LMFDB - Embedded newform 7616.2.a.bo.1.3

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:	$1$
Dimension:	$4$
Coefficient field:	4.4.13448.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} + 2 $ x^4 - 7*x^2 + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 952)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.58874$ 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)

$f(q)$	$=$	$q+1.44270 q^{3} +1.25886 q^{5} -1.00000 q^{7} -0.918614 q^{9} +5.17748 q^{11} -2.00000 q^{13} +1.81616 q^{15} +1.00000 q^{17} -1.44270 q^{21} -2.88540 q^{23} -3.41527 q^{25} -5.65339 q^{27} -0.454845 q^{29} -10.9541 q^{31} +7.46955 q^{33} -1.25886 q^{35} -7.17748 q^{37} -2.88540 q^{39} +3.21705 q^{41} -9.50558 q^{43} -1.15641 q^{45} -10.1500 q^{47} +1.00000 q^{49} +1.44270 q^{51} -6.59911 q^{53} +6.51772 q^{55} -10.3550 q^{59} -0.557299 q^{61} +0.918614 q^{63} -2.51772 q^{65} -0.966788 q^{67} -4.16277 q^{69} +2.68049 q^{71} +5.36709 q^{73} -4.92721 q^{75} -5.17748 q^{77} +10.1500 q^{79} -5.40031 q^{81} +4.37924 q^{83} +1.25886 q^{85} -0.656206 q^{87} -7.60803 q^{89} +2.00000 q^{91} -15.8034 q^{93} -1.48451 q^{97} -4.75610 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 3 q^{3} - 5 q^{5} - 4 q^{7} + 7 q^{9} - 8 q^{13} + 4 q^{17} - 3 q^{21} - 6 q^{23} + 3 q^{25} + 6 q^{27} - 8 q^{29} + 7 q^{31} - 6 q^{33} + 5 q^{35} - 8 q^{37} - 6 q^{39} + 15 q^{41} - 9 q^{43} + 2 q^{45}+ \cdots - 50 q^{99}+O(q^{100}) $ 4 * q + 3 * q^3 - 5 * q^5 - 4 * q^7 + 7 * q^9 - 8 * q^13 + 4 * q^17 - 3 * q^21 - 6 * q^23 + 3 * q^25 + 6 * q^27 - 8 * q^29 + 7 * q^31 - 6 * q^33 + 5 * q^35 - 8 * q^37 - 6 * q^39 + 15 * q^41 - 9 * q^43 + 2 * q^45 - 6 * q^47 + 4 * q^49 + 3 * q^51 - 17 * q^53 + 6 * q^55 - 5 * q^61 - 7 * q^63 + 10 * q^65 - 9 * q^67 - 38 * q^69 + 12 * q^71 - 11 * q^73 - 2 * q^75 + 6 * q^79 + 16 * q^81 - 6 * q^83 - 5 * q^85 + 14 * q^87 + 2 * q^89 + 8 * q^91 + 9 * q^97 - 50 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.44270	0.832944	0.416472	−	0.909149i	$-0.363266\pi$
$3$	1.44270	0.832944	0.416472	+	0.909149i	$0.363266\pi$
$4$	0	0
$5$	1.25886	0.562980	0.281490	−	0.959564i	$-0.409171\pi$
$5$	1.25886	0.562980	0.281490	+	0.959564i	$0.409171\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−0.918614	−0.306205
$10$	0	0
$11$	5.17748	1.56107	0.780534	−	0.625114i	$-0.214947\pi$
$11$	5.17748	1.56107	0.780534	+	0.625114i	$0.214947\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	1.81616	0.468931
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	−1.44270	−0.314823
$22$	0	0
$23$	−2.88540	−0.601648	−0.300824	−	0.953680i	$-0.597262\pi$
$23$	−2.88540	−0.601648	−0.300824	+	0.953680i	$0.597262\pi$
$24$	0	0
$25$	−3.41527	−0.683054
$26$	0	0
$27$	−5.65339	−1.08800
$28$	0	0
$29$	−0.454845	−0.0844627	−0.0422313	−	0.999108i	$-0.513447\pi$
$29$	−0.454845	−0.0844627	−0.0422313	+	0.999108i	$0.513447\pi$
$30$	0	0
$31$	−10.9541	−1.96741	−0.983704	−	0.179798i	$-0.942455\pi$
$31$	−10.9541	−1.96741	−0.983704	+	0.179798i	$0.942455\pi$
$32$	0	0
$33$	7.46955	1.30028
$34$	0	0
$35$	−1.25886	−0.212786
$36$	0	0
$37$	−7.17748	−1.17997	−0.589985	−	0.807414i	$-0.700866\pi$
$37$	−7.17748	−1.17997	−0.589985	+	0.807414i	$0.700866\pi$
$38$	0	0
$39$	−2.88540	−0.462034
$40$	0	0
$41$	3.21705	0.502419	0.251210	−	0.967933i	$-0.419172\pi$
$41$	3.21705	0.502419	0.251210	+	0.967933i	$0.419172\pi$
$42$	0	0
$43$	−9.50558	−1.44959	−0.724794	−	0.688966i	$-0.758065\pi$
$43$	−9.50558	−1.44959	−0.724794	+	0.688966i	$0.758065\pi$
$44$	0	0
$45$	−1.15641	−0.172387
$46$	0	0
$47$	−10.1500	−1.48054	−0.740268	−	0.672312i	$-0.765301\pi$
$47$	−10.1500	−1.48054	−0.740268	+	0.672312i	$0.765301\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.44270	0.202019
$52$	0	0
$53$	−6.59911	−0.906457	−0.453229	−	0.891394i	$-0.649728\pi$
$53$	−6.59911	−0.906457	−0.453229	+	0.891394i	$0.649728\pi$
$54$	0	0
$55$	6.51772	0.878849
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−10.3550	−1.34810	−0.674050	−	0.738686i	$-0.735447\pi$
$59$	−10.3550	−1.34810	−0.674050	+	0.738686i	$0.735447\pi$
$60$	0	0
$61$	−0.557299	−0.0713548	−0.0356774	−	0.999363i	$-0.511359\pi$
$61$	−0.557299	−0.0713548	−0.0356774	+	0.999363i	$0.511359\pi$
$62$	0	0
$63$	0.918614	0.115734
$64$	0	0
$65$	−2.51772	−0.312285
$66$	0	0
$67$	−0.966788	−0.118112	−0.0590560	−	0.998255i	$-0.518809\pi$
$67$	−0.966788	−0.118112	−0.0590560	+	0.998255i	$0.518809\pi$
$68$	0	0
$69$	−4.16277	−0.501139
$70$	0	0
$71$	2.68049	0.318116	0.159058	−	0.987269i	$-0.449154\pi$
$71$	2.68049	0.318116	0.159058	+	0.987269i	$0.449154\pi$
$72$	0	0
$73$	5.36709	0.628171	0.314085	−	0.949395i	$-0.398302\pi$
$73$	5.36709	0.628171	0.314085	+	0.949395i	$0.398302\pi$
$74$	0	0
$75$	−4.92721	−0.568945
$76$	0	0
$77$	−5.17748	−0.590028
$78$	0	0
$79$	10.1500	1.14197	0.570985	−	0.820961i	$-0.306562\pi$
$79$	10.1500	1.14197	0.570985	+	0.820961i	$0.306562\pi$
$80$	0	0
$81$	−5.40031	−0.600034
$82$	0	0
$83$	4.37924	0.480684	0.240342	−	0.970688i	$-0.422740\pi$
$83$	4.37924	0.480684	0.240342	+	0.970688i	$0.422740\pi$
$84$	0	0
$85$	1.25886	0.136543
$86$	0	0
$87$	−0.656206	−0.0703526
$88$	0	0
$89$	−7.60803	−0.806450	−0.403225	−	0.915101i	$-0.632111\pi$
$89$	−7.60803	−0.806450	−0.403225	+	0.915101i	$0.632111\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	0	0
$93$	−15.8034	−1.63874
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−1.48451	−0.150729	−0.0753646	−	0.997156i	$-0.524012\pi$
$97$	−1.48451	−0.150729	−0.0753646	+	0.997156i	$0.524012\pi$
$98$	0	0
$99$	−4.75610	−0.478006
$100$	0	0
$101$	1.63232	0.162422	0.0812110	−	0.996697i	$-0.474121\pi$
$101$	1.63232	0.162422	0.0812110	+	0.996697i	$0.474121\pi$
$102$	0	0
$103$	18.2758	1.80077	0.900384	−	0.435096i	$-0.143286\pi$
$103$	18.2758	1.80077	0.900384	+	0.435096i	$0.143286\pi$
$104$	0	0
$105$	−1.81616	−0.177239
$106$	0	0
$107$	7.85797	0.759659	0.379829	−	0.925057i	$-0.375983\pi$
$107$	7.85797	0.759659	0.379829	+	0.925057i	$0.375983\pi$
$108$	0	0
$109$	6.80980	0.652260	0.326130	−	0.945325i	$-0.394255\pi$
$109$	6.80980	0.652260	0.326130	+	0.945325i	$0.394255\pi$
$110$	0	0
$111$	−10.3550	−0.982848
$112$	0	0
$113$	−17.1019	−1.60881	−0.804404	−	0.594082i	$-0.797515\pi$
$113$	−17.1019	−1.60881	−0.804404	+	0.594082i	$0.797515\pi$
$114$	0	0
$115$	−3.63232	−0.338716
$116$	0	0
$117$	1.83723	0.169852
$118$	0	0
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	15.8062	1.43693
$122$	0	0
$123$	4.64124	0.418487
$124$	0	0
$125$	−10.5937	−0.947525
$126$	0	0
$127$	21.8395	1.93794	0.968969	−	0.247181i	$-0.0795042\pi$
$127$	21.8395	1.93794	0.968969	+	0.247181i	$0.0795042\pi$
$128$	0	0
$129$	−13.7137	−1.20742
$130$	0	0
$131$	−0.225649	−0.0197151	−0.00985753	−	0.999951i	$-0.503138\pi$
$131$	−0.225649	−0.0197151	−0.00985753	+	0.999951i	$0.503138\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−7.11683	−0.612519
$136$	0	0
$137$	17.6556	1.50842	0.754211	−	0.656632i	$-0.228020\pi$
$137$	17.6556	1.50842	0.754211	+	0.656632i	$0.228020\pi$
$138$	0	0
$139$	15.7639	1.33707	0.668536	−	0.743679i	$-0.266921\pi$
$139$	15.7639	1.33707	0.668536	+	0.743679i	$0.266921\pi$
$140$	0	0
$141$	−14.6435	−1.23320
$142$	0	0
$143$	−10.3550	−0.865924
$144$	0	0
$145$	−0.572587	−0.0475508
$146$	0	0
$147$	1.44270	0.118992
$148$	0	0
$149$	−21.6556	−1.77410	−0.887049	−	0.461676i	$-0.847248\pi$
$149$	−21.6556	−1.77410	−0.887049	+	0.461676i	$0.847248\pi$
$150$	0	0
$151$	21.1800	1.72361	0.861803	−	0.507243i	$-0.169335\pi$
$151$	21.1800	1.72361	0.861803	+	0.507243i	$0.169335\pi$
$152$	0	0
$153$	−0.918614	−0.0742655
$154$	0	0
$155$	−13.7896	−1.10761
$156$	0	0
$157$	12.7226	1.01538	0.507688	−	0.861541i	$-0.330500\pi$
$157$	12.7226	1.01538	0.507688	+	0.861541i	$0.330500\pi$
$158$	0	0
$159$	−9.52054	−0.755028
$160$	0	0
$161$	2.88540	0.227402
$162$	0	0
$163$	−21.8452	−1.71105	−0.855526	−	0.517760i	$-0.826766\pi$
$163$	−21.8452	−1.71105	−0.855526	+	0.517760i	$0.826766\pi$
$164$	0	0
$165$	9.40312	0.732032
$166$	0	0
$167$	14.4574	1.11875	0.559374	−	0.828916i	$-0.311042\pi$
$167$	14.4574	1.11875	0.559374	+	0.828916i	$0.311042\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2.12320	0.161424	0.0807118	−	0.996737i	$-0.474281\pi$
$173$	2.12320	0.161424	0.0807118	+	0.996737i	$0.474281\pi$
$174$	0	0
$175$	3.41527	0.258170
$176$	0	0
$177$	−14.9391	−1.12289
$178$	0	0
$179$	−6.62018	−0.494815	−0.247408	−	0.968911i	$-0.579579\pi$
$179$	−6.62018	−0.494815	−0.247408	+	0.968911i	$0.579579\pi$
$180$	0	0
$181$	−18.3514	−1.36405	−0.682025	−	0.731329i	$-0.738900\pi$
$181$	−18.3514	−1.36405	−0.682025	+	0.731329i	$0.738900\pi$
$182$	0	0
$183$	−0.804016	−0.0594346
$184$	0	0
$185$	−9.03544	−0.664299
$186$	0	0
$187$	5.17748	0.378614
$188$	0	0
$189$	5.65339	0.411223
$190$	0	0
$191$	−11.1590	−0.807434	−0.403717	−	0.914884i	$-0.632282\pi$
$191$	−11.1590	−0.807434	−0.403717	+	0.914884i	$0.632282\pi$
$192$	0	0
$193$	2.45130	0.176448	0.0882242	−	0.996101i	$-0.471881\pi$
$193$	2.45130	0.176448	0.0882242	+	0.996101i	$0.471881\pi$
$194$	0	0
$195$	−3.63232	−0.260116
$196$	0	0
$197$	−4.94828	−0.352550	−0.176275	−	0.984341i	$-0.556405\pi$
$197$	−4.94828	−0.352550	−0.176275	+	0.984341i	$0.556405\pi$
$198$	0	0
$199$	6.33165	0.448839	0.224419	−	0.974493i	$-0.427951\pi$
$199$	6.33165	0.448839	0.224419	+	0.974493i	$0.427951\pi$
$200$	0	0
$201$	−1.39479	−0.0983806
$202$	0	0
$203$	0.454845	0.0319239
$204$	0	0
$205$	4.04982	0.282852
$206$	0	0
$207$	2.65057	0.184227
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−12.4306	−0.855755	−0.427877	−	0.903837i	$-0.640739\pi$
$211$	−12.4306	−0.855755	−0.427877	+	0.903837i	$0.640739\pi$
$212$	0	0
$213$	3.86715	0.264973
$214$	0	0
$215$	−11.9662	−0.816088
$216$	0	0
$217$	10.9541	0.743610
$218$	0	0
$219$	7.74311	0.523231
$220$	0	0
$221$	−2.00000	−0.134535
$222$	0	0
$223$	−7.39039	−0.494897	−0.247449	−	0.968901i	$-0.579592\pi$
$223$	−7.39039	−0.494897	−0.247449	+	0.968901i	$0.579592\pi$
$224$	0	0
$225$	3.13731	0.209154
$226$	0	0
$227$	17.2346	1.14390	0.571949	−	0.820289i	$-0.306187\pi$
$227$	17.2346	1.14390	0.571949	+	0.820289i	$0.306187\pi$
$228$	0	0
$229$	2.58415	0.170765	0.0853826	−	0.996348i	$-0.472789\pi$
$229$	2.58415	0.170765	0.0853826	+	0.996348i	$0.472789\pi$
$230$	0	0
$231$	−7.46955	−0.491460
$232$	0	0
$233$	−14.5771	−0.954974	−0.477487	−	0.878639i	$-0.658452\pi$
$233$	−14.5771	−0.954974	−0.477487	+	0.878639i	$0.658452\pi$
$234$	0	0
$235$	−12.7775	−0.833512
$236$	0	0
$237$	14.6435	0.951196
$238$	0	0
$239$	0.921432	0.0596025	0.0298012	−	0.999556i	$-0.490513\pi$
$239$	0.921432	0.0596025	0.0298012	+	0.999556i	$0.490513\pi$
$240$	0	0
$241$	−19.3882	−1.24890	−0.624451	−	0.781064i	$-0.714677\pi$
$241$	−19.3882	−1.24890	−0.624451	+	0.781064i	$0.714677\pi$
$242$	0	0
$243$	9.16914	0.588200
$244$	0	0
$245$	1.25886	0.0804257
$246$	0	0
$247$	0	0
$248$	0	0
$249$	6.31793	0.400383
$250$	0	0
$251$	−22.6550	−1.42997	−0.714987	−	0.699138i	$-0.753567\pi$
$251$	−22.6550	−1.42997	−0.714987	+	0.699138i	$0.753567\pi$
$252$	0	0
$253$	−14.9391	−0.939213
$254$	0	0
$255$	1.81616	0.113732
$256$	0	0
$257$	−5.05973	−0.315617	−0.157809	−	0.987470i	$-0.550443\pi$
$257$	−5.05973	−0.315617	−0.157809	+	0.987470i	$0.550443\pi$
$258$	0	0
$259$	7.17748	0.445987
$260$	0	0
$261$	0.417827	0.0258629
$262$	0	0
$263$	−6.81334	−0.420129	−0.210064	−	0.977688i	$-0.567367\pi$
$263$	−6.81334	−0.420129	−0.210064	+	0.977688i	$0.567367\pi$
$264$	0	0
$265$	−8.30736	−0.510317
$266$	0	0
$267$	−10.9761	−0.671727
$268$	0	0
$269$	16.9904	1.03592	0.517962	−	0.855404i	$-0.326691\pi$
$269$	16.9904	1.03592	0.517962	+	0.855404i	$0.326691\pi$
$270$	0	0
$271$	−12.7891	−0.776880	−0.388440	−	0.921474i	$-0.626986\pi$
$271$	−12.7891	−0.776880	−0.388440	+	0.921474i	$0.626986\pi$
$272$	0	0
$273$	2.88540	0.174632
$274$	0	0
$275$	−17.6825	−1.06629
$276$	0	0
$277$	3.70793	0.222788	0.111394	−	0.993776i	$-0.464468\pi$
$277$	3.70793	0.222788	0.111394	+	0.993776i	$0.464468\pi$
$278$	0	0
$279$	10.0626	0.602429
$280$	0	0
$281$	−30.3993	−1.81347	−0.906736	−	0.421700i	$-0.861434\pi$
$281$	−30.3993	−1.81347	−0.906736	+	0.421700i	$0.861434\pi$
$282$	0	0
$283$	4.21069	0.250299	0.125150	−	0.992138i	$-0.460059\pi$
$283$	4.21069	0.250299	0.125150	+	0.992138i	$0.460059\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3.21705	−0.189897
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−2.14170	−0.125549
$292$	0	0
$293$	8.55986	0.500072	0.250036	−	0.968237i	$-0.419558\pi$
$293$	8.55986	0.500072	0.250036	+	0.968237i	$0.419558\pi$
$294$	0	0
$295$	−13.0354	−0.758953
$296$	0	0
$297$	−29.2703	−1.69843
$298$	0	0
$299$	5.77080	0.333734
$300$	0	0
$301$	9.50558	0.547892
$302$	0	0
$303$	2.35495	0.135288
$304$	0	0
$305$	−0.701562	−0.0401713
$306$	0	0
$307$	19.1019	1.09020	0.545101	−	0.838371i	$-0.316491\pi$
$307$	19.1019	1.09020	0.545101	+	0.838371i	$0.316491\pi$
$308$	0	0
$309$	26.3665	1.49994
$310$	0	0
$311$	−18.0814	−1.02530	−0.512651	−	0.858597i	$-0.671336\pi$
$311$	−18.0814	−1.02530	−0.512651	+	0.858597i	$0.671336\pi$
$312$	0	0
$313$	−17.1590	−0.969882	−0.484941	−	0.874547i	$-0.661159\pi$
$313$	−17.1590	−0.969882	−0.484941	+	0.874547i	$0.661159\pi$
$314$	0	0
$315$	1.15641	0.0651562
$316$	0	0
$317$	−2.60489	−0.146305	−0.0731525	−	0.997321i	$-0.523306\pi$
$317$	−2.60489	−0.146305	−0.0731525	+	0.997321i	$0.523306\pi$
$318$	0	0
$319$	−2.35495	−0.131852
$320$	0	0
$321$	11.3367	0.632753
$322$	0	0
$323$	0	0
$324$	0	0
$325$	6.83054	0.378890
$326$	0	0
$327$	9.82450	0.543296
$328$	0	0
$329$	10.1500	0.559590
$330$	0	0
$331$	−7.03321	−0.386580	−0.193290	−	0.981142i	$-0.561916\pi$
$331$	−7.03321	−0.386580	−0.193290	+	0.981142i	$0.561916\pi$
$332$	0	0
$333$	6.59333	0.361312
$334$	0	0
$335$	−1.21705	−0.0664946
$336$	0	0
$337$	−23.4031	−1.27485	−0.637425	−	0.770513i	$-0.720000\pi$
$337$	−23.4031	−1.27485	−0.637425	+	0.770513i	$0.720000\pi$
$338$	0	0
$339$	−24.6729	−1.34005
$340$	0	0
$341$	−56.7144	−3.07126
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−5.24035	−0.282131
$346$	0	0
$347$	2.82252	0.151521	0.0757605	−	0.997126i	$-0.475862\pi$
$347$	2.82252	0.151521	0.0757605	+	0.997126i	$0.475862\pi$
$348$	0	0
$349$	−24.7342	−1.32399	−0.661995	−	0.749508i	$-0.730290\pi$
$349$	−24.7342	−1.32399	−0.661995	+	0.749508i	$0.730290\pi$
$350$	0	0
$351$	11.3068	0.603511
$352$	0	0
$353$	25.8417	1.37541	0.687707	−	0.725988i	$-0.258617\pi$
$353$	25.8417	1.37541	0.687707	+	0.725988i	$0.258617\pi$
$354$	0	0
$355$	3.37437	0.179093
$356$	0	0
$357$	−1.44270	−0.0763558
$358$	0	0
$359$	28.6528	1.51224	0.756119	−	0.654435i	$-0.227093\pi$
$359$	28.6528	1.51224	0.756119	+	0.654435i	$0.227093\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	22.8037	1.19688
$364$	0	0
$365$	6.75643	0.353648
$366$	0	0
$367$	−1.34281	−0.0700939	−0.0350469	−	0.999386i	$-0.511158\pi$
$367$	−1.34281	−0.0700939	−0.0350469	+	0.999386i	$0.511158\pi$
$368$	0	0
$369$	−2.95523	−0.153843
$370$	0	0
$371$	6.59911	0.342609
$372$	0	0
$373$	4.74593	0.245735	0.122867	−	0.992423i	$-0.460791\pi$
$373$	4.74593	0.245735	0.122867	+	0.992423i	$0.460791\pi$
$374$	0	0
$375$	−15.2835	−0.789235
$376$	0	0
$377$	0.909691	0.0468514
$378$	0	0
$379$	−19.9908	−1.02686	−0.513430	−	0.858132i	$-0.671625\pi$
$379$	−19.9908	−1.02686	−0.513430	+	0.858132i	$0.671625\pi$
$380$	0	0
$381$	31.5078	1.61419
$382$	0	0
$383$	−20.3677	−1.04074	−0.520370	−	0.853941i	$-0.674206\pi$
$383$	−20.3677	−1.04074	−0.520370	+	0.853941i	$0.674206\pi$
$384$	0	0
$385$	−6.51772	−0.332174
$386$	0	0
$387$	8.73196	0.443870
$388$	0	0
$389$	1.44237	0.0731313	0.0365657	−	0.999331i	$-0.488358\pi$
$389$	1.44237	0.0731313	0.0365657	+	0.999331i	$0.488358\pi$
$390$	0	0
$391$	−2.88540	−0.145921
$392$	0	0
$393$	−0.325544	−0.0164215
$394$	0	0
$395$	12.7775	0.642906
$396$	0	0
$397$	−15.6023	−0.783055	−0.391527	−	0.920166i	$-0.628053\pi$
$397$	−15.6023	−0.783055	−0.391527	+	0.920166i	$0.628053\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	30.8357	1.53986	0.769930	−	0.638129i	$-0.220291\pi$
$401$	30.8357	1.53986	0.769930	+	0.638129i	$0.220291\pi$
$402$	0	0
$403$	21.9081	1.09132
$404$	0	0
$405$	−6.79824	−0.337807
$406$	0	0
$407$	−37.1612	−1.84201
$408$	0	0
$409$	8.64348	0.427392	0.213696	−	0.976900i	$-0.431450\pi$
$409$	8.64348	0.427392	0.213696	+	0.976900i	$0.431450\pi$
$410$	0	0
$411$	25.4718	1.25643
$412$	0	0
$413$	10.3550	0.509534
$414$	0	0
$415$	5.51285	0.270615
$416$	0	0
$417$	22.7425	1.11371
$418$	0	0
$419$	14.8912	0.727482	0.363741	−	0.931500i	$-0.381499\pi$
$419$	14.8912	0.727482	0.363741	+	0.931500i	$0.381499\pi$
$420$	0	0
$421$	−23.0472	−1.12325	−0.561626	−	0.827392i	$-0.689824\pi$
$421$	−23.0472	−1.12325	−0.561626	+	0.827392i	$0.689824\pi$
$422$	0	0
$423$	9.32397	0.453347
$424$	0	0
$425$	−3.41527	−0.165665
$426$	0	0
$427$	0.557299	0.0269696
$428$	0	0
$429$	−14.9391	−0.721266
$430$	0	0
$431$	−4.44683	−0.214196	−0.107098	−	0.994248i	$-0.534156\pi$
$431$	−4.44683	−0.214196	−0.107098	+	0.994248i	$0.534156\pi$
$432$	0	0
$433$	−9.92794	−0.477106	−0.238553	−	0.971129i	$-0.576673\pi$
$433$	−9.92794	−0.477106	−0.238553	+	0.971129i	$0.576673\pi$
$434$	0	0
$435$	−0.826072	−0.0396071
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−20.3540	−0.971442	−0.485721	−	0.874114i	$-0.661443\pi$
$439$	−20.3540	−0.971442	−0.485721	+	0.874114i	$0.661443\pi$
$440$	0	0
$441$	−0.918614	−0.0437435
$442$	0	0
$443$	9.31951	0.442783	0.221392	−	0.975185i	$-0.428940\pi$
$443$	9.31951	0.442783	0.221392	+	0.975185i	$0.428940\pi$
$444$	0	0
$445$	−9.57746	−0.454015
$446$	0	0
$447$	−31.2426	−1.47772
$448$	0	0
$449$	27.2033	1.28380	0.641902	−	0.766786i	$-0.278145\pi$
$449$	27.2033	1.28380	0.641902	+	0.766786i	$0.278145\pi$
$450$	0	0
$451$	16.6562	0.784310
$452$	0	0
$453$	30.5565	1.43567
$454$	0	0
$455$	2.51772	0.118033
$456$	0	0
$457$	37.4169	1.75029	0.875145	−	0.483861i	$-0.160766\pi$
$457$	37.4169	1.75029	0.875145	+	0.483861i	$0.160766\pi$
$458$	0	0
$459$	−5.65339	−0.263878
$460$	0	0
$461$	−29.0112	−1.35118	−0.675592	−	0.737276i	$-0.736112\pi$
$461$	−29.0112	−1.35118	−0.675592	+	0.737276i	$0.736112\pi$
$462$	0	0
$463$	10.9508	0.508929	0.254464	−	0.967082i	$-0.418101\pi$
$463$	10.9508	0.508929	0.254464	+	0.967082i	$0.418101\pi$
$464$	0	0
$465$	−19.8943	−0.922577
$466$	0	0
$467$	36.4143	1.68505	0.842526	−	0.538656i	$-0.181068\pi$
$467$	36.4143	1.68505	0.842526	+	0.538656i	$0.181068\pi$
$468$	0	0
$469$	0.966788	0.0446421
$470$	0	0
$471$	18.3550	0.845751
$472$	0	0
$473$	−49.2149	−2.26290
$474$	0	0
$475$	0	0
$476$	0	0
$477$	6.06203	0.277561
$478$	0	0
$479$	−22.4989	−1.02800	−0.514000	−	0.857790i	$-0.671837\pi$
$479$	−22.4989	−1.02800	−0.514000	+	0.857790i	$0.671837\pi$
$480$	0	0
$481$	14.3550	0.654529
$482$	0	0
$483$	4.16277	0.189413
$484$	0	0
$485$	−1.86879	−0.0848575
$486$	0	0
$487$	38.3594	1.73823	0.869116	−	0.494609i	$-0.164689\pi$
$487$	38.3594	1.73823	0.869116	+	0.494609i	$0.164689\pi$
$488$	0	0
$489$	−31.5161	−1.42521
$490$	0	0
$491$	−16.1899	−0.730642	−0.365321	−	0.930882i	$-0.619041\pi$
$491$	−16.1899	−0.730642	−0.365321	+	0.930882i	$0.619041\pi$
$492$	0	0
$493$	−0.454845	−0.0204852
$494$	0	0
$495$	−5.98727	−0.269108
$496$	0	0
$497$	−2.68049	−0.120237
$498$	0	0
$499$	6.11774	0.273868	0.136934	−	0.990580i	$-0.456275\pi$
$499$	6.11774	0.273868	0.136934	+	0.990580i	$0.456275\pi$
$500$	0	0
$501$	20.8577	0.931854
$502$	0	0
$503$	36.2780	1.61756	0.808779	−	0.588113i	$-0.200129\pi$
$503$	36.2780	1.61756	0.808779	+	0.588113i	$0.200129\pi$
$504$	0	0
$505$	2.05486	0.0914403
$506$	0	0
$507$	−12.9843	−0.576653
$508$	0	0
$509$	11.2519	0.498732	0.249366	−	0.968409i	$-0.419778\pi$
$509$	11.2519	0.498732	0.249366	+	0.968409i	$0.419778\pi$
$510$	0	0
$511$	−5.36709	−0.237426
$512$	0	0
$513$	0	0
$514$	0	0
$515$	23.0067	1.01380
$516$	0	0
$517$	−52.5516	−2.31122
$518$	0	0
$519$	3.06314	0.134457
$520$	0	0
$521$	5.05992	0.221679	0.110839	−	0.993838i	$-0.464646\pi$
$521$	5.05992	0.221679	0.110839	+	0.993838i	$0.464646\pi$
$522$	0	0
$523$	31.1855	1.36365	0.681823	−	0.731517i	$-0.261187\pi$
$523$	31.1855	1.36365	0.681823	+	0.731517i	$0.261187\pi$
$524$	0	0
$525$	4.92721	0.215041
$526$	0	0
$527$	−10.9541	−0.477166
$528$	0	0
$529$	−14.6745	−0.638020
$530$	0	0
$531$	9.51220	0.412794
$532$	0	0
$533$	−6.43410	−0.278692
$534$	0	0
$535$	9.89209	0.427672
$536$	0	0
$537$	−9.55093	−0.412153
$538$	0	0
$539$	5.17748	0.223010
$540$	0	0
$541$	8.13377	0.349698	0.174849	−	0.984595i	$-0.444056\pi$
$541$	8.13377	0.349698	0.174849	+	0.984595i	$0.444056\pi$
$542$	0	0
$543$	−26.4756	−1.13618
$544$	0	0
$545$	8.57259	0.367209
$546$	0	0
$547$	30.7064	1.31291	0.656454	−	0.754366i	$-0.272055\pi$
$547$	30.7064	1.31291	0.656454	+	0.754366i	$0.272055\pi$
$548$	0	0
$549$	0.511943	0.0218492
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−10.1500	−0.431624
$554$	0	0
$555$	−13.0354	−0.553324
$556$	0	0
$557$	17.3904	0.736855	0.368427	−	0.929657i	$-0.379896\pi$
$557$	17.3904	0.736855	0.368427	+	0.929657i	$0.379896\pi$
$558$	0	0
$559$	19.0112	0.804086
$560$	0	0
$561$	7.46955	0.315365
$562$	0	0
$563$	−2.26424	−0.0954262	−0.0477131	−	0.998861i	$-0.515193\pi$
$563$	−2.26424	−0.0954262	−0.0477131	+	0.998861i	$0.515193\pi$
$564$	0	0
$565$	−21.5289	−0.905727
$566$	0	0
$567$	5.40031	0.226792
$568$	0	0
$569$	−13.6090	−0.570520	−0.285260	−	0.958450i	$-0.592080\pi$
$569$	−13.6090	−0.570520	−0.285260	+	0.958450i	$0.592080\pi$
$570$	0	0
$571$	−14.0035	−0.586030	−0.293015	−	0.956108i	$-0.594659\pi$
$571$	−14.0035	−0.586030	−0.293015	+	0.956108i	$0.594659\pi$
$572$	0	0
$573$	−16.0991	−0.672548
$574$	0	0
$575$	9.85442	0.410958
$576$	0	0
$577$	22.6192	0.941649	0.470825	−	0.882227i	$-0.343956\pi$
$577$	22.6192	0.941649	0.470825	+	0.882227i	$0.343956\pi$
$578$	0	0
$579$	3.53649	0.146972
$580$	0	0
$581$	−4.37924	−0.181681
$582$	0	0
$583$	−34.1667	−1.41504
$584$	0	0
$585$	2.31281	0.0956231
$586$	0	0
$587$	−2.69205	−0.111113	−0.0555565	−	0.998456i	$-0.517693\pi$
$587$	−2.69205	−0.111113	−0.0555565	+	0.998456i	$0.517693\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−7.13889	−0.293655
$592$	0	0
$593$	−4.84838	−0.199099	−0.0995496	−	0.995033i	$-0.531740\pi$
$593$	−4.84838	−0.199099	−0.0995496	+	0.995033i	$0.531740\pi$
$594$	0	0
$595$	−1.25886	−0.0516083
$596$	0	0
$597$	9.13468	0.373857
$598$	0	0
$599$	3.48129	0.142242	0.0711208	−	0.997468i	$-0.477342\pi$
$599$	3.48129	0.142242	0.0711208	+	0.997468i	$0.477342\pi$
$600$	0	0
$601$	39.1612	1.59742	0.798709	−	0.601717i	$-0.205517\pi$
$601$	39.1612	1.59742	0.798709	+	0.601717i	$0.205517\pi$
$602$	0	0
$603$	0.888105	0.0361664
$604$	0	0
$605$	19.8979	0.808964
$606$	0	0
$607$	29.1833	1.18451	0.592256	−	0.805750i	$-0.298237\pi$
$607$	29.1833	1.18451	0.592256	+	0.805750i	$0.298237\pi$
$608$	0	0
$609$	0.656206	0.0265908
$610$	0	0
$611$	20.3001	0.821254
$612$	0	0
$613$	−14.1478	−0.571425	−0.285712	−	0.958315i	$-0.592230\pi$
$613$	−14.1478	−0.571425	−0.285712	+	0.958315i	$0.592230\pi$
$614$	0	0
$615$	5.84268	0.235600
$616$	0	0
$617$	−32.3224	−1.30125	−0.650625	−	0.759399i	$-0.725493\pi$
$617$	−32.3224	−1.30125	−0.650625	+	0.759399i	$0.725493\pi$
$618$	0	0
$619$	−33.3939	−1.34222	−0.671108	−	0.741360i	$-0.734181\pi$
$619$	−33.3939	−1.34222	−0.671108	+	0.741360i	$0.734181\pi$
$620$	0	0
$621$	16.3123	0.654590
$622$	0	0
$623$	7.60803	0.304809
$624$	0	0
$625$	3.74040	0.149616
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−7.17748	−0.286185
$630$	0	0
$631$	35.9474	1.43104	0.715521	−	0.698591i	$-0.246189\pi$
$631$	35.9474	1.43104	0.715521	+	0.698591i	$0.246189\pi$
$632$	0	0
$633$	−17.9336	−0.712796
$634$	0	0
$635$	27.4928	1.09102
$636$	0	0
$637$	−2.00000	−0.0792429
$638$	0	0
$639$	−2.46234	−0.0974086
$640$	0	0
$641$	10.6562	0.420895	0.210447	−	0.977605i	$-0.432508\pi$
$641$	10.6562	0.420895	0.210447	+	0.977605i	$0.432508\pi$
$642$	0	0
$643$	−35.2295	−1.38932	−0.694659	−	0.719340i	$-0.744445\pi$
$643$	−35.2295	−1.38932	−0.694659	+	0.719340i	$0.744445\pi$
$644$	0	0
$645$	−17.2637	−0.679756
$646$	0	0
$647$	−2.44566	−0.0961489	−0.0480745	−	0.998844i	$-0.515308\pi$
$647$	−2.44566	−0.0961489	−0.0480745	+	0.998844i	$0.515308\pi$
$648$	0	0
$649$	−53.6125	−2.10447
$650$	0	0
$651$	15.8034	0.619385
$652$	0	0
$653$	−26.7772	−1.04787	−0.523937	−	0.851757i	$-0.675537\pi$
$653$	−26.7772	−1.04787	−0.523937	+	0.851757i	$0.675537\pi$
$654$	0	0
$655$	−0.284061	−0.0110992
$656$	0	0
$657$	−4.93029	−0.192349
$658$	0	0
$659$	17.9652	0.699825	0.349913	−	0.936782i	$-0.386211\pi$
$659$	17.9652	0.699825	0.349913	+	0.936782i	$0.386211\pi$
$660$	0	0
$661$	−29.5416	−1.14904	−0.574518	−	0.818492i	$-0.694810\pi$
$661$	−29.5416	−1.14904	−0.574518	+	0.818492i	$0.694810\pi$
$662$	0	0
$663$	−2.88540	−0.112060
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.31241	0.0508168
$668$	0	0
$669$	−10.6621	−0.412222
$670$	0	0
$671$	−2.88540	−0.111390
$672$	0	0
$673$	−1.29522	−0.0499269	−0.0249635	−	0.999688i	$-0.507947\pi$
$673$	−1.29522	−0.0499269	−0.0249635	+	0.999688i	$0.507947\pi$
$674$	0	0
$675$	19.3078	0.743159
$676$	0	0
$677$	−49.5420	−1.90405	−0.952027	−	0.306014i	$-0.901005\pi$
$677$	−49.5420	−1.90405	−0.952027	+	0.306014i	$0.901005\pi$
$678$	0	0
$679$	1.48451	0.0569703
$680$	0	0
$681$	24.8643	0.952803
$682$	0	0
$683$	21.6780	0.829486	0.414743	−	0.909939i	$-0.363872\pi$
$683$	21.6780	0.829486	0.414743	+	0.909939i	$0.363872\pi$
$684$	0	0
$685$	22.2260	0.849211
$686$	0	0
$687$	3.72815	0.142238
$688$	0	0
$689$	13.1982	0.502812
$690$	0	0
$691$	−37.7479	−1.43600	−0.717999	−	0.696044i	$-0.754942\pi$
$691$	−37.7479	−1.43600	−0.717999	+	0.696044i	$0.754942\pi$
$692$	0	0
$693$	4.75610	0.180669
$694$	0	0
$695$	19.8445	0.752745
$696$	0	0
$697$	3.21705	0.121855
$698$	0	0
$699$	−21.0303	−0.795440
$700$	0	0
$701$	33.9674	1.28293	0.641466	−	0.767151i	$-0.278326\pi$
$701$	33.9674	1.28293	0.641466	+	0.767151i	$0.278326\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−18.4341	−0.694269
$706$	0	0
$707$	−1.63232	−0.0613897
$708$	0	0
$709$	4.99041	0.187419	0.0937095	−	0.995600i	$-0.470128\pi$
$709$	4.99041	0.187419	0.0937095	+	0.995600i	$0.470128\pi$
$710$	0	0
$711$	−9.32397	−0.349676
$712$	0	0
$713$	31.6069	1.18369
$714$	0	0
$715$	−13.0354	−0.487498
$716$	0	0
$717$	1.32935	0.0496455
$718$	0	0
$719$	18.8832	0.704226	0.352113	−	0.935957i	$-0.385463\pi$
$719$	18.8832	0.704226	0.352113	+	0.935957i	$0.385463\pi$
$720$	0	0
$721$	−18.2758	−0.680626
$722$	0	0
$723$	−27.9713	−1.04026
$724$	0	0
$725$	1.55342	0.0576925
$726$	0	0
$727$	44.2145	1.63982	0.819912	−	0.572489i	$-0.194022\pi$
$727$	44.2145	1.63982	0.819912	+	0.572489i	$0.194022\pi$
$728$	0	0
$729$	29.4292	1.08997
$730$	0	0
$731$	−9.50558	−0.351577
$732$	0	0
$733$	−18.2049	−0.672414	−0.336207	−	0.941788i	$-0.609144\pi$
$733$	−18.2049	−0.672414	−0.336207	+	0.941788i	$0.609144\pi$
$734$	0	0
$735$	1.81616	0.0669901
$736$	0	0
$737$	−5.00552	−0.184381
$738$	0	0
$739$	−17.5847	−0.646865	−0.323432	−	0.946251i	$-0.604837\pi$
$739$	−17.5847	−0.646865	−0.323432	+	0.946251i	$0.604837\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−27.6558	−1.01459	−0.507296	−	0.861772i	$-0.669355\pi$
$743$	−27.6558	−1.01459	−0.507296	+	0.861772i	$0.669355\pi$
$744$	0	0
$745$	−27.2614	−0.998781
$746$	0	0
$747$	−4.02283	−0.147188
$748$	0	0
$749$	−7.85797	−0.287124
$750$	0	0
$751$	29.5710	1.07906	0.539531	−	0.841966i	$-0.318602\pi$
$751$	29.5710	1.07906	0.539531	+	0.841966i	$0.318602\pi$
$752$	0	0
$753$	−32.6844	−1.19109
$754$	0	0
$755$	26.6627	0.970356
$756$	0	0
$757$	−18.6445	−0.677645	−0.338822	−	0.940850i	$-0.610029\pi$
$757$	−18.6445	−0.677645	−0.338822	+	0.940850i	$0.610029\pi$
$758$	0	0
$759$	−21.5526	−0.782312
$760$	0	0
$761$	−6.37214	−0.230990	−0.115495	−	0.993308i	$-0.536845\pi$
$761$	−6.37214	−0.230990	−0.115495	+	0.993308i	$0.536845\pi$
$762$	0	0
$763$	−6.80980	−0.246531
$764$	0	0
$765$	−1.15641	−0.0418100
$766$	0	0
$767$	20.7099	0.747791
$768$	0	0
$769$	−5.65778	−0.204025	−0.102012	−	0.994783i	$-0.532528\pi$
$769$	−5.65778	−0.204025	−0.102012	+	0.994783i	$0.532528\pi$
$770$	0	0
$771$	−7.29968	−0.262892
$772$	0	0
$773$	−2.99843	−0.107846	−0.0539230	−	0.998545i	$-0.517173\pi$
$773$	−2.99843	−0.107846	−0.0539230	+	0.998545i	$0.517173\pi$
$774$	0	0
$775$	37.4111	1.34384
$776$	0	0
$777$	10.3550	0.371482
$778$	0	0
$779$	0	0
$780$	0	0
$781$	13.8782	0.496601
$782$	0	0
$783$	2.57142	0.0918949
$784$	0	0
$785$	16.0160	0.571636
$786$	0	0
$787$	1.24507	0.0443819	0.0221910	−	0.999754i	$-0.492936\pi$
$787$	1.24507	0.0443819	0.0221910	+	0.999754i	$0.492936\pi$
$788$	0	0
$789$	−9.82962	−0.349944
$790$	0	0
$791$	17.1019	0.608072
$792$	0	0
$793$	1.11460	0.0395805
$794$	0	0
$795$	−11.9850	−0.425065
$796$	0	0
$797$	17.1019	0.605779	0.302890	−	0.953026i	$-0.402049\pi$
$797$	17.1019	0.605779	0.302890	+	0.953026i	$0.402049\pi$
$798$	0	0
$799$	−10.1500	−0.359083
$800$	0	0
$801$	6.98884	0.246939
$802$	0	0
$803$	27.7880	0.980617
$804$	0	0
$805$	3.63232	0.128022
$806$	0	0
$807$	24.5121	0.862866
$808$	0	0
$809$	−25.4325	−0.894160	−0.447080	−	0.894494i	$-0.647536\pi$
$809$	−25.4325	−0.894160	−0.447080	+	0.894494i	$0.647536\pi$
$810$	0	0
$811$	−3.95596	−0.138912	−0.0694562	−	0.997585i	$-0.522126\pi$
$811$	−3.95596	−0.138912	−0.0694562	+	0.997585i	$0.522126\pi$
$812$	0	0
$813$	−18.4508	−0.647097
$814$	0	0
$815$	−27.5001	−0.963287
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−1.83723	−0.0641979
$820$	0	0
$821$	−22.2838	−0.777710	−0.388855	−	0.921299i	$-0.627129\pi$
$821$	−22.2838	−0.777710	−0.388855	+	0.921299i	$0.627129\pi$
$822$	0	0
$823$	−28.3728	−0.989014	−0.494507	−	0.869174i	$-0.664651\pi$
$823$	−28.3728	−0.989014	−0.494507	+	0.869174i	$0.664651\pi$
$824$	0	0
$825$	−25.5105	−0.888162
$826$	0	0
$827$	−21.2566	−0.739165	−0.369583	−	0.929198i	$-0.620499\pi$
$827$	−21.2566	−0.739165	−0.369583	+	0.929198i	$0.620499\pi$
$828$	0	0
$829$	−28.1966	−0.979310	−0.489655	−	0.871916i	$-0.662877\pi$
$829$	−28.1966	−0.979310	−0.489655	+	0.871916i	$0.662877\pi$
$830$	0	0
$831$	5.34943	0.185570
$832$	0	0
$833$	1.00000	0.0346479
$834$	0	0
$835$	18.1999	0.629832
$836$	0	0
$837$	61.9275	2.14053
$838$	0	0
$839$	8.32554	0.287430	0.143715	−	0.989619i	$-0.454095\pi$
$839$	8.32554	0.287430	0.143715	+	0.989619i	$0.454095\pi$
$840$	0	0
$841$	−28.7931	−0.992866
$842$	0	0
$843$	−43.8571	−1.51052
$844$	0	0
$845$	−11.3298	−0.389755
$846$	0	0
$847$	−15.8062	−0.543109
$848$	0	0
$849$	6.07476	0.208485
$850$	0	0
$851$	20.7099	0.709926
$852$	0	0
$853$	54.9516	1.88151	0.940753	−	0.339093i	$-0.110120\pi$
$853$	54.9516	1.88151	0.940753	+	0.339093i	$0.110120\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	8.61630	0.294327	0.147164	−	0.989112i	$-0.452986\pi$
$857$	8.61630	0.294327	0.147164	+	0.989112i	$0.452986\pi$
$858$	0	0
$859$	27.8723	0.950990	0.475495	−	0.879719i	$-0.342269\pi$
$859$	27.8723	0.950990	0.475495	+	0.879719i	$0.342269\pi$
$860$	0	0
$861$	−4.64124	−0.158173
$862$	0	0
$863$	26.8749	0.914832	0.457416	−	0.889253i	$-0.348775\pi$
$863$	26.8749	0.914832	0.457416	+	0.889253i	$0.348775\pi$
$864$	0	0
$865$	2.67281	0.0908782
$866$	0	0
$867$	1.44270	0.0489967
$868$	0	0
$869$	52.5516	1.78269
$870$	0	0
$871$	1.93358	0.0655167
$872$	0	0
$873$	1.36369	0.0461540
$874$	0	0
$875$	10.5937	0.358131
$876$	0	0
$877$	37.8254	1.27727	0.638637	−	0.769508i	$-0.279499\pi$
$877$	37.8254	1.27727	0.638637	+	0.769508i	$0.279499\pi$
$878$	0	0
$879$	12.3493	0.416532
$880$	0	0
$881$	23.5139	0.792204	0.396102	−	0.918207i	$-0.370363\pi$
$881$	23.5139	0.792204	0.396102	+	0.918207i	$0.370363\pi$
$882$	0	0
$883$	28.5927	0.962220	0.481110	−	0.876660i	$-0.340234\pi$
$883$	28.5927	0.962220	0.481110	+	0.876660i	$0.340234\pi$
$884$	0	0
$885$	−18.8062	−0.632165
$886$	0	0
$887$	−15.9480	−0.535482	−0.267741	−	0.963491i	$-0.586277\pi$
$887$	−15.9480	−0.535482	−0.267741	+	0.963491i	$0.586277\pi$
$888$	0	0
$889$	−21.8395	−0.732472
$890$	0	0
$891$	−27.9600	−0.936694
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−8.33388	−0.278571
$896$	0	0
$897$	8.32554	0.277982
$898$	0	0
$899$	4.98240	0.166172
$900$	0	0
$901$	−6.59911	−0.219848
$902$	0	0
$903$	13.7137	0.456364
$904$	0	0
$905$	−23.1019	−0.767932
$906$	0	0
$907$	30.5130	1.01317	0.506584	−	0.862191i	$-0.330908\pi$
$907$	30.5130	1.01317	0.506584	+	0.862191i	$0.330908\pi$
$908$	0	0
$909$	−1.49947	−0.0497344
$910$	0	0
$911$	−12.0421	−0.398974	−0.199487	−	0.979900i	$-0.563928\pi$
$911$	−12.0421	−0.398974	−0.199487	+	0.979900i	$0.563928\pi$
$912$	0	0
$913$	22.6734	0.750380
$914$	0	0
$915$	−1.01214	−0.0334605
$916$	0	0
$917$	0.225649	0.00745159
$918$	0	0
$919$	22.7530	0.750553	0.375277	−	0.926913i	$-0.377548\pi$
$919$	22.7530	0.750553	0.375277	+	0.926913i	$0.377548\pi$
$920$	0	0
$921$	27.5583	0.908076
$922$	0	0
$923$	−5.36099	−0.176459
$924$	0	0
$925$	24.5130	0.805983
$926$	0	0
$927$	−16.7884	−0.551403
$928$	0	0
$929$	−29.7316	−0.975461	−0.487730	−	0.872994i	$-0.662175\pi$
$929$	−29.7316	−0.975461	−0.487730	+	0.872994i	$0.662175\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−26.0860	−0.854018
$934$	0	0
$935$	6.51772	0.213152
$936$	0	0
$937$	−0.138483	−0.00452406	−0.00226203	−	0.999997i	$-0.500720\pi$
$937$	−0.138483	−0.00452406	−0.00226203	+	0.999997i	$0.500720\pi$
$938$	0	0
$939$	−24.7553	−0.807857
$940$	0	0
$941$	−7.41212	−0.241628	−0.120814	−	0.992675i	$-0.538551\pi$
$941$	−7.41212	−0.241628	−0.120814	+	0.992675i	$0.538551\pi$
$942$	0	0
$943$	−9.28249	−0.302279
$944$	0	0
$945$	7.11683	0.231511
$946$	0	0
$947$	−60.7167	−1.97303	−0.986515	−	0.163674i	$-0.947666\pi$
$947$	−60.7167	−1.97303	−0.986515	+	0.163674i	$0.947666\pi$
$948$	0	0
$949$	−10.7342	−0.348447
$950$	0	0
$951$	−3.75807	−0.121864
$952$	0	0
$953$	48.3515	1.56626	0.783130	−	0.621858i	$-0.213622\pi$
$953$	48.3515	1.56626	0.783130	+	0.621858i	$0.213622\pi$
$954$	0	0
$955$	−14.0476	−0.454569
$956$	0	0
$957$	−3.39749	−0.109825
$958$	0	0
$959$	−17.6556	−0.570130
$960$	0	0
$961$	88.9914	2.87069
$962$	0	0
$963$	−7.21844	−0.232611
$964$	0	0
$965$	3.08584	0.0993368
$966$	0	0
$967$	−11.3555	−0.365169	−0.182585	−	0.983190i	$-0.558446\pi$
$967$	−11.3555	−0.365169	−0.182585	+	0.983190i	$0.558446\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−53.5034	−1.71701	−0.858503	−	0.512808i	$-0.828605\pi$
$971$	−53.5034	−1.71701	−0.858503	+	0.512808i	$0.828605\pi$
$972$	0	0
$973$	−15.7639	−0.505366
$974$	0	0
$975$	9.85442	0.315594
$976$	0	0
$977$	24.0559	0.769617	0.384809	−	0.922996i	$-0.374267\pi$
$977$	24.0559	0.769617	0.384809	+	0.922996i	$0.374267\pi$
$978$	0	0
$979$	−39.3904	−1.25892
$980$	0	0
$981$	−6.25557	−0.199725
$982$	0	0
$983$	−41.1762	−1.31332	−0.656658	−	0.754189i	$-0.728030\pi$
$983$	−41.1762	−1.31332	−0.656658	+	0.754189i	$0.728030\pi$
$984$	0	0
$985$	−6.22920	−0.198479
$986$	0	0
$987$	14.6435	0.466107
$988$	0	0
$989$	27.4274	0.872141
$990$	0	0
$991$	6.63836	0.210874	0.105437	−	0.994426i	$-0.466376\pi$
$991$	6.63836	0.210874	0.105437	+	0.994426i	$0.466376\pi$
$992$	0	0
$993$	−10.1468	−0.322000
$994$	0	0
$995$	7.97067	0.252687
$996$	0	0
$997$	−8.55730	−0.271012	−0.135506	−	0.990776i	$-0.543266\pi$
$997$	−8.55730	−0.271012	−0.135506	+	0.990776i	$0.543266\pi$
$998$	0	0
$999$	40.5771	1.28380

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7616.2.a.bo.1.3		4
4.3	odd	2		7616.2.a.bi.1.2		4
8.3	odd	2		952.2.a.h.1.3	✓	4
8.5	even	2		1904.2.a.r.1.2		4
24.11	even	2		8568.2.a.bg.1.4		4
56.27	even	2		6664.2.a.n.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
952.2.a.h.1.3	✓	4	8.3	odd	2
1904.2.a.r.1.2		4	8.5	even	2
6664.2.a.n.1.2		4	56.27	even	2
7616.2.a.bi.1.2		4	4.3	odd	2
7616.2.a.bo.1.3		4	1.1	even	1	trivial
8568.2.a.bg.1.4		4	24.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.44270 q^{3} +1.25886 q^{5} -1.00000 q^{7} -0.918614 q^{9} +5.17748 q^{11} -2.00000 q^{13} +1.81616 q^{15} +1.00000 q^{17} -1.44270 q^{21} -2.88540 q^{23} -3.41527 q^{25} -5.65339 q^{27} -0.454845 q^{29} -10.9541 q^{31} +7.46955 q^{33} -1.25886 q^{35} -7.17748 q^{37} -2.88540 q^{39} +3.21705 q^{41} -9.50558 q^{43} -1.15641 q^{45} -10.1500 q^{47} +1.00000 q^{49} +1.44270 q^{51} -6.59911 q^{53} +6.51772 q^{55} -10.3550 q^{59} -0.557299 q^{61} +0.918614 q^{63} -2.51772 q^{65} -0.966788 q^{67} -4.16277 q^{69} +2.68049 q^{71} +5.36709 q^{73} -4.92721 q^{75} -5.17748 q^{77} +10.1500 q^{79} -5.40031 q^{81} +4.37924 q^{83} +1.25886 q^{85} -0.656206 q^{87} -7.60803 q^{89} +2.00000 q^{91} -15.8034 q^{93} -1.48451 q^{97} -4.75610 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 3 q^{3} - 5 q^{5} - 4 q^{7} + 7 q^{9} - 8 q^{13} + 4 q^{17} - 3 q^{21} - 6 q^{23} + 3 q^{25} + 6 q^{27} - 8 q^{29} + 7 q^{31} - 6 q^{33} + 5 q^{35} - 8 q^{37} - 6 q^{39} + 15 q^{41} - 9 q^{43} + 2 q^{45}+ \cdots - 50 q^{99}+O(q^{100}) \) 4 * q + 3 * q^3 - 5 * q^5 - 4 * q^7 + 7 * q^9 - 8 * q^13 + 4 * q^17 - 3 * q^21 - 6 * q^23 + 3 * q^25 + 6 * q^27 - 8 * q^29 + 7 * q^31 - 6 * q^33 + 5 * q^35 - 8 * q^37 - 6 * q^39 + 15 * q^41 - 9 * q^43 + 2 * q^45 - 6 * q^47 + 4 * q^49 + 3 * q^51 - 17 * q^53 + 6 * q^55 - 5 * q^61 - 7 * q^63 + 10 * q^65 - 9 * q^67 - 38 * q^69 + 12 * q^71 - 11 * q^73 - 2 * q^75 + 6 * q^79 + 16 * q^81 - 6 * q^83 - 5 * q^85 + 14 * q^87 + 2 * q^89 + 8 * q^91 + 9 * q^97 - 50 * q^99

Introduction

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