LMFDB - Embedded newform 5070.2.b.q.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:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 390)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.4
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	5070.1351
Dual form			5070.2.b.q.1351.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} +2.82843i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} +2.82843i q^{7} -1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} -5.65685i q^{11} -1.00000 q^{12} -2.82843 q^{14} +1.00000i q^{15} +1.00000 q^{16} -0.828427 q^{17} +1.00000i q^{18} +2.82843i q^{19} -1.00000i q^{20} +2.82843i q^{21} +5.65685 q^{22} +8.48528 q^{23} -1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} -2.82843i q^{28} -8.82843 q^{29} -1.00000 q^{30} +4.00000i q^{31} +1.00000i q^{32} -5.65685i q^{33} -0.828427i q^{34} -2.82843 q^{35} -1.00000 q^{36} +11.6569i q^{37} -2.82843 q^{38} +1.00000 q^{40} -7.65685i q^{41} -2.82843 q^{42} -9.65685 q^{43} +5.65685i q^{44} +1.00000i q^{45} +8.48528i q^{46} +8.00000i q^{47} +1.00000 q^{48} -1.00000 q^{49} -1.00000i q^{50} -0.828427 q^{51} +13.3137 q^{53} +1.00000i q^{54} +5.65685 q^{55} +2.82843 q^{56} +2.82843i q^{57} -8.82843i q^{58} +2.34315i q^{59} -1.00000i q^{60} +6.00000 q^{61} -4.00000 q^{62} +2.82843i q^{63} -1.00000 q^{64} +5.65685 q^{66} +5.65685i q^{67} +0.828427 q^{68} +8.48528 q^{69} -2.82843i q^{70} -5.65685i q^{71} -1.00000i q^{72} +14.4853i q^{73} -11.6569 q^{74} -1.00000 q^{75} -2.82843i q^{76} +16.0000 q^{77} +2.34315 q^{79} +1.00000i q^{80} +1.00000 q^{81} +7.65685 q^{82} +6.34315i q^{83} -2.82843i q^{84} -0.828427i q^{85} -9.65685i q^{86} -8.82843 q^{87} -5.65685 q^{88} +15.6569i q^{89} -1.00000 q^{90} -8.48528 q^{92} +4.00000i q^{93} -8.00000 q^{94} -2.82843 q^{95} +1.00000i q^{96} -3.17157i q^{97} -1.00000i q^{98} -5.65685i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 4 q^{4} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 - 4 * q^4 + 4 * q^9	$ 4 q + 4 q^{3} - 4 q^{4} + 4 q^{9} - 4 q^{10} - 4 q^{12} + 4 q^{16} + 8 q^{17} - 4 q^{25} + 4 q^{27} - 24 q^{29} - 4 q^{30} - 4 q^{36} + 4 q^{40} - 16 q^{43} + 4 q^{48} - 4 q^{49} + 8 q^{51} + 8 q^{53} + 24 q^{61} - 16 q^{62} - 4 q^{64} - 8 q^{68} - 24 q^{74} - 4 q^{75} + 64 q^{77} + 32 q^{79} + 4 q^{81} + 8 q^{82} - 24 q^{87} - 4 q^{90} - 32 q^{94}+O(q^{100}) $ 4 * q + 4 * q^3 - 4 * q^4 + 4 * q^9 - 4 * q^10 - 4 * q^12 + 4 * q^16 + 8 * q^17 - 4 * q^25 + 4 * q^27 - 24 * q^29 - 4 * q^30 - 4 * q^36 + 4 * q^40 - 16 * q^43 + 4 * q^48 - 4 * q^49 + 8 * q^51 + 8 * q^53 + 24 * q^61 - 16 * q^62 - 4 * q^64 - 8 * q^68 - 24 * q^74 - 4 * q^75 + 64 * q^77 + 32 * q^79 + 4 * q^81 + 8 * q^82 - 24 * q^87 - 4 * q^90 - 32 * q^94

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
$7$	2.82843i	1.06904i	0.845154	+	0.534522i	$0.179509\pi$
$7$	2.82843i	1.06904i	−0.845154	+	0.534522i	$0.820491\pi$
$8$	− 1.00000i	− 0.353553i
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	− 5.65685i	− 1.70561i	−0.522233	−	0.852803i	$-0.674901\pi$
$11$	− 5.65685i	− 1.70561i	0.522233	−	0.852803i	$-0.325099\pi$
$12$	−1.00000	−0.288675
$13$	0	0
$13$	0	0
$14$	−2.82843	−0.755929
$15$	1.00000i	0.258199i
$16$	1.00000	0.250000
$17$	−0.828427	−0.200923	−0.100462	−	0.994941i	$-0.532032\pi$
$17$	−0.828427	−0.200923	−0.100462	+	0.994941i	$0.532032\pi$
$18$	1.00000i	0.235702i
$19$	2.82843i	0.648886i	0.945905	+	0.324443i	$0.105177\pi$
$19$	2.82843i	0.648886i	−0.945905	+	0.324443i	$0.894823\pi$
$20$	− 1.00000i	− 0.223607i
$21$	2.82843i	0.617213i
$22$	5.65685	1.20605
$23$	8.48528	1.76930	0.884652	−	0.466252i	$-0.154396\pi$
$23$	8.48528	1.76930	0.884652	+	0.466252i	$0.154396\pi$
$24$	− 1.00000i	− 0.204124i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	− 2.82843i	− 0.534522i
$29$	−8.82843	−1.63940	−0.819699	−	0.572795i	$-0.805859\pi$
$29$	−8.82843	−1.63940	−0.819699	+	0.572795i	$0.805859\pi$
$30$	−1.00000	−0.182574
$31$	4.00000i	0.718421i	0.933257	+	0.359211i	$0.116954\pi$
$31$	4.00000i	0.718421i	−0.933257	+	0.359211i	$0.883046\pi$
$32$	1.00000i	0.176777i
$33$	− 5.65685i	− 0.984732i
$34$	− 0.828427i	− 0.142074i
$35$	−2.82843	−0.478091
$36$	−1.00000	−0.166667
$37$	11.6569i	1.91638i	0.286141	+	0.958188i	$0.407627\pi$
$37$	11.6569i	1.91638i	−0.286141	+	0.958188i	$0.592373\pi$
$38$	−2.82843	−0.458831
$39$	0	0
$40$	1.00000	0.158114
$41$	− 7.65685i	− 1.19580i	−0.801571	−	0.597900i	$-0.796002\pi$
$41$	− 7.65685i	− 1.19580i	0.801571	−	0.597900i	$-0.203998\pi$
$42$	−2.82843	−0.436436
$43$	−9.65685	−1.47266	−0.736328	−	0.676625i	$-0.763442\pi$
$43$	−9.65685	−1.47266	−0.736328	+	0.676625i	$0.763442\pi$
$44$	5.65685i	0.852803i
$45$	1.00000i	0.149071i
$46$	8.48528i	1.25109i
$47$	8.00000i	1.16692i	0.812142	+	0.583460i	$0.198301\pi$
$47$	8.00000i	1.16692i	−0.812142	+	0.583460i	$0.801699\pi$
$48$	1.00000	0.144338
$49$	−1.00000	−0.142857
$50$	− 1.00000i	− 0.141421i
$51$	−0.828427	−0.116003
$52$	0	0
$53$	13.3137	1.82878	0.914389	−	0.404836i	$-0.132671\pi$
$53$	13.3137	1.82878	0.914389	+	0.404836i	$0.132671\pi$
$54$	1.00000i	0.136083i
$55$	5.65685	0.762770
$56$	2.82843	0.377964
$57$	2.82843i	0.374634i
$58$	− 8.82843i	− 1.15923i
$59$	2.34315i	0.305052i	0.988299	+	0.152526i	$0.0487407\pi$
$59$	2.34315i	0.305052i	−0.988299	+	0.152526i	$0.951259\pi$
$60$	− 1.00000i	− 0.129099i
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	−4.00000	−0.508001
$63$	2.82843i	0.356348i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	5.65685	0.696311
$67$	5.65685i	0.691095i	0.938401	+	0.345547i	$0.112307\pi$
$67$	5.65685i	0.691095i	−0.938401	+	0.345547i	$0.887693\pi$
$68$	0.828427	0.100462
$69$	8.48528	1.02151
$70$	− 2.82843i	− 0.338062i
$71$	− 5.65685i	− 0.671345i	−0.941979	−	0.335673i	$-0.891036\pi$
$71$	− 5.65685i	− 0.671345i	0.941979	−	0.335673i	$-0.108964\pi$
$72$	− 1.00000i	− 0.117851i
$73$	14.4853i	1.69537i	0.530497	+	0.847687i	$0.322005\pi$
$73$	14.4853i	1.69537i	−0.530497	+	0.847687i	$0.677995\pi$
$74$	−11.6569	−1.35508
$75$	−1.00000	−0.115470
$76$	− 2.82843i	− 0.324443i
$77$	16.0000	1.82337
$78$	0	0
$79$	2.34315	0.263624	0.131812	−	0.991275i	$-0.457920\pi$
$79$	2.34315	0.263624	0.131812	+	0.991275i	$0.457920\pi$
$80$	1.00000i	0.111803i
$81$	1.00000	0.111111
$82$	7.65685	0.845558
$83$	6.34315i	0.696251i	0.937448	+	0.348125i	$0.113182\pi$
$83$	6.34315i	0.696251i	−0.937448	+	0.348125i	$0.886818\pi$
$84$	− 2.82843i	− 0.308607i
$85$	− 0.828427i	− 0.0898555i
$86$	− 9.65685i	− 1.04133i
$87$	−8.82843	−0.946507
$88$	−5.65685	−0.603023
$89$	15.6569i	1.65962i	0.558044	+	0.829812i	$0.311552\pi$
$89$	15.6569i	1.65962i	−0.558044	+	0.829812i	$0.688448\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	−8.48528	−0.884652
$93$	4.00000i	0.414781i
$94$	−8.00000	−0.825137
$95$	−2.82843	−0.290191
$96$	1.00000i	0.102062i
$97$	− 3.17157i	− 0.322024i	−0.986952	−	0.161012i	$-0.948524\pi$
$97$	− 3.17157i	− 0.322024i	0.986952	−	0.161012i	$-0.0514759\pi$
$98$	− 1.00000i	− 0.101015i
$99$	− 5.65685i	− 0.568535i
$100$	1.00000	0.100000
$101$	−16.1421	−1.60620	−0.803101	−	0.595843i	$-0.796818\pi$
$101$	−16.1421	−1.60620	−0.803101	+	0.595843i	$0.796818\pi$
$102$	− 0.828427i	− 0.0820265i
$103$	1.65685	0.163255	0.0816274	−	0.996663i	$-0.473988\pi$
$103$	1.65685	0.163255	0.0816274	+	0.996663i	$0.473988\pi$
$104$	0	0
$105$	−2.82843	−0.276026
$106$	13.3137i	1.29314i
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	−1.00000	−0.0962250
$109$	8.82843i	0.845610i	0.906221	+	0.422805i	$0.138954\pi$
$109$	8.82843i	0.845610i	−0.906221	+	0.422805i	$0.861046\pi$
$110$	5.65685i	0.539360i
$111$	11.6569i	1.10642i
$112$	2.82843i	0.267261i
$113$	6.48528	0.610084	0.305042	−	0.952339i	$-0.401330\pi$
$113$	6.48528	0.610084	0.305042	+	0.952339i	$0.401330\pi$
$114$	−2.82843	−0.264906
$115$	8.48528i	0.791257i
$116$	8.82843	0.819699
$117$	0	0
$118$	−2.34315	−0.215704
$119$	− 2.34315i	− 0.214796i
$120$	1.00000	0.0912871
$121$	−21.0000	−1.90909
$122$	6.00000i	0.543214i
$123$	− 7.65685i	− 0.690395i
$124$	− 4.00000i	− 0.359211i
$125$	− 1.00000i	− 0.0894427i
$126$	−2.82843	−0.251976
$127$	−9.65685	−0.856907	−0.428454	−	0.903564i	$-0.640941\pi$
$127$	−9.65685	−0.856907	−0.428454	+	0.903564i	$0.640941\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	−9.65685	−0.850239
$130$	0	0
$131$	−6.14214	−0.536641	−0.268320	−	0.963330i	$-0.586469\pi$
$131$	−6.14214	−0.536641	−0.268320	+	0.963330i	$0.586469\pi$
$132$	5.65685i	0.492366i
$133$	−8.00000	−0.693688
$134$	−5.65685	−0.488678
$135$	1.00000i	0.0860663i
$136$	0.828427i	0.0710370i
$137$	17.3137i	1.47921i	0.673041	+	0.739605i	$0.264988\pi$
$137$	17.3137i	1.47921i	−0.673041	+	0.739605i	$0.735012\pi$
$138$	8.48528i	0.722315i
$139$	6.34315	0.538019	0.269009	−	0.963138i	$-0.413304\pi$
$139$	6.34315	0.538019	0.269009	+	0.963138i	$0.413304\pi$
$140$	2.82843	0.239046
$141$	8.00000i	0.673722i
$142$	5.65685	0.474713
$143$	0	0
$144$	1.00000	0.0833333
$145$	− 8.82843i	− 0.733161i
$146$	−14.4853	−1.19881
$147$	−1.00000	−0.0824786
$148$	− 11.6569i	− 0.958188i
$149$	3.65685i	0.299581i	0.988718	+	0.149791i	$0.0478599\pi$
$149$	3.65685i	0.299581i	−0.988718	+	0.149791i	$0.952140\pi$
$150$	− 1.00000i	− 0.0816497i
$151$	− 12.0000i	− 0.976546i	−0.872691	−	0.488273i	$-0.837627\pi$
$151$	− 12.0000i	− 0.976546i	0.872691	−	0.488273i	$-0.162373\pi$
$152$	2.82843	0.229416
$153$	−0.828427	−0.0669744
$154$	16.0000i	1.28932i
$155$	−4.00000	−0.321288
$156$	0	0
$157$	5.31371	0.424080	0.212040	−	0.977261i	$-0.431989\pi$
$157$	5.31371	0.424080	0.212040	+	0.977261i	$0.431989\pi$
$158$	2.34315i	0.186411i
$159$	13.3137	1.05585
$160$	−1.00000	−0.0790569
$161$	24.0000i	1.89146i
$162$	1.00000i	0.0785674i
$163$	− 11.3137i	− 0.886158i	−0.896483	−	0.443079i	$-0.853886\pi$
$163$	− 11.3137i	− 0.886158i	0.896483	−	0.443079i	$-0.146114\pi$
$164$	7.65685i	0.597900i
$165$	5.65685	0.440386
$166$	−6.34315	−0.492324
$167$	− 8.97056i	− 0.694163i	−0.937835	−	0.347081i	$-0.887173\pi$
$167$	− 8.97056i	− 0.694163i	0.937835	−	0.347081i	$-0.112827\pi$
$168$	2.82843	0.218218
$169$	0	0
$170$	0.828427	0.0635375
$171$	2.82843i	0.216295i
$172$	9.65685	0.736328
$173$	9.31371	0.708108	0.354054	−	0.935225i	$-0.384803\pi$
$173$	9.31371	0.708108	0.354054	+	0.935225i	$0.384803\pi$
$174$	− 8.82843i	− 0.669281i
$175$	− 2.82843i	− 0.213809i
$176$	− 5.65685i	− 0.426401i
$177$	2.34315i	0.176122i
$178$	−15.6569	−1.17353
$179$	−7.51472	−0.561676	−0.280838	−	0.959755i	$-0.590612\pi$
$179$	−7.51472	−0.561676	−0.280838	+	0.959755i	$0.590612\pi$
$180$	− 1.00000i	− 0.0745356i
$181$	7.65685	0.569129	0.284565	−	0.958657i	$-0.408151\pi$
$181$	7.65685	0.569129	0.284565	+	0.958657i	$0.408151\pi$
$182$	0	0
$183$	6.00000	0.443533
$184$	− 8.48528i	− 0.625543i
$185$	−11.6569	−0.857029
$186$	−4.00000	−0.293294
$187$	4.68629i	0.342696i
$188$	− 8.00000i	− 0.583460i
$189$	2.82843i	0.205738i
$190$	− 2.82843i	− 0.205196i
$191$	11.3137	0.818631	0.409316	−	0.912393i	$-0.365768\pi$
$191$	11.3137	0.818631	0.409316	+	0.912393i	$0.365768\pi$
$192$	−1.00000	−0.0721688
$193$	− 2.48528i	− 0.178894i	−0.995992	−	0.0894472i	$-0.971490\pi$
$193$	− 2.48528i	− 0.178894i	0.995992	−	0.0894472i	$-0.0285100\pi$
$194$	3.17157	0.227706
$195$	0	0
$196$	1.00000	0.0714286
$197$	− 13.3137i	− 0.948562i	−0.880373	−	0.474281i	$-0.842708\pi$
$197$	− 13.3137i	− 0.948562i	0.880373	−	0.474281i	$-0.157292\pi$
$198$	5.65685	0.402015
$199$	−10.3431	−0.733206	−0.366603	−	0.930377i	$-0.619479\pi$
$199$	−10.3431	−0.733206	−0.366603	+	0.930377i	$0.619479\pi$
$200$	1.00000i	0.0707107i
$201$	5.65685i	0.399004i
$202$	− 16.1421i	− 1.13576i
$203$	− 24.9706i	− 1.75259i
$204$	0.828427	0.0580015
$205$	7.65685	0.534778
$206$	1.65685i	0.115439i
$207$	8.48528	0.589768
$208$	0	0
$209$	16.0000	1.10674
$210$	− 2.82843i	− 0.195180i
$211$	0.686292	0.0472463	0.0236231	−	0.999721i	$-0.492480\pi$
$211$	0.686292	0.0472463	0.0236231	+	0.999721i	$0.492480\pi$
$212$	−13.3137	−0.914389
$213$	− 5.65685i	− 0.387601i
$214$	− 4.00000i	− 0.273434i
$215$	− 9.65685i	− 0.658592i
$216$	− 1.00000i	− 0.0680414i
$217$	−11.3137	−0.768025
$218$	−8.82843	−0.597937
$219$	14.4853i	0.978825i
$220$	−5.65685	−0.381385
$221$	0	0
$222$	−11.6569	−0.782357
$223$	10.8284i	0.725125i	0.931959	+	0.362563i	$0.118098\pi$
$223$	10.8284i	0.725125i	−0.931959	+	0.362563i	$0.881902\pi$
$224$	−2.82843	−0.188982
$225$	−1.00000	−0.0666667
$226$	6.48528i	0.431394i
$227$	4.00000i	0.265489i	0.991150	+	0.132745i	$0.0423790\pi$
$227$	4.00000i	0.265489i	−0.991150	+	0.132745i	$0.957621\pi$
$228$	− 2.82843i	− 0.187317i
$229$	− 4.14214i	− 0.273720i	−0.990590	−	0.136860i	$-0.956299\pi$
$229$	− 4.14214i	− 0.273720i	0.990590	−	0.136860i	$-0.0437011\pi$
$230$	−8.48528	−0.559503
$231$	16.0000	1.05272
$232$	8.82843i	0.579615i
$233$	−5.51472	−0.361281	−0.180641	−	0.983549i	$-0.557817\pi$
$233$	−5.51472	−0.361281	−0.180641	+	0.983549i	$0.557817\pi$
$234$	0	0
$235$	−8.00000	−0.521862
$236$	− 2.34315i	− 0.152526i
$237$	2.34315	0.152204
$238$	2.34315	0.151884
$239$	16.0000i	1.03495i	0.855697	+	0.517477i	$0.173129\pi$
$239$	16.0000i	1.03495i	−0.855697	+	0.517477i	$0.826871\pi$
$240$	1.00000i	0.0645497i
$241$	− 5.31371i	− 0.342286i	−0.985246	−	0.171143i	$-0.945254\pi$
$241$	− 5.31371i	− 0.342286i	0.985246	−	0.171143i	$-0.0547460\pi$
$242$	− 21.0000i	− 1.34993i
$243$	1.00000	0.0641500
$244$	−6.00000	−0.384111
$245$	− 1.00000i	− 0.0638877i
$246$	7.65685	0.488183
$247$	0	0
$248$	4.00000	0.254000
$249$	6.34315i	0.401981i
$250$	1.00000	0.0632456
$251$	−10.8284	−0.683484	−0.341742	−	0.939794i	$-0.611017\pi$
$251$	−10.8284	−0.683484	−0.341742	+	0.939794i	$0.611017\pi$
$252$	− 2.82843i	− 0.178174i
$253$	− 48.0000i	− 3.01773i
$254$	− 9.65685i	− 0.605925i
$255$	− 0.828427i	− 0.0518781i
$256$	1.00000	0.0625000
$257$	4.82843	0.301189	0.150595	−	0.988596i	$-0.451881\pi$
$257$	4.82843	0.301189	0.150595	+	0.988596i	$0.451881\pi$
$258$	− 9.65685i	− 0.601209i
$259$	−32.9706	−2.04869
$260$	0	0
$261$	−8.82843	−0.546466
$262$	− 6.14214i	− 0.379462i
$263$	16.4853	1.01653	0.508263	−	0.861202i	$-0.330288\pi$
$263$	16.4853	1.01653	0.508263	+	0.861202i	$0.330288\pi$
$264$	−5.65685	−0.348155
$265$	13.3137i	0.817855i
$266$	− 8.00000i	− 0.490511i
$267$	15.6569i	0.958184i
$268$	− 5.65685i	− 0.345547i
$269$	−14.4853	−0.883183	−0.441592	−	0.897216i	$-0.645586\pi$
$269$	−14.4853	−0.883183	−0.441592	+	0.897216i	$0.645586\pi$
$270$	−1.00000	−0.0608581
$271$	− 7.31371i	− 0.444276i	−0.975015	−	0.222138i	$-0.928696\pi$
$271$	− 7.31371i	− 0.444276i	0.975015	−	0.222138i	$-0.0713036\pi$
$272$	−0.828427	−0.0502308
$273$	0	0
$274$	−17.3137	−1.04596
$275$	5.65685i	0.341121i
$276$	−8.48528	−0.510754
$277$	−26.0000	−1.56219	−0.781094	−	0.624413i	$-0.785338\pi$
$277$	−26.0000	−1.56219	−0.781094	+	0.624413i	$0.785338\pi$
$278$	6.34315i	0.380437i
$279$	4.00000i	0.239474i
$280$	2.82843i	0.169031i
$281$	− 8.34315i	− 0.497710i	−0.968541	−	0.248855i	$-0.919946\pi$
$281$	− 8.34315i	− 0.497710i	0.968541	−	0.248855i	$-0.0800543\pi$
$282$	−8.00000	−0.476393
$283$	17.6569	1.04959	0.524796	−	0.851228i	$-0.324142\pi$
$283$	17.6569	1.04959	0.524796	+	0.851228i	$0.324142\pi$
$284$	5.65685i	0.335673i
$285$	−2.82843	−0.167542
$286$	0	0
$287$	21.6569	1.27836
$288$	1.00000i	0.0589256i
$289$	−16.3137	−0.959630
$290$	8.82843	0.518423
$291$	− 3.17157i	− 0.185921i
$292$	− 14.4853i	− 0.847687i
$293$	16.6274i	0.971384i	0.874130	+	0.485692i	$0.161432\pi$
$293$	16.6274i	0.971384i	−0.874130	+	0.485692i	$0.838568\pi$
$294$	− 1.00000i	− 0.0583212i
$295$	−2.34315	−0.136423
$296$	11.6569	0.677541
$297$	− 5.65685i	− 0.328244i
$298$	−3.65685	−0.211836
$299$	0	0
$300$	1.00000	0.0577350
$301$	− 27.3137i	− 1.57434i
$302$	12.0000	0.690522
$303$	−16.1421	−0.927341
$304$	2.82843i	0.162221i
$305$	6.00000i	0.343559i
$306$	− 0.828427i	− 0.0473580i
$307$	21.6569i	1.23602i	0.786169	+	0.618011i	$0.212061\pi$
$307$	21.6569i	1.23602i	−0.786169	+	0.618011i	$0.787939\pi$
$308$	−16.0000	−0.911685
$309$	1.65685	0.0942551
$310$	− 4.00000i	− 0.227185i
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	−30.9706	−1.75056	−0.875280	−	0.483617i	$-0.839323\pi$
$313$	−30.9706	−1.75056	−0.875280	+	0.483617i	$0.839323\pi$
$314$	5.31371i	0.299870i
$315$	−2.82843	−0.159364
$316$	−2.34315	−0.131812
$317$	25.3137i	1.42176i	0.703314	+	0.710880i	$0.251703\pi$
$317$	25.3137i	1.42176i	−0.703314	+	0.710880i	$0.748297\pi$
$318$	13.3137i	0.746596i
$319$	49.9411i	2.79617i
$320$	− 1.00000i	− 0.0559017i
$321$	−4.00000	−0.223258
$322$	−24.0000	−1.33747
$323$	− 2.34315i	− 0.130376i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	11.3137	0.626608
$327$	8.82843i	0.488213i
$328$	−7.65685	−0.422779
$329$	−22.6274	−1.24749
$330$	5.65685i	0.311400i
$331$	− 8.48528i	− 0.466393i	−0.972430	−	0.233197i	$-0.925081\pi$
$331$	− 8.48528i	− 0.466393i	0.972430	−	0.233197i	$-0.0749186\pi$
$332$	− 6.34315i	− 0.348125i
$333$	11.6569i	0.638792i
$334$	8.97056	0.490847
$335$	−5.65685	−0.309067
$336$	2.82843i	0.154303i
$337$	−10.9706	−0.597605	−0.298802	−	0.954315i	$-0.596587\pi$
$337$	−10.9706	−0.597605	−0.298802	+	0.954315i	$0.596587\pi$
$338$	0	0
$339$	6.48528	0.352232
$340$	0.828427i	0.0449278i
$341$	22.6274	1.22534
$342$	−2.82843	−0.152944
$343$	16.9706i	0.916324i
$344$	9.65685i	0.520663i
$345$	8.48528i	0.456832i
$346$	9.31371i	0.500708i
$347$	−9.65685	−0.518407	−0.259204	−	0.965823i	$-0.583460\pi$
$347$	−9.65685	−0.518407	−0.259204	+	0.965823i	$0.583460\pi$
$348$	8.82843	0.473253
$349$	− 12.1421i	− 0.649954i	−0.945722	−	0.324977i	$-0.894644\pi$
$349$	− 12.1421i	− 0.649954i	0.945722	−	0.324977i	$-0.105356\pi$
$350$	2.82843	0.151186
$351$	0	0
$352$	5.65685	0.301511
$353$	5.31371i	0.282820i	0.989951	+	0.141410i	$0.0451636\pi$
$353$	5.31371i	0.282820i	−0.989951	+	0.141410i	$0.954836\pi$
$354$	−2.34315	−0.124537
$355$	5.65685	0.300235
$356$	− 15.6569i	− 0.829812i
$357$	− 2.34315i	− 0.124012i
$358$	− 7.51472i	− 0.397165i
$359$	28.2843i	1.49279i	0.665505	+	0.746393i	$0.268216\pi$
$359$	28.2843i	1.49279i	−0.665505	+	0.746393i	$0.731784\pi$
$360$	1.00000	0.0527046
$361$	11.0000	0.578947
$362$	7.65685i	0.402435i
$363$	−21.0000	−1.10221
$364$	0	0
$365$	−14.4853	−0.758194
$366$	6.00000i	0.313625i
$367$	25.6569	1.33928	0.669638	−	0.742687i	$-0.266449\pi$
$367$	25.6569	1.33928	0.669638	+	0.742687i	$0.266449\pi$
$368$	8.48528	0.442326
$369$	− 7.65685i	− 0.398600i
$370$	− 11.6569i	− 0.606011i
$371$	37.6569i	1.95505i
$372$	− 4.00000i	− 0.207390i
$373$	−2.68629	−0.139091	−0.0695455	−	0.997579i	$-0.522155\pi$
$373$	−2.68629	−0.139091	−0.0695455	+	0.997579i	$0.522155\pi$
$374$	−4.68629	−0.242322
$375$	− 1.00000i	− 0.0516398i
$376$	8.00000	0.412568
$377$	0	0
$378$	−2.82843	−0.145479
$379$	− 7.51472i	− 0.386005i	−0.981198	−	0.193003i	$-0.938177\pi$
$379$	− 7.51472i	− 0.386005i	0.981198	−	0.193003i	$-0.0618226\pi$
$380$	2.82843	0.145095
$381$	−9.65685	−0.494736
$382$	11.3137i	0.578860i
$383$	− 29.6569i	− 1.51539i	−0.652606	−	0.757697i	$-0.726324\pi$
$383$	− 29.6569i	− 1.51539i	0.652606	−	0.757697i	$-0.273676\pi$
$384$	− 1.00000i	− 0.0510310i
$385$	16.0000i	0.815436i
$386$	2.48528	0.126497
$387$	−9.65685	−0.490885
$388$	3.17157i	0.161012i
$389$	6.48528	0.328817	0.164408	−	0.986392i	$-0.447429\pi$
$389$	6.48528	0.328817	0.164408	+	0.986392i	$0.447429\pi$
$390$	0	0
$391$	−7.02944	−0.355494
$392$	1.00000i	0.0505076i
$393$	−6.14214	−0.309830
$394$	13.3137	0.670735
$395$	2.34315i	0.117896i
$396$	5.65685i	0.284268i
$397$	− 30.2843i	− 1.51992i	−0.649968	−	0.759962i	$-0.725218\pi$
$397$	− 30.2843i	− 1.51992i	0.649968	−	0.759962i	$-0.274782\pi$
$398$	− 10.3431i	− 0.518455i
$399$	−8.00000	−0.400501
$400$	−1.00000	−0.0500000
$401$	26.9706i	1.34685i	0.739258	+	0.673423i	$0.235177\pi$
$401$	26.9706i	1.34685i	−0.739258	+	0.673423i	$0.764823\pi$
$402$	−5.65685	−0.282138
$403$	0	0
$404$	16.1421	0.803101
$405$	1.00000i	0.0496904i
$406$	24.9706	1.23927
$407$	65.9411	3.26858
$408$	0.828427i	0.0410133i
$409$	− 3.65685i	− 0.180820i	−0.995905	−	0.0904099i	$-0.971182\pi$
$409$	− 3.65685i	− 0.180820i	0.995905	−	0.0904099i	$-0.0288177\pi$
$410$	7.65685i	0.378145i
$411$	17.3137i	0.854022i
$412$	−1.65685	−0.0816274
$413$	−6.62742	−0.326114
$414$	8.48528i	0.417029i
$415$	−6.34315	−0.311373
$416$	0	0
$417$	6.34315	0.310625
$418$	16.0000i	0.782586i
$419$	−10.8284	−0.529003	−0.264502	−	0.964385i	$-0.585207\pi$
$419$	−10.8284	−0.529003	−0.264502	+	0.964385i	$0.585207\pi$
$420$	2.82843	0.138013
$421$	− 24.1421i	− 1.17662i	−0.808637	−	0.588308i	$-0.799794\pi$
$421$	− 24.1421i	− 1.17662i	0.808637	−	0.588308i	$-0.200206\pi$
$422$	0.686292i	0.0334081i
$423$	8.00000i	0.388973i
$424$	− 13.3137i	− 0.646571i
$425$	0.828427	0.0401846
$426$	5.65685	0.274075
$427$	16.9706i	0.821263i
$428$	4.00000	0.193347
$429$	0	0
$430$	9.65685	0.465695
$431$	− 16.0000i	− 0.770693i	−0.922772	−	0.385346i	$-0.874082\pi$
$431$	− 16.0000i	− 0.770693i	0.922772	−	0.385346i	$-0.125918\pi$
$432$	1.00000	0.0481125
$433$	22.9706	1.10389	0.551947	−	0.833879i	$-0.313885\pi$
$433$	22.9706	1.10389	0.551947	+	0.833879i	$0.313885\pi$
$434$	− 11.3137i	− 0.543075i
$435$	− 8.82843i	− 0.423291i
$436$	− 8.82843i	− 0.422805i
$437$	24.0000i	1.14808i
$438$	−14.4853	−0.692134
$439$	−22.6274	−1.07995	−0.539974	−	0.841682i	$-0.681566\pi$
$439$	−22.6274	−1.07995	−0.539974	+	0.841682i	$0.681566\pi$
$440$	− 5.65685i	− 0.269680i
$441$	−1.00000	−0.0476190
$442$	0	0
$443$	30.3431	1.44165	0.720823	−	0.693119i	$-0.243764\pi$
$443$	30.3431	1.44165	0.720823	+	0.693119i	$0.243764\pi$
$444$	− 11.6569i	− 0.553210i
$445$	−15.6569	−0.742206
$446$	−10.8284	−0.512741
$447$	3.65685i	0.172963i
$448$	− 2.82843i	− 0.133631i
$449$	− 26.2843i	− 1.24043i	−0.784431	−	0.620216i	$-0.787045\pi$
$449$	− 26.2843i	− 1.24043i	0.784431	−	0.620216i	$-0.212955\pi$
$450$	− 1.00000i	− 0.0471405i
$451$	−43.3137	−2.03956
$452$	−6.48528	−0.305042
$453$	− 12.0000i	− 0.563809i
$454$	−4.00000	−0.187729
$455$	0	0
$456$	2.82843	0.132453
$457$	20.8284i	0.974313i	0.873315	+	0.487156i	$0.161966\pi$
$457$	20.8284i	0.974313i	−0.873315	+	0.487156i	$0.838034\pi$
$458$	4.14214	0.193549
$459$	−0.828427	−0.0386677
$460$	− 8.48528i	− 0.395628i
$461$	14.0000i	0.652045i	0.945362	+	0.326023i	$0.105709\pi$
$461$	14.0000i	0.652045i	−0.945362	+	0.326023i	$0.894291\pi$
$462$	16.0000i	0.744387i
$463$	3.79899i	0.176554i	0.996096	+	0.0882770i	$0.0281361\pi$
$463$	3.79899i	0.176554i	−0.996096	+	0.0882770i	$0.971864\pi$
$464$	−8.82843	−0.409849
$465$	−4.00000	−0.185496
$466$	− 5.51472i	− 0.255464i
$467$	−7.31371	−0.338438	−0.169219	−	0.985578i	$-0.554125\pi$
$467$	−7.31371	−0.338438	−0.169219	+	0.985578i	$0.554125\pi$
$468$	0	0
$469$	−16.0000	−0.738811
$470$	− 8.00000i	− 0.369012i
$471$	5.31371	0.244843
$472$	2.34315	0.107852
$473$	54.6274i	2.51177i
$474$	2.34315i	0.107624i
$475$	− 2.82843i	− 0.129777i
$476$	2.34315i	0.107398i
$477$	13.3137	0.609593
$478$	−16.0000	−0.731823
$479$	11.3137i	0.516937i	0.966020	+	0.258468i	$0.0832177\pi$
$479$	11.3137i	0.516937i	−0.966020	+	0.258468i	$0.916782\pi$
$480$	−1.00000	−0.0456435
$481$	0	0
$482$	5.31371	0.242033
$483$	24.0000i	1.09204i
$484$	21.0000	0.954545
$485$	3.17157	0.144014
$486$	1.00000i	0.0453609i
$487$	16.4853i	0.747019i	0.927626	+	0.373510i	$0.121846\pi$
$487$	16.4853i	0.747019i	−0.927626	+	0.373510i	$0.878154\pi$
$488$	− 6.00000i	− 0.271607i
$489$	− 11.3137i	− 0.511624i
$490$	1.00000	0.0451754
$491$	38.1421	1.72133	0.860665	−	0.509171i	$-0.170048\pi$
$491$	38.1421	1.72133	0.860665	+	0.509171i	$0.170048\pi$
$492$	7.65685i	0.345198i
$493$	7.31371	0.329393
$494$	0	0
$495$	5.65685	0.254257
$496$	4.00000i	0.179605i
$497$	16.0000	0.717698
$498$	−6.34315	−0.284243
$499$	0.485281i	0.0217242i	0.999941	+	0.0108621i	$0.00345758\pi$
$499$	0.485281i	0.0217242i	−0.999941	+	0.0108621i	$0.996542\pi$
$500$	1.00000i	0.0447214i
$501$	− 8.97056i	− 0.400775i
$502$	− 10.8284i	− 0.483296i
$503$	−23.5147	−1.04847	−0.524235	−	0.851574i	$-0.675649\pi$
$503$	−23.5147	−1.04847	−0.524235	+	0.851574i	$0.675649\pi$
$504$	2.82843	0.125988
$505$	− 16.1421i	− 0.718316i
$506$	48.0000	2.13386
$507$	0	0
$508$	9.65685	0.428454
$509$	− 37.3137i	− 1.65390i	−0.562275	−	0.826951i	$-0.690074\pi$
$509$	− 37.3137i	− 1.65390i	0.562275	−	0.826951i	$-0.309926\pi$
$510$	0.828427	0.0366834
$511$	−40.9706	−1.81243
$512$	1.00000i	0.0441942i
$513$	2.82843i	0.124878i
$514$	4.82843i	0.212973i
$515$	1.65685i	0.0730097i
$516$	9.65685	0.425119
$517$	45.2548	1.99031
$518$	− 32.9706i	− 1.44864i
$519$	9.31371	0.408826
$520$	0	0
$521$	26.9706	1.18160	0.590801	−	0.806817i	$-0.298812\pi$
$521$	26.9706	1.18160	0.590801	+	0.806817i	$0.298812\pi$
$522$	− 8.82843i	− 0.386410i
$523$	−10.6274	−0.464704	−0.232352	−	0.972632i	$-0.574642\pi$
$523$	−10.6274	−0.464704	−0.232352	+	0.972632i	$0.574642\pi$
$524$	6.14214	0.268320
$525$	− 2.82843i	− 0.123443i
$526$	16.4853i	0.718792i
$527$	− 3.31371i	− 0.144347i
$528$	− 5.65685i	− 0.246183i
$529$	49.0000	2.13043
$530$	−13.3137	−0.578311
$531$	2.34315i	0.101684i
$532$	8.00000	0.346844
$533$	0	0
$534$	−15.6569	−0.677538
$535$	− 4.00000i	− 0.172935i
$536$	5.65685	0.244339
$537$	−7.51472	−0.324284
$538$	− 14.4853i	− 0.624505i
$539$	5.65685i	0.243658i
$540$	− 1.00000i	− 0.0430331i
$541$	− 14.4853i	− 0.622771i	−0.950284	−	0.311385i	$-0.899207\pi$
$541$	− 14.4853i	− 0.622771i	0.950284	−	0.311385i	$-0.100793\pi$
$542$	7.31371	0.314151
$543$	7.65685	0.328587
$544$	− 0.828427i	− 0.0355185i
$545$	−8.82843	−0.378168
$546$	0	0
$547$	0.686292	0.0293437	0.0146719	−	0.999892i	$-0.495330\pi$
$547$	0.686292	0.0293437	0.0146719	+	0.999892i	$0.495330\pi$
$548$	− 17.3137i	− 0.739605i
$549$	6.00000	0.256074
$550$	−5.65685	−0.241209
$551$	− 24.9706i	− 1.06378i
$552$	− 8.48528i	− 0.361158i
$553$	6.62742i	0.281826i
$554$	− 26.0000i	− 1.10463i
$555$	−11.6569	−0.494806
$556$	−6.34315	−0.269009
$557$	− 10.6863i	− 0.452793i	−0.974035	−	0.226396i	$-0.927306\pi$
$557$	− 10.6863i	− 0.452793i	0.974035	−	0.226396i	$-0.0726945\pi$
$558$	−4.00000	−0.169334
$559$	0	0
$560$	−2.82843	−0.119523
$561$	4.68629i	0.197855i
$562$	8.34315	0.351934
$563$	30.3431	1.27881	0.639406	−	0.768870i	$-0.279181\pi$
$563$	30.3431	1.27881	0.639406	+	0.768870i	$0.279181\pi$
$564$	− 8.00000i	− 0.336861i
$565$	6.48528i	0.272838i
$566$	17.6569i	0.742173i
$567$	2.82843i	0.118783i
$568$	−5.65685	−0.237356
$569$	−31.6569	−1.32712	−0.663562	−	0.748121i	$-0.730956\pi$
$569$	−31.6569	−1.32712	−0.663562	+	0.748121i	$0.730956\pi$
$570$	− 2.82843i	− 0.118470i
$571$	−20.9706	−0.877591	−0.438795	−	0.898587i	$-0.644595\pi$
$571$	−20.9706	−0.877591	−0.438795	+	0.898587i	$0.644595\pi$
$572$	0	0
$573$	11.3137	0.472637
$574$	21.6569i	0.903940i
$575$	−8.48528	−0.353861
$576$	−1.00000	−0.0416667
$577$	− 23.4558i	− 0.976480i	−0.872710	−	0.488240i	$-0.837639\pi$
$577$	− 23.4558i	− 0.976480i	0.872710	−	0.488240i	$-0.162361\pi$
$578$	− 16.3137i	− 0.678561i
$579$	− 2.48528i	− 0.103285i
$580$	8.82843i	0.366580i
$581$	−17.9411	−0.744323
$582$	3.17157	0.131466
$583$	− 75.3137i	− 3.11918i
$584$	14.4853	0.599405
$585$	0	0
$586$	−16.6274	−0.686872
$587$	2.62742i	0.108445i	0.998529	+	0.0542226i	$0.0172680\pi$
$587$	2.62742i	0.108445i	−0.998529	+	0.0542226i	$0.982732\pi$
$588$	1.00000	0.0412393
$589$	−11.3137	−0.466173
$590$	− 2.34315i	− 0.0964658i
$591$	− 13.3137i	− 0.547653i
$592$	11.6569i	0.479094i
$593$	0.343146i	0.0140913i	0.999975	+	0.00704565i	$0.00224272\pi$
$593$	0.343146i	0.0140913i	−0.999975	+	0.00704565i	$0.997757\pi$
$594$	5.65685	0.232104
$595$	2.34315	0.0960596
$596$	− 3.65685i	− 0.149791i
$597$	−10.3431	−0.423317
$598$	0	0
$599$	40.0000	1.63436	0.817178	−	0.576386i	$-0.195537\pi$
$599$	40.0000	1.63436	0.817178	+	0.576386i	$0.195537\pi$
$600$	1.00000i	0.0408248i
$601$	29.3137	1.19573	0.597866	−	0.801596i	$-0.296016\pi$
$601$	29.3137	1.19573	0.597866	+	0.801596i	$0.296016\pi$
$602$	27.3137	1.11322
$603$	5.65685i	0.230365i
$604$	12.0000i	0.488273i
$605$	− 21.0000i	− 0.853771i
$606$	− 16.1421i	− 0.655729i
$607$	28.9706	1.17588	0.587939	−	0.808905i	$-0.299939\pi$
$607$	28.9706	1.17588	0.587939	+	0.808905i	$0.299939\pi$
$608$	−2.82843	−0.114708
$609$	− 24.9706i	− 1.01186i
$610$	−6.00000	−0.242933
$611$	0	0
$612$	0.828427	0.0334872
$613$	22.2843i	0.900053i	0.893015	+	0.450027i	$0.148586\pi$
$613$	22.2843i	0.900053i	−0.893015	+	0.450027i	$0.851414\pi$
$614$	−21.6569	−0.874000
$615$	7.65685	0.308754
$616$	− 16.0000i	− 0.644658i
$617$	2.00000i	0.0805170i	0.999189	+	0.0402585i	$0.0128181\pi$
$617$	2.00000i	0.0805170i	−0.999189	+	0.0402585i	$0.987182\pi$
$618$	1.65685i	0.0666485i
$619$	34.8284i	1.39987i	0.714205	+	0.699936i	$0.246788\pi$
$619$	34.8284i	1.39987i	−0.714205	+	0.699936i	$0.753212\pi$
$620$	4.00000	0.160644
$621$	8.48528	0.340503
$622$	24.0000i	0.962312i
$623$	−44.2843	−1.77421
$624$	0	0
$625$	1.00000	0.0400000
$626$	− 30.9706i	− 1.23783i
$627$	16.0000	0.638978
$628$	−5.31371	−0.212040
$629$	− 9.65685i	− 0.385044i
$630$	− 2.82843i	− 0.112687i
$631$	33.6569i	1.33986i	0.742425	+	0.669929i	$0.233676\pi$
$631$	33.6569i	1.33986i	−0.742425	+	0.669929i	$0.766324\pi$
$632$	− 2.34315i	− 0.0932053i
$633$	0.686292	0.0272776
$634$	−25.3137	−1.00534
$635$	− 9.65685i	− 0.383221i
$636$	−13.3137	−0.527923
$637$	0	0
$638$	−49.9411	−1.97719
$639$	− 5.65685i	− 0.223782i
$640$	1.00000	0.0395285
$641$	4.62742	0.182772	0.0913860	−	0.995816i	$-0.470870\pi$
$641$	4.62742	0.182772	0.0913860	+	0.995816i	$0.470870\pi$
$642$	− 4.00000i	− 0.157867i
$643$	39.5980i	1.56159i	0.624786	+	0.780796i	$0.285186\pi$
$643$	39.5980i	1.56159i	−0.624786	+	0.780796i	$0.714814\pi$
$644$	− 24.0000i	− 0.945732i
$645$	− 9.65685i	− 0.380238i
$646$	2.34315	0.0921898
$647$	−8.48528	−0.333591	−0.166795	−	0.985992i	$-0.553342\pi$
$647$	−8.48528	−0.333591	−0.166795	+	0.985992i	$0.553342\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	13.2548	0.520298
$650$	0	0
$651$	−11.3137	−0.443419
$652$	11.3137i	0.443079i
$653$	−42.2843	−1.65471	−0.827356	−	0.561678i	$-0.810156\pi$
$653$	−42.2843	−1.65471	−0.827356	+	0.561678i	$0.810156\pi$
$654$	−8.82843	−0.345219
$655$	− 6.14214i	− 0.239993i
$656$	− 7.65685i	− 0.298950i
$657$	14.4853i	0.565125i
$658$	− 22.6274i	− 0.882109i
$659$	−7.51472	−0.292732	−0.146366	−	0.989231i	$-0.546758\pi$
$659$	−7.51472	−0.292732	−0.146366	+	0.989231i	$0.546758\pi$
$660$	−5.65685	−0.220193
$661$	8.14214i	0.316692i	0.987384	+	0.158346i	$0.0506162\pi$
$661$	8.14214i	0.316692i	−0.987384	+	0.158346i	$0.949384\pi$
$662$	8.48528	0.329790
$663$	0	0
$664$	6.34315	0.246162
$665$	− 8.00000i	− 0.310227i
$666$	−11.6569	−0.451694
$667$	−74.9117	−2.90059
$668$	8.97056i	0.347081i
$669$	10.8284i	0.418651i
$670$	− 5.65685i	− 0.218543i
$671$	− 33.9411i	− 1.31028i
$672$	−2.82843	−0.109109
$673$	−32.6274	−1.25769	−0.628847	−	0.777529i	$-0.716473\pi$
$673$	−32.6274	−1.25769	−0.628847	+	0.777529i	$0.716473\pi$
$674$	− 10.9706i	− 0.422570i
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	12.3431	0.474386	0.237193	−	0.971463i	$-0.423773\pi$
$677$	12.3431	0.474386	0.237193	+	0.971463i	$0.423773\pi$
$678$	6.48528i	0.249066i
$679$	8.97056	0.344259
$680$	−0.828427	−0.0317687
$681$	4.00000i	0.153280i
$682$	22.6274i	0.866449i
$683$	− 33.6569i	− 1.28784i	−0.765091	−	0.643922i	$-0.777306\pi$
$683$	− 33.6569i	− 1.28784i	0.765091	−	0.643922i	$-0.222694\pi$
$684$	− 2.82843i	− 0.108148i
$685$	−17.3137	−0.661523
$686$	−16.9706	−0.647939
$687$	− 4.14214i	− 0.158032i
$688$	−9.65685	−0.368164
$689$	0	0
$690$	−8.48528	−0.323029
$691$	− 27.7990i	− 1.05752i	−0.848770	−	0.528762i	$-0.822657\pi$
$691$	− 27.7990i	− 1.05752i	0.848770	−	0.528762i	$-0.177343\pi$
$692$	−9.31371	−0.354054
$693$	16.0000	0.607790
$694$	− 9.65685i	− 0.366569i
$695$	6.34315i	0.240609i
$696$	8.82843i	0.334641i
$697$	6.34315i	0.240264i
$698$	12.1421	0.459587
$699$	−5.51472	−0.208586
$700$	2.82843i	0.106904i
$701$	−0.142136	−0.00536839	−0.00268419	−	0.999996i	$-0.500854\pi$
$701$	−0.142136	−0.00536839	−0.00268419	+	0.999996i	$0.500854\pi$
$702$	0	0
$703$	−32.9706	−1.24351
$704$	5.65685i	0.213201i
$705$	−8.00000	−0.301297
$706$	−5.31371	−0.199984
$707$	− 45.6569i	− 1.71710i
$708$	− 2.34315i	− 0.0880608i
$709$	7.17157i	0.269334i	0.990891	+	0.134667i	$0.0429965\pi$
$709$	7.17157i	0.269334i	−0.990891	+	0.134667i	$0.957004\pi$
$710$	5.65685i	0.212298i
$711$	2.34315	0.0878748
$712$	15.6569	0.586765
$713$	33.9411i	1.27111i
$714$	2.34315	0.0876900
$715$	0	0
$716$	7.51472	0.280838
$717$	16.0000i	0.597531i
$718$	−28.2843	−1.05556
$719$	29.6569	1.10601	0.553007	−	0.833177i	$-0.313480\pi$
$719$	29.6569	1.10601	0.553007	+	0.833177i	$0.313480\pi$
$720$	1.00000i	0.0372678i
$721$	4.68629i	0.174527i
$722$	11.0000i	0.409378i
$723$	− 5.31371i	− 0.197619i
$724$	−7.65685	−0.284565
$725$	8.82843	0.327880
$726$	− 21.0000i	− 0.779383i
$727$	45.9411	1.70386	0.851931	−	0.523654i	$-0.175432\pi$
$727$	45.9411	1.70386	0.851931	+	0.523654i	$0.175432\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	− 14.4853i	− 0.536124i
$731$	8.00000	0.295891
$732$	−6.00000	−0.221766
$733$	− 0.343146i	− 0.0126744i	−0.999980	−	0.00633719i	$-0.997983\pi$
$733$	− 0.343146i	− 0.0126744i	0.999980	−	0.00633719i	$-0.00201720\pi$
$734$	25.6569i	0.947012i
$735$	− 1.00000i	− 0.0368856i
$736$	8.48528i	0.312772i
$737$	32.0000	1.17874
$738$	7.65685	0.281853
$739$	14.1421i	0.520227i	0.965578	+	0.260113i	$0.0837600\pi$
$739$	14.1421i	0.520227i	−0.965578	+	0.260113i	$0.916240\pi$
$740$	11.6569	0.428514
$741$	0	0
$742$	−37.6569	−1.38243
$743$	− 36.2843i	− 1.33114i	−0.746335	−	0.665570i	$-0.768188\pi$
$743$	− 36.2843i	− 1.33114i	0.746335	−	0.665570i	$-0.231812\pi$
$744$	4.00000	0.146647
$745$	−3.65685	−0.133977
$746$	− 2.68629i	− 0.0983521i
$747$	6.34315i	0.232084i
$748$	− 4.68629i	− 0.171348i
$749$	− 11.3137i	− 0.413394i
$750$	1.00000	0.0365148
$751$	−11.3137	−0.412843	−0.206422	−	0.978463i	$-0.566182\pi$
$751$	−11.3137	−0.412843	−0.206422	+	0.978463i	$0.566182\pi$
$752$	8.00000i	0.291730i
$753$	−10.8284	−0.394610
$754$	0	0
$755$	12.0000	0.436725
$756$	− 2.82843i	− 0.102869i
$757$	19.9411	0.724773	0.362386	−	0.932028i	$-0.381962\pi$
$757$	19.9411	0.724773	0.362386	+	0.932028i	$0.381962\pi$
$758$	7.51472	0.272947
$759$	− 48.0000i	− 1.74229i
$760$	2.82843i	0.102598i
$761$	− 27.6569i	− 1.00256i	−0.865285	−	0.501280i	$-0.832863\pi$
$761$	− 27.6569i	− 1.00256i	0.865285	−	0.501280i	$-0.167137\pi$
$762$	− 9.65685i	− 0.349831i
$763$	−24.9706	−0.903995
$764$	−11.3137	−0.409316
$765$	− 0.828427i	− 0.0299518i
$766$	29.6569	1.07155
$767$	0	0
$768$	1.00000	0.0360844
$769$	− 14.0000i	− 0.504853i	−0.967616	−	0.252426i	$-0.918771\pi$
$769$	− 14.0000i	− 0.504853i	0.967616	−	0.252426i	$-0.0812286\pi$
$770$	−16.0000	−0.576600
$771$	4.82843	0.173892
$772$	2.48528i	0.0894472i
$773$	− 53.3137i	− 1.91756i	−0.284148	−	0.958780i	$-0.591711\pi$
$773$	− 53.3137i	− 1.91756i	0.284148	−	0.958780i	$-0.408289\pi$
$774$	− 9.65685i	− 0.347108i
$775$	− 4.00000i	− 0.143684i
$776$	−3.17157	−0.113853
$777$	−32.9706	−1.18281
$778$	6.48528i	0.232509i
$779$	21.6569	0.775937
$780$	0	0
$781$	−32.0000	−1.14505
$782$	− 7.02944i	− 0.251372i
$783$	−8.82843	−0.315502
$784$	−1.00000	−0.0357143
$785$	5.31371i	0.189654i
$786$	− 6.14214i	− 0.219083i
$787$	24.0000i	0.855508i	0.903895	+	0.427754i	$0.140695\pi$
$787$	24.0000i	0.855508i	−0.903895	+	0.427754i	$0.859305\pi$
$788$	13.3137i	0.474281i
$789$	16.4853	0.586892
$790$	−2.34315	−0.0833654
$791$	18.3431i	0.652207i
$792$	−5.65685	−0.201008
$793$	0	0
$794$	30.2843	1.07475
$795$	13.3137i	0.472189i
$796$	10.3431	0.366603
$797$	−16.6274	−0.588973	−0.294487	−	0.955656i	$-0.595149\pi$
$797$	−16.6274	−0.588973	−0.294487	+	0.955656i	$0.595149\pi$
$798$	− 8.00000i	− 0.283197i
$799$	− 6.62742i	− 0.234461i
$800$	− 1.00000i	− 0.0353553i
$801$	15.6569i	0.553208i
$802$	−26.9706	−0.952364
$803$	81.9411	2.89164
$804$	− 5.65685i	− 0.199502i
$805$	−24.0000	−0.845889
$806$	0	0
$807$	−14.4853	−0.509906
$808$	16.1421i	0.567878i
$809$	13.3137	0.468085	0.234043	−	0.972226i	$-0.424804\pi$
$809$	13.3137	0.468085	0.234043	+	0.972226i	$0.424804\pi$
$810$	−1.00000	−0.0351364
$811$	− 1.85786i	− 0.0652384i	−0.999468	−	0.0326192i	$-0.989615\pi$
$811$	− 1.85786i	− 0.0652384i	0.999468	−	0.0326192i	$-0.0103849\pi$
$812$	24.9706i	0.876295i
$813$	− 7.31371i	− 0.256503i
$814$	65.9411i	2.31124i
$815$	11.3137	0.396302
$816$	−0.828427	−0.0290008
$817$	− 27.3137i	− 0.955586i
$818$	3.65685	0.127859
$819$	0	0
$820$	−7.65685	−0.267389
$821$	34.2843i	1.19653i	0.801299	+	0.598265i	$0.204143\pi$
$821$	34.2843i	1.19653i	−0.801299	+	0.598265i	$0.795857\pi$
$822$	−17.3137	−0.603885
$823$	52.9706	1.84644	0.923219	−	0.384275i	$-0.125548\pi$
$823$	52.9706	1.84644	0.923219	+	0.384275i	$0.125548\pi$
$824$	− 1.65685i	− 0.0577193i
$825$	5.65685i	0.196946i
$826$	− 6.62742i	− 0.230597i
$827$	− 9.65685i	− 0.335802i	−0.985804	−	0.167901i	$-0.946301\pi$
$827$	− 9.65685i	− 0.335802i	0.985804	−	0.167901i	$-0.0536989\pi$
$828$	−8.48528	−0.294884
$829$	53.3137	1.85166	0.925831	−	0.377938i	$-0.123367\pi$
$829$	53.3137	1.85166	0.925831	+	0.377938i	$0.123367\pi$
$830$	− 6.34315i	− 0.220174i
$831$	−26.0000	−0.901930
$832$	0	0
$833$	0.828427	0.0287033
$834$	6.34315i	0.219645i
$835$	8.97056	0.310439
$836$	−16.0000	−0.553372
$837$	4.00000i	0.138260i
$838$	− 10.8284i	− 0.374062i
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	2.82843i	0.0975900i
$841$	48.9411	1.68763
$842$	24.1421	0.831993
$843$	− 8.34315i	− 0.287353i
$844$	−0.686292	−0.0236231
$845$	0	0
$846$	−8.00000	−0.275046
$847$	− 59.3970i	− 2.04090i
$848$	13.3137	0.457195
$849$	17.6569	0.605982
$850$	0.828427i	0.0284148i
$851$	98.9117i	3.39065i
$852$	5.65685i	0.193801i
$853$	− 38.2843i	− 1.31083i	−0.755270	−	0.655414i	$-0.772494\pi$
$853$	− 38.2843i	− 1.31083i	0.755270	−	0.655414i	$-0.227506\pi$
$854$	−16.9706	−0.580721
$855$	−2.82843	−0.0967302
$856$	4.00000i	0.136717i
$857$	20.8284	0.711486	0.355743	−	0.934584i	$-0.384228\pi$
$857$	20.8284	0.711486	0.355743	+	0.934584i	$0.384228\pi$
$858$	0	0
$859$	37.9411	1.29453	0.647267	−	0.762263i	$-0.275912\pi$
$859$	37.9411	1.29453	0.647267	+	0.762263i	$0.275912\pi$
$860$	9.65685i	0.329296i
$861$	21.6569	0.738064
$862$	16.0000	0.544962
$863$	− 28.2843i	− 0.962808i	−0.876499	−	0.481404i	$-0.840127\pi$
$863$	− 28.2843i	− 0.962808i	0.876499	−	0.481404i	$-0.159873\pi$
$864$	1.00000i	0.0340207i
$865$	9.31371i	0.316676i
$866$	22.9706i	0.780571i
$867$	−16.3137	−0.554043
$868$	11.3137	0.384012
$869$	− 13.2548i	− 0.449639i
$870$	8.82843	0.299312
$871$	0	0
$872$	8.82843	0.298968
$873$	− 3.17157i	− 0.107341i
$874$	−24.0000	−0.811812
$875$	2.82843	0.0956183
$876$	− 14.4853i	− 0.489412i
$877$	− 51.2548i	− 1.73075i	−0.501122	−	0.865376i	$-0.667079\pi$
$877$	− 51.2548i	− 1.73075i	0.501122	−	0.865376i	$-0.332921\pi$
$878$	− 22.6274i	− 0.763638i
$879$	16.6274i	0.560829i
$880$	5.65685	0.190693
$881$	10.2843	0.346486	0.173243	−	0.984879i	$-0.444575\pi$
$881$	10.2843	0.346486	0.173243	+	0.984879i	$0.444575\pi$
$882$	− 1.00000i	− 0.0336718i
$883$	−31.3137	−1.05379	−0.526895	−	0.849930i	$-0.676644\pi$
$883$	−31.3137	−1.05379	−0.526895	+	0.849930i	$0.676644\pi$
$884$	0	0
$885$	−2.34315	−0.0787640
$886$	30.3431i	1.01940i
$887$	−40.4853	−1.35936	−0.679681	−	0.733508i	$-0.737882\pi$
$887$	−40.4853	−1.35936	−0.679681	+	0.733508i	$0.737882\pi$
$888$	11.6569	0.391178
$889$	− 27.3137i	− 0.916072i
$890$	− 15.6569i	− 0.524819i
$891$	− 5.65685i	− 0.189512i
$892$	− 10.8284i	− 0.362563i
$893$	−22.6274	−0.757198
$894$	−3.65685	−0.122304
$895$	− 7.51472i	− 0.251189i
$896$	2.82843	0.0944911
$897$	0	0
$898$	26.2843	0.877117
$899$	− 35.3137i	− 1.17778i
$900$	1.00000	0.0333333
$901$	−11.0294	−0.367444
$902$	− 43.3137i	− 1.44219i
$903$	− 27.3137i	− 0.908943i
$904$	− 6.48528i	− 0.215697i
$905$	7.65685i	0.254522i
$906$	12.0000	0.398673
$907$	−8.28427	−0.275075	−0.137537	−	0.990497i	$-0.543919\pi$
$907$	−8.28427	−0.275075	−0.137537	+	0.990497i	$0.543919\pi$
$908$	− 4.00000i	− 0.132745i
$909$	−16.1421	−0.535401
$910$	0	0
$911$	24.9706	0.827312	0.413656	−	0.910433i	$-0.364252\pi$
$911$	24.9706	0.827312	0.413656	+	0.910433i	$0.364252\pi$
$912$	2.82843i	0.0936586i
$913$	35.8823	1.18753
$914$	−20.8284	−0.688943
$915$	6.00000i	0.198354i
$916$	4.14214i	0.136860i
$917$	− 17.3726i	− 0.573693i
$918$	− 0.828427i	− 0.0273422i
$919$	41.9411	1.38351	0.691755	−	0.722132i	$-0.256838\pi$
$919$	41.9411	1.38351	0.691755	+	0.722132i	$0.256838\pi$
$920$	8.48528	0.279751
$921$	21.6569i	0.713618i
$922$	−14.0000	−0.461065
$923$	0	0
$924$	−16.0000	−0.526361
$925$	− 11.6569i	− 0.383275i
$926$	−3.79899	−0.124843
$927$	1.65685	0.0544182
$928$	− 8.82843i	− 0.289807i
$929$	− 33.5980i	− 1.10231i	−0.834402	−	0.551157i	$-0.814187\pi$
$929$	− 33.5980i	− 1.10231i	0.834402	−	0.551157i	$-0.185813\pi$
$930$	− 4.00000i	− 0.131165i
$931$	− 2.82843i	− 0.0926980i
$932$	5.51472	0.180641
$933$	24.0000	0.785725
$934$	− 7.31371i	− 0.239312i
$935$	−4.68629	−0.153258
$936$	0	0
$937$	16.6274	0.543194	0.271597	−	0.962411i	$-0.412448\pi$
$937$	16.6274	0.543194	0.271597	+	0.962411i	$0.412448\pi$
$938$	− 16.0000i	− 0.522419i
$939$	−30.9706	−1.01069
$940$	8.00000	0.260931
$941$	54.9706i	1.79199i	0.444065	+	0.895995i	$0.353536\pi$
$941$	54.9706i	1.79199i	−0.444065	+	0.895995i	$0.646464\pi$
$942$	5.31371i	0.173130i
$943$	− 64.9706i	− 2.11573i
$944$	2.34315i	0.0762629i
$945$	−2.82843	−0.0920087
$946$	−54.6274	−1.77609
$947$	− 30.3431i	− 0.986020i	−0.870024	−	0.493010i	$-0.835897\pi$
$947$	− 30.3431i	− 0.986020i	0.870024	−	0.493010i	$-0.164103\pi$
$948$	−2.34315	−0.0761018
$949$	0	0
$950$	2.82843	0.0917663
$951$	25.3137i	0.820853i
$952$	−2.34315	−0.0759418
$953$	27.8579	0.902405	0.451202	−	0.892422i	$-0.350995\pi$
$953$	27.8579	0.902405	0.451202	+	0.892422i	$0.350995\pi$
$954$	13.3137i	0.431047i
$955$	11.3137i	0.366103i
$956$	− 16.0000i	− 0.517477i
$957$	49.9411i	1.61437i
$958$	−11.3137	−0.365529
$959$	−48.9706	−1.58134
$960$	− 1.00000i	− 0.0322749i
$961$	15.0000	0.483871
$962$	0	0
$963$	−4.00000	−0.128898
$964$	5.31371i	0.171143i
$965$	2.48528	0.0800040
$966$	−24.0000	−0.772187
$967$	− 7.51472i	− 0.241657i	−0.992673	−	0.120829i	$-0.961445\pi$
$967$	− 7.51472i	− 0.241657i	0.992673	−	0.120829i	$-0.0385551\pi$
$968$	21.0000i	0.674966i
$969$	− 2.34315i	− 0.0752727i
$970$	3.17157i	0.101833i
$971$	15.5147	0.497891	0.248946	−	0.968517i	$-0.419916\pi$
$971$	15.5147	0.497891	0.248946	+	0.968517i	$0.419916\pi$
$972$	−1.00000	−0.0320750
$973$	17.9411i	0.575166i
$974$	−16.4853	−0.528222
$975$	0	0
$976$	6.00000	0.192055
$977$	− 8.34315i	− 0.266921i	−0.991054	−	0.133460i	$-0.957391\pi$
$977$	− 8.34315i	− 0.266921i	0.991054	−	0.133460i	$-0.0426089\pi$
$978$	11.3137	0.361773
$979$	88.5685	2.83066
$980$	1.00000i	0.0319438i
$981$	8.82843i	0.281870i
$982$	38.1421i	1.21716i
$983$	2.34315i	0.0747347i	0.999302	+	0.0373674i	$0.0118972\pi$
$983$	2.34315i	0.0747347i	−0.999302	+	0.0373674i	$0.988103\pi$
$984$	−7.65685	−0.244092
$985$	13.3137	0.424210
$986$	7.31371i	0.232916i
$987$	−22.6274	−0.720239
$988$	0	0
$989$	−81.9411	−2.60558
$990$	5.65685i	0.179787i
$991$	−42.9117	−1.36313	−0.681567	−	0.731755i	$-0.738701\pi$
$991$	−42.9117	−1.36313	−0.681567	+	0.731755i	$0.738701\pi$
$992$	−4.00000	−0.127000
$993$	− 8.48528i	− 0.269272i
$994$	16.0000i	0.507489i
$995$	− 10.3431i	− 0.327900i
$996$	− 6.34315i	− 0.200990i
$997$	61.3137	1.94182	0.970912	−	0.239435i	$-0.0769623\pi$
$997$	61.3137	1.94182	0.970912	+	0.239435i	$0.0769623\pi$
$998$	−0.485281	−0.0153613
$999$	11.6569i	0.368807i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.b.q.1351.4		4
13.5	odd	4		390.2.a.h.1.1	✓	2
13.8	odd	4		5070.2.a.bc.1.2		2
13.12	even	2	inner	5070.2.b.q.1351.1		4
39.5	even	4		1170.2.a.o.1.1		2
52.31	even	4		3120.2.a.bc.1.2		2
65.18	even	4		1950.2.e.o.1249.2		4
65.44	odd	4		1950.2.a.bd.1.2		2
65.57	even	4		1950.2.e.o.1249.3		4
156.83	odd	4		9360.2.a.ch.1.2		2
195.44	even	4		5850.2.a.cl.1.2		2
195.83	odd	4		5850.2.e.bk.5149.4		4
195.122	odd	4		5850.2.e.bk.5149.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.a.h.1.1	✓	2	13.5	odd	4
1170.2.a.o.1.1		2	39.5	even	4
1950.2.a.bd.1.2		2	65.44	odd	4
1950.2.e.o.1249.2		4	65.18	even	4
1950.2.e.o.1249.3		4	65.57	even	4
3120.2.a.bc.1.2		2	52.31	even	4
5070.2.a.bc.1.2		2	13.8	odd	4
5070.2.b.q.1351.1		4	13.12	even	2	inner
5070.2.b.q.1351.4		4	1.1	even	1	trivial
5850.2.a.cl.1.2		2	195.44	even	4
5850.2.e.bk.5149.1		4	195.122	odd	4
5850.2.e.bk.5149.4		4	195.83	odd	4
9360.2.a.ch.1.2		2	156.83	odd	4

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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} +2.82843i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} +2.82843i q^{7} -1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} -5.65685i q^{11} -1.00000 q^{12} -2.82843 q^{14} +1.00000i q^{15} +1.00000 q^{16} -0.828427 q^{17} +1.00000i q^{18} +2.82843i q^{19} -1.00000i q^{20} +2.82843i q^{21} +5.65685 q^{22} +8.48528 q^{23} -1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} -2.82843i q^{28} -8.82843 q^{29} -1.00000 q^{30} +4.00000i q^{31} +1.00000i q^{32} -5.65685i q^{33} -0.828427i q^{34} -2.82843 q^{35} -1.00000 q^{36} +11.6569i q^{37} -2.82843 q^{38} +1.00000 q^{40} -7.65685i q^{41} -2.82843 q^{42} -9.65685 q^{43} +5.65685i q^{44} +1.00000i q^{45} +8.48528i q^{46} +8.00000i q^{47} +1.00000 q^{48} -1.00000 q^{49} -1.00000i q^{50} -0.828427 q^{51} +13.3137 q^{53} +1.00000i q^{54} +5.65685 q^{55} +2.82843 q^{56} +2.82843i q^{57} -8.82843i q^{58} +2.34315i q^{59} -1.00000i q^{60} +6.00000 q^{61} -4.00000 q^{62} +2.82843i q^{63} -1.00000 q^{64} +5.65685 q^{66} +5.65685i q^{67} +0.828427 q^{68} +8.48528 q^{69} -2.82843i q^{70} -5.65685i q^{71} -1.00000i q^{72} +14.4853i q^{73} -11.6569 q^{74} -1.00000 q^{75} -2.82843i q^{76} +16.0000 q^{77} +2.34315 q^{79} +1.00000i q^{80} +1.00000 q^{81} +7.65685 q^{82} +6.34315i q^{83} -2.82843i q^{84} -0.828427i q^{85} -9.65685i q^{86} -8.82843 q^{87} -5.65685 q^{88} +15.6569i q^{89} -1.00000 q^{90} -8.48528 q^{92} +4.00000i q^{93} -8.00000 q^{94} -2.82843 q^{95} +1.00000i q^{96} -3.17157i q^{97} -1.00000i q^{98} -5.65685i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 4 q^{4} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 - 4 * q^4 + 4 * q^9	\( 4 q + 4 q^{3} - 4 q^{4} + 4 q^{9} - 4 q^{10} - 4 q^{12} + 4 q^{16} + 8 q^{17} - 4 q^{25} + 4 q^{27} - 24 q^{29} - 4 q^{30} - 4 q^{36} + 4 q^{40} - 16 q^{43} + 4 q^{48} - 4 q^{49} + 8 q^{51} + 8 q^{53} + 24 q^{61} - 16 q^{62} - 4 q^{64} - 8 q^{68} - 24 q^{74} - 4 q^{75} + 64 q^{77} + 32 q^{79} + 4 q^{81} + 8 q^{82} - 24 q^{87} - 4 q^{90} - 32 q^{94}+O(q^{100}) \) 4 * q + 4 * q^3 - 4 * q^4 + 4 * q^9 - 4 * q^10 - 4 * q^12 + 4 * q^16 + 8 * q^17 - 4 * q^25 + 4 * q^27 - 24 * q^29 - 4 * q^30 - 4 * q^36 + 4 * q^40 - 16 * q^43 + 4 * q^48 - 4 * q^49 + 8 * q^51 + 8 * q^53 + 24 * q^61 - 16 * q^62 - 4 * q^64 - 8 * q^68 - 24 * q^74 - 4 * q^75 + 64 * q^77 + 32 * q^79 + 4 * q^81 + 8 * q^82 - 24 * q^87 - 4 * q^90 - 32 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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