LMFDB - Embedded newform 2304.2.k.f.577.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2304,2,Mod(577,2304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2304, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("2304.577");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2304 = 2^{8} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2304.k (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$18.3975326257$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 256)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			577.3
Root			$-0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	2304.577
Dual form			2304.2.k.f.1729.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.73205 - 1.73205i) q^{5} -3.86370i q^{7} +O(q^{10})$	$q+(1.73205 - 1.73205i) q^{5} -3.86370i q^{7} +(-3.34607 + 3.34607i) q^{11} +(-0.267949 - 0.267949i) q^{13} -3.46410 q^{17} +(-3.34607 - 3.34607i) q^{19} -1.79315i q^{23} -1.00000i q^{25} +(1.73205 + 1.73205i) q^{29} -5.65685 q^{31} +(-6.69213 - 6.69213i) q^{35} +(3.73205 - 3.73205i) q^{37} -6.92820i q^{41} +(-1.55291 + 1.55291i) q^{43} -9.79796 q^{47} -7.92820 q^{49} +(-7.73205 + 7.73205i) q^{53} +11.5911i q^{55} +(5.13922 - 5.13922i) q^{59} +(3.73205 + 3.73205i) q^{61} -0.928203 q^{65} +(4.38134 + 4.38134i) q^{67} -1.79315i q^{71} -2.53590i q^{73} +(12.9282 + 12.9282i) q^{77} -4.14110 q^{79} +(-1.55291 - 1.55291i) q^{83} +(-6.00000 + 6.00000i) q^{85} -2.53590i q^{89} +(-1.03528 + 1.03528i) q^{91} -11.5911 q^{95} +3.46410 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 16 q^{13} + 16 q^{37} - 8 q^{49} - 48 q^{53} + 16 q^{61} + 48 q^{65} + 48 q^{77} - 48 q^{85}+O(q^{100}) $ 8 * q - 16 * q^13 + 16 * q^37 - 8 * q^49 - 48 * q^53 + 16 * q^61 + 48 * q^65 + 48 * q^77 - 48 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times$.

$n$	$1279$	$1793$	$2053$
$\chi(n)$	$1$	$1$	$e\left(\frac{3}{4}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.73205	−	1.73205i	0.774597	−	0.774597i	−0.204310	−	0.978906i	$-0.565495\pi$
$5$	1.73205	−	1.73205i	0.774597	−	0.774597i	0.978906	+	0.204310i	$0.0654949\pi$
$6$		0			0
$7$		−	3.86370i		−	1.46034i	−0.683264	−	0.730171i	$-0.739440\pi$
$7$		−	3.86370i		−	1.46034i	0.683264	−	0.730171i	$-0.260560\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.34607	+	3.34607i	−1.00888	+	1.00888i	−0.00891637	+	0.999960i	$0.502838\pi$
$11$	−3.34607	+	3.34607i	−1.00888	+	1.00888i	−0.999960	+	0.00891637i	$0.997162\pi$
$12$		0			0
$13$	−0.267949	−	0.267949i	−0.0743157	−	0.0743157i	0.668972	−	0.743288i	$-0.266735\pi$
$13$	−0.267949	−	0.267949i	−0.0743157	−	0.0743157i	−0.743288	+	0.668972i	$0.766735\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.46410			−0.840168			−0.420084	−	0.907485i	$-0.637999\pi$
$17$	−3.46410			−0.840168			−0.420084	+	0.907485i	$0.637999\pi$
$18$		0			0
$19$	−3.34607	−	3.34607i	−0.767640	−	0.767640i	0.210051	−	0.977691i	$-0.432637\pi$
$19$	−3.34607	−	3.34607i	−0.767640	−	0.767640i	−0.977691	+	0.210051i	$0.932637\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	1.79315i		−	0.373898i	−0.982370	−	0.186949i	$-0.940140\pi$
$23$		−	1.79315i		−	0.373898i	0.982370	−	0.186949i	$-0.0598599\pi$
$24$		0			0
$25$		−	1.00000i		−	0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.73205	+	1.73205i	0.321634	+	0.321634i	0.849394	−	0.527760i	$-0.176968\pi$
$29$	1.73205	+	1.73205i	0.321634	+	0.321634i	−0.527760	+	0.849394i	$0.676968\pi$
$30$		0			0
$31$	−5.65685			−1.01600			−0.508001	−	0.861357i	$-0.669615\pi$
$31$	−5.65685			−1.01600			−0.508001	+	0.861357i	$0.669615\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−6.69213	−	6.69213i	−1.13118	−	1.13118i
$36$		0			0
$37$	3.73205	−	3.73205i	0.613545	−	0.613545i	−0.330323	−	0.943868i	$-0.607158\pi$
$37$	3.73205	−	3.73205i	0.613545	−	0.613545i	0.943868	+	0.330323i	$0.107158\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	6.92820i		−	1.08200i	−0.841021	−	0.541002i	$-0.818045\pi$
$41$		−	6.92820i		−	1.08200i	0.841021	−	0.541002i	$-0.181955\pi$
$42$		0			0
$43$	−1.55291	+	1.55291i	−0.236817	+	0.236817i	−0.815531	−	0.578714i	$-0.803555\pi$
$43$	−1.55291	+	1.55291i	−0.236817	+	0.236817i	0.578714	+	0.815531i	$0.303555\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−9.79796			−1.42918			−0.714590	−	0.699544i	$-0.753387\pi$
$47$	−9.79796			−1.42918			−0.714590	+	0.699544i	$0.753387\pi$
$48$		0			0
$49$	−7.92820			−1.13260
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−7.73205	+	7.73205i	−1.06208	+	1.06208i	−0.0641378	+	0.997941i	$0.520430\pi$
$53$	−7.73205	+	7.73205i	−1.06208	+	1.06208i	−0.997941	+	0.0641378i	$0.979570\pi$
$54$		0			0
$55$			11.5911i			1.56294i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	5.13922	−	5.13922i	0.669069	−	0.669069i	−0.288432	−	0.957501i	$-0.593134\pi$
$59$	5.13922	−	5.13922i	0.669069	−	0.669069i	0.957501	+	0.288432i	$0.0931338\pi$
$60$		0			0
$61$	3.73205	+	3.73205i	0.477840	+	0.477840i	0.904440	−	0.426600i	$-0.140289\pi$
$61$	3.73205	+	3.73205i	0.477840	+	0.477840i	−0.426600	+	0.904440i	$0.640289\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−0.928203			−0.115129
$66$		0			0
$67$	4.38134	+	4.38134i	0.535266	+	0.535266i	0.922135	−	0.386869i	$-0.126443\pi$
$67$	4.38134	+	4.38134i	0.535266	+	0.535266i	−0.386869	+	0.922135i	$0.626443\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	1.79315i		−	0.212808i	−0.994323	−	0.106404i	$-0.966066\pi$
$71$		−	1.79315i		−	0.212808i	0.994323	−	0.106404i	$-0.0339337\pi$
$72$		0			0
$73$		−	2.53590i		−	0.296804i	−0.988927	−	0.148402i	$-0.952587\pi$
$73$		−	2.53590i		−	0.296804i	0.988927	−	0.148402i	$-0.0474130\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	12.9282	+	12.9282i	1.47331	+	1.47331i
$78$		0			0
$79$	−4.14110			−0.465911			−0.232955	−	0.972487i	$-0.574840\pi$
$79$	−4.14110			−0.465911			−0.232955	+	0.972487i	$0.574840\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−1.55291	−	1.55291i	−0.170454	−	0.170454i	0.616725	−	0.787179i	$-0.288459\pi$
$83$	−1.55291	−	1.55291i	−0.170454	−	0.170454i	−0.787179	+	0.616725i	$0.788459\pi$
$84$		0			0
$85$	−6.00000	+	6.00000i	−0.650791	+	0.650791i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	2.53590i		−	0.268805i	−0.990927	−	0.134402i	$-0.957089\pi$
$89$		−	2.53590i		−	0.268805i	0.990927	−	0.134402i	$-0.0429115\pi$
$90$		0			0
$91$	−1.03528	+	1.03528i	−0.108526	+	0.108526i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−11.5911			−1.18922
$96$		0			0
$97$	3.46410			0.351726			0.175863	−	0.984415i	$-0.443728\pi$
$97$	3.46410			0.351726			0.175863	+	0.984415i	$0.443728\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−0.803848	+	0.803848i	−0.0799858	+	0.0799858i	−0.745968	−	0.665982i	$-0.768013\pi$
$101$	−0.803848	+	0.803848i	−0.0799858	+	0.0799858i	0.665982	+	0.745968i	$0.268013\pi$
$102$		0			0
$103$		−	9.52056i		−	0.938088i	−0.883175	−	0.469044i	$-0.844598\pi$
$103$		−	9.52056i		−	0.938088i	0.883175	−	0.469044i	$-0.155402\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−13.1440	+	13.1440i	−1.27068	+	1.27068i	−0.324949	+	0.945731i	$0.605347\pi$
$107$	−13.1440	+	13.1440i	−1.27068	+	1.27068i	−0.945731	+	0.324949i	$0.894653\pi$
$108$		0			0
$109$	13.1962	+	13.1962i	1.26396	+	1.26396i	0.949156	+	0.314806i	$0.101940\pi$
$109$	13.1962	+	13.1962i	1.26396	+	1.26396i	0.314806	+	0.949156i	$0.398060\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−12.9282			−1.21618			−0.608092	−	0.793867i	$-0.708065\pi$
$113$	−12.9282			−1.21618			−0.608092	+	0.793867i	$0.708065\pi$
$114$		0			0
$115$	−3.10583	−	3.10583i	−0.289620	−	0.289620i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			13.3843i			1.22693i
$120$		0			0
$121$		−	11.3923i		−	1.03566i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	6.92820	+	6.92820i	0.619677	+	0.619677i
$126$		0			0
$127$	5.65685			0.501965			0.250982	−	0.967992i	$-0.419246\pi$
$127$	5.65685			0.501965			0.250982	+	0.967992i	$0.419246\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−0.240237	−	0.240237i	−0.0209896	−	0.0209896i	0.696534	−	0.717524i	$-0.254724\pi$
$131$	−0.240237	−	0.240237i	−0.0209896	−	0.0209896i	−0.717524	+	0.696534i	$0.754724\pi$
$132$		0			0
$133$	−12.9282	+	12.9282i	−1.12102	+	1.12102i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	12.0000i		−	1.02523i	−0.858619	−	0.512615i	$-0.828677\pi$
$137$		−	12.0000i		−	1.02523i	0.858619	−	0.512615i	$-0.171323\pi$
$138$		0			0
$139$	−7.20977	+	7.20977i	−0.611525	+	0.611525i	−0.943343	−	0.331819i	$-0.892338\pi$
$139$	−7.20977	+	7.20977i	−0.611525	+	0.611525i	0.331819	+	0.943343i	$0.392338\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	1.79315			0.149951
$144$		0			0
$145$	6.00000			0.498273
$146$		0			0
$147$		0			0
$148$		0			0
$149$	13.7321	−	13.7321i	1.12497	−	1.12497i	0.133991	−	0.990983i	$-0.457221\pi$
$149$	13.7321	−	13.7321i	1.12497	−	1.12497i	0.990983	−	0.133991i	$-0.0427793\pi$
$150$		0			0
$151$		−	9.52056i		−	0.774772i	−0.921918	−	0.387386i	$-0.873378\pi$
$151$		−	9.52056i		−	0.774772i	0.921918	−	0.387386i	$-0.126622\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−9.79796	+	9.79796i	−0.786991	+	0.786991i
$156$		0			0
$157$	−7.19615	−	7.19615i	−0.574315	−	0.574315i	0.359016	−	0.933331i	$-0.383112\pi$
$157$	−7.19615	−	7.19615i	−0.574315	−	0.574315i	−0.933331	+	0.359016i	$0.883112\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−6.92820			−0.546019
$162$		0			0
$163$	−11.8313	−	11.8313i	−0.926703	−	0.926703i	0.0707887	−	0.997491i	$-0.477448\pi$
$163$	−11.8313	−	11.8313i	−0.926703	−	0.926703i	−0.997491	+	0.0707887i	$0.977448\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	18.7637i		−	1.45198i	−0.687705	−	0.725990i	$-0.741382\pi$
$167$		−	18.7637i		−	1.45198i	0.687705	−	0.725990i	$-0.258618\pi$
$168$		0			0
$169$		−	12.8564i		−	0.988954i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−7.73205	−	7.73205i	−0.587857	−	0.587857i	0.349194	−	0.937051i	$-0.386456\pi$
$173$	−7.73205	−	7.73205i	−0.587857	−	0.587857i	−0.937051	+	0.349194i	$0.886456\pi$
$174$		0			0
$175$	−3.86370			−0.292069
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−1.55291	−	1.55291i	−0.116070	−	0.116070i	0.646686	−	0.762756i	$-0.276154\pi$
$179$	−1.55291	−	1.55291i	−0.116070	−	0.116070i	−0.762756	+	0.646686i	$0.776154\pi$
$180$		0			0
$181$	16.1244	−	16.1244i	1.19851	−	1.19851i	0.223902	−	0.974612i	$-0.428120\pi$
$181$	16.1244	−	16.1244i	1.19851	−	1.19851i	0.974612	−	0.223902i	$-0.0718797\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	12.9282i		−	0.950500i
$186$		0			0
$187$	11.5911	−	11.5911i	0.847626	−	0.847626i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−16.9706			−1.22795			−0.613973	−	0.789327i	$-0.710430\pi$
$191$	−16.9706			−1.22795			−0.613973	+	0.789327i	$0.710430\pi$
$192$		0			0
$193$	−18.3923			−1.32391			−0.661954	−	0.749545i	$-0.730272\pi$
$193$	−18.3923			−1.32391			−0.661954	+	0.749545i	$0.730272\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	1.73205	−	1.73205i	0.123404	−	0.123404i	−0.642708	−	0.766111i	$-0.722189\pi$
$197$	1.73205	−	1.73205i	0.123404	−	0.123404i	0.766111	+	0.642708i	$0.222189\pi$
$198$		0			0
$199$			15.7322i			1.11523i	0.830101	+	0.557614i	$0.188283\pi$
$199$			15.7322i			1.11523i	−0.830101	+	0.557614i	$0.811717\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	6.69213	−	6.69213i	0.469695	−	0.469695i
$204$		0			0
$205$	−12.0000	−	12.0000i	−0.838116	−	0.838116i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	22.3923			1.54891
$210$		0			0
$211$	7.20977	+	7.20977i	0.496341	+	0.496341i	0.910297	−	0.413956i	$-0.135853\pi$
$211$	7.20977	+	7.20977i	0.496341	+	0.496341i	−0.413956	+	0.910297i	$0.635853\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			5.37945i			0.366876i
$216$		0			0
$217$			21.8564i			1.48371i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	0.928203	+	0.928203i	0.0624377	+	0.0624377i
$222$		0			0
$223$	−5.65685			−0.378811			−0.189405	−	0.981899i	$-0.560656\pi$
$223$	−5.65685			−0.378811			−0.189405	+	0.981899i	$0.560656\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−14.9372	−	14.9372i	−0.991415	−	0.991415i	0.00854833	−	0.999963i	$-0.497279\pi$
$227$	−14.9372	−	14.9372i	−0.991415	−	0.991415i	−0.999963	+	0.00854833i	$0.997279\pi$
$228$		0			0
$229$	9.19615	−	9.19615i	0.607699	−	0.607699i	−0.334645	−	0.942344i	$-0.608617\pi$
$229$	9.19615	−	9.19615i	0.607699	−	0.607699i	0.942344	+	0.334645i	$0.108617\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			2.53590i			0.166132i	0.996544	+	0.0830661i	$0.0264713\pi$
$233$			2.53590i			0.166132i	−0.996544	+	0.0830661i	$0.973529\pi$
$234$		0			0
$235$	−16.9706	+	16.9706i	−1.10704	+	1.10704i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	26.7685			1.73151			0.865756	−	0.500467i	$-0.166838\pi$
$239$	26.7685			1.73151			0.865756	+	0.500467i	$0.166838\pi$
$240$		0			0
$241$	−2.39230			−0.154102			−0.0770510	−	0.997027i	$-0.524550\pi$
$241$	−2.39230			−0.154102			−0.0770510	+	0.997027i	$0.524550\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−13.7321	+	13.7321i	−0.877309	+	0.877309i
$246$		0			0
$247$			1.79315i			0.114095i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	14.9372	−	14.9372i	0.942826	−	0.942826i	−0.0556256	−	0.998452i	$-0.517715\pi$
$251$	14.9372	−	14.9372i	0.942826	−	0.942826i	0.998452	+	0.0556256i	$0.0177153\pi$
$252$		0			0
$253$	6.00000	+	6.00000i	0.377217	+	0.377217i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−12.9282			−0.806439			−0.403220	−	0.915103i	$-0.632109\pi$
$257$	−12.9282			−0.806439			−0.403220	+	0.915103i	$0.632109\pi$
$258$		0			0
$259$	−14.4195	−	14.4195i	−0.895986	−	0.895986i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	21.3891i		−	1.31891i	−0.751746	−	0.659453i	$-0.770788\pi$
$263$		−	21.3891i		−	1.31891i	0.751746	−	0.659453i	$-0.229212\pi$
$264$		0			0
$265$			26.7846i			1.64537i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−5.19615	−	5.19615i	−0.316815	−	0.316815i	0.530728	−	0.847543i	$-0.321919\pi$
$269$	−5.19615	−	5.19615i	−0.316815	−	0.316815i	−0.847543	+	0.530728i	$0.821919\pi$
$270$		0			0
$271$	−1.51575			−0.0920752			−0.0460376	−	0.998940i	$-0.514659\pi$
$271$	−1.51575			−0.0920752			−0.0460376	+	0.998940i	$0.514659\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	3.34607	+	3.34607i	0.201775	+	0.201775i
$276$		0			0
$277$	−3.19615	+	3.19615i	−0.192038	+	0.192038i	−0.796576	−	0.604538i	$-0.793358\pi$
$277$	−3.19615	+	3.19615i	−0.192038	+	0.192038i	0.604538	+	0.796576i	$0.293358\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			16.3923i			0.977883i	0.872317	+	0.488941i	$0.162617\pi$
$281$			16.3923i			0.977883i	−0.872317	+	0.488941i	$0.837383\pi$
$282$		0			0
$283$	1.27551	−	1.27551i	0.0758214	−	0.0758214i	−0.668179	−	0.744000i	$-0.732926\pi$
$283$	1.27551	−	1.27551i	0.0758214	−	0.0758214i	0.744000	+	0.668179i	$0.232926\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−26.7685			−1.58010
$288$		0			0
$289$	−5.00000			−0.294118
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−12.1244	+	12.1244i	−0.708312	+	0.708312i	−0.966180	−	0.257868i	$-0.916980\pi$
$293$	−12.1244	+	12.1244i	−0.708312	+	0.708312i	0.257868	+	0.966180i	$0.416980\pi$
$294$		0			0
$295$		−	17.8028i		−	1.03652i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−0.480473	+	0.480473i	−0.0277865	+	0.0277865i
$300$		0			0
$301$	6.00000	+	6.00000i	0.345834	+	0.345834i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	12.9282			0.740267
$306$		0			0
$307$	21.9067	+	21.9067i	1.25028	+	1.25028i	0.955593	+	0.294688i	$0.0952159\pi$
$307$	21.9067	+	21.9067i	1.25028	+	1.25028i	0.294688	+	0.955593i	$0.404784\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			15.1774i			0.860632i	0.902678	+	0.430316i	$0.141598\pi$
$311$			15.1774i			0.860632i	−0.902678	+	0.430316i	$0.858402\pi$
$312$		0			0
$313$		−	12.7846i		−	0.722629i	−0.932444	−	0.361314i	$-0.882328\pi$
$313$		−	12.7846i		−	0.722629i	0.932444	−	0.361314i	$-0.117672\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	6.80385	+	6.80385i	0.382142	+	0.382142i	0.871873	−	0.489731i	$-0.162905\pi$
$317$	6.80385	+	6.80385i	0.382142	+	0.382142i	−0.489731	+	0.871873i	$0.662905\pi$
$318$		0			0
$319$	−11.5911			−0.648978
$320$		0			0
$321$		0			0
$322$		0			0
$323$	11.5911	+	11.5911i	0.644947	+	0.644947i
$324$		0			0
$325$	−0.267949	+	0.267949i	−0.0148631	+	0.0148631i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			37.8564i			2.08709i
$330$		0			0
$331$	18.8009	−	18.8009i	1.03339	−	1.03339i	0.0339668	−	0.999423i	$-0.489186\pi$
$331$	18.8009	−	18.8009i	1.03339	−	1.03339i	0.999423	−	0.0339668i	$-0.0108140\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	15.1774			0.829231
$336$		0			0
$337$	21.7128			1.18277			0.591386	−	0.806389i	$-0.298581\pi$
$337$	21.7128			1.18277			0.591386	+	0.806389i	$0.298581\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	18.9282	−	18.9282i	1.02502	−	1.02502i
$342$		0			0
$343$			3.58630i			0.193642i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	5.13922	−	5.13922i	0.275888	−	0.275888i	−0.555577	−	0.831465i	$-0.687503\pi$
$347$	5.13922	−	5.13922i	0.275888	−	0.275888i	0.831465	+	0.555577i	$0.187503\pi$
$348$		0			0
$349$	4.12436	+	4.12436i	0.220772	+	0.220772i	0.808823	−	0.588052i	$-0.200105\pi$
$349$	4.12436	+	4.12436i	0.220772	+	0.220772i	−0.588052	+	0.808823i	$0.700105\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	0.928203			0.0494033			0.0247016	−	0.999695i	$-0.492136\pi$
$353$	0.928203			0.0494033			0.0247016	+	0.999695i	$0.492136\pi$
$354$		0			0
$355$	−3.10583	−	3.10583i	−0.164840	−	0.164840i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			17.8028i			0.939594i	0.882774	+	0.469797i	$0.155673\pi$
$359$			17.8028i			0.939594i	−0.882774	+	0.469797i	$0.844327\pi$
$360$		0			0
$361$			3.39230i			0.178542i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−4.39230	−	4.39230i	−0.229904	−	0.229904i
$366$		0			0
$367$	32.4254			1.69259			0.846295	−	0.532714i	$-0.178828\pi$
$367$	32.4254			1.69259			0.846295	+	0.532714i	$0.178828\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	29.8744	+	29.8744i	1.55100	+	1.55100i
$372$		0			0
$373$	−14.1244	+	14.1244i	−0.731331	+	0.731331i	−0.970884	−	0.239552i	$-0.922999\pi$
$373$	−14.1244	+	14.1244i	−0.731331	+	0.731331i	0.239552	+	0.970884i	$0.422999\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	0.928203i		−	0.0478049i
$378$		0			0
$379$	2.03339	−	2.03339i	0.104448	−	0.104448i	−0.652952	−	0.757400i	$-0.726469\pi$
$379$	2.03339	−	2.03339i	0.104448	−	0.104448i	0.757400	+	0.652952i	$0.226469\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	19.5959			1.00130			0.500652	−	0.865648i	$-0.333094\pi$
$383$	19.5959			1.00130			0.500652	+	0.865648i	$0.333094\pi$
$384$		0			0
$385$	44.7846			2.28244
$386$		0			0
$387$		0			0
$388$		0			0
$389$	4.26795	−	4.26795i	0.216394	−	0.216394i	−0.590583	−	0.806977i	$-0.701102\pi$
$389$	4.26795	−	4.26795i	0.216394	−	0.216394i	0.806977	+	0.590583i	$0.201102\pi$
$390$		0			0
$391$			6.21166i			0.314137i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−7.17260	+	7.17260i	−0.360893	+	0.360893i
$396$		0			0
$397$	−19.5885	−	19.5885i	−0.983116	−	0.983116i	0.0167433	−	0.999860i	$-0.494670\pi$
$397$	−19.5885	−	19.5885i	−0.983116	−	0.983116i	−0.999860	+	0.0167433i	$0.994670\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−13.6077			−0.679536			−0.339768	−	0.940509i	$-0.610349\pi$
$401$	−13.6077			−0.679536			−0.339768	+	0.940509i	$0.610349\pi$
$402$		0			0
$403$	1.51575	+	1.51575i	0.0755049	+	0.0755049i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			24.9754i			1.23798i
$408$		0			0
$409$		−	17.0718i		−	0.844146i	−0.906562	−	0.422073i	$-0.861303\pi$
$409$		−	17.0718i		−	0.844146i	0.906562	−	0.422073i	$-0.138697\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−19.8564	−	19.8564i	−0.977070	−	0.977070i
$414$		0			0
$415$	−5.37945			−0.264067
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−10.0382	−	10.0382i	−0.490398	−	0.490398i	0.418034	−	0.908432i	$-0.362719\pi$
$419$	−10.0382	−	10.0382i	−0.490398	−	0.490398i	−0.908432	+	0.418034i	$0.862719\pi$
$420$		0			0
$421$	−18.5167	+	18.5167i	−0.902447	+	0.902447i	−0.995647	−	0.0932005i	$-0.970290\pi$
$421$	−18.5167	+	18.5167i	−0.902447	+	0.902447i	0.0932005	+	0.995647i	$0.470290\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			3.46410i			0.168034i
$426$		0			0
$427$	14.4195	−	14.4195i	0.697810	−	0.697810i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	26.7685			1.28939			0.644697	−	0.764438i	$-0.276983\pi$
$431$	26.7685			1.28939			0.644697	+	0.764438i	$0.276983\pi$
$432$		0			0
$433$	−34.3923			−1.65279			−0.826394	−	0.563092i	$-0.809612\pi$
$433$	−34.3923			−1.65279			−0.826394	+	0.563092i	$0.809612\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−6.00000	+	6.00000i	−0.287019	+	0.287019i
$438$		0			0
$439$		−	12.1459i		−	0.579693i	−0.957073	−	0.289846i	$-0.906396\pi$
$439$		−	12.1459i		−	0.579693i	0.957073	−	0.289846i	$-0.0936042\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	10.0382	−	10.0382i	0.476929	−	0.476929i	−0.427219	−	0.904148i	$-0.640507\pi$
$443$	10.0382	−	10.0382i	0.476929	−	0.476929i	0.904148	+	0.427219i	$0.140507\pi$
$444$		0			0
$445$	−4.39230	−	4.39230i	−0.208215	−	0.208215i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	24.2487			1.14437			0.572184	−	0.820125i	$-0.306096\pi$
$449$	24.2487			1.14437			0.572184	+	0.820125i	$0.306096\pi$
$450$		0			0
$451$	23.1822	+	23.1822i	1.09161	+	1.09161i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			3.58630i			0.168128i
$456$		0			0
$457$		−	20.7846i		−	0.972263i	−0.873886	−	0.486132i	$-0.838408\pi$
$457$		−	20.7846i		−	0.972263i	0.873886	−	0.486132i	$-0.161592\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−24.1244	−	24.1244i	−1.12358	−	1.12358i	−0.991198	−	0.132385i	$-0.957737\pi$
$461$	−24.1244	−	24.1244i	−1.12358	−	1.12358i	−0.132385	−	0.991198i	$-0.542263\pi$
$462$		0			0
$463$	38.0822			1.76983			0.884916	−	0.465751i	$-0.154216\pi$
$463$	38.0822			1.76983			0.884916	+	0.465751i	$0.154216\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	20.3166	+	20.3166i	0.940141	+	0.940141i	0.998307	−	0.0581655i	$-0.0185251\pi$
$467$	20.3166	+	20.3166i	0.940141	+	0.940141i	−0.0581655	+	0.998307i	$0.518525\pi$
$468$		0			0
$469$	16.9282	−	16.9282i	0.781672	−	0.781672i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	10.3923i		−	0.477839i
$474$		0			0
$475$	−3.34607	+	3.34607i	−0.153528	+	0.153528i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	36.5665			1.67077			0.835383	−	0.549669i	$-0.185246\pi$
$479$	36.5665			1.67077			0.835383	+	0.549669i	$0.185246\pi$
$480$		0			0
$481$	−2.00000			−0.0911922
$482$		0			0
$483$		0			0
$484$		0			0
$485$	6.00000	−	6.00000i	0.272446	−	0.272446i
$486$		0			0
$487$			27.0459i			1.22557i	0.790251	+	0.612784i	$0.209950\pi$
$487$			27.0459i			1.22557i	−0.790251	+	0.612784i	$0.790050\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−15.4176	+	15.4176i	−0.695789	+	0.695789i	−0.963499	−	0.267711i	$-0.913733\pi$
$491$	−15.4176	+	15.4176i	−0.695789	+	0.695789i	0.267711	+	0.963499i	$0.413733\pi$
$492$		0			0
$493$	−6.00000	−	6.00000i	−0.270226	−	0.270226i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−6.92820			−0.310772
$498$		0			0
$499$	−4.10394	−	4.10394i	−0.183718	−	0.183718i	0.609256	−	0.792974i	$-0.291468\pi$
$499$	−4.10394	−	4.10394i	−0.183718	−	0.183718i	−0.792974	+	0.609256i	$0.791468\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	40.9850i		−	1.82743i	−0.406355	−	0.913715i	$-0.633200\pi$
$503$		−	40.9850i		−	1.82743i	0.406355	−	0.913715i	$-0.366800\pi$
$504$		0			0
$505$			2.78461i			0.123914i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−9.58846	−	9.58846i	−0.425001	−	0.425001i	0.461920	−	0.886921i	$-0.347161\pi$
$509$	−9.58846	−	9.58846i	−0.425001	−	0.425001i	−0.886921	+	0.461920i	$0.847161\pi$
$510$		0			0
$511$	−9.79796			−0.433436
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−16.4901	−	16.4901i	−0.726640	−	0.726640i
$516$		0			0
$517$	32.7846	−	32.7846i	1.44187	−	1.44187i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	12.0000i		−	0.525730i	−0.964833	−	0.262865i	$-0.915333\pi$
$521$		−	12.0000i		−	0.525730i	0.964833	−	0.262865i	$-0.0846673\pi$
$522$		0			0
$523$	−13.6245	+	13.6245i	−0.595758	+	0.595758i	−0.939181	−	0.343423i	$-0.888414\pi$
$523$	−13.6245	+	13.6245i	−0.595758	+	0.595758i	0.343423	+	0.939181i	$0.388414\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	19.5959			0.853612
$528$		0			0
$529$	19.7846			0.860200
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−1.85641	+	1.85641i	−0.0804099	+	0.0804099i
$534$		0			0
$535$			45.5322i			1.96853i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	26.5283	−	26.5283i	1.14265	−	1.14265i
$540$		0			0
$541$	−10.1244	−	10.1244i	−0.435280	−	0.435280i	0.455140	−	0.890420i	$-0.349589\pi$
$541$	−10.1244	−	10.1244i	−0.435280	−	0.435280i	−0.890420	+	0.455140i	$0.849589\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	45.7128			1.95812
$546$		0			0
$547$	0.240237	+	0.240237i	0.0102718	+	0.0102718i	0.712224	−	0.701952i	$-0.247688\pi$
$547$	0.240237	+	0.240237i	0.0102718	+	0.0102718i	−0.701952	+	0.712224i	$0.747688\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	11.5911i		−	0.493798i
$552$		0			0
$553$			16.0000i			0.680389i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	11.1962	+	11.1962i	0.474396	+	0.474396i	0.903334	−	0.428938i	$-0.141112\pi$
$557$	11.1962	+	11.1962i	0.474396	+	0.474396i	−0.428938	+	0.903334i	$0.641112\pi$
$558$		0			0
$559$	0.832204			0.0351985
$560$		0			0
$561$		0			0
$562$		0			0
$563$	10.5187	+	10.5187i	0.443309	+	0.443309i	0.893123	−	0.449813i	$-0.148510\pi$
$563$	10.5187	+	10.5187i	0.443309	+	0.443309i	−0.449813	+	0.893123i	$0.648510\pi$
$564$		0			0
$565$	−22.3923	+	22.3923i	−0.942051	+	0.942051i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			12.0000i			0.503066i	0.967849	+	0.251533i	$0.0809347\pi$
$569$			12.0000i			0.503066i	−0.967849	+	0.251533i	$0.919065\pi$
$570$		0			0
$571$	−21.3519	+	21.3519i	−0.893549	+	0.893549i	−0.994855	−	0.101306i	$-0.967698\pi$
$571$	−21.3519	+	21.3519i	−0.893549	+	0.893549i	0.101306	+	0.994855i	$0.467698\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1.79315			−0.0747796
$576$		0			0
$577$	−43.8564			−1.82577			−0.912883	−	0.408221i	$-0.866149\pi$
$577$	−43.8564			−1.82577			−0.912883	+	0.408221i	$0.866149\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−6.00000	+	6.00000i	−0.248922	+	0.248922i
$582$		0			0
$583$		−	51.7439i		−	2.14301i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	6.45189	−	6.45189i	0.266298	−	0.266298i	−0.561308	−	0.827607i	$-0.689702\pi$
$587$	6.45189	−	6.45189i	0.266298	−	0.266298i	0.827607	+	0.561308i	$0.189702\pi$
$588$		0			0
$589$	18.9282	+	18.9282i	0.779923	+	0.779923i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	19.8564			0.815405			0.407702	−	0.913115i	$-0.366330\pi$
$593$	19.8564			0.815405			0.407702	+	0.913115i	$0.366330\pi$
$594$		0			0
$595$	23.1822	+	23.1822i	0.950378	+	0.950378i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	4.41851i		−	0.180535i	−0.995918	−	0.0902676i	$-0.971228\pi$
$599$		−	4.41851i		−	0.180535i	0.995918	−	0.0902676i	$-0.0287722\pi$
$600$		0			0
$601$			40.3923i			1.64764i	0.566854	+	0.823818i	$0.308160\pi$
$601$			40.3923i			1.64764i	−0.566854	+	0.823818i	$0.691840\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−19.7321	−	19.7321i	−0.802222	−	0.802222i
$606$		0			0
$607$	8.28221			0.336165			0.168082	−	0.985773i	$-0.446243\pi$
$607$	8.28221			0.336165			0.168082	+	0.985773i	$0.446243\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	2.62536	+	2.62536i	0.106211	+	0.106211i
$612$		0			0
$613$	15.0526	−	15.0526i	0.607967	−	0.607967i	−0.334447	−	0.942414i	$-0.608550\pi$
$613$	15.0526	−	15.0526i	0.607967	−	0.607967i	0.942414	+	0.334447i	$0.108550\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			11.3205i			0.455746i	0.973691	+	0.227873i	$0.0731772\pi$
$617$			11.3205i			0.455746i	−0.973691	+	0.227873i	$0.926823\pi$
$618$		0			0
$619$	18.8009	−	18.8009i	0.755671	−	0.755671i	−0.219860	−	0.975531i	$-0.570560\pi$
$619$	18.8009	−	18.8009i	0.755671	−	0.755671i	0.975531	+	0.219860i	$0.0705601\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−9.79796			−0.392547
$624$		0			0
$625$	29.0000			1.16000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−12.9282	+	12.9282i	−0.515481	+	0.515481i
$630$		0			0
$631$		−	1.23835i		−	0.0492979i	−0.999696	−	0.0246489i	$-0.992153\pi$
$631$		−	1.23835i		−	0.0492979i	0.999696	−	0.0246489i	$-0.00784679\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	9.79796	−	9.79796i	0.388820	−	0.388820i
$636$		0			0
$637$	2.12436	+	2.12436i	0.0841700	+	0.0841700i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−13.6077			−0.537472			−0.268736	−	0.963214i	$-0.586606\pi$
$641$	−13.6077			−0.537472			−0.268736	+	0.963214i	$0.586606\pi$
$642$		0			0
$643$	−6.93237	−	6.93237i	−0.273386	−	0.273386i	0.557076	−	0.830462i	$-0.311923\pi$
$643$	−6.93237	−	6.93237i	−0.273386	−	0.273386i	−0.830462	+	0.557076i	$0.811923\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	1.79315i		−	0.0704960i	−0.999379	−	0.0352480i	$-0.988778\pi$
$647$		−	1.79315i		−	0.0704960i	0.999379	−	0.0352480i	$-0.0112221\pi$
$648$		0			0
$649$			34.3923i			1.35002i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−15.3397	−	15.3397i	−0.600291	−	0.600291i	0.340099	−	0.940390i	$-0.389539\pi$
$653$	−15.3397	−	15.3397i	−0.600291	−	0.600291i	−0.940390	+	0.340099i	$0.889539\pi$
$654$		0			0
$655$	−0.832204			−0.0325169
$656$		0			0
$657$		0			0
$658$		0			0
$659$	18.0430	+	18.0430i	0.702856	+	0.702856i	0.965023	−	0.262167i	$-0.0844371\pi$
$659$	18.0430	+	18.0430i	0.702856	+	0.702856i	−0.262167	+	0.965023i	$0.584437\pi$
$660$		0			0
$661$	18.2679	−	18.2679i	0.710541	−	0.710541i	−0.256107	−	0.966648i	$-0.582440\pi$
$661$	18.2679	−	18.2679i	0.710541	−	0.710541i	0.966648	+	0.256107i	$0.0824401\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			44.7846i			1.73667i
$666$		0			0
$667$	3.10583	−	3.10583i	0.120258	−	0.120258i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−24.9754			−0.964163
$672$		0			0
$673$	41.3205			1.59279			0.796394	−	0.604778i	$-0.206738\pi$
$673$	41.3205			1.59279			0.796394	+	0.604778i	$0.206738\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	28.2679	−	28.2679i	1.08643	−	1.08643i	0.0905320	−	0.995894i	$-0.471143\pi$
$677$	28.2679	−	28.2679i	1.08643	−	1.08643i	0.995894	−	0.0905320i	$-0.0288567\pi$
$678$		0			0
$679$		−	13.3843i		−	0.513641i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	8.72552	−	8.72552i	0.333873	−	0.333873i	−0.520182	−	0.854055i	$-0.674136\pi$
$683$	8.72552	−	8.72552i	0.333873	−	0.333873i	0.854055	+	0.520182i	$0.174136\pi$
$684$		0			0
$685$	−20.7846	−	20.7846i	−0.794139	−	0.794139i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	4.14359			0.157858
$690$		0			0
$691$	−27.8410	−	27.8410i	−1.05912	−	1.05912i	−0.998139	−	0.0609812i	$-0.980577\pi$
$691$	−27.8410	−	27.8410i	−1.05912	−	1.05912i	−0.0609812	−	0.998139i	$-0.519423\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			24.9754i			0.947370i
$696$		0			0
$697$			24.0000i			0.909065i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	9.33975	+	9.33975i	0.352757	+	0.352757i	0.861135	−	0.508377i	$-0.169754\pi$
$701$	9.33975	+	9.33975i	0.352757	+	0.352757i	−0.508377	+	0.861135i	$0.669754\pi$
$702$		0			0
$703$	−24.9754			−0.941964
$704$		0			0
$705$		0			0
$706$		0			0
$707$	3.10583	+	3.10583i	0.116807	+	0.116807i
$708$		0			0
$709$	−29.0526	+	29.0526i	−1.09109	+	1.09109i	−0.0956796	+	0.995412i	$0.530502\pi$
$709$	−29.0526	+	29.0526i	−1.09109	+	1.09109i	−0.995412	+	0.0956796i	$0.969498\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			10.1436i			0.379881i
$714$		0			0
$715$	3.10583	−	3.10583i	0.116151	−	0.116151i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	12.4233			0.463311			0.231656	−	0.972798i	$-0.425586\pi$
$719$	12.4233			0.463311			0.231656	+	0.972798i	$0.425586\pi$
$720$		0			0
$721$	−36.7846			−1.36993
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1.73205	−	1.73205i	0.0643268	−	0.0643268i
$726$		0			0
$727$		−	23.4596i		−	0.870069i	−0.900414	−	0.435035i	$-0.856736\pi$
$727$		−	23.4596i		−	0.870069i	0.900414	−	0.435035i	$-0.143264\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	5.37945	−	5.37945i	0.198966	−	0.198966i
$732$		0			0
$733$	−14.8038	−	14.8038i	−0.546793	−	0.546793i	0.378719	−	0.925512i	$-0.376365\pi$
$733$	−14.8038	−	14.8038i	−0.546793	−	0.546793i	−0.925512	+	0.378719i	$0.876365\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−29.3205			−1.08003
$738$		0			0
$739$	2.10772	+	2.10772i	0.0775336	+	0.0775336i	0.744810	−	0.667276i	$-0.232540\pi$
$739$	2.10772	+	2.10772i	0.0775336	+	0.0775336i	−0.667276	+	0.744810i	$0.732540\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			15.1774i			0.556805i	0.960464	+	0.278403i	$0.0898050\pi$
$743$			15.1774i			0.556805i	−0.960464	+	0.278403i	$0.910195\pi$
$744$		0			0
$745$		−	47.5692i		−	1.74280i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	50.7846	+	50.7846i	1.85563	+	1.85563i
$750$		0			0
$751$	4.14110			0.151111			0.0755555	−	0.997142i	$-0.475927\pi$
$751$	4.14110			0.151111			0.0755555	+	0.997142i	$0.475927\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−16.4901	−	16.4901i	−0.600136	−	0.600136i
$756$		0			0
$757$	−21.7321	+	21.7321i	−0.789865	+	0.789865i	−0.981472	−	0.191607i	$-0.938630\pi$
$757$	−21.7321	+	21.7321i	−0.789865	+	0.789865i	0.191607	+	0.981472i	$0.438630\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			34.6410i			1.25574i	0.778320	+	0.627868i	$0.216072\pi$
$761$			34.6410i			1.25574i	−0.778320	+	0.627868i	$0.783928\pi$
$762$		0			0
$763$	50.9860	−	50.9860i	1.84582	−	1.84582i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−2.75410			−0.0994447
$768$		0			0
$769$	−2.39230			−0.0862687			−0.0431344	−	0.999069i	$-0.513734\pi$
$769$	−2.39230			−0.0862687			−0.0431344	+	0.999069i	$0.513734\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−14.6603	+	14.6603i	−0.527293	+	0.527293i	−0.919764	−	0.392471i	$-0.871620\pi$
$773$	−14.6603	+	14.6603i	−0.527293	+	0.527293i	0.392471	+	0.919764i	$0.371620\pi$
$774$		0			0
$775$			5.65685i			0.203200i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−23.1822	+	23.1822i	−0.830589	+	0.830589i
$780$		0			0
$781$	6.00000	+	6.00000i	0.214697	+	0.214697i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−24.9282			−0.889726
$786$		0			0
$787$	−25.2156	−	25.2156i	−0.898839	−	0.898839i	0.0964942	−	0.995334i	$-0.469237\pi$
$787$	−25.2156	−	25.2156i	−0.898839	−	0.898839i	−0.995334	+	0.0964942i	$0.969237\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			49.9507i			1.77604i
$792$		0			0
$793$		−	2.00000i		−	0.0710221i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−33.5885	−	33.5885i	−1.18976	−	1.18976i	−0.977132	−	0.212632i	$-0.931797\pi$
$797$	−33.5885	−	33.5885i	−1.18976	−	1.18976i	−0.212632	−	0.977132i	$-0.568203\pi$
$798$		0			0
$799$	33.9411			1.20075
$800$		0			0
$801$		0			0
$802$		0			0
$803$	8.48528	+	8.48528i	0.299439	+	0.299439i
$804$		0			0
$805$	−12.0000	+	12.0000i	−0.422944	+	0.422944i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	17.0718i		−	0.600212i	−0.953906	−	0.300106i	$-0.902978\pi$
$809$		−	17.0718i		−	0.600212i	0.953906	−	0.300106i	$-0.0970221\pi$
$810$		0			0
$811$	−33.2204	+	33.2204i	−1.16653	+	1.16653i	−0.183508	+	0.983018i	$0.558745\pi$
$811$	−33.2204	+	33.2204i	−1.16653	+	1.16653i	−0.983018	+	0.183508i	$0.941255\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−40.9850			−1.43564
$816$		0			0
$817$	10.3923			0.363581
$818$		0			0
$819$		0			0
$820$		0			0
$821$	19.9808	−	19.9808i	0.697333	−	0.697333i	−0.266501	−	0.963835i	$-0.585868\pi$
$821$	19.9808	−	19.9808i	0.697333	−	0.697333i	0.963835	+	0.266501i	$0.0858676\pi$
$822$		0			0
$823$			15.7322i			0.548391i	0.961674	+	0.274195i	$0.0884115\pi$
$823$			15.7322i			0.548391i	−0.961674	+	0.274195i	$0.911589\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−16.7303	+	16.7303i	−0.581770	+	0.581770i	−0.935390	−	0.353619i	$-0.884951\pi$
$827$	−16.7303	+	16.7303i	−0.581770	+	0.581770i	0.353619	+	0.935390i	$0.384951\pi$
$828$		0			0
$829$	−2.12436	−	2.12436i	−0.0737819	−	0.0737819i	0.669253	−	0.743035i	$-0.266614\pi$
$829$	−2.12436	−	2.12436i	−0.0737819	−	0.0737819i	−0.743035	+	0.669253i	$0.766614\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	27.4641			0.951575
$834$		0			0
$835$	−32.4997	−	32.4997i	−1.12470	−	1.12470i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	38.3596i		−	1.32432i	−0.749362	−	0.662161i	$-0.769640\pi$
$839$		−	38.3596i		−	1.32432i	0.749362	−	0.662161i	$-0.230360\pi$
$840$		0			0
$841$		−	23.0000i		−	0.793103i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−22.2679	−	22.2679i	−0.766041	−	0.766041i
$846$		0			0
$847$	−44.0165			−1.51242
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−6.69213	−	6.69213i	−0.229403	−	0.229403i
$852$		0			0
$853$	9.87564	−	9.87564i	0.338136	−	0.338136i	−0.517530	−	0.855665i	$-0.673148\pi$
$853$	9.87564	−	9.87564i	0.338136	−	0.338136i	0.855665	+	0.517530i	$0.173148\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	25.8564i		−	0.883238i	−0.897203	−	0.441619i	$-0.854404\pi$
$857$		−	25.8564i		−	0.883238i	0.897203	−	0.441619i	$-0.145596\pi$
$858$		0			0
$859$	14.4567	−	14.4567i	0.493256	−	0.493256i	−0.416074	−	0.909331i	$-0.636594\pi$
$859$	14.4567	−	14.4567i	0.493256	−	0.493256i	0.909331	+	0.416074i	$0.136594\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−2.62536			−0.0893681			−0.0446841	−	0.999001i	$-0.514228\pi$
$863$	−2.62536			−0.0893681			−0.0446841	+	0.999001i	$0.514228\pi$
$864$		0			0
$865$	−26.7846			−0.910704
$866$		0			0
$867$		0			0
$868$		0			0
$869$	13.8564	−	13.8564i	0.470046	−	0.470046i
$870$		0			0
$871$		−	2.34795i		−	0.0795574i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	26.7685	−	26.7685i	0.904941	−	0.904941i
$876$		0			0
$877$	16.1244	+	16.1244i	0.544481	+	0.544481i	0.924839	−	0.380358i	$-0.124199\pi$
$877$	16.1244	+	16.1244i	0.544481	+	0.544481i	−0.380358	+	0.924839i	$0.624199\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−31.8564			−1.07327			−0.536635	−	0.843815i	$-0.680305\pi$
$881$	−31.8564			−1.07327			−0.536635	+	0.843815i	$0.680305\pi$
$882$		0			0
$883$	40.3930	+	40.3930i	1.35933	+	1.35933i	0.874739	+	0.484594i	$0.161033\pi$
$883$	40.3930	+	40.3930i	1.35933	+	1.35933i	0.484594	+	0.874739i	$0.338967\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			15.1774i			0.509608i	0.966993	+	0.254804i	$0.0820109\pi$
$887$			15.1774i			0.509608i	−0.966993	+	0.254804i	$0.917989\pi$
$888$		0			0
$889$		−	21.8564i		−	0.733040i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	32.7846	+	32.7846i	1.09710	+	1.09710i
$894$		0			0
$895$	−5.37945			−0.179815
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−9.79796	−	9.79796i	−0.326780	−	0.326780i
$900$		0			0
$901$	26.7846	−	26.7846i	0.892325	−	0.892325i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	55.8564i		−	1.85673i
$906$		0			0
$907$	−21.1488	+	21.1488i	−0.702235	+	0.702235i	−0.964890	−	0.262655i	$-0.915402\pi$
$907$	−21.1488	+	21.1488i	−0.702235	+	0.702235i	0.262655	+	0.964890i	$0.415402\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	24.1432			0.799899			0.399949	−	0.916537i	$-0.369028\pi$
$911$	24.1432			0.799899			0.399949	+	0.916537i	$0.369028\pi$
$912$		0			0
$913$	10.3923			0.343935
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−0.928203	+	0.928203i	−0.0306520	+	0.0306520i
$918$		0			0
$919$			18.3576i			0.605560i	0.953060	+	0.302780i	$0.0979148\pi$
$919$			18.3576i			0.605560i	−0.953060	+	0.302780i	$0.902085\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−0.480473	+	0.480473i	−0.0158150	+	0.0158150i
$924$		0			0
$925$	−3.73205	−	3.73205i	−0.122709	−	0.122709i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−3.46410			−0.113653			−0.0568267	−	0.998384i	$-0.518098\pi$
$929$	−3.46410			−0.113653			−0.0568267	+	0.998384i	$0.518098\pi$
$930$		0			0
$931$	26.5283	+	26.5283i	0.869429	+	0.869429i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	40.1528i		−	1.31314i
$936$		0			0
$937$			9.17691i			0.299797i	0.988701	+	0.149898i	$0.0478946\pi$
$937$			9.17691i			0.299797i	−0.988701	+	0.149898i	$0.952105\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	18.8038	+	18.8038i	0.612988	+	0.612988i	0.943723	−	0.330736i	$-0.107297\pi$
$941$	18.8038	+	18.8038i	0.612988	+	0.612988i	−0.330736	+	0.943723i	$0.607297\pi$
$942$		0			0
$943$	−12.4233			−0.404559
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−13.6245	−	13.6245i	−0.442737	−	0.442737i	0.450194	−	0.892931i	$-0.351355\pi$
$947$	−13.6245	−	13.6245i	−0.442737	−	0.442737i	−0.892931	+	0.450194i	$0.851355\pi$
$948$		0			0
$949$	−0.679492	+	0.679492i	−0.0220572	+	0.0220572i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	53.5692i		−	1.73528i	−0.497195	−	0.867639i	$-0.665637\pi$
$953$		−	53.5692i		−	1.73528i	0.497195	−	0.867639i	$-0.334363\pi$
$954$		0			0
$955$	−29.3939	+	29.3939i	−0.951164	+	0.951164i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−46.3644			−1.49719
$960$		0			0
$961$	1.00000			0.0322581
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−31.8564	+	31.8564i	−1.02549	+	1.02549i
$966$		0			0
$967$			27.0459i			0.869738i	0.900494	+	0.434869i	$0.143205\pi$
$967$			27.0459i			0.869738i	−0.900494	+	0.434869i	$0.856795\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−27.8410	+	27.8410i	−0.893459	+	0.893459i	−0.994847	−	0.101388i	$-0.967672\pi$
$971$	−27.8410	+	27.8410i	−0.893459	+	0.893459i	0.101388	+	0.994847i	$0.467672\pi$
$972$		0			0
$973$	27.8564	+	27.8564i	0.893035	+	0.893035i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	44.5359			1.42483			0.712415	−	0.701759i	$-0.247601\pi$
$977$	44.5359			1.42483			0.712415	+	0.701759i	$0.247601\pi$
$978$		0			0
$979$	8.48528	+	8.48528i	0.271191	+	0.271191i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	52.7048i		−	1.68102i	−0.541793	−	0.840512i	$-0.682255\pi$
$983$		−	52.7048i		−	1.68102i	0.541793	−	0.840512i	$-0.317745\pi$
$984$		0			0
$985$		−	6.00000i		−	0.191176i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	2.78461	+	2.78461i	0.0885454	+	0.0885454i
$990$		0			0
$991$	61.8193			1.96375			0.981877	−	0.189521i	$-0.0606936\pi$
$991$	61.8193			1.96375			0.981877	+	0.189521i	$0.0606936\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	27.2490	+	27.2490i	0.863851	+	0.863851i
$996$		0			0
$997$	−15.1962	+	15.1962i	−0.481267	+	0.481267i	−0.905536	−	0.424269i	$-0.860531\pi$
$997$	−15.1962	+	15.1962i	−0.481267	+	0.481267i	0.424269	+	0.905536i	$0.360531\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2304.2.k.f.577.3		8
3.2	odd	2		256.2.e.a.65.2	✓	8
4.3	odd	2	inner	2304.2.k.f.577.4		8
8.3	odd	2		2304.2.k.k.577.2		8
8.5	even	2		2304.2.k.k.577.1		8
12.11	even	2		256.2.e.a.65.3	yes	8
16.3	odd	4		2304.2.k.k.1729.1		8
16.5	even	4	inner	2304.2.k.f.1729.4		8
16.11	odd	4	inner	2304.2.k.f.1729.3		8
16.13	even	4		2304.2.k.k.1729.2		8
24.5	odd	2		256.2.e.b.65.3	yes	8
24.11	even	2		256.2.e.b.65.2	yes	8
32.5	even	8		9216.2.a.bk.1.4		4
32.11	odd	8		9216.2.a.bk.1.1		4
32.21	even	8		9216.2.a.bb.1.2		4
32.27	odd	8		9216.2.a.bb.1.3		4
48.5	odd	4		256.2.e.a.193.2	yes	8
48.11	even	4		256.2.e.a.193.3	yes	8
48.29	odd	4		256.2.e.b.193.3	yes	8
48.35	even	4		256.2.e.b.193.2	yes	8
96.5	odd	8		1024.2.a.g.1.3		4
96.11	even	8		1024.2.a.g.1.4		4
96.29	odd	8		1024.2.b.h.513.3		8
96.35	even	8		1024.2.b.h.513.5		8
96.53	odd	8		1024.2.a.j.1.2		4
96.59	even	8		1024.2.a.j.1.1		4
96.77	odd	8		1024.2.b.h.513.6		8
96.83	even	8		1024.2.b.h.513.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
256.2.e.a.65.2	✓	8	3.2	odd	2
256.2.e.a.65.3	yes	8	12.11	even	2
256.2.e.a.193.2	yes	8	48.5	odd	4
256.2.e.a.193.3	yes	8	48.11	even	4
256.2.e.b.65.2	yes	8	24.11	even	2
256.2.e.b.65.3	yes	8	24.5	odd	2
256.2.e.b.193.2	yes	8	48.35	even	4
256.2.e.b.193.3	yes	8	48.29	odd	4
1024.2.a.g.1.3		4	96.5	odd	8
1024.2.a.g.1.4		4	96.11	even	8
1024.2.a.j.1.1		4	96.59	even	8
1024.2.a.j.1.2		4	96.53	odd	8
1024.2.b.h.513.3		8	96.29	odd	8
1024.2.b.h.513.4		8	96.83	even	8
1024.2.b.h.513.5		8	96.35	even	8
1024.2.b.h.513.6		8	96.77	odd	8
2304.2.k.f.577.3		8	1.1	even	1	trivial
2304.2.k.f.577.4		8	4.3	odd	2	inner
2304.2.k.f.1729.3		8	16.11	odd	4	inner
2304.2.k.f.1729.4		8	16.5	even	4	inner
2304.2.k.k.577.1		8	8.5	even	2
2304.2.k.k.577.2		8	8.3	odd	2
2304.2.k.k.1729.1		8	16.3	odd	4
2304.2.k.k.1729.2		8	16.13	even	4
9216.2.a.bb.1.2		4	32.21	even	8
9216.2.a.bb.1.3		4	32.27	odd	8
9216.2.a.bk.1.1		4	32.11	odd	8
9216.2.a.bk.1.4		4	32.5	even	8

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:	\( N \)	\(=\)	\( 2304 = 2^{8} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2304.k (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.73205 - 1.73205i) q^{5} -3.86370i q^{7} +O(q^{10})\)	\(q+(1.73205 - 1.73205i) q^{5} -3.86370i q^{7} +(-3.34607 + 3.34607i) q^{11} +(-0.267949 - 0.267949i) q^{13} -3.46410 q^{17} +(-3.34607 - 3.34607i) q^{19} -1.79315i q^{23} -1.00000i q^{25} +(1.73205 + 1.73205i) q^{29} -5.65685 q^{31} +(-6.69213 - 6.69213i) q^{35} +(3.73205 - 3.73205i) q^{37} -6.92820i q^{41} +(-1.55291 + 1.55291i) q^{43} -9.79796 q^{47} -7.92820 q^{49} +(-7.73205 + 7.73205i) q^{53} +11.5911i q^{55} +(5.13922 - 5.13922i) q^{59} +(3.73205 + 3.73205i) q^{61} -0.928203 q^{65} +(4.38134 + 4.38134i) q^{67} -1.79315i q^{71} -2.53590i q^{73} +(12.9282 + 12.9282i) q^{77} -4.14110 q^{79} +(-1.55291 - 1.55291i) q^{83} +(-6.00000 + 6.00000i) q^{85} -2.53590i q^{89} +(-1.03528 + 1.03528i) q^{91} -11.5911 q^{95} +3.46410 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 16 q^{13} + 16 q^{37} - 8 q^{49} - 48 q^{53} + 16 q^{61} + 48 q^{65} + 48 q^{77} - 48 q^{85}+O(q^{100}) \) 8 * q - 16 * q^13 + 16 * q^37 - 8 * q^49 - 48 * q^53 + 16 * q^61 + 48 * q^65 + 48 * q^77 - 48 * q^85

Introduction

L-functions

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