LMFDB - Embedded newform 2520.2.k.b.1889.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2520,2,Mod(1889,2520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2520, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 1, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2520.1889");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2520.k (of order $2$, degree $1$, 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:	$20.1223013094$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1889.14
Character	$\chi$	$=$	2520.1889
Dual form			2520.2.k.b.1889.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.655755 + 2.13775i) q^{5} +(2.09441 - 1.61662i) q^{7} +O(q^{10})$	$q+(0.655755 + 2.13775i) q^{5} +(2.09441 - 1.61662i) q^{7} +6.16020i q^{11} -0.742412 q^{13} +3.80978i q^{17} -7.08718i q^{19} +5.47817 q^{23} +(-4.13997 + 2.80368i) q^{25} -3.48294i q^{29} +7.72681i q^{31} +(4.82935 + 3.41722i) q^{35} +4.20866i q^{37} +1.52402 q^{41} +12.0150i q^{43} -0.165189i q^{47} +(1.77309 - 6.77172i) q^{49} -1.52623 q^{53} +(-13.1690 + 4.03959i) q^{55} -11.2218 q^{59} -7.03535i q^{61} +(-0.486841 - 1.58709i) q^{65} -0.383095i q^{67} -5.81423i q^{71} -11.8823 q^{73} +(9.95870 + 12.9020i) q^{77} +10.5244 q^{79} +10.5797i q^{83} +(-8.14436 + 2.49828i) q^{85} -0.779764 q^{89} +(-1.55491 + 1.20020i) q^{91} +(15.1506 - 4.64745i) q^{95} +7.92966 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q + 8 q^{23} - 16 q^{25} - 4 q^{35} - 12 q^{49} - 24 q^{53} - 8 q^{65} - 4 q^{77} + 40 q^{79} + 24 q^{85} - 36 q^{91} + 48 q^{95}+O(q^{100}) $ 24 * q + 8 * q^23 - 16 * q^25 - 4 * q^35 - 12 * q^49 - 24 * q^53 - 8 * q^65 - 4 * q^77 + 40 * q^79 + 24 * q^85 - 36 * q^91 + 48 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2520\mathbb{Z}\right)^\times$.

$n$	$281$	$631$	$1081$	$1261$	$2017$
$\chi(n)$	$-1$	$1$	$-1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.655755	+	2.13775i	0.293263	+	0.956032i
$5$	0.655755	+	2.13775i	0.293263	+	0.956032i
$6$		0			0
$7$	2.09441	−	1.61662i	0.791612	−	0.611025i
$7$	2.09441	−	1.61662i	0.791612	−	0.611025i
$8$		0			0
$9$		0			0
$10$		0			0
$11$			6.16020i			1.85737i	0.370868	+	0.928686i	$0.379060\pi$
$11$			6.16020i			1.85737i	−0.370868	+	0.928686i	$0.620940\pi$
$12$		0			0
$13$	−0.742412			−0.205908			−0.102954	−	0.994686i	$-0.532829\pi$
$13$	−0.742412			−0.205908			−0.102954	+	0.994686i	$0.532829\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			3.80978i			0.924007i	0.886878	+	0.462004i	$0.152869\pi$
$17$			3.80978i			0.924007i	−0.886878	+	0.462004i	$0.847131\pi$
$18$		0			0
$19$		−	7.08718i		−	1.62591i	−0.582327	−	0.812955i	$-0.697858\pi$
$19$		−	7.08718i		−	1.62591i	0.582327	−	0.812955i	$-0.302142\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	5.47817			1.14228			0.571138	−	0.820854i	$-0.306502\pi$
$23$	5.47817			1.14228			0.571138	+	0.820854i	$0.306502\pi$
$24$		0			0
$25$	−4.13997	+	2.80368i	−0.827994	+	0.560737i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	3.48294i		−	0.646766i	−0.946268	−	0.323383i	$-0.895180\pi$
$29$		−	3.48294i		−	0.646766i	0.946268	−	0.323383i	$-0.104820\pi$
$30$		0			0
$31$			7.72681i			1.38778i	0.720083	+	0.693888i	$0.244104\pi$
$31$			7.72681i			1.38778i	−0.720083	+	0.693888i	$0.755896\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	4.82935	+	3.41722i	0.816309	+	0.577615i
$36$		0			0
$37$			4.20866i			0.691899i	0.938253	+	0.345950i	$0.112443\pi$
$37$			4.20866i			0.691899i	−0.938253	+	0.345950i	$0.887557\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	1.52402			0.238012			0.119006	−	0.992894i	$-0.462029\pi$
$41$	1.52402			0.238012			0.119006	+	0.992894i	$0.462029\pi$
$42$		0			0
$43$			12.0150i			1.83227i	0.400868	+	0.916136i	$0.368708\pi$
$43$			12.0150i			1.83227i	−0.400868	+	0.916136i	$0.631292\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	0.165189i		−	0.0240953i	−0.999927	−	0.0120477i	$-0.996165\pi$
$47$		−	0.165189i		−	0.0240953i	0.999927	−	0.0120477i	$-0.00383498\pi$
$48$		0			0
$49$	1.77309	−	6.77172i	0.253298	−	0.967388i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−1.52623			−0.209643			−0.104822	−	0.994491i	$-0.533427\pi$
$53$	−1.52623			−0.209643			−0.104822	+	0.994491i	$0.533427\pi$
$54$		0			0
$55$	−13.1690	+	4.03959i	−1.77571	+	0.544698i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−11.2218			−1.46095			−0.730477	−	0.682937i	$-0.760702\pi$
$59$	−11.2218			−1.46095			−0.730477	+	0.682937i	$0.760702\pi$
$60$		0			0
$61$		−	7.03535i		−	0.900785i	−0.892831	−	0.450392i	$-0.851284\pi$
$61$		−	7.03535i		−	0.900785i	0.892831	−	0.450392i	$-0.148716\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−0.486841	−	1.58709i	−0.0603852	−	0.196855i
$66$		0			0
$67$		−	0.383095i		−	0.0468025i	−0.999726	−	0.0234012i	$-0.992550\pi$
$67$		−	0.383095i		−	0.0468025i	0.999726	−	0.0234012i	$-0.00744952\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	5.81423i		−	0.690022i	−0.938599	−	0.345011i	$-0.887875\pi$
$71$		−	5.81423i		−	0.690022i	0.938599	−	0.345011i	$-0.112125\pi$
$72$		0			0
$73$	−11.8823			−1.39072			−0.695358	−	0.718663i	$-0.744754\pi$
$73$	−11.8823			−1.39072			−0.695358	+	0.718663i	$0.744754\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	9.95870	+	12.9020i	1.13490	+	1.47032i
$78$		0			0
$79$	10.5244			1.18409			0.592045	−	0.805905i	$-0.298321\pi$
$79$	10.5244			1.18409			0.592045	+	0.805905i	$0.298321\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			10.5797i			1.16127i	0.814164	+	0.580635i	$0.197196\pi$
$83$			10.5797i			1.16127i	−0.814164	+	0.580635i	$0.802804\pi$
$84$		0			0
$85$	−8.14436	+	2.49828i	−0.883380	+	0.270977i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−0.779764			−0.0826549			−0.0413274	−	0.999146i	$-0.513159\pi$
$89$	−0.779764			−0.0826549			−0.0413274	+	0.999146i	$0.513159\pi$
$90$		0			0
$91$	−1.55491	+	1.20020i	−0.162999	+	0.125815i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	15.1506	−	4.64745i	1.55442	−	0.476818i
$96$		0			0
$97$	7.92966			0.805135			0.402568	−	0.915390i	$-0.368118\pi$
$97$	7.92966			0.805135			0.402568	+	0.915390i	$0.368118\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−7.22645			−0.719058			−0.359529	−	0.933134i	$-0.617063\pi$
$101$	−7.22645			−0.719058			−0.359529	+	0.933134i	$0.617063\pi$
$102$		0			0
$103$	1.42399			0.140310			0.0701552	−	0.997536i	$-0.477651\pi$
$103$	1.42399			0.140310			0.0701552	+	0.997536i	$0.477651\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	8.40379			0.812425			0.406213	−	0.913779i	$-0.366849\pi$
$107$	8.40379			0.812425			0.406213	+	0.913779i	$0.366849\pi$
$108$		0			0
$109$	5.79066			0.554645			0.277322	−	0.960777i	$-0.410553\pi$
$109$	5.79066			0.554645			0.277322	+	0.960777i	$0.410553\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	16.3294			1.53614			0.768069	−	0.640367i	$-0.221218\pi$
$113$	16.3294			1.53614			0.768069	+	0.640367i	$0.221218\pi$
$114$		0			0
$115$	3.59234	+	11.7110i	0.334987	+	1.09205i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	6.15896	+	7.97923i	0.564591	+	0.731455i
$120$		0			0
$121$	−26.9481			−2.44983
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−8.70839	−	7.01170i	−0.778902	−	0.627146i
$126$		0			0
$127$			7.78787i			0.691062i	0.938407	+	0.345531i	$0.112301\pi$
$127$			7.78787i			0.691062i	−0.938407	+	0.345531i	$0.887699\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−10.7422			−0.938554			−0.469277	−	0.883051i	$-0.655485\pi$
$131$	−10.7422			−0.938554			−0.469277	+	0.883051i	$0.655485\pi$
$132$		0			0
$133$	−11.4573	−	14.8434i	−0.993471	−	1.28709i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	17.5443			1.49891			0.749457	−	0.662053i	$-0.230315\pi$
$137$	17.5443			1.49891			0.749457	+	0.662053i	$0.230315\pi$
$138$		0			0
$139$			5.01427i			0.425305i	0.977128	+	0.212652i	$0.0682102\pi$
$139$			5.01427i			0.425305i	−0.977128	+	0.212652i	$0.931790\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	4.57341i		−	0.382448i
$144$		0			0
$145$	7.44567	−	2.28396i	0.618329	−	0.189672i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			8.22666i			0.673955i	0.941513	+	0.336977i	$0.109405\pi$
$149$			8.22666i			0.673955i	−0.941513	+	0.336977i	$0.890595\pi$
$150$		0			0
$151$	11.6375			0.947044			0.473522	−	0.880782i	$-0.342982\pi$
$151$	11.6375			0.947044			0.473522	+	0.880782i	$0.342982\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−16.5180	+	5.06690i	−1.32676	+	0.406983i
$156$		0			0
$157$	−6.78889			−0.541812			−0.270906	−	0.962606i	$-0.587323\pi$
$157$	−6.78889			−0.541812			−0.270906	+	0.962606i	$0.587323\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	11.4735	−	8.85611i	0.904240	−	0.697959i
$162$		0			0
$163$			14.1055i			1.10482i	0.833571	+	0.552412i	$0.186293\pi$
$163$			14.1055i			1.10482i	−0.833571	+	0.552412i	$0.813707\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			18.7686i			1.45236i	0.687506	+	0.726179i	$0.258706\pi$
$167$			18.7686i			1.45236i	−0.687506	+	0.726179i	$0.741294\pi$
$168$		0			0
$169$	−12.4488			−0.957602
$170$		0			0
$171$		0			0
$172$		0			0
$173$			14.1094i			1.07272i	0.843989	+	0.536360i	$0.180201\pi$
$173$			14.1094i			1.07272i	−0.843989	+	0.536360i	$0.819799\pi$
$174$		0			0
$175$	−4.13830	+	12.5648i	−0.312826	+	0.949811i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			9.79825i			0.732356i	0.930545	+	0.366178i	$0.119334\pi$
$179$			9.79825i			0.732356i	−0.930545	+	0.366178i	$0.880666\pi$
$180$		0			0
$181$			7.49497i			0.557097i	0.960422	+	0.278548i	$0.0898532\pi$
$181$			7.49497i			0.557097i	−0.960422	+	0.278548i	$0.910147\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−8.99707	+	2.75985i	−0.661478	+	0.202908i
$186$		0			0
$187$	−23.4690			−1.71622
$188$		0			0
$189$		0			0
$190$		0			0
$191$			10.2912i			0.744643i	0.928104	+	0.372321i	$0.121438\pi$
$191$			10.2912i			0.744643i	−0.928104	+	0.372321i	$0.878562\pi$
$192$		0			0
$193$		−	9.26735i		−	0.667078i	−0.942736	−	0.333539i	$-0.891757\pi$
$193$		−	9.26735i		−	0.667078i	0.942736	−	0.333539i	$-0.108243\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−13.5136			−0.962805			−0.481403	−	0.876500i	$-0.659873\pi$
$197$	−13.5136			−0.962805			−0.481403	+	0.876500i	$0.659873\pi$
$198$		0			0
$199$		−	21.9602i		−	1.55672i	−0.627818	−	0.778360i	$-0.716052\pi$
$199$		−	21.9602i		−	1.55672i	0.627818	−	0.778360i	$-0.283948\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−5.63059	−	7.29470i	−0.395190	−	0.511987i
$204$		0			0
$205$	0.999385	+	3.25798i	0.0698001	+	0.227547i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	43.6585			3.01992
$210$		0			0
$211$	−10.2580			−0.706191			−0.353095	−	0.935587i	$-0.614871\pi$
$211$	−10.2580			−0.706191			−0.353095	+	0.935587i	$0.614871\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−25.6851	+	7.87890i	−1.75171	+	0.537337i
$216$		0			0
$217$	12.4913	+	16.1831i	0.847966	+	1.09858i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	2.82843i		−	0.190261i
$222$		0			0
$223$	3.95520			0.264860			0.132430	−	0.991192i	$-0.457722\pi$
$223$	3.95520			0.264860			0.132430	+	0.991192i	$0.457722\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	0.749393i		−	0.0497390i	−0.999691	−	0.0248695i	$-0.992083\pi$
$227$		−	0.749393i		−	0.0497390i	0.999691	−	0.0248695i	$-0.00791702\pi$
$228$		0			0
$229$		−	20.0597i		−	1.32558i	−0.748805	−	0.662790i	$-0.769372\pi$
$229$		−	20.0597i		−	1.32558i	0.748805	−	0.662790i	$-0.230628\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1.44302			0.0945352			0.0472676	−	0.998882i	$-0.484949\pi$
$233$	1.44302			0.0945352			0.0472676	+	0.998882i	$0.484949\pi$
$234$		0			0
$235$	0.353134	−	0.108324i	0.0230359	−	0.00706626i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			29.2334i			1.89095i	0.325689	+	0.945477i	$0.394404\pi$
$239$			29.2334i			1.89095i	−0.325689	+	0.945477i	$0.605596\pi$
$240$		0			0
$241$		−	24.1352i		−	1.55469i	−0.629076	−	0.777343i	$-0.716567\pi$
$241$		−	24.1352i		−	1.55469i	0.629076	−	0.777343i	$-0.283433\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	15.6390	−	0.650171i	0.999137	−	0.0415379i
$246$		0			0
$247$			5.26161i			0.334788i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−8.41107			−0.530902			−0.265451	−	0.964124i	$-0.585521\pi$
$251$	−8.41107			−0.530902			−0.265451	+	0.964124i	$0.585521\pi$
$252$		0			0
$253$			33.7466i			2.12163i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	23.0766i		−	1.43948i	−0.694246	−	0.719738i	$-0.744262\pi$
$257$		−	23.0766i		−	1.43948i	0.694246	−	0.719738i	$-0.255738\pi$
$258$		0			0
$259$	6.80380	+	8.81465i	0.422767	+	0.547715i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−9.82686			−0.605950			−0.302975	−	0.952998i	$-0.597980\pi$
$263$	−9.82686			−0.605950			−0.302975	+	0.952998i	$0.597980\pi$
$264$		0			0
$265$	−1.00083	−	3.26269i	−0.0614805	−	0.200426i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	29.2277			1.78205			0.891024	−	0.453957i	$-0.149988\pi$
$269$	29.2277			1.78205			0.891024	+	0.453957i	$0.149988\pi$
$270$		0			0
$271$			6.56863i			0.399016i	0.979896	+	0.199508i	$0.0639343\pi$
$271$			6.56863i			0.399016i	−0.979896	+	0.199508i	$0.936066\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−17.2713	−	25.5031i	−1.04150	−	1.53789i
$276$		0			0
$277$		−	14.7844i		−	0.888308i	−0.895951	−	0.444154i	$-0.853504\pi$
$277$		−	14.7844i		−	0.888308i	0.895951	−	0.444154i	$-0.146496\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	7.81272i		−	0.466068i	−0.972469	−	0.233034i	$-0.925135\pi$
$281$		−	7.81272i		−	0.466068i	0.972469	−	0.233034i	$-0.0748654\pi$
$282$		0			0
$283$	17.4722			1.03861			0.519306	−	0.854589i	$-0.326191\pi$
$283$	17.4722			1.03861			0.519306	+	0.854589i	$0.326191\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	3.19192	−	2.46376i	0.188413	−	0.145431i
$288$		0			0
$289$	2.48559			0.146211
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	1.21174i		−	0.0707908i	−0.999373	−	0.0353954i	$-0.988731\pi$
$293$		−	1.21174i		−	0.0707908i	0.999373	−	0.0353954i	$-0.0112691\pi$
$294$		0			0
$295$	−7.35876	−	23.9894i	−0.428443	−	1.39672i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−4.06706			−0.235204
$300$		0			0
$301$	19.4237	+	25.1643i	1.11956	+	1.45045i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	15.0398	−	4.61347i	0.861179	−	0.264166i
$306$		0			0
$307$	−8.83705			−0.504357			−0.252179	−	0.967681i	$-0.581147\pi$
$307$	−8.83705			−0.504357			−0.252179	+	0.967681i	$0.581147\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	8.22595			0.466451			0.233225	−	0.972423i	$-0.425072\pi$
$311$	8.22595			0.466451			0.233225	+	0.972423i	$0.425072\pi$
$312$		0			0
$313$	31.5697			1.78443			0.892213	−	0.451615i	$-0.149152\pi$
$313$	31.5697			1.78443			0.892213	+	0.451615i	$0.149152\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−15.4082			−0.865410			−0.432705	−	0.901536i	$-0.642441\pi$
$317$	−15.4082			−0.865410			−0.432705	+	0.901536i	$0.642441\pi$
$318$		0			0
$319$	21.4556			1.20128
$320$		0			0
$321$		0			0
$322$		0			0
$323$	27.0006			1.50235
$324$		0			0
$325$	3.07357	−	2.08149i	0.170491	−	0.115460i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−0.267048	−	0.345973i	−0.0147228	−	0.0190741i
$330$		0			0
$331$	33.5504			1.84410			0.922049	−	0.387072i	$-0.126514\pi$
$331$	33.5504			1.84410			0.922049	+	0.387072i	$0.126514\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	0.818961	−	0.251216i	0.0447446	−	0.0137254i
$336$		0			0
$337$		−	11.8785i		−	0.647062i	−0.946218	−	0.323531i	$-0.895130\pi$
$337$		−	11.8785i		−	0.647062i	0.946218	−	0.323531i	$-0.104870\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−47.5987			−2.57762
$342$		0			0
$343$	−7.23373	−	17.0491i	−0.390584	−	0.920567i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−6.05066			−0.324816			−0.162408	−	0.986724i	$-0.551926\pi$
$347$	−6.05066			−0.324816			−0.162408	+	0.986724i	$0.551926\pi$
$348$		0			0
$349$		−	4.09546i		−	0.219225i	−0.993974	−	0.109613i	$-0.965039\pi$
$349$		−	4.09546i		−	0.219225i	0.993974	−	0.109613i	$-0.0349610\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			16.2209i			0.863351i	0.902029	+	0.431676i	$0.142078\pi$
$353$			16.2209i			0.863351i	−0.902029	+	0.431676i	$0.857922\pi$
$354$		0			0
$355$	12.4294	−	3.81271i	0.659683	−	0.202358i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	2.86813i		−	0.151374i	−0.997132	−	0.0756870i	$-0.975885\pi$
$359$		−	2.86813i		−	0.151374i	0.997132	−	0.0756870i	$-0.0241150\pi$
$360$		0			0
$361$	−31.2281			−1.64358
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−7.79187	−	25.4014i	−0.407845	−	1.32957i
$366$		0			0
$367$	15.0600			0.786127			0.393064	−	0.919511i	$-0.371415\pi$
$367$	15.0600			0.786127			0.393064	+	0.919511i	$0.371415\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−3.19654	+	2.46733i	−0.165956	+	0.128097i
$372$		0			0
$373$		−	32.5151i		−	1.68357i	−0.539815	−	0.841784i	$-0.681506\pi$
$373$		−	32.5151i		−	1.68357i	0.539815	−	0.841784i	$-0.318494\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			2.58578i			0.133174i
$378$		0			0
$379$	31.3069			1.60813			0.804063	−	0.594544i	$-0.202667\pi$
$379$	31.3069			1.60813			0.804063	+	0.594544i	$0.202667\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	36.0216i		−	1.84062i	−0.391195	−	0.920308i	$-0.627938\pi$
$383$		−	36.0216i		−	1.84062i	0.391195	−	0.920308i	$-0.372062\pi$
$384$		0			0
$385$	−21.0508	+	29.7498i	−1.07285	+	1.51619i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			19.2443i			0.975726i	0.872920	+	0.487863i	$0.162223\pi$
$389$			19.2443i			0.975726i	−0.872920	+	0.487863i	$0.837777\pi$
$390$		0			0
$391$			20.8706i			1.05547i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	6.90145	+	22.4986i	0.347250	+	1.13203i
$396$		0			0
$397$	−7.11838			−0.357261			−0.178631	−	0.983916i	$-0.557167\pi$
$397$	−7.11838			−0.357261			−0.178631	+	0.983916i	$0.557167\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	8.68857i		−	0.433887i	−0.976184	−	0.216943i	$-0.930391\pi$
$401$		−	8.68857i		−	0.433887i	0.976184	−	0.216943i	$-0.0696087\pi$
$402$		0			0
$403$		−	5.73648i		−	0.285755i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−25.9262			−1.28511
$408$		0			0
$409$		−	32.5353i		−	1.60877i	−0.594111	−	0.804383i	$-0.702496\pi$
$409$		−	32.5353i		−	1.60877i	0.594111	−	0.804383i	$-0.297504\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−23.5030	+	18.1414i	−1.15651	+	0.892679i
$414$		0			0
$415$	−22.6167	+	6.93768i	−1.11021	+	0.340557i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	37.3737			1.82583			0.912913	−	0.408155i	$-0.133828\pi$
$419$	37.3737			1.82583			0.912913	+	0.408155i	$0.133828\pi$
$420$		0			0
$421$	8.55865			0.417123			0.208562	−	0.978009i	$-0.433122\pi$
$421$	8.55865			0.417123			0.208562	+	0.978009i	$0.433122\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−10.6814	−	15.7724i	−0.518125	−	0.765072i
$426$		0			0
$427$	−11.3735	−	14.7349i	−0.550402	−	0.713072i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	8.19339i		−	0.394662i	−0.980337	−	0.197331i	$-0.936773\pi$
$431$		−	8.19339i		−	0.394662i	0.980337	−	0.197331i	$-0.0632273\pi$
$432$		0			0
$433$	−30.8938			−1.48466			−0.742331	−	0.670034i	$-0.766280\pi$
$433$	−30.8938			−1.48466			−0.742331	+	0.670034i	$0.766280\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	38.8247i		−	1.85724i
$438$		0			0
$439$		−	16.5107i		−	0.788013i	−0.919108	−	0.394006i	$-0.871089\pi$
$439$		−	16.5107i		−	0.788013i	0.919108	−	0.394006i	$-0.128911\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	25.1430			1.19458			0.597290	−	0.802025i	$-0.296244\pi$
$443$	25.1430			1.19458			0.597290	+	0.802025i	$0.296244\pi$
$444$		0			0
$445$	−0.511335	−	1.66694i	−0.0242396	−	0.0790207i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	24.8317i		−	1.17188i	−0.810355	−	0.585940i	$-0.800725\pi$
$449$		−	24.8317i		−	1.17188i	0.810355	−	0.585940i	$-0.199275\pi$
$450$		0			0
$451$			9.38829i			0.442077i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−3.58537	−	2.53699i	−0.168085	−	0.118936i
$456$		0			0
$457$		−	10.3591i		−	0.484577i	−0.970204	−	0.242289i	$-0.922102\pi$
$457$		−	10.3591i		−	0.484577i	0.970204	−	0.242289i	$-0.0778981\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	35.0219			1.63113			0.815566	−	0.578664i	$-0.196426\pi$
$461$	35.0219			1.63113			0.815566	+	0.578664i	$0.196426\pi$
$462$		0			0
$463$			7.96723i			0.370269i	0.982713	+	0.185134i	$0.0592720\pi$
$463$			7.96723i			0.370269i	−0.982713	+	0.185134i	$0.940728\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	35.1007i		−	1.62427i	−0.583471	−	0.812134i	$-0.698306\pi$
$467$		−	35.1007i		−	1.62427i	0.583471	−	0.812134i	$-0.301694\pi$
$468$		0			0
$469$	−0.619318	−	0.802356i	−0.0285975	−	0.0370494i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−74.0149			−3.40321
$474$		0			0
$475$	19.8702	+	29.3407i	0.911707	+	1.34624i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−22.8584			−1.04443			−0.522213	−	0.852815i	$-0.674893\pi$
$479$	−22.8584			−1.04443			−0.522213	+	0.852815i	$0.674893\pi$
$480$		0			0
$481$		−	3.12456i		−	0.142468i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	5.19992	+	16.9517i	0.236116	+	0.769735i
$486$		0			0
$487$			22.4700i			1.01821i	0.860703	+	0.509107i	$0.170024\pi$
$487$			22.4700i			1.01821i	−0.860703	+	0.509107i	$0.829976\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	18.1659i		−	0.819814i	−0.912127	−	0.409907i	$-0.865561\pi$
$491$		−	18.1659i		−	0.819814i	0.912127	−	0.409907i	$-0.134439\pi$
$492$		0			0
$493$	13.2692			0.597616
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−9.39940	−	12.1774i	−0.421621	−	0.546230i
$498$		0			0
$499$	1.08913			0.0487563			0.0243781	−	0.999703i	$-0.492239\pi$
$499$	1.08913			0.0487563			0.0243781	+	0.999703i	$0.492239\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	38.9515i		−	1.73676i	−0.495898	−	0.868381i	$-0.665161\pi$
$503$		−	38.9515i		−	1.73676i	0.495898	−	0.868381i	$-0.334839\pi$
$504$		0			0
$505$	−4.73878	−	15.4484i	−0.210873	−	0.687443i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−32.5800			−1.44408			−0.722042	−	0.691849i	$-0.756796\pi$
$509$	−32.5800			−1.44408			−0.722042	+	0.691849i	$0.756796\pi$
$510$		0			0
$511$	−24.8864	+	19.2091i	−1.10091	+	0.849762i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	0.933792	+	3.04415i	0.0411478	+	0.134141i
$516$		0			0
$517$	1.01760			0.0447540
$518$		0			0
$519$		0			0
$520$		0			0
$521$	6.12007			0.268125			0.134063	−	0.990973i	$-0.457198\pi$
$521$	6.12007			0.268125			0.134063	+	0.990973i	$0.457198\pi$
$522$		0			0
$523$	−21.3379			−0.933042			−0.466521	−	0.884510i	$-0.654493\pi$
$523$	−21.3379			−0.933042			−0.466521	+	0.884510i	$0.654493\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−29.4374			−1.28232
$528$		0			0
$529$	7.01032			0.304796
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−1.13145			−0.0490087
$534$		0			0
$535$	5.51083	+	17.9652i	0.238254	+	0.776704i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	41.7152	+	10.9226i	1.79680	+	0.470468i
$540$		0			0
$541$	−17.7997			−0.765271			−0.382635	−	0.923899i	$-0.624983\pi$
$541$	−17.7997			−0.765271			−0.382635	+	0.923899i	$0.624983\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	3.79726	+	12.3790i	0.162657	+	0.530258i
$546$		0			0
$547$		−	31.4534i		−	1.34485i	−0.740165	−	0.672426i	$-0.765252\pi$
$547$		−	31.4534i		−	1.34485i	0.740165	−	0.672426i	$-0.234748\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−24.6842			−1.05158
$552$		0			0
$553$	22.0424	−	17.0140i	0.937340	−	0.723509i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	2.38569			0.101085			0.0505424	−	0.998722i	$-0.483905\pi$
$557$	2.38569			0.101085			0.0505424	+	0.998722i	$0.483905\pi$
$558$		0			0
$559$		−	8.92009i		−	0.377280i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	8.77574i		−	0.369853i	−0.982752	−	0.184927i	$-0.940795\pi$
$563$		−	8.77574i		−	0.369853i	0.982752	−	0.184927i	$-0.0592048\pi$
$564$		0			0
$565$	10.7081	+	34.9081i	0.450492	+	1.46860i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			7.65260i			0.320814i	0.987051	+	0.160407i	$0.0512806\pi$
$569$			7.65260i			0.320814i	−0.987051	+	0.160407i	$0.948719\pi$
$570$		0			0
$571$	−28.5424			−1.19446			−0.597231	−	0.802069i	$-0.703732\pi$
$571$	−28.5424			−1.19446			−0.597231	+	0.802069i	$0.703732\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−22.6795	+	15.3591i	−0.945798	+	0.640517i
$576$		0			0
$577$	−23.1289			−0.962869			−0.481435	−	0.876482i	$-0.659884\pi$
$577$	−23.1289			−0.962869			−0.481435	+	0.876482i	$0.659884\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	17.1033	+	22.1582i	0.709565	+	0.919275i
$582$		0			0
$583$		−	9.40186i		−	0.389385i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	14.9494i		−	0.617030i	−0.951220	−	0.308515i	$-0.900168\pi$
$587$		−	14.9494i		−	0.617030i	0.951220	−	0.308515i	$-0.0998319\pi$
$588$		0			0
$589$	54.7613			2.25640
$590$		0			0
$591$		0			0
$592$		0			0
$593$			20.3842i			0.837079i	0.908199	+	0.418539i	$0.137458\pi$
$593$			20.3842i			0.837079i	−0.908199	+	0.418539i	$0.862542\pi$
$594$		0			0
$595$	−13.0188	+	18.3988i	−0.533721	+	0.754275i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	22.9640i		−	0.938285i	−0.883122	−	0.469143i	$-0.844563\pi$
$599$		−	22.9640i		−	0.938285i	0.883122	−	0.469143i	$-0.155437\pi$
$600$		0			0
$601$			8.78139i			0.358200i	0.983831	+	0.179100i	$0.0573186\pi$
$601$			8.78139i			0.358200i	−0.983831	+	0.179100i	$0.942681\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−17.6714	−	57.6084i	−0.718443	−	2.34211i
$606$		0			0
$607$	35.4061			1.43709			0.718544	−	0.695481i	$-0.244809\pi$
$607$	35.4061			1.43709			0.718544	+	0.695481i	$0.244809\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			0.122639i			0.00496142i
$612$		0			0
$613$			18.4222i			0.744066i	0.928220	+	0.372033i	$0.121339\pi$
$613$			18.4222i			0.744066i	−0.928220	+	0.372033i	$0.878661\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	27.7729			1.11809			0.559047	−	0.829136i	$-0.311167\pi$
$617$	27.7729			1.11809			0.559047	+	0.829136i	$0.311167\pi$
$618$		0			0
$619$		−	24.5366i		−	0.986209i	−0.869970	−	0.493104i	$-0.835862\pi$
$619$		−	24.5366i		−	0.986209i	0.869970	−	0.493104i	$-0.164138\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−1.63314	+	1.26058i	−0.0654306	+	0.0505042i
$624$		0			0
$625$	9.27871	−	23.2143i	0.371148	−	0.928574i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−16.0341			−0.639320
$630$		0			0
$631$	−28.0362			−1.11610			−0.558052	−	0.829806i	$-0.688451\pi$
$631$	−28.0362			−1.11610			−0.558052	+	0.829806i	$0.688451\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−16.6485	+	5.10694i	−0.660677	+	0.202663i
$636$		0			0
$637$	−1.31636	+	5.02741i	−0.0521561	+	0.199193i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	43.7673i		−	1.72870i	−0.502889	−	0.864351i	$-0.667729\pi$
$641$		−	43.7673i		−	1.72870i	0.502889	−	0.864351i	$-0.332271\pi$
$642$		0			0
$643$	31.9683			1.26071			0.630353	−	0.776308i	$-0.282910\pi$
$643$	31.9683			1.26071			0.630353	+	0.776308i	$0.282910\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			30.5061i			1.19932i	0.800255	+	0.599660i	$0.204698\pi$
$647$			30.5061i			1.19932i	−0.800255	+	0.599660i	$0.795302\pi$
$648$		0			0
$649$		−	69.1286i		−	2.71354i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−12.6940			−0.496756			−0.248378	−	0.968663i	$-0.579898\pi$
$653$	−12.6940			−0.496756			−0.248378	+	0.968663i	$0.579898\pi$
$654$		0			0
$655$	−7.04428	−	22.9643i	−0.275243	−	0.897288i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			6.91969i			0.269553i	0.990876	+	0.134776i	$0.0430316\pi$
$659$			6.91969i			0.269553i	−0.990876	+	0.134776i	$0.956968\pi$
$660$		0			0
$661$			31.0138i			1.20630i	0.797629	+	0.603148i	$0.206087\pi$
$661$			31.0138i			1.20630i	−0.797629	+	0.603148i	$0.793913\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	24.2184	−	34.2265i	0.939150	−	1.32724i
$666$		0			0
$667$		−	19.0801i		−	0.738786i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	43.3392			1.67309
$672$		0			0
$673$		−	39.0349i		−	1.50469i	−0.658772	−	0.752343i	$-0.728924\pi$
$673$		−	39.0349i		−	1.50469i	0.658772	−	0.752343i	$-0.271076\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			32.2899i			1.24100i	0.784206	+	0.620500i	$0.213070\pi$
$677$			32.2899i			1.24100i	−0.784206	+	0.620500i	$0.786930\pi$
$678$		0			0
$679$	16.6079	−	12.8192i	0.637354	−	0.491957i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−33.6562			−1.28782			−0.643909	−	0.765102i	$-0.722688\pi$
$683$	−33.6562			−1.28782			−0.643909	+	0.765102i	$0.722688\pi$
$684$		0			0
$685$	11.5048	+	37.5054i	0.439575	+	1.43301i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1.13309			0.0431673
$690$		0			0
$691$		−	44.4617i		−	1.69140i	−0.533658	−	0.845701i	$-0.679183\pi$
$691$		−	44.4617i		−	1.69140i	0.533658	−	0.845701i	$-0.320817\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−10.7193	+	3.28813i	−0.406605	+	0.124726i
$696$		0			0
$697$			5.80619i			0.219925i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			41.7341i			1.57627i	0.615500	+	0.788137i	$0.288954\pi$
$701$			41.7341i			1.57627i	−0.615500	+	0.788137i	$0.711046\pi$
$702$		0			0
$703$	29.8275			1.12497
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−15.1351	+	11.6824i	−0.569215	+	0.439362i
$708$		0			0
$709$	44.9706			1.68891			0.844454	−	0.535629i	$-0.179925\pi$
$709$	44.9706			1.68891			0.844454	+	0.535629i	$0.179925\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			42.3288i			1.58522i
$714$		0			0
$715$	9.77682	−	2.99904i	0.365632	−	0.112158i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	3.49672			0.130405			0.0652027	−	0.997872i	$-0.479231\pi$
$719$	3.49672			0.130405			0.0652027	+	0.997872i	$0.479231\pi$
$720$		0			0
$721$	2.98242	−	2.30206i	0.111071	−	0.0857331i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	9.76507	+	14.4193i	0.362665	+	0.535518i
$726$		0			0
$727$	17.1577			0.636345			0.318172	−	0.948033i	$-0.396931\pi$
$727$	17.1577			0.636345			0.318172	+	0.948033i	$0.396931\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−45.7745			−1.69303
$732$		0			0
$733$	−3.18589			−0.117674			−0.0588368	−	0.998268i	$-0.518739\pi$
$733$	−3.18589			−0.117674			−0.0588368	+	0.998268i	$0.518739\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	2.35994			0.0869295
$738$		0			0
$739$	5.26746			0.193767			0.0968833	−	0.995296i	$-0.469113\pi$
$739$	5.26746			0.193767			0.0968833	+	0.995296i	$0.469113\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	18.3628			0.673667			0.336834	−	0.941564i	$-0.390644\pi$
$743$	18.3628			0.673667			0.336834	+	0.941564i	$0.390644\pi$
$744$		0			0
$745$	−17.5866	+	5.39468i	−0.644322	+	0.197646i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	17.6010	−	13.5857i	0.643125	−	0.496412i
$750$		0			0
$751$	34.5982			1.26250			0.631252	−	0.775578i	$-0.282541\pi$
$751$	34.5982			1.26250			0.631252	+	0.775578i	$0.282541\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	7.63133	+	24.8780i	0.277732	+	0.905404i
$756$		0			0
$757$		−	5.14201i		−	0.186889i	−0.995624	−	0.0934447i	$-0.970212\pi$
$757$		−	5.14201i		−	0.186889i	0.995624	−	0.0934447i	$-0.0297878\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	12.5024			0.453213			0.226606	−	0.973986i	$-0.427237\pi$
$761$	12.5024			0.453213			0.226606	+	0.973986i	$0.427237\pi$
$762$		0			0
$763$	12.1280	−	9.36129i	0.439063	−	0.338902i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	8.33121			0.300822
$768$		0			0
$769$			3.78532i			0.136502i	0.997668	+	0.0682510i	$0.0217419\pi$
$769$			3.78532i			0.136502i	−0.997668	+	0.0682510i	$0.978258\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	53.4480i		−	1.92239i	−0.275871	−	0.961195i	$-0.588966\pi$
$773$		−	53.4480i		−	1.92239i	0.275871	−	0.961195i	$-0.411034\pi$
$774$		0			0
$775$	−21.6635	−	31.9888i	−0.778177	−	1.14907i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	10.8010i		−	0.386986i
$780$		0			0
$781$	35.8169			1.28163
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−4.45185	−	14.5130i	−0.158893	−	0.517990i
$786$		0			0
$787$	22.3977			0.798392			0.399196	−	0.916866i	$-0.369289\pi$
$787$	22.3977			0.798392			0.399196	+	0.916866i	$0.369289\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	34.2003	−	26.3984i	1.21602	−	0.938618i
$792$		0			0
$793$			5.22313i			0.185479i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	8.25414i		−	0.292376i	−0.989257	−	0.146188i	$-0.953299\pi$
$797$		−	8.25414i		−	0.292376i	0.989257	−	0.146188i	$-0.0467005\pi$
$798$		0			0
$799$	0.629334			0.0222642
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	73.1973i		−	2.58308i
$804$		0			0
$805$	26.4560	+	18.7201i	0.932451	+	0.659797i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			14.0014i			0.492263i	0.969236	+	0.246132i	$0.0791595\pi$
$809$			14.0014i			0.492263i	−0.969236	+	0.246132i	$0.920840\pi$
$810$		0			0
$811$			23.8174i			0.836341i	0.908369	+	0.418170i	$0.137328\pi$
$811$			23.8174i			0.836341i	−0.908369	+	0.418170i	$0.862672\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−30.1540	+	9.24972i	−1.05625	+	0.324004i
$816$		0			0
$817$	85.1525			2.97911
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	34.0908i		−	1.18978i	−0.803808	−	0.594889i	$-0.797196\pi$
$821$		−	34.0908i		−	1.18978i	0.803808	−	0.594889i	$-0.202804\pi$
$822$		0			0
$823$			45.3733i			1.58161i	0.612065	+	0.790807i	$0.290339\pi$
$823$			45.3733i			1.58161i	−0.612065	+	0.790807i	$0.709661\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−9.14696			−0.318071			−0.159035	−	0.987273i	$-0.550838\pi$
$827$	−9.14696			−0.318071			−0.159035	+	0.987273i	$0.550838\pi$
$828$		0			0
$829$			22.3001i			0.774513i	0.921972	+	0.387257i	$0.126577\pi$
$829$			22.3001i			0.774513i	−0.921972	+	0.387257i	$0.873423\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	25.7987	+	6.75506i	0.893874	+	0.234049i
$834$		0			0
$835$	−40.1226	+	12.3076i	−1.38850	+	0.425922i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	35.7582			1.23451			0.617254	−	0.786764i	$-0.288245\pi$
$839$	35.7582			1.23451			0.617254	+	0.786764i	$0.288245\pi$
$840$		0			0
$841$	16.8691			0.581694
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−8.16338	−	26.6125i	−0.280829	−	0.915498i
$846$		0			0
$847$	−56.4403	+	43.5648i	−1.93931	+	1.49691i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			23.0557i			0.790340i
$852$		0			0
$853$	−44.0779			−1.50920			−0.754600	−	0.656186i	$-0.772169\pi$
$853$	−44.0779			−1.50920			−0.754600	+	0.656186i	$0.772169\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	18.1540i		−	0.620128i	−0.950716	−	0.310064i	$-0.899650\pi$
$857$		−	18.1540i		−	0.620128i	0.950716	−	0.310064i	$-0.100350\pi$
$858$		0			0
$859$		−	47.6928i		−	1.62726i	−0.581386	−	0.813628i	$-0.697489\pi$
$859$		−	47.6928i		−	1.62726i	0.581386	−	0.813628i	$-0.302511\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	35.2530			1.20003			0.600013	−	0.799990i	$-0.295162\pi$
$863$	35.2530			1.20003			0.600013	+	0.799990i	$0.295162\pi$
$864$		0			0
$865$	−30.1625	+	9.25233i	−1.02555	+	0.314589i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			64.8326i			2.19930i
$870$		0			0
$871$			0.284414i			0.00963701i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−29.5742	−	0.607212i	−0.999789	−	0.0205275i
$876$		0			0
$877$			39.4374i			1.33171i	0.746083	+	0.665853i	$0.231932\pi$
$877$			39.4374i			1.33171i	−0.746083	+	0.665853i	$0.768068\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−35.2494			−1.18758			−0.593791	−	0.804619i	$-0.702370\pi$
$881$	−35.2494			−1.18758			−0.593791	+	0.804619i	$0.702370\pi$
$882$		0			0
$883$			19.9932i			0.672825i	0.941715	+	0.336413i	$0.109214\pi$
$883$			19.9932i			0.672825i	−0.941715	+	0.336413i	$0.890786\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			33.0287i			1.10899i	0.832186	+	0.554497i	$0.187089\pi$
$887$			33.0287i			1.10899i	−0.832186	+	0.554497i	$0.812911\pi$
$888$		0			0
$889$	12.5900	+	16.3110i	0.422256	+	0.547052i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−1.17072			−0.0391768
$894$		0			0
$895$	−20.9462	+	6.42526i	−0.700156	+	0.214773i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	26.9120			0.897567
$900$		0			0
$901$		−	5.81458i		−	0.193712i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−16.0224	+	4.91486i	−0.532602	+	0.163376i
$906$		0			0
$907$		−	27.9418i		−	0.927793i	−0.885890	−	0.463896i	$-0.846451\pi$
$907$		−	27.9418i		−	0.927793i	0.885890	−	0.463896i	$-0.153549\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	7.40159i		−	0.245226i	−0.992455	−	0.122613i	$-0.960873\pi$
$911$		−	7.40159i		−	0.245226i	0.992455	−	0.122613i	$-0.0391274\pi$
$912$		0			0
$913$	−65.1730			−2.15691
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−22.4986	+	17.3661i	−0.742971	+	0.573480i
$918$		0			0
$919$	−24.0043			−0.791828			−0.395914	−	0.918288i	$-0.629572\pi$
$919$	−24.0043			−0.791828			−0.395914	+	0.918288i	$0.629572\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			4.31656i			0.142081i
$924$		0			0
$925$	−11.7997	−	17.4237i	−0.387973	−	0.572888i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−27.1637			−0.891211			−0.445606	−	0.895229i	$-0.647012\pi$
$929$	−27.1637			−0.891211			−0.445606	+	0.895229i	$0.647012\pi$
$930$		0			0
$931$	−47.9924	−	12.5662i	−1.57289	−	0.411840i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−15.3899	−	50.1709i	−0.503304	−	1.64077i
$936$		0			0
$937$	−24.7467			−0.808438			−0.404219	−	0.914662i	$-0.632457\pi$
$937$	−24.7467			−0.808438			−0.404219	+	0.914662i	$0.632457\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−36.8743			−1.20207			−0.601034	−	0.799223i	$-0.705244\pi$
$941$	−36.8743			−1.20207			−0.601034	+	0.799223i	$0.705244\pi$
$942$		0			0
$943$	8.34885			0.271876
$944$		0			0
$945$		0			0
$946$		0			0
$947$	14.8450			0.482398			0.241199	−	0.970476i	$-0.422459\pi$
$947$	14.8450			0.482398			0.241199	+	0.970476i	$0.422459\pi$
$948$		0			0
$949$	8.82156			0.286360
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−37.7817			−1.22387			−0.611934	−	0.790909i	$-0.709608\pi$
$953$	−37.7817			−1.22387			−0.611934	+	0.790909i	$0.709608\pi$
$954$		0			0
$955$	−22.0000	+	6.74849i	−0.711902	+	0.218376i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	36.7450	−	28.3625i	1.18656	−	0.915873i
$960$		0			0
$961$	−28.7036			−0.925924
$962$		0			0
$963$		0			0
$964$		0			0
$965$	19.8113	−	6.07711i	0.637748	−	0.195629i
$966$		0			0
$967$		−	42.2059i		−	1.35725i	−0.734484	−	0.678626i	$-0.762576\pi$
$967$		−	42.2059i		−	1.35725i	0.734484	−	0.678626i	$-0.237424\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−39.6791			−1.27336			−0.636682	−	0.771127i	$-0.719693\pi$
$971$	−39.6791			−1.27336			−0.636682	+	0.771127i	$0.719693\pi$
$972$		0			0
$973$	8.10616	+	10.5019i	0.259872	+	0.336676i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	15.7020			0.502350			0.251175	−	0.967942i	$-0.419183\pi$
$977$	15.7020			0.502350			0.251175	+	0.967942i	$0.419183\pi$
$978$		0			0
$979$		−	4.80351i		−	0.153521i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			41.9603i			1.33833i	0.743116	+	0.669163i	$0.233347\pi$
$983$			41.9603i			1.33833i	−0.743116	+	0.669163i	$0.766653\pi$
$984$		0			0
$985$	−8.86162	−	28.8888i	−0.282355	−	0.920473i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			65.8202i			2.09296i
$990$		0			0
$991$	19.2455			0.611352			0.305676	−	0.952136i	$-0.401118\pi$
$991$	19.2455			0.611352			0.305676	+	0.952136i	$0.401118\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	46.9456	−	14.4005i	1.48827	−	0.456528i
$996$		0			0
$997$	3.28186			0.103938			0.0519688	−	0.998649i	$-0.483450\pi$
$997$	3.28186			0.103938			0.0519688	+	0.998649i	$0.483450\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2520.2.k.b.1889.14	yes	24
3.2	odd	2		2520.2.k.a.1889.11	✓	24
4.3	odd	2		5040.2.k.h.1889.14		24
5.4	even	2		2520.2.k.a.1889.13	yes	24
7.6	odd	2	inner	2520.2.k.b.1889.11	yes	24
12.11	even	2		5040.2.k.i.1889.11		24
15.14	odd	2	inner	2520.2.k.b.1889.12	yes	24
20.19	odd	2		5040.2.k.i.1889.13		24
21.20	even	2		2520.2.k.a.1889.14	yes	24
28.27	even	2		5040.2.k.h.1889.11		24
35.34	odd	2		2520.2.k.a.1889.12	yes	24
60.59	even	2		5040.2.k.h.1889.12		24
84.83	odd	2		5040.2.k.i.1889.14		24
105.104	even	2	inner	2520.2.k.b.1889.13	yes	24
140.139	even	2		5040.2.k.i.1889.12		24
420.419	odd	2		5040.2.k.h.1889.13		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2520.2.k.a.1889.11	✓	24	3.2	odd	2
2520.2.k.a.1889.12	yes	24	35.34	odd	2
2520.2.k.a.1889.13	yes	24	5.4	even	2
2520.2.k.a.1889.14	yes	24	21.20	even	2
2520.2.k.b.1889.11	yes	24	7.6	odd	2	inner
2520.2.k.b.1889.12	yes	24	15.14	odd	2	inner
2520.2.k.b.1889.13	yes	24	105.104	even	2	inner
2520.2.k.b.1889.14	yes	24	1.1	even	1	trivial
5040.2.k.h.1889.11		24	28.27	even	2
5040.2.k.h.1889.12		24	60.59	even	2
5040.2.k.h.1889.13		24	420.419	odd	2
5040.2.k.h.1889.14		24	4.3	odd	2
5040.2.k.i.1889.11		24	12.11	even	2
5040.2.k.i.1889.12		24	140.139	even	2
5040.2.k.i.1889.13		24	20.19	odd	2
5040.2.k.i.1889.14		24	84.83	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:	\( N \)	\(=\)	\( 2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2520.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.655755 + 2.13775i) q^{5} +(2.09441 - 1.61662i) q^{7} +O(q^{10})\)	\(q+(0.655755 + 2.13775i) q^{5} +(2.09441 - 1.61662i) q^{7} +6.16020i q^{11} -0.742412 q^{13} +3.80978i q^{17} -7.08718i q^{19} +5.47817 q^{23} +(-4.13997 + 2.80368i) q^{25} -3.48294i q^{29} +7.72681i q^{31} +(4.82935 + 3.41722i) q^{35} +4.20866i q^{37} +1.52402 q^{41} +12.0150i q^{43} -0.165189i q^{47} +(1.77309 - 6.77172i) q^{49} -1.52623 q^{53} +(-13.1690 + 4.03959i) q^{55} -11.2218 q^{59} -7.03535i q^{61} +(-0.486841 - 1.58709i) q^{65} -0.383095i q^{67} -5.81423i q^{71} -11.8823 q^{73} +(9.95870 + 12.9020i) q^{77} +10.5244 q^{79} +10.5797i q^{83} +(-8.14436 + 2.49828i) q^{85} -0.779764 q^{89} +(-1.55491 + 1.20020i) q^{91} +(15.1506 - 4.64745i) q^{95} +7.92966 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q + 8 q^{23} - 16 q^{25} - 4 q^{35} - 12 q^{49} - 24 q^{53} - 8 q^{65} - 4 q^{77} + 40 q^{79} + 24 q^{85} - 36 q^{91} + 48 q^{95}+O(q^{100}) \) 24 * q + 8 * q^23 - 16 * q^25 - 4 * q^35 - 12 * q^49 - 24 * q^53 - 8 * q^65 - 4 * q^77 + 40 * q^79 + 24 * q^85 - 36 * q^91 + 48 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more