LMFDB - Embedded newform 1620.3.o.c.701.1

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:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 25 $ x^4 - 5*x^2 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 540)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			701.1
Root			$1.93649 + 1.11803i$ of defining polynomial
Character	$\chi$	$=$	1620.701
Dual form			1620.3.o.c.1241.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.93649 - 1.11803i) q^{5} +(2.00000 + 3.46410i) q^{7} +O(q^{10})$	$q+(-1.93649 - 1.11803i) q^{5} +(2.00000 + 3.46410i) q^{7} +(-5.80948 + 3.35410i) q^{11} +(3.50000 - 6.06218i) q^{13} -20.1246i q^{17} +8.00000 q^{19} +(5.80948 + 3.35410i) q^{23} +(2.50000 + 4.33013i) q^{25} +(-40.6663 + 23.4787i) q^{29} +(-14.5000 + 25.1147i) q^{31} -8.94427i q^{35} +2.00000 q^{37} +(11.6190 + 6.70820i) q^{41} +(3.50000 + 6.06218i) q^{43} +(-29.0474 + 16.7705i) q^{47} +(16.5000 - 28.5788i) q^{49} -93.9149i q^{53} +15.0000 q^{55} +(-34.8569 - 20.1246i) q^{59} +(-31.0000 - 53.6936i) q^{61} +(-13.5554 + 7.82624i) q^{65} +(29.0000 - 50.2295i) q^{67} -53.6656i q^{71} -52.0000 q^{73} +(-23.2379 - 13.4164i) q^{77} +(24.5000 + 42.4352i) q^{79} +(11.6190 - 6.70820i) q^{83} +(-22.5000 + 38.9711i) q^{85} -107.331i q^{89} +28.0000 q^{91} +(-15.4919 - 8.94427i) q^{95} +(17.0000 + 29.4449i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{7}+O(q^{10}) $ 4 * q + 8 * q^7	$ 4 q + 8 q^{7} + 14 q^{13} + 32 q^{19} + 10 q^{25} - 58 q^{31} + 8 q^{37} + 14 q^{43} + 66 q^{49} + 60 q^{55} - 124 q^{61} + 116 q^{67} - 208 q^{73} + 98 q^{79} - 90 q^{85} + 112 q^{91} + 68 q^{97}+O(q^{100}) $ 4 * q + 8 * q^7 + 14 * q^13 + 32 * q^19 + 10 * q^25 - 58 * q^31 + 8 * q^37 + 14 * q^43 + 66 * q^49 + 60 * q^55 - 124 * q^61 + 116 * q^67 - 208 * q^73 + 98 * q^79 - 90 * q^85 + 112 * q^91 + 68 * 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
$7$	2.00000	+	3.46410i	0.285714	+	0.494872i	0.972782	−	0.231722i	$-0.0744358\pi$
$7$	2.00000	+	3.46410i	0.285714	+	0.494872i	−0.687068	+	0.726593i	$0.741103\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−5.80948	+	3.35410i	−0.528134	+	0.304918i	−0.740256	−	0.672325i	$-0.765296\pi$
$11$	−5.80948	+	3.35410i	−0.528134	+	0.304918i	0.212122	+	0.977243i	$0.431963\pi$
$12$		0			0
$13$	3.50000	−	6.06218i	0.269231	−	0.466321i	−0.699433	−	0.714699i	$-0.746564\pi$
$13$	3.50000	−	6.06218i	0.269231	−	0.466321i	0.968663	+	0.248377i	$0.0798972\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	20.1246i		−	1.18380i	−0.806011	−	0.591900i	$-0.798378\pi$
$17$		−	20.1246i		−	1.18380i	0.806011	−	0.591900i	$-0.201622\pi$
$18$		0			0
$19$	8.00000			0.421053			0.210526	−	0.977588i	$-0.432482\pi$
$19$	8.00000			0.421053			0.210526	+	0.977588i	$0.432482\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	5.80948	+	3.35410i	0.252586	+	0.145831i	0.620948	−	0.783852i	$-0.286748\pi$
$23$	5.80948	+	3.35410i	0.252586	+	0.145831i	−0.368362	+	0.929682i	$0.620081\pi$
$24$		0			0
$25$	2.50000	+	4.33013i	0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−40.6663	+	23.4787i	−1.40229	+	0.809611i	−0.994627	−	0.103523i	$-0.966988\pi$
$29$	−40.6663	+	23.4787i	−1.40229	+	0.809611i	−0.407660	+	0.913134i	$0.633655\pi$
$30$		0			0
$31$	−14.5000	+	25.1147i	−0.467742	+	0.810153i	−0.999321	−	0.0368561i	$-0.988266\pi$
$31$	−14.5000	+	25.1147i	−0.467742	+	0.810153i	0.531579	+	0.847009i	$0.321599\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	8.94427i		−	0.255551i
$36$		0			0
$37$	2.00000			0.0540541			0.0270270	−	0.999635i	$-0.491396\pi$
$37$	2.00000			0.0540541			0.0270270	+	0.999635i	$0.491396\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	11.6190	+	6.70820i	0.283389	+	0.163615i	0.634957	−	0.772548i	$-0.281018\pi$
$41$	11.6190	+	6.70820i	0.283389	+	0.163615i	−0.351568	+	0.936162i	$0.614351\pi$
$42$		0			0
$43$	3.50000	+	6.06218i	0.0813953	+	0.140981i	0.903850	−	0.427851i	$-0.140729\pi$
$43$	3.50000	+	6.06218i	0.0813953	+	0.140981i	−0.822454	+	0.568831i	$0.807396\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−29.0474	+	16.7705i	−0.618029	+	0.356819i	−0.776101	−	0.630608i	$-0.782805\pi$
$47$	−29.0474	+	16.7705i	−0.618029	+	0.356819i	0.158072	+	0.987428i	$0.449472\pi$
$48$		0			0
$49$	16.5000	−	28.5788i	0.336735	−	0.583242i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	93.9149i		−	1.77198i	−0.463706	−	0.885989i	$-0.653481\pi$
$53$		−	93.9149i		−	1.77198i	0.463706	−	0.885989i	$-0.346519\pi$
$54$		0			0
$55$	15.0000			0.272727
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−34.8569	−	20.1246i	−0.590794	−	0.341095i	0.174617	−	0.984636i	$-0.444131\pi$
$59$	−34.8569	−	20.1246i	−0.590794	−	0.341095i	−0.765411	+	0.643541i	$0.777464\pi$
$60$		0			0
$61$	−31.0000	−	53.6936i	−0.508197	−	0.880223i	−0.999955	−	0.00949076i	$-0.996979\pi$
$61$	−31.0000	−	53.6936i	−0.508197	−	0.880223i	0.491758	−	0.870732i	$-0.336354\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−13.5554	+	7.82624i	−0.208545	+	0.120404i
$66$		0			0
$67$	29.0000	−	50.2295i	0.432836	−	0.749694i	−0.564280	−	0.825583i	$-0.690846\pi$
$67$	29.0000	−	50.2295i	0.432836	−	0.749694i	0.997116	+	0.0758896i	$0.0241796\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	53.6656i		−	0.755854i	−0.925835	−	0.377927i	$-0.876637\pi$
$71$		−	53.6656i		−	0.755854i	0.925835	−	0.377927i	$-0.123363\pi$
$72$		0			0
$73$	−52.0000			−0.712329			−0.356164	−	0.934423i	$-0.615916\pi$
$73$	−52.0000			−0.712329			−0.356164	+	0.934423i	$0.615916\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−23.2379	−	13.4164i	−0.301791	−	0.174239i
$78$		0			0
$79$	24.5000	+	42.4352i	0.310127	+	0.537155i	0.978390	−	0.206770i	$-0.0662952\pi$
$79$	24.5000	+	42.4352i	0.310127	+	0.537155i	−0.668263	+	0.743925i	$0.732962\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	11.6190	−	6.70820i	0.139987	−	0.0808217i	−0.428371	−	0.903603i	$-0.640912\pi$
$83$	11.6190	−	6.70820i	0.139987	−	0.0808217i	0.568358	+	0.822781i	$0.307579\pi$
$84$		0			0
$85$	−22.5000	+	38.9711i	−0.264706	+	0.458484i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	107.331i		−	1.20597i	−0.797753	−	0.602985i	$-0.793978\pi$
$89$		−	107.331i		−	1.20597i	0.797753	−	0.602985i	$-0.206022\pi$
$90$		0			0
$91$	28.0000			0.307692
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−15.4919	−	8.94427i	−0.163073	−	0.0941502i
$96$		0			0
$97$	17.0000	+	29.4449i	0.175258	+	0.303555i	0.940250	−	0.340484i	$-0.110591\pi$
$97$	17.0000	+	29.4449i	0.175258	+	0.303555i	−0.764993	+	0.644039i	$0.777257\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−87.1421	+	50.3115i	−0.862793	+	0.498134i	−0.864947	−	0.501864i	$-0.832648\pi$
$101$	−87.1421	+	50.3115i	−0.862793	+	0.498134i	0.00215338	+	0.999998i	$0.499315\pi$
$102$		0			0
$103$	−49.0000	+	84.8705i	−0.475728	+	0.823985i	−0.999613	−	0.0278035i	$-0.991149\pi$
$103$	−49.0000	+	84.8705i	−0.475728	+	0.823985i	0.523885	+	0.851789i	$0.324482\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		0			0
$109$	26.0000			0.238532			0.119266	−	0.992862i	$-0.461946\pi$
$109$	26.0000			0.238532			0.119266	+	0.992862i	$0.461946\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−156.856	−	90.5608i	−1.38810	−	0.801423i	−0.395003	−	0.918680i	$-0.629256\pi$
$113$	−156.856	−	90.5608i	−1.38810	−	0.801423i	−0.993102	+	0.117257i	$0.962590\pi$
$114$		0			0
$115$	−7.50000	−	12.9904i	−0.0652174	−	0.112960i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	69.7137	−	40.2492i	0.585829	−	0.338229i
$120$		0			0
$121$	−38.0000	+	65.8179i	−0.314050	+	0.543950i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	11.1803i		−	0.0894427i
$126$		0			0
$127$	−214.000			−1.68504			−0.842520	−	0.538666i	$-0.818929\pi$
$127$	−214.000			−1.68504			−0.842520	+	0.538666i	$0.818929\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−214.951	−	124.102i	−1.64084	−	0.947342i	−0.980534	−	0.196349i	$-0.937092\pi$
$131$	−214.951	−	124.102i	−1.64084	−	0.947342i	−0.660310	−	0.750993i	$-0.729575\pi$
$132$		0			0
$133$	16.0000	+	27.7128i	0.120301	+	0.208367i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−34.8569	+	20.1246i	−0.254430	+	0.146895i	−0.621791	−	0.783183i	$-0.713595\pi$
$137$	−34.8569	+	20.1246i	−0.254430	+	0.146895i	0.367361	+	0.930078i	$0.380261\pi$
$138$		0			0
$139$	32.0000	−	55.4256i	0.230216	−	0.398746i	−0.727656	−	0.685943i	$-0.759390\pi$
$139$	32.0000	−	55.4256i	0.230216	−	0.398746i	0.957872	+	0.287197i	$0.0927235\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			46.9574i			0.328374i
$144$		0			0
$145$	105.000			0.724138
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−133.618	−	77.1443i	−0.896765	−	0.517747i	−0.0206155	−	0.999787i	$-0.506563\pi$
$149$	−133.618	−	77.1443i	−0.896765	−	0.517747i	−0.876149	+	0.482040i	$0.839896\pi$
$150$		0			0
$151$	−143.500	−	248.549i	−0.950331	−	1.64602i	−0.744708	−	0.667391i	$-0.767411\pi$
$151$	−143.500	−	248.549i	−0.950331	−	1.64602i	−0.205623	−	0.978631i	$-0.565922\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	56.1583	−	32.4230i	0.362311	−	0.209181i
$156$		0			0
$157$	96.5000	−	167.143i	0.614650	−	1.06460i	−0.375796	−	0.926702i	$-0.622631\pi$
$157$	96.5000	−	167.143i	0.614650	−	1.06460i	0.990446	−	0.137902i	$-0.0440359\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			26.8328i			0.166663i
$162$		0			0
$163$	83.0000			0.509202			0.254601	−	0.967046i	$-0.418056\pi$
$163$	83.0000			0.509202			0.254601	+	0.967046i	$0.418056\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−267.236	−	154.289i	−1.60021	−	0.923884i	−0.991443	−	0.130539i	$-0.958329\pi$
$167$	−267.236	−	154.289i	−1.60021	−	0.923884i	−0.608771	−	0.793346i	$-0.708337\pi$
$168$		0			0
$169$	60.0000	+	103.923i	0.355030	+	0.614929i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−185.903	+	107.331i	−1.07458	+	0.620412i	−0.929431	−	0.368997i	$-0.879701\pi$
$173$	−185.903	+	107.331i	−1.07458	+	0.620412i	−0.145154	+	0.989409i	$0.546368\pi$
$174$		0			0
$175$	−10.0000	+	17.3205i	−0.0571429	+	0.0989743i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			93.9149i			0.524664i	0.964978	+	0.262332i	$0.0844916\pi$
$179$			93.9149i			0.524664i	−0.964978	+	0.262332i	$0.915508\pi$
$180$		0			0
$181$	164.000			0.906077			0.453039	−	0.891491i	$-0.350340\pi$
$181$	164.000			0.906077			0.453039	+	0.891491i	$0.350340\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−3.87298	−	2.23607i	−0.0209350	−	0.0120869i
$186$		0			0
$187$	67.5000	+	116.913i	0.360963	+	0.625206i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	185.903	−	107.331i	0.973315	−	0.561944i	0.0730699	−	0.997327i	$-0.476720\pi$
$191$	185.903	−	107.331i	0.973315	−	0.561944i	0.900245	+	0.435383i	$0.143387\pi$
$192$		0			0
$193$	8.00000	−	13.8564i	0.0414508	−	0.0717949i	−0.844556	−	0.535468i	$-0.820135\pi$
$193$	8.00000	−	13.8564i	0.0414508	−	0.0717949i	0.886006	+	0.463673i	$0.153469\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	187.830i		−	0.953450i	−0.879052	−	0.476725i	$-0.841824\pi$
$197$		−	187.830i		−	0.953450i	0.879052	−	0.476725i	$-0.158176\pi$
$198$		0			0
$199$	−139.000			−0.698492			−0.349246	−	0.937031i	$-0.613562\pi$
$199$	−139.000			−0.698492			−0.349246	+	0.937031i	$0.613562\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−162.665	−	93.9149i	−0.801307	−	0.462635i
$204$		0			0
$205$	−15.0000	−	25.9808i	−0.0731707	−	0.126735i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−46.4758	+	26.8328i	−0.222372	+	0.128387i
$210$		0			0
$211$	−37.0000	+	64.0859i	−0.175355	+	0.303725i	−0.940284	−	0.340390i	$-0.889441\pi$
$211$	−37.0000	+	64.0859i	−0.175355	+	0.303725i	0.764929	+	0.644115i	$0.222774\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	15.6525i		−	0.0728022i
$216$		0			0
$217$	−116.000			−0.534562
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−121.999	−	70.4361i	−0.552032	−	0.318716i
$222$		0			0
$223$	176.000	+	304.841i	0.789238	+	1.36700i	0.926435	+	0.376456i	$0.122857\pi$
$223$	176.000	+	304.841i	0.789238	+	1.36700i	−0.137197	+	0.990544i	$0.543809\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	185.903	−	107.331i	0.818957	−	0.472825i	−0.0310998	−	0.999516i	$-0.509901\pi$
$227$	185.903	−	107.331i	0.818957	−	0.472825i	0.850057	+	0.526691i	$0.176568\pi$
$228$		0			0
$229$	17.0000	−	29.4449i	0.0742358	−	0.128580i	−0.826518	−	0.562910i	$-0.809682\pi$
$229$	17.0000	−	29.4449i	0.0742358	−	0.128580i	0.900754	+	0.434330i	$0.143015\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			93.9149i			0.403068i	0.979482	+	0.201534i	$0.0645927\pi$
$233$			93.9149i			0.403068i	−0.979482	+	0.201534i	$0.935407\pi$
$234$		0			0
$235$	75.0000			0.319149
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−23.2379	−	13.4164i	−0.0972297	−	0.0561356i	0.450597	−	0.892728i	$-0.351211\pi$
$239$	−23.2379	−	13.4164i	−0.0972297	−	0.0561356i	−0.547826	+	0.836592i	$0.684545\pi$
$240$		0			0
$241$	−209.500	−	362.865i	−0.869295	−	1.50566i	−0.862719	−	0.505684i	$-0.831240\pi$
$241$	−209.500	−	362.865i	−0.869295	−	1.50566i	−0.00657585	−	0.999978i	$-0.502093\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−63.9042	+	36.8951i	−0.260834	+	0.150592i
$246$		0			0
$247$	28.0000	−	48.4974i	0.113360	−	0.196346i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	167.705i		−	0.668148i	−0.942547	−	0.334074i	$-0.891576\pi$
$251$		−	167.705i		−	0.668148i	0.942547	−	0.334074i	$-0.108424\pi$
$252$		0			0
$253$	−45.0000			−0.177866
$254$		0			0
$255$		0			0
$256$		0			0
$257$	17.4284	+	10.0623i	0.0678149	+	0.0391529i	0.533524	−	0.845785i	$-0.320867\pi$
$257$	17.4284	+	10.0623i	0.0678149	+	0.0391529i	−0.465709	+	0.884938i	$0.654201\pi$
$258$		0			0
$259$	4.00000	+	6.92820i	0.0154440	+	0.0267498i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	220.760	−	127.456i	0.839392	−	0.484623i	−0.0176656	−	0.999844i	$-0.505623\pi$
$263$	220.760	−	127.456i	0.839392	−	0.484623i	0.857057	+	0.515221i	$0.172290\pi$
$264$		0			0
$265$	−105.000	+	181.865i	−0.396226	+	0.686284i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	127.456i		−	0.473814i	−0.971532	−	0.236907i	$-0.923866\pi$
$269$		−	127.456i		−	0.473814i	0.971532	−	0.236907i	$-0.0761336\pi$
$270$		0			0
$271$	434.000			1.60148			0.800738	−	0.599015i	$-0.204441\pi$
$271$	434.000			1.60148			0.800738	+	0.599015i	$0.204441\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−29.0474	−	16.7705i	−0.105627	−	0.0609837i
$276$		0			0
$277$	59.0000	+	102.191i	0.212996	+	0.368921i	0.952651	−	0.304066i	$-0.0983444\pi$
$277$	59.0000	+	102.191i	0.212996	+	0.368921i	−0.739655	+	0.672987i	$0.765011\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−243.998	+	140.872i	−0.868320	+	0.501325i	−0.866790	−	0.498674i	$-0.833821\pi$
$281$	−243.998	+	140.872i	−0.868320	+	0.501325i	−0.00153048	+	0.999999i	$0.500487\pi$
$282$		0			0
$283$	−109.000	+	188.794i	−0.385159	+	0.667115i	−0.991791	−	0.127868i	$-0.959187\pi$
$283$	−109.000	+	188.794i	−0.385159	+	0.667115i	0.606632	+	0.794983i	$0.292520\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			53.6656i			0.186988i
$288$		0			0
$289$	−116.000			−0.401384
$290$		0			0
$291$		0			0
$292$		0			0
$293$	255.617	+	147.580i	0.872413	+	0.503688i	0.868149	−	0.496303i	$-0.165310\pi$
$293$	255.617	+	147.580i	0.872413	+	0.503688i	0.00426340	+	0.999991i	$0.498643\pi$
$294$		0			0
$295$	45.0000	+	77.9423i	0.152542	+	0.264211i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	40.6663	−	23.4787i	0.136008	−	0.0785241i
$300$		0			0
$301$	−14.0000	+	24.2487i	−0.0465116	+	0.0805605i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			138.636i			0.454545i
$306$		0			0
$307$	−223.000			−0.726384			−0.363192	−	0.931714i	$-0.618313\pi$
$307$	−223.000			−0.726384			−0.363192	+	0.931714i	$0.618313\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	383.425	+	221.371i	1.23288	+	0.711803i	0.967629	−	0.252377i	$-0.0812122\pi$
$311$	383.425	+	221.371i	1.23288	+	0.711803i	0.265250	+	0.964180i	$0.414546\pi$
$312$		0			0
$313$	281.000	+	486.706i	0.897764	+	1.55497i	0.830347	+	0.557247i	$0.188142\pi$
$313$	281.000	+	486.706i	0.897764	+	1.55497i	0.0674170	+	0.997725i	$0.478524\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−290.474	+	167.705i	−0.916321	+	0.529038i	−0.882460	−	0.470388i	$-0.844114\pi$
$317$	−290.474	+	167.705i	−0.916321	+	0.529038i	−0.0338615	+	0.999427i	$0.510781\pi$
$318$		0			0
$319$	157.500	−	272.798i	0.493730	−	0.855166i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	160.997i		−	0.498442i
$324$		0			0
$325$	35.0000			0.107692
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−116.190	−	67.0820i	−0.353160	−	0.203897i
$330$		0			0
$331$	−322.000	−	557.720i	−0.972810	−	1.68496i	−0.686982	−	0.726675i	$-0.741065\pi$
$331$	−322.000	−	557.720i	−0.972810	−	1.68496i	−0.285828	−	0.958281i	$-0.592269\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−112.317	+	64.8460i	−0.335273	+	0.193570i
$336$		0			0
$337$	239.000	−	413.960i	0.709199	−	1.22837i	−0.255956	−	0.966688i	$-0.582390\pi$
$337$	239.000	−	413.960i	0.709199	−	1.22837i	0.965155	−	0.261680i	$-0.0842765\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	194.538i		−	0.570492i
$342$		0			0
$343$	328.000			0.956268
$344$		0			0
$345$		0			0
$346$		0			0
$347$	429.901	+	248.204i	1.23891	+	0.715284i	0.968871	−	0.247566i	$-0.0796307\pi$
$347$	429.901	+	248.204i	1.23891	+	0.715284i	0.270037	+	0.962850i	$0.412964\pi$
$348$		0			0
$349$	−178.000	−	308.305i	−0.510029	−	0.883396i	−0.999932	−	0.0116191i	$-0.996301\pi$
$349$	−178.000	−	308.305i	−0.510029	−	0.883396i	0.489904	−	0.871776i	$-0.337032\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−121.999	+	70.4361i	−0.345606	+	0.199536i	−0.662748	−	0.748842i	$-0.730610\pi$
$353$	−121.999	+	70.4361i	−0.345606	+	0.199536i	0.317142	+	0.948378i	$0.397277\pi$
$354$		0			0
$355$	−60.0000	+	103.923i	−0.169014	+	0.292741i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	201.246i		−	0.560574i	−0.959916	−	0.280287i	$-0.909570\pi$
$359$		−	201.246i		−	0.560574i	0.959916	−	0.280287i	$-0.0904297\pi$
$360$		0			0
$361$	−297.000			−0.822715
$362$		0			0
$363$		0			0
$364$		0			0
$365$	100.698	+	58.1378i	0.275884	+	0.159282i
$366$		0			0
$367$	−46.0000	−	79.6743i	−0.125341	−	0.217096i	0.796525	−	0.604605i	$-0.206669\pi$
$367$	−46.0000	−	79.6743i	−0.125341	−	0.217096i	−0.921866	+	0.387509i	$0.873336\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	325.331	−	187.830i	0.876902	−	0.506280i
$372$		0			0
$373$	−119.500	+	206.980i	−0.320375	+	0.554906i	−0.980565	−	0.196192i	$-0.937142\pi$
$373$	−119.500	+	206.980i	−0.320375	+	0.554906i	0.660190	+	0.751099i	$0.270476\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			328.702i			0.871889i
$378$		0			0
$379$	−484.000			−1.27704			−0.638522	−	0.769603i	$-0.720454\pi$
$379$	−484.000			−1.27704			−0.638522	+	0.769603i	$0.720454\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	389.235	+	224.725i	1.01628	+	0.586749i	0.913024	−	0.407906i	$-0.133741\pi$
$383$	389.235	+	224.725i	1.01628	+	0.586749i	0.103255	+	0.994655i	$0.467074\pi$
$384$		0			0
$385$	30.0000	+	51.9615i	0.0779221	+	0.134965i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−540.281	+	311.931i	−1.38890	+	0.801880i	−0.993191	−	0.116497i	$-0.962834\pi$
$389$	−540.281	+	311.931i	−1.38890	+	0.801880i	−0.395707	+	0.918377i	$0.629500\pi$
$390$		0			0
$391$	67.5000	−	116.913i	0.172634	−	0.299011i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	109.567i		−	0.277386i
$396$		0			0
$397$	−193.000			−0.486146			−0.243073	−	0.970008i	$-0.578155\pi$
$397$	−193.000			−0.486146			−0.243073	+	0.970008i	$0.578155\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	11.6190	+	6.70820i	0.0289749	+	0.0167287i	0.514418	−	0.857540i	$-0.328008\pi$
$401$	11.6190	+	6.70820i	0.0289749	+	0.0167287i	−0.485443	+	0.874269i	$0.661342\pi$
$402$		0			0
$403$	101.500	+	175.803i	0.251861	+	0.436236i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−11.6190	+	6.70820i	−0.0285478	+	0.0164821i
$408$		0			0
$409$	−215.500	+	373.257i	−0.526895	+	0.912609i	0.472614	+	0.881270i	$0.343311\pi$
$409$	−215.500	+	373.257i	−0.526895	+	0.912609i	−0.999509	+	0.0313391i	$0.990023\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	160.997i		−	0.389823i
$414$		0			0
$415$	−30.0000			−0.0722892
$416$		0			0
$417$		0			0
$418$		0			0
$419$	586.757	+	338.764i	1.40037	+	0.808507i	0.994431	−	0.105391i	$-0.0336095\pi$
$419$	586.757	+	338.764i	1.40037	+	0.808507i	0.405944	+	0.913898i	$0.366943\pi$
$420$		0			0
$421$	−97.0000	−	168.009i	−0.230404	−	0.399071i	0.727523	−	0.686083i	$-0.240671\pi$
$421$	−97.0000	−	168.009i	−0.230404	−	0.399071i	−0.957927	+	0.287012i	$0.907338\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	87.1421	−	50.3115i	0.205040	−	0.118380i
$426$		0			0
$427$	124.000	−	214.774i	0.290398	−	0.502984i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			442.741i			1.02724i	0.858017	+	0.513621i	$0.171696\pi$
$431$			442.741i			1.02724i	−0.858017	+	0.513621i	$0.828304\pi$
$432$		0			0
$433$	278.000			0.642032			0.321016	−	0.947074i	$-0.395976\pi$
$433$	278.000			0.642032			0.321016	+	0.947074i	$0.395976\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	46.4758	+	26.8328i	0.106352	+	0.0614023i
$438$		0			0
$439$	137.000	+	237.291i	0.312073	+	0.540526i	0.978811	−	0.204766i	$-0.0656433\pi$
$439$	137.000	+	237.291i	0.312073	+	0.540526i	−0.666738	+	0.745292i	$0.732310\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	81.3327	−	46.9574i	0.183595	−	0.105999i	−0.405386	−	0.914146i	$-0.632863\pi$
$443$	81.3327	−	46.9574i	0.183595	−	0.105999i	0.588981	+	0.808147i	$0.299529\pi$
$444$		0			0
$445$	−120.000	+	207.846i	−0.269663	+	0.467070i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			643.988i			1.43427i	0.696934	+	0.717135i	$0.254547\pi$
$449$			643.988i			1.43427i	−0.696934	+	0.717135i	$0.745453\pi$
$450$		0			0
$451$	−90.0000			−0.199557
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−54.2218	−	31.3050i	−0.119169	−	0.0688021i
$456$		0			0
$457$	−133.000	−	230.363i	−0.291028	−	0.504076i	0.683025	−	0.730395i	$-0.260664\pi$
$457$	−133.000	−	230.363i	−0.291028	−	0.504076i	−0.974053	+	0.226319i	$0.927331\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−267.236	+	154.289i	−0.579687	+	0.334683i	−0.761009	−	0.648741i	$-0.775296\pi$
$461$	−267.236	+	154.289i	−0.579687	+	0.334683i	0.181322	+	0.983424i	$0.441962\pi$
$462$		0			0
$463$	−229.000	+	396.640i	−0.494600	+	0.856673i	−0.999981	−	0.00622370i	$-0.998019\pi$
$463$	−229.000	+	396.640i	−0.494600	+	0.856673i	0.505380	+	0.862897i	$0.331352\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	496.407i		−	1.06297i	−0.847068	−	0.531485i	$-0.821634\pi$
$467$		−	496.407i		−	1.06297i	0.847068	−	0.531485i	$-0.178366\pi$
$468$		0			0
$469$	232.000			0.494670
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−40.6663	−	23.4787i	−0.0859753	−	0.0496379i
$474$		0			0
$475$	20.0000	+	34.6410i	0.0421053	+	0.0729285i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	522.853	−	301.869i	1.09155	−	0.630207i	0.157562	−	0.987509i	$-0.449637\pi$
$479$	522.853	−	301.869i	1.09155	−	0.630207i	0.933989	+	0.357302i	$0.116303\pi$
$480$		0			0
$481$	7.00000	−	12.1244i	0.0145530	−	0.0252066i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	76.0263i		−	0.156755i
$486$		0			0
$487$	32.0000			0.0657084			0.0328542	−	0.999460i	$-0.489540\pi$
$487$	32.0000			0.0657084			0.0328542	+	0.999460i	$0.489540\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	406.663	+	234.787i	0.828235	+	0.478182i	0.853248	−	0.521506i	$-0.174629\pi$
$491$	406.663	+	234.787i	0.828235	+	0.478182i	−0.0250131	+	0.999687i	$0.507963\pi$
$492$		0			0
$493$	472.500	+	818.394i	0.958418	+	1.66003i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	185.903	−	107.331i	0.374051	−	0.215958i
$498$		0			0
$499$	257.000	−	445.137i	0.515030	−	0.892058i	−0.484818	−	0.874615i	$-0.661114\pi$
$499$	257.000	−	445.137i	0.515030	−	0.892058i	0.999848	−	0.0174431i	$-0.00555258\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			288.453i			0.573465i	0.958011	+	0.286732i	$0.0925690\pi$
$503$			288.453i			0.573465i	−0.958011	+	0.286732i	$0.907431\pi$
$504$		0			0
$505$	225.000			0.445545
$506$		0			0
$507$		0			0
$508$		0			0
$509$	412.473	+	238.141i	0.810359	+	0.467861i	0.847081	−	0.531465i	$-0.178358\pi$
$509$	412.473	+	238.141i	0.810359	+	0.467861i	−0.0367215	+	0.999326i	$0.511691\pi$
$510$		0			0
$511$	−104.000	−	180.133i	−0.203523	−	0.352511i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	189.776	−	109.567i	0.368497	−	0.212752i
$516$		0			0
$517$	112.500	−	194.856i	0.217602	−	0.376897i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			576.906i			1.10730i	0.832748	+	0.553652i	$0.186766\pi$
$521$			576.906i			1.10730i	−0.832748	+	0.553652i	$0.813234\pi$
$522$		0			0
$523$	689.000			1.31740			0.658700	−	0.752406i	$-0.271107\pi$
$523$	689.000			1.31740			0.658700	+	0.752406i	$0.271107\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	505.424	+	291.807i	0.959059	+	0.553713i
$528$		0			0
$529$	−242.000	−	419.156i	−0.457467	−	0.792356i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	81.3327	−	46.9574i	0.152594	−	0.0881002i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$			221.371i			0.410706i
$540$		0			0
$541$	−628.000			−1.16081			−0.580407	−	0.814327i	$-0.697106\pi$
$541$	−628.000			−1.16081			−0.580407	+	0.814327i	$0.697106\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−50.3488	−	29.0689i	−0.0923831	−	0.0533374i
$546$		0			0
$547$	−308.500	−	534.338i	−0.563985	−	0.976851i	−0.997143	−	0.0755333i	$-0.975934\pi$
$547$	−308.500	−	534.338i	−0.563985	−	0.976851i	0.433158	−	0.901318i	$-0.357399\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−325.331	+	187.830i	−0.590437	+	0.340889i
$552$		0			0
$553$	−98.0000	+	169.741i	−0.177215	+	0.306946i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			711.070i			1.27661i	0.769785	+	0.638303i	$0.220363\pi$
$557$			711.070i			1.27661i	−0.769785	+	0.638303i	$0.779637\pi$
$558$		0			0
$559$	49.0000			0.0876565
$560$		0			0
$561$		0			0
$562$		0			0
$563$	162.665	+	93.9149i	0.288926	+	0.166811i	0.637457	−	0.770486i	$-0.279986\pi$
$563$	162.665	+	93.9149i	0.288926	+	0.166811i	−0.348531	+	0.937297i	$0.613320\pi$
$564$		0			0
$565$	202.500	+	350.740i	0.358407	+	0.620779i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	580.948	−	335.410i	1.02100	−	0.589473i	0.106605	−	0.994301i	$-0.466002\pi$
$569$	580.948	−	335.410i	1.02100	−	0.589473i	0.914393	+	0.404828i	$0.132669\pi$
$570$		0			0
$571$	509.000	−	881.614i	0.891419	−	1.54398i	0.0532431	−	0.998582i	$-0.483044\pi$
$571$	509.000	−	881.614i	0.891419	−	1.54398i	0.838175	−	0.545401i	$-0.183622\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			33.5410i			0.0583322i
$576$		0			0
$577$	356.000			0.616984			0.308492	−	0.951227i	$-0.400176\pi$
$577$	356.000			0.616984			0.308492	+	0.951227i	$0.400176\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	46.4758	+	26.8328i	0.0799928	+	0.0461838i
$582$		0			0
$583$	315.000	+	545.596i	0.540309	+	0.935842i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	708.756	−	409.200i	1.20742	−	0.697105i	0.245226	−	0.969466i	$-0.421138\pi$
$587$	708.756	−	409.200i	1.20742	−	0.697105i	0.962195	+	0.272361i	$0.0878046\pi$
$588$		0			0
$589$	−116.000	+	200.918i	−0.196944	+	0.341117i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			1039.77i			1.75341i	0.481029	+	0.876705i	$0.340263\pi$
$593$			1039.77i			1.75341i	−0.481029	+	0.876705i	$0.659737\pi$
$594$		0			0
$595$	−180.000			−0.302521
$596$		0			0
$597$		0			0
$598$		0			0
$599$	139.427	+	80.4984i	0.232767	+	0.134388i	0.611848	−	0.790975i	$-0.290426\pi$
$599$	139.427	+	80.4984i	0.232767	+	0.134388i	−0.379081	+	0.925364i	$0.623760\pi$
$600$		0			0
$601$	−284.500	−	492.768i	−0.473378	−	0.819914i	0.526158	−	0.850387i	$-0.323632\pi$
$601$	−284.500	−	492.768i	−0.473378	−	0.819914i	−0.999536	+	0.0304727i	$0.990299\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	147.173	−	84.9706i	0.243262	−	0.140447i
$606$		0			0
$607$	−301.000	+	521.347i	−0.495881	+	0.858892i	−0.999989	−	0.00474926i	$-0.998488\pi$
$607$	−301.000	+	521.347i	−0.495881	+	0.858892i	0.504107	+	0.863641i	$0.331822\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			234.787i			0.384267i
$612$		0			0
$613$	−277.000			−0.451876			−0.225938	−	0.974142i	$-0.572545\pi$
$613$	−277.000			−0.451876			−0.225938	+	0.974142i	$0.572545\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−273.045	−	157.643i	−0.442537	−	0.255499i	0.262136	−	0.965031i	$-0.415573\pi$
$617$	−273.045	−	157.643i	−0.442537	−	0.255499i	−0.704673	+	0.709532i	$0.748906\pi$
$618$		0			0
$619$	311.000	+	538.668i	0.502423	+	0.870223i	0.999996	+	0.00280041i	$0.000891399\pi$
$619$	311.000	+	538.668i	0.502423	+	0.870223i	−0.497573	+	0.867422i	$0.665775\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	371.806	−	214.663i	0.596800	−	0.344563i
$624$		0			0
$625$	−12.5000	+	21.6506i	−0.0200000	+	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	40.2492i		−	0.0639892i
$630$		0			0
$631$	−106.000			−0.167987			−0.0839937	−	0.996466i	$-0.526768\pi$
$631$	−106.000			−0.167987			−0.0839937	+	0.996466i	$0.526768\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	414.409	+	239.259i	0.652613	+	0.376786i
$636$		0			0
$637$	−115.500	−	200.052i	−0.181319	−	0.314053i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	1045.71	−	603.738i	1.63137	−	0.941870i	0.647693	−	0.761901i	$-0.275734\pi$
$641$	1045.71	−	603.738i	1.63137	−	0.941870i	0.983672	−	0.179968i	$-0.0575996\pi$
$642$		0			0
$643$	−539.500	+	934.441i	−0.839036	+	1.45325i	0.0516660	+	0.998664i	$0.483547\pi$
$643$	−539.500	+	934.441i	−0.839036	+	1.45325i	−0.890702	+	0.454588i	$0.849786\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			737.902i			1.14050i	0.821472	+	0.570249i	$0.193153\pi$
$647$			737.902i			1.14050i	−0.821472	+	0.570249i	$0.806847\pi$
$648$		0			0
$649$	270.000			0.416025
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−639.042	−	368.951i	−0.978625	−	0.565010i	−0.0767704	−	0.997049i	$-0.524461\pi$
$653$	−639.042	−	368.951i	−0.978625	−	0.565010i	−0.901855	+	0.432039i	$0.857794\pi$
$654$		0			0
$655$	277.500	+	480.644i	0.423664	+	0.733808i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	267.236	−	154.289i	0.405517	−	0.234125i	−0.283345	−	0.959018i	$-0.591444\pi$
$659$	267.236	−	154.289i	0.405517	−	0.234125i	0.688862	+	0.724893i	$0.258111\pi$
$660$		0			0
$661$	563.000	−	975.145i	0.851740	−	1.47526i	−0.0278971	−	0.999611i	$-0.508881\pi$
$661$	563.000	−	975.145i	0.851740	−	1.47526i	0.879637	−	0.475646i	$-0.157786\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	71.5542i		−	0.107600i
$666$		0			0
$667$	−315.000			−0.472264
$668$		0			0
$669$		0			0
$670$		0			0
$671$	360.187	+	207.954i	0.536792	+	0.309917i
$672$		0			0
$673$	461.000	+	798.475i	0.684993	+	1.18644i	0.973439	+	0.228946i	$0.0735279\pi$
$673$	461.000	+	798.475i	0.684993	+	1.18644i	−0.288447	+	0.957496i	$0.593139\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−534.472	+	308.577i	−0.789471	+	0.455801i	−0.839776	−	0.542933i	$-0.817314\pi$
$677$	−534.472	+	308.577i	−0.789471	+	0.455801i	0.0503054	+	0.998734i	$0.483981\pi$
$678$		0			0
$679$	−68.0000	+	117.779i	−0.100147	+	0.173460i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	845.234i		−	1.23753i	−0.785576	−	0.618766i	$-0.787633\pi$
$683$		−	845.234i		−	1.23753i	0.785576	−	0.618766i	$-0.212367\pi$
$684$		0			0
$685$	90.0000			0.131387
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−569.329	−	328.702i	−0.826311	−	0.477071i
$690$		0			0
$691$	−376.000	−	651.251i	−0.544139	−	0.942476i	−0.998661	−	0.0517406i	$-0.983523\pi$
$691$	−376.000	−	651.251i	−0.544139	−	0.942476i	0.454522	−	0.890736i	$-0.349810\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−123.935	+	71.5542i	−0.178324	+	0.102956i
$696$		0			0
$697$	135.000	−	233.827i	0.193687	−	0.335476i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			181.122i			0.258376i	0.991620	+	0.129188i	$0.0412371\pi$
$701$			181.122i			0.258376i	−0.991620	+	0.129188i	$0.958763\pi$
$702$		0			0
$703$	16.0000			0.0227596
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−348.569	−	201.246i	−0.493025	−	0.284648i
$708$		0			0
$709$	−649.000	−	1124.10i	−0.915374	−	1.58547i	−0.806353	−	0.591434i	$-0.798562\pi$
$709$	−649.000	−	1124.10i	−0.915374	−	1.58547i	−0.109021	−	0.994039i	$-0.534771\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−168.475	+	97.2690i	−0.236290	+	0.136422i
$714$		0			0
$715$	52.5000	−	90.9327i	0.0734266	−	0.127179i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	67.0820i		−	0.0932991i	−0.998911	−	0.0466495i	$-0.985146\pi$
$719$		−	67.0820i		−	0.0932991i	0.998911	−	0.0466495i	$-0.0148544\pi$
$720$		0			0
$721$	−392.000			−0.543689
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−203.332	−	117.394i	−0.280457	−	0.161922i
$726$		0			0
$727$	−601.000	−	1040.96i	−0.826685	−	1.43186i	−0.900625	−	0.434598i	$-0.856891\pi$
$727$	−601.000	−	1040.96i	−0.826685	−	1.43186i	0.0739397	−	0.997263i	$-0.476443\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	121.999	−	70.4361i	0.166893	−	0.0963559i
$732$		0			0
$733$	−157.000	+	271.932i	−0.214188	+	0.370985i	−0.953021	−	0.302904i	$-0.902044\pi$
$733$	−157.000	+	271.932i	−0.214188	+	0.370985i	0.738833	+	0.673889i	$0.235377\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			389.076i			0.527918i
$738$		0			0
$739$	488.000			0.660352			0.330176	−	0.943919i	$-0.392892\pi$
$739$	488.000			0.660352			0.330176	+	0.943919i	$0.392892\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	1144.47	+	660.758i	1.54033	+	0.889311i	0.998817	+	0.0486196i	$0.0154822\pi$
$743$	1144.47	+	660.758i	1.54033	+	0.889311i	0.541515	+	0.840691i	$0.317851\pi$
$744$		0			0
$745$	172.500	+	298.779i	0.231544	+	0.401045i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	315.500	−	546.462i	0.420107	−	0.727646i	−0.575843	−	0.817560i	$-0.695326\pi$
$751$	315.500	−	546.462i	0.420107	−	0.727646i	0.995949	+	0.0899145i	$0.0286594\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			641.752i			0.850002i
$756$		0			0
$757$	941.000			1.24306			0.621532	−	0.783388i	$-0.286510\pi$
$757$	941.000			1.24306			0.621532	+	0.783388i	$0.286510\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−848.183	−	489.699i	−1.11456	−	0.643494i	−0.174556	−	0.984647i	$-0.555849\pi$
$761$	−848.183	−	489.699i	−1.11456	−	0.643494i	−0.940008	+	0.341153i	$0.889182\pi$
$762$		0			0
$763$	52.0000	+	90.0666i	0.0681520	+	0.118043i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−243.998	+	140.872i	−0.318120	+	0.183667i
$768$		0			0
$769$	−140.500	+	243.353i	−0.182705	+	0.316454i	−0.942801	−	0.333357i	$-0.891819\pi$
$769$	−140.500	+	243.353i	−0.182705	+	0.316454i	0.760096	+	0.649811i	$0.225152\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			214.663i			0.277701i	0.990313	+	0.138850i	$0.0443407\pi$
$773$			214.663i			0.277701i	−0.990313	+	0.138850i	$0.955659\pi$
$774$		0			0
$775$	−145.000			−0.187097
$776$		0			0
$777$		0			0
$778$		0			0
$779$	92.9516	+	53.6656i	0.119322	+	0.0688904i
$780$		0			0
$781$	180.000	+	311.769i	0.230474	+	0.399192i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−373.743	+	215.781i	−0.476106	+	0.274880i
$786$		0			0
$787$	−320.500	+	555.122i	−0.407243	+	0.705365i	−0.994580	−	0.103977i	$-0.966843\pi$
$787$	−320.500	+	555.122i	−0.407243	+	0.705365i	0.587337	+	0.809343i	$0.300176\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	724.486i		−	0.915912i
$792$		0			0
$793$	−434.000			−0.547289
$794$		0			0
$795$		0			0
$796$		0			0
$797$	348.569	+	201.246i	0.437351	+	0.252505i	0.702473	−	0.711710i	$-0.252079\pi$
$797$	348.569	+	201.246i	0.437351	+	0.252505i	−0.265123	+	0.964215i	$0.585412\pi$
$798$		0			0
$799$	337.500	+	584.567i	0.422403	+	0.731623i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	302.093	−	174.413i	0.376205	−	0.217202i
$804$		0			0
$805$	30.0000	−	51.9615i	0.0372671	−	0.0645485i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			657.404i			0.812613i	0.913737	+	0.406307i	$0.133183\pi$
$809$			657.404i			0.812613i	−0.913737	+	0.406307i	$0.866817\pi$
$810$		0			0
$811$	−1336.00			−1.64735			−0.823674	−	0.567063i	$-0.808080\pi$
$811$	−1336.00			−1.64735			−0.823674	+	0.567063i	$0.808080\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−160.729	−	92.7968i	−0.197213	−	0.113861i
$816$		0			0
$817$	28.0000	+	48.4974i	0.0342717	+	0.0593604i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−755.232	+	436.033i	−0.919893	+	0.531100i	−0.883601	−	0.468241i	$-0.844888\pi$
$821$	−755.232	+	436.033i	−0.919893	+	0.531100i	−0.0362917	+	0.999341i	$0.511555\pi$
$822$		0			0
$823$	−424.000	+	734.390i	−0.515188	+	0.892332i	0.484656	+	0.874705i	$0.338945\pi$
$823$	−424.000	+	734.390i	−0.515188	+	0.892332i	−0.999845	+	0.0176277i	$0.994389\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	1287.98i		−	1.55741i	−0.627392	−	0.778703i	$-0.715878\pi$
$827$		−	1287.98i		−	1.55741i	0.627392	−	0.778703i	$-0.284122\pi$
$828$		0			0
$829$	416.000			0.501809			0.250905	−	0.968012i	$-0.419272\pi$
$829$	416.000			0.501809			0.250905	+	0.968012i	$0.419272\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−575.138	−	332.056i	−0.690442	−	0.398627i
$834$		0			0
$835$	345.000	+	597.558i	0.413174	+	0.715638i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−1092.18	+	630.571i	−1.30177	+	0.751575i	−0.980707	−	0.195485i	$-0.937372\pi$
$839$	−1092.18	+	630.571i	−1.30177	+	0.751575i	−0.321059	+	0.947059i	$0.604039\pi$
$840$		0			0
$841$	682.000	−	1181.26i	0.810939	−	1.40459i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	268.328i		−	0.317548i
$846$		0			0
$847$	−304.000			−0.358914
$848$		0			0
$849$		0			0
$850$		0			0
$851$	11.6190	+	6.70820i	0.0136533	+	0.00788273i
$852$		0			0
$853$	498.500	+	863.427i	0.584408	+	1.01222i	0.994949	+	0.100382i	$0.0320065\pi$
$853$	498.500	+	863.427i	0.584408	+	1.01222i	−0.410541	+	0.911842i	$0.634660\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	429.901	−	248.204i	0.501635	−	0.289619i	−0.227754	−	0.973719i	$-0.573138\pi$
$857$	429.901	−	248.204i	0.501635	−	0.289619i	0.729388	+	0.684100i	$0.239805\pi$
$858$		0			0
$859$	167.000	−	289.252i	0.194412	−	0.336732i	−0.752296	−	0.658826i	$-0.771053\pi$
$859$	167.000	−	289.252i	0.194412	−	0.336732i	0.946708	+	0.322094i	$0.104387\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			597.030i			0.691808i	0.938270	+	0.345904i	$0.112428\pi$
$863$			597.030i			0.691808i	−0.938270	+	0.345904i	$0.887572\pi$
$864$		0			0
$865$	480.000			0.554913
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−284.664	−	164.351i	−0.327577	−	0.189127i
$870$		0			0
$871$	−203.000	−	351.606i	−0.233065	−	0.403681i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	38.7298	−	22.3607i	0.0442627	−	0.0255551i
$876$		0			0
$877$	−665.500	+	1152.68i	−0.758837	+	1.31434i	0.184607	+	0.982812i	$0.440899\pi$
$877$	−665.500	+	1152.68i	−0.758837	+	1.31434i	−0.943444	+	0.331532i	$0.892435\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	40.2492i		−	0.0456858i	−0.999739	−	0.0228429i	$-0.992728\pi$
$881$		−	40.2492i		−	0.0456858i	0.999739	−	0.0228429i	$-0.00727176\pi$
$882$		0			0
$883$	1214.00			1.37486			0.687429	−	0.726251i	$-0.258739\pi$
$883$	1214.00			1.37486			0.687429	+	0.726251i	$0.258739\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	761.041	+	439.387i	0.857995	+	0.495363i	0.863340	−	0.504622i	$-0.168368\pi$
$887$	761.041	+	439.387i	0.857995	+	0.495363i	−0.00534561	+	0.999986i	$0.501702\pi$
$888$		0			0
$889$	−428.000	−	741.318i	−0.481440	−	0.833878i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−232.379	+	134.164i	−0.260223	+	0.150240i
$894$		0			0
$895$	105.000	−	181.865i	0.117318	−	0.203201i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	1361.77i		−	1.51476i
$900$		0			0
$901$	−1890.00			−2.09767
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−317.585	−	183.358i	−0.350922	−	0.202605i
$906$		0			0
$907$	−623.500	−	1079.93i	−0.687431	−	1.19067i	−0.972666	−	0.232208i	$-0.925405\pi$
$907$	−623.500	−	1079.93i	−0.687431	−	1.19067i	0.285235	−	0.958458i	$-0.407928\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	220.760	−	127.456i	0.242327	−	0.139908i	−0.373919	−	0.927461i	$-0.621986\pi$
$911$	220.760	−	127.456i	0.242327	−	0.139908i	0.616246	+	0.787554i	$0.288653\pi$
$912$		0			0
$913$	−45.0000	+	77.9423i	−0.0492881	+	0.0853694i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	992.814i		−	1.08268i
$918$		0			0
$919$	−259.000			−0.281828			−0.140914	−	0.990022i	$-0.545004\pi$
$919$	−259.000			−0.281828			−0.140914	+	0.990022i	$0.545004\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−325.331	−	187.830i	−0.352471	−	0.203499i
$924$		0			0
$925$	5.00000	+	8.66025i	0.00540541	+	0.00936244i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1289.70	+	744.611i	−1.38827	+	0.801518i	−0.993120	−	0.117098i	$-0.962641\pi$
$929$	−1289.70	+	744.611i	−1.38827	+	0.801518i	−0.395150	+	0.918616i	$0.629307\pi$
$930$		0			0
$931$	132.000	−	228.631i	0.141783	−	0.245575i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	301.869i		−	0.322855i
$936$		0			0
$937$	236.000			0.251868			0.125934	−	0.992039i	$-0.459807\pi$
$937$	236.000			0.251868			0.125934	+	0.992039i	$0.459807\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	517.043	+	298.515i	0.549462	+	0.317232i	0.748905	−	0.662678i	$-0.230580\pi$
$941$	517.043	+	298.515i	0.549462	+	0.317232i	−0.199443	+	0.979909i	$0.563913\pi$
$942$		0			0
$943$	45.0000	+	77.9423i	0.0477200	+	0.0826535i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	81.3327	−	46.9574i	0.0858845	−	0.0495855i	−0.456443	−	0.889753i	$-0.650877\pi$
$947$	81.3327	−	46.9574i	0.0858845	−	0.0495855i	0.542327	+	0.840167i	$0.317543\pi$
$948$		0			0
$949$	−182.000	+	315.233i	−0.191781	+	0.332174i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	986.106i		−	1.03474i	−0.855762	−	0.517369i	$-0.826911\pi$
$953$		−	986.106i		−	1.03474i	0.855762	−	0.517369i	$-0.173089\pi$
$954$		0			0
$955$	−480.000			−0.502618
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−139.427	−	80.4984i	−0.145388	−	0.0839400i
$960$		0			0
$961$	60.0000	+	103.923i	0.0624350	+	0.108141i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−30.9839	+	17.8885i	−0.0321076	+	0.0185374i
$966$		0			0
$967$	284.000	−	491.902i	0.293692	−	0.508689i	−0.680988	−	0.732295i	$-0.738449\pi$
$967$	284.000	−	491.902i	0.293692	−	0.508689i	0.974680	+	0.223605i	$0.0717827\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	1428.85i		−	1.47152i	−0.677242	−	0.735761i	$-0.736825\pi$
$971$		−	1428.85i		−	1.47152i	0.677242	−	0.735761i	$-0.263175\pi$
$972$		0			0
$973$	256.000			0.263104
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1585.99	−	915.670i	−1.62332	−	0.937226i	−0.986023	−	0.166611i	$-0.946718\pi$
$977$	−1585.99	−	915.670i	−1.62332	−	0.937226i	−0.637300	−	0.770616i	$-0.719949\pi$
$978$		0			0
$979$	360.000	+	623.538i	0.367722	+	0.636913i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1214.18	+	701.007i	−1.23518	+	0.713131i	−0.968105	−	0.250546i	$-0.919390\pi$
$983$	−1214.18	+	701.007i	−1.23518	+	0.713131i	−0.267074	+	0.963676i	$0.586057\pi$
$984$		0			0
$985$	−210.000	+	363.731i	−0.213198	+	0.369270i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			46.9574i			0.0474797i
$990$		0			0
$991$	77.0000			0.0776993			0.0388496	−	0.999245i	$-0.487631\pi$
$991$	77.0000			0.0776993			0.0388496	+	0.999245i	$0.487631\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	269.172	+	155.407i	0.270525	+	0.156188i
$996$		0			0
$997$	−470.500	−	814.930i	−0.471916	−	0.817382i	0.527568	−	0.849513i	$-0.323104\pi$
$997$	−470.500	−	814.930i	−0.471916	−	0.817382i	−0.999484	+	0.0321308i	$0.989771\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.3.o.c.701.1		4
3.2	odd	2	inner	1620.3.o.c.701.2		4
9.2	odd	6	inner	1620.3.o.c.1241.1		4
9.4	even	3		540.3.g.b.161.2	yes	2
9.5	odd	6		540.3.g.b.161.1	✓	2
9.7	even	3	inner	1620.3.o.c.1241.2		4
36.23	even	6		2160.3.l.c.161.1		2
36.31	odd	6		2160.3.l.c.161.2		2
45.4	even	6		2700.3.g.k.701.1		2
45.13	odd	12		2700.3.b.i.1349.3		4
45.14	odd	6		2700.3.g.k.701.2		2
45.22	odd	12		2700.3.b.i.1349.1		4
45.23	even	12		2700.3.b.i.1349.4		4
45.32	even	12		2700.3.b.i.1349.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
540.3.g.b.161.1	✓	2	9.5	odd	6
540.3.g.b.161.2	yes	2	9.4	even	3
1620.3.o.c.701.1		4	1.1	even	1	trivial
1620.3.o.c.701.2		4	3.2	odd	2	inner
1620.3.o.c.1241.1		4	9.2	odd	6	inner
1620.3.o.c.1241.2		4	9.7	even	3	inner
2160.3.l.c.161.1		2	36.23	even	6
2160.3.l.c.161.2		2	36.31	odd	6
2700.3.b.i.1349.1		4	45.22	odd	12
2700.3.b.i.1349.2		4	45.32	even	12
2700.3.b.i.1349.3		4	45.13	odd	12
2700.3.b.i.1349.4		4	45.23	even	12
2700.3.g.k.701.1		2	45.4	even	6
2700.3.g.k.701.2		2	45.14	odd	6

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)

\(f(q)\)	\(=\)	\(q+(-1.93649 - 1.11803i) q^{5} +(2.00000 + 3.46410i) q^{7} +O(q^{10})\)	\(q+(-1.93649 - 1.11803i) q^{5} +(2.00000 + 3.46410i) q^{7} +(-5.80948 + 3.35410i) q^{11} +(3.50000 - 6.06218i) q^{13} -20.1246i q^{17} +8.00000 q^{19} +(5.80948 + 3.35410i) q^{23} +(2.50000 + 4.33013i) q^{25} +(-40.6663 + 23.4787i) q^{29} +(-14.5000 + 25.1147i) q^{31} -8.94427i q^{35} +2.00000 q^{37} +(11.6190 + 6.70820i) q^{41} +(3.50000 + 6.06218i) q^{43} +(-29.0474 + 16.7705i) q^{47} +(16.5000 - 28.5788i) q^{49} -93.9149i q^{53} +15.0000 q^{55} +(-34.8569 - 20.1246i) q^{59} +(-31.0000 - 53.6936i) q^{61} +(-13.5554 + 7.82624i) q^{65} +(29.0000 - 50.2295i) q^{67} -53.6656i q^{71} -52.0000 q^{73} +(-23.2379 - 13.4164i) q^{77} +(24.5000 + 42.4352i) q^{79} +(11.6190 - 6.70820i) q^{83} +(-22.5000 + 38.9711i) q^{85} -107.331i q^{89} +28.0000 q^{91} +(-15.4919 - 8.94427i) q^{95} +(17.0000 + 29.4449i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{7}+O(q^{10}) \) 4 * q + 8 * q^7	\( 4 q + 8 q^{7} + 14 q^{13} + 32 q^{19} + 10 q^{25} - 58 q^{31} + 8 q^{37} + 14 q^{43} + 66 q^{49} + 60 q^{55} - 124 q^{61} + 116 q^{67} - 208 q^{73} + 98 q^{79} - 90 q^{85} + 112 q^{91} + 68 q^{97}+O(q^{100}) \) 4 * q + 8 * q^7 + 14 * q^13 + 32 * q^19 + 10 * q^25 - 58 * q^31 + 8 * q^37 + 14 * q^43 + 66 * q^49 + 60 * q^55 - 124 * q^61 + 116 * q^67 - 208 * q^73 + 98 * q^79 - 90 * q^85 + 112 * q^91 + 68 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more