LMFDB - Embedded newform 1568.3.h.a.881.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1568,3,Mod(881,1568)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1568, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1568.881");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1568 = 2^{5} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1568.h (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:	$42.7249054517$
Analytic rank:	$0$
Dimension:	$28$
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.14
Character	$\chi$	$=$	1568.881
Dual form			1568.3.h.a.881.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.253256 q^{3} -3.57178 q^{5} -8.93586 q^{9} +O(q^{10})$	$q-0.253256 q^{3} -3.57178 q^{5} -8.93586 q^{9} +7.88285i q^{11} +18.1529 q^{13} +0.904575 q^{15} -9.53990i q^{17} -24.8189 q^{19} +4.29899 q^{23} -12.2424 q^{25} +4.54237 q^{27} -28.3630i q^{29} +32.6055i q^{31} -1.99638i q^{33} -29.9075i q^{37} -4.59733 q^{39} +45.2606i q^{41} +24.9109i q^{43} +31.9169 q^{45} +50.8567i q^{47} +2.41604i q^{51} -62.8076i q^{53} -28.1558i q^{55} +6.28554 q^{57} +74.0096 q^{59} +50.5987 q^{61} -64.8382 q^{65} -125.484i q^{67} -1.08875 q^{69} +5.33822 q^{71} -27.3166i q^{73} +3.10046 q^{75} +103.076 q^{79} +79.2724 q^{81} +51.5695 q^{83} +34.0744i q^{85} +7.18310i q^{87} -153.832i q^{89} -8.25753i q^{93} +88.6476 q^{95} -47.0436i q^{97} -70.4400i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 64 q^{9}+O(q^{10}) $ 28 * q + 64 * q^9	$ 28 q + 64 q^{9} - 28 q^{15} + 60 q^{23} + 64 q^{25} - 40 q^{39} + 124 q^{57} + 104 q^{65} + 136 q^{71} - 324 q^{79} + 36 q^{81} + 580 q^{95}+O(q^{100}) $ 28 * q + 64 * q^9 - 28 * q^15 + 60 * q^23 + 64 * q^25 - 40 * q^39 + 124 * q^57 + 104 * q^65 + 136 * q^71 - 324 * q^79 + 36 * q^81 + 580 * q^95

Display 100 coefficients 10 coefficients

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
$3$	−0.253256	−0.0844187	−0.0422093	−	0.999109i	$-0.513440\pi$
$3$	−0.253256	−0.0844187	−0.0422093	+	0.999109i	$0.513440\pi$
$4$	0	0
$5$	−3.57178	−0.714356	−0.357178	−	0.934036i	$-0.616261\pi$
$5$	−3.57178	−0.714356	−0.357178	+	0.934036i	$0.616261\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−8.93586	−0.992873
$10$	0	0
$11$	7.88285i	0.716623i	0.933602	+	0.358311i	$0.116647\pi$
$11$	7.88285i	0.716623i	−0.933602	+	0.358311i	$0.883353\pi$
$12$	0	0
$13$	18.1529	1.39638	0.698189	−	0.715914i	$-0.253990\pi$
$13$	18.1529	1.39638	0.698189	+	0.715914i	$0.253990\pi$
$14$	0	0
$15$	0.904575	0.0603050
$16$	0	0
$17$	− 9.53990i	− 0.561171i	−0.959829	−	0.280585i	$-0.909471\pi$
$17$	− 9.53990i	− 0.561171i	0.959829	−	0.280585i	$-0.0905286\pi$
$18$	0	0
$19$	−24.8189	−1.30626	−0.653129	−	0.757247i	$-0.726544\pi$
$19$	−24.8189	−1.30626	−0.653129	+	0.757247i	$0.726544\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.29899	0.186913	0.0934563	−	0.995623i	$-0.470208\pi$
$23$	4.29899	0.186913	0.0934563	+	0.995623i	$0.470208\pi$
$24$	0	0
$25$	−12.2424	−0.489696
$26$	0	0
$27$	4.54237	0.168236
$28$	0	0
$29$	− 28.3630i	− 0.978035i	−0.872274	−	0.489017i	$-0.837356\pi$
$29$	− 28.3630i	− 0.978035i	0.872274	−	0.489017i	$-0.162644\pi$
$30$	0	0
$31$	32.6055i	1.05179i	0.850550	+	0.525895i	$0.176269\pi$
$31$	32.6055i	1.05179i	−0.850550	+	0.525895i	$0.823731\pi$
$32$	0	0
$33$	− 1.99638i	− 0.0604963i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 29.9075i	− 0.808310i	−0.914690	−	0.404155i	$-0.867566\pi$
$37$	− 29.9075i	− 0.808310i	0.914690	−	0.404155i	$-0.132434\pi$
$38$	0	0
$39$	−4.59733	−0.117880
$40$	0	0
$41$	45.2606i	1.10392i	0.833872	+	0.551958i	$0.186119\pi$
$41$	45.2606i	1.10392i	−0.833872	+	0.551958i	$0.813881\pi$
$42$	0	0
$43$	24.9109i	0.579323i	0.957129	+	0.289661i	$0.0935427\pi$
$43$	24.9109i	0.579323i	−0.957129	+	0.289661i	$0.906457\pi$
$44$	0	0
$45$	31.9169	0.709265
$46$	0	0
$47$	50.8567i	1.08206i	0.841004	+	0.541029i	$0.181965\pi$
$47$	50.8567i	1.08206i	−0.841004	+	0.541029i	$0.818035\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	2.41604i	0.0473733i
$52$	0	0
$53$	− 62.8076i	− 1.18505i	−0.805552	−	0.592525i	$-0.798131\pi$
$53$	− 62.8076i	− 1.18505i	0.805552	−	0.592525i	$-0.201869\pi$
$54$	0	0
$55$	− 28.1558i	− 0.511924i
$56$	0	0
$57$	6.28554	0.110273
$58$	0	0
$59$	74.0096	1.25440	0.627200	−	0.778859i	$-0.284201\pi$
$59$	74.0096	1.25440	0.627200	+	0.778859i	$0.284201\pi$
$60$	0	0
$61$	50.5987	0.829487	0.414743	−	0.909938i	$-0.363871\pi$
$61$	50.5987	0.829487	0.414743	+	0.909938i	$0.363871\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−64.8382	−0.997510
$66$	0	0
$67$	− 125.484i	− 1.87290i	−0.350798	−	0.936451i	$-0.614089\pi$
$67$	− 125.484i	− 1.87290i	0.350798	−	0.936451i	$-0.385911\pi$
$68$	0	0
$69$	−1.08875	−0.0157789
$70$	0	0
$71$	5.33822	0.0751863	0.0375931	−	0.999293i	$-0.488031\pi$
$71$	5.33822	0.0751863	0.0375931	+	0.999293i	$0.488031\pi$
$72$	0	0
$73$	− 27.3166i	− 0.374200i	−0.982341	−	0.187100i	$-0.940091\pi$
$73$	− 27.3166i	− 0.374200i	0.982341	−	0.187100i	$-0.0599089\pi$
$74$	0	0
$75$	3.10046	0.0413395
$76$	0	0
$77$	0	0
$78$	0	0
$79$	103.076	1.30476	0.652380	−	0.757892i	$-0.273771\pi$
$79$	103.076	1.30476	0.652380	+	0.757892i	$0.273771\pi$
$80$	0	0
$81$	79.2724	0.978671
$82$	0	0
$83$	51.5695	0.621319	0.310660	−	0.950521i	$-0.399450\pi$
$83$	51.5695	0.621319	0.310660	+	0.950521i	$0.399450\pi$
$84$	0	0
$85$	34.0744i	0.400876i
$86$	0	0
$87$	7.18310i	0.0825644i
$88$	0	0
$89$	− 153.832i	− 1.72844i	−0.503111	−	0.864222i	$-0.667811\pi$
$89$	− 153.832i	− 1.72844i	0.503111	−	0.864222i	$-0.332189\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	− 8.25753i	− 0.0887907i
$94$	0	0
$95$	88.6476	0.933133
$96$	0	0
$97$	− 47.0436i	− 0.484986i	−0.970153	−	0.242493i	$-0.922035\pi$
$97$	− 47.0436i	− 0.484986i	0.970153	−	0.242493i	$-0.0779651\pi$
$98$	0	0
$99$	− 70.4400i	− 0.711516i
$100$	0	0
$101$	149.345	1.47867	0.739333	−	0.673339i	$-0.235141\pi$
$101$	149.345	1.47867	0.739333	+	0.673339i	$0.235141\pi$
$102$	0	0
$103$	− 19.8432i	− 0.192652i	−0.995350	−	0.0963262i	$-0.969291\pi$
$103$	− 19.8432i	− 0.192652i	0.995350	−	0.0963262i	$-0.0307092\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	9.14381i	0.0854561i	0.999087	+	0.0427281i	$0.0136049\pi$
$107$	9.14381i	0.0854561i	−0.999087	+	0.0427281i	$0.986395\pi$
$108$	0	0
$109$	− 119.198i	− 1.09356i	−0.837275	−	0.546781i	$-0.815853\pi$
$109$	− 119.198i	− 1.09356i	0.837275	−	0.546781i	$-0.184147\pi$
$110$	0	0
$111$	7.57425i	0.0682365i
$112$	0	0
$113$	−124.011	−1.09744	−0.548720	−	0.836006i	$-0.684885\pi$
$113$	−124.011	−1.09744	−0.548720	+	0.836006i	$0.684885\pi$
$114$	0	0
$115$	−15.3550	−0.133522
$116$	0	0
$117$	−162.212	−1.38643
$118$	0	0
$119$	0	0
$120$	0	0
$121$	58.8607	0.486452
$122$	0	0
$123$	− 11.4625i	− 0.0931912i
$124$	0	0
$125$	133.022	1.06417
$126$	0	0
$127$	57.6144	0.453656	0.226828	−	0.973935i	$-0.427164\pi$
$127$	57.6144	0.453656	0.226828	+	0.973935i	$0.427164\pi$
$128$	0	0
$129$	− 6.30883i	− 0.0489057i
$130$	0	0
$131$	−124.299	−0.948850	−0.474425	−	0.880296i	$-0.657344\pi$
$131$	−124.299	−0.948850	−0.474425	+	0.880296i	$0.657344\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−16.2243	−0.120180
$136$	0	0
$137$	−169.481	−1.23709	−0.618543	−	0.785751i	$-0.712277\pi$
$137$	−169.481	−1.23709	−0.618543	+	0.785751i	$0.712277\pi$
$138$	0	0
$139$	266.497	1.91725	0.958624	−	0.284677i	$-0.0918862\pi$
$139$	266.497	1.91725	0.958624	+	0.284677i	$0.0918862\pi$
$140$	0	0
$141$	− 12.8798i	− 0.0913459i
$142$	0	0
$143$	143.097i	1.00068i
$144$	0	0
$145$	101.306i	0.698665i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 30.8192i	− 0.206840i	−0.994638	−	0.103420i	$-0.967021\pi$
$149$	− 30.8192i	− 0.206840i	0.994638	−	0.103420i	$-0.0329786\pi$
$150$	0	0
$151$	−23.4895	−0.155560	−0.0777800	−	0.996971i	$-0.524783\pi$
$151$	−23.4895	−0.155560	−0.0777800	+	0.996971i	$0.524783\pi$
$152$	0	0
$153$	85.2473i	0.557172i
$154$	0	0
$155$	− 116.460i	− 0.751352i
$156$	0	0
$157$	127.629	0.812926	0.406463	−	0.913667i	$-0.366762\pi$
$157$	127.629	0.812926	0.406463	+	0.913667i	$0.366762\pi$
$158$	0	0
$159$	15.9064i	0.100040i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 159.684i	− 0.979658i	−0.871819	−	0.489829i	$-0.837059\pi$
$163$	− 159.684i	− 0.979658i	0.871819	−	0.489829i	$-0.162941\pi$
$164$	0	0
$165$	7.13063i	0.0432159i
$166$	0	0
$167$	142.792i	0.855042i	0.904005	+	0.427521i	$0.140613\pi$
$167$	142.792i	0.855042i	−0.904005	+	0.427521i	$0.859387\pi$
$168$	0	0
$169$	160.528	0.949869
$170$	0	0
$171$	221.778	1.29695
$172$	0	0
$173$	−195.780	−1.13167	−0.565837	−	0.824517i	$-0.691447\pi$
$173$	−195.780	−1.13167	−0.565837	+	0.824517i	$0.691447\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−18.7434	−0.105895
$178$	0	0
$179$	− 149.507i	− 0.835234i	−0.908623	−	0.417617i	$-0.862865\pi$
$179$	− 149.507i	− 0.835234i	0.908623	−	0.417617i	$-0.137135\pi$
$180$	0	0
$181$	91.2994	0.504417	0.252208	−	0.967673i	$-0.418843\pi$
$181$	91.2994	0.504417	0.252208	+	0.967673i	$0.418843\pi$
$182$	0	0
$183$	−12.8144	−0.0700242
$184$	0	0
$185$	106.823i	0.577421i
$186$	0	0
$187$	75.2016	0.402148
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−27.8279	−0.145696	−0.0728480	−	0.997343i	$-0.523209\pi$
$191$	−27.8279	−0.145696	−0.0728480	+	0.997343i	$0.523209\pi$
$192$	0	0
$193$	242.384	1.25588	0.627938	−	0.778264i	$-0.283899\pi$
$193$	242.384	1.25588	0.627938	+	0.778264i	$0.283899\pi$
$194$	0	0
$195$	16.4207	0.0842085
$196$	0	0
$197$	− 94.7050i	− 0.480736i	−0.970682	−	0.240368i	$-0.922732\pi$
$197$	− 94.7050i	− 0.480736i	0.970682	−	0.240368i	$-0.0772682\pi$
$198$	0	0
$199$	− 309.158i	− 1.55356i	−0.629774	−	0.776778i	$-0.716853\pi$
$199$	− 309.158i	− 1.55356i	0.629774	−	0.776778i	$-0.283147\pi$
$200$	0	0
$201$	31.7797i	0.158108i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	− 161.661i	− 0.788589i
$206$	0	0
$207$	−38.4152	−0.185581
$208$	0	0
$209$	− 195.644i	− 0.936094i
$210$	0	0
$211$	125.864i	0.596514i	0.954486	+	0.298257i	$0.0964052\pi$
$211$	125.864i	0.596514i	−0.954486	+	0.298257i	$0.903595\pi$
$212$	0	0
$213$	−1.35194	−0.00634712
$214$	0	0
$215$	− 88.9762i	− 0.413843i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	6.91810i	0.0315895i
$220$	0	0
$221$	− 173.177i	− 0.783606i
$222$	0	0
$223$	− 8.94619i	− 0.0401174i	−0.999799	−	0.0200587i	$-0.993615\pi$
$223$	− 8.94619i	− 0.0401174i	0.999799	−	0.0200587i	$-0.00638532\pi$
$224$	0	0
$225$	109.396	0.486206
$226$	0	0
$227$	272.694	1.20129	0.600647	−	0.799514i	$-0.294910\pi$
$227$	272.694	1.20129	0.600647	+	0.799514i	$0.294910\pi$
$228$	0	0
$229$	331.222	1.44638	0.723191	−	0.690648i	$-0.242675\pi$
$229$	331.222	1.44638	0.723191	+	0.690648i	$0.242675\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	158.237	0.679129	0.339564	−	0.940583i	$-0.389720\pi$
$233$	158.237	0.679129	0.339564	+	0.940583i	$0.389720\pi$
$234$	0	0
$235$	− 181.649i	− 0.772975i
$236$	0	0
$237$	−26.1046	−0.110146
$238$	0	0
$239$	−48.9981	−0.205013	−0.102507	−	0.994732i	$-0.532686\pi$
$239$	−48.9981	−0.205013	−0.102507	+	0.994732i	$0.532686\pi$
$240$	0	0
$241$	197.354i	0.818897i	0.912333	+	0.409449i	$0.134279\pi$
$241$	197.354i	0.818897i	−0.912333	+	0.409449i	$0.865721\pi$
$242$	0	0
$243$	−60.9575	−0.250854
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−450.535	−1.82403
$248$	0	0
$249$	−13.0603	−0.0524510
$250$	0	0
$251$	−315.497	−1.25696	−0.628480	−	0.777826i	$-0.716323\pi$
$251$	−315.497	−1.25696	−0.628480	+	0.777826i	$0.716323\pi$
$252$	0	0
$253$	33.8883i	0.133946i
$254$	0	0
$255$	− 8.62956i	− 0.0338414i
$256$	0	0
$257$	380.512i	1.48059i	0.672281	+	0.740296i	$0.265315\pi$
$257$	380.512i	1.48059i	−0.672281	+	0.740296i	$0.734685\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	253.448i	0.971065i
$262$	0	0
$263$	−196.327	−0.746491	−0.373246	−	0.927733i	$-0.621755\pi$
$263$	−196.327	−0.746491	−0.373246	+	0.927733i	$0.621755\pi$
$264$	0	0
$265$	224.335i	0.846547i
$266$	0	0
$267$	38.9588i	0.145913i
$268$	0	0
$269$	−102.311	−0.380340	−0.190170	−	0.981751i	$-0.560904\pi$
$269$	−102.311	−0.380340	−0.190170	+	0.981751i	$0.560904\pi$
$270$	0	0
$271$	− 256.321i	− 0.945836i	−0.881107	−	0.472918i	$-0.843201\pi$
$271$	− 256.321i	− 0.945836i	0.881107	−	0.472918i	$-0.156799\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 96.5049i	− 0.350927i
$276$	0	0
$277$	− 196.729i	− 0.710214i	−0.934826	−	0.355107i	$-0.884444\pi$
$277$	− 196.729i	− 0.710214i	0.934826	−	0.355107i	$-0.115556\pi$
$278$	0	0
$279$	− 291.358i	− 1.04429i
$280$	0	0
$281$	−70.2923	−0.250151	−0.125075	−	0.992147i	$-0.539917\pi$
$281$	−70.2923	−0.250151	−0.125075	+	0.992147i	$0.539917\pi$
$282$	0	0
$283$	−296.990	−1.04944	−0.524718	−	0.851276i	$-0.675829\pi$
$283$	−296.990	−1.04944	−0.524718	+	0.851276i	$0.675829\pi$
$284$	0	0
$285$	−22.4505	−0.0787738
$286$	0	0
$287$	0	0
$288$	0	0
$289$	197.990	0.685087
$290$	0	0
$291$	11.9141i	0.0409419i
$292$	0	0
$293$	135.561	0.462665	0.231333	−	0.972875i	$-0.425691\pi$
$293$	135.561	0.462665	0.231333	+	0.972875i	$0.425691\pi$
$294$	0	0
$295$	−264.346	−0.896087
$296$	0	0
$297$	35.8068i	0.120562i
$298$	0	0
$299$	78.0391	0.261000
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−37.8226	−0.124827
$304$	0	0
$305$	−180.727	−0.592549
$306$	0	0
$307$	−76.2052	−0.248225	−0.124113	−	0.992268i	$-0.539608\pi$
$307$	−76.2052	−0.248225	−0.124113	+	0.992268i	$0.539608\pi$
$308$	0	0
$309$	5.02541i	0.0162635i
$310$	0	0
$311$	198.094i	0.636957i	0.947930	+	0.318479i	$0.103172\pi$
$311$	198.094i	0.636957i	−0.947930	+	0.318479i	$0.896828\pi$
$312$	0	0
$313$	55.3902i	0.176965i	0.996078	+	0.0884827i	$0.0282018\pi$
$313$	55.3902i	0.176965i	−0.996078	+	0.0884827i	$0.971798\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	299.160i	0.943723i	0.881673	+	0.471862i	$0.156418\pi$
$317$	299.160i	0.943723i	−0.881673	+	0.471862i	$0.843582\pi$
$318$	0	0
$319$	223.581	0.700882
$320$	0	0
$321$	− 2.31572i	− 0.00721409i
$322$	0	0
$323$	236.770i	0.733034i
$324$	0	0
$325$	−222.235	−0.683800
$326$	0	0
$327$	30.1877i	0.0923171i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	375.378i	1.13407i	0.823693	+	0.567036i	$0.191910\pi$
$331$	375.378i	1.13407i	−0.823693	+	0.567036i	$0.808090\pi$
$332$	0	0
$333$	267.249i	0.802550i
$334$	0	0
$335$	448.203i	1.33792i
$336$	0	0
$337$	−4.99043	−0.0148084	−0.00740419	−	0.999973i	$-0.502357\pi$
$337$	−4.99043	−0.0148084	−0.00740419	+	0.999973i	$0.502357\pi$
$338$	0	0
$339$	31.4065	0.0926444
$340$	0	0
$341$	−257.024	−0.753736
$342$	0	0
$343$	0	0
$344$	0	0
$345$	3.88876	0.0112718
$346$	0	0
$347$	370.395i	1.06742i	0.845667	+	0.533711i	$0.179203\pi$
$347$	370.395i	1.06742i	−0.845667	+	0.533711i	$0.820797\pi$
$348$	0	0
$349$	25.6801	0.0735821	0.0367910	−	0.999323i	$-0.488286\pi$
$349$	25.6801	0.0735821	0.0367910	+	0.999323i	$0.488286\pi$
$350$	0	0
$351$	82.4571	0.234921
$352$	0	0
$353$	− 265.510i	− 0.752152i	−0.926589	−	0.376076i	$-0.877273\pi$
$353$	− 265.510i	− 0.752152i	0.926589	−	0.376076i	$-0.122727\pi$
$354$	0	0
$355$	−19.0670	−0.0537097
$356$	0	0
$357$	0	0
$358$	0	0
$359$	550.456	1.53330	0.766651	−	0.642064i	$-0.221921\pi$
$359$	550.456	1.53330	0.766651	+	0.642064i	$0.221921\pi$
$360$	0	0
$361$	254.978	0.706309
$362$	0	0
$363$	−14.9068	−0.0410656
$364$	0	0
$365$	97.5689i	0.267312i
$366$	0	0
$367$	− 207.960i	− 0.566649i	−0.959024	−	0.283324i	$-0.908563\pi$
$367$	− 207.960i	− 0.566649i	0.959024	−	0.283324i	$-0.0914373\pi$
$368$	0	0
$369$	− 404.442i	− 1.09605i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 454.420i	− 1.21828i	−0.793061	−	0.609142i	$-0.791514\pi$
$373$	− 454.420i	− 1.21828i	0.793061	−	0.609142i	$-0.208486\pi$
$374$	0	0
$375$	−33.6885	−0.0898361
$376$	0	0
$377$	− 514.871i	− 1.36570i
$378$	0	0
$379$	− 373.244i	− 0.984813i	−0.870365	−	0.492406i	$-0.836117\pi$
$379$	− 373.244i	− 0.984813i	0.870365	−	0.492406i	$-0.163883\pi$
$380$	0	0
$381$	−14.5912	−0.0382971
$382$	0	0
$383$	− 312.113i	− 0.814916i	−0.913224	−	0.407458i	$-0.866415\pi$
$383$	− 312.113i	− 0.814916i	0.913224	−	0.407458i	$-0.133585\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 222.600i	− 0.575194i
$388$	0	0
$389$	507.639i	1.30498i	0.757795	+	0.652492i	$0.226276\pi$
$389$	507.639i	1.30498i	−0.757795	+	0.652492i	$0.773724\pi$
$390$	0	0
$391$	− 41.0120i	− 0.104890i
$392$	0	0
$393$	31.4796	0.0801007
$394$	0	0
$395$	−368.165	−0.932063
$396$	0	0
$397$	191.297	0.481858	0.240929	−	0.970543i	$-0.422548\pi$
$397$	191.297	0.481858	0.240929	+	0.970543i	$0.422548\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−122.402	−0.305243	−0.152621	−	0.988285i	$-0.548771\pi$
$401$	−122.402	−0.305243	−0.152621	+	0.988285i	$0.548771\pi$
$402$	0	0
$403$	591.884i	1.46869i
$404$	0	0
$405$	−283.143	−0.699120
$406$	0	0
$407$	235.756	0.579254
$408$	0	0
$409$	− 5.28557i	− 0.0129232i	−0.999979	−	0.00646158i	$-0.997943\pi$
$409$	− 5.28557i	− 0.0129232i	0.999979	−	0.00646158i	$-0.00205680\pi$
$410$	0	0
$411$	42.9221	0.104433
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−184.195	−0.443843
$416$	0	0
$417$	−67.4921	−0.161851
$418$	0	0
$419$	34.7160	0.0828545	0.0414272	−	0.999142i	$-0.486810\pi$
$419$	34.7160	0.0828545	0.0414272	+	0.999142i	$0.486810\pi$
$420$	0	0
$421$	− 394.337i	− 0.936669i	−0.883551	−	0.468334i	$-0.844854\pi$
$421$	− 394.337i	− 0.936669i	0.883551	−	0.468334i	$-0.155146\pi$
$422$	0	0
$423$	− 454.449i	− 1.07435i
$424$	0	0
$425$	116.791i	0.274803i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	− 36.2401i	− 0.0844757i
$430$	0	0
$431$	431.743	1.00172	0.500862	−	0.865527i	$-0.333016\pi$
$431$	431.743	1.00172	0.500862	+	0.865527i	$0.333016\pi$
$432$	0	0
$433$	318.535i	0.735647i	0.929896	+	0.367823i	$0.119897\pi$
$433$	318.535i	0.735647i	−0.929896	+	0.367823i	$0.880103\pi$
$434$	0	0
$435$	− 25.6565i	− 0.0589803i
$436$	0	0
$437$	−106.696	−0.244156
$438$	0	0
$439$	615.223i	1.40142i	0.713446	+	0.700710i	$0.247133\pi$
$439$	615.223i	1.40142i	−0.713446	+	0.700710i	$0.752867\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 99.8300i	− 0.225350i	−0.993632	−	0.112675i	$-0.964058\pi$
$443$	− 99.8300i	− 0.225350i	0.993632	−	0.112675i	$-0.0359419\pi$
$444$	0	0
$445$	549.452i	1.23472i
$446$	0	0
$447$	7.80515i	0.0174612i
$448$	0	0
$449$	−75.3168	−0.167743	−0.0838717	−	0.996477i	$-0.526729\pi$
$449$	−75.3168	−0.167743	−0.0838717	+	0.996477i	$0.526729\pi$
$450$	0	0
$451$	−356.782	−0.791092
$452$	0	0
$453$	5.94887	0.0131322
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−208.893	−0.457097	−0.228549	−	0.973533i	$-0.573398\pi$
$457$	−208.893	−0.457097	−0.228549	+	0.973533i	$0.573398\pi$
$458$	0	0
$459$	− 43.3337i	− 0.0944090i
$460$	0	0
$461$	−751.461	−1.63007	−0.815034	−	0.579413i	$-0.803282\pi$
$461$	−751.461	−1.63007	−0.815034	+	0.579413i	$0.803282\pi$
$462$	0	0
$463$	3.56075	0.00769060	0.00384530	−	0.999993i	$-0.498776\pi$
$463$	3.56075	0.00769060	0.00384530	+	0.999993i	$0.498776\pi$
$464$	0	0
$465$	29.4941i	0.0634281i
$466$	0	0
$467$	413.891	0.886275	0.443138	−	0.896454i	$-0.353865\pi$
$467$	413.891	0.886275	0.443138	+	0.896454i	$0.353865\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−32.3229	−0.0686261
$472$	0	0
$473$	−196.369	−0.415156
$474$	0	0
$475$	303.843	0.639669
$476$	0	0
$477$	561.240i	1.17660i
$478$	0	0
$479$	907.362i	1.89428i	0.320815	+	0.947142i	$0.396043\pi$
$479$	907.362i	1.89428i	−0.320815	+	0.947142i	$0.603957\pi$
$480$	0	0
$481$	− 542.908i	− 1.12871i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	168.029i	0.346452i
$486$	0	0
$487$	−842.904	−1.73081	−0.865405	−	0.501073i	$-0.832939\pi$
$487$	−842.904	−1.73081	−0.865405	+	0.501073i	$0.832939\pi$
$488$	0	0
$489$	40.4410i	0.0827014i
$490$	0	0
$491$	144.126i	0.293535i	0.989171	+	0.146768i	$0.0468870\pi$
$491$	144.126i	0.293535i	−0.989171	+	0.146768i	$0.953113\pi$
$492$	0	0
$493$	−270.580	−0.548844
$494$	0	0
$495$	251.596i	0.508275i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	381.168i	0.763864i	0.924190	+	0.381932i	$0.124741\pi$
$499$	381.168i	0.763864i	−0.924190	+	0.381932i	$0.875259\pi$
$500$	0	0
$501$	− 36.1629i	− 0.0721815i
$502$	0	0
$503$	− 936.429i	− 1.86169i	−0.365418	−	0.930843i	$-0.619074\pi$
$503$	− 936.429i	− 1.86169i	0.365418	−	0.930843i	$-0.380926\pi$
$504$	0	0
$505$	−533.429	−1.05629
$506$	0	0
$507$	−40.6546	−0.0801867
$508$	0	0
$509$	335.184	0.658515	0.329258	−	0.944240i	$-0.393202\pi$
$509$	335.184	0.658515	0.329258	+	0.944240i	$0.393202\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−112.736	−0.219759
$514$	0	0
$515$	70.8755i	0.137622i
$516$	0	0
$517$	−400.896	−0.775427
$518$	0	0
$519$	49.5824	0.0955345
$520$	0	0
$521$	− 486.908i	− 0.934565i	−0.884108	−	0.467283i	$-0.845233\pi$
$521$	− 486.908i	− 0.934565i	0.884108	−	0.467283i	$-0.154767\pi$
$522$	0	0
$523$	172.626	0.330070	0.165035	−	0.986288i	$-0.447226\pi$
$523$	172.626	0.330070	0.165035	+	0.986288i	$0.447226\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	311.053	0.590234
$528$	0	0
$529$	−510.519	−0.965064
$530$	0	0
$531$	−661.339	−1.24546
$532$	0	0
$533$	821.611i	1.54148i
$534$	0	0
$535$	− 32.6597i	− 0.0610461i
$536$	0	0
$537$	37.8635i	0.0705094i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	578.200i	1.06876i	0.845244	+	0.534381i	$0.179455\pi$
$541$	578.200i	1.06876i	−0.845244	+	0.534381i	$0.820545\pi$
$542$	0	0
$543$	−23.1221	−0.0425822
$544$	0	0
$545$	425.750i	0.781193i
$546$	0	0
$547$	− 454.579i	− 0.831040i	−0.909584	−	0.415520i	$-0.863600\pi$
$547$	− 454.579i	− 0.831040i	0.909584	−	0.415520i	$-0.136400\pi$
$548$	0	0
$549$	−452.143	−0.823576
$550$	0	0
$551$	703.938i	1.27757i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	− 27.0536i	− 0.0487451i
$556$	0	0
$557$	− 29.1447i	− 0.0523244i	−0.999658	−	0.0261622i	$-0.991671\pi$
$557$	− 29.1447i	− 0.0523244i	0.999658	−	0.0261622i	$-0.00832864\pi$
$558$	0	0
$559$	452.205i	0.808953i
$560$	0	0
$561$	−19.0453	−0.0339488
$562$	0	0
$563$	−1028.01	−1.82595	−0.912975	−	0.408015i	$-0.866221\pi$
$563$	−1028.01	−1.82595	−0.912975	+	0.408015i	$0.866221\pi$
$564$	0	0
$565$	442.939	0.783963
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−819.705	−1.44061	−0.720303	−	0.693660i	$-0.755997\pi$
$569$	−819.705	−1.44061	−0.720303	+	0.693660i	$0.755997\pi$
$570$	0	0
$571$	162.605i	0.284772i	0.989811	+	0.142386i	$0.0454774\pi$
$571$	162.605i	0.284772i	−0.989811	+	0.142386i	$0.954523\pi$
$572$	0	0
$573$	7.04760	0.0122995
$574$	0	0
$575$	−52.6299	−0.0915303
$576$	0	0
$577$	152.280i	0.263917i	0.991255	+	0.131958i	$0.0421265\pi$
$577$	152.280i	0.263917i	−0.991255	+	0.131958i	$0.957873\pi$
$578$	0	0
$579$	−61.3852	−0.106019
$580$	0	0
$581$	0	0
$582$	0	0
$583$	495.103	0.849233
$584$	0	0
$585$	579.385	0.990401
$586$	0	0
$587$	−894.404	−1.52369	−0.761843	−	0.647761i	$-0.775705\pi$
$587$	−894.404	−1.52369	−0.761843	+	0.647761i	$0.775705\pi$
$588$	0	0
$589$	− 809.232i	− 1.37391i
$590$	0	0
$591$	23.9846i	0.0405831i
$592$	0	0
$593$	− 190.213i	− 0.320764i	−0.987055	−	0.160382i	$-0.948727\pi$
$593$	− 190.213i	− 0.320764i	0.987055	−	0.160382i	$-0.0512725\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	78.2961i	0.131149i
$598$	0	0
$599$	293.663	0.490256	0.245128	−	0.969491i	$-0.421170\pi$
$599$	293.663	0.490256	0.245128	+	0.969491i	$0.421170\pi$
$600$	0	0
$601$	− 597.574i	− 0.994299i	−0.867665	−	0.497150i	$-0.834380\pi$
$601$	− 597.574i	− 0.994299i	0.867665	−	0.497150i	$-0.165620\pi$
$602$	0	0
$603$	1121.31i	1.85956i
$604$	0	0
$605$	−210.237	−0.347500
$606$	0	0
$607$	− 12.3114i	− 0.0202823i	−0.999949	−	0.0101412i	$-0.996772\pi$
$607$	− 12.3114i	− 0.0202823i	0.999949	−	0.0101412i	$-0.00322809\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	923.197i	1.51096i
$612$	0	0
$613$	137.290i	0.223964i	0.993710	+	0.111982i	$0.0357200\pi$
$613$	137.290i	0.223964i	−0.993710	+	0.111982i	$0.964280\pi$
$614$	0	0
$615$	40.9416i	0.0665717i
$616$	0	0
$617$	290.516	0.470853	0.235427	−	0.971892i	$-0.424351\pi$
$617$	290.516	0.470853	0.235427	+	0.971892i	$0.424351\pi$
$618$	0	0
$619$	102.317	0.165294	0.0826471	−	0.996579i	$-0.473663\pi$
$619$	102.317	0.165294	0.0826471	+	0.996579i	$0.473663\pi$
$620$	0	0
$621$	19.5276	0.0314454
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−169.064	−0.270502
$626$	0	0
$627$	49.5479i	0.0790238i
$628$	0	0
$629$	−285.315	−0.453600
$630$	0	0
$631$	562.739	0.891820	0.445910	−	0.895078i	$-0.352880\pi$
$631$	562.739	0.891820	0.445910	+	0.895078i	$0.352880\pi$
$632$	0	0
$633$	− 31.8759i	− 0.0503569i
$634$	0	0
$635$	−205.786	−0.324072
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−47.7016	−0.0746504
$640$	0	0
$641$	−752.549	−1.17402	−0.587012	−	0.809578i	$-0.699696\pi$
$641$	−752.549	−1.17402	−0.587012	+	0.809578i	$0.699696\pi$
$642$	0	0
$643$	253.143	0.393690	0.196845	−	0.980435i	$-0.436930\pi$
$643$	253.143	0.393690	0.196845	+	0.980435i	$0.436930\pi$
$644$	0	0
$645$	22.5337i	0.0349360i
$646$	0	0
$647$	− 560.598i	− 0.866457i	−0.901284	−	0.433229i	$-0.857374\pi$
$647$	− 560.598i	− 0.866457i	0.901284	−	0.433229i	$-0.142626\pi$
$648$	0	0
$649$	583.406i	0.898931i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 695.587i	− 1.06522i	−0.846362	−	0.532609i	$-0.821212\pi$
$653$	− 695.587i	− 1.06522i	0.846362	−	0.532609i	$-0.178788\pi$
$654$	0	0
$655$	443.970	0.677817
$656$	0	0
$657$	244.097i	0.371533i
$658$	0	0
$659$	323.387i	0.490724i	0.969432	+	0.245362i	$0.0789068\pi$
$659$	323.387i	0.490724i	−0.969432	+	0.245362i	$0.921093\pi$
$660$	0	0
$661$	−31.0335	−0.0469493	−0.0234747	−	0.999724i	$-0.507473\pi$
$661$	−31.0335	−0.0469493	−0.0234747	+	0.999724i	$0.507473\pi$
$662$	0	0
$663$	43.8581i	0.0661510i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 121.932i	− 0.182807i
$668$	0	0
$669$	2.26568i	0.00338666i
$670$	0	0
$671$	398.862i	0.594429i
$672$	0	0
$673$	1011.75	1.50334	0.751670	−	0.659539i	$-0.229249\pi$
$673$	1011.75	1.50334	0.751670	+	0.659539i	$0.229249\pi$
$674$	0	0
$675$	−55.6094	−0.0823843
$676$	0	0
$677$	69.4753	0.102622	0.0513112	−	0.998683i	$-0.483660\pi$
$677$	69.4753	0.102622	0.0513112	+	0.998683i	$0.483660\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−69.0614	−0.101412
$682$	0	0
$683$	952.085i	1.39397i	0.717083	+	0.696987i	$0.245477\pi$
$683$	952.085i	1.39397i	−0.717083	+	0.696987i	$0.754523\pi$
$684$	0	0
$685$	605.348	0.883720
$686$	0	0
$687$	−83.8839	−0.122102
$688$	0	0
$689$	− 1140.14i	− 1.65478i
$690$	0	0
$691$	−68.1509	−0.0986265	−0.0493132	−	0.998783i	$-0.515703\pi$
$691$	−68.1509	−0.0986265	−0.0493132	+	0.998783i	$0.515703\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−951.870	−1.36960
$696$	0	0
$697$	431.782	0.619486
$698$	0	0
$699$	−40.0745	−0.0573311
$700$	0	0
$701$	1.67276i	0.00238625i	0.999999	+	0.00119312i	$0.000379783\pi$
$701$	1.67276i	0.00238625i	−0.999999	+	0.00119312i	$0.999620\pi$
$702$	0	0
$703$	742.271i	1.05586i
$704$	0	0
$705$	46.0037i	0.0652535i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	− 52.8817i	− 0.0745863i	−0.999304	−	0.0372931i	$-0.988126\pi$
$709$	− 52.8817i	− 0.0745863i	0.999304	−	0.0372931i	$-0.0118735\pi$
$710$	0	0
$711$	−921.073	−1.29546
$712$	0	0
$713$	140.171i	0.196593i
$714$	0	0
$715$	− 511.109i	− 0.714838i
$716$	0	0
$717$	12.4091	0.0173069
$718$	0	0
$719$	− 951.679i	− 1.32362i	−0.749674	−	0.661808i	$-0.769790\pi$
$719$	− 951.679i	− 1.32362i	0.749674	−	0.661808i	$-0.230210\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 49.9811i	− 0.0691302i
$724$	0	0
$725$	347.231i	0.478939i
$726$	0	0
$727$	− 1061.98i	− 1.46078i	−0.683032	−	0.730388i	$-0.739339\pi$
$727$	− 1061.98i	− 1.46078i	0.683032	−	0.730388i	$-0.260661\pi$
$728$	0	0
$729$	−698.013	−0.957494
$730$	0	0
$731$	237.647	0.325099
$732$	0	0
$733$	0.296203	0.000404097 0	0.000202048	−	1.00000i	$-0.499936\pi$
$733$	0.296203	0.000404097 0	0.000202048		1.00000i	$0.499936\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	989.175	1.34216
$738$	0	0
$739$	− 203.546i	− 0.275434i	−0.990472	−	0.137717i	$-0.956024\pi$
$739$	− 203.546i	− 0.275434i	0.990472	−	0.137717i	$-0.0439765\pi$
$740$	0	0
$741$	114.101	0.153982
$742$	0	0
$743$	−1142.13	−1.53718	−0.768592	−	0.639739i	$-0.779042\pi$
$743$	−1142.13	−1.53718	−0.768592	+	0.639739i	$0.779042\pi$
$744$	0	0
$745$	110.079i	0.147758i
$746$	0	0
$747$	−460.818	−0.616892
$748$	0	0
$749$	0	0
$750$	0	0
$751$	792.136	1.05477	0.527387	−	0.849625i	$-0.323172\pi$
$751$	792.136	1.05477	0.527387	+	0.849625i	$0.323172\pi$
$752$	0	0
$753$	79.9015	0.106111
$754$	0	0
$755$	83.8995	0.111125
$756$	0	0
$757$	− 1179.34i	− 1.55792i	−0.627076	−	0.778958i	$-0.715748\pi$
$757$	− 1179.34i	− 1.55792i	0.627076	−	0.778958i	$-0.284252\pi$
$758$	0	0
$759$	− 8.58241i	− 0.0113075i
$760$	0	0
$761$	− 228.479i	− 0.300235i	−0.988668	−	0.150118i	$-0.952035\pi$
$761$	− 228.479i	− 0.300235i	0.988668	−	0.150118i	$-0.0479652\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	− 304.484i	− 0.398019i
$766$	0	0
$767$	1343.49	1.75161
$768$	0	0
$769$	83.4232i	0.108483i	0.998528	+	0.0542414i	$0.0172740\pi$
$769$	83.4232i	0.108483i	−0.998528	+	0.0542414i	$0.982726\pi$
$770$	0	0
$771$	− 96.3670i	− 0.124990i
$772$	0	0
$773$	570.637	0.738210	0.369105	−	0.929388i	$-0.379664\pi$
$773$	570.637	0.738210	0.369105	+	0.929388i	$0.379664\pi$
$774$	0	0
$775$	− 399.169i	− 0.515057i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 1123.32i	− 1.44200i
$780$	0	0
$781$	42.0804i	0.0538802i
$782$	0	0
$783$	− 128.835i	− 0.164540i
$784$	0	0
$785$	−455.864	−0.580718
$786$	0	0
$787$	765.437	0.972601	0.486301	−	0.873792i	$-0.338346\pi$
$787$	765.437	0.972601	0.486301	+	0.873792i	$0.338346\pi$
$788$	0	0
$789$	49.7210	0.0630178
$790$	0	0
$791$	0	0
$792$	0	0
$793$	918.513	1.15828
$794$	0	0
$795$	− 56.8142i	− 0.0714644i
$796$	0	0
$797$	577.729	0.724880	0.362440	−	0.932007i	$-0.381944\pi$
$797$	577.729	0.724880	0.362440	+	0.932007i	$0.381944\pi$
$798$	0	0
$799$	485.168	0.607220
$800$	0	0
$801$	1374.62i	1.71613i
$802$	0	0
$803$	215.333	0.268160
$804$	0	0
$805$	0	0
$806$	0	0
$807$	25.9110	0.0321078
$808$	0	0
$809$	−82.9648	−0.102552	−0.0512761	−	0.998685i	$-0.516329\pi$
$809$	−82.9648	−0.102552	−0.0512761	+	0.998685i	$0.516329\pi$
$810$	0	0
$811$	525.164	0.647552	0.323776	−	0.946134i	$-0.395048\pi$
$811$	525.164	0.647552	0.323776	+	0.946134i	$0.395048\pi$
$812$	0	0
$813$	64.9150i	0.0798462i
$814$	0	0
$815$	570.357i	0.699824i
$816$	0	0
$817$	− 618.260i	− 0.756745i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	584.705i	0.712186i	0.934451	+	0.356093i	$0.115891\pi$
$821$	584.705i	0.712186i	−0.934451	+	0.356093i	$0.884109\pi$
$822$	0	0
$823$	−1180.97	−1.43496	−0.717478	−	0.696581i	$-0.754704\pi$
$823$	−1180.97	−1.43496	−0.717478	+	0.696581i	$0.754704\pi$
$824$	0	0
$825$	24.4405i	0.0296248i
$826$	0	0
$827$	336.806i	0.407262i	0.979048	+	0.203631i	$0.0652743\pi$
$827$	336.806i	0.407262i	−0.979048	+	0.203631i	$0.934726\pi$
$828$	0	0
$829$	−368.291	−0.444259	−0.222130	−	0.975017i	$-0.571301\pi$
$829$	−368.291	−0.444259	−0.222130	+	0.975017i	$0.571301\pi$
$830$	0	0
$831$	49.8229i	0.0599553i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	− 510.022i	− 0.610804i
$836$	0	0
$837$	148.106i	0.176949i
$838$	0	0
$839$	709.889i	0.846113i	0.906103	+	0.423056i	$0.139043\pi$
$839$	709.889i	0.846113i	−0.906103	+	0.423056i	$0.860957\pi$
$840$	0	0
$841$	36.5402	0.0434485
$842$	0	0
$843$	17.8020	0.0211174
$844$	0	0
$845$	−573.370	−0.678544
$846$	0	0
$847$	0	0
$848$	0	0
$849$	75.2146	0.0885919
$850$	0	0
$851$	− 128.572i	− 0.151083i
$852$	0	0
$853$	1136.65	1.33253	0.666267	−	0.745713i	$-0.267891\pi$
$853$	1136.65	1.33253	0.666267	+	0.745713i	$0.267891\pi$
$854$	0	0
$855$	−792.143	−0.926483
$856$	0	0
$857$	668.161i	0.779652i	0.920889	+	0.389826i	$0.127465\pi$
$857$	668.161i	0.779652i	−0.920889	+	0.389826i	$0.872535\pi$
$858$	0	0
$859$	200.755	0.233708	0.116854	−	0.993149i	$-0.462719\pi$
$859$	200.755	0.233708	0.116854	+	0.993149i	$0.462719\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−43.5711	−0.0504879	−0.0252440	−	0.999681i	$-0.508036\pi$
$863$	−43.5711	−0.0504879	−0.0252440	+	0.999681i	$0.508036\pi$
$864$	0	0
$865$	699.282	0.808418
$866$	0	0
$867$	−50.1422	−0.0578342
$868$	0	0
$869$	812.533i	0.935020i
$870$	0	0
$871$	− 2277.91i	− 2.61528i
$872$	0	0
$873$	420.375i	0.481530i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	149.236i	0.170166i	0.996374	+	0.0850832i	$0.0271156\pi$
$877$	149.236i	0.170166i	−0.996374	+	0.0850832i	$0.972884\pi$
$878$	0	0
$879$	−34.3316	−0.0390576
$880$	0	0
$881$	− 865.257i	− 0.982130i	−0.871123	−	0.491065i	$-0.836608\pi$
$881$	− 865.257i	− 0.982130i	0.871123	−	0.491065i	$-0.163392\pi$
$882$	0	0
$883$	− 1476.24i	− 1.67184i	−0.548850	−	0.835921i	$-0.684934\pi$
$883$	− 1476.24i	− 1.67184i	0.548850	−	0.835921i	$-0.315066\pi$
$884$	0	0
$885$	66.9472	0.0756465
$886$	0	0
$887$	− 598.326i	− 0.674550i	−0.941406	−	0.337275i	$-0.890495\pi$
$887$	− 598.326i	− 0.674550i	0.941406	−	0.337275i	$-0.109505\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	624.892i	0.701338i
$892$	0	0
$893$	− 1262.21i	− 1.41345i
$894$	0	0
$895$	534.006i	0.596655i
$896$	0	0
$897$	−19.7639	−0.0220333
$898$	0	0
$899$	924.789	1.02869
$900$	0	0
$901$	−599.179	−0.665015
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−326.101	−0.360333
$906$	0	0
$907$	− 1689.33i	− 1.86255i	−0.364318	−	0.931274i	$-0.618698\pi$
$907$	− 1689.33i	− 1.86255i	0.364318	−	0.931274i	$-0.381302\pi$
$908$	0	0
$909$	−1334.53	−1.46813
$910$	0	0
$911$	813.339	0.892798	0.446399	−	0.894834i	$-0.352706\pi$
$911$	813.339	0.892798	0.446399	+	0.894834i	$0.352706\pi$
$912$	0	0
$913$	406.515i	0.445252i
$914$	0	0
$915$	45.7703	0.0500222
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−1502.98	−1.63545	−0.817724	−	0.575610i	$-0.804765\pi$
$919$	−1502.98	−1.63545	−0.817724	+	0.575610i	$0.804765\pi$
$920$	0	0
$921$	19.2994	0.0209549
$922$	0	0
$923$	96.9043	0.104988
$924$	0	0
$925$	366.139i	0.395826i
$926$	0	0
$927$	177.316i	0.191280i
$928$	0	0
$929$	1503.09i	1.61796i	0.587835	+	0.808981i	$0.299980\pi$
$929$	1503.09i	1.61796i	−0.587835	+	0.808981i	$0.700020\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	− 50.1684i	− 0.0537711i
$934$	0	0
$935$	−268.604	−0.287277
$936$	0	0
$937$	419.349i	0.447545i	0.974641	+	0.223772i	$0.0718372\pi$
$937$	419.349i	0.447545i	−0.974641	+	0.223772i	$0.928163\pi$
$938$	0	0
$939$	− 14.0279i	− 0.0149392i
$940$	0	0
$941$	523.359	0.556174	0.278087	−	0.960556i	$-0.410300\pi$
$941$	523.359	0.556174	0.278087	+	0.960556i	$0.410300\pi$
$942$	0	0
$943$	194.575i	0.206336i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	360.208i	0.380368i	0.981748	+	0.190184i	$0.0609084\pi$
$947$	360.208i	0.380368i	−0.981748	+	0.190184i	$0.939092\pi$
$948$	0	0
$949$	− 495.876i	− 0.522525i
$950$	0	0
$951$	− 75.7642i	− 0.0796679i
$952$	0	0
$953$	1242.81	1.30410	0.652051	−	0.758175i	$-0.273909\pi$
$953$	1242.81	1.30410	0.652051	+	0.758175i	$0.273909\pi$
$954$	0	0
$955$	99.3953	0.104079
$956$	0	0
$957$	−56.6233	−0.0591675
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−102.117	−0.106261
$962$	0	0
$963$	− 81.7078i	− 0.0848471i
$964$	0	0
$965$	−865.742	−0.897142
$966$	0	0
$967$	−81.8793	−0.0846735	−0.0423368	−	0.999103i	$-0.513480\pi$
$967$	−81.8793	−0.0846735	−0.0423368	+	0.999103i	$0.513480\pi$
$968$	0	0
$969$	− 59.9634i	− 0.0618817i
$970$	0	0
$971$	−818.105	−0.842538	−0.421269	−	0.906936i	$-0.638415\pi$
$971$	−818.105	−0.842538	−0.421269	+	0.906936i	$0.638415\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	56.2823	0.0577255
$976$	0	0
$977$	896.310	0.917410	0.458705	−	0.888589i	$-0.348313\pi$
$977$	896.310	0.917410	0.458705	+	0.888589i	$0.348313\pi$
$978$	0	0
$979$	1212.63	1.23864
$980$	0	0
$981$	1065.14i	1.08577i
$982$	0	0
$983$	− 984.272i	− 1.00129i	−0.865652	−	0.500647i	$-0.833096\pi$
$983$	− 984.272i	− 1.00129i	0.865652	−	0.500647i	$-0.166904\pi$
$984$	0	0
$985$	338.265i	0.343417i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	107.092i	0.108283i
$990$	0	0
$991$	1224.07	1.23519	0.617596	−	0.786496i	$-0.288107\pi$
$991$	1224.07	1.23519	0.617596	+	0.786496i	$0.288107\pi$
$992$	0	0
$993$	− 95.0667i	− 0.0957368i
$994$	0	0
$995$	1104.24i	1.10979i
$996$	0	0
$997$	373.649	0.374773	0.187387	−	0.982286i	$-0.439998\pi$
$997$	373.649	0.374773	0.187387	+	0.982286i	$0.439998\pi$
$998$	0	0
$999$	− 135.851i	− 0.135987i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1568.3.h.a.881.14		28
4.3	odd	2		392.3.h.a.293.16		28
7.4	even	3		224.3.n.a.145.8		28
7.5	odd	6		224.3.n.a.17.7		28
7.6	odd	2	inner	1568.3.h.a.881.16		28
8.3	odd	2		392.3.h.a.293.13		28
8.5	even	2	inner	1568.3.h.a.881.15		28
28.3	even	6		392.3.j.e.117.2		28
28.11	odd	6		56.3.j.a.5.2	✓	28
28.19	even	6		56.3.j.a.45.12	yes	28
28.23	odd	6		392.3.j.e.325.12		28
28.27	even	2		392.3.h.a.293.15		28
56.3	even	6		392.3.j.e.117.12		28
56.5	odd	6		224.3.n.a.17.8		28
56.11	odd	6		56.3.j.a.5.12	yes	28
56.13	odd	2	inner	1568.3.h.a.881.13		28
56.19	even	6		56.3.j.a.45.2	yes	28
56.27	even	2		392.3.h.a.293.14		28
56.51	odd	6		392.3.j.e.325.2		28
56.53	even	6		224.3.n.a.145.7		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.3.j.a.5.2	✓	28	28.11	odd	6
56.3.j.a.5.12	yes	28	56.11	odd	6
56.3.j.a.45.2	yes	28	56.19	even	6
56.3.j.a.45.12	yes	28	28.19	even	6
224.3.n.a.17.7		28	7.5	odd	6
224.3.n.a.17.8		28	56.5	odd	6
224.3.n.a.145.7		28	56.53	even	6
224.3.n.a.145.8		28	7.4	even	3
392.3.h.a.293.13		28	8.3	odd	2
392.3.h.a.293.14		28	56.27	even	2
392.3.h.a.293.15		28	28.27	even	2
392.3.h.a.293.16		28	4.3	odd	2
392.3.j.e.117.2		28	28.3	even	6
392.3.j.e.117.12		28	56.3	even	6
392.3.j.e.325.2		28	56.51	odd	6
392.3.j.e.325.12		28	28.23	odd	6
1568.3.h.a.881.13		28	56.13	odd	2	inner
1568.3.h.a.881.14		28	1.1	even	1	trivial
1568.3.h.a.881.15		28	8.5	even	2	inner
1568.3.h.a.881.16		28	7.6	odd	2	inner

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:	\( N \)	\(=\)	\( 1568 = 2^{5} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1568.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.253256 q^{3} -3.57178 q^{5} -8.93586 q^{9} +O(q^{10})\)	\(q-0.253256 q^{3} -3.57178 q^{5} -8.93586 q^{9} +7.88285i q^{11} +18.1529 q^{13} +0.904575 q^{15} -9.53990i q^{17} -24.8189 q^{19} +4.29899 q^{23} -12.2424 q^{25} +4.54237 q^{27} -28.3630i q^{29} +32.6055i q^{31} -1.99638i q^{33} -29.9075i q^{37} -4.59733 q^{39} +45.2606i q^{41} +24.9109i q^{43} +31.9169 q^{45} +50.8567i q^{47} +2.41604i q^{51} -62.8076i q^{53} -28.1558i q^{55} +6.28554 q^{57} +74.0096 q^{59} +50.5987 q^{61} -64.8382 q^{65} -125.484i q^{67} -1.08875 q^{69} +5.33822 q^{71} -27.3166i q^{73} +3.10046 q^{75} +103.076 q^{79} +79.2724 q^{81} +51.5695 q^{83} +34.0744i q^{85} +7.18310i q^{87} -153.832i q^{89} -8.25753i q^{93} +88.6476 q^{95} -47.0436i q^{97} -70.4400i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 64 q^{9}+O(q^{10}) \) 28 * q + 64 * q^9	\( 28 q + 64 q^{9} - 28 q^{15} + 60 q^{23} + 64 q^{25} - 40 q^{39} + 124 q^{57} + 104 q^{65} + 136 q^{71} - 324 q^{79} + 36 q^{81} + 580 q^{95}+O(q^{100}) \) 28 * q + 64 * q^9 - 28 * q^15 + 60 * q^23 + 64 * q^25 - 40 * q^39 + 124 * q^57 + 104 * q^65 + 136 * q^71 - 324 * q^79 + 36 * q^81 + 580 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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