LMFDB - Embedded newform 1568.2.b.a.785.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1568,2,Mod(785,1568)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1568, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1568.785"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1568 = 2^{5} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1568.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,2,0,0,0,0,0,-4,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$12.5205430369$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			785.1
Root			$-1.41421i$ of defining polynomial
Character	$\chi$	$=$	1568.785
Dual form			1568.2.b.a.785.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{3} -1.41421i q^{5} +1.00000 q^{9} +2.82843i q^{11} -4.24264i q^{13} -2.00000 q^{15} +6.00000 q^{17} +4.24264i q^{19} +6.00000 q^{23} +3.00000 q^{25} -5.65685i q^{27} -2.82843i q^{29} -4.00000 q^{31} +4.00000 q^{33} -8.48528i q^{37} -6.00000 q^{39} -6.00000 q^{41} -8.48528i q^{43} -1.41421i q^{45} -8.48528i q^{51} +5.65685i q^{53} +4.00000 q^{55} +6.00000 q^{57} +1.41421i q^{59} +12.7279i q^{61} -6.00000 q^{65} -8.48528i q^{69} -2.00000 q^{73} -4.24264i q^{75} -8.00000 q^{79} -5.00000 q^{81} -15.5563i q^{83} -8.48528i q^{85} -4.00000 q^{87} -6.00000 q^{89} +5.65685i q^{93} +6.00000 q^{95} +10.0000 q^{97} +2.82843i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{9} - 4 q^{15} + 12 q^{17} + 12 q^{23} + 6 q^{25} - 8 q^{31} + 8 q^{33} - 12 q^{39} - 12 q^{41} + 8 q^{55} + 12 q^{57} - 12 q^{65} - 4 q^{73} - 16 q^{79} - 10 q^{81} - 8 q^{87} - 12 q^{89} + 12 q^{95}+ \cdots + 20 q^{97}+O(q^{100}) $ 2 * q + 2 * q^9 - 4 * q^15 + 12 * q^17 + 12 * q^23 + 6 * q^25 - 8 * q^31 + 8 * q^33 - 12 * q^39 - 12 * q^41 + 8 * q^55 + 12 * q^57 - 12 * q^65 - 4 * q^73 - 16 * q^79 - 10 * q^81 - 8 * q^87 - 12 * q^89 + 12 * q^95 + 20 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1568\mathbb{Z}\right)^\times$.

$n$	$197$	$1471$	$1473$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	− 1.41421i	− 0.816497i	−0.912871	−	0.408248i	$-0.866140\pi$
$3$	− 1.41421i	− 0.816497i	0.912871	−	0.408248i	$-0.133860\pi$
$4$	0	0
$5$	− 1.41421i	− 0.632456i	−0.948683	−	0.316228i	$-0.897584\pi$
$5$	− 1.41421i	− 0.632456i	0.948683	−	0.316228i	$-0.102416\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.82843i	0.852803i	0.904534	+	0.426401i	$0.140219\pi$
$11$	2.82843i	0.852803i	−0.904534	+	0.426401i	$0.859781\pi$
$12$	0	0
$13$	− 4.24264i	− 1.17670i	−0.808608	−	0.588348i	$-0.799778\pi$
$13$	− 4.24264i	− 1.17670i	0.808608	−	0.588348i	$-0.200222\pi$
$14$	0	0
$15$	−2.00000	−0.516398
$16$	0	0
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	0	0
$19$	4.24264i	0.973329i	0.873589	+	0.486664i	$0.161786\pi$
$19$	4.24264i	0.973329i	−0.873589	+	0.486664i	$0.838214\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	3.00000	0.600000
$26$	0	0
$27$	− 5.65685i	− 1.08866i
$28$	0	0
$29$	− 2.82843i	− 0.525226i	−0.964901	−	0.262613i	$-0.915416\pi$
$29$	− 2.82843i	− 0.525226i	0.964901	−	0.262613i	$-0.0845842\pi$
$30$	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
$33$	4.00000	0.696311
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 8.48528i	− 1.39497i	−0.716599	−	0.697486i	$-0.754302\pi$
$37$	− 8.48528i	− 1.39497i	0.716599	−	0.697486i	$-0.245698\pi$
$38$	0	0
$39$	−6.00000	−0.960769
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	− 8.48528i	− 1.29399i	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	− 8.48528i	− 1.29399i	0.762493	−	0.646997i	$-0.223975\pi$
$44$	0	0
$45$	− 1.41421i	− 0.210819i
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	− 8.48528i	− 1.18818i
$52$	0	0
$53$	5.65685i	0.777029i	0.921443	+	0.388514i	$0.127012\pi$
$53$	5.65685i	0.777029i	−0.921443	+	0.388514i	$0.872988\pi$
$54$	0	0
$55$	4.00000	0.539360
$56$	0	0
$57$	6.00000	0.794719
$58$	0	0
$59$	1.41421i	0.184115i	0.995754	+	0.0920575i	$0.0293443\pi$
$59$	1.41421i	0.184115i	−0.995754	+	0.0920575i	$0.970656\pi$
$60$	0	0
$61$	12.7279i	1.62964i	0.579712	+	0.814822i	$0.303165\pi$
$61$	12.7279i	1.62964i	−0.579712	+	0.814822i	$0.696835\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−6.00000	−0.744208
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	− 8.48528i	− 1.02151i
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	0	0
$75$	− 4.24264i	− 0.489898i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	0	0
$83$	− 15.5563i	− 1.70753i	−0.520658	−	0.853766i	$-0.674313\pi$
$83$	− 15.5563i	− 1.70753i	0.520658	−	0.853766i	$-0.325687\pi$
$84$	0	0
$85$	− 8.48528i	− 0.920358i
$86$	0	0
$87$	−4.00000	−0.428845
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	5.65685i	0.586588i
$94$	0	0
$95$	6.00000	0.615587
$96$	0	0
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	0	0
$99$	2.82843i	0.284268i
$100$	0	0
$101$	− 9.89949i	− 0.985037i	−0.870302	−	0.492518i	$-0.836076\pi$
$101$	− 9.89949i	− 0.985037i	0.870302	−	0.492518i	$-0.163924\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 5.65685i	− 0.546869i	−0.961891	−	0.273434i	$-0.911840\pi$
$107$	− 5.65685i	− 0.546869i	0.961891	−	0.273434i	$-0.0881596\pi$
$108$	0	0
$109$	8.48528i	0.812743i	0.913708	+	0.406371i	$0.133206\pi$
$109$	8.48528i	0.812743i	−0.913708	+	0.406371i	$0.866794\pi$
$110$	0	0
$111$	−12.0000	−1.13899
$112$	0	0
$113$	−12.0000	−1.12887	−0.564433	−	0.825479i	$-0.690905\pi$
$113$	−12.0000	−1.12887	−0.564433	+	0.825479i	$0.690905\pi$
$114$	0	0
$115$	− 8.48528i	− 0.791257i
$116$	0	0
$117$	− 4.24264i	− 0.392232i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	3.00000	0.272727
$122$	0	0
$123$	8.48528i	0.765092i
$124$	0	0
$125$	− 11.3137i	− 1.01193i
$126$	0	0
$127$	−2.00000	−0.177471	−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000	−0.177471	−0.0887357	+	0.996055i	$0.528283\pi$
$128$	0	0
$129$	−12.0000	−1.05654
$130$	0	0
$131$	1.41421i	0.123560i	0.998090	+	0.0617802i	$0.0196778\pi$
$131$	1.41421i	0.123560i	−0.998090	+	0.0617802i	$0.980322\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−8.00000	−0.688530
$136$	0	0
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	4.24264i	0.359856i	0.983680	+	0.179928i	$0.0575865\pi$
$139$	4.24264i	0.359856i	−0.983680	+	0.179928i	$0.942414\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	12.0000	1.00349
$144$	0	0
$145$	−4.00000	−0.332182
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 11.3137i	− 0.926855i	−0.886135	−	0.463428i	$-0.846619\pi$
$149$	− 11.3137i	− 0.926855i	0.886135	−	0.463428i	$-0.153381\pi$
$150$	0	0
$151$	10.0000	0.813788	0.406894	−	0.913475i	$-0.366612\pi$
$151$	10.0000	0.813788	0.406894	+	0.913475i	$0.366612\pi$
$152$	0	0
$153$	6.00000	0.485071
$154$	0	0
$155$	5.65685i	0.454369i
$156$	0	0
$157$	12.7279i	1.01580i	0.861416	+	0.507899i	$0.169578\pi$
$157$	12.7279i	1.01580i	−0.861416	+	0.507899i	$0.830422\pi$
$158$	0	0
$159$	8.00000	0.634441
$160$	0	0
$161$	0	0
$162$	0	0
$163$	8.48528i	0.664619i	0.943170	+	0.332309i	$0.107828\pi$
$163$	8.48528i	0.664619i	−0.943170	+	0.332309i	$0.892172\pi$
$164$	0	0
$165$	− 5.65685i	− 0.440386i
$166$	0	0
$167$	24.0000	1.85718	0.928588	−	0.371113i	$-0.121024\pi$
$167$	24.0000	1.85718	0.928588	+	0.371113i	$0.121024\pi$
$168$	0	0
$169$	−5.00000	−0.384615
$170$	0	0
$171$	4.24264i	0.324443i
$172$	0	0
$173$	− 9.89949i	− 0.752645i	−0.926489	−	0.376322i	$-0.877189\pi$
$173$	− 9.89949i	− 0.752645i	0.926489	−	0.376322i	$-0.122811\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2.00000	0.150329
$178$	0	0
$179$	− 5.65685i	− 0.422813i	−0.977398	−	0.211407i	$-0.932196\pi$
$179$	− 5.65685i	− 0.422813i	0.977398	−	0.211407i	$-0.0678044\pi$
$180$	0	0
$181$	12.7279i	0.946059i	0.881047	+	0.473029i	$0.156840\pi$
$181$	12.7279i	0.946059i	−0.881047	+	0.473029i	$0.843160\pi$
$182$	0	0
$183$	18.0000	1.33060
$184$	0	0
$185$	−12.0000	−0.882258
$186$	0	0
$187$	16.9706i	1.24101i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	0	0
$195$	8.48528i	0.607644i
$196$	0	0
$197$	5.65685i	0.403034i	0.979485	+	0.201517i	$0.0645872\pi$
$197$	5.65685i	0.403034i	−0.979485	+	0.201517i	$0.935413\pi$
$198$	0	0
$199$	20.0000	1.41776	0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000	1.41776	0.708881	+	0.705328i	$0.249200\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	8.48528i	0.592638i
$206$	0	0
$207$	6.00000	0.417029
$208$	0	0
$209$	−12.0000	−0.830057
$210$	0	0
$211$	− 16.9706i	− 1.16830i	−0.811645	−	0.584151i	$-0.801428\pi$
$211$	− 16.9706i	− 1.16830i	0.811645	−	0.584151i	$-0.198572\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−12.0000	−0.818393
$216$	0	0
$217$	0	0
$218$	0	0
$219$	2.82843i	0.191127i
$220$	0	0
$221$	− 25.4558i	− 1.71235i
$222$	0	0
$223$	−28.0000	−1.87502	−0.937509	−	0.347960i	$-0.886874\pi$
$223$	−28.0000	−1.87502	−0.937509	+	0.347960i	$0.886874\pi$
$224$	0	0
$225$	3.00000	0.200000
$226$	0	0
$227$	9.89949i	0.657053i	0.944495	+	0.328526i	$0.106552\pi$
$227$	9.89949i	0.657053i	−0.944495	+	0.328526i	$0.893448\pi$
$228$	0	0
$229$	4.24264i	0.280362i	0.990126	+	0.140181i	$0.0447684\pi$
$229$	4.24264i	0.280362i	−0.990126	+	0.140181i	$0.955232\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	11.3137i	0.734904i
$238$	0	0
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	0	0
$243$	− 9.89949i	− 0.635053i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	18.0000	1.14531
$248$	0	0
$249$	−22.0000	−1.39419
$250$	0	0
$251$	18.3848i	1.16044i	0.814461	+	0.580218i	$0.197033\pi$
$251$	18.3848i	1.16044i	−0.814461	+	0.580218i	$0.802967\pi$
$252$	0	0
$253$	16.9706i	1.06693i
$254$	0	0
$255$	−12.0000	−0.751469
$256$	0	0
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	− 2.82843i	− 0.175075i
$262$	0	0
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	0	0
$265$	8.00000	0.491436
$266$	0	0
$267$	8.48528i	0.519291i
$268$	0	0
$269$	7.07107i	0.431131i	0.976489	+	0.215565i	$0.0691594\pi$
$269$	7.07107i	0.431131i	−0.976489	+	0.215565i	$0.930841\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	8.48528i	0.511682i
$276$	0	0
$277$	16.9706i	1.01966i	0.860274	+	0.509831i	$0.170292\pi$
$277$	16.9706i	1.01966i	−0.860274	+	0.509831i	$0.829708\pi$
$278$	0	0
$279$	−4.00000	−0.239474
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	− 12.7279i	− 0.756596i	−0.925684	−	0.378298i	$-0.876509\pi$
$283$	− 12.7279i	− 0.756596i	0.925684	−	0.378298i	$-0.123491\pi$
$284$	0	0
$285$	− 8.48528i	− 0.502625i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	− 14.1421i	− 0.829027i
$292$	0	0
$293$	24.0416i	1.40453i	0.711917	+	0.702264i	$0.247827\pi$
$293$	24.0416i	1.40453i	−0.711917	+	0.702264i	$0.752173\pi$
$294$	0	0
$295$	2.00000	0.116445
$296$	0	0
$297$	16.0000	0.928414
$298$	0	0
$299$	− 25.4558i	− 1.47215i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−14.0000	−0.804279
$304$	0	0
$305$	18.0000	1.03068
$306$	0	0
$307$	12.7279i	0.726421i	0.931707	+	0.363210i	$0.118319\pi$
$307$	12.7279i	0.726421i	−0.931707	+	0.363210i	$0.881681\pi$
$308$	0	0
$309$	5.65685i	0.321807i
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	22.6274i	1.27088i	0.772149	+	0.635441i	$0.219182\pi$
$317$	22.6274i	1.27088i	−0.772149	+	0.635441i	$0.780818\pi$
$318$	0	0
$319$	8.00000	0.447914
$320$	0	0
$321$	−8.00000	−0.446516
$322$	0	0
$323$	25.4558i	1.41640i
$324$	0	0
$325$	− 12.7279i	− 0.706018i
$326$	0	0
$327$	12.0000	0.663602
$328$	0	0
$329$	0	0
$330$	0	0
$331$	25.4558i	1.39918i	0.714545	+	0.699590i	$0.246634\pi$
$331$	25.4558i	1.39918i	−0.714545	+	0.699590i	$0.753366\pi$
$332$	0	0
$333$	− 8.48528i	− 0.464991i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	32.0000	1.74315	0.871576	−	0.490261i	$-0.163099\pi$
$337$	32.0000	1.74315	0.871576	+	0.490261i	$0.163099\pi$
$338$	0	0
$339$	16.9706i	0.921714i
$340$	0	0
$341$	− 11.3137i	− 0.612672i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−12.0000	−0.646058
$346$	0	0
$347$	− 14.1421i	− 0.759190i	−0.925153	−	0.379595i	$-0.876063\pi$
$347$	− 14.1421i	− 0.759190i	0.925153	−	0.379595i	$-0.123937\pi$
$348$	0	0
$349$	4.24264i	0.227103i	0.993532	+	0.113552i	$0.0362227\pi$
$349$	4.24264i	0.227103i	−0.993532	+	0.113552i	$0.963777\pi$
$350$	0	0
$351$	−24.0000	−1.28103
$352$	0	0
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	30.0000	1.58334	0.791670	−	0.610949i	$-0.209212\pi$
$359$	30.0000	1.58334	0.791670	+	0.610949i	$0.209212\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	− 4.24264i	− 0.222681i
$364$	0	0
$365$	2.82843i	0.148047i
$366$	0	0
$367$	−28.0000	−1.46159	−0.730794	−	0.682598i	$-0.760850\pi$
$367$	−28.0000	−1.46159	−0.730794	+	0.682598i	$0.760850\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	0	0
$371$	0	0
$372$	0	0
$373$	33.9411i	1.75740i	0.477370	+	0.878702i	$0.341590\pi$
$373$	33.9411i	1.75740i	−0.477370	+	0.878702i	$0.658410\pi$
$374$	0	0
$375$	−16.0000	−0.826236
$376$	0	0
$377$	−12.0000	−0.618031
$378$	0	0
$379$	25.4558i	1.30758i	0.756677	+	0.653789i	$0.226822\pi$
$379$	25.4558i	1.30758i	−0.756677	+	0.653789i	$0.773178\pi$
$380$	0	0
$381$	2.82843i	0.144905i
$382$	0	0
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 8.48528i	− 0.431331i
$388$	0	0
$389$	− 2.82843i	− 0.143407i	−0.997426	−	0.0717035i	$-0.977156\pi$
$389$	− 2.82843i	− 0.143407i	0.997426	−	0.0717035i	$-0.0228435\pi$
$390$	0	0
$391$	36.0000	1.82060
$392$	0	0
$393$	2.00000	0.100887
$394$	0	0
$395$	11.3137i	0.569254i
$396$	0	0
$397$	21.2132i	1.06466i	0.846537	+	0.532330i	$0.178683\pi$
$397$	21.2132i	1.06466i	−0.846537	+	0.532330i	$0.821317\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−24.0000	−1.19850	−0.599251	−	0.800561i	$-0.704535\pi$
$401$	−24.0000	−1.19850	−0.599251	+	0.800561i	$0.704535\pi$
$402$	0	0
$403$	16.9706i	0.845364i
$404$	0	0
$405$	7.07107i	0.351364i
$406$	0	0
$407$	24.0000	1.18964
$408$	0	0
$409$	22.0000	1.08783	0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000	1.08783	0.543915	+	0.839140i	$0.316941\pi$
$410$	0	0
$411$	− 8.48528i	− 0.418548i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−22.0000	−1.07994
$416$	0	0
$417$	6.00000	0.293821
$418$	0	0
$419$	26.8701i	1.31269i	0.754462	+	0.656344i	$0.227898\pi$
$419$	26.8701i	1.31269i	−0.754462	+	0.656344i	$0.772102\pi$
$420$	0	0
$421$	− 16.9706i	− 0.827095i	−0.910483	−	0.413547i	$-0.864290\pi$
$421$	− 16.9706i	− 0.827095i	0.910483	−	0.413547i	$-0.135710\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	18.0000	0.873128
$426$	0	0
$427$	0	0
$428$	0	0
$429$	− 16.9706i	− 0.819346i
$430$	0	0
$431$	6.00000	0.289010	0.144505	−	0.989504i	$-0.453841\pi$
$431$	6.00000	0.289010	0.144505	+	0.989504i	$0.453841\pi$
$432$	0	0
$433$	−2.00000	−0.0961139	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000	−0.0961139	−0.0480569	+	0.998845i	$0.515303\pi$
$434$	0	0
$435$	5.65685i	0.271225i
$436$	0	0
$437$	25.4558i	1.21772i
$438$	0	0
$439$	−28.0000	−1.33637	−0.668184	−	0.743996i	$-0.732928\pi$
$439$	−28.0000	−1.33637	−0.668184	+	0.743996i	$0.732928\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 22.6274i	− 1.07506i	−0.843244	−	0.537531i	$-0.819357\pi$
$443$	− 22.6274i	− 1.07506i	0.843244	−	0.537531i	$-0.180643\pi$
$444$	0	0
$445$	8.48528i	0.402241i
$446$	0	0
$447$	−16.0000	−0.756774
$448$	0	0
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	0	0
$451$	− 16.9706i	− 0.799113i
$452$	0	0
$453$	− 14.1421i	− 0.664455i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−28.0000	−1.30978	−0.654892	−	0.755722i	$-0.727286\pi$
$457$	−28.0000	−1.30978	−0.654892	+	0.755722i	$0.727286\pi$
$458$	0	0
$459$	− 33.9411i	− 1.58424i
$460$	0	0
$461$	15.5563i	0.724531i	0.932075	+	0.362266i	$0.117997\pi$
$461$	15.5563i	0.724531i	−0.932075	+	0.362266i	$0.882003\pi$
$462$	0	0
$463$	−32.0000	−1.48717	−0.743583	−	0.668644i	$-0.766875\pi$
$463$	−32.0000	−1.48717	−0.743583	+	0.668644i	$0.766875\pi$
$464$	0	0
$465$	8.00000	0.370991
$466$	0	0
$467$	− 7.07107i	− 0.327210i	−0.986526	−	0.163605i	$-0.947688\pi$
$467$	− 7.07107i	− 0.327210i	0.986526	−	0.163605i	$-0.0523123\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	18.0000	0.829396
$472$	0	0
$473$	24.0000	1.10352
$474$	0	0
$475$	12.7279i	0.583997i
$476$	0	0
$477$	5.65685i	0.259010i
$478$	0	0
$479$	−12.0000	−0.548294	−0.274147	−	0.961688i	$-0.588395\pi$
$479$	−12.0000	−0.548294	−0.274147	+	0.961688i	$0.588395\pi$
$480$	0	0
$481$	−36.0000	−1.64146
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 14.1421i	− 0.642161i
$486$	0	0
$487$	−2.00000	−0.0906287	−0.0453143	−	0.998973i	$-0.514429\pi$
$487$	−2.00000	−0.0906287	−0.0453143	+	0.998973i	$0.514429\pi$
$488$	0	0
$489$	12.0000	0.542659
$490$	0	0
$491$	− 39.5980i	− 1.78703i	−0.449032	−	0.893516i	$-0.648231\pi$
$491$	− 39.5980i	− 1.78703i	0.449032	−	0.893516i	$-0.351769\pi$
$492$	0	0
$493$	− 16.9706i	− 0.764316i
$494$	0	0
$495$	4.00000	0.179787
$496$	0	0
$497$	0	0
$498$	0	0
$499$	16.9706i	0.759707i	0.925047	+	0.379853i	$0.124026\pi$
$499$	16.9706i	0.759707i	−0.925047	+	0.379853i	$0.875974\pi$
$500$	0	0
$501$	− 33.9411i	− 1.51638i
$502$	0	0
$503$	36.0000	1.60516	0.802580	−	0.596544i	$-0.203460\pi$
$503$	36.0000	1.60516	0.802580	+	0.596544i	$0.203460\pi$
$504$	0	0
$505$	−14.0000	−0.622992
$506$	0	0
$507$	7.07107i	0.314037i
$508$	0	0
$509$	− 1.41421i	− 0.0626839i	−0.999509	−	0.0313420i	$-0.990022\pi$
$509$	− 1.41421i	− 0.0626839i	0.999509	−	0.0313420i	$-0.00997809\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	24.0000	1.05963
$514$	0	0
$515$	5.65685i	0.249271i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−14.0000	−0.614532
$520$	0	0
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	0	0
$523$	29.6985i	1.29862i	0.760522	+	0.649312i	$0.224943\pi$
$523$	29.6985i	1.29862i	−0.760522	+	0.649312i	$0.775057\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−24.0000	−1.04546
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	1.41421i	0.0613716i
$532$	0	0
$533$	25.4558i	1.10262i
$534$	0	0
$535$	−8.00000	−0.345870
$536$	0	0
$537$	−8.00000	−0.345225
$538$	0	0
$539$	0	0
$540$	0	0
$541$	16.9706i	0.729621i	0.931082	+	0.364811i	$0.118866\pi$
$541$	16.9706i	0.729621i	−0.931082	+	0.364811i	$0.881134\pi$
$542$	0	0
$543$	18.0000	0.772454
$544$	0	0
$545$	12.0000	0.514024
$546$	0	0
$547$	− 8.48528i	− 0.362804i	−0.983409	−	0.181402i	$-0.941936\pi$
$547$	− 8.48528i	− 0.362804i	0.983409	−	0.181402i	$-0.0580636\pi$
$548$	0	0
$549$	12.7279i	0.543214i
$550$	0	0
$551$	12.0000	0.511217
$552$	0	0
$553$	0	0
$554$	0	0
$555$	16.9706i	0.720360i
$556$	0	0
$557$	5.65685i	0.239689i	0.992793	+	0.119844i	$0.0382395\pi$
$557$	5.65685i	0.239689i	−0.992793	+	0.119844i	$0.961760\pi$
$558$	0	0
$559$	−36.0000	−1.52264
$560$	0	0
$561$	24.0000	1.01328
$562$	0	0
$563$	1.41421i	0.0596020i	0.999556	+	0.0298010i	$0.00948736\pi$
$563$	1.41421i	0.0596020i	−0.999556	+	0.0298010i	$0.990513\pi$
$564$	0	0
$565$	16.9706i	0.713957i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	25.4558i	1.06529i	0.846338	+	0.532647i	$0.178803\pi$
$571$	25.4558i	1.06529i	−0.846338	+	0.532647i	$0.821197\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	18.0000	0.750652
$576$	0	0
$577$	−38.0000	−1.58196	−0.790980	−	0.611842i	$-0.790429\pi$
$577$	−38.0000	−1.58196	−0.790980	+	0.611842i	$0.790429\pi$
$578$	0	0
$579$	5.65685i	0.235091i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−16.0000	−0.662652
$584$	0	0
$585$	−6.00000	−0.248069
$586$	0	0
$587$	− 41.0122i	− 1.69275i	−0.532584	−	0.846377i	$-0.678779\pi$
$587$	− 41.0122i	− 1.69275i	0.532584	−	0.846377i	$-0.321221\pi$
$588$	0	0
$589$	− 16.9706i	− 0.699260i
$590$	0	0
$591$	8.00000	0.329076
$592$	0	0
$593$	30.0000	1.23195	0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000	1.23195	0.615976	+	0.787765i	$0.288762\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 28.2843i	− 1.15760i
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 4.24264i	− 0.172488i
$606$	0	0
$607$	32.0000	1.29884	0.649420	−	0.760430i	$-0.275012\pi$
$607$	32.0000	1.29884	0.649420	+	0.760430i	$0.275012\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	8.48528i	0.342717i	0.985209	+	0.171359i	$0.0548157\pi$
$613$	8.48528i	0.342717i	−0.985209	+	0.171359i	$0.945184\pi$
$614$	0	0
$615$	12.0000	0.483887
$616$	0	0
$617$	24.0000	0.966204	0.483102	−	0.875564i	$-0.339510\pi$
$617$	24.0000	0.966204	0.483102	+	0.875564i	$0.339510\pi$
$618$	0	0
$619$	4.24264i	0.170526i	0.996358	+	0.0852631i	$0.0271731\pi$
$619$	4.24264i	0.170526i	−0.996358	+	0.0852631i	$0.972827\pi$
$620$	0	0
$621$	− 33.9411i	− 1.36201i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	0	0
$627$	16.9706i	0.677739i
$628$	0	0
$629$	− 50.9117i	− 2.02998i
$630$	0	0
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	0	0
$633$	−24.0000	−0.953914
$634$	0	0
$635$	2.82843i	0.112243i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−48.0000	−1.89589	−0.947943	−	0.318440i	$-0.896841\pi$
$641$	−48.0000	−1.89589	−0.947943	+	0.318440i	$0.896841\pi$
$642$	0	0
$643$	− 21.2132i	− 0.836567i	−0.908317	−	0.418284i	$-0.862632\pi$
$643$	− 21.2132i	− 0.836567i	0.908317	−	0.418284i	$-0.137368\pi$
$644$	0	0
$645$	16.9706i	0.668215i
$646$	0	0
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	0	0
$649$	−4.00000	−0.157014
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 36.7696i	− 1.43890i	−0.694542	−	0.719452i	$-0.744393\pi$
$653$	− 36.7696i	− 1.43890i	0.694542	−	0.719452i	$-0.255607\pi$
$654$	0	0
$655$	2.00000	0.0781465
$656$	0	0
$657$	−2.00000	−0.0780274
$658$	0	0
$659$	2.82843i	0.110180i	0.998481	+	0.0550899i	$0.0175446\pi$
$659$	2.82843i	0.110180i	−0.998481	+	0.0550899i	$0.982455\pi$
$660$	0	0
$661$	− 38.1838i	− 1.48518i	−0.669748	−	0.742588i	$-0.733598\pi$
$661$	− 38.1838i	− 1.48518i	0.669748	−	0.742588i	$-0.266402\pi$
$662$	0	0
$663$	−36.0000	−1.39812
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 16.9706i	− 0.657103i
$668$	0	0
$669$	39.5980i	1.53095i
$670$	0	0
$671$	−36.0000	−1.38976
$672$	0	0
$673$	−46.0000	−1.77317	−0.886585	−	0.462566i	$-0.846929\pi$
$673$	−46.0000	−1.77317	−0.886585	+	0.462566i	$0.846929\pi$
$674$	0	0
$675$	− 16.9706i	− 0.653197i
$676$	0	0
$677$	− 9.89949i	− 0.380468i	−0.981739	−	0.190234i	$-0.939075\pi$
$677$	− 9.89949i	− 0.380468i	0.981739	−	0.190234i	$-0.0609248\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	14.0000	0.536481
$682$	0	0
$683$	− 5.65685i	− 0.216454i	−0.994126	−	0.108227i	$-0.965483\pi$
$683$	− 5.65685i	− 0.216454i	0.994126	−	0.108227i	$-0.0345173\pi$
$684$	0	0
$685$	− 8.48528i	− 0.324206i
$686$	0	0
$687$	6.00000	0.228914
$688$	0	0
$689$	24.0000	0.914327
$690$	0	0
$691$	12.7279i	0.484193i	0.970252	+	0.242096i	$0.0778351\pi$
$691$	12.7279i	0.484193i	−0.970252	+	0.242096i	$0.922165\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	6.00000	0.227593
$696$	0	0
$697$	−36.0000	−1.36360
$698$	0	0
$699$	− 8.48528i	− 0.320943i
$700$	0	0
$701$	− 19.7990i	− 0.747798i	−0.927470	−	0.373899i	$-0.878021\pi$
$701$	− 19.7990i	− 0.747798i	0.927470	−	0.373899i	$-0.121979\pi$
$702$	0	0
$703$	36.0000	1.35777
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	25.4558i	0.956014i	0.878356	+	0.478007i	$0.158641\pi$
$709$	25.4558i	0.956014i	−0.878356	+	0.478007i	$0.841359\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	0	0
$713$	−24.0000	−0.898807
$714$	0	0
$715$	− 16.9706i	− 0.634663i
$716$	0	0
$717$	8.48528i	0.316889i
$718$	0	0
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 14.1421i	− 0.525952i
$724$	0	0
$725$	− 8.48528i	− 0.315135i
$726$	0	0
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	0	0
$729$	−29.0000	−1.07407
$730$	0	0
$731$	− 50.9117i	− 1.88304i
$732$	0	0
$733$	29.6985i	1.09694i	0.836171	+	0.548469i	$0.184789\pi$
$733$	29.6985i	1.09694i	−0.836171	+	0.548469i	$0.815211\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	42.4264i	1.56068i	0.625355	+	0.780340i	$0.284954\pi$
$739$	42.4264i	1.56068i	−0.625355	+	0.780340i	$0.715046\pi$
$740$	0	0
$741$	− 25.4558i	− 0.935144i
$742$	0	0
$743$	30.0000	1.10059	0.550297	−	0.834969i	$-0.314515\pi$
$743$	30.0000	1.10059	0.550297	+	0.834969i	$0.314515\pi$
$744$	0	0
$745$	−16.0000	−0.586195
$746$	0	0
$747$	− 15.5563i	− 0.569177i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	22.0000	0.802791	0.401396	−	0.915905i	$-0.368525\pi$
$751$	22.0000	0.802791	0.401396	+	0.915905i	$0.368525\pi$
$752$	0	0
$753$	26.0000	0.947493
$754$	0	0
$755$	− 14.1421i	− 0.514685i
$756$	0	0
$757$	− 25.4558i	− 0.925208i	−0.886565	−	0.462604i	$-0.846915\pi$
$757$	− 25.4558i	− 0.925208i	0.886565	−	0.462604i	$-0.153085\pi$
$758$	0	0
$759$	24.0000	0.871145
$760$	0	0
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	− 8.48528i	− 0.306786i
$766$	0	0
$767$	6.00000	0.216647
$768$	0	0
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	0	0
$771$	− 8.48528i	− 0.305590i
$772$	0	0
$773$	32.5269i	1.16991i	0.811065	+	0.584956i	$0.198888\pi$
$773$	32.5269i	1.16991i	−0.811065	+	0.584956i	$0.801112\pi$
$774$	0	0
$775$	−12.0000	−0.431053
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 25.4558i	− 0.912050i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−16.0000	−0.571793
$784$	0	0
$785$	18.0000	0.642448
$786$	0	0
$787$	38.1838i	1.36110i	0.732700	+	0.680552i	$0.238260\pi$
$787$	38.1838i	1.36110i	−0.732700	+	0.680552i	$0.761740\pi$
$788$	0	0
$789$	33.9411i	1.20834i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	54.0000	1.91760
$794$	0	0
$795$	− 11.3137i	− 0.401256i
$796$	0	0
$797$	− 26.8701i	− 0.951786i	−0.879503	−	0.475893i	$-0.842125\pi$
$797$	− 26.8701i	− 0.951786i	0.879503	−	0.475893i	$-0.157875\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−6.00000	−0.212000
$802$	0	0
$803$	− 5.65685i	− 0.199626i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	10.0000	0.352017
$808$	0	0
$809$	12.0000	0.421898	0.210949	−	0.977497i	$-0.432345\pi$
$809$	12.0000	0.421898	0.210949	+	0.977497i	$0.432345\pi$
$810$	0	0
$811$	− 12.7279i	− 0.446938i	−0.974711	−	0.223469i	$-0.928262\pi$
$811$	− 12.7279i	− 0.446938i	0.974711	−	0.223469i	$-0.0717381\pi$
$812$	0	0
$813$	− 28.2843i	− 0.991973i
$814$	0	0
$815$	12.0000	0.420342
$816$	0	0
$817$	36.0000	1.25948
$818$	0	0
$819$	0	0
$820$	0	0
$821$	22.6274i	0.789702i	0.918745	+	0.394851i	$0.129204\pi$
$821$	22.6274i	0.789702i	−0.918745	+	0.394851i	$0.870796\pi$
$822$	0	0
$823$	−32.0000	−1.11545	−0.557725	−	0.830026i	$-0.688326\pi$
$823$	−32.0000	−1.11545	−0.557725	+	0.830026i	$0.688326\pi$
$824$	0	0
$825$	12.0000	0.417786
$826$	0	0
$827$	45.2548i	1.57366i	0.617167	+	0.786832i	$0.288280\pi$
$827$	45.2548i	1.57366i	−0.617167	+	0.786832i	$0.711720\pi$
$828$	0	0
$829$	− 21.2132i	− 0.736765i	−0.929674	−	0.368383i	$-0.879912\pi$
$829$	− 21.2132i	− 0.736765i	0.929674	−	0.368383i	$-0.120088\pi$
$830$	0	0
$831$	24.0000	0.832551
$832$	0	0
$833$	0	0
$834$	0	0
$835$	− 33.9411i	− 1.17458i
$836$	0	0
$837$	22.6274i	0.782118i
$838$	0	0
$839$	12.0000	0.414286	0.207143	−	0.978311i	$-0.433583\pi$
$839$	12.0000	0.414286	0.207143	+	0.978311i	$0.433583\pi$
$840$	0	0
$841$	21.0000	0.724138
$842$	0	0
$843$	8.48528i	0.292249i
$844$	0	0
$845$	7.07107i	0.243252i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−18.0000	−0.617758
$850$	0	0
$851$	− 50.9117i	− 1.74523i
$852$	0	0
$853$	4.24264i	0.145265i	0.997359	+	0.0726326i	$0.0231401\pi$
$853$	4.24264i	0.145265i	−0.997359	+	0.0726326i	$0.976860\pi$
$854$	0	0
$855$	6.00000	0.205196
$856$	0	0
$857$	−42.0000	−1.43469	−0.717346	−	0.696717i	$-0.754643\pi$
$857$	−42.0000	−1.43469	−0.717346	+	0.696717i	$0.754643\pi$
$858$	0	0
$859$	46.6690i	1.59233i	0.605081	+	0.796164i	$0.293141\pi$
$859$	46.6690i	1.59233i	−0.605081	+	0.796164i	$0.706859\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	0	0
$865$	−14.0000	−0.476014
$866$	0	0
$867$	− 26.8701i	− 0.912555i
$868$	0	0
$869$	− 22.6274i	− 0.767583i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	10.0000	0.338449
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 42.4264i	− 1.43264i	−0.697773	−	0.716319i	$-0.745826\pi$
$877$	− 42.4264i	− 1.43264i	0.697773	−	0.716319i	$-0.254174\pi$
$878$	0	0
$879$	34.0000	1.14679
$880$	0	0
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	0	0
$883$	− 50.9117i	− 1.71331i	−0.515886	−	0.856657i	$-0.672537\pi$
$883$	− 50.9117i	− 1.71331i	0.515886	−	0.856657i	$-0.327463\pi$
$884$	0	0
$885$	− 2.82843i	− 0.0950765i
$886$	0	0
$887$	48.0000	1.61168	0.805841	−	0.592132i	$-0.201714\pi$
$887$	48.0000	1.61168	0.805841	+	0.592132i	$0.201714\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	− 14.1421i	− 0.473779i
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−8.00000	−0.267411
$896$	0	0
$897$	−36.0000	−1.20201
$898$	0	0
$899$	11.3137i	0.377333i
$900$	0	0
$901$	33.9411i	1.13074i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	18.0000	0.598340
$906$	0	0
$907$	33.9411i	1.12700i	0.826117	+	0.563498i	$0.190545\pi$
$907$	33.9411i	1.12700i	−0.826117	+	0.563498i	$0.809455\pi$
$908$	0	0
$909$	− 9.89949i	− 0.328346i
$910$	0	0
$911$	−30.0000	−0.993944	−0.496972	−	0.867766i	$-0.665555\pi$
$911$	−30.0000	−0.993944	−0.496972	+	0.867766i	$0.665555\pi$
$912$	0	0
$913$	44.0000	1.45619
$914$	0	0
$915$	− 25.4558i	− 0.841544i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	0	0
$921$	18.0000	0.593120
$922$	0	0
$923$	0	0
$924$	0	0
$925$	− 25.4558i	− 0.836983i
$926$	0	0
$927$	−4.00000	−0.131377
$928$	0	0
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	24.0000	0.784884
$936$	0	0
$937$	−2.00000	−0.0653372	−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000	−0.0653372	−0.0326686	+	0.999466i	$0.510401\pi$
$938$	0	0
$939$	− 14.1421i	− 0.461511i
$940$	0	0
$941$	− 52.3259i	− 1.70578i	−0.522094	−	0.852888i	$-0.674849\pi$
$941$	− 52.3259i	− 1.70578i	0.522094	−	0.852888i	$-0.325151\pi$
$942$	0	0
$943$	−36.0000	−1.17232
$944$	0	0
$945$	0	0
$946$	0	0
$947$	2.82843i	0.0919115i	0.998943	+	0.0459558i	$0.0146333\pi$
$947$	2.82843i	0.0919115i	−0.998943	+	0.0459558i	$0.985367\pi$
$948$	0	0
$949$	8.48528i	0.275444i
$950$	0	0
$951$	32.0000	1.03767
$952$	0	0
$953$	18.0000	0.583077	0.291539	−	0.956559i	$-0.405833\pi$
$953$	18.0000	0.583077	0.291539	+	0.956559i	$0.405833\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 11.3137i	− 0.365720i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	− 5.65685i	− 0.182290i
$964$	0	0
$965$	5.65685i	0.182101i
$966$	0	0
$967$	22.0000	0.707472	0.353736	−	0.935345i	$-0.384911\pi$
$967$	22.0000	0.707472	0.353736	+	0.935345i	$0.384911\pi$
$968$	0	0
$969$	36.0000	1.15649
$970$	0	0
$971$	− 32.5269i	− 1.04384i	−0.852995	−	0.521919i	$-0.825216\pi$
$971$	− 32.5269i	− 1.04384i	0.852995	−	0.521919i	$-0.174784\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−18.0000	−0.576461
$976$	0	0
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	0	0
$979$	− 16.9706i	− 0.542382i
$980$	0	0
$981$	8.48528i	0.270914i
$982$	0	0
$983$	−12.0000	−0.382741	−0.191370	−	0.981518i	$-0.561293\pi$
$983$	−12.0000	−0.382741	−0.191370	+	0.981518i	$0.561293\pi$
$984$	0	0
$985$	8.00000	0.254901
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 50.9117i	− 1.61890i
$990$	0	0
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	0	0
$993$	36.0000	1.14243
$994$	0	0
$995$	− 28.2843i	− 0.896672i
$996$	0	0
$997$	21.2132i	0.671829i	0.941893	+	0.335914i	$0.109045\pi$
$997$	21.2132i	0.671829i	−0.941893	+	0.335914i	$0.890955\pi$
$998$	0	0
$999$	−48.0000	−1.51865

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1568.2.b.a.785.1		2
4.3	odd	2		392.2.b.b.197.1		2
7.2	even	3		1568.2.t.b.753.2		4
7.3	odd	6		1568.2.t.c.177.2		4
7.4	even	3		1568.2.t.b.177.1		4
7.5	odd	6		1568.2.t.c.753.1		4
7.6	odd	2		224.2.b.a.113.2		2
8.3	odd	2		392.2.b.b.197.2		2
8.5	even	2	inner	1568.2.b.a.785.2		2
21.20	even	2		2016.2.c.a.1009.1		2
28.3	even	6		392.2.p.a.373.2		4
28.11	odd	6		392.2.p.b.373.2		4
28.19	even	6		392.2.p.a.165.1		4
28.23	odd	6		392.2.p.b.165.1		4
28.27	even	2		56.2.b.a.29.1	✓	2
56.3	even	6		392.2.p.a.373.1		4
56.5	odd	6		1568.2.t.c.753.2		4
56.11	odd	6		392.2.p.b.373.1		4
56.13	odd	2		224.2.b.a.113.1		2
56.19	even	6		392.2.p.a.165.2		4
56.27	even	2		56.2.b.a.29.2	yes	2
56.37	even	6		1568.2.t.b.753.1		4
56.45	odd	6		1568.2.t.c.177.1		4
56.51	odd	6		392.2.p.b.165.2		4
56.53	even	6		1568.2.t.b.177.2		4
84.83	odd	2		504.2.c.a.253.2		2
112.13	odd	4		1792.2.a.p.1.1		2
112.27	even	4		1792.2.a.n.1.1		2
112.69	odd	4		1792.2.a.p.1.2		2
112.83	even	4		1792.2.a.n.1.2		2
168.83	odd	2		504.2.c.a.253.1		2
168.125	even	2		2016.2.c.a.1009.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.2.b.a.29.1	✓	2	28.27	even	2
56.2.b.a.29.2	yes	2	56.27	even	2
224.2.b.a.113.1		2	56.13	odd	2
224.2.b.a.113.2		2	7.6	odd	2
392.2.b.b.197.1		2	4.3	odd	2
392.2.b.b.197.2		2	8.3	odd	2
392.2.p.a.165.1		4	28.19	even	6
392.2.p.a.165.2		4	56.19	even	6
392.2.p.a.373.1		4	56.3	even	6
392.2.p.a.373.2		4	28.3	even	6
392.2.p.b.165.1		4	28.23	odd	6
392.2.p.b.165.2		4	56.51	odd	6
392.2.p.b.373.1		4	56.11	odd	6
392.2.p.b.373.2		4	28.11	odd	6
504.2.c.a.253.1		2	168.83	odd	2
504.2.c.a.253.2		2	84.83	odd	2
1568.2.b.a.785.1		2	1.1	even	1	trivial
1568.2.b.a.785.2		2	8.5	even	2	inner
1568.2.t.b.177.1		4	7.4	even	3
1568.2.t.b.177.2		4	56.53	even	6
1568.2.t.b.753.1		4	56.37	even	6
1568.2.t.b.753.2		4	7.2	even	3
1568.2.t.c.177.1		4	56.45	odd	6
1568.2.t.c.177.2		4	7.3	odd	6
1568.2.t.c.753.1		4	7.5	odd	6
1568.2.t.c.753.2		4	56.5	odd	6
1792.2.a.n.1.1		2	112.27	even	4
1792.2.a.n.1.2		2	112.83	even	4
1792.2.a.p.1.1		2	112.13	odd	4
1792.2.a.p.1.2		2	112.69	odd	4
2016.2.c.a.1009.1		2	21.20	even	2
2016.2.c.a.1009.2		2	168.125	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1568 = 2^{5} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1568.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{3} -1.41421i q^{5} +1.00000 q^{9} +2.82843i q^{11} -4.24264i q^{13} -2.00000 q^{15} +6.00000 q^{17} +4.24264i q^{19} +6.00000 q^{23} +3.00000 q^{25} -5.65685i q^{27} -2.82843i q^{29} -4.00000 q^{31} +4.00000 q^{33} -8.48528i q^{37} -6.00000 q^{39} -6.00000 q^{41} -8.48528i q^{43} -1.41421i q^{45} -8.48528i q^{51} +5.65685i q^{53} +4.00000 q^{55} +6.00000 q^{57} +1.41421i q^{59} +12.7279i q^{61} -6.00000 q^{65} -8.48528i q^{69} -2.00000 q^{73} -4.24264i q^{75} -8.00000 q^{79} -5.00000 q^{81} -15.5563i q^{83} -8.48528i q^{85} -4.00000 q^{87} -6.00000 q^{89} +5.65685i q^{93} +6.00000 q^{95} +10.0000 q^{97} +2.82843i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{9} - 4 q^{15} + 12 q^{17} + 12 q^{23} + 6 q^{25} - 8 q^{31} + 8 q^{33} - 12 q^{39} - 12 q^{41} + 8 q^{55} + 12 q^{57} - 12 q^{65} - 4 q^{73} - 16 q^{79} - 10 q^{81} - 8 q^{87} - 12 q^{89} + 12 q^{95}+ \cdots + 20 q^{97}+O(q^{100}) \) 2 * q + 2 * q^9 - 4 * q^15 + 12 * q^17 + 12 * q^23 + 6 * q^25 - 8 * q^31 + 8 * q^33 - 12 * q^39 - 12 * q^41 + 8 * q^55 + 12 * q^57 - 12 * q^65 - 4 * q^73 - 16 * q^79 - 10 * q^81 - 8 * q^87 - 12 * q^89 + 12 * q^95 + 20 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more