LMFDB - Embedded newform 5850.2.e.bk.5149.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5850,2,Mod(5149,5850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5850, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5850.5149");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5850.e (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:	$46.7124851824$
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			5149.4
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	5850.5149
Dual form			5850.2.e.bk.5149.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +2.82843i q^{7} -1.00000i q^{8} +O(q^{10})$	$q+1.00000i q^{2} -1.00000 q^{4} +2.82843i q^{7} -1.00000i q^{8} -5.65685 q^{11} -1.00000i q^{13} -2.82843 q^{14} +1.00000 q^{16} +0.828427i q^{17} -2.82843 q^{19} -5.65685i q^{22} +8.48528i q^{23} +1.00000 q^{26} -2.82843i q^{28} -8.82843 q^{29} +4.00000 q^{31} +1.00000i q^{32} -0.828427 q^{34} +11.6569i q^{37} -2.82843i q^{38} +7.65685 q^{41} +9.65685i q^{43} +5.65685 q^{44} -8.48528 q^{46} -8.00000i q^{47} -1.00000 q^{49} +1.00000i q^{52} -13.3137i q^{53} +2.82843 q^{56} -8.82843i q^{58} -2.34315 q^{59} +6.00000 q^{61} +4.00000i q^{62} -1.00000 q^{64} -5.65685i q^{67} -0.828427i q^{68} +5.65685 q^{71} -14.4853i q^{73} -11.6569 q^{74} +2.82843 q^{76} -16.0000i q^{77} -2.34315 q^{79} +7.65685i q^{82} -6.34315i q^{83} -9.65685 q^{86} +5.65685i q^{88} -15.6569 q^{89} +2.82843 q^{91} -8.48528i q^{92} +8.00000 q^{94} +3.17157i q^{97} -1.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4}+O(q^{10}) $ 4 * q - 4 * q^4	$ 4 q - 4 q^{4} + 4 q^{16} + 4 q^{26} - 24 q^{29} + 16 q^{31} + 8 q^{34} + 8 q^{41} - 4 q^{49} - 32 q^{59} + 24 q^{61} - 4 q^{64} - 24 q^{74} - 32 q^{79} - 16 q^{86} - 40 q^{89} + 32 q^{94}+O(q^{100}) $ 4 * q - 4 * q^4 + 4 * q^16 + 4 * q^26 - 24 * q^29 + 16 * q^31 + 8 * q^34 + 8 * q^41 - 4 * q^49 - 32 * q^59 + 24 * q^61 - 4 * q^64 - 24 * q^74 - 32 * q^79 - 16 * q^86 - 40 * q^89 + 32 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5850\mathbb{Z}\right)^\times$.

$n$	$2251$	$3251$	$3277$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	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
$9$	0	0
$10$	0	0
$11$	−5.65685	−1.70561	−0.852803	−	0.522233i	$-0.825099\pi$
$11$	−5.65685	−1.70561	−0.852803	+	0.522233i	$0.825099\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	−2.82843	−0.755929
$15$	0	0
$16$	1.00000	0.250000
$17$	0.828427i	0.200923i	0.994941	+	0.100462i	$0.0320319\pi$
$17$	0.828427i	0.200923i	−0.994941	+	0.100462i	$0.967968\pi$
$18$	0	0
$19$	−2.82843	−0.648886	−0.324443	−	0.945905i	$-0.605177\pi$
$19$	−2.82843	−0.648886	−0.324443	+	0.945905i	$0.605177\pi$
$20$	0	0
$21$	0	0
$22$	− 5.65685i	− 1.20605i
$23$	8.48528i	1.76930i	0.466252	+	0.884652i	$0.345604\pi$
$23$	8.48528i	1.76930i	−0.466252	+	0.884652i	$0.654396\pi$
$24$	0	0
$25$	0	0
$26$	1.00000	0.196116
$27$	0	0
$28$	− 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	$0.805859\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	−0.828427	−0.142074
$35$	0	0
$36$	0	0
$37$	11.6569i	1.91638i	0.286141	+	0.958188i	$0.407627\pi$
$37$	11.6569i	1.91638i	−0.286141	+	0.958188i	$0.592373\pi$
$38$	− 2.82843i	− 0.458831i
$39$	0	0
$40$	0	0
$41$	7.65685	1.19580	0.597900	−	0.801571i	$-0.296002\pi$
$41$	7.65685	1.19580	0.597900	+	0.801571i	$0.296002\pi$
$42$	0	0
$43$	9.65685i	1.47266i	0.676625	+	0.736328i	$0.263442\pi$
$43$	9.65685i	1.47266i	−0.676625	+	0.736328i	$0.736558\pi$
$44$	5.65685	0.852803
$45$	0	0
$46$	−8.48528	−1.25109
$47$	− 8.00000i	− 1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$	− 8.00000i	− 1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	1.00000i	0.138675i
$53$	− 13.3137i	− 1.82878i	−0.404836	−	0.914389i	$-0.632671\pi$
$53$	− 13.3137i	− 1.82878i	0.404836	−	0.914389i	$-0.367329\pi$
$54$	0	0
$55$	0	0
$56$	2.82843	0.377964
$57$	0	0
$58$	− 8.82843i	− 1.15923i
$59$	−2.34315	−0.305052	−0.152526	−	0.988299i	$-0.548741\pi$
$59$	−2.34315	−0.305052	−0.152526	+	0.988299i	$0.548741\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	4.00000i	0.508001i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 5.65685i	− 0.691095i	−0.938401	−	0.345547i	$-0.887693\pi$
$67$	− 5.65685i	− 0.691095i	0.938401	−	0.345547i	$-0.112307\pi$
$68$	− 0.828427i	− 0.100462i
$69$	0	0
$70$	0	0
$71$	5.65685	0.671345	0.335673	−	0.941979i	$-0.391036\pi$
$71$	5.65685	0.671345	0.335673	+	0.941979i	$0.391036\pi$
$72$	0	0
$73$	− 14.4853i	− 1.69537i	−0.530497	−	0.847687i	$-0.677995\pi$
$73$	− 14.4853i	− 1.69537i	0.530497	−	0.847687i	$-0.322005\pi$
$74$	−11.6569	−1.35508
$75$	0	0
$76$	2.82843	0.324443
$77$	− 16.0000i	− 1.82337i
$78$	0	0
$79$	−2.34315	−0.263624	−0.131812	−	0.991275i	$-0.542080\pi$
$79$	−2.34315	−0.263624	−0.131812	+	0.991275i	$0.542080\pi$
$80$	0	0
$81$	0	0
$82$	7.65685i	0.845558i
$83$	− 6.34315i	− 0.696251i	−0.937448	−	0.348125i	$-0.886818\pi$
$83$	− 6.34315i	− 0.696251i	0.937448	−	0.348125i	$-0.113182\pi$
$84$	0	0
$85$	0	0
$86$	−9.65685	−1.04133
$87$	0	0
$88$	5.65685i	0.603023i
$89$	−15.6569	−1.65962	−0.829812	−	0.558044i	$-0.811552\pi$
$89$	−15.6569	−1.65962	−0.829812	+	0.558044i	$0.811552\pi$
$90$	0	0
$91$	2.82843	0.296500
$92$	− 8.48528i	− 0.884652i
$93$	0	0
$94$	8.00000	0.825137
$95$	0	0
$96$	0	0
$97$	3.17157i	0.322024i	0.986952	+	0.161012i	$0.0514759\pi$
$97$	3.17157i	0.322024i	−0.986952	+	0.161012i	$0.948524\pi$
$98$	− 1.00000i	− 0.101015i
$99$	0	0
$100$	0	0
$101$	−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	0
$103$	− 1.65685i	− 0.163255i	−0.996663	−	0.0816274i	$-0.973988\pi$
$103$	− 1.65685i	− 0.163255i	0.996663	−	0.0816274i	$-0.0260117\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	13.3137	1.29314
$107$	− 4.00000i	− 0.386695i	−0.981130	−	0.193347i	$-0.938066\pi$
$107$	− 4.00000i	− 0.386695i	0.981130	−	0.193347i	$-0.0619344\pi$
$108$	0	0
$109$	−8.82843	−0.845610	−0.422805	−	0.906221i	$-0.638954\pi$
$109$	−8.82843	−0.845610	−0.422805	+	0.906221i	$0.638954\pi$
$110$	0	0
$111$	0	0
$112$	2.82843i	0.267261i
$113$	− 6.48528i	− 0.610084i	−0.952339	−	0.305042i	$-0.901330\pi$
$113$	− 6.48528i	− 0.610084i	0.952339	−	0.305042i	$-0.0986705\pi$
$114$	0	0
$115$	0	0
$116$	8.82843	0.819699
$117$	0	0
$118$	− 2.34315i	− 0.215704i
$119$	−2.34315	−0.214796
$120$	0	0
$121$	21.0000	1.90909
$122$	6.00000i	0.543214i
$123$	0	0
$124$	−4.00000	−0.359211
$125$	0	0
$126$	0	0
$127$	− 9.65685i	− 0.856907i	−0.903564	−	0.428454i	$-0.859059\pi$
$127$	− 9.65685i	− 0.856907i	0.903564	−	0.428454i	$-0.140941\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	0	0
$130$	0	0
$131$	6.14214	0.536641	0.268320	−	0.963330i	$-0.413531\pi$
$131$	6.14214	0.536641	0.268320	+	0.963330i	$0.413531\pi$
$132$	0	0
$133$	− 8.00000i	− 0.693688i
$134$	5.65685	0.488678
$135$	0	0
$136$	0.828427	0.0710370
$137$	− 17.3137i	− 1.47921i	−0.673041	−	0.739605i	$-0.735012\pi$
$137$	− 17.3137i	− 1.47921i	0.673041	−	0.739605i	$-0.264988\pi$
$138$	0	0
$139$	−6.34315	−0.538019	−0.269009	−	0.963138i	$-0.586696\pi$
$139$	−6.34315	−0.538019	−0.269009	+	0.963138i	$0.586696\pi$
$140$	0	0
$141$	0	0
$142$	5.65685i	0.474713i
$143$	5.65685i	0.473050i
$144$	0	0
$145$	0	0
$146$	14.4853	1.19881
$147$	0	0
$148$	− 11.6569i	− 0.958188i
$149$	3.65685	0.299581	0.149791	−	0.988718i	$-0.452140\pi$
$149$	3.65685	0.299581	0.149791	+	0.988718i	$0.452140\pi$
$150$	0	0
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	2.82843i	0.229416i
$153$	0	0
$154$	16.0000	1.28932
$155$	0	0
$156$	0	0
$157$	− 5.31371i	− 0.424080i	−0.977261	−	0.212040i	$-0.931989\pi$
$157$	− 5.31371i	− 0.424080i	0.977261	−	0.212040i	$-0.0680107\pi$
$158$	− 2.34315i	− 0.186411i
$159$	0	0
$160$	0	0
$161$	−24.0000	−1.89146
$162$	0	0
$163$	11.3137i	0.886158i	0.896483	+	0.443079i	$0.146114\pi$
$163$	11.3137i	0.886158i	−0.896483	+	0.443079i	$0.853886\pi$
$164$	−7.65685	−0.597900
$165$	0	0
$166$	6.34315	0.492324
$167$	8.97056i	0.694163i	0.937835	+	0.347081i	$0.112827\pi$
$167$	8.97056i	0.694163i	−0.937835	+	0.347081i	$0.887173\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	− 9.65685i	− 0.736328i
$173$	9.31371i	0.708108i	0.935225	+	0.354054i	$0.115197\pi$
$173$	9.31371i	0.708108i	−0.935225	+	0.354054i	$0.884803\pi$
$174$	0	0
$175$	0	0
$176$	−5.65685	−0.426401
$177$	0	0
$178$	− 15.6569i	− 1.17353i
$179$	7.51472	0.561676	0.280838	−	0.959755i	$-0.409388\pi$
$179$	7.51472	0.561676	0.280838	+	0.959755i	$0.409388\pi$
$180$	0	0
$181$	−7.65685	−0.569129	−0.284565	−	0.958657i	$-0.591849\pi$
$181$	−7.65685	−0.569129	−0.284565	+	0.958657i	$0.591849\pi$
$182$	2.82843i	0.209657i
$183$	0	0
$184$	8.48528	0.625543
$185$	0	0
$186$	0	0
$187$	− 4.68629i	− 0.342696i
$188$	8.00000i	0.583460i
$189$	0	0
$190$	0	0
$191$	−11.3137	−0.818631	−0.409316	−	0.912393i	$-0.634232\pi$
$191$	−11.3137	−0.818631	−0.409316	+	0.912393i	$0.634232\pi$
$192$	0	0
$193$	2.48528i	0.178894i	0.995992	+	0.0894472i	$0.0285100\pi$
$193$	2.48528i	0.178894i	−0.995992	+	0.0894472i	$0.971490\pi$
$194$	−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	$-0.157292\pi$
$198$	0	0
$199$	−10.3431	−0.733206	−0.366603	−	0.930377i	$-0.619479\pi$
$199$	−10.3431	−0.733206	−0.366603	+	0.930377i	$0.619479\pi$
$200$	0	0
$201$	0	0
$202$	− 16.1421i	− 1.13576i
$203$	− 24.9706i	− 1.75259i
$204$	0	0
$205$	0	0
$206$	1.65685	0.115439
$207$	0	0
$208$	− 1.00000i	− 0.0693375i
$209$	16.0000	1.10674
$210$	0	0
$211$	0.686292	0.0472463	0.0236231	−	0.999721i	$-0.492480\pi$
$211$	0.686292	0.0472463	0.0236231	+	0.999721i	$0.492480\pi$
$212$	13.3137i	0.914389i
$213$	0	0
$214$	4.00000	0.273434
$215$	0	0
$216$	0	0
$217$	11.3137i	0.768025i
$218$	− 8.82843i	− 0.597937i
$219$	0	0
$220$	0	0
$221$	0.828427	0.0557260
$222$	0	0
$223$	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
$225$	0	0
$226$	6.48528	0.431394
$227$	4.00000i	0.265489i	0.991150	+	0.132745i	$0.0423790\pi$
$227$	4.00000i	0.265489i	−0.991150	+	0.132745i	$0.957621\pi$
$228$	0	0
$229$	−4.14214	−0.273720	−0.136860	−	0.990590i	$-0.543701\pi$
$229$	−4.14214	−0.273720	−0.136860	+	0.990590i	$0.543701\pi$
$230$	0	0
$231$	0	0
$232$	8.82843i	0.579615i
$233$	− 5.51472i	− 0.361281i	−0.983549	−	0.180641i	$-0.942183\pi$
$233$	− 5.51472i	− 0.361281i	0.983549	−	0.180641i	$-0.0578171\pi$
$234$	0	0
$235$	0	0
$236$	2.34315	0.152526
$237$	0	0
$238$	− 2.34315i	− 0.151884i
$239$	16.0000	1.03495	0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000	1.03495	0.517477	+	0.855697i	$0.326871\pi$
$240$	0	0
$241$	5.31371	0.342286	0.171143	−	0.985246i	$-0.445254\pi$
$241$	5.31371	0.342286	0.171143	+	0.985246i	$0.445254\pi$
$242$	21.0000i	1.34993i
$243$	0	0
$244$	−6.00000	−0.384111
$245$	0	0
$246$	0	0
$247$	2.82843i	0.179969i
$248$	− 4.00000i	− 0.254000i
$249$	0	0
$250$	0	0
$251$	−10.8284	−0.683484	−0.341742	−	0.939794i	$-0.611017\pi$
$251$	−10.8284	−0.683484	−0.341742	+	0.939794i	$0.611017\pi$
$252$	0	0
$253$	− 48.0000i	− 3.01773i
$254$	9.65685	0.605925
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 4.82843i	− 0.301189i	−0.988596	−	0.150595i	$-0.951881\pi$
$257$	− 4.82843i	− 0.301189i	0.988596	−	0.150595i	$-0.0481188\pi$
$258$	0	0
$259$	−32.9706	−2.04869
$260$	0	0
$261$	0	0
$262$	6.14214i	0.379462i
$263$	− 16.4853i	− 1.01653i	−0.861202	−	0.508263i	$-0.830288\pi$
$263$	− 16.4853i	− 1.01653i	0.861202	−	0.508263i	$-0.169712\pi$
$264$	0	0
$265$	0	0
$266$	8.00000	0.490511
$267$	0	0
$268$	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	$0.645586\pi$
$270$	0	0
$271$	7.31371	0.444276	0.222138	−	0.975015i	$-0.428696\pi$
$271$	7.31371	0.444276	0.222138	+	0.975015i	$0.428696\pi$
$272$	0.828427i	0.0502308i
$273$	0	0
$274$	17.3137	1.04596
$275$	0	0
$276$	0	0
$277$	− 26.0000i	− 1.56219i	−0.624413	−	0.781094i	$-0.714662\pi$
$277$	− 26.0000i	− 1.56219i	0.624413	−	0.781094i	$-0.285338\pi$
$278$	− 6.34315i	− 0.380437i
$279$	0	0
$280$	0	0
$281$	−8.34315	−0.497710	−0.248855	−	0.968541i	$-0.580054\pi$
$281$	−8.34315	−0.497710	−0.248855	+	0.968541i	$0.580054\pi$
$282$	0	0
$283$	− 17.6569i	− 1.04959i	−0.851228	−	0.524796i	$-0.824142\pi$
$283$	− 17.6569i	− 1.04959i	0.851228	−	0.524796i	$-0.175858\pi$
$284$	−5.65685	−0.335673
$285$	0	0
$286$	−5.65685	−0.334497
$287$	21.6569i	1.27836i
$288$	0	0
$289$	16.3137	0.959630
$290$	0	0
$291$	0	0
$292$	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	$0.838568\pi$
$294$	0	0
$295$	0	0
$296$	11.6569	0.677541
$297$	0	0
$298$	3.65685i	0.211836i
$299$	8.48528	0.490716
$300$	0	0
$301$	−27.3137	−1.57434
$302$	12.0000i	0.690522i
$303$	0	0
$304$	−2.82843	−0.162221
$305$	0	0
$306$	0	0
$307$	21.6569i	1.23602i	0.786169	+	0.618011i	$0.212061\pi$
$307$	21.6569i	1.23602i	−0.786169	+	0.618011i	$0.787939\pi$
$308$	16.0000i	0.911685i
$309$	0	0
$310$	0	0
$311$	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
$313$	− 30.9706i	− 1.75056i	−0.483617	−	0.875280i	$-0.660677\pi$
$313$	− 30.9706i	− 1.75056i	0.483617	−	0.875280i	$-0.339323\pi$
$314$	5.31371	0.299870
$315$	0	0
$316$	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	$0.748297\pi$
$318$	0	0
$319$	49.9411	2.79617
$320$	0	0
$321$	0	0
$322$	− 24.0000i	− 1.33747i
$323$	− 2.34315i	− 0.130376i
$324$	0	0
$325$	0	0
$326$	−11.3137	−0.626608
$327$	0	0
$328$	− 7.65685i	− 0.422779i
$329$	22.6274	1.24749
$330$	0	0
$331$	−8.48528	−0.466393	−0.233197	−	0.972430i	$-0.574919\pi$
$331$	−8.48528	−0.466393	−0.233197	+	0.972430i	$0.574919\pi$
$332$	6.34315i	0.348125i
$333$	0	0
$334$	−8.97056	−0.490847
$335$	0	0
$336$	0	0
$337$	− 10.9706i	− 0.597605i	−0.954315	−	0.298802i	$-0.903413\pi$
$337$	− 10.9706i	− 0.597605i	0.954315	−	0.298802i	$-0.0965871\pi$
$338$	− 1.00000i	− 0.0543928i
$339$	0	0
$340$	0	0
$341$	−22.6274	−1.22534
$342$	0	0
$343$	16.9706i	0.916324i
$344$	9.65685	0.520663
$345$	0	0
$346$	−9.31371	−0.500708
$347$	− 9.65685i	− 0.518407i	−0.965823	−	0.259204i	$-0.916540\pi$
$347$	− 9.65685i	− 0.518407i	0.965823	−	0.259204i	$-0.0834600\pi$
$348$	0	0
$349$	−12.1421	−0.649954	−0.324977	−	0.945722i	$-0.605356\pi$
$349$	−12.1421	−0.649954	−0.324977	+	0.945722i	$0.605356\pi$
$350$	0	0
$351$	0	0
$352$	− 5.65685i	− 0.301511i
$353$	− 5.31371i	− 0.282820i	−0.989951	−	0.141410i	$-0.954836\pi$
$353$	− 5.31371i	− 0.282820i	0.989951	−	0.141410i	$-0.0451636\pi$
$354$	0	0
$355$	0	0
$356$	15.6569	0.829812
$357$	0	0
$358$	7.51472i	0.397165i
$359$	−28.2843	−1.49279	−0.746393	−	0.665505i	$-0.768216\pi$
$359$	−28.2843	−1.49279	−0.746393	+	0.665505i	$0.768216\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	− 7.65685i	− 0.402435i
$363$	0	0
$364$	−2.82843	−0.148250
$365$	0	0
$366$	0	0
$367$	− 25.6569i	− 1.33928i	−0.742687	−	0.669638i	$-0.766449\pi$
$367$	− 25.6569i	− 1.33928i	0.742687	−	0.669638i	$-0.233551\pi$
$368$	8.48528i	0.442326i
$369$	0	0
$370$	0	0
$371$	37.6569	1.95505
$372$	0	0
$373$	− 2.68629i	− 0.139091i	−0.997579	−	0.0695455i	$-0.977845\pi$
$373$	− 2.68629i	− 0.139091i	0.997579	−	0.0695455i	$-0.0221549\pi$
$374$	4.68629	0.242322
$375$	0	0
$376$	−8.00000	−0.412568
$377$	8.82843i	0.454687i
$378$	0	0
$379$	7.51472	0.386005	0.193003	−	0.981198i	$-0.438177\pi$
$379$	7.51472	0.386005	0.193003	+	0.981198i	$0.438177\pi$
$380$	0	0
$381$	0	0
$382$	− 11.3137i	− 0.578860i
$383$	29.6569i	1.51539i	0.652606	+	0.757697i	$0.273676\pi$
$383$	29.6569i	1.51539i	−0.652606	+	0.757697i	$0.726324\pi$
$384$	0	0
$385$	0	0
$386$	−2.48528	−0.126497
$387$	0	0
$388$	− 3.17157i	− 0.161012i
$389$	−6.48528	−0.328817	−0.164408	−	0.986392i	$-0.552571\pi$
$389$	−6.48528	−0.328817	−0.164408	+	0.986392i	$0.552571\pi$
$390$	0	0
$391$	−7.02944	−0.355494
$392$	1.00000i	0.0505076i
$393$	0	0
$394$	13.3137	0.670735
$395$	0	0
$396$	0	0
$397$	− 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
$399$	0	0
$400$	0	0
$401$	26.9706	1.34685	0.673423	−	0.739258i	$-0.264823\pi$
$401$	26.9706	1.34685	0.673423	+	0.739258i	$0.264823\pi$
$402$	0	0
$403$	− 4.00000i	− 0.199254i
$404$	16.1421	0.803101
$405$	0	0
$406$	24.9706	1.23927
$407$	− 65.9411i	− 3.26858i
$408$	0	0
$409$	3.65685	0.180820	0.0904099	−	0.995905i	$-0.471182\pi$
$409$	3.65685	0.180820	0.0904099	+	0.995905i	$0.471182\pi$
$410$	0	0
$411$	0	0
$412$	1.65685i	0.0816274i
$413$	− 6.62742i	− 0.326114i
$414$	0	0
$415$	0	0
$416$	1.00000	0.0490290
$417$	0	0
$418$	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	$0.585207\pi$
$420$	0	0
$421$	−24.1421	−1.17662	−0.588308	−	0.808637i	$-0.700206\pi$
$421$	−24.1421	−1.17662	−0.588308	+	0.808637i	$0.700206\pi$
$422$	0.686292i	0.0334081i
$423$	0	0
$424$	−13.3137	−0.646571
$425$	0	0
$426$	0	0
$427$	16.9706i	0.821263i
$428$	4.00000i	0.193347i
$429$	0	0
$430$	0	0
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	0	0
$433$	− 22.9706i	− 1.10389i	−0.833879	−	0.551947i	$-0.813885\pi$
$433$	− 22.9706i	− 1.10389i	0.833879	−	0.551947i	$-0.186115\pi$
$434$	−11.3137	−0.543075
$435$	0	0
$436$	8.82843	0.422805
$437$	− 24.0000i	− 1.14808i
$438$	0	0
$439$	−22.6274	−1.07995	−0.539974	−	0.841682i	$-0.681566\pi$
$439$	−22.6274	−1.07995	−0.539974	+	0.841682i	$0.681566\pi$
$440$	0	0
$441$	0	0
$442$	0.828427i	0.0394043i
$443$	− 30.3431i	− 1.44165i	−0.693119	−	0.720823i	$-0.743764\pi$
$443$	− 30.3431i	− 1.44165i	0.693119	−	0.720823i	$-0.256236\pi$
$444$	0	0
$445$	0	0
$446$	−10.8284	−0.512741
$447$	0	0
$448$	− 2.82843i	− 0.133631i
$449$	26.2843	1.24043	0.620216	−	0.784431i	$-0.287045\pi$
$449$	26.2843	1.24043	0.620216	+	0.784431i	$0.287045\pi$
$450$	0	0
$451$	−43.3137	−2.03956
$452$	6.48528i	0.305042i
$453$	0	0
$454$	−4.00000	−0.187729
$455$	0	0
$456$	0	0
$457$	− 20.8284i	− 0.974313i	−0.873315	−	0.487156i	$-0.838034\pi$
$457$	− 20.8284i	− 0.974313i	0.873315	−	0.487156i	$-0.161966\pi$
$458$	− 4.14214i	− 0.193549i
$459$	0	0
$460$	0	0
$461$	−14.0000	−0.652045	−0.326023	−	0.945362i	$-0.605709\pi$
$461$	−14.0000	−0.652045	−0.326023	+	0.945362i	$0.605709\pi$
$462$	0	0
$463$	− 3.79899i	− 0.176554i	−0.996096	−	0.0882770i	$-0.971864\pi$
$463$	− 3.79899i	− 0.176554i	0.996096	−	0.0882770i	$-0.0281361\pi$
$464$	−8.82843	−0.409849
$465$	0	0
$466$	5.51472	0.255464
$467$	7.31371i	0.338438i	0.985578	+	0.169219i	$0.0541245\pi$
$467$	7.31371i	0.338438i	−0.985578	+	0.169219i	$0.945875\pi$
$468$	0	0
$469$	16.0000	0.738811
$470$	0	0
$471$	0	0
$472$	2.34315i	0.107852i
$473$	− 54.6274i	− 2.51177i
$474$	0	0
$475$	0	0
$476$	2.34315	0.107398
$477$	0	0
$478$	16.0000i	0.731823i
$479$	−11.3137	−0.516937	−0.258468	−	0.966020i	$-0.583218\pi$
$479$	−11.3137	−0.516937	−0.258468	+	0.966020i	$0.583218\pi$
$480$	0	0
$481$	11.6569	0.531507
$482$	5.31371i	0.242033i
$483$	0	0
$484$	−21.0000	−0.954545
$485$	0	0
$486$	0	0
$487$	− 16.4853i	− 0.747019i	−0.927626	−	0.373510i	$-0.878154\pi$
$487$	− 16.4853i	− 0.747019i	0.927626	−	0.373510i	$-0.121846\pi$
$488$	− 6.00000i	− 0.271607i
$489$	0	0
$490$	0	0
$491$	38.1421	1.72133	0.860665	−	0.509171i	$-0.170048\pi$
$491$	38.1421	1.72133	0.860665	+	0.509171i	$0.170048\pi$
$492$	0	0
$493$	− 7.31371i	− 0.329393i
$494$	−2.82843	−0.127257
$495$	0	0
$496$	4.00000	0.179605
$497$	16.0000i	0.717698i
$498$	0	0
$499$	−0.485281	−0.0217242	−0.0108621	−	0.999941i	$-0.503458\pi$
$499$	−0.485281	−0.0217242	−0.0108621	+	0.999941i	$0.503458\pi$
$500$	0	0
$501$	0	0
$502$	− 10.8284i	− 0.483296i
$503$	23.5147i	1.04847i	0.851574	+	0.524235i	$0.175649\pi$
$503$	23.5147i	1.04847i	−0.851574	+	0.524235i	$0.824351\pi$
$504$	0	0
$505$	0	0
$506$	48.0000	2.13386
$507$	0	0
$508$	9.65685i	0.428454i
$509$	−37.3137	−1.65390	−0.826951	−	0.562275i	$-0.809926\pi$
$509$	−37.3137	−1.65390	−0.826951	+	0.562275i	$0.809926\pi$
$510$	0	0
$511$	40.9706	1.81243
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	4.82843	0.212973
$515$	0	0
$516$	0	0
$517$	45.2548i	1.99031i
$518$	− 32.9706i	− 1.44864i
$519$	0	0
$520$	0	0
$521$	−26.9706	−1.18160	−0.590801	−	0.806817i	$-0.701188\pi$
$521$	−26.9706	−1.18160	−0.590801	+	0.806817i	$0.701188\pi$
$522$	0	0
$523$	− 10.6274i	− 0.464704i	−0.972632	−	0.232352i	$-0.925358\pi$
$523$	− 10.6274i	− 0.464704i	0.972632	−	0.232352i	$-0.0746422\pi$
$524$	−6.14214	−0.268320
$525$	0	0
$526$	16.4853	0.718792
$527$	3.31371i	0.144347i
$528$	0	0
$529$	−49.0000	−2.13043
$530$	0	0
$531$	0	0
$532$	8.00000i	0.346844i
$533$	− 7.65685i	− 0.331655i
$534$	0	0
$535$	0	0
$536$	−5.65685	−0.244339
$537$	0	0
$538$	− 14.4853i	− 0.624505i
$539$	5.65685	0.243658
$540$	0	0
$541$	14.4853	0.622771	0.311385	−	0.950284i	$-0.399207\pi$
$541$	14.4853	0.622771	0.311385	+	0.950284i	$0.399207\pi$
$542$	7.31371i	0.314151i
$543$	0	0
$544$	−0.828427	−0.0355185
$545$	0	0
$546$	0	0
$547$	− 0.686292i	− 0.0293437i	−0.999892	−	0.0146719i	$-0.995330\pi$
$547$	− 0.686292i	− 0.0293437i	0.999892	−	0.0146719i	$-0.00467036\pi$
$548$	17.3137i	0.739605i
$549$	0	0
$550$	0	0
$551$	24.9706	1.06378
$552$	0	0
$553$	− 6.62742i	− 0.281826i
$554$	26.0000	1.10463
$555$	0	0
$556$	6.34315	0.269009
$557$	10.6863i	0.452793i	0.974035	+	0.226396i	$0.0726945\pi$
$557$	10.6863i	0.452793i	−0.974035	+	0.226396i	$0.927306\pi$
$558$	0	0
$559$	9.65685	0.408441
$560$	0	0
$561$	0	0
$562$	− 8.34315i	− 0.351934i
$563$	30.3431i	1.27881i	0.768870	+	0.639406i	$0.220819\pi$
$563$	30.3431i	1.27881i	−0.768870	+	0.639406i	$0.779181\pi$
$564$	0	0
$565$	0	0
$566$	17.6569	0.742173
$567$	0	0
$568$	− 5.65685i	− 0.237356i
$569$	31.6569	1.32712	0.663562	−	0.748121i	$-0.269044\pi$
$569$	31.6569	1.32712	0.663562	+	0.748121i	$0.269044\pi$
$570$	0	0
$571$	20.9706	0.877591	0.438795	−	0.898587i	$-0.355405\pi$
$571$	20.9706	0.877591	0.438795	+	0.898587i	$0.355405\pi$
$572$	− 5.65685i	− 0.236525i
$573$	0	0
$574$	−21.6569	−0.903940
$575$	0	0
$576$	0	0
$577$	23.4558i	0.976480i	0.872710	+	0.488240i	$0.162361\pi$
$577$	23.4558i	0.976480i	−0.872710	+	0.488240i	$0.837639\pi$
$578$	16.3137i	0.678561i
$579$	0	0
$580$	0	0
$581$	17.9411	0.744323
$582$	0	0
$583$	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	$0.982732\pi$
$588$	0	0
$589$	−11.3137	−0.466173
$590$	0	0
$591$	0	0
$592$	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	$0.997757\pi$
$594$	0	0
$595$	0	0
$596$	−3.65685	−0.149791
$597$	0	0
$598$	8.48528i	0.346989i
$599$	40.0000	1.63436	0.817178	−	0.576386i	$-0.195537\pi$
$599$	40.0000	1.63436	0.817178	+	0.576386i	$0.195537\pi$
$600$	0	0
$601$	29.3137	1.19573	0.597866	−	0.801596i	$-0.296016\pi$
$601$	29.3137	1.19573	0.597866	+	0.801596i	$0.296016\pi$
$602$	− 27.3137i	− 1.11322i
$603$	0	0
$604$	−12.0000	−0.488273
$605$	0	0
$606$	0	0
$607$	− 28.9706i	− 1.17588i	−0.808905	−	0.587939i	$-0.799939\pi$
$607$	− 28.9706i	− 1.17588i	0.808905	−	0.587939i	$-0.200061\pi$
$608$	− 2.82843i	− 0.114708i
$609$	0	0
$610$	0	0
$611$	−8.00000	−0.323645
$612$	0	0
$613$	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
$615$	0	0
$616$	−16.0000	−0.644658
$617$	2.00000i	0.0805170i	0.999189	+	0.0402585i	$0.0128181\pi$
$617$	2.00000i	0.0805170i	−0.999189	+	0.0402585i	$0.987182\pi$
$618$	0	0
$619$	34.8284	1.39987	0.699936	−	0.714205i	$-0.253212\pi$
$619$	34.8284	1.39987	0.699936	+	0.714205i	$0.253212\pi$
$620$	0	0
$621$	0	0
$622$	24.0000i	0.962312i
$623$	− 44.2843i	− 1.77421i
$624$	0	0
$625$	0	0
$626$	30.9706	1.23783
$627$	0	0
$628$	5.31371i	0.212040i
$629$	−9.65685	−0.385044
$630$	0	0
$631$	−33.6569	−1.33986	−0.669929	−	0.742425i	$-0.733676\pi$
$631$	−33.6569	−1.33986	−0.669929	+	0.742425i	$0.733676\pi$
$632$	2.34315i	0.0932053i
$633$	0	0
$634$	−25.3137	−1.00534
$635$	0	0
$636$	0	0
$637$	1.00000i	0.0396214i
$638$	49.9411i	1.97719i
$639$	0	0
$640$	0	0
$641$	4.62742	0.182772	0.0913860	−	0.995816i	$-0.470870\pi$
$641$	4.62742	0.182772	0.0913860	+	0.995816i	$0.470870\pi$
$642$	0	0
$643$	39.5980i	1.56159i	0.624786	+	0.780796i	$0.285186\pi$
$643$	39.5980i	1.56159i	−0.624786	+	0.780796i	$0.714814\pi$
$644$	24.0000	0.945732
$645$	0	0
$646$	2.34315	0.0921898
$647$	8.48528i	0.333591i	0.985992	+	0.166795i	$0.0533419\pi$
$647$	8.48528i	0.333591i	−0.985992	+	0.166795i	$0.946658\pi$
$648$	0	0
$649$	13.2548	0.520298
$650$	0	0
$651$	0	0
$652$	− 11.3137i	− 0.443079i
$653$	42.2843i	1.65471i	0.561678	+	0.827356i	$0.310156\pi$
$653$	42.2843i	1.65471i	−0.561678	+	0.827356i	$0.689844\pi$
$654$	0	0
$655$	0	0
$656$	7.65685	0.298950
$657$	0	0
$658$	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	$0.546758\pi$
$660$	0	0
$661$	−8.14214	−0.316692	−0.158346	−	0.987384i	$-0.550616\pi$
$661$	−8.14214	−0.316692	−0.158346	+	0.987384i	$0.550616\pi$
$662$	− 8.48528i	− 0.329790i
$663$	0	0
$664$	−6.34315	−0.246162
$665$	0	0
$666$	0	0
$667$	− 74.9117i	− 2.90059i
$668$	− 8.97056i	− 0.347081i
$669$	0	0
$670$	0	0
$671$	−33.9411	−1.31028
$672$	0	0
$673$	32.6274i	1.25769i	0.777529	+	0.628847i	$0.216473\pi$
$673$	32.6274i	1.25769i	−0.777529	+	0.628847i	$0.783527\pi$
$674$	10.9706	0.422570
$675$	0	0
$676$	1.00000	0.0384615
$677$	12.3431i	0.474386i	0.971463	+	0.237193i	$0.0762273\pi$
$677$	12.3431i	0.474386i	−0.971463	+	0.237193i	$0.923773\pi$
$678$	0	0
$679$	−8.97056	−0.344259
$680$	0	0
$681$	0	0
$682$	− 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	$-0.222694\pi$
$684$	0	0
$685$	0	0
$686$	−16.9706	−0.647939
$687$	0	0
$688$	9.65685i	0.368164i
$689$	−13.3137	−0.507212
$690$	0	0
$691$	−27.7990	−1.05752	−0.528762	−	0.848770i	$-0.677343\pi$
$691$	−27.7990	−1.05752	−0.528762	+	0.848770i	$0.677343\pi$
$692$	− 9.31371i	− 0.354054i
$693$	0	0
$694$	9.65685	0.366569
$695$	0	0
$696$	0	0
$697$	6.34315i	0.240264i
$698$	− 12.1421i	− 0.459587i
$699$	0	0
$700$	0	0
$701$	−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
$703$	− 32.9706i	− 1.24351i
$704$	5.65685	0.213201
$705$	0	0
$706$	5.31371	0.199984
$707$	− 45.6569i	− 1.71710i
$708$	0	0
$709$	7.17157	0.269334	0.134667	−	0.990891i	$-0.457004\pi$
$709$	7.17157	0.269334	0.134667	+	0.990891i	$0.457004\pi$
$710$	0	0
$711$	0	0
$712$	15.6569i	0.586765i
$713$	33.9411i	1.27111i
$714$	0	0
$715$	0	0
$716$	−7.51472	−0.280838
$717$	0	0
$718$	− 28.2843i	− 1.05556i
$719$	−29.6569	−1.10601	−0.553007	−	0.833177i	$-0.686520\pi$
$719$	−29.6569	−1.10601	−0.553007	+	0.833177i	$0.686520\pi$
$720$	0	0
$721$	4.68629	0.174527
$722$	− 11.0000i	− 0.409378i
$723$	0	0
$724$	7.65685	0.284565
$725$	0	0
$726$	0	0
$727$	45.9411i	1.70386i	0.523654	+	0.851931i	$0.324568\pi$
$727$	45.9411i	1.70386i	−0.523654	+	0.851931i	$0.675432\pi$
$728$	− 2.82843i	− 0.104828i
$729$	0	0
$730$	0	0
$731$	−8.00000	−0.295891
$732$	0	0
$733$	− 0.343146i	− 0.0126744i	−0.999980	−	0.00633719i	$-0.997983\pi$
$733$	− 0.343146i	− 0.0126744i	0.999980	−	0.00633719i	$-0.00201720\pi$
$734$	25.6569	0.947012
$735$	0	0
$736$	−8.48528	−0.312772
$737$	32.0000i	1.17874i
$738$	0	0
$739$	14.1421	0.520227	0.260113	−	0.965578i	$-0.416240\pi$
$739$	14.1421	0.520227	0.260113	+	0.965578i	$0.416240\pi$
$740$	0	0
$741$	0	0
$742$	37.6569i	1.38243i
$743$	36.2843i	1.33114i	0.746335	+	0.665570i	$0.231812\pi$
$743$	36.2843i	1.33114i	−0.746335	+	0.665570i	$0.768188\pi$
$744$	0	0
$745$	0	0
$746$	2.68629	0.0983521
$747$	0	0
$748$	4.68629i	0.171348i
$749$	11.3137	0.413394
$750$	0	0
$751$	11.3137	0.412843	0.206422	−	0.978463i	$-0.433818\pi$
$751$	11.3137	0.412843	0.206422	+	0.978463i	$0.433818\pi$
$752$	− 8.00000i	− 0.291730i
$753$	0	0
$754$	−8.82843	−0.321512
$755$	0	0
$756$	0	0
$757$	− 19.9411i	− 0.724773i	−0.932028	−	0.362386i	$-0.881962\pi$
$757$	− 19.9411i	− 0.724773i	0.932028	−	0.362386i	$-0.118038\pi$
$758$	7.51472i	0.272947i
$759$	0	0
$760$	0	0
$761$	−27.6569	−1.00256	−0.501280	−	0.865285i	$-0.667137\pi$
$761$	−27.6569	−1.00256	−0.501280	+	0.865285i	$0.667137\pi$
$762$	0	0
$763$	− 24.9706i	− 0.903995i
$764$	11.3137	0.409316
$765$	0	0
$766$	−29.6569	−1.07155
$767$	2.34315i	0.0846061i
$768$	0	0
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	0	0
$771$	0	0
$772$	− 2.48528i	− 0.0894472i
$773$	53.3137i	1.91756i	0.284148	+	0.958780i	$0.408289\pi$
$773$	53.3137i	1.91756i	−0.284148	+	0.958780i	$0.591711\pi$
$774$	0	0
$775$	0	0
$776$	3.17157	0.113853
$777$	0	0
$778$	− 6.48528i	− 0.232509i
$779$	−21.6569	−0.775937
$780$	0	0
$781$	−32.0000	−1.14505
$782$	− 7.02944i	− 0.251372i
$783$	0	0
$784$	−1.00000	−0.0357143
$785$	0	0
$786$	0	0
$787$	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
$789$	0	0
$790$	0	0
$791$	18.3431	0.652207
$792$	0	0
$793$	− 6.00000i	− 0.213066i
$794$	30.2843	1.07475
$795$	0	0
$796$	10.3431	0.366603
$797$	16.6274i	0.588973i	0.955656	+	0.294487i	$0.0951487\pi$
$797$	16.6274i	0.588973i	−0.955656	+	0.294487i	$0.904851\pi$
$798$	0	0
$799$	6.62742	0.234461
$800$	0	0
$801$	0	0
$802$	26.9706i	0.952364i
$803$	81.9411i	2.89164i
$804$	0	0
$805$	0	0
$806$	4.00000	0.140894
$807$	0	0
$808$	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	$0.424804\pi$
$810$	0	0
$811$	−1.85786	−0.0652384	−0.0326192	−	0.999468i	$-0.510385\pi$
$811$	−1.85786	−0.0652384	−0.0326192	+	0.999468i	$0.510385\pi$
$812$	24.9706i	0.876295i
$813$	0	0
$814$	65.9411	2.31124
$815$	0	0
$816$	0	0
$817$	− 27.3137i	− 0.955586i
$818$	3.65685i	0.127859i
$819$	0	0
$820$	0	0
$821$	−34.2843	−1.19653	−0.598265	−	0.801299i	$-0.704143\pi$
$821$	−34.2843	−1.19653	−0.598265	+	0.801299i	$0.704143\pi$
$822$	0	0
$823$	− 52.9706i	− 1.84644i	−0.384275	−	0.923219i	$-0.625548\pi$
$823$	− 52.9706i	− 1.84644i	0.384275	−	0.923219i	$-0.374452\pi$
$824$	−1.65685	−0.0577193
$825$	0	0
$826$	6.62742	0.230597
$827$	9.65685i	0.335802i	0.985804	+	0.167901i	$0.0536989\pi$
$827$	9.65685i	0.335802i	−0.985804	+	0.167901i	$0.946301\pi$
$828$	0	0
$829$	53.3137	1.85166	0.925831	−	0.377938i	$-0.123367\pi$
$829$	53.3137	1.85166	0.925831	+	0.377938i	$0.123367\pi$
$830$	0	0
$831$	0	0
$832$	1.00000i	0.0346688i
$833$	− 0.828427i	− 0.0287033i
$834$	0	0
$835$	0	0
$836$	−16.0000	−0.553372
$837$	0	0
$838$	− 10.8284i	− 0.374062i
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	48.9411	1.68763
$842$	− 24.1421i	− 0.831993i
$843$	0	0
$844$	−0.686292	−0.0236231
$845$	0	0
$846$	0	0
$847$	59.3970i	2.04090i
$848$	− 13.3137i	− 0.457195i
$849$	0	0
$850$	0	0
$851$	−98.9117	−3.39065
$852$	0	0
$853$	38.2843i	1.31083i	0.755270	+	0.655414i	$0.227506\pi$
$853$	38.2843i	1.31083i	−0.755270	+	0.655414i	$0.772494\pi$
$854$	−16.9706	−0.580721
$855$	0	0
$856$	−4.00000	−0.136717
$857$	− 20.8284i	− 0.711486i	−0.934584	−	0.355743i	$-0.884228\pi$
$857$	− 20.8284i	− 0.711486i	0.934584	−	0.355743i	$-0.115772\pi$
$858$	0	0
$859$	−37.9411	−1.29453	−0.647267	−	0.762263i	$-0.724088\pi$
$859$	−37.9411	−1.29453	−0.647267	+	0.762263i	$0.724088\pi$
$860$	0	0
$861$	0	0
$862$	16.0000i	0.544962i
$863$	28.2843i	0.962808i	0.876499	+	0.481404i	$0.159873\pi$
$863$	28.2843i	0.962808i	−0.876499	+	0.481404i	$0.840127\pi$
$864$	0	0
$865$	0	0
$866$	22.9706	0.780571
$867$	0	0
$868$	− 11.3137i	− 0.384012i
$869$	13.2548	0.449639
$870$	0	0
$871$	−5.65685	−0.191675
$872$	8.82843i	0.298968i
$873$	0	0
$874$	24.0000	0.811812
$875$	0	0
$876$	0	0
$877$	51.2548i	1.73075i	0.501122	+	0.865376i	$0.332921\pi$
$877$	51.2548i	1.73075i	−0.501122	+	0.865376i	$0.667079\pi$
$878$	− 22.6274i	− 0.763638i
$879$	0	0
$880$	0	0
$881$	10.2843	0.346486	0.173243	−	0.984879i	$-0.444575\pi$
$881$	10.2843	0.346486	0.173243	+	0.984879i	$0.444575\pi$
$882$	0	0
$883$	31.3137i	1.05379i	0.849930	+	0.526895i	$0.176644\pi$
$883$	31.3137i	1.05379i	−0.849930	+	0.526895i	$0.823356\pi$
$884$	−0.828427	−0.0278630
$885$	0	0
$886$	30.3431	1.01940
$887$	− 40.4853i	− 1.35936i	−0.733508	−	0.679681i	$-0.762118\pi$
$887$	− 40.4853i	− 1.35936i	0.733508	−	0.679681i	$-0.237882\pi$
$888$	0	0
$889$	27.3137	0.916072
$890$	0	0
$891$	0	0
$892$	− 10.8284i	− 0.362563i
$893$	22.6274i	0.757198i
$894$	0	0
$895$	0	0
$896$	2.82843	0.0944911
$897$	0	0
$898$	26.2843i	0.877117i
$899$	−35.3137	−1.17778
$900$	0	0
$901$	11.0294	0.367444
$902$	− 43.3137i	− 1.44219i
$903$	0	0
$904$	−6.48528	−0.215697
$905$	0	0
$906$	0	0
$907$	− 8.28427i	− 0.275075i	−0.990497	−	0.137537i	$-0.956081\pi$
$907$	− 8.28427i	− 0.275075i	0.990497	−	0.137537i	$-0.0439187\pi$
$908$	− 4.00000i	− 0.132745i
$909$	0	0
$910$	0	0
$911$	−24.9706	−0.827312	−0.413656	−	0.910433i	$-0.635748\pi$
$911$	−24.9706	−0.827312	−0.413656	+	0.910433i	$0.635748\pi$
$912$	0	0
$913$	35.8823i	1.18753i
$914$	20.8284	0.688943
$915$	0	0
$916$	4.14214	0.136860
$917$	17.3726i	0.573693i
$918$	0	0
$919$	−41.9411	−1.38351	−0.691755	−	0.722132i	$-0.743162\pi$
$919$	−41.9411	−1.38351	−0.691755	+	0.722132i	$0.743162\pi$
$920$	0	0
$921$	0	0
$922$	− 14.0000i	− 0.461065i
$923$	− 5.65685i	− 0.186198i
$924$	0	0
$925$	0	0
$926$	3.79899	0.124843
$927$	0	0
$928$	− 8.82843i	− 0.289807i
$929$	−33.5980	−1.10231	−0.551157	−	0.834402i	$-0.685813\pi$
$929$	−33.5980	−1.10231	−0.551157	+	0.834402i	$0.685813\pi$
$930$	0	0
$931$	2.82843	0.0926980
$932$	5.51472i	0.180641i
$933$	0	0
$934$	−7.31371	−0.239312
$935$	0	0
$936$	0	0
$937$	− 16.6274i	− 0.543194i	−0.962411	−	0.271597i	$-0.912448\pi$
$937$	− 16.6274i	− 0.543194i	0.962411	−	0.271597i	$-0.0875518\pi$
$938$	16.0000i	0.522419i
$939$	0	0
$940$	0	0
$941$	−54.9706	−1.79199	−0.895995	−	0.444065i	$-0.853536\pi$
$941$	−54.9706	−1.79199	−0.895995	+	0.444065i	$0.853536\pi$
$942$	0	0
$943$	64.9706i	2.11573i
$944$	−2.34315	−0.0762629
$945$	0	0
$946$	54.6274	1.77609
$947$	30.3431i	0.986020i	0.870024	+	0.493010i	$0.164103\pi$
$947$	30.3431i	0.986020i	−0.870024	+	0.493010i	$0.835897\pi$
$948$	0	0
$949$	−14.4853	−0.470212
$950$	0	0
$951$	0	0
$952$	2.34315i	0.0759418i
$953$	27.8579i	0.902405i	0.892422	+	0.451202i	$0.149005\pi$
$953$	27.8579i	0.902405i	−0.892422	+	0.451202i	$0.850995\pi$
$954$	0	0
$955$	0	0
$956$	−16.0000	−0.517477
$957$	0	0
$958$	− 11.3137i	− 0.365529i
$959$	48.9706	1.58134
$960$	0	0
$961$	−15.0000	−0.483871
$962$	11.6569i	0.375832i
$963$	0	0
$964$	−5.31371	−0.171143
$965$	0	0
$966$	0	0
$967$	7.51472i	0.241657i	0.992673	+	0.120829i	$0.0385551\pi$
$967$	7.51472i	0.241657i	−0.992673	+	0.120829i	$0.961445\pi$
$968$	− 21.0000i	− 0.674966i
$969$	0	0
$970$	0	0
$971$	−15.5147	−0.497891	−0.248946	−	0.968517i	$-0.580084\pi$
$971$	−15.5147	−0.497891	−0.248946	+	0.968517i	$0.580084\pi$
$972$	0	0
$973$	− 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	$-0.0426089\pi$
$978$	0	0
$979$	88.5685	2.83066
$980$	0	0
$981$	0	0
$982$	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	$0.988103\pi$
$984$	0	0
$985$	0	0
$986$	7.31371	0.232916
$987$	0	0
$988$	− 2.82843i	− 0.0899843i
$989$	−81.9411	−2.60558
$990$	0	0
$991$	−42.9117	−1.36313	−0.681567	−	0.731755i	$-0.738701\pi$
$991$	−42.9117	−1.36313	−0.681567	+	0.731755i	$0.738701\pi$
$992$	4.00000i	0.127000i
$993$	0	0
$994$	−16.0000	−0.507489
$995$	0	0
$996$	0	0
$997$	− 61.3137i	− 1.94182i	−0.239435	−	0.970912i	$-0.576962\pi$
$997$	− 61.3137i	− 1.94182i	0.239435	−	0.970912i	$-0.423038\pi$
$998$	− 0.485281i	− 0.0153613i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5850.2.e.bk.5149.4		4
3.2	odd	2		1950.2.e.o.1249.2		4
5.2	odd	4		1170.2.a.o.1.1		2
5.3	odd	4		5850.2.a.cl.1.2		2
5.4	even	2	inner	5850.2.e.bk.5149.1		4
15.2	even	4		390.2.a.h.1.1	✓	2
15.8	even	4		1950.2.a.bd.1.2		2
15.14	odd	2		1950.2.e.o.1249.3		4
20.7	even	4		9360.2.a.ch.1.2		2
60.47	odd	4		3120.2.a.bc.1.2		2
195.47	odd	4		5070.2.b.q.1351.4		4
195.77	even	4		5070.2.a.bc.1.2		2
195.122	odd	4		5070.2.b.q.1351.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.a.h.1.1	✓	2	15.2	even	4
1170.2.a.o.1.1		2	5.2	odd	4
1950.2.a.bd.1.2		2	15.8	even	4
1950.2.e.o.1249.2		4	3.2	odd	2
1950.2.e.o.1249.3		4	15.14	odd	2
3120.2.a.bc.1.2		2	60.47	odd	4
5070.2.a.bc.1.2		2	195.77	even	4
5070.2.b.q.1351.1		4	195.122	odd	4
5070.2.b.q.1351.4		4	195.47	odd	4
5850.2.a.cl.1.2		2	5.3	odd	4
5850.2.e.bk.5149.1		4	5.4	even	2	inner
5850.2.e.bk.5149.4		4	1.1	even	1	trivial
9360.2.a.ch.1.2		2	20.7	even	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:	\( N \)	\(=\)	\( 5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5850.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +2.82843i q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} +2.82843i q^{7} -1.00000i q^{8} -5.65685 q^{11} -1.00000i q^{13} -2.82843 q^{14} +1.00000 q^{16} +0.828427i q^{17} -2.82843 q^{19} -5.65685i q^{22} +8.48528i q^{23} +1.00000 q^{26} -2.82843i q^{28} -8.82843 q^{29} +4.00000 q^{31} +1.00000i q^{32} -0.828427 q^{34} +11.6569i q^{37} -2.82843i q^{38} +7.65685 q^{41} +9.65685i q^{43} +5.65685 q^{44} -8.48528 q^{46} -8.00000i q^{47} -1.00000 q^{49} +1.00000i q^{52} -13.3137i q^{53} +2.82843 q^{56} -8.82843i q^{58} -2.34315 q^{59} +6.00000 q^{61} +4.00000i q^{62} -1.00000 q^{64} -5.65685i q^{67} -0.828427i q^{68} +5.65685 q^{71} -14.4853i q^{73} -11.6569 q^{74} +2.82843 q^{76} -16.0000i q^{77} -2.34315 q^{79} +7.65685i q^{82} -6.34315i q^{83} -9.65685 q^{86} +5.65685i q^{88} -15.6569 q^{89} +2.82843 q^{91} -8.48528i q^{92} +8.00000 q^{94} +3.17157i q^{97} -1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4}+O(q^{10}) \) 4 * q - 4 * q^4	\( 4 q - 4 q^{4} + 4 q^{16} + 4 q^{26} - 24 q^{29} + 16 q^{31} + 8 q^{34} + 8 q^{41} - 4 q^{49} - 32 q^{59} + 24 q^{61} - 4 q^{64} - 24 q^{74} - 32 q^{79} - 16 q^{86} - 40 q^{89} + 32 q^{94}+O(q^{100}) \) 4 * q - 4 * q^4 + 4 * q^16 + 4 * q^26 - 24 * q^29 + 16 * q^31 + 8 * q^34 + 8 * q^41 - 4 * q^49 - 32 * q^59 + 24 * q^61 - 4 * q^64 - 24 * q^74 - 32 * q^79 - 16 * q^86 - 40 * q^89 + 32 * q^94

Introduction

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