LMFDB - Embedded newform 441.6.a.bc.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:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 187x^{4} + 9570x^{2} - 135576 $ x^6 - 187x^4 + 9570x^2 - 135576
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 63)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-7.08933$ 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)

$f(q)$	$=$	$q-7.08933 q^{2} +18.2586 q^{4} -82.2041 q^{5} +97.4172 q^{8} +O(q^{10})$	$q-7.08933 q^{2} +18.2586 q^{4} -82.2041 q^{5} +97.4172 q^{8} +582.772 q^{10} +352.237 q^{11} -885.257 q^{13} -1274.90 q^{16} -425.038 q^{17} -1562.38 q^{19} -1500.93 q^{20} -2497.12 q^{22} +2788.25 q^{23} +3632.51 q^{25} +6275.88 q^{26} +3678.79 q^{29} -3591.52 q^{31} +5920.83 q^{32} +3013.24 q^{34} +14289.2 q^{37} +11076.3 q^{38} -8008.09 q^{40} +14325.8 q^{41} +7589.72 q^{43} +6431.36 q^{44} -19766.8 q^{46} -5768.35 q^{47} -25752.1 q^{50} -16163.6 q^{52} -25388.4 q^{53} -28955.3 q^{55} -26080.2 q^{58} +43223.3 q^{59} +19447.5 q^{61} +25461.4 q^{62} -1177.95 q^{64} +72771.7 q^{65} -29441.1 q^{67} -7760.61 q^{68} -51664.4 q^{71} +37945.6 q^{73} -101301. q^{74} -28527.0 q^{76} +53799.5 q^{79} +104802. q^{80} -101560. q^{82} +85967.6 q^{83} +34939.9 q^{85} -53806.1 q^{86} +34313.9 q^{88} -20819.9 q^{89} +50909.6 q^{92} +40893.7 q^{94} +128434. q^{95} -97587.2 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 182 q^{4}+O(q^{10}) $ 6 * q + 182 * q^4	$ 6 q + 182 q^{4} - 686 q^{10} + 154 q^{13} + 1898 q^{16} - 9422 q^{19} - 9146 q^{22} + 7526 q^{25} - 23422 q^{31} - 27804 q^{34} + 18182 q^{37} - 69258 q^{40} - 43686 q^{43} - 25332 q^{46} - 34272 q^{52} - 48160 q^{55} + 89782 q^{58} + 16156 q^{61} + 190290 q^{64} - 144650 q^{67} + 100058 q^{73} - 342720 q^{76} - 101994 q^{79} - 75712 q^{82} + 301176 q^{85} - 752310 q^{88} + 120456 q^{94} - 433048 q^{97}+O(q^{100}) $ 6 * q + 182 * q^4 - 686 * q^10 + 154 * q^13 + 1898 * q^16 - 9422 * q^19 - 9146 * q^22 + 7526 * q^25 - 23422 * q^31 - 27804 * q^34 + 18182 * q^37 - 69258 * q^40 - 43686 * q^43 - 25332 * q^46 - 34272 * q^52 - 48160 * q^55 + 89782 * q^58 + 16156 * q^61 + 190290 * q^64 - 144650 * q^67 + 100058 * q^73 - 342720 * q^76 - 101994 * q^79 - 75712 * q^82 + 301176 * q^85 - 752310 * q^88 + 120456 * q^94 - 433048 * 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 $

$2$	−7.08933	−1.25323	−0.626614	−	0.779330i	$-0.715560\pi$
$2$	−7.08933	−1.25323	−0.626614	+	0.779330i	$0.715560\pi$
$3$	0	0
$3$	0	0
$4$	18.2586	0.570582
$5$	−82.2041	−1.47051	−0.735256	−	0.677790i	$-0.762938\pi$
$5$	−82.2041	−1.47051	−0.735256	+	0.677790i	$0.762938\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	97.4172	0.538159
$9$	0	0
$10$	582.772	1.84289
$11$	352.237	0.877714	0.438857	−	0.898557i	$-0.355383\pi$
$11$	352.237	0.877714	0.438857	+	0.898557i	$0.355383\pi$
$12$	0	0
$13$	−885.257	−1.45282	−0.726408	−	0.687263i	$-0.758812\pi$
$13$	−885.257	−1.45282	−0.726408	+	0.687263i	$0.758812\pi$
$14$	0	0
$15$	0	0
$16$	−1274.90	−1.24502
$17$	−425.038	−0.356702	−0.178351	−	0.983967i	$-0.557076\pi$
$17$	−425.038	−0.356702	−0.178351	+	0.983967i	$0.557076\pi$
$18$	0	0
$19$	−1562.38	−0.992895	−0.496448	−	0.868067i	$-0.665363\pi$
$19$	−1562.38	−0.992895	−0.496448	+	0.868067i	$0.665363\pi$
$20$	−1500.93	−0.839047
$21$	0	0
$22$	−2497.12	−1.09998
$23$	2788.25	1.09904	0.549518	−	0.835482i	$-0.314811\pi$
$23$	2788.25	1.09904	0.549518	+	0.835482i	$0.314811\pi$
$24$	0	0
$25$	3632.51	1.16240
$26$	6275.88	1.82071
$27$	0	0
$28$	0	0
$29$	3678.79	0.812289	0.406144	−	0.913809i	$-0.366873\pi$
$29$	3678.79	0.812289	0.406144	+	0.913809i	$0.366873\pi$
$30$	0	0
$31$	−3591.52	−0.671233	−0.335617	−	0.941999i	$-0.608945\pi$
$31$	−3591.52	−0.671233	−0.335617	+	0.941999i	$0.608945\pi$
$32$	5920.83	1.02213
$33$	0	0
$34$	3013.24	0.447029
$35$	0	0
$36$	0	0
$37$	14289.2	1.71594	0.857971	−	0.513698i	$-0.171725\pi$
$37$	14289.2	1.71594	0.857971	+	0.513698i	$0.171725\pi$
$38$	11076.3	1.24432
$39$	0	0
$40$	−8008.09	−0.791369
$41$	14325.8	1.33094	0.665471	−	0.746424i	$-0.268231\pi$
$41$	14325.8	1.33094	0.665471	+	0.746424i	$0.268231\pi$
$42$	0	0
$43$	7589.72	0.625972	0.312986	−	0.949758i	$-0.398671\pi$
$43$	7589.72	0.625972	0.312986	+	0.949758i	$0.398671\pi$
$44$	6431.36	0.500808
$45$	0	0
$46$	−19766.8	−1.37734
$47$	−5768.35	−0.380896	−0.190448	−	0.981697i	$-0.560994\pi$
$47$	−5768.35	−0.380896	−0.190448	+	0.981697i	$0.560994\pi$
$48$	0	0
$49$	0	0
$50$	−25752.1	−1.45676
$51$	0	0
$52$	−16163.6	−0.828951
$53$	−25388.4	−1.24149	−0.620747	−	0.784011i	$-0.713171\pi$
$53$	−25388.4	−1.24149	−0.620747	+	0.784011i	$0.713171\pi$
$54$	0	0
$55$	−28955.3	−1.29069
$56$	0	0
$57$	0	0
$58$	−26080.2	−1.01798
$59$	43223.3	1.61655	0.808273	−	0.588807i	$-0.200402\pi$
$59$	43223.3	1.61655	0.808273	+	0.588807i	$0.200402\pi$
$60$	0	0
$61$	19447.5	0.669173	0.334586	−	0.942365i	$-0.391403\pi$
$61$	19447.5	0.669173	0.334586	+	0.942365i	$0.391403\pi$
$62$	25461.4	0.841209
$63$	0	0
$64$	−1177.95	−0.0359481
$65$	72771.7	2.13638
$66$	0	0
$67$	−29441.1	−0.801248	−0.400624	−	0.916242i	$-0.631207\pi$
$67$	−29441.1	−0.801248	−0.400624	+	0.916242i	$0.631207\pi$
$68$	−7760.61	−0.203528
$69$	0	0
$70$	0	0
$71$	−51664.4	−1.21631	−0.608157	−	0.793817i	$-0.708091\pi$
$71$	−51664.4	−1.21631	−0.608157	+	0.793817i	$0.708091\pi$
$72$	0	0
$73$	37945.6	0.833401	0.416701	−	0.909044i	$-0.363186\pi$
$73$	37945.6	0.833401	0.416701	+	0.909044i	$0.363186\pi$
$74$	−101301.	−2.15047
$75$	0	0
$76$	−28527.0	−0.566528
$77$	0	0
$78$	0	0
$79$	53799.5	0.969862	0.484931	−	0.874552i	$-0.338845\pi$
$79$	53799.5	0.969862	0.484931	+	0.874552i	$0.338845\pi$
$80$	104802.	1.83081
$81$	0	0
$82$	−101560.	−1.66797
$83$	85967.6	1.36974	0.684872	−	0.728663i	$-0.259858\pi$
$83$	85967.6	1.36974	0.684872	+	0.728663i	$0.259858\pi$
$84$	0	0
$85$	34939.9	0.524535
$86$	−53806.1	−0.784486
$87$	0	0
$88$	34313.9	0.472350
$89$	−20819.9	−0.278614	−0.139307	−	0.990249i	$-0.544487\pi$
$89$	−20819.9	−0.278614	−0.139307	+	0.990249i	$0.544487\pi$
$90$	0	0
$91$	0	0
$92$	50909.6	0.627090
$93$	0	0
$94$	40893.7	0.477350
$95$	128434.	1.46006
$96$	0	0
$97$	−97587.2	−1.05309	−0.526543	−	0.850149i	$-0.676512\pi$
$97$	−97587.2	−1.05309	−0.526543	+	0.850149i	$0.676512\pi$
$98$	0	0
$99$	0	0
$100$	66324.7	0.663247
$101$	−39202.1	−0.382389	−0.191195	−	0.981552i	$-0.561236\pi$
$101$	−39202.1	−0.382389	−0.191195	+	0.981552i	$0.561236\pi$
$102$	0	0
$103$	−19228.2	−0.178586	−0.0892928	−	0.996005i	$-0.528461\pi$
$103$	−19228.2	−0.178586	−0.0892928	+	0.996005i	$0.528461\pi$
$104$	−86239.2	−0.781847
$105$	0	0
$106$	179986.	1.55588
$107$	189821.	1.60282	0.801411	−	0.598115i	$-0.204083\pi$
$107$	189821.	1.60282	0.801411	+	0.598115i	$0.204083\pi$
$108$	0	0
$109$	−228043.	−1.83844	−0.919222	−	0.393740i	$-0.871181\pi$
$109$	−228043.	−1.83844	−0.919222	+	0.393740i	$0.871181\pi$
$110$	205274.	1.61753
$111$	0	0
$112$	0	0
$113$	−50309.6	−0.370642	−0.185321	−	0.982678i	$-0.559332\pi$
$113$	−50309.6	−0.370642	−0.185321	+	0.982678i	$0.559332\pi$
$114$	0	0
$115$	−229205.	−1.61615
$116$	67169.7	0.463477
$117$	0	0
$118$	−306425.	−2.02590
$119$	0	0
$120$	0	0
$121$	−36980.2	−0.229618
$122$	−137869.	−0.838627
$123$	0	0
$124$	−65576.1	−0.382994
$125$	−41719.7	−0.238817
$126$	0	0
$127$	−235054.	−1.29318	−0.646588	−	0.762839i	$-0.723805\pi$
$127$	−235054.	−1.29318	−0.646588	+	0.762839i	$0.723805\pi$
$128$	−181116.	−0.977082
$129$	0	0
$130$	−515903.	−2.67738
$131$	−57379.9	−0.292134	−0.146067	−	0.989275i	$-0.546661\pi$
$131$	−57379.9	−0.292134	−0.146067	+	0.989275i	$0.546661\pi$
$132$	0	0
$133$	0	0
$134$	208718.	1.00415
$135$	0	0
$136$	−41406.1	−0.191963
$137$	218968.	0.996733	0.498367	−	0.866966i	$-0.333933\pi$
$137$	218968.	0.996733	0.498367	+	0.866966i	$0.333933\pi$
$138$	0	0
$139$	−37298.9	−0.163742	−0.0818708	−	0.996643i	$-0.526089\pi$
$139$	−37298.9	−0.163742	−0.0818708	+	0.996643i	$0.526089\pi$
$140$	0	0
$141$	0	0
$142$	366266.	1.52432
$143$	−311820.	−1.27516
$144$	0	0
$145$	−302412.	−1.19448
$146$	−269009.	−1.04444
$147$	0	0
$148$	260900.	0.979085
$149$	−396408.	−1.46277	−0.731387	−	0.681963i	$-0.761127\pi$
$149$	−396408.	−1.46277	−0.731387	+	0.681963i	$0.761127\pi$
$150$	0	0
$151$	276441.	0.986643	0.493321	−	0.869847i	$-0.335783\pi$
$151$	276441.	0.986643	0.493321	+	0.869847i	$0.335783\pi$
$152$	−152203.	−0.534336
$153$	0	0
$154$	0	0
$155$	295237.	0.987057
$156$	0	0
$157$	−108803.	−0.352285	−0.176142	−	0.984365i	$-0.556362\pi$
$157$	−108803.	−0.352285	−0.176142	+	0.984365i	$0.556362\pi$
$158$	−381402.	−1.21546
$159$	0	0
$160$	−486716.	−1.50306
$161$	0	0
$162$	0	0
$163$	131613.	0.387998	0.193999	−	0.981002i	$-0.437854\pi$
$163$	131613.	0.387998	0.193999	+	0.981002i	$0.437854\pi$
$164$	261569.	0.759411
$165$	0	0
$166$	−609453.	−1.71660
$167$	14833.5	0.0411580	0.0205790	−	0.999788i	$-0.493449\pi$
$167$	14833.5	0.0411580	0.0205790	+	0.999788i	$0.493449\pi$
$168$	0	0
$169$	412386.	1.11068
$170$	−247700.	−0.657362
$171$	0	0
$172$	138578.	0.357168
$173$	−718159.	−1.82434	−0.912169	−	0.409814i	$-0.865594\pi$
$173$	−718159.	−1.82434	−0.912169	+	0.409814i	$0.865594\pi$
$174$	0	0
$175$	0	0
$176$	−449066.	−1.09277
$177$	0	0
$178$	147599.	0.349167
$179$	−163893.	−0.382321	−0.191160	−	0.981559i	$-0.561225\pi$
$179$	−163893.	−0.382321	−0.191160	+	0.981559i	$0.561225\pi$
$180$	0	0
$181$	−414321.	−0.940027	−0.470013	−	0.882659i	$-0.655751\pi$
$181$	−414321.	−0.940027	−0.470013	+	0.882659i	$0.655751\pi$
$182$	0	0
$183$	0	0
$184$	271623.	0.591456
$185$	−1.17463e6	−2.52331
$186$	0	0
$187$	−149714.	−0.313083
$188$	−105322.	−0.217333
$189$	0	0
$190$	−910513.	−1.82979
$191$	102634.	0.203567	0.101783	−	0.994807i	$-0.467545\pi$
$191$	102634.	0.203567	0.101783	+	0.994807i	$0.467545\pi$
$192$	0	0
$193$	−320308.	−0.618976	−0.309488	−	0.950903i	$-0.600158\pi$
$193$	−320308.	−0.618976	−0.309488	+	0.950903i	$0.600158\pi$
$194$	691828.	1.31976
$195$	0	0
$196$	0	0
$197$	−495759.	−0.910134	−0.455067	−	0.890457i	$-0.650385\pi$
$197$	−495759.	−0.910134	−0.455067	+	0.890457i	$0.650385\pi$
$198$	0	0
$199$	520712.	0.932105	0.466052	−	0.884757i	$-0.345676\pi$
$199$	520712.	0.932105	0.466052	+	0.884757i	$0.345676\pi$
$200$	353869.	0.625559
$201$	0	0
$202$	277917.	0.479221
$203$	0	0
$204$	0	0
$205$	−1.17764e6	−1.95717
$206$	136315.	0.223809
$207$	0	0
$208$	1.12861e6	1.80878
$209$	−550329.	−0.871478
$210$	0	0
$211$	759493.	1.17440	0.587202	−	0.809440i	$-0.300229\pi$
$211$	759493.	1.17440	0.587202	+	0.809440i	$0.300229\pi$
$212$	−463556.	−0.708374
$213$	0	0
$214$	−1.34570e6	−2.00870
$215$	−623906.	−0.920499
$216$	0	0
$217$	0	0
$218$	1.61667e6	2.30399
$219$	0	0
$220$	−528684.	−0.736443
$221$	376268.	0.518223
$222$	0	0
$223$	−352354.	−0.474479	−0.237240	−	0.971451i	$-0.576243\pi$
$223$	−352354.	−0.474479	−0.237240	+	0.971451i	$0.576243\pi$
$224$	0	0
$225$	0	0
$226$	356661.	0.464499
$227$	1.17532e6	1.51388	0.756941	−	0.653483i	$-0.226693\pi$
$227$	1.17532e6	1.51388	0.756941	+	0.653483i	$0.226693\pi$
$228$	0	0
$229$	−34686.0	−0.0437085	−0.0218542	−	0.999761i	$-0.506957\pi$
$229$	−34686.0	−0.0437085	−0.0218542	+	0.999761i	$0.506957\pi$
$230$	1.62491e6	2.02540
$231$	0	0
$232$	358378.	0.437141
$233$	−66495.4	−0.0802421	−0.0401210	−	0.999195i	$-0.512774\pi$
$233$	−66495.4	−0.0802421	−0.0401210	+	0.999195i	$0.512774\pi$
$234$	0	0
$235$	474182.	0.560113
$236$	789198.	0.922372
$237$	0	0
$238$	0	0
$239$	−968532.	−1.09678	−0.548390	−	0.836223i	$-0.684759\pi$
$239$	−968532.	−1.09678	−0.548390	+	0.836223i	$0.684759\pi$
$240$	0	0
$241$	−634110.	−0.703270	−0.351635	−	0.936137i	$-0.614374\pi$
$241$	−634110.	−0.703270	−0.351635	+	0.936137i	$0.614374\pi$
$242$	262165.	0.287764
$243$	0	0
$244$	355084.	0.381818
$245$	0	0
$246$	0	0
$247$	1.38311e6	1.44250
$248$	−349876.	−0.361231
$249$	0	0
$250$	295764.	0.299293
$251$	−1.77716e6	−1.78050	−0.890249	−	0.455474i	$-0.849470\pi$
$251$	−1.77716e6	−1.78050	−0.890249	+	0.455474i	$0.849470\pi$
$252$	0	0
$253$	982124.	0.964639
$254$	1.66637e6	1.62064
$255$	0	0
$256$	1.32168e6	1.26046
$257$	1.44064e6	1.36058	0.680290	−	0.732943i	$-0.261854\pi$
$257$	1.44064e6	1.36058	0.680290	+	0.732943i	$0.261854\pi$
$258$	0	0
$259$	0	0
$260$	1.32871e6	1.21898
$261$	0	0
$262$	406785.	0.366110
$263$	98968.0	0.0882278	0.0441139	−	0.999027i	$-0.485954\pi$
$263$	98968.0	0.0882278	0.0441139	+	0.999027i	$0.485954\pi$
$264$	0	0
$265$	2.08703e6	1.82563
$266$	0	0
$267$	0	0
$268$	−537554.	−0.457178
$269$	75855.7	0.0639157	0.0319579	−	0.999489i	$-0.489826\pi$
$269$	75855.7	0.0639157	0.0319579	+	0.999489i	$0.489826\pi$
$270$	0	0
$271$	−194479.	−0.160860	−0.0804301	−	0.996760i	$-0.525629\pi$
$271$	−194479.	−0.160860	−0.0804301	+	0.996760i	$0.525629\pi$
$272$	541881.	0.444101
$273$	0	0
$274$	−1.55234e6	−1.24913
$275$	1.27951e6	1.02026
$276$	0	0
$277$	325356.	0.254776	0.127388	−	0.991853i	$-0.459341\pi$
$277$	325356.	0.254776	0.127388	+	0.991853i	$0.459341\pi$
$278$	264424.	0.205206
$279$	0	0
$280$	0	0
$281$	−2.17543e6	−1.64353	−0.821767	−	0.569824i	$-0.807011\pi$
$281$	−2.17543e6	−1.64353	−0.821767	+	0.569824i	$0.807011\pi$
$282$	0	0
$283$	571503.	0.424182	0.212091	−	0.977250i	$-0.431973\pi$
$283$	571503.	0.424182	0.212091	+	0.977250i	$0.431973\pi$
$284$	−943320.	−0.694006
$285$	0	0
$286$	2.21060e6	1.59806
$287$	0	0
$288$	0	0
$289$	−1.23920e6	−0.872763
$290$	2.14390e6	1.49696
$291$	0	0
$292$	692834.	0.475523
$293$	2.15439e6	1.46607	0.733037	−	0.680189i	$-0.238102\pi$
$293$	2.15439e6	1.46607	0.733037	+	0.680189i	$0.238102\pi$
$294$	0	0
$295$	−3.55314e6	−2.37715
$296$	1.39201e6	0.923450
$297$	0	0
$298$	2.81027e6	1.83319
$299$	−2.46832e6	−1.59670
$300$	0	0
$301$	0	0
$302$	−1.95978e6	−1.23649
$303$	0	0
$304$	1.99188e6	1.23617
$305$	−1.59866e6	−0.984026
$306$	0	0
$307$	−2.86577e6	−1.73539	−0.867693	−	0.497101i	$-0.834398\pi$
$307$	−2.86577e6	−1.73539	−0.867693	+	0.497101i	$0.834398\pi$
$308$	0	0
$309$	0	0
$310$	−2.09304e6	−1.23701
$311$	2.14232e6	1.25598	0.627992	−	0.778220i	$-0.283877\pi$
$311$	2.14232e6	1.25598	0.627992	+	0.778220i	$0.283877\pi$
$312$	0	0
$313$	249728.	0.144081	0.0720405	−	0.997402i	$-0.477049\pi$
$313$	249728.	0.144081	0.0720405	+	0.997402i	$0.477049\pi$
$314$	771344.	0.441493
$315$	0	0
$316$	982304.	0.553386
$317$	−858031.	−0.479573	−0.239787	−	0.970826i	$-0.577077\pi$
$317$	−858031.	−0.479573	−0.239787	+	0.970826i	$0.577077\pi$
$318$	0	0
$319$	1.29581e6	0.712957
$320$	96832.2	0.0528622
$321$	0	0
$322$	0	0
$323$	664073.	0.354168
$324$	0	0
$325$	−3.21571e6	−1.68876
$326$	−933048.	−0.486250
$327$	0	0
$328$	1.39558e6	0.716259
$329$	0	0
$330$	0	0
$331$	−2.32892e6	−1.16838	−0.584189	−	0.811617i	$-0.698588\pi$
$331$	−2.32892e6	−1.16838	−0.584189	+	0.811617i	$0.698588\pi$
$332$	1.56965e6	0.781551
$333$	0	0
$334$	−105160.	−0.0515803
$335$	2.42018e6	1.17824
$336$	0	0
$337$	−384940.	−0.184637	−0.0923185	−	0.995730i	$-0.529428\pi$
$337$	−384940.	−0.184637	−0.0923185	+	0.995730i	$0.529428\pi$
$338$	−2.92354e6	−1.39193
$339$	0	0
$340$	637954.	0.299290
$341$	−1.26506e6	−0.589151
$342$	0	0
$343$	0	0
$344$	739370.	0.336873
$345$	0	0
$346$	5.09127e6	2.28631
$347$	1.09215e6	0.486919	0.243460	−	0.969911i	$-0.421718\pi$
$347$	1.09215e6	0.486919	0.243460	+	0.969911i	$0.421718\pi$
$348$	0	0
$349$	1.96544e6	0.863765	0.431883	−	0.901930i	$-0.357849\pi$
$349$	1.96544e6	0.863765	0.431883	+	0.901930i	$0.357849\pi$
$350$	0	0
$351$	0	0
$352$	2.08553e6	0.897141
$353$	−2.59769e6	−1.10956	−0.554779	−	0.831998i	$-0.687197\pi$
$353$	−2.59769e6	−1.10956	−0.554779	+	0.831998i	$0.687197\pi$
$354$	0	0
$355$	4.24702e6	1.78860
$356$	−380142.	−0.158972
$357$	0	0
$358$	1.16189e6	0.479135
$359$	2.49188e6	1.02045	0.510225	−	0.860041i	$-0.329562\pi$
$359$	2.49188e6	1.02045	0.510225	+	0.860041i	$0.329562\pi$
$360$	0	0
$361$	−35058.1	−0.0141586
$362$	2.93726e6	1.17807
$363$	0	0
$364$	0	0
$365$	−3.11928e6	−1.22553
$366$	0	0
$367$	−3.60000e6	−1.39520	−0.697602	−	0.716486i	$-0.745749\pi$
$367$	−3.60000e6	−1.39520	−0.697602	+	0.716486i	$0.745749\pi$
$368$	−3.55473e6	−1.36832
$369$	0	0
$370$	8.32733e6	3.16229
$371$	0	0
$372$	0	0
$373$	1.88759e6	0.702484	0.351242	−	0.936285i	$-0.385759\pi$
$373$	1.88759e6	0.702484	0.351242	+	0.936285i	$0.385759\pi$
$374$	1.06137e6	0.392364
$375$	0	0
$376$	−561937.	−0.204983
$377$	−3.25668e6	−1.18011
$378$	0	0
$379$	753126.	0.269321	0.134660	−	0.990892i	$-0.457006\pi$
$379$	753126.	0.269321	0.134660	+	0.990892i	$0.457006\pi$
$380$	2.34503e6	0.833086
$381$	0	0
$382$	−727605.	−0.255116
$383$	−298258.	−0.103895	−0.0519475	−	0.998650i	$-0.516543\pi$
$383$	−298258.	−0.103895	−0.0519475	+	0.998650i	$0.516543\pi$
$384$	0	0
$385$	0	0
$386$	2.27077e6	0.775719
$387$	0	0
$388$	−1.78181e6	−0.600871
$389$	1.82418e6	0.611215	0.305608	−	0.952158i	$-0.401140\pi$
$389$	1.82418e6	0.611215	0.305608	+	0.952158i	$0.401140\pi$
$390$	0	0
$391$	−1.18511e6	−0.392029
$392$	0	0
$393$	0	0
$394$	3.51460e6	1.14061
$395$	−4.42254e6	−1.42619
$396$	0	0
$397$	1.61590e6	0.514563	0.257281	−	0.966337i	$-0.417173\pi$
$397$	1.61590e6	0.514563	0.257281	+	0.966337i	$0.417173\pi$
$398$	−3.69150e6	−1.16814
$399$	0	0
$400$	−4.63109e6	−1.44721
$401$	−3.45579e6	−1.07322	−0.536608	−	0.843832i	$-0.680294\pi$
$401$	−3.45579e6	−1.07322	−0.536608	+	0.843832i	$0.680294\pi$
$402$	0	0
$403$	3.17941e6	0.975179
$404$	−715776.	−0.218184
$405$	0	0
$406$	0	0
$407$	5.03317e6	1.50611
$408$	0	0
$409$	−1.42925e6	−0.422474	−0.211237	−	0.977435i	$-0.567749\pi$
$409$	−1.42925e6	−0.422474	−0.211237	+	0.977435i	$0.567749\pi$
$410$	8.34867e6	2.45278
$411$	0	0
$412$	−351081.	−0.101898
$413$	0	0
$414$	0	0
$415$	−7.06689e6	−2.01423
$416$	−5.24145e6	−1.48497
$417$	0	0
$418$	3.90146e6	1.09216
$419$	−2.27839e6	−0.634006	−0.317003	−	0.948425i	$-0.602677\pi$
$419$	−2.27839e6	−0.634006	−0.317003	+	0.948425i	$0.602677\pi$
$420$	0	0
$421$	6.36256e6	1.74955	0.874775	−	0.484529i	$-0.161009\pi$
$421$	6.36256e6	1.74955	0.874775	+	0.484529i	$0.161009\pi$
$422$	−5.38430e6	−1.47180
$423$	0	0
$424$	−2.47326e6	−0.668122
$425$	−1.54396e6	−0.414632
$426$	0	0
$427$	0	0
$428$	3.46587e6	0.914541
$429$	0	0
$430$	4.42308e6	1.15360
$431$	2.65149e6	0.687539	0.343770	−	0.939054i	$-0.388296\pi$
$431$	2.65149e6	0.687539	0.343770	+	0.939054i	$0.388296\pi$
$432$	0	0
$433$	688738.	0.176536	0.0882682	−	0.996097i	$-0.471867\pi$
$433$	688738.	0.176536	0.0882682	+	0.996097i	$0.471867\pi$
$434$	0	0
$435$	0	0
$436$	−4.16375e6	−1.04898
$437$	−4.35631e6	−1.09123
$438$	0	0
$439$	−232797.	−0.0576521	−0.0288261	−	0.999584i	$-0.509177\pi$
$439$	−232797.	−0.0576521	−0.0288261	+	0.999584i	$0.509177\pi$
$440$	−2.82075e6	−0.694596
$441$	0	0
$442$	−2.66749e6	−0.649452
$443$	−1.43748e6	−0.348011	−0.174006	−	0.984745i	$-0.555671\pi$
$443$	−1.43748e6	−0.348011	−0.174006	+	0.984745i	$0.555671\pi$
$444$	0	0
$445$	1.71148e6	0.409705
$446$	2.49795e6	0.594631
$447$	0	0
$448$	0	0
$449$	333630.	0.0780997	0.0390499	−	0.999237i	$-0.487567\pi$
$449$	333630.	0.0780997	0.0390499	+	0.999237i	$0.487567\pi$
$450$	0	0
$451$	5.04607e6	1.16819
$452$	−918583.	−0.211481
$453$	0	0
$454$	−8.33224e6	−1.89724
$455$	0	0
$456$	0	0
$457$	−2.54862e6	−0.570841	−0.285421	−	0.958402i	$-0.592133\pi$
$457$	−2.54862e6	−0.570841	−0.285421	+	0.958402i	$0.592133\pi$
$458$	245901.	0.0547767
$459$	0	0
$460$	−4.18497e6	−0.922143
$461$	1.54182e6	0.337894	0.168947	−	0.985625i	$-0.445963\pi$
$461$	1.54182e6	0.337894	0.168947	+	0.985625i	$0.445963\pi$
$462$	0	0
$463$	−1.05753e6	−0.229266	−0.114633	−	0.993408i	$-0.536569\pi$
$463$	−1.05753e6	−0.229266	−0.114633	+	0.993408i	$0.536569\pi$
$464$	−4.69009e6	−1.01131
$465$	0	0
$466$	471408.	0.100562
$467$	−823931.	−0.174823	−0.0874115	−	0.996172i	$-0.527860\pi$
$467$	−823931.	−0.174823	−0.0874115	+	0.996172i	$0.527860\pi$
$468$	0	0
$469$	0	0
$470$	−3.36163e6	−0.701949
$471$	0	0
$472$	4.21070e6	0.869960
$473$	2.67338e6	0.549424
$474$	0	0
$475$	−5.67538e6	−1.15415
$476$	0	0
$477$	0	0
$478$	6.86625e6	1.37452
$479$	1.22641e6	0.244228	0.122114	−	0.992516i	$-0.461033\pi$
$479$	1.22641e6	0.244228	0.122114	+	0.992516i	$0.461033\pi$
$480$	0	0
$481$	−1.26496e7	−2.49295
$482$	4.49541e6	0.881358
$483$	0	0
$484$	−675207.	−0.131016
$485$	8.02207e6	1.54857
$486$	0	0
$487$	−2.72189e6	−0.520053	−0.260027	−	0.965601i	$-0.583731\pi$
$487$	−2.72189e6	−0.520053	−0.260027	+	0.965601i	$0.583731\pi$
$488$	1.89452e6	0.360122
$489$	0	0
$490$	0	0
$491$	5.69198e6	1.06552	0.532758	−	0.846268i	$-0.321156\pi$
$491$	5.69198e6	1.06552	0.532758	+	0.846268i	$0.321156\pi$
$492$	0	0
$493$	−1.56363e6	−0.289745
$494$	−9.80532e6	−1.80778
$495$	0	0
$496$	4.57882e6	0.835698
$497$	0	0
$498$	0	0
$499$	4.86717e6	0.875034	0.437517	−	0.899210i	$-0.355858\pi$
$499$	4.86717e6	0.875034	0.437517	+	0.899210i	$0.355858\pi$
$500$	−761743.	−0.136265
$501$	0	0
$502$	1.25989e7	2.23137
$503$	−8.13233e6	−1.43316	−0.716581	−	0.697504i	$-0.754294\pi$
$503$	−8.13233e6	−1.43316	−0.716581	+	0.697504i	$0.754294\pi$
$504$	0	0
$505$	3.22257e6	0.562308
$506$	−6.96260e6	−1.20891
$507$	0	0
$508$	−4.29175e6	−0.737863
$509$	4.60250e6	0.787408	0.393704	−	0.919237i	$-0.371194\pi$
$509$	4.60250e6	0.787408	0.393704	+	0.919237i	$0.371194\pi$
$510$	0	0
$511$	0	0
$512$	−3.57415e6	−0.602556
$513$	0	0
$514$	−1.02132e7	−1.70512
$515$	1.58064e6	0.262612
$516$	0	0
$517$	−2.03183e6	−0.334318
$518$	0	0
$519$	0	0
$520$	7.08922e6	1.14971
$521$	−469648.	−0.0758016	−0.0379008	−	0.999282i	$-0.512067\pi$
$521$	−469648.	−0.0758016	−0.0379008	+	0.999282i	$0.512067\pi$
$522$	0	0
$523$	−1.20416e7	−1.92499	−0.962495	−	0.271300i	$-0.912546\pi$
$523$	−1.20416e7	−1.92499	−0.962495	+	0.271300i	$0.912546\pi$
$524$	−1.04768e6	−0.166686
$525$	0	0
$526$	−701617.	−0.110570
$527$	1.52653e6	0.239431
$528$	0	0
$529$	1.33799e6	0.207880
$530$	−1.47956e7	−2.28793
$531$	0	0
$532$	0	0
$533$	−1.26820e7	−1.93361
$534$	0	0
$535$	−1.56041e7	−2.35697
$536$	−2.86807e6	−0.431199
$537$	0	0
$538$	−537766.	−0.0801010
$539$	0	0
$540$	0	0
$541$	−4.86348e6	−0.714421	−0.357211	−	0.934024i	$-0.616272\pi$
$541$	−4.86348e6	−0.714421	−0.357211	+	0.934024i	$0.616272\pi$
$542$	1.37872e6	0.201595
$543$	0	0
$544$	−2.51658e6	−0.364597
$545$	1.87461e7	2.70345
$546$	0	0
$547$	5.47821e6	0.782836	0.391418	−	0.920213i	$-0.371985\pi$
$547$	5.47821e6	0.782836	0.391418	+	0.920213i	$0.371985\pi$
$548$	3.99805e6	0.568718
$549$	0	0
$550$	−9.07084e6	−1.27862
$551$	−5.74769e6	−0.806518
$552$	0	0
$553$	0	0
$554$	−2.30656e6	−0.319293
$555$	0	0
$556$	−681026.	−0.0934279
$557$	−1.23668e7	−1.68896	−0.844481	−	0.535586i	$-0.820091\pi$
$557$	−1.23668e7	−1.68896	−0.844481	+	0.535586i	$0.820091\pi$
$558$	0	0
$559$	−6.71885e6	−0.909422
$560$	0	0
$561$	0	0
$562$	1.54223e7	2.05972
$563$	1.43167e7	1.90359	0.951794	−	0.306737i	$-0.0992373\pi$
$563$	1.43167e7	1.90359	0.951794	+	0.306737i	$0.0992373\pi$
$564$	0	0
$565$	4.13565e6	0.545033
$566$	−4.05158e6	−0.531597
$567$	0	0
$568$	−5.03300e6	−0.654570
$569$	−9.11544e6	−1.18031	−0.590156	−	0.807289i	$-0.700934\pi$
$569$	−9.11544e6	−1.18031	−0.590156	+	0.807289i	$0.700934\pi$
$570$	0	0
$571$	−5.09542e6	−0.654019	−0.327009	−	0.945021i	$-0.606041\pi$
$571$	−5.09542e6	−0.654019	−0.327009	+	0.945021i	$0.606041\pi$
$572$	−5.69340e6	−0.727582
$573$	0	0
$574$	0	0
$575$	1.01283e7	1.27752
$576$	0	0
$577$	−175593.	−0.0219568	−0.0109784	−	0.999940i	$-0.503495\pi$
$577$	−175593.	−0.0219568	−0.0109784	+	0.999940i	$0.503495\pi$
$578$	8.78509e6	1.09377
$579$	0	0
$580$	−5.52162e6	−0.681549
$581$	0	0
$582$	0	0
$583$	−8.94272e6	−1.08968
$584$	3.69655e6	0.448502
$585$	0	0
$586$	−1.52732e7	−1.83733
$587$	6.35775e6	0.761567	0.380784	−	0.924664i	$-0.375654\pi$
$587$	6.35775e6	0.761567	0.380784	+	0.924664i	$0.375654\pi$
$588$	0	0
$589$	5.61132e6	0.666465
$590$	2.51894e7	2.97911
$591$	0	0
$592$	−1.82172e7	−2.13638
$593$	1.11879e7	1.30650	0.653251	−	0.757141i	$-0.273404\pi$
$593$	1.11879e7	1.30650	0.653251	+	0.757141i	$0.273404\pi$
$594$	0	0
$595$	0	0
$596$	−7.23787e6	−0.834632
$597$	0	0
$598$	1.74987e7	2.00103
$599$	−455702.	−0.0518936	−0.0259468	−	0.999663i	$-0.508260\pi$
$599$	−455702.	−0.0518936	−0.0259468	+	0.999663i	$0.508260\pi$
$600$	0	0
$601$	−1.72814e7	−1.95161	−0.975804	−	0.218649i	$-0.929835\pi$
$601$	−1.72814e7	−1.95161	−0.975804	+	0.218649i	$0.929835\pi$
$602$	0	0
$603$	0	0
$604$	5.04743e6	0.562960
$605$	3.03992e6	0.337656
$606$	0	0
$607$	1.92718e6	0.212300	0.106150	−	0.994350i	$-0.466148\pi$
$607$	1.92718e6	0.212300	0.106150	+	0.994350i	$0.466148\pi$
$608$	−9.25060e6	−1.01487
$609$	0	0
$610$	1.13334e7	1.23321
$611$	5.10647e6	0.553373
$612$	0	0
$613$	1.92778e6	0.207208	0.103604	−	0.994619i	$-0.466963\pi$
$613$	1.92778e6	0.207208	0.103604	+	0.994619i	$0.466963\pi$
$614$	2.03164e7	2.17484
$615$	0	0
$616$	0	0
$617$	8.90588e6	0.941812	0.470906	−	0.882183i	$-0.343927\pi$
$617$	8.90588e6	0.941812	0.470906	+	0.882183i	$0.343927\pi$
$618$	0	0
$619$	2.36438e6	0.248023	0.124011	−	0.992281i	$-0.460424\pi$
$619$	2.36438e6	0.248023	0.124011	+	0.992281i	$0.460424\pi$
$620$	5.39063e6	0.563196
$621$	0	0
$622$	−1.51876e7	−1.57403
$623$	0	0
$624$	0	0
$625$	−7.92208e6	−0.811221
$626$	−1.77041e6	−0.180566
$627$	0	0
$628$	−1.98660e6	−0.201007
$629$	−6.07344e6	−0.612080
$630$	0	0
$631$	−3.00892e6	−0.300841	−0.150420	−	0.988622i	$-0.548063\pi$
$631$	−3.00892e6	−0.300841	−0.150420	+	0.988622i	$0.548063\pi$
$632$	5.24099e6	0.521940
$633$	0	0
$634$	6.08287e6	0.601015
$635$	1.93224e7	1.90163
$636$	0	0
$637$	0	0
$638$	−9.18640e6	−0.893499
$639$	0	0
$640$	1.48884e7	1.43681
$641$	−7.06526e6	−0.679177	−0.339588	−	0.940574i	$-0.610288\pi$
$641$	−7.06526e6	−0.679177	−0.339588	+	0.940574i	$0.610288\pi$
$642$	0	0
$643$	1.11768e7	1.06608	0.533038	−	0.846091i	$-0.321050\pi$
$643$	1.11768e7	1.06608	0.533038	+	0.846091i	$0.321050\pi$
$644$	0	0
$645$	0	0
$646$	−4.70783e6	−0.443854
$647$	7.13354e6	0.669953	0.334977	−	0.942226i	$-0.391272\pi$
$647$	7.13354e6	0.669953	0.334977	+	0.942226i	$0.391272\pi$
$648$	0	0
$649$	1.52249e7	1.41887
$650$	2.27972e7	2.11640
$651$	0	0
$652$	2.40307e6	0.221385
$653$	−6.19171e6	−0.568235	−0.284117	−	0.958789i	$-0.591701\pi$
$653$	−6.19171e6	−0.568235	−0.284117	+	0.958789i	$0.591701\pi$
$654$	0	0
$655$	4.71687e6	0.429586
$656$	−1.82639e7	−1.65705
$657$	0	0
$658$	0	0
$659$	−747569.	−0.0670560	−0.0335280	−	0.999438i	$-0.510674\pi$
$659$	−747569.	−0.0670560	−0.0335280	+	0.999438i	$0.510674\pi$
$660$	0	0
$661$	−2.07762e7	−1.84954	−0.924769	−	0.380530i	$-0.875742\pi$
$661$	−2.07762e7	−1.84954	−0.924769	+	0.380530i	$0.875742\pi$
$662$	1.65105e7	1.46425
$663$	0	0
$664$	8.37473e6	0.737141
$665$	0	0
$666$	0	0
$667$	1.02574e7	0.892735
$668$	270840.	0.0234840
$669$	0	0
$670$	−1.71575e7	−1.47661
$671$	6.85011e6	0.587343
$672$	0	0
$673$	−1.20681e7	−1.02707	−0.513536	−	0.858068i	$-0.671665\pi$
$673$	−1.20681e7	−1.02707	−0.513536	+	0.858068i	$0.671665\pi$
$674$	2.72897e6	0.231392
$675$	0	0
$676$	7.52960e6	0.633732
$677$	1.13173e7	0.949013	0.474506	−	0.880252i	$-0.342627\pi$
$677$	1.13173e7	0.949013	0.474506	+	0.880252i	$0.342627\pi$
$678$	0	0
$679$	0	0
$680$	3.40375e6	0.282283
$681$	0	0
$682$	8.96846e6	0.738341
$683$	−5.68232e6	−0.466095	−0.233047	−	0.972465i	$-0.574870\pi$
$683$	−5.68232e6	−0.466095	−0.233047	+	0.972465i	$0.574870\pi$
$684$	0	0
$685$	−1.80001e7	−1.46571
$686$	0	0
$687$	0	0
$688$	−9.67613e6	−0.779346
$689$	2.24752e7	1.80366
$690$	0	0
$691$	−5.76562e6	−0.459357	−0.229679	−	0.973267i	$-0.573768\pi$
$691$	−5.76562e6	−0.459357	−0.229679	+	0.973267i	$0.573768\pi$
$692$	−1.31126e7	−1.04093
$693$	0	0
$694$	−7.74258e6	−0.610221
$695$	3.06612e6	0.240784
$696$	0	0
$697$	−6.08901e6	−0.474750
$698$	−1.39336e7	−1.08250
$699$	0	0
$700$	0	0
$701$	4.04459e6	0.310870	0.155435	−	0.987846i	$-0.450322\pi$
$701$	4.04459e6	0.310870	0.155435	+	0.987846i	$0.450322\pi$
$702$	0	0
$703$	−2.23251e7	−1.70375
$704$	−414917.	−0.0315522
$705$	0	0
$706$	1.84159e7	1.39053
$707$	0	0
$708$	0	0
$709$	7.15842e6	0.534812	0.267406	−	0.963584i	$-0.413833\pi$
$709$	7.15842e6	0.534812	0.267406	+	0.963584i	$0.413833\pi$
$710$	−3.01086e7	−2.24153
$711$	0	0
$712$	−2.02821e6	−0.149939
$713$	−1.00140e7	−0.737710
$714$	0	0
$715$	2.56329e7	1.87513
$716$	−2.99246e6	−0.218145
$717$	0	0
$718$	−1.76658e7	−1.27886
$719$	−9.82739e6	−0.708951	−0.354475	−	0.935065i	$-0.615340\pi$
$719$	−9.82739e6	−0.708951	−0.354475	+	0.935065i	$0.615340\pi$
$720$	0	0
$721$	0	0
$722$	248538.	0.0177439
$723$	0	0
$724$	−7.56492e6	−0.536362
$725$	1.33633e7	0.944208
$726$	0	0
$727$	1.63233e7	1.14544	0.572720	−	0.819751i	$-0.305888\pi$
$727$	1.63233e7	1.14544	0.572720	+	0.819751i	$0.305888\pi$
$728$	0	0
$729$	0	0
$730$	2.21136e7	1.53586
$731$	−3.22592e6	−0.223286
$732$	0	0
$733$	1.76033e7	1.21014	0.605069	−	0.796173i	$-0.293146\pi$
$733$	1.76033e7	1.21014	0.605069	+	0.796173i	$0.293146\pi$
$734$	2.55216e7	1.74851
$735$	0	0
$736$	1.65087e7	1.12336
$737$	−1.03702e7	−0.703267
$738$	0	0
$739$	6.92574e6	0.466504	0.233252	−	0.972416i	$-0.425063\pi$
$739$	6.92574e6	0.466504	0.233252	+	0.972416i	$0.425063\pi$
$740$	−2.14471e7	−1.43976
$741$	0	0
$742$	0	0
$743$	−1.38786e7	−0.922305	−0.461153	−	0.887321i	$-0.652564\pi$
$743$	−1.38786e7	−0.922305	−0.461153	+	0.887321i	$0.652564\pi$
$744$	0	0
$745$	3.25864e7	2.15103
$746$	−1.33818e7	−0.880373
$747$	0	0
$748$	−2.73357e6	−0.178639
$749$	0	0
$750$	0	0
$751$	−4.84817e6	−0.313674	−0.156837	−	0.987625i	$-0.550130\pi$
$751$	−4.84817e6	−0.313674	−0.156837	+	0.987625i	$0.550130\pi$
$752$	7.35406e6	0.474223
$753$	0	0
$754$	2.30877e7	1.47894
$755$	−2.27246e7	−1.45087
$756$	0	0
$757$	−2.41206e7	−1.52985	−0.764926	−	0.644119i	$-0.777224\pi$
$757$	−2.41206e7	−1.52985	−0.764926	+	0.644119i	$0.777224\pi$
$758$	−5.33916e6	−0.337520
$759$	0	0
$760$	1.25117e7	0.785747
$761$	2.06431e6	0.129215	0.0646074	−	0.997911i	$-0.479420\pi$
$761$	2.06431e6	0.129215	0.0646074	+	0.997911i	$0.479420\pi$
$762$	0	0
$763$	0	0
$764$	1.87395e6	0.116152
$765$	0	0
$766$	2.11445e6	0.130204
$767$	−3.82637e7	−2.34855
$768$	0	0
$769$	−1.80825e7	−1.10266	−0.551331	−	0.834287i	$-0.685880\pi$
$769$	−1.80825e7	−1.10266	−0.551331	+	0.834287i	$0.685880\pi$
$770$	0	0
$771$	0	0
$772$	−5.84837e6	−0.353177
$773$	−1.37481e7	−0.827550	−0.413775	−	0.910379i	$-0.635790\pi$
$773$	−1.37481e7	−0.827550	−0.413775	+	0.910379i	$0.635790\pi$
$774$	0	0
$775$	−1.30462e7	−0.780245
$776$	−9.50668e6	−0.566728
$777$	0	0
$778$	−1.29322e7	−0.765992
$779$	−2.23824e7	−1.32149
$780$	0	0
$781$	−1.81981e7	−1.06758
$782$	8.40166e6	0.491301
$783$	0	0
$784$	0	0
$785$	8.94409e6	0.518039
$786$	0	0
$787$	−2.48037e7	−1.42751	−0.713756	−	0.700395i	$-0.753007\pi$
$787$	−2.48037e7	−1.42751	−0.713756	+	0.700395i	$0.753007\pi$
$788$	−9.05188e6	−0.519306
$789$	0	0
$790$	3.13528e7	1.78735
$791$	0	0
$792$	0	0
$793$	−1.72160e7	−0.972185
$794$	−1.14557e7	−0.644865
$795$	0	0
$796$	9.50748e6	0.531842
$797$	−1.82261e7	−1.01636	−0.508181	−	0.861250i	$-0.669682\pi$
$797$	−1.82261e7	−1.01636	−0.508181	+	0.861250i	$0.669682\pi$
$798$	0	0
$799$	2.45177e6	0.135867
$800$	2.15075e7	1.18813
$801$	0	0
$802$	2.44993e7	1.34498
$803$	1.33658e7	0.731488
$804$	0	0
$805$	0	0
$806$	−2.25399e7	−1.22212
$807$	0	0
$808$	−3.81896e6	−0.205786
$809$	1.32390e6	0.0711189	0.0355594	−	0.999368i	$-0.488679\pi$
$809$	1.32390e6	0.0711189	0.0355594	+	0.999368i	$0.488679\pi$
$810$	0	0
$811$	−3.64227e6	−0.194455	−0.0972277	−	0.995262i	$-0.530998\pi$
$811$	−3.64227e6	−0.194455	−0.0972277	+	0.995262i	$0.530998\pi$
$812$	0	0
$813$	0	0
$814$	−3.56818e7	−1.88750
$815$	−1.08191e7	−0.570556
$816$	0	0
$817$	−1.18581e7	−0.621525
$818$	1.01324e7	0.529456
$819$	0	0
$820$	−2.15021e7	−1.11672
$821$	−9.32669e6	−0.482914	−0.241457	−	0.970412i	$-0.577625\pi$
$821$	−9.32669e6	−0.482914	−0.241457	+	0.970412i	$0.577625\pi$
$822$	0	0
$823$	8.48173e6	0.436501	0.218250	−	0.975893i	$-0.429965\pi$
$823$	8.48173e6	0.436501	0.218250	+	0.975893i	$0.429965\pi$
$824$	−1.87316e6	−0.0961075
$825$	0	0
$826$	0	0
$827$	2.70070e7	1.37313	0.686567	−	0.727066i	$-0.259117\pi$
$827$	2.70070e7	1.37313	0.686567	+	0.727066i	$0.259117\pi$
$828$	0	0
$829$	827844.	0.0418372	0.0209186	−	0.999781i	$-0.493341\pi$
$829$	827844.	0.0418372	0.0209186	+	0.999781i	$0.493341\pi$
$830$	5.00995e7	2.52428
$831$	0	0
$832$	1.04279e6	0.0522261
$833$	0	0
$834$	0	0
$835$	−1.21938e6	−0.0605233
$836$	−1.00482e7	−0.497250
$837$	0	0
$838$	1.61523e7	0.794555
$839$	1.01230e7	0.496483	0.248242	−	0.968698i	$-0.420147\pi$
$839$	1.01230e7	0.496483	0.248242	+	0.968698i	$0.420147\pi$
$840$	0	0
$841$	−6.97762e6	−0.340187
$842$	−4.51063e7	−2.19259
$843$	0	0
$844$	1.38673e7	0.670094
$845$	−3.38998e7	−1.63326
$846$	0	0
$847$	0	0
$848$	3.23676e7	1.54568
$849$	0	0
$850$	1.09456e7	0.519629
$851$	3.98417e7	1.88588
$852$	0	0
$853$	3.24119e7	1.52522	0.762609	−	0.646860i	$-0.223918\pi$
$853$	3.24119e7	1.52522	0.762609	+	0.646860i	$0.223918\pi$
$854$	0	0
$855$	0	0
$856$	1.84918e7	0.862573
$857$	−1.23513e7	−0.574460	−0.287230	−	0.957862i	$-0.592734\pi$
$857$	−1.23513e7	−0.574460	−0.287230	+	0.957862i	$0.592734\pi$
$858$	0	0
$859$	−2.46448e7	−1.13958	−0.569788	−	0.821792i	$-0.692975\pi$
$859$	−2.46448e7	−1.13958	−0.569788	+	0.821792i	$0.692975\pi$
$860$	−1.13917e7	−0.525220
$861$	0	0
$862$	−1.87973e7	−0.861644
$863$	−1.97433e7	−0.902386	−0.451193	−	0.892426i	$-0.649001\pi$
$863$	−1.97433e7	−0.902386	−0.451193	+	0.892426i	$0.649001\pi$
$864$	0	0
$865$	5.90356e7	2.68271
$866$	−4.88269e6	−0.221240
$867$	0	0
$868$	0	0
$869$	1.89502e7	0.851262
$870$	0	0
$871$	2.60629e7	1.16407
$872$	−2.22153e7	−0.989376
$873$	0	0
$874$	3.08833e7	1.36756
$875$	0	0
$876$	0	0
$877$	3.24241e6	0.142354	0.0711769	−	0.997464i	$-0.477325\pi$
$877$	3.24241e6	0.142354	0.0711769	+	0.997464i	$0.477325\pi$
$878$	1.65037e6	0.0722513
$879$	0	0
$880$	3.69151e7	1.60693
$881$	−1.10813e7	−0.481006	−0.240503	−	0.970648i	$-0.577312\pi$
$881$	−1.10813e7	−0.481006	−0.240503	+	0.970648i	$0.577312\pi$
$882$	0	0
$883$	2.35995e6	0.101859	0.0509296	−	0.998702i	$-0.483782\pi$
$883$	2.35995e6	0.101859	0.0509296	+	0.998702i	$0.483782\pi$
$884$	6.87013e6	0.295689
$885$	0	0
$886$	1.01908e7	0.436137
$887$	2.42382e7	1.03440	0.517202	−	0.855863i	$-0.326973\pi$
$887$	2.42382e7	1.03440	0.517202	+	0.855863i	$0.326973\pi$
$888$	0	0
$889$	0	0
$890$	−1.21332e7	−0.513454
$891$	0	0
$892$	−6.43350e6	−0.270729
$893$	9.01237e6	0.378190
$894$	0	0
$895$	1.34727e7	0.562207
$896$	0	0
$897$	0	0
$898$	−2.36521e6	−0.0978768
$899$	−1.32124e7	−0.545235
$900$	0	0
$901$	1.07910e7	0.442844
$902$	−3.57733e7	−1.46400
$903$	0	0
$904$	−4.90102e6	−0.199464
$905$	3.40589e7	1.38232
$906$	0	0
$907$	1.32928e7	0.536535	0.268268	−	0.963344i	$-0.413549\pi$
$907$	1.32928e7	0.536535	0.268268	+	0.963344i	$0.413549\pi$
$908$	2.14597e7	0.863793
$909$	0	0
$910$	0	0
$911$	−3.69288e7	−1.47424	−0.737122	−	0.675760i	$-0.763816\pi$
$911$	−3.69288e7	−1.47424	−0.737122	+	0.675760i	$0.763816\pi$
$912$	0	0
$913$	3.02810e7	1.20224
$914$	1.80680e7	0.715394
$915$	0	0
$916$	−633318.	−0.0249393
$917$	0	0
$918$	0	0
$919$	−1.01517e7	−0.396508	−0.198254	−	0.980151i	$-0.563527\pi$
$919$	−1.01517e7	−0.396508	−0.198254	+	0.980151i	$0.563527\pi$
$920$	−2.23286e7	−0.869743
$921$	0	0
$922$	−1.09305e7	−0.423459
$923$	4.57362e7	1.76708
$924$	0	0
$925$	5.19056e7	1.99462
$926$	7.49718e6	0.287323
$927$	0	0
$928$	2.17815e7	0.830267
$929$	6.26383e6	0.238123	0.119061	−	0.992887i	$-0.462011\pi$
$929$	6.26383e6	0.238123	0.119061	+	0.992887i	$0.462011\pi$
$930$	0	0
$931$	0	0
$932$	−1.21411e6	−0.0457846
$933$	0	0
$934$	5.84112e6	0.219093
$935$	1.23071e7	0.460392
$936$	0	0
$937$	2.51142e7	0.934480	0.467240	−	0.884131i	$-0.345248\pi$
$937$	2.51142e7	0.934480	0.467240	+	0.884131i	$0.345248\pi$
$938$	0	0
$939$	0	0
$940$	8.65791e6	0.319590
$941$	−3.07032e7	−1.13034	−0.565170	−	0.824974i	$-0.691190\pi$
$941$	−3.07032e7	−1.13034	−0.565170	+	0.824974i	$0.691190\pi$
$942$	0	0
$943$	3.99439e7	1.46275
$944$	−5.51054e7	−2.01263
$945$	0	0
$946$	−1.89525e7	−0.688554
$947$	2.43174e7	0.881133	0.440566	−	0.897720i	$-0.354778\pi$
$947$	2.43174e7	0.881133	0.440566	+	0.897720i	$0.354778\pi$
$948$	0	0
$949$	−3.35916e7	−1.21078
$950$	4.02346e7	1.44641
$951$	0	0
$952$	0	0
$953$	3.82087e7	1.36279	0.681397	−	0.731914i	$-0.261373\pi$
$953$	3.82087e7	1.36279	0.681397	+	0.731914i	$0.261373\pi$
$954$	0	0
$955$	−8.43692e6	−0.299347
$956$	−1.76841e7	−0.625802
$957$	0	0
$958$	−8.69441e6	−0.306074
$959$	0	0
$960$	0	0
$961$	−1.57302e7	−0.549446
$962$	8.96770e7	3.12423
$963$	0	0
$964$	−1.15780e7	−0.401273
$965$	2.63306e7	0.910212
$966$	0	0
$967$	−1.41624e7	−0.487048	−0.243524	−	0.969895i	$-0.578304\pi$
$967$	−1.41624e7	−0.487048	−0.243524	+	0.969895i	$0.578304\pi$
$968$	−3.60251e6	−0.123571
$969$	0	0
$970$	−5.68711e7	−1.94072
$971$	−6.98320e6	−0.237688	−0.118844	−	0.992913i	$-0.537919\pi$
$971$	−6.98320e6	−0.237688	−0.118844	+	0.992913i	$0.537919\pi$
$972$	0	0
$973$	0	0
$974$	1.92964e7	0.651746
$975$	0	0
$976$	−2.47935e7	−0.833132
$977$	−2.29623e7	−0.769625	−0.384813	−	0.922995i	$-0.625734\pi$
$977$	−2.29623e7	−0.769625	−0.384813	+	0.922995i	$0.625734\pi$
$978$	0	0
$979$	−7.33353e6	−0.244544
$980$	0	0
$981$	0	0
$982$	−4.03523e7	−1.33533
$983$	3.13975e7	1.03636	0.518181	−	0.855271i	$-0.326609\pi$
$983$	3.13975e7	1.03636	0.518181	+	0.855271i	$0.326609\pi$
$984$	0	0
$985$	4.07534e7	1.33836
$986$	1.10851e7	0.363117
$987$	0	0
$988$	2.52537e7	0.823061
$989$	2.11620e7	0.687966
$990$	0	0
$991$	−4.10735e7	−1.32855	−0.664274	−	0.747489i	$-0.731259\pi$
$991$	−4.10735e7	−1.32855	−0.664274	+	0.747489i	$0.731259\pi$
$992$	−2.12647e7	−0.686090
$993$	0	0
$994$	0	0
$995$	−4.28046e7	−1.37067
$996$	0	0
$997$	−3.30863e6	−0.105417	−0.0527085	−	0.998610i	$-0.516785\pi$
$997$	−3.30863e6	−0.105417	−0.0527085	+	0.998610i	$0.516785\pi$
$998$	−3.45050e7	−1.09662
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.6.a.bc.1.2		6
3.2	odd	2	inner	441.6.a.bc.1.5		6
7.2	even	3		63.6.e.f.46.5	yes	12
7.4	even	3		63.6.e.f.37.5	yes	12
7.6	odd	2		441.6.a.bd.1.2		6
21.2	odd	6		63.6.e.f.46.2	yes	12
21.11	odd	6		63.6.e.f.37.2	✓	12
21.20	even	2		441.6.a.bd.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.6.e.f.37.2	✓	12	21.11	odd	6
63.6.e.f.37.5	yes	12	7.4	even	3
63.6.e.f.46.2	yes	12	21.2	odd	6
63.6.e.f.46.5	yes	12	7.2	even	3
441.6.a.bc.1.2		6	1.1	even	1	trivial
441.6.a.bc.1.5		6	3.2	odd	2	inner
441.6.a.bd.1.2		6	7.6	odd	2
441.6.a.bd.1.5		6	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-7.08933 q^{2} +18.2586 q^{4} -82.2041 q^{5} +97.4172 q^{8} +O(q^{10})\)	\(q-7.08933 q^{2} +18.2586 q^{4} -82.2041 q^{5} +97.4172 q^{8} +582.772 q^{10} +352.237 q^{11} -885.257 q^{13} -1274.90 q^{16} -425.038 q^{17} -1562.38 q^{19} -1500.93 q^{20} -2497.12 q^{22} +2788.25 q^{23} +3632.51 q^{25} +6275.88 q^{26} +3678.79 q^{29} -3591.52 q^{31} +5920.83 q^{32} +3013.24 q^{34} +14289.2 q^{37} +11076.3 q^{38} -8008.09 q^{40} +14325.8 q^{41} +7589.72 q^{43} +6431.36 q^{44} -19766.8 q^{46} -5768.35 q^{47} -25752.1 q^{50} -16163.6 q^{52} -25388.4 q^{53} -28955.3 q^{55} -26080.2 q^{58} +43223.3 q^{59} +19447.5 q^{61} +25461.4 q^{62} -1177.95 q^{64} +72771.7 q^{65} -29441.1 q^{67} -7760.61 q^{68} -51664.4 q^{71} +37945.6 q^{73} -101301. q^{74} -28527.0 q^{76} +53799.5 q^{79} +104802. q^{80} -101560. q^{82} +85967.6 q^{83} +34939.9 q^{85} -53806.1 q^{86} +34313.9 q^{88} -20819.9 q^{89} +50909.6 q^{92} +40893.7 q^{94} +128434. q^{95} -97587.2 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 182 q^{4}+O(q^{10}) \) 6 * q + 182 * q^4	\( 6 q + 182 q^{4} - 686 q^{10} + 154 q^{13} + 1898 q^{16} - 9422 q^{19} - 9146 q^{22} + 7526 q^{25} - 23422 q^{31} - 27804 q^{34} + 18182 q^{37} - 69258 q^{40} - 43686 q^{43} - 25332 q^{46} - 34272 q^{52} - 48160 q^{55} + 89782 q^{58} + 16156 q^{61} + 190290 q^{64} - 144650 q^{67} + 100058 q^{73} - 342720 q^{76} - 101994 q^{79} - 75712 q^{82} + 301176 q^{85} - 752310 q^{88} + 120456 q^{94} - 433048 q^{97}+O(q^{100}) \) 6 * q + 182 * q^4 - 686 * q^10 + 154 * q^13 + 1898 * q^16 - 9422 * q^19 - 9146 * q^22 + 7526 * q^25 - 23422 * q^31 - 27804 * q^34 + 18182 * q^37 - 69258 * q^40 - 43686 * q^43 - 25332 * q^46 - 34272 * q^52 - 48160 * q^55 + 89782 * q^58 + 16156 * q^61 + 190290 * q^64 - 144650 * q^67 + 100058 * q^73 - 342720 * q^76 - 101994 * q^79 - 75712 * q^82 + 301176 * q^85 - 752310 * q^88 + 120456 * q^94 - 433048 * q^97

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