LMFDB - Embedded newform 448.4.j.b.335.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,4,Mod(111,448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(448, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3, 2]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("448.111");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	448.j (of order $4$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$26.4328556826$
Analytic rank:	$0$
Dimension:	$88$
Relative dimension:	$44$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			335.2
Character	$\chi$	$=$	448.335
Dual form			448.4.j.b.111.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-6.75145 + 6.75145i) q^{3} +(-10.7426 + 10.7426i) q^{5} +(-7.90955 + 16.7463i) q^{7} -64.1641i q^{9} +(-5.28964 + 5.28964i) q^{11} +(-50.4866 - 50.4866i) q^{13} -145.056i q^{15} -17.6955i q^{17} +(-63.5461 + 63.5461i) q^{19} +(-59.6609 - 166.463i) q^{21} -7.51181 q^{23} -105.807i q^{25} +(250.911 + 250.911i) q^{27} +(-153.894 + 153.894i) q^{29} -60.8015 q^{31} -71.4254i q^{33} +(-94.9298 - 264.868i) q^{35} +(129.137 + 129.137i) q^{37} +681.716 q^{39} +48.6217 q^{41} +(-123.020 + 123.020i) q^{43} +(689.289 + 689.289i) q^{45} -254.748 q^{47} +(-217.878 - 264.912i) q^{49} +(119.470 + 119.470i) q^{51} +(498.959 + 498.959i) q^{53} -113.649i q^{55} -858.056i q^{57} +(122.203 + 122.203i) q^{59} +(-228.300 - 228.300i) q^{61} +(1074.51 + 507.509i) q^{63} +1084.72 q^{65} +(-360.174 - 360.174i) q^{67} +(50.7156 - 50.7156i) q^{69} -605.544 q^{71} +913.990 q^{73} +(714.350 + 714.350i) q^{75} +(-46.7433 - 130.421i) q^{77} +885.306i q^{79} -1655.60 q^{81} +(-108.532 + 108.532i) q^{83} +(190.095 + 190.095i) q^{85} -2078.01i q^{87} -269.774 q^{89} +(1244.79 - 446.138i) q^{91} +(410.498 - 410.498i) q^{93} -1365.30i q^{95} +1452.58i q^{97} +(339.405 + 339.405i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 88 q + 4 q^{7} - 112 q^{11} + 52 q^{21} - 160 q^{23} + 528 q^{29} - 476 q^{35} - 896 q^{37} + 8 q^{39} - 40 q^{43} - 1376 q^{49} - 1504 q^{51} - 1560 q^{53} - 8 q^{65} - 648 q^{67} + 456 q^{71} + 292 q^{77}+ \cdots - 7592 q^{99}+O(q^{100}) $ 88 * q + 4 * q^7 - 112 * q^11 + 52 * q^21 - 160 * q^23 + 528 * q^29 - 476 * q^35 - 896 * q^37 + 8 * q^39 - 40 * q^43 - 1376 * q^49 - 1504 * q^51 - 1560 * q^53 - 8 * q^65 - 648 * q^67 + 456 * q^71 + 292 * q^77 - 1952 * q^81 + 496 * q^85 + 1964 * q^91 - 112 * q^93 - 7592 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/448\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$197$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{1}{4}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−6.75145	+	6.75145i	−1.29932	+	1.29932i	−0.370474	+	0.928843i	$0.620805\pi$
$3$	−6.75145	+	6.75145i	−1.29932	+	1.29932i	−0.928843	+	0.370474i	$0.879195\pi$
$4$		0			0
$5$	−10.7426	+	10.7426i	−0.960848	+	0.960848i	−0.999262	−	0.0384144i	$-0.987769\pi$
$5$	−10.7426	+	10.7426i	−0.960848	+	0.960848i	0.0384144	+	0.999262i	$0.487769\pi$
$6$		0			0
$7$	−7.90955	+	16.7463i	−0.427076	+	0.904216i
$7$	−7.90955	+	16.7463i	−0.427076	+	0.904216i
$8$		0			0
$9$		−	64.1641i		−	2.37645i
$10$		0			0
$11$	−5.28964	+	5.28964i	−0.144990	+	0.144990i	−0.775876	−	0.630886i	$-0.782692\pi$
$11$	−5.28964	+	5.28964i	−0.144990	+	0.144990i	0.630886	+	0.775876i	$0.282692\pi$
$12$		0			0
$13$	−50.4866	−	50.4866i	−1.07711	−	1.07711i	−0.996767	−	0.0803460i	$-0.974397\pi$
$13$	−50.4866	−	50.4866i	−1.07711	−	1.07711i	−0.0803460	−	0.996767i	$-0.525603\pi$
$14$		0			0
$15$		−	145.056i		−	2.49689i
$16$		0			0
$17$		−	17.6955i		−	0.252458i	−0.992001	−	0.126229i	$-0.959713\pi$
$17$		−	17.6955i		−	0.252458i	0.992001	−	0.126229i	$-0.0402874\pi$
$18$		0			0
$19$	−63.5461	+	63.5461i	−0.767288	+	0.767288i	−0.977628	−	0.210340i	$-0.932543\pi$
$19$	−63.5461	+	63.5461i	−0.767288	+	0.767288i	0.210340	+	0.977628i	$0.432543\pi$
$20$		0			0
$21$	−59.6609	−	166.463i	−0.619956	−	1.72977i
$22$		0			0
$23$	−7.51181			−0.0681009			−0.0340504	−	0.999420i	$-0.510841\pi$
$23$	−7.51181			−0.0681009			−0.0340504	+	0.999420i	$0.510841\pi$
$24$		0			0
$25$		−	105.807i		−	0.846456i
$26$		0			0
$27$	250.911	+	250.911i	1.78844	+	1.78844i
$28$		0			0
$29$	−153.894	+	153.894i	−0.985425	+	0.985425i	−0.999895	−	0.0144699i	$-0.995394\pi$
$29$	−153.894	+	153.894i	−0.985425	+	0.985425i	0.0144699	+	0.999895i	$0.495394\pi$
$30$		0			0
$31$	−60.8015			−0.352267			−0.176133	−	0.984366i	$-0.556359\pi$
$31$	−60.8015			−0.352267			−0.176133	+	0.984366i	$0.556359\pi$
$32$		0			0
$33$		−	71.4254i		−	0.376775i
$34$		0			0
$35$	−94.9298	−	264.868i	−0.458459	−	1.27917i
$36$		0			0
$37$	129.137	+	129.137i	0.573784	+	0.573784i	0.933184	−	0.359399i	$-0.117018\pi$
$37$	129.137	+	129.137i	0.573784	+	0.573784i	−0.359399	+	0.933184i	$0.617018\pi$
$38$		0			0
$39$	681.716			2.79902
$40$		0			0
$41$	48.6217			0.185206			0.0926029	−	0.995703i	$-0.470481\pi$
$41$	48.6217			0.185206			0.0926029	+	0.995703i	$0.470481\pi$
$42$		0			0
$43$	−123.020	+	123.020i	−0.436289	+	0.436289i	−0.890761	−	0.454472i	$-0.849828\pi$
$43$	−123.020	+	123.020i	−0.436289	+	0.436289i	0.454472	+	0.890761i	$0.349828\pi$
$44$		0			0
$45$	689.289	+	689.289i	2.28340	+	2.28340i
$46$		0			0
$47$	−254.748			−0.790612			−0.395306	−	0.918550i	$-0.629361\pi$
$47$	−254.748			−0.790612			−0.395306	+	0.918550i	$0.629361\pi$
$48$		0			0
$49$	−217.878	−	264.912i	−0.635213	−	0.772337i
$50$		0			0
$51$	119.470	+	119.470i	0.328023	+	0.328023i
$52$		0			0
$53$	498.959	+	498.959i	1.29316	+	1.29316i	0.932823	+	0.360334i	$0.117337\pi$
$53$	498.959	+	498.959i	1.29316	+	1.29316i	0.360334	+	0.932823i	$0.382663\pi$
$54$		0			0
$55$		−	113.649i		−	0.278626i
$56$		0			0
$57$		−	858.056i		−	1.99390i
$58$		0			0
$59$	122.203	+	122.203i	0.269652	+	0.269652i	0.828960	−	0.559308i	$-0.188933\pi$
$59$	122.203	+	122.203i	0.269652	+	0.269652i	−0.559308	+	0.828960i	$0.688933\pi$
$60$		0			0
$61$	−228.300	−	228.300i	−0.479194	−	0.479194i	0.425680	−	0.904874i	$-0.360035\pi$
$61$	−228.300	−	228.300i	−0.479194	−	0.479194i	−0.904874	+	0.425680i	$0.860035\pi$
$62$		0			0
$63$	1074.51	+	507.509i	2.14882	+	1.01492i
$64$		0			0
$65$	1084.72			2.06988
$66$		0			0
$67$	−360.174	−	360.174i	−0.656750	−	0.656750i	0.297860	−	0.954610i	$-0.403727\pi$
$67$	−360.174	−	360.174i	−0.656750	−	0.656750i	−0.954610	+	0.297860i	$0.903727\pi$
$68$		0			0
$69$	50.7156	−	50.7156i	0.0884846	−	0.0884846i
$70$		0			0
$71$	−605.544			−1.01218			−0.506091	−	0.862480i	$-0.668910\pi$
$71$	−605.544			−1.01218			−0.506091	+	0.862480i	$0.668910\pi$
$72$		0			0
$73$	913.990			1.46540			0.732702	−	0.680550i	$-0.238259\pi$
$73$	913.990			1.46540			0.732702	+	0.680550i	$0.238259\pi$
$74$		0			0
$75$	714.350	+	714.350i	1.09981	+	1.09981i
$76$		0			0
$77$	−46.7433	−	130.421i	−0.0691804	−	0.193024i
$78$		0			0
$79$			885.306i			1.26082i	0.776263	+	0.630410i	$0.217113\pi$
$79$			885.306i			1.26082i	−0.776263	+	0.630410i	$0.782887\pi$
$80$		0			0
$81$	−1655.60			−2.27105
$82$		0			0
$83$	−108.532	+	108.532i	−0.143530	+	0.143530i	−0.775221	−	0.631691i	$-0.782361\pi$
$83$	−108.532	+	108.532i	−0.143530	+	0.143530i	0.631691	+	0.775221i	$0.282361\pi$
$84$		0			0
$85$	190.095	+	190.095i	0.242573	+	0.242573i
$86$		0			0
$87$		−	2078.01i		−	2.56076i
$88$		0			0
$89$	−269.774			−0.321304			−0.160652	−	0.987011i	$-0.551360\pi$
$89$	−269.774			−0.321304			−0.160652	+	0.987011i	$0.551360\pi$
$90$		0			0
$91$	1244.79	−	446.138i	1.43395	−	0.513934i
$92$		0			0
$93$	410.498	−	410.498i	0.457706	−	0.457706i
$94$		0			0
$95$		−	1365.30i		−	1.47449i
$96$		0			0
$97$			1452.58i			1.52049i	0.649639	+	0.760243i	$0.274920\pi$
$97$			1452.58i			1.52049i	−0.649639	+	0.760243i	$0.725080\pi$
$98$		0			0
$99$	339.405	+	339.405i	0.344560	+	0.344560i
$100$		0			0
$101$	32.6448	−	32.6448i	0.0321612	−	0.0321612i	−0.690843	−	0.723004i	$-0.742761\pi$
$101$	32.6448	−	32.6448i	0.0321612	−	0.0321612i	0.723004	+	0.690843i	$0.242761\pi$
$102$		0			0
$103$			1486.69i			1.42221i	0.703086	+	0.711105i	$0.251805\pi$
$103$			1486.69i			1.42221i	−0.703086	+	0.711105i	$0.748195\pi$
$104$		0			0
$105$	2429.16	+	1147.33i	2.25773	+	1.06636i
$106$		0			0
$107$	973.114	−	973.114i	0.879201	−	0.879201i	−0.114251	−	0.993452i	$-0.536447\pi$
$107$	973.114	−	973.114i	0.879201	−	0.879201i	0.993452	+	0.114251i	$0.0364468\pi$
$108$		0			0
$109$	266.730	−	266.730i	0.234386	−	0.234386i	−0.580134	−	0.814521i	$-0.697000\pi$
$109$	266.730	−	266.730i	0.234386	−	0.234386i	0.814521	+	0.580134i	$0.197000\pi$
$110$		0			0
$111$	−1743.73			−1.49106
$112$		0			0
$113$	93.8639			0.0781414			0.0390707	−	0.999236i	$-0.487560\pi$
$113$	93.8639			0.0781414			0.0390707	+	0.999236i	$0.487560\pi$
$114$		0			0
$115$	80.6964	−	80.6964i	0.0654346	−	0.0654346i
$116$		0			0
$117$	−3239.43	+	3239.43i	−2.55970	+	2.55970i
$118$		0			0
$119$	296.334	+	139.963i	0.228276	+	0.107819i
$120$		0			0
$121$			1275.04i			0.957956i
$122$		0			0
$123$	−328.267	+	328.267i	−0.240641	+	0.240641i
$124$		0			0
$125$	−206.183	−	206.183i	−0.147532	−	0.147532i
$126$		0			0
$127$		−	18.1807i		−	0.0127030i	−0.999980	−	0.00635148i	$-0.997978\pi$
$127$		−	18.1807i		−	0.0127030i	0.999980	−	0.00635148i	$-0.00202175\pi$
$128$		0			0
$129$		−	1661.13i		−	1.13376i
$130$		0			0
$131$	989.922	−	989.922i	0.660228	−	0.660228i	−0.295206	−	0.955434i	$-0.595388\pi$
$131$	989.922	−	989.922i	0.660228	−	0.660228i	0.955434	+	0.295206i	$0.0953881\pi$
$132$		0			0
$133$	−561.541	−	1566.78i	−0.366104	−	1.02148i
$134$		0			0
$135$	−5390.88			−3.43684
$136$		0			0
$137$		−	2748.38i		−	1.71394i	−0.515363	−	0.856972i	$-0.672343\pi$
$137$		−	2748.38i		−	1.71394i	0.515363	−	0.856972i	$-0.327657\pi$
$138$		0			0
$139$	−1356.64	−	1356.64i	−0.827830	−	0.827830i	0.159387	−	0.987216i	$-0.449048\pi$
$139$	−1356.64	−	1356.64i	−0.827830	−	0.827830i	−0.987216	+	0.159387i	$0.949048\pi$
$140$		0			0
$141$	1719.91	−	1719.91i	1.02726	−	1.02726i
$142$		0			0
$143$	534.112			0.312341
$144$		0			0
$145$		−	3306.44i		−	1.89369i
$146$		0			0
$147$	3259.53	+	317.546i	1.82885	+	0.178169i
$148$		0			0
$149$	−1086.44	−	1086.44i	−0.597345	−	0.597345i	0.342260	−	0.939605i	$-0.388808\pi$
$149$	−1086.44	−	1086.44i	−0.597345	−	0.597345i	−0.939605	+	0.342260i	$0.888808\pi$
$150$		0			0
$151$	−270.825			−0.145957			−0.0729784	−	0.997334i	$-0.523250\pi$
$151$	−270.825			−0.145957			−0.0729784	+	0.997334i	$0.523250\pi$
$152$		0			0
$153$	−1135.41			−0.599953
$154$		0			0
$155$	653.166	−	653.166i	0.338475	−	0.338475i
$156$		0			0
$157$	360.348	+	360.348i	0.183178	+	0.183178i	0.792739	−	0.609561i	$-0.208654\pi$
$157$	360.348	+	360.348i	0.183178	+	0.183178i	−0.609561	+	0.792739i	$0.708654\pi$
$158$		0			0
$159$	−6737.39			−3.36044
$160$		0			0
$161$	59.4151	−	125.795i	0.0290842	−	0.0615779i
$162$		0			0
$163$	−1755.20	−	1755.20i	−0.843422	−	0.843422i	0.145880	−	0.989302i	$-0.453399\pi$
$163$	−1755.20	−	1755.20i	−0.843422	−	0.843422i	−0.989302	+	0.145880i	$0.953399\pi$
$164$		0			0
$165$	767.295	+	767.295i	0.362023	+	0.362023i
$166$		0			0
$167$		−	1281.85i		−	0.593968i	−0.954882	−	0.296984i	$-0.904019\pi$
$167$		−	1281.85i		−	0.593968i	0.954882	−	0.296984i	$-0.0959808\pi$
$168$		0			0
$169$			2900.80i			1.32035i
$170$		0			0
$171$	4077.37	+	4077.37i	1.82342	+	1.82342i
$172$		0			0
$173$	54.8132	+	54.8132i	0.0240888	+	0.0240888i	0.719049	−	0.694960i	$-0.244578\pi$
$173$	54.8132	+	54.8132i	0.0240888	+	0.0240888i	−0.694960	+	0.719049i	$0.744578\pi$
$174$		0			0
$175$	1771.88	+	836.886i	0.765379	+	0.361501i
$176$		0			0
$177$	−1650.10			−0.700727
$178$		0			0
$179$	1497.55	+	1497.55i	0.625320	+	0.625320i	0.946887	−	0.321567i	$-0.104210\pi$
$179$	1497.55	+	1497.55i	0.625320	+	0.625320i	−0.321567	+	0.946887i	$0.604210\pi$
$180$		0			0
$181$	−3092.20	+	3092.20i	−1.26984	+	1.26984i	−0.323675	+	0.946168i	$0.604918\pi$
$181$	−3092.20	+	3092.20i	−1.26984	+	1.26984i	−0.946168	+	0.323675i	$0.895082\pi$
$182$		0			0
$183$	3082.72			1.24525
$184$		0			0
$185$	−2774.54			−1.10264
$186$		0			0
$187$	93.6027	+	93.6027i	0.0366038	+	0.0366038i
$188$		0			0
$189$	−6186.44	+	2217.24i	−2.38094	+	0.853337i
$190$		0			0
$191$			880.871i			0.333705i	0.985982	+	0.166852i	$0.0533603\pi$
$191$			880.871i			0.333705i	−0.985982	+	0.166852i	$0.946640\pi$
$192$		0			0
$193$	4562.97			1.70181			0.850905	−	0.525319i	$-0.176054\pi$
$193$	4562.97			1.70181			0.850905	+	0.525319i	$0.176054\pi$
$194$		0			0
$195$	−7323.40	+	7323.40i	−2.68943	+	2.68943i
$196$		0			0
$197$	1121.88	+	1121.88i	0.405738	+	0.405738i	0.880249	−	0.474512i	$-0.157375\pi$
$197$	1121.88	+	1121.88i	0.405738	+	0.405738i	−0.474512	+	0.880249i	$0.657375\pi$
$198$		0			0
$199$		−	1396.02i		−	0.497293i	−0.968594	−	0.248646i	$-0.920014\pi$
$199$		−	1396.02i		−	0.497293i	0.968594	−	0.248646i	$-0.0799857\pi$
$200$		0			0
$201$	4863.39			1.70665
$202$		0			0
$203$	−1359.92	−	3794.38i	−0.470186	−	1.31189i
$204$		0			0
$205$	−522.324	+	522.324i	−0.177955	+	0.177955i
$206$		0			0
$207$			481.988i			0.161838i
$208$		0			0
$209$		−	672.272i		−	0.222498i
$210$		0			0
$211$	763.850	+	763.850i	0.249221	+	0.249221i	0.820651	−	0.571430i	$-0.193611\pi$
$211$	763.850	+	763.850i	0.249221	+	0.249221i	−0.571430	+	0.820651i	$0.693611\pi$
$212$		0			0
$213$	4088.30	−	4088.30i	1.31514	−	1.31514i
$214$		0			0
$215$		−	2643.12i		−	0.838415i
$216$		0			0
$217$	480.913	−	1018.20i	0.150445	−	0.318525i
$218$		0			0
$219$	−6170.76	+	6170.76i	−1.90402	+	1.90402i
$220$		0			0
$221$	−893.385	+	893.385i	−0.271926	+	0.271926i
$222$		0			0
$223$	3793.85			1.13926			0.569631	−	0.821901i	$-0.307086\pi$
$223$	3793.85			1.13926			0.569631	+	0.821901i	$0.307086\pi$
$224$		0			0
$225$	−6789.01			−2.01156
$226$		0			0
$227$	1146.33	−	1146.33i	0.335173	−	0.335173i	−0.519374	−	0.854547i	$-0.673835\pi$
$227$	1146.33	−	1146.33i	0.335173	−	0.335173i	0.854547	+	0.519374i	$0.173835\pi$
$228$		0			0
$229$	−2144.18	+	2144.18i	−0.618740	+	0.618740i	−0.945208	−	0.326468i	$-0.894141\pi$
$229$	−2144.18	+	2144.18i	−0.618740	+	0.618740i	0.326468	+	0.945208i	$0.394141\pi$
$230$		0			0
$231$	1196.11	+	564.943i	0.340686	+	0.160911i
$232$		0			0
$233$			614.860i			0.172879i	0.996257	+	0.0864395i	$0.0275489\pi$
$233$			614.860i			0.172879i	−0.996257	+	0.0864395i	$0.972451\pi$
$234$		0			0
$235$	2736.65	−	2736.65i	0.759657	−	0.759657i
$236$		0			0
$237$	−5977.10	−	5977.10i	−1.63820	−	1.63820i
$238$		0			0
$239$			4263.99i			1.15404i	0.816731	+	0.577018i	$0.195784\pi$
$239$			4263.99i			1.15404i	−0.816731	+	0.577018i	$0.804216\pi$
$240$		0			0
$241$		−	2731.19i		−	0.730006i	−0.931006	−	0.365003i	$-0.881068\pi$
$241$		−	2731.19i		−	0.730006i	0.931006	−	0.365003i	$-0.118932\pi$
$242$		0			0
$243$	4403.08	−	4403.08i	1.16238	−	1.16238i
$244$		0			0
$245$	5186.42	+	505.266i	1.35244	+	0.131756i
$246$		0			0
$247$	6416.45			1.65291
$248$		0			0
$249$		−	1465.50i		−	0.372981i
$250$		0			0
$251$	−975.121	−	975.121i	−0.245216	−	0.245216i	0.573788	−	0.819004i	$-0.305473\pi$
$251$	−975.121	−	975.121i	−0.245216	−	0.245216i	−0.819004	+	0.573788i	$0.805473\pi$
$252$		0			0
$253$	39.7348	−	39.7348i	0.00987393	−	0.00987393i
$254$		0			0
$255$	−2566.84			−0.630359
$256$		0			0
$257$		−	6513.55i		−	1.58095i	−0.612495	−	0.790475i	$-0.709834\pi$
$257$		−	6513.55i		−	1.58095i	0.612495	−	0.790475i	$-0.290166\pi$
$258$		0			0
$259$	−3183.99	+	1141.15i	−0.763874	+	0.273776i
$260$		0			0
$261$	9874.45	+	9874.45i	2.34181	+	2.34181i
$262$		0			0
$263$	−802.056			−0.188049			−0.0940245	−	0.995570i	$-0.529973\pi$
$263$	−802.056			−0.188049			−0.0940245	+	0.995570i	$0.529973\pi$
$264$		0			0
$265$	−10720.2			−2.48505
$266$		0			0
$267$	1821.37	−	1821.37i	0.417475	−	0.417475i
$268$		0			0
$269$	2029.90	+	2029.90i	0.460094	+	0.460094i	0.898686	−	0.438592i	$-0.144523\pi$
$269$	2029.90	+	2029.90i	0.460094	+	0.460094i	−0.438592	+	0.898686i	$0.644523\pi$
$270$		0			0
$271$	−2449.85			−0.549143			−0.274572	−	0.961567i	$-0.588536\pi$
$271$	−2449.85			−0.549143			−0.274572	+	0.961567i	$0.588536\pi$
$272$		0			0
$273$	−5392.07	+	11416.2i	−1.19539	+	2.53092i
$274$		0			0
$275$	559.681	+	559.681i	0.122727	+	0.122727i
$276$		0			0
$277$	−3804.74	−	3804.74i	−0.825288	−	0.825288i	0.161573	−	0.986861i	$-0.448343\pi$
$277$	−3804.74	−	3804.74i	−0.825288	−	0.825288i	−0.986861	+	0.161573i	$0.948343\pi$
$278$		0			0
$279$			3901.27i			0.837144i
$280$		0			0
$281$		−	5469.38i		−	1.16112i	−0.814216	−	0.580562i	$-0.802833\pi$
$281$		−	5469.38i		−	1.16112i	0.814216	−	0.580562i	$-0.197167\pi$
$282$		0			0
$283$	2997.42	+	2997.42i	0.629606	+	0.629606i	0.947969	−	0.318363i	$-0.103133\pi$
$283$	2997.42	+	2997.42i	0.629606	+	0.629606i	−0.318363	+	0.947969i	$0.603133\pi$
$284$		0			0
$285$	9217.75	+	9217.75i	1.91583	+	1.91583i
$286$		0			0
$287$	−384.576	+	814.235i	−0.0790970	+	0.167466i
$288$		0			0
$289$	4599.87			0.936265
$290$		0			0
$291$	−9807.02	−	9807.02i	−1.97559	−	1.97559i
$292$		0			0
$293$	4777.03	−	4777.03i	0.952482	−	0.952482i	−0.0464390	−	0.998921i	$-0.514787\pi$
$293$	4777.03	−	4777.03i	0.952482	−	0.952482i	0.998921	+	0.0464390i	$0.0147873\pi$
$294$		0			0
$295$	−2625.56			−0.518190
$296$		0			0
$297$	−2654.46			−0.518611
$298$		0			0
$299$	379.246	+	379.246i	0.0733524	+	0.0733524i
$300$		0			0
$301$	−1087.10	−	3033.18i	−0.208171	−	0.580828i
$302$		0			0
$303$			440.799i			0.0835751i
$304$		0			0
$305$	4905.08			0.920866
$306$		0			0
$307$	1382.11	−	1382.11i	0.256941	−	0.256941i	−0.566868	−	0.823809i	$-0.691845\pi$
$307$	1382.11	−	1382.11i	0.256941	−	0.256941i	0.823809	+	0.566868i	$0.191845\pi$
$308$		0			0
$309$	−10037.3	−	10037.3i	−1.84790	−	1.84790i
$310$		0			0
$311$		−	4113.03i		−	0.749931i	−0.927039	−	0.374966i	$-0.877654\pi$
$311$		−	4113.03i		−	0.749931i	0.927039	−	0.374966i	$-0.122346\pi$
$312$		0			0
$313$	−1183.14			−0.213659			−0.106829	−	0.994277i	$-0.534070\pi$
$313$	−1183.14			−0.213659			−0.106829	+	0.994277i	$0.534070\pi$
$314$		0			0
$315$	−16995.0	+	6091.08i	−3.03988	+	1.08950i
$316$		0			0
$317$	1538.47	−	1538.47i	0.272584	−	0.272584i	−0.557556	−	0.830140i	$-0.688261\pi$
$317$	1538.47	−	1538.47i	0.272584	−	0.272584i	0.830140	+	0.557556i	$0.188261\pi$
$318$		0			0
$319$		−	1628.08i		−	0.285753i
$320$		0			0
$321$			13139.9i			2.28472i
$322$		0			0
$323$	1124.48	+	1124.48i	0.193708	+	0.193708i
$324$		0			0
$325$	−5341.84	+	5341.84i	−0.911729	+	0.911729i
$326$		0			0
$327$			3601.63i			0.609084i
$328$		0			0
$329$	2014.94	−	4266.08i	0.337651	−	0.714884i
$330$		0			0
$331$	−1870.74	+	1870.74i	−0.310651	+	0.310651i	−0.845162	−	0.534511i	$-0.820496\pi$
$331$	−1870.74	+	1870.74i	−0.310651	+	0.310651i	0.534511	+	0.845162i	$0.320496\pi$
$332$		0			0
$333$	8285.97	−	8285.97i	1.36357	−	1.36357i
$334$		0			0
$335$	7738.41			1.26207
$336$		0			0
$337$	2561.19			0.413997			0.206999	−	0.978341i	$-0.433630\pi$
$337$	2561.19			0.413997			0.206999	+	0.978341i	$0.433630\pi$
$338$		0			0
$339$	−633.717	+	633.717i	−0.101530	+	0.101530i
$340$		0			0
$341$	321.618	−	321.618i	0.0510751	−	0.0510751i
$342$		0			0
$343$	6159.61	−	1553.32i	0.969644	−	0.244523i
$344$		0			0
$345$			1089.63i			0.170040i
$346$		0			0
$347$	7775.33	−	7775.33i	1.20289	−	1.20289i	0.229601	−	0.973285i	$-0.426258\pi$
$347$	7775.33	−	7775.33i	1.20289	−	1.20289i	0.973285	−	0.229601i	$-0.0737421\pi$
$348$		0			0
$349$	5078.97	+	5078.97i	0.778999	+	0.778999i	0.979661	−	0.200661i	$-0.0643090\pi$
$349$	5078.97	+	5078.97i	0.778999	+	0.778999i	−0.200661	+	0.979661i	$0.564309\pi$
$350$		0			0
$351$		−	25335.3i		−	3.85271i
$352$		0			0
$353$		−	321.509i		−	0.0484764i	−0.999706	−	0.0242382i	$-0.992284\pi$
$353$		−	321.509i		−	0.0484764i	0.999706	−	0.0242382i	$-0.00771602\pi$
$354$		0			0
$355$	6505.12	−	6505.12i	0.972552	−	0.972552i
$356$		0			0
$357$	−2945.64	+	1055.73i	−0.436694	+	0.156513i
$358$		0			0
$359$	−8216.04			−1.20787			−0.603936	−	0.797033i	$-0.706402\pi$
$359$	−8216.04			−1.20787			−0.603936	+	0.797033i	$0.706402\pi$
$360$		0			0
$361$		−	1217.21i		−	0.177461i
$362$		0			0
$363$	−8608.36	−	8608.36i	−1.24469	−	1.24469i
$364$		0			0
$365$	−9818.64	+	9818.64i	−1.40803	+	1.40803i
$366$		0			0
$367$	2600.73			0.369910			0.184955	−	0.982747i	$-0.440786\pi$
$367$	2600.73			0.369910			0.184955	+	0.982747i	$0.440786\pi$
$368$		0			0
$369$		−	3119.77i		−	0.440132i
$370$		0			0
$371$	−12302.3	+	4409.18i	−1.72157	+	0.617017i
$372$		0			0
$373$	−1438.00	−	1438.00i	−0.199616	−	0.199616i	0.600219	−	0.799835i	$-0.295080\pi$
$373$	−1438.00	−	1438.00i	−0.199616	−	0.199616i	−0.799835	+	0.600219i	$0.795080\pi$
$374$		0			0
$375$	2784.07			0.383383
$376$		0			0
$377$	15539.1			2.12283
$378$		0			0
$379$	5788.76	−	5788.76i	0.784561	−	0.784561i	−0.196036	−	0.980597i	$-0.562807\pi$
$379$	5788.76	−	5788.76i	0.784561	−	0.784561i	0.980597	+	0.196036i	$0.0628069\pi$
$380$		0			0
$381$	122.746	+	122.746i	0.0165052	+	0.0165052i
$382$		0			0
$383$	1861.45			0.248344			0.124172	−	0.992261i	$-0.460373\pi$
$383$	1861.45			0.248344			0.124172	+	0.992261i	$0.460373\pi$
$384$		0			0
$385$	1903.20	+	898.913i	0.251938	+	0.118994i
$386$		0			0
$387$	7893.49	+	7893.49i	1.03682	+	1.03682i
$388$		0			0
$389$	−9938.04	−	9938.04i	−1.29532	−	1.29532i	−0.931451	−	0.363866i	$-0.881457\pi$
$389$	−9938.04	−	9938.04i	−1.29532	−	1.29532i	−0.363866	−	0.931451i	$-0.618543\pi$
$390$		0			0
$391$			132.925i			0.0171926i
$392$		0			0
$393$			13366.8i			1.71569i
$394$		0			0
$395$	−9510.49	−	9510.49i	−1.21146	−	1.21146i
$396$		0			0
$397$	−2937.95	−	2937.95i	−0.371414	−	0.371414i	0.496578	−	0.867992i	$-0.334590\pi$
$397$	−2937.95	−	2937.95i	−0.371414	−	0.371414i	−0.867992	+	0.496578i	$0.834590\pi$
$398$		0			0
$399$	14369.3	+	6786.84i	1.80292	+	0.851546i
$400$		0			0
$401$	575.655			0.0716879			0.0358440	−	0.999357i	$-0.488588\pi$
$401$	575.655			0.0716879			0.0358440	+	0.999357i	$0.488588\pi$
$402$		0			0
$403$	3069.66	+	3069.66i	0.379431	+	0.379431i
$404$		0			0
$405$	17785.4	−	17785.4i	2.18214	−	2.18214i
$406$		0			0
$407$	−1366.18			−0.166386
$408$		0			0
$409$	1575.07			0.190421			0.0952105	−	0.995457i	$-0.469648\pi$
$409$	1575.07			0.190421			0.0952105	+	0.995457i	$0.469648\pi$
$410$		0			0
$411$	18555.6	+	18555.6i	2.22696	+	2.22696i
$412$		0			0
$413$	−3013.02	+	1079.88i	−0.358986	+	0.128662i
$414$		0			0
$415$		−	2331.84i		−	0.275821i
$416$		0			0
$417$	18318.5			2.15123
$418$		0			0
$419$	6534.27	−	6534.27i	0.761862	−	0.761862i	−0.214797	−	0.976659i	$-0.568909\pi$
$419$	6534.27	−	6534.27i	0.761862	−	0.761862i	0.976659	+	0.214797i	$0.0689089\pi$
$420$		0			0
$421$	5564.69	+	5564.69i	0.644196	+	0.644196i	0.951584	−	0.307388i	$-0.0994550\pi$
$421$	5564.69	+	5564.69i	0.644196	+	0.644196i	−0.307388	+	0.951584i	$0.599455\pi$
$422$		0			0
$423$			16345.6i			1.87885i
$424$		0			0
$425$	−1872.30			−0.213694
$426$		0			0
$427$	5628.94	−	2017.43i	0.637948	−	0.228643i
$428$		0			0
$429$	−3606.03	+	3606.03i	−0.405829	+	0.405829i
$430$		0			0
$431$			8245.08i			0.921465i	0.887539	+	0.460733i	$0.152413\pi$
$431$			8245.08i			0.921465i	−0.887539	+	0.460733i	$0.847587\pi$
$432$		0			0
$433$			6790.68i			0.753670i	0.926280	+	0.376835i	$0.122988\pi$
$433$			6790.68i			0.753670i	−0.926280	+	0.376835i	$0.877012\pi$
$434$		0			0
$435$	22323.2	+	22323.2i	2.46050	+	2.46050i
$436$		0			0
$437$	477.346	−	477.346i	0.0522530	−	0.0522530i
$438$		0			0
$439$			8798.44i			0.956553i	0.878209	+	0.478276i	$0.158738\pi$
$439$			8798.44i			0.956553i	−0.878209	+	0.478276i	$0.841262\pi$
$440$		0			0
$441$	−16997.8	+	13979.9i	−1.83542	+	1.50955i
$442$		0			0
$443$	−11899.1	+	11899.1i	−1.27617	+	1.27617i	−0.333374	+	0.942795i	$0.608187\pi$
$443$	−11899.1	+	11899.1i	−1.27617	+	1.27617i	−0.942795	+	0.333374i	$0.891813\pi$
$444$		0			0
$445$	2898.08	−	2898.08i	0.308724	−	0.308724i
$446$		0			0
$447$	14670.1			1.55228
$448$		0			0
$449$	−5778.61			−0.607370			−0.303685	−	0.952772i	$-0.598217\pi$
$449$	−5778.61			−0.607370			−0.303685	+	0.952772i	$0.598217\pi$
$450$		0			0
$451$	−257.192	+	257.192i	−0.0268529	+	0.0268529i
$452$		0			0
$453$	1828.46	−	1828.46i	0.189644	−	0.189644i
$454$		0			0
$455$	−8579.62	+	18165.0i	−0.883997	+	1.87162i
$456$		0			0
$457$			13073.2i			1.33816i	0.743190	+	0.669080i	$0.233312\pi$
$457$			13073.2i			1.33816i	−0.743190	+	0.669080i	$0.766688\pi$
$458$		0			0
$459$	4439.99	−	4439.99i	0.451506	−	0.451506i
$460$		0			0
$461$	−4275.73	−	4275.73i	−0.431976	−	0.431976i	0.457324	−	0.889300i	$-0.348808\pi$
$461$	−4275.73	−	4275.73i	−0.431976	−	0.431976i	−0.889300	+	0.457324i	$0.848808\pi$
$462$		0			0
$463$			7519.45i			0.754770i	0.926056	+	0.377385i	$0.123177\pi$
$463$			7519.45i			0.754770i	−0.926056	+	0.377385i	$0.876823\pi$
$464$		0			0
$465$			8819.64i			0.879572i
$466$		0			0
$467$	6745.11	−	6745.11i	0.668365	−	0.668365i	−0.288972	−	0.957337i	$-0.593314\pi$
$467$	6745.11	−	6745.11i	0.668365	−	0.668365i	0.957337	+	0.288972i	$0.0933135\pi$
$468$		0			0
$469$	8880.40	−	3182.77i	0.874326	−	0.313362i
$470$		0			0
$471$	−4865.74			−0.476012
$472$		0			0
$473$		−	1301.47i		−	0.126515i
$474$		0			0
$475$	6723.62	+	6723.62i	0.649475	+	0.649475i
$476$		0			0
$477$	32015.3	−	32015.3i	3.07312	−	3.07312i
$478$		0			0
$479$	−5722.39			−0.545851			−0.272925	−	0.962035i	$-0.587991\pi$
$479$	−5722.39			−0.545851			−0.272925	+	0.962035i	$0.587991\pi$
$480$		0			0
$481$		−	13039.4i		−	1.23606i
$482$		0			0
$483$	448.161	+	1250.44i	0.0422196	+	0.117799i
$484$		0			0
$485$	−15604.5	−	15604.5i	−1.46096	−	1.46096i
$486$		0			0
$487$	14286.2			1.32930			0.664651	−	0.747154i	$-0.268580\pi$
$487$	14286.2			1.32930			0.664651	+	0.747154i	$0.268580\pi$
$488$		0			0
$489$	23700.3			2.19175
$490$		0			0
$491$	2041.06	−	2041.06i	0.187600	−	0.187600i	−0.607057	−	0.794658i	$-0.707650\pi$
$491$	2041.06	−	2041.06i	0.187600	−	0.187600i	0.794658	+	0.607057i	$0.207650\pi$
$492$		0			0
$493$	2723.22	+	2723.22i	0.248778	+	0.248778i
$494$		0			0
$495$	−7292.18			−0.662140
$496$		0			0
$497$	4789.58	−	10140.6i	0.432278	−	0.915230i
$498$		0			0
$499$	−8527.48	−	8527.48i	−0.765015	−	0.765015i	0.212209	−	0.977224i	$-0.431934\pi$
$499$	−8527.48	−	8527.48i	−0.765015	−	0.765015i	−0.977224	+	0.212209i	$0.931934\pi$
$500$		0			0
$501$	8654.36	+	8654.36i	0.771753	+	0.771753i
$502$		0			0
$503$		−	5920.19i		−	0.524788i	−0.964961	−	0.262394i	$-0.915488\pi$
$503$		−	5920.19i		−	0.524788i	0.964961	−	0.262394i	$-0.0845120\pi$
$504$		0			0
$505$			701.380i			0.0618039i
$506$		0			0
$507$	−19584.6	−	19584.6i	−1.71555	−	1.71555i
$508$		0			0
$509$	6278.29	+	6278.29i	0.546719	+	0.546719i	0.925490	−	0.378771i	$-0.123653\pi$
$509$	6278.29	+	6278.29i	0.546719	+	0.546719i	−0.378771	+	0.925490i	$0.623653\pi$
$510$		0			0
$511$	−7229.26	+	15306.0i	−0.625839	+	1.32504i
$512$		0			0
$513$	−31888.9			−2.74450
$514$		0			0
$515$	−15970.9	−	15970.9i	−1.36653	−	1.36653i
$516$		0			0
$517$	1347.52	−	1347.52i	0.114631	−	0.114631i
$518$		0			0
$519$	−740.137			−0.0625981
$520$		0			0
$521$	6389.95			0.537330			0.268665	−	0.963234i	$-0.413418\pi$
$521$	6389.95			0.537330			0.268665	+	0.963234i	$0.413418\pi$
$522$		0			0
$523$	−14557.4	−	14557.4i	−1.21711	−	1.21711i	−0.968638	−	0.248475i	$-0.920070\pi$
$523$	−14557.4	−	14557.4i	−1.21711	−	1.21711i	−0.248475	−	0.968638i	$-0.579930\pi$
$524$		0			0
$525$	−17612.9	+	6312.54i	−1.46417	+	0.524765i
$526$		0			0
$527$			1075.91i			0.0889325i
$528$		0			0
$529$	−12110.6			−0.995362
$530$		0			0
$531$	7841.05	−	7841.05i	0.640815	−	0.640815i
$532$		0			0
$533$	−2454.75	−	2454.75i	−0.199488	−	0.199488i
$534$		0			0
$535$			20907.6i			1.68956i
$536$		0			0
$537$	−20221.3			−1.62498
$538$		0			0
$539$	2553.78	+	248.792i	0.204080	+	0.0198817i
$540$		0			0
$541$	−8616.54	+	8616.54i	−0.684758	+	0.684758i	−0.961068	−	0.276311i	$-0.910888\pi$
$541$	−8616.54	+	8616.54i	−0.684758	+	0.684758i	0.276311	+	0.961068i	$0.410888\pi$
$542$		0			0
$543$		−	41753.7i		−	3.29986i
$544$		0			0
$545$			5730.75i			0.450419i
$546$		0			0
$547$	−14849.2	−	14849.2i	−1.16071	−	1.16071i	−0.984321	−	0.176386i	$-0.943559\pi$
$547$	−14849.2	−	14849.2i	−1.16071	−	1.16071i	−0.176386	−	0.984321i	$-0.556441\pi$
$548$		0			0
$549$	−14648.7	+	14648.7i	−1.13878	+	1.13878i
$550$		0			0
$551$		−	19558.7i		−	1.51221i
$552$		0			0
$553$	−14825.6	−	7002.38i	−1.14005	−	0.538465i
$554$		0			0
$555$	18732.2	−	18732.2i	1.43268	−	1.43268i
$556$		0			0
$557$	−5671.99	+	5671.99i	−0.431472	+	0.431472i	−0.889129	−	0.457657i	$-0.848689\pi$
$557$	−5671.99	+	5671.99i	−0.431472	+	0.431472i	0.457657	+	0.889129i	$0.348689\pi$
$558$		0			0
$559$	12421.8			0.939866
$560$		0			0
$561$	−1263.91			−0.0951198
$562$		0			0
$563$	−9834.31	+	9834.31i	−0.736175	+	0.736175i	−0.971836	−	0.235660i	$-0.924275\pi$
$563$	−9834.31	+	9834.31i	−0.736175	+	0.736175i	0.235660	+	0.971836i	$0.424275\pi$
$564$		0			0
$565$	−1008.34	+	1008.34i	−0.0750819	+	0.0750819i
$566$		0			0
$567$	13095.0	−	27725.2i	0.969912	−	2.05352i
$568$		0			0
$569$			6475.74i			0.477112i	0.971129	+	0.238556i	$0.0766741\pi$
$569$			6475.74i			0.477112i	−0.971129	+	0.238556i	$0.923326\pi$
$570$		0			0
$571$	−8459.33	+	8459.33i	−0.619986	+	0.619986i	−0.945528	−	0.325542i	$-0.894453\pi$
$571$	−8459.33	+	8459.33i	−0.619986	+	0.619986i	0.325542	+	0.945528i	$0.394453\pi$
$572$		0			0
$573$	−5947.16	−	5947.16i	−0.433588	−	0.433588i
$574$		0			0
$575$			794.802i			0.0576444i
$576$		0			0
$577$			20187.2i			1.45650i	0.685310	+	0.728252i	$0.259667\pi$
$577$			20187.2i			1.45650i	−0.685310	+	0.728252i	$0.740333\pi$
$578$		0			0
$579$	−30806.6	+	30806.6i	−2.21119	+	2.21119i
$580$		0			0
$581$	−959.074	−	2675.96i	−0.0684838	−	0.191080i
$582$		0			0
$583$	−5278.63			−0.374989
$584$		0			0
$585$		−	69599.8i		−	4.91897i
$586$		0			0
$587$	1093.14	+	1093.14i	0.0768631	+	0.0768631i	0.744493	−	0.667630i	$-0.232691\pi$
$587$	1093.14	+	1093.14i	0.0768631	+	0.0768631i	−0.667630	+	0.744493i	$0.732691\pi$
$588$		0			0
$589$	3863.70	−	3863.70i	0.270290	−	0.270290i
$590$		0			0
$591$	−15148.6			−1.05436
$592$		0			0
$593$			7713.84i			0.534181i	0.963671	+	0.267091i	$0.0860623\pi$
$593$			7713.84i			0.534181i	−0.963671	+	0.267091i	$0.913938\pi$
$594$		0			0
$595$	−4686.97	+	1679.83i	−0.322936	+	0.115742i
$596$		0			0
$597$	9425.16	+	9425.16i	0.646141	+	0.646141i
$598$		0			0
$599$	4556.07			0.310778			0.155389	−	0.987853i	$-0.450337\pi$
$599$	4556.07			0.310778			0.155389	+	0.987853i	$0.450337\pi$
$600$		0			0
$601$	−21438.9			−1.45509			−0.727547	−	0.686058i	$-0.759340\pi$
$601$	−21438.9			−1.45509			−0.727547	+	0.686058i	$0.759340\pi$
$602$		0			0
$603$	−23110.2	+	23110.2i	−1.56073	+	1.56073i
$604$		0			0
$605$	−13697.2	−	13697.2i	−0.920450	−	0.920450i
$606$		0			0
$607$	−18799.0			−1.25705			−0.628523	−	0.777791i	$-0.716340\pi$
$607$	−18799.0			−1.25705			−0.628523	+	0.777791i	$0.716340\pi$
$608$		0			0
$609$	34799.0	+	16436.1i	2.31548	+	1.09364i
$610$		0			0
$611$	12861.3	+	12861.3i	0.851578	+	0.851578i
$612$		0			0
$613$	2748.66	+	2748.66i	0.181105	+	0.181105i	0.791837	−	0.610732i	$-0.209125\pi$
$613$	2748.66	+	2748.66i	0.181105	+	0.181105i	−0.610732	+	0.791837i	$0.709125\pi$
$614$		0			0
$615$		−	7052.89i		−	0.462439i
$616$		0			0
$617$			6315.24i			0.412061i	0.978546	+	0.206031i	$0.0660547\pi$
$617$			6315.24i			0.412061i	−0.978546	+	0.206031i	$0.933945\pi$
$618$		0			0
$619$	14106.2	+	14106.2i	0.915955	+	0.915955i	0.996732	−	0.0807768i	$-0.0257401\pi$
$619$	14106.2	+	14106.2i	0.915955	+	0.915955i	−0.0807768	+	0.996732i	$0.525740\pi$
$620$		0			0
$621$	−1884.80	−	1884.80i	−0.121794	−	0.121794i
$622$		0			0
$623$	2133.79	−	4517.72i	0.137221	−	0.290528i
$624$		0			0
$625$	17655.8			1.12997
$626$		0			0
$627$	4538.81	+	4538.81i	0.289095	+	0.289095i
$628$		0			0
$629$	2285.14	−	2285.14i	0.144856	−	0.144856i
$630$		0			0
$631$	−12921.9			−0.815232			−0.407616	−	0.913153i	$-0.633640\pi$
$631$	−12921.9			−0.815232			−0.407616	+	0.913153i	$0.633640\pi$
$632$		0			0
$633$	−10314.2			−0.647634
$634$		0			0
$635$	195.308	+	195.308i	0.0122056	+	0.0122056i
$636$		0			0
$637$	−2374.58	+	24374.4i	−0.147699	+	1.51609i
$638$		0			0
$639$			38854.2i			2.40540i
$640$		0			0
$641$	−2627.47			−0.161902			−0.0809509	−	0.996718i	$-0.525796\pi$
$641$	−2627.47			−0.161902			−0.0809509	+	0.996718i	$0.525796\pi$
$642$		0			0
$643$	−365.321	+	365.321i	−0.0224057	+	0.0224057i	−0.718221	−	0.695815i	$-0.755043\pi$
$643$	−365.321	+	365.321i	−0.0224057	+	0.0224057i	0.695815	+	0.718221i	$0.255043\pi$
$644$		0			0
$645$	17844.9	+	17844.9i	1.08937	+	1.08937i
$646$		0			0
$647$		−	28089.2i		−	1.70680i	−0.521254	−	0.853402i	$-0.674536\pi$
$647$		−	28089.2i		−	1.70680i	0.521254	−	0.853402i	$-0.325464\pi$
$648$		0			0
$649$	−1292.82			−0.0781936
$650$		0			0
$651$	3627.47	+	10121.2i	0.218390	+	0.609340i
$652$		0			0
$653$	−8988.84	+	8988.84i	−0.538684	+	0.538684i	−0.923142	−	0.384458i	$-0.874388\pi$
$653$	−8988.84	+	8988.84i	−0.538684	+	0.538684i	0.384458	+	0.923142i	$0.374388\pi$
$654$		0			0
$655$			21268.7i			1.26876i
$656$		0			0
$657$		−	58645.4i		−	3.48246i
$658$		0			0
$659$	−18384.3	−	18384.3i	−1.08672	−	1.08672i	−0.995864	−	0.0908590i	$-0.971039\pi$
$659$	−18384.3	−	18384.3i	−1.08672	−	1.08672i	−0.0908590	−	0.995864i	$-0.528961\pi$
$660$		0			0
$661$	−8728.20	+	8728.20i	−0.513597	+	0.513597i	−0.915627	−	0.402030i	$-0.868305\pi$
$661$	−8728.20	+	8728.20i	−0.513597	+	0.513597i	0.402030	+	0.915627i	$0.368305\pi$
$662$		0			0
$663$		−	12063.3i		−	0.706635i
$664$		0			0
$665$	22863.7	+	10798.9i	1.33326	+	0.629720i
$666$		0			0
$667$	1156.02	−	1156.02i	0.0671083	−	0.0671083i
$668$		0			0
$669$	−25614.0	+	25614.0i	−1.48026	+	1.48026i
$670$		0			0
$671$	2415.25			0.138956
$672$		0			0
$673$	−11763.2			−0.673758			−0.336879	−	0.941548i	$-0.609371\pi$
$673$	−11763.2			−0.673758			−0.336879	+	0.941548i	$0.609371\pi$
$674$		0			0
$675$	26548.2	−	26548.2i	1.51384	−	1.51384i
$676$		0			0
$677$	2299.45	−	2299.45i	0.130539	−	0.130539i	−0.638818	−	0.769358i	$-0.720577\pi$
$677$	2299.45	−	2299.45i	0.130539	−	0.130539i	0.769358	+	0.638818i	$0.220577\pi$
$678$		0			0
$679$	−24325.4	−	11489.3i	−1.37485	−	0.649363i
$680$		0			0
$681$			15478.7i			0.870992i
$682$		0			0
$683$	−22496.7	+	22496.7i	−1.26034	+	1.26034i	−0.309410	+	0.950929i	$0.600132\pi$
$683$	−22496.7	+	22496.7i	−1.26034	+	1.26034i	−0.950929	+	0.309410i	$0.899868\pi$
$684$		0			0
$685$	29524.8	+	29524.8i	1.64684	+	1.64684i
$686$		0			0
$687$		−	28952.6i		−	1.60788i
$688$		0			0
$689$		−	50381.5i		−	2.78575i
$690$		0			0
$691$	7136.02	−	7136.02i	0.392861	−	0.392861i	−0.482845	−	0.875706i	$-0.660397\pi$
$691$	7136.02	−	7136.02i	0.392861	−	0.392861i	0.875706	+	0.482845i	$0.160397\pi$
$692$		0			0
$693$	−8368.32	+	2999.24i	−0.458710	+	0.164404i
$694$		0			0
$695$	29147.6			1.59084
$696$		0			0
$697$		−	860.385i		−	0.0467567i
$698$		0			0
$699$	−4151.19	−	4151.19i	−0.224625	−	0.224625i
$700$		0			0
$701$	21007.7	−	21007.7i	1.13188	−	1.13188i	0.142017	−	0.989864i	$-0.454641\pi$
$701$	21007.7	−	21007.7i	1.13188	−	1.13188i	0.989864	−	0.142017i	$-0.0453588\pi$
$702$		0			0
$703$	−16412.3			−0.880515
$704$		0			0
$705$			36952.7i			1.97407i
$706$		0			0
$707$	288.474	+	804.885i	0.0153454	+	0.0428159i
$708$		0			0
$709$	920.048	+	920.048i	0.0487350	+	0.0487350i	0.731054	−	0.682319i	$-0.239029\pi$
$709$	920.048	+	920.048i	0.0487350	+	0.0487350i	−0.682319	+	0.731054i	$0.739029\pi$
$710$		0			0
$711$	56804.9			2.99627
$712$		0			0
$713$	456.729			0.0239897
$714$		0			0
$715$	−5737.75	+	5737.75i	−0.300112	+	0.300112i
$716$		0			0
$717$	−28788.1	−	28788.1i	−1.49946	−	1.49946i
$718$		0			0
$719$	27809.1			1.44242			0.721212	−	0.692714i	$-0.243585\pi$
$719$	27809.1			1.44242			0.721212	+	0.692714i	$0.243585\pi$
$720$		0			0
$721$	−24896.5	−	11759.0i	−1.28598	−	0.607391i
$722$		0			0
$723$	18439.5	+	18439.5i	0.948510	+	0.948510i
$724$		0			0
$725$	16283.0	+	16283.0i	0.834119	+	0.834119i
$726$		0			0
$727$			23899.8i			1.21925i	0.792691	+	0.609624i	$0.208680\pi$
$727$			23899.8i			1.21925i	−0.792691	+	0.609624i	$0.791320\pi$
$728$		0			0
$729$			14753.2i			0.749539i
$730$		0			0
$731$	2176.91	+	2176.91i	0.110145	+	0.110145i
$732$		0			0
$733$	799.842	+	799.842i	0.0403040	+	0.0403040i	0.726972	−	0.686668i	$-0.240927\pi$
$733$	799.842	+	799.842i	0.0403040	+	0.0403040i	−0.686668	+	0.726972i	$0.740927\pi$
$734$		0			0
$735$	−38427.1	+	31604.5i	−1.92844	+	1.58606i
$736$		0			0
$737$	3810.38			0.190444
$738$		0			0
$739$	−8049.79	−	8049.79i	−0.400699	−	0.400699i	0.477781	−	0.878479i	$-0.341441\pi$
$739$	−8049.79	−	8049.79i	−0.400699	−	0.400699i	−0.878479	+	0.477781i	$0.841441\pi$
$740$		0			0
$741$	−43320.3	+	43320.3i	−2.14766	+	2.14766i
$742$		0			0
$743$	−35083.7			−1.73229			−0.866147	−	0.499789i	$-0.833411\pi$
$743$	−35083.7			−1.73229			−0.866147	+	0.499789i	$0.833411\pi$
$744$		0			0
$745$	23342.3			1.14792
$746$		0			0
$747$	6963.88	+	6963.88i	0.341091	+	0.341091i
$748$		0			0
$749$	8599.17	+	23993.0i	0.419502	+	1.17047i
$750$		0			0
$751$		−	15178.6i		−	0.737518i	−0.929525	−	0.368759i	$-0.879783\pi$
$751$		−	15178.6i		−	0.737518i	0.929525	−	0.368759i	$-0.120217\pi$
$752$		0			0
$753$	13167.0			0.637225
$754$		0			0
$755$	2909.37	−	2909.37i	0.140242	−	0.140242i
$756$		0			0
$757$	4237.45	+	4237.45i	0.203451	+	0.203451i	0.801477	−	0.598026i	$-0.204048\pi$
$757$	4237.45	+	4237.45i	0.203451	+	0.203451i	−0.598026	+	0.801477i	$0.704048\pi$
$758$		0			0
$759$			536.534i			0.0256587i
$760$		0			0
$761$	−12456.8			−0.593378			−0.296689	−	0.954974i	$-0.595882\pi$
$761$	−12456.8			−0.593378			−0.296689	+	0.954974i	$0.595882\pi$
$762$		0			0
$763$	2357.03	+	6576.46i	0.111835	+	0.312037i
$764$		0			0
$765$	12197.3	−	12197.3i	0.576463	−	0.576463i
$766$		0			0
$767$		−	12339.2i		−	0.580892i
$768$		0			0
$769$		−	7576.59i		−	0.355291i	−0.984095	−	0.177645i	$-0.943152\pi$
$769$		−	7576.59i		−	0.355291i	0.984095	−	0.177645i	$-0.0568480\pi$
$770$		0			0
$771$	43975.9	+	43975.9i	2.05415	+	2.05415i
$772$		0			0
$773$	13423.2	−	13423.2i	0.624580	−	0.624580i	−0.322119	−	0.946699i	$-0.604395\pi$
$773$	13423.2	−	13423.2i	0.624580	−	0.624580i	0.946699	+	0.322119i	$0.104395\pi$
$774$		0			0
$775$			6433.22i			0.298178i
$776$		0			0
$777$	13792.1	−	29201.0i	0.636794	−	1.34824i
$778$		0			0
$779$	−3089.72	+	3089.72i	−0.142106	+	0.142106i
$780$		0			0
$781$	3203.11	−	3203.11i	0.146756	−	0.146756i
$782$		0			0
$783$	−77227.3			−3.52475
$784$		0			0
$785$	−7742.15			−0.352012
$786$		0			0
$787$	−9970.27	+	9970.27i	−0.451591	+	0.451591i	−0.895882	−	0.444292i	$-0.853455\pi$
$787$	−9970.27	+	9970.27i	−0.451591	+	0.451591i	0.444292	+	0.895882i	$0.353455\pi$
$788$		0			0
$789$	5415.04	−	5415.04i	0.244335	−	0.244335i
$790$		0			0
$791$	−742.422	+	1571.87i	−0.0333723	+	0.0706567i
$792$		0			0
$793$			23052.2i			1.03229i
$794$		0			0
$795$	72377.1	−	72377.1i	3.22887	−	3.22887i
$796$		0			0
$797$	−24511.9	−	24511.9i	−1.08941	−	1.08941i	−0.995590	−	0.0938157i	$-0.970094\pi$
$797$	−24511.9	−	24511.9i	−1.08941	−	1.08941i	−0.0938157	−	0.995590i	$-0.529906\pi$
$798$		0			0
$799$			4507.88i			0.199596i
$800$		0			0
$801$			17309.8i			0.763561i
$802$		0			0
$803$	−4834.68	+	4834.68i	−0.212468	+	0.212468i
$804$		0			0
$805$	713.094	+	1989.64i	0.0312215	+	0.0871125i
$806$		0			0
$807$	−27409.5			−1.19562
$808$		0			0
$809$			34908.0i			1.51706i	0.651640	+	0.758529i	$0.274081\pi$
$809$			34908.0i			1.51706i	−0.651640	+	0.758529i	$0.725919\pi$
$810$		0			0
$811$	12610.4	+	12610.4i	0.546004	+	0.546004i	0.925283	−	0.379278i	$-0.123828\pi$
$811$	12610.4	+	12610.4i	0.546004	+	0.546004i	−0.379278	+	0.925283i	$0.623828\pi$
$812$		0			0
$813$	16540.0	−	16540.0i	0.713511	−	0.713511i
$814$		0			0
$815$	37710.8			1.62080
$816$		0			0
$817$		−	15634.9i		−	0.669519i
$818$		0			0
$819$	−28626.0	−	79870.9i	−1.22134	−	3.40771i
$820$		0			0
$821$	−23212.1	−	23212.1i	−0.986731	−	0.986731i	0.0131819	−	0.999913i	$-0.495804\pi$
$821$	−23212.1	−	23212.1i	−0.986731	−	0.986731i	−0.999913	+	0.0131819i	$0.995804\pi$
$822$		0			0
$823$	−13666.7			−0.578849			−0.289425	−	0.957201i	$-0.593464\pi$
$823$	−13666.7			−0.578849			−0.289425	+	0.957201i	$0.593464\pi$
$824$		0			0
$825$	−7557.31			−0.318923
$826$		0			0
$827$	19896.8	−	19896.8i	0.836615	−	0.836615i	−0.151796	−	0.988412i	$-0.548506\pi$
$827$	19896.8	−	19896.8i	0.836615	−	0.836615i	0.988412	+	0.151796i	$0.0485058\pi$
$828$		0			0
$829$	29292.8	+	29292.8i	1.22724	+	1.22724i	0.965005	+	0.262233i	$0.0844588\pi$
$829$	29292.8	+	29292.8i	1.22724	+	1.22724i	0.262233	+	0.965005i	$0.415541\pi$
$830$		0			0
$831$	51375.0			2.14462
$832$		0			0
$833$	−4687.74	+	3855.45i	−0.194983	+	0.160364i
$834$		0			0
$835$	13770.4	+	13770.4i	0.570713	+	0.570713i
$836$		0			0
$837$	−15255.8	−	15255.8i	−0.630008	−	0.630008i
$838$		0			0
$839$		−	23024.9i		−	0.947447i	−0.880674	−	0.473723i	$-0.842910\pi$
$839$		−	23024.9i		−	0.947447i	0.880674	−	0.473723i	$-0.157090\pi$
$840$		0			0
$841$		−	22977.5i		−	0.942127i
$842$		0			0
$843$	36926.2	+	36926.2i	1.50867	+	1.50867i
$844$		0			0
$845$	−31162.1	−	31162.1i	−1.26865	−	1.26865i
$846$		0			0
$847$	−21352.2	−	10085.0i	−0.866199	−	0.409120i
$848$		0			0
$849$	−40473.9			−1.63611
$850$		0			0
$851$	−970.054	−	970.054i	−0.0390752	−	0.0390752i
$852$		0			0
$853$	−19763.3	+	19763.3i	−0.793298	+	0.793298i	−0.982029	−	0.188730i	$-0.939563\pi$
$853$	−19763.3	+	19763.3i	−0.793298	+	0.793298i	0.188730	+	0.982029i	$0.439563\pi$
$854$		0			0
$855$	−87603.2			−3.50406
$856$		0			0
$857$	20283.0			0.808464			0.404232	−	0.914656i	$-0.367539\pi$
$857$	20283.0			0.808464			0.404232	+	0.914656i	$0.367539\pi$
$858$		0			0
$859$	−30204.5	−	30204.5i	−1.19973	−	1.19973i	−0.974248	−	0.225478i	$-0.927606\pi$
$859$	−30204.5	−	30204.5i	−1.19973	−	1.19973i	−0.225478	−	0.974248i	$-0.572394\pi$
$860$		0			0
$861$	−2900.82	−	8093.71i	−0.114819	−	0.320363i
$862$		0			0
$863$			4343.51i			0.171326i	0.996324	+	0.0856632i	$0.0273009\pi$
$863$			4343.51i			0.171326i	−0.996324	+	0.0856632i	$0.972699\pi$
$864$		0			0
$865$	−1177.67			−0.0462914
$866$		0			0
$867$	−31055.8	+	31055.8i	−1.21650	+	1.21650i
$868$		0			0
$869$	−4682.95	−	4682.95i	−0.182806	−	0.182806i
$870$		0			0
$871$			36367.9i			1.41479i
$872$		0			0
$873$	93203.5			3.61336
$874$		0			0
$875$	5083.62	−	1821.99i	0.196409	−	0.0703936i
$876$		0			0
$877$	10248.6	−	10248.6i	0.394608	−	0.394608i	−0.481718	−	0.876326i	$-0.659987\pi$
$877$	10248.6	−	10248.6i	0.394608	−	0.394608i	0.876326	+	0.481718i	$0.159987\pi$
$878$		0			0
$879$			64503.8i			2.47515i
$880$		0			0
$881$			39067.0i			1.49399i	0.664832	+	0.746993i	$0.268503\pi$
$881$			39067.0i			1.49399i	−0.664832	+	0.746993i	$0.731497\pi$
$882$		0			0
$883$	33173.0	+	33173.0i	1.26428	+	1.26428i	0.948998	+	0.315283i	$0.102099\pi$
$883$	33173.0	+	33173.0i	1.26428	+	1.26428i	0.315283	+	0.948998i	$0.397901\pi$
$884$		0			0
$885$	17726.3	−	17726.3i	0.673292	−	0.673292i
$886$		0			0
$887$		−	31068.0i		−	1.17606i	−0.808841	−	0.588028i	$-0.799905\pi$
$887$		−	31068.0i		−	1.17606i	0.808841	−	0.588028i	$-0.200095\pi$
$888$		0			0
$889$	304.460	+	143.801i	0.0114862	+	0.00542513i
$890$		0			0
$891$	8757.52	−	8757.52i	0.329279	−	0.329279i
$892$		0			0
$893$	16188.2	−	16188.2i	0.606627	−	0.606627i
$894$		0			0
$895$	−32175.2			−1.20167
$896$		0			0
$897$	−5120.92			−0.190616
$898$		0			0
$899$	9356.97	−	9356.97i	0.347133	−	0.347133i
$900$		0			0
$901$	8829.32	−	8829.32i	0.326468	−	0.326468i
$902$		0			0
$903$	27817.8	+	13138.8i	1.02516	+	0.484200i
$904$		0			0
$905$		−	66436.6i		−	2.44025i
$906$		0			0
$907$	−7883.36	+	7883.36i	−0.288603	+	0.288603i	−0.836528	−	0.547925i	$-0.815418\pi$
$907$	−7883.36	+	7883.36i	−0.288603	+	0.288603i	0.547925	+	0.836528i	$0.315418\pi$
$908$		0			0
$909$	−2094.62	−	2094.62i	−0.0764293	−	0.0764293i
$910$		0			0
$911$		−	54680.2i		−	1.98862i	−0.106517	−	0.994311i	$-0.533970\pi$
$911$		−	54680.2i		−	1.98862i	0.106517	−	0.994311i	$-0.466030\pi$
$912$		0			0
$913$		−	1148.19i		−	0.0416207i
$914$		0			0
$915$	−33116.4	+	33116.4i	−1.19650	+	1.19650i
$916$		0			0
$917$	8747.70	+	24407.4i	0.315021	+	0.878956i
$918$		0			0
$919$	−3540.65			−0.127090			−0.0635448	−	0.997979i	$-0.520241\pi$
$919$	−3540.65			−0.127090			−0.0635448	+	0.997979i	$0.520241\pi$
$920$		0			0
$921$			18662.4i			0.667696i
$922$		0			0
$923$	30571.9	+	30571.9i	1.09023	+	1.09023i
$924$		0			0
$925$	13663.6	−	13663.6i	0.485683	−	0.485683i
$926$		0			0
$927$	95391.9			3.37981
$928$		0			0
$929$			16349.4i			0.577404i	0.957419	+	0.288702i	$0.0932236\pi$
$929$			16349.4i			0.577404i	−0.957419	+	0.288702i	$0.906776\pi$
$930$		0			0
$931$	30679.4	+	2988.81i	1.08000	+	0.105214i
$932$		0			0
$933$	27768.9	+	27768.9i	0.974398	+	0.974398i
$934$		0			0
$935$	−2011.07			−0.0703413
$936$		0			0
$937$	9033.40			0.314950			0.157475	−	0.987523i	$-0.449665\pi$
$937$	9033.40			0.314950			0.157475	+	0.987523i	$0.449665\pi$
$938$		0			0
$939$	7987.93	−	7987.93i	0.277611	−	0.277611i
$940$		0			0
$941$	−15317.5	−	15317.5i	−0.530644	−	0.530644i	0.390120	−	0.920764i	$-0.372434\pi$
$941$	−15317.5	−	15317.5i	−0.530644	−	0.530644i	−0.920764	+	0.390120i	$0.872434\pi$
$942$		0			0
$943$	−365.237			−0.0126127
$944$		0			0
$945$	42639.5	−	90277.4i	1.46779	−	3.10764i
$946$		0			0
$947$	−15099.6	−	15099.6i	−0.518131	−	0.518131i	0.398875	−	0.917005i	$-0.369401\pi$
$947$	−15099.6	−	15099.6i	−0.518131	−	0.518131i	−0.917005	+	0.398875i	$0.869401\pi$
$948$		0			0
$949$	−46144.3	−	46144.3i	−1.57841	−	1.57841i
$950$		0			0
$951$			20773.8i			0.708345i
$952$		0			0
$953$			29129.5i			0.990133i	0.868855	+	0.495067i	$0.164856\pi$
$953$			29129.5i			0.990133i	−0.868855	+	0.495067i	$0.835144\pi$
$954$		0			0
$955$	−9462.85	−	9462.85i	−0.320639	−	0.320639i
$956$		0			0
$957$	10991.9	+	10991.9i	0.371284	+	0.371284i
$958$		0			0
$959$	46025.3	+	21738.5i	1.54977	+	0.731984i
$960$		0			0
$961$	−26094.2			−0.875908
$962$		0			0
$963$	−62439.0	−	62439.0i	−2.08937	−	2.08937i
$964$		0			0
$965$	−49018.1	+	49018.1i	−1.63518	+	1.63518i
$966$		0			0
$967$	43742.0			1.45465			0.727325	−	0.686293i	$-0.240763\pi$
$967$	43742.0			1.45465			0.727325	+	0.686293i	$0.240763\pi$
$968$		0			0
$969$	−15183.7			−0.503375
$970$		0			0
$971$	−8883.28	−	8883.28i	−0.293592	−	0.293592i	0.544905	−	0.838498i	$-0.316566\pi$
$971$	−8883.28	−	8883.28i	−0.293592	−	0.293592i	−0.838498	+	0.544905i	$0.816566\pi$
$972$		0			0
$973$	33449.0	−	11988.3i	1.10208	−	0.394991i
$974$		0			0
$975$		−	72130.3i		−	2.36925i
$976$		0			0
$977$	307.596			0.0100725			0.00503627	−	0.999987i	$-0.498397\pi$
$977$	307.596			0.0100725			0.00503627	+	0.999987i	$0.498397\pi$
$978$		0			0
$979$	1427.01	−	1427.01i	0.0465857	−	0.0465857i
$980$		0			0
$981$	−17114.5	−	17114.5i	−0.557007	−	0.557007i
$982$		0			0
$983$		−	29915.4i		−	0.970653i	−0.874333	−	0.485327i	$-0.838701\pi$
$983$		−	29915.4i		−	0.970653i	0.874333	−	0.485327i	$-0.161299\pi$
$984$		0			0
$985$	−24103.7			−0.779704
$986$		0			0
$987$	15198.5	+	42406.0i	0.490144	+	1.36758i
$988$		0			0
$989$	924.106	−	924.106i	0.0297117	−	0.0297117i
$990$		0			0
$991$			40094.7i			1.28522i	0.766195	+	0.642609i	$0.222148\pi$
$991$			40094.7i			1.28522i	−0.766195	+	0.642609i	$0.777852\pi$
$992$		0			0
$993$		−	25260.5i		−	0.807267i
$994$		0			0
$995$	14996.9	+	14996.9i	0.477823	+	0.477823i
$996$		0			0
$997$	28430.3	−	28430.3i	0.903107	−	0.903107i	−0.0925966	−	0.995704i	$-0.529517\pi$
$997$	28430.3	−	28430.3i	0.903107	−	0.903107i	0.995704	+	0.0925966i	$0.0295167\pi$
$998$		0			0
$999$			64804.0i			2.05236i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.4.j.b.335.2		88
4.3	odd	2		112.4.j.b.27.6	yes	88
7.6	odd	2	inner	448.4.j.b.335.43		88
16.3	odd	4	inner	448.4.j.b.111.43		88
16.13	even	4		112.4.j.b.83.5	yes	88
28.27	even	2		112.4.j.b.27.5	✓	88
112.13	odd	4		112.4.j.b.83.6	yes	88
112.83	even	4	inner	448.4.j.b.111.2		88

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.4.j.b.27.5	✓	88	28.27	even	2
112.4.j.b.27.6	yes	88	4.3	odd	2
112.4.j.b.83.5	yes	88	16.13	even	4
112.4.j.b.83.6	yes	88	112.13	odd	4
448.4.j.b.111.2		88	112.83	even	4	inner
448.4.j.b.111.43		88	16.3	odd	4	inner
448.4.j.b.335.2		88	1.1	even	1	trivial
448.4.j.b.335.43		88	7.6	odd	2	inner

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:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	448.j (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-6.75145 + 6.75145i) q^{3} +(-10.7426 + 10.7426i) q^{5} +(-7.90955 + 16.7463i) q^{7} -64.1641i q^{9} +(-5.28964 + 5.28964i) q^{11} +(-50.4866 - 50.4866i) q^{13} -145.056i q^{15} -17.6955i q^{17} +(-63.5461 + 63.5461i) q^{19} +(-59.6609 - 166.463i) q^{21} -7.51181 q^{23} -105.807i q^{25} +(250.911 + 250.911i) q^{27} +(-153.894 + 153.894i) q^{29} -60.8015 q^{31} -71.4254i q^{33} +(-94.9298 - 264.868i) q^{35} +(129.137 + 129.137i) q^{37} +681.716 q^{39} +48.6217 q^{41} +(-123.020 + 123.020i) q^{43} +(689.289 + 689.289i) q^{45} -254.748 q^{47} +(-217.878 - 264.912i) q^{49} +(119.470 + 119.470i) q^{51} +(498.959 + 498.959i) q^{53} -113.649i q^{55} -858.056i q^{57} +(122.203 + 122.203i) q^{59} +(-228.300 - 228.300i) q^{61} +(1074.51 + 507.509i) q^{63} +1084.72 q^{65} +(-360.174 - 360.174i) q^{67} +(50.7156 - 50.7156i) q^{69} -605.544 q^{71} +913.990 q^{73} +(714.350 + 714.350i) q^{75} +(-46.7433 - 130.421i) q^{77} +885.306i q^{79} -1655.60 q^{81} +(-108.532 + 108.532i) q^{83} +(190.095 + 190.095i) q^{85} -2078.01i q^{87} -269.774 q^{89} +(1244.79 - 446.138i) q^{91} +(410.498 - 410.498i) q^{93} -1365.30i q^{95} +1452.58i q^{97} +(339.405 + 339.405i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q + 4 q^{7} - 112 q^{11} + 52 q^{21} - 160 q^{23} + 528 q^{29} - 476 q^{35} - 896 q^{37} + 8 q^{39} - 40 q^{43} - 1376 q^{49} - 1504 q^{51} - 1560 q^{53} - 8 q^{65} - 648 q^{67} + 456 q^{71} + 292 q^{77}+ \cdots - 7592 q^{99}+O(q^{100}) \) 88 * q + 4 * q^7 - 112 * q^11 + 52 * q^21 - 160 * q^23 + 528 * q^29 - 476 * q^35 - 896 * q^37 + 8 * q^39 - 40 * q^43 - 1376 * q^49 - 1504 * q^51 - 1560 * q^53 - 8 * q^65 - 648 * q^67 + 456 * q^71 + 292 * q^77 - 1952 * q^81 + 496 * q^85 + 1964 * q^91 - 112 * q^93 - 7592 * q^99

Introduction

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