LMFDB - Embedded newform 392.6.a.m.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [392,6,Mod(1,392)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(392, 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("392.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 392 = 2^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	392.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:	$62.8704573667$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 328x^{4} - 1328x^{3} + 25933x^{2} + 205840x + 390334 $ x^6 - 328x^4 - 1328x^3 + 25933x^2 + 205840x + 390334
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4}\cdot 7^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-10.9869$ of defining polynomial
Character	$\chi$	$=$	392.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+16.3662 q^{3} +51.0131 q^{5} +24.8528 q^{9} +O(q^{10})$	$q+16.3662 q^{3} +51.0131 q^{5} +24.8528 q^{9} -273.428 q^{11} -369.233 q^{13} +834.891 q^{15} -1036.23 q^{17} -2183.85 q^{19} -1013.90 q^{23} -522.662 q^{25} -3570.24 q^{27} -212.806 q^{29} +229.712 q^{31} -4474.99 q^{33} -9118.74 q^{37} -6042.94 q^{39} -16772.9 q^{41} +16121.9 q^{43} +1267.82 q^{45} +1125.47 q^{47} -16959.2 q^{51} +34164.1 q^{53} -13948.4 q^{55} -35741.3 q^{57} +11191.2 q^{59} +35993.1 q^{61} -18835.7 q^{65} -57518.6 q^{67} -16593.8 q^{69} -20951.5 q^{71} +56427.0 q^{73} -8554.00 q^{75} +48161.1 q^{79} -64470.6 q^{81} -64392.7 q^{83} -52861.3 q^{85} -3482.82 q^{87} +62496.2 q^{89} +3759.51 q^{93} -111405. q^{95} +35202.8 q^{97} -6795.46 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 122 q^{9}+O(q^{10}) $ 6 * q - 122 * q^9	$ 6 q - 122 q^{9} + 248 q^{11} - 376 q^{15} - 2904 q^{23} + 1510 q^{25} - 408 q^{29} - 18648 q^{37} - 15240 q^{39} - 10584 q^{43} + 3016 q^{51} - 66772 q^{53} - 56264 q^{57} - 55316 q^{65} - 169632 q^{67} - 54880 q^{71} + 50576 q^{79} - 270194 q^{81} - 406076 q^{85} - 164944 q^{93} - 551912 q^{95} - 173512 q^{99}+O(q^{100}) $ 6 * q - 122 * q^9 + 248 * q^11 - 376 * q^15 - 2904 * q^23 + 1510 * q^25 - 408 * q^29 - 18648 * q^37 - 15240 * q^39 - 10584 * q^43 + 3016 * q^51 - 66772 * q^53 - 56264 * q^57 - 55316 * q^65 - 169632 * q^67 - 54880 * q^71 + 50576 * q^79 - 270194 * q^81 - 406076 * q^85 - 164944 * q^93 - 551912 * q^95 - 173512 * 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
$3$	16.3662	1.04989	0.524946	−	0.851135i	$-0.324085\pi$
$3$	16.3662	1.04989	0.524946	+	0.851135i	$0.324085\pi$
$4$	0	0
$5$	51.0131	0.912550	0.456275	−	0.889839i	$-0.349183\pi$
$5$	51.0131	0.912550	0.456275	+	0.889839i	$0.349183\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	24.8528	0.102275
$10$	0	0
$11$	−273.428	−0.681337	−0.340668	−	0.940183i	$-0.610653\pi$
$11$	−273.428	−0.681337	−0.340668	+	0.940183i	$0.610653\pi$
$12$	0	0
$13$	−369.233	−0.605957	−0.302979	−	0.952997i	$-0.597981\pi$
$13$	−369.233	−0.605957	−0.302979	+	0.952997i	$0.597981\pi$
$14$	0	0
$15$	834.891	0.958080
$16$	0	0
$17$	−1036.23	−0.869628	−0.434814	−	0.900520i	$-0.643186\pi$
$17$	−1036.23	−0.869628	−0.434814	+	0.900520i	$0.643186\pi$
$18$	0	0
$19$	−2183.85	−1.38784	−0.693918	−	0.720054i	$-0.744117\pi$
$19$	−2183.85	−1.38784	−0.693918	+	0.720054i	$0.744117\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1013.90	−0.399648	−0.199824	−	0.979832i	$-0.564037\pi$
$23$	−1013.90	−0.399648	−0.199824	+	0.979832i	$0.564037\pi$
$24$	0	0
$25$	−522.662	−0.167252
$26$	0	0
$27$	−3570.24	−0.942515
$28$	0	0
$29$	−212.806	−0.0469881	−0.0234941	−	0.999724i	$-0.507479\pi$
$29$	−212.806	−0.0469881	−0.0234941	+	0.999724i	$0.507479\pi$
$30$	0	0
$31$	229.712	0.0429318	0.0214659	−	0.999770i	$-0.493167\pi$
$31$	229.712	0.0429318	0.0214659	+	0.999770i	$0.493167\pi$
$32$	0	0
$33$	−4474.99	−0.715331
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−9118.74	−1.09504	−0.547521	−	0.836792i	$-0.684428\pi$
$37$	−9118.74	−1.09504	−0.547521	+	0.836792i	$0.684428\pi$
$38$	0	0
$39$	−6042.94	−0.636190
$40$	0	0
$41$	−16772.9	−1.55829	−0.779144	−	0.626845i	$-0.784346\pi$
$41$	−16772.9	−1.55829	−0.779144	+	0.626845i	$0.784346\pi$
$42$	0	0
$43$	16121.9	1.32968	0.664838	−	0.746987i	$-0.268500\pi$
$43$	16121.9	1.32968	0.664838	+	0.746987i	$0.268500\pi$
$44$	0	0
$45$	1267.82	0.0933310
$46$	0	0
$47$	1125.47	0.0743170	0.0371585	−	0.999309i	$-0.488169\pi$
$47$	1125.47	0.0743170	0.0371585	+	0.999309i	$0.488169\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−16959.2	−0.913017
$52$	0	0
$53$	34164.1	1.67063	0.835315	−	0.549771i	$-0.185285\pi$
$53$	34164.1	1.67063	0.835315	+	0.549771i	$0.185285\pi$
$54$	0	0
$55$	−13948.4	−0.621754
$56$	0	0
$57$	−35741.3	−1.45708
$58$	0	0
$59$	11191.2	0.418549	0.209275	−	0.977857i	$-0.432890\pi$
$59$	11191.2	0.418549	0.209275	+	0.977857i	$0.432890\pi$
$60$	0	0
$61$	35993.1	1.23850	0.619248	−	0.785195i	$-0.287437\pi$
$61$	35993.1	1.23850	0.619248	+	0.785195i	$0.287437\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−18835.7	−0.552966
$66$	0	0
$67$	−57518.6	−1.56538	−0.782692	−	0.622409i	$-0.786154\pi$
$67$	−57518.6	−1.56538	−0.782692	+	0.622409i	$0.786154\pi$
$68$	0	0
$69$	−16593.8	−0.419587
$70$	0	0
$71$	−20951.5	−0.493253	−0.246627	−	0.969111i	$-0.579322\pi$
$71$	−20951.5	−0.493253	−0.246627	+	0.969111i	$0.579322\pi$
$72$	0	0
$73$	56427.0	1.23931	0.619655	−	0.784874i	$-0.287273\pi$
$73$	56427.0	1.23931	0.619655	+	0.784874i	$0.287273\pi$
$74$	0	0
$75$	−8554.00	−0.175597
$76$	0	0
$77$	0	0
$78$	0	0
$79$	48161.1	0.868217	0.434108	−	0.900861i	$-0.357063\pi$
$79$	48161.1	0.868217	0.434108	+	0.900861i	$0.357063\pi$
$80$	0	0
$81$	−64470.6	−1.09181
$82$	0	0
$83$	−64392.7	−1.02599	−0.512993	−	0.858393i	$-0.671463\pi$
$83$	−64392.7	−1.02599	−0.512993	+	0.858393i	$0.671463\pi$
$84$	0	0
$85$	−52861.3	−0.793580
$86$	0	0
$87$	−3482.82	−0.0493325
$88$	0	0
$89$	62496.2	0.836332	0.418166	−	0.908371i	$-0.362673\pi$
$89$	62496.2	0.836332	0.418166	+	0.908371i	$0.362673\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	3759.51	0.0450738
$94$	0	0
$95$	−111405.	−1.26647
$96$	0	0
$97$	35202.8	0.379881	0.189941	−	0.981796i	$-0.439170\pi$
$97$	35202.8	0.379881	0.189941	+	0.981796i	$0.439170\pi$
$98$	0	0
$99$	−6795.46	−0.0696837
$100$	0	0
$101$	82338.6	0.803156	0.401578	−	0.915825i	$-0.368462\pi$
$101$	82338.6	0.803156	0.401578	+	0.915825i	$0.368462\pi$
$102$	0	0
$103$	98841.9	0.918011	0.459006	−	0.888433i	$-0.348206\pi$
$103$	98841.9	0.918011	0.459006	+	0.888433i	$0.348206\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	216402.	1.82727	0.913633	−	0.406539i	$-0.133265\pi$
$107$	216402.	1.82727	0.913633	+	0.406539i	$0.133265\pi$
$108$	0	0
$109$	−181858.	−1.46611	−0.733056	−	0.680168i	$-0.761907\pi$
$109$	−181858.	−1.46611	−0.733056	+	0.680168i	$0.761907\pi$
$110$	0	0
$111$	−149239.	−1.14968
$112$	0	0
$113$	−191197.	−1.40859	−0.704296	−	0.709907i	$-0.748737\pi$
$113$	−191197.	−1.40859	−0.704296	+	0.709907i	$0.748737\pi$
$114$	0	0
$115$	−51722.4	−0.364699
$116$	0	0
$117$	−9176.47	−0.0619742
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−86287.9	−0.535780
$122$	0	0
$123$	−274508.	−1.63604
$124$	0	0
$125$	−186079.	−1.06518
$126$	0	0
$127$	−136702.	−0.752083	−0.376041	−	0.926603i	$-0.622715\pi$
$127$	−136702.	−0.752083	−0.376041	+	0.926603i	$0.622715\pi$
$128$	0	0
$129$	263855.	1.39602
$130$	0	0
$131$	85421.7	0.434900	0.217450	−	0.976071i	$-0.430226\pi$
$131$	85421.7	0.434900	0.217450	+	0.976071i	$0.430226\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−182129.	−0.860092
$136$	0	0
$137$	−41279.3	−0.187902	−0.0939508	−	0.995577i	$-0.529950\pi$
$137$	−41279.3	−0.187902	−0.0939508	+	0.995577i	$0.529950\pi$
$138$	0	0
$139$	−245593.	−1.07815	−0.539075	−	0.842258i	$-0.681226\pi$
$139$	−245593.	−1.07815	−0.539075	+	0.842258i	$0.681226\pi$
$140$	0	0
$141$	18419.6	0.0780249
$142$	0	0
$143$	100959.	0.412861
$144$	0	0
$145$	−10855.9	−0.0428790
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−327658.	−1.20908	−0.604539	−	0.796575i	$-0.706643\pi$
$149$	−327658.	−1.20908	−0.604539	+	0.796575i	$0.706643\pi$
$150$	0	0
$151$	372670.	1.33009	0.665047	−	0.746802i	$-0.268411\pi$
$151$	372670.	1.33009	0.665047	+	0.746802i	$0.268411\pi$
$152$	0	0
$153$	−25753.2	−0.0889412
$154$	0	0
$155$	11718.3	0.0391774
$156$	0	0
$157$	210988.	0.683139	0.341569	−	0.939857i	$-0.389042\pi$
$157$	210988.	0.683139	0.341569	+	0.939857i	$0.389042\pi$
$158$	0	0
$159$	559137.	1.75398
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−192821.	−0.568440	−0.284220	−	0.958759i	$-0.591735\pi$
$163$	−192821.	−0.568440	−0.284220	+	0.958759i	$0.591735\pi$
$164$	0	0
$165$	−228283.	−0.652775
$166$	0	0
$167$	−507788.	−1.40893	−0.704467	−	0.709736i	$-0.748814\pi$
$167$	−507788.	−1.40893	−0.704467	+	0.709736i	$0.748814\pi$
$168$	0	0
$169$	−234960.	−0.632816
$170$	0	0
$171$	−54274.7	−0.141941
$172$	0	0
$173$	−563675.	−1.43190	−0.715951	−	0.698150i	$-0.754007\pi$
$173$	−563675.	−1.43190	−0.715951	+	0.698150i	$0.754007\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	183157.	0.439432
$178$	0	0
$179$	190315.	0.443957	0.221978	−	0.975052i	$-0.428749\pi$
$179$	190315.	0.443957	0.221978	+	0.975052i	$0.428749\pi$
$180$	0	0
$181$	−428708.	−0.972670	−0.486335	−	0.873772i	$-0.661667\pi$
$181$	−428708.	−0.972670	−0.486335	+	0.873772i	$0.661667\pi$
$182$	0	0
$183$	589071.	1.30029
$184$	0	0
$185$	−465175.	−0.999280
$186$	0	0
$187$	283335.	0.592510
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−231675.	−0.459510	−0.229755	−	0.973248i	$-0.573793\pi$
$191$	−231675.	−0.459510	−0.229755	+	0.973248i	$0.573793\pi$
$192$	0	0
$193$	−421807.	−0.815118	−0.407559	−	0.913179i	$-0.633620\pi$
$193$	−421807.	−0.815118	−0.407559	+	0.913179i	$0.633620\pi$
$194$	0	0
$195$	−308269.	−0.580555
$196$	0	0
$197$	−708516.	−1.30072	−0.650360	−	0.759626i	$-0.725382\pi$
$197$	−708516.	−1.30072	−0.650360	+	0.759626i	$0.725382\pi$
$198$	0	0
$199$	642566.	1.15023	0.575116	−	0.818072i	$-0.304957\pi$
$199$	642566.	1.15023	0.575116	+	0.818072i	$0.304957\pi$
$200$	0	0
$201$	−941361.	−1.64349
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−855637.	−1.42202
$206$	0	0
$207$	−25198.4	−0.0408739
$208$	0	0
$209$	597126.	0.945584
$210$	0	0
$211$	916183.	1.41669	0.708347	−	0.705864i	$-0.249441\pi$
$211$	916183.	1.41669	0.708347	+	0.705864i	$0.249441\pi$
$212$	0	0
$213$	−342897.	−0.517863
$214$	0	0
$215$	822430.	1.21340
$216$	0	0
$217$	0	0
$218$	0	0
$219$	923496.	1.30114
$220$	0	0
$221$	382610.	0.526958
$222$	0	0
$223$	965140.	1.29966	0.649828	−	0.760081i	$-0.274841\pi$
$223$	965140.	1.29966	0.649828	+	0.760081i	$0.274841\pi$
$224$	0	0
$225$	−12989.6	−0.0171057
$226$	0	0
$227$	−893402.	−1.15075	−0.575377	−	0.817888i	$-0.695145\pi$
$227$	−893402.	−1.15075	−0.575377	+	0.817888i	$0.695145\pi$
$228$	0	0
$229$	−1.21753e6	−1.53423	−0.767117	−	0.641508i	$-0.778309\pi$
$229$	−1.21753e6	−1.53423	−0.767117	+	0.641508i	$0.778309\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.24507e6	1.50247	0.751234	−	0.660037i	$-0.229459\pi$
$233$	1.24507e6	1.50247	0.751234	+	0.660037i	$0.229459\pi$
$234$	0	0
$235$	57413.6	0.0678180
$236$	0	0
$237$	788214.	0.911535
$238$	0	0
$239$	−223953.	−0.253607	−0.126803	−	0.991928i	$-0.540472\pi$
$239$	−223953.	−0.253607	−0.126803	+	0.991928i	$0.540472\pi$
$240$	0	0
$241$	401523.	0.445316	0.222658	−	0.974897i	$-0.428527\pi$
$241$	401523.	0.445316	0.222658	+	0.974897i	$0.428527\pi$
$242$	0	0
$243$	−187570.	−0.203773
$244$	0	0
$245$	0	0
$246$	0	0
$247$	806348.	0.840970
$248$	0	0
$249$	−1.05386e6	−1.07718
$250$	0	0
$251$	406156.	0.406920	0.203460	−	0.979083i	$-0.434781\pi$
$251$	406156.	0.406920	0.203460	+	0.979083i	$0.434781\pi$
$252$	0	0
$253$	277230.	0.272295
$254$	0	0
$255$	−865139.	−0.833173
$256$	0	0
$257$	830971.	0.784790	0.392395	−	0.919797i	$-0.371647\pi$
$257$	830971.	0.784790	0.392395	+	0.919797i	$0.371647\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−5288.82	−0.00480571
$262$	0	0
$263$	1.65945e6	1.47936	0.739681	−	0.672958i	$-0.234976\pi$
$263$	1.65945e6	1.47936	0.739681	+	0.672958i	$0.234976\pi$
$264$	0	0
$265$	1.74282e6	1.52453
$266$	0	0
$267$	1.02283e6	0.878059
$268$	0	0
$269$	105517.	0.0889079	0.0444539	−	0.999011i	$-0.485845\pi$
$269$	105517.	0.0889079	0.0444539	+	0.999011i	$0.485845\pi$
$270$	0	0
$271$	1.75495e6	1.45158	0.725791	−	0.687915i	$-0.241474\pi$
$271$	1.75495e6	1.45158	0.725791	+	0.687915i	$0.241474\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	142911.	0.113955
$276$	0	0
$277$	2.15001e6	1.68361	0.841803	−	0.539785i	$-0.181494\pi$
$277$	2.15001e6	1.68361	0.841803	+	0.539785i	$0.181494\pi$
$278$	0	0
$279$	5708.98	0.00439084
$280$	0	0
$281$	−1.94418e6	−1.46883	−0.734415	−	0.678701i	$-0.762543\pi$
$281$	−1.94418e6	−1.46883	−0.734415	+	0.678701i	$0.762543\pi$
$282$	0	0
$283$	−428457.	−0.318010	−0.159005	−	0.987278i	$-0.550829\pi$
$283$	−428457.	−0.318010	−0.159005	+	0.987278i	$0.550829\pi$
$284$	0	0
$285$	−1.82328e6	−1.32966
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−346085.	−0.243747
$290$	0	0
$291$	576136.	0.398835
$292$	0	0
$293$	1.75383e6	1.19349	0.596746	−	0.802430i	$-0.296460\pi$
$293$	1.75383e6	1.19349	0.596746	+	0.802430i	$0.296460\pi$
$294$	0	0
$295$	570898.	0.381947
$296$	0	0
$297$	976206.	0.642170
$298$	0	0
$299$	374367.	0.242169
$300$	0	0
$301$	0	0
$302$	0	0
$303$	1.34757e6	0.843228
$304$	0	0
$305$	1.83612e6	1.13019
$306$	0	0
$307$	2.02249e6	1.22473	0.612364	−	0.790576i	$-0.290219\pi$
$307$	2.02249e6	1.22473	0.612364	+	0.790576i	$0.290219\pi$
$308$	0	0
$309$	1.61767e6	0.963813
$310$	0	0
$311$	−1.42625e6	−0.836169	−0.418085	−	0.908408i	$-0.637298\pi$
$311$	−1.42625e6	−0.836169	−0.418085	+	0.908408i	$0.637298\pi$
$312$	0	0
$313$	−53273.6	−0.0307363	−0.0153681	−	0.999882i	$-0.504892\pi$
$313$	−53273.6	−0.0307363	−0.0153681	+	0.999882i	$0.504892\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2.62481e6	−1.46707	−0.733533	−	0.679654i	$-0.762130\pi$
$317$	−2.62481e6	−1.46707	−0.733533	+	0.679654i	$0.762130\pi$
$318$	0	0
$319$	58187.1	0.0320147
$320$	0	0
$321$	3.54168e6	1.91843
$322$	0	0
$323$	2.26297e6	1.20690
$324$	0	0
$325$	192984.	0.101348
$326$	0	0
$327$	−2.97633e6	−1.53926
$328$	0	0
$329$	0	0
$330$	0	0
$331$	1.30348e6	0.653936	0.326968	−	0.945035i	$-0.393973\pi$
$331$	1.30348e6	0.653936	0.326968	+	0.945035i	$0.393973\pi$
$332$	0	0
$333$	−226626.	−0.111995
$334$	0	0
$335$	−2.93420e6	−1.42849
$336$	0	0
$337$	−1.33530e6	−0.640476	−0.320238	−	0.947337i	$-0.603763\pi$
$337$	−1.33530e6	−0.640476	−0.320238	+	0.947337i	$0.603763\pi$
$338$	0	0
$339$	−3.12917e6	−1.47887
$340$	0	0
$341$	−62809.7	−0.0292510
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−846500.	−0.382894
$346$	0	0
$347$	1.72459e6	0.768887	0.384443	−	0.923149i	$-0.374393\pi$
$347$	1.72459e6	0.768887	0.384443	+	0.923149i	$0.374393\pi$
$348$	0	0
$349$	−2.97845e6	−1.30896	−0.654481	−	0.756079i	$-0.727113\pi$
$349$	−2.97845e6	−1.30896	−0.654481	+	0.756079i	$0.727113\pi$
$350$	0	0
$351$	1.31825e6	0.571124
$352$	0	0
$353$	518454.	0.221449	0.110725	−	0.993851i	$-0.464683\pi$
$353$	518454.	0.221449	0.110725	+	0.993851i	$0.464683\pi$
$354$	0	0
$355$	−1.06880e6	−0.450118
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−253455.	−0.103792	−0.0518961	−	0.998652i	$-0.516526\pi$
$359$	−253455.	−0.103792	−0.0518961	+	0.998652i	$0.516526\pi$
$360$	0	0
$361$	2.29309e6	0.926091
$362$	0	0
$363$	−1.41221e6	−0.562512
$364$	0	0
$365$	2.87852e6	1.13093
$366$	0	0
$367$	−639610.	−0.247885	−0.123943	−	0.992289i	$-0.539554\pi$
$367$	−639610.	−0.247885	−0.123943	+	0.992289i	$0.539554\pi$
$368$	0	0
$369$	−416853.	−0.159374
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−3.10000e6	−1.15369	−0.576845	−	0.816854i	$-0.695716\pi$
$373$	−3.10000e6	−1.15369	−0.576845	+	0.816854i	$0.695716\pi$
$374$	0	0
$375$	−3.04540e6	−1.11832
$376$	0	0
$377$	78574.8	0.0284728
$378$	0	0
$379$	−746368.	−0.266904	−0.133452	−	0.991055i	$-0.542606\pi$
$379$	−746368.	−0.266904	−0.133452	+	0.991055i	$0.542606\pi$
$380$	0	0
$381$	−2.23729e6	−0.789606
$382$	0	0
$383$	−1.12449e6	−0.391703	−0.195852	−	0.980634i	$-0.562747\pi$
$383$	−1.12449e6	−0.391703	−0.195852	+	0.980634i	$0.562747\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	400675.	0.135993
$388$	0	0
$389$	885972.	0.296856	0.148428	−	0.988923i	$-0.452579\pi$
$389$	885972.	0.296856	0.148428	+	0.988923i	$0.452579\pi$
$390$	0	0
$391$	1.05064e6	0.347545
$392$	0	0
$393$	1.39803e6	0.456599
$394$	0	0
$395$	2.45685e6	0.792292
$396$	0	0
$397$	−1.77037e6	−0.563752	−0.281876	−	0.959451i	$-0.590957\pi$
$397$	−1.77037e6	−0.563752	−0.281876	+	0.959451i	$0.590957\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	4.07709e6	1.26616	0.633082	−	0.774085i	$-0.281790\pi$
$401$	4.07709e6	1.26616	0.633082	+	0.774085i	$0.281790\pi$
$402$	0	0
$403$	−84817.1	−0.0260148
$404$	0	0
$405$	−3.28884e6	−0.996336
$406$	0	0
$407$	2.49332e6	0.746092
$408$	0	0
$409$	−349030.	−0.103170	−0.0515852	−	0.998669i	$-0.516427\pi$
$409$	−349030.	−0.103170	−0.0515852	+	0.998669i	$0.516427\pi$
$410$	0	0
$411$	−675585.	−0.197277
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−3.28487e6	−0.936264
$416$	0	0
$417$	−4.01943e6	−1.13194
$418$	0	0
$419$	3.72353e6	1.03614	0.518072	−	0.855337i	$-0.326650\pi$
$419$	3.72353e6	1.03614	0.518072	+	0.855337i	$0.326650\pi$
$420$	0	0
$421$	−5.31261e6	−1.46084	−0.730421	−	0.682998i	$-0.760676\pi$
$421$	−5.31261e6	−1.46084	−0.730421	+	0.682998i	$0.760676\pi$
$422$	0	0
$423$	27971.0	0.00760077
$424$	0	0
$425$	541598.	0.145447
$426$	0	0
$427$	0	0
$428$	0	0
$429$	1.65231e6	0.433460
$430$	0	0
$431$	−3.73245e6	−0.967835	−0.483917	−	0.875114i	$-0.660786\pi$
$431$	−3.73245e6	−0.967835	−0.483917	+	0.875114i	$0.660786\pi$
$432$	0	0
$433$	1.39251e6	0.356927	0.178464	−	0.983947i	$-0.442887\pi$
$433$	1.39251e6	0.356927	0.178464	+	0.983947i	$0.442887\pi$
$434$	0	0
$435$	−177670.	−0.0450184
$436$	0	0
$437$	2.21421e6	0.554646
$438$	0	0
$439$	3.09041e6	0.765340	0.382670	−	0.923885i	$-0.375005\pi$
$439$	3.09041e6	0.765340	0.382670	+	0.923885i	$0.375005\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−4.17066e6	−1.00971	−0.504853	−	0.863205i	$-0.668453\pi$
$443$	−4.17066e6	−1.00971	−0.504853	+	0.863205i	$0.668453\pi$
$444$	0	0
$445$	3.18813e6	0.763195
$446$	0	0
$447$	−5.36251e6	−1.26940
$448$	0	0
$449$	−4.50893e6	−1.05550	−0.527749	−	0.849400i	$-0.676964\pi$
$449$	−4.50893e6	−1.05550	−0.527749	+	0.849400i	$0.676964\pi$
$450$	0	0
$451$	4.58618e6	1.06172
$452$	0	0
$453$	6.09920e6	1.39646
$454$	0	0
$455$	0	0
$456$	0	0
$457$	521324.	0.116766	0.0583831	−	0.998294i	$-0.481406\pi$
$457$	521324.	0.116766	0.0583831	+	0.998294i	$0.481406\pi$
$458$	0	0
$459$	3.69959e6	0.819638
$460$	0	0
$461$	−4.45956e6	−0.977327	−0.488663	−	0.872472i	$-0.662515\pi$
$461$	−4.45956e6	−0.977327	−0.488663	+	0.872472i	$0.662515\pi$
$462$	0	0
$463$	−7.41318e6	−1.60713	−0.803567	−	0.595214i	$-0.797067\pi$
$463$	−7.41318e6	−1.60713	−0.803567	+	0.595214i	$0.797067\pi$
$464$	0	0
$465$	191784.	0.0411321
$466$	0	0
$467$	6.07670e6	1.28936	0.644682	−	0.764451i	$-0.276990\pi$
$467$	6.07670e6	1.28936	0.644682	+	0.764451i	$0.276990\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	3.45308e6	0.717223
$472$	0	0
$473$	−4.40820e6	−0.905958
$474$	0	0
$475$	1.14142e6	0.232118
$476$	0	0
$477$	849074.	0.170864
$478$	0	0
$479$	7.82188e6	1.55766	0.778829	−	0.627236i	$-0.215814\pi$
$479$	7.82188e6	1.55766	0.778829	+	0.627236i	$0.215814\pi$
$480$	0	0
$481$	3.36694e6	0.663548
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.79580e6	0.346661
$486$	0	0
$487$	−7.64488e6	−1.46066	−0.730328	−	0.683097i	$-0.760633\pi$
$487$	−7.64488e6	−1.46066	−0.730328	+	0.683097i	$0.760633\pi$
$488$	0	0
$489$	−3.15575e6	−0.596801
$490$	0	0
$491$	−4.62867e6	−0.866468	−0.433234	−	0.901282i	$-0.642628\pi$
$491$	−4.62867e6	−0.866468	−0.433234	+	0.901282i	$0.642628\pi$
$492$	0	0
$493$	220516.	0.0408622
$494$	0	0
$495$	−346658.	−0.0635899
$496$	0	0
$497$	0	0
$498$	0	0
$499$	344781.	0.0619858	0.0309929	−	0.999520i	$-0.490133\pi$
$499$	344781.	0.0619858	0.0309929	+	0.999520i	$0.490133\pi$
$500$	0	0
$501$	−8.31056e6	−1.47923
$502$	0	0
$503$	2.44839e6	0.431479	0.215740	−	0.976451i	$-0.430784\pi$
$503$	2.44839e6	0.431479	0.215740	+	0.976451i	$0.430784\pi$
$504$	0	0
$505$	4.20035e6	0.732921
$506$	0	0
$507$	−3.84541e6	−0.664389
$508$	0	0
$509$	4.39844e6	0.752497	0.376248	−	0.926519i	$-0.377214\pi$
$509$	4.39844e6	0.752497	0.376248	+	0.926519i	$0.377214\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	7.79687e6	1.30806
$514$	0	0
$515$	5.04223e6	0.837731
$516$	0	0
$517$	−307735.	−0.0506349
$518$	0	0
$519$	−9.22522e6	−1.50334
$520$	0	0
$521$	−3.90229e6	−0.629833	−0.314916	−	0.949119i	$-0.601976\pi$
$521$	−3.90229e6	−0.629833	−0.314916	+	0.949119i	$0.601976\pi$
$522$	0	0
$523$	−2.95595e6	−0.472544	−0.236272	−	0.971687i	$-0.575926\pi$
$523$	−2.95595e6	−0.472544	−0.236272	+	0.971687i	$0.575926\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−238034.	−0.0373347
$528$	0	0
$529$	−5.40834e6	−0.840282
$530$	0	0
$531$	278133.	0.0428071
$532$	0	0
$533$	6.19310e6	0.944256
$534$	0	0
$535$	1.10393e7	1.66747
$536$	0	0
$537$	3.11474e6	0.466107
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−1.64995e6	−0.242369	−0.121185	−	0.992630i	$-0.538669\pi$
$541$	−1.64995e6	−0.242369	−0.121185	+	0.992630i	$0.538669\pi$
$542$	0	0
$543$	−7.01633e6	−1.02120
$544$	0	0
$545$	−9.27716e6	−1.33790
$546$	0	0
$547$	−8.86766e6	−1.26719	−0.633594	−	0.773666i	$-0.718421\pi$
$547$	−8.86766e6	−1.26719	−0.633594	+	0.773666i	$0.718421\pi$
$548$	0	0
$549$	894530.	0.126667
$550$	0	0
$551$	464735.	0.0652119
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−7.61316e6	−1.04914
$556$	0	0
$557$	−6.29582e6	−0.859833	−0.429917	−	0.902869i	$-0.641457\pi$
$557$	−6.29582e6	−0.859833	−0.429917	+	0.902869i	$0.641457\pi$
$558$	0	0
$559$	−5.95275e6	−0.805727
$560$	0	0
$561$	4.63711e6	0.622072
$562$	0	0
$563$	8.35369e6	1.11073	0.555364	−	0.831608i	$-0.312579\pi$
$563$	8.35369e6	1.11073	0.555364	+	0.831608i	$0.312579\pi$
$564$	0	0
$565$	−9.75355e6	−1.28541
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−2.48798e6	−0.322155	−0.161078	−	0.986942i	$-0.551497\pi$
$569$	−2.48798e6	−0.322155	−0.161078	+	0.986942i	$0.551497\pi$
$570$	0	0
$571$	2.44174e6	0.313408	0.156704	−	0.987646i	$-0.449913\pi$
$571$	2.44174e6	0.313408	0.156704	+	0.987646i	$0.449913\pi$
$572$	0	0
$573$	−3.79164e6	−0.482436
$574$	0	0
$575$	529930.	0.0668419
$576$	0	0
$577$	−2.14056e6	−0.267663	−0.133831	−	0.991004i	$-0.542728\pi$
$577$	−2.14056e6	−0.267663	−0.133831	+	0.991004i	$0.542728\pi$
$578$	0	0
$579$	−6.90338e6	−0.855786
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−9.34144e6	−1.13826
$584$	0	0
$585$	−468120.	−0.0565546
$586$	0	0
$587$	−16057.9	−0.00192350	−0.000961752	−	1.00000i	$-0.500306\pi$
$587$	−16057.9	−0.00192350	−0.000961752		1.00000i	$0.500306\pi$
$588$	0	0
$589$	−501655.	−0.0595823
$590$	0	0
$591$	−1.15957e7	−1.36562
$592$	0	0
$593$	1.21715e7	1.42137	0.710684	−	0.703511i	$-0.248386\pi$
$593$	1.21715e7	1.42137	0.710684	+	0.703511i	$0.248386\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.05164e7	1.20762
$598$	0	0
$599$	−7.60810e6	−0.866382	−0.433191	−	0.901302i	$-0.642612\pi$
$599$	−7.60810e6	−0.866382	−0.433191	+	0.901302i	$0.642612\pi$
$600$	0	0
$601$	−1.38834e7	−1.56786	−0.783932	−	0.620847i	$-0.786789\pi$
$601$	−1.38834e7	−1.56786	−0.783932	+	0.620847i	$0.786789\pi$
$602$	0	0
$603$	−1.42950e6	−0.160100
$604$	0	0
$605$	−4.40181e6	−0.488926
$606$	0	0
$607$	−9.28458e6	−1.02280	−0.511400	−	0.859343i	$-0.670873\pi$
$607$	−9.28458e6	−1.02280	−0.511400	+	0.859343i	$0.670873\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−415560.	−0.0450329
$612$	0	0
$613$	−1.63186e6	−0.175401	−0.0877006	−	0.996147i	$-0.527952\pi$
$613$	−1.63186e6	−0.175401	−0.0877006	+	0.996147i	$0.527952\pi$
$614$	0	0
$615$	−1.40035e7	−1.49297
$616$	0	0
$617$	8.30019e6	0.877758	0.438879	−	0.898546i	$-0.355376\pi$
$617$	8.30019e6	0.877758	0.438879	+	0.898546i	$0.355376\pi$
$618$	0	0
$619$	−9.66890e6	−1.01426	−0.507131	−	0.861869i	$-0.669294\pi$
$619$	−9.66890e6	−1.01426	−0.507131	+	0.861869i	$0.669294\pi$
$620$	0	0
$621$	3.61988e6	0.376674
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−7.85913e6	−0.804775
$626$	0	0
$627$	9.77269e6	0.992762
$628$	0	0
$629$	9.44911e6	0.952279
$630$	0	0
$631$	−8.91041e6	−0.890891	−0.445445	−	0.895309i	$-0.646955\pi$
$631$	−8.91041e6	−0.890891	−0.445445	+	0.895309i	$0.646955\pi$
$632$	0	0
$633$	1.49944e7	1.48738
$634$	0	0
$635$	−6.97359e6	−0.686313
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−520704.	−0.0504474
$640$	0	0
$641$	1.09470e7	1.05233	0.526164	−	0.850383i	$-0.323630\pi$
$641$	1.09470e7	1.05233	0.526164	+	0.850383i	$0.323630\pi$
$642$	0	0
$643$	1.81902e7	1.73504	0.867522	−	0.497399i	$-0.165712\pi$
$643$	1.81902e7	1.73504	0.867522	+	0.497399i	$0.165712\pi$
$644$	0	0
$645$	1.34601e7	1.27394
$646$	0	0
$647$	2.34882e6	0.220592	0.110296	−	0.993899i	$-0.464820\pi$
$647$	2.34882e6	0.220592	0.110296	+	0.993899i	$0.464820\pi$
$648$	0	0
$649$	−3.05999e6	−0.285173
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−5.10635e6	−0.468627	−0.234314	−	0.972161i	$-0.575284\pi$
$653$	−5.10635e6	−0.468627	−0.234314	+	0.972161i	$0.575284\pi$
$654$	0	0
$655$	4.35762e6	0.396868
$656$	0	0
$657$	1.40237e6	0.126750
$658$	0	0
$659$	1.94777e7	1.74713	0.873563	−	0.486711i	$-0.161803\pi$
$659$	1.94777e7	1.74713	0.873563	+	0.486711i	$0.161803\pi$
$660$	0	0
$661$	−1.63826e6	−0.145841	−0.0729204	−	0.997338i	$-0.523232\pi$
$661$	−1.63826e6	−0.145841	−0.0729204	+	0.997338i	$0.523232\pi$
$662$	0	0
$663$	6.26188e6	0.553249
$664$	0	0
$665$	0	0
$666$	0	0
$667$	215764.	0.0187787
$668$	0	0
$669$	1.57957e7	1.36450
$670$	0	0
$671$	−9.84154e6	−0.843834
$672$	0	0
$673$	−989610.	−0.0842222	−0.0421111	−	0.999113i	$-0.513408\pi$
$673$	−989610.	−0.0842222	−0.0421111	+	0.999113i	$0.513408\pi$
$674$	0	0
$675$	1.86603e6	0.157638
$676$	0	0
$677$	1.89639e7	1.59022	0.795108	−	0.606467i	$-0.207414\pi$
$677$	1.89639e7	1.59022	0.795108	+	0.606467i	$0.207414\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−1.46216e7	−1.20817
$682$	0	0
$683$	−3.61093e6	−0.296188	−0.148094	−	0.988973i	$-0.547314\pi$
$683$	−3.61093e6	−0.296188	−0.148094	+	0.988973i	$0.547314\pi$
$684$	0	0
$685$	−2.10578e6	−0.171470
$686$	0	0
$687$	−1.99264e7	−1.61078
$688$	0	0
$689$	−1.26145e7	−1.01233
$690$	0	0
$691$	−1.91835e7	−1.52838	−0.764192	−	0.644989i	$-0.776862\pi$
$691$	−1.91835e7	−1.52838	−0.764192	+	0.644989i	$0.776862\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.25285e7	−0.983866
$696$	0	0
$697$	1.73805e7	1.35513
$698$	0	0
$699$	2.03771e7	1.57743
$700$	0	0
$701$	−1.10240e7	−0.847314	−0.423657	−	0.905823i	$-0.639254\pi$
$701$	−1.10240e7	−0.847314	−0.423657	+	0.905823i	$0.639254\pi$
$702$	0	0
$703$	1.99139e7	1.51974
$704$	0	0
$705$	939643.	0.0712016
$706$	0	0
$707$	0	0
$708$	0	0
$709$	2.05337e7	1.53409	0.767047	−	0.641591i	$-0.221726\pi$
$709$	2.05337e7	1.53409	0.767047	+	0.641591i	$0.221726\pi$
$710$	0	0
$711$	1.19694e6	0.0887968
$712$	0	0
$713$	−232906.	−0.0171576
$714$	0	0
$715$	5.15022e6	0.376756
$716$	0	0
$717$	−3.66525e6	−0.266260
$718$	0	0
$719$	−8.09121e6	−0.583702	−0.291851	−	0.956464i	$-0.594271\pi$
$719$	−8.09121e6	−0.583702	−0.291851	+	0.956464i	$0.594271\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	6.57141e6	0.467534
$724$	0	0
$725$	111226.	0.00785886
$726$	0	0
$727$	1.58746e7	1.11396	0.556978	−	0.830527i	$-0.311961\pi$
$727$	1.58746e7	1.11396	0.556978	+	0.830527i	$0.311961\pi$
$728$	0	0
$729$	1.25965e7	0.877875
$730$	0	0
$731$	−1.67060e7	−1.15632
$732$	0	0
$733$	4.52547e6	0.311102	0.155551	−	0.987828i	$-0.450285\pi$
$733$	4.52547e6	0.311102	0.155551	+	0.987828i	$0.450285\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.57272e7	1.06655
$738$	0	0
$739$	−1.07043e7	−0.721022	−0.360511	−	0.932755i	$-0.617398\pi$
$739$	−1.07043e7	−0.721022	−0.360511	+	0.932755i	$0.617398\pi$
$740$	0	0
$741$	1.31969e7	0.882928
$742$	0	0
$743$	1.76704e7	1.17429	0.587144	−	0.809482i	$-0.300252\pi$
$743$	1.76704e7	1.17429	0.587144	+	0.809482i	$0.300252\pi$
$744$	0	0
$745$	−1.67148e7	−1.10334
$746$	0	0
$747$	−1.60034e6	−0.104933
$748$	0	0
$749$	0	0
$750$	0	0
$751$	1.17795e7	0.762125	0.381062	−	0.924549i	$-0.375558\pi$
$751$	1.17795e7	0.762125	0.381062	+	0.924549i	$0.375558\pi$
$752$	0	0
$753$	6.64724e6	0.427222
$754$	0	0
$755$	1.90111e7	1.21378
$756$	0	0
$757$	−2.89421e7	−1.83565	−0.917827	−	0.396981i	$-0.870058\pi$
$757$	−2.89421e7	−1.83565	−0.917827	+	0.396981i	$0.870058\pi$
$758$	0	0
$759$	4.53721e6	0.285880
$760$	0	0
$761$	−1.33672e7	−0.836715	−0.418358	−	0.908282i	$-0.637394\pi$
$761$	−1.33672e7	−0.836715	−0.418358	+	0.908282i	$0.637394\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−1.31375e6	−0.0811633
$766$	0	0
$767$	−4.13216e6	−0.253623
$768$	0	0
$769$	2.23666e7	1.36390	0.681952	−	0.731397i	$-0.261131\pi$
$769$	2.23666e7	1.36390	0.681952	+	0.731397i	$0.261131\pi$
$770$	0	0
$771$	1.35999e7	0.823945
$772$	0	0
$773$	−5.67731e6	−0.341738	−0.170869	−	0.985294i	$-0.554658\pi$
$773$	−5.67731e6	−0.341738	−0.170869	+	0.985294i	$0.554658\pi$
$774$	0	0
$775$	−120062.	−0.00718043
$776$	0	0
$777$	0	0
$778$	0	0
$779$	3.66294e7	2.16265
$780$	0	0
$781$	5.72874e6	0.336072
$782$	0	0
$783$	759768.	0.0442870
$784$	0	0
$785$	1.07632e7	0.623399
$786$	0	0
$787$	−1.56192e7	−0.898924	−0.449462	−	0.893300i	$-0.648384\pi$
$787$	−1.56192e7	−0.898924	−0.449462	+	0.893300i	$0.648384\pi$
$788$	0	0
$789$	2.71589e7	1.55317
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−1.32898e7	−0.750476
$794$	0	0
$795$	2.85233e7	1.60060
$796$	0	0
$797$	1.18054e7	0.658317	0.329159	−	0.944275i	$-0.393235\pi$
$797$	1.18054e7	0.658317	0.329159	+	0.944275i	$0.393235\pi$
$798$	0	0
$799$	−1.16624e6	−0.0646282
$800$	0	0
$801$	1.55321e6	0.0855358
$802$	0	0
$803$	−1.54287e7	−0.844388
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.72691e6	0.0933437
$808$	0	0
$809$	1.62838e7	0.874753	0.437376	−	0.899278i	$-0.355908\pi$
$809$	1.62838e7	0.874753	0.437376	+	0.899278i	$0.355908\pi$
$810$	0	0
$811$	2.34229e7	1.25051	0.625257	−	0.780419i	$-0.284994\pi$
$811$	2.34229e7	1.25051	0.625257	+	0.780419i	$0.284994\pi$
$812$	0	0
$813$	2.87219e7	1.52401
$814$	0	0
$815$	−9.83639e6	−0.518730
$816$	0	0
$817$	−3.52079e7	−1.84537
$818$	0	0
$819$	0	0
$820$	0	0
$821$	2.76011e7	1.42912	0.714559	−	0.699575i	$-0.246627\pi$
$821$	2.76011e7	1.42912	0.714559	+	0.699575i	$0.246627\pi$
$822$	0	0
$823$	−3.71262e7	−1.91065	−0.955325	−	0.295557i	$-0.904495\pi$
$823$	−3.71262e7	−1.91065	−0.955325	+	0.295557i	$0.904495\pi$
$824$	0	0
$825$	2.33891e6	0.119640
$826$	0	0
$827$	1.18296e7	0.601459	0.300729	−	0.953710i	$-0.402770\pi$
$827$	1.18296e7	0.601459	0.300729	+	0.953710i	$0.402770\pi$
$828$	0	0
$829$	2.56615e7	1.29687	0.648433	−	0.761271i	$-0.275424\pi$
$829$	2.56615e7	1.29687	0.648433	+	0.761271i	$0.275424\pi$
$830$	0	0
$831$	3.51875e7	1.76761
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−2.59038e7	−1.28572
$836$	0	0
$837$	−820127.	−0.0404639
$838$	0	0
$839$	−2.86405e7	−1.40467	−0.702336	−	0.711845i	$-0.747860\pi$
$839$	−2.86405e7	−1.40467	−0.702336	+	0.711845i	$0.747860\pi$
$840$	0	0
$841$	−2.04659e7	−0.997792
$842$	0	0
$843$	−3.18189e7	−1.54211
$844$	0	0
$845$	−1.19860e7	−0.577476
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−7.01222e6	−0.333877
$850$	0	0
$851$	9.24553e6	0.437631
$852$	0	0
$853$	3.22929e7	1.51962	0.759809	−	0.650146i	$-0.225292\pi$
$853$	3.22929e7	1.51962	0.759809	+	0.650146i	$0.225292\pi$
$854$	0	0
$855$	−2.76872e6	−0.129528
$856$	0	0
$857$	−1.72977e7	−0.804520	−0.402260	−	0.915525i	$-0.631775\pi$
$857$	−1.72977e7	−0.804520	−0.402260	+	0.915525i	$0.631775\pi$
$858$	0	0
$859$	−3.79054e7	−1.75274	−0.876372	−	0.481635i	$-0.840043\pi$
$859$	−3.79054e7	−1.75274	−0.876372	+	0.481635i	$0.840043\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	7.73660e6	0.353609	0.176804	−	0.984246i	$-0.443424\pi$
$863$	7.73660e6	0.353609	0.176804	+	0.984246i	$0.443424\pi$
$864$	0	0
$865$	−2.87548e7	−1.30668
$866$	0	0
$867$	−5.66410e6	−0.255908
$868$	0	0
$869$	−1.31686e7	−0.591548
$870$	0	0
$871$	2.12377e7	0.948556
$872$	0	0
$873$	874888.	0.0388523
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1.07875e6	0.0473612	0.0236806	−	0.999720i	$-0.492462\pi$
$877$	1.07875e6	0.0473612	0.0236806	+	0.999720i	$0.492462\pi$
$878$	0	0
$879$	2.87036e7	1.25304
$880$	0	0
$881$	−1.88089e7	−0.816440	−0.408220	−	0.912884i	$-0.633850\pi$
$881$	−1.88089e7	−0.816440	−0.408220	+	0.912884i	$0.633850\pi$
$882$	0	0
$883$	3.12763e7	1.34994	0.674968	−	0.737847i	$-0.264158\pi$
$883$	3.12763e7	1.34994	0.674968	+	0.737847i	$0.264158\pi$
$884$	0	0
$885$	9.34343e6	0.401004
$886$	0	0
$887$	2.69104e7	1.14845	0.574223	−	0.818699i	$-0.305304\pi$
$887$	2.69104e7	1.14845	0.574223	+	0.818699i	$0.305304\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	1.76281e7	0.743894
$892$	0	0
$893$	−2.45785e6	−0.103140
$894$	0	0
$895$	9.70857e6	0.405133
$896$	0	0
$897$	6.12696e6	0.254252
$898$	0	0
$899$	−48884.0	−0.00201728
$900$	0	0
$901$	−3.54019e7	−1.45283
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−2.18698e7	−0.887610
$906$	0	0
$907$	4.54725e7	1.83540	0.917699	−	0.397277i	$-0.130045\pi$
$907$	4.54725e7	1.83540	0.917699	+	0.397277i	$0.130045\pi$
$908$	0	0
$909$	2.04635e6	0.0821428
$910$	0	0
$911$	−4.44792e7	−1.77567	−0.887833	−	0.460165i	$-0.847790\pi$
$911$	−4.44792e7	−1.77567	−0.887833	+	0.460165i	$0.847790\pi$
$912$	0	0
$913$	1.76068e7	0.699042
$914$	0	0
$915$	3.00503e7	1.18658
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−4.51961e7	−1.76527	−0.882637	−	0.470055i	$-0.844234\pi$
$919$	−4.51961e7	−1.76527	−0.882637	+	0.470055i	$0.844234\pi$
$920$	0	0
$921$	3.31004e7	1.28583
$922$	0	0
$923$	7.73599e6	0.298890
$924$	0	0
$925$	4.76602e6	0.183148
$926$	0	0
$927$	2.45650e6	0.0938895
$928$	0	0
$929$	−197419.	−0.00750500	−0.00375250	−	0.999993i	$-0.501194\pi$
$929$	−197419.	−0.00750500	−0.00375250	+	0.999993i	$0.501194\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−2.33423e7	−0.877888
$934$	0	0
$935$	1.44538e7	0.540695
$936$	0	0
$937$	−2.35263e7	−0.875394	−0.437697	−	0.899122i	$-0.644206\pi$
$937$	−2.35263e7	−0.875394	−0.437697	+	0.899122i	$0.644206\pi$
$938$	0	0
$939$	−871888.	−0.0322698
$940$	0	0
$941$	−4.57924e7	−1.68585	−0.842925	−	0.538031i	$-0.819168\pi$
$941$	−4.57924e7	−1.68585	−0.842925	+	0.538031i	$0.819168\pi$
$942$	0	0
$943$	1.70061e7	0.622766
$944$	0	0
$945$	0	0
$946$	0	0
$947$	8.78958e6	0.318488	0.159244	−	0.987239i	$-0.449094\pi$
$947$	8.78958e6	0.318488	0.159244	+	0.987239i	$0.449094\pi$
$948$	0	0
$949$	−2.08347e7	−0.750969
$950$	0	0
$951$	−4.29582e7	−1.54026
$952$	0	0
$953$	2.97668e7	1.06170	0.530848	−	0.847467i	$-0.321874\pi$
$953$	2.97668e7	1.06170	0.530848	+	0.847467i	$0.321874\pi$
$954$	0	0
$955$	−1.18184e7	−0.419326
$956$	0	0
$957$	952302.	0.0336121
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−2.85764e7	−0.998157
$962$	0	0
$963$	5.37820e6	0.186884
$964$	0	0
$965$	−2.15177e7	−0.743836
$966$	0	0
$967$	1.40282e7	0.482433	0.241217	−	0.970471i	$-0.422454\pi$
$967$	1.40282e7	0.482433	0.241217	+	0.970471i	$0.422454\pi$
$968$	0	0
$969$	3.70362e7	1.26712
$970$	0	0
$971$	2.04446e7	0.695872	0.347936	−	0.937518i	$-0.386883\pi$
$971$	2.04446e7	0.695872	0.347936	+	0.937518i	$0.386883\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	3.15842e6	0.106404
$976$	0	0
$977$	−2.12878e7	−0.713500	−0.356750	−	0.934200i	$-0.616115\pi$
$977$	−2.12878e7	−0.713500	−0.356750	+	0.934200i	$0.616115\pi$
$978$	0	0
$979$	−1.70882e7	−0.569824
$980$	0	0
$981$	−4.51969e6	−0.149946
$982$	0	0
$983$	4.04387e7	1.33479	0.667396	−	0.744703i	$-0.267409\pi$
$983$	4.04387e7	1.33479	0.667396	+	0.744703i	$0.267409\pi$
$984$	0	0
$985$	−3.61436e7	−1.18697
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.63461e7	−0.531402
$990$	0	0
$991$	−2.47373e7	−0.800145	−0.400072	−	0.916484i	$-0.631015\pi$
$991$	−2.47373e7	−0.800145	−0.400072	+	0.916484i	$0.631015\pi$
$992$	0	0
$993$	2.13331e7	0.686562
$994$	0	0
$995$	3.27793e7	1.04964
$996$	0	0
$997$	2.09230e7	0.666631	0.333316	−	0.942815i	$-0.391833\pi$
$997$	2.09230e7	0.666631	0.333316	+	0.942815i	$0.391833\pi$
$998$	0	0
$999$	3.25561e7	1.03209

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.6.a.m.1.5	yes	6
4.3	odd	2		784.6.a.bn.1.2		6
7.2	even	3		392.6.i.q.361.2		12
7.3	odd	6		392.6.i.q.177.5		12
7.4	even	3		392.6.i.q.177.2		12
7.5	odd	6		392.6.i.q.361.5		12
7.6	odd	2	inner	392.6.a.m.1.2	✓	6
28.27	even	2		784.6.a.bn.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.6.a.m.1.2	✓	6	7.6	odd	2	inner
392.6.a.m.1.5	yes	6	1.1	even	1	trivial
392.6.i.q.177.2		12	7.4	even	3
392.6.i.q.177.5		12	7.3	odd	6
392.6.i.q.361.2		12	7.2	even	3
392.6.i.q.361.5		12	7.5	odd	6
784.6.a.bn.1.2		6	4.3	odd	2
784.6.a.bn.1.5		6	28.27	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 392 = 2^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	392.a (trivial)

\(f(q)\)	\(=\)	\(q+16.3662 q^{3} +51.0131 q^{5} +24.8528 q^{9} +O(q^{10})\)	\(q+16.3662 q^{3} +51.0131 q^{5} +24.8528 q^{9} -273.428 q^{11} -369.233 q^{13} +834.891 q^{15} -1036.23 q^{17} -2183.85 q^{19} -1013.90 q^{23} -522.662 q^{25} -3570.24 q^{27} -212.806 q^{29} +229.712 q^{31} -4474.99 q^{33} -9118.74 q^{37} -6042.94 q^{39} -16772.9 q^{41} +16121.9 q^{43} +1267.82 q^{45} +1125.47 q^{47} -16959.2 q^{51} +34164.1 q^{53} -13948.4 q^{55} -35741.3 q^{57} +11191.2 q^{59} +35993.1 q^{61} -18835.7 q^{65} -57518.6 q^{67} -16593.8 q^{69} -20951.5 q^{71} +56427.0 q^{73} -8554.00 q^{75} +48161.1 q^{79} -64470.6 q^{81} -64392.7 q^{83} -52861.3 q^{85} -3482.82 q^{87} +62496.2 q^{89} +3759.51 q^{93} -111405. q^{95} +35202.8 q^{97} -6795.46 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 122 q^{9}+O(q^{10}) \) 6 * q - 122 * q^9	\( 6 q - 122 q^{9} + 248 q^{11} - 376 q^{15} - 2904 q^{23} + 1510 q^{25} - 408 q^{29} - 18648 q^{37} - 15240 q^{39} - 10584 q^{43} + 3016 q^{51} - 66772 q^{53} - 56264 q^{57} - 55316 q^{65} - 169632 q^{67} - 54880 q^{71} + 50576 q^{79} - 270194 q^{81} - 406076 q^{85} - 164944 q^{93} - 551912 q^{95} - 173512 q^{99}+O(q^{100}) \) 6 * q - 122 * q^9 + 248 * q^11 - 376 * q^15 - 2904 * q^23 + 1510 * q^25 - 408 * q^29 - 18648 * q^37 - 15240 * q^39 - 10584 * q^43 + 3016 * q^51 - 66772 * q^53 - 56264 * q^57 - 55316 * q^65 - 169632 * q^67 - 54880 * q^71 + 50576 * q^79 - 270194 * q^81 - 406076 * q^85 - 164944 * q^93 - 551912 * q^95 - 173512 * q^99

Introduction

L-functions

Modular forms

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