LMFDB - Embedded newform 7616.2.a.cf.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7616,2,Mod(1,7616)]

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

lf = mfeigenbasis(mf)

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")

//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);

Level:	$ N $	$=$	$ 7616 = 2^{6} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7616.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:	$60.8140661794$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 17x^{6} + 7x^{5} + 64x^{4} - 11x^{3} - 87x^{2} + 9x + 37 $ x^8 - 2x^7 - 17x^6 + 7x^5 + 64x^4 - 11x^3 - 87x^2 + 9*x + 37
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 3808)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-0.794927$ 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)

$f(q)$	$=$	$q+3.15311 q^{3} +2.50649 q^{5} -1.00000 q^{7} +6.94208 q^{9} +O(q^{10})$	$q+3.15311 q^{3} +2.50649 q^{5} -1.00000 q^{7} +6.94208 q^{9} +3.88928 q^{11} +1.04643 q^{13} +7.90322 q^{15} +1.00000 q^{17} +1.93181 q^{19} -3.15311 q^{21} +0.142722 q^{23} +1.28248 q^{25} +12.4298 q^{27} +1.88928 q^{29} -7.05670 q^{31} +12.2633 q^{33} -2.50649 q^{35} +5.25942 q^{37} +3.29950 q^{39} -9.78021 q^{41} +3.49716 q^{43} +17.4002 q^{45} -6.69256 q^{47} +1.00000 q^{49} +3.15311 q^{51} -8.97011 q^{53} +9.74844 q^{55} +6.09120 q^{57} +12.5094 q^{59} +0.0384872 q^{61} -6.94208 q^{63} +2.62286 q^{65} +15.3398 q^{67} +0.450017 q^{69} -6.00461 q^{71} -11.1525 q^{73} +4.04381 q^{75} -3.88928 q^{77} +7.89836 q^{79} +18.3662 q^{81} -4.76075 q^{83} +2.50649 q^{85} +5.95710 q^{87} +12.3409 q^{89} -1.04643 q^{91} -22.2505 q^{93} +4.84206 q^{95} -15.0908 q^{97} +26.9997 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{3} - 6 q^{5} - 8 q^{7} + 14 q^{9}+O(q^{10}) $ 8 * q + 2 * q^3 - 6 * q^5 - 8 * q^7 + 14 * q^9	$ 8 q + 2 q^{3} - 6 q^{5} - 8 q^{7} + 14 q^{9} + 4 q^{11} - 10 q^{13} + 6 q^{15} + 8 q^{17} + 16 q^{19} - 2 q^{21} + 18 q^{25} + 8 q^{27} - 12 q^{29} + 4 q^{31} + 4 q^{33} + 6 q^{35} - 6 q^{37} + 14 q^{39} + 6 q^{41} + 8 q^{43} - 32 q^{45} + 2 q^{47} + 8 q^{49} + 2 q^{51} - 30 q^{53} + 18 q^{55} - 10 q^{57} + 24 q^{59} - 4 q^{61} - 14 q^{63} + 20 q^{65} - 2 q^{67} - 2 q^{69} - 8 q^{71} - 6 q^{73} - 30 q^{75} - 4 q^{77} + 30 q^{79} + 32 q^{81} + 18 q^{83} - 6 q^{85} + 18 q^{89} + 10 q^{91} - 40 q^{93} - 10 q^{95} - 4 q^{97} + 26 q^{99}+O(q^{100}) $ 8 * q + 2 * q^3 - 6 * q^5 - 8 * q^7 + 14 * q^9 + 4 * q^11 - 10 * q^13 + 6 * q^15 + 8 * q^17 + 16 * q^19 - 2 * q^21 + 18 * q^25 + 8 * q^27 - 12 * q^29 + 4 * q^31 + 4 * q^33 + 6 * q^35 - 6 * q^37 + 14 * q^39 + 6 * q^41 + 8 * q^43 - 32 * q^45 + 2 * q^47 + 8 * q^49 + 2 * q^51 - 30 * q^53 + 18 * q^55 - 10 * q^57 + 24 * q^59 - 4 * q^61 - 14 * q^63 + 20 * q^65 - 2 * q^67 - 2 * q^69 - 8 * q^71 - 6 * q^73 - 30 * q^75 - 4 * q^77 + 30 * q^79 + 32 * q^81 + 18 * q^83 - 6 * q^85 + 18 * q^89 + 10 * q^91 - 40 * q^93 - 10 * q^95 - 4 * q^97 + 26 * 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
$3$	3.15311	1.82045	0.910223	−	0.414118i	$-0.135910\pi$
$3$	3.15311	1.82045	0.910223	+	0.414118i	$0.135910\pi$
$4$	0	0
$5$	2.50649	1.12094	0.560468	−	0.828176i	$-0.310621\pi$
$5$	2.50649	1.12094	0.560468	+	0.828176i	$0.310621\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	6.94208	2.31403
$10$	0	0
$11$	3.88928	1.17266	0.586331	−	0.810071i	$-0.300572\pi$
$11$	3.88928	1.17266	0.586331	+	0.810071i	$0.300572\pi$
$12$	0	0
$13$	1.04643	0.290227	0.145114	−	0.989415i	$-0.453645\pi$
$13$	1.04643	0.290227	0.145114	+	0.989415i	$0.453645\pi$
$14$	0	0
$15$	7.90322	2.04060
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	1.93181	0.443188	0.221594	−	0.975139i	$-0.428874\pi$
$19$	1.93181	0.443188	0.221594	+	0.975139i	$0.428874\pi$
$20$	0	0
$21$	−3.15311	−0.688064
$22$	0	0
$23$	0.142722	0.0297595	0.0148798	−	0.999889i	$-0.495263\pi$
$23$	0.142722	0.0297595	0.0148798	+	0.999889i	$0.495263\pi$
$24$	0	0
$25$	1.28248	0.256497
$26$	0	0
$27$	12.4298	2.39211
$28$	0	0
$29$	1.88928	0.350831	0.175415	−	0.984495i	$-0.443873\pi$
$29$	1.88928	0.350831	0.175415	+	0.984495i	$0.443873\pi$
$30$	0	0
$31$	−7.05670	−1.26742	−0.633710	−	0.773571i	$-0.718469\pi$
$31$	−7.05670	−1.26742	−0.633710	+	0.773571i	$0.718469\pi$
$32$	0	0
$33$	12.2633	2.13477
$34$	0	0
$35$	−2.50649	−0.423674
$36$	0	0
$37$	5.25942	0.864643	0.432321	−	0.901720i	$-0.357695\pi$
$37$	5.25942	0.864643	0.432321	+	0.901720i	$0.357695\pi$
$38$	0	0
$39$	3.29950	0.528343
$40$	0	0
$41$	−9.78021	−1.52741	−0.763706	−	0.645564i	$-0.776622\pi$
$41$	−9.78021	−1.52741	−0.763706	+	0.645564i	$0.776622\pi$
$42$	0	0
$43$	3.49716	0.533312	0.266656	−	0.963792i	$-0.414081\pi$
$43$	3.49716	0.533312	0.266656	+	0.963792i	$0.414081\pi$
$44$	0	0
$45$	17.4002	2.59387
$46$	0	0
$47$	−6.69256	−0.976211	−0.488105	−	0.872785i	$-0.662312\pi$
$47$	−6.69256	−0.976211	−0.488105	+	0.872785i	$0.662312\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	3.15311	0.441523
$52$	0	0
$53$	−8.97011	−1.23214	−0.616069	−	0.787692i	$-0.711276\pi$
$53$	−8.97011	−1.23214	−0.616069	+	0.787692i	$0.711276\pi$
$54$	0	0
$55$	9.74844	1.31448
$56$	0	0
$57$	6.09120	0.806800
$58$	0	0
$59$	12.5094	1.62859	0.814294	−	0.580453i	$-0.197124\pi$
$59$	12.5094	1.62859	0.814294	+	0.580453i	$0.197124\pi$
$60$	0	0
$61$	0.0384872	0.00492779	0.00246389	−	0.999997i	$-0.499216\pi$
$61$	0.0384872	0.00492779	0.00246389	+	0.999997i	$0.499216\pi$
$62$	0	0
$63$	−6.94208	−0.874620
$64$	0	0
$65$	2.62286	0.325326
$66$	0	0
$67$	15.3398	1.87405	0.937024	−	0.349264i	$-0.113568\pi$
$67$	15.3398	1.87405	0.937024	+	0.349264i	$0.113568\pi$
$68$	0	0
$69$	0.450017	0.0541757
$70$	0	0
$71$	−6.00461	−0.712616	−0.356308	−	0.934368i	$-0.615965\pi$
$71$	−6.00461	−0.712616	−0.356308	+	0.934368i	$0.615965\pi$
$72$	0	0
$73$	−11.1525	−1.30530	−0.652651	−	0.757659i	$-0.726343\pi$
$73$	−11.1525	−1.30530	−0.652651	+	0.757659i	$0.726343\pi$
$74$	0	0
$75$	4.04381	0.466939
$76$	0	0
$77$	−3.88928	−0.443225
$78$	0	0
$79$	7.89836	0.888635	0.444317	−	0.895869i	$-0.353446\pi$
$79$	7.89836	0.888635	0.444317	+	0.895869i	$0.353446\pi$
$80$	0	0
$81$	18.3662	2.04069
$82$	0	0
$83$	−4.76075	−0.522561	−0.261280	−	0.965263i	$-0.584145\pi$
$83$	−4.76075	−0.522561	−0.261280	+	0.965263i	$0.584145\pi$
$84$	0	0
$85$	2.50649	0.271867
$86$	0	0
$87$	5.95710	0.638668
$88$	0	0
$89$	12.3409	1.30814	0.654069	−	0.756435i	$-0.273061\pi$
$89$	12.3409	1.30814	0.654069	+	0.756435i	$0.273061\pi$
$90$	0	0
$91$	−1.04643	−0.109696
$92$	0	0
$93$	−22.2505	−2.30727
$94$	0	0
$95$	4.84206	0.496785
$96$	0	0
$97$	−15.0908	−1.53224	−0.766118	−	0.642700i	$-0.777814\pi$
$97$	−15.0908	−1.53224	−0.766118	+	0.642700i	$0.777814\pi$
$98$	0	0
$99$	26.9997	2.71357
$100$	0	0
$101$	4.50481	0.448245	0.224123	−	0.974561i	$-0.428048\pi$
$101$	4.50481	0.448245	0.224123	+	0.974561i	$0.428048\pi$
$102$	0	0
$103$	−11.9483	−1.17730	−0.588651	−	0.808387i	$-0.700341\pi$
$103$	−11.9483	−1.17730	−0.588651	+	0.808387i	$0.700341\pi$
$104$	0	0
$105$	−7.90322	−0.771276
$106$	0	0
$107$	−12.4796	−1.20645	−0.603226	−	0.797571i	$-0.706118\pi$
$107$	−12.4796	−1.20645	−0.603226	+	0.797571i	$0.706118\pi$
$108$	0	0
$109$	7.57967	0.726001	0.363000	−	0.931789i	$-0.381752\pi$
$109$	7.57967	0.726001	0.363000	+	0.931789i	$0.381752\pi$
$110$	0	0
$111$	16.5835	1.57404
$112$	0	0
$113$	−6.88548	−0.647731	−0.323866	−	0.946103i	$-0.604983\pi$
$113$	−6.88548	−0.647731	−0.323866	+	0.946103i	$0.604983\pi$
$114$	0	0
$115$	0.357730	0.0333585
$116$	0	0
$117$	7.26440	0.671594
$118$	0	0
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	4.12651	0.375137
$122$	0	0
$123$	−30.8381	−2.78057
$124$	0	0
$125$	−9.31791	−0.833419
$126$	0	0
$127$	6.43492	0.571007	0.285503	−	0.958378i	$-0.407839\pi$
$127$	6.43492	0.571007	0.285503	+	0.958378i	$0.407839\pi$
$128$	0	0
$129$	11.0269	0.970866
$130$	0	0
$131$	14.2042	1.24103	0.620513	−	0.784196i	$-0.286924\pi$
$131$	14.2042	1.24103	0.620513	+	0.784196i	$0.286924\pi$
$132$	0	0
$133$	−1.93181	−0.167509
$134$	0	0
$135$	31.1551	2.68141
$136$	0	0
$137$	7.58114	0.647700	0.323850	−	0.946108i	$-0.395023\pi$
$137$	7.58114	0.647700	0.323850	+	0.946108i	$0.395023\pi$
$138$	0	0
$139$	−5.14031	−0.435995	−0.217998	−	0.975949i	$-0.569952\pi$
$139$	−5.14031	−0.435995	−0.217998	+	0.975949i	$0.569952\pi$
$140$	0	0
$141$	−21.1024	−1.77714
$142$	0	0
$143$	4.06986	0.340339
$144$	0	0
$145$	4.73546	0.393259
$146$	0	0
$147$	3.15311	0.260064
$148$	0	0
$149$	14.3140	1.17265	0.586323	−	0.810077i	$-0.300575\pi$
$149$	14.3140	1.17265	0.586323	+	0.810077i	$0.300575\pi$
$150$	0	0
$151$	2.89831	0.235861	0.117930	−	0.993022i	$-0.462374\pi$
$151$	2.89831	0.235861	0.117930	+	0.993022i	$0.462374\pi$
$152$	0	0
$153$	6.94208	0.561234
$154$	0	0
$155$	−17.6875	−1.42070
$156$	0	0
$157$	−10.4894	−0.837141	−0.418571	−	0.908184i	$-0.637469\pi$
$157$	−10.4894	−0.837141	−0.418571	+	0.908184i	$0.637469\pi$
$158$	0	0
$159$	−28.2837	−2.24304
$160$	0	0
$161$	−0.142722	−0.0112481
$162$	0	0
$163$	−21.9914	−1.72250	−0.861250	−	0.508182i	$-0.830318\pi$
$163$	−21.9914	−1.72250	−0.861250	+	0.508182i	$0.830318\pi$
$164$	0	0
$165$	30.7379	2.39294
$166$	0	0
$167$	−8.13306	−0.629355	−0.314678	−	0.949199i	$-0.601896\pi$
$167$	−8.13306	−0.629355	−0.314678	+	0.949199i	$0.601896\pi$
$168$	0	0
$169$	−11.9050	−0.915768
$170$	0	0
$171$	13.4108	1.02555
$172$	0	0
$173$	−23.2800	−1.76994	−0.884972	−	0.465645i	$-0.845822\pi$
$173$	−23.2800	−1.76994	−0.884972	+	0.465645i	$0.845822\pi$
$174$	0	0
$175$	−1.28248	−0.0969467
$176$	0	0
$177$	39.4435	2.96476
$178$	0	0
$179$	11.6512	0.870854	0.435427	−	0.900224i	$-0.356597\pi$
$179$	11.6512	0.870854	0.435427	+	0.900224i	$0.356597\pi$
$180$	0	0
$181$	−0.714947	−0.0531416	−0.0265708	−	0.999647i	$-0.508459\pi$
$181$	−0.714947	−0.0531416	−0.0265708	+	0.999647i	$0.508459\pi$
$182$	0	0
$183$	0.121354	0.00897077
$184$	0	0
$185$	13.1827	0.969209
$186$	0	0
$187$	3.88928	0.284412
$188$	0	0
$189$	−12.4298	−0.904134
$190$	0	0
$191$	−5.98251	−0.432879	−0.216440	−	0.976296i	$-0.569444\pi$
$191$	−5.98251	−0.432879	−0.216440	+	0.976296i	$0.569444\pi$
$192$	0	0
$193$	7.81422	0.562480	0.281240	−	0.959637i	$-0.409254\pi$
$193$	7.81422	0.562480	0.281240	+	0.959637i	$0.409254\pi$
$194$	0	0
$195$	8.27017	0.592239
$196$	0	0
$197$	8.83317	0.629338	0.314669	−	0.949202i	$-0.398107\pi$
$197$	8.83317	0.629338	0.314669	+	0.949202i	$0.398107\pi$
$198$	0	0
$199$	−17.8533	−1.26559	−0.632795	−	0.774320i	$-0.718092\pi$
$199$	−17.8533	−1.26559	−0.632795	+	0.774320i	$0.718092\pi$
$200$	0	0
$201$	48.3679	3.41161
$202$	0	0
$203$	−1.88928	−0.132602
$204$	0	0
$205$	−24.5140	−1.71213
$206$	0	0
$207$	0.990786	0.0688644
$208$	0	0
$209$	7.51335	0.519710
$210$	0	0
$211$	9.62441	0.662572	0.331286	−	0.943530i	$-0.392518\pi$
$211$	9.62441	0.662572	0.331286	+	0.943530i	$0.392518\pi$
$212$	0	0
$213$	−18.9332	−1.29728
$214$	0	0
$215$	8.76559	0.597808
$216$	0	0
$217$	7.05670	0.479040
$218$	0	0
$219$	−35.1650	−2.37623
$220$	0	0
$221$	1.04643	0.0703905
$222$	0	0
$223$	−9.20944	−0.616710	−0.308355	−	0.951271i	$-0.599778\pi$
$223$	−9.20944	−0.616710	−0.308355	+	0.951271i	$0.599778\pi$
$224$	0	0
$225$	8.90310	0.593540
$226$	0	0
$227$	−15.3328	−1.01768	−0.508838	−	0.860862i	$-0.669925\pi$
$227$	−15.3328	−1.01768	−0.508838	+	0.860862i	$0.669925\pi$
$228$	0	0
$229$	−15.5793	−1.02951	−0.514755	−	0.857338i	$-0.672117\pi$
$229$	−15.5793	−1.02951	−0.514755	+	0.857338i	$0.672117\pi$
$230$	0	0
$231$	−12.2633	−0.806867
$232$	0	0
$233$	23.4896	1.53886	0.769429	−	0.638732i	$-0.220541\pi$
$233$	23.4896	1.53886	0.769429	+	0.638732i	$0.220541\pi$
$234$	0	0
$235$	−16.7748	−1.09427
$236$	0	0
$237$	24.9044	1.61771
$238$	0	0
$239$	5.51582	0.356788	0.178394	−	0.983959i	$-0.442910\pi$
$239$	5.51582	0.356788	0.178394	+	0.983959i	$0.442910\pi$
$240$	0	0
$241$	−17.4333	−1.12298	−0.561489	−	0.827484i	$-0.689771\pi$
$241$	−17.4333	−1.12298	−0.561489	+	0.827484i	$0.689771\pi$
$242$	0	0
$243$	20.6213	1.32285
$244$	0	0
$245$	2.50649	0.160134
$246$	0	0
$247$	2.02150	0.128625
$248$	0	0
$249$	−15.0112	−0.951294
$250$	0	0
$251$	−11.3226	−0.714673	−0.357337	−	0.933976i	$-0.616315\pi$
$251$	−11.3226	−0.714673	−0.357337	+	0.933976i	$0.616315\pi$
$252$	0	0
$253$	0.555085	0.0348979
$254$	0	0
$255$	7.90322	0.494919
$256$	0	0
$257$	−16.4462	−1.02589	−0.512944	−	0.858422i	$-0.671445\pi$
$257$	−16.4462	−1.02589	−0.512944	+	0.858422i	$0.671445\pi$
$258$	0	0
$259$	−5.25942	−0.326804
$260$	0	0
$261$	13.1155	0.811831
$262$	0	0
$263$	21.8196	1.34545	0.672726	−	0.739892i	$-0.265123\pi$
$263$	21.8196	1.34545	0.672726	+	0.739892i	$0.265123\pi$
$264$	0	0
$265$	−22.4835	−1.38115
$266$	0	0
$267$	38.9123	2.38140
$268$	0	0
$269$	5.39919	0.329195	0.164597	−	0.986361i	$-0.447368\pi$
$269$	5.39919	0.329195	0.164597	+	0.986361i	$0.447368\pi$
$270$	0	0
$271$	27.5343	1.67259	0.836295	−	0.548280i	$-0.184717\pi$
$271$	27.5343	1.67259	0.836295	+	0.548280i	$0.184717\pi$
$272$	0	0
$273$	−3.29950	−0.199695
$274$	0	0
$275$	4.98794	0.300784
$276$	0	0
$277$	1.53967	0.0925101	0.0462550	−	0.998930i	$-0.485271\pi$
$277$	1.53967	0.0925101	0.0462550	+	0.998930i	$0.485271\pi$
$278$	0	0
$279$	−48.9881	−2.93284
$280$	0	0
$281$	16.2069	0.966825	0.483412	−	0.875393i	$-0.339397\pi$
$281$	16.2069	0.966825	0.483412	+	0.875393i	$0.339397\pi$
$282$	0	0
$283$	15.3916	0.914933	0.457467	−	0.889227i	$-0.348757\pi$
$283$	15.3916	0.914933	0.457467	+	0.889227i	$0.348757\pi$
$284$	0	0
$285$	15.2675	0.904370
$286$	0	0
$287$	9.78021	0.577308
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−47.5828	−2.78935
$292$	0	0
$293$	8.96532	0.523760	0.261880	−	0.965100i	$-0.415658\pi$
$293$	8.96532	0.523760	0.261880	+	0.965100i	$0.415658\pi$
$294$	0	0
$295$	31.3547	1.82554
$296$	0	0
$297$	48.3430	2.80514
$298$	0	0
$299$	0.149348	0.00863703
$300$	0	0
$301$	−3.49716	−0.201573
$302$	0	0
$303$	14.2041	0.816007
$304$	0	0
$305$	0.0964678	0.00552373
$306$	0	0
$307$	14.9101	0.850962	0.425481	−	0.904967i	$-0.360105\pi$
$307$	14.9101	0.850962	0.425481	+	0.904967i	$0.360105\pi$
$308$	0	0
$309$	−37.6743	−2.14322
$310$	0	0
$311$	24.9598	1.41534	0.707671	−	0.706542i	$-0.249746\pi$
$311$	24.9598	1.41534	0.707671	+	0.706542i	$0.249746\pi$
$312$	0	0
$313$	−6.44815	−0.364471	−0.182236	−	0.983255i	$-0.558333\pi$
$313$	−6.44815	−0.364471	−0.182236	+	0.983255i	$0.558333\pi$
$314$	0	0
$315$	−17.4002	−0.980392
$316$	0	0
$317$	−23.0881	−1.29676	−0.648378	−	0.761318i	$-0.724553\pi$
$317$	−23.0881	−1.29676	−0.648378	+	0.761318i	$0.724553\pi$
$318$	0	0
$319$	7.34794	0.411406
$320$	0	0
$321$	−39.3496	−2.19628
$322$	0	0
$323$	1.93181	0.107489
$324$	0	0
$325$	1.34203	0.0744424
$326$	0	0
$327$	23.8995	1.32165
$328$	0	0
$329$	6.69256	0.368973
$330$	0	0
$331$	6.29889	0.346218	0.173109	−	0.984903i	$-0.444619\pi$
$331$	6.29889	0.346218	0.173109	+	0.984903i	$0.444619\pi$
$332$	0	0
$333$	36.5113	2.00081
$334$	0	0
$335$	38.4489	2.10069
$336$	0	0
$337$	5.85647	0.319022	0.159511	−	0.987196i	$-0.449008\pi$
$337$	5.85647	0.319022	0.159511	+	0.987196i	$0.449008\pi$
$338$	0	0
$339$	−21.7106	−1.17916
$340$	0	0
$341$	−27.4455	−1.48626
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	1.12796	0.0607274
$346$	0	0
$347$	−20.3437	−1.09211	−0.546053	−	0.837751i	$-0.683870\pi$
$347$	−20.3437	−1.09211	−0.546053	+	0.837751i	$0.683870\pi$
$348$	0	0
$349$	−15.4883	−0.829069	−0.414534	−	0.910034i	$-0.636056\pi$
$349$	−15.4883	−0.829069	−0.414534	+	0.910034i	$0.636056\pi$
$350$	0	0
$351$	13.0069	0.694257
$352$	0	0
$353$	12.7359	0.677863	0.338931	−	0.940811i	$-0.389935\pi$
$353$	12.7359	0.677863	0.338931	+	0.940811i	$0.389935\pi$
$354$	0	0
$355$	−15.0505	−0.798797
$356$	0	0
$357$	−3.15311	−0.166880
$358$	0	0
$359$	30.4456	1.60686	0.803428	−	0.595402i	$-0.203007\pi$
$359$	30.4456	1.60686	0.803428	+	0.595402i	$0.203007\pi$
$360$	0	0
$361$	−15.2681	−0.803585
$362$	0	0
$363$	13.0113	0.682917
$364$	0	0
$365$	−27.9536	−1.46316
$366$	0	0
$367$	−20.7537	−1.08334	−0.541668	−	0.840593i	$-0.682207\pi$
$367$	−20.7537	−1.08334	−0.541668	+	0.840593i	$0.682207\pi$
$368$	0	0
$369$	−67.8950	−3.53447
$370$	0	0
$371$	8.97011	0.465705
$372$	0	0
$373$	26.4938	1.37180	0.685898	−	0.727698i	$-0.259410\pi$
$373$	26.4938	1.37180	0.685898	+	0.727698i	$0.259410\pi$
$374$	0	0
$375$	−29.3804	−1.51720
$376$	0	0
$377$	1.97700	0.101821
$378$	0	0
$379$	2.43421	0.125037	0.0625186	−	0.998044i	$-0.480087\pi$
$379$	2.43421	0.125037	0.0625186	+	0.998044i	$0.480087\pi$
$380$	0	0
$381$	20.2900	1.03949
$382$	0	0
$383$	26.4921	1.35368	0.676842	−	0.736128i	$-0.263348\pi$
$383$	26.4921	1.35368	0.676842	+	0.736128i	$0.263348\pi$
$384$	0	0
$385$	−9.74844	−0.496826
$386$	0	0
$387$	24.2776	1.23410
$388$	0	0
$389$	37.9546	1.92438	0.962188	−	0.272386i	$-0.0878128\pi$
$389$	37.9546	1.92438	0.962188	+	0.272386i	$0.0878128\pi$
$390$	0	0
$391$	0.142722	0.00721775
$392$	0	0
$393$	44.7874	2.25922
$394$	0	0
$395$	19.7971	0.996102
$396$	0	0
$397$	8.44743	0.423965	0.211982	−	0.977274i	$-0.432008\pi$
$397$	8.44743	0.423965	0.211982	+	0.977274i	$0.432008\pi$
$398$	0	0
$399$	−6.09120	−0.304942
$400$	0	0
$401$	−28.2308	−1.40978	−0.704889	−	0.709317i	$-0.749003\pi$
$401$	−28.2308	−1.40978	−0.704889	+	0.709317i	$0.749003\pi$
$402$	0	0
$403$	−7.38434	−0.367840
$404$	0	0
$405$	46.0347	2.28748
$406$	0	0
$407$	20.4553	1.01393
$408$	0	0
$409$	−24.2991	−1.20151	−0.600757	−	0.799432i	$-0.705134\pi$
$409$	−24.2991	−1.20151	−0.600757	+	0.799432i	$0.705134\pi$
$410$	0	0
$411$	23.9041	1.17910
$412$	0	0
$413$	−12.5094	−0.615548
$414$	0	0
$415$	−11.9328	−0.585757
$416$	0	0
$417$	−16.2079	−0.793706
$418$	0	0
$419$	8.24312	0.402703	0.201351	−	0.979519i	$-0.435467\pi$
$419$	8.24312	0.402703	0.201351	+	0.979519i	$0.435467\pi$
$420$	0	0
$421$	26.4577	1.28947	0.644734	−	0.764407i	$-0.276968\pi$
$421$	26.4577	1.28947	0.644734	+	0.764407i	$0.276968\pi$
$422$	0	0
$423$	−46.4603	−2.25898
$424$	0	0
$425$	1.28248	0.0622096
$426$	0	0
$427$	−0.0384872	−0.00186253
$428$	0	0
$429$	12.8327	0.619568
$430$	0	0
$431$	−23.1880	−1.11693	−0.558463	−	0.829529i	$-0.688609\pi$
$431$	−23.1880	−1.11693	−0.558463	+	0.829529i	$0.688609\pi$
$432$	0	0
$433$	−12.4159	−0.596669	−0.298335	−	0.954461i	$-0.596431\pi$
$433$	−12.4159	−0.596669	−0.298335	+	0.954461i	$0.596431\pi$
$434$	0	0
$435$	14.9314	0.715906
$436$	0	0
$437$	0.275711	0.0131891
$438$	0	0
$439$	32.4943	1.55087	0.775434	−	0.631429i	$-0.217531\pi$
$439$	32.4943	1.55087	0.775434	+	0.631429i	$0.217531\pi$
$440$	0	0
$441$	6.94208	0.330575
$442$	0	0
$443$	30.7152	1.45932	0.729661	−	0.683809i	$-0.239678\pi$
$443$	30.7152	1.45932	0.729661	+	0.683809i	$0.239678\pi$
$444$	0	0
$445$	30.9324	1.46634
$446$	0	0
$447$	45.1335	2.13474
$448$	0	0
$449$	−16.4958	−0.778484	−0.389242	−	0.921136i	$-0.627263\pi$
$449$	−16.4958	−0.778484	−0.389242	+	0.921136i	$0.627263\pi$
$450$	0	0
$451$	−38.0380	−1.79114
$452$	0	0
$453$	9.13867	0.429372
$454$	0	0
$455$	−2.62286	−0.122962
$456$	0	0
$457$	−41.8672	−1.95846	−0.979232	−	0.202742i	$-0.935015\pi$
$457$	−41.8672	−1.95846	−0.979232	+	0.202742i	$0.935015\pi$
$458$	0	0
$459$	12.4298	0.580173
$460$	0	0
$461$	−27.1356	−1.26383	−0.631915	−	0.775038i	$-0.717731\pi$
$461$	−27.1356	−1.26383	−0.631915	+	0.775038i	$0.717731\pi$
$462$	0	0
$463$	15.7087	0.730047	0.365023	−	0.930998i	$-0.381061\pi$
$463$	15.7087	0.730047	0.365023	+	0.930998i	$0.381061\pi$
$464$	0	0
$465$	−55.7707	−2.58630
$466$	0	0
$467$	29.8231	1.38005	0.690024	−	0.723786i	$-0.257600\pi$
$467$	29.8231	1.38005	0.690024	+	0.723786i	$0.257600\pi$
$468$	0	0
$469$	−15.3398	−0.708324
$470$	0	0
$471$	−33.0740	−1.52397
$472$	0	0
$473$	13.6014	0.625395
$474$	0	0
$475$	2.47752	0.113676
$476$	0	0
$477$	−62.2712	−2.85120
$478$	0	0
$479$	2.58755	0.118228	0.0591142	−	0.998251i	$-0.481172\pi$
$479$	2.58755	0.118228	0.0591142	+	0.998251i	$0.481172\pi$
$480$	0	0
$481$	5.50361	0.250943
$482$	0	0
$483$	−0.450017	−0.0204765
$484$	0	0
$485$	−37.8249	−1.71754
$486$	0	0
$487$	35.7733	1.62104	0.810521	−	0.585710i	$-0.199184\pi$
$487$	35.7733	1.62104	0.810521	+	0.585710i	$0.199184\pi$
$488$	0	0
$489$	−69.3412	−3.13572
$490$	0	0
$491$	19.2422	0.868388	0.434194	−	0.900819i	$-0.357033\pi$
$491$	19.2422	0.868388	0.434194	+	0.900819i	$0.357033\pi$
$492$	0	0
$493$	1.88928	0.0850889
$494$	0	0
$495$	67.6744	3.04174
$496$	0	0
$497$	6.00461	0.269344
$498$	0	0
$499$	−12.2663	−0.549114	−0.274557	−	0.961571i	$-0.588531\pi$
$499$	−12.2663	−0.549114	−0.274557	+	0.961571i	$0.588531\pi$
$500$	0	0
$501$	−25.6444	−1.14571
$502$	0	0
$503$	−40.8678	−1.82221	−0.911103	−	0.412179i	$-0.864768\pi$
$503$	−40.8678	−1.82221	−0.911103	+	0.412179i	$0.864768\pi$
$504$	0	0
$505$	11.2913	0.502454
$506$	0	0
$507$	−37.5377	−1.66711
$508$	0	0
$509$	34.8840	1.54621	0.773103	−	0.634281i	$-0.218704\pi$
$509$	34.8840	1.54621	0.773103	+	0.634281i	$0.218704\pi$
$510$	0	0
$511$	11.1525	0.493358
$512$	0	0
$513$	24.0120	1.06016
$514$	0	0
$515$	−29.9483	−1.31968
$516$	0	0
$517$	−26.0293	−1.14477
$518$	0	0
$519$	−73.4042	−3.22209
$520$	0	0
$521$	−31.0962	−1.36235	−0.681176	−	0.732120i	$-0.738531\pi$
$521$	−31.0962	−1.36235	−0.681176	+	0.732120i	$0.738531\pi$
$522$	0	0
$523$	14.1149	0.617202	0.308601	−	0.951192i	$-0.400139\pi$
$523$	14.1149	0.617202	0.308601	+	0.951192i	$0.400139\pi$
$524$	0	0
$525$	−4.04381	−0.176486
$526$	0	0
$527$	−7.05670	−0.307395
$528$	0	0
$529$	−22.9796	−0.999114
$530$	0	0
$531$	86.8414	3.76860
$532$	0	0
$533$	−10.2343	−0.443297
$534$	0	0
$535$	−31.2800	−1.35235
$536$	0	0
$537$	36.7376	1.58534
$538$	0	0
$539$	3.88928	0.167523
$540$	0	0
$541$	9.33209	0.401218	0.200609	−	0.979671i	$-0.435708\pi$
$541$	9.33209	0.401218	0.200609	+	0.979671i	$0.435708\pi$
$542$	0	0
$543$	−2.25431	−0.0967415
$544$	0	0
$545$	18.9984	0.813800
$546$	0	0
$547$	−23.8226	−1.01858	−0.509290	−	0.860595i	$-0.670092\pi$
$547$	−23.8226	−1.01858	−0.509290	+	0.860595i	$0.670092\pi$
$548$	0	0
$549$	0.267181	0.0114030
$550$	0	0
$551$	3.64973	0.155484
$552$	0	0
$553$	−7.89836	−0.335872
$554$	0	0
$555$	41.5663	1.76439
$556$	0	0
$557$	−28.3775	−1.20239	−0.601197	−	0.799101i	$-0.705309\pi$
$557$	−28.3775	−1.20239	−0.601197	+	0.799101i	$0.705309\pi$
$558$	0	0
$559$	3.65953	0.154782
$560$	0	0
$561$	12.2633	0.517758
$562$	0	0
$563$	−16.7069	−0.704111	−0.352055	−	0.935979i	$-0.614517\pi$
$563$	−16.7069	−0.704111	−0.352055	+	0.935979i	$0.614517\pi$
$564$	0	0
$565$	−17.2584	−0.726065
$566$	0	0
$567$	−18.3662	−0.771309
$568$	0	0
$569$	21.1799	0.887910	0.443955	−	0.896049i	$-0.353575\pi$
$569$	21.1799	0.887910	0.443955	+	0.896049i	$0.353575\pi$
$570$	0	0
$571$	−41.2194	−1.72498	−0.862490	−	0.506074i	$-0.831096\pi$
$571$	−41.2194	−1.72498	−0.862490	+	0.506074i	$0.831096\pi$
$572$	0	0
$573$	−18.8635	−0.788034
$574$	0	0
$575$	0.183038	0.00763323
$576$	0	0
$577$	5.51830	0.229730	0.114865	−	0.993381i	$-0.463356\pi$
$577$	5.51830	0.229730	0.114865	+	0.993381i	$0.463356\pi$
$578$	0	0
$579$	24.6391	1.02397
$580$	0	0
$581$	4.76075	0.197509
$582$	0	0
$583$	−34.8873	−1.44488
$584$	0	0
$585$	18.2081	0.752813
$586$	0	0
$587$	−17.4770	−0.721351	−0.360676	−	0.932691i	$-0.617454\pi$
$587$	−17.4770	−0.721351	−0.360676	+	0.932691i	$0.617454\pi$
$588$	0	0
$589$	−13.6322	−0.561705
$590$	0	0
$591$	27.8519	1.14568
$592$	0	0
$593$	−7.57213	−0.310950	−0.155475	−	0.987840i	$-0.549691\pi$
$593$	−7.57213	−0.310950	−0.155475	+	0.987840i	$0.549691\pi$
$594$	0	0
$595$	−2.50649	−0.102756
$596$	0	0
$597$	−56.2935	−2.30394
$598$	0	0
$599$	−14.7502	−0.602679	−0.301339	−	0.953517i	$-0.597434\pi$
$599$	−14.7502	−0.602679	−0.301339	+	0.953517i	$0.597434\pi$
$600$	0	0
$601$	−27.4646	−1.12031	−0.560153	−	0.828389i	$-0.689258\pi$
$601$	−27.4646	−1.12031	−0.560153	+	0.828389i	$0.689258\pi$
$602$	0	0
$603$	106.490	4.33660
$604$	0	0
$605$	10.3430	0.420504
$606$	0	0
$607$	24.8819	1.00992	0.504962	−	0.863141i	$-0.331506\pi$
$607$	24.8819	1.00992	0.504962	+	0.863141i	$0.331506\pi$
$608$	0	0
$609$	−5.95710	−0.241394
$610$	0	0
$611$	−7.00330	−0.283323
$612$	0	0
$613$	3.72893	0.150610	0.0753050	−	0.997161i	$-0.476007\pi$
$613$	3.72893	0.150610	0.0753050	+	0.997161i	$0.476007\pi$
$614$	0	0
$615$	−77.2952	−3.11684
$616$	0	0
$617$	−11.8402	−0.476668	−0.238334	−	0.971183i	$-0.576601\pi$
$617$	−11.8402	−0.476668	−0.238334	+	0.971183i	$0.576601\pi$
$618$	0	0
$619$	16.0561	0.645349	0.322675	−	0.946510i	$-0.395418\pi$
$619$	16.0561	0.645349	0.322675	+	0.946510i	$0.395418\pi$
$620$	0	0
$621$	1.77400	0.0711882
$622$	0	0
$623$	−12.3409	−0.494430
$624$	0	0
$625$	−29.7677	−1.19071
$626$	0	0
$627$	23.6904	0.946104
$628$	0	0
$629$	5.25942	0.209707
$630$	0	0
$631$	10.7028	0.426073	0.213037	−	0.977044i	$-0.431665\pi$
$631$	10.7028	0.426073	0.213037	+	0.977044i	$0.431665\pi$
$632$	0	0
$633$	30.3468	1.20618
$634$	0	0
$635$	16.1290	0.640062
$636$	0	0
$637$	1.04643	0.0414610
$638$	0	0
$639$	−41.6845	−1.64901
$640$	0	0
$641$	−18.5349	−0.732086	−0.366043	−	0.930598i	$-0.619288\pi$
$641$	−18.5349	−0.732086	−0.366043	+	0.930598i	$0.619288\pi$
$642$	0	0
$643$	−33.7068	−1.32927	−0.664633	−	0.747170i	$-0.731412\pi$
$643$	−33.7068	−1.32927	−0.664633	+	0.747170i	$0.731412\pi$
$644$	0	0
$645$	27.6388	1.08828
$646$	0	0
$647$	22.1715	0.871650	0.435825	−	0.900031i	$-0.356457\pi$
$647$	22.1715	0.871650	0.435825	+	0.900031i	$0.356457\pi$
$648$	0	0
$649$	48.6527	1.90978
$650$	0	0
$651$	22.2505	0.872066
$652$	0	0
$653$	−39.5457	−1.54754	−0.773770	−	0.633466i	$-0.781632\pi$
$653$	−39.5457	−1.54754	−0.773770	+	0.633466i	$0.781632\pi$
$654$	0	0
$655$	35.6027	1.39111
$656$	0	0
$657$	−77.4216	−3.02050
$658$	0	0
$659$	−18.7411	−0.730048	−0.365024	−	0.930998i	$-0.618939\pi$
$659$	−18.7411	−0.730048	−0.365024	+	0.930998i	$0.618939\pi$
$660$	0	0
$661$	17.9532	0.698297	0.349149	−	0.937067i	$-0.386471\pi$
$661$	17.9532	0.698297	0.349149	+	0.937067i	$0.386471\pi$
$662$	0	0
$663$	3.29950	0.128142
$664$	0	0
$665$	−4.84206	−0.187767
$666$	0	0
$667$	0.269641	0.0104406
$668$	0	0
$669$	−29.0383	−1.12269
$670$	0	0
$671$	0.149688	0.00577863
$672$	0	0
$673$	30.6027	1.17965	0.589823	−	0.807532i	$-0.299197\pi$
$673$	30.6027	1.17965	0.589823	+	0.807532i	$0.299197\pi$
$674$	0	0
$675$	15.9410	0.613570
$676$	0	0
$677$	−14.8064	−0.569055	−0.284528	−	0.958668i	$-0.591837\pi$
$677$	−14.8064	−0.569055	−0.284528	+	0.958668i	$0.591837\pi$
$678$	0	0
$679$	15.0908	0.579131
$680$	0	0
$681$	−48.3461	−1.85263
$682$	0	0
$683$	−13.4119	−0.513191	−0.256596	−	0.966519i	$-0.582601\pi$
$683$	−13.4119	−0.513191	−0.256596	+	0.966519i	$0.582601\pi$
$684$	0	0
$685$	19.0020	0.726030
$686$	0	0
$687$	−49.1232	−1.87417
$688$	0	0
$689$	−9.38658	−0.357600
$690$	0	0
$691$	21.7981	0.829238	0.414619	−	0.909995i	$-0.363915\pi$
$691$	21.7981	0.829238	0.414619	+	0.909995i	$0.363915\pi$
$692$	0	0
$693$	−26.9997	−1.02563
$694$	0	0
$695$	−12.8841	−0.488722
$696$	0	0
$697$	−9.78021	−0.370452
$698$	0	0
$699$	74.0653	2.80141
$700$	0	0
$701$	−35.6855	−1.34782	−0.673912	−	0.738811i	$-0.735387\pi$
$701$	−35.6855	−1.34782	−0.673912	+	0.738811i	$0.735387\pi$
$702$	0	0
$703$	10.1602	0.383199
$704$	0	0
$705$	−52.8928	−1.99206
$706$	0	0
$707$	−4.50481	−0.169421
$708$	0	0
$709$	23.0379	0.865208	0.432604	−	0.901584i	$-0.357595\pi$
$709$	23.0379	0.865208	0.432604	+	0.901584i	$0.357595\pi$
$710$	0	0
$711$	54.8310	2.05632
$712$	0	0
$713$	−1.00714	−0.0377179
$714$	0	0
$715$	10.2011	0.381498
$716$	0	0
$717$	17.3920	0.649514
$718$	0	0
$719$	4.96957	0.185334	0.0926668	−	0.995697i	$-0.470461\pi$
$719$	4.96957	0.185334	0.0926668	+	0.995697i	$0.470461\pi$
$720$	0	0
$721$	11.9483	0.444979
$722$	0	0
$723$	−54.9691	−2.04432
$724$	0	0
$725$	2.42297	0.0899869
$726$	0	0
$727$	51.3205	1.90337	0.951687	−	0.307071i	$-0.0993489\pi$
$727$	51.3205	1.90337	0.951687	+	0.307071i	$0.0993489\pi$
$728$	0	0
$729$	9.92236	0.367495
$730$	0	0
$731$	3.49716	0.129347
$732$	0	0
$733$	−24.1417	−0.891694	−0.445847	−	0.895109i	$-0.647097\pi$
$733$	−24.1417	−0.891694	−0.445847	+	0.895109i	$0.647097\pi$
$734$	0	0
$735$	7.90322	0.291515
$736$	0	0
$737$	59.6606	2.19763
$738$	0	0
$739$	21.9638	0.807951	0.403975	−	0.914770i	$-0.367628\pi$
$739$	21.9638	0.807951	0.403975	+	0.914770i	$0.367628\pi$
$740$	0	0
$741$	6.37401	0.234155
$742$	0	0
$743$	−16.9129	−0.620474	−0.310237	−	0.950659i	$-0.600408\pi$
$743$	−16.9129	−0.620474	−0.310237	+	0.950659i	$0.600408\pi$
$744$	0	0
$745$	35.8778	1.31446
$746$	0	0
$747$	−33.0495	−1.20922
$748$	0	0
$749$	12.4796	0.455996
$750$	0	0
$751$	−9.46138	−0.345251	−0.172625	−	0.984988i	$-0.555225\pi$
$751$	−9.46138	−0.345251	−0.172625	+	0.984988i	$0.555225\pi$
$752$	0	0
$753$	−35.7012	−1.30102
$754$	0	0
$755$	7.26457	0.264385
$756$	0	0
$757$	34.5109	1.25432	0.627159	−	0.778891i	$-0.284218\pi$
$757$	34.5109	1.25432	0.627159	+	0.778891i	$0.284218\pi$
$758$	0	0
$759$	1.75024	0.0635298
$760$	0	0
$761$	45.4235	1.64660	0.823300	−	0.567607i	$-0.192131\pi$
$761$	45.4235	1.64660	0.823300	+	0.567607i	$0.192131\pi$
$762$	0	0
$763$	−7.57967	−0.274403
$764$	0	0
$765$	17.4002	0.629107
$766$	0	0
$767$	13.0902	0.472661
$768$	0	0
$769$	−37.7862	−1.36261	−0.681303	−	0.732002i	$-0.738586\pi$
$769$	−37.7862	−1.36261	−0.681303	+	0.732002i	$0.738586\pi$
$770$	0	0
$771$	−51.8567	−1.86757
$772$	0	0
$773$	11.6222	0.418022	0.209011	−	0.977913i	$-0.432976\pi$
$773$	11.6222	0.418022	0.209011	+	0.977913i	$0.432976\pi$
$774$	0	0
$775$	−9.05010	−0.325089
$776$	0	0
$777$	−16.5835	−0.594930
$778$	0	0
$779$	−18.8935	−0.676931
$780$	0	0
$781$	−23.3536	−0.835658
$782$	0	0
$783$	23.4834	0.839227
$784$	0	0
$785$	−26.2914	−0.938382
$786$	0	0
$787$	7.54811	0.269061	0.134530	−	0.990909i	$-0.457047\pi$
$787$	7.54811	0.269061	0.134530	+	0.990909i	$0.457047\pi$
$788$	0	0
$789$	68.7994	2.44932
$790$	0	0
$791$	6.88548	0.244819
$792$	0	0
$793$	0.0402742	0.00143018
$794$	0	0
$795$	−70.8928	−2.51431
$796$	0	0
$797$	−30.9946	−1.09788	−0.548942	−	0.835860i	$-0.684969\pi$
$797$	−30.9946	−1.09788	−0.548942	+	0.835860i	$0.684969\pi$
$798$	0	0
$799$	−6.69256	−0.236766
$800$	0	0
$801$	85.6718	3.02707
$802$	0	0
$803$	−43.3752	−1.53068
$804$	0	0
$805$	−0.357730	−0.0126083
$806$	0	0
$807$	17.0242	0.599281
$808$	0	0
$809$	−21.0795	−0.741117	−0.370558	−	0.928809i	$-0.620834\pi$
$809$	−21.0795	−0.741117	−0.370558	+	0.928809i	$0.620834\pi$
$810$	0	0
$811$	39.8902	1.40074	0.700368	−	0.713782i	$-0.253019\pi$
$811$	39.8902	1.40074	0.700368	+	0.713782i	$0.253019\pi$
$812$	0	0
$813$	86.8186	3.04486
$814$	0	0
$815$	−55.1212	−1.93081
$816$	0	0
$817$	6.75585	0.236357
$818$	0	0
$819$	−7.26440	−0.253839
$820$	0	0
$821$	28.6531	0.999999	0.500000	−	0.866026i	$-0.333333\pi$
$821$	28.6531	0.999999	0.500000	+	0.866026i	$0.333333\pi$
$822$	0	0
$823$	−54.5692	−1.90216	−0.951082	−	0.308939i	$-0.900026\pi$
$823$	−54.5692	−1.90216	−0.951082	+	0.308939i	$0.900026\pi$
$824$	0	0
$825$	15.7275	0.547561
$826$	0	0
$827$	10.8720	0.378058	0.189029	−	0.981971i	$-0.439466\pi$
$827$	10.8720	0.378058	0.189029	+	0.981971i	$0.439466\pi$
$828$	0	0
$829$	−45.8596	−1.59277	−0.796386	−	0.604789i	$-0.793257\pi$
$829$	−45.8596	−1.59277	−0.796386	+	0.604789i	$0.793257\pi$
$830$	0	0
$831$	4.85476	0.168410
$832$	0	0
$833$	1.00000	0.0346479
$834$	0	0
$835$	−20.3854	−0.705467
$836$	0	0
$837$	−87.7133	−3.03181
$838$	0	0
$839$	−48.5642	−1.67662	−0.838311	−	0.545192i	$-0.816457\pi$
$839$	−48.5642	−1.67662	−0.838311	+	0.545192i	$0.816457\pi$
$840$	0	0
$841$	−25.4306	−0.876918
$842$	0	0
$843$	51.1022	1.76005
$844$	0	0
$845$	−29.8397	−1.02652
$846$	0	0
$847$	−4.12651	−0.141788
$848$	0	0
$849$	48.5312	1.66559
$850$	0	0
$851$	0.750633	0.0257314
$852$	0	0
$853$	34.0953	1.16740	0.583700	−	0.811969i	$-0.301604\pi$
$853$	34.0953	1.16740	0.583700	+	0.811969i	$0.301604\pi$
$854$	0	0
$855$	33.6140	1.14957
$856$	0	0
$857$	6.35176	0.216972	0.108486	−	0.994098i	$-0.465400\pi$
$857$	6.35176	0.216972	0.108486	+	0.994098i	$0.465400\pi$
$858$	0	0
$859$	−16.3941	−0.559359	−0.279679	−	0.960093i	$-0.590228\pi$
$859$	−16.3941	−0.559359	−0.279679	+	0.960093i	$0.590228\pi$
$860$	0	0
$861$	30.8381	1.05096
$862$	0	0
$863$	−27.3010	−0.929336	−0.464668	−	0.885485i	$-0.653826\pi$
$863$	−27.3010	−0.929336	−0.464668	+	0.885485i	$0.653826\pi$
$864$	0	0
$865$	−58.3510	−1.98399
$866$	0	0
$867$	3.15311	0.107085
$868$	0	0
$869$	30.7189	1.04207
$870$	0	0
$871$	16.0520	0.543900
$872$	0	0
$873$	−104.761	−3.54563
$874$	0	0
$875$	9.31791	0.315003
$876$	0	0
$877$	−2.30728	−0.0779113	−0.0389557	−	0.999241i	$-0.512403\pi$
$877$	−2.30728	−0.0779113	−0.0389557	+	0.999241i	$0.512403\pi$
$878$	0	0
$879$	28.2686	0.953477
$880$	0	0
$881$	3.89818	0.131333	0.0656666	−	0.997842i	$-0.479083\pi$
$881$	3.89818	0.131333	0.0656666	+	0.997842i	$0.479083\pi$
$882$	0	0
$883$	24.7932	0.834358	0.417179	−	0.908824i	$-0.363019\pi$
$883$	24.7932	0.834358	0.417179	+	0.908824i	$0.363019\pi$
$884$	0	0
$885$	98.8648	3.32330
$886$	0	0
$887$	57.2775	1.92319	0.961596	−	0.274470i	$-0.0885023\pi$
$887$	57.2775	1.92319	0.961596	+	0.274470i	$0.0885023\pi$
$888$	0	0
$889$	−6.43492	−0.215820
$890$	0	0
$891$	71.4314	2.39304
$892$	0	0
$893$	−12.9288	−0.432645
$894$	0	0
$895$	29.2037	0.976171
$896$	0	0
$897$	0.470911	0.0157233
$898$	0	0
$899$	−13.3321	−0.444650
$900$	0	0
$901$	−8.97011	−0.298838
$902$	0	0
$903$	−11.0269	−0.366953
$904$	0	0
$905$	−1.79201	−0.0595683
$906$	0	0
$907$	−12.4495	−0.413379	−0.206690	−	0.978407i	$-0.566269\pi$
$907$	−12.4495	−0.413379	−0.206690	+	0.978407i	$0.566269\pi$
$908$	0	0
$909$	31.2727	1.03725
$910$	0	0
$911$	40.0259	1.32612	0.663058	−	0.748568i	$-0.269258\pi$
$911$	40.0259	1.32612	0.663058	+	0.748568i	$0.269258\pi$
$912$	0	0
$913$	−18.5159	−0.612787
$914$	0	0
$915$	0.304173	0.0100557
$916$	0	0
$917$	−14.2042	−0.469064
$918$	0	0
$919$	−50.5744	−1.66830	−0.834149	−	0.551540i	$-0.814040\pi$
$919$	−50.5744	−1.66830	−0.834149	+	0.551540i	$0.814040\pi$
$920$	0	0
$921$	47.0130	1.54913
$922$	0	0
$923$	−6.28340	−0.206821
$924$	0	0
$925$	6.74512	0.221778
$926$	0	0
$927$	−82.9462	−2.72431
$928$	0	0
$929$	−8.12277	−0.266500	−0.133250	−	0.991082i	$-0.542541\pi$
$929$	−8.12277	−0.266500	−0.133250	+	0.991082i	$0.542541\pi$
$930$	0	0
$931$	1.93181	0.0633125
$932$	0	0
$933$	78.7010	2.57656
$934$	0	0
$935$	9.74844	0.318808
$936$	0	0
$937$	5.68123	0.185598	0.0927988	−	0.995685i	$-0.470419\pi$
$937$	5.68123	0.185598	0.0927988	+	0.995685i	$0.470419\pi$
$938$	0	0
$939$	−20.3317	−0.663500
$940$	0	0
$941$	37.6378	1.22696	0.613479	−	0.789711i	$-0.289769\pi$
$941$	37.6378	1.22696	0.613479	+	0.789711i	$0.289769\pi$
$942$	0	0
$943$	−1.39585	−0.0454551
$944$	0	0
$945$	−31.1551	−1.01348
$946$	0	0
$947$	−37.5030	−1.21868	−0.609342	−	0.792908i	$-0.708566\pi$
$947$	−37.5030	−1.21868	−0.609342	+	0.792908i	$0.708566\pi$
$948$	0	0
$949$	−11.6703	−0.378834
$950$	0	0
$951$	−72.7992	−2.36068
$952$	0	0
$953$	36.9025	1.19539	0.597694	−	0.801724i	$-0.296084\pi$
$953$	36.9025	1.19539	0.597694	+	0.801724i	$0.296084\pi$
$954$	0	0
$955$	−14.9951	−0.485230
$956$	0	0
$957$	23.1688	0.748942
$958$	0	0
$959$	−7.58114	−0.244808
$960$	0	0
$961$	18.7970	0.606354
$962$	0	0
$963$	−86.6346	−2.79176
$964$	0	0
$965$	19.5863	0.630504
$966$	0	0
$967$	11.9644	0.384750	0.192375	−	0.981321i	$-0.438381\pi$
$967$	11.9644	0.384750	0.192375	+	0.981321i	$0.438381\pi$
$968$	0	0
$969$	6.09120	0.195678
$970$	0	0
$971$	42.7951	1.37336	0.686680	−	0.726960i	$-0.259067\pi$
$971$	42.7951	1.37336	0.686680	+	0.726960i	$0.259067\pi$
$972$	0	0
$973$	5.14031	0.164791
$974$	0	0
$975$	4.23156	0.135518
$976$	0	0
$977$	−0.229023	−0.00732709	−0.00366354	−	0.999993i	$-0.501166\pi$
$977$	−0.229023	−0.00732709	−0.00366354	+	0.999993i	$0.501166\pi$
$978$	0	0
$979$	47.9974	1.53400
$980$	0	0
$981$	52.6187	1.67998
$982$	0	0
$983$	−3.48384	−0.111117	−0.0555586	−	0.998455i	$-0.517694\pi$
$983$	−3.48384	−0.111117	−0.0555586	+	0.998455i	$0.517694\pi$
$984$	0	0
$985$	22.1402	0.705447
$986$	0	0
$987$	21.1024	0.671696
$988$	0	0
$989$	0.499121	0.0158711
$990$	0	0
$991$	10.2005	0.324030	0.162015	−	0.986788i	$-0.448201\pi$
$991$	10.2005	0.324030	0.162015	+	0.986788i	$0.448201\pi$
$992$	0	0
$993$	19.8611	0.630272
$994$	0	0
$995$	−44.7492	−1.41864
$996$	0	0
$997$	2.56165	0.0811282	0.0405641	−	0.999177i	$-0.487085\pi$
$997$	2.56165	0.0811282	0.0405641	+	0.999177i	$0.487085\pi$
$998$	0	0
$999$	65.3734	2.06832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7616.2.a.cf.1.8		8
4.3	odd	2		7616.2.a.ce.1.1		8
8.3	odd	2		3808.2.a.r.1.8	yes	8
8.5	even	2		3808.2.a.q.1.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3808.2.a.q.1.1	✓	8	8.5	even	2
3808.2.a.r.1.8	yes	8	8.3	odd	2
7616.2.a.ce.1.1		8	4.3	odd	2
7616.2.a.cf.1.8		8	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:	\( N \)	\(=\)	\( 7616 = 2^{6} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7616.a (trivial)

\(f(q)\)	\(=\)	\(q+3.15311 q^{3} +2.50649 q^{5} -1.00000 q^{7} +6.94208 q^{9} +O(q^{10})\)	\(q+3.15311 q^{3} +2.50649 q^{5} -1.00000 q^{7} +6.94208 q^{9} +3.88928 q^{11} +1.04643 q^{13} +7.90322 q^{15} +1.00000 q^{17} +1.93181 q^{19} -3.15311 q^{21} +0.142722 q^{23} +1.28248 q^{25} +12.4298 q^{27} +1.88928 q^{29} -7.05670 q^{31} +12.2633 q^{33} -2.50649 q^{35} +5.25942 q^{37} +3.29950 q^{39} -9.78021 q^{41} +3.49716 q^{43} +17.4002 q^{45} -6.69256 q^{47} +1.00000 q^{49} +3.15311 q^{51} -8.97011 q^{53} +9.74844 q^{55} +6.09120 q^{57} +12.5094 q^{59} +0.0384872 q^{61} -6.94208 q^{63} +2.62286 q^{65} +15.3398 q^{67} +0.450017 q^{69} -6.00461 q^{71} -11.1525 q^{73} +4.04381 q^{75} -3.88928 q^{77} +7.89836 q^{79} +18.3662 q^{81} -4.76075 q^{83} +2.50649 q^{85} +5.95710 q^{87} +12.3409 q^{89} -1.04643 q^{91} -22.2505 q^{93} +4.84206 q^{95} -15.0908 q^{97} +26.9997 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{3} - 6 q^{5} - 8 q^{7} + 14 q^{9}+O(q^{10}) \) 8 * q + 2 * q^3 - 6 * q^5 - 8 * q^7 + 14 * q^9	\( 8 q + 2 q^{3} - 6 q^{5} - 8 q^{7} + 14 q^{9} + 4 q^{11} - 10 q^{13} + 6 q^{15} + 8 q^{17} + 16 q^{19} - 2 q^{21} + 18 q^{25} + 8 q^{27} - 12 q^{29} + 4 q^{31} + 4 q^{33} + 6 q^{35} - 6 q^{37} + 14 q^{39} + 6 q^{41} + 8 q^{43} - 32 q^{45} + 2 q^{47} + 8 q^{49} + 2 q^{51} - 30 q^{53} + 18 q^{55} - 10 q^{57} + 24 q^{59} - 4 q^{61} - 14 q^{63} + 20 q^{65} - 2 q^{67} - 2 q^{69} - 8 q^{71} - 6 q^{73} - 30 q^{75} - 4 q^{77} + 30 q^{79} + 32 q^{81} + 18 q^{83} - 6 q^{85} + 18 q^{89} + 10 q^{91} - 40 q^{93} - 10 q^{95} - 4 q^{97} + 26 q^{99}+O(q^{100}) \) 8 * q + 2 * q^3 - 6 * q^5 - 8 * q^7 + 14 * q^9 + 4 * q^11 - 10 * q^13 + 6 * q^15 + 8 * q^17 + 16 * q^19 - 2 * q^21 + 18 * q^25 + 8 * q^27 - 12 * q^29 + 4 * q^31 + 4 * q^33 + 6 * q^35 - 6 * q^37 + 14 * q^39 + 6 * q^41 + 8 * q^43 - 32 * q^45 + 2 * q^47 + 8 * q^49 + 2 * q^51 - 30 * q^53 + 18 * q^55 - 10 * q^57 + 24 * q^59 - 4 * q^61 - 14 * q^63 + 20 * q^65 - 2 * q^67 - 2 * q^69 - 8 * q^71 - 6 * q^73 - 30 * q^75 - 4 * q^77 + 30 * q^79 + 32 * q^81 + 18 * q^83 - 6 * q^85 + 18 * q^89 + 10 * q^91 - 40 * q^93 - 10 * q^95 - 4 * q^97 + 26 * q^99

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