 10 * q^9 + q^10 - 13 * q^11 - 5 * q^12 - 13 * q^14 - q^15 + 9 * q^16 - 12 * q^17 + 10 * q^18 + 9 * q^19 + q^20 + 18 * q^21 - 13 * q^22 - 22 * q^23 - 5 * q^24 + 4 * q^25 - 26 * q^27 - 13 * q^28 + 12 * q^29 - q^30 + q^31 + 9 * q^32 - 28 * q^33 - 12 * q^34 - 18 * q^35 + 10 * q^36 - 25 * q^37 + 9 * q^38 + q^40 + 11 * q^41 + 18 * q^42 - 10 * q^43 - 13 * q^44 - q^45 - 22 * q^46 - 12 * q^47 - 5 * q^48 + 2 * q^49 + 4 * q^50 + 35 * q^51 + 9 * q^53 - 26 * q^54 + 18 * q^55 - 13 * q^56 - 5 * q^57 + 12 * q^58 + 10 * q^59 - q^60 + 32 * q^61 + q^62 - 63 * q^63 + 9 * q^64 - 28 * q^66 - 73 * q^67 - 12 * q^68 + 2 * q^69 - 18 * q^70 - 51 * q^71 + 10 * q^72 - 14 * q^73 - 25 * q^74 - 49 * q^75 + 9 * q^76 + 18 * q^77 - 28 * q^79 + q^80 + 29 * q^81 + 11 * q^82 - 22 * q^83 + 18 * q^84 - 51 * q^85 - 10 * q^86 - 20 * q^87 - 13 * q^88 + 3 * q^89 - q^90 - 22 * q^92 - 59 * q^93 - 12 * q^94 + q^95 - 5 * q^96 + 2 * q^98 + 25 * 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
$3$	−1.60929	−0.929121	−0.464561	−	0.885541i	$-0.653788\pi$
$3$	−1.60929	−0.929121	−0.464561	+	0.885541i	$0.653788\pi$
$4$	1.00000	0.500000
$5$	−4.22571	−1.88979	−0.944897	−	0.327367i	$-0.893839\pi$
$5$	−4.22571	−1.88979	−0.944897	+	0.327367i	$0.893839\pi$
$6$	−1.60929	−0.656988
$7$	0.325660	0.123088	0.0615439	−	0.998104i	$-0.480398\pi$
$7$	0.325660	0.123088	0.0615439	+	0.998104i	$0.480398\pi$
$8$	1.00000	0.353553
$9$	−0.410201	−0.136734
$10$	−4.22571	−1.33629
$11$	−3.83154	−1.15525	−0.577627	−	0.816301i	$-0.696021\pi$
$11$	−3.83154	−1.15525	−0.577627	+	0.816301i	$0.696021\pi$
$12$	−1.60929	−0.464561
$13$	0	0
$13$	0	0
$14$	0.325660	0.0870362
$15$	6.80037	1.75585
$16$	1.00000	0.250000
$17$	2.40765	0.583942	0.291971	−	0.956427i	$-0.405689\pi$
$17$	2.40765	0.583942	0.291971	+	0.956427i	$0.405689\pi$
$18$	−0.410201	−0.0966853
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−4.22571	−0.944897
$21$	−0.524079	−0.114363
$22$	−3.83154	−0.816888
$23$	2.11166	0.440311	0.220155	−	0.975465i	$-0.429344\pi$
$23$	2.11166	0.440311	0.220155	+	0.975465i	$0.429344\pi$
$24$	−1.60929	−0.328494
$25$	12.8566	2.57132
$26$	0	0
$27$	5.48799	1.05616
$28$	0.325660	0.0615439
$29$	7.46895	1.38695	0.693474	−	0.720481i	$-0.256079\pi$
$29$	7.46895	1.38695	0.693474	+	0.720481i	$0.256079\pi$
$30$	6.80037	1.24157
$31$	−4.53149	−0.813879	−0.406940	−	0.913455i	$-0.633404\pi$
$31$	−4.53149	−0.813879	−0.406940	+	0.913455i	$0.633404\pi$
$32$	1.00000	0.176777
$33$	6.16605	1.07337
$34$	2.40765	0.412909
$35$	−1.37614	−0.232611
$36$	−0.410201	−0.0683668
$37$	2.51608	0.413641	0.206821	−	0.978379i	$-0.433688\pi$
$37$	2.51608	0.413641	0.206821	+	0.978379i	$0.433688\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	−4.22571	−0.668143
$41$	8.17536	1.27678	0.638389	−	0.769714i	$-0.279601\pi$
$41$	8.17536	1.27678	0.638389	+	0.769714i	$0.279601\pi$
$42$	−0.524079	−0.0808672
$43$	−8.05774	−1.22879	−0.614397	−	0.788997i	$-0.710601\pi$
$43$	−8.05774	−1.22879	−0.614397	+	0.788997i	$0.710601\pi$
$44$	−3.83154	−0.577627
$45$	1.73339	0.258399
$46$	2.11166	0.311347
$47$	−1.42557	−0.207941	−0.103971	−	0.994580i	$-0.533155\pi$
$47$	−1.42557	−0.207941	−0.103971	+	0.994580i	$0.533155\pi$
$48$	−1.60929	−0.232280
$49$	−6.89395	−0.984849
$50$	12.8566	1.81820
$51$	−3.87460	−0.542553
$52$	0	0
$53$	8.33758	1.14526	0.572628	−	0.819816i	$-0.305924\pi$
$53$	8.33758	1.14526	0.572628	+	0.819816i	$0.305924\pi$
$54$	5.48799	0.746820
$55$	16.1910	2.18319
$56$	0.325660	0.0435181
$57$	−1.60929	−0.213155
$58$	7.46895	0.980721
$59$	6.88289	0.896076	0.448038	−	0.894015i	$-0.352123\pi$
$59$	6.88289	0.896076	0.448038	+	0.894015i	$0.352123\pi$
$60$	6.80037	0.877924
$61$	9.16884	1.17395	0.586975	−	0.809605i	$-0.300319\pi$
$61$	9.16884	1.17395	0.586975	+	0.809605i	$0.300319\pi$
$62$	−4.53149	−0.575500
$63$	−0.133586	−0.0168302
$64$	1.00000	0.125000
$65$	0	0
$66$	6.16605	0.758988
$67$	−1.83754	−0.224491	−0.112246	−	0.993680i	$-0.535804\pi$
$67$	−1.83754	−0.224491	−0.112246	+	0.993680i	$0.535804\pi$
$68$	2.40765	0.291971
$69$	−3.39826	−0.409102
$70$	−1.37614	−0.164480
$71$	−11.6905	−1.38741	−0.693703	−	0.720261i	$-0.744022\pi$
$71$	−11.6905	−1.38741	−0.693703	+	0.720261i	$0.744022\pi$
$72$	−0.410201	−0.0483426
$73$	−12.3174	−1.44164	−0.720820	−	0.693122i	$-0.756235\pi$
$73$	−12.3174	−1.44164	−0.720820	+	0.693122i	$0.756235\pi$
$74$	2.51608	0.292488
$75$	−20.6900	−2.38907
$76$	1.00000	0.114708
$77$	−1.24778	−0.142198
$78$	0	0
$79$	−14.2281	−1.60078	−0.800392	−	0.599477i	$-0.795375\pi$
$79$	−14.2281	−1.60078	−0.800392	+	0.599477i	$0.795375\pi$
$80$	−4.22571	−0.472449
$81$	−7.60113	−0.844570
$82$	8.17536	0.902818
$83$	−6.05693	−0.664834	−0.332417	−	0.943132i	$-0.607864\pi$
$83$	−6.05693	−0.664834	−0.332417	+	0.943132i	$0.607864\pi$
$84$	−0.524079	−0.0571817
$85$	−10.1740	−1.10353
$86$	−8.05774	−0.868889
$87$	−12.0197	−1.28864
$88$	−3.83154	−0.408444
$89$	0.553738	0.0586961	0.0293481	−	0.999569i	$-0.490657\pi$
$89$	0.553738	0.0586961	0.0293481	+	0.999569i	$0.490657\pi$
$90$	1.73339	0.182715
$91$	0	0
$92$	2.11166	0.220155
$93$	7.29246	0.756193
$94$	−1.42557	−0.147037
$95$	−4.22571	−0.433549
$96$	−1.60929	−0.164247
$97$	−2.52157	−0.256027	−0.128013	−	0.991772i	$-0.540860\pi$
$97$	−2.52157	−0.256027	−0.128013	+	0.991772i	$0.540860\pi$
$98$	−6.89395	−0.696394
$99$	1.57170	0.157962
$100$	12.8566	1.28566
$101$	2.96309	0.294838	0.147419	−	0.989074i	$-0.452903\pi$
$101$	2.96309	0.294838	0.147419	+	0.989074i	$0.452903\pi$
$102$	−3.87460	−0.383643
$103$	5.15590	0.508026	0.254013	−	0.967201i	$-0.418249\pi$
$103$	5.15590	0.508026	0.254013	+	0.967201i	$0.418249\pi$
$104$	0	0
$105$	2.21461	0.216123
$106$	8.33758	0.809818
$107$	4.33755	0.419327	0.209663	−	0.977774i	$-0.432763\pi$
$107$	4.33755	0.419327	0.209663	+	0.977774i	$0.432763\pi$
$108$	5.48799	0.528082
$109$	15.5537	1.48977	0.744887	−	0.667191i	$-0.232503\pi$
$109$	15.5537	1.48977	0.744887	+	0.667191i	$0.232503\pi$
$110$	16.1910	1.54375
$111$	−4.04909	−0.384323
$112$	0.325660	0.0307719
$113$	−4.45253	−0.418859	−0.209430	−	0.977824i	$-0.567161\pi$
$113$	−4.45253	−0.418859	−0.209430	+	0.977824i	$0.567161\pi$
$114$	−1.60929	−0.150723
$115$	−8.92324	−0.832097
$116$	7.46895	0.693474
$117$	0	0
$118$	6.88289	0.633621
$119$	0.784075	0.0718761
$120$	6.80037	0.620786
$121$	3.68073	0.334612
$122$	9.16884	0.830108
$123$	−13.1565	−1.18628
$124$	−4.53149	−0.406940
$125$	−33.1998	−2.96948
$126$	−0.133586	−0.0119008
$127$	−9.13119	−0.810262	−0.405131	−	0.914259i	$-0.632774\pi$
$127$	−9.13119	−0.810262	−0.405131	+	0.914259i	$0.632774\pi$
$128$	1.00000	0.0883883
$129$	12.9672	1.14170
$130$	0	0
$131$	17.8533	1.55985	0.779923	−	0.625875i	$-0.215258\pi$
$131$	17.8533	1.55985	0.779923	+	0.625875i	$0.215258\pi$
$132$	6.16605	0.536686
$133$	0.325660	0.0282383
$134$	−1.83754	−0.158739
$135$	−23.1906	−1.99593
$136$	2.40765	0.206455
$137$	−16.1741	−1.38185	−0.690923	−	0.722929i	$-0.742795\pi$
$137$	−16.1741	−1.38185	−0.690923	+	0.722929i	$0.742795\pi$
$138$	−3.39826	−0.289279
$139$	−1.60624	−0.136240	−0.0681199	−	0.997677i	$-0.521700\pi$
$139$	−1.60624	−0.136240	−0.0681199	+	0.997677i	$0.521700\pi$
$140$	−1.37614	−0.116305
$141$	2.29416	0.193203
$142$	−11.6905	−0.981045
$143$	0	0
$144$	−0.410201	−0.0341834
$145$	−31.5616	−2.62105
$146$	−12.3174	−1.01939
$147$	11.0943	0.915045
$148$	2.51608	0.206821
$149$	−21.2606	−1.74174	−0.870868	−	0.491517i	$-0.836442\pi$
$149$	−21.2606	−1.74174	−0.870868	+	0.491517i	$0.836442\pi$
$150$	−20.6900	−1.68933
$151$	14.7653	1.20158	0.600790	−	0.799407i	$-0.294853\pi$
$151$	14.7653	1.20158	0.600790	+	0.799407i	$0.294853\pi$
$152$	1.00000	0.0811107
$153$	−0.987622	−0.0798445
$154$	−1.24778	−0.100549
$155$	19.1488	1.53807
$156$	0	0
$157$	−13.3640	−1.06656	−0.533282	−	0.845938i	$-0.679042\pi$
$157$	−13.3640	−1.06656	−0.533282	+	0.845938i	$0.679042\pi$
$158$	−14.2281	−1.13193
$159$	−13.4175	−1.06408
$160$	−4.22571	−0.334072
$161$	0.687681	0.0541968
$162$	−7.60113	−0.597201
$163$	19.5142	1.52847	0.764234	−	0.644939i	$-0.223117\pi$
$163$	19.5142	1.52847	0.764234	+	0.644939i	$0.223117\pi$
$164$	8.17536	0.638389
$165$	−26.0559	−2.02845
$166$	−6.05693	−0.470109
$167$	8.56080	0.662455	0.331227	−	0.943551i	$-0.392537\pi$
$167$	8.56080	0.662455	0.331227	+	0.943551i	$0.392537\pi$
$168$	−0.524079	−0.0404336
$169$	0	0
$170$	−10.1740	−0.780314
$171$	−0.410201	−0.0313689
$172$	−8.05774	−0.614397
$173$	9.96311	0.757481	0.378740	−	0.925503i	$-0.376357\pi$
$173$	9.96311	0.757481	0.378740	+	0.925503i	$0.376357\pi$
$174$	−12.0197	−0.911208
$175$	4.18688	0.316498
$176$	−3.83154	−0.288813
$177$	−11.0765	−0.832563
$178$	0.553738	0.0415044
$179$	−14.4182	−1.07766	−0.538832	−	0.842413i	$-0.681134\pi$
$179$	−14.4182	−1.07766	−0.538832	+	0.842413i	$0.681134\pi$
$180$	1.73339	0.129199
$181$	3.50318	0.260390	0.130195	−	0.991488i	$-0.458440\pi$
$181$	3.50318	0.260390	0.130195	+	0.991488i	$0.458440\pi$
$182$	0	0
$183$	−14.7553	−1.09074
$184$	2.11166	0.155673
$185$	−10.6322	−0.781697
$186$	7.29246	0.534709
$187$	−9.22503	−0.674601
$188$	−1.42557	−0.103971
$189$	1.78721	0.130001
$190$	−4.22571	−0.306565
$191$	−22.0384	−1.59464	−0.797322	−	0.603554i	$-0.793751\pi$
$191$	−22.0384	−1.59464	−0.797322	+	0.603554i	$0.793751\pi$
$192$	−1.60929	−0.116140
$193$	−14.6276	−1.05292	−0.526458	−	0.850201i	$-0.676480\pi$
$193$	−14.6276	−1.05292	−0.526458	+	0.850201i	$0.676480\pi$
$194$	−2.52157	−0.181038
$195$	0	0
$196$	−6.89395	−0.492425
$197$	−16.3289	−1.16338	−0.581692	−	0.813409i	$-0.697609\pi$
$197$	−16.3289	−1.16338	−0.581692	+	0.813409i	$0.697609\pi$
$198$	1.57170	0.111696
$199$	−11.6780	−0.827834	−0.413917	−	0.910315i	$-0.635840\pi$
$199$	−11.6780	−0.827834	−0.413917	+	0.910315i	$0.635840\pi$
$200$	12.8566	0.909100
$201$	2.95713	0.208580
$202$	2.96309	0.208482
$203$	2.43233	0.170716
$204$	−3.87460	−0.271276
$205$	−34.5467	−2.41285
$206$	5.15590	0.359228
$207$	−0.866203	−0.0602053
$208$	0	0
$209$	−3.83154	−0.265033
$210$	2.21461	0.152822
$211$	−13.4739	−0.927580	−0.463790	−	0.885945i	$-0.653511\pi$
$211$	−13.4739	−0.927580	−0.463790	+	0.885945i	$0.653511\pi$
$212$	8.33758	0.572628
$213$	18.8133	1.28907
$214$	4.33755	0.296509
$215$	34.0497	2.32217
$216$	5.48799	0.373410
$217$	−1.47572	−0.100179
$218$	15.5537	1.05343
$219$	19.8222	1.33946
$220$	16.1910	1.09160
$221$	0	0
$222$	−4.04909	−0.271757
$223$	−11.0855	−0.742341	−0.371171	−	0.928565i	$-0.621044\pi$
$223$	−11.0855	−0.742341	−0.371171	+	0.928565i	$0.621044\pi$
$224$	0.325660	0.0217590
$225$	−5.27380	−0.351587
$226$	−4.45253	−0.296178
$227$	24.2484	1.60942	0.804712	−	0.593665i	$-0.202320\pi$
$227$	24.2484	1.60942	0.804712	+	0.593665i	$0.202320\pi$
$228$	−1.60929	−0.106578
$229$	21.6584	1.43123	0.715613	−	0.698497i	$-0.246147\pi$
$229$	21.6584	1.43123	0.715613	+	0.698497i	$0.246147\pi$
$230$	−8.92324	−0.588381
$231$	2.00803	0.132119
$232$	7.46895	0.490360
$233$	23.9574	1.56950	0.784750	−	0.619812i	$-0.212791\pi$
$233$	23.9574	1.56950	0.784750	+	0.619812i	$0.212791\pi$
$234$	0	0
$235$	6.02406	0.392967
$236$	6.88289	0.448038
$237$	22.8970	1.48732
$238$	0.784075	0.0508241
$239$	−16.7636	−1.08435	−0.542173	−	0.840267i	$-0.682398\pi$
$239$	−16.7636	−1.08435	−0.542173	+	0.840267i	$0.682398\pi$
$240$	6.80037	0.438962
$241$	0.274712	0.0176958	0.00884789	−	0.999961i	$-0.497184\pi$
$241$	0.274712	0.0176958	0.00884789	+	0.999961i	$0.497184\pi$
$242$	3.68073	0.236606
$243$	−4.23157	−0.271455
$244$	9.16884	0.586975
$245$	29.1318	1.86116
$246$	−13.1565	−0.838827
$247$	0	0
$248$	−4.53149	−0.287750
$249$	9.74733	0.617712
$250$	−33.1998	−2.09974
$251$	10.8633	0.685688	0.342844	−	0.939392i	$-0.388610\pi$
$251$	10.8633	0.685688	0.342844	+	0.939392i	$0.388610\pi$
$252$	−0.133586	−0.00841512
$253$	−8.09090	−0.508671
$254$	−9.13119	−0.572942
$255$	16.3729	1.02531
$256$	1.00000	0.0625000
$257$	19.6530	1.22592	0.612961	−	0.790114i	$-0.289978\pi$
$257$	19.6530	1.22592	0.612961	+	0.790114i	$0.289978\pi$
$258$	12.9672	0.807303
$259$	0.819386	0.0509142
$260$	0	0
$261$	−3.06377	−0.189643
$262$	17.8533	1.10298
$263$	13.4522	0.829496	0.414748	−	0.909936i	$-0.363870\pi$
$263$	13.4522	0.829496	0.414748	+	0.909936i	$0.363870\pi$
$264$	6.16605	0.379494
$265$	−35.2322	−2.16430
$266$	0.325660	0.0199675
$267$	−0.891122	−0.0545358
$268$	−1.83754	−0.112246
$269$	−8.48951	−0.517615	−0.258807	−	0.965929i	$-0.583329\pi$
$269$	−8.48951	−0.517615	−0.258807	+	0.965929i	$0.583329\pi$
$270$	−23.1906	−1.41134
$271$	−4.74689	−0.288353	−0.144177	−	0.989552i	$-0.546053\pi$
$271$	−4.74689	−0.288353	−0.144177	+	0.989552i	$0.546053\pi$
$272$	2.40765	0.145985
$273$	0	0
$274$	−16.1741	−0.977112
$275$	−49.2607	−2.97053
$276$	−3.39826	−0.204551
$277$	−15.0479	−0.904141	−0.452070	−	0.891982i	$-0.649314\pi$
$277$	−15.0479	−0.904141	−0.452070	+	0.891982i	$0.649314\pi$
$278$	−1.60624	−0.0963360
$279$	1.85882	0.111285
$280$	−1.37614	−0.0822402
$281$	20.8067	1.24123	0.620613	−	0.784117i	$-0.286884\pi$
$281$	20.8067	1.24123	0.620613	+	0.784117i	$0.286884\pi$
$282$	2.29416	0.136615
$283$	5.34550	0.317757	0.158878	−	0.987298i	$-0.449212\pi$
$283$	5.34550	0.317757	0.158878	+	0.987298i	$0.449212\pi$
$284$	−11.6905	−0.693703
$285$	6.80037	0.402819
$286$	0	0
$287$	2.66239	0.157156
$288$	−0.410201	−0.0241713
$289$	−11.2032	−0.659012
$290$	−31.5616	−1.85336
$291$	4.05792	0.237880
$292$	−12.3174	−0.720820
$293$	−23.7099	−1.38515	−0.692573	−	0.721348i	$-0.743523\pi$
$293$	−23.7099	−1.38515	−0.692573	+	0.721348i	$0.743523\pi$
$294$	11.0943	0.647034
$295$	−29.0851	−1.69340
$296$	2.51608	0.146244
$297$	−21.0275	−1.22014
$298$	−21.2606	−1.23159
$299$	0	0
$300$	−20.6900	−1.19454
$301$	−2.62408	−0.151250
$302$	14.7653	0.849645
$303$	−4.76845	−0.273940
$304$	1.00000	0.0573539
$305$	−38.7448	−2.21852
$306$	−0.987622	−0.0564586
$307$	31.6659	1.80727	0.903634	−	0.428306i	$-0.140889\pi$
$307$	31.6659	1.80727	0.903634	+	0.428306i	$0.140889\pi$
$308$	−1.24778	−0.0710988
$309$	−8.29731	−0.472017
$310$	19.1488	1.08758
$311$	−23.0955	−1.30963	−0.654814	−	0.755790i	$-0.727253\pi$
$311$	−23.0955	−1.30963	−0.654814	+	0.755790i	$0.727253\pi$
$312$	0	0
$313$	10.0895	0.570292	0.285146	−	0.958484i	$-0.407958\pi$
$313$	10.0895	0.570292	0.285146	+	0.958484i	$0.407958\pi$
$314$	−13.3640	−0.754175
$315$	0.564495	0.0318057
$316$	−14.2281	−0.800392
$317$	16.9424	0.951578	0.475789	−	0.879560i	$-0.342163\pi$
$317$	16.9424	0.951578	0.475789	+	0.879560i	$0.342163\pi$
$318$	−13.4175	−0.752419
$319$	−28.6176	−1.60228
$320$	−4.22571	−0.236224
$321$	−6.98035	−0.389605
$322$	0.687681	0.0383230
$323$	2.40765	0.133965
$324$	−7.60113	−0.422285
$325$	0	0
$326$	19.5142	1.08079
$327$	−25.0303	−1.38418
$328$	8.17536	0.451409
$329$	−0.464252	−0.0255950
$330$	−26.0559	−1.43433
$331$	−27.4469	−1.50862	−0.754309	−	0.656519i	$-0.772028\pi$
$331$	−27.4469	−1.50862	−0.754309	+	0.656519i	$0.772028\pi$
$332$	−6.05693	−0.332417
$333$	−1.03210	−0.0565587
$334$	8.56080	0.468426
$335$	7.76491	0.424242
$336$	−0.524079	−0.0285909
$337$	14.4920	0.789432	0.394716	−	0.918803i	$-0.370843\pi$
$337$	14.4920	0.789432	0.394716	+	0.918803i	$0.370843\pi$
$338$	0	0
$339$	7.16540	0.389171
$340$	−10.1740	−0.551765
$341$	17.3626	0.940237
$342$	−0.410201	−0.0221811
$343$	−4.52470	−0.244311
$344$	−8.05774	−0.434444
$345$	14.3600	0.773119
$346$	9.96311	0.535620
$347$	−8.33842	−0.447630	−0.223815	−	0.974632i	$-0.571851\pi$
$347$	−8.33842	−0.447630	−0.223815	+	0.974632i	$0.571851\pi$
$348$	−12.0197	−0.644322
$349$	−29.1946	−1.56275	−0.781376	−	0.624060i	$-0.785482\pi$
$349$	−29.1946	−1.56275	−0.781376	+	0.624060i	$0.785482\pi$
$350$	4.18688	0.223798
$351$	0	0
$352$	−3.83154	−0.204222
$353$	7.79836	0.415065	0.207532	−	0.978228i	$-0.433457\pi$
$353$	7.79836	0.415065	0.207532	+	0.978228i	$0.433457\pi$
$354$	−11.0765	−0.588711
$355$	49.4006	2.62191
$356$	0.553738	0.0293481
$357$	−1.26180	−0.0667816
$358$	−14.4182	−0.762024
$359$	−5.86873	−0.309740	−0.154870	−	0.987935i	$-0.549496\pi$
$359$	−5.86873	−0.309740	−0.154870	+	0.987935i	$0.549496\pi$
$360$	1.73339	0.0913577
$361$	1.00000	0.0526316
$362$	3.50318	0.184123
$363$	−5.92334	−0.310895
$364$	0	0
$365$	52.0497	2.72440
$366$	−14.7553	−0.771271
$367$	−28.8228	−1.50454	−0.752268	−	0.658857i	$-0.771040\pi$
$367$	−28.8228	−1.50454	−0.752268	+	0.658857i	$0.771040\pi$
$368$	2.11166	0.110078
$369$	−3.35354	−0.174578
$370$	−10.6322	−0.552743
$371$	2.71521	0.140967
$372$	7.29246	0.378096
$373$	33.0722	1.71241	0.856207	−	0.516633i	$-0.172815\pi$
$373$	33.0722	1.71241	0.856207	+	0.516633i	$0.172815\pi$
$374$	−9.22503	−0.477015
$375$	53.4279	2.75901
$376$	−1.42557	−0.0735184
$377$	0	0
$378$	1.78721	0.0919244
$379$	−8.45318	−0.434211	−0.217105	−	0.976148i	$-0.569662\pi$
$379$	−8.45318	−0.434211	−0.217105	+	0.976148i	$0.569662\pi$
$380$	−4.22571	−0.216774
$381$	14.6947	0.752832
$382$	−22.0384	−1.12758
$383$	20.5157	1.04830	0.524152	−	0.851625i	$-0.324382\pi$
$383$	20.5157	1.04830	0.524152	+	0.851625i	$0.324382\pi$
$384$	−1.60929	−0.0821235
$385$	5.27275	0.268724
$386$	−14.6276	−0.744524
$387$	3.30529	0.168018
$388$	−2.52157	−0.128013
$389$	−21.3584	−1.08291	−0.541457	−	0.840728i	$-0.682127\pi$
$389$	−21.3584	−1.08291	−0.541457	+	0.840728i	$0.682127\pi$
$390$	0	0
$391$	5.08414	0.257116
$392$	−6.89395	−0.348197
$393$	−28.7310	−1.44929
$394$	−16.3289	−0.822637
$395$	60.1237	3.02515
$396$	1.57170	0.0789810
$397$	−17.2778	−0.867146	−0.433573	−	0.901118i	$-0.642747\pi$
$397$	−17.2778	−0.867146	−0.433573	+	0.901118i	$0.642747\pi$
$398$	−11.6780	−0.585367
$399$	−0.524079	−0.0262368
$400$	12.8566	0.642831
$401$	−18.3929	−0.918498	−0.459249	−	0.888308i	$-0.651881\pi$
$401$	−18.3929	−0.918498	−0.459249	+	0.888308i	$0.651881\pi$
$402$	2.95713	0.147488
$403$	0	0
$404$	2.96309	0.147419
$405$	32.1202	1.59606
$406$	2.43233	0.120715
$407$	−9.64048	−0.477861
$408$	−3.87460	−0.191821
$409$	−36.9686	−1.82798	−0.913989	−	0.405739i	$-0.867014\pi$
$409$	−36.9686	−1.82798	−0.913989	+	0.405739i	$0.867014\pi$
$410$	−34.5467	−1.70614
$411$	26.0287	1.28390
$412$	5.15590	0.254013
$413$	2.24148	0.110296
$414$	−0.866203	−0.0425716
$415$	25.5948	1.25640
$416$	0	0
$417$	2.58490	0.126583
$418$	−3.83154	−0.187407
$419$	−5.60299	−0.273724	−0.136862	−	0.990590i	$-0.543702\pi$
$419$	−5.60299	−0.273724	−0.136862	+	0.990590i	$0.543702\pi$
$420$	2.21461	0.108062
$421$	18.4644	0.899902	0.449951	−	0.893053i	$-0.351441\pi$
$421$	18.4644	0.899902	0.449951	+	0.893053i	$0.351441\pi$
$422$	−13.4739	−0.655898
$423$	0.584772	0.0284326
$424$	8.33758	0.404909
$425$	30.9543	1.50150
$426$	18.8133	0.911510
$427$	2.98592	0.144499
$428$	4.33755	0.209663
$429$	0	0
$430$	34.0497	1.64202
$431$	−19.9799	−0.962396	−0.481198	−	0.876612i	$-0.659798\pi$
$431$	−19.9799	−0.962396	−0.481198	+	0.876612i	$0.659798\pi$
$432$	5.48799	0.264041
$433$	−25.3519	−1.21834	−0.609168	−	0.793041i	$-0.708496\pi$
$433$	−25.3519	−1.21834	−0.609168	+	0.793041i	$0.708496\pi$
$434$	−1.47572	−0.0708370
$435$	50.7916	2.43527
$436$	15.5537	0.744887
$437$	2.11166	0.101014
$438$	19.8222	0.947140
$439$	16.1022	0.768518	0.384259	−	0.923225i	$-0.374457\pi$
$439$	16.1022	0.768518	0.384259	+	0.923225i	$0.374457\pi$
$440$	16.1910	0.771875
$441$	2.82790	0.134662
$442$	0	0
$443$	−4.43856	−0.210882	−0.105441	−	0.994426i	$-0.533625\pi$
$443$	−4.43856	−0.210882	−0.105441	+	0.994426i	$0.533625\pi$
$444$	−4.04909	−0.192161
$445$	−2.33994	−0.110924
$446$	−11.0855	−0.524915
$447$	34.2144	1.61828
$448$	0.325660	0.0153860
$449$	32.1020	1.51499	0.757493	−	0.652844i	$-0.226424\pi$
$449$	32.1020	1.51499	0.757493	+	0.652844i	$0.226424\pi$
$450$	−5.27380	−0.248609
$451$	−31.3243	−1.47500
$452$	−4.45253	−0.209430
$453$	−23.7615	−1.11641
$454$	24.2484	1.13803
$455$	0	0
$456$	−1.60929	−0.0753617
$457$	−25.4958	−1.19264	−0.596322	−	0.802745i	$-0.703372\pi$
$457$	−25.4958	−1.19264	−0.596322	+	0.802745i	$0.703372\pi$
$458$	21.6584	1.01203
$459$	13.2132	0.616738
$460$	−8.92324	−0.416048
$461$	−25.2496	−1.17599	−0.587995	−	0.808864i	$-0.700083\pi$
$461$	−25.2496	−1.17599	−0.587995	+	0.808864i	$0.700083\pi$
$462$	2.00803	0.0934221
$463$	−4.03431	−0.187490	−0.0937451	−	0.995596i	$-0.529884\pi$
$463$	−4.03431	−0.187490	−0.0937451	+	0.995596i	$0.529884\pi$
$464$	7.46895	0.346737
$465$	−30.8158	−1.42905
$466$	23.9574	1.10980
$467$	−17.5574	−0.812458	−0.406229	−	0.913771i	$-0.633156\pi$
$467$	−17.5574	−0.812458	−0.406229	+	0.913771i	$0.633156\pi$
$468$	0	0
$469$	−0.598413	−0.0276321
$470$	6.02406	0.277869
$471$	21.5065	0.990967
$472$	6.88289	0.316811
$473$	30.8736	1.41957
$474$	22.8970	1.05170
$475$	12.8566	0.589902
$476$	0.784075	0.0359380
$477$	−3.42008	−0.156595
$478$	−16.7636	−0.766749
$479$	−23.8961	−1.09184	−0.545921	−	0.837837i	$-0.683820\pi$
$479$	−23.8961	−1.09184	−0.545921	+	0.837837i	$0.683820\pi$
$480$	6.80037	0.310393
$481$	0	0
$482$	0.274712	0.0125128
$483$	−1.10667	−0.0503554
$484$	3.68073	0.167306
$485$	10.6554	0.483838
$486$	−4.23157	−0.191948
$487$	−27.7752	−1.25862	−0.629308	−	0.777156i	$-0.716662\pi$
$487$	−27.7752	−1.25862	−0.629308	+	0.777156i	$0.716662\pi$
$488$	9.16884	0.415054
$489$	−31.4039	−1.42013
$490$	29.1318	1.31604
$491$	−3.57205	−0.161204	−0.0806021	−	0.996746i	$-0.525684\pi$
$491$	−3.57205	−0.161204	−0.0806021	+	0.996746i	$0.525684\pi$
$492$	−13.1565	−0.593141
$493$	17.9826	0.809897
$494$	0	0
$495$	−6.64156	−0.298516
$496$	−4.53149	−0.203470
$497$	−3.80712	−0.170773
$498$	9.74733	0.436788
$499$	−39.7897	−1.78123	−0.890616	−	0.454757i	$-0.849726\pi$
$499$	−39.7897	−1.78123	−0.890616	+	0.454757i	$0.849726\pi$
$500$	−33.1998	−1.48474
$501$	−13.7768	−0.615501
$502$	10.8633	0.484854
$503$	31.2338	1.39265	0.696323	−	0.717729i	$-0.254818\pi$
$503$	31.2338	1.39265	0.696323	+	0.717729i	$0.254818\pi$
$504$	−0.133586	−0.00595039
$505$	−12.5211	−0.557184
$506$	−8.09090	−0.359684
$507$	0	0
$508$	−9.13119	−0.405131
$509$	28.5304	1.26459	0.632294	−	0.774729i	$-0.282113\pi$
$509$	28.5304	1.26459	0.632294	+	0.774729i	$0.282113\pi$
$510$	16.3729	0.725006
$511$	−4.01127	−0.177448
$512$	1.00000	0.0441942
$513$	5.48799	0.242301
$514$	19.6530	0.866857
$515$	−21.7873	−0.960064
$516$	12.9672	0.570850
$517$	5.46215	0.240225
$518$	0.819386	0.0360017
$519$	−16.0335	−0.703792
$520$	0	0
$521$	−1.48746	−0.0651667	−0.0325833	−	0.999469i	$-0.510373\pi$
$521$	−1.48746	−0.0651667	−0.0325833	+	0.999469i	$0.510373\pi$
$522$	−3.06377	−0.134098
$523$	−6.43369	−0.281326	−0.140663	−	0.990058i	$-0.544923\pi$
$523$	−6.43369	−0.281326	−0.140663	+	0.990058i	$0.544923\pi$
$524$	17.8533	0.779923
$525$	−6.73789	−0.294065
$526$	13.4522	0.586542
$527$	−10.9103	−0.475258
$528$	6.16605	0.268343
$529$	−18.5409	−0.806126
$530$	−35.2322	−1.53039
$531$	−2.82337	−0.122524
$532$	0.325660	0.0141191
$533$	0	0
$534$	−0.891122	−0.0385626
$535$	−18.3292	−0.792441
$536$	−1.83754	−0.0793697
$537$	23.2029	1.00128
$538$	−8.48951	−0.366009
$539$	26.4145	1.13775
$540$	−23.1906	−0.997966
$541$	1.70637	0.0733626	0.0366813	−	0.999327i	$-0.488321\pi$
$541$	1.70637	0.0733626	0.0366813	+	0.999327i	$0.488321\pi$
$542$	−4.74689	−0.203896
$543$	−5.63762	−0.241934
$544$	2.40765	0.103227
$545$	−65.7254	−2.81537
$546$	0	0
$547$	6.20005	0.265095	0.132548	−	0.991177i	$-0.457684\pi$
$547$	6.20005	0.265095	0.132548	+	0.991177i	$0.457684\pi$
$548$	−16.1741	−0.690923
$549$	−3.76107	−0.160518
$550$	−49.2607	−2.10048
$551$	7.46895	0.318188
$552$	−3.39826	−0.144639
$553$	−4.63351	−0.197037
$554$	−15.0479	−0.639324
$555$	17.1103	0.726291
$556$	−1.60624	−0.0681199
$557$	6.99061	0.296202	0.148101	−	0.988972i	$-0.452684\pi$
$557$	6.99061	0.296202	0.148101	+	0.988972i	$0.452684\pi$
$558$	1.85882	0.0786902
$559$	0	0
$560$	−1.37614	−0.0581526
$561$	14.8457	0.626786
$562$	20.8067	0.877679
$563$	37.0748	1.56252	0.781259	−	0.624207i	$-0.214578\pi$
$563$	37.0748	1.56252	0.781259	+	0.624207i	$0.214578\pi$
$564$	2.29416	0.0966014
$565$	18.8151	0.791558
$566$	5.34550	0.224688
$567$	−2.47538	−0.103956
$568$	−11.6905	−0.490522
$569$	1.69473	0.0710466	0.0355233	−	0.999369i	$-0.488690\pi$
$569$	1.69473	0.0710466	0.0355233	+	0.999369i	$0.488690\pi$
$570$	6.80037	0.284836
$571$	−32.1571	−1.34573	−0.672867	−	0.739764i	$-0.734937\pi$
$571$	−32.1571	−1.34573	−0.672867	+	0.739764i	$0.734937\pi$
$572$	0	0
$573$	35.4661	1.48162
$574$	2.66239	0.111126
$575$	27.1488	1.13218
$576$	−0.410201	−0.0170917
$577$	−24.4857	−1.01935	−0.509677	−	0.860366i	$-0.670235\pi$
$577$	−24.4857	−1.01935	−0.509677	+	0.860366i	$0.670235\pi$
$578$	−11.2032	−0.465992
$579$	23.5400	0.978287
$580$	−31.5616	−1.31052
$581$	−1.97250	−0.0818330
$582$	4.05792	0.168206
$583$	−31.9458	−1.32306
$584$	−12.3174	−0.509697
$585$	0	0
$586$	−23.7099	−0.979446
$587$	−6.05133	−0.249765	−0.124883	−	0.992172i	$-0.539855\pi$
$587$	−6.05133	−0.249765	−0.124883	+	0.992172i	$0.539855\pi$
$588$	11.0943	0.457522
$589$	−4.53149	−0.186717
$590$	−29.0851	−1.19741
$591$	26.2778	1.08093
$592$	2.51608	0.103410
$593$	6.12462	0.251508	0.125754	−	0.992061i	$-0.459865\pi$
$593$	6.12462	0.251508	0.125754	+	0.992061i	$0.459865\pi$
$594$	−21.0275	−0.862767
$595$	−3.31328	−0.135831
$596$	−21.2606	−0.870868
$597$	18.7933	0.769158
$598$	0	0
$599$	−5.15127	−0.210475	−0.105238	−	0.994447i	$-0.533560\pi$
$599$	−5.15127	−0.210475	−0.105238	+	0.994447i	$0.533560\pi$
$600$	−20.6900	−0.844665
$601$	−6.78519	−0.276774	−0.138387	−	0.990378i	$-0.544192\pi$
$601$	−6.78519	−0.276774	−0.138387	+	0.990378i	$0.544192\pi$
$602$	−2.62408	−0.106950
$603$	0.753761	0.0306955
$604$	14.7653	0.600790
$605$	−15.5537	−0.632347
$606$	−4.76845	−0.193705
$607$	−3.71494	−0.150785	−0.0753923	−	0.997154i	$-0.524021\pi$
$607$	−3.71494	−0.150785	−0.0753923	+	0.997154i	$0.524021\pi$
$608$	1.00000	0.0405554
$609$	−3.91432	−0.158616
$610$	−38.7448	−1.56873
$611$	0	0
$612$	−0.987622	−0.0399223
$613$	1.75561	0.0709085	0.0354543	−	0.999371i	$-0.488712\pi$
$613$	1.75561	0.0709085	0.0354543	+	0.999371i	$0.488712\pi$
$614$	31.6659	1.27793
$615$	55.5955	2.24183
$616$	−1.24778	−0.0502744
$617$	−34.2166	−1.37751	−0.688753	−	0.724996i	$-0.741842\pi$
$617$	−34.2166	−1.37751	−0.688753	+	0.724996i	$0.741842\pi$
$618$	−8.29731	−0.333767
$619$	16.3633	0.657695	0.328847	−	0.944383i	$-0.393340\pi$
$619$	16.3633	0.657695	0.328847	+	0.944383i	$0.393340\pi$
$620$	19.1488	0.769033
$621$	11.5887	0.465040
$622$	−23.0955	−0.926047
$623$	0.180330	0.00722477
$624$	0	0
$625$	76.0096	3.04038
$626$	10.0895	0.403257
$627$	6.16605	0.246248
$628$	−13.3640	−0.533282
$629$	6.05785	0.241542
$630$	0.564495	0.0224900
$631$	26.2842	1.04636	0.523180	−	0.852222i	$-0.324746\pi$
$631$	26.2842	1.04636	0.523180	+	0.852222i	$0.324746\pi$
$632$	−14.2281	−0.565963
$633$	21.6833	0.861834
$634$	16.9424	0.672867
$635$	38.5857	1.53123
$636$	−13.4175	−0.532040
$637$	0	0
$638$	−28.6176	−1.13298
$639$	4.79545	0.189705
$640$	−4.22571	−0.167036
$641$	19.9045	0.786180	0.393090	−	0.919500i	$-0.371406\pi$
$641$	19.9045	0.786180	0.393090	+	0.919500i	$0.371406\pi$
$642$	−6.98035	−0.275493
$643$	−41.0151	−1.61748	−0.808740	−	0.588167i	$-0.799850\pi$
$643$	−41.0151	−1.61748	−0.808740	+	0.588167i	$0.799850\pi$
$644$	0.687681	0.0270984
$645$	−54.7957	−2.15758
$646$	2.40765	0.0947279
$647$	−4.70023	−0.184785	−0.0923926	−	0.995723i	$-0.529451\pi$
$647$	−4.70023	−0.184785	−0.0923926	+	0.995723i	$0.529451\pi$
$648$	−7.60113	−0.298601
$649$	−26.3721	−1.03520
$650$	0	0
$651$	2.37486	0.0930781
$652$	19.5142	0.764234
$653$	11.1698	0.437107	0.218554	−	0.975825i	$-0.429866\pi$
$653$	11.1698	0.437107	0.218554	+	0.975825i	$0.429866\pi$
$654$	−25.0303	−0.978764
$655$	−75.4427	−2.94779
$656$	8.17536	0.319194
$657$	5.05260	0.197121
$658$	−0.464252	−0.0180984
$659$	24.3476	0.948449	0.474225	−	0.880404i	$-0.342728\pi$
$659$	24.3476	0.948449	0.474225	+	0.880404i	$0.342728\pi$
$660$	−26.0559	−1.01423
$661$	44.7213	1.73946	0.869729	−	0.493529i	$-0.164293\pi$
$661$	44.7213	1.73946	0.869729	+	0.493529i	$0.164293\pi$
$662$	−27.4469	−1.06675
$663$	0	0
$664$	−6.05693	−0.235054
$665$	−1.37614	−0.0533645
$666$	−1.03210	−0.0399930
$667$	15.7718	0.610688
$668$	8.56080	0.331227
$669$	17.8398	0.689725
$670$	7.76491	0.299985
$671$	−35.1308	−1.35621
$672$	−0.524079	−0.0202168
$673$	−42.6875	−1.64548	−0.822741	−	0.568417i	$-0.807556\pi$
$673$	−42.6875	−1.64548	−0.822741	+	0.568417i	$0.807556\pi$
$674$	14.4920	0.558213
$675$	70.5570	2.71574
$676$	0	0
$677$	5.75560	0.221206	0.110603	−	0.993865i	$-0.464722\pi$
$677$	5.75560	0.221206	0.110603	+	0.993865i	$0.464722\pi$
$678$	7.16540	0.275185
$679$	−0.821173	−0.0315137
$680$	−10.1740	−0.390157
$681$	−39.0226	−1.49535
$682$	17.3626	0.664848
$683$	−36.0088	−1.37784	−0.688919	−	0.724839i	$-0.741914\pi$
$683$	−36.0088	−1.37784	−0.688919	+	0.724839i	$0.741914\pi$
$684$	−0.410201	−0.0156844
$685$	68.3470	2.61140
$686$	−4.52470	−0.172754
$687$	−34.8545	−1.32978
$688$	−8.05774	−0.307199
$689$	0	0
$690$	14.3600	0.546678
$691$	−33.4774	−1.27354	−0.636770	−	0.771054i	$-0.719730\pi$
$691$	−33.4774	−1.27354	−0.636770	+	0.771054i	$0.719730\pi$
$692$	9.96311	0.378740
$693$	0.511840	0.0194432
$694$	−8.33842	−0.316522
$695$	6.78751	0.257465
$696$	−12.0197	−0.455604
$697$	19.6834	0.745564
$698$	−29.1946	−1.10503
$699$	−38.5543	−1.45826
$700$	4.18688	0.158249
$701$	47.0790	1.77815	0.889075	−	0.457762i	$-0.151349\pi$
$701$	47.0790	1.77815	0.889075	+	0.457762i	$0.151349\pi$
$702$	0	0
$703$	2.51608	0.0948958
$704$	−3.83154	−0.144407
$705$	−9.69444	−0.365114
$706$	7.79836	0.293495
$707$	0.964958	0.0362910
$708$	−11.0765	−0.416282
$709$	27.6223	1.03738	0.518688	−	0.854964i	$-0.326421\pi$
$709$	27.6223	1.03738	0.518688	+	0.854964i	$0.326421\pi$
$710$	49.4006	1.85397
$711$	5.83637	0.218881
$712$	0.553738	0.0207522
$713$	−9.56895	−0.358360
$714$	−1.26180	−0.0472217
$715$	0	0
$716$	−14.4182	−0.538832
$717$	26.9774	1.00749
$718$	−5.86873	−0.219019
$719$	−21.9726	−0.819439	−0.409720	−	0.912211i	$-0.634373\pi$
$719$	−21.9726	−0.819439	−0.409720	+	0.912211i	$0.634373\pi$
$720$	1.73339	0.0645996
$721$	1.67907	0.0625317
$722$	1.00000	0.0372161
$723$	−0.442090	−0.0164415
$724$	3.50318	0.130195
$725$	96.0254	3.56629
$726$	−5.92334	−0.219836
$727$	52.6625	1.95314	0.976572	−	0.215191i	$-0.0690375\pi$
$727$	52.6625	1.95314	0.976572	+	0.215191i	$0.0690375\pi$
$728$	0	0
$729$	29.6132	1.09679
$730$	52.0497	1.92644
$731$	−19.4003	−0.717545
$732$	−14.7553	−0.545371
$733$	49.8858	1.84257	0.921287	−	0.388883i	$-0.127139\pi$
$733$	49.8858	1.84257	0.921287	+	0.388883i	$0.127139\pi$
$734$	−28.8228	−1.06387
$735$	−46.8814	−1.72925
$736$	2.11166	0.0778367
$737$	7.04062	0.259344
$738$	−3.35354	−0.123446
$739$	−25.9498	−0.954580	−0.477290	−	0.878746i	$-0.658381\pi$
$739$	−25.9498	−0.954580	−0.477290	+	0.878746i	$0.658381\pi$
$740$	−10.6322	−0.390848
$741$	0	0
$742$	2.71521	0.0996786
$743$	−1.83470	−0.0673086	−0.0336543	−	0.999434i	$-0.510715\pi$
$743$	−1.83470	−0.0673086	−0.0336543	+	0.999434i	$0.510715\pi$
$744$	7.29246	0.267355
$745$	89.8411	3.29152
$746$	33.0722	1.21086
$747$	2.48456	0.0909052
$748$	−9.22503	−0.337301
$749$	1.41256	0.0516140
$750$	53.4279	1.95091
$751$	−15.8968	−0.580081	−0.290041	−	0.957014i	$-0.593669\pi$
$751$	−15.8968	−0.580081	−0.290041	+	0.957014i	$0.593669\pi$
$752$	−1.42557	−0.0519854
$753$	−17.4822	−0.637087
$754$	0	0
$755$	−62.3937	−2.27074
$756$	1.78721	0.0650004
$757$	13.0228	0.473321	0.236660	−	0.971592i	$-0.423947\pi$
$757$	13.0228	0.473321	0.236660	+	0.971592i	$0.423947\pi$
$758$	−8.45318	−0.307033
$759$	13.0206	0.472617
$760$	−4.22571	−0.153283
$761$	−7.52367	−0.272733	−0.136366	−	0.990658i	$-0.543542\pi$
$761$	−7.52367	−0.272733	−0.136366	+	0.990658i	$0.543542\pi$
$762$	14.6947	0.532332
$763$	5.06521	0.183373
$764$	−22.0384	−0.797322
$765$	4.17340	0.150890
$766$	20.5157	0.741263
$767$	0	0
$768$	−1.60929	−0.0580701
$769$	−28.4788	−1.02697	−0.513485	−	0.858099i	$-0.671646\pi$
$769$	−28.4788	−1.02697	−0.513485	+	0.858099i	$0.671646\pi$
$770$	5.27275	0.190017
$771$	−31.6273	−1.13903
$772$	−14.6276	−0.526458
$773$	−40.1971	−1.44579	−0.722894	−	0.690959i	$-0.757189\pi$
$773$	−40.1971	−1.44579	−0.722894	+	0.690959i	$0.757189\pi$
$774$	3.30529	0.118806
$775$	−58.2596	−2.09275
$776$	−2.52157	−0.0905190
$777$	−1.31863	−0.0473054
$778$	−21.3584	−0.765736
$779$	8.17536	0.292913
$780$	0	0
$781$	44.7927	1.60281
$782$	5.08414	0.181808
$783$	40.9895	1.46484
$784$	−6.89395	−0.246212
$785$	56.4724	2.01559
$786$	−28.7310	−1.02480
$787$	6.92338	0.246792	0.123396	−	0.992358i	$-0.460621\pi$
$787$	6.92338	0.246792	0.123396	+	0.992358i	$0.460621\pi$
$788$	−16.3289	−0.581692
$789$	−21.6484	−0.770702
$790$	60.1237	2.13911
$791$	−1.45001	−0.0515564
$792$	1.57170	0.0558480
$793$	0	0
$794$	−17.2778	−0.613165
$795$	56.6987	2.01089
$796$	−11.6780	−0.413917
$797$	−30.2079	−1.07002	−0.535010	−	0.844846i	$-0.679692\pi$
$797$	−30.2079	−1.07002	−0.535010	+	0.844846i	$0.679692\pi$
$798$	−0.524079	−0.0185522
$799$	−3.43229	−0.121426
$800$	12.8566	0.454550
$801$	−0.227144	−0.00802573
$802$	−18.3929	−0.649476
$803$	47.1946	1.66546
$804$	2.95713	0.104290
$805$	−2.90594	−0.102421
$806$	0	0
$807$	13.6620	0.480927
$808$	2.96309	0.104241
$809$	5.05888	0.177861	0.0889303	−	0.996038i	$-0.471655\pi$
$809$	5.05888	0.177861	0.0889303	+	0.996038i	$0.471655\pi$
$810$	32.1202	1.12859
$811$	12.2976	0.431829	0.215914	−	0.976412i	$-0.430727\pi$
$811$	12.2976	0.431829	0.215914	+	0.976412i	$0.430727\pi$
$812$	2.43233	0.0853582
$813$	7.63910	0.267915
$814$	−9.64048	−0.337898
$815$	−82.4613	−2.88849
$816$	−3.87460	−0.135638
$817$	−8.05774	−0.281905
$818$	−36.9686	−1.29258
$819$	0	0
$820$	−34.5467	−1.20642
$821$	37.8166	1.31981	0.659904	−	0.751350i	$-0.270597\pi$
$821$	37.8166	1.31981	0.659904	+	0.751350i	$0.270597\pi$
$822$	26.0287	0.907856
$823$	24.2121	0.843982	0.421991	−	0.906600i	$-0.361331\pi$
$823$	24.2121	0.843982	0.421991	+	0.906600i	$0.361331\pi$
$824$	5.15590	0.179614
$825$	79.2745	2.75998
$826$	2.24148	0.0779910
$827$	30.4235	1.05793	0.528965	−	0.848644i	$-0.322580\pi$
$827$	30.4235	1.05793	0.528965	+	0.848644i	$0.322580\pi$
$828$	−0.866203	−0.0301026
$829$	−20.9719	−0.728385	−0.364193	−	0.931324i	$-0.618655\pi$
$829$	−20.9719	−0.728385	−0.364193	+	0.931324i	$0.618655\pi$
$830$	25.5948	0.888409
$831$	24.2164	0.840057
$832$	0	0
$833$	−16.5982	−0.575095
$834$	2.58490	0.0895078
$835$	−36.1755	−1.25190
$836$	−3.83154	−0.132517
$837$	−24.8688	−0.859590
$838$	−5.60299	−0.193552
$839$	12.6366	0.436262	0.218131	−	0.975919i	$-0.430004\pi$
$839$	12.6366	0.436262	0.218131	+	0.975919i	$0.430004\pi$
$840$	2.21461	0.0764112
$841$	26.7852	0.923626
$842$	18.4644	0.636327
$843$	−33.4840	−1.15325
$844$	−13.4739	−0.463790
$845$	0	0
$846$	0.584772	0.0201049
$847$	1.19866	0.0411866
$848$	8.33758	0.286314
$849$	−8.60243	−0.295235
$850$	30.9543	1.06172
$851$	5.31310	0.182131
$852$	18.8133	0.644535
$853$	18.9428	0.648588	0.324294	−	0.945956i	$-0.394873\pi$
$853$	18.9428	0.648588	0.324294	+	0.945956i	$0.394873\pi$
$854$	2.98592	0.102176
$855$	1.73339	0.0592807
$856$	4.33755	0.148254
$857$	−17.0537	−0.582545	−0.291272	−	0.956640i	$-0.594079\pi$
$857$	−17.0537	−0.582545	−0.291272	+	0.956640i	$0.594079\pi$
$858$	0	0
$859$	−52.0106	−1.77458	−0.887290	−	0.461212i	$-0.847415\pi$
$859$	−52.0106	−1.77458	−0.887290	+	0.461212i	$0.847415\pi$
$860$	34.0497	1.16108
$861$	−4.28454	−0.146017
$862$	−19.9799	−0.680517
$863$	16.2207	0.552161	0.276080	−	0.961135i	$-0.410964\pi$
$863$	16.2207	0.552161	0.276080	+	0.961135i	$0.410964\pi$
$864$	5.48799	0.186705
$865$	−42.1012	−1.43148
$866$	−25.3519	−0.861493
$867$	18.0291	0.612302
$868$	−1.47572	−0.0500893
$869$	54.5155	1.84931
$870$	50.7916	1.72200
$871$	0	0
$872$	15.5537	0.526715
$873$	1.03435	0.0350074
$874$	2.11166	0.0714278
$875$	−10.8118	−0.365507
$876$	19.8222	0.669729
$877$	−54.8993	−1.85382	−0.926908	−	0.375288i	$-0.877544\pi$
$877$	−54.8993	−1.85382	−0.926908	+	0.375288i	$0.877544\pi$
$878$	16.1022	0.543424
$879$	38.1560	1.28697
$880$	16.1910	0.545798
$881$	46.0355	1.55098	0.775488	−	0.631362i	$-0.217504\pi$
$881$	46.0355	1.55098	0.775488	+	0.631362i	$0.217504\pi$
$882$	2.82790	0.0952205
$883$	−50.8715	−1.71196	−0.855981	−	0.517008i	$-0.827046\pi$
$883$	−50.8715	−1.71196	−0.855981	+	0.517008i	$0.827046\pi$
$884$	0	0
$885$	46.8062	1.57337
$886$	−4.43856	−0.149116
$887$	10.9630	0.368103	0.184052	−	0.982917i	$-0.441079\pi$
$887$	10.9630	0.368103	0.184052	+	0.982917i	$0.441079\pi$
$888$	−4.04909	−0.135879
$889$	−2.97366	−0.0997333
$890$	−2.33994	−0.0784348
$891$	29.1241	0.975693
$892$	−11.0855	−0.371171
$893$	−1.42557	−0.0477050
$894$	34.2144	1.14430
$895$	60.9270	2.03656
$896$	0.325660	0.0108795
$897$	0	0
$898$	32.1020	1.07126
$899$	−33.8454	−1.12881
$900$	−5.27380	−0.175793
$901$	20.0740	0.668762
$902$	−31.3243	−1.04298
$903$	4.22290	0.140529
$904$	−4.45253	−0.148089
$905$	−14.8034	−0.492083
$906$	−23.7615	−0.789423
$907$	−14.5982	−0.484724	−0.242362	−	0.970186i	$-0.577922\pi$
$907$	−14.5982	−0.484724	−0.242362	+	0.970186i	$0.577922\pi$
$908$	24.2484	0.804712
$909$	−1.21546	−0.0403143
$910$	0	0
$911$	31.2768	1.03625	0.518124	−	0.855306i	$-0.326631\pi$
$911$	31.2768	1.03625	0.518124	+	0.855306i	$0.326631\pi$
$912$	−1.60929	−0.0532888
$913$	23.2074	0.768053
$914$	−25.4958	−0.843327
$915$	62.3515	2.06128
$916$	21.6584	0.715613
$917$	5.81408	0.191998
$918$	13.2132	0.436100
$919$	11.9360	0.393732	0.196866	−	0.980430i	$-0.436924\pi$
$919$	11.9360	0.393732	0.196866	+	0.980430i	$0.436924\pi$
$920$	−8.92324	−0.294191
$921$	−50.9594	−1.67917
$922$	−25.2496	−0.831551
$923$	0	0
$924$	2.00803	0.0660594
$925$	32.3483	1.06361
$926$	−4.03431	−0.132576
$927$	−2.11495	−0.0694642
$928$	7.46895	0.245180
$929$	5.01131	0.164416	0.0822078	−	0.996615i	$-0.473803\pi$
$929$	5.01131	0.164416	0.0822078	+	0.996615i	$0.473803\pi$
$930$	−30.8158	−1.01049
$931$	−6.89395	−0.225940
$932$	23.9574	0.784750
$933$	37.1673	1.21680
$934$	−17.5574	−0.574494
$935$	38.9823	1.27486
$936$	0	0
$937$	5.31702	0.173699	0.0868497	−	0.996221i	$-0.472320\pi$
$937$	5.31702	0.173699	0.0868497	+	0.996221i	$0.472320\pi$
$938$	−0.598413	−0.0195389
$939$	−16.2369	−0.529870
$940$	6.02406	0.196483
$941$	−6.24673	−0.203638	−0.101819	−	0.994803i	$-0.532466\pi$
$941$	−6.24673	−0.203638	−0.101819	+	0.994803i	$0.532466\pi$
$942$	21.5065	0.700720
$943$	17.2636	0.562179
$944$	6.88289	0.224019
$945$	−7.55225	−0.245675
$946$	30.8736	1.00379
$947$	47.1283	1.53147	0.765733	−	0.643159i	$-0.222377\pi$
$947$	47.1283	1.53147	0.765733	+	0.643159i	$0.222377\pi$
$948$	22.8970	0.743661
$949$	0	0
$950$	12.8566	0.417124
$951$	−27.2651	−0.884131
$952$	0.784075	0.0254120
$953$	−10.9221	−0.353802	−0.176901	−	0.984229i	$-0.556607\pi$
$953$	−10.9221	−0.353802	−0.176901	+	0.984229i	$0.556607\pi$
$954$	−3.42008	−0.110729
$955$	93.1280	3.01355
$956$	−16.7636	−0.542173
$957$	46.0539	1.48871
$958$	−23.8961	−0.772048
$959$	−5.26724	−0.170088
$960$	6.80037	0.219481
$961$	−10.4656	−0.337600
$962$	0	0
$963$	−1.77927	−0.0573361
$964$	0.274712	0.00884789
$965$	61.8119	1.98980
$966$	−1.10667	−0.0356067
$967$	−26.3676	−0.847926	−0.423963	−	0.905679i	$-0.639361\pi$
$967$	−26.3676	−0.847926	−0.423963	+	0.905679i	$0.639361\pi$
$968$	3.68073	0.118303
$969$	−3.87460	−0.124470
$970$	10.6554	0.342125
$971$	−17.9746	−0.576831	−0.288416	−	0.957505i	$-0.593129\pi$
$971$	−17.9746	−0.576831	−0.288416	+	0.957505i	$0.593129\pi$
$972$	−4.23157	−0.135728
$973$	−0.523088	−0.0167694
$974$	−27.7752	−0.889976
$975$	0	0
$976$	9.16884	0.293487
$977$	2.48258	0.0794247	0.0397123	−	0.999211i	$-0.487356\pi$
$977$	2.48258	0.0794247	0.0397123	+	0.999211i	$0.487356\pi$
$978$	−31.4039	−1.00419
$979$	−2.12167	−0.0678089
$980$	29.1318	0.930582
$981$	−6.38014	−0.203702
$982$	−3.57205	−0.113989
$983$	−55.9275	−1.78381	−0.891905	−	0.452223i	$-0.850631\pi$
$983$	−55.9275	−1.78381	−0.891905	+	0.452223i	$0.850631\pi$
$984$	−13.1565	−0.419414
$985$	69.0011	2.19856
$986$	17.9826	0.572684
$987$	0.747114	0.0237809
$988$	0	0
$989$	−17.0152	−0.541051
$990$	−6.64156	−0.211083
$991$	50.5565	1.60598	0.802990	−	0.595992i	$-0.203241\pi$
$991$	50.5565	1.60598	0.802990	+	0.595992i	$0.203241\pi$
$992$	−4.53149	−0.143875
$993$	44.1699	1.40169
$994$	−3.80712	−0.120755
$995$	49.3480	1.56444
$996$	9.74733	0.308856
$997$	−47.8099	−1.51415	−0.757077	−	0.653326i	$-0.773373\pi$
$997$	−47.8099	−1.51415	−0.757077	+	0.653326i	$0.773373\pi$
$998$	−39.7897	−1.25952
$999$	13.8082	0.436873

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.bk.1.4	yes	9
13.12	even	2		6422.2.a.bi.1.4	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6422.2.a.bi.1.4	✓	9	13.12	even	2
6422.2.a.bk.1.4	yes	9	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6422.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.60929 q^{3} +1.00000 q^{4} -4.22571 q^{5} -1.60929 q^{6} +0.325660 q^{7} +1.00000 q^{8} -0.410201 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.60929 q^{3} +1.00000 q^{4} -4.22571 q^{5} -1.60929 q^{6} +0.325660 q^{7} +1.00000 q^{8} -0.410201 q^{9} -4.22571 q^{10} -3.83154 q^{11} -1.60929 q^{12} +0.325660 q^{14} +6.80037 q^{15} +1.00000 q^{16} +2.40765 q^{17} -0.410201 q^{18} +1.00000 q^{19} -4.22571 q^{20} -0.524079 q^{21} -3.83154 q^{22} +2.11166 q^{23} -1.60929 q^{24} +12.8566 q^{25} +5.48799 q^{27} +0.325660 q^{28} +7.46895 q^{29} +6.80037 q^{30} -4.53149 q^{31} +1.00000 q^{32} +6.16605 q^{33} +2.40765 q^{34} -1.37614 q^{35} -0.410201 q^{36} +2.51608 q^{37} +1.00000 q^{38} -4.22571 q^{40} +8.17536 q^{41} -0.524079 q^{42} -8.05774 q^{43} -3.83154 q^{44} +1.73339 q^{45} +2.11166 q^{46} -1.42557 q^{47} -1.60929 q^{48} -6.89395 q^{49} +12.8566 q^{50} -3.87460 q^{51} +8.33758 q^{53} +5.48799 q^{54} +16.1910 q^{55} +0.325660 q^{56} -1.60929 q^{57} +7.46895 q^{58} +6.88289 q^{59} +6.80037 q^{60} +9.16884 q^{61} -4.53149 q^{62} -0.133586 q^{63} +1.00000 q^{64} +6.16605 q^{66} -1.83754 q^{67} +2.40765 q^{68} -3.39826 q^{69} -1.37614 q^{70} -11.6905 q^{71} -0.410201 q^{72} -12.3174 q^{73} +2.51608 q^{74} -20.6900 q^{75} +1.00000 q^{76} -1.24778 q^{77} -14.2281 q^{79} -4.22571 q^{80} -7.60113 q^{81} +8.17536 q^{82} -6.05693 q^{83} -0.524079 q^{84} -10.1740 q^{85} -8.05774 q^{86} -12.0197 q^{87} -3.83154 q^{88} +0.553738 q^{89} +1.73339 q^{90} +2.11166 q^{92} +7.29246 q^{93} -1.42557 q^{94} -4.22571 q^{95} -1.60929 q^{96} -2.52157 q^{97} -6.89395 q^{98} +1.57170 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{2} - 5 q^{3} + 9 q^{4} + q^{5} - 5 q^{6} - 13 q^{7} + 9 q^{8} + 10 q^{9}+O(q^{10}) \) 9 * q + 9 * q^2 - 5 * q^3 + 9 * q^4 + q^5 - 5 * q^6 - 13 * q^7 + 9 * q^8 + 10 * q^9	\( 9 q + 9 q^{2} - 5 q^{3} + 9 q^{4} + q^{5} - 5 q^{6} - 13 q^{7} + 9 q^{8} + 10 q^{9} + q^{10} - 13 q^{11} - 5 q^{12} - 13 q^{14} - q^{15} + 9 q^{16} - 12 q^{17} + 10 q^{18} + 9 q^{19} + q^{20} + 18 q^{21} - 13 q^{22} - 22 q^{23} - 5 q^{24} + 4 q^{25} - 26 q^{27} - 13 q^{28} + 12 q^{29} - q^{30} + q^{31} + 9 q^{32} - 28 q^{33} - 12 q^{34} - 18 q^{35} + 10 q^{36} - 25 q^{37} + 9 q^{38} + q^{40} + 11 q^{41} + 18 q^{42} - 10 q^{43} - 13 q^{44} - q^{45} - 22 q^{46} - 12 q^{47} - 5 q^{48} + 2 q^{49} + 4 q^{50} + 35 q^{51} + 9 q^{53} - 26 q^{54} + 18 q^{55} - 13 q^{56} - 5 q^{57} + 12 q^{58} + 10 q^{59} - q^{60} + 32 q^{61} + q^{62} - 63 q^{63} + 9 q^{64} - 28 q^{66} - 73 q^{67} - 12 q^{68} + 2 q^{69} - 18 q^{70} - 51 q^{71} + 10 q^{72} - 14 q^{73} - 25 q^{74} - 49 q^{75} + 9 q^{76} + 18 q^{77} - 28 q^{79} + q^{80} + 29 q^{81} + 11 q^{82} - 22 q^{83} + 18 q^{84} - 51 q^{85} - 10 q^{86} - 20 q^{87} - 13 q^{88} + 3 q^{89} - q^{90} - 22 q^{92} - 59 q^{93} - 12 q^{94} + q^{95} - 5 q^{96} + 2 q^{98} + 25 q^{99}+O(q^{100}) \) 9 * q + 9 * q^2 - 5 * q^3 + 9 * q^4 + q^5 - 5 * q^6 - 13 * q^7 + 9 * q^8 + 10 * q^9 + q^10 - 13 * q^11 - 5 * q^12 - 13 * q^14 - q^15 + 9 * q^16 - 12 * q^17 + 10 * q^18 + 9 * q^19 + q^20 + 18 * q^21 - 13 * q^22 - 22 * q^23 - 5 * q^24 + 4 * q^25 - 26 * q^27 - 13 * q^28 + 12 * q^29 - q^30 + q^31 + 9 * q^32 - 28 * q^33 - 12 * q^34 - 18 * q^35 + 10 * q^36 - 25 * q^37 + 9 * q^38 + q^40 + 11 * q^41 + 18 * q^42 - 10 * q^43 - 13 * q^44 - q^45 - 22 * q^46 - 12 * q^47 - 5 * q^48 + 2 * q^49 + 4 * q^50 + 35 * q^51 + 9 * q^53 - 26 * q^54 + 18 * q^55 - 13 * q^56 - 5 * q^57 + 12 * q^58 + 10 * q^59 - q^60 + 32 * q^61 + q^62 - 63 * q^63 + 9 * q^64 - 28 * q^66 - 73 * q^67 - 12 * q^68 + 2 * q^69 - 18 * q^70 - 51 * q^71 + 10 * q^72 - 14 * q^73 - 25 * q^74 - 49 * q^75 + 9 * q^76 + 18 * q^77 - 28 * q^79 + q^80 + 29 * q^81 + 11 * q^82 - 22 * q^83 + 18 * q^84 - 51 * q^85 - 10 * q^86 - 20 * q^87 - 13 * q^88 + 3 * q^89 - q^90 - 22 * q^92 - 59 * q^93 - 12 * q^94 + q^95 - 5 * q^96 + 2 * q^98 + 25 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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