LMFDB - Embedded newform 624.4.bc.d.31.11

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [624,4,Mod(31,624)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("624.31"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(624, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 0, 0, 3])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 624 = 2^{4} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	624.bc (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [28,0,0,0,4,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$36.8171918436$
Analytic rank:	$0$
Dimension:	$28$
Relative dimension:	$14$ over $\Q(i)$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			31.11
Character	$\chi$	$=$	624.31
Dual form			624.4.bc.d.463.11

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+3.00000i q^{3} +(7.87509 - 7.87509i) q^{5} +(15.6298 - 15.6298i) q^{7} -9.00000 q^{9} +(35.2280 - 35.2280i) q^{11} +(39.8595 - 24.6622i) q^{13} +(23.6253 + 23.6253i) q^{15} -8.37246i q^{17} +(-3.94192 - 3.94192i) q^{19} +(46.8895 + 46.8895i) q^{21} -192.968 q^{23} +0.965900i q^{25} -27.0000i q^{27} -8.58178 q^{29} +(-214.896 - 214.896i) q^{31} +(105.684 + 105.684i) q^{33} -246.173i q^{35} +(-86.7849 - 86.7849i) q^{37} +(73.9866 + 119.578i) q^{39} +(210.866 - 210.866i) q^{41} -23.0221 q^{43} +(-70.8758 + 70.8758i) q^{45} +(-201.405 + 201.405i) q^{47} -145.584i q^{49} +25.1174 q^{51} +73.3118 q^{53} -554.848i q^{55} +(11.8258 - 11.8258i) q^{57} +(-240.499 + 240.499i) q^{59} -244.071 q^{61} +(-140.669 + 140.669i) q^{63} +(119.680 - 508.114i) q^{65} +(579.283 + 579.283i) q^{67} -578.905i q^{69} +(15.1455 + 15.1455i) q^{71} +(564.626 + 564.626i) q^{73} -2.89770 q^{75} -1101.22i q^{77} -507.987i q^{79} +81.0000 q^{81} +(-316.388 - 316.388i) q^{83} +(-65.9339 - 65.9339i) q^{85} -25.7453i q^{87} +(818.249 + 818.249i) q^{89} +(237.531 - 1008.46i) q^{91} +(644.688 - 644.688i) q^{93} -62.0860 q^{95} +(1161.89 - 1161.89i) q^{97} +(-317.052 + 317.052i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 4 q^{5} + 8 q^{7} - 252 q^{9} - 64 q^{11} - 32 q^{13} + 12 q^{15} - 56 q^{19} + 24 q^{21} - 384 q^{23} - 32 q^{29} + 168 q^{31} - 192 q^{33} + 412 q^{37} - 252 q^{39} + 1340 q^{41} - 624 q^{43} - 36 q^{45}+ \cdots + 576 q^{99}+O(q^{100}) $ 28 * q + 4 * q^5 + 8 * q^7 - 252 * q^9 - 64 * q^11 - 32 * q^13 + 12 * q^15 - 56 * q^19 + 24 * q^21 - 384 * q^23 - 32 * q^29 + 168 * q^31 - 192 * q^33 + 412 * q^37 - 252 * q^39 + 1340 * q^41 - 624 * q^43 - 36 * q^45 + 688 * q^47 - 312 * q^51 + 2152 * q^53 + 168 * q^57 - 624 * q^59 - 216 * q^61 - 72 * q^63 + 60 * q^65 + 2024 * q^67 - 448 * q^71 - 724 * q^73 + 1716 * q^75 + 2268 * q^81 - 2288 * q^83 - 1528 * q^85 - 1876 * q^89 + 3448 * q^91 - 504 * q^93 - 192 * q^95 + 76 * q^97 + 576 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/624\mathbb{Z}\right)^\times$.

$n$	$79$	$145$	$209$	$469$
$\chi(n)$	$-1$	$e\left(\frac{3}{4}\right)$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$			3.00000i			0.577350i
$3$			3.00000i			0.577350i
$4$		0			0
$5$	7.87509	−	7.87509i	0.704370	−	0.704370i	−0.260976	−	0.965345i	$-0.584044\pi$
$5$	7.87509	−	7.87509i	0.704370	−	0.704370i	0.965345	+	0.260976i	$0.0840443\pi$
$6$		0			0
$7$	15.6298	−	15.6298i	0.843933	−	0.843933i	−0.145435	−	0.989368i	$-0.546458\pi$
$7$	15.6298	−	15.6298i	0.843933	−	0.843933i	0.989368	+	0.145435i	$0.0464582\pi$
$8$		0			0
$9$	−9.00000			−0.333333
$10$		0			0
$11$	35.2280	−	35.2280i	0.965604	−	0.965604i	−0.0338234	−	0.999428i	$-0.510768\pi$
$11$	35.2280	−	35.2280i	0.965604	−	0.965604i	0.999428	+	0.0338234i	$0.0107684\pi$
$12$		0			0
$13$	39.8595	−	24.6622i	0.850386	−	0.526159i
$13$	39.8595	−	24.6622i	0.850386	−	0.526159i
$14$		0			0
$15$	23.6253	+	23.6253i	0.406668	+	0.406668i
$16$		0			0
$17$		−	8.37246i		−	0.119448i	−0.998215	−	0.0597241i	$-0.980978\pi$
$17$		−	8.37246i		−	0.119448i	0.998215	−	0.0597241i	$-0.0190221\pi$
$18$		0			0
$19$	−3.94192	−	3.94192i	−0.0475968	−	0.0475968i	0.682908	−	0.730505i	$-0.260715\pi$
$19$	−3.94192	−	3.94192i	−0.0475968	−	0.0475968i	−0.730505	+	0.682908i	$0.760715\pi$
$20$		0			0
$21$	46.8895	+	46.8895i	0.487245	+	0.487245i
$22$		0			0
$23$	−192.968			−1.74942			−0.874711	−	0.484645i	$-0.838949\pi$
$23$	−192.968			−1.74942			−0.874711	+	0.484645i	$0.838949\pi$
$24$		0			0
$25$			0.965900i			0.00772720i
$26$		0			0
$27$		−	27.0000i		−	0.192450i
$28$		0			0
$29$	−8.58178			−0.0549516			−0.0274758	−	0.999622i	$-0.508747\pi$
$29$	−8.58178			−0.0549516			−0.0274758	+	0.999622i	$0.508747\pi$
$30$		0			0
$31$	−214.896	−	214.896i	−1.24505	−	1.24505i	−0.957881	−	0.287166i	$-0.907287\pi$
$31$	−214.896	−	214.896i	−1.24505	−	1.24505i	−0.287166	−	0.957881i	$-0.592713\pi$
$32$		0			0
$33$	105.684	+	105.684i	0.557492	+	0.557492i
$34$		0			0
$35$		−	246.173i		−	1.18888i
$36$		0			0
$37$	−86.7849	−	86.7849i	−0.385604	−	0.385604i	0.487512	−	0.873116i	$-0.337904\pi$
$37$	−86.7849	−	86.7849i	−0.385604	−	0.385604i	−0.873116	+	0.487512i	$0.837904\pi$
$38$		0			0
$39$	73.9866	+	119.578i	0.303778	+	0.490971i
$40$		0			0
$41$	210.866	−	210.866i	0.803213	−	0.803213i	−0.180383	−	0.983596i	$-0.557734\pi$
$41$	210.866	−	210.866i	0.803213	−	0.803213i	0.983596	+	0.180383i	$0.0577338\pi$
$42$		0			0
$43$	−23.0221			−0.0816474			−0.0408237	−	0.999166i	$-0.512998\pi$
$43$	−23.0221			−0.0816474			−0.0408237	+	0.999166i	$0.512998\pi$
$44$		0			0
$45$	−70.8758	+	70.8758i	−0.234790	+	0.234790i
$46$		0			0
$47$	−201.405	+	201.405i	−0.625062	+	0.625062i	−0.946821	−	0.321760i	$-0.895726\pi$
$47$	−201.405	+	201.405i	−0.625062	+	0.625062i	0.321760	+	0.946821i	$0.395726\pi$
$48$		0			0
$49$		−	145.584i		−	0.424444i
$50$		0			0
$51$	25.1174			0.0689635
$52$		0			0
$53$	73.3118			0.190003			0.0950014	−	0.995477i	$-0.469714\pi$
$53$	73.3118			0.190003			0.0950014	+	0.995477i	$0.469714\pi$
$54$		0			0
$55$		−	554.848i		−	1.36028i
$56$		0			0
$57$	11.8258	−	11.8258i	0.0274800	−	0.0274800i
$58$		0			0
$59$	−240.499	+	240.499i	−0.530684	+	0.530684i	−0.920776	−	0.390092i	$-0.872443\pi$
$59$	−240.499	+	240.499i	−0.530684	+	0.530684i	0.390092	+	0.920776i	$0.372443\pi$
$60$		0			0
$61$	−244.071			−0.512296			−0.256148	−	0.966638i	$-0.582454\pi$
$61$	−244.071			−0.512296			−0.256148	+	0.966638i	$0.582454\pi$
$62$		0			0
$63$	−140.669	+	140.669i	−0.281311	+	0.281311i
$64$		0			0
$65$	119.680	−	508.114i	0.228376	−	0.969596i
$66$		0			0
$67$	579.283	+	579.283i	1.05628	+	1.05628i	0.998319	+	0.0579591i	$0.0184593\pi$
$67$	579.283	+	579.283i	1.05628	+	1.05628i	0.0579591	+	0.998319i	$0.481541\pi$
$68$		0			0
$69$		−	578.905i		−	1.01003i
$70$		0			0
$71$	15.1455	+	15.1455i	0.0253161	+	0.0253161i	0.719652	−	0.694335i	$-0.244302\pi$
$71$	15.1455	+	15.1455i	0.0253161	+	0.0253161i	−0.694335	+	0.719652i	$0.744302\pi$
$72$		0			0
$73$	564.626	+	564.626i	0.905266	+	0.905266i	0.995886	−	0.0906193i	$-0.0288846\pi$
$73$	564.626	+	564.626i	0.905266	+	0.905266i	−0.0906193	+	0.995886i	$0.528885\pi$
$74$		0			0
$75$	−2.89770			−0.00446130
$76$		0			0
$77$		−	1101.22i		−	1.62981i
$78$		0			0
$79$		−	507.987i		−	0.723456i	−0.932284	−	0.361728i	$-0.882187\pi$
$79$		−	507.987i		−	0.723456i	0.932284	−	0.361728i	$-0.117813\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$	−316.388	−	316.388i	−0.418411	−	0.418411i	0.466245	−	0.884656i	$-0.345607\pi$
$83$	−316.388	−	316.388i	−0.418411	−	0.418411i	−0.884656	+	0.466245i	$0.845607\pi$
$84$		0			0
$85$	−65.9339	−	65.9339i	−0.0841357	−	0.0841357i
$86$		0			0
$87$		−	25.7453i		−	0.0317263i
$88$		0			0
$89$	818.249	+	818.249i	0.974542	+	0.974542i	0.999684	−	0.0251422i	$-0.00800386\pi$
$89$	818.249	+	818.249i	0.974542	+	0.974542i	−0.0251422	+	0.999684i	$0.508004\pi$
$90$		0			0
$91$	237.531	−	1008.46i	0.273627	−	1.16171i
$92$		0			0
$93$	644.688	−	644.688i	0.718828	−	0.718828i
$94$		0			0
$95$	−62.0860			−0.0670515
$96$		0			0
$97$	1161.89	−	1161.89i	1.21621	−	1.21621i	0.247257	−	0.968950i	$-0.420471\pi$
$97$	1161.89	−	1161.89i	1.21621	−	1.21621i	0.968950	−	0.247257i	$-0.0795292\pi$
$98$		0			0
$99$	−317.052	+	317.052i	−0.321868	+	0.321868i
$100$		0			0
$101$			304.846i			0.300330i	0.988661	+	0.150165i	$0.0479804\pi$
$101$			304.846i			0.300330i	−0.988661	+	0.150165i	$0.952020\pi$
$102$		0			0
$103$	24.5435			0.0234791			0.0117395	−	0.999931i	$-0.496263\pi$
$103$	24.5435			0.0234791			0.0117395	+	0.999931i	$0.496263\pi$
$104$		0			0
$105$	738.519			0.686401
$106$		0			0
$107$		−	1262.45i		−	1.14062i	−0.821430	−	0.570309i	$-0.806824\pi$
$107$		−	1262.45i		−	1.14062i	0.821430	−	0.570309i	$-0.193176\pi$
$108$		0			0
$109$	1461.12	−	1461.12i	1.28394	−	1.28394i	0.345539	−	0.938405i	$-0.387696\pi$
$109$	1461.12	−	1461.12i	1.28394	−	1.28394i	0.938405	−	0.345539i	$-0.112304\pi$
$110$		0			0
$111$	260.355	−	260.355i	0.222629	−	0.222629i
$112$		0			0
$113$	598.613			0.498343			0.249171	−	0.968459i	$-0.419842\pi$
$113$	598.613			0.498343			0.249171	+	0.968459i	$0.419842\pi$
$114$		0			0
$115$	−1519.64	+	1519.64i	−1.23224	+	1.23224i
$116$		0			0
$117$	−358.735	+	221.960i	−0.283462	+	0.175386i
$118$		0			0
$119$	−130.860	−	130.860i	−0.100806	−	0.100806i
$120$		0			0
$121$		−	1151.03i		−	0.864784i
$122$		0			0
$123$	632.598	+	632.598i	0.463735	+	0.463735i
$124$		0			0
$125$	991.993	+	991.993i	0.709812	+	0.709812i
$126$		0			0
$127$	−246.923			−0.172527			−0.0862634	−	0.996272i	$-0.527493\pi$
$127$	−246.923			−0.172527			−0.0862634	+	0.996272i	$0.527493\pi$
$128$		0			0
$129$		−	69.0663i		−	0.0471392i
$130$		0			0
$131$		−	2212.73i		−	1.47578i	−0.674921	−	0.737890i	$-0.735822\pi$
$131$		−	2212.73i		−	1.47578i	0.674921	−	0.737890i	$-0.264178\pi$
$132$		0			0
$133$	−123.223			−0.0803370
$134$		0			0
$135$	−212.627	−	212.627i	−0.135556	−	0.135556i
$136$		0			0
$137$	−229.920	−	229.920i	−0.143383	−	0.143383i	0.631772	−	0.775154i	$-0.282328\pi$
$137$	−229.920	−	229.920i	−0.143383	−	0.143383i	−0.775154	+	0.631772i	$0.782328\pi$
$138$		0			0
$139$			1989.55i			1.21404i	0.794686	+	0.607020i	$0.207635\pi$
$139$			1989.55i			1.21404i	−0.794686	+	0.607020i	$0.792365\pi$
$140$		0			0
$141$	−604.214	−	604.214i	−0.360879	−	0.360879i
$142$		0			0
$143$	535.370	−	2272.97i	0.313076	−	1.32920i
$144$		0			0
$145$	−67.5823	+	67.5823i	−0.0387062	+	0.0387062i
$146$		0			0
$147$	436.753			0.245053
$148$		0			0
$149$	−2206.26	+	2206.26i	−1.21305	+	1.21305i	−0.243029	+	0.970019i	$0.578141\pi$
$149$	−2206.26	+	2206.26i	−1.21305	+	1.21305i	−0.970019	+	0.243029i	$0.921859\pi$
$150$		0			0
$151$	703.639	−	703.639i	0.379214	−	0.379214i	−0.491604	−	0.870819i	$-0.663589\pi$
$151$	703.639	−	703.639i	0.379214	−	0.379214i	0.870819	+	0.491604i	$0.163589\pi$
$152$		0			0
$153$			75.3522i			0.0398161i
$154$		0			0
$155$	−3384.65			−1.75395
$156$		0			0
$157$	746.179			0.379309			0.189655	−	0.981851i	$-0.439263\pi$
$157$	746.179			0.379309			0.189655	+	0.981851i	$0.439263\pi$
$158$		0			0
$159$			219.935i			0.109698i
$160$		0			0
$161$	−3016.07	+	3016.07i	−1.47639	+	1.47639i
$162$		0			0
$163$	−1255.72	+	1255.72i	−0.603408	+	0.603408i	−0.941215	−	0.337807i	$-0.890315\pi$
$163$	−1255.72	+	1255.72i	−0.603408	+	0.603408i	0.337807	+	0.941215i	$0.390315\pi$
$164$		0			0
$165$	1664.54			0.785361
$166$		0			0
$167$	−1777.63	+	1777.63i	−0.823697	+	0.823697i	−0.986636	−	0.162939i	$-0.947903\pi$
$167$	−1777.63	+	1777.63i	−0.823697	+	0.823697i	0.162939	+	0.986636i	$0.447903\pi$
$168$		0			0
$169$	980.553	−	1966.04i	0.446314	−	0.894876i
$170$		0			0
$171$	35.4773	+	35.4773i	0.0158656	+	0.0158656i
$172$		0			0
$173$		−	4341.26i		−	1.90786i	−0.300027	−	0.953931i	$-0.596996\pi$
$173$		−	4341.26i		−	1.90786i	0.300027	−	0.953931i	$-0.403004\pi$
$174$		0			0
$175$	15.0969	+	15.0969i	0.00652124	+	0.00652124i
$176$		0			0
$177$	−721.498	−	721.498i	−0.306391	−	0.306391i
$178$		0			0
$179$	1781.56			0.743910			0.371955	−	0.928251i	$-0.378688\pi$
$179$	1781.56			0.743910			0.371955	+	0.928251i	$0.378688\pi$
$180$		0			0
$181$			3016.00i			1.23855i	0.785174	+	0.619276i	$0.212574\pi$
$181$			3016.00i			1.23855i	−0.785174	+	0.619276i	$0.787426\pi$
$182$		0			0
$183$		−	732.213i		−	0.295774i
$184$		0			0
$185$	−1366.88			−0.543216
$186$		0			0
$187$	−294.945	−	294.945i	−0.115340	−	0.115340i
$188$		0			0
$189$	−422.006	−	422.006i	−0.162415	−	0.162415i
$190$		0			0
$191$		−	2870.78i		−	1.08755i	−0.839231	−	0.543775i	$-0.816994\pi$
$191$		−	2870.78i		−	1.08755i	0.839231	−	0.543775i	$-0.183006\pi$
$192$		0			0
$193$	919.381	+	919.381i	0.342894	+	0.342894i	0.857454	−	0.514560i	$-0.172045\pi$
$193$	919.381	+	919.381i	0.342894	+	0.342894i	−0.514560	+	0.857454i	$0.672045\pi$
$194$		0			0
$195$	1524.34	+	359.040i	0.559797	+	0.131853i
$196$		0			0
$197$	−394.046	+	394.046i	−0.142511	+	0.142511i	−0.774763	−	0.632252i	$-0.782131\pi$
$197$	−394.046	+	394.046i	−0.142511	+	0.142511i	0.632252	+	0.774763i	$0.282131\pi$
$198$		0			0
$199$	4471.48			1.59284			0.796419	−	0.604745i	$-0.206725\pi$
$199$	4471.48			1.59284			0.796419	+	0.604745i	$0.206725\pi$
$200$		0			0
$201$	−1737.85	+	1737.85i	−0.609842	+	0.609842i
$202$		0			0
$203$	−134.132	+	134.132i	−0.0463754	+	0.0463754i
$204$		0			0
$205$		−	3321.18i		−	1.13152i
$206$		0			0
$207$	1736.72			0.583141
$208$		0			0
$209$	−277.732			−0.0919194
$210$		0			0
$211$			1440.66i			0.470042i	0.971990	+	0.235021i	$0.0755159\pi$
$211$			1440.66i			0.470042i	−0.971990	+	0.235021i	$0.924484\pi$
$212$		0			0
$213$	−45.4366	+	45.4366i	−0.0146163	+	0.0146163i
$214$		0			0
$215$	−181.301	+	181.301i	−0.0575100	+	0.0575100i
$216$		0			0
$217$	−6717.58			−2.10147
$218$		0			0
$219$	−1693.88	+	1693.88i	−0.522656	+	0.522656i
$220$		0			0
$221$	−206.483	−	333.722i	−0.0628487	−	0.101577i
$222$		0			0
$223$	3840.12	+	3840.12i	1.15316	+	1.15316i	0.985916	+	0.167239i	$0.0534851\pi$
$223$	3840.12	+	3840.12i	1.15316	+	1.15316i	0.167239	+	0.985916i	$0.446515\pi$
$224$		0			0
$225$		−	8.69310i		−	0.00257573i
$226$		0			0
$227$	−3233.77	−	3233.77i	−0.945520	−	0.945520i	0.0530709	−	0.998591i	$-0.483099\pi$
$227$	−3233.77	−	3233.77i	−0.945520	−	0.945520i	−0.998591	+	0.0530709i	$0.983099\pi$
$228$		0			0
$229$	4729.09	+	4729.09i	1.36466	+	1.36466i	0.867872	+	0.496788i	$0.165487\pi$
$229$	4729.09	+	4729.09i	1.36466	+	1.36466i	0.496788	+	0.867872i	$0.334513\pi$
$230$		0			0
$231$	3303.65			0.940971
$232$		0			0
$233$			1546.78i			0.434907i	0.976071	+	0.217453i	$0.0697750\pi$
$233$			1546.78i			0.434907i	−0.976071	+	0.217453i	$0.930225\pi$
$234$		0			0
$235$			3172.16i			0.880549i
$236$		0			0
$237$	1523.96			0.417687
$238$		0			0
$239$	−871.277	−	871.277i	−0.235808	−	0.235808i	0.579304	−	0.815112i	$-0.303325\pi$
$239$	−871.277	−	871.277i	−0.235808	−	0.235808i	−0.815112	+	0.579304i	$0.803325\pi$
$240$		0			0
$241$	176.781	+	176.781i	0.0472510	+	0.0472510i	0.730337	−	0.683086i	$-0.239363\pi$
$241$	176.781	+	176.781i	0.0472510	+	0.0472510i	−0.683086	+	0.730337i	$0.739363\pi$
$242$		0			0
$243$			243.000i			0.0641500i
$244$		0			0
$245$	−1146.49	−	1146.49i	−0.298966	−	0.298966i
$246$		0			0
$247$	−254.339	−	59.9065i	−0.0655192	−	0.0154322i
$248$		0			0
$249$	949.165	−	949.165i	0.241570	−	0.241570i
$250$		0			0
$251$	−5266.78			−1.32445			−0.662223	−	0.749307i	$-0.730387\pi$
$251$	−5266.78			−1.32445			−0.662223	+	0.749307i	$0.730387\pi$
$252$		0			0
$253$	−6797.90	+	6797.90i	−1.68925	+	1.68925i
$254$		0			0
$255$	197.802	−	197.802i	0.0485758	−	0.0485758i
$256$		0			0
$257$			5569.67i			1.35185i	0.736969	+	0.675927i	$0.236257\pi$
$257$			5569.67i			1.35185i	−0.736969	+	0.675927i	$0.763743\pi$
$258$		0			0
$259$	−2712.87			−0.650848
$260$		0			0
$261$	77.2360			0.0183172
$262$		0			0
$263$			533.982i			0.125197i	0.998039	+	0.0625984i	$0.0199387\pi$
$263$			533.982i			0.125197i	−0.998039	+	0.0625984i	$0.980061\pi$
$264$		0			0
$265$	577.337	−	577.337i	0.133832	−	0.133832i
$266$		0			0
$267$	−2454.75	+	2454.75i	−0.562652	+	0.562652i
$268$		0			0
$269$	−4298.36			−0.974258			−0.487129	−	0.873330i	$-0.661956\pi$
$269$	−4298.36			−0.974258			−0.487129	+	0.873330i	$0.661956\pi$
$270$		0			0
$271$	4154.32	−	4154.32i	0.931207	−	0.931207i	−0.0665741	−	0.997781i	$-0.521207\pi$
$271$	4154.32	−	4154.32i	0.931207	−	0.931207i	0.997781	+	0.0665741i	$0.0212069\pi$
$272$		0			0
$273$	3025.39	+	712.593i	0.670714	+	0.157978i
$274$		0			0
$275$	34.0268	+	34.0268i	0.00746142	+	0.00746142i
$276$		0			0
$277$			1458.60i			0.316385i	0.987408	+	0.158192i	$0.0505666\pi$
$277$			1458.60i			0.316385i	−0.987408	+	0.158192i	$0.949433\pi$
$278$		0			0
$279$	1934.06	+	1934.06i	0.415015	+	0.415015i
$280$		0			0
$281$	−97.1163	−	97.1163i	−0.0206173	−	0.0206173i	0.696723	−	0.717340i	$-0.254641\pi$
$281$	−97.1163	−	97.1163i	−0.0206173	−	0.0206173i	−0.717340	+	0.696723i	$0.754641\pi$
$282$		0			0
$283$	6162.01			1.29432			0.647161	−	0.762353i	$-0.275956\pi$
$283$	6162.01			1.29432			0.647161	+	0.762353i	$0.275956\pi$
$284$		0			0
$285$		−	186.258i		−	0.0387122i
$286$		0			0
$287$		−	6591.61i		−	1.35572i
$288$		0			0
$289$	4842.90			0.985732
$290$		0			0
$291$	3485.67	+	3485.67i	0.702177	+	0.702177i
$292$		0			0
$293$	−625.073	−	625.073i	−0.124632	−	0.124632i	0.642040	−	0.766671i	$-0.278088\pi$
$293$	−625.073	−	625.073i	−0.124632	−	0.124632i	−0.766671	+	0.642040i	$0.778088\pi$
$294$		0			0
$295$			3787.91i			0.747596i
$296$		0			0
$297$	−951.157	−	951.157i	−0.185831	−	0.185831i
$298$		0			0
$299$	−7691.62	+	4759.03i	−1.48768	+	0.920473i
$300$		0			0
$301$	−359.832	+	359.832i	−0.0689049	+	0.0689049i
$302$		0			0
$303$	−914.538			−0.173395
$304$		0			0
$305$	−1922.08	+	1922.08i	−0.360846	+	0.360846i
$306$		0			0
$307$	−174.883	+	174.883i	−0.0325117	+	0.0325117i	−0.723176	−	0.690664i	$-0.757318\pi$
$307$	−174.883	+	174.883i	−0.0325117	+	0.0325117i	0.690664	+	0.723176i	$0.257318\pi$
$308$		0			0
$309$			73.6306i			0.0135557i
$310$		0			0
$311$	5212.31			0.950364			0.475182	−	0.879888i	$-0.342382\pi$
$311$	5212.31			0.950364			0.475182	+	0.879888i	$0.342382\pi$
$312$		0			0
$313$	1177.63			0.212664			0.106332	−	0.994331i	$-0.466089\pi$
$313$	1177.63			0.212664			0.106332	+	0.994331i	$0.466089\pi$
$314$		0			0
$315$			2215.56i			0.396294i
$316$		0			0
$317$	−3067.58	+	3067.58i	−0.543509	+	0.543509i	−0.924556	−	0.381047i	$-0.875564\pi$
$317$	−3067.58	+	3067.58i	−0.543509	+	0.543509i	0.381047	+	0.924556i	$0.375564\pi$
$318$		0			0
$319$	−302.319	+	302.319i	−0.0530615	+	0.0530615i
$320$		0			0
$321$	3787.36			0.658536
$322$		0			0
$323$	−33.0036	+	33.0036i	−0.00568536	+	0.00568536i
$324$		0			0
$325$	23.8212	+	38.5003i	0.00406573	+	0.00657111i
$326$		0			0
$327$	4383.36	+	4383.36i	0.741285	+	0.741285i
$328$		0			0
$329$			6295.85i			1.05502i
$330$		0			0
$331$	2828.55	+	2828.55i	0.469701	+	0.469701i	0.901818	−	0.432116i	$-0.142233\pi$
$331$	2828.55	+	2828.55i	0.469701	+	0.469701i	−0.432116	+	0.901818i	$0.642233\pi$
$332$		0			0
$333$	781.064	+	781.064i	0.128535	+	0.128535i
$334$		0			0
$335$	9123.81			1.48802
$336$		0			0
$337$		−	6960.82i		−	1.12516i	−0.826742	−	0.562582i	$-0.809808\pi$
$337$		−	6960.82i		−	1.12516i	0.826742	−	0.562582i	$-0.190192\pi$
$338$		0			0
$339$			1795.84i			0.287718i
$340$		0			0
$341$	−15140.7			−2.40444
$342$		0			0
$343$	3085.58	+	3085.58i	0.485730	+	0.485730i
$344$		0			0
$345$	−4558.93	−	4558.93i	−0.711434	−	0.711434i
$346$		0			0
$347$			5431.42i			0.840270i	0.907462	+	0.420135i	$0.138017\pi$
$347$			5431.42i			0.840270i	−0.907462	+	0.420135i	$0.861983\pi$
$348$		0			0
$349$	−3546.35	−	3546.35i	−0.543931	−	0.543931i	0.380748	−	0.924679i	$-0.375666\pi$
$349$	−3546.35	−	3546.35i	−0.543931	−	0.543931i	−0.924679	+	0.380748i	$0.875666\pi$
$350$		0			0
$351$	−665.879	−	1076.21i	−0.101259	−	0.163657i
$352$		0			0
$353$	−753.905	+	753.905i	−0.113672	+	0.113672i	−0.761655	−	0.647983i	$-0.775613\pi$
$353$	−753.905	+	753.905i	−0.113672	+	0.113672i	0.647983	+	0.761655i	$0.275613\pi$
$354$		0			0
$355$	238.545			0.0356638
$356$		0			0
$357$	392.581	−	392.581i	0.0582005	−	0.0582005i
$358$		0			0
$359$	4239.39	−	4239.39i	0.623249	−	0.623249i	−0.323112	−	0.946361i	$-0.604729\pi$
$359$	4239.39	−	4239.39i	0.623249	−	0.623249i	0.946361	+	0.323112i	$0.104729\pi$
$360$		0			0
$361$		−	6827.92i		−	0.995469i
$362$		0			0
$363$	3453.08			0.499283
$364$		0			0
$365$	8892.96			1.27528
$366$		0			0
$367$		−	6816.88i		−	0.969587i	−0.874629	−	0.484793i	$-0.838895\pi$
$367$		−	6816.88i		−	0.969587i	0.874629	−	0.484793i	$-0.161105\pi$
$368$		0			0
$369$	−1897.79	+	1897.79i	−0.267738	+	0.267738i
$370$		0			0
$371$	1145.85	−	1145.85i	0.160350	−	0.160350i
$372$		0			0
$373$	−13586.9			−1.88607			−0.943034	−	0.332696i	$-0.892042\pi$
$373$	−13586.9			−1.88607			−0.943034	+	0.332696i	$0.892042\pi$
$374$		0			0
$375$	−2975.98	+	2975.98i	−0.409810	+	0.409810i
$376$		0			0
$377$	−342.065	+	211.645i	−0.0467301	+	0.0289133i
$378$		0			0
$379$	3877.63	+	3877.63i	0.525542	+	0.525542i	0.919240	−	0.393698i	$-0.128804\pi$
$379$	3877.63	+	3877.63i	0.525542	+	0.525542i	−0.393698	+	0.919240i	$0.628804\pi$
$380$		0			0
$381$		−	740.770i		−	0.0996084i
$382$		0			0
$383$	−5793.77	−	5793.77i	−0.772971	−	0.772971i	0.205653	−	0.978625i	$-0.434068\pi$
$383$	−5793.77	−	5793.77i	−0.772971	−	0.772971i	−0.978625	+	0.205653i	$0.934068\pi$
$384$		0			0
$385$	−8672.19	−	8672.19i	−1.14799	−	1.14799i
$386$		0			0
$387$	207.199			0.0272158
$388$		0			0
$389$			10524.7i			1.37178i	0.727707	+	0.685888i	$0.240586\pi$
$389$			10524.7i			1.37178i	−0.727707	+	0.685888i	$0.759414\pi$
$390$		0			0
$391$			1615.62i			0.208965i
$392$		0			0
$393$	6638.19			0.852042
$394$		0			0
$395$	−4000.44	−	4000.44i	−0.509580	−	0.509580i
$396$		0			0
$397$	5328.39	+	5328.39i	0.673613	+	0.673613i	0.958547	−	0.284934i	$-0.0919717\pi$
$397$	5328.39	+	5328.39i	0.673613	+	0.673613i	−0.284934	+	0.958547i	$0.591972\pi$
$398$		0			0
$399$		−	369.670i		−	0.0463826i
$400$		0			0
$401$	5848.99	+	5848.99i	0.728391	+	0.728391i	0.970299	−	0.241908i	$-0.0777733\pi$
$401$	5848.99	+	5848.99i	0.728391	+	0.728391i	−0.241908	+	0.970299i	$0.577773\pi$
$402$		0			0
$403$	−13865.4	−	3265.83i	−1.71386	−	0.403679i
$404$		0			0
$405$	637.882	−	637.882i	0.0782633	−	0.0782633i
$406$		0			0
$407$	−6114.52			−0.744682
$408$		0			0
$409$	−6023.19	+	6023.19i	−0.728185	+	0.728185i	−0.970258	−	0.242073i	$-0.922173\pi$
$409$	−6023.19	+	6023.19i	−0.728185	+	0.728185i	0.242073	+	0.970258i	$0.422173\pi$
$410$		0			0
$411$	689.761	−	689.761i	0.0827820	−	0.0827820i
$412$		0			0
$413$			7517.94i			0.895723i
$414$		0			0
$415$	−4983.17			−0.589432
$416$		0			0
$417$	−5968.66			−0.700926
$418$		0			0
$419$			15642.7i			1.82386i	0.410350	+	0.911928i	$0.365407\pi$
$419$			15642.7i			1.82386i	−0.410350	+	0.911928i	$0.634593\pi$
$420$		0			0
$421$	11307.0	−	11307.0i	1.30895	−	1.30895i	0.386775	−	0.922174i	$-0.373589\pi$
$421$	11307.0	−	11307.0i	1.30895	−	1.30895i	0.922174	−	0.386775i	$-0.126411\pi$
$422$		0			0
$423$	1812.64	−	1812.64i	0.208354	−	0.208354i
$424$		0			0
$425$	8.08697			0.000923001
$426$		0			0
$427$	−3814.79	+	3814.79i	−0.432344	+	0.432344i
$428$		0			0
$429$	6818.91	+	1606.11i	0.767413	+	0.180754i
$430$		0			0
$431$	−6393.46	−	6393.46i	−0.714530	−	0.714530i	0.252950	−	0.967479i	$-0.418599\pi$
$431$	−6393.46	−	6393.46i	−0.714530	−	0.714530i	−0.967479	+	0.252950i	$0.918599\pi$
$432$		0			0
$433$			14458.3i			1.60467i	0.596876	+	0.802334i	$0.296409\pi$
$433$			14458.3i			1.60467i	−0.596876	+	0.802334i	$0.703591\pi$
$434$		0			0
$435$	−202.747	−	202.747i	−0.0223471	−	0.0223471i
$436$		0			0
$437$	760.667	+	760.667i	0.0832669	+	0.0832669i
$438$		0			0
$439$	−6741.89			−0.732968			−0.366484	−	0.930424i	$-0.619439\pi$
$439$	−6741.89			−0.732968			−0.366484	+	0.930424i	$0.619439\pi$
$440$		0			0
$441$			1310.26i			0.141481i
$442$		0			0
$443$		−	3196.98i		−	0.342874i	−0.985195	−	0.171437i	$-0.945159\pi$
$443$		−	3196.98i		−	0.342874i	0.985195	−	0.171437i	$-0.0548409\pi$
$444$		0			0
$445$	12887.6			1.37287
$446$		0			0
$447$	−6618.79	−	6618.79i	−0.700353	−	0.700353i
$448$		0			0
$449$	−8595.58	−	8595.58i	−0.903453	−	0.903453i	0.0922800	−	0.995733i	$-0.470585\pi$
$449$	−8595.58	−	8595.58i	−0.903453	−	0.903453i	−0.995733	+	0.0922800i	$0.970585\pi$
$450$		0			0
$451$		−	14856.8i		−	1.55117i
$452$		0			0
$453$	2110.92	+	2110.92i	0.218939	+	0.218939i
$454$		0			0
$455$	−6071.16	−	9812.32i	−0.625540	−	1.01101i
$456$		0			0
$457$	2439.24	−	2439.24i	0.249678	−	0.249678i	−0.571160	−	0.820838i	$-0.693507\pi$
$457$	2439.24	−	2439.24i	0.249678	−	0.249678i	0.820838	+	0.571160i	$0.193507\pi$
$458$		0			0
$459$	−226.057			−0.0229878
$460$		0			0
$461$	3601.15	−	3601.15i	0.363822	−	0.363822i	−0.501396	−	0.865218i	$-0.667180\pi$
$461$	3601.15	−	3601.15i	0.363822	−	0.363822i	0.865218	+	0.501396i	$0.167180\pi$
$462$		0			0
$463$	6014.74	−	6014.74i	0.603733	−	0.603733i	−0.337568	−	0.941301i	$-0.609604\pi$
$463$	6014.74	−	6014.74i	0.603733	−	0.603733i	0.941301	+	0.337568i	$0.109604\pi$
$464$		0			0
$465$		−	10153.9i		−	1.01264i
$466$		0			0
$467$	5617.63			0.556645			0.278322	−	0.960488i	$-0.410222\pi$
$467$	5617.63			0.556645			0.278322	+	0.960488i	$0.410222\pi$
$468$		0			0
$469$	18108.2			1.78285
$470$		0			0
$471$			2238.54i			0.218994i
$472$		0			0
$473$	−811.024	+	811.024i	−0.0788391	+	0.0788391i
$474$		0			0
$475$	3.80751	−	3.80751i	0.000367790	−	0.000367790i
$476$		0			0
$477$	−659.806			−0.0633343
$478$		0			0
$479$	29.5141	−	29.5141i	0.00281531	−	0.00281531i	−0.705698	−	0.708513i	$-0.749366\pi$
$479$	29.5141	−	29.5141i	0.00281531	−	0.00281531i	0.708513	+	0.705698i	$0.249366\pi$
$480$		0			0
$481$	−5599.51	−	1318.89i	−0.530801	−	0.125024i
$482$		0			0
$483$	−9048.20	−	9048.20i	−0.852397	−	0.852397i
$484$		0			0
$485$		−	18300.0i		−	1.71332i
$486$		0			0
$487$	12267.5	+	12267.5i	1.14147	+	1.14147i	0.988182	+	0.153288i	$0.0489863\pi$
$487$	12267.5	+	12267.5i	1.14147	+	1.14147i	0.153288	+	0.988182i	$0.451014\pi$
$488$		0			0
$489$	−3767.16	−	3767.16i	−0.348378	−	0.348378i
$490$		0			0
$491$	−7011.53			−0.644452			−0.322226	−	0.946663i	$-0.604431\pi$
$491$	−7011.53			−0.644452			−0.322226	+	0.946663i	$0.604431\pi$
$492$		0			0
$493$			71.8506i			0.00656387i
$494$		0			0
$495$			4993.63i			0.453428i
$496$		0			0
$497$	473.444			0.0427301
$498$		0			0
$499$	−648.254	−	648.254i	−0.0581560	−	0.0581560i	0.677431	−	0.735587i	$-0.263093\pi$
$499$	−648.254	−	648.254i	−0.0581560	−	0.0581560i	−0.735587	+	0.677431i	$0.763093\pi$
$500$		0			0
$501$	−5332.90	−	5332.90i	−0.475562	−	0.475562i
$502$		0			0
$503$		−	21365.7i		−	1.89394i	−0.321327	−	0.946968i	$-0.604129\pi$
$503$		−	21365.7i		−	1.89394i	0.321327	−	0.946968i	$-0.395871\pi$
$504$		0			0
$505$	2400.69	+	2400.69i	0.211543	+	0.211543i
$506$		0			0
$507$	5898.13	+	2941.66i	0.516657	+	0.257680i
$508$		0			0
$509$	8100.99	−	8100.99i	0.705442	−	0.705442i	−0.260131	−	0.965573i	$-0.583766\pi$
$509$	8100.99	−	8100.99i	0.705442	−	0.705442i	0.965573	+	0.260131i	$0.0837657\pi$
$510$		0			0
$511$	17650.0			1.52797
$512$		0			0
$513$	−106.432	+	106.432i	−0.00916001	+	0.00916001i
$514$		0			0
$515$	193.283	−	193.283i	0.0165380	−	0.0165380i
$516$		0			0
$517$			14190.2i			1.20712i
$518$		0			0
$519$	13023.8			1.10150
$520$		0			0
$521$	12194.6			1.02544			0.512722	−	0.858555i	$-0.328637\pi$
$521$	12194.6			1.02544			0.512722	+	0.858555i	$0.328637\pi$
$522$		0			0
$523$			6928.86i			0.579308i	0.957131	+	0.289654i	$0.0935402\pi$
$523$			6928.86i			0.579308i	−0.957131	+	0.289654i	$0.906460\pi$
$524$		0			0
$525$	−45.2906	+	45.2906i	−0.00376504	+	0.00376504i
$526$		0			0
$527$	−1799.21	+	1799.21i	−0.148719	+	0.148719i
$528$		0			0
$529$	25069.8			2.06048
$530$		0			0
$531$	2164.50	−	2164.50i	0.176895	−	0.176895i
$532$		0			0
$533$	3204.59	−	13605.4i	0.260424	−	1.10566i
$534$		0			0
$535$	−9941.94	−	9941.94i	−0.803416	−	0.803416i
$536$		0			0
$537$			5344.67i			0.429497i
$538$		0			0
$539$	−5128.65	−	5128.65i	−0.409845	−	0.409845i
$540$		0			0
$541$	−6864.60	−	6864.60i	−0.545531	−	0.545531i	0.379614	−	0.925145i	$-0.376057\pi$
$541$	−6864.60	−	6864.60i	−0.545531	−	0.545531i	−0.925145	+	0.379614i	$0.876057\pi$
$542$		0			0
$543$	−9048.01			−0.715078
$544$		0			0
$545$		−	23012.9i		−	1.80874i
$546$		0			0
$547$			19880.5i			1.55398i	0.629513	+	0.776990i	$0.283254\pi$
$547$			19880.5i			1.55398i	−0.629513	+	0.776990i	$0.716746\pi$
$548$		0			0
$549$	2196.64			0.170765
$550$		0			0
$551$	33.8287	+	33.8287i	0.00261552	+	0.00261552i
$552$		0			0
$553$	−7939.76	−	7939.76i	−0.610548	−	0.610548i
$554$		0			0
$555$		−	4100.63i		−	0.313626i
$556$		0			0
$557$	12147.8	+	12147.8i	0.924091	+	0.924091i	0.997315	−	0.0732243i	$-0.0233289\pi$
$557$	12147.8	+	12147.8i	0.924091	+	0.924091i	−0.0732243	+	0.997315i	$0.523329\pi$
$558$		0			0
$559$	−917.649	+	567.776i	−0.0694319	+	0.0429595i
$560$		0			0
$561$	884.836	−	884.836i	0.0665915	−	0.0665915i
$562$		0			0
$563$	8869.57			0.663957			0.331979	−	0.943287i	$-0.392284\pi$
$563$	8869.57			0.663957			0.331979	+	0.943287i	$0.392284\pi$
$564$		0			0
$565$	4714.13	−	4714.13i	0.351017	−	0.351017i
$566$		0			0
$567$	1266.02	−	1266.02i	0.0937703	−	0.0937703i
$568$		0			0
$569$		−	2730.47i		−	0.201173i	−0.994928	−	0.100586i	$-0.967928\pi$
$569$		−	2730.47i		−	0.201173i	0.994928	−	0.100586i	$-0.0320719\pi$
$570$		0			0
$571$	−10603.6			−0.777142			−0.388571	−	0.921419i	$-0.627031\pi$
$571$	−10603.6			−0.777142			−0.388571	+	0.921419i	$0.627031\pi$
$572$		0			0
$573$	8612.33			0.627898
$574$		0			0
$575$		−	186.388i		−	0.0135181i
$576$		0			0
$577$	8873.32	−	8873.32i	0.640210	−	0.640210i	−0.310397	−	0.950607i	$-0.600462\pi$
$577$	8873.32	−	8873.32i	0.640210	−	0.640210i	0.950607	+	0.310397i	$0.100462\pi$
$578$		0			0
$579$	−2758.14	+	2758.14i	−0.197970	+	0.197970i
$580$		0			0
$581$	−9890.20			−0.706222
$582$		0			0
$583$	2582.63	−	2582.63i	0.183468	−	0.183468i
$584$		0			0
$585$	−1077.12	+	4573.02i	−0.0761254	+	0.323199i
$586$		0			0
$587$	−12112.4	−	12112.4i	−0.851670	−	0.851670i	0.138669	−	0.990339i	$-0.455718\pi$
$587$	−12112.4	−	12112.4i	−0.851670	−	0.851670i	−0.990339	+	0.138669i	$0.955718\pi$
$588$		0			0
$589$			1694.21i			0.118520i
$590$		0			0
$591$	−1182.14	−	1182.14i	−0.0822786	−	0.0822786i
$592$		0			0
$593$	1189.13	+	1189.13i	0.0823471	+	0.0823471i	0.747081	−	0.664733i	$-0.231455\pi$
$593$	1189.13	+	1189.13i	0.0823471	+	0.0823471i	−0.664733	+	0.747081i	$0.731455\pi$
$594$		0			0
$595$	−2061.07			−0.142010
$596$		0			0
$597$			13414.4i			0.919625i
$598$		0			0
$599$			14970.3i			1.02115i	0.859833	+	0.510575i	$0.170567\pi$
$599$			14970.3i			1.02115i	−0.859833	+	0.510575i	$0.829433\pi$
$600$		0			0
$601$	−24874.2			−1.68826			−0.844128	−	0.536142i	$-0.819881\pi$
$601$	−24874.2			−1.68826			−0.844128	+	0.536142i	$0.819881\pi$
$602$		0			0
$603$	−5213.54	−	5213.54i	−0.352093	−	0.352093i
$604$		0			0
$605$	−9064.44	−	9064.44i	−0.609127	−	0.609127i
$606$		0			0
$607$		−	3789.57i		−	0.253400i	−0.991941	−	0.126700i	$-0.959561\pi$
$607$		−	3789.57i		−	0.253400i	0.991941	−	0.126700i	$-0.0404386\pi$
$608$		0			0
$609$	−402.396	−	402.396i	−0.0267749	−	0.0267749i
$610$		0			0
$611$	−3060.80	+	12995.0i	−0.202662	+	0.860425i
$612$		0			0
$613$	12063.2	−	12063.2i	0.794828	−	0.794828i	−0.187447	−	0.982275i	$-0.560021\pi$
$613$	12063.2	−	12063.2i	0.794828	−	0.794828i	0.982275	+	0.187447i	$0.0600212\pi$
$614$		0			0
$615$	9963.53			0.653282
$616$		0			0
$617$	16604.5	−	16604.5i	1.08342	−	1.08342i	0.0872335	−	0.996188i	$-0.472197\pi$
$617$	16604.5	−	16604.5i	1.08342	−	1.08342i	0.996188	−	0.0872335i	$-0.0278026\pi$
$618$		0			0
$619$	−6058.60	+	6058.60i	−0.393402	+	0.393402i	−0.875898	−	0.482496i	$-0.839730\pi$
$619$	−6058.60	+	6058.60i	−0.393402	+	0.393402i	0.482496	+	0.875898i	$0.339730\pi$
$620$		0			0
$621$			5210.15i			0.336676i
$622$		0			0
$623$	25578.2			1.64489
$624$		0			0
$625$	15503.3			0.992213
$626$		0			0
$627$		−	833.197i		−	0.0530697i
$628$		0			0
$629$	−726.604	+	726.604i	−0.0460597	+	0.0460597i
$630$		0			0
$631$	−16088.2	+	16088.2i	−1.01499	+	1.01499i	−0.0151063	+	0.999886i	$0.504809\pi$
$631$	−16088.2	+	16088.2i	−1.01499	+	1.01499i	−0.999886	+	0.0151063i	$0.995191\pi$
$632$		0			0
$633$	−4321.97			−0.271379
$634$		0			0
$635$	−1944.54	+	1944.54i	−0.121523	+	0.121523i
$636$		0			0
$637$	−3590.43	−	5802.91i	−0.223325	−	0.360942i
$638$		0			0
$639$	−136.310	−	136.310i	−0.00843870	−	0.00843870i
$640$		0			0
$641$			22122.1i			1.36314i	0.731754	+	0.681568i	$0.238702\pi$
$641$			22122.1i			1.36314i	−0.731754	+	0.681568i	$0.761298\pi$
$642$		0			0
$643$	1944.51	+	1944.51i	0.119260	+	0.119260i	0.764218	−	0.644958i	$-0.223125\pi$
$643$	1944.51	+	1944.51i	0.119260	+	0.119260i	−0.644958	+	0.764218i	$0.723125\pi$
$644$		0			0
$645$	−543.904	−	543.904i	−0.0332034	−	0.0332034i
$646$		0			0
$647$	2320.56			0.141006			0.0705028	−	0.997512i	$-0.477540\pi$
$647$	2320.56			0.141006			0.0705028	+	0.997512i	$0.477540\pi$
$648$		0			0
$649$			16944.6i			1.02486i
$650$		0			0
$651$		−	20152.7i		−	1.21328i
$652$		0			0
$653$	−18218.4			−1.09179			−0.545896	−	0.837853i	$-0.683811\pi$
$653$	−18218.4			−1.09179			−0.545896	+	0.837853i	$0.683811\pi$
$654$		0			0
$655$	−17425.5	−	17425.5i	−1.03949	−	1.03949i
$656$		0			0
$657$	−5081.63	−	5081.63i	−0.301755	−	0.301755i
$658$		0			0
$659$			17994.1i			1.06366i	0.846852	+	0.531829i	$0.178495\pi$
$659$			17994.1i			1.06366i	−0.846852	+	0.531829i	$0.821505\pi$
$660$		0			0
$661$	−11466.1	−	11466.1i	−0.674705	−	0.674705i	0.284092	−	0.958797i	$-0.408308\pi$
$661$	−11466.1	−	11466.1i	−0.674705	−	0.674705i	−0.958797	+	0.284092i	$0.908308\pi$
$662$		0			0
$663$	1001.17	−	619.450i	0.0586456	−	0.0362857i
$664$		0			0
$665$	−970.395	+	970.395i	−0.0565869	+	0.0565869i
$666$		0			0
$667$	1656.01			0.0961335
$668$		0			0
$669$	−11520.4	+	11520.4i	−0.665774	+	0.665774i
$670$		0			0
$671$	−8598.14	+	8598.14i	−0.494676	+	0.494676i
$672$		0			0
$673$		−	3575.50i		−	0.204792i	−0.994744	−	0.102396i	$-0.967349\pi$
$673$		−	3575.50i		−	0.204792i	0.994744	−	0.102396i	$-0.0326509\pi$
$674$		0			0
$675$	26.0793			0.00148710
$676$		0			0
$677$	9007.53			0.511355			0.255678	−	0.966762i	$-0.417701\pi$
$677$	9007.53			0.511355			0.255678	+	0.966762i	$0.417701\pi$
$678$		0			0
$679$		−	36320.3i		−	2.05279i
$680$		0			0
$681$	9701.32	−	9701.32i	0.545896	−	0.545896i
$682$		0			0
$683$	−825.076	+	825.076i	−0.0462235	+	0.0462235i	−0.729841	−	0.683617i	$-0.760406\pi$
$683$	−825.076	+	825.076i	−0.0462235	+	0.0462235i	0.683617	+	0.729841i	$0.260406\pi$
$684$		0			0
$685$	−3621.29			−0.201989
$686$		0			0
$687$	−14187.3	+	14187.3i	−0.787887	+	0.787887i
$688$		0			0
$689$	2922.17	−	1808.03i	0.161576	−	0.0999716i
$690$		0			0
$691$	10601.0	+	10601.0i	0.583617	+	0.583617i	0.935895	−	0.352278i	$-0.114593\pi$
$691$	10601.0	+	10601.0i	0.583617	+	0.583617i	−0.352278	+	0.935895i	$0.614593\pi$
$692$		0			0
$693$			9910.96i			0.543270i
$694$		0			0
$695$	15667.9	+	15667.9i	0.855133	+	0.855133i
$696$		0			0
$697$	−1765.47	−	1765.47i	−0.0959424	−	0.0959424i
$698$		0			0
$699$	−4640.35			−0.251093
$700$		0			0
$701$		−	32056.3i		−	1.72718i	−0.504197	−	0.863588i	$-0.668212\pi$
$701$		−	32056.3i		−	1.72718i	0.504197	−	0.863588i	$-0.331788\pi$
$702$		0			0
$703$			684.199i			0.0367071i
$704$		0			0
$705$	−9516.48			−0.508385
$706$		0			0
$707$	4764.69	+	4764.69i	0.253458	+	0.253458i
$708$		0			0
$709$	−3432.70	−	3432.70i	−0.181830	−	0.181830i	0.610323	−	0.792153i	$-0.291040\pi$
$709$	−3432.70	−	3432.70i	−0.181830	−	0.181830i	−0.792153	+	0.610323i	$0.791040\pi$
$710$		0			0
$711$			4571.88i			0.241152i
$712$		0			0
$713$	41468.1	+	41468.1i	2.17811	+	2.17811i
$714$		0			0
$715$	−13683.8	−	22115.9i	−0.715725	−	1.15677i
$716$		0			0
$717$	2613.83	−	2613.83i	0.136144	−	0.136144i
$718$		0			0
$719$	17064.7			0.885128			0.442564	−	0.896737i	$-0.354069\pi$
$719$	17064.7			0.885128			0.442564	+	0.896737i	$0.354069\pi$
$720$		0			0
$721$	383.612	−	383.612i	0.0198148	−	0.0198148i
$722$		0			0
$723$	−530.344	+	530.344i	−0.0272804	+	0.0272804i
$724$		0			0
$725$		−	8.28914i		−	0.000424622i
$726$		0			0
$727$	11157.5			0.569200			0.284600	−	0.958646i	$-0.408139\pi$
$727$	11157.5			0.569200			0.284600	+	0.958646i	$0.408139\pi$
$728$		0			0
$729$	−729.000			−0.0370370
$730$		0			0
$731$			192.752i			0.00975264i
$732$		0			0
$733$	7238.76	−	7238.76i	0.364761	−	0.364761i	−0.500801	−	0.865562i	$-0.666961\pi$
$733$	7238.76	−	7238.76i	0.364761	−	0.364761i	0.865562	+	0.500801i	$0.166961\pi$
$734$		0			0
$735$	3439.47	−	3439.47i	0.172608	−	0.172608i
$736$		0			0
$737$	40814.0			2.03989
$738$		0			0
$739$	15140.2	−	15140.2i	0.753643	−	0.753643i	−0.221514	−	0.975157i	$-0.571100\pi$
$739$	15140.2	−	15140.2i	0.753643	−	0.753643i	0.975157	+	0.221514i	$0.0710998\pi$
$740$		0			0
$741$	179.719	−	763.018i	0.00890980	−	0.0378275i
$742$		0			0
$743$	18909.8	+	18909.8i	0.933693	+	0.933693i	0.997934	−	0.0642418i	$-0.0204629\pi$
$743$	18909.8	+	18909.8i	0.933693	+	0.933693i	−0.0642418	+	0.997934i	$0.520463\pi$
$744$		0			0
$745$			34749.0i			1.70887i
$746$		0			0
$747$	2847.49	+	2847.49i	0.139470	+	0.139470i
$748$		0			0
$749$	−19732.0	−	19732.0i	−0.962604	−	0.962604i
$750$		0			0
$751$	−13628.9			−0.662220			−0.331110	−	0.943592i	$-0.607423\pi$
$751$	−13628.9			−0.662220			−0.331110	+	0.943592i	$0.607423\pi$
$752$		0			0
$753$		−	15800.3i		−	0.764669i
$754$		0			0
$755$		−	11082.4i		−	0.534214i
$756$		0			0
$757$	−31792.6			−1.52645			−0.763225	−	0.646133i	$-0.776385\pi$
$757$	−31792.6			−1.52645			−0.763225	+	0.646133i	$0.776385\pi$
$758$		0			0
$759$	−20393.7	−	20393.7i	−0.975289	−	0.975289i
$760$		0			0
$761$	2581.48	+	2581.48i	0.122968	+	0.122968i	0.765913	−	0.642945i	$-0.222287\pi$
$761$	2581.48	+	2581.48i	0.122968	+	0.122968i	−0.642945	+	0.765913i	$0.722287\pi$
$762$		0			0
$763$		−	45674.1i		−	2.16712i
$764$		0			0
$765$	593.405	+	593.405i	0.0280452	+	0.0280452i
$766$		0			0
$767$	−3654.93	+	15517.4i	−0.172063	+	0.730511i
$768$		0			0
$769$	3404.92	−	3404.92i	0.159668	−	0.159668i	−0.622752	−	0.782420i	$-0.713985\pi$
$769$	3404.92	−	3404.92i	0.159668	−	0.159668i	0.782420	+	0.622752i	$0.213985\pi$
$770$		0			0
$771$	−16709.0			−0.780493
$772$		0			0
$773$	−4722.42	+	4722.42i	−0.219733	+	0.219733i	−0.808386	−	0.588653i	$-0.799659\pi$
$773$	−4722.42	+	4722.42i	−0.219733	+	0.219733i	0.588653	+	0.808386i	$0.299659\pi$
$774$		0			0
$775$	207.568	−	207.568i	0.00962073	−	0.00962073i
$776$		0			0
$777$		−	8138.61i		−	0.375767i
$778$		0			0
$779$	−1662.44			−0.0764608
$780$		0			0
$781$	1067.09			0.0488907
$782$		0			0
$783$			231.708i			0.0105754i
$784$		0			0
$785$	5876.23	−	5876.23i	0.267174	−	0.267174i
$786$		0			0
$787$	−18437.3	+	18437.3i	−0.835095	+	0.835095i	−0.988209	−	0.153114i	$-0.951070\pi$
$787$	−18437.3	+	18437.3i	−0.835095	+	0.835095i	0.153114	+	0.988209i	$0.451070\pi$
$788$		0			0
$789$	−1601.95			−0.0722824
$790$		0			0
$791$	9356.22	−	9356.22i	0.420568	−	0.420568i
$792$		0			0
$793$	−9728.54	+	6019.32i	−0.435650	+	0.269549i
$794$		0			0
$795$	1732.01	+	1732.01i	0.0772680	+	0.0772680i
$796$		0			0
$797$			43195.3i			1.91977i	0.280393	+	0.959885i	$0.409535\pi$
$797$			43195.3i			1.91977i	−0.280393	+	0.959885i	$0.590465\pi$
$798$		0			0
$799$	1686.25	+	1686.25i	0.0746625	+	0.0746625i
$800$		0			0
$801$	−7364.24	−	7364.24i	−0.324847	−	0.324847i
$802$		0			0
$803$	39781.3			1.74826
$804$		0			0
$805$			47503.6i			2.07985i
$806$		0			0
$807$		−	12895.1i		−	0.562488i
$808$		0			0
$809$	−38190.2			−1.65970			−0.829850	−	0.557987i	$-0.811574\pi$
$809$	−38190.2			−1.65970			−0.829850	+	0.557987i	$0.811574\pi$
$810$		0			0
$811$	6568.48	+	6568.48i	0.284403	+	0.284403i	0.834862	−	0.550459i	$-0.185547\pi$
$811$	6568.48	+	6568.48i	0.284403	+	0.284403i	−0.550459	+	0.834862i	$0.685547\pi$
$812$		0			0
$813$	12463.0	+	12463.0i	0.537633	+	0.537633i
$814$		0			0
$815$			19777.8i			0.850045i
$816$		0			0
$817$	90.7514	+	90.7514i	0.00388616	+	0.00388616i
$818$		0			0
$819$	−2137.78	+	9076.17i	−0.0912088	+	0.387237i
$820$		0			0
$821$	−6419.09	+	6419.09i	−0.272872	+	0.272872i	−0.830255	−	0.557383i	$-0.811805\pi$
$821$	−6419.09	+	6419.09i	−0.272872	+	0.272872i	0.557383	+	0.830255i	$0.311805\pi$
$822$		0			0
$823$	2865.12			0.121351			0.0606755	−	0.998158i	$-0.480675\pi$
$823$	2865.12			0.121351			0.0606755	+	0.998158i	$0.480675\pi$
$824$		0			0
$825$	−102.080	+	102.080i	−0.00430785	+	0.00430785i
$826$		0			0
$827$	−1314.26	+	1314.26i	−0.0552615	+	0.0552615i	−0.734197	−	0.678936i	$-0.762441\pi$
$827$	−1314.26	+	1314.26i	−0.0552615	+	0.0552615i	0.678936	+	0.734197i	$0.262441\pi$
$828$		0			0
$829$		−	3356.65i		−	0.140629i	−0.997525	−	0.0703145i	$-0.977600\pi$
$829$		−	3356.65i		−	0.140629i	0.997525	−	0.0703145i	$-0.0224003\pi$
$830$		0			0
$831$	−4375.79			−0.182665
$832$		0			0
$833$	−1218.90			−0.0506991
$834$		0			0
$835$			27998.1i			1.16037i
$836$		0			0
$837$	−5802.19	+	5802.19i	−0.239609	+	0.239609i
$838$		0			0
$839$	−19905.6	+	19905.6i	−0.819091	+	0.819091i	−0.985976	−	0.166885i	$-0.946629\pi$
$839$	−19905.6	+	19905.6i	−0.819091	+	0.819091i	0.166885	+	0.985976i	$0.446629\pi$
$840$		0			0
$841$	−24315.4			−0.996980
$842$		0			0
$843$	291.349	−	291.349i	0.0119034	−	0.0119034i
$844$		0			0
$845$	−7760.83	−	23204.7i	−0.315953	−	0.944694i
$846$		0			0
$847$	−17990.4	−	17990.4i	−0.729819	−	0.729819i
$848$		0			0
$849$			18486.0i			0.747277i
$850$		0			0
$851$	16746.8	+	16746.8i	0.674584	+	0.674584i
$852$		0			0
$853$	1279.01	+	1279.01i	0.0513393	+	0.0513393i	0.732310	−	0.680971i	$-0.238442\pi$
$853$	1279.01	+	1279.01i	0.0513393	+	0.0513393i	−0.680971	+	0.732310i	$0.738442\pi$
$854$		0			0
$855$	558.774			0.0223505
$856$		0			0
$857$			23349.3i			0.930686i	0.885130	+	0.465343i	$0.154069\pi$
$857$			23349.3i			0.930686i	−0.885130	+	0.465343i	$0.845931\pi$
$858$		0			0
$859$		−	27894.6i		−	1.10798i	−0.832525	−	0.553988i	$-0.813106\pi$
$859$		−	27894.6i		−	1.10798i	0.832525	−	0.553988i	$-0.186894\pi$
$860$		0			0
$861$	19774.8			0.782723
$862$		0			0
$863$	4168.46	+	4168.46i	0.164422	+	0.164422i	0.784522	−	0.620100i	$-0.212908\pi$
$863$	4168.46	+	4168.46i	0.164422	+	0.164422i	−0.620100	+	0.784522i	$0.712908\pi$
$864$		0			0
$865$	−34187.8	−	34187.8i	−1.34384	−	1.34384i
$866$		0			0
$867$			14528.7i			0.569113i
$868$		0			0
$869$	−17895.4	−	17895.4i	−0.698572	−	0.698572i
$870$		0			0
$871$	37376.3	+	8803.51i	1.45401	+	0.342475i
$872$		0			0
$873$	−10457.0	+	10457.0i	−0.405402	+	0.405402i
$874$		0			0
$875$	31009.4			1.19807
$876$		0			0
$877$	−35863.9	+	35863.9i	−1.38089	+	1.38089i	−0.537845	+	0.843044i	$0.680761\pi$
$877$	−35863.9	+	35863.9i	−1.38089	+	1.38089i	−0.843044	+	0.537845i	$0.819239\pi$
$878$		0			0
$879$	1875.22	−	1875.22i	0.0719562	−	0.0719562i
$880$		0			0
$881$			7488.76i			0.286382i	0.989695	+	0.143191i	$0.0457364\pi$
$881$			7488.76i			0.286382i	−0.989695	+	0.143191i	$0.954264\pi$
$882$		0			0
$883$	−11940.9			−0.455090			−0.227545	−	0.973768i	$-0.573070\pi$
$883$	−11940.9			−0.455090			−0.227545	+	0.973768i	$0.573070\pi$
$884$		0			0
$885$	−11363.7			−0.431624
$886$		0			0
$887$			29433.4i			1.11418i	0.830452	+	0.557090i	$0.188082\pi$
$887$			29433.4i			1.11418i	−0.830452	+	0.557090i	$0.811918\pi$
$888$		0			0
$889$	−3859.38	+	3859.38i	−0.145601	+	0.145601i
$890$		0			0
$891$	2853.47	−	2853.47i	0.107289	−	0.107289i
$892$		0			0
$893$	1587.84			0.0595019
$894$		0			0
$895$	14029.9	−	14029.9i	0.523988	−	0.523988i
$896$		0			0
$897$	−14277.1	−	23074.9i	−0.531436	−	0.858915i
$898$		0			0
$899$	1844.19	+	1844.19i	0.0684173	+	0.0684173i
$900$		0			0
$901$		−	613.800i		−	0.0226955i
$902$		0			0
$903$	−1079.50	−	1079.50i	−0.0397823	−	0.0397823i
$904$		0			0
$905$	23751.3	+	23751.3i	0.872398	+	0.872398i
$906$		0			0
$907$	−1273.14			−0.0466085			−0.0233043	−	0.999728i	$-0.507419\pi$
$907$	−1273.14			−0.0466085			−0.0233043	+	0.999728i	$0.507419\pi$
$908$		0			0
$909$		−	2743.61i		−	0.100110i
$910$		0			0
$911$		−	10065.0i		−	0.366048i	−0.983108	−	0.183024i	$-0.941411\pi$
$911$		−	10065.0i		−	0.366048i	0.983108	−	0.183024i	$-0.0585885\pi$
$912$		0			0
$913$	−22291.5			−0.808039
$914$		0			0
$915$	−5766.24	−	5766.24i	−0.208335	−	0.208335i
$916$		0			0
$917$	−34584.6	−	34584.6i	−1.24546	−	1.24546i
$918$		0			0
$919$			26052.4i			0.935134i	0.883958	+	0.467567i	$0.154869\pi$
$919$			26052.4i			0.935134i	−0.883958	+	0.467567i	$0.845131\pi$
$920$		0			0
$921$	−524.648	−	524.648i	−0.0187706	−	0.0187706i
$922$		0			0
$923$	977.214	+	230.170i	0.0348487	+	0.00820818i
$924$		0			0
$925$	83.8256	−	83.8256i	0.00297964	−	0.00297964i
$926$		0			0
$927$	−220.892			−0.00782636
$928$		0			0
$929$	−30557.8	+	30557.8i	−1.07919	+	1.07919i	−0.0826101	+	0.996582i	$0.526326\pi$
$929$	−30557.8	+	30557.8i	−1.07919	+	1.07919i	−0.996582	+	0.0826101i	$0.973674\pi$
$930$		0			0
$931$	−573.883	+	573.883i	−0.0202022	+	0.0202022i
$932$		0			0
$933$			15636.9i			0.548693i
$934$		0			0
$935$	−4645.44			−0.162484
$936$		0			0
$937$	−18637.7			−0.649804			−0.324902	−	0.945748i	$-0.605331\pi$
$937$	−18637.7			−0.649804			−0.324902	+	0.945748i	$0.605331\pi$
$938$		0			0
$939$			3532.90i			0.122782i
$940$		0			0
$941$	35335.6	−	35335.6i	1.22413	−	1.22413i	0.257983	−	0.966149i	$-0.416942\pi$
$941$	35335.6	−	35335.6i	1.22413	−	1.22413i	0.966149	−	0.257983i	$-0.0830578\pi$
$942$		0			0
$943$	−40690.5	+	40690.5i	−1.40516	+	1.40516i
$944$		0			0
$945$	−6646.67			−0.228800
$946$		0			0
$947$	9672.77	−	9672.77i	0.331914	−	0.331914i	−0.521399	−	0.853313i	$-0.674590\pi$
$947$	9672.77	−	9672.77i	0.331914	−	0.331914i	0.853313	+	0.521399i	$0.174590\pi$
$948$		0			0
$949$	36430.6	+	8580.77i	1.24614	+	0.293513i
$950$		0			0
$951$	−9202.73	−	9202.73i	−0.313795	−	0.313795i
$952$		0			0
$953$			29068.9i			0.988073i	0.869441	+	0.494036i	$0.164479\pi$
$953$			29068.9i			0.988073i	−0.869441	+	0.494036i	$0.835521\pi$
$954$		0			0
$955$	−22607.6	−	22607.6i	−0.766038	−	0.766038i
$956$		0			0
$957$	−906.957	−	906.957i	−0.0306351	−	0.0306351i
$958$		0			0
$959$	−7187.24			−0.242011
$960$		0			0
$961$			62569.5i			2.10028i
$962$		0			0
$963$			11362.1i			0.380206i
$964$		0			0
$965$	14480.4			0.483048
$966$		0			0
$967$	9547.99	+	9547.99i	0.317521	+	0.317521i	0.847814	−	0.530293i	$-0.177918\pi$
$967$	9547.99	+	9547.99i	0.317521	+	0.317521i	−0.530293	+	0.847814i	$0.677918\pi$
$968$		0			0
$969$	−99.0109	−	99.0109i	−0.00328244	−	0.00328244i
$970$		0			0
$971$			52005.1i			1.71877i	0.511332	+	0.859383i	$0.329152\pi$
$971$			52005.1i			1.71877i	−0.511332	+	0.859383i	$0.670848\pi$
$972$		0			0
$973$	31096.4	+	31096.4i	1.02457	+	1.02457i
$974$		0			0
$975$	−115.501	+	71.4636i	−0.00379383	+	0.00234735i
$976$		0			0
$977$	29823.1	−	29823.1i	0.976588	−	0.976588i	−0.0231437	−	0.999732i	$-0.507368\pi$
$977$	29823.1	−	29823.1i	0.976588	−	0.976588i	0.999732	+	0.0231437i	$0.00736751\pi$
$978$		0			0
$979$	57650.6			1.88204
$980$		0			0
$981$	−13150.1	+	13150.1i	−0.427981	+	0.427981i
$982$		0			0
$983$	−40171.9	+	40171.9i	−1.30344	+	1.30344i	−0.377388	+	0.926055i	$0.623178\pi$
$983$	−40171.9	+	40171.9i	−1.30344	+	1.30344i	−0.926055	+	0.377388i	$0.876822\pi$
$984$		0			0
$985$			6206.29i			0.200760i
$986$		0			0
$987$	−18887.5			−0.609116
$988$		0			0
$989$	4442.54			0.142836
$990$		0			0
$991$			29902.3i			0.958506i	0.877677	+	0.479253i	$0.159092\pi$
$991$			29902.3i			0.958506i	−0.877677	+	0.479253i	$0.840908\pi$
$992$		0			0
$993$	−8485.65	+	8485.65i	−0.271182	+	0.271182i
$994$		0			0
$995$	35213.3	−	35213.3i	1.12195	−	1.12195i
$996$		0			0
$997$	11037.5			0.350613			0.175307	−	0.984514i	$-0.443908\pi$
$997$	11037.5			0.350613			0.175307	+	0.984514i	$0.443908\pi$
$998$		0			0
$999$	−2343.19	+	2343.19i	−0.0742095	+	0.0742095i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.4.bc.d.31.11	yes	28
4.3	odd	2		624.4.bc.c.31.11	✓	28
13.8	odd	4		624.4.bc.c.463.11	yes	28
52.47	even	4	inner	624.4.bc.d.463.11	yes	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
624.4.bc.c.31.11	✓	28	4.3	odd	2
624.4.bc.c.463.11	yes	28	13.8	odd	4
624.4.bc.d.31.11	yes	28	1.1	even	1	trivial
624.4.bc.d.463.11	yes	28	52.47	even	4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	624.bc (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+3.00000i q^{3} +(7.87509 - 7.87509i) q^{5} +(15.6298 - 15.6298i) q^{7} -9.00000 q^{9} +(35.2280 - 35.2280i) q^{11} +(39.8595 - 24.6622i) q^{13} +(23.6253 + 23.6253i) q^{15} -8.37246i q^{17} +(-3.94192 - 3.94192i) q^{19} +(46.8895 + 46.8895i) q^{21} -192.968 q^{23} +0.965900i q^{25} -27.0000i q^{27} -8.58178 q^{29} +(-214.896 - 214.896i) q^{31} +(105.684 + 105.684i) q^{33} -246.173i q^{35} +(-86.7849 - 86.7849i) q^{37} +(73.9866 + 119.578i) q^{39} +(210.866 - 210.866i) q^{41} -23.0221 q^{43} +(-70.8758 + 70.8758i) q^{45} +(-201.405 + 201.405i) q^{47} -145.584i q^{49} +25.1174 q^{51} +73.3118 q^{53} -554.848i q^{55} +(11.8258 - 11.8258i) q^{57} +(-240.499 + 240.499i) q^{59} -244.071 q^{61} +(-140.669 + 140.669i) q^{63} +(119.680 - 508.114i) q^{65} +(579.283 + 579.283i) q^{67} -578.905i q^{69} +(15.1455 + 15.1455i) q^{71} +(564.626 + 564.626i) q^{73} -2.89770 q^{75} -1101.22i q^{77} -507.987i q^{79} +81.0000 q^{81} +(-316.388 - 316.388i) q^{83} +(-65.9339 - 65.9339i) q^{85} -25.7453i q^{87} +(818.249 + 818.249i) q^{89} +(237.531 - 1008.46i) q^{91} +(644.688 - 644.688i) q^{93} -62.0860 q^{95} +(1161.89 - 1161.89i) q^{97} +(-317.052 + 317.052i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 4 q^{5} + 8 q^{7} - 252 q^{9} - 64 q^{11} - 32 q^{13} + 12 q^{15} - 56 q^{19} + 24 q^{21} - 384 q^{23} - 32 q^{29} + 168 q^{31} - 192 q^{33} + 412 q^{37} - 252 q^{39} + 1340 q^{41} - 624 q^{43} - 36 q^{45}+ \cdots + 576 q^{99}+O(q^{100}) \) 28 * q + 4 * q^5 + 8 * q^7 - 252 * q^9 - 64 * q^11 - 32 * q^13 + 12 * q^15 - 56 * q^19 + 24 * q^21 - 384 * q^23 - 32 * q^29 + 168 * q^31 - 192 * q^33 + 412 * q^37 - 252 * q^39 + 1340 * q^41 - 624 * q^43 - 36 * q^45 + 688 * q^47 - 312 * q^51 + 2152 * q^53 + 168 * q^57 - 624 * q^59 - 216 * q^61 - 72 * q^63 + 60 * q^65 + 2024 * q^67 - 448 * q^71 - 724 * q^73 + 1716 * q^75 + 2268 * q^81 - 2288 * q^83 - 1528 * q^85 - 1876 * q^89 + 3448 * q^91 - 504 * q^93 - 192 * q^95 + 76 * q^97 + 576 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more