LMFDB - Embedded newform 4608.2.c.p.4607.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4608,2,Mod(4607,4608)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4608, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4608.4607");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4608 = 2^{9} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4608.c (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:	$36.7950652514$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 1 $ x^4 + 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4607.2
Root			$-0.517638i$ of defining polynomial
Character	$\chi$	$=$	4608.4607
Dual form			4608.2.c.p.4607.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.03528i q^{5} +2.44949i q^{7} +O(q^{10})$	$q-1.03528i q^{5} +2.44949i q^{7} -1.46410 q^{11} +2.00000 q^{13} -3.48477i q^{17} +4.89898i q^{19} +0.535898 q^{23} +3.92820 q^{25} -3.86370i q^{29} +3.20736i q^{31} +2.53590 q^{35} -7.46410 q^{37} +0.656339i q^{41} -2.82843i q^{43} +7.46410 q^{47} +1.00000 q^{49} +11.5911i q^{53} +1.51575i q^{55} -6.92820 q^{59} +4.53590 q^{61} -2.07055i q^{65} -3.58630i q^{67} +10.3923 q^{71} +2.00000 q^{73} -3.58630i q^{77} +13.0053i q^{79} -13.4641 q^{83} -3.60770 q^{85} -1.41421i q^{89} +4.89898i q^{91} +5.07180 q^{95} -6.92820 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 8 q^{11} + 8 q^{13} + 16 q^{23} - 12 q^{25} + 24 q^{35} - 16 q^{37} + 16 q^{47} + 4 q^{49} + 32 q^{61} + 8 q^{73} - 40 q^{83} - 56 q^{85} + 48 q^{95}+O(q^{100}) $ 4 * q + 8 * q^11 + 8 * q^13 + 16 * q^23 - 12 * q^25 + 24 * q^35 - 16 * q^37 + 16 * q^47 + 4 * q^49 + 32 * q^61 + 8 * q^73 - 40 * q^83 - 56 * q^85 + 48 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4608\mathbb{Z}\right)^\times$.

$n$	$2053$	$3583$	$4097$
$\chi(n)$	$1$	$-1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 1.03528i	− 0.462990i	−0.972836	−	0.231495i	$-0.925638\pi$
$5$	− 1.03528i	− 0.462990i	0.972836	−	0.231495i	$-0.0743616\pi$
$6$	0	0
$7$	2.44949i	0.925820i	0.886405	+	0.462910i	$0.153195\pi$
$7$	2.44949i	0.925820i	−0.886405	+	0.462910i	$0.846805\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.46410	−0.441443	−0.220722	−	0.975337i	$-0.570841\pi$
$11$	−1.46410	−0.441443	−0.220722	+	0.975337i	$0.570841\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 3.48477i	− 0.845180i	−0.906321	−	0.422590i	$-0.861121\pi$
$17$	− 3.48477i	− 0.845180i	0.906321	−	0.422590i	$-0.138879\pi$
$18$	0	0
$19$	4.89898i	1.12390i	0.827170	+	0.561951i	$0.189949\pi$
$19$	4.89898i	1.12390i	−0.827170	+	0.561951i	$0.810051\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0.535898	0.111743	0.0558713	−	0.998438i	$-0.482206\pi$
$23$	0.535898	0.111743	0.0558713	+	0.998438i	$0.482206\pi$
$24$	0	0
$25$	3.92820	0.785641
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 3.86370i	− 0.717472i	−0.933439	−	0.358736i	$-0.883208\pi$
$29$	− 3.86370i	− 0.717472i	0.933439	−	0.358736i	$-0.116792\pi$
$30$	0	0
$31$	3.20736i	0.576060i	0.957621	+	0.288030i	$0.0930002\pi$
$31$	3.20736i	0.576060i	−0.957621	+	0.288030i	$0.907000\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.53590	0.428645
$36$	0	0
$37$	−7.46410	−1.22709	−0.613545	−	0.789659i	$-0.710257\pi$
$37$	−7.46410	−1.22709	−0.613545	+	0.789659i	$0.710257\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0.656339i	0.102503i	0.998686	+	0.0512514i	$0.0163210\pi$
$41$	0.656339i	0.102503i	−0.998686	+	0.0512514i	$0.983679\pi$
$42$	0	0
$43$	− 2.82843i	− 0.431331i	−0.976467	−	0.215666i	$-0.930808\pi$
$43$	− 2.82843i	− 0.431331i	0.976467	−	0.215666i	$-0.0691921\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	7.46410	1.08875	0.544376	−	0.838842i	$-0.316767\pi$
$47$	7.46410	1.08875	0.544376	+	0.838842i	$0.316767\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	11.5911i	1.59216i	0.605190	+	0.796081i	$0.293097\pi$
$53$	11.5911i	1.59216i	−0.605190	+	0.796081i	$0.706903\pi$
$54$	0	0
$55$	1.51575i	0.204384i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−6.92820	−0.901975	−0.450988	−	0.892530i	$-0.648928\pi$
$59$	−6.92820	−0.901975	−0.450988	+	0.892530i	$0.648928\pi$
$60$	0	0
$61$	4.53590	0.580762	0.290381	−	0.956911i	$-0.406218\pi$
$61$	4.53590	0.580762	0.290381	+	0.956911i	$0.406218\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 2.07055i	− 0.256820i
$66$	0	0
$67$	− 3.58630i	− 0.438137i	−0.975710	−	0.219068i	$-0.929698\pi$
$67$	− 3.58630i	− 0.438137i	0.975710	−	0.219068i	$-0.0703017\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	10.3923	1.23334	0.616670	−	0.787222i	$-0.288481\pi$
$71$	10.3923	1.23334	0.616670	+	0.787222i	$0.288481\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 3.58630i	− 0.408697i
$78$	0	0
$79$	13.0053i	1.46321i	0.681727	+	0.731607i	$0.261229\pi$
$79$	13.0053i	1.46321i	−0.681727	+	0.731607i	$0.738771\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−13.4641	−1.47788	−0.738939	−	0.673773i	$-0.764673\pi$
$83$	−13.4641	−1.47788	−0.738939	+	0.673773i	$0.764673\pi$
$84$	0	0
$85$	−3.60770	−0.391309
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 1.41421i	− 0.149906i	−0.997187	−	0.0749532i	$-0.976119\pi$
$89$	− 1.41421i	− 0.149906i	0.997187	−	0.0749532i	$-0.0238807\pi$
$90$	0	0
$91$	4.89898i	0.513553i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	5.07180	0.520355
$96$	0	0
$97$	−6.92820	−0.703452	−0.351726	−	0.936103i	$-0.614405\pi$
$97$	−6.92820	−0.703452	−0.351726	+	0.936103i	$0.614405\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1.79315i	0.178425i	0.996013	+	0.0892126i	$0.0284351\pi$
$101$	1.79315i	0.178425i	−0.996013	+	0.0892126i	$0.971565\pi$
$102$	0	0
$103$	17.9043i	1.76416i	0.471096	+	0.882082i	$0.343858\pi$
$103$	17.9043i	1.76416i	−0.471096	+	0.882082i	$0.656142\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	10.9282	1.05647	0.528235	−	0.849098i	$-0.322854\pi$
$107$	10.9282	1.05647	0.528235	+	0.849098i	$0.322854\pi$
$108$	0	0
$109$	16.9282	1.62143	0.810714	−	0.585443i	$-0.199079\pi$
$109$	16.9282	1.62143	0.810714	+	0.585443i	$0.199079\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	11.2122i	1.05475i	0.849632	+	0.527376i	$0.176824\pi$
$113$	11.2122i	1.05475i	−0.849632	+	0.527376i	$0.823176\pi$
$114$	0	0
$115$	− 0.554803i	− 0.0517356i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	8.53590	0.782485
$120$	0	0
$121$	−8.85641	−0.805128
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 9.24316i	− 0.826733i
$126$	0	0
$127$	8.86422i	0.786572i	0.919416	+	0.393286i	$0.128662\pi$
$127$	8.86422i	0.786572i	−0.919416	+	0.393286i	$0.871338\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	18.9282	1.65376	0.826882	−	0.562375i	$-0.190112\pi$
$131$	18.9282	1.65376	0.826882	+	0.562375i	$0.190112\pi$
$132$	0	0
$133$	−12.0000	−1.04053
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5.00052i	0.427223i	0.976919	+	0.213611i	$0.0685226\pi$
$137$	5.00052i	0.427223i	−0.976919	+	0.213611i	$0.931477\pi$
$138$	0	0
$139$	2.07055i	0.175622i	0.996137	+	0.0878110i	$0.0279871\pi$
$139$	2.07055i	0.175622i	−0.996137	+	0.0878110i	$0.972013\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−2.92820	−0.244869
$144$	0	0
$145$	−4.00000	−0.332182
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2.55103i	0.208988i	0.994526	+	0.104494i	$0.0333223\pi$
$149$	2.55103i	0.208988i	−0.994526	+	0.104494i	$0.966678\pi$
$150$	0	0
$151$	14.5211i	1.18171i	0.806778	+	0.590854i	$0.201209\pi$
$151$	14.5211i	1.18171i	−0.806778	+	0.590854i	$0.798791\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	3.32051	0.266710
$156$	0	0
$157$	18.3923	1.46787	0.733933	−	0.679222i	$-0.237683\pi$
$157$	18.3923	1.46787	0.733933	+	0.679222i	$0.237683\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.31268i	0.103453i
$162$	0	0
$163$	− 18.2832i	− 1.43205i	−0.698073	−	0.716027i	$-0.745959\pi$
$163$	− 18.2832i	− 1.43205i	0.698073	−	0.716027i	$-0.254041\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	5.07180	0.392467	0.196234	−	0.980557i	$-0.437129\pi$
$167$	5.07180	0.392467	0.196234	+	0.980557i	$0.437129\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 8.76268i	− 0.666214i	−0.942889	−	0.333107i	$-0.891903\pi$
$173$	− 8.76268i	− 0.666214i	0.942889	−	0.333107i	$-0.108097\pi$
$174$	0	0
$175$	9.62209i	0.727362i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−20.0000	−1.49487	−0.747435	−	0.664335i	$-0.768715\pi$
$179$	−20.0000	−1.49487	−0.747435	+	0.664335i	$0.768715\pi$
$180$	0	0
$181$	−22.7846	−1.69357	−0.846783	−	0.531938i	$-0.821464\pi$
$181$	−22.7846	−1.69357	−0.846783	+	0.531938i	$0.821464\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	7.72741i	0.568130i
$186$	0	0
$187$	5.10205i	0.373099i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	18.9282	1.36960	0.684798	−	0.728733i	$-0.259890\pi$
$191$	18.9282	1.36960	0.684798	+	0.728733i	$0.259890\pi$
$192$	0	0
$193$	9.85641	0.709480	0.354740	−	0.934965i	$-0.384569\pi$
$193$	9.85641	0.709480	0.354740	+	0.934965i	$0.384569\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	20.8343i	1.48438i	0.670190	+	0.742190i	$0.266213\pi$
$197$	20.8343i	1.48438i	−0.670190	+	0.742190i	$0.733787\pi$
$198$	0	0
$199$	3.96524i	0.281088i	0.990074	+	0.140544i	$0.0448852\pi$
$199$	3.96524i	0.281088i	−0.990074	+	0.140544i	$0.955115\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	9.46410	0.664250
$204$	0	0
$205$	0.679492	0.0474578
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 7.17260i	− 0.496139i
$210$	0	0
$211$	19.0411i	1.31084i	0.755263	+	0.655422i	$0.227509\pi$
$211$	19.0411i	1.31084i	−0.755263	+	0.655422i	$0.772491\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−2.92820	−0.199702
$216$	0	0
$217$	−7.85641	−0.533328
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 6.96953i	− 0.468821i
$222$	0	0
$223$	13.7632i	0.921652i	0.887491	+	0.460826i	$0.152447\pi$
$223$	13.7632i	0.921652i	−0.887491	+	0.460826i	$0.847553\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	21.4641	1.42462	0.712311	−	0.701864i	$-0.247648\pi$
$227$	21.4641	1.42462	0.712311	+	0.701864i	$0.247648\pi$
$228$	0	0
$229$	−7.07180	−0.467317	−0.233659	−	0.972319i	$-0.575070\pi$
$229$	−7.07180	−0.467317	−0.233659	+	0.972319i	$0.575070\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 21.0101i	− 1.37642i	−0.725512	−	0.688210i	$-0.758397\pi$
$233$	− 21.0101i	− 1.37642i	0.725512	−	0.688210i	$-0.241603\pi$
$234$	0	0
$235$	− 7.72741i	− 0.504080i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	24.7846	1.60318	0.801592	−	0.597872i	$-0.203987\pi$
$239$	24.7846	1.60318	0.801592	+	0.597872i	$0.203987\pi$
$240$	0	0
$241$	3.07180	0.197872	0.0989359	−	0.995094i	$-0.468456\pi$
$241$	3.07180	0.197872	0.0989359	+	0.995094i	$0.468456\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 1.03528i	− 0.0661414i
$246$	0	0
$247$	9.79796i	0.623429i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−6.53590	−0.412542	−0.206271	−	0.978495i	$-0.566133\pi$
$251$	−6.53590	−0.412542	−0.206271	+	0.978495i	$0.566133\pi$
$252$	0	0
$253$	−0.784610	−0.0493280
$254$	0	0
$255$	0	0
$256$	0	0
$257$	16.8690i	1.05226i	0.850404	+	0.526130i	$0.176358\pi$
$257$	16.8690i	1.05226i	−0.850404	+	0.526130i	$0.823642\pi$
$258$	0	0
$259$	− 18.2832i	− 1.13607i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	0	0
$265$	12.0000	0.737154
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 20.8343i	− 1.27029i	−0.772394	−	0.635144i	$-0.780941\pi$
$269$	− 20.8343i	− 1.27029i	0.772394	−	0.635144i	$-0.219059\pi$
$270$	0	0
$271$	13.7632i	0.836055i	0.908434	+	0.418027i	$0.137278\pi$
$271$	13.7632i	0.836055i	−0.908434	+	0.418027i	$0.862722\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−5.75129	−0.346816
$276$	0	0
$277$	−11.0718	−0.665240	−0.332620	−	0.943061i	$-0.607933\pi$
$277$	−11.0718	−0.665240	−0.332620	+	0.943061i	$0.607933\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 8.38375i	− 0.500132i	−0.968229	−	0.250066i	$-0.919548\pi$
$281$	− 8.38375i	− 0.500132i	0.968229	−	0.250066i	$-0.0804524\pi$
$282$	0	0
$283$	4.14110i	0.246163i	0.992397	+	0.123082i	$0.0392777\pi$
$283$	4.14110i	0.246163i	−0.992397	+	0.123082i	$0.960722\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.60770	−0.0948992
$288$	0	0
$289$	4.85641	0.285671
$290$	0	0
$291$	0	0
$292$	0	0
$293$	15.7322i	0.919086i	0.888156	+	0.459543i	$0.151987\pi$
$293$	15.7322i	0.919086i	−0.888156	+	0.459543i	$0.848013\pi$
$294$	0	0
$295$	7.17260i	0.417605i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.07180	0.0619836
$300$	0	0
$301$	6.92820	0.399335
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 4.69591i	− 0.268887i
$306$	0	0
$307$	28.8391i	1.64593i	0.568090	+	0.822966i	$0.307683\pi$
$307$	28.8391i	1.64593i	−0.568090	+	0.822966i	$0.692317\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	27.7128	1.57145	0.785725	−	0.618576i	$-0.212290\pi$
$311$	27.7128	1.57145	0.785725	+	0.618576i	$0.212290\pi$
$312$	0	0
$313$	23.7128	1.34033	0.670164	−	0.742213i	$-0.266224\pi$
$313$	23.7128	1.34033	0.670164	+	0.742213i	$0.266224\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	10.0754i	0.565889i	0.959136	+	0.282944i	$0.0913112\pi$
$317$	10.0754i	0.565889i	−0.959136	+	0.282944i	$0.908689\pi$
$318$	0	0
$319$	5.65685i	0.316723i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	17.0718	0.949900
$324$	0	0
$325$	7.85641	0.435795
$326$	0	0
$327$	0	0
$328$	0	0
$329$	18.2832i	1.00799i
$330$	0	0
$331$	− 32.9802i	− 1.81275i	−0.422469	−	0.906377i	$-0.638837\pi$
$331$	− 32.9802i	− 1.81275i	0.422469	−	0.906377i	$-0.361163\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−3.71281	−0.202853
$336$	0	0
$337$	−4.14359	−0.225716	−0.112858	−	0.993611i	$-0.536001\pi$
$337$	−4.14359	−0.225716	−0.112858	+	0.993611i	$0.536001\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 4.69591i	− 0.254298i
$342$	0	0
$343$	19.5959i	1.05808i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−30.2487	−1.62384	−0.811918	−	0.583772i	$-0.801576\pi$
$347$	−30.2487	−1.62384	−0.811918	+	0.583772i	$0.801576\pi$
$348$	0	0
$349$	27.1769	1.45475	0.727373	−	0.686242i	$-0.240741\pi$
$349$	27.1769	1.45475	0.727373	+	0.686242i	$0.240741\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 23.8386i	− 1.26880i	−0.773006	−	0.634399i	$-0.781248\pi$
$353$	− 23.8386i	− 1.26880i	0.773006	−	0.634399i	$-0.218752\pi$
$354$	0	0
$355$	− 10.7589i	− 0.571023i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−25.3205	−1.33637	−0.668183	−	0.743997i	$-0.732928\pi$
$359$	−25.3205	−1.33637	−0.668183	+	0.743997i	$0.732928\pi$
$360$	0	0
$361$	−5.00000	−0.263158
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 2.07055i	− 0.108378i
$366$	0	0
$367$	− 12.2474i	− 0.639312i	−0.947534	−	0.319656i	$-0.896433\pi$
$367$	− 12.2474i	− 0.639312i	0.947534	−	0.319656i	$-0.103567\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−28.3923	−1.47406
$372$	0	0
$373$	−10.3923	−0.538093	−0.269047	−	0.963127i	$-0.586709\pi$
$373$	−10.3923	−0.538093	−0.269047	+	0.963127i	$0.586709\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 7.72741i	− 0.397982i
$378$	0	0
$379$	− 0.757875i	− 0.0389294i	−0.999811	−	0.0194647i	$-0.993804\pi$
$379$	− 0.757875i	− 0.0389294i	0.999811	−	0.0194647i	$-0.00619620\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−8.78461	−0.448873	−0.224436	−	0.974489i	$-0.572054\pi$
$383$	−8.78461	−0.448873	−0.224436	+	0.974489i	$0.572054\pi$
$384$	0	0
$385$	−3.71281	−0.189222
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 2.55103i	− 0.129342i	−0.997907	−	0.0646711i	$-0.979400\pi$
$389$	− 2.55103i	− 0.129342i	0.997907	−	0.0646711i	$-0.0205998\pi$
$390$	0	0
$391$	− 1.86748i	− 0.0944425i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	13.4641	0.677452
$396$	0	0
$397$	14.3923	0.722329	0.361165	−	0.932502i	$-0.382379\pi$
$397$	14.3923	0.722329	0.361165	+	0.932502i	$0.382379\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 5.00052i	− 0.249714i	−0.992175	−	0.124857i	$-0.960153\pi$
$401$	− 5.00052i	− 0.249714i	0.992175	−	0.124857i	$-0.0398472\pi$
$402$	0	0
$403$	6.41473i	0.319540i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	10.9282	0.541691
$408$	0	0
$409$	−1.07180	−0.0529969	−0.0264985	−	0.999649i	$-0.508436\pi$
$409$	−1.07180	−0.0529969	−0.0264985	+	0.999649i	$0.508436\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 16.9706i	− 0.835067i
$414$	0	0
$415$	13.9391i	0.684242i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	26.5359	1.29636	0.648182	−	0.761486i	$-0.275530\pi$
$419$	26.5359	1.29636	0.648182	+	0.761486i	$0.275530\pi$
$420$	0	0
$421$	−3.85641	−0.187950	−0.0939749	−	0.995575i	$-0.529957\pi$
$421$	−3.85641	−0.187950	−0.0939749	+	0.995575i	$0.529957\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 13.6889i	− 0.664008i
$426$	0	0
$427$	11.1106i	0.537681i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−13.3205	−0.641626	−0.320813	−	0.947143i	$-0.603956\pi$
$431$	−13.3205	−0.641626	−0.320813	+	0.947143i	$0.603956\pi$
$432$	0	0
$433$	23.8564	1.14647	0.573233	−	0.819393i	$-0.305689\pi$
$433$	23.8564	1.14647	0.573233	+	0.819393i	$0.305689\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.62536i	0.125588i
$438$	0	0
$439$	− 3.96524i	− 0.189251i	−0.995513	−	0.0946253i	$-0.969835\pi$
$439$	− 3.96524i	− 0.189251i	0.995513	−	0.0946253i	$-0.0301653\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−5.46410	−0.259607	−0.129804	−	0.991540i	$-0.541435\pi$
$443$	−5.46410	−0.259607	−0.129804	+	0.991540i	$0.541435\pi$
$444$	0	0
$445$	−1.46410	−0.0694051
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 0.656339i	− 0.0309745i	−0.999880	−	0.0154873i	$-0.995070\pi$
$449$	− 0.656339i	− 0.0309745i	0.999880	−	0.0154873i	$-0.00492995\pi$
$450$	0	0
$451$	− 0.960947i	− 0.0452492i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	5.07180	0.237769
$456$	0	0
$457$	14.9282	0.698312	0.349156	−	0.937065i	$-0.386468\pi$
$457$	14.9282	0.698312	0.349156	+	0.937065i	$0.386468\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	29.8744i	1.39139i	0.718339	+	0.695694i	$0.244903\pi$
$461$	29.8744i	1.39139i	−0.718339	+	0.695694i	$0.755097\pi$
$462$	0	0
$463$	2.44949i	0.113837i	0.998379	+	0.0569187i	$0.0181276\pi$
$463$	2.44949i	0.113837i	−0.998379	+	0.0569187i	$0.981872\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−30.2487	−1.39974	−0.699872	−	0.714269i	$-0.746760\pi$
$467$	−30.2487	−1.39974	−0.699872	+	0.714269i	$0.746760\pi$
$468$	0	0
$469$	8.78461	0.405636
$470$	0	0
$471$	0	0
$472$	0	0
$473$	4.14110i	0.190408i
$474$	0	0
$475$	19.2442i	0.882984i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	10.3923	0.474837	0.237418	−	0.971408i	$-0.423699\pi$
$479$	10.3923	0.474837	0.237418	+	0.971408i	$0.423699\pi$
$480$	0	0
$481$	−14.9282	−0.680667
$482$	0	0
$483$	0	0
$484$	0	0
$485$	7.17260i	0.325691i
$486$	0	0
$487$	− 8.10634i	− 0.367334i	−0.982989	−	0.183667i	$-0.941203\pi$
$487$	− 8.10634i	− 0.367334i	0.982989	−	0.183667i	$-0.0587967\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−4.78461	−0.215926	−0.107963	−	0.994155i	$-0.534433\pi$
$491$	−4.78461	−0.215926	−0.107963	+	0.994155i	$0.534433\pi$
$492$	0	0
$493$	−13.4641	−0.606393
$494$	0	0
$495$	0	0
$496$	0	0
$497$	25.4558i	1.14185i
$498$	0	0
$499$	21.1117i	0.945088i	0.881307	+	0.472544i	$0.156664\pi$
$499$	21.1117i	0.945088i	−0.881307	+	0.472544i	$0.843336\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−27.4641	−1.22456	−0.612282	−	0.790640i	$-0.709748\pi$
$503$	−27.4641	−1.22456	−0.612282	+	0.790640i	$0.709748\pi$
$504$	0	0
$505$	1.85641	0.0826090
$506$	0	0
$507$	0	0
$508$	0	0
$509$	11.5911i	0.513767i	0.966442	+	0.256883i	$0.0826957\pi$
$509$	11.5911i	0.513767i	−0.966442	+	0.256883i	$0.917304\pi$
$510$	0	0
$511$	4.89898i	0.216718i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	18.5359	0.816789
$516$	0	0
$517$	−10.9282	−0.480622
$518$	0	0
$519$	0	0
$520$	0	0
$521$	38.7386i	1.69717i	0.529061	+	0.848584i	$0.322544\pi$
$521$	38.7386i	1.69717i	−0.529061	+	0.848584i	$0.677456\pi$
$522$	0	0
$523$	22.9791i	1.00481i	0.864633	+	0.502404i	$0.167551\pi$
$523$	22.9791i	1.00481i	−0.864633	+	0.502404i	$0.832449\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	11.1769	0.486874
$528$	0	0
$529$	−22.7128	−0.987514
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1.31268i	0.0568584i
$534$	0	0
$535$	− 11.3137i	− 0.489134i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1.46410	−0.0630633
$540$	0	0
$541$	−41.7128	−1.79337	−0.896687	−	0.442665i	$-0.854033\pi$
$541$	−41.7128	−1.79337	−0.896687	+	0.442665i	$0.854033\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 17.5254i	− 0.750704i
$546$	0	0
$547$	− 6.96953i	− 0.297996i	−0.988838	−	0.148998i	$-0.952395\pi$
$547$	− 6.96953i	− 0.297996i	0.988838	−	0.148998i	$-0.0476047\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	18.9282	0.806369
$552$	0	0
$553$	−31.8564	−1.35467
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 25.7332i	− 1.09035i	−0.838321	−	0.545176i	$-0.816463\pi$
$557$	− 25.7332i	− 1.09035i	0.838321	−	0.545176i	$-0.183537\pi$
$558$	0	0
$559$	− 5.65685i	− 0.239259i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−31.3205	−1.32000	−0.660001	−	0.751265i	$-0.729444\pi$
$563$	−31.3205	−1.32000	−0.660001	+	0.751265i	$0.729444\pi$
$564$	0	0
$565$	11.6077	0.488339
$566$	0	0
$567$	0	0
$568$	0	0
$569$	10.6574i	0.446780i	0.974729	+	0.223390i	$0.0717124\pi$
$569$	10.6574i	0.446780i	−0.974729	+	0.223390i	$0.928288\pi$
$570$	0	0
$571$	2.62536i	0.109868i	0.998490	+	0.0549338i	$0.0174948\pi$
$571$	2.62536i	0.109868i	−0.998490	+	0.0549338i	$0.982505\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	2.10512	0.0877895
$576$	0	0
$577$	−5.85641	−0.243805	−0.121903	−	0.992542i	$-0.538900\pi$
$577$	−5.85641	−0.243805	−0.121903	+	0.992542i	$0.538900\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 32.9802i	− 1.36825i
$582$	0	0
$583$	− 16.9706i	− 0.702849i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	20.0000	0.825488	0.412744	−	0.910847i	$-0.364570\pi$
$587$	20.0000	0.825488	0.412744	+	0.910847i	$0.364570\pi$
$588$	0	0
$589$	−15.7128	−0.647435
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 33.8396i	− 1.38963i	−0.719191	−	0.694813i	$-0.755487\pi$
$593$	− 33.8396i	− 1.38963i	0.719191	−	0.694813i	$-0.244513\pi$
$594$	0	0
$595$	− 8.83701i	− 0.362282i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−0.535898	−0.0218962	−0.0109481	−	0.999940i	$-0.503485\pi$
$599$	−0.535898	−0.0218962	−0.0109481	+	0.999940i	$0.503485\pi$
$600$	0	0
$601$	−21.8564	−0.891541	−0.445771	−	0.895147i	$-0.647070\pi$
$601$	−21.8564	−0.891541	−0.445771	+	0.895147i	$0.647070\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	9.16883i	0.372766i
$606$	0	0
$607$	− 29.2180i	− 1.18592i	−0.805231	−	0.592961i	$-0.797959\pi$
$607$	− 29.2180i	− 1.18592i	0.805231	−	0.592961i	$-0.202041\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	14.9282	0.603930
$612$	0	0
$613$	14.3923	0.581300	0.290650	−	0.956829i	$-0.406129\pi$
$613$	14.3923	0.581300	0.290650	+	0.956829i	$0.406129\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 15.5563i	− 0.626275i	−0.949708	−	0.313138i	$-0.898620\pi$
$617$	− 15.5563i	− 0.626275i	0.949708	−	0.313138i	$-0.101380\pi$
$618$	0	0
$619$	− 16.9706i	− 0.682105i	−0.940044	−	0.341052i	$-0.889217\pi$
$619$	− 16.9706i	− 0.682105i	0.940044	−	0.341052i	$-0.110783\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	3.46410	0.138786
$624$	0	0
$625$	10.0718	0.402872
$626$	0	0
$627$	0	0
$628$	0	0
$629$	26.0106i	1.03711i
$630$	0	0
$631$	7.34847i	0.292538i	0.989245	+	0.146269i	$0.0467265\pi$
$631$	7.34847i	0.292538i	−0.989245	+	0.146269i	$0.953274\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	9.17691	0.364175
$636$	0	0
$637$	2.00000	0.0792429
$638$	0	0
$639$	0	0
$640$	0	0
$641$	41.5670i	1.64180i	0.571074	+	0.820899i	$0.306527\pi$
$641$	41.5670i	1.64180i	−0.571074	+	0.820899i	$0.693473\pi$
$642$	0	0
$643$	− 33.7381i	− 1.33050i	−0.746621	−	0.665249i	$-0.768325\pi$
$643$	− 33.7381i	− 1.33050i	0.746621	−	0.665249i	$-0.231675\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−39.4641	−1.55149	−0.775747	−	0.631044i	$-0.782627\pi$
$647$	−39.4641	−1.55149	−0.775747	+	0.631044i	$0.782627\pi$
$648$	0	0
$649$	10.1436	0.398171
$650$	0	0
$651$	0	0
$652$	0	0
$653$	19.5216i	0.763939i	0.924175	+	0.381969i	$0.124754\pi$
$653$	19.5216i	0.763939i	−0.924175	+	0.381969i	$0.875246\pi$
$654$	0	0
$655$	− 19.5959i	− 0.765676i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	0	0
$661$	36.5359	1.42108	0.710541	−	0.703656i	$-0.248450\pi$
$661$	36.5359	1.42108	0.710541	+	0.703656i	$0.248450\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	12.4233i	0.481755i
$666$	0	0
$667$	− 2.07055i	− 0.0801721i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−6.64102	−0.256374
$672$	0	0
$673$	−0.143594	−0.00553512	−0.00276756	−	0.999996i	$-0.500881\pi$
$673$	−0.143594	−0.00553512	−0.00276756	+	0.999996i	$0.500881\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 33.4607i	− 1.28600i	−0.765867	−	0.642999i	$-0.777690\pi$
$677$	− 33.4607i	− 1.28600i	0.765867	−	0.642999i	$-0.222310\pi$
$678$	0	0
$679$	− 16.9706i	− 0.651270i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	8.39230	0.321123	0.160561	−	0.987026i	$-0.448670\pi$
$683$	8.39230	0.321123	0.160561	+	0.987026i	$0.448670\pi$
$684$	0	0
$685$	5.17691	0.197800
$686$	0	0
$687$	0	0
$688$	0	0
$689$	23.1822i	0.883172i
$690$	0	0
$691$	1.31268i	0.0499366i	0.999688	+	0.0249683i	$0.00794848\pi$
$691$	1.31268i	0.0499366i	−0.999688	+	0.0249683i	$0.992052\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2.14359	0.0813111
$696$	0	0
$697$	2.28719	0.0866334
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 25.5302i	− 0.964261i	−0.876099	−	0.482131i	$-0.839863\pi$
$701$	− 25.5302i	− 0.964261i	0.876099	−	0.482131i	$-0.160137\pi$
$702$	0	0
$703$	− 36.5665i	− 1.37913i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−4.39230	−0.165190
$708$	0	0
$709$	0.143594	0.00539277	0.00269638	−	0.999996i	$-0.499142\pi$
$709$	0.143594	0.00539277	0.00269638	+	0.999996i	$0.499142\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1.71882i	0.0643704i
$714$	0	0
$715$	3.03150i	0.113372i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−50.3923	−1.87932	−0.939658	−	0.342115i	$-0.888857\pi$
$719$	−50.3923	−1.87932	−0.939658	+	0.342115i	$0.888857\pi$
$720$	0	0
$721$	−43.8564	−1.63330
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 15.1774i	− 0.563675i
$726$	0	0
$727$	− 24.3190i	− 0.901943i	−0.892538	−	0.450971i	$-0.851078\pi$
$727$	− 24.3190i	− 0.901943i	0.892538	−	0.450971i	$-0.148922\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−9.85641	−0.364552
$732$	0	0
$733$	−4.14359	−0.153047	−0.0765236	−	0.997068i	$-0.524382\pi$
$733$	−4.14359	−0.153047	−0.0765236	+	0.997068i	$0.524382\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5.25071i	0.193412i
$738$	0	0
$739$	− 12.8295i	− 0.471939i	−0.971760	−	0.235970i	$-0.924173\pi$
$739$	− 12.8295i	− 0.471939i	0.971760	−	0.235970i	$-0.0758266\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−2.92820	−0.107425	−0.0537127	−	0.998556i	$-0.517106\pi$
$743$	−2.92820	−0.107425	−0.0537127	+	0.998556i	$0.517106\pi$
$744$	0	0
$745$	2.64102	0.0967593
$746$	0	0
$747$	0	0
$748$	0	0
$749$	26.7685i	0.978100i
$750$	0	0
$751$	5.07484i	0.185184i	0.995704	+	0.0925919i	$0.0295152\pi$
$751$	5.07484i	0.185184i	−0.995704	+	0.0925919i	$0.970485\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	15.0333	0.547119
$756$	0	0
$757$	14.7846	0.537356	0.268678	−	0.963230i	$-0.413413\pi$
$757$	14.7846	0.537356	0.268678	+	0.963230i	$0.413413\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 6.31319i	− 0.228853i	−0.993432	−	0.114427i	$-0.963497\pi$
$761$	− 6.31319i	− 0.228853i	0.993432	−	0.114427i	$-0.0365031\pi$
$762$	0	0
$763$	41.4655i	1.50115i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−13.8564	−0.500326
$768$	0	0
$769$	31.7128	1.14359	0.571797	−	0.820395i	$-0.306247\pi$
$769$	31.7128	1.14359	0.571797	+	0.820395i	$0.306247\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 5.93426i	− 0.213440i	−0.994289	−	0.106720i	$-0.965965\pi$
$773$	− 5.93426i	− 0.213440i	0.994289	−	0.106720i	$-0.0340349\pi$
$774$	0	0
$775$	12.5992i	0.452576i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−3.21539	−0.115203
$780$	0	0
$781$	−15.2154	−0.544449
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 19.0411i	− 0.679607i
$786$	0	0
$787$	20.3538i	0.725534i	0.931880	+	0.362767i	$0.118168\pi$
$787$	20.3538i	0.725534i	−0.931880	+	0.362767i	$0.881832\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−27.4641	−0.976511
$792$	0	0
$793$	9.07180	0.322149
$794$	0	0
$795$	0	0
$796$	0	0
$797$	26.8429i	0.950823i	0.879764	+	0.475411i	$0.157701\pi$
$797$	26.8429i	0.950823i	−0.879764	+	0.475411i	$0.842299\pi$
$798$	0	0
$799$	− 26.0106i	− 0.920191i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−2.92820	−0.103334
$804$	0	0
$805$	1.35898	0.0478979
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 51.3650i	− 1.80590i	−0.429750	−	0.902948i	$-0.641398\pi$
$809$	− 51.3650i	− 1.80590i	0.429750	−	0.902948i	$-0.358602\pi$
$810$	0	0
$811$	− 17.3223i	− 0.608268i	−0.952629	−	0.304134i	$-0.901633\pi$
$811$	− 17.3223i	− 0.608268i	0.952629	−	0.304134i	$-0.0983671\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−18.9282	−0.663026
$816$	0	0
$817$	13.8564	0.484774
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 43.4617i	− 1.51682i	−0.651776	−	0.758412i	$-0.725976\pi$
$821$	− 43.4617i	− 1.51682i	0.651776	−	0.758412i	$-0.274024\pi$
$822$	0	0
$823$	− 31.0855i	− 1.08357i	−0.840516	−	0.541786i	$-0.817748\pi$
$823$	− 31.0855i	− 1.08357i	0.840516	−	0.541786i	$-0.182252\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−16.0000	−0.556375	−0.278187	−	0.960527i	$-0.589734\pi$
$827$	−16.0000	−0.556375	−0.278187	+	0.960527i	$0.589734\pi$
$828$	0	0
$829$	−23.0718	−0.801317	−0.400658	−	0.916228i	$-0.631219\pi$
$829$	−23.0718	−0.801317	−0.400658	+	0.916228i	$0.631219\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 3.48477i	− 0.120740i
$834$	0	0
$835$	− 5.25071i	− 0.181708i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1.32051	0.0455890	0.0227945	−	0.999740i	$-0.492744\pi$
$839$	1.32051	0.0455890	0.0227945	+	0.999740i	$0.492744\pi$
$840$	0	0
$841$	14.0718	0.485234
$842$	0	0
$843$	0	0
$844$	0	0
$845$	9.31749i	0.320531i
$846$	0	0
$847$	− 21.6937i	− 0.745404i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−4.00000	−0.137118
$852$	0	0
$853$	24.2487	0.830260	0.415130	−	0.909762i	$-0.363736\pi$
$853$	24.2487	0.830260	0.415130	+	0.909762i	$0.363736\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 14.7985i	− 0.505506i	−0.967531	−	0.252753i	$-0.918664\pi$
$857$	− 14.7985i	− 0.505506i	0.967531	−	0.252753i	$-0.0813360\pi$
$858$	0	0
$859$	− 43.5360i	− 1.48543i	−0.669608	−	0.742715i	$-0.733538\pi$
$859$	− 43.5360i	− 1.48543i	0.669608	−	0.742715i	$-0.266462\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−37.0718	−1.26194	−0.630969	−	0.775808i	$-0.717343\pi$
$863$	−37.0718	−1.26194	−0.630969	+	0.775808i	$0.717343\pi$
$864$	0	0
$865$	−9.07180	−0.308450
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 19.0411i	− 0.645926i
$870$	0	0
$871$	− 7.17260i	− 0.243034i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	22.6410	0.765406
$876$	0	0
$877$	−28.2487	−0.953891	−0.476946	−	0.878933i	$-0.658256\pi$
$877$	−28.2487	−0.953891	−0.476946	+	0.878933i	$0.658256\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	47.7787i	1.60970i	0.593476	+	0.804852i	$0.297755\pi$
$881$	47.7787i	1.60970i	−0.593476	+	0.804852i	$0.702245\pi$
$882$	0	0
$883$	2.82843i	0.0951842i	0.998867	+	0.0475921i	$0.0151548\pi$
$883$	2.82843i	0.0951842i	−0.998867	+	0.0475921i	$0.984845\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−35.7128	−1.19912	−0.599559	−	0.800330i	$-0.704658\pi$
$887$	−35.7128	−1.19912	−0.599559	+	0.800330i	$0.704658\pi$
$888$	0	0
$889$	−21.7128	−0.728224
$890$	0	0
$891$	0	0
$892$	0	0
$893$	36.5665i	1.22365i
$894$	0	0
$895$	20.7055i	0.692109i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	12.3923	0.413307
$900$	0	0
$901$	40.3923	1.34566
$902$	0	0
$903$	0	0
$904$	0	0
$905$	23.5884i	0.784104i
$906$	0	0
$907$	− 42.4264i	− 1.40875i	−0.709830	−	0.704373i	$-0.751228\pi$
$907$	− 42.4264i	− 1.40875i	0.709830	−	0.704373i	$-0.248772\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	48.7846	1.61631	0.808153	−	0.588972i	$-0.200467\pi$
$911$	48.7846	1.61631	0.808153	+	0.588972i	$0.200467\pi$
$912$	0	0
$913$	19.7128	0.652399
$914$	0	0
$915$	0	0
$916$	0	0
$917$	46.3644i	1.53109i
$918$	0	0
$919$	− 49.5718i	− 1.63522i	−0.575770	−	0.817611i	$-0.695298\pi$
$919$	− 49.5718i	− 1.63522i	0.575770	−	0.817611i	$-0.304702\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	20.7846	0.684134
$924$	0	0
$925$	−29.3205	−0.964052
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 20.4553i	− 0.671118i	−0.942019	−	0.335559i	$-0.891075\pi$
$929$	− 20.4553i	− 0.671118i	0.942019	−	0.335559i	$-0.108925\pi$
$930$	0	0
$931$	4.89898i	0.160558i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	5.28203	0.172741
$936$	0	0
$937$	−26.0000	−0.849383	−0.424691	−	0.905338i	$-0.639617\pi$
$937$	−26.0000	−0.849383	−0.424691	+	0.905338i	$0.639617\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	42.7038i	1.39210i	0.717991	+	0.696052i	$0.245062\pi$
$941$	42.7038i	1.39210i	−0.717991	+	0.696052i	$0.754938\pi$
$942$	0	0
$943$	0.351731i	0.0114539i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	48.4974	1.57595	0.787977	−	0.615704i	$-0.211128\pi$
$947$	48.4974	1.57595	0.787977	+	0.615704i	$0.211128\pi$
$948$	0	0
$949$	4.00000	0.129845
$950$	0	0
$951$	0	0
$952$	0	0
$953$	10.8604i	0.351804i	0.984408	+	0.175902i	$0.0562842\pi$
$953$	10.8604i	0.351804i	−0.984408	+	0.175902i	$0.943716\pi$
$954$	0	0
$955$	− 19.5959i	− 0.634109i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−12.2487	−0.395532
$960$	0	0
$961$	20.7128	0.668155
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 10.2041i	− 0.328482i
$966$	0	0
$967$	− 26.5927i	− 0.855162i	−0.903977	−	0.427581i	$-0.859366\pi$
$967$	− 26.5927i	− 0.855162i	0.903977	−	0.427581i	$-0.140634\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−39.0333	−1.25264	−0.626319	−	0.779567i	$-0.715439\pi$
$971$	−39.0333	−1.25264	−0.626319	+	0.779567i	$0.715439\pi$
$972$	0	0
$973$	−5.07180	−0.162594
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 55.7091i	− 1.78229i	−0.453716	−	0.891147i	$-0.649902\pi$
$977$	− 55.7091i	− 1.78229i	0.453716	−	0.891147i	$-0.350098\pi$
$978$	0	0
$979$	2.07055i	0.0661751i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	21.5692	0.687252
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 1.51575i	− 0.0481980i
$990$	0	0
$991$	− 0.175865i	− 0.00558655i	−0.999996	−	0.00279328i	$-0.999111\pi$
$991$	− 0.175865i	− 0.00558655i	0.999996	−	0.00279328i	$-0.000889128\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	4.10512	0.130141
$996$	0	0
$997$	30.1051	0.953439	0.476719	−	0.879056i	$-0.341826\pi$
$997$	30.1051	0.953439	0.476719	+	0.879056i	$0.341826\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4608.2.c.p.4607.2	yes	4
3.2	odd	2		4608.2.c.j.4607.3	yes	4
4.3	odd	2		4608.2.c.j.4607.2	yes	4
8.3	odd	2		4608.2.c.o.4607.3	yes	4
8.5	even	2		4608.2.c.i.4607.3	yes	4
12.11	even	2	inner	4608.2.c.p.4607.3	yes	4
16.3	odd	4		4608.2.f.p.2303.4		8
16.5	even	4		4608.2.f.m.2303.5		8
16.11	odd	4		4608.2.f.p.2303.6		8
16.13	even	4		4608.2.f.m.2303.3		8
24.5	odd	2		4608.2.c.o.4607.2	yes	4
24.11	even	2		4608.2.c.i.4607.2	✓	4
48.5	odd	4		4608.2.f.p.2303.3		8
48.11	even	4		4608.2.f.m.2303.4		8
48.29	odd	4		4608.2.f.p.2303.5		8
48.35	even	4		4608.2.f.m.2303.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4608.2.c.i.4607.2	✓	4	24.11	even	2
4608.2.c.i.4607.3	yes	4	8.5	even	2
4608.2.c.j.4607.2	yes	4	4.3	odd	2
4608.2.c.j.4607.3	yes	4	3.2	odd	2
4608.2.c.o.4607.2	yes	4	24.5	odd	2
4608.2.c.o.4607.3	yes	4	8.3	odd	2
4608.2.c.p.4607.2	yes	4	1.1	even	1	trivial
4608.2.c.p.4607.3	yes	4	12.11	even	2	inner
4608.2.f.m.2303.3		8	16.13	even	4
4608.2.f.m.2303.4		8	48.11	even	4
4608.2.f.m.2303.5		8	16.5	even	4
4608.2.f.m.2303.6		8	48.35	even	4
4608.2.f.p.2303.3		8	48.5	odd	4
4608.2.f.p.2303.4		8	16.3	odd	4
4608.2.f.p.2303.5		8	48.29	odd	4
4608.2.f.p.2303.6		8	16.11	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4608 = 2^{9} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4608.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.03528i q^{5} +2.44949i q^{7} +O(q^{10})\)	\(q-1.03528i q^{5} +2.44949i q^{7} -1.46410 q^{11} +2.00000 q^{13} -3.48477i q^{17} +4.89898i q^{19} +0.535898 q^{23} +3.92820 q^{25} -3.86370i q^{29} +3.20736i q^{31} +2.53590 q^{35} -7.46410 q^{37} +0.656339i q^{41} -2.82843i q^{43} +7.46410 q^{47} +1.00000 q^{49} +11.5911i q^{53} +1.51575i q^{55} -6.92820 q^{59} +4.53590 q^{61} -2.07055i q^{65} -3.58630i q^{67} +10.3923 q^{71} +2.00000 q^{73} -3.58630i q^{77} +13.0053i q^{79} -13.4641 q^{83} -3.60770 q^{85} -1.41421i q^{89} +4.89898i q^{91} +5.07180 q^{95} -6.92820 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 8 q^{11} + 8 q^{13} + 16 q^{23} - 12 q^{25} + 24 q^{35} - 16 q^{37} + 16 q^{47} + 4 q^{49} + 32 q^{61} + 8 q^{73} - 40 q^{83} - 56 q^{85} + 48 q^{95}+O(q^{100}) \) 4 * q + 8 * q^11 + 8 * q^13 + 16 * q^23 - 12 * q^25 + 24 * q^35 - 16 * q^37 + 16 * q^47 + 4 * q^49 + 32 * q^61 + 8 * q^73 - 40 * q^83 - 56 * q^85 + 48 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more