LMFDB - Embedded newform 6171.2.a.bk.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6171,2,Mod(1,6171)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6171, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6171.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6171 = 3 \cdot 11^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6171.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:	$49.2756830873$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} - 16 x^{10} + 15 x^{9} + 89 x^{8} - 78 x^{7} - 201 x^{6} + 157 x^{5} + 159 x^{4} + \cdots - 1 $ x^12 - x^11 - 16x^10 + 15x^9 + 89x^8 - 78x^7 - 201x^6 + 157x^5 + 159x^4 - 92x^3 - 29x^2 + 17x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 561)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$0.0682851$ of defining polynomial
Character	$\chi$	$=$	6171.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.0682851 q^{2} +1.00000 q^{3} -1.99534 q^{4} -1.22357 q^{5} -0.0682851 q^{6} -4.50194 q^{7} +0.272822 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.0682851 q^{2} +1.00000 q^{3} -1.99534 q^{4} -1.22357 q^{5} -0.0682851 q^{6} -4.50194 q^{7} +0.272822 q^{8} +1.00000 q^{9} +0.0835515 q^{10} -1.99534 q^{12} -0.303468 q^{13} +0.307415 q^{14} -1.22357 q^{15} +3.97204 q^{16} +1.00000 q^{17} -0.0682851 q^{18} +3.84369 q^{19} +2.44143 q^{20} -4.50194 q^{21} -0.932914 q^{23} +0.272822 q^{24} -3.50288 q^{25} +0.0207223 q^{26} +1.00000 q^{27} +8.98288 q^{28} +5.06536 q^{29} +0.0835515 q^{30} +0.862115 q^{31} -0.816875 q^{32} -0.0682851 q^{34} +5.50843 q^{35} -1.99534 q^{36} -5.60415 q^{37} -0.262467 q^{38} -0.303468 q^{39} -0.333816 q^{40} +8.86200 q^{41} +0.307415 q^{42} +9.36261 q^{43} -1.22357 q^{45} +0.0637041 q^{46} +9.87093 q^{47} +3.97204 q^{48} +13.2674 q^{49} +0.239194 q^{50} +1.00000 q^{51} +0.605520 q^{52} -2.46273 q^{53} -0.0682851 q^{54} -1.22823 q^{56} +3.84369 q^{57} -0.345889 q^{58} -4.71310 q^{59} +2.44143 q^{60} -0.340285 q^{61} -0.0588696 q^{62} -4.50194 q^{63} -7.88831 q^{64} +0.371313 q^{65} -10.4800 q^{67} -1.99534 q^{68} -0.932914 q^{69} -0.376144 q^{70} -2.73114 q^{71} +0.272822 q^{72} -9.19569 q^{73} +0.382680 q^{74} -3.50288 q^{75} -7.66945 q^{76} +0.0207223 q^{78} -11.6693 q^{79} -4.86007 q^{80} +1.00000 q^{81} -0.605142 q^{82} -3.16228 q^{83} +8.98288 q^{84} -1.22357 q^{85} -0.639327 q^{86} +5.06536 q^{87} -5.45209 q^{89} +0.0835515 q^{90} +1.36619 q^{91} +1.86148 q^{92} +0.862115 q^{93} -0.674037 q^{94} -4.70302 q^{95} -0.816875 q^{96} +13.0877 q^{97} -0.905968 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - q^{2} + 12 q^{3} + 9 q^{4} - 14 q^{5} - q^{6} - 5 q^{7} + 12 q^{9}+O(q^{10}) $ 12 * q - q^2 + 12 * q^3 + 9 * q^4 - 14 * q^5 - q^6 - 5 * q^7 + 12 * q^9	$ 12 q - q^{2} + 12 q^{3} + 9 q^{4} - 14 q^{5} - q^{6} - 5 q^{7} + 12 q^{9} + 8 q^{10} + 9 q^{12} - 7 q^{13} - 13 q^{14} - 14 q^{15} + 11 q^{16} + 12 q^{17} - q^{18} + 3 q^{19} - 36 q^{20} - 5 q^{21} - 39 q^{23} + 8 q^{25} - 25 q^{26} + 12 q^{27} - 9 q^{28} + 10 q^{29} + 8 q^{30} - 25 q^{31} - 11 q^{32} - q^{34} + 12 q^{35} + 9 q^{36} - 10 q^{37} - 34 q^{38} - 7 q^{39} - 13 q^{40} + 17 q^{41} - 13 q^{42} - 18 q^{43} - 14 q^{45} + 19 q^{46} - 38 q^{47} + 11 q^{48} - 15 q^{49} - 18 q^{50} + 12 q^{51} + 32 q^{52} - 40 q^{53} - q^{54} - 25 q^{56} + 3 q^{57} + 3 q^{58} - 18 q^{59} - 36 q^{60} - 22 q^{61} + 3 q^{62} - 5 q^{63} - 20 q^{64} + 3 q^{65} - 9 q^{67} + 9 q^{68} - 39 q^{69} + 13 q^{70} - 24 q^{71} + q^{73} - 4 q^{74} + 8 q^{75} + 29 q^{76} - 25 q^{78} - 3 q^{79} - 43 q^{80} + 12 q^{81} + q^{82} + 4 q^{83} - 9 q^{84} - 14 q^{85} - 4 q^{86} + 10 q^{87} - 62 q^{89} + 8 q^{90} - 5 q^{91} - 52 q^{92} - 25 q^{93} - 16 q^{94} - 4 q^{95} - 11 q^{96} - 5 q^{97} - 7 q^{98}+O(q^{100}) $ 12 * q - q^2 + 12 * q^3 + 9 * q^4 - 14 * q^5 - q^6 - 5 * q^7 + 12 * q^9 + 8 * q^10 + 9 * q^12 - 7 * q^13 - 13 * q^14 - 14 * q^15 + 11 * q^16 + 12 * q^17 - q^18 + 3 * q^19 - 36 * q^20 - 5 * q^21 - 39 * q^23 + 8 * q^25 - 25 * q^26 + 12 * q^27 - 9 * q^28 + 10 * q^29 + 8 * q^30 - 25 * q^31 - 11 * q^32 - q^34 + 12 * q^35 + 9 * q^36 - 10 * q^37 - 34 * q^38 - 7 * q^39 - 13 * q^40 + 17 * q^41 - 13 * q^42 - 18 * q^43 - 14 * q^45 + 19 * q^46 - 38 * q^47 + 11 * q^48 - 15 * q^49 - 18 * q^50 + 12 * q^51 + 32 * q^52 - 40 * q^53 - q^54 - 25 * q^56 + 3 * q^57 + 3 * q^58 - 18 * q^59 - 36 * q^60 - 22 * q^61 + 3 * q^62 - 5 * q^63 - 20 * q^64 + 3 * q^65 - 9 * q^67 + 9 * q^68 - 39 * q^69 + 13 * q^70 - 24 * q^71 + q^73 - 4 * q^74 + 8 * q^75 + 29 * q^76 - 25 * q^78 - 3 * q^79 - 43 * q^80 + 12 * q^81 + q^82 + 4 * q^83 - 9 * q^84 - 14 * q^85 - 4 * q^86 + 10 * q^87 - 62 * q^89 + 8 * q^90 - 5 * q^91 - 52 * q^92 - 25 * q^93 - 16 * q^94 - 4 * q^95 - 11 * q^96 - 5 * q^97 - 7 * q^98

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 $

$2$	−0.0682851	−0.0482849	−0.0241424	−	0.999709i	$-0.507686\pi$
$2$	−0.0682851	−0.0482849	−0.0241424	+	0.999709i	$0.507686\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.99534	−0.997669
$5$	−1.22357	−0.547197	−0.273598	−	0.961844i	$-0.588214\pi$
$5$	−1.22357	−0.547197	−0.273598	+	0.961844i	$0.588214\pi$
$6$	−0.0682851	−0.0278773
$7$	−4.50194	−1.70157	−0.850786	−	0.525512i	$-0.823874\pi$
$7$	−4.50194	−1.70157	−0.850786	+	0.525512i	$0.823874\pi$
$8$	0.272822	0.0964571
$9$	1.00000	0.333333
$10$	0.0835515	0.0264213
$11$	0	0
$11$	0	0
$12$	−1.99534	−0.576004
$13$	−0.303468	−0.0841667	−0.0420834	−	0.999114i	$-0.513400\pi$
$13$	−0.303468	−0.0841667	−0.0420834	+	0.999114i	$0.513400\pi$
$14$	0.307415	0.0821602
$15$	−1.22357	−0.315924
$16$	3.97204	0.993011
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−0.0682851	−0.0160950
$19$	3.84369	0.881803	0.440901	−	0.897556i	$-0.354659\pi$
$19$	3.84369	0.881803	0.440901	+	0.897556i	$0.354659\pi$
$20$	2.44143	0.545921
$21$	−4.50194	−0.982403
$22$	0	0
$23$	−0.932914	−0.194526	−0.0972630	−	0.995259i	$-0.531009\pi$
$23$	−0.932914	−0.194526	−0.0972630	+	0.995259i	$0.531009\pi$
$24$	0.272822	0.0556896
$25$	−3.50288	−0.700576
$26$	0.0207223	0.00406398
$27$	1.00000	0.192450
$28$	8.98288	1.69761
$29$	5.06536	0.940613	0.470307	−	0.882503i	$-0.344143\pi$
$29$	5.06536	0.940613	0.470307	+	0.882503i	$0.344143\pi$
$30$	0.0835515	0.0152543
$31$	0.862115	0.154840	0.0774202	−	0.996999i	$-0.475332\pi$
$31$	0.862115	0.154840	0.0774202	+	0.996999i	$0.475332\pi$
$32$	−0.816875	−0.144405
$33$	0	0
$34$	−0.0682851	−0.0117108
$35$	5.50843	0.931094
$36$	−1.99534	−0.332556
$37$	−5.60415	−0.921317	−0.460659	−	0.887577i	$-0.652387\pi$
$37$	−5.60415	−0.921317	−0.460659	+	0.887577i	$0.652387\pi$
$38$	−0.262467	−0.0425777
$39$	−0.303468	−0.0485937
$40$	−0.333816	−0.0527810
$41$	8.86200	1.38401	0.692006	−	0.721892i	$-0.256727\pi$
$41$	8.86200	1.38401	0.692006	+	0.721892i	$0.256727\pi$
$42$	0.307415	0.0474352
$43$	9.36261	1.42778	0.713892	−	0.700256i	$-0.246931\pi$
$43$	9.36261	1.42778	0.713892	+	0.700256i	$0.246931\pi$
$44$	0	0
$45$	−1.22357	−0.182399
$46$	0.0637041	0.00939266
$47$	9.87093	1.43982	0.719911	−	0.694066i	$-0.244182\pi$
$47$	9.87093	1.43982	0.719911	+	0.694066i	$0.244182\pi$
$48$	3.97204	0.573315
$49$	13.2674	1.89535
$50$	0.239194	0.0338272
$51$	1.00000	0.140028
$52$	0.605520	0.0839705
$53$	−2.46273	−0.338282	−0.169141	−	0.985592i	$-0.554099\pi$
$53$	−2.46273	−0.338282	−0.169141	+	0.985592i	$0.554099\pi$
$54$	−0.0682851	−0.00929242
$55$	0	0
$56$	−1.22823	−0.164129
$57$	3.84369	0.509109
$58$	−0.345889	−0.0454174
$59$	−4.71310	−0.613593	−0.306797	−	0.951775i	$-0.599257\pi$
$59$	−4.71310	−0.613593	−0.306797	+	0.951775i	$0.599257\pi$
$60$	2.44143	0.315188
$61$	−0.340285	−0.0435691	−0.0217845	−	0.999763i	$-0.506935\pi$
$61$	−0.340285	−0.0435691	−0.0217845	+	0.999763i	$0.506935\pi$
$62$	−0.0588696	−0.00747645
$63$	−4.50194	−0.567191
$64$	−7.88831	−0.986039
$65$	0.371313	0.0460558
$66$	0	0
$67$	−10.4800	−1.28034	−0.640171	−	0.768233i	$-0.721136\pi$
$67$	−10.4800	−1.28034	−0.640171	+	0.768233i	$0.721136\pi$
$68$	−1.99534	−0.241970
$69$	−0.932914	−0.112310
$70$	−0.376144	−0.0449578
$71$	−2.73114	−0.324127	−0.162064	−	0.986780i	$-0.551815\pi$
$71$	−2.73114	−0.324127	−0.162064	+	0.986780i	$0.551815\pi$
$72$	0.272822	0.0321524
$73$	−9.19569	−1.07627	−0.538137	−	0.842857i	$-0.680872\pi$
$73$	−9.19569	−1.07627	−0.538137	+	0.842857i	$0.680872\pi$
$74$	0.382680	0.0444857
$75$	−3.50288	−0.404478
$76$	−7.66945	−0.879747
$77$	0	0
$78$	0.0207223	0.00234634
$79$	−11.6693	−1.31290	−0.656448	−	0.754371i	$-0.727942\pi$
$79$	−11.6693	−1.31290	−0.656448	+	0.754371i	$0.727942\pi$
$80$	−4.86007	−0.543372
$81$	1.00000	0.111111
$82$	−0.605142	−0.0668268
$83$	−3.16228	−0.347106	−0.173553	−	0.984825i	$-0.555525\pi$
$83$	−3.16228	−0.347106	−0.173553	+	0.984825i	$0.555525\pi$
$84$	8.98288	0.980113
$85$	−1.22357	−0.132715
$86$	−0.639327	−0.0689404
$87$	5.06536	0.543063
$88$	0	0
$89$	−5.45209	−0.577920	−0.288960	−	0.957341i	$-0.593309\pi$
$89$	−5.45209	−0.577920	−0.288960	+	0.957341i	$0.593309\pi$
$90$	0.0835515	0.00880710
$91$	1.36619	0.143216
$92$	1.86148	0.194073
$93$	0.862115	0.0893972
$94$	−0.674037	−0.0695216
$95$	−4.70302	−0.482519
$96$	−0.816875	−0.0833720
$97$	13.0877	1.32886	0.664428	−	0.747352i	$-0.268675\pi$
$97$	13.0877	1.32886	0.664428	+	0.747352i	$0.268675\pi$
$98$	−0.905968	−0.0915166
$99$	0	0
$100$	6.98943	0.698943
$101$	−17.9833	−1.78941	−0.894705	−	0.446658i	$-0.852614\pi$
$101$	−17.9833	−1.78941	−0.894705	+	0.446658i	$0.852614\pi$
$102$	−0.0682851	−0.00676123
$103$	8.16651	0.804670	0.402335	−	0.915492i	$-0.368199\pi$
$103$	8.16651	0.804670	0.402335	+	0.915492i	$0.368199\pi$
$104$	−0.0827926	−0.00811848
$105$	5.50843	0.537568
$106$	0.168168	0.0163339
$107$	17.1609	1.65901	0.829504	−	0.558501i	$-0.188623\pi$
$107$	17.1609	1.65901	0.829504	+	0.558501i	$0.188623\pi$
$108$	−1.99534	−0.192001
$109$	12.0798	1.15703	0.578516	−	0.815671i	$-0.303632\pi$
$109$	12.0798	1.15703	0.578516	+	0.815671i	$0.303632\pi$
$110$	0	0
$111$	−5.60415	−0.531923
$112$	−17.8819	−1.68968
$113$	−10.6036	−0.997507	−0.498754	−	0.866744i	$-0.666209\pi$
$113$	−10.6036	−0.997507	−0.498754	+	0.866744i	$0.666209\pi$
$114$	−0.262467	−0.0245823
$115$	1.14148	0.106444
$116$	−10.1071	−0.938420
$117$	−0.303468	−0.0280556
$118$	0.321834	0.0296273
$119$	−4.50194	−0.412692
$120$	−0.333816	−0.0304731
$121$	0	0
$122$	0.0232364	0.00210373
$123$	8.86200	0.799059
$124$	−1.72021	−0.154479
$125$	10.4039	0.930549
$126$	0.307415	0.0273867
$127$	−10.1554	−0.901142	−0.450571	−	0.892741i	$-0.648780\pi$
$127$	−10.1554	−0.901142	−0.450571	+	0.892741i	$0.648780\pi$
$128$	2.17240	0.192015
$129$	9.36261	0.824332
$130$	−0.0253552	−0.00222380
$131$	17.4368	1.52346	0.761729	−	0.647896i	$-0.224351\pi$
$131$	17.4368	1.52346	0.761729	+	0.647896i	$0.224351\pi$
$132$	0	0
$133$	−17.3040	−1.50045
$134$	0.715631	0.0618211
$135$	−1.22357	−0.105308
$136$	0.272822	0.0233943
$137$	−11.4622	−0.979280	−0.489640	−	0.871925i	$-0.662872\pi$
$137$	−11.4622	−0.979280	−0.489640	+	0.871925i	$0.662872\pi$
$138$	0.0637041	0.00542286
$139$	−6.34253	−0.537967	−0.268983	−	0.963145i	$-0.586688\pi$
$139$	−6.34253	−0.537967	−0.268983	+	0.963145i	$0.586688\pi$
$140$	−10.9912	−0.928924
$141$	9.87093	0.831282
$142$	0.186496	0.0156504
$143$	0	0
$144$	3.97204	0.331004
$145$	−6.19781	−0.514700
$146$	0.627929	0.0519678
$147$	13.2674	1.09428
$148$	11.1822	0.919169
$149$	19.4598	1.59421	0.797104	−	0.603842i	$-0.206364\pi$
$149$	19.4598	1.59421	0.797104	+	0.603842i	$0.206364\pi$
$150$	0.239194	0.0195301
$151$	−12.0355	−0.979431	−0.489716	−	0.871882i	$-0.662899\pi$
$151$	−12.0355	−0.979431	−0.489716	+	0.871882i	$0.662899\pi$
$152$	1.04864	0.0850562
$153$	1.00000	0.0808452
$154$	0	0
$155$	−1.05486	−0.0847282
$156$	0.605520	0.0484804
$157$	−9.35496	−0.746607	−0.373304	−	0.927709i	$-0.621775\pi$
$157$	−9.35496	−0.746607	−0.373304	+	0.927709i	$0.621775\pi$
$158$	0.796838	0.0633930
$159$	−2.46273	−0.195307
$160$	0.999503	0.0790177
$161$	4.19992	0.331000
$162$	−0.0682851	−0.00536498
$163$	−16.9330	−1.32629	−0.663146	−	0.748490i	$-0.730779\pi$
$163$	−16.9330	−1.32629	−0.663146	+	0.748490i	$0.730779\pi$
$164$	−17.6827	−1.38078
$165$	0	0
$166$	0.215937	0.0167600
$167$	−5.04347	−0.390276	−0.195138	−	0.980776i	$-0.562515\pi$
$167$	−5.04347	−0.390276	−0.195138	+	0.980776i	$0.562515\pi$
$168$	−1.22823	−0.0947598
$169$	−12.9079	−0.992916
$170$	0.0835515	0.00640811
$171$	3.84369	0.293934
$172$	−18.6816	−1.42446
$173$	−10.3090	−0.783777	−0.391888	−	0.920013i	$-0.628178\pi$
$173$	−10.3090	−0.783777	−0.391888	+	0.920013i	$0.628178\pi$
$174$	−0.345889	−0.0262217
$175$	15.7697	1.19208
$176$	0	0
$177$	−4.71310	−0.354258
$178$	0.372296	0.0279048
$179$	−15.5759	−1.16420	−0.582098	−	0.813119i	$-0.697768\pi$
$179$	−15.5759	−1.16420	−0.582098	+	0.813119i	$0.697768\pi$
$180$	2.44143	0.181974
$181$	−16.6554	−1.23798	−0.618992	−	0.785397i	$-0.712459\pi$
$181$	−16.6554	−1.23798	−0.618992	+	0.785397i	$0.712459\pi$
$182$	−0.0932905	−0.00691515
$183$	−0.340285	−0.0251546
$184$	−0.254520	−0.0187634
$185$	6.85707	0.504142
$186$	−0.0588696	−0.00431653
$187$	0	0
$188$	−19.6958	−1.43647
$189$	−4.50194	−0.327468
$190$	0.321146	0.0232984
$191$	−15.4943	−1.12113	−0.560566	−	0.828110i	$-0.689416\pi$
$191$	−15.4943	−1.12113	−0.560566	+	0.828110i	$0.689416\pi$
$192$	−7.88831	−0.569290
$193$	11.1322	0.801310	0.400655	−	0.916229i	$-0.368783\pi$
$193$	11.1322	0.801310	0.400655	+	0.916229i	$0.368783\pi$
$194$	−0.893696	−0.0641637
$195$	0.371313	0.0265903
$196$	−26.4730	−1.89093
$197$	16.6917	1.18924	0.594619	−	0.804008i	$-0.297303\pi$
$197$	16.6917	1.18924	0.594619	+	0.804008i	$0.297303\pi$
$198$	0	0
$199$	−10.1920	−0.722494	−0.361247	−	0.932470i	$-0.617649\pi$
$199$	−10.1920	−0.722494	−0.361247	+	0.932470i	$0.617649\pi$
$200$	−0.955663	−0.0675756
$201$	−10.4800	−0.739206
$202$	1.22799	0.0864014
$203$	−22.8039	−1.60052
$204$	−1.99534	−0.139702
$205$	−10.8433	−0.757326
$206$	−0.557651	−0.0388534
$207$	−0.932914	−0.0648420
$208$	−1.20539	−0.0835785
$209$	0	0
$210$	−0.376144	−0.0259564
$211$	−11.3644	−0.782355	−0.391178	−	0.920315i	$-0.627932\pi$
$211$	−11.3644	−0.782355	−0.391178	+	0.920315i	$0.627932\pi$
$212$	4.91398	0.337494
$213$	−2.73114	−0.187135
$214$	−1.17183	−0.0801050
$215$	−11.4558	−0.781279
$216$	0.272822	0.0185632
$217$	−3.88119	−0.263472
$218$	−0.824868	−0.0558671
$219$	−9.19569	−0.621387
$220$	0	0
$221$	−0.303468	−0.0204134
$222$	0.382680	0.0256838
$223$	20.8358	1.39527	0.697634	−	0.716455i	$-0.254236\pi$
$223$	20.8358	1.39527	0.697634	+	0.716455i	$0.254236\pi$
$224$	3.67752	0.245715
$225$	−3.50288	−0.233525
$226$	0.724071	0.0481645
$227$	2.66351	0.176784	0.0883918	−	0.996086i	$-0.471827\pi$
$227$	2.66351	0.176784	0.0883918	+	0.996086i	$0.471827\pi$
$228$	−7.66945	−0.507922
$229$	8.96242	0.592253	0.296127	−	0.955149i	$-0.404305\pi$
$229$	8.96242	0.592253	0.296127	+	0.955149i	$0.404305\pi$
$230$	−0.0779464	−0.00513963
$231$	0	0
$232$	1.38194	0.0907289
$233$	16.9844	1.11269	0.556343	−	0.830953i	$-0.312204\pi$
$233$	16.9844	1.11269	0.556343	+	0.830953i	$0.312204\pi$
$234$	0.0207223	0.00135466
$235$	−12.0778	−0.787866
$236$	9.40422	0.612163
$237$	−11.6693	−0.758001
$238$	0.307415	0.0199268
$239$	5.56317	0.359852	0.179926	−	0.983680i	$-0.442414\pi$
$239$	5.56317	0.359852	0.179926	+	0.983680i	$0.442414\pi$
$240$	−4.86007	−0.313716
$241$	−19.7746	−1.27380	−0.636898	−	0.770948i	$-0.719783\pi$
$241$	−19.7746	−1.27380	−0.636898	+	0.770948i	$0.719783\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0.678984	0.0434675
$245$	−16.2336	−1.03713
$246$	−0.605142	−0.0385825
$247$	−1.16643	−0.0742184
$248$	0.235204	0.0149355
$249$	−3.16228	−0.200402
$250$	−0.710428	−0.0449314
$251$	0.543336	0.0342951	0.0171475	−	0.999853i	$-0.494542\pi$
$251$	0.543336	0.0342951	0.0171475	+	0.999853i	$0.494542\pi$
$252$	8.98288	0.565868
$253$	0	0
$254$	0.693460	0.0435115
$255$	−1.22357	−0.0766228
$256$	15.6283	0.976767
$257$	20.4095	1.27311	0.636556	−	0.771231i	$-0.280359\pi$
$257$	20.4095	1.27311	0.636556	+	0.771231i	$0.280359\pi$
$258$	−0.639327	−0.0398027
$259$	25.2295	1.56769
$260$	−0.740895	−0.0459484
$261$	5.06536	0.313538
$262$	−1.19067	−0.0735599
$263$	−6.22797	−0.384033	−0.192017	−	0.981392i	$-0.561503\pi$
$263$	−6.22797	−0.384033	−0.192017	+	0.981392i	$0.561503\pi$
$264$	0	0
$265$	3.01332	0.185107
$266$	1.18161	0.0724491
$267$	−5.45209	−0.333662
$268$	20.9112	1.27736
$269$	3.48504	0.212486	0.106243	−	0.994340i	$-0.466118\pi$
$269$	3.48504	0.212486	0.106243	+	0.994340i	$0.466118\pi$
$270$	0.0835515	0.00508478
$271$	−15.0244	−0.912666	−0.456333	−	0.889809i	$-0.650837\pi$
$271$	−15.0244	−0.912666	−0.456333	+	0.889809i	$0.650837\pi$
$272$	3.97204	0.240841
$273$	1.36619	0.0826857
$274$	0.782696	0.0472844
$275$	0	0
$276$	1.86148	0.112048
$277$	−12.1303	−0.728837	−0.364419	−	0.931235i	$-0.618732\pi$
$277$	−12.1303	−0.728837	−0.364419	+	0.931235i	$0.618732\pi$
$278$	0.433101	0.0259756
$279$	0.862115	0.0516135
$280$	1.50282	0.0898107
$281$	−13.2485	−0.790342	−0.395171	−	0.918608i	$-0.629315\pi$
$281$	−13.2485	−0.790342	−0.395171	+	0.918608i	$0.629315\pi$
$282$	−0.674037	−0.0401383
$283$	17.2205	1.02365	0.511825	−	0.859090i	$-0.328969\pi$
$283$	17.2205	1.02365	0.511825	+	0.859090i	$0.328969\pi$
$284$	5.44955	0.323371
$285$	−4.70302	−0.278583
$286$	0	0
$287$	−39.8962	−2.35500
$288$	−0.816875	−0.0481348
$289$	1.00000	0.0588235
$290$	0.423218	0.0248522
$291$	13.0877	0.767216
$292$	18.3485	1.07377
$293$	−13.7941	−0.805861	−0.402931	−	0.915231i	$-0.632008\pi$
$293$	−13.7941	−0.805861	−0.402931	+	0.915231i	$0.632008\pi$
$294$	−0.905968	−0.0528371
$295$	5.76680	0.335756
$296$	−1.52894	−0.0888676
$297$	0	0
$298$	−1.32881	−0.0769761
$299$	0.283109	0.0163726
$300$	6.98943	0.403535
$301$	−42.1499	−2.42948
$302$	0.821842	0.0472917
$303$	−17.9833	−1.03312
$304$	15.2673	0.875640
$305$	0.416362	0.0238408
$306$	−0.0682851	−0.00390360
$307$	7.93592	0.452927	0.226463	−	0.974020i	$-0.427284\pi$
$307$	7.93592	0.452927	0.226463	+	0.974020i	$0.427284\pi$
$308$	0	0
$309$	8.16651	0.464577
$310$	0.0720310	0.00409109
$311$	−31.3996	−1.78051	−0.890253	−	0.455467i	$-0.849472\pi$
$311$	−31.3996	−1.78051	−0.890253	+	0.455467i	$0.849472\pi$
$312$	−0.0827926	−0.00468721
$313$	−8.87311	−0.501538	−0.250769	−	0.968047i	$-0.580683\pi$
$313$	−8.87311	−0.501538	−0.250769	+	0.968047i	$0.580683\pi$
$314$	0.638804	0.0360498
$315$	5.50843	0.310365
$316$	23.2842	1.30984
$317$	−13.1718	−0.739799	−0.369900	−	0.929072i	$-0.620608\pi$
$317$	−13.1718	−0.739799	−0.369900	+	0.929072i	$0.620608\pi$
$318$	0.168168	0.00943039
$319$	0	0
$320$	9.65189	0.539557
$321$	17.1609	0.957829
$322$	−0.286792	−0.0159823
$323$	3.84369	0.213869
$324$	−1.99534	−0.110852
$325$	1.06301	0.0589652
$326$	1.15627	0.0640398
$327$	12.0798	0.668013
$328$	2.41775	0.133498
$329$	−44.4383	−2.44996
$330$	0	0
$331$	−31.2764	−1.71911	−0.859554	−	0.511046i	$-0.829258\pi$
$331$	−31.2764	−1.71911	−0.859554	+	0.511046i	$0.829258\pi$
$332$	6.30982	0.346297
$333$	−5.60415	−0.307106
$334$	0.344394	0.0188444
$335$	12.8231	0.700599
$336$	−17.8819	−0.975537
$337$	−2.95914	−0.161194	−0.0805972	−	0.996747i	$-0.525683\pi$
$337$	−2.95914	−0.161194	−0.0805972	+	0.996747i	$0.525683\pi$
$338$	0.881418	0.0479428
$339$	−10.6036	−0.575911
$340$	2.44143	0.132405
$341$	0	0
$342$	−0.262467	−0.0141926
$343$	−28.2156	−1.52350
$344$	2.55433	0.137720
$345$	1.14148	0.0614555
$346$	0.703949	0.0378445
$347$	2.05256	0.110187	0.0550935	−	0.998481i	$-0.482454\pi$
$347$	2.05256	0.110187	0.0550935	+	0.998481i	$0.482454\pi$
$348$	−10.1071	−0.541797
$349$	−15.2436	−0.815971	−0.407986	−	0.912988i	$-0.633769\pi$
$349$	−15.2436	−0.815971	−0.407986	+	0.912988i	$0.633769\pi$
$350$	−1.07684	−0.0575594
$351$	−0.303468	−0.0161979
$352$	0	0
$353$	−6.51264	−0.346633	−0.173316	−	0.984866i	$-0.555448\pi$
$353$	−6.51264	−0.346633	−0.173316	+	0.984866i	$0.555448\pi$
$354$	0.321834	0.0171053
$355$	3.34174	0.177361
$356$	10.8788	0.576573
$357$	−4.50194	−0.238268
$358$	1.06360	0.0562130
$359$	19.0229	1.00399	0.501994	−	0.864871i	$-0.332600\pi$
$359$	19.0229	1.00399	0.501994	+	0.864871i	$0.332600\pi$
$360$	−0.333816	−0.0175937
$361$	−4.22606	−0.222424
$362$	1.13731	0.0597759
$363$	0	0
$364$	−2.72601	−0.142882
$365$	11.2516	0.588934
$366$	0.0232364	0.00121459
$367$	21.4149	1.11785	0.558924	−	0.829219i	$-0.311214\pi$
$367$	21.4149	1.11785	0.558924	+	0.829219i	$0.311214\pi$
$368$	−3.70558	−0.193167
$369$	8.86200	0.461337
$370$	−0.468235	−0.0243424
$371$	11.0871	0.575612
$372$	−1.72021	−0.0891888
$373$	−0.300687	−0.0155690	−0.00778448	−	0.999970i	$-0.502478\pi$
$373$	−0.300687	−0.0155690	−0.00778448	+	0.999970i	$0.502478\pi$
$374$	0	0
$375$	10.4039	0.537253
$376$	2.69301	0.138881
$377$	−1.53717	−0.0791684
$378$	0.307415	0.0158117
$379$	−27.5085	−1.41302	−0.706509	−	0.707704i	$-0.749731\pi$
$379$	−27.5085	−1.41302	−0.706509	+	0.707704i	$0.749731\pi$
$380$	9.38410	0.481394
$381$	−10.1554	−0.520275
$382$	1.05803	0.0541337
$383$	−32.7958	−1.67579	−0.837894	−	0.545833i	$-0.816213\pi$
$383$	−32.7958	−1.67579	−0.837894	+	0.545833i	$0.816213\pi$
$384$	2.17240	0.110860
$385$	0	0
$386$	−0.760160	−0.0386911
$387$	9.36261	0.475928
$388$	−26.1144	−1.32576
$389$	−13.0883	−0.663606	−0.331803	−	0.943349i	$-0.607657\pi$
$389$	−13.0883	−0.663606	−0.331803	+	0.943349i	$0.607657\pi$
$390$	−0.0253552	−0.00128391
$391$	−0.932914	−0.0471795
$392$	3.61965	0.182820
$393$	17.4368	0.879569
$394$	−1.13980	−0.0574222
$395$	14.2782	0.718413
$396$	0	0
$397$	24.8906	1.24922	0.624612	−	0.780935i	$-0.285257\pi$
$397$	24.8906	1.24922	0.624612	+	0.780935i	$0.285257\pi$
$398$	0.695964	0.0348855
$399$	−17.3040	−0.866286
$400$	−13.9136	−0.695680
$401$	6.92184	0.345660	0.172830	−	0.984952i	$-0.444709\pi$
$401$	6.92184	0.345660	0.172830	+	0.984952i	$0.444709\pi$
$402$	0.715631	0.0356924
$403$	−0.261624	−0.0130324
$404$	35.8828	1.78524
$405$	−1.22357	−0.0607996
$406$	1.55717	0.0772810
$407$	0	0
$408$	0.272822	0.0135067
$409$	31.8767	1.57620	0.788100	−	0.615547i	$-0.211065\pi$
$409$	31.8767	1.57620	0.788100	+	0.615547i	$0.211065\pi$
$410$	0.740433	0.0365674
$411$	−11.4622	−0.565387
$412$	−16.2949	−0.802794
$413$	21.2181	1.04407
$414$	0.0637041	0.00313089
$415$	3.86927	0.189935
$416$	0.247895	0.0121541
$417$	−6.34253	−0.310595
$418$	0	0
$419$	22.0040	1.07497	0.537483	−	0.843274i	$-0.319375\pi$
$419$	22.0040	1.07497	0.537483	+	0.843274i	$0.319375\pi$
$420$	−10.9912	−0.536314
$421$	−11.0872	−0.540359	−0.270179	−	0.962810i	$-0.587083\pi$
$421$	−11.0872	−0.540359	−0.270179	+	0.962810i	$0.587083\pi$
$422$	0.776017	0.0377759
$423$	9.87093	0.479941
$424$	−0.671887	−0.0326297
$425$	−3.50288	−0.169915
$426$	0.186496	0.00903578
$427$	1.53194	0.0741359
$428$	−34.2418	−1.65514
$429$	0	0
$430$	0.782260	0.0377239
$431$	−32.9627	−1.58776	−0.793879	−	0.608076i	$-0.791941\pi$
$431$	−32.9627	−1.58776	−0.793879	+	0.608076i	$0.791941\pi$
$432$	3.97204	0.191105
$433$	24.2855	1.16709	0.583543	−	0.812082i	$-0.301666\pi$
$433$	24.2855	1.16709	0.583543	+	0.812082i	$0.301666\pi$
$434$	0.265027	0.0127217
$435$	−6.19781	−0.297162
$436$	−24.1032	−1.15433
$437$	−3.58583	−0.171534
$438$	0.627929	0.0300036
$439$	20.4906	0.977965	0.488982	−	0.872294i	$-0.337368\pi$
$439$	20.4906	0.977965	0.488982	+	0.872294i	$0.337368\pi$
$440$	0	0
$441$	13.2674	0.631783
$442$	0.0207223	0.000985660 0
$443$	−26.1334	−1.24164	−0.620818	−	0.783954i	$-0.713200\pi$
$443$	−26.1334	−1.24164	−0.620818	+	0.783954i	$0.713200\pi$
$444$	11.1822	0.530683
$445$	6.67100	0.316236
$446$	−1.42277	−0.0673703
$447$	19.4598	0.920416
$448$	35.5127	1.67782
$449$	−23.4221	−1.10536	−0.552680	−	0.833394i	$-0.686395\pi$
$449$	−23.4221	−1.10536	−0.552680	+	0.833394i	$0.686395\pi$
$450$	0.239194	0.0112757
$451$	0	0
$452$	21.1579	0.995182
$453$	−12.0355	−0.565475
$454$	−0.181878	−0.00853597
$455$	−1.67163	−0.0783672
$456$	1.04864	0.0491072
$457$	19.6103	0.917332	0.458666	−	0.888609i	$-0.348328\pi$
$457$	19.6103	0.917332	0.458666	+	0.888609i	$0.348328\pi$
$458$	−0.612000	−0.0285969
$459$	1.00000	0.0466760
$460$	−2.27765	−0.106196
$461$	−39.6691	−1.84758	−0.923788	−	0.382905i	$-0.874924\pi$
$461$	−39.6691	−1.84758	−0.923788	+	0.382905i	$0.874924\pi$
$462$	0	0
$463$	−17.0875	−0.794125	−0.397062	−	0.917792i	$-0.629970\pi$
$463$	−17.0875	−0.794125	−0.397062	+	0.917792i	$0.629970\pi$
$464$	20.1198	0.934040
$465$	−1.05486	−0.0489178
$466$	−1.15978	−0.0537259
$467$	18.5882	0.860161	0.430081	−	0.902791i	$-0.358485\pi$
$467$	18.5882	0.860161	0.430081	+	0.902791i	$0.358485\pi$
$468$	0.605520	0.0279902
$469$	47.1805	2.17859
$470$	0.824731	0.0380420
$471$	−9.35496	−0.431054
$472$	−1.28584	−0.0591855
$473$	0	0
$474$	0.796838	0.0366000
$475$	−13.4640	−0.617770
$476$	8.98288	0.411730
$477$	−2.46273	−0.112761
$478$	−0.379882	−0.0173754
$479$	9.00176	0.411301	0.205651	−	0.978625i	$-0.434069\pi$
$479$	9.00176	0.411301	0.205651	+	0.978625i	$0.434069\pi$
$480$	0.999503	0.0456209
$481$	1.70068	0.0775443
$482$	1.35031	0.0615051
$483$	4.19992	0.191103
$484$	0	0
$485$	−16.0137	−0.727146
$486$	−0.0682851	−0.00309747
$487$	32.3438	1.46564	0.732818	−	0.680424i	$-0.238204\pi$
$487$	32.3438	1.46564	0.732818	+	0.680424i	$0.238204\pi$
$488$	−0.0928373	−0.00420255
$489$	−16.9330	−0.765735
$490$	1.10851	0.0500776
$491$	−8.75735	−0.395213	−0.197607	−	0.980281i	$-0.563317\pi$
$491$	−8.75735	−0.395213	−0.197607	+	0.980281i	$0.563317\pi$
$492$	−17.6827	−0.797196
$493$	5.06536	0.228132
$494$	0.0796501	0.00358363
$495$	0	0
$496$	3.42436	0.153758
$497$	12.2954	0.551526
$498$	0.215937	0.00967636
$499$	42.8857	1.91983	0.959914	−	0.280294i	$-0.0904319\pi$
$499$	42.8857	1.91983	0.959914	+	0.280294i	$0.0904319\pi$
$500$	−20.7592	−0.928380
$501$	−5.04347	−0.225326
$502$	−0.0371018	−0.00165593
$503$	13.4685	0.600533	0.300266	−	0.953855i	$-0.402924\pi$
$503$	13.4685	0.600533	0.300266	+	0.953855i	$0.402924\pi$
$504$	−1.22823	−0.0547096
$505$	22.0039	0.979159
$506$	0	0
$507$	−12.9079	−0.573260
$508$	20.2634	0.899041
$509$	−26.2686	−1.16433	−0.582167	−	0.813069i	$-0.697795\pi$
$509$	−26.2686	−1.16433	−0.582167	+	0.813069i	$0.697795\pi$
$510$	0.0835515	0.00369972
$511$	41.3984	1.83136
$512$	−5.41199	−0.239178
$513$	3.84369	0.169703
$514$	−1.39367	−0.0614720
$515$	−9.99229	−0.440313
$516$	−18.6816	−0.822410
$517$	0	0
$518$	−1.72280	−0.0756956
$519$	−10.3090	−0.452514
$520$	0.101302	0.00444241
$521$	20.2512	0.887221	0.443611	−	0.896220i	$-0.353697\pi$
$521$	20.2512	0.887221	0.443611	+	0.896220i	$0.353697\pi$
$522$	−0.345889	−0.0151391
$523$	24.7202	1.08094	0.540469	−	0.841364i	$-0.318247\pi$
$523$	24.7202	1.08094	0.540469	+	0.841364i	$0.318247\pi$
$524$	−34.7922	−1.51991
$525$	15.7697	0.688248
$526$	0.425278	0.0185430
$527$	0.862115	0.0375543
$528$	0	0
$529$	−22.1297	−0.962160
$530$	−0.205765	−0.00893786
$531$	−4.71310	−0.204531
$532$	34.5274	1.49695
$533$	−2.68933	−0.116488
$534$	0.372296	0.0161108
$535$	−20.9976	−0.907803
$536$	−2.85919	−0.123498
$537$	−15.5759	−0.672148
$538$	−0.237976	−0.0102599
$539$	0	0
$540$	2.44143	0.105063
$541$	−9.50205	−0.408525	−0.204263	−	0.978916i	$-0.565480\pi$
$541$	−9.50205	−0.408525	−0.204263	+	0.978916i	$0.565480\pi$
$542$	1.02594	0.0440679
$543$	−16.6554	−0.714751
$544$	−0.816875	−0.0350232
$545$	−14.7804	−0.633124
$546$	−0.0932905	−0.00399247
$547$	16.1630	0.691080	0.345540	−	0.938404i	$-0.387696\pi$
$547$	16.1630	0.691080	0.345540	+	0.938404i	$0.387696\pi$
$548$	22.8709	0.976997
$549$	−0.340285	−0.0145230
$550$	0	0
$551$	19.4697	0.829435
$552$	−0.254520	−0.0108331
$553$	52.5344	2.23399
$554$	0.828316	0.0351918
$555$	6.85707	0.291066
$556$	12.6555	0.536712
$557$	−3.91904	−0.166055	−0.0830275	−	0.996547i	$-0.526459\pi$
$557$	−3.91904	−0.166055	−0.0830275	+	0.996547i	$0.526459\pi$
$558$	−0.0588696	−0.00249215
$559$	−2.84125	−0.120172
$560$	21.8797	0.924587
$561$	0	0
$562$	0.904678	0.0381616
$563$	−20.4425	−0.861551	−0.430775	−	0.902459i	$-0.641760\pi$
$563$	−20.4425	−0.861551	−0.430775	+	0.902459i	$0.641760\pi$
$564$	−19.6958	−0.829344
$565$	12.9743	0.545833
$566$	−1.17590	−0.0494268
$567$	−4.50194	−0.189064
$568$	−0.745116	−0.0312644
$569$	−29.9102	−1.25390	−0.626951	−	0.779058i	$-0.715698\pi$
$569$	−29.9102	−1.25390	−0.626951	+	0.779058i	$0.715698\pi$
$570$	0.321146	0.0134513
$571$	9.68322	0.405230	0.202615	−	0.979258i	$-0.435056\pi$
$571$	9.68322	0.405230	0.202615	+	0.979258i	$0.435056\pi$
$572$	0	0
$573$	−15.4943	−0.647286
$574$	2.72431	0.113711
$575$	3.26789	0.136280
$576$	−7.88831	−0.328680
$577$	7.78640	0.324152	0.162076	−	0.986778i	$-0.448181\pi$
$577$	7.78640	0.324152	0.162076	+	0.986778i	$0.448181\pi$
$578$	−0.0682851	−0.00284029
$579$	11.1322	0.462637
$580$	12.3667	0.513500
$581$	14.2364	0.590626
$582$	−0.893696	−0.0370449
$583$	0	0
$584$	−2.50879	−0.103814
$585$	0.371313	0.0153519
$586$	0.941933	0.0389109
$587$	−41.6767	−1.72018	−0.860091	−	0.510141i	$-0.829593\pi$
$587$	−41.6767	−1.72018	−0.860091	+	0.510141i	$0.829593\pi$
$588$	−26.4730	−1.09173
$589$	3.31370	0.136539
$590$	−0.393787	−0.0162119
$591$	16.6917	0.686607
$592$	−22.2599	−0.914878
$593$	18.3334	0.752861	0.376431	−	0.926445i	$-0.377151\pi$
$593$	18.3334	0.752861	0.376431	+	0.926445i	$0.377151\pi$
$594$	0	0
$595$	5.50843	0.225824
$596$	−38.8288	−1.59049
$597$	−10.1920	−0.417132
$598$	−0.0193321	−0.000790550 0
$599$	3.75500	0.153425	0.0767126	−	0.997053i	$-0.475558\pi$
$599$	3.75500	0.153425	0.0767126	+	0.997053i	$0.475558\pi$
$600$	−0.955663	−0.0390148
$601$	−11.2072	−0.457151	−0.228576	−	0.973526i	$-0.573407\pi$
$601$	−11.2072	−0.457151	−0.228576	+	0.973526i	$0.573407\pi$
$602$	2.87821	0.117307
$603$	−10.4800	−0.426781
$604$	24.0148	0.977148
$605$	0	0
$606$	1.22799	0.0498839
$607$	3.09304	0.125543	0.0627713	−	0.998028i	$-0.480006\pi$
$607$	3.09304	0.125543	0.0627713	+	0.998028i	$0.480006\pi$
$608$	−3.13981	−0.127336
$609$	−22.8039	−0.924062
$610$	−0.0284313	−0.00115115
$611$	−2.99551	−0.121185
$612$	−1.99534	−0.0806567
$613$	−18.3294	−0.740317	−0.370159	−	0.928969i	$-0.620697\pi$
$613$	−18.3294	−0.740317	−0.370159	+	0.928969i	$0.620697\pi$
$614$	−0.541905	−0.0218695
$615$	−10.8433	−0.437243
$616$	0	0
$617$	35.3139	1.42168	0.710842	−	0.703351i	$-0.248314\pi$
$617$	35.3139	1.42168	0.710842	+	0.703351i	$0.248314\pi$
$618$	−0.557651	−0.0224320
$619$	8.31687	0.334283	0.167142	−	0.985933i	$-0.446546\pi$
$619$	8.31687	0.334283	0.167142	+	0.985933i	$0.446546\pi$
$620$	2.10480	0.0845306
$621$	−0.932914	−0.0374366
$622$	2.14412	0.0859715
$623$	24.5450	0.983373
$624$	−1.20539	−0.0482541
$625$	4.78456	0.191383
$626$	0.605901	0.0242167
$627$	0	0
$628$	18.6663	0.744866
$629$	−5.60415	−0.223452
$630$	−0.376144	−0.0149859
$631$	13.6522	0.543487	0.271744	−	0.962370i	$-0.412400\pi$
$631$	13.6522	0.543487	0.271744	+	0.962370i	$0.412400\pi$
$632$	−3.18364	−0.126638
$633$	−11.3644	−0.451693
$634$	0.899435	0.0357211
$635$	12.4258	0.493102
$636$	4.91398	0.194852
$637$	−4.02624	−0.159525
$638$	0	0
$639$	−2.73114	−0.108042
$640$	−2.65809	−0.105070
$641$	−45.6131	−1.80161	−0.900805	−	0.434224i	$-0.857023\pi$
$641$	−45.6131	−1.80161	−0.900805	+	0.434224i	$0.857023\pi$
$642$	−1.17183	−0.0462486
$643$	4.85972	0.191649	0.0958244	−	0.995398i	$-0.469451\pi$
$643$	4.85972	0.191649	0.0958244	+	0.995398i	$0.469451\pi$
$644$	−8.38026	−0.330229
$645$	−11.4558	−0.451071
$646$	−0.262467	−0.0103266
$647$	−39.6287	−1.55796	−0.778982	−	0.627046i	$-0.784264\pi$
$647$	−39.6287	−1.55796	−0.778982	+	0.627046i	$0.784264\pi$
$648$	0.272822	0.0107175
$649$	0	0
$650$	−0.0725878	−0.00284713
$651$	−3.88119	−0.152116
$652$	33.7870	1.32320
$653$	−33.7605	−1.32115	−0.660575	−	0.750760i	$-0.729687\pi$
$653$	−33.7605	−1.32115	−0.660575	+	0.750760i	$0.729687\pi$
$654$	−0.824868	−0.0322549
$655$	−21.3351	−0.833631
$656$	35.2002	1.37434
$657$	−9.19569	−0.358758
$658$	3.03447	0.118296
$659$	9.48851	0.369620	0.184810	−	0.982774i	$-0.440833\pi$
$659$	9.48851	0.369620	0.184810	+	0.982774i	$0.440833\pi$
$660$	0	0
$661$	−44.4936	−1.73060	−0.865299	−	0.501256i	$-0.832872\pi$
$661$	−44.4936	−1.73060	−0.865299	+	0.501256i	$0.832872\pi$
$662$	2.13571	0.0830068
$663$	−0.303468	−0.0117857
$664$	−0.862741	−0.0334808
$665$	21.1727	0.821041
$666$	0.382680	0.0148286
$667$	−4.72555	−0.182974
$668$	10.0634	0.389366
$669$	20.8358	0.805558
$670$	−0.875624	−0.0338283
$671$	0	0
$672$	3.67752	0.141863
$673$	−4.10794	−0.158349	−0.0791747	−	0.996861i	$-0.525229\pi$
$673$	−4.10794	−0.158349	−0.0791747	+	0.996861i	$0.525229\pi$
$674$	0.202065	0.00778325
$675$	−3.50288	−0.134826
$676$	25.7556	0.990601
$677$	−32.2597	−1.23984	−0.619921	−	0.784665i	$-0.712835\pi$
$677$	−32.2597	−1.23984	−0.619921	+	0.784665i	$0.712835\pi$
$678$	0.724071	0.0278078
$679$	−58.9201	−2.26115
$680$	−0.333816	−0.0128013
$681$	2.66351	0.102066
$682$	0	0
$683$	15.5608	0.595418	0.297709	−	0.954657i	$-0.403777\pi$
$683$	15.5608	0.595418	0.297709	+	0.954657i	$0.403777\pi$
$684$	−7.66945	−0.293249
$685$	14.0248	0.535859
$686$	1.92671	0.0735620
$687$	8.96242	0.341938
$688$	37.1887	1.41781
$689$	0.747359	0.0284721
$690$	−0.0779464	−0.00296737
$691$	40.1977	1.52919	0.764596	−	0.644510i	$-0.222939\pi$
$691$	40.1977	1.52919	0.764596	+	0.644510i	$0.222939\pi$
$692$	20.5699	0.781949
$693$	0	0
$694$	−0.140159	−0.00532036
$695$	7.76053	0.294374
$696$	1.38194	0.0523823
$697$	8.86200	0.335672
$698$	1.04091	0.0393991
$699$	16.9844	0.642410
$700$	−31.4660	−1.18930
$701$	−28.9131	−1.09203	−0.546017	−	0.837774i	$-0.683857\pi$
$701$	−28.9131	−1.09203	−0.546017	+	0.837774i	$0.683857\pi$
$702$	0.0207223	0.000782113 0
$703$	−21.5406	−0.812420
$704$	0	0
$705$	−12.0778	−0.454875
$706$	0.444716	0.0167371
$707$	80.9599	3.04481
$708$	9.40422	0.353432
$709$	5.46538	0.205257	0.102628	−	0.994720i	$-0.467275\pi$
$709$	5.46538	0.205257	0.102628	+	0.994720i	$0.467275\pi$
$710$	−0.228191	−0.00856386
$711$	−11.6693	−0.437632
$712$	−1.48745	−0.0557445
$713$	−0.804280	−0.0301205
$714$	0.307415	0.0115047
$715$	0	0
$716$	31.0791	1.16148
$717$	5.56317	0.207761
$718$	−1.29898	−0.0484775
$719$	29.6370	1.10527	0.552637	−	0.833422i	$-0.313622\pi$
$719$	29.6370	1.10527	0.552637	+	0.833422i	$0.313622\pi$
$720$	−4.86007	−0.181124
$721$	−36.7651	−1.36920
$722$	0.288577	0.0107397
$723$	−19.7746	−0.735427
$724$	33.2331	1.23510
$725$	−17.7433	−0.658971
$726$	0	0
$727$	9.96682	0.369649	0.184824	−	0.982772i	$-0.440828\pi$
$727$	9.96682	0.369649	0.184824	+	0.982772i	$0.440828\pi$
$728$	0.372727	0.0138142
$729$	1.00000	0.0370370
$730$	−0.768314	−0.0284366
$731$	9.36261	0.346289
$732$	0.678984	0.0250960
$733$	35.8520	1.32422	0.662112	−	0.749405i	$-0.269660\pi$
$733$	35.8520	1.32422	0.662112	+	0.749405i	$0.269660\pi$
$734$	−1.46232	−0.0539751
$735$	−16.2336	−0.598786
$736$	0.762075	0.0280905
$737$	0	0
$738$	−0.605142	−0.0222756
$739$	18.0803	0.665096	0.332548	−	0.943086i	$-0.392092\pi$
$739$	18.0803	0.665096	0.332548	+	0.943086i	$0.392092\pi$
$740$	−13.6822	−0.502966
$741$	−1.16643	−0.0428500
$742$	−0.757081	−0.0277933
$743$	6.54903	0.240260	0.120130	−	0.992758i	$-0.461669\pi$
$743$	6.54903	0.240260	0.120130	+	0.992758i	$0.461669\pi$
$744$	0.235204	0.00862300
$745$	−23.8104	−0.872345
$746$	0.0205324	0.000751745 0
$747$	−3.16228	−0.115702
$748$	0	0
$749$	−77.2573	−2.82292
$750$	−0.710428	−0.0259412
$751$	−21.2947	−0.777054	−0.388527	−	0.921437i	$-0.627016\pi$
$751$	−21.2947	−0.777054	−0.388527	+	0.921437i	$0.627016\pi$
$752$	39.2078	1.42976
$753$	0.543336	0.0198003
$754$	0.104966	0.00382263
$755$	14.7262	0.535941
$756$	8.98288	0.326704
$757$	48.4623	1.76139	0.880697	−	0.473680i	$-0.157075\pi$
$757$	48.4623	1.76139	0.880697	+	0.473680i	$0.157075\pi$
$758$	1.87842	0.0682273
$759$	0	0
$760$	−1.28309	−0.0465424
$761$	−32.4506	−1.17633	−0.588167	−	0.808739i	$-0.700150\pi$
$761$	−32.4506	−1.17633	−0.588167	+	0.808739i	$0.700150\pi$
$762$	0.693460	0.0251214
$763$	−54.3824	−1.96877
$764$	30.9164	1.11852
$765$	−1.22357	−0.0442382
$766$	2.23947	0.0809152
$767$	1.43027	0.0516442
$768$	15.6283	0.563937
$769$	−39.6970	−1.43151	−0.715755	−	0.698351i	$-0.753917\pi$
$769$	−39.6970	−1.43151	−0.715755	+	0.698351i	$0.753917\pi$
$770$	0	0
$771$	20.4095	0.735031
$772$	−22.2124	−0.799442
$773$	9.88304	0.355468	0.177734	−	0.984079i	$-0.443123\pi$
$773$	9.88304	0.355468	0.177734	+	0.984079i	$0.443123\pi$
$774$	−0.639327	−0.0229801
$775$	−3.01989	−0.108478
$776$	3.57062	0.128178
$777$	25.2295	0.905105
$778$	0.893739	0.0320421
$779$	34.0628	1.22042
$780$	−0.740895	−0.0265283
$781$	0	0
$782$	0.0637041	0.00227806
$783$	5.06536	0.181021
$784$	52.6989	1.88210
$785$	11.4464	0.408541
$786$	−1.19067	−0.0424699
$787$	8.96065	0.319413	0.159706	−	0.987165i	$-0.448945\pi$
$787$	8.96065	0.319413	0.159706	+	0.987165i	$0.448945\pi$
$788$	−33.3057	−1.18647
$789$	−6.22797	−0.221722
$790$	−0.974986	−0.0346884
$791$	47.7370	1.69733
$792$	0	0
$793$	0.103265	0.00366707
$794$	−1.69966	−0.0603186
$795$	3.01332	0.106871
$796$	20.3365	0.720810
$797$	−28.8535	−1.02204	−0.511021	−	0.859568i	$-0.670733\pi$
$797$	−28.8535	−1.02204	−0.511021	+	0.859568i	$0.670733\pi$
$798$	1.18161	0.0418285
$799$	9.87093	0.349208
$800$	2.86142	0.101166
$801$	−5.45209	−0.192640
$802$	−0.472658	−0.0166902
$803$	0	0
$804$	20.9112	0.737482
$805$	−5.13889	−0.181122
$806$	0.0178650	0.000629268 0
$807$	3.48504	0.122679
$808$	−4.90625	−0.172601
$809$	−8.81660	−0.309975	−0.154988	−	0.987916i	$-0.549534\pi$
$809$	−8.81660	−0.309975	−0.154988	+	0.987916i	$0.549534\pi$
$810$	0.0835515	0.00293570
$811$	15.6061	0.548006	0.274003	−	0.961729i	$-0.411652\pi$
$811$	15.6061	0.548006	0.274003	+	0.961729i	$0.411652\pi$
$812$	45.5015	1.59679
$813$	−15.0244	−0.526928
$814$	0	0
$815$	20.7186	0.725742
$816$	3.97204	0.139049
$817$	35.9869	1.25902
$818$	−2.17670	−0.0761066
$819$	1.36619	0.0477386
$820$	21.6360	0.755561
$821$	−31.2527	−1.09073	−0.545364	−	0.838200i	$-0.683608\pi$
$821$	−31.2527	−1.09073	−0.545364	+	0.838200i	$0.683608\pi$
$822$	0.782696	0.0272997
$823$	38.5625	1.34420	0.672102	−	0.740459i	$-0.265392\pi$
$823$	38.5625	1.34420	0.672102	+	0.740459i	$0.265392\pi$
$824$	2.22800	0.0776162
$825$	0	0
$826$	−1.44888	−0.0504129
$827$	43.0060	1.49546	0.747732	−	0.664000i	$-0.231143\pi$
$827$	43.0060	1.49546	0.747732	+	0.664000i	$0.231143\pi$
$828$	1.86148	0.0646909
$829$	3.11674	0.108249	0.0541245	−	0.998534i	$-0.482763\pi$
$829$	3.11674	0.108249	0.0541245	+	0.998534i	$0.482763\pi$
$830$	−0.264214	−0.00917099
$831$	−12.1303	−0.420794
$832$	2.39385	0.0829917
$833$	13.2674	0.459689
$834$	0.433101	0.0149970
$835$	6.17104	0.213558
$836$	0	0
$837$	0.862115	0.0297991
$838$	−1.50255	−0.0519046
$839$	9.05859	0.312737	0.156369	−	0.987699i	$-0.450021\pi$
$839$	9.05859	0.312737	0.156369	+	0.987699i	$0.450021\pi$
$840$	1.50282	0.0518522
$841$	−3.34214	−0.115246
$842$	0.757093	0.0260911
$843$	−13.2485	−0.456304
$844$	22.6757	0.780531
$845$	15.7937	0.543320
$846$	−0.674037	−0.0231739
$847$	0	0
$848$	−9.78208	−0.335918
$849$	17.2205	0.591005
$850$	0.239194	0.00820430
$851$	5.22820	0.179220
$852$	5.44955	0.186699
$853$	24.4010	0.835474	0.417737	−	0.908568i	$-0.362823\pi$
$853$	24.4010	0.835474	0.417737	+	0.908568i	$0.362823\pi$
$854$	−0.104609	−0.00357964
$855$	−4.70302	−0.160840
$856$	4.68187	0.160023
$857$	20.6633	0.705846	0.352923	−	0.935652i	$-0.385188\pi$
$857$	20.6633	0.705846	0.352923	+	0.935652i	$0.385188\pi$
$858$	0	0
$859$	−10.1686	−0.346949	−0.173474	−	0.984838i	$-0.555499\pi$
$859$	−10.1686	−0.346949	−0.173474	+	0.984838i	$0.555499\pi$
$860$	22.8582	0.779457
$861$	−39.8962	−1.35966
$862$	2.25086	0.0766646
$863$	−32.4326	−1.10402	−0.552009	−	0.833838i	$-0.686138\pi$
$863$	−32.4326	−1.10402	−0.552009	+	0.833838i	$0.686138\pi$
$864$	−0.816875	−0.0277907
$865$	12.6137	0.428880
$866$	−1.65834	−0.0563526
$867$	1.00000	0.0339618
$868$	7.74428	0.262858
$869$	0	0
$870$	0.423218	0.0143484
$871$	3.18035	0.107762
$872$	3.29563	0.111604
$873$	13.0877	0.442952
$874$	0.244859	0.00828248
$875$	−46.8375	−1.58340
$876$	18.3485	0.619939
$877$	−52.7922	−1.78267	−0.891333	−	0.453348i	$-0.850229\pi$
$877$	−52.7922	−1.78267	−0.891333	+	0.453348i	$0.850229\pi$
$878$	−1.39921	−0.0472209
$879$	−13.7941	−0.465264
$880$	0	0
$881$	−5.65290	−0.190451	−0.0952255	−	0.995456i	$-0.530357\pi$
$881$	−5.65290	−0.190451	−0.0952255	+	0.995456i	$0.530357\pi$
$882$	−0.905968	−0.0305055
$883$	−19.4379	−0.654138	−0.327069	−	0.945001i	$-0.606061\pi$
$883$	−19.4379	−0.654138	−0.327069	+	0.945001i	$0.606061\pi$
$884$	0.605520	0.0203658
$885$	5.76680	0.193849
$886$	1.78452	0.0599523
$887$	−38.2753	−1.28516	−0.642580	−	0.766219i	$-0.722136\pi$
$887$	−38.2753	−1.28516	−0.642580	+	0.766219i	$0.722136\pi$
$888$	−1.52894	−0.0513077
$889$	45.7188	1.53336
$890$	−0.455530	−0.0152694
$891$	0	0
$892$	−41.5744	−1.39201
$893$	37.9408	1.26964
$894$	−1.32881	−0.0444422
$895$	19.0581	0.637044
$896$	−9.78003	−0.326728
$897$	0.283109	0.00945274
$898$	1.59938	0.0533721
$899$	4.36692	0.145645
$900$	6.98943	0.232981
$901$	−2.46273	−0.0820455
$902$	0	0
$903$	−42.1499	−1.40266
$904$	−2.89291	−0.0962167
$905$	20.3790	0.677421
$906$	0.821842	0.0273039
$907$	11.6718	0.387557	0.193778	−	0.981045i	$-0.437926\pi$
$907$	11.6718	0.387557	0.193778	+	0.981045i	$0.437926\pi$
$908$	−5.31461	−0.176371
$909$	−17.9833	−0.596470
$910$	0.114147	0.00378395
$911$	−24.9726	−0.827378	−0.413689	−	0.910418i	$-0.635760\pi$
$911$	−24.9726	−0.827378	−0.413689	+	0.910418i	$0.635760\pi$
$912$	15.2673	0.505551
$913$	0	0
$914$	−1.33909	−0.0442932
$915$	0.416362	0.0137645
$916$	−17.8830	−0.590873
$917$	−78.4993	−2.59227
$918$	−0.0682851	−0.00225374
$919$	−17.9053	−0.590641	−0.295321	−	0.955398i	$-0.595426\pi$
$919$	−17.9053	−0.590641	−0.295321	+	0.955398i	$0.595426\pi$
$920$	0.311422	0.0102673
$921$	7.93592	0.261497
$922$	2.70881	0.0892099
$923$	0.828813	0.0272807
$924$	0	0
$925$	19.6307	0.645453
$926$	1.16682	0.0383442
$927$	8.16651	0.268223
$928$	−4.13777	−0.135829
$929$	−51.6852	−1.69574	−0.847868	−	0.530208i	$-0.822114\pi$
$929$	−51.6852	−1.69574	−0.847868	+	0.530208i	$0.822114\pi$
$930$	0.0720310	0.00236199
$931$	50.9959	1.67132
$932$	−33.8896	−1.11009
$933$	−31.3996	−1.02798
$934$	−1.26930	−0.0415328
$935$	0	0
$936$	−0.0827926	−0.00270616
$937$	−41.6852	−1.36179	−0.680897	−	0.732379i	$-0.738410\pi$
$937$	−41.6852	−1.36179	−0.680897	+	0.732379i	$0.738410\pi$
$938$	−3.22173	−0.105193
$939$	−8.87311	−0.289563
$940$	24.0992	0.786029
$941$	6.41178	0.209018	0.104509	−	0.994524i	$-0.466673\pi$
$941$	6.41178	0.209018	0.104509	+	0.994524i	$0.466673\pi$
$942$	0.638804	0.0208134
$943$	−8.26748	−0.269226
$944$	−18.7206	−0.609305
$945$	5.50843	0.179189
$946$	0	0
$947$	−28.6404	−0.930689	−0.465345	−	0.885130i	$-0.654070\pi$
$947$	−28.6404	−0.930689	−0.465345	+	0.885130i	$0.654070\pi$
$948$	23.2842	0.756234
$949$	2.79059	0.0905865
$950$	0.919389	0.0298289
$951$	−13.1718	−0.427123
$952$	−1.22823	−0.0398071
$953$	2.95711	0.0957901	0.0478950	−	0.998852i	$-0.484749\pi$
$953$	2.95711	0.0957901	0.0478950	+	0.998852i	$0.484749\pi$
$954$	0.168168	0.00544464
$955$	18.9584	0.613479
$956$	−11.1004	−0.359013
$957$	0	0
$958$	−0.614686	−0.0198596
$959$	51.6020	1.66632
$960$	9.65189	0.311513
$961$	−30.2568	−0.976024
$962$	−0.116131	−0.00374421
$963$	17.1609	0.553003
$964$	39.4571	1.27083
$965$	−13.6210	−0.438474
$966$	−0.286792	−0.00922738
$967$	24.0387	0.773032	0.386516	−	0.922283i	$-0.373678\pi$
$967$	24.0387	0.773032	0.386516	+	0.922283i	$0.373678\pi$
$968$	0	0
$969$	3.84369	0.123477
$970$	1.09350	0.0351101
$971$	23.3735	0.750091	0.375046	−	0.927006i	$-0.377627\pi$
$971$	23.3735	0.750091	0.375046	+	0.927006i	$0.377627\pi$
$972$	−1.99534	−0.0640005
$973$	28.5537	0.915389
$974$	−2.20860	−0.0707681
$975$	1.06301	0.0340436
$976$	−1.35163	−0.0432646
$977$	32.6871	1.04575	0.522877	−	0.852408i	$-0.324859\pi$
$977$	32.6871	1.04575	0.522877	+	0.852408i	$0.324859\pi$
$978$	1.15627	0.0369734
$979$	0	0
$980$	32.3915	1.03471
$981$	12.0798	0.385677
$982$	0.597996	0.0190828
$983$	−35.6745	−1.13784	−0.568920	−	0.822393i	$-0.692639\pi$
$983$	−35.6745	−1.13784	−0.568920	+	0.822393i	$0.692639\pi$
$984$	2.41775	0.0770750
$985$	−20.4235	−0.650747
$986$	−0.345889	−0.0110153
$987$	−44.4383	−1.41449
$988$	2.32743	0.0740454
$989$	−8.73451	−0.277741
$990$	0	0
$991$	10.1506	0.322444	0.161222	−	0.986918i	$-0.448456\pi$
$991$	10.1506	0.322444	0.161222	+	0.986918i	$0.448456\pi$
$992$	−0.704241	−0.0223597
$993$	−31.2764	−0.992527
$994$	−0.839595	−0.0266303
$995$	12.4707	0.395346
$996$	6.30982	0.199934
$997$	−32.9230	−1.04268	−0.521342	−	0.853348i	$-0.674568\pi$
$997$	−32.9230	−1.04268	−0.521342	+	0.853348i	$0.674568\pi$
$998$	−2.92846	−0.0926986
$999$	−5.60415	−0.177308

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6171.2.a.bk.1.7		12
11.5	even	5		561.2.m.d.256.3	yes	24
11.9	even	5		561.2.m.d.103.3	✓	24
11.10	odd	2		6171.2.a.bl.1.6		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
561.2.m.d.103.3	✓	24	11.9	even	5
561.2.m.d.256.3	yes	24	11.5	even	5
6171.2.a.bk.1.7		12	1.1	even	1	trivial
6171.2.a.bl.1.6		12	11.10	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6171 = 3 \cdot 11^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6171.a (trivial)

\(f(q)\)	\(=\)	\(q-0.0682851 q^{2} +1.00000 q^{3} -1.99534 q^{4} -1.22357 q^{5} -0.0682851 q^{6} -4.50194 q^{7} +0.272822 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.0682851 q^{2} +1.00000 q^{3} -1.99534 q^{4} -1.22357 q^{5} -0.0682851 q^{6} -4.50194 q^{7} +0.272822 q^{8} +1.00000 q^{9} +0.0835515 q^{10} -1.99534 q^{12} -0.303468 q^{13} +0.307415 q^{14} -1.22357 q^{15} +3.97204 q^{16} +1.00000 q^{17} -0.0682851 q^{18} +3.84369 q^{19} +2.44143 q^{20} -4.50194 q^{21} -0.932914 q^{23} +0.272822 q^{24} -3.50288 q^{25} +0.0207223 q^{26} +1.00000 q^{27} +8.98288 q^{28} +5.06536 q^{29} +0.0835515 q^{30} +0.862115 q^{31} -0.816875 q^{32} -0.0682851 q^{34} +5.50843 q^{35} -1.99534 q^{36} -5.60415 q^{37} -0.262467 q^{38} -0.303468 q^{39} -0.333816 q^{40} +8.86200 q^{41} +0.307415 q^{42} +9.36261 q^{43} -1.22357 q^{45} +0.0637041 q^{46} +9.87093 q^{47} +3.97204 q^{48} +13.2674 q^{49} +0.239194 q^{50} +1.00000 q^{51} +0.605520 q^{52} -2.46273 q^{53} -0.0682851 q^{54} -1.22823 q^{56} +3.84369 q^{57} -0.345889 q^{58} -4.71310 q^{59} +2.44143 q^{60} -0.340285 q^{61} -0.0588696 q^{62} -4.50194 q^{63} -7.88831 q^{64} +0.371313 q^{65} -10.4800 q^{67} -1.99534 q^{68} -0.932914 q^{69} -0.376144 q^{70} -2.73114 q^{71} +0.272822 q^{72} -9.19569 q^{73} +0.382680 q^{74} -3.50288 q^{75} -7.66945 q^{76} +0.0207223 q^{78} -11.6693 q^{79} -4.86007 q^{80} +1.00000 q^{81} -0.605142 q^{82} -3.16228 q^{83} +8.98288 q^{84} -1.22357 q^{85} -0.639327 q^{86} +5.06536 q^{87} -5.45209 q^{89} +0.0835515 q^{90} +1.36619 q^{91} +1.86148 q^{92} +0.862115 q^{93} -0.674037 q^{94} -4.70302 q^{95} -0.816875 q^{96} +13.0877 q^{97} -0.905968 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - q^{2} + 12 q^{3} + 9 q^{4} - 14 q^{5} - q^{6} - 5 q^{7} + 12 q^{9}+O(q^{10}) \) 12 * q - q^2 + 12 * q^3 + 9 * q^4 - 14 * q^5 - q^6 - 5 * q^7 + 12 * q^9	\( 12 q - q^{2} + 12 q^{3} + 9 q^{4} - 14 q^{5} - q^{6} - 5 q^{7} + 12 q^{9} + 8 q^{10} + 9 q^{12} - 7 q^{13} - 13 q^{14} - 14 q^{15} + 11 q^{16} + 12 q^{17} - q^{18} + 3 q^{19} - 36 q^{20} - 5 q^{21} - 39 q^{23} + 8 q^{25} - 25 q^{26} + 12 q^{27} - 9 q^{28} + 10 q^{29} + 8 q^{30} - 25 q^{31} - 11 q^{32} - q^{34} + 12 q^{35} + 9 q^{36} - 10 q^{37} - 34 q^{38} - 7 q^{39} - 13 q^{40} + 17 q^{41} - 13 q^{42} - 18 q^{43} - 14 q^{45} + 19 q^{46} - 38 q^{47} + 11 q^{48} - 15 q^{49} - 18 q^{50} + 12 q^{51} + 32 q^{52} - 40 q^{53} - q^{54} - 25 q^{56} + 3 q^{57} + 3 q^{58} - 18 q^{59} - 36 q^{60} - 22 q^{61} + 3 q^{62} - 5 q^{63} - 20 q^{64} + 3 q^{65} - 9 q^{67} + 9 q^{68} - 39 q^{69} + 13 q^{70} - 24 q^{71} + q^{73} - 4 q^{74} + 8 q^{75} + 29 q^{76} - 25 q^{78} - 3 q^{79} - 43 q^{80} + 12 q^{81} + q^{82} + 4 q^{83} - 9 q^{84} - 14 q^{85} - 4 q^{86} + 10 q^{87} - 62 q^{89} + 8 q^{90} - 5 q^{91} - 52 q^{92} - 25 q^{93} - 16 q^{94} - 4 q^{95} - 11 q^{96} - 5 q^{97} - 7 q^{98}+O(q^{100}) \) 12 * q - q^2 + 12 * q^3 + 9 * q^4 - 14 * q^5 - q^6 - 5 * q^7 + 12 * q^9 + 8 * q^10 + 9 * q^12 - 7 * q^13 - 13 * q^14 - 14 * q^15 + 11 * q^16 + 12 * q^17 - q^18 + 3 * q^19 - 36 * q^20 - 5 * q^21 - 39 * q^23 + 8 * q^25 - 25 * q^26 + 12 * q^27 - 9 * q^28 + 10 * q^29 + 8 * q^30 - 25 * q^31 - 11 * q^32 - q^34 + 12 * q^35 + 9 * q^36 - 10 * q^37 - 34 * q^38 - 7 * q^39 - 13 * q^40 + 17 * q^41 - 13 * q^42 - 18 * q^43 - 14 * q^45 + 19 * q^46 - 38 * q^47 + 11 * q^48 - 15 * q^49 - 18 * q^50 + 12 * q^51 + 32 * q^52 - 40 * q^53 - q^54 - 25 * q^56 + 3 * q^57 + 3 * q^58 - 18 * q^59 - 36 * q^60 - 22 * q^61 + 3 * q^62 - 5 * q^63 - 20 * q^64 + 3 * q^65 - 9 * q^67 + 9 * q^68 - 39 * q^69 + 13 * q^70 - 24 * q^71 + q^73 - 4 * q^74 + 8 * q^75 + 29 * q^76 - 25 * q^78 - 3 * q^79 - 43 * q^80 + 12 * q^81 + q^82 + 4 * q^83 - 9 * q^84 - 14 * q^85 - 4 * q^86 + 10 * q^87 - 62 * q^89 + 8 * q^90 - 5 * q^91 - 52 * q^92 - 25 * q^93 - 16 * q^94 - 4 * q^95 - 11 * q^96 - 5 * q^97 - 7 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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