LMFDB - Embedded newform 441.6.a.v.1.3

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:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 97x^{2} + 7x + 294 $ x^4 - x^3 - 97x^2 + 7x + 294
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 7 $
Twist minimal:	no (minimal twist has level 21)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.79080$ 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)

$f(q)$	$=$	$q+0.790805 q^{2} -31.3746 q^{4} -104.192 q^{5} -50.1170 q^{8} +O(q^{10})$	$q+0.790805 q^{2} -31.3746 q^{4} -104.192 q^{5} -50.1170 q^{8} -82.3953 q^{10} +497.660 q^{11} +206.551 q^{13} +964.355 q^{16} +63.1586 q^{17} -1323.95 q^{19} +3268.98 q^{20} +393.552 q^{22} +194.437 q^{23} +7730.91 q^{25} +163.342 q^{26} -4323.14 q^{29} +7525.31 q^{31} +2366.36 q^{32} +49.9461 q^{34} +10355.6 q^{37} -1046.99 q^{38} +5221.77 q^{40} -4180.92 q^{41} +5960.87 q^{43} -15613.9 q^{44} +153.762 q^{46} -4389.74 q^{47} +6113.64 q^{50} -6480.47 q^{52} -17784.8 q^{53} -51852.0 q^{55} -3418.76 q^{58} +3500.47 q^{59} +10632.5 q^{61} +5951.06 q^{62} -28988.0 q^{64} -21520.9 q^{65} -13274.7 q^{67} -1981.58 q^{68} -38811.1 q^{71} -31375.6 q^{73} +8189.30 q^{74} +41538.6 q^{76} +39491.5 q^{79} -100478. q^{80} -3306.29 q^{82} -102372. q^{83} -6580.60 q^{85} +4713.88 q^{86} -24941.2 q^{88} -112821. q^{89} -6100.40 q^{92} -3471.43 q^{94} +137945. q^{95} -30334.3 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 3 q^{2} + 69 q^{4} - 123 q^{8}+O(q^{10}) $ 4 * q - 3 * q^2 + 69 * q^4 - 123 * q^8	$ 4 q - 3 q^{2} + 69 q^{4} - 123 q^{8} - 283 q^{10} - 402 q^{11} - 462 q^{13} + 3273 q^{16} + 276 q^{17} - 510 q^{19} + 4719 q^{20} + 1375 q^{22} - 6900 q^{23} + 2814 q^{25} - 15138 q^{26} - 540 q^{29} + 6410 q^{31} - 15519 q^{32} - 21144 q^{34} + 15250 q^{37} - 41250 q^{38} + 8547 q^{40} + 4308 q^{41} + 29198 q^{43} - 70743 q^{44} + 61800 q^{46} - 15060 q^{47} + 7302 q^{50} + 47476 q^{52} - 13692 q^{53} - 73124 q^{55} + 52309 q^{58} + 34830 q^{59} + 5364 q^{61} + 16029 q^{62} - 73487 q^{64} - 66864 q^{65} - 5994 q^{67} - 58272 q^{68} - 89268 q^{71} - 59638 q^{73} + 185442 q^{74} - 21308 q^{76} - 44062 q^{79} - 33381 q^{80} - 57596 q^{82} - 208446 q^{83} + 36324 q^{85} + 136968 q^{86} + 87597 q^{88} - 77520 q^{89} - 158256 q^{92} + 73722 q^{94} + 221376 q^{95} + 188630 q^{97}+O(q^{100}) $ 4 * q - 3 * q^2 + 69 * q^4 - 123 * q^8 - 283 * q^10 - 402 * q^11 - 462 * q^13 + 3273 * q^16 + 276 * q^17 - 510 * q^19 + 4719 * q^20 + 1375 * q^22 - 6900 * q^23 + 2814 * q^25 - 15138 * q^26 - 540 * q^29 + 6410 * q^31 - 15519 * q^32 - 21144 * q^34 + 15250 * q^37 - 41250 * q^38 + 8547 * q^40 + 4308 * q^41 + 29198 * q^43 - 70743 * q^44 + 61800 * q^46 - 15060 * q^47 + 7302 * q^50 + 47476 * q^52 - 13692 * q^53 - 73124 * q^55 + 52309 * q^58 + 34830 * q^59 + 5364 * q^61 + 16029 * q^62 - 73487 * q^64 - 66864 * q^65 - 5994 * q^67 - 58272 * q^68 - 89268 * q^71 - 59638 * q^73 + 185442 * q^74 - 21308 * q^76 - 44062 * q^79 - 33381 * q^80 - 57596 * q^82 - 208446 * q^83 + 36324 * q^85 + 136968 * q^86 + 87597 * q^88 - 77520 * q^89 - 158256 * q^92 + 73722 * q^94 + 221376 * q^95 + 188630 * q^97

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 $

$2$	0.790805	0.139796	0.0698979	−	0.997554i	$-0.477733\pi$
$2$	0.790805	0.139796	0.0698979	+	0.997554i	$0.477733\pi$
$3$	0	0
$3$	0	0
$4$	−31.3746	−0.980457
$5$	−104.192	−1.86384	−0.931919	−	0.362667i	$-0.881866\pi$
$5$	−104.192	−1.86384	−0.931919	+	0.362667i	$0.881866\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−50.1170	−0.276860
$9$	0	0
$10$	−82.3953	−0.260557
$11$	497.660	1.24008	0.620041	−	0.784569i	$-0.287116\pi$
$11$	497.660	1.24008	0.620041	+	0.784569i	$0.287116\pi$
$12$	0	0
$13$	206.551	0.338977	0.169488	−	0.985532i	$-0.445789\pi$
$13$	206.551	0.338977	0.169488	+	0.985532i	$0.445789\pi$
$14$	0	0
$15$	0	0
$16$	964.355	0.941753
$17$	63.1586	0.0530042	0.0265021	−	0.999649i	$-0.491563\pi$
$17$	63.1586	0.0530042	0.0265021	+	0.999649i	$0.491563\pi$
$18$	0	0
$19$	−1323.95	−0.841374	−0.420687	−	0.907206i	$-0.638211\pi$
$19$	−1323.95	−0.841374	−0.420687	+	0.907206i	$0.638211\pi$
$20$	3268.98	1.82741
$21$	0	0
$22$	393.552	0.173358
$23$	194.437	0.0766408	0.0383204	−	0.999266i	$-0.487799\pi$
$23$	194.437	0.0766408	0.0383204	+	0.999266i	$0.487799\pi$
$24$	0	0
$25$	7730.91	2.47389
$26$	163.342	0.0473875
$27$	0	0
$28$	0	0
$29$	−4323.14	−0.954562	−0.477281	−	0.878751i	$-0.658378\pi$
$29$	−4323.14	−0.954562	−0.477281	+	0.878751i	$0.658378\pi$
$30$	0	0
$31$	7525.31	1.40644	0.703219	−	0.710974i	$-0.251745\pi$
$31$	7525.31	1.40644	0.703219	+	0.710974i	$0.251745\pi$
$32$	2366.36	0.408513
$33$	0	0
$34$	49.9461	0.00740977
$35$	0	0
$36$	0	0
$37$	10355.6	1.24358	0.621789	−	0.783185i	$-0.286406\pi$
$37$	10355.6	1.24358	0.621789	+	0.783185i	$0.286406\pi$
$38$	−1046.99	−0.117621
$39$	0	0
$40$	5221.77	0.516022
$41$	−4180.92	−0.388429	−0.194215	−	0.980959i	$-0.562216\pi$
$41$	−4180.92	−0.388429	−0.194215	+	0.980959i	$0.562216\pi$
$42$	0	0
$43$	5960.87	0.491630	0.245815	−	0.969317i	$-0.420944\pi$
$43$	5960.87	0.491630	0.245815	+	0.969317i	$0.420944\pi$
$44$	−15613.9	−1.21585
$45$	0	0
$46$	153.762	0.0107141
$47$	−4389.74	−0.289864	−0.144932	−	0.989442i	$-0.546296\pi$
$47$	−4389.74	−0.289864	−0.144932	+	0.989442i	$0.546296\pi$
$48$	0	0
$49$	0	0
$50$	6113.64	0.345840
$51$	0	0
$52$	−6480.47	−0.332352
$53$	−17784.8	−0.869679	−0.434839	−	0.900508i	$-0.643195\pi$
$53$	−17784.8	−0.869679	−0.434839	+	0.900508i	$0.643195\pi$
$54$	0	0
$55$	−51852.0	−2.31131
$56$	0	0
$57$	0	0
$58$	−3418.76	−0.133444
$59$	3500.47	0.130917	0.0654585	−	0.997855i	$-0.479149\pi$
$59$	3500.47	0.130917	0.0654585	+	0.997855i	$0.479149\pi$
$60$	0	0
$61$	10632.5	0.365855	0.182928	−	0.983126i	$-0.441443\pi$
$61$	10632.5	0.365855	0.182928	+	0.983126i	$0.441443\pi$
$62$	5951.06	0.196614
$63$	0	0
$64$	−28988.0	−0.884645
$65$	−21520.9	−0.631797
$66$	0	0
$67$	−13274.7	−0.361276	−0.180638	−	0.983550i	$-0.557816\pi$
$67$	−13274.7	−0.361276	−0.180638	+	0.983550i	$0.557816\pi$
$68$	−1981.58	−0.0519683
$69$	0	0
$70$	0	0
$71$	−38811.1	−0.913713	−0.456857	−	0.889540i	$-0.651025\pi$
$71$	−38811.1	−0.913713	−0.456857	+	0.889540i	$0.651025\pi$
$72$	0	0
$73$	−31375.6	−0.689105	−0.344552	−	0.938767i	$-0.611969\pi$
$73$	−31375.6	−0.689105	−0.344552	+	0.938767i	$0.611969\pi$
$74$	8189.30	0.173847
$75$	0	0
$76$	41538.6	0.824931
$77$	0	0
$78$	0	0
$79$	39491.5	0.711927	0.355964	−	0.934500i	$-0.384153\pi$
$79$	39491.5	0.711927	0.355964	+	0.934500i	$0.384153\pi$
$80$	−100478.	−1.75528
$81$	0	0
$82$	−3306.29	−0.0543008
$83$	−102372.	−1.63112	−0.815559	−	0.578675i	$-0.803570\pi$
$83$	−102372.	−1.63112	−0.815559	+	0.578675i	$0.803570\pi$
$84$	0	0
$85$	−6580.60	−0.0987912
$86$	4713.88	0.0687279
$87$	0	0
$88$	−24941.2	−0.343329
$89$	−112821.	−1.50978	−0.754892	−	0.655849i	$-0.772311\pi$
$89$	−112821.	−1.50978	−0.754892	+	0.655849i	$0.772311\pi$
$90$	0	0
$91$	0	0
$92$	−6100.40	−0.0751430
$93$	0	0
$94$	−3471.43	−0.0405218
$95$	137945.	1.56818
$96$	0	0
$97$	−30334.3	−0.327345	−0.163672	−	0.986515i	$-0.552334\pi$
$97$	−30334.3	−0.327345	−0.163672	+	0.986515i	$0.552334\pi$
$98$	0	0
$99$	0	0
$100$	−242554.	−2.42554
$101$	106796.	1.04172	0.520862	−	0.853641i	$-0.325610\pi$
$101$	106796.	1.04172	0.520862	+	0.853641i	$0.325610\pi$
$102$	0	0
$103$	−157536.	−1.46314	−0.731570	−	0.681767i	$-0.761212\pi$
$103$	−157536.	−1.46314	−0.731570	+	0.681767i	$0.761212\pi$
$104$	−10351.7	−0.0938490
$105$	0	0
$106$	−14064.3	−0.121578
$107$	89324.3	0.754241	0.377121	−	0.926164i	$-0.376914\pi$
$107$	89324.3	0.754241	0.377121	+	0.926164i	$0.376914\pi$
$108$	0	0
$109$	167605.	1.35120	0.675600	−	0.737268i	$-0.263885\pi$
$109$	167605.	1.35120	0.675600	+	0.737268i	$0.263885\pi$
$110$	−41004.8	−0.323112
$111$	0	0
$112$	0	0
$113$	115794.	0.853079	0.426539	−	0.904469i	$-0.359733\pi$
$113$	115794.	0.853079	0.426539	+	0.904469i	$0.359733\pi$
$114$	0	0
$115$	−20258.8	−0.142846
$116$	135637.	0.935907
$117$	0	0
$118$	2768.19	0.0183017
$119$	0	0
$120$	0	0
$121$	86614.1	0.537805
$122$	8408.20	0.0511451
$123$	0	0
$124$	−236104.	−1.37895
$125$	−479898.	−2.74709
$126$	0	0
$127$	201513.	1.10865	0.554325	−	0.832300i	$-0.312977\pi$
$127$	201513.	1.10865	0.554325	+	0.832300i	$0.312977\pi$
$128$	−98647.4	−0.532183
$129$	0	0
$130$	−17018.9	−0.0883227
$131$	38469.6	0.195857	0.0979285	−	0.995193i	$-0.468778\pi$
$131$	38469.6	0.195857	0.0979285	+	0.995193i	$0.468778\pi$
$132$	0	0
$133$	0	0
$134$	−10497.7	−0.0505049
$135$	0	0
$136$	−3165.32	−0.0146747
$137$	241722.	1.10031	0.550155	−	0.835063i	$-0.314569\pi$
$137$	241722.	1.10031	0.550155	+	0.835063i	$0.314569\pi$
$138$	0	0
$139$	53112.2	0.233162	0.116581	−	0.993181i	$-0.462807\pi$
$139$	53112.2	0.233162	0.116581	+	0.993181i	$0.462807\pi$
$140$	0	0
$141$	0	0
$142$	−30692.0	−0.127733
$143$	102792.	0.420359
$144$	0	0
$145$	450435.	1.77915
$146$	−24812.0	−0.0963340
$147$	0	0
$148$	−324905.	−1.21927
$149$	−129062.	−0.476248	−0.238124	−	0.971235i	$-0.576532\pi$
$149$	−129062.	−0.476248	−0.238124	+	0.971235i	$0.576532\pi$
$150$	0	0
$151$	153206.	0.546808	0.273404	−	0.961899i	$-0.411850\pi$
$151$	153206.	0.546808	0.273404	+	0.961899i	$0.411850\pi$
$152$	66352.6	0.232943
$153$	0	0
$154$	0	0
$155$	−784075.	−2.62137
$156$	0	0
$157$	151188.	0.489517	0.244758	−	0.969584i	$-0.421291\pi$
$157$	151188.	0.489517	0.244758	+	0.969584i	$0.421291\pi$
$158$	31230.0	0.0995245
$159$	0	0
$160$	−246555.	−0.761402
$161$	0	0
$162$	0	0
$163$	−32916.7	−0.0970392	−0.0485196	−	0.998822i	$-0.515450\pi$
$163$	−32916.7	−0.0970392	−0.0485196	+	0.998822i	$0.515450\pi$
$164$	131175.	0.380838
$165$	0	0
$166$	−80956.1	−0.228023
$167$	217586.	0.603725	0.301862	−	0.953352i	$-0.402392\pi$
$167$	217586.	0.603725	0.301862	+	0.953352i	$0.402392\pi$
$168$	0	0
$169$	−328630.	−0.885095
$170$	−5203.97	−0.0138106
$171$	0	0
$172$	−187020.	−0.482022
$173$	−421242.	−1.07008	−0.535041	−	0.844826i	$-0.679704\pi$
$173$	−421242.	−1.07008	−0.535041	+	0.844826i	$0.679704\pi$
$174$	0	0
$175$	0	0
$176$	479921.	1.16785
$177$	0	0
$178$	−89219.4	−0.211062
$179$	−3494.06	−0.00815075	−0.00407538	−	0.999992i	$-0.501297\pi$
$179$	−3494.06	−0.00815075	−0.00407538	+	0.999992i	$0.501297\pi$
$180$	0	0
$181$	−594611.	−1.34908	−0.674538	−	0.738240i	$-0.735657\pi$
$181$	−594611.	−1.34908	−0.674538	+	0.738240i	$0.735657\pi$
$182$	0	0
$183$	0	0
$184$	−9744.61	−0.0212188
$185$	−1.07897e6	−2.31783
$186$	0	0
$187$	31431.5	0.0657296
$188$	137726.	0.284199
$189$	0	0
$190$	109088.	0.219226
$191$	−828644.	−1.64356	−0.821778	−	0.569807i	$-0.807018\pi$
$191$	−828644.	−1.64356	−0.821778	+	0.569807i	$0.807018\pi$
$192$	0	0
$193$	219739.	0.424633	0.212316	−	0.977201i	$-0.431899\pi$
$193$	219739.	0.424633	0.212316	+	0.977201i	$0.431899\pi$
$194$	−23988.5	−0.0457614
$195$	0	0
$196$	0	0
$197$	−475612.	−0.873146	−0.436573	−	0.899669i	$-0.643808\pi$
$197$	−475612.	−0.873146	−0.436573	+	0.899669i	$0.643808\pi$
$198$	0	0
$199$	−627555.	−1.12336	−0.561681	−	0.827354i	$-0.689845\pi$
$199$	−627555.	−1.12336	−0.561681	+	0.827354i	$0.689845\pi$
$200$	−387450.	−0.684921
$201$	0	0
$202$	84455.1	0.145629
$203$	0	0
$204$	0	0
$205$	435617.	0.723969
$206$	−124580.	−0.204541
$207$	0	0
$208$	199189.	0.319232
$209$	−658879.	−1.04337
$210$	0	0
$211$	570989.	0.882920	0.441460	−	0.897281i	$-0.354461\pi$
$211$	570989.	0.882920	0.441460	+	0.897281i	$0.354461\pi$
$212$	557991.	0.852683
$213$	0	0
$214$	70638.1	0.105440
$215$	−621073.	−0.916319
$216$	0	0
$217$	0	0
$218$	132543.	0.188892
$219$	0	0
$220$	1.62684e6	2.26614
$221$	13045.5	0.0179672
$222$	0	0
$223$	−4233.11	−0.00570029	−0.00285015	−	0.999996i	$-0.500907\pi$
$223$	−4233.11	−0.00570029	−0.00285015	+	0.999996i	$0.500907\pi$
$224$	0	0
$225$	0	0
$226$	91570.3	0.119257
$227$	1.12986e6	1.45533	0.727664	−	0.685934i	$-0.240606\pi$
$227$	1.12986e6	1.45533	0.727664	+	0.685934i	$0.240606\pi$
$228$	0	0
$229$	−804643.	−1.01395	−0.506973	−	0.861962i	$-0.669236\pi$
$229$	−804643.	−1.01395	−0.506973	+	0.861962i	$0.669236\pi$
$230$	−16020.7	−0.0199693
$231$	0	0
$232$	216663.	0.264280
$233$	−1.16927e6	−1.41100	−0.705498	−	0.708712i	$-0.749277\pi$
$233$	−1.16927e6	−1.41100	−0.705498	+	0.708712i	$0.749277\pi$
$234$	0	0
$235$	457374.	0.540259
$236$	−109826.	−0.128358
$237$	0	0
$238$	0	0
$239$	1.70554e6	1.93138	0.965689	−	0.259700i	$-0.0836238\pi$
$239$	1.70554e6	1.93138	0.965689	+	0.259700i	$0.0836238\pi$
$240$	0	0
$241$	−951196.	−1.05494	−0.527470	−	0.849574i	$-0.676859\pi$
$241$	−951196.	−1.05494	−0.527470	+	0.849574i	$0.676859\pi$
$242$	68494.9	0.0751830
$243$	0	0
$244$	−333590.	−0.358705
$245$	0	0
$246$	0	0
$247$	−273465.	−0.285206
$248$	−377146.	−0.389386
$249$	0	0
$250$	−379505.	−0.384032
$251$	1.14498e6	1.14713	0.573566	−	0.819159i	$-0.305560\pi$
$251$	1.14498e6	1.14713	0.573566	+	0.819159i	$0.305560\pi$
$252$	0	0
$253$	96763.6	0.0950409
$254$	159358.	0.154985
$255$	0	0
$256$	849606.	0.810248
$257$	1.18374e6	1.11795	0.558976	−	0.829184i	$-0.311194\pi$
$257$	1.18374e6	1.11795	0.558976	+	0.829184i	$0.311194\pi$
$258$	0	0
$259$	0	0
$260$	675211.	0.619450
$261$	0	0
$262$	30421.9	0.0273800
$263$	−448316.	−0.399664	−0.199832	−	0.979830i	$-0.564040\pi$
$263$	−448316.	−0.399664	−0.199832	+	0.979830i	$0.564040\pi$
$264$	0	0
$265$	1.85303e6	1.62094
$266$	0	0
$267$	0	0
$268$	416490.	0.354216
$269$	747474.	0.629819	0.314909	−	0.949122i	$-0.398026\pi$
$269$	747474.	0.629819	0.314909	+	0.949122i	$0.398026\pi$
$270$	0	0
$271$	232637.	0.192423	0.0962114	−	0.995361i	$-0.469328\pi$
$271$	232637.	0.192423	0.0962114	+	0.995361i	$0.469328\pi$
$272$	60907.3	0.0499169
$273$	0	0
$274$	191155.	0.153819
$275$	3.84736e6	3.06783
$276$	0	0
$277$	−2.42924e6	−1.90227	−0.951134	−	0.308778i	$-0.900080\pi$
$277$	−2.42924e6	−1.90227	−0.951134	+	0.308778i	$0.900080\pi$
$278$	42001.4	0.0325951
$279$	0	0
$280$	0	0
$281$	−2.51704e6	−1.90163	−0.950813	−	0.309766i	$-0.899749\pi$
$281$	−2.51704e6	−1.90163	−0.950813	+	0.309766i	$0.899749\pi$
$282$	0	0
$283$	−260994.	−0.193716	−0.0968579	−	0.995298i	$-0.530879\pi$
$283$	−260994.	−0.193716	−0.0968579	+	0.995298i	$0.530879\pi$
$284$	1.21768e6	0.895857
$285$	0	0
$286$	81288.6	0.0587645
$287$	0	0
$288$	0	0
$289$	−1.41587e6	−0.997191
$290$	356206.	0.248718
$291$	0	0
$292$	984399.	0.675638
$293$	65011.7	0.0442408	0.0221204	−	0.999755i	$-0.492958\pi$
$293$	65011.7	0.0442408	0.0221204	+	0.999755i	$0.492958\pi$
$294$	0	0
$295$	−364720.	−0.244008
$296$	−518994.	−0.344297
$297$	0	0
$298$	−102063.	−0.0665775
$299$	40161.3	0.0259794
$300$	0	0
$301$	0	0
$302$	121156.	0.0764414
$303$	0	0
$304$	−1.27676e6	−0.792367
$305$	−1.10781e6	−0.681895
$306$	0	0
$307$	2.35599e6	1.42668	0.713342	−	0.700816i	$-0.247181\pi$
$307$	2.35599e6	1.42668	0.713342	+	0.700816i	$0.247181\pi$
$308$	0	0
$309$	0	0
$310$	−620051.	−0.366457
$311$	−2.11805e6	−1.24176	−0.620878	−	0.783907i	$-0.713224\pi$
$311$	−2.11805e6	−1.24176	−0.620878	+	0.783907i	$0.713224\pi$
$312$	0	0
$313$	187514.	0.108186	0.0540931	−	0.998536i	$-0.482773\pi$
$313$	187514.	0.108186	0.0540931	+	0.998536i	$0.482773\pi$
$314$	119560.	0.0684324
$315$	0	0
$316$	−1.23903e6	−0.698014
$317$	1.00541e6	0.561947	0.280974	−	0.959716i	$-0.409343\pi$
$317$	1.00541e6	0.561947	0.280974	+	0.959716i	$0.409343\pi$
$318$	0	0
$319$	−2.15145e6	−1.18374
$320$	3.02031e6	1.64883
$321$	0	0
$322$	0	0
$323$	−83619.1	−0.0445964
$324$	0	0
$325$	1.59683e6	0.838591
$326$	−26030.7	−0.0135657
$327$	0	0
$328$	209535.	0.107540
$329$	0	0
$330$	0	0
$331$	1.67984e6	0.842748	0.421374	−	0.906887i	$-0.361548\pi$
$331$	1.67984e6	0.842748	0.421374	+	0.906887i	$0.361548\pi$
$332$	3.21188e6	1.59924
$333$	0	0
$334$	172068.	0.0843982
$335$	1.38312e6	0.673360
$336$	0	0
$337$	−995036.	−0.477270	−0.238635	−	0.971109i	$-0.576700\pi$
$337$	−995036.	−0.477270	−0.238635	+	0.971109i	$0.576700\pi$
$338$	−259882.	−0.123733
$339$	0	0
$340$	206464.	0.0968605
$341$	3.74505e6	1.74410
$342$	0	0
$343$	0	0
$344$	−298741.	−0.136113
$345$	0	0
$346$	−333121.	−0.149593
$347$	−4.05665e6	−1.80861	−0.904304	−	0.426890i	$-0.859609\pi$
$347$	−4.05665e6	−1.80861	−0.904304	+	0.426890i	$0.859609\pi$
$348$	0	0
$349$	−2.86202e6	−1.25779	−0.628897	−	0.777488i	$-0.716493\pi$
$349$	−2.86202e6	−1.25779	−0.628897	+	0.777488i	$0.716493\pi$
$350$	0	0
$351$	0	0
$352$	1.17764e6	0.506590
$353$	832685.	0.355667	0.177834	−	0.984061i	$-0.443091\pi$
$353$	832685.	0.355667	0.177834	+	0.984061i	$0.443091\pi$
$354$	0	0
$355$	4.04379e6	1.70301
$356$	3.53972e6	1.48028
$357$	0	0
$358$	−2763.12	−0.00113944
$359$	2.69751e6	1.10465	0.552327	−	0.833627i	$-0.313740\pi$
$359$	2.69751e6	1.10465	0.552327	+	0.833627i	$0.313740\pi$
$360$	0	0
$361$	−723243.	−0.292090
$362$	−470221.	−0.188595
$363$	0	0
$364$	0	0
$365$	3.26908e6	1.28438
$366$	0	0
$367$	−1.45482e6	−0.563826	−0.281913	−	0.959440i	$-0.590969\pi$
$367$	−1.45482e6	−0.563826	−0.281913	+	0.959440i	$0.590969\pi$
$368$	187507.	0.0721767
$369$	0	0
$370$	−853257.	−0.324023
$371$	0	0
$372$	0	0
$373$	−2.04689e6	−0.761766	−0.380883	−	0.924623i	$-0.624380\pi$
$373$	−2.04689e6	−0.761766	−0.380883	+	0.924623i	$0.624380\pi$
$374$	24856.2	0.00918872
$375$	0	0
$376$	220000.	0.0802516
$377$	−892950.	−0.323574
$378$	0	0
$379$	−416898.	−0.149084	−0.0745421	−	0.997218i	$-0.523750\pi$
$379$	−416898.	−0.149084	−0.0745421	+	0.997218i	$0.523750\pi$
$380$	−4.32798e6	−1.53754
$381$	0	0
$382$	−655296.	−0.229762
$383$	−3.87626e6	−1.35025	−0.675127	−	0.737701i	$-0.735911\pi$
$383$	−3.87626e6	−1.35025	−0.675127	+	0.737701i	$0.735911\pi$
$384$	0	0
$385$	0	0
$386$	173771.	0.0593619
$387$	0	0
$388$	951729.	0.320947
$389$	−2.83802e6	−0.950915	−0.475458	−	0.879739i	$-0.657717\pi$
$389$	−2.83802e6	−0.950915	−0.475458	+	0.879739i	$0.657717\pi$
$390$	0	0
$391$	12280.4	0.00406228
$392$	0	0
$393$	0	0
$394$	−376116.	−0.122062
$395$	−4.11468e6	−1.32692
$396$	0	0
$397$	−4.34266e6	−1.38286	−0.691432	−	0.722442i	$-0.743020\pi$
$397$	−4.34266e6	−1.38286	−0.691432	+	0.722442i	$0.743020\pi$
$398$	−496274.	−0.157041
$399$	0	0
$400$	7.45534e6	2.32979
$401$	−3.40304e6	−1.05683	−0.528417	−	0.848985i	$-0.677214\pi$
$401$	−3.40304e6	−1.05683	−0.528417	+	0.848985i	$0.677214\pi$
$402$	0	0
$403$	1.55436e6	0.476749
$404$	−3.35070e6	−1.02137
$405$	0	0
$406$	0	0
$407$	5.15359e6	1.54214
$408$	0	0
$409$	5.29431e6	1.56495	0.782477	−	0.622680i	$-0.213956\pi$
$409$	5.29431e6	1.56495	0.782477	+	0.622680i	$0.213956\pi$
$410$	344488.	0.101208
$411$	0	0
$412$	4.94262e6	1.43455
$413$	0	0
$414$	0	0
$415$	1.06663e7	3.04014
$416$	488775.	0.138476
$417$	0	0
$418$	−521045.	−0.145859
$419$	2.87267e6	0.799376	0.399688	−	0.916651i	$-0.369119\pi$
$419$	2.87267e6	0.799376	0.399688	+	0.916651i	$0.369119\pi$
$420$	0	0
$421$	2.08688e6	0.573843	0.286921	−	0.957954i	$-0.407368\pi$
$421$	2.08688e6	0.573843	0.286921	+	0.957954i	$0.407368\pi$
$422$	451541.	0.123429
$423$	0	0
$424$	891319.	0.240779
$425$	488273.	0.131127
$426$	0	0
$427$	0	0
$428$	−2.80252e6	−0.739501
$429$	0	0
$430$	−491148.	−0.128098
$431$	2.42863e6	0.629751	0.314876	−	0.949133i	$-0.398037\pi$
$431$	2.42863e6	0.629751	0.314876	+	0.949133i	$0.398037\pi$
$432$	0	0
$433$	−956219.	−0.245097	−0.122548	−	0.992463i	$-0.539107\pi$
$433$	−956219.	−0.245097	−0.122548	+	0.992463i	$0.539107\pi$
$434$	0	0
$435$	0	0
$436$	−5.25853e6	−1.32479
$437$	−257426.	−0.0644836
$438$	0	0
$439$	−2.64205e6	−0.654304	−0.327152	−	0.944972i	$-0.606089\pi$
$439$	−2.64205e6	−0.654304	−0.327152	+	0.944972i	$0.606089\pi$
$440$	2.59867e6	0.639910
$441$	0	0
$442$	10316.4	0.00251174
$443$	3.43966e6	0.832733	0.416366	−	0.909197i	$-0.363303\pi$
$443$	3.43966e6	0.832733	0.416366	+	0.909197i	$0.363303\pi$
$444$	0	0
$445$	1.17550e7	2.81399
$446$	−3347.56	−0.000796878 0
$447$	0	0
$448$	0	0
$449$	−4.39903e6	−1.02977	−0.514886	−	0.857259i	$-0.672166\pi$
$449$	−4.39903e6	−1.02977	−0.514886	+	0.857259i	$0.672166\pi$
$450$	0	0
$451$	−2.08067e6	−0.481684
$452$	−3.63299e6	−0.836407
$453$	0	0
$454$	893501.	0.203449
$455$	0	0
$456$	0	0
$457$	−2.25110e6	−0.504202	−0.252101	−	0.967701i	$-0.581122\pi$
$457$	−2.25110e6	−0.504202	−0.252101	+	0.967701i	$0.581122\pi$
$458$	−636316.	−0.141745
$459$	0	0
$460$	635611.	0.140054
$461$	1.85307e6	0.406107	0.203053	−	0.979168i	$-0.434913\pi$
$461$	1.85307e6	0.406107	0.203053	+	0.979168i	$0.434913\pi$
$462$	0	0
$463$	−3.01089e6	−0.652744	−0.326372	−	0.945241i	$-0.605826\pi$
$463$	−3.01089e6	−0.652744	−0.326372	+	0.945241i	$0.605826\pi$
$464$	−4.16904e6	−0.898962
$465$	0	0
$466$	−924667.	−0.197252
$467$	−1.06725e6	−0.226452	−0.113226	−	0.993569i	$-0.536118\pi$
$467$	−1.06725e6	−0.226452	−0.113226	+	0.993569i	$0.536118\pi$
$468$	0	0
$469$	0	0
$470$	361694.	0.0755260
$471$	0	0
$472$	−175433.	−0.0362456
$473$	2.96648e6	0.609662
$474$	0	0
$475$	−1.02354e7	−2.08147
$476$	0	0
$477$	0	0
$478$	1.34875e6	0.269999
$479$	−5.68788e6	−1.13269	−0.566346	−	0.824168i	$-0.691643\pi$
$479$	−5.68788e6	−1.13269	−0.566346	+	0.824168i	$0.691643\pi$
$480$	0	0
$481$	2.13897e6	0.421544
$482$	−752211.	−0.147476
$483$	0	0
$484$	−2.71749e6	−0.527295
$485$	3.16059e6	0.610117
$486$	0	0
$487$	−6.84477e6	−1.30779	−0.653893	−	0.756587i	$-0.726865\pi$
$487$	−6.84477e6	−1.30779	−0.653893	+	0.756587i	$0.726865\pi$
$488$	−532867.	−0.101291
$489$	0	0
$490$	0	0
$491$	−5.60132e6	−1.04854	−0.524272	−	0.851551i	$-0.675662\pi$
$491$	−5.60132e6	−1.04854	−0.524272	+	0.851551i	$0.675662\pi$
$492$	0	0
$493$	−273043.	−0.0505958
$494$	−216257.	−0.0398706
$495$	0	0
$496$	7.25708e6	1.32452
$497$	0	0
$498$	0	0
$499$	−2.33529e6	−0.419845	−0.209923	−	0.977718i	$-0.567321\pi$
$499$	−2.33529e6	−0.419845	−0.209923	+	0.977718i	$0.567321\pi$
$500$	1.50566e7	2.69341
$501$	0	0
$502$	905455.	0.160364
$503$	−1.08278e7	−1.90819	−0.954093	−	0.299510i	$-0.903177\pi$
$503$	−1.08278e7	−1.90819	−0.954093	+	0.299510i	$0.903177\pi$
$504$	0	0
$505$	−1.11273e7	−1.94161
$506$	76521.1	0.0132863
$507$	0	0
$508$	−6.32240e6	−1.08698
$509$	3.17847e6	0.543781	0.271890	−	0.962328i	$-0.412351\pi$
$509$	3.17847e6	0.543781	0.271890	+	0.962328i	$0.412351\pi$
$510$	0	0
$511$	0	0
$512$	3.82859e6	0.645452
$513$	0	0
$514$	936107.	0.156285
$515$	1.64139e7	2.72705
$516$	0	0
$517$	−2.18460e6	−0.359455
$518$	0	0
$519$	0	0
$520$	1.07856e6	0.174919
$521$	−6.35837e6	−1.02625	−0.513123	−	0.858315i	$-0.671511\pi$
$521$	−6.35837e6	−1.02625	−0.513123	+	0.858315i	$0.671511\pi$
$522$	0	0
$523$	7.22187e6	1.15450	0.577252	−	0.816566i	$-0.304125\pi$
$523$	7.22187e6	1.15450	0.577252	+	0.816566i	$0.304125\pi$
$524$	−1.20697e6	−0.192029
$525$	0	0
$526$	−354531.	−0.0558714
$527$	475288.	0.0745471
$528$	0	0
$529$	−6.39854e6	−0.994126
$530$	1.46538e6	0.226601
$531$	0	0
$532$	0	0
$533$	−863574.	−0.131668
$534$	0	0
$535$	−9.30685e6	−1.40578
$536$	665290.	0.100023
$537$	0	0
$538$	591106.	0.0880461
$539$	0	0
$540$	0	0
$541$	−9.43165e6	−1.38546	−0.692731	−	0.721196i	$-0.743593\pi$
$541$	−9.43165e6	−1.38546	−0.692731	+	0.721196i	$0.743593\pi$
$542$	183971.	0.0268999
$543$	0	0
$544$	149456.	0.0216529
$545$	−1.74630e7	−2.51842
$546$	0	0
$547$	9.91568e6	1.41695	0.708474	−	0.705737i	$-0.249384\pi$
$547$	9.91568e6	1.41695	0.708474	+	0.705737i	$0.249384\pi$
$548$	−7.58394e6	−1.07881
$549$	0	0
$550$	3.04251e6	0.428870
$551$	5.72364e6	0.803144
$552$	0	0
$553$	0	0
$554$	−1.92106e6	−0.265929
$555$	0	0
$556$	−1.66638e6	−0.228605
$557$	−9.64613e6	−1.31739	−0.658696	−	0.752409i	$-0.728892\pi$
$557$	−9.64613e6	−1.31739	−0.658696	+	0.752409i	$0.728892\pi$
$558$	0	0
$559$	1.23123e6	0.166651
$560$	0	0
$561$	0	0
$562$	−1.99049e6	−0.265839
$563$	3.09504e6	0.411524	0.205762	−	0.978602i	$-0.434033\pi$
$563$	3.09504e6	0.411524	0.205762	+	0.978602i	$0.434033\pi$
$564$	0	0
$565$	−1.20648e7	−1.59000
$566$	−206396.	−0.0270807
$567$	0	0
$568$	1.94509e6	0.252970
$569$	1.23185e7	1.59507	0.797533	−	0.603275i	$-0.206138\pi$
$569$	1.23185e7	1.59507	0.797533	+	0.603275i	$0.206138\pi$
$570$	0	0
$571$	−3.23510e6	−0.415238	−0.207619	−	0.978210i	$-0.566571\pi$
$571$	−3.23510e6	−0.415238	−0.207619	+	0.978210i	$0.566571\pi$
$572$	−3.22507e6	−0.412144
$573$	0	0
$574$	0	0
$575$	1.50318e6	0.189601
$576$	0	0
$577$	9.05049e6	1.13170	0.565852	−	0.824507i	$-0.308547\pi$
$577$	9.05049e6	1.13170	0.565852	+	0.824507i	$0.308547\pi$
$578$	−1.11968e6	−0.139403
$579$	0	0
$580$	−1.41322e7	−1.74438
$581$	0	0
$582$	0	0
$583$	−8.85077e6	−1.07847
$584$	1.57245e6	0.190785
$585$	0	0
$586$	51411.6	0.00618468
$587$	−2.13170e6	−0.255347	−0.127673	−	0.991816i	$-0.540751\pi$
$587$	−2.13170e6	−0.255347	−0.127673	+	0.991816i	$0.540751\pi$
$588$	0	0
$589$	−9.96318e6	−1.18334
$590$	−288422.	−0.0341113
$591$	0	0
$592$	9.98652e6	1.17114
$593$	−7.26933e6	−0.848903	−0.424451	−	0.905451i	$-0.639533\pi$
$593$	−7.26933e6	−0.848903	−0.424451	+	0.905451i	$0.639533\pi$
$594$	0	0
$595$	0	0
$596$	4.04927e6	0.466941
$597$	0	0
$598$	31759.8	0.00363182
$599$	2.20565e6	0.251171	0.125585	−	0.992083i	$-0.459919\pi$
$599$	2.20565e6	0.251171	0.125585	+	0.992083i	$0.459919\pi$
$600$	0	0
$601$	1.00121e7	1.13067	0.565336	−	0.824860i	$-0.308746\pi$
$601$	1.00121e7	1.13067	0.565336	+	0.824860i	$0.308746\pi$
$602$	0	0
$603$	0	0
$604$	−4.80679e6	−0.536121
$605$	−9.02447e6	−1.00238
$606$	0	0
$607$	−6.32139e6	−0.696371	−0.348185	−	0.937426i	$-0.613202\pi$
$607$	−6.32139e6	−0.696371	−0.348185	+	0.937426i	$0.613202\pi$
$608$	−3.13295e6	−0.343712
$609$	0	0
$610$	−876065.	−0.0953261
$611$	−906706.	−0.0982570
$612$	0	0
$613$	1.43078e7	1.53788	0.768941	−	0.639319i	$-0.220784\pi$
$613$	1.43078e7	1.53788	0.768941	+	0.639319i	$0.220784\pi$
$614$	1.86313e6	0.199445
$615$	0	0
$616$	0	0
$617$	−1.73991e7	−1.83999	−0.919993	−	0.391936i	$-0.871806\pi$
$617$	−1.73991e7	−1.83999	−0.919993	+	0.391936i	$0.871806\pi$
$618$	0	0
$619$	−8.26736e6	−0.867242	−0.433621	−	0.901095i	$-0.642764\pi$
$619$	−8.26736e6	−0.867242	−0.433621	+	0.901095i	$0.642764\pi$
$620$	2.46001e7	2.57014
$621$	0	0
$622$	−1.67497e6	−0.173592
$623$	0	0
$624$	0	0
$625$	2.58422e7	2.64625
$626$	148287.	0.0151240
$627$	0	0
$628$	−4.74346e6	−0.479950
$629$	654048.	0.0659148
$630$	0	0
$631$	2.83238e6	0.283190	0.141595	−	0.989925i	$-0.454777\pi$
$631$	2.83238e6	0.283190	0.141595	+	0.989925i	$0.454777\pi$
$632$	−1.97919e6	−0.197104
$633$	0	0
$634$	795084.	0.0785579
$635$	−2.09960e7	−2.06634
$636$	0	0
$637$	0	0
$638$	−1.70138e6	−0.165481
$639$	0	0
$640$	1.02782e7	0.991902
$641$	3.57176e6	0.343350	0.171675	−	0.985154i	$-0.445082\pi$
$641$	3.57176e6	0.343350	0.171675	+	0.985154i	$0.445082\pi$
$642$	0	0
$643$	5.96911e6	0.569354	0.284677	−	0.958624i	$-0.408114\pi$
$643$	5.96911e6	0.569354	0.284677	+	0.958624i	$0.408114\pi$
$644$	0	0
$645$	0	0
$646$	−66126.4	−0.00623439
$647$	−1.68276e6	−0.158038	−0.0790189	−	0.996873i	$-0.525179\pi$
$647$	−1.68276e6	−0.158038	−0.0790189	+	0.996873i	$0.525179\pi$
$648$	0	0
$649$	1.74204e6	0.162348
$650$	1.26278e6	0.117232
$651$	0	0
$652$	1.03275e6	0.0951427
$653$	−1.70305e7	−1.56295	−0.781475	−	0.623937i	$-0.785532\pi$
$653$	−1.70305e7	−1.56295	−0.781475	+	0.623937i	$0.785532\pi$
$654$	0	0
$655$	−4.00821e6	−0.365046
$656$	−4.03189e6	−0.365804
$657$	0	0
$658$	0	0
$659$	1.99303e7	1.78772	0.893862	−	0.448342i	$-0.147985\pi$
$659$	1.99303e7	1.78772	0.893862	+	0.448342i	$0.147985\pi$
$660$	0	0
$661$	−9.55735e6	−0.850812	−0.425406	−	0.905003i	$-0.639869\pi$
$661$	−9.55735e6	−0.850812	−0.425406	+	0.905003i	$0.639869\pi$
$662$	1.32843e6	0.117813
$663$	0	0
$664$	5.13056e6	0.451591
$665$	0	0
$666$	0	0
$667$	−840579.	−0.0731584
$668$	−6.82666e6	−0.591926
$669$	0	0
$670$	1.09378e6	0.0941330
$671$	5.29135e6	0.453691
$672$	0	0
$673$	1.80397e7	1.53530	0.767648	−	0.640872i	$-0.221427\pi$
$673$	1.80397e7	1.53530	0.767648	+	0.640872i	$0.221427\pi$
$674$	−786879.	−0.0667204
$675$	0	0
$676$	1.03106e7	0.867798
$677$	−1.44741e7	−1.21372	−0.606861	−	0.794808i	$-0.707571\pi$
$677$	−1.44741e7	−1.21372	−0.606861	+	0.794808i	$0.707571\pi$
$678$	0	0
$679$	0	0
$680$	329800.	0.0273513
$681$	0	0
$682$	2.96160e6	0.243818
$683$	−2.68785e6	−0.220472	−0.110236	−	0.993905i	$-0.535161\pi$
$683$	−2.68785e6	−0.220472	−0.110236	+	0.993905i	$0.535161\pi$
$684$	0	0
$685$	−2.51854e7	−2.05080
$686$	0	0
$687$	0	0
$688$	5.74839e6	0.462994
$689$	−3.67347e6	−0.294801
$690$	0	0
$691$	−4.67627e6	−0.372567	−0.186284	−	0.982496i	$-0.559644\pi$
$691$	−4.67627e6	−0.372567	−0.186284	+	0.982496i	$0.559644\pi$
$692$	1.32163e7	1.04917
$693$	0	0
$694$	−3.20802e6	−0.252836
$695$	−5.53385e6	−0.434576
$696$	0	0
$697$	−264061.	−0.0205884
$698$	−2.26330e6	−0.175834
$699$	0	0
$700$	0	0
$701$	−6.34801e6	−0.487913	−0.243957	−	0.969786i	$-0.578445\pi$
$701$	−6.34801e6	−0.487913	−0.243957	+	0.969786i	$0.578445\pi$
$702$	0	0
$703$	−1.37104e7	−1.04631
$704$	−1.44262e7	−1.09703
$705$	0	0
$706$	658492.	0.0497208
$707$	0	0
$708$	0	0
$709$	−6.41052e6	−0.478936	−0.239468	−	0.970904i	$-0.576973\pi$
$709$	−6.41052e6	−0.478936	−0.239468	+	0.970904i	$0.576973\pi$
$710$	3.19785e6	0.238074
$711$	0	0
$712$	5.65425e6	0.417998
$713$	1.46320e6	0.107790
$714$	0	0
$715$	−1.07101e7	−0.783481
$716$	109625.	0.00799146
$717$	0	0
$718$	2.13320e6	0.154426
$719$	7.34505e6	0.529874	0.264937	−	0.964266i	$-0.414649\pi$
$719$	7.34505e6	0.529874	0.264937	+	0.964266i	$0.414649\pi$
$720$	0	0
$721$	0	0
$722$	−571944.	−0.0408329
$723$	0	0
$724$	1.86557e7	1.32271
$725$	−3.34218e7	−2.36148
$726$	0	0
$727$	−1.57839e7	−1.10759	−0.553793	−	0.832655i	$-0.686820\pi$
$727$	−1.57839e7	−1.10759	−0.553793	+	0.832655i	$0.686820\pi$
$728$	0	0
$729$	0	0
$730$	2.58520e6	0.179551
$731$	376480.	0.0260584
$732$	0	0
$733$	−1.35367e7	−0.930579	−0.465289	−	0.885159i	$-0.654050\pi$
$733$	−1.35367e7	−0.930579	−0.465289	+	0.885159i	$0.654050\pi$
$734$	−1.15048e6	−0.0788205
$735$	0	0
$736$	460109.	0.0313088
$737$	−6.60631e6	−0.448012
$738$	0	0
$739$	−7.33702e6	−0.494207	−0.247103	−	0.968989i	$-0.579479\pi$
$739$	−7.33702e6	−0.494207	−0.247103	+	0.968989i	$0.579479\pi$
$740$	3.38524e7	2.27253
$741$	0	0
$742$	0	0
$743$	−1.20844e7	−0.803068	−0.401534	−	0.915844i	$-0.631523\pi$
$743$	−1.20844e7	−0.803068	−0.401534	+	0.915844i	$0.631523\pi$
$744$	0	0
$745$	1.34472e7	0.887649
$746$	−1.61869e6	−0.106492
$747$	0	0
$748$	−986151.	−0.0644450
$749$	0	0
$750$	0	0
$751$	8.78176e6	0.568175	0.284087	−	0.958798i	$-0.408309\pi$
$751$	8.78176e6	0.568175	0.284087	+	0.958798i	$0.408309\pi$
$752$	−4.23327e6	−0.272980
$753$	0	0
$754$	−706149.	−0.0452343
$755$	−1.59628e7	−1.01916
$756$	0	0
$757$	465874.	0.0295480	0.0147740	−	0.999891i	$-0.495297\pi$
$757$	465874.	0.0295480	0.0147740	+	0.999891i	$0.495297\pi$
$758$	−329685.	−0.0208414
$759$	0	0
$760$	−6.91339e6	−0.434167
$761$	1.49978e7	0.938786	0.469393	−	0.882989i	$-0.344473\pi$
$761$	1.49978e7	0.938786	0.469393	+	0.882989i	$0.344473\pi$
$762$	0	0
$763$	0	0
$764$	2.59984e7	1.61144
$765$	0	0
$766$	−3.06536e6	−0.188760
$767$	723026.	0.0443778
$768$	0	0
$769$	1.85606e7	1.13181	0.565907	−	0.824469i	$-0.308526\pi$
$769$	1.85606e7	1.13181	0.565907	+	0.824469i	$0.308526\pi$
$770$	0	0
$771$	0	0
$772$	−6.89422e6	−0.416334
$773$	1.22760e7	0.738941	0.369470	−	0.929243i	$-0.379539\pi$
$773$	1.22760e7	0.738941	0.369470	+	0.929243i	$0.379539\pi$
$774$	0	0
$775$	5.81775e7	3.47937
$776$	1.52027e6	0.0906286
$777$	0	0
$778$	−2.24432e6	−0.132934
$779$	5.53534e6	0.326814
$780$	0	0
$781$	−1.93147e7	−1.13308
$782$	9711.39	0.000567890 0
$783$	0	0
$784$	0	0
$785$	−1.57525e7	−0.912380
$786$	0	0
$787$	−5.26991e6	−0.303296	−0.151648	−	0.988435i	$-0.548458\pi$
$787$	−5.26991e6	−0.303296	−0.151648	+	0.988435i	$0.548458\pi$
$788$	1.49221e7	0.856082
$789$	0	0
$790$	−3.25391e6	−0.185497
$791$	0	0
$792$	0	0
$793$	2.19615e6	0.124016
$794$	−3.43420e6	−0.193319
$795$	0	0
$796$	1.96893e7	1.10141
$797$	1.11889e7	0.623940	0.311970	−	0.950092i	$-0.399011\pi$
$797$	1.11889e7	0.623940	0.311970	+	0.950092i	$0.399011\pi$
$798$	0	0
$799$	−277250.	−0.0153640
$800$	1.82941e7	1.01062
$801$	0	0
$802$	−2.69114e6	−0.147741
$803$	−1.56144e7	−0.854547
$804$	0	0
$805$	0	0
$806$	1.22920e6	0.0666476
$807$	0	0
$808$	−5.35231e6	−0.288412
$809$	−7.05051e6	−0.378747	−0.189374	−	0.981905i	$-0.560646\pi$
$809$	−7.05051e6	−0.378747	−0.189374	+	0.981905i	$0.560646\pi$
$810$	0	0
$811$	2.54873e7	1.36073	0.680364	−	0.732875i	$-0.261822\pi$
$811$	2.54873e7	1.36073	0.680364	+	0.732875i	$0.261822\pi$
$812$	0	0
$813$	0	0
$814$	4.07548e6	0.215585
$815$	3.42964e6	0.180865
$816$	0	0
$817$	−7.89192e6	−0.413645
$818$	4.18677e6	0.218774
$819$	0	0
$820$	−1.36673e7	−0.709820
$821$	2.34879e7	1.21615	0.608073	−	0.793881i	$-0.291943\pi$
$821$	2.34879e7	1.21615	0.608073	+	0.793881i	$0.291943\pi$
$822$	0	0
$823$	−1.20903e7	−0.622213	−0.311107	−	0.950375i	$-0.600700\pi$
$823$	−1.20903e7	−0.622213	−0.311107	+	0.950375i	$0.600700\pi$
$824$	7.89521e6	0.405084
$825$	0	0
$826$	0	0
$827$	−3.33335e6	−0.169480	−0.0847398	−	0.996403i	$-0.527006\pi$
$827$	−3.33335e6	−0.169480	−0.0847398	+	0.996403i	$0.527006\pi$
$828$	0	0
$829$	7.70534e6	0.389408	0.194704	−	0.980862i	$-0.437625\pi$
$829$	7.70534e6	0.389408	0.194704	+	0.980862i	$0.437625\pi$
$830$	8.43496e6	0.424999
$831$	0	0
$832$	−5.98752e6	−0.299874
$833$	0	0
$834$	0	0
$835$	−2.26706e7	−1.12524
$836$	2.06721e7	1.02298
$837$	0	0
$838$	2.27172e6	0.111749
$839$	2.40843e7	1.18121	0.590607	−	0.806959i	$-0.298888\pi$
$839$	2.40843e7	1.18121	0.590607	+	0.806959i	$0.298888\pi$
$840$	0	0
$841$	−1.82163e6	−0.0888116
$842$	1.65032e6	0.0802208
$843$	0	0
$844$	−1.79146e7	−0.865665
$845$	3.42405e7	1.64967
$846$	0	0
$847$	0	0
$848$	−1.71509e7	−0.819023
$849$	0	0
$850$	386129.	0.0183310
$851$	2.01352e6	0.0953088
$852$	0	0
$853$	2.82938e7	1.33143	0.665716	−	0.746206i	$-0.268126\pi$
$853$	2.82938e7	1.33143	0.665716	+	0.746206i	$0.268126\pi$
$854$	0	0
$855$	0	0
$856$	−4.47666e6	−0.208819
$857$	2.16176e7	1.00544	0.502719	−	0.864450i	$-0.332333\pi$
$857$	2.16176e7	1.00544	0.502719	+	0.864450i	$0.332333\pi$
$858$	0	0
$859$	−3.25382e7	−1.50456	−0.752282	−	0.658841i	$-0.771047\pi$
$859$	−3.25382e7	−1.50456	−0.752282	+	0.658841i	$0.771047\pi$
$860$	1.94859e7	0.898411
$861$	0	0
$862$	1.92058e6	0.0880366
$863$	−8.06384e6	−0.368566	−0.184283	−	0.982873i	$-0.558996\pi$
$863$	−8.06384e6	−0.368566	−0.184283	+	0.982873i	$0.558996\pi$
$864$	0	0
$865$	4.38900e7	1.99446
$866$	−756183.	−0.0342635
$867$	0	0
$868$	0	0
$869$	1.96533e7	0.882849
$870$	0	0
$871$	−2.74192e6	−0.122464
$872$	−8.39983e6	−0.374093
$873$	0	0
$874$	−203574.	−0.00901454
$875$	0	0
$876$	0	0
$877$	1.70312e7	0.747732	0.373866	−	0.927483i	$-0.378032\pi$
$877$	1.70312e7	0.747732	0.373866	+	0.927483i	$0.378032\pi$
$878$	−2.08934e6	−0.0914690
$879$	0	0
$880$	−5.00038e7	−2.17669
$881$	1.24740e7	0.541459	0.270729	−	0.962655i	$-0.412735\pi$
$881$	1.24740e7	0.541459	0.270729	+	0.962655i	$0.412735\pi$
$882$	0	0
$883$	7.28955e6	0.314629	0.157315	−	0.987549i	$-0.449716\pi$
$883$	7.28955e6	0.314629	0.157315	+	0.987549i	$0.449716\pi$
$884$	−409298.	−0.0176160
$885$	0	0
$886$	2.72010e6	0.116413
$887$	1.87675e7	0.800937	0.400469	−	0.916310i	$-0.368847\pi$
$887$	1.87675e7	0.800937	0.400469	+	0.916310i	$0.368847\pi$
$888$	0	0
$889$	0	0
$890$	9.29592e6	0.393385
$891$	0	0
$892$	132812.	0.00558889
$893$	5.81181e6	0.243884
$894$	0	0
$895$	364052.	0.0151917
$896$	0	0
$897$	0	0
$898$	−3.47878e6	−0.143958
$899$	−3.25330e7	−1.34253
$900$	0	0
$901$	−1.12326e6	−0.0460966
$902$	−1.64541e6	−0.0673375
$903$	0	0
$904$	−5.80323e6	−0.236183
$905$	6.19535e7	2.51446
$906$	0	0
$907$	1.26367e6	0.0510052	0.0255026	−	0.999675i	$-0.491881\pi$
$907$	1.26367e6	0.0510052	0.0255026	+	0.999675i	$0.491881\pi$
$908$	−3.54490e7	−1.42689
$909$	0	0
$910$	0	0
$911$	2.47718e7	0.988921	0.494461	−	0.869200i	$-0.335366\pi$
$911$	2.47718e7	0.988921	0.494461	+	0.869200i	$0.335366\pi$
$912$	0	0
$913$	−5.09463e7	−2.02272
$914$	−1.78018e6	−0.0704854
$915$	0	0
$916$	2.52454e7	0.994130
$917$	0	0
$918$	0	0
$919$	−9.36676e6	−0.365848	−0.182924	−	0.983127i	$-0.558556\pi$
$919$	−9.36676e6	−0.365848	−0.182924	+	0.983127i	$0.558556\pi$
$920$	1.01531e6	0.0395483
$921$	0	0
$922$	1.46542e6	0.0567721
$923$	−8.01648e6	−0.309727
$924$	0	0
$925$	8.00586e7	3.07648
$926$	−2.38103e6	−0.0912509
$927$	0	0
$928$	−1.02301e7	−0.389951
$929$	−6.87350e6	−0.261300	−0.130650	−	0.991429i	$-0.541706\pi$
$929$	−6.87350e6	−0.261300	−0.130650	+	0.991429i	$0.541706\pi$
$930$	0	0
$931$	0	0
$932$	3.66855e7	1.38342
$933$	0	0
$934$	−843990.	−0.0316570
$935$	−3.27490e6	−0.122509
$936$	0	0
$937$	4.26197e7	1.58585	0.792923	−	0.609322i	$-0.208558\pi$
$937$	4.26197e7	1.58585	0.792923	+	0.609322i	$0.208558\pi$
$938$	0	0
$939$	0	0
$940$	−1.43499e7	−0.529701
$941$	4.50226e6	0.165751	0.0828755	−	0.996560i	$-0.473590\pi$
$941$	4.50226e6	0.165751	0.0828755	+	0.996560i	$0.473590\pi$
$942$	0	0
$943$	−812926.	−0.0297695
$944$	3.37569e6	0.123291
$945$	0	0
$946$	2.34591e6	0.0852282
$947$	−3.37725e7	−1.22374	−0.611870	−	0.790959i	$-0.709582\pi$
$947$	−3.37725e7	−1.22374	−0.611870	+	0.790959i	$0.709582\pi$
$948$	0	0
$949$	−6.48068e6	−0.233590
$950$	−8.09419e6	−0.290981
$951$	0	0
$952$	0	0
$953$	−4.81813e7	−1.71849	−0.859244	−	0.511566i	$-0.829066\pi$
$953$	−4.81813e7	−1.71849	−0.859244	+	0.511566i	$0.829066\pi$
$954$	0	0
$955$	8.63379e7	3.06332
$956$	−5.35107e7	−1.89363
$957$	0	0
$958$	−4.49800e6	−0.158346
$959$	0	0
$960$	0	0
$961$	2.80012e7	0.978066
$962$	1.69151e6	0.0589301
$963$	0	0
$964$	2.98434e7	1.03432
$965$	−2.28950e7	−0.791446
$966$	0	0
$967$	−3.54640e7	−1.21961	−0.609805	−	0.792551i	$-0.708752\pi$
$967$	−3.54640e7	−1.21961	−0.609805	+	0.792551i	$0.708752\pi$
$968$	−4.34084e6	−0.148897
$969$	0	0
$970$	2.49941e6	0.0852919
$971$	−7.84296e6	−0.266951	−0.133476	−	0.991052i	$-0.542614\pi$
$971$	−7.84296e6	−0.266951	−0.133476	+	0.991052i	$0.542614\pi$
$972$	0	0
$973$	0	0
$974$	−5.41288e6	−0.182823
$975$	0	0
$976$	1.02535e7	0.344545
$977$	2.47534e7	0.829658	0.414829	−	0.909899i	$-0.363841\pi$
$977$	2.47534e7	0.829658	0.414829	+	0.909899i	$0.363841\pi$
$978$	0	0
$979$	−5.61464e7	−1.87226
$980$	0	0
$981$	0	0
$982$	−4.42955e6	−0.146582
$983$	4.53129e7	1.49568	0.747838	−	0.663881i	$-0.231092\pi$
$983$	4.53129e7	1.49568	0.747838	+	0.663881i	$0.231092\pi$
$984$	0	0
$985$	4.95548e7	1.62740
$986$	−215924.	−0.00707308
$987$	0	0
$988$	8.57985e6	0.279632
$989$	1.15902e6	0.0376789
$990$	0	0
$991$	−3.74143e7	−1.21019	−0.605095	−	0.796154i	$-0.706865\pi$
$991$	−3.74143e7	−1.21019	−0.605095	+	0.796154i	$0.706865\pi$
$992$	1.78076e7	0.574548
$993$	0	0
$994$	0	0
$995$	6.53861e7	2.09376
$996$	0	0
$997$	−4.65278e7	−1.48243	−0.741216	−	0.671267i	$-0.765751\pi$
$997$	−4.65278e7	−1.48243	−0.741216	+	0.671267i	$0.765751\pi$
$998$	−1.84676e6	−0.0586926
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.6.a.v.1.3		4
3.2	odd	2		147.6.a.l.1.2		4
7.3	odd	6		63.6.e.e.37.2		8
7.5	odd	6		63.6.e.e.46.2		8
7.6	odd	2		441.6.a.w.1.3		4
21.2	odd	6		147.6.e.o.67.3		8
21.5	even	6		21.6.e.c.4.3	✓	8
21.11	odd	6		147.6.e.o.79.3		8
21.17	even	6		21.6.e.c.16.3	yes	8
21.20	even	2		147.6.a.m.1.2		4
84.47	odd	6		336.6.q.j.193.4		8
84.59	odd	6		336.6.q.j.289.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.6.e.c.4.3	✓	8	21.5	even	6
21.6.e.c.16.3	yes	8	21.17	even	6
63.6.e.e.37.2		8	7.3	odd	6
63.6.e.e.46.2		8	7.5	odd	6
147.6.a.l.1.2		4	3.2	odd	2
147.6.a.m.1.2		4	21.20	even	2
147.6.e.o.67.3		8	21.2	odd	6
147.6.e.o.79.3		8	21.11	odd	6
336.6.q.j.193.4		8	84.47	odd	6
336.6.q.j.289.4		8	84.59	odd	6
441.6.a.v.1.3		4	1.1	even	1	trivial
441.6.a.w.1.3		4	7.6	odd	2

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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)

\(f(q)\)	\(=\)	\(q+0.790805 q^{2} -31.3746 q^{4} -104.192 q^{5} -50.1170 q^{8} +O(q^{10})\)	\(q+0.790805 q^{2} -31.3746 q^{4} -104.192 q^{5} -50.1170 q^{8} -82.3953 q^{10} +497.660 q^{11} +206.551 q^{13} +964.355 q^{16} +63.1586 q^{17} -1323.95 q^{19} +3268.98 q^{20} +393.552 q^{22} +194.437 q^{23} +7730.91 q^{25} +163.342 q^{26} -4323.14 q^{29} +7525.31 q^{31} +2366.36 q^{32} +49.9461 q^{34} +10355.6 q^{37} -1046.99 q^{38} +5221.77 q^{40} -4180.92 q^{41} +5960.87 q^{43} -15613.9 q^{44} +153.762 q^{46} -4389.74 q^{47} +6113.64 q^{50} -6480.47 q^{52} -17784.8 q^{53} -51852.0 q^{55} -3418.76 q^{58} +3500.47 q^{59} +10632.5 q^{61} +5951.06 q^{62} -28988.0 q^{64} -21520.9 q^{65} -13274.7 q^{67} -1981.58 q^{68} -38811.1 q^{71} -31375.6 q^{73} +8189.30 q^{74} +41538.6 q^{76} +39491.5 q^{79} -100478. q^{80} -3306.29 q^{82} -102372. q^{83} -6580.60 q^{85} +4713.88 q^{86} -24941.2 q^{88} -112821. q^{89} -6100.40 q^{92} -3471.43 q^{94} +137945. q^{95} -30334.3 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3 q^{2} + 69 q^{4} - 123 q^{8}+O(q^{10}) \) 4 * q - 3 * q^2 + 69 * q^4 - 123 * q^8	\( 4 q - 3 q^{2} + 69 q^{4} - 123 q^{8} - 283 q^{10} - 402 q^{11} - 462 q^{13} + 3273 q^{16} + 276 q^{17} - 510 q^{19} + 4719 q^{20} + 1375 q^{22} - 6900 q^{23} + 2814 q^{25} - 15138 q^{26} - 540 q^{29} + 6410 q^{31} - 15519 q^{32} - 21144 q^{34} + 15250 q^{37} - 41250 q^{38} + 8547 q^{40} + 4308 q^{41} + 29198 q^{43} - 70743 q^{44} + 61800 q^{46} - 15060 q^{47} + 7302 q^{50} + 47476 q^{52} - 13692 q^{53} - 73124 q^{55} + 52309 q^{58} + 34830 q^{59} + 5364 q^{61} + 16029 q^{62} - 73487 q^{64} - 66864 q^{65} - 5994 q^{67} - 58272 q^{68} - 89268 q^{71} - 59638 q^{73} + 185442 q^{74} - 21308 q^{76} - 44062 q^{79} - 33381 q^{80} - 57596 q^{82} - 208446 q^{83} + 36324 q^{85} + 136968 q^{86} + 87597 q^{88} - 77520 q^{89} - 158256 q^{92} + 73722 q^{94} + 221376 q^{95} + 188630 q^{97}+O(q^{100}) \) 4 * q - 3 * q^2 + 69 * q^4 - 123 * q^8 - 283 * q^10 - 402 * q^11 - 462 * q^13 + 3273 * q^16 + 276 * q^17 - 510 * q^19 + 4719 * q^20 + 1375 * q^22 - 6900 * q^23 + 2814 * q^25 - 15138 * q^26 - 540 * q^29 + 6410 * q^31 - 15519 * q^32 - 21144 * q^34 + 15250 * q^37 - 41250 * q^38 + 8547 * q^40 + 4308 * q^41 + 29198 * q^43 - 70743 * q^44 + 61800 * q^46 - 15060 * q^47 + 7302 * q^50 + 47476 * q^52 - 13692 * q^53 - 73124 * q^55 + 52309 * q^58 + 34830 * q^59 + 5364 * q^61 + 16029 * q^62 - 73487 * q^64 - 66864 * q^65 - 5994 * q^67 - 58272 * q^68 - 89268 * q^71 - 59638 * q^73 + 185442 * q^74 - 21308 * q^76 - 44062 * q^79 - 33381 * q^80 - 57596 * q^82 - 208446 * q^83 + 36324 * q^85 + 136968 * q^86 + 87597 * q^88 - 77520 * q^89 - 158256 * q^92 + 73722 * q^94 + 221376 * q^95 + 188630 * q^97

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