LMFDB - Embedded newform 672.4.i.c.209.14

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [672,4,Mod(209,672)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(672, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1, 1])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("672.209"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 672 = 2^{5} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	672.i (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [80] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$39.6492835239$
Analytic rank:	$0$
Dimension:	$80$
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			209.14
Character	$\chi$	$=$	672.209
Dual form			672.4.i.c.209.16

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-4.72061 - 2.17161i) q^{3} -2.27823i q^{5} +(13.7109 + 12.4503i) q^{7} +(17.5682 + 20.5026i) q^{9} -33.8293 q^{11} +41.0309 q^{13} +(-4.94743 + 10.7546i) q^{15} -28.4181 q^{17} -109.654 q^{19} +(-37.6866 - 88.5478i) q^{21} +142.918i q^{23} +119.810 q^{25} +(-38.4090 - 134.936i) q^{27} +147.654 q^{29} -293.446i q^{31} +(159.695 + 73.4640i) q^{33} +(28.3647 - 31.2366i) q^{35} -162.028i q^{37} +(-193.691 - 89.1031i) q^{39} -144.009 q^{41} -71.1121i q^{43} +(46.7097 - 40.0245i) q^{45} -367.170 q^{47} +(32.9788 + 341.411i) q^{49} +(134.151 + 61.7130i) q^{51} -508.040 q^{53} +77.0710i q^{55} +(517.635 + 238.127i) q^{57} -202.128i q^{59} -363.737 q^{61} +(-14.3877 + 499.840i) q^{63} -93.4779i q^{65} +531.804i q^{67} +(310.362 - 674.659i) q^{69} +796.705i q^{71} +510.267i q^{73} +(-565.574 - 260.180i) q^{75} +(-463.831 - 421.186i) q^{77} +38.1516 q^{79} +(-111.715 + 720.389i) q^{81} +973.913i q^{83} +64.7430i q^{85} +(-697.016 - 320.647i) q^{87} -1638.53 q^{89} +(562.572 + 510.848i) q^{91} +(-637.249 + 1385.24i) q^{93} +249.818i q^{95} -133.473i q^{97} +(-594.321 - 693.589i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 80 q - 64 q^{7} + 104 q^{9} + 8 q^{15} - 976 q^{25} - 568 q^{39} - 4048 q^{49} - 1448 q^{57} + 2152 q^{63} - 4992 q^{79} + 1568 q^{81}+O(q^{100}) $ 80 * q - 64 * q^7 + 104 * q^9 + 8 * q^15 - 976 * q^25 - 568 * q^39 - 4048 * q^49 - 1448 * q^57 + 2152 * q^63 - 4992 * q^79 + 1568 * q^81

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/672\mathbb{Z}\right)^\times$.

$n$	$127$	$421$	$449$	$577$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

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
$3$	−4.72061	−	2.17161i	−0.908481	−	0.417926i
$3$	−4.72061	−	2.17161i	−0.908481	−	0.417926i
$4$		0			0
$5$		−	2.27823i		−	0.203771i	−0.994796	−	0.101886i	$-0.967512\pi$
$5$		−	2.27823i		−	0.203771i	0.994796	−	0.101886i	$-0.0324876\pi$
$6$		0			0
$7$	13.7109	+	12.4503i	0.740320	+	0.672254i
$7$	13.7109	+	12.4503i	0.740320	+	0.672254i
$8$		0			0
$9$	17.5682	+	20.5026i	0.650675	+	0.759356i
$10$		0			0
$11$	−33.8293			−0.927265			−0.463633	−	0.886028i	$-0.653454\pi$
$11$	−33.8293			−0.927265			−0.463633	+	0.886028i	$0.653454\pi$
$12$		0			0
$13$	41.0309			0.875379			0.437689	−	0.899126i	$-0.355797\pi$
$13$	41.0309			0.875379			0.437689	+	0.899126i	$0.355797\pi$
$14$		0			0
$15$	−4.94743	+	10.7546i	−0.0851614	+	0.185122i
$16$		0			0
$17$	−28.4181			−0.405436			−0.202718	−	0.979237i	$-0.564977\pi$
$17$	−28.4181			−0.405436			−0.202718	+	0.979237i	$0.564977\pi$
$18$		0			0
$19$	−109.654			−1.32402			−0.662012	−	0.749493i	$-0.730297\pi$
$19$	−109.654			−1.32402			−0.662012	+	0.749493i	$0.730297\pi$
$20$		0			0
$21$	−37.6866	−	88.5478i	−0.391614	−	0.920130i
$22$		0			0
$23$			142.918i			1.29567i	0.761780	+	0.647835i	$0.224326\pi$
$23$			142.918i			1.29567i	−0.761780	+	0.647835i	$0.775674\pi$
$24$		0			0
$25$	119.810			0.958477
$26$		0			0
$27$	−38.4090	−	134.936i	−0.273771	−	0.961795i
$28$		0			0
$29$	147.654			0.945471			0.472735	−	0.881204i	$-0.343267\pi$
$29$	147.654			0.945471			0.472735	+	0.881204i	$0.343267\pi$
$30$		0			0
$31$		−	293.446i		−	1.70014i	−0.526669	−	0.850071i	$-0.676559\pi$
$31$		−	293.446i		−	1.70014i	0.526669	−	0.850071i	$-0.323441\pi$
$32$		0			0
$33$	159.695	+	73.4640i	0.842403	+	0.387529i
$34$		0			0
$35$	28.3647	−	31.2366i	0.136986	−	0.150856i
$36$		0			0
$37$		−	162.028i		−	0.719926i	−0.932967	−	0.359963i	$-0.882789\pi$
$37$		−	162.028i		−	0.719926i	0.932967	−	0.359963i	$-0.117211\pi$
$38$		0			0
$39$	−193.691	−	89.1031i	−0.795265	−	0.365844i
$40$		0			0
$41$	−144.009			−0.548545			−0.274273	−	0.961652i	$-0.588437\pi$
$41$	−144.009			−0.548545			−0.274273	+	0.961652i	$0.588437\pi$
$42$		0			0
$43$		−	71.1121i		−	0.252197i	−0.992018	−	0.126099i	$-0.959754\pi$
$43$		−	71.1121i		−	0.252197i	0.992018	−	0.126099i	$-0.0402456\pi$
$44$		0			0
$45$	46.7097	−	40.0245i	0.154735	−	0.132589i
$46$		0			0
$47$	−367.170			−1.13951			−0.569757	−	0.821813i	$-0.692963\pi$
$47$	−367.170			−1.13951			−0.569757	+	0.821813i	$0.692963\pi$
$48$		0			0
$49$	32.9788	+	341.411i	0.0961480	+	0.995367i
$50$		0			0
$51$	134.151	+	61.7130i	0.368331	+	0.169442i
$52$		0			0
$53$	−508.040			−1.31669			−0.658345	−	0.752716i	$-0.728743\pi$
$53$	−508.040			−1.31669			−0.658345	+	0.752716i	$0.728743\pi$
$54$		0			0
$55$			77.0710i			0.188950i
$56$		0			0
$57$	517.635	+	238.127i	1.20285	+	0.553345i
$58$		0			0
$59$		−	202.128i		−	0.446013i	−0.974817	−	0.223007i	$-0.928413\pi$
$59$		−	202.128i		−	0.446013i	0.974817	−	0.223007i	$-0.0715872\pi$
$60$		0			0
$61$	−363.737			−0.763472			−0.381736	−	0.924271i	$-0.624674\pi$
$61$	−363.737			−0.763472			−0.381736	+	0.924271i	$0.624674\pi$
$62$		0			0
$63$	−14.3877	+	499.840i	−0.0287727	+	0.999586i
$64$		0			0
$65$		−	93.4779i		−	0.178377i
$66$		0			0
$67$			531.804i			0.969705i	0.874596	+	0.484852i	$0.161127\pi$
$67$			531.804i			0.969705i	−0.874596	+	0.484852i	$0.838873\pi$
$68$		0			0
$69$	310.362	−	674.659i	0.541495	−	1.17709i
$70$		0			0
$71$			796.705i			1.33171i	0.746081	+	0.665855i	$0.231933\pi$
$71$			796.705i			1.33171i	−0.746081	+	0.665855i	$0.768067\pi$
$72$		0			0
$73$			510.267i			0.818114i	0.912509	+	0.409057i	$0.134142\pi$
$73$			510.267i			0.818114i	−0.912509	+	0.409057i	$0.865858\pi$
$74$		0			0
$75$	−565.574	−	260.180i	−0.870758	−	0.400573i
$76$		0			0
$77$	−463.831	−	421.186i	−0.686473	−	0.623358i
$78$		0			0
$79$	38.1516			0.0543340			0.0271670	−	0.999631i	$-0.491351\pi$
$79$	38.1516			0.0543340			0.0271670	+	0.999631i	$0.491351\pi$
$80$		0			0
$81$	−111.715	+	720.389i	−0.153244	+	0.988188i
$82$		0			0
$83$			973.913i			1.28796i	0.765042	+	0.643981i	$0.222718\pi$
$83$			973.913i			1.28796i	−0.765042	+	0.643981i	$0.777282\pi$
$84$		0			0
$85$			64.7430i			0.0826161i
$86$		0			0
$87$	−697.016	−	320.647i	−0.858942	−	0.395137i
$88$		0			0
$89$	−1638.53			−1.95150			−0.975750	−	0.218889i	$-0.929757\pi$
$89$	−1638.53			−1.95150			−0.975750	+	0.218889i	$0.929757\pi$
$90$		0			0
$91$	562.572	+	510.848i	0.648061	+	0.588477i
$92$		0			0
$93$	−637.249	+	1385.24i	−0.710534	+	1.54455i
$94$		0			0
$95$			249.818i			0.269798i
$96$		0			0
$97$		−	133.473i		−	0.139713i	−0.997557	−	0.0698565i	$-0.977746\pi$
$97$		−	133.473i		−	0.139713i	0.997557	−	0.0698565i	$-0.0222541\pi$
$98$		0			0
$99$	−594.321	−	693.589i	−0.603348	−	0.704125i
$100$		0			0
$101$			169.167i			0.166661i	0.996522	+	0.0833306i	$0.0265557\pi$
$101$			169.167i			0.166661i	−0.996522	+	0.0833306i	$0.973444\pi$
$102$		0			0
$103$			939.850i			0.899089i	0.893258	+	0.449544i	$0.148414\pi$
$103$			939.850i			0.899089i	−0.893258	+	0.449544i	$0.851586\pi$
$104$		0			0
$105$	−201.732	+	85.8588i	−0.187496	+	0.0797996i
$106$		0			0
$107$	−2048.48			−1.85078			−0.925392	−	0.379012i	$-0.876264\pi$
$107$	−2048.48			−1.85078			−0.925392	+	0.379012i	$0.876264\pi$
$108$		0			0
$109$			238.831i			0.209871i	0.994479	+	0.104935i	$0.0334635\pi$
$109$			238.831i			0.209871i	−0.994479	+	0.104935i	$0.966536\pi$
$110$		0			0
$111$	−351.862	+	764.871i	−0.300876	+	0.654039i
$112$		0			0
$113$		−	0.845945i		−	0.000704246i	−1.00000		0.000352123i	$-0.999888\pi$
$113$		−	0.845945i		−	0.000704246i	1.00000		0.000352123i	$-0.000112084\pi$
$114$		0			0
$115$	325.600			0.264020
$116$		0			0
$117$	720.840	+	841.241i	0.569587	+	0.664725i
$118$		0			0
$119$	−389.639	−	353.815i	−0.300152	−	0.272556i
$120$		0			0
$121$	−186.578			−0.140179
$122$		0			0
$123$	679.808	+	312.730i	0.498343	+	0.229252i
$124$		0			0
$125$		−	557.733i		−	0.399081i
$126$		0			0
$127$	−46.6406			−0.0325881			−0.0162940	−	0.999867i	$-0.505187\pi$
$127$	−46.6406			−0.0325881			−0.0162940	+	0.999867i	$0.505187\pi$
$128$		0			0
$129$	−154.428	+	335.692i	−0.105400	+	0.229117i
$130$		0			0
$131$		−	2494.50i		−	1.66371i	−0.554995	−	0.831854i	$-0.687280\pi$
$131$		−	2494.50i		−	1.66371i	0.554995	−	0.831854i	$-0.312720\pi$
$132$		0			0
$133$	−1503.46	−	1365.23i	−0.980202	−	0.890081i
$134$		0			0
$135$	−307.416	+	87.5046i	−0.195986	+	0.0557866i
$136$		0			0
$137$			201.774i			0.125830i	0.998019	+	0.0629151i	$0.0200397\pi$
$137$			201.774i			0.125830i	−0.998019	+	0.0629151i	$0.979960\pi$
$138$		0			0
$139$	309.414			0.188807			0.0944034	−	0.995534i	$-0.469906\pi$
$139$	309.414			0.188807			0.0944034	+	0.995534i	$0.469906\pi$
$140$		0			0
$141$	1733.26	+	797.349i	1.03523	+	0.476233i
$142$		0			0
$143$	−1388.05			−0.811709
$144$		0			0
$145$		−	336.390i		−	0.192660i
$146$		0			0
$147$	585.731	−	1683.28i	0.328642	−	0.944455i
$148$		0			0
$149$	−1207.81			−0.664078			−0.332039	−	0.943266i	$-0.607737\pi$
$149$	−1207.81			−0.664078			−0.332039	+	0.943266i	$0.607737\pi$
$150$		0			0
$151$	2555.54			1.37726			0.688631	−	0.725112i	$-0.258212\pi$
$151$	2555.54			1.37726			0.688631	+	0.725112i	$0.258212\pi$
$152$		0			0
$153$	−499.256	−	582.646i	−0.263807	−	0.307870i
$154$		0			0
$155$	−668.537			−0.346440
$156$		0			0
$157$	38.2182			0.0194276			0.00971382	−	0.999953i	$-0.496908\pi$
$157$	38.2182			0.0194276			0.00971382	+	0.999953i	$0.496908\pi$
$158$		0			0
$159$	2398.25	+	1103.26i	1.19619	+	0.550280i
$160$		0			0
$161$	−1779.37	+	1959.54i	−0.871020	+	0.959211i
$162$		0			0
$163$			2887.96i			1.38775i	0.720097	+	0.693873i	$0.244097\pi$
$163$			2887.96i			1.38775i	−0.720097	+	0.693873i	$0.755903\pi$
$164$		0			0
$165$	167.368	−	363.822i	0.0789672	−	0.171657i
$166$		0			0
$167$	307.719			0.142587			0.0712935	−	0.997455i	$-0.477287\pi$
$167$	307.719			0.142587			0.0712935	+	0.997455i	$0.477287\pi$
$168$		0			0
$169$	−513.464			−0.233712
$170$		0			0
$171$	−1926.43	−	2248.20i	−0.861509	−	1.00541i
$172$		0			0
$173$			2741.35i			1.20475i	0.798215	+	0.602373i	$0.205778\pi$
$173$			2741.35i			1.20475i	−0.798215	+	0.602373i	$0.794222\pi$
$174$		0			0
$175$	1642.70	+	1491.67i	0.709580	+	0.644341i
$176$		0			0
$177$	−438.942	+	954.165i	−0.186401	+	0.405195i
$178$		0			0
$179$	−729.734			−0.304709			−0.152354	−	0.988326i	$-0.548686\pi$
$179$	−729.734			−0.304709			−0.152354	+	0.988326i	$0.548686\pi$
$180$		0			0
$181$	−2056.86			−0.844671			−0.422336	−	0.906440i	$-0.638790\pi$
$181$	−2056.86			−0.844671			−0.422336	+	0.906440i	$0.638790\pi$
$182$		0			0
$183$	1717.06	+	789.895i	0.693600	+	0.319075i
$184$		0			0
$185$	−369.138			−0.146700
$186$		0			0
$187$	961.365			0.375946
$188$		0			0
$189$	1153.38	−	2328.30i	0.443893	−	0.896080i
$190$		0			0
$191$		−	2290.89i		−	0.867868i	−0.900945	−	0.433934i	$-0.857125\pi$
$191$		−	2290.89i		−	0.867868i	0.900945	−	0.433934i	$-0.142875\pi$
$192$		0			0
$193$	−3149.48			−1.17464			−0.587318	−	0.809356i	$-0.699816\pi$
$193$	−3149.48			−1.17464			−0.587318	+	0.809356i	$0.699816\pi$
$194$		0			0
$195$	−202.997	+	441.272i	−0.0745485	+	0.162052i
$196$		0			0
$197$	−415.958			−0.150435			−0.0752177	−	0.997167i	$-0.523965\pi$
$197$	−415.958			−0.150435			−0.0752177	+	0.997167i	$0.523965\pi$
$198$		0			0
$199$		−	1202.25i		−	0.428266i	−0.976804	−	0.214133i	$-0.931307\pi$
$199$		−	1202.25i		−	0.428266i	0.976804	−	0.214133i	$-0.0686927\pi$
$200$		0			0
$201$	1154.87	−	2510.44i	0.405265	−	0.880958i
$202$		0			0
$203$	2024.47	+	1838.34i	0.699951	+	0.635597i
$204$		0			0
$205$			328.085i			0.111778i
$206$		0			0
$207$	−2930.19	+	2510.81i	−0.983876	+	0.843061i
$208$		0			0
$209$	3709.53			1.22772
$210$		0			0
$211$		−	4107.87i		−	1.34027i	−0.742239	−	0.670135i	$-0.766236\pi$
$211$		−	4107.87i		−	1.34027i	0.742239	−	0.670135i	$-0.233764\pi$
$212$		0			0
$213$	1730.13	−	3760.93i	0.556557	−	1.20983i
$214$		0			0
$215$	−162.010			−0.0513906
$216$		0			0
$217$	3653.49	−	4023.41i	1.14293	−	1.25865i
$218$		0			0
$219$	1108.10	−	2408.77i	0.341911	−	0.743240i
$220$		0			0
$221$	−1166.02			−0.354910
$222$		0			0
$223$		−	3886.29i		−	1.16702i	−0.812106	−	0.583509i	$-0.801679\pi$
$223$		−	3886.29i		−	1.16702i	0.812106	−	0.583509i	$-0.198321\pi$
$224$		0			0
$225$	2104.84	+	2456.41i	0.623657	+	0.727826i
$226$		0			0
$227$			2560.28i			0.748598i	0.927308	+	0.374299i	$0.122117\pi$
$227$			2560.28i			0.748598i	−0.927308	+	0.374299i	$0.877883\pi$
$228$		0			0
$229$	1212.09			0.349770			0.174885	−	0.984589i	$-0.444045\pi$
$229$	1212.09			0.349770			0.174885	+	0.984589i	$0.444045\pi$
$230$		0			0
$231$	1274.91	+	2995.51i	0.363130	+	0.853204i
$232$		0			0
$233$		−	4448.69i		−	1.25083i	−0.780292	−	0.625415i	$-0.784930\pi$
$233$		−	4448.69i		−	1.25083i	0.780292	−	0.625415i	$-0.215070\pi$
$234$		0			0
$235$			836.497i			0.232200i
$236$		0			0
$237$	−180.099	−	82.8503i	−0.0493614	−	0.0227076i
$238$		0			0
$239$		−	475.148i		−	0.128597i	−0.997931	−	0.0642987i	$-0.979519\pi$
$239$		−	475.148i		−	0.128597i	0.997931	−	0.0642987i	$-0.0204810\pi$
$240$		0			0
$241$			7311.01i			1.95412i	0.212960	+	0.977061i	$0.431690\pi$
$241$			7311.01i			1.95412i	−0.212960	+	0.977061i	$0.568310\pi$
$242$		0			0
$243$	2091.77	−	3158.07i	0.552209	−	0.833706i
$244$		0			0
$245$	777.813	−	75.1332i	0.202827	−	0.0195922i
$246$		0			0
$247$	−4499.22			−1.15902
$248$		0			0
$249$	2114.96	−	4597.46i	0.538273	−	1.17009i
$250$		0			0
$251$			5996.84i			1.50804i	0.656853	+	0.754018i	$0.271887\pi$
$251$			5996.84i			1.50804i	−0.656853	+	0.754018i	$0.728113\pi$
$252$		0			0
$253$		−	4834.81i		−	1.20143i
$254$		0			0
$255$	140.597	−	305.626i	0.0345274	−	0.0750551i
$256$		0			0
$257$	2528.53			0.613719			0.306859	−	0.951755i	$-0.400722\pi$
$257$	2528.53			0.613719			0.306859	+	0.951755i	$0.400722\pi$
$258$		0			0
$259$	2017.30	−	2221.56i	0.483974	−	0.532976i
$260$		0			0
$261$	2594.02	+	3027.29i	0.615194	+	0.717949i
$262$		0			0
$263$		−	6792.30i		−	1.59251i	−0.604958	−	0.796257i	$-0.706810\pi$
$263$		−	6792.30i		−	1.59251i	0.604958	−	0.796257i	$-0.293190\pi$
$264$		0			0
$265$			1157.43i			0.268304i
$266$		0			0
$267$	7734.84	+	3558.24i	1.77290	+	0.815583i
$268$		0			0
$269$			2766.06i			0.626950i	0.949597	+	0.313475i	$0.101493\pi$
$269$			2766.06i			0.626950i	−0.949597	+	0.313475i	$0.898507\pi$
$270$		0			0
$271$		−	923.677i		−	0.207046i	−0.994627	−	0.103523i	$-0.966989\pi$
$271$		−	923.677i		−	0.207046i	0.994627	−	0.103523i	$-0.0330115\pi$
$272$		0			0
$273$	−1546.32	−	3633.20i	−0.342811	−	0.805462i
$274$		0			0
$275$	−4053.08			−0.888763
$276$		0			0
$277$			3713.69i			0.805537i	0.915302	+	0.402768i	$0.131952\pi$
$277$			3713.69i			0.805537i	−0.915302	+	0.402768i	$0.868048\pi$
$278$		0			0
$279$	6016.40	−	5155.32i	1.29101	−	1.10624i
$280$		0			0
$281$		−	4766.56i		−	1.01192i	−0.862557	−	0.505960i	$-0.831138\pi$
$281$		−	4766.56i		−	1.01192i	0.862557	−	0.505960i	$-0.168862\pi$
$282$		0			0
$283$	−8241.05			−1.73102			−0.865512	−	0.500888i	$-0.833007\pi$
$283$	−8241.05			−1.73102			−0.865512	+	0.500888i	$0.833007\pi$
$284$		0			0
$285$	542.507	−	1179.29i	0.112756	−	0.245106i
$286$		0			0
$287$	−1974.49	−	1792.95i	−0.406099	−	0.368762i
$288$		0			0
$289$	−4105.41			−0.835622
$290$		0			0
$291$	−289.852	+	630.075i	−0.0583898	+	0.126927i
$292$		0			0
$293$			4955.76i			0.988118i	0.869428	+	0.494059i	$0.164487\pi$
$293$			4955.76i			0.988118i	−0.869428	+	0.494059i	$0.835513\pi$
$294$		0			0
$295$	−460.494			−0.0908847
$296$		0			0
$297$	1299.35	+	4564.79i	0.253858	+	0.891839i
$298$		0			0
$299$			5864.05i			1.13420i
$300$		0			0
$301$	885.369	−	975.012i	0.169541	−	0.186707i
$302$		0			0
$303$	367.365	−	798.572i	0.0696521	−	0.151408i
$304$		0			0
$305$			828.678i			0.155574i
$306$		0			0
$307$	6152.25			1.14374			0.571869	−	0.820345i	$-0.306219\pi$
$307$	6152.25			1.14374			0.571869	+	0.820345i	$0.306219\pi$
$308$		0			0
$309$	2040.99	−	4436.66i	0.375753	−	0.816805i
$310$		0			0
$311$	−4710.11			−0.858798			−0.429399	−	0.903115i	$-0.641275\pi$
$311$	−4710.11			−0.858798			−0.429399	+	0.903115i	$0.641275\pi$
$312$		0			0
$313$		−	6006.89i		−	1.08476i	−0.840134	−	0.542379i	$-0.817524\pi$
$313$		−	6006.89i		−	1.08476i	0.840134	−	0.542379i	$-0.182476\pi$
$314$		0			0
$315$	1138.75	+	32.7785i	0.203687	+	0.00586305i
$316$		0			0
$317$	−8487.77			−1.50385			−0.751926	−	0.659248i	$-0.770875\pi$
$317$	−8487.77			−1.50385			−0.751926	+	0.659248i	$0.770875\pi$
$318$		0			0
$319$	−4995.03			−0.876702
$320$		0			0
$321$	9670.06	+	4448.49i	1.68140	+	0.773491i
$322$		0			0
$323$	3116.17			0.536806
$324$		0			0
$325$	4915.90			0.839031
$326$		0			0
$327$	518.648	−	1127.43i	0.0877104	−	0.190663i
$328$		0			0
$329$	−5034.23	−	4571.38i	−0.843606	−	0.766044i
$330$		0			0
$331$			10975.0i			1.82247i	0.411885	+	0.911236i	$0.364870\pi$
$331$			10975.0i			1.82247i	−0.411885	+	0.911236i	$0.635130\pi$
$332$		0			0
$333$	3322.00	−	2846.55i	0.546680	−	0.468438i
$334$		0			0
$335$	1211.57			0.197598
$336$		0			0
$337$	1629.69			0.263427			0.131714	−	0.991288i	$-0.457952\pi$
$337$	1629.69			0.263427			0.131714	+	0.991288i	$0.457952\pi$
$338$		0			0
$339$	−1.83706	+	3.99337i	−0.000294323	+	0.000639794i
$340$		0			0
$341$			9927.06i			1.57648i
$342$		0			0
$343$	−3798.51	+	5091.65i	−0.597960	+	0.801526i
$344$		0			0
$345$	−1537.03	−	707.076i	−0.239857	−	0.110341i
$346$		0			0
$347$	−3090.79			−0.478162			−0.239081	−	0.971000i	$-0.576846\pi$
$347$	−3090.79			−0.478162			−0.239081	+	0.971000i	$0.576846\pi$
$348$		0			0
$349$	8418.13			1.29115			0.645576	−	0.763696i	$-0.276617\pi$
$349$	8418.13			1.29115			0.645576	+	0.763696i	$0.276617\pi$
$350$		0			0
$351$	−1575.96	−	5536.55i	−0.239653	−	0.841935i
$352$		0			0
$353$	694.292			0.104684			0.0523420	−	0.998629i	$-0.483331\pi$
$353$	694.292			0.104684			0.0523420	+	0.998629i	$0.483331\pi$
$354$		0			0
$355$	1815.08			0.271364
$356$		0			0
$357$	1070.98	+	2516.36i	0.158774	+	0.373053i
$358$		0			0
$359$		−	3336.44i		−	0.490503i	−0.969460	−	0.245251i	$-0.921129\pi$
$359$		−	3336.44i		−	0.490503i	0.969460	−	0.245251i	$-0.0788705\pi$
$360$		0			0
$361$	5165.10			0.753040
$362$		0			0
$363$	880.762	+	405.175i	0.127350	+	0.0585845i
$364$		0			0
$365$	1162.51			0.166708
$366$		0			0
$367$		−	8875.98i		−	1.26246i	−0.775596	−	0.631229i	$-0.782551\pi$
$367$		−	8875.98i		−	1.26246i	0.775596	−	0.631229i	$-0.217449\pi$
$368$		0			0
$369$	−2529.98	−	2952.55i	−0.356925	−	0.416541i
$370$		0			0
$371$	−6965.69	−	6325.26i	−0.974773	−	0.885151i
$372$		0			0
$373$		−	1584.56i		−	0.219960i	−0.993934	−	0.109980i	$-0.964921\pi$
$373$		−	1584.56i		−	0.219960i	0.993934	−	0.109980i	$-0.0350788\pi$
$374$		0			0
$375$	−1211.18	+	2632.84i	−0.166787	+	0.362558i
$376$		0			0
$377$	6058.37			0.827645
$378$		0			0
$379$		−	1454.58i		−	0.197142i	−0.995130	−	0.0985711i	$-0.968573\pi$
$379$		−	1454.58i		−	0.197142i	0.995130	−	0.0985711i	$-0.0314272\pi$
$380$		0			0
$381$	220.172	+	101.285i	0.0296056	+	0.0136194i
$382$		0			0
$383$	14348.5			1.91429			0.957147	−	0.289603i	$-0.0935233\pi$
$383$	14348.5			1.91429			0.957147	+	0.289603i	$0.0935233\pi$
$384$		0			0
$385$	−959.559	+	1056.71i	−0.127022	+	0.139883i
$386$		0			0
$387$	1457.98	−	1249.31i	0.191508	−	0.164099i
$388$		0			0
$389$	2531.67			0.329977			0.164988	−	0.986296i	$-0.447241\pi$
$389$	2531.67			0.329977			0.164988	+	0.986296i	$0.447241\pi$
$390$		0			0
$391$		−	4061.46i		−	0.525311i
$392$		0			0
$393$	−5417.09	+	11775.6i	−0.695307	+	1.51145i
$394$		0			0
$395$		−	86.9181i		−	0.0110717i
$396$		0			0
$397$	−6962.76			−0.880229			−0.440115	−	0.897942i	$-0.645062\pi$
$397$	−6962.76			−0.880229			−0.440115	+	0.897942i	$0.645062\pi$
$398$		0			0
$399$	4132.50	+	9709.66i	0.518506	+	1.21827i
$400$		0			0
$401$		−	6420.79i		−	0.799598i	−0.916603	−	0.399799i	$-0.869080\pi$
$401$		−	6420.79i		−	0.799598i	0.916603	−	0.399799i	$-0.130920\pi$
$402$		0			0
$403$		−	12040.3i		−	1.48827i
$404$		0			0
$405$	1641.21	+	254.512i	0.201364	+	0.0312267i
$406$		0			0
$407$			5481.30i			0.667563i
$408$		0			0
$409$			650.008i			0.0785840i	0.999228	+	0.0392920i	$0.0125102\pi$
$409$			650.008i			0.0785840i	−0.999228	+	0.0392920i	$0.987490\pi$
$410$		0			0
$411$	438.175	−	952.497i	0.0525878	−	0.114314i
$412$		0			0
$413$	2516.56	−	2771.36i	0.299834	−	0.330193i
$414$		0			0
$415$	2218.80			0.262449
$416$		0			0
$417$	−1460.62	−	671.926i	−0.171527	−	0.0789074i
$418$		0			0
$419$			2741.55i			0.319651i	0.987145	+	0.159825i	$0.0510931\pi$
$419$			2741.55i			0.319651i	−0.987145	+	0.159825i	$0.948907\pi$
$420$		0			0
$421$			5647.52i			0.653784i	0.945062	+	0.326892i	$0.106001\pi$
$421$			5647.52i			0.653784i	−0.945062	+	0.326892i	$0.893999\pi$
$422$		0			0
$423$	−6450.52	−	7527.94i	−0.741454	−	0.865298i
$424$		0			0
$425$	−3404.76			−0.388601
$426$		0			0
$427$	−4987.17	−	4528.65i	−0.565214	−	0.513247i
$428$		0			0
$429$	6552.42	+	3014.30i	0.737422	+	0.339234i
$430$		0			0
$431$		−	1269.04i		−	0.141828i	−0.997482	−	0.0709139i	$-0.977408\pi$
$431$		−	1269.04i		−	0.141828i	0.997482	−	0.0709139i	$-0.0225915\pi$
$432$		0			0
$433$		−	9965.68i		−	1.10605i	−0.833164	−	0.553026i	$-0.813473\pi$
$433$		−	9965.68i		−	1.10605i	0.833164	−	0.553026i	$-0.186527\pi$
$434$		0			0
$435$	−730.507	+	1587.96i	−0.0805176	+	0.175028i
$436$		0			0
$437$		−	15671.6i		−	1.71550i
$438$		0			0
$439$			17103.9i			1.85951i	0.368180	+	0.929754i	$0.379981\pi$
$439$			17103.9i			1.85951i	−0.368180	+	0.929754i	$0.620019\pi$
$440$		0			0
$441$	−6420.44	+	6674.14i	−0.693277	+	0.720671i
$442$		0			0
$443$	6755.15			0.724485			0.362242	−	0.932084i	$-0.382011\pi$
$443$	6755.15			0.724485			0.362242	+	0.932084i	$0.382011\pi$
$444$		0			0
$445$			3732.94i			0.397659i
$446$		0			0
$447$	5701.59	+	2622.89i	0.603302	+	0.277536i
$448$		0			0
$449$		−	6557.46i		−	0.689233i	−0.938744	−	0.344617i	$-0.888009\pi$
$449$		−	6557.46i		−	0.689233i	0.938744	−	0.344617i	$-0.111991\pi$
$450$		0			0
$451$	4871.71			0.508647
$452$		0			0
$453$	−12063.7	−	5549.63i	−1.25122	−	0.575594i
$454$		0			0
$455$	1163.83	−	1281.67i	0.119915	−	0.132056i
$456$		0			0
$457$	428.629			0.0438740			0.0219370	−	0.999759i	$-0.493017\pi$
$457$	428.629			0.0438740			0.0219370	+	0.999759i	$0.493017\pi$
$458$		0			0
$459$	1091.51	+	3834.63i	0.110996	+	0.389946i
$460$		0			0
$461$			14233.2i			1.43798i	0.695022	+	0.718988i	$0.255395\pi$
$461$			14233.2i			1.43798i	−0.695022	+	0.718988i	$0.744605\pi$
$462$		0			0
$463$	6905.29			0.693123			0.346562	−	0.938027i	$-0.387349\pi$
$463$	6905.29			0.693123			0.346562	+	0.938027i	$0.387349\pi$
$464$		0			0
$465$	3155.90	+	1451.80i	0.314734	+	0.144786i
$466$		0			0
$467$			9126.22i			0.904307i	0.891940	+	0.452153i	$0.149344\pi$
$467$			9126.22i			0.904307i	−0.891940	+	0.452153i	$0.850656\pi$
$468$		0			0
$469$	−6621.13	+	7291.52i	−0.651888	+	0.717892i
$470$		0			0
$471$	−180.413	−	82.9949i	−0.0176496	−	0.00811933i
$472$		0			0
$473$			2405.67i			0.233854i
$474$		0			0
$475$	−13137.7			−1.26905
$476$		0			0
$477$	−8925.36	−	10416.1i	−0.856738	−	0.999837i
$478$		0			0
$479$	−17657.1			−1.68429			−0.842143	−	0.539254i	$-0.818706\pi$
$479$	−17657.1			−1.68429			−0.842143	+	0.539254i	$0.818706\pi$
$480$		0			0
$481$		−	6648.16i		−	0.630208i
$482$		0			0
$483$	12655.1	−	5386.09i	1.19219	−	0.507403i
$484$		0			0
$485$	−304.083			−0.0284695
$486$		0			0
$487$	9567.19			0.890206			0.445103	−	0.895479i	$-0.353167\pi$
$487$	9567.19			0.890206			0.445103	+	0.895479i	$0.353167\pi$
$488$		0			0
$489$	6271.53	−	13632.9i	0.579976	−	1.26074i
$490$		0			0
$491$	−7159.61			−0.658063			−0.329031	−	0.944319i	$-0.606722\pi$
$491$	−7159.61			−0.658063			−0.329031	+	0.944319i	$0.606722\pi$
$492$		0			0
$493$	−4196.05			−0.383327
$494$		0			0
$495$	−1580.16	+	1354.00i	−0.143480	+	0.122945i
$496$		0			0
$497$	−9919.23	+	10923.6i	−0.895248	+	0.985892i
$498$		0			0
$499$		−	17819.0i		−	1.59858i	−0.600947	−	0.799289i	$-0.705210\pi$
$499$		−	17819.0i		−	1.59858i	0.600947	−	0.799289i	$-0.294790\pi$
$500$		0			0
$501$	−1452.62	−	668.246i	−0.129538	−	0.0595909i
$502$		0			0
$503$	20477.4			1.81520			0.907599	−	0.419839i	$-0.137913\pi$
$503$	20477.4			1.81520			0.907599	+	0.419839i	$0.137913\pi$
$504$		0			0
$505$	385.402			0.0339607
$506$		0			0
$507$	2423.86	+	1115.04i	0.212323	+	0.0976742i
$508$		0			0
$509$		−	18659.8i		−	1.62491i	−0.583022	−	0.812457i	$-0.698130\pi$
$509$		−	18659.8i		−	1.62491i	0.583022	−	0.812457i	$-0.301870\pi$
$510$		0			0
$511$	−6353.00	+	6996.24i	−0.549980	+	0.605666i
$512$		0			0
$513$	4211.72	+	14796.3i	0.362479	+	1.27344i
$514$		0			0
$515$	2141.19			0.183208
$516$		0			0
$517$	12421.1			1.05663
$518$		0			0
$519$	5953.15	−	12940.8i	0.503495	−	1.09449i
$520$		0			0
$521$	−17496.1			−1.47125			−0.735623	−	0.677391i	$-0.763110\pi$
$521$	−17496.1			−1.47125			−0.735623	+	0.677391i	$0.763110\pi$
$522$		0			0
$523$	−1593.34			−0.133216			−0.0666079	−	0.997779i	$-0.521218\pi$
$523$	−1593.34			−0.133216			−0.0666079	+	0.997779i	$0.521218\pi$
$524$		0			0
$525$	−4515.22	−	10608.9i	−0.375353	−	0.881923i
$526$		0			0
$527$			8339.17i			0.689298i
$528$		0			0
$529$	−8258.51			−0.678763
$530$		0			0
$531$	4144.15	−	3551.03i	0.338683	−	0.290210i
$532$		0			0
$533$	−5908.80			−0.480185
$534$		0			0
$535$			4666.91i			0.377136i
$536$		0			0
$537$	3444.79	+	1584.70i	0.276822	+	0.127346i
$538$		0			0
$539$	−1115.65	−	11549.7i	−0.0891547	−	0.922969i
$540$		0			0
$541$		−	8592.00i		−	0.682807i	−0.939917	−	0.341404i	$-0.889098\pi$
$541$		−	8592.00i		−	0.682807i	0.939917	−	0.341404i	$-0.110902\pi$
$542$		0			0
$543$	9709.64	+	4466.70i	0.767368	+	0.353010i
$544$		0			0
$545$	544.113			0.0427656
$546$		0			0
$547$			468.066i			0.0365870i	0.999833	+	0.0182935i	$0.00582332\pi$
$547$			468.066i			0.0365870i	−0.999833	+	0.0182935i	$0.994177\pi$
$548$		0			0
$549$	−6390.22	−	7457.57i	−0.496772	−	0.579747i
$550$		0			0
$551$	−16190.9			−1.25183
$552$		0			0
$553$	523.093	+	475.000i	0.0402246	+	0.0365263i
$554$		0			0
$555$	1742.55	+	801.623i	0.133274	+	0.0613099i
$556$		0			0
$557$	−7441.55			−0.566084			−0.283042	−	0.959108i	$-0.591344\pi$
$557$	−7441.55			−0.566084			−0.283042	+	0.959108i	$0.591344\pi$
$558$		0			0
$559$		−	2917.79i		−	0.220768i
$560$		0			0
$561$	−4538.22	−	2087.71i	−0.341540	−	0.157118i
$562$		0			0
$563$		−	14562.5i		−	1.09012i	−0.838397	−	0.545059i	$-0.816507\pi$
$563$		−	14562.5i		−	1.09012i	0.838397	−	0.545059i	$-0.183493\pi$
$564$		0			0
$565$	−1.92726			−0.000143505
$566$		0			0
$567$	−10500.8	+	8486.32i	−0.777764	+	0.628557i
$568$		0			0
$569$			19130.4i			1.40947i	0.709470	+	0.704735i	$0.248934\pi$
$569$			19130.4i			1.40947i	−0.709470	+	0.704735i	$0.751066\pi$
$570$		0			0
$571$			13903.0i			1.01896i	0.860483	+	0.509479i	$0.170162\pi$
$571$			13903.0i			1.01896i	−0.860483	+	0.509479i	$0.829838\pi$
$572$		0			0
$573$	−4974.91	+	10814.4i	−0.362705	+	0.788441i
$574$		0			0
$575$			17122.9i			1.24187i
$576$		0			0
$577$			20711.2i			1.49431i	0.664648	+	0.747157i	$0.268582\pi$
$577$			20711.2i			1.49431i	−0.664648	+	0.747157i	$0.731418\pi$
$578$		0			0
$579$	14867.5	+	6839.45i	1.06713	+	0.490911i
$580$		0			0
$581$	−12125.5	+	13353.2i	−0.865838	+	0.953504i
$582$		0			0
$583$	17186.6			1.22092
$584$		0			0
$585$	1916.54	−	1642.24i	0.135452	−	0.116065i
$586$		0			0
$587$			4831.23i			0.339704i	0.985470	+	0.169852i	$0.0543290\pi$
$587$			4831.23i			0.339704i	−0.985470	+	0.169852i	$0.945671\pi$
$588$		0			0
$589$			32177.6i			2.25103i
$590$		0			0
$591$	1963.57	+	903.298i	0.136668	+	0.0628709i
$592$		0			0
$593$	637.576			0.0441519			0.0220760	−	0.999756i	$-0.492972\pi$
$593$	637.576			0.0441519			0.0220760	+	0.999756i	$0.492972\pi$
$594$		0			0
$595$	−806.072	+	887.687i	−0.0555390	+	0.0611624i
$596$		0			0
$597$	−2610.81	+	5675.33i	−0.178984	+	0.389072i
$598$		0			0
$599$			10757.3i			0.733774i	0.930266	+	0.366887i	$0.119576\pi$
$599$			10757.3i			0.733774i	−0.930266	+	0.366887i	$0.880424\pi$
$600$		0			0
$601$		−	594.275i		−	0.0403344i	−0.999797	−	0.0201672i	$-0.993580\pi$
$601$		−	594.275i		−	0.0403344i	0.999797	−	0.0201672i	$-0.00641985\pi$
$602$		0			0
$603$	−10903.4	+	9342.86i	−0.736351	+	0.630963i
$604$		0			0
$605$			425.068i			0.0285644i
$606$		0			0
$607$		−	1028.34i		−	0.0687628i	−0.999409	−	0.0343814i	$-0.989054\pi$
$607$		−	1028.34i		−	0.0687628i	0.999409	−	0.0343814i	$-0.0109461\pi$
$608$		0			0
$609$	−5564.58	−	13074.4i	−0.370259	−	0.869955i
$610$		0			0
$611$	−15065.3			−0.997507
$612$		0			0
$613$		−	20648.4i		−	1.36049i	−0.732983	−	0.680247i	$-0.761873\pi$
$613$		−	20648.4i		−	1.36049i	0.732983	−	0.680247i	$-0.238127\pi$
$614$		0			0
$615$	712.472	−	1548.76i	0.0467149	−	0.101548i
$616$		0			0
$617$		−	8883.01i		−	0.579605i	−0.957086	−	0.289803i	$-0.906410\pi$
$617$		−	8883.01i		−	0.579605i	0.957086	−	0.289803i	$-0.0935897\pi$
$618$		0			0
$619$	−20038.9			−1.30118			−0.650589	−	0.759430i	$-0.725478\pi$
$619$	−20038.9			−1.30118			−0.650589	+	0.759430i	$0.725478\pi$
$620$		0			0
$621$	19284.8	−	5489.33i	1.24617	−	0.354717i
$622$		0			0
$623$	−22465.7	−	20400.2i	−1.44473	−	1.31190i
$624$		0			0
$625$	13705.6			0.877156
$626$		0			0
$627$	−17511.2	−	8055.66i	−1.11536	−	0.513097i
$628$		0			0
$629$			4604.54i			0.291884i
$630$		0			0
$631$	−24780.4			−1.56338			−0.781689	−	0.623669i	$-0.785641\pi$
$631$	−24780.4			−1.56338			−0.781689	+	0.623669i	$0.785641\pi$
$632$		0			0
$633$	−8920.68	+	19391.6i	−0.560134	+	1.21761i
$634$		0			0
$635$			106.258i			0.00664051i
$636$		0			0
$637$	1353.15	+	14008.4i	0.0841659	+	0.871323i
$638$		0			0
$639$	−16334.5	+	13996.7i	−1.01124	+	0.866511i
$640$		0			0
$641$			17191.8i			1.05934i	0.848205	+	0.529668i	$0.177684\pi$
$641$			17191.8i			1.05934i	−0.848205	+	0.529668i	$0.822316\pi$
$642$		0			0
$643$	6251.36			0.383405			0.191703	−	0.981453i	$-0.438599\pi$
$643$	6251.36			0.383405			0.191703	+	0.981453i	$0.438599\pi$
$644$		0			0
$645$	764.784	+	351.822i	0.0466874	+	0.0214775i
$646$		0			0
$647$	−24046.1			−1.46113			−0.730565	−	0.682843i	$-0.760743\pi$
$647$	−24046.1			−1.46113			−0.730565	+	0.682843i	$0.760743\pi$
$648$		0			0
$649$			6837.84i			0.413573i
$650$		0			0
$651$	−25984.0	+	11059.0i	−1.56435	+	0.665799i
$652$		0			0
$653$	15862.2			0.950593			0.475296	−	0.879826i	$-0.342341\pi$
$653$	15862.2			0.950593			0.475296	+	0.879826i	$0.342341\pi$
$654$		0			0
$655$	−5683.05			−0.339016
$656$		0			0
$657$	−10461.8	+	8964.49i	−0.621240	+	0.532326i
$658$		0			0
$659$	−16357.6			−0.966922			−0.483461	−	0.875366i	$-0.660620\pi$
$659$	−16357.6			−0.966922			−0.483461	+	0.875366i	$0.660620\pi$
$660$		0			0
$661$	1599.08			0.0940954			0.0470477	−	0.998893i	$-0.485019\pi$
$661$	1599.08			0.0940954			0.0470477	+	0.998893i	$0.485019\pi$
$662$		0			0
$663$	5504.33	+	2532.14i	0.322429	+	0.148326i
$664$		0			0
$665$	−3110.32	+	3425.24i	−0.181373	+	0.199737i
$666$		0			0
$667$			21102.4i			1.22502i
$668$		0			0
$669$	−8439.50	+	18345.6i	−0.487728	+	1.06021i
$670$		0			0
$671$	12305.0			0.707941
$672$		0			0
$673$	22120.1			1.26696			0.633481	−	0.773758i	$-0.281625\pi$
$673$	22120.1			1.26696			0.633481	+	0.773758i	$0.281625\pi$
$674$		0			0
$675$	−4601.77	−	16166.6i	−0.262403	−	0.921859i
$676$		0			0
$677$		−	1541.71i		−	0.0875227i	−0.999042	−	0.0437613i	$-0.986066\pi$
$677$		−	1541.71i		−	0.0875227i	0.999042	−	0.0437613i	$-0.0139341\pi$
$678$		0			0
$679$	1661.79	−	1830.04i	0.0939227	−	0.103432i
$680$		0			0
$681$	5559.93	−	12086.1i	0.312859	−	0.680087i
$682$		0			0
$683$	25827.4			1.44694			0.723468	−	0.690357i	$-0.242547\pi$
$683$	25827.4			1.44694			0.723468	+	0.690357i	$0.242547\pi$
$684$		0			0
$685$	459.689			0.0256406
$686$		0			0
$687$	−5721.81	−	2632.19i	−0.317759	−	0.146178i
$688$		0			0
$689$	−20845.3			−1.15260
$690$		0			0
$691$	21254.8			1.17015			0.585073	−	0.810981i	$-0.301066\pi$
$691$	21254.8			1.17015			0.585073	+	0.810981i	$0.301066\pi$
$692$		0			0
$693$	486.726	−	16909.2i	0.0266799	−	0.926881i
$694$		0			0
$695$		−	704.917i		−	0.0384734i
$696$		0			0
$697$	4092.45			0.222400
$698$		0			0
$699$	−9660.81	+	21000.5i	−0.522755	+	1.13635i
$700$		0			0
$701$	12172.2			0.655829			0.327914	−	0.944707i	$-0.393654\pi$
$701$	12172.2			0.655829			0.327914	+	0.944707i	$0.393654\pi$
$702$		0			0
$703$			17767.1i			0.953200i
$704$		0			0
$705$	1816.55	−	3948.77i	0.0970426	−	0.210950i
$706$		0			0
$707$	−2106.19	+	2319.44i	−0.112039	+	0.123383i
$708$		0			0
$709$		−	24941.7i		−	1.32117i	−0.750753	−	0.660583i	$-0.770309\pi$
$709$		−	24941.7i		−	1.32117i	0.750753	−	0.660583i	$-0.229691\pi$
$710$		0			0
$711$	670.256	+	782.207i	0.0353538	+	0.0412589i
$712$		0			0
$713$	41938.6			2.20282
$714$		0			0
$715$			3162.29i			0.165403i
$716$		0			0
$717$	−1031.84	+	2242.99i	−0.0537442	+	0.116828i
$718$		0			0
$719$	9926.70			0.514887			0.257443	−	0.966293i	$-0.417120\pi$
$719$	9926.70			0.514887			0.257443	+	0.966293i	$0.417120\pi$
$720$		0			0
$721$	−11701.4	+	12886.2i	−0.604416	+	0.665613i
$722$		0			0
$723$	15876.6	−	34512.4i	0.816679	−	1.77528i
$724$		0			0
$725$	17690.4			0.906212
$726$		0			0
$727$		−	18517.3i		−	0.944659i	−0.881422	−	0.472330i	$-0.843413\pi$
$727$		−	18517.3i		−	0.944659i	0.881422	−	0.472330i	$-0.156587\pi$
$728$		0			0
$729$	−16732.5	+	10365.5i	−0.850099	+	0.526623i
$730$		0			0
$731$			2020.87i			0.102250i
$732$		0			0
$733$	2791.96			0.140687			0.0703434	−	0.997523i	$-0.477591\pi$
$733$	2791.96			0.140687			0.0703434	+	0.997523i	$0.477591\pi$
$734$		0			0
$735$	−3834.91	−	1334.43i	−0.192453	−	0.0669677i
$736$		0			0
$737$		−	17990.6i		−	0.899174i
$738$		0			0
$739$			7989.50i			0.397698i	0.980030	+	0.198849i	$0.0637203\pi$
$739$			7989.50i			0.397698i	−0.980030	+	0.198849i	$0.936280\pi$
$740$		0			0
$741$	21239.1	+	9770.55i	1.05295	+	0.484386i
$742$		0			0
$743$			10189.8i			0.503131i	0.967840	+	0.251565i	$0.0809454\pi$
$743$			10189.8i			0.503131i	−0.967840	+	0.251565i	$0.919055\pi$
$744$		0			0
$745$			2751.67i			0.135320i
$746$		0			0
$747$	−19967.8	+	17109.9i	−0.978022	+	0.838044i
$748$		0			0
$749$	−28086.5	−	25504.2i	−1.37017	−	1.24420i
$750$		0			0
$751$	7615.28			0.370020			0.185010	−	0.982737i	$-0.440768\pi$
$751$	7615.28			0.370020			0.185010	+	0.982737i	$0.440768\pi$
$752$		0			0
$753$	13022.8	−	28308.7i	0.630248	−	1.37002i
$754$		0			0
$755$		−	5822.10i		−	0.280646i
$756$		0			0
$757$			3586.57i			0.172201i	0.996286	+	0.0861004i	$0.0274406\pi$
$757$			3586.57i			0.172201i	−0.996286	+	0.0861004i	$0.972559\pi$
$758$		0			0
$759$	−10499.3	+	22823.2i	−0.502110	+	1.09148i
$760$		0			0
$761$	17846.0			0.850089			0.425045	−	0.905172i	$-0.360258\pi$
$761$	17846.0			0.850089			0.425045	+	0.905172i	$0.360258\pi$
$762$		0			0
$763$	−2973.53	+	3274.60i	−0.141086	+	0.155371i
$764$		0			0
$765$	−1327.40	+	1137.42i	−0.0627351	+	0.0537562i
$766$		0			0
$767$		−	8293.49i		−	0.390431i
$768$		0			0
$769$		−	16449.0i		−	0.771346i	−0.922636	−	0.385673i	$-0.873969\pi$
$769$		−	16449.0i		−	0.771346i	0.922636	−	0.385673i	$-0.126031\pi$
$770$		0			0
$771$	−11936.2	−	5490.99i	−0.557552	−	0.256489i
$772$		0			0
$773$		−	34278.9i		−	1.59499i	−0.603328	−	0.797493i	$-0.706159\pi$
$773$		−	34278.9i		−	1.59499i	0.603328	−	0.797493i	$-0.293841\pi$
$774$		0			0
$775$		−	35157.6i		−	1.62955i
$776$		0			0
$777$	−14347.2	+	6106.29i	−0.662425	+	0.281933i
$778$		0			0
$779$	15791.2			0.726287
$780$		0			0
$781$		−	26952.0i		−	1.23485i
$782$		0			0
$783$	−5671.24	−	19923.8i	−0.258842	−	0.909349i
$784$		0			0
$785$		−	87.0698i		−	0.00395880i
$786$		0			0
$787$	17734.8			0.803273			0.401636	−	0.915799i	$-0.368442\pi$
$787$	17734.8			0.803273			0.401636	+	0.915799i	$0.368442\pi$
$788$		0			0
$789$	−14750.2	+	32063.8i	−0.665554	+	1.44677i
$790$		0			0
$791$	10.5323	−	11.5987i	0.000473433	−	0.000521368i
$792$		0			0
$793$	−14924.5			−0.668327
$794$		0			0
$795$	2513.49	−	5463.78i	0.112131	−	0.243749i
$796$		0			0
$797$			10674.8i			0.474428i	0.971457	+	0.237214i	$0.0762343\pi$
$797$			10674.8i			0.474428i	−0.971457	+	0.237214i	$0.923766\pi$
$798$		0			0
$799$	10434.3			0.462000
$800$		0			0
$801$	−28786.0	−	33594.1i	−1.26979	−	1.48188i
$802$		0			0
$803$		−	17262.0i		−	0.758608i
$804$		0			0
$805$	4464.27	+	4053.82i	0.195460	+	0.177489i
$806$		0			0
$807$	6006.79	−	13057.5i	0.262019	−	0.569572i
$808$		0			0
$809$			40808.4i			1.77348i	0.462265	+	0.886742i	$0.347037\pi$
$809$			40808.4i			1.77348i	−0.462265	+	0.886742i	$0.652963\pi$
$810$		0			0
$811$	18140.2			0.785436			0.392718	−	0.919659i	$-0.371535\pi$
$811$	18140.2			0.785436			0.392718	+	0.919659i	$0.371535\pi$
$812$		0			0
$813$	−2005.87	+	4360.32i	−0.0865299	+	0.188097i
$814$		0			0
$815$	6579.45			0.282783
$816$		0			0
$817$			7797.76i			0.333915i
$818$		0			0
$819$	−590.341	+	20508.9i	−0.0251870	+	0.875017i
$820$		0			0
$821$	7359.71			0.312857			0.156429	−	0.987689i	$-0.450002\pi$
$821$	7359.71			0.312857			0.156429	+	0.987689i	$0.450002\pi$
$822$		0			0
$823$	−23969.6			−1.01522			−0.507611	−	0.861586i	$-0.669471\pi$
$823$	−23969.6			−1.01522			−0.507611	+	0.861586i	$0.669471\pi$
$824$		0			0
$825$	19133.0	+	8801.70i	0.807424	+	0.371437i
$826$		0			0
$827$	35380.2			1.48765			0.743827	−	0.668373i	$-0.233009\pi$
$827$	35380.2			1.48765			0.743827	+	0.668373i	$0.233009\pi$
$828$		0			0
$829$	−29620.3			−1.24096			−0.620479	−	0.784223i	$-0.713062\pi$
$829$	−29620.3			−1.24096			−0.620479	+	0.784223i	$0.713062\pi$
$830$		0			0
$831$	8064.67	−	17530.8i	0.336655	−	0.731815i
$832$		0			0
$833$	−937.194	−	9702.25i	−0.0389818	−	0.403557i
$834$		0			0
$835$		−	701.056i		−	0.0290551i
$836$		0			0
$837$	−39596.4	+	11270.9i	−1.63519	+	0.465449i
$838$		0			0
$839$	25895.0			1.06555			0.532774	−	0.846257i	$-0.321149\pi$
$839$	25895.0			1.06555			0.532774	+	0.846257i	$0.321149\pi$
$840$		0			0
$841$	−2587.32			−0.106085
$842$		0			0
$843$	−10351.1	+	22501.1i	−0.422908	+	0.919309i
$844$		0			0
$845$			1169.79i			0.0476237i
$846$		0			0
$847$	−2558.16	−	2322.96i	−0.103777	−	0.0942359i
$848$		0			0
$849$	38902.8	+	17896.3i	1.57260	+	0.723441i
$850$		0			0
$851$	23156.7			0.932787
$852$		0			0
$853$	−37192.4			−1.49290			−0.746449	−	0.665442i	$-0.768243\pi$
$853$	−37192.4			−1.49290			−0.746449	+	0.665442i	$0.768243\pi$
$854$		0			0
$855$	−5121.93	+	4388.86i	−0.204873	+	0.175551i
$856$		0			0
$857$	16970.1			0.676414			0.338207	−	0.941072i	$-0.390180\pi$
$857$	16970.1			0.676414			0.338207	+	0.941072i	$0.390180\pi$
$858$		0			0
$859$	−31307.1			−1.24352			−0.621760	−	0.783208i	$-0.713582\pi$
$859$	−31307.1			−1.24352			−0.621760	+	0.783208i	$0.713582\pi$
$860$		0			0
$861$	5427.20	+	12751.6i	0.214818	+	0.504733i
$862$		0			0
$863$			35041.9i			1.38220i	0.722759	+	0.691101i	$0.242874\pi$
$863$			35041.9i			1.38220i	−0.722759	+	0.691101i	$0.757126\pi$
$864$		0			0
$865$	6245.43			0.245493
$866$		0			0
$867$	19380.0	+	8915.35i	0.759147	+	0.349228i
$868$		0			0
$869$	−1290.64			−0.0503821
$870$		0			0
$871$			21820.4i			0.848859i
$872$		0			0
$873$	2736.55	−	2344.89i	0.106092	−	0.0909078i
$874$		0			0
$875$	6943.96	−	7647.03i	0.268284	−	0.295448i
$876$		0			0
$877$		−	19272.8i		−	0.742070i	−0.928619	−	0.371035i	$-0.879003\pi$
$877$		−	19272.8i		−	0.742070i	0.928619	−	0.371035i	$-0.120997\pi$
$878$		0			0
$879$	10762.0	−	23394.2i	0.412961	−	0.897686i
$880$		0			0
$881$	40919.7			1.56484			0.782418	−	0.622754i	$-0.213986\pi$
$881$	40919.7			1.56484			0.782418	+	0.622754i	$0.213986\pi$
$882$		0			0
$883$		−	31577.4i		−	1.20347i	−0.798696	−	0.601735i	$-0.794476\pi$
$883$		−	31577.4i		−	1.20347i	0.798696	−	0.601735i	$-0.205524\pi$
$884$		0			0
$885$	2173.81	+	1000.01i	0.0825670	+	0.0379831i
$886$		0			0
$887$	17298.4			0.654817			0.327409	−	0.944883i	$-0.393825\pi$
$887$	17298.4			0.654817			0.327409	+	0.944883i	$0.393825\pi$
$888$		0			0
$889$	−639.486	−	580.691i	−0.0241256	−	0.0219075i
$890$		0			0
$891$	3779.23	−	24370.3i	0.142098	−	0.916313i
$892$		0			0
$893$	40261.8			1.50875
$894$		0			0
$895$			1662.50i			0.0620909i
$896$		0			0
$897$	12734.4	−	27681.9i	0.474013	−	1.03040i
$898$		0			0
$899$		−	43328.4i		−	1.60743i
$900$		0			0
$901$	14437.5			0.533833
$902$		0			0
$903$	−6296.82	+	2679.97i	−0.232054	+	0.0987640i
$904$		0			0
$905$			4686.01i			0.172120i
$906$		0			0
$907$			28199.6i			1.03236i	0.856480	+	0.516181i	$0.172647\pi$
$907$			28199.6i			1.03236i	−0.856480	+	0.516181i	$0.827353\pi$
$908$		0			0
$909$	−3468.37	+	2971.97i	−0.126555	+	0.108442i
$910$		0			0
$911$		−	37866.6i		−	1.37714i	−0.725168	−	0.688571i	$-0.758238\pi$
$911$		−	37866.6i		−	1.37714i	0.725168	−	0.688571i	$-0.241762\pi$
$912$		0			0
$913$		−	32946.8i		−	1.19428i
$914$		0			0
$915$	1799.56	−	3911.86i	0.0650183	−	0.141336i
$916$		0			0
$917$	31057.4	−	34201.9i	1.11844	−	1.23168i
$918$		0			0
$919$	47333.4			1.69901			0.849503	−	0.527584i	$-0.176902\pi$
$919$	47333.4			1.69901			0.849503	+	0.527584i	$0.176902\pi$
$920$		0			0
$921$	−29042.3	−	13360.3i	−1.03906	−	0.477998i
$922$		0			0
$923$			32689.5i			1.16575i
$924$		0			0
$925$		−	19412.5i		−	0.690033i
$926$		0			0
$927$	−19269.4	+	16511.5i	−0.682729	+	0.585015i
$928$		0			0
$929$	3499.09			0.123575			0.0617877	−	0.998089i	$-0.480320\pi$
$929$	3499.09			0.123575			0.0617877	+	0.998089i	$0.480320\pi$
$930$		0			0
$931$	−3616.27	−	37437.2i	−0.127302	−	1.31789i
$932$		0			0
$933$	22234.6	+	10228.5i	0.780201	+	0.358914i
$934$		0			0
$935$		−	2190.21i		−	0.0766070i
$936$		0			0
$937$		−	5794.61i		−	0.202029i	−0.994885	−	0.101015i	$-0.967791\pi$
$937$		−	5794.61i		−	0.202029i	0.994885	−	0.101015i	$-0.0322089\pi$
$938$		0			0
$939$	−13044.6	+	28356.2i	−0.453349	+	0.985483i
$940$		0			0
$941$			20620.2i			0.714345i	0.934038	+	0.357173i	$0.116259\pi$
$941$			20620.2i			0.714345i	−0.934038	+	0.357173i	$0.883741\pi$
$942$		0			0
$943$		−	20581.4i		−	0.710734i
$944$		0			0
$945$	−5304.41	−	2627.66i	−0.182595	−	0.0904526i
$946$		0			0
$947$	2984.78			0.102421			0.0512103	−	0.998688i	$-0.483692\pi$
$947$	2984.78			0.102421			0.0512103	+	0.998688i	$0.483692\pi$
$948$		0			0
$949$			20936.7i			0.716159i
$950$		0			0
$951$	40067.4	+	18432.1i	1.36622	+	0.628499i
$952$		0			0
$953$		−	1650.21i		−	0.0560920i	−0.999607	−	0.0280460i	$-0.991072\pi$
$953$		−	1650.21i		−	0.0560920i	0.999607	−	0.0280460i	$-0.00892849\pi$
$954$		0			0
$955$	−5219.17			−0.176846
$956$		0			0
$957$	23579.6	+	10847.3i	0.796467	+	0.366397i
$958$		0			0
$959$	−2512.16	+	2766.51i	−0.0845899	+	0.0931547i
$960$		0			0
$961$	−56319.3			−1.89048
$962$		0			0
$963$	−35988.1	−	41999.2i	−1.20426	−	1.40540i
$964$		0			0
$965$			7175.25i			0.239357i
$966$		0			0
$967$	−19631.7			−0.652858			−0.326429	−	0.945222i	$-0.605845\pi$
$967$	−19631.7			−0.652858			−0.326429	+	0.945222i	$0.605845\pi$
$968$		0			0
$969$	−14710.2	−	6767.11i	−0.487678	−	0.224346i
$970$		0			0
$971$			2477.59i			0.0818841i	0.999162	+	0.0409421i	$0.0130359\pi$
$971$			2477.59i			0.0818841i	−0.999162	+	0.0409421i	$0.986964\pi$
$972$		0			0
$973$	4242.35	+	3852.31i	0.139778	+	0.126926i
$974$		0			0
$975$	−23206.0	−	10675.4i	−0.762244	−	0.350653i
$976$		0			0
$977$		−	27876.8i		−	0.912853i	−0.889761	−	0.456427i	$-0.849129\pi$
$977$		−	27876.8i		−	0.912853i	0.889761	−	0.456427i	$-0.150871\pi$
$978$		0			0
$979$	55430.2			1.80956
$980$		0			0
$981$	−4896.67	+	4195.84i	−0.159367	+	0.136558i
$982$		0			0
$983$	55169.7			1.79007			0.895035	−	0.445996i	$-0.147150\pi$
$983$	55169.7			1.79007			0.895035	+	0.445996i	$0.147150\pi$
$984$		0			0
$985$			947.648i			0.0306544i
$986$		0			0
$987$	13837.4	+	32512.1i	0.446250	+	1.04850i
$988$		0			0
$989$	10163.2			0.326765
$990$		0			0
$991$	24768.7			0.793949			0.396975	−	0.917830i	$-0.370060\pi$
$991$	24768.7			0.793949			0.396975	+	0.917830i	$0.370060\pi$
$992$		0			0
$993$	23833.3	−	51808.4i	0.761659	−	1.65568i
$994$		0			0
$995$	−2739.00			−0.0872683
$996$		0			0
$997$	3203.48			0.101760			0.0508802	−	0.998705i	$-0.483797\pi$
$997$	3203.48			0.101760			0.0508802	+	0.998705i	$0.483797\pi$
$998$		0			0
$999$	−21863.5	+	6223.34i	−0.692421	+	0.197095i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.4.i.c.209.14		80
3.2	odd	2	inner	672.4.i.c.209.15		80
4.3	odd	2		168.4.i.c.125.16	yes	80
7.6	odd	2	inner	672.4.i.c.209.68		80
8.3	odd	2		168.4.i.c.125.67	yes	80
8.5	even	2	inner	672.4.i.c.209.67		80
12.11	even	2		168.4.i.c.125.66	yes	80
21.20	even	2	inner	672.4.i.c.209.65		80
24.5	odd	2	inner	672.4.i.c.209.66		80
24.11	even	2		168.4.i.c.125.13	✓	80
28.27	even	2		168.4.i.c.125.15	yes	80
56.13	odd	2	inner	672.4.i.c.209.13		80
56.27	even	2		168.4.i.c.125.68	yes	80
84.83	odd	2		168.4.i.c.125.65	yes	80
168.83	odd	2		168.4.i.c.125.14	yes	80
168.125	even	2	inner	672.4.i.c.209.16		80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.4.i.c.125.13	✓	80	24.11	even	2
168.4.i.c.125.14	yes	80	168.83	odd	2
168.4.i.c.125.15	yes	80	28.27	even	2
168.4.i.c.125.16	yes	80	4.3	odd	2
168.4.i.c.125.65	yes	80	84.83	odd	2
168.4.i.c.125.66	yes	80	12.11	even	2
168.4.i.c.125.67	yes	80	8.3	odd	2
168.4.i.c.125.68	yes	80	56.27	even	2
672.4.i.c.209.13		80	56.13	odd	2	inner
672.4.i.c.209.14		80	1.1	even	1	trivial
672.4.i.c.209.15		80	3.2	odd	2	inner
672.4.i.c.209.16		80	168.125	even	2	inner
672.4.i.c.209.65		80	21.20	even	2	inner
672.4.i.c.209.66		80	24.5	odd	2	inner
672.4.i.c.209.67		80	8.5	even	2	inner
672.4.i.c.209.68		80	7.6	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	672.i (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-4.72061 - 2.17161i) q^{3} -2.27823i q^{5} +(13.7109 + 12.4503i) q^{7} +(17.5682 + 20.5026i) q^{9} -33.8293 q^{11} +41.0309 q^{13} +(-4.94743 + 10.7546i) q^{15} -28.4181 q^{17} -109.654 q^{19} +(-37.6866 - 88.5478i) q^{21} +142.918i q^{23} +119.810 q^{25} +(-38.4090 - 134.936i) q^{27} +147.654 q^{29} -293.446i q^{31} +(159.695 + 73.4640i) q^{33} +(28.3647 - 31.2366i) q^{35} -162.028i q^{37} +(-193.691 - 89.1031i) q^{39} -144.009 q^{41} -71.1121i q^{43} +(46.7097 - 40.0245i) q^{45} -367.170 q^{47} +(32.9788 + 341.411i) q^{49} +(134.151 + 61.7130i) q^{51} -508.040 q^{53} +77.0710i q^{55} +(517.635 + 238.127i) q^{57} -202.128i q^{59} -363.737 q^{61} +(-14.3877 + 499.840i) q^{63} -93.4779i q^{65} +531.804i q^{67} +(310.362 - 674.659i) q^{69} +796.705i q^{71} +510.267i q^{73} +(-565.574 - 260.180i) q^{75} +(-463.831 - 421.186i) q^{77} +38.1516 q^{79} +(-111.715 + 720.389i) q^{81} +973.913i q^{83} +64.7430i q^{85} +(-697.016 - 320.647i) q^{87} -1638.53 q^{89} +(562.572 + 510.848i) q^{91} +(-637.249 + 1385.24i) q^{93} +249.818i q^{95} -133.473i q^{97} +(-594.321 - 693.589i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 64 q^{7} + 104 q^{9} + 8 q^{15} - 976 q^{25} - 568 q^{39} - 4048 q^{49} - 1448 q^{57} + 2152 q^{63} - 4992 q^{79} + 1568 q^{81}+O(q^{100}) \) 80 * q - 64 * q^7 + 104 * q^9 + 8 * q^15 - 976 * q^25 - 568 * q^39 - 4048 * q^49 - 1448 * q^57 + 2152 * q^63 - 4992 * q^79 + 1568 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more