LMFDB - Embedded newform 76.6.a.b.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [76,6,Mod(1,76)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(76, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("76.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 76 = 2^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	76.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:	$12.1891703058$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 140x^{2} - 84x + 3103 $ x^4 - 140x^2 - 84x + 3103
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$11.0547$ of defining polynomial
Character	$\chi$	$=$	76.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+17.5489 q^{3} +87.4671 q^{5} +76.9277 q^{7} +64.9636 q^{9} +O(q^{10})$	$q+17.5489 q^{3} +87.4671 q^{5} +76.9277 q^{7} +64.9636 q^{9} -99.2130 q^{11} -79.5656 q^{13} +1534.95 q^{15} -465.981 q^{17} -361.000 q^{19} +1350.00 q^{21} +2231.89 q^{23} +4525.49 q^{25} -3124.34 q^{27} +757.584 q^{29} +2718.02 q^{31} -1741.08 q^{33} +6728.64 q^{35} +10309.9 q^{37} -1396.29 q^{39} -6945.22 q^{41} -2107.29 q^{43} +5682.17 q^{45} -20564.6 q^{47} -10889.1 q^{49} -8177.45 q^{51} -17567.4 q^{53} -8677.87 q^{55} -6335.15 q^{57} -34514.7 q^{59} -40975.0 q^{61} +4997.50 q^{63} -6959.37 q^{65} +68248.5 q^{67} +39167.2 q^{69} +38876.4 q^{71} -83648.9 q^{73} +79417.3 q^{75} -7632.23 q^{77} +39001.5 q^{79} -70614.9 q^{81} +99454.0 q^{83} -40758.0 q^{85} +13294.8 q^{87} -88908.3 q^{89} -6120.80 q^{91} +47698.3 q^{93} -31575.6 q^{95} -87484.2 q^{97} -6445.23 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 10 q^{3} - 110 q^{5} + 30 q^{7} + 878 q^{9}+O(q^{10}) $ 4 * q + 10 * q^3 - 110 * q^5 + 30 * q^7 + 878 * q^9	$ 4 q + 10 q^{3} - 110 q^{5} + 30 q^{7} + 878 q^{9} + 706 q^{11} + 788 q^{13} + 2792 q^{15} + 240 q^{17} - 1444 q^{19} + 7128 q^{21} + 5884 q^{23} + 11774 q^{25} + 4426 q^{27} + 5240 q^{29} - 860 q^{31} + 10748 q^{33} + 20322 q^{35} - 20732 q^{37} + 15404 q^{39} - 10204 q^{41} - 12554 q^{43} - 71306 q^{45} - 4826 q^{47} - 21376 q^{49} + 34518 q^{51} - 76484 q^{53} - 72914 q^{55} - 3610 q^{57} + 23898 q^{59} - 32482 q^{61} - 43566 q^{63} - 6076 q^{65} + 5022 q^{67} - 149524 q^{69} + 121300 q^{71} - 104700 q^{73} - 95230 q^{75} - 10002 q^{77} + 117128 q^{79} + 44924 q^{81} + 92832 q^{83} + 80322 q^{85} + 300148 q^{87} + 5988 q^{89} + 165618 q^{91} + 16180 q^{93} + 39710 q^{95} + 22972 q^{97} + 441418 q^{99}+O(q^{100}) $ 4 * q + 10 * q^3 - 110 * q^5 + 30 * q^7 + 878 * q^9 + 706 * q^11 + 788 * q^13 + 2792 * q^15 + 240 * q^17 - 1444 * q^19 + 7128 * q^21 + 5884 * q^23 + 11774 * q^25 + 4426 * q^27 + 5240 * q^29 - 860 * q^31 + 10748 * q^33 + 20322 * q^35 - 20732 * q^37 + 15404 * q^39 - 10204 * q^41 - 12554 * q^43 - 71306 * q^45 - 4826 * q^47 - 21376 * q^49 + 34518 * q^51 - 76484 * q^53 - 72914 * q^55 - 3610 * q^57 + 23898 * q^59 - 32482 * q^61 - 43566 * q^63 - 6076 * q^65 + 5022 * q^67 - 149524 * q^69 + 121300 * q^71 - 104700 * q^73 - 95230 * q^75 - 10002 * q^77 + 117128 * q^79 + 44924 * q^81 + 92832 * q^83 + 80322 * q^85 + 300148 * q^87 + 5988 * q^89 + 165618 * q^91 + 16180 * q^93 + 39710 * q^95 + 22972 * q^97 + 441418 * 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
$3$	17.5489	1.12576	0.562881	−	0.826538i	$-0.309693\pi$
$3$	17.5489	1.12576	0.562881	+	0.826538i	$0.309693\pi$
$4$	0	0
$5$	87.4671	1.56466	0.782329	−	0.622865i	$-0.214031\pi$
$5$	87.4671	1.56466	0.782329	+	0.622865i	$0.214031\pi$
$6$	0	0
$7$	76.9277	0.593386	0.296693	−	0.954973i	$-0.404116\pi$
$7$	76.9277	0.593386	0.296693	+	0.954973i	$0.404116\pi$
$8$	0	0
$9$	64.9636	0.267340
$10$	0	0
$11$	−99.2130	−0.247222	−0.123611	−	0.992331i	$-0.539447\pi$
$11$	−99.2130	−0.247222	−0.123611	+	0.992331i	$0.539447\pi$
$12$	0	0
$13$	−79.5656	−0.130577	−0.0652886	−	0.997866i	$-0.520797\pi$
$13$	−79.5656	−0.130577	−0.0652886	+	0.997866i	$0.520797\pi$
$14$	0	0
$15$	1534.95	1.76143
$16$	0	0
$17$	−465.981	−0.391062	−0.195531	−	0.980698i	$-0.562643\pi$
$17$	−465.981	−0.391062	−0.195531	+	0.980698i	$0.562643\pi$
$18$	0	0
$19$	−361.000	−0.229416
$19$	−361.000	−0.229416
$20$	0	0
$21$	1350.00	0.668012
$22$	0	0
$23$	2231.89	0.879738	0.439869	−	0.898062i	$-0.355025\pi$
$23$	2231.89	0.879738	0.439869	+	0.898062i	$0.355025\pi$
$24$	0	0
$25$	4525.49	1.44816
$26$	0	0
$27$	−3124.34	−0.824801
$28$	0	0
$29$	757.584	0.167277	0.0836384	−	0.996496i	$-0.473346\pi$
$29$	757.584	0.167277	0.0836384	+	0.996496i	$0.473346\pi$
$30$	0	0
$31$	2718.02	0.507983	0.253991	−	0.967206i	$-0.418257\pi$
$31$	2718.02	0.507983	0.253991	+	0.967206i	$0.418257\pi$
$32$	0	0
$33$	−1741.08	−0.278313
$34$	0	0
$35$	6728.64	0.928447
$36$	0	0
$37$	10309.9	1.23808	0.619040	−	0.785360i	$-0.287522\pi$
$37$	10309.9	1.23808	0.619040	+	0.785360i	$0.287522\pi$
$38$	0	0
$39$	−1396.29	−0.146999
$40$	0	0
$41$	−6945.22	−0.645248	−0.322624	−	0.946527i	$-0.604565\pi$
$41$	−6945.22	−0.645248	−0.322624	+	0.946527i	$0.604565\pi$
$42$	0	0
$43$	−2107.29	−0.173802	−0.0869009	−	0.996217i	$-0.527696\pi$
$43$	−2107.29	−0.173802	−0.0869009	+	0.996217i	$0.527696\pi$
$44$	0	0
$45$	5682.17	0.418296
$46$	0	0
$47$	−20564.6	−1.35793	−0.678963	−	0.734173i	$-0.737570\pi$
$47$	−20564.6	−1.35793	−0.678963	+	0.734173i	$0.737570\pi$
$48$	0	0
$49$	−10889.1	−0.647893
$50$	0	0
$51$	−8177.45	−0.440243
$52$	0	0
$53$	−17567.4	−0.859050	−0.429525	−	0.903055i	$-0.641319\pi$
$53$	−17567.4	−0.859050	−0.429525	+	0.903055i	$0.641319\pi$
$54$	0	0
$55$	−8677.87	−0.386818
$56$	0	0
$57$	−6335.15	−0.258267
$58$	0	0
$59$	−34514.7	−1.29084	−0.645422	−	0.763827i	$-0.723318\pi$
$59$	−34514.7	−1.29084	−0.645422	+	0.763827i	$0.723318\pi$
$60$	0	0
$61$	−40975.0	−1.40992	−0.704959	−	0.709248i	$-0.749035\pi$
$61$	−40975.0	−1.40992	−0.704959	+	0.709248i	$0.749035\pi$
$62$	0	0
$63$	4997.50	0.158636
$64$	0	0
$65$	−6959.37	−0.204309
$66$	0	0
$67$	68248.5	1.85740	0.928701	−	0.370829i	$-0.120926\pi$
$67$	68248.5	1.85740	0.928701	+	0.370829i	$0.120926\pi$
$68$	0	0
$69$	39167.2	0.990376
$70$	0	0
$71$	38876.4	0.915251	0.457626	−	0.889145i	$-0.348700\pi$
$71$	38876.4	0.915251	0.457626	+	0.889145i	$0.348700\pi$
$72$	0	0
$73$	−83648.9	−1.83719	−0.918593	−	0.395205i	$-0.870674\pi$
$73$	−83648.9	−1.83719	−0.918593	+	0.395205i	$0.870674\pi$
$74$	0	0
$75$	79417.3	1.63028
$76$	0	0
$77$	−7632.23	−0.146698
$78$	0	0
$79$	39001.5	0.703094	0.351547	−	0.936170i	$-0.385656\pi$
$79$	39001.5	0.703094	0.351547	+	0.936170i	$0.385656\pi$
$80$	0	0
$81$	−70614.9	−1.19587
$82$	0	0
$83$	99454.0	1.58463	0.792314	−	0.610114i	$-0.208876\pi$
$83$	99454.0	1.58463	0.792314	+	0.610114i	$0.208876\pi$
$84$	0	0
$85$	−40758.0	−0.611879
$86$	0	0
$87$	13294.8	0.188314
$88$	0	0
$89$	−88908.3	−1.18978	−0.594891	−	0.803806i	$-0.702805\pi$
$89$	−88908.3	−1.18978	−0.594891	+	0.803806i	$0.702805\pi$
$90$	0	0
$91$	−6120.80	−0.0774827
$92$	0	0
$93$	47698.3	0.571868
$94$	0	0
$95$	−31575.6	−0.358957
$96$	0	0
$97$	−87484.2	−0.944061	−0.472031	−	0.881582i	$-0.656479\pi$
$97$	−87484.2	−0.944061	−0.472031	+	0.881582i	$0.656479\pi$
$98$	0	0
$99$	−6445.23	−0.0660923
$100$	0	0
$101$	124712.	1.21648	0.608238	−	0.793754i	$-0.291876\pi$
$101$	124712.	1.21648	0.608238	+	0.793754i	$0.291876\pi$
$102$	0	0
$103$	−28096.0	−0.260947	−0.130473	−	0.991452i	$-0.541650\pi$
$103$	−28096.0	−0.260947	−0.130473	+	0.991452i	$0.541650\pi$
$104$	0	0
$105$	118080.	1.04521
$106$	0	0
$107$	106686.	0.900839	0.450419	−	0.892817i	$-0.351274\pi$
$107$	106686.	0.900839	0.450419	+	0.892817i	$0.351274\pi$
$108$	0	0
$109$	−81738.7	−0.658964	−0.329482	−	0.944162i	$-0.606874\pi$
$109$	−81738.7	−0.658964	−0.329482	+	0.944162i	$0.606874\pi$
$110$	0	0
$111$	180927.	1.39378
$112$	0	0
$113$	265828.	1.95841	0.979206	−	0.202871i	$-0.0650271\pi$
$113$	265828.	1.95841	0.979206	+	0.202871i	$0.0650271\pi$
$114$	0	0
$115$	195217.	1.37649
$116$	0	0
$117$	−5168.87	−0.0349085
$118$	0	0
$119$	−35846.8	−0.232051
$120$	0	0
$121$	−151208.	−0.938881
$122$	0	0
$123$	−121881.	−0.726395
$124$	0	0
$125$	122497.	0.701213
$126$	0	0
$127$	−320548.	−1.76353	−0.881766	−	0.471688i	$-0.843645\pi$
$127$	−320548.	−1.76353	−0.881766	+	0.471688i	$0.843645\pi$
$128$	0	0
$129$	−36980.7	−0.195659
$130$	0	0
$131$	−104893.	−0.534034	−0.267017	−	0.963692i	$-0.586038\pi$
$131$	−104893.	−0.534034	−0.267017	+	0.963692i	$0.586038\pi$
$132$	0	0
$133$	−27770.9	−0.136132
$134$	0	0
$135$	−273277.	−1.29053
$136$	0	0
$137$	276152.	1.25704	0.628518	−	0.777795i	$-0.283662\pi$
$137$	276152.	1.25704	0.628518	+	0.777795i	$0.283662\pi$
$138$	0	0
$139$	51956.7	0.228089	0.114045	−	0.993476i	$-0.463619\pi$
$139$	51956.7	0.228089	0.114045	+	0.993476i	$0.463619\pi$
$140$	0	0
$141$	−360886.	−1.52870
$142$	0	0
$143$	7893.94	0.0322815
$144$	0	0
$145$	66263.7	0.261731
$146$	0	0
$147$	−191092.	−0.729373
$148$	0	0
$149$	−420951.	−1.55334	−0.776670	−	0.629908i	$-0.783093\pi$
$149$	−420951.	−1.55334	−0.776670	+	0.629908i	$0.783093\pi$
$150$	0	0
$151$	−56718.6	−0.202434	−0.101217	−	0.994864i	$-0.532274\pi$
$151$	−56718.6	−0.202434	−0.101217	+	0.994864i	$0.532274\pi$
$152$	0	0
$153$	−30271.8	−0.104547
$154$	0	0
$155$	237738.	0.794820
$156$	0	0
$157$	−65573.0	−0.212313	−0.106156	−	0.994349i	$-0.533854\pi$
$157$	−65573.0	−0.212313	−0.106156	+	0.994349i	$0.533854\pi$
$158$	0	0
$159$	−308289.	−0.967085
$160$	0	0
$161$	171694.	0.522025
$162$	0	0
$163$	445737.	1.31404	0.657022	−	0.753872i	$-0.271816\pi$
$163$	445737.	1.31404	0.657022	+	0.753872i	$0.271816\pi$
$164$	0	0
$165$	−152287.	−0.435465
$166$	0	0
$167$	−150553.	−0.417734	−0.208867	−	0.977944i	$-0.566978\pi$
$167$	−150553.	−0.417734	−0.208867	+	0.977944i	$0.566978\pi$
$168$	0	0
$169$	−364962.	−0.982950
$170$	0	0
$171$	−23451.9	−0.0613320
$172$	0	0
$173$	−106599.	−0.270793	−0.135397	−	0.990791i	$-0.543231\pi$
$173$	−106599.	−0.270793	−0.135397	+	0.990791i	$0.543231\pi$
$174$	0	0
$175$	348135.	0.859316
$176$	0	0
$177$	−605694.	−1.45318
$178$	0	0
$179$	416262.	0.971034	0.485517	−	0.874227i	$-0.338631\pi$
$179$	416262.	0.971034	0.485517	+	0.874227i	$0.338631\pi$
$180$	0	0
$181$	100405.	0.227803	0.113901	−	0.993492i	$-0.463665\pi$
$181$	100405.	0.227803	0.113901	+	0.993492i	$0.463665\pi$
$182$	0	0
$183$	−719065.	−1.58723
$184$	0	0
$185$	901774.	1.93717
$186$	0	0
$187$	46231.4	0.0966791
$188$	0	0
$189$	−240348.	−0.489426
$190$	0	0
$191$	597176.	1.18446	0.592228	−	0.805770i	$-0.298248\pi$
$191$	597176.	1.18446	0.592228	+	0.805770i	$0.298248\pi$
$192$	0	0
$193$	714364.	1.38047	0.690234	−	0.723586i	$-0.257507\pi$
$193$	714364.	1.38047	0.690234	+	0.723586i	$0.257507\pi$
$194$	0	0
$195$	−122129.	−0.230003
$196$	0	0
$197$	−11062.3	−0.0203085	−0.0101543	−	0.999948i	$-0.503232\pi$
$197$	−11062.3	−0.0203085	−0.0101543	+	0.999948i	$0.503232\pi$
$198$	0	0
$199$	357858.	0.640587	0.320294	−	0.947318i	$-0.396218\pi$
$199$	357858.	0.640587	0.320294	+	0.947318i	$0.396218\pi$
$200$	0	0
$201$	1.19769e6	2.09099
$202$	0	0
$203$	58279.2	0.0992598
$204$	0	0
$205$	−607478.	−1.00959
$206$	0	0
$207$	144992.	0.235189
$208$	0	0
$209$	35815.9	0.0567166
$210$	0	0
$211$	−284130.	−0.439350	−0.219675	−	0.975573i	$-0.570500\pi$
$211$	−284130.	−0.439350	−0.219675	+	0.975573i	$0.570500\pi$
$212$	0	0
$213$	682238.	1.03035
$214$	0	0
$215$	−184319.	−0.271940
$216$	0	0
$217$	209091.	0.301430
$218$	0	0
$219$	−1.46795e6	−2.06823
$220$	0	0
$221$	37076.1	0.0510638
$222$	0	0
$223$	870495.	1.17221	0.586103	−	0.810236i	$-0.300661\pi$
$223$	870495.	1.17221	0.586103	+	0.810236i	$0.300661\pi$
$224$	0	0
$225$	293992.	0.387150
$226$	0	0
$227$	484247.	0.623738	0.311869	−	0.950125i	$-0.399045\pi$
$227$	484247.	0.623738	0.311869	+	0.950125i	$0.399045\pi$
$228$	0	0
$229$	−512807.	−0.646197	−0.323099	−	0.946365i	$-0.604725\pi$
$229$	−512807.	−0.646197	−0.323099	+	0.946365i	$0.604725\pi$
$230$	0	0
$231$	−133937.	−0.165147
$232$	0	0
$233$	1.14046e6	1.37623	0.688115	−	0.725601i	$-0.258438\pi$
$233$	1.14046e6	1.37623	0.688115	+	0.725601i	$0.258438\pi$
$234$	0	0
$235$	−1.79873e6	−2.12469
$236$	0	0
$237$	684433.	0.791516
$238$	0	0
$239$	−564259.	−0.638975	−0.319488	−	0.947590i	$-0.603511\pi$
$239$	−564259.	−0.638975	−0.319488	+	0.947590i	$0.603511\pi$
$240$	0	0
$241$	−499520.	−0.554001	−0.277000	−	0.960870i	$-0.589340\pi$
$241$	−499520.	−0.554001	−0.277000	+	0.960870i	$0.589340\pi$
$242$	0	0
$243$	−479998.	−0.521463
$244$	0	0
$245$	−952441.	−1.01373
$246$	0	0
$247$	28723.2	0.0299564
$248$	0	0
$249$	1.74531e6	1.78391
$250$	0	0
$251$	1.32259e6	1.32508	0.662539	−	0.749027i	$-0.269479\pi$
$251$	1.32259e6	1.32508	0.662539	+	0.749027i	$0.269479\pi$
$252$	0	0
$253$	−221433.	−0.217491
$254$	0	0
$255$	−715258.	−0.688830
$256$	0	0
$257$	29088.9	0.0274722	0.0137361	−	0.999906i	$-0.495628\pi$
$257$	29088.9	0.0274722	0.0137361	+	0.999906i	$0.495628\pi$
$258$	0	0
$259$	793114.	0.734660
$260$	0	0
$261$	49215.4	0.0447198
$262$	0	0
$263$	1.39828e6	1.24653	0.623266	−	0.782010i	$-0.285806\pi$
$263$	1.39828e6	1.24653	0.623266	+	0.782010i	$0.285806\pi$
$264$	0	0
$265$	−1.53657e6	−1.34412
$266$	0	0
$267$	−1.56024e6	−1.33941
$268$	0	0
$269$	1.62054e6	1.36546	0.682728	−	0.730672i	$-0.260793\pi$
$269$	1.62054e6	1.36546	0.682728	+	0.730672i	$0.260793\pi$
$270$	0	0
$271$	−311537.	−0.257683	−0.128842	−	0.991665i	$-0.541126\pi$
$271$	−311537.	−0.257683	−0.128842	+	0.991665i	$0.541126\pi$
$272$	0	0
$273$	−107413.	−0.0872270
$274$	0	0
$275$	−448988.	−0.358016
$276$	0	0
$277$	−801500.	−0.627631	−0.313815	−	0.949484i	$-0.601607\pi$
$277$	−801500.	−0.627631	−0.313815	+	0.949484i	$0.601607\pi$
$278$	0	0
$279$	176573.	0.135804
$280$	0	0
$281$	1.59535e6	1.20528	0.602642	−	0.798012i	$-0.294115\pi$
$281$	1.59535e6	1.20528	0.602642	+	0.798012i	$0.294115\pi$
$282$	0	0
$283$	−1.04144e6	−0.772980	−0.386490	−	0.922294i	$-0.626313\pi$
$283$	−1.04144e6	−0.772980	−0.386490	+	0.922294i	$0.626313\pi$
$284$	0	0
$285$	−554117.	−0.404100
$286$	0	0
$287$	−534280.	−0.382881
$288$	0	0
$289$	−1.20272e6	−0.847070
$290$	0	0
$291$	−1.53525e6	−1.06279
$292$	0	0
$293$	−391025.	−0.266094	−0.133047	−	0.991110i	$-0.542476\pi$
$293$	−391025.	−0.266094	−0.133047	+	0.991110i	$0.542476\pi$
$294$	0	0
$295$	−3.01890e6	−2.01973
$296$	0	0
$297$	309975.	0.203909
$298$	0	0
$299$	−177582.	−0.114874
$300$	0	0
$301$	−162109.	−0.103132
$302$	0	0
$303$	2.18855e6	1.36946
$304$	0	0
$305$	−3.58396e6	−2.20604
$306$	0	0
$307$	−504246.	−0.305349	−0.152675	−	0.988277i	$-0.548789\pi$
$307$	−504246.	−0.305349	−0.152675	+	0.988277i	$0.548789\pi$
$308$	0	0
$309$	−493054.	−0.293764
$310$	0	0
$311$	−80004.3	−0.0469042	−0.0234521	−	0.999725i	$-0.507466\pi$
$311$	−80004.3	−0.0469042	−0.0234521	+	0.999725i	$0.507466\pi$
$312$	0	0
$313$	−99620.5	−0.0574762	−0.0287381	−	0.999587i	$-0.509149\pi$
$313$	−99620.5	−0.0574762	−0.0287381	+	0.999587i	$0.509149\pi$
$314$	0	0
$315$	437117.	0.248211
$316$	0	0
$317$	−177148.	−0.0990121	−0.0495060	−	0.998774i	$-0.515765\pi$
$317$	−177148.	−0.0990121	−0.0495060	+	0.998774i	$0.515765\pi$
$318$	0	0
$319$	−75162.2	−0.0413545
$320$	0	0
$321$	1.87222e6	1.01413
$322$	0	0
$323$	168219.	0.0897158
$324$	0	0
$325$	−360073.	−0.189096
$326$	0	0
$327$	−1.43442e6	−0.741837
$328$	0	0
$329$	−1.58199e6	−0.805774
$330$	0	0
$331$	1.05867e6	0.531116	0.265558	−	0.964095i	$-0.414444\pi$
$331$	1.05867e6	0.531116	0.265558	+	0.964095i	$0.414444\pi$
$332$	0	0
$333$	669766.	0.330988
$334$	0	0
$335$	5.96950e6	2.90620
$336$	0	0
$337$	3.24019e6	1.55416	0.777081	−	0.629401i	$-0.216700\pi$
$337$	3.24019e6	1.55416	0.777081	+	0.629401i	$0.216700\pi$
$338$	0	0
$339$	4.66498e6	2.20470
$340$	0	0
$341$	−269663.	−0.125584
$342$	0	0
$343$	−2.13060e6	−0.977837
$344$	0	0
$345$	3.42584e6	1.54960
$346$	0	0
$347$	−2.67241e6	−1.19146	−0.595729	−	0.803185i	$-0.703137\pi$
$347$	−2.67241e6	−1.19146	−0.595729	+	0.803185i	$0.703137\pi$
$348$	0	0
$349$	3.02667e6	1.33015	0.665076	−	0.746775i	$-0.268399\pi$
$349$	3.02667e6	1.33015	0.665076	+	0.746775i	$0.268399\pi$
$350$	0	0
$351$	248590.	0.107700
$352$	0	0
$353$	790142.	0.337496	0.168748	−	0.985659i	$-0.446028\pi$
$353$	790142.	0.337496	0.168748	+	0.985659i	$0.446028\pi$
$354$	0	0
$355$	3.40041e6	1.43206
$356$	0	0
$357$	−629072.	−0.261234
$358$	0	0
$359$	1.80303e6	0.738359	0.369179	−	0.929358i	$-0.379639\pi$
$359$	1.80303e6	0.738359	0.369179	+	0.929358i	$0.379639\pi$
$360$	0	0
$361$	130321.	0.0526316
$362$	0	0
$363$	−2.65353e6	−1.05696
$364$	0	0
$365$	−7.31653e6	−2.87457
$366$	0	0
$367$	1.68799e6	0.654192	0.327096	−	0.944991i	$-0.393930\pi$
$367$	1.68799e6	0.654192	0.327096	+	0.944991i	$0.393930\pi$
$368$	0	0
$369$	−451186.	−0.172500
$370$	0	0
$371$	−1.35142e6	−0.509748
$372$	0	0
$373$	4.48620e6	1.66958	0.834788	−	0.550571i	$-0.185590\pi$
$373$	4.48620e6	1.66958	0.834788	+	0.550571i	$0.185590\pi$
$374$	0	0
$375$	2.14968e6	0.789398
$376$	0	0
$377$	−60277.6	−0.0218425
$378$	0	0
$379$	103084.	0.0368634	0.0184317	−	0.999830i	$-0.494133\pi$
$379$	103084.	0.0368634	0.0184317	+	0.999830i	$0.494133\pi$
$380$	0	0
$381$	−5.62525e6	−1.98532
$382$	0	0
$383$	2.21030e6	0.769934	0.384967	−	0.922930i	$-0.374213\pi$
$383$	2.21030e6	0.769934	0.384967	+	0.922930i	$0.374213\pi$
$384$	0	0
$385$	−667569.	−0.229532
$386$	0	0
$387$	−136897.	−0.0464641
$388$	0	0
$389$	3.56273e6	1.19374	0.596870	−	0.802338i	$-0.296411\pi$
$389$	3.56273e6	1.19374	0.596870	+	0.802338i	$0.296411\pi$
$390$	0	0
$391$	−1.04002e6	−0.344032
$392$	0	0
$393$	−1.84076e6	−0.601196
$394$	0	0
$395$	3.41135e6	1.10010
$396$	0	0
$397$	−4.69992e6	−1.49663	−0.748315	−	0.663344i	$-0.769137\pi$
$397$	−4.69992e6	−1.49663	−0.748315	+	0.663344i	$0.769137\pi$
$398$	0	0
$399$	−487348.	−0.153252
$400$	0	0
$401$	−2.76391e6	−0.858347	−0.429173	−	0.903222i	$-0.641195\pi$
$401$	−2.76391e6	−0.858347	−0.429173	+	0.903222i	$0.641195\pi$
$402$	0	0
$403$	−216261.	−0.0663309
$404$	0	0
$405$	−6.17648e6	−1.87113
$406$	0	0
$407$	−1.02287e6	−0.306080
$408$	0	0
$409$	4.43668e6	1.31144	0.655722	−	0.755002i	$-0.272364\pi$
$409$	4.43668e6	1.31144	0.655722	+	0.755002i	$0.272364\pi$
$410$	0	0
$411$	4.84617e6	1.41512
$412$	0	0
$413$	−2.65513e6	−0.765969
$414$	0	0
$415$	8.69895e6	2.47940
$416$	0	0
$417$	911783.	0.256774
$418$	0	0
$419$	−3.67747e6	−1.02333	−0.511664	−	0.859186i	$-0.670971\pi$
$419$	−3.67747e6	−1.02333	−0.511664	+	0.859186i	$0.670971\pi$
$420$	0	0
$421$	2.97195e6	0.817216	0.408608	−	0.912710i	$-0.366014\pi$
$421$	2.97195e6	0.817216	0.408608	+	0.912710i	$0.366014\pi$
$422$	0	0
$423$	−1.33595e6	−0.363028
$424$	0	0
$425$	−2.10879e6	−0.566319
$426$	0	0
$427$	−3.15211e6	−0.836626
$428$	0	0
$429$	138530.	0.0363413
$430$	0	0
$431$	−2.90998e6	−0.754565	−0.377283	−	0.926098i	$-0.623141\pi$
$431$	−2.90998e6	−0.754565	−0.377283	+	0.926098i	$0.623141\pi$
$432$	0	0
$433$	3.06888e6	0.786611	0.393305	−	0.919408i	$-0.371332\pi$
$433$	3.06888e6	0.786611	0.393305	+	0.919408i	$0.371332\pi$
$434$	0	0
$435$	1.16285e6	0.294647
$436$	0	0
$437$	−805713.	−0.201826
$438$	0	0
$439$	2.87829e6	0.712809	0.356404	−	0.934332i	$-0.384003\pi$
$439$	2.87829e6	0.712809	0.356404	+	0.934332i	$0.384003\pi$
$440$	0	0
$441$	−707397.	−0.173208
$442$	0	0
$443$	1.29112e6	0.312578	0.156289	−	0.987711i	$-0.450047\pi$
$443$	1.29112e6	0.312578	0.156289	+	0.987711i	$0.450047\pi$
$444$	0	0
$445$	−7.77655e6	−1.86160
$446$	0	0
$447$	−7.38723e6	−1.74869
$448$	0	0
$449$	4.24274e6	0.993186	0.496593	−	0.867984i	$-0.334584\pi$
$449$	4.24274e6	0.993186	0.496593	+	0.867984i	$0.334584\pi$
$450$	0	0
$451$	689056.	0.159519
$452$	0	0
$453$	−995349.	−0.227892
$454$	0	0
$455$	−535368.	−0.121234
$456$	0	0
$457$	−6.19374e6	−1.38727	−0.693637	−	0.720324i	$-0.743993\pi$
$457$	−6.19374e6	−1.38727	−0.693637	+	0.720324i	$0.743993\pi$
$458$	0	0
$459$	1.45588e6	0.322548
$460$	0	0
$461$	−5.87032e6	−1.28650	−0.643250	−	0.765656i	$-0.722414\pi$
$461$	−5.87032e6	−1.28650	−0.643250	+	0.765656i	$0.722414\pi$
$462$	0	0
$463$	−8.40538e6	−1.82224	−0.911119	−	0.412144i	$-0.864780\pi$
$463$	−8.40538e6	−1.82224	−0.911119	+	0.412144i	$0.864780\pi$
$464$	0	0
$465$	4.17203e6	0.894778
$466$	0	0
$467$	−148549.	−0.0315193	−0.0157597	−	0.999876i	$-0.505017\pi$
$467$	−148549.	−0.0315193	−0.0157597	+	0.999876i	$0.505017\pi$
$468$	0	0
$469$	5.25020e6	1.10216
$470$	0	0
$471$	−1.15073e6	−0.239013
$472$	0	0
$473$	209071.	0.0429676
$474$	0	0
$475$	−1.63370e6	−0.332230
$476$	0	0
$477$	−1.14124e6	−0.229658
$478$	0	0
$479$	5.35414e6	1.06623	0.533115	−	0.846043i	$-0.321021\pi$
$479$	5.35414e6	1.06623	0.533115	+	0.846043i	$0.321021\pi$
$480$	0	0
$481$	−820311.	−0.161665
$482$	0	0
$483$	3.01304e6	0.587675
$484$	0	0
$485$	−7.65198e6	−1.47713
$486$	0	0
$487$	−2.86815e6	−0.547998	−0.273999	−	0.961730i	$-0.588346\pi$
$487$	−2.86815e6	−0.547998	−0.273999	+	0.961730i	$0.588346\pi$
$488$	0	0
$489$	7.82219e6	1.47930
$490$	0	0
$491$	7.73298e6	1.44758	0.723791	−	0.690019i	$-0.242398\pi$
$491$	7.73298e6	1.44758	0.723791	+	0.690019i	$0.242398\pi$
$492$	0	0
$493$	−353020.	−0.0654156
$494$	0	0
$495$	−563746.	−0.103412
$496$	0	0
$497$	2.99067e6	0.543097
$498$	0	0
$499$	1.93075e6	0.347116	0.173558	−	0.984824i	$-0.444474\pi$
$499$	1.93075e6	0.347116	0.173558	+	0.984824i	$0.444474\pi$
$500$	0	0
$501$	−2.64205e6	−0.470269
$502$	0	0
$503$	−5.50103e6	−0.969447	−0.484723	−	0.874668i	$-0.661080\pi$
$503$	−5.50103e6	−0.969447	−0.484723	+	0.874668i	$0.661080\pi$
$504$	0	0
$505$	1.09082e7	1.90337
$506$	0	0
$507$	−6.40468e6	−1.10657
$508$	0	0
$509$	−4.93464e6	−0.844230	−0.422115	−	0.906542i	$-0.638712\pi$
$509$	−4.93464e6	−0.844230	−0.422115	+	0.906542i	$0.638712\pi$
$510$	0	0
$511$	−6.43492e6	−1.09016
$512$	0	0
$513$	1.12789e6	0.189222
$514$	0	0
$515$	−2.45748e6	−0.408292
$516$	0	0
$517$	2.04028e6	0.335709
$518$	0	0
$519$	−1.87069e6	−0.304849
$520$	0	0
$521$	5.98044e6	0.965248	0.482624	−	0.875828i	$-0.339684\pi$
$521$	5.98044e6	0.965248	0.482624	+	0.875828i	$0.339684\pi$
$522$	0	0
$523$	−3.74282e6	−0.598336	−0.299168	−	0.954200i	$-0.596709\pi$
$523$	−3.74282e6	−0.598336	−0.299168	+	0.954200i	$0.596709\pi$
$524$	0	0
$525$	6.10939e6	0.967386
$526$	0	0
$527$	−1.26655e6	−0.198653
$528$	0	0
$529$	−1.45500e6	−0.226061
$530$	0	0
$531$	−2.24220e6	−0.345094
$532$	0	0
$533$	552601.	0.0842546
$534$	0	0
$535$	9.33149e6	1.40951
$536$	0	0
$537$	7.30494e6	1.09315
$538$	0	0
$539$	1.08034e6	0.160173
$540$	0	0
$541$	7.66502e6	1.12595	0.562977	−	0.826473i	$-0.309656\pi$
$541$	7.66502e6	1.12595	0.562977	+	0.826473i	$0.309656\pi$
$542$	0	0
$543$	1.76200e6	0.256452
$544$	0	0
$545$	−7.14945e6	−1.03105
$546$	0	0
$547$	−8.02578e6	−1.14688	−0.573441	−	0.819247i	$-0.694392\pi$
$547$	−8.02578e6	−1.14688	−0.573441	+	0.819247i	$0.694392\pi$
$548$	0	0
$549$	−2.66188e6	−0.376927
$550$	0	0
$551$	−273488.	−0.0383759
$552$	0	0
$553$	3.00029e6	0.417206
$554$	0	0
$555$	1.58251e7	2.18079
$556$	0	0
$557$	−1.39685e7	−1.90771	−0.953855	−	0.300266i	$-0.902924\pi$
$557$	−1.39685e7	−1.90771	−0.953855	+	0.300266i	$0.902924\pi$
$558$	0	0
$559$	167668.	0.0226945
$560$	0	0
$561$	811309.	0.108838
$562$	0	0
$563$	−5.14218e6	−0.683717	−0.341858	−	0.939751i	$-0.611056\pi$
$563$	−5.14218e6	−0.683717	−0.341858	+	0.939751i	$0.611056\pi$
$564$	0	0
$565$	2.32512e7	3.06425
$566$	0	0
$567$	−5.43224e6	−0.709612
$568$	0	0
$569$	1.38063e7	1.78771	0.893853	−	0.448359i	$-0.147991\pi$
$569$	1.38063e7	1.78771	0.893853	+	0.448359i	$0.147991\pi$
$570$	0	0
$571$	1.91974e6	0.246406	0.123203	−	0.992381i	$-0.460683\pi$
$571$	1.91974e6	0.246406	0.123203	+	0.992381i	$0.460683\pi$
$572$	0	0
$573$	1.04798e7	1.33342
$574$	0	0
$575$	1.01004e7	1.27400
$576$	0	0
$577$	−7.86662e6	−0.983669	−0.491834	−	0.870689i	$-0.663673\pi$
$577$	−7.86662e6	−0.983669	−0.491834	+	0.870689i	$0.663673\pi$
$578$	0	0
$579$	1.25363e7	1.55408
$580$	0	0
$581$	7.65077e6	0.940296
$582$	0	0
$583$	1.74292e6	0.212376
$584$	0	0
$585$	−452106.	−0.0546198
$586$	0	0
$587$	−5.61716e6	−0.672856	−0.336428	−	0.941709i	$-0.609219\pi$
$587$	−5.61716e6	−0.672856	−0.336428	+	0.941709i	$0.609219\pi$
$588$	0	0
$589$	−981206.	−0.116539
$590$	0	0
$591$	−194130.	−0.0228625
$592$	0	0
$593$	−3.37137e6	−0.393704	−0.196852	−	0.980433i	$-0.563072\pi$
$593$	−3.37137e6	−0.393704	−0.196852	+	0.980433i	$0.563072\pi$
$594$	0	0
$595$	−3.13542e6	−0.363081
$596$	0	0
$597$	6.28002e6	0.721149
$598$	0	0
$599$	−4.63713e6	−0.528059	−0.264030	−	0.964515i	$-0.585052\pi$
$599$	−4.63713e6	−0.528059	−0.264030	+	0.964515i	$0.585052\pi$
$600$	0	0
$601$	−1.32857e7	−1.50037	−0.750184	−	0.661229i	$-0.770035\pi$
$601$	−1.32857e7	−1.50037	−0.750184	+	0.661229i	$0.770035\pi$
$602$	0	0
$603$	4.43367e6	0.496558
$604$	0	0
$605$	−1.32257e7	−1.46903
$606$	0	0
$607$	−5.36627e6	−0.591154	−0.295577	−	0.955319i	$-0.595512\pi$
$607$	−5.36627e6	−0.591154	−0.295577	+	0.955319i	$0.595512\pi$
$608$	0	0
$609$	1.02273e6	0.111743
$610$	0	0
$611$	1.63624e6	0.177314
$612$	0	0
$613$	−1.19912e7	−1.28888	−0.644440	−	0.764655i	$-0.722909\pi$
$613$	−1.19912e7	−1.28888	−0.644440	+	0.764655i	$0.722909\pi$
$614$	0	0
$615$	−1.06606e7	−1.13656
$616$	0	0
$617$	−5.57116e6	−0.589159	−0.294579	−	0.955627i	$-0.595180\pi$
$617$	−5.57116e6	−0.589159	−0.294579	+	0.955627i	$0.595180\pi$
$618$	0	0
$619$	1.06940e7	1.12179	0.560897	−	0.827885i	$-0.310456\pi$
$619$	1.06940e7	1.12179	0.560897	+	0.827885i	$0.310456\pi$
$620$	0	0
$621$	−6.97319e6	−0.725609
$622$	0	0
$623$	−6.83951e6	−0.706000
$624$	0	0
$625$	−3.42772e6	−0.350999
$626$	0	0
$627$	628529.	0.0638494
$628$	0	0
$629$	−4.80420e6	−0.484166
$630$	0	0
$631$	1.27937e7	1.27915	0.639576	−	0.768728i	$-0.279110\pi$
$631$	1.27937e7	1.27915	0.639576	+	0.768728i	$0.279110\pi$
$632$	0	0
$633$	−4.98617e6	−0.494604
$634$	0	0
$635$	−2.80374e7	−2.75932
$636$	0	0
$637$	866401.	0.0846000
$638$	0	0
$639$	2.52555e6	0.244683
$640$	0	0
$641$	−3.64712e6	−0.350594	−0.175297	−	0.984516i	$-0.556089\pi$
$641$	−3.64712e6	−0.350594	−0.175297	+	0.984516i	$0.556089\pi$
$642$	0	0
$643$	9.41881e6	0.898398	0.449199	−	0.893432i	$-0.351709\pi$
$643$	9.41881e6	0.898398	0.449199	+	0.893432i	$0.351709\pi$
$644$	0	0
$645$	−3.23459e6	−0.306140
$646$	0	0
$647$	−8.34880e6	−0.784085	−0.392043	−	0.919947i	$-0.628231\pi$
$647$	−8.34880e6	−0.784085	−0.392043	+	0.919947i	$0.628231\pi$
$648$	0	0
$649$	3.42430e6	0.319125
$650$	0	0
$651$	3.66932e6	0.339338
$652$	0	0
$653$	7.48058e6	0.686518	0.343259	−	0.939241i	$-0.388469\pi$
$653$	7.48058e6	0.686518	0.343259	+	0.939241i	$0.388469\pi$
$654$	0	0
$655$	−9.17471e6	−0.835582
$656$	0	0
$657$	−5.43413e6	−0.491153
$658$	0	0
$659$	−1.39427e7	−1.25064	−0.625320	−	0.780368i	$-0.715032\pi$
$659$	−1.39427e7	−1.25064	−0.625320	+	0.780368i	$0.715032\pi$
$660$	0	0
$661$	−1.06152e7	−0.944986	−0.472493	−	0.881334i	$-0.656646\pi$
$661$	−1.06152e7	−0.944986	−0.472493	+	0.881334i	$0.656646\pi$
$662$	0	0
$663$	650644.	0.0574856
$664$	0	0
$665$	−2.42904e6	−0.213000
$666$	0	0
$667$	1.69085e6	0.147160
$668$	0	0
$669$	1.52762e7	1.31963
$670$	0	0
$671$	4.06525e6	0.348563
$672$	0	0
$673$	9.42379e6	0.802025	0.401012	−	0.916073i	$-0.368658\pi$
$673$	9.42379e6	0.802025	0.401012	+	0.916073i	$0.368658\pi$
$674$	0	0
$675$	−1.41392e7	−1.19444
$676$	0	0
$677$	1.80040e6	0.150972	0.0754862	−	0.997147i	$-0.475949\pi$
$677$	1.80040e6	0.150972	0.0754862	+	0.997147i	$0.475949\pi$
$678$	0	0
$679$	−6.72995e6	−0.560193
$680$	0	0
$681$	8.49799e6	0.702180
$682$	0	0
$683$	1.73202e7	1.42070	0.710348	−	0.703850i	$-0.248537\pi$
$683$	1.73202e7	1.42070	0.710348	+	0.703850i	$0.248537\pi$
$684$	0	0
$685$	2.41543e7	1.96683
$686$	0	0
$687$	−8.99919e6	−0.727464
$688$	0	0
$689$	1.39776e6	0.112172
$690$	0	0
$691$	−1.68765e7	−1.34458	−0.672292	−	0.740286i	$-0.734690\pi$
$691$	−1.68765e7	−1.34458	−0.672292	+	0.740286i	$0.734690\pi$
$692$	0	0
$693$	−495817.	−0.0392182
$694$	0	0
$695$	4.54450e6	0.356882
$696$	0	0
$697$	3.23634e6	0.252332
$698$	0	0
$699$	2.00138e7	1.54931
$700$	0	0
$701$	5.40330e6	0.415302	0.207651	−	0.978203i	$-0.433418\pi$
$701$	5.40330e6	0.415302	0.207651	+	0.978203i	$0.433418\pi$
$702$	0	0
$703$	−3.72186e6	−0.284035
$704$	0	0
$705$	−3.15657e7	−2.39189
$706$	0	0
$707$	9.59378e6	0.721841
$708$	0	0
$709$	2.05223e7	1.53324	0.766619	−	0.642103i	$-0.221938\pi$
$709$	2.05223e7	1.53324	0.766619	+	0.642103i	$0.221938\pi$
$710$	0	0
$711$	2.53368e6	0.187965
$712$	0	0
$713$	6.06633e6	0.446892
$714$	0	0
$715$	690460.	0.0505096
$716$	0	0
$717$	−9.90213e6	−0.719334
$718$	0	0
$719$	−4.37778e6	−0.315815	−0.157907	−	0.987454i	$-0.550475\pi$
$719$	−4.37778e6	−0.315815	−0.157907	+	0.987454i	$0.550475\pi$
$720$	0	0
$721$	−2.16136e6	−0.154842
$722$	0	0
$723$	−8.76602e6	−0.623673
$724$	0	0
$725$	3.42844e6	0.242243
$726$	0	0
$727$	−1.75812e7	−1.23371	−0.616853	−	0.787078i	$-0.711593\pi$
$727$	−1.75812e7	−1.23371	−0.616853	+	0.787078i	$0.711593\pi$
$728$	0	0
$729$	8.73599e6	0.608826
$730$	0	0
$731$	981959.	0.0679673
$732$	0	0
$733$	5.07428e6	0.348831	0.174415	−	0.984672i	$-0.444196\pi$
$733$	5.07428e6	0.348831	0.174415	+	0.984672i	$0.444196\pi$
$734$	0	0
$735$	−1.67143e7	−1.14122
$736$	0	0
$737$	−6.77114e6	−0.459191
$738$	0	0
$739$	−2.35559e7	−1.58668	−0.793339	−	0.608780i	$-0.791659\pi$
$739$	−2.35559e7	−1.58668	−0.793339	+	0.608780i	$0.791659\pi$
$740$	0	0
$741$	504060.	0.0337238
$742$	0	0
$743$	2.16206e7	1.43680	0.718399	−	0.695631i	$-0.244875\pi$
$743$	2.16206e7	1.43680	0.718399	+	0.695631i	$0.244875\pi$
$744$	0	0
$745$	−3.68194e7	−2.43045
$746$	0	0
$747$	6.46089e6	0.423634
$748$	0	0
$749$	8.20709e6	0.534545
$750$	0	0
$751$	2.87245e7	1.85845	0.929227	−	0.369508i	$-0.120474\pi$
$751$	2.87245e7	1.85845	0.929227	+	0.369508i	$0.120474\pi$
$752$	0	0
$753$	2.32100e7	1.49172
$754$	0	0
$755$	−4.96101e6	−0.316740
$756$	0	0
$757$	1.24115e7	0.787199	0.393599	−	0.919282i	$-0.371230\pi$
$757$	1.24115e7	0.787199	0.393599	+	0.919282i	$0.371230\pi$
$758$	0	0
$759$	−3.88590e6	−0.244843
$760$	0	0
$761$	766287.	0.0479656	0.0239828	−	0.999712i	$-0.492365\pi$
$761$	766287.	0.0479656	0.0239828	+	0.999712i	$0.492365\pi$
$762$	0	0
$763$	−6.28797e6	−0.391020
$764$	0	0
$765$	−2.64779e6	−0.163580
$766$	0	0
$767$	2.74618e6	0.168555
$768$	0	0
$769$	−1.49872e7	−0.913914	−0.456957	−	0.889489i	$-0.651061\pi$
$769$	−1.49872e7	−0.913914	−0.456957	+	0.889489i	$0.651061\pi$
$770$	0	0
$771$	510477.	0.0309272
$772$	0	0
$773$	1.16435e7	0.700867	0.350434	−	0.936588i	$-0.386034\pi$
$773$	1.16435e7	0.700867	0.350434	+	0.936588i	$0.386034\pi$
$774$	0	0
$775$	1.23004e7	0.735639
$776$	0	0
$777$	1.39183e7	0.827052
$778$	0	0
$779$	2.50722e6	0.148030
$780$	0	0
$781$	−3.85705e6	−0.226270
$782$	0	0
$783$	−2.36695e6	−0.137970
$784$	0	0
$785$	−5.73547e6	−0.332197
$786$	0	0
$787$	−1.80280e7	−1.03756	−0.518778	−	0.854909i	$-0.673613\pi$
$787$	−1.80280e7	−1.03756	−0.518778	+	0.854909i	$0.673613\pi$
$788$	0	0
$789$	2.45382e7	1.40330
$790$	0	0
$791$	2.04495e7	1.16209
$792$	0	0
$793$	3.26020e6	0.184103
$794$	0	0
$795$	−2.69651e7	−1.51316
$796$	0	0
$797$	1.97016e7	1.09864	0.549319	−	0.835612i	$-0.314887\pi$
$797$	1.97016e7	1.09864	0.549319	+	0.835612i	$0.314887\pi$
$798$	0	0
$799$	9.58272e6	0.531033
$800$	0	0
$801$	−5.77580e6	−0.318076
$802$	0	0
$803$	8.29906e6	0.454193
$804$	0	0
$805$	1.50176e7	0.816790
$806$	0	0
$807$	2.84386e7	1.53718
$808$	0	0
$809$	−6.78469e6	−0.364468	−0.182234	−	0.983255i	$-0.558333\pi$
$809$	−6.78469e6	−0.364468	−0.182234	+	0.983255i	$0.558333\pi$
$810$	0	0
$811$	3.30503e7	1.76450	0.882252	−	0.470777i	$-0.156026\pi$
$811$	3.30503e7	1.76450	0.882252	+	0.470777i	$0.156026\pi$
$812$	0	0
$813$	−5.46713e6	−0.290090
$814$	0	0
$815$	3.89873e7	2.05603
$816$	0	0
$817$	760733.	0.0398729
$818$	0	0
$819$	−397629.	−0.0207142
$820$	0	0
$821$	−2.99319e7	−1.54980	−0.774901	−	0.632083i	$-0.782200\pi$
$821$	−2.99319e7	−1.54980	−0.774901	+	0.632083i	$0.782200\pi$
$822$	0	0
$823$	1.11456e7	0.573591	0.286796	−	0.957992i	$-0.407410\pi$
$823$	1.11456e7	0.573591	0.286796	+	0.957992i	$0.407410\pi$
$824$	0	0
$825$	−7.87923e6	−0.403041
$826$	0	0
$827$	1.66420e7	0.846141	0.423070	−	0.906097i	$-0.360952\pi$
$827$	1.66420e7	0.846141	0.423070	+	0.906097i	$0.360952\pi$
$828$	0	0
$829$	−1.67963e7	−0.848844	−0.424422	−	0.905465i	$-0.639523\pi$
$829$	−1.67963e7	−0.848844	−0.424422	+	0.905465i	$0.639523\pi$
$830$	0	0
$831$	−1.40654e7	−0.706563
$832$	0	0
$833$	5.07413e6	0.253366
$834$	0	0
$835$	−1.31685e7	−0.653611
$836$	0	0
$837$	−8.49203e6	−0.418985
$838$	0	0
$839$	2.44304e7	1.19819	0.599095	−	0.800678i	$-0.295527\pi$
$839$	2.44304e7	1.19819	0.599095	+	0.800678i	$0.295527\pi$
$840$	0	0
$841$	−1.99372e7	−0.972018
$842$	0	0
$843$	2.79966e7	1.35686
$844$	0	0
$845$	−3.19222e7	−1.53798
$846$	0	0
$847$	−1.16321e7	−0.557119
$848$	0	0
$849$	−1.82761e7	−0.870191
$850$	0	0
$851$	2.30105e7	1.08919
$852$	0	0
$853$	−3.68577e6	−0.173442	−0.0867212	−	0.996233i	$-0.527639\pi$
$853$	−3.68577e6	−0.173442	−0.0867212	+	0.996233i	$0.527639\pi$
$854$	0	0
$855$	−2.05127e6	−0.0959636
$856$	0	0
$857$	−3.18729e7	−1.48241	−0.741207	−	0.671276i	$-0.765746\pi$
$857$	−3.18729e7	−1.48241	−0.741207	+	0.671276i	$0.765746\pi$
$858$	0	0
$859$	−3.56678e7	−1.64928	−0.824638	−	0.565661i	$-0.808621\pi$
$859$	−3.56678e7	−1.64928	−0.824638	+	0.565661i	$0.808621\pi$
$860$	0	0
$861$	−9.37602e6	−0.431033
$862$	0	0
$863$	2.38642e7	1.09074	0.545369	−	0.838196i	$-0.316390\pi$
$863$	2.38642e7	1.09074	0.545369	+	0.838196i	$0.316390\pi$
$864$	0	0
$865$	−9.32391e6	−0.423699
$866$	0	0
$867$	−2.11064e7	−0.953600
$868$	0	0
$869$	−3.86945e6	−0.173820
$870$	0	0
$871$	−5.43023e6	−0.242534
$872$	0	0
$873$	−5.68328e6	−0.252385
$874$	0	0
$875$	9.42339e6	0.416090
$876$	0	0
$877$	−1.78983e7	−0.785800	−0.392900	−	0.919581i	$-0.628528\pi$
$877$	−1.78983e7	−0.785800	−0.392900	+	0.919581i	$0.628528\pi$
$878$	0	0
$879$	−6.86206e6	−0.299559
$880$	0	0
$881$	4.19383e7	1.82042	0.910209	−	0.414148i	$-0.135921\pi$
$881$	4.19383e7	1.82042	0.910209	+	0.414148i	$0.135921\pi$
$882$	0	0
$883$	−5.72425e6	−0.247068	−0.123534	−	0.992340i	$-0.539423\pi$
$883$	−5.72425e6	−0.247068	−0.123534	+	0.992340i	$0.539423\pi$
$884$	0	0
$885$	−5.29783e7	−2.27373
$886$	0	0
$887$	5.28478e6	0.225537	0.112769	−	0.993621i	$-0.464028\pi$
$887$	5.28478e6	0.225537	0.112769	+	0.993621i	$0.464028\pi$
$888$	0	0
$889$	−2.46590e7	−1.04646
$890$	0	0
$891$	7.00592e6	0.295645
$892$	0	0
$893$	7.42383e6	0.311529
$894$	0	0
$895$	3.64092e7	1.51934
$896$	0	0
$897$	−3.11636e6	−0.129320
$898$	0	0
$899$	2.05913e6	0.0849737
$900$	0	0
$901$	8.18608e6	0.335942
$902$	0	0
$903$	−2.84484e6	−0.116102
$904$	0	0
$905$	8.78213e6	0.356433
$906$	0	0
$907$	−1.88055e6	−0.0759045	−0.0379523	−	0.999280i	$-0.512083\pi$
$907$	−1.88055e6	−0.0759045	−0.0379523	+	0.999280i	$0.512083\pi$
$908$	0	0
$909$	8.10172e6	0.325213
$910$	0	0
$911$	−1.81410e7	−0.724210	−0.362105	−	0.932137i	$-0.617942\pi$
$911$	−1.81410e7	−0.724210	−0.362105	+	0.932137i	$0.617942\pi$
$912$	0	0
$913$	−9.86713e6	−0.391755
$914$	0	0
$915$	−6.28945e7	−2.48348
$916$	0	0
$917$	−8.06919e6	−0.316889
$918$	0	0
$919$	−3.25793e7	−1.27249	−0.636243	−	0.771489i	$-0.719512\pi$
$919$	−3.25793e7	−1.27249	−0.636243	+	0.771489i	$0.719512\pi$
$920$	0	0
$921$	−8.84897e6	−0.343751
$922$	0	0
$923$	−3.09323e6	−0.119511
$924$	0	0
$925$	4.66572e7	1.79293
$926$	0	0
$927$	−1.82522e6	−0.0697614
$928$	0	0
$929$	5.80658e6	0.220740	0.110370	−	0.993891i	$-0.464796\pi$
$929$	5.80658e6	0.220740	0.110370	+	0.993891i	$0.464796\pi$
$930$	0	0
$931$	3.93098e6	0.148637
$932$	0	0
$933$	−1.40399e6	−0.0528030
$934$	0	0
$935$	4.04372e6	0.151270
$936$	0	0
$937$	3.50025e7	1.30242	0.651209	−	0.758899i	$-0.274262\pi$
$937$	3.50025e7	1.30242	0.651209	+	0.758899i	$0.274262\pi$
$938$	0	0
$939$	−1.74823e6	−0.0647045
$940$	0	0
$941$	−3.99448e7	−1.47057	−0.735285	−	0.677758i	$-0.762952\pi$
$941$	−3.99448e7	−1.47057	−0.735285	+	0.677758i	$0.762952\pi$
$942$	0	0
$943$	−1.55010e7	−0.567649
$944$	0	0
$945$	−2.10226e7	−0.765784
$946$	0	0
$947$	−6.80996e6	−0.246757	−0.123379	−	0.992360i	$-0.539373\pi$
$947$	−6.80996e6	−0.246757	−0.123379	+	0.992360i	$0.539373\pi$
$948$	0	0
$949$	6.65558e6	0.239894
$950$	0	0
$951$	−3.10875e6	−0.111464
$952$	0	0
$953$	−3.48768e7	−1.24395	−0.621976	−	0.783036i	$-0.713670\pi$
$953$	−3.48768e7	−1.24395	−0.621976	+	0.783036i	$0.713670\pi$
$954$	0	0
$955$	5.22332e7	1.85327
$956$	0	0
$957$	−1.31901e6	−0.0465553
$958$	0	0
$959$	2.12438e7	0.745908
$960$	0	0
$961$	−2.12415e7	−0.741954
$962$	0	0
$963$	6.93069e6	0.240830
$964$	0	0
$965$	6.24834e7	2.15996
$966$	0	0
$967$	−4.78004e7	−1.64386	−0.821931	−	0.569587i	$-0.807103\pi$
$967$	−4.78004e7	−1.64386	−0.821931	+	0.569587i	$0.807103\pi$
$968$	0	0
$969$	2.95206e6	0.100999
$970$	0	0
$971$	3.82374e7	1.30149	0.650744	−	0.759298i	$-0.274457\pi$
$971$	3.82374e7	1.30149	0.650744	+	0.759298i	$0.274457\pi$
$972$	0	0
$973$	3.99691e6	0.135345
$974$	0	0
$975$	−6.31889e6	−0.212877
$976$	0	0
$977$	1.14899e7	0.385107	0.192553	−	0.981287i	$-0.438323\pi$
$977$	1.14899e7	0.385107	0.192553	+	0.981287i	$0.438323\pi$
$978$	0	0
$979$	8.82086e6	0.294140
$980$	0	0
$981$	−5.31004e6	−0.176167
$982$	0	0
$983$	−5.75149e7	−1.89844	−0.949220	−	0.314614i	$-0.898125\pi$
$983$	−5.75149e7	−1.89844	−0.949220	+	0.314614i	$0.898125\pi$
$984$	0	0
$985$	−967583.	−0.0317759
$986$	0	0
$987$	−2.77621e7	−0.907110
$988$	0	0
$989$	−4.70325e6	−0.152900
$990$	0	0
$991$	−1.07593e7	−0.348016	−0.174008	−	0.984744i	$-0.555672\pi$
$991$	−1.07593e7	−0.348016	−0.174008	+	0.984744i	$0.555672\pi$
$992$	0	0
$993$	1.85784e7	0.597910
$994$	0	0
$995$	3.13008e7	1.00230
$996$	0	0
$997$	−9.88575e6	−0.314972	−0.157486	−	0.987521i	$-0.550339\pi$
$997$	−9.88575e6	−0.314972	−0.157486	+	0.987521i	$0.550339\pi$
$998$	0	0
$999$	−3.22115e7	−1.02117

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.6.a.b.1.3	✓	4
3.2	odd	2		684.6.a.e.1.1		4
4.3	odd	2		304.6.a.k.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.6.a.b.1.3	✓	4	1.1	even	1	trivial
304.6.a.k.1.2		4	4.3	odd	2
684.6.a.e.1.1		4	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	76.a (trivial)

\(f(q)\)	\(=\)	\(q+17.5489 q^{3} +87.4671 q^{5} +76.9277 q^{7} +64.9636 q^{9} +O(q^{10})\)	\(q+17.5489 q^{3} +87.4671 q^{5} +76.9277 q^{7} +64.9636 q^{9} -99.2130 q^{11} -79.5656 q^{13} +1534.95 q^{15} -465.981 q^{17} -361.000 q^{19} +1350.00 q^{21} +2231.89 q^{23} +4525.49 q^{25} -3124.34 q^{27} +757.584 q^{29} +2718.02 q^{31} -1741.08 q^{33} +6728.64 q^{35} +10309.9 q^{37} -1396.29 q^{39} -6945.22 q^{41} -2107.29 q^{43} +5682.17 q^{45} -20564.6 q^{47} -10889.1 q^{49} -8177.45 q^{51} -17567.4 q^{53} -8677.87 q^{55} -6335.15 q^{57} -34514.7 q^{59} -40975.0 q^{61} +4997.50 q^{63} -6959.37 q^{65} +68248.5 q^{67} +39167.2 q^{69} +38876.4 q^{71} -83648.9 q^{73} +79417.3 q^{75} -7632.23 q^{77} +39001.5 q^{79} -70614.9 q^{81} +99454.0 q^{83} -40758.0 q^{85} +13294.8 q^{87} -88908.3 q^{89} -6120.80 q^{91} +47698.3 q^{93} -31575.6 q^{95} -87484.2 q^{97} -6445.23 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 10 q^{3} - 110 q^{5} + 30 q^{7} + 878 q^{9}+O(q^{10}) \) 4 * q + 10 * q^3 - 110 * q^5 + 30 * q^7 + 878 * q^9	\( 4 q + 10 q^{3} - 110 q^{5} + 30 q^{7} + 878 q^{9} + 706 q^{11} + 788 q^{13} + 2792 q^{15} + 240 q^{17} - 1444 q^{19} + 7128 q^{21} + 5884 q^{23} + 11774 q^{25} + 4426 q^{27} + 5240 q^{29} - 860 q^{31} + 10748 q^{33} + 20322 q^{35} - 20732 q^{37} + 15404 q^{39} - 10204 q^{41} - 12554 q^{43} - 71306 q^{45} - 4826 q^{47} - 21376 q^{49} + 34518 q^{51} - 76484 q^{53} - 72914 q^{55} - 3610 q^{57} + 23898 q^{59} - 32482 q^{61} - 43566 q^{63} - 6076 q^{65} + 5022 q^{67} - 149524 q^{69} + 121300 q^{71} - 104700 q^{73} - 95230 q^{75} - 10002 q^{77} + 117128 q^{79} + 44924 q^{81} + 92832 q^{83} + 80322 q^{85} + 300148 q^{87} + 5988 q^{89} + 165618 q^{91} + 16180 q^{93} + 39710 q^{95} + 22972 q^{97} + 441418 q^{99}+O(q^{100}) \) 4 * q + 10 * q^3 - 110 * q^5 + 30 * q^7 + 878 * q^9 + 706 * q^11 + 788 * q^13 + 2792 * q^15 + 240 * q^17 - 1444 * q^19 + 7128 * q^21 + 5884 * q^23 + 11774 * q^25 + 4426 * q^27 + 5240 * q^29 - 860 * q^31 + 10748 * q^33 + 20322 * q^35 - 20732 * q^37 + 15404 * q^39 - 10204 * q^41 - 12554 * q^43 - 71306 * q^45 - 4826 * q^47 - 21376 * q^49 + 34518 * q^51 - 76484 * q^53 - 72914 * q^55 - 3610 * q^57 + 23898 * q^59 - 32482 * q^61 - 43566 * q^63 - 6076 * q^65 + 5022 * q^67 - 149524 * q^69 + 121300 * q^71 - 104700 * q^73 - 95230 * q^75 - 10002 * q^77 + 117128 * q^79 + 44924 * q^81 + 92832 * q^83 + 80322 * q^85 + 300148 * q^87 + 5988 * q^89 + 165618 * q^91 + 16180 * q^93 + 39710 * q^95 + 22972 * q^97 + 441418 * q^99

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