LMFDB - Embedded newform 8034.2.a.bc.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8034,2,Mod(1,8034)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8034, 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("8034.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8034 = 2 \cdot 3 \cdot 13 \cdot 103 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8034.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:	$64.1518129839$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - x^{14} - 48 x^{13} + 44 x^{12} + 872 x^{11} - 707 x^{10} - 7580 x^{9} + 5112 x^{8} + \cdots + 2048 $ x^15 - x^14 - 48x^13 + 44x^12 + 872x^11 - 707x^10 - 7580x^9 + 5112x^8 + 33191x^7 - 16428x^6 - 71361x^5 + 21747x^4 + 65434x^3 - 11840x^2 - 17600*x + 2048
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$-1.40985$ of defining polynomial
Character	$\chi$	$=$	8034.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.40985 q^{5} -1.00000 q^{6} +4.86981 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.40985 q^{5} -1.00000 q^{6} +4.86981 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.40985 q^{10} -2.33063 q^{11} -1.00000 q^{12} -1.00000 q^{13} +4.86981 q^{14} -1.40985 q^{15} +1.00000 q^{16} +5.38125 q^{17} +1.00000 q^{18} +5.46412 q^{19} +1.40985 q^{20} -4.86981 q^{21} -2.33063 q^{22} -1.00935 q^{23} -1.00000 q^{24} -3.01231 q^{25} -1.00000 q^{26} -1.00000 q^{27} +4.86981 q^{28} -2.01110 q^{29} -1.40985 q^{30} +0.445300 q^{31} +1.00000 q^{32} +2.33063 q^{33} +5.38125 q^{34} +6.86572 q^{35} +1.00000 q^{36} +9.09660 q^{37} +5.46412 q^{38} +1.00000 q^{39} +1.40985 q^{40} -0.952506 q^{41} -4.86981 q^{42} -3.56484 q^{43} -2.33063 q^{44} +1.40985 q^{45} -1.00935 q^{46} +0.884177 q^{47} -1.00000 q^{48} +16.7150 q^{49} -3.01231 q^{50} -5.38125 q^{51} -1.00000 q^{52} -7.50659 q^{53} -1.00000 q^{54} -3.28585 q^{55} +4.86981 q^{56} -5.46412 q^{57} -2.01110 q^{58} +4.52096 q^{59} -1.40985 q^{60} +6.62115 q^{61} +0.445300 q^{62} +4.86981 q^{63} +1.00000 q^{64} -1.40985 q^{65} +2.33063 q^{66} +2.00724 q^{67} +5.38125 q^{68} +1.00935 q^{69} +6.86572 q^{70} -9.19397 q^{71} +1.00000 q^{72} -6.35844 q^{73} +9.09660 q^{74} +3.01231 q^{75} +5.46412 q^{76} -11.3497 q^{77} +1.00000 q^{78} +3.70385 q^{79} +1.40985 q^{80} +1.00000 q^{81} -0.952506 q^{82} +14.4610 q^{83} -4.86981 q^{84} +7.58678 q^{85} -3.56484 q^{86} +2.01110 q^{87} -2.33063 q^{88} +2.48255 q^{89} +1.40985 q^{90} -4.86981 q^{91} -1.00935 q^{92} -0.445300 q^{93} +0.884177 q^{94} +7.70361 q^{95} -1.00000 q^{96} +6.09866 q^{97} +16.7150 q^{98} -2.33063 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 15 q^{2} - 15 q^{3} + 15 q^{4} - q^{5} - 15 q^{6} + 5 q^{7} + 15 q^{8} + 15 q^{9}+O(q^{10}) $ 15 * q + 15 * q^2 - 15 * q^3 + 15 * q^4 - q^5 - 15 * q^6 + 5 * q^7 + 15 * q^8 + 15 * q^9	\( 15 q + 15 q^{2} - 15 q^{3} + 15 q^{4} - q^{5} - 15 q^{6} + 5 q^{7} + 15 q^{8} + 15 q^{9} - q^{10} + 3 q^{11} - 15 q^{12} - 15 q^{13} + 5 q^{14} + q^{15} + 15 q^{16} - 2 q^{17} + 15 q^{18} + 8 q^{19} - q^{20} - 5 q^{21} + 3 q^{22} + 3 q^{23} - 15 q^{24} + 22 q^{25} - 15 q^{26} - 15 q^{27} + 5 q^{28} + 26 q^{29} + q^{30} + 15 q^{32} - 3 q^{33} - 2 q^{34} - 8 q^{35} + 15 q^{36} + 25 q^{37} + 8 q^{38} + 15 q^{39} - q^{40} - q^{41} - 5 q^{42} + 10 q^{43} + 3 q^{44} - q^{45} + 3 q^{46} - 3 q^{47} - 15 q^{48} + 32 q^{49} + 22 q^{50} + 2 q^{51} - 15 q^{52} + 13 q^{53} - 15 q^{54} - 2 q^{55} + 5 q^{56} - 8 q^{57} + 26 q^{58} + 28 q^{59} + q^{60} + 22 q^{61} + 5 q^{63} + 15 q^{64} + q^{65} - 3 q^{66} + 29 q^{67} - 2 q^{68} - 3 q^{69} - 8 q^{70} + 18 q^{71} + 15 q^{72} + 23 q^{73} + 25 q^{74} - 22 q^{75} + 8 q^{76} + 17 q^{77} + 15 q^{78} + 27 q^{79} - q^{80} + 15 q^{81} - q^{82} + 7 q^{83} - 5 q^{84} + 43 q^{85} + 10 q^{86} - 26 q^{87} + 3 q^{88} + 35 q^{89} - q^{90} - 5 q^{91} + 3 q^{92} - 3 q^{94} + 6 q^{95} - 15 q^{96} + 19 q^{97} + 32 q^{98} + 3 q^{99}+O(q^{100}) \) 15 * q + 15 * q^2 - 15 * q^3 + 15 * q^4 - q^5 - 15 * q^6 + 5 * q^7 + 15 * q^8 + 15 * q^9 - q^10 + 3 * q^11 - 15 * q^12 - 15 * q^13 + 5 * q^14 + q^15 + 15 * q^16 - 2 * q^17 + 15 * q^18 + 8 * q^19 - q^20 - 5 * q^21 + 3 * q^22 + 3 * q^23 - 15 * q^24 + 22 * q^25 - 15 * q^26 - 15 * q^27 + 5 * q^28 + 26 * q^29 + q^30 + 15 * q^32 - 3 * q^33 - 2 * q^34 - 8 * q^35 + 15 * q^36 + 25 * q^37 + 8 * q^38 + 15 * q^39 - q^40 - q^41 - 5 * q^42 + 10 * q^43 + 3 * q^44 - q^45 + 3 * q^46 - 3 * q^47 - 15 * q^48 + 32 * q^49 + 22 * q^50 + 2 * q^51 - 15 * q^52 + 13 * q^53 - 15 * q^54 - 2 * q^55 + 5 * q^56 - 8 * q^57 + 26 * q^58 + 28 * q^59 + q^60 + 22 * q^61 + 5 * q^63 + 15 * q^64 + q^65 - 3 * q^66 + 29 * q^67 - 2 * q^68 - 3 * q^69 - 8 * q^70 + 18 * q^71 + 15 * q^72 + 23 * q^73 + 25 * q^74 - 22 * q^75 + 8 * q^76 + 17 * q^77 + 15 * q^78 + 27 * q^79 - q^80 + 15 * q^81 - q^82 + 7 * q^83 - 5 * q^84 + 43 * q^85 + 10 * q^86 - 26 * q^87 + 3 * q^88 + 35 * q^89 - q^90 - 5 * q^91 + 3 * q^92 - 3 * q^94 + 6 * q^95 - 15 * q^96 + 19 * q^97 + 32 * q^98 + 3 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	1.40985	0.630506	0.315253	−	0.949008i	$-0.397911\pi$
$5$	1.40985	0.630506	0.315253	+	0.949008i	$0.397911\pi$
$6$	−1.00000	−0.408248
$7$	4.86981	1.84061	0.920307	−	0.391196i	$-0.127939\pi$
$7$	4.86981	1.84061	0.920307	+	0.391196i	$0.127939\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	1.40985	0.445835
$11$	−2.33063	−0.702712	−0.351356	−	0.936242i	$-0.614279\pi$
$11$	−2.33063	−0.702712	−0.351356	+	0.936242i	$0.614279\pi$
$12$	−1.00000	−0.288675
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	4.86981	1.30151
$15$	−1.40985	−0.364023
$16$	1.00000	0.250000
$17$	5.38125	1.30515	0.652573	−	0.757726i	$-0.273690\pi$
$17$	5.38125	1.30515	0.652573	+	0.757726i	$0.273690\pi$
$18$	1.00000	0.235702
$19$	5.46412	1.25355	0.626777	−	0.779199i	$-0.284374\pi$
$19$	5.46412	1.25355	0.626777	+	0.779199i	$0.284374\pi$
$20$	1.40985	0.315253
$21$	−4.86981	−1.06268
$22$	−2.33063	−0.496892
$23$	−1.00935	−0.210464	−0.105232	−	0.994448i	$-0.533558\pi$
$23$	−1.00935	−0.210464	−0.105232	+	0.994448i	$0.533558\pi$
$24$	−1.00000	−0.204124
$25$	−3.01231	−0.602462
$26$	−1.00000	−0.196116
$27$	−1.00000	−0.192450
$28$	4.86981	0.920307
$29$	−2.01110	−0.373452	−0.186726	−	0.982412i	$-0.559788\pi$
$29$	−2.01110	−0.373452	−0.186726	+	0.982412i	$0.559788\pi$
$30$	−1.40985	−0.257403
$31$	0.445300	0.0799783	0.0399891	−	0.999200i	$-0.487268\pi$
$31$	0.445300	0.0799783	0.0399891	+	0.999200i	$0.487268\pi$
$32$	1.00000	0.176777
$33$	2.33063	0.405711
$34$	5.38125	0.922877
$35$	6.86572	1.16052
$36$	1.00000	0.166667
$37$	9.09660	1.49547	0.747736	−	0.663996i	$-0.231141\pi$
$37$	9.09660	1.49547	0.747736	+	0.663996i	$0.231141\pi$
$38$	5.46412	0.886397
$39$	1.00000	0.160128
$40$	1.40985	0.222918
$41$	−0.952506	−0.148756	−0.0743782	−	0.997230i	$-0.523697\pi$
$41$	−0.952506	−0.148756	−0.0743782	+	0.997230i	$0.523697\pi$
$42$	−4.86981	−0.751428
$43$	−3.56484	−0.543633	−0.271817	−	0.962349i	$-0.587624\pi$
$43$	−3.56484	−0.543633	−0.271817	+	0.962349i	$0.587624\pi$
$44$	−2.33063	−0.351356
$45$	1.40985	0.210169
$46$	−1.00935	−0.148820
$47$	0.884177	0.128971	0.0644853	−	0.997919i	$-0.479459\pi$
$47$	0.884177	0.128971	0.0644853	+	0.997919i	$0.479459\pi$
$48$	−1.00000	−0.144338
$49$	16.7150	2.38786
$50$	−3.01231	−0.426005
$51$	−5.38125	−0.753526
$52$	−1.00000	−0.138675
$53$	−7.50659	−1.03111	−0.515555	−	0.856857i	$-0.672414\pi$
$53$	−7.50659	−1.03111	−0.515555	+	0.856857i	$0.672414\pi$
$54$	−1.00000	−0.136083
$55$	−3.28585	−0.443064
$56$	4.86981	0.650756
$57$	−5.46412	−0.723740
$58$	−2.01110	−0.264071
$59$	4.52096	0.588579	0.294289	−	0.955716i	$-0.404917\pi$
$59$	4.52096	0.588579	0.294289	+	0.955716i	$0.404917\pi$
$60$	−1.40985	−0.182011
$61$	6.62115	0.847752	0.423876	−	0.905720i	$-0.360669\pi$
$61$	6.62115	0.847752	0.423876	+	0.905720i	$0.360669\pi$
$62$	0.445300	0.0565532
$63$	4.86981	0.613538
$64$	1.00000	0.125000
$65$	−1.40985	−0.174871
$66$	2.33063	0.286881
$67$	2.00724	0.245224	0.122612	−	0.992455i	$-0.460873\pi$
$67$	2.00724	0.245224	0.122612	+	0.992455i	$0.460873\pi$
$68$	5.38125	0.652573
$69$	1.00935	0.121511
$70$	6.86572	0.820611
$71$	−9.19397	−1.09112	−0.545562	−	0.838070i	$-0.683684\pi$
$71$	−9.19397	−1.09112	−0.545562	+	0.838070i	$0.683684\pi$
$72$	1.00000	0.117851
$73$	−6.35844	−0.744199	−0.372099	−	0.928193i	$-0.621362\pi$
$73$	−6.35844	−0.744199	−0.372099	+	0.928193i	$0.621362\pi$
$74$	9.09660	1.05746
$75$	3.01231	0.347832
$76$	5.46412	0.626777
$77$	−11.3497	−1.29342
$78$	1.00000	0.113228
$79$	3.70385	0.416715	0.208358	−	0.978053i	$-0.433188\pi$
$79$	3.70385	0.416715	0.208358	+	0.978053i	$0.433188\pi$
$80$	1.40985	0.157627
$81$	1.00000	0.111111
$82$	−0.952506	−0.105187
$83$	14.4610	1.58730	0.793648	−	0.608377i	$-0.208179\pi$
$83$	14.4610	1.58730	0.793648	+	0.608377i	$0.208179\pi$
$84$	−4.86981	−0.531340
$85$	7.58678	0.822902
$86$	−3.56484	−0.384407
$87$	2.01110	0.215613
$88$	−2.33063	−0.248446
$89$	2.48255	0.263150	0.131575	−	0.991306i	$-0.457997\pi$
$89$	2.48255	0.263150	0.131575	+	0.991306i	$0.457997\pi$
$90$	1.40985	0.148612
$91$	−4.86981	−0.510495
$92$	−1.00935	−0.105232
$93$	−0.445300	−0.0461755
$94$	0.884177	0.0911959
$95$	7.70361	0.790373
$96$	−1.00000	−0.102062
$97$	6.09866	0.619225	0.309613	−	0.950863i	$-0.399801\pi$
$97$	6.09866	0.619225	0.309613	+	0.950863i	$0.399801\pi$
$98$	16.7150	1.68847
$99$	−2.33063	−0.234237
$100$	−3.01231	−0.301231
$101$	7.35330	0.731681	0.365840	−	0.930678i	$-0.380782\pi$
$101$	7.35330	0.731681	0.365840	+	0.930678i	$0.380782\pi$
$102$	−5.38125	−0.532823
$103$	−1.00000	−0.0985329
$103$	−1.00000	−0.0985329
$104$	−1.00000	−0.0980581
$105$	−6.86572	−0.670026
$106$	−7.50659	−0.729104
$107$	−0.00520205	−0.000502901 0	−0.000251450	−	1.00000i	$-0.500080\pi$
$107$	−0.00520205	−0.000502901 0	−0.000251450		1.00000i	$0.500080\pi$
$108$	−1.00000	−0.0962250
$109$	−12.1170	−1.16060	−0.580299	−	0.814404i	$-0.697064\pi$
$109$	−12.1170	−1.16060	−0.580299	+	0.814404i	$0.697064\pi$
$110$	−3.28585	−0.313294
$111$	−9.09660	−0.863411
$112$	4.86981	0.460154
$113$	−3.32058	−0.312373	−0.156187	−	0.987728i	$-0.549920\pi$
$113$	−3.32058	−0.312373	−0.156187	+	0.987728i	$0.549920\pi$
$114$	−5.46412	−0.511761
$115$	−1.42303	−0.132699
$116$	−2.01110	−0.186726
$117$	−1.00000	−0.0924500
$118$	4.52096	0.416188
$119$	26.2057	2.40227
$120$	−1.40985	−0.128702
$121$	−5.56816	−0.506196
$122$	6.62115	0.599451
$123$	0.952506	0.0858845
$124$	0.445300	0.0399891
$125$	−11.2962	−1.01036
$126$	4.86981	0.433837
$127$	−3.48520	−0.309261	−0.154631	−	0.987972i	$-0.549419\pi$
$127$	−3.48520	−0.309261	−0.154631	+	0.987972i	$0.549419\pi$
$128$	1.00000	0.0883883
$129$	3.56484	0.313867
$130$	−1.40985	−0.123652
$131$	0.539376	0.0471255	0.0235628	−	0.999722i	$-0.492499\pi$
$131$	0.539376	0.0471255	0.0235628	+	0.999722i	$0.492499\pi$
$132$	2.33063	0.202855
$133$	26.6092	2.30731
$134$	2.00724	0.173399
$135$	−1.40985	−0.121341
$136$	5.38125	0.461439
$137$	1.94980	0.166583	0.0832915	−	0.996525i	$-0.473457\pi$
$137$	1.94980	0.166583	0.0832915	+	0.996525i	$0.473457\pi$
$138$	1.00935	0.0859214
$139$	10.6357	0.902110	0.451055	−	0.892496i	$-0.351048\pi$
$139$	10.6357	0.902110	0.451055	+	0.892496i	$0.351048\pi$
$140$	6.86572	0.580259
$141$	−0.884177	−0.0744612
$142$	−9.19397	−0.771541
$143$	2.33063	0.194897
$144$	1.00000	0.0833333
$145$	−2.83536	−0.235464
$146$	−6.35844	−0.526228
$147$	−16.7150	−1.37863
$148$	9.09660	0.747736
$149$	−5.69498	−0.466551	−0.233276	−	0.972411i	$-0.574944\pi$
$149$	−5.69498	−0.466551	−0.233276	+	0.972411i	$0.574944\pi$
$150$	3.01231	0.245954
$151$	−4.77259	−0.388388	−0.194194	−	0.980963i	$-0.562209\pi$
$151$	−4.77259	−0.388388	−0.194194	+	0.980963i	$0.562209\pi$
$152$	5.46412	0.443198
$153$	5.38125	0.435048
$154$	−11.3497	−0.914587
$155$	0.627809	0.0504268
$156$	1.00000	0.0800641
$157$	4.58778	0.366145	0.183073	−	0.983099i	$-0.441396\pi$
$157$	4.58778	0.366145	0.183073	+	0.983099i	$0.441396\pi$
$158$	3.70385	0.294662
$159$	7.50659	0.595311
$160$	1.40985	0.111459
$161$	−4.91533	−0.387383
$162$	1.00000	0.0785674
$163$	−3.09389	−0.242332	−0.121166	−	0.992632i	$-0.538663\pi$
$163$	−3.09389	−0.242332	−0.121166	+	0.992632i	$0.538663\pi$
$164$	−0.952506	−0.0743782
$165$	3.28585	0.255803
$166$	14.4610	1.12239
$167$	−11.7557	−0.909680	−0.454840	−	0.890573i	$-0.650304\pi$
$167$	−11.7557	−0.909680	−0.454840	+	0.890573i	$0.650304\pi$
$168$	−4.86981	−0.375714
$169$	1.00000	0.0769231
$170$	7.58678	0.581880
$171$	5.46412	0.417851
$172$	−3.56484	−0.271817
$173$	3.99859	0.304007	0.152004	−	0.988380i	$-0.451427\pi$
$173$	3.99859	0.304007	0.152004	+	0.988380i	$0.451427\pi$
$174$	2.01110	0.152461
$175$	−14.6694	−1.10890
$176$	−2.33063	−0.175678
$177$	−4.52096	−0.339816
$178$	2.48255	0.186075
$179$	−12.3417	−0.922465	−0.461233	−	0.887279i	$-0.652593\pi$
$179$	−12.3417	−0.922465	−0.461233	+	0.887279i	$0.652593\pi$
$180$	1.40985	0.105084
$181$	22.4793	1.67087	0.835437	−	0.549586i	$-0.185215\pi$
$181$	22.4793	1.67087	0.835437	+	0.549586i	$0.185215\pi$
$182$	−4.86981	−0.360974
$183$	−6.62115	−0.489450
$184$	−1.00935	−0.0744102
$185$	12.8249	0.942904
$186$	−0.445300	−0.0326510
$187$	−12.5417	−0.917141
$188$	0.884177	0.0644853
$189$	−4.86981	−0.354226
$190$	7.70361	0.558878
$191$	−19.7321	−1.42776	−0.713882	−	0.700266i	$-0.753065\pi$
$191$	−19.7321	−1.42776	−0.713882	+	0.700266i	$0.753065\pi$
$192$	−1.00000	−0.0721688
$193$	17.3031	1.24550	0.622751	−	0.782420i	$-0.286015\pi$
$193$	17.3031	1.24550	0.622751	+	0.782420i	$0.286015\pi$
$194$	6.09866	0.437858
$195$	1.40985	0.100962
$196$	16.7150	1.19393
$197$	4.26407	0.303803	0.151901	−	0.988396i	$-0.451460\pi$
$197$	4.26407	0.303803	0.151901	+	0.988396i	$0.451460\pi$
$198$	−2.33063	−0.165631
$199$	−20.1794	−1.43048	−0.715238	−	0.698881i	$-0.753682\pi$
$199$	−20.1794	−1.43048	−0.715238	+	0.698881i	$0.753682\pi$
$200$	−3.01231	−0.213003
$201$	−2.00724	−0.141580
$202$	7.35330	0.517376
$203$	−9.79368	−0.687382
$204$	−5.38125	−0.376763
$205$	−1.34289	−0.0937918
$206$	−1.00000	−0.0696733
$207$	−1.00935	−0.0701546
$208$	−1.00000	−0.0693375
$209$	−12.7348	−0.880887
$210$	−6.86572	−0.473780
$211$	−0.460401	−0.0316953	−0.0158477	−	0.999874i	$-0.505045\pi$
$211$	−0.460401	−0.0316953	−0.0158477	+	0.999874i	$0.505045\pi$
$212$	−7.50659	−0.515555
$213$	9.19397	0.629961
$214$	−0.00520205	−0.000355605 0
$215$	−5.02591	−0.342764
$216$	−1.00000	−0.0680414
$217$	2.16853	0.147209
$218$	−12.1170	−0.820666
$219$	6.35844	0.429663
$220$	−3.28585	−0.221532
$221$	−5.38125	−0.361982
$222$	−9.09660	−0.610524
$223$	19.9215	1.33404	0.667021	−	0.745039i	$-0.267569\pi$
$223$	19.9215	1.33404	0.667021	+	0.745039i	$0.267569\pi$
$224$	4.86981	0.325378
$225$	−3.01231	−0.200821
$226$	−3.32058	−0.220881
$227$	−20.1728	−1.33892	−0.669458	−	0.742850i	$-0.733473\pi$
$227$	−20.1728	−1.33892	−0.669458	+	0.742850i	$0.733473\pi$
$228$	−5.46412	−0.361870
$229$	−16.5160	−1.09140	−0.545702	−	0.837979i	$-0.683737\pi$
$229$	−16.5160	−1.09140	−0.545702	+	0.837979i	$0.683737\pi$
$230$	−1.42303	−0.0938321
$231$	11.3497	0.746757
$232$	−2.01110	−0.132035
$233$	−11.7843	−0.772016	−0.386008	−	0.922495i	$-0.626146\pi$
$233$	−11.7843	−0.772016	−0.386008	+	0.922495i	$0.626146\pi$
$234$	−1.00000	−0.0653720
$235$	1.24656	0.0813167
$236$	4.52096	0.294289
$237$	−3.70385	−0.240591
$238$	26.2057	1.69866
$239$	8.32974	0.538806	0.269403	−	0.963028i	$-0.413174\pi$
$239$	8.32974	0.538806	0.269403	+	0.963028i	$0.413174\pi$
$240$	−1.40985	−0.0910057
$241$	−10.9822	−0.707426	−0.353713	−	0.935354i	$-0.615081\pi$
$241$	−10.9822	−0.707426	−0.353713	+	0.935354i	$0.615081\pi$
$242$	−5.56816	−0.357935
$243$	−1.00000	−0.0641500
$244$	6.62115	0.423876
$245$	23.5658	1.50556
$246$	0.952506	0.0607295
$247$	−5.46412	−0.347673
$248$	0.445300	0.0282766
$249$	−14.4610	−0.916426
$250$	−11.2962	−0.714434
$251$	8.73482	0.551337	0.275669	−	0.961253i	$-0.411101\pi$
$251$	8.73482	0.551337	0.275669	+	0.961253i	$0.411101\pi$
$252$	4.86981	0.306769
$253$	2.35242	0.147895
$254$	−3.48520	−0.218681
$255$	−7.58678	−0.475103
$256$	1.00000	0.0625000
$257$	−9.60029	−0.598850	−0.299425	−	0.954120i	$-0.596795\pi$
$257$	−9.60029	−0.598850	−0.299425	+	0.954120i	$0.596795\pi$
$258$	3.56484	0.221937
$259$	44.2987	2.75259
$260$	−1.40985	−0.0874355
$261$	−2.01110	−0.124484
$262$	0.539376	0.0333228
$263$	−22.0405	−1.35908	−0.679538	−	0.733640i	$-0.737820\pi$
$263$	−22.0405	−1.35908	−0.679538	+	0.733640i	$0.737820\pi$
$264$	2.33063	0.143440
$265$	−10.5832	−0.650120
$266$	26.6092	1.63151
$267$	−2.48255	−0.151930
$268$	2.00724	0.122612
$269$	13.0683	0.796788	0.398394	−	0.917214i	$-0.369568\pi$
$269$	13.0683	0.796788	0.398394	+	0.917214i	$0.369568\pi$
$270$	−1.40985	−0.0858010
$271$	9.85282	0.598516	0.299258	−	0.954172i	$-0.403261\pi$
$271$	9.85282	0.598516	0.299258	+	0.954172i	$0.403261\pi$
$272$	5.38125	0.326286
$273$	4.86981	0.294734
$274$	1.94980	0.117792
$275$	7.02059	0.423357
$276$	1.00935	0.0607556
$277$	−28.2551	−1.69769	−0.848843	−	0.528644i	$-0.822701\pi$
$277$	−28.2551	−1.69769	−0.848843	+	0.528644i	$0.822701\pi$
$278$	10.6357	0.637888
$279$	0.445300	0.0266594
$280$	6.86572	0.410305
$281$	1.71433	0.102268	0.0511341	−	0.998692i	$-0.483716\pi$
$281$	1.71433	0.102268	0.0511341	+	0.998692i	$0.483716\pi$
$282$	−0.884177	−0.0526520
$283$	16.3986	0.974793	0.487396	−	0.873181i	$-0.337947\pi$
$283$	16.3986	0.974793	0.487396	+	0.873181i	$0.337947\pi$
$284$	−9.19397	−0.545562
$285$	−7.70361	−0.456322
$286$	2.33063	0.137813
$287$	−4.63852	−0.273803
$288$	1.00000	0.0589256
$289$	11.9579	0.703404
$290$	−2.83536	−0.166498
$291$	−6.09866	−0.357510
$292$	−6.35844	−0.372099
$293$	22.5655	1.31829	0.659146	−	0.752015i	$-0.270918\pi$
$293$	22.5655	1.31829	0.659146	+	0.752015i	$0.270918\pi$
$294$	−16.7150	−0.974841
$295$	6.37389	0.371102
$296$	9.09660	0.528729
$297$	2.33063	0.135237
$298$	−5.69498	−0.329901
$299$	1.00935	0.0583721
$300$	3.01231	0.173916
$301$	−17.3601	−1.00062
$302$	−4.77259	−0.274632
$303$	−7.35330	−0.422436
$304$	5.46412	0.313389
$305$	9.33486	0.534513
$306$	5.38125	0.307626
$307$	26.4261	1.50822	0.754109	−	0.656749i	$-0.228069\pi$
$307$	26.4261	1.50822	0.754109	+	0.656749i	$0.228069\pi$
$308$	−11.3497	−0.646711
$309$	1.00000	0.0568880
$310$	0.627809	0.0356571
$311$	−5.10993	−0.289758	−0.144879	−	0.989449i	$-0.546279\pi$
$311$	−5.10993	−0.289758	−0.144879	+	0.989449i	$0.546279\pi$
$312$	1.00000	0.0566139
$313$	26.4548	1.49531	0.747657	−	0.664085i	$-0.231179\pi$
$313$	26.4548	1.49531	0.747657	+	0.664085i	$0.231179\pi$
$314$	4.58778	0.258904
$315$	6.86572	0.386840
$316$	3.70385	0.208358
$317$	−6.08045	−0.341512	−0.170756	−	0.985313i	$-0.554621\pi$
$317$	−6.08045	−0.341512	−0.170756	+	0.985313i	$0.554621\pi$
$318$	7.50659	0.420949
$319$	4.68714	0.262429
$320$	1.40985	0.0788133
$321$	0.00520205	0.000290350 0
$322$	−4.91533	−0.273921
$323$	29.4038	1.63607
$324$	1.00000	0.0555556
$325$	3.01231	0.167093
$326$	−3.09389	−0.171354
$327$	12.1170	0.670071
$328$	−0.952506	−0.0525933
$329$	4.30577	0.237385
$330$	3.28585	0.180880
$331$	3.03497	0.166817	0.0834085	−	0.996515i	$-0.473419\pi$
$331$	3.03497	0.166817	0.0834085	+	0.996515i	$0.473419\pi$
$332$	14.4610	0.793648
$333$	9.09660	0.498491
$334$	−11.7557	−0.643241
$335$	2.82992	0.154615
$336$	−4.86981	−0.265670
$337$	13.8840	0.756310	0.378155	−	0.925742i	$-0.376559\pi$
$337$	13.8840	0.756310	0.378155	+	0.925742i	$0.376559\pi$
$338$	1.00000	0.0543928
$339$	3.32058	0.180349
$340$	7.58678	0.411451
$341$	−1.03783	−0.0562017
$342$	5.46412	0.295466
$343$	47.3104	2.55452
$344$	−3.56484	−0.192203
$345$	1.42303	0.0766136
$346$	3.99859	0.214966
$347$	25.4955	1.36867	0.684336	−	0.729167i	$-0.260092\pi$
$347$	25.4955	1.36867	0.684336	+	0.729167i	$0.260092\pi$
$348$	2.01110	0.107806
$349$	−19.0588	−1.02019	−0.510096	−	0.860117i	$-0.670390\pi$
$349$	−19.0588	−1.02019	−0.510096	+	0.860117i	$0.670390\pi$
$350$	−14.6694	−0.784111
$351$	1.00000	0.0533761
$352$	−2.33063	−0.124223
$353$	−28.7383	−1.52959	−0.764793	−	0.644276i	$-0.777159\pi$
$353$	−28.7383	−1.52959	−0.764793	+	0.644276i	$0.777159\pi$
$354$	−4.52096	−0.240286
$355$	−12.9622	−0.687960
$356$	2.48255	0.131575
$357$	−26.2057	−1.38695
$358$	−12.3417	−0.652281
$359$	−8.50496	−0.448874	−0.224437	−	0.974489i	$-0.572054\pi$
$359$	−8.50496	−0.448874	−0.224437	+	0.974489i	$0.572054\pi$
$360$	1.40985	0.0743058
$361$	10.8566	0.571398
$362$	22.4793	1.18149
$363$	5.56816	0.292252
$364$	−4.86981	−0.255247
$365$	−8.96447	−0.469222
$366$	−6.62115	−0.346093
$367$	−9.47425	−0.494552	−0.247276	−	0.968945i	$-0.579535\pi$
$367$	−9.47425	−0.494552	−0.247276	+	0.968945i	$0.579535\pi$
$368$	−1.00935	−0.0526159
$369$	−0.952506	−0.0495855
$370$	12.8249	0.666734
$371$	−36.5556	−1.89787
$372$	−0.445300	−0.0230877
$373$	−4.61157	−0.238778	−0.119389	−	0.992848i	$-0.538094\pi$
$373$	−4.61157	−0.238778	−0.119389	+	0.992848i	$0.538094\pi$
$374$	−12.5417	−0.648517
$375$	11.2962	0.583333
$376$	0.884177	0.0455980
$377$	2.01110	0.103577
$378$	−4.86981	−0.250476
$379$	21.9061	1.12524	0.562620	−	0.826716i	$-0.309793\pi$
$379$	21.9061	1.12524	0.562620	+	0.826716i	$0.309793\pi$
$380$	7.70361	0.395187
$381$	3.48520	0.178552
$382$	−19.7321	−1.00958
$383$	−8.61706	−0.440311	−0.220156	−	0.975465i	$-0.570657\pi$
$383$	−8.61706	−0.440311	−0.220156	+	0.975465i	$0.570657\pi$
$384$	−1.00000	−0.0510310
$385$	−16.0015	−0.815510
$386$	17.3031	0.880703
$387$	−3.56484	−0.181211
$388$	6.09866	0.309613
$389$	31.6874	1.60661	0.803307	−	0.595566i	$-0.203072\pi$
$389$	31.6874	1.60661	0.803307	+	0.595566i	$0.203072\pi$
$390$	1.40985	0.0713907
$391$	−5.43156	−0.274686
$392$	16.7150	0.844237
$393$	−0.539376	−0.0272079
$394$	4.26407	0.214821
$395$	5.22188	0.262741
$396$	−2.33063	−0.117119
$397$	−32.9613	−1.65428	−0.827141	−	0.561994i	$-0.810034\pi$
$397$	−32.9613	−1.65428	−0.827141	+	0.561994i	$0.810034\pi$
$398$	−20.1794	−1.01150
$399$	−26.6092	−1.33213
$400$	−3.01231	−0.150616
$401$	23.0372	1.15042	0.575211	−	0.818005i	$-0.304920\pi$
$401$	23.0372	1.15042	0.575211	+	0.818005i	$0.304920\pi$
$402$	−2.00724	−0.100112
$403$	−0.445300	−0.0221820
$404$	7.35330	0.365840
$405$	1.40985	0.0700562
$406$	−9.79368	−0.486052
$407$	−21.2008	−1.05089
$408$	−5.38125	−0.266412
$409$	2.78238	0.137580	0.0687900	−	0.997631i	$-0.478086\pi$
$409$	2.78238	0.137580	0.0687900	+	0.997631i	$0.478086\pi$
$410$	−1.34289	−0.0663208
$411$	−1.94980	−0.0961768
$412$	−1.00000	−0.0492665
$413$	22.0162	1.08335
$414$	−1.00935	−0.0496068
$415$	20.3879	1.00080
$416$	−1.00000	−0.0490290
$417$	−10.6357	−0.520833
$418$	−12.7348	−0.622881
$419$	6.97312	0.340659	0.170330	−	0.985387i	$-0.445517\pi$
$419$	6.97312	0.340659	0.170330	+	0.985387i	$0.445517\pi$
$420$	−6.86572	−0.335013
$421$	4.97348	0.242393	0.121196	−	0.992629i	$-0.461327\pi$
$421$	4.97348	0.242393	0.121196	+	0.992629i	$0.461327\pi$
$422$	−0.460401	−0.0224120
$423$	0.884177	0.0429902
$424$	−7.50659	−0.364552
$425$	−16.2100	−0.786301
$426$	9.19397	0.445450
$427$	32.2437	1.56038
$428$	−0.00520205	−0.000251450 0
$429$	−2.33063	−0.112524
$430$	−5.02591	−0.242371
$431$	−10.6617	−0.513554	−0.256777	−	0.966471i	$-0.582661\pi$
$431$	−10.6617	−0.513554	−0.256777	+	0.966471i	$0.582661\pi$
$432$	−1.00000	−0.0481125
$433$	36.5166	1.75487	0.877437	−	0.479692i	$-0.159252\pi$
$433$	36.5166	1.75487	0.877437	+	0.479692i	$0.159252\pi$
$434$	2.16853	0.104093
$435$	2.83536	0.135945
$436$	−12.1170	−0.580299
$437$	−5.51520	−0.263828
$438$	6.35844	0.303818
$439$	28.5065	1.36054	0.680271	−	0.732961i	$-0.261862\pi$
$439$	28.5065	1.36054	0.680271	+	0.732961i	$0.261862\pi$
$440$	−3.28585	−0.156647
$441$	16.7150	0.795954
$442$	−5.38125	−0.255960
$443$	−18.8447	−0.895340	−0.447670	−	0.894199i	$-0.647746\pi$
$443$	−18.8447	−0.895340	−0.447670	+	0.894199i	$0.647746\pi$
$444$	−9.09660	−0.431706
$445$	3.50004	0.165918
$446$	19.9215	0.943310
$447$	5.69498	0.269363
$448$	4.86981	0.230077
$449$	26.6224	1.25639	0.628195	−	0.778056i	$-0.283794\pi$
$449$	26.6224	1.25639	0.628195	+	0.778056i	$0.283794\pi$
$450$	−3.01231	−0.142002
$451$	2.21994	0.104533
$452$	−3.32058	−0.156187
$453$	4.77259	0.224236
$454$	−20.1728	−0.946756
$455$	−6.86572	−0.321870
$456$	−5.46412	−0.255881
$457$	−2.05556	−0.0961550	−0.0480775	−	0.998844i	$-0.515309\pi$
$457$	−2.05556	−0.0961550	−0.0480775	+	0.998844i	$0.515309\pi$
$458$	−16.5160	−0.771740
$459$	−5.38125	−0.251175
$460$	−1.42303	−0.0663493
$461$	12.0356	0.560553	0.280277	−	0.959919i	$-0.409574\pi$
$461$	12.0356	0.560553	0.280277	+	0.959919i	$0.409574\pi$
$462$	11.3497	0.528037
$463$	−26.5610	−1.23440	−0.617198	−	0.786808i	$-0.711732\pi$
$463$	−26.5610	−1.23440	−0.617198	+	0.786808i	$0.711732\pi$
$464$	−2.01110	−0.0933631
$465$	−0.627809	−0.0291139
$466$	−11.7843	−0.545898
$467$	−25.8182	−1.19472	−0.597361	−	0.801973i	$-0.703784\pi$
$467$	−25.8182	−1.19472	−0.597361	+	0.801973i	$0.703784\pi$
$468$	−1.00000	−0.0462250
$469$	9.77489	0.451363
$470$	1.24656	0.0574996
$471$	−4.58778	−0.211394
$472$	4.52096	0.208094
$473$	8.30833	0.382017
$474$	−3.70385	−0.170123
$475$	−16.4596	−0.755219
$476$	26.2057	1.20113
$477$	−7.50659	−0.343703
$478$	8.32974	0.380993
$479$	−4.91205	−0.224437	−0.112219	−	0.993684i	$-0.535796\pi$
$479$	−4.91205	−0.224437	−0.112219	+	0.993684i	$0.535796\pi$
$480$	−1.40985	−0.0643508
$481$	−9.09660	−0.414769
$482$	−10.9822	−0.500226
$483$	4.91533	0.223655
$484$	−5.56816	−0.253098
$485$	8.59822	0.390425
$486$	−1.00000	−0.0453609
$487$	−33.5943	−1.52230	−0.761152	−	0.648573i	$-0.775366\pi$
$487$	−33.5943	−1.52230	−0.761152	+	0.648573i	$0.775366\pi$
$488$	6.62115	0.299725
$489$	3.09389	0.139910
$490$	23.5658	1.06459
$491$	20.7294	0.935503	0.467752	−	0.883860i	$-0.345064\pi$
$491$	20.7294	0.935503	0.467752	+	0.883860i	$0.345064\pi$
$492$	0.952506	0.0429423
$493$	−10.8222	−0.487410
$494$	−5.46412	−0.245842
$495$	−3.28585	−0.147688
$496$	0.445300	0.0199946
$497$	−44.7729	−2.00834
$498$	−14.4610	−0.648011
$499$	1.53739	0.0688231	0.0344116	−	0.999408i	$-0.489044\pi$
$499$	1.53739	0.0688231	0.0344116	+	0.999408i	$0.489044\pi$
$500$	−11.2962	−0.505181
$501$	11.7557	0.525204
$502$	8.73482	0.389854
$503$	−23.1448	−1.03198	−0.515988	−	0.856596i	$-0.672575\pi$
$503$	−23.1448	−1.03198	−0.515988	+	0.856596i	$0.672575\pi$
$504$	4.86981	0.216919
$505$	10.3671	0.461329
$506$	2.35242	0.104578
$507$	−1.00000	−0.0444116
$508$	−3.48520	−0.154631
$509$	−15.4322	−0.684020	−0.342010	−	0.939696i	$-0.611108\pi$
$509$	−15.4322	−0.684020	−0.342010	+	0.939696i	$0.611108\pi$
$510$	−7.58678	−0.335948
$511$	−30.9644	−1.36978
$512$	1.00000	0.0441942
$513$	−5.46412	−0.241247
$514$	−9.60029	−0.423451
$515$	−1.40985	−0.0621256
$516$	3.56484	0.156933
$517$	−2.06069	−0.0906291
$518$	44.2987	1.94637
$519$	−3.99859	−0.175519
$520$	−1.40985	−0.0618262
$521$	−15.5549	−0.681471	−0.340736	−	0.940159i	$-0.610676\pi$
$521$	−15.5549	−0.681471	−0.340736	+	0.940159i	$0.610676\pi$
$522$	−2.01110	−0.0880236
$523$	25.8172	1.12891	0.564455	−	0.825464i	$-0.309086\pi$
$523$	25.8172	1.12891	0.564455	+	0.825464i	$0.309086\pi$
$524$	0.539376	0.0235628
$525$	14.6694	0.640224
$526$	−22.0405	−0.961012
$527$	2.39627	0.104383
$528$	2.33063	0.101428
$529$	−21.9812	−0.955705
$530$	−10.5832	−0.459705
$531$	4.52096	0.196193
$532$	26.6092	1.15365
$533$	0.952506	0.0412576
$534$	−2.48255	−0.107431
$535$	−0.00733413	−0.000317082 0
$536$	2.00724	0.0866997
$537$	12.3417	0.532586
$538$	13.0683	0.563414
$539$	−38.9566	−1.67798
$540$	−1.40985	−0.0606705
$541$	−23.3074	−1.00206	−0.501032	−	0.865429i	$-0.667046\pi$
$541$	−23.3074	−1.00206	−0.501032	+	0.865429i	$0.667046\pi$
$542$	9.85282	0.423215
$543$	−22.4793	−0.964680
$544$	5.38125	0.230719
$545$	−17.0832	−0.731764
$546$	4.86981	0.208409
$547$	40.4603	1.72996	0.864978	−	0.501810i	$-0.167332\pi$
$547$	40.4603	1.72996	0.864978	+	0.501810i	$0.167332\pi$
$548$	1.94980	0.0832915
$549$	6.62115	0.282584
$550$	7.02059	0.299359
$551$	−10.9889	−0.468143
$552$	1.00935	0.0429607
$553$	18.0370	0.767012
$554$	−28.2551	−1.20045
$555$	−12.8249	−0.544386
$556$	10.6357	0.451055
$557$	35.6694	1.51136	0.755680	−	0.654941i	$-0.227307\pi$
$557$	35.6694	1.51136	0.755680	+	0.654941i	$0.227307\pi$
$558$	0.445300	0.0188511
$559$	3.56484	0.150777
$560$	6.86572	0.290130
$561$	12.5417	0.529512
$562$	1.71433	0.0723146
$563$	−11.0723	−0.466641	−0.233321	−	0.972400i	$-0.574959\pi$
$563$	−11.0723	−0.466641	−0.233321	+	0.972400i	$0.574959\pi$
$564$	−0.884177	−0.0372306
$565$	−4.68153	−0.196953
$566$	16.3986	0.689283
$567$	4.86981	0.204513
$568$	−9.19397	−0.385771
$569$	9.70010	0.406649	0.203325	−	0.979111i	$-0.434825\pi$
$569$	9.70010	0.406649	0.203325	+	0.979111i	$0.434825\pi$
$570$	−7.70361	−0.322669
$571$	−34.8614	−1.45890	−0.729451	−	0.684033i	$-0.760224\pi$
$571$	−34.8614	−1.45890	−0.729451	+	0.684033i	$0.760224\pi$
$572$	2.33063	0.0974486
$573$	19.7321	0.824320
$574$	−4.63852	−0.193608
$575$	3.04047	0.126796
$576$	1.00000	0.0416667
$577$	−3.62525	−0.150921	−0.0754605	−	0.997149i	$-0.524043\pi$
$577$	−3.62525	−0.150921	−0.0754605	+	0.997149i	$0.524043\pi$
$578$	11.9579	0.497382
$579$	−17.3031	−0.719091
$580$	−2.83536	−0.117732
$581$	70.4221	2.92160
$582$	−6.09866	−0.252798
$583$	17.4951	0.724573
$584$	−6.35844	−0.263114
$585$	−1.40985	−0.0582903
$586$	22.5655	0.932174
$587$	−29.6395	−1.22335	−0.611676	−	0.791109i	$-0.709504\pi$
$587$	−29.6395	−1.22335	−0.611676	+	0.791109i	$0.709504\pi$
$588$	−16.7150	−0.689316
$589$	2.43317	0.100257
$590$	6.37389	0.262409
$591$	−4.26407	−0.175401
$592$	9.09660	0.373868
$593$	−9.58200	−0.393485	−0.196743	−	0.980455i	$-0.563036\pi$
$593$	−9.58200	−0.393485	−0.196743	+	0.980455i	$0.563036\pi$
$594$	2.33063	0.0956270
$595$	36.9462	1.51465
$596$	−5.69498	−0.233276
$597$	20.1794	0.825886
$598$	1.00935	0.0412753
$599$	−2.64077	−0.107899	−0.0539496	−	0.998544i	$-0.517181\pi$
$599$	−2.64077	−0.107899	−0.0539496	+	0.998544i	$0.517181\pi$
$600$	3.01231	0.122977
$601$	36.7215	1.49790	0.748952	−	0.662625i	$-0.230558\pi$
$601$	36.7215	1.49790	0.748952	+	0.662625i	$0.230558\pi$
$602$	−17.3601	−0.707544
$603$	2.00724	0.0817413
$604$	−4.77259	−0.194194
$605$	−7.85029	−0.319160
$606$	−7.35330	−0.298707
$607$	−13.5559	−0.550217	−0.275108	−	0.961413i	$-0.588714\pi$
$607$	−13.5559	−0.550217	−0.275108	+	0.961413i	$0.588714\pi$
$608$	5.46412	0.221599
$609$	9.79368	0.396860
$610$	9.33486	0.377957
$611$	−0.884177	−0.0357700
$612$	5.38125	0.217524
$613$	6.82023	0.275467	0.137733	−	0.990469i	$-0.456018\pi$
$613$	6.82023	0.275467	0.137733	+	0.990469i	$0.456018\pi$
$614$	26.4261	1.06647
$615$	1.34289	0.0541507
$616$	−11.3497	−0.457294
$617$	27.9917	1.12690	0.563452	−	0.826149i	$-0.309473\pi$
$617$	27.9917	1.12690	0.563452	+	0.826149i	$0.309473\pi$
$618$	1.00000	0.0402259
$619$	−18.5850	−0.746995	−0.373498	−	0.927631i	$-0.621842\pi$
$619$	−18.5850	−0.746995	−0.373498	+	0.927631i	$0.621842\pi$
$620$	0.627809	0.0252134
$621$	1.00935	0.0405038
$622$	−5.10993	−0.204890
$623$	12.0896	0.484358
$624$	1.00000	0.0400320
$625$	−0.864429	−0.0345772
$626$	26.4548	1.05735
$627$	12.7348	0.508581
$628$	4.58778	0.183073
$629$	48.9511	1.95181
$630$	6.86572	0.273537
$631$	12.6504	0.503602	0.251801	−	0.967779i	$-0.418977\pi$
$631$	12.6504	0.503602	0.251801	+	0.967779i	$0.418977\pi$
$632$	3.70385	0.147331
$633$	0.460401	0.0182993
$634$	−6.08045	−0.241485
$635$	−4.91362	−0.194991
$636$	7.50659	0.297656
$637$	−16.7150	−0.662274
$638$	4.68714	0.185566
$639$	−9.19397	−0.363708
$640$	1.40985	0.0557294
$641$	8.11914	0.320687	0.160343	−	0.987061i	$-0.448740\pi$
$641$	8.11914	0.320687	0.160343	+	0.987061i	$0.448740\pi$
$642$	0.00520205	0.000205308 0
$643$	20.6268	0.813442	0.406721	−	0.913552i	$-0.366672\pi$
$643$	20.6268	0.813442	0.406721	+	0.913552i	$0.366672\pi$
$644$	−4.91533	−0.193691
$645$	5.02591	0.197895
$646$	29.4038	1.15688
$647$	48.7601	1.91696	0.958478	−	0.285166i	$-0.0920487\pi$
$647$	48.7601	1.91696	0.958478	+	0.285166i	$0.0920487\pi$
$648$	1.00000	0.0392837
$649$	−10.5367	−0.413601
$650$	3.01231	0.118153
$651$	−2.16853	−0.0849913
$652$	−3.09389	−0.121166
$653$	16.8596	0.659766	0.329883	−	0.944022i	$-0.392991\pi$
$653$	16.8596	0.659766	0.329883	+	0.944022i	$0.392991\pi$
$654$	12.1170	0.473812
$655$	0.760442	0.0297129
$656$	−0.952506	−0.0371891
$657$	−6.35844	−0.248066
$658$	4.30577	0.167857
$659$	−22.1441	−0.862613	−0.431306	−	0.902206i	$-0.641947\pi$
$659$	−22.1441	−0.862613	−0.431306	+	0.902206i	$0.641947\pi$
$660$	3.28585	0.127902
$661$	−12.0035	−0.466881	−0.233441	−	0.972371i	$-0.574998\pi$
$661$	−12.0035	−0.466881	−0.233441	+	0.972371i	$0.574998\pi$
$662$	3.03497	0.117957
$663$	5.38125	0.208991
$664$	14.4610	0.561194
$665$	37.5151	1.45477
$666$	9.09660	0.352486
$667$	2.02990	0.0785982
$668$	−11.7557	−0.454840
$669$	−19.9215	−0.770210
$670$	2.82992	0.109329
$671$	−15.4315	−0.595725
$672$	−4.86981	−0.187857
$673$	−8.44454	−0.325513	−0.162757	−	0.986666i	$-0.552039\pi$
$673$	−8.44454	−0.325513	−0.162757	+	0.986666i	$0.552039\pi$
$674$	13.8840	0.534792
$675$	3.01231	0.115944
$676$	1.00000	0.0384615
$677$	24.4985	0.941554	0.470777	−	0.882252i	$-0.343974\pi$
$677$	24.4985	0.941554	0.470777	+	0.882252i	$0.343974\pi$
$678$	3.32058	0.127526
$679$	29.6993	1.13975
$680$	7.58678	0.290940
$681$	20.1728	0.773023
$682$	−1.03783	−0.0397406
$683$	44.0120	1.68407	0.842037	−	0.539420i	$-0.181356\pi$
$683$	44.0120	1.68407	0.842037	+	0.539420i	$0.181356\pi$
$684$	5.46412	0.208926
$685$	2.74894	0.105032
$686$	47.3104	1.80632
$687$	16.5160	0.630123
$688$	−3.56484	−0.135908
$689$	7.50659	0.285978
$690$	1.42303	0.0541740
$691$	−39.9976	−1.52158	−0.760790	−	0.648998i	$-0.775188\pi$
$691$	−39.9976	−1.52158	−0.760790	+	0.648998i	$0.775188\pi$
$692$	3.99859	0.152004
$693$	−11.3497	−0.431141
$694$	25.4955	0.967797
$695$	14.9948	0.568786
$696$	2.01110	0.0762306
$697$	−5.12567	−0.194149
$698$	−19.0588	−0.721385
$699$	11.7843	0.445724
$700$	−14.6694	−0.554450
$701$	−43.4942	−1.64276	−0.821378	−	0.570385i	$-0.806794\pi$
$701$	−43.4942	−1.64276	−0.821378	+	0.570385i	$0.806794\pi$
$702$	1.00000	0.0377426
$703$	49.7049	1.87466
$704$	−2.33063	−0.0878390
$705$	−1.24656	−0.0469482
$706$	−28.7383	−1.08158
$707$	35.8092	1.34674
$708$	−4.52096	−0.169908
$709$	30.5582	1.14764	0.573819	−	0.818982i	$-0.305461\pi$
$709$	30.5582	1.14764	0.573819	+	0.818982i	$0.305461\pi$
$710$	−12.9622	−0.486461
$711$	3.70385	0.138905
$712$	2.48255	0.0930376
$713$	−0.449463	−0.0168325
$714$	−26.2057	−0.980723
$715$	3.28585	0.122884
$716$	−12.3417	−0.461233
$717$	−8.32974	−0.311080
$718$	−8.50496	−0.317402
$719$	13.7827	0.514008	0.257004	−	0.966410i	$-0.417265\pi$
$719$	13.7827	0.514008	0.257004	+	0.966410i	$0.417265\pi$
$720$	1.40985	0.0525422
$721$	−4.86981	−0.181361
$722$	10.8566	0.404039
$723$	10.9822	0.408433
$724$	22.4793	0.835437
$725$	6.05807	0.224991
$726$	5.56816	0.206654
$727$	−1.87934	−0.0697007	−0.0348504	−	0.999393i	$-0.511095\pi$
$727$	−1.87934	−0.0697007	−0.0348504	+	0.999393i	$0.511095\pi$
$728$	−4.86981	−0.180487
$729$	1.00000	0.0370370
$730$	−8.96447	−0.331790
$731$	−19.1833	−0.709520
$732$	−6.62115	−0.244725
$733$	13.4501	0.496791	0.248395	−	0.968659i	$-0.420097\pi$
$733$	13.4501	0.496791	0.248395	+	0.968659i	$0.420097\pi$
$734$	−9.47425	−0.349701
$735$	−23.5658	−0.869236
$736$	−1.00935	−0.0372051
$737$	−4.67814	−0.172322
$738$	−0.952506	−0.0350622
$739$	−22.0485	−0.811069	−0.405534	−	0.914080i	$-0.632915\pi$
$739$	−22.0485	−0.811069	−0.405534	+	0.914080i	$0.632915\pi$
$740$	12.8249	0.471452
$741$	5.46412	0.200729
$742$	−36.5556	−1.34200
$743$	7.21093	0.264543	0.132272	−	0.991214i	$-0.457773\pi$
$743$	7.21093	0.264543	0.132272	+	0.991214i	$0.457773\pi$
$744$	−0.445300	−0.0163255
$745$	−8.02909	−0.294163
$746$	−4.61157	−0.168842
$747$	14.4610	0.529099
$748$	−12.5417	−0.458571
$749$	−0.0253330	−0.000925647 0
$750$	11.2962	0.412479
$751$	−41.0774	−1.49894	−0.749469	−	0.662040i	$-0.769691\pi$
$751$	−41.0774	−1.49894	−0.749469	+	0.662040i	$0.769691\pi$
$752$	0.884177	0.0322426
$753$	−8.73482	−0.318315
$754$	2.01110	0.0732400
$755$	−6.72866	−0.244881
$756$	−4.86981	−0.177113
$757$	6.35811	0.231089	0.115545	−	0.993302i	$-0.463139\pi$
$757$	6.35811	0.231089	0.115545	+	0.993302i	$0.463139\pi$
$758$	21.9061	0.795665
$759$	−2.35242	−0.0853874
$760$	7.70361	0.279439
$761$	15.9103	0.576747	0.288373	−	0.957518i	$-0.406886\pi$
$761$	15.9103	0.576747	0.288373	+	0.957518i	$0.406886\pi$
$762$	3.48520	0.126255
$763$	−59.0074	−2.13621
$764$	−19.7321	−0.713882
$765$	7.58678	0.274301
$766$	−8.61706	−0.311347
$767$	−4.52096	−0.163242
$768$	−1.00000	−0.0360844
$769$	22.9658	0.828167	0.414083	−	0.910239i	$-0.364102\pi$
$769$	22.9658	0.828167	0.414083	+	0.910239i	$0.364102\pi$
$770$	−16.0015	−0.576653
$771$	9.60029	0.345746
$772$	17.3031	0.622751
$773$	15.0931	0.542861	0.271431	−	0.962458i	$-0.412503\pi$
$773$	15.0931	0.542861	0.271431	+	0.962458i	$0.412503\pi$
$774$	−3.56484	−0.128136
$775$	−1.34138	−0.0481839
$776$	6.09866	0.218929
$777$	−44.2987	−1.58921
$778$	31.6874	1.13605
$779$	−5.20460	−0.186474
$780$	1.40985	0.0504809
$781$	21.4278	0.766746
$782$	−5.43156	−0.194232
$783$	2.01110	0.0718709
$784$	16.7150	0.596966
$785$	6.46811	0.230857
$786$	−0.539376	−0.0192389
$787$	17.0156	0.606539	0.303269	−	0.952905i	$-0.401922\pi$
$787$	17.0156	0.606539	0.303269	+	0.952905i	$0.401922\pi$
$788$	4.26407	0.151901
$789$	22.0405	0.784663
$790$	5.22188	0.185786
$791$	−16.1706	−0.574959
$792$	−2.33063	−0.0828154
$793$	−6.62115	−0.235124
$794$	−32.9613	−1.16975
$795$	10.5832	0.375347
$796$	−20.1794	−0.715238
$797$	−8.33831	−0.295358	−0.147679	−	0.989035i	$-0.547180\pi$
$797$	−8.33831	−0.295358	−0.147679	+	0.989035i	$0.547180\pi$
$798$	−26.6092	−0.941955
$799$	4.75798	0.168325
$800$	−3.01231	−0.106501
$801$	2.48255	0.0877167
$802$	23.0372	0.813472
$803$	14.8192	0.522957
$804$	−2.00724	−0.0707900
$805$	−6.92990	−0.244247
$806$	−0.445300	−0.0156850
$807$	−13.0683	−0.460026
$808$	7.35330	0.258688
$809$	3.28317	0.115430	0.0577151	−	0.998333i	$-0.481619\pi$
$809$	3.28317	0.115430	0.0577151	+	0.998333i	$0.481619\pi$
$810$	1.40985	0.0495372
$811$	33.7600	1.18547	0.592736	−	0.805397i	$-0.298048\pi$
$811$	33.7600	1.18547	0.592736	+	0.805397i	$0.298048\pi$
$812$	−9.79368	−0.343691
$813$	−9.85282	−0.345553
$814$	−21.2008	−0.743089
$815$	−4.36193	−0.152792
$816$	−5.38125	−0.188382
$817$	−19.4787	−0.681473
$818$	2.78238	0.0972837
$819$	−4.86981	−0.170165
$820$	−1.34289	−0.0468959
$821$	−15.4035	−0.537587	−0.268793	−	0.963198i	$-0.586625\pi$
$821$	−15.4035	−0.537587	−0.268793	+	0.963198i	$0.586625\pi$
$822$	−1.94980	−0.0680072
$823$	5.10990	0.178120	0.0890599	−	0.996026i	$-0.471614\pi$
$823$	5.10990	0.178120	0.0890599	+	0.996026i	$0.471614\pi$
$824$	−1.00000	−0.0348367
$825$	−7.02059	−0.244425
$826$	22.0162	0.766041
$827$	−33.9926	−1.18204	−0.591020	−	0.806657i	$-0.701274\pi$
$827$	−33.9926	−1.18204	−0.591020	+	0.806657i	$0.701274\pi$
$828$	−1.00935	−0.0350773
$829$	−14.5425	−0.505082	−0.252541	−	0.967586i	$-0.581266\pi$
$829$	−14.5425	−0.505082	−0.252541	+	0.967586i	$0.581266\pi$
$830$	20.3879	0.707673
$831$	28.2551	0.980160
$832$	−1.00000	−0.0346688
$833$	89.9478	3.11651
$834$	−10.6357	−0.368285
$835$	−16.5738	−0.573559
$836$	−12.7348	−0.440444
$837$	−0.445300	−0.0153918
$838$	6.97312	0.240882
$839$	−38.8377	−1.34083	−0.670414	−	0.741987i	$-0.733883\pi$
$839$	−38.8377	−1.34083	−0.670414	+	0.741987i	$0.733883\pi$
$840$	−6.86572	−0.236890
$841$	−24.9555	−0.860533
$842$	4.97348	0.171398
$843$	−1.71433	−0.0590446
$844$	−0.460401	−0.0158477
$845$	1.40985	0.0485005
$846$	0.884177	0.0303986
$847$	−27.1159	−0.931712
$848$	−7.50659	−0.257777
$849$	−16.3986	−0.562797
$850$	−16.2100	−0.555999
$851$	−9.18164	−0.314743
$852$	9.19397	0.314980
$853$	37.5994	1.28738	0.643690	−	0.765286i	$-0.277403\pi$
$853$	37.5994	1.28738	0.643690	+	0.765286i	$0.277403\pi$
$854$	32.2437	1.10336
$855$	7.70361	0.263458
$856$	−0.00520205	−0.000177802 0
$857$	−53.1598	−1.81590	−0.907952	−	0.419073i	$-0.862355\pi$
$857$	−53.1598	−1.81590	−0.907952	+	0.419073i	$0.862355\pi$
$858$	−2.33063	−0.0795665
$859$	0.402154	0.0137213	0.00686067	−	0.999976i	$-0.497816\pi$
$859$	0.402154	0.0137213	0.00686067	+	0.999976i	$0.497816\pi$
$860$	−5.02591	−0.171382
$861$	4.63852	0.158080
$862$	−10.6617	−0.363137
$863$	25.1477	0.856039	0.428019	−	0.903770i	$-0.359211\pi$
$863$	25.1477	0.856039	0.428019	+	0.903770i	$0.359211\pi$
$864$	−1.00000	−0.0340207
$865$	5.63743	0.191678
$866$	36.5166	1.24088
$867$	−11.9579	−0.406111
$868$	2.16853	0.0736046
$869$	−8.63230	−0.292831
$870$	2.83536	0.0961277
$871$	−2.00724	−0.0680128
$872$	−12.1170	−0.410333
$873$	6.09866	0.206408
$874$	−5.51520	−0.186554
$875$	−55.0103	−1.85969
$876$	6.35844	0.214832
$877$	−24.7664	−0.836302	−0.418151	−	0.908378i	$-0.637322\pi$
$877$	−24.7664	−0.836302	−0.418151	+	0.908378i	$0.637322\pi$
$878$	28.5065	0.962048
$879$	−22.5655	−0.761117
$880$	−3.28585	−0.110766
$881$	3.38245	0.113958	0.0569788	−	0.998375i	$-0.481853\pi$
$881$	3.38245	0.113958	0.0569788	+	0.998375i	$0.481853\pi$
$882$	16.7150	0.562825
$883$	17.7294	0.596643	0.298322	−	0.954465i	$-0.403573\pi$
$883$	17.7294	0.596643	0.298322	+	0.954465i	$0.403573\pi$
$884$	−5.38125	−0.180991
$885$	−6.37389	−0.214256
$886$	−18.8447	−0.633101
$887$	−23.9345	−0.803641	−0.401820	−	0.915719i	$-0.631622\pi$
$887$	−23.9345	−0.803641	−0.401820	+	0.915719i	$0.631622\pi$
$888$	−9.09660	−0.305262
$889$	−16.9722	−0.569231
$890$	3.50004	0.117322
$891$	−2.33063	−0.0780791
$892$	19.9215	0.667021
$893$	4.83125	0.161671
$894$	5.69498	0.190469
$895$	−17.4001	−0.581620
$896$	4.86981	0.162689
$897$	−1.00935	−0.0337012
$898$	26.6224	0.888402
$899$	−0.895544	−0.0298681
$900$	−3.01231	−0.100410
$901$	−40.3948	−1.34575
$902$	2.21994	0.0739159
$903$	17.3601	0.577708
$904$	−3.32058	−0.110441
$905$	31.6926	1.05350
$906$	4.77259	0.158559
$907$	17.8828	0.593789	0.296894	−	0.954910i	$-0.404049\pi$
$907$	17.8828	0.593789	0.296894	+	0.954910i	$0.404049\pi$
$908$	−20.1728	−0.669458
$909$	7.35330	0.243894
$910$	−6.86572	−0.227596
$911$	−29.8178	−0.987906	−0.493953	−	0.869489i	$-0.664449\pi$
$911$	−29.8178	−0.987906	−0.493953	+	0.869489i	$0.664449\pi$
$912$	−5.46412	−0.180935
$913$	−33.7032	−1.11541
$914$	−2.05556	−0.0679919
$915$	−9.33486	−0.308601
$916$	−16.5160	−0.545702
$917$	2.62666	0.0867399
$918$	−5.38125	−0.177608
$919$	−6.38995	−0.210785	−0.105392	−	0.994431i	$-0.533610\pi$
$919$	−6.38995	−0.210785	−0.105392	+	0.994431i	$0.533610\pi$
$920$	−1.42303	−0.0469161
$921$	−26.4261	−0.870770
$922$	12.0356	0.396371
$923$	9.19397	0.302623
$924$	11.3497	0.373379
$925$	−27.4018	−0.900965
$926$	−26.5610	−0.872850
$927$	−1.00000	−0.0328443
$928$	−2.01110	−0.0660177
$929$	−38.2288	−1.25425	−0.627124	−	0.778920i	$-0.715768\pi$
$929$	−38.2288	−1.25425	−0.627124	+	0.778920i	$0.715768\pi$
$930$	−0.627809	−0.0205867
$931$	91.3329	2.99331
$932$	−11.7843	−0.386008
$933$	5.10993	0.167292
$934$	−25.8182	−0.844796
$935$	−17.6820	−0.578263
$936$	−1.00000	−0.0326860
$937$	6.86973	0.224424	0.112212	−	0.993684i	$-0.464206\pi$
$937$	6.86973	0.224424	0.112212	+	0.993684i	$0.464206\pi$
$938$	9.77489	0.319161
$939$	−26.4548	−0.863320
$940$	1.24656	0.0406583
$941$	−11.5963	−0.378027	−0.189014	−	0.981974i	$-0.560529\pi$
$941$	−11.5963	−0.378027	−0.189014	+	0.981974i	$0.560529\pi$
$942$	−4.58778	−0.149478
$943$	0.961410	0.0313078
$944$	4.52096	0.147145
$945$	−6.86572	−0.223342
$946$	8.30833	0.270127
$947$	8.74704	0.284241	0.142120	−	0.989849i	$-0.454608\pi$
$947$	8.74704	0.284241	0.142120	+	0.989849i	$0.454608\pi$
$948$	−3.70385	−0.120295
$949$	6.35844	0.206404
$950$	−16.4596	−0.534020
$951$	6.08045	0.197172
$952$	26.2057	0.849331
$953$	−8.36637	−0.271013	−0.135507	−	0.990776i	$-0.543266\pi$
$953$	−8.36637	−0.271013	−0.135507	+	0.990776i	$0.543266\pi$
$954$	−7.50659	−0.243035
$955$	−27.8194	−0.900214
$956$	8.32974	0.269403
$957$	−4.68714	−0.151514
$958$	−4.91205	−0.158701
$959$	9.49518	0.306615
$960$	−1.40985	−0.0455029
$961$	−30.8017	−0.993603
$962$	−9.09660	−0.293286
$963$	−0.00520205	−0.000167634 0
$964$	−10.9822	−0.353713
$965$	24.3948	0.785296
$966$	4.91533	0.158148
$967$	−3.64903	−0.117345	−0.0586725	−	0.998277i	$-0.518687\pi$
$967$	−3.64903	−0.117345	−0.0586725	+	0.998277i	$0.518687\pi$
$968$	−5.56816	−0.178967
$969$	−29.4038	−0.944586
$970$	8.59822	0.276072
$971$	−26.8560	−0.861850	−0.430925	−	0.902388i	$-0.641813\pi$
$971$	−26.8560	−0.861850	−0.430925	+	0.902388i	$0.641813\pi$
$972$	−1.00000	−0.0320750
$973$	51.7939	1.66044
$974$	−33.5943	−1.07643
$975$	−3.01231	−0.0964712
$976$	6.62115	0.211938
$977$	−1.16535	−0.0372828	−0.0186414	−	0.999826i	$-0.505934\pi$
$977$	−1.16535	−0.0372828	−0.0186414	+	0.999826i	$0.505934\pi$
$978$	3.09389	0.0989315
$979$	−5.78592	−0.184919
$980$	23.5658	0.752781
$981$	−12.1170	−0.386866
$982$	20.7294	0.661501
$983$	−36.4004	−1.16099	−0.580496	−	0.814263i	$-0.697141\pi$
$983$	−36.4004	−1.16099	−0.580496	+	0.814263i	$0.697141\pi$
$984$	0.952506	0.0303648
$985$	6.01172	0.191549
$986$	−10.8222	−0.344651
$987$	−4.30577	−0.137054
$988$	−5.46412	−0.173837
$989$	3.59817	0.114415
$990$	−3.28585	−0.104431
$991$	−41.3784	−1.31443	−0.657214	−	0.753704i	$-0.728265\pi$
$991$	−41.3784	−1.31443	−0.657214	+	0.753704i	$0.728265\pi$
$992$	0.445300	0.0141383
$993$	−3.03497	−0.0963119
$994$	−44.7729	−1.42011
$995$	−28.4499	−0.901924
$996$	−14.4610	−0.458213
$997$	5.37352	0.170181	0.0850906	−	0.996373i	$-0.472882\pi$
$997$	5.37352	0.170181	0.0850906	+	0.996373i	$0.472882\pi$
$998$	1.53739	0.0486653
$999$	−9.09660	−0.287804

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8034.2.a.bc.1.11	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8034.2.a.bc.1.11	✓	15	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:	\( N \)	\(=\)	\( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8034.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.40985 q^{5} -1.00000 q^{6} +4.86981 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.40985 q^{5} -1.00000 q^{6} +4.86981 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.40985 q^{10} -2.33063 q^{11} -1.00000 q^{12} -1.00000 q^{13} +4.86981 q^{14} -1.40985 q^{15} +1.00000 q^{16} +5.38125 q^{17} +1.00000 q^{18} +5.46412 q^{19} +1.40985 q^{20} -4.86981 q^{21} -2.33063 q^{22} -1.00935 q^{23} -1.00000 q^{24} -3.01231 q^{25} -1.00000 q^{26} -1.00000 q^{27} +4.86981 q^{28} -2.01110 q^{29} -1.40985 q^{30} +0.445300 q^{31} +1.00000 q^{32} +2.33063 q^{33} +5.38125 q^{34} +6.86572 q^{35} +1.00000 q^{36} +9.09660 q^{37} +5.46412 q^{38} +1.00000 q^{39} +1.40985 q^{40} -0.952506 q^{41} -4.86981 q^{42} -3.56484 q^{43} -2.33063 q^{44} +1.40985 q^{45} -1.00935 q^{46} +0.884177 q^{47} -1.00000 q^{48} +16.7150 q^{49} -3.01231 q^{50} -5.38125 q^{51} -1.00000 q^{52} -7.50659 q^{53} -1.00000 q^{54} -3.28585 q^{55} +4.86981 q^{56} -5.46412 q^{57} -2.01110 q^{58} +4.52096 q^{59} -1.40985 q^{60} +6.62115 q^{61} +0.445300 q^{62} +4.86981 q^{63} +1.00000 q^{64} -1.40985 q^{65} +2.33063 q^{66} +2.00724 q^{67} +5.38125 q^{68} +1.00935 q^{69} +6.86572 q^{70} -9.19397 q^{71} +1.00000 q^{72} -6.35844 q^{73} +9.09660 q^{74} +3.01231 q^{75} +5.46412 q^{76} -11.3497 q^{77} +1.00000 q^{78} +3.70385 q^{79} +1.40985 q^{80} +1.00000 q^{81} -0.952506 q^{82} +14.4610 q^{83} -4.86981 q^{84} +7.58678 q^{85} -3.56484 q^{86} +2.01110 q^{87} -2.33063 q^{88} +2.48255 q^{89} +1.40985 q^{90} -4.86981 q^{91} -1.00935 q^{92} -0.445300 q^{93} +0.884177 q^{94} +7.70361 q^{95} -1.00000 q^{96} +6.09866 q^{97} +16.7150 q^{98} -2.33063 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 15 q^{2} - 15 q^{3} + 15 q^{4} - q^{5} - 15 q^{6} + 5 q^{7} + 15 q^{8} + 15 q^{9}+O(q^{10}) \) 15 * q + 15 * q^2 - 15 * q^3 + 15 * q^4 - q^5 - 15 * q^6 + 5 * q^7 + 15 * q^8 + 15 * q^9	\( 15 q + 15 q^{2} - 15 q^{3} + 15 q^{4} - q^{5} - 15 q^{6} + 5 q^{7} + 15 q^{8} + 15 q^{9} - q^{10} + 3 q^{11} - 15 q^{12} - 15 q^{13} + 5 q^{14} + q^{15} + 15 q^{16} - 2 q^{17} + 15 q^{18} + 8 q^{19} - q^{20} - 5 q^{21} + 3 q^{22} + 3 q^{23} - 15 q^{24} + 22 q^{25} - 15 q^{26} - 15 q^{27} + 5 q^{28} + 26 q^{29} + q^{30} + 15 q^{32} - 3 q^{33} - 2 q^{34} - 8 q^{35} + 15 q^{36} + 25 q^{37} + 8 q^{38} + 15 q^{39} - q^{40} - q^{41} - 5 q^{42} + 10 q^{43} + 3 q^{44} - q^{45} + 3 q^{46} - 3 q^{47} - 15 q^{48} + 32 q^{49} + 22 q^{50} + 2 q^{51} - 15 q^{52} + 13 q^{53} - 15 q^{54} - 2 q^{55} + 5 q^{56} - 8 q^{57} + 26 q^{58} + 28 q^{59} + q^{60} + 22 q^{61} + 5 q^{63} + 15 q^{64} + q^{65} - 3 q^{66} + 29 q^{67} - 2 q^{68} - 3 q^{69} - 8 q^{70} + 18 q^{71} + 15 q^{72} + 23 q^{73} + 25 q^{74} - 22 q^{75} + 8 q^{76} + 17 q^{77} + 15 q^{78} + 27 q^{79} - q^{80} + 15 q^{81} - q^{82} + 7 q^{83} - 5 q^{84} + 43 q^{85} + 10 q^{86} - 26 q^{87} + 3 q^{88} + 35 q^{89} - q^{90} - 5 q^{91} + 3 q^{92} - 3 q^{94} + 6 q^{95} - 15 q^{96} + 19 q^{97} + 32 q^{98} + 3 q^{99}+O(q^{100}) \) 15 * q + 15 * q^2 - 15 * q^3 + 15 * q^4 - q^5 - 15 * q^6 + 5 * q^7 + 15 * q^8 + 15 * q^9 - q^10 + 3 * q^11 - 15 * q^12 - 15 * q^13 + 5 * q^14 + q^15 + 15 * q^16 - 2 * q^17 + 15 * q^18 + 8 * q^19 - q^20 - 5 * q^21 + 3 * q^22 + 3 * q^23 - 15 * q^24 + 22 * q^25 - 15 * q^26 - 15 * q^27 + 5 * q^28 + 26 * q^29 + q^30 + 15 * q^32 - 3 * q^33 - 2 * q^34 - 8 * q^35 + 15 * q^36 + 25 * q^37 + 8 * q^38 + 15 * q^39 - q^40 - q^41 - 5 * q^42 + 10 * q^43 + 3 * q^44 - q^45 + 3 * q^46 - 3 * q^47 - 15 * q^48 + 32 * q^49 + 22 * q^50 + 2 * q^51 - 15 * q^52 + 13 * q^53 - 15 * q^54 - 2 * q^55 + 5 * q^56 - 8 * q^57 + 26 * q^58 + 28 * q^59 + q^60 + 22 * q^61 + 5 * q^63 + 15 * q^64 + q^65 - 3 * q^66 + 29 * q^67 - 2 * q^68 - 3 * q^69 - 8 * q^70 + 18 * q^71 + 15 * q^72 + 23 * q^73 + 25 * q^74 - 22 * q^75 + 8 * q^76 + 17 * q^77 + 15 * q^78 + 27 * q^79 - q^80 + 15 * q^81 - q^82 + 7 * q^83 - 5 * q^84 + 43 * q^85 + 10 * q^86 - 26 * q^87 + 3 * q^88 + 35 * q^89 - q^90 - 5 * q^91 + 3 * q^92 - 3 * q^94 + 6 * q^95 - 15 * q^96 + 19 * q^97 + 32 * q^98 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more