LMFDB - Embedded newform 2925.2.c.v.2224.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2925,2,Mod(2224,2925)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2925, 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("2925.2224");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2925 = 3^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2925.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:	$23.3562425912$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 65)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2224.1
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2925.2224
Dual form			2925.2.c.v.2224.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{2} -1.00000 q^{4} -2.00000i q^{7} -1.73205i q^{8} +O(q^{10})$	$q-1.73205i q^{2} -1.00000 q^{4} -2.00000i q^{7} -1.73205i q^{8} +4.73205 q^{11} +1.00000i q^{13} -3.46410 q^{14} -5.00000 q^{16} -3.46410i q^{17} +6.19615 q^{19} -8.19615i q^{22} -1.26795i q^{23} +1.73205 q^{26} +2.00000i q^{28} -2.53590 q^{29} +10.1962 q^{31} +5.19615i q^{32} -6.00000 q^{34} +4.00000i q^{37} -10.7321i q^{38} -3.46410 q^{41} -0.196152i q^{43} -4.73205 q^{44} -2.19615 q^{46} +6.00000i q^{47} +3.00000 q^{49} -1.00000i q^{52} -10.3923i q^{53} -3.46410 q^{56} +4.39230i q^{58} +9.12436 q^{59} -8.39230 q^{61} -17.6603i q^{62} -1.00000 q^{64} -6.39230i q^{67} +3.46410i q^{68} -4.73205 q^{71} -4.00000i q^{73} +6.92820 q^{74} -6.19615 q^{76} -9.46410i q^{77} +8.39230 q^{79} +6.00000i q^{82} +6.00000i q^{83} -0.339746 q^{86} -8.19615i q^{88} -12.9282 q^{89} +2.00000 q^{91} +1.26795i q^{92} +10.3923 q^{94} -2.00000i q^{97} -5.19615i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4}+O(q^{10}) $ 4 * q - 4 * q^4	$ 4 q - 4 q^{4} + 12 q^{11} - 20 q^{16} + 4 q^{19} - 24 q^{29} + 20 q^{31} - 24 q^{34} - 12 q^{44} + 12 q^{46} + 12 q^{49} - 12 q^{59} + 8 q^{61} - 4 q^{64} - 12 q^{71} - 4 q^{76} - 8 q^{79} - 36 q^{86} - 24 q^{89} + 8 q^{91}+O(q^{100}) $ 4 * q - 4 * q^4 + 12 * q^11 - 20 * q^16 + 4 * q^19 - 24 * q^29 + 20 * q^31 - 24 * q^34 - 12 * q^44 + 12 * q^46 + 12 * q^49 - 12 * q^59 + 8 * q^61 - 4 * q^64 - 12 * q^71 - 4 * q^76 - 8 * q^79 - 36 * q^86 - 24 * q^89 + 8 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2925\mathbb{Z}\right)^\times$.

$n$	$326$	$352$	$2251$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$	− 1.73205i	− 1.22474i	−0.790569	−	0.612372i	$-0.790215\pi$
$2$	− 1.73205i	− 1.22474i	0.790569	−	0.612372i	$-0.209785\pi$
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.00000i	− 0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	− 2.00000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	− 1.73205i	− 0.612372i
$9$	0	0
$10$	0	0
$11$	4.73205	1.42677	0.713384	−	0.700774i	$-0.247162\pi$
$11$	4.73205	1.42677	0.713384	+	0.700774i	$0.247162\pi$
$12$	0	0
$13$	1.00000i	0.277350i
$13$	1.00000i	0.277350i
$14$	−3.46410	−0.925820
$15$	0	0
$16$	−5.00000	−1.25000
$17$	− 3.46410i	− 0.840168i	−0.907485	−	0.420084i	$-0.862001\pi$
$17$	− 3.46410i	− 0.840168i	0.907485	−	0.420084i	$-0.137999\pi$
$18$	0	0
$19$	6.19615	1.42149	0.710747	−	0.703447i	$-0.248357\pi$
$19$	6.19615	1.42149	0.710747	+	0.703447i	$0.248357\pi$
$20$	0	0
$21$	0	0
$22$	− 8.19615i	− 1.74743i
$23$	− 1.26795i	− 0.264386i	−0.991224	−	0.132193i	$-0.957798\pi$
$23$	− 1.26795i	− 0.264386i	0.991224	−	0.132193i	$-0.0422018\pi$
$24$	0	0
$25$	0	0
$26$	1.73205	0.339683
$27$	0	0
$28$	2.00000i	0.377964i
$29$	−2.53590	−0.470905	−0.235452	−	0.971886i	$-0.575657\pi$
$29$	−2.53590	−0.470905	−0.235452	+	0.971886i	$0.575657\pi$
$30$	0	0
$31$	10.1962	1.83128	0.915642	−	0.401996i	$-0.131683\pi$
$31$	10.1962	1.83128	0.915642	+	0.401996i	$0.131683\pi$
$32$	5.19615i	0.918559i
$33$	0	0
$34$	−6.00000	−1.02899
$35$	0	0
$36$	0	0
$37$	4.00000i	0.657596i	0.944400	+	0.328798i	$0.106644\pi$
$37$	4.00000i	0.657596i	−0.944400	+	0.328798i	$0.893356\pi$
$38$	− 10.7321i	− 1.74097i
$39$	0	0
$40$	0	0
$41$	−3.46410	−0.541002	−0.270501	−	0.962720i	$-0.587189\pi$
$41$	−3.46410	−0.541002	−0.270501	+	0.962720i	$0.587189\pi$
$42$	0	0
$43$	− 0.196152i	− 0.0299130i	−0.999888	−	0.0149565i	$-0.995239\pi$
$43$	− 0.196152i	− 0.0299130i	0.999888	−	0.0149565i	$-0.00476097\pi$
$44$	−4.73205	−0.713384
$45$	0	0
$46$	−2.19615	−0.323805
$47$	6.00000i	0.875190i	0.899172	+	0.437595i	$0.144170\pi$
$47$	6.00000i	0.875190i	−0.899172	+	0.437595i	$0.855830\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	0	0
$52$	− 1.00000i	− 0.138675i
$53$	− 10.3923i	− 1.42749i	−0.700404	−	0.713746i	$-0.746997\pi$
$53$	− 10.3923i	− 1.42749i	0.700404	−	0.713746i	$-0.253003\pi$
$54$	0	0
$55$	0	0
$56$	−3.46410	−0.462910
$57$	0	0
$58$	4.39230i	0.576738i
$59$	9.12436	1.18789	0.593945	−	0.804506i	$-0.297570\pi$
$59$	9.12436	1.18789	0.593945	+	0.804506i	$0.297570\pi$
$60$	0	0
$61$	−8.39230	−1.07452	−0.537262	−	0.843415i	$-0.680541\pi$
$61$	−8.39230	−1.07452	−0.537262	+	0.843415i	$0.680541\pi$
$62$	− 17.6603i	− 2.24285i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 6.39230i	− 0.780944i	−0.920615	−	0.390472i	$-0.872312\pi$
$67$	− 6.39230i	− 0.780944i	0.920615	−	0.390472i	$-0.127688\pi$
$68$	3.46410i	0.420084i
$69$	0	0
$70$	0	0
$71$	−4.73205	−0.561591	−0.280796	−	0.959768i	$-0.590598\pi$
$71$	−4.73205	−0.561591	−0.280796	+	0.959768i	$0.590598\pi$
$72$	0	0
$73$	− 4.00000i	− 0.468165i	−0.972217	−	0.234082i	$-0.924791\pi$
$73$	− 4.00000i	− 0.468165i	0.972217	−	0.234082i	$-0.0752085\pi$
$74$	6.92820	0.805387
$75$	0	0
$76$	−6.19615	−0.710747
$77$	− 9.46410i	− 1.07853i
$78$	0	0
$79$	8.39230	0.944208	0.472104	−	0.881543i	$-0.343495\pi$
$79$	8.39230	0.944208	0.472104	+	0.881543i	$0.343495\pi$
$80$	0	0
$81$	0	0
$82$	6.00000i	0.662589i
$83$	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
$86$	−0.339746	−0.0366357
$87$	0	0
$88$	− 8.19615i	− 0.873713i
$89$	−12.9282	−1.37039	−0.685193	−	0.728361i	$-0.740282\pi$
$89$	−12.9282	−1.37039	−0.685193	+	0.728361i	$0.740282\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	1.26795i	0.132193i
$93$	0	0
$94$	10.3923	1.07188
$95$	0	0
$96$	0	0
$97$	− 2.00000i	− 0.203069i	−0.994832	−	0.101535i	$-0.967625\pi$
$97$	− 2.00000i	− 0.203069i	0.994832	−	0.101535i	$-0.0323753\pi$
$98$	− 5.19615i	− 0.524891i
$99$	0	0
$100$	0	0
$101$	0.928203	0.0923597	0.0461798	−	0.998933i	$-0.485295\pi$
$101$	0.928203	0.0923597	0.0461798	+	0.998933i	$0.485295\pi$
$102$	0	0
$103$	− 0.196152i	− 0.0193275i	−0.999953	−	0.00966374i	$-0.996924\pi$
$103$	− 0.196152i	− 0.0193275i	0.999953	−	0.00966374i	$-0.00307611\pi$
$104$	1.73205	0.169842
$105$	0	0
$106$	−18.0000	−1.74831
$107$	17.6603i	1.70728i	0.520862	+	0.853641i	$0.325610\pi$
$107$	17.6603i	1.70728i	−0.520862	+	0.853641i	$0.674390\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	0	0
$112$	10.0000i	0.944911i
$113$	− 8.53590i	− 0.802990i	−0.915861	−	0.401495i	$-0.868491\pi$
$113$	− 8.53590i	− 0.802990i	0.915861	−	0.401495i	$-0.131509\pi$
$114$	0	0
$115$	0	0
$116$	2.53590	0.235452
$117$	0	0
$118$	− 15.8038i	− 1.45486i
$119$	−6.92820	−0.635107
$120$	0	0
$121$	11.3923	1.03566
$122$	14.5359i	1.31602i
$123$	0	0
$124$	−10.1962	−0.915642
$125$	0	0
$126$	0	0
$127$	− 16.1962i	− 1.43718i	−0.695436	−	0.718588i	$-0.744789\pi$
$127$	− 16.1962i	− 1.43718i	0.695436	−	0.718588i	$-0.255211\pi$
$128$	12.1244i	1.07165i
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	− 12.3923i	− 1.07455i
$134$	−11.0718	−0.956458
$135$	0	0
$136$	−6.00000	−0.514496
$137$	0.928203i	0.0793018i	0.999214	+	0.0396509i	$0.0126246\pi$
$137$	0.928203i	0.0793018i	−0.999214	+	0.0396509i	$0.987375\pi$
$138$	0	0
$139$	−12.3923	−1.05110	−0.525551	−	0.850762i	$-0.676141\pi$
$139$	−12.3923	−1.05110	−0.525551	+	0.850762i	$0.676141\pi$
$140$	0	0
$141$	0	0
$142$	8.19615i	0.687806i
$143$	4.73205i	0.395714i
$144$	0	0
$145$	0	0
$146$	−6.92820	−0.573382
$147$	0	0
$148$	− 4.00000i	− 0.328798i
$149$	−7.85641	−0.643622	−0.321811	−	0.946804i	$-0.604292\pi$
$149$	−7.85641	−0.643622	−0.321811	+	0.946804i	$0.604292\pi$
$150$	0	0
$151$	−1.80385	−0.146795	−0.0733975	−	0.997303i	$-0.523384\pi$
$151$	−1.80385	−0.146795	−0.0733975	+	0.997303i	$0.523384\pi$
$152$	− 10.7321i	− 0.870484i
$153$	0	0
$154$	−16.3923	−1.32093
$155$	0	0
$156$	0	0
$157$	10.0000i	0.798087i	0.916932	+	0.399043i	$0.130658\pi$
$157$	10.0000i	0.798087i	−0.916932	+	0.399043i	$0.869342\pi$
$158$	− 14.5359i	− 1.15641i
$159$	0	0
$160$	0	0
$161$	−2.53590	−0.199857
$162$	0	0
$163$	− 14.3923i	− 1.12729i	−0.826016	−	0.563646i	$-0.809398\pi$
$163$	− 14.3923i	− 1.12729i	0.826016	−	0.563646i	$-0.190602\pi$
$164$	3.46410	0.270501
$165$	0	0
$166$	10.3923	0.806599
$167$	− 0.928203i	− 0.0718265i	−0.999355	−	0.0359133i	$-0.988566\pi$
$167$	− 0.928203i	− 0.0718265i	0.999355	−	0.0359133i	$-0.0114340\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	0.196152i	0.0149565i
$173$	− 8.53590i	− 0.648972i	−0.945890	−	0.324486i	$-0.894809\pi$
$173$	− 8.53590i	− 0.648972i	0.945890	−	0.324486i	$-0.105191\pi$
$174$	0	0
$175$	0	0
$176$	−23.6603	−1.78346
$177$	0	0
$178$	22.3923i	1.67837i
$179$	−18.9282	−1.41476	−0.707380	−	0.706833i	$-0.750123\pi$
$179$	−18.9282	−1.41476	−0.707380	+	0.706833i	$0.750123\pi$
$180$	0	0
$181$	0.392305	0.0291598	0.0145799	−	0.999894i	$-0.495359\pi$
$181$	0.392305	0.0291598	0.0145799	+	0.999894i	$0.495359\pi$
$182$	− 3.46410i	− 0.256776i
$183$	0	0
$184$	−2.19615	−0.161903
$185$	0	0
$186$	0	0
$187$	− 16.3923i	− 1.19872i
$188$	− 6.00000i	− 0.437595i
$189$	0	0
$190$	0	0
$191$	5.07180	0.366982	0.183491	−	0.983021i	$-0.441260\pi$
$191$	5.07180	0.366982	0.183491	+	0.983021i	$0.441260\pi$
$192$	0	0
$193$	− 10.0000i	− 0.719816i	−0.932988	−	0.359908i	$-0.882808\pi$
$193$	− 10.0000i	− 0.719816i	0.932988	−	0.359908i	$-0.117192\pi$
$194$	−3.46410	−0.248708
$195$	0	0
$196$	−3.00000	−0.214286
$197$	− 12.9282i	− 0.921096i	−0.887635	−	0.460548i	$-0.847653\pi$
$197$	− 12.9282i	− 0.921096i	0.887635	−	0.460548i	$-0.152347\pi$
$198$	0	0
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	0	0
$201$	0	0
$202$	− 1.60770i	− 0.113117i
$203$	5.07180i	0.355970i
$204$	0	0
$205$	0	0
$206$	−0.339746	−0.0236712
$207$	0	0
$208$	− 5.00000i	− 0.346688i
$209$	29.3205	2.02814
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	10.3923i	0.713746i
$213$	0	0
$214$	30.5885	2.09098
$215$	0	0
$216$	0	0
$217$	− 20.3923i	− 1.38432i
$218$	3.46410i	0.234619i
$219$	0	0
$220$	0	0
$221$	3.46410	0.233021
$222$	0	0
$223$	2.00000i	0.133930i	0.997755	+	0.0669650i	$0.0213316\pi$
$223$	2.00000i	0.133930i	−0.997755	+	0.0669650i	$0.978668\pi$
$224$	10.3923	0.694365
$225$	0	0
$226$	−14.7846	−0.983458
$227$	− 3.46410i	− 0.229920i	−0.993370	−	0.114960i	$-0.963326\pi$
$227$	− 3.46410i	− 0.229920i	0.993370	−	0.114960i	$-0.0366741\pi$
$228$	0	0
$229$	−6.39230	−0.422415	−0.211208	−	0.977441i	$-0.567740\pi$
$229$	−6.39230	−0.422415	−0.211208	+	0.977441i	$0.567740\pi$
$230$	0	0
$231$	0	0
$232$	4.39230i	0.288369i
$233$	6.00000i	0.393073i	0.980497	+	0.196537i	$0.0629694\pi$
$233$	6.00000i	0.393073i	−0.980497	+	0.196537i	$0.937031\pi$
$234$	0	0
$235$	0	0
$236$	−9.12436	−0.593945
$237$	0	0
$238$	12.0000i	0.777844i
$239$	−14.1962	−0.918273	−0.459136	−	0.888366i	$-0.651841\pi$
$239$	−14.1962	−0.918273	−0.459136	+	0.888366i	$0.651841\pi$
$240$	0	0
$241$	−2.39230	−0.154102	−0.0770510	−	0.997027i	$-0.524550\pi$
$241$	−2.39230	−0.154102	−0.0770510	+	0.997027i	$0.524550\pi$
$242$	− 19.7321i	− 1.26842i
$243$	0	0
$244$	8.39230	0.537262
$245$	0	0
$246$	0	0
$247$	6.19615i	0.394252i
$248$	− 17.6603i	− 1.12143i
$249$	0	0
$250$	0	0
$251$	21.4641	1.35480	0.677401	−	0.735614i	$-0.263106\pi$
$251$	21.4641	1.35480	0.677401	+	0.735614i	$0.263106\pi$
$252$	0	0
$253$	− 6.00000i	− 0.377217i
$254$	−28.0526	−1.76017
$255$	0	0
$256$	19.0000	1.18750
$257$	19.8564i	1.23861i	0.785151	+	0.619304i	$0.212585\pi$
$257$	19.8564i	1.23861i	−0.785151	+	0.619304i	$0.787415\pi$
$258$	0	0
$259$	8.00000	0.497096
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 1.26795i	− 0.0781851i	−0.999236	−	0.0390925i	$-0.987553\pi$
$263$	− 1.26795i	− 0.0781851i	0.999236	−	0.0390925i	$-0.0124467\pi$
$264$	0	0
$265$	0	0
$266$	−21.4641	−1.31605
$267$	0	0
$268$	6.39230i	0.390472i
$269$	−19.8564	−1.21067	−0.605333	−	0.795972i	$-0.706960\pi$
$269$	−19.8564	−1.21067	−0.605333	+	0.795972i	$0.706960\pi$
$270$	0	0
$271$	30.9808	1.88195	0.940974	−	0.338480i	$-0.109913\pi$
$271$	30.9808	1.88195	0.940974	+	0.338480i	$0.109913\pi$
$272$	17.3205i	1.05021i
$273$	0	0
$274$	1.60770	0.0971244
$275$	0	0
$276$	0	0
$277$	26.3923i	1.58576i	0.609378	+	0.792880i	$0.291419\pi$
$277$	26.3923i	1.58576i	−0.609378	+	0.792880i	$0.708581\pi$
$278$	21.4641i	1.28733i
$279$	0	0
$280$	0	0
$281$	−22.3923	−1.33581	−0.667906	−	0.744245i	$-0.732809\pi$
$281$	−22.3923	−1.33581	−0.667906	+	0.744245i	$0.732809\pi$
$282$	0	0
$283$	32.5885i	1.93718i	0.248658	+	0.968591i	$0.420010\pi$
$283$	32.5885i	1.93718i	−0.248658	+	0.968591i	$0.579990\pi$
$284$	4.73205	0.280796
$285$	0	0
$286$	8.19615	0.484649
$287$	6.92820i	0.408959i
$288$	0	0
$289$	5.00000	0.294118
$290$	0	0
$291$	0	0
$292$	4.00000i	0.234082i
$293$	5.07180i	0.296298i	0.988965	+	0.148149i	$0.0473314\pi$
$293$	5.07180i	0.296298i	−0.988965	+	0.148149i	$0.952669\pi$
$294$	0	0
$295$	0	0
$296$	6.92820	0.402694
$297$	0	0
$298$	13.6077i	0.788273i
$299$	1.26795	0.0733274
$300$	0	0
$301$	−0.392305	−0.0226121
$302$	3.12436i	0.179786i
$303$	0	0
$304$	−30.9808	−1.77687
$305$	0	0
$306$	0	0
$307$	18.7846i	1.07209i	0.844188	+	0.536047i	$0.180083\pi$
$307$	18.7846i	1.07209i	−0.844188	+	0.536047i	$0.819917\pi$
$308$	9.46410i	0.539267i
$309$	0	0
$310$	0	0
$311$	16.3923	0.929522	0.464761	−	0.885436i	$-0.346140\pi$
$311$	16.3923	0.929522	0.464761	+	0.885436i	$0.346140\pi$
$312$	0	0
$313$	− 14.3923i	− 0.813501i	−0.913539	−	0.406751i	$-0.866662\pi$
$313$	− 14.3923i	− 0.813501i	0.913539	−	0.406751i	$-0.133338\pi$
$314$	17.3205	0.977453
$315$	0	0
$316$	−8.39230	−0.472104
$317$	24.0000i	1.34797i	0.738743	+	0.673987i	$0.235420\pi$
$317$	24.0000i	1.34797i	−0.738743	+	0.673987i	$0.764580\pi$
$318$	0	0
$319$	−12.0000	−0.671871
$320$	0	0
$321$	0	0
$322$	4.39230i	0.244774i
$323$	− 21.4641i	− 1.19429i
$324$	0	0
$325$	0	0
$326$	−24.9282	−1.38065
$327$	0	0
$328$	6.00000i	0.331295i
$329$	12.0000	0.661581
$330$	0	0
$331$	2.58846	0.142274	0.0711372	−	0.997467i	$-0.477337\pi$
$331$	2.58846	0.142274	0.0711372	+	0.997467i	$0.477337\pi$
$332$	− 6.00000i	− 0.329293i
$333$	0	0
$334$	−1.60770	−0.0879692
$335$	0	0
$336$	0	0
$337$	26.3923i	1.43768i	0.695175	+	0.718840i	$0.255327\pi$
$337$	26.3923i	1.43768i	−0.695175	+	0.718840i	$0.744673\pi$
$338$	1.73205i	0.0942111i
$339$	0	0
$340$	0	0
$341$	48.2487	2.61281
$342$	0	0
$343$	− 20.0000i	− 1.07990i
$344$	−0.339746	−0.0183179
$345$	0	0
$346$	−14.7846	−0.794826
$347$	− 5.66025i	− 0.303858i	−0.988391	−	0.151929i	$-0.951451\pi$
$347$	− 5.66025i	− 0.303858i	0.988391	−	0.151929i	$-0.0485486\pi$
$348$	0	0
$349$	14.3923	0.770402	0.385201	−	0.922833i	$-0.374132\pi$
$349$	14.3923	0.770402	0.385201	+	0.922833i	$0.374132\pi$
$350$	0	0
$351$	0	0
$352$	24.5885i	1.31057i
$353$	27.7128i	1.47500i	0.675345	+	0.737502i	$0.263995\pi$
$353$	27.7128i	1.47500i	−0.675345	+	0.737502i	$0.736005\pi$
$354$	0	0
$355$	0	0
$356$	12.9282	0.685193
$357$	0	0
$358$	32.7846i	1.73272i
$359$	−2.19615	−0.115908	−0.0579542	−	0.998319i	$-0.518458\pi$
$359$	−2.19615	−0.115908	−0.0579542	+	0.998319i	$0.518458\pi$
$360$	0	0
$361$	19.3923	1.02065
$362$	− 0.679492i	− 0.0357133i
$363$	0	0
$364$	−2.00000	−0.104828
$365$	0	0
$366$	0	0
$367$	− 11.8038i	− 0.616156i	−0.951361	−	0.308078i	$-0.900314\pi$
$367$	− 11.8038i	− 0.616156i	0.951361	−	0.308078i	$-0.0996857\pi$
$368$	6.33975i	0.330482i
$369$	0	0
$370$	0	0
$371$	−20.7846	−1.07908
$372$	0	0
$373$	− 10.0000i	− 0.517780i	−0.965907	−	0.258890i	$-0.916643\pi$
$373$	− 10.0000i	− 0.517780i	0.965907	−	0.258890i	$-0.0833568\pi$
$374$	−28.3923	−1.46813
$375$	0	0
$376$	10.3923	0.535942
$377$	− 2.53590i	− 0.130605i
$378$	0	0
$379$	−18.9808	−0.974976	−0.487488	−	0.873130i	$-0.662087\pi$
$379$	−18.9808	−0.974976	−0.487488	+	0.873130i	$0.662087\pi$
$380$	0	0
$381$	0	0
$382$	− 8.78461i	− 0.449460i
$383$	− 12.9282i	− 0.660600i	−0.943876	−	0.330300i	$-0.892850\pi$
$383$	− 12.9282i	− 0.660600i	0.943876	−	0.330300i	$-0.107150\pi$
$384$	0	0
$385$	0	0
$386$	−17.3205	−0.881591
$387$	0	0
$388$	2.00000i	0.101535i
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	−4.39230	−0.222128
$392$	− 5.19615i	− 0.262445i
$393$	0	0
$394$	−22.3923	−1.12811
$395$	0	0
$396$	0	0
$397$	− 28.7846i	− 1.44466i	−0.691549	−	0.722329i	$-0.743072\pi$
$397$	− 28.7846i	− 1.44466i	0.691549	−	0.722329i	$-0.256928\pi$
$398$	34.6410i	1.73640i
$399$	0	0
$400$	0	0
$401$	36.9282	1.84411	0.922053	−	0.387063i	$-0.126510\pi$
$401$	36.9282	1.84411	0.922053	+	0.387063i	$0.126510\pi$
$402$	0	0
$403$	10.1962i	0.507907i
$404$	−0.928203	−0.0461798
$405$	0	0
$406$	8.78461	0.435973
$407$	18.9282i	0.938236i
$408$	0	0
$409$	17.6077	0.870644	0.435322	−	0.900275i	$-0.356634\pi$
$409$	17.6077	0.870644	0.435322	+	0.900275i	$0.356634\pi$
$410$	0	0
$411$	0	0
$412$	0.196152i	0.00966374i
$413$	− 18.2487i	− 0.897960i
$414$	0	0
$415$	0	0
$416$	−5.19615	−0.254762
$417$	0	0
$418$	− 50.7846i	− 2.48396i
$419$	−2.53590	−0.123887	−0.0619434	−	0.998080i	$-0.519730\pi$
$419$	−2.53590	−0.123887	−0.0619434	+	0.998080i	$0.519730\pi$
$420$	0	0
$421$	−30.7846	−1.50035	−0.750175	−	0.661239i	$-0.770031\pi$
$421$	−30.7846	−1.50035	−0.750175	+	0.661239i	$0.770031\pi$
$422$	− 13.8564i	− 0.674519i
$423$	0	0
$424$	−18.0000	−0.874157
$425$	0	0
$426$	0	0
$427$	16.7846i	0.812264i
$428$	− 17.6603i	− 0.853641i
$429$	0	0
$430$	0	0
$431$	25.5167	1.22909	0.614547	−	0.788880i	$-0.289339\pi$
$431$	25.5167	1.22909	0.614547	+	0.788880i	$0.289339\pi$
$432$	0	0
$433$	34.7846i	1.67164i	0.549002	+	0.835821i	$0.315008\pi$
$433$	34.7846i	1.67164i	−0.549002	+	0.835821i	$0.684992\pi$
$434$	−35.3205	−1.69544
$435$	0	0
$436$	2.00000	0.0957826
$437$	− 7.85641i	− 0.375823i
$438$	0	0
$439$	−32.0000	−1.52728	−0.763638	−	0.645644i	$-0.776589\pi$
$439$	−32.0000	−1.52728	−0.763638	+	0.645644i	$0.776589\pi$
$440$	0	0
$441$	0	0
$442$	− 6.00000i	− 0.285391i
$443$	16.9808i	0.806780i	0.915028	+	0.403390i	$0.132168\pi$
$443$	16.9808i	0.806780i	−0.915028	+	0.403390i	$0.867832\pi$
$444$	0	0
$445$	0	0
$446$	3.46410	0.164030
$447$	0	0
$448$	2.00000i	0.0944911i
$449$	20.5359	0.969149	0.484574	−	0.874750i	$-0.338974\pi$
$449$	20.5359	0.969149	0.484574	+	0.874750i	$0.338974\pi$
$450$	0	0
$451$	−16.3923	−0.771883
$452$	8.53590i	0.401495i
$453$	0	0
$454$	−6.00000	−0.281594
$455$	0	0
$456$	0	0
$457$	− 10.7846i	− 0.504483i	−0.967664	−	0.252241i	$-0.918832\pi$
$457$	− 10.7846i	− 0.504483i	0.967664	−	0.252241i	$-0.0811677\pi$
$458$	11.0718i	0.517351i
$459$	0	0
$460$	0	0
$461$	3.46410	0.161339	0.0806696	−	0.996741i	$-0.474294\pi$
$461$	3.46410	0.161339	0.0806696	+	0.996741i	$0.474294\pi$
$462$	0	0
$463$	− 2.39230i	− 0.111180i	−0.998454	−	0.0555899i	$-0.982296\pi$
$463$	− 2.39230i	− 0.111180i	0.998454	−	0.0555899i	$-0.0177039\pi$
$464$	12.6795	0.588631
$465$	0	0
$466$	10.3923	0.481414
$467$	27.8038i	1.28661i	0.765611	+	0.643304i	$0.222437\pi$
$467$	27.8038i	1.28661i	−0.765611	+	0.643304i	$0.777563\pi$
$468$	0	0
$469$	−12.7846	−0.590338
$470$	0	0
$471$	0	0
$472$	− 15.8038i	− 0.727431i
$473$	− 0.928203i	− 0.0426788i
$474$	0	0
$475$	0	0
$476$	6.92820	0.317554
$477$	0	0
$478$	24.5885i	1.12465i
$479$	35.6603	1.62936	0.814679	−	0.579912i	$-0.196913\pi$
$479$	35.6603	1.62936	0.814679	+	0.579912i	$0.196913\pi$
$480$	0	0
$481$	−4.00000	−0.182384
$482$	4.14359i	0.188736i
$483$	0	0
$484$	−11.3923	−0.517832
$485$	0	0
$486$	0	0
$487$	26.3923i	1.19595i	0.801515	+	0.597975i	$0.204028\pi$
$487$	26.3923i	1.19595i	−0.801515	+	0.597975i	$0.795972\pi$
$488$	14.5359i	0.658009i
$489$	0	0
$490$	0	0
$491$	2.53590	0.114443	0.0572217	−	0.998361i	$-0.481776\pi$
$491$	2.53590	0.114443	0.0572217	+	0.998361i	$0.481776\pi$
$492$	0	0
$493$	8.78461i	0.395639i
$494$	10.7321	0.482858
$495$	0	0
$496$	−50.9808	−2.28910
$497$	9.46410i	0.424523i
$498$	0	0
$499$	38.9808	1.74502	0.872509	−	0.488598i	$-0.162491\pi$
$499$	38.9808	1.74502	0.872509	+	0.488598i	$0.162491\pi$
$500$	0	0
$501$	0	0
$502$	− 37.1769i	− 1.65929i
$503$	− 19.5167i	− 0.870205i	−0.900381	−	0.435102i	$-0.856712\pi$
$503$	− 19.5167i	− 0.870205i	0.900381	−	0.435102i	$-0.143288\pi$
$504$	0	0
$505$	0	0
$506$	−10.3923	−0.461994
$507$	0	0
$508$	16.1962i	0.718588i
$509$	39.4641	1.74922	0.874608	−	0.484831i	$-0.161119\pi$
$509$	39.4641	1.74922	0.874608	+	0.484831i	$0.161119\pi$
$510$	0	0
$511$	−8.00000	−0.353899
$512$	− 8.66025i	− 0.382733i
$513$	0	0
$514$	34.3923	1.51698
$515$	0	0
$516$	0	0
$517$	28.3923i	1.24869i
$518$	− 13.8564i	− 0.608816i
$519$	0	0
$520$	0	0
$521$	28.3923	1.24389	0.621945	−	0.783061i	$-0.286343\pi$
$521$	28.3923	1.24389	0.621945	+	0.783061i	$0.286343\pi$
$522$	0	0
$523$	− 24.1962i	− 1.05802i	−0.848614	−	0.529012i	$-0.822563\pi$
$523$	− 24.1962i	− 1.05802i	0.848614	−	0.529012i	$-0.177437\pi$
$524$	0	0
$525$	0	0
$526$	−2.19615	−0.0957568
$527$	− 35.3205i	− 1.53859i
$528$	0	0
$529$	21.3923	0.930100
$530$	0	0
$531$	0	0
$532$	12.3923i	0.537275i
$533$	− 3.46410i	− 0.150047i
$534$	0	0
$535$	0	0
$536$	−11.0718	−0.478229
$537$	0	0
$538$	34.3923i	1.48276i
$539$	14.1962	0.611472
$540$	0	0
$541$	−26.3923	−1.13469	−0.567347	−	0.823479i	$-0.692030\pi$
$541$	−26.3923	−1.13469	−0.567347	+	0.823479i	$0.692030\pi$
$542$	− 53.6603i	− 2.30491i
$543$	0	0
$544$	18.0000	0.771744
$545$	0	0
$546$	0	0
$547$	12.1962i	0.521470i	0.965410	+	0.260735i	$0.0839649\pi$
$547$	12.1962i	0.521470i	−0.965410	+	0.260735i	$0.916035\pi$
$548$	− 0.928203i	− 0.0396509i
$549$	0	0
$550$	0	0
$551$	−15.7128	−0.669388
$552$	0	0
$553$	− 16.7846i	− 0.713754i
$554$	45.7128	1.94215
$555$	0	0
$556$	12.3923	0.525551
$557$	1.85641i	0.0786585i	0.999226	+	0.0393292i	$0.0125221\pi$
$557$	1.85641i	0.0786585i	−0.999226	+	0.0393292i	$0.987478\pi$
$558$	0	0
$559$	0.196152	0.00829636
$560$	0	0
$561$	0	0
$562$	38.7846i	1.63603i
$563$	22.0526i	0.929405i	0.885467	+	0.464702i	$0.153839\pi$
$563$	22.0526i	0.929405i	−0.885467	+	0.464702i	$0.846161\pi$
$564$	0	0
$565$	0	0
$566$	56.4449	2.37255
$567$	0	0
$568$	8.19615i	0.343903i
$569$	−2.53590	−0.106310	−0.0531552	−	0.998586i	$-0.516928\pi$
$569$	−2.53590	−0.106310	−0.0531552	+	0.998586i	$0.516928\pi$
$570$	0	0
$571$	36.3923	1.52297	0.761485	−	0.648182i	$-0.224470\pi$
$571$	36.3923	1.52297	0.761485	+	0.648182i	$0.224470\pi$
$572$	− 4.73205i	− 0.197857i
$573$	0	0
$574$	12.0000	0.500870
$575$	0	0
$576$	0	0
$577$	4.00000i	0.166522i	0.996528	+	0.0832611i	$0.0265335\pi$
$577$	4.00000i	0.166522i	−0.996528	+	0.0832611i	$0.973466\pi$
$578$	− 8.66025i	− 0.360219i
$579$	0	0
$580$	0	0
$581$	12.0000	0.497844
$582$	0	0
$583$	− 49.1769i	− 2.03670i
$584$	−6.92820	−0.286691
$585$	0	0
$586$	8.78461	0.362889
$587$	− 8.53590i	− 0.352314i	−0.984362	−	0.176157i	$-0.943633\pi$
$587$	− 8.53590i	− 0.352314i	0.984362	−	0.176157i	$-0.0563667\pi$
$588$	0	0
$589$	63.1769	2.60316
$590$	0	0
$591$	0	0
$592$	− 20.0000i	− 0.821995i
$593$	− 26.7846i	− 1.09991i	−0.835194	−	0.549956i	$-0.814644\pi$
$593$	− 26.7846i	− 1.09991i	0.835194	−	0.549956i	$-0.185356\pi$
$594$	0	0
$595$	0	0
$596$	7.85641	0.321811
$597$	0	0
$598$	− 2.19615i	− 0.0898074i
$599$	−7.60770	−0.310842	−0.155421	−	0.987848i	$-0.549673\pi$
$599$	−7.60770	−0.310842	−0.155421	+	0.987848i	$0.549673\pi$
$600$	0	0
$601$	43.5692	1.77723	0.888613	−	0.458658i	$-0.151670\pi$
$601$	43.5692	1.77723	0.888613	+	0.458658i	$0.151670\pi$
$602$	0.679492i	0.0276940i
$603$	0	0
$604$	1.80385	0.0733975
$605$	0	0
$606$	0	0
$607$	− 24.9808i	− 1.01394i	−0.861964	−	0.506969i	$-0.830766\pi$
$607$	− 24.9808i	− 1.01394i	0.861964	−	0.506969i	$-0.169234\pi$
$608$	32.1962i	1.30573i
$609$	0	0
$610$	0	0
$611$	−6.00000	−0.242734
$612$	0	0
$613$	26.0000i	1.05013i	0.851062	+	0.525065i	$0.175959\pi$
$613$	26.0000i	1.05013i	−0.851062	+	0.525065i	$0.824041\pi$
$614$	32.5359	1.31304
$615$	0	0
$616$	−16.3923	−0.660465
$617$	33.7128i	1.35723i	0.734496	+	0.678613i	$0.237419\pi$
$617$	33.7128i	1.35723i	−0.734496	+	0.678613i	$0.762581\pi$
$618$	0	0
$619$	−6.98076	−0.280581	−0.140290	−	0.990110i	$-0.544804\pi$
$619$	−6.98076	−0.280581	−0.140290	+	0.990110i	$0.544804\pi$
$620$	0	0
$621$	0	0
$622$	− 28.3923i	− 1.13843i
$623$	25.8564i	1.03592i
$624$	0	0
$625$	0	0
$626$	−24.9282	−0.996331
$627$	0	0
$628$	− 10.0000i	− 0.399043i
$629$	13.8564	0.552491
$630$	0	0
$631$	5.80385	0.231048	0.115524	−	0.993305i	$-0.463145\pi$
$631$	5.80385	0.231048	0.115524	+	0.993305i	$0.463145\pi$
$632$	− 14.5359i	− 0.578207i
$633$	0	0
$634$	41.5692	1.65092
$635$	0	0
$636$	0	0
$637$	3.00000i	0.118864i
$638$	20.7846i	0.822871i
$639$	0	0
$640$	0	0
$641$	−12.9282	−0.510633	−0.255317	−	0.966857i	$-0.582180\pi$
$641$	−12.9282	−0.510633	−0.255317	+	0.966857i	$0.582180\pi$
$642$	0	0
$643$	− 6.78461i	− 0.267559i	−0.991011	−	0.133779i	$-0.957289\pi$
$643$	− 6.78461i	− 0.267559i	0.991011	−	0.133779i	$-0.0427114\pi$
$644$	2.53590	0.0999284
$645$	0	0
$646$	−37.1769	−1.46271
$647$	22.0526i	0.866976i	0.901159	+	0.433488i	$0.142717\pi$
$647$	22.0526i	0.866976i	−0.901159	+	0.433488i	$0.857283\pi$
$648$	0	0
$649$	43.1769	1.69484
$650$	0	0
$651$	0	0
$652$	14.3923i	0.563646i
$653$	− 7.85641i	− 0.307445i	−0.988114	−	0.153722i	$-0.950874\pi$
$653$	− 7.85641i	− 0.307445i	0.988114	−	0.153722i	$-0.0491262\pi$
$654$	0	0
$655$	0	0
$656$	17.3205	0.676252
$657$	0	0
$658$	− 20.7846i	− 0.810268i
$659$	−21.4641	−0.836123	−0.418061	−	0.908419i	$-0.637290\pi$
$659$	−21.4641	−0.836123	−0.418061	+	0.908419i	$0.637290\pi$
$660$	0	0
$661$	10.7846	0.419473	0.209736	−	0.977758i	$-0.432739\pi$
$661$	10.7846	0.419473	0.209736	+	0.977758i	$0.432739\pi$
$662$	− 4.48334i	− 0.174250i
$663$	0	0
$664$	10.3923	0.403300
$665$	0	0
$666$	0	0
$667$	3.21539i	0.124500i
$668$	0.928203i	0.0359133i
$669$	0	0
$670$	0	0
$671$	−39.7128	−1.53310
$672$	0	0
$673$	− 14.3923i	− 0.554783i	−0.960757	−	0.277391i	$-0.910530\pi$
$673$	− 14.3923i	− 0.554783i	0.960757	−	0.277391i	$-0.0894698\pi$
$674$	45.7128	1.76079
$675$	0	0
$676$	1.00000	0.0384615
$677$	10.3923i	0.399409i	0.979856	+	0.199704i	$0.0639982\pi$
$677$	10.3923i	0.399409i	−0.979856	+	0.199704i	$0.936002\pi$
$678$	0	0
$679$	−4.00000	−0.153506
$680$	0	0
$681$	0	0
$682$	− 83.5692i	− 3.20003i
$683$	− 32.5359i	− 1.24495i	−0.782639	−	0.622476i	$-0.786127\pi$
$683$	− 32.5359i	− 1.24495i	0.782639	−	0.622476i	$-0.213873\pi$
$684$	0	0
$685$	0	0
$686$	−34.6410	−1.32260
$687$	0	0
$688$	0.980762i	0.0373912i
$689$	10.3923	0.395915
$690$	0	0
$691$	−47.7654	−1.81708	−0.908540	−	0.417797i	$-0.862802\pi$
$691$	−47.7654	−1.81708	−0.908540	+	0.417797i	$0.862802\pi$
$692$	8.53590i	0.324486i
$693$	0	0
$694$	−9.80385	−0.372149
$695$	0	0
$696$	0	0
$697$	12.0000i	0.454532i
$698$	− 24.9282i	− 0.943546i
$699$	0	0
$700$	0	0
$701$	−42.0000	−1.58632	−0.793159	−	0.609015i	$-0.791565\pi$
$701$	−42.0000	−1.58632	−0.793159	+	0.609015i	$0.791565\pi$
$702$	0	0
$703$	24.7846i	0.934769i
$704$	−4.73205	−0.178346
$705$	0	0
$706$	48.0000	1.80650
$707$	− 1.85641i	− 0.0698174i
$708$	0	0
$709$	−30.3923	−1.14141	−0.570703	−	0.821156i	$-0.693329\pi$
$709$	−30.3923	−1.14141	−0.570703	+	0.821156i	$0.693329\pi$
$710$	0	0
$711$	0	0
$712$	22.3923i	0.839187i
$713$	− 12.9282i	− 0.484165i
$714$	0	0
$715$	0	0
$716$	18.9282	0.707380
$717$	0	0
$718$	3.80385i	0.141958i
$719$	−25.8564	−0.964281	−0.482141	−	0.876094i	$-0.660141\pi$
$719$	−25.8564	−0.964281	−0.482141	+	0.876094i	$0.660141\pi$
$720$	0	0
$721$	−0.392305	−0.0146102
$722$	− 33.5885i	− 1.25003i
$723$	0	0
$724$	−0.392305	−0.0145799
$725$	0	0
$726$	0	0
$727$	− 44.5885i	− 1.65369i	−0.562427	−	0.826847i	$-0.690132\pi$
$727$	− 44.5885i	− 1.65369i	0.562427	−	0.826847i	$-0.309868\pi$
$728$	− 3.46410i	− 0.128388i
$729$	0	0
$730$	0	0
$731$	−0.679492	−0.0251319
$732$	0	0
$733$	38.0000i	1.40356i	0.712393	+	0.701781i	$0.247612\pi$
$733$	38.0000i	1.40356i	−0.712393	+	0.701781i	$0.752388\pi$
$734$	−20.4449	−0.754634
$735$	0	0
$736$	6.58846	0.242854
$737$	− 30.2487i	− 1.11423i
$738$	0	0
$739$	18.1962	0.669356	0.334678	−	0.942332i	$-0.391372\pi$
$739$	18.1962	0.669356	0.334678	+	0.942332i	$0.391372\pi$
$740$	0	0
$741$	0	0
$742$	36.0000i	1.32160i
$743$	16.1436i	0.592251i	0.955149	+	0.296126i	$0.0956947\pi$
$743$	16.1436i	0.592251i	−0.955149	+	0.296126i	$0.904305\pi$
$744$	0	0
$745$	0	0
$746$	−17.3205	−0.634149
$747$	0	0
$748$	16.3923i	0.599362i
$749$	35.3205	1.29058
$750$	0	0
$751$	36.3923	1.32797	0.663987	−	0.747744i	$-0.268863\pi$
$751$	36.3923	1.32797	0.663987	+	0.747744i	$0.268863\pi$
$752$	− 30.0000i	− 1.09399i
$753$	0	0
$754$	−4.39230	−0.159958
$755$	0	0
$756$	0	0
$757$	2.39230i	0.0869498i	0.999055	+	0.0434749i	$0.0138429\pi$
$757$	2.39230i	0.0869498i	−0.999055	+	0.0434749i	$0.986157\pi$
$758$	32.8756i	1.19410i
$759$	0	0
$760$	0	0
$761$	19.8564	0.719794	0.359897	−	0.932992i	$-0.382812\pi$
$761$	19.8564	0.719794	0.359897	+	0.932992i	$0.382812\pi$
$762$	0	0
$763$	4.00000i	0.144810i
$764$	−5.07180	−0.183491
$765$	0	0
$766$	−22.3923	−0.809067
$767$	9.12436i	0.329461i
$768$	0	0
$769$	−34.7846	−1.25437	−0.627183	−	0.778872i	$-0.715792\pi$
$769$	−34.7846	−1.25437	−0.627183	+	0.778872i	$0.715792\pi$
$770$	0	0
$771$	0	0
$772$	10.0000i	0.359908i
$773$	− 6.92820i	− 0.249190i	−0.992208	−	0.124595i	$-0.960237\pi$
$773$	− 6.92820i	− 0.249190i	0.992208	−	0.124595i	$-0.0397632\pi$
$774$	0	0
$775$	0	0
$776$	−3.46410	−0.124354
$777$	0	0
$778$	− 10.3923i	− 0.372582i
$779$	−21.4641	−0.769031
$780$	0	0
$781$	−22.3923	−0.801260
$782$	7.60770i	0.272051i
$783$	0	0
$784$	−15.0000	−0.535714
$785$	0	0
$786$	0	0
$787$	− 31.5692i	− 1.12532i	−0.826688	−	0.562661i	$-0.809778\pi$
$787$	− 31.5692i	− 1.12532i	0.826688	−	0.562661i	$-0.190222\pi$
$788$	12.9282i	0.460548i
$789$	0	0
$790$	0	0
$791$	−17.0718	−0.607003
$792$	0	0
$793$	− 8.39230i	− 0.298019i
$794$	−49.8564	−1.76934
$795$	0	0
$796$	20.0000	0.708881
$797$	− 40.6410i	− 1.43958i	−0.694193	−	0.719789i	$-0.744238\pi$
$797$	− 40.6410i	− 1.43958i	0.694193	−	0.719789i	$-0.255762\pi$
$798$	0	0
$799$	20.7846	0.735307
$800$	0	0
$801$	0	0
$802$	− 63.9615i	− 2.25856i
$803$	− 18.9282i	− 0.667962i
$804$	0	0
$805$	0	0
$806$	17.6603	0.622056
$807$	0	0
$808$	− 1.60770i	− 0.0565585i
$809$	−2.53590	−0.0891574	−0.0445787	−	0.999006i	$-0.514195\pi$
$809$	−2.53590	−0.0891574	−0.0445787	+	0.999006i	$0.514195\pi$
$810$	0	0
$811$	17.8038	0.625178	0.312589	−	0.949889i	$-0.398804\pi$
$811$	17.8038	0.625178	0.312589	+	0.949889i	$0.398804\pi$
$812$	− 5.07180i	− 0.177985i
$813$	0	0
$814$	32.7846	1.14910
$815$	0	0
$816$	0	0
$817$	− 1.21539i	− 0.0425211i
$818$	− 30.4974i	− 1.06632i
$819$	0	0
$820$	0	0
$821$	−28.6410	−0.999578	−0.499789	−	0.866147i	$-0.666589\pi$
$821$	−28.6410	−0.999578	−0.499789	+	0.866147i	$0.666589\pi$
$822$	0	0
$823$	− 15.4115i	− 0.537213i	−0.963250	−	0.268606i	$-0.913437\pi$
$823$	− 15.4115i	− 0.537213i	0.963250	−	0.268606i	$-0.0865631\pi$
$824$	−0.339746	−0.0118356
$825$	0	0
$826$	−31.6077	−1.09977
$827$	18.0000i	0.625921i	0.949766	+	0.312961i	$0.101321\pi$
$827$	18.0000i	0.625921i	−0.949766	+	0.312961i	$0.898679\pi$
$828$	0	0
$829$	−0.392305	−0.0136253	−0.00681266	−	0.999977i	$-0.502169\pi$
$829$	−0.392305	−0.0136253	−0.00681266	+	0.999977i	$0.502169\pi$
$830$	0	0
$831$	0	0
$832$	− 1.00000i	− 0.0346688i
$833$	− 10.3923i	− 0.360072i
$834$	0	0
$835$	0	0
$836$	−29.3205	−1.01407
$837$	0	0
$838$	4.39230i	0.151730i
$839$	0.339746	0.0117293	0.00586467	−	0.999983i	$-0.498133\pi$
$839$	0.339746	0.0117293	0.00586467	+	0.999983i	$0.498133\pi$
$840$	0	0
$841$	−22.5692	−0.778249
$842$	53.3205i	1.83755i
$843$	0	0
$844$	−8.00000	−0.275371
$845$	0	0
$846$	0	0
$847$	− 22.7846i	− 0.782888i
$848$	51.9615i	1.78437i
$849$	0	0
$850$	0	0
$851$	5.07180	0.173859
$852$	0	0
$853$	8.00000i	0.273915i	0.990577	+	0.136957i	$0.0437323\pi$
$853$	8.00000i	0.273915i	−0.990577	+	0.136957i	$0.956268\pi$
$854$	29.0718	0.994816
$855$	0	0
$856$	30.5885	1.04549
$857$	− 35.5692i	− 1.21502i	−0.794311	−	0.607511i	$-0.792168\pi$
$857$	− 35.5692i	− 1.21502i	0.794311	−	0.607511i	$-0.207832\pi$
$858$	0	0
$859$	17.1769	0.586069	0.293034	−	0.956102i	$-0.405335\pi$
$859$	17.1769	0.586069	0.293034	+	0.956102i	$0.405335\pi$
$860$	0	0
$861$	0	0
$862$	− 44.1962i	− 1.50533i
$863$	38.7846i	1.32024i	0.751159	+	0.660122i	$0.229495\pi$
$863$	38.7846i	1.32024i	−0.751159	+	0.660122i	$0.770505\pi$
$864$	0	0
$865$	0	0
$866$	60.2487	2.04733
$867$	0	0
$868$	20.3923i	0.692160i
$869$	39.7128	1.34716
$870$	0	0
$871$	6.39230	0.216595
$872$	3.46410i	0.117309i
$873$	0	0
$874$	−13.6077	−0.460287
$875$	0	0
$876$	0	0
$877$	− 2.00000i	− 0.0675352i	−0.999430	−	0.0337676i	$-0.989249\pi$
$877$	− 2.00000i	− 0.0675352i	0.999430	−	0.0337676i	$-0.0107506\pi$
$878$	55.4256i	1.87052i
$879$	0	0
$880$	0	0
$881$	47.3205	1.59427	0.797134	−	0.603802i	$-0.206348\pi$
$881$	47.3205	1.59427	0.797134	+	0.603802i	$0.206348\pi$
$882$	0	0
$883$	23.8038i	0.801063i	0.916283	+	0.400532i	$0.131175\pi$
$883$	23.8038i	0.801063i	−0.916283	+	0.400532i	$0.868825\pi$
$884$	−3.46410	−0.116510
$885$	0	0
$886$	29.4115	0.988100
$887$	− 47.9090i	− 1.60863i	−0.594206	−	0.804313i	$-0.702534\pi$
$887$	− 47.9090i	− 1.60863i	0.594206	−	0.804313i	$-0.297466\pi$
$888$	0	0
$889$	−32.3923	−1.08640
$890$	0	0
$891$	0	0
$892$	− 2.00000i	− 0.0669650i
$893$	37.1769i	1.24408i
$894$	0	0
$895$	0	0
$896$	24.2487	0.810093
$897$	0	0
$898$	− 35.5692i	− 1.18696i
$899$	−25.8564	−0.862359
$900$	0	0
$901$	−36.0000	−1.19933
$902$	28.3923i	0.945360i
$903$	0	0
$904$	−14.7846	−0.491729
$905$	0	0
$906$	0	0
$907$	53.7654i	1.78525i	0.450800	+	0.892625i	$0.351139\pi$
$907$	53.7654i	1.78525i	−0.450800	+	0.892625i	$0.648861\pi$
$908$	3.46410i	0.114960i
$909$	0	0
$910$	0	0
$911$	36.0000	1.19273	0.596367	−	0.802712i	$-0.296610\pi$
$911$	36.0000	1.19273	0.596367	+	0.802712i	$0.296610\pi$
$912$	0	0
$913$	28.3923i	0.939648i
$914$	−18.6795	−0.617863
$915$	0	0
$916$	6.39230	0.211208
$917$	0	0
$918$	0	0
$919$	−9.17691	−0.302718	−0.151359	−	0.988479i	$-0.548365\pi$
$919$	−9.17691	−0.302718	−0.151359	+	0.988479i	$0.548365\pi$
$920$	0	0
$921$	0	0
$922$	− 6.00000i	− 0.197599i
$923$	− 4.73205i	− 0.155757i
$924$	0	0
$925$	0	0
$926$	−4.14359	−0.136167
$927$	0	0
$928$	− 13.1769i	− 0.432553i
$929$	44.5359	1.46118	0.730588	−	0.682819i	$-0.239246\pi$
$929$	44.5359	1.46118	0.730588	+	0.682819i	$0.239246\pi$
$930$	0	0
$931$	18.5885	0.609212
$932$	− 6.00000i	− 0.196537i
$933$	0	0
$934$	48.1577	1.57577
$935$	0	0
$936$	0	0
$937$	− 34.7846i	− 1.13636i	−0.822903	−	0.568182i	$-0.807647\pi$
$937$	− 34.7846i	− 1.13636i	0.822903	−	0.568182i	$-0.192353\pi$
$938$	22.1436i	0.723014i
$939$	0	0
$940$	0	0
$941$	−31.1769	−1.01634	−0.508169	−	0.861257i	$-0.669678\pi$
$941$	−31.1769	−1.01634	−0.508169	+	0.861257i	$0.669678\pi$
$942$	0	0
$943$	4.39230i	0.143033i
$944$	−45.6218	−1.48486
$945$	0	0
$946$	−1.60770	−0.0522707
$947$	40.6410i	1.32066i	0.750978	+	0.660328i	$0.229583\pi$
$947$	40.6410i	1.32066i	−0.750978	+	0.660328i	$0.770417\pi$
$948$	0	0
$949$	4.00000	0.129845
$950$	0	0
$951$	0	0
$952$	12.0000i	0.388922i
$953$	− 0.928203i	− 0.0300675i	−0.999887	−	0.0150337i	$-0.995214\pi$
$953$	− 0.928203i	− 0.0300675i	0.999887	−	0.0150337i	$-0.00478556\pi$
$954$	0	0
$955$	0	0
$956$	14.1962	0.459136
$957$	0	0
$958$	− 61.7654i	− 1.99555i
$959$	1.85641	0.0599465
$960$	0	0
$961$	72.9615	2.35360
$962$	6.92820i	0.223374i
$963$	0	0
$964$	2.39230	0.0770510
$965$	0	0
$966$	0	0
$967$	50.3923i	1.62051i	0.586079	+	0.810254i	$0.300671\pi$
$967$	50.3923i	1.62051i	−0.586079	+	0.810254i	$0.699329\pi$
$968$	− 19.7321i	− 0.634212i
$969$	0	0
$970$	0	0
$971$	−18.9282	−0.607435	−0.303717	−	0.952762i	$-0.598228\pi$
$971$	−18.9282	−0.607435	−0.303717	+	0.952762i	$0.598228\pi$
$972$	0	0
$973$	24.7846i	0.794558i
$974$	45.7128	1.46473
$975$	0	0
$976$	41.9615	1.34316
$977$	− 15.7128i	− 0.502697i	−0.967897	−	0.251349i	$-0.919126\pi$
$977$	− 15.7128i	− 0.502697i	0.967897	−	0.251349i	$-0.0808741\pi$
$978$	0	0
$979$	−61.1769	−1.95522
$980$	0	0
$981$	0	0
$982$	− 4.39230i	− 0.140164i
$983$	34.3923i	1.09694i	0.836169	+	0.548472i	$0.184790\pi$
$983$	34.3923i	1.09694i	−0.836169	+	0.548472i	$0.815210\pi$
$984$	0	0
$985$	0	0
$986$	15.2154	0.484557
$987$	0	0
$988$	− 6.19615i	− 0.197126i
$989$	−0.248711	−0.00790856
$990$	0	0
$991$	8.00000	0.254128	0.127064	−	0.991894i	$-0.459445\pi$
$991$	8.00000	0.254128	0.127064	+	0.991894i	$0.459445\pi$
$992$	52.9808i	1.68214i
$993$	0	0
$994$	16.3923	0.519932
$995$	0	0
$996$	0	0
$997$	− 33.6077i	− 1.06437i	−0.846629	−	0.532183i	$-0.821372\pi$
$997$	− 33.6077i	− 1.06437i	0.846629	−	0.532183i	$-0.178628\pi$
$998$	− 67.5167i	− 2.13720i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2925.2.c.v.2224.1		4
3.2	odd	2		325.2.b.e.274.4		4
5.2	odd	4		585.2.a.k.1.2		2
5.3	odd	4		2925.2.a.z.1.1		2
5.4	even	2	inner	2925.2.c.v.2224.4		4
15.2	even	4		65.2.a.c.1.1	✓	2
15.8	even	4		325.2.a.g.1.2		2
15.14	odd	2		325.2.b.e.274.1		4
20.7	even	4		9360.2.a.cm.1.1		2
60.23	odd	4		5200.2.a.ca.1.2		2
60.47	odd	4		1040.2.a.h.1.1		2
65.12	odd	4		7605.2.a.be.1.1		2
105.62	odd	4		3185.2.a.k.1.1		2
120.77	even	4		4160.2.a.y.1.1		2
120.107	odd	4		4160.2.a.bj.1.2		2
165.32	odd	4		7865.2.a.h.1.2		2
195.2	odd	12		845.2.m.a.316.1		4
195.17	even	12		845.2.e.f.146.1		4
195.32	odd	12		845.2.m.c.361.1		4
195.38	even	4		4225.2.a.w.1.1		2
195.47	odd	4		845.2.c.e.506.2		4
195.62	even	12		845.2.e.f.191.1		4
195.77	even	4		845.2.a.d.1.2		2
195.107	even	12		845.2.e.e.191.2		4
195.122	odd	4		845.2.c.e.506.4		4
195.137	odd	12		845.2.m.a.361.1		4
195.152	even	12		845.2.e.e.146.2		4
195.167	odd	12		845.2.m.c.316.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.a.c.1.1	✓	2	15.2	even	4
325.2.a.g.1.2		2	15.8	even	4
325.2.b.e.274.1		4	15.14	odd	2
325.2.b.e.274.4		4	3.2	odd	2
585.2.a.k.1.2		2	5.2	odd	4
845.2.a.d.1.2		2	195.77	even	4
845.2.c.e.506.2		4	195.47	odd	4
845.2.c.e.506.4		4	195.122	odd	4
845.2.e.e.146.2		4	195.152	even	12
845.2.e.e.191.2		4	195.107	even	12
845.2.e.f.146.1		4	195.17	even	12
845.2.e.f.191.1		4	195.62	even	12
845.2.m.a.316.1		4	195.2	odd	12
845.2.m.a.361.1		4	195.137	odd	12
845.2.m.c.316.1		4	195.167	odd	12
845.2.m.c.361.1		4	195.32	odd	12
1040.2.a.h.1.1		2	60.47	odd	4
2925.2.a.z.1.1		2	5.3	odd	4
2925.2.c.v.2224.1		4	1.1	even	1	trivial
2925.2.c.v.2224.4		4	5.4	even	2	inner
3185.2.a.k.1.1		2	105.62	odd	4
4160.2.a.y.1.1		2	120.77	even	4
4160.2.a.bj.1.2		2	120.107	odd	4
4225.2.a.w.1.1		2	195.38	even	4
5200.2.a.ca.1.2		2	60.23	odd	4
7605.2.a.be.1.1		2	65.12	odd	4
7865.2.a.h.1.2		2	165.32	odd	4
9360.2.a.cm.1.1		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:	\( N \)	\(=\)	\( 2925 = 3^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2925.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{2} -1.00000 q^{4} -2.00000i q^{7} -1.73205i q^{8} +O(q^{10})\)	\(q-1.73205i q^{2} -1.00000 q^{4} -2.00000i q^{7} -1.73205i q^{8} +4.73205 q^{11} +1.00000i q^{13} -3.46410 q^{14} -5.00000 q^{16} -3.46410i q^{17} +6.19615 q^{19} -8.19615i q^{22} -1.26795i q^{23} +1.73205 q^{26} +2.00000i q^{28} -2.53590 q^{29} +10.1962 q^{31} +5.19615i q^{32} -6.00000 q^{34} +4.00000i q^{37} -10.7321i q^{38} -3.46410 q^{41} -0.196152i q^{43} -4.73205 q^{44} -2.19615 q^{46} +6.00000i q^{47} +3.00000 q^{49} -1.00000i q^{52} -10.3923i q^{53} -3.46410 q^{56} +4.39230i q^{58} +9.12436 q^{59} -8.39230 q^{61} -17.6603i q^{62} -1.00000 q^{64} -6.39230i q^{67} +3.46410i q^{68} -4.73205 q^{71} -4.00000i q^{73} +6.92820 q^{74} -6.19615 q^{76} -9.46410i q^{77} +8.39230 q^{79} +6.00000i q^{82} +6.00000i q^{83} -0.339746 q^{86} -8.19615i q^{88} -12.9282 q^{89} +2.00000 q^{91} +1.26795i q^{92} +10.3923 q^{94} -2.00000i q^{97} -5.19615i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4}+O(q^{10}) \) 4 * q - 4 * q^4	\( 4 q - 4 q^{4} + 12 q^{11} - 20 q^{16} + 4 q^{19} - 24 q^{29} + 20 q^{31} - 24 q^{34} - 12 q^{44} + 12 q^{46} + 12 q^{49} - 12 q^{59} + 8 q^{61} - 4 q^{64} - 12 q^{71} - 4 q^{76} - 8 q^{79} - 36 q^{86} - 24 q^{89} + 8 q^{91}+O(q^{100}) \) 4 * q - 4 * q^4 + 12 * q^11 - 20 * q^16 + 4 * q^19 - 24 * q^29 + 20 * q^31 - 24 * q^34 - 12 * q^44 + 12 * q^46 + 12 * q^49 - 12 * q^59 + 8 * q^61 - 4 * q^64 - 12 * q^71 - 4 * q^76 - 8 * q^79 - 36 * q^86 - 24 * q^89 + 8 * q^91

Introduction

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