LMFDB - Embedded newform 672.4.b.a.223.19

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [672,4,Mod(223,672)] 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("672.223"); S:= CuspForms(chi, 4); N := Newforms(S);

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

Level:	$ N $	$=$	$ 672 = 2^{5} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	672.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [24,0,-72] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$39.6492835239$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			223.19
Character	$\chi$	$=$	672.223
Dual form			672.4.b.a.223.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +12.7173i q^{5} +(-2.54263 - 18.3449i) q^{7} +9.00000 q^{9} -2.42308i q^{11} +26.6669i q^{13} -38.1518i q^{15} -10.6363i q^{17} +30.0792 q^{19} +(7.62790 + 55.0347i) q^{21} -120.218i q^{23} -36.7290 q^{25} -27.0000 q^{27} +43.1872 q^{29} +6.30036 q^{31} +7.26925i q^{33} +(233.297 - 32.3354i) q^{35} -61.6192 q^{37} -80.0008i q^{39} -75.9491i q^{41} +212.731i q^{43} +114.455i q^{45} +330.294 q^{47} +(-330.070 + 93.2887i) q^{49} +31.9089i q^{51} -4.87616 q^{53} +30.8150 q^{55} -90.2376 q^{57} +481.332 q^{59} +236.177i q^{61} +(-22.8837 - 165.104i) q^{63} -339.130 q^{65} +661.939i q^{67} +360.653i q^{69} +547.254i q^{71} +968.026i q^{73} +110.187 q^{75} +(-44.4512 + 6.16102i) q^{77} -64.6485i q^{79} +81.0000 q^{81} +665.281 q^{83} +135.265 q^{85} -129.561 q^{87} +384.181i q^{89} +(489.202 - 67.8042i) q^{91} -18.9011 q^{93} +382.525i q^{95} +1226.09i q^{97} -21.8078i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 72 q^{3} - 20 q^{7} + 216 q^{9} + 56 q^{19} + 60 q^{21} - 432 q^{25} - 648 q^{27} + 464 q^{31} - 568 q^{35} + 504 q^{37} + 560 q^{47} - 128 q^{49} - 784 q^{53} + 424 q^{55} - 168 q^{57} - 800 q^{59}+ \cdots - 1392 q^{93}+O(q^{100}) $ 24 * q - 72 * q^3 - 20 * q^7 + 216 * q^9 + 56 * q^19 + 60 * q^21 - 432 * q^25 - 648 * q^27 + 464 * q^31 - 568 * q^35 + 504 * q^37 + 560 * q^47 - 128 * q^49 - 784 * q^53 + 424 * q^55 - 168 * q^57 - 800 * q^59 - 180 * q^63 + 560 * q^65 + 1296 * q^75 - 1568 * q^77 + 1944 * q^81 - 1936 * q^83 + 3000 * q^85 + 496 * q^91 - 1392 * q^93

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
$3$	−3.00000			−0.577350
$3$	−3.00000			−0.577350
$4$		0			0
$5$			12.7173i			1.13747i	0.822522	+	0.568734i	$0.192566\pi$
$5$			12.7173i			1.13747i	−0.822522	+	0.568734i	$0.807434\pi$
$6$		0			0
$7$	−2.54263	−	18.3449i	−0.137289	−	0.990531i
$7$	−2.54263	−	18.3449i	−0.137289	−	0.990531i
$8$		0			0
$9$	9.00000			0.333333
$10$		0			0
$11$		−	2.42308i		−	0.0664171i	−0.999448	−	0.0332085i	$-0.989427\pi$
$11$		−	2.42308i		−	0.0664171i	0.999448	−	0.0332085i	$-0.0105725\pi$
$12$		0			0
$13$			26.6669i			0.568929i	0.958687	+	0.284464i	$0.0918157\pi$
$13$			26.6669i			0.568929i	−0.958687	+	0.284464i	$0.908184\pi$
$14$		0			0
$15$		−	38.1518i		−	0.656717i
$16$		0			0
$17$		−	10.6363i		−	0.151746i	−0.997117	−	0.0758731i	$-0.975826\pi$
$17$		−	10.6363i		−	0.151746i	0.997117	−	0.0758731i	$-0.0241744\pi$
$18$		0			0
$19$	30.0792			0.363192			0.181596	−	0.983373i	$-0.441874\pi$
$19$	30.0792			0.363192			0.181596	+	0.983373i	$0.441874\pi$
$20$		0			0
$21$	7.62790	+	55.0347i	0.0792640	+	0.571883i
$22$		0			0
$23$		−	120.218i		−	1.08987i	−0.838477	−	0.544937i	$-0.816554\pi$
$23$		−	120.218i		−	1.08987i	0.838477	−	0.544937i	$-0.183446\pi$
$24$		0			0
$25$	−36.7290			−0.293832
$26$		0			0
$27$	−27.0000			−0.192450
$28$		0			0
$29$	43.1872			0.276540			0.138270	−	0.990395i	$-0.455846\pi$
$29$	43.1872			0.276540			0.138270	+	0.990395i	$0.455846\pi$
$30$		0			0
$31$	6.30036			0.0365025			0.0182513	−	0.999833i	$-0.494190\pi$
$31$	6.30036			0.0365025			0.0182513	+	0.999833i	$0.494190\pi$
$32$		0			0
$33$			7.26925i			0.0383459i
$34$		0			0
$35$	233.297	−	32.3354i	1.12670	−	0.156162i
$36$		0			0
$37$	−61.6192			−0.273788			−0.136894	−	0.990586i	$-0.543712\pi$
$37$	−61.6192			−0.273788			−0.136894	+	0.990586i	$0.543712\pi$
$38$		0			0
$39$		−	80.0008i		−	0.328471i
$40$		0			0
$41$		−	75.9491i		−	0.289299i	−0.989483	−	0.144650i	$-0.953795\pi$
$41$		−	75.9491i		−	0.289299i	0.989483	−	0.144650i	$-0.0462055\pi$
$42$		0			0
$43$			212.731i			0.754446i	0.926122	+	0.377223i	$0.123121\pi$
$43$			212.731i			0.754446i	−0.926122	+	0.377223i	$0.876879\pi$
$44$		0			0
$45$			114.455i			0.379156i
$46$		0			0
$47$	330.294			1.02507			0.512536	−	0.858666i	$-0.328706\pi$
$47$	330.294			1.02507			0.512536	+	0.858666i	$0.328706\pi$
$48$		0			0
$49$	−330.070	+	93.2887i	−0.962303	+	0.271979i
$50$		0			0
$51$			31.9089i			0.0876107i
$52$		0			0
$53$	−4.87616			−0.0126376			−0.00631879	−	0.999980i	$-0.502011\pi$
$53$	−4.87616			−0.0126376			−0.00631879	+	0.999980i	$0.502011\pi$
$54$		0			0
$55$	30.8150			0.0755472
$56$		0			0
$57$	−90.2376			−0.209689
$58$		0			0
$59$	481.332			1.06210			0.531052	−	0.847339i	$-0.321797\pi$
$59$	481.332			1.06210			0.531052	+	0.847339i	$0.321797\pi$
$60$		0			0
$61$			236.177i			0.495728i	0.968795	+	0.247864i	$0.0797287\pi$
$61$			236.177i			0.495728i	−0.968795	+	0.247864i	$0.920271\pi$
$62$		0			0
$63$	−22.8837	−	165.104i	−0.0457631	−	0.330177i
$64$		0			0
$65$	−339.130			−0.647138
$66$		0			0
$67$			661.939i			1.20700i	0.797365	+	0.603498i	$0.206227\pi$
$67$			661.939i			1.20700i	−0.797365	+	0.603498i	$0.793773\pi$
$68$		0			0
$69$			360.653i			0.629239i
$70$		0			0
$71$			547.254i			0.914748i	0.889274	+	0.457374i	$0.151210\pi$
$71$			547.254i			0.914748i	−0.889274	+	0.457374i	$0.848790\pi$
$72$		0			0
$73$			968.026i			1.55204i	0.630709	+	0.776020i	$0.282764\pi$
$73$			968.026i			1.55204i	−0.630709	+	0.776020i	$0.717236\pi$
$74$		0			0
$75$	110.187			0.169644
$76$		0			0
$77$	−44.4512	+	6.16102i	−0.0657881	+	0.00911835i
$78$		0			0
$79$		−	64.6485i		−	0.0920700i	−0.998940	−	0.0460350i	$-0.985341\pi$
$79$		−	64.6485i		−	0.0920700i	0.998940	−	0.0460350i	$-0.0146586\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$	665.281			0.879808			0.439904	−	0.898045i	$-0.355012\pi$
$83$	665.281			0.879808			0.439904	+	0.898045i	$0.355012\pi$
$84$		0			0
$85$	135.265			0.172606
$86$		0			0
$87$	−129.561			−0.159660
$88$		0			0
$89$			384.181i			0.457563i	0.973478	+	0.228781i	$0.0734741\pi$
$89$			384.181i			0.457563i	−0.973478	+	0.228781i	$0.926526\pi$
$90$		0			0
$91$	489.202	−	67.8042i	0.563541	−	0.0781078i
$92$		0			0
$93$	−18.9011			−0.0210747
$94$		0			0
$95$			382.525i			0.413119i
$96$		0			0
$97$			1226.09i			1.28341i	0.766952	+	0.641705i	$0.221773\pi$
$97$			1226.09i			1.28341i	−0.766952	+	0.641705i	$0.778227\pi$
$98$		0			0
$99$		−	21.8078i		−	0.0221390i
$100$		0			0
$101$			647.043i			0.637458i	0.947846	+	0.318729i	$0.103256\pi$
$101$			647.043i			0.637458i	−0.947846	+	0.318729i	$0.896744\pi$
$102$		0			0
$103$	−344.493			−0.329553			−0.164776	−	0.986331i	$-0.552690\pi$
$103$	−344.493			−0.329553			−0.164776	+	0.986331i	$0.552690\pi$
$104$		0			0
$105$	−699.891	+	97.0061i	−0.650499	+	0.0901603i
$106$		0			0
$107$		−	1332.50i		−	1.20390i	−0.798532	−	0.601952i	$-0.794390\pi$
$107$		−	1332.50i		−	1.20390i	0.798532	−	0.601952i	$-0.205610\pi$
$108$		0			0
$109$	2230.40			1.95994			0.979971	−	0.199141i	$-0.0638152\pi$
$109$	2230.40			1.95994			0.979971	+	0.199141i	$0.0638152\pi$
$110$		0			0
$111$	184.858			0.158071
$112$		0			0
$113$	247.508			0.206049			0.103025	−	0.994679i	$-0.467148\pi$
$113$	247.508			0.206049			0.103025	+	0.994679i	$0.467148\pi$
$114$		0			0
$115$	1528.84			1.23970
$116$		0			0
$117$			240.002i			0.189643i
$118$		0			0
$119$	−195.122	+	27.0443i	−0.150309	+	0.0208331i
$120$		0			0
$121$	1325.13			0.995589
$122$		0			0
$123$			227.847i			0.167027i
$124$		0			0
$125$			1122.57i			0.803243i
$126$		0			0
$127$			423.075i			0.295605i	0.989017	+	0.147802i	$0.0472200\pi$
$127$			423.075i			0.295605i	−0.989017	+	0.147802i	$0.952780\pi$
$128$		0			0
$129$		−	638.193i		−	0.435580i
$130$		0			0
$131$	2231.23			1.48812			0.744058	−	0.668115i	$-0.232898\pi$
$131$	2231.23			1.48812			0.744058	+	0.668115i	$0.232898\pi$
$132$		0			0
$133$	−76.4804	−	551.800i	−0.0498623	−	0.359753i
$134$		0			0
$135$		−	343.366i		−	0.218906i
$136$		0			0
$137$	1269.91			0.791939			0.395970	−	0.918264i	$-0.370409\pi$
$137$	1269.91			0.791939			0.395970	+	0.918264i	$0.370409\pi$
$138$		0			0
$139$	1993.14			1.21623			0.608114	−	0.793850i	$-0.291926\pi$
$139$	1993.14			1.21623			0.608114	+	0.793850i	$0.291926\pi$
$140$		0			0
$141$	−990.883			−0.591826
$142$		0			0
$143$	64.6162			0.0377866
$144$		0			0
$145$			549.223i			0.314555i
$146$		0			0
$147$	990.210	−	279.866i	0.555586	−	0.157027i
$148$		0			0
$149$	−2054.75			−1.12974			−0.564871	−	0.825179i	$-0.691074\pi$
$149$	−2054.75			−1.12974			−0.564871	+	0.825179i	$0.691074\pi$
$150$		0			0
$151$			1622.32i			0.874323i	0.899383	+	0.437161i	$0.144016\pi$
$151$			1622.32i			0.874323i	−0.899383	+	0.437161i	$0.855984\pi$
$152$		0			0
$153$		−	95.7268i		−	0.0505821i
$154$		0			0
$155$			80.1234i			0.0415204i
$156$		0			0
$157$		−	694.512i		−	0.353045i	−0.984297	−	0.176523i	$-0.943515\pi$
$157$		−	694.512i		−	0.353045i	0.984297	−	0.176523i	$-0.0564849\pi$
$158$		0			0
$159$	14.6285			0.00729631
$160$		0			0
$161$	−2205.38	+	305.669i	−1.07955	+	0.149628i
$162$		0			0
$163$		−	506.509i		−	0.243392i	−0.992567	−	0.121696i	$-0.961167\pi$
$163$		−	506.509i		−	0.243392i	0.992567	−	0.121696i	$-0.0388332\pi$
$164$		0			0
$165$	−92.4451			−0.0436172
$166$		0			0
$167$	−3689.21			−1.70946			−0.854730	−	0.519074i	$-0.826277\pi$
$167$	−3689.21			−1.70946			−0.854730	+	0.519074i	$0.826277\pi$
$168$		0			0
$169$	1485.88			0.676320
$170$		0			0
$171$	270.713			0.121064
$172$		0			0
$173$			2559.84i			1.12498i	0.826806	+	0.562488i	$0.190156\pi$
$173$			2559.84i			1.12498i	−0.826806	+	0.562488i	$0.809844\pi$
$174$		0			0
$175$	93.3883	+	673.789i	0.0403400	+	0.291049i
$176$		0			0
$177$	−1444.00			−0.613206
$178$		0			0
$179$			1429.61i			0.596949i	0.954418	+	0.298475i	$0.0964778\pi$
$179$			1429.61i			0.596949i	−0.954418	+	0.298475i	$0.903522\pi$
$180$		0			0
$181$			981.980i			0.403260i	0.979462	+	0.201630i	$0.0646238\pi$
$181$			981.980i			0.403260i	−0.979462	+	0.201630i	$0.935376\pi$
$182$		0			0
$183$		−	708.532i		−	0.286209i
$184$		0			0
$185$		−	783.628i		−	0.311424i
$186$		0			0
$187$	−25.7727			−0.0100785
$188$		0			0
$189$	68.6511	+	495.312i	0.0264213	+	0.190628i
$190$		0			0
$191$		−	3321.25i		−	1.25820i	−0.777323	−	0.629102i	$-0.783423\pi$
$191$		−	3321.25i		−	1.25820i	0.777323	−	0.629102i	$-0.216577\pi$
$192$		0			0
$193$	−173.898			−0.0648573			−0.0324286	−	0.999474i	$-0.510324\pi$
$193$	−173.898			−0.0648573			−0.0324286	+	0.999474i	$0.510324\pi$
$194$		0			0
$195$	1017.39			0.373625
$196$		0			0
$197$	−5383.96			−1.94716			−0.973581	−	0.228340i	$-0.926670\pi$
$197$	−5383.96			−1.94716			−0.973581	+	0.228340i	$0.926670\pi$
$198$		0			0
$199$	−1695.29			−0.603899			−0.301949	−	0.953324i	$-0.597637\pi$
$199$	−1695.29			−0.603899			−0.301949	+	0.953324i	$0.597637\pi$
$200$		0			0
$201$		−	1985.82i		−	0.696859i
$202$		0			0
$203$	−109.809	−	792.264i	−0.0379660	−	0.273921i
$204$		0			0
$205$	965.866			0.329068
$206$		0			0
$207$		−	1081.96i		−	0.363291i
$208$		0			0
$209$		−	72.8844i		−	0.0241221i
$210$		0			0
$211$			4735.60i			1.54508i	0.634965	+	0.772541i	$0.281014\pi$
$211$			4735.60i			1.54508i	−0.634965	+	0.772541i	$0.718986\pi$
$212$		0			0
$213$		−	1641.76i		−	0.528130i
$214$		0			0
$215$	−2705.36			−0.858158
$216$		0			0
$217$	−16.0195	−	115.579i	−0.00501141	−	0.0361569i
$218$		0			0
$219$		−	2904.08i		−	0.896070i
$220$		0			0
$221$	283.638			0.0863328
$222$		0			0
$223$	−3440.90			−1.03327			−0.516636	−	0.856205i	$-0.672816\pi$
$223$	−3440.90			−1.03327			−0.516636	+	0.856205i	$0.672816\pi$
$224$		0			0
$225$	−330.561			−0.0979439
$226$		0			0
$227$	949.355			0.277581			0.138791	−	0.990322i	$-0.455679\pi$
$227$	949.355			0.277581			0.138791	+	0.990322i	$0.455679\pi$
$228$		0			0
$229$		−	2303.60i		−	0.664742i	−0.943149	−	0.332371i	$-0.892151\pi$
$229$		−	2303.60i		−	0.664742i	0.943149	−	0.332371i	$-0.107849\pi$
$230$		0			0
$231$	133.354	−	18.4831i	0.0379828	−	0.00526448i
$232$		0			0
$233$	3268.21			0.918918			0.459459	−	0.888199i	$-0.348043\pi$
$233$	3268.21			0.918918			0.459459	+	0.888199i	$0.348043\pi$
$234$		0			0
$235$			4200.44i			1.16599i
$236$		0			0
$237$			193.946i			0.0531566i
$238$		0			0
$239$		−	3116.13i		−	0.843370i	−0.906742	−	0.421685i	$-0.861439\pi$
$239$		−	3116.13i		−	0.843370i	0.906742	−	0.421685i	$-0.138561\pi$
$240$		0			0
$241$		−	2290.95i		−	0.612337i	−0.951977	−	0.306169i	$-0.900953\pi$
$241$		−	2290.95i		−	0.612337i	0.951977	−	0.306169i	$-0.0990471\pi$
$242$		0			0
$243$	−243.000			−0.0641500
$244$		0			0
$245$	−1186.38	−	4197.59i	−0.309367	−	1.09459i
$246$		0			0
$247$			802.119i			0.206630i
$248$		0			0
$249$	−1995.84			−0.507958
$250$		0			0
$251$	−2307.47			−0.580264			−0.290132	−	0.956987i	$-0.593699\pi$
$251$	−2307.47			−0.580264			−0.290132	+	0.956987i	$0.593699\pi$
$252$		0			0
$253$	−291.297			−0.0723862
$254$		0			0
$255$	−405.795			−0.0996543
$256$		0			0
$257$			2832.31i			0.687449i	0.939071	+	0.343725i	$0.111689\pi$
$257$			2832.31i			0.687449i	−0.939071	+	0.343725i	$0.888311\pi$
$258$		0			0
$259$	156.675	+	1130.40i	0.0375881	+	0.271195i
$260$		0			0
$261$	388.684			0.0921799
$262$		0			0
$263$		−	6588.14i		−	1.54465i	−0.635230	−	0.772323i	$-0.719095\pi$
$263$		−	6588.14i		−	1.54465i	0.635230	−	0.772323i	$-0.280905\pi$
$264$		0			0
$265$		−	62.0114i		−	0.0143748i
$266$		0			0
$267$		−	1152.54i		−	0.264174i
$268$		0			0
$269$		−	3744.39i		−	0.848698i	−0.905499	−	0.424349i	$-0.860503\pi$
$269$		−	3744.39i		−	0.848698i	0.905499	−	0.424349i	$-0.139497\pi$
$270$		0			0
$271$	−7505.17			−1.68231			−0.841156	−	0.540792i	$-0.818125\pi$
$271$	−7505.17			−1.68231			−0.841156	+	0.540792i	$0.818125\pi$
$272$		0			0
$273$	−1467.61	+	203.413i	−0.325361	+	0.0450956i
$274$		0			0
$275$			88.9974i			0.0195154i
$276$		0			0
$277$	7500.64			1.62697			0.813483	−	0.581589i	$-0.197569\pi$
$277$	7500.64			1.62697			0.813483	+	0.581589i	$0.197569\pi$
$278$		0			0
$279$	56.7033			0.0121675
$280$		0			0
$281$	5803.98			1.23216			0.616079	−	0.787684i	$-0.288720\pi$
$281$	5803.98			1.23216			0.616079	+	0.787684i	$0.288720\pi$
$282$		0			0
$283$	5915.55			1.24255			0.621277	−	0.783591i	$-0.286614\pi$
$283$	5915.55			1.24255			0.621277	+	0.783591i	$0.286614\pi$
$284$		0			0
$285$		−	1147.58i		−	0.238514i
$286$		0			0
$287$	−1393.28	+	193.111i	−0.286560	+	0.0397177i
$288$		0			0
$289$	4799.87			0.976973
$290$		0			0
$291$		−	3678.28i		−	0.740977i
$292$		0			0
$293$		−	6960.27i		−	1.38779i	−0.720075	−	0.693896i	$-0.755893\pi$
$293$		−	6960.27i		−	1.38779i	0.720075	−	0.693896i	$-0.244107\pi$
$294$		0			0
$295$			6121.23i			1.20811i
$296$		0			0
$297$			65.4233i			0.0127820i
$298$		0			0
$299$	3205.83			0.620061
$300$		0			0
$301$	3902.53	−	540.898i	0.747303	−	0.103577i
$302$		0			0
$303$		−	1941.13i		−	0.368036i
$304$		0			0
$305$	−3003.53			−0.563875
$306$		0			0
$307$	−8249.52			−1.53363			−0.766816	−	0.641868i	$-0.778160\pi$
$307$	−8249.52			−1.53363			−0.766816	+	0.641868i	$0.778160\pi$
$308$		0			0
$309$	1033.48			0.190267
$310$		0			0
$311$	2780.95			0.507052			0.253526	−	0.967329i	$-0.418410\pi$
$311$	2780.95			0.507052			0.253526	+	0.967329i	$0.418410\pi$
$312$		0			0
$313$		−	8209.60i		−	1.48254i	−0.671209	−	0.741268i	$-0.734225\pi$
$313$		−	8209.60i		−	1.48254i	0.671209	−	0.741268i	$-0.265775\pi$
$314$		0			0
$315$	2099.67	−	291.018i	0.375566	−	0.0520540i
$316$		0			0
$317$	1813.06			0.321235			0.160617	−	0.987017i	$-0.448652\pi$
$317$	1813.06			0.321235			0.160617	+	0.987017i	$0.448652\pi$
$318$		0			0
$319$		−	104.646i		−	0.0183670i
$320$		0			0
$321$			3997.50i			0.695074i
$322$		0			0
$323$		−	319.932i		−	0.0551129i
$324$		0			0
$325$		−	979.448i		−	0.167169i
$326$		0			0
$327$	−6691.20			−1.13157
$328$		0			0
$329$	−839.818	−	6059.21i	−0.140731	−	1.01537i
$330$		0			0
$331$		−	81.4819i		−	0.0135307i	−0.999977	−	0.00676534i	$-0.997847\pi$
$331$		−	81.4819i		−	0.0135307i	0.999977	−	0.00676534i	$-0.00215349\pi$
$332$		0			0
$333$	−554.573			−0.0912625
$334$		0			0
$335$	−8418.06			−1.37292
$336$		0			0
$337$	5750.09			0.929459			0.464729	−	0.885453i	$-0.346152\pi$
$337$	5750.09			0.929459			0.464729	+	0.885453i	$0.346152\pi$
$338$		0			0
$339$	−742.524			−0.118963
$340$		0			0
$341$		−	15.2663i		−	0.00242439i
$342$		0			0
$343$	2550.62	+	5817.90i	0.401517	+	0.915851i
$344$		0			0
$345$	−4586.52			−0.715739
$346$		0			0
$347$		−	2013.58i		−	0.311512i	−0.987796	−	0.155756i	$-0.950219\pi$
$347$		−	2013.58i		−	0.311512i	0.987796	−	0.155756i	$-0.0497814\pi$
$348$		0			0
$349$			7540.08i			1.15648i	0.815867	+	0.578239i	$0.196260\pi$
$349$			7540.08i			1.15648i	−0.815867	+	0.578239i	$0.803740\pi$
$350$		0			0
$351$		−	720.007i		−	0.109490i
$352$		0			0
$353$		−	3936.08i		−	0.593474i	−0.954959	−	0.296737i	$-0.904102\pi$
$353$		−	3936.08i		−	0.593474i	0.954959	−	0.296737i	$-0.0958985\pi$
$354$		0			0
$355$	−6959.58			−1.04050
$356$		0			0
$357$	585.366	−	81.1328i	0.0867811	−	0.0120280i
$358$		0			0
$359$			7300.80i			1.07332i	0.843799	+	0.536659i	$0.180314\pi$
$359$			7300.80i			1.07332i	−0.843799	+	0.536659i	$0.819686\pi$
$360$		0			0
$361$	−5954.24			−0.868092
$362$		0			0
$363$	−3975.39			−0.574803
$364$		0			0
$365$	−12310.6			−1.76539
$366$		0			0
$367$	−9457.31			−1.34514			−0.672572	−	0.740032i	$-0.734810\pi$
$367$	−9457.31			−1.34514			−0.672572	+	0.740032i	$0.734810\pi$
$368$		0			0
$369$		−	683.542i		−	0.0964330i
$370$		0			0
$371$	12.3983	+	89.4526i	0.00173500	+	0.0125179i
$372$		0			0
$373$	12084.4			1.67750			0.838750	−	0.544516i	$-0.183287\pi$
$373$	12084.4			1.67750			0.838750	+	0.544516i	$0.183287\pi$
$374$		0			0
$375$		−	3367.70i		−	0.463753i
$376$		0			0
$377$			1151.67i			0.157331i
$378$		0			0
$379$			642.992i			0.0871458i	0.999050	+	0.0435729i	$0.0138741\pi$
$379$			642.992i			0.0871458i	−0.999050	+	0.0435729i	$0.986126\pi$
$380$		0			0
$381$		−	1269.22i		−	0.170667i
$382$		0			0
$383$	6360.56			0.848589			0.424294	−	0.905524i	$-0.360522\pi$
$383$	6360.56			0.848589			0.424294	+	0.905524i	$0.360522\pi$
$384$		0			0
$385$	−78.3513	−	565.298i	−0.0103718	−	0.0748319i
$386$		0			0
$387$			1914.58i			0.251482i
$388$		0			0
$389$	−5469.46			−0.712886			−0.356443	−	0.934317i	$-0.616011\pi$
$389$	−5469.46			−0.712886			−0.356443	+	0.934317i	$0.616011\pi$
$390$		0			0
$391$	−1278.67			−0.165384
$392$		0			0
$393$	−6693.68			−0.859165
$394$		0			0
$395$	822.153			0.104727
$396$		0			0
$397$			10426.0i			1.31806i	0.752118	+	0.659028i	$0.229032\pi$
$397$			10426.0i			1.31806i	−0.752118	+	0.659028i	$0.770968\pi$
$398$		0			0
$399$	229.441	+	1655.40i	0.0287880	+	0.207703i
$400$		0			0
$401$	2207.60			0.274919			0.137459	−	0.990507i	$-0.456106\pi$
$401$	2207.60			0.274919			0.137459	+	0.990507i	$0.456106\pi$
$402$		0			0
$403$			168.011i			0.0207673i
$404$		0			0
$405$			1030.10i			0.126385i
$406$		0			0
$407$			149.309i			0.0181842i
$408$		0			0
$409$		−	14152.7i		−	1.71102i	−0.517790	−	0.855508i	$-0.673245\pi$
$409$		−	14152.7i		−	1.71102i	0.517790	−	0.855508i	$-0.326755\pi$
$410$		0			0
$411$	−3809.73			−0.457226
$412$		0			0
$413$	−1223.85	−	8829.98i	−0.145815	−	1.05205i
$414$		0			0
$415$			8460.56i			1.00075i
$416$		0			0
$417$	−5979.41			−0.702190
$418$		0			0
$419$	−7423.51			−0.865542			−0.432771	−	0.901504i	$-0.642464\pi$
$419$	−7423.51			−0.865542			−0.432771	+	0.901504i	$0.642464\pi$
$420$		0			0
$421$	2998.76			0.347151			0.173575	−	0.984821i	$-0.444468\pi$
$421$	2998.76			0.347151			0.173575	+	0.984821i	$0.444468\pi$
$422$		0			0
$423$	2972.65			0.341691
$424$		0			0
$425$			390.661i			0.0445879i
$426$		0			0
$427$	4332.65	−	600.513i	0.491034	−	0.0680582i
$428$		0			0
$429$	−193.849			−0.0218161
$430$		0			0
$431$		−	4484.82i		−	0.501221i	−0.968088	−	0.250610i	$-0.919369\pi$
$431$		−	4484.82i		−	0.501221i	0.968088	−	0.250610i	$-0.0806313\pi$
$432$		0			0
$433$			15808.8i			1.75456i	0.479979	+	0.877280i	$0.340644\pi$
$433$			15808.8i			1.75456i	−0.479979	+	0.877280i	$0.659356\pi$
$434$		0			0
$435$		−	1647.67i		−	0.181608i
$436$		0			0
$437$		−	3616.05i		−	0.395833i
$438$		0			0
$439$	10718.1			1.16525			0.582626	−	0.812740i	$-0.302025\pi$
$439$	10718.1			1.16525			0.582626	+	0.812740i	$0.302025\pi$
$440$		0			0
$441$	−2970.63	+	839.598i	−0.320768	+	0.0906596i
$442$		0			0
$443$		−	9253.69i		−	0.992452i	−0.868193	−	0.496226i	$-0.834719\pi$
$443$		−	9253.69i		−	0.992452i	0.868193	−	0.496226i	$-0.165281\pi$
$444$		0			0
$445$	−4885.73			−0.520462
$446$		0			0
$447$	6164.25			0.652257
$448$		0			0
$449$	3008.54			0.316218			0.158109	−	0.987422i	$-0.449460\pi$
$449$	3008.54			0.316218			0.158109	+	0.987422i	$0.449460\pi$
$450$		0			0
$451$	−184.031			−0.0192144
$452$		0			0
$453$		−	4866.97i		−	0.504791i
$454$		0			0
$455$	862.285	+	6221.31i	0.0888451	+	0.641010i
$456$		0			0
$457$	1784.99			0.182710			0.0913548	−	0.995818i	$-0.470880\pi$
$457$	1784.99			0.182710			0.0913548	+	0.995818i	$0.470880\pi$
$458$		0			0
$459$			287.181i			0.0292036i
$460$		0			0
$461$			6144.65i			0.620791i	0.950607	+	0.310396i	$0.100462\pi$
$461$			6144.65i			0.620791i	−0.950607	+	0.310396i	$0.899538\pi$
$462$		0			0
$463$		−	6052.20i		−	0.607494i	−0.952753	−	0.303747i	$-0.901762\pi$
$463$		−	6052.20i		−	0.607494i	0.952753	−	0.303747i	$-0.0982378\pi$
$464$		0			0
$465$		−	240.370i		−	0.0239718i
$466$		0			0
$467$	6026.63			0.597172			0.298586	−	0.954383i	$-0.403485\pi$
$467$	6026.63			0.597172			0.298586	+	0.954383i	$0.403485\pi$
$468$		0			0
$469$	12143.2	−	1683.07i	1.19557	−	0.165708i
$470$		0			0
$471$			2083.54i			0.203831i
$472$		0			0
$473$	515.466			0.0501081
$474$		0			0
$475$	−1104.78			−0.106717
$476$		0			0
$477$	−43.8854			−0.00421253
$478$		0			0
$479$	18418.6			1.75693			0.878465	−	0.477807i	$-0.158568\pi$
$479$	18418.6			1.75693			0.878465	+	0.477807i	$0.158568\pi$
$480$		0			0
$481$		−	1643.20i		−	0.155766i
$482$		0			0
$483$	6616.14	−	917.008i	0.623281	−	0.0863878i
$484$		0			0
$485$	−15592.5			−1.45984
$486$		0			0
$487$		−	9759.61i		−	0.908111i	−0.890973	−	0.454055i	$-0.849977\pi$
$487$		−	9759.61i		−	0.908111i	0.890973	−	0.454055i	$-0.150023\pi$
$488$		0			0
$489$			1519.53i			0.140522i
$490$		0			0
$491$		−	6108.89i		−	0.561487i	−0.959783	−	0.280744i	$-0.909419\pi$
$491$		−	6108.89i		−	0.561487i	0.959783	−	0.280744i	$-0.0905811\pi$
$492$		0			0
$493$		−	459.352i		−	0.0419639i
$494$		0			0
$495$	277.335			0.0251824
$496$		0			0
$497$	10039.3	−	1391.47i	0.906086	−	0.125585i
$498$		0			0
$499$			3588.93i			0.321969i	0.986957	+	0.160985i	$0.0514669\pi$
$499$			3588.93i			0.321969i	−0.986957	+	0.160985i	$0.948533\pi$
$500$		0			0
$501$	11067.6			0.986957
$502$		0			0
$503$	−6279.88			−0.556672			−0.278336	−	0.960484i	$-0.589783\pi$
$503$	−6279.88			−0.556672			−0.278336	+	0.960484i	$0.589783\pi$
$504$		0			0
$505$	−8228.62			−0.725087
$506$		0			0
$507$	−4457.63			−0.390474
$508$		0			0
$509$			19976.3i			1.73956i	0.493441	+	0.869779i	$0.335739\pi$
$509$			19976.3i			1.73956i	−0.493441	+	0.869779i	$0.664261\pi$
$510$		0			0
$511$	17758.3	−	2461.34i	1.53734	−	0.213078i
$512$		0			0
$513$	−812.138			−0.0698963
$514$		0			0
$515$		−	4381.02i		−	0.374856i
$516$		0			0
$517$		−	800.331i		−	0.0680823i
$518$		0			0
$519$		−	7679.51i		−	0.649505i
$520$		0			0
$521$			888.540i			0.0747172i	0.999302	+	0.0373586i	$0.0118944\pi$
$521$			888.540i			0.0747172i	−0.999302	+	0.0373586i	$0.988106\pi$
$522$		0			0
$523$	−13103.6			−1.09556			−0.547781	−	0.836622i	$-0.684527\pi$
$523$	−13103.6			−1.09556			−0.547781	+	0.836622i	$0.684527\pi$
$524$		0			0
$525$	−280.165	−	2021.37i	−0.0232903	−	0.168037i
$526$		0			0
$527$		−	67.0126i		−	0.00553912i
$528$		0			0
$529$	−2285.28			−0.187826
$530$		0			0
$531$	4331.99			0.354034
$532$		0			0
$533$	2025.33			0.164591
$534$		0			0
$535$	16945.8			1.36940
$536$		0			0
$537$		−	4288.82i		−	0.344649i
$538$		0			0
$539$	226.046	+	799.788i	0.0180640	+	0.0639133i
$540$		0			0
$541$	7441.18			0.591352			0.295676	−	0.955288i	$-0.404455\pi$
$541$	7441.18			0.591352			0.295676	+	0.955288i	$0.404455\pi$
$542$		0			0
$543$		−	2945.94i		−	0.232822i
$544$		0			0
$545$			28364.6i			2.22937i
$546$		0			0
$547$		−	1330.37i		−	0.103990i	−0.998647	−	0.0519949i	$-0.983442\pi$
$547$		−	1330.37i		−	0.103990i	0.998647	−	0.0519949i	$-0.0165580\pi$
$548$		0			0
$549$			2125.60i			0.165243i
$550$		0			0
$551$	1299.03			0.100437
$552$		0			0
$553$	−1185.97	+	164.378i	−0.0911981	+	0.0126402i
$554$		0			0
$555$			2350.89i			0.179801i
$556$		0			0
$557$	−12680.7			−0.964630			−0.482315	−	0.875998i	$-0.660204\pi$
$557$	−12680.7			−0.964630			−0.482315	+	0.875998i	$0.660204\pi$
$558$		0			0
$559$	−5672.88			−0.429226
$560$		0			0
$561$	77.3181			0.00581885
$562$		0			0
$563$	−1318.32			−0.0986864			−0.0493432	−	0.998782i	$-0.515713\pi$
$563$	−1318.32			−0.0986864			−0.0493432	+	0.998782i	$0.515713\pi$
$564$		0			0
$565$			3147.63i			0.234375i
$566$		0			0
$567$	−205.953	−	1485.94i	−0.0152544	−	0.110059i
$568$		0			0
$569$	10877.6			0.801430			0.400715	−	0.916203i	$-0.368762\pi$
$569$	10877.6			0.801430			0.400715	+	0.916203i	$0.368762\pi$
$570$		0			0
$571$			22316.3i			1.63557i	0.575525	+	0.817784i	$0.304798\pi$
$571$			22316.3i			1.63557i	−0.575525	+	0.817784i	$0.695202\pi$
$572$		0			0
$573$			9963.74i			0.726424i
$574$		0			0
$575$			4415.47i			0.320240i
$576$		0			0
$577$			8864.03i			0.639540i	0.947495	+	0.319770i	$0.103606\pi$
$577$			8864.03i			0.639540i	−0.947495	+	0.319770i	$0.896394\pi$
$578$		0			0
$579$	521.694			0.0374454
$580$		0			0
$581$	−1691.57	−	12204.5i	−0.120788	−	0.871477i
$582$		0			0
$583$			11.8153i			0.000839351i
$584$		0			0
$585$	−3052.17			−0.215713
$586$		0			0
$587$	−22108.0			−1.55450			−0.777251	−	0.629190i	$-0.783387\pi$
$587$	−22108.0			−1.55450			−0.777251	+	0.629190i	$0.783387\pi$
$588$		0			0
$589$	189.510			0.0132574
$590$		0			0
$591$	16151.9			1.12420
$592$		0			0
$593$			15594.6i			1.07992i	0.841690	+	0.539961i	$0.181561\pi$
$593$			15594.6i			1.07992i	−0.841690	+	0.539961i	$0.818439\pi$
$594$		0			0
$595$	−343.929	−	2481.42i	−0.0236970	−	0.170972i
$596$		0			0
$597$	5085.87			0.348661
$598$		0			0
$599$		−	9127.45i		−	0.622600i	−0.950312	−	0.311300i	$-0.899236\pi$
$599$		−	9127.45i		−	0.622600i	0.950312	−	0.311300i	$-0.100764\pi$
$600$		0			0
$601$			16498.5i			1.11978i	0.828567	+	0.559889i	$0.189156\pi$
$601$			16498.5i			1.11978i	−0.828567	+	0.559889i	$0.810844\pi$
$602$		0			0
$603$			5957.45i			0.402332i
$604$		0			0
$605$			16852.0i			1.13245i
$606$		0			0
$607$	16019.7			1.07120			0.535601	−	0.844471i	$-0.320085\pi$
$607$	16019.7			1.07120			0.535601	+	0.844471i	$0.320085\pi$
$608$		0			0
$609$	329.427	+	2376.79i	0.0219197	+	0.158148i
$610$		0			0
$611$			8807.93i			0.583193i
$612$		0			0
$613$	−16964.5			−1.11776			−0.558882	−	0.829247i	$-0.688769\pi$
$613$	−16964.5			−1.11776			−0.558882	+	0.829247i	$0.688769\pi$
$614$		0			0
$615$	−2897.60			−0.189988
$616$		0			0
$617$	9609.72			0.627023			0.313511	−	0.949584i	$-0.398495\pi$
$617$	9609.72			0.627023			0.313511	+	0.949584i	$0.398495\pi$
$618$		0			0
$619$	20764.8			1.34832			0.674158	−	0.738587i	$-0.264507\pi$
$619$	20764.8			1.34832			0.674158	+	0.738587i	$0.264507\pi$
$620$		0			0
$621$			3245.88i			0.209746i
$622$		0			0
$623$	7047.75	−	976.831i	0.453230	−	0.0628185i
$624$		0			0
$625$	−18867.1			−1.20749
$626$		0			0
$627$			218.653i			0.0139269i
$628$		0			0
$629$			655.402i			0.0415462i
$630$		0			0
$631$		−	2283.73i		−	0.144079i	−0.997402	−	0.0720396i	$-0.977049\pi$
$631$		−	2283.73i		−	0.144079i	0.997402	−	0.0720396i	$-0.0229508\pi$
$632$		0			0
$633$		−	14206.8i		−	0.892053i
$634$		0			0
$635$	−5380.35			−0.336241
$636$		0			0
$637$	−2487.72	−	8801.95i	−0.154736	−	0.547482i
$638$		0			0
$639$			4925.29i			0.304916i
$640$		0			0
$641$	25207.0			1.55322			0.776611	−	0.629980i	$-0.216937\pi$
$641$	25207.0			1.55322			0.776611	+	0.629980i	$0.216937\pi$
$642$		0			0
$643$	−2293.63			−0.140672			−0.0703358	−	0.997523i	$-0.522407\pi$
$643$	−2293.63			−0.140672			−0.0703358	+	0.997523i	$0.522407\pi$
$644$		0			0
$645$	8116.08			0.495458
$646$		0			0
$647$	25545.0			1.55221			0.776103	−	0.630606i	$-0.217194\pi$
$647$	25545.0			1.55221			0.776103	+	0.630606i	$0.217194\pi$
$648$		0			0
$649$		−	1166.31i		−	0.0705418i
$650$		0			0
$651$	48.0585	+	346.738i	0.00289334	+	0.0208752i
$652$		0			0
$653$	−25797.1			−1.54597			−0.772986	−	0.634423i	$-0.781238\pi$
$653$	−25797.1			−1.54597			−0.772986	+	0.634423i	$0.781238\pi$
$654$		0			0
$655$			28375.1i			1.69268i
$656$		0			0
$657$			8712.23i			0.517346i
$658$		0			0
$659$			4912.12i			0.290363i	0.989405	+	0.145181i	$0.0463766\pi$
$659$			4912.12i			0.290363i	−0.989405	+	0.145181i	$0.953623\pi$
$660$		0			0
$661$		−	21077.8i		−	1.24029i	−0.784488	−	0.620144i	$-0.787074\pi$
$661$		−	21077.8i		−	1.24029i	0.784488	−	0.620144i	$-0.212926\pi$
$662$		0			0
$663$	−850.913			−0.0498442
$664$		0			0
$665$	7017.38	−	972.622i	0.409207	−	0.0567168i
$666$		0			0
$667$		−	5191.86i		−	0.301394i
$668$		0			0
$669$	10322.7			0.596560
$670$		0			0
$671$	572.278			0.0329248
$672$		0			0
$673$	−12861.0			−0.736634			−0.368317	−	0.929700i	$-0.620066\pi$
$673$	−12861.0			−0.736634			−0.368317	+	0.929700i	$0.620066\pi$
$674$		0			0
$675$	991.682			0.0565480
$676$		0			0
$677$			18883.1i			1.07199i	0.844222	+	0.535994i	$0.180063\pi$
$677$			18883.1i			1.07199i	−0.844222	+	0.535994i	$0.819937\pi$
$678$		0			0
$679$	22492.5	−	3117.50i	1.27126	−	0.176199i
$680$		0			0
$681$	−2848.07			−0.160262
$682$		0			0
$683$			15053.4i			0.843343i	0.906749	+	0.421671i	$0.138556\pi$
$683$			15053.4i			0.843343i	−0.906749	+	0.421671i	$0.861444\pi$
$684$		0			0
$685$			16149.8i			0.900805i
$686$		0			0
$687$			6910.79i			0.383789i
$688$		0			0
$689$		−	130.032i		−	0.00718988i
$690$		0			0
$691$	−18526.0			−1.01992			−0.509958	−	0.860199i	$-0.670339\pi$
$691$	−18526.0			−1.01992			−0.509958	+	0.860199i	$0.670339\pi$
$692$		0			0
$693$	−400.061	+	55.4492i	−0.0219294	+	0.00303945i
$694$		0			0
$695$			25347.3i			1.38342i
$696$		0			0
$697$	−807.819			−0.0439000
$698$		0			0
$699$	−9804.64			−0.530537
$700$		0			0
$701$	14508.6			0.781714			0.390857	−	0.920451i	$-0.372179\pi$
$701$	14508.6			0.781714			0.390857	+	0.920451i	$0.372179\pi$
$702$		0			0
$703$	−1853.46			−0.0994373
$704$		0			0
$705$		−	12601.3i		−	0.673182i
$706$		0			0
$707$	11869.9	−	1645.19i	0.631421	−	0.0875161i
$708$		0			0
$709$	−13290.8			−0.704015			−0.352008	−	0.935997i	$-0.614501\pi$
$709$	−13290.8			−0.704015			−0.352008	+	0.935997i	$0.614501\pi$
$710$		0			0
$711$		−	581.837i		−	0.0306900i
$712$		0			0
$713$		−	757.414i		−	0.0397832i
$714$		0			0
$715$			821.742i			0.0429810i
$716$		0			0
$717$			9348.38i			0.486920i
$718$		0			0
$719$	8693.96			0.450946			0.225473	−	0.974249i	$-0.427607\pi$
$719$	8693.96			0.450946			0.225473	+	0.974249i	$0.427607\pi$
$720$		0			0
$721$	875.921	+	6319.69i	0.0452441	+	0.326432i
$722$		0			0
$723$			6872.86i			0.353533i
$724$		0			0
$725$	−1586.22			−0.0812562
$726$		0			0
$727$	−7864.03			−0.401184			−0.200592	−	0.979675i	$-0.564287\pi$
$727$	−7864.03			−0.401184			−0.200592	+	0.979675i	$0.564287\pi$
$728$		0			0
$729$	729.000			0.0370370
$730$		0			0
$731$	2262.68			0.114484
$732$		0			0
$733$			15026.2i			0.757170i	0.925566	+	0.378585i	$0.123589\pi$
$733$			15026.2i			0.757170i	−0.925566	+	0.378585i	$0.876411\pi$
$734$		0			0
$735$	3559.13	+	12592.8i	0.178613	+	0.631961i
$736$		0			0
$737$	1603.93			0.0801651
$738$		0			0
$739$		−	21552.5i		−	1.07283i	−0.843954	−	0.536416i	$-0.819778\pi$
$739$		−	21552.5i		−	1.07283i	0.843954	−	0.536416i	$-0.180222\pi$
$740$		0			0
$741$		−	2406.36i		−	0.119298i
$742$		0			0
$743$		−	5867.60i		−	0.289719i	−0.989452	−	0.144860i	$-0.953727\pi$
$743$		−	5867.60i		−	0.289719i	0.989452	−	0.144860i	$-0.0462730\pi$
$744$		0			0
$745$		−	26130.8i		−	1.28504i
$746$		0			0
$747$	5987.53			0.293269
$748$		0			0
$749$	−24444.6	+	3388.06i	−1.19250	+	0.165283i
$750$		0			0
$751$		−	16037.5i		−	0.779252i	−0.920973	−	0.389626i	$-0.872604\pi$
$751$		−	16037.5i		−	0.779252i	0.920973	−	0.389626i	$-0.127396\pi$
$752$		0			0
$753$	6922.41			0.335016
$754$		0			0
$755$	−20631.5			−0.994514
$756$		0			0
$757$	20457.7			0.982230			0.491115	−	0.871095i	$-0.336590\pi$
$757$	20457.7			0.982230			0.491115	+	0.871095i	$0.336590\pi$
$758$		0			0
$759$	873.892			0.0417922
$760$		0			0
$761$		−	35072.4i		−	1.67066i	−0.549746	−	0.835332i	$-0.685276\pi$
$761$		−	35072.4i		−	1.67066i	0.549746	−	0.835332i	$-0.314724\pi$
$762$		0			0
$763$	−5671.09	−	40916.5i	−0.269079	−	1.94138i
$764$		0			0
$765$	1217.38			0.0575354
$766$		0			0
$767$			12835.6i			0.604261i
$768$		0			0
$769$			25720.7i			1.20613i	0.797692	+	0.603065i	$0.206054\pi$
$769$			25720.7i			1.20613i	−0.797692	+	0.603065i	$0.793946\pi$
$770$		0			0
$771$		−	8496.92i		−	0.396899i
$772$		0			0
$773$		−	38851.8i		−	1.80776i	−0.427783	−	0.903881i	$-0.640705\pi$
$773$		−	38851.8i		−	1.80776i	0.427783	−	0.903881i	$-0.359295\pi$
$774$		0			0
$775$	−231.406			−0.0107256
$776$		0			0
$777$	−470.026	−	3391.19i	−0.0217015	−	0.156575i
$778$		0			0
$779$		−	2284.49i		−	0.105071i
$780$		0			0
$781$	1326.04			0.0607549
$782$		0			0
$783$	−1166.05			−0.0532201
$784$		0			0
$785$	8832.30			0.401578
$786$		0			0
$787$	15643.8			0.708567			0.354284	−	0.935138i	$-0.384725\pi$
$787$	15643.8			0.708567			0.354284	+	0.935138i	$0.384725\pi$
$788$		0			0
$789$			19764.4i			0.891802i
$790$		0			0
$791$	−629.322	−	4540.51i	−0.0282884	−	0.204098i
$792$		0			0
$793$	−6298.13			−0.282034
$794$		0			0
$795$			186.034i			0.00829931i
$796$		0			0
$797$			2473.22i			0.109920i	0.998489	+	0.0549599i	$0.0175031\pi$
$797$			2473.22i			0.109920i	−0.998489	+	0.0549599i	$0.982497\pi$
$798$		0			0
$799$		−	3513.12i		−	0.155551i
$800$		0			0
$801$			3457.63i			0.152521i
$802$		0			0
$803$	2345.61			0.103082
$804$		0			0
$805$	−3887.28	−	28046.4i	−0.170197	−	1.22796i
$806$		0			0
$807$			11233.2i			0.489996i
$808$		0			0
$809$	38282.3			1.66370			0.831850	−	0.555001i	$-0.187282\pi$
$809$	38282.3			1.66370			0.831850	+	0.555001i	$0.187282\pi$
$810$		0			0
$811$	−24461.8			−1.05915			−0.529574	−	0.848264i	$-0.677648\pi$
$811$	−24461.8			−1.05915			−0.529574	+	0.848264i	$0.677648\pi$
$812$		0			0
$813$	22515.5			0.971284
$814$		0			0
$815$	6441.41			0.276850
$816$		0			0
$817$			6398.78i			0.274009i
$818$		0			0
$819$	4402.82	−	610.238i	0.187847	−	0.0260359i
$820$		0			0
$821$	3528.10			0.149977			0.0749887	−	0.997184i	$-0.476108\pi$
$821$	3528.10			0.149977			0.0749887	+	0.997184i	$0.476108\pi$
$822$		0			0
$823$		−	19151.7i		−	0.811162i	−0.914059	−	0.405581i	$-0.867069\pi$
$823$		−	19151.7i		−	0.811162i	0.914059	−	0.405581i	$-0.132931\pi$
$824$		0			0
$825$		−	266.992i		−	0.0112672i
$826$		0			0
$827$		−	4356.94i		−	0.183199i	−0.995796	−	0.0915996i	$-0.970802\pi$
$827$		−	4356.94i		−	0.183199i	0.995796	−	0.0915996i	$-0.0291980\pi$
$828$		0			0
$829$			16788.9i			0.703382i	0.936116	+	0.351691i	$0.114393\pi$
$829$			16788.9i			0.703382i	−0.936116	+	0.351691i	$0.885607\pi$
$830$		0			0
$831$	−22501.9			−0.939329
$832$		0			0
$833$	992.248	+	3510.73i	0.0412717	+	0.146026i
$834$		0			0
$835$		−	46916.7i		−	1.94445i
$836$		0			0
$837$	−170.110			−0.00702491
$838$		0			0
$839$	−7922.27			−0.325992			−0.162996	−	0.986627i	$-0.552116\pi$
$839$	−7922.27			−0.325992			−0.162996	+	0.986627i	$0.552116\pi$
$840$		0			0
$841$	−22523.9			−0.923526
$842$		0			0
$843$	−17412.0			−0.711387
$844$		0			0
$845$			18896.3i			0.769292i
$846$		0			0
$847$	−3369.32	−	24309.3i	−0.136684	−	0.986162i
$848$		0			0
$849$	−17746.7			−0.717389
$850$		0			0
$851$			7407.72i			0.298394i
$852$		0			0
$853$		−	20448.3i		−	0.820792i	−0.911907	−	0.410396i	$-0.865390\pi$
$853$		−	20448.3i		−	0.820792i	0.911907	−	0.410396i	$-0.134610\pi$
$854$		0			0
$855$			3442.73i			0.137706i
$856$		0			0
$857$		−	41194.3i		−	1.64197i	−0.570948	−	0.820986i	$-0.693424\pi$
$857$		−	41194.3i		−	1.64197i	0.570948	−	0.820986i	$-0.306576\pi$
$858$		0			0
$859$	−13867.8			−0.550830			−0.275415	−	0.961325i	$-0.588815\pi$
$859$	−13867.8			−0.550830			−0.275415	+	0.961325i	$0.588815\pi$
$860$		0			0
$861$	4179.84	−	579.333i	0.165445	−	0.0229310i
$862$		0			0
$863$		−	13175.0i		−	0.519678i	−0.965652	−	0.259839i	$-0.916331\pi$
$863$		−	13175.0i		−	0.519678i	0.965652	−	0.259839i	$-0.0836694\pi$
$864$		0			0
$865$	−32554.1			−1.27962
$866$		0			0
$867$	−14399.6			−0.564056
$868$		0			0
$869$	−156.649			−0.00611502
$870$		0			0
$871$	−17651.9			−0.686695
$872$		0			0
$873$			11034.8i			0.427803i
$874$		0			0
$875$	20593.4	−	2854.28i	0.795637	−	0.110277i
$876$		0			0
$877$	3551.84			0.136759			0.0683793	−	0.997659i	$-0.478217\pi$
$877$	3551.84			0.136759			0.0683793	+	0.997659i	$0.478217\pi$
$878$		0			0
$879$			20880.8i			0.801242i
$880$		0			0
$881$			34701.2i			1.32703i	0.748164	+	0.663514i	$0.230936\pi$
$881$			34701.2i			1.32703i	−0.748164	+	0.663514i	$0.769064\pi$
$882$		0			0
$883$		−	12801.7i		−	0.487895i	−0.969788	−	0.243947i	$-0.921558\pi$
$883$		−	12801.7i		−	0.487895i	0.969788	−	0.243947i	$-0.0784425\pi$
$884$		0			0
$885$		−	18363.7i		−	0.697501i
$886$		0			0
$887$	−23073.8			−0.873443			−0.436721	−	0.899597i	$-0.643860\pi$
$887$	−23073.8			−0.873443			−0.436721	+	0.899597i	$0.643860\pi$
$888$		0			0
$889$	7761.26	−	1075.72i	0.292806	−	0.0405834i
$890$		0			0
$891$		−	196.270i		−	0.00737967i
$892$		0			0
$893$	9934.99			0.372298
$894$		0			0
$895$	−18180.7			−0.679010
$896$		0			0
$897$	−9617.50			−0.357992
$898$		0			0
$899$	272.095			0.0100944
$900$		0			0
$901$			51.8643i			0.00191770i
$902$		0			0
$903$	−11707.6	+	1622.69i	−0.431455	+	0.0598005i
$904$		0			0
$905$	−12488.1			−0.458695
$906$		0			0
$907$			25106.5i			0.919126i	0.888145	+	0.459563i	$0.151994\pi$
$907$			25106.5i			0.919126i	−0.888145	+	0.459563i	$0.848006\pi$
$908$		0			0
$909$			5823.39i			0.212486i
$910$		0			0
$911$			16256.1i			0.591205i	0.955311	+	0.295602i	$0.0955204\pi$
$911$			16256.1i			0.591205i	−0.955311	+	0.295602i	$0.904480\pi$
$912$		0			0
$913$		−	1612.03i		−	0.0584343i
$914$		0			0
$915$	9010.60			0.325553
$916$		0			0
$917$	−5673.20	−	40931.6i	−0.204303	−	1.47403i
$918$		0			0
$919$		−	27941.1i		−	1.00293i	−0.865178	−	0.501465i	$-0.832795\pi$
$919$		−	27941.1i		−	1.00293i	0.865178	−	0.501465i	$-0.167205\pi$
$920$		0			0
$921$	24748.5			0.885442
$922$		0			0
$923$	−14593.6			−0.520426
$924$		0			0
$925$	2263.21			0.0804475
$926$		0			0
$927$	−3100.44			−0.109851
$928$		0			0
$929$		−	35855.2i		−	1.26628i	−0.774039	−	0.633138i	$-0.781766\pi$
$929$		−	35855.2i		−	1.26628i	0.774039	−	0.633138i	$-0.218234\pi$
$930$		0			0
$931$	−9928.24	+	2806.05i	−0.349500	+	0.0987804i
$932$		0			0
$933$	−8342.85			−0.292747
$934$		0			0
$935$		−	327.758i		−	0.0114640i
$936$		0			0
$937$			7058.56i			0.246097i	0.992401	+	0.123049i	$0.0392671\pi$
$937$			7058.56i			0.246097i	−0.992401	+	0.123049i	$0.960733\pi$
$938$		0			0
$939$			24628.8i			0.855943i
$940$		0			0
$941$		−	18308.8i		−	0.634270i	−0.948380	−	0.317135i	$-0.897279\pi$
$941$		−	18308.8i		−	0.634270i	0.948380	−	0.317135i	$-0.102721\pi$
$942$		0			0
$943$	−9130.43			−0.315300
$944$		0			0
$945$	−6299.02	+	873.055i	−0.216833	+	0.0300534i
$946$		0			0
$947$			52972.7i			1.81772i	0.417099	+	0.908861i	$0.363047\pi$
$947$			52972.7i			1.81772i	−0.417099	+	0.908861i	$0.636953\pi$
$948$		0			0
$949$	−25814.3			−0.883000
$950$		0			0
$951$	−5439.17			−0.185465
$952$		0			0
$953$	−456.374			−0.0155125			−0.00775625	−	0.999970i	$-0.502469\pi$
$953$	−456.374			−0.0155125			−0.00775625	+	0.999970i	$0.502469\pi$
$954$		0			0
$955$	42237.2			1.43117
$956$		0			0
$957$			313.938i			0.0106042i
$958$		0			0
$959$	−3228.91	−	23296.3i	−0.108725	−	0.784440i
$960$		0			0
$961$	−29751.3			−0.998668
$962$		0			0
$963$		−	11992.5i		−	0.401301i
$964$		0			0
$965$		−	2211.51i		−	0.0737730i
$966$		0			0
$967$		−	29461.9i		−	0.979762i	−0.871789	−	0.489881i	$-0.837040\pi$
$967$		−	29461.9i		−	0.979762i	0.871789	−	0.489881i	$-0.162960\pi$
$968$		0			0
$969$			959.795i			0.0318195i
$970$		0			0
$971$	−14529.2			−0.480190			−0.240095	−	0.970749i	$-0.577179\pi$
$971$	−14529.2			−0.480190			−0.240095	+	0.970749i	$0.577179\pi$
$972$		0			0
$973$	−5067.82	−	36563.9i	−0.166975	−	1.20471i
$974$		0			0
$975$			2938.35i			0.0965152i
$976$		0			0
$977$	8595.02			0.281452			0.140726	−	0.990049i	$-0.455056\pi$
$977$	8595.02			0.281452			0.140726	+	0.990049i	$0.455056\pi$
$978$		0			0
$979$	930.902			0.0303900
$980$		0			0
$981$	20073.6			0.653314
$982$		0			0
$983$	−16799.7			−0.545094			−0.272547	−	0.962142i	$-0.587866\pi$
$983$	−16799.7			−0.545094			−0.272547	+	0.962142i	$0.587866\pi$
$984$		0			0
$985$		−	68469.2i		−	2.21483i
$986$		0			0
$987$	2519.45	+	18177.6i	0.0812514	+	0.586222i
$988$		0			0
$989$	25574.0			0.822252
$990$		0			0
$991$			37597.1i			1.20516i	0.798059	+	0.602579i	$0.205860\pi$
$991$			37597.1i			1.20516i	−0.798059	+	0.602579i	$0.794140\pi$
$992$		0			0
$993$			244.446i			0.00781194i
$994$		0			0
$995$		−	21559.5i		−	0.686915i
$996$		0			0
$997$		−	23547.0i		−	0.747985i	−0.927432	−	0.373992i	$-0.877989\pi$
$997$		−	23547.0i		−	0.747985i	0.927432	−	0.373992i	$-0.122011\pi$
$998$		0			0
$999$	1663.72			0.0526904

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.4.b.a.223.19	yes	24
4.3	odd	2		672.4.b.b.223.19	yes	24
7.6	odd	2		672.4.b.b.223.6	yes	24
8.3	odd	2		1344.4.b.i.895.6		24
8.5	even	2		1344.4.b.j.895.6		24
28.27	even	2	inner	672.4.b.a.223.6	✓	24
56.13	odd	2		1344.4.b.i.895.19		24
56.27	even	2		1344.4.b.j.895.19		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.4.b.a.223.6	✓	24	28.27	even	2	inner
672.4.b.a.223.19	yes	24	1.1	even	1	trivial
672.4.b.b.223.6	yes	24	7.6	odd	2
672.4.b.b.223.19	yes	24	4.3	odd	2
1344.4.b.i.895.6		24	8.3	odd	2
1344.4.b.i.895.19		24	56.13	odd	2
1344.4.b.j.895.6		24	8.5	even	2
1344.4.b.j.895.19		24	56.27	even	2

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.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +12.7173i q^{5} +(-2.54263 - 18.3449i) q^{7} +9.00000 q^{9} -2.42308i q^{11} +26.6669i q^{13} -38.1518i q^{15} -10.6363i q^{17} +30.0792 q^{19} +(7.62790 + 55.0347i) q^{21} -120.218i q^{23} -36.7290 q^{25} -27.0000 q^{27} +43.1872 q^{29} +6.30036 q^{31} +7.26925i q^{33} +(233.297 - 32.3354i) q^{35} -61.6192 q^{37} -80.0008i q^{39} -75.9491i q^{41} +212.731i q^{43} +114.455i q^{45} +330.294 q^{47} +(-330.070 + 93.2887i) q^{49} +31.9089i q^{51} -4.87616 q^{53} +30.8150 q^{55} -90.2376 q^{57} +481.332 q^{59} +236.177i q^{61} +(-22.8837 - 165.104i) q^{63} -339.130 q^{65} +661.939i q^{67} +360.653i q^{69} +547.254i q^{71} +968.026i q^{73} +110.187 q^{75} +(-44.4512 + 6.16102i) q^{77} -64.6485i q^{79} +81.0000 q^{81} +665.281 q^{83} +135.265 q^{85} -129.561 q^{87} +384.181i q^{89} +(489.202 - 67.8042i) q^{91} -18.9011 q^{93} +382.525i q^{95} +1226.09i q^{97} -21.8078i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 72 q^{3} - 20 q^{7} + 216 q^{9} + 56 q^{19} + 60 q^{21} - 432 q^{25} - 648 q^{27} + 464 q^{31} - 568 q^{35} + 504 q^{37} + 560 q^{47} - 128 q^{49} - 784 q^{53} + 424 q^{55} - 168 q^{57} - 800 q^{59}+ \cdots - 1392 q^{93}+O(q^{100}) \) 24 * q - 72 * q^3 - 20 * q^7 + 216 * q^9 + 56 * q^19 + 60 * q^21 - 432 * q^25 - 648 * q^27 + 464 * q^31 - 568 * q^35 + 504 * q^37 + 560 * q^47 - 128 * q^49 - 784 * q^53 + 424 * q^55 - 168 * q^57 - 800 * q^59 - 180 * q^63 + 560 * q^65 + 1296 * q^75 - 1568 * q^77 + 1944 * q^81 - 1936 * q^83 + 3000 * q^85 + 496 * q^91 - 1392 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more