LMFDB - Embedded newform 675.4.a.m.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [675,4,Mod(1,675)] 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("675.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(675, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	675.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,10,0,0,18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$39.8262892539$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.30278$ of defining polynomial
Character	$\chi$	$=$	675.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.60555 q^{2} +5.00000 q^{4} +9.00000 q^{7} +10.8167 q^{8} +14.4222 q^{11} -34.0000 q^{13} -32.4500 q^{14} -79.0000 q^{16} +7.21110 q^{17} -101.000 q^{19} -52.0000 q^{22} +108.167 q^{23} +122.589 q^{26} +45.0000 q^{28} -165.855 q^{29} -3.00000 q^{31} +198.305 q^{32} -26.0000 q^{34} +67.0000 q^{37} +364.161 q^{38} +201.911 q^{41} +137.000 q^{43} +72.1110 q^{44} -390.000 q^{46} +115.378 q^{47} -262.000 q^{49} -170.000 q^{52} +677.844 q^{53} +97.3499 q^{56} +598.000 q^{58} -872.543 q^{59} -563.000 q^{61} +10.8167 q^{62} -83.0000 q^{64} -1044.00 q^{67} +36.0555 q^{68} +786.010 q^{71} +503.000 q^{73} -241.572 q^{74} -505.000 q^{76} +129.800 q^{77} +615.000 q^{79} -728.000 q^{82} -237.966 q^{83} -493.961 q^{86} +156.000 q^{88} +973.499 q^{89} -306.000 q^{91} +540.833 q^{92} -416.000 q^{94} +19.0000 q^{97} +944.654 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 10 q^{4} + 18 q^{7} - 68 q^{13} - 158 q^{16} - 202 q^{19} - 104 q^{22} + 90 q^{28} - 6 q^{31} - 52 q^{34} + 134 q^{37} + 274 q^{43} - 780 q^{46} - 524 q^{49} - 340 q^{52} + 1196 q^{58} - 1126 q^{61}+ \cdots + 38 q^{97}+O(q^{100}) $ 2 * q + 10 * q^4 + 18 * q^7 - 68 * q^13 - 158 * q^16 - 202 * q^19 - 104 * q^22 + 90 * q^28 - 6 * q^31 - 52 * q^34 + 134 * q^37 + 274 * q^43 - 780 * q^46 - 524 * q^49 - 340 * q^52 + 1196 * q^58 - 1126 * q^61 - 166 * q^64 - 2088 * q^67 + 1006 * q^73 - 1010 * q^76 + 1230 * q^79 - 1456 * q^82 + 312 * q^88 - 612 * q^91 - 832 * q^94 + 38 * q^97

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 $

$2$	−3.60555	−1.27475	−0.637377	−	0.770552i	$-0.719981\pi$
$2$	−3.60555	−1.27475	−0.637377	+	0.770552i	$0.719981\pi$
$3$	0	0
$3$	0	0
$4$	5.00000	0.625000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	9.00000	0.485954	0.242977	−	0.970032i	$-0.421876\pi$
$7$	9.00000	0.485954	0.242977	+	0.970032i	$0.421876\pi$
$8$	10.8167	0.478033
$9$	0	0
$10$	0	0
$11$	14.4222	0.395314	0.197657	−	0.980271i	$-0.436667\pi$
$11$	14.4222	0.395314	0.197657	+	0.980271i	$0.436667\pi$
$12$	0	0
$13$	−34.0000	−0.725377	−0.362689	−	0.931910i	$-0.618141\pi$
$13$	−34.0000	−0.725377	−0.362689	+	0.931910i	$0.618141\pi$
$14$	−32.4500	−0.619473
$15$	0	0
$16$	−79.0000	−1.23438
$17$	7.21110	0.102879	0.0514397	−	0.998676i	$-0.483619\pi$
$17$	7.21110	0.102879	0.0514397	+	0.998676i	$0.483619\pi$
$18$	0	0
$19$	−101.000	−1.21953	−0.609763	−	0.792584i	$-0.708735\pi$
$19$	−101.000	−1.21953	−0.609763	+	0.792584i	$0.708735\pi$
$20$	0	0
$21$	0	0
$22$	−52.0000	−0.503929
$23$	108.167	0.980621	0.490310	−	0.871548i	$-0.336883\pi$
$23$	108.167	0.980621	0.490310	+	0.871548i	$0.336883\pi$
$24$	0	0
$25$	0	0
$26$	122.589	0.924678
$27$	0	0
$28$	45.0000	0.303721
$29$	−165.855	−1.06202	−0.531010	−	0.847366i	$-0.678187\pi$
$29$	−165.855	−1.06202	−0.531010	+	0.847366i	$0.678187\pi$
$30$	0	0
$31$	−3.00000	−0.0173812	−0.00869058	−	0.999962i	$-0.502766\pi$
$31$	−3.00000	−0.0173812	−0.00869058	+	0.999962i	$0.502766\pi$
$32$	198.305	1.09549
$33$	0	0
$34$	−26.0000	−0.131146
$35$	0	0
$36$	0	0
$37$	67.0000	0.297695	0.148848	−	0.988860i	$-0.452444\pi$
$37$	67.0000	0.297695	0.148848	+	0.988860i	$0.452444\pi$
$38$	364.161	1.55460
$39$	0	0
$40$	0	0
$41$	201.911	0.769102	0.384551	−	0.923104i	$-0.374356\pi$
$41$	201.911	0.769102	0.384551	+	0.923104i	$0.374356\pi$
$42$	0	0
$43$	137.000	0.485868	0.242934	−	0.970043i	$-0.421890\pi$
$43$	137.000	0.485868	0.242934	+	0.970043i	$0.421890\pi$
$44$	72.1110	0.247072
$45$	0	0
$46$	−390.000	−1.25005
$47$	115.378	0.358076	0.179038	−	0.983842i	$-0.442702\pi$
$47$	115.378	0.358076	0.179038	+	0.983842i	$0.442702\pi$
$48$	0	0
$49$	−262.000	−0.763848
$50$	0	0
$51$	0	0
$52$	−170.000	−0.453361
$53$	677.844	1.75677	0.878387	−	0.477951i	$-0.158620\pi$
$53$	677.844	1.75677	0.878387	+	0.477951i	$0.158620\pi$
$54$	0	0
$55$	0	0
$56$	97.3499	0.232302
$57$	0	0
$58$	598.000	1.35381
$59$	−872.543	−1.92535	−0.962674	−	0.270665i	$-0.912756\pi$
$59$	−872.543	−1.92535	−0.962674	+	0.270665i	$0.912756\pi$
$60$	0	0
$61$	−563.000	−1.18172	−0.590859	−	0.806775i	$-0.701211\pi$
$61$	−563.000	−1.18172	−0.590859	+	0.806775i	$0.701211\pi$
$62$	10.8167	0.0221567
$63$	0	0
$64$	−83.0000	−0.162109
$65$	0	0
$66$	0	0
$67$	−1044.00	−1.90366	−0.951828	−	0.306634i	$-0.900797\pi$
$67$	−1044.00	−1.90366	−0.951828	+	0.306634i	$0.900797\pi$
$68$	36.0555	0.0642996
$69$	0	0
$70$	0	0
$71$	786.010	1.31383	0.656917	−	0.753963i	$-0.271860\pi$
$71$	786.010	1.31383	0.656917	+	0.753963i	$0.271860\pi$
$72$	0	0
$73$	503.000	0.806462	0.403231	−	0.915098i	$-0.367887\pi$
$73$	503.000	0.806462	0.403231	+	0.915098i	$0.367887\pi$
$74$	−241.572	−0.379489
$75$	0	0
$76$	−505.000	−0.762204
$77$	129.800	0.192105
$78$	0	0
$79$	615.000	0.875860	0.437930	−	0.899009i	$-0.355712\pi$
$79$	615.000	0.875860	0.437930	+	0.899009i	$0.355712\pi$
$80$	0	0
$81$	0	0
$82$	−728.000	−0.980416
$83$	−237.966	−0.314701	−0.157351	−	0.987543i	$-0.550295\pi$
$83$	−237.966	−0.314701	−0.157351	+	0.987543i	$0.550295\pi$
$84$	0	0
$85$	0	0
$86$	−493.961	−0.619362
$87$	0	0
$88$	156.000	0.188973
$89$	973.499	1.15945	0.579723	−	0.814814i	$-0.303161\pi$
$89$	973.499	1.15945	0.579723	+	0.814814i	$0.303161\pi$
$90$	0	0
$91$	−306.000	−0.352500
$92$	540.833	0.612888
$93$	0	0
$94$	−416.000	−0.456459
$95$	0	0
$96$	0	0
$97$	19.0000	0.0198882	0.00994411	−	0.999951i	$-0.496835\pi$
$97$	19.0000	0.0198882	0.00994411	+	0.999951i	$0.496835\pi$
$98$	944.654	0.973719
$99$	0	0
$100$	0	0
$101$	259.600	0.255754	0.127877	−	0.991790i	$-0.459184\pi$
$101$	259.600	0.255754	0.127877	+	0.991790i	$0.459184\pi$
$102$	0	0
$103$	−149.000	−0.142538	−0.0712690	−	0.997457i	$-0.522705\pi$
$103$	−149.000	−0.142538	−0.0712690	+	0.997457i	$0.522705\pi$
$104$	−367.766	−0.346754
$105$	0	0
$106$	−2444.00	−2.23946
$107$	−2055.16	−1.85682	−0.928412	−	0.371552i	$-0.878826\pi$
$107$	−2055.16	−1.85682	−0.928412	+	0.371552i	$0.878826\pi$
$108$	0	0
$109$	−1177.00	−1.03428	−0.517138	−	0.855902i	$-0.673003\pi$
$109$	−1177.00	−1.03428	−0.517138	+	0.855902i	$0.673003\pi$
$110$	0	0
$111$	0	0
$112$	−711.000	−0.599850
$113$	−2004.69	−1.66889	−0.834447	−	0.551088i	$-0.814213\pi$
$113$	−2004.69	−1.66889	−0.834447	+	0.551088i	$0.814213\pi$
$114$	0	0
$115$	0	0
$116$	−829.277	−0.663762
$117$	0	0
$118$	3146.00	2.45435
$119$	64.8999	0.0499947
$120$	0	0
$121$	−1123.00	−0.843727
$122$	2029.93	1.50640
$123$	0	0
$124$	−15.0000	−0.0108632
$125$	0	0
$126$	0	0
$127$	296.000	0.206817	0.103408	−	0.994639i	$-0.467025\pi$
$127$	296.000	0.206817	0.103408	+	0.994639i	$0.467025\pi$
$128$	−1287.18	−0.888843
$129$	0	0
$130$	0	0
$131$	−1665.76	−1.11098	−0.555491	−	0.831523i	$-0.687470\pi$
$131$	−1665.76	−1.11098	−0.555491	+	0.831523i	$0.687470\pi$
$132$	0	0
$133$	−909.000	−0.592634
$134$	3764.20	2.42669
$135$	0	0
$136$	78.0000	0.0491797
$137$	−1211.47	−0.755492	−0.377746	−	0.925909i	$-0.623301\pi$
$137$	−1211.47	−0.755492	−0.377746	+	0.925909i	$0.623301\pi$
$138$	0	0
$139$	−1873.00	−1.14292	−0.571460	−	0.820630i	$-0.693623\pi$
$139$	−1873.00	−1.14292	−0.571460	+	0.820630i	$0.693623\pi$
$140$	0	0
$141$	0	0
$142$	−2834.00	−1.67482
$143$	−490.355	−0.286752
$144$	0	0
$145$	0	0
$146$	−1813.59	−1.02804
$147$	0	0
$148$	335.000	0.186060
$149$	−2329.19	−1.28063	−0.640317	−	0.768111i	$-0.721197\pi$
$149$	−2329.19	−1.28063	−0.640317	+	0.768111i	$0.721197\pi$
$150$	0	0
$151$	−1031.00	−0.555640	−0.277820	−	0.960633i	$-0.589612\pi$
$151$	−1031.00	−0.555640	−0.277820	+	0.960633i	$0.589612\pi$
$152$	−1092.48	−0.582974
$153$	0	0
$154$	−468.000	−0.244886
$155$	0	0
$156$	0	0
$157$	2443.00	1.24186	0.620932	−	0.783864i	$-0.286754\pi$
$157$	2443.00	1.24186	0.620932	+	0.783864i	$0.286754\pi$
$158$	−2217.41	−1.11651
$159$	0	0
$160$	0	0
$161$	973.499	0.476537
$162$	0	0
$163$	−188.000	−0.0903392	−0.0451696	−	0.998979i	$-0.514383\pi$
$163$	−188.000	−0.0903392	−0.0451696	+	0.998979i	$0.514383\pi$
$164$	1009.55	0.480689
$165$	0	0
$166$	858.000	0.401167
$167$	−1492.70	−0.691667	−0.345834	−	0.938296i	$-0.612404\pi$
$167$	−1492.70	−0.691667	−0.345834	+	0.938296i	$0.612404\pi$
$168$	0	0
$169$	−1041.00	−0.473828
$170$	0	0
$171$	0	0
$172$	685.000	0.303667
$173$	−3136.83	−1.37855	−0.689274	−	0.724501i	$-0.742070\pi$
$173$	−3136.83	−1.37855	−0.689274	+	0.724501i	$0.742070\pi$
$174$	0	0
$175$	0	0
$176$	−1139.35	−0.487966
$177$	0	0
$178$	−3510.00	−1.47801
$179$	2913.29	1.21648	0.608238	−	0.793755i	$-0.291877\pi$
$179$	2913.29	1.21648	0.608238	+	0.793755i	$0.291877\pi$
$180$	0	0
$181$	−286.000	−0.117449	−0.0587243	−	0.998274i	$-0.518703\pi$
$181$	−286.000	−0.117449	−0.0587243	+	0.998274i	$0.518703\pi$
$182$	1103.30	0.449351
$183$	0	0
$184$	1170.00	0.468769
$185$	0	0
$186$	0	0
$187$	104.000	0.0406697
$188$	576.888	0.223797
$189$	0	0
$190$	0	0
$191$	850.910	0.322354	0.161177	−	0.986925i	$-0.448471\pi$
$191$	850.910	0.322354	0.161177	+	0.986925i	$0.448471\pi$
$192$	0	0
$193$	3001.00	1.11926	0.559629	−	0.828743i	$-0.310944\pi$
$193$	3001.00	1.11926	0.559629	+	0.828743i	$0.310944\pi$
$194$	−68.5055	−0.0253526
$195$	0	0
$196$	−1310.00	−0.477405
$197$	−3807.46	−1.37701	−0.688504	−	0.725233i	$-0.741732\pi$
$197$	−3807.46	−1.37701	−0.688504	+	0.725233i	$0.741732\pi$
$198$	0	0
$199$	−1544.00	−0.550006	−0.275003	−	0.961443i	$-0.588679\pi$
$199$	−1544.00	−0.550006	−0.275003	+	0.961443i	$0.588679\pi$
$200$	0	0
$201$	0	0
$202$	−936.000	−0.326023
$203$	−1492.70	−0.516093
$204$	0	0
$205$	0	0
$206$	537.227	0.181701
$207$	0	0
$208$	2686.00	0.895387
$209$	−1456.64	−0.482096
$210$	0	0
$211$	−1612.00	−0.525946	−0.262973	−	0.964803i	$-0.584703\pi$
$211$	−1612.00	−0.525946	−0.262973	+	0.964803i	$0.584703\pi$
$212$	3389.22	1.09798
$213$	0	0
$214$	7410.00	2.36700
$215$	0	0
$216$	0	0
$217$	−27.0000	−0.00844645
$218$	4243.73	1.31845
$219$	0	0
$220$	0	0
$221$	−245.177	−0.0746263
$222$	0	0
$223$	−6113.00	−1.83568	−0.917840	−	0.396950i	$-0.870069\pi$
$223$	−6113.00	−1.83568	−0.917840	+	0.396950i	$0.870069\pi$
$224$	1784.75	0.532359
$225$	0	0
$226$	7228.00	2.12743
$227$	187.489	0.0548196	0.0274098	−	0.999624i	$-0.491274\pi$
$227$	187.489	0.0548196	0.0274098	+	0.999624i	$0.491274\pi$
$228$	0	0
$229$	2199.00	0.634559	0.317279	−	0.948332i	$-0.397231\pi$
$229$	2199.00	0.634559	0.317279	+	0.948332i	$0.397231\pi$
$230$	0	0
$231$	0	0
$232$	−1794.00	−0.507680
$233$	1492.70	0.419699	0.209850	−	0.977734i	$-0.432703\pi$
$233$	1492.70	0.419699	0.209850	+	0.977734i	$0.432703\pi$
$234$	0	0
$235$	0	0
$236$	−4362.72	−1.20334
$237$	0	0
$238$	−234.000	−0.0637310
$239$	−5350.64	−1.44813	−0.724067	−	0.689730i	$-0.757729\pi$
$239$	−5350.64	−1.44813	−0.724067	+	0.689730i	$0.757729\pi$
$240$	0	0
$241$	1930.00	0.515860	0.257930	−	0.966164i	$-0.416960\pi$
$241$	1930.00	0.515860	0.257930	+	0.966164i	$0.416960\pi$
$242$	4049.03	1.07554
$243$	0	0
$244$	−2815.00	−0.738573
$245$	0	0
$246$	0	0
$247$	3434.00	0.884616
$248$	−32.4500	−0.00830877
$249$	0	0
$250$	0	0
$251$	3526.23	0.886747	0.443374	−	0.896337i	$-0.353781\pi$
$251$	3526.23	0.886747	0.443374	+	0.896337i	$0.353781\pi$
$252$	0	0
$253$	1560.00	0.387654
$254$	−1067.24	−0.263641
$255$	0	0
$256$	5305.00	1.29517
$257$	−995.132	−0.241536	−0.120768	−	0.992681i	$-0.538536\pi$
$257$	−995.132	−0.241536	−0.120768	+	0.992681i	$0.538536\pi$
$258$	0	0
$259$	603.000	0.144666
$260$	0	0
$261$	0	0
$262$	6006.00	1.41623
$263$	2026.32	0.475088	0.237544	−	0.971377i	$-0.423658\pi$
$263$	2026.32	0.475088	0.237544	+	0.971377i	$0.423658\pi$
$264$	0	0
$265$	0	0
$266$	3277.45	0.755463
$267$	0	0
$268$	−5220.00	−1.18978
$269$	−2040.74	−0.462551	−0.231276	−	0.972888i	$-0.574290\pi$
$269$	−2040.74	−0.462551	−0.231276	+	0.972888i	$0.574290\pi$
$270$	0	0
$271$	6691.00	1.49981	0.749906	−	0.661544i	$-0.230098\pi$
$271$	6691.00	1.49981	0.749906	+	0.661544i	$0.230098\pi$
$272$	−569.677	−0.126992
$273$	0	0
$274$	4368.00	0.963068
$275$	0	0
$276$	0	0
$277$	−2139.00	−0.463971	−0.231986	−	0.972719i	$-0.574522\pi$
$277$	−2139.00	−0.463971	−0.231986	+	0.972719i	$0.574522\pi$
$278$	6753.20	1.45694
$279$	0	0
$280$	0	0
$281$	7614.92	1.61661	0.808307	−	0.588762i	$-0.200384\pi$
$281$	7614.92	1.61661	0.808307	+	0.588762i	$0.200384\pi$
$282$	0	0
$283$	−3297.00	−0.692531	−0.346266	−	0.938137i	$-0.612550\pi$
$283$	−3297.00	−0.692531	−0.346266	+	0.938137i	$0.612550\pi$
$284$	3930.05	0.821147
$285$	0	0
$286$	1768.00	0.365539
$287$	1817.20	0.373748
$288$	0	0
$289$	−4861.00	−0.989416
$290$	0	0
$291$	0	0
$292$	2515.00	0.504039
$293$	7247.16	1.44499	0.722497	−	0.691374i	$-0.242994\pi$
$293$	7247.16	1.44499	0.722497	+	0.691374i	$0.242994\pi$
$294$	0	0
$295$	0	0
$296$	724.716	0.142308
$297$	0	0
$298$	8398.00	1.63249
$299$	−3677.66	−0.711320
$300$	0	0
$301$	1233.00	0.236109
$302$	3717.32	0.708304
$303$	0	0
$304$	7979.00	1.50535
$305$	0	0
$306$	0	0
$307$	6335.00	1.17771	0.588856	−	0.808238i	$-0.299579\pi$
$307$	6335.00	1.17771	0.588856	+	0.808238i	$0.299579\pi$
$308$	648.999	0.120065
$309$	0	0
$310$	0	0
$311$	173.066	0.0315553	0.0157777	−	0.999876i	$-0.494978\pi$
$311$	173.066	0.0315553	0.0157777	+	0.999876i	$0.494978\pi$
$312$	0	0
$313$	−9542.00	−1.72315	−0.861575	−	0.507631i	$-0.830521\pi$
$313$	−9542.00	−1.72315	−0.861575	+	0.507631i	$0.830521\pi$
$314$	−8808.36	−1.58307
$315$	0	0
$316$	3075.00	0.547412
$317$	1723.45	0.305359	0.152679	−	0.988276i	$-0.451210\pi$
$317$	1723.45	0.305359	0.152679	+	0.988276i	$0.451210\pi$
$318$	0	0
$319$	−2392.00	−0.419832
$320$	0	0
$321$	0	0
$322$	−3510.00	−0.607468
$323$	−728.321	−0.125464
$324$	0	0
$325$	0	0
$326$	677.844	0.115160
$327$	0	0
$328$	2184.00	0.367656
$329$	1038.40	0.174008
$330$	0	0
$331$	−6273.00	−1.04168	−0.520839	−	0.853655i	$-0.674381\pi$
$331$	−6273.00	−1.04168	−0.520839	+	0.853655i	$0.674381\pi$
$332$	−1189.83	−0.196688
$333$	0	0
$334$	5382.00	0.881706
$335$	0	0
$336$	0	0
$337$	6854.00	1.10790	0.553948	−	0.832551i	$-0.313121\pi$
$337$	6854.00	1.10790	0.553948	+	0.832551i	$0.313121\pi$
$338$	3753.38	0.604014
$339$	0	0
$340$	0	0
$341$	−43.2666	−0.00687102
$342$	0	0
$343$	−5445.00	−0.857150
$344$	1481.88	0.232261
$345$	0	0
$346$	11310.0	1.75731
$347$	−5530.92	−0.855663	−0.427832	−	0.903858i	$-0.640722\pi$
$347$	−5530.92	−0.855663	−0.427832	+	0.903858i	$0.640722\pi$
$348$	0	0
$349$	−5495.00	−0.842810	−0.421405	−	0.906873i	$-0.638463\pi$
$349$	−5495.00	−0.842810	−0.421405	+	0.906873i	$0.638463\pi$
$350$	0	0
$351$	0	0
$352$	2860.00	0.433064
$353$	432.666	0.0652365	0.0326183	−	0.999468i	$-0.489615\pi$
$353$	432.666	0.0652365	0.0326183	+	0.999468i	$0.489615\pi$
$354$	0	0
$355$	0	0
$356$	4867.49	0.724654
$357$	0	0
$358$	−10504.0	−1.55071
$359$	−6857.76	−1.00819	−0.504093	−	0.863649i	$-0.668173\pi$
$359$	−6857.76	−1.00819	−0.504093	+	0.863649i	$0.668173\pi$
$360$	0	0
$361$	3342.00	0.487243
$362$	1031.19	0.149718
$363$	0	0
$364$	−1530.00	−0.220313
$365$	0	0
$366$	0	0
$367$	9180.00	1.30570	0.652850	−	0.757487i	$-0.273573\pi$
$367$	9180.00	1.30570	0.652850	+	0.757487i	$0.273573\pi$
$368$	−8545.16	−1.21045
$369$	0	0
$370$	0	0
$371$	6100.59	0.853712
$372$	0	0
$373$	−7393.00	−1.02626	−0.513130	−	0.858311i	$-0.671514\pi$
$373$	−7393.00	−1.02626	−0.513130	+	0.858311i	$0.671514\pi$
$374$	−374.977	−0.0518439
$375$	0	0
$376$	1248.00	0.171172
$377$	5639.08	0.770365
$378$	0	0
$379$	−8480.00	−1.14931	−0.574655	−	0.818396i	$-0.694864\pi$
$379$	−8480.00	−1.14931	−0.574655	+	0.818396i	$0.694864\pi$
$380$	0	0
$381$	0	0
$382$	−3068.00	−0.410923
$383$	7160.62	0.955329	0.477664	−	0.878542i	$-0.341484\pi$
$383$	7160.62	0.955329	0.477664	+	0.878542i	$0.341484\pi$
$384$	0	0
$385$	0	0
$386$	−10820.3	−1.42678
$387$	0	0
$388$	95.0000	0.0124301
$389$	6446.73	0.840262	0.420131	−	0.907463i	$-0.361984\pi$
$389$	6446.73	0.840262	0.420131	+	0.907463i	$0.361984\pi$
$390$	0	0
$391$	780.000	0.100886
$392$	−2833.96	−0.365145
$393$	0	0
$394$	13728.0	1.75535
$395$	0	0
$396$	0	0
$397$	−4957.00	−0.626662	−0.313331	−	0.949644i	$-0.601445\pi$
$397$	−4957.00	−0.626662	−0.313331	+	0.949644i	$0.601445\pi$
$398$	5566.97	0.701123
$399$	0	0
$400$	0	0
$401$	−8307.19	−1.03452	−0.517258	−	0.855829i	$-0.673047\pi$
$401$	−8307.19	−1.03452	−0.517258	+	0.855829i	$0.673047\pi$
$402$	0	0
$403$	102.000	0.0126079
$404$	1298.00	0.159846
$405$	0	0
$406$	5382.00	0.657892
$407$	966.288	0.117683
$408$	0	0
$409$	−10630.0	−1.28513	−0.642567	−	0.766230i	$-0.722130\pi$
$409$	−10630.0	−1.28513	−0.642567	+	0.766230i	$0.722130\pi$
$410$	0	0
$411$	0	0
$412$	−745.000	−0.0890862
$413$	−7852.89	−0.935631
$414$	0	0
$415$	0	0
$416$	−6742.38	−0.794645
$417$	0	0
$418$	5252.00	0.614554
$419$	15366.9	1.79170	0.895848	−	0.444362i	$-0.146569\pi$
$419$	15366.9	1.79170	0.895848	+	0.444362i	$0.146569\pi$
$420$	0	0
$421$	−10277.0	−1.18972	−0.594858	−	0.803831i	$-0.702792\pi$
$421$	−10277.0	−1.18972	−0.594858	+	0.803831i	$0.702792\pi$
$422$	5812.15	0.670453
$423$	0	0
$424$	7332.00	0.839796
$425$	0	0
$426$	0	0
$427$	−5067.00	−0.574261
$428$	−10275.8	−1.16052
$429$	0	0
$430$	0	0
$431$	−7636.56	−0.853457	−0.426729	−	0.904380i	$-0.640334\pi$
$431$	−7636.56	−0.853457	−0.426729	+	0.904380i	$0.640334\pi$
$432$	0	0
$433$	12173.0	1.35103	0.675516	−	0.737345i	$-0.263921\pi$
$433$	12173.0	1.35103	0.675516	+	0.737345i	$0.263921\pi$
$434$	97.3499	0.0107672
$435$	0	0
$436$	−5885.00	−0.646423
$437$	−10924.8	−1.19589
$438$	0	0
$439$	8989.00	0.977270	0.488635	−	0.872488i	$-0.337495\pi$
$439$	8989.00	0.977270	0.488635	+	0.872488i	$0.337495\pi$
$440$	0	0
$441$	0	0
$442$	884.000	0.0951303
$443$	−11141.2	−1.19488	−0.597440	−	0.801913i	$-0.703816\pi$
$443$	−11141.2	−1.19488	−0.597440	+	0.801913i	$0.703816\pi$
$444$	0	0
$445$	0	0
$446$	22040.7	2.34004
$447$	0	0
$448$	−747.000	−0.0787778
$449$	−7968.27	−0.837519	−0.418759	−	0.908097i	$-0.637535\pi$
$449$	−7968.27	−0.837519	−0.418759	+	0.908097i	$0.637535\pi$
$450$	0	0
$451$	2912.00	0.304037
$452$	−10023.4	−1.04306
$453$	0	0
$454$	−676.000	−0.0698816
$455$	0	0
$456$	0	0
$457$	3566.00	0.365012	0.182506	−	0.983205i	$-0.441579\pi$
$457$	3566.00	0.365012	0.182506	+	0.983205i	$0.441579\pi$
$458$	−7928.61	−0.808907
$459$	0	0
$460$	0	0
$461$	−9280.69	−0.937624	−0.468812	−	0.883298i	$-0.655318\pi$
$461$	−9280.69	−0.937624	−0.468812	+	0.883298i	$0.655318\pi$
$462$	0	0
$463$	15.0000	0.00150564	0.000752818	−	1.00000i	$-0.499760\pi$
$463$	15.0000	0.00150564	0.000752818		1.00000i	$0.499760\pi$
$464$	13102.6	1.31093
$465$	0	0
$466$	−5382.00	−0.535014
$467$	6396.25	0.633797	0.316898	−	0.948460i	$-0.397359\pi$
$467$	6396.25	0.633797	0.316898	+	0.948460i	$0.397359\pi$
$468$	0	0
$469$	−9396.00	−0.925089
$470$	0	0
$471$	0	0
$472$	−9438.00	−0.920380
$473$	1975.84	0.192070
$474$	0	0
$475$	0	0
$476$	324.500	0.0312467
$477$	0	0
$478$	19292.0	1.84602
$479$	12525.7	1.19481	0.597404	−	0.801940i	$-0.296199\pi$
$479$	12525.7	1.19481	0.597404	+	0.801940i	$0.296199\pi$
$480$	0	0
$481$	−2278.00	−0.215941
$482$	−6958.71	−0.657595
$483$	0	0
$484$	−5615.00	−0.527329
$485$	0	0
$486$	0	0
$487$	6088.00	0.566476	0.283238	−	0.959050i	$-0.408591\pi$
$487$	6088.00	0.566476	0.283238	+	0.959050i	$0.408591\pi$
$488$	−6089.78	−0.564900
$489$	0	0
$490$	0	0
$491$	−15071.2	−1.38524	−0.692621	−	0.721302i	$-0.743544\pi$
$491$	−15071.2	−1.38524	−0.692621	+	0.721302i	$0.743544\pi$
$492$	0	0
$493$	−1196.00	−0.109260
$494$	−12381.5	−1.12767
$495$	0	0
$496$	237.000	0.0214549
$497$	7074.09	0.638464
$498$	0	0
$499$	2389.00	0.214321	0.107161	−	0.994242i	$-0.465824\pi$
$499$	2389.00	0.214321	0.107161	+	0.994242i	$0.465824\pi$
$500$	0	0
$501$	0	0
$502$	−12714.0	−1.13039
$503$	−17054.3	−1.51175	−0.755877	−	0.654714i	$-0.772789\pi$
$503$	−17054.3	−1.51175	−0.755877	+	0.654714i	$0.772789\pi$
$504$	0	0
$505$	0	0
$506$	−5624.66	−0.494163
$507$	0	0
$508$	1480.00	0.129261
$509$	8624.48	0.751028	0.375514	−	0.926817i	$-0.377466\pi$
$509$	8624.48	0.751028	0.375514	+	0.926817i	$0.377466\pi$
$510$	0	0
$511$	4527.00	0.391904
$512$	−8830.00	−0.762176
$513$	0	0
$514$	3588.00	0.307899
$515$	0	0
$516$	0	0
$517$	1664.00	0.141552
$518$	−2174.15	−0.184414
$519$	0	0
$520$	0	0
$521$	−7888.95	−0.663380	−0.331690	−	0.943388i	$-0.607619\pi$
$521$	−7888.95	−0.663380	−0.331690	+	0.943388i	$0.607619\pi$
$522$	0	0
$523$	−6163.00	−0.515276	−0.257638	−	0.966242i	$-0.582944\pi$
$523$	−6163.00	−0.515276	−0.257638	+	0.966242i	$0.582944\pi$
$524$	−8328.82	−0.694363
$525$	0	0
$526$	−7306.00	−0.605621
$527$	−21.6333	−0.00178816
$528$	0	0
$529$	−467.000	−0.0383825
$530$	0	0
$531$	0	0
$532$	−4545.00	−0.370396
$533$	−6864.97	−0.557889
$534$	0	0
$535$	0	0
$536$	−11292.6	−0.910010
$537$	0	0
$538$	7358.00	0.589639
$539$	−3778.62	−0.301960
$540$	0	0
$541$	−15765.0	−1.25285	−0.626424	−	0.779483i	$-0.715482\pi$
$541$	−15765.0	−1.25285	−0.626424	+	0.779483i	$0.715482\pi$
$542$	−24124.7	−1.91189
$543$	0	0
$544$	1430.00	0.112704
$545$	0	0
$546$	0	0
$547$	−7591.00	−0.593360	−0.296680	−	0.954977i	$-0.595879\pi$
$547$	−7591.00	−0.593360	−0.296680	+	0.954977i	$0.595879\pi$
$548$	−6057.33	−0.472183
$549$	0	0
$550$	0	0
$551$	16751.4	1.29516
$552$	0	0
$553$	5535.00	0.425628
$554$	7712.27	0.591450
$555$	0	0
$556$	−9365.00	−0.714325
$557$	18366.7	1.39717	0.698583	−	0.715529i	$-0.253814\pi$
$557$	18366.7	1.39717	0.698583	+	0.715529i	$0.253814\pi$
$558$	0	0
$559$	−4658.00	−0.352437
$560$	0	0
$561$	0	0
$562$	−27456.0	−2.06079
$563$	14018.4	1.04939	0.524693	−	0.851291i	$-0.324180\pi$
$563$	14018.4	1.04939	0.524693	+	0.851291i	$0.324180\pi$
$564$	0	0
$565$	0	0
$566$	11887.5	0.882807
$567$	0	0
$568$	8502.00	0.628056
$569$	15035.1	1.10774	0.553872	−	0.832602i	$-0.313150\pi$
$569$	15035.1	1.10774	0.553872	+	0.832602i	$0.313150\pi$
$570$	0	0
$571$	−2725.00	−0.199716	−0.0998579	−	0.995002i	$-0.531839\pi$
$571$	−2725.00	−0.199716	−0.0998579	+	0.995002i	$0.531839\pi$
$572$	−2451.77	−0.179220
$573$	0	0
$574$	−6552.00	−0.476438
$575$	0	0
$576$	0	0
$577$	−12815.0	−0.924602	−0.462301	−	0.886723i	$-0.652976\pi$
$577$	−12815.0	−0.924602	−0.462301	+	0.886723i	$0.652976\pi$
$578$	17526.6	1.26126
$579$	0	0
$580$	0	0
$581$	−2141.70	−0.152930
$582$	0	0
$583$	9776.00	0.694478
$584$	5440.78	0.385515
$585$	0	0
$586$	−26130.0	−1.84201
$587$	10189.3	0.716451	0.358226	−	0.933635i	$-0.383382\pi$
$587$	10189.3	0.716451	0.358226	+	0.933635i	$0.383382\pi$
$588$	0	0
$589$	303.000	0.0211968
$590$	0	0
$591$	0	0
$592$	−5293.00	−0.367468
$593$	3454.12	0.239197	0.119598	−	0.992822i	$-0.461839\pi$
$593$	3454.12	0.239197	0.119598	+	0.992822i	$0.461839\pi$
$594$	0	0
$595$	0	0
$596$	−11645.9	−0.800396
$597$	0	0
$598$	13260.0	0.906759
$599$	−6374.61	−0.434824	−0.217412	−	0.976080i	$-0.569762\pi$
$599$	−6374.61	−0.434824	−0.217412	+	0.976080i	$0.569762\pi$
$600$	0	0
$601$	−25645.0	−1.74057	−0.870284	−	0.492550i	$-0.836065\pi$
$601$	−25645.0	−1.74057	−0.870284	+	0.492550i	$0.836065\pi$
$602$	−4445.64	−0.300982
$603$	0	0
$604$	−5155.00	−0.347275
$605$	0	0
$606$	0	0
$607$	27923.0	1.86715	0.933575	−	0.358383i	$-0.116672\pi$
$607$	27923.0	1.86715	0.933575	+	0.358383i	$0.116672\pi$
$608$	−20028.8	−1.33598
$609$	0	0
$610$	0	0
$611$	−3922.84	−0.259740
$612$	0	0
$613$	−27817.0	−1.83282	−0.916410	−	0.400242i	$-0.868926\pi$
$613$	−27817.0	−1.83282	−0.916410	+	0.400242i	$0.868926\pi$
$614$	−22841.2	−1.50129
$615$	0	0
$616$	1404.00	0.0918324
$617$	−3677.66	−0.239963	−0.119981	−	0.992776i	$-0.538284\pi$
$617$	−3677.66	−0.239963	−0.119981	+	0.992776i	$0.538284\pi$
$618$	0	0
$619$	−16729.0	−1.08626	−0.543130	−	0.839648i	$-0.682761\pi$
$619$	−16729.0	−1.08626	−0.543130	+	0.839648i	$0.682761\pi$
$620$	0	0
$621$	0	0
$622$	−624.000	−0.0402253
$623$	8761.49	0.563438
$624$	0	0
$625$	0	0
$626$	34404.2	2.19659
$627$	0	0
$628$	12215.0	0.776165
$629$	483.144	0.0306267
$630$	0	0
$631$	−1748.00	−0.110280	−0.0551401	−	0.998479i	$-0.517561\pi$
$631$	−1748.00	−0.110280	−0.0551401	+	0.998479i	$0.517561\pi$
$632$	6652.24	0.418690
$633$	0	0
$634$	−6214.00	−0.389258
$635$	0	0
$636$	0	0
$637$	8908.00	0.554078
$638$	8624.48	0.535182
$639$	0	0
$640$	0	0
$641$	−1492.70	−0.0919782	−0.0459891	−	0.998942i	$-0.514644\pi$
$641$	−1492.70	−0.0919782	−0.0459891	+	0.998942i	$0.514644\pi$
$642$	0	0
$643$	2076.00	0.127324	0.0636621	−	0.997972i	$-0.479722\pi$
$643$	2076.00	0.127324	0.0636621	+	0.997972i	$0.479722\pi$
$644$	4867.49	0.297836
$645$	0	0
$646$	2626.00	0.159936
$647$	11710.8	0.711592	0.355796	−	0.934564i	$-0.384210\pi$
$647$	11710.8	0.711592	0.355796	+	0.934564i	$0.384210\pi$
$648$	0	0
$649$	−12584.0	−0.761117
$650$	0	0
$651$	0	0
$652$	−940.000	−0.0564620
$653$	20825.7	1.24804	0.624021	−	0.781408i	$-0.285498\pi$
$653$	20825.7	1.24804	0.624021	+	0.781408i	$0.285498\pi$
$654$	0	0
$655$	0	0
$656$	−15951.0	−0.949360
$657$	0	0
$658$	−3744.00	−0.221818
$659$	4276.18	0.252772	0.126386	−	0.991981i	$-0.459662\pi$
$659$	4276.18	0.252772	0.126386	+	0.991981i	$0.459662\pi$
$660$	0	0
$661$	−23987.0	−1.41148	−0.705738	−	0.708473i	$-0.749385\pi$
$661$	−23987.0	−1.41148	−0.705738	+	0.708473i	$0.749385\pi$
$662$	22617.6	1.32788
$663$	0	0
$664$	−2574.00	−0.150438
$665$	0	0
$666$	0	0
$667$	−17940.0	−1.04144
$668$	−7463.49	−0.432292
$669$	0	0
$670$	0	0
$671$	−8119.70	−0.467150
$672$	0	0
$673$	10965.0	0.628038	0.314019	−	0.949417i	$-0.398324\pi$
$673$	10965.0	0.628038	0.314019	+	0.949417i	$0.398324\pi$
$674$	−24712.4	−1.41230
$675$	0	0
$676$	−5205.00	−0.296142
$677$	15273.1	0.867051	0.433525	−	0.901141i	$-0.357269\pi$
$677$	15273.1	0.867051	0.433525	+	0.901141i	$0.357269\pi$
$678$	0	0
$679$	171.000	0.00966477
$680$	0	0
$681$	0	0
$682$	156.000	0.00875887
$683$	33985.9	1.90400	0.952002	−	0.306090i	$-0.0990210\pi$
$683$	33985.9	1.90400	0.952002	+	0.306090i	$0.0990210\pi$
$684$	0	0
$685$	0	0
$686$	19632.2	1.09266
$687$	0	0
$688$	−10823.0	−0.599743
$689$	−23046.7	−1.27432
$690$	0	0
$691$	23500.0	1.29375	0.646876	−	0.762596i	$-0.276075\pi$
$691$	23500.0	1.29375	0.646876	+	0.762596i	$0.276075\pi$
$692$	−15684.1	−0.861592
$693$	0	0
$694$	19942.0	1.09076
$695$	0	0
$696$	0	0
$697$	1456.00	0.0791247
$698$	19812.5	1.07438
$699$	0	0
$700$	0	0
$701$	28173.8	1.51799	0.758994	−	0.651098i	$-0.225691\pi$
$701$	28173.8	1.51799	0.758994	+	0.651098i	$0.225691\pi$
$702$	0	0
$703$	−6767.00	−0.363047
$704$	−1197.04	−0.0640842
$705$	0	0
$706$	−1560.00	−0.0831606
$707$	2336.40	0.124285
$708$	0	0
$709$	−14213.0	−0.752864	−0.376432	−	0.926444i	$-0.622849\pi$
$709$	−14213.0	−0.752864	−0.376432	+	0.926444i	$0.622849\pi$
$710$	0	0
$711$	0	0
$712$	10530.0	0.554253
$713$	−324.500	−0.0170443
$714$	0	0
$715$	0	0
$716$	14566.4	0.760297
$717$	0	0
$718$	24726.0	1.28519
$719$	28058.4	1.45536	0.727679	−	0.685918i	$-0.240599\pi$
$719$	28058.4	1.45536	0.727679	+	0.685918i	$0.240599\pi$
$720$	0	0
$721$	−1341.00	−0.0692669
$722$	−12049.8	−0.621115
$723$	0	0
$724$	−1430.00	−0.0734054
$725$	0	0
$726$	0	0
$727$	−13541.0	−0.690795	−0.345397	−	0.938456i	$-0.612256\pi$
$727$	−13541.0	−0.690795	−0.345397	+	0.938456i	$0.612256\pi$
$728$	−3309.90	−0.168507
$729$	0	0
$730$	0	0
$731$	987.921	0.0499857
$732$	0	0
$733$	23482.0	1.18326	0.591629	−	0.806211i	$-0.298485\pi$
$733$	23482.0	1.18326	0.591629	+	0.806211i	$0.298485\pi$
$734$	−33099.0	−1.66445
$735$	0	0
$736$	21450.0	1.07426
$737$	−15056.8	−0.752542
$738$	0	0
$739$	−11420.0	−0.568459	−0.284230	−	0.958756i	$-0.591738\pi$
$739$	−11420.0	−0.568459	−0.284230	+	0.958756i	$0.591738\pi$
$740$	0	0
$741$	0	0
$742$	−21996.0	−1.08827
$743$	−4038.22	−0.199391	−0.0996957	−	0.995018i	$-0.531787\pi$
$743$	−4038.22	−0.199391	−0.0996957	+	0.995018i	$0.531787\pi$
$744$	0	0
$745$	0	0
$746$	26655.8	1.30823
$747$	0	0
$748$	520.000	0.0254186
$749$	−18496.5	−0.902332
$750$	0	0
$751$	−15737.0	−0.764649	−0.382324	−	0.924028i	$-0.624876\pi$
$751$	−15737.0	−0.764649	−0.382324	+	0.924028i	$0.624876\pi$
$752$	−9114.83	−0.442000
$753$	0	0
$754$	−20332.0	−0.982026
$755$	0	0
$756$	0	0
$757$	−23539.0	−1.13017	−0.565086	−	0.825032i	$-0.691157\pi$
$757$	−23539.0	−1.13017	−0.565086	+	0.825032i	$0.691157\pi$
$758$	30575.1	1.46509
$759$	0	0
$760$	0	0
$761$	−31238.5	−1.48803	−0.744017	−	0.668160i	$-0.767082\pi$
$761$	−31238.5	−1.48803	−0.744017	+	0.668160i	$0.767082\pi$
$762$	0	0
$763$	−10593.0	−0.502611
$764$	4254.55	0.201472
$765$	0	0
$766$	−25818.0	−1.21781
$767$	29666.5	1.39660
$768$	0	0
$769$	26126.0	1.22513	0.612567	−	0.790419i	$-0.290137\pi$
$769$	26126.0	1.22513	0.612567	+	0.790419i	$0.290137\pi$
$770$	0	0
$771$	0	0
$772$	15005.0	0.699536
$773$	9929.69	0.462026	0.231013	−	0.972951i	$-0.425796\pi$
$773$	9929.69	0.462026	0.231013	+	0.972951i	$0.425796\pi$
$774$	0	0
$775$	0	0
$776$	205.516	0.00950723
$777$	0	0
$778$	−23244.0	−1.07113
$779$	−20393.0	−0.937940
$780$	0	0
$781$	11336.0	0.519378
$782$	−2812.33	−0.128604
$783$	0	0
$784$	20698.0	0.942875
$785$	0	0
$786$	0	0
$787$	34985.0	1.58460	0.792300	−	0.610131i	$-0.208883\pi$
$787$	34985.0	1.58460	0.792300	+	0.610131i	$0.208883\pi$
$788$	−19037.3	−0.860630
$789$	0	0
$790$	0	0
$791$	−18042.2	−0.811006
$792$	0	0
$793$	19142.0	0.857191
$794$	17872.7	0.798840
$795$	0	0
$796$	−7720.00	−0.343754
$797$	11696.4	0.519834	0.259917	−	0.965631i	$-0.416305\pi$
$797$	11696.4	0.519834	0.259917	+	0.965631i	$0.416305\pi$
$798$	0	0
$799$	832.000	0.0368386
$800$	0	0
$801$	0	0
$802$	29952.0	1.31876
$803$	7254.37	0.318806
$804$	0	0
$805$	0	0
$806$	−367.766	−0.0160720
$807$	0	0
$808$	2808.00	0.122259
$809$	33409.0	1.45191	0.725957	−	0.687740i	$-0.241397\pi$
$809$	33409.0	1.45191	0.725957	+	0.687740i	$0.241397\pi$
$810$	0	0
$811$	44817.0	1.94049	0.970245	−	0.242124i	$-0.0778442\pi$
$811$	44817.0	1.94049	0.970245	+	0.242124i	$0.0778442\pi$
$812$	−7463.49	−0.322558
$813$	0	0
$814$	−3484.00	−0.150017
$815$	0	0
$816$	0	0
$817$	−13837.0	−0.592528
$818$	38327.0	1.63823
$819$	0	0
$820$	0	0
$821$	42660.9	1.81349	0.906745	−	0.421680i	$-0.138559\pi$
$821$	42660.9	1.81349	0.906745	+	0.421680i	$0.138559\pi$
$822$	0	0
$823$	−16952.0	−0.717995	−0.358997	−	0.933339i	$-0.616881\pi$
$823$	−16952.0	−0.717995	−0.358997	+	0.933339i	$0.616881\pi$
$824$	−1611.68	−0.0681378
$825$	0	0
$826$	28314.0	1.19270
$827$	−11566.6	−0.486349	−0.243174	−	0.969983i	$-0.578189\pi$
$827$	−11566.6	−0.486349	−0.243174	+	0.969983i	$0.578189\pi$
$828$	0	0
$829$	39723.0	1.66422	0.832109	−	0.554612i	$-0.187133\pi$
$829$	39723.0	1.66422	0.832109	+	0.554612i	$0.187133\pi$
$830$	0	0
$831$	0	0
$832$	2822.00	0.117590
$833$	−1889.31	−0.0785842
$834$	0	0
$835$	0	0
$836$	−7283.21	−0.301310
$837$	0	0
$838$	−55406.0	−2.28397
$839$	−27943.0	−1.14982	−0.574911	−	0.818216i	$-0.694963\pi$
$839$	−27943.0	−1.14982	−0.574911	+	0.818216i	$0.694963\pi$
$840$	0	0
$841$	3119.00	0.127886
$842$	37054.3	1.51660
$843$	0	0
$844$	−8060.00	−0.328716
$845$	0	0
$846$	0	0
$847$	−10107.0	−0.410013
$848$	−53549.6	−2.16852
$849$	0	0
$850$	0	0
$851$	7247.16	0.291926
$852$	0	0
$853$	26046.0	1.04548	0.522742	−	0.852491i	$-0.324909\pi$
$853$	26046.0	1.04548	0.522742	+	0.852491i	$0.324909\pi$
$854$	18269.3	0.732042
$855$	0	0
$856$	−22230.0	−0.887624
$857$	−1312.42	−0.0523121	−0.0261560	−	0.999658i	$-0.508327\pi$
$857$	−1312.42	−0.0523121	−0.0261560	+	0.999658i	$0.508327\pi$
$858$	0	0
$859$	34207.0	1.35871	0.679353	−	0.733812i	$-0.262261\pi$
$859$	34207.0	1.35871	0.679353	+	0.733812i	$0.262261\pi$
$860$	0	0
$861$	0	0
$862$	27534.0	1.08795
$863$	−9915.27	−0.391100	−0.195550	−	0.980694i	$-0.562649\pi$
$863$	−9915.27	−0.391100	−0.195550	+	0.980694i	$0.562649\pi$
$864$	0	0
$865$	0	0
$866$	−43890.4	−1.72224
$867$	0	0
$868$	−135.000	−0.00527903
$869$	8869.66	0.346240
$870$	0	0
$871$	35496.0	1.38087
$872$	−12731.2	−0.494418
$873$	0	0
$874$	39390.0	1.52447
$875$	0	0
$876$	0	0
$877$	−19965.0	−0.768723	−0.384362	−	0.923183i	$-0.625578\pi$
$877$	−19965.0	−0.768723	−0.384362	+	0.923183i	$0.625578\pi$
$878$	−32410.3	−1.24578
$879$	0	0
$880$	0	0
$881$	20025.2	0.765797	0.382899	−	0.923790i	$-0.374926\pi$
$881$	20025.2	0.765797	0.382899	+	0.923790i	$0.374926\pi$
$882$	0	0
$883$	−36041.0	−1.37359	−0.686793	−	0.726853i	$-0.740982\pi$
$883$	−36041.0	−1.37359	−0.686793	+	0.726853i	$0.740982\pi$
$884$	−1225.89	−0.0466415
$885$	0	0
$886$	40170.0	1.52318
$887$	−3634.40	−0.137577	−0.0687886	−	0.997631i	$-0.521913\pi$
$887$	−3634.40	−0.137577	−0.0687886	+	0.997631i	$0.521913\pi$
$888$	0	0
$889$	2664.00	0.100504
$890$	0	0
$891$	0	0
$892$	−30565.0	−1.14730
$893$	−11653.1	−0.436683
$894$	0	0
$895$	0	0
$896$	−11584.6	−0.431937
$897$	0	0
$898$	28730.0	1.06763
$899$	497.566	0.0184591
$900$	0	0
$901$	4888.00	0.180736
$902$	−10499.4	−0.387573
$903$	0	0
$904$	−21684.0	−0.797787
$905$	0	0
$906$	0	0
$907$	45339.0	1.65982	0.829910	−	0.557897i	$-0.188392\pi$
$907$	45339.0	1.65982	0.829910	+	0.557897i	$0.188392\pi$
$908$	937.443	0.0342623
$909$	0	0
$910$	0	0
$911$	−30034.2	−1.09229	−0.546146	−	0.837690i	$-0.683906\pi$
$911$	−30034.2	−1.09229	−0.546146	+	0.837690i	$0.683906\pi$
$912$	0	0
$913$	−3432.00	−0.124406
$914$	−12857.4	−0.465301
$915$	0	0
$916$	10995.0	0.396599
$917$	−14991.9	−0.539886
$918$	0	0
$919$	−8693.00	−0.312030	−0.156015	−	0.987755i	$-0.549865\pi$
$919$	−8693.00	−0.312030	−0.156015	+	0.987755i	$0.549865\pi$
$920$	0	0
$921$	0	0
$922$	33462.0	1.19524
$923$	−26724.3	−0.953026
$924$	0	0
$925$	0	0
$926$	−54.0833	−0.00191932
$927$	0	0
$928$	−32890.0	−1.16343
$929$	30164.0	1.06529	0.532643	−	0.846340i	$-0.321199\pi$
$929$	30164.0	1.06529	0.532643	+	0.846340i	$0.321199\pi$
$930$	0	0
$931$	26462.0	0.931533
$932$	7463.49	0.262312
$933$	0	0
$934$	−23062.0	−0.807935
$935$	0	0
$936$	0	0
$937$	−22767.0	−0.793773	−0.396887	−	0.917868i	$-0.629909\pi$
$937$	−22767.0	−0.793773	−0.396887	+	0.917868i	$0.629909\pi$
$938$	33877.8	1.17926
$939$	0	0
$940$	0	0
$941$	−47737.5	−1.65377	−0.826885	−	0.562371i	$-0.809889\pi$
$941$	−47737.5	−1.65377	−0.826885	+	0.562371i	$0.809889\pi$
$942$	0	0
$943$	21840.0	0.754198
$944$	68930.9	2.37660
$945$	0	0
$946$	−7124.00	−0.244843
$947$	−27669.0	−0.949442	−0.474721	−	0.880136i	$-0.657451\pi$
$947$	−27669.0	−0.949442	−0.474721	+	0.880136i	$0.657451\pi$
$948$	0	0
$949$	−17102.0	−0.584989
$950$	0	0
$951$	0	0
$952$	702.000	0.0238991
$953$	−2177.75	−0.0740234	−0.0370117	−	0.999315i	$-0.511784\pi$
$953$	−2177.75	−0.0740234	−0.0370117	+	0.999315i	$0.511784\pi$
$954$	0	0
$955$	0	0
$956$	−26753.2	−0.905084
$957$	0	0
$958$	−45162.0	−1.52309
$959$	−10903.2	−0.367135
$960$	0	0
$961$	−29782.0	−0.999698
$962$	8213.45	0.275272
$963$	0	0
$964$	9650.00	0.322412
$965$	0	0
$966$	0	0
$967$	5389.00	0.179213	0.0896063	−	0.995977i	$-0.471439\pi$
$967$	5389.00	0.179213	0.0896063	+	0.995977i	$0.471439\pi$
$968$	−12147.1	−0.403329
$969$	0	0
$970$	0	0
$971$	36863.2	1.21833	0.609163	−	0.793045i	$-0.291505\pi$
$971$	36863.2	1.21833	0.609163	+	0.793045i	$0.291505\pi$
$972$	0	0
$973$	−16857.0	−0.555407
$974$	−21950.6	−0.722118
$975$	0	0
$976$	44477.0	1.45868
$977$	6973.14	0.228342	0.114171	−	0.993461i	$-0.463579\pi$
$977$	6973.14	0.228342	0.114171	+	0.993461i	$0.463579\pi$
$978$	0	0
$979$	14040.0	0.458346
$980$	0	0
$981$	0	0
$982$	54340.0	1.76584
$983$	−50232.5	−1.62988	−0.814939	−	0.579547i	$-0.803229\pi$
$983$	−50232.5	−1.62988	−0.814939	+	0.579547i	$0.803229\pi$
$984$	0	0
$985$	0	0
$986$	4312.24	0.139280
$987$	0	0
$988$	17170.0	0.552885
$989$	14818.8	0.476452
$990$	0	0
$991$	5303.00	0.169985	0.0849926	−	0.996382i	$-0.472913\pi$
$991$	5303.00	0.169985	0.0849926	+	0.996382i	$0.472913\pi$
$992$	−594.916	−0.0190409
$993$	0	0
$994$	−25506.0	−0.813885
$995$	0	0
$996$	0	0
$997$	38274.0	1.21580	0.607899	−	0.794015i	$-0.292013\pi$
$997$	38274.0	1.21580	0.607899	+	0.794015i	$0.292013\pi$
$998$	−8613.66	−0.273207
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.4.a.m.1.1	yes	2
3.2	odd	2	inner	675.4.a.m.1.2	yes	2
5.2	odd	4		675.4.b.j.649.2		4
5.3	odd	4		675.4.b.j.649.3		4
5.4	even	2		675.4.a.l.1.2	yes	2
15.2	even	4		675.4.b.j.649.4		4
15.8	even	4		675.4.b.j.649.1		4
15.14	odd	2		675.4.a.l.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
675.4.a.l.1.1	✓	2	15.14	odd	2
675.4.a.l.1.2	yes	2	5.4	even	2
675.4.a.m.1.1	yes	2	1.1	even	1	trivial
675.4.a.m.1.2	yes	2	3.2	odd	2	inner
675.4.b.j.649.1		4	15.8	even	4
675.4.b.j.649.2		4	5.2	odd	4
675.4.b.j.649.3		4	5.3	odd	4
675.4.b.j.649.4		4	15.2	even	4

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:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	675.a (trivial)

\(f(q)\)	\(=\)	\(q-3.60555 q^{2} +5.00000 q^{4} +9.00000 q^{7} +10.8167 q^{8} +14.4222 q^{11} -34.0000 q^{13} -32.4500 q^{14} -79.0000 q^{16} +7.21110 q^{17} -101.000 q^{19} -52.0000 q^{22} +108.167 q^{23} +122.589 q^{26} +45.0000 q^{28} -165.855 q^{29} -3.00000 q^{31} +198.305 q^{32} -26.0000 q^{34} +67.0000 q^{37} +364.161 q^{38} +201.911 q^{41} +137.000 q^{43} +72.1110 q^{44} -390.000 q^{46} +115.378 q^{47} -262.000 q^{49} -170.000 q^{52} +677.844 q^{53} +97.3499 q^{56} +598.000 q^{58} -872.543 q^{59} -563.000 q^{61} +10.8167 q^{62} -83.0000 q^{64} -1044.00 q^{67} +36.0555 q^{68} +786.010 q^{71} +503.000 q^{73} -241.572 q^{74} -505.000 q^{76} +129.800 q^{77} +615.000 q^{79} -728.000 q^{82} -237.966 q^{83} -493.961 q^{86} +156.000 q^{88} +973.499 q^{89} -306.000 q^{91} +540.833 q^{92} -416.000 q^{94} +19.0000 q^{97} +944.654 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 10 q^{4} + 18 q^{7} - 68 q^{13} - 158 q^{16} - 202 q^{19} - 104 q^{22} + 90 q^{28} - 6 q^{31} - 52 q^{34} + 134 q^{37} + 274 q^{43} - 780 q^{46} - 524 q^{49} - 340 q^{52} + 1196 q^{58} - 1126 q^{61}+ \cdots + 38 q^{97}+O(q^{100}) \) 2 * q + 10 * q^4 + 18 * q^7 - 68 * q^13 - 158 * q^16 - 202 * q^19 - 104 * q^22 + 90 * q^28 - 6 * q^31 - 52 * q^34 + 134 * q^37 + 274 * q^43 - 780 * q^46 - 524 * q^49 - 340 * q^52 + 1196 * q^58 - 1126 * q^61 - 166 * q^64 - 2088 * q^67 + 1006 * q^73 - 1010 * q^76 + 1230 * q^79 - 1456 * q^82 + 312 * q^88 - 612 * q^91 - 832 * q^94 + 38 * q^97

Introduction

L-functions

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