LMFDB - Embedded newform 441.6.a.bd.1.4

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:	$6$
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			1.4
Root			$4.88952$ 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)

$f(q)$	$=$	$q+4.88952 q^{2} -8.09260 q^{4} +42.7504 q^{5} -196.034 q^{8} +O(q^{10})$	$q+4.88952 q^{2} -8.09260 q^{4} +42.7504 q^{5} -196.034 q^{8} +209.029 q^{10} +710.878 q^{11} -885.624 q^{13} -699.547 q^{16} +701.317 q^{17} +1255.89 q^{19} -345.962 q^{20} +3475.85 q^{22} -1046.18 q^{23} -1297.40 q^{25} -4330.27 q^{26} +6150.88 q^{29} -2294.24 q^{31} +2852.63 q^{32} +3429.10 q^{34} +404.109 q^{37} +6140.71 q^{38} -8380.51 q^{40} +17891.4 q^{41} -14604.8 q^{43} -5752.85 q^{44} -5115.34 q^{46} +21137.3 q^{47} -6343.67 q^{50} +7167.00 q^{52} +107.213 q^{53} +30390.3 q^{55} +30074.9 q^{58} +44498.8 q^{59} +23212.4 q^{61} -11217.7 q^{62} +36333.5 q^{64} -37860.8 q^{65} -6671.32 q^{67} -5675.48 q^{68} +25110.5 q^{71} -9474.97 q^{73} +1975.90 q^{74} -10163.4 q^{76} -26476.1 q^{79} -29905.9 q^{80} +87480.3 q^{82} +7494.03 q^{83} +29981.6 q^{85} -71410.5 q^{86} -139356. q^{88} -32171.3 q^{89} +8466.35 q^{92} +103351. q^{94} +53689.9 q^{95} +155070. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	4.88952	0.864353	0.432177	−	0.901789i	$-0.357746\pi$
$2$	4.88952	0.864353	0.432177	+	0.901789i	$0.357746\pi$
$3$	0	0
$3$	0	0
$4$	−8.09260	−0.252894
$5$	42.7504	0.764743	0.382371	−	0.924009i	$-0.375108\pi$
$5$	42.7504	0.764743	0.382371	+	0.924009i	$0.375108\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−196.034	−1.08294
$9$	0	0
$10$	209.029	0.661008
$11$	710.878	1.77139	0.885693	−	0.464271i	$-0.153684\pi$
$11$	710.878	1.77139	0.885693	+	0.464271i	$0.153684\pi$
$12$	0	0
$13$	−885.624	−1.45342	−0.726709	−	0.686945i	$-0.758951\pi$
$13$	−885.624	−1.45342	−0.726709	+	0.686945i	$0.758951\pi$
$14$	0	0
$15$	0	0
$16$	−699.547	−0.683151
$17$	701.317	0.588562	0.294281	−	0.955719i	$-0.404920\pi$
$17$	701.317	0.588562	0.294281	+	0.955719i	$0.404920\pi$
$18$	0	0
$19$	1255.89	0.798120	0.399060	−	0.916925i	$-0.369336\pi$
$19$	1255.89	0.798120	0.399060	+	0.916925i	$0.369336\pi$
$20$	−345.962	−0.193399
$21$	0	0
$22$	3475.85	1.53110
$23$	−1046.18	−0.412371	−0.206186	−	0.978513i	$-0.566105\pi$
$23$	−1046.18	−0.412371	−0.206186	+	0.978513i	$0.566105\pi$
$24$	0	0
$25$	−1297.40	−0.415169
$26$	−4330.27	−1.25627
$27$	0	0
$28$	0	0
$29$	6150.88	1.35813	0.679067	−	0.734077i	$-0.262385\pi$
$29$	6150.88	1.35813	0.679067	+	0.734077i	$0.262385\pi$
$30$	0	0
$31$	−2294.24	−0.428780	−0.214390	−	0.976748i	$-0.568776\pi$
$31$	−2294.24	−0.428780	−0.214390	+	0.976748i	$0.568776\pi$
$32$	2852.63	0.492459
$33$	0	0
$34$	3429.10	0.508725
$35$	0	0
$36$	0	0
$37$	404.109	0.0485282	0.0242641	−	0.999706i	$-0.492276\pi$
$37$	404.109	0.0485282	0.0242641	+	0.999706i	$0.492276\pi$
$38$	6140.71	0.689858
$39$	0	0
$40$	−8380.51	−0.828172
$41$	17891.4	1.66221	0.831103	−	0.556119i	$-0.187710\pi$
$41$	17891.4	1.66221	0.831103	+	0.556119i	$0.187710\pi$
$42$	0	0
$43$	−14604.8	−1.20455	−0.602275	−	0.798289i	$-0.705739\pi$
$43$	−14604.8	−1.20455	−0.602275	+	0.798289i	$0.705739\pi$
$44$	−5752.85	−0.447973
$45$	0	0
$46$	−5115.34	−0.356434
$47$	21137.3	1.39574	0.697870	−	0.716225i	$-0.254132\pi$
$47$	21137.3	1.39574	0.697870	+	0.716225i	$0.254132\pi$
$48$	0	0
$49$	0	0
$50$	−6343.67	−0.358852
$51$	0	0
$52$	7167.00	0.367560
$53$	107.213	0.00524272	0.00262136	−	0.999997i	$-0.499166\pi$
$53$	107.213	0.00524272	0.00262136	+	0.999997i	$0.499166\pi$
$54$	0	0
$55$	30390.3	1.35465
$56$	0	0
$57$	0	0
$58$	30074.9	1.17391
$59$	44498.8	1.66425	0.832124	−	0.554590i	$-0.187125\pi$
$59$	44498.8	1.66425	0.832124	+	0.554590i	$0.187125\pi$
$60$	0	0
$61$	23212.4	0.798720	0.399360	−	0.916794i	$-0.369232\pi$
$61$	23212.4	0.798720	0.399360	+	0.916794i	$0.369232\pi$
$62$	−11217.7	−0.370617
$63$	0	0
$64$	36333.5	1.10881
$65$	−37860.8	−1.11149
$66$	0	0
$67$	−6671.32	−0.181562	−0.0907809	−	0.995871i	$-0.528936\pi$
$67$	−6671.32	−0.181562	−0.0907809	+	0.995871i	$0.528936\pi$
$68$	−5675.48	−0.148844
$69$	0	0
$70$	0	0
$71$	25110.5	0.591166	0.295583	−	0.955317i	$-0.404486\pi$
$71$	25110.5	0.591166	0.295583	+	0.955317i	$0.404486\pi$
$72$	0	0
$73$	−9474.97	−0.208099	−0.104050	−	0.994572i	$-0.533180\pi$
$73$	−9474.97	−0.208099	−0.104050	+	0.994572i	$0.533180\pi$
$74$	1975.90	0.0419455
$75$	0	0
$76$	−10163.4	−0.201840
$77$	0	0
$78$	0	0
$79$	−26476.1	−0.477295	−0.238647	−	0.971106i	$-0.576704\pi$
$79$	−26476.1	−0.477295	−0.238647	+	0.971106i	$0.576704\pi$
$80$	−29905.9	−0.522435
$81$	0	0
$82$	87480.3	1.43673
$83$	7494.03	0.119404	0.0597022	−	0.998216i	$-0.480985\pi$
$83$	7494.03	0.119404	0.0597022	+	0.998216i	$0.480985\pi$
$84$	0	0
$85$	29981.6	0.450098
$86$	−71410.5	−1.04116
$87$	0	0
$88$	−139356.	−1.91831
$89$	−32171.3	−0.430520	−0.215260	−	0.976557i	$-0.569060\pi$
$89$	−32171.3	−0.430520	−0.215260	+	0.976557i	$0.569060\pi$
$90$	0	0
$91$	0	0
$92$	8466.35	0.104286
$93$	0	0
$94$	103351.	1.20641
$95$	53689.9	0.610357
$96$	0	0
$97$	155070.	1.67340	0.836699	−	0.547664i	$-0.184483\pi$
$97$	155070.	1.67340	0.836699	+	0.547664i	$0.184483\pi$
$98$	0	0
$99$	0	0
$100$	10499.4	0.104994
$101$	111922.	1.09172	0.545860	−	0.837876i	$-0.316203\pi$
$101$	111922.	1.09172	0.545860	+	0.837876i	$0.316203\pi$
$102$	0	0
$103$	60684.3	0.563616	0.281808	−	0.959471i	$-0.409066\pi$
$103$	60684.3	0.563616	0.281808	+	0.959471i	$0.409066\pi$
$104$	173612.	1.57397
$105$	0	0
$106$	524.219	0.00453156
$107$	112533.	0.950210	0.475105	−	0.879929i	$-0.342410\pi$
$107$	112533.	0.950210	0.475105	+	0.879929i	$0.342410\pi$
$108$	0	0
$109$	−7623.35	−0.0614581	−0.0307291	−	0.999528i	$-0.509783\pi$
$109$	−7623.35	−0.0614581	−0.0307291	+	0.999528i	$0.509783\pi$
$110$	148594.	1.17090
$111$	0	0
$112$	0	0
$113$	41498.7	0.305730	0.152865	−	0.988247i	$-0.451150\pi$
$113$	41498.7	0.305730	0.152865	+	0.988247i	$0.451150\pi$
$114$	0	0
$115$	−44724.8	−0.315358
$116$	−49776.6	−0.343463
$117$	0	0
$118$	217578.	1.43850
$119$	0	0
$120$	0	0
$121$	344297.	2.13781
$122$	113497.	0.690376
$123$	0	0
$124$	18566.3	0.108436
$125$	−189060.	−1.08224
$126$	0	0
$127$	94796.7	0.521535	0.260768	−	0.965402i	$-0.416024\pi$
$127$	94796.7	0.521535	0.260768	+	0.965402i	$0.416024\pi$
$128$	86369.2	0.465944
$129$	0	0
$130$	−185121.	−0.960721
$131$	−211095.	−1.07473	−0.537365	−	0.843350i	$-0.680580\pi$
$131$	−211095.	−1.07473	−0.537365	+	0.843350i	$0.680580\pi$
$132$	0	0
$133$	0	0
$134$	−32619.5	−0.156934
$135$	0	0
$136$	−137482.	−0.637379
$137$	96193.1	0.437867	0.218934	−	0.975740i	$-0.429742\pi$
$137$	96193.1	0.437867	0.218934	+	0.975740i	$0.429742\pi$
$138$	0	0
$139$	401994.	1.76475	0.882375	−	0.470548i	$-0.155944\pi$
$139$	401994.	1.76475	0.882375	+	0.470548i	$0.155944\pi$
$140$	0	0
$141$	0	0
$142$	122778.	0.510976
$143$	−629570.	−2.57457
$144$	0	0
$145$	262953.	1.03862
$146$	−46328.1	−0.179871
$147$	0	0
$148$	−3270.29	−0.0122725
$149$	−75300.1	−0.277862	−0.138931	−	0.990302i	$-0.544367\pi$
$149$	−75300.1	−0.277862	−0.138931	+	0.990302i	$0.544367\pi$
$150$	0	0
$151$	−490594.	−1.75098	−0.875488	−	0.483240i	$-0.839460\pi$
$151$	−490594.	−1.75098	−0.875488	+	0.483240i	$0.839460\pi$
$152$	−246197.	−0.864318
$153$	0	0
$154$	0	0
$155$	−98079.6	−0.327906
$156$	0	0
$157$	461015.	1.49268	0.746339	−	0.665566i	$-0.231810\pi$
$157$	461015.	1.49268	0.746339	+	0.665566i	$0.231810\pi$
$158$	−129455.	−0.412551
$159$	0	0
$160$	121951.	0.376604
$161$	0	0
$162$	0	0
$163$	−478444.	−1.41046	−0.705232	−	0.708977i	$-0.749157\pi$
$163$	−478444.	−1.41046	−0.705232	+	0.708977i	$0.749157\pi$
$164$	−144788.	−0.420361
$165$	0	0
$166$	36642.2	0.103208
$167$	−498852.	−1.38414	−0.692070	−	0.721830i	$-0.743301\pi$
$167$	−498852.	−1.38414	−0.692070	+	0.721830i	$0.743301\pi$
$168$	0	0
$169$	413036.	1.11243
$170$	146596.	0.389044
$171$	0	0
$172$	118191.	0.304623
$173$	260643.	0.662112	0.331056	−	0.943611i	$-0.392595\pi$
$173$	260643.	0.662112	0.331056	+	0.943611i	$0.392595\pi$
$174$	0	0
$175$	0	0
$176$	−497292.	−1.21012
$177$	0	0
$178$	−157302.	−0.372121
$179$	−533418.	−1.24433	−0.622164	−	0.782887i	$-0.713746\pi$
$179$	−533418.	−1.24433	−0.622164	+	0.782887i	$0.713746\pi$
$180$	0	0
$181$	−185798.	−0.421546	−0.210773	−	0.977535i	$-0.567598\pi$
$181$	−185798.	−0.421546	−0.210773	+	0.977535i	$0.567598\pi$
$182$	0	0
$183$	0	0
$184$	205087.	0.446574
$185$	17275.8	0.0371115
$186$	0	0
$187$	498551.	1.04257
$188$	−171055.	−0.352974
$189$	0	0
$190$	262518.	0.527564
$191$	708837.	1.40593	0.702964	−	0.711226i	$-0.251860\pi$
$191$	708837.	1.40593	0.702964	+	0.711226i	$0.251860\pi$
$192$	0	0
$193$	−723504.	−1.39813	−0.699065	−	0.715058i	$-0.746400\pi$
$193$	−723504.	−1.39813	−0.699065	+	0.715058i	$0.746400\pi$
$194$	758219.	1.44641
$195$	0	0
$196$	0	0
$197$	147313.	0.270443	0.135222	−	0.990815i	$-0.456825\pi$
$197$	147313.	0.270443	0.135222	+	0.990815i	$0.456825\pi$
$198$	0	0
$199$	−1420.98	−0.00254364	−0.00127182	−	0.999999i	$-0.500405\pi$
$199$	−1420.98	−0.00254364	−0.00127182	+	0.999999i	$0.500405\pi$
$200$	254334.	0.449604
$201$	0	0
$202$	547244.	0.943632
$203$	0	0
$204$	0	0
$205$	764865.	1.27116
$206$	296717.	0.487163
$207$	0	0
$208$	619535.	0.992905
$209$	892786.	1.41378
$210$	0	0
$211$	−30296.2	−0.0468470	−0.0234235	−	0.999726i	$-0.507457\pi$
$211$	−30296.2	−0.0468470	−0.0234235	+	0.999726i	$0.507457\pi$
$212$	−867.630	−0.00132585
$213$	0	0
$214$	550231.	0.821317
$215$	−624362.	−0.921171
$216$	0	0
$217$	0	0
$218$	−37274.5	−0.0531215
$219$	0	0
$220$	−245937.	−0.342584
$221$	−621103.	−0.855427
$222$	0	0
$223$	147864.	0.199114	0.0995570	−	0.995032i	$-0.468257\pi$
$223$	147864.	0.199114	0.0995570	+	0.995032i	$0.468257\pi$
$224$	0	0
$225$	0	0
$226$	202908.	0.264259
$227$	−813445.	−1.04776	−0.523882	−	0.851791i	$-0.675517\pi$
$227$	−813445.	−1.04776	−0.523882	+	0.851791i	$0.675517\pi$
$228$	0	0
$229$	−1.28219e6	−1.61571	−0.807854	−	0.589382i	$-0.799371\pi$
$229$	−1.28219e6	−1.61571	−0.807854	+	0.589382i	$0.799371\pi$
$230$	−218683.	−0.272581
$231$	0	0
$232$	−1.20578e6	−1.47078
$233$	−1.23570e6	−1.49116	−0.745578	−	0.666418i	$-0.767827\pi$
$233$	−1.23570e6	−1.49116	−0.745578	+	0.666418i	$0.767827\pi$
$234$	0	0
$235$	903627.	1.06738
$236$	−360111.	−0.420878
$237$	0	0
$238$	0	0
$239$	874240.	0.990001	0.495001	−	0.868893i	$-0.335168\pi$
$239$	874240.	0.990001	0.495001	+	0.868893i	$0.335168\pi$
$240$	0	0
$241$	979610.	1.08645	0.543226	−	0.839586i	$-0.317203\pi$
$241$	979610.	1.08645	0.543226	+	0.839586i	$0.317203\pi$
$242$	1.68344e6	1.84782
$243$	0	0
$244$	−187848.	−0.201991
$245$	0	0
$246$	0	0
$247$	−1.11225e6	−1.16000
$248$	449747.	0.464344
$249$	0	0
$250$	−924410.	−0.935437
$251$	213005.	0.213406	0.106703	−	0.994291i	$-0.465971\pi$
$251$	213005.	0.213406	0.106703	+	0.994291i	$0.465971\pi$
$252$	0	0
$253$	−743709.	−0.730469
$254$	463510.	0.450791
$255$	0	0
$256$	−740367.	−0.706069
$257$	−1.57901e6	−1.49126	−0.745629	−	0.666362i	$-0.767851\pi$
$257$	−1.57901e6	−1.49126	−0.745629	+	0.666362i	$0.767851\pi$
$258$	0	0
$259$	0	0
$260$	306392.	0.281089
$261$	0	0
$262$	−1.03215e6	−0.928947
$263$	289087.	0.257715	0.128857	−	0.991663i	$-0.458869\pi$
$263$	289087.	0.257715	0.128857	+	0.991663i	$0.458869\pi$
$264$	0	0
$265$	4583.39	0.00400933
$266$	0	0
$267$	0	0
$268$	53988.3	0.0459159
$269$	2.04489e6	1.72301	0.861506	−	0.507747i	$-0.169521\pi$
$269$	2.04489e6	1.72301	0.861506	+	0.507747i	$0.169521\pi$
$270$	0	0
$271$	605963.	0.501213	0.250607	−	0.968089i	$-0.419370\pi$
$271$	605963.	0.501213	0.250607	+	0.968089i	$0.419370\pi$
$272$	−490604.	−0.402077
$273$	0	0
$274$	470338.	0.378472
$275$	−922295.	−0.735424
$276$	0	0
$277$	1.22275e6	0.957499	0.478750	−	0.877951i	$-0.341090\pi$
$277$	1.22275e6	0.957499	0.478750	+	0.877951i	$0.341090\pi$
$278$	1.96556e6	1.52537
$279$	0	0
$280$	0	0
$281$	639205.	0.482919	0.241460	−	0.970411i	$-0.422374\pi$
$281$	639205.	0.482919	0.241460	+	0.970411i	$0.422374\pi$
$282$	0	0
$283$	−473655.	−0.351557	−0.175779	−	0.984430i	$-0.556244\pi$
$283$	−473655.	−0.351557	−0.175779	+	0.984430i	$0.556244\pi$
$284$	−203209.	−0.149502
$285$	0	0
$286$	−3.07830e6	−2.22533
$287$	0	0
$288$	0	0
$289$	−928011.	−0.653595
$290$	1.28571e6	0.897737
$291$	0	0
$292$	76677.1	0.0526270
$293$	829294.	0.564338	0.282169	−	0.959365i	$-0.408946\pi$
$293$	829294.	0.564338	0.282169	+	0.959365i	$0.408946\pi$
$294$	0	0
$295$	1.90234e6	1.27272
$296$	−79218.8	−0.0525532
$297$	0	0
$298$	−368181.	−0.240171
$299$	926525.	0.599348
$300$	0	0
$301$	0	0
$302$	−2.39877e6	−1.51346
$303$	0	0
$304$	−878555.	−0.545237
$305$	992338.	0.610815
$306$	0	0
$307$	−1.63715e6	−0.991385	−0.495693	−	0.868498i	$-0.665086\pi$
$307$	−1.63715e6	−0.991385	−0.495693	+	0.868498i	$0.665086\pi$
$308$	0	0
$309$	0	0
$310$	−479562.	−0.283427
$311$	−607458.	−0.356135	−0.178068	−	0.984018i	$-0.556985\pi$
$311$	−607458.	−0.356135	−0.178068	+	0.984018i	$0.556985\pi$
$312$	0	0
$313$	−1.57802e6	−0.910443	−0.455222	−	0.890378i	$-0.650440\pi$
$313$	−1.57802e6	−0.910443	−0.455222	+	0.890378i	$0.650440\pi$
$314$	2.25414e6	1.29020
$315$	0	0
$316$	214261.	0.120705
$317$	−2.18686e6	−1.22228	−0.611142	−	0.791521i	$-0.709290\pi$
$317$	−2.18686e6	−1.22228	−0.611142	+	0.791521i	$0.709290\pi$
$318$	0	0
$319$	4.37253e6	2.40578
$320$	1.55327e6	0.847954
$321$	0	0
$322$	0	0
$323$	880779.	0.469743
$324$	0	0
$325$	1.14901e6	0.603414
$326$	−2.33936e6	−1.21914
$327$	0	0
$328$	−3.50731e6	−1.80007
$329$	0	0
$330$	0	0
$331$	−2.76869e6	−1.38901	−0.694503	−	0.719490i	$-0.744376\pi$
$331$	−2.76869e6	−1.38901	−0.694503	+	0.719490i	$0.744376\pi$
$332$	−60646.2	−0.0301966
$333$	0	0
$334$	−2.43915e6	−1.19639
$335$	−285202.	−0.138848
$336$	0	0
$337$	−2.45187e6	−1.17604	−0.588021	−	0.808846i	$-0.700093\pi$
$337$	−2.45187e6	−1.17604	−0.588021	+	0.808846i	$0.700093\pi$
$338$	2.01955e6	0.961529
$339$	0	0
$340$	−242629.	−0.113827
$341$	−1.63092e6	−0.759534
$342$	0	0
$343$	0	0
$344$	2.86303e6	1.30446
$345$	0	0
$346$	1.27442e6	0.572298
$347$	−1.48100e6	−0.660284	−0.330142	−	0.943931i	$-0.607097\pi$
$347$	−1.48100e6	−0.660284	−0.330142	+	0.943931i	$0.607097\pi$
$348$	0	0
$349$	1.28643e6	0.565357	0.282678	−	0.959215i	$-0.408777\pi$
$349$	1.28643e6	0.565357	0.282678	+	0.959215i	$0.408777\pi$
$350$	0	0
$351$	0	0
$352$	2.02787e6	0.872335
$353$	−2.25124e6	−0.961580	−0.480790	−	0.876836i	$-0.659650\pi$
$353$	−2.25124e6	−0.961580	−0.480790	+	0.876836i	$0.659650\pi$
$354$	0	0
$355$	1.07348e6	0.452090
$356$	260349.	0.108876
$357$	0	0
$358$	−2.60816e6	−1.07554
$359$	−725864.	−0.297248	−0.148624	−	0.988894i	$-0.547484\pi$
$359$	−725864.	−0.297248	−0.148624	+	0.988894i	$0.547484\pi$
$360$	0	0
$361$	−898834.	−0.363004
$362$	−908464.	−0.364365
$363$	0	0
$364$	0	0
$365$	−405059.	−0.159142
$366$	0	0
$367$	−3.42764e6	−1.32840	−0.664202	−	0.747553i	$-0.731229\pi$
$367$	−3.42764e6	−1.32840	−0.664202	+	0.747553i	$0.731229\pi$
$368$	731854.	0.281712
$369$	0	0
$370$	84470.4	0.0320775
$371$	0	0
$372$	0	0
$373$	−1.09304e6	−0.406786	−0.203393	−	0.979097i	$-0.565197\pi$
$373$	−1.09304e6	−0.406786	−0.203393	+	0.979097i	$0.565197\pi$
$374$	2.43767e6	0.901149
$375$	0	0
$376$	−4.14361e6	−1.51151
$377$	−5.44737e6	−1.97394
$378$	0	0
$379$	−2.92579e6	−1.04627	−0.523137	−	0.852249i	$-0.675238\pi$
$379$	−2.92579e6	−1.04627	−0.523137	+	0.852249i	$0.675238\pi$
$380$	−434491.	−0.154355
$381$	0	0
$382$	3.46587e6	1.21522
$383$	103933.	0.0362038	0.0181019	−	0.999836i	$-0.494238\pi$
$383$	103933.	0.0362038	0.0181019	+	0.999836i	$0.494238\pi$
$384$	0	0
$385$	0	0
$386$	−3.53759e6	−1.20848
$387$	0	0
$388$	−1.25492e6	−0.423192
$389$	172931.	0.0579428	0.0289714	−	0.999580i	$-0.490777\pi$
$389$	172931.	0.0579428	0.0289714	+	0.999580i	$0.490777\pi$
$390$	0	0
$391$	−733707.	−0.242706
$392$	0	0
$393$	0	0
$394$	720291.	0.233759
$395$	−1.13186e6	−0.365008
$396$	0	0
$397$	−2.59965e6	−0.827826	−0.413913	−	0.910316i	$-0.635838\pi$
$397$	−2.59965e6	−0.827826	−0.413913	+	0.910316i	$0.635838\pi$
$398$	−6947.91	−0.00219860
$399$	0	0
$400$	907594.	0.283623
$401$	3.45011e6	1.07145	0.535726	−	0.844392i	$-0.320038\pi$
$401$	3.45011e6	1.07145	0.535726	+	0.844392i	$0.320038\pi$
$402$	0	0
$403$	2.03183e6	0.623196
$404$	−905738.	−0.276089
$405$	0	0
$406$	0	0
$407$	287272.	0.0859621
$408$	0	0
$409$	1.44275e6	0.426464	0.213232	−	0.977002i	$-0.431601\pi$
$409$	1.44275e6	0.426464	0.213232	+	0.977002i	$0.431601\pi$
$410$	3.73982e6	1.09873
$411$	0	0
$412$	−491093.	−0.142535
$413$	0	0
$414$	0	0
$415$	320373.	0.0913136
$416$	−2.52635e6	−0.715749
$417$	0	0
$418$	4.36530e6	1.22200
$419$	5.10039e6	1.41928	0.709641	−	0.704564i	$-0.248857\pi$
$419$	5.10039e6	1.41928	0.709641	+	0.704564i	$0.248857\pi$
$420$	0	0
$421$	5.08241e6	1.39754	0.698771	−	0.715346i	$-0.253731\pi$
$421$	5.08241e6	1.39754	0.698771	+	0.715346i	$0.253731\pi$
$422$	−148134.	−0.0404923
$423$	0	0
$424$	−21017.3	−0.00567757
$425$	−909891.	−0.244353
$426$	0	0
$427$	0	0
$428$	−910683.	−0.240302
$429$	0	0
$430$	−3.05283e6	−0.796217
$431$	2.47441e6	0.641620	0.320810	−	0.947144i	$-0.396045\pi$
$431$	2.47441e6	0.641620	0.320810	+	0.947144i	$0.396045\pi$
$432$	0	0
$433$	331952.	0.0850855	0.0425428	−	0.999095i	$-0.486454\pi$
$433$	331952.	0.0850855	0.0425428	+	0.999095i	$0.486454\pi$
$434$	0	0
$435$	0	0
$436$	61692.7	0.0155424
$437$	−1.31389e6	−0.329122
$438$	0	0
$439$	4.11220e6	1.01839	0.509193	−	0.860652i	$-0.329944\pi$
$439$	4.11220e6	1.01839	0.509193	+	0.860652i	$0.329944\pi$
$440$	−5.95752e6	−1.46701
$441$	0	0
$442$	−3.03690e6	−0.739391
$443$	5.85348e6	1.41712	0.708558	−	0.705653i	$-0.249346\pi$
$443$	5.85348e6	1.41712	0.708558	+	0.705653i	$0.249346\pi$
$444$	0	0
$445$	−1.37534e6	−0.329237
$446$	722986.	0.172105
$447$	0	0
$448$	0	0
$449$	−3.86497e6	−0.904754	−0.452377	−	0.891827i	$-0.649424\pi$
$449$	−3.86497e6	−0.904754	−0.452377	+	0.891827i	$0.649424\pi$
$450$	0	0
$451$	1.27186e7	2.94441
$452$	−335832.	−0.0773172
$453$	0	0
$454$	−3.97736e6	−0.905638
$455$	0	0
$456$	0	0
$457$	−5.39046e6	−1.20736	−0.603678	−	0.797228i	$-0.706299\pi$
$457$	−5.39046e6	−1.20736	−0.603678	+	0.797228i	$0.706299\pi$
$458$	−6.26928e6	−1.39654
$459$	0	0
$460$	361940.	0.0797520
$461$	−1.17429e6	−0.257350	−0.128675	−	0.991687i	$-0.541072\pi$
$461$	−1.17429e6	−0.257350	−0.128675	+	0.991687i	$0.541072\pi$
$462$	0	0
$463$	4.03702e6	0.875203	0.437601	−	0.899169i	$-0.355828\pi$
$463$	4.03702e6	0.875203	0.437601	+	0.899169i	$0.355828\pi$
$464$	−4.30283e6	−0.927810
$465$	0	0
$466$	−6.04198e6	−1.28889
$467$	−1.43710e6	−0.304926	−0.152463	−	0.988309i	$-0.548721\pi$
$467$	−1.43710e6	−0.304926	−0.152463	+	0.988309i	$0.548721\pi$
$468$	0	0
$469$	0	0
$470$	4.41830e6	0.922594
$471$	0	0
$472$	−8.72325e6	−1.80229
$473$	−1.03822e7	−2.13372
$474$	0	0
$475$	−1.62940e6	−0.331355
$476$	0	0
$477$	0	0
$478$	4.27461e6	0.855711
$479$	2.37217e6	0.472396	0.236198	−	0.971705i	$-0.424099\pi$
$479$	2.37217e6	0.472396	0.236198	+	0.971705i	$0.424099\pi$
$480$	0	0
$481$	−357888.	−0.0705317
$482$	4.78982e6	0.939079
$483$	0	0
$484$	−2.78625e6	−0.540639
$485$	6.62932e6	1.27972
$486$	0	0
$487$	669068.	0.127834	0.0639172	−	0.997955i	$-0.479641\pi$
$487$	669068.	0.127834	0.0639172	+	0.997955i	$0.479641\pi$
$488$	−4.55040e6	−0.864968
$489$	0	0
$490$	0	0
$491$	9.58134e6	1.79359	0.896794	−	0.442448i	$-0.145890\pi$
$491$	9.58134e6	1.79359	0.896794	+	0.442448i	$0.145890\pi$
$492$	0	0
$493$	4.31372e6	0.799346
$494$	−5.43836e6	−1.00265
$495$	0	0
$496$	1.60493e6	0.292921
$497$	0	0
$498$	0	0
$499$	−2.17805e6	−0.391576	−0.195788	−	0.980646i	$-0.562726\pi$
$499$	−2.17805e6	−0.391576	−0.195788	+	0.980646i	$0.562726\pi$
$500$	1.52998e6	0.273692
$501$	0	0
$502$	1.04149e6	0.184458
$503$	−5.11108e6	−0.900726	−0.450363	−	0.892846i	$-0.648705\pi$
$503$	−5.11108e6	−0.900726	−0.450363	+	0.892846i	$0.648705\pi$
$504$	0	0
$505$	4.78470e6	0.834885
$506$	−3.63638e6	−0.631383
$507$	0	0
$508$	−767151.	−0.131893
$509$	7.08669e6	1.21241	0.606204	−	0.795309i	$-0.292691\pi$
$509$	7.08669e6	1.21241	0.606204	+	0.795309i	$0.292691\pi$
$510$	0	0
$511$	0	0
$512$	−6.38385e6	−1.07624
$513$	0	0
$514$	−7.72061e6	−1.28897
$515$	2.59428e6	0.431021
$516$	0	0
$517$	1.50260e7	2.47239
$518$	0	0
$519$	0	0
$520$	7.42198e6	1.20368
$521$	−3.45387e6	−0.557457	−0.278729	−	0.960370i	$-0.589913\pi$
$521$	−3.45387e6	−0.557457	−0.278729	+	0.960370i	$0.589913\pi$
$522$	0	0
$523$	557199.	0.0890751	0.0445375	−	0.999008i	$-0.485819\pi$
$523$	557199.	0.0890751	0.0445375	+	0.999008i	$0.485819\pi$
$524$	1.70831e6	0.271793
$525$	0	0
$526$	1.41350e6	0.222757
$527$	−1.60899e6	−0.252363
$528$	0	0
$529$	−5.34184e6	−0.829950
$530$	22410.6	0.00346548
$531$	0	0
$532$	0	0
$533$	−1.58450e7	−2.41588
$534$	0	0
$535$	4.81082e6	0.726666
$536$	1.30780e6	0.196621
$537$	0	0
$538$	9.99851e6	1.48929
$539$	0	0
$540$	0	0
$541$	4.78728e6	0.703227	0.351614	−	0.936145i	$-0.385633\pi$
$541$	4.78728e6	0.703227	0.351614	+	0.936145i	$0.385633\pi$
$542$	2.96287e6	0.433225
$543$	0	0
$544$	2.00060e6	0.289843
$545$	−325901.	−0.0469997
$546$	0	0
$547$	2.30360e6	0.329184	0.164592	−	0.986362i	$-0.447369\pi$
$547$	2.30360e6	0.329184	0.164592	+	0.986362i	$0.447369\pi$
$548$	−778452.	−0.110734
$549$	0	0
$550$	−4.50958e6	−0.635666
$551$	7.72485e6	1.08395
$552$	0	0
$553$	0	0
$554$	5.97866e6	0.827617
$555$	0	0
$556$	−3.25318e6	−0.446294
$557$	1.38851e7	1.89631	0.948156	−	0.317805i	$-0.102946\pi$
$557$	1.38851e7	1.89631	0.948156	+	0.317805i	$0.102946\pi$
$558$	0	0
$559$	1.29344e7	1.75072
$560$	0	0
$561$	0	0
$562$	3.12541e6	0.417413
$563$	3.95924e6	0.526430	0.263215	−	0.964737i	$-0.415217\pi$
$563$	3.95924e6	0.526430	0.263215	+	0.964737i	$0.415217\pi$
$564$	0	0
$565$	1.77408e6	0.233805
$566$	−2.31595e6	−0.303870
$567$	0	0
$568$	−4.92250e6	−0.640199
$569$	−9.51079e6	−1.23150	−0.615752	−	0.787940i	$-0.711148\pi$
$569$	−9.51079e6	−1.23150	−0.615752	+	0.787940i	$0.711148\pi$
$570$	0	0
$571$	−9.07955e6	−1.16540	−0.582699	−	0.812688i	$-0.698003\pi$
$571$	−9.07955e6	−1.16540	−0.582699	+	0.812688i	$0.698003\pi$
$572$	5.09486e6	0.651092
$573$	0	0
$574$	0	0
$575$	1.35732e6	0.171204
$576$	0	0
$577$	3.58386e6	0.448138	0.224069	−	0.974573i	$-0.428066\pi$
$577$	3.58386e6	0.448138	0.224069	+	0.974573i	$0.428066\pi$
$578$	−4.53753e6	−0.564937
$579$	0	0
$580$	−2.12797e6	−0.262661
$581$	0	0
$582$	0	0
$583$	76215.2	0.00928689
$584$	1.85741e6	0.225360
$585$	0	0
$586$	4.05485e6	0.487788
$587$	6.14848e6	0.736499	0.368250	−	0.929727i	$-0.379957\pi$
$587$	6.14848e6	0.736499	0.368250	+	0.929727i	$0.379957\pi$
$588$	0	0
$589$	−2.88131e6	−0.342218
$590$	9.30153e6	1.10008
$591$	0	0
$592$	−282693.	−0.0331521
$593$	−1.32996e7	−1.55311	−0.776554	−	0.630051i	$-0.783034\pi$
$593$	−1.32996e7	−1.55311	−0.776554	+	0.630051i	$0.783034\pi$
$594$	0	0
$595$	0	0
$596$	609373.	0.0702697
$597$	0	0
$598$	4.53026e6	0.518048
$599$	4.61463e6	0.525497	0.262748	−	0.964864i	$-0.415371\pi$
$599$	4.61463e6	0.525497	0.262748	+	0.964864i	$0.415371\pi$
$600$	0	0
$601$	−8.17563e6	−0.923284	−0.461642	−	0.887066i	$-0.652740\pi$
$601$	−8.17563e6	−0.923284	−0.461642	+	0.887066i	$0.652740\pi$
$602$	0	0
$603$	0	0
$604$	3.97018e6	0.442811
$605$	1.47188e7	1.63487
$606$	0	0
$607$	4.36988e6	0.481391	0.240696	−	0.970601i	$-0.422624\pi$
$607$	4.36988e6	0.481391	0.240696	+	0.970601i	$0.422624\pi$
$608$	3.58259e6	0.393041
$609$	0	0
$610$	4.85205e6	0.527960
$611$	−1.87197e7	−2.02859
$612$	0	0
$613$	5.44920e6	0.585708	0.292854	−	0.956157i	$-0.405395\pi$
$613$	5.44920e6	0.585708	0.292854	+	0.956157i	$0.405395\pi$
$614$	−8.00488e6	−0.856907
$615$	0	0
$616$	0	0
$617$	1.32758e7	1.40394	0.701969	−	0.712208i	$-0.252305\pi$
$617$	1.32758e7	1.40394	0.701969	+	0.712208i	$0.252305\pi$
$618$	0	0
$619$	4.46285e6	0.468150	0.234075	−	0.972219i	$-0.424794\pi$
$619$	4.46285e6	0.468150	0.234075	+	0.972219i	$0.424794\pi$
$620$	793719.	0.0829254
$621$	0	0
$622$	−2.97018e6	−0.307827
$623$	0	0
$624$	0	0
$625$	−4.02799e6	−0.412466
$626$	−7.71578e6	−0.786945
$627$	0	0
$628$	−3.73081e6	−0.377489
$629$	283408.	0.0285618
$630$	0	0
$631$	8.00421e6	0.800286	0.400143	−	0.916453i	$-0.368960\pi$
$631$	8.00421e6	0.800286	0.400143	+	0.916453i	$0.368960\pi$
$632$	5.19021e6	0.516883
$633$	0	0
$634$	−1.06927e7	−1.05649
$635$	4.05260e6	0.398840
$636$	0	0
$637$	0	0
$638$	2.13796e7	2.07944
$639$	0	0
$640$	3.69232e6	0.356327
$641$	−3.88910e6	−0.373855	−0.186928	−	0.982374i	$-0.559853\pi$
$641$	−3.88910e6	−0.373855	−0.186928	+	0.982374i	$0.559853\pi$
$642$	0	0
$643$	1.77478e7	1.69285	0.846423	−	0.532511i	$-0.178751\pi$
$643$	1.77478e7	1.69285	0.846423	+	0.532511i	$0.178751\pi$
$644$	0	0
$645$	0	0
$646$	4.30659e6	0.406024
$647$	8.72276e6	0.819206	0.409603	−	0.912264i	$-0.365667\pi$
$647$	8.72276e6	0.819206	0.409603	+	0.912264i	$0.365667\pi$
$648$	0	0
$649$	3.16332e7	2.94803
$650$	5.61811e6	0.521563
$651$	0	0
$652$	3.87185e6	0.356697
$653$	1.42992e6	0.131228	0.0656141	−	0.997845i	$-0.479099\pi$
$653$	1.42992e6	0.131228	0.0656141	+	0.997845i	$0.479099\pi$
$654$	0	0
$655$	−9.02440e6	−0.821892
$656$	−1.25159e7	−1.13554
$657$	0	0
$658$	0	0
$659$	−9.20182e6	−0.825392	−0.412696	−	0.910869i	$-0.635413\pi$
$659$	−9.20182e6	−0.825392	−0.412696	+	0.910869i	$0.635413\pi$
$660$	0	0
$661$	−1.43513e6	−0.127758	−0.0638790	−	0.997958i	$-0.520347\pi$
$661$	−1.43513e6	−0.127758	−0.0638790	+	0.997958i	$0.520347\pi$
$662$	−1.35376e7	−1.20059
$663$	0	0
$664$	−1.46908e6	−0.129308
$665$	0	0
$666$	0	0
$667$	−6.43495e6	−0.560055
$668$	4.03701e6	0.350040
$669$	0	0
$670$	−1.39450e6	−0.120014
$671$	1.65012e7	1.41484
$672$	0	0
$673$	2.10985e7	1.79562	0.897808	−	0.440388i	$-0.145159\pi$
$673$	2.10985e7	1.79562	0.897808	+	0.440388i	$0.145159\pi$
$674$	−1.19885e7	−1.01652
$675$	0	0
$676$	−3.34254e6	−0.281326
$677$	2.33670e7	1.95944	0.979720	−	0.200372i	$-0.0642151\pi$
$677$	2.33670e7	1.95944	0.979720	+	0.200372i	$0.0642151\pi$
$678$	0	0
$679$	0	0
$680$	−5.87740e6	−0.487431
$681$	0	0
$682$	−7.97443e6	−0.656506
$683$	1.33489e7	1.09495	0.547474	−	0.836823i	$-0.315589\pi$
$683$	1.33489e7	1.09495	0.547474	+	0.836823i	$0.315589\pi$
$684$	0	0
$685$	4.11229e6	0.334856
$686$	0	0
$687$	0	0
$688$	1.02168e7	0.822890
$689$	−94950.1	−0.00761987
$690$	0	0
$691$	−1.31469e7	−1.04743	−0.523717	−	0.851892i	$-0.675455\pi$
$691$	−1.31469e7	−1.04743	−0.523717	+	0.851892i	$0.675455\pi$
$692$	−2.10928e6	−0.167444
$693$	0	0
$694$	−7.24137e6	−0.570718
$695$	1.71854e7	1.34958
$696$	0	0
$697$	1.25475e7	0.978311
$698$	6.29002e6	0.488668
$699$	0	0
$700$	0	0
$701$	−7.87304e6	−0.605128	−0.302564	−	0.953129i	$-0.597843\pi$
$701$	−7.87304e6	−0.605128	−0.302564	+	0.953129i	$0.597843\pi$
$702$	0	0
$703$	507517.	0.0387313
$704$	2.58287e7	1.96413
$705$	0	0
$706$	−1.10075e7	−0.831144
$707$	0	0
$708$	0	0
$709$	−1.49458e7	−1.11661	−0.558306	−	0.829635i	$-0.688548\pi$
$709$	−1.49458e7	−1.11661	−0.558306	+	0.829635i	$0.688548\pi$
$710$	5.24882e6	0.390765
$711$	0	0
$712$	6.30665e6	0.466229
$713$	2.40019e6	0.176816
$714$	0	0
$715$	−2.69144e7	−1.96888
$716$	4.31674e6	0.314683
$717$	0	0
$718$	−3.54913e6	−0.256927
$719$	−355825.	−0.0256693	−0.0128347	−	0.999918i	$-0.504086\pi$
$719$	−355825.	−0.0256693	−0.0128347	+	0.999918i	$0.504086\pi$
$720$	0	0
$721$	0	0
$722$	−4.39487e6	−0.313764
$723$	0	0
$724$	1.50359e6	0.106606
$725$	−7.98017e6	−0.563855
$726$	0	0
$727$	1.16451e7	0.817160	0.408580	−	0.912722i	$-0.366024\pi$
$727$	1.16451e7	0.817160	0.408580	+	0.912722i	$0.366024\pi$
$728$	0	0
$729$	0	0
$730$	−1.98054e6	−0.137555
$731$	−1.02426e7	−0.708953
$732$	0	0
$733$	−1.56892e7	−1.07855	−0.539275	−	0.842130i	$-0.681302\pi$
$733$	−1.56892e7	−1.07855	−0.539275	+	0.842130i	$0.681302\pi$
$734$	−1.67595e7	−1.14821
$735$	0	0
$736$	−2.98437e6	−0.203076
$737$	−4.74249e6	−0.321616
$738$	0	0
$739$	1.49707e7	1.00839	0.504197	−	0.863589i	$-0.331789\pi$
$739$	1.49707e7	1.00839	0.504197	+	0.863589i	$0.331789\pi$
$740$	−139806.	−0.00938528
$741$	0	0
$742$	0	0
$743$	−1.51503e7	−1.00682	−0.503408	−	0.864049i	$-0.667921\pi$
$743$	−1.51503e7	−1.00682	−0.503408	+	0.864049i	$0.667921\pi$
$744$	0	0
$745$	−3.21911e6	−0.212493
$746$	−5.34446e6	−0.351606
$747$	0	0
$748$	−4.03457e6	−0.263660
$749$	0	0
$750$	0	0
$751$	2.13714e7	1.38272	0.691359	−	0.722512i	$-0.257013\pi$
$751$	2.13714e7	1.38272	0.691359	+	0.722512i	$0.257013\pi$
$752$	−1.47865e7	−0.953501
$753$	0	0
$754$	−2.66350e7	−1.70618
$755$	−2.09731e7	−1.33905
$756$	0	0
$757$	−1.74578e7	−1.10726	−0.553632	−	0.832762i	$-0.686758\pi$
$757$	−1.74578e7	−1.10726	−0.553632	+	0.832762i	$0.686758\pi$
$758$	−1.43057e7	−0.904350
$759$	0	0
$760$	−1.05250e7	−0.660981
$761$	−1.22517e7	−0.766895	−0.383448	−	0.923563i	$-0.625263\pi$
$761$	−1.22517e7	−0.766895	−0.383448	+	0.923563i	$0.625263\pi$
$762$	0	0
$763$	0	0
$764$	−5.73633e6	−0.355550
$765$	0	0
$766$	508180.	0.0312929
$767$	−3.94092e7	−2.41885
$768$	0	0
$769$	−7.26941e6	−0.443285	−0.221643	−	0.975128i	$-0.571142\pi$
$769$	−7.26941e6	−0.443285	−0.221643	+	0.975128i	$0.571142\pi$
$770$	0	0
$771$	0	0
$772$	5.85503e6	0.353578
$773$	−2.02144e6	−0.121678	−0.0608389	−	0.998148i	$-0.519378\pi$
$773$	−2.02144e6	−0.121678	−0.0608389	+	0.998148i	$0.519378\pi$
$774$	0	0
$775$	2.97655e6	0.178016
$776$	−3.03990e7	−1.81219
$777$	0	0
$778$	845550.	0.0500830
$779$	2.24697e7	1.32664
$780$	0	0
$781$	1.78505e7	1.04718
$782$	−3.58747e6	−0.209784
$783$	0	0
$784$	0	0
$785$	1.97086e7	1.14151
$786$	0	0
$787$	1.21113e7	0.697035	0.348517	−	0.937302i	$-0.386685\pi$
$787$	1.21113e7	0.697035	0.348517	+	0.937302i	$0.386685\pi$
$788$	−1.19215e6	−0.0683934
$789$	0	0
$790$	−5.53428e6	−0.315495
$791$	0	0
$792$	0	0
$793$	−2.05574e7	−1.16087
$794$	−1.27111e7	−0.715534
$795$	0	0
$796$	11499.4	0.000643270 0
$797$	2.45062e7	1.36657	0.683283	−	0.730154i	$-0.260552\pi$
$797$	2.45062e7	1.36657	0.683283	+	0.730154i	$0.260552\pi$
$798$	0	0
$799$	1.48239e7	0.821479
$800$	−3.70100e6	−0.204454
$801$	0	0
$802$	1.68694e7	0.926112
$803$	−6.73555e6	−0.368624
$804$	0	0
$805$	0	0
$806$	9.93468e6	0.538662
$807$	0	0
$808$	−2.19404e7	−1.18227
$809$	5.98754e6	0.321645	0.160823	−	0.986983i	$-0.448585\pi$
$809$	5.98754e6	0.321645	0.160823	+	0.986983i	$0.448585\pi$
$810$	0	0
$811$	−1.57094e7	−0.838699	−0.419350	−	0.907825i	$-0.637742\pi$
$811$	−1.57094e7	−0.838699	−0.419350	+	0.907825i	$0.637742\pi$
$812$	0	0
$813$	0	0
$814$	1.40462e6	0.0743016
$815$	−2.04537e7	−1.07864
$816$	0	0
$817$	−1.83421e7	−0.961376
$818$	7.05435e6	0.368615
$819$	0	0
$820$	−6.18974e6	−0.321468
$821$	111158.	0.00575547	0.00287774	−	0.999996i	$-0.499084\pi$
$821$	111158.	0.00575547	0.00287774	+	0.999996i	$0.499084\pi$
$822$	0	0
$823$	3.43892e6	0.176980	0.0884898	−	0.996077i	$-0.471796\pi$
$823$	3.43892e6	0.176980	0.0884898	+	0.996077i	$0.471796\pi$
$824$	−1.18961e7	−0.610363
$825$	0	0
$826$	0	0
$827$	−1.87782e7	−0.954751	−0.477375	−	0.878700i	$-0.658412\pi$
$827$	−1.87782e7	−0.954751	−0.477375	+	0.878700i	$0.658412\pi$
$828$	0	0
$829$	−7.53778e6	−0.380940	−0.190470	−	0.981693i	$-0.561001\pi$
$829$	−7.53778e6	−0.380940	−0.190470	+	0.981693i	$0.561001\pi$
$830$	1.56647e6	0.0789272
$831$	0	0
$832$	−3.21778e7	−1.61156
$833$	0	0
$834$	0	0
$835$	−2.13261e7	−1.05851
$836$	−7.22496e6	−0.357536
$837$	0	0
$838$	2.49385e7	1.22676
$839$	−1.65147e7	−0.809962	−0.404981	−	0.914325i	$-0.632722\pi$
$839$	−1.65147e7	−0.809962	−0.404981	+	0.914325i	$0.632722\pi$
$840$	0	0
$841$	1.73222e7	0.844527
$842$	2.48506e7	1.20797
$843$	0	0
$844$	245175.	0.0118473
$845$	1.76575e7	0.850720
$846$	0	0
$847$	0	0
$848$	−75000.3	−0.00358157
$849$	0	0
$850$	−4.44893e6	−0.211207
$851$	−422772.	−0.0200116
$852$	0	0
$853$	−2.36772e7	−1.11418	−0.557092	−	0.830451i	$-0.688083\pi$
$853$	−2.36772e7	−1.11418	−0.557092	+	0.830451i	$0.688083\pi$
$854$	0	0
$855$	0	0
$856$	−2.20602e7	−1.02902
$857$	296551.	0.0137927	0.00689633	−	0.999976i	$-0.497805\pi$
$857$	296551.	0.0137927	0.00689633	+	0.999976i	$0.497805\pi$
$858$	0	0
$859$	−1.21295e7	−0.560866	−0.280433	−	0.959874i	$-0.590478\pi$
$859$	−1.21295e7	−0.560866	−0.280433	+	0.959874i	$0.590478\pi$
$860$	5.05271e6	0.232958
$861$	0	0
$862$	1.20987e7	0.554586
$863$	1.30180e7	0.595000	0.297500	−	0.954722i	$-0.403847\pi$
$863$	1.30180e7	0.595000	0.297500	+	0.954722i	$0.403847\pi$
$864$	0	0
$865$	1.11426e7	0.506345
$866$	1.62309e6	0.0735440
$867$	0	0
$868$	0	0
$869$	−1.88213e7	−0.845473
$870$	0	0
$871$	5.90828e6	0.263885
$872$	1.49443e6	0.0665556
$873$	0	0
$874$	−6.42431e6	−0.284478
$875$	0	0
$876$	0	0
$877$	−2.08926e6	−0.0917260	−0.0458630	−	0.998948i	$-0.514604\pi$
$877$	−2.08926e6	−0.0917260	−0.0458630	+	0.998948i	$0.514604\pi$
$878$	2.01067e7	0.880245
$879$	0	0
$880$	−2.12595e7	−0.925434
$881$	−2.84897e6	−0.123665	−0.0618327	−	0.998087i	$-0.519695\pi$
$881$	−2.84897e6	−0.123665	−0.0618327	+	0.998087i	$0.519695\pi$
$882$	0	0
$883$	−1.75998e7	−0.759636	−0.379818	−	0.925061i	$-0.624013\pi$
$883$	−1.75998e7	−0.759636	−0.379818	+	0.925061i	$0.624013\pi$
$884$	5.02634e6	0.216332
$885$	0	0
$886$	2.86207e7	1.22489
$887$	2.37318e7	1.01279	0.506397	−	0.862300i	$-0.330977\pi$
$887$	2.37318e7	1.01279	0.506397	+	0.862300i	$0.330977\pi$
$888$	0	0
$889$	0	0
$890$	−6.72473e6	−0.284577
$891$	0	0
$892$	−1.19661e6	−0.0503547
$893$	2.65461e7	1.11397
$894$	0	0
$895$	−2.28038e7	−0.951591
$896$	0	0
$897$	0	0
$898$	−1.88979e7	−0.782027
$899$	−1.41116e7	−0.582340
$900$	0	0
$901$	75190.1	0.00308567
$902$	6.21878e7	2.54501
$903$	0	0
$904$	−8.13513e6	−0.331088
$905$	−7.94295e6	−0.322374
$906$	0	0
$907$	−2.47626e7	−0.999489	−0.499744	−	0.866173i	$-0.666573\pi$
$907$	−2.47626e7	−0.999489	−0.499744	+	0.866173i	$0.666573\pi$
$908$	6.58288e6	0.264973
$909$	0	0
$910$	0	0
$911$	1.67991e7	0.670639	0.335320	−	0.942104i	$-0.391156\pi$
$911$	1.67991e7	0.670639	0.335320	+	0.942104i	$0.391156\pi$
$912$	0	0
$913$	5.32734e6	0.211511
$914$	−2.63568e7	−1.04358
$915$	0	0
$916$	1.03762e7	0.408602
$917$	0	0
$918$	0	0
$919$	1.41649e7	0.553256	0.276628	−	0.960977i	$-0.410783\pi$
$919$	1.41649e7	0.553256	0.276628	+	0.960977i	$0.410783\pi$
$920$	8.76756e6	0.341514
$921$	0	0
$922$	−5.74174e6	−0.222442
$923$	−2.22384e7	−0.859212
$924$	0	0
$925$	−524292.	−0.0201474
$926$	1.97391e7	0.756484
$927$	0	0
$928$	1.75462e7	0.668825
$929$	−2.95554e7	−1.12356	−0.561781	−	0.827286i	$-0.689884\pi$
$929$	−2.95554e7	−1.12356	−0.561781	+	0.827286i	$0.689884\pi$
$930$	0	0
$931$	0	0
$932$	1.00000e7	0.377104
$933$	0	0
$934$	−7.02672e6	−0.263564
$935$	2.13133e7	0.797298
$936$	0	0
$937$	−3.20477e7	−1.19247	−0.596235	−	0.802810i	$-0.703337\pi$
$937$	−3.20477e7	−1.19247	−0.596235	+	0.802810i	$0.703337\pi$
$938$	0	0
$939$	0	0
$940$	−7.31269e6	−0.269934
$941$	3.68947e7	1.35828	0.679141	−	0.734008i	$-0.262353\pi$
$941$	3.68947e7	1.35828	0.679141	+	0.734008i	$0.262353\pi$
$942$	0	0
$943$	−1.87177e7	−0.685446
$944$	−3.11290e7	−1.13693
$945$	0	0
$946$	−5.07642e7	−1.84429
$947$	−1.45741e7	−0.528087	−0.264044	−	0.964511i	$-0.585056\pi$
$947$	−1.45741e7	−0.528087	−0.264044	+	0.964511i	$0.585056\pi$
$948$	0	0
$949$	8.39126e6	0.302456
$950$	−7.96697e6	−0.286407
$951$	0	0
$952$	0	0
$953$	2.34882e7	0.837757	0.418878	−	0.908042i	$-0.362423\pi$
$953$	2.34882e7	0.837757	0.418878	+	0.908042i	$0.362423\pi$
$954$	0	0
$955$	3.03031e7	1.07517
$956$	−7.07487e6	−0.250365
$957$	0	0
$958$	1.15988e7	0.408317
$959$	0	0
$960$	0	0
$961$	−2.33656e7	−0.816148
$962$	−1.74990e6	−0.0609643
$963$	0	0
$964$	−7.92759e6	−0.274757
$965$	−3.09301e7	−1.06921
$966$	0	0
$967$	−2.25237e7	−0.774593	−0.387296	−	0.921955i	$-0.626591\pi$
$967$	−2.25237e7	−0.774593	−0.387296	+	0.921955i	$0.626591\pi$
$968$	−6.74937e7	−2.31513
$969$	0	0
$970$	3.24142e7	1.10613
$971$	−5.31726e7	−1.80984	−0.904920	−	0.425583i	$-0.860069\pi$
$971$	−5.31726e7	−1.80984	−0.904920	+	0.425583i	$0.860069\pi$
$972$	0	0
$973$	0	0
$974$	3.27142e6	0.110494
$975$	0	0
$976$	−1.62381e7	−0.545646
$977$	2.82218e7	0.945905	0.472953	−	0.881088i	$-0.343188\pi$
$977$	2.82218e7	0.945905	0.472953	+	0.881088i	$0.343188\pi$
$978$	0	0
$979$	−2.28699e7	−0.762618
$980$	0	0
$981$	0	0
$982$	4.68482e7	1.55029
$983$	−1.71614e7	−0.566461	−0.283230	−	0.959052i	$-0.591406\pi$
$983$	−1.71614e7	−0.566461	−0.283230	+	0.959052i	$0.591406\pi$
$984$	0	0
$985$	6.29771e6	0.206820
$986$	2.10920e7	0.690917
$987$	0	0
$988$	9.00097e6	0.293357
$989$	1.52793e7	0.496722
$990$	0	0
$991$	3.05742e7	0.988941	0.494471	−	0.869194i	$-0.335362\pi$
$991$	3.05742e7	0.988941	0.494471	+	0.869194i	$0.335362\pi$
$992$	−6.54460e6	−0.211156
$993$	0	0
$994$	0	0
$995$	−60747.5	−0.00194523
$996$	0	0
$997$	−4.75481e7	−1.51494	−0.757470	−	0.652870i	$-0.773565\pi$
$997$	−4.75481e7	−1.51494	−0.757470	+	0.652870i	$0.773565\pi$
$998$	−1.06496e7	−0.338460
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.6.a.bd.1.4		6
3.2	odd	2	inner	441.6.a.bd.1.3		6
7.3	odd	6		63.6.e.f.37.3	✓	12
7.5	odd	6		63.6.e.f.46.3	yes	12
7.6	odd	2		441.6.a.bc.1.4		6
21.5	even	6		63.6.e.f.46.4	yes	12
21.17	even	6		63.6.e.f.37.4	yes	12
21.20	even	2		441.6.a.bc.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.6.e.f.37.3	✓	12	7.3	odd	6
63.6.e.f.37.4	yes	12	21.17	even	6
63.6.e.f.46.3	yes	12	7.5	odd	6
63.6.e.f.46.4	yes	12	21.5	even	6
441.6.a.bc.1.3		6	21.20	even	2
441.6.a.bc.1.4		6	7.6	odd	2
441.6.a.bd.1.3		6	3.2	odd	2	inner
441.6.a.bd.1.4		6	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)

\(f(q)\)	\(=\)	\(q+4.88952 q^{2} -8.09260 q^{4} +42.7504 q^{5} -196.034 q^{8} +O(q^{10})\)	\(q+4.88952 q^{2} -8.09260 q^{4} +42.7504 q^{5} -196.034 q^{8} +209.029 q^{10} +710.878 q^{11} -885.624 q^{13} -699.547 q^{16} +701.317 q^{17} +1255.89 q^{19} -345.962 q^{20} +3475.85 q^{22} -1046.18 q^{23} -1297.40 q^{25} -4330.27 q^{26} +6150.88 q^{29} -2294.24 q^{31} +2852.63 q^{32} +3429.10 q^{34} +404.109 q^{37} +6140.71 q^{38} -8380.51 q^{40} +17891.4 q^{41} -14604.8 q^{43} -5752.85 q^{44} -5115.34 q^{46} +21137.3 q^{47} -6343.67 q^{50} +7167.00 q^{52} +107.213 q^{53} +30390.3 q^{55} +30074.9 q^{58} +44498.8 q^{59} +23212.4 q^{61} -11217.7 q^{62} +36333.5 q^{64} -37860.8 q^{65} -6671.32 q^{67} -5675.48 q^{68} +25110.5 q^{71} -9474.97 q^{73} +1975.90 q^{74} -10163.4 q^{76} -26476.1 q^{79} -29905.9 q^{80} +87480.3 q^{82} +7494.03 q^{83} +29981.6 q^{85} -71410.5 q^{86} -139356. q^{88} -32171.3 q^{89} +8466.35 q^{92} +103351. q^{94} +53689.9 q^{95} +155070. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 182 q^{4}+O(q^{10}) \) 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

Introduction

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