LMFDB - Embedded newform 6422.2.a.bc.1.5

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:	$6$
Coefficient field:	6.6.316645497.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 12x^{4} + 24x^{3} + 13x^{2} - 24x + 5 $ x^6 - 2x^5 - 12x^4 + 24x^3 + 13x^2 - 24*x + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 494)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.19765$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +2.19765 q^{3} +1.00000 q^{4} -3.12703 q^{5} -2.19765 q^{6} -2.17944 q^{7} -1.00000 q^{8} +1.82965 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +2.19765 q^{3} +1.00000 q^{4} -3.12703 q^{5} -2.19765 q^{6} -2.17944 q^{7} -1.00000 q^{8} +1.82965 q^{9} +3.12703 q^{10} -3.12703 q^{11} +2.19765 q^{12} +2.17944 q^{14} -6.87211 q^{15} +1.00000 q^{16} -7.10300 q^{17} -1.82965 q^{18} +1.00000 q^{19} -3.12703 q^{20} -4.78963 q^{21} +3.12703 q^{22} -0.644393 q^{23} -2.19765 q^{24} +4.77832 q^{25} -2.57201 q^{27} -2.17944 q^{28} +0.426223 q^{29} +6.87211 q^{30} +4.85159 q^{31} -1.00000 q^{32} -6.87211 q^{33} +7.10300 q^{34} +6.81517 q^{35} +1.82965 q^{36} -7.52510 q^{37} -1.00000 q^{38} +3.12703 q^{40} +0.117941 q^{41} +4.78963 q^{42} +2.42854 q^{43} -3.12703 q^{44} -5.72138 q^{45} +0.644393 q^{46} -6.12703 q^{47} +2.19765 q^{48} -2.25005 q^{49} -4.77832 q^{50} -15.6099 q^{51} +4.11436 q^{53} +2.57201 q^{54} +9.77832 q^{55} +2.17944 q^{56} +2.19765 q^{57} -0.426223 q^{58} +10.6322 q^{59} -6.87211 q^{60} +8.65802 q^{61} -4.85159 q^{62} -3.98761 q^{63} +1.00000 q^{64} +6.87211 q^{66} -1.75463 q^{67} -7.10300 q^{68} -1.41615 q^{69} -6.81517 q^{70} -10.2346 q^{71} -1.82965 q^{72} -13.3753 q^{73} +7.52510 q^{74} +10.5011 q^{75} +1.00000 q^{76} +6.81517 q^{77} +11.3371 q^{79} -3.12703 q^{80} -11.1413 q^{81} -0.117941 q^{82} +12.5884 q^{83} -4.78963 q^{84} +22.2113 q^{85} -2.42854 q^{86} +0.936688 q^{87} +3.12703 q^{88} -1.43889 q^{89} +5.72138 q^{90} -0.644393 q^{92} +10.6621 q^{93} +6.12703 q^{94} -3.12703 q^{95} -2.19765 q^{96} +11.9346 q^{97} +2.25005 q^{98} -5.72138 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{2} + 2 q^{3} + 6 q^{4} - 2 q^{5} - 2 q^{6} - q^{7} - 6 q^{8} + 10 q^{9}+O(q^{10}) $ 6 * q - 6 * q^2 + 2 * q^3 + 6 * q^4 - 2 * q^5 - 2 * q^6 - q^7 - 6 * q^8 + 10 * q^9	$ 6 q - 6 q^{2} + 2 q^{3} + 6 q^{4} - 2 q^{5} - 2 q^{6} - q^{7} - 6 q^{8} + 10 q^{9} + 2 q^{10} - 2 q^{11} + 2 q^{12} + q^{14} + 9 q^{15} + 6 q^{16} - 2 q^{17} - 10 q^{18} + 6 q^{19} - 2 q^{20} - q^{21} + 2 q^{22} + 8 q^{23} - 2 q^{24} + 16 q^{25} - 4 q^{27} - q^{28} + 20 q^{29} - 9 q^{30} - 3 q^{31} - 6 q^{32} + 9 q^{33} + 2 q^{34} + 7 q^{35} + 10 q^{36} + 9 q^{37} - 6 q^{38} + 2 q^{40} - 3 q^{41} + q^{42} + 13 q^{43} - 2 q^{44} - 38 q^{45} - 8 q^{46} - 20 q^{47} + 2 q^{48} - 7 q^{49} - 16 q^{50} + 2 q^{51} + 25 q^{53} + 4 q^{54} + 46 q^{55} + q^{56} + 2 q^{57} - 20 q^{58} + 9 q^{60} + 6 q^{61} + 3 q^{62} - 46 q^{63} + 6 q^{64} - 9 q^{66} + 32 q^{67} - 2 q^{68} - 29 q^{69} - 7 q^{70} - 39 q^{71} - 10 q^{72} - 7 q^{73} - 9 q^{74} - 15 q^{75} + 6 q^{76} + 7 q^{77} + 18 q^{79} - 2 q^{80} + 54 q^{81} + 3 q^{82} + 7 q^{83} - q^{84} + 2 q^{85} - 13 q^{86} + 12 q^{87} + 2 q^{88} - 9 q^{89} + 38 q^{90} + 8 q^{92} + 47 q^{93} + 20 q^{94} - 2 q^{95} - 2 q^{96} + 4 q^{97} + 7 q^{98} - 38 q^{99}+O(q^{100}) $ 6 * q - 6 * q^2 + 2 * q^3 + 6 * q^4 - 2 * q^5 - 2 * q^6 - q^7 - 6 * q^8 + 10 * q^9 + 2 * q^10 - 2 * q^11 + 2 * q^12 + q^14 + 9 * q^15 + 6 * q^16 - 2 * q^17 - 10 * q^18 + 6 * q^19 - 2 * q^20 - q^21 + 2 * q^22 + 8 * q^23 - 2 * q^24 + 16 * q^25 - 4 * q^27 - q^28 + 20 * q^29 - 9 * q^30 - 3 * q^31 - 6 * q^32 + 9 * q^33 + 2 * q^34 + 7 * q^35 + 10 * q^36 + 9 * q^37 - 6 * q^38 + 2 * q^40 - 3 * q^41 + q^42 + 13 * q^43 - 2 * q^44 - 38 * q^45 - 8 * q^46 - 20 * q^47 + 2 * q^48 - 7 * q^49 - 16 * q^50 + 2 * q^51 + 25 * q^53 + 4 * q^54 + 46 * q^55 + q^56 + 2 * q^57 - 20 * q^58 + 9 * q^60 + 6 * q^61 + 3 * q^62 - 46 * q^63 + 6 * q^64 - 9 * q^66 + 32 * q^67 - 2 * q^68 - 29 * q^69 - 7 * q^70 - 39 * q^71 - 10 * q^72 - 7 * q^73 - 9 * q^74 - 15 * q^75 + 6 * q^76 + 7 * q^77 + 18 * q^79 - 2 * q^80 + 54 * q^81 + 3 * q^82 + 7 * q^83 - q^84 + 2 * q^85 - 13 * q^86 + 12 * q^87 + 2 * q^88 - 9 * q^89 + 38 * q^90 + 8 * q^92 + 47 * q^93 + 20 * q^94 - 2 * q^95 - 2 * q^96 + 4 * q^97 + 7 * q^98 - 38 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	2.19765	1.26881	0.634406	−	0.773000i	$-0.281245\pi$
$3$	2.19765	1.26881	0.634406	+	0.773000i	$0.281245\pi$
$4$	1.00000	0.500000
$5$	−3.12703	−1.39845	−0.699225	−	0.714901i	$-0.746472\pi$
$5$	−3.12703	−1.39845	−0.699225	+	0.714901i	$0.746472\pi$
$6$	−2.19765	−0.897186
$7$	−2.17944	−0.823750	−0.411875	−	0.911240i	$-0.635126\pi$
$7$	−2.17944	−0.823750	−0.411875	+	0.911240i	$0.635126\pi$
$8$	−1.00000	−0.353553
$9$	1.82965	0.609884
$10$	3.12703	0.988854
$11$	−3.12703	−0.942835	−0.471418	−	0.881910i	$-0.656258\pi$
$11$	−3.12703	−0.942835	−0.471418	+	0.881910i	$0.656258\pi$
$12$	2.19765	0.634406
$13$	0	0
$13$	0	0
$14$	2.17944	0.582479
$15$	−6.87211	−1.77437
$16$	1.00000	0.250000
$17$	−7.10300	−1.72273	−0.861365	−	0.507986i	$-0.830390\pi$
$17$	−7.10300	−1.72273	−0.861365	+	0.507986i	$0.830390\pi$
$18$	−1.82965	−0.431253
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−3.12703	−0.699225
$21$	−4.78963	−1.04518
$22$	3.12703	0.666685
$23$	−0.644393	−0.134365	−0.0671826	−	0.997741i	$-0.521401\pi$
$23$	−0.644393	−0.134365	−0.0671826	+	0.997741i	$0.521401\pi$
$24$	−2.19765	−0.448593
$25$	4.77832	0.955664
$26$	0	0
$27$	−2.57201	−0.494984
$28$	−2.17944	−0.411875
$29$	0.426223	0.0791477	0.0395738	−	0.999217i	$-0.487400\pi$
$29$	0.426223	0.0791477	0.0395738	+	0.999217i	$0.487400\pi$
$30$	6.87211	1.25467
$31$	4.85159	0.871371	0.435685	−	0.900099i	$-0.356506\pi$
$31$	4.85159	0.871371	0.435685	+	0.900099i	$0.356506\pi$
$32$	−1.00000	−0.176777
$33$	−6.87211	−1.19628
$34$	7.10300	1.21815
$35$	6.81517	1.15197
$36$	1.82965	0.304942
$37$	−7.52510	−1.23712	−0.618559	−	0.785738i	$-0.712283\pi$
$37$	−7.52510	−1.23712	−0.618559	+	0.785738i	$0.712283\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	3.12703	0.494427
$41$	0.117941	0.0184193	0.00920967	−	0.999958i	$-0.497068\pi$
$41$	0.117941	0.0184193	0.00920967	+	0.999958i	$0.497068\pi$
$42$	4.78963	0.739057
$43$	2.42854	0.370348	0.185174	−	0.982706i	$-0.440715\pi$
$43$	2.42854	0.370348	0.185174	+	0.982706i	$0.440715\pi$
$44$	−3.12703	−0.471418
$45$	−5.72138	−0.852893
$46$	0.644393	0.0950106
$47$	−6.12703	−0.893719	−0.446860	−	0.894604i	$-0.647458\pi$
$47$	−6.12703	−0.893719	−0.446860	+	0.894604i	$0.647458\pi$
$48$	2.19765	0.317203
$49$	−2.25005	−0.321436
$50$	−4.77832	−0.675757
$51$	−15.6099	−2.18582
$52$	0	0
$53$	4.11436	0.565151	0.282575	−	0.959245i	$-0.408811\pi$
$53$	4.11436	0.565151	0.282575	+	0.959245i	$0.408811\pi$
$54$	2.57201	0.350006
$55$	9.77832	1.31851
$56$	2.17944	0.291240
$57$	2.19765	0.291085
$58$	−0.426223	−0.0559658
$59$	10.6322	1.38420	0.692099	−	0.721803i	$-0.256686\pi$
$59$	10.6322	1.38420	0.692099	+	0.721803i	$0.256686\pi$
$60$	−6.87211	−0.887186
$61$	8.65802	1.10855	0.554273	−	0.832335i	$-0.312996\pi$
$61$	8.65802	1.10855	0.554273	+	0.832335i	$0.312996\pi$
$62$	−4.85159	−0.616152
$63$	−3.98761	−0.502392
$64$	1.00000	0.125000
$65$	0	0
$66$	6.87211	0.845898
$67$	−1.75463	−0.214362	−0.107181	−	0.994240i	$-0.534182\pi$
$67$	−1.75463	−0.214362	−0.107181	+	0.994240i	$0.534182\pi$
$68$	−7.10300	−0.861365
$69$	−1.41615	−0.170484
$70$	−6.81517	−0.814568
$71$	−10.2346	−1.21463	−0.607313	−	0.794463i	$-0.707752\pi$
$71$	−10.2346	−1.21463	−0.607313	+	0.794463i	$0.707752\pi$
$72$	−1.82965	−0.215627
$73$	−13.3753	−1.56546	−0.782728	−	0.622363i	$-0.786173\pi$
$73$	−13.3753	−1.56546	−0.782728	+	0.622363i	$0.786173\pi$
$74$	7.52510	0.874775
$75$	10.5011	1.21256
$76$	1.00000	0.114708
$77$	6.81517	0.776660
$78$	0	0
$79$	11.3371	1.27552	0.637760	−	0.770235i	$-0.279861\pi$
$79$	11.3371	1.27552	0.637760	+	0.770235i	$0.279861\pi$
$80$	−3.12703	−0.349613
$81$	−11.1413	−1.23793
$82$	−0.117941	−0.0130244
$83$	12.5884	1.38175	0.690876	−	0.722973i	$-0.257225\pi$
$83$	12.5884	1.38175	0.690876	+	0.722973i	$0.257225\pi$
$84$	−4.78963	−0.522592
$85$	22.2113	2.40915
$86$	−2.42854	−0.261876
$87$	0.936688	0.100423
$88$	3.12703	0.333343
$89$	−1.43889	−0.152522	−0.0762612	−	0.997088i	$-0.524298\pi$
$89$	−1.43889	−0.152522	−0.0762612	+	0.997088i	$0.524298\pi$
$90$	5.72138	0.603086
$91$	0	0
$92$	−0.644393	−0.0671826
$93$	10.6621	1.10561
$94$	6.12703	0.631955
$95$	−3.12703	−0.320827
$96$	−2.19765	−0.224296
$97$	11.9346	1.21178	0.605888	−	0.795550i	$-0.292818\pi$
$97$	11.9346	1.21178	0.605888	+	0.795550i	$0.292818\pi$
$98$	2.25005	0.227290
$99$	−5.72138	−0.575020
$100$	4.77832	0.477832
$101$	−12.1844	−1.21239	−0.606197	−	0.795315i	$-0.707306\pi$
$101$	−12.1844	−1.21239	−0.606197	+	0.795315i	$0.707306\pi$
$102$	15.6099	1.54561
$103$	10.5338	1.03792	0.518961	−	0.854798i	$-0.326319\pi$
$103$	10.5338	1.03792	0.518961	+	0.854798i	$0.326319\pi$
$104$	0	0
$105$	14.9773	1.46164
$106$	−4.11436	−0.399622
$107$	5.67809	0.548922	0.274461	−	0.961598i	$-0.411501\pi$
$107$	5.67809	0.548922	0.274461	+	0.961598i	$0.411501\pi$
$108$	−2.57201	−0.247492
$109$	11.7186	1.12243	0.561217	−	0.827669i	$-0.310333\pi$
$109$	11.7186	1.12243	0.561217	+	0.827669i	$0.310333\pi$
$110$	−9.77832	−0.932326
$111$	−16.5375	−1.56967
$112$	−2.17944	−0.205937
$113$	−0.945828	−0.0889760	−0.0444880	−	0.999010i	$-0.514166\pi$
$113$	−0.945828	−0.0889760	−0.0444880	+	0.999010i	$0.514166\pi$
$114$	−2.19765	−0.205828
$115$	2.01504	0.187903
$116$	0.426223	0.0395738
$117$	0	0
$118$	−10.6322	−0.978775
$119$	15.4805	1.41910
$120$	6.87211	0.627335
$121$	−1.22168	−0.111062
$122$	−8.65802	−0.783860
$123$	0.259193	0.0233707
$124$	4.85159	0.435685
$125$	0.693196	0.0620013
$126$	3.98761	0.355245
$127$	9.51474	0.844296	0.422148	−	0.906527i	$-0.361276\pi$
$127$	9.51474	0.844296	0.422148	+	0.906527i	$0.361276\pi$
$128$	−1.00000	−0.0883883
$129$	5.33707	0.469902
$130$	0	0
$131$	2.94454	0.257266	0.128633	−	0.991692i	$-0.458941\pi$
$131$	2.94454	0.257266	0.128633	+	0.991692i	$0.458941\pi$
$132$	−6.87211	−0.598140
$133$	−2.17944	−0.188981
$134$	1.75463	0.151577
$135$	8.04276	0.692210
$136$	7.10300	0.609077
$137$	4.54962	0.388701	0.194350	−	0.980932i	$-0.437740\pi$
$137$	4.54962	0.388701	0.194350	+	0.980932i	$0.437740\pi$
$138$	1.41615	0.120551
$139$	−14.9108	−1.26472	−0.632359	−	0.774675i	$-0.717913\pi$
$139$	−14.9108	−1.26472	−0.632359	+	0.774675i	$0.717913\pi$
$140$	6.81517	0.575987
$141$	−13.4651	−1.13396
$142$	10.2346	0.858870
$143$	0	0
$144$	1.82965	0.152471
$145$	−1.33281	−0.110684
$146$	13.3753	1.10695
$147$	−4.94482	−0.407842
$148$	−7.52510	−0.618559
$149$	−22.5037	−1.84358	−0.921788	−	0.387694i	$-0.873272\pi$
$149$	−22.5037	−1.84358	−0.921788	+	0.387694i	$0.873272\pi$
$150$	−10.5011	−0.857408
$151$	−9.30065	−0.756876	−0.378438	−	0.925627i	$-0.623539\pi$
$151$	−9.30065	−0.756876	−0.378438	+	0.925627i	$0.623539\pi$
$152$	−1.00000	−0.0811107
$153$	−12.9960	−1.05067
$154$	−6.81517	−0.549182
$155$	−15.1711	−1.21857
$156$	0	0
$157$	2.58803	0.206547	0.103274	−	0.994653i	$-0.467068\pi$
$157$	2.58803	0.206547	0.103274	+	0.994653i	$0.467068\pi$
$158$	−11.3371	−0.901929
$159$	9.04191	0.717070
$160$	3.12703	0.247213
$161$	1.40441	0.110683
$162$	11.1413	0.875346
$163$	22.3172	1.74802	0.874008	−	0.485911i	$-0.161512\pi$
$163$	22.3172	1.74802	0.874008	+	0.485911i	$0.161512\pi$
$164$	0.117941	0.00920967
$165$	21.4893	1.67294
$166$	−12.5884	−0.977046
$167$	23.2517	1.79927	0.899636	−	0.436640i	$-0.143832\pi$
$167$	23.2517	1.79927	0.899636	+	0.436640i	$0.143832\pi$
$168$	4.78963	0.369528
$169$	0	0
$170$	−22.2113	−1.70353
$171$	1.82965	0.139917
$172$	2.42854	0.185174
$173$	−21.7696	−1.65511	−0.827557	−	0.561381i	$-0.810270\pi$
$173$	−21.7696	−1.65511	−0.827557	+	0.561381i	$0.810270\pi$
$174$	−0.936688	−0.0710101
$175$	−10.4141	−0.787228
$176$	−3.12703	−0.235709
$177$	23.3659	1.75629
$178$	1.43889	0.107850
$179$	−24.6147	−1.83979	−0.919893	−	0.392170i	$-0.871725\pi$
$179$	−24.6147	−1.83979	−0.919893	+	0.392170i	$0.871725\pi$
$180$	−5.72138	−0.426446
$181$	−3.87802	−0.288251	−0.144126	−	0.989559i	$-0.546037\pi$
$181$	−3.87802	−0.288251	−0.144126	+	0.989559i	$0.546037\pi$
$182$	0	0
$183$	19.0273	1.40654
$184$	0.644393	0.0475053
$185$	23.5312	1.73005
$186$	−10.6621	−0.781781
$187$	22.2113	1.62425
$188$	−6.12703	−0.446860
$189$	5.60554	0.407743
$190$	3.12703	0.226859
$191$	14.5288	1.05127	0.525634	−	0.850711i	$-0.323828\pi$
$191$	14.5288	1.05127	0.525634	+	0.850711i	$0.323828\pi$
$192$	2.19765	0.158602
$193$	25.5087	1.83616	0.918080	−	0.396396i	$-0.129739\pi$
$193$	25.5087	1.83616	0.918080	+	0.396396i	$0.129739\pi$
$194$	−11.9346	−0.856855
$195$	0	0
$196$	−2.25005	−0.160718
$197$	5.12980	0.365483	0.182742	−	0.983161i	$-0.441503\pi$
$197$	5.12980	0.365483	0.182742	+	0.983161i	$0.441503\pi$
$198$	5.72138	0.406601
$199$	−6.58770	−0.466989	−0.233495	−	0.972358i	$-0.575016\pi$
$199$	−6.58770	−0.466989	−0.233495	+	0.972358i	$0.575016\pi$
$200$	−4.77832	−0.337878
$201$	−3.85605	−0.271985
$202$	12.1844	0.857291
$203$	−0.928927	−0.0651979
$204$	−15.6099	−1.09291
$205$	−0.368806	−0.0257585
$206$	−10.5338	−0.733922
$207$	−1.17902	−0.0819472
$208$	0	0
$209$	−3.12703	−0.216301
$210$	−14.9773	−1.03353
$211$	−4.98682	−0.343307	−0.171654	−	0.985157i	$-0.554911\pi$
$211$	−4.98682	−0.343307	−0.171654	+	0.985157i	$0.554911\pi$
$212$	4.11436	0.282575
$213$	−22.4921	−1.54113
$214$	−5.67809	−0.388146
$215$	−7.59411	−0.517914
$216$	2.57201	0.175003
$217$	−10.5737	−0.717791
$218$	−11.7186	−0.793681
$219$	−29.3941	−1.98627
$220$	9.77832	0.659254
$221$	0	0
$222$	16.5375	1.10992
$223$	22.9805	1.53889	0.769445	−	0.638713i	$-0.220533\pi$
$223$	22.9805	1.53889	0.769445	+	0.638713i	$0.220533\pi$
$224$	2.17944	0.145620
$225$	8.74267	0.582844
$226$	0.945828	0.0629155
$227$	−5.49722	−0.364863	−0.182432	−	0.983219i	$-0.558397\pi$
$227$	−5.49722	−0.364863	−0.182432	+	0.983219i	$0.558397\pi$
$228$	2.19765	0.145543
$229$	1.52318	0.100655	0.0503274	−	0.998733i	$-0.483974\pi$
$229$	1.52318	0.100655	0.0503274	+	0.998733i	$0.483974\pi$
$230$	−2.01504	−0.132868
$231$	14.9773	0.985436
$232$	−0.426223	−0.0279829
$233$	25.9377	1.69924	0.849618	−	0.527399i	$-0.176833\pi$
$233$	25.9377	1.69924	0.849618	+	0.527399i	$0.176833\pi$
$234$	0	0
$235$	19.1594	1.24982
$236$	10.6322	0.692099
$237$	24.9149	1.61839
$238$	−15.4805	−1.00345
$239$	5.55924	0.359597	0.179799	−	0.983703i	$-0.442455\pi$
$239$	5.55924	0.359597	0.179799	+	0.983703i	$0.442455\pi$
$240$	−6.87211	−0.443593
$241$	−11.9583	−0.770302	−0.385151	−	0.922853i	$-0.625851\pi$
$241$	−11.9583	−0.770302	−0.385151	+	0.922853i	$0.625851\pi$
$242$	1.22168	0.0785325
$243$	−16.7687	−1.07571
$244$	8.65802	0.554273
$245$	7.03599	0.449513
$246$	−0.259193	−0.0165256
$247$	0	0
$248$	−4.85159	−0.308076
$249$	27.6648	1.75318
$250$	−0.693196	−0.0438416
$251$	−22.4814	−1.41902	−0.709508	−	0.704697i	$-0.751083\pi$
$251$	−22.4814	−1.41902	−0.709508	+	0.704697i	$0.751083\pi$
$252$	−3.98761	−0.251196
$253$	2.01504	0.126684
$254$	−9.51474	−0.597008
$255$	48.8126	3.05676
$256$	1.00000	0.0625000
$257$	4.85032	0.302555	0.151277	−	0.988491i	$-0.451661\pi$
$257$	4.85032	0.302555	0.151277	+	0.988491i	$0.451661\pi$
$258$	−5.33707	−0.332271
$259$	16.4005	1.01908
$260$	0	0
$261$	0.779840	0.0482709
$262$	−2.94454	−0.181914
$263$	−10.1214	−0.624109	−0.312055	−	0.950064i	$-0.601017\pi$
$263$	−10.1214	−0.624109	−0.312055	+	0.950064i	$0.601017\pi$
$264$	6.87211	0.422949
$265$	−12.8657	−0.790335
$266$	2.17944	0.133630
$267$	−3.16218	−0.193522
$268$	−1.75463	−0.107181
$269$	7.99747	0.487614	0.243807	−	0.969824i	$-0.421604\pi$
$269$	7.99747	0.487614	0.243807	+	0.969824i	$0.421604\pi$
$270$	−8.04276	−0.489467
$271$	−9.26475	−0.562794	−0.281397	−	0.959591i	$-0.590798\pi$
$271$	−9.26475	−0.562794	−0.281397	+	0.959591i	$0.590798\pi$
$272$	−7.10300	−0.430683
$273$	0	0
$274$	−4.54962	−0.274853
$275$	−14.9420	−0.901034
$276$	−1.41615	−0.0852421
$277$	14.1772	0.851823	0.425912	−	0.904765i	$-0.359953\pi$
$277$	14.1772	0.851823	0.425912	+	0.904765i	$0.359953\pi$
$278$	14.9108	0.894291
$279$	8.87672	0.531435
$280$	−6.81517	−0.407284
$281$	27.3738	1.63299	0.816493	−	0.577355i	$-0.195915\pi$
$281$	27.3738	1.63299	0.816493	+	0.577355i	$0.195915\pi$
$282$	13.4651	0.801832
$283$	23.5136	1.39774	0.698868	−	0.715250i	$-0.253687\pi$
$283$	23.5136	1.39774	0.698868	+	0.715250i	$0.253687\pi$
$284$	−10.2346	−0.607313
$285$	−6.87211	−0.407069
$286$	0	0
$287$	−0.257046	−0.0151729
$288$	−1.82965	−0.107813
$289$	33.4526	1.96780
$290$	1.33281	0.0782655
$291$	26.2281	1.53752
$292$	−13.3753	−0.782728
$293$	−11.1201	−0.649645	−0.324823	−	0.945775i	$-0.605305\pi$
$293$	−11.1201	−0.649645	−0.324823	+	0.945775i	$0.605305\pi$
$294$	4.94482	0.288388
$295$	−33.2473	−1.93573
$296$	7.52510	0.437387
$297$	8.04276	0.466688
$298$	22.5037	1.30361
$299$	0	0
$300$	10.5011	0.606279
$301$	−5.29284	−0.305074
$302$	9.30065	0.535192
$303$	−26.7770	−1.53830
$304$	1.00000	0.0573539
$305$	−27.0739	−1.55025
$306$	12.9960	0.742933
$307$	10.2579	0.585450	0.292725	−	0.956197i	$-0.405438\pi$
$307$	10.2579	0.585450	0.292725	+	0.956197i	$0.405438\pi$
$308$	6.81517	0.388330
$309$	23.1495	1.31693
$310$	15.1711	0.861658
$311$	13.8693	0.786455	0.393228	−	0.919441i	$-0.371358\pi$
$311$	13.8693	0.786455	0.393228	+	0.919441i	$0.371358\pi$
$312$	0	0
$313$	−3.37717	−0.190889	−0.0954446	−	0.995435i	$-0.530427\pi$
$313$	−3.37717	−0.190889	−0.0954446	+	0.995435i	$0.530427\pi$
$314$	−2.58803	−0.146051
$315$	12.4694	0.702570
$316$	11.3371	0.637760
$317$	−6.12834	−0.344202	−0.172101	−	0.985079i	$-0.555056\pi$
$317$	−6.12834	−0.344202	−0.172101	+	0.985079i	$0.555056\pi$
$318$	−9.04191	−0.507045
$319$	−1.33281	−0.0746232
$320$	−3.12703	−0.174806
$321$	12.4784	0.696479
$322$	−1.40441	−0.0782649
$323$	−7.10300	−0.395221
$324$	−11.1413	−0.618963
$325$	0	0
$326$	−22.3172	−1.23603
$327$	25.7533	1.42416
$328$	−0.117941	−0.00651222
$329$	13.3535	0.736201
$330$	−21.4893	−1.18295
$331$	−0.0943337	−0.00518505	−0.00259253	−	0.999997i	$-0.500825\pi$
$331$	−0.0943337	−0.00518505	−0.00259253	+	0.999997i	$0.500825\pi$
$332$	12.5884	0.690876
$333$	−13.7683	−0.754499
$334$	−23.2517	−1.27228
$335$	5.48677	0.299774
$336$	−4.78963	−0.261296
$337$	−9.25406	−0.504101	−0.252050	−	0.967714i	$-0.581105\pi$
$337$	−9.25406	−0.504101	−0.252050	+	0.967714i	$0.581105\pi$
$338$	0	0
$339$	−2.07860	−0.112894
$340$	22.2113	1.20458
$341$	−15.1711	−0.821559
$342$	−1.82965	−0.0989363
$343$	20.1599	1.08853
$344$	−2.42854	−0.130938
$345$	4.42834	0.238414
$346$	21.7696	1.17034
$347$	−29.1677	−1.56580	−0.782902	−	0.622145i	$-0.786261\pi$
$347$	−29.1677	−1.56580	−0.782902	+	0.622145i	$0.786261\pi$
$348$	0.936688	0.0502117
$349$	29.5303	1.58072	0.790361	−	0.612641i	$-0.209893\pi$
$349$	29.5303	1.58072	0.790361	+	0.612641i	$0.209893\pi$
$350$	10.4141	0.556654
$351$	0	0
$352$	3.12703	0.166671
$353$	29.0884	1.54822	0.774110	−	0.633052i	$-0.218198\pi$
$353$	29.0884	1.54822	0.774110	+	0.633052i	$0.218198\pi$
$354$	−23.3659	−1.24188
$355$	32.0040	1.69859
$356$	−1.43889	−0.0762612
$357$	34.0208	1.80057
$358$	24.6147	1.30092
$359$	19.0423	1.00502	0.502508	−	0.864573i	$-0.332411\pi$
$359$	19.0423	1.00502	0.502508	+	0.864573i	$0.332411\pi$
$360$	5.72138	0.301543
$361$	1.00000	0.0526316
$362$	3.87802	0.203824
$363$	−2.68482	−0.140916
$364$	0	0
$365$	41.8249	2.18921
$366$	−19.0273	−0.994571
$367$	1.02770	0.0536456	0.0268228	−	0.999640i	$-0.491461\pi$
$367$	1.02770	0.0536456	0.0268228	+	0.999640i	$0.491461\pi$
$368$	−0.644393	−0.0335913
$369$	0.215792	0.0112337
$370$	−23.5312	−1.22333
$371$	−8.96699	−0.465543
$372$	10.6621	0.552803
$373$	15.9433	0.825512	0.412756	−	0.910842i	$-0.364566\pi$
$373$	15.9433	0.825512	0.412756	+	0.910842i	$0.364566\pi$
$374$	−22.2113	−1.14852
$375$	1.52340	0.0786681
$376$	6.12703	0.315977
$377$	0	0
$378$	−5.60554	−0.288318
$379$	−4.79302	−0.246201	−0.123100	−	0.992394i	$-0.539284\pi$
$379$	−4.79302	−0.246201	−0.123100	+	0.992394i	$0.539284\pi$
$380$	−3.12703	−0.160413
$381$	20.9100	1.07125
$382$	−14.5288	−0.743359
$383$	21.8340	1.11567	0.557833	−	0.829953i	$-0.311633\pi$
$383$	21.8340	1.11567	0.557833	+	0.829953i	$0.311633\pi$
$384$	−2.19765	−0.112148
$385$	−21.3112	−1.08612
$386$	−25.5087	−1.29836
$387$	4.44338	0.225870
$388$	11.9346	0.605888
$389$	16.3944	0.831231	0.415615	−	0.909540i	$-0.363566\pi$
$389$	16.3944	0.831231	0.415615	+	0.909540i	$0.363566\pi$
$390$	0	0
$391$	4.57712	0.231475
$392$	2.25005	0.113645
$393$	6.47106	0.326422
$394$	−5.12980	−0.258436
$395$	−35.4514	−1.78375
$396$	−5.72138	−0.287510
$397$	−8.32544	−0.417842	−0.208921	−	0.977933i	$-0.566995\pi$
$397$	−8.32544	−0.417842	−0.208921	+	0.977933i	$0.566995\pi$
$398$	6.58770	0.330211
$399$	−4.78963	−0.239782
$400$	4.77832	0.238916
$401$	−35.8203	−1.78878	−0.894390	−	0.447289i	$-0.852390\pi$
$401$	−35.8203	−1.78878	−0.894390	+	0.447289i	$0.852390\pi$
$402$	3.85605	0.192322
$403$	0	0
$404$	−12.1844	−0.606197
$405$	34.8393	1.73118
$406$	0.928927	0.0461019
$407$	23.5312	1.16640
$408$	15.6099	0.772804
$409$	−28.5608	−1.41224	−0.706121	−	0.708091i	$-0.749556\pi$
$409$	−28.5608	−1.41224	−0.706121	+	0.708091i	$0.749556\pi$
$410$	0.368806	0.0182140
$411$	9.99847	0.493188
$412$	10.5338	0.518961
$413$	−23.1723	−1.14023
$414$	1.17902	0.0579454
$415$	−39.3642	−1.93231
$416$	0	0
$417$	−32.7687	−1.60469
$418$	3.12703	0.152948
$419$	−39.0336	−1.90692	−0.953459	−	0.301522i	$-0.902505\pi$
$419$	−39.0336	−1.90692	−0.953459	+	0.301522i	$0.902505\pi$
$420$	14.9773	0.730819
$421$	−35.6361	−1.73680	−0.868398	−	0.495867i	$-0.834850\pi$
$421$	−35.6361	−1.73680	−0.868398	+	0.495867i	$0.834850\pi$
$422$	4.98682	0.242755
$423$	−11.2103	−0.545065
$424$	−4.11436	−0.199811
$425$	−33.9404	−1.64635
$426$	22.4921	1.08974
$427$	−18.8696	−0.913164
$428$	5.67809	0.274461
$429$	0	0
$430$	7.59411	0.366220
$431$	−28.5835	−1.37682	−0.688408	−	0.725323i	$-0.741690\pi$
$431$	−28.5835	−1.37682	−0.688408	+	0.725323i	$0.741690\pi$
$432$	−2.57201	−0.123746
$433$	−24.3228	−1.16888	−0.584440	−	0.811437i	$-0.698686\pi$
$433$	−24.3228	−1.16888	−0.584440	+	0.811437i	$0.698686\pi$
$434$	10.5737	0.507555
$435$	−2.92905	−0.140437
$436$	11.7186	0.561217
$437$	−0.644393	−0.0308255
$438$	29.3941	1.40451
$439$	−3.40022	−0.162283	−0.0811417	−	0.996703i	$-0.525857\pi$
$439$	−3.40022	−0.162283	−0.0811417	+	0.996703i	$0.525857\pi$
$440$	−9.77832	−0.466163
$441$	−4.11682	−0.196039
$442$	0	0
$443$	16.5029	0.784075	0.392038	−	0.919949i	$-0.371770\pi$
$443$	16.5029	0.784075	0.392038	+	0.919949i	$0.371770\pi$
$444$	−16.5375	−0.784835
$445$	4.49947	0.213295
$446$	−22.9805	−1.08816
$447$	−49.4552	−2.33915
$448$	−2.17944	−0.102969
$449$	−7.72011	−0.364335	−0.182167	−	0.983268i	$-0.558311\pi$
$449$	−7.72011	−0.364335	−0.182167	+	0.983268i	$0.558311\pi$
$450$	−8.74267	−0.412133
$451$	−0.368806	−0.0173664
$452$	−0.945828	−0.0444880
$453$	−20.4395	−0.960333
$454$	5.49722	0.257997
$455$	0	0
$456$	−2.19765	−0.102914
$457$	−35.4938	−1.66033	−0.830165	−	0.557518i	$-0.811754\pi$
$457$	−35.4938	−1.66033	−0.830165	+	0.557518i	$0.811754\pi$
$458$	−1.52318	−0.0711737
$459$	18.2690	0.852724
$460$	2.01504	0.0939516
$461$	14.3955	0.670467	0.335233	−	0.942135i	$-0.391185\pi$
$461$	14.3955	0.670467	0.335233	+	0.942135i	$0.391185\pi$
$462$	−14.9773	−0.696809
$463$	−2.13703	−0.0993164	−0.0496582	−	0.998766i	$-0.515813\pi$
$463$	−2.13703	−0.0993164	−0.0496582	+	0.998766i	$0.515813\pi$
$464$	0.426223	0.0197869
$465$	−33.3406	−1.54613
$466$	−25.9377	−1.20154
$467$	22.3298	1.03330	0.516651	−	0.856196i	$-0.327179\pi$
$467$	22.3298	1.03330	0.516651	+	0.856196i	$0.327179\pi$
$468$	0	0
$469$	3.82410	0.176580
$470$	−19.1594	−0.883758
$471$	5.68757	0.262070
$472$	−10.6322	−0.489388
$473$	−7.59411	−0.349177
$474$	−24.9149	−1.14438
$475$	4.77832	0.219244
$476$	15.4805	0.709549
$477$	7.52785	0.344676
$478$	−5.55924	−0.254274
$479$	−6.43997	−0.294250	−0.147125	−	0.989118i	$-0.547002\pi$
$479$	−6.43997	−0.294250	−0.147125	+	0.989118i	$0.547002\pi$
$480$	6.87211	0.313667
$481$	0	0
$482$	11.9583	0.544686
$483$	3.08641	0.140436
$484$	−1.22168	−0.0555308
$485$	−37.3199	−1.69461
$486$	16.7687	0.760643
$487$	34.7098	1.57285	0.786427	−	0.617684i	$-0.211929\pi$
$487$	34.7098	1.57285	0.786427	+	0.617684i	$0.211929\pi$
$488$	−8.65802	−0.391930
$489$	49.0453	2.21790
$490$	−7.03599	−0.317853
$491$	27.6996	1.25006	0.625032	−	0.780599i	$-0.285086\pi$
$491$	27.6996	1.25006	0.625032	+	0.780599i	$0.285086\pi$
$492$	0.259193	0.0116853
$493$	−3.02746	−0.136350
$494$	0	0
$495$	17.8909	0.804137
$496$	4.85159	0.217843
$497$	22.3057	1.00055
$498$	−27.6648	−1.23969
$499$	10.0199	0.448551	0.224276	−	0.974526i	$-0.427998\pi$
$499$	10.0199	0.448551	0.224276	+	0.974526i	$0.427998\pi$
$500$	0.693196	0.0310007
$501$	51.0991	2.28294
$502$	22.4814	1.00340
$503$	−0.0625293	−0.00278804	−0.00139402	−	0.999999i	$-0.500444\pi$
$503$	−0.0625293	−0.00278804	−0.00139402	+	0.999999i	$0.500444\pi$
$504$	3.98761	0.177622
$505$	38.1010	1.69547
$506$	−2.01504	−0.0895793
$507$	0	0
$508$	9.51474	0.422148
$509$	26.8041	1.18807	0.594035	−	0.804439i	$-0.297534\pi$
$509$	26.8041	1.18807	0.594035	+	0.804439i	$0.297534\pi$
$510$	−48.8126	−2.16146
$511$	29.1506	1.28954
$512$	−1.00000	−0.0441942
$513$	−2.57201	−0.113557
$514$	−4.85032	−0.213938
$515$	−32.9394	−1.45148
$516$	5.33707	0.234951
$517$	19.1594	0.842630
$518$	−16.4005	−0.720595
$519$	−47.8420	−2.10003
$520$	0	0
$521$	−21.2646	−0.931619	−0.465810	−	0.884885i	$-0.654237\pi$
$521$	−21.2646	−0.931619	−0.465810	+	0.884885i	$0.654237\pi$
$522$	−0.779840	−0.0341327
$523$	−11.6367	−0.508838	−0.254419	−	0.967094i	$-0.581884\pi$
$523$	−11.6367	−0.508838	−0.254419	+	0.967094i	$0.581884\pi$
$524$	2.94454	0.128633
$525$	−22.8864	−0.998845
$526$	10.1214	0.441312
$527$	−34.4608	−1.50114
$528$	−6.87211	−0.299070
$529$	−22.5848	−0.981946
$530$	12.8657	0.558852
$531$	19.4533	0.844200
$532$	−2.17944	−0.0944906
$533$	0	0
$534$	3.16218	0.136841
$535$	−17.7556	−0.767640
$536$	1.75463	0.0757883
$537$	−54.0943	−2.33434
$538$	−7.99747	−0.344795
$539$	7.03599	0.303061
$540$	8.04276	0.346105
$541$	−11.0025	−0.473033	−0.236517	−	0.971627i	$-0.576006\pi$
$541$	−11.0025	−0.473033	−0.236517	+	0.971627i	$0.576006\pi$
$542$	9.26475	0.397955
$543$	−8.52252	−0.365736
$544$	7.10300	0.304539
$545$	−36.6443	−1.56967
$546$	0	0
$547$	−42.3946	−1.81266	−0.906332	−	0.422566i	$-0.861129\pi$
$547$	−42.3946	−1.81266	−0.906332	+	0.422566i	$0.861129\pi$
$548$	4.54962	0.194350
$549$	15.8412	0.676084
$550$	14.9420	0.637127
$551$	0.426223	0.0181577
$552$	1.41615	0.0602753
$553$	−24.7084	−1.05071
$554$	−14.1772	−0.602330
$555$	51.7133	2.19511
$556$	−14.9108	−0.632359
$557$	−21.7486	−0.921519	−0.460760	−	0.887525i	$-0.652423\pi$
$557$	−21.7486	−0.921519	−0.460760	+	0.887525i	$0.652423\pi$
$558$	−8.87672	−0.375781
$559$	0	0
$560$	6.81517	0.287993
$561$	48.8126	2.06087
$562$	−27.3738	−1.15470
$563$	−6.23601	−0.262817	−0.131408	−	0.991328i	$-0.541950\pi$
$563$	−6.23601	−0.262817	−0.131408	+	0.991328i	$0.541950\pi$
$564$	−13.4651	−0.566981
$565$	2.95763	0.124429
$566$	−23.5136	−0.988349
$567$	24.2818	1.01974
$568$	10.2346	0.429435
$569$	−24.7089	−1.03585	−0.517925	−	0.855426i	$-0.673295\pi$
$569$	−24.7089	−1.03585	−0.517925	+	0.855426i	$0.673295\pi$
$570$	6.87211	0.287841
$571$	12.5056	0.523342	0.261671	−	0.965157i	$-0.415727\pi$
$571$	12.5056	0.523342	0.261671	+	0.965157i	$0.415727\pi$
$572$	0	0
$573$	31.9292	1.33386
$574$	0.257046	0.0107289
$575$	−3.07912	−0.128408
$576$	1.82965	0.0762355
$577$	23.2185	0.966599	0.483300	−	0.875455i	$-0.339438\pi$
$577$	23.2185	0.966599	0.483300	+	0.875455i	$0.339438\pi$
$578$	−33.4526	−1.39144
$579$	56.0592	2.32974
$580$	−1.33281	−0.0553420
$581$	−27.4355	−1.13822
$582$	−26.2281	−1.08719
$583$	−12.8657	−0.532844
$584$	13.3753	0.553473
$585$	0	0
$586$	11.1201	0.459368
$587$	32.3213	1.33404	0.667022	−	0.745038i	$-0.267569\pi$
$587$	32.3213	1.33404	0.667022	+	0.745038i	$0.267569\pi$
$588$	−4.94482	−0.203921
$589$	4.85159	0.199906
$590$	33.2473	1.36877
$591$	11.2735	0.463729
$592$	−7.52510	−0.309280
$593$	19.5435	0.802556	0.401278	−	0.915956i	$-0.368566\pi$
$593$	19.5435	0.802556	0.401278	+	0.915956i	$0.368566\pi$
$594$	−8.04276	−0.329998
$595$	−48.4081	−1.98454
$596$	−22.5037	−0.921788
$597$	−14.4774	−0.592522
$598$	0	0
$599$	23.8796	0.975692	0.487846	−	0.872930i	$-0.337783\pi$
$599$	23.8796	0.975692	0.487846	+	0.872930i	$0.337783\pi$
$600$	−10.5011	−0.428704
$601$	−1.52829	−0.0623402	−0.0311701	−	0.999514i	$-0.509923\pi$
$601$	−1.52829	−0.0623402	−0.0311701	+	0.999514i	$0.509923\pi$
$602$	5.29284	0.215720
$603$	−3.21035	−0.130736
$604$	−9.30065	−0.378438
$605$	3.82023	0.155314
$606$	26.7770	1.08774
$607$	43.4059	1.76179	0.880896	−	0.473309i	$-0.156941\pi$
$607$	43.4059	1.76179	0.880896	+	0.473309i	$0.156941\pi$
$608$	−1.00000	−0.0405554
$609$	−2.04145	−0.0827238
$610$	27.0739	1.09619
$611$	0	0
$612$	−12.9960	−0.525333
$613$	−36.6253	−1.47928	−0.739641	−	0.673001i	$-0.765005\pi$
$613$	−36.6253	−1.47928	−0.739641	+	0.673001i	$0.765005\pi$
$614$	−10.2579	−0.413975
$615$	−0.810506	−0.0326827
$616$	−6.81517	−0.274591
$617$	−37.2515	−1.49969	−0.749844	−	0.661614i	$-0.769872\pi$
$617$	−37.2515	−1.49969	−0.749844	+	0.661614i	$0.769872\pi$
$618$	−23.1495	−0.931209
$619$	−16.3273	−0.656249	−0.328124	−	0.944635i	$-0.606416\pi$
$619$	−16.3273	−0.656249	−0.328124	+	0.944635i	$0.606416\pi$
$620$	−15.1711	−0.609284
$621$	1.65739	0.0665086
$622$	−13.8693	−0.556108
$623$	3.13598	0.125640
$624$	0	0
$625$	−26.0593	−1.04237
$626$	3.37717	0.134979
$627$	−6.87211	−0.274446
$628$	2.58803	0.103274
$629$	53.4507	2.13122
$630$	−12.4694	−0.496792
$631$	6.00903	0.239216	0.119608	−	0.992821i	$-0.461836\pi$
$631$	6.00903	0.239216	0.119608	+	0.992821i	$0.461836\pi$
$632$	−11.3371	−0.450964
$633$	−10.9593	−0.435592
$634$	6.12834	0.243387
$635$	−29.7529	−1.18071
$636$	9.04191	0.358535
$637$	0	0
$638$	1.33281	0.0527666
$639$	−18.7258	−0.740781
$640$	3.12703	0.123607
$641$	−14.9927	−0.592178	−0.296089	−	0.955160i	$-0.595682\pi$
$641$	−14.9927	−0.592178	−0.296089	+	0.955160i	$0.595682\pi$
$642$	−12.4784	−0.492485
$643$	−48.0012	−1.89298	−0.946492	−	0.322727i	$-0.895400\pi$
$643$	−48.0012	−1.89298	−0.946492	+	0.322727i	$0.895400\pi$
$644$	1.40441	0.0553417
$645$	−16.6892	−0.657135
$646$	7.10300	0.279464
$647$	37.2312	1.46371	0.731855	−	0.681460i	$-0.238655\pi$
$647$	37.2312	1.46371	0.731855	+	0.681460i	$0.238655\pi$
$648$	11.1413	0.437673
$649$	−33.2473	−1.30507
$650$	0	0
$651$	−23.2373	−0.910742
$652$	22.3172	0.874008
$653$	36.6613	1.43467	0.717334	−	0.696730i	$-0.245362\pi$
$653$	36.6613	1.43467	0.717334	+	0.696730i	$0.245362\pi$
$654$	−25.7533	−1.00703
$655$	−9.20767	−0.359773
$656$	0.117941	0.00460483
$657$	−24.4721	−0.954747
$658$	−13.3535	−0.520573
$659$	22.8356	0.889550	0.444775	−	0.895642i	$-0.353284\pi$
$659$	22.8356	0.889550	0.444775	+	0.895642i	$0.353284\pi$
$660$	21.4893	0.836470
$661$	−44.8720	−1.74532	−0.872660	−	0.488329i	$-0.837607\pi$
$661$	−44.8720	−1.74532	−0.872660	+	0.488329i	$0.837607\pi$
$662$	0.0943337	0.00366638
$663$	0	0
$664$	−12.5884	−0.488523
$665$	6.81517	0.264281
$666$	13.7683	0.533511
$667$	−0.274655	−0.0106347
$668$	23.2517	0.899636
$669$	50.5031	1.95256
$670$	−5.48677	−0.211972
$671$	−27.0739	−1.04518
$672$	4.78963	0.184764
$673$	−7.38846	−0.284804	−0.142402	−	0.989809i	$-0.545483\pi$
$673$	−7.38846	−0.284804	−0.142402	+	0.989809i	$0.545483\pi$
$674$	9.25406	0.356453
$675$	−12.2899	−0.473038
$676$	0	0
$677$	12.5549	0.482526	0.241263	−	0.970460i	$-0.422438\pi$
$677$	12.5549	0.482526	0.241263	+	0.970460i	$0.422438\pi$
$678$	2.07860	0.0798280
$679$	−26.0107	−0.998200
$680$	−22.2113	−0.851764
$681$	−12.0809	−0.462943
$682$	15.1711	0.580930
$683$	−30.5047	−1.16723	−0.583616	−	0.812030i	$-0.698363\pi$
$683$	−30.5047	−1.16723	−0.583616	+	0.812030i	$0.698363\pi$
$684$	1.82965	0.0699585
$685$	−14.2268	−0.543579
$686$	−20.1599	−0.769709
$687$	3.34742	0.127712
$688$	2.42854	0.0925871
$689$	0	0
$690$	−4.42834	−0.168584
$691$	4.48550	0.170637	0.0853183	−	0.996354i	$-0.472809\pi$
$691$	4.48550	0.170637	0.0853183	+	0.996354i	$0.472809\pi$
$692$	−21.7696	−0.827557
$693$	12.4694	0.473673
$694$	29.1677	1.10719
$695$	46.6266	1.76865
$696$	−0.936688	−0.0355051
$697$	−0.837737	−0.0317315
$698$	−29.5303	−1.11774
$699$	57.0019	2.15601
$700$	−10.4141	−0.393614
$701$	−6.07918	−0.229607	−0.114804	−	0.993388i	$-0.536624\pi$
$701$	−6.07918	−0.229607	−0.114804	+	0.993388i	$0.536624\pi$
$702$	0	0
$703$	−7.52510	−0.283814
$704$	−3.12703	−0.117854
$705$	42.1056	1.58579
$706$	−29.0884	−1.09476
$707$	26.5551	0.998709
$708$	23.3659	0.878143
$709$	45.0758	1.69286	0.846429	−	0.532502i	$-0.178748\pi$
$709$	45.0758	1.69286	0.846429	+	0.532502i	$0.178748\pi$
$710$	−32.0040	−1.20109
$711$	20.7429	0.777919
$712$	1.43889	0.0539248
$713$	−3.12633	−0.117082
$714$	−34.0208	−1.27320
$715$	0	0
$716$	−24.6147	−0.919893
$717$	12.2172	0.456262
$718$	−19.0423	−0.710654
$719$	24.2298	0.903621	0.451810	−	0.892114i	$-0.350778\pi$
$719$	24.2298	0.903621	0.451810	+	0.892114i	$0.350778\pi$
$720$	−5.72138	−0.213223
$721$	−22.9577	−0.854988
$722$	−1.00000	−0.0372161
$723$	−26.2801	−0.977369
$724$	−3.87802	−0.144126
$725$	2.03663	0.0756386
$726$	2.68482	0.0996430
$727$	−10.3596	−0.384218	−0.192109	−	0.981374i	$-0.561533\pi$
$727$	−10.3596	−0.384218	−0.192109	+	0.981374i	$0.561533\pi$
$728$	0	0
$729$	−3.42764	−0.126950
$730$	−41.8249	−1.54801
$731$	−17.2499	−0.638010
$732$	19.0273	0.703268
$733$	24.3843	0.900654	0.450327	−	0.892864i	$-0.351307\pi$
$733$	24.3843	0.900654	0.450327	+	0.892864i	$0.351307\pi$
$734$	−1.02770	−0.0379332
$735$	15.4626	0.570347
$736$	0.644393	0.0237526
$737$	5.48677	0.202108
$738$	−0.215792	−0.00794340
$739$	1.07875	0.0396825	0.0198413	−	0.999803i	$-0.493684\pi$
$739$	1.07875	0.0396825	0.0198413	+	0.999803i	$0.493684\pi$
$740$	23.5312	0.865024
$741$	0	0
$742$	8.96699	0.329188
$743$	3.38672	0.124247	0.0621233	−	0.998068i	$-0.480213\pi$
$743$	3.38672	0.124247	0.0621233	+	0.998068i	$0.480213\pi$
$744$	−10.6621	−0.390891
$745$	70.3698	2.57815
$746$	−15.9433	−0.583725
$747$	23.0323	0.842708
$748$	22.2113	0.812125
$749$	−12.3750	−0.452174
$750$	−1.52340	−0.0556267
$751$	25.1897	0.919186	0.459593	−	0.888130i	$-0.347995\pi$
$751$	25.1897	0.919186	0.459593	+	0.888130i	$0.347995\pi$
$752$	−6.12703	−0.223430
$753$	−49.4063	−1.80046
$754$	0	0
$755$	29.0834	1.05845
$756$	5.60554	0.203871
$757$	42.8857	1.55871	0.779353	−	0.626585i	$-0.215548\pi$
$757$	42.8857	1.55871	0.779353	+	0.626585i	$0.215548\pi$
$758$	4.79302	0.174090
$759$	4.42834	0.160739
$760$	3.12703	0.113429
$761$	−22.8112	−0.826906	−0.413453	−	0.910525i	$-0.635677\pi$
$761$	−22.8112	−0.826906	−0.413453	+	0.910525i	$0.635677\pi$
$762$	−20.9100	−0.757491
$763$	−25.5399	−0.924605
$764$	14.5288	0.525634
$765$	40.6389	1.46930
$766$	−21.8340	−0.788895
$767$	0	0
$768$	2.19765	0.0793008
$769$	−16.1128	−0.581042	−0.290521	−	0.956869i	$-0.593829\pi$
$769$	−16.1128	−0.581042	−0.290521	+	0.956869i	$0.593829\pi$
$770$	21.3112	0.768004
$771$	10.6593	0.383885
$772$	25.5087	0.918080
$773$	−33.3582	−1.19981	−0.599905	−	0.800071i	$-0.704795\pi$
$773$	−33.3582	−1.19981	−0.599905	+	0.800071i	$0.704795\pi$
$774$	−4.44338	−0.159714
$775$	23.1824	0.832738
$776$	−11.9346	−0.428428
$777$	36.0424	1.29302
$778$	−16.3944	−0.587769
$779$	0.117941	0.00422569
$780$	0	0
$781$	32.0040	1.14519
$782$	−4.57712	−0.163678
$783$	−1.09625	−0.0391768
$784$	−2.25005	−0.0803591
$785$	−8.09285	−0.288846
$786$	−6.47106	−0.230815
$787$	14.1846	0.505628	0.252814	−	0.967515i	$-0.418644\pi$
$787$	14.1846	0.505628	0.252814	+	0.967515i	$0.418644\pi$
$788$	5.12980	0.182742
$789$	−22.2432	−0.791877
$790$	35.4514	1.26130
$791$	2.06137	0.0732940
$792$	5.72138	0.203300
$793$	0	0
$794$	8.32544	0.295459
$795$	−28.2743	−1.00279
$796$	−6.58770	−0.233495
$797$	−16.5599	−0.586581	−0.293291	−	0.956023i	$-0.594750\pi$
$797$	−16.5599	−0.586581	−0.293291	+	0.956023i	$0.594750\pi$
$798$	4.78963	0.169551
$799$	43.5203	1.53964
$800$	−4.77832	−0.168939
$801$	−2.63268	−0.0930210
$802$	35.8203	1.26486
$803$	41.8249	1.47597
$804$	−3.85605	−0.135992
$805$	−4.39165	−0.154785
$806$	0	0
$807$	17.5756	0.618691
$808$	12.1844	0.428646
$809$	22.3953	0.787378	0.393689	−	0.919244i	$-0.371199\pi$
$809$	22.3953	0.787378	0.393689	+	0.919244i	$0.371199\pi$
$810$	−34.8393	−1.22413
$811$	28.1667	0.989066	0.494533	−	0.869159i	$-0.335339\pi$
$811$	28.1667	0.989066	0.494533	+	0.869159i	$0.335339\pi$
$812$	−0.928927	−0.0325989
$813$	−20.3607	−0.714080
$814$	−23.5312	−0.824768
$815$	−69.7865	−2.44452
$816$	−15.6099	−0.546455
$817$	2.42854	0.0849637
$818$	28.5608	0.998606
$819$	0	0
$820$	−0.368806	−0.0128793
$821$	10.2960	0.359334	0.179667	−	0.983727i	$-0.442498\pi$
$821$	10.2960	0.359334	0.179667	+	0.983727i	$0.442498\pi$
$822$	−9.99847	−0.348737
$823$	18.1042	0.631074	0.315537	−	0.948913i	$-0.397815\pi$
$823$	18.1042	0.631074	0.315537	+	0.948913i	$0.397815\pi$
$824$	−10.5338	−0.366961
$825$	−32.8371	−1.14324
$826$	23.1723	0.806266
$827$	0.466529	0.0162228	0.00811139	−	0.999967i	$-0.497418\pi$
$827$	0.466529	0.0162228	0.00811139	+	0.999967i	$0.497418\pi$
$828$	−1.17902	−0.0409736
$829$	−14.4832	−0.503023	−0.251511	−	0.967854i	$-0.580928\pi$
$829$	−14.4832	−0.503023	−0.251511	+	0.967854i	$0.580928\pi$
$830$	39.3642	1.36635
$831$	31.1564	1.08080
$832$	0	0
$833$	15.9821	0.553748
$834$	32.7687	1.13469
$835$	−72.7089	−2.51619
$836$	−3.12703	−0.108151
$837$	−12.4783	−0.431314
$838$	39.0336	1.34839
$839$	0.938352	0.0323955	0.0161978	−	0.999869i	$-0.494844\pi$
$839$	0.938352	0.0323955	0.0161978	+	0.999869i	$0.494844\pi$
$840$	−14.9773	−0.516767
$841$	−28.8183	−0.993736
$842$	35.6361	1.22810
$843$	60.1580	2.07195
$844$	−4.98682	−0.171654
$845$	0	0
$846$	11.2103	0.385419
$847$	2.66257	0.0914871
$848$	4.11436	0.141288
$849$	51.6745	1.77346
$850$	33.9404	1.16415
$851$	4.84912	0.166226
$852$	−22.4921	−0.770566
$853$	16.5020	0.565018	0.282509	−	0.959265i	$-0.408833\pi$
$853$	16.5020	0.565018	0.282509	+	0.959265i	$0.408833\pi$
$854$	18.8696	0.645705
$855$	−5.72138	−0.195667
$856$	−5.67809	−0.194073
$857$	−1.55202	−0.0530159	−0.0265079	−	0.999649i	$-0.508439\pi$
$857$	−1.55202	−0.0530159	−0.0265079	+	0.999649i	$0.508439\pi$
$858$	0	0
$859$	18.1217	0.618306	0.309153	−	0.951012i	$-0.399955\pi$
$859$	18.1217	0.618306	0.309153	+	0.951012i	$0.399955\pi$
$860$	−7.59411	−0.258957
$861$	−0.564896	−0.0192516
$862$	28.5835	0.973557
$863$	−47.5374	−1.61819	−0.809096	−	0.587676i	$-0.800043\pi$
$863$	−47.5374	−1.61819	−0.809096	+	0.587676i	$0.800043\pi$
$864$	2.57201	0.0875016
$865$	68.0743	2.31460
$866$	24.3228	0.826524
$867$	73.5170	2.49677
$868$	−10.5737	−0.358896
$869$	−35.4514	−1.20260
$870$	2.92905	0.0993042
$871$	0	0
$872$	−11.7186	−0.396841
$873$	21.8362	0.739043
$874$	0.644393	0.0217969
$875$	−1.51078	−0.0510736
$876$	−29.3941	−0.993135
$877$	53.0121	1.79009	0.895046	−	0.445973i	$-0.147142\pi$
$877$	53.0121	1.79009	0.895046	+	0.445973i	$0.147142\pi$
$878$	3.40022	0.114752
$879$	−24.4381	−0.824278
$880$	9.77832	0.329627
$881$	−26.0499	−0.877645	−0.438822	−	0.898574i	$-0.644604\pi$
$881$	−26.0499	−0.877645	−0.438822	+	0.898574i	$0.644604\pi$
$882$	4.11682	0.138620
$883$	36.9504	1.24348	0.621740	−	0.783224i	$-0.286426\pi$
$883$	36.9504	1.24348	0.621740	+	0.783224i	$0.286426\pi$
$884$	0	0
$885$	−73.0658	−2.45608
$886$	−16.5029	−0.554425
$887$	42.5093	1.42732	0.713661	−	0.700491i	$-0.247036\pi$
$887$	42.5093	1.42732	0.713661	+	0.700491i	$0.247036\pi$
$888$	16.5375	0.554962
$889$	−20.7368	−0.695489
$890$	−4.49947	−0.150822
$891$	34.8393	1.16716
$892$	22.9805	0.769445
$893$	−6.12703	−0.205033
$894$	49.4552	1.65403
$895$	76.9708	2.57285
$896$	2.17944	0.0728099
$897$	0	0
$898$	7.72011	0.257623
$899$	2.06786	0.0689669
$900$	8.74267	0.291422
$901$	−29.2243	−0.973602
$902$	0.368806	0.0122799
$903$	−11.6318	−0.387082
$904$	0.945828	0.0314578
$905$	12.1267	0.403105
$906$	20.4395	0.679058
$907$	4.12649	0.137018	0.0685090	−	0.997651i	$-0.478176\pi$
$907$	4.12649	0.137018	0.0685090	+	0.997651i	$0.478176\pi$
$908$	−5.49722	−0.182432
$909$	−22.2932	−0.739419
$910$	0	0
$911$	14.4605	0.479097	0.239549	−	0.970884i	$-0.423001\pi$
$911$	14.4605	0.479097	0.239549	+	0.970884i	$0.423001\pi$
$912$	2.19765	0.0727714
$913$	−39.3642	−1.30276
$914$	35.4938	1.17403
$915$	−59.4989	−1.96697
$916$	1.52318	0.0503274
$917$	−6.41744	−0.211923
$918$	−18.2690	−0.602967
$919$	59.7790	1.97193	0.985963	−	0.166965i	$-0.0533967\pi$
$919$	59.7790	1.97193	0.985963	+	0.166965i	$0.0533967\pi$
$920$	−2.01504	−0.0664338
$921$	22.5433	0.742826
$922$	−14.3955	−0.474092
$923$	0	0
$924$	14.9773	0.492718
$925$	−35.9573	−1.18227
$926$	2.13703	0.0702273
$927$	19.2731	0.633012
$928$	−0.426223	−0.0139915
$929$	−5.85030	−0.191942	−0.0959711	−	0.995384i	$-0.530596\pi$
$929$	−5.85030	−0.191942	−0.0959711	+	0.995384i	$0.530596\pi$
$930$	33.3406	1.09328
$931$	−2.25005	−0.0737425
$932$	25.9377	0.849618
$933$	30.4798	0.997864
$934$	−22.3298	−0.730654
$935$	−69.4554	−2.27143
$936$	0	0
$937$	−28.4140	−0.928243	−0.464122	−	0.885771i	$-0.653630\pi$
$937$	−28.4140	−0.928243	−0.464122	+	0.885771i	$0.653630\pi$
$938$	−3.82410	−0.124861
$939$	−7.42184	−0.242203
$940$	19.1594	0.624911
$941$	−5.81060	−0.189420	−0.0947101	−	0.995505i	$-0.530192\pi$
$941$	−5.81060	−0.189420	−0.0947101	+	0.995505i	$0.530192\pi$
$942$	−5.68757	−0.185311
$943$	−0.0760006	−0.00247492
$944$	10.6322	0.346049
$945$	−17.5287	−0.570208
$946$	7.59411	0.246906
$947$	−8.52929	−0.277165	−0.138582	−	0.990351i	$-0.544255\pi$
$947$	−8.52929	−0.277165	−0.138582	+	0.990351i	$0.544255\pi$
$948$	24.9149	0.809197
$949$	0	0
$950$	−4.77832	−0.155029
$951$	−13.4679	−0.436727
$952$	−15.4805	−0.501727
$953$	37.1660	1.20392	0.601962	−	0.798525i	$-0.294386\pi$
$953$	37.1660	1.20392	0.601962	+	0.798525i	$0.294386\pi$
$954$	−7.52785	−0.243723
$955$	−45.4321	−1.47015
$956$	5.55924	0.179799
$957$	−2.92905	−0.0946828
$958$	6.43997	0.208066
$959$	−9.91562	−0.320192
$960$	−6.87211	−0.221796
$961$	−7.46211	−0.240713
$962$	0	0
$963$	10.3889	0.334779
$964$	−11.9583	−0.385151
$965$	−79.7666	−2.56778
$966$	−3.08641	−0.0993035
$967$	38.7782	1.24702	0.623511	−	0.781815i	$-0.285706\pi$
$967$	38.7782	1.24702	0.623511	+	0.781815i	$0.285706\pi$
$968$	1.22168	0.0392662
$969$	−15.6099	−0.501462
$970$	37.3199	1.19827
$971$	4.84313	0.155424	0.0777118	−	0.996976i	$-0.475239\pi$
$971$	4.84313	0.155424	0.0777118	+	0.996976i	$0.475239\pi$
$972$	−16.7687	−0.537856
$973$	32.4972	1.04181
$974$	−34.7098	−1.11218
$975$	0	0
$976$	8.65802	0.277136
$977$	20.1405	0.644352	0.322176	−	0.946680i	$-0.395586\pi$
$977$	20.1405	0.644352	0.322176	+	0.946680i	$0.395586\pi$
$978$	−49.0453	−1.56830
$979$	4.49947	0.143804
$980$	7.03599	0.224756
$981$	21.4409	0.684555
$982$	−27.6996	−0.883929
$983$	−11.3342	−0.361506	−0.180753	−	0.983529i	$-0.557853\pi$
$983$	−11.3342	−0.361506	−0.180753	+	0.983529i	$0.557853\pi$
$984$	−0.259193	−0.00826278
$985$	−16.0410	−0.511110
$986$	3.02746	0.0964140
$987$	29.3462	0.934101
$988$	0	0
$989$	−1.56493	−0.0497619
$990$	−17.8909	−0.568611
$991$	−52.9777	−1.68289	−0.841447	−	0.540340i	$-0.818296\pi$
$991$	−52.9777	−1.68289	−0.841447	+	0.540340i	$0.818296\pi$
$992$	−4.85159	−0.154038
$993$	−0.207312	−0.00657885
$994$	−22.3057	−0.707494
$995$	20.5999	0.653062
$996$	27.6648	0.876592
$997$	54.6400	1.73047	0.865233	−	0.501369i	$-0.167170\pi$
$997$	54.6400	1.73047	0.865233	+	0.501369i	$0.167170\pi$
$998$	−10.0199	−0.317174
$999$	19.3546	0.612353

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.bc.1.5		6
13.4	even	6		494.2.g.f.419.2	yes	12
13.10	even	6		494.2.g.f.191.2	✓	12
13.12	even	2		6422.2.a.bd.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
494.2.g.f.191.2	✓	12	13.10	even	6
494.2.g.f.419.2	yes	12	13.4	even	6
6422.2.a.bc.1.5		6	1.1	even	1	trivial
6422.2.a.bd.1.5		6	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +2.19765 q^{3} +1.00000 q^{4} -3.12703 q^{5} -2.19765 q^{6} -2.17944 q^{7} -1.00000 q^{8} +1.82965 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +2.19765 q^{3} +1.00000 q^{4} -3.12703 q^{5} -2.19765 q^{6} -2.17944 q^{7} -1.00000 q^{8} +1.82965 q^{9} +3.12703 q^{10} -3.12703 q^{11} +2.19765 q^{12} +2.17944 q^{14} -6.87211 q^{15} +1.00000 q^{16} -7.10300 q^{17} -1.82965 q^{18} +1.00000 q^{19} -3.12703 q^{20} -4.78963 q^{21} +3.12703 q^{22} -0.644393 q^{23} -2.19765 q^{24} +4.77832 q^{25} -2.57201 q^{27} -2.17944 q^{28} +0.426223 q^{29} +6.87211 q^{30} +4.85159 q^{31} -1.00000 q^{32} -6.87211 q^{33} +7.10300 q^{34} +6.81517 q^{35} +1.82965 q^{36} -7.52510 q^{37} -1.00000 q^{38} +3.12703 q^{40} +0.117941 q^{41} +4.78963 q^{42} +2.42854 q^{43} -3.12703 q^{44} -5.72138 q^{45} +0.644393 q^{46} -6.12703 q^{47} +2.19765 q^{48} -2.25005 q^{49} -4.77832 q^{50} -15.6099 q^{51} +4.11436 q^{53} +2.57201 q^{54} +9.77832 q^{55} +2.17944 q^{56} +2.19765 q^{57} -0.426223 q^{58} +10.6322 q^{59} -6.87211 q^{60} +8.65802 q^{61} -4.85159 q^{62} -3.98761 q^{63} +1.00000 q^{64} +6.87211 q^{66} -1.75463 q^{67} -7.10300 q^{68} -1.41615 q^{69} -6.81517 q^{70} -10.2346 q^{71} -1.82965 q^{72} -13.3753 q^{73} +7.52510 q^{74} +10.5011 q^{75} +1.00000 q^{76} +6.81517 q^{77} +11.3371 q^{79} -3.12703 q^{80} -11.1413 q^{81} -0.117941 q^{82} +12.5884 q^{83} -4.78963 q^{84} +22.2113 q^{85} -2.42854 q^{86} +0.936688 q^{87} +3.12703 q^{88} -1.43889 q^{89} +5.72138 q^{90} -0.644393 q^{92} +10.6621 q^{93} +6.12703 q^{94} -3.12703 q^{95} -2.19765 q^{96} +11.9346 q^{97} +2.25005 q^{98} -5.72138 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{2} + 2 q^{3} + 6 q^{4} - 2 q^{5} - 2 q^{6} - q^{7} - 6 q^{8} + 10 q^{9}+O(q^{10}) \) 6 * q - 6 * q^2 + 2 * q^3 + 6 * q^4 - 2 * q^5 - 2 * q^6 - q^7 - 6 * q^8 + 10 * q^9	\( 6 q - 6 q^{2} + 2 q^{3} + 6 q^{4} - 2 q^{5} - 2 q^{6} - q^{7} - 6 q^{8} + 10 q^{9} + 2 q^{10} - 2 q^{11} + 2 q^{12} + q^{14} + 9 q^{15} + 6 q^{16} - 2 q^{17} - 10 q^{18} + 6 q^{19} - 2 q^{20} - q^{21} + 2 q^{22} + 8 q^{23} - 2 q^{24} + 16 q^{25} - 4 q^{27} - q^{28} + 20 q^{29} - 9 q^{30} - 3 q^{31} - 6 q^{32} + 9 q^{33} + 2 q^{34} + 7 q^{35} + 10 q^{36} + 9 q^{37} - 6 q^{38} + 2 q^{40} - 3 q^{41} + q^{42} + 13 q^{43} - 2 q^{44} - 38 q^{45} - 8 q^{46} - 20 q^{47} + 2 q^{48} - 7 q^{49} - 16 q^{50} + 2 q^{51} + 25 q^{53} + 4 q^{54} + 46 q^{55} + q^{56} + 2 q^{57} - 20 q^{58} + 9 q^{60} + 6 q^{61} + 3 q^{62} - 46 q^{63} + 6 q^{64} - 9 q^{66} + 32 q^{67} - 2 q^{68} - 29 q^{69} - 7 q^{70} - 39 q^{71} - 10 q^{72} - 7 q^{73} - 9 q^{74} - 15 q^{75} + 6 q^{76} + 7 q^{77} + 18 q^{79} - 2 q^{80} + 54 q^{81} + 3 q^{82} + 7 q^{83} - q^{84} + 2 q^{85} - 13 q^{86} + 12 q^{87} + 2 q^{88} - 9 q^{89} + 38 q^{90} + 8 q^{92} + 47 q^{93} + 20 q^{94} - 2 q^{95} - 2 q^{96} + 4 q^{97} + 7 q^{98} - 38 q^{99}+O(q^{100}) \) 6 * q - 6 * q^2 + 2 * q^3 + 6 * q^4 - 2 * q^5 - 2 * q^6 - q^7 - 6 * q^8 + 10 * q^9 + 2 * q^10 - 2 * q^11 + 2 * q^12 + q^14 + 9 * q^15 + 6 * q^16 - 2 * q^17 - 10 * q^18 + 6 * q^19 - 2 * q^20 - q^21 + 2 * q^22 + 8 * q^23 - 2 * q^24 + 16 * q^25 - 4 * q^27 - q^28 + 20 * q^29 - 9 * q^30 - 3 * q^31 - 6 * q^32 + 9 * q^33 + 2 * q^34 + 7 * q^35 + 10 * q^36 + 9 * q^37 - 6 * q^38 + 2 * q^40 - 3 * q^41 + q^42 + 13 * q^43 - 2 * q^44 - 38 * q^45 - 8 * q^46 - 20 * q^47 + 2 * q^48 - 7 * q^49 - 16 * q^50 + 2 * q^51 + 25 * q^53 + 4 * q^54 + 46 * q^55 + q^56 + 2 * q^57 - 20 * q^58 + 9 * q^60 + 6 * q^61 + 3 * q^62 - 46 * q^63 + 6 * q^64 - 9 * q^66 + 32 * q^67 - 2 * q^68 - 29 * q^69 - 7 * q^70 - 39 * q^71 - 10 * q^72 - 7 * q^73 - 9 * q^74 - 15 * q^75 + 6 * q^76 + 7 * q^77 + 18 * q^79 - 2 * q^80 + 54 * q^81 + 3 * q^82 + 7 * q^83 - q^84 + 2 * q^85 - 13 * q^86 + 12 * q^87 + 2 * q^88 - 9 * q^89 + 38 * q^90 + 8 * q^92 + 47 * q^93 + 20 * q^94 - 2 * q^95 - 2 * q^96 + 4 * q^97 + 7 * q^98 - 38 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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