LMFDB - Embedded newform 2600.2.d.f.1249.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2600,2,Mod(1249,2600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2600.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2600 = 2^{3} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2600.d (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:	$20.7611045255$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 104)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	2600.1249
Dual form			2600.2.d.f.1249.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +5.00000i q^{7} +2.00000 q^{9} -2.00000 q^{11} +1.00000i q^{13} -3.00000i q^{17} +2.00000 q^{19} +5.00000 q^{21} -4.00000i q^{23} -5.00000i q^{27} +6.00000 q^{29} -4.00000 q^{31} +2.00000i q^{33} +11.0000i q^{37} +1.00000 q^{39} +8.00000 q^{41} +1.00000i q^{43} +9.00000i q^{47} -18.0000 q^{49} -3.00000 q^{51} +12.0000i q^{53} -2.00000i q^{57} -6.00000 q^{59} +10.0000i q^{63} +6.00000i q^{67} -4.00000 q^{69} +7.00000 q^{71} +2.00000i q^{73} -10.0000i q^{77} -12.0000 q^{79} +1.00000 q^{81} +16.0000i q^{83} -6.00000i q^{87} +10.0000 q^{89} -5.00000 q^{91} +4.00000i q^{93} -10.0000i q^{97} -4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{9} - 4 q^{11} + 4 q^{19} + 10 q^{21} + 12 q^{29} - 8 q^{31} + 2 q^{39} + 16 q^{41} - 36 q^{49} - 6 q^{51} - 12 q^{59} - 8 q^{69} + 14 q^{71} - 24 q^{79} + 2 q^{81} + 20 q^{89} - 10 q^{91} - 8 q^{99}+O(q^{100}) $ 2 * q + 4 * q^9 - 4 * q^11 + 4 * q^19 + 10 * q^21 + 12 * q^29 - 8 * q^31 + 2 * q^39 + 16 * q^41 - 36 * q^49 - 6 * q^51 - 12 * q^59 - 8 * q^69 + 14 * q^71 - 24 * q^79 + 2 * q^81 + 20 * q^89 - 10 * q^91 - 8 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2600\mathbb{Z}\right)^\times$.

$n$	$1301$	$1601$	$1951$	$1977$
$\chi(n)$	$1$	$1$	$1$	$-1$

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	−0.957427	−	0.288675i	$-0.906785\pi$
$3$	− 1.00000i	− 0.577350i	0.957427	−	0.288675i	$-0.0932147\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	5.00000i	1.88982i	0.327327	+	0.944911i	$0.393852\pi$
$7$	5.00000i	1.88982i	−0.327327	+	0.944911i	$0.606148\pi$
$8$	0	0
$9$	2.00000	0.666667
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	1.00000i	0.277350i
$13$	1.00000i	0.277350i
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 3.00000i	− 0.727607i	−0.931476	−	0.363803i	$-0.881478\pi$
$17$	− 3.00000i	− 0.727607i	0.931476	−	0.363803i	$-0.118522\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	5.00000	1.09109
$22$	0	0
$23$	− 4.00000i	− 0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$	− 4.00000i	− 0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 5.00000i	− 0.962250i
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	2.00000i	0.348155i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	11.0000i	1.80839i	0.427121	+	0.904194i	$0.359528\pi$
$37$	11.0000i	1.80839i	−0.427121	+	0.904194i	$0.640472\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	8.00000	1.24939	0.624695	−	0.780869i	$-0.285223\pi$
$41$	8.00000	1.24939	0.624695	+	0.780869i	$0.285223\pi$
$42$	0	0
$43$	1.00000i	0.152499i	0.997089	+	0.0762493i	$0.0242945\pi$
$43$	1.00000i	0.152499i	−0.997089	+	0.0762493i	$0.975706\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.00000i	1.31278i	0.754420	+	0.656392i	$0.227918\pi$
$47$	9.00000i	1.31278i	−0.754420	+	0.656392i	$0.772082\pi$
$48$	0	0
$49$	−18.0000	−2.57143
$50$	0	0
$51$	−3.00000	−0.420084
$52$	0	0
$53$	12.0000i	1.64833i	0.566352	+	0.824163i	$0.308354\pi$
$53$	12.0000i	1.64833i	−0.566352	+	0.824163i	$0.691646\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 2.00000i	− 0.264906i
$58$	0	0
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	0	0
$63$	10.0000i	1.25988i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	6.00000i	0.733017i	0.930415	+	0.366508i	$0.119447\pi$
$67$	6.00000i	0.733017i	−0.930415	+	0.366508i	$0.880553\pi$
$68$	0	0
$69$	−4.00000	−0.481543
$70$	0	0
$71$	7.00000	0.830747	0.415374	−	0.909651i	$-0.363651\pi$
$71$	7.00000	0.830747	0.415374	+	0.909651i	$0.363651\pi$
$72$	0	0
$73$	2.00000i	0.234082i	0.993127	+	0.117041i	$0.0373409\pi$
$73$	2.00000i	0.234082i	−0.993127	+	0.117041i	$0.962659\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 10.0000i	− 1.13961i
$78$	0	0
$79$	−12.0000	−1.35011	−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000	−1.35011	−0.675053	+	0.737769i	$0.735879\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	16.0000i	1.75623i	0.478451	+	0.878114i	$0.341198\pi$
$83$	16.0000i	1.75623i	−0.478451	+	0.878114i	$0.658802\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 6.00000i	− 0.643268i
$88$	0	0
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	−5.00000	−0.524142
$92$	0	0
$93$	4.00000i	0.414781i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 10.0000i	− 1.01535i	−0.861550	−	0.507673i	$-0.830506\pi$
$97$	− 10.0000i	− 1.01535i	0.861550	−	0.507673i	$-0.169494\pi$
$98$	0	0
$99$	−4.00000	−0.402015
$100$	0	0
$101$	4.00000	0.398015	0.199007	−	0.979998i	$-0.436228\pi$
$101$	4.00000	0.398015	0.199007	+	0.979998i	$0.436228\pi$
$102$	0	0
$103$	8.00000i	0.788263i	0.919054	+	0.394132i	$0.128955\pi$
$103$	8.00000i	0.788263i	−0.919054	+	0.394132i	$0.871045\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	20.0000i	1.93347i	0.255774	+	0.966736i	$0.417670\pi$
$107$	20.0000i	1.93347i	−0.255774	+	0.966736i	$0.582330\pi$
$108$	0	0
$109$	5.00000	0.478913	0.239457	−	0.970907i	$-0.423031\pi$
$109$	5.00000	0.478913	0.239457	+	0.970907i	$0.423031\pi$
$110$	0	0
$111$	11.0000	1.04407
$112$	0	0
$113$	− 2.00000i	− 0.188144i	−0.995565	−	0.0940721i	$-0.970012\pi$
$113$	− 2.00000i	− 0.188144i	0.995565	−	0.0940721i	$-0.0299884\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.00000i	0.184900i
$118$	0	0
$119$	15.0000	1.37505
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	− 8.00000i	− 0.721336i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 8.00000i	− 0.709885i	−0.934888	−	0.354943i	$-0.884500\pi$
$127$	− 8.00000i	− 0.709885i	0.934888	−	0.354943i	$-0.115500\pi$
$128$	0	0
$129$	1.00000	0.0880451
$130$	0	0
$131$	3.00000	0.262111	0.131056	−	0.991375i	$-0.458163\pi$
$131$	3.00000	0.262111	0.131056	+	0.991375i	$0.458163\pi$
$132$	0	0
$133$	10.0000i	0.867110i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 12.0000i	− 1.02523i	−0.858619	−	0.512615i	$-0.828677\pi$
$137$	− 12.0000i	− 1.02523i	0.858619	−	0.512615i	$-0.171323\pi$
$138$	0	0
$139$	−3.00000	−0.254457	−0.127228	−	0.991873i	$-0.540608\pi$
$139$	−3.00000	−0.254457	−0.127228	+	0.991873i	$0.540608\pi$
$140$	0	0
$141$	9.00000	0.757937
$142$	0	0
$143$	− 2.00000i	− 0.167248i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	18.0000i	1.48461i
$148$	0	0
$149$	−14.0000	−1.14692	−0.573462	−	0.819232i	$-0.694400\pi$
$149$	−14.0000	−1.14692	−0.573462	+	0.819232i	$0.694400\pi$
$150$	0	0
$151$	−5.00000	−0.406894	−0.203447	−	0.979086i	$-0.565214\pi$
$151$	−5.00000	−0.406894	−0.203447	+	0.979086i	$0.565214\pi$
$152$	0	0
$153$	− 6.00000i	− 0.485071i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 2.00000i	− 0.159617i	−0.996810	−	0.0798087i	$-0.974569\pi$
$157$	− 2.00000i	− 0.159617i	0.996810	−	0.0798087i	$-0.0254309\pi$
$158$	0	0
$159$	12.0000	0.951662
$160$	0	0
$161$	20.0000	1.57622
$162$	0	0
$163$	− 4.00000i	− 0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$	− 4.00000i	− 0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.00000i	0.619059i	0.950890	+	0.309529i	$0.100171\pi$
$167$	8.00000i	0.619059i	−0.950890	+	0.309529i	$0.899829\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	4.00000	0.305888
$172$	0	0
$173$	12.0000i	0.912343i	0.889892	+	0.456172i	$0.150780\pi$
$173$	12.0000i	0.912343i	−0.889892	+	0.456172i	$0.849220\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	6.00000i	0.450988i
$178$	0	0
$179$	−3.00000	−0.224231	−0.112115	−	0.993695i	$-0.535763\pi$
$179$	−3.00000	−0.224231	−0.112115	+	0.993695i	$0.535763\pi$
$180$	0	0
$181$	16.0000	1.18927	0.594635	−	0.803996i	$-0.297296\pi$
$181$	16.0000	1.18927	0.594635	+	0.803996i	$0.297296\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	6.00000i	0.438763i
$188$	0	0
$189$	25.0000	1.81848
$190$	0	0
$191$	10.0000	0.723575	0.361787	−	0.932261i	$-0.382167\pi$
$191$	10.0000	0.723575	0.361787	+	0.932261i	$0.382167\pi$
$192$	0	0
$193$	− 16.0000i	− 1.15171i	−0.817554	−	0.575853i	$-0.804670\pi$
$193$	− 16.0000i	− 1.15171i	0.817554	−	0.575853i	$-0.195330\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	17.0000i	1.21120i	0.795769	+	0.605600i	$0.207067\pi$
$197$	17.0000i	1.21120i	−0.795769	+	0.605600i	$0.792933\pi$
$198$	0	0
$199$	18.0000	1.27599	0.637993	−	0.770042i	$-0.279765\pi$
$199$	18.0000	1.27599	0.637993	+	0.770042i	$0.279765\pi$
$200$	0	0
$201$	6.00000	0.423207
$202$	0	0
$203$	30.0000i	2.10559i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 8.00000i	− 0.556038i
$208$	0	0
$209$	−4.00000	−0.276686
$210$	0	0
$211$	−5.00000	−0.344214	−0.172107	−	0.985078i	$-0.555058\pi$
$211$	−5.00000	−0.344214	−0.172107	+	0.985078i	$0.555058\pi$
$212$	0	0
$213$	− 7.00000i	− 0.479632i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 20.0000i	− 1.35769i
$218$	0	0
$219$	2.00000	0.135147
$220$	0	0
$221$	3.00000	0.201802
$222$	0	0
$223$	17.0000i	1.13840i	0.822198	+	0.569202i	$0.192748\pi$
$223$	17.0000i	1.13840i	−0.822198	+	0.569202i	$0.807252\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 16.0000i	− 1.06196i	−0.847385	−	0.530979i	$-0.821824\pi$
$227$	− 16.0000i	− 1.06196i	0.847385	−	0.530979i	$-0.178176\pi$
$228$	0	0
$229$	15.0000	0.991228	0.495614	−	0.868543i	$-0.334943\pi$
$229$	15.0000	0.991228	0.495614	+	0.868543i	$0.334943\pi$
$230$	0	0
$231$	−10.0000	−0.657952
$232$	0	0
$233$	− 13.0000i	− 0.851658i	−0.904804	−	0.425829i	$-0.859982\pi$
$233$	− 13.0000i	− 0.851658i	0.904804	−	0.425829i	$-0.140018\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	12.0000i	0.779484i
$238$	0	0
$239$	3.00000	0.194054	0.0970269	−	0.995282i	$-0.469067\pi$
$239$	3.00000	0.194054	0.0970269	+	0.995282i	$0.469067\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	0	0
$243$	− 16.0000i	− 1.02640i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.00000i	0.127257i
$248$	0	0
$249$	16.0000	1.01396
$250$	0	0
$251$	8.00000	0.504956	0.252478	−	0.967603i	$-0.418755\pi$
$251$	8.00000	0.504956	0.252478	+	0.967603i	$0.418755\pi$
$252$	0	0
$253$	8.00000i	0.502956i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 31.0000i	− 1.93373i	−0.255294	−	0.966863i	$-0.582172\pi$
$257$	− 31.0000i	− 1.93373i	0.255294	−	0.966863i	$-0.417828\pi$
$258$	0	0
$259$	−55.0000	−3.41753
$260$	0	0
$261$	12.0000	0.742781
$262$	0	0
$263$	− 20.0000i	− 1.23325i	−0.787256	−	0.616626i	$-0.788499\pi$
$263$	− 20.0000i	− 1.23325i	0.787256	−	0.616626i	$-0.211501\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 10.0000i	− 0.611990i
$268$	0	0
$269$	−16.0000	−0.975537	−0.487769	−	0.872973i	$-0.662189\pi$
$269$	−16.0000	−0.975537	−0.487769	+	0.872973i	$0.662189\pi$
$270$	0	0
$271$	−7.00000	−0.425220	−0.212610	−	0.977137i	$-0.568196\pi$
$271$	−7.00000	−0.425220	−0.212610	+	0.977137i	$0.568196\pi$
$272$	0	0
$273$	5.00000i	0.302614i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 12.0000i	− 0.721010i	−0.932757	−	0.360505i	$-0.882604\pi$
$277$	− 12.0000i	− 0.721010i	0.932757	−	0.360505i	$-0.117396\pi$
$278$	0	0
$279$	−8.00000	−0.478947
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	20.0000i	1.18888i	0.804141	+	0.594438i	$0.202626\pi$
$283$	20.0000i	1.18888i	−0.804141	+	0.594438i	$0.797374\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	40.0000i	2.36113i
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	−10.0000	−0.586210
$292$	0	0
$293$	− 15.0000i	− 0.876309i	−0.898900	−	0.438155i	$-0.855632\pi$
$293$	− 15.0000i	− 0.876309i	0.898900	−	0.438155i	$-0.144368\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	10.0000i	0.580259i
$298$	0	0
$299$	4.00000	0.231326
$300$	0	0
$301$	−5.00000	−0.288195
$302$	0	0
$303$	− 4.00000i	− 0.229794i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 2.00000i	− 0.114146i	−0.998370	−	0.0570730i	$-0.981823\pi$
$307$	− 2.00000i	− 0.114146i	0.998370	−	0.0570730i	$-0.0181768\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	0	0
$311$	−22.0000	−1.24751	−0.623753	−	0.781622i	$-0.714393\pi$
$311$	−22.0000	−1.24751	−0.623753	+	0.781622i	$0.714393\pi$
$312$	0	0
$313$	9.00000i	0.508710i	0.967111	+	0.254355i	$0.0818632\pi$
$313$	9.00000i	0.508710i	−0.967111	+	0.254355i	$0.918137\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 2.00000i	− 0.112331i	−0.998421	−	0.0561656i	$-0.982113\pi$
$317$	− 2.00000i	− 0.112331i	0.998421	−	0.0561656i	$-0.0178875\pi$
$318$	0	0
$319$	−12.0000	−0.671871
$320$	0	0
$321$	20.0000	1.11629
$322$	0	0
$323$	− 6.00000i	− 0.333849i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 5.00000i	− 0.276501i
$328$	0	0
$329$	−45.0000	−2.48093
$330$	0	0
$331$	12.0000	0.659580	0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000	0.659580	0.329790	+	0.944054i	$0.393022\pi$
$332$	0	0
$333$	22.0000i	1.20559i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	15.0000i	0.817102i	0.912735	+	0.408551i	$0.133966\pi$
$337$	15.0000i	0.817102i	−0.912735	+	0.408551i	$0.866034\pi$
$338$	0	0
$339$	−2.00000	−0.108625
$340$	0	0
$341$	8.00000	0.433224
$342$	0	0
$343$	− 55.0000i	− 2.96972i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	3.00000i	0.161048i	0.996753	+	0.0805242i	$0.0256594\pi$
$347$	3.00000i	0.161048i	−0.996753	+	0.0805242i	$0.974341\pi$
$348$	0	0
$349$	9.00000	0.481759	0.240879	−	0.970555i	$-0.422564\pi$
$349$	9.00000	0.481759	0.240879	+	0.970555i	$0.422564\pi$
$350$	0	0
$351$	5.00000	0.266880
$352$	0	0
$353$	28.0000i	1.49029i	0.666903	+	0.745145i	$0.267620\pi$
$353$	28.0000i	1.49029i	−0.666903	+	0.745145i	$0.732380\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 15.0000i	− 0.793884i
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	7.00000i	0.367405i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 18.0000i	− 0.939592i	−0.882775	−	0.469796i	$-0.844327\pi$
$367$	− 18.0000i	− 0.939592i	0.882775	−	0.469796i	$-0.155673\pi$
$368$	0	0
$369$	16.0000	0.832927
$370$	0	0
$371$	−60.0000	−3.11504
$372$	0	0
$373$	4.00000i	0.207112i	0.994624	+	0.103556i	$0.0330221\pi$
$373$	4.00000i	0.207112i	−0.994624	+	0.103556i	$0.966978\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6.00000i	0.309016i
$378$	0	0
$379$	−24.0000	−1.23280	−0.616399	−	0.787434i	$-0.711409\pi$
$379$	−24.0000	−1.23280	−0.616399	+	0.787434i	$0.711409\pi$
$380$	0	0
$381$	−8.00000	−0.409852
$382$	0	0
$383$	− 15.0000i	− 0.766464i	−0.923652	−	0.383232i	$-0.874811\pi$
$383$	− 15.0000i	− 0.766464i	0.923652	−	0.383232i	$-0.125189\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2.00000i	0.101666i
$388$	0	0
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	−12.0000	−0.606866
$392$	0	0
$393$	− 3.00000i	− 0.151330i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	10.0000i	0.501886i	0.968002	+	0.250943i	$0.0807406\pi$
$397$	10.0000i	0.501886i	−0.968002	+	0.250943i	$0.919259\pi$
$398$	0	0
$399$	10.0000	0.500626
$400$	0	0
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	− 4.00000i	− 0.199254i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 22.0000i	− 1.09050i
$408$	0	0
$409$	4.00000	0.197787	0.0988936	−	0.995098i	$-0.468470\pi$
$409$	4.00000	0.197787	0.0988936	+	0.995098i	$0.468470\pi$
$410$	0	0
$411$	−12.0000	−0.591916
$412$	0	0
$413$	− 30.0000i	− 1.47620i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	3.00000i	0.146911i
$418$	0	0
$419$	39.0000	1.90527	0.952637	−	0.304109i	$-0.0983586\pi$
$419$	39.0000	1.90527	0.952637	+	0.304109i	$0.0983586\pi$
$420$	0	0
$421$	11.0000	0.536107	0.268054	−	0.963404i	$-0.413620\pi$
$421$	11.0000	0.536107	0.268054	+	0.963404i	$0.413620\pi$
$422$	0	0
$423$	18.0000i	0.875190i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−2.00000	−0.0965609
$430$	0	0
$431$	−11.0000	−0.529851	−0.264926	−	0.964269i	$-0.585347\pi$
$431$	−11.0000	−0.529851	−0.264926	+	0.964269i	$0.585347\pi$
$432$	0	0
$433$	17.0000i	0.816968i	0.912766	+	0.408484i	$0.133942\pi$
$433$	17.0000i	0.816968i	−0.912766	+	0.408484i	$0.866058\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 8.00000i	− 0.382692i
$438$	0	0
$439$	−10.0000	−0.477274	−0.238637	−	0.971109i	$-0.576701\pi$
$439$	−10.0000	−0.477274	−0.238637	+	0.971109i	$0.576701\pi$
$440$	0	0
$441$	−36.0000	−1.71429
$442$	0	0
$443$	27.0000i	1.28281i	0.767203	+	0.641404i	$0.221648\pi$
$443$	27.0000i	1.28281i	−0.767203	+	0.641404i	$0.778352\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	14.0000i	0.662177i
$448$	0	0
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	0	0
$451$	−16.0000	−0.753411
$452$	0	0
$453$	5.00000i	0.234920i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	10.0000i	0.467780i	0.972263	+	0.233890i	$0.0751456\pi$
$457$	10.0000i	0.467780i	−0.972263	+	0.233890i	$0.924854\pi$
$458$	0	0
$459$	−15.0000	−0.700140
$460$	0	0
$461$	3.00000	0.139724	0.0698620	−	0.997557i	$-0.477744\pi$
$461$	3.00000	0.139724	0.0698620	+	0.997557i	$0.477744\pi$
$462$	0	0
$463$	− 24.0000i	− 1.11537i	−0.830051	−	0.557687i	$-0.811689\pi$
$463$	− 24.0000i	− 1.11537i	0.830051	−	0.557687i	$-0.188311\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 20.0000i	− 0.925490i	−0.886492	−	0.462745i	$-0.846865\pi$
$467$	− 20.0000i	− 0.925490i	0.886492	−	0.462745i	$-0.153135\pi$
$468$	0	0
$469$	−30.0000	−1.38527
$470$	0	0
$471$	−2.00000	−0.0921551
$472$	0	0
$473$	− 2.00000i	− 0.0919601i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	24.0000i	1.09888i
$478$	0	0
$479$	−9.00000	−0.411220	−0.205610	−	0.978634i	$-0.565918\pi$
$479$	−9.00000	−0.411220	−0.205610	+	0.978634i	$0.565918\pi$
$480$	0	0
$481$	−11.0000	−0.501557
$482$	0	0
$483$	− 20.0000i	− 0.910032i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	8.00000i	0.362515i	0.983436	+	0.181257i	$0.0580167\pi$
$487$	8.00000i	0.362515i	−0.983436	+	0.181257i	$0.941983\pi$
$488$	0	0
$489$	−4.00000	−0.180886
$490$	0	0
$491$	15.0000	0.676941	0.338470	−	0.940977i	$-0.390091\pi$
$491$	15.0000	0.676941	0.338470	+	0.940977i	$0.390091\pi$
$492$	0	0
$493$	− 18.0000i	− 0.810679i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	35.0000i	1.56996i
$498$	0	0
$499$	24.0000	1.07439	0.537194	−	0.843459i	$-0.319484\pi$
$499$	24.0000	1.07439	0.537194	+	0.843459i	$0.319484\pi$
$500$	0	0
$501$	8.00000	0.357414
$502$	0	0
$503$	− 26.0000i	− 1.15928i	−0.814872	−	0.579641i	$-0.803193\pi$
$503$	− 26.0000i	− 1.15928i	0.814872	−	0.579641i	$-0.196807\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.00000i	0.0444116i
$508$	0	0
$509$	−2.00000	−0.0886484	−0.0443242	−	0.999017i	$-0.514113\pi$
$509$	−2.00000	−0.0886484	−0.0443242	+	0.999017i	$0.514113\pi$
$510$	0	0
$511$	−10.0000	−0.442374
$512$	0	0
$513$	− 10.0000i	− 0.441511i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 18.0000i	− 0.791639i
$518$	0	0
$519$	12.0000	0.526742
$520$	0	0
$521$	15.0000	0.657162	0.328581	−	0.944476i	$-0.393430\pi$
$521$	15.0000	0.657162	0.328581	+	0.944476i	$0.393430\pi$
$522$	0	0
$523$	− 20.0000i	− 0.874539i	−0.899331	−	0.437269i	$-0.855946\pi$
$523$	− 20.0000i	− 0.874539i	0.899331	−	0.437269i	$-0.144054\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	12.0000i	0.522728i
$528$	0	0
$529$	7.00000	0.304348
$530$	0	0
$531$	−12.0000	−0.520756
$532$	0	0
$533$	8.00000i	0.346518i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	3.00000i	0.129460i
$538$	0	0
$539$	36.0000	1.55063
$540$	0	0
$541$	9.00000	0.386940	0.193470	−	0.981106i	$-0.438026\pi$
$541$	9.00000	0.386940	0.193470	+	0.981106i	$0.438026\pi$
$542$	0	0
$543$	− 16.0000i	− 0.686626i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 39.0000i	− 1.66752i	−0.552127	−	0.833760i	$-0.686184\pi$
$547$	− 39.0000i	− 1.66752i	0.552127	−	0.833760i	$-0.313816\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	12.0000	0.511217
$552$	0	0
$553$	− 60.0000i	− 2.55146i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 23.0000i	− 0.974541i	−0.873251	−	0.487271i	$-0.837993\pi$
$557$	− 23.0000i	− 0.974541i	0.873251	−	0.487271i	$-0.162007\pi$
$558$	0	0
$559$	−1.00000	−0.0422955
$560$	0	0
$561$	6.00000	0.253320
$562$	0	0
$563$	− 39.0000i	− 1.64365i	−0.569737	−	0.821827i	$-0.692955\pi$
$563$	− 39.0000i	− 1.64365i	0.569737	−	0.821827i	$-0.307045\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	5.00000i	0.209980i
$568$	0	0
$569$	17.0000	0.712677	0.356339	−	0.934357i	$-0.384025\pi$
$569$	17.0000	0.712677	0.356339	+	0.934357i	$0.384025\pi$
$570$	0	0
$571$	37.0000	1.54840	0.774201	−	0.632940i	$-0.218152\pi$
$571$	37.0000	1.54840	0.774201	+	0.632940i	$0.218152\pi$
$572$	0	0
$573$	− 10.0000i	− 0.417756i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 38.0000i	− 1.58196i	−0.611842	−	0.790980i	$-0.709571\pi$
$577$	− 38.0000i	− 1.58196i	0.611842	−	0.790980i	$-0.290429\pi$
$578$	0	0
$579$	−16.0000	−0.664937
$580$	0	0
$581$	−80.0000	−3.31896
$582$	0	0
$583$	− 24.0000i	− 0.993978i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	12.0000i	0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$	12.0000i	0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$	0	0
$589$	−8.00000	−0.329634
$590$	0	0
$591$	17.0000	0.699287
$592$	0	0
$593$	− 2.00000i	− 0.0821302i	−0.999156	−	0.0410651i	$-0.986925\pi$
$593$	− 2.00000i	− 0.0821302i	0.999156	−	0.0410651i	$-0.0130751\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 18.0000i	− 0.736691i
$598$	0	0
$599$	2.00000	0.0817178	0.0408589	−	0.999165i	$-0.486991\pi$
$599$	2.00000	0.0817178	0.0408589	+	0.999165i	$0.486991\pi$
$600$	0	0
$601$	−3.00000	−0.122373	−0.0611863	−	0.998126i	$-0.519488\pi$
$601$	−3.00000	−0.122373	−0.0611863	+	0.998126i	$0.519488\pi$
$602$	0	0
$603$	12.0000i	0.488678i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 10.0000i	− 0.405887i	−0.979190	−	0.202944i	$-0.934949\pi$
$607$	− 10.0000i	− 0.405887i	0.979190	−	0.202944i	$-0.0650509\pi$
$608$	0	0
$609$	30.0000	1.21566
$610$	0	0
$611$	−9.00000	−0.364101
$612$	0	0
$613$	6.00000i	0.242338i	0.992632	+	0.121169i	$0.0386643\pi$
$613$	6.00000i	0.242338i	−0.992632	+	0.121169i	$0.961336\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	32.0000i	1.28827i	0.764911	+	0.644136i	$0.222783\pi$
$617$	32.0000i	1.28827i	−0.764911	+	0.644136i	$0.777217\pi$
$618$	0	0
$619$	−28.0000	−1.12542	−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000	−1.12542	−0.562708	+	0.826656i	$0.690240\pi$
$620$	0	0
$621$	−20.0000	−0.802572
$622$	0	0
$623$	50.0000i	2.00321i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	4.00000i	0.159745i
$628$	0	0
$629$	33.0000	1.31580
$630$	0	0
$631$	15.0000	0.597141	0.298570	−	0.954388i	$-0.403490\pi$
$631$	15.0000	0.597141	0.298570	+	0.954388i	$0.403490\pi$
$632$	0	0
$633$	5.00000i	0.198732i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 18.0000i	− 0.713186i
$638$	0	0
$639$	14.0000	0.553831
$640$	0	0
$641$	−34.0000	−1.34292	−0.671460	−	0.741041i	$-0.734332\pi$
$641$	−34.0000	−1.34292	−0.671460	+	0.741041i	$0.734332\pi$
$642$	0	0
$643$	26.0000i	1.02534i	0.858586	+	0.512670i	$0.171344\pi$
$643$	26.0000i	1.02534i	−0.858586	+	0.512670i	$0.828656\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	42.0000i	1.65119i	0.564263	+	0.825595i	$0.309160\pi$
$647$	42.0000i	1.65119i	−0.564263	+	0.825595i	$0.690840\pi$
$648$	0	0
$649$	12.0000	0.471041
$650$	0	0
$651$	−20.0000	−0.783862
$652$	0	0
$653$	− 24.0000i	− 0.939193i	−0.882881	−	0.469596i	$-0.844399\pi$
$653$	− 24.0000i	− 0.939193i	0.882881	−	0.469596i	$-0.155601\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	4.00000i	0.156055i
$658$	0	0
$659$	−28.0000	−1.09073	−0.545363	−	0.838200i	$-0.683608\pi$
$659$	−28.0000	−1.09073	−0.545363	+	0.838200i	$0.683608\pi$
$660$	0	0
$661$	38.0000	1.47803	0.739014	−	0.673690i	$-0.235292\pi$
$661$	38.0000	1.47803	0.739014	+	0.673690i	$0.235292\pi$
$662$	0	0
$663$	− 3.00000i	− 0.116510i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 24.0000i	− 0.929284i
$668$	0	0
$669$	17.0000	0.657258
$670$	0	0
$671$	0	0
$672$	0	0
$673$	− 21.0000i	− 0.809491i	−0.914429	−	0.404745i	$-0.867360\pi$
$673$	− 21.0000i	− 0.809491i	0.914429	−	0.404745i	$-0.132640\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 36.0000i	− 1.38359i	−0.722093	−	0.691796i	$-0.756820\pi$
$677$	− 36.0000i	− 1.38359i	0.722093	−	0.691796i	$-0.243180\pi$
$678$	0	0
$679$	50.0000	1.91882
$680$	0	0
$681$	−16.0000	−0.613121
$682$	0	0
$683$	12.0000i	0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$	12.0000i	0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 15.0000i	− 0.572286i
$688$	0	0
$689$	−12.0000	−0.457164
$690$	0	0
$691$	40.0000	1.52167	0.760836	−	0.648944i	$-0.224789\pi$
$691$	40.0000	1.52167	0.760836	+	0.648944i	$0.224789\pi$
$692$	0	0
$693$	− 20.0000i	− 0.759737i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 24.0000i	− 0.909065i
$698$	0	0
$699$	−13.0000	−0.491705
$700$	0	0
$701$	−4.00000	−0.151078	−0.0755390	−	0.997143i	$-0.524068\pi$
$701$	−4.00000	−0.151078	−0.0755390	+	0.997143i	$0.524068\pi$
$702$	0	0
$703$	22.0000i	0.829746i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	20.0000i	0.752177i
$708$	0	0
$709$	−22.0000	−0.826227	−0.413114	−	0.910679i	$-0.635559\pi$
$709$	−22.0000	−0.826227	−0.413114	+	0.910679i	$0.635559\pi$
$710$	0	0
$711$	−24.0000	−0.900070
$712$	0	0
$713$	16.0000i	0.599205i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 3.00000i	− 0.112037i
$718$	0	0
$719$	30.0000	1.11881	0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000	1.11881	0.559406	+	0.828894i	$0.311029\pi$
$720$	0	0
$721$	−40.0000	−1.48968
$722$	0	0
$723$	− 2.00000i	− 0.0743808i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 14.0000i	− 0.519231i	−0.965712	−	0.259616i	$-0.916404\pi$
$727$	− 14.0000i	− 0.519231i	0.965712	−	0.259616i	$-0.0835959\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	3.00000	0.110959
$732$	0	0
$733$	− 37.0000i	− 1.36663i	−0.730125	−	0.683313i	$-0.760538\pi$
$733$	− 37.0000i	− 1.36663i	0.730125	−	0.683313i	$-0.239462\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 12.0000i	− 0.442026i
$738$	0	0
$739$	−52.0000	−1.91285	−0.956425	−	0.291977i	$-0.905687\pi$
$739$	−52.0000	−1.91285	−0.956425	+	0.291977i	$0.905687\pi$
$740$	0	0
$741$	2.00000	0.0734718
$742$	0	0
$743$	− 29.0000i	− 1.06391i	−0.846774	−	0.531953i	$-0.821458\pi$
$743$	− 29.0000i	− 1.06391i	0.846774	−	0.531953i	$-0.178542\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	32.0000i	1.17082i
$748$	0	0
$749$	−100.000	−3.65392
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	0	0
$753$	− 8.00000i	− 0.291536i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 52.0000i	− 1.88997i	−0.327111	−	0.944986i	$-0.606075\pi$
$757$	− 52.0000i	− 1.88997i	0.327111	−	0.944986i	$-0.393925\pi$
$758$	0	0
$759$	8.00000	0.290382
$760$	0	0
$761$	30.0000	1.08750	0.543750	−	0.839248i	$-0.317004\pi$
$761$	30.0000	1.08750	0.543750	+	0.839248i	$0.317004\pi$
$762$	0	0
$763$	25.0000i	0.905061i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 6.00000i	− 0.216647i
$768$	0	0
$769$	−16.0000	−0.576975	−0.288487	−	0.957484i	$-0.593152\pi$
$769$	−16.0000	−0.576975	−0.288487	+	0.957484i	$0.593152\pi$
$770$	0	0
$771$	−31.0000	−1.11644
$772$	0	0
$773$	− 27.0000i	− 0.971123i	−0.874203	−	0.485561i	$-0.838615\pi$
$773$	− 27.0000i	− 0.971123i	0.874203	−	0.485561i	$-0.161385\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	55.0000i	1.97311i
$778$	0	0
$779$	16.0000	0.573259
$780$	0	0
$781$	−14.0000	−0.500959
$782$	0	0
$783$	− 30.0000i	− 1.07211i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 24.0000i	− 0.855508i	−0.903895	−	0.427754i	$-0.859305\pi$
$787$	− 24.0000i	− 0.855508i	0.903895	−	0.427754i	$-0.140695\pi$
$788$	0	0
$789$	−20.0000	−0.712019
$790$	0	0
$791$	10.0000	0.355559
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	38.0000i	1.34603i	0.739629	+	0.673015i	$0.235001\pi$
$797$	38.0000i	1.34603i	−0.739629	+	0.673015i	$0.764999\pi$
$798$	0	0
$799$	27.0000	0.955191
$800$	0	0
$801$	20.0000	0.706665
$802$	0	0
$803$	− 4.00000i	− 0.141157i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	16.0000i	0.563227i
$808$	0	0
$809$	−15.0000	−0.527372	−0.263686	−	0.964609i	$-0.584938\pi$
$809$	−15.0000	−0.527372	−0.263686	+	0.964609i	$0.584938\pi$
$810$	0	0
$811$	−28.0000	−0.983213	−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000	−0.983213	−0.491606	+	0.870817i	$0.663590\pi$
$812$	0	0
$813$	7.00000i	0.245501i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	2.00000i	0.0699711i
$818$	0	0
$819$	−10.0000	−0.349428
$820$	0	0
$821$	−41.0000	−1.43091	−0.715455	−	0.698659i	$-0.753781\pi$
$821$	−41.0000	−1.43091	−0.715455	+	0.698659i	$0.753781\pi$
$822$	0	0
$823$	26.0000i	0.906303i	0.891434	+	0.453152i	$0.149700\pi$
$823$	26.0000i	0.906303i	−0.891434	+	0.453152i	$0.850300\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 10.0000i	− 0.347734i	−0.984769	−	0.173867i	$-0.944374\pi$
$827$	− 10.0000i	− 0.347734i	0.984769	−	0.173867i	$-0.0556263\pi$
$828$	0	0
$829$	6.00000	0.208389	0.104194	−	0.994557i	$-0.466774\pi$
$829$	6.00000	0.208389	0.104194	+	0.994557i	$0.466774\pi$
$830$	0	0
$831$	−12.0000	−0.416275
$832$	0	0
$833$	54.0000i	1.87099i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	20.0000i	0.691301i
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	10.0000i	0.344418i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 35.0000i	− 1.20261i
$848$	0	0
$849$	20.0000	0.686398
$850$	0	0
$851$	44.0000	1.50830
$852$	0	0
$853$	47.0000i	1.60925i	0.593784	+	0.804625i	$0.297633\pi$
$853$	47.0000i	1.60925i	−0.593784	+	0.804625i	$0.702367\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	30.0000i	1.02478i	0.858753	+	0.512390i	$0.171240\pi$
$857$	30.0000i	1.02478i	−0.858753	+	0.512390i	$0.828760\pi$
$858$	0	0
$859$	−44.0000	−1.50126	−0.750630	−	0.660722i	$-0.770250\pi$
$859$	−44.0000	−1.50126	−0.750630	+	0.660722i	$0.770250\pi$
$860$	0	0
$861$	40.0000	1.36320
$862$	0	0
$863$	7.00000i	0.238283i	0.992877	+	0.119141i	$0.0380142\pi$
$863$	7.00000i	0.238283i	−0.992877	+	0.119141i	$0.961986\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 8.00000i	− 0.271694i
$868$	0	0
$869$	24.0000	0.814144
$870$	0	0
$871$	−6.00000	−0.203302
$872$	0	0
$873$	− 20.0000i	− 0.676897i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 7.00000i	− 0.236373i	−0.992991	−	0.118187i	$-0.962292\pi$
$877$	− 7.00000i	− 0.236373i	0.992991	−	0.118187i	$-0.0377081\pi$
$878$	0	0
$879$	−15.0000	−0.505937
$880$	0	0
$881$	−27.0000	−0.909653	−0.454827	−	0.890580i	$-0.650299\pi$
$881$	−27.0000	−0.909653	−0.454827	+	0.890580i	$0.650299\pi$
$882$	0	0
$883$	51.0000i	1.71629i	0.513410	+	0.858143i	$0.328382\pi$
$883$	51.0000i	1.71629i	−0.513410	+	0.858143i	$0.671618\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 48.0000i	− 1.61168i	−0.592132	−	0.805841i	$-0.701714\pi$
$887$	− 48.0000i	− 1.61168i	0.592132	−	0.805841i	$-0.298286\pi$
$888$	0	0
$889$	40.0000	1.34156
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	0	0
$893$	18.0000i	0.602347i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 4.00000i	− 0.133556i
$898$	0	0
$899$	−24.0000	−0.800445
$900$	0	0
$901$	36.0000	1.19933
$902$	0	0
$903$	5.00000i	0.166390i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 5.00000i	− 0.166022i	−0.996549	−	0.0830111i	$-0.973546\pi$
$907$	− 5.00000i	− 0.166022i	0.996549	−	0.0830111i	$-0.0264537\pi$
$908$	0	0
$909$	8.00000	0.265343
$910$	0	0
$911$	26.0000	0.861418	0.430709	−	0.902491i	$-0.358263\pi$
$911$	26.0000	0.861418	0.430709	+	0.902491i	$0.358263\pi$
$912$	0	0
$913$	− 32.0000i	− 1.05905i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	15.0000i	0.495344i
$918$	0	0
$919$	24.0000	0.791687	0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000	0.791687	0.395843	+	0.918318i	$0.370452\pi$
$920$	0	0
$921$	−2.00000	−0.0659022
$922$	0	0
$923$	7.00000i	0.230408i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	16.0000i	0.525509i
$928$	0	0
$929$	20.0000	0.656179	0.328089	−	0.944647i	$-0.393595\pi$
$929$	20.0000	0.656179	0.328089	+	0.944647i	$0.393595\pi$
$930$	0	0
$931$	−36.0000	−1.17985
$932$	0	0
$933$	22.0000i	0.720248i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	22.0000i	0.718709i	0.933201	+	0.359354i	$0.117003\pi$
$937$	22.0000i	0.718709i	−0.933201	+	0.359354i	$0.882997\pi$
$938$	0	0
$939$	9.00000	0.293704
$940$	0	0
$941$	−7.00000	−0.228193	−0.114097	−	0.993470i	$-0.536397\pi$
$941$	−7.00000	−0.228193	−0.114097	+	0.993470i	$0.536397\pi$
$942$	0	0
$943$	− 32.0000i	− 1.04206i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 18.0000i	− 0.584921i	−0.956278	−	0.292461i	$-0.905526\pi$
$947$	− 18.0000i	− 0.584921i	0.956278	−	0.292461i	$-0.0944741\pi$
$948$	0	0
$949$	−2.00000	−0.0649227
$950$	0	0
$951$	−2.00000	−0.0648544
$952$	0	0
$953$	9.00000i	0.291539i	0.989319	+	0.145769i	$0.0465657\pi$
$953$	9.00000i	0.291539i	−0.989319	+	0.145769i	$0.953434\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	12.0000i	0.387905i
$958$	0	0
$959$	60.0000	1.93750
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	40.0000i	1.28898i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 13.0000i	− 0.418052i	−0.977910	−	0.209026i	$-0.932971\pi$
$967$	− 13.0000i	− 0.418052i	0.977910	−	0.209026i	$-0.0670293\pi$
$968$	0	0
$969$	−6.00000	−0.192748
$970$	0	0
$971$	45.0000	1.44412	0.722059	−	0.691831i	$-0.243196\pi$
$971$	45.0000	1.44412	0.722059	+	0.691831i	$0.243196\pi$
$972$	0	0
$973$	− 15.0000i	− 0.480878i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 30.0000i	− 0.959785i	−0.877327	−	0.479893i	$-0.840676\pi$
$977$	− 30.0000i	− 0.959785i	0.877327	−	0.479893i	$-0.159324\pi$
$978$	0	0
$979$	−20.0000	−0.639203
$980$	0	0
$981$	10.0000	0.319275
$982$	0	0
$983$	− 5.00000i	− 0.159475i	−0.996816	−	0.0797376i	$-0.974592\pi$
$983$	− 5.00000i	− 0.159475i	0.996816	−	0.0797376i	$-0.0254082\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	45.0000i	1.43237i
$988$	0	0
$989$	4.00000	0.127193
$990$	0	0
$991$	58.0000	1.84243	0.921215	−	0.389053i	$-0.127198\pi$
$991$	58.0000	1.84243	0.921215	+	0.389053i	$0.127198\pi$
$992$	0	0
$993$	− 12.0000i	− 0.380808i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	14.0000i	0.443384i	0.975117	+	0.221692i	$0.0711580\pi$
$997$	14.0000i	0.443384i	−0.975117	+	0.221692i	$0.928842\pi$
$998$	0	0
$999$	55.0000	1.74012

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2600.2.d.f.1249.1		2
5.2	odd	4		2600.2.a.e.1.1		1
5.3	odd	4		104.2.a.a.1.1	✓	1
5.4	even	2	inner	2600.2.d.f.1249.2		2
15.8	even	4		936.2.a.f.1.1		1
20.3	even	4		208.2.a.b.1.1		1
20.7	even	4		5200.2.a.bb.1.1		1
35.13	even	4		5096.2.a.c.1.1		1
40.3	even	4		832.2.a.h.1.1		1
40.13	odd	4		832.2.a.c.1.1		1
60.23	odd	4		1872.2.a.l.1.1		1
65.3	odd	12		1352.2.i.b.529.1		2
65.8	even	4		1352.2.f.b.337.1		2
65.18	even	4		1352.2.f.b.337.2		2
65.23	odd	12		1352.2.i.c.529.1		2
65.28	even	12		1352.2.o.a.1161.2		4
65.33	even	12		1352.2.o.a.361.1		4
65.38	odd	4		1352.2.a.b.1.1		1
65.43	odd	12		1352.2.i.c.1329.1		2
65.48	odd	12		1352.2.i.b.1329.1		2
65.58	even	12		1352.2.o.a.361.2		4
65.63	even	12		1352.2.o.a.1161.1		4
80.3	even	4		3328.2.b.t.1665.1		2
80.13	odd	4		3328.2.b.a.1665.2		2
80.43	even	4		3328.2.b.t.1665.2		2
80.53	odd	4		3328.2.b.a.1665.1		2
120.53	even	4		7488.2.a.x.1.1		1
120.83	odd	4		7488.2.a.u.1.1		1
260.83	odd	4		2704.2.f.e.337.2		2
260.103	even	4		2704.2.a.d.1.1		1
260.203	odd	4		2704.2.f.e.337.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.2.a.a.1.1	✓	1	5.3	odd	4
208.2.a.b.1.1		1	20.3	even	4
832.2.a.c.1.1		1	40.13	odd	4
832.2.a.h.1.1		1	40.3	even	4
936.2.a.f.1.1		1	15.8	even	4
1352.2.a.b.1.1		1	65.38	odd	4
1352.2.f.b.337.1		2	65.8	even	4
1352.2.f.b.337.2		2	65.18	even	4
1352.2.i.b.529.1		2	65.3	odd	12
1352.2.i.b.1329.1		2	65.48	odd	12
1352.2.i.c.529.1		2	65.23	odd	12
1352.2.i.c.1329.1		2	65.43	odd	12
1352.2.o.a.361.1		4	65.33	even	12
1352.2.o.a.361.2		4	65.58	even	12
1352.2.o.a.1161.1		4	65.63	even	12
1352.2.o.a.1161.2		4	65.28	even	12
1872.2.a.l.1.1		1	60.23	odd	4
2600.2.a.e.1.1		1	5.2	odd	4
2600.2.d.f.1249.1		2	1.1	even	1	trivial
2600.2.d.f.1249.2		2	5.4	even	2	inner
2704.2.a.d.1.1		1	260.103	even	4
2704.2.f.e.337.1		2	260.203	odd	4
2704.2.f.e.337.2		2	260.83	odd	4
3328.2.b.a.1665.1		2	80.53	odd	4
3328.2.b.a.1665.2		2	80.13	odd	4
3328.2.b.t.1665.1		2	80.3	even	4
3328.2.b.t.1665.2		2	80.43	even	4
5096.2.a.c.1.1		1	35.13	even	4
5200.2.a.bb.1.1		1	20.7	even	4
7488.2.a.u.1.1		1	120.83	odd	4
7488.2.a.x.1.1		1	120.53	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2600 = 2^{3} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2600.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +5.00000i q^{7} +2.00000 q^{9} -2.00000 q^{11} +1.00000i q^{13} -3.00000i q^{17} +2.00000 q^{19} +5.00000 q^{21} -4.00000i q^{23} -5.00000i q^{27} +6.00000 q^{29} -4.00000 q^{31} +2.00000i q^{33} +11.0000i q^{37} +1.00000 q^{39} +8.00000 q^{41} +1.00000i q^{43} +9.00000i q^{47} -18.0000 q^{49} -3.00000 q^{51} +12.0000i q^{53} -2.00000i q^{57} -6.00000 q^{59} +10.0000i q^{63} +6.00000i q^{67} -4.00000 q^{69} +7.00000 q^{71} +2.00000i q^{73} -10.0000i q^{77} -12.0000 q^{79} +1.00000 q^{81} +16.0000i q^{83} -6.00000i q^{87} +10.0000 q^{89} -5.00000 q^{91} +4.00000i q^{93} -10.0000i q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{9} - 4 q^{11} + 4 q^{19} + 10 q^{21} + 12 q^{29} - 8 q^{31} + 2 q^{39} + 16 q^{41} - 36 q^{49} - 6 q^{51} - 12 q^{59} - 8 q^{69} + 14 q^{71} - 24 q^{79} + 2 q^{81} + 20 q^{89} - 10 q^{91} - 8 q^{99}+O(q^{100}) \) 2 * q + 4 * q^9 - 4 * q^11 + 4 * q^19 + 10 * q^21 + 12 * q^29 - 8 * q^31 + 2 * q^39 + 16 * q^41 - 36 * q^49 - 6 * q^51 - 12 * q^59 - 8 * q^69 + 14 * q^71 - 24 * q^79 + 2 * q^81 + 20 * q^89 - 10 * q^91 - 8 * q^99

Introduction

L-functions

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