LMFDB - Embedded newform 6422.2.a.v.1.1

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:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.80194$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -1.80194 q^{3} +1.00000 q^{4} -2.44504 q^{5} -1.80194 q^{6} -4.49396 q^{7} +1.00000 q^{8} +0.246980 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.80194 q^{3} +1.00000 q^{4} -2.44504 q^{5} -1.80194 q^{6} -4.49396 q^{7} +1.00000 q^{8} +0.246980 q^{9} -2.44504 q^{10} -2.49396 q^{11} -1.80194 q^{12} -4.49396 q^{14} +4.40581 q^{15} +1.00000 q^{16} +7.74094 q^{17} +0.246980 q^{18} +1.00000 q^{19} -2.44504 q^{20} +8.09783 q^{21} -2.49396 q^{22} +4.27413 q^{23} -1.80194 q^{24} +0.978230 q^{25} +4.96077 q^{27} -4.49396 q^{28} -0.713792 q^{29} +4.40581 q^{30} -0.911854 q^{31} +1.00000 q^{32} +4.49396 q^{33} +7.74094 q^{34} +10.9879 q^{35} +0.246980 q^{36} +6.81163 q^{37} +1.00000 q^{38} -2.44504 q^{40} -3.28621 q^{41} +8.09783 q^{42} +7.20775 q^{43} -2.49396 q^{44} -0.603875 q^{45} +4.27413 q^{46} +5.20775 q^{47} -1.80194 q^{48} +13.1957 q^{49} +0.978230 q^{50} -13.9487 q^{51} -8.89008 q^{53} +4.96077 q^{54} +6.09783 q^{55} -4.49396 q^{56} -1.80194 q^{57} -0.713792 q^{58} -11.2567 q^{59} +4.40581 q^{60} +3.43296 q^{61} -0.911854 q^{62} -1.10992 q^{63} +1.00000 q^{64} +4.49396 q^{66} -1.40581 q^{67} +7.74094 q^{68} -7.70171 q^{69} +10.9879 q^{70} +3.86294 q^{71} +0.246980 q^{72} -7.74094 q^{73} +6.81163 q^{74} -1.76271 q^{75} +1.00000 q^{76} +11.2078 q^{77} -1.91723 q^{79} -2.44504 q^{80} -9.67994 q^{81} -3.28621 q^{82} -0.396125 q^{83} +8.09783 q^{84} -18.9269 q^{85} +7.20775 q^{86} +1.28621 q^{87} -2.49396 q^{88} -16.7681 q^{89} -0.603875 q^{90} +4.27413 q^{92} +1.64310 q^{93} +5.20775 q^{94} -2.44504 q^{95} -1.80194 q^{96} +3.16421 q^{97} +13.1957 q^{98} -0.615957 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} - q^{3} + 3 q^{4} - 7 q^{5} - q^{6} - 4 q^{7} + 3 q^{8} - 4 q^{9}+O(q^{10}) $ 3 * q + 3 * q^2 - q^3 + 3 * q^4 - 7 * q^5 - q^6 - 4 * q^7 + 3 * q^8 - 4 * q^9	$ 3 q + 3 q^{2} - q^{3} + 3 q^{4} - 7 q^{5} - q^{6} - 4 q^{7} + 3 q^{8} - 4 q^{9} - 7 q^{10} + 2 q^{11} - q^{12} - 4 q^{14} + 3 q^{16} + 9 q^{17} - 4 q^{18} + 3 q^{19} - 7 q^{20} + 6 q^{21} + 2 q^{22} + 2 q^{23} - q^{24} + 6 q^{25} + 2 q^{27} - 4 q^{28} + 6 q^{29} + q^{31} + 3 q^{32} + 4 q^{33} + 9 q^{34} + 14 q^{35} - 4 q^{36} - 6 q^{37} + 3 q^{38} - 7 q^{40} - 18 q^{41} + 6 q^{42} + 4 q^{43} + 2 q^{44} + 7 q^{45} + 2 q^{46} - 2 q^{47} - q^{48} + 3 q^{49} + 6 q^{50} - 10 q^{51} - 26 q^{53} + 2 q^{54} - 4 q^{56} - q^{57} + 6 q^{58} - 7 q^{59} - 9 q^{61} + q^{62} - 4 q^{63} + 3 q^{64} + 4 q^{66} + 9 q^{67} + 9 q^{68} + 4 q^{69} + 14 q^{70} + 17 q^{71} - 4 q^{72} - 9 q^{73} - 6 q^{74} + 12 q^{75} + 3 q^{76} + 16 q^{77} + q^{79} - 7 q^{80} - 5 q^{81} - 18 q^{82} - 10 q^{83} + 6 q^{84} - 28 q^{85} + 4 q^{86} + 12 q^{87} + 2 q^{88} - 30 q^{89} + 7 q^{90} + 2 q^{92} + 9 q^{93} - 2 q^{94} - 7 q^{95} - q^{96} - 2 q^{97} + 3 q^{98} - 12 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 - q^3 + 3 * q^4 - 7 * q^5 - q^6 - 4 * q^7 + 3 * q^8 - 4 * q^9 - 7 * q^10 + 2 * q^11 - q^12 - 4 * q^14 + 3 * q^16 + 9 * q^17 - 4 * q^18 + 3 * q^19 - 7 * q^20 + 6 * q^21 + 2 * q^22 + 2 * q^23 - q^24 + 6 * q^25 + 2 * q^27 - 4 * q^28 + 6 * q^29 + q^31 + 3 * q^32 + 4 * q^33 + 9 * q^34 + 14 * q^35 - 4 * q^36 - 6 * q^37 + 3 * q^38 - 7 * q^40 - 18 * q^41 + 6 * q^42 + 4 * q^43 + 2 * q^44 + 7 * q^45 + 2 * q^46 - 2 * q^47 - q^48 + 3 * q^49 + 6 * q^50 - 10 * q^51 - 26 * q^53 + 2 * q^54 - 4 * q^56 - q^57 + 6 * q^58 - 7 * q^59 - 9 * q^61 + q^62 - 4 * q^63 + 3 * q^64 + 4 * q^66 + 9 * q^67 + 9 * q^68 + 4 * q^69 + 14 * q^70 + 17 * q^71 - 4 * q^72 - 9 * q^73 - 6 * q^74 + 12 * q^75 + 3 * q^76 + 16 * q^77 + q^79 - 7 * q^80 - 5 * q^81 - 18 * q^82 - 10 * q^83 + 6 * q^84 - 28 * q^85 + 4 * q^86 + 12 * q^87 + 2 * q^88 - 30 * q^89 + 7 * q^90 + 2 * q^92 + 9 * q^93 - 2 * q^94 - 7 * q^95 - q^96 - 2 * q^97 + 3 * q^98 - 12 * 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
$3$	−1.80194	−1.04035	−0.520175	−	0.854060i	$-0.674133\pi$
$3$	−1.80194	−1.04035	−0.520175	+	0.854060i	$0.674133\pi$
$4$	1.00000	0.500000
$5$	−2.44504	−1.09346	−0.546728	−	0.837310i	$-0.684127\pi$
$5$	−2.44504	−1.09346	−0.546728	+	0.837310i	$0.684127\pi$
$6$	−1.80194	−0.735638
$7$	−4.49396	−1.69856	−0.849278	−	0.527945i	$-0.822963\pi$
$7$	−4.49396	−1.69856	−0.849278	+	0.527945i	$0.822963\pi$
$8$	1.00000	0.353553
$9$	0.246980	0.0823265
$10$	−2.44504	−0.773190
$11$	−2.49396	−0.751957	−0.375978	−	0.926628i	$-0.622693\pi$
$11$	−2.49396	−0.751957	−0.375978	+	0.926628i	$0.622693\pi$
$12$	−1.80194	−0.520175
$13$	0	0
$13$	0	0
$14$	−4.49396	−1.20106
$15$	4.40581	1.13758
$16$	1.00000	0.250000
$17$	7.74094	1.87745	0.938727	−	0.344662i	$-0.112007\pi$
$17$	7.74094	1.87745	0.938727	+	0.344662i	$0.112007\pi$
$18$	0.246980	0.0582137
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−2.44504	−0.546728
$21$	8.09783	1.76709
$22$	−2.49396	−0.531714
$23$	4.27413	0.891217	0.445609	−	0.895228i	$-0.352987\pi$
$23$	4.27413	0.891217	0.445609	+	0.895228i	$0.352987\pi$
$24$	−1.80194	−0.367819
$25$	0.978230	0.195646
$26$	0	0
$27$	4.96077	0.954701
$28$	−4.49396	−0.849278
$29$	−0.713792	−0.132548	−0.0662739	−	0.997801i	$-0.521111\pi$
$29$	−0.713792	−0.132548	−0.0662739	+	0.997801i	$0.521111\pi$
$30$	4.40581	0.804388
$31$	−0.911854	−0.163774	−0.0818869	−	0.996642i	$-0.526095\pi$
$31$	−0.911854	−0.163774	−0.0818869	+	0.996642i	$0.526095\pi$
$32$	1.00000	0.176777
$33$	4.49396	0.782298
$34$	7.74094	1.32756
$35$	10.9879	1.85730
$36$	0.246980	0.0411633
$37$	6.81163	1.11982	0.559912	−	0.828552i	$-0.310835\pi$
$37$	6.81163	1.11982	0.559912	+	0.828552i	$0.310835\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	−2.44504	−0.386595
$41$	−3.28621	−0.513220	−0.256610	−	0.966515i	$-0.582606\pi$
$41$	−3.28621	−0.513220	−0.256610	+	0.966515i	$0.582606\pi$
$42$	8.09783	1.24952
$43$	7.20775	1.09917	0.549586	−	0.835437i	$-0.314786\pi$
$43$	7.20775	1.09917	0.549586	+	0.835437i	$0.314786\pi$
$44$	−2.49396	−0.375978
$45$	−0.603875	−0.0900204
$46$	4.27413	0.630186
$47$	5.20775	0.759629	0.379814	−	0.925063i	$-0.375988\pi$
$47$	5.20775	0.759629	0.379814	+	0.925063i	$0.375988\pi$
$48$	−1.80194	−0.260087
$49$	13.1957	1.88510
$50$	0.978230	0.138343
$51$	−13.9487	−1.95321
$52$	0	0
$53$	−8.89008	−1.22115	−0.610573	−	0.791960i	$-0.709061\pi$
$53$	−8.89008	−1.22115	−0.610573	+	0.791960i	$0.709061\pi$
$54$	4.96077	0.675075
$55$	6.09783	0.822232
$56$	−4.49396	−0.600531
$57$	−1.80194	−0.238672
$58$	−0.713792	−0.0937254
$59$	−11.2567	−1.46549	−0.732747	−	0.680501i	$-0.761762\pi$
$59$	−11.2567	−1.46549	−0.732747	+	0.680501i	$0.761762\pi$
$60$	4.40581	0.568788
$61$	3.43296	0.439546	0.219773	−	0.975551i	$-0.429468\pi$
$61$	3.43296	0.439546	0.219773	+	0.975551i	$0.429468\pi$
$62$	−0.911854	−0.115806
$63$	−1.10992	−0.139836
$64$	1.00000	0.125000
$65$	0	0
$66$	4.49396	0.553168
$67$	−1.40581	−0.171747	−0.0858737	−	0.996306i	$-0.527368\pi$
$67$	−1.40581	−0.171747	−0.0858737	+	0.996306i	$0.527368\pi$
$68$	7.74094	0.938727
$69$	−7.70171	−0.927177
$70$	10.9879	1.31331
$71$	3.86294	0.458446	0.229223	−	0.973374i	$-0.426381\pi$
$71$	3.86294	0.458446	0.229223	+	0.973374i	$0.426381\pi$
$72$	0.246980	0.0291068
$73$	−7.74094	−0.906008	−0.453004	−	0.891508i	$-0.649648\pi$
$73$	−7.74094	−0.906008	−0.453004	+	0.891508i	$0.649648\pi$
$74$	6.81163	0.791835
$75$	−1.76271	−0.203540
$76$	1.00000	0.114708
$77$	11.2078	1.27724
$78$	0	0
$79$	−1.91723	−0.215705	−0.107853	−	0.994167i	$-0.534397\pi$
$79$	−1.91723	−0.215705	−0.107853	+	0.994167i	$0.534397\pi$
$80$	−2.44504	−0.273364
$81$	−9.67994	−1.07555
$82$	−3.28621	−0.362901
$83$	−0.396125	−0.0434803	−0.0217402	−	0.999764i	$-0.506921\pi$
$83$	−0.396125	−0.0434803	−0.0217402	+	0.999764i	$0.506921\pi$
$84$	8.09783	0.883546
$85$	−18.9269	−2.05291
$86$	7.20775	0.777232
$87$	1.28621	0.137896
$88$	−2.49396	−0.265857
$89$	−16.7681	−1.77741	−0.888707	−	0.458476i	$-0.848395\pi$
$89$	−16.7681	−1.77741	−0.888707	+	0.458476i	$0.848395\pi$
$90$	−0.603875	−0.0636541
$91$	0	0
$92$	4.27413	0.445609
$93$	1.64310	0.170382
$94$	5.20775	0.537138
$95$	−2.44504	−0.250856
$96$	−1.80194	−0.183910
$97$	3.16421	0.321277	0.160638	−	0.987013i	$-0.448645\pi$
$97$	3.16421	0.321277	0.160638	+	0.987013i	$0.448645\pi$
$98$	13.1957	1.33296
$99$	−0.615957	−0.0619060
$100$	0.978230	0.0978230
$101$	15.5308	1.54537	0.772686	−	0.634789i	$-0.218913\pi$
$101$	15.5308	1.54537	0.772686	+	0.634789i	$0.218913\pi$
$102$	−13.9487	−1.38113
$103$	−3.03684	−0.299228	−0.149614	−	0.988744i	$-0.547803\pi$
$103$	−3.03684	−0.299228	−0.149614	+	0.988744i	$0.547803\pi$
$104$	0	0
$105$	−19.7995	−1.93224
$106$	−8.89008	−0.863481
$107$	−2.53319	−0.244893	−0.122446	−	0.992475i	$-0.539074\pi$
$107$	−2.53319	−0.244893	−0.122446	+	0.992475i	$0.539074\pi$
$108$	4.96077	0.477350
$109$	−10.2198	−0.978882	−0.489441	−	0.872036i	$-0.662799\pi$
$109$	−10.2198	−0.978882	−0.489441	+	0.872036i	$0.662799\pi$
$110$	6.09783	0.581406
$111$	−12.2741	−1.16501
$112$	−4.49396	−0.424639
$113$	7.60388	0.715312	0.357656	−	0.933853i	$-0.383576\pi$
$113$	7.60388	0.715312	0.357656	+	0.933853i	$0.383576\pi$
$114$	−1.80194	−0.168767
$115$	−10.4504	−0.974507
$116$	−0.713792	−0.0662739
$117$	0	0
$118$	−11.2567	−1.03626
$119$	−34.7875	−3.18896
$120$	4.40581	0.402194
$121$	−4.78017	−0.434561
$122$	3.43296	0.310806
$123$	5.92154	0.533928
$124$	−0.911854	−0.0818869
$125$	9.83340	0.879526
$126$	−1.10992	−0.0988792
$127$	−18.6920	−1.65865	−0.829324	−	0.558768i	$-0.811274\pi$
$127$	−18.6920	−1.65865	−0.829324	+	0.558768i	$0.811274\pi$
$128$	1.00000	0.0883883
$129$	−12.9879	−1.14352
$130$	0	0
$131$	−8.49396	−0.742121	−0.371060	−	0.928609i	$-0.621006\pi$
$131$	−8.49396	−0.742121	−0.371060	+	0.928609i	$0.621006\pi$
$132$	4.49396	0.391149
$133$	−4.49396	−0.389676
$134$	−1.40581	−0.121444
$135$	−12.1293	−1.04392
$136$	7.74094	0.663780
$137$	19.7506	1.68741	0.843705	−	0.536807i	$-0.180370\pi$
$137$	19.7506	1.68741	0.843705	+	0.536807i	$0.180370\pi$
$138$	−7.70171	−0.655613
$139$	8.15213	0.691455	0.345727	−	0.938335i	$-0.387632\pi$
$139$	8.15213	0.691455	0.345727	+	0.938335i	$0.387632\pi$
$140$	10.9879	0.928649
$141$	−9.38404	−0.790279
$142$	3.86294	0.324170
$143$	0	0
$144$	0.246980	0.0205816
$145$	1.74525	0.144935
$146$	−7.74094	−0.640645
$147$	−23.7778	−1.96116
$148$	6.81163	0.559912
$149$	8.11290	0.664635	0.332317	−	0.943168i	$-0.392170\pi$
$149$	8.11290	0.664635	0.332317	+	0.943168i	$0.392170\pi$
$150$	−1.76271	−0.143925
$151$	5.56465	0.452845	0.226422	−	0.974029i	$-0.427297\pi$
$151$	5.56465	0.452845	0.226422	+	0.974029i	$0.427297\pi$
$152$	1.00000	0.0811107
$153$	1.91185	0.154564
$154$	11.2078	0.903146
$155$	2.22952	0.179079
$156$	0	0
$157$	−6.77240	−0.540496	−0.270248	−	0.962791i	$-0.587106\pi$
$157$	−6.77240	−0.540496	−0.270248	+	0.962791i	$0.587106\pi$
$158$	−1.91723	−0.152527
$159$	16.0194	1.27042
$160$	−2.44504	−0.193298
$161$	−19.2078	−1.51378
$162$	−9.67994	−0.760528
$163$	−8.27413	−0.648080	−0.324040	−	0.946043i	$-0.605041\pi$
$163$	−8.27413	−0.648080	−0.324040	+	0.946043i	$0.605041\pi$
$164$	−3.28621	−0.256610
$165$	−10.9879	−0.855408
$166$	−0.396125	−0.0307452
$167$	−6.25236	−0.483822	−0.241911	−	0.970298i	$-0.577774\pi$
$167$	−6.25236	−0.483822	−0.241911	+	0.970298i	$0.577774\pi$
$168$	8.09783	0.624762
$169$	0	0
$170$	−18.9269	−1.45163
$171$	0.246980	0.0188870
$172$	7.20775	0.549586
$173$	24.5133	1.86371	0.931857	−	0.362825i	$-0.118188\pi$
$173$	24.5133	1.86371	0.931857	+	0.362825i	$0.118188\pi$
$174$	1.28621	0.0975072
$175$	−4.39612	−0.332316
$176$	−2.49396	−0.187989
$177$	20.2838	1.52462
$178$	−16.7681	−1.25682
$179$	−25.9366	−1.93859	−0.969297	−	0.245895i	$-0.920918\pi$
$179$	−25.9366	−1.93859	−0.969297	+	0.245895i	$0.920918\pi$
$180$	−0.603875	−0.0450102
$181$	5.23191	0.388885	0.194443	−	0.980914i	$-0.437710\pi$
$181$	5.23191	0.388885	0.194443	+	0.980914i	$0.437710\pi$
$182$	0	0
$183$	−6.18598	−0.457281
$184$	4.27413	0.315093
$185$	−16.6547	−1.22448
$186$	1.64310	0.120478
$187$	−19.3056	−1.41176
$188$	5.20775	0.379814
$189$	−22.2935	−1.62161
$190$	−2.44504	−0.177382
$191$	−17.4819	−1.26494	−0.632472	−	0.774583i	$-0.717960\pi$
$191$	−17.4819	−1.26494	−0.632472	+	0.774583i	$0.717960\pi$
$192$	−1.80194	−0.130044
$193$	23.7754	1.71139	0.855695	−	0.517481i	$-0.173130\pi$
$193$	23.7754	1.71139	0.855695	+	0.517481i	$0.173130\pi$
$194$	3.16421	0.227177
$195$	0	0
$196$	13.1957	0.942548
$197$	22.6069	1.61067	0.805336	−	0.592819i	$-0.201985\pi$
$197$	22.6069	1.61067	0.805336	+	0.592819i	$0.201985\pi$
$198$	−0.615957	−0.0437742
$199$	−16.7332	−1.18618	−0.593091	−	0.805135i	$-0.702093\pi$
$199$	−16.7332	−1.18618	−0.593091	+	0.805135i	$0.702093\pi$
$200$	0.978230	0.0691713
$201$	2.53319	0.178677
$202$	15.5308	1.09274
$203$	3.20775	0.225140
$204$	−13.9487	−0.976604
$205$	8.03492	0.561183
$206$	−3.03684	−0.211586
$207$	1.05562	0.0733708
$208$	0	0
$209$	−2.49396	−0.172511
$210$	−19.7995	−1.36630
$211$	15.3817	1.05892	0.529458	−	0.848336i	$-0.322395\pi$
$211$	15.3817	1.05892	0.529458	+	0.848336i	$0.322395\pi$
$212$	−8.89008	−0.610573
$213$	−6.96077	−0.476944
$214$	−2.53319	−0.173165
$215$	−17.6233	−1.20190
$216$	4.96077	0.337538
$217$	4.09783	0.278179
$218$	−10.2198	−0.692174
$219$	13.9487	0.942565
$220$	6.09783	0.411116
$221$	0	0
$222$	−12.2741	−0.823785
$223$	−28.4198	−1.90313	−0.951566	−	0.307445i	$-0.900526\pi$
$223$	−28.4198	−1.90313	−0.951566	+	0.307445i	$0.900526\pi$
$224$	−4.49396	−0.300265
$225$	0.241603	0.0161069
$226$	7.60388	0.505802
$227$	−10.1981	−0.676869	−0.338435	−	0.940990i	$-0.609897\pi$
$227$	−10.1981	−0.676869	−0.338435	+	0.940990i	$0.609897\pi$
$228$	−1.80194	−0.119336
$229$	−14.5429	−0.961020	−0.480510	−	0.876989i	$-0.659548\pi$
$229$	−14.5429	−0.961020	−0.480510	+	0.876989i	$0.659548\pi$
$230$	−10.4504	−0.689080
$231$	−20.1957	−1.32878
$232$	−0.713792	−0.0468627
$233$	−26.0683	−1.70779	−0.853895	−	0.520445i	$-0.825766\pi$
$233$	−26.0683	−1.70779	−0.853895	+	0.520445i	$0.825766\pi$
$234$	0	0
$235$	−12.7332	−0.830620
$236$	−11.2567	−0.732747
$237$	3.45473	0.224409
$238$	−34.7875	−2.25494
$239$	16.2198	1.04917	0.524587	−	0.851357i	$-0.324220\pi$
$239$	16.2198	1.04917	0.524587	+	0.851357i	$0.324220\pi$
$240$	4.40581	0.284394
$241$	27.3250	1.76016	0.880078	−	0.474830i	$-0.157490\pi$
$241$	27.3250	1.76016	0.880078	+	0.474830i	$0.157490\pi$
$242$	−4.78017	−0.307281
$243$	2.56033	0.164246
$244$	3.43296	0.219773
$245$	−32.2640	−2.06127
$246$	5.92154	0.377544
$247$	0	0
$248$	−0.911854	−0.0579028
$249$	0.713792	0.0452347
$250$	9.83340	0.621919
$251$	14.7922	0.933678	0.466839	−	0.884342i	$-0.345393\pi$
$251$	14.7922	0.933678	0.466839	+	0.884342i	$0.345393\pi$
$252$	−1.10992	−0.0699182
$253$	−10.6595	−0.670157
$254$	−18.6920	−1.17284
$255$	34.1051	2.13575
$256$	1.00000	0.0625000
$257$	−26.5918	−1.65875	−0.829375	−	0.558692i	$-0.811303\pi$
$257$	−26.5918	−1.65875	−0.829375	+	0.558692i	$0.811303\pi$
$258$	−12.9879	−0.808592
$259$	−30.6112	−1.90209
$260$	0	0
$261$	−0.176292	−0.0109122
$262$	−8.49396	−0.524759
$263$	−24.7332	−1.52511	−0.762556	−	0.646922i	$-0.776056\pi$
$263$	−24.7332	−1.52511	−0.762556	+	0.646922i	$0.776056\pi$
$264$	4.49396	0.276584
$265$	21.7366	1.33527
$266$	−4.49396	−0.275542
$267$	30.2150	1.84913
$268$	−1.40581	−0.0858737
$269$	16.2935	0.993432	0.496716	−	0.867913i	$-0.334539\pi$
$269$	16.2935	0.993432	0.496716	+	0.867913i	$0.334539\pi$
$270$	−12.1293	−0.738165
$271$	25.7560	1.56457	0.782283	−	0.622923i	$-0.214055\pi$
$271$	25.7560	1.56457	0.782283	+	0.622923i	$0.214055\pi$
$272$	7.74094	0.469363
$273$	0	0
$274$	19.7506	1.19318
$275$	−2.43967	−0.147117
$276$	−7.70171	−0.463588
$277$	2.19029	0.131602	0.0658010	−	0.997833i	$-0.479040\pi$
$277$	2.19029	0.131602	0.0658010	+	0.997833i	$0.479040\pi$
$278$	8.15213	0.488932
$279$	−0.225209	−0.0134829
$280$	10.9879	0.656654
$281$	−24.6353	−1.46962	−0.734810	−	0.678273i	$-0.762729\pi$
$281$	−24.6353	−1.46962	−0.734810	+	0.678273i	$0.762729\pi$
$282$	−9.38404	−0.558812
$283$	−4.48321	−0.266499	−0.133249	−	0.991083i	$-0.542541\pi$
$283$	−4.48321	−0.266499	−0.133249	+	0.991083i	$0.542541\pi$
$284$	3.86294	0.229223
$285$	4.40581	0.260978
$286$	0	0
$287$	14.7681	0.871733
$288$	0.246980	0.0145534
$289$	42.9221	2.52483
$290$	1.74525	0.102485
$291$	−5.70171	−0.334240
$292$	−7.74094	−0.453004
$293$	−3.32975	−0.194526	−0.0972630	−	0.995259i	$-0.531009\pi$
$293$	−3.32975	−0.194526	−0.0972630	+	0.995259i	$0.531009\pi$
$294$	−23.7778	−1.38675
$295$	27.5230	1.60245
$296$	6.81163	0.395918
$297$	−12.3720	−0.717894
$298$	8.11290	0.469968
$299$	0	0
$300$	−1.76271	−0.101770
$301$	−32.3913	−1.86701
$302$	5.56465	0.320209
$303$	−27.9855	−1.60773
$304$	1.00000	0.0573539
$305$	−8.39373	−0.480624
$306$	1.91185	0.109293
$307$	−2.74632	−0.156741	−0.0783703	−	0.996924i	$-0.524972\pi$
$307$	−2.74632	−0.156741	−0.0783703	+	0.996924i	$0.524972\pi$
$308$	11.2078	0.638621
$309$	5.47219	0.311302
$310$	2.22952	0.126628
$311$	−13.6775	−0.775583	−0.387791	−	0.921747i	$-0.626762\pi$
$311$	−13.6775	−0.775583	−0.387791	+	0.921747i	$0.626762\pi$
$312$	0	0
$313$	16.1903	0.915129	0.457565	−	0.889176i	$-0.348722\pi$
$313$	16.1903	0.915129	0.457565	+	0.889176i	$0.348722\pi$
$314$	−6.77240	−0.382189
$315$	2.71379	0.152905
$316$	−1.91723	−0.107853
$317$	−19.8538	−1.11510	−0.557551	−	0.830142i	$-0.688259\pi$
$317$	−19.8538	−1.11510	−0.557551	+	0.830142i	$0.688259\pi$
$318$	16.0194	0.898322
$319$	1.78017	0.0996702
$320$	−2.44504	−0.136682
$321$	4.56465	0.254774
$322$	−19.2078	−1.07041
$323$	7.74094	0.430717
$324$	−9.67994	−0.537774
$325$	0	0
$326$	−8.27413	−0.458261
$327$	18.4155	1.01838
$328$	−3.28621	−0.181450
$329$	−23.4034	−1.29027
$330$	−10.9879	−0.604865
$331$	19.2295	1.05695	0.528475	−	0.848949i	$-0.322764\pi$
$331$	19.2295	1.05695	0.528475	+	0.848949i	$0.322764\pi$
$332$	−0.396125	−0.0217402
$333$	1.68233	0.0921913
$334$	−6.25236	−0.342114
$335$	3.43727	0.187798
$336$	8.09783	0.441773
$337$	−10.8465	−0.590849	−0.295424	−	0.955366i	$-0.595461\pi$
$337$	−10.8465	−0.590849	−0.295424	+	0.955366i	$0.595461\pi$
$338$	0	0
$339$	−13.7017	−0.744175
$340$	−18.9269	−1.02646
$341$	2.27413	0.123151
$342$	0.246980	0.0133551
$343$	−27.8431	−1.50339
$344$	7.20775	0.388616
$345$	18.8310	1.01383
$346$	24.5133	1.31785
$347$	13.8974	0.746050	0.373025	−	0.927821i	$-0.378320\pi$
$347$	13.8974	0.746050	0.373025	+	0.927821i	$0.378320\pi$
$348$	1.28621	0.0689480
$349$	1.89977	0.101692	0.0508462	−	0.998706i	$-0.483808\pi$
$349$	1.89977	0.101692	0.0508462	+	0.998706i	$0.483808\pi$
$350$	−4.39612	−0.234983
$351$	0	0
$352$	−2.49396	−0.132928
$353$	17.1618	0.913431	0.456716	−	0.889613i	$-0.349026\pi$
$353$	17.1618	0.913431	0.456716	+	0.889613i	$0.349026\pi$
$354$	20.2838	1.07807
$355$	−9.44504	−0.501291
$356$	−16.7681	−0.888707
$357$	62.6848	3.31763
$358$	−25.9366	−1.37079
$359$	18.5181	0.977349	0.488675	−	0.872466i	$-0.337481\pi$
$359$	18.5181	0.977349	0.488675	+	0.872466i	$0.337481\pi$
$360$	−0.603875	−0.0318270
$361$	1.00000	0.0526316
$362$	5.23191	0.274983
$363$	8.61356	0.452095
$364$	0	0
$365$	18.9269	0.990680
$366$	−6.18598	−0.323346
$367$	8.39612	0.438274	0.219137	−	0.975694i	$-0.429676\pi$
$367$	8.39612	0.438274	0.219137	+	0.975694i	$0.429676\pi$
$368$	4.27413	0.222804
$369$	−0.811626	−0.0422516
$370$	−16.6547	−0.865837
$371$	39.9517	2.07419
$372$	1.64310	0.0851910
$373$	29.5991	1.53258	0.766291	−	0.642493i	$-0.222100\pi$
$373$	29.5991	1.53258	0.766291	+	0.642493i	$0.222100\pi$
$374$	−19.3056	−0.998268
$375$	−17.7192	−0.915014
$376$	5.20775	0.268569
$377$	0	0
$378$	−22.2935	−1.14665
$379$	18.9638	0.974103	0.487051	−	0.873373i	$-0.338072\pi$
$379$	18.9638	0.974103	0.487051	+	0.873373i	$0.338072\pi$
$380$	−2.44504	−0.125428
$381$	33.6819	1.72557
$382$	−17.4819	−0.894451
$383$	−28.4644	−1.45446	−0.727232	−	0.686392i	$-0.759194\pi$
$383$	−28.4644	−1.45446	−0.727232	+	0.686392i	$0.759194\pi$
$384$	−1.80194	−0.0919548
$385$	−27.4034	−1.39661
$386$	23.7754	1.21014
$387$	1.78017	0.0904910
$388$	3.16421	0.160638
$389$	6.90084	0.349886	0.174943	−	0.984579i	$-0.444026\pi$
$389$	6.90084	0.349886	0.174943	+	0.984579i	$0.444026\pi$
$390$	0	0
$391$	33.0858	1.67322
$392$	13.1957	0.666482
$393$	15.3056	0.772065
$394$	22.6069	1.13892
$395$	4.68771	0.235864
$396$	−0.615957	−0.0309530
$397$	27.8726	1.39889	0.699443	−	0.714688i	$-0.253431\pi$
$397$	27.8726	1.39889	0.699443	+	0.714688i	$0.253431\pi$
$398$	−16.7332	−0.838758
$399$	8.09783	0.405399
$400$	0.978230	0.0489115
$401$	−27.5120	−1.37388	−0.686942	−	0.726712i	$-0.741047\pi$
$401$	−27.5120	−1.37388	−0.686942	+	0.726712i	$0.741047\pi$
$402$	2.53319	0.126344
$403$	0	0
$404$	15.5308	0.772686
$405$	23.6679	1.17607
$406$	3.20775	0.159198
$407$	−16.9879	−0.842060
$408$	−13.9487	−0.690563
$409$	−10.8358	−0.535795	−0.267898	−	0.963447i	$-0.586329\pi$
$409$	−10.8358	−0.535795	−0.267898	+	0.963447i	$0.586329\pi$
$410$	8.03492	0.396816
$411$	−35.5894	−1.75550
$412$	−3.03684	−0.149614
$413$	50.5870	2.48922
$414$	1.05562	0.0518810
$415$	0.968541	0.0475438
$416$	0	0
$417$	−14.6896	−0.719354
$418$	−2.49396	−0.121984
$419$	−28.4892	−1.39179	−0.695894	−	0.718145i	$-0.744991\pi$
$419$	−28.4892	−1.39179	−0.695894	+	0.718145i	$0.744991\pi$
$420$	−19.7995	−0.966119
$421$	−28.1957	−1.37417	−0.687086	−	0.726576i	$-0.741111\pi$
$421$	−28.1957	−1.37417	−0.687086	+	0.726576i	$0.741111\pi$
$422$	15.3817	0.748767
$423$	1.28621	0.0625376
$424$	−8.89008	−0.431741
$425$	7.57242	0.367316
$426$	−6.96077	−0.337250
$427$	−15.4276	−0.746593
$428$	−2.53319	−0.122446
$429$	0	0
$430$	−17.6233	−0.849869
$431$	20.4722	0.986111	0.493055	−	0.869998i	$-0.335880\pi$
$431$	20.4722	0.986111	0.493055	+	0.869998i	$0.335880\pi$
$432$	4.96077	0.238675
$433$	9.69096	0.465718	0.232859	−	0.972511i	$-0.425192\pi$
$433$	9.69096	0.465718	0.232859	+	0.972511i	$0.425192\pi$
$434$	4.09783	0.196702
$435$	−3.14483	−0.150783
$436$	−10.2198	−0.489441
$437$	4.27413	0.204459
$438$	13.9487	0.666494
$439$	−7.67994	−0.366544	−0.183272	−	0.983062i	$-0.558669\pi$
$439$	−7.67994	−0.366544	−0.183272	+	0.983062i	$0.558669\pi$
$440$	6.09783	0.290703
$441$	3.25906	0.155193
$442$	0	0
$443$	−11.5168	−0.547179	−0.273590	−	0.961846i	$-0.588211\pi$
$443$	−11.5168	−0.547179	−0.273590	+	0.961846i	$0.588211\pi$
$444$	−12.2741	−0.582504
$445$	40.9987	1.94352
$446$	−28.4198	−1.34572
$447$	−14.6189	−0.691452
$448$	−4.49396	−0.212320
$449$	1.16421	0.0549425	0.0274712	−	0.999623i	$-0.491255\pi$
$449$	1.16421	0.0549425	0.0274712	+	0.999623i	$0.491255\pi$
$450$	0.241603	0.0113893
$451$	8.19567	0.385919
$452$	7.60388	0.357656
$453$	−10.0271	−0.471116
$454$	−10.1981	−0.478619
$455$	0	0
$456$	−1.80194	−0.0843835
$457$	−30.3230	−1.41845	−0.709226	−	0.704981i	$-0.750955\pi$
$457$	−30.3230	−1.41845	−0.709226	+	0.704981i	$0.750955\pi$
$458$	−14.5429	−0.679544
$459$	38.4010	1.79241
$460$	−10.4504	−0.487253
$461$	0.992230	0.0462128	0.0231064	−	0.999733i	$-0.492644\pi$
$461$	0.992230	0.0462128	0.0231064	+	0.999733i	$0.492644\pi$
$462$	−20.1957	−0.939588
$463$	−28.8767	−1.34201	−0.671006	−	0.741452i	$-0.734138\pi$
$463$	−28.8767	−1.34201	−0.671006	+	0.741452i	$0.734138\pi$
$464$	−0.713792	−0.0331369
$465$	−4.01746	−0.186305
$466$	−26.0683	−1.20759
$467$	−19.8189	−0.917110	−0.458555	−	0.888666i	$-0.651633\pi$
$467$	−19.8189	−0.917110	−0.458555	+	0.888666i	$0.651633\pi$
$468$	0	0
$469$	6.31767	0.291723
$470$	−12.7332	−0.587337
$471$	12.2034	0.562305
$472$	−11.2567	−0.518130
$473$	−17.9758	−0.826530
$474$	3.45473	0.158681
$475$	0.978230	0.0448843
$476$	−34.7875	−1.59448
$477$	−2.19567	−0.100533
$478$	16.2198	0.741878
$479$	21.9758	1.00410	0.502051	−	0.864838i	$-0.332579\pi$
$479$	21.9758	1.00410	0.502051	+	0.864838i	$0.332579\pi$
$480$	4.40581	0.201097
$481$	0	0
$482$	27.3250	1.24462
$483$	34.6112	1.57486
$484$	−4.78017	−0.217280
$485$	−7.73663	−0.351302
$486$	2.56033	0.116139
$487$	−36.1148	−1.63652	−0.818259	−	0.574849i	$-0.805061\pi$
$487$	−36.1148	−1.63652	−0.818259	+	0.574849i	$0.805061\pi$
$488$	3.43296	0.155403
$489$	14.9095	0.674229
$490$	−32.2640	−1.45754
$491$	−21.2814	−0.960417	−0.480209	−	0.877154i	$-0.659439\pi$
$491$	−21.2814	−0.960417	−0.480209	+	0.877154i	$0.659439\pi$
$492$	5.92154	0.266964
$493$	−5.52542	−0.248852
$494$	0	0
$495$	1.50604	0.0676915
$496$	−0.911854	−0.0409435
$497$	−17.3599	−0.778697
$498$	0.713792	0.0319858
$499$	−5.00730	−0.224157	−0.112079	−	0.993699i	$-0.535751\pi$
$499$	−5.00730	−0.224157	−0.112079	+	0.993699i	$0.535751\pi$
$500$	9.83340	0.439763
$501$	11.2664	0.503344
$502$	14.7922	0.660210
$503$	44.0930	1.96601	0.983006	−	0.183574i	$-0.0587666\pi$
$503$	44.0930	1.96601	0.983006	+	0.183574i	$0.0587666\pi$
$504$	−1.10992	−0.0494396
$505$	−37.9734	−1.68980
$506$	−10.6595	−0.473872
$507$	0	0
$508$	−18.6920	−0.829324
$509$	24.8901	1.10323	0.551617	−	0.834098i	$-0.314011\pi$
$509$	24.8901	1.10323	0.551617	+	0.834098i	$0.314011\pi$
$510$	34.1051	1.51020
$511$	34.7875	1.53891
$512$	1.00000	0.0441942
$513$	4.96077	0.219023
$514$	−26.5918	−1.17291
$515$	7.42519	0.327193
$516$	−12.9879	−0.571761
$517$	−12.9879	−0.571208
$518$	−30.6112	−1.34498
$519$	−44.1715	−1.93891
$520$	0	0
$521$	−13.8974	−0.608855	−0.304428	−	0.952535i	$-0.598465\pi$
$521$	−13.8974	−0.608855	−0.304428	+	0.952535i	$0.598465\pi$
$522$	−0.176292	−0.00771609
$523$	1.78017	0.0778413	0.0389206	−	0.999242i	$-0.487608\pi$
$523$	1.78017	0.0778413	0.0389206	+	0.999242i	$0.487608\pi$
$524$	−8.49396	−0.371060
$525$	7.92154	0.345724
$526$	−24.7332	−1.07842
$527$	−7.05861	−0.307478
$528$	4.49396	0.195574
$529$	−4.73184	−0.205732
$530$	21.7366	0.944179
$531$	−2.78017	−0.120649
$532$	−4.49396	−0.194838
$533$	0	0
$534$	30.2150	1.30753
$535$	6.19375	0.267779
$536$	−1.40581	−0.0607219
$537$	46.7362	2.01681
$538$	16.2935	0.702463
$539$	−32.9095	−1.41751
$540$	−12.1293	−0.521962
$541$	−38.7308	−1.66517	−0.832583	−	0.553900i	$-0.813139\pi$
$541$	−38.7308	−1.66517	−0.832583	+	0.553900i	$0.813139\pi$
$542$	25.7560	1.10632
$543$	−9.42758	−0.404576
$544$	7.74094	0.331890
$545$	24.9879	1.07036
$546$	0	0
$547$	−6.25906	−0.267618	−0.133809	−	0.991007i	$-0.542721\pi$
$547$	−6.25906	−0.267618	−0.133809	+	0.991007i	$0.542721\pi$
$548$	19.7506	0.843705
$549$	0.847871	0.0361863
$550$	−2.43967	−0.104028
$551$	−0.713792	−0.0304086
$552$	−7.70171	−0.327807
$553$	8.61596	0.366388
$554$	2.19029	0.0930566
$555$	30.0108	1.27389
$556$	8.15213	0.345727
$557$	−25.3980	−1.07615	−0.538075	−	0.842897i	$-0.680848\pi$
$557$	−25.3980	−1.07615	−0.538075	+	0.842897i	$0.680848\pi$
$558$	−0.225209	−0.00953387
$559$	0	0
$560$	10.9879	0.464324
$561$	34.7875	1.46873
$562$	−24.6353	−1.03918
$563$	8.61835	0.363220	0.181610	−	0.983371i	$-0.441869\pi$
$563$	8.61835	0.363220	0.181610	+	0.983371i	$0.441869\pi$
$564$	−9.38404	−0.395139
$565$	−18.5918	−0.782163
$566$	−4.48321	−0.188443
$567$	43.5013	1.82688
$568$	3.86294	0.162085
$569$	−3.10992	−0.130374	−0.0651872	−	0.997873i	$-0.520764\pi$
$569$	−3.10992	−0.130374	−0.0651872	+	0.997873i	$0.520764\pi$
$570$	4.40581	0.184539
$571$	−44.6741	−1.86955	−0.934776	−	0.355237i	$-0.884400\pi$
$571$	−44.6741	−1.86955	−0.934776	+	0.355237i	$0.884400\pi$
$572$	0	0
$573$	31.5013	1.31598
$574$	14.7681	0.616408
$575$	4.18108	0.174363
$576$	0.246980	0.0102908
$577$	−35.3817	−1.47296	−0.736479	−	0.676461i	$-0.763513\pi$
$577$	−35.3817	−1.47296	−0.736479	+	0.676461i	$0.763513\pi$
$578$	42.9221	1.78533
$579$	−42.8418	−1.78044
$580$	1.74525	0.0724676
$581$	1.78017	0.0738538
$582$	−5.70171	−0.236343
$583$	22.1715	0.918250
$584$	−7.74094	−0.320322
$585$	0	0
$586$	−3.32975	−0.137551
$587$	−19.4711	−0.803659	−0.401830	−	0.915714i	$-0.631626\pi$
$587$	−19.4711	−0.803659	−0.401830	+	0.915714i	$0.631626\pi$
$588$	−23.7778	−0.980579
$589$	−0.911854	−0.0375723
$590$	27.5230	1.13311
$591$	−40.7362	−1.67566
$592$	6.81163	0.279956
$593$	5.31634	0.218316	0.109158	−	0.994024i	$-0.465185\pi$
$593$	5.31634	0.218316	0.109158	+	0.994024i	$0.465185\pi$
$594$	−12.3720	−0.507628
$595$	85.0568	3.48699
$596$	8.11290	0.332317
$597$	30.1521	1.23404
$598$	0	0
$599$	−13.2644	−0.541970	−0.270985	−	0.962584i	$-0.587349\pi$
$599$	−13.2644	−0.541970	−0.270985	+	0.962584i	$0.587349\pi$
$600$	−1.76271	−0.0719623
$601$	32.0388	1.30689	0.653444	−	0.756975i	$-0.273323\pi$
$601$	32.0388	1.30689	0.653444	+	0.756975i	$0.273323\pi$
$602$	−32.3913	−1.32017
$603$	−0.347207	−0.0141394
$604$	5.56465	0.226422
$605$	11.6877	0.475173
$606$	−27.9855	−1.13683
$607$	−24.0355	−0.975571	−0.487786	−	0.872963i	$-0.662195\pi$
$607$	−24.0355	−0.975571	−0.487786	+	0.872963i	$0.662195\pi$
$608$	1.00000	0.0405554
$609$	−5.78017	−0.234224
$610$	−8.39373	−0.339852
$611$	0	0
$612$	1.91185	0.0772821
$613$	−41.6122	−1.68070	−0.840351	−	0.542042i	$-0.817651\pi$
$613$	−41.6122	−1.68070	−0.840351	+	0.542042i	$0.817651\pi$
$614$	−2.74632	−0.110832
$615$	−14.4784	−0.583826
$616$	11.2078	0.451573
$617$	−2.20477	−0.0887606	−0.0443803	−	0.999015i	$-0.514131\pi$
$617$	−2.20477	−0.0887606	−0.0443803	+	0.999015i	$0.514131\pi$
$618$	5.47219	0.220124
$619$	19.1207	0.768525	0.384262	−	0.923224i	$-0.374456\pi$
$619$	19.1207	0.768525	0.384262	+	0.923224i	$0.374456\pi$
$620$	2.22952	0.0895397
$621$	21.2030	0.850846
$622$	−13.6775	−0.548420
$623$	75.3551	3.01904
$624$	0	0
$625$	−28.9342	−1.15737
$626$	16.1903	0.647094
$627$	4.49396	0.179471
$628$	−6.77240	−0.270248
$629$	52.7284	2.10242
$630$	2.71379	0.108120
$631$	−40.7332	−1.62156	−0.810781	−	0.585350i	$-0.800957\pi$
$631$	−40.7332	−1.62156	−0.810781	+	0.585350i	$0.800957\pi$
$632$	−1.91723	−0.0762633
$633$	−27.7168	−1.10164
$634$	−19.8538	−0.788497
$635$	45.7028	1.81366
$636$	16.0194	0.635210
$637$	0	0
$638$	1.78017	0.0704775
$639$	0.954067	0.0377423
$640$	−2.44504	−0.0966488
$641$	11.3491	0.448264	0.224132	−	0.974559i	$-0.428045\pi$
$641$	11.3491	0.448264	0.224132	+	0.974559i	$0.428045\pi$
$642$	4.56465	0.180152
$643$	−35.9409	−1.41737	−0.708686	−	0.705524i	$-0.750712\pi$
$643$	−35.9409	−1.41737	−0.708686	+	0.705524i	$0.750712\pi$
$644$	−19.2078	−0.756891
$645$	31.7560	1.25039
$646$	7.74094	0.304563
$647$	−22.7788	−0.895529	−0.447764	−	0.894152i	$-0.647780\pi$
$647$	−22.7788	−0.895529	−0.447764	+	0.894152i	$0.647780\pi$
$648$	−9.67994	−0.380264
$649$	28.0737	1.10199
$650$	0	0
$651$	−7.38404	−0.289403
$652$	−8.27413	−0.324040
$653$	22.2543	0.870877	0.435439	−	0.900218i	$-0.356593\pi$
$653$	22.2543	0.870877	0.435439	+	0.900218i	$0.356593\pi$
$654$	18.4155	0.720103
$655$	20.7681	0.811476
$656$	−3.28621	−0.128305
$657$	−1.91185	−0.0745885
$658$	−23.4034	−0.912360
$659$	−46.3666	−1.80619	−0.903093	−	0.429445i	$-0.858709\pi$
$659$	−46.3666	−1.80619	−0.903093	+	0.429445i	$0.858709\pi$
$660$	−10.9879	−0.427704
$661$	−31.0315	−1.20698	−0.603492	−	0.797369i	$-0.706224\pi$
$661$	−31.0315	−1.20698	−0.603492	+	0.797369i	$0.706224\pi$
$662$	19.2295	0.747377
$663$	0	0
$664$	−0.396125	−0.0153726
$665$	10.9879	0.426093
$666$	1.68233	0.0651891
$667$	−3.05084	−0.118129
$668$	−6.25236	−0.241911
$669$	51.2107	1.97992
$670$	3.43727	0.132793
$671$	−8.56166	−0.330519
$672$	8.09783	0.312381
$673$	−22.6461	−0.872943	−0.436471	−	0.899718i	$-0.643772\pi$
$673$	−22.6461	−0.872943	−0.436471	+	0.899718i	$0.643772\pi$
$674$	−10.8465	−0.417793
$675$	4.85277	0.186783
$676$	0	0
$677$	24.7138	0.949828	0.474914	−	0.880032i	$-0.342479\pi$
$677$	24.7138	0.949828	0.474914	+	0.880032i	$0.342479\pi$
$678$	−13.7017	−0.526211
$679$	−14.2198	−0.545707
$680$	−18.9269	−0.725814
$681$	18.3763	0.704180
$682$	2.27413	0.0870808
$683$	−30.9536	−1.18441	−0.592203	−	0.805789i	$-0.701742\pi$
$683$	−30.9536	−1.18441	−0.592203	+	0.805789i	$0.701742\pi$
$684$	0.246980	0.00944350
$685$	−48.2911	−1.84511
$686$	−27.8431	−1.06305
$687$	26.2054	0.999797
$688$	7.20775	0.274793
$689$	0	0
$690$	18.8310	0.716884
$691$	5.94092	0.226003	0.113002	−	0.993595i	$-0.463953\pi$
$691$	5.94092	0.226003	0.113002	+	0.993595i	$0.463953\pi$
$692$	24.5133	0.931857
$693$	2.76809	0.105151
$694$	13.8974	0.527537
$695$	−19.9323	−0.756075
$696$	1.28621	0.0487536
$697$	−25.4383	−0.963546
$698$	1.89977	0.0719074
$699$	46.9734	1.77670
$700$	−4.39612	−0.166158
$701$	−7.48427	−0.282677	−0.141338	−	0.989961i	$-0.545141\pi$
$701$	−7.48427	−0.282677	−0.141338	+	0.989961i	$0.545141\pi$
$702$	0	0
$703$	6.81163	0.256905
$704$	−2.49396	−0.0939946
$705$	22.9444	0.864135
$706$	17.1618	0.645894
$707$	−69.7948	−2.62490
$708$	20.2838	0.762312
$709$	20.7966	0.781031	0.390516	−	0.920596i	$-0.372297\pi$
$709$	20.7966	0.781031	0.390516	+	0.920596i	$0.372297\pi$
$710$	−9.44504	−0.354466
$711$	−0.473517	−0.0177583
$712$	−16.7681	−0.628411
$713$	−3.89738	−0.145958
$714$	62.6848	2.34592
$715$	0	0
$716$	−25.9366	−0.969297
$717$	−29.2271	−1.09151
$718$	18.5181	0.691090
$719$	−36.3913	−1.35717	−0.678584	−	0.734523i	$-0.737406\pi$
$719$	−36.3913	−1.35717	−0.678584	+	0.734523i	$0.737406\pi$
$720$	−0.603875	−0.0225051
$721$	13.6474	0.508256
$722$	1.00000	0.0372161
$723$	−49.2379	−1.83118
$724$	5.23191	0.194443
$725$	−0.698252	−0.0259324
$726$	8.61356	0.319679
$727$	34.4698	1.27841	0.639207	−	0.769035i	$-0.279263\pi$
$727$	34.4698	1.27841	0.639207	+	0.769035i	$0.279263\pi$
$728$	0	0
$729$	24.4263	0.904676
$730$	18.9269	0.700517
$731$	55.7948	2.06364
$732$	−6.18598	−0.228640
$733$	−15.6732	−0.578904	−0.289452	−	0.957193i	$-0.593473\pi$
$733$	−15.6732	−0.578904	−0.289452	+	0.957193i	$0.593473\pi$
$734$	8.39612	0.309907
$735$	58.1377	2.14444
$736$	4.27413	0.157546
$737$	3.50604	0.129147
$738$	−0.811626	−0.0298764
$739$	−31.7861	−1.16927	−0.584636	−	0.811296i	$-0.698763\pi$
$739$	−31.7861	−1.16927	−0.584636	+	0.811296i	$0.698763\pi$
$740$	−16.6547	−0.612239
$741$	0	0
$742$	39.9517	1.46667
$743$	26.7308	0.980657	0.490329	−	0.871538i	$-0.336877\pi$
$743$	26.7308	0.980657	0.490329	+	0.871538i	$0.336877\pi$
$744$	1.64310	0.0602391
$745$	−19.8364	−0.726749
$746$	29.5991	1.08370
$747$	−0.0978347	−0.00357958
$748$	−19.3056	−0.705882
$749$	11.3840	0.415964
$750$	−17.7192	−0.647013
$751$	3.09916	0.113090	0.0565450	−	0.998400i	$-0.481992\pi$
$751$	3.09916	0.113090	0.0565450	+	0.998400i	$0.481992\pi$
$752$	5.20775	0.189907
$753$	−26.6547	−0.971352
$754$	0	0
$755$	−13.6058	−0.495166
$756$	−22.2935	−0.810807
$757$	−25.0670	−0.911074	−0.455537	−	0.890217i	$-0.650553\pi$
$757$	−25.0670	−0.911074	−0.455537	+	0.890217i	$0.650553\pi$
$758$	18.9638	0.688795
$759$	19.2078	0.697197
$760$	−2.44504	−0.0886910
$761$	32.7375	1.18673	0.593366	−	0.804933i	$-0.297799\pi$
$761$	32.7375	1.18673	0.593366	+	0.804933i	$0.297799\pi$
$762$	33.6819	1.22017
$763$	45.9275	1.66269
$764$	−17.4819	−0.632472
$765$	−4.67456	−0.169009
$766$	−28.4644	−1.02846
$767$	0	0
$768$	−1.80194	−0.0650218
$769$	0.836381	0.0301607	0.0150803	−	0.999886i	$-0.495200\pi$
$769$	0.836381	0.0301607	0.0150803	+	0.999886i	$0.495200\pi$
$770$	−27.4034	−0.987551
$771$	47.9168	1.72568
$772$	23.7754	0.855695
$773$	−9.52542	−0.342605	−0.171303	−	0.985218i	$-0.554798\pi$
$773$	−9.52542	−0.342605	−0.171303	+	0.985218i	$0.554798\pi$
$774$	1.78017	0.0639868
$775$	−0.892003	−0.0320417
$776$	3.16421	0.113589
$777$	55.1594	1.97883
$778$	6.90084	0.247407
$779$	−3.28621	−0.117741
$780$	0	0
$781$	−9.63401	−0.344732
$782$	33.0858	1.18314
$783$	−3.54096	−0.126544
$784$	13.1957	0.471274
$785$	16.5588	0.591009
$786$	15.3056	0.545932
$787$	27.4040	0.976848	0.488424	−	0.872607i	$-0.337572\pi$
$787$	27.4040	0.976848	0.488424	+	0.872607i	$0.337572\pi$
$788$	22.6069	0.805336
$789$	44.5676	1.58665
$790$	4.68771	0.166781
$791$	−34.1715	−1.21500
$792$	−0.615957	−0.0218871
$793$	0	0
$794$	27.8726	0.989162
$795$	−39.1680	−1.38915
$796$	−16.7332	−0.593091
$797$	−28.4241	−1.00683	−0.503417	−	0.864044i	$-0.667924\pi$
$797$	−28.4241	−1.00683	−0.503417	+	0.864044i	$0.667924\pi$
$798$	8.09783	0.286660
$799$	40.3129	1.42617
$800$	0.978230	0.0345856
$801$	−4.14138	−0.146328
$802$	−27.5120	−0.971483
$803$	19.3056	0.681279
$804$	2.53319	0.0893386
$805$	46.9638	1.65525
$806$	0	0
$807$	−29.3599	−1.03352
$808$	15.5308	0.546371
$809$	−34.9855	−1.23003	−0.615013	−	0.788517i	$-0.710849\pi$
$809$	−34.9855	−1.23003	−0.615013	+	0.788517i	$0.710849\pi$
$810$	23.6679	0.831604
$811$	−6.08383	−0.213632	−0.106816	−	0.994279i	$-0.534066\pi$
$811$	−6.08383	−0.213632	−0.106816	+	0.994279i	$0.534066\pi$
$812$	3.20775	0.112570
$813$	−46.4107	−1.62769
$814$	−16.9879	−0.595426
$815$	20.2306	0.708647
$816$	−13.9487	−0.488302
$817$	7.20775	0.252167
$818$	−10.8358	−0.378864
$819$	0	0
$820$	8.03492	0.280591
$821$	15.0804	0.526309	0.263154	−	0.964754i	$-0.415237\pi$
$821$	15.0804	0.526309	0.263154	+	0.964754i	$0.415237\pi$
$822$	−35.5894	−1.24132
$823$	9.20297	0.320795	0.160398	−	0.987052i	$-0.448722\pi$
$823$	9.20297	0.320795	0.160398	+	0.987052i	$0.448722\pi$
$824$	−3.03684	−0.105793
$825$	4.39612	0.153053
$826$	50.5870	1.76015
$827$	38.4849	1.33825	0.669125	−	0.743150i	$-0.266669\pi$
$827$	38.4849	1.33825	0.669125	+	0.743150i	$0.266669\pi$
$828$	1.05562	0.0366854
$829$	46.1366	1.60239	0.801195	−	0.598403i	$-0.204198\pi$
$829$	46.1366	1.60239	0.801195	+	0.598403i	$0.204198\pi$
$830$	0.968541	0.0336186
$831$	−3.94677	−0.136912
$832$	0	0
$833$	102.147	3.53918
$834$	−14.6896	−0.508660
$835$	15.2873	0.529038
$836$	−2.49396	−0.0862554
$837$	−4.52350	−0.156355
$838$	−28.4892	−0.984142
$839$	−27.7808	−0.959098	−0.479549	−	0.877515i	$-0.659200\pi$
$839$	−27.7808	−0.959098	−0.479549	+	0.877515i	$0.659200\pi$
$840$	−19.7995	−0.683149
$841$	−28.4905	−0.982431
$842$	−28.1957	−0.971687
$843$	44.3913	1.52892
$844$	15.3817	0.529458
$845$	0	0
$846$	1.28621	0.0442208
$847$	21.4819	0.738126
$848$	−8.89008	−0.305287
$849$	8.07846	0.277252
$850$	7.57242	0.259732
$851$	29.1138	0.998007
$852$	−6.96077	−0.238472
$853$	−7.16746	−0.245409	−0.122705	−	0.992443i	$-0.539157\pi$
$853$	−7.16746	−0.245409	−0.122705	+	0.992443i	$0.539157\pi$
$854$	−15.4276	−0.527921
$855$	−0.603875	−0.0206521
$856$	−2.53319	−0.0865826
$857$	6.47458	0.221168	0.110584	−	0.993867i	$-0.464728\pi$
$857$	6.47458	0.221168	0.110584	+	0.993867i	$0.464728\pi$
$858$	0	0
$859$	4.64874	0.158613	0.0793065	−	0.996850i	$-0.474729\pi$
$859$	4.64874	0.158613	0.0793065	+	0.996850i	$0.474729\pi$
$860$	−17.6233	−0.600948
$861$	−26.6112	−0.906906
$862$	20.4722	0.697286
$863$	3.16900	0.107874	0.0539369	−	0.998544i	$-0.482823\pi$
$863$	3.16900	0.107874	0.0539369	+	0.998544i	$0.482823\pi$
$864$	4.96077	0.168769
$865$	−59.9361	−2.03789
$866$	9.69096	0.329312
$867$	−77.3430	−2.62671
$868$	4.09783	0.139090
$869$	4.78150	0.162201
$870$	−3.14483	−0.106620
$871$	0	0
$872$	−10.2198	−0.346087
$873$	0.781495	0.0264496
$874$	4.27413	0.144574
$875$	−44.1909	−1.49392
$876$	13.9487	0.471283
$877$	−12.1763	−0.411164	−0.205582	−	0.978640i	$-0.565909\pi$
$877$	−12.1763	−0.411164	−0.205582	+	0.978640i	$0.565909\pi$
$878$	−7.67994	−0.259185
$879$	6.00000	0.202375
$880$	6.09783	0.205558
$881$	−21.2704	−0.716618	−0.358309	−	0.933603i	$-0.616647\pi$
$881$	−21.2704	−0.716618	−0.358309	+	0.933603i	$0.616647\pi$
$882$	3.25906	0.109738
$883$	−36.0978	−1.21479	−0.607394	−	0.794400i	$-0.707785\pi$
$883$	−36.0978	−1.21479	−0.607394	+	0.794400i	$0.707785\pi$
$884$	0	0
$885$	−49.5948	−1.66711
$886$	−11.5168	−0.386914
$887$	−2.97956	−0.100044	−0.0500219	−	0.998748i	$-0.515929\pi$
$887$	−2.97956	−0.100044	−0.0500219	+	0.998748i	$0.515929\pi$
$888$	−12.2741	−0.411893
$889$	84.0012	2.81731
$890$	40.9987	1.37428
$891$	24.1414	0.808766
$892$	−28.4198	−0.951566
$893$	5.20775	0.174271
$894$	−14.6189	−0.488931
$895$	63.4161	2.11977
$896$	−4.49396	−0.150133
$897$	0	0
$898$	1.16421	0.0388502
$899$	0.650874	0.0217079
$900$	0.241603	0.00805343
$901$	−68.8176	−2.29265
$902$	8.19567	0.272886
$903$	58.3672	1.94234
$904$	7.60388	0.252901
$905$	−12.7922	−0.425229
$906$	−10.0271	−0.333130
$907$	−11.1594	−0.370543	−0.185271	−	0.982687i	$-0.559316\pi$
$907$	−11.1594	−0.370543	−0.185271	+	0.982687i	$0.559316\pi$
$908$	−10.1981	−0.338435
$909$	3.83579	0.127225
$910$	0	0
$911$	−19.3787	−0.642044	−0.321022	−	0.947072i	$-0.604026\pi$
$911$	−19.3787	−0.642044	−0.321022	+	0.947072i	$0.604026\pi$
$912$	−1.80194	−0.0596681
$913$	0.987918	0.0326953
$914$	−30.3230	−1.00300
$915$	15.1250	0.500016
$916$	−14.5429	−0.480510
$917$	38.1715	1.26053
$918$	38.4010	1.26742
$919$	16.1763	0.533607	0.266803	−	0.963751i	$-0.414033\pi$
$919$	16.1763	0.533607	0.266803	+	0.963751i	$0.414033\pi$
$920$	−10.4504	−0.344540
$921$	4.94869	0.163065
$922$	0.992230	0.0326774
$923$	0	0
$924$	−20.1957	−0.664389
$925$	6.66334	0.219089
$926$	−28.8767	−0.948946
$927$	−0.750036	−0.0246344
$928$	−0.713792	−0.0234314
$929$	−20.2127	−0.663156	−0.331578	−	0.943428i	$-0.607581\pi$
$929$	−20.2127	−0.663156	−0.331578	+	0.943428i	$0.607581\pi$
$930$	−4.01746	−0.131738
$931$	13.1957	0.432471
$932$	−26.0683	−0.853895
$933$	24.6461	0.806877
$934$	−19.8189	−0.648495
$935$	47.2030	1.54370
$936$	0	0
$937$	30.8769	1.00871	0.504353	−	0.863498i	$-0.331731\pi$
$937$	30.8769	1.00871	0.504353	+	0.863498i	$0.331731\pi$
$938$	6.31767	0.206279
$939$	−29.1739	−0.952054
$940$	−12.7332	−0.415310
$941$	−17.5749	−0.572926	−0.286463	−	0.958091i	$-0.592480\pi$
$941$	−17.5749	−0.572926	−0.286463	+	0.958091i	$0.592480\pi$
$942$	12.2034	0.397610
$943$	−14.0457	−0.457390
$944$	−11.2567	−0.366373
$945$	54.5086	1.77316
$946$	−17.9758	−0.584445
$947$	−20.0435	−0.651328	−0.325664	−	0.945486i	$-0.605588\pi$
$947$	−20.0435	−0.651328	−0.325664	+	0.945486i	$0.605588\pi$
$948$	3.45473	0.112204
$949$	0	0
$950$	0.978230	0.0317380
$951$	35.7754	1.16010
$952$	−34.7875	−1.12747
$953$	−11.3685	−0.368262	−0.184131	−	0.982902i	$-0.558947\pi$
$953$	−11.3685	−0.368262	−0.184131	+	0.982902i	$0.558947\pi$
$954$	−2.19567	−0.0710874
$955$	42.7439	1.38316
$956$	16.2198	0.524587
$957$	−3.20775	−0.103692
$958$	21.9758	0.710007
$959$	−88.7585	−2.86616
$960$	4.40581	0.142197
$961$	−30.1685	−0.973178
$962$	0	0
$963$	−0.625646	−0.0201612
$964$	27.3250	0.880078
$965$	−58.1318	−1.87133
$966$	34.6112	1.11360
$967$	20.2741	0.651972	0.325986	−	0.945375i	$-0.394304\pi$
$967$	20.2741	0.651972	0.325986	+	0.945375i	$0.394304\pi$
$968$	−4.78017	−0.153640
$969$	−13.9487	−0.448096
$970$	−7.73663	−0.248408
$971$	20.8528	0.669197	0.334599	−	0.942361i	$-0.391399\pi$
$971$	20.8528	0.669197	0.334599	+	0.942361i	$0.391399\pi$
$972$	2.56033	0.0821228
$973$	−36.6353	−1.17447
$974$	−36.1148	−1.15719
$975$	0	0
$976$	3.43296	0.109886
$977$	−16.2452	−0.519729	−0.259865	−	0.965645i	$-0.583678\pi$
$977$	−16.2452	−0.519729	−0.259865	+	0.965645i	$0.583678\pi$
$978$	14.9095	0.476752
$979$	41.8189	1.33654
$980$	−32.2640	−1.03063
$981$	−2.52409	−0.0805880
$982$	−21.2814	−0.679117
$983$	6.02582	0.192194	0.0960969	−	0.995372i	$-0.469364\pi$
$983$	6.02582	0.192194	0.0960969	+	0.995372i	$0.469364\pi$
$984$	5.92154	0.188772
$985$	−55.2747	−1.76120
$986$	−5.52542	−0.175965
$987$	42.1715	1.34233
$988$	0	0
$989$	30.8068	0.979601
$990$	1.50604	0.0478651
$991$	−39.6334	−1.25900	−0.629498	−	0.777002i	$-0.716740\pi$
$991$	−39.6334	−1.25900	−0.629498	+	0.777002i	$0.716740\pi$
$992$	−0.911854	−0.0289514
$993$	−34.6504	−1.09960
$994$	−17.3599	−0.550622
$995$	40.9133	1.29704
$996$	0.713792	0.0226174
$997$	−15.9627	−0.505543	−0.252772	−	0.967526i	$-0.581342\pi$
$997$	−15.9627	−0.505543	−0.252772	+	0.967526i	$0.581342\pi$
$998$	−5.00730	−0.158503
$999$	33.7909	1.06910

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6422.2.a.n.1.1	✓	3	13.12	even	2
6422.2.a.v.1.1	yes	3	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.80194 q^{3} +1.00000 q^{4} -2.44504 q^{5} -1.80194 q^{6} -4.49396 q^{7} +1.00000 q^{8} +0.246980 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.80194 q^{3} +1.00000 q^{4} -2.44504 q^{5} -1.80194 q^{6} -4.49396 q^{7} +1.00000 q^{8} +0.246980 q^{9} -2.44504 q^{10} -2.49396 q^{11} -1.80194 q^{12} -4.49396 q^{14} +4.40581 q^{15} +1.00000 q^{16} +7.74094 q^{17} +0.246980 q^{18} +1.00000 q^{19} -2.44504 q^{20} +8.09783 q^{21} -2.49396 q^{22} +4.27413 q^{23} -1.80194 q^{24} +0.978230 q^{25} +4.96077 q^{27} -4.49396 q^{28} -0.713792 q^{29} +4.40581 q^{30} -0.911854 q^{31} +1.00000 q^{32} +4.49396 q^{33} +7.74094 q^{34} +10.9879 q^{35} +0.246980 q^{36} +6.81163 q^{37} +1.00000 q^{38} -2.44504 q^{40} -3.28621 q^{41} +8.09783 q^{42} +7.20775 q^{43} -2.49396 q^{44} -0.603875 q^{45} +4.27413 q^{46} +5.20775 q^{47} -1.80194 q^{48} +13.1957 q^{49} +0.978230 q^{50} -13.9487 q^{51} -8.89008 q^{53} +4.96077 q^{54} +6.09783 q^{55} -4.49396 q^{56} -1.80194 q^{57} -0.713792 q^{58} -11.2567 q^{59} +4.40581 q^{60} +3.43296 q^{61} -0.911854 q^{62} -1.10992 q^{63} +1.00000 q^{64} +4.49396 q^{66} -1.40581 q^{67} +7.74094 q^{68} -7.70171 q^{69} +10.9879 q^{70} +3.86294 q^{71} +0.246980 q^{72} -7.74094 q^{73} +6.81163 q^{74} -1.76271 q^{75} +1.00000 q^{76} +11.2078 q^{77} -1.91723 q^{79} -2.44504 q^{80} -9.67994 q^{81} -3.28621 q^{82} -0.396125 q^{83} +8.09783 q^{84} -18.9269 q^{85} +7.20775 q^{86} +1.28621 q^{87} -2.49396 q^{88} -16.7681 q^{89} -0.603875 q^{90} +4.27413 q^{92} +1.64310 q^{93} +5.20775 q^{94} -2.44504 q^{95} -1.80194 q^{96} +3.16421 q^{97} +13.1957 q^{98} -0.615957 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} - q^{3} + 3 q^{4} - 7 q^{5} - q^{6} - 4 q^{7} + 3 q^{8} - 4 q^{9}+O(q^{10}) \) 3 * q + 3 * q^2 - q^3 + 3 * q^4 - 7 * q^5 - q^6 - 4 * q^7 + 3 * q^8 - 4 * q^9	\( 3 q + 3 q^{2} - q^{3} + 3 q^{4} - 7 q^{5} - q^{6} - 4 q^{7} + 3 q^{8} - 4 q^{9} - 7 q^{10} + 2 q^{11} - q^{12} - 4 q^{14} + 3 q^{16} + 9 q^{17} - 4 q^{18} + 3 q^{19} - 7 q^{20} + 6 q^{21} + 2 q^{22} + 2 q^{23} - q^{24} + 6 q^{25} + 2 q^{27} - 4 q^{28} + 6 q^{29} + q^{31} + 3 q^{32} + 4 q^{33} + 9 q^{34} + 14 q^{35} - 4 q^{36} - 6 q^{37} + 3 q^{38} - 7 q^{40} - 18 q^{41} + 6 q^{42} + 4 q^{43} + 2 q^{44} + 7 q^{45} + 2 q^{46} - 2 q^{47} - q^{48} + 3 q^{49} + 6 q^{50} - 10 q^{51} - 26 q^{53} + 2 q^{54} - 4 q^{56} - q^{57} + 6 q^{58} - 7 q^{59} - 9 q^{61} + q^{62} - 4 q^{63} + 3 q^{64} + 4 q^{66} + 9 q^{67} + 9 q^{68} + 4 q^{69} + 14 q^{70} + 17 q^{71} - 4 q^{72} - 9 q^{73} - 6 q^{74} + 12 q^{75} + 3 q^{76} + 16 q^{77} + q^{79} - 7 q^{80} - 5 q^{81} - 18 q^{82} - 10 q^{83} + 6 q^{84} - 28 q^{85} + 4 q^{86} + 12 q^{87} + 2 q^{88} - 30 q^{89} + 7 q^{90} + 2 q^{92} + 9 q^{93} - 2 q^{94} - 7 q^{95} - q^{96} - 2 q^{97} + 3 q^{98} - 12 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 - q^3 + 3 * q^4 - 7 * q^5 - q^6 - 4 * q^7 + 3 * q^8 - 4 * q^9 - 7 * q^10 + 2 * q^11 - q^12 - 4 * q^14 + 3 * q^16 + 9 * q^17 - 4 * q^18 + 3 * q^19 - 7 * q^20 + 6 * q^21 + 2 * q^22 + 2 * q^23 - q^24 + 6 * q^25 + 2 * q^27 - 4 * q^28 + 6 * q^29 + q^31 + 3 * q^32 + 4 * q^33 + 9 * q^34 + 14 * q^35 - 4 * q^36 - 6 * q^37 + 3 * q^38 - 7 * q^40 - 18 * q^41 + 6 * q^42 + 4 * q^43 + 2 * q^44 + 7 * q^45 + 2 * q^46 - 2 * q^47 - q^48 + 3 * q^49 + 6 * q^50 - 10 * q^51 - 26 * q^53 + 2 * q^54 - 4 * q^56 - q^57 + 6 * q^58 - 7 * q^59 - 9 * q^61 + q^62 - 4 * q^63 + 3 * q^64 + 4 * q^66 + 9 * q^67 + 9 * q^68 + 4 * q^69 + 14 * q^70 + 17 * q^71 - 4 * q^72 - 9 * q^73 - 6 * q^74 + 12 * q^75 + 3 * q^76 + 16 * q^77 + q^79 - 7 * q^80 - 5 * q^81 - 18 * q^82 - 10 * q^83 + 6 * q^84 - 28 * q^85 + 4 * q^86 + 12 * q^87 + 2 * q^88 - 30 * q^89 + 7 * q^90 + 2 * q^92 + 9 * q^93 - 2 * q^94 - 7 * q^95 - q^96 - 2 * q^97 + 3 * q^98 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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