LMFDB - Embedded newform 672.4.i.c.209.19

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			209.19
Character	$\chi$	$=$	672.209
Dual form			672.4.i.c.209.17

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-4.05635 + 3.24747i) q^{3} -13.2397i q^{5} +(-18.4748 + 1.29729i) q^{7} +(5.90788 - 26.3457i) q^{9} +49.5250 q^{11} -28.7924 q^{13} +(42.9954 + 53.7046i) q^{15} +69.2560 q^{17} -24.7890 q^{19} +(70.7271 - 65.2585i) q^{21} -85.5006i q^{23} -50.2885 q^{25} +(61.5925 + 126.053i) q^{27} +111.594 q^{29} -236.812i q^{31} +(-200.891 + 160.831i) q^{33} +(17.1756 + 244.600i) q^{35} +261.434i q^{37} +(116.792 - 93.5026i) q^{39} -471.089 q^{41} +261.133i q^{43} +(-348.808 - 78.2183i) q^{45} -217.418 q^{47} +(339.634 - 47.9342i) q^{49} +(-280.926 + 224.907i) q^{51} +11.8711 q^{53} -655.694i q^{55} +(100.553 - 80.5016i) q^{57} -236.528i q^{59} -754.519 q^{61} +(-74.9688 + 494.395i) q^{63} +381.202i q^{65} -163.127i q^{67} +(277.661 + 346.820i) q^{69} -478.338i q^{71} -1038.37i q^{73} +(203.988 - 163.310i) q^{75} +(-914.963 + 64.2482i) q^{77} +148.084 q^{79} +(-659.194 - 311.295i) q^{81} +739.797i q^{83} -916.925i q^{85} +(-452.664 + 362.398i) q^{87} -1347.82 q^{89} +(531.934 - 37.3521i) q^{91} +(769.039 + 960.590i) q^{93} +328.198i q^{95} +183.115i q^{97} +(292.588 - 1304.77i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−4.05635	+	3.24747i	−0.780644	+	0.624976i
$3$	−4.05635	+	3.24747i	−0.780644	+	0.624976i
$4$		0			0
$5$		−	13.2397i		−	1.18419i	−0.805868	−	0.592095i	$-0.798301\pi$
$5$		−	13.2397i		−	1.18419i	0.805868	−	0.592095i	$-0.201699\pi$
$6$		0			0
$7$	−18.4748	+	1.29729i	−0.997544	+	0.0700470i
$7$	−18.4748	+	1.29729i	−0.997544	+	0.0700470i
$8$		0			0
$9$	5.90788	−	26.3457i	0.218810	−	0.975767i
$10$		0			0
$11$	49.5250			1.35749			0.678743	−	0.734376i	$-0.262525\pi$
$11$	49.5250			1.35749			0.678743	+	0.734376i	$0.262525\pi$
$12$		0			0
$13$	−28.7924			−0.614276			−0.307138	−	0.951665i	$-0.599371\pi$
$13$	−28.7924			−0.614276			−0.307138	+	0.951665i	$0.599371\pi$
$14$		0			0
$15$	42.9954	+	53.7046i	0.740091	+	0.924432i
$16$		0			0
$17$	69.2560			0.988061			0.494031	−	0.869445i	$-0.335523\pi$
$17$	69.2560			0.988061			0.494031	+	0.869445i	$0.335523\pi$
$18$		0			0
$19$	−24.7890			−0.299315			−0.149658	−	0.988738i	$-0.547817\pi$
$19$	−24.7890			−0.299315			−0.149658	+	0.988738i	$0.547817\pi$
$20$		0			0
$21$	70.7271	−	65.2585i	0.734949	−	0.678122i
$22$		0			0
$23$		−	85.5006i		−	0.775135i	−0.921841	−	0.387568i	$-0.873315\pi$
$23$		−	85.5006i		−	0.775135i	0.921841	−	0.387568i	$-0.126685\pi$
$24$		0			0
$25$	−50.2885			−0.402308
$26$		0			0
$27$	61.5925	+	126.053i	0.439018	+	0.898478i
$28$		0			0
$29$	111.594			0.714569			0.357284	−	0.933996i	$-0.383703\pi$
$29$	111.594			0.714569			0.357284	+	0.933996i	$0.383703\pi$
$30$		0			0
$31$		−	236.812i		−	1.37202i	−0.727592	−	0.686010i	$-0.759360\pi$
$31$		−	236.812i		−	1.37202i	0.727592	−	0.686010i	$-0.240640\pi$
$32$		0			0
$33$	−200.891	+	160.831i	−1.05971	+	0.848396i
$34$		0			0
$35$	17.1756	+	244.600i	0.0829490	+	1.18128i
$36$		0			0
$37$			261.434i			1.16161i	0.814044	+	0.580804i	$0.197262\pi$
$37$			261.434i			1.16161i	−0.814044	+	0.580804i	$0.802738\pi$
$38$		0			0
$39$	116.792	−	93.5026i	0.479531	−	0.383908i
$40$		0			0
$41$	−471.089			−1.79443			−0.897217	−	0.441590i	$-0.854415\pi$
$41$	−471.089			−1.79443			−0.897217	+	0.441590i	$0.854415\pi$
$42$		0			0
$43$			261.133i			0.926101i	0.886332	+	0.463051i	$0.153245\pi$
$43$			261.133i			0.926101i	−0.886332	+	0.463051i	$0.846755\pi$
$44$		0			0
$45$	−348.808	−	78.2183i	−1.15549	−	0.259113i
$46$		0			0
$47$	−217.418			−0.674757			−0.337379	−	0.941369i	$-0.609540\pi$
$47$	−217.418			−0.674757			−0.337379	+	0.941369i	$0.609540\pi$
$48$		0			0
$49$	339.634	−	47.9342i	0.990187	−	0.139750i
$50$		0			0
$51$	−280.926	+	224.907i	−0.771324	+	0.617514i
$52$		0			0
$53$	11.8711			0.0307665			0.0153832	−	0.999882i	$-0.495103\pi$
$53$	11.8711			0.0307665			0.0153832	+	0.999882i	$0.495103\pi$
$54$		0			0
$55$		−	655.694i		−	1.60752i
$56$		0			0
$57$	100.553	−	80.5016i	0.233659	−	0.187065i
$58$		0			0
$59$		−	236.528i		−	0.521921i	−0.965350	−	0.260960i	$-0.915961\pi$
$59$		−	236.528i		−	0.521921i	0.965350	−	0.260960i	$-0.0840392\pi$
$60$		0			0
$61$	−754.519			−1.58371			−0.791855	−	0.610709i	$-0.790884\pi$
$61$	−754.519			−1.58371			−0.791855	+	0.610709i	$0.790884\pi$
$62$		0			0
$63$	−74.9688	+	494.395i	−0.149923	+	0.988698i
$64$		0			0
$65$			381.202i			0.727420i
$66$		0			0
$67$		−	163.127i		−	0.297449i	−0.988879	−	0.148725i	$-0.952483\pi$
$67$		−	163.127i		−	0.297449i	0.988879	−	0.148725i	$-0.0475168\pi$
$68$		0			0
$69$	277.661	+	346.820i	0.484441	+	0.605105i
$70$		0			0
$71$		−	478.338i		−	0.799554i	−0.916613	−	0.399777i	$-0.869088\pi$
$71$		−	478.338i		−	0.799554i	0.916613	−	0.399777i	$-0.130912\pi$
$72$		0			0
$73$		−	1038.37i		−	1.66482i	−0.554157	−	0.832412i	$-0.686959\pi$
$73$		−	1038.37i		−	1.66482i	0.554157	−	0.832412i	$-0.313041\pi$
$74$		0			0
$75$	203.988	−	163.310i	0.314059	−	0.251433i
$76$		0			0
$77$	−914.963	+	64.2482i	−1.35415	+	0.0950878i
$78$		0			0
$79$	148.084			0.210895			0.105448	−	0.994425i	$-0.466372\pi$
$79$	148.084			0.210895			0.105448	+	0.994425i	$0.466372\pi$
$80$		0			0
$81$	−659.194	−	311.295i	−0.904244	−	0.427016i
$82$		0			0
$83$			739.797i			0.978353i	0.872185	+	0.489176i	$0.162703\pi$
$83$			739.797i			0.978353i	−0.872185	+	0.489176i	$0.837297\pi$
$84$		0			0
$85$		−	916.925i		−	1.17005i
$86$		0			0
$87$	−452.664	+	362.398i	−0.557824	+	0.446588i
$88$		0			0
$89$	−1347.82			−1.60527			−0.802635	−	0.596470i	$-0.796569\pi$
$89$	−1347.82			−1.60527			−0.802635	+	0.596470i	$0.796569\pi$
$90$		0			0
$91$	531.934	−	37.3521i	0.612767	−	0.0430282i
$92$		0			0
$93$	769.039	+	960.590i	0.857480	+	1.07106i
$94$		0			0
$95$			328.198i			0.354447i
$96$		0			0
$97$			183.115i			0.191675i	0.995397	+	0.0958375i	$0.0305529\pi$
$97$			183.115i			0.191675i	−0.995397	+	0.0958375i	$0.969447\pi$
$98$		0			0
$99$	292.588	−	1304.77i	0.297032	−	1.32459i
$100$		0			0
$101$			857.904i			0.845194i	0.906318	+	0.422597i	$0.138882\pi$
$101$			857.904i			0.845194i	−0.906318	+	0.422597i	$0.861118\pi$
$102$		0			0
$103$		−	493.737i		−	0.472324i	−0.971714	−	0.236162i	$-0.924110\pi$
$103$		−	493.737i		−	0.472324i	0.971714	−	0.236162i	$-0.0758896\pi$
$104$		0			0
$105$	−864.000	−	936.403i	−0.803026	−	0.870320i
$106$		0			0
$107$	−583.614			−0.527290			−0.263645	−	0.964620i	$-0.584925\pi$
$107$	−583.614			−0.527290			−0.263645	+	0.964620i	$0.584925\pi$
$108$		0			0
$109$		−	1856.22i		−	1.63113i	−0.578665	−	0.815565i	$-0.696426\pi$
$109$		−	1856.22i		−	1.63113i	0.578665	−	0.815565i	$-0.303574\pi$
$110$		0			0
$111$	−848.999	−	1060.47i	−0.725976	−	0.906802i
$112$		0			0
$113$			951.492i			0.792114i	0.918226	+	0.396057i	$0.129622\pi$
$113$			951.492i			0.792114i	−0.918226	+	0.396057i	$0.870378\pi$
$114$		0			0
$115$	−1132.00			−0.917908
$116$		0			0
$117$	−170.102	+	758.558i	−0.134410	+	0.599390i
$118$		0			0
$119$	−1279.49	+	89.8449i	−0.985634	+	0.0692107i
$120$		0			0
$121$	1121.73			0.842770
$122$		0			0
$123$	1910.90	−	1529.85i	1.40081	−	1.12148i
$124$		0			0
$125$		−	989.155i		−	0.707781i
$126$		0			0
$127$	−2550.61			−1.78213			−0.891065	−	0.453876i	$-0.850041\pi$
$127$	−2550.61			−1.78213			−0.891065	+	0.453876i	$0.850041\pi$
$128$		0			0
$129$	−848.020	−	1059.24i	−0.578791	−	0.722956i
$130$		0			0
$131$		−	439.670i		−	0.293238i	−0.989193	−	0.146619i	$-0.953161\pi$
$131$		−	439.670i		−	0.293238i	0.989193	−	0.146619i	$-0.0468391\pi$
$132$		0			0
$133$	457.972	−	32.1585i	0.298580	−	0.0209661i
$134$		0			0
$135$	1668.90	−	815.464i	1.06397	−	0.519881i
$136$		0			0
$137$		−	1126.88i		−	0.702745i	−0.936236	−	0.351373i	$-0.885715\pi$
$137$		−	1126.88i		−	0.702745i	0.936236	−	0.351373i	$-0.114285\pi$
$138$		0			0
$139$	−2419.55			−1.47643			−0.738215	−	0.674565i	$-0.764331\pi$
$139$	−2419.55			−1.47643			−0.738215	+	0.674565i	$0.764331\pi$
$140$		0			0
$141$	881.921	−	706.057i	0.526745	−	0.421707i
$142$		0			0
$143$	−1425.95			−0.833871
$144$		0			0
$145$		−	1477.47i		−	0.846186i
$146$		0			0
$147$	−1222.01	+	1297.39i	−0.685643	+	0.727938i
$148$		0			0
$149$	−2049.54			−1.12688			−0.563439	−	0.826158i	$-0.690522\pi$
$149$	−2049.54			−1.12688			−0.563439	+	0.826158i	$0.690522\pi$
$150$		0			0
$151$	−288.181			−0.155310			−0.0776551	−	0.996980i	$-0.524743\pi$
$151$	−288.181			−0.155310			−0.0776551	+	0.996980i	$0.524743\pi$
$152$		0			0
$153$	409.156	−	1824.60i	0.216198	−	0.964118i
$154$		0			0
$155$	−3135.31			−1.62473
$156$		0			0
$157$	2088.48			1.06165			0.530824	−	0.847482i	$-0.321883\pi$
$157$	2088.48			1.06165			0.530824	+	0.847482i	$0.321883\pi$
$158$		0			0
$159$	−48.1534	+	38.5511i	−0.0240177	+	0.0192283i
$160$		0			0
$161$	110.919	+	1579.60i	0.0542959	+	0.773231i
$162$		0			0
$163$			4111.75i			1.97581i	0.155057	+	0.987906i	$0.450444\pi$
$163$			4111.75i			1.97581i	−0.155057	+	0.987906i	$0.549556\pi$
$164$		0			0
$165$	2129.35	+	2659.72i	1.00466	+	1.25490i
$166$		0			0
$167$	−1224.94			−0.567598			−0.283799	−	0.958884i	$-0.591595\pi$
$167$	−1224.94			−0.567598			−0.283799	+	0.958884i	$0.591595\pi$
$168$		0			0
$169$	−1368.00			−0.622665
$170$		0			0
$171$	−146.451	+	653.085i	−0.0654934	+	0.292062i
$172$		0			0
$173$			2451.51i			1.07737i	0.842508	+	0.538685i	$0.181079\pi$
$173$			2451.51i			1.07737i	−0.842508	+	0.538685i	$0.818921\pi$
$174$		0			0
$175$	929.068	−	65.2386i	0.401320	−	0.0281804i
$176$		0			0
$177$	768.118	+	959.439i	0.326188	+	0.407434i
$178$		0			0
$179$	926.214			0.386751			0.193376	−	0.981125i	$-0.438056\pi$
$179$	926.214			0.386751			0.193376	+	0.981125i	$0.438056\pi$
$180$		0			0
$181$	876.720			0.360034			0.180017	−	0.983664i	$-0.442385\pi$
$181$	876.720			0.360034			0.180017	+	0.983664i	$0.442385\pi$
$182$		0			0
$183$	3060.59	−	2450.28i	1.23631	−	0.989780i
$184$		0			0
$185$	3461.29			1.37556
$186$		0			0
$187$	3429.90			1.34128
$188$		0			0
$189$	−1301.43	−	2248.90i	−0.500875	−	0.865520i
$190$		0			0
$191$			2476.56i			0.938208i	0.883143	+	0.469104i	$0.155423\pi$
$191$			2476.56i			0.938208i	−0.883143	+	0.469104i	$0.844577\pi$
$192$		0			0
$193$	3130.49			1.16755			0.583776	−	0.811915i	$-0.301575\pi$
$193$	3130.49			1.16755			0.583776	+	0.811915i	$0.301575\pi$
$194$		0			0
$195$	−1237.94	−	1546.29i	−0.454620	−	0.567856i
$196$		0			0
$197$	−85.4470			−0.0309028			−0.0154514	−	0.999881i	$-0.504919\pi$
$197$	−85.4470			−0.0309028			−0.0154514	+	0.999881i	$0.504919\pi$
$198$		0			0
$199$			3536.74i			1.25986i	0.776650	+	0.629932i	$0.216917\pi$
$199$			3536.74i			1.25986i	−0.776650	+	0.629932i	$0.783083\pi$
$200$		0			0
$201$	529.749	+	661.699i	0.185899	+	0.232202i
$202$		0			0
$203$	−2061.67	+	144.770i	−0.712814	+	0.0500534i
$204$		0			0
$205$			6237.06i			2.12495i
$206$		0			0
$207$	−2252.58	−	505.128i	−0.756352	−	0.169608i
$208$		0			0
$209$	−1227.68			−0.406317
$210$		0			0
$211$		−	1696.15i		−	0.553403i	−0.960956	−	0.276701i	$-0.910759\pi$
$211$		−	1696.15i		−	0.553403i	0.960956	−	0.276701i	$-0.0892413\pi$
$212$		0			0
$213$	1553.39	+	1940.31i	0.499702	+	0.624167i
$214$		0			0
$215$	3457.31			1.09668
$216$		0			0
$217$	307.213	+	4375.04i	0.0961059	+	1.36865i
$218$		0			0
$219$	3372.08	+	4211.99i	1.04047	+	1.29964i
$220$		0			0
$221$	−1994.05			−0.606942
$222$		0			0
$223$			3936.51i			1.18210i	0.806635	+	0.591050i	$0.201286\pi$
$223$			3936.51i			1.18210i	−0.806635	+	0.591050i	$0.798714\pi$
$224$		0			0
$225$	−297.099	+	1324.89i	−0.0880292	+	0.392559i
$226$		0			0
$227$		−	6229.85i		−	1.82154i	−0.412912	−	0.910771i	$-0.635488\pi$
$227$		−	6229.85i		−	1.82154i	0.412912	−	0.910771i	$-0.364512\pi$
$228$		0			0
$229$	−3973.83			−1.14672			−0.573359	−	0.819304i	$-0.694360\pi$
$229$	−3973.83			−1.14672			−0.573359	+	0.819304i	$0.694360\pi$
$230$		0			0
$231$	3502.76	−	3231.93i	0.997683	−	0.920542i
$232$		0			0
$233$		−	3401.07i		−	0.956273i	−0.878285	−	0.478137i	$-0.841312\pi$
$233$		−	3401.07i		−	0.956273i	0.878285	−	0.478137i	$-0.158688\pi$
$234$		0			0
$235$			2878.53i			0.799042i
$236$		0			0
$237$	−600.679	+	480.898i	−0.164634	+	0.131804i
$238$		0			0
$239$			4477.57i			1.21184i	0.795525	+	0.605920i	$0.207195\pi$
$239$			4477.57i			1.21184i	−0.795525	+	0.605920i	$0.792805\pi$
$240$		0			0
$241$			2312.70i			0.618151i	0.951037	+	0.309076i	$0.100020\pi$
$241$			2312.70i			0.618151i	−0.951037	+	0.309076i	$0.899980\pi$
$242$		0			0
$243$	3684.84	−	877.992i	0.972768	−	0.231783i
$244$		0			0
$245$	−634.632	−	4496.64i	−0.165490	−	1.17257i
$246$		0			0
$247$	713.737			0.183862
$248$		0			0
$249$	−2402.47	−	3000.87i	−0.611447	−	0.763745i
$250$		0			0
$251$			4493.21i			1.12992i	0.825119	+	0.564958i	$0.191108\pi$
$251$			4493.21i			1.12992i	−0.825119	+	0.564958i	$0.808892\pi$
$252$		0			0
$253$		−	4234.42i		−	1.05224i
$254$		0			0
$255$	2977.69	+	3719.37i	0.731255	+	0.913395i
$256$		0			0
$257$	−1431.96			−0.347562			−0.173781	−	0.984784i	$-0.555598\pi$
$257$	−1431.96			−0.347562			−0.173781	+	0.984784i	$0.555598\pi$
$258$		0			0
$259$	−339.155	−	4829.93i	−0.0813670	−	1.15875i
$260$		0			0
$261$	659.285	−	2940.03i	0.156355	−	0.697253i
$262$		0			0
$263$		−	902.626i		−	0.211629i	−0.994386	−	0.105814i	$-0.966255\pi$
$263$		−	902.626i		−	0.211629i	0.994386	−	0.105814i	$-0.0337449\pi$
$264$		0			0
$265$		−	157.170i		−	0.0364334i
$266$		0			0
$267$	5467.24	−	4377.02i	1.25314	−	1.00325i
$268$		0			0
$269$			1889.72i			0.428321i	0.976798	+	0.214160i	$0.0687015\pi$
$269$			1889.72i			0.428321i	−0.976798	+	0.214160i	$0.931298\pi$
$270$		0			0
$271$		−	7620.90i		−	1.70825i	−0.520065	−	0.854127i	$-0.674092\pi$
$271$		−	7620.90i		−	1.70825i	0.520065	−	0.854127i	$-0.325908\pi$
$272$		0			0
$273$	−2036.41	+	1878.95i	−0.451461	+	0.416554i
$274$		0			0
$275$	−2490.54			−0.546128
$276$		0			0
$277$		−	2983.80i		−	0.647218i	−0.946191	−	0.323609i	$-0.895104\pi$
$277$		−	2983.80i		−	0.647218i	0.946191	−	0.323609i	$-0.104896\pi$
$278$		0			0
$279$	−6238.98	−	1399.06i	−1.33877	−	0.300213i
$280$		0			0
$281$		−	2988.80i		−	0.634508i	−0.948341	−	0.317254i	$-0.897239\pi$
$281$		−	2988.80i		−	0.634508i	0.948341	−	0.317254i	$-0.102761\pi$
$282$		0			0
$283$	3150.99			0.661861			0.330930	−	0.943655i	$-0.392637\pi$
$283$	3150.99			0.661861			0.330930	+	0.943655i	$0.392637\pi$
$284$		0			0
$285$	−1065.81	−	1331.29i	−0.221521	−	0.276697i
$286$		0			0
$287$	8703.27	−	611.138i	1.79003	−	0.125695i
$288$		0			0
$289$	−116.610			−0.0237350
$290$		0			0
$291$	−594.659	−	742.776i	−0.119792	−	0.149630i
$292$		0			0
$293$		−	1635.98i		−	0.326194i	−0.986610	−	0.163097i	$-0.947852\pi$
$293$		−	1635.98i		−	0.326194i	0.986610	−	0.163097i	$-0.0521484\pi$
$294$		0			0
$295$	−3131.55			−0.618054
$296$		0			0
$297$	3050.37	+	6242.78i	0.595961	+	1.21967i
$298$		0			0
$299$			2461.77i			0.476147i
$300$		0			0
$301$	−338.764	−	4824.37i	−0.0648706	−	0.923827i
$302$		0			0
$303$	−2786.02	−	3479.96i	−0.528226	−	0.659796i
$304$		0			0
$305$			9989.58i			1.87541i
$306$		0			0
$307$	3788.51			0.704305			0.352152	−	0.935943i	$-0.385450\pi$
$307$	3788.51			0.704305			0.352152	+	0.935943i	$0.385450\pi$
$308$		0			0
$309$	1603.40	+	2002.77i	0.295191	+	0.368717i
$310$		0			0
$311$	4998.37			0.911356			0.455678	−	0.890145i	$-0.349397\pi$
$311$	4998.37			0.911356			0.455678	+	0.890145i	$0.349397\pi$
$312$		0			0
$313$			1152.65i			0.208152i	0.994569	+	0.104076i	$0.0331885\pi$
$313$			1152.65i			0.208152i	−0.994569	+	0.104076i	$0.966812\pi$
$314$		0			0
$315$	6545.62	+	992.561i	1.17081	+	0.177538i
$316$		0			0
$317$	2738.80			0.485256			0.242628	−	0.970119i	$-0.421991\pi$
$317$	2738.80			0.485256			0.242628	+	0.970119i	$0.421991\pi$
$318$		0			0
$319$	5526.70			0.970018
$320$		0			0
$321$	2367.34	−	1895.27i	0.411626	−	0.329544i
$322$		0			0
$323$	−1716.79			−0.295742
$324$		0			0
$325$	1447.93			0.247128
$326$		0			0
$327$	6028.01	+	7529.46i	1.01942	+	1.27333i
$328$		0			0
$329$	4016.74	−	282.053i	0.673100	−	0.0472647i
$330$		0			0
$331$		−	5222.20i		−	0.867184i	−0.901109	−	0.433592i	$-0.857246\pi$
$331$		−	5222.20i		−	0.867184i	0.901109	−	0.433592i	$-0.142754\pi$
$332$		0			0
$333$	6887.66	+	1544.52i	1.13346	+	0.254172i
$334$		0			0
$335$	−2159.74			−0.352237
$336$		0			0
$337$	−2898.62			−0.468540			−0.234270	−	0.972172i	$-0.575270\pi$
$337$	−2898.62			−0.468540			−0.234270	+	0.972172i	$0.575270\pi$
$338$		0			0
$339$	−3089.94	−	3859.58i	−0.495052	−	0.618359i
$340$		0			0
$341$		−	11728.1i		−	1.86250i
$342$		0			0
$343$	−6212.48	+	1326.18i	−0.977966	+	0.208766i
$344$		0			0
$345$	4591.78	−	3676.13i	0.716560	−	0.573670i
$346$		0			0
$347$	−10559.0			−1.63353			−0.816765	−	0.576970i	$-0.804235\pi$
$347$	−10559.0			−1.63353			−0.816765	+	0.576970i	$0.804235\pi$
$348$		0			0
$349$	4344.86			0.666404			0.333202	−	0.942855i	$-0.391871\pi$
$349$	4344.86			0.666404			0.333202	+	0.942855i	$0.391871\pi$
$350$		0			0
$351$	−1773.40	−	3629.37i	−0.269678	−	0.551914i
$352$		0			0
$353$	9819.24			1.48053			0.740263	−	0.672318i	$-0.234701\pi$
$353$	9819.24			1.48053			0.740263	+	0.672318i	$0.234701\pi$
$354$		0			0
$355$	−6333.03			−0.946824
$356$		0			0
$357$	4898.28	−	4519.54i	0.726175	−	0.670026i
$358$		0			0
$359$			728.937i			0.107164i	0.998563	+	0.0535820i	$0.0170638\pi$
$359$			728.937i			0.107164i	−0.998563	+	0.0535820i	$0.982936\pi$
$360$		0			0
$361$	−6244.50			−0.910410
$362$		0			0
$363$	−4550.11	+	3642.77i	−0.657903	+	0.526711i
$364$		0			0
$365$	−13747.7			−1.97147
$366$		0			0
$367$			8207.43i			1.16737i	0.811981	+	0.583685i	$0.198390\pi$
$367$			8207.43i			1.16737i	−0.811981	+	0.583685i	$0.801610\pi$
$368$		0			0
$369$	−2783.14	+	12411.2i	−0.392641	+	1.75095i
$370$		0			0
$371$	−219.316	+	15.4003i	−0.0306909	+	0.00215510i
$372$		0			0
$373$			6620.48i			0.919023i	0.888172	+	0.459511i	$0.151975\pi$
$373$			6620.48i			0.919023i	−0.888172	+	0.459511i	$0.848025\pi$
$374$		0			0
$375$	3212.25	+	4012.35i	0.442346	+	0.552525i
$376$		0			0
$377$	−3213.07			−0.438942
$378$		0			0
$379$			2808.73i			0.380672i	0.981719	+	0.190336i	$0.0609578\pi$
$379$			2808.73i			0.380672i	−0.981719	+	0.190336i	$0.939042\pi$
$380$		0			0
$381$	10346.2	−	8283.04i	1.39121	−	1.11379i
$382$		0			0
$383$	−2395.34			−0.319573			−0.159786	−	0.987152i	$-0.551081\pi$
$383$	−2395.34			−0.319573			−0.159786	+	0.987152i	$0.551081\pi$
$384$		0			0
$385$	850.624	+	12113.8i	0.112602	+	1.60357i
$386$		0			0
$387$	6879.73	+	1542.74i	0.903660	+	0.202641i
$388$		0			0
$389$	−12078.7			−1.57432			−0.787162	−	0.616746i	$-0.788450\pi$
$389$	−12078.7			−1.57432			−0.787162	+	0.616746i	$0.788450\pi$
$390$		0			0
$391$		−	5921.43i		−	0.765881i
$392$		0			0
$393$	1427.81	+	1783.45i	0.183266	+	0.228914i
$394$		0			0
$395$		−	1960.58i		−	0.249740i
$396$		0			0
$397$	−5240.22			−0.662466			−0.331233	−	0.943549i	$-0.607465\pi$
$397$	−5240.22			−0.662466			−0.331233	+	0.943549i	$0.607465\pi$
$398$		0			0
$399$	−1753.26	+	1617.69i	−0.219982	+	0.202973i
$400$		0			0
$401$			10649.8i			1.32625i	0.748509	+	0.663125i	$0.230770\pi$
$401$			10649.8i			1.32625i	−0.748509	+	0.663125i	$0.769230\pi$
$402$		0			0
$403$			6818.39i			0.842799i
$404$		0			0
$405$	−4121.44	+	8727.50i	−0.505669	+	1.07080i
$406$		0			0
$407$			12947.5i			1.57687i
$408$		0			0
$409$		−	10197.5i		−	1.23284i	−0.787416	−	0.616421i	$-0.788582\pi$
$409$		−	10197.5i		−	1.23284i	0.787416	−	0.616421i	$-0.211418\pi$
$410$		0			0
$411$	3659.52	+	4571.03i	0.439199	+	0.548594i
$412$		0			0
$413$	306.845	+	4369.80i	0.0365590	+	0.520639i
$414$		0			0
$415$	9794.66			1.15856
$416$		0			0
$417$	9814.54	−	7857.43i	1.15257	−	0.922733i
$418$		0			0
$419$		−	9402.75i		−	1.09631i	−0.836376	−	0.548155i	$-0.815330\pi$
$419$		−	9402.75i		−	1.09631i	0.836376	−	0.548155i	$-0.184670\pi$
$420$		0			0
$421$			1525.15i			0.176559i	0.996096	+	0.0882793i	$0.0281368\pi$
$421$			1525.15i			0.176559i	−0.996096	+	0.0882793i	$0.971863\pi$
$422$		0			0
$423$	−1284.48	+	5728.02i	−0.147644	+	0.658406i
$424$		0			0
$425$	−3482.78			−0.397505
$426$		0			0
$427$	13939.6	−	978.829i	1.57982	−	0.110934i
$428$		0			0
$429$	5784.13	−	4630.72i	0.650957	−	0.521149i
$430$		0			0
$431$		−	7819.69i		−	0.873924i	−0.899480	−	0.436962i	$-0.856054\pi$
$431$		−	7819.69i		−	0.873924i	0.899480	−	0.436962i	$-0.143946\pi$
$432$		0			0
$433$		−	11246.6i		−	1.24821i	−0.781340	−	0.624106i	$-0.785464\pi$
$433$		−	11246.6i		−	1.24821i	0.781340	−	0.624106i	$-0.214536\pi$
$434$		0			0
$435$	4798.03	+	5993.12i	0.528846	+	0.660570i
$436$		0			0
$437$			2119.48i			0.232010i
$438$		0			0
$439$		−	5899.62i		−	0.641397i	−0.947181	−	0.320698i	$-0.896082\pi$
$439$		−	5899.62i		−	0.641397i	0.947181	−	0.320698i	$-0.103918\pi$
$440$		0			0
$441$	743.658	−	9231.09i	0.0803000	−	0.996771i
$442$		0			0
$443$	991.808			0.106371			0.0531854	−	0.998585i	$-0.483063\pi$
$443$	991.808			0.106371			0.0531854	+	0.998585i	$0.483063\pi$
$444$		0			0
$445$			17844.7i			1.90095i
$446$		0			0
$447$	8313.64	−	6655.81i	0.879690	−	0.704271i
$448$		0			0
$449$		−	13929.7i		−	1.46411i	−0.681248	−	0.732053i	$-0.738562\pi$
$449$		−	13929.7i		−	1.46411i	0.681248	−	0.732053i	$-0.261438\pi$
$450$		0			0
$451$	−23330.7			−2.43592
$452$		0			0
$453$	1168.96	−	935.859i	0.121242	−	0.0970651i
$454$		0			0
$455$	−494.529	−	7042.62i	−0.0509535	−	0.725633i
$456$		0			0
$457$	−10005.2			−1.02412			−0.512060	−	0.858950i	$-0.671117\pi$
$457$	−10005.2			−1.02412			−0.512060	+	0.858950i	$0.671117\pi$
$458$		0			0
$459$	4265.65	+	8729.92i	0.433777	+	0.887752i
$460$		0			0
$461$			15566.9i			1.57272i	0.617770	+	0.786359i	$0.288036\pi$
$461$			15566.9i			1.57272i	−0.617770	+	0.786359i	$0.711964\pi$
$462$		0			0
$463$	10866.3			1.09071			0.545357	−	0.838204i	$-0.316394\pi$
$463$	10866.3			1.09071			0.545357	+	0.838204i	$0.316394\pi$
$464$		0			0
$465$	12717.9	−	10181.8i	1.26834	−	1.01542i
$466$		0			0
$467$			2184.04i			0.216414i	0.994128	+	0.108207i	$0.0345109\pi$
$467$			2184.04i			0.216414i	−0.994128	+	0.108207i	$0.965489\pi$
$468$		0			0
$469$	211.622	+	3013.73i	0.0208354	+	0.296719i
$470$		0			0
$471$	−8471.59	+	6782.27i	−0.828769	+	0.663504i
$472$		0			0
$473$			12932.6i			1.25717i
$474$		0			0
$475$	1246.60			0.120417
$476$		0			0
$477$	70.1332	−	312.753i	0.00673203	−	0.0300209i
$478$		0			0
$479$	−7028.43			−0.670433			−0.335216	−	0.942141i	$-0.608809\pi$
$479$	−7028.43			−0.670433			−0.335216	+	0.942141i	$0.608809\pi$
$480$		0			0
$481$		−	7527.32i		−	0.713547i
$482$		0			0
$483$	−5579.64	−	6047.22i	−0.525637	−	0.569685i
$484$		0			0
$485$	2424.37			0.226980
$486$		0			0
$487$	−3566.36			−0.331842			−0.165921	−	0.986139i	$-0.553060\pi$
$487$	−3566.36			−0.331842			−0.165921	+	0.986139i	$0.553060\pi$
$488$		0			0
$489$	−13352.8	−	16678.7i	−1.23483	−	1.54241i
$490$		0			0
$491$	−2231.72			−0.205124			−0.102562	−	0.994727i	$-0.532704\pi$
$491$	−2231.72			−0.205124			−0.102562	+	0.994727i	$0.532704\pi$
$492$		0			0
$493$	7728.56			0.706038
$494$		0			0
$495$	−17274.7	−	3873.76i	−1.56857	−	0.351743i
$496$		0			0
$497$	620.542	+	8837.19i	0.0560063	+	0.797590i
$498$		0			0
$499$		−	20669.1i		−	1.85426i	−0.374743	−	0.927129i	$-0.622269\pi$
$499$		−	20669.1i		−	1.85426i	0.374743	−	0.927129i	$-0.377731\pi$
$500$		0			0
$501$	4968.79	−	3977.96i	0.443092	−	0.354735i
$502$		0			0
$503$	−17146.1			−1.51990			−0.759948	−	0.649984i	$-0.774775\pi$
$503$	−17146.1			−1.51990			−0.759948	+	0.649984i	$0.774775\pi$
$504$		0			0
$505$	11358.4			1.00087
$506$		0			0
$507$	5549.06	−	4442.52i	0.486080	−	0.389151i
$508$		0			0
$509$			7036.98i			0.612787i	0.951905	+	0.306393i	$0.0991223\pi$
$509$			7036.98i			0.612787i	−0.951905	+	0.306393i	$0.900878\pi$
$510$		0			0
$511$	1347.07	+	19183.7i	0.116616	+	1.66074i
$512$		0			0
$513$	−1526.82	−	3124.73i	−0.131405	−	0.268928i
$514$		0			0
$515$	−6536.91			−0.559322
$516$		0			0
$517$	−10767.6			−0.915974
$518$		0			0
$519$	−7961.20	−	9944.17i	−0.673330	−	0.841042i
$520$		0			0
$521$	5554.16			0.467048			0.233524	−	0.972351i	$-0.424974\pi$
$521$	5554.16			0.467048			0.233524	+	0.972351i	$0.424974\pi$
$522$		0			0
$523$	1129.07			0.0943994			0.0471997	−	0.998885i	$-0.484970\pi$
$523$	1129.07			0.0943994			0.0471997	+	0.998885i	$0.484970\pi$
$524$		0			0
$525$	−3556.76	+	3281.75i	−0.295676	+	0.272814i
$526$		0			0
$527$		−	16400.6i		−	1.35564i
$528$		0			0
$529$	4856.64			0.399165
$530$		0			0
$531$	−6231.50	−	1397.38i	−0.509273	−	0.114202i
$532$		0			0
$533$	13563.8			1.10228
$534$		0			0
$535$			7726.84i			0.624412i
$536$		0			0
$537$	−3757.04	+	3007.85i	−0.301915	+	0.241710i
$538$		0			0
$539$	16820.4	−	2373.94i	1.34417	−	0.189708i
$540$		0			0
$541$		−	13088.7i		−	1.04016i	−0.854118	−	0.520079i	$-0.825903\pi$
$541$		−	13088.7i		−	1.04016i	0.854118	−	0.520079i	$-0.174097\pi$
$542$		0			0
$543$	−3556.28	+	2847.12i	−0.281058	+	0.225012i
$544$		0			0
$545$	−24575.7			−1.93157
$546$		0			0
$547$		−	134.806i		−	0.0105373i	−0.999986	−	0.00526863i	$-0.998323\pi$
$547$		−	134.806i		−	0.0105373i	0.999986	−	0.00526863i	$-0.00167706\pi$
$548$		0			0
$549$	−4457.61	+	19878.4i	−0.346532	+	1.54533i
$550$		0			0
$551$	−2766.31			−0.213881
$552$		0			0
$553$	−2735.81	+	192.107i	−0.210377	+	0.0147726i
$554$		0			0
$555$	−14040.2	+	11240.4i	−1.07383	+	0.859695i
$556$		0			0
$557$	−949.790			−0.0722512			−0.0361256	−	0.999347i	$-0.511502\pi$
$557$	−949.790			−0.0722512			−0.0361256	+	0.999347i	$0.511502\pi$
$558$		0			0
$559$		−	7518.65i		−	0.568882i
$560$		0			0
$561$	−13912.9	+	11138.5i	−1.04706	+	0.838267i
$562$		0			0
$563$		−	3122.65i		−	0.233755i	−0.993146	−	0.116877i	$-0.962712\pi$
$563$		−	3122.65i		−	0.233755i	0.993146	−	0.116877i	$-0.0372884\pi$
$564$		0			0
$565$	12597.4			0.938014
$566$		0			0
$567$	12582.3	+	4895.94i	0.931934	+	0.362628i
$568$		0			0
$569$		−	15497.7i		−	1.14182i	−0.821011	−	0.570912i	$-0.806590\pi$
$569$		−	15497.7i		−	1.14182i	0.821011	−	0.570912i	$-0.193410\pi$
$570$		0			0
$571$		−	11912.4i		−	0.873063i	−0.899689	−	0.436531i	$-0.856207\pi$
$571$		−	11912.4i		−	0.873063i	0.899689	−	0.436531i	$-0.143793\pi$
$572$		0			0
$573$	−8042.56	−	10045.8i	−0.586357	−	0.732406i
$574$		0			0
$575$			4299.70i			0.311843i
$576$		0			0
$577$			5650.41i			0.407677i	0.979005	+	0.203838i	$0.0653417\pi$
$577$			5650.41i			0.407677i	−0.979005	+	0.203838i	$0.934658\pi$
$578$		0			0
$579$	−12698.3	+	10166.2i	−0.911443	+	0.729692i
$580$		0			0
$581$	−959.729	−	13667.6i	−0.0685306	−	0.975949i
$582$		0			0
$583$	587.917			0.0417651
$584$		0			0
$585$	10043.0	+	2252.10i	0.709793	+	0.159167i
$586$		0			0
$587$			9396.05i			0.660675i	0.943863	+	0.330338i	$0.107163\pi$
$587$			9396.05i			0.660675i	−0.943863	+	0.330338i	$0.892837\pi$
$588$		0			0
$589$			5870.33i			0.410667i
$590$		0			0
$591$	346.603	−	277.487i	0.0241241	−	0.0193135i
$592$		0			0
$593$	−5680.51			−0.393374			−0.196687	−	0.980466i	$-0.563018\pi$
$593$	−5680.51			−0.393374			−0.196687	+	0.980466i	$0.563018\pi$
$594$		0			0
$595$	1189.52	+	16940.0i	0.0819587	+	1.16718i
$596$		0			0
$597$	−11485.5	−	14346.2i	−0.787385	−	0.983506i
$598$		0			0
$599$			11213.1i			0.764866i	0.923983	+	0.382433i	$0.124914\pi$
$599$			11213.1i			0.764866i	−0.923983	+	0.382433i	$0.875086\pi$
$600$		0			0
$601$			4599.55i			0.312179i	0.987743	+	0.156089i	$0.0498888\pi$
$601$			4599.55i			0.312179i	−0.987743	+	0.156089i	$0.950111\pi$
$602$		0			0
$603$	−4297.69	−	963.734i	−0.290241	−	0.0650850i
$604$		0			0
$605$		−	14851.3i		−	0.998000i
$606$		0			0
$607$		−	27115.8i		−	1.81318i	−0.422017	−	0.906588i	$-0.638678\pi$
$607$		−	27115.8i		−	1.81318i	0.422017	−	0.906588i	$-0.361322\pi$
$608$		0			0
$609$	7892.73	−	7282.46i	0.525172	−	0.484565i
$610$		0			0
$611$	6259.98			0.414487
$612$		0			0
$613$		−	2010.26i		−	0.132453i	−0.997805	−	0.0662263i	$-0.978904\pi$
$613$		−	2010.26i		−	0.132453i	0.997805	−	0.0662263i	$-0.0210959\pi$
$614$		0			0
$615$	−20254.7	−	25299.7i	−1.32804	−	1.65883i
$616$		0			0
$617$			15659.8i			1.02178i	0.859646	+	0.510891i	$0.170684\pi$
$617$			15659.8i			1.02178i	−0.859646	+	0.510891i	$0.829316\pi$
$618$		0			0
$619$	27135.8			1.76200			0.881001	−	0.473115i	$-0.156870\pi$
$619$	27135.8			1.76200			0.881001	+	0.473115i	$0.156870\pi$
$620$		0			0
$621$	10777.6	−	5266.20i	0.696442	−	0.340298i
$622$		0			0
$623$	24900.7	−	1748.52i	1.60133	−	0.112444i
$624$		0			0
$625$	−19382.1			−1.24046
$626$		0			0
$627$	4979.88	−	3986.84i	0.317189	−	0.253938i
$628$		0			0
$629$			18105.9i			1.14774i
$630$		0			0
$631$	−854.200			−0.0538909			−0.0269455	−	0.999637i	$-0.508578\pi$
$631$	−854.200			−0.0538909			−0.0269455	+	0.999637i	$0.508578\pi$
$632$		0			0
$633$	5508.20	+	6880.18i	0.345863	+	0.432011i
$634$		0			0
$635$			33769.3i			2.11038i
$636$		0			0
$637$	−9778.89	+	1380.14i	−0.608248	+	0.0858449i
$638$		0			0
$639$	−12602.2	−	2825.97i	−0.780178	−	0.174951i
$640$		0			0
$641$		−	29337.9i		−	1.80777i	−0.427779	−	0.903884i	$-0.640704\pi$
$641$		−	29337.9i		−	1.80777i	0.427779	−	0.903884i	$-0.359296\pi$
$642$		0			0
$643$	−24076.1			−1.47662			−0.738310	−	0.674461i	$-0.764376\pi$
$643$	−24076.1			−1.47662			−0.738310	+	0.674461i	$0.764376\pi$
$644$		0			0
$645$	−14024.0	+	11227.5i	−0.856117	+	0.685399i
$646$		0			0
$647$	9606.99			0.583755			0.291878	−	0.956456i	$-0.405720\pi$
$647$	9606.99			0.583755			0.291878	+	0.956456i	$0.405720\pi$
$648$		0			0
$649$		−	11714.1i		−	0.708500i
$650$		0			0
$651$	−15454.0	−	16749.0i	−0.930398	−	1.00837i
$652$		0			0
$653$	−5450.16			−0.326618			−0.163309	−	0.986575i	$-0.552217\pi$
$653$	−5450.16			−0.326618			−0.163309	+	0.986575i	$0.552217\pi$
$654$		0			0
$655$	−5821.08			−0.347249
$656$		0			0
$657$	−27356.6	−	6134.58i	−1.62448	−	0.364281i
$658$		0			0
$659$	−15534.8			−0.918287			−0.459144	−	0.888362i	$-0.651844\pi$
$659$	−15534.8			−0.918287			−0.459144	+	0.888362i	$0.651844\pi$
$660$		0			0
$661$	10381.1			0.610858			0.305429	−	0.952215i	$-0.401200\pi$
$661$	10381.1			0.610858			0.305429	+	0.952215i	$0.401200\pi$
$662$		0			0
$663$	8088.55	−	6475.61i	0.473806	−	0.379324i
$664$		0			0
$665$	−425.768	−	6063.39i	−0.0248279	−	0.353576i
$666$		0			0
$667$		−	9541.36i		−	0.553888i
$668$		0			0
$669$	−12783.7	−	15967.9i	−0.738784	−	0.922799i
$670$		0			0
$671$	−37367.6			−2.14986
$672$		0			0
$673$	29748.6			1.70390			0.851948	−	0.523626i	$-0.175421\pi$
$673$	29748.6			1.70390			0.851948	+	0.523626i	$0.175421\pi$
$674$		0			0
$675$	−3097.39	−	6339.02i	−0.176620	−	0.361465i
$676$		0			0
$677$		−	6845.80i		−	0.388634i	−0.980939	−	0.194317i	$-0.937751\pi$
$677$		−	6845.80i		−	0.388634i	0.980939	−	0.194317i	$-0.0622491\pi$
$678$		0			0
$679$	−237.552	−	3383.00i	−0.0134262	−	0.191204i
$680$		0			0
$681$	20231.3	+	25270.4i	1.13842	+	1.42198i
$682$		0			0
$683$	13901.9			0.778833			0.389417	−	0.921062i	$-0.372677\pi$
$683$	13901.9			0.778833			0.389417	+	0.921062i	$0.372677\pi$
$684$		0			0
$685$	−14919.5			−0.832184
$686$		0			0
$687$	16119.2	−	12904.9i	0.895178	−	0.716671i
$688$		0			0
$689$	−341.799			−0.0188991
$690$		0			0
$691$	−7713.20			−0.424637			−0.212318	−	0.977201i	$-0.568101\pi$
$691$	−7713.20			−0.424637			−0.212318	+	0.977201i	$0.568101\pi$
$692$		0			0
$693$	−3712.83	+	24484.9i	−0.203519	+	1.34214i
$694$		0			0
$695$			32034.1i			1.74838i
$696$		0			0
$697$	−32625.7			−1.77301
$698$		0			0
$699$	11044.9	+	13795.9i	0.597648	+	0.746509i
$700$		0			0
$701$	28716.9			1.54725			0.773626	−	0.633643i	$-0.218441\pi$
$701$	28716.9			1.54725			0.773626	+	0.633643i	$0.218441\pi$
$702$		0			0
$703$		−	6480.69i		−	0.347687i
$704$		0			0
$705$	−9347.95	−	11676.3i	−0.499382	−	0.623767i
$706$		0			0
$707$	−1112.95	−	15849.6i	−0.0592033	−	0.843118i
$708$		0			0
$709$		−	6971.08i		−	0.369259i	−0.982808	−	0.184629i	$-0.940892\pi$
$709$		−	6971.08i		−	0.369259i	0.982808	−	0.184629i	$-0.0591085\pi$
$710$		0			0
$711$	874.862	−	3901.38i	0.0461461	−	0.205785i
$712$		0			0
$713$	−20247.6			−1.06350
$714$		0			0
$715$			18879.0i			0.987463i
$716$		0			0
$717$	−14540.8	−	18162.6i	−0.757371	−	0.946016i
$718$		0			0
$719$	−19292.8			−1.00070			−0.500348	−	0.865825i	$-0.666795\pi$
$719$	−19292.8			−1.00070			−0.500348	+	0.865825i	$0.666795\pi$
$720$		0			0
$721$	640.519	+	9121.68i	0.0330849	+	0.471164i
$722$		0			0
$723$	−7510.44	−	9381.13i	−0.386329	−	0.482556i
$724$		0			0
$725$	−5611.90			−0.287477
$726$		0			0
$727$		−	12461.6i		−	0.635732i	−0.948136	−	0.317866i	$-0.897034\pi$
$727$		−	12461.6i		−	0.635732i	0.948136	−	0.317866i	$-0.102966\pi$
$728$		0			0
$729$	−12095.7	+	15527.8i	−0.614527	+	0.788896i
$730$		0			0
$731$			18085.0i			0.915045i
$732$		0			0
$733$	−26437.4			−1.33218			−0.666090	−	0.745872i	$-0.732033\pi$
$733$	−26437.4			−1.33218			−0.666090	+	0.745872i	$0.732033\pi$
$734$		0			0
$735$	17177.0	+	16179.0i	0.862017	+	0.811933i
$736$		0			0
$737$		−	8078.86i		−	0.403784i
$738$		0			0
$739$		−	24642.5i		−	1.22664i	−0.789834	−	0.613321i	$-0.789833\pi$
$739$		−	24642.5i		−	1.22664i	0.789834	−	0.613321i	$-0.210167\pi$
$740$		0			0
$741$	−2895.16	+	2317.84i	−0.143531	+	0.114909i
$742$		0			0
$743$		−	29736.2i		−	1.46826i	−0.679012	−	0.734128i	$-0.737591\pi$
$743$		−	29736.2i		−	1.46826i	0.679012	−	0.734128i	$-0.262409\pi$
$744$		0			0
$745$			27135.2i			1.33444i
$746$		0			0
$747$	19490.5	+	4370.63i	0.954645	+	0.214074i
$748$		0			0
$749$	10782.1	−	757.115i	0.525995	−	0.0369351i
$750$		0			0
$751$	−8068.95			−0.392064			−0.196032	−	0.980597i	$-0.562806\pi$
$751$	−8068.95			−0.392064			−0.196032	+	0.980597i	$0.562806\pi$
$752$		0			0
$753$	−14591.6	−	18226.0i	−0.706171	−	0.882063i
$754$		0			0
$755$			3815.42i			0.183917i
$756$		0			0
$757$		−	10634.4i		−	0.510587i	−0.966864	−	0.255293i	$-0.917828\pi$
$757$		−	10634.4i		−	0.510587i	0.966864	−	0.255293i	$-0.0821720\pi$
$758$		0			0
$759$	13751.1	+	17176.3i	0.657622	+	0.821422i
$760$		0			0
$761$	3296.31			0.157019			0.0785094	−	0.996913i	$-0.474984\pi$
$761$	3296.31			0.157019			0.0785094	+	0.996913i	$0.474984\pi$
$762$		0			0
$763$	2408.05	+	34293.2i	0.114256	+	1.62712i
$764$		0			0
$765$	−24157.1	−	5417.09i	−1.14170	−	0.256020i
$766$		0			0
$767$			6810.22i			0.320603i
$768$		0			0
$769$		−	29295.6i		−	1.37376i	−0.726769	−	0.686882i	$-0.758979\pi$
$769$		−	29295.6i		−	1.37376i	0.726769	−	0.686882i	$-0.241021\pi$
$770$		0			0
$771$	5808.53	−	4650.25i	0.271322	−	0.217218i
$772$		0			0
$773$		−	3337.03i		−	0.155271i	−0.996982	−	0.0776356i	$-0.975263\pi$
$773$		−	3337.03i		−	0.155271i	0.996982	−	0.0776356i	$-0.0247371\pi$
$774$		0			0
$775$			11908.9i			0.551975i
$776$		0			0
$777$	17060.8	+	18490.5i	0.787712	+	0.853722i
$778$		0			0
$779$	11677.8			0.537102
$780$		0			0
$781$		−	23689.7i		−	1.08538i
$782$		0			0
$783$	6873.36	+	14066.8i	0.313709	+	0.642025i
$784$		0			0
$785$		−	27650.7i		−	1.25719i
$786$		0			0
$787$	643.131			0.0291298			0.0145649	−	0.999894i	$-0.495364\pi$
$787$	643.131			0.0291298			0.0145649	+	0.999894i	$0.495364\pi$
$788$		0			0
$789$	2931.25	+	3661.37i	0.132263	+	0.165207i
$790$		0			0
$791$	−1234.36	−	17578.6i	−0.0554851	−	0.790168i
$792$		0			0
$793$	21724.5			0.972835
$794$		0			0
$795$	510.403	+	637.534i	0.0227700	+	0.0284415i
$796$		0			0
$797$			12470.7i			0.554248i	0.960834	+	0.277124i	$0.0893813\pi$
$797$			12470.7i			0.554248i	−0.960834	+	0.277124i	$0.910619\pi$
$798$		0			0
$799$	−15057.5			−0.666702
$800$		0			0
$801$	−7962.79	+	35509.4i	−0.351250	+	1.56637i
$802$		0			0
$803$		−	51425.4i		−	2.25998i
$804$		0			0
$805$	20913.4	−	1468.53i	0.915654	−	0.0642967i
$806$		0			0
$807$	−6136.81	−	7665.36i	−0.267690	−	0.334366i
$808$		0			0
$809$			36823.8i			1.60032i	0.599789	+	0.800158i	$0.295251\pi$
$809$			36823.8i			1.60032i	−0.599789	+	0.800158i	$0.704749\pi$
$810$		0			0
$811$	28331.7			1.22671			0.613354	−	0.789808i	$-0.289820\pi$
$811$	28331.7			1.22671			0.613354	+	0.789808i	$0.289820\pi$
$812$		0			0
$813$	24748.6	+	30913.0i	1.06762	+	1.33354i
$814$		0			0
$815$	54438.2			2.33974
$816$		0			0
$817$		−	6473.23i		−	0.277196i
$818$		0			0
$819$	2158.53	−	14234.8i	0.0920944	−	0.607333i
$820$		0			0
$821$	24314.1			1.03358			0.516789	−	0.856113i	$-0.327127\pi$
$821$	24314.1			1.03358			0.516789	+	0.856113i	$0.327127\pi$
$822$		0			0
$823$	7780.71			0.329549			0.164774	−	0.986331i	$-0.447310\pi$
$823$	7780.71			0.329549			0.164774	+	0.986331i	$0.447310\pi$
$824$		0			0
$825$	10102.5	−	8087.95i	0.426331	−	0.341317i
$826$		0			0
$827$	9658.38			0.406112			0.203056	−	0.979167i	$-0.434913\pi$
$827$	9658.38			0.406112			0.203056	+	0.979167i	$0.434913\pi$
$828$		0			0
$829$	4178.46			0.175059			0.0875296	−	0.996162i	$-0.472103\pi$
$829$	4178.46			0.175059			0.0875296	+	0.996162i	$0.472103\pi$
$830$		0			0
$831$	9689.81	+	12103.3i	0.404496	+	0.505247i
$832$		0			0
$833$	23521.7	−	3319.73i	0.978365	−	0.138081i
$834$		0			0
$835$			16217.8i			0.672144i
$836$		0			0
$837$	29850.8	−	14585.8i	1.23273	−	0.602342i
$838$		0			0
$839$	937.651			0.0385832			0.0192916	−	0.999814i	$-0.493859\pi$
$839$	937.651			0.0385832			0.0192916	+	0.999814i	$0.493859\pi$
$840$		0			0
$841$	−11935.8			−0.489391
$842$		0			0
$843$	9706.04	+	12123.6i	0.396552	+	0.495325i
$844$		0			0
$845$			18111.8i			0.737354i
$846$		0			0
$847$	−20723.6	+	1455.20i	−0.840700	+	0.0590335i
$848$		0			0
$849$	−12781.5	+	10232.7i	−0.516678	+	0.413647i
$850$		0			0
$851$	22352.8			0.900403
$852$		0			0
$853$	−5791.89			−0.232486			−0.116243	−	0.993221i	$-0.537085\pi$
$853$	−5791.89			−0.232486			−0.116243	+	0.993221i	$0.537085\pi$
$854$		0			0
$855$	8646.62	+	1938.96i	0.345857	+	0.0775566i
$856$		0			0
$857$	1880.84			0.0749687			0.0374843	−	0.999297i	$-0.488066\pi$
$857$	1880.84			0.0749687			0.0374843	+	0.999297i	$0.488066\pi$
$858$		0			0
$859$	−9503.79			−0.377491			−0.188746	−	0.982026i	$-0.560442\pi$
$859$	−9503.79			−0.377491			−0.188746	+	0.982026i	$0.560442\pi$
$860$		0			0
$861$	−33318.8	+	30742.6i	−1.31882	+	1.21685i
$862$		0			0
$863$		−	5322.63i		−	0.209947i	−0.994475	−	0.104974i	$-0.966524\pi$
$863$		−	5322.63i		−	0.209947i	0.994475	−	0.104974i	$-0.0334758\pi$
$864$		0			0
$865$	32457.2			1.27581
$866$		0			0
$867$	473.011	−	378.688i	0.0185286	−	0.0148338i
$868$		0			0
$869$	7333.85			0.286288
$870$		0			0
$871$			4696.82i			0.182716i
$872$		0			0
$873$	4824.29	+	1081.82i	0.187030	+	0.0419405i
$874$		0			0
$875$	1283.22	+	18274.4i	0.0495779	+	0.706043i
$876$		0			0
$877$			15099.1i			0.581369i	0.956819	+	0.290684i	$0.0938829\pi$
$877$			15099.1i			0.581369i	−0.956819	+	0.290684i	$0.906117\pi$
$878$		0			0
$879$	5312.79	+	6636.09i	0.203863	+	0.254641i
$880$		0			0
$881$	25460.6			0.973654			0.486827	−	0.873498i	$-0.338154\pi$
$881$	25460.6			0.973654			0.486827	+	0.873498i	$0.338154\pi$
$882$		0			0
$883$			29492.7i			1.12402i	0.827131	+	0.562009i	$0.189972\pi$
$883$			29492.7i			1.12402i	−0.827131	+	0.562009i	$0.810028\pi$
$884$		0			0
$885$	12702.6	−	10169.6i	0.482480	−	0.386269i
$886$		0			0
$887$	11558.8			0.437550			0.218775	−	0.975775i	$-0.429794\pi$
$887$	11558.8			0.437550			0.218775	+	0.975775i	$0.429794\pi$
$888$		0			0
$889$	47122.0	−	3308.88i	1.77775	−	0.124833i
$890$		0			0
$891$	−32646.6	−	15416.9i	−1.22750	−	0.579669i
$892$		0			0
$893$	5389.57			0.201965
$894$		0			0
$895$		−	12262.8i		−	0.457987i
$896$		0			0
$897$	−7994.53	−	9985.80i	−0.297580	−	0.371701i
$898$		0			0
$899$		−	26426.8i		−	0.980403i
$900$		0			0
$901$	822.146			0.0303992
$902$		0			0
$903$	17041.1	+	18469.2i	0.628010	+	0.680637i
$904$		0			0
$905$		−	11607.5i		−	0.426348i
$906$		0			0
$907$		−	10190.2i		−	0.373053i	−0.982450	−	0.186527i	$-0.940277\pi$
$907$		−	10190.2i		−	0.373053i	0.982450	−	0.186527i	$-0.0597231\pi$
$908$		0			0
$909$	22602.1	+	5068.40i	0.824713	+	0.184937i
$910$		0			0
$911$			33432.8i			1.21589i	0.793978	+	0.607947i	$0.208007\pi$
$911$			33432.8i			1.21589i	−0.793978	+	0.607947i	$0.791993\pi$
$912$		0			0
$913$			36638.5i			1.32810i
$914$		0			0
$915$	−32440.8	−	40521.2i	−1.17209	−	1.46403i
$916$		0			0
$917$	570.378	+	8122.80i	0.0205404	+	0.292517i
$918$		0			0
$919$	38879.2			1.39554			0.697772	−	0.716320i	$-0.254175\pi$
$919$	38879.2			1.39554			0.697772	+	0.716320i	$0.254175\pi$
$920$		0			0
$921$	−15367.5	+	12303.1i	−0.549812	+	0.440174i
$922$		0			0
$923$			13772.5i			0.491146i
$924$		0			0
$925$		−	13147.1i		−	0.467324i
$926$		0			0
$927$	−13007.9	−	2916.94i	−0.460878	−	0.103349i
$928$		0			0
$929$	19916.3			0.703370			0.351685	−	0.936118i	$-0.385609\pi$
$929$	19916.3			0.703370			0.351685	+	0.936118i	$0.385609\pi$
$930$		0			0
$931$	−8419.20	+	1188.24i	−0.296378	+	0.0418293i
$932$		0			0
$933$	−20275.1	+	16232.1i	−0.711444	+	0.569575i
$934$		0			0
$935$		−	45410.7i		−	1.58833i
$936$		0			0
$937$		−	27614.4i		−	0.962779i	−0.876507	−	0.481390i	$-0.840132\pi$
$937$		−	27614.4i		−	0.962779i	0.876507	−	0.481390i	$-0.159868\pi$
$938$		0			0
$939$	−3743.19	−	4675.54i	−0.130090	−	0.162492i
$940$		0			0
$941$		−	33189.4i		−	1.14978i	−0.818231	−	0.574890i	$-0.805045\pi$
$941$		−	33189.4i		−	1.14978i	0.818231	−	0.574890i	$-0.194955\pi$
$942$		0			0
$943$			40278.4i			1.39093i
$944$		0			0
$945$	−29774.6	+	17230.5i	−1.02494	+	0.593132i
$946$		0			0
$947$	−6141.76			−0.210750			−0.105375	−	0.994433i	$-0.533604\pi$
$947$	−6141.76			−0.210750			−0.105375	+	0.994433i	$0.533604\pi$
$948$		0			0
$949$			29897.2i			1.02266i
$950$		0			0
$951$	−11109.5	+	8894.16i	−0.378812	+	0.303273i
$952$		0			0
$953$			7097.85i			0.241261i	0.992697	+	0.120631i	$0.0384916\pi$
$953$			7097.85i			0.241261i	−0.992697	+	0.120631i	$0.961508\pi$
$954$		0			0
$955$	32788.8			1.11102
$956$		0			0
$957$	−22418.2	+	17947.8i	−0.757239	+	0.606238i
$958$		0			0
$959$	1461.89	+	20818.9i	0.0492252	+	0.701019i
$960$		0			0
$961$	−26288.8			−0.882441
$962$		0			0
$963$	−3447.92	+	15375.7i	−0.115377	+	0.514513i
$964$		0			0
$965$		−	41446.6i		−	1.38260i
$966$		0			0
$967$	−10526.5			−0.350063			−0.175031	−	0.984563i	$-0.556003\pi$
$967$	−10526.5			−0.350063			−0.175031	+	0.984563i	$0.556003\pi$
$968$		0			0
$969$	6963.89	−	5575.22i	0.230869	−	0.184832i
$970$		0			0
$971$		−	10918.3i		−	0.360848i	−0.983589	−	0.180424i	$-0.942253\pi$
$971$		−	10918.3i		−	0.360848i	0.983589	−	0.180424i	$-0.0577470\pi$
$972$		0			0
$973$	44700.7	−	3138.86i	1.47280	−	0.103419i
$974$		0			0
$975$	−5873.30	+	4702.10i	−0.192919	+	0.154449i
$976$		0			0
$977$		−	23312.8i		−	0.763400i	−0.924286	−	0.381700i	$-0.875339\pi$
$977$		−	23312.8i		−	0.763400i	0.924286	−	0.381700i	$-0.124661\pi$
$978$		0			0
$979$	−66751.0			−2.17913
$980$		0			0
$981$	−48903.4	−	10966.3i	−1.59160	−	0.356909i
$982$		0			0
$983$	17319.7			0.561965			0.280982	−	0.959713i	$-0.409340\pi$
$983$	17319.7			0.561965			0.280982	+	0.959713i	$0.409340\pi$
$984$		0			0
$985$			1131.29i			0.0365948i
$986$		0			0
$987$	−15377.3	+	14188.3i	−0.495912	+	0.457568i
$988$		0			0
$989$	22327.0			0.717854
$990$		0			0
$991$	−20423.3			−0.654660			−0.327330	−	0.944910i	$-0.606149\pi$
$991$	−20423.3			−0.654660			−0.327330	+	0.944910i	$0.606149\pi$
$992$		0			0
$993$	16958.9	+	21183.0i	0.541969	+	0.676962i
$994$		0			0
$995$	46825.2			1.49192
$996$		0			0
$997$	−47651.1			−1.51367			−0.756833	−	0.653608i	$-0.773255\pi$
$997$	−47651.1			−1.51367			−0.756833	+	0.653608i	$0.773255\pi$
$998$		0			0
$999$	−32954.5	+	16102.4i	−1.04368	+	0.509966i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.4.i.c.209.19		80
3.2	odd	2	inner	672.4.i.c.209.18		80
4.3	odd	2		168.4.i.c.125.72	yes	80
7.6	odd	2	inner	672.4.i.c.209.61		80
8.3	odd	2		168.4.i.c.125.11	yes	80
8.5	even	2	inner	672.4.i.c.209.62		80
12.11	even	2		168.4.i.c.125.10	yes	80
21.20	even	2	inner	672.4.i.c.209.64		80
24.5	odd	2	inner	672.4.i.c.209.63		80
24.11	even	2		168.4.i.c.125.69	yes	80
28.27	even	2		168.4.i.c.125.71	yes	80
56.13	odd	2	inner	672.4.i.c.209.20		80
56.27	even	2		168.4.i.c.125.12	yes	80
84.83	odd	2		168.4.i.c.125.9	✓	80
168.83	odd	2		168.4.i.c.125.70	yes	80
168.125	even	2	inner	672.4.i.c.209.17		80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.4.i.c.125.9	✓	80	84.83	odd	2
168.4.i.c.125.10	yes	80	12.11	even	2
168.4.i.c.125.11	yes	80	8.3	odd	2
168.4.i.c.125.12	yes	80	56.27	even	2
168.4.i.c.125.69	yes	80	24.11	even	2
168.4.i.c.125.70	yes	80	168.83	odd	2
168.4.i.c.125.71	yes	80	28.27	even	2
168.4.i.c.125.72	yes	80	4.3	odd	2
672.4.i.c.209.17		80	168.125	even	2	inner
672.4.i.c.209.18		80	3.2	odd	2	inner
672.4.i.c.209.19		80	1.1	even	1	trivial
672.4.i.c.209.20		80	56.13	odd	2	inner
672.4.i.c.209.61		80	7.6	odd	2	inner
672.4.i.c.209.62		80	8.5	even	2	inner
672.4.i.c.209.63		80	24.5	odd	2	inner
672.4.i.c.209.64		80	21.20	even	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)

\(f(q)\)	\(=\)	\(q+(-4.05635 + 3.24747i) q^{3} -13.2397i q^{5} +(-18.4748 + 1.29729i) q^{7} +(5.90788 - 26.3457i) q^{9} +49.5250 q^{11} -28.7924 q^{13} +(42.9954 + 53.7046i) q^{15} +69.2560 q^{17} -24.7890 q^{19} +(70.7271 - 65.2585i) q^{21} -85.5006i q^{23} -50.2885 q^{25} +(61.5925 + 126.053i) q^{27} +111.594 q^{29} -236.812i q^{31} +(-200.891 + 160.831i) q^{33} +(17.1756 + 244.600i) q^{35} +261.434i q^{37} +(116.792 - 93.5026i) q^{39} -471.089 q^{41} +261.133i q^{43} +(-348.808 - 78.2183i) q^{45} -217.418 q^{47} +(339.634 - 47.9342i) q^{49} +(-280.926 + 224.907i) q^{51} +11.8711 q^{53} -655.694i q^{55} +(100.553 - 80.5016i) q^{57} -236.528i q^{59} -754.519 q^{61} +(-74.9688 + 494.395i) q^{63} +381.202i q^{65} -163.127i q^{67} +(277.661 + 346.820i) q^{69} -478.338i q^{71} -1038.37i q^{73} +(203.988 - 163.310i) q^{75} +(-914.963 + 64.2482i) q^{77} +148.084 q^{79} +(-659.194 - 311.295i) q^{81} +739.797i q^{83} -916.925i q^{85} +(-452.664 + 362.398i) q^{87} -1347.82 q^{89} +(531.934 - 37.3521i) q^{91} +(769.039 + 960.590i) q^{93} +328.198i q^{95} +183.115i q^{97} +(292.588 - 1304.77i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 64 q^{7} + 104 q^{9} + 8 q^{15} - 976 q^{25} - 568 q^{39} - 4048 q^{49} - 1448 q^{57} + 2152 q^{63} - 4992 q^{79} + 1568 q^{81}+O(q^{100}) \) 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

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