LMFDB - Embedded newform 392.6.a.n.1.7

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:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 425x^{6} + 1312x^{5} + 56757x^{4} - 195610x^{3} - 2560079x^{2} + 6060020x + 37289602 $ x^8 - 2x^7 - 425x^6 + 1312x^5 + 56757x^4 - 195610x^3 - 2560079x^2 + 6060020*x + 37289602
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{10}\cdot 7^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$10.1213$ 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)

$f(q)$	$=$	$q+23.5268 q^{3} +30.4003 q^{5} +310.511 q^{9} +O(q^{10})$	$q+23.5268 q^{3} +30.4003 q^{5} +310.511 q^{9} +356.893 q^{11} +761.116 q^{13} +715.223 q^{15} -502.764 q^{17} +1731.46 q^{19} -3568.72 q^{23} -2200.82 q^{25} +1588.32 q^{27} +4526.22 q^{29} +7135.84 q^{31} +8396.56 q^{33} -13464.5 q^{37} +17906.6 q^{39} +12233.6 q^{41} +5420.80 q^{43} +9439.64 q^{45} -3871.10 q^{47} -11828.4 q^{51} +6892.06 q^{53} +10849.7 q^{55} +40735.8 q^{57} -36610.3 q^{59} +45330.1 q^{61} +23138.2 q^{65} +69357.5 q^{67} -83960.6 q^{69} -36534.0 q^{71} +50102.3 q^{73} -51778.3 q^{75} -20562.6 q^{79} -38086.0 q^{81} -100683. q^{83} -15284.2 q^{85} +106487. q^{87} +53759.8 q^{89} +167884. q^{93} +52637.0 q^{95} -94978.9 q^{97} +110819. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 932 q^{9}+O(q^{10}) $ 8 * q + 932 * q^9	$ 8 q + 932 q^{9} + 1204 q^{11} + 864 q^{15} - 2368 q^{23} + 2188 q^{25} - 9684 q^{29} + 60 q^{37} + 16088 q^{39} + 43564 q^{43} + 76684 q^{51} + 94472 q^{53} + 61004 q^{57} + 41900 q^{65} + 136504 q^{67} + 192328 q^{71} - 29816 q^{79} + 457860 q^{81} + 111132 q^{85} - 166504 q^{93} + 670800 q^{95} + 857404 q^{99}+O(q^{100}) $ 8 * q + 932 * q^9 + 1204 * q^11 + 864 * q^15 - 2368 * q^23 + 2188 * q^25 - 9684 * q^29 + 60 * q^37 + 16088 * q^39 + 43564 * q^43 + 76684 * q^51 + 94472 * q^53 + 61004 * q^57 + 41900 * q^65 + 136504 * q^67 + 192328 * q^71 - 29816 * q^79 + 457860 * q^81 + 111132 * q^85 - 166504 * q^93 + 670800 * q^95 + 857404 * 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
$3$	23.5268	1.50925	0.754623	−	0.656159i	$-0.227820\pi$
$3$	23.5268	1.50925	0.754623	+	0.656159i	$0.227820\pi$
$4$	0	0
$5$	30.4003	0.543818	0.271909	−	0.962323i	$-0.412345\pi$
$5$	30.4003	0.543818	0.271909	+	0.962323i	$0.412345\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	310.511	1.27782
$10$	0	0
$11$	356.893	0.889317	0.444658	−	0.895700i	$-0.353325\pi$
$11$	356.893	0.889317	0.444658	+	0.895700i	$0.353325\pi$
$12$	0	0
$13$	761.116	1.24909	0.624543	−	0.780990i	$-0.285285\pi$
$13$	761.116	1.24909	0.624543	+	0.780990i	$0.285285\pi$
$14$	0	0
$15$	715.223	0.820755
$16$	0	0
$17$	−502.764	−0.421932	−0.210966	−	0.977493i	$-0.567661\pi$
$17$	−502.764	−0.421932	−0.210966	+	0.977493i	$0.567661\pi$
$18$	0	0
$19$	1731.46	1.10035	0.550173	−	0.835051i	$-0.314562\pi$
$19$	1731.46	1.10035	0.550173	+	0.835051i	$0.314562\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3568.72	−1.40667	−0.703336	−	0.710857i	$-0.748307\pi$
$23$	−3568.72	−1.40667	−0.703336	+	0.710857i	$0.748307\pi$
$24$	0	0
$25$	−2200.82	−0.704262
$26$	0	0
$27$	1588.32	0.419304
$28$	0	0
$29$	4526.22	0.999402	0.499701	−	0.866198i	$-0.333443\pi$
$29$	4526.22	0.999402	0.499701	+	0.866198i	$0.333443\pi$
$30$	0	0
$31$	7135.84	1.33365	0.666824	−	0.745216i	$-0.267653\pi$
$31$	7135.84	1.33365	0.666824	+	0.745216i	$0.267653\pi$
$32$	0	0
$33$	8396.56	1.34220
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−13464.5	−1.61691	−0.808454	−	0.588559i	$-0.799695\pi$
$37$	−13464.5	−1.61691	−0.808454	+	0.588559i	$0.799695\pi$
$38$	0	0
$39$	17906.6	1.88518
$40$	0	0
$41$	12233.6	1.13656	0.568281	−	0.822835i	$-0.307609\pi$
$41$	12233.6	1.13656	0.568281	+	0.822835i	$0.307609\pi$
$42$	0	0
$43$	5420.80	0.447087	0.223543	−	0.974694i	$-0.428238\pi$
$43$	5420.80	0.447087	0.223543	+	0.974694i	$0.428238\pi$
$44$	0	0
$45$	9439.64	0.694903
$46$	0	0
$47$	−3871.10	−0.255617	−0.127808	−	0.991799i	$-0.540794\pi$
$47$	−3871.10	−0.255617	−0.127808	+	0.991799i	$0.540794\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−11828.4	−0.636799
$52$	0	0
$53$	6892.06	0.337023	0.168511	−	0.985700i	$-0.446104\pi$
$53$	6892.06	0.337023	0.168511	+	0.985700i	$0.446104\pi$
$54$	0	0
$55$	10849.7	0.483626
$56$	0	0
$57$	40735.8	1.66069
$58$	0	0
$59$	−36610.3	−1.36922	−0.684610	−	0.728910i	$-0.740027\pi$
$59$	−36610.3	−1.36922	−0.684610	+	0.728910i	$0.740027\pi$
$60$	0	0
$61$	45330.1	1.55978	0.779889	−	0.625918i	$-0.215276\pi$
$61$	45330.1	1.55978	0.779889	+	0.625918i	$0.215276\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	23138.2	0.679276
$66$	0	0
$67$	69357.5	1.88759	0.943793	−	0.330537i	$-0.107230\pi$
$67$	69357.5	1.88759	0.943793	+	0.330537i	$0.107230\pi$
$68$	0	0
$69$	−83960.6	−2.12301
$70$	0	0
$71$	−36534.0	−0.860105	−0.430053	−	0.902804i	$-0.641505\pi$
$71$	−36534.0	−0.860105	−0.430053	+	0.902804i	$0.641505\pi$
$72$	0	0
$73$	50102.3	1.10040	0.550200	−	0.835033i	$-0.314551\pi$
$73$	50102.3	1.10040	0.550200	+	0.835033i	$0.314551\pi$
$74$	0	0
$75$	−51778.3	−1.06291
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−20562.6	−0.370690	−0.185345	−	0.982674i	$-0.559340\pi$
$79$	−20562.6	−0.370690	−0.185345	+	0.982674i	$0.559340\pi$
$80$	0	0
$81$	−38086.0	−0.644990
$82$	0	0
$83$	−100683.	−1.60421	−0.802104	−	0.597184i	$-0.796286\pi$
$83$	−100683.	−1.60421	−0.802104	+	0.597184i	$0.796286\pi$
$84$	0	0
$85$	−15284.2	−0.229454
$86$	0	0
$87$	106487.	1.50834
$88$	0	0
$89$	53759.8	0.719420	0.359710	−	0.933064i	$-0.382876\pi$
$89$	53759.8	0.719420	0.359710	+	0.933064i	$0.382876\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	167884.	2.01280
$94$	0	0
$95$	52637.0	0.598387
$96$	0	0
$97$	−94978.9	−1.02494	−0.512469	−	0.858705i	$-0.671269\pi$
$97$	−94978.9	−1.02494	−0.512469	+	0.858705i	$0.671269\pi$
$98$	0	0
$99$	110819.	1.13639
$100$	0	0
$101$	32444.2	0.316470	0.158235	−	0.987401i	$-0.449420\pi$
$101$	32444.2	0.316470	0.158235	+	0.987401i	$0.449420\pi$
$102$	0	0
$103$	−120951.	−1.12335	−0.561676	−	0.827358i	$-0.689843\pi$
$103$	−120951.	−1.12335	−0.561676	+	0.827358i	$0.689843\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	194989.	1.64646	0.823228	−	0.567711i	$-0.192171\pi$
$107$	194989.	1.64646	0.823228	+	0.567711i	$0.192171\pi$
$108$	0	0
$109$	−60028.5	−0.483939	−0.241970	−	0.970284i	$-0.577793\pi$
$109$	−60028.5	−0.483939	−0.241970	+	0.970284i	$0.577793\pi$
$110$	0	0
$111$	−316776.	−2.44031
$112$	0	0
$113$	−10103.2	−0.0744324	−0.0372162	−	0.999307i	$-0.511849\pi$
$113$	−10103.2	−0.0744324	−0.0372162	+	0.999307i	$0.511849\pi$
$114$	0	0
$115$	−108490.	−0.764973
$116$	0	0
$117$	236335.	1.59611
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−33678.3	−0.209116
$122$	0	0
$123$	287817.	1.71535
$124$	0	0
$125$	−161907.	−0.926808
$126$	0	0
$127$	190889.	1.05020	0.525099	−	0.851041i	$-0.324028\pi$
$127$	190889.	1.05020	0.525099	+	0.851041i	$0.324028\pi$
$128$	0	0
$129$	127534.	0.674764
$130$	0	0
$131$	68572.4	0.349117	0.174559	−	0.984647i	$-0.444150\pi$
$131$	68572.4	0.349117	0.174559	+	0.984647i	$0.444150\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	48285.5	0.228025
$136$	0	0
$137$	386552.	1.75957	0.879785	−	0.475372i	$-0.157686\pi$
$137$	386552.	1.75957	0.879785	+	0.475372i	$0.157686\pi$
$138$	0	0
$139$	61895.8	0.271722	0.135861	−	0.990728i	$-0.456620\pi$
$139$	61895.8	0.271722	0.135861	+	0.990728i	$0.456620\pi$
$140$	0	0
$141$	−91074.6	−0.385789
$142$	0	0
$143$	271637.	1.11083
$144$	0	0
$145$	137599.	0.543493
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−285307.	−1.05280	−0.526401	−	0.850237i	$-0.676459\pi$
$149$	−285307.	−1.05280	−0.526401	+	0.850237i	$0.676459\pi$
$150$	0	0
$151$	−3543.34	−0.0126465	−0.00632325	−	0.999980i	$-0.502013\pi$
$151$	−3543.34	−0.0126465	−0.00632325	+	0.999980i	$0.502013\pi$
$152$	0	0
$153$	−156114.	−0.539154
$154$	0	0
$155$	216932.	0.725261
$156$	0	0
$157$	−3181.81	−0.0103021	−0.00515104	−	0.999987i	$-0.501640\pi$
$157$	−3181.81	−0.0103021	−0.00515104	+	0.999987i	$0.501640\pi$
$158$	0	0
$159$	162148.	0.508650
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−287113.	−0.846417	−0.423208	−	0.906032i	$-0.639096\pi$
$163$	−287113.	−0.846417	−0.423208	+	0.906032i	$0.639096\pi$
$164$	0	0
$165$	255258.	0.729911
$166$	0	0
$167$	529817.	1.47006	0.735030	−	0.678035i	$-0.237168\pi$
$167$	529817.	1.47006	0.735030	+	0.678035i	$0.237168\pi$
$168$	0	0
$169$	208005.	0.560218
$170$	0	0
$171$	537638.	1.40605
$172$	0	0
$173$	−266157.	−0.676117	−0.338059	−	0.941125i	$-0.609770\pi$
$173$	−266157.	−0.676117	−0.338059	+	0.941125i	$0.609770\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−861324.	−2.06649
$178$	0	0
$179$	169107.	0.394484	0.197242	−	0.980355i	$-0.436802\pi$
$179$	169107.	0.394484	0.197242	+	0.980355i	$0.436802\pi$
$180$	0	0
$181$	−643489.	−1.45997	−0.729986	−	0.683462i	$-0.760474\pi$
$181$	−643489.	−1.45997	−0.729986	+	0.683462i	$0.760474\pi$
$182$	0	0
$183$	1.06647e6	2.35409
$184$	0	0
$185$	−409325.	−0.879303
$186$	0	0
$187$	−179433.	−0.375231
$188$	0	0
$189$	0	0
$190$	0	0
$191$	329131.	0.652808	0.326404	−	0.945230i	$-0.394163\pi$
$191$	329131.	0.652808	0.326404	+	0.945230i	$0.394163\pi$
$192$	0	0
$193$	818337.	1.58139	0.790694	−	0.612211i	$-0.209720\pi$
$193$	818337.	1.58139	0.790694	+	0.612211i	$0.209720\pi$
$194$	0	0
$195$	544368.	1.02519
$196$	0	0
$197$	−687276.	−1.26173	−0.630864	−	0.775894i	$-0.717299\pi$
$197$	−687276.	−1.26173	−0.630864	+	0.775894i	$0.717299\pi$
$198$	0	0
$199$	84087.7	0.150522	0.0752609	−	0.997164i	$-0.476021\pi$
$199$	84087.7	0.150522	0.0752609	+	0.997164i	$0.476021\pi$
$200$	0	0
$201$	1.63176e6	2.84883
$202$	0	0
$203$	0	0
$204$	0	0
$205$	371904.	0.618083
$206$	0	0
$207$	−1.10813e6	−1.79748
$208$	0	0
$209$	617947.	0.978556
$210$	0	0
$211$	−215848.	−0.333765	−0.166883	−	0.985977i	$-0.553370\pi$
$211$	−215848.	−0.333765	−0.166883	+	0.985977i	$0.553370\pi$
$212$	0	0
$213$	−859529.	−1.29811
$214$	0	0
$215$	164794.	0.243134
$216$	0	0
$217$	0	0
$218$	0	0
$219$	1.17875e6	1.66078
$220$	0	0
$221$	−382662.	−0.527029
$222$	0	0
$223$	205819.	0.277155	0.138578	−	0.990352i	$-0.455747\pi$
$223$	205819.	0.277155	0.138578	+	0.990352i	$0.455747\pi$
$224$	0	0
$225$	−683379.	−0.899923
$226$	0	0
$227$	950681.	1.22453	0.612266	−	0.790652i	$-0.290258\pi$
$227$	950681.	1.22453	0.612266	+	0.790652i	$0.290258\pi$
$228$	0	0
$229$	−1.28724e6	−1.62207	−0.811037	−	0.584995i	$-0.801097\pi$
$229$	−1.28724e6	−1.62207	−0.811037	+	0.584995i	$0.801097\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.41402e6	−1.70634	−0.853169	−	0.521634i	$-0.825323\pi$
$233$	−1.41402e6	−1.70634	−0.853169	+	0.521634i	$0.825323\pi$
$234$	0	0
$235$	−117683.	−0.139009
$236$	0	0
$237$	−483773.	−0.559463
$238$	0	0
$239$	−334549.	−0.378848	−0.189424	−	0.981895i	$-0.560662\pi$
$239$	−334549.	−0.378848	−0.189424	+	0.981895i	$0.560662\pi$
$240$	0	0
$241$	−730975.	−0.810700	−0.405350	−	0.914162i	$-0.632850\pi$
$241$	−730975.	−0.810700	−0.405350	+	0.914162i	$0.632850\pi$
$242$	0	0
$243$	−1.28201e6	−1.39275
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.31784e6	1.37443
$248$	0	0
$249$	−2.36875e6	−2.42114
$250$	0	0
$251$	−929251.	−0.930998	−0.465499	−	0.885048i	$-0.654125\pi$
$251$	−929251.	−0.930998	−0.465499	+	0.885048i	$0.654125\pi$
$252$	0	0
$253$	−1.27365e6	−1.25098
$254$	0	0
$255$	−359589.	−0.346303
$256$	0	0
$257$	−917481.	−0.866492	−0.433246	−	0.901276i	$-0.642632\pi$
$257$	−917481.	−0.866492	−0.433246	+	0.901276i	$0.642632\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	1.40544e6	1.27706
$262$	0	0
$263$	269858.	0.240572	0.120286	−	0.992739i	$-0.461619\pi$
$263$	269858.	0.240572	0.120286	+	0.992739i	$0.461619\pi$
$264$	0	0
$265$	209521.	0.183279
$266$	0	0
$267$	1.26480e6	1.08578
$268$	0	0
$269$	−421136.	−0.354848	−0.177424	−	0.984135i	$-0.556776\pi$
$269$	−421136.	−0.354848	−0.177424	+	0.984135i	$0.556776\pi$
$270$	0	0
$271$	−1.49149e6	−1.23367	−0.616834	−	0.787093i	$-0.711585\pi$
$271$	−1.49149e6	−1.23367	−0.616834	+	0.787093i	$0.711585\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−785457.	−0.626312
$276$	0	0
$277$	−1.06922e6	−0.837277	−0.418639	−	0.908153i	$-0.637493\pi$
$277$	−1.06922e6	−0.837277	−0.418639	+	0.908153i	$0.637493\pi$
$278$	0	0
$279$	2.21576e6	1.70417
$280$	0	0
$281$	−1.81283e6	−1.36959	−0.684795	−	0.728736i	$-0.740108\pi$
$281$	−1.81283e6	−1.36959	−0.684795	+	0.728736i	$0.740108\pi$
$282$	0	0
$283$	−1.29416e6	−0.960552	−0.480276	−	0.877117i	$-0.659464\pi$
$283$	−1.29416e6	−0.960552	−0.480276	+	0.877117i	$0.659464\pi$
$284$	0	0
$285$	1.23838e6	0.903114
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−1.16708e6	−0.821974
$290$	0	0
$291$	−2.23455e6	−1.54689
$292$	0	0
$293$	−1.90998e6	−1.29975	−0.649876	−	0.760040i	$-0.725179\pi$
$293$	−1.90998e6	−1.29975	−0.649876	+	0.760040i	$0.725179\pi$
$294$	0	0
$295$	−1.11297e6	−0.744606
$296$	0	0
$297$	566861.	0.372894
$298$	0	0
$299$	−2.71621e6	−1.75706
$300$	0	0
$301$	0	0
$302$	0	0
$303$	763308.	0.477632
$304$	0	0
$305$	1.37805e6	0.848235
$306$	0	0
$307$	1.90843e6	1.15566	0.577830	−	0.816157i	$-0.303900\pi$
$307$	1.90843e6	1.15566	0.577830	+	0.816157i	$0.303900\pi$
$308$	0	0
$309$	−2.84559e6	−1.69541
$310$	0	0
$311$	−2.17477e6	−1.27501	−0.637504	−	0.770447i	$-0.720033\pi$
$311$	−2.17477e6	−1.27501	−0.637504	+	0.770447i	$0.720033\pi$
$312$	0	0
$313$	2.22620e6	1.28441	0.642205	−	0.766533i	$-0.278020\pi$
$313$	2.22620e6	1.28441	0.642205	+	0.766533i	$0.278020\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.24158e6	1.25287	0.626435	−	0.779474i	$-0.284514\pi$
$317$	2.24158e6	1.25287	0.626435	+	0.779474i	$0.284514\pi$
$318$	0	0
$319$	1.61538e6	0.888785
$320$	0	0
$321$	4.58747e6	2.48491
$322$	0	0
$323$	−870518.	−0.464271
$324$	0	0
$325$	−1.67508e6	−0.879685
$326$	0	0
$327$	−1.41228e6	−0.730384
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−3.07900e6	−1.54468	−0.772341	−	0.635208i	$-0.780914\pi$
$331$	−3.07900e6	−1.54468	−0.772341	+	0.635208i	$0.780914\pi$
$332$	0	0
$333$	−4.18087e6	−2.06612
$334$	0	0
$335$	2.10849e6	1.02650
$336$	0	0
$337$	3.86773e6	1.85516	0.927579	−	0.373626i	$-0.121886\pi$
$337$	3.86773e6	1.85516	0.927579	+	0.373626i	$0.121886\pi$
$338$	0	0
$339$	−237696.	−0.112337
$340$	0	0
$341$	2.54673e6	1.18604
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−2.55243e6	−1.15453
$346$	0	0
$347$	2.34566e6	1.04578	0.522892	−	0.852399i	$-0.324853\pi$
$347$	2.34566e6	1.04578	0.522892	+	0.852399i	$0.324853\pi$
$348$	0	0
$349$	67561.0	0.0296915	0.0148458	−	0.999890i	$-0.495274\pi$
$349$	67561.0	0.0296915	0.0148458	+	0.999890i	$0.495274\pi$
$350$	0	0
$351$	1.20890e6	0.523747
$352$	0	0
$353$	588968.	0.251568	0.125784	−	0.992058i	$-0.459855\pi$
$353$	588968.	0.251568	0.125784	+	0.992058i	$0.459855\pi$
$354$	0	0
$355$	−1.11065e6	−0.467740
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2.39605e6	−0.981203	−0.490601	−	0.871384i	$-0.663223\pi$
$359$	−2.39605e6	−0.981203	−0.490601	+	0.871384i	$0.663223\pi$
$360$	0	0
$361$	521863.	0.210760
$362$	0	0
$363$	−792342.	−0.315607
$364$	0	0
$365$	1.52313e6	0.598417
$366$	0	0
$367$	2.27514e6	0.881746	0.440873	−	0.897570i	$-0.354669\pi$
$367$	2.27514e6	0.881746	0.440873	+	0.897570i	$0.354669\pi$
$368$	0	0
$369$	3.79866e6	1.45233
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−1.71235e6	−0.637264	−0.318632	−	0.947879i	$-0.603223\pi$
$373$	−1.71235e6	−0.637264	−0.318632	+	0.947879i	$0.603223\pi$
$374$	0	0
$375$	−3.80915e6	−1.39878
$376$	0	0
$377$	3.44498e6	1.24834
$378$	0	0
$379$	2.10602e6	0.753122	0.376561	−	0.926392i	$-0.377107\pi$
$379$	2.10602e6	0.753122	0.376561	+	0.926392i	$0.377107\pi$
$380$	0	0
$381$	4.49100e6	1.58501
$382$	0	0
$383$	−1.91759e6	−0.667974	−0.333987	−	0.942578i	$-0.608394\pi$
$383$	−1.91759e6	−0.667974	−0.333987	+	0.942578i	$0.608394\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.68322e6	0.571298
$388$	0	0
$389$	−2.28742e6	−0.766430	−0.383215	−	0.923659i	$-0.625183\pi$
$389$	−2.28742e6	−0.766430	−0.383215	+	0.923659i	$0.625183\pi$
$390$	0	0
$391$	1.79423e6	0.593520
$392$	0	0
$393$	1.61329e6	0.526904
$394$	0	0
$395$	−625111.	−0.201588
$396$	0	0
$397$	−5.25263e6	−1.67263	−0.836316	−	0.548248i	$-0.815295\pi$
$397$	−5.25263e6	−1.67263	−0.836316	+	0.548248i	$0.815295\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−2.80141e6	−0.869991	−0.434996	−	0.900433i	$-0.643250\pi$
$401$	−2.80141e6	−0.869991	−0.434996	+	0.900433i	$0.643250\pi$
$402$	0	0
$403$	5.43121e6	1.66584
$404$	0	0
$405$	−1.15783e6	−0.350757
$406$	0	0
$407$	−4.80538e6	−1.43794
$408$	0	0
$409$	−800112.	−0.236506	−0.118253	−	0.992983i	$-0.537729\pi$
$409$	−800112.	−0.236506	−0.118253	+	0.992983i	$0.537729\pi$
$410$	0	0
$411$	9.09434e6	2.65562
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−3.06080e6	−0.872397
$416$	0	0
$417$	1.45621e6	0.410095
$418$	0	0
$419$	−1.03096e6	−0.286884	−0.143442	−	0.989659i	$-0.545817\pi$
$419$	−1.03096e6	−0.286884	−0.143442	+	0.989659i	$0.545817\pi$
$420$	0	0
$421$	−5.51368e6	−1.51613	−0.758065	−	0.652179i	$-0.773855\pi$
$421$	−5.51368e6	−1.51613	−0.758065	+	0.652179i	$0.773855\pi$
$422$	0	0
$423$	−1.20202e6	−0.326633
$424$	0	0
$425$	1.10649e6	0.297151
$426$	0	0
$427$	0	0
$428$	0	0
$429$	6.39076e6	1.67652
$430$	0	0
$431$	3.44656e6	0.893702	0.446851	−	0.894608i	$-0.352545\pi$
$431$	3.44656e6	0.893702	0.446851	+	0.894608i	$0.352545\pi$
$432$	0	0
$433$	−1.93763e6	−0.496650	−0.248325	−	0.968677i	$-0.579880\pi$
$433$	−1.93763e6	−0.496650	−0.248325	+	0.968677i	$0.579880\pi$
$434$	0	0
$435$	3.23726e6	0.820264
$436$	0	0
$437$	−6.17911e6	−1.54783
$438$	0	0
$439$	−2.12354e6	−0.525896	−0.262948	−	0.964810i	$-0.584695\pi$
$439$	−2.12354e6	−0.525896	−0.262948	+	0.964810i	$0.584695\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	6.14514e6	1.48772	0.743862	−	0.668333i	$-0.232992\pi$
$443$	6.14514e6	1.48772	0.743862	+	0.668333i	$0.232992\pi$
$444$	0	0
$445$	1.63431e6	0.391233
$446$	0	0
$447$	−6.71236e6	−1.58894
$448$	0	0
$449$	4.43540e6	1.03829	0.519143	−	0.854688i	$-0.326251\pi$
$449$	4.43540e6	1.03829	0.519143	+	0.854688i	$0.326251\pi$
$450$	0	0
$451$	4.36607e6	1.01076
$452$	0	0
$453$	−83363.4	−0.0190867
$454$	0	0
$455$	0	0
$456$	0	0
$457$	3.39654e6	0.760757	0.380379	−	0.924831i	$-0.375794\pi$
$457$	3.39654e6	0.760757	0.380379	+	0.924831i	$0.375794\pi$
$458$	0	0
$459$	−798552.	−0.176918
$460$	0	0
$461$	−7.24894e6	−1.58863	−0.794314	−	0.607508i	$-0.792169\pi$
$461$	−7.24894e6	−1.58863	−0.794314	+	0.607508i	$0.792169\pi$
$462$	0	0
$463$	2.82567e6	0.612588	0.306294	−	0.951937i	$-0.400911\pi$
$463$	2.82567e6	0.612588	0.306294	+	0.951937i	$0.400911\pi$
$464$	0	0
$465$	5.10372e6	1.09460
$466$	0	0
$467$	1.78235e6	0.378183	0.189091	−	0.981959i	$-0.439446\pi$
$467$	1.78235e6	0.378183	0.189091	+	0.981959i	$0.439446\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−74857.8	−0.0155484
$472$	0	0
$473$	1.93465e6	0.397602
$474$	0	0
$475$	−3.81064e6	−0.774932
$476$	0	0
$477$	2.14006e6	0.430656
$478$	0	0
$479$	−2.41877e6	−0.481678	−0.240839	−	0.970565i	$-0.577423\pi$
$479$	−2.41877e6	−0.481678	−0.240839	+	0.970565i	$0.577423\pi$
$480$	0	0
$481$	−1.02480e7	−2.01966
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2.88739e6	−0.557380
$486$	0	0
$487$	−2.87289e6	−0.548905	−0.274452	−	0.961601i	$-0.588497\pi$
$487$	−2.87289e6	−0.548905	−0.274452	+	0.961601i	$0.588497\pi$
$488$	0	0
$489$	−6.75486e6	−1.27745
$490$	0	0
$491$	−5.02817e6	−0.941253	−0.470626	−	0.882333i	$-0.655972\pi$
$491$	−5.02817e6	−0.941253	−0.470626	+	0.882333i	$0.655972\pi$
$492$	0	0
$493$	−2.27562e6	−0.421680
$494$	0	0
$495$	3.36894e6	0.617989
$496$	0	0
$497$	0	0
$498$	0	0
$499$	9.88348e6	1.77688	0.888441	−	0.458991i	$-0.151789\pi$
$499$	9.88348e6	1.77688	0.888441	+	0.458991i	$0.151789\pi$
$500$	0	0
$501$	1.24649e7	2.21868
$502$	0	0
$503$	−4.42533e6	−0.779876	−0.389938	−	0.920841i	$-0.627504\pi$
$503$	−4.42533e6	−0.779876	−0.389938	+	0.920841i	$0.627504\pi$
$504$	0	0
$505$	986313.	0.172102
$506$	0	0
$507$	4.89369e6	0.845506
$508$	0	0
$509$	−7.62207e6	−1.30400	−0.652002	−	0.758218i	$-0.726070\pi$
$509$	−7.62207e6	−1.30400	−0.652002	+	0.758218i	$0.726070\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	2.75012e6	0.461380
$514$	0	0
$515$	−3.67694e6	−0.610898
$516$	0	0
$517$	−1.38157e6	−0.227324
$518$	0	0
$519$	−6.26182e6	−1.02043
$520$	0	0
$521$	1.58598e6	0.255978	0.127989	−	0.991776i	$-0.459148\pi$
$521$	1.58598e6	0.255978	0.127989	+	0.991776i	$0.459148\pi$
$522$	0	0
$523$	4.35734e6	0.696574	0.348287	−	0.937388i	$-0.386763\pi$
$523$	4.35734e6	0.696574	0.348287	+	0.937388i	$0.386763\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3.58765e6	−0.562708
$528$	0	0
$529$	6.29943e6	0.978728
$530$	0	0
$531$	−1.13679e7	−1.74962
$532$	0	0
$533$	9.31116e6	1.41966
$534$	0	0
$535$	5.92772e6	0.895372
$536$	0	0
$537$	3.97855e6	0.595373
$538$	0	0
$539$	0	0
$540$	0	0
$541$	8.31407e6	1.22129	0.610647	−	0.791903i	$-0.290909\pi$
$541$	8.31407e6	1.22129	0.610647	+	0.791903i	$0.290909\pi$
$542$	0	0
$543$	−1.51392e7	−2.20346
$544$	0	0
$545$	−1.82489e6	−0.263175
$546$	0	0
$547$	−2.31787e6	−0.331223	−0.165611	−	0.986191i	$-0.552960\pi$
$547$	−2.31787e6	−0.331223	−0.165611	+	0.986191i	$0.552960\pi$
$548$	0	0
$549$	1.40755e7	1.99312
$550$	0	0
$551$	7.83697e6	1.09969
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−9.63011e6	−1.32708
$556$	0	0
$557$	7.34710e6	1.00341	0.501705	−	0.865039i	$-0.332706\pi$
$557$	7.34710e6	1.00341	0.501705	+	0.865039i	$0.332706\pi$
$558$	0	0
$559$	4.12586e6	0.558450
$560$	0	0
$561$	−4.22149e6	−0.566316
$562$	0	0
$563$	−529427.	−0.0703939	−0.0351970	−	0.999380i	$-0.511206\pi$
$563$	−529427.	−0.0703939	−0.0351970	+	0.999380i	$0.511206\pi$
$564$	0	0
$565$	−307140.	−0.0404777
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−6.76861e6	−0.876433	−0.438216	−	0.898869i	$-0.644390\pi$
$569$	−6.76861e6	−0.876433	−0.438216	+	0.898869i	$0.644390\pi$
$570$	0	0
$571$	7.89961e6	1.01395	0.506974	−	0.861961i	$-0.330764\pi$
$571$	7.89961e6	1.01395	0.506974	+	0.861961i	$0.330764\pi$
$572$	0	0
$573$	7.74340e6	0.985247
$574$	0	0
$575$	7.85411e6	0.990666
$576$	0	0
$577$	−2.21815e6	−0.277365	−0.138682	−	0.990337i	$-0.544287\pi$
$577$	−2.21815e6	−0.277365	−0.138682	+	0.990337i	$0.544287\pi$
$578$	0	0
$579$	1.92529e7	2.38670
$580$	0	0
$581$	0	0
$582$	0	0
$583$	2.45973e6	0.299720
$584$	0	0
$585$	7.18467e6	0.867994
$586$	0	0
$587$	−7.93512e6	−0.950514	−0.475257	−	0.879847i	$-0.657645\pi$
$587$	−7.93512e6	−0.950514	−0.475257	+	0.879847i	$0.657645\pi$
$588$	0	0
$589$	1.23554e7	1.46747
$590$	0	0
$591$	−1.61694e7	−1.90426
$592$	0	0
$593$	−1.58495e6	−0.185089	−0.0925444	−	0.995709i	$-0.529500\pi$
$593$	−1.58495e6	−0.185089	−0.0925444	+	0.995709i	$0.529500\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.97832e6	0.227175
$598$	0	0
$599$	8.79410e6	1.00144	0.500719	−	0.865610i	$-0.333069\pi$
$599$	8.79410e6	1.00144	0.500719	+	0.865610i	$0.333069\pi$
$600$	0	0
$601$	4.49495e6	0.507620	0.253810	−	0.967254i	$-0.418316\pi$
$601$	4.49495e6	0.507620	0.253810	+	0.967254i	$0.418316\pi$
$602$	0	0
$603$	2.15363e7	2.41200
$604$	0	0
$605$	−1.02383e6	−0.113721
$606$	0	0
$607$	3.22078e6	0.354804	0.177402	−	0.984138i	$-0.443231\pi$
$607$	3.22078e6	0.354804	0.177402	+	0.984138i	$0.443231\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−2.94636e6	−0.319288
$612$	0	0
$613$	−9.58068e6	−1.02978	−0.514891	−	0.857256i	$-0.672168\pi$
$613$	−9.58068e6	−1.02978	−0.514891	+	0.857256i	$0.672168\pi$
$614$	0	0
$615$	8.74972e6	0.932839
$616$	0	0
$617$	−5.91459e6	−0.625477	−0.312739	−	0.949839i	$-0.601246\pi$
$617$	−5.91459e6	−0.625477	−0.312739	+	0.949839i	$0.601246\pi$
$618$	0	0
$619$	4.65799e6	0.488621	0.244311	−	0.969697i	$-0.421438\pi$
$619$	4.65799e6	0.488621	0.244311	+	0.969697i	$0.421438\pi$
$620$	0	0
$621$	−5.66828e6	−0.589824
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.95554e6	0.200248
$626$	0	0
$627$	1.45383e7	1.47688
$628$	0	0
$629$	6.76946e6	0.682225
$630$	0	0
$631$	−1.16468e7	−1.16448	−0.582239	−	0.813017i	$-0.697823\pi$
$631$	−1.16468e7	−1.16448	−0.582239	+	0.813017i	$0.697823\pi$
$632$	0	0
$633$	−5.07821e6	−0.503734
$634$	0	0
$635$	5.80308e6	0.571116
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−1.13442e7	−1.09906
$640$	0	0
$641$	−1.20670e7	−1.15999	−0.579995	−	0.814620i	$-0.696946\pi$
$641$	−1.20670e7	−1.15999	−0.579995	+	0.814620i	$0.696946\pi$
$642$	0	0
$643$	1.20635e6	0.115065	0.0575327	−	0.998344i	$-0.481677\pi$
$643$	1.20635e6	0.115065	0.0575327	+	0.998344i	$0.481677\pi$
$644$	0	0
$645$	3.87708e6	0.366949
$646$	0	0
$647$	7.84856e6	0.737104	0.368552	−	0.929607i	$-0.379854\pi$
$647$	7.84856e6	0.737104	0.368552	+	0.929607i	$0.379854\pi$
$648$	0	0
$649$	−1.30660e7	−1.21767
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1.94745e7	−1.78724	−0.893620	−	0.448824i	$-0.851843\pi$
$653$	−1.94745e7	−1.78724	−0.893620	+	0.448824i	$0.851843\pi$
$654$	0	0
$655$	2.08462e6	0.189856
$656$	0	0
$657$	1.55573e7	1.40612
$658$	0	0
$659$	−1.26512e7	−1.13480	−0.567400	−	0.823443i	$-0.692051\pi$
$659$	−1.26512e7	−1.13480	−0.567400	+	0.823443i	$0.692051\pi$
$660$	0	0
$661$	1.67926e7	1.49491	0.747454	−	0.664314i	$-0.231276\pi$
$661$	1.67926e7	1.49491	0.747454	+	0.664314i	$0.231276\pi$
$662$	0	0
$663$	−9.00282e6	−0.795417
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.61528e7	−1.40583
$668$	0	0
$669$	4.84226e6	0.418295
$670$	0	0
$671$	1.61780e7	1.38714
$672$	0	0
$673$	2.20315e7	1.87502	0.937509	−	0.347960i	$-0.113125\pi$
$673$	2.20315e7	1.87502	0.937509	+	0.347960i	$0.113125\pi$
$674$	0	0
$675$	−3.49561e6	−0.295300
$676$	0	0
$677$	1.60565e7	1.34641	0.673207	−	0.739454i	$-0.264916\pi$
$677$	1.60565e7	1.34641	0.673207	+	0.739454i	$0.264916\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	2.23665e7	1.84812
$682$	0	0
$683$	−1.01604e7	−0.833409	−0.416704	−	0.909042i	$-0.636815\pi$
$683$	−1.01604e7	−0.833409	−0.416704	+	0.909042i	$0.636815\pi$
$684$	0	0
$685$	1.17513e7	0.956885
$686$	0	0
$687$	−3.02847e7	−2.44811
$688$	0	0
$689$	5.24566e6	0.420971
$690$	0	0
$691$	−1.23600e7	−0.984747	−0.492374	−	0.870384i	$-0.663871\pi$
$691$	−1.23600e7	−0.984747	−0.492374	+	0.870384i	$0.663871\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.88165e6	0.147767
$696$	0	0
$697$	−6.15060e6	−0.479552
$698$	0	0
$699$	−3.32674e7	−2.57528
$700$	0	0
$701$	−5.58795e6	−0.429495	−0.214747	−	0.976670i	$-0.568893\pi$
$701$	−5.58795e6	−0.429495	−0.214747	+	0.976670i	$0.568893\pi$
$702$	0	0
$703$	−2.33132e7	−1.77916
$704$	0	0
$705$	−2.76870e6	−0.209799
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−8.46936e6	−0.632754	−0.316377	−	0.948633i	$-0.602467\pi$
$709$	−8.46936e6	−0.632754	−0.316377	+	0.948633i	$0.602467\pi$
$710$	0	0
$711$	−6.38493e6	−0.473677
$712$	0	0
$713$	−2.54658e7	−1.87601
$714$	0	0
$715$	8.25786e6	0.604091
$716$	0	0
$717$	−7.87088e6	−0.571775
$718$	0	0
$719$	5.05165e6	0.364428	0.182214	−	0.983259i	$-0.441674\pi$
$719$	5.05165e6	0.364428	0.182214	+	0.983259i	$0.441674\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−1.71975e7	−1.22355
$724$	0	0
$725$	−9.96139e6	−0.703841
$726$	0	0
$727$	1.67234e7	1.17352	0.586758	−	0.809762i	$-0.300404\pi$
$727$	1.67234e7	1.17352	0.586758	+	0.809762i	$0.300404\pi$
$728$	0	0
$729$	−2.09066e7	−1.45702
$730$	0	0
$731$	−2.72538e6	−0.188640
$732$	0	0
$733$	2.85393e6	0.196193	0.0980964	−	0.995177i	$-0.468725\pi$
$733$	2.85393e6	0.196193	0.0980964	+	0.995177i	$0.468725\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2.47532e7	1.67866
$738$	0	0
$739$	7.76014e6	0.522707	0.261354	−	0.965243i	$-0.415831\pi$
$739$	7.76014e6	0.522707	0.261354	+	0.965243i	$0.415831\pi$
$740$	0	0
$741$	3.10047e7	2.07435
$742$	0	0
$743$	−1.73146e7	−1.15065	−0.575323	−	0.817927i	$-0.695124\pi$
$743$	−1.73146e7	−1.15065	−0.575323	+	0.817927i	$0.695124\pi$
$744$	0	0
$745$	−8.67342e6	−0.572532
$746$	0	0
$747$	−3.12632e7	−2.04989
$748$	0	0
$749$	0	0
$750$	0	0
$751$	2.40730e7	1.55751	0.778754	−	0.627329i	$-0.215852\pi$
$751$	2.40730e7	1.55751	0.778754	+	0.627329i	$0.215852\pi$
$752$	0	0
$753$	−2.18623e7	−1.40510
$754$	0	0
$755$	−107719.	−0.00687739
$756$	0	0
$757$	2.53942e7	1.61062	0.805312	−	0.592851i	$-0.201998\pi$
$757$	2.53942e7	1.61062	0.805312	+	0.592851i	$0.201998\pi$
$758$	0	0
$759$	−2.99650e7	−1.88803
$760$	0	0
$761$	−1.69944e7	−1.06376	−0.531881	−	0.846819i	$-0.678515\pi$
$761$	−1.69944e7	−1.06376	−0.531881	+	0.846819i	$0.678515\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−4.74592e6	−0.293202
$766$	0	0
$767$	−2.78647e7	−1.71027
$768$	0	0
$769$	−7.49043e6	−0.456763	−0.228382	−	0.973572i	$-0.573343\pi$
$769$	−7.49043e6	−0.456763	−0.228382	+	0.973572i	$0.573343\pi$
$770$	0	0
$771$	−2.15854e7	−1.30775
$772$	0	0
$773$	−2.39473e7	−1.44148	−0.720739	−	0.693207i	$-0.756197\pi$
$773$	−2.39473e7	−1.44148	−0.720739	+	0.693207i	$0.756197\pi$
$774$	0	0
$775$	−1.57047e7	−0.939238
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2.11819e7	1.25061
$780$	0	0
$781$	−1.30387e7	−0.764906
$782$	0	0
$783$	7.18909e6	0.419054
$784$	0	0
$785$	−96728.0	−0.00560245
$786$	0	0
$787$	3.12247e7	1.79705	0.898526	−	0.438919i	$-0.144639\pi$
$787$	3.12247e7	1.79705	0.898526	+	0.438919i	$0.144639\pi$
$788$	0	0
$789$	6.34889e6	0.363082
$790$	0	0
$791$	0	0
$792$	0	0
$793$	3.45015e7	1.94830
$794$	0	0
$795$	4.92936e6	0.276613
$796$	0	0
$797$	−588976.	−0.0328437	−0.0164218	−	0.999865i	$-0.505227\pi$
$797$	−588976.	−0.0328437	−0.0164218	+	0.999865i	$0.505227\pi$
$798$	0	0
$799$	1.94625e6	0.107853
$800$	0	0
$801$	1.66930e7	0.919291
$802$	0	0
$803$	1.78812e7	0.978605
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−9.90800e6	−0.535553
$808$	0	0
$809$	−3.27865e7	−1.76126	−0.880630	−	0.473804i	$-0.842881\pi$
$809$	−3.27865e7	−1.76126	−0.880630	+	0.473804i	$0.842881\pi$
$810$	0	0
$811$	2.40790e7	1.28554	0.642772	−	0.766057i	$-0.277784\pi$
$811$	2.40790e7	1.28554	0.642772	+	0.766057i	$0.277784\pi$
$812$	0	0
$813$	−3.50901e7	−1.86191
$814$	0	0
$815$	−8.72834e6	−0.460296
$816$	0	0
$817$	9.38591e6	0.491950
$818$	0	0
$819$	0	0
$820$	0	0
$821$	3.75376e6	0.194361	0.0971804	−	0.995267i	$-0.469018\pi$
$821$	3.75376e6	0.194361	0.0971804	+	0.995267i	$0.469018\pi$
$822$	0	0
$823$	−2.06177e7	−1.06106	−0.530530	−	0.847666i	$-0.678007\pi$
$823$	−2.06177e7	−1.06106	−0.530530	+	0.847666i	$0.678007\pi$
$824$	0	0
$825$	−1.84793e7	−0.945259
$826$	0	0
$827$	−1.16708e7	−0.593384	−0.296692	−	0.954973i	$-0.595883\pi$
$827$	−1.16708e7	−0.593384	−0.296692	+	0.954973i	$0.595883\pi$
$828$	0	0
$829$	7.23524e6	0.365651	0.182825	−	0.983145i	$-0.441476\pi$
$829$	7.23524e6	0.365651	0.182825	+	0.983145i	$0.441476\pi$
$830$	0	0
$831$	−2.51554e7	−1.26366
$832$	0	0
$833$	0	0
$834$	0	0
$835$	1.61066e7	0.799444
$836$	0	0
$837$	1.13340e7	0.559204
$838$	0	0
$839$	2.74254e7	1.34508	0.672539	−	0.740062i	$-0.265204\pi$
$839$	2.74254e7	1.34508	0.672539	+	0.740062i	$0.265204\pi$
$840$	0	0
$841$	−24507.5	−0.00119484
$842$	0	0
$843$	−4.26501e7	−2.06705
$844$	0	0
$845$	6.32342e6	0.304656
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−3.04474e7	−1.44971
$850$	0	0
$851$	4.80510e7	2.27446
$852$	0	0
$853$	6.67554e6	0.314133	0.157067	−	0.987588i	$-0.449796\pi$
$853$	6.67554e6	0.314133	0.157067	+	0.987588i	$0.449796\pi$
$854$	0	0
$855$	1.63444e7	0.764634
$856$	0	0
$857$	3.76654e6	0.175183	0.0875913	−	0.996156i	$-0.472083\pi$
$857$	3.76654e6	0.175183	0.0875913	+	0.996156i	$0.472083\pi$
$858$	0	0
$859$	−1.23895e7	−0.572891	−0.286446	−	0.958096i	$-0.592474\pi$
$859$	−1.23895e7	−0.572891	−0.286446	+	0.958096i	$0.592474\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−2.35320e7	−1.07555	−0.537776	−	0.843088i	$-0.680735\pi$
$863$	−2.35320e7	−1.07555	−0.537776	+	0.843088i	$0.680735\pi$
$864$	0	0
$865$	−8.09125e6	−0.367685
$866$	0	0
$867$	−2.74578e7	−1.24056
$868$	0	0
$869$	−7.33866e6	−0.329661
$870$	0	0
$871$	5.27892e7	2.35776
$872$	0	0
$873$	−2.94920e7	−1.30969
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−2.82967e7	−1.24233	−0.621165	−	0.783679i	$-0.713341\pi$
$877$	−2.82967e7	−1.24233	−0.621165	+	0.783679i	$0.713341\pi$
$878$	0	0
$879$	−4.49358e7	−1.96165
$880$	0	0
$881$	5.55341e6	0.241057	0.120529	−	0.992710i	$-0.461541\pi$
$881$	5.55341e6	0.241057	0.120529	+	0.992710i	$0.461541\pi$
$882$	0	0
$883$	3.53912e7	1.52754	0.763771	−	0.645487i	$-0.223346\pi$
$883$	3.53912e7	1.52754	0.763771	+	0.645487i	$0.223346\pi$
$884$	0	0
$885$	−2.61845e7	−1.12379
$886$	0	0
$887$	2.75726e7	1.17671	0.588353	−	0.808604i	$-0.299777\pi$
$887$	2.75726e7	1.17671	0.588353	+	0.808604i	$0.299777\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−1.35926e7	−0.573601
$892$	0	0
$893$	−6.70266e6	−0.281267
$894$	0	0
$895$	5.14091e6	0.214527
$896$	0	0
$897$	−6.39038e7	−2.65183
$898$	0	0
$899$	3.22984e7	1.33285
$900$	0	0
$901$	−3.46508e6	−0.142201
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1.95623e7	−0.793959
$906$	0	0
$907$	−9.69350e6	−0.391257	−0.195629	−	0.980678i	$-0.562675\pi$
$907$	−9.69350e6	−0.391257	−0.195629	+	0.980678i	$0.562675\pi$
$908$	0	0
$909$	1.00743e7	0.404393
$910$	0	0
$911$	1.88317e7	0.751786	0.375893	−	0.926663i	$-0.377336\pi$
$911$	1.88317e7	0.751786	0.375893	+	0.926663i	$0.377336\pi$
$912$	0	0
$913$	−3.59331e7	−1.42665
$914$	0	0
$915$	3.24212e7	1.28019
$916$	0	0
$917$	0	0
$918$	0	0
$919$	1.33496e7	0.521409	0.260704	−	0.965419i	$-0.416045\pi$
$919$	1.33496e7	0.521409	0.260704	+	0.965419i	$0.416045\pi$
$920$	0	0
$921$	4.48993e7	1.74418
$922$	0	0
$923$	−2.78066e7	−1.07435
$924$	0	0
$925$	2.96329e7	1.13873
$926$	0	0
$927$	−3.75566e7	−1.43544
$928$	0	0
$929$	2.19144e7	0.833086	0.416543	−	0.909116i	$-0.363242\pi$
$929$	2.19144e7	0.833086	0.416543	+	0.909116i	$0.363242\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−5.11655e7	−1.92430
$934$	0	0
$935$	−5.45483e6	−0.204057
$936$	0	0
$937$	4.77785e7	1.77780	0.888902	−	0.458097i	$-0.151469\pi$
$937$	4.77785e7	1.77780	0.888902	+	0.458097i	$0.151469\pi$
$938$	0	0
$939$	5.23754e7	1.93849
$940$	0	0
$941$	2.34167e7	0.862090	0.431045	−	0.902331i	$-0.358145\pi$
$941$	2.34167e7	0.862090	0.431045	+	0.902331i	$0.358145\pi$
$942$	0	0
$943$	−4.36582e7	−1.59877
$944$	0	0
$945$	0	0
$946$	0	0
$947$	4.54963e7	1.64855	0.824274	−	0.566191i	$-0.191584\pi$
$947$	4.54963e7	1.64855	0.824274	+	0.566191i	$0.191584\pi$
$948$	0	0
$949$	3.81337e7	1.37450
$950$	0	0
$951$	5.27372e7	1.89089
$952$	0	0
$953$	2.11167e7	0.753172	0.376586	−	0.926382i	$-0.377098\pi$
$953$	2.11167e7	0.753172	0.376586	+	0.926382i	$0.377098\pi$
$954$	0	0
$955$	1.00057e7	0.355008
$956$	0	0
$957$	3.80047e7	1.34140
$958$	0	0
$959$	0	0
$960$	0	0
$961$	2.22911e7	0.778615
$962$	0	0
$963$	6.05462e7	2.10388
$964$	0	0
$965$	2.48777e7	0.859987
$966$	0	0
$967$	−1.51303e7	−0.520334	−0.260167	−	0.965564i	$-0.583778\pi$
$967$	−1.51303e7	−0.520334	−0.260167	+	0.965564i	$0.583778\pi$
$968$	0	0
$969$	−2.04805e7	−0.700699
$970$	0	0
$971$	4.15896e7	1.41559	0.707794	−	0.706418i	$-0.249690\pi$
$971$	4.15896e7	1.41559	0.707794	+	0.706418i	$0.249690\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−3.94093e7	−1.32766
$976$	0	0
$977$	3.79756e7	1.27282	0.636411	−	0.771350i	$-0.280418\pi$
$977$	3.79756e7	1.27282	0.636411	+	0.771350i	$0.280418\pi$
$978$	0	0
$979$	1.91865e7	0.639792
$980$	0	0
$981$	−1.86395e7	−0.618389
$982$	0	0
$983$	−841958.	−0.0277911	−0.0138956	−	0.999903i	$-0.504423\pi$
$983$	−841958.	−0.0277911	−0.0138956	+	0.999903i	$0.504423\pi$
$984$	0	0
$985$	−2.08934e7	−0.686150
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.93453e7	−0.628905
$990$	0	0
$991$	−3.66434e7	−1.18526	−0.592628	−	0.805476i	$-0.701910\pi$
$991$	−3.66434e7	−1.18526	−0.592628	+	0.805476i	$0.701910\pi$
$992$	0	0
$993$	−7.24390e7	−2.33131
$994$	0	0
$995$	2.55629e6	0.0818565
$996$	0	0
$997$	3.43869e7	1.09561	0.547803	−	0.836607i	$-0.315464\pi$
$997$	3.43869e7	1.09561	0.547803	+	0.836607i	$0.315464\pi$
$998$	0	0
$999$	−2.13859e7	−0.677976

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.6.a.n.1.7	yes	8
4.3	odd	2		784.6.a.bo.1.2		8
7.2	even	3		392.6.i.r.361.2		16
7.3	odd	6		392.6.i.r.177.7		16
7.4	even	3		392.6.i.r.177.2		16
7.5	odd	6		392.6.i.r.361.7		16
7.6	odd	2	inner	392.6.a.n.1.2	✓	8
28.27	even	2		784.6.a.bo.1.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.6.a.n.1.2	✓	8	7.6	odd	2	inner
392.6.a.n.1.7	yes	8	1.1	even	1	trivial
392.6.i.r.177.2		16	7.4	even	3
392.6.i.r.177.7		16	7.3	odd	6
392.6.i.r.361.2		16	7.2	even	3
392.6.i.r.361.7		16	7.5	odd	6
784.6.a.bo.1.2		8	4.3	odd	2
784.6.a.bo.1.7		8	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)

\(f(q)\)	\(=\)	\(q+23.5268 q^{3} +30.4003 q^{5} +310.511 q^{9} +O(q^{10})\)	\(q+23.5268 q^{3} +30.4003 q^{5} +310.511 q^{9} +356.893 q^{11} +761.116 q^{13} +715.223 q^{15} -502.764 q^{17} +1731.46 q^{19} -3568.72 q^{23} -2200.82 q^{25} +1588.32 q^{27} +4526.22 q^{29} +7135.84 q^{31} +8396.56 q^{33} -13464.5 q^{37} +17906.6 q^{39} +12233.6 q^{41} +5420.80 q^{43} +9439.64 q^{45} -3871.10 q^{47} -11828.4 q^{51} +6892.06 q^{53} +10849.7 q^{55} +40735.8 q^{57} -36610.3 q^{59} +45330.1 q^{61} +23138.2 q^{65} +69357.5 q^{67} -83960.6 q^{69} -36534.0 q^{71} +50102.3 q^{73} -51778.3 q^{75} -20562.6 q^{79} -38086.0 q^{81} -100683. q^{83} -15284.2 q^{85} +106487. q^{87} +53759.8 q^{89} +167884. q^{93} +52637.0 q^{95} -94978.9 q^{97} +110819. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 932 q^{9}+O(q^{10}) \) 8 * q + 932 * q^9	\( 8 q + 932 q^{9} + 1204 q^{11} + 864 q^{15} - 2368 q^{23} + 2188 q^{25} - 9684 q^{29} + 60 q^{37} + 16088 q^{39} + 43564 q^{43} + 76684 q^{51} + 94472 q^{53} + 61004 q^{57} + 41900 q^{65} + 136504 q^{67} + 192328 q^{71} - 29816 q^{79} + 457860 q^{81} + 111132 q^{85} - 166504 q^{93} + 670800 q^{95} + 857404 q^{99}+O(q^{100}) \) 8 * q + 932 * q^9 + 1204 * q^11 + 864 * q^15 - 2368 * q^23 + 2188 * q^25 - 9684 * q^29 + 60 * q^37 + 16088 * q^39 + 43564 * q^43 + 76684 * q^51 + 94472 * q^53 + 61004 * q^57 + 41900 * q^65 + 136504 * q^67 + 192328 * q^71 - 29816 * q^79 + 457860 * q^81 + 111132 * q^85 - 166504 * q^93 + 670800 * q^95 + 857404 * q^99

Introduction

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