LMFDB - Embedded newform 448.4.j.b.335.36

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			335.36
Character	$\chi$	$=$	448.335
Dual form			448.4.j.b.111.36

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(4.73059 - 4.73059i) q^{3} +(-5.14976 + 5.14976i) q^{5} +(-10.9791 - 14.9150i) q^{7} -17.7570i q^{9} +(-16.1657 + 16.1657i) q^{11} +(-23.8297 - 23.8297i) q^{13} +48.7228i q^{15} +15.9094i q^{17} +(-3.84826 + 3.84826i) q^{19} +(-122.495 - 18.6192i) q^{21} -38.6350 q^{23} +71.9599i q^{25} +(43.7248 + 43.7248i) q^{27} +(-201.996 + 201.996i) q^{29} -25.5884 q^{31} +152.947i q^{33} +(133.349 + 20.2690i) q^{35} +(-149.092 - 149.092i) q^{37} -225.457 q^{39} -275.363 q^{41} +(180.535 - 180.535i) q^{43} +(91.4444 + 91.4444i) q^{45} -360.435 q^{47} +(-101.917 + 327.509i) q^{49} +(75.2609 + 75.2609i) q^{51} +(-278.986 - 278.986i) q^{53} -166.499i q^{55} +36.4091i q^{57} +(478.902 + 478.902i) q^{59} +(622.588 + 622.588i) q^{61} +(-264.847 + 194.957i) q^{63} +245.434 q^{65} +(-421.140 - 421.140i) q^{67} +(-182.767 + 182.767i) q^{69} -204.915 q^{71} +182.486 q^{73} +(340.413 + 340.413i) q^{75} +(418.599 + 63.6270i) q^{77} -906.929i q^{79} +893.128 q^{81} +(-266.605 + 266.605i) q^{83} +(-81.9296 - 81.9296i) q^{85} +1911.12i q^{87} -490.288 q^{89} +(-93.7917 + 617.050i) q^{91} +(-121.048 + 121.048i) q^{93} -39.6353i q^{95} -1660.46i q^{97} +(287.055 + 287.055i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	4.73059	−	4.73059i	0.910403	−	0.910403i	−0.0859007	−	0.996304i	$-0.527377\pi$
$3$	4.73059	−	4.73059i	0.910403	−	0.910403i	0.996304	+	0.0859007i	$0.0273768\pi$
$4$		0			0
$5$	−5.14976	+	5.14976i	−0.460609	+	0.460609i	−0.898855	−	0.438246i	$-0.855600\pi$
$5$	−5.14976	+	5.14976i	−0.460609	+	0.460609i	0.438246	+	0.898855i	$0.355600\pi$
$6$		0			0
$7$	−10.9791	−	14.9150i	−0.592817	−	0.805337i
$7$	−10.9791	−	14.9150i	−0.592817	−	0.805337i
$8$		0			0
$9$		−	17.7570i		−	0.657667i
$10$		0			0
$11$	−16.1657	+	16.1657i	−0.443105	+	0.443105i	−0.893054	−	0.449949i	$-0.851442\pi$
$11$	−16.1657	+	16.1657i	−0.443105	+	0.443105i	0.449949	+	0.893054i	$0.351442\pi$
$12$		0			0
$13$	−23.8297	−	23.8297i	−0.508398	−	0.508398i	0.405637	−	0.914034i	$-0.367050\pi$
$13$	−23.8297	−	23.8297i	−0.508398	−	0.508398i	−0.914034	+	0.405637i	$0.867050\pi$
$14$		0			0
$15$			48.7228i			0.838679i
$16$		0			0
$17$			15.9094i			0.226976i	0.993539	+	0.113488i	$0.0362024\pi$
$17$			15.9094i			0.226976i	−0.993539	+	0.113488i	$0.963798\pi$
$18$		0			0
$19$	−3.84826	+	3.84826i	−0.0464659	+	0.0464659i	−0.729958	−	0.683492i	$-0.760460\pi$
$19$	−3.84826	+	3.84826i	−0.0464659	+	0.0464659i	0.683492	+	0.729958i	$0.260460\pi$
$20$		0			0
$21$	−122.495	−	18.6192i	−1.27288	−	0.193478i
$22$		0			0
$23$	−38.6350			−0.350259			−0.175130	−	0.984545i	$-0.556034\pi$
$23$	−38.6350			−0.350259			−0.175130	+	0.984545i	$0.556034\pi$
$24$		0			0
$25$			71.9599i			0.575680i
$26$		0			0
$27$	43.7248	+	43.7248i	0.311661	+	0.311661i
$28$		0			0
$29$	−201.996	+	201.996i	−1.29344	+	1.29344i	−0.360795	+	0.932645i	$0.617495\pi$
$29$	−201.996	+	201.996i	−1.29344	+	1.29344i	−0.932645	+	0.360795i	$0.882505\pi$
$30$		0			0
$31$	−25.5884			−0.148252			−0.0741260	−	0.997249i	$-0.523617\pi$
$31$	−25.5884			−0.148252			−0.0741260	+	0.997249i	$0.523617\pi$
$32$		0			0
$33$			152.947i			0.806808i
$34$		0			0
$35$	133.349	+	20.2690i	0.644002	+	0.0978883i
$36$		0			0
$37$	−149.092	−	149.092i	−0.662448	−	0.662448i	0.293508	−	0.955957i	$-0.405177\pi$
$37$	−149.092	−	149.092i	−0.662448	−	0.662448i	−0.955957	+	0.293508i	$0.905177\pi$
$38$		0			0
$39$	−225.457			−0.925693
$40$		0			0
$41$	−275.363			−1.04889			−0.524445	−	0.851445i	$-0.675727\pi$
$41$	−275.363			−1.04889			−0.524445	+	0.851445i	$0.675727\pi$
$42$		0			0
$43$	180.535	−	180.535i	0.640264	−	0.640264i	−0.310356	−	0.950620i	$-0.600448\pi$
$43$	180.535	−	180.535i	0.640264	−	0.640264i	0.950620	+	0.310356i	$0.100448\pi$
$44$		0			0
$45$	91.4444	+	91.4444i	0.302927	+	0.302927i
$46$		0			0
$47$	−360.435			−1.11861			−0.559307	−	0.828960i	$-0.688933\pi$
$47$	−360.435			−1.11861			−0.559307	+	0.828960i	$0.688933\pi$
$48$		0			0
$49$	−101.917	+	327.509i	−0.297135	+	0.954835i
$50$		0			0
$51$	75.2609	+	75.2609i	0.206640	+	0.206640i
$52$		0			0
$53$	−278.986	−	278.986i	−0.723050	−	0.723050i	0.246176	−	0.969225i	$-0.420826\pi$
$53$	−278.986	−	278.986i	−0.723050	−	0.723050i	−0.969225	+	0.246176i	$0.920826\pi$
$54$		0			0
$55$		−	166.499i		−	0.408196i
$56$		0			0
$57$			36.4091i			0.0846054i
$58$		0			0
$59$	478.902	+	478.902i	1.05674	+	1.05674i	0.998290	+	0.0584501i	$0.0186159\pi$
$59$	478.902	+	478.902i	1.05674	+	1.05674i	0.0584501	+	0.998290i	$0.481384\pi$
$60$		0			0
$61$	622.588	+	622.588i	1.30679	+	1.30679i	0.923719	+	0.383071i	$0.125134\pi$
$61$	622.588	+	622.588i	1.30679	+	1.30679i	0.383071	+	0.923719i	$0.374866\pi$
$62$		0			0
$63$	−264.847	+	194.957i	−0.529644	+	0.389877i
$64$		0			0
$65$	245.434			0.468344
$66$		0			0
$67$	−421.140	−	421.140i	−0.767918	−	0.767918i	0.209822	−	0.977740i	$-0.432712\pi$
$67$	−421.140	−	421.140i	−0.767918	−	0.767918i	−0.977740	+	0.209822i	$0.932712\pi$
$68$		0			0
$69$	−182.767	+	182.767i	−0.318877	+	0.318877i
$70$		0			0
$71$	−204.915			−0.342520			−0.171260	−	0.985226i	$-0.554784\pi$
$71$	−204.915			−0.342520			−0.171260	+	0.985226i	$0.554784\pi$
$72$		0			0
$73$	182.486			0.292580			0.146290	−	0.989242i	$-0.453267\pi$
$73$	182.486			0.292580			0.146290	+	0.989242i	$0.453267\pi$
$74$		0			0
$75$	340.413	+	340.413i	0.524100	+	0.524100i
$76$		0			0
$77$	418.599	+	63.6270i	0.619529	+	0.0941684i
$78$		0			0
$79$		−	906.929i		−	1.29161i	−0.763501	−	0.645807i	$-0.776521\pi$
$79$		−	906.929i		−	1.29161i	0.763501	−	0.645807i	$-0.223479\pi$
$80$		0			0
$81$	893.128			1.22514
$82$		0			0
$83$	−266.605	+	266.605i	−0.352574	+	0.352574i	−0.861066	−	0.508492i	$-0.830203\pi$
$83$	−266.605	+	266.605i	−0.352574	+	0.352574i	0.508492	+	0.861066i	$0.330203\pi$
$84$		0			0
$85$	−81.9296	−	81.9296i	−0.104547	−	0.104547i
$86$		0			0
$87$			1911.12i			2.35510i
$88$		0			0
$89$	−490.288			−0.583938			−0.291969	−	0.956428i	$-0.594310\pi$
$89$	−490.288			−0.583938			−0.291969	+	0.956428i	$0.594310\pi$
$90$		0			0
$91$	−93.7917	+	617.050i	−0.108044	+	0.710818i
$92$		0			0
$93$	−121.048	+	121.048i	−0.134969	+	0.134969i
$94$		0			0
$95$		−	39.6353i		−	0.0428052i
$96$		0			0
$97$		−	1660.46i		−	1.73808i	−0.494741	−	0.869040i	$-0.664737\pi$
$97$		−	1660.46i		−	1.73808i	0.494741	−	0.869040i	$-0.335263\pi$
$98$		0			0
$99$	287.055	+	287.055i	0.291416	+	0.291416i
$100$		0			0
$101$	−780.879	+	780.879i	−0.769311	+	0.769311i	−0.977985	−	0.208674i	$-0.933085\pi$
$101$	−780.879	+	780.879i	−0.769311	+	0.769311i	0.208674	+	0.977985i	$0.433085\pi$
$102$		0			0
$103$			403.749i			0.386238i	0.981175	+	0.193119i	$0.0618604\pi$
$103$			403.749i			0.386238i	−0.981175	+	0.193119i	$0.938140\pi$
$104$		0			0
$105$	726.703	−	534.934i	0.675419	−	0.497183i
$106$		0			0
$107$	−796.715	+	796.715i	−0.719825	+	0.719825i	−0.968569	−	0.248744i	$-0.919982\pi$
$107$	−796.715	+	796.715i	−0.719825	+	0.719825i	0.248744	+	0.968569i	$0.419982\pi$
$108$		0			0
$109$	−441.435	+	441.435i	−0.387907	+	0.387907i	−0.873940	−	0.486033i	$-0.838443\pi$
$109$	−441.435	+	441.435i	−0.387907	+	0.387907i	0.486033	+	0.873940i	$0.338443\pi$
$110$		0			0
$111$	−1410.59			−1.20619
$112$		0			0
$113$	−1222.51			−1.01773			−0.508867	−	0.860845i	$-0.669935\pi$
$113$	−1222.51			−1.01773			−0.508867	+	0.860845i	$0.669935\pi$
$114$		0			0
$115$	198.961	−	198.961i	0.161332	−	0.161332i
$116$		0			0
$117$	−423.144	+	423.144i	−0.334356	+	0.334356i
$118$		0			0
$119$	237.289	−	174.671i	0.182792	−	0.134555i
$120$		0			0
$121$			808.338i			0.607316i
$122$		0			0
$123$	−1302.63	+	1302.63i	−0.954912	+	0.954912i
$124$		0			0
$125$	−1014.30	−	1014.30i	−0.725771	−	0.725771i
$126$		0			0
$127$		−	1991.93i		−	1.39178i	−0.718150	−	0.695888i	$-0.755011\pi$
$127$		−	1991.93i		−	1.39178i	0.718150	−	0.695888i	$-0.244989\pi$
$128$		0			0
$129$		−	1708.08i		−	1.16580i
$130$		0			0
$131$	1890.69	−	1890.69i	1.26099	−	1.26099i	0.310381	−	0.950612i	$-0.399543\pi$
$131$	1890.69	−	1890.69i	1.26099	−	1.26099i	0.950612	−	0.310381i	$-0.100457\pi$
$132$		0			0
$133$	99.6476	+	15.1464i	0.0649665	+	0.00987491i
$134$		0			0
$135$	−450.344			−0.287107
$136$		0			0
$137$		−	1122.77i		−	0.700180i	−0.936716	−	0.350090i	$-0.886151\pi$
$137$		−	1122.77i		−	0.700180i	0.936716	−	0.350090i	$-0.113849\pi$
$138$		0			0
$139$	−1578.34	−	1578.34i	−0.963118	−	0.963118i	0.0362256	−	0.999344i	$-0.488467\pi$
$139$	−1578.34	−	1578.34i	−0.963118	−	0.963118i	−0.999344	+	0.0362256i	$0.988467\pi$
$140$		0			0
$141$	−1705.07	+	1705.07i	−1.01839	+	1.01839i
$142$		0			0
$143$	770.449			0.450547
$144$		0			0
$145$		−	2080.46i		−	1.19154i
$146$		0			0
$147$	1067.18	+	2031.44i	0.598772	+	1.13980i
$148$		0			0
$149$	2027.35	+	2027.35i	1.11468	+	1.11468i	0.992510	+	0.122166i	$0.0389840\pi$
$149$	2027.35	+	2027.35i	1.11468	+	1.11468i	0.122166	+	0.992510i	$0.461016\pi$
$150$		0			0
$151$	−1631.87			−0.879470			−0.439735	−	0.898127i	$-0.644928\pi$
$151$	−1631.87			−0.879470			−0.439735	+	0.898127i	$0.644928\pi$
$152$		0			0
$153$	282.503			0.149275
$154$		0			0
$155$	131.774	−	131.774i	0.0682861	−	0.0682861i
$156$		0			0
$157$	2254.09	+	2254.09i	1.14584	+	1.14584i	0.987363	+	0.158473i	$0.0506571\pi$
$157$	2254.09	+	2254.09i	1.14584	+	1.14584i	0.158473	+	0.987363i	$0.449343\pi$
$158$		0			0
$159$	−2639.54			−1.31653
$160$		0			0
$161$	424.179	+	576.243i	0.207640	+	0.282077i
$162$		0			0
$163$	−286.759	−	286.759i	−0.137796	−	0.137796i	0.634844	−	0.772640i	$-0.281064\pi$
$163$	−286.759	−	286.759i	−0.137796	−	0.137796i	−0.772640	+	0.634844i	$0.781064\pi$
$164$		0			0
$165$	−787.641	−	787.641i	−0.371623	−	0.371623i
$166$		0			0
$167$		−	3940.83i		−	1.82605i	−0.407904	−	0.913025i	$-0.633740\pi$
$167$		−	3940.83i		−	1.82605i	0.407904	−	0.913025i	$-0.366260\pi$
$168$		0			0
$169$		−	1061.29i		−	0.483064i
$170$		0			0
$171$	68.3337	+	68.3337i	0.0305591	+	0.0305591i
$172$		0			0
$173$	1329.23	+	1329.23i	0.584158	+	0.584158i	0.936043	−	0.351885i	$-0.114459\pi$
$173$	1329.23	+	1329.23i	0.584158	+	0.584158i	−0.351885	+	0.936043i	$0.614459\pi$
$174$		0			0
$175$	1073.29	−	790.058i	0.463616	−	0.341273i
$176$		0			0
$177$	4530.98			1.92412
$178$		0			0
$179$	1865.77	+	1865.77i	0.779072	+	0.779072i	0.979673	−	0.200601i	$-0.0642895\pi$
$179$	1865.77	+	1865.77i	0.779072	+	0.779072i	−0.200601	+	0.979673i	$0.564290\pi$
$180$		0			0
$181$	−817.801	+	817.801i	−0.335838	+	0.335838i	−0.854798	−	0.518960i	$-0.826319\pi$
$181$	−817.801	+	817.801i	−0.335838	+	0.335838i	0.518960	+	0.854798i	$0.326319\pi$
$182$		0			0
$183$	5890.42			2.37941
$184$		0			0
$185$	1535.58			0.610259
$186$		0			0
$187$	−257.187	−	257.187i	−0.100574	−	0.100574i
$188$		0			0
$189$	172.097	−	1132.22i	0.0662340	−	0.435750i
$190$		0			0
$191$		−	2062.36i		−	0.781293i	−0.920541	−	0.390646i	$-0.872252\pi$
$191$		−	2062.36i		−	0.781293i	0.920541	−	0.390646i	$-0.127748\pi$
$192$		0			0
$193$	−1093.60			−0.407872			−0.203936	−	0.978984i	$-0.565373\pi$
$193$	−1093.60			−0.407872			−0.203936	+	0.978984i	$0.565373\pi$
$194$		0			0
$195$	1161.05	−	1161.05i	0.426382	−	0.426382i
$196$		0			0
$197$	−2236.95	−	2236.95i	−0.809017	−	0.809017i	0.175468	−	0.984485i	$-0.443856\pi$
$197$	−2236.95	−	2236.95i	−0.809017	−	0.809017i	−0.984485	+	0.175468i	$0.943856\pi$
$198$		0			0
$199$		−	1535.84i		−	0.547100i	−0.961858	−	0.273550i	$-0.911802\pi$
$199$		−	1535.84i		−	0.547100i	0.961858	−	0.273550i	$-0.0881979\pi$
$200$		0			0
$201$	−3984.49			−1.39823
$202$		0			0
$203$	5230.53	+	795.040i	1.80843	+	0.274881i
$204$		0			0
$205$	1418.05	−	1418.05i	0.483127	−	0.483127i
$206$		0			0
$207$			686.043i			0.230354i
$208$		0			0
$209$		−	124.420i		−	0.0411785i
$210$		0			0
$211$	721.942	+	721.942i	0.235548	+	0.235548i	0.815004	−	0.579456i	$-0.196735\pi$
$211$	721.942	+	721.942i	0.235548	+	0.235548i	−0.579456	+	0.815004i	$0.696735\pi$
$212$		0			0
$213$	−969.368	+	969.368i	−0.311831	+	0.311831i
$214$		0			0
$215$			1859.43i			0.589822i
$216$		0			0
$217$	280.938	+	381.652i	0.0878864	+	0.119393i
$218$		0			0
$219$	863.267	−	863.267i	0.266366	−	0.266366i
$220$		0			0
$221$	379.116	−	379.116i	0.115394	−	0.115394i
$222$		0			0
$223$	1558.81			0.468098			0.234049	−	0.972225i	$-0.424802\pi$
$223$	1558.81			0.468098			0.234049	+	0.972225i	$0.424802\pi$
$224$		0			0
$225$	1277.79			0.378606
$226$		0			0
$227$	2832.12	−	2832.12i	0.828082	−	0.828082i	−0.159170	−	0.987251i	$-0.550882\pi$
$227$	2832.12	−	2832.12i	0.828082	−	0.828082i	0.987251	+	0.159170i	$0.0508816\pi$
$228$		0			0
$229$	3341.32	−	3341.32i	0.964194	−	0.964194i	−0.0351867	−	0.999381i	$-0.511203\pi$
$229$	3341.32	−	3341.32i	0.964194	−	0.964194i	0.999381	+	0.0351867i	$0.0112026\pi$
$230$		0			0
$231$	2281.21	−	1679.23i	0.649752	−	0.478290i
$232$		0			0
$233$			691.664i			0.194474i	0.995261	+	0.0972370i	$0.0310005\pi$
$233$			691.664i			0.194474i	−0.995261	+	0.0972370i	$0.969000\pi$
$234$		0			0
$235$	1856.15	−	1856.15i	0.515243	−	0.515243i
$236$		0			0
$237$	−4290.31	−	4290.31i	−1.17589	−	1.17589i
$238$		0			0
$239$			4229.73i			1.14476i	0.819988	+	0.572381i	$0.193980\pi$
$239$			4229.73i			1.14476i	−0.819988	+	0.572381i	$0.806020\pi$
$240$		0			0
$241$			972.399i			0.259908i	0.991520	+	0.129954i	$0.0414829\pi$
$241$			972.399i			0.259908i	−0.991520	+	0.129954i	$0.958517\pi$
$242$		0			0
$243$	3044.45	−	3044.45i	0.803711	−	0.803711i
$244$		0			0
$245$	−1161.74	−	2211.44i	−0.302942	−	0.576668i
$246$		0			0
$247$	183.406			0.0472463
$248$		0			0
$249$			2522.39i			0.641969i
$250$		0			0
$251$	490.000	+	490.000i	0.123221	+	0.123221i	0.766028	−	0.642807i	$-0.222230\pi$
$251$	490.000	+	490.000i	0.123221	+	0.123221i	−0.642807	+	0.766028i	$0.722230\pi$
$252$		0			0
$253$	624.564	−	624.564i	0.155202	−	0.155202i
$254$		0			0
$255$	−775.151			−0.190360
$256$		0			0
$257$		−	5361.50i		−	1.30133i	−0.759366	−	0.650664i	$-0.774491\pi$
$257$		−	5361.50i		−	1.30133i	0.759366	−	0.650664i	$-0.225509\pi$
$258$		0			0
$259$	−586.814	+	3860.62i	−0.140783	+	0.926205i
$260$		0			0
$261$	3586.85	+	3586.85i	0.850653	+	0.850653i
$262$		0			0
$263$	−6070.76			−1.42334			−0.711671	−	0.702513i	$-0.752061\pi$
$263$	−6070.76			−1.42334			−0.711671	+	0.702513i	$0.752061\pi$
$264$		0			0
$265$	2873.42			0.666086
$266$		0			0
$267$	−2319.35	+	2319.35i	−0.531619	+	0.531619i
$268$		0			0
$269$	2788.04	+	2788.04i	0.631933	+	0.631933i	0.948552	−	0.316620i	$-0.102548\pi$
$269$	2788.04	+	2788.04i	0.631933	+	0.631933i	−0.316620	+	0.948552i	$0.602548\pi$
$270$		0			0
$271$	−6455.58			−1.44704			−0.723522	−	0.690302i	$-0.757478\pi$
$271$	−6455.58			−1.44704			−0.723522	+	0.690302i	$0.757478\pi$
$272$		0			0
$273$	2475.32	+	3362.70i	0.548767	+	0.745495i
$274$		0			0
$275$	−1163.29	−	1163.29i	−0.255086	−	0.255086i
$276$		0			0
$277$	−2686.97	−	2686.97i	−0.582832	−	0.582832i	0.352848	−	0.935681i	$-0.385213\pi$
$277$	−2686.97	−	2686.97i	−0.582832	−	0.582832i	−0.935681	+	0.352848i	$0.885213\pi$
$278$		0			0
$279$			454.374i			0.0975005i
$280$		0			0
$281$		−	503.134i		−	0.106813i	−0.998573	−	0.0534065i	$-0.982992\pi$
$281$		−	503.134i		−	0.106813i	0.998573	−	0.0534065i	$-0.0170079\pi$
$282$		0			0
$283$	−4641.91	−	4641.91i	−0.975028	−	0.975028i	0.0246674	−	0.999696i	$-0.492147\pi$
$283$	−4641.91	−	4641.91i	−0.975028	−	0.975028i	−0.999696	+	0.0246674i	$0.992147\pi$
$284$		0			0
$285$	−187.498	−	187.498i	−0.0389700	−	0.0389700i
$286$		0			0
$287$	3023.25	+	4107.05i	0.621800	+	0.844709i
$288$		0			0
$289$	4659.89			0.948482
$290$		0			0
$291$	−7854.94	−	7854.94i	−1.58235	−	1.58235i
$292$		0			0
$293$	2038.37	−	2038.37i	0.406425	−	0.406425i	−0.474065	−	0.880490i	$-0.657214\pi$
$293$	2038.37	−	2038.37i	0.406425	−	0.406425i	0.880490	+	0.474065i	$0.157214\pi$
$294$		0			0
$295$	−4932.46			−0.973487
$296$		0			0
$297$	−1413.69			−0.276197
$298$		0			0
$299$	920.661	+	920.661i	0.178071	+	0.178071i
$300$		0			0
$301$	−4674.81	−	710.571i	−0.895188	−	0.136069i
$302$		0			0
$303$			7388.05i			1.40077i
$304$		0			0
$305$	−6412.35			−1.20384
$306$		0			0
$307$	−6109.18	+	6109.18i	−1.13573	+	1.13573i	−0.146523	+	0.989207i	$0.546808\pi$
$307$	−6109.18	+	6109.18i	−1.13573	+	1.13573i	−0.989207	+	0.146523i	$0.953192\pi$
$308$		0			0
$309$	1909.97	+	1909.97i	0.351633	+	0.351633i
$310$		0			0
$311$			1081.07i			0.197112i	0.995131	+	0.0985562i	$0.0314224\pi$
$311$			1081.07i			0.197112i	−0.995131	+	0.0985562i	$0.968578\pi$
$312$		0			0
$313$	−6343.09			−1.14547			−0.572736	−	0.819740i	$-0.694118\pi$
$313$	−6343.09			−1.14547			−0.572736	+	0.819740i	$0.694118\pi$
$314$		0			0
$315$	359.917	−	2367.88i	0.0643779	−	0.423539i
$316$		0			0
$317$	2324.60	−	2324.60i	0.411869	−	0.411869i	−0.470520	−	0.882389i	$-0.655934\pi$
$317$	2324.60	−	2324.60i	0.411869	−	0.411869i	0.882389	+	0.470520i	$0.155934\pi$
$318$		0			0
$319$		−	6530.84i		−	1.14626i
$320$		0			0
$321$			7537.86i			1.31066i
$322$		0			0
$323$	−61.2235	−	61.2235i	−0.0105467	−	0.0105467i
$324$		0			0
$325$	1714.78	−	1714.78i	0.292674	−	0.292674i
$326$		0			0
$327$			4176.50i			0.706303i
$328$		0			0
$329$	3957.27	+	5375.91i	0.663134	+	0.900862i
$330$		0			0
$331$	3988.06	−	3988.06i	0.662246	−	0.662246i	−0.293663	−	0.955909i	$-0.594874\pi$
$331$	3988.06	−	3988.06i	0.662246	−	0.662246i	0.955909	+	0.293663i	$0.0948743\pi$
$332$		0			0
$333$	−2647.43	+	2647.43i	−0.435671	+	0.435671i
$334$		0			0
$335$	4337.54			0.707419
$336$		0			0
$337$	−8978.60			−1.45132			−0.725661	−	0.688053i	$-0.758466\pi$
$337$	−8978.60			−1.45132			−0.725661	+	0.688053i	$0.758466\pi$
$338$		0			0
$339$	−5783.19	+	5783.19i	−0.926548	+	0.926548i
$340$		0			0
$341$	413.655	−	413.655i	0.0656912	−	0.0656912i
$342$		0			0
$343$	6003.77	−	2075.66i	0.945111	−	0.326749i
$344$		0			0
$345$		−	1882.41i		−	0.293755i
$346$		0			0
$347$	−3062.89	+	3062.89i	−0.473847	+	0.473847i	−0.903157	−	0.429310i	$-0.858757\pi$
$347$	−3062.89	+	3062.89i	−0.473847	+	0.473847i	0.429310	+	0.903157i	$0.358757\pi$
$348$		0			0
$349$	542.447	+	542.447i	0.0831993	+	0.0831993i	0.747482	−	0.664282i	$-0.231263\pi$
$349$	542.447	+	542.447i	0.0831993	+	0.0831993i	−0.664282	+	0.747482i	$0.731263\pi$
$350$		0			0
$351$		−	2083.90i		−	0.316895i
$352$		0			0
$353$			1222.13i			0.184271i	0.995746	+	0.0921355i	$0.0293693\pi$
$353$			1222.13i			0.184271i	−0.995746	+	0.0921355i	$0.970631\pi$
$354$		0			0
$355$	1055.26	−	1055.26i	0.157767	−	0.157767i
$356$		0			0
$357$	296.220	−	1948.82i	0.0439150	−	0.288914i
$358$		0			0
$359$	−929.162			−0.136600			−0.0682998	−	0.997665i	$-0.521757\pi$
$359$	−929.162			−0.136600			−0.0682998	+	0.997665i	$0.521757\pi$
$360$		0			0
$361$			6829.38i			0.995682i
$362$		0			0
$363$	3823.92	+	3823.92i	0.552902	+	0.552902i
$364$		0			0
$365$	−939.759	+	939.759i	−0.134765	+	0.134765i
$366$		0			0
$367$	−20.3599			−0.00289586			−0.00144793	−	0.999999i	$-0.500461\pi$
$367$	−20.3599			−0.00289586			−0.00144793	+	0.999999i	$0.500461\pi$
$368$		0			0
$369$			4889.62i			0.689820i
$370$		0			0
$371$	−1098.06	+	7224.10i	−0.153662	+	1.01093i
$372$		0			0
$373$	1666.13	+	1666.13i	0.231284	+	0.231284i	0.813228	−	0.581945i	$-0.197708\pi$
$373$	1666.13	+	1666.13i	0.231284	+	0.231284i	−0.581945	+	0.813228i	$0.697708\pi$
$374$		0			0
$375$	−9596.45			−1.32149
$376$		0			0
$377$	9627.02			1.31516
$378$		0			0
$379$	−5852.10	+	5852.10i	−0.793146	+	0.793146i	−0.982004	−	0.188858i	$-0.939521\pi$
$379$	−5852.10	+	5852.10i	−0.793146	+	0.793146i	0.188858	+	0.982004i	$0.439521\pi$
$380$		0			0
$381$	−9423.03	−	9423.03i	−1.26708	−	1.26708i
$382$		0			0
$383$	10176.6			1.35771			0.678854	−	0.734273i	$-0.262477\pi$
$383$	10176.6			1.35771			0.678854	+	0.734273i	$0.262477\pi$
$384$		0			0
$385$	−2483.35	+	1828.02i	−0.328735	+	0.241986i
$386$		0			0
$387$	−3205.77	−	3205.77i	−0.421081	−	0.421081i
$388$		0			0
$389$	622.058	+	622.058i	0.0810786	+	0.0810786i	0.746483	−	0.665404i	$-0.231741\pi$
$389$	622.058	+	622.058i	0.0810786	+	0.0810786i	−0.665404	+	0.746483i	$0.731741\pi$
$390$		0			0
$391$		−	614.660i		−	0.0795005i
$392$		0			0
$393$		−	17888.1i		−	2.29602i
$394$		0			0
$395$	4670.47	+	4670.47i	0.594928	+	0.594928i
$396$		0			0
$397$	8725.65	+	8725.65i	1.10309	+	1.10309i	0.994035	+	0.109058i	$0.0347833\pi$
$397$	8725.65	+	8725.65i	1.10309	+	1.10309i	0.109058	+	0.994035i	$0.465217\pi$
$398$		0			0
$399$	543.044	−	399.741i	0.0681358	−	0.0501555i
$400$		0			0
$401$	765.185			0.0952906			0.0476453	−	0.998864i	$-0.484828\pi$
$401$	765.185			0.0952906			0.0476453	+	0.998864i	$0.484828\pi$
$402$		0			0
$403$	609.764	+	609.764i	0.0753710	+	0.0753710i
$404$		0			0
$405$	−4599.39	+	4599.39i	−0.564310	+	0.564310i
$406$		0			0
$407$	4820.37			0.587068
$408$		0			0
$409$	−5690.42			−0.687954			−0.343977	−	0.938978i	$-0.611774\pi$
$409$	−5690.42			−0.687954			−0.343977	+	0.938978i	$0.611774\pi$
$410$		0			0
$411$	−5311.37	−	5311.37i	−0.637446	−	0.637446i
$412$		0			0
$413$	1884.92	−	12400.8i	0.224578	−	1.47749i
$414$		0			0
$415$		−	2745.90i		−	0.324797i
$416$		0			0
$417$	−14933.0			−1.75365
$418$		0			0
$419$	−11135.9	+	11135.9i	−1.29839	+	1.29839i	−0.368935	+	0.929455i	$0.620277\pi$
$419$	−11135.9	+	11135.9i	−1.29839	+	1.29839i	−0.929455	+	0.368935i	$0.879723\pi$
$420$		0			0
$421$	4414.00	+	4414.00i	0.510986	+	0.510986i	0.914828	−	0.403843i	$-0.132326\pi$
$421$	4414.00	+	4414.00i	0.510986	+	0.510986i	−0.403843	+	0.914828i	$0.632326\pi$
$422$		0			0
$423$			6400.26i			0.735676i
$424$		0			0
$425$	−1144.84			−0.130666
$426$		0			0
$427$	2450.45	−	16121.4i	0.277718	−	1.82709i
$428$		0			0
$429$	3644.68	−	3644.68i	0.410179	−	0.410179i
$430$		0			0
$431$			2766.42i			0.309173i	0.987979	+	0.154587i	$0.0494045\pi$
$431$			2766.42i			0.309173i	−0.987979	+	0.154587i	$0.950595\pi$
$432$		0			0
$433$			14228.8i			1.57919i	0.613627	+	0.789596i	$0.289710\pi$
$433$			14228.8i			1.57919i	−0.613627	+	0.789596i	$0.710290\pi$
$434$		0			0
$435$	−9841.83	−	9841.83i	−1.08478	−	1.08478i
$436$		0			0
$437$	148.678	−	148.678i	0.0162751	−	0.0162751i
$438$		0			0
$439$			8483.17i			0.922277i	0.887328	+	0.461139i	$0.152559\pi$
$439$			8483.17i			0.922277i	−0.887328	+	0.461139i	$0.847441\pi$
$440$		0			0
$441$	5815.58	+	1809.75i	0.627964	+	0.195416i
$442$		0			0
$443$	−7329.19	+	7329.19i	−0.786051	+	0.786051i	−0.980844	−	0.194793i	$-0.937596\pi$
$443$	−7329.19	+	7329.19i	−0.786051	+	0.786051i	0.194793	+	0.980844i	$0.437596\pi$
$444$		0			0
$445$	2524.87	−	2524.87i	0.268967	−	0.268967i
$446$		0			0
$447$	19181.1			2.02961
$448$		0			0
$449$	−4339.50			−0.456111			−0.228055	−	0.973648i	$-0.573237\pi$
$449$	−4339.50			−0.456111			−0.228055	+	0.973648i	$0.573237\pi$
$450$		0			0
$451$	4451.45	−	4451.45i	0.464768	−	0.464768i
$452$		0			0
$453$	−7719.73	+	7719.73i	−0.800672	+	0.800672i
$454$		0			0
$455$	−2694.66	−	3660.67i	−0.277643	−	0.377175i
$456$		0			0
$457$		−	8607.23i		−	0.881027i	−0.897746	−	0.440514i	$-0.854796\pi$
$457$		−	8607.23i		−	0.881027i	0.897746	−	0.440514i	$-0.145204\pi$
$458$		0			0
$459$	−695.635	+	695.635i	−0.0707395	+	0.0707395i
$460$		0			0
$461$	6006.00	+	6006.00i	0.606783	+	0.606783i	0.942104	−	0.335321i	$-0.108845\pi$
$461$	6006.00	+	6006.00i	0.606783	+	0.606783i	−0.335321	+	0.942104i	$0.608845\pi$
$462$		0			0
$463$			7562.97i			0.759138i	0.925163	+	0.379569i	$0.123928\pi$
$463$			7562.97i			0.759138i	−0.925163	+	0.379569i	$0.876072\pi$
$464$		0			0
$465$		−	1246.74i		−	0.124336i
$466$		0			0
$467$	2253.37	−	2253.37i	0.223284	−	0.223284i	−0.586596	−	0.809880i	$-0.699532\pi$
$467$	2253.37	−	2253.37i	0.223284	−	0.223284i	0.809880	+	0.586596i	$0.199532\pi$
$468$		0			0
$469$	−1657.57	+	10905.1i	−0.163197	+	1.07367i
$470$		0			0
$471$	21326.4			2.08635
$472$		0			0
$473$			5836.97i			0.567408i
$474$		0			0
$475$	−276.921	−	276.921i	−0.0267495	−	0.0267495i
$476$		0			0
$477$	−4953.95	+	4953.95i	−0.475526	+	0.475526i
$478$		0			0
$479$	4768.48			0.454859			0.227430	−	0.973795i	$-0.426968\pi$
$479$	4768.48			0.454859			0.227430	+	0.973795i	$0.426968\pi$
$480$		0			0
$481$			7105.64i			0.673574i
$482$		0			0
$483$	4732.59	+	719.354i	0.445839	+	0.0677676i
$484$		0			0
$485$	8550.95	+	8550.95i	0.800575	+	0.800575i
$486$		0			0
$487$	−9882.70			−0.919565			−0.459782	−	0.888032i	$-0.652073\pi$
$487$	−9882.70			−0.919565			−0.459782	+	0.888032i	$0.652073\pi$
$488$		0			0
$489$	−2713.08			−0.250899
$490$		0			0
$491$	−2353.08	+	2353.08i	−0.216279	+	0.216279i	−0.806929	−	0.590649i	$-0.798872\pi$
$491$	−2353.08	+	2353.08i	−0.216279	+	0.216279i	0.590649	+	0.806929i	$0.298872\pi$
$492$		0			0
$493$	−3213.64	−	3213.64i	−0.293580	−	0.293580i
$494$		0			0
$495$	−2956.53			−0.268457
$496$		0			0
$497$	2249.78	+	3056.31i	0.203052	+	0.275844i
$498$		0			0
$499$	10238.7	+	10238.7i	0.918530	+	0.918530i	0.996923	−	0.0783925i	$-0.0249787\pi$
$499$	10238.7	+	10238.7i	0.918530	+	0.918530i	−0.0783925	+	0.996923i	$0.524979\pi$
$500$		0			0
$501$	−18642.4	−	18642.4i	−1.66244	−	1.66244i
$502$		0			0
$503$			5116.51i			0.453547i	0.973948	+	0.226773i	$0.0728177\pi$
$503$			5116.51i			0.453547i	−0.973948	+	0.226773i	$0.927182\pi$
$504$		0			0
$505$		−	8042.68i		−	0.708702i
$506$		0			0
$507$	−5020.54	−	5020.54i	−0.439783	−	0.439783i
$508$		0			0
$509$	4950.17	+	4950.17i	0.431065	+	0.431065i	0.888991	−	0.457925i	$-0.151407\pi$
$509$	4950.17	+	4950.17i	0.431065	+	0.431065i	−0.457925	+	0.888991i	$0.651407\pi$
$510$		0			0
$511$	−2003.54	−	2721.79i	−0.173447	−	0.235626i
$512$		0			0
$513$	−336.529			−0.0289632
$514$		0			0
$515$	−2079.21	−	2079.21i	−0.177905	−	0.177905i
$516$		0			0
$517$	5826.70	−	5826.70i	0.495664	−	0.495664i
$518$		0			0
$519$	12576.1			1.06364
$520$		0			0
$521$	−5752.30			−0.483710			−0.241855	−	0.970312i	$-0.577756\pi$
$521$	−5752.30			−0.483710			−0.241855	+	0.970312i	$0.577756\pi$
$522$		0			0
$523$	−3872.88	−	3872.88i	−0.323804	−	0.323804i	0.526421	−	0.850224i	$-0.323534\pi$
$523$	−3872.88	−	3872.88i	−0.323804	−	0.323804i	−0.850224	+	0.526421i	$0.823534\pi$
$524$		0			0
$525$	1339.84	−	8814.72i	0.111382	−	0.732773i
$526$		0			0
$527$		−	407.096i		−	0.0336497i
$528$		0			0
$529$	−10674.3			−0.877319
$530$		0			0
$531$	8503.87	−	8503.87i	0.694984	−	0.694984i
$532$		0			0
$533$	6561.81	+	6561.81i	0.533253	+	0.533253i
$534$		0			0
$535$		−	8205.78i		−	0.663115i
$536$		0			0
$537$	17652.4			1.41854
$538$		0			0
$539$	−3646.85	−	6941.99i	−0.291430	−	0.554754i
$540$		0			0
$541$	−2609.55	+	2609.55i	−0.207382	+	0.207382i	−0.803154	−	0.595772i	$-0.796846\pi$
$541$	−2609.55	+	2609.55i	−0.207382	+	0.207382i	0.595772	+	0.803154i	$0.296846\pi$
$542$		0			0
$543$			7737.37i			0.611496i
$544$		0			0
$545$		−	4546.57i		−	0.357346i
$546$		0			0
$547$	−12196.5	−	12196.5i	−0.953352	−	0.953352i	0.0456071	−	0.998959i	$-0.485478\pi$
$547$	−12196.5	−	12196.5i	−0.953352	−	0.953352i	−0.998959	+	0.0456071i	$0.985478\pi$
$548$		0			0
$549$	11055.3	−	11055.3i	0.859433	−	0.859433i
$550$		0			0
$551$		−	1554.67i		−	0.120202i
$552$		0			0
$553$	−13526.9	+	9957.29i	−1.04018	+	0.765691i
$554$		0			0
$555$	7264.19	−	7264.19i	0.555581	−	0.555581i
$556$		0			0
$557$	−7286.28	+	7286.28i	−0.554272	+	0.554272i	−0.927671	−	0.373399i	$-0.878192\pi$
$557$	−7286.28	+	7286.28i	−0.554272	+	0.554272i	0.373399	+	0.927671i	$0.378192\pi$
$558$		0			0
$559$	−8604.20			−0.651017
$560$		0			0
$561$	−2433.30			−0.183126
$562$		0			0
$563$	521.072	−	521.072i	0.0390064	−	0.0390064i	−0.687335	−	0.726341i	$-0.741219\pi$
$563$	521.072	−	521.072i	0.0390064	−	0.0390064i	0.726341	+	0.687335i	$0.241219\pi$
$564$		0			0
$565$	6295.63	−	6295.63i	0.468777	−	0.468777i
$566$		0			0
$567$	−9805.77	−	13321.0i	−0.726285	−	0.986651i
$568$		0			0
$569$			25701.7i			1.89362i	0.321792	+	0.946811i	$0.395715\pi$
$569$			25701.7i			1.89362i	−0.321792	+	0.946811i	$0.604285\pi$
$570$		0			0
$571$	10099.1	−	10099.1i	0.740167	−	0.740167i	−0.232443	−	0.972610i	$-0.574672\pi$
$571$	10099.1	−	10099.1i	0.740167	−	0.740167i	0.972610	+	0.232443i	$0.0746720\pi$
$572$		0			0
$573$	−9756.17	−	9756.17i	−0.711291	−	0.711291i
$574$		0			0
$575$		−	2780.17i		−	0.201637i
$576$		0			0
$577$		−	7848.90i		−	0.566298i	−0.959076	−	0.283149i	$-0.908621\pi$
$577$		−	7848.90i		−	0.566298i	0.959076	−	0.283149i	$-0.0913791\pi$
$578$		0			0
$579$	−5173.39	+	5173.39i	−0.371328	+	0.371328i
$580$		0			0
$581$	6903.51	+	1049.33i	0.492953	+	0.0749289i
$582$		0			0
$583$	9020.02			0.640774
$584$		0			0
$585$		−	4358.18i		−	0.308015i
$586$		0			0
$587$	7598.36	+	7598.36i	0.534272	+	0.534272i	0.921841	−	0.387569i	$-0.126685\pi$
$587$	7598.36	+	7598.36i	0.534272	+	0.534272i	−0.387569	+	0.921841i	$0.626685\pi$
$588$		0			0
$589$	98.4709	−	98.4709i	0.00688866	−	0.00688866i
$590$		0			0
$591$	−21164.2			−1.47306
$592$		0			0
$593$		−	7410.34i		−	0.513164i	−0.966522	−	0.256582i	$-0.917404\pi$
$593$		−	7410.34i		−	0.513164i	0.966522	−	0.256582i	$-0.0825963\pi$
$594$		0			0
$595$	−322.468	+	2121.50i	−0.0222183	+	0.146173i
$596$		0			0
$597$	−7265.44	−	7265.44i	−0.498081	−	0.498081i
$598$		0			0
$599$	−19708.9			−1.34438			−0.672190	−	0.740379i	$-0.734646\pi$
$599$	−19708.9			−1.34438			−0.672190	+	0.740379i	$0.734646\pi$
$600$		0			0
$601$	−2893.01			−0.196354			−0.0981768	−	0.995169i	$-0.531301\pi$
$601$	−2893.01			−0.196354			−0.0981768	+	0.995169i	$0.531301\pi$
$602$		0			0
$603$	−7478.20	+	7478.20i	−0.505034	+	0.505034i
$604$		0			0
$605$	−4162.74	−	4162.74i	−0.279735	−	0.279735i
$606$		0			0
$607$	−3597.17			−0.240535			−0.120267	−	0.992742i	$-0.538375\pi$
$607$	−3597.17			−0.240535			−0.120267	+	0.992742i	$0.538375\pi$
$608$		0			0
$609$	28504.5	−	20982.5i	1.89665	−	1.39615i
$610$		0			0
$611$	8589.06	+	8589.06i	0.568701	+	0.568701i
$612$		0			0
$613$	−10075.9	−	10075.9i	−0.663886	−	0.663886i	0.292408	−	0.956294i	$-0.405544\pi$
$613$	−10075.9	−	10075.9i	−0.663886	−	0.663886i	−0.956294	+	0.292408i	$0.905544\pi$
$614$		0			0
$615$		−	13416.5i		−	0.879681i
$616$		0			0
$617$			11242.5i			0.733561i	0.930308	+	0.366780i	$0.119540\pi$
$617$			11242.5i			0.733561i	−0.930308	+	0.366780i	$0.880460\pi$
$618$		0			0
$619$	6197.22	+	6197.22i	0.402403	+	0.402403i	0.879079	−	0.476676i	$-0.158159\pi$
$619$	6197.22	+	6197.22i	0.402403	+	0.402403i	−0.476676	+	0.879079i	$0.658159\pi$
$620$		0			0
$621$	−1689.31	−	1689.31i	−0.109162	−	0.109162i
$622$		0			0
$623$	5382.94	+	7312.67i	0.346168	+	0.470267i
$624$		0			0
$625$	1451.77			0.0929134
$626$		0			0
$627$	−588.581	−	588.581i	−0.0374891	−	0.0374891i
$628$		0			0
$629$	2371.97	−	2371.97i	0.150360	−	0.150360i
$630$		0			0
$631$	−1652.18			−0.104235			−0.0521174	−	0.998641i	$-0.516597\pi$
$631$	−1652.18			−0.104235			−0.0521174	+	0.998641i	$0.516597\pi$
$632$		0			0
$633$	6830.43			0.428886
$634$		0			0
$635$	10258.0	+	10258.0i	0.641064	+	0.641064i
$636$		0			0
$637$	10233.1	−	5375.77i	0.636499	−	0.334373i
$638$		0			0
$639$			3638.67i			0.225264i
$640$		0			0
$641$	21643.0			1.33362			0.666808	−	0.745229i	$-0.267660\pi$
$641$	21643.0			1.33362			0.666808	+	0.745229i	$0.267660\pi$
$642$		0			0
$643$	7451.23	−	7451.23i	0.456995	−	0.456995i	−0.440673	−	0.897668i	$-0.645260\pi$
$643$	7451.23	−	7451.23i	0.456995	−	0.456995i	0.897668	+	0.440673i	$0.145260\pi$
$644$		0			0
$645$	8796.19	+	8796.19i	0.536976	+	0.536976i
$646$		0			0
$647$		−	20059.5i		−	1.21889i	−0.792828	−	0.609445i	$-0.791392\pi$
$647$		−	20059.5i		−	1.21889i	0.792828	−	0.609445i	$-0.208608\pi$
$648$		0			0
$649$	−15483.6			−0.936494
$650$		0			0
$651$	3134.45	+	476.436i	0.188708	+	0.0286836i
$652$		0			0
$653$	−794.802	+	794.802i	−0.0476310	+	0.0476310i	−0.730521	−	0.682890i	$-0.760723\pi$
$653$	−794.802	+	794.802i	−0.0476310	+	0.0476310i	0.682890	+	0.730521i	$0.260723\pi$
$654$		0			0
$655$			19473.2i			1.16165i
$656$		0			0
$657$		−	3240.41i		−	0.192421i
$658$		0			0
$659$	5170.21	+	5170.21i	0.305619	+	0.305619i	0.843207	−	0.537588i	$-0.180665\pi$
$659$	5170.21	+	5170.21i	0.305619	+	0.305619i	−0.537588	+	0.843207i	$0.680665\pi$
$660$		0			0
$661$	2888.45	−	2888.45i	0.169967	−	0.169967i	−0.616998	−	0.786965i	$-0.711651\pi$
$661$	2888.45	−	2888.45i	0.169967	−	0.169967i	0.786965	+	0.616998i	$0.211651\pi$
$662$		0			0
$663$		−	3586.89i		−	0.210110i
$664$		0			0
$665$	−591.162	+	435.161i	−0.0344726	+	0.0253757i
$666$		0			0
$667$	7804.13	−	7804.13i	0.453039	−	0.453039i
$668$		0			0
$669$	7374.11	−	7374.11i	0.426158	−	0.426158i
$670$		0			0
$671$	−20129.2			−1.15809
$672$		0			0
$673$	30030.6			1.72005			0.860026	−	0.510251i	$-0.170447\pi$
$673$	30030.6			1.72005			0.860026	+	0.510251i	$0.170447\pi$
$674$		0			0
$675$	−3146.43	+	3146.43i	−0.179417	+	0.179417i
$676$		0			0
$677$	6465.93	−	6465.93i	0.367069	−	0.367069i	−0.499338	−	0.866407i	$-0.666423\pi$
$677$	6465.93	−	6465.93i	0.367069	−	0.367069i	0.866407	+	0.499338i	$0.166423\pi$
$678$		0			0
$679$	−24765.8	+	18230.4i	−1.39974	+	1.03036i
$680$		0			0
$681$		−	26795.2i		−	1.50778i
$682$		0			0
$683$	−5685.87	+	5685.87i	−0.318541	+	0.318541i	−0.848207	−	0.529665i	$-0.822318\pi$
$683$	−5685.87	+	5685.87i	−0.318541	+	0.318541i	0.529665	+	0.848207i	$0.322318\pi$
$684$		0			0
$685$	5781.99	+	5781.99i	0.322509	+	0.322509i
$686$		0			0
$687$		−	31612.8i		−	1.75561i
$688$		0			0
$689$			13296.3i			0.735193i
$690$		0			0
$691$	4300.39	−	4300.39i	0.236751	−	0.236751i	−0.578753	−	0.815503i	$-0.696460\pi$
$691$	4300.39	−	4300.39i	0.236751	−	0.236751i	0.815503	+	0.578753i	$0.196460\pi$
$692$		0			0
$693$	1129.83	−	7433.06i	0.0619315	−	0.407444i
$694$		0			0
$695$	16256.2			0.887241
$696$		0			0
$697$		−	4380.86i		−	0.238073i
$698$		0			0
$699$	3271.98	+	3271.98i	0.177050	+	0.177050i
$700$		0			0
$701$	−18283.2	+	18283.2i	−0.985088	+	0.985088i	−0.999890	−	0.0148024i	$-0.995288\pi$
$701$	−18283.2	+	18283.2i	−0.985088	+	0.985088i	0.0148024	+	0.999890i	$0.495288\pi$
$702$		0			0
$703$	1147.49			0.0615625
$704$		0			0
$705$		−	17561.4i		−	0.938158i
$706$		0			0
$707$	20220.2	+	3073.48i	1.07562	+	0.163494i
$708$		0			0
$709$	5068.76	+	5068.76i	0.268493	+	0.268493i	0.828493	−	0.560000i	$-0.189199\pi$
$709$	5068.76	+	5068.76i	0.268493	+	0.268493i	−0.560000	+	0.828493i	$0.689199\pi$
$710$		0			0
$711$	−16104.4			−0.849452
$712$		0			0
$713$	988.609			0.0519266
$714$		0			0
$715$	−3967.63	+	3967.63i	−0.207526	+	0.207526i
$716$		0			0
$717$	20009.1	+	20009.1i	1.04220	+	1.04220i
$718$		0			0
$719$	26087.1			1.35311			0.676555	−	0.736392i	$-0.263472\pi$
$719$	26087.1			1.35311			0.676555	+	0.736392i	$0.263472\pi$
$720$		0			0
$721$	6021.93	−	4432.81i	0.311052	−	0.228969i
$722$		0			0
$723$	4600.02	+	4600.02i	0.236621	+	0.236621i
$724$		0			0
$725$	−14535.6	−	14535.6i	−0.744607	−	0.744607i
$726$		0			0
$727$			25168.8i			1.28399i	0.766709	+	0.641995i	$0.221893\pi$
$727$			25168.8i			1.28399i	−0.766709	+	0.641995i	$0.778107\pi$
$728$		0			0
$729$		−	4689.70i		−	0.238262i
$730$		0			0
$731$	2872.21	+	2872.21i	0.145325	+	0.145325i
$732$		0			0
$733$	−27526.8	−	27526.8i	−1.38708	−	1.38708i	−0.831395	−	0.555682i	$-0.812457\pi$
$733$	−27526.8	−	27526.8i	−1.38708	−	1.38708i	−0.555682	−	0.831395i	$-0.687543\pi$
$734$		0			0
$735$	−15957.1	−	4965.70i	−0.800800	−	0.249201i
$736$		0			0
$737$	13616.1			0.680536
$738$		0			0
$739$	14687.0	+	14687.0i	0.731083	+	0.731083i	0.970834	−	0.239752i	$-0.0770660\pi$
$739$	14687.0	+	14687.0i	0.731083	+	0.731083i	−0.239752	+	0.970834i	$0.577066\pi$
$740$		0			0
$741$	867.618	−	867.618i	0.0430132	−	0.0430132i
$742$		0			0
$743$	38504.7			1.90121			0.950606	−	0.310402i	$-0.100463\pi$
$743$	38504.7			1.90121			0.950606	+	0.310402i	$0.100463\pi$
$744$		0			0
$745$	−20880.7			−1.02686
$746$		0			0
$747$	4734.10	+	4734.10i	0.231876	+	0.231876i
$748$		0			0
$749$	20630.3	+	3135.80i	1.00643	+	0.152977i
$750$		0			0
$751$		−	18878.3i		−	0.917284i	−0.888621	−	0.458642i	$-0.848336\pi$
$751$		−	18878.3i		−	0.917284i	0.888621	−	0.458642i	$-0.151664\pi$
$752$		0			0
$753$	4635.98			0.224362
$754$		0			0
$755$	8403.76	−	8403.76i	0.405092	−	0.405092i
$756$		0			0
$757$	16826.2	+	16826.2i	0.807872	+	0.807872i	0.984311	−	0.176440i	$-0.0564581\pi$
$757$	16826.2	+	16826.2i	0.807872	+	0.807872i	−0.176440	+	0.984311i	$0.556458\pi$
$758$		0			0
$759$		−	5909.12i		−	0.282592i
$760$		0			0
$761$	32996.0			1.57175			0.785877	−	0.618382i	$-0.212212\pi$
$761$	32996.0			1.57175			0.785877	+	0.618382i	$0.212212\pi$
$762$		0			0
$763$	11430.6	+	1737.45i	0.542354	+	0.0824378i
$764$		0			0
$765$	−1454.82	+	1454.82i	−0.0687573	+	0.0687573i
$766$		0			0
$767$		−	22824.2i		−	1.07449i
$768$		0			0
$769$		−	40080.7i		−	1.87952i	−0.341840	−	0.939758i	$-0.611050\pi$
$769$		−	40080.7i		−	1.87952i	0.341840	−	0.939758i	$-0.388950\pi$
$770$		0			0
$771$	−25363.1	−	25363.1i	−1.18473	−	1.18473i
$772$		0			0
$773$	−14826.6	+	14826.6i	−0.689877	+	0.689877i	−0.962205	−	0.272327i	$-0.912207\pi$
$773$	−14826.6	+	14826.6i	−0.689877	+	0.689877i	0.272327	+	0.962205i	$0.412207\pi$
$774$		0			0
$775$		−	1841.34i		−	0.0853457i
$776$		0			0
$777$	15487.0	+	21039.0i	0.715050	+	0.971389i
$778$		0			0
$779$	1059.67	−	1059.67i	0.0487376	−	0.0487376i
$780$		0			0
$781$	3312.60	−	3312.60i	0.151772	−	0.151772i
$782$		0			0
$783$	−17664.5			−0.806229
$784$		0			0
$785$	−23216.1			−1.05556
$786$		0			0
$787$	10969.0	−	10969.0i	0.496825	−	0.496825i	−0.413623	−	0.910448i	$-0.635737\pi$
$787$	10969.0	−	10969.0i	0.496825	−	0.496825i	0.910448	+	0.413623i	$0.135737\pi$
$788$		0			0
$789$	−28718.3	+	28718.3i	−1.29581	+	1.29581i
$790$		0			0
$791$	13422.1	+	18233.8i	0.603330	+	0.819619i
$792$		0			0
$793$		−	29672.1i		−	1.32874i
$794$		0			0
$795$	13593.0	−	13593.0i	0.606406	−	0.606406i
$796$		0			0
$797$	13341.7	+	13341.7i	0.592956	+	0.592956i	0.938429	−	0.345473i	$-0.112281\pi$
$797$	13341.7	+	13341.7i	0.592956	+	0.592956i	−0.345473	+	0.938429i	$0.612281\pi$
$798$		0			0
$799$		−	5734.31i		−	0.253899i
$800$		0			0
$801$			8706.06i			0.384037i
$802$		0			0
$803$	−2950.02	+	2950.02i	−0.129644	+	0.129644i
$804$		0			0
$805$	−5151.94	−	783.094i	−0.225568	−	0.0342863i
$806$		0			0
$807$	26378.2			1.15063
$808$		0			0
$809$			32335.9i			1.40528i	0.711547	+	0.702639i	$0.247995\pi$
$809$			32335.9i			1.40528i	−0.711547	+	0.702639i	$0.752005\pi$
$810$		0			0
$811$	−28797.0	−	28797.0i	−1.24685	−	1.24685i	−0.957102	−	0.289751i	$-0.906427\pi$
$811$	−28797.0	−	28797.0i	−1.24685	−	1.24685i	−0.289751	−	0.957102i	$-0.593573\pi$
$812$		0			0
$813$	−30538.7	+	30538.7i	−1.31739	+	1.31739i
$814$		0			0
$815$	2953.48			0.126940
$816$		0			0
$817$			1389.49i			0.0595009i
$818$		0			0
$819$	10957.0	+	1665.46i	0.467482	+	0.0710573i
$820$		0			0
$821$	−15051.0	−	15051.0i	−0.639810	−	0.639810i	0.310699	−	0.950508i	$-0.399437\pi$
$821$	−15051.0	−	15051.0i	−0.639810	−	0.639810i	−0.950508	+	0.310699i	$0.899437\pi$
$822$		0			0
$823$	−22273.8			−0.943396			−0.471698	−	0.881760i	$-0.656359\pi$
$823$	−22273.8			−0.943396			−0.471698	+	0.881760i	$0.656359\pi$
$824$		0			0
$825$	−11006.1			−0.464463
$826$		0			0
$827$	−27641.1	+	27641.1i	−1.16224	+	1.16224i	−0.178260	+	0.983983i	$0.557047\pi$
$827$	−27641.1	+	27641.1i	−1.16224	+	1.16224i	−0.983983	+	0.178260i	$0.942953\pi$
$828$		0			0
$829$	13182.6	+	13182.6i	0.552291	+	0.552291i	0.927101	−	0.374811i	$-0.122292\pi$
$829$	13182.6	+	13182.6i	0.552291	+	0.552291i	−0.374811	+	0.927101i	$0.622292\pi$
$830$		0			0
$831$	−25421.9			−1.06122
$832$		0			0
$833$	−5210.46	−	1621.44i	−0.216725	−	0.0674426i
$834$		0			0
$835$	20294.3	+	20294.3i	0.841094	+	0.841094i
$836$		0			0
$837$	−1118.85	−	1118.85i	−0.0462043	−	0.0462043i
$838$		0			0
$839$		−	1761.39i		−	0.0724791i	−0.999343	−	0.0362396i	$-0.988462\pi$
$839$		−	1761.39i		−	0.0724791i	0.999343	−	0.0362396i	$-0.0115379\pi$
$840$		0			0
$841$		−	57216.0i		−	2.34598i
$842$		0			0
$843$	−2380.12	−	2380.12i	−0.0972429	−	0.0972429i
$844$		0			0
$845$	5465.40	+	5465.40i	0.222503	+	0.222503i
$846$		0			0
$847$	12056.4	−	8874.84i	0.489094	−	0.360027i
$848$		0			0
$849$	−43918.0			−1.77534
$850$		0			0
$851$	5760.18	+	5760.18i	0.232029	+	0.232029i
$852$		0			0
$853$	−19594.2	+	19594.2i	−0.786508	+	0.786508i	−0.980920	−	0.194412i	$-0.937720\pi$
$853$	−19594.2	+	19594.2i	−0.786508	+	0.786508i	0.194412	+	0.980920i	$0.437720\pi$
$854$		0			0
$855$	−703.804			−0.0281516
$856$		0			0
$857$	49926.5			1.99003			0.995016	−	0.0997152i	$-0.0317932\pi$
$857$	49926.5			1.99003			0.995016	+	0.0997152i	$0.0317932\pi$
$858$		0			0
$859$	267.320	+	267.320i	0.0106180	+	0.0106180i	0.712396	−	0.701778i	$-0.247610\pi$
$859$	267.320	+	267.320i	0.0106180	+	0.0106180i	−0.701778	+	0.712396i	$0.747610\pi$
$860$		0			0
$861$	33730.5	+	5127.04i	1.33511	+	0.202937i
$862$		0			0
$863$			28802.7i			1.13610i	0.822993	+	0.568051i	$0.192302\pi$
$863$			28802.7i			1.13610i	−0.822993	+	0.568051i	$0.807698\pi$
$864$		0			0
$865$	−13690.4			−0.538136
$866$		0			0
$867$	22044.0	−	22044.0i	0.863501	−	0.863501i
$868$		0			0
$869$	14661.2	+	14661.2i	0.572320	+	0.572320i
$870$		0			0
$871$			20071.3i			0.780815i
$872$		0			0
$873$	−29484.8			−1.14308
$874$		0			0
$875$	−3992.19	+	26264.4i	−0.154241	+	1.01474i
$876$		0			0
$877$	−22654.2	+	22654.2i	−0.872266	+	0.872266i	−0.992719	−	0.120453i	$-0.961565\pi$
$877$	−22654.2	+	22654.2i	−0.872266	+	0.872266i	0.120453	+	0.992719i	$0.461565\pi$
$878$		0			0
$879$		−	19285.4i		−	0.740022i
$880$		0			0
$881$			12370.9i			0.473082i	0.971621	+	0.236541i	$0.0760138\pi$
$881$			12370.9i			0.473082i	−0.971621	+	0.236541i	$0.923986\pi$
$882$		0			0
$883$	−11486.6	−	11486.6i	−0.437773	−	0.437773i	0.453489	−	0.891262i	$-0.350179\pi$
$883$	−11486.6	−	11486.6i	−0.437773	−	0.437773i	−0.891262	+	0.453489i	$0.850179\pi$
$884$		0			0
$885$	−23333.4	+	23333.4i	−0.886266	+	0.886266i
$886$		0			0
$887$		−	39896.1i		−	1.51024i	−0.655588	−	0.755119i	$-0.727579\pi$
$887$		−	39896.1i		−	1.51024i	0.655588	−	0.755119i	$-0.272421\pi$
$888$		0			0
$889$	−29709.8	+	21869.7i	−1.12085	+	0.825069i
$890$		0			0
$891$	−14438.1	+	14438.1i	−0.542866	+	0.542866i
$892$		0			0
$893$	1387.05	−	1387.05i	0.0519774	−	0.0519774i
$894$		0			0
$895$	−19216.5			−0.717694
$896$		0			0
$897$	8710.54			0.324233
$898$		0			0
$899$	5168.76	−	5168.76i	0.191755	−	0.191755i
$900$		0			0
$901$	4438.49	−	4438.49i	0.164115	−	0.164115i
$902$		0			0
$903$	−25476.0	+	18753.2i	−0.938859	+	0.691105i
$904$		0			0
$905$		−	8422.96i		−	0.309380i
$906$		0			0
$907$	−25295.4	+	25295.4i	−0.926041	+	0.926041i	−0.997447	−	0.0714064i	$-0.977251\pi$
$907$	−25295.4	+	25295.4i	−0.926041	+	0.926041i	0.0714064	+	0.997447i	$0.477251\pi$
$908$		0			0
$909$	13866.1	+	13866.1i	0.505951	+	0.505951i
$910$		0			0
$911$		−	7335.63i		−	0.266784i	−0.991063	−	0.133392i	$-0.957413\pi$
$911$		−	7335.63i		−	0.266784i	0.991063	−	0.133392i	$-0.0425870\pi$
$912$		0			0
$913$		−	8619.72i		−	0.312455i
$914$		0			0
$915$	−30334.2	+	30334.2i	−1.09598	+	1.09598i
$916$		0			0
$917$	−48957.8	−	7441.59i	−1.76306	−	0.267986i
$918$		0			0
$919$	−8970.16			−0.321979			−0.160989	−	0.986956i	$-0.551468\pi$
$919$	−8970.16			−0.321979			−0.160989	+	0.986956i	$0.551468\pi$
$920$		0			0
$921$			57800.0i			2.06794i
$922$		0			0
$923$	4883.05	+	4883.05i	0.174136	+	0.174136i
$924$		0			0
$925$	10728.7	−	10728.7i	0.381358	−	0.381358i
$926$		0			0
$927$	7169.38			0.254016
$928$		0			0
$929$			20647.8i			0.729205i	0.931163	+	0.364603i	$0.118795\pi$
$929$			20647.8i			0.729205i	−0.931163	+	0.364603i	$0.881205\pi$
$930$		0			0
$931$	−868.134	−	1652.54i	−0.0305606	−	0.0581739i
$932$		0			0
$933$	5114.11	+	5114.11i	0.179452	+	0.179452i
$934$		0			0
$935$	2648.90			0.0926507
$936$		0			0
$937$	−42203.8			−1.47144			−0.735720	−	0.677286i	$-0.763156\pi$
$937$	−42203.8			−1.47144			−0.735720	+	0.677286i	$0.763156\pi$
$938$		0			0
$939$	−30006.6	+	30006.6i	−1.04284	+	1.04284i
$940$		0			0
$941$	−3986.63	−	3986.63i	−0.138109	−	0.138109i	0.634672	−	0.772781i	$-0.281135\pi$
$941$	−3986.63	−	3986.63i	−0.138109	−	0.138109i	−0.772781	+	0.634672i	$0.781135\pi$
$942$		0			0
$943$	10638.7			0.367383
$944$		0			0
$945$	4944.39	+	6716.91i	0.170202	+	0.231218i
$946$		0			0
$947$	−32201.4	−	32201.4i	−1.10497	−	1.10497i	−0.993802	−	0.111166i	$-0.964542\pi$
$947$	−32201.4	−	32201.4i	−1.10497	−	1.10497i	−0.111166	−	0.993802i	$-0.535458\pi$
$948$		0			0
$949$	−4348.59	−	4348.59i	−0.148747	−	0.148747i
$950$		0			0
$951$		−	21993.5i		−	0.749933i
$952$		0			0
$953$		−	28099.9i		−	0.955135i	−0.878595	−	0.477568i	$-0.841519\pi$
$953$		−	28099.9i		−	0.955135i	0.878595	−	0.477568i	$-0.158481\pi$
$954$		0			0
$955$	10620.6	+	10620.6i	0.359870	+	0.359870i
$956$		0			0
$957$	−30894.7	−	30894.7i	−1.04356	−	1.04356i
$958$		0			0
$959$	−16746.2	+	12327.0i	−0.563881	+	0.415079i
$960$		0			0
$961$	−29136.2			−0.978021
$962$		0			0
$963$	14147.3	+	14147.3i	0.473406	+	0.473406i
$964$		0			0
$965$	5631.79	−	5631.79i	0.187869	−	0.187869i
$966$		0			0
$967$	8908.61			0.296258			0.148129	−	0.988968i	$-0.452675\pi$
$967$	8908.61			0.296258			0.148129	+	0.988968i	$0.452675\pi$
$968$		0			0
$969$	−579.247			−0.0192034
$970$		0			0
$971$	−20802.3	−	20802.3i	−0.687516	−	0.687516i	0.274166	−	0.961682i	$-0.411598\pi$
$971$	−20802.3	−	20802.3i	−0.687516	−	0.687516i	−0.961682	+	0.274166i	$0.911598\pi$
$972$		0			0
$973$	−6212.23	+	40869.9i	−0.204681	+	1.34659i
$974$		0			0
$975$		−	16223.9i		−	0.532903i
$976$		0			0
$977$	−10844.3			−0.355108			−0.177554	−	0.984111i	$-0.556818\pi$
$977$	−10844.3			−0.355108			−0.177554	+	0.984111i	$0.556818\pi$
$978$		0			0
$979$	7925.87	−	7925.87i	0.258746	−	0.258746i
$980$		0			0
$981$	7838.58	+	7838.58i	0.255114	+	0.255114i
$982$		0			0
$983$			42864.7i			1.39082i	0.718615	+	0.695408i	$0.244776\pi$
$983$			42864.7i			1.39082i	−0.718615	+	0.695408i	$0.755224\pi$
$984$		0			0
$985$	23039.6			0.745280
$986$		0			0
$987$	44151.5	+	6711.02i	1.42387	+	0.216428i
$988$		0			0
$989$	−6974.98	+	6974.98i	−0.224258	+	0.224258i
$990$		0			0
$991$			12792.6i			0.410060i	0.978756	+	0.205030i	$0.0657292\pi$
$991$			12792.6i			0.410060i	−0.978756	+	0.205030i	$0.934271\pi$
$992$		0			0
$993$		−	37731.8i		−	1.20582i
$994$		0			0
$995$	7909.21	+	7909.21i	0.251999	+	0.251999i
$996$		0			0
$997$	12421.8	−	12421.8i	0.394587	−	0.394587i	−0.481732	−	0.876319i	$-0.659992\pi$
$997$	12421.8	−	12421.8i	0.394587	−	0.394587i	0.876319	+	0.481732i	$0.159992\pi$
$998$		0			0
$999$		−	13038.0i		−	0.412918i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.4.j.b.335.36		88
4.3	odd	2		112.4.j.b.27.27	✓	88
7.6	odd	2	inner	448.4.j.b.335.9		88
16.3	odd	4	inner	448.4.j.b.111.9		88
16.13	even	4		112.4.j.b.83.28	yes	88
28.27	even	2		112.4.j.b.27.28	yes	88
112.13	odd	4		112.4.j.b.83.27	yes	88
112.83	even	4	inner	448.4.j.b.111.36		88

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.4.j.b.27.27	✓	88	4.3	odd	2
112.4.j.b.27.28	yes	88	28.27	even	2
112.4.j.b.83.27	yes	88	112.13	odd	4
112.4.j.b.83.28	yes	88	16.13	even	4
448.4.j.b.111.9		88	16.3	odd	4	inner
448.4.j.b.111.36		88	112.83	even	4	inner
448.4.j.b.335.9		88	7.6	odd	2	inner
448.4.j.b.335.36		88	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+(4.73059 - 4.73059i) q^{3} +(-5.14976 + 5.14976i) q^{5} +(-10.9791 - 14.9150i) q^{7} -17.7570i q^{9} +(-16.1657 + 16.1657i) q^{11} +(-23.8297 - 23.8297i) q^{13} +48.7228i q^{15} +15.9094i q^{17} +(-3.84826 + 3.84826i) q^{19} +(-122.495 - 18.6192i) q^{21} -38.6350 q^{23} +71.9599i q^{25} +(43.7248 + 43.7248i) q^{27} +(-201.996 + 201.996i) q^{29} -25.5884 q^{31} +152.947i q^{33} +(133.349 + 20.2690i) q^{35} +(-149.092 - 149.092i) q^{37} -225.457 q^{39} -275.363 q^{41} +(180.535 - 180.535i) q^{43} +(91.4444 + 91.4444i) q^{45} -360.435 q^{47} +(-101.917 + 327.509i) q^{49} +(75.2609 + 75.2609i) q^{51} +(-278.986 - 278.986i) q^{53} -166.499i q^{55} +36.4091i q^{57} +(478.902 + 478.902i) q^{59} +(622.588 + 622.588i) q^{61} +(-264.847 + 194.957i) q^{63} +245.434 q^{65} +(-421.140 - 421.140i) q^{67} +(-182.767 + 182.767i) q^{69} -204.915 q^{71} +182.486 q^{73} +(340.413 + 340.413i) q^{75} +(418.599 + 63.6270i) q^{77} -906.929i q^{79} +893.128 q^{81} +(-266.605 + 266.605i) q^{83} +(-81.9296 - 81.9296i) q^{85} +1911.12i q^{87} -490.288 q^{89} +(-93.7917 + 617.050i) q^{91} +(-121.048 + 121.048i) q^{93} -39.6353i q^{95} -1660.46i q^{97} +(287.055 + 287.055i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q + 4 q^{7} - 112 q^{11} + 52 q^{21} - 160 q^{23} + 528 q^{29} - 476 q^{35} - 896 q^{37} + 8 q^{39} - 40 q^{43} - 1376 q^{49} - 1504 q^{51} - 1560 q^{53} - 8 q^{65} - 648 q^{67} + 456 q^{71} + 292 q^{77}+ \cdots - 7592 q^{99}+O(q^{100}) \) 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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