LMFDB - Embedded newform 256.2.e.b.65.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [256,2,Mod(65,256)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(256, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("256.65");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 256 = 2^{8} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	256.e (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:	$2.04417029174$
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:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			65.2
Root			$0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	256.65
Dual form			256.2.e.b.193.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.517638 - 0.517638i) q^{3} +(1.73205 - 1.73205i) q^{5} +3.86370i q^{7} -2.46410i q^{9} +O(q^{10})$	\(q+(-0.517638 - 0.517638i) q^{3} +(1.73205 - 1.73205i) q^{5} +3.86370i q^{7} -2.46410i q^{9} +(3.34607 - 3.34607i) q^{11} +(0.267949 + 0.267949i) q^{13} -1.79315 q^{15} +3.46410 q^{17} +(-3.34607 - 3.34607i) q^{19} +(2.00000 - 2.00000i) q^{21} -1.79315i q^{23} -1.00000i q^{25} +(-2.82843 + 2.82843i) q^{27} +(1.73205 + 1.73205i) q^{29} +5.65685 q^{31} -3.46410 q^{33} +(6.69213 + 6.69213i) q^{35} +(-3.73205 + 3.73205i) q^{37} -0.277401i q^{39} +6.92820i q^{41} +(-1.55291 + 1.55291i) q^{43} +(-4.26795 - 4.26795i) q^{45} -9.79796 q^{47} -7.92820 q^{49} +(-1.79315 - 1.79315i) q^{51} +(-7.73205 + 7.73205i) q^{53} -11.5911i q^{55} +3.46410i q^{57} +(-5.13922 + 5.13922i) q^{59} +(-3.73205 - 3.73205i) q^{61} +9.52056 q^{63} +0.928203 q^{65} +(4.38134 + 4.38134i) q^{67} +(-0.928203 + 0.928203i) q^{69} -1.79315i q^{71} -2.53590i q^{73} +(-0.517638 + 0.517638i) q^{75} +(12.9282 + 12.9282i) q^{77} +4.14110 q^{79} -4.46410 q^{81} +(1.55291 + 1.55291i) q^{83} +(6.00000 - 6.00000i) q^{85} -1.79315i q^{87} +2.53590i q^{89} +(-1.03528 + 1.03528i) q^{91} +(-2.92820 - 2.92820i) q^{93} -11.5911 q^{95} +3.46410 q^{97} +(-8.24504 - 8.24504i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 16 q^{13} + 16 q^{21} - 16 q^{37} - 48 q^{45} - 8 q^{49} - 48 q^{53} - 16 q^{61} - 48 q^{65} + 48 q^{69} + 48 q^{77} - 8 q^{81} + 48 q^{85} + 32 q^{93}+O(q^{100}) $ 8 * q + 16 * q^13 + 16 * q^21 - 16 * q^37 - 48 * q^45 - 8 * q^49 - 48 * q^53 - 16 * q^61 - 48 * q^65 + 48 * q^69 + 48 * q^77 - 8 * q^81 + 48 * q^85 + 32 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/256\mathbb{Z}\right)^\times$.

$n$	$5$	$255$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−0.517638	−	0.517638i	−0.298858	−	0.298858i	0.541708	−	0.840567i	$-0.317778\pi$
$3$	−0.517638	−	0.517638i	−0.298858	−	0.298858i	−0.840567	+	0.541708i	$0.817778\pi$
$4$		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	$0.260560\pi$
$7$			3.86370i			1.46034i	−0.683264	+	0.730171i	$0.739440\pi$
$8$		0			0
$9$		−	2.46410i		−	0.821367i
$10$		0			0
$11$	3.34607	−	3.34607i	1.00888	−	1.00888i	0.00891637	−	0.999960i	$-0.497162\pi$
$11$	3.34607	−	3.34607i	1.00888	−	1.00888i	0.999960	−	0.00891637i	$-0.00283821\pi$
$12$		0			0
$13$	0.267949	+	0.267949i	0.0743157	+	0.0743157i	0.743288	−	0.668972i	$-0.233265\pi$
$13$	0.267949	+	0.267949i	0.0743157	+	0.0743157i	−0.668972	+	0.743288i	$0.733265\pi$
$14$		0			0
$15$	−1.79315			−0.462990
$16$		0			0
$17$	3.46410			0.840168			0.420084	−	0.907485i	$-0.362001\pi$
$17$	3.46410			0.840168			0.420084	+	0.907485i	$0.362001\pi$
$18$		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
$21$	2.00000	−	2.00000i	0.436436	−	0.436436i
$22$		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
$27$	−2.82843	+	2.82843i	−0.544331	+	0.544331i
$28$		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	$-0.330385\pi$
$31$	5.65685			1.01600			0.508001	+	0.861357i	$0.330385\pi$
$32$		0			0
$33$	−3.46410			−0.603023
$34$		0			0
$35$	6.69213	+	6.69213i	1.13118	+	1.13118i
$36$		0			0
$37$	−3.73205	+	3.73205i	−0.613545	+	0.613545i	−0.943868	−	0.330323i	$-0.892842\pi$
$37$	−3.73205	+	3.73205i	−0.613545	+	0.613545i	0.330323	+	0.943868i	$0.392842\pi$
$38$		0			0
$39$		−	0.277401i		−	0.0444198i
$40$		0			0
$41$			6.92820i			1.08200i	0.841021	+	0.541002i	$0.181955\pi$
$41$			6.92820i			1.08200i	−0.841021	+	0.541002i	$0.818045\pi$
$42$		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
$45$	−4.26795	−	4.26795i	−0.636228	−	0.636228i
$46$		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
$51$	−1.79315	−	1.79315i	−0.251091	−	0.251091i
$52$		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
$57$			3.46410i			0.458831i
$58$		0			0
$59$	−5.13922	+	5.13922i	−0.669069	+	0.669069i	−0.957501	−	0.288432i	$-0.906866\pi$
$59$	−5.13922	+	5.13922i	−0.669069	+	0.669069i	0.288432	+	0.957501i	$0.406866\pi$
$60$		0			0
$61$	−3.73205	−	3.73205i	−0.477840	−	0.477840i	0.426600	−	0.904440i	$-0.359711\pi$
$61$	−3.73205	−	3.73205i	−0.477840	−	0.477840i	−0.904440	+	0.426600i	$0.859711\pi$
$62$		0			0
$63$	9.52056			1.19948
$64$		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
$69$	−0.928203	+	0.928203i	−0.111743	+	0.111743i
$70$		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
$75$	−0.517638	+	0.517638i	−0.0597717	+	0.0597717i
$76$		0			0
$77$	12.9282	+	12.9282i	1.47331	+	1.47331i
$78$		0			0
$79$	4.14110			0.465911			0.232955	−	0.972487i	$-0.425160\pi$
$79$	4.14110			0.465911			0.232955	+	0.972487i	$0.425160\pi$
$80$		0			0
$81$	−4.46410			−0.496011
$82$		0			0
$83$	1.55291	+	1.55291i	0.170454	+	0.170454i	0.787179	−	0.616725i	$-0.211541\pi$
$83$	1.55291	+	1.55291i	0.170454	+	0.170454i	−0.616725	+	0.787179i	$0.711541\pi$
$84$		0			0
$85$	6.00000	−	6.00000i	0.650791	−	0.650791i
$86$		0			0
$87$		−	1.79315i		−	0.192246i
$88$		0			0
$89$			2.53590i			0.268805i	0.990927	+	0.134402i	$0.0429115\pi$
$89$			2.53590i			0.268805i	−0.990927	+	0.134402i	$0.957089\pi$
$90$		0			0
$91$	−1.03528	+	1.03528i	−0.108526	+	0.108526i
$92$		0			0
$93$	−2.92820	−	2.92820i	−0.303641	−	0.303641i
$94$		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
$99$	−8.24504	−	8.24504i	−0.828658	−	0.828658i
$100$		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	$0.155402\pi$
$103$			9.52056i			0.938088i	−0.883175	+	0.469044i	$0.844598\pi$
$104$		0			0
$105$		−	6.92820i		−	0.676123i
$106$		0			0
$107$	13.1440	−	13.1440i	1.27068	−	1.27068i	0.324949	−	0.945731i	$-0.394653\pi$
$107$	13.1440	−	13.1440i	1.27068	−	1.27068i	0.945731	−	0.324949i	$-0.105347\pi$
$108$		0			0
$109$	−13.1962	−	13.1962i	−1.26396	−	1.26396i	−0.949156	−	0.314806i	$-0.898060\pi$
$109$	−13.1962	−	13.1962i	−1.26396	−	1.26396i	−0.314806	−	0.949156i	$-0.601940\pi$
$110$		0			0
$111$	3.86370			0.366726
$112$		0			0
$113$	12.9282			1.21618			0.608092	−	0.793867i	$-0.291935\pi$
$113$	12.9282			1.21618			0.608092	+	0.793867i	$0.291935\pi$
$114$		0			0
$115$	−3.10583	−	3.10583i	−0.289620	−	0.289620i
$116$		0			0
$117$	0.660254	−	0.660254i	0.0610405	−	0.0610405i
$118$		0			0
$119$			13.3843i			1.22693i
$120$		0			0
$121$		−	11.3923i		−	1.03566i
$122$		0			0
$123$	3.58630	−	3.58630i	0.323366	−	0.323366i
$124$		0			0
$125$	6.92820	+	6.92820i	0.619677	+	0.619677i
$126$		0			0
$127$	−5.65685			−0.501965			−0.250982	−	0.967992i	$-0.580754\pi$
$127$	−5.65685			−0.501965			−0.250982	+	0.967992i	$0.580754\pi$
$128$		0			0
$129$	1.60770			0.141550
$130$		0			0
$131$	0.240237	+	0.240237i	0.0209896	+	0.0209896i	0.717524	−	0.696534i	$-0.245276\pi$
$131$	0.240237	+	0.240237i	0.0209896	+	0.0209896i	−0.696534	+	0.717524i	$0.745276\pi$
$132$		0			0
$133$	12.9282	−	12.9282i	1.12102	−	1.12102i
$134$		0			0
$135$			9.79796i			0.843274i
$136$		0			0
$137$			12.0000i			1.02523i	0.858619	+	0.512615i	$0.171323\pi$
$137$			12.0000i			1.02523i	−0.858619	+	0.512615i	$0.828677\pi$
$138$		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
$141$	5.07180	+	5.07180i	0.427122	+	0.427122i
$142$		0			0
$143$	1.79315			0.149951
$144$		0			0
$145$	6.00000			0.498273
$146$		0			0
$147$	4.10394	+	4.10394i	0.338487	+	0.338487i
$148$		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	$0.126622\pi$
$151$			9.52056i			0.774772i	−0.921918	+	0.387386i	$0.873378\pi$
$152$		0			0
$153$		−	8.53590i		−	0.690086i
$154$		0			0
$155$	9.79796	−	9.79796i	0.786991	−	0.786991i
$156$		0			0
$157$	7.19615	+	7.19615i	0.574315	+	0.574315i	0.933331	−	0.359016i	$-0.116888\pi$
$157$	7.19615	+	7.19615i	0.574315	+	0.574315i	−0.359016	+	0.933331i	$0.616888\pi$
$158$		0			0
$159$	8.00481			0.634823
$160$		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
$165$	−6.00000	+	6.00000i	−0.467099	+	0.467099i
$166$		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
$171$	−8.24504	+	8.24504i	−0.630514	+	0.630514i
$172$		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
$177$	5.32051			0.399914
$178$		0			0
$179$	1.55291	+	1.55291i	0.116070	+	0.116070i	0.762756	−	0.646686i	$-0.223846\pi$
$179$	1.55291	+	1.55291i	0.116070	+	0.116070i	−0.646686	+	0.762756i	$0.723846\pi$
$180$		0			0
$181$	−16.1244	+	16.1244i	−1.19851	+	1.19851i	−0.223902	+	0.974612i	$0.571880\pi$
$181$	−16.1244	+	16.1244i	−1.19851	+	1.19851i	−0.974612	+	0.223902i	$0.928120\pi$
$182$		0			0
$183$			3.86370i			0.285613i
$184$		0			0
$185$			12.9282i			0.950500i
$186$		0			0
$187$	11.5911	−	11.5911i	0.847626	−	0.847626i
$188$		0			0
$189$	−10.9282	−	10.9282i	−0.794910	−	0.794910i
$190$		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
$195$	−0.480473	−	0.480473i	−0.0344074	−	0.0344074i
$196$		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	$-0.811717\pi$
$199$		−	15.7322i		−	1.11523i	0.830101	−	0.557614i	$-0.188283\pi$
$200$		0			0
$201$		−	4.53590i		−	0.319938i
$202$		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
$207$	−4.41851			−0.307107
$208$		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
$213$	−0.928203	+	0.928203i	−0.0635994	+	0.0635994i
$214$		0			0
$215$			5.37945i			0.366876i
$216$		0			0
$217$			21.8564i			1.48371i
$218$		0			0
$219$	−1.31268	+	1.31268i	−0.0887025	+	0.0887025i
$220$		0			0
$221$	0.928203	+	0.928203i	0.0624377	+	0.0624377i
$222$		0			0
$223$	5.65685			0.378811			0.189405	−	0.981899i	$-0.439344\pi$
$223$	5.65685			0.378811			0.189405	+	0.981899i	$0.439344\pi$
$224$		0			0
$225$	−2.46410			−0.164273
$226$		0			0
$227$	14.9372	+	14.9372i	0.991415	+	0.991415i	0.999963	−	0.00854833i	$-0.00272105\pi$
$227$	14.9372	+	14.9372i	0.991415	+	0.991415i	−0.00854833	+	0.999963i	$0.502721\pi$
$228$		0			0
$229$	−9.19615	+	9.19615i	−0.607699	+	0.607699i	−0.942344	−	0.334645i	$-0.891383\pi$
$229$	−9.19615	+	9.19615i	−0.607699	+	0.607699i	0.334645	+	0.942344i	$0.391383\pi$
$230$		0			0
$231$		−	13.3843i		−	0.880620i
$232$		0			0
$233$		−	2.53590i		−	0.166132i	−0.996544	−	0.0830661i	$-0.973529\pi$
$233$		−	2.53590i		−	0.166132i	0.996544	−	0.0830661i	$-0.0264713\pi$
$234$		0			0
$235$	−16.9706	+	16.9706i	−1.10704	+	1.10704i
$236$		0			0
$237$	−2.14359	−	2.14359i	−0.139241	−	0.139241i
$238$		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
$243$	10.7961	+	10.7961i	0.692568	+	0.692568i
$244$		0			0
$245$	−13.7321	+	13.7321i	−0.877309	+	0.877309i
$246$		0			0
$247$		−	1.79315i		−	0.114095i
$248$		0			0
$249$		−	1.60770i		−	0.101884i
$250$		0			0
$251$	−14.9372	+	14.9372i	−0.942826	+	0.942826i	−0.998452	−	0.0556256i	$-0.982285\pi$
$251$	−14.9372	+	14.9372i	−0.942826	+	0.942826i	0.0556256	+	0.998452i	$0.482285\pi$
$252$		0			0
$253$	−6.00000	−	6.00000i	−0.377217	−	0.377217i
$254$		0			0
$255$	−6.21166			−0.388989
$256$		0			0
$257$	12.9282			0.806439			0.403220	−	0.915103i	$-0.367891\pi$
$257$	12.9282			0.806439			0.403220	+	0.915103i	$0.367891\pi$
$258$		0			0
$259$	−14.4195	−	14.4195i	−0.895986	−	0.895986i
$260$		0			0
$261$	4.26795	−	4.26795i	0.264179	−	0.264179i
$262$		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
$267$	1.31268	−	1.31268i	0.0803346	−	0.0803346i
$268$		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	$-0.485341\pi$
$271$	1.51575			0.0920752			0.0460376	+	0.998940i	$0.485341\pi$
$272$		0			0
$273$	1.07180			0.0648681
$274$		0			0
$275$	−3.34607	−	3.34607i	−0.201775	−	0.201775i
$276$		0			0
$277$	3.19615	−	3.19615i	0.192038	−	0.192038i	−0.604538	−	0.796576i	$-0.706642\pi$
$277$	3.19615	−	3.19615i	0.192038	−	0.192038i	0.796576	+	0.604538i	$0.206642\pi$
$278$		0			0
$279$		−	13.9391i		−	0.834510i
$280$		0			0
$281$		−	16.3923i		−	0.977883i	−0.872317	−	0.488941i	$-0.837383\pi$
$281$		−	16.3923i		−	0.977883i	0.872317	−	0.488941i	$-0.162617\pi$
$282$		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
$285$	6.00000	+	6.00000i	0.355409	+	0.355409i
$286$		0			0
$287$	−26.7685			−1.58010
$288$		0			0
$289$	−5.00000			−0.294118
$290$		0			0
$291$	−1.79315	−	1.79315i	−0.105116	−	0.105116i
$292$		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
$297$			18.9282i			1.09833i
$298$		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
$303$	0.832204			0.0478089
$304$		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
$309$	4.92820	−	4.92820i	0.280356	−	0.280356i
$310$		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
$315$	16.4901	−	16.4901i	0.929111	−	0.929111i
$316$		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
$321$	−13.6077			−0.759507
$322$		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
$327$			13.6617i			0.755492i
$328$		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
$333$	9.19615	+	9.19615i	0.503946	+	0.503946i
$334$		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
$339$	−6.69213	−	6.69213i	−0.363467	−	0.363467i
$340$		0			0
$341$	18.9282	−	18.9282i	1.02502	−	1.02502i
$342$		0			0
$343$		−	3.58630i		−	0.193642i
$344$		0			0
$345$			3.21539i			0.173111i
$346$		0			0
$347$	−5.13922	+	5.13922i	−0.275888	+	0.275888i	−0.831465	−	0.555577i	$-0.812497\pi$
$347$	−5.13922	+	5.13922i	−0.275888	+	0.275888i	0.555577	+	0.831465i	$0.312497\pi$
$348$		0			0
$349$	−4.12436	−	4.12436i	−0.220772	−	0.220772i	0.588052	−	0.808823i	$-0.299895\pi$
$349$	−4.12436	−	4.12436i	−0.220772	−	0.220772i	−0.808823	+	0.588052i	$0.799895\pi$
$350$		0			0
$351$	−1.51575			−0.0809047
$352$		0			0
$353$	−0.928203			−0.0494033			−0.0247016	−	0.999695i	$-0.507864\pi$
$353$	−0.928203			−0.0494033			−0.0247016	+	0.999695i	$0.507864\pi$
$354$		0			0
$355$	−3.10583	−	3.10583i	−0.164840	−	0.164840i
$356$		0			0
$357$	6.92820	−	6.92820i	0.366679	−	0.366679i
$358$		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
$363$	−5.89709	+	5.89709i	−0.309517	+	0.309517i
$364$		0			0
$365$	−4.39230	−	4.39230i	−0.229904	−	0.229904i
$366$		0			0
$367$	−32.4254			−1.69259			−0.846295	−	0.532714i	$-0.821172\pi$
$367$	−32.4254			−1.69259			−0.846295	+	0.532714i	$0.821172\pi$
$368$		0			0
$369$	17.0718			0.888722
$370$		0			0
$371$	−29.8744	−	29.8744i	−1.55100	−	1.55100i
$372$		0			0
$373$	14.1244	−	14.1244i	0.731331	−	0.731331i	−0.239552	−	0.970884i	$-0.577001\pi$
$373$	14.1244	−	14.1244i	0.731331	−	0.731331i	0.970884	+	0.239552i	$0.0770006\pi$
$374$		0			0
$375$		−	7.17260i		−	0.370392i
$376$		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
$381$	2.92820	+	2.92820i	0.150016	+	0.150016i
$382$		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
$387$	3.82654	+	3.82654i	0.194514	+	0.194514i
$388$		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
$393$		−	0.248711i		−	0.0125458i
$394$		0			0
$395$	7.17260	−	7.17260i	0.360893	−	0.360893i
$396$		0			0
$397$	19.5885	+	19.5885i	0.983116	+	0.983116i	0.999860	−	0.0167433i	$-0.00532982\pi$
$397$	19.5885	+	19.5885i	0.983116	+	0.983116i	−0.0167433	+	0.999860i	$0.505330\pi$
$398$		0			0
$399$	−13.3843			−0.670051
$400$		0			0
$401$	13.6077			0.679536			0.339768	−	0.940509i	$-0.389651\pi$
$401$	13.6077			0.679536			0.339768	+	0.940509i	$0.389651\pi$
$402$		0			0
$403$	1.51575	+	1.51575i	0.0755049	+	0.0755049i
$404$		0			0
$405$	−7.73205	+	7.73205i	−0.384209	+	0.384209i
$406$		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
$411$	6.21166	−	6.21166i	0.306398	−	0.306398i
$412$		0			0
$413$	−19.8564	−	19.8564i	−0.977070	−	0.977070i
$414$		0			0
$415$	5.37945			0.264067
$416$		0			0
$417$	7.46410			0.365519
$418$		0			0
$419$	10.0382	+	10.0382i	0.490398	+	0.490398i	0.908432	−	0.418034i	$-0.137281\pi$
$419$	10.0382	+	10.0382i	0.490398	+	0.490398i	−0.418034	+	0.908432i	$0.637281\pi$
$420$		0			0
$421$	18.5167	−	18.5167i	0.902447	−	0.902447i	−0.0932005	−	0.995647i	$-0.529710\pi$
$421$	18.5167	−	18.5167i	0.902447	−	0.902447i	0.995647	+	0.0932005i	$0.0297098\pi$
$422$		0			0
$423$			24.1432i			1.17388i
$424$		0			0
$425$		−	3.46410i		−	0.168034i
$426$		0			0
$427$	14.4195	−	14.4195i	0.697810	−	0.697810i
$428$		0			0
$429$	−0.928203	−	0.928203i	−0.0448141	−	0.0448141i
$430$		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
$435$	−3.10583	−	3.10583i	−0.148913	−	0.148913i
$436$		0			0
$437$	−6.00000	+	6.00000i	−0.287019	+	0.287019i
$438$		0			0
$439$			12.1459i			0.579693i	0.957073	+	0.289846i	$0.0936042\pi$
$439$			12.1459i			0.579693i	−0.957073	+	0.289846i	$0.906396\pi$
$440$		0			0
$441$			19.5359i			0.930281i
$442$		0			0
$443$	−10.0382	+	10.0382i	−0.476929	+	0.476929i	−0.904148	−	0.427219i	$-0.859493\pi$
$443$	−10.0382	+	10.0382i	−0.476929	+	0.476929i	0.427219	+	0.904148i	$0.359493\pi$
$444$		0			0
$445$	4.39230	+	4.39230i	0.208215	+	0.208215i
$446$		0			0
$447$	−14.2165			−0.672416
$448$		0			0
$449$	−24.2487			−1.14437			−0.572184	−	0.820125i	$-0.693904\pi$
$449$	−24.2487			−1.14437			−0.572184	+	0.820125i	$0.693904\pi$
$450$		0			0
$451$	23.1822	+	23.1822i	1.09161	+	1.09161i
$452$		0			0
$453$	4.92820	−	4.92820i	0.231547	−	0.231547i
$454$		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
$459$	−9.79796	+	9.79796i	−0.457330	+	0.457330i
$460$		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	$-0.845784\pi$
$463$	−38.0822			−1.76983			−0.884916	+	0.465751i	$0.845784\pi$
$464$		0			0
$465$	−10.1436			−0.470398
$466$		0			0
$467$	−20.3166	−	20.3166i	−0.940141	−	0.940141i	0.0581655	−	0.998307i	$-0.481475\pi$
$467$	−20.3166	−	20.3166i	−0.940141	−	0.940141i	−0.998307	+	0.0581655i	$0.981475\pi$
$468$		0			0
$469$	−16.9282	+	16.9282i	−0.781672	+	0.781672i
$470$		0			0
$471$		−	7.45001i		−	0.343278i
$472$		0			0
$473$			10.3923i			0.477839i
$474$		0			0
$475$	−3.34607	+	3.34607i	−0.153528	+	0.153528i
$476$		0			0
$477$	19.0526	+	19.0526i	0.872357	+	0.872357i
$478$		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
$483$	−3.58630	−	3.58630i	−0.163182	−	0.163182i
$484$		0			0
$485$	6.00000	−	6.00000i	0.272446	−	0.272446i
$486$		0			0
$487$		−	27.0459i		−	1.22557i	−0.790251	−	0.612784i	$-0.790050\pi$
$487$		−	27.0459i		−	1.22557i	0.790251	−	0.612784i	$-0.209950\pi$
$488$		0			0
$489$			12.2487i			0.553906i
$490$		0			0
$491$	15.4176	−	15.4176i	0.695789	−	0.695789i	−0.267711	−	0.963499i	$-0.586267\pi$
$491$	15.4176	−	15.4176i	0.695789	−	0.695789i	0.963499	+	0.267711i	$0.0862671\pi$
$492$		0			0
$493$	6.00000	+	6.00000i	0.270226	+	0.270226i
$494$		0			0
$495$	−28.5617			−1.28375
$496$		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
$501$	−9.71281	+	9.71281i	−0.433936	+	0.433936i
$502$		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
$507$	−6.65497	+	6.65497i	−0.295557	+	0.295557i
$508$		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
$513$	18.9282			0.835701
$514$		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
$519$			8.00481i			0.351372i
$520$		0			0
$521$			12.0000i			0.525730i	0.964833	+	0.262865i	$0.0846673\pi$
$521$			12.0000i			0.525730i	−0.964833	+	0.262865i	$0.915333\pi$
$522$		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
$525$	−2.00000	−	2.00000i	−0.0872872	−	0.0872872i
$526$		0			0
$527$	19.5959			0.853612
$528$		0			0
$529$	19.7846			0.860200
$530$		0			0
$531$	12.6636	+	12.6636i	0.549551	+	0.549551i
$532$		0			0
$533$	−1.85641	+	1.85641i	−0.0804099	+	0.0804099i
$534$		0			0
$535$		−	45.5322i		−	1.96853i
$536$		0			0
$537$		−	1.60770i		−	0.0693772i
$538$		0			0
$539$	−26.5283	+	26.5283i	−1.14265	+	1.14265i
$540$		0			0
$541$	10.1244	+	10.1244i	0.435280	+	0.435280i	0.890420	−	0.455140i	$-0.150411\pi$
$541$	10.1244	+	10.1244i	0.435280	+	0.435280i	−0.455140	+	0.890420i	$0.650411\pi$
$542$		0			0
$543$	16.6932			0.716372
$544$		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
$549$	−9.19615	+	9.19615i	−0.392482	+	0.392482i
$550$		0			0
$551$		−	11.5911i		−	0.493798i
$552$		0			0
$553$			16.0000i			0.680389i
$554$		0			0
$555$	6.69213	−	6.69213i	0.284065	−	0.284065i
$556$		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
$561$	−12.0000			−0.506640
$562$		0			0
$563$	−10.5187	−	10.5187i	−0.443309	−	0.443309i	0.449813	−	0.893123i	$-0.351490\pi$
$563$	−10.5187	−	10.5187i	−0.443309	−	0.443309i	−0.893123	+	0.449813i	$0.851490\pi$
$564$		0			0
$565$	22.3923	−	22.3923i	0.942051	−	0.942051i
$566$		0			0
$567$		−	17.2480i		−	0.724346i
$568$		0			0
$569$		−	12.0000i		−	0.503066i	−0.967849	−	0.251533i	$-0.919065\pi$
$569$		−	12.0000i		−	0.503066i	0.967849	−	0.251533i	$-0.0809347\pi$
$570$		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
$573$	8.78461	+	8.78461i	0.366982	+	0.366982i
$574$		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
$579$	9.52056	+	9.52056i	0.395661	+	0.395661i
$580$		0			0
$581$	−6.00000	+	6.00000i	−0.248922	+	0.248922i
$582$		0			0
$583$			51.7439i			2.14301i
$584$		0			0
$585$		−	2.28719i		−	0.0945635i
$586$		0			0
$587$	−6.45189	+	6.45189i	−0.266298	+	0.266298i	−0.827607	−	0.561308i	$-0.810298\pi$
$587$	−6.45189	+	6.45189i	−0.266298	+	0.266298i	0.561308	+	0.827607i	$0.310298\pi$
$588$		0			0
$589$	−18.9282	−	18.9282i	−0.779923	−	0.779923i
$590$		0			0
$591$	−1.79315			−0.0737604
$592$		0			0
$593$	−19.8564			−0.815405			−0.407702	−	0.913115i	$-0.633670\pi$
$593$	−19.8564			−0.815405			−0.407702	+	0.913115i	$0.633670\pi$
$594$		0			0
$595$	23.1822	+	23.1822i	0.950378	+	0.950378i
$596$		0			0
$597$	−8.14359	+	8.14359i	−0.333295	+	0.333295i
$598$		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
$603$	10.7961	−	10.7961i	0.439650	−	0.439650i
$604$		0			0
$605$	−19.7321	−	19.7321i	−0.802222	−	0.802222i
$606$		0			0
$607$	−8.28221			−0.336165			−0.168082	−	0.985773i	$-0.553757\pi$
$607$	−8.28221			−0.336165			−0.168082	+	0.985773i	$0.553757\pi$
$608$		0			0
$609$	6.92820			0.280745
$610$		0			0
$611$	−2.62536	−	2.62536i	−0.106211	−	0.106211i
$612$		0			0
$613$	−15.0526	+	15.0526i	−0.607967	+	0.607967i	−0.942414	−	0.334447i	$-0.891450\pi$
$613$	−15.0526	+	15.0526i	−0.607967	+	0.607967i	0.334447	+	0.942414i	$0.391450\pi$
$614$		0			0
$615$		−	12.4233i		−	0.500956i
$616$		0			0
$617$		−	11.3205i		−	0.455746i	−0.973691	−	0.227873i	$-0.926823\pi$
$617$		−	11.3205i		−	0.455746i	0.973691	−	0.227873i	$-0.0731772\pi$
$618$		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
$621$	5.07180	+	5.07180i	0.203524	+	0.203524i
$622$		0			0
$623$	−9.79796			−0.392547
$624$		0			0
$625$	29.0000			1.16000
$626$		0			0
$627$	11.5911	+	11.5911i	0.462904	+	0.462904i
$628$		0			0
$629$	−12.9282	+	12.9282i	−0.515481	+	0.515481i
$630$		0			0
$631$			1.23835i			0.0492979i	0.999696	+	0.0246489i	$0.00784679\pi$
$631$			1.23835i			0.0492979i	−0.999696	+	0.0246489i	$0.992153\pi$
$632$		0			0
$633$		−	7.46410i		−	0.296671i
$634$		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
$639$	−4.41851			−0.174793
$640$		0			0
$641$	13.6077			0.537472			0.268736	−	0.963214i	$-0.413394\pi$
$641$	13.6077			0.537472			0.268736	+	0.963214i	$0.413394\pi$
$642$		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
$645$	2.78461	−	2.78461i	0.109644	−	0.109644i
$646$		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
$651$	11.3137	−	11.3137i	0.443419	−	0.443419i
$652$		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
$657$	−6.24871			−0.243785
$658$		0			0
$659$	−18.0430	−	18.0430i	−0.702856	−	0.702856i	0.262167	−	0.965023i	$-0.415563\pi$
$659$	−18.0430	−	18.0430i	−0.702856	−	0.702856i	−0.965023	+	0.262167i	$0.915563\pi$
$660$		0			0
$661$	−18.2679	+	18.2679i	−0.710541	+	0.710541i	−0.966648	−	0.256107i	$-0.917560\pi$
$661$	−18.2679	+	18.2679i	−0.710541	+	0.710541i	0.256107	+	0.966648i	$0.417560\pi$
$662$		0			0
$663$		−	0.960947i		−	0.0373201i
$664$		0			0
$665$		−	44.7846i		−	1.73667i
$666$		0			0
$667$	3.10583	−	3.10583i	0.120258	−	0.120258i
$668$		0			0
$669$	−2.92820	−	2.92820i	−0.113211	−	0.113211i
$670$		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
$675$	2.82843	+	2.82843i	0.108866	+	0.108866i
$676$		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
$681$		−	15.4641i		−	0.592586i
$682$		0			0
$683$	−8.72552	+	8.72552i	−0.333873	+	0.333873i	−0.854055	−	0.520182i	$-0.825864\pi$
$683$	−8.72552	+	8.72552i	−0.333873	+	0.333873i	0.520182	+	0.854055i	$0.325864\pi$
$684$		0			0
$685$	20.7846	+	20.7846i	0.794139	+	0.794139i
$686$		0			0
$687$	9.52056			0.363232
$688$		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
$693$	31.8564	−	31.8564i	1.21012	−	1.21012i
$694$		0			0
$695$			24.9754i			0.947370i
$696$		0			0
$697$			24.0000i			0.909065i
$698$		0			0
$699$	−1.31268	+	1.31268i	−0.0496500	+	0.0496500i
$700$		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
$705$	17.5692			0.661695
$706$		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	$-0.469498\pi$
$709$	29.0526	−	29.0526i	1.09109	−	1.09109i	0.995412	−	0.0956796i	$-0.0305024\pi$
$710$		0			0
$711$		−	10.2041i		−	0.382684i
$712$		0			0
$713$		−	10.1436i		−	0.379881i
$714$		0			0
$715$	3.10583	−	3.10583i	0.116151	−	0.116151i
$716$		0			0
$717$	−13.8564	−	13.8564i	−0.517477	−	0.517477i
$718$		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
$723$	1.23835	+	1.23835i	0.0460547	+	0.0460547i
$724$		0			0
$725$	1.73205	−	1.73205i	0.0643268	−	0.0643268i
$726$		0			0
$727$			23.4596i			0.870069i	0.900414	+	0.435035i	$0.143264\pi$
$727$			23.4596i			0.870069i	−0.900414	+	0.435035i	$0.856736\pi$
$728$		0			0
$729$			2.21539i			0.0820515i
$730$		0			0
$731$	−5.37945	+	5.37945i	−0.198966	+	0.198966i
$732$		0			0
$733$	14.8038	+	14.8038i	0.546793	+	0.546793i	0.925512	−	0.378719i	$-0.123635\pi$
$733$	14.8038	+	14.8038i	0.546793	+	0.546793i	−0.378719	+	0.925512i	$0.623635\pi$
$734$		0			0
$735$	14.2165			0.524382
$736$		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
$741$	−0.928203	+	0.928203i	−0.0340984	+	0.0340984i
$742$		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
$747$	3.82654	−	3.82654i	0.140006	−	0.140006i
$748$		0			0
$749$	50.7846	+	50.7846i	1.85563	+	1.85563i
$750$		0			0
$751$	−4.14110			−0.151111			−0.0755555	−	0.997142i	$-0.524073\pi$
$751$	−4.14110			−0.151111			−0.0755555	+	0.997142i	$0.524073\pi$
$752$		0			0
$753$	15.4641			0.563543
$754$		0			0
$755$	16.4901	+	16.4901i	0.600136	+	0.600136i
$756$		0			0
$757$	21.7321	−	21.7321i	0.789865	−	0.789865i	−0.191607	−	0.981472i	$-0.561370\pi$
$757$	21.7321	−	21.7321i	0.789865	−	0.789865i	0.981472	+	0.191607i	$0.0613699\pi$
$758$		0			0
$759$			6.21166i			0.225469i
$760$		0			0
$761$		−	34.6410i		−	1.25574i	−0.778320	−	0.627868i	$-0.783928\pi$
$761$		−	34.6410i		−	1.25574i	0.778320	−	0.627868i	$-0.216072\pi$
$762$		0			0
$763$	50.9860	−	50.9860i	1.84582	−	1.84582i
$764$		0			0
$765$	−14.7846	−	14.7846i	−0.534539	−	0.534539i
$766$		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
$771$	−6.69213	−	6.69213i	−0.241011	−	0.241011i
$772$		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
$777$			14.9282i			0.535546i
$778$		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
$783$	−9.79796			−0.350150
$784$		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
$789$	−11.0718	+	11.0718i	−0.394166	+	0.394166i
$790$		0			0
$791$			49.9507i			1.77604i
$792$		0			0
$793$		−	2.00000i		−	0.0710221i
$794$		0			0
$795$	13.8647	−	13.8647i	0.491731	−	0.491731i
$796$		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
$801$	6.24871			0.220787
$802$		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
$807$			5.37945i			0.189366i
$808$		0			0
$809$			17.0718i			0.600212i	0.953906	+	0.300106i	$0.0970221\pi$
$809$			17.0718i			0.600212i	−0.953906	+	0.300106i	$0.902978\pi$
$810$		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
$813$	−0.784610	−	0.784610i	−0.0275175	−	0.0275175i
$814$		0			0
$815$	−40.9850			−1.43564
$816$		0			0
$817$	10.3923			0.363581
$818$		0			0
$819$	2.55103	+	2.55103i	0.0891401	+	0.0891401i
$820$		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	$-0.911589\pi$
$823$		−	15.7322i		−	0.548391i	0.961674	−	0.274195i	$-0.0884115\pi$
$824$		0			0
$825$			3.46410i			0.120605i
$826$		0			0
$827$	16.7303	−	16.7303i	0.581770	−	0.581770i	−0.353619	−	0.935390i	$-0.615049\pi$
$827$	16.7303	−	16.7303i	0.581770	−	0.581770i	0.935390	+	0.353619i	$0.115049\pi$
$828$		0			0
$829$	2.12436	+	2.12436i	0.0737819	+	0.0737819i	0.743035	−	0.669253i	$-0.233386\pi$
$829$	2.12436	+	2.12436i	0.0737819	+	0.0737819i	−0.669253	+	0.743035i	$0.733386\pi$
$830$		0			0
$831$	−3.30890			−0.114784
$832$		0			0
$833$	−27.4641			−0.951575
$834$		0			0
$835$	−32.4997	−	32.4997i	−1.12470	−	1.12470i
$836$		0			0
$837$	−16.0000	+	16.0000i	−0.553041	+	0.553041i
$838$		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
$843$	−8.48528	+	8.48528i	−0.292249	+	0.292249i
$844$		0			0
$845$	−22.2679	−	22.2679i	−0.766041	−	0.766041i
$846$		0			0
$847$	44.0165			1.51242
$848$		0			0
$849$	−1.32051			−0.0453197
$850$		0			0
$851$	6.69213	+	6.69213i	0.229403	+	0.229403i
$852$		0			0
$853$	−9.87564	+	9.87564i	−0.338136	+	0.338136i	−0.855665	−	0.517530i	$-0.826852\pi$
$853$	−9.87564	+	9.87564i	−0.338136	+	0.338136i	0.517530	+	0.855665i	$0.326852\pi$
$854$		0			0
$855$			28.5617i			0.976789i
$856$		0			0
$857$			25.8564i			0.883238i	0.897203	+	0.441619i	$0.145596\pi$
$857$			25.8564i			0.883238i	−0.897203	+	0.441619i	$0.854404\pi$
$858$		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
$861$	13.8564	+	13.8564i	0.472225	+	0.472225i
$862$		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
$867$	2.58819	+	2.58819i	0.0878996	+	0.0878996i
$868$		0			0
$869$	13.8564	−	13.8564i	0.470046	−	0.470046i
$870$		0			0
$871$			2.34795i			0.0795574i
$872$		0			0
$873$		−	8.53590i		−	0.288896i
$874$		0			0
$875$	−26.7685	+	26.7685i	−0.904941	+	0.904941i
$876$		0			0
$877$	−16.1244	−	16.1244i	−0.544481	−	0.544481i	0.380358	−	0.924839i	$-0.375801\pi$
$877$	−16.1244	−	16.1244i	−0.544481	−	0.544481i	−0.924839	+	0.380358i	$0.875801\pi$
$878$		0			0
$879$	12.5521			0.423370
$880$		0			0
$881$	31.8564			1.07327			0.536635	−	0.843815i	$-0.319695\pi$
$881$	31.8564			1.07327			0.536635	+	0.843815i	$0.319695\pi$
$882$		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
$885$	9.21539	−	9.21539i	0.309772	−	0.309772i
$886$		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
$891$	−14.9372	+	14.9372i	−0.500414	+	0.500414i
$892$		0			0
$893$	32.7846	+	32.7846i	1.09710	+	1.09710i
$894$		0			0
$895$	5.37945			0.179815
$896$		0			0
$897$	−0.497423			−0.0166085
$898$		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
$903$			6.21166i			0.206711i
$904$		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
$909$	1.98076	+	1.98076i	0.0656977	+	0.0656977i
$910$		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
$915$	6.69213	+	6.69213i	0.221235	+	0.221235i
$916$		0			0
$917$	−0.928203	+	0.928203i	−0.0306520	+	0.0306520i
$918$		0			0
$919$		−	18.3576i		−	0.605560i	−0.953060	−	0.302780i	$-0.902085\pi$
$919$		−	18.3576i		−	0.605560i	0.953060	−	0.302780i	$-0.0979148\pi$
$920$		0			0
$921$		−	22.6795i		−	0.747315i
$922$		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
$927$	23.4596			0.770515
$928$		0			0
$929$	3.46410			0.113653			0.0568267	−	0.998384i	$-0.481902\pi$
$929$	3.46410			0.113653			0.0568267	+	0.998384i	$0.481902\pi$
$930$		0			0
$931$	26.5283	+	26.5283i	0.869429	+	0.869429i
$932$		0			0
$933$	7.85641	−	7.85641i	0.257207	−	0.257207i
$934$		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
$939$	−6.61780	+	6.61780i	−0.215964	+	0.215964i
$940$		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
$945$	−37.8564			−1.23147
$946$		0			0
$947$	13.6245	+	13.6245i	0.442737	+	0.442737i	0.892931	−	0.450194i	$-0.148645\pi$
$947$	13.6245	+	13.6245i	0.442737	+	0.442737i	−0.450194	+	0.892931i	$0.648645\pi$
$948$		0			0
$949$	0.679492	−	0.679492i	0.0220572	−	0.0220572i
$950$		0			0
$951$		−	7.04386i		−	0.228413i
$952$		0			0
$953$			53.5692i			1.73528i	0.497195	+	0.867639i	$0.334363\pi$
$953$			53.5692i			1.73528i	−0.497195	+	0.867639i	$0.665637\pi$
$954$		0			0
$955$	−29.3939	+	29.3939i	−0.951164	+	0.951164i
$956$		0			0
$957$	−6.00000	−	6.00000i	−0.193952	−	0.193952i
$958$		0			0
$959$	−46.3644			−1.49719
$960$		0			0
$961$	1.00000			0.0322581
$962$		0			0
$963$	−32.3882	−	32.3882i	−1.04370	−	1.04370i
$964$		0			0
$965$	−31.8564	+	31.8564i	−1.02549	+	1.02549i
$966$		0			0
$967$		−	27.0459i		−	0.869738i	−0.900494	−	0.434869i	$-0.856795\pi$
$967$		−	27.0459i		−	0.869738i	0.900494	−	0.434869i	$-0.143205\pi$
$968$		0			0
$969$			12.0000i			0.385496i
$970$		0			0
$971$	27.8410	−	27.8410i	0.893459	−	0.893459i	−0.101388	−	0.994847i	$-0.532328\pi$
$971$	27.8410	−	27.8410i	0.893459	−	0.893459i	0.994847	+	0.101388i	$0.0323283\pi$
$972$		0			0
$973$	−27.8564	−	27.8564i	−0.893035	−	0.893035i
$974$		0			0
$975$	−0.277401			−0.00888396
$976$		0			0
$977$	−44.5359			−1.42483			−0.712415	−	0.701759i	$-0.752399\pi$
$977$	−44.5359			−1.42483			−0.712415	+	0.701759i	$0.752399\pi$
$978$		0			0
$979$	8.48528	+	8.48528i	0.271191	+	0.271191i
$980$		0			0
$981$	−32.5167	+	32.5167i	−1.03818	+	1.03818i
$982$		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
$987$	−19.5959	+	19.5959i	−0.623745	+	0.623745i
$988$		0			0
$989$	2.78461	+	2.78461i	0.0885454	+	0.0885454i
$990$		0			0
$991$	−61.8193			−1.96375			−0.981877	−	0.189521i	$-0.939306\pi$
$991$	−61.8193			−1.96375			−0.981877	+	0.189521i	$0.939306\pi$
$992$		0			0
$993$	−19.4641			−0.617675
$994$		0			0
$995$	−27.2490	−	27.2490i	−0.863851	−	0.863851i
$996$		0			0
$997$	15.1962	−	15.1962i	0.481267	−	0.481267i	−0.424269	−	0.905536i	$-0.639469\pi$
$997$	15.1962	−	15.1962i	0.481267	−	0.481267i	0.905536	+	0.424269i	$0.139469\pi$
$998$		0			0
$999$		−	21.1117i		−	0.667944i

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
256.2.e.a.65.2	✓	8	8.3	odd	2
256.2.e.a.65.3	yes	8	8.5	even	2
256.2.e.a.193.2	yes	8	16.3	odd	4
256.2.e.a.193.3	yes	8	16.13	even	4
256.2.e.b.65.2	yes	8	1.1	even	1	trivial
256.2.e.b.65.3	yes	8	4.3	odd	2	inner
256.2.e.b.193.2	yes	8	16.5	even	4	inner
256.2.e.b.193.3	yes	8	16.11	odd	4	inner
1024.2.a.g.1.3		4	32.11	odd	8
1024.2.a.g.1.4		4	32.5	even	8
1024.2.a.j.1.1		4	32.21	even	8
1024.2.a.j.1.2		4	32.27	odd	8
1024.2.b.h.513.3		8	32.19	odd	8
1024.2.b.h.513.4		8	32.29	even	8
1024.2.b.h.513.5		8	32.13	even	8
1024.2.b.h.513.6		8	32.3	odd	8
2304.2.k.f.577.3		8	24.11	even	2
2304.2.k.f.577.4		8	24.5	odd	2
2304.2.k.f.1729.3		8	48.29	odd	4
2304.2.k.f.1729.4		8	48.35	even	4
2304.2.k.k.577.1		8	12.11	even	2
2304.2.k.k.577.2		8	3.2	odd	2
2304.2.k.k.1729.1		8	48.5	odd	4
2304.2.k.k.1729.2		8	48.11	even	4
9216.2.a.bb.1.2		4	96.59	even	8
9216.2.a.bb.1.3		4	96.53	odd	8
9216.2.a.bk.1.1		4	96.5	odd	8
9216.2.a.bk.1.4		4	96.11	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:	\( N \)	\(=\)	\( 256 = 2^{8} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	256.e (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.517638 - 0.517638i) q^{3} +(1.73205 - 1.73205i) q^{5} +3.86370i q^{7} -2.46410i q^{9} +O(q^{10})\)	\(q+(-0.517638 - 0.517638i) q^{3} +(1.73205 - 1.73205i) q^{5} +3.86370i q^{7} -2.46410i q^{9} +(3.34607 - 3.34607i) q^{11} +(0.267949 + 0.267949i) q^{13} -1.79315 q^{15} +3.46410 q^{17} +(-3.34607 - 3.34607i) q^{19} +(2.00000 - 2.00000i) q^{21} -1.79315i q^{23} -1.00000i q^{25} +(-2.82843 + 2.82843i) q^{27} +(1.73205 + 1.73205i) q^{29} +5.65685 q^{31} -3.46410 q^{33} +(6.69213 + 6.69213i) q^{35} +(-3.73205 + 3.73205i) q^{37} -0.277401i q^{39} +6.92820i q^{41} +(-1.55291 + 1.55291i) q^{43} +(-4.26795 - 4.26795i) q^{45} -9.79796 q^{47} -7.92820 q^{49} +(-1.79315 - 1.79315i) q^{51} +(-7.73205 + 7.73205i) q^{53} -11.5911i q^{55} +3.46410i q^{57} +(-5.13922 + 5.13922i) q^{59} +(-3.73205 - 3.73205i) q^{61} +9.52056 q^{63} +0.928203 q^{65} +(4.38134 + 4.38134i) q^{67} +(-0.928203 + 0.928203i) q^{69} -1.79315i q^{71} -2.53590i q^{73} +(-0.517638 + 0.517638i) q^{75} +(12.9282 + 12.9282i) q^{77} +4.14110 q^{79} -4.46410 q^{81} +(1.55291 + 1.55291i) q^{83} +(6.00000 - 6.00000i) q^{85} -1.79315i q^{87} +2.53590i q^{89} +(-1.03528 + 1.03528i) q^{91} +(-2.92820 - 2.92820i) q^{93} -11.5911 q^{95} +3.46410 q^{97} +(-8.24504 - 8.24504i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 16 q^{13} + 16 q^{21} - 16 q^{37} - 48 q^{45} - 8 q^{49} - 48 q^{53} - 16 q^{61} - 48 q^{65} + 48 q^{69} + 48 q^{77} - 8 q^{81} + 48 q^{85} + 32 q^{93}+O(q^{100}) \) 8 * q + 16 * q^13 + 16 * q^21 - 16 * q^37 - 48 * q^45 - 8 * q^49 - 48 * q^53 - 16 * q^61 - 48 * q^65 + 48 * q^69 + 48 * q^77 - 8 * q^81 + 48 * q^85 + 32 * q^93

Introduction

L-functions

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