LMFDB - Embedded newform 784.2.f.d.783.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,2,Mod(783,784)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("784.783");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	784.f (of order $2$, degree $1$, 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:	$6.26027151847$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 7x^{2} + 49 $ x^4 + 7*x^2 + 49
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			783.1
Root			$1.32288 + 2.29129i$ of defining polynomial
Character	$\chi$	$=$	784.783
Dual form			784.2.f.d.783.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.64575 q^{3} -1.73205i q^{5} +4.00000 q^{9} +O(q^{10})$	$q-2.64575 q^{3} -1.73205i q^{5} +4.00000 q^{9} +4.58258i q^{11} +3.46410i q^{13} +4.58258i q^{15} -5.19615i q^{17} -2.64575 q^{19} -4.58258i q^{23} +2.00000 q^{25} -2.64575 q^{27} +2.64575 q^{31} -12.1244i q^{33} +7.00000 q^{37} -9.16515i q^{39} -3.46410i q^{41} -9.16515i q^{43} -6.92820i q^{45} +7.93725 q^{47} +13.7477i q^{51} +3.00000 q^{53} +7.93725 q^{55} +7.00000 q^{57} +7.93725 q^{59} +1.73205i q^{61} +6.00000 q^{65} -4.58258i q^{67} +12.1244i q^{69} -9.16515i q^{71} +5.19615i q^{73} -5.29150 q^{75} -4.58258i q^{79} -5.00000 q^{81} -9.00000 q^{85} +1.73205i q^{89} -7.00000 q^{93} +4.58258i q^{95} -3.46410i q^{97} +18.3303i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 16 q^{9}+O(q^{10}) $ 4 * q + 16 * q^9	$ 4 q + 16 q^{9} + 8 q^{25} + 28 q^{37} + 12 q^{53} + 28 q^{57} + 24 q^{65} - 20 q^{81} - 36 q^{85} - 28 q^{93}+O(q^{100}) $ 4 * q + 16 * q^9 + 8 * q^25 + 28 * q^37 + 12 * q^53 + 28 * q^57 + 24 * q^65 - 20 * q^81 - 36 * q^85 - 28 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/784\mathbb{Z}\right)^\times$.

$n$	$197$	$687$	$689$
$\chi(n)$	$1$	$-1$	$-1$

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
$3$	−2.64575	−1.52753	−0.763763	−	0.645497i	$-0.776650\pi$
$3$	−2.64575	−1.52753	−0.763763	+	0.645497i	$0.776650\pi$
$4$	0	0
$5$	− 1.73205i	− 0.774597i	−0.921954	−	0.387298i	$-0.873408\pi$
$5$	− 1.73205i	− 0.774597i	0.921954	−	0.387298i	$-0.126592\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	4.00000	1.33333
$10$	0	0
$11$	4.58258i	1.38170i	0.722999	+	0.690849i	$0.242763\pi$
$11$	4.58258i	1.38170i	−0.722999	+	0.690849i	$0.757237\pi$
$12$	0	0
$13$	3.46410i	0.960769i	0.877058	+	0.480384i	$0.159503\pi$
$13$	3.46410i	0.960769i	−0.877058	+	0.480384i	$0.840497\pi$
$14$	0	0
$15$	4.58258i	1.18322i
$16$	0	0
$17$	− 5.19615i	− 1.26025i	−0.776493	−	0.630126i	$-0.783003\pi$
$17$	− 5.19615i	− 1.26025i	0.776493	−	0.630126i	$-0.216997\pi$
$18$	0	0
$19$	−2.64575	−0.606977	−0.303488	−	0.952835i	$-0.598151\pi$
$19$	−2.64575	−0.606977	−0.303488	+	0.952835i	$0.598151\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 4.58258i	− 0.955533i	−0.878487	−	0.477767i	$-0.841446\pi$
$23$	− 4.58258i	− 0.955533i	0.878487	−	0.477767i	$-0.158554\pi$
$24$	0	0
$25$	2.00000	0.400000
$26$	0	0
$27$	−2.64575	−0.509175
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	2.64575	0.475191	0.237595	−	0.971364i	$-0.423641\pi$
$31$	2.64575	0.475191	0.237595	+	0.971364i	$0.423641\pi$
$32$	0	0
$33$	− 12.1244i	− 2.11058i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.00000	1.15079	0.575396	−	0.817875i	$-0.304848\pi$
$37$	7.00000	1.15079	0.575396	+	0.817875i	$0.304848\pi$
$38$	0	0
$39$	− 9.16515i	− 1.46760i
$40$	0	0
$41$	− 3.46410i	− 0.541002i	−0.962720	−	0.270501i	$-0.912811\pi$
$41$	− 3.46410i	− 0.541002i	0.962720	−	0.270501i	$-0.0871893\pi$
$42$	0	0
$43$	− 9.16515i	− 1.39767i	−0.715282	−	0.698836i	$-0.753702\pi$
$43$	− 9.16515i	− 1.39767i	0.715282	−	0.698836i	$-0.246298\pi$
$44$	0	0
$45$	− 6.92820i	− 1.03280i
$46$	0	0
$47$	7.93725	1.15777	0.578884	−	0.815410i	$-0.303489\pi$
$47$	7.93725	1.15777	0.578884	+	0.815410i	$0.303489\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	13.7477i	1.92507i
$52$	0	0
$53$	3.00000	0.412082	0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000	0.412082	0.206041	+	0.978543i	$0.433942\pi$
$54$	0	0
$55$	7.93725	1.07026
$56$	0	0
$57$	7.00000	0.927173
$58$	0	0
$59$	7.93725	1.03334	0.516671	−	0.856184i	$-0.327171\pi$
$59$	7.93725	1.03334	0.516671	+	0.856184i	$0.327171\pi$
$60$	0	0
$61$	1.73205i	0.221766i	0.993833	+	0.110883i	$0.0353679\pi$
$61$	1.73205i	0.221766i	−0.993833	+	0.110883i	$0.964632\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	6.00000	0.744208
$66$	0	0
$67$	− 4.58258i	− 0.559851i	−0.960022	−	0.279925i	$-0.909690\pi$
$67$	− 4.58258i	− 0.559851i	0.960022	−	0.279925i	$-0.0903097\pi$
$68$	0	0
$69$	12.1244i	1.45960i
$70$	0	0
$71$	− 9.16515i	− 1.08770i	−0.839181	−	0.543852i	$-0.816965\pi$
$71$	− 9.16515i	− 1.08770i	0.839181	−	0.543852i	$-0.183035\pi$
$72$	0	0
$73$	5.19615i	0.608164i	0.952646	+	0.304082i	$0.0983496\pi$
$73$	5.19615i	0.608164i	−0.952646	+	0.304082i	$0.901650\pi$
$74$	0	0
$75$	−5.29150	−0.611010
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 4.58258i	− 0.515580i	−0.966201	−	0.257790i	$-0.917006\pi$
$79$	− 4.58258i	− 0.515580i	0.966201	−	0.257790i	$-0.0829943\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	−9.00000	−0.976187
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.73205i	0.183597i	0.995778	+	0.0917985i	$0.0292616\pi$
$89$	1.73205i	0.183597i	−0.995778	+	0.0917985i	$0.970738\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−7.00000	−0.725866
$94$	0	0
$95$	4.58258i	0.470162i
$96$	0	0
$97$	− 3.46410i	− 0.351726i	−0.984415	−	0.175863i	$-0.943728\pi$
$97$	− 3.46410i	− 0.351726i	0.984415	−	0.175863i	$-0.0562716\pi$
$98$	0	0
$99$	18.3303i	1.84226i
$100$	0	0
$101$	− 5.19615i	− 0.517036i	−0.966006	−	0.258518i	$-0.916766\pi$
$101$	− 5.19615i	− 0.517036i	0.966006	−	0.258518i	$-0.0832342\pi$
$102$	0	0
$103$	2.64575	0.260694	0.130347	−	0.991468i	$-0.458391\pi$
$103$	2.64575	0.260694	0.130347	+	0.991468i	$0.458391\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 13.7477i	− 1.32904i	−0.747269	−	0.664521i	$-0.768635\pi$
$107$	− 13.7477i	− 1.32904i	0.747269	−	0.664521i	$-0.231365\pi$
$108$	0	0
$109$	−7.00000	−0.670478	−0.335239	−	0.942133i	$-0.608817\pi$
$109$	−7.00000	−0.670478	−0.335239	+	0.942133i	$0.608817\pi$
$110$	0	0
$111$	−18.5203	−1.75787
$112$	0	0
$113$	12.0000	1.12887	0.564433	−	0.825479i	$-0.309095\pi$
$113$	12.0000	1.12887	0.564433	+	0.825479i	$0.309095\pi$
$114$	0	0
$115$	−7.93725	−0.740153
$116$	0	0
$117$	13.8564i	1.28103i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	9.16515i	0.826394i
$124$	0	0
$125$	− 12.1244i	− 1.08444i
$126$	0	0
$127$	9.16515i	0.813276i	0.913589	+	0.406638i	$0.133299\pi$
$127$	9.16515i	0.813276i	−0.913589	+	0.406638i	$0.866701\pi$
$128$	0	0
$129$	24.2487i	2.13498i
$130$	0	0
$131$	−7.93725	−0.693481	−0.346741	−	0.937961i	$-0.612712\pi$
$131$	−7.93725	−0.693481	−0.346741	+	0.937961i	$0.612712\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	4.58258i	0.394405i
$136$	0	0
$137$	21.0000	1.79415	0.897076	−	0.441877i	$-0.145687\pi$
$137$	21.0000	1.79415	0.897076	+	0.441877i	$0.145687\pi$
$138$	0	0
$139$	10.5830	0.897639	0.448819	−	0.893622i	$-0.351845\pi$
$139$	10.5830	0.897639	0.448819	+	0.893622i	$0.351845\pi$
$140$	0	0
$141$	−21.0000	−1.76852
$142$	0	0
$143$	−15.8745	−1.32749
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−15.0000	−1.22885	−0.614424	−	0.788976i	$-0.710612\pi$
$149$	−15.0000	−1.22885	−0.614424	+	0.788976i	$0.710612\pi$
$150$	0	0
$151$	22.9129i	1.86462i	0.361656	+	0.932312i	$0.382212\pi$
$151$	22.9129i	1.86462i	−0.361656	+	0.932312i	$0.617788\pi$
$152$	0	0
$153$	− 20.7846i	− 1.68034i
$154$	0	0
$155$	− 4.58258i	− 0.368081i
$156$	0	0
$157$	5.19615i	0.414698i	0.978267	+	0.207349i	$0.0664836\pi$
$157$	5.19615i	0.414698i	−0.978267	+	0.207349i	$0.933516\pi$
$158$	0	0
$159$	−7.93725	−0.629465
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 4.58258i	− 0.358935i	−0.983764	−	0.179468i	$-0.942563\pi$
$163$	− 4.58258i	− 0.358935i	0.983764	−	0.179468i	$-0.0574375\pi$
$164$	0	0
$165$	−21.0000	−1.63485
$166$	0	0
$167$	15.8745	1.22841	0.614203	−	0.789148i	$-0.289478\pi$
$167$	15.8745	1.22841	0.614203	+	0.789148i	$0.289478\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−10.5830	−0.809303
$172$	0	0
$173$	22.5167i	1.71191i	0.517050	+	0.855955i	$0.327030\pi$
$173$	22.5167i	1.71191i	−0.517050	+	0.855955i	$0.672970\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−21.0000	−1.57846
$178$	0	0
$179$	− 13.7477i	− 1.02755i	−0.857924	−	0.513777i	$-0.828246\pi$
$179$	− 13.7477i	− 1.02755i	0.857924	−	0.513777i	$-0.171754\pi$
$180$	0	0
$181$	20.7846i	1.54491i	0.635071	+	0.772454i	$0.280971\pi$
$181$	20.7846i	1.54491i	−0.635071	+	0.772454i	$0.719029\pi$
$182$	0	0
$183$	− 4.58258i	− 0.338754i
$184$	0	0
$185$	− 12.1244i	− 0.891400i
$186$	0	0
$187$	23.8118	1.74129
$188$	0	0
$189$	0	0
$190$	0	0
$191$	22.9129i	1.65792i	0.559310	+	0.828959i	$0.311066\pi$
$191$	22.9129i	1.65792i	−0.559310	+	0.828959i	$0.688934\pi$
$192$	0	0
$193$	−11.0000	−0.791797	−0.395899	−	0.918294i	$-0.629567\pi$
$193$	−11.0000	−0.791797	−0.395899	+	0.918294i	$0.629567\pi$
$194$	0	0
$195$	−15.8745	−1.13680
$196$	0	0
$197$	−12.0000	−0.854965	−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000	−0.854965	−0.427482	+	0.904024i	$0.640599\pi$
$198$	0	0
$199$	13.2288	0.937762	0.468881	−	0.883261i	$-0.344657\pi$
$199$	13.2288	0.937762	0.468881	+	0.883261i	$0.344657\pi$
$200$	0	0
$201$	12.1244i	0.855186i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−6.00000	−0.419058
$206$	0	0
$207$	− 18.3303i	− 1.27404i
$208$	0	0
$209$	− 12.1244i	− 0.838659i
$210$	0	0
$211$	− 9.16515i	− 0.630955i	−0.948933	−	0.315478i	$-0.897835\pi$
$211$	− 9.16515i	− 0.630955i	0.948933	−	0.315478i	$-0.102165\pi$
$212$	0	0
$213$	24.2487i	1.66149i
$214$	0	0
$215$	−15.8745	−1.08263
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 13.7477i	− 0.928985i
$220$	0	0
$221$	18.0000	1.21081
$222$	0	0
$223$	−10.5830	−0.708690	−0.354345	−	0.935115i	$-0.615296\pi$
$223$	−10.5830	−0.708690	−0.354345	+	0.935115i	$0.615296\pi$
$224$	0	0
$225$	8.00000	0.533333
$226$	0	0
$227$	23.8118	1.58044	0.790221	−	0.612822i	$-0.209966\pi$
$227$	23.8118	1.58044	0.790221	+	0.612822i	$0.209966\pi$
$228$	0	0
$229$	− 1.73205i	− 0.114457i	−0.998361	−	0.0572286i	$-0.981774\pi$
$229$	− 1.73205i	− 0.114457i	0.998361	−	0.0572286i	$-0.0182264\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	21.0000	1.37576	0.687878	−	0.725826i	$-0.258542\pi$
$233$	21.0000	1.37576	0.687878	+	0.725826i	$0.258542\pi$
$234$	0	0
$235$	− 13.7477i	− 0.896803i
$236$	0	0
$237$	12.1244i	0.787562i
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	19.0526i	1.22728i	0.789585	+	0.613642i	$0.210296\pi$
$241$	19.0526i	1.22728i	−0.789585	+	0.613642i	$0.789704\pi$
$242$	0	0
$243$	21.1660	1.35780
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 9.16515i	− 0.583165i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	15.8745	1.00199	0.500995	−	0.865450i	$-0.332967\pi$
$251$	15.8745	1.00199	0.500995	+	0.865450i	$0.332967\pi$
$252$	0	0
$253$	21.0000	1.32026
$254$	0	0
$255$	23.8118	1.49115
$256$	0	0
$257$	− 1.73205i	− 0.108042i	−0.998540	−	0.0540212i	$-0.982796\pi$
$257$	− 1.73205i	− 0.108042i	0.998540	−	0.0540212i	$-0.0172039\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 4.58258i	− 0.282574i	−0.989969	−	0.141287i	$-0.954876\pi$
$263$	− 4.58258i	− 0.282574i	0.989969	−	0.141287i	$-0.0451240\pi$
$264$	0	0
$265$	− 5.19615i	− 0.319197i
$266$	0	0
$267$	− 4.58258i	− 0.280449i
$268$	0	0
$269$	− 29.4449i	− 1.79529i	−0.440724	−	0.897643i	$-0.645278\pi$
$269$	− 29.4449i	− 1.79529i	0.440724	−	0.897643i	$-0.354722\pi$
$270$	0	0
$271$	−29.1033	−1.76790	−0.883949	−	0.467584i	$-0.845125\pi$
$271$	−29.1033	−1.76790	−0.883949	+	0.467584i	$0.845125\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	9.16515i	0.552679i
$276$	0	0
$277$	−17.0000	−1.02143	−0.510716	−	0.859750i	$-0.670619\pi$
$277$	−17.0000	−1.02143	−0.510716	+	0.859750i	$0.670619\pi$
$278$	0	0
$279$	10.5830	0.633588
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	13.2288	0.786368	0.393184	−	0.919460i	$-0.371374\pi$
$283$	13.2288	0.786368	0.393184	+	0.919460i	$0.371374\pi$
$284$	0	0
$285$	− 12.1244i	− 0.718185i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−10.0000	−0.588235
$290$	0	0
$291$	9.16515i	0.537271i
$292$	0	0
$293$	− 20.7846i	− 1.21425i	−0.794606	−	0.607125i	$-0.792323\pi$
$293$	− 20.7846i	− 1.21425i	0.794606	−	0.607125i	$-0.207677\pi$
$294$	0	0
$295$	− 13.7477i	− 0.800424i
$296$	0	0
$297$	− 12.1244i	− 0.703526i
$298$	0	0
$299$	15.8745	0.918046
$300$	0	0
$301$	0	0
$302$	0	0
$303$	13.7477i	0.789786i
$304$	0	0
$305$	3.00000	0.171780
$306$	0	0
$307$	−10.5830	−0.604004	−0.302002	−	0.953307i	$-0.597655\pi$
$307$	−10.5830	−0.604004	−0.302002	+	0.953307i	$0.597655\pi$
$308$	0	0
$309$	−7.00000	−0.398216
$310$	0	0
$311$	−23.8118	−1.35024	−0.675121	−	0.737707i	$-0.735908\pi$
$311$	−23.8118	−1.35024	−0.675121	+	0.737707i	$0.735908\pi$
$312$	0	0
$313$	− 25.9808i	− 1.46852i	−0.678869	−	0.734260i	$-0.737529\pi$
$313$	− 25.9808i	− 1.46852i	0.678869	−	0.734260i	$-0.262471\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	15.0000	0.842484	0.421242	−	0.906948i	$-0.361594\pi$
$317$	15.0000	0.842484	0.421242	+	0.906948i	$0.361594\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	36.3731i	2.03015i
$322$	0	0
$323$	13.7477i	0.764944i
$324$	0	0
$325$	6.92820i	0.384308i
$326$	0	0
$327$	18.5203	1.02417
$328$	0	0
$329$	0	0
$330$	0	0
$331$	13.7477i	0.755643i	0.925878	+	0.377822i	$0.123327\pi$
$331$	13.7477i	0.755643i	−0.925878	+	0.377822i	$0.876673\pi$
$332$	0	0
$333$	28.0000	1.53439
$334$	0	0
$335$	−7.93725	−0.433659
$336$	0	0
$337$	−28.0000	−1.52526	−0.762629	−	0.646837i	$-0.776092\pi$
$337$	−28.0000	−1.52526	−0.762629	+	0.646837i	$0.776092\pi$
$338$	0	0
$339$	−31.7490	−1.72437
$340$	0	0
$341$	12.1244i	0.656571i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	21.0000	1.13060
$346$	0	0
$347$	− 4.58258i	− 0.246006i	−0.992406	−	0.123003i	$-0.960748\pi$
$347$	− 4.58258i	− 0.246006i	0.992406	−	0.123003i	$-0.0392524\pi$
$348$	0	0
$349$	− 3.46410i	− 0.185429i	−0.995693	−	0.0927146i	$-0.970446\pi$
$349$	− 3.46410i	− 0.185429i	0.995693	−	0.0927146i	$-0.0295544\pi$
$350$	0	0
$351$	− 9.16515i	− 0.489200i
$352$	0	0
$353$	− 19.0526i	− 1.01407i	−0.861927	−	0.507033i	$-0.830742\pi$
$353$	− 19.0526i	− 1.01407i	0.861927	−	0.507033i	$-0.169258\pi$
$354$	0	0
$355$	−15.8745	−0.842531
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 13.7477i	− 0.725577i	−0.931871	−	0.362789i	$-0.881825\pi$
$359$	− 13.7477i	− 0.725577i	0.931871	−	0.362789i	$-0.118175\pi$
$360$	0	0
$361$	−12.0000	−0.631579
$362$	0	0
$363$	26.4575	1.38866
$364$	0	0
$365$	9.00000	0.471082
$366$	0	0
$367$	−34.3948	−1.79539	−0.897696	−	0.440615i	$-0.854760\pi$
$367$	−34.3948	−1.79539	−0.897696	+	0.440615i	$0.854760\pi$
$368$	0	0
$369$	− 13.8564i	− 0.721336i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−1.00000	−0.0517780	−0.0258890	−	0.999665i	$-0.508242\pi$
$373$	−1.00000	−0.0517780	−0.0258890	+	0.999665i	$0.508242\pi$
$374$	0	0
$375$	32.0780i	1.65650i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	18.3303i	0.941564i	0.882249	+	0.470782i	$0.156028\pi$
$379$	18.3303i	0.941564i	−0.882249	+	0.470782i	$0.843972\pi$
$380$	0	0
$381$	− 24.2487i	− 1.24230i
$382$	0	0
$383$	7.93725	0.405575	0.202787	−	0.979223i	$-0.435000\pi$
$383$	7.93725	0.405575	0.202787	+	0.979223i	$0.435000\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 36.6606i	− 1.86356i
$388$	0	0
$389$	21.0000	1.06474	0.532371	−	0.846511i	$-0.321301\pi$
$389$	21.0000	1.06474	0.532371	+	0.846511i	$0.321301\pi$
$390$	0	0
$391$	−23.8118	−1.20421
$392$	0	0
$393$	21.0000	1.05931
$394$	0	0
$395$	−7.93725	−0.399367
$396$	0	0
$397$	− 22.5167i	− 1.13008i	−0.825064	−	0.565039i	$-0.808861\pi$
$397$	− 22.5167i	− 1.13008i	0.825064	−	0.565039i	$-0.191139\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	21.0000	1.04869	0.524345	−	0.851506i	$-0.324310\pi$
$401$	21.0000	1.04869	0.524345	+	0.851506i	$0.324310\pi$
$402$	0	0
$403$	9.16515i	0.456549i
$404$	0	0
$405$	8.66025i	0.430331i
$406$	0	0
$407$	32.0780i	1.59005i
$408$	0	0
$409$	− 19.0526i	− 0.942088i	−0.882110	−	0.471044i	$-0.843877\pi$
$409$	− 19.0526i	− 0.942088i	0.882110	−	0.471044i	$-0.156123\pi$
$410$	0	0
$411$	−55.5608	−2.74061
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−28.0000	−1.37117
$418$	0	0
$419$	−31.7490	−1.55104	−0.775520	−	0.631322i	$-0.782512\pi$
$419$	−31.7490	−1.55104	−0.775520	+	0.631322i	$0.782512\pi$
$420$	0	0
$421$	16.0000	0.779792	0.389896	−	0.920859i	$-0.372511\pi$
$421$	16.0000	0.779792	0.389896	+	0.920859i	$0.372511\pi$
$422$	0	0
$423$	31.7490	1.54369
$424$	0	0
$425$	− 10.3923i	− 0.504101i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	42.0000	2.02778
$430$	0	0
$431$	22.9129i	1.10367i	0.833952	+	0.551837i	$0.186073\pi$
$431$	22.9129i	1.10367i	−0.833952	+	0.551837i	$0.813927\pi$
$432$	0	0
$433$	3.46410i	0.166474i	0.996530	+	0.0832370i	$0.0265259\pi$
$433$	3.46410i	0.166474i	−0.996530	+	0.0832370i	$0.973474\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	12.1244i	0.579987i
$438$	0	0
$439$	29.1033	1.38902	0.694512	−	0.719482i	$-0.255621\pi$
$439$	29.1033	1.38902	0.694512	+	0.719482i	$0.255621\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	4.58258i	0.217725i	0.994057	+	0.108862i	$0.0347208\pi$
$443$	4.58258i	0.217725i	−0.994057	+	0.108862i	$0.965279\pi$
$444$	0	0
$445$	3.00000	0.142214
$446$	0	0
$447$	39.6863	1.87710
$448$	0	0
$449$	12.0000	0.566315	0.283158	−	0.959073i	$-0.408618\pi$
$449$	12.0000	0.566315	0.283158	+	0.959073i	$0.408618\pi$
$450$	0	0
$451$	15.8745	0.747501
$452$	0	0
$453$	− 60.6218i	− 2.84826i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−1.00000	−0.0467780	−0.0233890	−	0.999726i	$-0.507446\pi$
$457$	−1.00000	−0.0467780	−0.0233890	+	0.999726i	$0.507446\pi$
$458$	0	0
$459$	13.7477i	0.641689i
$460$	0	0
$461$	− 3.46410i	− 0.161339i	−0.996741	−	0.0806696i	$-0.974294\pi$
$461$	− 3.46410i	− 0.161339i	0.996741	−	0.0806696i	$-0.0257059\pi$
$462$	0	0
$463$	27.4955i	1.27782i	0.769281	+	0.638911i	$0.220615\pi$
$463$	27.4955i	1.27782i	−0.769281	+	0.638911i	$0.779385\pi$
$464$	0	0
$465$	12.1244i	0.562254i
$466$	0	0
$467$	−7.93725	−0.367292	−0.183646	−	0.982992i	$-0.558790\pi$
$467$	−7.93725	−0.367292	−0.183646	+	0.982992i	$0.558790\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	− 13.7477i	− 0.633462i
$472$	0	0
$473$	42.0000	1.93116
$474$	0	0
$475$	−5.29150	−0.242791
$476$	0	0
$477$	12.0000	0.549442
$478$	0	0
$479$	7.93725	0.362662	0.181331	−	0.983422i	$-0.441959\pi$
$479$	7.93725	0.362662	0.181331	+	0.983422i	$0.441959\pi$
$480$	0	0
$481$	24.2487i	1.10565i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−6.00000	−0.272446
$486$	0	0
$487$	− 41.2432i	− 1.86891i	−0.356085	−	0.934453i	$-0.615889\pi$
$487$	− 41.2432i	− 1.86891i	0.356085	−	0.934453i	$-0.384111\pi$
$488$	0	0
$489$	12.1244i	0.548282i
$490$	0	0
$491$	− 18.3303i	− 0.827235i	−0.910451	−	0.413617i	$-0.864265\pi$
$491$	− 18.3303i	− 0.827235i	0.910451	−	0.413617i	$-0.135735\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	31.7490	1.42701
$496$	0	0
$497$	0	0
$498$	0	0
$499$	32.0780i	1.43601i	0.696038	+	0.718005i	$0.254944\pi$
$499$	32.0780i	1.43601i	−0.696038	+	0.718005i	$0.745056\pi$
$500$	0	0
$501$	−42.0000	−1.87642
$502$	0	0
$503$	−31.7490	−1.41562	−0.707809	−	0.706404i	$-0.750316\pi$
$503$	−31.7490	−1.41562	−0.707809	+	0.706404i	$0.750316\pi$
$504$	0	0
$505$	−9.00000	−0.400495
$506$	0	0
$507$	−2.64575	−0.117502
$508$	0	0
$509$	25.9808i	1.15158i	0.817599	+	0.575789i	$0.195305\pi$
$509$	25.9808i	1.15158i	−0.817599	+	0.575789i	$0.804695\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	7.00000	0.309058
$514$	0	0
$515$	− 4.58258i	− 0.201932i
$516$	0	0
$517$	36.3731i	1.59969i
$518$	0	0
$519$	− 59.5735i	− 2.61499i
$520$	0	0
$521$	− 43.3013i	− 1.89706i	−0.316683	−	0.948532i	$-0.602569\pi$
$521$	− 43.3013i	− 1.89706i	0.316683	−	0.948532i	$-0.397431\pi$
$522$	0	0
$523$	−13.2288	−0.578453	−0.289227	−	0.957261i	$-0.593398\pi$
$523$	−13.2288	−0.578453	−0.289227	+	0.957261i	$0.593398\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 13.7477i	− 0.598860i
$528$	0	0
$529$	2.00000	0.0869565
$530$	0	0
$531$	31.7490	1.37779
$532$	0	0
$533$	12.0000	0.519778
$534$	0	0
$535$	−23.8118	−1.02947
$536$	0	0
$537$	36.3731i	1.56961i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	13.0000	0.558914	0.279457	−	0.960158i	$-0.409846\pi$
$541$	13.0000	0.558914	0.279457	+	0.960158i	$0.409846\pi$
$542$	0	0
$543$	− 54.9909i	− 2.35989i
$544$	0	0
$545$	12.1244i	0.519350i
$546$	0	0
$547$	− 18.3303i	− 0.783747i	−0.920019	−	0.391874i	$-0.871827\pi$
$547$	− 18.3303i	− 0.783747i	0.920019	−	0.391874i	$-0.128173\pi$
$548$	0	0
$549$	6.92820i	0.295689i
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	32.0780i	1.36164i
$556$	0	0
$557$	−3.00000	−0.127114	−0.0635570	−	0.997978i	$-0.520244\pi$
$557$	−3.00000	−0.127114	−0.0635570	+	0.997978i	$0.520244\pi$
$558$	0	0
$559$	31.7490	1.34284
$560$	0	0
$561$	−63.0000	−2.65986
$562$	0	0
$563$	7.93725	0.334515	0.167258	−	0.985913i	$-0.446509\pi$
$563$	7.93725	0.334515	0.167258	+	0.985913i	$0.446509\pi$
$564$	0	0
$565$	− 20.7846i	− 0.874415i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−21.0000	−0.880366	−0.440183	−	0.897908i	$-0.645086\pi$
$569$	−21.0000	−0.880366	−0.440183	+	0.897908i	$0.645086\pi$
$570$	0	0
$571$	− 13.7477i	− 0.575324i	−0.957732	−	0.287662i	$-0.907122\pi$
$571$	− 13.7477i	− 0.575324i	0.957732	−	0.287662i	$-0.0928781\pi$
$572$	0	0
$573$	− 60.6218i	− 2.53251i
$574$	0	0
$575$	− 9.16515i	− 0.382213i
$576$	0	0
$577$	19.0526i	0.793168i	0.917998	+	0.396584i	$0.129805\pi$
$577$	19.0526i	0.793168i	−0.917998	+	0.396584i	$0.870195\pi$
$578$	0	0
$579$	29.1033	1.20949
$580$	0	0
$581$	0	0
$582$	0	0
$583$	13.7477i	0.569373i
$584$	0	0
$585$	24.0000	0.992278
$586$	0	0
$587$	−31.7490	−1.31042	−0.655211	−	0.755446i	$-0.727420\pi$
$587$	−31.7490	−1.31042	−0.655211	+	0.755446i	$0.727420\pi$
$588$	0	0
$589$	−7.00000	−0.288430
$590$	0	0
$591$	31.7490	1.30598
$592$	0	0
$593$	− 22.5167i	− 0.924648i	−0.886711	−	0.462324i	$-0.847016\pi$
$593$	− 22.5167i	− 0.924648i	0.886711	−	0.462324i	$-0.152984\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−35.0000	−1.43245
$598$	0	0
$599$	− 4.58258i	− 0.187239i	−0.995608	−	0.0936195i	$-0.970156\pi$
$599$	− 4.58258i	− 0.187239i	0.995608	−	0.0936195i	$-0.0298437\pi$
$600$	0	0
$601$	− 3.46410i	− 0.141304i	−0.997501	−	0.0706518i	$-0.977492\pi$
$601$	− 3.46410i	− 0.141304i	0.997501	−	0.0706518i	$-0.0225079\pi$
$602$	0	0
$603$	− 18.3303i	− 0.746468i
$604$	0	0
$605$	17.3205i	0.704179i
$606$	0	0
$607$	−2.64575	−0.107388	−0.0536939	−	0.998557i	$-0.517100\pi$
$607$	−2.64575	−0.107388	−0.0536939	+	0.998557i	$0.517100\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	27.4955i	1.11235i
$612$	0	0
$613$	−7.00000	−0.282727	−0.141364	−	0.989958i	$-0.545149\pi$
$613$	−7.00000	−0.282727	−0.141364	+	0.989958i	$0.545149\pi$
$614$	0	0
$615$	15.8745	0.640122
$616$	0	0
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	0	0
$619$	18.5203	0.744392	0.372196	−	0.928154i	$-0.378605\pi$
$619$	18.5203	0.744392	0.372196	+	0.928154i	$0.378605\pi$
$620$	0	0
$621$	12.1244i	0.486534i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−11.0000	−0.440000
$626$	0	0
$627$	32.0780i	1.28107i
$628$	0	0
$629$	− 36.3731i	− 1.45029i
$630$	0	0
$631$	45.8258i	1.82429i	0.409863	+	0.912147i	$0.365577\pi$
$631$	45.8258i	1.82429i	−0.409863	+	0.912147i	$0.634423\pi$
$632$	0	0
$633$	24.2487i	0.963800i
$634$	0	0
$635$	15.8745	0.629961
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 36.6606i	− 1.45027i
$640$	0	0
$641$	−21.0000	−0.829450	−0.414725	−	0.909947i	$-0.636122\pi$
$641$	−21.0000	−0.829450	−0.414725	+	0.909947i	$0.636122\pi$
$642$	0	0
$643$	21.1660	0.834706	0.417353	−	0.908744i	$-0.362958\pi$
$643$	21.1660	0.834706	0.417353	+	0.908744i	$0.362958\pi$
$644$	0	0
$645$	42.0000	1.65375
$646$	0	0
$647$	39.6863	1.56023	0.780114	−	0.625637i	$-0.215161\pi$
$647$	39.6863	1.56023	0.780114	+	0.625637i	$0.215161\pi$
$648$	0	0
$649$	36.3731i	1.42777i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−21.0000	−0.821794	−0.410897	−	0.911682i	$-0.634784\pi$
$653$	−21.0000	−0.821794	−0.410897	+	0.911682i	$0.634784\pi$
$654$	0	0
$655$	13.7477i	0.537168i
$656$	0	0
$657$	20.7846i	0.810885i
$658$	0	0
$659$	− 9.16515i	− 0.357024i	−0.983938	−	0.178512i	$-0.942872\pi$
$659$	− 9.16515i	− 0.357024i	0.983938	−	0.178512i	$-0.0571283\pi$
$660$	0	0
$661$	5.19615i	0.202107i	0.994881	+	0.101053i	$0.0322213\pi$
$661$	5.19615i	0.202107i	−0.994881	+	0.101053i	$0.967779\pi$
$662$	0	0
$663$	−47.6235	−1.84954
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	28.0000	1.08254
$670$	0	0
$671$	−7.93725	−0.306414
$672$	0	0
$673$	28.0000	1.07932	0.539660	−	0.841883i	$-0.318553\pi$
$673$	28.0000	1.07932	0.539660	+	0.841883i	$0.318553\pi$
$674$	0	0
$675$	−5.29150	−0.203670
$676$	0	0
$677$	− 1.73205i	− 0.0665681i	−0.999446	−	0.0332841i	$-0.989403\pi$
$677$	− 1.73205i	− 0.0665681i	0.999446	−	0.0332841i	$-0.0105966\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−63.0000	−2.41417
$682$	0	0
$683$	4.58258i	0.175347i	0.996149	+	0.0876737i	$0.0279433\pi$
$683$	4.58258i	0.175347i	−0.996149	+	0.0876737i	$0.972057\pi$
$684$	0	0
$685$	− 36.3731i	− 1.38974i
$686$	0	0
$687$	4.58258i	0.174836i
$688$	0	0
$689$	10.3923i	0.395915i
$690$	0	0
$691$	2.64575	0.100649	0.0503246	−	0.998733i	$-0.483974\pi$
$691$	2.64575	0.100649	0.0503246	+	0.998733i	$0.483974\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 18.3303i	− 0.695308i
$696$	0	0
$697$	−18.0000	−0.681799
$698$	0	0
$699$	−55.5608	−2.10150
$700$	0	0
$701$	−42.0000	−1.58632	−0.793159	−	0.609015i	$-0.791565\pi$
$701$	−42.0000	−1.58632	−0.793159	+	0.609015i	$0.791565\pi$
$702$	0	0
$703$	−18.5203	−0.698505
$704$	0	0
$705$	36.3731i	1.36989i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	35.0000	1.31445	0.657226	−	0.753693i	$-0.271730\pi$
$709$	35.0000	1.31445	0.657226	+	0.753693i	$0.271730\pi$
$710$	0	0
$711$	− 18.3303i	− 0.687440i
$712$	0	0
$713$	− 12.1244i	− 0.454061i
$714$	0	0
$715$	27.4955i	1.02827i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−7.93725	−0.296010	−0.148005	−	0.988987i	$-0.547285\pi$
$719$	−7.93725	−0.296010	−0.148005	+	0.988987i	$0.547285\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 50.4083i	− 1.87471i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−21.1660	−0.785004	−0.392502	−	0.919751i	$-0.628390\pi$
$727$	−21.1660	−0.785004	−0.392502	+	0.919751i	$0.628390\pi$
$728$	0	0
$729$	−41.0000	−1.51852
$730$	0	0
$731$	−47.6235	−1.76142
$732$	0	0
$733$	22.5167i	0.831672i	0.909440	+	0.415836i	$0.136511\pi$
$733$	22.5167i	0.831672i	−0.909440	+	0.415836i	$0.863489\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	21.0000	0.773545
$738$	0	0
$739$	22.9129i	0.842864i	0.906860	+	0.421432i	$0.138472\pi$
$739$	22.9129i	0.842864i	−0.906860	+	0.421432i	$0.861528\pi$
$740$	0	0
$741$	24.2487i	0.890799i
$742$	0	0
$743$	45.8258i	1.68118i	0.541669	+	0.840592i	$0.317793\pi$
$743$	45.8258i	1.68118i	−0.541669	+	0.840592i	$0.682207\pi$
$744$	0	0
$745$	25.9808i	0.951861i
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	− 13.7477i	− 0.501662i	−0.968031	−	0.250831i	$-0.919296\pi$
$751$	− 13.7477i	− 0.501662i	0.968031	−	0.250831i	$-0.0807038\pi$
$752$	0	0
$753$	−42.0000	−1.53057
$754$	0	0
$755$	39.6863	1.44433
$756$	0	0
$757$	−28.0000	−1.01768	−0.508839	−	0.860862i	$-0.669925\pi$
$757$	−28.0000	−1.01768	−0.508839	+	0.860862i	$0.669925\pi$
$758$	0	0
$759$	−55.5608	−2.01673
$760$	0	0
$761$	25.9808i	0.941802i	0.882186	+	0.470901i	$0.156071\pi$
$761$	25.9808i	0.941802i	−0.882186	+	0.470901i	$0.843929\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−36.0000	−1.30158
$766$	0	0
$767$	27.4955i	0.992803i
$768$	0	0
$769$	51.9615i	1.87378i	0.349624	+	0.936890i	$0.386309\pi$
$769$	51.9615i	1.87378i	−0.349624	+	0.936890i	$0.613691\pi$
$770$	0	0
$771$	4.58258i	0.165037i
$772$	0	0
$773$	29.4449i	1.05906i	0.848292	+	0.529529i	$0.177631\pi$
$773$	29.4449i	1.05906i	−0.848292	+	0.529529i	$0.822369\pi$
$774$	0	0
$775$	5.29150	0.190076
$776$	0	0
$777$	0	0
$778$	0	0
$779$	9.16515i	0.328376i
$780$	0	0
$781$	42.0000	1.50288
$782$	0	0
$783$	0	0
$784$	0	0
$785$	9.00000	0.321224
$786$	0	0
$787$	−13.2288	−0.471554	−0.235777	−	0.971807i	$-0.575764\pi$
$787$	−13.2288	−0.471554	−0.235777	+	0.971807i	$0.575764\pi$
$788$	0	0
$789$	12.1244i	0.431638i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−6.00000	−0.213066
$794$	0	0
$795$	13.7477i	0.487582i
$796$	0	0
$797$	3.46410i	0.122705i	0.998116	+	0.0613524i	$0.0195413\pi$
$797$	3.46410i	0.122705i	−0.998116	+	0.0613524i	$0.980459\pi$
$798$	0	0
$799$	− 41.2432i	− 1.45908i
$800$	0	0
$801$	6.92820i	0.244796i
$802$	0	0
$803$	−23.8118	−0.840299
$804$	0	0
$805$	0	0
$806$	0	0
$807$	77.9038i	2.74234i
$808$	0	0
$809$	−45.0000	−1.58212	−0.791058	−	0.611741i	$-0.790469\pi$
$809$	−45.0000	−1.58212	−0.791058	+	0.611741i	$0.790469\pi$
$810$	0	0
$811$	−21.1660	−0.743239	−0.371620	−	0.928385i	$-0.621197\pi$
$811$	−21.1660	−0.743239	−0.371620	+	0.928385i	$0.621197\pi$
$812$	0	0
$813$	77.0000	2.70051
$814$	0	0
$815$	−7.93725	−0.278030
$816$	0	0
$817$	24.2487i	0.848355i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−27.0000	−0.942306	−0.471153	−	0.882051i	$-0.656162\pi$
$821$	−27.0000	−0.942306	−0.471153	+	0.882051i	$0.656162\pi$
$822$	0	0
$823$	32.0780i	1.11817i	0.829110	+	0.559085i	$0.188847\pi$
$823$	32.0780i	1.11817i	−0.829110	+	0.559085i	$0.811153\pi$
$824$	0	0
$825$	− 24.2487i	− 0.844232i
$826$	0	0
$827$	9.16515i	0.318704i	0.987222	+	0.159352i	$0.0509404\pi$
$827$	9.16515i	0.318704i	−0.987222	+	0.159352i	$0.949060\pi$
$828$	0	0
$829$	− 5.19615i	− 0.180470i	−0.995921	−	0.0902349i	$-0.971238\pi$
$829$	− 5.19615i	− 0.180470i	0.995921	−	0.0902349i	$-0.0287618\pi$
$830$	0	0
$831$	44.9778	1.56026
$832$	0	0
$833$	0	0
$834$	0	0
$835$	− 27.4955i	− 0.951519i
$836$	0	0
$837$	−7.00000	−0.241955
$838$	0	0
$839$	31.7490	1.09610	0.548049	−	0.836446i	$-0.315371\pi$
$839$	31.7490	1.09610	0.548049	+	0.836446i	$0.315371\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 1.73205i	− 0.0595844i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−35.0000	−1.20120
$850$	0	0
$851$	− 32.0780i	− 1.09962i
$852$	0	0
$853$	− 51.9615i	− 1.77913i	−0.456810	−	0.889564i	$-0.651008\pi$
$853$	− 51.9615i	− 1.77913i	0.456810	−	0.889564i	$-0.348992\pi$
$854$	0	0
$855$	18.3303i	0.626883i
$856$	0	0
$857$	− 29.4449i	− 1.00582i	−0.864340	−	0.502909i	$-0.832263\pi$
$857$	− 29.4449i	− 1.00582i	0.864340	−	0.502909i	$-0.167737\pi$
$858$	0	0
$859$	−2.64575	−0.0902719	−0.0451359	−	0.998981i	$-0.514372\pi$
$859$	−2.64575	−0.0902719	−0.0451359	+	0.998981i	$0.514372\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	13.7477i	0.467978i	0.972239	+	0.233989i	$0.0751780\pi$
$863$	13.7477i	0.467978i	−0.972239	+	0.233989i	$0.924822\pi$
$864$	0	0
$865$	39.0000	1.32604
$866$	0	0
$867$	26.4575	0.898544
$868$	0	0
$869$	21.0000	0.712376
$870$	0	0
$871$	15.8745	0.537887
$872$	0	0
$873$	− 13.8564i	− 0.468968i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−7.00000	−0.236373	−0.118187	−	0.992991i	$-0.537708\pi$
$877$	−7.00000	−0.236373	−0.118187	+	0.992991i	$0.537708\pi$
$878$	0	0
$879$	54.9909i	1.85480i
$880$	0	0
$881$	27.7128i	0.933668i	0.884345	+	0.466834i	$0.154606\pi$
$881$	27.7128i	0.933668i	−0.884345	+	0.466834i	$0.845394\pi$
$882$	0	0
$883$	− 27.4955i	− 0.925296i	−0.886542	−	0.462648i	$-0.846899\pi$
$883$	− 27.4955i	− 0.925296i	0.886542	−	0.462648i	$-0.153101\pi$
$884$	0	0
$885$	36.3731i	1.22267i
$886$	0	0
$887$	−23.8118	−0.799521	−0.399760	−	0.916620i	$-0.630907\pi$
$887$	−23.8118	−0.799521	−0.399760	+	0.916620i	$0.630907\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	− 22.9129i	− 0.767610i
$892$	0	0
$893$	−21.0000	−0.702738
$894$	0	0
$895$	−23.8118	−0.795939
$896$	0	0
$897$	−42.0000	−1.40234
$898$	0	0
$899$	0	0
$900$	0	0
$901$	− 15.5885i	− 0.519327i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	36.0000	1.19668
$906$	0	0
$907$	− 22.9129i	− 0.760810i	−0.924820	−	0.380405i	$-0.875785\pi$
$907$	− 22.9129i	− 0.760810i	0.924820	−	0.380405i	$-0.124215\pi$
$908$	0	0
$909$	− 20.7846i	− 0.689382i
$910$	0	0
$911$	− 27.4955i	− 0.910965i	−0.890245	−	0.455483i	$-0.849467\pi$
$911$	− 27.4955i	− 0.910965i	0.890245	−	0.455483i	$-0.150533\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−7.93725	−0.262398
$916$	0	0
$917$	0	0
$918$	0	0
$919$	22.9129i	0.755826i	0.925841	+	0.377913i	$0.123358\pi$
$919$	22.9129i	0.755826i	−0.925841	+	0.377913i	$0.876642\pi$
$920$	0	0
$921$	28.0000	0.922631
$922$	0	0
$923$	31.7490	1.04503
$924$	0	0
$925$	14.0000	0.460317
$926$	0	0
$927$	10.5830	0.347591
$928$	0	0
$929$	− 46.7654i	− 1.53432i	−0.641455	−	0.767161i	$-0.721669\pi$
$929$	− 46.7654i	− 1.53432i	0.641455	−	0.767161i	$-0.278331\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	63.0000	2.06253
$934$	0	0
$935$	− 41.2432i	− 1.34880i
$936$	0	0
$937$	− 27.7128i	− 0.905338i	−0.891679	−	0.452669i	$-0.850472\pi$
$937$	− 27.7128i	− 0.905338i	0.891679	−	0.452669i	$-0.149528\pi$
$938$	0	0
$939$	68.7386i	2.24320i
$940$	0	0
$941$	19.0526i	0.621096i	0.950558	+	0.310548i	$0.100512\pi$
$941$	19.0526i	0.621096i	−0.950558	+	0.310548i	$0.899488\pi$
$942$	0	0
$943$	−15.8745	−0.516945
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 4.58258i	− 0.148914i	−0.997224	−	0.0744569i	$-0.976278\pi$
$947$	− 4.58258i	− 0.148914i	0.997224	−	0.0744569i	$-0.0237223\pi$
$948$	0	0
$949$	−18.0000	−0.584305
$950$	0	0
$951$	−39.6863	−1.28692
$952$	0	0
$953$	−12.0000	−0.388718	−0.194359	−	0.980930i	$-0.562263\pi$
$953$	−12.0000	−0.388718	−0.194359	+	0.980930i	$0.562263\pi$
$954$	0	0
$955$	39.6863	1.28422
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−24.0000	−0.774194
$962$	0	0
$963$	− 54.9909i	− 1.77206i
$964$	0	0
$965$	19.0526i	0.613324i
$966$	0	0
$967$	− 45.8258i	− 1.47366i	−0.676080	−	0.736828i	$-0.736323\pi$
$967$	− 45.8258i	− 1.47366i	0.676080	−	0.736828i	$-0.263677\pi$
$968$	0	0
$969$	− 36.3731i	− 1.16847i
$970$	0	0
$971$	55.5608	1.78303	0.891515	−	0.452991i	$-0.149643\pi$
$971$	55.5608	1.78303	0.891515	+	0.452991i	$0.149643\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	− 18.3303i	− 0.587040i
$976$	0	0
$977$	21.0000	0.671850	0.335925	−	0.941889i	$-0.390951\pi$
$977$	21.0000	0.671850	0.335925	+	0.941889i	$0.390951\pi$
$978$	0	0
$979$	−7.93725	−0.253676
$980$	0	0
$981$	−28.0000	−0.893971
$982$	0	0
$983$	−7.93725	−0.253159	−0.126580	−	0.991956i	$-0.540400\pi$
$983$	−7.93725	−0.253159	−0.126580	+	0.991956i	$0.540400\pi$
$984$	0	0
$985$	20.7846i	0.662253i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−42.0000	−1.33552
$990$	0	0
$991$	− 4.58258i	− 0.145570i	−0.997348	−	0.0727852i	$-0.976811\pi$
$991$	− 4.58258i	− 0.145570i	0.997348	−	0.0727852i	$-0.0231887\pi$
$992$	0	0
$993$	− 36.3731i	− 1.15426i
$994$	0	0
$995$	− 22.9129i	− 0.726387i
$996$	0	0
$997$	− 43.3013i	− 1.37136i	−0.727901	−	0.685682i	$-0.759504\pi$
$997$	− 43.3013i	− 1.37136i	0.727901	−	0.685682i	$-0.240496\pi$
$998$	0	0
$999$	−18.5203	−0.585955

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.2.f.d.783.1		4
3.2	odd	2		7056.2.b.s.1567.3		4
4.3	odd	2	inner	784.2.f.d.783.3		4
7.2	even	3		112.2.p.c.31.2	yes	4
7.3	odd	6		112.2.p.c.47.1	yes	4
7.4	even	3		784.2.p.g.607.2		4
7.5	odd	6		784.2.p.g.31.1		4
7.6	odd	2	inner	784.2.f.d.783.4		4
8.3	odd	2		3136.2.f.f.3135.2		4
8.5	even	2		3136.2.f.f.3135.4		4
12.11	even	2		7056.2.b.s.1567.4		4
21.2	odd	6		1008.2.cs.q.703.2		4
21.17	even	6		1008.2.cs.q.271.1		4
21.20	even	2		7056.2.b.s.1567.1		4
28.3	even	6		112.2.p.c.47.2	yes	4
28.11	odd	6		784.2.p.g.607.1		4
28.19	even	6		784.2.p.g.31.2		4
28.23	odd	6		112.2.p.c.31.1	✓	4
28.27	even	2	inner	784.2.f.d.783.2		4
56.3	even	6		448.2.p.c.383.1		4
56.13	odd	2		3136.2.f.f.3135.1		4
56.27	even	2		3136.2.f.f.3135.3		4
56.37	even	6		448.2.p.c.255.1		4
56.45	odd	6		448.2.p.c.383.2		4
56.51	odd	6		448.2.p.c.255.2		4
84.23	even	6		1008.2.cs.q.703.1		4
84.59	odd	6		1008.2.cs.q.271.2		4
84.83	odd	2		7056.2.b.s.1567.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.2.p.c.31.1	✓	4	28.23	odd	6
112.2.p.c.31.2	yes	4	7.2	even	3
112.2.p.c.47.1	yes	4	7.3	odd	6
112.2.p.c.47.2	yes	4	28.3	even	6
448.2.p.c.255.1		4	56.37	even	6
448.2.p.c.255.2		4	56.51	odd	6
448.2.p.c.383.1		4	56.3	even	6
448.2.p.c.383.2		4	56.45	odd	6
784.2.f.d.783.1		4	1.1	even	1	trivial
784.2.f.d.783.2		4	28.27	even	2	inner
784.2.f.d.783.3		4	4.3	odd	2	inner
784.2.f.d.783.4		4	7.6	odd	2	inner
784.2.p.g.31.1		4	7.5	odd	6
784.2.p.g.31.2		4	28.19	even	6
784.2.p.g.607.1		4	28.11	odd	6
784.2.p.g.607.2		4	7.4	even	3
1008.2.cs.q.271.1		4	21.17	even	6
1008.2.cs.q.271.2		4	84.59	odd	6
1008.2.cs.q.703.1		4	84.23	even	6
1008.2.cs.q.703.2		4	21.2	odd	6
3136.2.f.f.3135.1		4	56.13	odd	2
3136.2.f.f.3135.2		4	8.3	odd	2
3136.2.f.f.3135.3		4	56.27	even	2
3136.2.f.f.3135.4		4	8.5	even	2
7056.2.b.s.1567.1		4	21.20	even	2
7056.2.b.s.1567.2		4	84.83	odd	2
7056.2.b.s.1567.3		4	3.2	odd	2
7056.2.b.s.1567.4		4	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:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	784.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.64575 q^{3} -1.73205i q^{5} +4.00000 q^{9} +O(q^{10})\)	\(q-2.64575 q^{3} -1.73205i q^{5} +4.00000 q^{9} +4.58258i q^{11} +3.46410i q^{13} +4.58258i q^{15} -5.19615i q^{17} -2.64575 q^{19} -4.58258i q^{23} +2.00000 q^{25} -2.64575 q^{27} +2.64575 q^{31} -12.1244i q^{33} +7.00000 q^{37} -9.16515i q^{39} -3.46410i q^{41} -9.16515i q^{43} -6.92820i q^{45} +7.93725 q^{47} +13.7477i q^{51} +3.00000 q^{53} +7.93725 q^{55} +7.00000 q^{57} +7.93725 q^{59} +1.73205i q^{61} +6.00000 q^{65} -4.58258i q^{67} +12.1244i q^{69} -9.16515i q^{71} +5.19615i q^{73} -5.29150 q^{75} -4.58258i q^{79} -5.00000 q^{81} -9.00000 q^{85} +1.73205i q^{89} -7.00000 q^{93} +4.58258i q^{95} -3.46410i q^{97} +18.3303i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 16 q^{9}+O(q^{10}) \) 4 * q + 16 * q^9	\( 4 q + 16 q^{9} + 8 q^{25} + 28 q^{37} + 12 q^{53} + 28 q^{57} + 24 q^{65} - 20 q^{81} - 36 q^{85} - 28 q^{93}+O(q^{100}) \) 4 * q + 16 * q^9 + 8 * q^25 + 28 * q^37 + 12 * q^53 + 28 * q^57 + 24 * q^65 - 20 * q^81 - 36 * q^85 - 28 * q^93

Introduction

L-functions

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