LMFDB - Embedded newform 3850.2.c.x.1849.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3850,2,Mod(1849,3850)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3850.1849"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3850, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3850.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,-4,0,4,0,0,-4,0,4,0,0,-4,0,4,0,0,-20,0,4,0,0,-4,0,8,0, 0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)

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

Self dual:	no
Analytic conductor:	$30.7424047782$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 770)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1849.1
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	3850.1849
Dual form			3850.2.c.x.1849.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000i q^{2} -0.732051i q^{3} -1.00000 q^{4} -0.732051 q^{6} -1.00000i q^{7} +1.00000i q^{8} +2.46410 q^{9} +1.00000 q^{11} +0.732051i q^{12} +5.46410i q^{13} -1.00000 q^{14} +1.00000 q^{16} -3.46410i q^{17} -2.46410i q^{18} -3.26795 q^{19} -0.732051 q^{21} -1.00000i q^{22} +2.19615i q^{23} +0.732051 q^{24} +5.46410 q^{26} -4.00000i q^{27} +1.00000i q^{28} +1.26795 q^{29} +2.00000 q^{31} -1.00000i q^{32} -0.732051i q^{33} -3.46410 q^{34} -2.46410 q^{36} +2.73205i q^{37} +3.26795i q^{38} +4.00000 q^{39} +8.19615 q^{41} +0.732051i q^{42} +2.00000i q^{43} -1.00000 q^{44} +2.19615 q^{46} +6.92820i q^{47} -0.732051i q^{48} -1.00000 q^{49} -2.53590 q^{51} -5.46410i q^{52} -10.7321i q^{53} -4.00000 q^{54} +1.00000 q^{56} +2.39230i q^{57} -1.26795i q^{58} +6.92820 q^{59} +8.92820 q^{61} -2.00000i q^{62} -2.46410i q^{63} -1.00000 q^{64} -0.732051 q^{66} +4.00000i q^{67} +3.46410i q^{68} +1.60770 q^{69} +2.53590 q^{71} +2.46410i q^{72} +6.39230i q^{73} +2.73205 q^{74} +3.26795 q^{76} -1.00000i q^{77} -4.00000i q^{78} +1.80385 q^{79} +4.46410 q^{81} -8.19615i q^{82} +4.39230i q^{83} +0.732051 q^{84} +2.00000 q^{86} -0.928203i q^{87} +1.00000i q^{88} +3.46410 q^{89} +5.46410 q^{91} -2.19615i q^{92} -1.46410i q^{93} +6.92820 q^{94} -0.732051 q^{96} +16.5885i q^{97} +1.00000i q^{98} +2.46410 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9} + 4 q^{11} - 4 q^{14} + 4 q^{16} - 20 q^{19} + 4 q^{21} - 4 q^{24} + 8 q^{26} + 12 q^{29} + 8 q^{31} + 4 q^{36} + 16 q^{39} + 12 q^{41} - 4 q^{44} - 12 q^{46} - 4 q^{49}+ \cdots - 4 q^{99}+O(q^{100}) $ 4 * q - 4 * q^4 + 4 * q^6 - 4 * q^9 + 4 * q^11 - 4 * q^14 + 4 * q^16 - 20 * q^19 + 4 * q^21 - 4 * q^24 + 8 * q^26 + 12 * q^29 + 8 * q^31 + 4 * q^36 + 16 * q^39 + 12 * q^41 - 4 * q^44 - 12 * q^46 - 4 * q^49 - 24 * q^51 - 16 * q^54 + 4 * q^56 + 8 * q^61 - 4 * q^64 + 4 * q^66 + 48 * q^69 + 24 * q^71 + 4 * q^74 + 20 * q^76 + 28 * q^79 + 4 * q^81 - 4 * q^84 + 8 * q^86 + 8 * q^91 + 4 * q^96 - 4 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3850\mathbb{Z}\right)^\times$.

$n$	$1751$	$2201$	$2927$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	− 0.732051i	− 0.422650i	−0.977416	−	0.211325i	$-0.932222\pi$
$3$	− 0.732051i	− 0.422650i	0.977416	−	0.211325i	$-0.0677778\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−0.732051	−0.298858
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	1.00000i	0.353553i
$9$	2.46410	0.821367
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0.732051i	0.211325i
$13$	5.46410i	1.51547i	0.652563	+	0.757735i	$0.273694\pi$
$13$	5.46410i	1.51547i	−0.652563	+	0.757735i	$0.726306\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	− 3.46410i	− 0.840168i	−0.907485	−	0.420084i	$-0.862001\pi$
$17$	− 3.46410i	− 0.840168i	0.907485	−	0.420084i	$-0.137999\pi$
$18$	− 2.46410i	− 0.580794i
$19$	−3.26795	−0.749719	−0.374859	−	0.927082i	$-0.622309\pi$
$19$	−3.26795	−0.749719	−0.374859	+	0.927082i	$0.622309\pi$
$20$	0	0
$21$	−0.732051	−0.159747
$22$	− 1.00000i	− 0.213201i
$23$	2.19615i	0.457929i	0.973435	+	0.228965i	$0.0735340\pi$
$23$	2.19615i	0.457929i	−0.973435	+	0.228965i	$0.926466\pi$
$24$	0.732051	0.149429
$25$	0	0
$26$	5.46410	1.07160
$27$	− 4.00000i	− 0.769800i
$28$	1.00000i	0.188982i
$29$	1.26795	0.235452	0.117726	−	0.993046i	$-0.462440\pi$
$29$	1.26795	0.235452	0.117726	+	0.993046i	$0.462440\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	− 1.00000i	− 0.176777i
$33$	− 0.732051i	− 0.127434i
$34$	−3.46410	−0.594089
$35$	0	0
$36$	−2.46410	−0.410684
$37$	2.73205i	0.449146i	0.974457	+	0.224573i	$0.0720988\pi$
$37$	2.73205i	0.449146i	−0.974457	+	0.224573i	$0.927901\pi$
$38$	3.26795i	0.530131i
$39$	4.00000	0.640513
$40$	0	0
$41$	8.19615	1.28002	0.640012	−	0.768365i	$-0.278929\pi$
$41$	8.19615	1.28002	0.640012	+	0.768365i	$0.278929\pi$
$42$	0.732051i	0.112958i
$43$	2.00000i	0.304997i	0.988304	+	0.152499i	$0.0487319\pi$
$43$	2.00000i	0.304997i	−0.988304	+	0.152499i	$0.951268\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	2.19615	0.323805
$47$	6.92820i	1.01058i	0.862949	+	0.505291i	$0.168615\pi$
$47$	6.92820i	1.01058i	−0.862949	+	0.505291i	$0.831385\pi$
$48$	− 0.732051i	− 0.105662i
$49$	−1.00000	−0.142857
$50$	0	0
$51$	−2.53590	−0.355097
$52$	− 5.46410i	− 0.757735i
$53$	− 10.7321i	− 1.47416i	−0.675805	−	0.737080i	$-0.736204\pi$
$53$	− 10.7321i	− 1.47416i	0.675805	−	0.737080i	$-0.263796\pi$
$54$	−4.00000	−0.544331
$55$	0	0
$56$	1.00000	0.133631
$57$	2.39230i	0.316869i
$58$	− 1.26795i	− 0.166490i
$59$	6.92820	0.901975	0.450988	−	0.892530i	$-0.351072\pi$
$59$	6.92820	0.901975	0.450988	+	0.892530i	$0.351072\pi$
$60$	0	0
$61$	8.92820	1.14314	0.571570	−	0.820554i	$-0.306335\pi$
$61$	8.92820	1.14314	0.571570	+	0.820554i	$0.306335\pi$
$62$	− 2.00000i	− 0.254000i
$63$	− 2.46410i	− 0.310448i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−0.732051	−0.0901092
$67$	4.00000i	0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$	4.00000i	0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	3.46410i	0.420084i
$69$	1.60770	0.193544
$70$	0	0
$71$	2.53590	0.300956	0.150478	−	0.988613i	$-0.451919\pi$
$71$	2.53590	0.300956	0.150478	+	0.988613i	$0.451919\pi$
$72$	2.46410i	0.290397i
$73$	6.39230i	0.748163i	0.927396	+	0.374081i	$0.122042\pi$
$73$	6.39230i	0.748163i	−0.927396	+	0.374081i	$0.877958\pi$
$74$	2.73205	0.317594
$75$	0	0
$76$	3.26795	0.374859
$77$	− 1.00000i	− 0.113961i
$78$	− 4.00000i	− 0.452911i
$79$	1.80385	0.202949	0.101474	−	0.994838i	$-0.467644\pi$
$79$	1.80385	0.202949	0.101474	+	0.994838i	$0.467644\pi$
$80$	0	0
$81$	4.46410	0.496011
$82$	− 8.19615i	− 0.905114i
$83$	4.39230i	0.482118i	0.970510	+	0.241059i	$0.0774947\pi$
$83$	4.39230i	0.482118i	−0.970510	+	0.241059i	$0.922505\pi$
$84$	0.732051	0.0798733
$85$	0	0
$86$	2.00000	0.215666
$87$	− 0.928203i	− 0.0995138i
$88$	1.00000i	0.106600i
$89$	3.46410	0.367194	0.183597	−	0.983002i	$-0.441226\pi$
$89$	3.46410	0.367194	0.183597	+	0.983002i	$0.441226\pi$
$90$	0	0
$91$	5.46410	0.572793
$92$	− 2.19615i	− 0.228965i
$93$	− 1.46410i	− 0.151820i
$94$	6.92820	0.714590
$95$	0	0
$96$	−0.732051	−0.0747146
$97$	16.5885i	1.68430i	0.539241	+	0.842151i	$0.318711\pi$
$97$	16.5885i	1.68430i	−0.539241	+	0.842151i	$0.681289\pi$
$98$	1.00000i	0.101015i
$99$	2.46410	0.247652
$100$	0	0
$101$	19.8564	1.97579	0.987893	−	0.155136i	$-0.0495815\pi$
$101$	19.8564	1.97579	0.987893	+	0.155136i	$0.0495815\pi$
$102$	2.53590i	0.251091i
$103$	− 8.39230i	− 0.826918i	−0.910523	−	0.413459i	$-0.864320\pi$
$103$	− 8.39230i	− 0.826918i	0.910523	−	0.413459i	$-0.135680\pi$
$104$	−5.46410	−0.535799
$105$	0	0
$106$	−10.7321	−1.04239
$107$	19.8564i	1.91959i	0.280700	+	0.959796i	$0.409433\pi$
$107$	19.8564i	1.91959i	−0.280700	+	0.959796i	$0.590567\pi$
$108$	4.00000i	0.384900i
$109$	−1.66025	−0.159023	−0.0795117	−	0.996834i	$-0.525336\pi$
$109$	−1.66025	−0.159023	−0.0795117	+	0.996834i	$0.525336\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	− 1.00000i	− 0.0944911i
$113$	− 19.8564i	− 1.86793i	−0.357360	−	0.933967i	$-0.616323\pi$
$113$	− 19.8564i	− 1.86793i	0.357360	−	0.933967i	$-0.383677\pi$
$114$	2.39230	0.224060
$115$	0	0
$116$	−1.26795	−0.117726
$117$	13.4641i	1.24476i
$118$	− 6.92820i	− 0.637793i
$119$	−3.46410	−0.317554
$120$	0	0
$121$	1.00000	0.0909091
$122$	− 8.92820i	− 0.808322i
$123$	− 6.00000i	− 0.541002i
$124$	−2.00000	−0.179605
$125$	0	0
$126$	−2.46410	−0.219520
$127$	− 14.9282i	− 1.32466i	−0.749211	−	0.662332i	$-0.769567\pi$
$127$	− 14.9282i	− 1.32466i	0.749211	−	0.662332i	$-0.230433\pi$
$128$	1.00000i	0.0883883i
$129$	1.46410	0.128907
$130$	0	0
$131$	−11.6603	−1.01876	−0.509381	−	0.860541i	$-0.670125\pi$
$131$	−11.6603	−1.01876	−0.509381	+	0.860541i	$0.670125\pi$
$132$	0.732051i	0.0637168i
$133$	3.26795i	0.283367i
$134$	4.00000	0.345547
$135$	0	0
$136$	3.46410	0.297044
$137$	− 12.9282i	− 1.10453i	−0.833668	−	0.552265i	$-0.813763\pi$
$137$	− 12.9282i	− 1.10453i	0.833668	−	0.552265i	$-0.186237\pi$
$138$	− 1.60770i	− 0.136856i
$139$	3.66025	0.310459	0.155229	−	0.987878i	$-0.450388\pi$
$139$	3.66025	0.310459	0.155229	+	0.987878i	$0.450388\pi$
$140$	0	0
$141$	5.07180	0.427122
$142$	− 2.53590i	− 0.212808i
$143$	5.46410i	0.456931i
$144$	2.46410	0.205342
$145$	0	0
$146$	6.39230	0.529031
$147$	0.732051i	0.0603785i
$148$	− 2.73205i	− 0.224573i
$149$	10.7321	0.879204	0.439602	−	0.898193i	$-0.355120\pi$
$149$	10.7321	0.879204	0.439602	+	0.898193i	$0.355120\pi$
$150$	0	0
$151$	−13.1244	−1.06804	−0.534022	−	0.845470i	$-0.679320\pi$
$151$	−13.1244	−1.06804	−0.534022	+	0.845470i	$0.679320\pi$
$152$	− 3.26795i	− 0.265066i
$153$	− 8.53590i	− 0.690086i
$154$	−1.00000	−0.0805823
$155$	0	0
$156$	−4.00000	−0.320256
$157$	− 11.4641i	− 0.914935i	−0.889226	−	0.457467i	$-0.848757\pi$
$157$	− 11.4641i	− 0.914935i	0.889226	−	0.457467i	$-0.151243\pi$
$158$	− 1.80385i	− 0.143506i
$159$	−7.85641	−0.623054
$160$	0	0
$161$	2.19615	0.173081
$162$	− 4.46410i	− 0.350733i
$163$	− 4.00000i	− 0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$	− 4.00000i	− 0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$	−8.19615	−0.640012
$165$	0	0
$166$	4.39230	0.340909
$167$	13.8564i	1.07224i	0.844141	+	0.536120i	$0.180111\pi$
$167$	13.8564i	1.07224i	−0.844141	+	0.536120i	$0.819889\pi$
$168$	− 0.732051i	− 0.0564789i
$169$	−16.8564	−1.29665
$170$	0	0
$171$	−8.05256	−0.615795
$172$	− 2.00000i	− 0.152499i
$173$	− 12.9282i	− 0.982913i	−0.870902	−	0.491457i	$-0.836465\pi$
$173$	− 12.9282i	− 0.982913i	0.870902	−	0.491457i	$-0.163535\pi$
$174$	−0.928203	−0.0703669
$175$	0	0
$176$	1.00000	0.0753778
$177$	− 5.07180i	− 0.381220i
$178$	− 3.46410i	− 0.259645i
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	0	0
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	− 5.46410i	− 0.405026i
$183$	− 6.53590i	− 0.483148i
$184$	−2.19615	−0.161903
$185$	0	0
$186$	−1.46410	−0.107353
$187$	− 3.46410i	− 0.253320i
$188$	− 6.92820i	− 0.505291i
$189$	−4.00000	−0.290957
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	0.732051i	0.0528312i
$193$	− 8.39230i	− 0.604091i	−0.953293	−	0.302046i	$-0.902330\pi$
$193$	− 8.39230i	− 0.604091i	0.953293	−	0.302046i	$-0.0976695\pi$
$194$	16.5885	1.19098
$195$	0	0
$196$	1.00000	0.0714286
$197$	− 24.2487i	− 1.72765i	−0.503793	−	0.863825i	$-0.668062\pi$
$197$	− 24.2487i	− 1.72765i	0.503793	−	0.863825i	$-0.331938\pi$
$198$	− 2.46410i	− 0.175116i
$199$	10.9282	0.774680	0.387340	−	0.921937i	$-0.373394\pi$
$199$	10.9282	0.774680	0.387340	+	0.921937i	$0.373394\pi$
$200$	0	0
$201$	2.92820	0.206540
$202$	− 19.8564i	− 1.39709i
$203$	− 1.26795i	− 0.0889926i
$204$	2.53590	0.177548
$205$	0	0
$206$	−8.39230	−0.584720
$207$	5.41154i	0.376128i
$208$	5.46410i	0.378867i
$209$	−3.26795	−0.226049
$210$	0	0
$211$	13.0718	0.899900	0.449950	−	0.893054i	$-0.351442\pi$
$211$	13.0718	0.899900	0.449950	+	0.893054i	$0.351442\pi$
$212$	10.7321i	0.737080i
$213$	− 1.85641i	− 0.127199i
$214$	19.8564	1.35736
$215$	0	0
$216$	4.00000	0.272166
$217$	− 2.00000i	− 0.135769i
$218$	1.66025i	0.112447i
$219$	4.67949	0.316211
$220$	0	0
$221$	18.9282	1.27325
$222$	− 2.00000i	− 0.134231i
$223$	− 18.5359i	− 1.24126i	−0.784105	−	0.620628i	$-0.786878\pi$
$223$	− 18.5359i	− 1.24126i	0.784105	−	0.620628i	$-0.213122\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	−19.8564	−1.32083
$227$	6.92820i	0.459841i	0.973209	+	0.229920i	$0.0738466\pi$
$227$	6.92820i	0.459841i	−0.973209	+	0.229920i	$0.926153\pi$
$228$	− 2.39230i	− 0.158434i
$229$	−3.60770	−0.238403	−0.119202	−	0.992870i	$-0.538033\pi$
$229$	−3.60770	−0.238403	−0.119202	+	0.992870i	$0.538033\pi$
$230$	0	0
$231$	−0.732051	−0.0481654
$232$	1.26795i	0.0832449i
$233$	19.8564i	1.30084i	0.759576	+	0.650418i	$0.225406\pi$
$233$	19.8564i	1.30084i	−0.759576	+	0.650418i	$0.774594\pi$
$234$	13.4641	0.880176
$235$	0	0
$236$	−6.92820	−0.450988
$237$	− 1.32051i	− 0.0857762i
$238$	3.46410i	0.224544i
$239$	−4.73205	−0.306091	−0.153045	−	0.988219i	$-0.548908\pi$
$239$	−4.73205	−0.306091	−0.153045	+	0.988219i	$0.548908\pi$
$240$	0	0
$241$	6.73205	0.433650	0.216825	−	0.976211i	$-0.430430\pi$
$241$	6.73205	0.433650	0.216825	+	0.976211i	$0.430430\pi$
$242$	− 1.00000i	− 0.0642824i
$243$	− 15.2679i	− 0.979439i
$244$	−8.92820	−0.571570
$245$	0	0
$246$	−6.00000	−0.382546
$247$	− 17.8564i	− 1.13618i
$248$	2.00000i	0.127000i
$249$	3.21539	0.203767
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	2.46410i	0.155224i
$253$	2.19615i	0.138071i
$254$	−14.9282	−0.936679
$255$	0	0
$256$	1.00000	0.0625000
$257$	6.33975i	0.395462i	0.980256	+	0.197731i	$0.0633573\pi$
$257$	6.33975i	0.395462i	−0.980256	+	0.197731i	$0.936643\pi$
$258$	− 1.46410i	− 0.0911510i
$259$	2.73205	0.169761
$260$	0	0
$261$	3.12436	0.193393
$262$	11.6603i	0.720373i
$263$	− 24.0000i	− 1.47990i	−0.672660	−	0.739952i	$-0.734848\pi$
$263$	− 24.0000i	− 1.47990i	0.672660	−	0.739952i	$-0.265152\pi$
$264$	0.732051	0.0450546
$265$	0	0
$266$	3.26795	0.200371
$267$	− 2.53590i	− 0.155194i
$268$	− 4.00000i	− 0.244339i
$269$	−7.60770	−0.463849	−0.231925	−	0.972734i	$-0.574502\pi$
$269$	−7.60770	−0.463849	−0.231925	+	0.972734i	$0.574502\pi$
$270$	0	0
$271$	−20.3923	−1.23874	−0.619372	−	0.785098i	$-0.712613\pi$
$271$	−20.3923	−1.23874	−0.619372	+	0.785098i	$0.712613\pi$
$272$	− 3.46410i	− 0.210042i
$273$	− 4.00000i	− 0.242091i
$274$	−12.9282	−0.781021
$275$	0	0
$276$	−1.60770	−0.0967719
$277$	− 20.9282i	− 1.25745i	−0.777626	−	0.628727i	$-0.783576\pi$
$277$	− 20.9282i	− 1.25745i	0.777626	−	0.628727i	$-0.216424\pi$
$278$	− 3.66025i	− 0.219527i
$279$	4.92820	0.295044
$280$	0	0
$281$	1.60770	0.0959071	0.0479535	−	0.998850i	$-0.484730\pi$
$281$	1.60770	0.0959071	0.0479535	+	0.998850i	$0.484730\pi$
$282$	− 5.07180i	− 0.302021i
$283$	23.7128i	1.40958i	0.709416	+	0.704790i	$0.248959\pi$
$283$	23.7128i	1.40958i	−0.709416	+	0.704790i	$0.751041\pi$
$284$	−2.53590	−0.150478
$285$	0	0
$286$	5.46410	0.323099
$287$	− 8.19615i	− 0.483804i
$288$	− 2.46410i	− 0.145199i
$289$	5.00000	0.294118
$290$	0	0
$291$	12.1436	0.711870
$292$	− 6.39230i	− 0.374081i
$293$	14.5359i	0.849196i	0.905382	+	0.424598i	$0.139585\pi$
$293$	14.5359i	0.849196i	−0.905382	+	0.424598i	$0.860415\pi$
$294$	0.732051	0.0426941
$295$	0	0
$296$	−2.73205	−0.158797
$297$	− 4.00000i	− 0.232104i
$298$	− 10.7321i	− 0.621691i
$299$	−12.0000	−0.693978
$300$	0	0
$301$	2.00000	0.115278
$302$	13.1244i	0.755222i
$303$	− 14.5359i	− 0.835066i
$304$	−3.26795	−0.187430
$305$	0	0
$306$	−8.53590	−0.487965
$307$	− 3.60770i	− 0.205902i	−0.994686	−	0.102951i	$-0.967172\pi$
$307$	− 3.60770i	− 0.205902i	0.994686	−	0.102951i	$-0.0328285\pi$
$308$	1.00000i	0.0569803i
$309$	−6.14359	−0.349497
$310$	0	0
$311$	11.0718	0.627824	0.313912	−	0.949452i	$-0.398360\pi$
$311$	11.0718	0.627824	0.313912	+	0.949452i	$0.398360\pi$
$312$	4.00000i	0.226455i
$313$	11.8038i	0.667193i	0.942716	+	0.333596i	$0.108262\pi$
$313$	11.8038i	0.667193i	−0.942716	+	0.333596i	$0.891738\pi$
$314$	−11.4641	−0.646957
$315$	0	0
$316$	−1.80385	−0.101474
$317$	0.588457i	0.0330511i	0.999863	+	0.0165255i	$0.00526048\pi$
$317$	0.588457i	0.0330511i	−0.999863	+	0.0165255i	$0.994740\pi$
$318$	7.85641i	0.440565i
$319$	1.26795	0.0709915
$320$	0	0
$321$	14.5359	0.811315
$322$	− 2.19615i	− 0.122387i
$323$	11.3205i	0.629890i
$324$	−4.46410	−0.248006
$325$	0	0
$326$	−4.00000	−0.221540
$327$	1.21539i	0.0672112i
$328$	8.19615i	0.452557i
$329$	6.92820	0.381964
$330$	0	0
$331$	22.7846	1.25236	0.626178	−	0.779680i	$-0.284618\pi$
$331$	22.7846	1.25236	0.626178	+	0.779680i	$0.284618\pi$
$332$	− 4.39230i	− 0.241059i
$333$	6.73205i	0.368914i
$334$	13.8564	0.758189
$335$	0	0
$336$	−0.732051	−0.0399366
$337$	18.7846i	1.02326i	0.859205	+	0.511631i	$0.170959\pi$
$337$	18.7846i	1.02326i	−0.859205	+	0.511631i	$0.829041\pi$
$338$	16.8564i	0.916868i
$339$	−14.5359	−0.789482
$340$	0	0
$341$	2.00000	0.108306
$342$	8.05256i	0.435433i
$343$	1.00000i	0.0539949i
$344$	−2.00000	−0.107833
$345$	0	0
$346$	−12.9282	−0.695025
$347$	36.9282i	1.98241i	0.132336	+	0.991205i	$0.457752\pi$
$347$	36.9282i	1.98241i	−0.132336	+	0.991205i	$0.542248\pi$
$348$	0.928203i	0.0497569i
$349$	26.3923	1.41275	0.706374	−	0.707839i	$-0.250330\pi$
$349$	26.3923	1.41275	0.706374	+	0.707839i	$0.250330\pi$
$350$	0	0
$351$	21.8564	1.16661
$352$	− 1.00000i	− 0.0533002i
$353$	3.80385i	0.202458i	0.994863	+	0.101229i	$0.0322775\pi$
$353$	3.80385i	0.202458i	−0.994863	+	0.101229i	$0.967722\pi$
$354$	−5.07180	−0.269563
$355$	0	0
$356$	−3.46410	−0.183597
$357$	2.53590i	0.134214i
$358$	− 6.00000i	− 0.317110i
$359$	−4.05256	−0.213886	−0.106943	−	0.994265i	$-0.534106\pi$
$359$	−4.05256	−0.213886	−0.106943	+	0.994265i	$0.534106\pi$
$360$	0	0
$361$	−8.32051	−0.437921
$362$	− 14.0000i	− 0.735824i
$363$	− 0.732051i	− 0.0384227i
$364$	−5.46410	−0.286397
$365$	0	0
$366$	−6.53590	−0.341637
$367$	8.39230i	0.438075i	0.975716	+	0.219037i	$0.0702917\pi$
$367$	8.39230i	0.438075i	−0.975716	+	0.219037i	$0.929708\pi$
$368$	2.19615i	0.114482i
$369$	20.1962	1.05137
$370$	0	0
$371$	−10.7321	−0.557180
$372$	1.46410i	0.0759101i
$373$	− 2.39230i	− 0.123869i	−0.998080	−	0.0619344i	$-0.980273\pi$
$373$	− 2.39230i	− 0.123869i	0.998080	−	0.0619344i	$-0.0197270\pi$
$374$	−3.46410	−0.179124
$375$	0	0
$376$	−6.92820	−0.357295
$377$	6.92820i	0.356821i
$378$	4.00000i	0.205738i
$379$	−33.8564	−1.73909	−0.869543	−	0.493857i	$-0.835587\pi$
$379$	−33.8564	−1.73909	−0.869543	+	0.493857i	$0.835587\pi$
$380$	0	0
$381$	−10.9282	−0.559869
$382$	12.0000i	0.613973i
$383$	− 26.5359i	− 1.35592i	−0.735098	−	0.677961i	$-0.762864\pi$
$383$	− 26.5359i	− 1.35592i	0.735098	−	0.677961i	$-0.237136\pi$
$384$	0.732051	0.0373573
$385$	0	0
$386$	−8.39230	−0.427157
$387$	4.92820i	0.250515i
$388$	− 16.5885i	− 0.842151i
$389$	22.3923	1.13533	0.567667	−	0.823258i	$-0.307846\pi$
$389$	22.3923	1.13533	0.567667	+	0.823258i	$0.307846\pi$
$390$	0	0
$391$	7.60770	0.384738
$392$	− 1.00000i	− 0.0505076i
$393$	8.53590i	0.430579i
$394$	−24.2487	−1.22163
$395$	0	0
$396$	−2.46410	−0.123826
$397$	− 9.60770i	− 0.482196i	−0.970501	−	0.241098i	$-0.922492\pi$
$397$	− 9.60770i	− 0.482196i	0.970501	−	0.241098i	$-0.0775076\pi$
$398$	− 10.9282i	− 0.547781i
$399$	2.39230	0.119765
$400$	0	0
$401$	9.46410	0.472615	0.236307	−	0.971678i	$-0.424063\pi$
$401$	9.46410	0.472615	0.236307	+	0.971678i	$0.424063\pi$
$402$	− 2.92820i	− 0.146046i
$403$	10.9282i	0.544373i
$404$	−19.8564	−0.987893
$405$	0	0
$406$	−1.26795	−0.0629273
$407$	2.73205i	0.135423i
$408$	− 2.53590i	− 0.125546i
$409$	−35.1244	−1.73679	−0.868394	−	0.495875i	$-0.834847\pi$
$409$	−35.1244	−1.73679	−0.868394	+	0.495875i	$0.834847\pi$
$410$	0	0
$411$	−9.46410	−0.466830
$412$	8.39230i	0.413459i
$413$	− 6.92820i	− 0.340915i
$414$	5.41154	0.265963
$415$	0	0
$416$	5.46410	0.267900
$417$	− 2.67949i	− 0.131215i
$418$	3.26795i	0.159841i
$419$	−17.0718	−0.834012	−0.417006	−	0.908904i	$-0.636921\pi$
$419$	−17.0718	−0.834012	−0.417006	+	0.908904i	$0.636921\pi$
$420$	0	0
$421$	−8.14359	−0.396894	−0.198447	−	0.980112i	$-0.563590\pi$
$421$	−8.14359	−0.396894	−0.198447	+	0.980112i	$0.563590\pi$
$422$	− 13.0718i	− 0.636325i
$423$	17.0718i	0.830059i
$424$	10.7321	0.521194
$425$	0	0
$426$	−1.85641	−0.0899432
$427$	− 8.92820i	− 0.432066i
$428$	− 19.8564i	− 0.959796i
$429$	4.00000	0.193122
$430$	0	0
$431$	−21.1244	−1.01752	−0.508762	−	0.860907i	$-0.669897\pi$
$431$	−21.1244	−1.01752	−0.508762	+	0.860907i	$0.669897\pi$
$432$	− 4.00000i	− 0.192450i
$433$	− 7.80385i	− 0.375029i	−0.982262	−	0.187514i	$-0.939957\pi$
$433$	− 7.80385i	− 0.375029i	0.982262	−	0.187514i	$-0.0600432\pi$
$434$	−2.00000	−0.0960031
$435$	0	0
$436$	1.66025	0.0795117
$437$	− 7.17691i	− 0.343318i
$438$	− 4.67949i	− 0.223595i
$439$	22.2487	1.06187	0.530937	−	0.847412i	$-0.321840\pi$
$439$	22.2487	1.06187	0.530937	+	0.847412i	$0.321840\pi$
$440$	0	0
$441$	−2.46410	−0.117338
$442$	− 18.9282i	− 0.900323i
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	−2.00000	−0.0949158
$445$	0	0
$446$	−18.5359	−0.877700
$447$	− 7.85641i	− 0.371595i
$448$	1.00000i	0.0472456i
$449$	−19.6077	−0.925344	−0.462672	−	0.886529i	$-0.653109\pi$
$449$	−19.6077	−0.925344	−0.462672	+	0.886529i	$0.653109\pi$
$450$	0	0
$451$	8.19615	0.385942
$452$	19.8564i	0.933967i
$453$	9.60770i	0.451409i
$454$	6.92820	0.325157
$455$	0	0
$456$	−2.39230	−0.112030
$457$	− 2.00000i	− 0.0935561i	−0.998905	−	0.0467780i	$-0.985105\pi$
$457$	− 2.00000i	− 0.0935561i	0.998905	−	0.0467780i	$-0.0148953\pi$
$458$	3.60770i	0.168577i
$459$	−13.8564	−0.646762
$460$	0	0
$461$	1.60770	0.0748778	0.0374389	−	0.999299i	$-0.488080\pi$
$461$	1.60770	0.0748778	0.0374389	+	0.999299i	$0.488080\pi$
$462$	0.732051i	0.0340581i
$463$	− 17.5167i	− 0.814068i	−0.913413	−	0.407034i	$-0.866563\pi$
$463$	− 17.5167i	− 0.814068i	0.913413	−	0.407034i	$-0.133437\pi$
$464$	1.26795	0.0588631
$465$	0	0
$466$	19.8564	0.919830
$467$	− 4.05256i	− 0.187530i	−0.995594	−	0.0937650i	$-0.970110\pi$
$467$	− 4.05256i	− 0.187530i	0.995594	−	0.0937650i	$-0.0298902\pi$
$468$	− 13.4641i	− 0.622378i
$469$	4.00000	0.184703
$470$	0	0
$471$	−8.39230	−0.386697
$472$	6.92820i	0.318896i
$473$	2.00000i	0.0919601i
$474$	−1.32051	−0.0606529
$475$	0	0
$476$	3.46410	0.158777
$477$	− 26.4449i	− 1.21083i
$478$	4.73205i	0.216439i
$479$	−8.78461	−0.401379	−0.200690	−	0.979655i	$-0.564318\pi$
$479$	−8.78461	−0.401379	−0.200690	+	0.979655i	$0.564318\pi$
$480$	0	0
$481$	−14.9282	−0.680667
$482$	− 6.73205i	− 0.306637i
$483$	− 1.60770i	− 0.0731527i
$484$	−1.00000	−0.0454545
$485$	0	0
$486$	−15.2679	−0.692568
$487$	6.19615i	0.280774i	0.990097	+	0.140387i	$0.0448347\pi$
$487$	6.19615i	0.280774i	−0.990097	+	0.140387i	$0.955165\pi$
$488$	8.92820i	0.404161i
$489$	−2.92820	−0.132418
$490$	0	0
$491$	27.7128	1.25066	0.625331	−	0.780360i	$-0.284964\pi$
$491$	27.7128	1.25066	0.625331	+	0.780360i	$0.284964\pi$
$492$	6.00000i	0.270501i
$493$	− 4.39230i	− 0.197819i
$494$	−17.8564	−0.803398
$495$	0	0
$496$	2.00000	0.0898027
$497$	− 2.53590i	− 0.113751i
$498$	− 3.21539i	− 0.144085i
$499$	−39.8564	−1.78422	−0.892109	−	0.451820i	$-0.850775\pi$
$499$	−39.8564	−1.78422	−0.892109	+	0.451820i	$0.850775\pi$
$500$	0	0
$501$	10.1436	0.453182
$502$	− 12.0000i	− 0.535586i
$503$	− 32.7846i	− 1.46179i	−0.682488	−	0.730897i	$-0.739102\pi$
$503$	− 32.7846i	− 1.46179i	0.682488	−	0.730897i	$-0.260898\pi$
$504$	2.46410	0.109760
$505$	0	0
$506$	2.19615	0.0976309
$507$	12.3397i	0.548027i
$508$	14.9282i	0.662332i
$509$	−24.9282	−1.10492	−0.552462	−	0.833538i	$-0.686311\pi$
$509$	−24.9282	−1.10492	−0.552462	+	0.833538i	$0.686311\pi$
$510$	0	0
$511$	6.39230	0.282779
$512$	− 1.00000i	− 0.0441942i
$513$	13.0718i	0.577134i
$514$	6.33975	0.279634
$515$	0	0
$516$	−1.46410	−0.0644535
$517$	6.92820i	0.304702i
$518$	− 2.73205i	− 0.120039i
$519$	−9.46410	−0.415428
$520$	0	0
$521$	−10.3923	−0.455295	−0.227648	−	0.973744i	$-0.573103\pi$
$521$	−10.3923	−0.455295	−0.227648	+	0.973744i	$0.573103\pi$
$522$	− 3.12436i	− 0.136749i
$523$	33.1769i	1.45073i	0.688367	+	0.725363i	$0.258328\pi$
$523$	33.1769i	1.45073i	−0.688367	+	0.725363i	$0.741672\pi$
$524$	11.6603	0.509381
$525$	0	0
$526$	−24.0000	−1.04645
$527$	− 6.92820i	− 0.301797i
$528$	− 0.732051i	− 0.0318584i
$529$	18.1769	0.790301
$530$	0	0
$531$	17.0718	0.740853
$532$	− 3.26795i	− 0.141684i
$533$	44.7846i	1.93984i
$534$	−2.53590	−0.109739
$535$	0	0
$536$	−4.00000	−0.172774
$537$	− 4.39230i	− 0.189542i
$538$	7.60770i	0.327991i
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−17.2679	−0.742407	−0.371204	−	0.928552i	$-0.621055\pi$
$541$	−17.2679	−0.742407	−0.371204	+	0.928552i	$0.621055\pi$
$542$	20.3923i	0.875924i
$543$	− 10.2487i	− 0.439814i
$544$	−3.46410	−0.148522
$545$	0	0
$546$	−4.00000	−0.171184
$547$	12.7846i	0.546630i	0.961925	+	0.273315i	$0.0881202\pi$
$547$	12.7846i	0.546630i	−0.961925	+	0.273315i	$0.911880\pi$
$548$	12.9282i	0.552265i
$549$	22.0000	0.938937
$550$	0	0
$551$	−4.14359	−0.176523
$552$	1.60770i	0.0684280i
$553$	− 1.80385i	− 0.0767074i
$554$	−20.9282	−0.889154
$555$	0	0
$556$	−3.66025	−0.155229
$557$	46.3923i	1.96571i	0.184393	+	0.982853i	$0.440968\pi$
$557$	46.3923i	1.96571i	−0.184393	+	0.982853i	$0.559032\pi$
$558$	− 4.92820i	− 0.208627i
$559$	−10.9282	−0.462214
$560$	0	0
$561$	−2.53590	−0.107066
$562$	− 1.60770i	− 0.0678165i
$563$	18.9282i	0.797729i	0.917010	+	0.398864i	$0.130596\pi$
$563$	18.9282i	0.797729i	−0.917010	+	0.398864i	$0.869404\pi$
$564$	−5.07180	−0.213561
$565$	0	0
$566$	23.7128	0.996724
$567$	− 4.46410i	− 0.187475i
$568$	2.53590i	0.106404i
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	24.3923	1.02079	0.510393	−	0.859941i	$-0.329500\pi$
$571$	24.3923	1.02079	0.510393	+	0.859941i	$0.329500\pi$
$572$	− 5.46410i	− 0.228466i
$573$	8.78461i	0.366982i
$574$	−8.19615	−0.342101
$575$	0	0
$576$	−2.46410	−0.102671
$577$	3.41154i	0.142024i	0.997475	+	0.0710122i	$0.0226229\pi$
$577$	3.41154i	0.142024i	−0.997475	+	0.0710122i	$0.977377\pi$
$578$	− 5.00000i	− 0.207973i
$579$	−6.14359	−0.255319
$580$	0	0
$581$	4.39230	0.182224
$582$	− 12.1436i	− 0.503368i
$583$	− 10.7321i	− 0.444476i
$584$	−6.39230	−0.264515
$585$	0	0
$586$	14.5359	0.600472
$587$	42.5885i	1.75781i	0.476993	+	0.878907i	$0.341727\pi$
$587$	42.5885i	1.75781i	−0.476993	+	0.878907i	$0.658273\pi$
$588$	− 0.732051i	− 0.0301893i
$589$	−6.53590	−0.269307
$590$	0	0
$591$	−17.7513	−0.730190
$592$	2.73205i	0.112287i
$593$	− 48.2487i	− 1.98134i	−0.136293	−	0.990669i	$-0.543519\pi$
$593$	− 48.2487i	− 1.98134i	0.136293	−	0.990669i	$-0.456481\pi$
$594$	−4.00000	−0.164122
$595$	0	0
$596$	−10.7321	−0.439602
$597$	− 8.00000i	− 0.327418i
$598$	12.0000i	0.490716i
$599$	25.1769	1.02870	0.514350	−	0.857580i	$-0.328033\pi$
$599$	25.1769	1.02870	0.514350	+	0.857580i	$0.328033\pi$
$600$	0	0
$601$	39.5167	1.61192	0.805959	−	0.591971i	$-0.201650\pi$
$601$	39.5167	1.61192	0.805959	+	0.591971i	$0.201650\pi$
$602$	− 2.00000i	− 0.0815139i
$603$	9.85641i	0.401384i
$604$	13.1244	0.534022
$605$	0	0
$606$	−14.5359	−0.590481
$607$	− 7.07180i	− 0.287035i	−0.989648	−	0.143518i	$-0.954159\pi$
$607$	− 7.07180i	− 0.287035i	0.989648	−	0.143518i	$-0.0458414\pi$
$608$	3.26795i	0.132533i
$609$	−0.928203	−0.0376127
$610$	0	0
$611$	−37.8564	−1.53151
$612$	8.53590i	0.345043i
$613$	27.1769i	1.09767i	0.835932	+	0.548833i	$0.184928\pi$
$613$	27.1769i	1.09767i	−0.835932	+	0.548833i	$0.815072\pi$
$614$	−3.60770	−0.145595
$615$	0	0
$616$	1.00000	0.0402911
$617$	17.3205i	0.697297i	0.937253	+	0.348649i	$0.113359\pi$
$617$	17.3205i	0.697297i	−0.937253	+	0.348649i	$0.886641\pi$
$618$	6.14359i	0.247132i
$619$	12.7846	0.513857	0.256928	−	0.966430i	$-0.417290\pi$
$619$	12.7846	0.513857	0.256928	+	0.966430i	$0.417290\pi$
$620$	0	0
$621$	8.78461	0.352514
$622$	− 11.0718i	− 0.443939i
$623$	− 3.46410i	− 0.138786i
$624$	4.00000	0.160128
$625$	0	0
$626$	11.8038	0.471777
$627$	2.39230i	0.0955395i
$628$	11.4641i	0.457467i
$629$	9.46410	0.377358
$630$	0	0
$631$	−33.0718	−1.31657	−0.658284	−	0.752770i	$-0.728717\pi$
$631$	−33.0718	−1.31657	−0.658284	+	0.752770i	$0.728717\pi$
$632$	1.80385i	0.0717532i
$633$	− 9.56922i	− 0.380342i
$634$	0.588457	0.0233706
$635$	0	0
$636$	7.85641	0.311527
$637$	− 5.46410i	− 0.216496i
$638$	− 1.26795i	− 0.0501986i
$639$	6.24871	0.247195
$640$	0	0
$641$	47.5692	1.87887	0.939436	−	0.342725i	$-0.111350\pi$
$641$	47.5692	1.87887	0.939436	+	0.342725i	$0.111350\pi$
$642$	− 14.5359i	− 0.573686i
$643$	36.7321i	1.44857i	0.689500	+	0.724285i	$0.257830\pi$
$643$	36.7321i	1.44857i	−0.689500	+	0.724285i	$0.742170\pi$
$644$	−2.19615	−0.0865405
$645$	0	0
$646$	11.3205	0.445399
$647$	33.4641i	1.31561i	0.753188	+	0.657805i	$0.228515\pi$
$647$	33.4641i	1.31561i	−0.753188	+	0.657805i	$0.771485\pi$
$648$	4.46410i	0.175366i
$649$	6.92820	0.271956
$650$	0	0
$651$	−1.46410	−0.0573827
$652$	4.00000i	0.156652i
$653$	− 5.66025i	− 0.221503i	−0.993848	−	0.110751i	$-0.964674\pi$
$653$	− 5.66025i	− 0.221503i	0.993848	−	0.110751i	$-0.0353257\pi$
$654$	1.21539	0.0475255
$655$	0	0
$656$	8.19615	0.320006
$657$	15.7513i	0.614516i
$658$	− 6.92820i	− 0.270089i
$659$	18.2487	0.710869	0.355434	−	0.934701i	$-0.384333\pi$
$659$	18.2487	0.710869	0.355434	+	0.934701i	$0.384333\pi$
$660$	0	0
$661$	−46.0000	−1.78919	−0.894596	−	0.446875i	$-0.852537\pi$
$661$	−46.0000	−1.78919	−0.894596	+	0.446875i	$0.852537\pi$
$662$	− 22.7846i	− 0.885549i
$663$	− 13.8564i	− 0.538138i
$664$	−4.39230	−0.170454
$665$	0	0
$666$	6.73205	0.260862
$667$	2.78461i	0.107821i
$668$	− 13.8564i	− 0.536120i
$669$	−13.5692	−0.524616
$670$	0	0
$671$	8.92820	0.344669
$672$	0.732051i	0.0282395i
$673$	2.00000i	0.0770943i	0.999257	+	0.0385472i	$0.0122730\pi$
$673$	2.00000i	0.0770943i	−0.999257	+	0.0385472i	$0.987727\pi$
$674$	18.7846	0.723556
$675$	0	0
$676$	16.8564	0.648323
$677$	31.8564i	1.22434i	0.790726	+	0.612171i	$0.209703\pi$
$677$	31.8564i	1.22434i	−0.790726	+	0.612171i	$0.790297\pi$
$678$	14.5359i	0.558248i
$679$	16.5885	0.636607
$680$	0	0
$681$	5.07180	0.194352
$682$	− 2.00000i	− 0.0765840i
$683$	− 42.2487i	− 1.61660i	−0.588769	−	0.808301i	$-0.700387\pi$
$683$	− 42.2487i	− 1.61660i	0.588769	−	0.808301i	$-0.299613\pi$
$684$	8.05256	0.307897
$685$	0	0
$686$	1.00000	0.0381802
$687$	2.64102i	0.100761i
$688$	2.00000i	0.0762493i
$689$	58.6410	2.23404
$690$	0	0
$691$	−6.53590	−0.248637	−0.124319	−	0.992242i	$-0.539675\pi$
$691$	−6.53590	−0.248637	−0.124319	+	0.992242i	$0.539675\pi$
$692$	12.9282i	0.491457i
$693$	− 2.46410i	− 0.0936035i
$694$	36.9282	1.40178
$695$	0	0
$696$	0.928203	0.0351835
$697$	− 28.3923i	− 1.07544i
$698$	− 26.3923i	− 0.998963i
$699$	14.5359	0.549798
$700$	0	0
$701$	−23.9090	−0.903029	−0.451515	−	0.892264i	$-0.649116\pi$
$701$	−23.9090	−0.903029	−0.451515	+	0.892264i	$0.649116\pi$
$702$	− 21.8564i	− 0.824917i
$703$	− 8.92820i	− 0.336734i
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	3.80385	0.143160
$707$	− 19.8564i	− 0.746777i
$708$	5.07180i	0.190610i
$709$	9.32051	0.350039	0.175020	−	0.984565i	$-0.444001\pi$
$709$	9.32051	0.350039	0.175020	+	0.984565i	$0.444001\pi$
$710$	0	0
$711$	4.44486	0.166695
$712$	3.46410i	0.129823i
$713$	4.39230i	0.164493i
$714$	2.53590	0.0949036
$715$	0	0
$716$	−6.00000	−0.224231
$717$	3.46410i	0.129369i
$718$	4.05256i	0.151240i
$719$	−27.7128	−1.03351	−0.516757	−	0.856132i	$-0.672861\pi$
$719$	−27.7128	−1.03351	−0.516757	+	0.856132i	$0.672861\pi$
$720$	0	0
$721$	−8.39230	−0.312546
$722$	8.32051i	0.309657i
$723$	− 4.92820i	− 0.183282i
$724$	−14.0000	−0.520306
$725$	0	0
$726$	−0.732051	−0.0271690
$727$	4.00000i	0.148352i	0.997245	+	0.0741759i	$0.0236326\pi$
$727$	4.00000i	0.148352i	−0.997245	+	0.0741759i	$0.976367\pi$
$728$	5.46410i	0.202513i
$729$	2.21539	0.0820515
$730$	0	0
$731$	6.92820	0.256249
$732$	6.53590i	0.241574i
$733$	26.0000i	0.960332i	0.877178	+	0.480166i	$0.159424\pi$
$733$	26.0000i	0.960332i	−0.877178	+	0.480166i	$0.840576\pi$
$734$	8.39230	0.309766
$735$	0	0
$736$	2.19615	0.0809513
$737$	4.00000i	0.147342i
$738$	− 20.1962i	− 0.743431i
$739$	43.7128	1.60800	0.804001	−	0.594628i	$-0.202701\pi$
$739$	43.7128	1.60800	0.804001	+	0.594628i	$0.202701\pi$
$740$	0	0
$741$	−13.0718	−0.480204
$742$	10.7321i	0.393986i
$743$	8.78461i	0.322276i	0.986932	+	0.161138i	$0.0515164\pi$
$743$	8.78461i	0.322276i	−0.986932	+	0.161138i	$0.948484\pi$
$744$	1.46410	0.0536766
$745$	0	0
$746$	−2.39230	−0.0875885
$747$	10.8231i	0.395996i
$748$	3.46410i	0.126660i
$749$	19.8564	0.725537
$750$	0	0
$751$	−32.3923	−1.18201	−0.591006	−	0.806667i	$-0.701269\pi$
$751$	−32.3923	−1.18201	−0.591006	+	0.806667i	$0.701269\pi$
$752$	6.92820i	0.252646i
$753$	− 8.78461i	− 0.320129i
$754$	6.92820	0.252310
$755$	0	0
$756$	4.00000	0.145479
$757$	− 29.3731i	− 1.06758i	−0.845616	−	0.533791i	$-0.820767\pi$
$757$	− 29.3731i	− 1.06758i	0.845616	−	0.533791i	$-0.179233\pi$
$758$	33.8564i	1.22972i
$759$	1.60770	0.0583556
$760$	0	0
$761$	−29.6603	−1.07518	−0.537592	−	0.843205i	$-0.680666\pi$
$761$	−29.6603	−1.07518	−0.537592	+	0.843205i	$0.680666\pi$
$762$	10.9282i	0.395887i
$763$	1.66025i	0.0601052i
$764$	12.0000	0.434145
$765$	0	0
$766$	−26.5359	−0.958781
$767$	37.8564i	1.36692i
$768$	− 0.732051i	− 0.0264156i
$769$	7.80385	0.281414	0.140707	−	0.990051i	$-0.455062\pi$
$769$	7.80385	0.281414	0.140707	+	0.990051i	$0.455062\pi$
$770$	0	0
$771$	4.64102	0.167142
$772$	8.39230i	0.302046i
$773$	12.9282i	0.464995i	0.972597	+	0.232498i	$0.0746898\pi$
$773$	12.9282i	0.464995i	−0.972597	+	0.232498i	$0.925310\pi$
$774$	4.92820	0.177141
$775$	0	0
$776$	−16.5885	−0.595491
$777$	− 2.00000i	− 0.0717496i
$778$	− 22.3923i	− 0.802803i
$779$	−26.7846	−0.959658
$780$	0	0
$781$	2.53590	0.0907416
$782$	− 7.60770i	− 0.272051i
$783$	− 5.07180i	− 0.181251i
$784$	−1.00000	−0.0357143
$785$	0	0
$786$	8.53590	0.304465
$787$	− 45.8564i	− 1.63460i	−0.576209	−	0.817302i	$-0.695469\pi$
$787$	− 45.8564i	− 1.63460i	0.576209	−	0.817302i	$-0.304531\pi$
$788$	24.2487i	0.863825i
$789$	−17.5692	−0.625481
$790$	0	0
$791$	−19.8564	−0.706013
$792$	2.46410i	0.0875580i
$793$	48.7846i	1.73239i
$794$	−9.60770	−0.340964
$795$	0	0
$796$	−10.9282	−0.387340
$797$	46.3923i	1.64330i	0.569993	+	0.821650i	$0.306946\pi$
$797$	46.3923i	1.64330i	−0.569993	+	0.821650i	$0.693054\pi$
$798$	− 2.39230i	− 0.0846867i
$799$	24.0000	0.849059
$800$	0	0
$801$	8.53590	0.301601
$802$	− 9.46410i	− 0.334189i
$803$	6.39230i	0.225580i
$804$	−2.92820	−0.103270
$805$	0	0
$806$	10.9282	0.384930
$807$	5.56922i	0.196046i
$808$	19.8564i	0.698546i
$809$	−31.1769	−1.09612	−0.548061	−	0.836438i	$-0.684634\pi$
$809$	−31.1769	−1.09612	−0.548061	+	0.836438i	$0.684634\pi$
$810$	0	0
$811$	−18.8756	−0.662814	−0.331407	−	0.943488i	$-0.607523\pi$
$811$	−18.8756	−0.662814	−0.331407	+	0.943488i	$0.607523\pi$
$812$	1.26795i	0.0444963i
$813$	14.9282i	0.523555i
$814$	2.73205	0.0957583
$815$	0	0
$816$	−2.53590	−0.0887742
$817$	− 6.53590i	− 0.228662i
$818$	35.1244i	1.22809i
$819$	13.4641	0.470474
$820$	0	0
$821$	−47.9090	−1.67203	−0.836017	−	0.548703i	$-0.815122\pi$
$821$	−47.9090	−1.67203	−0.836017	+	0.548703i	$0.815122\pi$
$822$	9.46410i	0.330098i
$823$	− 25.1244i	− 0.875780i	−0.899029	−	0.437890i	$-0.855726\pi$
$823$	− 25.1244i	− 0.875780i	0.899029	−	0.437890i	$-0.144274\pi$
$824$	8.39230	0.292360
$825$	0	0
$826$	−6.92820	−0.241063
$827$	− 1.85641i	− 0.0645536i	−0.999479	−	0.0322768i	$-0.989724\pi$
$827$	− 1.85641i	− 0.0645536i	0.999479	−	0.0322768i	$-0.0102758\pi$
$828$	− 5.41154i	− 0.188064i
$829$	46.2487	1.60628	0.803142	−	0.595788i	$-0.203160\pi$
$829$	46.2487	1.60628	0.803142	+	0.595788i	$0.203160\pi$
$830$	0	0
$831$	−15.3205	−0.531463
$832$	− 5.46410i	− 0.189434i
$833$	3.46410i	0.120024i
$834$	−2.67949	−0.0927832
$835$	0	0
$836$	3.26795	0.113024
$837$	− 8.00000i	− 0.276520i
$838$	17.0718i	0.589735i
$839$	4.14359	0.143053	0.0715264	−	0.997439i	$-0.477213\pi$
$839$	4.14359	0.143053	0.0715264	+	0.997439i	$0.477213\pi$
$840$	0	0
$841$	−27.3923	−0.944562
$842$	8.14359i	0.280647i
$843$	− 1.17691i	− 0.0405351i
$844$	−13.0718	−0.449950
$845$	0	0
$846$	17.0718	0.586940
$847$	− 1.00000i	− 0.0343604i
$848$	− 10.7321i	− 0.368540i
$849$	17.3590	0.595759
$850$	0	0
$851$	−6.00000	−0.205677
$852$	1.85641i	0.0635994i
$853$	− 13.2154i	− 0.452486i	−0.974071	−	0.226243i	$-0.927356\pi$
$853$	− 13.2154i	− 0.452486i	0.974071	−	0.226243i	$-0.0726444\pi$
$854$	−8.92820	−0.305517
$855$	0	0
$856$	−19.8564	−0.678678
$857$	20.5359i	0.701493i	0.936470	+	0.350746i	$0.114072\pi$
$857$	20.5359i	0.701493i	−0.936470	+	0.350746i	$0.885928\pi$
$858$	− 4.00000i	− 0.136558i
$859$	−28.7846	−0.982118	−0.491059	−	0.871126i	$-0.663390\pi$
$859$	−28.7846	−0.982118	−0.491059	+	0.871126i	$0.663390\pi$
$860$	0	0
$861$	−6.00000	−0.204479
$862$	21.1244i	0.719498i
$863$	37.5167i	1.27708i	0.769588	+	0.638541i	$0.220462\pi$
$863$	37.5167i	1.27708i	−0.769588	+	0.638541i	$0.779538\pi$
$864$	−4.00000	−0.136083
$865$	0	0
$866$	−7.80385	−0.265186
$867$	− 3.66025i	− 0.124309i
$868$	2.00000i	0.0678844i
$869$	1.80385	0.0611913
$870$	0	0
$871$	−21.8564	−0.740576
$872$	− 1.66025i	− 0.0562233i
$873$	40.8756i	1.38343i
$874$	−7.17691	−0.242763
$875$	0	0
$876$	−4.67949	−0.158105
$877$	− 13.3205i	− 0.449802i	−0.974382	−	0.224901i	$-0.927794\pi$
$877$	− 13.3205i	− 0.449802i	0.974382	−	0.224901i	$-0.0722058\pi$
$878$	− 22.2487i	− 0.750858i
$879$	10.6410	0.358913
$880$	0	0
$881$	24.9282	0.839853	0.419926	−	0.907558i	$-0.362056\pi$
$881$	24.9282	0.839853	0.419926	+	0.907558i	$0.362056\pi$
$882$	2.46410i	0.0829706i
$883$	− 56.3923i	− 1.89775i	−0.315649	−	0.948876i	$-0.602222\pi$
$883$	− 56.3923i	− 1.89775i	0.315649	−	0.948876i	$-0.397778\pi$
$884$	−18.9282	−0.636624
$885$	0	0
$886$	0	0
$887$	13.8564i	0.465253i	0.972566	+	0.232626i	$0.0747319\pi$
$887$	13.8564i	0.465253i	−0.972566	+	0.232626i	$0.925268\pi$
$888$	2.00000i	0.0671156i
$889$	−14.9282	−0.500676
$890$	0	0
$891$	4.46410	0.149553
$892$	18.5359i	0.620628i
$893$	− 22.6410i	− 0.757653i
$894$	−7.85641	−0.262758
$895$	0	0
$896$	1.00000	0.0334077
$897$	8.78461i	0.293310i
$898$	19.6077i	0.654317i
$899$	2.53590	0.0845769
$900$	0	0
$901$	−37.1769	−1.23854
$902$	− 8.19615i	− 0.272902i
$903$	− 1.46410i	− 0.0487223i
$904$	19.8564	0.660414
$905$	0	0
$906$	9.60770	0.319194
$907$	− 59.0333i	− 1.96017i	−0.198580	−	0.980085i	$-0.563633\pi$
$907$	− 59.0333i	− 1.96017i	0.198580	−	0.980085i	$-0.436367\pi$
$908$	− 6.92820i	− 0.229920i
$909$	48.9282	1.62285
$910$	0	0
$911$	2.53590	0.0840181	0.0420090	−	0.999117i	$-0.486624\pi$
$911$	2.53590	0.0840181	0.0420090	+	0.999117i	$0.486624\pi$
$912$	2.39230i	0.0792171i
$913$	4.39230i	0.145364i
$914$	−2.00000	−0.0661541
$915$	0	0
$916$	3.60770	0.119202
$917$	11.6603i	0.385056i
$918$	13.8564i	0.457330i
$919$	−47.3731	−1.56269	−0.781347	−	0.624097i	$-0.785467\pi$
$919$	−47.3731	−1.56269	−0.781347	+	0.624097i	$0.785467\pi$
$920$	0	0
$921$	−2.64102	−0.0870244
$922$	− 1.60770i	− 0.0529466i
$923$	13.8564i	0.456089i
$924$	0.732051	0.0240827
$925$	0	0
$926$	−17.5167	−0.575633
$927$	− 20.6795i	− 0.679204i
$928$	− 1.26795i	− 0.0416225i
$929$	46.3923	1.52208	0.761041	−	0.648704i	$-0.224689\pi$
$929$	46.3923	1.52208	0.761041	+	0.648704i	$0.224689\pi$
$930$	0	0
$931$	3.26795	0.107103
$932$	− 19.8564i	− 0.650418i
$933$	− 8.10512i	− 0.265350i
$934$	−4.05256	−0.132604
$935$	0	0
$936$	−13.4641	−0.440088
$937$	42.7846i	1.39771i	0.715262	+	0.698856i	$0.246307\pi$
$937$	42.7846i	1.39771i	−0.715262	+	0.698856i	$0.753693\pi$
$938$	− 4.00000i	− 0.130605i
$939$	8.64102	0.281989
$940$	0	0
$941$	26.7846	0.873153	0.436577	−	0.899667i	$-0.356191\pi$
$941$	26.7846	0.873153	0.436577	+	0.899667i	$0.356191\pi$
$942$	8.39230i	0.273436i
$943$	18.0000i	0.586161i
$944$	6.92820	0.225494
$945$	0	0
$946$	2.00000	0.0650256
$947$	− 12.6795i	− 0.412028i	−0.978549	−	0.206014i	$-0.933951\pi$
$947$	− 12.6795i	− 0.412028i	0.978549	−	0.206014i	$-0.0660493\pi$
$948$	1.32051i	0.0428881i
$949$	−34.9282	−1.13382
$950$	0	0
$951$	0.430781	0.0139690
$952$	− 3.46410i	− 0.112272i
$953$	33.4641i	1.08401i	0.840376	+	0.542004i	$0.182334\pi$
$953$	33.4641i	1.08401i	−0.840376	+	0.542004i	$0.817666\pi$
$954$	−26.4449	−0.856184
$955$	0	0
$956$	4.73205	0.153045
$957$	− 0.928203i	− 0.0300045i
$958$	8.78461i	0.283818i
$959$	−12.9282	−0.417473
$960$	0	0
$961$	−27.0000	−0.870968
$962$	14.9282i	0.481305i
$963$	48.9282i	1.57669i
$964$	−6.73205	−0.216825
$965$	0	0
$966$	−1.60770	−0.0517267
$967$	− 13.0718i	− 0.420361i	−0.977663	−	0.210180i	$-0.932595\pi$
$967$	− 13.0718i	− 0.420361i	0.977663	−	0.210180i	$-0.0674051\pi$
$968$	1.00000i	0.0321412i
$969$	8.28719	0.266223
$970$	0	0
$971$	29.0718	0.932958	0.466479	−	0.884532i	$-0.345522\pi$
$971$	29.0718	0.932958	0.466479	+	0.884532i	$0.345522\pi$
$972$	15.2679i	0.489720i
$973$	− 3.66025i	− 0.117342i
$974$	6.19615	0.198538
$975$	0	0
$976$	8.92820	0.285785
$977$	− 21.7128i	− 0.694654i	−0.937744	−	0.347327i	$-0.887089\pi$
$977$	− 21.7128i	− 0.694654i	0.937744	−	0.347327i	$-0.112911\pi$
$978$	2.92820i	0.0936336i
$979$	3.46410	0.110713
$980$	0	0
$981$	−4.09103	−0.130617
$982$	− 27.7128i	− 0.884351i
$983$	25.1769i	0.803019i	0.915855	+	0.401509i	$0.131514\pi$
$983$	25.1769i	0.803019i	−0.915855	+	0.401509i	$0.868486\pi$
$984$	6.00000	0.191273
$985$	0	0
$986$	−4.39230	−0.139879
$987$	− 5.07180i	− 0.161437i
$988$	17.8564i	0.568088i
$989$	−4.39230	−0.139667
$990$	0	0
$991$	−4.00000	−0.127064	−0.0635321	−	0.997980i	$-0.520237\pi$
$991$	−4.00000	−0.127064	−0.0635321	+	0.997980i	$0.520237\pi$
$992$	− 2.00000i	− 0.0635001i
$993$	− 16.6795i	− 0.529308i
$994$	−2.53590	−0.0804338
$995$	0	0
$996$	−3.21539	−0.101884
$997$	37.7128i	1.19438i	0.802101	+	0.597188i	$0.203716\pi$
$997$	37.7128i	1.19438i	−0.802101	+	0.597188i	$0.796284\pi$
$998$	39.8564i	1.26163i
$999$	10.9282	0.345753

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3850.2.c.x.1849.1		4
5.2	odd	4		770.2.a.j.1.1	✓	2
5.3	odd	4		3850.2.a.bd.1.2		2
5.4	even	2	inner	3850.2.c.x.1849.4		4
15.2	even	4		6930.2.a.bv.1.2		2
20.7	even	4		6160.2.a.t.1.2		2
35.27	even	4		5390.2.a.bs.1.2		2
55.32	even	4		8470.2.a.br.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.a.j.1.1	✓	2	5.2	odd	4
3850.2.a.bd.1.2		2	5.3	odd	4
3850.2.c.x.1849.1		4	1.1	even	1	trivial
3850.2.c.x.1849.4		4	5.4	even	2	inner
5390.2.a.bs.1.2		2	35.27	even	4
6160.2.a.t.1.2		2	20.7	even	4
6930.2.a.bv.1.2		2	15.2	even	4
8470.2.a.br.1.1		2	55.32	even	4

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:	\( N \)	\(=\)	\( 3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3850.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -0.732051i q^{3} -1.00000 q^{4} -0.732051 q^{6} -1.00000i q^{7} +1.00000i q^{8} +2.46410 q^{9} +1.00000 q^{11} +0.732051i q^{12} +5.46410i q^{13} -1.00000 q^{14} +1.00000 q^{16} -3.46410i q^{17} -2.46410i q^{18} -3.26795 q^{19} -0.732051 q^{21} -1.00000i q^{22} +2.19615i q^{23} +0.732051 q^{24} +5.46410 q^{26} -4.00000i q^{27} +1.00000i q^{28} +1.26795 q^{29} +2.00000 q^{31} -1.00000i q^{32} -0.732051i q^{33} -3.46410 q^{34} -2.46410 q^{36} +2.73205i q^{37} +3.26795i q^{38} +4.00000 q^{39} +8.19615 q^{41} +0.732051i q^{42} +2.00000i q^{43} -1.00000 q^{44} +2.19615 q^{46} +6.92820i q^{47} -0.732051i q^{48} -1.00000 q^{49} -2.53590 q^{51} -5.46410i q^{52} -10.7321i q^{53} -4.00000 q^{54} +1.00000 q^{56} +2.39230i q^{57} -1.26795i q^{58} +6.92820 q^{59} +8.92820 q^{61} -2.00000i q^{62} -2.46410i q^{63} -1.00000 q^{64} -0.732051 q^{66} +4.00000i q^{67} +3.46410i q^{68} +1.60770 q^{69} +2.53590 q^{71} +2.46410i q^{72} +6.39230i q^{73} +2.73205 q^{74} +3.26795 q^{76} -1.00000i q^{77} -4.00000i q^{78} +1.80385 q^{79} +4.46410 q^{81} -8.19615i q^{82} +4.39230i q^{83} +0.732051 q^{84} +2.00000 q^{86} -0.928203i q^{87} +1.00000i q^{88} +3.46410 q^{89} +5.46410 q^{91} -2.19615i q^{92} -1.46410i q^{93} +6.92820 q^{94} -0.732051 q^{96} +16.5885i q^{97} +1.00000i q^{98} +2.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9} + 4 q^{11} - 4 q^{14} + 4 q^{16} - 20 q^{19} + 4 q^{21} - 4 q^{24} + 8 q^{26} + 12 q^{29} + 8 q^{31} + 4 q^{36} + 16 q^{39} + 12 q^{41} - 4 q^{44} - 12 q^{46} - 4 q^{49}+ \cdots - 4 q^{99}+O(q^{100}) \) 4 * q - 4 * q^4 + 4 * q^6 - 4 * q^9 + 4 * q^11 - 4 * q^14 + 4 * q^16 - 20 * q^19 + 4 * q^21 - 4 * q^24 + 8 * q^26 + 12 * q^29 + 8 * q^31 + 4 * q^36 + 16 * q^39 + 12 * q^41 - 4 * q^44 - 12 * q^46 - 4 * q^49 - 24 * q^51 - 16 * q^54 + 4 * q^56 + 8 * q^61 - 4 * q^64 + 4 * q^66 + 48 * q^69 + 24 * q^71 + 4 * q^74 + 20 * q^76 + 28 * q^79 + 4 * q^81 - 4 * q^84 + 8 * q^86 + 8 * q^91 + 4 * q^96 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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