LMFDB - Embedded newform 1620.3.l.f.1297.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,3,Mod(973,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 3]))

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

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

chi := DirichletCharacter("1620.973");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1620.l (of order $4$, degree $2$, 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:	$24$
Relative dimension:	$12$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 180)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1297.7
Character	$\chi$	$=$	1620.1297
Dual form			1620.3.l.f.973.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.555312 + 4.96907i) q^{5} +(3.23469 + 3.23469i) q^{7} +O(q^{10})$	$q+(0.555312 + 4.96907i) q^{5} +(3.23469 + 3.23469i) q^{7} -6.14054 q^{11} +(-16.3264 + 16.3264i) q^{13} +(6.88593 + 6.88593i) q^{17} -20.9567i q^{19} +(-25.3892 + 25.3892i) q^{23} +(-24.3833 + 5.51877i) q^{25} -39.5192i q^{29} +15.0554 q^{31} +(-14.2771 + 17.8697i) q^{35} +(-47.5915 - 47.5915i) q^{37} +31.1344 q^{41} +(6.13777 - 6.13777i) q^{43} +(13.3376 + 13.3376i) q^{47} -28.0735i q^{49} +(-58.4005 + 58.4005i) q^{53} +(-3.40992 - 30.5128i) q^{55} +8.96449i q^{59} -10.8317 q^{61} +(-90.1930 - 72.0606i) q^{65} +(-24.3968 - 24.3968i) q^{67} +89.8205 q^{71} +(42.9177 - 42.9177i) q^{73} +(-19.8628 - 19.8628i) q^{77} +35.9196i q^{79} +(31.5474 - 31.5474i) q^{83} +(-30.3928 + 38.0405i) q^{85} -75.7246i q^{89} -105.622 q^{91} +(104.135 - 11.6375i) q^{95} +(-21.5403 - 21.5403i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q - 12 q^{11} + 18 q^{17} + 12 q^{23} + 6 q^{25} + 36 q^{35} - 42 q^{37} - 72 q^{41} + 78 q^{47} - 156 q^{53} + 66 q^{55} + 96 q^{61} + 132 q^{65} + 78 q^{67} - 156 q^{71} + 240 q^{77} - 132 q^{83} - 96 q^{85} + 84 q^{91} + 168 q^{95} + 90 q^{97}+O(q^{100}) $ 24 * q - 12 * q^11 + 18 * q^17 + 12 * q^23 + 6 * q^25 + 36 * q^35 - 42 * q^37 - 72 * q^41 + 78 * q^47 - 156 * q^53 + 66 * q^55 + 96 * q^61 + 132 * q^65 + 78 * q^67 - 156 * q^71 + 240 * q^77 - 132 * q^83 - 96 * q^85 + 84 * q^91 + 168 * q^95 + 90 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$e\left(\frac{1}{4}\right)$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.555312	+	4.96907i	0.111062	+	0.993813i
$5$	0.555312	+	4.96907i	0.111062	+	0.993813i
$6$		0			0
$7$	3.23469	+	3.23469i	0.462099	+	0.462099i	0.899343	−	0.437244i	$-0.144045\pi$
$7$	3.23469	+	3.23469i	0.462099	+	0.462099i	−0.437244	+	0.899343i	$0.644045\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−6.14054			−0.558231			−0.279116	−	0.960258i	$-0.590041\pi$
$11$	−6.14054			−0.558231			−0.279116	+	0.960258i	$0.590041\pi$
$12$		0			0
$13$	−16.3264	+	16.3264i	−1.25587	+	1.25587i	−0.302830	+	0.953045i	$0.597931\pi$
$13$	−16.3264	+	16.3264i	−1.25587	+	1.25587i	−0.953045	+	0.302830i	$0.902069\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	6.88593	+	6.88593i	0.405055	+	0.405055i	0.880010	−	0.474955i	$-0.157536\pi$
$17$	6.88593	+	6.88593i	0.405055	+	0.405055i	−0.474955	+	0.880010i	$0.657536\pi$
$18$		0			0
$19$		−	20.9567i		−	1.10299i	−0.834179	−	0.551493i	$-0.814058\pi$
$19$		−	20.9567i		−	1.10299i	0.834179	−	0.551493i	$-0.185942\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−25.3892	+	25.3892i	−1.10388	+	1.10388i	−0.109939	+	0.993938i	$0.535065\pi$
$23$	−25.3892	+	25.3892i	−1.10388	+	1.10388i	−0.993938	+	0.109939i	$0.964935\pi$
$24$		0			0
$25$	−24.3833	+	5.51877i	−0.975330	+	0.220751i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	39.5192i		−	1.36273i	−0.731943	−	0.681366i	$-0.761386\pi$
$29$		−	39.5192i		−	1.36273i	0.731943	−	0.681366i	$-0.238614\pi$
$30$		0			0
$31$	15.0554			0.485658			0.242829	−	0.970069i	$-0.421925\pi$
$31$	15.0554			0.485658			0.242829	+	0.970069i	$0.421925\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−14.2771	+	17.8697i	−0.407918	+	0.510562i
$36$		0			0
$37$	−47.5915	−	47.5915i	−1.28626	−	1.28626i	−0.937043	−	0.349214i	$-0.886449\pi$
$37$	−47.5915	−	47.5915i	−1.28626	−	1.28626i	−0.349214	−	0.937043i	$-0.613551\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	31.1344			0.759377			0.379688	−	0.925114i	$-0.376031\pi$
$41$	31.1344			0.759377			0.379688	+	0.925114i	$0.376031\pi$
$42$		0			0
$43$	6.13777	−	6.13777i	0.142739	−	0.142739i	−0.632126	−	0.774865i	$-0.717818\pi$
$43$	6.13777	−	6.13777i	0.142739	−	0.142739i	0.774865	+	0.632126i	$0.217818\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	13.3376	+	13.3376i	0.283778	+	0.283778i	0.834614	−	0.550836i	$-0.185691\pi$
$47$	13.3376	+	13.3376i	0.283778	+	0.283778i	−0.550836	+	0.834614i	$0.685691\pi$
$48$		0			0
$49$		−	28.0735i		−	0.572929i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−58.4005	+	58.4005i	−1.10190	+	1.10190i	−0.107715	+	0.994182i	$0.534353\pi$
$53$	−58.4005	+	58.4005i	−1.10190	+	1.10190i	−0.994182	+	0.107715i	$0.965647\pi$
$54$		0			0
$55$	−3.40992	−	30.5128i	−0.0619985	−	0.554778i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			8.96449i			0.151940i	0.997110	+	0.0759702i	$0.0242054\pi$
$59$			8.96449i			0.151940i	−0.997110	+	0.0759702i	$0.975795\pi$
$60$		0			0
$61$	−10.8317			−0.177569			−0.0887846	−	0.996051i	$-0.528298\pi$
$61$	−10.8317			−0.177569			−0.0887846	+	0.996051i	$0.528298\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−90.1930	−	72.0606i	−1.38759	−	1.10862i
$66$		0			0
$67$	−24.3968	−	24.3968i	−0.364131	−	0.364131i	0.501200	−	0.865331i	$-0.332892\pi$
$67$	−24.3968	−	24.3968i	−0.364131	−	0.364131i	−0.865331	+	0.501200i	$0.832892\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	89.8205			1.26508			0.632538	−	0.774529i	$-0.282013\pi$
$71$	89.8205			1.26508			0.632538	+	0.774529i	$0.282013\pi$
$72$		0			0
$73$	42.9177	−	42.9177i	0.587914	−	0.587914i	−0.349152	−	0.937066i	$-0.613530\pi$
$73$	42.9177	−	42.9177i	0.587914	−	0.587914i	0.937066	+	0.349152i	$0.113530\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−19.8628	−	19.8628i	−0.257958	−	0.257958i
$78$		0			0
$79$			35.9196i			0.454678i	0.973816	+	0.227339i	$0.0730026\pi$
$79$			35.9196i			0.454678i	−0.973816	+	0.227339i	$0.926997\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	31.5474	−	31.5474i	0.380089	−	0.380089i	−0.491045	−	0.871134i	$-0.663385\pi$
$83$	31.5474	−	31.5474i	0.380089	−	0.380089i	0.871134	+	0.491045i	$0.163385\pi$
$84$		0			0
$85$	−30.3928	+	38.0405i	−0.357563	+	0.447535i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	75.7246i		−	0.850838i	−0.904996	−	0.425419i	$-0.860127\pi$
$89$		−	75.7246i		−	0.850838i	0.904996	−	0.425419i	$-0.139873\pi$
$90$		0			0
$91$	−105.622			−1.16068
$92$		0			0
$93$		0			0
$94$		0			0
$95$	104.135	−	11.6375i	1.09616	−	0.122500i
$96$		0			0
$97$	−21.5403	−	21.5403i	−0.222065	−	0.222065i	0.587302	−	0.809368i	$-0.300190\pi$
$97$	−21.5403	−	21.5403i	−0.222065	−	0.222065i	−0.809368	+	0.587302i	$0.800190\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	109.067			1.07987			0.539936	−	0.841706i	$-0.318448\pi$
$101$	109.067			1.07987			0.539936	+	0.841706i	$0.318448\pi$
$102$		0			0
$103$	−23.3667	+	23.3667i	−0.226862	+	0.226862i	−0.811380	−	0.584519i	$-0.801283\pi$
$103$	−23.3667	+	23.3667i	−0.226862	+	0.226862i	0.584519	+	0.811380i	$0.301283\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	52.4218	+	52.4218i	0.489924	+	0.489924i	0.908282	−	0.418358i	$-0.137394\pi$
$107$	52.4218	+	52.4218i	0.489924	+	0.489924i	−0.418358	+	0.908282i	$0.637394\pi$
$108$		0			0
$109$			65.1785i			0.597968i	0.954258	+	0.298984i	$0.0966477\pi$
$109$			65.1785i			0.597968i	−0.954258	+	0.298984i	$0.903352\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	6.58011	−	6.58011i	0.0582311	−	0.0582311i	−0.677392	−	0.735623i	$-0.736890\pi$
$113$	6.58011	−	6.58011i	0.0582311	−	0.0582311i	0.735623	+	0.677392i	$0.236890\pi$
$114$		0			0
$115$	−140.259	−	112.062i	−1.21965	−	0.974449i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			44.5478i			0.374351i
$120$		0			0
$121$	−83.2937			−0.688378
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−40.9634	−	118.097i	−0.327708	−	0.944779i
$126$		0			0
$127$	−139.889	−	139.889i	−1.10149	−	1.10149i	−0.994232	−	0.107254i	$-0.965794\pi$
$127$	−139.889	−	139.889i	−1.10149	−	1.10149i	−0.107254	−	0.994232i	$-0.534206\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−142.901			−1.09085			−0.545423	−	0.838161i	$-0.683631\pi$
$131$	−142.901			−1.09085			−0.545423	+	0.838161i	$0.683631\pi$
$132$		0			0
$133$	67.7886	−	67.7886i	0.509689	−	0.509689i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	94.5304	+	94.5304i	0.690003	+	0.690003i	0.962232	−	0.272230i	$-0.0877610\pi$
$137$	94.5304	+	94.5304i	0.690003	+	0.690003i	−0.272230	+	0.962232i	$0.587761\pi$
$138$		0			0
$139$		−	205.522i		−	1.47858i	−0.673389	−	0.739289i	$-0.735162\pi$
$139$		−	205.522i		−	1.47858i	0.673389	−	0.739289i	$-0.264838\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	100.253	−	100.253i	0.701068	−	0.701068i
$144$		0			0
$145$	196.374	−	21.9455i	1.35430	−	0.151348i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			113.093i			0.759016i	0.925188	+	0.379508i	$0.123907\pi$
$149$			113.093i			0.759016i	−0.925188	+	0.379508i	$0.876093\pi$
$150$		0			0
$151$	−190.448			−1.26124			−0.630622	−	0.776090i	$-0.717200\pi$
$151$	−190.448			−1.26124			−0.630622	+	0.776090i	$0.717200\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	8.36045	+	74.8113i	0.0539384	+	0.482654i
$156$		0			0
$157$	−172.332	−	172.332i	−1.09766	−	1.09766i	−0.994684	−	0.102975i	$-0.967164\pi$
$157$	−172.332	−	172.332i	−1.09766	−	1.09766i	−0.102975	−	0.994684i	$-0.532836\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−164.252			−1.02020
$162$		0			0
$163$	−88.0504	+	88.0504i	−0.540186	+	0.540186i	−0.923584	−	0.383397i	$-0.874754\pi$
$163$	−88.0504	+	88.0504i	−0.540186	+	0.540186i	0.383397	+	0.923584i	$0.374754\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−121.904	−	121.904i	−0.729965	−	0.729965i	0.240648	−	0.970612i	$-0.422640\pi$
$167$	−121.904	−	121.904i	−0.729965	−	0.729965i	−0.970612	+	0.240648i	$0.922640\pi$
$168$		0			0
$169$		−	364.101i		−	2.15444i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−109.784	+	109.784i	−0.634592	+	0.634592i	−0.949216	−	0.314624i	$-0.898122\pi$
$173$	−109.784	+	109.784i	−0.634592	+	0.634592i	0.314624	+	0.949216i	$0.398122\pi$
$174$		0			0
$175$	−96.7239	−	61.0208i	−0.552708	−	0.348690i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	19.2958i		−	0.107798i	−0.998546	−	0.0538990i	$-0.982835\pi$
$179$		−	19.2958i		−	0.107798i	0.998546	−	0.0538990i	$-0.0171649\pi$
$180$		0			0
$181$	−285.133			−1.57532			−0.787659	−	0.616111i	$-0.788707\pi$
$181$	−285.133			−1.57532			−0.787659	+	0.616111i	$0.788707\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	210.057	−	262.914i	1.13544	−	1.42115i
$186$		0			0
$187$	−42.2834	−	42.2834i	−0.226114	−	0.226114i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−261.967			−1.37155			−0.685777	−	0.727812i	$-0.740537\pi$
$191$	−261.967			−1.37155			−0.685777	+	0.727812i	$0.740537\pi$
$192$		0			0
$193$	−20.2192	+	20.2192i	−0.104763	+	0.104763i	−0.757545	−	0.652783i	$-0.773602\pi$
$193$	−20.2192	+	20.2192i	−0.104763	+	0.104763i	0.652783	+	0.757545i	$0.273602\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−2.31516	−	2.31516i	−0.0117521	−	0.0117521i	0.701206	−	0.712958i	$-0.252645\pi$
$197$	−2.31516	−	2.31516i	−0.0117521	−	0.0117521i	−0.712958	+	0.701206i	$0.752645\pi$
$198$		0			0
$199$			371.306i			1.86586i	0.360057	+	0.932930i	$0.382757\pi$
$199$			371.306i			1.86586i	−0.360057	+	0.932930i	$0.617243\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	127.833	−	127.833i	0.629717	−	0.629717i
$204$		0			0
$205$	17.2893	+	154.709i	0.0843382	+	0.754679i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			128.686i			0.615721i
$210$		0			0
$211$	57.0239			0.270255			0.135128	−	0.990828i	$-0.456856\pi$
$211$	57.0239			0.270255			0.135128	+	0.990828i	$0.456856\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	33.9074	+	27.0906i	0.157709	+	0.126003i
$216$		0			0
$217$	48.6996	+	48.6996i	0.224422	+	0.224422i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−224.845			−1.01740
$222$		0			0
$223$	−275.841	+	275.841i	−1.23696	+	1.23696i	−0.275718	+	0.961238i	$0.588916\pi$
$223$	−275.841	+	275.841i	−1.23696	+	1.23696i	−0.961238	+	0.275718i	$0.911084\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	170.461	+	170.461i	0.750927	+	0.750927i	0.974652	−	0.223725i	$-0.0718217\pi$
$227$	170.461	+	170.461i	0.750927	+	0.750927i	−0.223725	+	0.974652i	$0.571822\pi$
$228$		0			0
$229$		−	234.916i		−	1.02583i	−0.858439	−	0.512916i	$-0.828565\pi$
$229$		−	234.916i		−	1.02583i	0.858439	−	0.512916i	$-0.171435\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−290.176	+	290.176i	−1.24539	+	1.24539i	−0.287655	+	0.957734i	$0.592876\pi$
$233$	−290.176	+	290.176i	−1.24539	+	1.24539i	−0.957734	+	0.287655i	$0.907124\pi$
$234$		0			0
$235$	−58.8687	+	73.6818i	−0.250505	+	0.313539i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			472.656i			1.97764i	0.149114	+	0.988820i	$0.452358\pi$
$239$			472.656i			1.97764i	−0.149114	+	0.988820i	$0.547642\pi$
$240$		0			0
$241$	308.462			1.27993			0.639964	−	0.768405i	$-0.278949\pi$
$241$	308.462			1.27993			0.639964	+	0.768405i	$0.278949\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	139.499	−	15.5896i	0.569385	−	0.0636309i
$246$		0			0
$247$	342.147	+	342.147i	1.38521	+	1.38521i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	108.785			0.433407			0.216703	−	0.976237i	$-0.430470\pi$
$251$	108.785			0.433407			0.216703	+	0.976237i	$0.430470\pi$
$252$		0			0
$253$	155.903	−	155.903i	0.616219	−	0.616219i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−44.9357	−	44.9357i	−0.174847	−	0.174847i	0.614258	−	0.789105i	$-0.289455\pi$
$257$	−44.9357	−	44.9357i	−0.174847	−	0.174847i	−0.789105	+	0.614258i	$0.789455\pi$
$258$		0			0
$259$		−	307.888i		−	1.18876i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−282.009	+	282.009i	−1.07228	+	1.07228i	−0.0750997	+	0.997176i	$0.523928\pi$
$263$	−282.009	+	282.009i	−1.07228	+	1.07228i	−0.997176	+	0.0750997i	$0.976072\pi$
$264$		0			0
$265$	−322.627	−	257.766i	−1.21746	−	0.972701i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	184.278i		−	0.685049i	−0.939509	−	0.342525i	$-0.888718\pi$
$269$		−	184.278i		−	0.685049i	0.939509	−	0.342525i	$-0.111282\pi$
$270$		0			0
$271$	35.0581			0.129366			0.0646828	−	0.997906i	$-0.479396\pi$
$271$	35.0581			0.129366			0.0646828	+	0.997906i	$0.479396\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	149.726	−	33.8882i	0.544460	−	0.123230i
$276$		0			0
$277$	−63.3774	−	63.3774i	−0.228799	−	0.228799i	0.583392	−	0.812191i	$-0.301725\pi$
$277$	−63.3774	−	63.3774i	−0.228799	−	0.228799i	−0.812191	+	0.583392i	$0.801725\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	360.230			1.28196			0.640978	−	0.767559i	$-0.278529\pi$
$281$	360.230			1.28196			0.640978	+	0.767559i	$0.278529\pi$
$282$		0			0
$283$	29.6196	−	29.6196i	0.104663	−	0.104663i	−0.652836	−	0.757499i	$-0.726421\pi$
$283$	29.6196	−	29.6196i	0.104663	−	0.104663i	0.757499	+	0.652836i	$0.226421\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	100.710	+	100.710i	0.350907	+	0.350907i
$288$		0			0
$289$		−	194.168i		−	0.671861i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	35.5959	−	35.5959i	0.121488	−	0.121488i	−0.643749	−	0.765237i	$-0.722622\pi$
$293$	35.5959	−	35.5959i	0.121488	−	0.121488i	0.765237	+	0.643749i	$0.222622\pi$
$294$		0			0
$295$	−44.5451	+	4.97809i	−0.151000	+	0.0168749i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	829.026i		−	2.77266i
$300$		0			0
$301$	39.7076			0.131919
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−6.01499	−	53.8236i	−0.0197213	−	0.176471i
$306$		0			0
$307$	−304.741	−	304.741i	−0.992641	−	0.992641i	0.00733168	−	0.999973i	$-0.497666\pi$
$307$	−304.741	−	304.741i	−0.992641	−	0.992641i	−0.999973	+	0.00733168i	$0.997666\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−434.249			−1.39630			−0.698150	−	0.715952i	$-0.745993\pi$
$311$	−434.249			−1.39630			−0.698150	+	0.715952i	$0.745993\pi$
$312$		0			0
$313$	33.0591	−	33.0591i	0.105620	−	0.105620i	−0.652322	−	0.757942i	$-0.726205\pi$
$313$	33.0591	−	33.0591i	0.105620	−	0.105620i	0.757942	+	0.652322i	$0.226205\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−182.516	−	182.516i	−0.575759	−	0.575759i	0.357973	−	0.933732i	$-0.383468\pi$
$317$	−182.516	−	182.516i	−0.575759	−	0.575759i	−0.933732	+	0.357973i	$0.883468\pi$
$318$		0			0
$319$			242.670i			0.760720i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	144.307	−	144.307i	0.446770	−	0.446770i
$324$		0			0
$325$	307.989	−	488.191i	0.947657	−	1.50213i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			86.2859i			0.262267i
$330$		0			0
$331$	136.588			0.412653			0.206327	−	0.978483i	$-0.433849\pi$
$331$	136.588			0.412653			0.206327	+	0.978483i	$0.433849\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	107.681	−	134.777i	0.321437	−	0.402320i
$336$		0			0
$337$	170.334	+	170.334i	0.505441	+	0.505441i	0.913124	−	0.407683i	$-0.133663\pi$
$337$	170.334	+	170.334i	0.505441	+	0.505441i	−0.407683	+	0.913124i	$0.633663\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−92.4484			−0.271110
$342$		0			0
$343$	249.309	−	249.309i	0.726849	−	0.726849i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	223.636	+	223.636i	0.644483	+	0.644483i	0.951654	−	0.307171i	$-0.0993823\pi$
$347$	223.636	+	223.636i	0.644483	+	0.644483i	−0.307171	+	0.951654i	$0.599382\pi$
$348$		0			0
$349$			398.637i			1.14223i	0.820871	+	0.571113i	$0.193488\pi$
$349$			398.637i			1.14223i	−0.820871	+	0.571113i	$0.806512\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−57.6943	+	57.6943i	−0.163440	+	0.163440i	−0.784089	−	0.620649i	$-0.786869\pi$
$353$	−57.6943	+	57.6943i	−0.163440	+	0.163440i	0.620649	+	0.784089i	$0.286869\pi$
$354$		0			0
$355$	49.8784	+	446.324i	0.140503	+	1.25725i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	180.854i		−	0.503770i	−0.967757	−	0.251885i	$-0.918949\pi$
$359$		−	180.854i		−	0.503770i	0.967757	−	0.251885i	$-0.0810505\pi$
$360$		0			0
$361$	−78.1849			−0.216579
$362$		0			0
$363$		0			0
$364$		0			0
$365$	237.094	+	189.428i	0.649572	+	0.518982i
$366$		0			0
$367$	359.673	+	359.673i	0.980034	+	0.980034i	0.999805	−	0.0197701i	$-0.00629342\pi$
$367$	359.673	+	359.673i	0.980034	+	0.980034i	−0.0197701	+	0.999805i	$0.506293\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−377.816			−1.01837
$372$		0			0
$373$	−76.0873	+	76.0873i	−0.203987	+	0.203987i	−0.801706	−	0.597719i	$-0.796074\pi$
$373$	−76.0873	+	76.0873i	−0.203987	+	0.203987i	0.597719	+	0.801706i	$0.296074\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	645.206	+	645.206i	1.71142	+	1.71142i
$378$		0			0
$379$			490.992i			1.29549i	0.761856	+	0.647747i	$0.224289\pi$
$379$			490.992i			1.29549i	−0.761856	+	0.647747i	$0.775711\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−25.5585	+	25.5585i	−0.0667325	+	0.0667325i	−0.739685	−	0.672953i	$-0.765026\pi$
$383$	−25.5585	+	25.5585i	−0.0667325	+	0.0667325i	0.672953	+	0.739685i	$0.265026\pi$
$384$		0			0
$385$	87.6694	−	109.729i	0.227713	−	0.285012i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	674.943i		−	1.73507i	−0.497375	−	0.867536i	$-0.665703\pi$
$389$		−	674.943i		−	1.73507i	0.497375	−	0.867536i	$-0.334297\pi$
$390$		0			0
$391$	−349.656			−0.894261
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−178.487	+	19.9466i	−0.451866	+	0.0504977i
$396$		0			0
$397$	−13.2163	−	13.2163i	−0.0332903	−	0.0332903i	0.690266	−	0.723556i	$-0.257494\pi$
$397$	−13.2163	−	13.2163i	−0.0332903	−	0.0332903i	−0.723556	+	0.690266i	$0.757494\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−56.9136			−0.141929			−0.0709646	−	0.997479i	$-0.522608\pi$
$401$	−56.9136			−0.141929			−0.0709646	+	0.997479i	$0.522608\pi$
$402$		0			0
$403$	−245.800	+	245.800i	−0.609926	+	0.609926i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	292.238	+	292.238i	0.718029	+	0.718029i
$408$		0			0
$409$			376.755i			0.921162i	0.887618	+	0.460581i	$0.152359\pi$
$409$			376.755i			0.921162i	−0.887618	+	0.460581i	$0.847641\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−28.9974	+	28.9974i	−0.0702115	+	0.0702115i
$414$		0			0
$415$	174.280	+	139.242i	0.419951	+	0.335524i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	624.292i		−	1.48996i	−0.667089	−	0.744978i	$-0.732460\pi$
$419$		−	624.292i		−	1.48996i	0.667089	−	0.744978i	$-0.267540\pi$
$420$		0			0
$421$	38.6859			0.0918905			0.0459452	−	0.998944i	$-0.485370\pi$
$421$	38.6859			0.0918905			0.0459452	+	0.998944i	$0.485370\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−205.903	−	129.900i	−0.484478	−	0.305646i
$426$		0			0
$427$	−35.0373	−	35.0373i	−0.0820546	−	0.0820546i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−342.236			−0.794050			−0.397025	−	0.917808i	$-0.629957\pi$
$431$	−342.236			−0.794050			−0.397025	+	0.917808i	$0.629957\pi$
$432$		0			0
$433$	−66.7070	+	66.7070i	−0.154058	+	0.154058i	−0.779928	−	0.625870i	$-0.784744\pi$
$433$	−66.7070	+	66.7070i	−0.154058	+	0.154058i	0.625870	+	0.779928i	$0.284744\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	532.074	+	532.074i	1.21756	+	1.21756i
$438$		0			0
$439$			473.966i			1.07965i	0.841778	+	0.539824i	$0.181509\pi$
$439$			473.966i			1.07965i	−0.841778	+	0.539824i	$0.818491\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	398.334	−	398.334i	0.899173	−	0.899173i	−0.0961898	−	0.995363i	$-0.530666\pi$
$443$	398.334	−	398.334i	0.899173	−	0.899173i	0.995363	+	0.0961898i	$0.0306656\pi$
$444$		0			0
$445$	376.281	−	42.0508i	0.845574	−	0.0944962i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	357.683i		−	0.796621i	−0.917251	−	0.398310i	$-0.869597\pi$
$449$		−	357.683i		−	0.796621i	0.917251	−	0.398310i	$-0.130403\pi$
$450$		0			0
$451$	−191.182			−0.423908
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−58.6529	−	524.841i	−0.128908	−	1.15350i
$456$		0			0
$457$	−2.62813	−	2.62813i	−0.00575084	−	0.00575084i	0.704226	−	0.709976i	$-0.251294\pi$
$457$	−2.62813	−	2.62813i	−0.00575084	−	0.00575084i	−0.709976	+	0.704226i	$0.751294\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	200.498			0.434919			0.217460	−	0.976069i	$-0.430223\pi$
$461$	200.498			0.434919			0.217460	+	0.976069i	$0.430223\pi$
$462$		0			0
$463$	236.729	−	236.729i	0.511293	−	0.511293i	−0.403629	−	0.914923i	$-0.632252\pi$
$463$	236.729	−	236.729i	0.511293	−	0.511293i	0.914923	+	0.403629i	$0.132252\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	66.1570	+	66.1570i	0.141664	+	0.141664i	0.774382	−	0.632718i	$-0.218061\pi$
$467$	66.1570	+	66.1570i	0.141664	+	0.141664i	−0.632718	+	0.774382i	$0.718061\pi$
$468$		0			0
$469$		−	157.832i		−	0.336529i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−37.6893	+	37.6893i	−0.0796813	+	0.0796813i
$474$		0			0
$475$	115.655	+	510.994i	0.243485	+	1.07578i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			216.308i			0.451582i	0.974176	+	0.225791i	$0.0724966\pi$
$479$			216.308i			0.451582i	−0.974176	+	0.225791i	$0.927503\pi$
$480$		0			0
$481$	1553.99			3.23075
$482$		0			0
$483$		0			0
$484$		0			0
$485$	95.0738	−	118.997i	0.196028	−	0.245355i
$486$		0			0
$487$	131.415	+	131.415i	0.269847	+	0.269847i	0.829038	−	0.559192i	$-0.188888\pi$
$487$	131.415	+	131.415i	0.269847	+	0.269847i	−0.559192	+	0.829038i	$0.688888\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−155.504			−0.316709			−0.158354	−	0.987382i	$-0.550619\pi$
$491$	−155.504			−0.316709			−0.158354	+	0.987382i	$0.550619\pi$
$492$		0			0
$493$	272.127	−	272.127i	0.551981	−	0.551981i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	290.542	+	290.542i	0.584591	+	0.584591i
$498$		0			0
$499$			30.8100i			0.0617435i	0.999523	+	0.0308718i	$0.00982835\pi$
$499$			30.8100i			0.0617435i	−0.999523	+	0.0308718i	$0.990172\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−682.523	+	682.523i	−1.35690	+	1.35690i	−0.479196	+	0.877708i	$0.659072\pi$
$503$	−682.523	+	682.523i	−1.35690	+	1.35690i	−0.877708	+	0.479196i	$0.840928\pi$
$504$		0			0
$505$	60.5663	+	541.962i	0.119933	+	1.07319i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	192.205i		−	0.377614i	−0.982014	−	0.188807i	$-0.939538\pi$
$509$		−	192.205i		−	0.377614i	0.982014	−	0.188807i	$-0.0604620\pi$
$510$		0			0
$511$	277.651			0.543349
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−129.087	−	103.135i	−0.250654	−	0.200262i
$516$		0			0
$517$	−81.8999	−	81.8999i	−0.158414	−	0.158414i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	28.8098			0.0552971			0.0276485	−	0.999618i	$-0.491198\pi$
$521$	28.8098			0.0552971			0.0276485	+	0.999618i	$0.491198\pi$
$522$		0			0
$523$	−80.4717	+	80.4717i	−0.153866	+	0.153866i	−0.779842	−	0.625976i	$-0.784701\pi$
$523$	−80.4717	+	80.4717i	−0.153866	+	0.153866i	0.625976	+	0.779842i	$0.284701\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	103.670	+	103.670i	0.196718	+	0.196718i
$528$		0			0
$529$		−	760.220i		−	1.43709i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−508.312	+	508.312i	−0.953682	+	0.953682i
$534$		0			0
$535$	−231.377	+	289.598i	−0.432481	+	0.541305i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			172.387i			0.319827i
$540$		0			0
$541$	895.353			1.65500			0.827498	−	0.561469i	$-0.189764\pi$
$541$	895.353			1.65500			0.827498	+	0.561469i	$0.189764\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−323.876	+	36.1944i	−0.594268	+	0.0664118i
$546$		0			0
$547$	112.258	+	112.258i	0.205224	+	0.205224i	0.802234	−	0.597010i	$-0.203645\pi$
$547$	112.258	+	112.258i	0.205224	+	0.205224i	−0.597010	+	0.802234i	$0.703645\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−828.195			−1.50308
$552$		0			0
$553$	−116.189	+	116.189i	−0.210106	+	0.210106i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	216.572	+	216.572i	0.388819	+	0.388819i	0.874266	−	0.485447i	$-0.161343\pi$
$557$	216.572	+	216.572i	0.388819	+	0.388819i	−0.485447	+	0.874266i	$0.661343\pi$
$558$		0			0
$559$			200.415i			0.358524i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−584.918	+	584.918i	−1.03893	+	1.03893i	−0.0397201	+	0.999211i	$0.512647\pi$
$563$	−584.918	+	584.918i	−1.03893	+	1.03893i	−0.999211	+	0.0397201i	$0.987353\pi$
$564$		0			0
$565$	36.3510	+	29.0430i	0.0643381	+	0.0514036i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			881.720i			1.54960i	0.632209	+	0.774798i	$0.282148\pi$
$569$			881.720i			1.54960i	−0.632209	+	0.774798i	$0.717852\pi$
$570$		0			0
$571$	−873.802			−1.53030			−0.765151	−	0.643851i	$-0.777336\pi$
$571$	−873.802			−1.53030			−0.765151	+	0.643851i	$0.777336\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	478.954	−	759.188i	0.832963	−	1.32033i
$576$		0			0
$577$	564.076	+	564.076i	0.977601	+	0.977601i	0.999755	−	0.0221536i	$-0.00705228\pi$
$577$	564.076	+	564.076i	0.977601	+	0.977601i	−0.0221536	+	0.999755i	$0.507052\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	204.092			0.351278
$582$		0			0
$583$	358.611	−	358.611i	0.615113	−	0.615113i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	743.311	+	743.311i	1.26629	+	1.26629i	0.947991	+	0.318297i	$0.103111\pi$
$587$	743.311	+	743.311i	1.26629	+	1.26629i	0.318297	+	0.947991i	$0.396889\pi$
$588$		0			0
$589$		−	315.512i		−	0.535674i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	288.378	−	288.378i	0.486304	−	0.486304i	−0.420834	−	0.907138i	$-0.638263\pi$
$593$	288.378	−	288.378i	0.486304	−	0.486304i	0.907138	+	0.420834i	$0.138263\pi$
$594$		0			0
$595$	−221.361	+	24.7379i	−0.372035	+	0.0415763i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			503.655i			0.840826i	0.907333	+	0.420413i	$0.138115\pi$
$599$			503.655i			0.840826i	−0.907333	+	0.420413i	$0.861885\pi$
$600$		0			0
$601$	−48.7083			−0.0810454			−0.0405227	−	0.999179i	$-0.512902\pi$
$601$	−48.7083			−0.0810454			−0.0405227	+	0.999179i	$0.512902\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−46.2540	−	413.892i	−0.0764529	−	0.684119i
$606$		0			0
$607$	83.4775	+	83.4775i	0.137525	+	0.137525i	0.772518	−	0.634993i	$-0.218997\pi$
$607$	83.4775	+	83.4775i	0.137525	+	0.137525i	−0.634993	+	0.772518i	$0.718997\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−435.508			−0.712779
$612$		0			0
$613$	87.0923	−	87.0923i	0.142076	−	0.142076i	−0.632492	−	0.774567i	$-0.717968\pi$
$613$	87.0923	−	87.0923i	0.142076	−	0.142076i	0.774567	+	0.632492i	$0.217968\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−382.471	−	382.471i	−0.619889	−	0.619889i	0.325614	−	0.945503i	$-0.394429\pi$
$617$	−382.471	−	382.471i	−0.619889	−	0.619889i	−0.945503	+	0.325614i	$0.894429\pi$
$618$		0			0
$619$			785.763i			1.26941i	0.772755	+	0.634704i	$0.218878\pi$
$619$			785.763i			1.26941i	−0.772755	+	0.634704i	$0.781122\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	244.946	−	244.946i	0.393171	−	0.393171i
$624$		0			0
$625$	564.086	−	269.131i	0.902538	−	0.430610i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	655.424i		−	1.04201i
$630$		0			0
$631$	696.429			1.10369			0.551845	−	0.833947i	$-0.313924\pi$
$631$	696.429			1.10369			0.551845	+	0.833947i	$0.313924\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	617.434	−	772.798i	0.972338	−	1.21701i
$636$		0			0
$637$	458.339	+	458.339i	0.719527	+	0.719527i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−706.802			−1.10266			−0.551328	−	0.834289i	$-0.685879\pi$
$641$	−706.802			−1.10266			−0.551328	+	0.834289i	$0.685879\pi$
$642$		0			0
$643$	−579.371	+	579.371i	−0.901043	+	0.901043i	−0.995526	−	0.0944833i	$-0.969880\pi$
$643$	−579.371	+	579.371i	−0.901043	+	0.901043i	0.0944833	+	0.995526i	$0.469880\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−633.831	−	633.831i	−0.979647	−	0.979647i	0.0201503	−	0.999797i	$-0.493586\pi$
$647$	−633.831	−	633.831i	−0.979647	−	0.979647i	−0.999797	+	0.0201503i	$0.993586\pi$
$648$		0			0
$649$		−	55.0468i		−	0.0848179i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−713.218	+	713.218i	−1.09222	+	1.09222i	−0.0969264	+	0.995292i	$0.530901\pi$
$653$	−713.218	+	713.218i	−1.09222	+	1.09222i	−0.995292	+	0.0969264i	$0.969099\pi$
$654$		0			0
$655$	−79.3545	−	710.083i	−0.121152	−	1.08410i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			702.063i			1.06535i	0.846321	+	0.532673i	$0.178812\pi$
$659$			702.063i			1.06535i	−0.846321	+	0.532673i	$0.821188\pi$
$660$		0			0
$661$	558.795			0.845378			0.422689	−	0.906275i	$-0.361086\pi$
$661$	558.795			0.845378			0.422689	+	0.906275i	$0.361086\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	374.490	+	299.202i	0.563143	+	0.449928i
$666$		0			0
$667$	1003.36	+	1003.36i	1.50429	+	1.50429i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	66.5127			0.0991247
$672$		0			0
$673$	755.646	−	755.646i	1.12280	−	1.12280i	0.131485	−	0.991318i	$-0.458026\pi$
$673$	755.646	−	755.646i	1.12280	−	1.12280i	0.991318	−	0.131485i	$-0.0419744\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	54.5762	+	54.5762i	0.0806148	+	0.0806148i	0.746264	−	0.665650i	$-0.231845\pi$
$677$	54.5762	+	54.5762i	0.0806148	+	0.0806148i	−0.665650	+	0.746264i	$0.731845\pi$
$678$		0			0
$679$		−	139.353i		−	0.205232i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	183.477	−	183.477i	0.268633	−	0.268633i	−0.559916	−	0.828549i	$-0.689167\pi$
$683$	183.477	−	183.477i	0.268633	−	0.268633i	0.828549	+	0.559916i	$0.189167\pi$
$684$		0			0
$685$	−417.234	+	522.222i	−0.609101	+	0.762367i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	1906.94i		−	2.76769i
$690$		0			0
$691$	−311.723			−0.451119			−0.225559	−	0.974229i	$-0.572421\pi$
$691$	−311.723			−0.451119			−0.225559	+	0.974229i	$0.572421\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	1021.25	−	114.129i	1.46943	−	0.164214i
$696$		0			0
$697$	214.390	+	214.390i	0.307589	+	0.307589i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−597.041			−0.851699			−0.425850	−	0.904794i	$-0.640025\pi$
$701$	−597.041			−0.851699			−0.425850	+	0.904794i	$0.640025\pi$
$702$		0			0
$703$	−997.363	+	997.363i	−1.41872	+	1.41872i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	352.798	+	352.798i	0.499008	+	0.499008i
$708$		0			0
$709$			908.757i			1.28174i	0.767648	+	0.640872i	$0.221427\pi$
$709$			908.757i			1.28174i	−0.767648	+	0.640872i	$0.778573\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−382.244	+	382.244i	−0.536107	+	0.536107i
$714$		0			0
$715$	553.834	+	442.491i	0.774593	+	0.618869i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			1127.59i			1.56828i	0.620583	+	0.784141i	$0.286896\pi$
$719$			1127.59i			1.56828i	−0.620583	+	0.784141i	$0.713104\pi$
$720$		0			0
$721$	−151.168			−0.209665
$722$		0			0
$723$		0			0
$724$		0			0
$725$	218.098	+	963.608i	0.300824	+	1.32911i
$726$		0			0
$727$	−356.458	−	356.458i	−0.490313	−	0.490313i	0.418092	−	0.908405i	$-0.362699\pi$
$727$	−356.458	−	356.458i	−0.490313	−	0.490313i	−0.908405	+	0.418092i	$0.862699\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	84.5286			0.115634
$732$		0			0
$733$	809.684	−	809.684i	1.10462	−	1.10462i	0.110771	−	0.993846i	$-0.464668\pi$
$733$	809.684	−	809.684i	1.10462	−	1.10462i	0.993846	−	0.110771i	$-0.0353320\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	149.810	+	149.810i	0.203269	+	0.203269i
$738$		0			0
$739$		−	247.344i		−	0.334700i	−0.985898	−	0.167350i	$-0.946479\pi$
$739$		−	247.344i		−	0.334700i	0.985898	−	0.167350i	$-0.0535211\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	428.003	−	428.003i	0.576047	−	0.576047i	−0.357765	−	0.933812i	$-0.616461\pi$
$743$	428.003	−	428.003i	0.576047	−	0.576047i	0.933812	+	0.357765i	$0.116461\pi$
$744$		0			0
$745$	−561.969	+	62.8021i	−0.754321	+	0.0842982i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			339.137i			0.452787i
$750$		0			0
$751$	−585.241			−0.779282			−0.389641	−	0.920967i	$-0.627401\pi$
$751$	−585.241			−0.779282			−0.389641	+	0.920967i	$0.627401\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−105.758	−	946.347i	−0.140077	−	1.25344i
$756$		0			0
$757$	−401.140	−	401.140i	−0.529907	−	0.529907i	0.390637	−	0.920545i	$-0.372255\pi$
$757$	−401.140	−	401.140i	−0.529907	−	0.529907i	−0.920545	+	0.390637i	$0.872255\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1064.98			−1.39944			−0.699722	−	0.714415i	$-0.746693\pi$
$761$	−1064.98			−1.39944			−0.699722	+	0.714415i	$0.746693\pi$
$762$		0			0
$763$	−210.832	+	210.832i	−0.276320	+	0.276320i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−146.357	−	146.357i	−0.190818	−	0.190818i
$768$		0			0
$769$			246.381i			0.320391i	0.987085	+	0.160196i	$0.0512125\pi$
$769$			246.381i			0.320391i	−0.987085	+	0.160196i	$0.948788\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	59.6027	−	59.6027i	0.0771056	−	0.0771056i	−0.667502	−	0.744608i	$-0.732637\pi$
$773$	59.6027	−	59.6027i	0.0771056	−	0.0771056i	0.744608	+	0.667502i	$0.232637\pi$
$774$		0			0
$775$	−367.100	+	83.0873i	−0.473677	+	0.107209i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	652.476i		−	0.837582i
$780$		0			0
$781$	−551.546			−0.706205
$782$		0			0
$783$		0			0
$784$		0			0
$785$	760.633	−	952.030i	0.968960	−	1.21278i
$786$		0			0
$787$	−463.102	−	463.102i	−0.588439	−	0.588439i	0.348769	−	0.937209i	$-0.386600\pi$
$787$	−463.102	−	463.102i	−0.588439	−	0.588439i	−0.937209	+	0.348769i	$0.886600\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	42.5693			0.0538171
$792$		0			0
$793$	176.843	−	176.843i	0.223005	−	0.223005i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	530.408	+	530.408i	0.665506	+	0.665506i	0.956672	−	0.291166i	$-0.0940434\pi$
$797$	530.408	+	530.408i	0.665506	+	0.665506i	−0.291166	+	0.956672i	$0.594043\pi$
$798$		0			0
$799$			183.683i			0.229891i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−263.538	+	263.538i	−0.328192	+	0.328192i
$804$		0			0
$805$	−91.2113	−	816.181i	−0.113306	−	1.01389i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			401.260i			0.495996i	0.968761	+	0.247998i	$0.0797726\pi$
$809$			401.260i			0.495996i	−0.968761	+	0.247998i	$0.920227\pi$
$810$		0			0
$811$	280.389			0.345732			0.172866	−	0.984945i	$-0.444697\pi$
$811$	280.389			0.345732			0.172866	+	0.984945i	$0.444697\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−486.424	−	388.633i	−0.596839	−	0.476850i
$816$		0			0
$817$	−128.628	−	128.628i	−0.157439	−	0.157439i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	86.0398			0.104799			0.0523994	−	0.998626i	$-0.483313\pi$
$821$	86.0398			0.104799			0.0523994	+	0.998626i	$0.483313\pi$
$822$		0			0
$823$	−3.66736	+	3.66736i	−0.00445609	+	0.00445609i	−0.709331	−	0.704875i	$-0.751003\pi$
$823$	−3.66736	+	3.66736i	−0.00445609	+	0.00445609i	0.704875	+	0.709331i	$0.251003\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	411.003	+	411.003i	0.496980	+	0.496980i	0.910497	−	0.413516i	$-0.135700\pi$
$827$	411.003	+	411.003i	0.496980	+	0.496980i	−0.413516	+	0.910497i	$0.635700\pi$
$828$		0			0
$829$		−	924.508i		−	1.11521i	−0.830107	−	0.557604i	$-0.811721\pi$
$829$		−	924.508i		−	1.11521i	0.830107	−	0.557604i	$-0.188279\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	193.312	−	193.312i	0.232068	−	0.232068i
$834$		0			0
$835$	538.055	−	673.444i	0.644377	−	0.806520i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	257.167i		−	0.306516i	−0.988186	−	0.153258i	$-0.951023\pi$
$839$		−	257.167i		−	0.306516i	0.988186	−	0.153258i	$-0.0489766\pi$
$840$		0			0
$841$	−720.771			−0.857041
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1809.24	−	202.189i	2.14111	−	0.239277i
$846$		0			0
$847$	−269.430	−	269.430i	−0.318099	−	0.318099i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	2416.62			2.83974
$852$		0			0
$853$	−588.869	+	588.869i	−0.690351	+	0.690351i	−0.962309	−	0.271958i	$-0.912329\pi$
$853$	−588.869	+	588.869i	−0.690351	+	0.690351i	0.271958	+	0.962309i	$0.412329\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−321.427	−	321.427i	−0.375061	−	0.375061i	0.494256	−	0.869317i	$-0.335441\pi$
$857$	−321.427	−	321.427i	−0.375061	−	0.375061i	−0.869317	+	0.494256i	$0.835441\pi$
$858$		0			0
$859$			355.578i			0.413944i	0.978347	+	0.206972i	$0.0663609\pi$
$859$			355.578i			0.413944i	−0.978347	+	0.206972i	$0.933639\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−1011.60	+	1011.60i	−1.17219	+	1.17219i	−0.190509	+	0.981685i	$0.561014\pi$
$863$	−1011.60	+	1011.60i	−1.17219	+	1.17219i	−0.981685	+	0.190509i	$0.938986\pi$
$864$		0			0
$865$	−606.491	−	484.562i	−0.701146	−	0.560187i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	220.566i		−	0.253816i
$870$		0			0
$871$	796.622			0.914606
$872$		0			0
$873$		0			0
$874$		0			0
$875$	249.505	−	514.513i	0.285148	−	0.588015i
$876$		0			0
$877$	−597.564	−	597.564i	−0.681373	−	0.681373i	0.278937	−	0.960309i	$-0.410018\pi$
$877$	−597.564	−	597.564i	−0.681373	−	0.681373i	−0.960309	+	0.278937i	$0.910018\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−714.094			−0.810549			−0.405274	−	0.914195i	$-0.632824\pi$
$881$	−714.094			−0.810549			−0.405274	+	0.914195i	$0.632824\pi$
$882$		0			0
$883$	49.1955	−	49.1955i	0.0557140	−	0.0557140i	−0.678701	−	0.734415i	$-0.737457\pi$
$883$	49.1955	−	49.1955i	0.0557140	−	0.0557140i	0.734415	+	0.678701i	$0.237457\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	999.116	+	999.116i	1.12640	+	1.12640i	0.990758	+	0.135641i	$0.0433093\pi$
$887$	999.116	+	999.116i	1.12640	+	1.12640i	0.135641	+	0.990758i	$0.456691\pi$
$888$		0			0
$889$		−	904.994i		−	1.01799i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	279.512	−	279.512i	0.313003	−	0.313003i
$894$		0			0
$895$	95.8823	−	10.7152i	0.107131	−	0.0119723i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	594.978i		−	0.661822i
$900$		0			0
$901$	−804.284			−0.892657
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−158.338	−	1416.84i	−0.174959	−	1.56557i
$906$		0			0
$907$	−760.074	−	760.074i	−0.838009	−	0.838009i	0.150587	−	0.988597i	$-0.451884\pi$
$907$	−760.074	−	760.074i	−0.838009	−	0.838009i	−0.988597	+	0.150587i	$0.951884\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−1073.63			−1.17852			−0.589258	−	0.807945i	$-0.700580\pi$
$911$	−1073.63			−1.17852			−0.589258	+	0.807945i	$0.700580\pi$
$912$		0			0
$913$	−193.718	+	193.718i	−0.212178	+	0.212178i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−462.240	−	462.240i	−0.504079	−	0.504079i
$918$		0			0
$919$		−	1294.24i		−	1.40832i	−0.710043	−	0.704158i	$-0.751325\pi$
$919$		−	1294.24i		−	1.40832i	0.710043	−	0.704158i	$-0.248675\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−1466.44	+	1466.44i	−1.58878	+	1.58878i
$924$		0			0
$925$	1423.08	+	897.789i	1.53847	+	0.970583i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	947.765i		−	1.02020i	−0.860115	−	0.510100i	$-0.829609\pi$
$929$		−	947.765i		−	1.02020i	0.860115	−	0.510100i	$-0.170391\pi$
$930$		0			0
$931$	−588.329			−0.631933
$932$		0			0
$933$		0			0
$934$		0			0
$935$	186.628	−	233.589i	0.199603	−	0.249828i
$936$		0			0
$937$	210.468	+	210.468i	0.224619	+	0.224619i	0.810440	−	0.585821i	$-0.199228\pi$
$937$	210.468	+	210.468i	0.224619	+	0.224619i	−0.585821	+	0.810440i	$0.699228\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1175.11			−1.24879			−0.624393	−	0.781111i	$-0.714653\pi$
$941$	−1175.11			−1.24879			−0.624393	+	0.781111i	$0.714653\pi$
$942$		0			0
$943$	−790.478	+	790.478i	−0.838258	+	0.838258i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−585.786	−	585.786i	−0.618571	−	0.618571i	0.326594	−	0.945165i	$-0.394099\pi$
$947$	−585.786	−	585.786i	−0.618571	−	0.618571i	−0.945165	+	0.326594i	$0.894099\pi$
$948$		0			0
$949$			1401.38i			1.47669i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	269.593	−	269.593i	0.282889	−	0.282889i	−0.551371	−	0.834260i	$-0.685895\pi$
$953$	269.593	−	269.593i	0.282889	−	0.282889i	0.834260	+	0.551371i	$0.185895\pi$
$954$		0			0
$955$	−145.473	−	1301.73i	−0.152328	−	1.36307i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			611.553i			0.637699i
$960$		0			0
$961$	−734.335			−0.764136
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−111.699	−	89.2426i	−0.115750	−	0.0924794i
$966$		0			0
$967$	480.381	+	480.381i	0.496774	+	0.496774i	0.910432	−	0.413658i	$-0.135749\pi$
$967$	480.381	+	480.381i	0.496774	+	0.496774i	−0.413658	+	0.910432i	$0.635749\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	666.958			0.686878			0.343439	−	0.939175i	$-0.388408\pi$
$971$	666.958			0.686878			0.343439	+	0.939175i	$0.388408\pi$
$972$		0			0
$973$	664.801	−	664.801i	0.683249	−	0.683249i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	908.564	+	908.564i	0.929953	+	0.929953i	0.997702	−	0.0677491i	$-0.0215818\pi$
$977$	908.564	+	908.564i	0.929953	+	0.929953i	−0.0677491	+	0.997702i	$0.521582\pi$
$978$		0			0
$979$			464.990i			0.474964i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1016.90	−	1016.90i	1.03448	−	1.03448i	0.0350994	−	0.999384i	$-0.488825\pi$
$983$	1016.90	−	1016.90i	1.03448	−	1.03448i	0.999384	−	0.0350994i	$-0.0111748\pi$
$984$		0			0
$985$	10.2186	−	12.7898i	0.0103742	−	0.0129846i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			311.666i			0.315132i
$990$		0			0
$991$	1138.40			1.14874			0.574371	−	0.818595i	$-0.305247\pi$
$991$	1138.40			1.14874			0.574371	+	0.818595i	$0.305247\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−1845.05	+	206.191i	−1.85432	+	0.207227i
$996$		0			0
$997$	−447.129	−	447.129i	−0.448475	−	0.448475i	0.446372	−	0.894847i	$-0.352716\pi$
$997$	−447.129	−	447.129i	−0.448475	−	0.448475i	−0.894847	+	0.446372i	$0.852716\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.3.l.f.1297.7		24
3.2	odd	2		1620.3.l.g.1297.6		24
5.3	odd	4	inner	1620.3.l.f.973.7		24
9.2	odd	6		540.3.v.a.37.11		48
9.4	even	3		180.3.u.a.97.2	yes	48
9.5	odd	6		540.3.v.a.397.3		48
9.7	even	3		180.3.u.a.157.5	yes	48
15.8	even	4		1620.3.l.g.973.6		24
45.13	odd	12		180.3.u.a.133.5	yes	48
45.23	even	12		540.3.v.a.73.11		48
45.38	even	12		540.3.v.a.253.3		48
45.43	odd	12		180.3.u.a.13.2	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.3.u.a.13.2	✓	48	45.43	odd	12
180.3.u.a.97.2	yes	48	9.4	even	3
180.3.u.a.133.5	yes	48	45.13	odd	12
180.3.u.a.157.5	yes	48	9.7	even	3
540.3.v.a.37.11		48	9.2	odd	6
540.3.v.a.73.11		48	45.23	even	12
540.3.v.a.253.3		48	45.38	even	12
540.3.v.a.397.3		48	9.5	odd	6
1620.3.l.f.973.7		24	5.3	odd	4	inner
1620.3.l.f.1297.7		24	1.1	even	1	trivial
1620.3.l.g.973.6		24	15.8	even	4
1620.3.l.g.1297.6		24	3.2	odd	2

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.l (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.555312 + 4.96907i) q^{5} +(3.23469 + 3.23469i) q^{7} +O(q^{10})\)	\(q+(0.555312 + 4.96907i) q^{5} +(3.23469 + 3.23469i) q^{7} -6.14054 q^{11} +(-16.3264 + 16.3264i) q^{13} +(6.88593 + 6.88593i) q^{17} -20.9567i q^{19} +(-25.3892 + 25.3892i) q^{23} +(-24.3833 + 5.51877i) q^{25} -39.5192i q^{29} +15.0554 q^{31} +(-14.2771 + 17.8697i) q^{35} +(-47.5915 - 47.5915i) q^{37} +31.1344 q^{41} +(6.13777 - 6.13777i) q^{43} +(13.3376 + 13.3376i) q^{47} -28.0735i q^{49} +(-58.4005 + 58.4005i) q^{53} +(-3.40992 - 30.5128i) q^{55} +8.96449i q^{59} -10.8317 q^{61} +(-90.1930 - 72.0606i) q^{65} +(-24.3968 - 24.3968i) q^{67} +89.8205 q^{71} +(42.9177 - 42.9177i) q^{73} +(-19.8628 - 19.8628i) q^{77} +35.9196i q^{79} +(31.5474 - 31.5474i) q^{83} +(-30.3928 + 38.0405i) q^{85} -75.7246i q^{89} -105.622 q^{91} +(104.135 - 11.6375i) q^{95} +(-21.5403 - 21.5403i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q - 12 q^{11} + 18 q^{17} + 12 q^{23} + 6 q^{25} + 36 q^{35} - 42 q^{37} - 72 q^{41} + 78 q^{47} - 156 q^{53} + 66 q^{55} + 96 q^{61} + 132 q^{65} + 78 q^{67} - 156 q^{71} + 240 q^{77} - 132 q^{83} - 96 q^{85} + 84 q^{91} + 168 q^{95} + 90 q^{97}+O(q^{100}) \) 24 * q - 12 * q^11 + 18 * q^17 + 12 * q^23 + 6 * q^25 + 36 * q^35 - 42 * q^37 - 72 * q^41 + 78 * q^47 - 156 * q^53 + 66 * q^55 + 96 * q^61 + 132 * q^65 + 78 * q^67 - 156 * q^71 + 240 * q^77 - 132 * q^83 - 96 * q^85 + 84 * q^91 + 168 * q^95 + 90 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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