LMFDB - Embedded newform 5712.2.a.bz.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5712,2,Mod(1,5712)]

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

lf = mfeigenbasis(mf)

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")

//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: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$45.6105496346$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.7232.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 5x^{2} + 4x + 4 $ x^4 - 2x^3 - 5x^2 + 4*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 2856)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.63640$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +1.19202 q^{5} +1.00000 q^{7} +1.00000 q^{9} -1.95063 q^{11} -4.46483 q^{13} +1.19202 q^{15} +1.00000 q^{17} +3.82047 q^{19} +1.00000 q^{21} -5.22344 q^{23} -3.57908 q^{25} +1.00000 q^{27} +5.27281 q^{29} +4.51420 q^{31} -1.95063 q^{33} +1.19202 q^{35} +0.758609 q^{37} -4.46483 q^{39} -3.83638 q^{41} +7.60749 q^{43} +1.19202 q^{45} +10.0314 q^{47} +1.00000 q^{49} +1.00000 q^{51} +8.03142 q^{53} -2.32520 q^{55} +3.82047 q^{57} +10.6599 q^{59} +3.77111 q^{61} +1.00000 q^{63} -5.32218 q^{65} -7.90126 q^{67} -5.22344 q^{69} +11.9013 q^{71} +3.02840 q^{73} -3.57908 q^{75} -1.95063 q^{77} +7.15856 q^{79} +1.00000 q^{81} +2.24441 q^{83} +1.19202 q^{85} +5.27281 q^{87} +2.16158 q^{89} -4.46483 q^{91} +4.51420 q^{93} +4.55409 q^{95} +12.4469 q^{97} -1.95063 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 2 q^{5} + 4 q^{7} + 4 q^{9} + 6 q^{11} + 2 q^{13} + 2 q^{15} + 4 q^{17} + 2 q^{19} + 4 q^{21} + 10 q^{23} + 10 q^{25} + 4 q^{27} + 4 q^{29} + 12 q^{31} + 6 q^{33} + 2 q^{35} - 8 q^{37} + 2 q^{39}+ \cdots + 6 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + 2 * q^5 + 4 * q^7 + 4 * q^9 + 6 * q^11 + 2 * q^13 + 2 * q^15 + 4 * q^17 + 2 * q^19 + 4 * q^21 + 10 * q^23 + 10 * q^25 + 4 * q^27 + 4 * q^29 + 12 * q^31 + 6 * q^33 + 2 * q^35 - 8 * q^37 + 2 * q^39 - 6 * q^41 - 6 * q^43 + 2 * q^45 + 12 * q^47 + 4 * q^49 + 4 * q^51 + 4 * q^53 + 10 * q^55 + 2 * q^57 + 4 * q^59 - 12 * q^61 + 4 * q^63 - 18 * q^65 - 4 * q^67 + 10 * q^69 + 20 * q^71 + 10 * q^75 + 6 * q^77 + 16 * q^79 + 4 * q^81 + 4 * q^83 + 2 * q^85 + 4 * q^87 - 20 * q^89 + 2 * q^91 + 12 * q^93 + 26 * q^95 - 12 * q^97 + 6 * 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
$5$	1.19202	0.533089	0.266544	−	0.963823i	$-0.414118\pi$
$5$	1.19202	0.533089	0.266544	+	0.963823i	$0.414118\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.95063	−0.588137	−0.294069	−	0.955784i	$-0.595009\pi$
$11$	−1.95063	−0.588137	−0.294069	+	0.955784i	$0.595009\pi$
$12$	0	0
$13$	−4.46483	−1.23832	−0.619161	−	0.785264i	$-0.712527\pi$
$13$	−4.46483	−1.23832	−0.619161	+	0.785264i	$0.712527\pi$
$14$	0	0
$15$	1.19202	0.307779
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	3.82047	0.876477	0.438238	−	0.898859i	$-0.355603\pi$
$19$	3.82047	0.876477	0.438238	+	0.898859i	$0.355603\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−5.22344	−1.08916	−0.544581	−	0.838708i	$-0.683311\pi$
$23$	−5.22344	−1.08916	−0.544581	+	0.838708i	$0.683311\pi$
$24$	0	0
$25$	−3.57908	−0.715817
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	5.27281	0.979136	0.489568	−	0.871965i	$-0.337154\pi$
$29$	5.27281	0.979136	0.489568	+	0.871965i	$0.337154\pi$
$30$	0	0
$31$	4.51420	0.810774	0.405387	−	0.914145i	$-0.367137\pi$
$31$	4.51420	0.810774	0.405387	+	0.914145i	$0.367137\pi$
$32$	0	0
$33$	−1.95063	−0.339561
$34$	0	0
$35$	1.19202	0.201489
$36$	0	0
$37$	0.758609	0.124715	0.0623573	−	0.998054i	$-0.480138\pi$
$37$	0.758609	0.124715	0.0623573	+	0.998054i	$0.480138\pi$
$38$	0	0
$39$	−4.46483	−0.714945
$40$	0	0
$41$	−3.83638	−0.599142	−0.299571	−	0.954074i	$-0.596844\pi$
$41$	−3.83638	−0.599142	−0.299571	+	0.954074i	$0.596844\pi$
$42$	0	0
$43$	7.60749	1.16013	0.580065	−	0.814570i	$-0.303027\pi$
$43$	7.60749	1.16013	0.580065	+	0.814570i	$0.303027\pi$
$44$	0	0
$45$	1.19202	0.177696
$46$	0	0
$47$	10.0314	1.46323	0.731616	−	0.681717i	$-0.238766\pi$
$47$	10.0314	1.46323	0.731616	+	0.681717i	$0.238766\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.00000	0.140028
$52$	0	0
$53$	8.03142	1.10320	0.551600	−	0.834109i	$-0.314017\pi$
$53$	8.03142	1.10320	0.551600	+	0.834109i	$0.314017\pi$
$54$	0	0
$55$	−2.32520	−0.313529
$56$	0	0
$57$	3.82047	0.506034
$58$	0	0
$59$	10.6599	1.38780	0.693898	−	0.720073i	$-0.255892\pi$
$59$	10.6599	1.38780	0.693898	+	0.720073i	$0.255892\pi$
$60$	0	0
$61$	3.77111	0.482841	0.241420	−	0.970421i	$-0.422387\pi$
$61$	3.77111	0.482841	0.241420	+	0.970421i	$0.422387\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	−5.32218	−0.660135
$66$	0	0
$67$	−7.90126	−0.965293	−0.482646	−	0.875815i	$-0.660324\pi$
$67$	−7.90126	−0.965293	−0.482646	+	0.875815i	$0.660324\pi$
$68$	0	0
$69$	−5.22344	−0.628828
$70$	0	0
$71$	11.9013	1.41242	0.706210	−	0.708002i	$-0.250403\pi$
$71$	11.9013	1.41242	0.706210	+	0.708002i	$0.250403\pi$
$72$	0	0
$73$	3.02840	0.354448	0.177224	−	0.984171i	$-0.443288\pi$
$73$	3.02840	0.354448	0.177224	+	0.984171i	$0.443288\pi$
$74$	0	0
$75$	−3.57908	−0.413277
$76$	0	0
$77$	−1.95063	−0.222295
$78$	0	0
$79$	7.15856	0.805401	0.402700	−	0.915332i	$-0.368072\pi$
$79$	7.15856	0.805401	0.402700	+	0.915332i	$0.368072\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	2.24441	0.246356	0.123178	−	0.992385i	$-0.460691\pi$
$83$	2.24441	0.246356	0.123178	+	0.992385i	$0.460691\pi$
$84$	0	0
$85$	1.19202	0.129293
$86$	0	0
$87$	5.27281	0.565305
$88$	0	0
$89$	2.16158	0.229127	0.114563	−	0.993416i	$-0.463453\pi$
$89$	2.16158	0.229127	0.114563	+	0.993416i	$0.463453\pi$
$90$	0	0
$91$	−4.46483	−0.468042
$92$	0	0
$93$	4.51420	0.468101
$94$	0	0
$95$	4.55409	0.467240
$96$	0	0
$97$	12.4469	1.26379	0.631895	−	0.775054i	$-0.282278\pi$
$97$	12.4469	1.26379	0.631895	+	0.775054i	$0.282278\pi$
$98$	0	0
$99$	−1.95063	−0.196046
$100$	0	0
$101$	1.62544	0.161737	0.0808684	−	0.996725i	$-0.474231\pi$
$101$	1.62544	0.161737	0.0808684	+	0.996725i	$0.474231\pi$
$102$	0	0
$103$	7.68084	0.756815	0.378408	−	0.925639i	$-0.376472\pi$
$103$	7.68084	0.756815	0.378408	+	0.925639i	$0.376472\pi$
$104$	0	0
$105$	1.19202	0.116329
$106$	0	0
$107$	4.43341	0.428594	0.214297	−	0.976769i	$-0.431254\pi$
$107$	4.43341	0.428594	0.214297	+	0.976769i	$0.431254\pi$
$108$	0	0
$109$	−0.498295	−0.0477280	−0.0238640	−	0.999715i	$-0.507597\pi$
$109$	−0.498295	−0.0477280	−0.0238640	+	0.999715i	$0.507597\pi$
$110$	0	0
$111$	0.758609	0.0720040
$112$	0	0
$113$	1.70622	0.160508	0.0802540	−	0.996774i	$-0.474427\pi$
$113$	1.70622	0.160508	0.0802540	+	0.996774i	$0.474427\pi$
$114$	0	0
$115$	−6.22646	−0.580620
$116$	0	0
$117$	−4.46483	−0.412774
$118$	0	0
$119$	1.00000	0.0916698
$120$	0	0
$121$	−7.19504	−0.654094
$122$	0	0
$123$	−3.83638	−0.345915
$124$	0	0
$125$	−10.2265	−0.914682
$126$	0	0
$127$	1.32218	0.117324	0.0586622	−	0.998278i	$-0.481317\pi$
$127$	1.32218	0.117324	0.0586622	+	0.998278i	$0.481317\pi$
$128$	0	0
$129$	7.60749	0.669802
$130$	0	0
$131$	−8.02295	−0.700968	−0.350484	−	0.936569i	$-0.613983\pi$
$131$	−8.02295	−0.700968	−0.350484	+	0.936569i	$0.613983\pi$
$132$	0	0
$133$	3.82047	0.331277
$134$	0	0
$135$	1.19202	0.102593
$136$	0	0
$137$	−9.05982	−0.774033	−0.387016	−	0.922073i	$-0.626494\pi$
$137$	−9.05982	−0.774033	−0.387016	+	0.922073i	$0.626494\pi$
$138$	0	0
$139$	19.4435	1.64917	0.824587	−	0.565735i	$-0.191408\pi$
$139$	19.4435	1.64917	0.824587	+	0.565735i	$0.191408\pi$
$140$	0	0
$141$	10.0314	0.844798
$142$	0	0
$143$	8.70924	0.728303
$144$	0	0
$145$	6.28531	0.521966
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−17.0534	−1.39707	−0.698534	−	0.715577i	$-0.746164\pi$
$149$	−17.0534	−1.39707	−0.698534	+	0.715577i	$0.746164\pi$
$150$	0	0
$151$	8.16158	0.664180	0.332090	−	0.943248i	$-0.392246\pi$
$151$	8.16158	0.664180	0.332090	+	0.943248i	$0.392246\pi$
$152$	0	0
$153$	1.00000	0.0808452
$154$	0	0
$155$	5.38103	0.432215
$156$	0	0
$157$	−12.6264	−1.00770	−0.503849	−	0.863792i	$-0.668083\pi$
$157$	−12.6264	−1.00770	−0.503849	+	0.863792i	$0.668083\pi$
$158$	0	0
$159$	8.03142	0.636933
$160$	0	0
$161$	−5.22344	−0.411665
$162$	0	0
$163$	2.24441	0.175796	0.0878978	−	0.996129i	$-0.471985\pi$
$163$	2.24441	0.175796	0.0878978	+	0.996129i	$0.471985\pi$
$164$	0	0
$165$	−2.32520	−0.181016
$166$	0	0
$167$	8.54766	0.661438	0.330719	−	0.943729i	$-0.392709\pi$
$167$	8.54766	0.661438	0.330719	+	0.943729i	$0.392709\pi$
$168$	0	0
$169$	6.93473	0.533440
$170$	0	0
$171$	3.82047	0.292159
$172$	0	0
$173$	15.5402	1.18150	0.590748	−	0.806856i	$-0.298833\pi$
$173$	15.5402	1.18150	0.590748	+	0.806856i	$0.298833\pi$
$174$	0	0
$175$	−3.57908	−0.270553
$176$	0	0
$177$	10.6599	0.801245
$178$	0	0
$179$	2.90467	0.217105	0.108553	−	0.994091i	$-0.465378\pi$
$179$	2.90467	0.217105	0.108553	+	0.994091i	$0.465378\pi$
$180$	0	0
$181$	−17.0534	−1.26757	−0.633784	−	0.773510i	$-0.718499\pi$
$181$	−17.0534	−1.26757	−0.633784	+	0.773510i	$0.718499\pi$
$182$	0	0
$183$	3.77111	0.278768
$184$	0	0
$185$	0.904279	0.0664839
$186$	0	0
$187$	−1.95063	−0.142644
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	21.1582	1.53095	0.765476	−	0.643465i	$-0.222504\pi$
$191$	21.1582	1.53095	0.765476	+	0.643465i	$0.222504\pi$
$192$	0	0
$193$	6.51420	0.468902	0.234451	−	0.972128i	$-0.424671\pi$
$193$	6.51420	0.468902	0.234451	+	0.972128i	$0.424671\pi$
$194$	0	0
$195$	−5.32218	−0.381129
$196$	0	0
$197$	−0.979033	−0.0697532	−0.0348766	−	0.999392i	$-0.511104\pi$
$197$	−0.979033	−0.0697532	−0.0348766	+	0.999392i	$0.511104\pi$
$198$	0	0
$199$	−5.18355	−0.367452	−0.183726	−	0.982977i	$-0.558816\pi$
$199$	−5.18355	−0.367452	−0.183726	+	0.982977i	$0.558816\pi$
$200$	0	0
$201$	−7.90126	−0.557312
$202$	0	0
$203$	5.27281	0.370079
$204$	0	0
$205$	−4.57305	−0.319396
$206$	0	0
$207$	−5.22344	−0.363054
$208$	0	0
$209$	−7.45234	−0.515489
$210$	0	0
$211$	−22.5611	−1.55317	−0.776586	−	0.630011i	$-0.783050\pi$
$211$	−22.5611	−1.55317	−0.776586	+	0.630011i	$0.783050\pi$
$212$	0	0
$213$	11.9013	0.815461
$214$	0	0
$215$	9.06829	0.618452
$216$	0	0
$217$	4.51420	0.306444
$218$	0	0
$219$	3.02840	0.204640
$220$	0	0
$221$	−4.46483	−0.300337
$222$	0	0
$223$	8.12169	0.543868	0.271934	−	0.962316i	$-0.412337\pi$
$223$	8.12169	0.543868	0.271934	+	0.962316i	$0.412337\pi$
$224$	0	0
$225$	−3.57908	−0.238606
$226$	0	0
$227$	2.42393	0.160882	0.0804411	−	0.996759i	$-0.474367\pi$
$227$	2.42393	0.160882	0.0804411	+	0.996759i	$0.474367\pi$
$228$	0	0
$229$	−15.8184	−1.04531	−0.522656	−	0.852544i	$-0.675059\pi$
$229$	−15.8184	−1.04531	−0.522656	+	0.852544i	$0.675059\pi$
$230$	0	0
$231$	−1.95063	−0.128342
$232$	0	0
$233$	−23.6325	−1.54822	−0.774108	−	0.633054i	$-0.781801\pi$
$233$	−23.6325	−1.54822	−0.774108	+	0.633054i	$0.781801\pi$
$234$	0	0
$235$	11.9577	0.780033
$236$	0	0
$237$	7.15856	0.464998
$238$	0	0
$239$	−8.57704	−0.554802	−0.277401	−	0.960754i	$-0.589473\pi$
$239$	−8.57704	−0.554802	−0.277401	+	0.960754i	$0.589473\pi$
$240$	0	0
$241$	−24.4469	−1.57476	−0.787381	−	0.616467i	$-0.788564\pi$
$241$	−24.4469	−1.57476	−0.787381	+	0.616467i	$0.788564\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	1.19202	0.0761555
$246$	0	0
$247$	−17.0578	−1.08536
$248$	0	0
$249$	2.24441	0.142234
$250$	0	0
$251$	−3.75520	−0.237026	−0.118513	−	0.992952i	$-0.537813\pi$
$251$	−3.75520	−0.237026	−0.118513	+	0.992952i	$0.537813\pi$
$252$	0	0
$253$	10.1890	0.640577
$254$	0	0
$255$	1.19202	0.0746473
$256$	0	0
$257$	−2.51420	−0.156832	−0.0784158	−	0.996921i	$-0.524986\pi$
$257$	−2.51420	−0.156832	−0.0784158	+	0.996921i	$0.524986\pi$
$258$	0	0
$259$	0.758609	0.0471377
$260$	0	0
$261$	5.27281	0.326379
$262$	0	0
$263$	−7.05340	−0.434931	−0.217465	−	0.976068i	$-0.569779\pi$
$263$	−7.05340	−0.434931	−0.217465	+	0.976068i	$0.569779\pi$
$264$	0	0
$265$	9.57363	0.588103
$266$	0	0
$267$	2.16158	0.132286
$268$	0	0
$269$	−12.3252	−0.751480	−0.375740	−	0.926725i	$-0.622611\pi$
$269$	−12.3252	−0.751480	−0.375740	+	0.926725i	$0.622611\pi$
$270$	0	0
$271$	−19.6639	−1.19450	−0.597248	−	0.802056i	$-0.703739\pi$
$271$	−19.6639	−1.19450	−0.597248	+	0.802056i	$0.703739\pi$
$272$	0	0
$273$	−4.46483	−0.270224
$274$	0	0
$275$	6.98147	0.420999
$276$	0	0
$277$	9.37155	0.563082	0.281541	−	0.959549i	$-0.409154\pi$
$277$	9.37155	0.563082	0.281541	+	0.959549i	$0.409154\pi$
$278$	0	0
$279$	4.51420	0.270258
$280$	0	0
$281$	8.90467	0.531208	0.265604	−	0.964082i	$-0.414429\pi$
$281$	8.90467	0.531208	0.265604	+	0.964082i	$0.414429\pi$
$282$	0	0
$283$	4.25389	0.252867	0.126434	−	0.991975i	$-0.459647\pi$
$283$	4.25389	0.252867	0.126434	+	0.991975i	$0.459647\pi$
$284$	0	0
$285$	4.55409	0.269761
$286$	0	0
$287$	−3.83638	−0.226454
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	12.4469	0.729649
$292$	0	0
$293$	23.8813	1.39516	0.697579	−	0.716508i	$-0.254261\pi$
$293$	23.8813	1.39516	0.697579	+	0.716508i	$0.254261\pi$
$294$	0	0
$295$	12.7068	0.739819
$296$	0	0
$297$	−1.95063	−0.113187
$298$	0	0
$299$	23.3218	1.34873
$300$	0	0
$301$	7.60749	0.438488
$302$	0	0
$303$	1.62544	0.0933788
$304$	0	0
$305$	4.49524	0.257397
$306$	0	0
$307$	−29.9482	−1.70923	−0.854617	−	0.519259i	$-0.826208\pi$
$307$	−29.9482	−1.70923	−0.854617	+	0.519259i	$0.826208\pi$
$308$	0	0
$309$	7.68084	0.436948
$310$	0	0
$311$	6.74310	0.382366	0.191183	−	0.981554i	$-0.438768\pi$
$311$	6.74310	0.382366	0.191183	+	0.981554i	$0.438768\pi$
$312$	0	0
$313$	−26.3985	−1.49213	−0.746066	−	0.665872i	$-0.768060\pi$
$313$	−26.3985	−1.49213	−0.746066	+	0.665872i	$0.768060\pi$
$314$	0	0
$315$	1.19202	0.0671628
$316$	0	0
$317$	−17.4344	−0.979213	−0.489606	−	0.871944i	$-0.662859\pi$
$317$	−17.4344	−0.979213	−0.489606	+	0.871944i	$0.662859\pi$
$318$	0	0
$319$	−10.2853	−0.575867
$320$	0	0
$321$	4.43341	0.247449
$322$	0	0
$323$	3.82047	0.212577
$324$	0	0
$325$	15.9800	0.886411
$326$	0	0
$327$	−0.498295	−0.0275558
$328$	0	0
$329$	10.0314	0.553050
$330$	0	0
$331$	−12.9631	−0.712518	−0.356259	−	0.934387i	$-0.615948\pi$
$331$	−12.9631	−0.712518	−0.356259	+	0.934387i	$0.615948\pi$
$332$	0	0
$333$	0.758609	0.0415715
$334$	0	0
$335$	−9.41848	−0.514587
$336$	0	0
$337$	34.6588	1.88799	0.943994	−	0.329964i	$-0.107037\pi$
$337$	34.6588	1.88799	0.943994	+	0.329964i	$0.107037\pi$
$338$	0	0
$339$	1.70622	0.0926693
$340$	0	0
$341$	−8.80554	−0.476847
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−6.22646	−0.335221
$346$	0	0
$347$	−3.47029	−0.186295	−0.0931474	−	0.995652i	$-0.529693\pi$
$347$	−3.47029	−0.186295	−0.0931474	+	0.995652i	$0.529693\pi$
$348$	0	0
$349$	−19.2390	−1.02984	−0.514919	−	0.857239i	$-0.672178\pi$
$349$	−19.2390	−1.02984	−0.514919	+	0.857239i	$0.672178\pi$
$350$	0	0
$351$	−4.46483	−0.238315
$352$	0	0
$353$	25.9895	1.38328	0.691640	−	0.722242i	$-0.256888\pi$
$353$	25.9895	1.38328	0.691640	+	0.722242i	$0.256888\pi$
$354$	0	0
$355$	14.1866	0.752945
$356$	0	0
$357$	1.00000	0.0529256
$358$	0	0
$359$	36.3167	1.91672	0.958362	−	0.285557i	$-0.0921786\pi$
$359$	36.3167	1.91672	0.958362	+	0.285557i	$0.0921786\pi$
$360$	0	0
$361$	−4.40398	−0.231788
$362$	0	0
$363$	−7.19504	−0.377642
$364$	0	0
$365$	3.60992	0.188952
$366$	0	0
$367$	−8.08180	−0.421866	−0.210933	−	0.977500i	$-0.567650\pi$
$367$	−8.08180	−0.421866	−0.210933	+	0.977500i	$0.567650\pi$
$368$	0	0
$369$	−3.83638	−0.199714
$370$	0	0
$371$	8.03142	0.416970
$372$	0	0
$373$	−18.9925	−0.983394	−0.491697	−	0.870766i	$-0.663623\pi$
$373$	−18.9925	−0.983394	−0.491697	+	0.870766i	$0.663623\pi$
$374$	0	0
$375$	−10.2265	−0.528092
$376$	0	0
$377$	−23.5422	−1.21249
$378$	0	0
$379$	−4.56561	−0.234520	−0.117260	−	0.993101i	$-0.537411\pi$
$379$	−4.56561	−0.234520	−0.117260	+	0.993101i	$0.537411\pi$
$380$	0	0
$381$	1.32218	0.0677373
$382$	0	0
$383$	3.92665	0.200642	0.100321	−	0.994955i	$-0.468013\pi$
$383$	3.92665	0.200642	0.100321	+	0.994955i	$0.468013\pi$
$384$	0	0
$385$	−2.32520	−0.118503
$386$	0	0
$387$	7.60749	0.386710
$388$	0	0
$389$	12.1866	0.617884	0.308942	−	0.951081i	$-0.400025\pi$
$389$	12.1866	0.617884	0.308942	+	0.951081i	$0.400025\pi$
$390$	0	0
$391$	−5.22344	−0.264161
$392$	0	0
$393$	−8.02295	−0.404704
$394$	0	0
$395$	8.53316	0.429350
$396$	0	0
$397$	17.9936	0.903072	0.451536	−	0.892253i	$-0.350876\pi$
$397$	17.9936	0.903072	0.451536	+	0.892253i	$0.350876\pi$
$398$	0	0
$399$	3.82047	0.191263
$400$	0	0
$401$	−15.9915	−0.798579	−0.399289	−	0.916825i	$-0.630743\pi$
$401$	−15.9915	−0.798579	−0.399289	+	0.916825i	$0.630743\pi$
$402$	0	0
$403$	−20.1551	−1.00400
$404$	0	0
$405$	1.19202	0.0592321
$406$	0	0
$407$	−1.47977	−0.0733493
$408$	0	0
$409$	−21.0264	−1.03969	−0.519843	−	0.854262i	$-0.674010\pi$
$409$	−21.0264	−1.03969	−0.519843	+	0.854262i	$0.674010\pi$
$410$	0	0
$411$	−9.05982	−0.446888
$412$	0	0
$413$	10.6599	0.524538
$414$	0	0
$415$	2.67538	0.131329
$416$	0	0
$417$	19.4435	0.952151
$418$	0	0
$419$	−2.15554	−0.105305	−0.0526526	−	0.998613i	$-0.516768\pi$
$419$	−2.15554	−0.105305	−0.0526526	+	0.998613i	$0.516768\pi$
$420$	0	0
$421$	19.9726	0.973403	0.486702	−	0.873568i	$-0.338200\pi$
$421$	19.9726	0.973403	0.486702	+	0.873568i	$0.338200\pi$
$422$	0	0
$423$	10.0314	0.487744
$424$	0	0
$425$	−3.57908	−0.173611
$426$	0	0
$427$	3.77111	0.182497
$428$	0	0
$429$	8.70924	0.420486
$430$	0	0
$431$	−21.0912	−1.01593	−0.507965	−	0.861378i	$-0.669602\pi$
$431$	−21.0912	−1.01593	−0.507965	+	0.861378i	$0.669602\pi$
$432$	0	0
$433$	26.4320	1.27024	0.635120	−	0.772413i	$-0.280951\pi$
$433$	26.4320	1.27024	0.635120	+	0.772413i	$0.280951\pi$
$434$	0	0
$435$	6.28531	0.301357
$436$	0	0
$437$	−19.9560	−0.954626
$438$	0	0
$439$	30.7636	1.46827	0.734133	−	0.679005i	$-0.237589\pi$
$439$	30.7636	1.46827	0.734133	+	0.679005i	$0.237589\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	34.7636	1.65167	0.825834	−	0.563913i	$-0.190705\pi$
$443$	34.7636	1.65167	0.825834	+	0.563913i	$0.190705\pi$
$444$	0	0
$445$	2.57665	0.122145
$446$	0	0
$447$	−17.0534	−0.806598
$448$	0	0
$449$	−8.44688	−0.398633	−0.199316	−	0.979935i	$-0.563872\pi$
$449$	−8.44688	−0.398633	−0.199316	+	0.979935i	$0.563872\pi$
$450$	0	0
$451$	7.48336	0.352378
$452$	0	0
$453$	8.16158	0.383464
$454$	0	0
$455$	−5.32218	−0.249508
$456$	0	0
$457$	31.4668	1.47196	0.735978	−	0.677006i	$-0.236723\pi$
$457$	31.4668	1.47196	0.735978	+	0.677006i	$0.236723\pi$
$458$	0	0
$459$	1.00000	0.0466760
$460$	0	0
$461$	29.8370	1.38965	0.694824	−	0.719180i	$-0.255482\pi$
$461$	29.8370	1.38965	0.694824	+	0.719180i	$0.255482\pi$
$462$	0	0
$463$	−20.1298	−0.935509	−0.467755	−	0.883858i	$-0.654937\pi$
$463$	−20.1298	−0.935509	−0.467755	+	0.883858i	$0.654937\pi$
$464$	0	0
$465$	5.38103	0.249539
$466$	0	0
$467$	−17.5392	−0.811615	−0.405808	−	0.913958i	$-0.633010\pi$
$467$	−17.5392	−0.811615	−0.405808	+	0.913958i	$0.633010\pi$
$468$	0	0
$469$	−7.90126	−0.364846
$470$	0	0
$471$	−12.6264	−0.581794
$472$	0	0
$473$	−14.8394	−0.682316
$474$	0	0
$475$	−13.6738	−0.627397
$476$	0	0
$477$	8.03142	0.367733
$478$	0	0
$479$	−5.80048	−0.265031	−0.132515	−	0.991181i	$-0.542305\pi$
$479$	−5.80048	−0.265031	−0.132515	+	0.991181i	$0.542305\pi$
$480$	0	0
$481$	−3.38706	−0.154437
$482$	0	0
$483$	−5.22344	−0.237675
$484$	0	0
$485$	14.8370	0.673712
$486$	0	0
$487$	15.2150	0.689456	0.344728	−	0.938703i	$-0.387971\pi$
$487$	15.2150	0.689456	0.344728	+	0.938703i	$0.387971\pi$
$488$	0	0
$489$	2.24441	0.101496
$490$	0	0
$491$	−40.3062	−1.81899	−0.909497	−	0.415711i	$-0.863533\pi$
$491$	−40.3062	−1.81899	−0.909497	+	0.415711i	$0.863533\pi$
$492$	0	0
$493$	5.27281	0.237475
$494$	0	0
$495$	−2.32520	−0.104510
$496$	0	0
$497$	11.9013	0.533845
$498$	0	0
$499$	−15.2664	−0.683417	−0.341709	−	0.939806i	$-0.611006\pi$
$499$	−15.2664	−0.683417	−0.341709	+	0.939806i	$0.611006\pi$
$500$	0	0
$501$	8.54766	0.381881
$502$	0	0
$503$	13.8364	0.616934	0.308467	−	0.951235i	$-0.400184\pi$
$503$	13.8364	0.616934	0.308467	+	0.951235i	$0.400184\pi$
$504$	0	0
$505$	1.93756	0.0862201
$506$	0	0
$507$	6.93473	0.307982
$508$	0	0
$509$	26.8718	1.19107	0.595536	−	0.803328i	$-0.296940\pi$
$509$	26.8718	1.19107	0.595536	+	0.803328i	$0.296940\pi$
$510$	0	0
$511$	3.02840	0.133969
$512$	0	0
$513$	3.82047	0.168678
$514$	0	0
$515$	9.15573	0.403450
$516$	0	0
$517$	−19.5676	−0.860582
$518$	0	0
$519$	15.5402	0.682138
$520$	0	0
$521$	2.28326	0.100032	0.0500158	−	0.998748i	$-0.484073\pi$
$521$	2.28326	0.100032	0.0500158	+	0.998748i	$0.484073\pi$
$522$	0	0
$523$	7.69470	0.336466	0.168233	−	0.985747i	$-0.446194\pi$
$523$	7.69470	0.336466	0.168233	+	0.985747i	$0.446194\pi$
$524$	0	0
$525$	−3.57908	−0.156204
$526$	0	0
$527$	4.51420	0.196642
$528$	0	0
$529$	4.28433	0.186275
$530$	0	0
$531$	10.6599	0.462599
$532$	0	0
$533$	17.1288	0.741930
$534$	0	0
$535$	5.28473	0.228479
$536$	0	0
$537$	2.90467	0.125346
$538$	0	0
$539$	−1.95063	−0.0840196
$540$	0	0
$541$	−0.993537	−0.0427155	−0.0213577	−	0.999772i	$-0.506799\pi$
$541$	−0.993537	−0.0427155	−0.0213577	+	0.999772i	$0.506799\pi$
$542$	0	0
$543$	−17.0534	−0.731831
$544$	0	0
$545$	−0.593979	−0.0254433
$546$	0	0
$547$	−1.07982	−0.0461696	−0.0230848	−	0.999734i	$-0.507349\pi$
$547$	−1.07982	−0.0461696	−0.0230848	+	0.999734i	$0.507349\pi$
$548$	0	0
$549$	3.77111	0.160947
$550$	0	0
$551$	20.1446	0.858190
$552$	0	0
$553$	7.15856	0.304413
$554$	0	0
$555$	0.904279	0.0383845
$556$	0	0
$557$	34.0882	1.44436	0.722182	−	0.691703i	$-0.243139\pi$
$557$	34.0882	1.44436	0.722182	+	0.691703i	$0.243139\pi$
$558$	0	0
$559$	−33.9661	−1.43661
$560$	0	0
$561$	−1.95063	−0.0823557
$562$	0	0
$563$	31.1071	1.31101	0.655505	−	0.755191i	$-0.272456\pi$
$563$	31.1071	1.31101	0.655505	+	0.755191i	$0.272456\pi$
$564$	0	0
$565$	2.03386	0.0855650
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	4.95466	0.207710	0.103855	−	0.994592i	$-0.466882\pi$
$569$	4.95466	0.207710	0.103855	+	0.994592i	$0.466882\pi$
$570$	0	0
$571$	−7.68185	−0.321475	−0.160738	−	0.986997i	$-0.551387\pi$
$571$	−7.68185	−0.321475	−0.160738	+	0.986997i	$0.551387\pi$
$572$	0	0
$573$	21.1582	0.883895
$574$	0	0
$575$	18.6951	0.779641
$576$	0	0
$577$	−0.671395	−0.0279506	−0.0139753	−	0.999902i	$-0.504449\pi$
$577$	−0.671395	−0.0279506	−0.0139753	+	0.999902i	$0.504449\pi$
$578$	0	0
$579$	6.51420	0.270721
$580$	0	0
$581$	2.24441	0.0931137
$582$	0	0
$583$	−15.6663	−0.648833
$584$	0	0
$585$	−5.32218	−0.220045
$586$	0	0
$587$	−13.7302	−0.566706	−0.283353	−	0.959016i	$-0.591447\pi$
$587$	−13.7302	−0.566706	−0.283353	+	0.959016i	$0.591447\pi$
$588$	0	0
$589$	17.2464	0.710625
$590$	0	0
$591$	−0.979033	−0.0402720
$592$	0	0
$593$	−12.4533	−0.511396	−0.255698	−	0.966757i	$-0.582305\pi$
$593$	−12.4533	−0.511396	−0.255698	+	0.966757i	$0.582305\pi$
$594$	0	0
$595$	1.19202	0.0488681
$596$	0	0
$597$	−5.18355	−0.212149
$598$	0	0
$599$	21.2400	0.867841	0.433921	−	0.900951i	$-0.357130\pi$
$599$	21.2400	0.867841	0.433921	+	0.900951i	$0.357130\pi$
$600$	0	0
$601$	22.6758	0.924964	0.462482	−	0.886629i	$-0.346959\pi$
$601$	22.6758	0.924964	0.462482	+	0.886629i	$0.346959\pi$
$602$	0	0
$603$	−7.90126	−0.321764
$604$	0	0
$605$	−8.57665	−0.348690
$606$	0	0
$607$	−1.05379	−0.0427719	−0.0213860	−	0.999771i	$-0.506808\pi$
$607$	−1.05379	−0.0427719	−0.0213860	+	0.999771i	$0.506808\pi$
$608$	0	0
$609$	5.27281	0.213665
$610$	0	0
$611$	−44.7886	−1.81195
$612$	0	0
$613$	13.6393	0.550886	0.275443	−	0.961317i	$-0.411175\pi$
$613$	13.6393	0.550886	0.275443	+	0.961317i	$0.411175\pi$
$614$	0	0
$615$	−4.57305	−0.184403
$616$	0	0
$617$	15.3867	0.619444	0.309722	−	0.950827i	$-0.399764\pi$
$617$	15.3867	0.619444	0.309722	+	0.950827i	$0.399764\pi$
$618$	0	0
$619$	−48.4114	−1.94582	−0.972909	−	0.231190i	$-0.925738\pi$
$619$	−48.4114	−1.94582	−0.972909	+	0.231190i	$0.925738\pi$
$620$	0	0
$621$	−5.22344	−0.209609
$622$	0	0
$623$	2.16158	0.0866017
$624$	0	0
$625$	5.70525	0.228210
$626$	0	0
$627$	−7.45234	−0.297618
$628$	0	0
$629$	0.758609	0.0302477
$630$	0	0
$631$	−43.3072	−1.72403	−0.862016	−	0.506881i	$-0.830798\pi$
$631$	−43.3072	−1.72403	−0.862016	+	0.506881i	$0.830798\pi$
$632$	0	0
$633$	−22.5611	−0.896725
$634$	0	0
$635$	1.57607	0.0625443
$636$	0	0
$637$	−4.46483	−0.176903
$638$	0	0
$639$	11.9013	0.470807
$640$	0	0
$641$	−24.6250	−0.972628	−0.486314	−	0.873784i	$-0.661659\pi$
$641$	−24.6250	−0.972628	−0.486314	+	0.873784i	$0.661659\pi$
$642$	0	0
$643$	−13.1751	−0.519575	−0.259788	−	0.965666i	$-0.583653\pi$
$643$	−13.1751	−0.519575	−0.259788	+	0.965666i	$0.583653\pi$
$644$	0	0
$645$	9.06829	0.357064
$646$	0	0
$647$	−6.18657	−0.243219	−0.121610	−	0.992578i	$-0.538806\pi$
$647$	−6.18657	−0.243219	−0.121610	+	0.992578i	$0.538806\pi$
$648$	0	0
$649$	−20.7935	−0.816215
$650$	0	0
$651$	4.51420	0.176925
$652$	0	0
$653$	31.8978	1.24826	0.624128	−	0.781322i	$-0.285454\pi$
$653$	31.8978	1.24826	0.624128	+	0.781322i	$0.285454\pi$
$654$	0	0
$655$	−9.56353	−0.373678
$656$	0	0
$657$	3.02840	0.118149
$658$	0	0
$659$	−21.6724	−0.844236	−0.422118	−	0.906541i	$-0.638713\pi$
$659$	−21.6724	−0.844236	−0.422118	+	0.906541i	$0.638713\pi$
$660$	0	0
$661$	−17.1511	−0.667101	−0.333551	−	0.942732i	$-0.608247\pi$
$661$	−17.1511	−0.667101	−0.333551	+	0.942732i	$0.608247\pi$
$662$	0	0
$663$	−4.46483	−0.173400
$664$	0	0
$665$	4.55409	0.176600
$666$	0	0
$667$	−27.5422	−1.06644
$668$	0	0
$669$	8.12169	0.314003
$670$	0	0
$671$	−7.35603	−0.283977
$672$	0	0
$673$	−15.0594	−0.580498	−0.290249	−	0.956951i	$-0.593738\pi$
$673$	−15.0594	−0.580498	−0.290249	+	0.956951i	$0.593738\pi$
$674$	0	0
$675$	−3.57908	−0.137759
$676$	0	0
$677$	27.0514	1.03967	0.519834	−	0.854267i	$-0.325994\pi$
$677$	27.0514	1.03967	0.519834	+	0.854267i	$0.325994\pi$
$678$	0	0
$679$	12.4469	0.477667
$680$	0	0
$681$	2.42393	0.0928853
$682$	0	0
$683$	−31.9087	−1.22095	−0.610476	−	0.792035i	$-0.709022\pi$
$683$	−31.9087	−1.22095	−0.610476	+	0.792035i	$0.709022\pi$
$684$	0	0
$685$	−10.7995	−0.412628
$686$	0	0
$687$	−15.8184	−0.603511
$688$	0	0
$689$	−35.8589	−1.36612
$690$	0	0
$691$	−33.5885	−1.27777	−0.638883	−	0.769304i	$-0.720603\pi$
$691$	−33.5885	−1.27777	−0.638883	+	0.769304i	$0.720603\pi$
$692$	0	0
$693$	−1.95063	−0.0740983
$694$	0	0
$695$	23.1771	0.879156
$696$	0	0
$697$	−3.83638	−0.145313
$698$	0	0
$699$	−23.6325	−0.893862
$700$	0	0
$701$	−35.8234	−1.35303	−0.676516	−	0.736428i	$-0.736511\pi$
$701$	−35.8234	−1.35303	−0.676516	+	0.736428i	$0.736511\pi$
$702$	0	0
$703$	2.89825	0.109309
$704$	0	0
$705$	11.9577	0.450352
$706$	0	0
$707$	1.62544	0.0611308
$708$	0	0
$709$	4.63449	0.174052	0.0870259	−	0.996206i	$-0.472264\pi$
$709$	4.63449	0.174052	0.0870259	+	0.996206i	$0.472264\pi$
$710$	0	0
$711$	7.15856	0.268467
$712$	0	0
$713$	−23.5797	−0.883065
$714$	0	0
$715$	10.3816	0.388250
$716$	0	0
$717$	−8.57704	−0.320315
$718$	0	0
$719$	8.57869	0.319931	0.159966	−	0.987123i	$-0.448862\pi$
$719$	8.57869	0.319931	0.159966	+	0.987123i	$0.448862\pi$
$720$	0	0
$721$	7.68084	0.286049
$722$	0	0
$723$	−24.4469	−0.909189
$724$	0	0
$725$	−18.8718	−0.700882
$726$	0	0
$727$	−9.74650	−0.361478	−0.180739	−	0.983531i	$-0.557849\pi$
$727$	−9.74650	−0.361478	−0.180739	+	0.983531i	$0.557849\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	7.60749	0.281373
$732$	0	0
$733$	41.5790	1.53576	0.767878	−	0.640596i	$-0.221313\pi$
$733$	41.5790	1.53576	0.767878	+	0.640596i	$0.221313\pi$
$734$	0	0
$735$	1.19202	0.0439684
$736$	0	0
$737$	15.4124	0.567725
$738$	0	0
$739$	−20.0963	−0.739254	−0.369627	−	0.929180i	$-0.620515\pi$
$739$	−20.0963	−0.739254	−0.369627	+	0.929180i	$0.620515\pi$
$740$	0	0
$741$	−17.0578	−0.626633
$742$	0	0
$743$	30.7890	1.12954	0.564769	−	0.825249i	$-0.308965\pi$
$743$	30.7890	1.12954	0.564769	+	0.825249i	$0.308965\pi$
$744$	0	0
$745$	−20.3280	−0.744761
$746$	0	0
$747$	2.24441	0.0821186
$748$	0	0
$749$	4.43341	0.161993
$750$	0	0
$751$	46.7326	1.70530	0.852648	−	0.522486i	$-0.174995\pi$
$751$	46.7326	1.70530	0.852648	+	0.522486i	$0.174995\pi$
$752$	0	0
$753$	−3.75520	−0.136847
$754$	0	0
$755$	9.72878	0.354067
$756$	0	0
$757$	−7.51557	−0.273158	−0.136579	−	0.990629i	$-0.543611\pi$
$757$	−7.51557	−0.273158	−0.136579	+	0.990629i	$0.543611\pi$
$758$	0	0
$759$	10.1890	0.369837
$760$	0	0
$761$	18.9776	0.687939	0.343969	−	0.938981i	$-0.388228\pi$
$761$	18.9776	0.687939	0.343969	+	0.938981i	$0.388228\pi$
$762$	0	0
$763$	−0.498295	−0.0180395
$764$	0	0
$765$	1.19202	0.0430977
$766$	0	0
$767$	−47.5945	−1.71854
$768$	0	0
$769$	−26.2515	−0.946652	−0.473326	−	0.880887i	$-0.656947\pi$
$769$	−26.2515	−0.946652	−0.473326	+	0.880887i	$0.656947\pi$
$770$	0	0
$771$	−2.51420	−0.0905467
$772$	0	0
$773$	12.1082	0.435502	0.217751	−	0.976004i	$-0.430128\pi$
$773$	12.1082	0.435502	0.217751	+	0.976004i	$0.430128\pi$
$774$	0	0
$775$	−16.1567	−0.580366
$776$	0	0
$777$	0.758609	0.0272149
$778$	0	0
$779$	−14.6568	−0.525134
$780$	0	0
$781$	−23.2150	−0.830697
$782$	0	0
$783$	5.27281	0.188435
$784$	0	0
$785$	−15.0510	−0.537192
$786$	0	0
$787$	36.0713	1.28580	0.642901	−	0.765949i	$-0.277731\pi$
$787$	36.0713	1.28580	0.642901	+	0.765949i	$0.277731\pi$
$788$	0	0
$789$	−7.05340	−0.251108
$790$	0	0
$791$	1.70622	0.0606663
$792$	0	0
$793$	−16.8374	−0.597912
$794$	0	0
$795$	9.57363	0.339542
$796$	0	0
$797$	−39.3925	−1.39535	−0.697676	−	0.716413i	$-0.745783\pi$
$797$	−39.3925	−1.39535	−0.697676	+	0.716413i	$0.745783\pi$
$798$	0	0
$799$	10.0314	0.354886
$800$	0	0
$801$	2.16158	0.0763755
$802$	0	0
$803$	−5.90730	−0.208464
$804$	0	0
$805$	−6.22646	−0.219454
$806$	0	0
$807$	−12.3252	−0.433867
$808$	0	0
$809$	−3.92869	−0.138125	−0.0690627	−	0.997612i	$-0.522001\pi$
$809$	−3.92869	−0.138125	−0.0690627	+	0.997612i	$0.522001\pi$
$810$	0	0
$811$	−15.5781	−0.547021	−0.273511	−	0.961869i	$-0.588185\pi$
$811$	−15.5781	−0.547021	−0.273511	+	0.961869i	$0.588185\pi$
$812$	0	0
$813$	−19.6639	−0.689643
$814$	0	0
$815$	2.67538	0.0937146
$816$	0	0
$817$	29.0642	1.01683
$818$	0	0
$819$	−4.46483	−0.156014
$820$	0	0
$821$	11.5247	0.402213	0.201107	−	0.979569i	$-0.435546\pi$
$821$	11.5247	0.402213	0.201107	+	0.979569i	$0.435546\pi$
$822$	0	0
$823$	3.99357	0.139207	0.0696036	−	0.997575i	$-0.477827\pi$
$823$	3.99357	0.139207	0.0696036	+	0.997575i	$0.477827\pi$
$824$	0	0
$825$	6.98147	0.243064
$826$	0	0
$827$	−48.0215	−1.66987	−0.834936	−	0.550347i	$-0.814495\pi$
$827$	−48.0215	−1.66987	−0.834936	+	0.550347i	$0.814495\pi$
$828$	0	0
$829$	−28.7481	−0.998463	−0.499231	−	0.866469i	$-0.666384\pi$
$829$	−28.7481	−0.998463	−0.499231	+	0.866469i	$0.666384\pi$
$830$	0	0
$831$	9.37155	0.325095
$832$	0	0
$833$	1.00000	0.0346479
$834$	0	0
$835$	10.1890	0.352605
$836$	0	0
$837$	4.51420	0.156034
$838$	0	0
$839$	26.8478	0.926890	0.463445	−	0.886126i	$-0.346613\pi$
$839$	26.8478	0.926890	0.463445	+	0.886126i	$0.346613\pi$
$840$	0	0
$841$	−1.19748	−0.0412923
$842$	0	0
$843$	8.90467	0.306693
$844$	0	0
$845$	8.26635	0.284371
$846$	0	0
$847$	−7.19504	−0.247224
$848$	0	0
$849$	4.25389	0.145993
$850$	0	0
$851$	−3.96255	−0.135834
$852$	0	0
$853$	−50.9232	−1.74358	−0.871789	−	0.489881i	$-0.837040\pi$
$853$	−50.9232	−1.74358	−0.871789	+	0.489881i	$0.837040\pi$
$854$	0	0
$855$	4.55409	0.155747
$856$	0	0
$857$	8.77412	0.299718	0.149859	−	0.988707i	$-0.452118\pi$
$857$	8.77412	0.299718	0.149859	+	0.988707i	$0.452118\pi$
$858$	0	0
$859$	23.9740	0.817981	0.408991	−	0.912539i	$-0.365881\pi$
$859$	23.9740	0.817981	0.408991	+	0.912539i	$0.365881\pi$
$860$	0	0
$861$	−3.83638	−0.130743
$862$	0	0
$863$	16.0564	0.546567	0.273283	−	0.961934i	$-0.411890\pi$
$863$	16.0564	0.546567	0.273283	+	0.961934i	$0.411890\pi$
$864$	0	0
$865$	18.5242	0.629842
$866$	0	0
$867$	1.00000	0.0339618
$868$	0	0
$869$	−13.9637	−0.473686
$870$	0	0
$871$	35.2778	1.19534
$872$	0	0
$873$	12.4469	0.421263
$874$	0	0
$875$	−10.2265	−0.345717
$876$	0	0
$877$	7.43478	0.251055	0.125527	−	0.992090i	$-0.459938\pi$
$877$	7.43478	0.251055	0.125527	+	0.992090i	$0.459938\pi$
$878$	0	0
$879$	23.8813	0.805495
$880$	0	0
$881$	10.6335	0.358250	0.179125	−	0.983826i	$-0.442673\pi$
$881$	10.6335	0.358250	0.179125	+	0.983826i	$0.442673\pi$
$882$	0	0
$883$	−53.0578	−1.78554	−0.892768	−	0.450516i	$-0.851240\pi$
$883$	−53.0578	−1.78554	−0.892768	+	0.450516i	$0.851240\pi$
$884$	0	0
$885$	12.7068	0.427134
$886$	0	0
$887$	33.0832	1.11082	0.555412	−	0.831575i	$-0.312561\pi$
$887$	33.0832	1.11082	0.555412	+	0.831575i	$0.312561\pi$
$888$	0	0
$889$	1.32218	0.0443445
$890$	0	0
$891$	−1.95063	−0.0653486
$892$	0	0
$893$	38.3248	1.28249
$894$	0	0
$895$	3.46243	0.115736
$896$	0	0
$897$	23.3218	0.778692
$898$	0	0
$899$	23.8025	0.793859
$900$	0	0
$901$	8.03142	0.267565
$902$	0	0
$903$	7.60749	0.253161
$904$	0	0
$905$	−20.3280	−0.675726
$906$	0	0
$907$	−28.0091	−0.930026	−0.465013	−	0.885304i	$-0.653950\pi$
$907$	−28.0091	−0.930026	−0.465013	+	0.885304i	$0.653950\pi$
$908$	0	0
$909$	1.62544	0.0539123
$910$	0	0
$911$	51.2484	1.69794	0.848968	−	0.528444i	$-0.177224\pi$
$911$	51.2484	1.69794	0.848968	+	0.528444i	$0.177224\pi$
$912$	0	0
$913$	−4.37801	−0.144891
$914$	0	0
$915$	4.49524	0.148608
$916$	0	0
$917$	−8.02295	−0.264941
$918$	0	0
$919$	12.4743	0.411490	0.205745	−	0.978606i	$-0.434038\pi$
$919$	12.4743	0.411490	0.205745	+	0.978606i	$0.434038\pi$
$920$	0	0
$921$	−29.9482	−0.986827
$922$	0	0
$923$	−53.1371	−1.74903
$924$	0	0
$925$	−2.71512	−0.0892727
$926$	0	0
$927$	7.68084	0.252272
$928$	0	0
$929$	9.37256	0.307504	0.153752	−	0.988109i	$-0.450864\pi$
$929$	9.37256	0.307504	0.153752	+	0.988109i	$0.450864\pi$
$930$	0	0
$931$	3.82047	0.125211
$932$	0	0
$933$	6.74310	0.220759
$934$	0	0
$935$	−2.32520	−0.0760420
$936$	0	0
$937$	−13.0303	−0.425683	−0.212841	−	0.977087i	$-0.568272\pi$
$937$	−13.0303	−0.425683	−0.212841	+	0.977087i	$0.568272\pi$
$938$	0	0
$939$	−26.3985	−0.861483
$940$	0	0
$941$	7.97501	0.259978	0.129989	−	0.991515i	$-0.458506\pi$
$941$	7.97501	0.259978	0.129989	+	0.991515i	$0.458506\pi$
$942$	0	0
$943$	20.0391	0.652563
$944$	0	0
$945$	1.19202	0.0387765
$946$	0	0
$947$	36.1348	1.17422	0.587111	−	0.809506i	$-0.300265\pi$
$947$	36.1348	1.17422	0.587111	+	0.809506i	$0.300265\pi$
$948$	0	0
$949$	−13.5213	−0.438920
$950$	0	0
$951$	−17.4344	−0.565349
$952$	0	0
$953$	52.7108	1.70747	0.853735	−	0.520708i	$-0.174332\pi$
$953$	52.7108	1.70747	0.853735	+	0.520708i	$0.174332\pi$
$954$	0	0
$955$	25.2210	0.816133
$956$	0	0
$957$	−10.2853	−0.332477
$958$	0	0
$959$	−9.05982	−0.292557
$960$	0	0
$961$	−10.6220	−0.342645
$962$	0	0
$963$	4.43341	0.142865
$964$	0	0
$965$	7.76507	0.249967
$966$	0	0
$967$	23.2295	0.747010	0.373505	−	0.927628i	$-0.378156\pi$
$967$	23.2295	0.747010	0.373505	+	0.927628i	$0.378156\pi$
$968$	0	0
$969$	3.82047	0.122731
$970$	0	0
$971$	48.1537	1.54533	0.772663	−	0.634816i	$-0.218924\pi$
$971$	48.1537	1.54533	0.772663	+	0.634816i	$0.218924\pi$
$972$	0	0
$973$	19.4435	0.623329
$974$	0	0
$975$	15.9800	0.511770
$976$	0	0
$977$	−17.3447	−0.554907	−0.277454	−	0.960739i	$-0.589490\pi$
$977$	−17.3447	−0.554907	−0.277454	+	0.960739i	$0.589490\pi$
$978$	0	0
$979$	−4.21644	−0.134758
$980$	0	0
$981$	−0.498295	−0.0159093
$982$	0	0
$983$	8.07579	0.257578	0.128789	−	0.991672i	$-0.458891\pi$
$983$	8.07579	0.257578	0.128789	+	0.991672i	$0.458891\pi$
$984$	0	0
$985$	−1.16703	−0.0371846
$986$	0	0
$987$	10.0314	0.319304
$988$	0	0
$989$	−39.7372	−1.26357
$990$	0	0
$991$	34.7213	1.10296	0.551479	−	0.834189i	$-0.314064\pi$
$991$	34.7213	1.10296	0.551479	+	0.834189i	$0.314064\pi$
$992$	0	0
$993$	−12.9631	−0.411372
$994$	0	0
$995$	−6.17891	−0.195885
$996$	0	0
$997$	−41.0844	−1.30116	−0.650578	−	0.759439i	$-0.725473\pi$
$997$	−41.0844	−1.30116	−0.650578	+	0.759439i	$0.725473\pi$
$998$	0	0
$999$	0.758609	0.0240013

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2856.2.a.u.1.3	✓	4	4.3	odd	2
5712.2.a.bz.1.3		4	1.1	even	1	trivial
8568.2.a.bi.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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.19202 q^{5} +1.00000 q^{7} +1.00000 q^{9} -1.95063 q^{11} -4.46483 q^{13} +1.19202 q^{15} +1.00000 q^{17} +3.82047 q^{19} +1.00000 q^{21} -5.22344 q^{23} -3.57908 q^{25} +1.00000 q^{27} +5.27281 q^{29} +4.51420 q^{31} -1.95063 q^{33} +1.19202 q^{35} +0.758609 q^{37} -4.46483 q^{39} -3.83638 q^{41} +7.60749 q^{43} +1.19202 q^{45} +10.0314 q^{47} +1.00000 q^{49} +1.00000 q^{51} +8.03142 q^{53} -2.32520 q^{55} +3.82047 q^{57} +10.6599 q^{59} +3.77111 q^{61} +1.00000 q^{63} -5.32218 q^{65} -7.90126 q^{67} -5.22344 q^{69} +11.9013 q^{71} +3.02840 q^{73} -3.57908 q^{75} -1.95063 q^{77} +7.15856 q^{79} +1.00000 q^{81} +2.24441 q^{83} +1.19202 q^{85} +5.27281 q^{87} +2.16158 q^{89} -4.46483 q^{91} +4.51420 q^{93} +4.55409 q^{95} +12.4469 q^{97} -1.95063 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 2 q^{5} + 4 q^{7} + 4 q^{9} + 6 q^{11} + 2 q^{13} + 2 q^{15} + 4 q^{17} + 2 q^{19} + 4 q^{21} + 10 q^{23} + 10 q^{25} + 4 q^{27} + 4 q^{29} + 12 q^{31} + 6 q^{33} + 2 q^{35} - 8 q^{37} + 2 q^{39}+ \cdots + 6 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + 2 * q^5 + 4 * q^7 + 4 * q^9 + 6 * q^11 + 2 * q^13 + 2 * q^15 + 4 * q^17 + 2 * q^19 + 4 * q^21 + 10 * q^23 + 10 * q^25 + 4 * q^27 + 4 * q^29 + 12 * q^31 + 6 * q^33 + 2 * q^35 - 8 * q^37 + 2 * q^39 - 6 * q^41 - 6 * q^43 + 2 * q^45 + 12 * q^47 + 4 * q^49 + 4 * q^51 + 4 * q^53 + 10 * q^55 + 2 * q^57 + 4 * q^59 - 12 * q^61 + 4 * q^63 - 18 * q^65 - 4 * q^67 + 10 * q^69 + 20 * q^71 + 10 * q^75 + 6 * q^77 + 16 * q^79 + 4 * q^81 + 4 * q^83 + 2 * q^85 + 4 * q^87 - 20 * q^89 + 2 * q^91 + 12 * q^93 + 26 * q^95 - 12 * q^97 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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