LMFDB - Embedded newform 5070.2.b.s.1351.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5070,2,Mod(1351,5070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5070, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("5070.1351");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5070.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$40.4841538248$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.153664.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 5x^{4} + 6x^{2} + 1 $ x^6 + 5x^4 + 6x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.4
Root			$-1.24698i$ of defining polynomial
Character	$\chi$	$=$	5070.1351
Dual form			5070.2.b.s.1351.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} -3.04892i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} -3.04892i q^{7} -1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} +6.24698i q^{11} +1.00000 q^{12} +3.04892 q^{14} -1.00000i q^{15} +1.00000 q^{16} -2.69202 q^{17} +1.00000i q^{18} -5.82908i q^{19} -1.00000i q^{20} +3.04892i q^{21} -6.24698 q^{22} +5.62565 q^{23} +1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} +3.04892i q^{28} -5.14675 q^{29} +1.00000 q^{30} -3.53319i q^{31} +1.00000i q^{32} -6.24698i q^{33} -2.69202i q^{34} +3.04892 q^{35} -1.00000 q^{36} +4.65279i q^{37} +5.82908 q^{38} +1.00000 q^{40} -3.77479i q^{41} -3.04892 q^{42} -2.85086 q^{43} -6.24698i q^{44} +1.00000i q^{45} +5.62565i q^{46} +3.61596i q^{47} -1.00000 q^{48} -2.29590 q^{49} -1.00000i q^{50} +2.69202 q^{51} +0.664874 q^{53} -1.00000i q^{54} -6.24698 q^{55} -3.04892 q^{56} +5.82908i q^{57} -5.14675i q^{58} +12.9487i q^{59} +1.00000i q^{60} -7.91723 q^{61} +3.53319 q^{62} -3.04892i q^{63} -1.00000 q^{64} +6.24698 q^{66} -0.198062i q^{67} +2.69202 q^{68} -5.62565 q^{69} +3.04892i q^{70} -0.374354i q^{71} -1.00000i q^{72} -14.6136i q^{73} -4.65279 q^{74} +1.00000 q^{75} +5.82908i q^{76} +19.0465 q^{77} -11.2567 q^{79} +1.00000i q^{80} +1.00000 q^{81} +3.77479 q^{82} +6.83877i q^{83} -3.04892i q^{84} -2.69202i q^{85} -2.85086i q^{86} +5.14675 q^{87} +6.24698 q^{88} -9.50365i q^{89} -1.00000 q^{90} -5.62565 q^{92} +3.53319i q^{93} -3.61596 q^{94} +5.82908 q^{95} -1.00000i q^{96} -0.335126i q^{97} -2.29590i q^{98} +6.24698i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{3} - 6 q^{4} + 6 q^{9}+O(q^{10}) $ 6 * q - 6 * q^3 - 6 * q^4 + 6 * q^9	$ 6 q - 6 q^{3} - 6 q^{4} + 6 q^{9} - 6 q^{10} + 6 q^{12} + 6 q^{16} - 6 q^{17} - 28 q^{22} + 10 q^{23} - 6 q^{25} - 6 q^{27} + 24 q^{29} + 6 q^{30} - 6 q^{36} + 14 q^{38} + 6 q^{40} + 10 q^{43} - 6 q^{48} + 14 q^{49} + 6 q^{51} + 6 q^{53} - 28 q^{55} - 34 q^{61} + 28 q^{62} - 6 q^{64} + 28 q^{66} + 6 q^{68} - 10 q^{69} + 8 q^{74} + 6 q^{75} + 14 q^{77} - 14 q^{79} + 6 q^{81} + 26 q^{82} - 24 q^{87} + 28 q^{88} - 6 q^{90} - 10 q^{92} - 42 q^{94} + 14 q^{95}+O(q^{100}) $ 6 * q - 6 * q^3 - 6 * q^4 + 6 * q^9 - 6 * q^10 + 6 * q^12 + 6 * q^16 - 6 * q^17 - 28 * q^22 + 10 * q^23 - 6 * q^25 - 6 * q^27 + 24 * q^29 + 6 * q^30 - 6 * q^36 + 14 * q^38 + 6 * q^40 + 10 * q^43 - 6 * q^48 + 14 * q^49 + 6 * q^51 + 6 * q^53 - 28 * q^55 - 34 * q^61 + 28 * q^62 - 6 * q^64 + 28 * q^66 + 6 * q^68 - 10 * q^69 + 8 * q^74 + 6 * q^75 + 14 * q^77 - 14 * q^79 + 6 * q^81 + 26 * q^82 - 24 * q^87 + 28 * q^88 - 6 * q^90 - 10 * q^92 - 42 * q^94 + 14 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times$.

$n$	$1691$	$1861$	$4057$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.00000	−0.500000
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	− 1.00000i	− 0.408248i
$7$	− 3.04892i	− 1.15238i	−0.817315	−	0.576191i	$-0.804538\pi$
$7$	− 3.04892i	− 1.15238i	0.817315	−	0.576191i	$-0.195462\pi$
$8$	− 1.00000i	− 0.353553i
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	6.24698i	1.88354i	0.336264	+	0.941768i	$0.390836\pi$
$11$	6.24698i	1.88354i	−0.336264	+	0.941768i	$0.609164\pi$
$12$	1.00000	0.288675
$13$	0	0
$13$	0	0
$14$	3.04892	0.814857
$15$	− 1.00000i	− 0.258199i
$16$	1.00000	0.250000
$17$	−2.69202	−0.652911	−0.326456	−	0.945213i	$-0.605854\pi$
$17$	−2.69202	−0.652911	−0.326456	+	0.945213i	$0.605854\pi$
$18$	1.00000i	0.235702i
$19$	− 5.82908i	− 1.33728i	−0.743585	−	0.668642i	$-0.766876\pi$
$19$	− 5.82908i	− 1.33728i	0.743585	−	0.668642i	$-0.233124\pi$
$20$	− 1.00000i	− 0.223607i
$21$	3.04892i	0.665328i
$22$	−6.24698	−1.33186
$23$	5.62565	1.17303	0.586514	−	0.809939i	$-0.300500\pi$
$23$	5.62565	1.17303	0.586514	+	0.809939i	$0.300500\pi$
$24$	1.00000i	0.204124i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	3.04892i	0.576191i
$29$	−5.14675	−0.955728	−0.477864	−	0.878434i	$-0.658589\pi$
$29$	−5.14675	−0.955728	−0.477864	+	0.878434i	$0.658589\pi$
$30$	1.00000	0.182574
$31$	− 3.53319i	− 0.634579i	−0.948329	−	0.317290i	$-0.897227\pi$
$31$	− 3.53319i	− 0.634579i	0.948329	−	0.317290i	$-0.102773\pi$
$32$	1.00000i	0.176777i
$33$	− 6.24698i	− 1.08746i
$34$	− 2.69202i	− 0.461678i
$35$	3.04892	0.515361
$36$	−1.00000	−0.166667
$37$	4.65279i	0.764914i	0.923973	+	0.382457i	$0.124922\pi$
$37$	4.65279i	0.764914i	−0.923973	+	0.382457i	$0.875078\pi$
$38$	5.82908	0.945602
$39$	0	0
$40$	1.00000	0.158114
$41$	− 3.77479i	− 0.589523i	−0.955571	−	0.294762i	$-0.904760\pi$
$41$	− 3.77479i	− 0.589523i	0.955571	−	0.294762i	$-0.0952402\pi$
$42$	−3.04892	−0.470458
$43$	−2.85086	−0.434751	−0.217376	−	0.976088i	$-0.569750\pi$
$43$	−2.85086	−0.434751	−0.217376	+	0.976088i	$0.569750\pi$
$44$	− 6.24698i	− 0.941768i
$45$	1.00000i	0.149071i
$46$	5.62565i	0.829456i
$47$	3.61596i	0.527442i	0.964599	+	0.263721i	$0.0849497\pi$
$47$	3.61596i	0.527442i	−0.964599	+	0.263721i	$0.915050\pi$
$48$	−1.00000	−0.144338
$49$	−2.29590	−0.327985
$50$	− 1.00000i	− 0.141421i
$51$	2.69202	0.376958
$52$	0	0
$53$	0.664874	0.0913275	0.0456638	−	0.998957i	$-0.485460\pi$
$53$	0.664874	0.0913275	0.0456638	+	0.998957i	$0.485460\pi$
$54$	− 1.00000i	− 0.136083i
$55$	−6.24698	−0.842343
$56$	−3.04892	−0.407429
$57$	5.82908i	0.772081i
$58$	− 5.14675i	− 0.675802i
$59$	12.9487i	1.68578i	0.538089	+	0.842888i	$0.319146\pi$
$59$	12.9487i	1.68578i	−0.538089	+	0.842888i	$0.680854\pi$
$60$	1.00000i	0.129099i
$61$	−7.91723	−1.01370	−0.506849	−	0.862035i	$-0.669190\pi$
$61$	−7.91723	−1.01370	−0.506849	+	0.862035i	$0.669190\pi$
$62$	3.53319	0.448715
$63$	− 3.04892i	− 0.384127i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	6.24698	0.768950
$67$	− 0.198062i	− 0.0241972i	−0.999927	−	0.0120986i	$-0.996149\pi$
$67$	− 0.198062i	− 0.0241972i	0.999927	−	0.0120986i	$-0.00385119\pi$
$68$	2.69202	0.326456
$69$	−5.62565	−0.677248
$70$	3.04892i	0.364415i
$71$	− 0.374354i	− 0.0444277i	−0.999753	−	0.0222138i	$-0.992929\pi$
$71$	− 0.374354i	− 0.0444277i	0.999753	−	0.0222138i	$-0.00707147\pi$
$72$	− 1.00000i	− 0.117851i
$73$	− 14.6136i	− 1.71039i	−0.518308	−	0.855194i	$-0.673438\pi$
$73$	− 14.6136i	− 1.71039i	0.518308	−	0.855194i	$-0.326562\pi$
$74$	−4.65279	−0.540876
$75$	1.00000	0.115470
$76$	5.82908i	0.668642i
$77$	19.0465	2.17055
$78$	0	0
$79$	−11.2567	−1.26647	−0.633237	−	0.773958i	$-0.718274\pi$
$79$	−11.2567	−1.26647	−0.633237	+	0.773958i	$0.718274\pi$
$80$	1.00000i	0.111803i
$81$	1.00000	0.111111
$82$	3.77479	0.416856
$83$	6.83877i	0.750653i	0.926893	+	0.375326i	$0.122469\pi$
$83$	6.83877i	0.750653i	−0.926893	+	0.375326i	$0.877531\pi$
$84$	− 3.04892i	− 0.332664i
$85$	− 2.69202i	− 0.291991i
$86$	− 2.85086i	− 0.307416i
$87$	5.14675	0.551790
$88$	6.24698	0.665930
$89$	− 9.50365i	− 1.00738i	−0.863883	−	0.503692i	$-0.831975\pi$
$89$	− 9.50365i	− 1.00738i	0.863883	−	0.503692i	$-0.168025\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	−5.62565	−0.586514
$93$	3.53319i	0.366375i
$94$	−3.61596	−0.372957
$95$	5.82908	0.598051
$96$	− 1.00000i	− 0.102062i
$97$	− 0.335126i	− 0.0340268i	−0.999855	−	0.0170134i	$-0.994584\pi$
$97$	− 0.335126i	− 0.0340268i	0.999855	−	0.0170134i	$-0.00541580\pi$
$98$	− 2.29590i	− 0.231921i
$99$	6.24698i	0.627845i
$100$	1.00000	0.100000
$101$	−0.274127	−0.0272766	−0.0136383	−	0.999907i	$-0.504341\pi$
$101$	−0.274127	−0.0272766	−0.0136383	+	0.999907i	$0.504341\pi$
$102$	2.69202i	0.266550i
$103$	9.00000	0.886796	0.443398	−	0.896325i	$-0.353773\pi$
$103$	9.00000	0.886796	0.443398	+	0.896325i	$0.353773\pi$
$104$	0	0
$105$	−3.04892	−0.297544
$106$	0.664874i	0.0645783i
$107$	−10.8509	−1.04899	−0.524496	−	0.851413i	$-0.675746\pi$
$107$	−10.8509	−1.04899	−0.524496	+	0.851413i	$0.675746\pi$
$108$	1.00000	0.0962250
$109$	− 1.02177i	− 0.0978678i	−0.998802	−	0.0489339i	$-0.984418\pi$
$109$	− 1.02177i	− 0.0978678i	0.998802	−	0.0489339i	$-0.0155824\pi$
$110$	− 6.24698i	− 0.595626i
$111$	− 4.65279i	− 0.441624i
$112$	− 3.04892i	− 0.288096i
$113$	19.6383	1.84742	0.923709	−	0.383095i	$-0.125142\pi$
$113$	19.6383	1.84742	0.923709	+	0.383095i	$0.125142\pi$
$114$	−5.82908	−0.545944
$115$	5.62565i	0.524594i
$116$	5.14675	0.477864
$117$	0	0
$118$	−12.9487	−1.19202
$119$	8.20775i	0.752403i
$120$	−1.00000	−0.0912871
$121$	−28.0248	−2.54770
$122$	− 7.91723i	− 0.716792i
$123$	3.77479i	0.340361i
$124$	3.53319i	0.317290i
$125$	− 1.00000i	− 0.0894427i
$126$	3.04892	0.271619
$127$	−18.5187	−1.64327	−0.821635	−	0.570014i	$-0.806938\pi$
$127$	−18.5187	−1.64327	−0.821635	+	0.570014i	$0.806938\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	2.85086	0.251004
$130$	0	0
$131$	−12.2349	−1.06897	−0.534484	−	0.845179i	$-0.679494\pi$
$131$	−12.2349	−1.06897	−0.534484	+	0.845179i	$0.679494\pi$
$132$	6.24698i	0.543730i
$133$	−17.7724	−1.54106
$134$	0.198062	0.0171100
$135$	− 1.00000i	− 0.0860663i
$136$	2.69202i	0.230839i
$137$	19.3002i	1.64893i	0.565914	+	0.824464i	$0.308523\pi$
$137$	19.3002i	1.64893i	−0.565914	+	0.824464i	$0.691477\pi$
$138$	− 5.62565i	− 0.478887i
$139$	19.5133	1.65510	0.827550	−	0.561392i	$-0.189734\pi$
$139$	19.5133	1.65510	0.827550	+	0.561392i	$0.189734\pi$
$140$	−3.04892	−0.257681
$141$	− 3.61596i	− 0.304519i
$142$	0.374354	0.0314151
$143$	0	0
$144$	1.00000	0.0833333
$145$	− 5.14675i	− 0.427414i
$146$	14.6136	1.20943
$147$	2.29590	0.189362
$148$	− 4.65279i	− 0.382457i
$149$	− 18.0248i	− 1.47665i	−0.674448	−	0.738323i	$-0.735618\pi$
$149$	− 18.0248i	− 1.47665i	0.674448	−	0.738323i	$-0.264382\pi$
$150$	1.00000i	0.0816497i
$151$	− 5.30127i	− 0.431412i	−0.976458	−	0.215706i	$-0.930795\pi$
$151$	− 5.30127i	− 0.431412i	0.976458	−	0.215706i	$-0.0692052\pi$
$152$	−5.82908	−0.472801
$153$	−2.69202	−0.217637
$154$	19.0465i	1.53481i
$155$	3.53319	0.283792
$156$	0	0
$157$	7.41789	0.592012	0.296006	−	0.955186i	$-0.404345\pi$
$157$	7.41789	0.592012	0.296006	+	0.955186i	$0.404345\pi$
$158$	− 11.2567i	− 0.895532i
$159$	−0.664874	−0.0527280
$160$	−1.00000	−0.0790569
$161$	− 17.1521i	− 1.35178i
$162$	1.00000i	0.0785674i
$163$	− 24.0151i	− 1.88101i	−0.339787	−	0.940503i	$-0.610355\pi$
$163$	− 24.0151i	− 1.88101i	0.339787	−	0.940503i	$-0.389645\pi$
$164$	3.77479i	0.294762i
$165$	6.24698	0.486327
$166$	−6.83877	−0.530792
$167$	− 11.9922i	− 0.927987i	−0.885839	−	0.463993i	$-0.846416\pi$
$167$	− 11.9922i	− 0.927987i	0.885839	−	0.463993i	$-0.153584\pi$
$168$	3.04892	0.235229
$169$	0	0
$170$	2.69202	0.206469
$171$	− 5.82908i	− 0.445761i
$172$	2.85086	0.217376
$173$	−19.8823	−1.51162	−0.755812	−	0.654788i	$-0.772758\pi$
$173$	−19.8823	−1.51162	−0.755812	+	0.654788i	$0.772758\pi$
$174$	5.14675i	0.390174i
$175$	3.04892i	0.230476i
$176$	6.24698i	0.470884i
$177$	− 12.9487i	− 0.973283i
$178$	9.50365	0.712329
$179$	−16.7453	−1.25160	−0.625799	−	0.779984i	$-0.715227\pi$
$179$	−16.7453	−1.25160	−0.625799	+	0.779984i	$0.715227\pi$
$180$	− 1.00000i	− 0.0745356i
$181$	3.93900	0.292784	0.146392	−	0.989227i	$-0.453234\pi$
$181$	3.93900	0.292784	0.146392	+	0.989227i	$0.453234\pi$
$182$	0	0
$183$	7.91723	0.585259
$184$	− 5.62565i	− 0.414728i
$185$	−4.65279	−0.342080
$186$	−3.53319	−0.259066
$187$	− 16.8170i	− 1.22978i
$188$	− 3.61596i	− 0.263721i
$189$	3.04892i	0.221776i
$190$	5.82908i	0.422886i
$191$	−27.1540	−1.96480	−0.982399	−	0.186795i	$-0.940190\pi$
$191$	−27.1540	−1.96480	−0.982399	+	0.186795i	$0.940190\pi$
$192$	1.00000	0.0721688
$193$	− 15.3937i	− 1.10807i	−0.832495	−	0.554033i	$-0.813088\pi$
$193$	− 15.3937i	− 1.10807i	0.832495	−	0.554033i	$-0.186912\pi$
$194$	0.335126	0.0240606
$195$	0	0
$196$	2.29590	0.163993
$197$	− 23.5036i	− 1.67457i	−0.546770	−	0.837283i	$-0.684143\pi$
$197$	− 23.5036i	− 1.67457i	0.546770	−	0.837283i	$-0.315857\pi$
$198$	−6.24698	−0.443954
$199$	20.8442	1.47760	0.738801	−	0.673923i	$-0.235392\pi$
$199$	20.8442	1.47760	0.738801	+	0.673923i	$0.235392\pi$
$200$	1.00000i	0.0707107i
$201$	0.198062i	0.0139702i
$202$	− 0.274127i	− 0.0192875i
$203$	15.6920i	1.10136i
$204$	−2.69202	−0.188479
$205$	3.77479	0.263643
$206$	9.00000i	0.627060i
$207$	5.62565	0.391009
$208$	0	0
$209$	36.4142	2.51882
$210$	− 3.04892i	− 0.210395i
$211$	−19.0073	−1.30852	−0.654258	−	0.756271i	$-0.727019\pi$
$211$	−19.0073	−1.30852	−0.654258	+	0.756271i	$0.727019\pi$
$212$	−0.664874	−0.0456638
$213$	0.374354i	0.0256503i
$214$	− 10.8509i	− 0.741749i
$215$	− 2.85086i	− 0.194427i
$216$	1.00000i	0.0680414i
$217$	−10.7724	−0.731278
$218$	1.02177	0.0692030
$219$	14.6136i	0.987493i
$220$	6.24698	0.421171
$221$	0	0
$222$	4.65279	0.312275
$223$	− 20.9433i	− 1.40247i	−0.712931	−	0.701234i	$-0.752633\pi$
$223$	− 20.9433i	− 1.40247i	0.712931	−	0.701234i	$-0.247367\pi$
$224$	3.04892	0.203714
$225$	−1.00000	−0.0666667
$226$	19.6383i	1.30632i
$227$	− 19.0978i	− 1.26757i	−0.773510	−	0.633784i	$-0.781501\pi$
$227$	− 19.0978i	− 1.26757i	0.773510	−	0.633784i	$-0.218499\pi$
$228$	− 5.82908i	− 0.386041i
$229$	9.33513i	0.616882i	0.951243	+	0.308441i	$0.0998073\pi$
$229$	9.33513i	0.616882i	−0.951243	+	0.308441i	$0.900193\pi$
$230$	−5.62565	−0.370944
$231$	−19.0465	−1.25317
$232$	5.14675i	0.337901i
$233$	14.3599	0.940747	0.470374	−	0.882467i	$-0.344119\pi$
$233$	14.3599	0.940747	0.470374	+	0.882467i	$0.344119\pi$
$234$	0	0
$235$	−3.61596	−0.235879
$236$	− 12.9487i	− 0.842888i
$237$	11.2567	0.731199
$238$	−8.20775	−0.532029
$239$	− 27.7482i	− 1.79488i	−0.441132	−	0.897442i	$-0.645423\pi$
$239$	− 27.7482i	− 1.79488i	0.441132	−	0.897442i	$-0.354577\pi$
$240$	− 1.00000i	− 0.0645497i
$241$	− 14.5579i	− 0.937759i	−0.883262	−	0.468880i	$-0.844658\pi$
$241$	− 14.5579i	− 0.937759i	0.883262	−	0.468880i	$-0.155342\pi$
$242$	− 28.0248i	− 1.80150i
$243$	−1.00000	−0.0641500
$244$	7.91723	0.506849
$245$	− 2.29590i	− 0.146679i
$246$	−3.77479	−0.240672
$247$	0	0
$248$	−3.53319	−0.224358
$249$	− 6.83877i	− 0.433390i
$250$	1.00000	0.0632456
$251$	−27.8756	−1.75949	−0.879746	−	0.475443i	$-0.842288\pi$
$251$	−27.8756	−1.75949	−0.879746	+	0.475443i	$0.842288\pi$
$252$	3.04892i	0.192064i
$253$	35.1433i	2.20944i
$254$	− 18.5187i	− 1.16197i
$255$	2.69202i	0.168581i
$256$	1.00000	0.0625000
$257$	5.64848	0.352343	0.176171	−	0.984360i	$-0.443629\pi$
$257$	5.64848	0.352343	0.176171	+	0.984360i	$0.443629\pi$
$258$	2.85086i	0.177486i
$259$	14.1860	0.881474
$260$	0	0
$261$	−5.14675	−0.318576
$262$	− 12.2349i	− 0.755875i
$263$	6.06100	0.373737	0.186869	−	0.982385i	$-0.440166\pi$
$263$	6.06100	0.373737	0.186869	+	0.982385i	$0.440166\pi$
$264$	−6.24698	−0.384475
$265$	0.664874i	0.0408429i
$266$	− 17.7724i	− 1.08970i
$267$	9.50365i	0.581614i
$268$	0.198062i	0.0120986i
$269$	8.19136	0.499436	0.249718	−	0.968319i	$-0.419662\pi$
$269$	8.19136	0.499436	0.249718	+	0.968319i	$0.419662\pi$
$270$	1.00000	0.0608581
$271$	− 12.8006i	− 0.777582i	−0.921326	−	0.388791i	$-0.872893\pi$
$271$	− 12.8006i	− 0.777582i	0.921326	−	0.388791i	$-0.127107\pi$
$272$	−2.69202	−0.163228
$273$	0	0
$274$	−19.3002	−1.16597
$275$	− 6.24698i	− 0.376707i
$276$	5.62565	0.338624
$277$	7.88338	0.473666	0.236833	−	0.971550i	$-0.423891\pi$
$277$	7.88338	0.473666	0.236833	+	0.971550i	$0.423891\pi$
$278$	19.5133i	1.17033i
$279$	− 3.53319i	− 0.211526i
$280$	− 3.04892i	− 0.182208i
$281$	− 7.16182i	− 0.427238i	−0.976917	−	0.213619i	$-0.931475\pi$
$281$	− 7.16182i	− 0.427238i	0.976917	−	0.213619i	$-0.0685251\pi$
$282$	3.61596	0.215327
$283$	−11.7342	−0.697528	−0.348764	−	0.937211i	$-0.613398\pi$
$283$	−11.7342	−0.697528	−0.348764	+	0.937211i	$0.613398\pi$
$284$	0.374354i	0.0222138i
$285$	−5.82908	−0.345285
$286$	0	0
$287$	−11.5090	−0.679356
$288$	1.00000i	0.0589256i
$289$	−9.75302	−0.573707
$290$	5.14675	0.302228
$291$	0.335126i	0.0196454i
$292$	14.6136i	0.855194i
$293$	26.4456i	1.54497i	0.635033	+	0.772485i	$0.280987\pi$
$293$	26.4456i	1.54497i	−0.635033	+	0.772485i	$0.719013\pi$
$294$	2.29590i	0.133899i
$295$	−12.9487	−0.753902
$296$	4.65279	0.270438
$297$	− 6.24698i	− 0.362487i
$298$	18.0248	1.04415
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	8.69202i	0.501000i
$302$	5.30127	0.305054
$303$	0.274127	0.0157482
$304$	− 5.82908i	− 0.334321i
$305$	− 7.91723i	− 0.453339i
$306$	− 2.69202i	− 0.153893i
$307$	10.8562i	0.619598i	0.950802	+	0.309799i	$0.100262\pi$
$307$	10.8562i	0.619598i	−0.950802	+	0.309799i	$0.899738\pi$
$308$	−19.0465	−1.08528
$309$	−9.00000	−0.511992
$310$	3.53319i	0.200672i
$311$	−11.5453	−0.654672	−0.327336	−	0.944908i	$-0.606151\pi$
$311$	−11.5453	−0.654672	−0.327336	+	0.944908i	$0.606151\pi$
$312$	0	0
$313$	18.4862	1.04490	0.522451	−	0.852670i	$-0.325018\pi$
$313$	18.4862	1.04490	0.522451	+	0.852670i	$0.325018\pi$
$314$	7.41789i	0.418616i
$315$	3.04892	0.171787
$316$	11.2567	0.633237
$317$	− 3.05131i	− 0.171379i	−0.996322	−	0.0856893i	$-0.972691\pi$
$317$	− 3.05131i	− 0.171379i	0.996322	−	0.0856893i	$-0.0273093\pi$
$318$	− 0.664874i	− 0.0372843i
$319$	− 32.1517i	− 1.80015i
$320$	− 1.00000i	− 0.0559017i
$321$	10.8509	0.605636
$322$	17.1521	0.955851
$323$	15.6920i	0.873127i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	24.0151	1.33007
$327$	1.02177i	0.0565040i
$328$	−3.77479	−0.208428
$329$	11.0248	0.607814
$330$	6.24698i	0.343885i
$331$	− 12.1575i	− 0.668237i	−0.942531	−	0.334118i	$-0.891561\pi$
$331$	− 12.1575i	− 0.668237i	0.942531	−	0.334118i	$-0.108439\pi$
$332$	− 6.83877i	− 0.375326i
$333$	4.65279i	0.254971i
$334$	11.9922	0.656186
$335$	0.198062	0.0108213
$336$	3.04892i	0.166332i
$337$	−6.48188	−0.353090	−0.176545	−	0.984293i	$-0.556492\pi$
$337$	−6.48188	−0.353090	−0.176545	+	0.984293i	$0.556492\pi$
$338$	0	0
$339$	−19.6383	−1.06661
$340$	2.69202i	0.145995i
$341$	22.0718	1.19525
$342$	5.82908	0.315201
$343$	− 14.3424i	− 0.774418i
$344$	2.85086i	0.153708i
$345$	− 5.62565i	− 0.302875i
$346$	− 19.8823i	− 1.06888i
$347$	19.1933	1.03035	0.515175	−	0.857085i	$-0.327727\pi$
$347$	19.1933	1.03035	0.515175	+	0.857085i	$0.327727\pi$
$348$	−5.14675	−0.275895
$349$	11.5767i	0.619688i	0.950787	+	0.309844i	$0.100277\pi$
$349$	11.5767i	0.619688i	−0.950787	+	0.309844i	$0.899723\pi$
$350$	−3.04892	−0.162971
$351$	0	0
$352$	−6.24698	−0.332965
$353$	− 21.6963i	− 1.15478i	−0.816469	−	0.577390i	$-0.804071\pi$
$353$	− 21.6963i	− 1.15478i	0.816469	−	0.577390i	$-0.195929\pi$
$354$	12.9487	0.688215
$355$	0.374354	0.0198687
$356$	9.50365i	0.503692i
$357$	− 8.20775i	− 0.434400i
$358$	− 16.7453i	− 0.885014i
$359$	− 13.0476i	− 0.688625i	−0.938855	−	0.344313i	$-0.888112\pi$
$359$	− 13.0476i	− 0.688625i	0.938855	−	0.344313i	$-0.111888\pi$
$360$	1.00000	0.0527046
$361$	−14.9782	−0.788328
$362$	3.93900i	0.207029i
$363$	28.0248	1.47092
$364$	0	0
$365$	14.6136	0.764909
$366$	7.91723i	0.413840i
$367$	1.76377	0.0920683	0.0460341	−	0.998940i	$-0.485342\pi$
$367$	1.76377	0.0920683	0.0460341	+	0.998940i	$0.485342\pi$
$368$	5.62565	0.293257
$369$	− 3.77479i	− 0.196508i
$370$	− 4.65279i	− 0.241887i
$371$	− 2.02715i	− 0.105244i
$372$	− 3.53319i	− 0.183187i
$373$	15.5013	0.802625	0.401312	−	0.915941i	$-0.368554\pi$
$373$	15.5013	0.802625	0.401312	+	0.915941i	$0.368554\pi$
$374$	16.8170	0.869587
$375$	1.00000i	0.0516398i
$376$	3.61596	0.186479
$377$	0	0
$378$	−3.04892	−0.156819
$379$	8.62863i	0.443223i	0.975135	+	0.221611i	$0.0711317\pi$
$379$	8.62863i	0.443223i	−0.975135	+	0.221611i	$0.928868\pi$
$380$	−5.82908	−0.299026
$381$	18.5187	0.948742
$382$	− 27.1540i	− 1.38932i
$383$	31.1008i	1.58918i	0.607148	+	0.794589i	$0.292314\pi$
$383$	31.1008i	1.58918i	−0.607148	+	0.794589i	$0.707686\pi$
$384$	1.00000i	0.0510310i
$385$	19.0465i	0.970701i
$386$	15.3937	0.783520
$387$	−2.85086	−0.144917
$388$	0.335126i	0.0170134i
$389$	−7.07069	−0.358498	−0.179249	−	0.983804i	$-0.557367\pi$
$389$	−7.07069	−0.358498	−0.179249	+	0.983804i	$0.557367\pi$
$390$	0	0
$391$	−15.1444	−0.765883
$392$	2.29590i	0.115960i
$393$	12.2349	0.617169
$394$	23.5036	1.18410
$395$	− 11.2567i	− 0.566384i
$396$	− 6.24698i	− 0.313923i
$397$	28.5948i	1.43513i	0.696491	+	0.717565i	$0.254744\pi$
$397$	28.5948i	1.43513i	−0.696491	+	0.717565i	$0.745256\pi$
$398$	20.8442i	1.04482i
$399$	17.7724	0.889733
$400$	−1.00000	−0.0500000
$401$	11.8267i	0.590597i	0.955405	+	0.295298i	$0.0954191\pi$
$401$	11.8267i	0.590597i	−0.955405	+	0.295298i	$0.904581\pi$
$402$	−0.198062	−0.00987845
$403$	0	0
$404$	0.274127	0.0136383
$405$	1.00000i	0.0496904i
$406$	−15.6920	−0.778782
$407$	−29.0659	−1.44074
$408$	− 2.69202i	− 0.133275i
$409$	− 9.00000i	− 0.445021i	−0.974930	−	0.222511i	$-0.928575\pi$
$409$	− 9.00000i	− 0.445021i	0.974930	−	0.222511i	$-0.0714252\pi$
$410$	3.77479i	0.186424i
$411$	− 19.3002i	− 0.952009i
$412$	−9.00000	−0.443398
$413$	39.4795	1.94266
$414$	5.62565i	0.276485i
$415$	−6.83877	−0.335702
$416$	0	0
$417$	−19.5133	−0.955572
$418$	36.4142i	1.78108i
$419$	19.7735	0.965997	0.482998	−	0.875621i	$-0.339548\pi$
$419$	19.7735	0.965997	0.482998	+	0.875621i	$0.339548\pi$
$420$	3.04892	0.148772
$421$	− 21.0127i	− 1.02409i	−0.858957	−	0.512047i	$-0.828887\pi$
$421$	− 21.0127i	− 1.02409i	0.858957	−	0.512047i	$-0.171113\pi$
$422$	− 19.0073i	− 0.925261i
$423$	3.61596i	0.175814i
$424$	− 0.664874i	− 0.0322892i
$425$	2.69202	0.130582
$426$	−0.374354	−0.0181375
$427$	24.1390i	1.16817i
$428$	10.8509	0.524496
$429$	0	0
$430$	2.85086	0.137480
$431$	− 26.5633i	− 1.27951i	−0.768579	−	0.639755i	$-0.779036\pi$
$431$	− 26.5633i	− 1.27951i	0.768579	−	0.639755i	$-0.220964\pi$
$432$	−1.00000	−0.0481125
$433$	14.7047	0.706662	0.353331	−	0.935498i	$-0.385049\pi$
$433$	14.7047	0.706662	0.353331	+	0.935498i	$0.385049\pi$
$434$	− 10.7724i	− 0.517092i
$435$	5.14675i	0.246768i
$436$	1.02177i	0.0489339i
$437$	− 32.7924i	− 1.56867i
$438$	−14.6136	−0.698263
$439$	28.2640	1.34897	0.674483	−	0.738291i	$-0.264367\pi$
$439$	28.2640	1.34897	0.674483	+	0.738291i	$0.264367\pi$
$440$	6.24698i	0.297813i
$441$	−2.29590	−0.109328
$442$	0	0
$443$	20.5109	0.974504	0.487252	−	0.873261i	$-0.337999\pi$
$443$	20.5109	0.974504	0.487252	+	0.873261i	$0.337999\pi$
$444$	4.65279i	0.220812i
$445$	9.50365	0.450516
$446$	20.9433	0.991695
$447$	18.0248i	0.852542i
$448$	3.04892i	0.144048i
$449$	3.19806i	0.150926i	0.997149	+	0.0754629i	$0.0240435\pi$
$449$	3.19806i	0.150926i	−0.997149	+	0.0754629i	$0.975957\pi$
$450$	− 1.00000i	− 0.0471405i
$451$	23.5810	1.11039
$452$	−19.6383	−0.923709
$453$	5.30127i	0.249076i
$454$	19.0978	0.896306
$455$	0	0
$456$	5.82908	0.272972
$457$	8.65710i	0.404962i	0.979286	+	0.202481i	$0.0649005\pi$
$457$	8.65710i	0.404962i	−0.979286	+	0.202481i	$0.935100\pi$
$458$	−9.33513	−0.436202
$459$	2.69202	0.125653
$460$	− 5.62565i	− 0.262297i
$461$	− 11.1239i	− 0.518092i	−0.965865	−	0.259046i	$-0.916592\pi$
$461$	− 11.1239i	− 0.518092i	0.965865	−	0.259046i	$-0.0834082\pi$
$462$	− 19.0465i	− 0.886125i
$463$	33.0659i	1.53670i	0.640028	+	0.768351i	$0.278923\pi$
$463$	33.0659i	1.53670i	−0.640028	+	0.768351i	$0.721077\pi$
$464$	−5.14675	−0.238932
$465$	−3.53319	−0.163848
$466$	14.3599i	0.665209i
$467$	−15.3937	−0.712337	−0.356168	−	0.934422i	$-0.615917\pi$
$467$	−15.3937	−0.712337	−0.356168	+	0.934422i	$0.615917\pi$
$468$	0	0
$469$	−0.603875	−0.0278844
$470$	− 3.61596i	− 0.166792i
$471$	−7.41789	−0.341799
$472$	12.9487	0.596012
$473$	− 17.8092i	− 0.818869i
$474$	11.2567i	0.517036i
$475$	5.82908i	0.267457i
$476$	− 8.20775i	− 0.376202i
$477$	0.664874	0.0304425
$478$	27.7482	1.26917
$479$	− 42.2137i	− 1.92879i	−0.264459	−	0.964397i	$-0.585193\pi$
$479$	− 42.2137i	− 1.92879i	0.264459	−	0.964397i	$-0.414807\pi$
$480$	1.00000	0.0456435
$481$	0	0
$482$	14.5579	0.663096
$483$	17.1521i	0.780449i
$484$	28.0248	1.27385
$485$	0.335126	0.0152173
$486$	− 1.00000i	− 0.0453609i
$487$	14.9903i	0.679276i	0.940556	+	0.339638i	$0.110305\pi$
$487$	14.9903i	0.679276i	−0.940556	+	0.339638i	$0.889695\pi$
$488$	7.91723i	0.358396i
$489$	24.0151i	1.08600i
$490$	2.29590	0.103718
$491$	21.3706	0.964443	0.482222	−	0.876049i	$-0.339830\pi$
$491$	21.3706	0.964443	0.482222	+	0.876049i	$0.339830\pi$
$492$	− 3.77479i	− 0.170181i
$493$	13.8552	0.624005
$494$	0	0
$495$	−6.24698	−0.280781
$496$	− 3.53319i	− 0.158645i
$497$	−1.14138	−0.0511977
$498$	6.83877	0.306453
$499$	− 26.6160i	− 1.19149i	−0.803172	−	0.595747i	$-0.796856\pi$
$499$	− 26.6160i	− 1.19149i	0.803172	−	0.595747i	$-0.203144\pi$
$500$	1.00000i	0.0447214i
$501$	11.9922i	0.535773i
$502$	− 27.8756i	− 1.24415i
$503$	−25.2218	−1.12458	−0.562291	−	0.826939i	$-0.690080\pi$
$503$	−25.2218	−1.12458	−0.562291	+	0.826939i	$0.690080\pi$
$504$	−3.04892	−0.135810
$505$	− 0.274127i	− 0.0121985i
$506$	−35.1433	−1.56231
$507$	0	0
$508$	18.5187	0.821635
$509$	− 17.3797i	− 0.770343i	−0.922845	−	0.385172i	$-0.874142\pi$
$509$	− 17.3797i	− 0.770343i	0.922845	−	0.385172i	$-0.125858\pi$
$510$	−2.69202	−0.119205
$511$	−44.5555	−1.97102
$512$	1.00000i	0.0441942i
$513$	5.82908i	0.257360i
$514$	5.64848i	0.249144i
$515$	9.00000i	0.396587i
$516$	−2.85086	−0.125502
$517$	−22.5888	−0.993455
$518$	14.1860i	0.623296i
$519$	19.8823	0.872737
$520$	0	0
$521$	8.27114	0.362365	0.181183	−	0.983449i	$-0.442007\pi$
$521$	8.27114	0.362365	0.181183	+	0.983449i	$0.442007\pi$
$522$	− 5.14675i	− 0.225267i
$523$	−40.9154	−1.78911	−0.894553	−	0.446961i	$-0.852506\pi$
$523$	−40.9154	−1.78911	−0.894553	+	0.446961i	$0.852506\pi$
$524$	12.2349	0.534484
$525$	− 3.04892i	− 0.133066i
$526$	6.06100i	0.264272i
$527$	9.51142i	0.414324i
$528$	− 6.24698i	− 0.271865i
$529$	8.64789	0.375995
$530$	−0.664874	−0.0288803
$531$	12.9487i	0.561925i
$532$	17.7724	0.770531
$533$	0	0
$534$	−9.50365	−0.411263
$535$	− 10.8509i	− 0.469123i
$536$	−0.198062	−0.00855499
$537$	16.7453	0.722611
$538$	8.19136i	0.353154i
$539$	− 14.3424i	− 0.617772i
$540$	1.00000i	0.0430331i
$541$	− 27.2252i	− 1.17050i	−0.810852	−	0.585252i	$-0.800996\pi$
$541$	− 27.2252i	− 1.17050i	0.810852	−	0.585252i	$-0.199004\pi$
$542$	12.8006	0.549833
$543$	−3.93900	−0.169039
$544$	− 2.69202i	− 0.115419i
$545$	1.02177	0.0437678
$546$	0	0
$547$	−32.7023	−1.39825	−0.699125	−	0.715000i	$-0.746427\pi$
$547$	−32.7023	−1.39825	−0.699125	+	0.715000i	$0.746427\pi$
$548$	− 19.3002i	− 0.824464i
$549$	−7.91723	−0.337899
$550$	6.24698	0.266372
$551$	30.0009i	1.27808i
$552$	5.62565i	0.239443i
$553$	34.3207i	1.45946i
$554$	7.88338i	0.334933i
$555$	4.65279	0.197500
$556$	−19.5133	−0.827550
$557$	− 0.225209i	− 0.00954243i	−0.999989	−	0.00477121i	$-0.998481\pi$
$557$	− 0.225209i	− 0.00954243i	0.999989	−	0.00477121i	$-0.00151873\pi$
$558$	3.53319	0.149572
$559$	0	0
$560$	3.04892	0.128840
$561$	16.8170i	0.710014i
$562$	7.16182	0.302103
$563$	9.55735	0.402794	0.201397	−	0.979510i	$-0.435452\pi$
$563$	9.55735	0.402794	0.201397	+	0.979510i	$0.435452\pi$
$564$	3.61596i	0.152259i
$565$	19.6383i	0.826190i
$566$	− 11.7342i	− 0.493227i
$567$	− 3.04892i	− 0.128042i
$568$	−0.374354	−0.0157076
$569$	−34.8455	−1.46080	−0.730399	−	0.683020i	$-0.760666\pi$
$569$	−34.8455	−1.46080	−0.730399	+	0.683020i	$0.760666\pi$
$570$	− 5.82908i	− 0.244153i
$571$	−36.5472	−1.52945	−0.764726	−	0.644355i	$-0.777126\pi$
$571$	−36.5472	−1.52945	−0.764726	+	0.644355i	$0.777126\pi$
$572$	0	0
$573$	27.1540	1.13438
$574$	− 11.5090i	− 0.480377i
$575$	−5.62565	−0.234606
$576$	−1.00000	−0.0416667
$577$	18.3284i	0.763022i	0.924364	+	0.381511i	$0.124596\pi$
$577$	18.3284i	0.763022i	−0.924364	+	0.381511i	$0.875404\pi$
$578$	− 9.75302i	− 0.405672i
$579$	15.3937i	0.639742i
$580$	5.14675i	0.213707i
$581$	20.8509	0.865039
$582$	−0.335126	−0.0138914
$583$	4.15346i	0.172019i
$584$	−14.6136	−0.604714
$585$	0	0
$586$	−26.4456	−1.09246
$587$	− 17.3937i	− 0.717916i	−0.933354	−	0.358958i	$-0.883132\pi$
$587$	− 17.3937i	− 0.717916i	0.933354	−	0.358958i	$-0.116868\pi$
$588$	−2.29590	−0.0946812
$589$	−20.5953	−0.848613
$590$	− 12.9487i	− 0.533089i
$591$	23.5036i	0.966811i
$592$	4.65279i	0.191229i
$593$	19.5362i	0.802254i	0.916022	+	0.401127i	$0.131381\pi$
$593$	19.5362i	0.802254i	−0.916022	+	0.401127i	$0.868619\pi$
$594$	6.24698	0.256317
$595$	−8.20775	−0.336485
$596$	18.0248i	0.738323i
$597$	−20.8442	−0.853094
$598$	0	0
$599$	−14.6364	−0.598027	−0.299014	−	0.954249i	$-0.596658\pi$
$599$	−14.6364	−0.598027	−0.299014	+	0.954249i	$0.596658\pi$
$600$	− 1.00000i	− 0.0408248i
$601$	33.3980	1.36233	0.681167	−	0.732128i	$-0.261473\pi$
$601$	33.3980	1.36233	0.681167	+	0.732128i	$0.261473\pi$
$602$	−8.69202	−0.354260
$603$	− 0.198062i	− 0.00806572i
$604$	5.30127i	0.215706i
$605$	− 28.0248i	− 1.13937i
$606$	0.274127i	0.0111356i
$607$	29.9694	1.21642	0.608210	−	0.793776i	$-0.291888\pi$
$607$	29.9694	1.21642	0.608210	+	0.793776i	$0.291888\pi$
$608$	5.82908	0.236401
$609$	− 15.6920i	− 0.635873i
$610$	7.91723	0.320559
$611$	0	0
$612$	2.69202	0.108819
$613$	1.58748i	0.0641178i	0.999486	+	0.0320589i	$0.0102064\pi$
$613$	1.58748i	0.0641178i	−0.999486	+	0.0320589i	$0.989794\pi$
$614$	−10.8562	−0.438122
$615$	−3.77479	−0.152214
$616$	− 19.0465i	− 0.767406i
$617$	− 12.9511i	− 0.521391i	−0.965421	−	0.260695i	$-0.916048\pi$
$617$	− 12.9511i	− 0.521391i	0.965421	−	0.260695i	$-0.0839519\pi$
$618$	− 9.00000i	− 0.362033i
$619$	− 23.9554i	− 0.962849i	−0.876488	−	0.481424i	$-0.840120\pi$
$619$	− 23.9554i	− 0.962849i	0.876488	−	0.481424i	$-0.159880\pi$
$620$	−3.53319	−0.141896
$621$	−5.62565	−0.225749
$622$	− 11.5453i	− 0.462923i
$623$	−28.9758	−1.16089
$624$	0	0
$625$	1.00000	0.0400000
$626$	18.4862i	0.738857i
$627$	−36.4142	−1.45424
$628$	−7.41789	−0.296006
$629$	− 12.5254i	− 0.499421i
$630$	3.04892i	0.121472i
$631$	− 35.4601i	− 1.41164i	−0.708389	−	0.705822i	$-0.750578\pi$
$631$	− 35.4601i	− 1.41164i	0.708389	−	0.705822i	$-0.249422\pi$
$632$	11.2567i	0.447766i
$633$	19.0073	0.755472
$634$	3.05131	0.121183
$635$	− 18.5187i	− 0.734893i
$636$	0.664874	0.0263640
$637$	0	0
$638$	32.1517	1.27290
$639$	− 0.374354i	− 0.0148092i
$640$	1.00000	0.0395285
$641$	−16.8358	−0.664974	−0.332487	−	0.943108i	$-0.607888\pi$
$641$	−16.8358	−0.664974	−0.332487	+	0.943108i	$0.607888\pi$
$642$	10.8509i	0.428249i
$643$	18.4838i	0.728930i	0.931217	+	0.364465i	$0.118748\pi$
$643$	18.4838i	0.728930i	−0.931217	+	0.364465i	$0.881252\pi$
$644$	17.1521i	0.675889i
$645$	2.85086i	0.112252i
$646$	−15.6920	−0.617394
$647$	1.20237	0.0472702	0.0236351	−	0.999721i	$-0.492476\pi$
$647$	1.20237	0.0472702	0.0236351	+	0.999721i	$0.492476\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	−80.8902	−3.17522
$650$	0	0
$651$	10.7724	0.422204
$652$	24.0151i	0.940503i
$653$	−36.9530	−1.44608	−0.723041	−	0.690805i	$-0.757256\pi$
$653$	−36.9530	−1.44608	−0.723041	+	0.690805i	$0.757256\pi$
$654$	−1.02177	−0.0399544
$655$	− 12.2349i	− 0.478057i
$656$	− 3.77479i	− 0.147381i
$657$	− 14.6136i	− 0.570129i
$658$	11.0248i	0.429790i
$659$	−26.4494	−1.03032	−0.515160	−	0.857094i	$-0.672268\pi$
$659$	−26.4494	−1.03032	−0.515160	+	0.857094i	$0.672268\pi$
$660$	−6.24698	−0.243163
$661$	2.21073i	0.0859876i	0.999075	+	0.0429938i	$0.0136896\pi$
$661$	2.21073i	0.0859876i	−0.999075	+	0.0429938i	$0.986310\pi$
$662$	12.1575	0.472515
$663$	0	0
$664$	6.83877	0.265396
$665$	− 17.7724i	− 0.689184i
$666$	−4.65279	−0.180292
$667$	−28.9538	−1.12110
$668$	11.9922i	0.463993i
$669$	20.9433i	0.809715i
$670$	0.198062i	0.00765181i
$671$	− 49.4588i	− 1.90934i
$672$	−3.04892	−0.117615
$673$	−16.5555	−0.638170	−0.319085	−	0.947726i	$-0.603375\pi$
$673$	−16.5555	−0.638170	−0.319085	+	0.947726i	$0.603375\pi$
$674$	− 6.48188i	− 0.249673i
$675$	1.00000	0.0384900
$676$	0	0
$677$	−8.53617	−0.328072	−0.164036	−	0.986454i	$-0.552451\pi$
$677$	−8.53617	−0.328072	−0.164036	+	0.986454i	$0.552451\pi$
$678$	− 19.6383i	− 0.754205i
$679$	−1.02177	−0.0392119
$680$	−2.69202	−0.103234
$681$	19.0978i	0.731831i
$682$	22.0718i	0.845171i
$683$	− 0.204767i	− 0.00783519i	−0.999992	−	0.00391760i	$-0.998753\pi$
$683$	− 0.204767i	− 0.00783519i	0.999992	−	0.00391760i	$-0.00124701\pi$
$684$	5.82908i	0.222881i
$685$	−19.3002	−0.737423
$686$	14.3424	0.547596
$687$	− 9.33513i	− 0.356157i
$688$	−2.85086	−0.108688
$689$	0	0
$690$	5.62565	0.214165
$691$	33.5187i	1.27511i	0.770404	+	0.637556i	$0.220055\pi$
$691$	33.5187i	1.27511i	−0.770404	+	0.637556i	$0.779945\pi$
$692$	19.8823	0.755812
$693$	19.0465	0.723518
$694$	19.1933i	0.728567i
$695$	19.5133i	0.740183i
$696$	− 5.14675i	− 0.195087i
$697$	10.1618i	0.384906i
$698$	−11.5767	−0.438186
$699$	−14.3599	−0.543141
$700$	− 3.04892i	− 0.115238i
$701$	−19.0271	−0.718645	−0.359323	−	0.933213i	$-0.616992\pi$
$701$	−19.0271	−0.718645	−0.359323	+	0.933213i	$0.616992\pi$
$702$	0	0
$703$	27.1215	1.02291
$704$	− 6.24698i	− 0.235442i
$705$	3.61596	0.136185
$706$	21.6963	0.816552
$707$	0.835790i	0.0314331i
$708$	12.9487i	0.486642i
$709$	− 42.5760i	− 1.59897i	−0.600683	−	0.799487i	$-0.705105\pi$
$709$	− 42.5760i	− 1.59897i	0.600683	−	0.799487i	$-0.294895\pi$
$710$	0.374354i	0.0140493i
$711$	−11.2567	−0.422158
$712$	−9.50365	−0.356164
$713$	− 19.8765i	− 0.744379i
$714$	8.20775	0.307167
$715$	0	0
$716$	16.7453	0.625799
$717$	27.7482i	1.03628i
$718$	13.0476	0.486932
$719$	−7.51632	−0.280311	−0.140156	−	0.990129i	$-0.544760\pi$
$719$	−7.51632	−0.280311	−0.140156	+	0.990129i	$0.544760\pi$
$720$	1.00000i	0.0372678i
$721$	− 27.4403i	− 1.02193i
$722$	− 14.9782i	− 0.557432i
$723$	14.5579i	0.541416i
$724$	−3.93900	−0.146392
$725$	5.14675	0.191146
$726$	28.0248i	1.04010i
$727$	−39.6558	−1.47075	−0.735376	−	0.677660i	$-0.762994\pi$
$727$	−39.6558	−1.47075	−0.735376	+	0.677660i	$0.762994\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	14.6136i	0.540872i
$731$	7.67456	0.283854
$732$	−7.91723	−0.292629
$733$	24.4886i	0.904506i	0.891890	+	0.452253i	$0.149380\pi$
$733$	24.4886i	0.904506i	−0.891890	+	0.452253i	$0.850620\pi$
$734$	1.76377i	0.0651021i
$735$	2.29590i	0.0846854i
$736$	5.62565i	0.207364i
$737$	1.23729	0.0455762
$738$	3.77479	0.138952
$739$	41.9842i	1.54441i	0.635371	+	0.772207i	$0.280847\pi$
$739$	41.9842i	1.54441i	−0.635371	+	0.772207i	$0.719153\pi$
$740$	4.65279	0.171040
$741$	0	0
$742$	2.02715	0.0744189
$743$	29.7711i	1.09219i	0.837722	+	0.546097i	$0.183887\pi$
$743$	29.7711i	1.09219i	−0.837722	+	0.546097i	$0.816113\pi$
$744$	3.53319	0.129533
$745$	18.0248	0.660376
$746$	15.5013i	0.567541i
$747$	6.83877i	0.250218i
$748$	16.8170i	0.614891i
$749$	33.0834i	1.20884i
$750$	−1.00000	−0.0365148
$751$	−35.4999	−1.29541	−0.647705	−	0.761891i	$-0.724271\pi$
$751$	−35.4999	−1.29541	−0.647705	+	0.761891i	$0.724271\pi$
$752$	3.61596i	0.131860i
$753$	27.8756	1.01584
$754$	0	0
$755$	5.30127	0.192933
$756$	− 3.04892i	− 0.110888i
$757$	2.73258	0.0993172	0.0496586	−	0.998766i	$-0.484187\pi$
$757$	2.73258	0.0993172	0.0496586	+	0.998766i	$0.484187\pi$
$758$	−8.62863	−0.313406
$759$	− 35.1433i	− 1.27562i
$760$	− 5.82908i	− 0.211443i
$761$	30.2349i	1.09601i	0.836474	+	0.548007i	$0.184613\pi$
$761$	30.2349i	1.09601i	−0.836474	+	0.548007i	$0.815387\pi$
$762$	18.5187i	0.670862i
$763$	−3.11529	−0.112781
$764$	27.1540	0.982399
$765$	− 2.69202i	− 0.0973302i
$766$	−31.1008	−1.12372
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	− 35.3183i	− 1.27361i	−0.771025	−	0.636804i	$-0.780256\pi$
$769$	− 35.3183i	− 1.27361i	0.771025	−	0.636804i	$-0.219744\pi$
$770$	−19.0465	−0.686389
$771$	−5.64848	−0.203425
$772$	15.3937i	0.554033i
$773$	− 32.0551i	− 1.15294i	−0.817117	−	0.576472i	$-0.804429\pi$
$773$	− 32.0551i	− 1.15294i	0.817117	−	0.576472i	$-0.195571\pi$
$774$	− 2.85086i	− 0.102472i
$775$	3.53319i	0.126916i
$776$	−0.335126	−0.0120303
$777$	−14.1860	−0.508919
$778$	− 7.07069i	− 0.253496i
$779$	−22.0036	−0.788360
$780$	0	0
$781$	2.33858	0.0836811
$782$	− 15.1444i	− 0.541561i
$783$	5.14675	0.183930
$784$	−2.29590	−0.0819963
$785$	7.41789i	0.264756i
$786$	12.2349i	0.436404i
$787$	43.9235i	1.56570i	0.622209	+	0.782851i	$0.286235\pi$
$787$	43.9235i	1.56570i	−0.622209	+	0.782851i	$0.713765\pi$
$788$	23.5036i	0.837283i
$789$	−6.06100	−0.215777
$790$	11.2567	0.400494
$791$	− 59.8756i	− 2.12893i
$792$	6.24698	0.221977
$793$	0	0
$794$	−28.5948	−1.01479
$795$	− 0.664874i	− 0.0235807i
$796$	−20.8442	−0.738801
$797$	26.4523	0.936990	0.468495	−	0.883466i	$-0.344796\pi$
$797$	26.4523	0.936990	0.468495	+	0.883466i	$0.344796\pi$
$798$	17.7724i	0.629136i
$799$	− 9.73423i	− 0.344372i
$800$	− 1.00000i	− 0.0353553i
$801$	− 9.50365i	− 0.335795i
$802$	−11.8267	−0.417615
$803$	91.2906	3.22158
$804$	− 0.198062i	− 0.00698512i
$805$	17.1521	0.604533
$806$	0	0
$807$	−8.19136	−0.288349
$808$	0.274127i	0.00964374i
$809$	41.4634	1.45777	0.728887	−	0.684634i	$-0.240038\pi$
$809$	41.4634	1.45777	0.728887	+	0.684634i	$0.240038\pi$
$810$	−1.00000	−0.0351364
$811$	24.9095i	0.874689i	0.899294	+	0.437345i	$0.144081\pi$
$811$	24.9095i	0.874689i	−0.899294	+	0.437345i	$0.855919\pi$
$812$	− 15.6920i	− 0.550682i
$813$	12.8006i	0.448937i
$814$	− 29.0659i	− 1.01876i
$815$	24.0151	0.841211
$816$	2.69202	0.0942396
$817$	16.6179i	0.581386i
$818$	9.00000	0.314678
$819$	0	0
$820$	−3.77479	−0.131821
$821$	3.11901i	0.108854i	0.998518	+	0.0544272i	$0.0173333\pi$
$821$	3.11901i	0.108854i	−0.998518	+	0.0544272i	$0.982667\pi$
$822$	19.3002	0.673172
$823$	−36.4077	−1.26909	−0.634547	−	0.772884i	$-0.718813\pi$
$823$	−36.4077	−1.26909	−0.634547	+	0.772884i	$0.718813\pi$
$824$	− 9.00000i	− 0.313530i
$825$	6.24698i	0.217492i
$826$	39.4795i	1.37367i
$827$	8.04785i	0.279851i	0.990162	+	0.139926i	$0.0446863\pi$
$827$	8.04785i	0.279851i	−0.990162	+	0.139926i	$0.955314\pi$
$828$	−5.62565	−0.195505
$829$	−28.7006	−0.996815	−0.498407	−	0.866943i	$-0.666082\pi$
$829$	−28.7006	−0.996815	−0.498407	+	0.866943i	$0.666082\pi$
$830$	− 6.83877i	− 0.237377i
$831$	−7.88338	−0.273471
$832$	0	0
$833$	6.18060	0.214145
$834$	− 19.5133i	− 0.675692i
$835$	11.9922	0.415008
$836$	−36.4142	−1.25941
$837$	3.53319i	0.122125i
$838$	19.7735i	0.683063i
$839$	34.8491i	1.20312i	0.798827	+	0.601561i	$0.205455\pi$
$839$	34.8491i	1.20312i	−0.798827	+	0.601561i	$0.794545\pi$
$840$	3.04892i	0.105198i
$841$	−2.51094	−0.0865843
$842$	21.0127	0.724145
$843$	7.16182i	0.246666i
$844$	19.0073	0.654258
$845$	0	0
$846$	−3.61596	−0.124319
$847$	85.4452i	2.93593i
$848$	0.664874	0.0228319
$849$	11.7342	0.402718
$850$	2.69202i	0.0923356i
$851$	26.1750i	0.897266i
$852$	− 0.374354i	− 0.0128252i
$853$	− 0.0779834i	− 0.00267010i	−0.999999	−	0.00133505i	$-0.999575\pi$
$853$	− 0.0779834i	− 0.00267010i	0.999999	−	0.00133505i	$-0.000424960\pi$
$854$	−24.1390	−0.826019
$855$	5.82908	0.199350
$856$	10.8509i	0.370875i
$857$	−18.3811	−0.627885	−0.313943	−	0.949442i	$-0.601650\pi$
$857$	−18.3811	−0.627885	−0.313943	+	0.949442i	$0.601650\pi$
$858$	0	0
$859$	0.871297	0.0297283	0.0148641	−	0.999890i	$-0.495268\pi$
$859$	0.871297	0.0297283	0.0148641	+	0.999890i	$0.495268\pi$
$860$	2.85086i	0.0972134i
$861$	11.5090	0.392227
$862$	26.5633	0.904750
$863$	1.88172i	0.0640546i	0.999487	+	0.0320273i	$0.0101963\pi$
$863$	1.88172i	0.0640546i	−0.999487	+	0.0320273i	$0.989804\pi$
$864$	− 1.00000i	− 0.0340207i
$865$	− 19.8823i	− 0.676019i
$866$	14.7047i	0.499686i
$867$	9.75302	0.331230
$868$	10.7724	0.365639
$869$	− 70.3202i	− 2.38545i
$870$	−5.14675	−0.174491
$871$	0	0
$872$	−1.02177	−0.0346015
$873$	− 0.335126i	− 0.0113423i
$874$	32.7924	1.10922
$875$	−3.04892	−0.103072
$876$	− 14.6136i	− 0.493747i
$877$	46.6859i	1.57647i	0.615374	+	0.788236i	$0.289005\pi$
$877$	46.6859i	1.57647i	−0.615374	+	0.788236i	$0.710995\pi$
$878$	28.2640i	0.953863i
$879$	− 26.4456i	− 0.891989i
$880$	−6.24698	−0.210586
$881$	−51.9821	−1.75132	−0.875660	−	0.482928i	$-0.839573\pi$
$881$	−51.9821	−1.75132	−0.875660	+	0.482928i	$0.839573\pi$
$882$	− 2.29590i	− 0.0773069i
$883$	−22.3889	−0.753448	−0.376724	−	0.926326i	$-0.622949\pi$
$883$	−22.3889	−0.753448	−0.376724	+	0.926326i	$0.622949\pi$
$884$	0	0
$885$	12.9487	0.435265
$886$	20.5109i	0.689079i
$887$	−20.7332	−0.696152	−0.348076	−	0.937466i	$-0.613165\pi$
$887$	−20.7332	−0.696152	−0.348076	+	0.937466i	$0.613165\pi$
$888$	−4.65279	−0.156138
$889$	56.4620i	1.89368i
$890$	9.50365i	0.318563i
$891$	6.24698i	0.209282i
$892$	20.9433i	0.701234i
$893$	21.0777	0.705339
$894$	−18.0248	−0.602838
$895$	− 16.7453i	− 0.559732i
$896$	−3.04892	−0.101857
$897$	0	0
$898$	−3.19806	−0.106721
$899$	18.1844i	0.606485i
$900$	1.00000	0.0333333
$901$	−1.78986	−0.0596288
$902$	23.5810i	0.785163i
$903$	− 8.69202i	− 0.289252i
$904$	− 19.6383i	− 0.653161i
$905$	3.93900i	0.130937i
$906$	−5.30127	−0.176123
$907$	9.27365	0.307927	0.153963	−	0.988077i	$-0.450796\pi$
$907$	9.27365	0.307927	0.153963	+	0.988077i	$0.450796\pi$
$908$	19.0978i	0.633784i
$909$	−0.274127	−0.00909221
$910$	0	0
$911$	−3.87071	−0.128242	−0.0641211	−	0.997942i	$-0.520424\pi$
$911$	−3.87071	−0.128242	−0.0641211	+	0.997942i	$0.520424\pi$
$912$	5.82908i	0.193020i
$913$	−42.7217	−1.41388
$914$	−8.65710	−0.286352
$915$	7.91723i	0.261736i
$916$	− 9.33513i	− 0.308441i
$917$	37.3032i	1.23186i
$918$	2.69202i	0.0888499i
$919$	−33.2644	−1.09729	−0.548646	−	0.836055i	$-0.684857\pi$
$919$	−33.2644	−1.09729	−0.548646	+	0.836055i	$0.684857\pi$
$920$	5.62565	0.185472
$921$	− 10.8562i	− 0.357725i
$922$	11.1239	0.366347
$923$	0	0
$924$	19.0465	0.626585
$925$	− 4.65279i	− 0.152983i
$926$	−33.0659	−1.08661
$927$	9.00000	0.295599
$928$	− 5.14675i	− 0.168950i
$929$	− 1.92095i	− 0.0630244i	−0.999503	−	0.0315122i	$-0.989968\pi$
$929$	− 1.92095i	− 0.0630244i	0.999503	−	0.0315122i	$-0.0100323\pi$
$930$	− 3.53319i	− 0.115858i
$931$	13.3830i	0.438609i
$932$	−14.3599	−0.470374
$933$	11.5453	0.377975
$934$	− 15.3937i	− 0.503698i
$935$	16.8170	0.549975
$936$	0	0
$937$	26.6886	0.871877	0.435939	−	0.899976i	$-0.356416\pi$
$937$	26.6886	0.871877	0.435939	+	0.899976i	$0.356416\pi$
$938$	− 0.603875i	− 0.0197172i
$939$	−18.4862	−0.603274
$940$	3.61596	0.117940
$941$	2.69309i	0.0877921i	0.999036	+	0.0438961i	$0.0139770\pi$
$941$	2.69309i	0.0877921i	−0.999036	+	0.0438961i	$0.986023\pi$
$942$	− 7.41789i	− 0.241688i
$943$	− 21.2356i	− 0.691527i
$944$	12.9487i	0.421444i
$945$	−3.04892	−0.0991813
$946$	17.8092	0.579028
$947$	23.1226i	0.751383i	0.926745	+	0.375692i	$0.122595\pi$
$947$	23.1226i	0.751383i	−0.926745	+	0.375692i	$0.877405\pi$
$948$	−11.2567	−0.365600
$949$	0	0
$950$	−5.82908	−0.189120
$951$	3.05131i	0.0989455i
$952$	8.20775	0.266015
$953$	31.7313	1.02788	0.513938	−	0.857827i	$-0.328186\pi$
$953$	31.7313	1.02788	0.513938	+	0.857827i	$0.328186\pi$
$954$	0.664874i	0.0215261i
$955$	− 27.1540i	− 0.878684i
$956$	27.7482i	0.897442i
$957$	32.1517i	1.03932i
$958$	42.2137	1.36386
$959$	58.8447	1.90020
$960$	1.00000i	0.0322749i
$961$	18.5166	0.597309
$962$	0	0
$963$	−10.8509	−0.349664
$964$	14.5579i	0.468880i
$965$	15.3937	0.495542
$966$	−17.1521	−0.551861
$967$	− 29.7095i	− 0.955392i	−0.878525	−	0.477696i	$-0.841472\pi$
$967$	− 29.7095i	− 0.955392i	0.878525	−	0.477696i	$-0.158528\pi$
$968$	28.0248i	0.900750i
$969$	− 15.6920i	− 0.504100i
$970$	0.335126i	0.0107602i
$971$	−50.0428	−1.60595	−0.802975	−	0.596013i	$-0.796751\pi$
$971$	−50.0428	−1.60595	−0.802975	+	0.596013i	$0.796751\pi$
$972$	1.00000	0.0320750
$973$	− 59.4946i	− 1.90731i
$974$	−14.9903	−0.480321
$975$	0	0
$976$	−7.91723	−0.253424
$977$	− 15.4800i	− 0.495248i	−0.968856	−	0.247624i	$-0.920350\pi$
$977$	− 15.4800i	− 0.495248i	0.968856	−	0.247624i	$-0.0796497\pi$
$978$	−24.0151	−0.767917
$979$	59.3691	1.89744
$980$	2.29590i	0.0733397i
$981$	− 1.02177i	− 0.0326226i
$982$	21.3706i	0.681964i
$983$	− 24.3080i	− 0.775304i	−0.921806	−	0.387652i	$-0.873286\pi$
$983$	− 24.3080i	− 0.775304i	0.921806	−	0.387652i	$-0.126714\pi$
$984$	3.77479	0.120336
$985$	23.5036	0.748888
$986$	13.8552i	0.441238i
$987$	−11.0248	−0.350922
$988$	0	0
$989$	−16.0379	−0.509976
$990$	− 6.24698i	− 0.198542i
$991$	41.1196	1.30621	0.653104	−	0.757269i	$-0.273467\pi$
$991$	41.1196	1.30621	0.653104	+	0.757269i	$0.273467\pi$
$992$	3.53319	0.112179
$993$	12.1575i	0.385807i
$994$	− 1.14138i	− 0.0362022i
$995$	20.8442i	0.660804i
$996$	6.83877i	0.216695i
$997$	8.75063	0.277135	0.138568	−	0.990353i	$-0.455750\pi$
$997$	8.75063	0.277135	0.138568	+	0.990353i	$0.455750\pi$
$998$	26.6160	0.842513
$999$	− 4.65279i	− 0.147208i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.b.s.1351.4		6
13.5	odd	4		5070.2.a.bt.1.3	yes	3
13.8	odd	4		5070.2.a.bk.1.1	✓	3
13.12	even	2	inner	5070.2.b.s.1351.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5070.2.a.bk.1.1	✓	3	13.8	odd	4
5070.2.a.bt.1.3	yes	3	13.5	odd	4
5070.2.b.s.1351.3		6	13.12	even	2	inner
5070.2.b.s.1351.4		6	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:	\( N \)	\(=\)	\( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5070.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} -3.04892i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} -3.04892i q^{7} -1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} +6.24698i q^{11} +1.00000 q^{12} +3.04892 q^{14} -1.00000i q^{15} +1.00000 q^{16} -2.69202 q^{17} +1.00000i q^{18} -5.82908i q^{19} -1.00000i q^{20} +3.04892i q^{21} -6.24698 q^{22} +5.62565 q^{23} +1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} +3.04892i q^{28} -5.14675 q^{29} +1.00000 q^{30} -3.53319i q^{31} +1.00000i q^{32} -6.24698i q^{33} -2.69202i q^{34} +3.04892 q^{35} -1.00000 q^{36} +4.65279i q^{37} +5.82908 q^{38} +1.00000 q^{40} -3.77479i q^{41} -3.04892 q^{42} -2.85086 q^{43} -6.24698i q^{44} +1.00000i q^{45} +5.62565i q^{46} +3.61596i q^{47} -1.00000 q^{48} -2.29590 q^{49} -1.00000i q^{50} +2.69202 q^{51} +0.664874 q^{53} -1.00000i q^{54} -6.24698 q^{55} -3.04892 q^{56} +5.82908i q^{57} -5.14675i q^{58} +12.9487i q^{59} +1.00000i q^{60} -7.91723 q^{61} +3.53319 q^{62} -3.04892i q^{63} -1.00000 q^{64} +6.24698 q^{66} -0.198062i q^{67} +2.69202 q^{68} -5.62565 q^{69} +3.04892i q^{70} -0.374354i q^{71} -1.00000i q^{72} -14.6136i q^{73} -4.65279 q^{74} +1.00000 q^{75} +5.82908i q^{76} +19.0465 q^{77} -11.2567 q^{79} +1.00000i q^{80} +1.00000 q^{81} +3.77479 q^{82} +6.83877i q^{83} -3.04892i q^{84} -2.69202i q^{85} -2.85086i q^{86} +5.14675 q^{87} +6.24698 q^{88} -9.50365i q^{89} -1.00000 q^{90} -5.62565 q^{92} +3.53319i q^{93} -3.61596 q^{94} +5.82908 q^{95} -1.00000i q^{96} -0.335126i q^{97} -2.29590i q^{98} +6.24698i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} - 6 q^{4} + 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^3 - 6 * q^4 + 6 * q^9	\( 6 q - 6 q^{3} - 6 q^{4} + 6 q^{9} - 6 q^{10} + 6 q^{12} + 6 q^{16} - 6 q^{17} - 28 q^{22} + 10 q^{23} - 6 q^{25} - 6 q^{27} + 24 q^{29} + 6 q^{30} - 6 q^{36} + 14 q^{38} + 6 q^{40} + 10 q^{43} - 6 q^{48} + 14 q^{49} + 6 q^{51} + 6 q^{53} - 28 q^{55} - 34 q^{61} + 28 q^{62} - 6 q^{64} + 28 q^{66} + 6 q^{68} - 10 q^{69} + 8 q^{74} + 6 q^{75} + 14 q^{77} - 14 q^{79} + 6 q^{81} + 26 q^{82} - 24 q^{87} + 28 q^{88} - 6 q^{90} - 10 q^{92} - 42 q^{94} + 14 q^{95}+O(q^{100}) \) 6 * q - 6 * q^3 - 6 * q^4 + 6 * q^9 - 6 * q^10 + 6 * q^12 + 6 * q^16 - 6 * q^17 - 28 * q^22 + 10 * q^23 - 6 * q^25 - 6 * q^27 + 24 * q^29 + 6 * q^30 - 6 * q^36 + 14 * q^38 + 6 * q^40 + 10 * q^43 - 6 * q^48 + 14 * q^49 + 6 * q^51 + 6 * q^53 - 28 * q^55 - 34 * q^61 + 28 * q^62 - 6 * q^64 + 28 * q^66 + 6 * q^68 - 10 * q^69 + 8 * q^74 + 6 * q^75 + 14 * q^77 - 14 * q^79 + 6 * q^81 + 26 * q^82 - 24 * q^87 + 28 * q^88 - 6 * q^90 - 10 * q^92 - 42 * q^94 + 14 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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