LMFDB - Embedded newform 441.6.a.t.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,6,Mod(1,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("441.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	441.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:	$70.7292645375$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{249}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 62 $ x^2 - x - 62
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 21)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$8.38987$ of defining polynomial
Character	$\chi$	$=$	441.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.38987 q^{2} +56.1696 q^{4} +71.7291 q^{5} +226.949 q^{8} +673.526 q^{10} +560.610 q^{11} +533.509 q^{13} +333.597 q^{16} -1005.70 q^{17} +1368.53 q^{19} +4028.99 q^{20} +5264.05 q^{22} -3228.08 q^{23} +2020.06 q^{25} +5009.58 q^{26} +753.456 q^{29} +8206.42 q^{31} -4129.95 q^{32} -9443.39 q^{34} -2808.66 q^{37} +12850.3 q^{38} +16278.9 q^{40} -245.827 q^{41} -17504.5 q^{43} +31489.2 q^{44} -30311.2 q^{46} +16345.5 q^{47} +18968.1 q^{50} +29967.0 q^{52} +29641.7 q^{53} +40212.0 q^{55} +7074.85 q^{58} +10356.1 q^{59} +954.179 q^{61} +77057.2 q^{62} -49454.8 q^{64} +38268.1 q^{65} -19815.2 q^{67} -56489.8 q^{68} -62125.4 q^{71} +27109.6 q^{73} -26373.0 q^{74} +76869.6 q^{76} +44687.4 q^{79} +23928.6 q^{80} -2308.29 q^{82} -15606.6 q^{83} -72137.9 q^{85} -164365. q^{86} +127230. q^{88} -13635.3 q^{89} -181320. q^{92} +153482. q^{94} +98163.1 q^{95} -12919.5 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{2} + 65 q^{4} + 33 q^{5} + 375 q^{8} + 921 q^{10} + 1137 q^{11} + 925 q^{13} - 895 q^{16} + 324 q^{17} + 2311 q^{19} + 3687 q^{20} + 1581 q^{22} - 1596 q^{23} + 395 q^{25} + 2508 q^{26} + 2217 q^{29}+ \cdots - 9791 q^{97}+O(q^{100}) $ 2 * q + 3 * q^2 + 65 * q^4 + 33 * q^5 + 375 * q^8 + 921 * q^10 + 1137 * q^11 + 925 * q^13 - 895 * q^16 + 324 * q^17 + 2311 * q^19 + 3687 * q^20 + 1581 * q^22 - 1596 * q^23 + 395 * q^25 + 2508 * q^26 + 2217 * q^29 + 4294 * q^31 - 1017 * q^32 - 17940 * q^34 - 19109 * q^37 + 6828 * q^38 + 10545 * q^40 + 12858 * q^41 - 2771 * q^43 + 36579 * q^44 - 40740 * q^46 + 23160 * q^47 + 29352 * q^50 + 33424 * q^52 + 31653 * q^53 + 17889 * q^55 - 2277 * q^58 - 41097 * q^59 + 42052 * q^61 + 102057 * q^62 - 30031 * q^64 + 23106 * q^65 + 30763 * q^67 - 44748 * q^68 - 102096 * q^71 - 28577 * q^73 + 77784 * q^74 + 85192 * q^76 - 18464 * q^79 + 71511 * q^80 - 86040 * q^82 - 61179 * q^83 - 123636 * q^85 - 258510 * q^86 + 212565 * q^88 - 29322 * q^89 - 166908 * q^92 + 109938 * q^94 + 61662 * q^95 - 9791 * q^97

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	9.38987	1.65991	0.829955	−	0.557831i	$-0.188366\pi$
$2$	9.38987	1.65991	0.829955	+	0.557831i	$0.188366\pi$
$3$	0	0
$3$	0	0
$4$	56.1696	1.75530
$5$	71.7291	1.28313	0.641564	−	0.767069i	$-0.278286\pi$
$5$	71.7291	1.28313	0.641564	+	0.767069i	$0.278286\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	226.949	1.25373
$9$	0	0
$10$	673.526	2.12988
$11$	560.610	1.39694	0.698472	−	0.715637i	$-0.253863\pi$
$11$	560.610	1.39694	0.698472	+	0.715637i	$0.253863\pi$
$12$	0	0
$13$	533.509	0.875555	0.437777	−	0.899083i	$-0.355766\pi$
$13$	533.509	0.875555	0.437777	+	0.899083i	$0.355766\pi$
$14$	0	0
$15$	0	0
$16$	333.597	0.325778
$17$	−1005.70	−0.844007	−0.422004	−	0.906594i	$-0.638673\pi$
$17$	−1005.70	−0.844007	−0.422004	+	0.906594i	$0.638673\pi$
$18$	0	0
$19$	1368.53	0.869699	0.434850	−	0.900503i	$-0.356801\pi$
$19$	1368.53	0.869699	0.434850	+	0.900503i	$0.356801\pi$
$20$	4028.99	2.25228
$21$	0	0
$22$	5264.05	2.31880
$23$	−3228.08	−1.27240	−0.636201	−	0.771523i	$-0.719495\pi$
$23$	−3228.08	−1.27240	−0.636201	+	0.771523i	$0.719495\pi$
$24$	0	0
$25$	2020.06	0.646419
$26$	5009.58	1.45334
$27$	0	0
$28$	0	0
$29$	753.456	0.166365	0.0831827	−	0.996534i	$-0.473492\pi$
$29$	753.456	0.166365	0.0831827	+	0.996534i	$0.473492\pi$
$30$	0	0
$31$	8206.42	1.53373	0.766866	−	0.641807i	$-0.221815\pi$
$31$	8206.42	1.53373	0.766866	+	0.641807i	$0.221815\pi$
$32$	−4129.95	−0.712968
$33$	0	0
$34$	−9443.39	−1.40098
$35$	0	0
$36$	0	0
$37$	−2808.66	−0.337284	−0.168642	−	0.985677i	$-0.553938\pi$
$37$	−2808.66	−0.337284	−0.168642	+	0.985677i	$0.553938\pi$
$38$	12850.3	1.44362
$39$	0	0
$40$	16278.9	1.60870
$41$	−245.827	−0.0228387	−0.0114193	−	0.999935i	$-0.503635\pi$
$41$	−245.827	−0.0228387	−0.0114193	+	0.999935i	$0.503635\pi$
$42$	0	0
$43$	−17504.5	−1.44371	−0.721853	−	0.692047i	$-0.756709\pi$
$43$	−17504.5	−1.44371	−0.721853	+	0.692047i	$0.756709\pi$
$44$	31489.2	2.45206
$45$	0	0
$46$	−30311.2	−2.11207
$47$	16345.5	1.07933	0.539663	−	0.841881i	$-0.318551\pi$
$47$	16345.5	1.07933	0.539663	+	0.841881i	$0.318551\pi$
$48$	0	0
$49$	0	0
$50$	18968.1	1.07300
$51$	0	0
$52$	29967.0	1.53686
$53$	29641.7	1.44948	0.724741	−	0.689021i	$-0.241959\pi$
$53$	29641.7	1.44948	0.724741	+	0.689021i	$0.241959\pi$
$54$	0	0
$55$	40212.0	1.79246
$56$	0	0
$57$	0	0
$58$	7074.85	0.276151
$59$	10356.1	0.387317	0.193659	−	0.981069i	$-0.437965\pi$
$59$	10356.1	0.387317	0.193659	+	0.981069i	$0.437965\pi$
$60$	0	0
$61$	954.179	0.0328326	0.0164163	−	0.999865i	$-0.494774\pi$
$61$	954.179	0.0328326	0.0164163	+	0.999865i	$0.494774\pi$
$62$	77057.2	2.54586
$63$	0	0
$64$	−49454.8	−1.50924
$65$	38268.1	1.12345
$66$	0	0
$67$	−19815.2	−0.539276	−0.269638	−	0.962962i	$-0.586904\pi$
$67$	−19815.2	−0.539276	−0.269638	+	0.962962i	$0.586904\pi$
$68$	−56489.8	−1.48149
$69$	0	0
$70$	0	0
$71$	−62125.4	−1.46259	−0.731296	−	0.682060i	$-0.761084\pi$
$71$	−62125.4	−1.46259	−0.731296	+	0.682060i	$0.761084\pi$
$72$	0	0
$73$	27109.6	0.595410	0.297705	−	0.954658i	$-0.403779\pi$
$73$	27109.6	0.595410	0.297705	+	0.954658i	$0.403779\pi$
$74$	−26373.0	−0.559861
$75$	0	0
$76$	76869.6	1.52658
$77$	0	0
$78$	0	0
$79$	44687.4	0.805595	0.402798	−	0.915289i	$-0.368038\pi$
$79$	44687.4	0.805595	0.402798	+	0.915289i	$0.368038\pi$
$80$	23928.6	0.418015
$81$	0	0
$82$	−2308.29	−0.0379101
$83$	−15606.6	−0.248665	−0.124332	−	0.992241i	$-0.539679\pi$
$83$	−15606.6	−0.248665	−0.124332	+	0.992241i	$0.539679\pi$
$84$	0	0
$85$	−72137.9	−1.08297
$86$	−164365.	−2.39642
$87$	0	0
$88$	127230.	1.75139
$89$	−13635.3	−0.182469	−0.0912347	−	0.995829i	$-0.529081\pi$
$89$	−13635.3	−0.182469	−0.0912347	+	0.995829i	$0.529081\pi$
$90$	0	0
$91$	0	0
$92$	−181320.	−2.23345
$93$	0	0
$94$	153482.	1.79159
$95$	98163.1	1.11594
$96$	0	0
$97$	−12919.5	−0.139417	−0.0697086	−	0.997567i	$-0.522207\pi$
$97$	−12919.5	−0.139417	−0.0697086	+	0.997567i	$0.522207\pi$
$98$	0	0
$99$	0	0
$100$	113466.	1.13466
$101$	−25142.9	−0.245252	−0.122626	−	0.992453i	$-0.539132\pi$
$101$	−25142.9	−0.245252	−0.122626	+	0.992453i	$0.539132\pi$
$102$	0	0
$103$	−160753.	−1.49302	−0.746511	−	0.665373i	$-0.768273\pi$
$103$	−160753.	−1.49302	−0.746511	+	0.665373i	$0.768273\pi$
$104$	121079.	1.09771
$105$	0	0
$106$	278331.	2.40601
$107$	94375.5	0.796893	0.398446	−	0.917192i	$-0.369549\pi$
$107$	94375.5	0.796893	0.398446	+	0.917192i	$0.369549\pi$
$108$	0	0
$109$	−83393.5	−0.672304	−0.336152	−	0.941808i	$-0.609126\pi$
$109$	−83393.5	−0.672304	−0.336152	+	0.941808i	$0.609126\pi$
$110$	377586.	2.97532
$111$	0	0
$112$	0	0
$113$	179254.	1.32060	0.660301	−	0.751001i	$-0.270429\pi$
$113$	179254.	1.32060	0.660301	+	0.751001i	$0.270429\pi$
$114$	0	0
$115$	−231547.	−1.63266
$116$	42321.3	0.292021
$117$	0	0
$118$	97242.5	0.642911
$119$	0	0
$120$	0	0
$121$	153233.	0.951455
$122$	8959.61	0.0544991
$123$	0	0
$124$	460951.	2.69216
$125$	−79256.4	−0.453690
$126$	0	0
$127$	−143674.	−0.790440	−0.395220	−	0.918586i	$-0.629332\pi$
$127$	−143674.	−0.790440	−0.395220	+	0.918586i	$0.629332\pi$
$128$	−332215.	−1.79223
$129$	0	0
$130$	359332.	1.86482
$131$	−52289.9	−0.266219	−0.133110	−	0.991101i	$-0.542496\pi$
$131$	−52289.9	−0.266219	−0.133110	+	0.991101i	$0.542496\pi$
$132$	0	0
$133$	0	0
$134$	−186062.	−0.895150
$135$	0	0
$136$	−228243.	−1.05816
$137$	−9410.10	−0.0428344	−0.0214172	−	0.999771i	$-0.506818\pi$
$137$	−9410.10	−0.0428344	−0.0214172	+	0.999771i	$0.506818\pi$
$138$	0	0
$139$	183094.	0.803781	0.401890	−	0.915688i	$-0.368353\pi$
$139$	183094.	0.803781	0.401890	+	0.915688i	$0.368353\pi$
$140$	0	0
$141$	0	0
$142$	−583349.	−2.42777
$143$	299090.	1.22310
$144$	0	0
$145$	54044.7	0.213468
$146$	254556.	0.988328
$147$	0	0
$148$	−157762.	−0.592034
$149$	167002.	0.616247	0.308123	−	0.951346i	$-0.400299\pi$
$149$	167002.	0.616247	0.308123	+	0.951346i	$0.400299\pi$
$150$	0	0
$151$	376264.	1.34292	0.671461	−	0.741040i	$-0.265667\pi$
$151$	376264.	1.34292	0.671461	+	0.741040i	$0.265667\pi$
$152$	310586.	1.09037
$153$	0	0
$154$	0	0
$155$	588639.	1.96797
$156$	0	0
$157$	39075.0	0.126517	0.0632587	−	0.997997i	$-0.479851\pi$
$157$	39075.0	0.126517	0.0632587	+	0.997997i	$0.479851\pi$
$158$	419608.	1.33722
$159$	0	0
$160$	−296237.	−0.914829
$161$	0	0
$162$	0	0
$163$	−477919.	−1.40892	−0.704458	−	0.709745i	$-0.748810\pi$
$163$	−477919.	−1.40892	−0.704458	+	0.709745i	$0.748810\pi$
$164$	−13808.0	−0.0400887
$165$	0	0
$166$	−146544.	−0.412761
$167$	−39793.4	−0.110413	−0.0552064	−	0.998475i	$-0.517582\pi$
$167$	−39793.4	−0.110413	−0.0552064	+	0.998475i	$0.517582\pi$
$168$	0	0
$169$	−86661.4	−0.233404
$170$	−677366.	−1.79763
$171$	0	0
$172$	−983221.	−2.53414
$173$	−48338.2	−0.122794	−0.0613968	−	0.998113i	$-0.519555\pi$
$173$	−48338.2	−0.122794	−0.0613968	+	0.998113i	$0.519555\pi$
$174$	0	0
$175$	0	0
$176$	187018.	0.455094
$177$	0	0
$178$	−128034.	−0.302883
$179$	142911.	0.333374	0.166687	−	0.986010i	$-0.446693\pi$
$179$	142911.	0.333374	0.166687	+	0.986010i	$0.446693\pi$
$180$	0	0
$181$	−77245.3	−0.175257	−0.0876285	−	0.996153i	$-0.527929\pi$
$181$	−77245.3	−0.175257	−0.0876285	+	0.996153i	$0.527929\pi$
$182$	0	0
$183$	0	0
$184$	−732610.	−1.59525
$185$	−201463.	−0.432778
$186$	0	0
$187$	−563806.	−1.17903
$188$	918119.	1.89454
$189$	0	0
$190$	921739.	1.85235
$191$	−272054.	−0.539600	−0.269800	−	0.962916i	$-0.586958\pi$
$191$	−272054.	−0.539600	−0.269800	+	0.962916i	$0.586958\pi$
$192$	0	0
$193$	−16033.8	−0.0309844	−0.0154922	−	0.999880i	$-0.504932\pi$
$193$	−16033.8	−0.0309844	−0.0154922	+	0.999880i	$0.504932\pi$
$194$	−121312.	−0.231420
$195$	0	0
$196$	0	0
$197$	−1.03228e6	−1.89510	−0.947552	−	0.319603i	$-0.896451\pi$
$197$	−1.03228e6	−1.89510	−0.947552	+	0.319603i	$0.896451\pi$
$198$	0	0
$199$	−881736.	−1.57836	−0.789180	−	0.614162i	$-0.789494\pi$
$199$	−881736.	−1.57836	−0.789180	+	0.614162i	$0.789494\pi$
$200$	458451.	0.810435
$201$	0	0
$202$	−236089.	−0.407096
$203$	0	0
$204$	0	0
$205$	−17633.0	−0.0293049
$206$	−1.50945e6	−2.47828
$207$	0	0
$208$	177977.	0.285237
$209$	767210.	1.21492
$210$	0	0
$211$	−372813.	−0.576480	−0.288240	−	0.957558i	$-0.593070\pi$
$211$	−372813.	−0.576480	−0.288240	+	0.957558i	$0.593070\pi$
$212$	1.66496e6	2.54428
$213$	0	0
$214$	886174.	1.32277
$215$	−1.25558e6	−1.85246
$216$	0	0
$217$	0	0
$218$	−783054.	−1.11596
$219$	0	0
$220$	2.25869e6	3.14630
$221$	−536550.	−0.738975
$222$	0	0
$223$	−1.08205e6	−1.45708	−0.728541	−	0.685002i	$-0.759801\pi$
$223$	−1.08205e6	−1.45708	−0.728541	+	0.685002i	$0.759801\pi$
$224$	0	0
$225$	0	0
$226$	1.68317e6	2.19208
$227$	−553049.	−0.712359	−0.356179	−	0.934418i	$-0.615921\pi$
$227$	−553049.	−0.712359	−0.356179	+	0.934418i	$0.615921\pi$
$228$	0	0
$229$	523024.	0.659072	0.329536	−	0.944143i	$-0.393108\pi$
$229$	523024.	0.659072	0.329536	+	0.944143i	$0.393108\pi$
$230$	−2.17420e6	−2.71006
$231$	0	0
$232$	170996.	0.208577
$233$	364181.	0.439468	0.219734	−	0.975560i	$-0.429481\pi$
$233$	364181.	0.439468	0.219734	+	0.975560i	$0.429481\pi$
$234$	0	0
$235$	1.17245e6	1.38492
$236$	581698.	0.679858
$237$	0	0
$238$	0	0
$239$	−371841.	−0.421078	−0.210539	−	0.977585i	$-0.567522\pi$
$239$	−371841.	−0.421078	−0.210539	+	0.977585i	$0.567522\pi$
$240$	0	0
$241$	−1.71147e6	−1.89814	−0.949069	−	0.315067i	$-0.897973\pi$
$241$	−1.71147e6	−1.89814	−0.949069	+	0.315067i	$0.897973\pi$
$242$	1.43883e6	1.57933
$243$	0	0
$244$	53595.8	0.0576310
$245$	0	0
$246$	0	0
$247$	730121.	0.761469
$248$	1.86244e6	1.92289
$249$	0	0
$250$	−744207.	−0.753084
$251$	−58134.1	−0.0582434	−0.0291217	−	0.999576i	$-0.509271\pi$
$251$	−58134.1	−0.0582434	−0.0291217	+	0.999576i	$0.509271\pi$
$252$	0	0
$253$	−1.80969e6	−1.77748
$254$	−1.34908e6	−1.31206
$255$	0	0
$256$	−1.53691e6	−1.46571
$257$	311839.	0.294509	0.147254	−	0.989099i	$-0.452956\pi$
$257$	311839.	0.294509	0.147254	+	0.989099i	$0.452956\pi$
$258$	0	0
$259$	0	0
$260$	2.14950e6	1.97199
$261$	0	0
$262$	−490995.	−0.441900
$263$	−863965.	−0.770206	−0.385103	−	0.922874i	$-0.625834\pi$
$263$	−863965.	−0.770206	−0.385103	+	0.922874i	$0.625834\pi$
$264$	0	0
$265$	2.12617e6	1.85987
$266$	0	0
$267$	0	0
$268$	−1.11301e6	−0.946592
$269$	1.12069e6	0.944290	0.472145	−	0.881521i	$-0.343480\pi$
$269$	1.12069e6	0.944290	0.472145	+	0.881521i	$0.343480\pi$
$270$	0	0
$271$	1.14012e6	0.943030	0.471515	−	0.881858i	$-0.343707\pi$
$271$	1.14012e6	0.943030	0.471515	+	0.881858i	$0.343707\pi$
$272$	−335498.	−0.274959
$273$	0	0
$274$	−88359.6	−0.0711013
$275$	1.13247e6	0.903012
$276$	0	0
$277$	−1.98801e6	−1.55675	−0.778375	−	0.627799i	$-0.783956\pi$
$277$	−1.98801e6	−1.55675	−0.778375	+	0.627799i	$0.783956\pi$
$278$	1.71923e6	1.33420
$279$	0	0
$280$	0	0
$281$	−532321.	−0.402168	−0.201084	−	0.979574i	$-0.564446\pi$
$281$	−532321.	−0.402168	−0.201084	+	0.979574i	$0.564446\pi$
$282$	0	0
$283$	2.62473e6	1.94813	0.974067	−	0.226259i	$-0.0726497\pi$
$283$	2.62473e6	1.94813	0.974067	+	0.226259i	$0.0726497\pi$
$284$	−3.48956e6	−2.56729
$285$	0	0
$286$	2.80842e6	2.03024
$287$	0	0
$288$	0	0
$289$	−408424.	−0.287651
$290$	507473.	0.354338
$291$	0	0
$292$	1.52274e6	1.04512
$293$	−609962.	−0.415082	−0.207541	−	0.978226i	$-0.566546\pi$
$293$	−609962.	−0.415082	−0.207541	+	0.978226i	$0.566546\pi$
$294$	0	0
$295$	742834.	0.496978
$296$	−637424.	−0.422863
$297$	0	0
$298$	1.56812e6	1.02291
$299$	−1.72221e6	−1.11406
$300$	0	0
$301$	0	0
$302$	3.53307e6	2.22913
$303$	0	0
$304$	456536.	0.283329
$305$	68442.3	0.0421284
$306$	0	0
$307$	−1.34843e6	−0.816551	−0.408275	−	0.912859i	$-0.633870\pi$
$307$	−1.34843e6	−0.816551	−0.408275	+	0.912859i	$0.633870\pi$
$308$	0	0
$309$	0	0
$310$	5.52724e6	3.26666
$311$	3.33061e6	1.95264	0.976320	−	0.216330i	$-0.0694087\pi$
$311$	3.33061e6	1.95264	0.976320	+	0.216330i	$0.0694087\pi$
$312$	0	0
$313$	−2.83670e6	−1.63664	−0.818320	−	0.574763i	$-0.805094\pi$
$313$	−2.83670e6	−1.63664	−0.818320	+	0.574763i	$0.805094\pi$
$314$	366909.	0.210007
$315$	0	0
$316$	2.51007e6	1.41406
$317$	1.08839e6	0.608326	0.304163	−	0.952620i	$-0.401623\pi$
$317$	1.08839e6	0.608326	0.304163	+	0.952620i	$0.401623\pi$
$318$	0	0
$319$	422395.	0.232403
$320$	−3.54734e6	−1.93655
$321$	0	0
$322$	0	0
$323$	−1.37633e6	−0.734033
$324$	0	0
$325$	1.07772e6	0.565975
$326$	−4.48760e6	−2.33867
$327$	0	0
$328$	−55790.4	−0.0286335
$329$	0	0
$330$	0	0
$331$	1.30555e6	0.654971	0.327485	−	0.944856i	$-0.393799\pi$
$331$	1.30555e6	0.654971	0.327485	+	0.944856i	$0.393799\pi$
$332$	−876619.	−0.436481
$333$	0	0
$334$	−373654.	−0.183275
$335$	−1.42133e6	−0.691961
$336$	0	0
$337$	−3.17016e6	−1.52057	−0.760285	−	0.649590i	$-0.774941\pi$
$337$	−3.17016e6	−1.52057	−0.760285	+	0.649590i	$0.774941\pi$
$338$	−813739.	−0.387430
$339$	0	0
$340$	−4.05196e6	−1.90094
$341$	4.60060e6	2.14254
$342$	0	0
$343$	0	0
$344$	−3.97263e6	−1.81002
$345$	0	0
$346$	−453889.	−0.203826
$347$	1.71592e6	0.765019	0.382510	−	0.923951i	$-0.375060\pi$
$347$	1.71592e6	0.765019	0.382510	+	0.923951i	$0.375060\pi$
$348$	0	0
$349$	−2.95822e6	−1.30007	−0.650034	−	0.759905i	$-0.725245\pi$
$349$	−2.95822e6	−1.30007	−0.650034	+	0.759905i	$0.725245\pi$
$350$	0	0
$351$	0	0
$352$	−2.31529e6	−0.995976
$353$	−3.76980e6	−1.61021	−0.805103	−	0.593135i	$-0.797890\pi$
$353$	−3.76980e6	−1.61021	−0.805103	+	0.593135i	$0.797890\pi$
$354$	0	0
$355$	−4.45620e6	−1.87669
$356$	−765890.	−0.320289
$357$	0	0
$358$	1.34191e6	0.553371
$359$	−1.92987e6	−0.790300	−0.395150	−	0.918617i	$-0.629307\pi$
$359$	−1.92987e6	−0.790300	−0.395150	+	0.918617i	$0.629307\pi$
$360$	0	0
$361$	−603234.	−0.243623
$362$	−725323.	−0.290911
$363$	0	0
$364$	0	0
$365$	1.94455e6	0.763988
$366$	0	0
$367$	2.36742e6	0.917509	0.458754	−	0.888563i	$-0.348296\pi$
$367$	2.36742e6	0.917509	0.458754	+	0.888563i	$0.348296\pi$
$368$	−1.07688e6	−0.414521
$369$	0	0
$370$	−1.89171e6	−0.718373
$371$	0	0
$372$	0	0
$373$	3.53829e6	1.31680	0.658402	−	0.752666i	$-0.271233\pi$
$373$	3.53829e6	1.31680	0.658402	+	0.752666i	$0.271233\pi$
$374$	−5.29406e6	−1.95709
$375$	0	0
$376$	3.70960e6	1.35318
$377$	401975.	0.145662
$378$	0	0
$379$	1.79847e6	0.643139	0.321569	−	0.946886i	$-0.395790\pi$
$379$	1.79847e6	0.643139	0.321569	+	0.946886i	$0.395790\pi$
$380$	5.51378e6	1.95880
$381$	0	0
$382$	−2.55455e6	−0.895686
$383$	−2.60815e6	−0.908521	−0.454261	−	0.890869i	$-0.650096\pi$
$383$	−2.60815e6	−0.908521	−0.454261	+	0.890869i	$0.650096\pi$
$384$	0	0
$385$	0	0
$386$	−150555.	−0.0514313
$387$	0	0
$388$	−725683.	−0.244719
$389$	2.82995e6	0.948211	0.474106	−	0.880468i	$-0.342772\pi$
$389$	2.82995e6	0.948211	0.474106	+	0.880468i	$0.342772\pi$
$390$	0	0
$391$	3.24648e6	1.07392
$392$	0	0
$393$	0	0
$394$	−9.69299e6	−3.14570
$395$	3.20538e6	1.03368
$396$	0	0
$397$	−2.43062e6	−0.773999	−0.387000	−	0.922080i	$-0.626489\pi$
$397$	−2.43062e6	−0.773999	−0.387000	+	0.922080i	$0.626489\pi$
$398$	−8.27939e6	−2.61993
$399$	0	0
$400$	673885.	0.210589
$401$	2.43184e6	0.755222	0.377611	−	0.925964i	$-0.376746\pi$
$401$	2.43184e6	0.755222	0.377611	+	0.925964i	$0.376746\pi$
$402$	0	0
$403$	4.37820e6	1.34287
$404$	−1.41227e6	−0.430491
$405$	0	0
$406$	0	0
$407$	−1.57457e6	−0.471167
$408$	0	0
$409$	−4.77466e6	−1.41135	−0.705674	−	0.708537i	$-0.749356\pi$
$409$	−4.77466e6	−1.41135	−0.705674	+	0.708537i	$0.749356\pi$
$410$	−165571.	−0.0486436
$411$	0	0
$412$	−9.02944e6	−2.62070
$413$	0	0
$414$	0	0
$415$	−1.11945e6	−0.319069
$416$	−2.20336e6	−0.624242
$417$	0	0
$418$	7.20400e6	2.01666
$419$	457181.	0.127219	0.0636097	−	0.997975i	$-0.479739\pi$
$419$	457181.	0.127219	0.0636097	+	0.997975i	$0.479739\pi$
$420$	0	0
$421$	−1.82396e6	−0.501545	−0.250773	−	0.968046i	$-0.580685\pi$
$421$	−1.82396e6	−0.501545	−0.250773	+	0.968046i	$0.580685\pi$
$422$	−3.50066e6	−0.956905
$423$	0	0
$424$	6.72715e6	1.81726
$425$	−2.03157e6	−0.545582
$426$	0	0
$427$	0	0
$428$	5.30104e6	1.39879
$429$	0	0
$430$	−1.17897e7	−3.07492
$431$	3.38249e6	0.877087	0.438544	−	0.898710i	$-0.355494\pi$
$431$	3.38249e6	0.877087	0.438544	+	0.898710i	$0.355494\pi$
$432$	0	0
$433$	−285266.	−0.0731190	−0.0365595	−	0.999331i	$-0.511640\pi$
$433$	−285266.	−0.0731190	−0.0365595	+	0.999331i	$0.511640\pi$
$434$	0	0
$435$	0	0
$436$	−4.68418e6	−1.18010
$437$	−4.41771e6	−1.10661
$438$	0	0
$439$	4.35220e6	1.07782	0.538911	−	0.842363i	$-0.318836\pi$
$439$	4.35220e6	1.07782	0.538911	+	0.842363i	$0.318836\pi$
$440$	9.12610e6	2.24726
$441$	0	0
$442$	−5.03813e6	−1.22663
$443$	5.10560e6	1.23605	0.618027	−	0.786157i	$-0.287932\pi$
$443$	5.10560e6	1.23605	0.618027	+	0.786157i	$0.287932\pi$
$444$	0	0
$445$	−978049.	−0.234132
$446$	−1.01603e7	−2.41862
$447$	0	0
$448$	0	0
$449$	−3.04163e6	−0.712016	−0.356008	−	0.934483i	$-0.615862\pi$
$449$	−3.04163e6	−0.712016	−0.356008	+	0.934483i	$0.615862\pi$
$450$	0	0
$451$	−137813.	−0.0319043
$452$	1.00686e7	2.31805
$453$	0	0
$454$	−5.19305e6	−1.18245
$455$	0	0
$456$	0	0
$457$	−1.74432e6	−0.390694	−0.195347	−	0.980734i	$-0.562583\pi$
$457$	−1.74432e6	−0.390694	−0.195347	+	0.980734i	$0.562583\pi$
$458$	4.91113e6	1.09400
$459$	0	0
$460$	−1.30059e7	−2.86580
$461$	6.85701e6	1.50273	0.751367	−	0.659884i	$-0.229395\pi$
$461$	6.85701e6	1.50273	0.751367	+	0.659884i	$0.229395\pi$
$462$	0	0
$463$	5.13844e6	1.11398	0.556992	−	0.830518i	$-0.311955\pi$
$463$	5.13844e6	1.11398	0.556992	+	0.830518i	$0.311955\pi$
$464$	251351.	0.0541982
$465$	0	0
$466$	3.41961e6	0.729477
$467$	4.58171e6	0.972154	0.486077	−	0.873916i	$-0.338427\pi$
$467$	4.58171e6	0.972154	0.486077	+	0.873916i	$0.338427\pi$
$468$	0	0
$469$	0	0
$470$	1.10091e7	2.29883
$471$	0	0
$472$	2.35031e6	0.485591
$473$	−9.81320e6	−2.01678
$474$	0	0
$475$	2.76450e6	0.562190
$476$	0	0
$477$	0	0
$478$	−3.49154e6	−0.698952
$479$	288523.	0.0574568	0.0287284	−	0.999587i	$-0.490854\pi$
$479$	288523.	0.0574568	0.0287284	+	0.999587i	$0.490854\pi$
$480$	0	0
$481$	−1.49845e6	−0.295310
$482$	−1.60705e7	−3.15074
$483$	0	0
$484$	8.60702e6	1.67009
$485$	−926703.	−0.178890
$486$	0	0
$487$	−7.81685e6	−1.49351	−0.746757	−	0.665097i	$-0.768390\pi$
$487$	−7.81685e6	−1.49351	−0.746757	+	0.665097i	$0.768390\pi$
$488$	216550.	0.0411632
$489$	0	0
$490$	0	0
$491$	−3.14467e6	−0.588669	−0.294335	−	0.955702i	$-0.595098\pi$
$491$	−3.14467e6	−0.588669	−0.294335	+	0.955702i	$0.595098\pi$
$492$	0	0
$493$	−757751.	−0.140414
$494$	6.85574e6	1.26397
$495$	0	0
$496$	2.73763e6	0.499656
$497$	0	0
$498$	0	0
$499$	6.72563e6	1.20915	0.604577	−	0.796547i	$-0.293342\pi$
$499$	6.72563e6	1.20915	0.604577	+	0.796547i	$0.293342\pi$
$500$	−4.45180e6	−0.796362
$501$	0	0
$502$	−545871.	−0.0966787
$503$	9.45056e6	1.66547	0.832737	−	0.553669i	$-0.186773\pi$
$503$	9.45056e6	1.66547	0.832737	+	0.553669i	$0.186773\pi$
$504$	0	0
$505$	−1.80348e6	−0.314690
$506$	−1.69928e7	−2.95045
$507$	0	0
$508$	−8.07011e6	−1.38746
$509$	8.83702e6	1.51186	0.755930	−	0.654653i	$-0.227185\pi$
$509$	8.83702e6	1.51186	0.755930	+	0.654653i	$0.227185\pi$
$510$	0	0
$511$	0	0
$512$	−3.80044e6	−0.640707
$513$	0	0
$514$	2.92813e6	0.488858
$515$	−1.15307e7	−1.91574
$516$	0	0
$517$	9.16344e6	1.50776
$518$	0	0
$519$	0	0
$520$	8.68492e6	1.40850
$521$	−694985.	−0.112171	−0.0560855	−	0.998426i	$-0.517862\pi$
$521$	−694985.	−0.112171	−0.0560855	+	0.998426i	$0.517862\pi$
$522$	0	0
$523$	3.58210e6	0.572643	0.286321	−	0.958134i	$-0.407568\pi$
$523$	3.58210e6	0.572643	0.286321	+	0.958134i	$0.407568\pi$
$524$	−2.93710e6	−0.467294
$525$	0	0
$526$	−8.11252e6	−1.27847
$527$	−8.25320e6	−1.29448
$528$	0	0
$529$	3.98415e6	0.619009
$530$	1.99644e7	3.08722
$531$	0	0
$532$	0	0
$533$	−131151.	−0.0199965
$534$	0	0
$535$	6.76947e6	1.02252
$536$	−4.49705e6	−0.676107
$537$	0	0
$538$	1.05231e7	1.56744
$539$	0	0
$540$	0	0
$541$	−5.16846e6	−0.759220	−0.379610	−	0.925147i	$-0.623942\pi$
$541$	−5.16846e6	−0.759220	−0.379610	+	0.925147i	$0.623942\pi$
$542$	1.07055e7	1.56535
$543$	0	0
$544$	4.15349e6	0.601750
$545$	−5.98174e6	−0.862653
$546$	0	0
$547$	8.47489e6	1.21106	0.605530	−	0.795822i	$-0.292961\pi$
$547$	8.47489e6	1.21106	0.605530	+	0.795822i	$0.292961\pi$
$548$	−528562.	−0.0751873
$549$	0	0
$550$	1.06337e7	1.49892
$551$	1.03112e6	0.144688
$552$	0	0
$553$	0	0
$554$	−1.86671e7	−2.58407
$555$	0	0
$556$	1.02843e7	1.41088
$557$	1.04213e7	1.42325	0.711627	−	0.702557i	$-0.247959\pi$
$557$	1.04213e7	1.42325	0.711627	+	0.702557i	$0.247959\pi$
$558$	0	0
$559$	−9.33880e6	−1.26404
$560$	0	0
$561$	0	0
$562$	−4.99842e6	−0.667562
$563$	7.24077e6	0.962751	0.481375	−	0.876515i	$-0.340137\pi$
$563$	7.24077e6	0.962751	0.481375	+	0.876515i	$0.340137\pi$
$564$	0	0
$565$	1.28577e7	1.69450
$566$	2.46459e7	3.23373
$567$	0	0
$568$	−1.40993e7	−1.83369
$569$	−916536.	−0.118678	−0.0593388	−	0.998238i	$-0.518899\pi$
$569$	−916536.	−0.118678	−0.0593388	+	0.998238i	$0.518899\pi$
$570$	0	0
$571$	1.00708e7	1.29262	0.646312	−	0.763073i	$-0.276311\pi$
$571$	1.00708e7	1.29262	0.646312	+	0.763073i	$0.276311\pi$
$572$	1.67998e7	2.14691
$573$	0	0
$574$	0	0
$575$	−6.52091e6	−0.822505
$576$	0	0
$577$	1.16997e7	1.46298	0.731488	−	0.681855i	$-0.238826\pi$
$577$	1.16997e7	1.46298	0.731488	+	0.681855i	$0.238826\pi$
$578$	−3.83505e6	−0.477475
$579$	0	0
$580$	3.03567e6	0.374701
$581$	0	0
$582$	0	0
$583$	1.66174e7	2.02485
$584$	6.15251e6	0.746484
$585$	0	0
$586$	−5.72747e6	−0.688999
$587$	6.92367e6	0.829357	0.414678	−	0.909968i	$-0.363894\pi$
$587$	6.92367e6	0.829357	0.414678	+	0.909968i	$0.363894\pi$
$588$	0	0
$589$	1.12307e7	1.33389
$590$	6.97511e6	0.824938
$591$	0	0
$592$	−936961.	−0.109880
$593$	−1.57770e7	−1.84242	−0.921208	−	0.389069i	$-0.872797\pi$
$593$	−1.57770e7	−1.84242	−0.921208	+	0.389069i	$0.872797\pi$
$594$	0	0
$595$	0	0
$596$	9.38041e6	1.08170
$597$	0	0
$598$	−1.61713e7	−1.84924
$599$	−1.42708e7	−1.62511	−0.812553	−	0.582887i	$-0.801923\pi$
$599$	−1.42708e7	−1.62511	−0.812553	+	0.582887i	$0.801923\pi$
$600$	0	0
$601$	7.63222e6	0.861916	0.430958	−	0.902372i	$-0.358176\pi$
$601$	7.63222e6	0.861916	0.430958	+	0.902372i	$0.358176\pi$
$602$	0	0
$603$	0	0
$604$	2.11346e7	2.35723
$605$	1.09912e7	1.22084
$606$	0	0
$607$	−3.56035e6	−0.392212	−0.196106	−	0.980583i	$-0.562830\pi$
$607$	−3.56035e6	−0.392212	−0.196106	+	0.980583i	$0.562830\pi$
$608$	−5.65194e6	−0.620067
$609$	0	0
$610$	642664.	0.0699294
$611$	8.72046e6	0.945010
$612$	0	0
$613$	−1.37284e7	−1.47560	−0.737800	−	0.675019i	$-0.764135\pi$
$613$	−1.37284e7	−1.47560	−0.737800	+	0.675019i	$0.764135\pi$
$614$	−1.26616e7	−1.35540
$615$	0	0
$616$	0	0
$617$	6.51173e6	0.688626	0.344313	−	0.938855i	$-0.388112\pi$
$617$	6.51173e6	0.688626	0.344313	+	0.938855i	$0.388112\pi$
$618$	0	0
$619$	8.85110e6	0.928476	0.464238	−	0.885711i	$-0.346328\pi$
$619$	8.85110e6	0.928476	0.464238	+	0.885711i	$0.346328\pi$
$620$	3.30636e7	3.45439
$621$	0	0
$622$	3.12739e7	3.24121
$623$	0	0
$624$	0	0
$625$	−1.19977e7	−1.22856
$626$	−2.66363e7	−2.71667
$627$	0	0
$628$	2.19483e6	0.222076
$629$	2.82467e6	0.284670
$630$	0	0
$631$	−6.89663e6	−0.689546	−0.344773	−	0.938686i	$-0.612044\pi$
$631$	−6.89663e6	−0.689546	−0.344773	+	0.938686i	$0.612044\pi$
$632$	1.01418e7	1.01000
$633$	0	0
$634$	1.02198e7	1.00977
$635$	−1.03056e7	−1.01424
$636$	0	0
$637$	0	0
$638$	3.96623e6	0.385768
$639$	0	0
$640$	−2.38295e7	−2.29967
$641$	1.66695e7	1.60242	0.801210	−	0.598383i	$-0.204190\pi$
$641$	1.66695e7	1.60242	0.801210	+	0.598383i	$0.204190\pi$
$642$	0	0
$643$	−1.28697e7	−1.22756	−0.613779	−	0.789478i	$-0.710351\pi$
$643$	−1.28697e7	−1.22756	−0.613779	+	0.789478i	$0.710351\pi$
$644$	0	0
$645$	0	0
$646$	−1.29235e7	−1.21843
$647$	1.14731e7	1.07751	0.538754	−	0.842463i	$-0.318895\pi$
$647$	1.14731e7	1.07751	0.538754	+	0.842463i	$0.318895\pi$
$648$	0	0
$649$	5.80574e6	0.541060
$650$	1.01196e7	0.939467
$651$	0	0
$652$	−2.68445e7	−2.47307
$653$	1.31801e7	1.20958	0.604791	−	0.796384i	$-0.293257\pi$
$653$	1.31801e7	1.20958	0.604791	+	0.796384i	$0.293257\pi$
$654$	0	0
$655$	−3.75070e6	−0.341593
$656$	−82007.2	−0.00744034
$657$	0	0
$658$	0	0
$659$	1.13068e7	1.01421	0.507104	−	0.861885i	$-0.330716\pi$
$659$	1.13068e7	1.01421	0.507104	+	0.861885i	$0.330716\pi$
$660$	0	0
$661$	2.20319e6	0.196132	0.0980661	−	0.995180i	$-0.468734\pi$
$661$	2.20319e6	0.196132	0.0980661	+	0.995180i	$0.468734\pi$
$662$	1.22589e7	1.08719
$663$	0	0
$664$	−3.54192e6	−0.311758
$665$	0	0
$666$	0	0
$667$	−2.43222e6	−0.211684
$668$	−2.23518e6	−0.193808
$669$	0	0
$670$	−1.33461e7	−1.14859
$671$	534922.	0.0458653
$672$	0	0
$673$	−1.89787e7	−1.61521	−0.807606	−	0.589723i	$-0.799237\pi$
$673$	−1.89787e7	−1.61521	−0.807606	+	0.589723i	$0.799237\pi$
$674$	−2.97674e7	−2.52401
$675$	0	0
$676$	−4.86773e6	−0.409694
$677$	1.96475e7	1.64754	0.823771	−	0.566923i	$-0.191866\pi$
$677$	1.96475e7	1.64754	0.823771	+	0.566923i	$0.191866\pi$
$678$	0	0
$679$	0	0
$680$	−1.63717e7	−1.35775
$681$	0	0
$682$	4.31990e7	3.55642
$683$	−1.62705e7	−1.33459	−0.667295	−	0.744793i	$-0.732548\pi$
$683$	−1.62705e7	−1.33459	−0.667295	+	0.744793i	$0.732548\pi$
$684$	0	0
$685$	−674978.	−0.0549621
$686$	0	0
$687$	0	0
$688$	−5.83944e6	−0.470328
$689$	1.58141e7	1.26910
$690$	0	0
$691$	1.94467e7	1.54935	0.774677	−	0.632357i	$-0.217912\pi$
$691$	1.94467e7	1.54935	0.774677	+	0.632357i	$0.217912\pi$
$692$	−2.71514e6	−0.215539
$693$	0	0
$694$	1.61122e7	1.26986
$695$	1.31332e7	1.03135
$696$	0	0
$697$	247229.	0.0192760
$698$	−2.77772e7	−2.15800
$699$	0	0
$700$	0	0
$701$	−1.49625e7	−1.15003	−0.575014	−	0.818144i	$-0.695003\pi$
$701$	−1.49625e7	−1.15003	−0.575014	+	0.818144i	$0.695003\pi$
$702$	0	0
$703$	−3.84373e6	−0.293335
$704$	−2.77248e7	−2.10832
$705$	0	0
$706$	−3.53979e7	−2.67280
$707$	0	0
$708$	0	0
$709$	1.58639e6	0.118521	0.0592603	−	0.998243i	$-0.481126\pi$
$709$	1.58639e6	0.118521	0.0592603	+	0.998243i	$0.481126\pi$
$710$	−4.18431e7	−3.11514
$711$	0	0
$712$	−3.09453e6	−0.228767
$713$	−2.64910e7	−1.95152
$714$	0	0
$715$	2.14535e7	1.56940
$716$	8.02723e6	0.585172
$717$	0	0
$718$	−1.81212e7	−1.31183
$719$	1.81675e7	1.31061	0.655305	−	0.755364i	$-0.272540\pi$
$719$	1.81675e7	1.31061	0.655305	+	0.755364i	$0.272540\pi$
$720$	0	0
$721$	0	0
$722$	−5.66429e6	−0.404392
$723$	0	0
$724$	−4.33884e6	−0.307629
$725$	1.52203e6	0.107542
$726$	0	0
$727$	1.26903e7	0.890506	0.445253	−	0.895405i	$-0.353114\pi$
$727$	1.26903e7	0.890506	0.445253	+	0.895405i	$0.353114\pi$
$728$	0	0
$729$	0	0
$730$	1.82591e7	1.26815
$731$	1.76043e7	1.21850
$732$	0	0
$733$	−3.11401e6	−0.214072	−0.107036	−	0.994255i	$-0.534136\pi$
$733$	−3.11401e6	−0.214072	−0.107036	+	0.994255i	$0.534136\pi$
$734$	2.22298e7	1.52298
$735$	0	0
$736$	1.33318e7	0.907182
$737$	−1.11086e7	−0.753339
$738$	0	0
$739$	9.09899e6	0.612890	0.306445	−	0.951888i	$-0.400861\pi$
$739$	9.09899e6	0.612890	0.306445	+	0.951888i	$0.400861\pi$
$740$	−1.13161e7	−0.759656
$741$	0	0
$742$	0	0
$743$	8.79218e6	0.584285	0.292142	−	0.956375i	$-0.405632\pi$
$743$	8.79218e6	0.584285	0.292142	+	0.956375i	$0.405632\pi$
$744$	0	0
$745$	1.19789e7	0.790724
$746$	3.32241e7	2.18578
$747$	0	0
$748$	−3.16687e7	−2.06955
$749$	0	0
$750$	0	0
$751$	−2.80918e7	−1.81752	−0.908759	−	0.417320i	$-0.862969\pi$
$751$	−2.80918e7	−1.81752	−0.908759	+	0.417320i	$0.862969\pi$
$752$	5.45280e6	0.351621
$753$	0	0
$754$	3.77450e6	0.241786
$755$	2.69891e7	1.72314
$756$	0	0
$757$	−4.18815e6	−0.265634	−0.132817	−	0.991141i	$-0.542402\pi$
$757$	−4.18815e6	−0.265634	−0.132817	+	0.991141i	$0.542402\pi$
$758$	1.68874e7	1.06755
$759$	0	0
$760$	2.22781e7	1.39908
$761$	393182.	0.0246111	0.0123056	−	0.999924i	$-0.496083\pi$
$761$	393182.	0.0246111	0.0123056	+	0.999924i	$0.496083\pi$
$762$	0	0
$763$	0	0
$764$	−1.52812e7	−0.947159
$765$	0	0
$766$	−2.44901e7	−1.50806
$767$	5.52508e6	0.339117
$768$	0	0
$769$	1.57517e7	0.960530	0.480265	−	0.877124i	$-0.340541\pi$
$769$	1.57517e7	0.960530	0.480265	+	0.877124i	$0.340541\pi$
$770$	0	0
$771$	0	0
$772$	−900611.	−0.0543869
$773$	−2.88472e6	−0.173642	−0.0868210	−	0.996224i	$-0.527671\pi$
$773$	−2.88472e6	−0.173642	−0.0868210	+	0.996224i	$0.527671\pi$
$774$	0	0
$775$	1.65775e7	0.991433
$776$	−2.93207e6	−0.174791
$777$	0	0
$778$	2.65729e7	1.57394
$779$	−336421.	−0.0198628
$780$	0	0
$781$	−3.48281e7	−2.04316
$782$	3.04840e7	1.78261
$783$	0	0
$784$	0	0
$785$	2.80281e6	0.162338
$786$	0	0
$787$	−2.97866e7	−1.71429	−0.857145	−	0.515076i	$-0.827764\pi$
$787$	−2.97866e7	−1.71429	−0.857145	+	0.515076i	$0.827764\pi$
$788$	−5.79829e7	−3.32647
$789$	0	0
$790$	3.00981e7	1.71582
$791$	0	0
$792$	0	0
$793$	509063.	0.0287467
$794$	−2.28232e7	−1.28477
$795$	0	0
$796$	−4.95268e7	−2.77049
$797$	−8.11321e6	−0.452425	−0.226213	−	0.974078i	$-0.572634\pi$
$797$	−8.11321e6	−0.452425	−0.226213	+	0.974078i	$0.572634\pi$
$798$	0	0
$799$	−1.64387e7	−0.910960
$800$	−8.34274e6	−0.460876
$801$	0	0
$802$	2.28347e7	1.25360
$803$	1.51979e7	0.831756
$804$	0	0
$805$	0	0
$806$	4.11107e7	2.22904
$807$	0	0
$808$	−5.70617e6	−0.307480
$809$	−111597.	−0.00599489	−0.00299744	−	0.999996i	$-0.500954\pi$
$809$	−111597.	−0.00599489	−0.00299744	+	0.999996i	$0.500954\pi$
$810$	0	0
$811$	−209250.	−0.0111716	−0.00558578	−	0.999984i	$-0.501778\pi$
$811$	−209250.	−0.0111716	−0.00558578	+	0.999984i	$0.501778\pi$
$812$	0	0
$813$	0	0
$814$	−1.47850e7	−0.782094
$815$	−3.42807e7	−1.80782
$816$	0	0
$817$	−2.39554e7	−1.25559
$818$	−4.48334e7	−2.34271
$819$	0	0
$820$	−990437.	−0.0514390
$821$	−6.84614e6	−0.354477	−0.177238	−	0.984168i	$-0.556716\pi$
$821$	−6.84614e6	−0.354477	−0.177238	+	0.984168i	$0.556716\pi$
$822$	0	0
$823$	−5.61210e6	−0.288819	−0.144409	−	0.989518i	$-0.546128\pi$
$823$	−5.61210e6	−0.288819	−0.144409	+	0.989518i	$0.546128\pi$
$824$	−3.64828e7	−1.87185
$825$	0	0
$826$	0	0
$827$	2.15868e7	1.09755	0.548776	−	0.835969i	$-0.315094\pi$
$827$	2.15868e7	1.09755	0.548776	+	0.835969i	$0.315094\pi$
$828$	0	0
$829$	1.92427e7	0.972478	0.486239	−	0.873826i	$-0.338368\pi$
$829$	1.92427e7	0.972478	0.486239	+	0.873826i	$0.338368\pi$
$830$	−1.05115e7	−0.529625
$831$	0	0
$832$	−2.63846e7	−1.32142
$833$	0	0
$834$	0	0
$835$	−2.85434e6	−0.141674
$836$	4.30939e7	2.13255
$837$	0	0
$838$	4.29287e6	0.211173
$839$	−2.47678e7	−1.21474	−0.607369	−	0.794420i	$-0.707775\pi$
$839$	−2.47678e7	−1.21474	−0.607369	+	0.794420i	$0.707775\pi$
$840$	0	0
$841$	−1.99435e7	−0.972323
$842$	−1.71267e7	−0.832520
$843$	0	0
$844$	−2.09407e7	−1.01190
$845$	−6.21614e6	−0.299488
$846$	0	0
$847$	0	0
$848$	9.88836e6	0.472210
$849$	0	0
$850$	−1.90762e7	−0.905618
$851$	9.06659e6	0.429161
$852$	0	0
$853$	−2.91267e7	−1.37062	−0.685312	−	0.728250i	$-0.740334\pi$
$853$	−2.91267e7	−1.37062	−0.685312	+	0.728250i	$0.740334\pi$
$854$	0	0
$855$	0	0
$856$	2.14185e7	0.999088
$857$	3.22586e6	0.150035	0.0750175	−	0.997182i	$-0.476099\pi$
$857$	3.22586e6	0.150035	0.0750175	+	0.997182i	$0.476099\pi$
$858$	0	0
$859$	2.64671e7	1.22384	0.611918	−	0.790921i	$-0.290398\pi$
$859$	2.64671e7	1.22384	0.611918	+	0.790921i	$0.290398\pi$
$860$	−7.05255e7	−3.25162
$861$	0	0
$862$	3.17611e7	1.45589
$863$	−1.58510e7	−0.724487	−0.362243	−	0.932084i	$-0.617989\pi$
$863$	−1.58510e7	−0.724487	−0.362243	+	0.932084i	$0.617989\pi$
$864$	0	0
$865$	−3.46726e6	−0.157560
$866$	−2.67861e6	−0.121371
$867$	0	0
$868$	0	0
$869$	2.50522e7	1.12537
$870$	0	0
$871$	−1.05716e7	−0.472166
$872$	−1.89261e7	−0.842888
$873$	0	0
$874$	−4.14817e7	−1.83687
$875$	0	0
$876$	0	0
$877$	−1.49132e7	−0.654743	−0.327372	−	0.944896i	$-0.606163\pi$
$877$	−1.49132e7	−0.654743	−0.327372	+	0.944896i	$0.606163\pi$
$878$	4.08665e7	1.78909
$879$	0	0
$880$	1.34146e7	0.583944
$881$	3.70679e7	1.60901	0.804504	−	0.593947i	$-0.202431\pi$
$881$	3.70679e7	1.60901	0.804504	+	0.593947i	$0.202431\pi$
$882$	0	0
$883$	1.50440e7	0.649325	0.324663	−	0.945830i	$-0.394749\pi$
$883$	1.50440e7	0.649325	0.324663	+	0.945830i	$0.394749\pi$
$884$	−3.01378e7	−1.29712
$885$	0	0
$886$	4.79409e7	2.05174
$887$	−1.78101e7	−0.760075	−0.380038	−	0.924971i	$-0.624089\pi$
$887$	−1.78101e7	−0.760075	−0.380038	+	0.924971i	$0.624089\pi$
$888$	0	0
$889$	0	0
$890$	−9.18375e6	−0.388638
$891$	0	0
$892$	−6.07782e7	−2.55762
$893$	2.23692e7	0.938690
$894$	0	0
$895$	1.02508e7	0.427762
$896$	0	0
$897$	0	0
$898$	−2.85605e7	−1.18188
$899$	6.18317e6	0.255160
$900$	0	0
$901$	−2.98106e7	−1.22337
$902$	−1.29405e6	−0.0529583
$903$	0	0
$904$	4.06815e7	1.65568
$905$	−5.54073e6	−0.224877
$906$	0	0
$907$	−1.16936e6	−0.0471987	−0.0235993	−	0.999721i	$-0.507513\pi$
$907$	−1.16936e6	−0.0471987	−0.0235993	+	0.999721i	$0.507513\pi$
$908$	−3.10645e7	−1.25040
$909$	0	0
$910$	0	0
$911$	−2.74389e7	−1.09539	−0.547697	−	0.836677i	$-0.684495\pi$
$911$	−2.74389e7	−1.09539	−0.547697	+	0.836677i	$0.684495\pi$
$912$	0	0
$913$	−8.74924e6	−0.347371
$914$	−1.63790e7	−0.648516
$915$	0	0
$916$	2.93781e7	1.15687
$917$	0	0
$918$	0	0
$919$	1.10218e7	0.430492	0.215246	−	0.976560i	$-0.430945\pi$
$919$	1.10218e7	0.430492	0.215246	+	0.976560i	$0.430945\pi$
$920$	−5.25495e7	−2.04691
$921$	0	0
$922$	6.43864e7	2.49440
$923$	−3.31444e7	−1.28058
$924$	0	0
$925$	−5.67367e6	−0.218027
$926$	4.82493e7	1.84911
$927$	0	0
$928$	−3.11173e6	−0.118613
$929$	−4.25051e6	−0.161585	−0.0807927	−	0.996731i	$-0.525745\pi$
$929$	−4.25051e6	−0.161585	−0.0807927	+	0.996731i	$0.525745\pi$
$930$	0	0
$931$	0	0
$932$	2.04559e7	0.771398
$933$	0	0
$934$	4.30216e7	1.61369
$935$	−4.04413e7	−1.51285
$936$	0	0
$937$	3.75738e7	1.39809	0.699046	−	0.715077i	$-0.253608\pi$
$937$	3.75738e7	1.39809	0.699046	+	0.715077i	$0.253608\pi$
$938$	0	0
$939$	0	0
$940$	6.58558e7	2.43094
$941$	−6.87043e6	−0.252936	−0.126468	−	0.991971i	$-0.540364\pi$
$941$	−6.87043e6	−0.252936	−0.126468	+	0.991971i	$0.540364\pi$
$942$	0	0
$943$	793550.	0.0290600
$944$	3.45476e6	0.126179
$945$	0	0
$946$	−9.21446e7	−3.34767
$947$	2.30923e7	0.836744	0.418372	−	0.908276i	$-0.362601\pi$
$947$	2.30923e7	0.836744	0.418372	+	0.908276i	$0.362601\pi$
$948$	0	0
$949$	1.44632e7	0.521314
$950$	2.59583e7	0.933185
$951$	0	0
$952$	0	0
$953$	2.95399e7	1.05360	0.526802	−	0.849988i	$-0.323391\pi$
$953$	2.95399e7	1.05360	0.526802	+	0.849988i	$0.323391\pi$
$954$	0	0
$955$	−1.95142e7	−0.692376
$956$	−2.08862e7	−0.739119
$957$	0	0
$958$	2.70919e6	0.0953730
$959$	0	0
$960$	0	0
$961$	3.87161e7	1.35233
$962$	−1.40702e7	−0.490188
$963$	0	0
$964$	−9.61329e7	−3.33180
$965$	−1.15009e6	−0.0397569
$966$	0	0
$967$	1.49151e7	0.512931	0.256466	−	0.966553i	$-0.417442\pi$
$967$	1.49151e7	0.512931	0.256466	+	0.966553i	$0.417442\pi$
$968$	3.47761e7	1.19287
$969$	0	0
$970$	−8.70162e6	−0.296941
$971$	−1.54830e7	−0.526995	−0.263498	−	0.964660i	$-0.584876\pi$
$971$	−1.54830e7	−0.526995	−0.263498	+	0.964660i	$0.584876\pi$
$972$	0	0
$973$	0	0
$974$	−7.33992e7	−2.47910
$975$	0	0
$976$	318311.	0.0106961
$977$	2.83428e7	0.949961	0.474980	−	0.879996i	$-0.342455\pi$
$977$	2.83428e7	0.949961	0.474980	+	0.879996i	$0.342455\pi$
$978$	0	0
$979$	−7.64410e6	−0.254900
$980$	0	0
$981$	0	0
$982$	−2.95280e7	−0.977138
$983$	−2.56993e7	−0.848278	−0.424139	−	0.905597i	$-0.639423\pi$
$983$	−2.56993e7	−0.848278	−0.424139	+	0.905597i	$0.639423\pi$
$984$	0	0
$985$	−7.40446e7	−2.43166
$986$	−7.11518e6	−0.233074
$987$	0	0
$988$	4.10106e7	1.33661
$989$	5.65059e7	1.83697
$990$	0	0
$991$	4.46405e7	1.44392	0.721962	−	0.691933i	$-0.243240\pi$
$991$	4.46405e7	1.44392	0.721962	+	0.691933i	$0.243240\pi$
$992$	−3.38921e7	−1.09350
$993$	0	0
$994$	0	0
$995$	−6.32461e7	−2.02524
$996$	0	0
$997$	2.30554e7	0.734572	0.367286	−	0.930108i	$-0.380287\pi$
$997$	2.30554e7	0.734572	0.367286	+	0.930108i	$0.380287\pi$
$998$	6.31528e7	2.00709
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.6.a.t.1.2		2
3.2	odd	2		147.6.a.i.1.1		2
7.2	even	3		63.6.e.c.46.1		4
7.4	even	3		63.6.e.c.37.1		4
7.6	odd	2		441.6.a.s.1.2		2
21.2	odd	6		21.6.e.b.4.2	✓	4
21.5	even	6		147.6.e.l.67.2		4
21.11	odd	6		21.6.e.b.16.2	yes	4
21.17	even	6		147.6.e.l.79.2		4
21.20	even	2		147.6.a.k.1.1		2
84.11	even	6		336.6.q.e.289.2		4
84.23	even	6		336.6.q.e.193.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.6.e.b.4.2	✓	4	21.2	odd	6
21.6.e.b.16.2	yes	4	21.11	odd	6
63.6.e.c.37.1		4	7.4	even	3
63.6.e.c.46.1		4	7.2	even	3
147.6.a.i.1.1		2	3.2	odd	2
147.6.a.k.1.1		2	21.20	even	2
147.6.e.l.67.2		4	21.5	even	6
147.6.e.l.79.2		4	21.17	even	6
336.6.q.e.193.2		4	84.23	even	6
336.6.q.e.289.2		4	84.11	even	6
441.6.a.s.1.2		2	7.6	odd	2
441.6.a.t.1.2		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	441.a (trivial)

\(f(q)\)	\(=\)	\(q+9.38987 q^{2} +56.1696 q^{4} +71.7291 q^{5} +226.949 q^{8} +673.526 q^{10} +560.610 q^{11} +533.509 q^{13} +333.597 q^{16} -1005.70 q^{17} +1368.53 q^{19} +4028.99 q^{20} +5264.05 q^{22} -3228.08 q^{23} +2020.06 q^{25} +5009.58 q^{26} +753.456 q^{29} +8206.42 q^{31} -4129.95 q^{32} -9443.39 q^{34} -2808.66 q^{37} +12850.3 q^{38} +16278.9 q^{40} -245.827 q^{41} -17504.5 q^{43} +31489.2 q^{44} -30311.2 q^{46} +16345.5 q^{47} +18968.1 q^{50} +29967.0 q^{52} +29641.7 q^{53} +40212.0 q^{55} +7074.85 q^{58} +10356.1 q^{59} +954.179 q^{61} +77057.2 q^{62} -49454.8 q^{64} +38268.1 q^{65} -19815.2 q^{67} -56489.8 q^{68} -62125.4 q^{71} +27109.6 q^{73} -26373.0 q^{74} +76869.6 q^{76} +44687.4 q^{79} +23928.6 q^{80} -2308.29 q^{82} -15606.6 q^{83} -72137.9 q^{85} -164365. q^{86} +127230. q^{88} -13635.3 q^{89} -181320. q^{92} +153482. q^{94} +98163.1 q^{95} -12919.5 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{2} + 65 q^{4} + 33 q^{5} + 375 q^{8} + 921 q^{10} + 1137 q^{11} + 925 q^{13} - 895 q^{16} + 324 q^{17} + 2311 q^{19} + 3687 q^{20} + 1581 q^{22} - 1596 q^{23} + 395 q^{25} + 2508 q^{26} + 2217 q^{29}+ \cdots - 9791 q^{97}+O(q^{100}) \) 2 * q + 3 * q^2 + 65 * q^4 + 33 * q^5 + 375 * q^8 + 921 * q^10 + 1137 * q^11 + 925 * q^13 - 895 * q^16 + 324 * q^17 + 2311 * q^19 + 3687 * q^20 + 1581 * q^22 - 1596 * q^23 + 395 * q^25 + 2508 * q^26 + 2217 * q^29 + 4294 * q^31 - 1017 * q^32 - 17940 * q^34 - 19109 * q^37 + 6828 * q^38 + 10545 * q^40 + 12858 * q^41 - 2771 * q^43 + 36579 * q^44 - 40740 * q^46 + 23160 * q^47 + 29352 * q^50 + 33424 * q^52 + 31653 * q^53 + 17889 * q^55 - 2277 * q^58 - 41097 * q^59 + 42052 * q^61 + 102057 * q^62 - 30031 * q^64 + 23106 * q^65 + 30763 * q^67 - 44748 * q^68 - 102096 * q^71 - 28577 * q^73 + 77784 * q^74 + 85192 * q^76 - 18464 * q^79 + 71511 * q^80 - 86040 * q^82 - 61179 * q^83 - 123636 * q^85 - 258510 * q^86 + 212565 * q^88 - 29322 * q^89 - 166908 * q^92 + 109938 * q^94 + 61662 * q^95 - 9791 * q^97

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