LMFDB - Embedded newform 672.4.i.c.209.75

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			209.75
Character	$\chi$	$=$	672.209
Dual form			672.4.i.c.209.73

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(5.10794 + 0.953394i) q^{3} +17.0023i q^{5} +(5.04990 + 17.8185i) q^{7} +(25.1821 + 9.73975i) q^{9} -37.8286 q^{11} +77.9029 q^{13} +(-16.2099 + 86.8468i) q^{15} +34.6479 q^{17} +37.7811 q^{19} +(8.80655 + 95.8303i) q^{21} -47.8900i q^{23} -164.079 q^{25} +(119.343 + 73.7585i) q^{27} +180.790 q^{29} +163.846i q^{31} +(-193.226 - 36.0656i) q^{33} +(-302.956 + 85.8600i) q^{35} +159.677i q^{37} +(397.923 + 74.2721i) q^{39} -81.5544 q^{41} -241.294i q^{43} +(-165.598 + 428.154i) q^{45} -356.246 q^{47} +(-291.997 + 179.963i) q^{49} +(176.979 + 33.0331i) q^{51} -585.689 q^{53} -643.174i q^{55} +(192.984 + 36.0203i) q^{57} -172.591i q^{59} -572.943 q^{61} +(-46.3807 + 497.891i) q^{63} +1324.53i q^{65} -765.089i q^{67} +(45.6580 - 244.619i) q^{69} +925.379i q^{71} -590.913i q^{73} +(-838.105 - 156.432i) q^{75} +(-191.031 - 674.049i) q^{77} +28.4235 q^{79} +(539.274 + 490.534i) q^{81} -2.66288i q^{83} +589.094i q^{85} +(923.466 + 172.364i) q^{87} +600.037 q^{89} +(393.402 + 1388.11i) q^{91} +(-156.210 + 836.917i) q^{93} +642.367i q^{95} +703.593i q^{97} +(-952.603 - 368.441i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	5.10794	+	0.953394i	0.983023	+	0.183481i
$3$	5.10794	+	0.953394i	0.983023	+	0.183481i
$4$		0			0
$5$			17.0023i			1.52073i	0.649494	+	0.760367i	$0.274981\pi$
$5$			17.0023i			1.52073i	−0.649494	+	0.760367i	$0.725019\pi$
$6$		0			0
$7$	5.04990	+	17.8185i	0.272669	+	0.962108i
$7$	5.04990	+	17.8185i	0.272669	+	0.962108i
$8$		0			0
$9$	25.1821	+	9.73975i	0.932670	+	0.360732i
$10$		0			0
$11$	−37.8286			−1.03689			−0.518443	−	0.855112i	$-0.673488\pi$
$11$	−37.8286			−1.03689			−0.518443	+	0.855112i	$0.673488\pi$
$12$		0			0
$13$	77.9029			1.66203			0.831015	−	0.556250i	$-0.187760\pi$
$13$	77.9029			1.66203			0.831015	+	0.556250i	$0.187760\pi$
$14$		0			0
$15$	−16.2099	+	86.8468i	−0.279025	+	1.49492i
$16$		0			0
$17$	34.6479			0.494314			0.247157	−	0.968975i	$-0.420504\pi$
$17$	34.6479			0.494314			0.247157	+	0.968975i	$0.420504\pi$
$18$		0			0
$19$	37.7811			0.456189			0.228094	−	0.973639i	$-0.426751\pi$
$19$	37.7811			0.456189			0.228094	+	0.973639i	$0.426751\pi$
$20$		0			0
$21$	8.80655	+	95.8303i	0.0915117	+	0.995804i
$22$		0			0
$23$		−	47.8900i		−	0.434163i	−0.976153	−	0.217081i	$-0.930346\pi$
$23$		−	47.8900i		−	0.434163i	0.976153	−	0.217081i	$-0.0696537\pi$
$24$		0			0
$25$	−164.079			−1.31263
$26$		0			0
$27$	119.343	+	73.7585i	0.850649	+	0.525734i
$28$		0			0
$29$	180.790			1.15765			0.578826	−	0.815451i	$-0.303511\pi$
$29$	180.790			1.15765			0.578826	+	0.815451i	$0.303511\pi$
$30$		0			0
$31$			163.846i			0.949280i	0.880180	+	0.474640i	$0.157422\pi$
$31$			163.846i			0.949280i	−0.880180	+	0.474640i	$0.842578\pi$
$32$		0			0
$33$	−193.226	−	36.0656i	−1.01928	−	0.190249i
$34$		0			0
$35$	−302.956	+	85.8600i	−1.46311	+	0.414657i
$36$		0			0
$37$			159.677i			0.709481i	0.934965	+	0.354741i	$0.115431\pi$
$37$			159.677i			0.709481i	−0.934965	+	0.354741i	$0.884569\pi$
$38$		0			0
$39$	397.923	+	74.2721i	1.63381	+	0.304950i
$40$		0			0
$41$	−81.5544			−0.310650			−0.155325	−	0.987863i	$-0.549642\pi$
$41$	−81.5544			−0.310650			−0.155325	+	0.987863i	$0.549642\pi$
$42$		0			0
$43$		−	241.294i		−	0.855745i	−0.903839	−	0.427873i	$-0.859263\pi$
$43$		−	241.294i		−	0.855745i	0.903839	−	0.427873i	$-0.140737\pi$
$44$		0			0
$45$	−165.598	+	428.154i	−0.548577	+	1.41834i
$46$		0			0
$47$	−356.246			−1.10561			−0.552807	−	0.833309i	$-0.686443\pi$
$47$	−356.246			−1.10561			−0.552807	+	0.833309i	$0.686443\pi$
$48$		0			0
$49$	−291.997	+	179.963i	−0.851303	+	0.524674i
$50$		0			0
$51$	176.979	+	33.0331i	0.485923	+	0.0906971i
$52$		0			0
$53$	−585.689			−1.51794			−0.758968	−	0.651128i	$-0.774296\pi$
$53$	−585.689			−1.51794			−0.758968	+	0.651128i	$0.774296\pi$
$54$		0			0
$55$		−	643.174i		−	1.57683i
$56$		0			0
$57$	192.984	+	36.0203i	0.448444	+	0.0837018i
$58$		0			0
$59$		−	172.591i		−	0.380839i	−0.981703	−	0.190419i	$-0.939015\pi$
$59$		−	172.591i		−	0.380839i	0.981703	−	0.190419i	$-0.0609848\pi$
$60$		0			0
$61$	−572.943			−1.20259			−0.601294	−	0.799028i	$-0.705348\pi$
$61$	−572.943			−1.20259			−0.601294	+	0.799028i	$0.705348\pi$
$62$		0			0
$63$	−46.3807	+	497.891i	−0.0927527	+	0.995689i
$64$		0			0
$65$			1324.53i			2.52750i
$66$		0			0
$67$		−	765.089i		−	1.39508i	−0.716545	−	0.697541i	$-0.754277\pi$
$67$		−	765.089i		−	1.39508i	0.716545	−	0.697541i	$-0.245723\pi$
$68$		0			0
$69$	45.6580	−	244.619i	0.0796605	−	0.426792i
$70$		0			0
$71$			925.379i			1.54679i	0.633923	+	0.773396i	$0.281444\pi$
$71$			925.379i			1.54679i	−0.633923	+	0.773396i	$0.718556\pi$
$72$		0			0
$73$		−	590.913i		−	0.947413i	−0.880683	−	0.473706i	$-0.842916\pi$
$73$		−	590.913i		−	0.947413i	0.880683	−	0.473706i	$-0.157084\pi$
$74$		0			0
$75$	−838.105	−	156.432i	−1.29035	−	0.240843i
$76$		0			0
$77$	−191.031	−	674.049i	−0.282727	−	0.997597i
$78$		0			0
$79$	28.4235			0.0404797			0.0202399	−	0.999795i	$-0.493557\pi$
$79$	28.4235			0.0404797			0.0202399	+	0.999795i	$0.493557\pi$
$80$		0			0
$81$	539.274	+	490.534i	0.739745	+	0.672887i
$82$		0			0
$83$		−	2.66288i		−	0.00352155i	−0.999998	−	0.00176077i	$-0.999440\pi$
$83$		−	2.66288i		−	0.00352155i	0.999998	−	0.00176077i	$-0.000560472\pi$
$84$		0			0
$85$			589.094i			0.751721i
$86$		0			0
$87$	923.466	+	172.364i	1.13800	+	0.212407i
$88$		0			0
$89$	600.037			0.714649			0.357324	−	0.933980i	$-0.383689\pi$
$89$	600.037			0.714649			0.357324	+	0.933980i	$0.383689\pi$
$90$		0			0
$91$	393.402	+	1388.11i	0.453184	+	1.59905i
$92$		0			0
$93$	−156.210	+	836.917i	−0.174175	+	0.933164i
$94$		0			0
$95$			642.367i			0.693742i
$96$		0			0
$97$			703.593i			0.736485i	0.929730	+	0.368243i	$0.120040\pi$
$97$			703.593i			0.736485i	−0.929730	+	0.368243i	$0.879960\pi$
$98$		0			0
$99$	−952.603	−	368.441i	−0.967073	−	0.374038i
$100$		0			0
$101$		−	1007.27i		−	0.992350i	−0.868223	−	0.496175i	$-0.834737\pi$
$101$		−	1007.27i		−	0.992350i	0.868223	−	0.496175i	$-0.165263\pi$
$102$		0			0
$103$		−	274.314i		−	0.262417i	−0.991355	−	0.131209i	$-0.958114\pi$
$103$		−	274.314i		−	0.262417i	0.991355	−	0.131209i	$-0.0418857\pi$
$104$		0			0
$105$	−1629.34	+	149.732i	−1.51435	+	0.139165i
$106$		0			0
$107$	1132.06			1.02281			0.511405	−	0.859340i	$-0.329125\pi$
$107$	1132.06			1.02281			0.511405	+	0.859340i	$0.329125\pi$
$108$		0			0
$109$		−	2168.32i		−	1.90539i	−0.303934	−	0.952693i	$-0.598300\pi$
$109$		−	2168.32i		−	1.90539i	0.303934	−	0.952693i	$-0.401700\pi$
$110$		0			0
$111$	−152.235	+	815.622i	−0.130176	+	0.697436i
$112$		0			0
$113$			42.7131i			0.0355585i	0.999842	+	0.0177793i	$0.00565961\pi$
$113$			42.7131i			0.0355585i	−0.999842	+	0.0177793i	$0.994340\pi$
$114$		0			0
$115$	814.240			0.660246
$116$		0			0
$117$	1961.76	+	758.755i	1.55012	+	0.599546i
$118$		0			0
$119$	174.968	+	617.373i	0.134784	+	0.475584i
$120$		0			0
$121$	100.004			0.0751345
$122$		0			0
$123$	−416.575	−	77.7534i	−0.305376	−	0.0569983i
$124$		0			0
$125$		−	664.433i		−	0.475430i
$126$		0			0
$127$	57.0406			0.0398546			0.0199273	−	0.999801i	$-0.493657\pi$
$127$	57.0406			0.0398546			0.0199273	+	0.999801i	$0.493657\pi$
$128$		0			0
$129$	230.048	−	1232.52i	0.157013	−	0.841217i
$130$		0			0
$131$		−	2244.81i		−	1.49717i	−0.663036	−	0.748587i	$-0.730732\pi$
$131$		−	2244.81i		−	1.49717i	0.663036	−	0.748587i	$-0.269268\pi$
$132$		0			0
$133$	190.791	+	673.203i	0.124389	+	0.438903i
$134$		0			0
$135$	−1254.07	+	2029.10i	−0.799502	+	1.29361i
$136$		0			0
$137$			1244.36i			0.776003i	0.921659	+	0.388002i	$0.126835\pi$
$137$			1244.36i			0.776003i	−0.921659	+	0.388002i	$0.873165\pi$
$138$		0			0
$139$	502.246			0.306475			0.153237	−	0.988189i	$-0.451030\pi$
$139$	502.246			0.306475			0.153237	+	0.988189i	$0.451030\pi$
$140$		0			0
$141$	−1819.69	−	339.643i	−1.08684	−	0.202859i
$142$		0			0
$143$	−2946.96			−1.72334
$144$		0			0
$145$			3073.86i			1.76048i
$146$		0			0
$147$	−1663.08	+	640.853i	−0.933119	+	0.359569i
$148$		0			0
$149$	2671.10			1.46863			0.734313	−	0.678811i	$-0.237504\pi$
$149$	2671.10			1.46863			0.734313	+	0.678811i	$0.237504\pi$
$150$		0			0
$151$	1518.45			0.818345			0.409172	−	0.912457i	$-0.365817\pi$
$151$	1518.45			0.818345			0.409172	+	0.912457i	$0.365817\pi$
$152$		0			0
$153$	872.506	+	337.462i	0.461032	+	0.178315i
$154$		0			0
$155$	−2785.77			−1.44360
$156$		0			0
$157$	−1311.73			−0.666800			−0.333400	−	0.942785i	$-0.608196\pi$
$157$	−1311.73			−0.666800			−0.333400	+	0.942785i	$0.608196\pi$
$158$		0			0
$159$	−2991.66	−	558.392i	−1.49217	−	0.278512i
$160$		0			0
$161$	853.327	−	241.839i	0.417712	−	0.118383i
$162$		0			0
$163$			3119.89i			1.49920i	0.661894	+	0.749598i	$0.269753\pi$
$163$			3119.89i			1.49920i	−0.661894	+	0.749598i	$0.730247\pi$
$164$		0			0
$165$	613.198	−	3285.29i	0.289318	−	1.55006i
$166$		0			0
$167$	3622.79			1.67868			0.839342	−	0.543604i	$-0.182941\pi$
$167$	3622.79			1.67868			0.839342	+	0.543604i	$0.182941\pi$
$168$		0			0
$169$	3871.87			1.76234
$170$		0			0
$171$	951.408	+	367.979i	0.425474	+	0.164562i
$172$		0			0
$173$			1901.49i			0.835653i	0.908527	+	0.417826i	$0.137208\pi$
$173$			1901.49i			0.835653i	−0.908527	+	0.417826i	$0.862792\pi$
$174$		0			0
$175$	−828.582	−	2923.64i	−0.357914	−	1.26289i
$176$		0			0
$177$	164.548	−	881.586i	0.0698766	−	0.374373i
$178$		0			0
$179$	1329.17			0.555010			0.277505	−	0.960724i	$-0.410493\pi$
$179$	1329.17			0.555010			0.277505	+	0.960724i	$0.410493\pi$
$180$		0			0
$181$	−2346.73			−0.963709			−0.481855	−	0.876251i	$-0.660037\pi$
$181$	−2346.73			−0.963709			−0.481855	+	0.876251i	$0.660037\pi$
$182$		0			0
$183$	−2926.56	−	546.240i	−1.18217	−	0.220652i
$184$		0			0
$185$	−2714.89			−1.07893
$186$		0			0
$187$	−1310.68			−0.512548
$188$		0			0
$189$	−711.596	+	2498.98i	−0.273868	+	0.961767i
$190$		0			0
$191$			2129.66i			0.806791i	0.915026	+	0.403395i	$0.132170\pi$
$191$			2129.66i			0.806791i	−0.915026	+	0.403395i	$0.867830\pi$
$192$		0			0
$193$	878.502			0.327647			0.163824	−	0.986490i	$-0.447617\pi$
$193$	878.502			0.327647			0.163824	+	0.986490i	$0.447617\pi$
$194$		0			0
$195$	−1262.80	+	6765.62i	−0.463748	+	2.48460i
$196$		0			0
$197$	−3405.38			−1.23159			−0.615796	−	0.787906i	$-0.711165\pi$
$197$	−3405.38			−1.23159			−0.615796	+	0.787906i	$0.711165\pi$
$198$		0			0
$199$			2603.72i			0.927503i	0.885965	+	0.463752i	$0.153497\pi$
$199$			2603.72i			0.927503i	−0.885965	+	0.463752i	$0.846503\pi$
$200$		0			0
$201$	729.431	−	3908.03i	0.255971	−	1.37140i
$202$		0			0
$203$	912.973	+	3221.41i	0.315656	+	1.11379i
$204$		0			0
$205$		−	1386.61i		−	0.472416i
$206$		0			0
$207$	466.436	−	1205.97i	0.156616	−	0.404931i
$208$		0			0
$209$	−1429.21			−0.473016
$210$		0			0
$211$			987.223i			0.322101i	0.986946	+	0.161050i	$0.0514881\pi$
$211$			987.223i			0.322101i	−0.986946	+	0.161050i	$0.948512\pi$
$212$		0			0
$213$	−882.250	+	4726.78i	−0.283807	+	1.52053i
$214$		0			0
$215$	4102.56			1.30136
$216$		0			0
$217$	−2919.50	+	827.408i	−0.913310	+	0.258839i
$218$		0			0
$219$	563.373	−	3018.35i	0.173832	−	0.931329i
$220$		0			0
$221$	2699.17			0.821565
$222$		0			0
$223$		−	1430.30i		−	0.429507i	−0.976668	−	0.214754i	$-0.931105\pi$
$223$		−	1430.30i		−	0.429507i	0.976668	−	0.214754i	$-0.0688949\pi$
$224$		0			0
$225$	−4131.85	−	1598.09i	−1.22425	−	0.473508i
$226$		0			0
$227$		−	5026.34i		−	1.46965i	−0.678258	−	0.734824i	$-0.737265\pi$
$227$		−	5026.34i		−	1.46965i	0.678258	−	0.734824i	$-0.262735\pi$
$228$		0			0
$229$	4464.57			1.28833			0.644164	−	0.764887i	$-0.277205\pi$
$229$	4464.57			1.28833			0.644164	+	0.764887i	$0.277205\pi$
$230$		0			0
$231$	−333.139	−	3625.13i	−0.0948873	−	1.03254i
$232$		0			0
$233$		−	2131.34i		−	0.599265i	−0.954055	−	0.299632i	$-0.903136\pi$
$233$		−	2131.34i		−	0.599265i	0.954055	−	0.299632i	$-0.0968641\pi$
$234$		0			0
$235$		−	6057.02i		−	1.68135i
$236$		0			0
$237$	145.186	+	27.0988i	0.0397925	+	0.00742725i
$238$		0			0
$239$		−	6763.79i		−	1.83060i	−0.402774	−	0.915299i	$-0.631954\pi$
$239$		−	6763.79i		−	1.83060i	0.402774	−	0.915299i	$-0.368046\pi$
$240$		0			0
$241$			2721.77i			0.727489i	0.931499	+	0.363744i	$0.118502\pi$
$241$			2721.77i			0.727489i	−0.931499	+	0.363744i	$0.881498\pi$
$242$		0			0
$243$	2286.91	+	3019.76i	0.603725	+	0.797192i
$244$		0			0
$245$	−3059.79	−	4964.63i	−0.797889	−	1.29461i
$246$		0			0
$247$	2943.26			0.758199
$248$		0			0
$249$	2.53877	−	13.6018i	0.000646136	−	0.00346177i
$250$		0			0
$251$		−	5463.02i		−	1.37380i	−0.726754	−	0.686898i	$-0.758972\pi$
$251$		−	5463.02i		−	1.37380i	0.726754	−	0.686898i	$-0.241028\pi$
$252$		0			0
$253$			1811.61i			0.450178i
$254$		0			0
$255$	−561.639	+	3009.06i	−0.137926	+	0.738959i
$256$		0			0
$257$	6828.16			1.65731			0.828655	−	0.559760i	$-0.189107\pi$
$257$	6828.16			1.65731			0.828655	+	0.559760i	$0.189107\pi$
$258$		0			0
$259$	−2845.21	+	806.355i	−0.682597	+	0.193453i
$260$		0			0
$261$	4552.68	+	1760.85i	1.07971	+	0.417602i
$262$		0			0
$263$		−	665.207i		−	0.155964i	−0.996955	−	0.0779818i	$-0.975152\pi$
$263$		−	665.207i		−	0.155964i	0.996955	−	0.0779818i	$-0.0248476\pi$
$264$		0			0
$265$		−	9958.08i		−	2.30838i
$266$		0			0
$267$	3064.95	+	572.071i	0.702517	+	0.131124i
$268$		0			0
$269$		−	4857.98i		−	1.10110i	−0.834802	−	0.550550i	$-0.814418\pi$
$269$		−	4857.98i		−	1.10110i	0.834802	−	0.550550i	$-0.185582\pi$
$270$		0			0
$271$			1075.74i			0.241131i	0.992705	+	0.120566i	$0.0384709\pi$
$271$			1075.74i			0.241131i	−0.992705	+	0.120566i	$0.961529\pi$
$272$		0			0
$273$	686.056	+	7465.46i	0.152095	+	1.65506i
$274$		0			0
$275$	6206.88			1.36105
$276$		0			0
$277$			6119.99i			1.32749i	0.747959	+	0.663745i	$0.231034\pi$
$277$			6119.99i			1.32749i	−0.747959	+	0.663745i	$0.768966\pi$
$278$		0			0
$279$	−1595.82	+	4125.99i	−0.342435	+	0.885365i
$280$		0			0
$281$			580.494i			0.123236i	0.998100	+	0.0616181i	$0.0196261\pi$
$281$			580.494i			0.123236i	−0.998100	+	0.0616181i	$0.980374\pi$
$282$		0			0
$283$	5973.23			1.25467			0.627335	−	0.778750i	$-0.284146\pi$
$283$	5973.23			1.25467			0.627335	+	0.778750i	$0.284146\pi$
$284$		0			0
$285$	−612.429	+	3281.17i	−0.127288	+	0.681964i
$286$		0			0
$287$	−411.841	−	1453.18i	−0.0847046	−	0.298879i
$288$		0			0
$289$	−3712.52			−0.755653
$290$		0			0
$291$	−670.801	+	3593.91i	−0.135131	+	0.723982i
$292$		0			0
$293$			522.434i			0.104167i	0.998643	+	0.0520835i	$0.0165862\pi$
$293$			522.434i			0.104167i	−0.998643	+	0.0520835i	$0.983414\pi$
$294$		0			0
$295$	2934.45			0.579154
$296$		0			0
$297$	−4514.57	−	2790.18i	−0.882027	−	0.545127i
$298$		0			0
$299$		−	3730.77i		−	0.721591i
$300$		0			0
$301$	4299.50	−	1218.51i	0.823319	−	0.233335i
$302$		0			0
$303$	960.327	−	5145.08i	0.182077	−	0.975503i
$304$		0			0
$305$		−	9741.37i		−	1.82882i
$306$		0			0
$307$	7907.97			1.47014			0.735068	−	0.677994i	$-0.237150\pi$
$307$	7907.97			1.47014			0.735068	+	0.677994i	$0.237150\pi$
$308$		0			0
$309$	261.529	−	1401.18i	0.0481485	−	0.257962i
$310$		0			0
$311$	−1459.26			−0.266068			−0.133034	−	0.991111i	$-0.542472\pi$
$311$	−1459.26			−0.266068			−0.133034	+	0.991111i	$0.542472\pi$
$312$		0			0
$313$			5437.91i			0.982008i	0.871157	+	0.491004i	$0.163370\pi$
$313$			5437.91i			0.982008i	−0.871157	+	0.491004i	$0.836630\pi$
$314$		0			0
$315$	−8465.31	−	788.579i	−1.51418	−	0.141052i
$316$		0			0
$317$	7991.76			1.41597			0.707984	−	0.706228i	$-0.249605\pi$
$317$	7991.76			1.41597			0.707984	+	0.706228i	$0.249605\pi$
$318$		0			0
$319$	−6839.05			−1.20035
$320$		0			0
$321$	5782.51	+	1079.30i	1.00545	+	0.187666i
$322$		0			0
$323$	1309.04			0.225501
$324$		0			0
$325$	−12782.2			−2.18163
$326$		0			0
$327$	2067.26	−	11075.6i	0.349602	−	1.87304i
$328$		0			0
$329$	−1799.01	−	6347.77i	−0.301467	−	1.06372i
$330$		0			0
$331$		−	5913.18i		−	0.981927i	−0.871180	−	0.490963i	$-0.836645\pi$
$331$		−	5913.18i		−	0.981927i	0.871180	−	0.490963i	$-0.163355\pi$
$332$		0			0
$333$	−1555.22	+	4021.01i	−0.255932	+	0.661712i
$334$		0			0
$335$	13008.3			2.12155
$336$		0			0
$337$	−10946.5			−1.76942			−0.884712	−	0.466138i	$-0.845645\pi$
$337$	−10946.5			−1.76942			−0.884712	+	0.466138i	$0.845645\pi$
$338$		0			0
$339$	−40.7224	+	218.176i	−0.00652430	+	0.0349549i
$340$		0			0
$341$		−	6198.08i		−	0.984296i
$342$		0			0
$343$	−4681.23	−	4294.15i	−0.736917	−	0.675983i
$344$		0			0
$345$	4159.09	+	776.292i	0.649037	+	0.121142i
$346$		0			0
$347$	1637.21			0.253285			0.126643	−	0.991948i	$-0.459580\pi$
$347$	1637.21			0.253285			0.126643	+	0.991948i	$0.459580\pi$
$348$		0			0
$349$	4492.59			0.689062			0.344531	−	0.938775i	$-0.388038\pi$
$349$	4492.59			0.689062			0.344531	+	0.938775i	$0.388038\pi$
$350$		0			0
$351$	9297.15	+	5746.00i	1.41380	+	0.873786i
$352$		0			0
$353$	1372.22			0.206901			0.103451	−	0.994635i	$-0.467012\pi$
$353$	1372.22			0.206901			0.103451	+	0.994635i	$0.467012\pi$
$354$		0			0
$355$	−15733.6			−2.35226
$356$		0			0
$357$	305.128	+	3320.32i	0.0452355	+	0.492240i
$358$		0			0
$359$		−	5118.09i		−	0.752430i	−0.926532	−	0.376215i	$-0.877225\pi$
$359$		−	5118.09i		−	0.752430i	0.926532	−	0.376215i	$-0.122775\pi$
$360$		0			0
$361$	−5431.59			−0.791892
$362$		0			0
$363$	510.814	+	95.3432i	0.0738590	+	0.0137857i
$364$		0			0
$365$	10046.9			1.44076
$366$		0			0
$367$			6151.73i			0.874980i	0.899223	+	0.437490i	$0.144133\pi$
$367$			6151.73i			0.874980i	−0.899223	+	0.437490i	$0.855867\pi$
$368$		0			0
$369$	−2053.71	−	794.320i	−0.289734	−	0.112061i
$370$		0			0
$371$	−2957.67	−	10436.1i	−0.413894	−	1.46042i
$372$		0			0
$373$		−	12854.7i		−	1.78443i	−0.451614	−	0.892213i	$-0.649152\pi$
$373$		−	12854.7i		−	1.78443i	0.451614	−	0.892213i	$-0.350848\pi$
$374$		0			0
$375$	633.466	−	3393.88i	0.0872321	−	0.467358i
$376$		0			0
$377$	14084.1			1.92405
$378$		0			0
$379$		−	2284.79i		−	0.309662i	−0.987941	−	0.154831i	$-0.950517\pi$
$379$		−	2284.79i		−	0.309662i	0.987941	−	0.154831i	$-0.0494832\pi$
$380$		0			0
$381$	291.360	+	54.3821i	0.0391780	+	0.00731255i
$382$		0			0
$383$	−3061.90			−0.408501			−0.204251	−	0.978919i	$-0.565476\pi$
$383$	−3061.90			−0.408501			−0.204251	+	0.978919i	$0.565476\pi$
$384$		0			0
$385$	11460.4	−	3247.97i	1.51708	−	0.429952i
$386$		0			0
$387$	2350.15	−	6076.29i	0.308694	−	0.798128i
$388$		0			0
$389$	−2350.30			−0.306336			−0.153168	−	0.988200i	$-0.548948\pi$
$389$	−2350.30			−0.306336			−0.153168	+	0.988200i	$0.548948\pi$
$390$		0			0
$391$		−	1659.29i		−	0.214613i
$392$		0			0
$393$	2140.19	−	11466.3i	0.274703	−	1.47176i
$394$		0			0
$395$			483.266i			0.0615589i
$396$		0			0
$397$	−1338.92			−0.169266			−0.0846329	−	0.996412i	$-0.526972\pi$
$397$	−1338.92			−0.169266			−0.0846329	+	0.996412i	$0.526972\pi$
$398$		0			0
$399$	332.721	+	3620.58i	0.0417466	+	0.454275i
$400$		0			0
$401$			7728.45i			0.962444i	0.876599	+	0.481222i	$0.159807\pi$
$401$			7728.45i			0.962444i	−0.876599	+	0.481222i	$0.840193\pi$
$402$		0			0
$403$			12764.1i			1.57773i
$404$		0			0
$405$	−8340.23	+	9168.92i	−1.02328	+	1.12496i
$406$		0			0
$407$		−	6040.37i		−	0.735652i
$408$		0			0
$409$		−	13294.9i		−	1.60731i	−0.595093	−	0.803657i	$-0.702885\pi$
$409$		−	13294.9i		−	1.60731i	0.595093	−	0.803657i	$-0.297115\pi$
$410$		0			0
$411$	−1186.36	+	6356.09i	−0.142382	+	0.762829i
$412$		0			0
$413$	3075.32	−	871.569i	0.366408	−	0.103843i
$414$		0			0
$415$	45.2751			0.00535534
$416$		0			0
$417$	2565.44	+	478.838i	0.301272	+	0.0562322i
$418$		0			0
$419$			8465.25i			0.987003i	0.869745	+	0.493502i	$0.164283\pi$
$419$			8465.25i			0.987003i	−0.869745	+	0.493502i	$0.835717\pi$
$420$		0			0
$421$			1141.45i			0.132140i	0.997815	+	0.0660701i	$0.0210461\pi$
$421$			1141.45i			0.132140i	−0.997815	+	0.0660701i	$0.978954\pi$
$422$		0			0
$423$	−8971.03	−	3469.75i	−1.03117	−	0.398830i
$424$		0			0
$425$	−5684.99			−0.648853
$426$		0			0
$427$	−2893.31	−	10209.0i	−0.327908	−	1.15702i
$428$		0			0
$429$	−15052.9	−	2809.61i	−1.69408	−	0.316199i
$430$		0			0
$431$			6623.29i			0.740215i	0.928989	+	0.370108i	$0.120679\pi$
$431$			6623.29i			0.740215i	−0.928989	+	0.370108i	$0.879321\pi$
$432$		0			0
$433$		−	151.928i		−	0.0168619i	−0.999964	−	0.00843095i	$-0.997316\pi$
$433$		−	151.928i		−	0.0168619i	0.999964	−	0.00843095i	$-0.00268369\pi$
$434$		0			0
$435$	−2930.59	+	15701.1i	−0.323014	+	1.73059i
$436$		0			0
$437$		−	1809.34i		−	0.198060i
$438$		0			0
$439$			9828.43i			1.06853i	0.845317	+	0.534266i	$0.179412\pi$
$439$			9828.43i			1.06853i	−0.845317	+	0.534266i	$0.820588\pi$
$440$		0			0
$441$	−9105.89	+	1687.87i	−0.983251	+	0.182255i
$442$		0			0
$443$	2207.57			0.236760			0.118380	−	0.992968i	$-0.462230\pi$
$443$	2207.57			0.236760			0.118380	+	0.992968i	$0.462230\pi$
$444$		0			0
$445$			10202.0i			1.08679i
$446$		0			0
$447$	13643.8	+	2546.61i	1.44369	+	0.269465i
$448$		0			0
$449$			7960.02i			0.836651i	0.908297	+	0.418326i	$0.137383\pi$
$449$			7960.02i			0.836651i	−0.908297	+	0.418326i	$0.862617\pi$
$450$		0			0
$451$	3085.09			0.322109
$452$		0			0
$453$	7756.17	+	1447.68i	0.804452	+	0.150150i
$454$		0			0
$455$	−23601.1	+	6688.75i	−2.43173	+	0.689172i
$456$		0			0
$457$	3624.43			0.370993			0.185496	−	0.982645i	$-0.440611\pi$
$457$	3624.43			0.370993			0.185496	+	0.982645i	$0.440611\pi$
$458$		0			0
$459$	4134.97	+	2555.58i	0.420488	+	0.259878i
$460$		0			0
$461$			5885.29i			0.594589i	0.954786	+	0.297294i	$0.0960843\pi$
$461$			5885.29i			0.594589i	−0.954786	+	0.297294i	$0.903916\pi$
$462$		0			0
$463$	−13856.2			−1.39083			−0.695415	−	0.718609i	$-0.744779\pi$
$463$	−13856.2			−1.39083			−0.695415	+	0.718609i	$0.744779\pi$
$464$		0			0
$465$	−14229.5	−	2655.93i	−1.41909	−	0.264873i
$466$		0			0
$467$			12318.9i			1.22066i	0.792147	+	0.610330i	$0.208963\pi$
$467$			12318.9i			1.22066i	−0.792147	+	0.610330i	$0.791037\pi$
$468$		0			0
$469$	13632.7	−	3863.62i	1.34222	−	0.380396i
$470$		0			0
$471$	−6700.24	−	1250.60i	−0.655480	−	0.122345i
$472$		0			0
$473$			9127.83i			0.887311i
$474$		0			0
$475$	−6199.09			−0.598808
$476$		0			0
$477$	−14748.9	−	5704.47i	−1.41573	−	0.547567i
$478$		0			0
$479$	5683.40			0.542132			0.271066	−	0.962561i	$-0.412624\pi$
$479$	5683.40			0.542132			0.271066	+	0.962561i	$0.412624\pi$
$480$		0			0
$481$			12439.3i			1.17918i
$482$		0			0
$483$	4589.31	−	421.745i	0.432341	−	0.0397310i
$484$		0			0
$485$	−11962.7			−1.12000
$486$		0			0
$487$	−3252.77			−0.302663			−0.151332	−	0.988483i	$-0.548356\pi$
$487$	−3252.77			−0.302663			−0.151332	+	0.988483i	$0.548356\pi$
$488$		0			0
$489$	−2974.49	+	15936.2i	−0.275073	+	1.47374i
$490$		0			0
$491$	−8991.73			−0.826459			−0.413229	−	0.910627i	$-0.635599\pi$
$491$	−8991.73			−0.826459			−0.413229	+	0.910627i	$0.635599\pi$
$492$		0			0
$493$	6264.00			0.572244
$494$		0			0
$495$	6264.36	−	16196.5i	0.568812	−	1.47066i
$496$		0			0
$497$	−16488.9	+	4673.07i	−1.48818	+	0.421762i
$498$		0			0
$499$			10529.3i			0.944599i	0.881438	+	0.472300i	$0.156576\pi$
$499$			10529.3i			0.944599i	−0.881438	+	0.472300i	$0.843424\pi$
$500$		0			0
$501$	18505.0	+	3453.95i	1.65019	+	0.308006i
$502$		0			0
$503$	−1886.42			−0.167220			−0.0836098	−	0.996499i	$-0.526645\pi$
$503$	−1886.42			−0.167220			−0.0836098	+	0.996499i	$0.526645\pi$
$504$		0			0
$505$	17126.0			1.50910
$506$		0			0
$507$	19777.3	+	3691.41i	1.73242	+	0.323356i
$508$		0			0
$509$		−	10860.9i		−	0.945781i	−0.881121	−	0.472891i	$-0.843211\pi$
$509$		−	10860.9i		−	0.945781i	0.881121	−	0.472891i	$-0.156789\pi$
$510$		0			0
$511$	10529.2	−	2984.05i	0.911513	−	0.258330i
$512$		0			0
$513$	4508.90	+	2786.68i	0.388057	+	0.239834i
$514$		0			0
$515$	4663.97			0.399067
$516$		0			0
$517$	13476.3			1.14640
$518$		0			0
$519$	−1812.87	+	9712.71i	−0.153326	+	0.821466i
$520$		0			0
$521$	−6930.38			−0.582774			−0.291387	−	0.956605i	$-0.594117\pi$
$521$	−6930.38			−0.582774			−0.291387	+	0.956605i	$0.594117\pi$
$522$		0			0
$523$	9140.75			0.764239			0.382119	−	0.924113i	$-0.375194\pi$
$523$	9140.75			0.764239			0.382119	+	0.924113i	$0.375194\pi$
$524$		0			0
$525$	−1444.97	−	15723.7i	−0.120121	−	1.30712i
$526$		0			0
$527$			5676.93i			0.469243i
$528$		0			0
$529$	9873.55			0.811503
$530$		0			0
$531$	1681.00	−	4346.21i	0.137381	−	0.355197i
$532$		0			0
$533$	−6353.33			−0.516310
$534$		0			0
$535$			19247.7i			1.55542i
$536$		0			0
$537$	6789.31	+	1267.22i	0.545587	+	0.101834i
$538$		0			0
$539$	11045.8	−	6807.76i	0.882705	−	0.544027i
$540$		0			0
$541$		−	1327.06i		−	0.105461i	−0.998609	−	0.0527307i	$-0.983207\pi$
$541$		−	1327.06i		−	0.105461i	0.998609	−	0.0527307i	$-0.0167925\pi$
$542$		0			0
$543$	−11987.0	−	2237.36i	−0.947348	−	0.176822i
$544$		0			0
$545$	36866.4			2.89759
$546$		0			0
$547$			6943.34i			0.542735i	0.962476	+	0.271367i	$0.0874758\pi$
$547$			6943.34i			0.542735i	−0.962476	+	0.271367i	$0.912524\pi$
$548$		0			0
$549$	−14427.9	−	5580.33i	−1.12162	−	0.433811i
$550$		0			0
$551$	6830.47			0.528108
$552$		0			0
$553$	143.536	+	506.465i	0.0110376	+	0.0389459i
$554$		0			0
$555$	−13867.5	−	2588.36i	−1.06062	−	0.197963i
$556$		0			0
$557$	−12038.4			−0.915770			−0.457885	−	0.889012i	$-0.651393\pi$
$557$	−12038.4			−0.915770			−0.457885	+	0.889012i	$0.651393\pi$
$558$		0			0
$559$		−	18797.5i		−	1.42227i
$560$		0			0
$561$	−6694.88	−	1249.59i	−0.503847	−	0.0940427i
$562$		0			0
$563$		−	10849.1i		−	0.812139i	−0.913842	−	0.406070i	$-0.866899\pi$
$563$		−	10849.1i		−	0.812139i	0.913842	−	0.406070i	$-0.133101\pi$
$564$		0			0
$565$	−726.222			−0.0540750
$566$		0			0
$567$	−6017.30	+	12086.2i	−0.445684	+	0.895190i
$568$		0			0
$569$		−	10335.0i		−	0.761450i	−0.924688	−	0.380725i	$-0.875674\pi$
$569$		−	10335.0i		−	0.761450i	0.924688	−	0.380725i	$-0.124326\pi$
$570$		0			0
$571$			2025.81i			0.148472i	0.997241	+	0.0742360i	$0.0236518\pi$
$571$			2025.81i			0.148472i	−0.997241	+	0.0742360i	$0.976348\pi$
$572$		0			0
$573$	−2030.41	+	10878.2i	−0.148030	+	0.793094i
$574$		0			0
$575$			7857.73i			0.569896i
$576$		0			0
$577$		−	3778.32i		−	0.272606i	−0.990667	−	0.136303i	$-0.956478\pi$
$577$		−	3778.32i		−	0.272606i	0.990667	−	0.136303i	$-0.0435220\pi$
$578$		0			0
$579$	4487.33	+	837.558i	0.322085	+	0.0601170i
$580$		0			0
$581$	47.4484	−	13.4473i	0.00338811	−	0.000960217i
$582$		0			0
$583$	22155.8			1.57393
$584$		0			0
$585$	−12900.6	+	33354.4i	−0.911751	+	2.35733i
$586$		0			0
$587$		−	1814.70i		−	0.127599i	−0.997963	−	0.0637997i	$-0.979678\pi$
$587$		−	1814.70i		−	0.127599i	0.997963	−	0.0637997i	$-0.0203219\pi$
$588$		0			0
$589$			6190.30i			0.433051i
$590$		0			0
$591$	−17394.5	−	3246.67i	−1.21068	−	0.225973i
$592$		0			0
$593$	−9793.33			−0.678185			−0.339093	−	0.940753i	$-0.610120\pi$
$593$	−9793.33			−0.678185			−0.339093	+	0.940753i	$0.610120\pi$
$594$		0			0
$595$	−10496.8	+	2974.87i	−0.723236	+	0.204971i
$596$		0			0
$597$	−2482.37	+	13299.7i	−0.170179	+	0.911757i
$598$		0			0
$599$			10824.1i			0.738332i	0.929363	+	0.369166i	$0.120357\pi$
$599$			10824.1i			0.738332i	−0.929363	+	0.369166i	$0.879643\pi$
$600$		0			0
$601$		−	16512.2i		−	1.12071i	−0.828252	−	0.560355i	$-0.810665\pi$
$601$		−	16512.2i		−	1.12071i	0.828252	−	0.560355i	$-0.189335\pi$
$602$		0			0
$603$	7451.78	−	19266.5i	0.503250	−	1.30115i
$604$		0			0
$605$			1700.30i			0.114260i
$606$		0			0
$607$			7537.39i			0.504009i	0.967726	+	0.252004i	$0.0810898\pi$
$607$			7537.39i			0.504009i	−0.967726	+	0.252004i	$0.918910\pi$
$608$		0			0
$609$	1592.14	+	17325.2i	0.105939	+	1.15279i
$610$		0			0
$611$	−27752.6			−1.83756
$612$		0			0
$613$		−	10916.9i		−	0.719297i	−0.933088	−	0.359648i	$-0.882897\pi$
$613$		−	10916.9i		−	0.719297i	0.933088	−	0.359648i	$-0.117103\pi$
$614$		0			0
$615$	1321.99	−	7082.74i	0.0866792	−	0.464396i
$616$		0			0
$617$			27154.5i			1.77180i	0.463881	+	0.885898i	$0.346457\pi$
$617$			27154.5i			1.77180i	−0.463881	+	0.885898i	$0.653543\pi$
$618$		0			0
$619$	−12483.7			−0.810602			−0.405301	−	0.914183i	$-0.632833\pi$
$619$	−12483.7			−0.810602			−0.405301	+	0.914183i	$0.632833\pi$
$620$		0			0
$621$	3532.29	−	5715.32i	0.228254	−	0.369320i
$622$		0			0
$623$	3030.12	+	10691.7i	0.194863	+	0.687569i
$624$		0			0
$625$	−9212.97			−0.589630
$626$		0			0
$627$	−7300.31	−	1362.60i	−0.464986	−	0.0867894i
$628$		0			0
$629$			5532.48i			0.350707i
$630$		0			0
$631$	−5011.50			−0.316172			−0.158086	−	0.987425i	$-0.550532\pi$
$631$	−5011.50			−0.316172			−0.158086	+	0.987425i	$0.550532\pi$
$632$		0			0
$633$	−941.212	+	5042.68i	−0.0590992	+	0.316632i
$634$		0			0
$635$			969.822i			0.0606082i
$636$		0			0
$637$	−22747.4	+	14019.7i	−1.41489	+	0.872023i
$638$		0			0
$639$	−9012.96	+	23303.0i	−0.557977	+	1.44265i
$640$		0			0
$641$		−	20433.1i		−	1.25906i	−0.776975	−	0.629531i	$-0.783247\pi$
$641$		−	20433.1i		−	1.25906i	0.776975	−	0.629531i	$-0.216753\pi$
$642$		0			0
$643$	−7784.75			−0.477450			−0.238725	−	0.971087i	$-0.576730\pi$
$643$	−7784.75			−0.477450			−0.238725	+	0.971087i	$0.576730\pi$
$644$		0			0
$645$	20955.6	+	3911.36i	1.27927	+	0.238775i
$646$		0			0
$647$	−13288.6			−0.807463			−0.403732	−	0.914877i	$-0.632287\pi$
$647$	−13288.6			−0.807463			−0.403732	+	0.914877i	$0.632287\pi$
$648$		0			0
$649$			6528.89i			0.394887i
$650$		0			0
$651$	−15701.4	+	1442.92i	−0.945297	+	0.0868702i
$652$		0			0
$653$	6596.26			0.395301			0.197651	−	0.980273i	$-0.436669\pi$
$653$	6596.26			0.395301			0.197651	+	0.980273i	$0.436669\pi$
$654$		0			0
$655$	38167.0			2.27680
$656$		0			0
$657$	5755.35	−	14880.4i	0.341762	−	0.883623i
$658$		0			0
$659$	8372.03			0.494883			0.247442	−	0.968903i	$-0.420410\pi$
$659$	8372.03			0.494883			0.247442	+	0.968903i	$0.420410\pi$
$660$		0			0
$661$	−6191.40			−0.364323			−0.182162	−	0.983269i	$-0.558309\pi$
$661$	−6191.40			−0.364323			−0.182162	+	0.983269i	$0.558309\pi$
$662$		0			0
$663$	13787.2	+	2573.37i	0.807618	+	0.150741i
$664$		0			0
$665$	−11446.0	+	3243.89i	−0.667455	+	0.189162i
$666$		0			0
$667$		−	8658.04i		−	0.502610i
$668$		0			0
$669$	1363.64	−	7305.90i	0.0788063	−	0.422216i
$670$		0			0
$671$	21673.7			1.24695
$672$		0			0
$673$	1317.91			0.0754855			0.0377428	−	0.999287i	$-0.487983\pi$
$673$	1317.91			0.0754855			0.0377428	+	0.999287i	$0.487983\pi$
$674$		0			0
$675$	−19581.6	−	12102.2i	−1.11659	−	0.690096i
$676$		0			0
$677$		−	11076.8i		−	0.628826i	−0.949286	−	0.314413i	$-0.898192\pi$
$677$		−	11076.8i		−	0.628826i	0.949286	−	0.314413i	$-0.101808\pi$
$678$		0			0
$679$	−12537.0	+	3553.08i	−0.708578	+	0.200817i
$680$		0			0
$681$	4792.08	−	25674.2i	0.269652	−	1.44470i
$682$		0			0
$683$	−31967.1			−1.79090			−0.895451	−	0.445161i	$-0.853147\pi$
$683$	−31967.1			−1.79090			−0.895451	+	0.445161i	$0.853147\pi$
$684$		0			0
$685$	−21156.9			−1.18009
$686$		0			0
$687$	22804.8	+	4256.50i	1.26646	+	0.236383i
$688$		0			0
$689$	−45626.9			−2.52285
$690$		0			0
$691$	4700.10			0.258756			0.129378	−	0.991595i	$-0.458702\pi$
$691$	4700.10			0.258756			0.129378	+	0.991595i	$0.458702\pi$
$692$		0			0
$693$	1754.52	−	18834.5i	0.0961740	−	1.03242i
$694$		0			0
$695$			8539.35i			0.466066i
$696$		0			0
$697$	−2825.69			−0.153559
$698$		0			0
$699$	2032.01	−	10886.8i	0.109954	−	0.589091i
$700$		0			0
$701$	22855.2			1.23143			0.615714	−	0.787970i	$-0.288868\pi$
$701$	22855.2			1.23143			0.615714	+	0.787970i	$0.288868\pi$
$702$		0			0
$703$			6032.79i			0.323657i
$704$		0			0
$705$	5774.72	−	30938.9i	0.308494	−	1.65280i
$706$		0			0
$707$	17948.1	−	5086.62i	0.954748	−	0.270583i
$708$		0			0
$709$			20814.1i			1.10252i	0.834333	+	0.551261i	$0.185853\pi$
$709$			20814.1i			1.10252i	−0.834333	+	0.551261i	$0.814147\pi$
$710$		0			0
$711$	715.764	+	276.838i	0.0377542	+	0.0146023i
$712$		0			0
$713$	7846.60			0.412142
$714$		0			0
$715$		−	50105.2i		−	2.62074i
$716$		0			0
$717$	6448.55	−	34549.0i	0.335880	−	1.79952i
$718$		0			0
$719$	−5088.93			−0.263957			−0.131979	−	0.991253i	$-0.542133\pi$
$719$	−5088.93			−0.263957			−0.131979	+	0.991253i	$0.542133\pi$
$720$		0			0
$721$	4887.86	−	1385.26i	0.252474	−	0.0715530i
$722$		0			0
$723$	−2594.92	+	13902.7i	−0.133480	+	0.715139i
$724$		0			0
$725$	−29663.9			−1.51957
$726$		0			0
$727$			1716.37i			0.0875609i	0.999041	+	0.0437804i	$0.0139402\pi$
$727$			1716.37i			0.0875609i	−0.999041	+	0.0437804i	$0.986060\pi$
$728$		0			0
$729$	8802.37	+	17605.1i	0.447207	+	0.894431i
$730$		0			0
$731$		−	8360.34i		−	0.423007i
$732$		0			0
$733$	−2487.51			−0.125346			−0.0626728	−	0.998034i	$-0.519962\pi$
$733$	−2487.51			−0.125346			−0.0626728	+	0.998034i	$0.519962\pi$
$734$		0			0
$735$	−10896.0	−	28276.2i	−0.546809	−	1.41902i
$736$		0			0
$737$			28942.3i			1.44654i
$738$		0			0
$739$			16167.7i			0.804786i	0.915467	+	0.402393i	$0.131821\pi$
$739$			16167.7i			0.804786i	−0.915467	+	0.402393i	$0.868179\pi$
$740$		0			0
$741$	15034.0	+	2806.09i	0.745328	+	0.139115i
$742$		0			0
$743$		−	6261.24i		−	0.309156i	−0.987981	−	0.154578i	$-0.950598\pi$
$743$		−	6261.24i		−	0.309156i	0.987981	−	0.154578i	$-0.0494017\pi$
$744$		0			0
$745$			45415.0i			2.23339i
$746$		0			0
$747$	25.9358	−	67.0568i	0.00127033	−	0.00328444i
$748$		0			0
$749$	5716.80	+	20171.6i	0.278888	+	0.984053i
$750$		0			0
$751$	−26131.4			−1.26971			−0.634853	−	0.772633i	$-0.718939\pi$
$751$	−26131.4			−1.26971			−0.634853	+	0.772633i	$0.718939\pi$
$752$		0			0
$753$	5208.41	−	27904.8i	0.252065	−	1.35047i
$754$		0			0
$755$			25817.3i			1.24448i
$756$		0			0
$757$			2301.88i			0.110519i	0.998472	+	0.0552596i	$0.0175987\pi$
$757$			2301.88i			0.110519i	−0.998472	+	0.0552596i	$0.982401\pi$
$758$		0			0
$759$	−1727.18	+	9253.60i	−0.0825989	+	0.442535i
$760$		0			0
$761$	22624.8			1.07772			0.538862	−	0.842394i	$-0.318854\pi$
$761$	22624.8			1.07772			0.538862	+	0.842394i	$0.318854\pi$
$762$		0			0
$763$	38636.1	−	10949.8i	1.83319	−	0.519540i
$764$		0			0
$765$	−5737.63	+	14834.6i	−0.271169	+	0.701107i
$766$		0			0
$767$		−	13445.4i		−	0.632965i
$768$		0			0
$769$		−	12312.9i		−	0.577393i	−0.957421	−	0.288697i	$-0.906778\pi$
$769$		−	12312.9i		−	0.577393i	0.957421	−	0.288697i	$-0.0932219\pi$
$770$		0			0
$771$	34877.8	+	6509.92i	1.62917	+	0.304084i
$772$		0			0
$773$			18738.9i			0.871916i	0.899967	+	0.435958i	$0.143590\pi$
$773$			18738.9i			0.871916i	−0.899967	+	0.435958i	$0.856410\pi$
$774$		0			0
$775$		−	26883.7i		−	1.24606i
$776$		0			0
$777$	−15301.9	+	1406.21i	−0.706504	+	0.0649258i
$778$		0			0
$779$	−3081.22			−0.141715
$780$		0			0
$781$		−	35005.8i		−	1.60385i
$782$		0			0
$783$	21576.0	+	13334.8i	0.984756	+	0.608618i
$784$		0			0
$785$		−	22302.5i		−	1.01403i
$786$		0			0
$787$	7115.83			0.322302			0.161151	−	0.986930i	$-0.448479\pi$
$787$	7115.83			0.322302			0.161151	+	0.986930i	$0.448479\pi$
$788$		0			0
$789$	634.204	−	3397.84i	0.0286163	−	0.153316i
$790$		0			0
$791$	−761.083	+	215.697i	−0.0342111	+	0.00969570i
$792$		0			0
$793$	−44634.0			−1.99874
$794$		0			0
$795$	9493.96	−	50865.2i	0.423542	−	2.26919i
$796$		0			0
$797$			12278.2i			0.545693i	0.962058	+	0.272846i	$0.0879650\pi$
$797$			12278.2i			0.545693i	−0.962058	+	0.272846i	$0.912035\pi$
$798$		0			0
$799$	−12343.2			−0.546521
$800$		0			0
$801$	15110.2	+	5844.21i	0.666531	+	0.257796i
$802$		0			0
$803$			22353.4i			0.982360i
$804$		0			0
$805$	4111.83	+	14508.5i	0.180029	+	0.635228i
$806$		0			0
$807$	4631.57	−	24814.3i	0.202031	−	1.08241i
$808$		0			0
$809$			9226.15i			0.400957i	0.979698	+	0.200479i	$0.0642497\pi$
$809$			9226.15i			0.400957i	−0.979698	+	0.200479i	$0.935750\pi$
$810$		0			0
$811$	36172.6			1.56621			0.783103	−	0.621892i	$-0.213636\pi$
$811$	36172.6			1.56621			0.783103	+	0.621892i	$0.213636\pi$
$812$		0			0
$813$	−1025.60	+	5494.82i	−0.0442430	+	0.237038i
$814$		0			0
$815$	−53045.4			−2.27988
$816$		0			0
$817$		−	9116.38i		−	0.390381i
$818$		0			0
$819$	−3613.19	+	38787.2i	−0.154158	+	1.65486i
$820$		0			0
$821$	−7181.84			−0.305296			−0.152648	−	0.988281i	$-0.548780\pi$
$821$	−7181.84			−0.305296			−0.152648	+	0.988281i	$0.548780\pi$
$822$		0			0
$823$	2266.38			0.0959916			0.0479958	−	0.998848i	$-0.484717\pi$
$823$	2266.38			0.0959916			0.0479958	+	0.998848i	$0.484717\pi$
$824$		0			0
$825$	31704.4	+	5917.60i	1.33794	+	0.249726i
$826$		0			0
$827$	5411.98			0.227561			0.113781	−	0.993506i	$-0.463704\pi$
$827$	5411.98			0.227561			0.113781	+	0.993506i	$0.463704\pi$
$828$		0			0
$829$	27630.7			1.15760			0.578802	−	0.815468i	$-0.303520\pi$
$829$	27630.7			1.15760			0.578802	+	0.815468i	$0.303520\pi$
$830$		0			0
$831$	−5834.76	+	31260.5i	−0.243569	+	1.30495i
$832$		0			0
$833$	−10117.1	+	6235.34i	−0.420811	+	0.259354i
$834$		0			0
$835$			61595.9i			2.55283i
$836$		0			0
$837$	−12085.1	+	19553.9i	−0.499069	+	0.807504i
$838$		0			0
$839$	38177.4			1.57095			0.785476	−	0.618892i	$-0.212418\pi$
$839$	38177.4			1.57095			0.785476	+	0.618892i	$0.212418\pi$
$840$		0			0
$841$	8296.14			0.340159
$842$		0			0
$843$	−553.439	+	2965.13i	−0.0226115	+	0.121144i
$844$		0			0
$845$			65830.7i			2.68005i
$846$		0			0
$847$	505.010	+	1781.92i	0.0204868	+	0.0722875i
$848$		0			0
$849$	30510.9	+	5694.84i	1.23337	+	0.230208i
$850$		0			0
$851$	7646.94			0.308030
$852$		0			0
$853$	5549.77			0.222767			0.111384	−	0.993777i	$-0.464472\pi$
$853$	5549.77			0.222767			0.111384	+	0.993777i	$0.464472\pi$
$854$		0			0
$855$	−6256.50	+	16176.1i	−0.250255	+	0.647032i
$856$		0			0
$857$	−22763.4			−0.907332			−0.453666	−	0.891172i	$-0.649884\pi$
$857$	−22763.4			−0.907332			−0.453666	+	0.891172i	$0.649884\pi$
$858$		0			0
$859$	−672.946			−0.0267295			−0.0133647	−	0.999911i	$-0.504254\pi$
$859$	−672.946			−0.0267295			−0.0133647	+	0.999911i	$0.504254\pi$
$860$		0			0
$861$	−718.212	−	7815.38i	−0.0284281	−	0.309347i
$862$		0			0
$863$		−	38773.2i		−	1.52938i	−0.644397	−	0.764691i	$-0.722892\pi$
$863$		−	38773.2i		−	1.52938i	0.644397	−	0.764691i	$-0.277108\pi$
$864$		0			0
$865$	−32329.8			−1.27081
$866$		0			0
$867$	−18963.4	−	3539.50i	−0.742825	−	0.138648i
$868$		0			0
$869$	−1075.22			−0.0419729
$870$		0			0
$871$		−	59602.7i		−	2.31867i
$872$		0			0
$873$	−6852.83	+	17717.9i	−0.265674	+	0.686898i
$874$		0			0
$875$	11839.2	−	3355.32i	0.457415	−	0.129635i
$876$		0			0
$877$			5911.60i			0.227617i	0.993503	+	0.113809i	$0.0363051\pi$
$877$			5911.60i			0.227617i	−0.993503	+	0.113809i	$0.963695\pi$
$878$		0			0
$879$	−498.085	+	2668.56i	−0.0191126	+	0.102399i
$880$		0			0
$881$	−1628.16			−0.0622635			−0.0311317	−	0.999515i	$-0.509911\pi$
$881$	−1628.16			−0.0622635			−0.0311317	+	0.999515i	$0.509911\pi$
$882$		0			0
$883$			13348.7i			0.508743i	0.967107	+	0.254371i	$0.0818686\pi$
$883$			13348.7i			0.508743i	−0.967107	+	0.254371i	$0.918131\pi$
$884$		0			0
$885$	14989.0	+	2797.69i	0.569322	+	0.106264i
$886$		0			0
$887$	35230.3			1.33362			0.666808	−	0.745230i	$-0.267660\pi$
$887$	35230.3			1.33362			0.666808	+	0.745230i	$0.267660\pi$
$888$		0			0
$889$	288.049	+	1016.38i	0.0108671	+	0.0383444i
$890$		0			0
$891$	−20400.0	−	18556.2i	−0.767032	−	0.697708i
$892$		0			0
$893$	−13459.4			−0.504369
$894$		0			0
$895$			22598.9i			0.844022i
$896$		0			0
$897$	3556.89	−	19056.5i	0.132398	−	0.709341i
$898$		0			0
$899$			29621.8i			1.09894i
$900$		0			0
$901$	−20292.9			−0.750337
$902$		0			0
$903$	23123.3	−	2124.97i	0.852154	−	0.0783107i
$904$		0			0
$905$		−	39899.9i		−	1.46554i
$906$		0			0
$907$		−	41269.7i		−	1.51085i	−0.655237	−	0.755423i	$-0.727431\pi$
$907$		−	41269.7i		−	1.51085i	0.655237	−	0.755423i	$-0.272569\pi$
$908$		0			0
$909$	9810.58	−	25365.2i	0.357972	−	0.925535i
$910$		0			0
$911$		−	11899.0i		−	0.432747i	−0.976311	−	0.216373i	$-0.930577\pi$
$911$		−	11899.0i		−	0.432747i	0.976311	−	0.216373i	$-0.0694228\pi$
$912$		0			0
$913$			100.733i			0.00365145i
$914$		0			0
$915$	9287.36	−	49758.3i	0.335552	−	1.79777i
$916$		0			0
$917$	39999.1	−	11336.1i	1.44044	−	0.408233i
$918$		0			0
$919$	19268.9			0.691647			0.345823	−	0.938300i	$-0.387600\pi$
$919$	19268.9			0.691647			0.345823	+	0.938300i	$0.387600\pi$
$920$		0			0
$921$	40393.4	+	7539.40i	1.44518	+	0.269741i
$922$		0			0
$923$			72089.7i			2.57081i
$924$		0			0
$925$		−	26199.7i		−	0.931287i
$926$		0			0
$927$	2671.75	−	6907.80i	0.0946621	−	0.244748i
$928$		0			0
$929$	22796.3			0.805083			0.402542	−	0.915402i	$-0.368127\pi$
$929$	22796.3			0.805083			0.402542	+	0.915402i	$0.368127\pi$
$930$		0			0
$931$	−11032.0	+	6799.21i	−0.388355	+	0.239350i
$932$		0			0
$933$	−7453.81	−	1391.25i	−0.261551	−	0.0488183i
$934$		0			0
$935$		−	22284.6i		−	0.779449i
$936$		0			0
$937$			34878.2i			1.21603i	0.793925	+	0.608016i	$0.208034\pi$
$937$			34878.2i			1.21603i	−0.793925	+	0.608016i	$0.791966\pi$
$938$		0			0
$939$	−5184.46	+	27776.5i	−0.180180	+	0.965337i
$940$		0			0
$941$		−	34823.7i		−	1.20640i	−0.797591	−	0.603198i	$-0.793893\pi$
$941$		−	34823.7i		−	1.20640i	0.797591	−	0.603198i	$-0.206107\pi$
$942$		0			0
$943$			3905.64i			0.134873i
$944$		0			0
$945$	−42488.5	−	12098.8i	−1.46259	−	0.416480i
$946$		0			0
$947$	44753.6			1.53569			0.767844	−	0.640637i	$-0.221329\pi$
$947$	44753.6			1.53569			0.767844	+	0.640637i	$0.221329\pi$
$948$		0			0
$949$		−	46033.8i		−	1.57463i
$950$		0			0
$951$	40821.4	+	7619.29i	1.39193	+	0.259803i
$952$		0			0
$953$		−	27410.7i		−	0.931709i	−0.884861	−	0.465854i	$-0.845747\pi$
$953$		−	27410.7i		−	0.931709i	0.884861	−	0.465854i	$-0.154253\pi$
$954$		0			0
$955$	−36209.2			−1.22691
$956$		0			0
$957$	−34933.4	−	6520.30i	−1.17998	−	0.220242i
$958$		0			0
$959$	−22172.5	+	6283.87i	−0.746599	+	0.211592i
$960$		0			0
$961$	2945.36			0.0988674
$962$		0			0
$963$	28507.7	+	11026.0i	0.953943	+	0.368960i
$964$		0			0
$965$			14936.6i			0.498264i
$966$		0			0
$967$	−19242.8			−0.639923			−0.319961	−	0.947431i	$-0.603670\pi$
$967$	−19242.8			−0.639923			−0.319961	+	0.947431i	$0.603670\pi$
$968$		0			0
$969$	6686.48	+	1248.03i	0.221672	+	0.0413750i
$970$		0			0
$971$		−	9065.66i		−	0.299620i	−0.988715	−	0.149810i	$-0.952134\pi$
$971$		−	9065.66i		−	0.299620i	0.988715	−	0.149810i	$-0.0478661\pi$
$972$		0			0
$973$	2536.29	+	8949.27i	0.0835661	+	0.294862i
$974$		0			0
$975$	−65290.9	−	12186.5i	−2.14460	−	0.400287i
$976$		0			0
$977$		−	59374.1i		−	1.94426i	−0.234435	−	0.972132i	$-0.575324\pi$
$977$		−	59374.1i		−	1.94426i	0.234435	−	0.972132i	$-0.424676\pi$
$978$		0			0
$979$	−22698.6			−0.741010
$980$		0			0
$981$	21118.9	−	54602.7i	0.687333	−	1.77710i
$982$		0			0
$983$	−22834.0			−0.740886			−0.370443	−	0.928855i	$-0.620794\pi$
$983$	−22834.0			−0.740886			−0.370443	+	0.928855i	$0.620794\pi$
$984$		0			0
$985$		−	57899.4i		−	1.87292i
$986$		0			0
$987$	−3137.30	−	34139.2i	−0.101177	−	1.10098i
$988$		0			0
$989$	−11555.6			−0.371533
$990$		0			0
$991$	−33338.9			−1.06866			−0.534331	−	0.845275i	$-0.679437\pi$
$991$	−33338.9			−1.06866			−0.534331	+	0.845275i	$0.679437\pi$
$992$		0			0
$993$	5637.59	−	30204.2i	0.180165	−	0.965257i
$994$		0			0
$995$	−44269.4			−1.41049
$996$		0			0
$997$	−17170.3			−0.545427			−0.272713	−	0.962095i	$-0.587921\pi$
$997$	−17170.3			−0.545427			−0.272713	+	0.962095i	$0.587921\pi$
$998$		0			0
$999$	−11777.6	+	19056.3i	−0.372999	+	0.603519i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.4.i.c.209.75		80
3.2	odd	2	inner	672.4.i.c.209.74		80
4.3	odd	2		168.4.i.c.125.73	yes	80
7.6	odd	2	inner	672.4.i.c.209.5		80
8.3	odd	2		168.4.i.c.125.6	yes	80
8.5	even	2	inner	672.4.i.c.209.6		80
12.11	even	2		168.4.i.c.125.7	yes	80
21.20	even	2	inner	672.4.i.c.209.8		80
24.5	odd	2	inner	672.4.i.c.209.7		80
24.11	even	2		168.4.i.c.125.76	yes	80
28.27	even	2		168.4.i.c.125.74	yes	80
56.13	odd	2	inner	672.4.i.c.209.76		80
56.27	even	2		168.4.i.c.125.5	✓	80
84.83	odd	2		168.4.i.c.125.8	yes	80
168.83	odd	2		168.4.i.c.125.75	yes	80
168.125	even	2	inner	672.4.i.c.209.73		80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.4.i.c.125.5	✓	80	56.27	even	2
168.4.i.c.125.6	yes	80	8.3	odd	2
168.4.i.c.125.7	yes	80	12.11	even	2
168.4.i.c.125.8	yes	80	84.83	odd	2
168.4.i.c.125.73	yes	80	4.3	odd	2
168.4.i.c.125.74	yes	80	28.27	even	2
168.4.i.c.125.75	yes	80	168.83	odd	2
168.4.i.c.125.76	yes	80	24.11	even	2
672.4.i.c.209.5		80	7.6	odd	2	inner
672.4.i.c.209.6		80	8.5	even	2	inner
672.4.i.c.209.7		80	24.5	odd	2	inner
672.4.i.c.209.8		80	21.20	even	2	inner
672.4.i.c.209.73		80	168.125	even	2	inner
672.4.i.c.209.74		80	3.2	odd	2	inner
672.4.i.c.209.75		80	1.1	even	1	trivial
672.4.i.c.209.76		80	56.13	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(5.10794 + 0.953394i) q^{3} +17.0023i q^{5} +(5.04990 + 17.8185i) q^{7} +(25.1821 + 9.73975i) q^{9} -37.8286 q^{11} +77.9029 q^{13} +(-16.2099 + 86.8468i) q^{15} +34.6479 q^{17} +37.7811 q^{19} +(8.80655 + 95.8303i) q^{21} -47.8900i q^{23} -164.079 q^{25} +(119.343 + 73.7585i) q^{27} +180.790 q^{29} +163.846i q^{31} +(-193.226 - 36.0656i) q^{33} +(-302.956 + 85.8600i) q^{35} +159.677i q^{37} +(397.923 + 74.2721i) q^{39} -81.5544 q^{41} -241.294i q^{43} +(-165.598 + 428.154i) q^{45} -356.246 q^{47} +(-291.997 + 179.963i) q^{49} +(176.979 + 33.0331i) q^{51} -585.689 q^{53} -643.174i q^{55} +(192.984 + 36.0203i) q^{57} -172.591i q^{59} -572.943 q^{61} +(-46.3807 + 497.891i) q^{63} +1324.53i q^{65} -765.089i q^{67} +(45.6580 - 244.619i) q^{69} +925.379i q^{71} -590.913i q^{73} +(-838.105 - 156.432i) q^{75} +(-191.031 - 674.049i) q^{77} +28.4235 q^{79} +(539.274 + 490.534i) q^{81} -2.66288i q^{83} +589.094i q^{85} +(923.466 + 172.364i) q^{87} +600.037 q^{89} +(393.402 + 1388.11i) q^{91} +(-156.210 + 836.917i) q^{93} +642.367i q^{95} +703.593i q^{97} +(-952.603 - 368.441i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 64 q^{7} + 104 q^{9} + 8 q^{15} - 976 q^{25} - 568 q^{39} - 4048 q^{49} - 1448 q^{57} + 2152 q^{63} - 4992 q^{79} + 1568 q^{81}+O(q^{100}) \) 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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