LMFDB - Embedded newform 416.2.b.b.209.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [416,2,Mod(209,416)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(416, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("416.209");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 416 = 2^{5} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	416.b (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:	$3.32177672409$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 104)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			209.2
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	416.209
Dual form			416.2.b.b.209.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{3} +3.46410i q^{5} +1.26795 q^{7} -1.00000 q^{9} +O(q^{10})$	$q-2.00000i q^{3} +3.46410i q^{5} +1.26795 q^{7} -1.00000 q^{9} +4.73205i q^{11} +1.00000i q^{13} +6.92820 q^{15} +5.46410 q^{17} -0.732051i q^{19} -2.53590i q^{21} -4.00000 q^{23} -7.00000 q^{25} -4.00000i q^{27} +2.00000i q^{29} +6.73205 q^{31} +9.46410 q^{33} +4.39230i q^{35} -8.92820i q^{37} +2.00000 q^{39} +8.92820 q^{41} +0.535898i q^{43} -3.46410i q^{45} -6.73205 q^{47} -5.39230 q^{49} -10.9282i q^{51} -2.92820i q^{53} -16.3923 q^{55} -1.46410 q^{57} +10.1962i q^{59} +2.92820i q^{61} -1.26795 q^{63} -3.46410 q^{65} -0.732051i q^{67} +8.00000i q^{69} +8.19615 q^{71} -7.46410 q^{73} +14.0000i q^{75} +6.00000i q^{77} -5.46410 q^{79} -11.0000 q^{81} -3.26795i q^{83} +18.9282i q^{85} +4.00000 q^{87} -17.3205 q^{89} +1.26795i q^{91} -13.4641i q^{93} +2.53590 q^{95} +6.39230 q^{97} -4.73205i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{7} - 4 q^{9}+O(q^{10}) $ 4 * q + 12 * q^7 - 4 * q^9	$ 4 q + 12 q^{7} - 4 q^{9} + 8 q^{17} - 16 q^{23} - 28 q^{25} + 20 q^{31} + 24 q^{33} + 8 q^{39} + 8 q^{41} - 20 q^{47} + 20 q^{49} - 24 q^{55} + 8 q^{57} - 12 q^{63} + 12 q^{71} - 16 q^{73} - 8 q^{79} - 44 q^{81} + 16 q^{87} + 24 q^{95} - 16 q^{97}+O(q^{100}) $ 4 * q + 12 * q^7 - 4 * q^9 + 8 * q^17 - 16 * q^23 - 28 * q^25 + 20 * q^31 + 24 * q^33 + 8 * q^39 + 8 * q^41 - 20 * q^47 + 20 * q^49 - 24 * q^55 + 8 * q^57 - 12 * q^63 + 12 * q^71 - 16 * q^73 - 8 * q^79 - 44 * q^81 + 16 * q^87 + 24 * q^95 - 16 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/416\mathbb{Z}\right)^\times$.

$n$	$261$	$287$	$353$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	0	0
$2$	0	0
$3$	− 2.00000i	− 1.15470i	−0.816497	−	0.577350i	$-0.804087\pi$
$3$	− 2.00000i	− 1.15470i	0.816497	−	0.577350i	$-0.195913\pi$
$4$	0	0
$5$	3.46410i	1.54919i	0.632456	+	0.774597i	$0.282047\pi$
$5$	3.46410i	1.54919i	−0.632456	+	0.774597i	$0.717953\pi$
$6$	0	0
$7$	1.26795	0.479240	0.239620	−	0.970867i	$-0.422977\pi$
$7$	1.26795	0.479240	0.239620	+	0.970867i	$0.422977\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	4.73205i	1.42677i	0.700774	+	0.713384i	$0.252838\pi$
$11$	4.73205i	1.42677i	−0.700774	+	0.713384i	$0.747162\pi$
$12$	0	0
$13$	1.00000i	0.277350i
$13$	1.00000i	0.277350i
$14$	0	0
$15$	6.92820	1.78885
$16$	0	0
$17$	5.46410	1.32524	0.662620	−	0.748956i	$-0.269445\pi$
$17$	5.46410	1.32524	0.662620	+	0.748956i	$0.269445\pi$
$18$	0	0
$19$	− 0.732051i	− 0.167944i	−0.996468	−	0.0839720i	$-0.973239\pi$
$19$	− 0.732051i	− 0.167944i	0.996468	−	0.0839720i	$-0.0267606\pi$
$20$	0	0
$21$	− 2.53590i	− 0.553378i
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	−7.00000	−1.40000
$26$	0	0
$27$	− 4.00000i	− 0.769800i
$28$	0	0
$29$	2.00000i	0.371391i	0.982607	+	0.185695i	$0.0594537\pi$
$29$	2.00000i	0.371391i	−0.982607	+	0.185695i	$0.940546\pi$
$30$	0	0
$31$	6.73205	1.20911	0.604556	−	0.796563i	$-0.293351\pi$
$31$	6.73205	1.20911	0.604556	+	0.796563i	$0.293351\pi$
$32$	0	0
$33$	9.46410	1.64749
$34$	0	0
$35$	4.39230i	0.742435i
$36$	0	0
$37$	− 8.92820i	− 1.46779i	−0.679264	−	0.733894i	$-0.737701\pi$
$37$	− 8.92820i	− 1.46779i	0.679264	−	0.733894i	$-0.262299\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	0	0
$41$	8.92820	1.39435	0.697176	−	0.716900i	$-0.254440\pi$
$41$	8.92820	1.39435	0.697176	+	0.716900i	$0.254440\pi$
$42$	0	0
$43$	0.535898i	0.0817237i	0.999165	+	0.0408619i	$0.0130104\pi$
$43$	0.535898i	0.0817237i	−0.999165	+	0.0408619i	$0.986990\pi$
$44$	0	0
$45$	− 3.46410i	− 0.516398i
$46$	0	0
$47$	−6.73205	−0.981971	−0.490985	−	0.871168i	$-0.663363\pi$
$47$	−6.73205	−0.981971	−0.490985	+	0.871168i	$0.663363\pi$
$48$	0	0
$49$	−5.39230	−0.770329
$50$	0	0
$51$	− 10.9282i	− 1.53025i
$52$	0	0
$53$	− 2.92820i	− 0.402220i	−0.979569	−	0.201110i	$-0.935545\pi$
$53$	− 2.92820i	− 0.402220i	0.979569	−	0.201110i	$-0.0644548\pi$
$54$	0	0
$55$	−16.3923	−2.21034
$56$	0	0
$57$	−1.46410	−0.193925
$58$	0	0
$59$	10.1962i	1.32743i	0.747987	+	0.663713i	$0.231020\pi$
$59$	10.1962i	1.32743i	−0.747987	+	0.663713i	$0.768980\pi$
$60$	0	0
$61$	2.92820i	0.374918i	0.982272	+	0.187459i	$0.0600252\pi$
$61$	2.92820i	0.374918i	−0.982272	+	0.187459i	$0.939975\pi$
$62$	0	0
$63$	−1.26795	−0.159747
$64$	0	0
$65$	−3.46410	−0.429669
$66$	0	0
$67$	− 0.732051i	− 0.0894342i	−0.999000	−	0.0447171i	$-0.985761\pi$
$67$	− 0.732051i	− 0.0894342i	0.999000	−	0.0447171i	$-0.0142386\pi$
$68$	0	0
$69$	8.00000i	0.963087i
$70$	0	0
$71$	8.19615	0.972704	0.486352	−	0.873763i	$-0.338327\pi$
$71$	8.19615	0.972704	0.486352	+	0.873763i	$0.338327\pi$
$72$	0	0
$73$	−7.46410	−0.873607	−0.436804	−	0.899557i	$-0.643889\pi$
$73$	−7.46410	−0.873607	−0.436804	+	0.899557i	$0.643889\pi$
$74$	0	0
$75$	14.0000i	1.61658i
$76$	0	0
$77$	6.00000i	0.683763i
$78$	0	0
$79$	−5.46410	−0.614759	−0.307380	−	0.951587i	$-0.599452\pi$
$79$	−5.46410	−0.614759	−0.307380	+	0.951587i	$0.599452\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	− 3.26795i	− 0.358704i	−0.983785	−	0.179352i	$-0.942600\pi$
$83$	− 3.26795i	− 0.358704i	0.983785	−	0.179352i	$-0.0574001\pi$
$84$	0	0
$85$	18.9282i	2.05305i
$86$	0	0
$87$	4.00000	0.428845
$88$	0	0
$89$	−17.3205	−1.83597	−0.917985	−	0.396615i	$-0.870185\pi$
$89$	−17.3205	−1.83597	−0.917985	+	0.396615i	$0.870185\pi$
$90$	0	0
$91$	1.26795i	0.132917i
$92$	0	0
$93$	− 13.4641i	− 1.39616i
$94$	0	0
$95$	2.53590	0.260178
$96$	0	0
$97$	6.39230	0.649040	0.324520	−	0.945879i	$-0.394797\pi$
$97$	6.39230	0.649040	0.324520	+	0.945879i	$0.394797\pi$
$98$	0	0
$99$	− 4.73205i	− 0.475589i
$100$	0	0
$101$	− 12.0000i	− 1.19404i	−0.802225	−	0.597022i	$-0.796350\pi$
$101$	− 12.0000i	− 1.19404i	0.802225	−	0.597022i	$-0.203650\pi$
$102$	0	0
$103$	−6.92820	−0.682656	−0.341328	−	0.939944i	$-0.610877\pi$
$103$	−6.92820	−0.682656	−0.341328	+	0.939944i	$0.610877\pi$
$104$	0	0
$105$	8.78461	0.857290
$106$	0	0
$107$	− 4.92820i	− 0.476427i	−0.971213	−	0.238214i	$-0.923438\pi$
$107$	− 4.92820i	− 0.476427i	0.971213	−	0.238214i	$-0.0765619\pi$
$108$	0	0
$109$	− 2.00000i	− 0.191565i	−0.995402	−	0.0957826i	$-0.969465\pi$
$109$	− 2.00000i	− 0.191565i	0.995402	−	0.0957826i	$-0.0305354\pi$
$110$	0	0
$111$	−17.8564	−1.69486
$112$	0	0
$113$	2.53590	0.238557	0.119279	−	0.992861i	$-0.461942\pi$
$113$	2.53590	0.238557	0.119279	+	0.992861i	$0.461942\pi$
$114$	0	0
$115$	− 13.8564i	− 1.29212i
$116$	0	0
$117$	− 1.00000i	− 0.0924500i
$118$	0	0
$119$	6.92820	0.635107
$120$	0	0
$121$	−11.3923	−1.03566
$122$	0	0
$123$	− 17.8564i	− 1.61006i
$124$	0	0
$125$	− 6.92820i	− 0.619677i
$126$	0	0
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	0	0
$129$	1.07180	0.0943664
$130$	0	0
$131$	− 19.8564i	− 1.73486i	−0.497557	−	0.867431i	$-0.665770\pi$
$131$	− 19.8564i	− 1.73486i	0.497557	−	0.867431i	$-0.334230\pi$
$132$	0	0
$133$	− 0.928203i	− 0.0804854i
$134$	0	0
$135$	13.8564	1.19257
$136$	0	0
$137$	12.9282	1.10453	0.552265	−	0.833668i	$-0.313763\pi$
$137$	12.9282	1.10453	0.552265	+	0.833668i	$0.313763\pi$
$138$	0	0
$139$	10.0000i	0.848189i	0.905618	+	0.424094i	$0.139408\pi$
$139$	10.0000i	0.848189i	−0.905618	+	0.424094i	$0.860592\pi$
$140$	0	0
$141$	13.4641i	1.13388i
$142$	0	0
$143$	−4.73205	−0.395714
$144$	0	0
$145$	−6.92820	−0.575356
$146$	0	0
$147$	10.7846i	0.889500i
$148$	0	0
$149$	− 12.9282i	− 1.05912i	−0.848273	−	0.529560i	$-0.822357\pi$
$149$	− 12.9282i	− 1.05912i	0.848273	−	0.529560i	$-0.177643\pi$
$150$	0	0
$151$	7.12436	0.579772	0.289886	−	0.957061i	$-0.406383\pi$
$151$	7.12436	0.579772	0.289886	+	0.957061i	$0.406383\pi$
$152$	0	0
$153$	−5.46410	−0.441746
$154$	0	0
$155$	23.3205i	1.87315i
$156$	0	0
$157$	− 16.9282i	− 1.35102i	−0.737352	−	0.675509i	$-0.763924\pi$
$157$	− 16.9282i	− 1.35102i	0.737352	−	0.675509i	$-0.236076\pi$
$158$	0	0
$159$	−5.85641	−0.464443
$160$	0	0
$161$	−5.07180	−0.399714
$162$	0	0
$163$	− 16.7321i	− 1.31056i	−0.755388	−	0.655278i	$-0.772552\pi$
$163$	− 16.7321i	− 1.31056i	0.755388	−	0.655278i	$-0.227448\pi$
$164$	0	0
$165$	32.7846i	2.55228i
$166$	0	0
$167$	5.66025	0.438004	0.219002	−	0.975724i	$-0.429720\pi$
$167$	5.66025	0.438004	0.219002	+	0.975724i	$0.429720\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0.732051i	0.0559813i
$172$	0	0
$173$	6.92820i	0.526742i	0.964695	+	0.263371i	$0.0848343\pi$
$173$	6.92820i	0.526742i	−0.964695	+	0.263371i	$0.915166\pi$
$174$	0	0
$175$	−8.87564	−0.670936
$176$	0	0
$177$	20.3923	1.53278
$178$	0	0
$179$	10.3923i	0.776757i	0.921500	+	0.388379i	$0.126965\pi$
$179$	10.3923i	0.776757i	−0.921500	+	0.388379i	$0.873035\pi$
$180$	0	0
$181$	− 8.92820i	− 0.663628i	−0.943345	−	0.331814i	$-0.892339\pi$
$181$	− 8.92820i	− 0.663628i	0.943345	−	0.331814i	$-0.107661\pi$
$182$	0	0
$183$	5.85641	0.432918
$184$	0	0
$185$	30.9282	2.27389
$186$	0	0
$187$	25.8564i	1.89081i
$188$	0	0
$189$	− 5.07180i	− 0.368919i
$190$	0	0
$191$	14.5359	1.05178	0.525890	−	0.850552i	$-0.323732\pi$
$191$	14.5359	1.05178	0.525890	+	0.850552i	$0.323732\pi$
$192$	0	0
$193$	−1.60770	−0.115724	−0.0578622	−	0.998325i	$-0.518428\pi$
$193$	−1.60770	−0.115724	−0.0578622	+	0.998325i	$0.518428\pi$
$194$	0	0
$195$	6.92820i	0.496139i
$196$	0	0
$197$	− 3.07180i	− 0.218856i	−0.993995	−	0.109428i	$-0.965098\pi$
$197$	− 3.07180i	− 0.218856i	0.993995	−	0.109428i	$-0.0349020\pi$
$198$	0	0
$199$	−16.7846	−1.18983	−0.594915	−	0.803789i	$-0.702814\pi$
$199$	−16.7846	−1.18983	−0.594915	+	0.803789i	$0.702814\pi$
$200$	0	0
$201$	−1.46410	−0.103270
$202$	0	0
$203$	2.53590i	0.177985i
$204$	0	0
$205$	30.9282i	2.16012i
$206$	0	0
$207$	4.00000	0.278019
$208$	0	0
$209$	3.46410	0.239617
$210$	0	0
$211$	7.85641i	0.540857i	0.962740	+	0.270429i	$0.0871654\pi$
$211$	7.85641i	0.540857i	−0.962740	+	0.270429i	$0.912835\pi$
$212$	0	0
$213$	− 16.3923i	− 1.12318i
$214$	0	0
$215$	−1.85641	−0.126606
$216$	0	0
$217$	8.53590	0.579455
$218$	0	0
$219$	14.9282i	1.00875i
$220$	0	0
$221$	5.46410i	0.367555i
$222$	0	0
$223$	0.196152	0.0131353	0.00656767	−	0.999978i	$-0.497909\pi$
$223$	0.196152	0.0131353	0.00656767	+	0.999978i	$0.497909\pi$
$224$	0	0
$225$	7.00000	0.466667
$226$	0	0
$227$	− 2.87564i	− 0.190863i	−0.995436	−	0.0954316i	$-0.969577\pi$
$227$	− 2.87564i	− 0.190863i	0.995436	−	0.0954316i	$-0.0304231\pi$
$228$	0	0
$229$	− 5.32051i	− 0.351589i	−0.984427	−	0.175795i	$-0.943751\pi$
$229$	− 5.32051i	− 0.351589i	0.984427	−	0.175795i	$-0.0562494\pi$
$230$	0	0
$231$	12.0000	0.789542
$232$	0	0
$233$	16.9282	1.10900	0.554502	−	0.832183i	$-0.312909\pi$
$233$	16.9282	1.10900	0.554502	+	0.832183i	$0.312909\pi$
$234$	0	0
$235$	− 23.3205i	− 1.52126i
$236$	0	0
$237$	10.9282i	0.709863i
$238$	0	0
$239$	−18.7321	−1.21168	−0.605838	−	0.795588i	$-0.707162\pi$
$239$	−18.7321	−1.21168	−0.605838	+	0.795588i	$0.707162\pi$
$240$	0	0
$241$	−30.3923	−1.95774	−0.978870	−	0.204482i	$-0.934449\pi$
$241$	−30.3923	−1.95774	−0.978870	+	0.204482i	$0.934449\pi$
$242$	0	0
$243$	10.0000i	0.641500i
$244$	0	0
$245$	− 18.6795i	− 1.19339i
$246$	0	0
$247$	0.732051	0.0465793
$248$	0	0
$249$	−6.53590	−0.414196
$250$	0	0
$251$	− 6.39230i	− 0.403479i	−0.979439	−	0.201739i	$-0.935341\pi$
$251$	− 6.39230i	− 0.403479i	0.979439	−	0.201739i	$-0.0646594\pi$
$252$	0	0
$253$	− 18.9282i	− 1.19001i
$254$	0	0
$255$	37.8564	2.37066
$256$	0	0
$257$	−23.8564	−1.48812	−0.744061	−	0.668112i	$-0.767103\pi$
$257$	−23.8564	−1.48812	−0.744061	+	0.668112i	$0.767103\pi$
$258$	0	0
$259$	− 11.3205i	− 0.703422i
$260$	0	0
$261$	− 2.00000i	− 0.123797i
$262$	0	0
$263$	27.3205	1.68465	0.842327	−	0.538966i	$-0.181185\pi$
$263$	27.3205	1.68465	0.842327	+	0.538966i	$0.181185\pi$
$264$	0	0
$265$	10.1436	0.623116
$266$	0	0
$267$	34.6410i	2.12000i
$268$	0	0
$269$	− 7.85641i	− 0.479014i	−0.970895	−	0.239507i	$-0.923014\pi$
$269$	− 7.85641i	− 0.479014i	0.970895	−	0.239507i	$-0.0769857\pi$
$270$	0	0
$271$	20.1962	1.22683	0.613414	−	0.789761i	$-0.289796\pi$
$271$	20.1962	1.22683	0.613414	+	0.789761i	$0.289796\pi$
$272$	0	0
$273$	2.53590	0.153480
$274$	0	0
$275$	− 33.1244i	− 1.99747i
$276$	0	0
$277$	1.85641i	0.111541i	0.998444	+	0.0557703i	$0.0177615\pi$
$277$	1.85641i	0.111541i	−0.998444	+	0.0557703i	$0.982239\pi$
$278$	0	0
$279$	−6.73205	−0.403037
$280$	0	0
$281$	9.32051	0.556015	0.278007	−	0.960579i	$-0.410326\pi$
$281$	9.32051	0.556015	0.278007	+	0.960579i	$0.410326\pi$
$282$	0	0
$283$	− 19.4641i	− 1.15702i	−0.815675	−	0.578510i	$-0.803634\pi$
$283$	− 19.4641i	− 1.15702i	0.815675	−	0.578510i	$-0.196366\pi$
$284$	0	0
$285$	− 5.07180i	− 0.300427i
$286$	0	0
$287$	11.3205	0.668228
$288$	0	0
$289$	12.8564	0.756259
$290$	0	0
$291$	− 12.7846i	− 0.749447i
$292$	0	0
$293$	32.9282i	1.92369i	0.273602	+	0.961843i	$0.411785\pi$
$293$	32.9282i	1.92369i	−0.273602	+	0.961843i	$0.588215\pi$
$294$	0	0
$295$	−35.3205	−2.05644
$296$	0	0
$297$	18.9282	1.09833
$298$	0	0
$299$	− 4.00000i	− 0.231326i
$300$	0	0
$301$	0.679492i	0.0391653i
$302$	0	0
$303$	−24.0000	−1.37876
$304$	0	0
$305$	−10.1436	−0.580820
$306$	0	0
$307$	− 0.732051i	− 0.0417803i	−0.999782	−	0.0208902i	$-0.993350\pi$
$307$	− 0.732051i	− 0.0417803i	0.999782	−	0.0208902i	$-0.00665003\pi$
$308$	0	0
$309$	13.8564i	0.788263i
$310$	0	0
$311$	−1.07180	−0.0607760	−0.0303880	−	0.999538i	$-0.509674\pi$
$311$	−1.07180	−0.0607760	−0.0303880	+	0.999538i	$0.509674\pi$
$312$	0	0
$313$	0.392305	0.0221744	0.0110872	−	0.999939i	$-0.496471\pi$
$313$	0.392305	0.0221744	0.0110872	+	0.999939i	$0.496471\pi$
$314$	0	0
$315$	− 4.39230i	− 0.247478i
$316$	0	0
$317$	3.46410i	0.194563i	0.995257	+	0.0972817i	$0.0310148\pi$
$317$	3.46410i	0.194563i	−0.995257	+	0.0972817i	$0.968985\pi$
$318$	0	0
$319$	−9.46410	−0.529888
$320$	0	0
$321$	−9.85641	−0.550131
$322$	0	0
$323$	− 4.00000i	− 0.222566i
$324$	0	0
$325$	− 7.00000i	− 0.388290i
$326$	0	0
$327$	−4.00000	−0.221201
$328$	0	0
$329$	−8.53590	−0.470599
$330$	0	0
$331$	− 17.5167i	− 0.962803i	−0.876500	−	0.481401i	$-0.840128\pi$
$331$	− 17.5167i	− 0.962803i	0.876500	−	0.481401i	$-0.159872\pi$
$332$	0	0
$333$	8.92820i	0.489263i
$334$	0	0
$335$	2.53590	0.138551
$336$	0	0
$337$	1.46410	0.0797547	0.0398773	−	0.999205i	$-0.487303\pi$
$337$	1.46410	0.0797547	0.0398773	+	0.999205i	$0.487303\pi$
$338$	0	0
$339$	− 5.07180i	− 0.275462i
$340$	0	0
$341$	31.8564i	1.72512i
$342$	0	0
$343$	−15.7128	−0.848412
$344$	0	0
$345$	−27.7128	−1.49201
$346$	0	0
$347$	7.07180i	0.379634i	0.981819	+	0.189817i	$0.0607895\pi$
$347$	7.07180i	0.379634i	−0.981819	+	0.189817i	$0.939211\pi$
$348$	0	0
$349$	9.60770i	0.514288i	0.966373	+	0.257144i	$0.0827815\pi$
$349$	9.60770i	0.514288i	−0.966373	+	0.257144i	$0.917219\pi$
$350$	0	0
$351$	4.00000	0.213504
$352$	0	0
$353$	−11.0718	−0.589292	−0.294646	−	0.955606i	$-0.595202\pi$
$353$	−11.0718	−0.589292	−0.294646	+	0.955606i	$0.595202\pi$
$354$	0	0
$355$	28.3923i	1.50691i
$356$	0	0
$357$	− 13.8564i	− 0.733359i
$358$	0	0
$359$	−31.5167	−1.66339	−0.831693	−	0.555236i	$-0.812628\pi$
$359$	−31.5167	−1.66339	−0.831693	+	0.555236i	$0.812628\pi$
$360$	0	0
$361$	18.4641	0.971795
$362$	0	0
$363$	22.7846i	1.19588i
$364$	0	0
$365$	− 25.8564i	− 1.35339i
$366$	0	0
$367$	22.2487	1.16137	0.580687	−	0.814127i	$-0.302784\pi$
$367$	22.2487	1.16137	0.580687	+	0.814127i	$0.302784\pi$
$368$	0	0
$369$	−8.92820	−0.464784
$370$	0	0
$371$	− 3.71281i	− 0.192760i
$372$	0	0
$373$	− 14.7846i	− 0.765518i	−0.923848	−	0.382759i	$-0.874974\pi$
$373$	− 14.7846i	− 0.765518i	0.923848	−	0.382759i	$-0.125026\pi$
$374$	0	0
$375$	−13.8564	−0.715542
$376$	0	0
$377$	−2.00000	−0.103005
$378$	0	0
$379$	− 14.5885i	− 0.749359i	−0.927154	−	0.374679i	$-0.877753\pi$
$379$	− 14.5885i	− 0.749359i	0.927154	−	0.374679i	$-0.122247\pi$
$380$	0	0
$381$	− 8.00000i	− 0.409852i
$382$	0	0
$383$	10.3397	0.528336	0.264168	−	0.964477i	$-0.414903\pi$
$383$	10.3397	0.528336	0.264168	+	0.964477i	$0.414903\pi$
$384$	0	0
$385$	−20.7846	−1.05928
$386$	0	0
$387$	− 0.535898i	− 0.0272412i
$388$	0	0
$389$	2.00000i	0.101404i	0.998714	+	0.0507020i	$0.0161459\pi$
$389$	2.00000i	0.101404i	−0.998714	+	0.0507020i	$0.983854\pi$
$390$	0	0
$391$	−21.8564	−1.10533
$392$	0	0
$393$	−39.7128	−2.00325
$394$	0	0
$395$	− 18.9282i	− 0.952381i
$396$	0	0
$397$	− 4.53590i	− 0.227650i	−0.993501	−	0.113825i	$-0.963690\pi$
$397$	− 4.53590i	− 0.227650i	0.993501	−	0.113825i	$-0.0363104\pi$
$398$	0	0
$399$	−1.85641	−0.0929366
$400$	0	0
$401$	4.53590	0.226512	0.113256	−	0.993566i	$-0.463872\pi$
$401$	4.53590	0.226512	0.113256	+	0.993566i	$0.463872\pi$
$402$	0	0
$403$	6.73205i	0.335347i
$404$	0	0
$405$	− 38.1051i	− 1.89346i
$406$	0	0
$407$	42.2487	2.09419
$408$	0	0
$409$	12.9282	0.639259	0.319629	−	0.947543i	$-0.396442\pi$
$409$	12.9282	0.639259	0.319629	+	0.947543i	$0.396442\pi$
$410$	0	0
$411$	− 25.8564i	− 1.27540i
$412$	0	0
$413$	12.9282i	0.636155i
$414$	0	0
$415$	11.3205	0.555702
$416$	0	0
$417$	20.0000	0.979404
$418$	0	0
$419$	30.7846i	1.50393i	0.659205	+	0.751963i	$0.270893\pi$
$419$	30.7846i	1.50393i	−0.659205	+	0.751963i	$0.729107\pi$
$420$	0	0
$421$	24.2487i	1.18181i	0.806741	+	0.590905i	$0.201229\pi$
$421$	24.2487i	1.18181i	−0.806741	+	0.590905i	$0.798771\pi$
$422$	0	0
$423$	6.73205	0.327324
$424$	0	0
$425$	−38.2487	−1.85534
$426$	0	0
$427$	3.71281i	0.179676i
$428$	0	0
$429$	9.46410i	0.456931i
$430$	0	0
$431$	−39.1244	−1.88455	−0.942277	−	0.334835i	$-0.891320\pi$
$431$	−39.1244	−1.88455	−0.942277	+	0.334835i	$0.891320\pi$
$432$	0	0
$433$	−6.78461	−0.326048	−0.163024	−	0.986622i	$-0.552125\pi$
$433$	−6.78461	−0.326048	−0.163024	+	0.986622i	$0.552125\pi$
$434$	0	0
$435$	13.8564i	0.664364i
$436$	0	0
$437$	2.92820i	0.140075i
$438$	0	0
$439$	−2.92820	−0.139756	−0.0698778	−	0.997556i	$-0.522261\pi$
$439$	−2.92820	−0.139756	−0.0698778	+	0.997556i	$0.522261\pi$
$440$	0	0
$441$	5.39230	0.256776
$442$	0	0
$443$	30.0000i	1.42534i	0.701498	+	0.712672i	$0.252515\pi$
$443$	30.0000i	1.42534i	−0.701498	+	0.712672i	$0.747485\pi$
$444$	0	0
$445$	− 60.0000i	− 2.84427i
$446$	0	0
$447$	−25.8564	−1.22297
$448$	0	0
$449$	17.6077	0.830959	0.415479	−	0.909603i	$-0.363614\pi$
$449$	17.6077	0.830959	0.415479	+	0.909603i	$0.363614\pi$
$450$	0	0
$451$	42.2487i	1.98941i
$452$	0	0
$453$	− 14.2487i	− 0.669463i
$454$	0	0
$455$	−4.39230	−0.205914
$456$	0	0
$457$	−14.0000	−0.654892	−0.327446	−	0.944870i	$-0.606188\pi$
$457$	−14.0000	−0.654892	−0.327446	+	0.944870i	$0.606188\pi$
$458$	0	0
$459$	− 21.8564i	− 1.02017i
$460$	0	0
$461$	20.9282i	0.974724i	0.873200	+	0.487362i	$0.162041\pi$
$461$	20.9282i	0.974724i	−0.873200	+	0.487362i	$0.837959\pi$
$462$	0	0
$463$	−1.66025	−0.0771585	−0.0385793	−	0.999256i	$-0.512283\pi$
$463$	−1.66025	−0.0771585	−0.0385793	+	0.999256i	$0.512283\pi$
$464$	0	0
$465$	46.6410	2.16293
$466$	0	0
$467$	− 36.2487i	− 1.67739i	−0.544601	−	0.838695i	$-0.683319\pi$
$467$	− 36.2487i	− 1.67739i	0.544601	−	0.838695i	$-0.316681\pi$
$468$	0	0
$469$	− 0.928203i	− 0.0428604i
$470$	0	0
$471$	−33.8564	−1.56002
$472$	0	0
$473$	−2.53590	−0.116601
$474$	0	0
$475$	5.12436i	0.235122i
$476$	0	0
$477$	2.92820i	0.134073i
$478$	0	0
$479$	−21.2679	−0.971757	−0.485879	−	0.874026i	$-0.661500\pi$
$479$	−21.2679	−0.971757	−0.485879	+	0.874026i	$0.661500\pi$
$480$	0	0
$481$	8.92820	0.407091
$482$	0	0
$483$	10.1436i	0.461549i
$484$	0	0
$485$	22.1436i	1.00549i
$486$	0	0
$487$	42.4449	1.92336	0.961680	−	0.274174i	$-0.0884043\pi$
$487$	42.4449	1.92336	0.961680	+	0.274174i	$0.0884043\pi$
$488$	0	0
$489$	−33.4641	−1.51330
$490$	0	0
$491$	24.2487i	1.09433i	0.837025	+	0.547165i	$0.184293\pi$
$491$	24.2487i	1.09433i	−0.837025	+	0.547165i	$0.815707\pi$
$492$	0	0
$493$	10.9282i	0.492182i
$494$	0	0
$495$	16.3923	0.736779
$496$	0	0
$497$	10.3923	0.466159
$498$	0	0
$499$	7.26795i	0.325358i	0.986679	+	0.162679i	$0.0520135\pi$
$499$	7.26795i	0.325358i	−0.986679	+	0.162679i	$0.947986\pi$
$500$	0	0
$501$	− 11.3205i	− 0.505763i
$502$	0	0
$503$	−30.5359	−1.36153	−0.680764	−	0.732503i	$-0.738352\pi$
$503$	−30.5359	−1.36153	−0.680764	+	0.732503i	$0.738352\pi$
$504$	0	0
$505$	41.5692	1.84981
$506$	0	0
$507$	2.00000i	0.0888231i
$508$	0	0
$509$	14.3923i	0.637928i	0.947767	+	0.318964i	$0.103335\pi$
$509$	14.3923i	0.637928i	−0.947767	+	0.318964i	$0.896665\pi$
$510$	0	0
$511$	−9.46410	−0.418667
$512$	0	0
$513$	−2.92820	−0.129283
$514$	0	0
$515$	− 24.0000i	− 1.05757i
$516$	0	0
$517$	− 31.8564i	− 1.40104i
$518$	0	0
$519$	13.8564	0.608229
$520$	0	0
$521$	−25.1769	−1.10302	−0.551510	−	0.834168i	$-0.685948\pi$
$521$	−25.1769	−1.10302	−0.551510	+	0.834168i	$0.685948\pi$
$522$	0	0
$523$	− 14.0000i	− 0.612177i	−0.952003	−	0.306089i	$-0.900980\pi$
$523$	− 14.0000i	− 0.612177i	0.952003	−	0.306089i	$-0.0990204\pi$
$524$	0	0
$525$	17.7513i	0.774730i
$526$	0	0
$527$	36.7846	1.60236
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	− 10.1962i	− 0.442475i
$532$	0	0
$533$	8.92820i	0.386723i
$534$	0	0
$535$	17.0718	0.738078
$536$	0	0
$537$	20.7846	0.896922
$538$	0	0
$539$	− 25.5167i	− 1.09908i
$540$	0	0
$541$	− 3.07180i	− 0.132067i	−0.997817	−	0.0660334i	$-0.978966\pi$
$541$	− 3.07180i	− 0.132067i	0.997817	−	0.0660334i	$-0.0210344\pi$
$542$	0	0
$543$	−17.8564	−0.766292
$544$	0	0
$545$	6.92820	0.296772
$546$	0	0
$547$	11.8564i	0.506943i	0.967343	+	0.253472i	$0.0815725\pi$
$547$	11.8564i	0.506943i	−0.967343	+	0.253472i	$0.918428\pi$
$548$	0	0
$549$	− 2.92820i	− 0.124973i
$550$	0	0
$551$	1.46410	0.0623728
$552$	0	0
$553$	−6.92820	−0.294617
$554$	0	0
$555$	− 61.8564i	− 2.62566i
$556$	0	0
$557$	34.7846i	1.47387i	0.675963	+	0.736936i	$0.263728\pi$
$557$	34.7846i	1.47387i	−0.675963	+	0.736936i	$0.736272\pi$
$558$	0	0
$559$	−0.535898	−0.0226661
$560$	0	0
$561$	51.7128	2.18332
$562$	0	0
$563$	7.46410i	0.314574i	0.987553	+	0.157287i	$0.0502748\pi$
$563$	7.46410i	0.314574i	−0.987553	+	0.157287i	$0.949725\pi$
$564$	0	0
$565$	8.78461i	0.369571i
$566$	0	0
$567$	−13.9474	−0.585737
$568$	0	0
$569$	−14.0000	−0.586911	−0.293455	−	0.955973i	$-0.594805\pi$
$569$	−14.0000	−0.586911	−0.293455	+	0.955973i	$0.594805\pi$
$570$	0	0
$571$	− 2.67949i	− 0.112133i	−0.998427	−	0.0560666i	$-0.982144\pi$
$571$	− 2.67949i	− 0.112133i	0.998427	−	0.0560666i	$-0.0178559\pi$
$572$	0	0
$573$	− 29.0718i	− 1.21449i
$574$	0	0
$575$	28.0000	1.16768
$576$	0	0
$577$	−7.07180	−0.294403	−0.147201	−	0.989107i	$-0.547027\pi$
$577$	−7.07180	−0.294403	−0.147201	+	0.989107i	$0.547027\pi$
$578$	0	0
$579$	3.21539i	0.133627i
$580$	0	0
$581$	− 4.14359i	− 0.171905i
$582$	0	0
$583$	13.8564	0.573874
$584$	0	0
$585$	3.46410	0.143223
$586$	0	0
$587$	4.33975i	0.179120i	0.995981	+	0.0895602i	$0.0285462\pi$
$587$	4.33975i	0.179120i	−0.995981	+	0.0895602i	$0.971454\pi$
$588$	0	0
$589$	− 4.92820i	− 0.203063i
$590$	0	0
$591$	−6.14359	−0.252714
$592$	0	0
$593$	−7.85641	−0.322624	−0.161312	−	0.986903i	$-0.551573\pi$
$593$	−7.85641	−0.322624	−0.161312	+	0.986903i	$0.551573\pi$
$594$	0	0
$595$	24.0000i	0.983904i
$596$	0	0
$597$	33.5692i	1.37390i
$598$	0	0
$599$	45.1769	1.84588	0.922939	−	0.384945i	$-0.125780\pi$
$599$	45.1769	1.84588	0.922939	+	0.384945i	$0.125780\pi$
$600$	0	0
$601$	−4.39230	−0.179166	−0.0895829	−	0.995979i	$-0.528553\pi$
$601$	−4.39230	−0.179166	−0.0895829	+	0.995979i	$0.528553\pi$
$602$	0	0
$603$	0.732051i	0.0298114i
$604$	0	0
$605$	− 39.4641i	− 1.60444i
$606$	0	0
$607$	−7.32051	−0.297130	−0.148565	−	0.988903i	$-0.547465\pi$
$607$	−7.32051	−0.297130	−0.148565	+	0.988903i	$0.547465\pi$
$608$	0	0
$609$	5.07180	0.205520
$610$	0	0
$611$	− 6.73205i	− 0.272350i
$612$	0	0
$613$	24.6410i	0.995241i	0.867395	+	0.497621i	$0.165793\pi$
$613$	24.6410i	0.995241i	−0.867395	+	0.497621i	$0.834207\pi$
$614$	0	0
$615$	61.8564	2.49429
$616$	0	0
$617$	41.3205	1.66350	0.831751	−	0.555150i	$-0.187339\pi$
$617$	41.3205	1.66350	0.831751	+	0.555150i	$0.187339\pi$
$618$	0	0
$619$	18.5885i	0.747133i	0.927603	+	0.373567i	$0.121865\pi$
$619$	18.5885i	0.747133i	−0.927603	+	0.373567i	$0.878135\pi$
$620$	0	0
$621$	16.0000i	0.642058i
$622$	0	0
$623$	−21.9615	−0.879870
$624$	0	0
$625$	−11.0000	−0.440000
$626$	0	0
$627$	− 6.92820i	− 0.276686i
$628$	0	0
$629$	− 48.7846i	− 1.94517i
$630$	0	0
$631$	42.0526	1.67409	0.837043	−	0.547137i	$-0.184282\pi$
$631$	42.0526	1.67409	0.837043	+	0.547137i	$0.184282\pi$
$632$	0	0
$633$	15.7128	0.624528
$634$	0	0
$635$	13.8564i	0.549875i
$636$	0	0
$637$	− 5.39230i	− 0.213651i
$638$	0	0
$639$	−8.19615	−0.324235
$640$	0	0
$641$	−22.2487	−0.878771	−0.439386	−	0.898299i	$-0.644804\pi$
$641$	−22.2487	−0.878771	−0.439386	+	0.898299i	$0.644804\pi$
$642$	0	0
$643$	12.7321i	0.502103i	0.967974	+	0.251052i	$0.0807764\pi$
$643$	12.7321i	0.502103i	−0.967974	+	0.251052i	$0.919224\pi$
$644$	0	0
$645$	3.71281i	0.146192i
$646$	0	0
$647$	−37.8564	−1.48829	−0.744144	−	0.668019i	$-0.767143\pi$
$647$	−37.8564	−1.48829	−0.744144	+	0.668019i	$0.767143\pi$
$648$	0	0
$649$	−48.2487	−1.89393
$650$	0	0
$651$	− 17.0718i	− 0.669096i
$652$	0	0
$653$	− 3.07180i	− 0.120209i	−0.998192	−	0.0601043i	$-0.980857\pi$
$653$	− 3.07180i	− 0.120209i	0.998192	−	0.0601043i	$-0.0191433\pi$
$654$	0	0
$655$	68.7846	2.68764
$656$	0	0
$657$	7.46410	0.291202
$658$	0	0
$659$	− 14.0000i	− 0.545363i	−0.962104	−	0.272681i	$-0.912090\pi$
$659$	− 14.0000i	− 0.545363i	0.962104	−	0.272681i	$-0.0879105\pi$
$660$	0	0
$661$	17.3205i	0.673690i	0.941560	+	0.336845i	$0.109360\pi$
$661$	17.3205i	0.673690i	−0.941560	+	0.336845i	$0.890640\pi$
$662$	0	0
$663$	10.9282	0.424416
$664$	0	0
$665$	3.21539	0.124687
$666$	0	0
$667$	− 8.00000i	− 0.309761i
$668$	0	0
$669$	− 0.392305i	− 0.0151674i
$670$	0	0
$671$	−13.8564	−0.534921
$672$	0	0
$673$	33.1769	1.27888	0.639438	−	0.768843i	$-0.279167\pi$
$673$	33.1769	1.27888	0.639438	+	0.768843i	$0.279167\pi$
$674$	0	0
$675$	28.0000i	1.07772i
$676$	0	0
$677$	21.0718i	0.809855i	0.914349	+	0.404927i	$0.132703\pi$
$677$	21.0718i	0.809855i	−0.914349	+	0.404927i	$0.867297\pi$
$678$	0	0
$679$	8.10512	0.311046
$680$	0	0
$681$	−5.75129	−0.220390
$682$	0	0
$683$	17.8038i	0.681245i	0.940200	+	0.340623i	$0.110638\pi$
$683$	17.8038i	0.681245i	−0.940200	+	0.340623i	$0.889362\pi$
$684$	0	0
$685$	44.7846i	1.71113i
$686$	0	0
$687$	−10.6410	−0.405980
$688$	0	0
$689$	2.92820	0.111556
$690$	0	0
$691$	32.8372i	1.24918i	0.780951	+	0.624592i	$0.214735\pi$
$691$	32.8372i	1.24918i	−0.780951	+	0.624592i	$0.785265\pi$
$692$	0	0
$693$	− 6.00000i	− 0.227921i
$694$	0	0
$695$	−34.6410	−1.31401
$696$	0	0
$697$	48.7846	1.84785
$698$	0	0
$699$	− 33.8564i	− 1.28057i
$700$	0	0
$701$	− 40.6410i	− 1.53499i	−0.641055	−	0.767495i	$-0.721503\pi$
$701$	− 40.6410i	− 1.53499i	0.641055	−	0.767495i	$-0.278497\pi$
$702$	0	0
$703$	−6.53590	−0.246506
$704$	0	0
$705$	−46.6410	−1.75660
$706$	0	0
$707$	− 15.2154i	− 0.572234i
$708$	0	0
$709$	29.3205i	1.10115i	0.834784	+	0.550577i	$0.185592\pi$
$709$	29.3205i	1.10115i	−0.834784	+	0.550577i	$0.814408\pi$
$710$	0	0
$711$	5.46410	0.204920
$712$	0	0
$713$	−26.9282	−1.00847
$714$	0	0
$715$	− 16.3923i	− 0.613037i
$716$	0	0
$717$	37.4641i	1.39912i
$718$	0	0
$719$	−30.9282	−1.15343	−0.576714	−	0.816946i	$-0.695665\pi$
$719$	−30.9282	−1.15343	−0.576714	+	0.816946i	$0.695665\pi$
$720$	0	0
$721$	−8.78461	−0.327156
$722$	0	0
$723$	60.7846i	2.26060i
$724$	0	0
$725$	− 14.0000i	− 0.519947i
$726$	0	0
$727$	22.9282	0.850360	0.425180	−	0.905109i	$-0.360211\pi$
$727$	22.9282	0.850360	0.425180	+	0.905109i	$0.360211\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	2.92820i	0.108304i
$732$	0	0
$733$	37.3205i	1.37846i	0.724541	+	0.689232i	$0.242052\pi$
$733$	37.3205i	1.37846i	−0.724541	+	0.689232i	$0.757948\pi$
$734$	0	0
$735$	−37.3590	−1.37801
$736$	0	0
$737$	3.46410	0.127602
$738$	0	0
$739$	− 39.7654i	− 1.46279i	−0.681953	−	0.731396i	$-0.738869\pi$
$739$	− 39.7654i	− 1.46279i	0.681953	−	0.731396i	$-0.261131\pi$
$740$	0	0
$741$	− 1.46410i	− 0.0537851i
$742$	0	0
$743$	32.1962	1.18116	0.590581	−	0.806978i	$-0.298899\pi$
$743$	32.1962	1.18116	0.590581	+	0.806978i	$0.298899\pi$
$744$	0	0
$745$	44.7846	1.64078
$746$	0	0
$747$	3.26795i	0.119568i
$748$	0	0
$749$	− 6.24871i	− 0.228323i
$750$	0	0
$751$	−1.07180	−0.0391104	−0.0195552	−	0.999809i	$-0.506225\pi$
$751$	−1.07180	−0.0391104	−0.0195552	+	0.999809i	$0.506225\pi$
$752$	0	0
$753$	−12.7846	−0.465897
$754$	0	0
$755$	24.6795i	0.898179i
$756$	0	0
$757$	40.7846i	1.48234i	0.671316	+	0.741171i	$0.265729\pi$
$757$	40.7846i	1.48234i	−0.671316	+	0.741171i	$0.734271\pi$
$758$	0	0
$759$	−37.8564	−1.37410
$760$	0	0
$761$	18.7846	0.680942	0.340471	−	0.940255i	$-0.389414\pi$
$761$	18.7846	0.680942	0.340471	+	0.940255i	$0.389414\pi$
$762$	0	0
$763$	− 2.53590i	− 0.0918057i
$764$	0	0
$765$	− 18.9282i	− 0.684351i
$766$	0	0
$767$	−10.1962	−0.368162
$768$	0	0
$769$	31.4641	1.13462	0.567312	−	0.823503i	$-0.307983\pi$
$769$	31.4641	1.13462	0.567312	+	0.823503i	$0.307983\pi$
$770$	0	0
$771$	47.7128i	1.71833i
$772$	0	0
$773$	27.4641i	0.987815i	0.869514	+	0.493908i	$0.164432\pi$
$773$	27.4641i	0.987815i	−0.869514	+	0.493908i	$0.835568\pi$
$774$	0	0
$775$	−47.1244	−1.69276
$776$	0	0
$777$	−22.6410	−0.812242
$778$	0	0
$779$	− 6.53590i	− 0.234173i
$780$	0	0
$781$	38.7846i	1.38782i
$782$	0	0
$783$	8.00000	0.285897
$784$	0	0
$785$	58.6410	2.09299
$786$	0	0
$787$	29.1244i	1.03817i	0.854722	+	0.519086i	$0.173727\pi$
$787$	29.1244i	1.03817i	−0.854722	+	0.519086i	$0.826273\pi$
$788$	0	0
$789$	− 54.6410i	− 1.94527i
$790$	0	0
$791$	3.21539	0.114326
$792$	0	0
$793$	−2.92820	−0.103984
$794$	0	0
$795$	− 20.2872i	− 0.719512i
$796$	0	0
$797$	− 30.0000i	− 1.06265i	−0.847167	−	0.531327i	$-0.821693\pi$
$797$	− 30.0000i	− 1.06265i	0.847167	−	0.531327i	$-0.178307\pi$
$798$	0	0
$799$	−36.7846	−1.30135
$800$	0	0
$801$	17.3205	0.611990
$802$	0	0
$803$	− 35.3205i	− 1.24643i
$804$	0	0
$805$	− 17.5692i	− 0.619234i
$806$	0	0
$807$	−15.7128	−0.553117
$808$	0	0
$809$	−5.46410	−0.192108	−0.0960538	−	0.995376i	$-0.530622\pi$
$809$	−5.46410	−0.192108	−0.0960538	+	0.995376i	$0.530622\pi$
$810$	0	0
$811$	− 21.4115i	− 0.751861i	−0.926648	−	0.375930i	$-0.877323\pi$
$811$	− 21.4115i	− 0.751861i	0.926648	−	0.375930i	$-0.122677\pi$
$812$	0	0
$813$	− 40.3923i	− 1.41662i
$814$	0	0
$815$	57.9615	2.03030
$816$	0	0
$817$	0.392305	0.0137250
$818$	0	0
$819$	− 1.26795i	− 0.0443057i
$820$	0	0
$821$	− 48.2487i	− 1.68389i	−0.539562	−	0.841946i	$-0.681410\pi$
$821$	− 48.2487i	− 1.68389i	0.539562	−	0.841946i	$-0.318590\pi$
$822$	0	0
$823$	−20.0000	−0.697156	−0.348578	−	0.937280i	$-0.613335\pi$
$823$	−20.0000	−0.697156	−0.348578	+	0.937280i	$0.613335\pi$
$824$	0	0
$825$	−66.2487	−2.30648
$826$	0	0
$827$	− 8.73205i	− 0.303643i	−0.988408	−	0.151822i	$-0.951486\pi$
$827$	− 8.73205i	− 0.303643i	0.988408	−	0.151822i	$-0.0485139\pi$
$828$	0	0
$829$	− 28.7846i	− 0.999731i	−0.866103	−	0.499865i	$-0.833383\pi$
$829$	− 28.7846i	− 0.999731i	0.866103	−	0.499865i	$-0.166617\pi$
$830$	0	0
$831$	3.71281	0.128796
$832$	0	0
$833$	−29.4641	−1.02087
$834$	0	0
$835$	19.6077i	0.678552i
$836$	0	0
$837$	− 26.9282i	− 0.930775i
$838$	0	0
$839$	20.9808	0.724336	0.362168	−	0.932113i	$-0.382037\pi$
$839$	20.9808	0.724336	0.362168	+	0.932113i	$0.382037\pi$
$840$	0	0
$841$	25.0000	0.862069
$842$	0	0
$843$	− 18.6410i	− 0.642031i
$844$	0	0
$845$	− 3.46410i	− 0.119169i
$846$	0	0
$847$	−14.4449	−0.496331
$848$	0	0
$849$	−38.9282	−1.33601
$850$	0	0
$851$	35.7128i	1.22422i
$852$	0	0
$853$	− 55.1769i	− 1.88922i	−0.328193	−	0.944611i	$-0.606440\pi$
$853$	− 55.1769i	− 1.88922i	0.328193	−	0.944611i	$-0.393560\pi$
$854$	0	0
$855$	−2.53590	−0.0867259
$856$	0	0
$857$	−7.85641	−0.268370	−0.134185	−	0.990956i	$-0.542842\pi$
$857$	−7.85641	−0.268370	−0.134185	+	0.990956i	$0.542842\pi$
$858$	0	0
$859$	− 18.0000i	− 0.614152i	−0.951685	−	0.307076i	$-0.900649\pi$
$859$	− 18.0000i	− 0.614152i	0.951685	−	0.307076i	$-0.0993506\pi$
$860$	0	0
$861$	− 22.6410i	− 0.771604i
$862$	0	0
$863$	−1.26795	−0.0431615	−0.0215807	−	0.999767i	$-0.506870\pi$
$863$	−1.26795	−0.0431615	−0.0215807	+	0.999767i	$0.506870\pi$
$864$	0	0
$865$	−24.0000	−0.816024
$866$	0	0
$867$	− 25.7128i	− 0.873253i
$868$	0	0
$869$	− 25.8564i	− 0.877119i
$870$	0	0
$871$	0.732051	0.0248046
$872$	0	0
$873$	−6.39230	−0.216347
$874$	0	0
$875$	− 8.78461i	− 0.296974i
$876$	0	0
$877$	− 23.4641i	− 0.792326i	−0.918180	−	0.396163i	$-0.870341\pi$
$877$	− 23.4641i	− 0.792326i	0.918180	−	0.396163i	$-0.129659\pi$
$878$	0	0
$879$	65.8564	2.22128
$880$	0	0
$881$	−33.7128	−1.13581	−0.567907	−	0.823093i	$-0.692247\pi$
$881$	−33.7128	−1.13581	−0.567907	+	0.823093i	$0.692247\pi$
$882$	0	0
$883$	− 31.4641i	− 1.05885i	−0.848356	−	0.529426i	$-0.822407\pi$
$883$	− 31.4641i	− 1.05885i	0.848356	−	0.529426i	$-0.177593\pi$
$884$	0	0
$885$	70.6410i	2.37457i
$886$	0	0
$887$	−30.2487	−1.01565	−0.507826	−	0.861460i	$-0.669551\pi$
$887$	−30.2487	−1.01565	−0.507826	+	0.861460i	$0.669551\pi$
$888$	0	0
$889$	5.07180	0.170103
$890$	0	0
$891$	− 52.0526i	− 1.74383i
$892$	0	0
$893$	4.92820i	0.164916i
$894$	0	0
$895$	−36.0000	−1.20335
$896$	0	0
$897$	−8.00000	−0.267112
$898$	0	0
$899$	13.4641i	0.449053i
$900$	0	0
$901$	− 16.0000i	− 0.533037i
$902$	0	0
$903$	1.35898	0.0452242
$904$	0	0
$905$	30.9282	1.02809
$906$	0	0
$907$	− 22.1051i	− 0.733988i	−0.930223	−	0.366994i	$-0.880387\pi$
$907$	− 22.1051i	− 0.733988i	0.930223	−	0.366994i	$-0.119613\pi$
$908$	0	0
$909$	12.0000i	0.398015i
$910$	0	0
$911$	−3.32051	−0.110013	−0.0550067	−	0.998486i	$-0.517518\pi$
$911$	−3.32051	−0.110013	−0.0550067	+	0.998486i	$0.517518\pi$
$912$	0	0
$913$	15.4641	0.511787
$914$	0	0
$915$	20.2872i	0.670674i
$916$	0	0
$917$	− 25.1769i	− 0.831415i
$918$	0	0
$919$	35.3205	1.16512	0.582558	−	0.812789i	$-0.302052\pi$
$919$	35.3205	1.16512	0.582558	+	0.812789i	$0.302052\pi$
$920$	0	0
$921$	−1.46410	−0.0482438
$922$	0	0
$923$	8.19615i	0.269780i
$924$	0	0
$925$	62.4974i	2.05490i
$926$	0	0
$927$	6.92820	0.227552
$928$	0	0
$929$	20.5359	0.673761	0.336880	−	0.941547i	$-0.390628\pi$
$929$	20.5359	0.673761	0.336880	+	0.941547i	$0.390628\pi$
$930$	0	0
$931$	3.94744i	0.129372i
$932$	0	0
$933$	2.14359i	0.0701781i
$934$	0	0
$935$	−89.5692	−2.92923
$936$	0	0
$937$	−13.7128	−0.447978	−0.223989	−	0.974592i	$-0.571908\pi$
$937$	−13.7128	−0.447978	−0.223989	+	0.974592i	$0.571908\pi$
$938$	0	0
$939$	− 0.784610i	− 0.0256048i
$940$	0	0
$941$	32.2487i	1.05128i	0.850708	+	0.525639i	$0.176174\pi$
$941$	32.2487i	1.05128i	−0.850708	+	0.525639i	$0.823826\pi$
$942$	0	0
$943$	−35.7128	−1.16297
$944$	0	0
$945$	17.5692	0.571527
$946$	0	0
$947$	48.4449i	1.57425i	0.616796	+	0.787123i	$0.288430\pi$
$947$	48.4449i	1.57425i	−0.616796	+	0.787123i	$0.711570\pi$
$948$	0	0
$949$	− 7.46410i	− 0.242295i
$950$	0	0
$951$	6.92820	0.224662
$952$	0	0
$953$	−31.8564	−1.03193	−0.515965	−	0.856610i	$-0.672567\pi$
$953$	−31.8564	−1.03193	−0.515965	+	0.856610i	$0.672567\pi$
$954$	0	0
$955$	50.3538i	1.62941i
$956$	0	0
$957$	18.9282i	0.611862i
$958$	0	0
$959$	16.3923	0.529335
$960$	0	0
$961$	14.3205	0.461952
$962$	0	0
$963$	4.92820i	0.158809i
$964$	0	0
$965$	− 5.56922i	− 0.179280i
$966$	0	0
$967$	8.98076	0.288802	0.144401	−	0.989519i	$-0.453875\pi$
$967$	8.98076	0.288802	0.144401	+	0.989519i	$0.453875\pi$
$968$	0	0
$969$	−8.00000	−0.256997
$970$	0	0
$971$	16.1436i	0.518073i	0.965868	+	0.259036i	$0.0834049\pi$
$971$	16.1436i	0.518073i	−0.965868	+	0.259036i	$0.916595\pi$
$972$	0	0
$973$	12.6795i	0.406486i
$974$	0	0
$975$	−14.0000	−0.448359
$976$	0	0
$977$	−58.3923	−1.86814	−0.934068	−	0.357096i	$-0.883767\pi$
$977$	−58.3923	−1.86814	−0.934068	+	0.357096i	$0.883767\pi$
$978$	0	0
$979$	− 81.9615i	− 2.61950i
$980$	0	0
$981$	2.00000i	0.0638551i
$982$	0	0
$983$	−16.1962	−0.516577	−0.258289	−	0.966068i	$-0.583159\pi$
$983$	−16.1962	−0.516577	−0.258289	+	0.966068i	$0.583159\pi$
$984$	0	0
$985$	10.6410	0.339051
$986$	0	0
$987$	17.0718i	0.543401i
$988$	0	0
$989$	− 2.14359i	− 0.0681623i
$990$	0	0
$991$	8.67949	0.275713	0.137857	−	0.990452i	$-0.455979\pi$
$991$	8.67949	0.275713	0.137857	+	0.990452i	$0.455979\pi$
$992$	0	0
$993$	−35.0333	−1.11175
$994$	0	0
$995$	− 58.1436i	− 1.84328i
$996$	0	0
$997$	33.5692i	1.06315i	0.847012	+	0.531574i	$0.178399\pi$
$997$	33.5692i	1.06315i	−0.847012	+	0.531574i	$0.821601\pi$
$998$	0	0
$999$	−35.7128	−1.12990

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	416.2.b.b.209.2		4
3.2	odd	2		3744.2.g.b.1873.1		4
4.3	odd	2		104.2.b.b.53.1	✓	4
8.3	odd	2		104.2.b.b.53.2	yes	4
8.5	even	2	inner	416.2.b.b.209.3		4
12.11	even	2		936.2.g.b.469.4		4
16.3	odd	4		3328.2.a.n.1.2		2
16.5	even	4		3328.2.a.m.1.1		2
16.11	odd	4		3328.2.a.bd.1.1		2
16.13	even	4		3328.2.a.bc.1.2		2
24.5	odd	2		3744.2.g.b.1873.3		4
24.11	even	2		936.2.g.b.469.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.2.b.b.53.1	✓	4	4.3	odd	2
104.2.b.b.53.2	yes	4	8.3	odd	2
416.2.b.b.209.2		4	1.1	even	1	trivial
416.2.b.b.209.3		4	8.5	even	2	inner
936.2.g.b.469.3		4	24.11	even	2
936.2.g.b.469.4		4	12.11	even	2
3328.2.a.m.1.1		2	16.5	even	4
3328.2.a.n.1.2		2	16.3	odd	4
3328.2.a.bc.1.2		2	16.13	even	4
3328.2.a.bd.1.1		2	16.11	odd	4
3744.2.g.b.1873.1		4	3.2	odd	2
3744.2.g.b.1873.3		4	24.5	odd	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:	\( N \)	\(=\)	\( 416 = 2^{5} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	416.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{3} +3.46410i q^{5} +1.26795 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-2.00000i q^{3} +3.46410i q^{5} +1.26795 q^{7} -1.00000 q^{9} +4.73205i q^{11} +1.00000i q^{13} +6.92820 q^{15} +5.46410 q^{17} -0.732051i q^{19} -2.53590i q^{21} -4.00000 q^{23} -7.00000 q^{25} -4.00000i q^{27} +2.00000i q^{29} +6.73205 q^{31} +9.46410 q^{33} +4.39230i q^{35} -8.92820i q^{37} +2.00000 q^{39} +8.92820 q^{41} +0.535898i q^{43} -3.46410i q^{45} -6.73205 q^{47} -5.39230 q^{49} -10.9282i q^{51} -2.92820i q^{53} -16.3923 q^{55} -1.46410 q^{57} +10.1962i q^{59} +2.92820i q^{61} -1.26795 q^{63} -3.46410 q^{65} -0.732051i q^{67} +8.00000i q^{69} +8.19615 q^{71} -7.46410 q^{73} +14.0000i q^{75} +6.00000i q^{77} -5.46410 q^{79} -11.0000 q^{81} -3.26795i q^{83} +18.9282i q^{85} +4.00000 q^{87} -17.3205 q^{89} +1.26795i q^{91} -13.4641i q^{93} +2.53590 q^{95} +6.39230 q^{97} -4.73205i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{7} - 4 q^{9}+O(q^{10}) \) 4 * q + 12 * q^7 - 4 * q^9	\( 4 q + 12 q^{7} - 4 q^{9} + 8 q^{17} - 16 q^{23} - 28 q^{25} + 20 q^{31} + 24 q^{33} + 8 q^{39} + 8 q^{41} - 20 q^{47} + 20 q^{49} - 24 q^{55} + 8 q^{57} - 12 q^{63} + 12 q^{71} - 16 q^{73} - 8 q^{79} - 44 q^{81} + 16 q^{87} + 24 q^{95} - 16 q^{97}+O(q^{100}) \) 4 * q + 12 * q^7 - 4 * q^9 + 8 * q^17 - 16 * q^23 - 28 * q^25 + 20 * q^31 + 24 * q^33 + 8 * q^39 + 8 * q^41 - 20 * q^47 + 20 * q^49 - 24 * q^55 + 8 * q^57 - 12 * q^63 + 12 * q^71 - 16 * q^73 - 8 * q^79 - 44 * q^81 + 16 * q^87 + 24 * q^95 - 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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