LMFDB - Embedded newform 5776.2.a.bz.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5776,2,Mod(1,5776)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5776, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("5776.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5776 = 2^{4} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5776.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:	$46.1215922075$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.3022625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 7x^{4} + 7x^{3} + 17x^{2} + 4x - 1 $ x^6 - 2x^5 - 7x^4 + 7x^3 + 17x^2 + 4*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 2888)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.149742$ of defining polynomial
Character	$\chi$	$=$	5776.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.03477 q^{3} +1.96231 q^{5} -1.69050 q^{7} +6.20985 q^{9} +O(q^{10})$	$q+3.03477 q^{3} +1.96231 q^{5} -1.69050 q^{7} +6.20985 q^{9} +3.51935 q^{11} -2.77721 q^{13} +5.95515 q^{15} +7.67138 q^{17} -5.13029 q^{21} +6.70640 q^{23} -1.14936 q^{25} +9.74117 q^{27} -9.86023 q^{29} -6.78596 q^{31} +10.6804 q^{33} -3.31728 q^{35} +0.811988 q^{37} -8.42822 q^{39} +1.74832 q^{41} +12.7435 q^{43} +12.1856 q^{45} +0.564137 q^{47} -4.14220 q^{49} +23.2809 q^{51} +6.60559 q^{53} +6.90604 q^{55} -4.28493 q^{59} +7.64249 q^{61} -10.4978 q^{63} -5.44974 q^{65} +4.89603 q^{67} +20.3524 q^{69} -1.90800 q^{71} +3.83790 q^{73} -3.48803 q^{75} -5.94947 q^{77} -4.37662 q^{79} +10.9327 q^{81} -11.0958 q^{83} +15.0536 q^{85} -29.9236 q^{87} +1.71592 q^{89} +4.69489 q^{91} -20.5939 q^{93} -7.02908 q^{97} +21.8546 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 3 q^{3} + 2 q^{5} + 2 q^{7} + 9 q^{9}+O(q^{10}) $ 6 * q + 3 * q^3 + 2 * q^5 + 2 * q^7 + 9 * q^9	$ 6 q + 3 q^{3} + 2 q^{5} + 2 q^{7} + 9 q^{9} + 5 q^{11} - 10 q^{13} + 4 q^{17} - 23 q^{21} + 15 q^{23} + 12 q^{25} + 18 q^{27} + 7 q^{29} + 5 q^{31} + q^{33} + 38 q^{35} - 17 q^{37} - 37 q^{39} - 4 q^{41} + 11 q^{43} + 6 q^{45} - 18 q^{47} + 20 q^{49} + 48 q^{51} + 7 q^{53} + 42 q^{55} + q^{59} - 4 q^{61} - 19 q^{63} - 6 q^{65} + 20 q^{67} - 6 q^{69} - 6 q^{71} - 2 q^{73} + 19 q^{75} + 41 q^{77} + 8 q^{79} - 6 q^{81} + 22 q^{83} + 32 q^{85} - 29 q^{87} - 14 q^{89} + 26 q^{91} - 5 q^{93} - 41 q^{97} + 47 q^{99}+O(q^{100}) $ 6 * q + 3 * q^3 + 2 * q^5 + 2 * q^7 + 9 * q^9 + 5 * q^11 - 10 * q^13 + 4 * q^17 - 23 * q^21 + 15 * q^23 + 12 * q^25 + 18 * q^27 + 7 * q^29 + 5 * q^31 + q^33 + 38 * q^35 - 17 * q^37 - 37 * q^39 - 4 * q^41 + 11 * q^43 + 6 * q^45 - 18 * q^47 + 20 * q^49 + 48 * q^51 + 7 * q^53 + 42 * q^55 + q^59 - 4 * q^61 - 19 * q^63 - 6 * q^65 + 20 * q^67 - 6 * q^69 - 6 * q^71 - 2 * q^73 + 19 * q^75 + 41 * q^77 + 8 * q^79 - 6 * q^81 + 22 * q^83 + 32 * q^85 - 29 * q^87 - 14 * q^89 + 26 * q^91 - 5 * q^93 - 41 * q^97 + 47 * q^99

Display 100 coefficients 10 coefficients

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
$3$	3.03477	1.75213	0.876064	−	0.482195i	$-0.160160\pi$
$3$	3.03477	1.75213	0.876064	+	0.482195i	$0.160160\pi$
$4$	0	0
$5$	1.96231	0.877570	0.438785	−	0.898592i	$-0.355409\pi$
$5$	1.96231	0.877570	0.438785	+	0.898592i	$0.355409\pi$
$6$	0	0
$7$	−1.69050	−0.638950	−0.319475	−	0.947595i	$-0.603506\pi$
$7$	−1.69050	−0.638950	−0.319475	+	0.947595i	$0.603506\pi$
$8$	0	0
$9$	6.20985	2.06995
$10$	0	0
$11$	3.51935	1.06112	0.530562	−	0.847646i	$-0.321981\pi$
$11$	3.51935	1.06112	0.530562	+	0.847646i	$0.321981\pi$
$12$	0	0
$13$	−2.77721	−0.770261	−0.385130	−	0.922862i	$-0.625844\pi$
$13$	−2.77721	−0.770261	−0.385130	+	0.922862i	$0.625844\pi$
$14$	0	0
$15$	5.95515	1.53761
$16$	0	0
$17$	7.67138	1.86058	0.930291	−	0.366822i	$-0.119554\pi$
$17$	7.67138	1.86058	0.930291	+	0.366822i	$0.119554\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	−5.13029	−1.11952
$22$	0	0
$23$	6.70640	1.39838	0.699190	−	0.714935i	$-0.253544\pi$
$23$	6.70640	1.39838	0.699190	+	0.714935i	$0.253544\pi$
$24$	0	0
$25$	−1.14936	−0.229871
$26$	0	0
$27$	9.74117	1.87469
$28$	0	0
$29$	−9.86023	−1.83100	−0.915500	−	0.402319i	$-0.868204\pi$
$29$	−9.86023	−1.83100	−0.915500	+	0.402319i	$0.868204\pi$
$30$	0	0
$31$	−6.78596	−1.21879	−0.609397	−	0.792865i	$-0.708589\pi$
$31$	−6.78596	−1.21879	−0.609397	+	0.792865i	$0.708589\pi$
$32$	0	0
$33$	10.6804	1.85922
$34$	0	0
$35$	−3.31728	−0.560723
$36$	0	0
$37$	0.811988	0.133490	0.0667450	−	0.997770i	$-0.478739\pi$
$37$	0.811988	0.133490	0.0667450	+	0.997770i	$0.478739\pi$
$38$	0	0
$39$	−8.42822	−1.34959
$40$	0	0
$41$	1.74832	0.273042	0.136521	−	0.990637i	$-0.456408\pi$
$41$	1.74832	0.273042	0.136521	+	0.990637i	$0.456408\pi$
$42$	0	0
$43$	12.7435	1.94337	0.971686	−	0.236276i	$-0.0759271\pi$
$43$	12.7435	1.94337	0.971686	+	0.236276i	$0.0759271\pi$
$44$	0	0
$45$	12.1856	1.81653
$46$	0	0
$47$	0.564137	0.0822879	0.0411439	−	0.999153i	$-0.486900\pi$
$47$	0.564137	0.0822879	0.0411439	+	0.999153i	$0.486900\pi$
$48$	0	0
$49$	−4.14220	−0.591743
$50$	0	0
$51$	23.2809	3.25998
$52$	0	0
$53$	6.60559	0.907348	0.453674	−	0.891168i	$-0.350113\pi$
$53$	6.60559	0.907348	0.453674	+	0.891168i	$0.350113\pi$
$54$	0	0
$55$	6.90604	0.931210
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−4.28493	−0.557851	−0.278925	−	0.960313i	$-0.589978\pi$
$59$	−4.28493	−0.557851	−0.278925	+	0.960313i	$0.589978\pi$
$60$	0	0
$61$	7.64249	0.978520	0.489260	−	0.872138i	$-0.337267\pi$
$61$	7.64249	0.978520	0.489260	+	0.872138i	$0.337267\pi$
$62$	0	0
$63$	−10.4978	−1.32259
$64$	0	0
$65$	−5.44974	−0.675958
$66$	0	0
$67$	4.89603	0.598145	0.299072	−	0.954230i	$-0.403323\pi$
$67$	4.89603	0.598145	0.299072	+	0.954230i	$0.403323\pi$
$68$	0	0
$69$	20.3524	2.45014
$70$	0	0
$71$	−1.90800	−0.226438	−0.113219	−	0.993570i	$-0.536116\pi$
$71$	−1.90800	−0.226438	−0.113219	+	0.993570i	$0.536116\pi$
$72$	0	0
$73$	3.83790	0.449192	0.224596	−	0.974452i	$-0.427894\pi$
$73$	3.83790	0.449192	0.224596	+	0.974452i	$0.427894\pi$
$74$	0	0
$75$	−3.48803	−0.402763
$76$	0	0
$77$	−5.94947	−0.678005
$78$	0	0
$79$	−4.37662	−0.492408	−0.246204	−	0.969218i	$-0.579183\pi$
$79$	−4.37662	−0.492408	−0.246204	+	0.969218i	$0.579183\pi$
$80$	0	0
$81$	10.9327	1.21474
$82$	0	0
$83$	−11.0958	−1.21792	−0.608962	−	0.793200i	$-0.708414\pi$
$83$	−11.0958	−1.21792	−0.608962	+	0.793200i	$0.708414\pi$
$84$	0	0
$85$	15.0536	1.63279
$86$	0	0
$87$	−29.9236	−3.20814
$88$	0	0
$89$	1.71592	0.181887	0.0909435	−	0.995856i	$-0.471012\pi$
$89$	1.71592	0.181887	0.0909435	+	0.995856i	$0.471012\pi$
$90$	0	0
$91$	4.69489	0.492158
$92$	0	0
$93$	−20.5939	−2.13548
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−7.02908	−0.713695	−0.356847	−	0.934163i	$-0.616148\pi$
$97$	−7.02908	−0.713695	−0.356847	+	0.934163i	$0.616148\pi$
$98$	0	0
$99$	21.8546	2.19647
$100$	0	0
$101$	−10.3490	−1.02977	−0.514884	−	0.857260i	$-0.672165\pi$
$101$	−10.3490	−1.02977	−0.514884	+	0.857260i	$0.672165\pi$
$102$	0	0
$103$	2.73948	0.269929	0.134964	−	0.990850i	$-0.456908\pi$
$103$	2.73948	0.269929	0.134964	+	0.990850i	$0.456908\pi$
$104$	0	0
$105$	−10.0672	−0.982458
$106$	0	0
$107$	7.94514	0.768086	0.384043	−	0.923315i	$-0.374531\pi$
$107$	7.94514	0.768086	0.384043	+	0.923315i	$0.374531\pi$
$108$	0	0
$109$	−1.30188	−0.124697	−0.0623487	−	0.998054i	$-0.519859\pi$
$109$	−1.30188	−0.124697	−0.0623487	+	0.998054i	$0.519859\pi$
$110$	0	0
$111$	2.46420	0.233892
$112$	0	0
$113$	−2.20664	−0.207583	−0.103792	−	0.994599i	$-0.533098\pi$
$113$	−2.20664	−0.207583	−0.103792	+	0.994599i	$0.533098\pi$
$114$	0	0
$115$	13.1600	1.22718
$116$	0	0
$117$	−17.2461	−1.59440
$118$	0	0
$119$	−12.9685	−1.18882
$120$	0	0
$121$	1.38582	0.125984
$122$	0	0
$123$	5.30577	0.478405
$124$	0	0
$125$	−12.0669	−1.07930
$126$	0	0
$127$	5.29298	0.469676	0.234838	−	0.972035i	$-0.424544\pi$
$127$	5.29298	0.469676	0.234838	+	0.972035i	$0.424544\pi$
$128$	0	0
$129$	38.6738	3.40503
$130$	0	0
$131$	5.68658	0.496838	0.248419	−	0.968653i	$-0.420089\pi$
$131$	5.68658	0.496838	0.248419	+	0.968653i	$0.420089\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	19.1152	1.64517
$136$	0	0
$137$	−18.4310	−1.57467	−0.787334	−	0.616526i	$-0.788539\pi$
$137$	−18.4310	−1.57467	−0.787334	+	0.616526i	$0.788539\pi$
$138$	0	0
$139$	6.60329	0.560084	0.280042	−	0.959988i	$-0.409652\pi$
$139$	6.60329	0.560084	0.280042	+	0.959988i	$0.409652\pi$
$140$	0	0
$141$	1.71203	0.144179
$142$	0	0
$143$	−9.77399	−0.817342
$144$	0	0
$145$	−19.3488	−1.60683
$146$	0	0
$147$	−12.5707	−1.03681
$148$	0	0
$149$	−2.34879	−0.192420	−0.0962101	−	0.995361i	$-0.530672\pi$
$149$	−2.34879	−0.192420	−0.0962101	+	0.995361i	$0.530672\pi$
$150$	0	0
$151$	19.8411	1.61465	0.807323	−	0.590109i	$-0.200915\pi$
$151$	19.8411	1.61465	0.807323	+	0.590109i	$0.200915\pi$
$152$	0	0
$153$	47.6381	3.85131
$154$	0	0
$155$	−13.3161	−1.06958
$156$	0	0
$157$	14.6459	1.16887	0.584437	−	0.811439i	$-0.301315\pi$
$157$	14.6459	1.16887	0.584437	+	0.811439i	$0.301315\pi$
$158$	0	0
$159$	20.0465	1.58979
$160$	0	0
$161$	−11.3372	−0.893495
$162$	0	0
$163$	2.58699	0.202629	0.101314	−	0.994854i	$-0.467695\pi$
$163$	2.58699	0.202629	0.101314	+	0.994854i	$0.467695\pi$
$164$	0	0
$165$	20.9583	1.63160
$166$	0	0
$167$	17.3546	1.34294	0.671471	−	0.741031i	$-0.265663\pi$
$167$	17.3546	1.34294	0.671471	+	0.741031i	$0.265663\pi$
$168$	0	0
$169$	−5.28708	−0.406698
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−3.09703	−0.235463	−0.117731	−	0.993045i	$-0.537562\pi$
$173$	−3.09703	−0.235463	−0.117731	+	0.993045i	$0.537562\pi$
$174$	0	0
$175$	1.94299	0.146876
$176$	0	0
$177$	−13.0038	−0.977426
$178$	0	0
$179$	−15.5295	−1.16073	−0.580365	−	0.814357i	$-0.697090\pi$
$179$	−15.5295	−1.16073	−0.580365	+	0.814357i	$0.697090\pi$
$180$	0	0
$181$	−13.9551	−1.03727	−0.518637	−	0.854995i	$-0.673560\pi$
$181$	−13.9551	−1.03727	−0.518637	+	0.854995i	$0.673560\pi$
$182$	0	0
$183$	23.1932	1.71449
$184$	0	0
$185$	1.59337	0.117147
$186$	0	0
$187$	26.9983	1.97431
$188$	0	0
$189$	−16.4675	−1.19783
$190$	0	0
$191$	−1.90047	−0.137513	−0.0687567	−	0.997633i	$-0.521903\pi$
$191$	−1.90047	−0.137513	−0.0687567	+	0.997633i	$0.521903\pi$
$192$	0	0
$193$	6.54703	0.471266	0.235633	−	0.971842i	$-0.424284\pi$
$193$	6.54703	0.471266	0.235633	+	0.971842i	$0.424284\pi$
$194$	0	0
$195$	−16.5387	−1.18436
$196$	0	0
$197$	−4.44417	−0.316634	−0.158317	−	0.987388i	$-0.550607\pi$
$197$	−4.44417	−0.316634	−0.158317	+	0.987388i	$0.550607\pi$
$198$	0	0
$199$	−10.0842	−0.714852	−0.357426	−	0.933941i	$-0.616346\pi$
$199$	−10.0842	−0.714852	−0.357426	+	0.933941i	$0.616346\pi$
$200$	0	0
$201$	14.8583	1.04803
$202$	0	0
$203$	16.6687	1.16992
$204$	0	0
$205$	3.43075	0.239614
$206$	0	0
$207$	41.6457	2.89458
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−16.1930	−1.11477	−0.557387	−	0.830253i	$-0.688196\pi$
$211$	−16.1930	−1.11477	−0.557387	+	0.830253i	$0.688196\pi$
$212$	0	0
$213$	−5.79034	−0.396748
$214$	0	0
$215$	25.0067	1.70544
$216$	0	0
$217$	11.4717	0.778748
$218$	0	0
$219$	11.6472	0.787042
$220$	0	0
$221$	−21.3051	−1.43313
$222$	0	0
$223$	24.0771	1.61232	0.806162	−	0.591695i	$-0.201541\pi$
$223$	24.0771	1.61232	0.806162	+	0.591695i	$0.201541\pi$
$224$	0	0
$225$	−7.13733	−0.475822
$226$	0	0
$227$	4.08772	0.271311	0.135656	−	0.990756i	$-0.456686\pi$
$227$	4.08772	0.271311	0.135656	+	0.990756i	$0.456686\pi$
$228$	0	0
$229$	−20.0652	−1.32594	−0.662972	−	0.748644i	$-0.730705\pi$
$229$	−20.0652	−1.32594	−0.662972	+	0.748644i	$0.730705\pi$
$230$	0	0
$231$	−18.0553	−1.18795
$232$	0	0
$233$	−20.3765	−1.33491	−0.667454	−	0.744651i	$-0.732616\pi$
$233$	−20.3765	−1.33491	−0.667454	+	0.744651i	$0.732616\pi$
$234$	0	0
$235$	1.10701	0.0722134
$236$	0	0
$237$	−13.2820	−0.862762
$238$	0	0
$239$	3.04418	0.196911	0.0984557	−	0.995141i	$-0.468610\pi$
$239$	3.04418	0.196911	0.0984557	+	0.995141i	$0.468610\pi$
$240$	0	0
$241$	−21.3534	−1.37549	−0.687745	−	0.725952i	$-0.741399\pi$
$241$	−21.3534	−1.37549	−0.687745	+	0.725952i	$0.741399\pi$
$242$	0	0
$243$	3.95475	0.253698
$244$	0	0
$245$	−8.12827	−0.519296
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−33.6733	−2.13396
$250$	0	0
$251$	−2.03363	−0.128361	−0.0641807	−	0.997938i	$-0.520443\pi$
$251$	−2.03363	−0.128361	−0.0641807	+	0.997938i	$0.520443\pi$
$252$	0	0
$253$	23.6022	1.48386
$254$	0	0
$255$	45.6842	2.86086
$256$	0	0
$257$	15.0545	0.939074	0.469537	−	0.882913i	$-0.344421\pi$
$257$	15.0545	0.939074	0.469537	+	0.882913i	$0.344421\pi$
$258$	0	0
$259$	−1.37267	−0.0852934
$260$	0	0
$261$	−61.2306	−3.79008
$262$	0	0
$263$	14.4037	0.888167	0.444084	−	0.895985i	$-0.353529\pi$
$263$	14.4037	0.888167	0.444084	+	0.895985i	$0.353529\pi$
$264$	0	0
$265$	12.9622	0.796261
$266$	0	0
$267$	5.20743	0.318689
$268$	0	0
$269$	−16.3795	−0.998677	−0.499339	−	0.866407i	$-0.666424\pi$
$269$	−16.3795	−0.998677	−0.499339	+	0.866407i	$0.666424\pi$
$270$	0	0
$271$	4.65357	0.282684	0.141342	−	0.989961i	$-0.454858\pi$
$271$	4.65357	0.282684	0.141342	+	0.989961i	$0.454858\pi$
$272$	0	0
$273$	14.2479	0.862323
$274$	0	0
$275$	−4.04498	−0.243922
$276$	0	0
$277$	5.62331	0.337872	0.168936	−	0.985627i	$-0.445967\pi$
$277$	5.62331	0.337872	0.168936	+	0.985627i	$0.445967\pi$
$278$	0	0
$279$	−42.1398	−2.52284
$280$	0	0
$281$	15.6361	0.932774	0.466387	−	0.884581i	$-0.345555\pi$
$281$	15.6361	0.932774	0.466387	+	0.884581i	$0.345555\pi$
$282$	0	0
$283$	5.26714	0.313099	0.156549	−	0.987670i	$-0.449963\pi$
$283$	5.26714	0.313099	0.156549	+	0.987670i	$0.449963\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.95555	−0.174460
$288$	0	0
$289$	41.8501	2.46177
$290$	0	0
$291$	−21.3317	−1.25048
$292$	0	0
$293$	28.1628	1.64529	0.822643	−	0.568558i	$-0.192498\pi$
$293$	28.1628	1.64529	0.822643	+	0.568558i	$0.192498\pi$
$294$	0	0
$295$	−8.40835	−0.489553
$296$	0	0
$297$	34.2826	1.98928
$298$	0	0
$299$	−18.6251	−1.07712
$300$	0	0
$301$	−21.5430	−1.24172
$302$	0	0
$303$	−31.4070	−1.80428
$304$	0	0
$305$	14.9969	0.858720
$306$	0	0
$307$	10.7524	0.613673	0.306837	−	0.951762i	$-0.400729\pi$
$307$	10.7524	0.613673	0.306837	+	0.951762i	$0.400729\pi$
$308$	0	0
$309$	8.31369	0.472950
$310$	0	0
$311$	3.37059	0.191129	0.0955643	−	0.995423i	$-0.469534\pi$
$311$	3.37059	0.191129	0.0955643	+	0.995423i	$0.469534\pi$
$312$	0	0
$313$	−30.9446	−1.74909	−0.874545	−	0.484944i	$-0.838840\pi$
$313$	−30.9446	−1.74909	−0.874545	+	0.484944i	$0.838840\pi$
$314$	0	0
$315$	−20.5998	−1.16067
$316$	0	0
$317$	−31.7571	−1.78365	−0.891827	−	0.452377i	$-0.850576\pi$
$317$	−31.7571	−1.78365	−0.891827	+	0.452377i	$0.850576\pi$
$318$	0	0
$319$	−34.7016	−1.94292
$320$	0	0
$321$	24.1117	1.34578
$322$	0	0
$323$	0	0
$324$	0	0
$325$	3.19201	0.177061
$326$	0	0
$327$	−3.95091	−0.218486
$328$	0	0
$329$	−0.953675	−0.0525778
$330$	0	0
$331$	25.4140	1.39688	0.698441	−	0.715668i	$-0.253878\pi$
$331$	25.4140	1.39688	0.698441	+	0.715668i	$0.253878\pi$
$332$	0	0
$333$	5.04233	0.276318
$334$	0	0
$335$	9.60750	0.524914
$336$	0	0
$337$	10.6752	0.581518	0.290759	−	0.956796i	$-0.406092\pi$
$337$	10.6752	0.581518	0.290759	+	0.956796i	$0.406092\pi$
$338$	0	0
$339$	−6.69666	−0.363712
$340$	0	0
$341$	−23.8822	−1.29329
$342$	0	0
$343$	18.8359	1.01704
$344$	0	0
$345$	39.9376	2.15017
$346$	0	0
$347$	7.00806	0.376212	0.188106	−	0.982149i	$-0.439765\pi$
$347$	7.00806	0.376212	0.188106	+	0.982149i	$0.439765\pi$
$348$	0	0
$349$	28.2756	1.51356	0.756780	−	0.653669i	$-0.226771\pi$
$349$	28.2756	1.51356	0.756780	+	0.653669i	$0.226771\pi$
$350$	0	0
$351$	−27.0533	−1.44400
$352$	0	0
$353$	5.97938	0.318250	0.159125	−	0.987258i	$-0.449133\pi$
$353$	5.97938	0.318250	0.159125	+	0.987258i	$0.449133\pi$
$354$	0	0
$355$	−3.74408	−0.198715
$356$	0	0
$357$	−39.3564	−2.08296
$358$	0	0
$359$	−20.2638	−1.06949	−0.534743	−	0.845015i	$-0.679591\pi$
$359$	−20.2638	−1.06949	−0.534743	+	0.845015i	$0.679591\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	4.20566	0.220740
$364$	0	0
$365$	7.53113	0.394197
$366$	0	0
$367$	12.2838	0.641209	0.320604	−	0.947213i	$-0.396114\pi$
$367$	12.2838	0.641209	0.320604	+	0.947213i	$0.396114\pi$
$368$	0	0
$369$	10.8568	0.565184
$370$	0	0
$371$	−11.1668	−0.579750
$372$	0	0
$373$	14.5172	0.751673	0.375836	−	0.926686i	$-0.377355\pi$
$373$	14.5172	0.751673	0.375836	+	0.926686i	$0.377355\pi$
$374$	0	0
$375$	−36.6204	−1.89107
$376$	0	0
$377$	27.3840	1.41035
$378$	0	0
$379$	−30.0001	−1.54100	−0.770500	−	0.637439i	$-0.779994\pi$
$379$	−30.0001	−1.54100	−0.770500	+	0.637439i	$0.779994\pi$
$380$	0	0
$381$	16.0630	0.822932
$382$	0	0
$383$	−24.6149	−1.25776	−0.628881	−	0.777501i	$-0.716487\pi$
$383$	−24.6149	−1.25776	−0.628881	+	0.777501i	$0.716487\pi$
$384$	0	0
$385$	−11.6747	−0.594996
$386$	0	0
$387$	79.1355	4.02268
$388$	0	0
$389$	27.0417	1.37107	0.685534	−	0.728040i	$-0.259569\pi$
$389$	27.0417	1.37107	0.685534	+	0.728040i	$0.259569\pi$
$390$	0	0
$391$	51.4473	2.60180
$392$	0	0
$393$	17.2575	0.870524
$394$	0	0
$395$	−8.58827	−0.432123
$396$	0	0
$397$	−13.4000	−0.672525	−0.336263	−	0.941768i	$-0.609163\pi$
$397$	−13.4000	−0.672525	−0.336263	+	0.941768i	$0.609163\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	10.6752	0.533096	0.266548	−	0.963822i	$-0.414117\pi$
$401$	10.6752	0.533096	0.266548	+	0.963822i	$0.414117\pi$
$402$	0	0
$403$	18.8461	0.938789
$404$	0	0
$405$	21.4533	1.06602
$406$	0	0
$407$	2.85767	0.141649
$408$	0	0
$409$	−17.4105	−0.860894	−0.430447	−	0.902616i	$-0.641644\pi$
$409$	−17.4105	−0.860894	−0.430447	+	0.902616i	$0.641644\pi$
$410$	0	0
$411$	−55.9340	−2.75902
$412$	0	0
$413$	7.24369	0.356439
$414$	0	0
$415$	−21.7734	−1.06881
$416$	0	0
$417$	20.0395	0.981339
$418$	0	0
$419$	−11.5555	−0.564523	−0.282262	−	0.959337i	$-0.591085\pi$
$419$	−11.5555	−0.564523	−0.282262	+	0.959337i	$0.591085\pi$
$420$	0	0
$421$	−25.0006	−1.21846	−0.609228	−	0.792995i	$-0.708521\pi$
$421$	−25.0006	−1.21846	−0.609228	+	0.792995i	$0.708521\pi$
$422$	0	0
$423$	3.50321	0.170332
$424$	0	0
$425$	−8.81714	−0.427694
$426$	0	0
$427$	−12.9196	−0.625225
$428$	0	0
$429$	−29.6618	−1.43209
$430$	0	0
$431$	−8.73603	−0.420800	−0.210400	−	0.977615i	$-0.567477\pi$
$431$	−8.73603	−0.420800	−0.210400	+	0.977615i	$0.567477\pi$
$432$	0	0
$433$	−24.9146	−1.19732	−0.598659	−	0.801004i	$-0.704300\pi$
$433$	−24.9146	−1.19732	−0.598659	+	0.801004i	$0.704300\pi$
$434$	0	0
$435$	−58.7192	−2.81537
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−26.5516	−1.26724	−0.633619	−	0.773646i	$-0.718431\pi$
$439$	−26.5516	−1.26724	−0.633619	+	0.773646i	$0.718431\pi$
$440$	0	0
$441$	−25.7225	−1.22488
$442$	0	0
$443$	4.75321	0.225832	0.112916	−	0.993605i	$-0.463981\pi$
$443$	4.75321	0.225832	0.112916	+	0.993605i	$0.463981\pi$
$444$	0	0
$445$	3.36716	0.159619
$446$	0	0
$447$	−7.12804	−0.337145
$448$	0	0
$449$	−29.9397	−1.41294	−0.706472	−	0.707741i	$-0.749714\pi$
$449$	−29.9397	−1.41294	−0.706472	+	0.707741i	$0.749714\pi$
$450$	0	0
$451$	6.15296	0.289732
$452$	0	0
$453$	60.2133	2.82907
$454$	0	0
$455$	9.21280	0.431903
$456$	0	0
$457$	−31.5276	−1.47480	−0.737400	−	0.675456i	$-0.763947\pi$
$457$	−31.5276	−1.47480	−0.737400	+	0.675456i	$0.763947\pi$
$458$	0	0
$459$	74.7282	3.48801
$460$	0	0
$461$	−24.2116	−1.12765	−0.563824	−	0.825895i	$-0.690670\pi$
$461$	−24.2116	−1.12765	−0.563824	+	0.825895i	$0.690670\pi$
$462$	0	0
$463$	−40.2983	−1.87282	−0.936411	−	0.350904i	$-0.885874\pi$
$463$	−40.2983	−1.87282	−0.936411	+	0.350904i	$0.885874\pi$
$464$	0	0
$465$	−40.4114	−1.87404
$466$	0	0
$467$	24.5486	1.13597	0.567986	−	0.823038i	$-0.307723\pi$
$467$	24.5486	1.13597	0.567986	+	0.823038i	$0.307723\pi$
$468$	0	0
$469$	−8.27674	−0.382184
$470$	0	0
$471$	44.4471	2.04802
$472$	0	0
$473$	44.8490	2.06216
$474$	0	0
$475$	0	0
$476$	0	0
$477$	41.0198	1.87817
$478$	0	0
$479$	−29.6581	−1.35511	−0.677556	−	0.735471i	$-0.736961\pi$
$479$	−29.6581	−1.35511	−0.677556	+	0.735471i	$0.736961\pi$
$480$	0	0
$481$	−2.25507	−0.102822
$482$	0	0
$483$	−34.4058	−1.56552
$484$	0	0
$485$	−13.7932	−0.626317
$486$	0	0
$487$	21.9641	0.995288	0.497644	−	0.867381i	$-0.334199\pi$
$487$	21.9641	0.995288	0.497644	+	0.867381i	$0.334199\pi$
$488$	0	0
$489$	7.85093	0.355031
$490$	0	0
$491$	6.49953	0.293320	0.146660	−	0.989187i	$-0.453148\pi$
$491$	6.49953	0.293320	0.146660	+	0.989187i	$0.453148\pi$
$492$	0	0
$493$	−75.6416	−3.40673
$494$	0	0
$495$	42.8855	1.92756
$496$	0	0
$497$	3.22548	0.144682
$498$	0	0
$499$	−44.2360	−1.98028	−0.990139	−	0.140092i	$-0.955260\pi$
$499$	−44.2360	−1.98028	−0.990139	+	0.140092i	$0.955260\pi$
$500$	0	0
$501$	52.6674	2.35301
$502$	0	0
$503$	−13.8944	−0.619519	−0.309759	−	0.950815i	$-0.600248\pi$
$503$	−13.8944	−0.619519	−0.309759	+	0.950815i	$0.600248\pi$
$504$	0	0
$505$	−20.3080	−0.903693
$506$	0	0
$507$	−16.0451	−0.712588
$508$	0	0
$509$	10.9348	0.484677	0.242339	−	0.970192i	$-0.422086\pi$
$509$	10.9348	0.484677	0.242339	+	0.970192i	$0.422086\pi$
$510$	0	0
$511$	−6.48798	−0.287011
$512$	0	0
$513$	0	0
$514$	0	0
$515$	5.37569	0.236881
$516$	0	0
$517$	1.98540	0.0873176
$518$	0	0
$519$	−9.39879	−0.412561
$520$	0	0
$521$	−6.91246	−0.302840	−0.151420	−	0.988469i	$-0.548385\pi$
$521$	−6.91246	−0.302840	−0.151420	+	0.988469i	$0.548385\pi$
$522$	0	0
$523$	−2.04262	−0.0893176	−0.0446588	−	0.999002i	$-0.514220\pi$
$523$	−2.04262	−0.0893176	−0.0446588	+	0.999002i	$0.514220\pi$
$524$	0	0
$525$	5.89653	0.257346
$526$	0	0
$527$	−52.0577	−2.26767
$528$	0	0
$529$	21.9758	0.955469
$530$	0	0
$531$	−26.6088	−1.15472
$532$	0	0
$533$	−4.85547	−0.210314
$534$	0	0
$535$	15.5908	0.674049
$536$	0	0
$537$	−47.1285	−2.03375
$538$	0	0
$539$	−14.5779	−0.627913
$540$	0	0
$541$	10.9075	0.468949	0.234474	−	0.972122i	$-0.424663\pi$
$541$	10.9075	0.468949	0.234474	+	0.972122i	$0.424663\pi$
$542$	0	0
$543$	−42.3505	−1.81744
$544$	0	0
$545$	−2.55468	−0.109431
$546$	0	0
$547$	38.3276	1.63877	0.819386	−	0.573243i	$-0.194315\pi$
$547$	38.3276	1.63877	0.819386	+	0.573243i	$0.194315\pi$
$548$	0	0
$549$	47.4587	2.02549
$550$	0	0
$551$	0	0
$552$	0	0
$553$	7.39868	0.314624
$554$	0	0
$555$	4.83551	0.205256
$556$	0	0
$557$	22.4581	0.951580	0.475790	−	0.879559i	$-0.342162\pi$
$557$	22.4581	0.951580	0.475790	+	0.879559i	$0.342162\pi$
$558$	0	0
$559$	−35.3915	−1.49690
$560$	0	0
$561$	81.9336	3.45924
$562$	0	0
$563$	−20.0537	−0.845161	−0.422581	−	0.906325i	$-0.638876\pi$
$563$	−20.0537	−0.845161	−0.422581	+	0.906325i	$0.638876\pi$
$564$	0	0
$565$	−4.33010	−0.182169
$566$	0	0
$567$	−18.4817	−0.776161
$568$	0	0
$569$	24.7810	1.03887	0.519436	−	0.854509i	$-0.326142\pi$
$569$	24.7810	1.03887	0.519436	+	0.854509i	$0.326142\pi$
$570$	0	0
$571$	10.2935	0.430771	0.215385	−	0.976529i	$-0.430899\pi$
$571$	10.2935	0.430771	0.215385	+	0.976529i	$0.430899\pi$
$572$	0	0
$573$	−5.76751	−0.240941
$574$	0	0
$575$	−7.70804	−0.321447
$576$	0	0
$577$	−16.2328	−0.675782	−0.337891	−	0.941185i	$-0.609713\pi$
$577$	−16.2328	−0.675782	−0.337891	+	0.941185i	$0.609713\pi$
$578$	0	0
$579$	19.8688	0.825717
$580$	0	0
$581$	18.7575	0.778191
$582$	0	0
$583$	23.2474	0.962809
$584$	0	0
$585$	−33.8421	−1.39920
$586$	0	0
$587$	−8.25843	−0.340862	−0.170431	−	0.985370i	$-0.554516\pi$
$587$	−8.25843	−0.340862	−0.170431	+	0.985370i	$0.554516\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−13.4871	−0.554784
$592$	0	0
$593$	9.25020	0.379860	0.189930	−	0.981798i	$-0.439174\pi$
$593$	9.25020	0.379860	0.189930	+	0.981798i	$0.439174\pi$
$594$	0	0
$595$	−25.4481	−1.04327
$596$	0	0
$597$	−30.6033	−1.25251
$598$	0	0
$599$	23.2218	0.948818	0.474409	−	0.880305i	$-0.342662\pi$
$599$	23.2218	0.948818	0.474409	+	0.880305i	$0.342662\pi$
$600$	0	0
$601$	16.5448	0.674876	0.337438	−	0.941348i	$-0.390440\pi$
$601$	16.5448	0.674876	0.337438	+	0.941348i	$0.390440\pi$
$602$	0	0
$603$	30.4036	1.23813
$604$	0	0
$605$	2.71941	0.110560
$606$	0	0
$607$	7.93908	0.322237	0.161119	−	0.986935i	$-0.448490\pi$
$607$	7.93908	0.322237	0.161119	+	0.986935i	$0.448490\pi$
$608$	0	0
$609$	50.5859	2.04984
$610$	0	0
$611$	−1.56673	−0.0633831
$612$	0	0
$613$	−43.7334	−1.76638	−0.883188	−	0.469018i	$-0.844608\pi$
$613$	−43.7334	−1.76638	−0.883188	+	0.469018i	$0.844608\pi$
$614$	0	0
$615$	10.4115	0.419834
$616$	0	0
$617$	16.1292	0.649339	0.324669	−	0.945828i	$-0.394747\pi$
$617$	16.1292	0.649339	0.324669	+	0.945828i	$0.394747\pi$
$618$	0	0
$619$	−16.1980	−0.651051	−0.325525	−	0.945533i	$-0.605541\pi$
$619$	−16.1980	−0.651051	−0.325525	+	0.945533i	$0.605541\pi$
$620$	0	0
$621$	65.3282	2.62153
$622$	0	0
$623$	−2.90076	−0.116217
$624$	0	0
$625$	−17.9322	−0.717288
$626$	0	0
$627$	0	0
$628$	0	0
$629$	6.22907	0.248369
$630$	0	0
$631$	−14.7659	−0.587821	−0.293910	−	0.955833i	$-0.594957\pi$
$631$	−14.7659	−0.587821	−0.293910	+	0.955833i	$0.594957\pi$
$632$	0	0
$633$	−49.1422	−1.95323
$634$	0	0
$635$	10.3864	0.412174
$636$	0	0
$637$	11.5038	0.455797
$638$	0	0
$639$	−11.8484	−0.468715
$640$	0	0
$641$	8.71148	0.344083	0.172041	−	0.985090i	$-0.444964\pi$
$641$	8.71148	0.344083	0.172041	+	0.985090i	$0.444964\pi$
$642$	0	0
$643$	19.2403	0.758762	0.379381	−	0.925241i	$-0.376137\pi$
$643$	19.2403	0.758762	0.379381	+	0.925241i	$0.376137\pi$
$644$	0	0
$645$	75.8898	2.98816
$646$	0	0
$647$	37.5193	1.47503	0.737517	−	0.675328i	$-0.235998\pi$
$647$	37.5193	1.47503	0.737517	+	0.675328i	$0.235998\pi$
$648$	0	0
$649$	−15.0802	−0.591949
$650$	0	0
$651$	34.8139	1.36447
$652$	0	0
$653$	−37.4490	−1.46549	−0.732746	−	0.680502i	$-0.761762\pi$
$653$	−37.4490	−1.46549	−0.732746	+	0.680502i	$0.761762\pi$
$654$	0	0
$655$	11.1588	0.436010
$656$	0	0
$657$	23.8328	0.929805
$658$	0	0
$659$	−9.31174	−0.362734	−0.181367	−	0.983416i	$-0.558052\pi$
$659$	−9.31174	−0.362734	−0.181367	+	0.983416i	$0.558052\pi$
$660$	0	0
$661$	−6.38908	−0.248507	−0.124253	−	0.992251i	$-0.539654\pi$
$661$	−6.38908	−0.248507	−0.124253	+	0.992251i	$0.539654\pi$
$662$	0	0
$663$	−64.6560	−2.51103
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−66.1267	−2.56043
$668$	0	0
$669$	73.0686	2.82500
$670$	0	0
$671$	26.8966	1.03833
$672$	0	0
$673$	22.3511	0.861571	0.430786	−	0.902454i	$-0.358236\pi$
$673$	22.3511	0.861571	0.430786	+	0.902454i	$0.358236\pi$
$674$	0	0
$675$	−11.1961	−0.430937
$676$	0	0
$677$	−11.5572	−0.444181	−0.222091	−	0.975026i	$-0.571288\pi$
$677$	−11.5572	−0.444181	−0.222091	+	0.975026i	$0.571288\pi$
$678$	0	0
$679$	11.8827	0.456015
$680$	0	0
$681$	12.4053	0.475372
$682$	0	0
$683$	−38.8321	−1.48587	−0.742935	−	0.669364i	$-0.766567\pi$
$683$	−38.8321	−1.48587	−0.742935	+	0.669364i	$0.766567\pi$
$684$	0	0
$685$	−36.1673	−1.38188
$686$	0	0
$687$	−60.8933	−2.32322
$688$	0	0
$689$	−18.3452	−0.698895
$690$	0	0
$691$	−0.531406	−0.0202156	−0.0101078	−	0.999949i	$-0.503217\pi$
$691$	−0.531406	−0.0202156	−0.0101078	+	0.999949i	$0.503217\pi$
$692$	0	0
$693$	−36.9453	−1.40344
$694$	0	0
$695$	12.9577	0.491513
$696$	0	0
$697$	13.4121	0.508018
$698$	0	0
$699$	−61.8380	−2.33893
$700$	0	0
$701$	−31.3535	−1.18421	−0.592103	−	0.805862i	$-0.701702\pi$
$701$	−31.3535	−1.18421	−0.592103	+	0.805862i	$0.701702\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	3.35952	0.126527
$706$	0	0
$707$	17.4951	0.657970
$708$	0	0
$709$	−0.551425	−0.0207092	−0.0103546	−	0.999946i	$-0.503296\pi$
$709$	−0.551425	−0.0207092	−0.0103546	+	0.999946i	$0.503296\pi$
$710$	0	0
$711$	−27.1782	−1.01926
$712$	0	0
$713$	−45.5094	−1.70434
$714$	0	0
$715$	−19.1796	−0.717275
$716$	0	0
$717$	9.23839	0.345014
$718$	0	0
$719$	−9.84056	−0.366991	−0.183496	−	0.983021i	$-0.558741\pi$
$719$	−9.84056	−0.366991	−0.183496	+	0.983021i	$0.558741\pi$
$720$	0	0
$721$	−4.63109	−0.172471
$722$	0	0
$723$	−64.8026	−2.41003
$724$	0	0
$725$	11.3329	0.420894
$726$	0	0
$727$	4.46877	0.165737	0.0828687	−	0.996560i	$-0.473592\pi$
$727$	4.46877	0.165737	0.0828687	+	0.996560i	$0.473592\pi$
$728$	0	0
$729$	−20.7963	−0.770234
$730$	0	0
$731$	97.7605	3.61580
$732$	0	0
$733$	2.39964	0.0886328	0.0443164	−	0.999018i	$-0.485889\pi$
$733$	2.39964	0.0886328	0.0443164	+	0.999018i	$0.485889\pi$
$734$	0	0
$735$	−24.6675	−0.909873
$736$	0	0
$737$	17.2308	0.634706
$738$	0	0
$739$	24.4313	0.898718	0.449359	−	0.893351i	$-0.351652\pi$
$739$	24.4313	0.898718	0.449359	+	0.893351i	$0.351652\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	36.7301	1.34750	0.673749	−	0.738961i	$-0.264683\pi$
$743$	36.7301	1.34750	0.673749	+	0.738961i	$0.264683\pi$
$744$	0	0
$745$	−4.60904	−0.168862
$746$	0	0
$747$	−68.9033	−2.52104
$748$	0	0
$749$	−13.4313	−0.490768
$750$	0	0
$751$	16.8657	0.615438	0.307719	−	0.951477i	$-0.400434\pi$
$751$	16.8657	0.615438	0.307719	+	0.951477i	$0.400434\pi$
$752$	0	0
$753$	−6.17160	−0.224906
$754$	0	0
$755$	38.9343	1.41697
$756$	0	0
$757$	−40.2063	−1.46132	−0.730661	−	0.682741i	$-0.760788\pi$
$757$	−40.2063	−1.46132	−0.730661	+	0.682741i	$0.760788\pi$
$758$	0	0
$759$	71.6272	2.59990
$760$	0	0
$761$	−21.4896	−0.778999	−0.389499	−	0.921027i	$-0.627352\pi$
$761$	−21.4896	−0.778999	−0.389499	+	0.921027i	$0.627352\pi$
$762$	0	0
$763$	2.20083	0.0796753
$764$	0	0
$765$	93.4806	3.37980
$766$	0	0
$767$	11.9002	0.429691
$768$	0	0
$769$	36.2910	1.30869	0.654343	−	0.756198i	$-0.272945\pi$
$769$	36.2910	1.30869	0.654343	+	0.756198i	$0.272945\pi$
$770$	0	0
$771$	45.6870	1.64538
$772$	0	0
$773$	9.36739	0.336922	0.168461	−	0.985708i	$-0.446120\pi$
$773$	9.36739	0.336922	0.168461	+	0.985708i	$0.446120\pi$
$774$	0	0
$775$	7.79948	0.280166
$776$	0	0
$777$	−4.16573	−0.149445
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−6.71491	−0.240279
$782$	0	0
$783$	−96.0502	−3.43256
$784$	0	0
$785$	28.7398	1.02577
$786$	0	0
$787$	11.4559	0.408357	0.204179	−	0.978934i	$-0.434548\pi$
$787$	11.4559	0.408357	0.204179	+	0.978934i	$0.434548\pi$
$788$	0	0
$789$	43.7118	1.55618
$790$	0	0
$791$	3.73033	0.132635
$792$	0	0
$793$	−21.2248	−0.753716
$794$	0	0
$795$	39.3373	1.39515
$796$	0	0
$797$	34.8172	1.23329	0.616644	−	0.787242i	$-0.288492\pi$
$797$	34.8172	1.23329	0.616644	+	0.787242i	$0.288492\pi$
$798$	0	0
$799$	4.32771	0.153103
$800$	0	0
$801$	10.6556	0.376497
$802$	0	0
$803$	13.5069	0.476648
$804$	0	0
$805$	−22.2470	−0.784104
$806$	0	0
$807$	−49.7082	−1.74981
$808$	0	0
$809$	−32.5180	−1.14327	−0.571637	−	0.820507i	$-0.693691\pi$
$809$	−32.5180	−1.14327	−0.571637	+	0.820507i	$0.693691\pi$
$810$	0	0
$811$	−27.8693	−0.978624	−0.489312	−	0.872109i	$-0.662752\pi$
$811$	−27.8693	−0.978624	−0.489312	+	0.872109i	$0.662752\pi$
$812$	0	0
$813$	14.1225	0.495298
$814$	0	0
$815$	5.07647	0.177821
$816$	0	0
$817$	0	0
$818$	0	0
$819$	29.1545	1.01874
$820$	0	0
$821$	6.66304	0.232542	0.116271	−	0.993218i	$-0.462906\pi$
$821$	6.66304	0.232542	0.116271	+	0.993218i	$0.462906\pi$
$822$	0	0
$823$	10.7259	0.373882	0.186941	−	0.982371i	$-0.440143\pi$
$823$	10.7259	0.373882	0.186941	+	0.982371i	$0.440143\pi$
$824$	0	0
$825$	−12.2756	−0.427382
$826$	0	0
$827$	−36.8383	−1.28099	−0.640497	−	0.767961i	$-0.721272\pi$
$827$	−36.8383	−1.28099	−0.640497	+	0.767961i	$0.721272\pi$
$828$	0	0
$829$	3.87402	0.134550	0.0672751	−	0.997734i	$-0.478569\pi$
$829$	3.87402	0.134550	0.0672751	+	0.997734i	$0.478569\pi$
$830$	0	0
$831$	17.0655	0.591994
$832$	0	0
$833$	−31.7764	−1.10099
$834$	0	0
$835$	34.0551	1.17853
$836$	0	0
$837$	−66.1032	−2.28486
$838$	0	0
$839$	−32.1250	−1.10908	−0.554538	−	0.832158i	$-0.687105\pi$
$839$	−32.1250	−1.10908	−0.554538	+	0.832158i	$0.687105\pi$
$840$	0	0
$841$	68.2242	2.35256
$842$	0	0
$843$	47.4521	1.63434
$844$	0	0
$845$	−10.3749	−0.356906
$846$	0	0
$847$	−2.34274	−0.0804973
$848$	0	0
$849$	15.9846	0.548589
$850$	0	0
$851$	5.44552	0.186670
$852$	0	0
$853$	−13.2578	−0.453938	−0.226969	−	0.973902i	$-0.572882\pi$
$853$	−13.2578	−0.453938	−0.226969	+	0.973902i	$0.572882\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−12.9515	−0.442413	−0.221207	−	0.975227i	$-0.571000\pi$
$857$	−12.9515	−0.442413	−0.221207	+	0.975227i	$0.571000\pi$
$858$	0	0
$859$	2.12077	0.0723596	0.0361798	−	0.999345i	$-0.488481\pi$
$859$	2.12077	0.0723596	0.0361798	+	0.999345i	$0.488481\pi$
$860$	0	0
$861$	−8.96941	−0.305677
$862$	0	0
$863$	10.1256	0.344680	0.172340	−	0.985038i	$-0.444867\pi$
$863$	10.1256	0.344680	0.172340	+	0.985038i	$0.444867\pi$
$864$	0	0
$865$	−6.07732	−0.206635
$866$	0	0
$867$	127.005	4.31333
$868$	0	0
$869$	−15.4029	−0.522506
$870$	0	0
$871$	−13.5973	−0.460728
$872$	0	0
$873$	−43.6495	−1.47731
$874$	0	0
$875$	20.3991	0.689617
$876$	0	0
$877$	−10.9122	−0.368479	−0.184240	−	0.982881i	$-0.558982\pi$
$877$	−10.9122	−0.368479	−0.184240	+	0.982881i	$0.558982\pi$
$878$	0	0
$879$	85.4677	2.88275
$880$	0	0
$881$	0.566163	0.0190745	0.00953725	−	0.999955i	$-0.496964\pi$
$881$	0.566163	0.0190745	0.00953725	+	0.999955i	$0.496964\pi$
$882$	0	0
$883$	−3.05761	−0.102897	−0.0514484	−	0.998676i	$-0.516384\pi$
$883$	−3.05761	−0.102897	−0.0514484	+	0.998676i	$0.516384\pi$
$884$	0	0
$885$	−25.5174	−0.857760
$886$	0	0
$887$	14.0083	0.470352	0.235176	−	0.971953i	$-0.424433\pi$
$887$	14.0083	0.470352	0.235176	+	0.971953i	$0.424433\pi$
$888$	0	0
$889$	−8.94779	−0.300099
$890$	0	0
$891$	38.4760	1.28899
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−30.4736	−1.01862
$896$	0	0
$897$	−56.5230	−1.88725
$898$	0	0
$899$	66.9111	2.23161
$900$	0	0
$901$	50.6740	1.68820
$902$	0	0
$903$	−65.3781	−2.17565
$904$	0	0
$905$	−27.3842	−0.910280
$906$	0	0
$907$	38.2984	1.27168	0.635839	−	0.771822i	$-0.280654\pi$
$907$	38.2984	1.27168	0.635839	+	0.771822i	$0.280654\pi$
$908$	0	0
$909$	−64.2660	−2.13157
$910$	0	0
$911$	−26.3001	−0.871361	−0.435680	−	0.900101i	$-0.643492\pi$
$911$	−26.3001	−0.871361	−0.435680	+	0.900101i	$0.643492\pi$
$912$	0	0
$913$	−39.0500	−1.29237
$914$	0	0
$915$	45.5122	1.50459
$916$	0	0
$917$	−9.61317	−0.317455
$918$	0	0
$919$	−42.5901	−1.40492	−0.702460	−	0.711724i	$-0.747915\pi$
$919$	−42.5901	−1.40492	−0.702460	+	0.711724i	$0.747915\pi$
$920$	0	0
$921$	32.6312	1.07523
$922$	0	0
$923$	5.29892	0.174416
$924$	0	0
$925$	−0.933263	−0.0306855
$926$	0	0
$927$	17.0117	0.558739
$928$	0	0
$929$	32.5784	1.06886	0.534432	−	0.845212i	$-0.320526\pi$
$929$	32.5784	1.06886	0.534432	+	0.845212i	$0.320526\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	10.2290	0.334882
$934$	0	0
$935$	52.9789	1.73259
$936$	0	0
$937$	4.53022	0.147996	0.0739978	−	0.997258i	$-0.476424\pi$
$937$	4.53022	0.147996	0.0739978	+	0.997258i	$0.476424\pi$
$938$	0	0
$939$	−93.9098	−3.06463
$940$	0	0
$941$	33.8555	1.10366	0.551829	−	0.833957i	$-0.313930\pi$
$941$	33.8555	1.10366	0.551829	+	0.833957i	$0.313930\pi$
$942$	0	0
$943$	11.7250	0.381817
$944$	0	0
$945$	−32.3142	−1.05118
$946$	0	0
$947$	−25.4994	−0.828618	−0.414309	−	0.910136i	$-0.635977\pi$
$947$	−25.4994	−0.828618	−0.414309	+	0.910136i	$0.635977\pi$
$948$	0	0
$949$	−10.6587	−0.345995
$950$	0	0
$951$	−96.3755	−3.12519
$952$	0	0
$953$	17.9905	0.582769	0.291385	−	0.956606i	$-0.405884\pi$
$953$	17.9905	0.582769	0.291385	+	0.956606i	$0.405884\pi$
$954$	0	0
$955$	−3.72931	−0.120678
$956$	0	0
$957$	−105.312	−3.40424
$958$	0	0
$959$	31.1577	1.00613
$960$	0	0
$961$	15.0493	0.485460
$962$	0	0
$963$	49.3381	1.58990
$964$	0	0
$965$	12.8473	0.413568
$966$	0	0
$967$	47.6513	1.53236	0.766182	−	0.642624i	$-0.222154\pi$
$967$	47.6513	1.53236	0.766182	+	0.642624i	$0.222154\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	22.3427	0.717013	0.358506	−	0.933527i	$-0.383286\pi$
$971$	22.3427	0.717013	0.358506	+	0.933527i	$0.383286\pi$
$972$	0	0
$973$	−11.1629	−0.357866
$974$	0	0
$975$	9.68702	0.310233
$976$	0	0
$977$	4.02708	0.128838	0.0644188	−	0.997923i	$-0.479481\pi$
$977$	4.02708	0.128838	0.0644188	+	0.997923i	$0.479481\pi$
$978$	0	0
$979$	6.03892	0.193005
$980$	0	0
$981$	−8.08447	−0.258117
$982$	0	0
$983$	50.9650	1.62553	0.812766	−	0.582591i	$-0.197961\pi$
$983$	50.9650	1.62553	0.812766	+	0.582591i	$0.197961\pi$
$984$	0	0
$985$	−8.72083	−0.277869
$986$	0	0
$987$	−2.89419	−0.0921230
$988$	0	0
$989$	85.4633	2.71757
$990$	0	0
$991$	−7.06825	−0.224530	−0.112265	−	0.993678i	$-0.535811\pi$
$991$	−7.06825	−0.224530	−0.112265	+	0.993678i	$0.535811\pi$
$992$	0	0
$993$	77.1258	2.44751
$994$	0	0
$995$	−19.7883	−0.627333
$996$	0	0
$997$	43.7207	1.38465	0.692324	−	0.721586i	$-0.256587\pi$
$997$	43.7207	1.38465	0.692324	+	0.721586i	$0.256587\pi$
$998$	0	0
$999$	7.90972	0.250252

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5776.2.a.bz.1.6		6
4.3	odd	2		2888.2.a.t.1.1	✓	6
19.18	odd	2		5776.2.a.bx.1.1		6
76.75	even	2		2888.2.a.u.1.6	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2888.2.a.t.1.1	✓	6	4.3	odd	2
2888.2.a.u.1.6	yes	6	76.75	even	2
5776.2.a.bx.1.1		6	19.18	odd	2
5776.2.a.bz.1.6		6	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 5776 = 2^{4} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5776.a (trivial)

\(f(q)\)	\(=\)	\(q+3.03477 q^{3} +1.96231 q^{5} -1.69050 q^{7} +6.20985 q^{9} +O(q^{10})\)	\(q+3.03477 q^{3} +1.96231 q^{5} -1.69050 q^{7} +6.20985 q^{9} +3.51935 q^{11} -2.77721 q^{13} +5.95515 q^{15} +7.67138 q^{17} -5.13029 q^{21} +6.70640 q^{23} -1.14936 q^{25} +9.74117 q^{27} -9.86023 q^{29} -6.78596 q^{31} +10.6804 q^{33} -3.31728 q^{35} +0.811988 q^{37} -8.42822 q^{39} +1.74832 q^{41} +12.7435 q^{43} +12.1856 q^{45} +0.564137 q^{47} -4.14220 q^{49} +23.2809 q^{51} +6.60559 q^{53} +6.90604 q^{55} -4.28493 q^{59} +7.64249 q^{61} -10.4978 q^{63} -5.44974 q^{65} +4.89603 q^{67} +20.3524 q^{69} -1.90800 q^{71} +3.83790 q^{73} -3.48803 q^{75} -5.94947 q^{77} -4.37662 q^{79} +10.9327 q^{81} -11.0958 q^{83} +15.0536 q^{85} -29.9236 q^{87} +1.71592 q^{89} +4.69489 q^{91} -20.5939 q^{93} -7.02908 q^{97} +21.8546 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{3} + 2 q^{5} + 2 q^{7} + 9 q^{9}+O(q^{10}) \) 6 * q + 3 * q^3 + 2 * q^5 + 2 * q^7 + 9 * q^9	\( 6 q + 3 q^{3} + 2 q^{5} + 2 q^{7} + 9 q^{9} + 5 q^{11} - 10 q^{13} + 4 q^{17} - 23 q^{21} + 15 q^{23} + 12 q^{25} + 18 q^{27} + 7 q^{29} + 5 q^{31} + q^{33} + 38 q^{35} - 17 q^{37} - 37 q^{39} - 4 q^{41} + 11 q^{43} + 6 q^{45} - 18 q^{47} + 20 q^{49} + 48 q^{51} + 7 q^{53} + 42 q^{55} + q^{59} - 4 q^{61} - 19 q^{63} - 6 q^{65} + 20 q^{67} - 6 q^{69} - 6 q^{71} - 2 q^{73} + 19 q^{75} + 41 q^{77} + 8 q^{79} - 6 q^{81} + 22 q^{83} + 32 q^{85} - 29 q^{87} - 14 q^{89} + 26 q^{91} - 5 q^{93} - 41 q^{97} + 47 q^{99}+O(q^{100}) \) 6 * q + 3 * q^3 + 2 * q^5 + 2 * q^7 + 9 * q^9 + 5 * q^11 - 10 * q^13 + 4 * q^17 - 23 * q^21 + 15 * q^23 + 12 * q^25 + 18 * q^27 + 7 * q^29 + 5 * q^31 + q^33 + 38 * q^35 - 17 * q^37 - 37 * q^39 - 4 * q^41 + 11 * q^43 + 6 * q^45 - 18 * q^47 + 20 * q^49 + 48 * q^51 + 7 * q^53 + 42 * q^55 + q^59 - 4 * q^61 - 19 * q^63 - 6 * q^65 + 20 * q^67 - 6 * q^69 - 6 * q^71 - 2 * q^73 + 19 * q^75 + 41 * q^77 + 8 * q^79 - 6 * q^81 + 22 * q^83 + 32 * q^85 - 29 * q^87 - 14 * q^89 + 26 * q^91 - 5 * q^93 - 41 * q^97 + 47 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more