LMFDB - Embedded newform 5712.2.a.ca.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5712,2,Mod(1,5712)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5712, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 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("5712.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 5712 = 2^{4} \cdot 3 \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5712.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,4,0,5,0,4,0,4,0,-1,0,5,0,5,0,4,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	yes
Analytic conductor:	$45.6105496346$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.183064.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 10x^{2} + 6x + 8 $ x^4 - x^3 - 10x^2 + 6x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2856)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.27274$ of defining polynomial
Character	$\chi$	$=$	5712.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +2.27274 q^{5} +1.00000 q^{7} +1.00000 q^{9} +3.38012 q^{11} +2.27274 q^{13} +2.27274 q^{15} +1.00000 q^{17} -2.87007 q^{19} +1.00000 q^{21} -6.52294 q^{23} +0.165367 q^{25} +1.00000 q^{27} -3.74014 q^{29} +9.39301 q^{31} +3.38012 q^{33} +2.27274 q^{35} +2.51005 q^{37} +2.27274 q^{39} +10.6102 q^{41} -9.66575 q^{43} +2.27274 q^{45} +2.08728 q^{47} +1.00000 q^{49} +1.00000 q^{51} +8.25019 q^{53} +7.68215 q^{55} -2.87007 q^{57} +14.4840 q^{59} +10.3020 q^{61} +1.00000 q^{63} +5.16537 q^{65} +2.59733 q^{67} -6.52294 q^{69} -13.0459 q^{71} +5.14282 q^{73} +0.165367 q^{75} +3.38012 q^{77} -3.48995 q^{79} +1.00000 q^{81} -2.59733 q^{83} +2.27274 q^{85} -3.74014 q^{87} -3.90306 q^{89} +2.27274 q^{91} +9.39301 q^{93} -6.52294 q^{95} -0.597327 q^{97} +3.38012 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 5 q^{5} + 4 q^{7} + 4 q^{9} - q^{11} + 5 q^{13} + 5 q^{15} + 4 q^{17} + 4 q^{21} + 4 q^{23} + 7 q^{25} + 4 q^{27} + 8 q^{29} - 4 q^{31} - q^{33} + 5 q^{35} + 7 q^{37} + 5 q^{39} + 8 q^{41}+ \cdots - q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + 5 * q^5 + 4 * q^7 + 4 * q^9 - q^11 + 5 * q^13 + 5 * q^15 + 4 * q^17 + 4 * q^21 + 4 * q^23 + 7 * q^25 + 4 * q^27 + 8 * q^29 - 4 * q^31 - q^33 + 5 * q^35 + 7 * q^37 + 5 * q^39 + 8 * q^41 + 7 * q^43 + 5 * q^45 + 4 * q^47 + 4 * q^49 + 4 * q^51 + 7 * q^53 - 9 * q^55 + 16 * q^61 + 4 * q^63 + 27 * q^65 + 3 * q^67 + 4 * q^69 + 8 * q^71 + 5 * q^73 + 7 * q^75 - q^77 - 17 * q^79 + 4 * q^81 - 3 * q^83 + 5 * q^85 + 8 * q^87 + 29 * q^89 + 5 * q^91 - 4 * q^93 + 4 * q^95 + 5 * q^97 - q^99

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	2.27274	1.01640	0.508201	−	0.861238i	$-0.330311\pi$
$5$	2.27274	1.01640	0.508201	+	0.861238i	$0.330311\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.38012	1.01915	0.509573	−	0.860428i	$-0.329804\pi$
$11$	3.38012	1.01915	0.509573	+	0.860428i	$0.329804\pi$
$12$	0	0
$13$	2.27274	0.630346	0.315173	−	0.949034i	$-0.397937\pi$
$13$	2.27274	0.630346	0.315173	+	0.949034i	$0.397937\pi$
$14$	0	0
$15$	2.27274	0.586820
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−2.87007	−0.658439	−0.329220	−	0.944253i	$-0.606786\pi$
$19$	−2.87007	−0.658439	−0.329220	+	0.944253i	$0.606786\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−6.52294	−1.36013	−0.680063	−	0.733153i	$-0.738048\pi$
$23$	−6.52294	−1.36013	−0.680063	+	0.733153i	$0.738048\pi$
$24$	0	0
$25$	0.165367	0.0330733
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−3.74014	−0.694527	−0.347263	−	0.937768i	$-0.612889\pi$
$29$	−3.74014	−0.694527	−0.347263	+	0.937768i	$0.612889\pi$
$30$	0	0
$31$	9.39301	1.68703	0.843517	−	0.537102i	$-0.180481\pi$
$31$	9.39301	1.68703	0.843517	+	0.537102i	$0.180481\pi$
$32$	0	0
$33$	3.38012	0.588404
$34$	0	0
$35$	2.27274	0.384164
$36$	0	0
$37$	2.51005	0.412650	0.206325	−	0.978484i	$-0.433850\pi$
$37$	2.51005	0.412650	0.206325	+	0.978484i	$0.433850\pi$
$38$	0	0
$39$	2.27274	0.363930
$40$	0	0
$41$	10.6102	1.65704	0.828518	−	0.559962i	$-0.189184\pi$
$41$	10.6102	1.65704	0.828518	+	0.559962i	$0.189184\pi$
$42$	0	0
$43$	−9.66575	−1.47401	−0.737007	−	0.675885i	$-0.763761\pi$
$43$	−9.66575	−1.47401	−0.737007	+	0.675885i	$0.763761\pi$
$44$	0	0
$45$	2.27274	0.338801
$46$	0	0
$47$	2.08728	0.304460	0.152230	−	0.988345i	$-0.451354\pi$
$47$	2.08728	0.304460	0.152230	+	0.988345i	$0.451354\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.00000	0.140028
$52$	0	0
$53$	8.25019	1.13325	0.566626	−	0.823975i	$-0.308249\pi$
$53$	8.25019	1.13325	0.566626	+	0.823975i	$0.308249\pi$
$54$	0	0
$55$	7.68215	1.03586
$56$	0	0
$57$	−2.87007	−0.380150
$58$	0	0
$59$	14.4840	1.88565	0.942827	−	0.333282i	$-0.108156\pi$
$59$	14.4840	1.88565	0.942827	+	0.333282i	$0.108156\pi$
$60$	0	0
$61$	10.3020	1.31904	0.659520	−	0.751687i	$-0.270760\pi$
$61$	10.3020	1.31904	0.659520	+	0.751687i	$0.270760\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	5.16537	0.640685
$66$	0	0
$67$	2.59733	0.317314	0.158657	−	0.987334i	$-0.449284\pi$
$67$	2.59733	0.317314	0.158657	+	0.987334i	$0.449284\pi$
$68$	0	0
$69$	−6.52294	−0.785269
$70$	0	0
$71$	−13.0459	−1.54826	−0.774130	−	0.633026i	$-0.781812\pi$
$71$	−13.0459	−1.54826	−0.774130	+	0.633026i	$0.781812\pi$
$72$	0	0
$73$	5.14282	0.601921	0.300960	−	0.953637i	$-0.402693\pi$
$73$	5.14282	0.601921	0.300960	+	0.953637i	$0.402693\pi$
$74$	0	0
$75$	0.165367	0.0190949
$76$	0	0
$77$	3.38012	0.385201
$78$	0	0
$79$	−3.48995	−0.392650	−0.196325	−	0.980539i	$-0.562901\pi$
$79$	−3.48995	−0.392650	−0.196325	+	0.980539i	$0.562901\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−2.59733	−0.285094	−0.142547	−	0.989788i	$-0.545529\pi$
$83$	−2.59733	−0.285094	−0.142547	+	0.989788i	$0.545529\pi$
$84$	0	0
$85$	2.27274	0.246514
$86$	0	0
$87$	−3.74014	−0.400985
$88$	0	0
$89$	−3.90306	−0.413723	−0.206862	−	0.978370i	$-0.566325\pi$
$89$	−3.90306	−0.413723	−0.206862	+	0.978370i	$0.566325\pi$
$90$	0	0
$91$	2.27274	0.238248
$92$	0	0
$93$	9.39301	0.974010
$94$	0	0
$95$	−6.52294	−0.669239
$96$	0	0
$97$	−0.597327	−0.0606493	−0.0303247	−	0.999540i	$-0.509654\pi$
$97$	−0.597327	−0.0606493	−0.0303247	+	0.999540i	$0.509654\pi$
$98$	0	0
$99$	3.38012	0.339715
$100$	0	0
$101$	0.0164006	0.00163192	0.000815962	−	1.00000i	$-0.499740\pi$
$101$	0.0164006	0.00163192	0.000815962		1.00000i	$0.499740\pi$
$102$	0	0
$103$	0.272744	0.0268743	0.0134371	−	0.999910i	$-0.495723\pi$
$103$	0.272744	0.0268743	0.0134371	+	0.999910i	$0.495723\pi$
$104$	0	0
$105$	2.27274	0.221797
$106$	0	0
$107$	−4.23731	−0.409636	−0.204818	−	0.978800i	$-0.565660\pi$
$107$	−4.23731	−0.409636	−0.204818	+	0.978800i	$0.565660\pi$
$108$	0	0
$109$	−5.65287	−0.541446	−0.270723	−	0.962657i	$-0.587263\pi$
$109$	−5.65287	−0.541446	−0.270723	+	0.962657i	$0.587263\pi$
$110$	0	0
$111$	2.51005	0.238244
$112$	0	0
$113$	10.5748	0.994791	0.497396	−	0.867524i	$-0.334290\pi$
$113$	10.5748	0.994791	0.497396	+	0.867524i	$0.334290\pi$
$114$	0	0
$115$	−14.8250	−1.38244
$116$	0	0
$117$	2.27274	0.210115
$118$	0	0
$119$	1.00000	0.0916698
$120$	0	0
$121$	0.425225	0.0386568
$122$	0	0
$123$	10.6102	0.956691
$124$	0	0
$125$	−10.9879	−0.982786
$126$	0	0
$127$	2.56804	0.227877	0.113938	−	0.993488i	$-0.463653\pi$
$127$	2.56804	0.227877	0.113938	+	0.993488i	$0.463653\pi$
$128$	0	0
$129$	−9.66575	−0.851022
$130$	0	0
$131$	−1.12993	−0.0987224	−0.0493612	−	0.998781i	$-0.515719\pi$
$131$	−1.12993	−0.0987224	−0.0493612	+	0.998781i	$0.515719\pi$
$132$	0	0
$133$	−2.87007	−0.248867
$134$	0	0
$135$	2.27274	0.195607
$136$	0	0
$137$	−22.7696	−1.94534	−0.972670	−	0.232193i	$-0.925410\pi$
$137$	−22.7696	−1.94534	−0.972670	+	0.232193i	$0.925410\pi$
$138$	0	0
$139$	−16.8121	−1.42598	−0.712991	−	0.701173i	$-0.752660\pi$
$139$	−16.8121	−1.42598	−0.712991	+	0.701173i	$0.752660\pi$
$140$	0	0
$141$	2.08728	0.175780
$142$	0	0
$143$	7.68215	0.642414
$144$	0	0
$145$	−8.50039	−0.705919
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	6.85718	0.561762	0.280881	−	0.959743i	$-0.409373\pi$
$149$	6.85718	0.561762	0.280881	+	0.959743i	$0.409373\pi$
$150$	0	0
$151$	4.64916	0.378344	0.189172	−	0.981944i	$-0.439420\pi$
$151$	4.64916	0.378344	0.189172	+	0.981944i	$0.439420\pi$
$152$	0	0
$153$	1.00000	0.0808452
$154$	0	0
$155$	21.3479	1.71471
$156$	0	0
$157$	22.4614	1.79262	0.896309	−	0.443430i	$-0.146239\pi$
$157$	22.4614	1.79262	0.896309	+	0.443430i	$0.146239\pi$
$158$	0	0
$159$	8.25019	0.654283
$160$	0	0
$161$	−6.52294	−0.514079
$162$	0	0
$163$	14.2405	1.11540	0.557702	−	0.830041i	$-0.311683\pi$
$163$	14.2405	1.11540	0.557702	+	0.830041i	$0.311683\pi$
$164$	0	0
$165$	7.68215	0.598055
$166$	0	0
$167$	−21.9678	−1.69992	−0.849959	−	0.526849i	$-0.823373\pi$
$167$	−21.9678	−1.69992	−0.849959	+	0.526849i	$0.823373\pi$
$168$	0	0
$169$	−7.83463	−0.602664
$170$	0	0
$171$	−2.87007	−0.219480
$172$	0	0
$173$	10.1098	0.768636	0.384318	−	0.923201i	$-0.374437\pi$
$173$	10.1098	0.768636	0.384318	+	0.923201i	$0.374437\pi$
$174$	0	0
$175$	0.165367	0.0125005
$176$	0	0
$177$	14.4840	1.08868
$178$	0	0
$179$	−21.1167	−1.57834	−0.789170	−	0.614175i	$-0.789489\pi$
$179$	−21.1167	−1.57834	−0.789170	+	0.614175i	$0.789489\pi$
$180$	0	0
$181$	4.11108	0.305574	0.152787	−	0.988259i	$-0.451175\pi$
$181$	4.11108	0.305574	0.152787	+	0.988259i	$0.451175\pi$
$182$	0	0
$183$	10.3020	0.761548
$184$	0	0
$185$	5.70470	0.419418
$186$	0	0
$187$	3.38012	0.247179
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−25.9739	−1.87941	−0.939704	−	0.341989i	$-0.888899\pi$
$191$	−25.9739	−1.87941	−0.939704	+	0.341989i	$0.888899\pi$
$192$	0	0
$193$	7.49668	0.539623	0.269812	−	0.962913i	$-0.413039\pi$
$193$	7.49668	0.539623	0.269812	+	0.962913i	$0.413039\pi$
$194$	0	0
$195$	5.16537	0.369900
$196$	0	0
$197$	16.5229	1.17721	0.588605	−	0.808421i	$-0.299677\pi$
$197$	16.5229	1.17721	0.588605	+	0.808421i	$0.299677\pi$
$198$	0	0
$199$	−22.4389	−1.59065	−0.795325	−	0.606183i	$-0.792700\pi$
$199$	−22.4389	−1.59065	−0.795325	+	0.606183i	$0.792700\pi$
$200$	0	0
$201$	2.59733	0.183201
$202$	0	0
$203$	−3.74014	−0.262506
$204$	0	0
$205$	24.1143	1.68422
$206$	0	0
$207$	−6.52294	−0.453375
$208$	0	0
$209$	−9.70119	−0.671045
$210$	0	0
$211$	0.418009	0.0287769	0.0143884	−	0.999896i	$-0.495420\pi$
$211$	0.418009	0.0287769	0.0143884	+	0.999896i	$0.495420\pi$
$212$	0	0
$213$	−13.0459	−0.893889
$214$	0	0
$215$	−21.9678	−1.49819
$216$	0	0
$217$	9.39301	0.637639
$218$	0	0
$219$	5.14282	0.347519
$220$	0	0
$221$	2.27274	0.152881
$222$	0	0
$223$	11.3705	0.761422	0.380711	−	0.924694i	$-0.375679\pi$
$223$	11.3705	0.761422	0.380711	+	0.924694i	$0.375679\pi$
$224$	0	0
$225$	0.165367	0.0110244
$226$	0	0
$227$	3.24101	0.215113	0.107557	−	0.994199i	$-0.465697\pi$
$227$	3.24101	0.215113	0.107557	+	0.994199i	$0.465697\pi$
$228$	0	0
$229$	−8.99404	−0.594343	−0.297171	−	0.954824i	$-0.596043\pi$
$229$	−8.99404	−0.594343	−0.297171	+	0.954824i	$0.596043\pi$
$230$	0	0
$231$	3.38012	0.222396
$232$	0	0
$233$	19.6658	1.28835	0.644173	−	0.764880i	$-0.277202\pi$
$233$	19.6658	1.28835	0.644173	+	0.764880i	$0.277202\pi$
$234$	0	0
$235$	4.74384	0.309454
$236$	0	0
$237$	−3.48995	−0.226697
$238$	0	0
$239$	18.7506	1.21287	0.606437	−	0.795131i	$-0.292598\pi$
$239$	18.7506	1.21287	0.606437	+	0.795131i	$0.292598\pi$
$240$	0	0
$241$	6.47461	0.417067	0.208533	−	0.978015i	$-0.433131\pi$
$241$	6.47461	0.417067	0.208533	+	0.978015i	$0.433131\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	2.27274	0.145200
$246$	0	0
$247$	−6.52294	−0.415045
$248$	0	0
$249$	−2.59733	−0.164599
$250$	0	0
$251$	−1.53015	−0.0965824	−0.0482912	−	0.998833i	$-0.515378\pi$
$251$	−1.53015	−0.0965824	−0.0482912	+	0.998833i	$0.515378\pi$
$252$	0	0
$253$	−22.0483	−1.38617
$254$	0	0
$255$	2.27274	0.142325
$256$	0	0
$257$	25.7707	1.60753	0.803765	−	0.594946i	$-0.202827\pi$
$257$	25.7707	1.60753	0.803765	+	0.594946i	$0.202827\pi$
$258$	0	0
$259$	2.51005	0.155967
$260$	0	0
$261$	−3.74014	−0.231509
$262$	0	0
$263$	30.9489	1.90839	0.954196	−	0.299181i	$-0.0967133\pi$
$263$	30.9489	1.90839	0.954196	+	0.299181i	$0.0967133\pi$
$264$	0	0
$265$	18.7506	1.15184
$266$	0	0
$267$	−3.90306	−0.238863
$268$	0	0
$269$	25.4156	1.54961	0.774807	−	0.632198i	$-0.217847\pi$
$269$	25.4156	1.54961	0.774807	+	0.632198i	$0.217847\pi$
$270$	0	0
$271$	11.6822	0.709640	0.354820	−	0.934935i	$-0.384542\pi$
$271$	11.6822	0.709640	0.354820	+	0.934935i	$0.384542\pi$
$272$	0	0
$273$	2.27274	0.137553
$274$	0	0
$275$	0.558959	0.0337065
$276$	0	0
$277$	−26.3967	−1.58602	−0.793012	−	0.609206i	$-0.791488\pi$
$277$	−26.3967	−1.58602	−0.793012	+	0.609206i	$0.791488\pi$
$278$	0	0
$279$	9.39301	0.562345
$280$	0	0
$281$	8.83112	0.526820	0.263410	−	0.964684i	$-0.415153\pi$
$281$	8.83112	0.526820	0.263410	+	0.964684i	$0.415153\pi$
$282$	0	0
$283$	4.53582	0.269627	0.134813	−	0.990871i	$-0.456957\pi$
$283$	4.53582	0.269627	0.134813	+	0.990871i	$0.456957\pi$
$284$	0	0
$285$	−6.52294	−0.386385
$286$	0	0
$287$	10.6102	0.626301
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−0.597327	−0.0350159
$292$	0	0
$293$	22.9422	1.34030	0.670149	−	0.742227i	$-0.266230\pi$
$293$	22.9422	1.34030	0.670149	+	0.742227i	$0.266230\pi$
$294$	0	0
$295$	32.9184	1.91658
$296$	0	0
$297$	3.38012	0.196135
$298$	0	0
$299$	−14.8250	−0.857350
$300$	0	0
$301$	−9.66575	−0.557125
$302$	0	0
$303$	0.0164006	0.000942191 0
$304$	0	0
$305$	23.4139	1.34067
$306$	0	0
$307$	−21.6622	−1.23633	−0.618165	−	0.786049i	$-0.712123\pi$
$307$	−21.6622	−1.23633	−0.618165	+	0.786049i	$0.712123\pi$
$308$	0	0
$309$	0.272744	0.0155159
$310$	0	0
$311$	−18.3449	−1.04024	−0.520121	−	0.854093i	$-0.674113\pi$
$311$	−18.3449	−1.04024	−0.520121	+	0.854093i	$0.674113\pi$
$312$	0	0
$313$	−30.8733	−1.74506	−0.872531	−	0.488559i	$-0.837523\pi$
$313$	−30.8733	−1.74506	−0.872531	+	0.488559i	$0.837523\pi$
$314$	0	0
$315$	2.27274	0.128055
$316$	0	0
$317$	−11.6622	−0.655017	−0.327508	−	0.944848i	$-0.606209\pi$
$317$	−11.6622	−0.655017	−0.327508	+	0.944848i	$0.606209\pi$
$318$	0	0
$319$	−12.6421	−0.707824
$320$	0	0
$321$	−4.23731	−0.236503
$322$	0	0
$323$	−2.87007	−0.159695
$324$	0	0
$325$	0.375836	0.0208476
$326$	0	0
$327$	−5.65287	−0.312604
$328$	0	0
$329$	2.08728	0.115075
$330$	0	0
$331$	−1.33992	−0.0736485	−0.0368243	−	0.999322i	$-0.511724\pi$
$331$	−1.33992	−0.0736485	−0.0368243	+	0.999322i	$0.511724\pi$
$332$	0	0
$333$	2.51005	0.137550
$334$	0	0
$335$	5.90306	0.322519
$336$	0	0
$337$	−8.34224	−0.454431	−0.227215	−	0.973845i	$-0.572962\pi$
$337$	−8.34224	−0.454431	−0.227215	+	0.973845i	$0.572962\pi$
$338$	0	0
$339$	10.5748	0.574343
$340$	0	0
$341$	31.7495	1.71933
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−14.8250	−0.798149
$346$	0	0
$347$	−15.0977	−0.810488	−0.405244	−	0.914209i	$-0.632813\pi$
$347$	−15.0977	−0.810488	−0.405244	+	0.914209i	$0.632813\pi$
$348$	0	0
$349$	−6.74736	−0.361178	−0.180589	−	0.983559i	$-0.557800\pi$
$349$	−6.74736	−0.361178	−0.180589	+	0.983559i	$0.557800\pi$
$350$	0	0
$351$	2.27274	0.121310
$352$	0	0
$353$	−2.51005	−0.133597	−0.0667983	−	0.997767i	$-0.521278\pi$
$353$	−2.51005	−0.133597	−0.0667983	+	0.997767i	$0.521278\pi$
$354$	0	0
$355$	−29.6499	−1.57366
$356$	0	0
$357$	1.00000	0.0529256
$358$	0	0
$359$	−17.2185	−0.908755	−0.454378	−	0.890809i	$-0.650138\pi$
$359$	−17.2185	−0.908755	−0.454378	+	0.890809i	$0.650138\pi$
$360$	0	0
$361$	−10.7627	−0.566458
$362$	0	0
$363$	0.425225	0.0223185
$364$	0	0
$365$	11.6883	0.611794
$366$	0	0
$367$	1.09098	0.0569485	0.0284743	−	0.999595i	$-0.490935\pi$
$367$	1.09098	0.0569485	0.0284743	+	0.999595i	$0.490935\pi$
$368$	0	0
$369$	10.6102	0.552346
$370$	0	0
$371$	8.25019	0.428329
$372$	0	0
$373$	−15.4619	−0.800588	−0.400294	−	0.916387i	$-0.631092\pi$
$373$	−15.4619	−0.800588	−0.400294	+	0.916387i	$0.631092\pi$
$374$	0	0
$375$	−10.9879	−0.567412
$376$	0	0
$377$	−8.50039	−0.437792
$378$	0	0
$379$	5.71437	0.293527	0.146764	−	0.989172i	$-0.453114\pi$
$379$	5.71437	0.293527	0.146764	+	0.989172i	$0.453114\pi$
$380$	0	0
$381$	2.56804	0.131565
$382$	0	0
$383$	28.3987	1.45110	0.725552	−	0.688167i	$-0.241584\pi$
$383$	28.3987	1.45110	0.725552	+	0.688167i	$0.241584\pi$
$384$	0	0
$385$	7.68215	0.391519
$386$	0	0
$387$	−9.66575	−0.491338
$388$	0	0
$389$	−19.0131	−0.964001	−0.482001	−	0.876171i	$-0.660090\pi$
$389$	−19.0131	−0.964001	−0.482001	+	0.876171i	$0.660090\pi$
$390$	0	0
$391$	−6.52294	−0.329879
$392$	0	0
$393$	−1.12993	−0.0569974
$394$	0	0
$395$	−7.93176	−0.399090
$396$	0	0
$397$	−36.9844	−1.85619	−0.928096	−	0.372341i	$-0.878555\pi$
$397$	−36.9844	−1.85619	−0.928096	+	0.372341i	$0.878555\pi$
$398$	0	0
$399$	−2.87007	−0.143683
$400$	0	0
$401$	1.19143	0.0594973	0.0297487	−	0.999557i	$-0.490529\pi$
$401$	1.19143	0.0594973	0.0297487	+	0.999557i	$0.490529\pi$
$402$	0	0
$403$	21.3479	1.06341
$404$	0	0
$405$	2.27274	0.112934
$406$	0	0
$407$	8.48428	0.420550
$408$	0	0
$409$	−19.1106	−0.944958	−0.472479	−	0.881342i	$-0.656641\pi$
$409$	−19.1106	−0.944958	−0.472479	+	0.881342i	$0.656641\pi$
$410$	0	0
$411$	−22.7696	−1.12314
$412$	0	0
$413$	14.4840	0.712710
$414$	0	0
$415$	−5.90306	−0.289770
$416$	0	0
$417$	−16.8121	−0.823291
$418$	0	0
$419$	−8.37094	−0.408947	−0.204474	−	0.978872i	$-0.565548\pi$
$419$	−8.37094	−0.408947	−0.204474	+	0.978872i	$0.565548\pi$
$420$	0	0
$421$	−26.1594	−1.27493	−0.637466	−	0.770479i	$-0.720017\pi$
$421$	−26.1594	−1.27493	−0.637466	+	0.770479i	$0.720017\pi$
$422$	0	0
$423$	2.08728	0.101487
$424$	0	0
$425$	0.165367	0.00802146
$426$	0	0
$427$	10.3020	0.498550
$428$	0	0
$429$	7.68215	0.370898
$430$	0	0
$431$	28.5713	1.37623	0.688115	−	0.725602i	$-0.258439\pi$
$431$	28.5713	1.37623	0.688115	+	0.725602i	$0.258439\pi$
$432$	0	0
$433$	3.48154	0.167312	0.0836560	−	0.996495i	$-0.473340\pi$
$433$	3.48154	0.167312	0.0836560	+	0.996495i	$0.473340\pi$
$434$	0	0
$435$	−8.50039	−0.407562
$436$	0	0
$437$	18.7213	0.895561
$438$	0	0
$439$	6.52909	0.311616	0.155808	−	0.987787i	$-0.450202\pi$
$439$	6.52909	0.311616	0.155808	+	0.987787i	$0.450202\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−33.5225	−1.59270	−0.796350	−	0.604836i	$-0.793239\pi$
$443$	−33.5225	−1.59270	−0.796350	+	0.604836i	$0.793239\pi$
$444$	0	0
$445$	−8.87066	−0.420509
$446$	0	0
$447$	6.85718	0.324334
$448$	0	0
$449$	−30.7928	−1.45320	−0.726600	−	0.687061i	$-0.758901\pi$
$449$	−30.7928	−1.45320	−0.726600	+	0.687061i	$0.758901\pi$
$450$	0	0
$451$	35.8638	1.68876
$452$	0	0
$453$	4.64916	0.218437
$454$	0	0
$455$	5.16537	0.242156
$456$	0	0
$457$	−18.3674	−0.859192	−0.429596	−	0.903021i	$-0.641344\pi$
$457$	−18.3674	−0.859192	−0.429596	+	0.903021i	$0.641344\pi$
$458$	0	0
$459$	1.00000	0.0466760
$460$	0	0
$461$	23.5225	1.09555	0.547775	−	0.836626i	$-0.315475\pi$
$461$	23.5225	1.09555	0.547775	+	0.836626i	$0.315475\pi$
$462$	0	0
$463$	−0.129448	−0.00601597	−0.00300798	−	0.999995i	$-0.500957\pi$
$463$	−0.129448	−0.00601597	−0.00300798	+	0.999995i	$0.500957\pi$
$464$	0	0
$465$	21.3479	0.989985
$466$	0	0
$467$	22.1697	1.02589	0.512945	−	0.858422i	$-0.328555\pi$
$467$	22.1697	1.02589	0.512945	+	0.858422i	$0.328555\pi$
$468$	0	0
$469$	2.59733	0.119933
$470$	0	0
$471$	22.4614	1.03497
$472$	0	0
$473$	−32.6714	−1.50223
$474$	0	0
$475$	−0.474614	−0.0217768
$476$	0	0
$477$	8.25019	0.377750
$478$	0	0
$479$	−18.0715	−0.825706	−0.412853	−	0.910798i	$-0.635468\pi$
$479$	−18.0715	−0.825706	−0.412853	+	0.910798i	$0.635468\pi$
$480$	0	0
$481$	5.70470	0.260112
$482$	0	0
$483$	−6.52294	−0.296804
$484$	0	0
$485$	−1.35757	−0.0616441
$486$	0	0
$487$	34.4034	1.55897	0.779484	−	0.626422i	$-0.215481\pi$
$487$	34.4034	1.55897	0.779484	+	0.626422i	$0.215481\pi$
$488$	0	0
$489$	14.2405	0.643979
$490$	0	0
$491$	38.4627	1.73580	0.867898	−	0.496742i	$-0.165470\pi$
$491$	38.4627	1.73580	0.867898	+	0.496742i	$0.165470\pi$
$492$	0	0
$493$	−3.74014	−0.168448
$494$	0	0
$495$	7.68215	0.345287
$496$	0	0
$497$	−13.0459	−0.585187
$498$	0	0
$499$	30.0094	1.34340	0.671702	−	0.740821i	$-0.265564\pi$
$499$	30.0094	1.34340	0.671702	+	0.740821i	$0.265564\pi$
$500$	0	0
$501$	−21.9678	−0.981448
$502$	0	0
$503$	20.2869	0.904547	0.452274	−	0.891879i	$-0.350613\pi$
$503$	20.2869	0.904547	0.452274	+	0.891879i	$0.350613\pi$
$504$	0	0
$505$	0.0372744	0.00165869
$506$	0	0
$507$	−7.83463	−0.347948
$508$	0	0
$509$	3.46191	0.153447	0.0767233	−	0.997052i	$-0.475554\pi$
$509$	3.46191	0.153447	0.0767233	+	0.997052i	$0.475554\pi$
$510$	0	0
$511$	5.14282	0.227505
$512$	0	0
$513$	−2.87007	−0.126717
$514$	0	0
$515$	0.619878	0.0273151
$516$	0	0
$517$	7.05525	0.310289
$518$	0	0
$519$	10.1098	0.443772
$520$	0	0
$521$	−3.74629	−0.164128	−0.0820640	−	0.996627i	$-0.526151\pi$
$521$	−3.74629	−0.164128	−0.0820640	+	0.996627i	$0.526151\pi$
$522$	0	0
$523$	4.03280	0.176342	0.0881710	−	0.996105i	$-0.471898\pi$
$523$	4.03280	0.176342	0.0881710	+	0.996105i	$0.471898\pi$
$524$	0	0
$525$	0.165367	0.00721719
$526$	0	0
$527$	9.39301	0.409166
$528$	0	0
$529$	19.5487	0.849944
$530$	0	0
$531$	14.4840	0.628551
$532$	0	0
$533$	24.1143	1.04451
$534$	0	0
$535$	−9.63031	−0.416355
$536$	0	0
$537$	−21.1167	−0.911255
$538$	0	0
$539$	3.38012	0.145592
$540$	0	0
$541$	15.3833	0.661382	0.330691	−	0.943739i	$-0.392718\pi$
$541$	15.3833	0.661382	0.330691	+	0.943739i	$0.392718\pi$
$542$	0	0
$543$	4.11108	0.176423
$544$	0	0
$545$	−12.8475	−0.550327
$546$	0	0
$547$	9.89339	0.423011	0.211505	−	0.977377i	$-0.432163\pi$
$547$	9.89339	0.423011	0.211505	+	0.977377i	$0.432163\pi$
$548$	0	0
$549$	10.3020	0.439680
$550$	0	0
$551$	10.7345	0.457304
$552$	0	0
$553$	−3.48995	−0.148408
$554$	0	0
$555$	5.70470	0.242151
$556$	0	0
$557$	−23.9127	−1.01321	−0.506607	−	0.862177i	$-0.669101\pi$
$557$	−23.9127	−1.01321	−0.506607	+	0.862177i	$0.669101\pi$
$558$	0	0
$559$	−21.9678	−0.929138
$560$	0	0
$561$	3.38012	0.142709
$562$	0	0
$563$	−23.7726	−1.00190	−0.500949	−	0.865477i	$-0.667016\pi$
$563$	−23.7726	−1.00190	−0.500949	+	0.865477i	$0.667016\pi$
$564$	0	0
$565$	24.0338	1.01111
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	21.4024	0.897234	0.448617	−	0.893724i	$-0.351917\pi$
$569$	21.4024	0.897234	0.448617	+	0.893724i	$0.351917\pi$
$570$	0	0
$571$	−32.1320	−1.34468	−0.672340	−	0.740242i	$-0.734711\pi$
$571$	−32.1320	−1.34468	−0.672340	+	0.740242i	$0.734711\pi$
$572$	0	0
$573$	−25.9739	−1.08508
$574$	0	0
$575$	−1.07868	−0.0449839
$576$	0	0
$577$	−24.7213	−1.02916	−0.514580	−	0.857442i	$-0.672052\pi$
$577$	−24.7213	−1.02916	−0.514580	+	0.857442i	$0.672052\pi$
$578$	0	0
$579$	7.49668	0.311552
$580$	0	0
$581$	−2.59733	−0.107755
$582$	0	0
$583$	27.8867	1.15495
$584$	0	0
$585$	5.16537	0.213562
$586$	0	0
$587$	1.27519	0.0526329	0.0263164	−	0.999654i	$-0.491622\pi$
$587$	1.27519	0.0526329	0.0263164	+	0.999654i	$0.491622\pi$
$588$	0	0
$589$	−26.9586	−1.11081
$590$	0	0
$591$	16.5229	0.679663
$592$	0	0
$593$	−26.6536	−1.09453	−0.547267	−	0.836958i	$-0.684332\pi$
$593$	−26.6536	−1.09453	−0.547267	+	0.836958i	$0.684332\pi$
$594$	0	0
$595$	2.27274	0.0931734
$596$	0	0
$597$	−22.4389	−0.918362
$598$	0	0
$599$	−31.6573	−1.29348	−0.646742	−	0.762709i	$-0.723869\pi$
$599$	−31.6573	−1.29348	−0.646742	+	0.762709i	$0.723869\pi$
$600$	0	0
$601$	−13.2562	−0.540730	−0.270365	−	0.962758i	$-0.587144\pi$
$601$	−13.2562	−0.540730	−0.270365	+	0.962758i	$0.587144\pi$
$602$	0	0
$603$	2.59733	0.105771
$604$	0	0
$605$	0.966427	0.0392909
$606$	0	0
$607$	28.8733	1.17193	0.585965	−	0.810336i	$-0.300715\pi$
$607$	28.8733	1.17193	0.585965	+	0.810336i	$0.300715\pi$
$608$	0	0
$609$	−3.74014	−0.151558
$610$	0	0
$611$	4.74384	0.191915
$612$	0	0
$613$	4.30328	0.173808	0.0869040	−	0.996217i	$-0.472303\pi$
$613$	4.30328	0.173808	0.0869040	+	0.996217i	$0.472303\pi$
$614$	0	0
$615$	24.1143	0.972382
$616$	0	0
$617$	9.42355	0.379378	0.189689	−	0.981844i	$-0.439252\pi$
$617$	9.42355	0.379378	0.189689	+	0.981844i	$0.439252\pi$
$618$	0	0
$619$	−39.6670	−1.59435	−0.797176	−	0.603747i	$-0.793674\pi$
$619$	−39.6670	−1.59435	−0.797176	+	0.603747i	$0.793674\pi$
$620$	0	0
$621$	−6.52294	−0.261756
$622$	0	0
$623$	−3.90306	−0.156373
$624$	0	0
$625$	−25.7995	−1.03198
$626$	0	0
$627$	−9.70119	−0.387428
$628$	0	0
$629$	2.51005	0.100082
$630$	0	0
$631$	−0.763465	−0.0303930	−0.0151965	−	0.999885i	$-0.504837\pi$
$631$	−0.763465	−0.0303930	−0.0151965	+	0.999885i	$0.504837\pi$
$632$	0	0
$633$	0.418009	0.0166143
$634$	0	0
$635$	5.83650	0.231614
$636$	0	0
$637$	2.27274	0.0900494
$638$	0	0
$639$	−13.0459	−0.516087
$640$	0	0
$641$	2.10016	0.0829514	0.0414757	−	0.999140i	$-0.486794\pi$
$641$	2.10016	0.0829514	0.0414757	+	0.999140i	$0.486794\pi$
$642$	0	0
$643$	−21.2539	−0.838172	−0.419086	−	0.907947i	$-0.637649\pi$
$643$	−21.2539	−0.838172	−0.419086	+	0.907947i	$0.637649\pi$
$644$	0	0
$645$	−21.9678	−0.864981
$646$	0	0
$647$	35.4024	1.39181	0.695906	−	0.718133i	$-0.255003\pi$
$647$	35.4024	1.39181	0.695906	+	0.718133i	$0.255003\pi$
$648$	0	0
$649$	48.9576	1.92176
$650$	0	0
$651$	9.39301	0.368141
$652$	0	0
$653$	−0.419261	−0.0164070	−0.00820348	−	0.999966i	$-0.502611\pi$
$653$	−0.419261	−0.0164070	−0.00820348	+	0.999966i	$0.502611\pi$
$654$	0	0
$655$	−2.56804	−0.100342
$656$	0	0
$657$	5.14282	0.200640
$658$	0	0
$659$	31.1331	1.21278	0.606388	−	0.795169i	$-0.292618\pi$
$659$	31.1331	1.21278	0.606388	+	0.795169i	$0.292618\pi$
$660$	0	0
$661$	43.2523	1.68232	0.841161	−	0.540785i	$-0.181873\pi$
$661$	43.2523	1.68232	0.841161	+	0.540785i	$0.181873\pi$
$662$	0	0
$663$	2.27274	0.0882661
$664$	0	0
$665$	−6.52294	−0.252949
$666$	0	0
$667$	24.3967	0.944644
$668$	0	0
$669$	11.3705	0.439607
$670$	0	0
$671$	34.8221	1.34429
$672$	0	0
$673$	−5.23486	−0.201789	−0.100894	−	0.994897i	$-0.532170\pi$
$673$	−5.23486	−0.201789	−0.100894	+	0.994897i	$0.532170\pi$
$674$	0	0
$675$	0.165367	0.00636496
$676$	0	0
$677$	11.2220	0.431295	0.215648	−	0.976471i	$-0.430814\pi$
$677$	11.2220	0.431295	0.215648	+	0.976471i	$0.430814\pi$
$678$	0	0
$679$	−0.597327	−0.0229233
$680$	0	0
$681$	3.24101	0.124196
$682$	0	0
$683$	−3.63324	−0.139022	−0.0695111	−	0.997581i	$-0.522144\pi$
$683$	−3.63324	−0.139022	−0.0695111	+	0.997581i	$0.522144\pi$
$684$	0	0
$685$	−51.7495	−1.97725
$686$	0	0
$687$	−8.99404	−0.343144
$688$	0	0
$689$	18.7506	0.714340
$690$	0	0
$691$	−9.27519	−0.352845	−0.176422	−	0.984315i	$-0.556452\pi$
$691$	−9.27519	−0.352845	−0.176422	+	0.984315i	$0.556452\pi$
$692$	0	0
$693$	3.38012	0.128400
$694$	0	0
$695$	−38.2096	−1.44937
$696$	0	0
$697$	10.6102	0.401890
$698$	0	0
$699$	19.6658	0.743827
$700$	0	0
$701$	13.4101	0.506492	0.253246	−	0.967402i	$-0.418502\pi$
$701$	13.4101	0.506492	0.253246	+	0.967402i	$0.418502\pi$
$702$	0	0
$703$	−7.20402	−0.271705
$704$	0	0
$705$	4.74384	0.178663
$706$	0	0
$707$	0.0164006	0.000616809 0
$708$	0	0
$709$	−47.0600	−1.76738	−0.883688	−	0.468077i	$-0.844947\pi$
$709$	−47.0600	−1.76738	−0.883688	+	0.468077i	$0.844947\pi$
$710$	0	0
$711$	−3.48995	−0.130883
$712$	0	0
$713$	−61.2700	−2.29458
$714$	0	0
$715$	17.4596	0.652951
$716$	0	0
$717$	18.7506	0.700253
$718$	0	0
$719$	−10.0696	−0.375534	−0.187767	−	0.982214i	$-0.560125\pi$
$719$	−10.0696	−0.375534	−0.187767	+	0.982214i	$0.560125\pi$
$720$	0	0
$721$	0.272744	0.0101575
$722$	0	0
$723$	6.47461	0.240793
$724$	0	0
$725$	−0.618494	−0.0229703
$726$	0	0
$727$	28.9087	1.07217	0.536083	−	0.844165i	$-0.319904\pi$
$727$	28.9087	1.07217	0.536083	+	0.844165i	$0.319904\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−9.66575	−0.357501
$732$	0	0
$733$	18.9164	0.698694	0.349347	−	0.936993i	$-0.386403\pi$
$733$	18.9164	0.698694	0.349347	+	0.936993i	$0.386403\pi$
$734$	0	0
$735$	2.27274	0.0838314
$736$	0	0
$737$	8.77928	0.323389
$738$	0	0
$739$	−15.7955	−0.581047	−0.290523	−	0.956868i	$-0.593829\pi$
$739$	−15.7955	−0.581047	−0.290523	+	0.956868i	$0.593829\pi$
$740$	0	0
$741$	−6.52294	−0.239626
$742$	0	0
$743$	10.7930	0.395958	0.197979	−	0.980206i	$-0.436562\pi$
$743$	10.7930	0.395958	0.197979	+	0.980206i	$0.436562\pi$
$744$	0	0
$745$	15.5846	0.570977
$746$	0	0
$747$	−2.59733	−0.0950312
$748$	0	0
$749$	−4.23731	−0.154828
$750$	0	0
$751$	−45.3479	−1.65477	−0.827384	−	0.561636i	$-0.810172\pi$
$751$	−45.3479	−1.65477	−0.827384	+	0.561636i	$0.810172\pi$
$752$	0	0
$753$	−1.53015	−0.0557619
$754$	0	0
$755$	10.5664	0.384549
$756$	0	0
$757$	−26.3082	−0.956187	−0.478094	−	0.878309i	$-0.658672\pi$
$757$	−26.3082	−0.956187	−0.478094	+	0.878309i	$0.658672\pi$
$758$	0	0
$759$	−22.0483	−0.800303
$760$	0	0
$761$	−18.0592	−0.654647	−0.327323	−	0.944912i	$-0.606147\pi$
$761$	−18.0592	−0.654647	−0.327323	+	0.944912i	$0.606147\pi$
$762$	0	0
$763$	−5.65287	−0.204647
$764$	0	0
$765$	2.27274	0.0821712
$766$	0	0
$767$	32.9184	1.18861
$768$	0	0
$769$	−43.0204	−1.55136	−0.775678	−	0.631129i	$-0.782592\pi$
$769$	−43.0204	−1.55136	−0.775678	+	0.631129i	$0.782592\pi$
$770$	0	0
$771$	25.7707	0.928108
$772$	0	0
$773$	23.3221	0.838839	0.419419	−	0.907793i	$-0.362234\pi$
$773$	23.3221	0.838839	0.419419	+	0.907793i	$0.362234\pi$
$774$	0	0
$775$	1.55329	0.0557958
$776$	0	0
$777$	2.51005	0.0900476
$778$	0	0
$779$	−30.4521	−1.09106
$780$	0	0
$781$	−44.0966	−1.57790
$782$	0	0
$783$	−3.74014	−0.133662
$784$	0	0
$785$	51.0491	1.82202
$786$	0	0
$787$	−7.09324	−0.252847	−0.126423	−	0.991976i	$-0.540350\pi$
$787$	−7.09324	−0.252847	−0.126423	+	0.991976i	$0.540350\pi$
$788$	0	0
$789$	30.9489	1.10181
$790$	0	0
$791$	10.5748	0.375996
$792$	0	0
$793$	23.4139	0.831451
$794$	0	0
$795$	18.7506	0.665014
$796$	0	0
$797$	−13.9549	−0.494308	−0.247154	−	0.968976i	$-0.579495\pi$
$797$	−13.9549	−0.494308	−0.247154	+	0.968976i	$0.579495\pi$
$798$	0	0
$799$	2.08728	0.0738425
$800$	0	0
$801$	−3.90306	−0.137908
$802$	0	0
$803$	17.3833	0.613445
$804$	0	0
$805$	−14.8250	−0.522511
$806$	0	0
$807$	25.4156	0.894670
$808$	0	0
$809$	−0.244710	−0.00860355	−0.00430177	−	0.999991i	$-0.501369\pi$
$809$	−0.244710	−0.00860355	−0.00430177	+	0.999991i	$0.501369\pi$
$810$	0	0
$811$	−51.7398	−1.81683	−0.908416	−	0.418068i	$-0.862707\pi$
$811$	−51.7398	−1.81683	−0.908416	+	0.418068i	$0.862707\pi$
$812$	0	0
$813$	11.6822	0.409711
$814$	0	0
$815$	32.3651	1.13370
$816$	0	0
$817$	27.7414	0.970549
$818$	0	0
$819$	2.27274	0.0794161
$820$	0	0
$821$	−1.51024	−0.0527077	−0.0263538	−	0.999653i	$-0.508390\pi$
$821$	−1.51024	−0.0527077	−0.0263538	+	0.999653i	$0.508390\pi$
$822$	0	0
$823$	−19.8088	−0.690490	−0.345245	−	0.938513i	$-0.612204\pi$
$823$	−19.8088	−0.690490	−0.345245	+	0.938513i	$0.612204\pi$
$824$	0	0
$825$	0.558959	0.0194605
$826$	0	0
$827$	−6.78953	−0.236095	−0.118048	−	0.993008i	$-0.537664\pi$
$827$	−6.78953	−0.236095	−0.118048	+	0.993008i	$0.537664\pi$
$828$	0	0
$829$	45.7752	1.58984	0.794918	−	0.606716i	$-0.207514\pi$
$829$	45.7752	1.58984	0.794918	+	0.606716i	$0.207514\pi$
$830$	0	0
$831$	−26.3967	−0.915692
$832$	0	0
$833$	1.00000	0.0346479
$834$	0	0
$835$	−49.9272	−1.72780
$836$	0	0
$837$	9.39301	0.324670
$838$	0	0
$839$	41.4481	1.43095	0.715473	−	0.698640i	$-0.246211\pi$
$839$	41.4481	1.43095	0.715473	+	0.698640i	$0.246211\pi$
$840$	0	0
$841$	−15.0113	−0.517632
$842$	0	0
$843$	8.83112	0.304160
$844$	0	0
$845$	−17.8061	−0.612549
$846$	0	0
$847$	0.425225	0.0146109
$848$	0	0
$849$	4.53582	0.155669
$850$	0	0
$851$	−16.3729	−0.561256
$852$	0	0
$853$	−55.0175	−1.88376	−0.941882	−	0.335943i	$-0.890945\pi$
$853$	−55.0175	−1.88376	−0.941882	+	0.335943i	$0.890945\pi$
$854$	0	0
$855$	−6.52294	−0.223080
$856$	0	0
$857$	−23.0201	−0.786352	−0.393176	−	0.919463i	$-0.628624\pi$
$857$	−23.0201	−0.786352	−0.393176	+	0.919463i	$0.628624\pi$
$858$	0	0
$859$	−3.49961	−0.119405	−0.0597026	−	0.998216i	$-0.519015\pi$
$859$	−3.49961	−0.119405	−0.0597026	+	0.998216i	$0.519015\pi$
$860$	0	0
$861$	10.6102	0.361595
$862$	0	0
$863$	−18.8923	−0.643102	−0.321551	−	0.946892i	$-0.604204\pi$
$863$	−18.8923	−0.643102	−0.321551	+	0.946892i	$0.604204\pi$
$864$	0	0
$865$	22.9771	0.781243
$866$	0	0
$867$	1.00000	0.0339618
$868$	0	0
$869$	−11.7965	−0.400167
$870$	0	0
$871$	5.90306	0.200018
$872$	0	0
$873$	−0.597327	−0.0202164
$874$	0	0
$875$	−10.9879	−0.371458
$876$	0	0
$877$	−31.8722	−1.07625	−0.538124	−	0.842865i	$-0.680867\pi$
$877$	−31.8722	−1.07625	−0.538124	+	0.842865i	$0.680867\pi$
$878$	0	0
$879$	22.9422	0.773821
$880$	0	0
$881$	22.9680	0.773811	0.386905	−	0.922119i	$-0.373544\pi$
$881$	22.9680	0.773811	0.386905	+	0.922119i	$0.373544\pi$
$882$	0	0
$883$	−14.6266	−0.492225	−0.246112	−	0.969241i	$-0.579153\pi$
$883$	−14.6266	−0.492225	−0.246112	+	0.969241i	$0.579153\pi$
$884$	0	0
$885$	32.9184	1.10654
$886$	0	0
$887$	14.6621	0.492303	0.246152	−	0.969231i	$-0.420834\pi$
$887$	14.6621	0.492303	0.246152	+	0.969231i	$0.420834\pi$
$888$	0	0
$889$	2.56804	0.0861293
$890$	0	0
$891$	3.38012	0.113238
$892$	0	0
$893$	−5.99063	−0.200469
$894$	0	0
$895$	−47.9930	−1.60423
$896$	0	0
$897$	−14.8250	−0.494991
$898$	0	0
$899$	−35.1312	−1.17169
$900$	0	0
$901$	8.25019	0.274854
$902$	0	0
$903$	−9.66575	−0.321656
$904$	0	0
$905$	9.34343	0.310586
$906$	0	0
$907$	−51.5343	−1.71117	−0.855585	−	0.517663i	$-0.826802\pi$
$907$	−51.5343	−1.71117	−0.855585	+	0.517663i	$0.826802\pi$
$908$	0	0
$909$	0.0164006	0.000543974 0
$910$	0	0
$911$	35.8593	1.18807	0.594036	−	0.804438i	$-0.297533\pi$
$911$	35.8593	1.18807	0.594036	+	0.804438i	$0.297533\pi$
$912$	0	0
$913$	−8.77928	−0.290552
$914$	0	0
$915$	23.4139	0.774039
$916$	0	0
$917$	−1.12993	−0.0373135
$918$	0	0
$919$	54.5552	1.79961	0.899804	−	0.436294i	$-0.143709\pi$
$919$	54.5552	1.79961	0.899804	+	0.436294i	$0.143709\pi$
$920$	0	0
$921$	−21.6622	−0.713795
$922$	0	0
$923$	−29.6499	−0.975940
$924$	0	0
$925$	0.415078	0.0136477
$926$	0	0
$927$	0.272744	0.00895810
$928$	0	0
$929$	3.57174	0.117185	0.0585925	−	0.998282i	$-0.481339\pi$
$929$	3.57174	0.117185	0.0585925	+	0.998282i	$0.481339\pi$
$930$	0	0
$931$	−2.87007	−0.0940628
$932$	0	0
$933$	−18.3449	−0.600584
$934$	0	0
$935$	7.68215	0.251233
$936$	0	0
$937$	−47.5463	−1.55327	−0.776634	−	0.629952i	$-0.783075\pi$
$937$	−47.5463	−1.55327	−0.776634	+	0.629952i	$0.783075\pi$
$938$	0	0
$939$	−30.8733	−1.00751
$940$	0	0
$941$	−16.0325	−0.522645	−0.261322	−	0.965252i	$-0.584159\pi$
$941$	−16.0325	−0.522645	−0.261322	+	0.965252i	$0.584159\pi$
$942$	0	0
$943$	−69.2098	−2.25378
$944$	0	0
$945$	2.27274	0.0739324
$946$	0	0
$947$	−26.8311	−0.871894	−0.435947	−	0.899972i	$-0.643587\pi$
$947$	−26.8311	−0.871894	−0.435947	+	0.899972i	$0.643587\pi$
$948$	0	0
$949$	11.6883	0.379418
$950$	0	0
$951$	−11.6622	−0.378174
$952$	0	0
$953$	29.5627	0.957629	0.478814	−	0.877916i	$-0.341067\pi$
$953$	29.5627	0.957629	0.478814	+	0.877916i	$0.341067\pi$
$954$	0	0
$955$	−59.0321	−1.91023
$956$	0	0
$957$	−12.6421	−0.408662
$958$	0	0
$959$	−22.7696	−0.735269
$960$	0	0
$961$	57.2286	1.84608
$962$	0	0
$963$	−4.23731	−0.136545
$964$	0	0
$965$	17.0380	0.548474
$966$	0	0
$967$	54.2060	1.74315	0.871575	−	0.490263i	$-0.163099\pi$
$967$	54.2060	1.74315	0.871575	+	0.490263i	$0.163099\pi$
$968$	0	0
$969$	−2.87007	−0.0922000
$970$	0	0
$971$	12.3117	0.395101	0.197551	−	0.980293i	$-0.436701\pi$
$971$	12.3117	0.395101	0.197551	+	0.980293i	$0.436701\pi$
$972$	0	0
$973$	−16.8121	−0.538971
$974$	0	0
$975$	0.375836	0.0120364
$976$	0	0
$977$	−2.44884	−0.0783454	−0.0391727	−	0.999232i	$-0.512472\pi$
$977$	−2.44884	−0.0783454	−0.0391727	+	0.999232i	$0.512472\pi$
$978$	0	0
$979$	−13.1928	−0.421644
$980$	0	0
$981$	−5.65287	−0.180482
$982$	0	0
$983$	41.3843	1.31995	0.659977	−	0.751286i	$-0.270566\pi$
$983$	41.3843	1.31995	0.659977	+	0.751286i	$0.270566\pi$
$984$	0	0
$985$	37.5524	1.19652
$986$	0	0
$987$	2.08728	0.0664387
$988$	0	0
$989$	63.0491	2.00484
$990$	0	0
$991$	37.8252	1.20156	0.600778	−	0.799416i	$-0.294858\pi$
$991$	37.8252	1.20156	0.600778	+	0.799416i	$0.294858\pi$
$992$	0	0
$993$	−1.33992	−0.0425210
$994$	0	0
$995$	−50.9978	−1.61674
$996$	0	0
$997$	−1.80612	−0.0572003	−0.0286002	−	0.999591i	$-0.509105\pi$
$997$	−1.80612	−0.0572003	−0.0286002	+	0.999591i	$0.509105\pi$
$998$	0	0
$999$	2.51005	0.0794145

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5712.2.a.ca.1.3		4
4.3	odd	2		2856.2.a.v.1.3	✓	4
12.11	even	2		8568.2.a.bf.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2856.2.a.v.1.3	✓	4	4.3	odd	2
5712.2.a.ca.1.3		4	1.1	even	1	trivial
8568.2.a.bf.1.2		4	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5712 = 2^{4} \cdot 3 \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5712.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.27274 q^{5} +1.00000 q^{7} +1.00000 q^{9} +3.38012 q^{11} +2.27274 q^{13} +2.27274 q^{15} +1.00000 q^{17} -2.87007 q^{19} +1.00000 q^{21} -6.52294 q^{23} +0.165367 q^{25} +1.00000 q^{27} -3.74014 q^{29} +9.39301 q^{31} +3.38012 q^{33} +2.27274 q^{35} +2.51005 q^{37} +2.27274 q^{39} +10.6102 q^{41} -9.66575 q^{43} +2.27274 q^{45} +2.08728 q^{47} +1.00000 q^{49} +1.00000 q^{51} +8.25019 q^{53} +7.68215 q^{55} -2.87007 q^{57} +14.4840 q^{59} +10.3020 q^{61} +1.00000 q^{63} +5.16537 q^{65} +2.59733 q^{67} -6.52294 q^{69} -13.0459 q^{71} +5.14282 q^{73} +0.165367 q^{75} +3.38012 q^{77} -3.48995 q^{79} +1.00000 q^{81} -2.59733 q^{83} +2.27274 q^{85} -3.74014 q^{87} -3.90306 q^{89} +2.27274 q^{91} +9.39301 q^{93} -6.52294 q^{95} -0.597327 q^{97} +3.38012 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 5 q^{5} + 4 q^{7} + 4 q^{9} - q^{11} + 5 q^{13} + 5 q^{15} + 4 q^{17} + 4 q^{21} + 4 q^{23} + 7 q^{25} + 4 q^{27} + 8 q^{29} - 4 q^{31} - q^{33} + 5 q^{35} + 7 q^{37} + 5 q^{39} + 8 q^{41}+ \cdots - q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + 5 * q^5 + 4 * q^7 + 4 * q^9 - q^11 + 5 * q^13 + 5 * q^15 + 4 * q^17 + 4 * q^21 + 4 * q^23 + 7 * q^25 + 4 * q^27 + 8 * q^29 - 4 * q^31 - q^33 + 5 * q^35 + 7 * q^37 + 5 * q^39 + 8 * q^41 + 7 * q^43 + 5 * q^45 + 4 * q^47 + 4 * q^49 + 4 * q^51 + 7 * q^53 - 9 * q^55 + 16 * q^61 + 4 * q^63 + 27 * q^65 + 3 * q^67 + 4 * q^69 + 8 * q^71 + 5 * q^73 + 7 * q^75 - q^77 - 17 * q^79 + 4 * q^81 - 3 * q^83 + 5 * q^85 + 8 * q^87 + 29 * q^89 + 5 * q^91 - 4 * q^93 + 4 * q^95 + 5 * q^97 - q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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