LMFDB - Embedded newform 2450.2.c.o.99.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2450,2,Mod(99,2450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2450, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2450.99");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2450 = 2 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2450.c (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:	$19.5633484952$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 350)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			99.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	2450.99
Dual form			2450.2.c.o.99.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{2} +2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} -2.00000i q^{12} -2.00000i q^{13} +1.00000 q^{16} -3.00000i q^{17} +1.00000i q^{18} -8.00000 q^{19} +9.00000i q^{23} -2.00000 q^{24} -2.00000 q^{26} +4.00000i q^{27} +6.00000 q^{29} +5.00000 q^{31} -1.00000i q^{32} -3.00000 q^{34} +1.00000 q^{36} +8.00000i q^{37} +8.00000i q^{38} +4.00000 q^{39} -3.00000 q^{41} +10.0000i q^{43} +9.00000 q^{46} -3.00000i q^{47} +2.00000i q^{48} +6.00000 q^{51} +2.00000i q^{52} -6.00000i q^{53} +4.00000 q^{54} -16.0000i q^{57} -6.00000i q^{58} -12.0000 q^{59} -4.00000 q^{61} -5.00000i q^{62} -1.00000 q^{64} +2.00000i q^{67} +3.00000i q^{68} -18.0000 q^{69} -9.00000 q^{71} -1.00000i q^{72} +10.0000i q^{73} +8.00000 q^{74} +8.00000 q^{76} -4.00000i q^{78} -5.00000 q^{79} -11.0000 q^{81} +3.00000i q^{82} +6.00000i q^{83} +10.0000 q^{86} +12.0000i q^{87} -3.00000 q^{89} -9.00000i q^{92} +10.0000i q^{93} -3.00000 q^{94} +2.00000 q^{96} +5.00000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 4 q^{6} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 + 4 * q^6 - 2 * q^9	$ 2 q - 2 q^{4} + 4 q^{6} - 2 q^{9} + 2 q^{16} - 16 q^{19} - 4 q^{24} - 4 q^{26} + 12 q^{29} + 10 q^{31} - 6 q^{34} + 2 q^{36} + 8 q^{39} - 6 q^{41} + 18 q^{46} + 12 q^{51} + 8 q^{54} - 24 q^{59} - 8 q^{61} - 2 q^{64} - 36 q^{69} - 18 q^{71} + 16 q^{74} + 16 q^{76} - 10 q^{79} - 22 q^{81} + 20 q^{86} - 6 q^{89} - 6 q^{94} + 4 q^{96}+O(q^{100}) $ 2 * q - 2 * q^4 + 4 * q^6 - 2 * q^9 + 2 * q^16 - 16 * q^19 - 4 * q^24 - 4 * q^26 + 12 * q^29 + 10 * q^31 - 6 * q^34 + 2 * q^36 + 8 * q^39 - 6 * q^41 + 18 * q^46 + 12 * q^51 + 8 * q^54 - 24 * q^59 - 8 * q^61 - 2 * q^64 - 36 * q^69 - 18 * q^71 + 16 * q^74 + 16 * q^76 - 10 * q^79 - 22 * q^81 + 20 * q^86 - 6 * q^89 - 6 * q^94 + 4 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2450\mathbb{Z}\right)^\times$.

$n$	$101$	$1177$
$\chi(n)$	$1$	$-1$

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
$3$	2.00000i	1.15470i	0.816497	+	0.577350i	$0.195913\pi$
$3$	2.00000i	1.15470i	−0.816497	+	0.577350i	$0.804087\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	2.00000	0.816497
$7$	0	0
$7$	0	0
$8$	1.00000i	0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	− 2.00000i	− 0.577350i
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	− 3.00000i	− 0.727607i	−0.931476	−	0.363803i	$-0.881478\pi$
$17$	− 3.00000i	− 0.727607i	0.931476	−	0.363803i	$-0.118522\pi$
$18$	1.00000i	0.235702i
$19$	−8.00000	−1.83533	−0.917663	−	0.397360i	$-0.869927\pi$
$19$	−8.00000	−1.83533	−0.917663	+	0.397360i	$0.869927\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	9.00000i	1.87663i	0.345782	+	0.938315i	$0.387614\pi$
$23$	9.00000i	1.87663i	−0.345782	+	0.938315i	$0.612386\pi$
$24$	−2.00000	−0.408248
$25$	0	0
$26$	−2.00000	−0.392232
$27$	4.00000i	0.769800i
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	−3.00000	−0.514496
$35$	0	0
$36$	1.00000	0.166667
$37$	8.00000i	1.31519i	0.753371	+	0.657596i	$0.228427\pi$
$37$	8.00000i	1.31519i	−0.753371	+	0.657596i	$0.771573\pi$
$38$	8.00000i	1.29777i
$39$	4.00000	0.640513
$40$	0	0
$41$	−3.00000	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000	−0.468521	−0.234261	+	0.972174i	$0.575267\pi$
$42$	0	0
$43$	10.0000i	1.52499i	0.646997	+	0.762493i	$0.276025\pi$
$43$	10.0000i	1.52499i	−0.646997	+	0.762493i	$0.723975\pi$
$44$	0	0
$45$	0	0
$46$	9.00000	1.32698
$47$	− 3.00000i	− 0.437595i	−0.975770	−	0.218797i	$-0.929787\pi$
$47$	− 3.00000i	− 0.437595i	0.975770	−	0.218797i	$-0.0702134\pi$
$48$	2.00000i	0.288675i
$49$	0	0
$50$	0	0
$51$	6.00000	0.840168
$52$	2.00000i	0.277350i
$53$	− 6.00000i	− 0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	4.00000	0.544331
$55$	0	0
$56$	0	0
$57$	− 16.0000i	− 2.11925i
$58$	− 6.00000i	− 0.787839i
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	−4.00000	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$62$	− 5.00000i	− 0.635001i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	2.00000i	0.244339i	0.992509	+	0.122169i	$0.0389851\pi$
$67$	2.00000i	0.244339i	−0.992509	+	0.122169i	$0.961015\pi$
$68$	3.00000i	0.363803i
$69$	−18.0000	−2.16695
$70$	0	0
$71$	−9.00000	−1.06810	−0.534052	−	0.845452i	$-0.679331\pi$
$71$	−9.00000	−1.06810	−0.534052	+	0.845452i	$0.679331\pi$
$72$	− 1.00000i	− 0.117851i
$73$	10.0000i	1.17041i	0.810885	+	0.585206i	$0.198986\pi$
$73$	10.0000i	1.17041i	−0.810885	+	0.585206i	$0.801014\pi$
$74$	8.00000	0.929981
$75$	0	0
$76$	8.00000	0.917663
$77$	0	0
$78$	− 4.00000i	− 0.452911i
$79$	−5.00000	−0.562544	−0.281272	−	0.959628i	$-0.590756\pi$
$79$	−5.00000	−0.562544	−0.281272	+	0.959628i	$0.590756\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	3.00000i	0.331295i
$83$	6.00000i	0.658586i	0.944228	+	0.329293i	$0.106810\pi$
$83$	6.00000i	0.658586i	−0.944228	+	0.329293i	$0.893190\pi$
$84$	0	0
$85$	0	0
$86$	10.0000	1.07833
$87$	12.0000i	1.28654i
$88$	0	0
$89$	−3.00000	−0.317999	−0.159000	−	0.987279i	$-0.550827\pi$
$89$	−3.00000	−0.317999	−0.159000	+	0.987279i	$0.550827\pi$
$90$	0	0
$91$	0	0
$92$	− 9.00000i	− 0.938315i
$93$	10.0000i	1.03695i
$94$	−3.00000	−0.309426
$95$	0	0
$96$	2.00000	0.204124
$97$	5.00000i	0.507673i	0.967247	+	0.253837i	$0.0816925\pi$
$97$	5.00000i	0.507673i	−0.967247	+	0.253837i	$0.918307\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	− 6.00000i	− 0.594089i
$103$	− 11.0000i	− 1.08386i	−0.840423	−	0.541931i	$-0.817693\pi$
$103$	− 11.0000i	− 1.08386i	0.840423	−	0.541931i	$-0.182307\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	−6.00000	−0.582772
$107$	12.0000i	1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$	12.0000i	1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$	− 4.00000i	− 0.384900i
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	−16.0000	−1.51865
$112$	0	0
$113$	− 15.0000i	− 1.41108i	−0.708669	−	0.705541i	$-0.750704\pi$
$113$	− 15.0000i	− 1.41108i	0.708669	−	0.705541i	$-0.249296\pi$
$114$	−16.0000	−1.49854
$115$	0	0
$116$	−6.00000	−0.557086
$117$	2.00000i	0.184900i
$118$	12.0000i	1.10469i
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	4.00000i	0.362143i
$123$	− 6.00000i	− 0.541002i
$124$	−5.00000	−0.449013
$125$	0	0
$126$	0	0
$127$	8.00000i	0.709885i	0.934888	+	0.354943i	$0.115500\pi$
$127$	8.00000i	0.709885i	−0.934888	+	0.354943i	$0.884500\pi$
$128$	1.00000i	0.0883883i
$129$	−20.0000	−1.76090
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	2.00000	0.172774
$135$	0	0
$136$	3.00000	0.257248
$137$	9.00000i	0.768922i	0.923141	+	0.384461i	$0.125613\pi$
$137$	9.00000i	0.768922i	−0.923141	+	0.384461i	$0.874387\pi$
$138$	18.0000i	1.53226i
$139$	−2.00000	−0.169638	−0.0848189	−	0.996396i	$-0.527031\pi$
$139$	−2.00000	−0.169638	−0.0848189	+	0.996396i	$0.527031\pi$
$140$	0	0
$141$	6.00000	0.505291
$142$	9.00000i	0.755263i
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	10.0000	0.827606
$147$	0	0
$148$	− 8.00000i	− 0.657596i
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	− 8.00000i	− 0.648886i
$153$	3.00000i	0.242536i
$154$	0	0
$155$	0	0
$156$	−4.00000	−0.320256
$157$	2.00000i	0.159617i	0.996810	+	0.0798087i	$0.0254309\pi$
$157$	2.00000i	0.159617i	−0.996810	+	0.0798087i	$0.974569\pi$
$158$	5.00000i	0.397779i
$159$	12.0000	0.951662
$160$	0	0
$161$	0	0
$162$	11.0000i	0.864242i
$163$	− 8.00000i	− 0.626608i	−0.949653	−	0.313304i	$-0.898564\pi$
$163$	− 8.00000i	− 0.626608i	0.949653	−	0.313304i	$-0.101436\pi$
$164$	3.00000	0.234261
$165$	0	0
$166$	6.00000	0.465690
$167$	24.0000i	1.85718i	0.371113	+	0.928588i	$0.378976\pi$
$167$	24.0000i	1.85718i	−0.371113	+	0.928588i	$0.621024\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	8.00000	0.611775
$172$	− 10.0000i	− 0.762493i
$173$	12.0000i	0.912343i	0.889892	+	0.456172i	$0.150780\pi$
$173$	12.0000i	0.912343i	−0.889892	+	0.456172i	$0.849220\pi$
$174$	12.0000	0.909718
$175$	0	0
$176$	0	0
$177$	− 24.0000i	− 1.80395i
$178$	3.00000i	0.224860i
$179$	−24.0000	−1.79384	−0.896922	−	0.442189i	$-0.854202\pi$
$179$	−24.0000	−1.79384	−0.896922	+	0.442189i	$0.854202\pi$
$180$	0	0
$181$	20.0000	1.48659	0.743294	−	0.668965i	$-0.233262\pi$
$181$	20.0000	1.48659	0.743294	+	0.668965i	$0.233262\pi$
$182$	0	0
$183$	− 8.00000i	− 0.591377i
$184$	−9.00000	−0.663489
$185$	0	0
$186$	10.0000	0.733236
$187$	0	0
$188$	3.00000i	0.218797i
$189$	0	0
$190$	0	0
$191$	15.0000	1.08536	0.542681	−	0.839939i	$-0.317409\pi$
$191$	15.0000	1.08536	0.542681	+	0.839939i	$0.317409\pi$
$192$	− 2.00000i	− 0.144338i
$193$	− 5.00000i	− 0.359908i	−0.983675	−	0.179954i	$-0.942405\pi$
$193$	− 5.00000i	− 0.359908i	0.983675	−	0.179954i	$-0.0575949\pi$
$194$	5.00000	0.358979
$195$	0	0
$196$	0	0
$197$	18.0000i	1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$	18.0000i	1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$	0	0
$199$	−17.0000	−1.20510	−0.602549	−	0.798082i	$-0.705848\pi$
$199$	−17.0000	−1.20510	−0.602549	+	0.798082i	$0.705848\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	0	0
$203$	0	0
$204$	−6.00000	−0.420084
$205$	0	0
$206$	−11.0000	−0.766406
$207$	− 9.00000i	− 0.625543i
$208$	− 2.00000i	− 0.138675i
$209$	0	0
$210$	0	0
$211$	2.00000	0.137686	0.0688428	−	0.997628i	$-0.478069\pi$
$211$	2.00000	0.137686	0.0688428	+	0.997628i	$0.478069\pi$
$212$	6.00000i	0.412082i
$213$	− 18.0000i	− 1.23334i
$214$	12.0000	0.820303
$215$	0	0
$216$	−4.00000	−0.272166
$217$	0	0
$218$	− 10.0000i	− 0.677285i
$219$	−20.0000	−1.35147
$220$	0	0
$221$	−6.00000	−0.403604
$222$	16.0000i	1.07385i
$223$	− 23.0000i	− 1.54019i	−0.637927	−	0.770097i	$-0.720208\pi$
$223$	− 23.0000i	− 1.54019i	0.637927	−	0.770097i	$-0.279792\pi$
$224$	0	0
$225$	0	0
$226$	−15.0000	−0.997785
$227$	18.0000i	1.19470i	0.801980	+	0.597351i	$0.203780\pi$
$227$	18.0000i	1.19470i	−0.801980	+	0.597351i	$0.796220\pi$
$228$	16.0000i	1.05963i
$229$	−20.0000	−1.32164	−0.660819	−	0.750546i	$-0.729791\pi$
$229$	−20.0000	−1.32164	−0.660819	+	0.750546i	$0.729791\pi$
$230$	0	0
$231$	0	0
$232$	6.00000i	0.393919i
$233$	6.00000i	0.393073i	0.980497	+	0.196537i	$0.0629694\pi$
$233$	6.00000i	0.393073i	−0.980497	+	0.196537i	$0.937031\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	12.0000	0.781133
$237$	− 10.0000i	− 0.649570i
$238$	0	0
$239$	−15.0000	−0.970269	−0.485135	−	0.874439i	$-0.661229\pi$
$239$	−15.0000	−0.970269	−0.485135	+	0.874439i	$0.661229\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	11.0000i	0.707107i
$243$	− 10.0000i	− 0.641500i
$244$	4.00000	0.256074
$245$	0	0
$246$	−6.00000	−0.382546
$247$	16.0000i	1.01806i
$248$	5.00000i	0.317500i
$249$	−12.0000	−0.760469
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	18.0000i	1.12281i	0.827541	+	0.561405i	$0.189739\pi$
$257$	18.0000i	1.12281i	−0.827541	+	0.561405i	$0.810261\pi$
$258$	20.0000i	1.24515i
$259$	0	0
$260$	0	0
$261$	−6.00000	−0.371391
$262$	0	0
$263$	− 9.00000i	− 0.554964i	−0.960731	−	0.277482i	$-0.910500\pi$
$263$	− 9.00000i	− 0.554964i	0.960731	−	0.277482i	$-0.0894999\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 6.00000i	− 0.367194i
$268$	− 2.00000i	− 0.122169i
$269$	30.0000	1.82913	0.914566	−	0.404436i	$-0.132532\pi$
$269$	30.0000	1.82913	0.914566	+	0.404436i	$0.132532\pi$
$270$	0	0
$271$	11.0000	0.668202	0.334101	−	0.942537i	$-0.391567\pi$
$271$	11.0000	0.668202	0.334101	+	0.942537i	$0.391567\pi$
$272$	− 3.00000i	− 0.181902i
$273$	0	0
$274$	9.00000	0.543710
$275$	0	0
$276$	18.0000	1.08347
$277$	− 4.00000i	− 0.240337i	−0.992754	−	0.120168i	$-0.961657\pi$
$277$	− 4.00000i	− 0.240337i	0.992754	−	0.120168i	$-0.0383434\pi$
$278$	2.00000i	0.119952i
$279$	−5.00000	−0.299342
$280$	0	0
$281$	3.00000	0.178965	0.0894825	−	0.995988i	$-0.471479\pi$
$281$	3.00000	0.178965	0.0894825	+	0.995988i	$0.471479\pi$
$282$	− 6.00000i	− 0.357295i
$283$	− 2.00000i	− 0.118888i	−0.998232	−	0.0594438i	$-0.981067\pi$
$283$	− 2.00000i	− 0.118888i	0.998232	−	0.0594438i	$-0.0189327\pi$
$284$	9.00000	0.534052
$285$	0	0
$286$	0	0
$287$	0	0
$288$	1.00000i	0.0589256i
$289$	8.00000	0.470588
$290$	0	0
$291$	−10.0000	−0.586210
$292$	− 10.0000i	− 0.585206i
$293$	− 6.00000i	− 0.350524i	−0.984522	−	0.175262i	$-0.943923\pi$
$293$	− 6.00000i	− 0.350524i	0.984522	−	0.175262i	$-0.0560772\pi$
$294$	0	0
$295$	0	0
$296$	−8.00000	−0.464991
$297$	0	0
$298$	0	0
$299$	18.0000	1.04097
$300$	0	0
$301$	0	0
$302$	4.00000i	0.230174i
$303$	0	0
$304$	−8.00000	−0.458831
$305$	0	0
$306$	3.00000	0.171499
$307$	− 16.0000i	− 0.913168i	−0.889680	−	0.456584i	$-0.849073\pi$
$307$	− 16.0000i	− 0.913168i	0.889680	−	0.456584i	$-0.150927\pi$
$308$	0	0
$309$	22.0000	1.25154
$310$	0	0
$311$	21.0000	1.19080	0.595400	−	0.803429i	$-0.296993\pi$
$311$	21.0000	1.19080	0.595400	+	0.803429i	$0.296993\pi$
$312$	4.00000i	0.226455i
$313$	− 17.0000i	− 0.960897i	−0.877023	−	0.480448i	$-0.840474\pi$
$313$	− 17.0000i	− 0.960897i	0.877023	−	0.480448i	$-0.159526\pi$
$314$	2.00000	0.112867
$315$	0	0
$316$	5.00000	0.281272
$317$	− 12.0000i	− 0.673987i	−0.941507	−	0.336994i	$-0.890590\pi$
$317$	− 12.0000i	− 0.673987i	0.941507	−	0.336994i	$-0.109410\pi$
$318$	− 12.0000i	− 0.672927i
$319$	0	0
$320$	0	0
$321$	−24.0000	−1.33955
$322$	0	0
$323$	24.0000i	1.33540i
$324$	11.0000	0.611111
$325$	0	0
$326$	−8.00000	−0.443079
$327$	20.0000i	1.10600i
$328$	− 3.00000i	− 0.165647i
$329$	0	0
$330$	0	0
$331$	14.0000	0.769510	0.384755	−	0.923019i	$-0.374286\pi$
$331$	14.0000	0.769510	0.384755	+	0.923019i	$0.374286\pi$
$332$	− 6.00000i	− 0.329293i
$333$	− 8.00000i	− 0.438397i
$334$	24.0000	1.31322
$335$	0	0
$336$	0	0
$337$	− 13.0000i	− 0.708155i	−0.935216	−	0.354078i	$-0.884795\pi$
$337$	− 13.0000i	− 0.708155i	0.935216	−	0.354078i	$-0.115205\pi$
$338$	− 9.00000i	− 0.489535i
$339$	30.0000	1.62938
$340$	0	0
$341$	0	0
$342$	− 8.00000i	− 0.432590i
$343$	0	0
$344$	−10.0000	−0.539164
$345$	0	0
$346$	12.0000	0.645124
$347$	− 6.00000i	− 0.322097i	−0.986947	−	0.161048i	$-0.948512\pi$
$347$	− 6.00000i	− 0.322097i	0.986947	−	0.161048i	$-0.0514875\pi$
$348$	− 12.0000i	− 0.643268i
$349$	−26.0000	−1.39175	−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000	−1.39175	−0.695874	+	0.718164i	$0.744983\pi$
$350$	0	0
$351$	8.00000	0.427008
$352$	0	0
$353$	9.00000i	0.479022i	0.970894	+	0.239511i	$0.0769871\pi$
$353$	9.00000i	0.479022i	−0.970894	+	0.239511i	$0.923013\pi$
$354$	−24.0000	−1.27559
$355$	0	0
$356$	3.00000	0.159000
$357$	0	0
$358$	24.0000i	1.26844i
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	− 20.0000i	− 1.05118i
$363$	− 22.0000i	− 1.15470i
$364$	0	0
$365$	0	0
$366$	−8.00000	−0.418167
$367$	8.00000i	0.417597i	0.977959	+	0.208798i	$0.0669552\pi$
$367$	8.00000i	0.417597i	−0.977959	+	0.208798i	$0.933045\pi$
$368$	9.00000i	0.469157i
$369$	3.00000	0.156174
$370$	0	0
$371$	0	0
$372$	− 10.0000i	− 0.518476i
$373$	− 2.00000i	− 0.103556i	−0.998659	−	0.0517780i	$-0.983511\pi$
$373$	− 2.00000i	− 0.103556i	0.998659	−	0.0517780i	$-0.0164888\pi$
$374$	0	0
$375$	0	0
$376$	3.00000	0.154713
$377$	− 12.0000i	− 0.618031i
$378$	0	0
$379$	−8.00000	−0.410932	−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000	−0.410932	−0.205466	+	0.978664i	$0.565871\pi$
$380$	0	0
$381$	−16.0000	−0.819705
$382$	− 15.0000i	− 0.767467i
$383$	− 9.00000i	− 0.459879i	−0.973205	−	0.229939i	$-0.926147\pi$
$383$	− 9.00000i	− 0.459879i	0.973205	−	0.229939i	$-0.0738528\pi$
$384$	−2.00000	−0.102062
$385$	0	0
$386$	−5.00000	−0.254493
$387$	− 10.0000i	− 0.508329i
$388$	− 5.00000i	− 0.253837i
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	0	0
$391$	27.0000	1.36545
$392$	0	0
$393$	0	0
$394$	18.0000	0.906827
$395$	0	0
$396$	0	0
$397$	2.00000i	0.100377i	0.998740	+	0.0501886i	$0.0159822\pi$
$397$	2.00000i	0.100377i	−0.998740	+	0.0501886i	$0.984018\pi$
$398$	17.0000i	0.852133i
$399$	0	0
$400$	0	0
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	4.00000i	0.199502i
$403$	− 10.0000i	− 0.498135i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	6.00000i	0.297044i
$409$	7.00000	0.346128	0.173064	−	0.984911i	$-0.444633\pi$
$409$	7.00000	0.346128	0.173064	+	0.984911i	$0.444633\pi$
$410$	0	0
$411$	−18.0000	−0.887875
$412$	11.0000i	0.541931i
$413$	0	0
$414$	−9.00000	−0.442326
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	− 4.00000i	− 0.195881i
$418$	0	0
$419$	30.0000	1.46560	0.732798	−	0.680446i	$-0.238214\pi$
$419$	30.0000	1.46560	0.732798	+	0.680446i	$0.238214\pi$
$420$	0	0
$421$	−4.00000	−0.194948	−0.0974740	−	0.995238i	$-0.531076\pi$
$421$	−4.00000	−0.194948	−0.0974740	+	0.995238i	$0.531076\pi$
$422$	− 2.00000i	− 0.0973585i
$423$	3.00000i	0.145865i
$424$	6.00000	0.291386
$425$	0	0
$426$	−18.0000	−0.872103
$427$	0	0
$428$	− 12.0000i	− 0.580042i
$429$	0	0
$430$	0	0
$431$	−3.00000	−0.144505	−0.0722525	−	0.997386i	$-0.523019\pi$
$431$	−3.00000	−0.144505	−0.0722525	+	0.997386i	$0.523019\pi$
$432$	4.00000i	0.192450i
$433$	− 29.0000i	− 1.39365i	−0.717241	−	0.696826i	$-0.754595\pi$
$433$	− 29.0000i	− 1.39365i	0.717241	−	0.696826i	$-0.245405\pi$
$434$	0	0
$435$	0	0
$436$	−10.0000	−0.478913
$437$	− 72.0000i	− 3.44423i
$438$	20.0000i	0.955637i
$439$	−23.0000	−1.09773	−0.548865	−	0.835911i	$-0.684940\pi$
$439$	−23.0000	−1.09773	−0.548865	+	0.835911i	$0.684940\pi$
$440$	0	0
$441$	0	0
$442$	6.00000i	0.285391i
$443$	12.0000i	0.570137i	0.958507	+	0.285069i	$0.0920164\pi$
$443$	12.0000i	0.570137i	−0.958507	+	0.285069i	$0.907984\pi$
$444$	16.0000	0.759326
$445$	0	0
$446$	−23.0000	−1.08908
$447$	0	0
$448$	0	0
$449$	27.0000	1.27421	0.637104	−	0.770778i	$-0.280132\pi$
$449$	27.0000	1.27421	0.637104	+	0.770778i	$0.280132\pi$
$450$	0	0
$451$	0	0
$452$	15.0000i	0.705541i
$453$	− 8.00000i	− 0.375873i
$454$	18.0000	0.844782
$455$	0	0
$456$	16.0000	0.749269
$457$	− 22.0000i	− 1.02912i	−0.857455	−	0.514558i	$-0.827956\pi$
$457$	− 22.0000i	− 1.02912i	0.857455	−	0.514558i	$-0.172044\pi$
$458$	20.0000i	0.934539i
$459$	12.0000	0.560112
$460$	0	0
$461$	−24.0000	−1.11779	−0.558896	−	0.829238i	$-0.688775\pi$
$461$	−24.0000	−1.11779	−0.558896	+	0.829238i	$0.688775\pi$
$462$	0	0
$463$	25.0000i	1.16185i	0.813958	+	0.580924i	$0.197309\pi$
$463$	25.0000i	1.16185i	−0.813958	+	0.580924i	$0.802691\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	6.00000	0.277945
$467$	− 6.00000i	− 0.277647i	−0.990317	−	0.138823i	$-0.955668\pi$
$467$	− 6.00000i	− 0.277647i	0.990317	−	0.138823i	$-0.0443321\pi$
$468$	− 2.00000i	− 0.0924500i
$469$	0	0
$470$	0	0
$471$	−4.00000	−0.184310
$472$	− 12.0000i	− 0.552345i
$473$	0	0
$474$	−10.0000	−0.459315
$475$	0	0
$476$	0	0
$477$	6.00000i	0.274721i
$478$	15.0000i	0.686084i
$479$	27.0000	1.23366	0.616831	−	0.787096i	$-0.288416\pi$
$479$	27.0000	1.23366	0.616831	+	0.787096i	$0.288416\pi$
$480$	0	0
$481$	16.0000	0.729537
$482$	22.0000i	1.00207i
$483$	0	0
$484$	11.0000	0.500000
$485$	0	0
$486$	−10.0000	−0.453609
$487$	11.0000i	0.498458i	0.968445	+	0.249229i	$0.0801771\pi$
$487$	11.0000i	0.498458i	−0.968445	+	0.249229i	$0.919823\pi$
$488$	− 4.00000i	− 0.181071i
$489$	16.0000	0.723545
$490$	0	0
$491$	42.0000	1.89543	0.947717	−	0.319113i	$-0.103385\pi$
$491$	42.0000	1.89543	0.947717	+	0.319113i	$0.103385\pi$
$492$	6.00000i	0.270501i
$493$	− 18.0000i	− 0.810679i
$494$	16.0000	0.719874
$495$	0	0
$496$	5.00000	0.224507
$497$	0	0
$498$	12.0000i	0.537733i
$499$	34.0000	1.52205	0.761025	−	0.648723i	$-0.224697\pi$
$499$	34.0000	1.52205	0.761025	+	0.648723i	$0.224697\pi$
$500$	0	0
$501$	−48.0000	−2.14448
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	18.0000i	0.799408i
$508$	− 8.00000i	− 0.354943i
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	− 1.00000i	− 0.0441942i
$513$	− 32.0000i	− 1.41283i
$514$	18.0000	0.793946
$515$	0	0
$516$	20.0000	0.880451
$517$	0	0
$518$	0	0
$519$	−24.0000	−1.05348
$520$	0	0
$521$	−3.00000	−0.131432	−0.0657162	−	0.997838i	$-0.520933\pi$
$521$	−3.00000	−0.131432	−0.0657162	+	0.997838i	$0.520933\pi$
$522$	6.00000i	0.262613i
$523$	34.0000i	1.48672i	0.668894	+	0.743358i	$0.266768\pi$
$523$	34.0000i	1.48672i	−0.668894	+	0.743358i	$0.733232\pi$
$524$	0	0
$525$	0	0
$526$	−9.00000	−0.392419
$527$	− 15.0000i	− 0.653410i
$528$	0	0
$529$	−58.0000	−2.52174
$530$	0	0
$531$	12.0000	0.520756
$532$	0	0
$533$	6.00000i	0.259889i
$534$	−6.00000	−0.259645
$535$	0	0
$536$	−2.00000	−0.0863868
$537$	− 48.0000i	− 2.07135i
$538$	− 30.0000i	− 1.29339i
$539$	0	0
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	− 11.0000i	− 0.472490i
$543$	40.0000i	1.71656i
$544$	−3.00000	−0.128624
$545$	0	0
$546$	0	0
$547$	8.00000i	0.342055i	0.985266	+	0.171028i	$0.0547087\pi$
$547$	8.00000i	0.342055i	−0.985266	+	0.171028i	$0.945291\pi$
$548$	− 9.00000i	− 0.384461i
$549$	4.00000	0.170716
$550$	0	0
$551$	−48.0000	−2.04487
$552$	− 18.0000i	− 0.766131i
$553$	0	0
$554$	−4.00000	−0.169944
$555$	0	0
$556$	2.00000	0.0848189
$557$	24.0000i	1.01691i	0.861088	+	0.508456i	$0.169784\pi$
$557$	24.0000i	1.01691i	−0.861088	+	0.508456i	$0.830216\pi$
$558$	5.00000i	0.211667i
$559$	20.0000	0.845910
$560$	0	0
$561$	0	0
$562$	− 3.00000i	− 0.126547i
$563$	6.00000i	0.252870i	0.991975	+	0.126435i	$0.0403535\pi$
$563$	6.00000i	0.252870i	−0.991975	+	0.126435i	$0.959647\pi$
$564$	−6.00000	−0.252646
$565$	0	0
$566$	−2.00000	−0.0840663
$567$	0	0
$568$	− 9.00000i	− 0.377632i
$569$	3.00000	0.125767	0.0628833	−	0.998021i	$-0.479970\pi$
$569$	3.00000	0.125767	0.0628833	+	0.998021i	$0.479970\pi$
$570$	0	0
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	0	0
$573$	30.0000i	1.25327i
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 34.0000i	− 1.41544i	−0.706494	−	0.707719i	$-0.749724\pi$
$577$	− 34.0000i	− 1.41544i	0.706494	−	0.707719i	$-0.250276\pi$
$578$	− 8.00000i	− 0.332756i
$579$	10.0000	0.415586
$580$	0	0
$581$	0	0
$582$	10.0000i	0.414513i
$583$	0	0
$584$	−10.0000	−0.413803
$585$	0	0
$586$	−6.00000	−0.247858
$587$	12.0000i	0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$	12.0000i	0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$	0	0
$589$	−40.0000	−1.64817
$590$	0	0
$591$	−36.0000	−1.48084
$592$	8.00000i	0.328798i
$593$	− 3.00000i	− 0.123195i	−0.998101	−	0.0615976i	$-0.980380\pi$
$593$	− 3.00000i	− 0.123195i	0.998101	−	0.0615976i	$-0.0196196\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 34.0000i	− 1.39153i
$598$	− 18.0000i	− 0.736075i
$599$	−21.0000	−0.858037	−0.429018	−	0.903296i	$-0.641140\pi$
$599$	−21.0000	−0.858037	−0.429018	+	0.903296i	$0.641140\pi$
$600$	0	0
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	0	0
$603$	− 2.00000i	− 0.0814463i
$604$	4.00000	0.162758
$605$	0	0
$606$	0	0
$607$	− 25.0000i	− 1.01472i	−0.861735	−	0.507359i	$-0.830622\pi$
$607$	− 25.0000i	− 1.01472i	0.861735	−	0.507359i	$-0.169378\pi$
$608$	8.00000i	0.324443i
$609$	0	0
$610$	0	0
$611$	−6.00000	−0.242734
$612$	− 3.00000i	− 0.121268i
$613$	− 2.00000i	− 0.0807792i	−0.999184	−	0.0403896i	$-0.987140\pi$
$613$	− 2.00000i	− 0.0807792i	0.999184	−	0.0403896i	$-0.0128599\pi$
$614$	−16.0000	−0.645707
$615$	0	0
$616$	0	0
$617$	15.0000i	0.603877i	0.953327	+	0.301939i	$0.0976338\pi$
$617$	15.0000i	0.603877i	−0.953327	+	0.301939i	$0.902366\pi$
$618$	− 22.0000i	− 0.884970i
$619$	10.0000	0.401934	0.200967	−	0.979598i	$-0.435592\pi$
$619$	10.0000	0.401934	0.200967	+	0.979598i	$0.435592\pi$
$620$	0	0
$621$	−36.0000	−1.44463
$622$	− 21.0000i	− 0.842023i
$623$	0	0
$624$	4.00000	0.160128
$625$	0	0
$626$	−17.0000	−0.679457
$627$	0	0
$628$	− 2.00000i	− 0.0798087i
$629$	24.0000	0.956943
$630$	0	0
$631$	−19.0000	−0.756378	−0.378189	−	0.925728i	$-0.623453\pi$
$631$	−19.0000	−0.756378	−0.378189	+	0.925728i	$0.623453\pi$
$632$	− 5.00000i	− 0.198889i
$633$	4.00000i	0.158986i
$634$	−12.0000	−0.476581
$635$	0	0
$636$	−12.0000	−0.475831
$637$	0	0
$638$	0	0
$639$	9.00000	0.356034
$640$	0	0
$641$	45.0000	1.77739	0.888697	−	0.458496i	$-0.151612\pi$
$641$	45.0000	1.77739	0.888697	+	0.458496i	$0.151612\pi$
$642$	24.0000i	0.947204i
$643$	− 20.0000i	− 0.788723i	−0.918955	−	0.394362i	$-0.870966\pi$
$643$	− 20.0000i	− 0.788723i	0.918955	−	0.394362i	$-0.129034\pi$
$644$	0	0
$645$	0	0
$646$	24.0000	0.944267
$647$	24.0000i	0.943537i	0.881722	+	0.471769i	$0.156384\pi$
$647$	24.0000i	0.943537i	−0.881722	+	0.471769i	$0.843616\pi$
$648$	− 11.0000i	− 0.432121i
$649$	0	0
$650$	0	0
$651$	0	0
$652$	8.00000i	0.313304i
$653$	− 18.0000i	− 0.704394i	−0.935926	−	0.352197i	$-0.885435\pi$
$653$	− 18.0000i	− 0.704394i	0.935926	−	0.352197i	$-0.114565\pi$
$654$	20.0000	0.782062
$655$	0	0
$656$	−3.00000	−0.117130
$657$	− 10.0000i	− 0.390137i
$658$	0	0
$659$	−30.0000	−1.16863	−0.584317	−	0.811525i	$-0.698638\pi$
$659$	−30.0000	−1.16863	−0.584317	+	0.811525i	$0.698638\pi$
$660$	0	0
$661$	−16.0000	−0.622328	−0.311164	−	0.950356i	$-0.600719\pi$
$661$	−16.0000	−0.622328	−0.311164	+	0.950356i	$0.600719\pi$
$662$	− 14.0000i	− 0.544125i
$663$	− 12.0000i	− 0.466041i
$664$	−6.00000	−0.232845
$665$	0	0
$666$	−8.00000	−0.309994
$667$	54.0000i	2.09089i
$668$	− 24.0000i	− 0.928588i
$669$	46.0000	1.77846
$670$	0	0
$671$	0	0
$672$	0	0
$673$	− 5.00000i	− 0.192736i	−0.995346	−	0.0963679i	$-0.969277\pi$
$673$	− 5.00000i	− 0.192736i	0.995346	−	0.0963679i	$-0.0307225\pi$
$674$	−13.0000	−0.500741
$675$	0	0
$676$	−9.00000	−0.346154
$677$	42.0000i	1.61419i	0.590421	+	0.807096i	$0.298962\pi$
$677$	42.0000i	1.61419i	−0.590421	+	0.807096i	$0.701038\pi$
$678$	− 30.0000i	− 1.15214i
$679$	0	0
$680$	0	0
$681$	−36.0000	−1.37952
$682$	0	0
$683$	30.0000i	1.14792i	0.818884	+	0.573959i	$0.194593\pi$
$683$	30.0000i	1.14792i	−0.818884	+	0.573959i	$0.805407\pi$
$684$	−8.00000	−0.305888
$685$	0	0
$686$	0	0
$687$	− 40.0000i	− 1.52610i
$688$	10.0000i	0.381246i
$689$	−12.0000	−0.457164
$690$	0	0
$691$	50.0000	1.90209	0.951045	−	0.309053i	$-0.100012\pi$
$691$	50.0000	1.90209	0.951045	+	0.309053i	$0.100012\pi$
$692$	− 12.0000i	− 0.456172i
$693$	0	0
$694$	−6.00000	−0.227757
$695$	0	0
$696$	−12.0000	−0.454859
$697$	9.00000i	0.340899i
$698$	26.0000i	0.984115i
$699$	−12.0000	−0.453882
$700$	0	0
$701$	−36.0000	−1.35970	−0.679851	−	0.733351i	$-0.737955\pi$
$701$	−36.0000	−1.35970	−0.679851	+	0.733351i	$0.737955\pi$
$702$	− 8.00000i	− 0.301941i
$703$	− 64.0000i	− 2.41381i
$704$	0	0
$705$	0	0
$706$	9.00000	0.338719
$707$	0	0
$708$	24.0000i	0.901975i
$709$	4.00000	0.150223	0.0751116	−	0.997175i	$-0.476069\pi$
$709$	4.00000	0.150223	0.0751116	+	0.997175i	$0.476069\pi$
$710$	0	0
$711$	5.00000	0.187515
$712$	− 3.00000i	− 0.112430i
$713$	45.0000i	1.68526i
$714$	0	0
$715$	0	0
$716$	24.0000	0.896922
$717$	− 30.0000i	− 1.12037i
$718$	− 24.0000i	− 0.895672i
$719$	15.0000	0.559406	0.279703	−	0.960087i	$-0.409764\pi$
$719$	15.0000	0.559406	0.279703	+	0.960087i	$0.409764\pi$
$720$	0	0
$721$	0	0
$722$	− 45.0000i	− 1.67473i
$723$	− 44.0000i	− 1.63638i
$724$	−20.0000	−0.743294
$725$	0	0
$726$	−22.0000	−0.816497
$727$	− 31.0000i	− 1.14973i	−0.818250	−	0.574863i	$-0.805055\pi$
$727$	− 31.0000i	− 1.14973i	0.818250	−	0.574863i	$-0.194945\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	30.0000	1.10959
$732$	8.00000i	0.295689i
$733$	34.0000i	1.25582i	0.778287	+	0.627909i	$0.216089\pi$
$733$	34.0000i	1.25582i	−0.778287	+	0.627909i	$0.783911\pi$
$734$	8.00000	0.295285
$735$	0	0
$736$	9.00000	0.331744
$737$	0	0
$738$	− 3.00000i	− 0.110432i
$739$	−8.00000	−0.294285	−0.147142	−	0.989115i	$-0.547008\pi$
$739$	−8.00000	−0.294285	−0.147142	+	0.989115i	$0.547008\pi$
$740$	0	0
$741$	−32.0000	−1.17555
$742$	0	0
$743$	− 33.0000i	− 1.21065i	−0.795977	−	0.605326i	$-0.793043\pi$
$743$	− 33.0000i	− 1.21065i	0.795977	−	0.605326i	$-0.206957\pi$
$744$	−10.0000	−0.366618
$745$	0	0
$746$	−2.00000	−0.0732252
$747$	− 6.00000i	− 0.219529i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−16.0000	−0.583848	−0.291924	−	0.956441i	$-0.594295\pi$
$751$	−16.0000	−0.583848	−0.291924	+	0.956441i	$0.594295\pi$
$752$	− 3.00000i	− 0.109399i
$753$	0	0
$754$	−12.0000	−0.437014
$755$	0	0
$756$	0	0
$757$	26.0000i	0.944986i	0.881334	+	0.472493i	$0.156646\pi$
$757$	26.0000i	0.944986i	−0.881334	+	0.472493i	$0.843354\pi$
$758$	8.00000i	0.290573i
$759$	0	0
$760$	0	0
$761$	45.0000	1.63125	0.815624	−	0.578582i	$-0.196394\pi$
$761$	45.0000	1.63125	0.815624	+	0.578582i	$0.196394\pi$
$762$	16.0000i	0.579619i
$763$	0	0
$764$	−15.0000	−0.542681
$765$	0	0
$766$	−9.00000	−0.325183
$767$	24.0000i	0.866590i
$768$	2.00000i	0.0721688i
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	0	0
$771$	−36.0000	−1.29651
$772$	5.00000i	0.179954i
$773$	12.0000i	0.431610i	0.976436	+	0.215805i	$0.0692376\pi$
$773$	12.0000i	0.431610i	−0.976436	+	0.215805i	$0.930762\pi$
$774$	−10.0000	−0.359443
$775$	0	0
$776$	−5.00000	−0.179490
$777$	0	0
$778$	− 18.0000i	− 0.645331i
$779$	24.0000	0.859889
$780$	0	0
$781$	0	0
$782$	− 27.0000i	− 0.965518i
$783$	24.0000i	0.857690i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 16.0000i	− 0.570338i	−0.958477	−	0.285169i	$-0.907950\pi$
$787$	− 16.0000i	− 0.570338i	0.958477	−	0.285169i	$-0.0920498\pi$
$788$	− 18.0000i	− 0.641223i
$789$	18.0000	0.640817
$790$	0	0
$791$	0	0
$792$	0	0
$793$	8.00000i	0.284088i
$794$	2.00000	0.0709773
$795$	0	0
$796$	17.0000	0.602549
$797$	− 12.0000i	− 0.425062i	−0.977154	−	0.212531i	$-0.931829\pi$
$797$	− 12.0000i	− 0.425062i	0.977154	−	0.212531i	$-0.0681706\pi$
$798$	0	0
$799$	−9.00000	−0.318397
$800$	0	0
$801$	3.00000	0.106000
$802$	− 18.0000i	− 0.635602i
$803$	0	0
$804$	4.00000	0.141069
$805$	0	0
$806$	−10.0000	−0.352235
$807$	60.0000i	2.11210i
$808$	0	0
$809$	30.0000	1.05474	0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000	1.05474	0.527372	+	0.849635i	$0.323177\pi$
$810$	0	0
$811$	26.0000	0.912983	0.456492	−	0.889728i	$-0.349106\pi$
$811$	26.0000	0.912983	0.456492	+	0.889728i	$0.349106\pi$
$812$	0	0
$813$	22.0000i	0.771574i
$814$	0	0
$815$	0	0
$816$	6.00000	0.210042
$817$	− 80.0000i	− 2.79885i
$818$	− 7.00000i	− 0.244749i
$819$	0	0
$820$	0	0
$821$	−12.0000	−0.418803	−0.209401	−	0.977830i	$-0.567152\pi$
$821$	−12.0000	−0.418803	−0.209401	+	0.977830i	$0.567152\pi$
$822$	18.0000i	0.627822i
$823$	− 20.0000i	− 0.697156i	−0.937280	−	0.348578i	$-0.886665\pi$
$823$	− 20.0000i	− 0.697156i	0.937280	−	0.348578i	$-0.113335\pi$
$824$	11.0000	0.383203
$825$	0	0
$826$	0	0
$827$	− 18.0000i	− 0.625921i	−0.949766	−	0.312961i	$-0.898679\pi$
$827$	− 18.0000i	− 0.625921i	0.949766	−	0.312961i	$-0.101321\pi$
$828$	9.00000i	0.312772i
$829$	−44.0000	−1.52818	−0.764092	−	0.645108i	$-0.776812\pi$
$829$	−44.0000	−1.52818	−0.764092	+	0.645108i	$0.776812\pi$
$830$	0	0
$831$	8.00000	0.277517
$832$	2.00000i	0.0693375i
$833$	0	0
$834$	−4.00000	−0.138509
$835$	0	0
$836$	0	0
$837$	20.0000i	0.691301i
$838$	− 30.0000i	− 1.03633i
$839$	33.0000	1.13929	0.569643	−	0.821892i	$-0.307081\pi$
$839$	33.0000	1.13929	0.569643	+	0.821892i	$0.307081\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	4.00000i	0.137849i
$843$	6.00000i	0.206651i
$844$	−2.00000	−0.0688428
$845$	0	0
$846$	3.00000	0.103142
$847$	0	0
$848$	− 6.00000i	− 0.206041i
$849$	4.00000	0.137280
$850$	0	0
$851$	−72.0000	−2.46813
$852$	18.0000i	0.616670i
$853$	10.0000i	0.342393i	0.985237	+	0.171197i	$0.0547634\pi$
$853$	10.0000i	0.342393i	−0.985237	+	0.171197i	$0.945237\pi$
$854$	0	0
$855$	0	0
$856$	−12.0000	−0.410152
$857$	54.0000i	1.84460i	0.386469	+	0.922302i	$0.373695\pi$
$857$	54.0000i	1.84460i	−0.386469	+	0.922302i	$0.626305\pi$
$858$	0	0
$859$	40.0000	1.36478	0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000	1.36478	0.682391	+	0.730987i	$0.260940\pi$
$860$	0	0
$861$	0	0
$862$	3.00000i	0.102180i
$863$	− 21.0000i	− 0.714848i	−0.933942	−	0.357424i	$-0.883655\pi$
$863$	− 21.0000i	− 0.714848i	0.933942	−	0.357424i	$-0.116345\pi$
$864$	4.00000	0.136083
$865$	0	0
$866$	−29.0000	−0.985460
$867$	16.0000i	0.543388i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	4.00000	0.135535
$872$	10.0000i	0.338643i
$873$	− 5.00000i	− 0.169224i
$874$	−72.0000	−2.43544
$875$	0	0
$876$	20.0000	0.675737
$877$	− 16.0000i	− 0.540282i	−0.962821	−	0.270141i	$-0.912930\pi$
$877$	− 16.0000i	− 0.540282i	0.962821	−	0.270141i	$-0.0870703\pi$
$878$	23.0000i	0.776212i
$879$	12.0000	0.404750
$880$	0	0
$881$	27.0000	0.909653	0.454827	−	0.890580i	$-0.349701\pi$
$881$	27.0000	0.909653	0.454827	+	0.890580i	$0.349701\pi$
$882$	0	0
$883$	− 26.0000i	− 0.874970i	−0.899226	−	0.437485i	$-0.855869\pi$
$883$	− 26.0000i	− 0.874970i	0.899226	−	0.437485i	$-0.144131\pi$
$884$	6.00000	0.201802
$885$	0	0
$886$	12.0000	0.403148
$887$	36.0000i	1.20876i	0.796696	+	0.604381i	$0.206579\pi$
$887$	36.0000i	1.20876i	−0.796696	+	0.604381i	$0.793421\pi$
$888$	− 16.0000i	− 0.536925i
$889$	0	0
$890$	0	0
$891$	0	0
$892$	23.0000i	0.770097i
$893$	24.0000i	0.803129i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	36.0000i	1.20201i
$898$	− 27.0000i	− 0.901002i
$899$	30.0000	1.00056
$900$	0	0
$901$	−18.0000	−0.599667
$902$	0	0
$903$	0	0
$904$	15.0000	0.498893
$905$	0	0
$906$	−8.00000	−0.265782
$907$	− 40.0000i	− 1.32818i	−0.747653	−	0.664089i	$-0.768820\pi$
$907$	− 40.0000i	− 1.32818i	0.747653	−	0.664089i	$-0.231180\pi$
$908$	− 18.0000i	− 0.597351i
$909$	0	0
$910$	0	0
$911$	3.00000	0.0993944	0.0496972	−	0.998764i	$-0.484174\pi$
$911$	3.00000	0.0993944	0.0496972	+	0.998764i	$0.484174\pi$
$912$	− 16.0000i	− 0.529813i
$913$	0	0
$914$	−22.0000	−0.727695
$915$	0	0
$916$	20.0000	0.660819
$917$	0	0
$918$	− 12.0000i	− 0.396059i
$919$	−11.0000	−0.362857	−0.181428	−	0.983404i	$-0.558072\pi$
$919$	−11.0000	−0.362857	−0.181428	+	0.983404i	$0.558072\pi$
$920$	0	0
$921$	32.0000	1.05444
$922$	24.0000i	0.790398i
$923$	18.0000i	0.592477i
$924$	0	0
$925$	0	0
$926$	25.0000	0.821551
$927$	11.0000i	0.361287i
$928$	− 6.00000i	− 0.196960i
$929$	−42.0000	−1.37798	−0.688988	−	0.724773i	$-0.741945\pi$
$929$	−42.0000	−1.37798	−0.688988	+	0.724773i	$0.741945\pi$
$930$	0	0
$931$	0	0
$932$	− 6.00000i	− 0.196537i
$933$	42.0000i	1.37502i
$934$	−6.00000	−0.196326
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	− 34.0000i	− 1.11073i	−0.831606	−	0.555366i	$-0.812578\pi$
$937$	− 34.0000i	− 1.11073i	0.831606	−	0.555366i	$-0.187422\pi$
$938$	0	0
$939$	34.0000	1.10955
$940$	0	0
$941$	−6.00000	−0.195594	−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000	−0.195594	−0.0977972	+	0.995206i	$0.531180\pi$
$942$	4.00000i	0.130327i
$943$	− 27.0000i	− 0.879241i
$944$	−12.0000	−0.390567
$945$	0	0
$946$	0	0
$947$	− 12.0000i	− 0.389948i	−0.980808	−	0.194974i	$-0.937538\pi$
$947$	− 12.0000i	− 0.389948i	0.980808	−	0.194974i	$-0.0624622\pi$
$948$	10.0000i	0.324785i
$949$	20.0000	0.649227
$950$	0	0
$951$	24.0000	0.778253
$952$	0	0
$953$	− 54.0000i	− 1.74923i	−0.484817	−	0.874616i	$-0.661114\pi$
$953$	− 54.0000i	− 1.74923i	0.484817	−	0.874616i	$-0.338886\pi$
$954$	6.00000	0.194257
$955$	0	0
$956$	15.0000	0.485135
$957$	0	0
$958$	− 27.0000i	− 0.872330i
$959$	0	0
$960$	0	0
$961$	−6.00000	−0.193548
$962$	− 16.0000i	− 0.515861i
$963$	− 12.0000i	− 0.386695i
$964$	22.0000	0.708572
$965$	0	0
$966$	0	0
$967$	− 7.00000i	− 0.225105i	−0.993646	−	0.112552i	$-0.964097\pi$
$967$	− 7.00000i	− 0.225105i	0.993646	−	0.112552i	$-0.0359026\pi$
$968$	− 11.0000i	− 0.353553i
$969$	−48.0000	−1.54198
$970$	0	0
$971$	30.0000	0.962746	0.481373	−	0.876516i	$-0.340138\pi$
$971$	30.0000	0.962746	0.481373	+	0.876516i	$0.340138\pi$
$972$	10.0000i	0.320750i
$973$	0	0
$974$	11.0000	0.352463
$975$	0	0
$976$	−4.00000	−0.128037
$977$	− 51.0000i	− 1.63163i	−0.578310	−	0.815817i	$-0.696287\pi$
$977$	− 51.0000i	− 1.63163i	0.578310	−	0.815817i	$-0.303713\pi$
$978$	− 16.0000i	− 0.511624i
$979$	0	0
$980$	0	0
$981$	−10.0000	−0.319275
$982$	− 42.0000i	− 1.34027i
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	6.00000	0.191273
$985$	0	0
$986$	−18.0000	−0.573237
$987$	0	0
$988$	− 16.0000i	− 0.509028i
$989$	−90.0000	−2.86183
$990$	0	0
$991$	29.0000	0.921215	0.460608	−	0.887604i	$-0.347632\pi$
$991$	29.0000	0.921215	0.460608	+	0.887604i	$0.347632\pi$
$992$	− 5.00000i	− 0.158750i
$993$	28.0000i	0.888553i
$994$	0	0
$995$	0	0
$996$	12.0000	0.380235
$997$	− 4.00000i	− 0.126681i	−0.997992	−	0.0633406i	$-0.979825\pi$
$997$	− 4.00000i	− 0.126681i	0.997992	−	0.0633406i	$-0.0201755\pi$
$998$	− 34.0000i	− 1.07625i
$999$	−32.0000	−1.01244

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2450.2.c.o.99.1		2
5.2	odd	4		2450.2.a.be.1.1		1
5.3	odd	4		2450.2.a.e.1.1		1
5.4	even	2	inner	2450.2.c.o.99.2		2
7.2	even	3		350.2.j.e.249.1		4
7.4	even	3		350.2.j.e.149.2		4
7.6	odd	2		2450.2.c.d.99.1		2
35.2	odd	12		350.2.e.a.151.1	yes	2
35.4	even	6		350.2.j.e.149.1		4
35.9	even	6		350.2.j.e.249.2		4
35.13	even	4		2450.2.a.o.1.1		1
35.18	odd	12		350.2.e.k.51.1	yes	2
35.23	odd	12		350.2.e.k.151.1	yes	2
35.27	even	4		2450.2.a.u.1.1		1
35.32	odd	12		350.2.e.a.51.1	✓	2
35.34	odd	2		2450.2.c.d.99.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.2.e.a.51.1	✓	2	35.32	odd	12
350.2.e.a.151.1	yes	2	35.2	odd	12
350.2.e.k.51.1	yes	2	35.18	odd	12
350.2.e.k.151.1	yes	2	35.23	odd	12
350.2.j.e.149.1		4	35.4	even	6
350.2.j.e.149.2		4	7.4	even	3
350.2.j.e.249.1		4	7.2	even	3
350.2.j.e.249.2		4	35.9	even	6
2450.2.a.e.1.1		1	5.3	odd	4
2450.2.a.o.1.1		1	35.13	even	4
2450.2.a.u.1.1		1	35.27	even	4
2450.2.a.be.1.1		1	5.2	odd	4
2450.2.c.d.99.1		2	7.6	odd	2
2450.2.c.d.99.2		2	35.34	odd	2
2450.2.c.o.99.1		2	1.1	even	1	trivial
2450.2.c.o.99.2		2	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2450.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} -2.00000i q^{12} -2.00000i q^{13} +1.00000 q^{16} -3.00000i q^{17} +1.00000i q^{18} -8.00000 q^{19} +9.00000i q^{23} -2.00000 q^{24} -2.00000 q^{26} +4.00000i q^{27} +6.00000 q^{29} +5.00000 q^{31} -1.00000i q^{32} -3.00000 q^{34} +1.00000 q^{36} +8.00000i q^{37} +8.00000i q^{38} +4.00000 q^{39} -3.00000 q^{41} +10.0000i q^{43} +9.00000 q^{46} -3.00000i q^{47} +2.00000i q^{48} +6.00000 q^{51} +2.00000i q^{52} -6.00000i q^{53} +4.00000 q^{54} -16.0000i q^{57} -6.00000i q^{58} -12.0000 q^{59} -4.00000 q^{61} -5.00000i q^{62} -1.00000 q^{64} +2.00000i q^{67} +3.00000i q^{68} -18.0000 q^{69} -9.00000 q^{71} -1.00000i q^{72} +10.0000i q^{73} +8.00000 q^{74} +8.00000 q^{76} -4.00000i q^{78} -5.00000 q^{79} -11.0000 q^{81} +3.00000i q^{82} +6.00000i q^{83} +10.0000 q^{86} +12.0000i q^{87} -3.00000 q^{89} -9.00000i q^{92} +10.0000i q^{93} -3.00000 q^{94} +2.00000 q^{96} +5.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 4 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 4 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} + 4 q^{6} - 2 q^{9} + 2 q^{16} - 16 q^{19} - 4 q^{24} - 4 q^{26} + 12 q^{29} + 10 q^{31} - 6 q^{34} + 2 q^{36} + 8 q^{39} - 6 q^{41} + 18 q^{46} + 12 q^{51} + 8 q^{54} - 24 q^{59} - 8 q^{61} - 2 q^{64} - 36 q^{69} - 18 q^{71} + 16 q^{74} + 16 q^{76} - 10 q^{79} - 22 q^{81} + 20 q^{86} - 6 q^{89} - 6 q^{94} + 4 q^{96}+O(q^{100}) \) 2 * q - 2 * q^4 + 4 * q^6 - 2 * q^9 + 2 * q^16 - 16 * q^19 - 4 * q^24 - 4 * q^26 + 12 * q^29 + 10 * q^31 - 6 * q^34 + 2 * q^36 + 8 * q^39 - 6 * q^41 + 18 * q^46 + 12 * q^51 + 8 * q^54 - 24 * q^59 - 8 * q^61 - 2 * q^64 - 36 * q^69 - 18 * q^71 + 16 * q^74 + 16 * q^76 - 10 * q^79 - 22 * q^81 + 20 * q^86 - 6 * q^89 - 6 * q^94 + 4 * q^96

Introduction

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