LMFDB - Embedded newform 256.4.e.c.193.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [256,4,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, 4, names="a")

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

chi := DirichletCharacter("256.65");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 256 = 2^{8} $
Weight:	$ k $	$=$	$ 4 $
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:	$15.1044889615$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 242x^{12} + 13297x^{8} + 201600x^{4} + 331776 $ x^16 + 242x^12 + 13297x^8 + 201600*x^4 + 331776
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 2^{40} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			193.5
Root			$1.53379 - 1.53379i$ of defining polynomial
Character	$\chi$	$=$	256.193
Dual form			256.4.e.c.65.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.13572 - 1.13572i) q^{3} +(10.4583 + 10.4583i) q^{5} +10.8978i q^{7} +24.4203i q^{9} +O(q^{10})$	\(q+(1.13572 - 1.13572i) q^{3} +(10.4583 + 10.4583i) q^{5} +10.8978i q^{7} +24.4203i q^{9} +(-12.0933 - 12.0933i) q^{11} +(-16.2588 + 16.2588i) q^{13} +23.7555 q^{15} -30.9788 q^{17} +(-103.230 + 103.230i) q^{19} +(12.3769 + 12.3769i) q^{21} -186.521i q^{23} +93.7538i q^{25} +(58.3991 + 58.3991i) q^{27} +(-120.517 + 120.517i) q^{29} +263.876 q^{31} -27.4693 q^{33} +(-113.973 + 113.973i) q^{35} +(169.573 + 169.573i) q^{37} +36.9309i q^{39} +469.193i q^{41} +(-251.253 - 251.253i) q^{43} +(-255.396 + 255.396i) q^{45} +202.196 q^{47} +224.238 q^{49} +(-35.1833 + 35.1833i) q^{51} +(159.870 + 159.870i) q^{53} -252.952i q^{55} +234.482i q^{57} +(229.462 + 229.462i) q^{59} +(115.041 - 115.041i) q^{61} -266.128 q^{63} -340.080 q^{65} +(-523.429 + 523.429i) q^{67} +(-211.835 - 211.835i) q^{69} -108.833i q^{71} -771.343i q^{73} +(106.478 + 106.478i) q^{75} +(131.791 - 131.791i) q^{77} +647.309 q^{79} -526.697 q^{81} +(702.088 - 702.088i) q^{83} +(-323.987 - 323.987i) q^{85} +273.748i q^{87} -603.673i q^{89} +(-177.185 - 177.185i) q^{91} +(299.690 - 299.690i) q^{93} -2159.24 q^{95} +741.131 q^{97} +(295.323 - 295.323i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 32 q^{5}+O(q^{10}) $ 16 * q - 32 * q^5	$ 16 q - 32 q^{5} - 320 q^{13} - 384 q^{17} - 224 q^{21} - 928 q^{29} - 2432 q^{33} - 640 q^{37} - 896 q^{45} - 2832 q^{49} - 64 q^{53} - 1024 q^{61} + 2208 q^{65} - 32 q^{69} - 1056 q^{77} + 4208 q^{81} - 1824 q^{85} + 3776 q^{93} + 4480 q^{97}+O(q^{100}) $ 16 * q - 32 * q^5 - 320 * q^13 - 384 * q^17 - 224 * q^21 - 928 * q^29 - 2432 * q^33 - 640 * q^37 - 896 * q^45 - 2832 * q^49 - 64 * q^53 - 1024 * q^61 + 2208 * q^65 - 32 * q^69 - 1056 * q^77 + 4208 * q^81 - 1824 * q^85 + 3776 * q^93 + 4480 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

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
$3$	1.13572	−	1.13572i	0.218570	−	0.218570i	−0.589326	−	0.807895i	$-0.700607\pi$
$3$	1.13572	−	1.13572i	0.218570	−	0.218570i	0.807895	+	0.589326i	$0.200607\pi$
$4$		0			0
$5$	10.4583	+	10.4583i	0.935422	+	0.935422i	0.998038	−	0.0626154i	$-0.0199441\pi$
$5$	10.4583	+	10.4583i	0.935422	+	0.935422i	−0.0626154	+	0.998038i	$0.519944\pi$
$6$		0			0
$7$			10.8978i			0.588427i	0.955740	+	0.294213i	$0.0950577\pi$
$7$			10.8978i			0.588427i	−0.955740	+	0.294213i	$0.904942\pi$
$8$		0			0
$9$			24.4203i			0.904455i
$10$		0			0
$11$	−12.0933	−	12.0933i	−0.331480	−	0.331480i	0.521669	−	0.853148i	$-0.325310\pi$
$11$	−12.0933	−	12.0933i	−0.331480	−	0.331480i	−0.853148	+	0.521669i	$0.825310\pi$
$12$		0			0
$13$	−16.2588	+	16.2588i	−0.346875	+	0.346875i	−0.858944	−	0.512069i	$-0.828879\pi$
$13$	−16.2588	+	16.2588i	−0.346875	+	0.346875i	0.512069	+	0.858944i	$0.328879\pi$
$14$		0			0
$15$	23.7555			0.408910
$16$		0			0
$17$	−30.9788			−0.441968			−0.220984	−	0.975277i	$-0.570927\pi$
$17$	−30.9788			−0.441968			−0.220984	+	0.975277i	$0.570927\pi$
$18$		0			0
$19$	−103.230	+	103.230i	−1.24646	+	1.24646i	−0.289182	+	0.957274i	$0.593383\pi$
$19$	−103.230	+	103.230i	−1.24646	+	1.24646i	−0.957274	+	0.289182i	$0.906617\pi$
$20$		0			0
$21$	12.3769	+	12.3769i	0.128612	+	0.128612i
$22$		0			0
$23$		−	186.521i		−	1.69097i	−0.534001	−	0.845484i	$-0.679312\pi$
$23$		−	186.521i		−	1.69097i	0.534001	−	0.845484i	$-0.320688\pi$
$24$		0			0
$25$			93.7538i			0.750030i
$26$		0			0
$27$	58.3991	+	58.3991i	0.416256	+	0.416256i
$28$		0			0
$29$	−120.517	+	120.517i	−0.771706	+	0.771706i	−0.978405	−	0.206699i	$-0.933728\pi$
$29$	−120.517	+	120.517i	−0.771706	+	0.771706i	0.206699	+	0.978405i	$0.433728\pi$
$30$		0			0
$31$	263.876			1.52883			0.764413	−	0.644727i	$-0.223029\pi$
$31$	263.876			1.52883			0.764413	+	0.644727i	$0.223029\pi$
$32$		0			0
$33$	−27.4693			−0.144903
$34$		0			0
$35$	−113.973	+	113.973i	−0.550428	+	0.550428i
$36$		0			0
$37$	169.573	+	169.573i	0.753449	+	0.753449i	0.975121	−	0.221672i	$-0.0711515\pi$
$37$	169.573	+	169.573i	0.753449	+	0.753449i	−0.221672	+	0.975121i	$0.571151\pi$
$38$		0			0
$39$			36.9309i			0.151633i
$40$		0			0
$41$			469.193i			1.78721i	0.448853	+	0.893606i	$0.351833\pi$
$41$			469.193i			1.78721i	−0.448853	+	0.893606i	$0.648167\pi$
$42$		0			0
$43$	−251.253	−	251.253i	−0.891064	−	0.891064i	0.103560	−	0.994623i	$-0.466977\pi$
$43$	−251.253	−	251.253i	−0.891064	−	0.891064i	−0.994623	+	0.103560i	$0.966977\pi$
$44$		0			0
$45$	−255.396	+	255.396i	−0.846047	+	0.846047i
$46$		0			0
$47$	202.196			0.627518			0.313759	−	0.949503i	$-0.398412\pi$
$47$	202.196			0.627518			0.313759	+	0.949503i	$0.398412\pi$
$48$		0			0
$49$	224.238			0.653754
$50$		0			0
$51$	−35.1833	+	35.1833i	−0.0966008	+	0.0966008i
$52$		0			0
$53$	159.870	+	159.870i	0.414335	+	0.414335i	0.883246	−	0.468910i	$-0.155353\pi$
$53$	159.870	+	159.870i	0.414335	+	0.414335i	−0.468910	+	0.883246i	$0.655353\pi$
$54$		0			0
$55$		−	252.952i		−	0.620147i
$56$		0			0
$57$			234.482i			0.544875i
$58$		0			0
$59$	229.462	+	229.462i	0.506328	+	0.506328i	0.913397	−	0.407069i	$-0.133449\pi$
$59$	229.462	+	229.462i	0.506328	+	0.506328i	−0.407069	+	0.913397i	$0.633449\pi$
$60$		0			0
$61$	115.041	−	115.041i	0.241467	−	0.241467i	−0.575990	−	0.817457i	$-0.695383\pi$
$61$	115.041	−	115.041i	0.241467	−	0.241467i	0.817457	+	0.575990i	$0.195383\pi$
$62$		0			0
$63$	−266.128			−0.532205
$64$		0			0
$65$	−340.080			−0.648950
$66$		0			0
$67$	−523.429	+	523.429i	−0.954433	+	0.954433i	−0.999006	−	0.0445736i	$-0.985807\pi$
$67$	−523.429	+	523.429i	−0.954433	+	0.954433i	0.0445736	+	0.999006i	$0.485807\pi$
$68$		0			0
$69$	−211.835	−	211.835i	−0.369594	−	0.369594i
$70$		0			0
$71$		−	108.833i		−	0.181917i	−0.995855	−	0.0909584i	$-0.971007\pi$
$71$		−	108.833i		−	0.181917i	0.995855	−	0.0909584i	$-0.0289930\pi$
$72$		0			0
$73$		−	771.343i		−	1.23670i	−0.785904	−	0.618348i	$-0.787802\pi$
$73$		−	771.343i		−	1.23670i	0.785904	−	0.618348i	$-0.212198\pi$
$74$		0			0
$75$	106.478	+	106.478i	0.163934	+	0.163934i
$76$		0			0
$77$	131.791	−	131.791i	0.195052	−	0.195052i
$78$		0			0
$79$	647.309			0.921873			0.460937	−	0.887433i	$-0.347513\pi$
$79$	647.309			0.921873			0.460937	+	0.887433i	$0.347513\pi$
$80$		0			0
$81$	−526.697			−0.722493
$82$		0			0
$83$	702.088	−	702.088i	0.928485	−	0.928485i	−0.0691235	−	0.997608i	$-0.522020\pi$
$83$	702.088	−	702.088i	0.928485	−	0.928485i	0.997608	+	0.0691235i	$0.0220203\pi$
$84$		0			0
$85$	−323.987	−	323.987i	−0.413427	−	0.413427i
$86$		0			0
$87$			273.748i			0.337343i
$88$		0			0
$89$		−	603.673i		−	0.718980i	−0.933149	−	0.359490i	$-0.882951\pi$
$89$		−	603.673i		−	0.718980i	0.933149	−	0.359490i	$-0.117049\pi$
$90$		0			0
$91$	−177.185	−	177.185i	−0.204111	−	0.204111i
$92$		0			0
$93$	299.690	−	299.690i	0.334155	−	0.334155i
$94$		0			0
$95$	−2159.24			−2.33193
$96$		0			0
$97$	741.131			0.775777			0.387889	−	0.921706i	$-0.373204\pi$
$97$	741.131			0.775777			0.387889	+	0.921706i	$0.373204\pi$
$98$		0			0
$99$	295.323	−	295.323i	0.299808	−	0.299808i
$100$		0			0
$101$	159.461	+	159.461i	0.157099	+	0.157099i	0.781280	−	0.624181i	$-0.214567\pi$
$101$	159.461	+	159.461i	0.157099	+	0.157099i	−0.624181	+	0.781280i	$0.714567\pi$
$102$		0			0
$103$			1654.11i			1.58237i	0.611577	+	0.791185i	$0.290536\pi$
$103$			1654.11i			1.58237i	−0.611577	+	0.791185i	$0.709464\pi$
$104$		0			0
$105$			258.883i			0.240613i
$106$		0			0
$107$	−112.838	−	112.838i	−0.101949	−	0.101949i	0.654293	−	0.756241i	$-0.272966\pi$
$107$	−112.838	−	112.838i	−0.101949	−	0.101949i	−0.756241	+	0.654293i	$0.772966\pi$
$108$		0			0
$109$	748.876	−	748.876i	0.658067	−	0.658067i	−0.296855	−	0.954922i	$-0.595938\pi$
$109$	748.876	−	748.876i	0.658067	−	0.658067i	0.954922	+	0.296855i	$0.0959379\pi$
$110$		0			0
$111$	385.175			0.329362
$112$		0			0
$113$	1476.61			1.22927			0.614635	−	0.788812i	$-0.289303\pi$
$113$	1476.61			1.22927			0.614635	+	0.788812i	$0.289303\pi$
$114$		0			0
$115$	1950.70	−	1950.70i	1.58177	−	1.58177i
$116$		0			0
$117$	−397.044	−	397.044i	−0.313733	−	0.313733i
$118$		0			0
$119$		−	337.601i		−	0.260066i
$120$		0			0
$121$		−	1038.50i		−	0.780242i
$122$		0			0
$123$	532.872	+	532.872i	0.390630	+	0.390630i
$124$		0			0
$125$	326.784	−	326.784i	0.233828	−	0.233828i
$126$		0			0
$127$	−1153.10			−0.805679			−0.402840	−	0.915271i	$-0.631977\pi$
$127$	−1153.10			−0.805679			−0.402840	+	0.915271i	$0.631977\pi$
$128$		0			0
$129$	−570.707			−0.389519
$130$		0			0
$131$	−281.478	+	281.478i	−0.187731	+	0.187731i	−0.794715	−	0.606983i	$-0.792379\pi$
$131$	−281.478	+	281.478i	−0.187731	+	0.187731i	0.606983	+	0.794715i	$0.292379\pi$
$132$		0			0
$133$	−1124.99	−	1124.99i	−0.733448	−	0.733448i
$134$		0			0
$135$			1221.51i			0.778750i
$136$		0			0
$137$		−	2237.83i		−	1.39556i	−0.716315	−	0.697778i	$-0.754172\pi$
$137$		−	2237.83i		−	1.39556i	0.716315	−	0.697778i	$-0.245828\pi$
$138$		0			0
$139$	−929.419	−	929.419i	−0.567139	−	0.567139i	0.364187	−	0.931326i	$-0.381347\pi$
$139$	−929.419	−	929.419i	−0.567139	−	0.567139i	−0.931326	+	0.364187i	$0.881347\pi$
$140$		0			0
$141$	229.638	−	229.638i	0.137156	−	0.137156i
$142$		0			0
$143$	393.246			0.229964
$144$		0			0
$145$	−2520.82			−1.44374
$146$		0			0
$147$	254.671	−	254.671i	0.142891	−	0.142891i
$148$		0			0
$149$	−231.031	−	231.031i	−0.127025	−	0.127025i	0.640736	−	0.767761i	$-0.278629\pi$
$149$	−231.031	−	231.031i	−0.127025	−	0.127025i	−0.767761	+	0.640736i	$0.778629\pi$
$150$		0			0
$151$		−	2178.67i		−	1.17416i	−0.809529	−	0.587079i	$-0.800278\pi$
$151$		−	2178.67i		−	1.17416i	0.809529	−	0.587079i	$-0.199722\pi$
$152$		0			0
$153$		−	756.511i		−	0.399740i
$154$		0			0
$155$	2759.71	+	2759.71i	1.43010	+	1.43010i
$156$		0			0
$157$	−555.307	+	555.307i	−0.282282	+	0.282282i	−0.834019	−	0.551736i	$-0.813965\pi$
$157$	−555.307	+	555.307i	−0.282282	+	0.282282i	0.551736	+	0.834019i	$0.313965\pi$
$158$		0			0
$159$	363.134			0.181122
$160$		0			0
$161$	2032.67			0.995011
$162$		0			0
$163$	−1057.53	+	1057.53i	−0.508174	+	0.508174i	−0.913966	−	0.405791i	$-0.866996\pi$
$163$	−1057.53	+	1057.53i	−0.508174	+	0.508174i	0.405791	+	0.913966i	$0.366996\pi$
$164$		0			0
$165$	−287.283	−	287.283i	−0.135545	−	0.135545i
$166$		0			0
$167$			336.977i			0.156144i	0.996948	+	0.0780721i	$0.0248764\pi$
$167$			336.977i			0.156144i	−0.996948	+	0.0780721i	$0.975124\pi$
$168$		0			0
$169$			1668.30i			0.759355i
$170$		0			0
$171$	−2520.91	−	2520.91i	−1.12736	−	1.12736i
$172$		0			0
$173$	1291.91	−	1291.91i	0.567760	−	0.567760i	−0.363741	−	0.931500i	$-0.618501\pi$
$173$	1291.91	−	1291.91i	0.567760	−	0.567760i	0.931500	+	0.363741i	$0.118501\pi$
$174$		0			0
$175$	−1021.71			−0.441338
$176$		0			0
$177$	521.209			0.221336
$178$		0			0
$179$	−1037.43	+	1037.43i	−0.433190	+	0.433190i	−0.889712	−	0.456522i	$-0.849095\pi$
$179$	−1037.43	+	1037.43i	−0.433190	+	0.433190i	0.456522	+	0.889712i	$0.349095\pi$
$180$		0			0
$181$	1650.79	+	1650.79i	0.677914	+	0.677914i	0.959528	−	0.281614i	$-0.0908697\pi$
$181$	1650.79	+	1650.79i	0.677914	+	0.677914i	−0.281614	+	0.959528i	$0.590870\pi$
$182$		0			0
$183$		−	261.309i		−	0.105555i
$184$		0			0
$185$			3546.90i			1.40959i
$186$		0			0
$187$	374.637	+	374.637i	0.146504	+	0.146504i
$188$		0			0
$189$	−636.422	+	636.422i	−0.244936	+	0.244936i
$190$		0			0
$191$	102.447			0.0388106			0.0194053	−	0.999812i	$-0.493823\pi$
$191$	102.447			0.0388106			0.0194053	+	0.999812i	$0.493823\pi$
$192$		0			0
$193$	−2158.90			−0.805185			−0.402593	−	0.915379i	$-0.631891\pi$
$193$	−2158.90			−0.805185			−0.402593	+	0.915379i	$0.631891\pi$
$194$		0			0
$195$	−386.236	+	386.236i	−0.141841	+	0.141841i
$196$		0			0
$197$	3022.32	+	3022.32i	1.09305	+	1.09305i	0.995201	+	0.0978506i	$0.0311967\pi$
$197$	3022.32	+	3022.32i	1.09305	+	1.09305i	0.0978506	+	0.995201i	$0.468803\pi$
$198$		0			0
$199$			3300.90i			1.17585i	0.808914	+	0.587926i	$0.200056\pi$
$199$			3300.90i			1.17585i	−0.808914	+	0.587926i	$0.799944\pi$
$200$		0			0
$201$			1188.94i			0.417220i
$202$		0			0
$203$	−1313.37	−	1313.37i	−0.454093	−	0.454093i
$204$		0			0
$205$	−4906.98	+	4906.98i	−1.67180	+	1.67180i
$206$		0			0
$207$	4554.89			1.52940
$208$		0			0
$209$	2496.80			0.826350
$210$		0			0
$211$	77.0963	−	77.0963i	0.0251541	−	0.0251541i	−0.694418	−	0.719572i	$-0.744338\pi$
$211$	77.0963	−	77.0963i	0.0251541	−	0.0251541i	0.719572	+	0.694418i	$0.244338\pi$
$212$		0			0
$213$	−123.604	−	123.604i	−0.0397615	−	0.0397615i
$214$		0			0
$215$		−	5255.38i		−	1.66704i
$216$		0			0
$217$			2875.68i			0.899602i
$218$		0			0
$219$	−876.030	−	876.030i	−0.270304	−	0.270304i
$220$		0			0
$221$	503.678	−	503.678i	0.153308	−	0.153308i
$222$		0			0
$223$	1824.45			0.547865			0.273933	−	0.961749i	$-0.411675\pi$
$223$	1824.45			0.547865			0.273933	+	0.961749i	$0.411675\pi$
$224$		0			0
$225$	−2289.49			−0.678368
$226$		0			0
$227$	−2967.42	+	2967.42i	−0.867641	+	0.867641i	−0.992211	−	0.124569i	$-0.960245\pi$
$227$	−2967.42	+	2967.42i	−0.867641	+	0.867641i	0.124569	+	0.992211i	$0.460245\pi$
$228$		0			0
$229$	−792.161	−	792.161i	−0.228592	−	0.228592i	0.583513	−	0.812104i	$-0.301678\pi$
$229$	−792.161	−	792.161i	−0.228592	−	0.228592i	−0.812104	+	0.583513i	$0.801678\pi$
$230$		0			0
$231$		−	299.355i		−	0.0852646i
$232$		0			0
$233$			892.148i			0.250844i	0.992104	+	0.125422i	$0.0400284\pi$
$233$			892.148i			0.250844i	−0.992104	+	0.125422i	$0.959972\pi$
$234$		0			0
$235$	2114.64	+	2114.64i	0.586994	+	0.586994i
$236$		0			0
$237$	735.163	−	735.163i	0.201493	−	0.201493i
$238$		0			0
$239$	2647.81			0.716623			0.358311	−	0.933602i	$-0.383353\pi$
$239$	2647.81			0.716623			0.358311	+	0.933602i	$0.383353\pi$
$240$		0			0
$241$	−2718.29			−0.726559			−0.363279	−	0.931680i	$-0.618343\pi$
$241$	−2718.29			−0.726559			−0.363279	+	0.931680i	$0.618343\pi$
$242$		0			0
$243$	−2174.96	+	2174.96i	−0.574171	+	0.574171i
$244$		0			0
$245$	2345.15	+	2345.15i	0.611536	+	0.611536i
$246$		0			0
$247$		−	3356.80i		−	0.864730i
$248$		0			0
$249$		−	1594.75i		−	0.405877i
$250$		0			0
$251$	2278.00	+	2278.00i	0.572853	+	0.572853i	0.932925	−	0.360071i	$-0.117248\pi$
$251$	2278.00	+	2278.00i	0.572853	+	0.572853i	−0.360071	+	0.932925i	$0.617248\pi$
$252$		0			0
$253$	−2255.66	+	2255.66i	−0.560521	+	0.560521i
$254$		0			0
$255$	−735.917			−0.180725
$256$		0			0
$257$	5058.41			1.22776			0.613881	−	0.789398i	$-0.289607\pi$
$257$	5058.41			1.22776			0.613881	+	0.789398i	$0.289607\pi$
$258$		0			0
$259$	−1847.97	+	1847.97i	−0.443350	+	0.443350i
$260$		0			0
$261$	−2943.06	−	2943.06i	−0.697973	−	0.697973i
$262$		0			0
$263$		−	7942.44i		−	1.86217i	−0.364797	−	0.931087i	$-0.618862\pi$
$263$		−	7942.44i		−	1.86217i	0.364797	−	0.931087i	$-0.381138\pi$
$264$		0			0
$265$			3343.94i			0.775157i
$266$		0			0
$267$	−685.604	−	685.604i	−0.157147	−	0.157147i
$268$		0			0
$269$	67.1308	−	67.1308i	0.0152158	−	0.0152158i	−0.699458	−	0.714674i	$-0.746575\pi$
$269$	67.1308	−	67.1308i	0.0152158	−	0.0152158i	0.714674	+	0.699458i	$0.246575\pi$
$270$		0			0
$271$	2237.68			0.501585			0.250793	−	0.968041i	$-0.419309\pi$
$271$	2237.68			0.501585			0.250793	+	0.968041i	$0.419309\pi$
$272$		0			0
$273$	−402.466			−0.0892248
$274$		0			0
$275$	1133.80	−	1133.80i	0.248620	−	0.248620i
$276$		0			0
$277$	−3646.89	−	3646.89i	−0.791048	−	0.791048i	0.190617	−	0.981665i	$-0.438951\pi$
$277$	−3646.89	−	3646.89i	−0.791048	−	0.791048i	−0.981665	+	0.190617i	$0.938951\pi$
$278$		0			0
$279$			6443.93i			1.38275i
$280$		0			0
$281$			367.919i			0.0781075i	0.999237	+	0.0390538i	$0.0124344\pi$
$281$			367.919i			0.0781075i	−0.999237	+	0.0390538i	$0.987566\pi$
$282$		0			0
$283$	−4994.53	−	4994.53i	−1.04909	−	1.04909i	−0.998731	−	0.0503634i	$-0.983962\pi$
$283$	−4994.53	−	4994.53i	−1.04909	−	1.04909i	−0.0503634	−	0.998731i	$-0.516038\pi$
$284$		0			0
$285$	−2452.29	+	2452.29i	−0.509688	+	0.509688i
$286$		0			0
$287$	−5113.18			−1.05164
$288$		0			0
$289$	−3953.31			−0.804664
$290$		0			0
$291$	841.717	−	841.717i	0.169561	−	0.169561i
$292$		0			0
$293$	6070.97	+	6070.97i	1.21048	+	1.21048i	0.970870	+	0.239608i	$0.0770191\pi$
$293$	6070.97	+	6070.97i	1.21048	+	1.21048i	0.239608	+	0.970870i	$0.422981\pi$
$294$		0			0
$295$			4799.58i			0.947261i
$296$		0			0
$297$		−	1412.48i		−	0.275961i
$298$		0			0
$299$	3032.60	+	3032.60i	0.586555	+	0.586555i
$300$		0			0
$301$	2738.11	−	2738.11i	0.524326	−	0.524326i
$302$		0			0
$303$	362.207			0.0686741
$304$		0			0
$305$	2406.27			0.451747
$306$		0			0
$307$	4007.99	−	4007.99i	0.745108	−	0.745108i	−0.228448	−	0.973556i	$-0.573365\pi$
$307$	4007.99	−	4007.99i	0.745108	−	0.745108i	0.973556	+	0.228448i	$0.0733650\pi$
$308$		0			0
$309$	1878.61	+	1878.61i	0.345858	+	0.345858i
$310$		0			0
$311$			6413.20i			1.16932i	0.811278	+	0.584661i	$0.198773\pi$
$311$			6413.20i			1.16932i	−0.811278	+	0.584661i	$0.801227\pi$
$312$		0			0
$313$		−	4044.09i		−	0.730305i	−0.930948	−	0.365153i	$-0.881017\pi$
$313$		−	4044.09i		−	0.730305i	0.930948	−	0.365153i	$-0.118983\pi$
$314$		0			0
$315$	−2783.25	−	2783.25i	−0.497837	−	0.497837i
$316$		0			0
$317$	6785.49	−	6785.49i	1.20224	−	1.20224i	0.228760	−	0.973483i	$-0.426533\pi$
$317$	6785.49	−	6785.49i	1.20224	−	1.20224i	0.973483	−	0.228760i	$-0.0734670\pi$
$318$		0			0
$319$	2914.91			0.511610
$320$		0			0
$321$	−256.306			−0.0445657
$322$		0			0
$323$	3197.95	−	3197.95i	0.550894	−	0.550894i
$324$		0			0
$325$	−1524.32	−	1524.32i	−0.260167	−	0.260167i
$326$		0			0
$327$		−	1701.03i		−	0.287667i
$328$		0			0
$329$			2203.50i			0.369248i
$330$		0			0
$331$	490.326	+	490.326i	0.0814222	+	0.0814222i	0.746645	−	0.665223i	$-0.231663\pi$
$331$	490.326	+	490.326i	0.0814222	+	0.0814222i	−0.665223	+	0.746645i	$0.731663\pi$
$332$		0			0
$333$	−4141.02	+	4141.02i	−0.681461	+	0.681461i
$334$		0			0
$335$	−10948.4			−1.78560
$336$		0			0
$337$	4617.22			0.746338			0.373169	−	0.927763i	$-0.378271\pi$
$337$	4617.22			0.746338			0.373169	+	0.927763i	$0.378271\pi$
$338$		0			0
$339$	1677.01	−	1677.01i	0.268681	−	0.268681i
$340$		0			0
$341$	−3191.14	−	3191.14i	−0.506775	−	0.506775i
$342$		0			0
$343$			6181.65i			0.973113i
$344$		0			0
$345$		−	4430.89i		−	0.691453i
$346$		0			0
$347$	7690.50	+	7690.50i	1.18976	+	1.18976i	0.977133	+	0.212629i	$0.0682027\pi$
$347$	7690.50	+	7690.50i	1.18976	+	1.18976i	0.212629	+	0.977133i	$0.431797\pi$
$348$		0			0
$349$	3203.48	−	3203.48i	0.491342	−	0.491342i	−0.417387	−	0.908729i	$-0.637054\pi$
$349$	3203.48	−	3203.48i	0.491342	−	0.491342i	0.908729	+	0.417387i	$0.137054\pi$
$350$		0			0
$351$	−1899.00			−0.288778
$352$		0			0
$353$	914.110			0.137828			0.0689139	−	0.997623i	$-0.478047\pi$
$353$	914.110			0.137828			0.0689139	+	0.997623i	$0.478047\pi$
$354$		0			0
$355$	1138.21	−	1138.21i	0.170169	−	0.170169i
$356$		0			0
$357$	−383.421	−	383.421i	−0.0568425	−	0.0568425i
$358$		0			0
$359$		−	6252.16i		−	0.919155i	−0.888138	−	0.459577i	$-0.848001\pi$
$359$		−	6252.16i		−	0.919155i	0.888138	−	0.459577i	$-0.151999\pi$
$360$		0			0
$361$		−	14454.0i		−	2.10731i
$362$		0			0
$363$	−1179.45	−	1179.45i	−0.170537	−	0.170537i
$364$		0			0
$365$	8066.97	−	8066.97i	1.15683	−	1.15683i
$366$		0			0
$367$	−6025.24			−0.856988			−0.428494	−	0.903545i	$-0.640956\pi$
$367$	−6025.24			−0.856988			−0.428494	+	0.903545i	$0.640956\pi$
$368$		0			0
$369$	−11457.8			−1.61645
$370$		0			0
$371$	−1742.23	+	1742.23i	−0.243806	+	0.243806i
$372$		0			0
$373$	−3971.27	−	3971.27i	−0.551273	−	0.551273i	0.375535	−	0.926808i	$-0.377459\pi$
$373$	−3971.27	−	3971.27i	−0.551273	−	0.551273i	−0.926808	+	0.375535i	$0.877459\pi$
$374$		0			0
$375$		−	742.270i		−	0.102215i
$376$		0			0
$377$		−	3918.93i		−	0.535372i
$378$		0			0
$379$	−7117.10	−	7117.10i	−0.964593	−	0.964593i	0.0348013	−	0.999394i	$-0.488920\pi$
$379$	−7117.10	−	7117.10i	−0.964593	−	0.964593i	−0.999394	+	0.0348013i	$0.988920\pi$
$380$		0			0
$381$	−1309.60	+	1309.60i	−0.176097	+	0.176097i
$382$		0			0
$383$	−779.648			−0.104016			−0.0520080	−	0.998647i	$-0.516562\pi$
$383$	−779.648			−0.104016			−0.0520080	+	0.998647i	$0.516562\pi$
$384$		0			0
$385$	2756.63			0.364911
$386$		0			0
$387$	6135.67	−	6135.67i	0.805927	−	0.805927i
$388$		0			0
$389$	−10296.3	−	10296.3i	−1.34202	−	1.34202i	−0.894052	−	0.447963i	$-0.852150\pi$
$389$	−10296.3	−	10296.3i	−1.34202	−	1.34202i	−0.447963	−	0.894052i	$-0.647850\pi$
$390$		0			0
$391$			5778.19i			0.747354i
$392$		0			0
$393$			639.360i			0.0820647i
$394$		0			0
$395$	6769.78	+	6769.78i	0.862341	+	0.862341i
$396$		0			0
$397$	795.225	−	795.225i	0.100532	−	0.100532i	−0.655052	−	0.755584i	$-0.727353\pi$
$397$	795.225	−	795.225i	0.100532	−	0.100532i	0.755584	+	0.655052i	$0.227353\pi$
$398$		0			0
$399$	−2555.34			−0.320619
$400$		0			0
$401$	8140.19			1.01372			0.506860	−	0.862028i	$-0.330806\pi$
$401$	8140.19			1.01372			0.506860	+	0.862028i	$0.330806\pi$
$402$		0			0
$403$	−4290.31	+	4290.31i	−0.530312	+	0.530312i
$404$		0			0
$405$	−5508.38	−	5508.38i	−0.675836	−	0.675836i
$406$		0			0
$407$		−	4101.40i		−	0.499506i
$408$		0			0
$409$		−	243.511i		−	0.0294397i	−0.999892	−	0.0147198i	$-0.995314\pi$
$409$		−	243.511i		−	0.0294397i	0.999892	−	0.0147198i	$-0.00468564\pi$
$410$		0			0
$411$	−2541.55	−	2541.55i	−0.305026	−	0.305026i
$412$		0			0
$413$	−2500.63	+	2500.63i	−0.297937	+	0.297937i
$414$		0			0
$415$	14685.4			1.73705
$416$		0			0
$417$	−2111.12			−0.247919
$418$		0			0
$419$	10950.1	−	10950.1i	1.27672	−	1.27672i	0.334233	−	0.942490i	$-0.391523\pi$
$419$	10950.1	−	10950.1i	1.27672	−	1.27672i	0.942490	−	0.334233i	$-0.108477\pi$
$420$		0			0
$421$	4623.28	+	4623.28i	0.535214	+	0.535214i	0.922119	−	0.386906i	$-0.126456\pi$
$421$	4623.28	+	4623.28i	0.535214	+	0.535214i	−0.386906	+	0.922119i	$0.626456\pi$
$422$		0			0
$423$			4937.69i			0.567562i
$424$		0			0
$425$		−	2904.38i		−	0.331490i
$426$		0			0
$427$	1253.69	+	1253.69i	0.142086	+	0.142086i
$428$		0			0
$429$	446.618	−	446.618i	0.0502632	−	0.0502632i
$430$		0			0
$431$	−12116.9			−1.35418			−0.677090	−	0.735900i	$-0.736759\pi$
$431$	−12116.9			−1.35418			−0.677090	+	0.735900i	$0.736759\pi$
$432$		0			0
$433$	−939.963			−0.104323			−0.0521614	−	0.998639i	$-0.516611\pi$
$433$	−939.963			−0.104323			−0.0521614	+	0.998639i	$0.516611\pi$
$434$		0			0
$435$	−2862.95	+	2862.95i	−0.315558	+	0.315558i
$436$		0			0
$437$	19254.6	+	19254.6i	2.10772	+	2.10772i
$438$		0			0
$439$			6773.26i			0.736378i	0.929751	+	0.368189i	$0.120022\pi$
$439$			6773.26i			0.736378i	−0.929751	+	0.368189i	$0.879978\pi$
$440$		0			0
$441$			5475.94i			0.591291i
$442$		0			0
$443$	3342.04	+	3342.04i	0.358431	+	0.358431i	0.863234	−	0.504803i	$-0.168435\pi$
$443$	3342.04	+	3342.04i	0.358431	+	0.358431i	−0.504803	+	0.863234i	$0.668435\pi$
$444$		0			0
$445$	6313.42	−	6313.42i	0.672550	−	0.672550i
$446$		0			0
$447$	−524.773			−0.0555278
$448$		0			0
$449$	5035.15			0.529228			0.264614	−	0.964354i	$-0.414755\pi$
$449$	5035.15			0.529228			0.264614	+	0.964354i	$0.414755\pi$
$450$		0			0
$451$	5674.11	−	5674.11i	0.592424	−	0.592424i
$452$		0			0
$453$	−2474.36	−	2474.36i	−0.256635	−	0.256635i
$454$		0			0
$455$		−	3706.13i		−	0.381860i
$456$		0			0
$457$			7652.79i			0.783331i	0.920108	+	0.391666i	$0.128101\pi$
$457$			7652.79i			0.783331i	−0.920108	+	0.391666i	$0.871899\pi$
$458$		0			0
$459$	−1809.13	−	1809.13i	−0.183972	−	0.183972i
$460$		0			0
$461$	−10300.0	+	10300.0i	−1.04061	+	1.04061i	−0.0414659	+	0.999140i	$0.513203\pi$
$461$	−10300.0	+	10300.0i	−1.04061	+	1.04061i	−0.999140	+	0.0414659i	$0.986797\pi$
$462$		0			0
$463$	1213.28			0.121784			0.0608919	−	0.998144i	$-0.480606\pi$
$463$	1213.28			0.121784			0.0608919	+	0.998144i	$0.480606\pi$
$464$		0			0
$465$	6268.51			0.625151
$466$		0			0
$467$	−8124.34	+	8124.34i	−0.805031	+	0.805031i	−0.983877	−	0.178846i	$-0.942764\pi$
$467$	−8124.34	+	8124.34i	−0.805031	+	0.805031i	0.178846	+	0.983877i	$0.442764\pi$
$468$		0			0
$469$	−5704.23	−	5704.23i	−0.561614	−	0.561614i
$470$		0			0
$471$			1261.35i			0.123397i
$472$		0			0
$473$			6076.97i			0.590739i
$474$		0			0
$475$	−9678.23	−	9678.23i	−0.934880	−	0.934880i
$476$		0			0
$477$	−3904.06	+	3904.06i	−0.374747	+	0.374747i
$478$		0			0
$479$	8755.05			0.835133			0.417566	−	0.908646i	$-0.362883\pi$
$479$	8755.05			0.835133			0.417566	+	0.908646i	$0.362883\pi$
$480$		0			0
$481$	−5514.10			−0.522706
$482$		0			0
$483$	2308.54	−	2308.54i	0.217479	−	0.217479i
$484$		0			0
$485$	7751.00	+	7751.00i	0.725680	+	0.725680i
$486$		0			0
$487$		−	14346.3i		−	1.33489i	−0.744658	−	0.667446i	$-0.767387\pi$
$487$		−	14346.3i		−	1.33489i	0.744658	−	0.667446i	$-0.232613\pi$
$488$		0			0
$489$			2402.13i			0.222143i
$490$		0			0
$491$	933.557	+	933.557i	0.0858062	+	0.0858062i	0.748707	−	0.662901i	$-0.230675\pi$
$491$	933.557	+	933.557i	0.0858062	+	0.0858062i	−0.662901	+	0.748707i	$0.730675\pi$
$492$		0			0
$493$	3733.48	−	3733.48i	0.341070	−	0.341070i
$494$		0			0
$495$	6177.17			0.560895
$496$		0			0
$497$	1186.04			0.107045
$498$		0			0
$499$	−11103.6	+	11103.6i	−0.996120	+	0.996120i	−0.999993	−	0.00387261i	$-0.998767\pi$
$499$	−11103.6	+	11103.6i	−0.996120	+	0.996120i	0.00387261	+	0.999993i	$0.498767\pi$
$500$		0			0
$501$	382.712	+	382.712i	0.0341284	+	0.0341284i
$502$		0			0
$503$			3340.94i			0.296153i	0.988976	+	0.148077i	$0.0473082\pi$
$503$			3340.94i			0.296153i	−0.988976	+	0.148077i	$0.952692\pi$
$504$		0			0
$505$			3335.40i			0.293908i
$506$		0			0
$507$	1894.73	+	1894.73i	0.165972	+	0.165972i
$508$		0			0
$509$	−6399.94	+	6399.94i	−0.557313	+	0.557313i	−0.928541	−	0.371229i	$-0.878937\pi$
$509$	−6399.94	+	6399.94i	−0.557313	+	0.557313i	0.371229	+	0.928541i	$0.378937\pi$
$510$		0			0
$511$	8405.95			0.727705
$512$		0			0
$513$	−12057.1			−1.03769
$514$		0			0
$515$	−17299.2	+	17299.2i	−1.48018	+	1.48018i
$516$		0			0
$517$	−2445.23	−	2445.23i	−0.208009	−	0.208009i
$518$		0			0
$519$		−	2934.51i		−	0.248190i
$520$		0			0
$521$			5430.32i			0.456635i	0.973587	+	0.228317i	$0.0733224\pi$
$521$			5430.32i			0.456635i	−0.973587	+	0.228317i	$0.926678\pi$
$522$		0			0
$523$	4251.48	+	4251.48i	0.355458	+	0.355458i	0.862135	−	0.506678i	$-0.169127\pi$
$523$	4251.48	+	4251.48i	0.355458	+	0.355458i	−0.506678	+	0.862135i	$0.669127\pi$
$524$		0			0
$525$	−1160.38	+	1160.38i	−0.0964630	+	0.0964630i
$526$		0			0
$527$	−8174.57			−0.675693
$528$		0			0
$529$	−22623.0			−1.85937
$530$		0			0
$531$	−5603.52	+	5603.52i	−0.457951	+	0.457951i
$532$		0			0
$533$	−7628.52	−	7628.52i	−0.619940	−	0.619940i
$534$		0			0
$535$		−	2360.20i		−	0.190730i
$536$		0			0
$537$			2356.46i			0.189364i
$538$		0			0
$539$	−2711.78	−	2711.78i	−0.216706	−	0.216706i
$540$		0			0
$541$	−12763.5	+	12763.5i	−1.01432	+	1.01432i	−0.0144213	+	0.999896i	$0.504591\pi$
$541$	−12763.5	+	12763.5i	−1.01432	+	1.01432i	−0.999896	+	0.0144213i	$0.995409\pi$
$542$		0			0
$543$	3749.68			0.296343
$544$		0			0
$545$	15664.0			1.23114
$546$		0			0
$547$	−5360.42	+	5360.42i	−0.419004	+	0.419004i	−0.884860	−	0.465857i	$-0.845746\pi$
$547$	−5360.42	+	5360.42i	−0.419004	+	0.419004i	0.465857	+	0.884860i	$0.345746\pi$
$548$		0			0
$549$	2809.33	+	2809.33i	0.218396	+	0.218396i
$550$		0			0
$551$		−	24882.1i		−	1.92380i
$552$		0			0
$553$			7054.26i			0.542455i
$554$		0			0
$555$	4028.29	+	4028.29i	0.308093	+	0.308093i
$556$		0			0
$557$	12284.0	−	12284.0i	0.934456	−	0.934456i	−0.0635245	−	0.997980i	$-0.520234\pi$
$557$	12284.0	−	12284.0i	0.934456	−	0.934456i	0.997980	+	0.0635245i	$0.0202341\pi$
$558$		0			0
$559$	8170.15			0.618176
$560$		0			0
$561$	850.966			0.0640424
$562$		0			0
$563$	5592.31	−	5592.31i	0.418628	−	0.418628i	−0.466103	−	0.884731i	$-0.654342\pi$
$563$	5592.31	−	5592.31i	0.418628	−	0.418628i	0.884731	+	0.466103i	$0.154342\pi$
$564$		0			0
$565$	15442.9	+	15442.9i	1.14989	+	1.14989i
$566$		0			0
$567$		−	5739.85i		−	0.425134i
$568$		0			0
$569$			5368.07i			0.395503i	0.980252	+	0.197751i	$0.0633639\pi$
$569$			5368.07i			0.395503i	−0.980252	+	0.197751i	$0.936636\pi$
$570$		0			0
$571$	−1135.50	−	1135.50i	−0.0832209	−	0.0832209i	0.664271	−	0.747492i	$-0.268742\pi$
$571$	−1135.50	−	1135.50i	−0.0832209	−	0.0832209i	−0.747492	+	0.664271i	$0.768742\pi$
$572$		0			0
$573$	116.351	−	116.351i	0.00848281	−	0.00848281i
$574$		0			0
$575$	17487.0			1.26828
$576$		0			0
$577$	11551.0			0.833407			0.416704	−	0.909042i	$-0.363185\pi$
$577$	11551.0			0.833407			0.416704	+	0.909042i	$0.363185\pi$
$578$		0			0
$579$	−2451.90	+	2451.90i	−0.175989	+	0.175989i
$580$		0			0
$581$	7651.23	+	7651.23i	0.546345	+	0.546345i
$582$		0			0
$583$		−	3866.71i		−	0.274687i
$584$		0			0
$585$		−	8304.85i		−	0.586946i
$586$		0			0
$587$	7688.04	+	7688.04i	0.540578	+	0.540578i	0.923698	−	0.383120i	$-0.125151\pi$
$587$	7688.04	+	7688.04i	0.540578	+	0.540578i	−0.383120	+	0.923698i	$0.625151\pi$
$588$		0			0
$589$	−27240.1	+	27240.1i	−1.90561	+	1.90561i
$590$		0			0
$591$	6865.02			0.477816
$592$		0			0
$593$	−6202.98			−0.429554			−0.214777	−	0.976663i	$-0.568903\pi$
$593$	−6202.98			−0.429554			−0.214777	+	0.976663i	$0.568903\pi$
$594$		0			0
$595$	3530.75	−	3530.75i	0.243272	−	0.243272i
$596$		0			0
$597$	3748.90	+	3748.90i	0.257006	+	0.257006i
$598$		0			0
$599$			25399.0i			1.73251i	0.499601	+	0.866256i	$0.333480\pi$
$599$			25399.0i			1.73251i	−0.499601	+	0.866256i	$0.666520\pi$
$600$		0			0
$601$			10592.6i			0.718937i	0.933157	+	0.359468i	$0.117042\pi$
$601$			10592.6i			0.718937i	−0.933157	+	0.359468i	$0.882958\pi$
$602$		0			0
$603$	−12782.3	−	12782.3i	−0.863241	−	0.863241i
$604$		0			0
$605$	10861.0	−	10861.0i	0.729856	−	0.729856i
$606$		0			0
$607$	−22149.9			−1.48112			−0.740559	−	0.671991i	$-0.765439\pi$
$607$	−22149.9			−1.48112			−0.740559	+	0.671991i	$0.765439\pi$
$608$		0			0
$609$	−2983.25			−0.198502
$610$		0			0
$611$	−3287.47	+	3287.47i	−0.217671	+	0.217671i
$612$		0			0
$613$	−706.775	−	706.775i	−0.0465683	−	0.0465683i	0.683439	−	0.730007i	$-0.260483\pi$
$613$	−706.775	−	706.775i	−0.0465683	−	0.0465683i	−0.730007	+	0.683439i	$0.760483\pi$
$614$		0			0
$615$			11145.9i			0.730808i
$616$		0			0
$617$			19532.2i			1.27445i	0.770676	+	0.637227i	$0.219919\pi$
$617$			19532.2i			1.27445i	−0.770676	+	0.637227i	$0.780081\pi$
$618$		0			0
$619$	−2955.47	−	2955.47i	−0.191907	−	0.191907i	0.604613	−	0.796520i	$-0.293328\pi$
$619$	−2955.47	−	2955.47i	−0.191907	−	0.191907i	−0.796520	+	0.604613i	$0.793328\pi$
$620$		0			0
$621$	10892.6	−	10892.6i	0.703875	−	0.703875i
$622$		0			0
$623$	6578.72			0.423067
$624$		0			0
$625$	18554.5			1.18748
$626$		0			0
$627$	2835.66	−	2835.66i	0.180615	−	0.180615i
$628$		0			0
$629$	−5253.17	−	5253.17i	−0.333001	−	0.333001i
$630$		0			0
$631$			3807.03i			0.240183i	0.992763	+	0.120092i	$0.0383188\pi$
$631$			3807.03i			0.240183i	−0.992763	+	0.120092i	$0.961681\pi$
$632$		0			0
$633$		−	175.120i		−	0.0109959i
$634$		0			0
$635$	−12059.5	−	12059.5i	−0.753650	−	0.753650i
$636$		0			0
$637$	−3645.83	+	3645.83i	−0.226771	+	0.226771i
$638$		0			0
$639$	2657.73			0.164536
$640$		0			0
$641$	−13614.0			−0.838876			−0.419438	−	0.907784i	$-0.637773\pi$
$641$	−13614.0			−0.838876			−0.419438	+	0.907784i	$0.637773\pi$
$642$		0			0
$643$	17427.0	−	17427.0i	1.06882	−	1.06882i	0.0713737	−	0.997450i	$-0.477262\pi$
$643$	17427.0	−	17427.0i	1.06882	−	1.06882i	0.997450	−	0.0713737i	$-0.0227383\pi$
$644$		0			0
$645$	−5968.64	−	5968.64i	−0.364364	−	0.364364i
$646$		0			0
$647$		−	8973.38i		−	0.545255i	−0.962120	−	0.272628i	$-0.912107\pi$
$647$		−	8973.38i		−	0.545255i	0.962120	−	0.272628i	$-0.0878927\pi$
$648$		0			0
$649$		−	5549.91i		−	0.335675i
$650$		0			0
$651$	3265.96	+	3265.96i	0.196626	+	0.196626i
$652$		0			0
$653$	−12077.2	+	12077.2i	−0.723765	+	0.723765i	−0.969370	−	0.245605i	$-0.921013\pi$
$653$	−12077.2	+	12077.2i	−0.723765	+	0.723765i	0.245605	+	0.969370i	$0.421013\pi$
$654$		0			0
$655$	−5887.57			−0.351216
$656$		0			0
$657$	18836.4			1.11854
$658$		0			0
$659$	7420.23	−	7420.23i	0.438621	−	0.438621i	−0.452927	−	0.891548i	$-0.649620\pi$
$659$	7420.23	−	7420.23i	0.438621	−	0.438621i	0.891548	+	0.452927i	$0.149620\pi$
$660$		0			0
$661$	−14935.9	−	14935.9i	−0.878880	−	0.878880i	0.114539	−	0.993419i	$-0.463461\pi$
$661$	−14935.9	−	14935.9i	−0.878880	−	0.878880i	−0.993419	+	0.114539i	$0.963461\pi$
$662$		0			0
$663$		−	1144.08i		−	0.0670169i
$664$		0			0
$665$		−	23531.0i		−	1.37217i
$666$		0			0
$667$	22478.9	+	22478.9i	1.30493	+	1.30493i
$668$		0			0
$669$	2072.06	−	2072.06i	0.119747	−	0.119747i
$670$		0			0
$671$	−2782.46			−0.160083
$672$		0			0
$673$	−10148.2			−0.581257			−0.290628	−	0.956836i	$-0.593864\pi$
$673$	−10148.2			−0.581257			−0.290628	+	0.956836i	$0.593864\pi$
$674$		0			0
$675$	−5475.13	+	5475.13i	−0.312204	+	0.312204i
$676$		0			0
$677$	3810.25	+	3810.25i	0.216307	+	0.216307i	0.806940	−	0.590633i	$-0.201122\pi$
$677$	3810.25	+	3810.25i	0.216307	+	0.216307i	−0.590633	+	0.806940i	$0.701122\pi$
$678$		0			0
$679$			8076.71i			0.456488i
$680$		0			0
$681$			6740.32i			0.379280i
$682$		0			0
$683$	−9878.47	−	9878.47i	−0.553425	−	0.553425i	0.374003	−	0.927428i	$-0.377985\pi$
$683$	−9878.47	−	9878.47i	−0.553425	−	0.553425i	−0.927428	+	0.374003i	$0.877985\pi$
$684$		0			0
$685$	23404.0	−	23404.0i	1.30543	−	1.30543i
$686$		0			0
$687$	−1799.35			−0.0999263
$688$		0			0
$689$	−5198.57			−0.287445
$690$		0			0
$691$	−4045.86	+	4045.86i	−0.222737	+	0.222737i	−0.809650	−	0.586913i	$-0.800343\pi$
$691$	−4045.86	+	4045.86i	−0.222737	+	0.222737i	0.586913	+	0.809650i	$0.300343\pi$
$692$		0			0
$693$	3218.37	+	3218.37i	0.176415	+	0.176415i
$694$		0			0
$695$		−	19440.4i		−	1.06103i
$696$		0			0
$697$		−	14535.0i		−	0.789891i
$698$		0			0
$699$	1013.23	+	1013.23i	0.0548267	+	0.0548267i
$700$		0			0
$701$	11492.0	−	11492.0i	0.619181	−	0.619181i	−0.326140	−	0.945321i	$-0.605748\pi$
$701$	11492.0	−	11492.0i	0.619181	−	0.619181i	0.945321	+	0.326140i	$0.105748\pi$
$702$		0			0
$703$	−35010.1			−1.87828
$704$		0			0
$705$	4803.27			0.256598
$706$		0			0
$707$	−1737.78	+	1737.78i	−0.0924413	+	0.0924413i
$708$		0			0
$709$	−11844.5	−	11844.5i	−0.627402	−	0.627402i	0.320012	−	0.947414i	$-0.396313\pi$
$709$	−11844.5	−	11844.5i	−0.627402	−	0.627402i	−0.947414	+	0.320012i	$0.896313\pi$
$710$		0			0
$711$			15807.5i			0.833793i
$712$		0			0
$713$		−	49218.4i		−	2.58519i
$714$		0			0
$715$	4112.70	+	4112.70i	0.215114	+	0.215114i
$716$		0			0
$717$	3007.18	−	3007.18i	0.156632	−	0.156632i
$718$		0			0
$719$	−4208.96			−0.218314			−0.109157	−	0.994025i	$-0.534815\pi$
$719$	−4208.96			−0.218314			−0.109157	+	0.994025i	$0.534815\pi$
$720$		0			0
$721$	−18026.2			−0.931109
$722$		0			0
$723$	−3087.22	+	3087.22i	−0.158804	+	0.158804i
$724$		0			0
$725$	−11298.9	−	11298.9i	−0.578803	−	0.578803i
$726$		0			0
$727$		−	17761.1i		−	0.906083i	−0.891490	−	0.453041i	$-0.850339\pi$
$727$		−	17761.1i		−	0.906083i	0.891490	−	0.453041i	$-0.149661\pi$
$728$		0			0
$729$		−	9280.55i		−	0.471501i
$730$		0			0
$731$	7783.52	+	7783.52i	0.393822	+	0.393822i
$732$		0			0
$733$	3081.13	−	3081.13i	0.155258	−	0.155258i	−0.625204	−	0.780462i	$-0.714984\pi$
$733$	3081.13	−	3081.13i	0.155258	−	0.155258i	0.780462	+	0.625204i	$0.214984\pi$
$734$		0			0
$735$	5326.88			0.267326
$736$		0			0
$737$	12660.0			0.632750
$738$		0			0
$739$	18416.2	−	18416.2i	0.916714	−	0.916714i	−0.0800745	−	0.996789i	$-0.525516\pi$
$739$	18416.2	−	18416.2i	0.916714	−	0.916714i	0.996789	+	0.0800745i	$0.0255158\pi$
$740$		0			0
$741$	−3812.39	−	3812.39i	−0.189004	−	0.189004i
$742$		0			0
$743$			29048.2i			1.43429i	0.696925	+	0.717144i	$0.254551\pi$
$743$			29048.2i			1.43429i	−0.696925	+	0.717144i	$0.745449\pi$
$744$		0			0
$745$		−	4832.40i		−	0.237645i
$746$		0			0
$747$	17145.2	+	17145.2i	0.839772	+	0.839772i
$748$		0			0
$749$	1229.69	−	1229.69i	0.0599893	−	0.0599893i
$750$		0			0
$751$	2569.49			0.124850			0.0624248	−	0.998050i	$-0.480117\pi$
$751$	2569.49			0.124850			0.0624248	+	0.998050i	$0.480117\pi$
$752$		0			0
$753$	5174.35			0.250417
$754$		0			0
$755$	22785.3	−	22785.3i	1.09833	−	1.09833i
$756$		0			0
$757$	16400.4	+	16400.4i	0.787429	+	0.787429i	0.981072	−	0.193643i	$-0.0620305\pi$
$757$	16400.4	+	16400.4i	0.787429	+	0.787429i	−0.193643	+	0.981072i	$0.562030\pi$
$758$		0			0
$759$			5123.59i			0.245026i
$760$		0			0
$761$			4393.44i			0.209280i	0.994510	+	0.104640i	$0.0333691\pi$
$761$			4393.44i			0.209280i	−0.994510	+	0.104640i	$0.966631\pi$
$762$		0			0
$763$	8161.11	+	8161.11i	0.387224	+	0.387224i
$764$		0			0
$765$	7911.85	−	7911.85i	0.373926	−	0.373926i
$766$		0			0
$767$	−7461.54			−0.351266
$768$		0			0
$769$	−22868.9			−1.07240			−0.536200	−	0.844091i	$-0.680141\pi$
$769$	−22868.9			−1.07240			−0.536200	+	0.844091i	$0.680141\pi$
$770$		0			0
$771$	5744.94	−	5744.94i	0.268352	−	0.268352i
$772$		0			0
$773$	−17846.6	−	17846.6i	−0.830396	−	0.830396i	0.157175	−	0.987571i	$-0.449761\pi$
$773$	−17846.6	−	17846.6i	−0.830396	−	0.830396i	−0.987571	+	0.157175i	$0.949761\pi$
$774$		0			0
$775$			24739.4i			1.14666i
$776$		0			0
$777$			4197.57i			0.193805i
$778$		0			0
$779$	−48435.0	−	48435.0i	−2.22768	−	2.22768i
$780$		0			0
$781$	−1316.15	+	1316.15i	−0.0603017	+	0.0603017i
$782$		0			0
$783$	−14076.2			−0.642454
$784$		0			0
$785$	−11615.2			−0.528106
$786$		0			0
$787$	2229.13	−	2229.13i	0.100966	−	0.100966i	−0.654820	−	0.755785i	$-0.727255\pi$
$787$	2229.13	−	2229.13i	0.100966	−	0.100966i	0.755785	+	0.654820i	$0.227255\pi$
$788$		0			0
$789$	−9020.39	−	9020.39i	−0.407014	−	0.407014i
$790$		0			0
$791$			16091.8i			0.723336i
$792$		0			0
$793$			3740.85i			0.167518i
$794$		0			0
$795$	3797.78	+	3797.78i	0.169426	+	0.169426i
$796$		0			0
$797$	7260.72	−	7260.72i	0.322695	−	0.322695i	−0.527105	−	0.849800i	$-0.676723\pi$
$797$	7260.72	−	7260.72i	0.322695	−	0.322695i	0.849800	+	0.527105i	$0.176723\pi$
$798$		0			0
$799$	−6263.80			−0.277343
$800$		0			0
$801$	14741.9			0.650285
$802$		0			0
$803$	−9328.10	+	9328.10i	−0.409940	+	0.409940i
$804$		0			0
$805$	21258.3	+	21258.3i	0.930755	+	0.930755i
$806$		0			0
$807$		−	152.484i		−	0.00665140i
$808$		0			0
$809$			6066.32i			0.263635i	0.991274	+	0.131817i	$0.0420813\pi$
$809$			6066.32i			0.263635i	−0.991274	+	0.131817i	$0.957919\pi$
$810$		0			0
$811$	−16174.7	−	16174.7i	−0.700334	−	0.700334i	0.264148	−	0.964482i	$-0.414909\pi$
$811$	−16174.7	−	16174.7i	−0.700334	−	0.700334i	−0.964482	+	0.264148i	$0.914909\pi$
$812$		0			0
$813$	2541.38	−	2541.38i	0.109631	−	0.109631i
$814$		0			0
$815$	−22120.1			−0.950715
$816$		0			0
$817$	51873.9			2.22134
$818$		0			0
$819$	4326.92	−	4326.92i	0.184609	−	0.184609i
$820$		0			0
$821$	1109.32	+	1109.32i	0.0471566	+	0.0471566i	0.730292	−	0.683135i	$-0.239384\pi$
$821$	1109.32	+	1109.32i	0.0471566	+	0.0471566i	−0.683135	+	0.730292i	$0.739384\pi$
$822$		0			0
$823$			2875.65i			0.121797i	0.998144	+	0.0608985i	$0.0193966\pi$
$823$			2875.65i			0.121797i	−0.998144	+	0.0608985i	$0.980603\pi$
$824$		0			0
$825$		−	2575.35i		−	0.108681i
$826$		0			0
$827$	24943.8	+	24943.8i	1.04883	+	1.04883i	0.998745	+	0.0500833i	$0.0159487\pi$
$827$	24943.8	+	24943.8i	1.04883	+	1.04883i	0.0500833	+	0.998745i	$0.484051\pi$
$828$		0			0
$829$	4884.59	−	4884.59i	0.204643	−	0.204643i	−0.597343	−	0.801986i	$-0.703777\pi$
$829$	4884.59	−	4884.59i	0.204643	−	0.204643i	0.801986	+	0.597343i	$0.203777\pi$
$830$		0			0
$831$	−8283.69			−0.345798
$832$		0			0
$833$	−6946.61			−0.288939
$834$		0			0
$835$	−3524.22	+	3524.22i	−0.146061	+	0.146061i
$836$		0			0
$837$	15410.1	+	15410.1i	0.636382	+	0.636382i
$838$		0			0
$839$			24039.4i			0.989193i	0.869123	+	0.494596i	$0.164684\pi$
$839$			24039.4i			0.989193i	−0.869123	+	0.494596i	$0.835316\pi$
$840$		0			0
$841$		−	4659.77i		−	0.191060i
$842$		0			0
$843$	417.853	+	417.853i	0.0170719	+	0.0170719i
$844$		0			0
$845$	−17447.7	+	17447.7i	−0.710318	+	0.710318i
$846$		0			0
$847$	11317.4			0.459116
$848$		0			0
$849$	−11344.8			−0.458600
$850$		0			0
$851$	31628.8	−	31628.8i	1.27406	−	1.27406i
$852$		0			0
$853$	−15724.1	−	15724.1i	−0.631163	−	0.631163i	0.317197	−	0.948360i	$-0.397258\pi$
$853$	−15724.1	−	15724.1i	−0.631163	−	0.631163i	−0.948360	+	0.317197i	$0.897258\pi$
$854$		0			0
$855$		−	52729.2i		−	2.10912i
$856$		0			0
$857$		−	1796.16i		−	0.0715935i	−0.999359	−	0.0357968i	$-0.988603\pi$
$857$		−	1796.16i		−	0.0715935i	0.999359	−	0.0357968i	$-0.0113969\pi$
$858$		0			0
$859$	−863.555	−	863.555i	−0.0343005	−	0.0343005i	0.689749	−	0.724049i	$-0.257721\pi$
$859$	−863.555	−	863.555i	−0.0343005	−	0.0343005i	−0.724049	+	0.689749i	$0.757721\pi$
$860$		0			0
$861$	−5807.15	+	5807.15i	−0.229857	+	0.229857i
$862$		0			0
$863$	39589.1			1.56156			0.780781	−	0.624805i	$-0.214822\pi$
$863$	39589.1			1.56156			0.780781	+	0.624805i	$0.214822\pi$
$864$		0			0
$865$	27022.6			1.06219
$866$		0			0
$867$	−4489.86	+	4489.86i	−0.175875	+	0.175875i
$868$		0			0
$869$	−7828.13	−	7828.13i	−0.305582	−	0.305582i
$870$		0			0
$871$		−	17020.6i		−	0.662138i
$872$		0			0
$873$			18098.6i			0.701656i
$874$		0			0
$875$	3561.23	+	3561.23i	0.137590	+	0.137590i
$876$		0			0
$877$	18503.2	−	18503.2i	0.712438	−	0.712438i	−0.254607	−	0.967045i	$-0.581946\pi$
$877$	18503.2	−	18503.2i	0.712438	−	0.712438i	0.967045	+	0.254607i	$0.0819460\pi$
$878$		0			0
$879$	13789.9			0.529147
$880$		0			0
$881$	−35346.0			−1.35169			−0.675844	−	0.737045i	$-0.736221\pi$
$881$	−35346.0			−1.35169			−0.675844	+	0.737045i	$0.736221\pi$
$882$		0			0
$883$	−24633.1	+	24633.1i	−0.938811	+	0.938811i	−0.998233	−	0.0594221i	$-0.981074\pi$
$883$	−24633.1	+	24633.1i	−0.938811	+	0.938811i	0.0594221	+	0.998233i	$0.481074\pi$
$884$		0			0
$885$	5450.98	+	5450.98i	0.207042	+	0.207042i
$886$		0			0
$887$		−	7317.89i		−	0.277013i	−0.990362	−	0.138507i	$-0.955770\pi$
$887$		−	7317.89i		−	0.277013i	0.990362	−	0.138507i	$-0.0442302\pi$
$888$		0			0
$889$		−	12566.3i		−	0.474083i
$890$		0			0
$891$	6369.53	+	6369.53i	0.239492	+	0.239492i
$892$		0			0
$893$	−20872.8	+	20872.8i	−0.782174	+	0.782174i
$894$		0			0
$895$	−21699.6			−0.810431
$896$		0			0
$897$	6888.38			0.256406
$898$		0			0
$899$	−31801.6	+	31801.6i	−1.17980	+	1.17980i
$900$		0			0
$901$	−4952.57	−	4952.57i	−0.183123	−	0.183123i
$902$		0			0
$903$		−	6219.46i		−	0.229203i
$904$		0			0
$905$			34529.1i			1.26827i
$906$		0			0
$907$	7815.82	+	7815.82i	0.286130	+	0.286130i	0.835548	−	0.549418i	$-0.185150\pi$
$907$	7815.82	+	7815.82i	0.286130	+	0.286130i	−0.549418	+	0.835548i	$0.685150\pi$
$908$		0			0
$909$	−3894.09	+	3894.09i	−0.142089	+	0.142089i
$910$		0			0
$911$	39108.7			1.42232			0.711158	−	0.703032i	$-0.248171\pi$
$911$	39108.7			1.42232			0.711158	+	0.703032i	$0.248171\pi$
$912$		0			0
$913$	−16981.2			−0.615548
$914$		0			0
$915$	2732.85	−	2732.85i	0.0987381	−	0.0987381i
$916$		0			0
$917$	−3067.49	−	3067.49i	−0.110466	−	0.110466i
$918$		0			0
$919$		−	15863.3i		−	0.569402i	−0.958616	−	0.284701i	$-0.908106\pi$
$919$		−	15863.3i		−	0.569402i	0.958616	−	0.284701i	$-0.0918944\pi$
$920$		0			0
$921$		−	9103.92i		−	0.325716i
$922$		0			0
$923$	1769.49	+	1769.49i	0.0631025	+	0.0631025i
$924$		0			0
$925$	−15898.1	+	15898.1i	−0.565109	+	0.565109i
$926$		0			0
$927$	−40393.8			−1.43118
$928$		0			0
$929$	−47226.0			−1.66785			−0.833927	−	0.551875i	$-0.813912\pi$
$929$	−47226.0			−1.66785			−0.833927	+	0.551875i	$0.813912\pi$
$930$		0			0
$931$	−23148.1	+	23148.1i	−0.814876	+	0.814876i
$932$		0			0
$933$	7283.60	+	7283.60i	0.255578	+	0.255578i
$934$		0			0
$935$			7836.16i			0.274085i
$936$		0			0
$937$		−	24564.3i		−	0.856437i	−0.903675	−	0.428219i	$-0.859141\pi$
$937$		−	24564.3i		−	0.856437i	0.903675	−	0.428219i	$-0.140859\pi$
$938$		0			0
$939$	−4592.96	−	4592.96i	−0.159622	−	0.159622i
$940$		0			0
$941$	−12767.7	+	12767.7i	−0.442310	+	0.442310i	−0.892788	−	0.450478i	$-0.851254\pi$
$941$	−12767.7	+	12767.7i	−0.442310	+	0.442310i	0.450478	+	0.892788i	$0.351254\pi$
$942$		0			0
$943$	87514.2			3.02212
$944$		0			0
$945$	−13311.8			−0.458237
$946$		0			0
$947$	348.162	−	348.162i	0.0119469	−	0.0119469i	−0.701108	−	0.713055i	$-0.747311\pi$
$947$	348.162	−	348.162i	0.0119469	−	0.0119469i	0.713055	+	0.701108i	$0.247311\pi$
$948$		0			0
$949$	12541.1	+	12541.1i	0.428980	+	0.428980i
$950$		0			0
$951$		−	15412.8i		−	0.525547i
$952$		0			0
$953$			37577.4i			1.27728i	0.769504	+	0.638642i	$0.220504\pi$
$953$			37577.4i			1.27728i	−0.769504	+	0.638642i	$0.779496\pi$
$954$		0			0
$955$	1071.43	+	1071.43i	0.0363043	+	0.0363043i
$956$		0			0
$957$	3310.52	−	3310.52i	0.111822	−	0.111822i
$958$		0			0
$959$	24387.5			0.821182
$960$		0			0
$961$	39839.7			1.33731
$962$		0			0
$963$	2755.55	−	2755.55i	0.0922079	−	0.0922079i
$964$		0			0
$965$	−22578.5	−	22578.5i	−0.753188	−	0.753188i
$966$		0			0
$967$		−	41535.5i		−	1.38127i	−0.723202	−	0.690637i	$-0.757330\pi$
$967$		−	41535.5i		−	1.38127i	0.723202	−	0.690637i	$-0.242670\pi$
$968$		0			0
$969$		−	7263.96i		−	0.240817i
$970$		0			0
$971$	−26608.0	−	26608.0i	−0.879393	−	0.879393i	0.114079	−	0.993472i	$-0.463608\pi$
$971$	−26608.0	−	26608.0i	−0.879393	−	0.879393i	−0.993472	+	0.114079i	$0.963608\pi$
$972$		0			0
$973$	10128.6	−	10128.6i	0.333720	−	0.333720i
$974$		0			0
$975$	−3462.41			−0.113729
$976$		0			0
$977$	41599.6			1.36222			0.681110	−	0.732181i	$-0.261498\pi$
$977$	41599.6			1.36222			0.681110	+	0.732181i	$0.261498\pi$
$978$		0			0
$979$	−7300.42	+	7300.42i	−0.238327	+	0.238327i
$980$		0			0
$981$	18287.8	+	18287.8i	0.595192	+	0.595192i
$982$		0			0
$983$			28284.2i			0.917726i	0.888507	+	0.458863i	$0.151743\pi$
$983$			28284.2i			0.917726i	−0.888507	+	0.458863i	$0.848257\pi$
$984$		0			0
$985$			63216.8i			2.04493i
$986$		0			0
$987$	2502.56	+	2502.56i	0.0807065	+	0.0807065i
$988$		0			0
$989$	−46863.9	+	46863.9i	−1.50676	+	1.50676i
$990$		0			0
$991$	−40254.3			−1.29033			−0.645167	−	0.764042i	$-0.723212\pi$
$991$	−40254.3			−1.29033			−0.645167	+	0.764042i	$0.723212\pi$
$992$		0			0
$993$	1113.75			0.0355928
$994$		0			0
$995$	−34522.0	+	34522.0i	−1.09992	+	1.09992i
$996$		0			0
$997$	34773.0	+	34773.0i	1.10459	+	1.10459i	0.993850	+	0.110737i	$0.0353212\pi$
$997$	34773.0	+	34773.0i	1.10459	+	1.10459i	0.110737	+	0.993850i	$0.464679\pi$
$998$		0			0
$999$			19805.8i			0.627255i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.4.e.c.193.5	yes	16
4.3	odd	2	inner	256.4.e.c.193.4	yes	16
8.3	odd	2		256.4.e.d.193.5	yes	16
8.5	even	2		256.4.e.d.193.4	yes	16
16.3	odd	4	inner	256.4.e.c.65.4	✓	16
16.5	even	4		256.4.e.d.65.4	yes	16
16.11	odd	4		256.4.e.d.65.5	yes	16
16.13	even	4	inner	256.4.e.c.65.5	yes	16
32.3	odd	8		1024.4.a.l.1.3		8
32.5	even	8		1024.4.b.l.513.10		16
32.11	odd	8		1024.4.b.l.513.9		16
32.13	even	8		1024.4.a.l.1.4		8
32.19	odd	8		1024.4.a.k.1.6		8
32.21	even	8		1024.4.b.l.513.7		16
32.27	odd	8		1024.4.b.l.513.8		16
32.29	even	8		1024.4.a.k.1.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
256.4.e.c.65.4	✓	16	16.3	odd	4	inner
256.4.e.c.65.5	yes	16	16.13	even	4	inner
256.4.e.c.193.4	yes	16	4.3	odd	2	inner
256.4.e.c.193.5	yes	16	1.1	even	1	trivial
256.4.e.d.65.4	yes	16	16.5	even	4
256.4.e.d.65.5	yes	16	16.11	odd	4
256.4.e.d.193.4	yes	16	8.5	even	2
256.4.e.d.193.5	yes	16	8.3	odd	2
1024.4.a.k.1.5		8	32.29	even	8
1024.4.a.k.1.6		8	32.19	odd	8
1024.4.a.l.1.3		8	32.3	odd	8
1024.4.a.l.1.4		8	32.13	even	8
1024.4.b.l.513.7		16	32.21	even	8
1024.4.b.l.513.8		16	32.27	odd	8
1024.4.b.l.513.9		16	32.11	odd	8
1024.4.b.l.513.10		16	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:	\( N \)	\(=\)	\( 256 = 2^{8} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	256.e (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.13572 - 1.13572i) q^{3} +(10.4583 + 10.4583i) q^{5} +10.8978i q^{7} +24.4203i q^{9} +O(q^{10})\)	\(q+(1.13572 - 1.13572i) q^{3} +(10.4583 + 10.4583i) q^{5} +10.8978i q^{7} +24.4203i q^{9} +(-12.0933 - 12.0933i) q^{11} +(-16.2588 + 16.2588i) q^{13} +23.7555 q^{15} -30.9788 q^{17} +(-103.230 + 103.230i) q^{19} +(12.3769 + 12.3769i) q^{21} -186.521i q^{23} +93.7538i q^{25} +(58.3991 + 58.3991i) q^{27} +(-120.517 + 120.517i) q^{29} +263.876 q^{31} -27.4693 q^{33} +(-113.973 + 113.973i) q^{35} +(169.573 + 169.573i) q^{37} +36.9309i q^{39} +469.193i q^{41} +(-251.253 - 251.253i) q^{43} +(-255.396 + 255.396i) q^{45} +202.196 q^{47} +224.238 q^{49} +(-35.1833 + 35.1833i) q^{51} +(159.870 + 159.870i) q^{53} -252.952i q^{55} +234.482i q^{57} +(229.462 + 229.462i) q^{59} +(115.041 - 115.041i) q^{61} -266.128 q^{63} -340.080 q^{65} +(-523.429 + 523.429i) q^{67} +(-211.835 - 211.835i) q^{69} -108.833i q^{71} -771.343i q^{73} +(106.478 + 106.478i) q^{75} +(131.791 - 131.791i) q^{77} +647.309 q^{79} -526.697 q^{81} +(702.088 - 702.088i) q^{83} +(-323.987 - 323.987i) q^{85} +273.748i q^{87} -603.673i q^{89} +(-177.185 - 177.185i) q^{91} +(299.690 - 299.690i) q^{93} -2159.24 q^{95} +741.131 q^{97} +(295.323 - 295.323i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 32 q^{5}+O(q^{10}) \) 16 * q - 32 * q^5	\( 16 q - 32 q^{5} - 320 q^{13} - 384 q^{17} - 224 q^{21} - 928 q^{29} - 2432 q^{33} - 640 q^{37} - 896 q^{45} - 2832 q^{49} - 64 q^{53} - 1024 q^{61} + 2208 q^{65} - 32 q^{69} - 1056 q^{77} + 4208 q^{81} - 1824 q^{85} + 3776 q^{93} + 4480 q^{97}+O(q^{100}) \) 16 * q - 32 * q^5 - 320 * q^13 - 384 * q^17 - 224 * q^21 - 928 * q^29 - 2432 * q^33 - 640 * q^37 - 896 * q^45 - 2832 * q^49 - 64 * q^53 - 1024 * q^61 + 2208 * q^65 - 32 * q^69 - 1056 * q^77 + 4208 * q^81 - 1824 * q^85 + 3776 * q^93 + 4480 * q^97

Introduction

L-functions

Modular forms

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