LMFDB - Embedded newform 968.6.a.p.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [968,6,Mod(1,968)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(968, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("968.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 968 = 2^{3} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	968.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:	$155.251537579$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 2635 x^{14} + 10644 x^{13} + 2721739 x^{12} - 11107836 x^{11} + \cdots + 53\!\cdots\!20 $ x^16 - 4x^15 - 2635x^14 + 10644x^13 + 2721739x^12 - 11107836x^11 - 1388443249x^10 + 5594180788x^9 + 362606893936x^8 - 1331049359080x^7 - 46300877961632x^6 + 122447405593024x^5 + 2801116058142464x^4 - 3524304397180928x^3 - 71736438677110016x^2 + 11226318210301440*x + 533879212527569920
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{26}\cdot 11^{8} $
Twist minimal:	no (minimal twist has level 88)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Root			$26.9484$ of defining polynomial
Character	$\chi$	$=$	968.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+28.5665 q^{3} -7.81027 q^{5} -226.871 q^{7} +573.042 q^{9} +103.782 q^{13} -223.112 q^{15} -743.834 q^{17} -505.644 q^{19} -6480.91 q^{21} +198.898 q^{23} -3064.00 q^{25} +9428.13 q^{27} +6772.60 q^{29} +4660.09 q^{31} +1771.93 q^{35} +15273.7 q^{37} +2964.69 q^{39} +6410.43 q^{41} -17173.7 q^{43} -4475.61 q^{45} +11370.9 q^{47} +34663.6 q^{49} -21248.7 q^{51} +7986.68 q^{53} -14444.5 q^{57} -29343.7 q^{59} +39236.8 q^{61} -130007. q^{63} -810.568 q^{65} +50764.1 q^{67} +5681.82 q^{69} -49625.8 q^{71} -10301.7 q^{73} -87527.6 q^{75} +8842.71 q^{79} +130079. q^{81} +118939. q^{83} +5809.54 q^{85} +193469. q^{87} +113732. q^{89} -23545.3 q^{91} +133122. q^{93} +3949.22 q^{95} -98770.0 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 12 q^{3} + 81 q^{5} - 47 q^{7} + 1416 q^{9} + 859 q^{13} - 738 q^{15} + 1226 q^{17} + 616 q^{19} + 1141 q^{21} + 2258 q^{23} + 10307 q^{25} + 564 q^{27} + 1613 q^{29} + 18511 q^{31} - 23544 q^{35}+ \cdots + 171314 q^{97}+O(q^{100}) $ 16 * q + 12 * q^3 + 81 * q^5 - 47 * q^7 + 1416 * q^9 + 859 * q^13 - 738 * q^15 + 1226 * q^17 + 616 * q^19 + 1141 * q^21 + 2258 * q^23 + 10307 * q^25 + 564 * q^27 + 1613 * q^29 + 18511 * q^31 - 23544 * q^35 + 8553 * q^37 + 12020 * q^39 + 25514 * q^41 - 18931 * q^43 + 23278 * q^45 + 38781 * q^47 + 94523 * q^49 - 68584 * q^51 + 779 * q^53 + 19293 * q^57 + 50316 * q^59 + 53153 * q^61 - 54911 * q^63 + 51517 * q^65 + 10491 * q^67 - 95356 * q^69 - 91677 * q^71 + 97660 * q^73 - 3771 * q^75 - 96003 * q^79 + 120400 * q^81 + 4552 * q^83 - 129679 * q^85 + 77219 * q^87 + 6697 * q^89 + 217380 * q^91 + 132586 * q^93 - 54168 * q^95 + 171314 * 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 $

$2$	0	0
$2$	0	0
$3$	28.5665	1.83254	0.916269	−	0.400563i	$-0.131185\pi$
$3$	28.5665	1.83254	0.916269	+	0.400563i	$0.131185\pi$
$4$	0	0
$5$	−7.81027	−0.139714	−0.0698572	−	0.997557i	$-0.522254\pi$
$5$	−7.81027	−0.139714	−0.0698572	+	0.997557i	$0.522254\pi$
$6$	0	0
$7$	−226.871	−1.74999	−0.874993	−	0.484135i	$-0.839134\pi$
$7$	−226.871	−1.74999	−0.874993	+	0.484135i	$0.839134\pi$
$8$	0	0
$9$	573.042	2.35820
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	103.782	0.170320	0.0851599	−	0.996367i	$-0.472860\pi$
$13$	103.782	0.170320	0.0851599	+	0.996367i	$0.472860\pi$
$14$	0	0
$15$	−223.112	−0.256032
$16$	0	0
$17$	−743.834	−0.624243	−0.312122	−	0.950042i	$-0.601040\pi$
$17$	−743.834	−0.624243	−0.312122	+	0.950042i	$0.601040\pi$
$18$	0	0
$19$	−505.644	−0.321337	−0.160669	−	0.987008i	$-0.551365\pi$
$19$	−505.644	−0.321337	−0.160669	+	0.987008i	$0.551365\pi$
$20$	0	0
$21$	−6480.91	−3.20692
$22$	0	0
$23$	198.898	0.0783991	0.0391996	−	0.999231i	$-0.487519\pi$
$23$	198.898	0.0783991	0.0391996	+	0.999231i	$0.487519\pi$
$24$	0	0
$25$	−3064.00	−0.980480
$26$	0	0
$27$	9428.13	2.48895
$28$	0	0
$29$	6772.60	1.49541	0.747705	−	0.664031i	$-0.231156\pi$
$29$	6772.60	1.49541	0.747705	+	0.664031i	$0.231156\pi$
$30$	0	0
$31$	4660.09	0.870944	0.435472	−	0.900202i	$-0.356581\pi$
$31$	4660.09	0.870944	0.435472	+	0.900202i	$0.356581\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1771.93	0.244498
$36$	0	0
$37$	15273.7	1.83418	0.917089	−	0.398684i	$-0.130533\pi$
$37$	15273.7	1.83418	0.917089	+	0.398684i	$0.130533\pi$
$38$	0	0
$39$	2964.69	0.312118
$40$	0	0
$41$	6410.43	0.595563	0.297781	−	0.954634i	$-0.403753\pi$
$41$	6410.43	0.595563	0.297781	+	0.954634i	$0.403753\pi$
$42$	0	0
$43$	−17173.7	−1.41642	−0.708209	−	0.706003i	$-0.750497\pi$
$43$	−17173.7	−1.41642	−0.708209	+	0.706003i	$0.750497\pi$
$44$	0	0
$45$	−4475.61	−0.329474
$46$	0	0
$47$	11370.9	0.750844	0.375422	−	0.926854i	$-0.377498\pi$
$47$	11370.9	0.750844	0.375422	+	0.926854i	$0.377498\pi$
$48$	0	0
$49$	34663.6	2.06245
$50$	0	0
$51$	−21248.7	−1.14395
$52$	0	0
$53$	7986.68	0.390550	0.195275	−	0.980749i	$-0.437440\pi$
$53$	7986.68	0.390550	0.195275	+	0.980749i	$0.437440\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−14444.5	−0.588863
$58$	0	0
$59$	−29343.7	−1.09745	−0.548724	−	0.836003i	$-0.684886\pi$
$59$	−29343.7	−1.09745	−0.548724	+	0.836003i	$0.684886\pi$
$60$	0	0
$61$	39236.8	1.35011	0.675055	−	0.737768i	$-0.264120\pi$
$61$	39236.8	1.35011	0.675055	+	0.737768i	$0.264120\pi$
$62$	0	0
$63$	−130007.	−4.12681
$64$	0	0
$65$	−810.568	−0.0237961
$66$	0	0
$67$	50764.1	1.38156	0.690781	−	0.723065i	$-0.257267\pi$
$67$	50764.1	1.38156	0.690781	+	0.723065i	$0.257267\pi$
$68$	0	0
$69$	5681.82	0.143669
$70$	0	0
$71$	−49625.8	−1.16832	−0.584159	−	0.811639i	$-0.698576\pi$
$71$	−49625.8	−1.16832	−0.584159	+	0.811639i	$0.698576\pi$
$72$	0	0
$73$	−10301.7	−0.226257	−0.113129	−	0.993580i	$-0.536087\pi$
$73$	−10301.7	−0.226257	−0.113129	+	0.993580i	$0.536087\pi$
$74$	0	0
$75$	−87527.6	−1.79677
$76$	0	0
$77$	0	0
$78$	0	0
$79$	8842.71	0.159411	0.0797054	−	0.996818i	$-0.474602\pi$
$79$	8842.71	0.159411	0.0797054	+	0.996818i	$0.474602\pi$
$80$	0	0
$81$	130079.	2.20290
$82$	0	0
$83$	118939.	1.89508	0.947540	−	0.319636i	$-0.103561\pi$
$83$	118939.	1.89508	0.947540	+	0.319636i	$0.103561\pi$
$84$	0	0
$85$	5809.54	0.0872157
$86$	0	0
$87$	193469.	2.74040
$88$	0	0
$89$	113732.	1.52198	0.760988	−	0.648766i	$-0.224714\pi$
$89$	113732.	1.52198	0.760988	+	0.648766i	$0.224714\pi$
$90$	0	0
$91$	−23545.3	−0.298057
$92$	0	0
$93$	133122.	1.59604
$94$	0	0
$95$	3949.22	0.0448954
$96$	0	0
$97$	−98770.0	−1.06585	−0.532924	−	0.846163i	$-0.678907\pi$
$97$	−98770.0	−1.06585	−0.532924	+	0.846163i	$0.678907\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	110176.	1.07469	0.537346	−	0.843362i	$-0.319427\pi$
$101$	110176.	1.07469	0.537346	+	0.843362i	$0.319427\pi$
$102$	0	0
$103$	1092.79	0.0101495	0.00507474	−	0.999987i	$-0.498385\pi$
$103$	1092.79	0.0101495	0.00507474	+	0.999987i	$0.498385\pi$
$104$	0	0
$105$	50617.6	0.448052
$106$	0	0
$107$	102670.	0.866929	0.433465	−	0.901171i	$-0.357291\pi$
$107$	102670.	0.866929	0.433465	+	0.901171i	$0.357291\pi$
$108$	0	0
$109$	−165410.	−1.33351	−0.666755	−	0.745277i	$-0.732317\pi$
$109$	−165410.	−1.33351	−0.666755	+	0.745277i	$0.732317\pi$
$110$	0	0
$111$	436317.	3.36120
$112$	0	0
$113$	20724.5	0.152682	0.0763410	−	0.997082i	$-0.475676\pi$
$113$	20724.5	0.152682	0.0763410	+	0.997082i	$0.475676\pi$
$114$	0	0
$115$	−1553.45	−0.0109535
$116$	0	0
$117$	59471.7	0.401648
$118$	0	0
$119$	168755.	1.09242
$120$	0	0
$121$	0	0
$122$	0	0
$123$	183123.	1.09139
$124$	0	0
$125$	48337.7	0.276701
$126$	0	0
$127$	120534.	0.663135	0.331567	−	0.943432i	$-0.392423\pi$
$127$	120534.	0.663135	0.331567	+	0.943432i	$0.392423\pi$
$128$	0	0
$129$	−490590.	−2.59564
$130$	0	0
$131$	−194173.	−0.988579	−0.494289	−	0.869297i	$-0.664572\pi$
$131$	−194173.	−0.988579	−0.494289	+	0.869297i	$0.664572\pi$
$132$	0	0
$133$	114716.	0.562336
$134$	0	0
$135$	−73636.2	−0.347742
$136$	0	0
$137$	203753.	0.927476	0.463738	−	0.885972i	$-0.346508\pi$
$137$	203753.	0.927476	0.463738	+	0.885972i	$0.346508\pi$
$138$	0	0
$139$	−133442.	−0.585808	−0.292904	−	0.956142i	$-0.594622\pi$
$139$	−133442.	−0.585808	−0.292904	+	0.956142i	$0.594622\pi$
$140$	0	0
$141$	324826.	1.37595
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−52895.8	−0.208930
$146$	0	0
$147$	990217.	3.77952
$148$	0	0
$149$	161735.	0.596812	0.298406	−	0.954439i	$-0.403545\pi$
$149$	161735.	0.596812	0.298406	+	0.954439i	$0.403545\pi$
$150$	0	0
$151$	328389.	1.17205	0.586025	−	0.810293i	$-0.300692\pi$
$151$	328389.	1.17205	0.586025	+	0.810293i	$0.300692\pi$
$152$	0	0
$153$	−426248.	−1.47209
$154$	0	0
$155$	−36396.6	−0.121683
$156$	0	0
$157$	436449.	1.41314	0.706569	−	0.707644i	$-0.250242\pi$
$157$	436449.	1.41314	0.706569	+	0.707644i	$0.250242\pi$
$158$	0	0
$159$	228151.	0.715698
$160$	0	0
$161$	−45124.3	−0.137197
$162$	0	0
$163$	−420387.	−1.23931	−0.619655	−	0.784874i	$-0.712728\pi$
$163$	−420387.	−1.23931	−0.619655	+	0.784874i	$0.712728\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−59123.7	−0.164048	−0.0820239	−	0.996630i	$-0.526138\pi$
$167$	−59123.7	−0.164048	−0.0820239	+	0.996630i	$0.526138\pi$
$168$	0	0
$169$	−360522.	−0.970991
$170$	0	0
$171$	−289756.	−0.757777
$172$	0	0
$173$	15782.3	0.0400917	0.0200458	−	0.999799i	$-0.493619\pi$
$173$	15782.3	0.0400917	0.0200458	+	0.999799i	$0.493619\pi$
$174$	0	0
$175$	695134.	1.71583
$176$	0	0
$177$	−838244.	−2.01112
$178$	0	0
$179$	246375.	0.574731	0.287365	−	0.957821i	$-0.407221\pi$
$179$	246375.	0.574731	0.287365	+	0.957821i	$0.407221\pi$
$180$	0	0
$181$	461119.	1.04621	0.523103	−	0.852270i	$-0.324774\pi$
$181$	461119.	1.04621	0.523103	+	0.852270i	$0.324774\pi$
$182$	0	0
$183$	1.12086e6	2.47413
$184$	0	0
$185$	−119292.	−0.256261
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−2.13897e6	−4.35563
$190$	0	0
$191$	55118.9	0.109324	0.0546622	−	0.998505i	$-0.482592\pi$
$191$	55118.9	0.109324	0.0546622	+	0.998505i	$0.482592\pi$
$192$	0	0
$193$	334715.	0.646818	0.323409	−	0.946259i	$-0.395171\pi$
$193$	334715.	0.646818	0.323409	+	0.946259i	$0.395171\pi$
$194$	0	0
$195$	−23155.1	−0.0436073
$196$	0	0
$197$	−448550.	−0.823466	−0.411733	−	0.911305i	$-0.635076\pi$
$197$	−448550.	−0.823466	−0.411733	+	0.911305i	$0.635076\pi$
$198$	0	0
$199$	−248626.	−0.445055	−0.222527	−	0.974926i	$-0.571431\pi$
$199$	−248626.	−0.445055	−0.222527	+	0.974926i	$0.571431\pi$
$200$	0	0
$201$	1.45015e6	2.53176
$202$	0	0
$203$	−1.53651e6	−2.61695
$204$	0	0
$205$	−50067.2	−0.0832086
$206$	0	0
$207$	113977.	0.184881
$208$	0	0
$209$	0	0
$210$	0	0
$211$	108300.	0.167464	0.0837320	−	0.996488i	$-0.473316\pi$
$211$	108300.	0.167464	0.0837320	+	0.996488i	$0.473316\pi$
$212$	0	0
$213$	−1.41763e6	−2.14099
$214$	0	0
$215$	134131.	0.197894
$216$	0	0
$217$	−1.05724e6	−1.52414
$218$	0	0
$219$	−294284.	−0.414625
$220$	0	0
$221$	−77196.9	−0.106321
$222$	0	0
$223$	−416787.	−0.561244	−0.280622	−	0.959818i	$-0.590541\pi$
$223$	−416787.	−0.561244	−0.280622	+	0.959818i	$0.590541\pi$
$224$	0	0
$225$	−1.75580e6	−2.31217
$226$	0	0
$227$	−120823.	−0.155627	−0.0778135	−	0.996968i	$-0.524794\pi$
$227$	−120823.	−0.155627	−0.0778135	+	0.996968i	$0.524794\pi$
$228$	0	0
$229$	−727597.	−0.916858	−0.458429	−	0.888731i	$-0.651588\pi$
$229$	−727597.	−0.916858	−0.458429	+	0.888731i	$0.651588\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.28248e6	−1.54761	−0.773803	−	0.633426i	$-0.781648\pi$
$233$	−1.28248e6	−1.54761	−0.773803	+	0.633426i	$0.781648\pi$
$234$	0	0
$235$	−88809.6	−0.104904
$236$	0	0
$237$	252605.	0.292126
$238$	0	0
$239$	1.32934e6	1.50536	0.752680	−	0.658387i	$-0.228761\pi$
$239$	1.32934e6	1.50536	0.752680	+	0.658387i	$0.228761\pi$
$240$	0	0
$241$	141314.	0.156727	0.0783635	−	0.996925i	$-0.475031\pi$
$241$	141314.	0.156727	0.0783635	+	0.996925i	$0.475031\pi$
$242$	0	0
$243$	1.42486e6	1.54795
$244$	0	0
$245$	−270732.	−0.288154
$246$	0	0
$247$	−52477.0	−0.0547301
$248$	0	0
$249$	3.39765e6	3.47281
$250$	0	0
$251$	1.04399e6	1.04595	0.522975	−	0.852348i	$-0.324822\pi$
$251$	1.04399e6	1.04595	0.522975	+	0.852348i	$0.324822\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	165958.	0.159826
$256$	0	0
$257$	458476.	0.432996	0.216498	−	0.976283i	$-0.430536\pi$
$257$	458476.	0.432996	0.216498	+	0.976283i	$0.430536\pi$
$258$	0	0
$259$	−3.46518e6	−3.20978
$260$	0	0
$261$	3.88098e6	3.52647
$262$	0	0
$263$	670400.	0.597647	0.298824	−	0.954308i	$-0.403406\pi$
$263$	670400.	0.597647	0.298824	+	0.954308i	$0.403406\pi$
$264$	0	0
$265$	−62378.1	−0.0545654
$266$	0	0
$267$	3.24892e6	2.78908
$268$	0	0
$269$	420624.	0.354416	0.177208	−	0.984173i	$-0.443294\pi$
$269$	420624.	0.354416	0.177208	+	0.984173i	$0.443294\pi$
$270$	0	0
$271$	−247861.	−0.205015	−0.102507	−	0.994732i	$-0.532687\pi$
$271$	−247861.	−0.205015	−0.102507	+	0.994732i	$0.532687\pi$
$272$	0	0
$273$	−672604.	−0.546202
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−824737.	−0.645827	−0.322913	−	0.946429i	$-0.604662\pi$
$277$	−824737.	−0.645827	−0.322913	+	0.946429i	$0.604662\pi$
$278$	0	0
$279$	2.67043e6	2.05386
$280$	0	0
$281$	55290.1	0.0417717	0.0208858	−	0.999782i	$-0.493351\pi$
$281$	55290.1	0.0417717	0.0208858	+	0.999782i	$0.493351\pi$
$282$	0	0
$283$	402749.	0.298929	0.149465	−	0.988767i	$-0.452245\pi$
$283$	402749.	0.298929	0.149465	+	0.988767i	$0.452245\pi$
$284$	0	0
$285$	112815.	0.0822726
$286$	0	0
$287$	−1.45434e6	−1.04223
$288$	0	0
$289$	−866568.	−0.610320
$290$	0	0
$291$	−2.82151e6	−1.95321
$292$	0	0
$293$	1.08012e6	0.735028	0.367514	−	0.930018i	$-0.380209\pi$
$293$	1.08012e6	0.735028	0.367514	+	0.930018i	$0.380209\pi$
$294$	0	0
$295$	229182.	0.153329
$296$	0	0
$297$	0	0
$298$	0	0
$299$	20642.1	0.0133529
$300$	0	0
$301$	3.89621e6	2.47871
$302$	0	0
$303$	3.14734e6	1.96941
$304$	0	0
$305$	−306450.	−0.188630
$306$	0	0
$307$	−1.62198e6	−0.982199	−0.491100	−	0.871103i	$-0.663405\pi$
$307$	−1.62198e6	−0.982199	−0.491100	+	0.871103i	$0.663405\pi$
$308$	0	0
$309$	31217.1	0.0185993
$310$	0	0
$311$	−2.89346e6	−1.69635	−0.848177	−	0.529713i	$-0.822300\pi$
$311$	−2.89346e6	−1.69635	−0.848177	+	0.529713i	$0.822300\pi$
$312$	0	0
$313$	51329.5	0.0296146	0.0148073	−	0.999890i	$-0.495287\pi$
$313$	51329.5	0.0296146	0.0148073	+	0.999890i	$0.495287\pi$
$314$	0	0
$315$	1.01539e6	0.576575
$316$	0	0
$317$	2.03903e6	1.13966	0.569829	−	0.821763i	$-0.307009\pi$
$317$	2.03903e6	1.13966	0.569829	+	0.821763i	$0.307009\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	2.93291e6	1.58868
$322$	0	0
$323$	376116.	0.200593
$324$	0	0
$325$	−317989.	−0.166995
$326$	0	0
$327$	−4.72519e6	−2.44371
$328$	0	0
$329$	−2.57973e6	−1.31397
$330$	0	0
$331$	3.07847e6	1.54442	0.772210	−	0.635368i	$-0.219151\pi$
$331$	3.07847e6	1.54442	0.772210	+	0.635368i	$0.219151\pi$
$332$	0	0
$333$	8.75250e6	4.32535
$334$	0	0
$335$	−396482.	−0.193024
$336$	0	0
$337$	−3.75093e6	−1.79914	−0.899569	−	0.436779i	$-0.856119\pi$
$337$	−3.75093e6	−1.79914	−0.899569	+	0.436779i	$0.856119\pi$
$338$	0	0
$339$	592025.	0.279796
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−4.05116e6	−1.85928
$344$	0	0
$345$	−44376.5	−0.0200727
$346$	0	0
$347$	1.41792e6	0.632162	0.316081	−	0.948732i	$-0.397633\pi$
$347$	1.41792e6	0.632162	0.316081	+	0.948732i	$0.397633\pi$
$348$	0	0
$349$	1.47969e6	0.650289	0.325144	−	0.945664i	$-0.394587\pi$
$349$	1.47969e6	0.650289	0.325144	+	0.945664i	$0.394587\pi$
$350$	0	0
$351$	978474.	0.423918
$352$	0	0
$353$	1.03495e6	0.442059	0.221030	−	0.975267i	$-0.429058\pi$
$353$	1.03495e6	0.442059	0.221030	+	0.975267i	$0.429058\pi$
$354$	0	0
$355$	387591.	0.163231
$356$	0	0
$357$	4.82072e6	2.00190
$358$	0	0
$359$	2.56769e6	1.05149	0.525746	−	0.850641i	$-0.323786\pi$
$359$	2.56769e6	1.05149	0.525746	+	0.850641i	$0.323786\pi$
$360$	0	0
$361$	−2.22042e6	−0.896742
$362$	0	0
$363$	0	0
$364$	0	0
$365$	80459.2	0.0316114
$366$	0	0
$367$	−1.02419e6	−0.396931	−0.198465	−	0.980108i	$-0.563596\pi$
$367$	−1.02419e6	−0.396931	−0.198465	+	0.980108i	$0.563596\pi$
$368$	0	0
$369$	3.67345e6	1.40445
$370$	0	0
$371$	−1.81195e6	−0.683457
$372$	0	0
$373$	1.22803e6	0.457022	0.228511	−	0.973541i	$-0.426614\pi$
$373$	1.22803e6	0.457022	0.228511	+	0.973541i	$0.426614\pi$
$374$	0	0
$375$	1.38084e6	0.507066
$376$	0	0
$377$	702877.	0.254698
$378$	0	0
$379$	3.32029e6	1.18735	0.593675	−	0.804705i	$-0.297677\pi$
$379$	3.32029e6	1.18735	0.593675	+	0.804705i	$0.297677\pi$
$380$	0	0
$381$	3.44324e6	1.21522
$382$	0	0
$383$	−2.15177e6	−0.749547	−0.374774	−	0.927116i	$-0.622280\pi$
$383$	−2.15177e6	−0.749547	−0.374774	+	0.927116i	$0.622280\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−9.84123e6	−3.34020
$388$	0	0
$389$	−131576.	−0.0440862	−0.0220431	−	0.999757i	$-0.507017\pi$
$389$	−131576.	−0.0440862	−0.0220431	+	0.999757i	$0.507017\pi$
$390$	0	0
$391$	−147947.	−0.0489401
$392$	0	0
$393$	−5.54684e6	−1.81161
$394$	0	0
$395$	−69063.9	−0.0222720
$396$	0	0
$397$	2.18510e6	0.695818	0.347909	−	0.937528i	$-0.386892\pi$
$397$	2.18510e6	0.695818	0.347909	+	0.937528i	$0.386892\pi$
$398$	0	0
$399$	3.27704e6	1.03050
$400$	0	0
$401$	−2.82287e6	−0.876659	−0.438329	−	0.898814i	$-0.644430\pi$
$401$	−2.82287e6	−0.876659	−0.438329	+	0.898814i	$0.644430\pi$
$402$	0	0
$403$	483636.	0.148339
$404$	0	0
$405$	−1.01595e6	−0.307777
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−2.03753e6	−0.602275	−0.301138	−	0.953581i	$-0.597366\pi$
$409$	−2.03753e6	−0.602275	−0.301138	+	0.953581i	$0.597366\pi$
$410$	0	0
$411$	5.82050e6	1.69964
$412$	0	0
$413$	6.65724e6	1.92052
$414$	0	0
$415$	−928943.	−0.264770
$416$	0	0
$417$	−3.81196e6	−1.07352
$418$	0	0
$419$	−6.34804e6	−1.76646	−0.883232	−	0.468937i	$-0.844637\pi$
$419$	−6.34804e6	−1.76646	−0.883232	+	0.468937i	$0.844637\pi$
$420$	0	0
$421$	−4.62038e6	−1.27049	−0.635247	−	0.772309i	$-0.719102\pi$
$421$	−4.62038e6	−1.27049	−0.635247	+	0.772309i	$0.719102\pi$
$422$	0	0
$423$	6.51599e6	1.77064
$424$	0	0
$425$	2.27911e6	0.612058
$426$	0	0
$427$	−8.90171e6	−2.36267
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−239506.	−0.0621045	−0.0310522	−	0.999518i	$-0.509886\pi$
$431$	−239506.	−0.0621045	−0.0310522	+	0.999518i	$0.509886\pi$
$432$	0	0
$433$	−3.99866e6	−1.02493	−0.512466	−	0.858708i	$-0.671268\pi$
$433$	−3.99866e6	−1.02493	−0.512466	+	0.858708i	$0.671268\pi$
$434$	0	0
$435$	−1.51105e6	−0.382873
$436$	0	0
$437$	−100572.	−0.0251926
$438$	0	0
$439$	−1.89124e6	−0.468367	−0.234183	−	0.972192i	$-0.575242\pi$
$439$	−1.89124e6	−0.468367	−0.234183	+	0.972192i	$0.575242\pi$
$440$	0	0
$441$	1.98637e7	4.86367
$442$	0	0
$443$	−325767.	−0.0788675	−0.0394338	−	0.999222i	$-0.512555\pi$
$443$	−325767.	−0.0788675	−0.0394338	+	0.999222i	$0.512555\pi$
$444$	0	0
$445$	−888278.	−0.212642
$446$	0	0
$447$	4.62019e6	1.09368
$448$	0	0
$449$	4.04299e6	0.946426	0.473213	−	0.880948i	$-0.343094\pi$
$449$	4.04299e6	0.946426	0.473213	+	0.880948i	$0.343094\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	9.38091e6	2.14783
$454$	0	0
$455$	183895.	0.0416429
$456$	0	0
$457$	5.69414e6	1.27537	0.637687	−	0.770296i	$-0.279891\pi$
$457$	5.69414e6	1.27537	0.637687	+	0.770296i	$0.279891\pi$
$458$	0	0
$459$	−7.01297e6	−1.55371
$460$	0	0
$461$	3.16150e6	0.692853	0.346426	−	0.938077i	$-0.387395\pi$
$461$	3.16150e6	0.692853	0.346426	+	0.938077i	$0.387395\pi$
$462$	0	0
$463$	5.10293e6	1.10628	0.553142	−	0.833087i	$-0.313429\pi$
$463$	5.10293e6	1.10628	0.553142	+	0.833087i	$0.313429\pi$
$464$	0	0
$465$	−1.03972e6	−0.222990
$466$	0	0
$467$	8.63661e6	1.83253	0.916265	−	0.400573i	$-0.131189\pi$
$467$	8.63661e6	1.83253	0.916265	+	0.400573i	$0.131189\pi$
$468$	0	0
$469$	−1.15169e7	−2.41771
$470$	0	0
$471$	1.24678e7	2.58963
$472$	0	0
$473$	0	0
$474$	0	0
$475$	1.54929e6	0.315065
$476$	0	0
$477$	4.57670e6	0.920994
$478$	0	0
$479$	−2.76656e6	−0.550936	−0.275468	−	0.961310i	$-0.588833\pi$
$479$	−2.76656e6	−0.550936	−0.275468	+	0.961310i	$0.588833\pi$
$480$	0	0
$481$	1.58515e6	0.312397
$482$	0	0
$483$	−1.28904e6	−0.251420
$484$	0	0
$485$	771420.	0.148914
$486$	0	0
$487$	1.28730e6	0.245955	0.122978	−	0.992409i	$-0.460756\pi$
$487$	1.28730e6	0.245955	0.122978	+	0.992409i	$0.460756\pi$
$488$	0	0
$489$	−1.20090e7	−2.27108
$490$	0	0
$491$	−4.50003e6	−0.842386	−0.421193	−	0.906971i	$-0.638389\pi$
$491$	−4.50003e6	−0.842386	−0.421193	+	0.906971i	$0.638389\pi$
$492$	0	0
$493$	−5.03769e6	−0.933500
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.12587e7	2.04454
$498$	0	0
$499$	6.62509e6	1.19108	0.595539	−	0.803326i	$-0.296939\pi$
$499$	6.62509e6	1.19108	0.595539	+	0.803326i	$0.296939\pi$
$500$	0	0
$501$	−1.68896e6	−0.300624
$502$	0	0
$503$	−2.53815e6	−0.447298	−0.223649	−	0.974670i	$-0.571797\pi$
$503$	−2.53815e6	−0.447298	−0.223649	+	0.974670i	$0.571797\pi$
$504$	0	0
$505$	−860504.	−0.150150
$506$	0	0
$507$	−1.02988e7	−1.77938
$508$	0	0
$509$	1.07741e7	1.84325	0.921627	−	0.388076i	$-0.126860\pi$
$509$	1.07741e7	1.84325	0.921627	+	0.388076i	$0.126860\pi$
$510$	0	0
$511$	2.33717e6	0.395947
$512$	0	0
$513$	−4.76728e6	−0.799793
$514$	0	0
$515$	−8534.99	−0.00141803
$516$	0	0
$517$	0	0
$518$	0	0
$519$	450843.	0.0734695
$520$	0	0
$521$	2.33552e6	0.376955	0.188477	−	0.982078i	$-0.439645\pi$
$521$	2.33552e6	0.376955	0.188477	+	0.982078i	$0.439645\pi$
$522$	0	0
$523$	5.82595e6	0.931349	0.465675	−	0.884956i	$-0.345812\pi$
$523$	5.82595e6	0.931349	0.465675	+	0.884956i	$0.345812\pi$
$524$	0	0
$525$	1.98575e7	3.14432
$526$	0	0
$527$	−3.46634e6	−0.543681
$528$	0	0
$529$	−6.39678e6	−0.993854
$530$	0	0
$531$	−1.68152e7	−2.58800
$532$	0	0
$533$	665290.	0.101436
$534$	0	0
$535$	−801879.	−0.121122
$536$	0	0
$537$	7.03806e6	1.05322
$538$	0	0
$539$	0	0
$540$	0	0
$541$	1.43227e6	0.210394	0.105197	−	0.994451i	$-0.466453\pi$
$541$	1.43227e6	0.210394	0.105197	+	0.994451i	$0.466453\pi$
$542$	0	0
$543$	1.31725e7	1.91721
$544$	0	0
$545$	1.29190e6	0.186311
$546$	0	0
$547$	−2.04920e6	−0.292831	−0.146415	−	0.989223i	$-0.546774\pi$
$547$	−2.04920e6	−0.292831	−0.146415	+	0.989223i	$0.546774\pi$
$548$	0	0
$549$	2.24843e7	3.18383
$550$	0	0
$551$	−3.42453e6	−0.480531
$552$	0	0
$553$	−2.00616e6	−0.278967
$554$	0	0
$555$	−3.40775e6	−0.469608
$556$	0	0
$557$	1.31288e7	1.79302	0.896511	−	0.443022i	$-0.146094\pi$
$557$	1.31288e7	1.79302	0.896511	+	0.443022i	$0.146094\pi$
$558$	0	0
$559$	−1.78232e6	−0.241244
$560$	0	0
$561$	0	0
$562$	0	0
$563$	9.67304e6	1.28615	0.643076	−	0.765803i	$-0.277658\pi$
$563$	9.67304e6	1.28615	0.643076	+	0.765803i	$0.277658\pi$
$564$	0	0
$565$	−161864.	−0.0213319
$566$	0	0
$567$	−2.95112e7	−3.85504
$568$	0	0
$569$	7.77016e6	1.00612	0.503059	−	0.864252i	$-0.332208\pi$
$569$	7.77016e6	1.00612	0.503059	+	0.864252i	$0.332208\pi$
$570$	0	0
$571$	−7.11020e6	−0.912624	−0.456312	−	0.889820i	$-0.650830\pi$
$571$	−7.11020e6	−0.912624	−0.456312	+	0.889820i	$0.650830\pi$
$572$	0	0
$573$	1.57455e6	0.200341
$574$	0	0
$575$	−609424.	−0.0768688
$576$	0	0
$577$	−6.57427e6	−0.822068	−0.411034	−	0.911620i	$-0.634832\pi$
$577$	−6.57427e6	−0.822068	−0.411034	+	0.911620i	$0.634832\pi$
$578$	0	0
$579$	9.56162e6	1.18532
$580$	0	0
$581$	−2.69838e7	−3.31636
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−464490.	−0.0561160
$586$	0	0
$587$	−1.32751e7	−1.59016	−0.795082	−	0.606502i	$-0.792572\pi$
$587$	−1.32751e7	−1.59016	−0.795082	+	0.606502i	$0.792572\pi$
$588$	0	0
$589$	−2.35635e6	−0.279867
$590$	0	0
$591$	−1.28135e7	−1.50903
$592$	0	0
$593$	−3.41554e6	−0.398862	−0.199431	−	0.979912i	$-0.563909\pi$
$593$	−3.41554e6	−0.398862	−0.199431	+	0.979912i	$0.563909\pi$
$594$	0	0
$595$	−1.31802e6	−0.152626
$596$	0	0
$597$	−7.10235e6	−0.815580
$598$	0	0
$599$	−1.28063e7	−1.45833	−0.729167	−	0.684336i	$-0.760092\pi$
$599$	−1.28063e7	−1.45833	−0.729167	+	0.684336i	$0.760092\pi$
$600$	0	0
$601$	−5.36209e6	−0.605547	−0.302774	−	0.953063i	$-0.597913\pi$
$601$	−5.36209e6	−0.605547	−0.302774	+	0.953063i	$0.597913\pi$
$602$	0	0
$603$	2.90900e7	3.25799
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−1.11943e7	−1.23318	−0.616589	−	0.787285i	$-0.711486\pi$
$607$	−1.11943e7	−1.23318	−0.616589	+	0.787285i	$0.711486\pi$
$608$	0	0
$609$	−4.38926e7	−4.79566
$610$	0	0
$611$	1.18010e6	0.127884
$612$	0	0
$613$	1.00749e7	1.08291	0.541453	−	0.840731i	$-0.317874\pi$
$613$	1.00749e7	1.08291	0.541453	+	0.840731i	$0.317874\pi$
$614$	0	0
$615$	−1.43024e6	−0.152483
$616$	0	0
$617$	−6.05167e6	−0.639974	−0.319987	−	0.947422i	$-0.603679\pi$
$617$	−6.05167e6	−0.639974	−0.319987	+	0.947422i	$0.603679\pi$
$618$	0	0
$619$	−3.05589e6	−0.320561	−0.160281	−	0.987072i	$-0.551240\pi$
$619$	−3.05589e6	−0.320561	−0.160281	+	0.987072i	$0.551240\pi$
$620$	0	0
$621$	1.87524e6	0.195132
$622$	0	0
$623$	−2.58026e7	−2.66344
$624$	0	0
$625$	9.19747e6	0.941821
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1.13611e7	−1.14497
$630$	0	0
$631$	−1.70588e7	−1.70559	−0.852797	−	0.522242i	$-0.825096\pi$
$631$	−1.70588e7	−1.70559	−0.852797	+	0.522242i	$0.825096\pi$
$632$	0	0
$633$	3.09374e6	0.306884
$634$	0	0
$635$	−941406.	−0.0926494
$636$	0	0
$637$	3.59747e6	0.351276
$638$	0	0
$639$	−2.84377e7	−2.75513
$640$	0	0
$641$	4.69770e6	0.451586	0.225793	−	0.974175i	$-0.427503\pi$
$641$	4.69770e6	0.451586	0.225793	+	0.974175i	$0.427503\pi$
$642$	0	0
$643$	1.22911e7	1.17237	0.586184	−	0.810178i	$-0.300630\pi$
$643$	1.22911e7	1.17237	0.586184	+	0.810178i	$0.300630\pi$
$644$	0	0
$645$	3.83164e6	0.362648
$646$	0	0
$647$	−1.25331e7	−1.17706	−0.588530	−	0.808475i	$-0.700293\pi$
$647$	−1.25331e7	−1.17706	−0.588530	+	0.808475i	$0.700293\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−3.02016e7	−2.79305
$652$	0	0
$653$	−1.71823e6	−0.157688	−0.0788438	−	0.996887i	$-0.525123\pi$
$653$	−1.71823e6	−0.157688	−0.0788438	+	0.996887i	$0.525123\pi$
$654$	0	0
$655$	1.51655e6	0.138119
$656$	0	0
$657$	−5.90332e6	−0.533560
$658$	0	0
$659$	4.08412e6	0.366341	0.183170	−	0.983081i	$-0.441364\pi$
$659$	4.08412e6	0.366341	0.183170	+	0.983081i	$0.441364\pi$
$660$	0	0
$661$	−1.93674e7	−1.72412	−0.862062	−	0.506803i	$-0.830827\pi$
$661$	−1.93674e7	−1.72412	−0.862062	+	0.506803i	$0.830827\pi$
$662$	0	0
$663$	−2.20524e6	−0.194837
$664$	0	0
$665$	−895965.	−0.0785664
$666$	0	0
$667$	1.34706e6	0.117239
$668$	0	0
$669$	−1.19061e7	−1.02850
$670$	0	0
$671$	0	0
$672$	0	0
$673$	7.66812e6	0.652606	0.326303	−	0.945265i	$-0.394197\pi$
$673$	7.66812e6	0.652606	0.326303	+	0.945265i	$0.394197\pi$
$674$	0	0
$675$	−2.88878e7	−2.44037
$676$	0	0
$677$	−9.43644e6	−0.791292	−0.395646	−	0.918403i	$-0.629479\pi$
$677$	−9.43644e6	−0.791292	−0.395646	+	0.918403i	$0.629479\pi$
$678$	0	0
$679$	2.24081e7	1.86522
$680$	0	0
$681$	−3.45149e6	−0.285193
$682$	0	0
$683$	−3.74584e6	−0.307254	−0.153627	−	0.988129i	$-0.549095\pi$
$683$	−3.74584e6	−0.307254	−0.153627	+	0.988129i	$0.549095\pi$
$684$	0	0
$685$	−1.59137e6	−0.129582
$686$	0	0
$687$	−2.07849e7	−1.68018
$688$	0	0
$689$	828877.	0.0665184
$690$	0	0
$691$	6.77469e6	0.539752	0.269876	−	0.962895i	$-0.413017\pi$
$691$	6.77469e6	0.539752	0.269876	+	0.962895i	$0.413017\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.04222e6	0.0818458
$696$	0	0
$697$	−4.76830e6	−0.371776
$698$	0	0
$699$	−3.66359e7	−2.83605
$700$	0	0
$701$	4.84402e6	0.372316	0.186158	−	0.982520i	$-0.440396\pi$
$701$	4.84402e6	0.372316	0.186158	+	0.982520i	$0.440396\pi$
$702$	0	0
$703$	−7.72308e6	−0.589390
$704$	0	0
$705$	−2.53698e6	−0.192240
$706$	0	0
$707$	−2.49958e7	−1.88069
$708$	0	0
$709$	−5.68600e6	−0.424807	−0.212403	−	0.977182i	$-0.568129\pi$
$709$	−5.68600e6	−0.424807	−0.212403	+	0.977182i	$0.568129\pi$
$710$	0	0
$711$	5.06724e6	0.375922
$712$	0	0
$713$	926884.	0.0682813
$714$	0	0
$715$	0	0
$716$	0	0
$717$	3.79744e7	2.75863
$718$	0	0
$719$	1.53777e7	1.10935	0.554675	−	0.832067i	$-0.312843\pi$
$719$	1.53777e7	1.10935	0.554675	+	0.832067i	$0.312843\pi$
$720$	0	0
$721$	−247923.	−0.0177614
$722$	0	0
$723$	4.03685e6	0.287208
$724$	0	0
$725$	−2.07512e7	−1.46622
$726$	0	0
$727$	−5.66695e6	−0.397662	−0.198831	−	0.980034i	$-0.563714\pi$
$727$	−5.66695e6	−0.397662	−0.198831	+	0.980034i	$0.563714\pi$
$728$	0	0
$729$	9.09400e6	0.633776
$730$	0	0
$731$	1.27744e7	0.884190
$732$	0	0
$733$	2.05277e7	1.41117	0.705587	−	0.708623i	$-0.250683\pi$
$733$	2.05277e7	1.41117	0.705587	+	0.708623i	$0.250683\pi$
$734$	0	0
$735$	−7.73386e6	−0.528053
$736$	0	0
$737$	0	0
$738$	0	0
$739$	7.30063e6	0.491755	0.245878	−	0.969301i	$-0.420924\pi$
$739$	7.30063e6	0.491755	0.245878	+	0.969301i	$0.420924\pi$
$740$	0	0
$741$	−1.49908e6	−0.100295
$742$	0	0
$743$	5.31052e6	0.352911	0.176455	−	0.984309i	$-0.443537\pi$
$743$	5.31052e6	0.352911	0.176455	+	0.984309i	$0.443537\pi$
$744$	0	0
$745$	−1.26319e6	−0.0833832
$746$	0	0
$747$	6.81568e7	4.46898
$748$	0	0
$749$	−2.32929e7	−1.51711
$750$	0	0
$751$	−1.84307e7	−1.19246	−0.596228	−	0.802815i	$-0.703335\pi$
$751$	−1.84307e7	−1.19246	−0.596228	+	0.802815i	$0.703335\pi$
$752$	0	0
$753$	2.98230e7	1.91674
$754$	0	0
$755$	−2.56481e6	−0.163752
$756$	0	0
$757$	1.44043e7	0.913594	0.456797	−	0.889571i	$-0.348997\pi$
$757$	1.44043e7	0.913594	0.456797	+	0.889571i	$0.348997\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1.32563e7	−0.829777	−0.414888	−	0.909872i	$-0.636179\pi$
$761$	−1.32563e7	−0.829777	−0.414888	+	0.909872i	$0.636179\pi$
$762$	0	0
$763$	3.75269e7	2.33362
$764$	0	0
$765$	3.32911e6	0.205672
$766$	0	0
$767$	−3.04536e6	−0.186917
$768$	0	0
$769$	−1.42172e7	−0.866958	−0.433479	−	0.901164i	$-0.642714\pi$
$769$	−1.42172e7	−0.866958	−0.433479	+	0.901164i	$0.642714\pi$
$770$	0	0
$771$	1.30970e7	0.793482
$772$	0	0
$773$	−6.54253e6	−0.393819	−0.196910	−	0.980422i	$-0.563091\pi$
$773$	−6.54253e6	−0.393819	−0.196910	+	0.980422i	$0.563091\pi$
$774$	0	0
$775$	−1.42785e7	−0.853943
$776$	0	0
$777$	−9.89878e7	−5.88205
$778$	0	0
$779$	−3.24140e6	−0.191376
$780$	0	0
$781$	0	0
$782$	0	0
$783$	6.38530e7	3.72200
$784$	0	0
$785$	−3.40878e6	−0.197436
$786$	0	0
$787$	1.06942e7	0.615474	0.307737	−	0.951471i	$-0.400428\pi$
$787$	1.06942e7	0.615474	0.307737	+	0.951471i	$0.400428\pi$
$788$	0	0
$789$	1.91510e7	1.09521
$790$	0	0
$791$	−4.70179e6	−0.267191
$792$	0	0
$793$	4.07209e6	0.229950
$794$	0	0
$795$	−1.78192e6	−0.0999933
$796$	0	0
$797$	6.10148e6	0.340243	0.170122	−	0.985423i	$-0.445584\pi$
$797$	6.10148e6	0.340243	0.170122	+	0.985423i	$0.445584\pi$
$798$	0	0
$799$	−8.45805e6	−0.468709
$800$	0	0
$801$	6.51733e7	3.58912
$802$	0	0
$803$	0	0
$804$	0	0
$805$	352433.	0.0191684
$806$	0	0
$807$	1.20157e7	0.649480
$808$	0	0
$809$	4.48277e6	0.240811	0.120405	−	0.992725i	$-0.461581\pi$
$809$	4.48277e6	0.240811	0.120405	+	0.992725i	$0.461581\pi$
$810$	0	0
$811$	−2.72220e7	−1.45334	−0.726672	−	0.686985i	$-0.758934\pi$
$811$	−2.72220e7	−1.45334	−0.726672	+	0.686985i	$0.758934\pi$
$812$	0	0
$813$	−7.08051e6	−0.375697
$814$	0	0
$815$	3.28333e6	0.173149
$816$	0	0
$817$	8.68376e6	0.455148
$818$	0	0
$819$	−1.34924e7	−0.702878
$820$	0	0
$821$	1.09162e7	0.565213	0.282607	−	0.959236i	$-0.408801\pi$
$821$	1.09162e7	0.565213	0.282607	+	0.959236i	$0.408801\pi$
$822$	0	0
$823$	−2.17066e7	−1.11710	−0.558551	−	0.829470i	$-0.688643\pi$
$823$	−2.17066e7	−1.11710	−0.558551	+	0.829470i	$0.688643\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3.15427e7	1.60374	0.801872	−	0.597496i	$-0.203837\pi$
$827$	3.15427e7	1.60374	0.801872	+	0.597496i	$0.203837\pi$
$828$	0	0
$829$	−3.15118e6	−0.159253	−0.0796264	−	0.996825i	$-0.525373\pi$
$829$	−3.15118e6	−0.159253	−0.0796264	+	0.996825i	$0.525373\pi$
$830$	0	0
$831$	−2.35598e7	−1.18350
$832$	0	0
$833$	−2.57840e7	−1.28747
$834$	0	0
$835$	461772.	0.0229198
$836$	0	0
$837$	4.39360e7	2.16774
$838$	0	0
$839$	−2.65837e7	−1.30380	−0.651900	−	0.758305i	$-0.726028\pi$
$839$	−2.65837e7	−1.30380	−0.651900	+	0.758305i	$0.726028\pi$
$840$	0	0
$841$	2.53570e7	1.23625
$842$	0	0
$843$	1.57944e6	0.0765482
$844$	0	0
$845$	2.81578e6	0.135661
$846$	0	0
$847$	0	0
$848$	0	0
$849$	1.15051e7	0.547799
$850$	0	0
$851$	3.03792e6	0.143798
$852$	0	0
$853$	3.74756e7	1.76350	0.881751	−	0.471716i	$-0.156365\pi$
$853$	3.74756e7	1.76350	0.881751	+	0.471716i	$0.156365\pi$
$854$	0	0
$855$	2.26307e6	0.105872
$856$	0	0
$857$	−1.33372e7	−0.620314	−0.310157	−	0.950685i	$-0.600382\pi$
$857$	−1.33372e7	−0.620314	−0.310157	+	0.950685i	$0.600382\pi$
$858$	0	0
$859$	−42617.6	−0.00197064	−0.000985318	−	1.00000i	$-0.500314\pi$
$859$	−42617.6	−0.00197064	−0.000985318		1.00000i	$0.500314\pi$
$860$	0	0
$861$	−4.15454e7	−1.90992
$862$	0	0
$863$	916590.	0.0418936	0.0209468	−	0.999781i	$-0.493332\pi$
$863$	916590.	0.0418936	0.0209468	+	0.999781i	$0.493332\pi$
$864$	0	0
$865$	−123264.	−0.00560138
$866$	0	0
$867$	−2.47548e7	−1.11844
$868$	0	0
$869$	0	0
$870$	0	0
$871$	5.26842e6	0.235307
$872$	0	0
$873$	−5.65993e7	−2.51348
$874$	0	0
$875$	−1.09665e7	−0.484224
$876$	0	0
$877$	−978998.	−0.0429816	−0.0214908	−	0.999769i	$-0.506841\pi$
$877$	−978998.	−0.0429816	−0.0214908	+	0.999769i	$0.506841\pi$
$878$	0	0
$879$	3.08553e7	1.34697
$880$	0	0
$881$	7.30408e6	0.317049	0.158524	−	0.987355i	$-0.449326\pi$
$881$	7.30408e6	0.317049	0.158524	+	0.987355i	$0.449326\pi$
$882$	0	0
$883$	7.42858e6	0.320630	0.160315	−	0.987066i	$-0.448749\pi$
$883$	7.42858e6	0.320630	0.160315	+	0.987066i	$0.448749\pi$
$884$	0	0
$885$	6.54691e6	0.280982
$886$	0	0
$887$	−3.16930e6	−0.135255	−0.0676277	−	0.997711i	$-0.521543\pi$
$887$	−3.16930e6	−0.135255	−0.0676277	+	0.997711i	$0.521543\pi$
$888$	0	0
$889$	−2.73458e7	−1.16048
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−5.74962e6	−0.241274
$894$	0	0
$895$	−1.92426e6	−0.0802981
$896$	0	0
$897$	589673.	0.0244698
$898$	0	0
$899$	3.15609e7	1.30242
$900$	0	0
$901$	−5.94077e6	−0.243798
$902$	0	0
$903$	1.11301e8	4.54234
$904$	0	0
$905$	−3.60147e6	−0.146170
$906$	0	0
$907$	3.41859e6	0.137984	0.0689919	−	0.997617i	$-0.478022\pi$
$907$	3.41859e6	0.137984	0.0689919	+	0.997617i	$0.478022\pi$
$908$	0	0
$909$	6.31355e7	2.53433
$910$	0	0
$911$	−9.37087e6	−0.374097	−0.187048	−	0.982351i	$-0.559892\pi$
$911$	−9.37087e6	−0.374097	−0.187048	+	0.982351i	$0.559892\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−8.75419e6	−0.345671
$916$	0	0
$917$	4.40524e7	1.73000
$918$	0	0
$919$	1.18620e7	0.463308	0.231654	−	0.972798i	$-0.425586\pi$
$919$	1.18620e7	0.463308	0.231654	+	0.972798i	$0.425586\pi$
$920$	0	0
$921$	−4.63342e7	−1.79992
$922$	0	0
$923$	−5.15028e6	−0.198988
$924$	0	0
$925$	−4.67988e7	−1.79837
$926$	0	0
$927$	626215.	0.0239345
$928$	0	0
$929$	−1.79278e7	−0.681535	−0.340767	−	0.940148i	$-0.610687\pi$
$929$	−1.79278e7	−0.681535	−0.340767	+	0.940148i	$0.610687\pi$
$930$	0	0
$931$	−1.75275e7	−0.662743
$932$	0	0
$933$	−8.26559e7	−3.10863
$934$	0	0
$935$	0	0
$936$	0	0
$937$	2.50430e7	0.931833	0.465916	−	0.884829i	$-0.345725\pi$
$937$	2.50430e7	0.931833	0.465916	+	0.884829i	$0.345725\pi$
$938$	0	0
$939$	1.46630e6	0.0542699
$940$	0	0
$941$	4.11569e7	1.51520	0.757598	−	0.652722i	$-0.226373\pi$
$941$	4.11569e7	1.51520	0.757598	+	0.652722i	$0.226373\pi$
$942$	0	0
$943$	1.27502e6	0.0466916
$944$	0	0
$945$	1.67060e7	0.608544
$946$	0	0
$947$	−2.36116e7	−0.855559	−0.427780	−	0.903883i	$-0.640704\pi$
$947$	−2.36116e7	−0.855559	−0.427780	+	0.903883i	$0.640704\pi$
$948$	0	0
$949$	−1.06914e6	−0.0385361
$950$	0	0
$951$	5.82477e7	2.08847
$952$	0	0
$953$	−3.62112e7	−1.29155	−0.645775	−	0.763528i	$-0.723465\pi$
$953$	−3.62112e7	−1.29155	−0.645775	+	0.763528i	$0.723465\pi$
$954$	0	0
$955$	−430493.	−0.0152742
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−4.62257e7	−1.62307
$960$	0	0
$961$	−6.91268e6	−0.241456
$962$	0	0
$963$	5.88342e7	2.04439
$964$	0	0
$965$	−2.61421e6	−0.0903697
$966$	0	0
$967$	3.93226e7	1.35231	0.676154	−	0.736760i	$-0.263645\pi$
$967$	3.93226e7	1.35231	0.676154	+	0.736760i	$0.263645\pi$
$968$	0	0
$969$	1.07443e7	0.367594
$970$	0	0
$971$	−1.24936e7	−0.425244	−0.212622	−	0.977135i	$-0.568200\pi$
$971$	−1.24936e7	−0.425244	−0.212622	+	0.977135i	$0.568200\pi$
$972$	0	0
$973$	3.02742e7	1.02516
$974$	0	0
$975$	−9.08382e6	−0.306025
$976$	0	0
$977$	1.71837e6	0.0575943	0.0287971	−	0.999585i	$-0.490832\pi$
$977$	1.71837e6	0.0575943	0.0287971	+	0.999585i	$0.490832\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−9.47871e7	−3.14468
$982$	0	0
$983$	−3.45031e7	−1.13887	−0.569435	−	0.822036i	$-0.692838\pi$
$983$	−3.45031e7	−1.13887	−0.569435	+	0.822036i	$0.692838\pi$
$984$	0	0
$985$	3.50330e6	0.115050
$986$	0	0
$987$	−7.36937e7	−2.40789
$988$	0	0
$989$	−3.41581e6	−0.111046
$990$	0	0
$991$	−5.18091e7	−1.67580	−0.837899	−	0.545825i	$-0.816216\pi$
$991$	−5.18091e7	−1.67580	−0.837899	+	0.545825i	$0.816216\pi$
$992$	0	0
$993$	8.79410e7	2.83021
$994$	0	0
$995$	1.94183e6	0.0621805
$996$	0	0
$997$	7.05101e6	0.224654	0.112327	−	0.993671i	$-0.464170\pi$
$997$	7.05101e6	0.224654	0.112327	+	0.993671i	$0.464170\pi$
$998$	0	0
$999$	1.44003e8	4.56518

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	968.6.a.p.1.16		16
11.7	odd	10		88.6.i.b.49.1	yes	32
11.8	odd	10		88.6.i.b.9.1	✓	32
11.10	odd	2		968.6.a.q.1.16		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.6.i.b.9.1	✓	32	11.8	odd	10
88.6.i.b.49.1	yes	32	11.7	odd	10
968.6.a.p.1.16		16	1.1	even	1	trivial
968.6.a.q.1.16		16	11.10	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:	\( N \)	\(=\)	\( 968 = 2^{3} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	968.a (trivial)

\(f(q)\)	\(=\)	\(q+28.5665 q^{3} -7.81027 q^{5} -226.871 q^{7} +573.042 q^{9} +103.782 q^{13} -223.112 q^{15} -743.834 q^{17} -505.644 q^{19} -6480.91 q^{21} +198.898 q^{23} -3064.00 q^{25} +9428.13 q^{27} +6772.60 q^{29} +4660.09 q^{31} +1771.93 q^{35} +15273.7 q^{37} +2964.69 q^{39} +6410.43 q^{41} -17173.7 q^{43} -4475.61 q^{45} +11370.9 q^{47} +34663.6 q^{49} -21248.7 q^{51} +7986.68 q^{53} -14444.5 q^{57} -29343.7 q^{59} +39236.8 q^{61} -130007. q^{63} -810.568 q^{65} +50764.1 q^{67} +5681.82 q^{69} -49625.8 q^{71} -10301.7 q^{73} -87527.6 q^{75} +8842.71 q^{79} +130079. q^{81} +118939. q^{83} +5809.54 q^{85} +193469. q^{87} +113732. q^{89} -23545.3 q^{91} +133122. q^{93} +3949.22 q^{95} -98770.0 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 12 q^{3} + 81 q^{5} - 47 q^{7} + 1416 q^{9} + 859 q^{13} - 738 q^{15} + 1226 q^{17} + 616 q^{19} + 1141 q^{21} + 2258 q^{23} + 10307 q^{25} + 564 q^{27} + 1613 q^{29} + 18511 q^{31} - 23544 q^{35}+ \cdots + 171314 q^{97}+O(q^{100}) \) 16 * q + 12 * q^3 + 81 * q^5 - 47 * q^7 + 1416 * q^9 + 859 * q^13 - 738 * q^15 + 1226 * q^17 + 616 * q^19 + 1141 * q^21 + 2258 * q^23 + 10307 * q^25 + 564 * q^27 + 1613 * q^29 + 18511 * q^31 - 23544 * q^35 + 8553 * q^37 + 12020 * q^39 + 25514 * q^41 - 18931 * q^43 + 23278 * q^45 + 38781 * q^47 + 94523 * q^49 - 68584 * q^51 + 779 * q^53 + 19293 * q^57 + 50316 * q^59 + 53153 * q^61 - 54911 * q^63 + 51517 * q^65 + 10491 * q^67 - 95356 * q^69 - 91677 * q^71 + 97660 * q^73 - 3771 * q^75 - 96003 * q^79 + 120400 * q^81 + 4552 * q^83 - 129679 * q^85 + 77219 * q^87 + 6697 * q^89 + 217380 * q^91 + 132586 * q^93 - 54168 * q^95 + 171314 * q^97

Introduction

L-functions

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Database

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