LMFDB - Embedded newform 392.6.a.n.1.2

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			1.2
Root			$12.9497$ 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	$-0.772180\pi$
$3$	−23.5268	−1.50925	−0.754623	+	0.656159i	$0.772180\pi$
$4$	0	0
$5$	−30.4003	−0.543818	−0.271909	−	0.962323i	$-0.587655\pi$
$5$	−30.4003	−0.543818	−0.271909	+	0.962323i	$0.587655\pi$
$6$	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	$-0.714715\pi$
$13$	−761.116	−1.24909	−0.624543	+	0.780990i	$0.714715\pi$
$14$	0	0
$15$	715.223	0.820755
$16$	0	0
$17$	502.764	0.421932	0.210966	−	0.977493i	$-0.432339\pi$
$17$	502.764	0.421932	0.210966	+	0.977493i	$0.432339\pi$
$18$	0	0
$19$	−1731.46	−1.10035	−0.550173	−	0.835051i	$-0.685438\pi$
$19$	−1731.46	−1.10035	−0.550173	+	0.835051i	$0.685438\pi$
$20$	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	$-0.732347\pi$
$31$	−7135.84	−1.33365	−0.666824	+	0.745216i	$0.732347\pi$
$32$	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	$-0.692391\pi$
$41$	−12233.6	−1.13656	−0.568281	+	0.822835i	$0.692391\pi$
$42$	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	$-0.459206\pi$
$47$	3871.10	0.255617	0.127808	+	0.991799i	$0.459206\pi$
$48$	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	$-0.259973\pi$
$59$	36610.3	1.36922	0.684610	+	0.728910i	$0.259973\pi$
$60$	0	0
$61$	−45330.1	−1.55978	−0.779889	−	0.625918i	$-0.784724\pi$
$61$	−45330.1	−1.55978	−0.779889	+	0.625918i	$0.784724\pi$
$62$	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	$-0.685449\pi$
$73$	−50102.3	−1.10040	−0.550200	+	0.835033i	$0.685449\pi$
$74$	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	$-0.203714\pi$
$83$	100683.	1.60421	0.802104	+	0.597184i	$0.203714\pi$
$84$	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	$-0.617124\pi$
$89$	−53759.8	−0.719420	−0.359710	+	0.933064i	$0.617124\pi$
$90$	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	$-0.328731\pi$
$97$	94978.9	1.02494	0.512469	+	0.858705i	$0.328731\pi$
$98$	0	0
$99$	110819.	1.13639
$100$	0	0
$101$	−32444.2	−0.316470	−0.158235	−	0.987401i	$-0.550580\pi$
$101$	−32444.2	−0.316470	−0.158235	+	0.987401i	$0.550580\pi$
$102$	0	0
$103$	120951.	1.12335	0.561676	−	0.827358i	$-0.310157\pi$
$103$	120951.	1.12335	0.561676	+	0.827358i	$0.310157\pi$
$104$	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	$-0.555850\pi$
$131$	−68572.4	−0.349117	−0.174559	+	0.984647i	$0.555850\pi$
$132$	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	$-0.543380\pi$
$139$	−61895.8	−0.271722	−0.135861	+	0.990728i	$0.543380\pi$
$140$	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	$-0.498360\pi$
$157$	3181.81	0.0103021	0.00515104	+	0.999987i	$0.498360\pi$
$158$	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	$-0.762832\pi$
$167$	−529817.	−1.47006	−0.735030	+	0.678035i	$0.762832\pi$
$168$	0	0
$169$	208005.	0.560218
$170$	0	0
$171$	−537638.	−1.40605
$172$	0	0
$173$	266157.	0.676117	0.338059	−	0.941125i	$-0.390230\pi$
$173$	266157.	0.676117	0.338059	+	0.941125i	$0.390230\pi$
$174$	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	$-0.239526\pi$
$181$	643489.	1.45997	0.729986	+	0.683462i	$0.239526\pi$
$182$	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	$-0.523979\pi$
$199$	−84087.7	−0.150522	−0.0752609	+	0.997164i	$0.523979\pi$
$200$	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	$-0.544253\pi$
$223$	−205819.	−0.277155	−0.138578	+	0.990352i	$0.544253\pi$
$224$	0	0
$225$	−683379.	−0.899923
$226$	0	0
$227$	−950681.	−1.22453	−0.612266	−	0.790652i	$-0.709742\pi$
$227$	−950681.	−1.22453	−0.612266	+	0.790652i	$0.709742\pi$
$228$	0	0
$229$	1.28724e6	1.62207	0.811037	−	0.584995i	$-0.198903\pi$
$229$	1.28724e6	1.62207	0.811037	+	0.584995i	$0.198903\pi$
$230$	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	$-0.367150\pi$
$241$	730975.	0.810700	0.405350	+	0.914162i	$0.367150\pi$
$242$	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	$-0.345875\pi$
$251$	929251.	0.930998	0.465499	+	0.885048i	$0.345875\pi$
$252$	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	$-0.357368\pi$
$257$	917481.	0.866492	0.433246	+	0.901276i	$0.357368\pi$
$258$	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	$-0.443224\pi$
$269$	421136.	0.354848	0.177424	+	0.984135i	$0.443224\pi$
$270$	0	0
$271$	1.49149e6	1.23367	0.616834	−	0.787093i	$-0.288415\pi$
$271$	1.49149e6	1.23367	0.616834	+	0.787093i	$0.288415\pi$
$272$	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	$-0.340536\pi$
$283$	1.29416e6	0.960552	0.480276	+	0.877117i	$0.340536\pi$
$284$	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	$-0.274821\pi$
$293$	1.90998e6	1.29975	0.649876	+	0.760040i	$0.274821\pi$
$294$	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	$-0.696100\pi$
$307$	−1.90843e6	−1.15566	−0.577830	+	0.816157i	$0.696100\pi$
$308$	0	0
$309$	−2.84559e6	−1.69541
$310$	0	0
$311$	2.17477e6	1.27501	0.637504	−	0.770447i	$-0.279967\pi$
$311$	2.17477e6	1.27501	0.637504	+	0.770447i	$0.279967\pi$
$312$	0	0
$313$	−2.22620e6	−1.28441	−0.642205	−	0.766533i	$-0.721980\pi$
$313$	−2.22620e6	−1.28441	−0.642205	+	0.766533i	$0.721980\pi$
$314$	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	$-0.504726\pi$
$349$	−67561.0	−0.0296915	−0.0148458	+	0.999890i	$0.504726\pi$
$350$	0	0
$351$	1.20890e6	0.523747
$352$	0	0
$353$	−588968.	−0.251568	−0.125784	−	0.992058i	$-0.540145\pi$
$353$	−588968.	−0.251568	−0.125784	+	0.992058i	$0.540145\pi$
$354$	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	$-0.645331\pi$
$367$	−2.27514e6	−0.881746	−0.440873	+	0.897570i	$0.645331\pi$
$368$	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	$-0.391606\pi$
$383$	1.91759e6	0.667974	0.333987	+	0.942578i	$0.391606\pi$
$384$	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	$-0.184705\pi$
$397$	5.25263e6	1.67263	0.836316	+	0.548248i	$0.184705\pi$
$398$	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	$-0.462271\pi$
$409$	800112.	0.236506	0.118253	+	0.992983i	$0.462271\pi$
$410$	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	$-0.454183\pi$
$419$	1.03096e6	0.286884	0.143442	+	0.989659i	$0.454183\pi$
$420$	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	$-0.420120\pi$
$433$	1.93763e6	0.496650	0.248325	+	0.968677i	$0.420120\pi$
$434$	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	$-0.415305\pi$
$439$	2.12354e6	0.525896	0.262948	+	0.964810i	$0.415305\pi$
$440$	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	$-0.207831\pi$
$461$	7.24894e6	1.58863	0.794314	+	0.607508i	$0.207831\pi$
$462$	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	$-0.560554\pi$
$467$	−1.78235e6	−0.378183	−0.189091	+	0.981959i	$0.560554\pi$
$468$	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	$-0.422577\pi$
$479$	2.41877e6	0.481678	0.240839	+	0.970565i	$0.422577\pi$
$480$	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	$-0.372496\pi$
$503$	4.42533e6	0.779876	0.389938	+	0.920841i	$0.372496\pi$
$504$	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	$-0.273930\pi$
$509$	7.62207e6	1.30400	0.652002	+	0.758218i	$0.273930\pi$
$510$	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	$-0.540852\pi$
$521$	−1.58598e6	−0.255978	−0.127989	+	0.991776i	$0.540852\pi$
$522$	0	0
$523$	−4.35734e6	−0.696574	−0.348287	−	0.937388i	$-0.613237\pi$
$523$	−4.35734e6	−0.696574	−0.348287	+	0.937388i	$0.613237\pi$
$524$	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	$-0.488794\pi$
$563$	529427.	0.0703939	0.0351970	+	0.999380i	$0.488794\pi$
$564$	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	$-0.455713\pi$
$577$	2.21815e6	0.277365	0.138682	+	0.990337i	$0.455713\pi$
$578$	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	$-0.342355\pi$
$587$	7.93512e6	0.950514	0.475257	+	0.879847i	$0.342355\pi$
$588$	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	$-0.470500\pi$
$593$	1.58495e6	0.185089	0.0925444	+	0.995709i	$0.470500\pi$
$594$	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	$-0.581684\pi$
$601$	−4.49495e6	−0.507620	−0.253810	+	0.967254i	$0.581684\pi$
$602$	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	$-0.556769\pi$
$607$	−3.22078e6	−0.354804	−0.177402	+	0.984138i	$0.556769\pi$
$608$	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	$-0.578562\pi$
$619$	−4.65799e6	−0.488621	−0.244311	+	0.969697i	$0.578562\pi$
$620$	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	$-0.518323\pi$
$643$	−1.20635e6	−0.115065	−0.0575327	+	0.998344i	$0.518323\pi$
$644$	0	0
$645$	3.87708e6	0.366949
$646$	0	0
$647$	−7.84856e6	−0.737104	−0.368552	−	0.929607i	$-0.620146\pi$
$647$	−7.84856e6	−0.737104	−0.368552	+	0.929607i	$0.620146\pi$
$648$	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	$-0.768724\pi$
$661$	−1.67926e7	−1.49491	−0.747454	+	0.664314i	$0.768724\pi$
$662$	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	$-0.735084\pi$
$677$	−1.60565e7	−1.34641	−0.673207	+	0.739454i	$0.735084\pi$
$678$	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	$-0.336129\pi$
$691$	1.23600e7	0.984747	0.492374	+	0.870384i	$0.336129\pi$
$692$	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	$-0.558326\pi$
$719$	−5.05165e6	−0.364428	−0.182214	+	0.983259i	$0.558326\pi$
$720$	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	$-0.699596\pi$
$727$	−1.67234e7	−1.17352	−0.586758	+	0.809762i	$0.699596\pi$
$728$	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	$-0.531275\pi$
$733$	−2.85393e6	−0.196193	−0.0980964	+	0.995177i	$0.531275\pi$
$734$	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	$-0.321485\pi$
$761$	1.69944e7	1.06376	0.531881	+	0.846819i	$0.321485\pi$
$762$	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	$-0.426657\pi$
$769$	7.49043e6	0.456763	0.228382	+	0.973572i	$0.426657\pi$
$770$	0	0
$771$	−2.15854e7	−1.30775
$772$	0	0
$773$	2.39473e7	1.44148	0.720739	−	0.693207i	$-0.243803\pi$
$773$	2.39473e7	1.44148	0.720739	+	0.693207i	$0.243803\pi$
$774$	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	$-0.855361\pi$
$787$	−3.12247e7	−1.79705	−0.898526	+	0.438919i	$0.855361\pi$
$788$	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	$-0.494773\pi$
$797$	588976.	0.0328437	0.0164218	+	0.999865i	$0.494773\pi$
$798$	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	$-0.722216\pi$
$811$	−2.40790e7	−1.28554	−0.642772	+	0.766057i	$0.722216\pi$
$812$	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	$-0.558524\pi$
$829$	−7.23524e6	−0.365651	−0.182825	+	0.983145i	$0.558524\pi$
$830$	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	$-0.734796\pi$
$839$	−2.74254e7	−1.34508	−0.672539	+	0.740062i	$0.734796\pi$
$840$	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	$-0.550204\pi$
$853$	−6.67554e6	−0.314133	−0.157067	+	0.987588i	$0.550204\pi$
$854$	0	0
$855$	1.63444e7	0.764634
$856$	0	0
$857$	−3.76654e6	−0.175183	−0.0875913	−	0.996156i	$-0.527917\pi$
$857$	−3.76654e6	−0.175183	−0.0875913	+	0.996156i	$0.527917\pi$
$858$	0	0
$859$	1.23895e7	0.572891	0.286446	−	0.958096i	$-0.407526\pi$
$859$	1.23895e7	0.572891	0.286446	+	0.958096i	$0.407526\pi$
$860$	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	$-0.538459\pi$
$881$	−5.55341e6	−0.241057	−0.120529	+	0.992710i	$0.538459\pi$
$882$	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	$-0.700223\pi$
$887$	−2.75726e7	−1.17671	−0.588353	+	0.808604i	$0.700223\pi$
$888$	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	$-0.636758\pi$
$929$	−2.19144e7	−0.833086	−0.416543	+	0.909116i	$0.636758\pi$
$930$	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	$-0.848531\pi$
$937$	−4.77785e7	−1.77780	−0.888902	+	0.458097i	$0.848531\pi$
$938$	0	0
$939$	5.23754e7	1.93849
$940$	0	0
$941$	−2.34167e7	−0.862090	−0.431045	−	0.902331i	$-0.641855\pi$
$941$	−2.34167e7	−0.862090	−0.431045	+	0.902331i	$0.641855\pi$
$942$	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	$-0.750310\pi$
$971$	−4.15896e7	−1.41559	−0.707794	+	0.706418i	$0.750310\pi$
$972$	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	$-0.495577\pi$
$983$	841958.	0.0277911	0.0138956	+	0.999903i	$0.495577\pi$
$984$	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	$-0.684536\pi$
$997$	−3.43869e7	−1.09561	−0.547803	+	0.836607i	$0.684536\pi$
$998$	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.1.2	✓	8
4.3	odd	2		784.6.a.bo.1.7		8
7.2	even	3		392.6.i.r.361.7		16
7.3	odd	6		392.6.i.r.177.2		16
7.4	even	3		392.6.i.r.177.7		16
7.5	odd	6		392.6.i.r.361.2		16
7.6	odd	2	inner	392.6.a.n.1.7	yes	8
28.27	even	2		784.6.a.bo.1.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.6.a.n.1.2	✓	8	1.1	even	1	trivial
392.6.a.n.1.7	yes	8	7.6	odd	2	inner
392.6.i.r.177.2		16	7.3	odd	6
392.6.i.r.177.7		16	7.4	even	3
392.6.i.r.361.2		16	7.5	odd	6
392.6.i.r.361.7		16	7.2	even	3
784.6.a.bo.1.2		8	28.27	even	2
784.6.a.bo.1.7		8	4.3	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:	\( N \)	\(=\)	\( 392 = 2^{3} \cdot 7^{2} \)
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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