LMFDB - Embedded newform 1968.2.j.f.1393.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1968,2,Mod(1393,1968)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1968, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1968.1393");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1968 = 2^{4} \cdot 3 \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1968.j (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$15.7145591178$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.265727878144.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 15x^{6} + 67x^{4} + 77x^{2} + 4 $ x^8 + 15x^6 + 67x^4 + 77*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 984)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1393.7
Root			$1.29051i$ of defining polynomial
Character	$\chi$	$=$	1968.1393
Dual form			1968.2.j.f.1393.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +2.29051 q^{5} -1.04406i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +2.29051 q^{5} -1.04406i q^{7} -1.00000 q^{9} +5.59383i q^{11} +2.59383i q^{13} +2.29051i q^{15} -2.25926i q^{17} +5.17486i q^{19} +1.04406 q^{21} -3.92840 q^{23} +0.246456 q^{25} -1.00000i q^{27} +0.987200i q^{29} -10.5535 q^{31} -5.59383 q^{33} -2.39143i q^{35} -1.75354 q^{37} -2.59383 q^{39} +(5.84028 - 2.62509i) q^{41} +9.15640 q^{43} -2.29051 q^{45} +3.22800i q^{47} +5.90994 q^{49} +2.25926 q^{51} +6.88434i q^{53} +12.8127i q^{55} -5.17486 q^{57} -1.61229 q^{59} -1.88062 q^{61} +1.04406i q^{63} +5.94120i q^{65} +14.8881i q^{67} -3.92840i q^{69} -4.92840i q^{71} -2.76869 q^{73} +0.246456i q^{75} +5.84028 q^{77} -3.50709i q^{79} +1.00000 q^{81} +4.89714 q^{83} -5.17486i q^{85} -0.987200 q^{87} -2.45023i q^{89} +2.70811 q^{91} -10.5535i q^{93} +11.8531i q^{95} -0.374913i q^{97} -5.59383i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 10 q^{5} - 8 q^{9}+O(q^{10}) $ 8 * q + 10 * q^5 - 8 * q^9	$ 8 q + 10 q^{5} - 8 q^{9} + 20 q^{23} + 2 q^{25} - 8 q^{31} - 10 q^{33} - 14 q^{37} + 14 q^{39} + 12 q^{41} - 6 q^{43} - 10 q^{45} - 32 q^{49} + 10 q^{57} - 6 q^{59} - 22 q^{61} + 64 q^{73} + 12 q^{77} + 8 q^{81} - 22 q^{83} - 26 q^{87} + 12 q^{91}+O(q^{100}) $ 8 * q + 10 * q^5 - 8 * q^9 + 20 * q^23 + 2 * q^25 - 8 * q^31 - 10 * q^33 - 14 * q^37 + 14 * q^39 + 12 * q^41 - 6 * q^43 - 10 * q^45 - 32 * q^49 + 10 * q^57 - 6 * q^59 - 22 * q^61 + 64 * q^73 + 12 * q^77 + 8 * q^81 - 22 * q^83 - 26 * q^87 + 12 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1968\mathbb{Z}\right)^\times$.

$n$	$1231$	$1313$	$1441$	$1477$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$		0			0
$5$	2.29051			1.02435			0.512175	−	0.858881i	$-0.328840\pi$
$5$	2.29051			1.02435			0.512175	+	0.858881i	$0.328840\pi$
$6$		0			0
$7$		−	1.04406i		−	0.394617i	−0.980341	−	0.197309i	$-0.936780\pi$
$7$		−	1.04406i		−	0.394617i	0.980341	−	0.197309i	$-0.0632201\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$			5.59383i			1.68660i	0.537441	+	0.843301i	$0.319391\pi$
$11$			5.59383i			1.68660i	−0.537441	+	0.843301i	$0.680609\pi$
$12$		0			0
$13$			2.59383i			0.719399i	0.933068	+	0.359699i	$0.117121\pi$
$13$			2.59383i			0.719399i	−0.933068	+	0.359699i	$0.882879\pi$
$14$		0			0
$15$			2.29051i			0.591408i
$16$		0			0
$17$		−	2.25926i		−	0.547950i	−0.961737	−	0.273975i	$-0.911661\pi$
$17$		−	2.25926i		−	0.547950i	0.961737	−	0.273975i	$-0.0883385\pi$
$18$		0			0
$19$			5.17486i			1.18719i	0.804763	+	0.593597i	$0.202293\pi$
$19$			5.17486i			1.18719i	−0.804763	+	0.593597i	$0.797707\pi$
$20$		0			0
$21$	1.04406			0.227832
$22$		0			0
$23$	−3.92840			−0.819128			−0.409564	−	0.912281i	$-0.634319\pi$
$23$	−3.92840			−0.819128			−0.409564	+	0.912281i	$0.634319\pi$
$24$		0			0
$25$	0.246456			0.0492912
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$			0.987200i			0.183319i	0.995790	+	0.0916593i	$0.0292171\pi$
$29$			0.987200i			0.183319i	−0.995790	+	0.0916593i	$0.970783\pi$
$30$		0			0
$31$	−10.5535			−1.89546			−0.947731	−	0.319070i	$-0.896630\pi$
$31$	−10.5535			−1.89546			−0.947731	+	0.319070i	$0.896630\pi$
$32$		0			0
$33$	−5.59383			−0.973761
$34$		0			0
$35$		−	2.39143i		−	0.404226i
$36$		0			0
$37$	−1.75354			−0.288281			−0.144140	−	0.989557i	$-0.546042\pi$
$37$	−1.75354			−0.288281			−0.144140	+	0.989557i	$0.546042\pi$
$38$		0			0
$39$	−2.59383			−0.415345
$40$		0			0
$41$	5.84028	−	2.62509i	0.912099	−	0.409970i
$41$	5.84028	−	2.62509i	0.912099	−	0.409970i
$42$		0			0
$43$	9.15640			1.39634			0.698169	−	0.715933i	$-0.253999\pi$
$43$	9.15640			1.39634			0.698169	+	0.715933i	$0.253999\pi$
$44$		0			0
$45$	−2.29051			−0.341450
$46$		0			0
$47$			3.22800i			0.470852i	0.971892	+	0.235426i	$0.0756485\pi$
$47$			3.22800i			0.470852i	−0.971892	+	0.235426i	$0.924352\pi$
$48$		0			0
$49$	5.90994			0.844277
$50$		0			0
$51$	2.25926			0.316359
$52$		0			0
$53$			6.88434i			0.945637i	0.881160	+	0.472819i	$0.156763\pi$
$53$			6.88434i			0.945637i	−0.881160	+	0.472819i	$0.843237\pi$
$54$		0			0
$55$			12.8127i			1.72767i
$56$		0			0
$57$	−5.17486			−0.685427
$58$		0			0
$59$	−1.61229			−0.209902			−0.104951	−	0.994477i	$-0.533469\pi$
$59$	−1.61229			−0.209902			−0.104951	+	0.994477i	$0.533469\pi$
$60$		0			0
$61$	−1.88062			−0.240789			−0.120395	−	0.992726i	$-0.538416\pi$
$61$	−1.88062			−0.240789			−0.120395	+	0.992726i	$0.538416\pi$
$62$		0			0
$63$			1.04406i			0.131539i
$64$		0			0
$65$			5.94120i			0.736915i
$66$		0			0
$67$			14.8881i			1.81887i	0.415850	+	0.909433i	$0.363484\pi$
$67$			14.8881i			1.81887i	−0.415850	+	0.909433i	$0.636516\pi$
$68$		0			0
$69$		−	3.92840i		−	0.472924i
$70$		0			0
$71$		−	4.92840i		−	0.584894i	−0.956282	−	0.292447i	$-0.905531\pi$
$71$		−	4.92840i		−	0.584894i	0.956282	−	0.292447i	$-0.0944694\pi$
$72$		0			0
$73$	−2.76869			−0.324050			−0.162025	−	0.986787i	$-0.551803\pi$
$73$	−2.76869			−0.324050			−0.162025	+	0.986787i	$0.551803\pi$
$74$		0			0
$75$			0.246456i			0.0284583i
$76$		0			0
$77$	5.84028			0.665562
$78$		0			0
$79$		−	3.50709i		−	0.394578i	−0.980345	−	0.197289i	$-0.936786\pi$
$79$		−	3.50709i		−	0.394578i	0.980345	−	0.197289i	$-0.0632138\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	4.89714			0.537531			0.268766	−	0.963206i	$-0.413384\pi$
$83$	4.89714			0.537531			0.268766	+	0.963206i	$0.413384\pi$
$84$		0			0
$85$		−	5.17486i		−	0.561292i
$86$		0			0
$87$	−0.987200			−0.105839
$88$		0			0
$89$		−	2.45023i		−	0.259724i	−0.991532	−	0.129862i	$-0.958547\pi$
$89$		−	2.45023i		−	0.259724i	0.991532	−	0.129862i	$-0.0414534\pi$
$90$		0			0
$91$	2.70811			0.283887
$92$		0			0
$93$		−	10.5535i		−	1.09435i
$94$		0			0
$95$			11.8531i			1.21610i
$96$		0			0
$97$		−	0.374913i		−	0.0380666i	−0.999819	−	0.0190333i	$-0.993941\pi$
$97$		−	0.374913i		−	0.0380666i	0.999819	−	0.0190333i	$-0.00605886\pi$
$98$		0			0
$99$		−	5.59383i		−	0.562201i
$100$		0			0
$101$		−	1.20240i		−	0.119643i	−0.998209	−	0.0598215i	$-0.980947\pi$
$101$		−	1.20240i		−	0.119643i	0.998209	−	0.0598215i	$-0.0190532\pi$
$102$		0			0
$103$	18.9449			1.86670			0.933349	−	0.358970i	$-0.116872\pi$
$103$	18.9449			1.86670			0.933349	+	0.358970i	$0.116872\pi$
$104$		0			0
$105$	2.39143			0.233380
$106$		0			0
$107$	−11.8995			−1.15037			−0.575183	−	0.818025i	$-0.695069\pi$
$107$	−11.8995			−1.15037			−0.575183	+	0.818025i	$0.695069\pi$
$108$		0			0
$109$			6.20612i			0.594438i	0.954809	+	0.297219i	$0.0960592\pi$
$109$			6.20612i			0.594438i	−0.954809	+	0.297219i	$0.903941\pi$
$110$		0			0
$111$		−	1.75354i		−	0.166439i
$112$		0			0
$113$	11.8659			1.11625			0.558124	−	0.829757i	$-0.311521\pi$
$113$	11.8659			1.11625			0.558124	+	0.829757i	$0.311521\pi$
$114$		0			0
$115$	−8.99806			−0.839073
$116$		0			0
$117$		−	2.59383i		−	0.239800i
$118$		0			0
$119$	−2.35879			−0.216230
$120$		0			0
$121$	−20.2909			−1.84463
$122$		0			0
$123$	2.62509	+	5.84028i	0.236696	+	0.526601i
$124$		0			0
$125$	−10.8881			−0.973858
$126$		0			0
$127$	−6.00372			−0.532744			−0.266372	−	0.963870i	$-0.585825\pi$
$127$	−6.00372			−0.532744			−0.266372	+	0.963870i	$0.585825\pi$
$128$		0			0
$129$			9.15640i			0.806176i
$130$		0			0
$131$	−18.9398			−1.65478			−0.827390	−	0.561628i	$-0.810175\pi$
$131$	−18.9398			−1.65478			−0.827390	+	0.561628i	$0.810175\pi$
$132$		0			0
$133$	5.40285			0.468487
$134$		0			0
$135$		−	2.29051i		−	0.197136i
$136$		0			0
$137$			13.8554i			1.18375i	0.806030	+	0.591874i	$0.201612\pi$
$137$			13.8554i			1.18375i	−0.806030	+	0.591874i	$0.798388\pi$
$138$		0			0
$139$	10.8843			0.923198			0.461599	−	0.887089i	$-0.347276\pi$
$139$	10.8843			0.923198			0.461599	+	0.887089i	$0.347276\pi$
$140$		0			0
$141$	−3.22800			−0.271846
$142$		0			0
$143$	−14.5094			−1.21334
$144$		0			0
$145$			2.26120i			0.187782i
$146$		0			0
$147$			5.90994i			0.487444i
$148$		0			0
$149$			8.66915i			0.710204i	0.934828	+	0.355102i	$0.115554\pi$
$149$			8.66915i			0.710204i	−0.934828	+	0.355102i	$0.884446\pi$
$150$		0			0
$151$		−	20.9307i		−	1.70332i	−0.524095	−	0.851660i	$-0.675596\pi$
$151$		−	20.9307i		−	1.70332i	0.524095	−	0.851660i	$-0.324404\pi$
$152$		0			0
$153$			2.25926i			0.182650i
$154$		0			0
$155$	−24.1729			−1.94162
$156$		0			0
$157$		−	1.39337i		−	0.111203i	−0.998453	−	0.0556016i	$-0.982292\pi$
$157$		−	1.39337i		−	0.111203i	0.998453	−	0.0556016i	$-0.0177077\pi$
$158$		0			0
$159$	−6.88434			−0.545964
$160$		0			0
$161$			4.10148i			0.323242i
$162$		0			0
$163$	21.4805			1.68248			0.841242	−	0.540659i	$-0.181825\pi$
$163$	21.4805			1.68248			0.841242	+	0.540659i	$0.181825\pi$
$164$		0			0
$165$	−12.8127			−0.997471
$166$		0			0
$167$		−	15.1033i		−	1.16873i	−0.811493	−	0.584363i	$-0.801345\pi$
$167$		−	15.1033i		−	1.16873i	0.811493	−	0.584363i	$-0.198655\pi$
$168$		0			0
$169$	6.27206			0.482466
$170$		0			0
$171$		−	5.17486i		−	0.395731i
$172$		0			0
$173$	−5.86588			−0.445975			−0.222987	−	0.974821i	$-0.571581\pi$
$173$	−5.86588			−0.445975			−0.222987	+	0.974821i	$0.571581\pi$
$174$		0			0
$175$		−	0.257314i		−	0.0194511i
$176$		0			0
$177$		−	1.61229i		−	0.121187i
$178$		0			0
$179$		−	19.7027i		−	1.47265i	−0.676627	−	0.736326i	$-0.736559\pi$
$179$		−	19.7027i		−	1.47265i	0.676627	−	0.736326i	$-0.263441\pi$
$180$		0			0
$181$			9.57537i			0.711732i	0.934537	+	0.355866i	$0.115814\pi$
$181$			9.57537i			0.711732i	−0.934537	+	0.355866i	$0.884186\pi$
$182$		0			0
$183$		−	1.88062i		−	0.139020i
$184$		0			0
$185$	−4.01652			−0.295300
$186$		0			0
$187$	12.6379			0.924174
$188$		0			0
$189$	−1.04406			−0.0759441
$190$		0			0
$191$			12.6066i			0.912184i	0.889933	+	0.456092i	$0.150751\pi$
$191$			12.6066i			0.912184i	−0.889933	+	0.456092i	$0.849249\pi$
$192$		0			0
$193$			24.9862i			1.79855i	0.437386	+	0.899274i	$0.355904\pi$
$193$			24.9862i			1.79855i	−0.437386	+	0.899274i	$0.644096\pi$
$194$		0			0
$195$	−5.94120			−0.425458
$196$		0			0
$197$	−7.79760			−0.555556			−0.277778	−	0.960645i	$-0.589598\pi$
$197$	−7.79760			−0.555556			−0.277778	+	0.960645i	$0.589598\pi$
$198$		0			0
$199$			12.9009i			0.914518i	0.889334	+	0.457259i	$0.151169\pi$
$199$			12.9009i			0.914518i	−0.889334	+	0.457259i	$0.848831\pi$
$200$		0			0
$201$	−14.8881			−1.05012
$202$		0			0
$203$	1.03069			0.0723406
$204$		0			0
$205$	13.3773	−	6.01280i	0.934308	−	0.419952i
$206$		0			0
$207$	3.92840			0.273043
$208$		0			0
$209$	−28.9473			−2.00232
$210$		0			0
$211$		−	2.81811i		−	0.194006i	−0.995284	−	0.0970032i	$-0.969074\pi$
$211$		−	2.81811i		−	0.194006i	0.995284	−	0.0970032i	$-0.0309257\pi$
$212$		0			0
$213$	4.92840			0.337688
$214$		0			0
$215$	20.9729			1.43034
$216$		0			0
$217$			11.0185i			0.747982i
$218$		0			0
$219$		−	2.76869i		−	0.187090i
$220$		0			0
$221$	5.86012			0.394194
$222$		0			0
$223$	4.88806			0.327329			0.163664	−	0.986516i	$-0.447669\pi$
$223$	4.88806			0.327329			0.163664	+	0.986516i	$0.447669\pi$
$224$		0			0
$225$	−0.246456			−0.0164304
$226$		0			0
$227$		−	12.9321i		−	0.858335i	−0.903225	−	0.429167i	$-0.858807\pi$
$227$		−	12.9321i		−	0.858335i	0.903225	−	0.429167i	$-0.141193\pi$
$228$		0			0
$229$		−	8.48863i		−	0.560944i	−0.959862	−	0.280472i	$-0.909509\pi$
$229$		−	8.48863i		−	0.560944i	0.959862	−	0.280472i	$-0.0904911\pi$
$230$		0			0
$231$			5.84028i			0.384262i
$232$		0			0
$233$			20.1933i			1.32291i	0.749986	+	0.661454i	$0.230060\pi$
$233$			20.1933i			1.32291i	−0.749986	+	0.661454i	$0.769940\pi$
$234$		0			0
$235$			7.39377i			0.482317i
$236$		0			0
$237$	3.50709			0.227810
$238$		0			0
$239$			12.5686i			0.812997i	0.913651	+	0.406499i	$0.133250\pi$
$239$			12.5686i			0.812997i	−0.913651	+	0.406499i	$0.866750\pi$
$240$		0			0
$241$	−0.291892			−0.0188024			−0.00940119	−	0.999956i	$-0.502993\pi$
$241$	−0.291892			−0.0188024			−0.00940119	+	0.999956i	$0.502993\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	13.5368			0.864835
$246$		0			0
$247$	−13.4227			−0.854065
$248$		0			0
$249$			4.89714i			0.310344i
$250$		0			0
$251$	−2.02560			−0.127855			−0.0639273	−	0.997955i	$-0.520363\pi$
$251$	−2.02560			−0.127855			−0.0639273	+	0.997955i	$0.520363\pi$
$252$		0			0
$253$		−	21.9748i		−	1.38154i
$254$		0			0
$255$	5.17486			0.324062
$256$		0			0
$257$			15.1729i			0.946460i	0.880939	+	0.473230i	$0.156912\pi$
$257$			15.1729i			0.946460i	−0.880939	+	0.473230i	$0.843088\pi$
$258$		0			0
$259$			1.83080i			0.113761i
$260$		0			0
$261$		−	0.987200i		−	0.0611062i
$262$		0			0
$263$		−	22.8867i		−	1.41125i	−0.708584	−	0.705627i	$-0.750666\pi$
$263$		−	22.8867i		−	1.41125i	0.708584	−	0.705627i	$-0.249334\pi$
$264$		0			0
$265$			15.7687i			0.968663i
$266$		0			0
$267$	2.45023			0.149952
$268$		0			0
$269$	25.5828			1.55981			0.779906	−	0.625897i	$-0.215267\pi$
$269$	25.5828			1.55981			0.779906	+	0.625897i	$0.215267\pi$
$270$		0			0
$271$	14.8275			0.900706			0.450353	−	0.892851i	$-0.351298\pi$
$271$	14.8275			0.900706			0.450353	+	0.892851i	$0.351298\pi$
$272$		0			0
$273$			2.70811i			0.163902i
$274$		0			0
$275$			1.37863i			0.0831346i
$276$		0			0
$277$	24.1033			1.44822			0.724112	−	0.689682i	$-0.242250\pi$
$277$	24.1033			1.44822			0.724112	+	0.689682i	$0.242250\pi$
$278$		0			0
$279$	10.5535			0.631821
$280$		0			0
$281$		−	27.2942i		−	1.62824i	−0.580698	−	0.814119i	$-0.697220\pi$
$281$		−	27.2942i		−	1.62824i	0.580698	−	0.814119i	$-0.302780\pi$
$282$		0			0
$283$	8.01514			0.476451			0.238225	−	0.971210i	$-0.423434\pi$
$283$	8.01514			0.476451			0.238225	+	0.971210i	$0.423434\pi$
$284$		0			0
$285$	−11.8531			−0.702116
$286$		0			0
$287$	−2.74074	−	6.09760i	−0.161781	−	0.359930i
$288$		0			0
$289$	11.8958			0.699751
$290$		0			0
$291$	0.374913			0.0219778
$292$		0			0
$293$		−	15.2536i		−	0.891125i	−0.895251	−	0.445562i	$-0.853004\pi$
$293$		−	15.2536i		−	0.891125i	0.895251	−	0.445562i	$-0.146996\pi$
$294$		0			0
$295$	−3.69297			−0.215013
$296$		0			0
$297$	5.59383			0.324587
$298$		0			0
$299$		−	10.1896i		−	0.589280i
$300$		0			0
$301$		−	9.55981i		−	0.551019i
$302$		0			0
$303$	1.20240			0.0690759
$304$		0			0
$305$	−4.30760			−0.246652
$306$		0			0
$307$	19.1772			1.09450			0.547250	−	0.836969i	$-0.315675\pi$
$307$	19.1772			1.09450			0.547250	+	0.836969i	$0.315675\pi$
$308$		0			0
$309$			18.9449i			1.07774i
$310$		0			0
$311$		−	31.0871i		−	1.76279i	−0.472380	−	0.881395i	$-0.656605\pi$
$311$		−	31.0871i		−	1.76279i	0.472380	−	0.881395i	$-0.343395\pi$
$312$		0			0
$313$			18.0004i			1.01744i	0.860931	+	0.508721i	$0.169882\pi$
$313$			18.0004i			1.01744i	−0.860931	+	0.508721i	$0.830118\pi$
$314$		0			0
$315$			2.39143i			0.134742i
$316$		0			0
$317$		−	19.1681i		−	1.07659i	−0.842757	−	0.538294i	$-0.819069\pi$
$317$		−	19.1681i		−	1.07659i	0.842757	−	0.538294i	$-0.180931\pi$
$318$		0			0
$319$	−5.52223			−0.309185
$320$		0			0
$321$		−	11.8995i		−	0.664164i
$322$		0			0
$323$	11.6913			0.650523
$324$		0			0
$325$			0.639264i			0.0354600i
$326$		0			0
$327$	−6.20612			−0.343199
$328$		0			0
$329$	3.37022			0.185806
$330$		0			0
$331$			12.6695i			0.696381i	0.937424	+	0.348191i	$0.113204\pi$
$331$			12.6695i			0.696381i	−0.937424	+	0.348191i	$0.886796\pi$
$332$		0			0
$333$	1.75354			0.0960936
$334$		0			0
$335$			34.1013i			1.86315i
$336$		0			0
$337$	−1.21520			−0.0661960			−0.0330980	−	0.999452i	$-0.510537\pi$
$337$	−1.21520			−0.0661960			−0.0330980	+	0.999452i	$0.510537\pi$
$338$		0			0
$339$			11.8659i			0.644466i
$340$		0			0
$341$		−	59.0344i		−	3.19689i
$342$		0			0
$343$		−	13.4787i		−	0.727783i
$344$		0			0
$345$		−	8.99806i		−	0.484439i
$346$		0			0
$347$		−	2.06594i		−	0.110905i	−0.998461	−	0.0554527i	$-0.982340\pi$
$347$		−	2.06594i		−	0.110905i	0.998461	−	0.0554527i	$-0.0176602\pi$
$348$		0			0
$349$	7.79994			0.417521			0.208761	−	0.977967i	$-0.433057\pi$
$349$	7.79994			0.417521			0.208761	+	0.977967i	$0.433057\pi$
$350$		0			0
$351$	2.59383			0.138448
$352$		0			0
$353$	−34.1809			−1.81927			−0.909633	−	0.415412i	$-0.863637\pi$
$353$	−34.1809			−1.81927			−0.909633	+	0.415412i	$0.863637\pi$
$354$		0			0
$355$		−	11.2886i		−	0.599135i
$356$		0			0
$357$		−	2.35879i		−	0.124841i
$358$		0			0
$359$	15.4327			0.814509			0.407254	−	0.913315i	$-0.366486\pi$
$359$	15.4327			0.814509			0.407254	+	0.913315i	$0.366486\pi$
$360$		0			0
$361$	−7.77914			−0.409429
$362$		0			0
$363$		−	20.2909i		−	1.06500i
$364$		0			0
$365$	−6.34171			−0.331940
$366$		0			0
$367$	−18.8370			−0.983282			−0.491641	−	0.870798i	$-0.663603\pi$
$367$	−18.8370			−0.983282			−0.491641	+	0.870798i	$0.663603\pi$
$368$		0			0
$369$	−5.84028	+	2.62509i	−0.304033	+	0.136657i
$370$		0			0
$371$	7.18766			0.373165
$372$		0			0
$373$	3.13080			0.162107			0.0810533	−	0.996710i	$-0.474172\pi$
$373$	3.13080			0.162107			0.0810533	+	0.996710i	$0.474172\pi$
$374$		0			0
$375$		−	10.8881i		−	0.562257i
$376$		0			0
$377$	−2.56063			−0.131879
$378$		0			0
$379$	20.6712			1.06181			0.530904	−	0.847432i	$-0.321853\pi$
$379$	20.6712			1.06181			0.530904	+	0.847432i	$0.321853\pi$
$380$		0			0
$381$		−	6.00372i		−	0.307580i
$382$		0			0
$383$		−	6.05354i		−	0.309321i	−0.987968	−	0.154661i	$-0.950572\pi$
$383$		−	6.05354i		−	0.309321i	0.987968	−	0.154661i	$-0.0494284\pi$
$384$		0			0
$385$	13.3773			0.681768
$386$		0			0
$387$	−9.15640			−0.465446
$388$		0			0
$389$	−8.25360			−0.418474			−0.209237	−	0.977865i	$-0.567098\pi$
$389$	−8.25360			−0.418474			−0.209237	+	0.977865i	$0.567098\pi$
$390$		0			0
$391$			8.87526i			0.448841i
$392$		0			0
$393$		−	18.9398i		−	0.955388i
$394$		0			0
$395$		−	8.03304i		−	0.404186i
$396$		0			0
$397$			29.1454i			1.46276i	0.681968	+	0.731382i	$0.261124\pi$
$397$			29.1454i			1.46276i	−0.681968	+	0.731382i	$0.738876\pi$
$398$		0			0
$399$			5.40285i			0.270481i
$400$		0			0
$401$	−2.36788			−0.118246			−0.0591230	−	0.998251i	$-0.518830\pi$
$401$	−2.36788			−0.118246			−0.0591230	+	0.998251i	$0.518830\pi$
$402$		0			0
$403$		−	27.3739i		−	1.36359i
$404$		0			0
$405$	2.29051			0.113817
$406$		0			0
$407$		−	9.80903i		−	0.486215i
$408$		0			0
$409$	26.6880			1.31964			0.659818	−	0.751425i	$-0.270633\pi$
$409$	26.6880			1.31964			0.659818	+	0.751425i	$0.270633\pi$
$410$		0			0
$411$	−13.8554			−0.683438
$412$		0			0
$413$			1.68332i			0.0828309i
$414$		0			0
$415$	11.2170			0.550620
$416$		0			0
$417$			10.8843i			0.533009i
$418$		0			0
$419$	−19.1541			−0.935737			−0.467868	−	0.883798i	$-0.654978\pi$
$419$	−19.1541			−0.935737			−0.467868	+	0.883798i	$0.654978\pi$
$420$		0			0
$421$		−	39.7744i		−	1.93849i	−0.246102	−	0.969244i	$-0.579150\pi$
$421$		−	39.7744i		−	1.93849i	0.246102	−	0.969244i	$-0.420850\pi$
$422$		0			0
$423$		−	3.22800i		−	0.156951i
$424$		0			0
$425$		−	0.556807i		−	0.0270091i
$426$		0			0
$427$			1.96348i			0.0950195i
$428$		0			0
$429$		−	14.5094i		−	0.700522i
$430$		0			0
$431$	19.6668			0.947317			0.473658	−	0.880709i	$-0.342933\pi$
$431$	19.6668			0.947317			0.473658	+	0.880709i	$0.342933\pi$
$432$		0			0
$433$	11.4170			0.548667			0.274334	−	0.961635i	$-0.411543\pi$
$433$	11.4170			0.548667			0.274334	+	0.961635i	$0.411543\pi$
$434$		0			0
$435$	−2.26120			−0.108416
$436$		0			0
$437$		−	20.3289i		−	0.972464i
$438$		0			0
$439$		−	4.38663i		−	0.209363i	−0.994506	−	0.104681i	$-0.966618\pi$
$439$		−	4.38663i		−	0.209363i	0.994506	−	0.104681i	$-0.0333822\pi$
$440$		0			0
$441$	−5.90994			−0.281426
$442$		0			0
$443$	−32.4879			−1.54355			−0.771775	−	0.635896i	$-0.780631\pi$
$443$	−32.4879			−1.54355			−0.771775	+	0.635896i	$0.780631\pi$
$444$		0			0
$445$		−	5.61229i		−	0.266048i
$446$		0			0
$447$	−8.66915			−0.410037
$448$		0			0
$449$	16.0700			0.758388			0.379194	−	0.925317i	$-0.376201\pi$
$449$	16.0700			0.758388			0.379194	+	0.925317i	$0.376201\pi$
$450$		0			0
$451$	14.6843	+	32.6695i	0.691456	+	1.53835i
$452$		0			0
$453$	20.9307			0.983412
$454$		0			0
$455$	6.20296			0.290799
$456$		0			0
$457$		−	24.7203i		−	1.15637i	−0.815906	−	0.578184i	$-0.803761\pi$
$457$		−	24.7203i		−	1.15637i	0.815906	−	0.578184i	$-0.196239\pi$
$458$		0			0
$459$	−2.25926			−0.105453
$460$		0			0
$461$	19.1326			0.891093			0.445546	−	0.895259i	$-0.353009\pi$
$461$	19.1326			0.891093			0.445546	+	0.895259i	$0.353009\pi$
$462$		0			0
$463$		−	22.9634i		−	1.06720i	−0.845737	−	0.533599i	$-0.820839\pi$
$463$		−	22.9634i		−	1.06720i	0.845737	−	0.533599i	$-0.179161\pi$
$464$		0			0
$465$		−	24.1729i		−	1.12099i
$466$		0			0
$467$	−28.2492			−1.30722			−0.653608	−	0.756833i	$-0.726746\pi$
$467$	−28.2492			−1.30722			−0.653608	+	0.756833i	$0.726746\pi$
$468$		0			0
$469$	15.5440			0.717756
$470$		0			0
$471$	1.39337			0.0642032
$472$		0			0
$473$			51.2193i			2.35507i
$474$		0			0
$475$			1.27537i			0.0585181i
$476$		0			0
$477$		−	6.88434i		−	0.315212i
$478$		0			0
$479$			4.24314i			0.193874i	0.995291	+	0.0969369i	$0.0309045\pi$
$479$			4.24314i			0.193874i	−0.995291	+	0.0969369i	$0.969095\pi$
$480$		0			0
$481$		−	4.54839i		−	0.207389i
$482$		0			0
$483$	−4.10148			−0.186624
$484$		0			0
$485$		−	0.858743i		−	0.0389935i
$486$		0			0
$487$	−10.7270			−0.486085			−0.243043	−	0.970016i	$-0.578146\pi$
$487$	−10.7270			−0.486085			−0.243043	+	0.970016i	$0.578146\pi$
$488$		0			0
$489$			21.4805i			0.971382i
$490$		0			0
$491$	33.2725			1.50156			0.750782	−	0.660550i	$-0.229677\pi$
$491$	33.2725			1.50156			0.750782	+	0.660550i	$0.229677\pi$
$492$		0			0
$493$	2.23034			0.100449
$494$		0			0
$495$		−	12.8127i		−	0.575890i
$496$		0			0
$497$	−5.14554			−0.230809
$498$		0			0
$499$		−	20.1013i		−	0.899859i	−0.893064	−	0.449929i	$-0.851449\pi$
$499$		−	20.1013i		−	0.899859i	0.893064	−	0.449929i	$-0.148551\pi$
$500$		0			0
$501$	15.1033			0.674764
$502$		0			0
$503$			30.4012i			1.35552i	0.735282	+	0.677761i	$0.237050\pi$
$503$			30.4012i			1.35552i	−0.735282	+	0.677761i	$0.762950\pi$
$504$		0			0
$505$		−	2.75411i		−	0.122556i
$506$		0			0
$507$			6.27206i			0.278552i
$508$		0			0
$509$		−	7.59851i		−	0.336798i	−0.985719	−	0.168399i	$-0.946140\pi$
$509$		−	7.59851i		−	0.336798i	0.985719	−	0.168399i	$-0.0538597\pi$
$510$		0			0
$511$			2.89067i			0.127876i
$512$		0			0
$513$	5.17486			0.228476
$514$		0			0
$515$	43.3936			1.91215
$516$		0			0
$517$	−18.0569			−0.794140
$518$		0			0
$519$		−	5.86588i		−	0.257484i
$520$		0			0
$521$		−	6.73605i		−	0.295112i	−0.989054	−	0.147556i	$-0.952859\pi$
$521$		−	6.73605i		−	0.295112i	0.989054	−	0.147556i	$-0.0471406\pi$
$522$		0			0
$523$	−6.21795			−0.271892			−0.135946	−	0.990716i	$-0.543407\pi$
$523$	−6.21795			−0.271892			−0.135946	+	0.990716i	$0.543407\pi$
$524$		0			0
$525$	0.257314			0.0112301
$526$		0			0
$527$			23.8430i			1.03862i
$528$		0			0
$529$	−7.56767			−0.329029
$530$		0			0
$531$	1.61229			0.0699673
$532$		0			0
$533$	6.80903	+	15.1487i	0.294932	+	0.656163i
$534$		0			0
$535$	−27.2559			−1.17838
$536$		0			0
$537$	19.7027			0.850236
$538$		0			0
$539$			33.0592i			1.42396i
$540$		0			0
$541$	−14.2798			−0.613935			−0.306967	−	0.951720i	$-0.599314\pi$
$541$	−14.2798			−0.613935			−0.306967	+	0.951720i	$0.599314\pi$
$542$		0			0
$543$	−9.57537			−0.410918
$544$		0			0
$545$			14.2152i			0.608912i
$546$		0			0
$547$			33.1934i			1.41925i	0.704581	+	0.709624i	$0.251135\pi$
$547$			33.1934i			1.41925i	−0.704581	+	0.709624i	$0.748865\pi$
$548$		0			0
$549$	1.88062			0.0802631
$550$		0			0
$551$	−5.10862			−0.217635
$552$		0			0
$553$	−3.66161			−0.155707
$554$		0			0
$555$		−	4.01652i		−	0.170492i
$556$		0			0
$557$		−	36.7889i		−	1.55880i	−0.626529	−	0.779398i	$-0.715525\pi$
$557$		−	36.7889i		−	1.55880i	0.626529	−	0.779398i	$-0.284475\pi$
$558$		0			0
$559$			23.7501i			1.00452i
$560$		0			0
$561$			12.6379i			0.533572i
$562$		0			0
$563$		−	0.794848i		−	0.0334989i	−0.999860	−	0.0167494i	$-0.994668\pi$
$563$		−	0.794848i		−	0.0334989i	0.999860	−	0.0167494i	$-0.00533176\pi$
$564$		0			0
$565$	27.1790			1.14343
$566$		0			0
$567$		−	1.04406i		−	0.0438463i
$568$		0			0
$569$	−8.03936			−0.337028			−0.168514	−	0.985699i	$-0.553897\pi$
$569$	−8.03936			−0.337028			−0.168514	+	0.985699i	$0.553897\pi$
$570$		0			0
$571$		−	10.1688i		−	0.425551i	−0.977101	−	0.212775i	$-0.931750\pi$
$571$		−	10.1688i		−	0.425551i	0.977101	−	0.212775i	$-0.0682503\pi$
$572$		0			0
$573$	−12.6066			−0.526649
$574$		0			0
$575$	−0.968177			−0.0403758
$576$		0			0
$577$		−	43.1400i		−	1.79594i	−0.440054	−	0.897971i	$-0.645041\pi$
$577$		−	43.1400i		−	1.79594i	0.440054	−	0.897971i	$-0.354959\pi$
$578$		0			0
$579$	−24.9862			−1.03839
$580$		0			0
$581$		−	5.11290i		−	0.212119i
$582$		0			0
$583$	−38.5098			−1.59491
$584$		0			0
$585$		−	5.94120i		−	0.245638i
$586$		0			0
$587$			11.2831i			0.465702i	0.972512	+	0.232851i	$0.0748055\pi$
$587$			11.2831i			0.465702i	−0.972512	+	0.232851i	$0.925194\pi$
$588$		0			0
$589$		−	54.6128i		−	2.25028i
$590$		0			0
$591$		−	7.79760i		−	0.320751i
$592$		0			0
$593$		−	7.71126i		−	0.316664i	−0.987386	−	0.158332i	$-0.949388\pi$
$593$		−	7.71126i		−	0.316664i	0.987386	−	0.158332i	$-0.0506116\pi$
$594$		0			0
$595$	−5.40285			−0.221495
$596$		0			0
$597$	−12.9009			−0.527997
$598$		0			0
$599$	24.4772			1.00011			0.500056	−	0.865993i	$-0.333313\pi$
$599$	24.4772			1.00011			0.500056	+	0.865993i	$0.333313\pi$
$600$		0			0
$601$			30.7950i			1.25615i	0.778151	+	0.628077i	$0.216158\pi$
$601$			30.7950i			1.25615i	−0.778151	+	0.628077i	$0.783842\pi$
$602$		0			0
$603$		−	14.8881i		−	0.606289i
$604$		0			0
$605$	−46.4766			−1.88954
$606$		0			0
$607$	26.7667			1.08643			0.543214	−	0.839594i	$-0.317207\pi$
$607$	26.7667			1.08643			0.543214	+	0.839594i	$0.317207\pi$
$608$		0			0
$609$			1.03069i			0.0417659i
$610$		0			0
$611$	−8.37287			−0.338730
$612$		0			0
$613$	11.0237			0.445241			0.222621	−	0.974905i	$-0.428539\pi$
$613$	11.0237			0.445241			0.222621	+	0.974905i	$0.428539\pi$
$614$		0			0
$615$	6.01280	+	13.3773i	0.242459	+	0.539423i
$616$		0			0
$617$	25.1417			1.01216			0.506082	−	0.862485i	$-0.331093\pi$
$617$	25.1417			1.01216			0.506082	+	0.862485i	$0.331093\pi$
$618$		0			0
$619$	−8.59617			−0.345509			−0.172755	−	0.984965i	$-0.555267\pi$
$619$	−8.59617			−0.345509			−0.172755	+	0.984965i	$0.555267\pi$
$620$		0			0
$621$			3.92840i			0.157641i
$622$		0			0
$623$	−2.55818			−0.102491
$624$		0			0
$625$	−26.1715			−1.04686
$626$		0			0
$627$		−	28.9473i		−	1.15604i
$628$		0			0
$629$			3.96170i			0.157963i
$630$		0			0
$631$	28.6389			1.14009			0.570047	−	0.821612i	$-0.306925\pi$
$631$	28.6389			1.14009			0.570047	+	0.821612i	$0.306925\pi$
$632$		0			0
$633$	2.81811			0.112010
$634$		0			0
$635$	−13.7516			−0.545716
$636$		0			0
$637$			15.3294i			0.607372i
$638$		0			0
$639$			4.92840i			0.194965i
$640$		0			0
$641$		−	34.2382i		−	1.35233i	−0.736752	−	0.676163i	$-0.763641\pi$
$641$		−	34.2382i		−	1.35233i	0.736752	−	0.676163i	$-0.236359\pi$
$642$		0			0
$643$			7.54977i			0.297734i	0.988857	+	0.148867i	$0.0475626\pi$
$643$			7.54977i			0.297734i	−0.988857	+	0.148867i	$0.952437\pi$
$644$		0			0
$645$			20.9729i			0.825806i
$646$		0			0
$647$	−1.11994			−0.0440294			−0.0220147	−	0.999758i	$-0.507008\pi$
$647$	−1.11994			−0.0440294			−0.0220147	+	0.999758i	$0.507008\pi$
$648$		0			0
$649$		−	9.01886i		−	0.354021i
$650$		0			0
$651$	−11.0185			−0.431847
$652$		0			0
$653$		−	12.8440i		−	0.502625i	−0.967906	−	0.251312i	$-0.919138\pi$
$653$		−	12.8440i		−	0.502625i	0.967906	−	0.251312i	$-0.0808621\pi$
$654$		0			0
$655$	−43.3819			−1.69507
$656$		0			0
$657$	2.76869			0.108017
$658$		0			0
$659$		−	3.24274i		−	0.126319i	−0.998003	−	0.0631596i	$-0.979882\pi$
$659$		−	3.24274i		−	0.126319i	0.998003	−	0.0631596i	$-0.0201177\pi$
$660$		0			0
$661$	8.98378			0.349428			0.174714	−	0.984619i	$-0.444100\pi$
$661$	8.98378			0.349428			0.174714	+	0.984619i	$0.444100\pi$
$662$		0			0
$663$			5.86012i			0.227588i
$664$		0			0
$665$	12.3753			0.479894
$666$		0			0
$667$		−	3.87812i		−	0.150161i
$668$		0			0
$669$			4.88806i			0.188983i
$670$		0			0
$671$		−	10.5199i		−	0.406116i
$672$		0			0
$673$			11.7318i			0.452226i	0.974101	+	0.226113i	$0.0726019\pi$
$673$			11.7318i			0.452226i	−0.974101	+	0.226113i	$0.927398\pi$
$674$		0			0
$675$		−	0.246456i		−	0.00948609i
$676$		0			0
$677$	11.1359			0.427987			0.213994	−	0.976835i	$-0.431353\pi$
$677$	11.1359			0.427987			0.213994	+	0.976835i	$0.431353\pi$
$678$		0			0
$679$	−0.391431			−0.0150217
$680$		0			0
$681$	12.9321			0.495560
$682$		0			0
$683$			19.7553i			0.755916i	0.925823	+	0.377958i	$0.123374\pi$
$683$			19.7553i			0.755916i	−0.925823	+	0.377958i	$0.876626\pi$
$684$		0			0
$685$			31.7360i			1.21257i
$686$		0			0
$687$	8.48863			0.323861
$688$		0			0
$689$	−17.8568			−0.680290
$690$		0			0
$691$		−	9.30566i		−	0.354004i	−0.984211	−	0.177002i	$-0.943360\pi$
$691$		−	9.30566i		−	0.354004i	0.984211	−	0.177002i	$-0.0566399\pi$
$692$		0			0
$693$	−5.84028			−0.221854
$694$		0			0
$695$	24.9307			0.945677
$696$		0			0
$697$	−5.93074	−	13.1947i	−0.224643	−	0.499785i
$698$		0			0
$699$	−20.1933			−0.763781
$700$		0			0
$701$	−0.168797			−0.00637538			−0.00318769	−	0.999995i	$-0.501015\pi$
$701$	−0.168797			−0.00637538			−0.00318769	+	0.999995i	$0.501015\pi$
$702$		0			0
$703$		−	9.07434i		−	0.342245i
$704$		0			0
$705$	−7.39377			−0.278466
$706$		0			0
$707$	−1.25537			−0.0472132
$708$		0			0
$709$			29.2331i			1.09787i	0.835864	+	0.548936i	$0.184967\pi$
$709$			29.2331i			1.09787i	−0.835864	+	0.548936i	$0.815033\pi$
$710$		0			0
$711$			3.50709i			0.131526i
$712$		0			0
$713$	41.4583			1.55263
$714$		0			0
$715$	−33.2341			−1.24288
$716$		0			0
$717$	−12.5686			−0.469384
$718$		0			0
$719$			29.7879i			1.11090i	0.831549	+	0.555451i	$0.187455\pi$
$719$			29.7879i			1.11090i	−0.831549	+	0.555451i	$0.812545\pi$
$720$		0			0
$721$		−	19.7796i		−	0.736631i
$722$		0			0
$723$		−	0.291892i		−	0.0108556i
$724$		0			0
$725$			0.243301i			0.00903598i
$726$		0			0
$727$		−	45.0193i		−	1.66967i	−0.550499	−	0.834836i	$-0.685562\pi$
$727$		−	45.0193i		−	1.66967i	0.550499	−	0.834836i	$-0.314438\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$		−	20.6866i		−	0.765123i
$732$		0			0
$733$	−30.6938			−1.13370			−0.566850	−	0.823821i	$-0.691838\pi$
$733$	−30.6938			−1.13370			−0.566850	+	0.823821i	$0.691838\pi$
$734$		0			0
$735$			13.5368i			0.499313i
$736$		0			0
$737$	−83.2813			−3.06770
$738$		0			0
$739$	17.1638			0.631382			0.315691	−	0.948862i	$-0.397764\pi$
$739$	17.1638			0.631382			0.315691	+	0.948862i	$0.397764\pi$
$740$		0			0
$741$		−	13.4227i		−	0.493095i
$742$		0			0
$743$	44.7325			1.64107			0.820537	−	0.571593i	$-0.193674\pi$
$743$	44.7325			1.64107			0.820537	+	0.571593i	$0.193674\pi$
$744$		0			0
$745$			19.8568i			0.727497i
$746$		0			0
$747$	−4.89714			−0.179177
$748$		0			0
$749$			12.4238i			0.453954i
$750$		0			0
$751$			39.9336i			1.45720i	0.684941	+	0.728599i	$0.259828\pi$
$751$			39.9336i			1.45720i	−0.684941	+	0.728599i	$0.740172\pi$
$752$		0			0
$753$		−	2.02560i		−	0.0738169i
$754$		0			0
$755$		−	47.9422i		−	1.74479i
$756$		0			0
$757$			10.7199i			0.389623i	0.980841	+	0.194811i	$0.0624095\pi$
$757$			10.7199i			0.389623i	−0.980841	+	0.194811i	$0.937590\pi$
$758$		0			0
$759$	21.9748			0.797635
$760$		0			0
$761$	−10.7754			−0.390609			−0.195304	−	0.980743i	$-0.562569\pi$
$761$	−10.7754			−0.390609			−0.195304	+	0.980743i	$0.562569\pi$
$762$		0			0
$763$	6.47955			0.234575
$764$		0			0
$765$			5.17486i			0.187097i
$766$		0			0
$767$		−	4.18200i		−	0.151003i
$768$		0			0
$769$	−22.6329			−0.816164			−0.408082	−	0.912945i	$-0.633802\pi$
$769$	−22.6329			−0.816164			−0.408082	+	0.912945i	$0.633802\pi$
$770$		0			0
$771$	−15.1729			−0.546439
$772$		0			0
$773$			31.5178i			1.13362i	0.823849	+	0.566809i	$0.191822\pi$
$773$			31.5178i			1.13362i	−0.823849	+	0.566809i	$0.808178\pi$
$774$		0			0
$775$	−2.60097			−0.0934295
$776$		0			0
$777$	−1.83080			−0.0656797
$778$		0			0
$779$	13.5845	+	30.2226i	0.486713	+	1.08284i
$780$		0			0
$781$	27.5686			0.986483
$782$		0			0
$783$	0.987200			0.0352797
$784$		0			0
$785$		−	3.19154i		−	0.113911i
$786$		0			0
$787$	16.4578			0.586656			0.293328	−	0.956012i	$-0.405237\pi$
$787$	16.4578			0.586656			0.293328	+	0.956012i	$0.405237\pi$
$788$		0			0
$789$	22.8867			0.814788
$790$		0			0
$791$		−	12.3887i		−	0.440491i
$792$		0			0
$793$		−	4.87802i		−	0.173223i
$794$		0			0
$795$	−15.7687			−0.559258
$796$		0			0
$797$	−15.0615			−0.533507			−0.266754	−	0.963765i	$-0.585951\pi$
$797$	−15.0615			−0.533507			−0.266754	+	0.963765i	$0.585951\pi$
$798$		0			0
$799$	7.29287			0.258003
$800$		0			0
$801$			2.45023i			0.0865746i
$802$		0			0
$803$		−	15.4875i		−	0.546544i
$804$		0			0
$805$			9.39450i			0.331113i
$806$		0			0
$807$			25.5828i			0.900557i
$808$		0			0
$809$			1.43137i			0.0503243i	0.999683	+	0.0251622i	$0.00801021\pi$
$809$			1.43137i			0.0503243i	−0.999683	+	0.0251622i	$0.991990\pi$
$810$		0			0
$811$	35.9345			1.26183			0.630915	−	0.775852i	$-0.282680\pi$
$811$	35.9345			1.26183			0.630915	+	0.775852i	$0.282680\pi$
$812$		0			0
$813$			14.8275i			0.520023i
$814$		0			0
$815$	49.2014			1.72345
$816$		0			0
$817$			47.3830i			1.65772i
$818$		0			0
$819$	−2.70811			−0.0946290
$820$		0			0
$821$	−40.2214			−1.40374			−0.701868	−	0.712307i	$-0.747650\pi$
$821$	−40.2214			−1.40374			−0.701868	+	0.712307i	$0.747650\pi$
$822$		0			0
$823$		−	37.8642i		−	1.31986i	−0.751325	−	0.659932i	$-0.770585\pi$
$823$		−	37.8642i		−	1.31986i	0.751325	−	0.659932i	$-0.229415\pi$
$824$		0			0
$825$	−1.37863			−0.0479978
$826$		0			0
$827$			27.4298i			0.953829i	0.878950	+	0.476914i	$0.158245\pi$
$827$			27.4298i			0.953829i	−0.878950	+	0.476914i	$0.841755\pi$
$828$		0			0
$829$	−18.0461			−0.626768			−0.313384	−	0.949626i	$-0.601463\pi$
$829$	−18.0461			−0.626768			−0.313384	+	0.949626i	$0.601463\pi$
$830$		0			0
$831$			24.1033i			0.836133i
$832$		0			0
$833$		−	13.3521i		−	0.462622i
$834$		0			0
$835$		−	34.5942i		−	1.19718i
$836$		0			0
$837$			10.5535i			0.364782i
$838$		0			0
$839$		−	34.6749i		−	1.19711i	−0.801081	−	0.598555i	$-0.795742\pi$
$839$		−	34.6749i		−	1.19711i	0.801081	−	0.598555i	$-0.204258\pi$
$840$		0			0
$841$	28.0254			0.966394
$842$		0			0
$843$	27.2942			0.940063
$844$		0			0
$845$	14.3662			0.494213
$846$		0			0
$847$			21.1849i			0.727922i
$848$		0			0
$849$			8.01514i			0.275079i
$850$		0			0
$851$	6.88862			0.236139
$852$		0			0
$853$	−40.4366			−1.38452			−0.692261	−	0.721647i	$-0.743385\pi$
$853$	−40.4366			−1.38452			−0.692261	+	0.721647i	$0.743385\pi$
$854$		0			0
$855$		−	11.8531i		−	0.405367i
$856$		0			0
$857$	27.1763			0.928326			0.464163	−	0.885750i	$-0.346355\pi$
$857$	27.1763			0.928326			0.464163	+	0.885750i	$0.346355\pi$
$858$		0			0
$859$	−45.9762			−1.56869			−0.784343	−	0.620327i	$-0.787000\pi$
$859$	−45.9762			−1.56869			−0.784343	+	0.620327i	$0.787000\pi$
$860$		0			0
$861$	6.09760	−	2.74074i	0.207806	−	0.0934043i
$862$		0			0
$863$	−21.9796			−0.748194			−0.374097	−	0.927390i	$-0.622047\pi$
$863$	−21.9796			−0.748194			−0.374097	+	0.927390i	$0.622047\pi$
$864$		0			0
$865$	−13.4359			−0.456834
$866$		0			0
$867$			11.8958i			0.404001i
$868$		0			0
$869$	19.6181			0.665497
$870$		0			0
$871$	−38.6171			−1.30849
$872$		0			0
$873$			0.374913i			0.0126889i
$874$		0			0
$875$			11.3678i			0.384301i
$876$		0			0
$877$	11.9393			0.403160			0.201580	−	0.979472i	$-0.435392\pi$
$877$	11.9393			0.403160			0.201580	+	0.979472i	$0.435392\pi$
$878$		0			0
$879$	15.2536			0.514491
$880$		0			0
$881$	2.11096			0.0711201			0.0355601	−	0.999368i	$-0.488678\pi$
$881$	2.11096			0.0711201			0.0355601	+	0.999368i	$0.488678\pi$
$882$		0			0
$883$		−	31.9475i		−	1.07512i	−0.843226	−	0.537560i	$-0.819346\pi$
$883$		−	31.9475i		−	1.07512i	0.843226	−	0.537560i	$-0.180654\pi$
$884$		0			0
$885$		−	3.69297i		−	0.124138i
$886$		0			0
$887$			4.72657i			0.158703i	0.996847	+	0.0793513i	$0.0252849\pi$
$887$			4.72657i			0.158703i	−0.996847	+	0.0793513i	$0.974715\pi$
$888$		0			0
$889$			6.26823i			0.210230i
$890$		0			0
$891$			5.59383i			0.187400i
$892$		0			0
$893$	−16.7044			−0.558992
$894$		0			0
$895$		−	45.1294i		−	1.50851i
$896$		0			0
$897$	10.1896			0.340221
$898$		0			0
$899$		−	10.4184i		−	0.347473i
$900$		0			0
$901$	15.5535			0.518162
$902$		0			0
$903$	9.55981			0.318131
$904$		0			0
$905$			21.9325i			0.729062i
$906$		0			0
$907$	−51.2974			−1.70330			−0.851651	−	0.524110i	$-0.824398\pi$
$907$	−51.2974			−1.70330			−0.851651	+	0.524110i	$0.824398\pi$
$908$		0			0
$909$			1.20240i			0.0398810i
$910$		0			0
$911$	−40.7898			−1.35143			−0.675713	−	0.737165i	$-0.736164\pi$
$911$	−40.7898			−1.35143			−0.675713	+	0.737165i	$0.736164\pi$
$912$		0			0
$913$			27.3938i			0.906602i
$914$		0			0
$915$		−	4.30760i		−	0.142405i
$916$		0			0
$917$			19.7743i			0.653004i
$918$		0			0
$919$		−	41.8828i		−	1.38159i	−0.723053	−	0.690793i	$-0.757262\pi$
$919$		−	41.8828i		−	1.38159i	0.723053	−	0.690793i	$-0.242738\pi$
$920$		0			0
$921$			19.1772i			0.631910i
$922$		0			0
$923$	12.7834			0.420772
$924$		0			0
$925$	−0.432171			−0.0142097
$926$		0			0
$927$	−18.9449			−0.622233
$928$		0			0
$929$		−	42.3018i		−	1.38788i	−0.720035	−	0.693938i	$-0.755874\pi$
$929$		−	42.3018i		−	1.38788i	0.720035	−	0.693938i	$-0.244126\pi$
$930$		0			0
$931$			30.5831i			1.00232i
$932$		0			0
$933$	31.0871			1.01775
$934$		0			0
$935$	28.9473			0.946677
$936$		0			0
$937$		−	17.3383i		−	0.566417i	−0.959058	−	0.283209i	$-0.908601\pi$
$937$		−	17.3383i		−	0.566417i	0.959058	−	0.283209i	$-0.0913989\pi$
$938$		0			0
$939$	−18.0004			−0.587421
$940$		0			0
$941$	48.4739			1.58020			0.790102	−	0.612976i	$-0.210028\pi$
$941$	48.4739			1.58020			0.790102	+	0.612976i	$0.210028\pi$
$942$		0			0
$943$	−22.9430	+	10.3124i	−0.747126	+	0.335818i
$944$		0			0
$945$	−2.39143			−0.0777933
$946$		0			0
$947$	37.7078			1.22534			0.612670	−	0.790339i	$-0.290095\pi$
$947$	37.7078			1.22534			0.612670	+	0.790339i	$0.290095\pi$
$948$		0			0
$949$		−	7.18149i		−	0.233121i
$950$		0			0
$951$	19.1681			0.621569
$952$		0			0
$953$	−9.06098			−0.293514			−0.146757	−	0.989173i	$-0.546884\pi$
$953$	−9.06098			−0.293514			−0.146757	+	0.989173i	$0.546884\pi$
$954$		0			0
$955$			28.8757i			0.934394i
$956$		0			0
$957$		−	5.52223i		−	0.178508i
$958$		0			0
$959$	14.4659			0.467127
$960$		0			0
$961$	80.3761			2.59278
$962$		0			0
$963$	11.8995			0.383455
$964$		0			0
$965$			57.2313i			1.84234i
$966$		0			0
$967$			48.0702i			1.54583i	0.634508	+	0.772916i	$0.281203\pi$
$967$			48.0702i			1.54583i	−0.634508	+	0.772916i	$0.718797\pi$
$968$		0			0
$969$			11.6913i			0.375579i
$970$		0			0
$971$		−	32.7359i		−	1.05054i	−0.850934	−	0.525272i	$-0.823963\pi$
$971$		−	32.7359i		−	1.05054i	0.850934	−	0.525272i	$-0.176037\pi$
$972$		0			0
$973$		−	11.3639i		−	0.364310i
$974$		0			0
$975$	−0.639264			−0.0204728
$976$		0			0
$977$			38.2601i			1.22405i	0.790839	+	0.612024i	$0.209644\pi$
$977$			38.2601i			1.22405i	−0.790839	+	0.612024i	$0.790356\pi$
$978$		0			0
$979$	13.7062			0.438051
$980$		0			0
$981$		−	6.20612i		−	0.198146i
$982$		0			0
$983$	−43.2157			−1.37837			−0.689183	−	0.724587i	$-0.742031\pi$
$983$	−43.2157			−1.37837			−0.689183	+	0.724587i	$0.742031\pi$
$984$		0			0
$985$	−17.8605			−0.569084
$986$		0			0
$987$			3.37022i			0.107275i
$988$		0			0
$989$	−35.9700			−1.14378
$990$		0			0
$991$		−	5.02630i		−	0.159666i	−0.996808	−	0.0798328i	$-0.974561\pi$
$991$		−	5.02630i		−	0.159666i	0.996808	−	0.0798328i	$-0.0254386\pi$
$992$		0			0
$993$	−12.6695			−0.402056
$994$		0			0
$995$			29.5496i			0.936785i
$996$		0			0
$997$			43.9767i			1.39276i	0.717675	+	0.696378i	$0.245206\pi$
$997$			43.9767i			1.39276i	−0.717675	+	0.696378i	$0.754794\pi$
$998$		0			0
$999$			1.75354i			0.0554797i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1968.2.j.f.1393.7		8
4.3	odd	2		984.2.j.b.409.3	✓	8
12.11	even	2		2952.2.j.d.2377.4		8
41.40	even	2	inner	1968.2.j.f.1393.3		8
164.163	odd	2		984.2.j.b.409.7	yes	8
492.491	even	2		2952.2.j.d.2377.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
984.2.j.b.409.3	✓	8	4.3	odd	2
984.2.j.b.409.7	yes	8	164.163	odd	2
1968.2.j.f.1393.3		8	41.40	even	2	inner
1968.2.j.f.1393.7		8	1.1	even	1	trivial
2952.2.j.d.2377.3		8	492.491	even	2
2952.2.j.d.2377.4		8	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1968 = 2^{4} \cdot 3 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1968.j (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +2.29051 q^{5} -1.04406i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +2.29051 q^{5} -1.04406i q^{7} -1.00000 q^{9} +5.59383i q^{11} +2.59383i q^{13} +2.29051i q^{15} -2.25926i q^{17} +5.17486i q^{19} +1.04406 q^{21} -3.92840 q^{23} +0.246456 q^{25} -1.00000i q^{27} +0.987200i q^{29} -10.5535 q^{31} -5.59383 q^{33} -2.39143i q^{35} -1.75354 q^{37} -2.59383 q^{39} +(5.84028 - 2.62509i) q^{41} +9.15640 q^{43} -2.29051 q^{45} +3.22800i q^{47} +5.90994 q^{49} +2.25926 q^{51} +6.88434i q^{53} +12.8127i q^{55} -5.17486 q^{57} -1.61229 q^{59} -1.88062 q^{61} +1.04406i q^{63} +5.94120i q^{65} +14.8881i q^{67} -3.92840i q^{69} -4.92840i q^{71} -2.76869 q^{73} +0.246456i q^{75} +5.84028 q^{77} -3.50709i q^{79} +1.00000 q^{81} +4.89714 q^{83} -5.17486i q^{85} -0.987200 q^{87} -2.45023i q^{89} +2.70811 q^{91} -10.5535i q^{93} +11.8531i q^{95} -0.374913i q^{97} -5.59383i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 10 q^{5} - 8 q^{9}+O(q^{10}) \) 8 * q + 10 * q^5 - 8 * q^9	\( 8 q + 10 q^{5} - 8 q^{9} + 20 q^{23} + 2 q^{25} - 8 q^{31} - 10 q^{33} - 14 q^{37} + 14 q^{39} + 12 q^{41} - 6 q^{43} - 10 q^{45} - 32 q^{49} + 10 q^{57} - 6 q^{59} - 22 q^{61} + 64 q^{73} + 12 q^{77} + 8 q^{81} - 22 q^{83} - 26 q^{87} + 12 q^{91}+O(q^{100}) \) 8 * q + 10 * q^5 - 8 * q^9 + 20 * q^23 + 2 * q^25 - 8 * q^31 - 10 * q^33 - 14 * q^37 + 14 * q^39 + 12 * q^41 - 6 * q^43 - 10 * q^45 - 32 * q^49 + 10 * q^57 - 6 * q^59 - 22 * q^61 + 64 * q^73 + 12 * q^77 + 8 * q^81 - 22 * q^83 - 26 * q^87 + 12 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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