LMFDB - Embedded newform 1620.3.o.g.701.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,3,Mod(701,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 5, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1620.701");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1620.o (of order $6$, degree $2$, 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:	$44.1418028264$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$16$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			701.6
Character	$\chi$	$=$	1620.701
Dual form			1620.3.o.g.1241.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.93649 - 1.11803i) q^{5} +(-6.81144 - 11.7978i) q^{7} +O(q^{10})$	$q+(-1.93649 - 1.11803i) q^{5} +(-6.81144 - 11.7978i) q^{7} +(0.693463 - 0.400371i) q^{11} +(9.67459 - 16.7569i) q^{13} -29.8774i q^{17} -4.17445 q^{19} +(31.9007 + 18.4179i) q^{23} +(2.50000 + 4.33013i) q^{25} +(-23.2382 + 13.4166i) q^{29} +(22.0690 - 38.2246i) q^{31} +30.4617i q^{35} -29.4644 q^{37} +(34.2153 + 19.7542i) q^{41} +(-3.31174 - 5.73611i) q^{43} +(2.21019 - 1.27605i) q^{47} +(-68.2915 + 118.284i) q^{49} -95.6164i q^{53} -1.79051 q^{55} +(-60.1876 - 34.7493i) q^{59} +(25.0523 + 43.3919i) q^{61} +(-37.4695 + 21.6330i) q^{65} +(-30.2143 + 52.3328i) q^{67} +27.4501i q^{71} -106.033 q^{73} +(-9.44697 - 5.45421i) q^{77} +(-61.3438 - 106.251i) q^{79} +(125.271 - 72.3255i) q^{83} +(-33.4040 + 57.8574i) q^{85} +110.143i q^{89} -263.592 q^{91} +(8.08378 + 4.66717i) q^{95} +(-1.06754 - 1.84903i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 8 q^{7}+O(q^{10}) $ 32 * q - 8 * q^7	$ 32 q - 8 q^{7} + 40 q^{13} + 112 q^{19} + 80 q^{25} + 64 q^{31} - 176 q^{37} - 128 q^{43} - 216 q^{49} - 8 q^{61} + 40 q^{67} + 112 q^{73} + 136 q^{79} - 784 q^{91} - 128 q^{97}+O(q^{100}) $ 32 * q - 8 * q^7 + 40 * q^13 + 112 * q^19 + 80 * q^25 + 64 * q^31 - 176 * q^37 - 128 * q^43 - 216 * q^49 - 8 * q^61 + 40 * q^67 + 112 * q^73 + 136 * q^79 - 784 * q^91 - 128 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$.

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$1$	$e\left(\frac{5}{6}\right)$

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 $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.93649	−	1.11803i	−0.387298	−	0.223607i
$5$	−1.93649	−	1.11803i	−0.387298	−	0.223607i
$6$		0			0
$7$	−6.81144	−	11.7978i	−0.973063	−	1.68539i	−0.686184	−	0.727428i	$-0.740716\pi$
$7$	−6.81144	−	11.7978i	−0.973063	−	1.68539i	−0.286879	−	0.957967i	$-0.592618\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.693463	−	0.400371i	0.0630421	−	0.0363974i	−0.468148	−	0.883650i	$-0.655078\pi$
$11$	0.693463	−	0.400371i	0.0630421	−	0.0363974i	0.531190	+	0.847253i	$0.321745\pi$
$12$		0			0
$13$	9.67459	−	16.7569i	0.744199	−	1.28899i	−0.206369	−	0.978474i	$-0.566165\pi$
$13$	9.67459	−	16.7569i	0.744199	−	1.28899i	0.950568	−	0.310517i	$-0.100502\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	29.8774i		−	1.75750i	−0.477286	−	0.878748i	$-0.658379\pi$
$17$		−	29.8774i		−	1.75750i	0.477286	−	0.878748i	$-0.341621\pi$
$18$		0			0
$19$	−4.17445			−0.219708			−0.109854	−	0.993948i	$-0.535038\pi$
$19$	−4.17445			−0.219708			−0.109854	+	0.993948i	$0.535038\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	31.9007	+	18.4179i	1.38699	+	0.800777i	0.992975	−	0.118328i	$-0.0377533\pi$
$23$	31.9007	+	18.4179i	1.38699	+	0.800777i	0.394012	+	0.919105i	$0.371087\pi$
$24$		0			0
$25$	2.50000	+	4.33013i	0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−23.2382	+	13.4166i	−0.801317	+	0.462641i	−0.843931	−	0.536451i	$-0.819765\pi$
$29$	−23.2382	+	13.4166i	−0.801317	+	0.462641i	0.0426145	+	0.999092i	$0.486431\pi$
$30$		0			0
$31$	22.0690	−	38.2246i	0.711903	−	1.23305i	−0.252239	−	0.967665i	$-0.581167\pi$
$31$	22.0690	−	38.2246i	0.711903	−	1.23305i	0.964142	−	0.265388i	$-0.0854999\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			30.4617i			0.870334i
$36$		0			0
$37$	−29.4644			−0.796334			−0.398167	−	0.917313i	$-0.630354\pi$
$37$	−29.4644			−0.796334			−0.398167	+	0.917313i	$0.630354\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	34.2153	+	19.7542i	0.834519	+	0.481810i	0.855397	−	0.517972i	$-0.173313\pi$
$41$	34.2153	+	19.7542i	0.834519	+	0.481810i	−0.0208783	+	0.999782i	$0.506646\pi$
$42$		0			0
$43$	−3.31174	−	5.73611i	−0.0770173	−	0.133398i	0.824944	−	0.565214i	$-0.191206\pi$
$43$	−3.31174	−	5.73611i	−0.0770173	−	0.133398i	−0.901962	+	0.431816i	$0.857873\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	2.21019	−	1.27605i	0.0470254	−	0.0271501i	−0.476303	−	0.879281i	$-0.658023\pi$
$47$	2.21019	−	1.27605i	0.0470254	−	0.0271501i	0.523328	+	0.852131i	$0.324690\pi$
$48$		0			0
$49$	−68.2915	+	118.284i	−1.39370	+	2.41396i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	95.6164i		−	1.80408i	−0.431649	−	0.902042i	$-0.642068\pi$
$53$		−	95.6164i		−	1.80408i	0.431649	−	0.902042i	$-0.357932\pi$
$54$		0			0
$55$	−1.79051			−0.0325548
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−60.1876	−	34.7493i	−1.02013	−	0.588971i	−0.105987	−	0.994367i	$-0.533800\pi$
$59$	−60.1876	−	34.7493i	−1.02013	−	0.588971i	−0.914141	+	0.405396i	$0.867134\pi$
$60$		0			0
$61$	25.0523	+	43.3919i	0.410694	+	0.711342i	0.994966	−	0.100215i	$-0.0319532\pi$
$61$	25.0523	+	43.3919i	0.410694	+	0.711342i	−0.584272	+	0.811558i	$0.698620\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−37.4695	+	21.6330i	−0.576454	+	0.332816i
$66$		0			0
$67$	−30.2143	+	52.3328i	−0.450960	+	0.781086i	−0.998446	−	0.0557288i	$-0.982252\pi$
$67$	−30.2143	+	52.3328i	−0.450960	+	0.781086i	0.547486	+	0.836815i	$0.315585\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			27.4501i			0.386622i	0.981138	+	0.193311i	$0.0619226\pi$
$71$			27.4501i			0.386622i	−0.981138	+	0.193311i	$0.938077\pi$
$72$		0			0
$73$	−106.033			−1.45250			−0.726252	−	0.687429i	$-0.758739\pi$
$73$	−106.033			−1.45250			−0.726252	+	0.687429i	$0.758739\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−9.44697	−	5.45421i	−0.122688	−	0.0708339i
$78$		0			0
$79$	−61.3438	−	106.251i	−0.776504	−	1.34494i	−0.933945	−	0.357416i	$-0.883658\pi$
$79$	−61.3438	−	106.251i	−0.776504	−	1.34494i	0.157442	−	0.987528i	$-0.449675\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	125.271	−	72.3255i	1.50929	−	0.871391i	0.509353	−	0.860558i	$-0.329885\pi$
$83$	125.271	−	72.3255i	1.50929	−	0.871391i	0.999941	−	0.0108335i	$-0.00344848\pi$
$84$		0			0
$85$	−33.4040	+	57.8574i	−0.392988	+	0.680675i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			110.143i			1.23756i	0.785566	+	0.618778i	$0.212372\pi$
$89$			110.143i			1.23756i	−0.785566	+	0.618778i	$0.787628\pi$
$90$		0			0
$91$	−263.592			−2.89661
$92$		0			0
$93$		0			0
$94$		0			0
$95$	8.08378	+	4.66717i	0.0850924	+	0.0491281i
$96$		0			0
$97$	−1.06754	−	1.84903i	−0.0110055	−	0.0190622i	0.860470	−	0.509501i	$-0.170170\pi$
$97$	−1.06754	−	1.84903i	−0.0110055	−	0.0190622i	−0.871476	+	0.490439i	$0.836837\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−47.2258	+	27.2658i	−0.467582	+	0.269959i	−0.715227	−	0.698892i	$-0.753677\pi$
$101$	−47.2258	+	27.2658i	−0.467582	+	0.269959i	0.247645	+	0.968851i	$0.420343\pi$
$102$		0			0
$103$	−23.3306	+	40.4098i	−0.226511	+	0.392328i	−0.956772	−	0.290840i	$-0.906065\pi$
$103$	−23.3306	+	40.4098i	−0.226511	+	0.392328i	0.730261	+	0.683168i	$0.239398\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	109.623i		−	1.02452i	−0.858831	−	0.512259i	$-0.828809\pi$
$107$		−	109.623i		−	1.02452i	0.858831	−	0.512259i	$-0.171191\pi$
$108$		0			0
$109$	35.5273			0.325938			0.162969	−	0.986631i	$-0.447893\pi$
$109$	35.5273			0.325938			0.162969	+	0.986631i	$0.447893\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	129.347	+	74.6784i	1.14466	+	0.660871i	0.947581	−	0.319516i	$-0.103520\pi$
$113$	129.347	+	74.6784i	1.14466	+	0.660871i	0.197081	+	0.980387i	$0.436854\pi$
$114$		0			0
$115$	−41.1836	−	71.3321i	−0.358119	−	0.620279i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−352.487	+	203.508i	−2.96208	+	1.71015i
$120$		0			0
$121$	−60.1794	+	104.234i	−0.497350	+	0.861436i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	11.1803i		−	0.0894427i
$126$		0			0
$127$	−43.7900			−0.344803			−0.172401	−	0.985027i	$-0.555153\pi$
$127$	−43.7900			−0.344803			−0.172401	+	0.985027i	$0.555153\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	64.7512	+	37.3842i	0.494284	+	0.285375i	0.726350	−	0.687325i	$-0.241215\pi$
$131$	64.7512	+	37.3842i	0.494284	+	0.285375i	−0.232066	+	0.972700i	$0.574548\pi$
$132$		0			0
$133$	28.4340	+	49.2491i	0.213789	+	0.370294i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−48.3162	+	27.8954i	−0.352673	+	0.203616i	−0.665862	−	0.746075i	$-0.731936\pi$
$137$	−48.3162	+	27.8954i	−0.352673	+	0.203616i	0.313189	+	0.949691i	$0.398603\pi$
$138$		0			0
$139$	43.9754	−	76.1677i	0.316370	−	0.547969i	−0.663358	−	0.748302i	$-0.730869\pi$
$139$	43.9754	−	76.1677i	0.316370	−	0.547969i	0.979728	+	0.200333i	$0.0642025\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	15.4937i		−	0.108348i
$144$		0			0
$145$	60.0007			0.413798
$146$		0			0
$147$		0			0
$148$		0			0
$149$	75.5887	+	43.6412i	0.507307	+	0.292894i	0.731726	−	0.681599i	$-0.238715\pi$
$149$	75.5887	+	43.6412i	0.507307	+	0.292894i	−0.224419	+	0.974493i	$0.572048\pi$
$150$		0			0
$151$	−59.3560	−	102.808i	−0.393086	−	0.680845i	0.599769	−	0.800173i	$-0.295259\pi$
$151$	−59.3560	−	102.808i	−0.393086	−	0.680845i	−0.992855	+	0.119328i	$0.961926\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−85.4729	+	49.3478i	−0.551438	+	0.318373i
$156$		0			0
$157$	42.2080	−	73.1064i	0.268841	−	0.465646i	−0.699722	−	0.714415i	$-0.746693\pi$
$157$	42.2080	−	73.1064i	0.268841	−	0.465646i	0.968563	+	0.248769i	$0.0800262\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	501.809i		−	3.11683i
$162$		0			0
$163$	−166.213			−1.01971			−0.509855	−	0.860260i	$-0.670301\pi$
$163$	−166.213			−1.01971			−0.509855	+	0.860260i	$0.670301\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	115.939	+	66.9373i	0.694244	+	0.400822i	0.805200	−	0.593003i	$-0.202058\pi$
$167$	115.939	+	66.9373i	0.694244	+	0.400822i	−0.110956	+	0.993825i	$0.535391\pi$
$168$		0			0
$169$	−102.695	−	177.874i	−0.607665	−	1.05251i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	101.211	−	58.4341i	0.585034	−	0.337769i	−0.178098	−	0.984013i	$-0.556994\pi$
$173$	101.211	−	58.4341i	0.585034	−	0.337769i	0.763131	+	0.646243i	$0.223661\pi$
$174$		0			0
$175$	34.0572	−	58.9888i	0.194613	−	0.337079i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	275.951i		−	1.54163i	−0.637062	−	0.770813i	$-0.719850\pi$
$179$		−	275.951i		−	1.54163i	0.637062	−	0.770813i	$-0.280150\pi$
$180$		0			0
$181$	32.8036			0.181235			0.0906176	−	0.995886i	$-0.471116\pi$
$181$	32.8036			0.181235			0.0906176	+	0.995886i	$0.471116\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	57.0575	+	32.9422i	0.308419	+	0.178066i
$186$		0			0
$187$	−11.9621	−	20.7189i	−0.0639683	−	0.110796i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	45.7718	−	26.4264i	0.239643	−	0.138358i	−0.375370	−	0.926875i	$-0.622484\pi$
$191$	45.7718	−	26.4264i	0.239643	−	0.138358i	0.615013	+	0.788517i	$0.289151\pi$
$192$		0			0
$193$	−88.7329	+	153.690i	−0.459756	+	0.796321i	−0.998948	−	0.0458622i	$-0.985396\pi$
$193$	−88.7329	+	153.690i	−0.459756	+	0.796321i	0.539192	+	0.842183i	$0.318730\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			258.911i			1.31427i	0.753772	+	0.657135i	$0.228232\pi$
$197$			258.911i			1.31427i	−0.753772	+	0.657135i	$0.771768\pi$
$198$		0			0
$199$	−78.3174			−0.393555			−0.196777	−	0.980448i	$-0.563048\pi$
$199$	−78.3174			−0.393555			−0.196777	+	0.980448i	$0.563048\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	316.571	+	182.772i	1.55946	+	0.900357i
$204$		0			0
$205$	−44.1717	−	76.5077i	−0.215472	−	0.373208i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−2.89482	+	1.67133i	−0.0138508	+	0.00799678i
$210$		0			0
$211$	35.1342	−	60.8542i	0.166513	−	0.288408i	−0.770679	−	0.637224i	$-0.780083\pi$
$211$	35.1342	−	60.8542i	0.166513	−	0.288408i	0.937191	+	0.348815i	$0.113416\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			14.8106i			0.0688864i
$216$		0			0
$217$	−601.287			−2.77091
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−500.653	−	289.052i	−2.26540	−	1.30793i
$222$		0			0
$223$	−88.4628	−	153.222i	−0.396694	−	0.687094i	0.596622	−	0.802523i	$-0.296509\pi$
$223$	−88.4628	−	153.222i	−0.396694	−	0.687094i	−0.993316	+	0.115428i	$0.963176\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−199.792	+	115.350i	−0.880141	+	0.508149i	−0.870705	−	0.491806i	$-0.836337\pi$
$227$	−199.792	+	115.350i	−0.880141	+	0.508149i	−0.00943585	+	0.999955i	$0.503004\pi$
$228$		0			0
$229$	−56.8245	+	98.4229i	−0.248142	+	0.429795i	−0.963010	−	0.269465i	$-0.913153\pi$
$229$	−56.8245	+	98.4229i	−0.248142	+	0.429795i	0.714868	+	0.699259i	$0.246487\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			166.380i			0.714077i	0.934090	+	0.357038i	$0.116213\pi$
$233$			166.380i			0.714077i	−0.934090	+	0.357038i	$0.883787\pi$
$234$		0			0
$235$	−5.70669			−0.0242838
$236$		0			0
$237$		0			0
$238$		0			0
$239$	122.383	+	70.6578i	0.512062	+	0.295639i	0.733681	−	0.679494i	$-0.237801\pi$
$239$	122.383	+	70.6578i	0.512062	+	0.295639i	−0.221619	+	0.975133i	$0.571134\pi$
$240$		0			0
$241$	88.3635	+	153.050i	0.366653	+	0.635062i	0.989040	−	0.147648i	$-0.0471702\pi$
$241$	88.3635	+	153.050i	0.366653	+	0.635062i	−0.622387	+	0.782710i	$0.713837\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	264.492	−	152.704i	1.07956	−	0.623283i
$246$		0			0
$247$	−40.3861	+	69.9507i	−0.163506	+	0.283201i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			229.886i			0.915881i	0.888983	+	0.457941i	$0.151413\pi$
$251$			229.886i			0.915881i	−0.888983	+	0.457941i	$0.848587\pi$
$252$		0			0
$253$	29.4960			0.116585
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−21.7332	−	12.5477i	−0.0845649	−	0.0488236i	0.457121	−	0.889404i	$-0.348881\pi$
$257$	−21.7332	−	12.5477i	−0.0845649	−	0.0488236i	−0.541686	+	0.840581i	$0.682214\pi$
$258$		0			0
$259$	200.695	+	347.614i	0.774883	+	1.34214i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−108.004	+	62.3559i	−0.410660	+	0.237095i	−0.691073	−	0.722785i	$-0.742862\pi$
$263$	−108.004	+	62.3559i	−0.410660	+	0.237095i	0.280413	+	0.959879i	$0.409529\pi$
$264$		0			0
$265$	−106.902	+	185.160i	−0.403405	+	0.698718i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			348.909i			1.29706i	0.761190	+	0.648529i	$0.224616\pi$
$269$			348.909i			1.29706i	−0.761190	+	0.648529i	$0.775384\pi$
$270$		0			0
$271$	58.7176			0.216670			0.108335	−	0.994114i	$-0.465448\pi$
$271$	58.7176			0.216670			0.108335	+	0.994114i	$0.465448\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	3.46732	+	2.00186i	0.0126084	+	0.00727948i
$276$		0			0
$277$	87.3991	+	151.380i	0.315520	+	0.546497i	0.979548	−	0.201211i	$-0.0644877\pi$
$277$	87.3991	+	151.380i	0.315520	+	0.546497i	−0.664028	+	0.747708i	$0.731154\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−81.0464	+	46.7922i	−0.288421	+	0.166520i	−0.637230	−	0.770674i	$-0.719920\pi$
$281$	−81.0464	+	46.7922i	−0.288421	+	0.166520i	0.348808	+	0.937194i	$0.386586\pi$
$282$		0			0
$283$	−138.572	+	240.013i	−0.489652	+	0.848102i	−0.999929	−	0.0119078i	$-0.996210\pi$
$283$	−138.572	+	240.013i	−0.489652	+	0.848102i	0.510277	+	0.860010i	$0.329543\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	538.218i		−	1.87533i
$288$		0			0
$289$	−603.661			−2.08879
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−95.5912	−	55.1896i	−0.326250	−	0.188360i	0.327925	−	0.944704i	$-0.393651\pi$
$293$	−95.5912	−	55.1896i	−0.326250	−	0.188360i	−0.654175	+	0.756343i	$0.726984\pi$
$294$		0			0
$295$	77.7018	+	134.584i	0.263396	+	0.456215i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	617.253	−	356.371i	2.06439	−	1.19188i
$300$		0			0
$301$	−45.1155	+	78.1423i	−0.149885	+	0.259609i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	112.037i		−	0.367336i
$306$		0			0
$307$	458.389			1.49312			0.746562	−	0.665316i	$-0.231703\pi$
$307$	458.389			1.49312			0.746562	+	0.665316i	$0.231703\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	36.1060	+	20.8458i	0.116097	+	0.0670284i	0.556924	−	0.830564i	$-0.311982\pi$
$311$	36.1060	+	20.8458i	0.116097	+	0.0670284i	−0.440827	+	0.897592i	$0.645315\pi$
$312$		0			0
$313$	−256.667	−	444.560i	−0.820022	−	1.42032i	−0.905665	−	0.423994i	$-0.860628\pi$
$313$	−256.667	−	444.560i	−0.820022	−	1.42032i	0.0856426	−	0.996326i	$-0.472706\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−42.3553	+	24.4539i	−0.133613	+	0.0771415i	−0.565317	−	0.824874i	$-0.691246\pi$
$317$	−42.3553	+	24.4539i	−0.133613	+	0.0771415i	0.431704	+	0.902016i	$0.357913\pi$
$318$		0			0
$319$	−10.7432	+	18.6078i	−0.0336778	+	0.0583317i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			124.722i			0.386135i
$324$		0			0
$325$	96.7459			0.297680
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−30.1092	−	17.3835i	−0.0915173	−	0.0528375i
$330$		0			0
$331$	110.006	+	190.536i	0.332345	+	0.575638i	0.982971	−	0.183760i	$-0.0588268\pi$
$331$	110.006	+	190.536i	0.332345	+	0.575638i	−0.650626	+	0.759398i	$0.725494\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	117.020	−	67.5613i	0.349312	−	0.201676i
$336$		0			0
$337$	−9.87472	+	17.1035i	−0.0293018	+	0.0507523i	−0.880304	−	0.474409i	$-0.842662\pi$
$337$	−9.87472	+	17.1035i	−0.0293018	+	0.0507523i	0.851003	+	0.525161i	$0.175995\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	35.3432i		−	0.103646i
$342$		0			0
$343$	1193.13			3.47852
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−485.120	−	280.084i	−1.39804	−	0.807159i	−0.403854	−	0.914824i	$-0.632330\pi$
$347$	−485.120	−	280.084i	−1.39804	−	0.807159i	−0.994187	+	0.107664i	$0.965663\pi$
$348$		0			0
$349$	213.485	+	369.767i	0.611704	+	1.05950i	0.990953	+	0.134208i	$0.0428491\pi$
$349$	213.485	+	369.767i	0.611704	+	1.05950i	−0.379249	+	0.925295i	$0.623818\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−163.970	+	94.6679i	−0.464503	+	0.268181i	−0.713936	−	0.700211i	$-0.753089\pi$
$353$	−163.970	+	94.6679i	−0.464503	+	0.268181i	0.249433	+	0.968392i	$0.419756\pi$
$354$		0			0
$355$	30.6902	−	53.1570i	0.0864512	−	0.149738i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	548.290i		−	1.52727i	−0.645648	−	0.763635i	$-0.723413\pi$
$359$		−	548.290i		−	1.52727i	0.645648	−	0.763635i	$-0.276587\pi$
$360$		0			0
$361$	−343.574			−0.951729
$362$		0			0
$363$		0			0
$364$		0			0
$365$	205.332	+	118.548i	0.562552	+	0.324790i
$366$		0			0
$367$	−4.17562	−	7.23238i	−0.0113777	−	0.0197068i	0.860280	−	0.509821i	$-0.170288\pi$
$367$	−4.17562	−	7.23238i	−0.0113777	−	0.0197068i	−0.871658	+	0.490114i	$0.836955\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−1128.06	+	651.286i	−3.04059	+	1.75549i
$372$		0			0
$373$	−31.5875	+	54.7112i	−0.0846850	+	0.146679i	−0.905257	−	0.424864i	$-0.860322\pi$
$373$	−31.5875	+	54.7112i	−0.0846850	+	0.146679i	0.820572	+	0.571543i	$0.193655\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			519.200i			1.37719i
$378$		0			0
$379$	147.019			0.387912			0.193956	−	0.981010i	$-0.437868\pi$
$379$	147.019			0.387912			0.193956	+	0.981010i	$0.437868\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	616.045	+	355.674i	1.60847	+	0.928652i	0.989712	+	0.143072i	$0.0456981\pi$
$383$	616.045	+	355.674i	1.60847	+	0.928652i	0.618760	+	0.785580i	$0.287635\pi$
$384$		0			0
$385$	12.1960	+	21.1241i	0.0316779	+	0.0548677i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	540.215	−	311.893i	1.38873	−	0.801782i	0.395556	−	0.918442i	$-0.370552\pi$
$389$	540.215	−	311.893i	1.38873	−	0.801782i	0.993172	+	0.116660i	$0.0372186\pi$
$390$		0			0
$391$	550.279	−	953.111i	1.40736	−	2.43762i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			274.338i			0.694526i
$396$		0			0
$397$	310.815			0.782908			0.391454	−	0.920198i	$-0.371972\pi$
$397$	310.815			0.782908			0.391454	+	0.920198i	$0.371972\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−370.900	−	214.139i	−0.924938	−	0.534013i	−0.0397311	−	0.999210i	$-0.512650\pi$
$401$	−370.900	−	214.139i	−0.924938	−	0.534013i	−0.885207	+	0.465197i	$0.845983\pi$
$402$		0			0
$403$	−427.017	−	739.615i	−1.05960	−	1.83527i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−20.4325	+	11.7967i	−0.0502026	+	0.0289845i
$408$		0			0
$409$	275.705	−	477.535i	0.674095	−	1.16757i	−0.302637	−	0.953106i	$-0.597867\pi$
$409$	275.705	−	477.535i	0.674095	−	1.16757i	0.976732	−	0.214461i	$-0.0687996\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			946.772i			2.29243i
$414$		0			0
$415$	−323.449			−0.779396
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−465.293	−	268.637i	−1.11048	−	0.641138i	−0.171529	−	0.985179i	$-0.554871\pi$
$419$	−465.293	−	268.637i	−1.11048	−	0.641138i	−0.938955	+	0.344041i	$0.888204\pi$
$420$		0			0
$421$	−234.895	−	406.850i	−0.557945	−	0.966388i	−0.997668	−	0.0682551i	$-0.978257\pi$
$421$	−234.895	−	406.850i	−0.557945	−	0.966388i	0.439723	−	0.898133i	$-0.355077\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	129.373	−	74.6936i	0.304407	−	0.175750i
$426$		0			0
$427$	341.285	−	591.123i	0.799262	−	1.38436i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			256.875i			0.595997i	0.954566	+	0.297999i	$0.0963191\pi$
$431$			256.875i			0.595997i	−0.954566	+	0.297999i	$0.903681\pi$
$432$		0			0
$433$	514.912			1.18917			0.594587	−	0.804032i	$-0.297316\pi$
$433$	514.912			1.18917			0.594587	+	0.804032i	$0.297316\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−133.168	−	76.8844i	−0.304732	−	0.175937i
$438$		0			0
$439$	−90.0727	−	156.010i	−0.205177	−	0.355377i	0.745012	−	0.667051i	$-0.232444\pi$
$439$	−90.0727	−	156.010i	−0.205177	−	0.355377i	−0.950189	+	0.311674i	$0.899110\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	43.7339	−	25.2498i	0.0987220	−	0.0569972i	−0.449826	−	0.893116i	$-0.648514\pi$
$443$	43.7339	−	25.2498i	0.0987220	−	0.0569972i	0.548548	+	0.836119i	$0.315181\pi$
$444$		0			0
$445$	123.143	−	213.290i	0.276726	−	0.479304i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	434.135i		−	0.966893i	−0.875374	−	0.483446i	$-0.839385\pi$
$449$		−	434.135i		−	0.966893i	0.875374	−	0.483446i	$-0.160615\pi$
$450$		0			0
$451$	31.6361			0.0701465
$452$		0			0
$453$		0			0
$454$		0			0
$455$	510.443	+	294.704i	1.12185	+	0.647702i
$456$		0			0
$457$	124.696	+	215.980i	0.272858	+	0.472604i	0.969592	−	0.244725i	$-0.0786978\pi$
$457$	124.696	+	215.980i	0.272858	+	0.472604i	−0.696735	+	0.717329i	$0.745364\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	231.180	−	133.472i	0.501475	−	0.289527i	−0.227847	−	0.973697i	$-0.573169\pi$
$461$	231.180	−	133.472i	0.501475	−	0.289527i	0.729322	+	0.684170i	$0.239835\pi$
$462$		0			0
$463$	−1.49652	+	2.59206i	−0.00323223	+	0.00559839i	−0.867637	−	0.497198i	$-0.834362\pi$
$463$	−1.49652	+	2.59206i	−0.00323223	+	0.00559839i	0.864405	+	0.502797i	$0.167696\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	524.277i		−	1.12265i	−0.827596	−	0.561324i	$-0.810292\pi$
$467$		−	524.277i		−	1.12265i	0.827596	−	0.561324i	$-0.189708\pi$
$468$		0			0
$469$	823.213			1.75525
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−4.59315	−	2.65185i	−0.00971067	−	0.00560646i
$474$		0			0
$475$	−10.4361	−	18.0759i	−0.0219708	−	0.0380545i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	253.073	−	146.111i	0.528335	−	0.305034i	−0.212003	−	0.977269i	$-0.567999\pi$
$479$	253.073	−	146.111i	0.528335	−	0.305034i	0.740338	+	0.672235i	$0.234665\pi$
$480$		0			0
$481$	−285.056	+	493.731i	−0.592631	+	1.02647i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			4.77417i			0.00984366i
$486$		0			0
$487$	547.614			1.12446			0.562232	−	0.826980i	$-0.309943\pi$
$487$	547.614			1.12446			0.562232	+	0.826980i	$0.309943\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−402.992	−	232.668i	−0.820758	−	0.473865i	0.0299195	−	0.999552i	$-0.490475\pi$
$491$	−402.992	−	232.668i	−0.820758	−	0.473865i	−0.850678	+	0.525687i	$0.823808\pi$
$492$		0			0
$493$	400.853	+	694.298i	0.813089	+	1.40831i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	323.850	−	186.975i	0.651610	−	0.376207i
$498$		0			0
$499$	427.591	−	740.609i	0.856896	−	1.48419i	−0.0179798	−	0.999838i	$-0.505723\pi$
$499$	427.591	−	740.609i	0.856896	−	1.48419i	0.874875	−	0.484348i	$-0.160943\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			224.344i			0.446011i	0.974817	+	0.223006i	$0.0715868\pi$
$503$			224.344i			0.446011i	−0.974817	+	0.223006i	$0.928413\pi$
$504$		0			0
$505$	121.937			0.241458
$506$		0			0
$507$		0			0
$508$		0			0
$509$	374.609	+	216.281i	0.735971	+	0.424913i	0.820603	−	0.571499i	$-0.193638\pi$
$509$	374.609	+	216.281i	0.735971	+	0.424913i	−0.0846313	+	0.996412i	$0.526971\pi$
$510$		0			0
$511$	722.236	+	1250.95i	1.41338	+	2.44804i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	90.3590	−	52.1688i	0.175454	−	0.101299i
$516$		0			0
$517$	1.02179	−	1.76979i	0.00197639	−	0.00342320i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			116.573i			0.223748i	0.993722	+	0.111874i	$0.0356853\pi$
$521$			116.573i			0.223748i	−0.993722	+	0.111874i	$0.964315\pi$
$522$		0			0
$523$	179.110			0.342467			0.171233	−	0.985231i	$-0.445225\pi$
$523$	179.110			0.342467			0.171233	+	0.985231i	$0.445225\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1142.05	−	659.365i	−2.16709	−	1.25117i
$528$		0			0
$529$	413.937	+	716.959i	0.782489	+	1.35531i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	662.038	−	382.228i	1.24210	−	0.717125i
$534$		0			0
$535$	−122.563	+	212.285i	−0.229089	+	0.396794i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			109.368i			0.202909i
$540$		0			0
$541$	253.105			0.467846			0.233923	−	0.972255i	$-0.424844\pi$
$541$	253.105			0.467846			0.233923	+	0.972255i	$0.424844\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−68.7983	−	39.7207i	−0.126235	−	0.0728820i
$546$		0			0
$547$	−439.564	−	761.347i	−0.803590	−	1.39186i	−0.917239	−	0.398338i	$-0.869587\pi$
$547$	−439.564	−	761.347i	−0.803590	−	1.39186i	0.113649	−	0.993521i	$-0.463746\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	97.0065	−	56.0068i	0.176055	−	0.101646i
$552$		0			0
$553$	−835.679	+	1447.44i	−1.51117	+	2.61743i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	206.443i		−	0.370635i	−0.982679	−	0.185317i	$-0.940669\pi$
$557$		−	206.443i		−	0.370635i	0.982679	−	0.185317i	$-0.0593313\pi$
$558$		0			0
$559$	−128.159			−0.229265
$560$		0			0
$561$		0			0
$562$		0			0
$563$	434.977	+	251.134i	0.772605	+	0.446064i	0.833803	−	0.552062i	$-0.186159\pi$
$563$	434.977	+	251.134i	0.772605	+	0.446064i	−0.0611981	+	0.998126i	$0.519492\pi$
$564$		0			0
$565$	−166.986	−	289.228i	−0.295551	−	0.511908i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−619.776	+	357.828i	−1.08924	+	0.628871i	−0.933373	−	0.358908i	$-0.883149\pi$
$569$	−619.776	+	357.828i	−1.08924	+	0.628871i	−0.155863	+	0.987779i	$0.549816\pi$
$570$		0			0
$571$	363.664	−	629.885i	0.636890	−	1.10313i	−0.349221	−	0.937040i	$-0.613554\pi$
$571$	363.664	−	629.885i	0.636890	−	1.10313i	0.986111	−	0.166086i	$-0.0531129\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			184.179i			0.320311i
$576$		0			0
$577$	−302.432			−0.524146			−0.262073	−	0.965048i	$-0.584406\pi$
$577$	−302.432			−0.524146			−0.262073	+	0.965048i	$0.584406\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−1706.56	−	985.282i	−2.93728	−	1.69584i
$582$		0			0
$583$	−38.2821	−	66.3065i	−0.0656639	−	0.113733i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	171.308	−	98.9046i	0.291836	−	0.168492i	−0.346933	−	0.937890i	$-0.612777\pi$
$587$	171.308	−	98.9046i	0.291836	−	0.168492i	0.638770	+	0.769398i	$0.279444\pi$
$588$		0			0
$589$	−92.1258	+	159.567i	−0.156411	+	0.270911i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			937.126i			1.58031i	0.612905	+	0.790157i	$0.290001\pi$
$593$			937.126i			1.58031i	−0.612905	+	0.790157i	$0.709999\pi$
$594$		0			0
$595$	910.117			1.52961
$596$		0			0
$597$		0			0
$598$		0			0
$599$	385.795	+	222.739i	0.644065	+	0.371851i	0.786179	−	0.617999i	$-0.212056\pi$
$599$	385.795	+	222.739i	0.644065	+	0.371851i	−0.142114	+	0.989850i	$0.545390\pi$
$600$		0			0
$601$	513.644	+	889.657i	0.854649	+	1.48030i	0.876970	+	0.480544i	$0.159561\pi$
$601$	513.644	+	889.657i	0.854649	+	1.48030i	−0.0223214	+	0.999751i	$0.507106\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	233.074	−	134.565i	0.385246	−	0.222422i
$606$		0			0
$607$	−292.100	+	505.932i	−0.481219	+	0.833496i	−0.999768	−	0.0215519i	$-0.993139\pi$
$607$	−292.100	+	505.932i	−0.481219	+	0.833496i	0.518548	+	0.855048i	$0.326473\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	49.3812i		−	0.0808203i
$612$		0			0
$613$	482.798			0.787599			0.393799	−	0.919196i	$-0.371160\pi$
$613$	482.798			0.787599			0.393799	+	0.919196i	$0.371160\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	343.653	+	198.408i	0.556975	+	0.321570i	0.751930	−	0.659243i	$-0.229123\pi$
$617$	343.653	+	198.408i	0.556975	+	0.321570i	−0.194956	+	0.980812i	$0.562456\pi$
$618$		0			0
$619$	−390.523	−	676.406i	−0.630894	−	1.09274i	−0.987369	−	0.158435i	$-0.949355\pi$
$619$	−390.523	−	676.406i	−0.630894	−	1.09274i	0.356476	−	0.934305i	$-0.383978\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	1299.44	−	750.230i	2.08577	−	1.20422i
$624$		0			0
$625$	−12.5000	+	21.6506i	−0.0200000	+	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			880.320i			1.39955i
$630$		0			0
$631$	−565.808			−0.896685			−0.448343	−	0.893862i	$-0.647986\pi$
$631$	−565.808			−0.896685			−0.448343	+	0.893862i	$0.647986\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	84.7989	+	48.9587i	0.133542	+	0.0771003i
$636$		0			0
$637$	1321.38	+	2288.70i	2.07439	+	3.59294i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	355.533	−	205.267i	0.554654	−	0.320230i	−0.196343	−	0.980535i	$-0.562907\pi$
$641$	355.533	−	205.267i	0.554654	−	0.320230i	0.750997	+	0.660306i	$0.229573\pi$
$642$		0			0
$643$	411.455	−	712.661i	0.639899	−	1.10834i	−0.345556	−	0.938398i	$-0.612310\pi$
$643$	411.455	−	712.661i	0.639899	−	1.10834i	0.985455	−	0.169939i	$-0.0543572\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			939.786i			1.45253i	0.687415	+	0.726265i	$0.258745\pi$
$647$			939.786i			1.45253i	−0.687415	+	0.726265i	$0.741255\pi$
$648$		0			0
$649$	−55.6505			−0.0857481
$650$		0			0
$651$		0			0
$652$		0			0
$653$	854.214	+	493.181i	1.30814	+	0.755254i	0.981785	−	0.189994i	$-0.0608468\pi$
$653$	854.214	+	493.181i	1.30814	+	0.755254i	0.326353	+	0.945248i	$0.394180\pi$
$654$		0			0
$655$	−83.5935	−	144.788i	−0.127624	−	0.221051i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−171.408	+	98.9624i	−0.260103	+	0.150171i	−0.624382	−	0.781120i	$-0.714649\pi$
$659$	−171.408	+	98.9624i	−0.260103	+	0.150171i	0.364279	+	0.931290i	$0.381316\pi$
$660$		0			0
$661$	465.837	−	806.853i	0.704746	−	1.22066i	−0.262038	−	0.965058i	$-0.584394\pi$
$661$	465.837	−	806.853i	0.704746	−	1.22066i	0.966783	−	0.255598i	$-0.0822722\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	127.161i		−	0.191219i
$666$		0			0
$667$	−988.419			−1.48189
$668$		0			0
$669$		0			0
$670$		0			0
$671$	34.7457	+	20.0605i	0.0517820	+	0.0298964i
$672$		0			0
$673$	−171.466	−	296.989i	−0.254779	−	0.441291i	0.710056	−	0.704145i	$-0.248669\pi$
$673$	−171.466	−	296.989i	−0.254779	−	0.441291i	−0.964836	+	0.262854i	$0.915336\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−509.963	+	294.427i	−0.753269	+	0.434900i	−0.826874	−	0.562387i	$-0.809883\pi$
$677$	−509.963	+	294.427i	−0.753269	+	0.434900i	0.0736046	+	0.997288i	$0.476550\pi$
$678$		0			0
$679$	−14.5429	+	25.1891i	−0.0214182	+	0.0370974i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	393.277i		−	0.575808i	−0.957659	−	0.287904i	$-0.907042\pi$
$683$		−	393.277i		−	0.575808i	0.957659	−	0.287904i	$-0.0929584\pi$
$684$		0			0
$685$	124.752			0.182120
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−1602.23	−	925.050i	−2.32545	−	1.34260i
$690$		0			0
$691$	−448.363	−	776.588i	−0.648861	−	1.12386i	−0.983395	−	0.181477i	$-0.941912\pi$
$691$	−448.363	−	776.588i	−0.648861	−	1.12386i	0.334534	−	0.942384i	$-0.391421\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−170.316	+	98.3321i	−0.245059	+	0.141485i
$696$		0			0
$697$	590.205	−	1022.27i	0.846779	−	1.46666i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	722.580i		−	1.03078i	−0.856954	−	0.515392i	$-0.827646\pi$
$701$		−	722.580i		−	1.03078i	0.856954	−	0.515392i	$-0.172354\pi$
$702$		0			0
$703$	122.997			0.174961
$704$		0			0
$705$		0			0
$706$		0			0
$707$	643.352	+	371.439i	0.909974	+	0.525374i
$708$		0			0
$709$	−301.738	−	522.625i	−0.425582	−	0.737130i	0.570892	−	0.821025i	$-0.306597\pi$
$709$	−301.738	−	522.625i	−0.425582	−	0.737130i	−0.996475	+	0.0838948i	$0.973264\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	1408.03	−	812.928i	1.97480	−	1.14015i
$714$		0			0
$715$	−17.3225	+	30.0034i	−0.0242273	+	0.0419629i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			122.661i			0.170599i	0.996355	+	0.0852996i	$0.0271848\pi$
$719$			122.661i			0.170599i	−0.996355	+	0.0852996i	$0.972815\pi$
$720$		0			0
$721$	635.660			0.881636
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−116.191	−	67.0829i	−0.160263	−	0.0925281i
$726$		0			0
$727$	−482.148	−	835.105i	−0.663202	−	1.14870i	−0.979769	−	0.200130i	$-0.935864\pi$
$727$	−482.148	−	835.105i	−0.663202	−	1.14870i	0.316567	−	0.948570i	$-0.397470\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−171.380	+	98.9464i	−0.234446	+	0.135358i
$732$		0			0
$733$	−124.695	+	215.978i	−0.170116	+	0.294649i	−0.938460	−	0.345387i	$-0.887748\pi$
$733$	−124.695	+	215.978i	−0.170116	+	0.294649i	0.768344	+	0.640037i	$0.221081\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			48.3878i			0.0656551i
$738$		0			0
$739$	1144.68			1.54896			0.774478	−	0.632601i	$-0.218013\pi$
$739$	1144.68			1.54896			0.774478	+	0.632601i	$0.218013\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	209.289	+	120.833i	0.281681	+	0.162629i	0.634184	−	0.773182i	$-0.281336\pi$
$743$	209.289	+	120.833i	0.281681	+	0.162629i	−0.352503	+	0.935811i	$0.614669\pi$
$744$		0			0
$745$	−97.5846	−	169.021i	−0.130986	−	0.226874i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1293.31	+	746.694i	−1.72672	+	0.996921i
$750$		0			0
$751$	74.5373	−	129.102i	0.0992507	−	0.171907i	−0.812124	−	0.583485i	$-0.801689\pi$
$751$	74.5373	−	129.102i	0.0992507	−	0.171907i	0.911375	+	0.411578i	$0.135022\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			265.448i			0.351587i
$756$		0			0
$757$	−618.284			−0.816756			−0.408378	−	0.912813i	$-0.633906\pi$
$757$	−618.284			−0.816756			−0.408378	+	0.912813i	$0.633906\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−306.195	−	176.782i	−0.402358	−	0.232302i	0.285143	−	0.958485i	$-0.407959\pi$
$761$	−306.195	−	176.782i	−0.402358	−	0.232302i	−0.687501	+	0.726183i	$0.741292\pi$
$762$		0			0
$763$	−241.992	−	419.142i	−0.317158	−	0.549335i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1164.58	+	672.371i	−1.51836	+	0.876624i
$768$		0			0
$769$	−498.004	+	862.568i	−0.647599	+	1.12167i	0.336095	+	0.941828i	$0.390894\pi$
$769$	−498.004	+	862.568i	−0.647599	+	1.12167i	−0.983695	+	0.179847i	$0.942440\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	1049.09i		−	1.35716i	−0.734526	−	0.678580i	$-0.762596\pi$
$773$		−	1049.09i		−	1.35716i	0.734526	−	0.678580i	$-0.237404\pi$
$774$		0			0
$775$	220.690			0.284761
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−142.830	−	82.4628i	−0.183350	−	0.105857i
$780$		0			0
$781$	10.9902	+	19.0357i	0.0140720	+	0.0243735i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−163.471	+	94.3799i	−0.208243	+	0.120229i
$786$		0			0
$787$	642.200	−	1112.32i	0.816010	−	1.41337i	−0.0925902	−	0.995704i	$-0.529515\pi$
$787$	642.200	−	1112.32i	0.816010	−	1.41337i	0.908600	−	0.417667i	$-0.137152\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	2034.67i		−	2.57228i
$792$		0			0
$793$	969.484			1.22255
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−637.579	−	368.106i	−0.799973	−	0.461865i	0.0434884	−	0.999054i	$-0.486153\pi$
$797$	−637.579	−	368.106i	−0.799973	−	0.461865i	−0.843462	+	0.537189i	$0.819486\pi$
$798$		0			0
$799$	−38.1252	−	66.0349i	−0.0477162	−	0.0826469i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−73.5298	+	42.4525i	−0.0915689	+	0.0528673i
$804$		0			0
$805$	−561.040	+	971.749i	−0.696944	+	1.20714i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			920.879i			1.13829i	0.822236	+	0.569147i	$0.192726\pi$
$809$			920.879i			1.13829i	−0.822236	+	0.569147i	$0.807274\pi$
$810$		0			0
$811$	15.6840			0.0193391			0.00966956	−	0.999953i	$-0.496922\pi$
$811$	15.6840			0.0193391			0.00966956	+	0.999953i	$0.496922\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	321.869	+	185.831i	0.394932	+	0.228014i
$816$		0			0
$817$	13.8247	+	23.9451i	0.0169213	+	0.0293085i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−316.094	+	182.497i	−0.385011	+	0.222286i	−0.679996	−	0.733216i	$-0.738019\pi$
$821$	−316.094	+	182.497i	−0.385011	+	0.222286i	0.294985	+	0.955502i	$0.404685\pi$
$822$		0			0
$823$	622.411	−	1078.05i	0.756271	−	1.30990i	−0.188469	−	0.982079i	$-0.560352\pi$
$823$	622.411	−	1078.05i	0.756271	−	1.30990i	0.944740	−	0.327821i	$-0.106314\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	492.788i		−	0.595875i	−0.954585	−	0.297937i	$-0.903701\pi$
$827$		−	492.788i		−	0.595875i	0.954585	−	0.297937i	$-0.0962987\pi$
$828$		0			0
$829$	−359.326			−0.433445			−0.216723	−	0.976233i	$-0.569537\pi$
$829$	−359.326			−0.433445			−0.216723	+	0.976233i	$0.569537\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	3534.03	+	2040.37i	4.24253	+	2.44943i
$834$		0			0
$835$	−149.676	−	259.247i	−0.179253	−	0.310475i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−679.096	+	392.076i	−0.809411	+	0.467314i	−0.846751	−	0.531989i	$-0.821445\pi$
$839$	−679.096	+	392.076i	−0.809411	+	0.467314i	0.0373400	+	0.999303i	$0.488112\pi$
$840$		0			0
$841$	−60.4910	+	104.774i	−0.0719275	+	0.124582i
$842$		0			0
$843$		0			0
$844$		0			0
$845$			459.268i			0.543512i
$846$		0			0
$847$	1639.63			1.93581
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−939.934	−	542.671i	−1.10451	−	0.637686i
$852$		0			0
$853$	−577.196	−	999.733i	−0.676666	−	1.17202i	−0.975979	−	0.217865i	$-0.930091\pi$
$853$	−577.196	−	999.733i	−0.676666	−	1.17202i	0.299313	−	0.954155i	$-0.403243\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	207.818	−	119.983i	0.242494	−	0.140004i	−0.373828	−	0.927498i	$-0.621955\pi$
$857$	207.818	−	119.983i	0.242494	−	0.140004i	0.616323	+	0.787494i	$0.288622\pi$
$858$		0			0
$859$	−10.6795	+	18.4974i	−0.0124325	+	0.0215337i	−0.872175	−	0.489195i	$-0.837291\pi$
$859$	−10.6795	+	18.4974i	−0.0124325	+	0.0215337i	0.859742	+	0.510728i	$0.170624\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			632.227i			0.732592i	0.930498	+	0.366296i	$0.119374\pi$
$863$			632.227i			0.732592i	−0.930498	+	0.366296i	$0.880626\pi$
$864$		0			0
$865$	−261.325			−0.302110
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−85.0794	−	49.1206i	−0.0979049	−	0.0565254i
$870$		0			0
$871$	584.623	+	1012.60i	0.671209	+	1.16257i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−131.903	+	76.1542i	−0.150746	+	0.0870334i
$876$		0			0
$877$	−358.608	+	621.127i	−0.408903	+	0.708240i	−0.994767	−	0.102169i	$-0.967422\pi$
$877$	−358.608	+	621.127i	−0.408903	+	0.708240i	0.585864	+	0.810409i	$0.300755\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	1308.02i		−	1.48470i	−0.670010	−	0.742352i	$-0.733710\pi$
$881$		−	1308.02i		−	1.48470i	0.670010	−	0.742352i	$-0.266290\pi$
$882$		0			0
$883$	−1031.04			−1.16765			−0.583827	−	0.811878i	$-0.698446\pi$
$883$	−1031.04			−1.16765			−0.583827	+	0.811878i	$0.698446\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−203.754	−	117.637i	−0.229711	−	0.132624i	0.380728	−	0.924687i	$-0.375674\pi$
$887$	−203.754	−	117.637i	−0.229711	−	0.132624i	−0.610439	+	0.792063i	$0.709007\pi$
$888$		0			0
$889$	298.273	+	516.623i	0.335515	+	0.581129i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−9.22632	+	5.32682i	−0.0103318	+	0.00596508i
$894$		0			0
$895$	−308.523	+	534.377i	−0.344718	+	0.597069i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			1184.36i			1.31742i
$900$		0			0
$901$	−2856.77			−3.17067
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−63.5238	−	36.6755i	−0.0701921	−	0.0405254i
$906$		0			0
$907$	451.548	+	782.103i	0.497847	+	0.862297i	0.999997	−	0.00248388i	$-0.000790645\pi$
$907$	451.548	+	782.103i	0.497847	+	0.862297i	−0.502150	+	0.864781i	$0.667457\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−738.999	+	426.661i	−0.811195	+	0.468344i	−0.847371	−	0.531002i	$-0.821816\pi$
$911$	−738.999	+	426.661i	−0.811195	+	0.468344i	0.0361754	+	0.999345i	$0.488482\pi$
$912$		0			0
$913$	57.9141	−	100.310i	0.0634327	−	0.109869i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	1018.56i		−	1.11075i
$918$		0			0
$919$	−386.159			−0.420195			−0.210097	−	0.977680i	$-0.567378\pi$
$919$	−386.159			−0.420195			−0.210097	+	0.977680i	$0.567378\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	459.979	+	265.569i	0.498352	+	0.287724i
$924$		0			0
$925$	−73.6609	−	127.584i	−0.0796334	−	0.137929i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−511.615	+	295.381i	−0.550715	+	0.317956i	−0.749410	−	0.662106i	$-0.769663\pi$
$929$	−511.615	+	295.381i	−0.550715	+	0.317956i	0.198695	+	0.980061i	$0.436330\pi$
$930$		0			0
$931$	285.079	−	493.771i	0.306207	−	0.530367i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			53.4960i			0.0572150i
$936$		0			0
$937$	−320.489			−0.342037			−0.171019	−	0.985268i	$-0.554706\pi$
$937$	−320.489			−0.342037			−0.171019	+	0.985268i	$0.554706\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	356.838	+	206.020i	0.379211	+	0.218938i	0.677475	−	0.735546i	$-0.263074\pi$
$941$	356.838	+	206.020i	0.379211	+	0.218938i	−0.298264	+	0.954483i	$0.596408\pi$
$942$		0			0
$943$	727.661	+	1260.35i	0.771645	+	1.33653i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	937.566	−	541.304i	0.990038	−	0.571599i	0.0847521	−	0.996402i	$-0.472990\pi$
$947$	937.566	−	541.304i	0.990038	−	0.571599i	0.905286	+	0.424804i	$0.139657\pi$
$948$		0			0
$949$	−1025.82	+	1776.78i	−1.08095	+	1.87226i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			823.933i			0.864567i	0.901738	+	0.432284i	$0.142292\pi$
$953$			823.933i			0.864567i	−0.901738	+	0.432284i	$0.857708\pi$
$954$		0			0
$955$	−118.182			−0.123751
$956$		0			0
$957$		0			0
$958$		0			0
$959$	658.206	+	380.015i	0.686346	+	0.396262i
$960$		0			0
$961$	−493.582	−	854.909i	−0.513613	−	0.889603i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	343.661	−	198.413i	0.356126	−	0.205609i
$966$		0			0
$967$	21.3125	−	36.9144i	0.0220398	−	0.0381741i	−0.854795	−	0.518966i	$-0.826317\pi$
$967$	21.3125	−	36.9144i	0.0220398	−	0.0381741i	0.876835	+	0.480791i	$0.159651\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			641.156i			0.660305i	0.943928	+	0.330153i	$0.107100\pi$
$971$			641.156i			0.660305i	−0.943928	+	0.330153i	$0.892900\pi$
$972$		0			0
$973$	−1198.14			−1.23139
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1223.05	−	706.129i	−1.25184	−	0.722753i	−0.280369	−	0.959892i	$-0.590457\pi$
$977$	−1223.05	−	706.129i	−1.25184	−	0.722753i	−0.971476	+	0.237140i	$0.923790\pi$
$978$		0			0
$979$	44.0979	+	76.3798i	0.0450438	+	0.0780182i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	380.785	−	219.846i	0.387370	−	0.223648i	−0.293650	−	0.955913i	$-0.594870\pi$
$983$	380.785	−	219.846i	0.387370	−	0.223648i	0.681020	+	0.732265i	$0.261537\pi$
$984$		0			0
$985$	289.472	−	501.380i	0.293880	−	0.509015i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	243.981i		−	0.246695i
$990$		0			0
$991$	−938.347			−0.946869			−0.473434	−	0.880829i	$-0.656986\pi$
$991$	−938.347			−0.946869			−0.473434	+	0.880829i	$0.656986\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	151.661	+	87.5615i	0.152423	+	0.0880015i
$996$		0			0
$997$	−986.454	−	1708.59i	−0.989422	−	1.71373i	−0.620340	−	0.784333i	$-0.713006\pi$
$997$	−986.454	−	1708.59i	−0.989422	−	1.71373i	−0.369082	−	0.929397i	$-0.620328\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.3.o.g.701.6		32
3.2	odd	2	inner	1620.3.o.g.701.11		32
9.2	odd	6	inner	1620.3.o.g.1241.7		32
9.4	even	3		1620.3.g.c.161.16	yes	16
9.5	odd	6		1620.3.g.c.161.8	✓	16
9.7	even	3	inner	1620.3.o.g.1241.11		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1620.3.g.c.161.8	✓	16	9.5	odd	6
1620.3.g.c.161.16	yes	16	9.4	even	3
1620.3.o.g.701.6		32	1.1	even	1	trivial
1620.3.o.g.701.11		32	3.2	odd	2	inner
1620.3.o.g.1241.7		32	9.2	odd	6	inner
1620.3.o.g.1241.11		32	9.7	even	3	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:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1620.o (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.93649 - 1.11803i) q^{5} +(-6.81144 - 11.7978i) q^{7} +O(q^{10})\)	\(q+(-1.93649 - 1.11803i) q^{5} +(-6.81144 - 11.7978i) q^{7} +(0.693463 - 0.400371i) q^{11} +(9.67459 - 16.7569i) q^{13} -29.8774i q^{17} -4.17445 q^{19} +(31.9007 + 18.4179i) q^{23} +(2.50000 + 4.33013i) q^{25} +(-23.2382 + 13.4166i) q^{29} +(22.0690 - 38.2246i) q^{31} +30.4617i q^{35} -29.4644 q^{37} +(34.2153 + 19.7542i) q^{41} +(-3.31174 - 5.73611i) q^{43} +(2.21019 - 1.27605i) q^{47} +(-68.2915 + 118.284i) q^{49} -95.6164i q^{53} -1.79051 q^{55} +(-60.1876 - 34.7493i) q^{59} +(25.0523 + 43.3919i) q^{61} +(-37.4695 + 21.6330i) q^{65} +(-30.2143 + 52.3328i) q^{67} +27.4501i q^{71} -106.033 q^{73} +(-9.44697 - 5.45421i) q^{77} +(-61.3438 - 106.251i) q^{79} +(125.271 - 72.3255i) q^{83} +(-33.4040 + 57.8574i) q^{85} +110.143i q^{89} -263.592 q^{91} +(8.08378 + 4.66717i) q^{95} +(-1.06754 - 1.84903i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 8 q^{7}+O(q^{10}) \) 32 * q - 8 * q^7	\( 32 q - 8 q^{7} + 40 q^{13} + 112 q^{19} + 80 q^{25} + 64 q^{31} - 176 q^{37} - 128 q^{43} - 216 q^{49} - 8 q^{61} + 40 q^{67} + 112 q^{73} + 136 q^{79} - 784 q^{91} - 128 q^{97}+O(q^{100}) \) 32 * q - 8 * q^7 + 40 * q^13 + 112 * q^19 + 80 * q^25 + 64 * q^31 - 176 * q^37 - 128 * q^43 - 216 * q^49 - 8 * q^61 + 40 * q^67 + 112 * q^73 + 136 * q^79 - 784 * q^91 - 128 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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