LMFDB - Embedded newform 7616.2.a.bu.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7616,2,Mod(1,7616)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7616.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7616, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7616 = 2^{6} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7616.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,-4,0,-4,0,6,0,8,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$60.8140661794$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.109859312.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 7x^{4} + 15x^{3} + 13x^{2} - 9x - 2 $ x^6 - 3x^5 - 7x^4 + 15x^3 + 13x^2 - 9*x - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 3808)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.13809$ of defining polynomial
Character	$\chi$	$=$	7616.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.59220 q^{3} +3.54237 q^{5} +1.00000 q^{7} +3.71950 q^{9} -4.51681 q^{11} -3.73108 q^{13} -9.18252 q^{15} -1.00000 q^{17} -1.43935 q^{19} -2.59220 q^{21} +3.43935 q^{23} +7.54836 q^{25} -1.86508 q^{27} +5.39551 q^{29} -9.51250 q^{31} +11.7085 q^{33} +3.54237 q^{35} +1.46886 q^{37} +9.67170 q^{39} +10.4770 q^{41} +2.53439 q^{43} +13.1758 q^{45} +5.60860 q^{47} +1.00000 q^{49} +2.59220 q^{51} -10.4283 q^{53} -16.0002 q^{55} +3.73108 q^{57} -1.74541 q^{59} -6.45540 q^{61} +3.71950 q^{63} -13.2169 q^{65} +5.62100 q^{67} -8.91548 q^{69} -8.96226 q^{71} +16.8458 q^{73} -19.5669 q^{75} -4.51681 q^{77} +14.8075 q^{79} -6.32384 q^{81} -11.6314 q^{83} -3.54237 q^{85} -13.9862 q^{87} +2.54632 q^{89} -3.73108 q^{91} +24.6583 q^{93} -5.09870 q^{95} -16.8522 q^{97} -16.8003 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{3} - 4 q^{5} + 6 q^{7} + 8 q^{9} - 8 q^{11} - 8 q^{13} - 6 q^{17} - 6 q^{19} - 4 q^{21} + 18 q^{23} + 12 q^{25} - 22 q^{27} + 8 q^{29} - 4 q^{31} - 6 q^{33} - 4 q^{35} + 8 q^{37} + 14 q^{39}+ \cdots - 28 q^{99}+O(q^{100}) $ 6 * q - 4 * q^3 - 4 * q^5 + 6 * q^7 + 8 * q^9 - 8 * q^11 - 8 * q^13 - 6 * q^17 - 6 * q^19 - 4 * q^21 + 18 * q^23 + 12 * q^25 - 22 * q^27 + 8 * q^29 - 4 * q^31 - 6 * q^33 - 4 * q^35 + 8 * q^37 + 14 * q^39 + 16 * q^41 - 12 * q^43 - 4 * q^45 + 18 * q^47 + 6 * q^49 + 4 * q^51 + 2 * q^53 - 8 * q^55 + 8 * q^57 - 16 * q^59 - 6 * q^61 + 8 * q^63 - 22 * q^65 - 12 * q^67 - 16 * q^69 - 2 * q^71 + 8 * q^73 - 24 * q^75 - 8 * q^77 - 18 * q^79 + 18 * q^81 - 28 * q^83 + 4 * q^85 - 2 * q^87 - 2 * q^89 - 8 * q^91 + 32 * q^93 + 26 * q^95 - 18 * q^97 - 28 * 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
$3$	−2.59220	−1.49661	−0.748303	−	0.663357i	$-0.769131\pi$
$3$	−2.59220	−1.49661	−0.748303	+	0.663357i	$0.769131\pi$
$4$	0	0
$5$	3.54237	1.58419	0.792097	−	0.610395i	$-0.208989\pi$
$5$	3.54237	1.58419	0.792097	+	0.610395i	$0.208989\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	3.71950	1.23983
$10$	0	0
$11$	−4.51681	−1.36187	−0.680935	−	0.732344i	$-0.738427\pi$
$11$	−4.51681	−1.36187	−0.680935	+	0.732344i	$0.738427\pi$
$12$	0	0
$13$	−3.73108	−1.03482	−0.517408	−	0.855739i	$-0.673103\pi$
$13$	−3.73108	−1.03482	−0.517408	+	0.855739i	$0.673103\pi$
$14$	0	0
$15$	−9.18252	−2.37092
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−1.43935	−0.330209	−0.165105	−	0.986276i	$-0.552796\pi$
$19$	−1.43935	−0.330209	−0.165105	+	0.986276i	$0.552796\pi$
$20$	0	0
$21$	−2.59220	−0.565664
$22$	0	0
$23$	3.43935	0.717154	0.358577	−	0.933500i	$-0.383262\pi$
$23$	3.43935	0.717154	0.358577	+	0.933500i	$0.383262\pi$
$24$	0	0
$25$	7.54836	1.50967
$26$	0	0
$27$	−1.86508	−0.358934
$28$	0	0
$29$	5.39551	1.00192	0.500961	−	0.865470i	$-0.332980\pi$
$29$	5.39551	1.00192	0.500961	+	0.865470i	$0.332980\pi$
$30$	0	0
$31$	−9.51250	−1.70850	−0.854248	−	0.519866i	$-0.825982\pi$
$31$	−9.51250	−1.70850	−0.854248	+	0.519866i	$0.825982\pi$
$32$	0	0
$33$	11.7085	2.03818
$34$	0	0
$35$	3.54237	0.598769
$36$	0	0
$37$	1.46886	0.241479	0.120740	−	0.992684i	$-0.461473\pi$
$37$	1.46886	0.241479	0.120740	+	0.992684i	$0.461473\pi$
$38$	0	0
$39$	9.67170	1.54871
$40$	0	0
$41$	10.4770	1.63623	0.818116	−	0.575054i	$-0.195019\pi$
$41$	10.4770	1.63623	0.818116	+	0.575054i	$0.195019\pi$
$42$	0	0
$43$	2.53439	0.386491	0.193246	−	0.981150i	$-0.438099\pi$
$43$	2.53439	0.386491	0.193246	+	0.981150i	$0.438099\pi$
$44$	0	0
$45$	13.1758	1.96414
$46$	0	0
$47$	5.60860	0.818099	0.409050	−	0.912512i	$-0.365860\pi$
$47$	5.60860	0.818099	0.409050	+	0.912512i	$0.365860\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	2.59220	0.362980
$52$	0	0
$53$	−10.4283	−1.43244	−0.716221	−	0.697874i	$-0.754130\pi$
$53$	−10.4283	−1.43244	−0.716221	+	0.697874i	$0.754130\pi$
$54$	0	0
$55$	−16.0002	−2.15747
$56$	0	0
$57$	3.73108	0.494194
$58$	0	0
$59$	−1.74541	−0.227232	−0.113616	−	0.993525i	$-0.536243\pi$
$59$	−1.74541	−0.227232	−0.113616	+	0.993525i	$0.536243\pi$
$60$	0	0
$61$	−6.45540	−0.826529	−0.413265	−	0.910611i	$-0.635612\pi$
$61$	−6.45540	−0.826529	−0.413265	+	0.910611i	$0.635612\pi$
$62$	0	0
$63$	3.71950	0.468612
$64$	0	0
$65$	−13.2169	−1.63935
$66$	0	0
$67$	5.62100	0.686715	0.343358	−	0.939205i	$-0.388436\pi$
$67$	5.62100	0.686715	0.343358	+	0.939205i	$0.388436\pi$
$68$	0	0
$69$	−8.91548	−1.07330
$70$	0	0
$71$	−8.96226	−1.06362	−0.531812	−	0.846862i	$-0.678489\pi$
$71$	−8.96226	−1.06362	−0.531812	+	0.846862i	$0.678489\pi$
$72$	0	0
$73$	16.8458	1.97165	0.985825	−	0.167775i	$-0.0536583\pi$
$73$	16.8458	1.97165	0.985825	+	0.167775i	$0.0536583\pi$
$74$	0	0
$75$	−19.5669	−2.25939
$76$	0	0
$77$	−4.51681	−0.514739
$78$	0	0
$79$	14.8075	1.66598	0.832989	−	0.553289i	$-0.186628\pi$
$79$	14.8075	1.66598	0.832989	+	0.553289i	$0.186628\pi$
$80$	0	0
$81$	−6.32384	−0.702649
$82$	0	0
$83$	−11.6314	−1.27671	−0.638357	−	0.769741i	$-0.720386\pi$
$83$	−11.6314	−1.27671	−0.638357	+	0.769741i	$0.720386\pi$
$84$	0	0
$85$	−3.54237	−0.384224
$86$	0	0
$87$	−13.9862	−1.49948
$88$	0	0
$89$	2.54632	0.269910	0.134955	−	0.990852i	$-0.456911\pi$
$89$	2.54632	0.269910	0.134955	+	0.990852i	$0.456911\pi$
$90$	0	0
$91$	−3.73108	−0.391123
$92$	0	0
$93$	24.6583	2.55695
$94$	0	0
$95$	−5.09870	−0.523116
$96$	0	0
$97$	−16.8522	−1.71108	−0.855540	−	0.517737i	$-0.826775\pi$
$97$	−16.8522	−1.71108	−0.855540	+	0.517737i	$0.826775\pi$
$98$	0	0
$99$	−16.8003	−1.68849
$100$	0	0
$101$	−9.94062	−0.989129	−0.494564	−	0.869141i	$-0.664672\pi$
$101$	−9.94062	−0.989129	−0.494564	+	0.869141i	$0.664672\pi$
$102$	0	0
$103$	−8.28346	−0.816193	−0.408097	−	0.912939i	$-0.633807\pi$
$103$	−8.28346	−0.816193	−0.408097	+	0.912939i	$0.633807\pi$
$104$	0	0
$105$	−9.18252	−0.896122
$106$	0	0
$107$	−10.2339	−0.989351	−0.494675	−	0.869078i	$-0.664713\pi$
$107$	−10.2339	−0.989351	−0.494675	+	0.869078i	$0.664713\pi$
$108$	0	0
$109$	4.68155	0.448412	0.224206	−	0.974542i	$-0.428021\pi$
$109$	4.68155	0.448412	0.224206	+	0.974542i	$0.428021\pi$
$110$	0	0
$111$	−3.80758	−0.361399
$112$	0	0
$113$	6.10023	0.573862	0.286931	−	0.957951i	$-0.407365\pi$
$113$	6.10023	0.573862	0.286931	+	0.957951i	$0.407365\pi$
$114$	0	0
$115$	12.1834	1.13611
$116$	0	0
$117$	−13.8777	−1.28300
$118$	0	0
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	9.40161	0.854692
$122$	0	0
$123$	−27.1585	−2.44880
$124$	0	0
$125$	9.02723	0.807420
$126$	0	0
$127$	−14.6546	−1.30039	−0.650194	−	0.759768i	$-0.725312\pi$
$127$	−14.6546	−1.30039	−0.650194	+	0.759768i	$0.725312\pi$
$128$	0	0
$129$	−6.56965	−0.578426
$130$	0	0
$131$	−18.2045	−1.59053	−0.795266	−	0.606261i	$-0.792669\pi$
$131$	−18.2045	−1.59053	−0.795266	+	0.606261i	$0.792669\pi$
$132$	0	0
$133$	−1.43935	−0.124807
$134$	0	0
$135$	−6.60679	−0.568622
$136$	0	0
$137$	−16.7465	−1.43075	−0.715374	−	0.698742i	$-0.753744\pi$
$137$	−16.7465	−1.43075	−0.715374	+	0.698742i	$0.753744\pi$
$138$	0	0
$139$	10.7185	0.909131	0.454565	−	0.890713i	$-0.349795\pi$
$139$	10.7185	0.909131	0.454565	+	0.890713i	$0.349795\pi$
$140$	0	0
$141$	−14.5386	−1.22437
$142$	0	0
$143$	16.8526	1.40928
$144$	0	0
$145$	19.1129	1.58724
$146$	0	0
$147$	−2.59220	−0.213801
$148$	0	0
$149$	10.2651	0.840951	0.420476	−	0.907304i	$-0.361863\pi$
$149$	10.2651	0.840951	0.420476	+	0.907304i	$0.361863\pi$
$150$	0	0
$151$	5.32937	0.433698	0.216849	−	0.976205i	$-0.430422\pi$
$151$	5.32937	0.433698	0.216849	+	0.976205i	$0.430422\pi$
$152$	0	0
$153$	−3.71950	−0.300703
$154$	0	0
$155$	−33.6968	−2.70659
$156$	0	0
$157$	4.99488	0.398635	0.199317	−	0.979935i	$-0.436128\pi$
$157$	4.99488	0.398635	0.199317	+	0.979935i	$0.436128\pi$
$158$	0	0
$159$	27.0323	2.14380
$160$	0	0
$161$	3.43935	0.271059
$162$	0	0
$163$	−12.3615	−0.968230	−0.484115	−	0.875005i	$-0.660858\pi$
$163$	−12.3615	−0.968230	−0.484115	+	0.875005i	$0.660858\pi$
$164$	0	0
$165$	41.4757	3.22888
$166$	0	0
$167$	11.0465	0.854803	0.427401	−	0.904062i	$-0.359429\pi$
$167$	11.0465	0.854803	0.427401	+	0.904062i	$0.359429\pi$
$168$	0	0
$169$	0.920956	0.0708428
$170$	0	0
$171$	−5.35365	−0.409404
$172$	0	0
$173$	−4.54166	−0.345296	−0.172648	−	0.984984i	$-0.555232\pi$
$173$	−4.54166	−0.345296	−0.172648	+	0.984984i	$0.555232\pi$
$174$	0	0
$175$	7.54836	0.570603
$176$	0	0
$177$	4.52444	0.340078
$178$	0	0
$179$	14.5026	1.08397	0.541987	−	0.840387i	$-0.317672\pi$
$179$	14.5026	1.08397	0.541987	+	0.840387i	$0.317672\pi$
$180$	0	0
$181$	−8.28410	−0.615753	−0.307876	−	0.951426i	$-0.599618\pi$
$181$	−8.28410	−0.615753	−0.307876	+	0.951426i	$0.599618\pi$
$182$	0	0
$183$	16.7337	1.23699
$184$	0	0
$185$	5.20324	0.382550
$186$	0	0
$187$	4.51681	0.330302
$188$	0	0
$189$	−1.86508	−0.135664
$190$	0	0
$191$	8.46455	0.612473	0.306236	−	0.951955i	$-0.400930\pi$
$191$	8.46455	0.612473	0.306236	+	0.951955i	$0.400930\pi$
$192$	0	0
$193$	−13.7456	−0.989431	−0.494716	−	0.869055i	$-0.664728\pi$
$193$	−13.7456	−0.989431	−0.494716	+	0.869055i	$0.664728\pi$
$194$	0	0
$195$	34.2607	2.45346
$196$	0	0
$197$	−19.0451	−1.35690	−0.678452	−	0.734645i	$-0.737349\pi$
$197$	−19.0451	−1.35690	−0.678452	+	0.734645i	$0.737349\pi$
$198$	0	0
$199$	12.6562	0.897171	0.448585	−	0.893740i	$-0.351928\pi$
$199$	12.6562	0.897171	0.448585	+	0.893740i	$0.351928\pi$
$200$	0	0
$201$	−14.5708	−1.02774
$202$	0	0
$203$	5.39551	0.378691
$204$	0	0
$205$	37.1134	2.59211
$206$	0	0
$207$	12.7926	0.889150
$208$	0	0
$209$	6.50127	0.449702
$210$	0	0
$211$	−19.7503	−1.35967	−0.679835	−	0.733365i	$-0.737948\pi$
$211$	−19.7503	−1.35967	−0.679835	+	0.733365i	$0.737948\pi$
$212$	0	0
$213$	23.2320	1.59183
$214$	0	0
$215$	8.97775	0.612277
$216$	0	0
$217$	−9.51250	−0.645751
$218$	0	0
$219$	−43.6676	−2.95079
$220$	0	0
$221$	3.73108	0.250980
$222$	0	0
$223$	−3.79615	−0.254209	−0.127105	−	0.991889i	$-0.540568\pi$
$223$	−3.79615	−0.254209	−0.127105	+	0.991889i	$0.540568\pi$
$224$	0	0
$225$	28.0761	1.87174
$226$	0	0
$227$	−6.03992	−0.400884	−0.200442	−	0.979706i	$-0.564238\pi$
$227$	−6.03992	−0.400884	−0.200442	+	0.979706i	$0.564238\pi$
$228$	0	0
$229$	−14.6959	−0.971131	−0.485566	−	0.874200i	$-0.661386\pi$
$229$	−14.6959	−0.971131	−0.485566	+	0.874200i	$0.661386\pi$
$230$	0	0
$231$	11.7085	0.770361
$232$	0	0
$233$	1.77786	0.116472	0.0582358	−	0.998303i	$-0.481452\pi$
$233$	1.77786	0.116472	0.0582358	+	0.998303i	$0.481452\pi$
$234$	0	0
$235$	19.8677	1.29603
$236$	0	0
$237$	−38.3841	−2.49331
$238$	0	0
$239$	−6.74043	−0.436002	−0.218001	−	0.975949i	$-0.569954\pi$
$239$	−6.74043	−0.436002	−0.218001	+	0.975949i	$0.569954\pi$
$240$	0	0
$241$	9.20167	0.592732	0.296366	−	0.955074i	$-0.404225\pi$
$241$	9.20167	0.592732	0.296366	+	0.955074i	$0.404225\pi$
$242$	0	0
$243$	21.9879	1.41052
$244$	0	0
$245$	3.54237	0.226314
$246$	0	0
$247$	5.37033	0.341706
$248$	0	0
$249$	30.1509	1.91074
$250$	0	0
$251$	−11.6222	−0.733588	−0.366794	−	0.930302i	$-0.619545\pi$
$251$	−11.6222	−0.733588	−0.366794	+	0.930302i	$0.619545\pi$
$252$	0	0
$253$	−15.5349	−0.976671
$254$	0	0
$255$	9.18252	0.575032
$256$	0	0
$257$	26.0183	1.62298	0.811490	−	0.584367i	$-0.198657\pi$
$257$	26.0183	1.62298	0.811490	+	0.584367i	$0.198657\pi$
$258$	0	0
$259$	1.46886	0.0912705
$260$	0	0
$261$	20.0686	1.24221
$262$	0	0
$263$	−8.26913	−0.509897	−0.254948	−	0.966955i	$-0.582058\pi$
$263$	−8.26913	−0.509897	−0.254948	+	0.966955i	$0.582058\pi$
$264$	0	0
$265$	−36.9410	−2.26927
$266$	0	0
$267$	−6.60058	−0.403949
$268$	0	0
$269$	−10.7447	−0.655113	−0.327556	−	0.944832i	$-0.606225\pi$
$269$	−10.7447	−0.655113	−0.327556	+	0.944832i	$0.606225\pi$
$270$	0	0
$271$	8.79067	0.533995	0.266998	−	0.963697i	$-0.413968\pi$
$271$	8.79067	0.533995	0.266998	+	0.963697i	$0.413968\pi$
$272$	0	0
$273$	9.67170	0.585358
$274$	0	0
$275$	−34.0945	−2.05598
$276$	0	0
$277$	24.2113	1.45472	0.727359	−	0.686257i	$-0.240748\pi$
$277$	24.2113	1.45472	0.727359	+	0.686257i	$0.240748\pi$
$278$	0	0
$279$	−35.3817	−2.11825
$280$	0	0
$281$	−19.2067	−1.14577	−0.572886	−	0.819635i	$-0.694176\pi$
$281$	−19.2067	−1.14577	−0.572886	+	0.819635i	$0.694176\pi$
$282$	0	0
$283$	0.821301	0.0488213	0.0244106	−	0.999702i	$-0.492229\pi$
$283$	0.821301	0.0488213	0.0244106	+	0.999702i	$0.492229\pi$
$284$	0	0
$285$	13.2169	0.782899
$286$	0	0
$287$	10.4770	0.618437
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	43.6842	2.56081
$292$	0	0
$293$	−10.1753	−0.594448	−0.297224	−	0.954808i	$-0.596061\pi$
$293$	−10.1753	−0.594448	−0.297224	+	0.954808i	$0.596061\pi$
$294$	0	0
$295$	−6.18287	−0.359980
$296$	0	0
$297$	8.42421	0.488822
$298$	0	0
$299$	−12.8325	−0.742122
$300$	0	0
$301$	2.53439	0.146080
$302$	0	0
$303$	25.7681	1.48034
$304$	0	0
$305$	−22.8674	−1.30938
$306$	0	0
$307$	9.76145	0.557115	0.278558	−	0.960420i	$-0.410144\pi$
$307$	9.76145	0.557115	0.278558	+	0.960420i	$0.410144\pi$
$308$	0	0
$309$	21.4724	1.22152
$310$	0	0
$311$	−28.7977	−1.63297	−0.816484	−	0.577368i	$-0.804080\pi$
$311$	−28.7977	−1.63297	−0.816484	+	0.577368i	$0.804080\pi$
$312$	0	0
$313$	−30.0549	−1.69880	−0.849401	−	0.527748i	$-0.823037\pi$
$313$	−30.0549	−1.69880	−0.849401	+	0.527748i	$0.823037\pi$
$314$	0	0
$315$	13.1758	0.742373
$316$	0	0
$317$	27.7111	1.55641	0.778204	−	0.628011i	$-0.216131\pi$
$317$	27.7111	1.55641	0.778204	+	0.628011i	$0.216131\pi$
$318$	0	0
$319$	−24.3705	−1.36449
$320$	0	0
$321$	26.5284	1.48067
$322$	0	0
$323$	1.43935	0.0800875
$324$	0	0
$325$	−28.1635	−1.56223
$326$	0	0
$327$	−12.1355	−0.671096
$328$	0	0
$329$	5.60860	0.309212
$330$	0	0
$331$	36.2511	1.99254	0.996270	−	0.0862866i	$-0.0275001\pi$
$331$	36.2511	1.99254	0.996270	+	0.0862866i	$0.0275001\pi$
$332$	0	0
$333$	5.46342	0.299393
$334$	0	0
$335$	19.9117	1.08789
$336$	0	0
$337$	−24.5223	−1.33582	−0.667908	−	0.744244i	$-0.732810\pi$
$337$	−24.5223	−1.33582	−0.667908	+	0.744244i	$0.732810\pi$
$338$	0	0
$339$	−15.8130	−0.858845
$340$	0	0
$341$	42.9662	2.32675
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−31.5819	−1.70031
$346$	0	0
$347$	−4.53870	−0.243650	−0.121825	−	0.992552i	$-0.538875\pi$
$347$	−4.53870	−0.243650	−0.121825	+	0.992552i	$0.538875\pi$
$348$	0	0
$349$	−14.0234	−0.750654	−0.375327	−	0.926892i	$-0.622470\pi$
$349$	−14.0234	−0.750654	−0.375327	+	0.926892i	$0.622470\pi$
$350$	0	0
$351$	6.95875	0.371431
$352$	0	0
$353$	−18.5746	−0.988625	−0.494313	−	0.869284i	$-0.664580\pi$
$353$	−18.5746	−0.988625	−0.494313	+	0.869284i	$0.664580\pi$
$354$	0	0
$355$	−31.7476	−1.68499
$356$	0	0
$357$	2.59220	0.137194
$358$	0	0
$359$	−12.2116	−0.644502	−0.322251	−	0.946654i	$-0.604439\pi$
$359$	−12.2116	−0.644502	−0.322251	+	0.946654i	$0.604439\pi$
$360$	0	0
$361$	−16.9283	−0.890962
$362$	0	0
$363$	−24.3708	−1.27914
$364$	0	0
$365$	59.6740	3.12348
$366$	0	0
$367$	−2.86036	−0.149309	−0.0746547	−	0.997209i	$-0.523785\pi$
$367$	−2.86036	−0.149309	−0.0746547	+	0.997209i	$0.523785\pi$
$368$	0	0
$369$	38.9691	2.02865
$370$	0	0
$371$	−10.4283	−0.541412
$372$	0	0
$373$	−13.3536	−0.691423	−0.345712	−	0.938341i	$-0.612362\pi$
$373$	−13.3536	−0.691423	−0.345712	+	0.938341i	$0.612362\pi$
$374$	0	0
$375$	−23.4004	−1.20839
$376$	0	0
$377$	−20.1311	−1.03680
$378$	0	0
$379$	11.7199	0.602011	0.301006	−	0.953622i	$-0.402678\pi$
$379$	11.7199	0.602011	0.301006	+	0.953622i	$0.402678\pi$
$380$	0	0
$381$	37.9877	1.94617
$382$	0	0
$383$	−9.30474	−0.475450	−0.237725	−	0.971333i	$-0.576402\pi$
$383$	−9.30474	−0.475450	−0.237725	+	0.971333i	$0.576402\pi$
$384$	0	0
$385$	−16.0002	−0.815446
$386$	0	0
$387$	9.42666	0.479184
$388$	0	0
$389$	4.62846	0.234672	0.117336	−	0.993092i	$-0.462565\pi$
$389$	4.62846	0.234672	0.117336	+	0.993092i	$0.462565\pi$
$390$	0	0
$391$	−3.43935	−0.173935
$392$	0	0
$393$	47.1896	2.38040
$394$	0	0
$395$	52.4537	2.63923
$396$	0	0
$397$	7.50815	0.376823	0.188412	−	0.982090i	$-0.439666\pi$
$397$	7.50815	0.376823	0.188412	+	0.982090i	$0.439666\pi$
$398$	0	0
$399$	3.73108	0.186788
$400$	0	0
$401$	4.40815	0.220133	0.110066	−	0.993924i	$-0.464894\pi$
$401$	4.40815	0.220133	0.110066	+	0.993924i	$0.464894\pi$
$402$	0	0
$403$	35.4919	1.76798
$404$	0	0
$405$	−22.4013	−1.11313
$406$	0	0
$407$	−6.63457	−0.328863
$408$	0	0
$409$	−14.9817	−0.740799	−0.370399	−	0.928873i	$-0.620779\pi$
$409$	−14.9817	−0.740799	−0.370399	+	0.928873i	$0.620779\pi$
$410$	0	0
$411$	43.4102	2.14127
$412$	0	0
$413$	−1.74541	−0.0858858
$414$	0	0
$415$	−41.2027	−2.02256
$416$	0	0
$417$	−27.7845	−1.36061
$418$	0	0
$419$	0.548311	0.0267868	0.0133934	−	0.999910i	$-0.495737\pi$
$419$	0.548311	0.0267868	0.0133934	+	0.999910i	$0.495737\pi$
$420$	0	0
$421$	−35.6285	−1.73643	−0.868214	−	0.496189i	$-0.834732\pi$
$421$	−35.6285	−1.73643	−0.868214	+	0.496189i	$0.834732\pi$
$422$	0	0
$423$	20.8612	1.01431
$424$	0	0
$425$	−7.54836	−0.366149
$426$	0	0
$427$	−6.45540	−0.312399
$428$	0	0
$429$	−43.6853	−2.10914
$430$	0	0
$431$	−20.0338	−0.964996	−0.482498	−	0.875897i	$-0.660270\pi$
$431$	−20.0338	−0.964996	−0.482498	+	0.875897i	$0.660270\pi$
$432$	0	0
$433$	−35.9905	−1.72959	−0.864797	−	0.502121i	$-0.832553\pi$
$433$	−35.9905	−1.72959	−0.864797	+	0.502121i	$0.832553\pi$
$434$	0	0
$435$	−49.5444	−2.37547
$436$	0	0
$437$	−4.95042	−0.236811
$438$	0	0
$439$	−33.6939	−1.60812	−0.804060	−	0.594548i	$-0.797331\pi$
$439$	−33.6939	−1.60812	−0.804060	+	0.594548i	$0.797331\pi$
$440$	0	0
$441$	3.71950	0.177119
$442$	0	0
$443$	25.8734	1.22928	0.614642	−	0.788806i	$-0.289301\pi$
$443$	25.8734	1.22928	0.614642	+	0.788806i	$0.289301\pi$
$444$	0	0
$445$	9.02002	0.427590
$446$	0	0
$447$	−26.6092	−1.25857
$448$	0	0
$449$	−5.10618	−0.240975	−0.120488	−	0.992715i	$-0.538446\pi$
$449$	−5.10618	−0.240975	−0.120488	+	0.992715i	$0.538446\pi$
$450$	0	0
$451$	−47.3226	−2.22834
$452$	0	0
$453$	−13.8148	−0.649076
$454$	0	0
$455$	−13.2169	−0.619616
$456$	0	0
$457$	−4.46562	−0.208893	−0.104446	−	0.994531i	$-0.533307\pi$
$457$	−4.46562	−0.208893	−0.104446	+	0.994531i	$0.533307\pi$
$458$	0	0
$459$	1.86508	0.0870543
$460$	0	0
$461$	36.1862	1.68536	0.842679	−	0.538416i	$-0.180977\pi$
$461$	36.1862	1.68536	0.842679	+	0.538416i	$0.180977\pi$
$462$	0	0
$463$	34.1567	1.58740	0.793698	−	0.608312i	$-0.208153\pi$
$463$	34.1567	1.58740	0.793698	+	0.608312i	$0.208153\pi$
$464$	0	0
$465$	87.3487	4.05070
$466$	0	0
$467$	−28.8993	−1.33730	−0.668650	−	0.743577i	$-0.733127\pi$
$467$	−28.8993	−1.33730	−0.668650	+	0.743577i	$0.733127\pi$
$468$	0	0
$469$	5.62100	0.259554
$470$	0	0
$471$	−12.9477	−0.596599
$472$	0	0
$473$	−11.4474	−0.526351
$474$	0	0
$475$	−10.8647	−0.498508
$476$	0	0
$477$	−38.7881	−1.77599
$478$	0	0
$479$	6.51566	0.297708	0.148854	−	0.988859i	$-0.452442\pi$
$479$	6.51566	0.297708	0.148854	+	0.988859i	$0.452442\pi$
$480$	0	0
$481$	−5.48043	−0.249886
$482$	0	0
$483$	−8.91548	−0.405668
$484$	0	0
$485$	−59.6966	−2.71068
$486$	0	0
$487$	31.9526	1.44791	0.723954	−	0.689848i	$-0.242323\pi$
$487$	31.9526	1.44791	0.723954	+	0.689848i	$0.242323\pi$
$488$	0	0
$489$	32.0435	1.44906
$490$	0	0
$491$	−21.3657	−0.964220	−0.482110	−	0.876111i	$-0.660130\pi$
$491$	−21.3657	−0.964220	−0.482110	+	0.876111i	$0.660130\pi$
$492$	0	0
$493$	−5.39551	−0.243002
$494$	0	0
$495$	−59.5127	−2.67490
$496$	0	0
$497$	−8.96226	−0.402012
$498$	0	0
$499$	23.7954	1.06523	0.532613	−	0.846359i	$-0.321210\pi$
$499$	23.7954	1.06523	0.532613	+	0.846359i	$0.321210\pi$
$500$	0	0
$501$	−28.6347	−1.27930
$502$	0	0
$503$	−17.3116	−0.771884	−0.385942	−	0.922523i	$-0.626124\pi$
$503$	−17.3116	−0.771884	−0.385942	+	0.922523i	$0.626124\pi$
$504$	0	0
$505$	−35.2133	−1.56697
$506$	0	0
$507$	−2.38730	−0.106024
$508$	0	0
$509$	−30.9619	−1.37236	−0.686181	−	0.727431i	$-0.740714\pi$
$509$	−30.9619	−1.37236	−0.686181	+	0.727431i	$0.740714\pi$
$510$	0	0
$511$	16.8458	0.745214
$512$	0	0
$513$	2.68450	0.118523
$514$	0	0
$515$	−29.3430	−1.29301
$516$	0	0
$517$	−25.3330	−1.11415
$518$	0	0
$519$	11.7729	0.516773
$520$	0	0
$521$	−30.6504	−1.34282	−0.671410	−	0.741086i	$-0.734311\pi$
$521$	−30.6504	−1.34282	−0.671410	+	0.741086i	$0.734311\pi$
$522$	0	0
$523$	−28.5317	−1.24760	−0.623802	−	0.781582i	$-0.714413\pi$
$523$	−28.5317	−1.24760	−0.623802	+	0.781582i	$0.714413\pi$
$524$	0	0
$525$	−19.5669	−0.853968
$526$	0	0
$527$	9.51250	0.414371
$528$	0	0
$529$	−11.1709	−0.485690
$530$	0	0
$531$	−6.49203	−0.281730
$532$	0	0
$533$	−39.0905	−1.69320
$534$	0	0
$535$	−36.2523	−1.56732
$536$	0	0
$537$	−37.5936	−1.62228
$538$	0	0
$539$	−4.51681	−0.194553
$540$	0	0
$541$	−41.9235	−1.80243	−0.901217	−	0.433369i	$-0.857325\pi$
$541$	−41.9235	−1.80243	−0.901217	+	0.433369i	$0.857325\pi$
$542$	0	0
$543$	21.4740	0.921540
$544$	0	0
$545$	16.5838	0.710371
$546$	0	0
$547$	−1.79915	−0.0769260	−0.0384630	−	0.999260i	$-0.512246\pi$
$547$	−1.79915	−0.0769260	−0.0384630	+	0.999260i	$0.512246\pi$
$548$	0	0
$549$	−24.0108	−1.02476
$550$	0	0
$551$	−7.76603	−0.330844
$552$	0	0
$553$	14.8075	0.629681
$554$	0	0
$555$	−13.4878	−0.572527
$556$	0	0
$557$	−24.9829	−1.05856	−0.529281	−	0.848447i	$-0.677538\pi$
$557$	−24.9829	−1.05856	−0.529281	+	0.848447i	$0.677538\pi$
$558$	0	0
$559$	−9.45602	−0.399947
$560$	0	0
$561$	−11.7085	−0.494332
$562$	0	0
$563$	35.5109	1.49661	0.748303	−	0.663357i	$-0.230869\pi$
$563$	35.5109	1.49661	0.748303	+	0.663357i	$0.230869\pi$
$564$	0	0
$565$	21.6093	0.909109
$566$	0	0
$567$	−6.32384	−0.265576
$568$	0	0
$569$	3.43122	0.143844	0.0719221	−	0.997410i	$-0.477087\pi$
$569$	3.43122	0.143844	0.0719221	+	0.997410i	$0.477087\pi$
$570$	0	0
$571$	−13.6591	−0.571618	−0.285809	−	0.958287i	$-0.592262\pi$
$571$	−13.6591	−0.571618	−0.285809	+	0.958287i	$0.592262\pi$
$572$	0	0
$573$	−21.9418	−0.916631
$574$	0	0
$575$	25.9615	1.08267
$576$	0	0
$577$	−11.3265	−0.471530	−0.235765	−	0.971810i	$-0.575759\pi$
$577$	−11.3265	−0.471530	−0.235765	+	0.971810i	$0.575759\pi$
$578$	0	0
$579$	35.6314	1.48079
$580$	0	0
$581$	−11.6314	−0.482552
$582$	0	0
$583$	47.1028	1.95080
$584$	0	0
$585$	−49.1600	−2.03252
$586$	0	0
$587$	34.2746	1.41466	0.707332	−	0.706882i	$-0.249899\pi$
$587$	34.2746	1.41466	0.707332	+	0.706882i	$0.249899\pi$
$588$	0	0
$589$	13.6918	0.564161
$590$	0	0
$591$	49.3686	2.03075
$592$	0	0
$593$	25.4883	1.04668	0.523340	−	0.852124i	$-0.324686\pi$
$593$	25.4883	1.04668	0.523340	+	0.852124i	$0.324686\pi$
$594$	0	0
$595$	−3.54237	−0.145223
$596$	0	0
$597$	−32.8073	−1.34271
$598$	0	0
$599$	−36.2250	−1.48012	−0.740058	−	0.672544i	$-0.765202\pi$
$599$	−36.2250	−1.48012	−0.740058	+	0.672544i	$0.765202\pi$
$600$	0	0
$601$	−29.6118	−1.20789	−0.603946	−	0.797025i	$-0.706406\pi$
$601$	−29.6118	−1.20789	−0.603946	+	0.797025i	$0.706406\pi$
$602$	0	0
$603$	20.9073	0.851411
$604$	0	0
$605$	33.3039	1.35400
$606$	0	0
$607$	7.44392	0.302139	0.151070	−	0.988523i	$-0.451728\pi$
$607$	7.44392	0.302139	0.151070	+	0.988523i	$0.451728\pi$
$608$	0	0
$609$	−13.9862	−0.566751
$610$	0	0
$611$	−20.9262	−0.846581
$612$	0	0
$613$	44.1924	1.78492	0.892458	−	0.451131i	$-0.148979\pi$
$613$	44.1924	1.78492	0.892458	+	0.451131i	$0.148979\pi$
$614$	0	0
$615$	−96.2052	−3.87937
$616$	0	0
$617$	12.2256	0.492185	0.246092	−	0.969246i	$-0.420853\pi$
$617$	12.2256	0.492185	0.246092	+	0.969246i	$0.420853\pi$
$618$	0	0
$619$	22.5860	0.907807	0.453903	−	0.891051i	$-0.350031\pi$
$619$	22.5860	0.907807	0.453903	+	0.891051i	$0.350031\pi$
$620$	0	0
$621$	−6.41465	−0.257411
$622$	0	0
$623$	2.54632	0.102016
$624$	0	0
$625$	−5.76404	−0.230562
$626$	0	0
$627$	−16.8526	−0.673028
$628$	0	0
$629$	−1.46886	−0.0585673
$630$	0	0
$631$	−42.9781	−1.71093	−0.855466	−	0.517859i	$-0.826729\pi$
$631$	−42.9781	−1.71093	−0.855466	+	0.517859i	$0.826729\pi$
$632$	0	0
$633$	51.1968	2.03489
$634$	0	0
$635$	−51.9121	−2.06007
$636$	0	0
$637$	−3.73108	−0.147831
$638$	0	0
$639$	−33.3351	−1.31872
$640$	0	0
$641$	0.431882	0.0170583	0.00852915	−	0.999964i	$-0.497285\pi$
$641$	0.431882	0.0170583	0.00852915	+	0.999964i	$0.497285\pi$
$642$	0	0
$643$	32.8009	1.29354	0.646771	−	0.762685i	$-0.276119\pi$
$643$	32.8009	1.29354	0.646771	+	0.762685i	$0.276119\pi$
$644$	0	0
$645$	−23.2721	−0.916339
$646$	0	0
$647$	33.8690	1.33153	0.665764	−	0.746163i	$-0.268106\pi$
$647$	33.8690	1.33153	0.665764	+	0.746163i	$0.268106\pi$
$648$	0	0
$649$	7.88367	0.309461
$650$	0	0
$651$	24.6583	0.966435
$652$	0	0
$653$	20.6324	0.807408	0.403704	−	0.914890i	$-0.367723\pi$
$653$	20.6324	0.807408	0.403704	+	0.914890i	$0.367723\pi$
$654$	0	0
$655$	−64.4869	−2.51971
$656$	0	0
$657$	62.6578	2.44452
$658$	0	0
$659$	−13.0024	−0.506501	−0.253251	−	0.967401i	$-0.581500\pi$
$659$	−13.0024	−0.506501	−0.253251	+	0.967401i	$0.581500\pi$
$660$	0	0
$661$	13.4954	0.524911	0.262456	−	0.964944i	$-0.415468\pi$
$661$	13.4954	0.524911	0.262456	+	0.964944i	$0.415468\pi$
$662$	0	0
$663$	−9.67170	−0.375618
$664$	0	0
$665$	−5.09870	−0.197719
$666$	0	0
$667$	18.5571	0.718532
$668$	0	0
$669$	9.84039	0.380451
$670$	0	0
$671$	29.1578	1.12563
$672$	0	0
$673$	18.5966	0.716848	0.358424	−	0.933559i	$-0.383314\pi$
$673$	18.5966	0.716848	0.358424	+	0.933559i	$0.383314\pi$
$674$	0	0
$675$	−14.0783	−0.541873
$676$	0	0
$677$	−42.5433	−1.63507	−0.817536	−	0.575877i	$-0.804661\pi$
$677$	−42.5433	−1.63507	−0.817536	+	0.575877i	$0.804661\pi$
$678$	0	0
$679$	−16.8522	−0.646727
$680$	0	0
$681$	15.6567	0.599965
$682$	0	0
$683$	−19.1730	−0.733634	−0.366817	−	0.930293i	$-0.619552\pi$
$683$	−19.1730	−0.733634	−0.366817	+	0.930293i	$0.619552\pi$
$684$	0	0
$685$	−59.3222	−2.26658
$686$	0	0
$687$	38.0947	1.45340
$688$	0	0
$689$	38.9089	1.48231
$690$	0	0
$691$	−17.5445	−0.667423	−0.333711	−	0.942675i	$-0.608301\pi$
$691$	−17.5445	−0.667423	−0.333711	+	0.942675i	$0.608301\pi$
$692$	0	0
$693$	−16.8003	−0.638190
$694$	0	0
$695$	37.9688	1.44024
$696$	0	0
$697$	−10.4770	−0.396844
$698$	0	0
$699$	−4.60857	−0.174312
$700$	0	0
$701$	24.9866	0.943731	0.471865	−	0.881671i	$-0.343581\pi$
$701$	24.9866	0.943731	0.471865	+	0.881671i	$0.343581\pi$
$702$	0	0
$703$	−2.11420	−0.0797386
$704$	0	0
$705$	−51.5011	−1.93964
$706$	0	0
$707$	−9.94062	−0.373856
$708$	0	0
$709$	5.26419	0.197701	0.0988504	−	0.995102i	$-0.468483\pi$
$709$	5.26419	0.197701	0.0988504	+	0.995102i	$0.468483\pi$
$710$	0	0
$711$	55.0766	2.06553
$712$	0	0
$713$	−32.7168	−1.22525
$714$	0	0
$715$	59.6981	2.23258
$716$	0	0
$717$	17.4725	0.652524
$718$	0	0
$719$	8.34650	0.311272	0.155636	−	0.987814i	$-0.450257\pi$
$719$	8.34650	0.311272	0.155636	+	0.987814i	$0.450257\pi$
$720$	0	0
$721$	−8.28346	−0.308492
$722$	0	0
$723$	−23.8526	−0.887086
$724$	0	0
$725$	40.7273	1.51257
$726$	0	0
$727$	−26.8333	−0.995192	−0.497596	−	0.867409i	$-0.665784\pi$
$727$	−26.8333	−0.995192	−0.497596	+	0.867409i	$0.665784\pi$
$728$	0	0
$729$	−38.0254	−1.40835
$730$	0	0
$731$	−2.53439	−0.0937379
$732$	0	0
$733$	34.6937	1.28144	0.640721	−	0.767774i	$-0.278635\pi$
$733$	34.6937	1.28144	0.640721	+	0.767774i	$0.278635\pi$
$734$	0	0
$735$	−9.18252	−0.338702
$736$	0	0
$737$	−25.3890	−0.935217
$738$	0	0
$739$	21.0430	0.774079	0.387039	−	0.922063i	$-0.373498\pi$
$739$	21.0430	0.774079	0.387039	+	0.922063i	$0.373498\pi$
$740$	0	0
$741$	−13.9210	−0.511399
$742$	0	0
$743$	38.3303	1.40620	0.703101	−	0.711090i	$-0.251798\pi$
$743$	38.3303	1.40620	0.703101	+	0.711090i	$0.251798\pi$
$744$	0	0
$745$	36.3628	1.33223
$746$	0	0
$747$	−43.2630	−1.58291
$748$	0	0
$749$	−10.2339	−0.373939
$750$	0	0
$751$	−8.07533	−0.294673	−0.147336	−	0.989086i	$-0.547070\pi$
$751$	−8.07533	−0.294673	−0.147336	+	0.989086i	$0.547070\pi$
$752$	0	0
$753$	30.1271	1.09789
$754$	0	0
$755$	18.8786	0.687063
$756$	0	0
$757$	46.7790	1.70021	0.850106	−	0.526612i	$-0.176538\pi$
$757$	46.7790	1.70021	0.850106	+	0.526612i	$0.176538\pi$
$758$	0	0
$759$	40.2696	1.46169
$760$	0	0
$761$	2.34259	0.0849187	0.0424593	−	0.999098i	$-0.486481\pi$
$761$	2.34259	0.0849187	0.0424593	+	0.999098i	$0.486481\pi$
$762$	0	0
$763$	4.68155	0.169484
$764$	0	0
$765$	−13.1758	−0.476373
$766$	0	0
$767$	6.51225	0.235144
$768$	0	0
$769$	−0.736412	−0.0265557	−0.0132778	−	0.999912i	$-0.504227\pi$
$769$	−0.736412	−0.0265557	−0.0132778	+	0.999912i	$0.504227\pi$
$770$	0	0
$771$	−67.4447	−2.42896
$772$	0	0
$773$	22.0224	0.792092	0.396046	−	0.918231i	$-0.370382\pi$
$773$	22.0224	0.792092	0.396046	+	0.918231i	$0.370382\pi$
$774$	0	0
$775$	−71.8038	−2.57927
$776$	0	0
$777$	−3.80758	−0.136596
$778$	0	0
$779$	−15.0801	−0.540299
$780$	0	0
$781$	40.4809	1.44852
$782$	0	0
$783$	−10.0630	−0.359624
$784$	0	0
$785$	17.6937	0.631515
$786$	0	0
$787$	20.8151	0.741977	0.370988	−	0.928637i	$-0.379019\pi$
$787$	20.8151	0.741977	0.370988	+	0.928637i	$0.379019\pi$
$788$	0	0
$789$	21.4352	0.763115
$790$	0	0
$791$	6.10023	0.216899
$792$	0	0
$793$	24.0856	0.855305
$794$	0	0
$795$	95.7584	3.39620
$796$	0	0
$797$	50.1164	1.77522	0.887608	−	0.460600i	$-0.152366\pi$
$797$	50.1164	1.77522	0.887608	+	0.460600i	$0.152366\pi$
$798$	0	0
$799$	−5.60860	−0.198418
$800$	0	0
$801$	9.47104	0.334643
$802$	0	0
$803$	−76.0893	−2.68513
$804$	0	0
$805$	12.1834	0.429410
$806$	0	0
$807$	27.8523	0.980446
$808$	0	0
$809$	10.4099	0.365991	0.182995	−	0.983114i	$-0.441421\pi$
$809$	10.4099	0.365991	0.182995	+	0.983114i	$0.441421\pi$
$810$	0	0
$811$	−16.8637	−0.592164	−0.296082	−	0.955163i	$-0.595680\pi$
$811$	−16.8637	−0.592164	−0.296082	+	0.955163i	$0.595680\pi$
$812$	0	0
$813$	−22.7872	−0.799181
$814$	0	0
$815$	−43.7891	−1.53386
$816$	0	0
$817$	−3.64788	−0.127623
$818$	0	0
$819$	−13.8777	−0.484927
$820$	0	0
$821$	−10.7195	−0.374113	−0.187057	−	0.982349i	$-0.559895\pi$
$821$	−10.7195	−0.374113	−0.187057	+	0.982349i	$0.559895\pi$
$822$	0	0
$823$	36.7974	1.28268	0.641338	−	0.767259i	$-0.278380\pi$
$823$	36.7974	1.28268	0.641338	+	0.767259i	$0.278380\pi$
$824$	0	0
$825$	88.3799	3.07699
$826$	0	0
$827$	−6.32892	−0.220078	−0.110039	−	0.993927i	$-0.535098\pi$
$827$	−6.32892	−0.220078	−0.110039	+	0.993927i	$0.535098\pi$
$828$	0	0
$829$	−33.9253	−1.17828	−0.589138	−	0.808033i	$-0.700533\pi$
$829$	−33.9253	−1.17828	−0.589138	+	0.808033i	$0.700533\pi$
$830$	0	0
$831$	−62.7606	−2.17714
$832$	0	0
$833$	−1.00000	−0.0346479
$834$	0	0
$835$	39.1307	1.35417
$836$	0	0
$837$	17.7415	0.613238
$838$	0	0
$839$	−39.6470	−1.36877	−0.684383	−	0.729123i	$-0.739928\pi$
$839$	−39.6470	−1.36877	−0.684383	+	0.729123i	$0.739928\pi$
$840$	0	0
$841$	0.111553	0.00384666
$842$	0	0
$843$	49.7875	1.71477
$844$	0	0
$845$	3.26236	0.112229
$846$	0	0
$847$	9.40161	0.323043
$848$	0	0
$849$	−2.12898	−0.0730663
$850$	0	0
$851$	5.05192	0.173178
$852$	0	0
$853$	−46.8928	−1.60558	−0.802789	−	0.596263i	$-0.796652\pi$
$853$	−46.8928	−1.60558	−0.802789	+	0.596263i	$0.796652\pi$
$854$	0	0
$855$	−18.9646	−0.648576
$856$	0	0
$857$	−43.6201	−1.49003	−0.745016	−	0.667046i	$-0.767558\pi$
$857$	−43.6201	−1.49003	−0.745016	+	0.667046i	$0.767558\pi$
$858$	0	0
$859$	−56.7054	−1.93476	−0.967381	−	0.253326i	$-0.918475\pi$
$859$	−56.7054	−1.93476	−0.967381	+	0.253326i	$0.918475\pi$
$860$	0	0
$861$	−27.1585	−0.925558
$862$	0	0
$863$	13.5523	0.461326	0.230663	−	0.973034i	$-0.425911\pi$
$863$	13.5523	0.461326	0.230663	+	0.973034i	$0.425911\pi$
$864$	0	0
$865$	−16.0882	−0.547016
$866$	0	0
$867$	−2.59220	−0.0880357
$868$	0	0
$869$	−66.8829	−2.26885
$870$	0	0
$871$	−20.9724	−0.710623
$872$	0	0
$873$	−62.6816	−2.12145
$874$	0	0
$875$	9.02723	0.305176
$876$	0	0
$877$	41.6525	1.40651	0.703253	−	0.710940i	$-0.251730\pi$
$877$	41.6525	1.40651	0.703253	+	0.710940i	$0.251730\pi$
$878$	0	0
$879$	26.3764	0.889655
$880$	0	0
$881$	16.7683	0.564937	0.282469	−	0.959277i	$-0.408847\pi$
$881$	16.7683	0.564937	0.282469	+	0.959277i	$0.408847\pi$
$882$	0	0
$883$	−2.06620	−0.0695331	−0.0347665	−	0.999395i	$-0.511069\pi$
$883$	−2.06620	−0.0695331	−0.0347665	+	0.999395i	$0.511069\pi$
$884$	0	0
$885$	16.0272	0.538749
$886$	0	0
$887$	−34.6135	−1.16221	−0.581104	−	0.813829i	$-0.697379\pi$
$887$	−34.6135	−1.16221	−0.581104	+	0.813829i	$0.697379\pi$
$888$	0	0
$889$	−14.6546	−0.491501
$890$	0	0
$891$	28.5636	0.956916
$892$	0	0
$893$	−8.07274	−0.270144
$894$	0	0
$895$	51.3735	1.71723
$896$	0	0
$897$	33.2644	1.11066
$898$	0	0
$899$	−51.3248	−1.71178
$900$	0	0
$901$	10.4283	0.347418
$902$	0	0
$903$	−6.56965	−0.218624
$904$	0	0
$905$	−29.3453	−0.975472
$906$	0	0
$907$	−14.0657	−0.467043	−0.233522	−	0.972352i	$-0.575025\pi$
$907$	−14.0657	−0.467043	−0.233522	+	0.972352i	$0.575025\pi$
$908$	0	0
$909$	−36.9741	−1.22635
$910$	0	0
$911$	−31.9222	−1.05763	−0.528814	−	0.848738i	$-0.677363\pi$
$911$	−31.9222	−1.05763	−0.528814	+	0.848738i	$0.677363\pi$
$912$	0	0
$913$	52.5369	1.73872
$914$	0	0
$915$	59.2768	1.95963
$916$	0	0
$917$	−18.2045	−0.601164
$918$	0	0
$919$	15.8206	0.521874	0.260937	−	0.965356i	$-0.415969\pi$
$919$	15.8206	0.521874	0.260937	+	0.965356i	$0.415969\pi$
$920$	0	0
$921$	−25.3036	−0.833783
$922$	0	0
$923$	33.4389	1.10065
$924$	0	0
$925$	11.0875	0.364554
$926$	0	0
$927$	−30.8103	−1.01194
$928$	0	0
$929$	−34.0082	−1.11577	−0.557887	−	0.829917i	$-0.688388\pi$
$929$	−34.0082	−1.11577	−0.557887	+	0.829917i	$0.688388\pi$
$930$	0	0
$931$	−1.43935	−0.0471728
$932$	0	0
$933$	74.6494	2.44391
$934$	0	0
$935$	16.0002	0.523263
$936$	0	0
$937$	29.0018	0.947449	0.473725	−	0.880673i	$-0.342909\pi$
$937$	29.0018	0.947449	0.473725	+	0.880673i	$0.342909\pi$
$938$	0	0
$939$	77.9082	2.54244
$940$	0	0
$941$	6.82875	0.222611	0.111305	−	0.993786i	$-0.464497\pi$
$941$	6.82875	0.222611	0.111305	+	0.993786i	$0.464497\pi$
$942$	0	0
$943$	36.0340	1.17343
$944$	0	0
$945$	−6.60679	−0.214919
$946$	0	0
$947$	23.6225	0.767628	0.383814	−	0.923410i	$-0.374610\pi$
$947$	23.6225	0.767628	0.383814	+	0.923410i	$0.374610\pi$
$948$	0	0
$949$	−62.8530	−2.04029
$950$	0	0
$951$	−71.8326	−2.32933
$952$	0	0
$953$	11.7506	0.380639	0.190319	−	0.981722i	$-0.439048\pi$
$953$	11.7506	0.380639	0.190319	+	0.981722i	$0.439048\pi$
$954$	0	0
$955$	29.9845	0.970276
$956$	0	0
$957$	63.1733	2.04210
$958$	0	0
$959$	−16.7465	−0.540772
$960$	0	0
$961$	59.4876	1.91896
$962$	0	0
$963$	−38.0650	−1.22663
$964$	0	0
$965$	−48.6920	−1.56745
$966$	0	0
$967$	−22.4946	−0.723376	−0.361688	−	0.932299i	$-0.617800\pi$
$967$	−22.4946	−0.723376	−0.361688	+	0.932299i	$0.617800\pi$
$968$	0	0
$969$	−3.73108	−0.119860
$970$	0	0
$971$	−24.1947	−0.776445	−0.388223	−	0.921566i	$-0.626911\pi$
$971$	−24.1947	−0.776445	−0.388223	+	0.921566i	$0.626911\pi$
$972$	0	0
$973$	10.7185	0.343619
$974$	0	0
$975$	73.0055	2.33805
$976$	0	0
$977$	−45.3568	−1.45109	−0.725547	−	0.688173i	$-0.758413\pi$
$977$	−45.3568	−1.45109	−0.725547	+	0.688173i	$0.758413\pi$
$978$	0	0
$979$	−11.5013	−0.367582
$980$	0	0
$981$	17.4130	0.555955
$982$	0	0
$983$	−4.89307	−0.156065	−0.0780323	−	0.996951i	$-0.524864\pi$
$983$	−4.89307	−0.156065	−0.0780323	+	0.996951i	$0.524864\pi$
$984$	0	0
$985$	−67.4646	−2.14960
$986$	0	0
$987$	−14.5386	−0.462769
$988$	0	0
$989$	8.71666	0.277174
$990$	0	0
$991$	10.0697	0.319875	0.159938	−	0.987127i	$-0.448871\pi$
$991$	10.0697	0.319875	0.159938	+	0.987127i	$0.448871\pi$
$992$	0	0
$993$	−93.9701	−2.98205
$994$	0	0
$995$	44.8327	1.42129
$996$	0	0
$997$	−15.8627	−0.502377	−0.251188	−	0.967938i	$-0.580821\pi$
$997$	−15.8627	−0.502377	−0.251188	+	0.967938i	$0.580821\pi$
$998$	0	0
$999$	−2.73954	−0.0866751

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7616.2.a.bu.1.2		6
4.3	odd	2		7616.2.a.cc.1.5		6
8.3	odd	2		3808.2.a.h.1.2	✓	6
8.5	even	2		3808.2.a.p.1.5	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3808.2.a.h.1.2	✓	6	8.3	odd	2
3808.2.a.p.1.5	yes	6	8.5	even	2
7616.2.a.bu.1.2		6	1.1	even	1	trivial
7616.2.a.cc.1.5		6	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7616 = 2^{6} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7616.a (trivial)

\(f(q)\)	\(=\)	\(q-2.59220 q^{3} +3.54237 q^{5} +1.00000 q^{7} +3.71950 q^{9} -4.51681 q^{11} -3.73108 q^{13} -9.18252 q^{15} -1.00000 q^{17} -1.43935 q^{19} -2.59220 q^{21} +3.43935 q^{23} +7.54836 q^{25} -1.86508 q^{27} +5.39551 q^{29} -9.51250 q^{31} +11.7085 q^{33} +3.54237 q^{35} +1.46886 q^{37} +9.67170 q^{39} +10.4770 q^{41} +2.53439 q^{43} +13.1758 q^{45} +5.60860 q^{47} +1.00000 q^{49} +2.59220 q^{51} -10.4283 q^{53} -16.0002 q^{55} +3.73108 q^{57} -1.74541 q^{59} -6.45540 q^{61} +3.71950 q^{63} -13.2169 q^{65} +5.62100 q^{67} -8.91548 q^{69} -8.96226 q^{71} +16.8458 q^{73} -19.5669 q^{75} -4.51681 q^{77} +14.8075 q^{79} -6.32384 q^{81} -11.6314 q^{83} -3.54237 q^{85} -13.9862 q^{87} +2.54632 q^{89} -3.73108 q^{91} +24.6583 q^{93} -5.09870 q^{95} -16.8522 q^{97} -16.8003 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{3} - 4 q^{5} + 6 q^{7} + 8 q^{9} - 8 q^{11} - 8 q^{13} - 6 q^{17} - 6 q^{19} - 4 q^{21} + 18 q^{23} + 12 q^{25} - 22 q^{27} + 8 q^{29} - 4 q^{31} - 6 q^{33} - 4 q^{35} + 8 q^{37} + 14 q^{39}+ \cdots - 28 q^{99}+O(q^{100}) \) 6 * q - 4 * q^3 - 4 * q^5 + 6 * q^7 + 8 * q^9 - 8 * q^11 - 8 * q^13 - 6 * q^17 - 6 * q^19 - 4 * q^21 + 18 * q^23 + 12 * q^25 - 22 * q^27 + 8 * q^29 - 4 * q^31 - 6 * q^33 - 4 * q^35 + 8 * q^37 + 14 * q^39 + 16 * q^41 - 12 * q^43 - 4 * q^45 + 18 * q^47 + 6 * q^49 + 4 * q^51 + 2 * q^53 - 8 * q^55 + 8 * q^57 - 16 * q^59 - 6 * q^61 + 8 * q^63 - 22 * q^65 - 12 * q^67 - 16 * q^69 - 2 * q^71 + 8 * q^73 - 24 * q^75 - 8 * q^77 - 18 * q^79 + 18 * q^81 - 28 * q^83 + 4 * q^85 - 2 * q^87 - 2 * q^89 - 8 * q^91 + 32 * q^93 + 26 * q^95 - 18 * q^97 - 28 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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