LMFDB - Embedded newform 3528.2.k.b.881.12

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3528,2,Mod(881,3528)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3528.881"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3528, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3528.k (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)

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

Self dual:	no
Analytic conductor:	$28.1712218331$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 10x^{14} + 61x^{12} + 266x^{10} + 852x^{8} + 1438x^{6} + 1933x^{4} + 3038x^{2} + 2401 $ x^16 + 10x^14 + 61x^12 + 266x^10 + 852x^8 + 1438x^6 + 1933x^4 + 3038*x^2 + 2401
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{18} $
Twist minimal:	no (minimal twist has level 504)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.12
Root			$1.45333 + 1.51725i$ of defining polynomial
Character	$\chi$	$=$	3528.881
Dual form			3528.2.k.b.881.11

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.17462 q^{5} +1.67114i q^{11} -1.14525i q^{13} -4.15218 q^{17} +2.11833i q^{19} -4.87721i q^{23} -3.62028 q^{25} -8.32591i q^{29} -8.29219i q^{31} +4.39993 q^{37} +2.67577 q^{41} +4.08536 q^{43} -3.50563 q^{47} -1.84480i q^{53} +1.96295i q^{55} +5.97333 q^{59} -14.7814i q^{61} -1.34523i q^{65} +8.44871 q^{67} +5.48320i q^{71} -0.977702i q^{73} +11.1219 q^{79} +10.2258 q^{83} -4.87721 q^{85} -12.5997 q^{89} +2.48823i q^{95} +9.52049i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 24 q^{25} - 8 q^{37} + 8 q^{43} + 56 q^{67} + 64 q^{79} - 32 q^{85}+O(q^{100}) $ 16 * q - 24 * q^25 - 8 * q^37 + 8 * q^43 + 56 * q^67 + 64 * q^79 - 32 * q^85

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3528\mathbb{Z}\right)^\times$.

$n$	$785$	$1081$	$1765$	$2647$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.17462	0.525304	0.262652	−	0.964891i	$-0.415403\pi$
$5$	1.17462	0.525304	0.262652	+	0.964891i	$0.415403\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.67114i	0.503868i	0.967744	+	0.251934i	$0.0810666\pi$
$11$	1.67114i	0.503868i	−0.967744	+	0.251934i	$0.918933\pi$
$12$	0	0
$13$	− 1.14525i	− 0.317635i	−0.987308	−	0.158818i	$-0.949232\pi$
$13$	− 1.14525i	− 0.317635i	0.987308	−	0.158818i	$-0.0507682\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.15218	−1.00705	−0.503525	−	0.863981i	$-0.667964\pi$
$17$	−4.15218	−1.00705	−0.503525	+	0.863981i	$0.667964\pi$
$18$	0	0
$19$	2.11833i	0.485979i	0.970029	+	0.242989i	$0.0781280\pi$
$19$	2.11833i	0.485979i	−0.970029	+	0.242989i	$0.921872\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 4.87721i	− 1.01697i	−0.861071	−	0.508484i	$-0.830206\pi$
$23$	− 4.87721i	− 1.01697i	0.861071	−	0.508484i	$-0.169794\pi$
$24$	0	0
$25$	−3.62028	−0.724056
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 8.32591i	− 1.54608i	−0.634355	−	0.773042i	$-0.718734\pi$
$29$	− 8.32591i	− 1.54608i	0.634355	−	0.773042i	$-0.281266\pi$
$30$	0	0
$31$	− 8.29219i	− 1.48932i	−0.667444	−	0.744660i	$-0.732611\pi$
$31$	− 8.29219i	− 1.48932i	0.667444	−	0.744660i	$-0.267389\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.39993	0.723343	0.361672	−	0.932306i	$-0.382206\pi$
$37$	4.39993	0.723343	0.361672	+	0.932306i	$0.382206\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.67577	0.417885	0.208942	−	0.977928i	$-0.432998\pi$
$41$	2.67577	0.417885	0.208942	+	0.977928i	$0.432998\pi$
$42$	0	0
$43$	4.08536	0.623011	0.311505	−	0.950244i	$-0.399167\pi$
$43$	4.08536	0.623011	0.311505	+	0.950244i	$0.399167\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.50563	−0.511349	−0.255674	−	0.966763i	$-0.582297\pi$
$47$	−3.50563	−0.511349	−0.255674	+	0.966763i	$0.582297\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 1.84480i	− 0.253402i	−0.991941	−	0.126701i	$-0.959561\pi$
$53$	− 1.84480i	− 0.253402i	0.991941	−	0.126701i	$-0.0404389\pi$
$54$	0	0
$55$	1.96295i	0.264684i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	5.97333	0.777662	0.388831	−	0.921309i	$-0.372879\pi$
$59$	5.97333	0.777662	0.388831	+	0.921309i	$0.372879\pi$
$60$	0	0
$61$	− 14.7814i	− 1.89257i	−0.323337	−	0.946284i	$-0.604805\pi$
$61$	− 14.7814i	− 1.89257i	0.323337	−	0.946284i	$-0.395195\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 1.34523i	− 0.166855i
$66$	0	0
$67$	8.44871	1.03217	0.516087	−	0.856536i	$-0.327388\pi$
$67$	8.44871	1.03217	0.516087	+	0.856536i	$0.327388\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	5.48320i	0.650736i	0.945587	+	0.325368i	$0.105488\pi$
$71$	5.48320i	0.650736i	−0.945587	+	0.325368i	$0.894512\pi$
$72$	0	0
$73$	− 0.977702i	− 0.114431i	−0.998362	−	0.0572157i	$-0.981778\pi$
$73$	− 0.977702i	− 0.114431i	0.998362	−	0.0572157i	$-0.0182223\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	11.1219	1.25131	0.625657	−	0.780098i	$-0.284831\pi$
$79$	11.1219	1.25131	0.625657	+	0.780098i	$0.284831\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	10.2258	1.12242	0.561212	−	0.827672i	$-0.310335\pi$
$83$	10.2258	1.12242	0.561212	+	0.827672i	$0.310335\pi$
$84$	0	0
$85$	−4.87721	−0.529007
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−12.5997	−1.33557	−0.667785	−	0.744354i	$-0.732758\pi$
$89$	−12.5997	−1.33557	−0.667785	+	0.744354i	$0.732758\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2.48823i	0.255287i
$96$	0	0
$97$	9.52049i	0.966660i	0.875438	+	0.483330i	$0.160573\pi$
$97$	9.52049i	0.966660i	−0.875438	+	0.483330i	$0.839427\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−19.3189	−1.92230	−0.961150	−	0.276027i	$-0.910982\pi$
$101$	−19.3189	−1.92230	−0.961150	+	0.276027i	$0.910982\pi$
$102$	0	0
$103$	− 15.0954i	− 1.48739i	−0.668518	−	0.743696i	$-0.733071\pi$
$103$	− 15.0954i	− 1.48739i	0.668518	−	0.743696i	$-0.266929\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 10.9349i	− 1.05711i	−0.848898	−	0.528556i	$-0.822734\pi$
$107$	− 10.9349i	− 1.05711i	0.848898	−	0.528556i	$-0.177266\pi$
$108$	0	0
$109$	6.51769	0.624282	0.312141	−	0.950036i	$-0.398954\pi$
$109$	6.51769	0.624282	0.312141	+	0.950036i	$0.398954\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	10.8743i	1.02296i	0.859294	+	0.511482i	$0.170903\pi$
$113$	10.8743i	1.02296i	−0.859294	+	0.511482i	$0.829097\pi$
$114$	0	0
$115$	− 5.72884i	− 0.534217i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	8.20729	0.746117
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−10.1255	−0.905653
$126$	0	0
$127$	−1.56264	−0.138662	−0.0693308	−	0.997594i	$-0.522086\pi$
$127$	−1.56264	−0.138662	−0.0693308	+	0.997594i	$0.522086\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−20.5937	−1.79928	−0.899642	−	0.436628i	$-0.856173\pi$
$131$	−20.5937	−1.79928	−0.899642	+	0.436628i	$0.856173\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 21.2810i	− 1.81816i	−0.416626	−	0.909078i	$-0.636788\pi$
$137$	− 21.2810i	− 1.81816i	0.416626	−	0.909078i	$-0.363212\pi$
$138$	0	0
$139$	6.50746i	0.551955i	0.961164	+	0.275978i	$0.0890016\pi$
$139$	6.50746i	0.551955i	−0.961164	+	0.275978i	$0.910998\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.91387	0.160046
$144$	0	0
$145$	− 9.77974i	− 0.812164i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 14.3836i	− 1.17835i	−0.808007	−	0.589173i	$-0.799453\pi$
$149$	− 14.3836i	− 1.17835i	0.808007	−	0.589173i	$-0.200547\pi$
$150$	0	0
$151$	10.2569	0.834697	0.417348	−	0.908747i	$-0.362959\pi$
$151$	10.2569	0.834697	0.417348	+	0.908747i	$0.362959\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 9.74013i	− 0.782346i
$156$	0	0
$157$	− 0.461795i	− 0.0368552i	−0.999830	−	0.0184276i	$-0.994134\pi$
$157$	− 0.461795i	− 0.0368552i	0.999830	−	0.0184276i	$-0.00586603\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	22.3289	1.74893	0.874467	−	0.485085i	$-0.161212\pi$
$163$	22.3289	1.74893	0.874467	+	0.485085i	$0.161212\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	5.42604	0.419879	0.209940	−	0.977714i	$-0.432673\pi$
$167$	5.42604	0.419879	0.209940	+	0.977714i	$0.432673\pi$
$168$	0	0
$169$	11.6884	0.899108
$170$	0	0
$171$	0	0
$172$	0	0
$173$	9.51438	0.723365	0.361683	−	0.932301i	$-0.382202\pi$
$173$	9.51438	0.723365	0.361683	+	0.932301i	$0.382202\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	2.26285i	0.169133i	0.996418	+	0.0845665i	$0.0269506\pi$
$179$	2.26285i	0.169133i	−0.996418	+	0.0845665i	$0.973049\pi$
$180$	0	0
$181$	21.7987i	1.62029i	0.586231	+	0.810144i	$0.300611\pi$
$181$	21.7987i	1.62029i	−0.586231	+	0.810144i	$0.699389\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	5.16822	0.379975
$186$	0	0
$187$	− 6.93887i	− 0.507421i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	21.7431i	1.57328i	0.617415	+	0.786638i	$0.288180\pi$
$191$	21.7431i	1.57328i	−0.617415	+	0.786638i	$0.711820\pi$
$192$	0	0
$193$	−21.6518	−1.55853	−0.779266	−	0.626693i	$-0.784408\pi$
$193$	−21.6518	−1.55853	−0.779266	+	0.626693i	$0.784408\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 5.58787i	− 0.398119i	−0.979987	−	0.199060i	$-0.936211\pi$
$197$	− 5.58787i	− 0.398119i	0.979987	−	0.199060i	$-0.0637887\pi$
$198$	0	0
$199$	− 16.4062i	− 1.16300i	−0.813546	−	0.581501i	$-0.802466\pi$
$199$	− 16.4062i	− 1.16300i	0.813546	−	0.581501i	$-0.197534\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	3.14300	0.219516
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−3.54003	−0.244869
$210$	0	0
$211$	8.11777	0.558850	0.279425	−	0.960168i	$-0.409856\pi$
$211$	8.11777	0.558850	0.279425	+	0.960168i	$0.409856\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4.79872	0.327270
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	4.75528i	0.319875i
$222$	0	0
$223$	− 21.1486i	− 1.41621i	−0.706105	−	0.708107i	$-0.749549\pi$
$223$	− 21.1486i	− 1.41621i	0.706105	−	0.708107i	$-0.250451\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−23.5981	−1.56626	−0.783130	−	0.621858i	$-0.786378\pi$
$227$	−23.5981	−1.56626	−0.783130	+	0.621858i	$0.786378\pi$
$228$	0	0
$229$	− 1.11949i	− 0.0739780i	−0.999316	−	0.0369890i	$-0.988223\pi$
$229$	− 1.11949i	− 0.0739780i	0.999316	−	0.0369890i	$-0.0117767\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 24.1678i	− 1.58328i	−0.610986	−	0.791642i	$-0.709227\pi$
$233$	− 24.1678i	− 1.58328i	0.610986	−	0.791642i	$-0.290773\pi$
$234$	0	0
$235$	−4.11777	−0.268613
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 0.484062i	− 0.0313114i	−0.999877	−	0.0156557i	$-0.995016\pi$
$239$	− 0.484062i	− 0.0313114i	0.999877	−	0.0156557i	$-0.00498356\pi$
$240$	0	0
$241$	− 13.1385i	− 0.846324i	−0.906054	−	0.423162i	$-0.860920\pi$
$241$	− 13.1385i	− 0.846324i	0.906054	−	0.423162i	$-0.139080\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.42602	0.154364
$248$	0	0
$249$	0	0
$250$	0	0
$251$	13.4221	0.847193	0.423597	−	0.905851i	$-0.360767\pi$
$251$	13.4221	0.847193	0.423597	+	0.905851i	$0.360767\pi$
$252$	0	0
$253$	8.15051	0.512418
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−10.4596	−0.652452	−0.326226	−	0.945292i	$-0.605777\pi$
$257$	−10.4596	−0.652452	−0.326226	+	0.945292i	$0.605777\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	23.6712i	1.45963i	0.683646	+	0.729813i	$0.260393\pi$
$263$	23.6712i	1.45963i	−0.683646	+	0.729813i	$0.739607\pi$
$264$	0	0
$265$	− 2.16693i	− 0.133113i
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−18.1013	−1.10366	−0.551828	−	0.833958i	$-0.686069\pi$
$269$	−18.1013	−1.10366	−0.551828	+	0.833958i	$0.686069\pi$
$270$	0	0
$271$	6.68574i	0.406130i	0.979165	+	0.203065i	$0.0650902\pi$
$271$	6.68574i	0.406130i	−0.979165	+	0.203065i	$0.934910\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 6.05000i	− 0.364829i
$276$	0	0
$277$	−3.44957	−0.207265	−0.103632	−	0.994616i	$-0.533047\pi$
$277$	−3.44957	−0.207265	−0.103632	+	0.994616i	$0.533047\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 9.58198i	− 0.571613i	−0.958287	−	0.285807i	$-0.907739\pi$
$281$	− 9.58198i	− 0.571613i	0.958287	−	0.285807i	$-0.0922615\pi$
$282$	0	0
$283$	− 8.10786i	− 0.481962i	−0.970530	−	0.240981i	$-0.922531\pi$
$283$	− 8.10786i	− 0.481962i	0.970530	−	0.240981i	$-0.0774692\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	0.240559	0.0141505
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−12.9122	−0.754339	−0.377170	−	0.926144i	$-0.623103\pi$
$293$	−12.9122	−0.754339	−0.377170	+	0.926144i	$0.623103\pi$
$294$	0	0
$295$	7.01637	0.408509
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−5.58562	−0.323025
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 17.3625i	− 0.994173i
$306$	0	0
$307$	− 11.9163i	− 0.680099i	−0.940408	−	0.340049i	$-0.889556\pi$
$307$	− 11.9163i	− 0.680099i	0.940408	−	0.340049i	$-0.110444\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	30.6591	1.73852	0.869259	−	0.494357i	$-0.164596\pi$
$311$	30.6591	1.73852	0.869259	+	0.494357i	$0.164596\pi$
$312$	0	0
$313$	9.40849i	0.531799i	0.964001	+	0.265900i	$0.0856690\pi$
$313$	9.40849i	0.531799i	−0.964001	+	0.265900i	$0.914331\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 1.42850i	− 0.0802326i	−0.999195	−	0.0401163i	$-0.987227\pi$
$317$	− 1.42850i	− 0.0802326i	0.999195	−	0.0401163i	$-0.0127728\pi$
$318$	0	0
$319$	13.9138	0.779022
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 8.79569i	− 0.489405i
$324$	0	0
$325$	4.14612i	0.229986i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−14.4160	−0.792373	−0.396187	−	0.918170i	$-0.629667\pi$
$331$	−14.4160	−0.792373	−0.396187	+	0.918170i	$0.629667\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	9.92398	0.542205
$336$	0	0
$337$	3.62914	0.197692	0.0988460	−	0.995103i	$-0.468485\pi$
$337$	3.62914	0.197692	0.0988460	+	0.995103i	$0.468485\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	13.8574	0.750421
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	29.7258i	1.59577i	0.602812	+	0.797883i	$0.294047\pi$
$347$	29.7258i	1.59577i	−0.602812	+	0.797883i	$0.705953\pi$
$348$	0	0
$349$	5.41027i	0.289605i	0.989461	+	0.144802i	$0.0462547\pi$
$349$	5.41027i	0.289605i	−0.989461	+	0.144802i	$0.953745\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−14.0504	−0.747827	−0.373914	−	0.927464i	$-0.621984\pi$
$353$	−14.0504	−0.747827	−0.373914	+	0.927464i	$0.621984\pi$
$354$	0	0
$355$	6.44065i	0.341834i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	7.73000i	0.407974i	0.978974	+	0.203987i	$0.0653900\pi$
$359$	7.73000i	0.407974i	−0.978974	+	0.203987i	$0.934610\pi$
$360$	0	0
$361$	14.5127	0.763825
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 1.14842i	− 0.0601112i
$366$	0	0
$367$	− 7.85472i	− 0.410013i	−0.978761	−	0.205006i	$-0.934278\pi$
$367$	− 7.85472i	− 0.410013i	0.978761	−	0.205006i	$-0.0657216\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	7.88855	0.408454	0.204227	−	0.978924i	$-0.434532\pi$
$373$	7.88855	0.408454	0.204227	+	0.978924i	$0.434532\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−9.53525	−0.491090
$378$	0	0
$379$	−11.9349	−0.613052	−0.306526	−	0.951862i	$-0.599167\pi$
$379$	−11.9349	−0.613052	−0.306526	+	0.951862i	$0.599167\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−2.71873	−0.138921	−0.0694603	−	0.997585i	$-0.522128\pi$
$383$	−2.71873	−0.138921	−0.0694603	+	0.997585i	$0.522128\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 19.9197i	− 1.00997i	−0.863128	−	0.504984i	$-0.831498\pi$
$389$	− 19.9197i	− 1.00997i	0.863128	−	0.504984i	$-0.168502\pi$
$390$	0	0
$391$	20.2510i	1.02414i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	13.0640	0.657321
$396$	0	0
$397$	− 16.8046i	− 0.843399i	−0.906736	−	0.421699i	$-0.861434\pi$
$397$	− 16.8046i	− 0.843399i	0.906736	−	0.421699i	$-0.138566\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	8.44487i	0.421717i	0.977517	+	0.210858i	$0.0676259\pi$
$401$	8.44487i	0.421717i	−0.977517	+	0.210858i	$0.932374\pi$
$402$	0	0
$403$	−9.49662	−0.473061
$404$	0	0
$405$	0	0
$406$	0	0
$407$	7.35290i	0.364470i
$408$	0	0
$409$	24.9367i	1.23304i	0.787339	+	0.616521i	$0.211458\pi$
$409$	24.9367i	1.23304i	−0.787339	+	0.616521i	$0.788542\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	12.0113	0.589614
$416$	0	0
$417$	0	0
$418$	0	0
$419$	32.2386	1.57496	0.787480	−	0.616340i	$-0.211385\pi$
$419$	32.2386	1.57496	0.787480	+	0.616340i	$0.211385\pi$
$420$	0	0
$421$	−36.4958	−1.77869	−0.889347	−	0.457232i	$-0.848841\pi$
$421$	−36.4958	−1.77869	−0.889347	+	0.457232i	$0.848841\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	15.0320	0.729161
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 18.6708i	− 0.899341i	−0.893194	−	0.449671i	$-0.851541\pi$
$431$	− 18.6708i	− 0.899341i	0.893194	−	0.449671i	$-0.148459\pi$
$432$	0	0
$433$	− 12.4261i	− 0.597163i	−0.954384	−	0.298581i	$-0.903487\pi$
$433$	− 12.4261i	− 0.597163i	0.954384	−	0.298581i	$-0.0965134\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	10.3315	0.494225
$438$	0	0
$439$	− 19.9386i	− 0.951615i	−0.879549	−	0.475808i	$-0.842156\pi$
$439$	− 19.9386i	− 0.951615i	0.879549	−	0.475808i	$-0.157844\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 29.3125i	− 1.39268i	−0.717712	−	0.696340i	$-0.754811\pi$
$443$	− 29.3125i	− 1.39268i	0.717712	−	0.696340i	$-0.245189\pi$
$444$	0	0
$445$	−14.7999	−0.701580
$446$	0	0
$447$	0	0
$448$	0	0
$449$	3.58492i	0.169183i	0.996416	+	0.0845915i	$0.0269585\pi$
$449$	3.58492i	0.169183i	−0.996416	+	0.0845915i	$0.973041\pi$
$450$	0	0
$451$	4.47159i	0.210559i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−16.7948	−0.785629	−0.392814	−	0.919618i	$-0.628499\pi$
$457$	−16.7948	−0.785629	−0.392814	+	0.919618i	$0.628499\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	11.3775	0.529903	0.264951	−	0.964262i	$-0.414644\pi$
$461$	11.3775	0.529903	0.264951	+	0.964262i	$0.414644\pi$
$462$	0	0
$463$	2.48145	0.115323	0.0576613	−	0.998336i	$-0.481636\pi$
$463$	2.48145	0.115323	0.0576613	+	0.998336i	$0.481636\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−23.9678	−1.10910	−0.554550	−	0.832151i	$-0.687109\pi$
$467$	−23.9678	−1.10910	−0.554550	+	0.832151i	$0.687109\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	6.82721i	0.313915i
$474$	0	0
$475$	− 7.66896i	− 0.351876i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	22.4894	1.02757	0.513784	−	0.857920i	$-0.328243\pi$
$479$	22.4894	1.02757	0.513784	+	0.857920i	$0.328243\pi$
$480$	0	0
$481$	− 5.03901i	− 0.229759i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	11.1829i	0.507790i
$486$	0	0
$487$	21.9135	0.992993	0.496497	−	0.868039i	$-0.334619\pi$
$487$	21.9135	0.992993	0.496497	+	0.868039i	$0.334619\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	22.5875i	1.01936i	0.860364	+	0.509681i	$0.170236\pi$
$491$	22.5875i	1.01936i	−0.860364	+	0.509681i	$0.829764\pi$
$492$	0	0
$493$	34.5707i	1.55698i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−28.6598	−1.28299	−0.641495	−	0.767127i	$-0.721686\pi$
$499$	−28.6598	−1.28299	−0.641495	+	0.767127i	$0.721686\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−39.4762	−1.76016	−0.880078	−	0.474830i	$-0.842510\pi$
$503$	−39.4762	−1.76016	−0.880078	+	0.474830i	$0.842510\pi$
$504$	0	0
$505$	−22.6922	−1.00979
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−5.47001	−0.242454	−0.121227	−	0.992625i	$-0.538683\pi$
$509$	−5.47001	−0.242454	−0.121227	+	0.992625i	$0.538683\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 17.7313i	− 0.781333i
$516$	0	0
$517$	− 5.85841i	− 0.257652i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−30.3592	−1.33006	−0.665029	−	0.746817i	$-0.731581\pi$
$521$	−30.3592	−1.33006	−0.665029	+	0.746817i	$0.731581\pi$
$522$	0	0
$523$	37.5168i	1.64050i	0.572008	+	0.820248i	$0.306165\pi$
$523$	37.5168i	1.64050i	−0.572008	+	0.820248i	$0.693835\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	34.4306i	1.49982i
$528$	0	0
$529$	−0.787154	−0.0342241
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 3.06442i	− 0.132735i
$534$	0	0
$535$	− 12.8442i	− 0.555305i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−21.0518	−0.905086	−0.452543	−	0.891743i	$-0.649483\pi$
$541$	−21.0518	−0.905086	−0.452543	+	0.891743i	$0.649483\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	7.65578	0.327938
$546$	0	0
$547$	2.57620	0.110150	0.0550751	−	0.998482i	$-0.482460\pi$
$547$	2.57620	0.110150	0.0550751	+	0.998482i	$0.482460\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	17.6371	0.751364
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 22.7612i	− 0.964423i	−0.876055	−	0.482211i	$-0.839834\pi$
$557$	− 22.7612i	− 0.964423i	0.876055	−	0.482211i	$-0.160166\pi$
$558$	0	0
$559$	− 4.67875i	− 0.197890i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	26.4809	1.11604	0.558018	−	0.829829i	$-0.311562\pi$
$563$	26.4809	1.11604	0.558018	+	0.829829i	$0.311562\pi$
$564$	0	0
$565$	12.7731i	0.537367i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	17.5808i	0.737024i	0.929623	+	0.368512i	$0.120133\pi$
$569$	17.5808i	0.737024i	−0.929623	+	0.368512i	$0.879867\pi$
$570$	0	0
$571$	24.2233	1.01371	0.506857	−	0.862030i	$-0.330807\pi$
$571$	24.2233	1.01371	0.506857	+	0.862030i	$0.330807\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	17.6569i	0.736342i
$576$	0	0
$577$	21.4698i	0.893798i	0.894584	+	0.446899i	$0.147472\pi$
$577$	21.4698i	0.893798i	−0.894584	+	0.446899i	$0.852528\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	3.08292	0.127681
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−30.2492	−1.24852	−0.624260	−	0.781217i	$-0.714599\pi$
$587$	−30.2492	−1.24852	−0.624260	+	0.781217i	$0.714599\pi$
$588$	0	0
$589$	17.5656	0.723778
$590$	0	0
$591$	0	0
$592$	0	0
$593$	33.8046	1.38819	0.694095	−	0.719884i	$-0.255805\pi$
$593$	33.8046	1.38819	0.694095	+	0.719884i	$0.255805\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	38.1707i	1.55961i	0.626021	+	0.779806i	$0.284683\pi$
$599$	38.1707i	1.55961i	−0.626021	+	0.779806i	$0.715317\pi$
$600$	0	0
$601$	37.6195i	1.53453i	0.641329	+	0.767266i	$0.278384\pi$
$601$	37.6195i	1.53453i	−0.641329	+	0.767266i	$0.721616\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	9.64040	0.391938
$606$	0	0
$607$	− 0.865759i	− 0.0351401i	−0.999846	−	0.0175700i	$-0.994407\pi$
$607$	− 0.865759i	− 0.0351401i	0.999846	−	0.0175700i	$-0.00559301\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	4.01482i	0.162422i
$612$	0	0
$613$	−21.1127	−0.852736	−0.426368	−	0.904550i	$-0.640207\pi$
$613$	−21.1127	−0.852736	−0.426368	+	0.904550i	$0.640207\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 10.6262i	− 0.427795i	−0.976856	−	0.213897i	$-0.931384\pi$
$617$	− 10.6262i	− 0.427795i	0.976856	−	0.213897i	$-0.0686158\pi$
$618$	0	0
$619$	22.9365i	0.921896i	0.887427	+	0.460948i	$0.152491\pi$
$619$	22.9365i	0.921896i	−0.887427	+	0.460948i	$0.847509\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	6.20782	0.248313
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−18.2693	−0.728443
$630$	0	0
$631$	−4.19211	−0.166885	−0.0834425	−	0.996513i	$-0.526592\pi$
$631$	−4.19211	−0.166885	−0.0834425	+	0.996513i	$0.526592\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−1.83550	−0.0728395
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	17.8857i	0.706444i	0.935539	+	0.353222i	$0.114914\pi$
$641$	17.8857i	0.706444i	−0.935539	+	0.353222i	$0.885086\pi$
$642$	0	0
$643$	− 30.8980i	− 1.21850i	−0.792980	−	0.609248i	$-0.791471\pi$
$643$	− 30.8980i	− 1.21850i	0.792980	−	0.609248i	$-0.208529\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	7.63407	0.300126	0.150063	−	0.988676i	$-0.452052\pi$
$647$	7.63407	0.300126	0.150063	+	0.988676i	$0.452052\pi$
$648$	0	0
$649$	9.98229i	0.391839i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	22.2473i	0.870606i	0.900284	+	0.435303i	$0.143359\pi$
$653$	22.2473i	0.870606i	−0.900284	+	0.435303i	$0.856641\pi$
$654$	0	0
$655$	−24.1897	−0.945171
$656$	0	0
$657$	0	0
$658$	0	0
$659$	8.74939i	0.340828i	0.985373	+	0.170414i	$0.0545105\pi$
$659$	8.74939i	0.340828i	−0.985373	+	0.170414i	$0.945490\pi$
$660$	0	0
$661$	− 7.46987i	− 0.290544i	−0.989392	−	0.145272i	$-0.953594\pi$
$661$	− 7.46987i	− 0.290544i	0.989392	−	0.145272i	$-0.0464058\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−40.6072	−1.57232
$668$	0	0
$669$	0	0
$670$	0	0
$671$	24.7019	0.953605
$672$	0	0
$673$	−0.388583	−0.0149788	−0.00748938	−	0.999972i	$-0.502384\pi$
$673$	−0.388583	−0.0149788	−0.00748938	+	0.999972i	$0.502384\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−18.9188	−0.727110	−0.363555	−	0.931573i	$-0.618437\pi$
$677$	−18.9188	−0.727110	−0.363555	+	0.931573i	$0.618437\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	5.71698i	0.218754i	0.994000	+	0.109377i	$0.0348856\pi$
$683$	5.71698i	0.218754i	−0.994000	+	0.109377i	$0.965114\pi$
$684$	0	0
$685$	− 24.9969i	− 0.955084i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−2.11275	−0.0804895
$690$	0	0
$691$	50.5303i	1.92226i	0.276090	+	0.961132i	$0.410961\pi$
$691$	50.5303i	1.92226i	−0.276090	+	0.961132i	$0.589039\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	7.64376i	0.289944i
$696$	0	0
$697$	−11.1103	−0.420831
$698$	0	0
$699$	0	0
$700$	0	0
$701$	14.5025i	0.547752i	0.961765	+	0.273876i	$0.0883058\pi$
$701$	14.5025i	0.547752i	−0.961765	+	0.273876i	$0.911694\pi$
$702$	0	0
$703$	9.32051i	0.351530i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−18.0731	−0.678751	−0.339376	−	0.940651i	$-0.610216\pi$
$709$	−18.0731	−0.678751	−0.339376	+	0.940651i	$0.610216\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−40.4427	−1.51459
$714$	0	0
$715$	2.24807	0.0840729
$716$	0	0
$717$	0	0
$718$	0	0
$719$	50.7440	1.89243	0.946216	−	0.323536i	$-0.104872\pi$
$719$	50.7440	1.89243	0.946216	+	0.323536i	$0.104872\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	30.1421i	1.11945i
$726$	0	0
$727$	− 6.77139i	− 0.251137i	−0.992085	−	0.125568i	$-0.959925\pi$
$727$	− 6.77139i	− 0.251137i	0.992085	−	0.125568i	$-0.0400755\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−16.9631	−0.627403
$732$	0	0
$733$	50.6148i	1.86950i	0.355306	+	0.934750i	$0.384377\pi$
$733$	50.6148i	1.86950i	−0.355306	+	0.934750i	$0.615623\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	14.1190i	0.520080i
$738$	0	0
$739$	−30.7020	−1.12939	−0.564695	−	0.825300i	$-0.691006\pi$
$739$	−30.7020	−1.12939	−0.564695	+	0.825300i	$0.691006\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 5.18883i	− 0.190360i	−0.995460	−	0.0951799i	$-0.969657\pi$
$743$	− 5.18883i	− 0.190360i	0.995460	−	0.0951799i	$-0.0303426\pi$
$744$	0	0
$745$	− 16.8951i	− 0.618990i
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−1.23722	−0.0451466	−0.0225733	−	0.999745i	$-0.507186\pi$
$751$	−1.23722	−0.0451466	−0.0225733	+	0.999745i	$0.507186\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	12.0479	0.438470
$756$	0	0
$757$	51.3970	1.86806	0.934028	−	0.357200i	$-0.116268\pi$
$757$	51.3970	1.86806	0.934028	+	0.357200i	$0.116268\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	29.6381	1.07438	0.537190	−	0.843461i	$-0.319486\pi$
$761$	29.6381	1.07438	0.537190	+	0.843461i	$0.319486\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 6.84096i	− 0.247013i
$768$	0	0
$769$	− 28.1387i	− 1.01471i	−0.861738	−	0.507354i	$-0.830624\pi$
$769$	− 28.1387i	− 1.01471i	0.861738	−	0.507354i	$-0.169376\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	11.9045	0.428177	0.214088	−	0.976814i	$-0.431322\pi$
$773$	11.9045	0.428177	0.214088	+	0.976814i	$0.431322\pi$
$774$	0	0
$775$	30.0200i	1.07835i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	5.66817i	0.203083i
$780$	0	0
$781$	−9.16320	−0.327885
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 0.542431i	− 0.0193602i
$786$	0	0
$787$	34.0573i	1.21401i	0.794697	+	0.607007i	$0.207630\pi$
$787$	34.0573i	1.21401i	−0.794697	+	0.607007i	$0.792370\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−16.9284	−0.601146
$794$	0	0
$795$	0	0
$796$	0	0
$797$	32.8088	1.16215	0.581074	−	0.813851i	$-0.302633\pi$
$797$	32.8088	1.16215	0.581074	+	0.813851i	$0.302633\pi$
$798$	0	0
$799$	14.5560	0.514954
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1.63388	0.0576583
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 7.95416i	− 0.279654i	−0.990176	−	0.139827i	$-0.955345\pi$
$809$	− 7.95416i	− 0.279654i	0.990176	−	0.139827i	$-0.0446546\pi$
$810$	0	0
$811$	22.4200i	0.787273i	0.919266	+	0.393636i	$0.128783\pi$
$811$	22.4200i	0.787273i	−0.919266	+	0.393636i	$0.871217\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	26.2278	0.918721
$816$	0	0
$817$	8.65414i	0.302770i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 5.37678i	− 0.187651i	−0.995589	−	0.0938254i	$-0.970090\pi$
$821$	− 5.37678i	− 0.187651i	0.995589	−	0.0938254i	$-0.0299095\pi$
$822$	0	0
$823$	−7.10259	−0.247581	−0.123790	−	0.992308i	$-0.539505\pi$
$823$	−7.10259	−0.247581	−0.123790	+	0.992308i	$0.539505\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 49.1228i	− 1.70817i	−0.520135	−	0.854084i	$-0.674118\pi$
$827$	− 49.1228i	− 1.70817i	0.520135	−	0.854084i	$-0.325882\pi$
$828$	0	0
$829$	0.170141i	0.00590924i	0.999996	+	0.00295462i	$0.000940486\pi$
$829$	0.170141i	0.00590924i	−0.999996	+	0.00295462i	$0.999060\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	6.37350	0.220564
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−43.2416	−1.49287	−0.746434	−	0.665460i	$-0.768235\pi$
$839$	−43.2416	−1.49287	−0.746434	+	0.665460i	$0.768235\pi$
$840$	0	0
$841$	−40.3208	−1.39037
$842$	0	0
$843$	0	0
$844$	0	0
$845$	13.7294	0.472305
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 21.4594i	− 0.735617i
$852$	0	0
$853$	− 19.3097i	− 0.661153i	−0.943779	−	0.330577i	$-0.892757\pi$
$853$	− 19.3097i	− 0.661153i	0.943779	−	0.330577i	$-0.107243\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	53.5994	1.83092	0.915461	−	0.402408i	$-0.131827\pi$
$857$	53.5994	1.83092	0.915461	+	0.402408i	$0.131827\pi$
$858$	0	0
$859$	3.00087i	0.102389i	0.998689	+	0.0511943i	$0.0163028\pi$
$859$	3.00087i	0.102389i	−0.998689	+	0.0511943i	$0.983697\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	41.5766i	1.41528i	0.706572	+	0.707641i	$0.250241\pi$
$863$	41.5766i	1.41528i	−0.706572	+	0.707641i	$0.749759\pi$
$864$	0	0
$865$	11.1757	0.379986
$866$	0	0
$867$	0	0
$868$	0	0
$869$	18.5863i	0.630498i
$870$	0	0
$871$	− 9.67588i	− 0.327855i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−35.4921	−1.19848	−0.599241	−	0.800568i	$-0.704531\pi$
$877$	−35.4921	−1.19848	−0.599241	+	0.800568i	$0.704531\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−12.7343	−0.429030	−0.214515	−	0.976721i	$-0.568817\pi$
$881$	−12.7343	−0.429030	−0.214515	+	0.976721i	$0.568817\pi$
$882$	0	0
$883$	−29.0274	−0.976849	−0.488425	−	0.872606i	$-0.662428\pi$
$883$	−29.0274	−0.976849	−0.488425	+	0.872606i	$0.662428\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	44.6485	1.49915	0.749575	−	0.661919i	$-0.230258\pi$
$887$	44.6485	1.49915	0.749575	+	0.661919i	$0.230258\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 7.42609i	− 0.248505i
$894$	0	0
$895$	2.65797i	0.0888462i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−69.0400	−2.30261
$900$	0	0
$901$	7.65992i	0.255189i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	25.6051i	0.851143i
$906$	0	0
$907$	29.7726	0.988582	0.494291	−	0.869297i	$-0.335428\pi$
$907$	29.7726	0.988582	0.494291	+	0.869297i	$0.335428\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	33.4911i	1.10961i	0.831981	+	0.554804i	$0.187207\pi$
$911$	33.4911i	1.10961i	−0.831981	+	0.554804i	$0.812793\pi$
$912$	0	0
$913$	17.0887i	0.565554i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	13.5840	0.448096	0.224048	−	0.974578i	$-0.428073\pi$
$919$	13.5840	0.448096	0.224048	+	0.974578i	$0.428073\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	6.27963	0.206697
$924$	0	0
$925$	−15.9290	−0.523741
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−0.0536313	−0.00175959	−0.000879793	−	1.00000i	$-0.500280\pi$
$929$	−0.0536313	−0.00175959	−0.000879793		1.00000i	$0.500280\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 8.15051i	− 0.266550i
$936$	0	0
$937$	− 14.3633i	− 0.469230i	−0.972088	−	0.234615i	$-0.924617\pi$
$937$	− 14.3633i	− 0.469230i	0.972088	−	0.234615i	$-0.0753829\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	33.9124	1.10551	0.552757	−	0.833343i	$-0.313576\pi$
$941$	33.9124	1.10551	0.552757	+	0.833343i	$0.313576\pi$
$942$	0	0
$943$	− 13.0503i	− 0.424976i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 32.7927i	− 1.06562i	−0.846235	−	0.532810i	$-0.821136\pi$
$947$	− 32.7927i	− 1.06562i	0.846235	−	0.532810i	$-0.178864\pi$
$948$	0	0
$949$	−1.11971	−0.0363474
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 25.2186i	− 0.816912i	−0.912778	−	0.408456i	$-0.866067\pi$
$953$	− 25.2186i	− 0.816912i	0.912778	−	0.408456i	$-0.133933\pi$
$954$	0	0
$955$	25.5398i	0.826448i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−37.7604	−1.21808
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−25.4326	−0.818703
$966$	0	0
$967$	−15.0565	−0.484183	−0.242091	−	0.970253i	$-0.577833\pi$
$967$	−15.0565	−0.484183	−0.242091	+	0.970253i	$0.577833\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−12.6049	−0.404510	−0.202255	−	0.979333i	$-0.564827\pi$
$971$	−12.6049	−0.404510	−0.202255	+	0.979333i	$0.564827\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	53.1669i	1.70096i	0.526007	+	0.850480i	$0.323688\pi$
$977$	53.1669i	1.70096i	−0.526007	+	0.850480i	$0.676312\pi$
$978$	0	0
$979$	− 21.0560i	− 0.672951i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	8.86131	0.282632	0.141316	−	0.989965i	$-0.454867\pi$
$983$	8.86131	0.282632	0.141316	+	0.989965i	$0.454867\pi$
$984$	0	0
$985$	− 6.56359i	− 0.209134i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 19.9251i	− 0.633582i
$990$	0	0
$991$	−29.8469	−0.948118	−0.474059	−	0.880493i	$-0.657212\pi$
$991$	−29.8469	−0.948118	−0.474059	+	0.880493i	$0.657212\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 19.2709i	− 0.610929i
$996$	0	0
$997$	− 12.7612i	− 0.404153i	−0.979370	−	0.202076i	$-0.935231\pi$
$997$	− 12.7612i	− 0.404153i	0.979370	−	0.202076i	$-0.0647689\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3528.2.k.b.881.12		16
3.2	odd	2	inner	3528.2.k.b.881.5		16
4.3	odd	2		7056.2.k.h.881.11		16
7.2	even	3		504.2.bl.a.17.3	✓	16
7.3	odd	6		504.2.bl.a.89.6	yes	16
7.4	even	3		3528.2.bl.a.1097.3		16
7.5	odd	6		3528.2.bl.a.521.6		16
7.6	odd	2	inner	3528.2.k.b.881.6		16
12.11	even	2		7056.2.k.h.881.6		16
21.2	odd	6		504.2.bl.a.17.6	yes	16
21.5	even	6		3528.2.bl.a.521.3		16
21.11	odd	6		3528.2.bl.a.1097.6		16
21.17	even	6		504.2.bl.a.89.3	yes	16
21.20	even	2	inner	3528.2.k.b.881.11		16
28.3	even	6		1008.2.bt.d.593.6		16
28.23	odd	6		1008.2.bt.d.17.3		16
28.27	even	2		7056.2.k.h.881.5		16
84.23	even	6		1008.2.bt.d.17.6		16
84.59	odd	6		1008.2.bt.d.593.3		16
84.83	odd	2		7056.2.k.h.881.12		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.bl.a.17.3	✓	16	7.2	even	3
504.2.bl.a.17.6	yes	16	21.2	odd	6
504.2.bl.a.89.3	yes	16	21.17	even	6
504.2.bl.a.89.6	yes	16	7.3	odd	6
1008.2.bt.d.17.3		16	28.23	odd	6
1008.2.bt.d.17.6		16	84.23	even	6
1008.2.bt.d.593.3		16	84.59	odd	6
1008.2.bt.d.593.6		16	28.3	even	6
3528.2.k.b.881.5		16	3.2	odd	2	inner
3528.2.k.b.881.6		16	7.6	odd	2	inner
3528.2.k.b.881.11		16	21.20	even	2	inner
3528.2.k.b.881.12		16	1.1	even	1	trivial
3528.2.bl.a.521.3		16	21.5	even	6
3528.2.bl.a.521.6		16	7.5	odd	6
3528.2.bl.a.1097.3		16	7.4	even	3
3528.2.bl.a.1097.6		16	21.11	odd	6
7056.2.k.h.881.5		16	28.27	even	2
7056.2.k.h.881.6		16	12.11	even	2
7056.2.k.h.881.11		16	4.3	odd	2
7056.2.k.h.881.12		16	84.83	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3528.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.17462 q^{5} +1.67114i q^{11} -1.14525i q^{13} -4.15218 q^{17} +2.11833i q^{19} -4.87721i q^{23} -3.62028 q^{25} -8.32591i q^{29} -8.29219i q^{31} +4.39993 q^{37} +2.67577 q^{41} +4.08536 q^{43} -3.50563 q^{47} -1.84480i q^{53} +1.96295i q^{55} +5.97333 q^{59} -14.7814i q^{61} -1.34523i q^{65} +8.44871 q^{67} +5.48320i q^{71} -0.977702i q^{73} +11.1219 q^{79} +10.2258 q^{83} -4.87721 q^{85} -12.5997 q^{89} +2.48823i q^{95} +9.52049i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 24 q^{25} - 8 q^{37} + 8 q^{43} + 56 q^{67} + 64 q^{79} - 32 q^{85}+O(q^{100}) \) 16 * q - 24 * q^25 - 8 * q^37 + 8 * q^43 + 56 * q^67 + 64 * q^79 - 32 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more