LMFDB - Embedded newform 4608.2.d.i.2305.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4608,2,Mod(2305,4608)]

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([0, 1, 0]))

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

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

chi := DirichletCharacter("4608.2305");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4608 = 2^{9} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4608.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$36.7950652514$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 9 $ x^4 - 4*x^2 + 9
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			2305.1
Root			$-1.58114 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	4608.2305
Dual form			4608.2.d.i.2305.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{5} -3.16228 q^{7} +O(q^{10})$	$q-1.41421i q^{5} -3.16228 q^{7} +4.47214i q^{11} -4.47214i q^{13} +6.32456 q^{17} +2.82843i q^{19} -4.00000 q^{23} +3.00000 q^{25} -4.24264i q^{29} -3.16228 q^{31} +4.47214i q^{35} +4.47214i q^{37} -6.32456 q^{41} -8.48528i q^{43} +12.0000 q^{47} +3.00000 q^{49} -7.07107i q^{53} +6.32456 q^{55} +13.4164i q^{61} -6.32456 q^{65} -8.00000 q^{71} +4.00000 q^{73} -14.1421i q^{77} +3.16228 q^{79} +4.47214i q^{83} -8.94427i q^{85} +14.1421i q^{91} +4.00000 q^{95} +2.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 16 q^{23} + 12 q^{25} + 48 q^{47} + 12 q^{49} - 32 q^{71} + 16 q^{73} + 16 q^{95} + 8 q^{97}+O(q^{100}) $ 4 * q - 16 * q^23 + 12 * q^25 + 48 * q^47 + 12 * q^49 - 32 * q^71 + 16 * q^73 + 16 * q^95 + 8 * q^97

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
$5$	− 1.41421i	− 0.632456i	−0.948683	−	0.316228i	$-0.897584\pi$
$5$	− 1.41421i	− 0.632456i	0.948683	−	0.316228i	$-0.102416\pi$
$6$	0	0
$7$	−3.16228	−1.19523	−0.597614	−	0.801784i	$-0.703885\pi$
$7$	−3.16228	−1.19523	−0.597614	+	0.801784i	$0.703885\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.47214i	1.34840i	0.738549	+	0.674200i	$0.235511\pi$
$11$	4.47214i	1.34840i	−0.738549	+	0.674200i	$0.764489\pi$
$12$	0	0
$13$	− 4.47214i	− 1.24035i	−0.784465	−	0.620174i	$-0.787062\pi$
$13$	− 4.47214i	− 1.24035i	0.784465	−	0.620174i	$-0.212938\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.32456	1.53393	0.766965	−	0.641689i	$-0.221766\pi$
$17$	6.32456	1.53393	0.766965	+	0.641689i	$0.221766\pi$
$18$	0	0
$19$	2.82843i	0.648886i	0.945905	+	0.324443i	$0.105177\pi$
$19$	2.82843i	0.648886i	−0.945905	+	0.324443i	$0.894823\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	3.00000	0.600000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 4.24264i	− 0.787839i	−0.919145	−	0.393919i	$-0.871119\pi$
$29$	− 4.24264i	− 0.787839i	0.919145	−	0.393919i	$-0.128881\pi$
$30$	0	0
$31$	−3.16228	−0.567962	−0.283981	−	0.958830i	$-0.591655\pi$
$31$	−3.16228	−0.567962	−0.283981	+	0.958830i	$0.591655\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.47214i	0.755929i
$36$	0	0
$37$	4.47214i	0.735215i	0.929981	+	0.367607i	$0.119823\pi$
$37$	4.47214i	0.735215i	−0.929981	+	0.367607i	$0.880177\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.32456	−0.987730	−0.493865	−	0.869539i	$-0.664416\pi$
$41$	−6.32456	−0.987730	−0.493865	+	0.869539i	$0.664416\pi$
$42$	0	0
$43$	− 8.48528i	− 1.29399i	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	− 8.48528i	− 1.29399i	0.762493	−	0.646997i	$-0.223975\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 7.07107i	− 0.971286i	−0.874157	−	0.485643i	$-0.838586\pi$
$53$	− 7.07107i	− 0.971286i	0.874157	−	0.485643i	$-0.161414\pi$
$54$	0	0
$55$	6.32456	0.852803
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	13.4164i	1.71780i	0.512148	+	0.858898i	$0.328850\pi$
$61$	13.4164i	1.71780i	−0.512148	+	0.858898i	$0.671150\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−6.32456	−0.784465
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 14.1421i	− 1.61165i
$78$	0	0
$79$	3.16228	0.355784	0.177892	−	0.984050i	$-0.443072\pi$
$79$	3.16228	0.355784	0.177892	+	0.984050i	$0.443072\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	4.47214i	0.490881i	0.969412	+	0.245440i	$0.0789325\pi$
$83$	4.47214i	0.490881i	−0.969412	+	0.245440i	$0.921067\pi$
$84$	0	0
$85$	− 8.94427i	− 0.970143i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	14.1421i	1.48250i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	4.00000	0.410391
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 15.5563i	− 1.54791i	−0.633238	−	0.773957i	$-0.718274\pi$
$101$	− 15.5563i	− 1.54791i	0.633238	−	0.773957i	$-0.281726\pi$
$102$	0	0
$103$	9.48683	0.934765	0.467383	−	0.884055i	$-0.345197\pi$
$103$	9.48683	0.934765	0.467383	+	0.884055i	$0.345197\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 17.8885i	− 1.72935i	−0.502331	−	0.864675i	$-0.667524\pi$
$107$	− 17.8885i	− 1.72935i	0.502331	−	0.864675i	$-0.332476\pi$
$108$	0	0
$109$	− 4.47214i	− 0.428353i	−0.976795	−	0.214176i	$-0.931293\pi$
$109$	− 4.47214i	− 0.428353i	0.976795	−	0.214176i	$-0.0687068\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−12.6491	−1.18993	−0.594964	−	0.803752i	$-0.702834\pi$
$113$	−12.6491	−1.18993	−0.594964	+	0.803752i	$0.702834\pi$
$114$	0	0
$115$	5.65685i	0.527504i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−20.0000	−1.83340
$120$	0	0
$121$	−9.00000	−0.818182
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 11.3137i	− 1.01193i
$126$	0	0
$127$	15.8114	1.40303	0.701517	−	0.712653i	$-0.252506\pi$
$127$	15.8114	1.40303	0.701517	+	0.712653i	$0.252506\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 17.8885i	− 1.56293i	−0.623949	−	0.781465i	$-0.714473\pi$
$131$	− 17.8885i	− 1.56293i	0.623949	−	0.781465i	$-0.285527\pi$
$132$	0	0
$133$	− 8.94427i	− 0.775567i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	18.9737	1.62103	0.810515	−	0.585718i	$-0.199187\pi$
$137$	18.9737	1.62103	0.810515	+	0.585718i	$0.199187\pi$
$138$	0	0
$139$	− 16.9706i	− 1.43942i	−0.694273	−	0.719712i	$-0.744274\pi$
$139$	− 16.9706i	− 1.43942i	0.694273	−	0.719712i	$-0.255726\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	20.0000	1.67248
$144$	0	0
$145$	−6.00000	−0.498273
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 18.3848i	− 1.50614i	−0.657941	−	0.753070i	$-0.728572\pi$
$149$	− 18.3848i	− 1.50614i	0.657941	−	0.753070i	$-0.271428\pi$
$150$	0	0
$151$	−22.1359	−1.80140	−0.900699	−	0.434444i	$-0.856945\pi$
$151$	−22.1359	−1.80140	−0.900699	+	0.434444i	$0.856945\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4.47214i	0.359211i
$156$	0	0
$157$	− 13.4164i	− 1.07075i	−0.844616	−	0.535373i	$-0.820171\pi$
$157$	− 13.4164i	− 1.07075i	0.844616	−	0.535373i	$-0.179829\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	12.6491	0.996890
$162$	0	0
$163$	− 19.7990i	− 1.55078i	−0.631485	−	0.775388i	$-0.717554\pi$
$163$	− 19.7990i	− 1.55078i	0.631485	−	0.775388i	$-0.282446\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	−7.00000	−0.538462
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 7.07107i	− 0.537603i	−0.963196	−	0.268802i	$-0.913372\pi$
$173$	− 7.07107i	− 0.537603i	0.963196	−	0.268802i	$-0.0866276\pi$
$174$	0	0
$175$	−9.48683	−0.717137
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 17.8885i	− 1.33705i	−0.743689	−	0.668526i	$-0.766925\pi$
$179$	− 17.8885i	− 1.33705i	0.743689	−	0.668526i	$-0.233075\pi$
$180$	0	0
$181$	13.4164i	0.997234i	0.866822	+	0.498617i	$0.166159\pi$
$181$	13.4164i	0.997234i	−0.866822	+	0.498617i	$0.833841\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	6.32456	0.464991
$186$	0	0
$187$	28.2843i	2.06835i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	20.0000	1.44715	0.723575	−	0.690246i	$-0.242498\pi$
$191$	20.0000	1.44715	0.723575	+	0.690246i	$0.242498\pi$
$192$	0	0
$193$	−16.0000	−1.15171	−0.575853	−	0.817554i	$-0.695330\pi$
$193$	−16.0000	−1.15171	−0.575853	+	0.817554i	$0.695330\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 7.07107i	− 0.503793i	−0.967754	−	0.251896i	$-0.918946\pi$
$197$	− 7.07107i	− 0.503793i	0.967754	−	0.251896i	$-0.0810542\pi$
$198$	0	0
$199$	−3.16228	−0.224168	−0.112084	−	0.993699i	$-0.535753\pi$
$199$	−3.16228	−0.224168	−0.112084	+	0.993699i	$0.535753\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	13.4164i	0.941647i
$204$	0	0
$205$	8.94427i	0.624695i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−12.6491	−0.874957
$210$	0	0
$211$	5.65685i	0.389434i	0.980859	+	0.194717i	$0.0623788\pi$
$211$	5.65685i	0.389434i	−0.980859	+	0.194717i	$0.937621\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−12.0000	−0.818393
$216$	0	0
$217$	10.0000	0.678844
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 28.2843i	− 1.90261i
$222$	0	0
$223$	−28.4605	−1.90586	−0.952928	−	0.303197i	$-0.901946\pi$
$223$	−28.4605	−1.90586	−0.952928	+	0.303197i	$0.901946\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	13.4164i	0.890478i	0.895412	+	0.445239i	$0.146881\pi$
$227$	13.4164i	0.890478i	−0.895412	+	0.445239i	$0.853119\pi$
$228$	0	0
$229$	13.4164i	0.886581i	0.896378	+	0.443291i	$0.146189\pi$
$229$	13.4164i	0.886581i	−0.896378	+	0.443291i	$0.853811\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	− 16.9706i	− 1.10704i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	4.00000	0.258738	0.129369	−	0.991596i	$-0.458705\pi$
$239$	4.00000	0.258738	0.129369	+	0.991596i	$0.458705\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 4.24264i	− 0.271052i
$246$	0	0
$247$	12.6491	0.804844
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 13.4164i	− 0.846836i	−0.905934	−	0.423418i	$-0.860830\pi$
$251$	− 13.4164i	− 0.846836i	0.905934	−	0.423418i	$-0.139170\pi$
$252$	0	0
$253$	− 17.8885i	− 1.12464i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	12.6491	0.789030	0.394515	−	0.918890i	$-0.370913\pi$
$257$	12.6491	0.789030	0.394515	+	0.918890i	$0.370913\pi$
$258$	0	0
$259$	− 14.1421i	− 0.878750i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−16.0000	−0.986602	−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000	−0.986602	−0.493301	+	0.869859i	$0.664210\pi$
$264$	0	0
$265$	−10.0000	−0.614295
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 18.3848i	− 1.12094i	−0.828175	−	0.560470i	$-0.810621\pi$
$269$	− 18.3848i	− 1.12094i	0.828175	−	0.560470i	$-0.189379\pi$
$270$	0	0
$271$	−3.16228	−0.192095	−0.0960473	−	0.995377i	$-0.530620\pi$
$271$	−3.16228	−0.192095	−0.0960473	+	0.995377i	$0.530620\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	13.4164i	0.809040i
$276$	0	0
$277$	13.4164i	0.806114i	0.915175	+	0.403057i	$0.132052\pi$
$277$	13.4164i	0.806114i	−0.915175	+	0.403057i	$0.867948\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−12.6491	−0.754583	−0.377291	−	0.926095i	$-0.623144\pi$
$281$	−12.6491	−0.754583	−0.377291	+	0.926095i	$0.623144\pi$
$282$	0	0
$283$	28.2843i	1.68133i	0.541559	+	0.840663i	$0.317834\pi$
$283$	28.2843i	1.68133i	−0.541559	+	0.840663i	$0.682166\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	20.0000	1.18056
$288$	0	0
$289$	23.0000	1.35294
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 29.6985i	− 1.73500i	−0.497434	−	0.867502i	$-0.665724\pi$
$293$	− 29.6985i	− 1.73500i	0.497434	−	0.867502i	$-0.334276\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	17.8885i	1.03452i
$300$	0	0
$301$	26.8328i	1.54662i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	18.9737	1.08643
$306$	0	0
$307$	11.3137i	0.645707i	0.946449	+	0.322854i	$0.104642\pi$
$307$	11.3137i	0.645707i	−0.946449	+	0.322854i	$0.895358\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	0	0
$313$	−16.0000	−0.904373	−0.452187	−	0.891923i	$-0.649356\pi$
$313$	−16.0000	−0.904373	−0.452187	+	0.891923i	$0.649356\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	9.89949i	0.556011i	0.960579	+	0.278006i	$0.0896734\pi$
$317$	9.89949i	0.556011i	−0.960579	+	0.278006i	$0.910327\pi$
$318$	0	0
$319$	18.9737	1.06232
$320$	0	0
$321$	0	0
$322$	0	0
$323$	17.8885i	0.995345i
$324$	0	0
$325$	− 13.4164i	− 0.744208i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−37.9473	−2.09210
$330$	0	0
$331$	− 33.9411i	− 1.86557i	−0.360429	−	0.932786i	$-0.617370\pi$
$331$	− 33.9411i	− 1.86557i	0.360429	−	0.932786i	$-0.382630\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−12.0000	−0.653682	−0.326841	−	0.945079i	$-0.605984\pi$
$337$	−12.0000	−0.653682	−0.326841	+	0.945079i	$0.605984\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 14.1421i	− 0.765840i
$342$	0	0
$343$	12.6491	0.682988
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 4.47214i	− 0.240077i	−0.992769	−	0.120038i	$-0.961698\pi$
$347$	− 4.47214i	− 0.240077i	0.992769	−	0.120038i	$-0.0383018\pi$
$348$	0	0
$349$	− 4.47214i	− 0.239388i	−0.992811	−	0.119694i	$-0.961809\pi$
$349$	− 4.47214i	− 0.239388i	0.992811	−	0.119694i	$-0.0381913\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−25.2982	−1.34649	−0.673244	−	0.739420i	$-0.735100\pi$
$353$	−25.2982	−1.34649	−0.673244	+	0.739420i	$0.735100\pi$
$354$	0	0
$355$	11.3137i	0.600469i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	11.0000	0.578947
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 5.65685i	− 0.296093i
$366$	0	0
$367$	22.1359	1.15549	0.577743	−	0.816218i	$-0.303933\pi$
$367$	22.1359	1.15549	0.577743	+	0.816218i	$0.303933\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	22.3607i	1.16091i
$372$	0	0
$373$	− 22.3607i	− 1.15779i	−0.815401	−	0.578896i	$-0.803484\pi$
$373$	− 22.3607i	− 1.15779i	0.815401	−	0.578896i	$-0.196516\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−18.9737	−0.977194
$378$	0	0
$379$	25.4558i	1.30758i	0.756677	+	0.653789i	$0.226822\pi$
$379$	25.4558i	1.30758i	−0.756677	+	0.653789i	$0.773178\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	4.00000	0.204390	0.102195	−	0.994764i	$-0.467413\pi$
$383$	4.00000	0.204390	0.102195	+	0.994764i	$0.467413\pi$
$384$	0	0
$385$	−20.0000	−1.01929
$386$	0	0
$387$	0	0
$388$	0	0
$389$	24.0416i	1.21896i	0.792802	+	0.609480i	$0.208622\pi$
$389$	24.0416i	1.21896i	−0.792802	+	0.609480i	$0.791378\pi$
$390$	0	0
$391$	−25.2982	−1.27939
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 4.47214i	− 0.225018i
$396$	0	0
$397$	4.47214i	0.224450i	0.993683	+	0.112225i	$0.0357978\pi$
$397$	4.47214i	0.224450i	−0.993683	+	0.112225i	$0.964202\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	18.9737	0.947500	0.473750	−	0.880659i	$-0.342900\pi$
$401$	18.9737	0.947500	0.473750	+	0.880659i	$0.342900\pi$
$402$	0	0
$403$	14.1421i	0.704470i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−20.0000	−0.991363
$408$	0	0
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	6.32456	0.310460
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 4.47214i	− 0.218478i	−0.994016	−	0.109239i	$-0.965159\pi$
$419$	− 4.47214i	− 0.218478i	0.994016	−	0.109239i	$-0.0348414\pi$
$420$	0	0
$421$	− 40.2492i	− 1.96163i	−0.194948	−	0.980814i	$-0.562454\pi$
$421$	− 40.2492i	− 1.96163i	0.194948	−	0.980814i	$-0.437546\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	18.9737	0.920358
$426$	0	0
$427$	− 42.4264i	− 2.05316i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	0	0
$433$	34.0000	1.63394	0.816968	−	0.576683i	$-0.195653\pi$
$433$	34.0000	1.63394	0.816968	+	0.576683i	$0.195653\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 11.3137i	− 0.541208i
$438$	0	0
$439$	22.1359	1.05649	0.528245	−	0.849092i	$-0.322850\pi$
$439$	22.1359	1.05649	0.528245	+	0.849092i	$0.322850\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	31.3050i	1.48734i	0.668545	+	0.743672i	$0.266917\pi$
$443$	31.3050i	1.48734i	−0.668545	+	0.743672i	$0.733083\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−6.32456	−0.298474	−0.149237	−	0.988801i	$-0.547682\pi$
$449$	−6.32456	−0.298474	−0.149237	+	0.988801i	$0.547682\pi$
$450$	0	0
$451$	− 28.2843i	− 1.33185i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	20.0000	0.937614
$456$	0	0
$457$	22.0000	1.02912	0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000	1.02912	0.514558	+	0.857455i	$0.327956\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 1.41421i	− 0.0658665i	−0.999458	−	0.0329332i	$-0.989515\pi$
$461$	− 1.41421i	− 0.0658665i	0.999458	−	0.0329332i	$-0.0104849\pi$
$462$	0	0
$463$	15.8114	0.734818	0.367409	−	0.930060i	$-0.380245\pi$
$463$	15.8114	0.734818	0.367409	+	0.930060i	$0.380245\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 22.3607i	− 1.03473i	−0.855765	−	0.517364i	$-0.826913\pi$
$467$	− 22.3607i	− 1.03473i	0.855765	−	0.517364i	$-0.173087\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	37.9473	1.74482
$474$	0	0
$475$	8.48528i	0.389331i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	20.0000	0.911922
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 2.82843i	− 0.128432i
$486$	0	0
$487$	9.48683	0.429889	0.214945	−	0.976626i	$-0.431043\pi$
$487$	9.48683	0.429889	0.214945	+	0.976626i	$0.431043\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 17.8885i	− 0.807299i	−0.914914	−	0.403649i	$-0.867742\pi$
$491$	− 17.8885i	− 0.807299i	0.914914	−	0.403649i	$-0.132258\pi$
$492$	0	0
$493$	− 26.8328i	− 1.20849i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	25.2982	1.13478
$498$	0	0
$499$	16.9706i	0.759707i	0.925047	+	0.379853i	$0.124026\pi$
$499$	16.9706i	0.759707i	−0.925047	+	0.379853i	$0.875974\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	16.0000	0.713405	0.356702	−	0.934218i	$-0.383901\pi$
$503$	16.0000	0.713405	0.356702	+	0.934218i	$0.383901\pi$
$504$	0	0
$505$	−22.0000	−0.978987
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 9.89949i	− 0.438787i	−0.975636	−	0.219394i	$-0.929592\pi$
$509$	− 9.89949i	− 0.438787i	0.975636	−	0.219394i	$-0.0704079\pi$
$510$	0	0
$511$	−12.6491	−0.559564
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 13.4164i	− 0.591198i
$516$	0	0
$517$	53.6656i	2.36021i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−6.32456	−0.277084	−0.138542	−	0.990357i	$-0.544242\pi$
$521$	−6.32456	−0.277084	−0.138542	+	0.990357i	$0.544242\pi$
$522$	0	0
$523$	− 14.1421i	− 0.618392i	−0.950998	−	0.309196i	$-0.899940\pi$
$523$	− 14.1421i	− 0.618392i	0.950998	−	0.309196i	$-0.100060\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−20.0000	−0.871214
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	0	0
$532$	0	0
$533$	28.2843i	1.22513i
$534$	0	0
$535$	−25.2982	−1.09374
$536$	0	0
$537$	0	0
$538$	0	0
$539$	13.4164i	0.577886i
$540$	0	0
$541$	40.2492i	1.73045i	0.501384	+	0.865225i	$0.332824\pi$
$541$	40.2492i	1.73045i	−0.501384	+	0.865225i	$0.667176\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−6.32456	−0.270914
$546$	0	0
$547$	2.82843i	0.120935i	0.998170	+	0.0604674i	$0.0192591\pi$
$547$	2.82843i	0.120935i	−0.998170	+	0.0604674i	$0.980741\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	12.0000	0.511217
$552$	0	0
$553$	−10.0000	−0.425243
$554$	0	0
$555$	0	0
$556$	0	0
$557$	24.0416i	1.01868i	0.860566	+	0.509338i	$0.170110\pi$
$557$	24.0416i	1.01868i	−0.860566	+	0.509338i	$0.829890\pi$
$558$	0	0
$559$	−37.9473	−1.60500
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 4.47214i	− 0.188478i	−0.995550	−	0.0942390i	$-0.969958\pi$
$563$	− 4.47214i	− 0.188478i	0.995550	−	0.0942390i	$-0.0300418\pi$
$564$	0	0
$565$	17.8885i	0.752577i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	18.9737	0.795417	0.397709	−	0.917512i	$-0.369805\pi$
$569$	18.9737	0.795417	0.397709	+	0.917512i	$0.369805\pi$
$570$	0	0
$571$	22.6274i	0.946928i	0.880813	+	0.473464i	$0.156997\pi$
$571$	22.6274i	0.946928i	−0.880813	+	0.473464i	$0.843003\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−12.0000	−0.500435
$576$	0	0
$577$	−12.0000	−0.499567	−0.249783	−	0.968302i	$-0.580359\pi$
$577$	−12.0000	−0.499567	−0.249783	+	0.968302i	$0.580359\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 14.1421i	− 0.586715i
$582$	0	0
$583$	31.6228	1.30968
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 35.7771i	− 1.47668i	−0.674430	−	0.738339i	$-0.735610\pi$
$587$	− 35.7771i	− 1.47668i	0.674430	−	0.738339i	$-0.264390\pi$
$588$	0	0
$589$	− 8.94427i	− 0.368542i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−37.9473	−1.55831	−0.779155	−	0.626831i	$-0.784352\pi$
$593$	−37.9473	−1.55831	−0.779155	+	0.626831i	$0.784352\pi$
$594$	0	0
$595$	28.2843i	1.15954i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	36.0000	1.47092	0.735460	−	0.677568i	$-0.236966\pi$
$599$	36.0000	1.47092	0.735460	+	0.677568i	$0.236966\pi$
$600$	0	0
$601$	28.0000	1.14214	0.571072	−	0.820900i	$-0.306528\pi$
$601$	28.0000	1.14214	0.571072	+	0.820900i	$0.306528\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	12.7279i	0.517464i
$606$	0	0
$607$	−9.48683	−0.385059	−0.192529	−	0.981291i	$-0.561669\pi$
$607$	−9.48683	−0.385059	−0.192529	+	0.981291i	$0.561669\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 53.6656i	− 2.17108i
$612$	0	0
$613$	13.4164i	0.541884i	0.962596	+	0.270942i	$0.0873351\pi$
$613$	13.4164i	0.541884i	−0.962596	+	0.270942i	$0.912665\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	25.2982	1.01847	0.509234	−	0.860628i	$-0.329929\pi$
$617$	25.2982	1.01847	0.509234	+	0.860628i	$0.329929\pi$
$618$	0	0
$619$	− 11.3137i	− 0.454736i	−0.973809	−	0.227368i	$-0.926988\pi$
$619$	− 11.3137i	− 0.454736i	0.973809	−	0.227368i	$-0.0730121\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	28.2843i	1.12777i
$630$	0	0
$631$	−41.1096	−1.63655	−0.818274	−	0.574829i	$-0.805069\pi$
$631$	−41.1096	−1.63655	−0.818274	+	0.574829i	$0.805069\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 22.3607i	− 0.887357i
$636$	0	0
$637$	− 13.4164i	− 0.531577i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	6.32456	0.249805	0.124902	−	0.992169i	$-0.460138\pi$
$641$	6.32456	0.249805	0.124902	+	0.992169i	$0.460138\pi$
$642$	0	0
$643$	8.48528i	0.334627i	0.985904	+	0.167313i	$0.0535092\pi$
$643$	8.48528i	0.334627i	−0.985904	+	0.167313i	$0.946491\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−12.0000	−0.471769	−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000	−0.471769	−0.235884	+	0.971781i	$0.575799\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 1.41421i	− 0.0553425i	−0.999617	−	0.0276712i	$-0.991191\pi$
$653$	− 1.41421i	− 0.0553425i	0.999617	−	0.0276712i	$-0.00880915\pi$
$654$	0	0
$655$	−25.2982	−0.988483
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 17.8885i	− 0.696839i	−0.937339	−	0.348419i	$-0.886719\pi$
$659$	− 17.8885i	− 0.696839i	0.937339	−	0.348419i	$-0.113281\pi$
$660$	0	0
$661$	− 4.47214i	− 0.173946i	−0.996211	−	0.0869730i	$-0.972281\pi$
$661$	− 4.47214i	− 0.173946i	0.996211	−	0.0869730i	$-0.0277194\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−12.6491	−0.490511
$666$	0	0
$667$	16.9706i	0.657103i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−60.0000	−2.31627
$672$	0	0
$673$	34.0000	1.31060	0.655302	−	0.755367i	$-0.272541\pi$
$673$	34.0000	1.31060	0.655302	+	0.755367i	$0.272541\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	38.1838i	1.46752i	0.679408	+	0.733761i	$0.262237\pi$
$677$	38.1838i	1.46752i	−0.679408	+	0.733761i	$0.737763\pi$
$678$	0	0
$679$	−6.32456	−0.242714
$680$	0	0
$681$	0	0
$682$	0	0
$683$	40.2492i	1.54009i	0.637987	+	0.770047i	$0.279767\pi$
$683$	40.2492i	1.54009i	−0.637987	+	0.770047i	$0.720233\pi$
$684$	0	0
$685$	− 26.8328i	− 1.02523i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−31.6228	−1.20473
$690$	0	0
$691$	− 8.48528i	− 0.322795i	−0.986889	−	0.161398i	$-0.948400\pi$
$691$	− 8.48528i	− 0.322795i	0.986889	−	0.161398i	$-0.0516002\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−24.0000	−0.910372
$696$	0	0
$697$	−40.0000	−1.51511
$698$	0	0
$699$	0	0
$700$	0	0
$701$	26.8701i	1.01487i	0.861691	+	0.507434i	$0.169406\pi$
$701$	26.8701i	1.01487i	−0.861691	+	0.507434i	$0.830594\pi$
$702$	0	0
$703$	−12.6491	−0.477070
$704$	0	0
$705$	0	0
$706$	0	0
$707$	49.1935i	1.85011i
$708$	0	0
$709$	13.4164i	0.503864i	0.967745	+	0.251932i	$0.0810659\pi$
$709$	13.4164i	0.503864i	−0.967745	+	0.251932i	$0.918934\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	12.6491	0.473713
$714$	0	0
$715$	− 28.2843i	− 1.05777i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	−30.0000	−1.11726
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 12.7279i	− 0.472703i
$726$	0	0
$727$	15.8114	0.586412	0.293206	−	0.956049i	$-0.405278\pi$
$727$	15.8114	0.586412	0.293206	+	0.956049i	$0.405278\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 53.6656i	− 1.98490i
$732$	0	0
$733$	22.3607i	0.825911i	0.910751	+	0.412955i	$0.135503\pi$
$733$	22.3607i	0.825911i	−0.910751	+	0.412955i	$0.864497\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	− 45.2548i	− 1.66473i	−0.554231	−	0.832363i	$-0.686988\pi$
$739$	− 45.2548i	− 1.66473i	0.554231	−	0.832363i	$-0.313012\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	44.0000	1.61420	0.807102	−	0.590412i	$-0.201035\pi$
$743$	44.0000	1.61420	0.807102	+	0.590412i	$0.201035\pi$
$744$	0	0
$745$	−26.0000	−0.952566
$746$	0	0
$747$	0	0
$748$	0	0
$749$	56.5685i	2.06697i
$750$	0	0
$751$	15.8114	0.576966	0.288483	−	0.957485i	$-0.406849\pi$
$751$	15.8114	0.576966	0.288483	+	0.957485i	$0.406849\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	31.3050i	1.13930i
$756$	0	0
$757$	− 40.2492i	− 1.46288i	−0.681904	−	0.731441i	$-0.738848\pi$
$757$	− 40.2492i	− 1.46288i	0.681904	−	0.731441i	$-0.261152\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−18.9737	−0.687795	−0.343897	−	0.939007i	$-0.611747\pi$
$761$	−18.9737	−0.687795	−0.343897	+	0.939007i	$0.611747\pi$
$762$	0	0
$763$	14.1421i	0.511980i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−40.0000	−1.44244	−0.721218	−	0.692708i	$-0.756418\pi$
$769$	−40.0000	−1.44244	−0.721218	+	0.692708i	$0.756418\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	29.6985i	1.06818i	0.845428	+	0.534090i	$0.179346\pi$
$773$	29.6985i	1.06818i	−0.845428	+	0.534090i	$0.820654\pi$
$774$	0	0
$775$	−9.48683	−0.340777
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 17.8885i	− 0.640924i
$780$	0	0
$781$	− 35.7771i	− 1.28020i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−18.9737	−0.677199
$786$	0	0
$787$	− 42.4264i	− 1.51234i	−0.654376	−	0.756169i	$-0.727069\pi$
$787$	− 42.4264i	− 1.51234i	0.654376	−	0.756169i	$-0.272931\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	40.0000	1.42224
$792$	0	0
$793$	60.0000	2.13066
$794$	0	0
$795$	0	0
$796$	0	0
$797$	18.3848i	0.651222i	0.945504	+	0.325611i	$0.105570\pi$
$797$	18.3848i	0.651222i	−0.945504	+	0.325611i	$0.894430\pi$
$798$	0	0
$799$	75.8947	2.68496
$800$	0	0
$801$	0	0
$802$	0	0
$803$	17.8885i	0.631273i
$804$	0	0
$805$	− 17.8885i	− 0.630488i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−18.9737	−0.667079	−0.333539	−	0.942736i	$-0.608243\pi$
$809$	−18.9737	−0.667079	−0.333539	+	0.942736i	$0.608243\pi$
$810$	0	0
$811$	19.7990i	0.695237i	0.937636	+	0.347618i	$0.113009\pi$
$811$	19.7990i	0.695237i	−0.937636	+	0.347618i	$0.886991\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−28.0000	−0.980797
$816$	0	0
$817$	24.0000	0.839654
$818$	0	0
$819$	0	0
$820$	0	0
$821$	12.7279i	0.444208i	0.975023	+	0.222104i	$0.0712924\pi$
$821$	12.7279i	0.444208i	−0.975023	+	0.222104i	$0.928708\pi$
$822$	0	0
$823$	−15.8114	−0.551150	−0.275575	−	0.961280i	$-0.588868\pi$
$823$	−15.8114	−0.551150	−0.275575	+	0.961280i	$0.588868\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	13.4164i	0.465971i	0.972480	+	0.232986i	$0.0748495\pi$
$829$	13.4164i	0.465971i	−0.972480	+	0.232986i	$0.925151\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	18.9737	0.657399
$834$	0	0
$835$	16.9706i	0.587291i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	16.0000	0.552381	0.276191	−	0.961103i	$-0.410928\pi$
$839$	16.0000	0.552381	0.276191	+	0.961103i	$0.410928\pi$
$840$	0	0
$841$	11.0000	0.379310
$842$	0	0
$843$	0	0
$844$	0	0
$845$	9.89949i	0.340553i
$846$	0	0
$847$	28.4605	0.977914
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 17.8885i	− 0.613211i
$852$	0	0
$853$	13.4164i	0.459369i	0.973265	+	0.229685i	$0.0737694\pi$
$853$	13.4164i	0.459369i	−0.973265	+	0.229685i	$0.926231\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−44.2719	−1.51230	−0.756149	−	0.654399i	$-0.772922\pi$
$857$	−44.2719	−1.51230	−0.756149	+	0.654399i	$0.772922\pi$
$858$	0	0
$859$	25.4558i	0.868542i	0.900782	+	0.434271i	$0.142994\pi$
$859$	25.4558i	0.868542i	−0.900782	+	0.434271i	$0.857006\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−36.0000	−1.22545	−0.612727	−	0.790295i	$-0.709928\pi$
$863$	−36.0000	−1.22545	−0.612727	+	0.790295i	$0.709928\pi$
$864$	0	0
$865$	−10.0000	−0.340010
$866$	0	0
$867$	0	0
$868$	0	0
$869$	14.1421i	0.479739i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	35.7771i	1.20949i
$876$	0	0
$877$	13.4164i	0.453040i	0.974007	+	0.226520i	$0.0727348\pi$
$877$	13.4164i	0.453040i	−0.974007	+	0.226520i	$0.927265\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−12.6491	−0.426159	−0.213080	−	0.977035i	$-0.568349\pi$
$881$	−12.6491	−0.426159	−0.213080	+	0.977035i	$0.568349\pi$
$882$	0	0
$883$	19.7990i	0.666289i	0.942876	+	0.333145i	$0.108110\pi$
$883$	19.7990i	0.666289i	−0.942876	+	0.333145i	$0.891890\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	32.0000	1.07445	0.537227	−	0.843437i	$-0.319472\pi$
$887$	32.0000	1.07445	0.537227	+	0.843437i	$0.319472\pi$
$888$	0	0
$889$	−50.0000	−1.67695
$890$	0	0
$891$	0	0
$892$	0	0
$893$	33.9411i	1.13580i
$894$	0	0
$895$	−25.2982	−0.845626
$896$	0	0
$897$	0	0
$898$	0	0
$899$	13.4164i	0.447462i
$900$	0	0
$901$	− 44.7214i	− 1.48988i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	18.9737	0.630706
$906$	0	0
$907$	14.1421i	0.469582i	0.972046	+	0.234791i	$0.0754406\pi$
$907$	14.1421i	0.469582i	−0.972046	+	0.234791i	$0.924559\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	0	0
$913$	−20.0000	−0.661903
$914$	0	0
$915$	0	0
$916$	0	0
$917$	56.5685i	1.86806i
$918$	0	0
$919$	34.7851	1.14745	0.573727	−	0.819047i	$-0.305497\pi$
$919$	34.7851	1.14745	0.573727	+	0.819047i	$0.305497\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	35.7771i	1.17762i
$924$	0	0
$925$	13.4164i	0.441129i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	56.9210	1.86752	0.933759	−	0.357903i	$-0.116508\pi$
$929$	56.9210	1.86752	0.933759	+	0.357903i	$0.116508\pi$
$930$	0	0
$931$	8.48528i	0.278094i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	40.0000	1.30814
$936$	0	0
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 57.9828i	− 1.89018i	−0.326805	−	0.945092i	$-0.605972\pi$
$941$	− 57.9828i	− 1.89018i	0.326805	−	0.945092i	$-0.394028\pi$
$942$	0	0
$943$	25.2982	0.823823
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 17.8885i	− 0.581300i	−0.956830	−	0.290650i	$-0.906129\pi$
$947$	− 17.8885i	− 0.581300i	0.956830	−	0.290650i	$-0.0938715\pi$
$948$	0	0
$949$	− 17.8885i	− 0.580687i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	18.9737	0.614617	0.307309	−	0.951610i	$-0.400572\pi$
$953$	18.9737	0.614617	0.307309	+	0.951610i	$0.400572\pi$
$954$	0	0
$955$	− 28.2843i	− 0.915258i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−60.0000	−1.93750
$960$	0	0
$961$	−21.0000	−0.677419
$962$	0	0
$963$	0	0
$964$	0	0
$965$	22.6274i	0.728402i
$966$	0	0
$967$	41.1096	1.32200	0.660998	−	0.750388i	$-0.270133\pi$
$967$	41.1096	1.32200	0.660998	+	0.750388i	$0.270133\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	22.3607i	0.717588i	0.933417	+	0.358794i	$0.116812\pi$
$971$	22.3607i	0.717588i	−0.933417	+	0.358794i	$0.883188\pi$
$972$	0	0
$973$	53.6656i	1.72044i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	56.9210	1.82106	0.910532	−	0.413439i	$-0.135672\pi$
$977$	56.9210	1.82106	0.910532	+	0.413439i	$0.135672\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−24.0000	−0.765481	−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000	−0.765481	−0.382741	+	0.923856i	$0.625020\pi$
$984$	0	0
$985$	−10.0000	−0.318626
$986$	0	0
$987$	0	0
$988$	0	0
$989$	33.9411i	1.07927i
$990$	0	0
$991$	−34.7851	−1.10498	−0.552492	−	0.833518i	$-0.686323\pi$
$991$	−34.7851	−1.10498	−0.552492	+	0.833518i	$0.686323\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	4.47214i	0.141776i
$996$	0	0
$997$	22.3607i	0.708170i	0.935213	+	0.354085i	$0.115208\pi$
$997$	22.3607i	0.708170i	−0.935213	+	0.354085i	$0.884792\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4608.2.d.i.2305.1		4
3.2	odd	2		4608.2.d.l.2305.3		4
4.3	odd	2		4608.2.d.l.2305.2		4
8.3	odd	2		4608.2.d.l.2305.4		4
8.5	even	2	inner	4608.2.d.i.2305.3		4
12.11	even	2	inner	4608.2.d.i.2305.4		4
16.3	odd	4		4608.2.a.x.1.1	✓	4
16.5	even	4		4608.2.a.y.1.4	yes	4
16.11	odd	4		4608.2.a.x.1.3	yes	4
16.13	even	4		4608.2.a.y.1.2	yes	4
24.5	odd	2		4608.2.d.l.2305.1		4
24.11	even	2	inner	4608.2.d.i.2305.2		4
48.5	odd	4		4608.2.a.x.1.2	yes	4
48.11	even	4		4608.2.a.y.1.1	yes	4
48.29	odd	4		4608.2.a.x.1.4	yes	4
48.35	even	4		4608.2.a.y.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4608.2.a.x.1.1	✓	4	16.3	odd	4
4608.2.a.x.1.2	yes	4	48.5	odd	4
4608.2.a.x.1.3	yes	4	16.11	odd	4
4608.2.a.x.1.4	yes	4	48.29	odd	4
4608.2.a.y.1.1	yes	4	48.11	even	4
4608.2.a.y.1.2	yes	4	16.13	even	4
4608.2.a.y.1.3	yes	4	48.35	even	4
4608.2.a.y.1.4	yes	4	16.5	even	4
4608.2.d.i.2305.1		4	1.1	even	1	trivial
4608.2.d.i.2305.2		4	24.11	even	2	inner
4608.2.d.i.2305.3		4	8.5	even	2	inner
4608.2.d.i.2305.4		4	12.11	even	2	inner
4608.2.d.l.2305.1		4	24.5	odd	2
4608.2.d.l.2305.2		4	4.3	odd	2
4608.2.d.l.2305.3		4	3.2	odd	2
4608.2.d.l.2305.4		4	8.3	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:	\( N \)	\(=\)	\( 4608 = 2^{9} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4608.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{5} -3.16228 q^{7} +O(q^{10})\)	\(q-1.41421i q^{5} -3.16228 q^{7} +4.47214i q^{11} -4.47214i q^{13} +6.32456 q^{17} +2.82843i q^{19} -4.00000 q^{23} +3.00000 q^{25} -4.24264i q^{29} -3.16228 q^{31} +4.47214i q^{35} +4.47214i q^{37} -6.32456 q^{41} -8.48528i q^{43} +12.0000 q^{47} +3.00000 q^{49} -7.07107i q^{53} +6.32456 q^{55} +13.4164i q^{61} -6.32456 q^{65} -8.00000 q^{71} +4.00000 q^{73} -14.1421i q^{77} +3.16228 q^{79} +4.47214i q^{83} -8.94427i q^{85} +14.1421i q^{91} +4.00000 q^{95} +2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 16 q^{23} + 12 q^{25} + 48 q^{47} + 12 q^{49} - 32 q^{71} + 16 q^{73} + 16 q^{95} + 8 q^{97}+O(q^{100}) \) 4 * q - 16 * q^23 + 12 * q^25 + 48 * q^47 + 12 * q^49 - 32 * q^71 + 16 * q^73 + 16 * q^95 + 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more