LMFDB - Embedded newform 441.4.a.x.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,4,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, 4, names="a")

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

chi := DirichletCharacter("441.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
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:	$26.0198423125$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 74x^{6} + 1469x^{4} - 8828x^{2} + 2500 $ x^8 - 74x^6 + 1469x^4 - 8828*x^2 + 2500
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4}\cdot 7^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$3.37406$ 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)

$f(q)$	$=$	$q+1.95985 q^{2} -4.15899 q^{4} +11.8897 q^{5} -23.8298 q^{8} +O(q^{10})$	$q+1.95985 q^{2} -4.15899 q^{4} +11.8897 q^{5} -23.8298 q^{8} +23.3019 q^{10} +36.4130 q^{11} +0.964525 q^{13} -13.4309 q^{16} +98.2514 q^{17} -106.001 q^{19} -49.4490 q^{20} +71.3640 q^{22} +54.3633 q^{23} +16.3640 q^{25} +1.89032 q^{26} +229.725 q^{29} +127.729 q^{31} +164.316 q^{32} +192.558 q^{34} +311.816 q^{37} -207.746 q^{38} -283.328 q^{40} -419.919 q^{41} +523.180 q^{43} -151.441 q^{44} +106.544 q^{46} +270.328 q^{47} +32.0711 q^{50} -4.01145 q^{52} -251.082 q^{53} +432.938 q^{55} +450.226 q^{58} +408.029 q^{59} -860.546 q^{61} +250.329 q^{62} +429.481 q^{64} +11.4679 q^{65} +506.360 q^{67} -408.626 q^{68} -523.702 q^{71} +629.964 q^{73} +611.112 q^{74} +440.856 q^{76} +319.548 q^{79} -159.689 q^{80} -822.978 q^{82} +1309.16 q^{83} +1168.18 q^{85} +1025.35 q^{86} -867.715 q^{88} -348.581 q^{89} -226.096 q^{92} +529.803 q^{94} -1260.31 q^{95} -161.996 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 68 q^{4}+O(q^{10}) $ 8 * q + 68 * q^4	$ 8 q + 68 q^{4} + 804 q^{16} + 976 q^{22} + 536 q^{25} + 64 q^{37} + 2160 q^{43} - 768 q^{46} + 2184 q^{58} + 7588 q^{64} + 5392 q^{79} + 2864 q^{85} + 5616 q^{88}+O(q^{100}) $ 8 * q + 68 * q^4 + 804 * q^16 + 976 * q^22 + 536 * q^25 + 64 * q^37 + 2160 * q^43 - 768 * q^46 + 2184 * q^58 + 7588 * q^64 + 5392 * q^79 + 2864 * q^85 + 5616 * q^88

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 $

$2$	1.95985	0.692912	0.346456	−	0.938066i	$-0.387385\pi$
$2$	1.95985	0.692912	0.346456	+	0.938066i	$0.387385\pi$
$3$	0	0
$3$	0	0
$4$	−4.15899	−0.519874
$5$	11.8897	1.06344	0.531722	−	0.846919i	$-0.321545\pi$
$5$	11.8897	1.06344	0.531722	+	0.846919i	$0.321545\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−23.8298	−1.05314
$9$	0	0
$10$	23.3019	0.736872
$11$	36.4130	0.998085	0.499043	−	0.866577i	$-0.333685\pi$
$11$	36.4130	0.998085	0.499043	+	0.866577i	$0.333685\pi$
$12$	0	0
$13$	0.964525	0.0205778	0.0102889	−	0.999947i	$-0.496725\pi$
$13$	0.964525	0.0205778	0.0102889	+	0.999947i	$0.496725\pi$
$14$	0	0
$15$	0	0
$16$	−13.4309	−0.209858
$17$	98.2514	1.40173	0.700866	−	0.713293i	$-0.252797\pi$
$17$	98.2514	1.40173	0.700866	+	0.713293i	$0.252797\pi$
$18$	0	0
$19$	−106.001	−1.27991	−0.639954	−	0.768413i	$-0.721047\pi$
$19$	−106.001	−1.27991	−0.639954	+	0.768413i	$0.721047\pi$
$20$	−49.4490	−0.552856
$21$	0	0
$22$	71.3640	0.691585
$23$	54.3633	0.492849	0.246424	−	0.969162i	$-0.420744\pi$
$23$	54.3633	0.492849	0.246424	+	0.969162i	$0.420744\pi$
$24$	0	0
$25$	16.3640	0.130912
$26$	1.89032	0.0142586
$27$	0	0
$28$	0	0
$29$	229.725	1.47099	0.735497	−	0.677528i	$-0.236949\pi$
$29$	229.725	1.47099	0.735497	+	0.677528i	$0.236949\pi$
$30$	0	0
$31$	127.729	0.740025	0.370013	−	0.929027i	$-0.379353\pi$
$31$	127.729	0.740025	0.370013	+	0.929027i	$0.379353\pi$
$32$	164.316	0.907725
$33$	0	0
$34$	192.558	0.971277
$35$	0	0
$36$	0	0
$37$	311.816	1.38546	0.692732	−	0.721195i	$-0.256407\pi$
$37$	311.816	1.38546	0.692732	+	0.721195i	$0.256407\pi$
$38$	−207.746	−0.886864
$39$	0	0
$40$	−283.328	−1.11995
$41$	−419.919	−1.59952	−0.799760	−	0.600320i	$-0.795040\pi$
$41$	−419.919	−1.59952	−0.799760	+	0.600320i	$0.795040\pi$
$42$	0	0
$43$	523.180	1.85545	0.927723	−	0.373270i	$-0.121763\pi$
$43$	523.180	1.85545	0.927723	+	0.373270i	$0.121763\pi$
$44$	−151.441	−0.518878
$45$	0	0
$46$	106.544	0.341501
$47$	270.328	0.838966	0.419483	−	0.907763i	$-0.362211\pi$
$47$	270.328	0.838966	0.419483	+	0.907763i	$0.362211\pi$
$48$	0	0
$49$	0	0
$50$	32.0711	0.0907107
$51$	0	0
$52$	−4.01145	−0.0106978
$53$	−251.082	−0.650732	−0.325366	−	0.945588i	$-0.605487\pi$
$53$	−251.082	−0.650732	−0.325366	+	0.945588i	$0.605487\pi$
$54$	0	0
$55$	432.938	1.06141
$56$	0	0
$57$	0	0
$58$	450.226	1.01927
$59$	408.029	0.900354	0.450177	−	0.892939i	$-0.351361\pi$
$59$	408.029	0.900354	0.450177	+	0.892939i	$0.351361\pi$
$60$	0	0
$61$	−860.546	−1.80626	−0.903128	−	0.429372i	$-0.858735\pi$
$61$	−860.546	−1.80626	−0.903128	+	0.429372i	$0.858735\pi$
$62$	250.329	0.512772
$63$	0	0
$64$	429.481	0.838831
$65$	11.4679	0.0218833
$66$	0	0
$67$	506.360	0.923308	0.461654	−	0.887060i	$-0.347256\pi$
$67$	506.360	0.923308	0.461654	+	0.887060i	$0.347256\pi$
$68$	−408.626	−0.728724
$69$	0	0
$70$	0	0
$71$	−523.702	−0.875380	−0.437690	−	0.899126i	$-0.644203\pi$
$71$	−523.702	−0.875380	−0.437690	+	0.899126i	$0.644203\pi$
$72$	0	0
$73$	629.964	1.01002	0.505012	−	0.863113i	$-0.331488\pi$
$73$	629.964	1.01002	0.505012	+	0.863113i	$0.331488\pi$
$74$	611.112	0.960004
$75$	0	0
$76$	440.856	0.665391
$77$	0	0
$78$	0	0
$79$	319.548	0.455089	0.227544	−	0.973768i	$-0.426930\pi$
$79$	319.548	0.455089	0.227544	+	0.973768i	$0.426930\pi$
$80$	−159.689	−0.223172
$81$	0	0
$82$	−822.978	−1.10833
$83$	1309.16	1.73131	0.865654	−	0.500643i	$-0.166903\pi$
$83$	1309.16	1.73131	0.865654	+	0.500643i	$0.166903\pi$
$84$	0	0
$85$	1168.18	1.49066
$86$	1025.35	1.28566
$87$	0	0
$88$	−867.715	−1.05112
$89$	−348.581	−0.415163	−0.207581	−	0.978218i	$-0.566559\pi$
$89$	−348.581	−0.415163	−0.207581	+	0.978218i	$0.566559\pi$
$90$	0	0
$91$	0	0
$92$	−226.096	−0.256219
$93$	0	0
$94$	529.803	0.581329
$95$	−1260.31	−1.36111
$96$	0	0
$97$	−161.996	−0.169569	−0.0847844	−	0.996399i	$-0.527020\pi$
$97$	−161.996	−0.169569	−0.0847844	+	0.996399i	$0.527020\pi$
$98$	0	0
$99$	0	0
$100$	−68.0579	−0.0680579
$101$	−218.342	−0.215108	−0.107554	−	0.994199i	$-0.534302\pi$
$101$	−218.342	−0.215108	−0.107554	+	0.994199i	$0.534302\pi$
$102$	0	0
$103$	−231.090	−0.221068	−0.110534	−	0.993872i	$-0.535256\pi$
$103$	−231.090	−0.221068	−0.110534	+	0.993872i	$0.535256\pi$
$104$	−22.9844	−0.0216712
$105$	0	0
$106$	−492.083	−0.450899
$107$	−1865.12	−1.68513	−0.842563	−	0.538598i	$-0.818954\pi$
$107$	−1865.12	−1.68513	−0.842563	+	0.538598i	$0.818954\pi$
$108$	0	0
$109$	601.004	0.528127	0.264063	−	0.964505i	$-0.414937\pi$
$109$	601.004	0.528127	0.264063	+	0.964505i	$0.414937\pi$
$110$	848.494	0.735461
$111$	0	0
$112$	0	0
$113$	−475.349	−0.395727	−0.197863	−	0.980230i	$-0.563400\pi$
$113$	−475.349	−0.395727	−0.197863	+	0.980230i	$0.563400\pi$
$114$	0	0
$115$	646.361	0.524117
$116$	−955.422	−0.764731
$117$	0	0
$118$	799.676	0.623865
$119$	0	0
$120$	0	0
$121$	−5.09213	−0.00382580
$122$	−1686.54	−1.25158
$123$	0	0
$124$	−531.223	−0.384720
$125$	−1291.64	−0.924226
$126$	0	0
$127$	−29.0876	−0.0203237	−0.0101619	−	0.999948i	$-0.503235\pi$
$127$	−29.0876	−0.0203237	−0.0101619	+	0.999948i	$0.503235\pi$
$128$	−472.807	−0.326489
$129$	0	0
$130$	22.4753	0.0151632
$131$	−2115.26	−1.41077	−0.705387	−	0.708822i	$-0.749227\pi$
$131$	−2115.26	−1.41077	−0.705387	+	0.708822i	$0.749227\pi$
$132$	0	0
$133$	0	0
$134$	992.389	0.639771
$135$	0	0
$136$	−2341.31	−1.47622
$137$	177.342	0.110594	0.0552968	−	0.998470i	$-0.482390\pi$
$137$	177.342	0.110594	0.0552968	+	0.998470i	$0.482390\pi$
$138$	0	0
$139$	−2369.18	−1.44569	−0.722846	−	0.691009i	$-0.757166\pi$
$139$	−2369.18	−1.44569	−0.722846	+	0.691009i	$0.757166\pi$
$140$	0	0
$141$	0	0
$142$	−1026.38	−0.606561
$143$	35.1213	0.0205384
$144$	0	0
$145$	2731.35	1.56432
$146$	1234.63	0.699857
$147$	0	0
$148$	−1296.84	−0.720266
$149$	1491.03	0.819801	0.409900	−	0.912130i	$-0.365563\pi$
$149$	1491.03	0.819801	0.409900	+	0.912130i	$0.365563\pi$
$150$	0	0
$151$	410.461	0.221211	0.110605	−	0.993864i	$-0.464721\pi$
$151$	410.461	0.221211	0.110605	+	0.993864i	$0.464721\pi$
$152$	2525.98	1.34792
$153$	0	0
$154$	0	0
$155$	1518.65	0.786975
$156$	0	0
$157$	−453.500	−0.230530	−0.115265	−	0.993335i	$-0.536772\pi$
$157$	−453.500	−0.230530	−0.115265	+	0.993335i	$0.536772\pi$
$158$	626.267	0.315336
$159$	0	0
$160$	1953.66	0.965314
$161$	0	0
$162$	0	0
$163$	748.166	0.359515	0.179757	−	0.983711i	$-0.442469\pi$
$163$	748.166	0.359515	0.179757	+	0.983711i	$0.442469\pi$
$164$	1746.44	0.831548
$165$	0	0
$166$	2565.75	1.19964
$167$	−518.269	−0.240149	−0.120074	−	0.992765i	$-0.538313\pi$
$167$	−518.269	−0.240149	−0.120074	+	0.992765i	$0.538313\pi$
$168$	0	0
$169$	−2196.07	−0.999577
$170$	2289.45	1.03290
$171$	0	0
$172$	−2175.90	−0.964597
$173$	−1313.48	−0.577239	−0.288620	−	0.957444i	$-0.593196\pi$
$173$	−1313.48	−0.577239	−0.288620	+	0.957444i	$0.593196\pi$
$174$	0	0
$175$	0	0
$176$	−489.060	−0.209456
$177$	0	0
$178$	−683.166	−0.287671
$179$	4133.15	1.72584	0.862921	−	0.505339i	$-0.168632\pi$
$179$	4133.15	1.72584	0.862921	+	0.505339i	$0.168632\pi$
$180$	0	0
$181$	−3714.04	−1.52521	−0.762603	−	0.646867i	$-0.776079\pi$
$181$	−3714.04	−1.52521	−0.762603	+	0.646867i	$0.776079\pi$
$182$	0	0
$183$	0	0
$184$	−1295.47	−0.519038
$185$	3707.38	1.47336
$186$	0	0
$187$	3577.63	1.39905
$188$	−1124.29	−0.436156
$189$	0	0
$190$	−2470.03	−0.943129
$191$	−3068.37	−1.16241	−0.581203	−	0.813758i	$-0.697418\pi$
$191$	−3068.37	−1.16241	−0.581203	+	0.813758i	$0.697418\pi$
$192$	0	0
$193$	2089.53	0.779314	0.389657	−	0.920960i	$-0.372593\pi$
$193$	2089.53	0.779314	0.389657	+	0.920960i	$0.372593\pi$
$194$	−317.487	−0.117496
$195$	0	0
$196$	0	0
$197$	−3729.89	−1.34895	−0.674476	−	0.738297i	$-0.735630\pi$
$197$	−3729.89	−1.34895	−0.674476	+	0.738297i	$0.735630\pi$
$198$	0	0
$199$	−2108.85	−0.751217	−0.375608	−	0.926778i	$-0.622566\pi$
$199$	−2108.85	−0.751217	−0.375608	+	0.926778i	$0.622566\pi$
$200$	−389.952	−0.137869
$201$	0	0
$202$	−427.918	−0.149051
$203$	0	0
$204$	0	0
$205$	−4992.69	−1.70100
$206$	−452.903	−0.153181
$207$	0	0
$208$	−12.9544	−0.00431841
$209$	−3859.81	−1.27746
$210$	0	0
$211$	−1546.53	−0.504584	−0.252292	−	0.967651i	$-0.581184\pi$
$211$	−1546.53	−0.504584	−0.252292	+	0.967651i	$0.581184\pi$
$212$	1044.25	0.338298
$213$	0	0
$214$	−3655.36	−1.16764
$215$	6220.43	1.97316
$216$	0	0
$217$	0	0
$218$	1177.88	0.365945
$219$	0	0
$220$	−1800.59	−0.551798
$221$	94.7659	0.0288445
$222$	0	0
$223$	−4860.40	−1.45954	−0.729768	−	0.683695i	$-0.760372\pi$
$223$	−4860.40	−1.45954	−0.729768	+	0.683695i	$0.760372\pi$
$224$	0	0
$225$	0	0
$226$	−931.614	−0.274204
$227$	−700.942	−0.204948	−0.102474	−	0.994736i	$-0.532676\pi$
$227$	−700.942	−0.204948	−0.102474	+	0.994736i	$0.532676\pi$
$228$	0	0
$229$	3694.24	1.06604	0.533018	−	0.846104i	$-0.321058\pi$
$229$	3694.24	1.06604	0.533018	+	0.846104i	$0.321058\pi$
$230$	1266.77	0.363167
$231$	0	0
$232$	−5474.29	−1.54916
$233$	344.322	0.0968124	0.0484062	−	0.998828i	$-0.484586\pi$
$233$	344.322	0.0968124	0.0484062	+	0.998828i	$0.484586\pi$
$234$	0	0
$235$	3214.11	0.892193
$236$	−1696.99	−0.468070
$237$	0	0
$238$	0	0
$239$	6144.21	1.66291	0.831456	−	0.555590i	$-0.187508\pi$
$239$	6144.21	1.66291	0.831456	+	0.555590i	$0.187508\pi$
$240$	0	0
$241$	5626.11	1.50378	0.751888	−	0.659291i	$-0.229144\pi$
$241$	5626.11	1.50378	0.751888	+	0.659291i	$0.229144\pi$
$242$	−9.97982	−0.00265094
$243$	0	0
$244$	3579.00	0.939025
$245$	0	0
$246$	0	0
$247$	−102.240	−0.0263377
$248$	−3043.75	−0.779349
$249$	0	0
$250$	−2531.43	−0.640407
$251$	520.460	0.130881	0.0654405	−	0.997856i	$-0.479155\pi$
$251$	520.460	0.130881	0.0654405	+	0.997856i	$0.479155\pi$
$252$	0	0
$253$	1979.53	0.491905
$254$	−57.0074	−0.0140825
$255$	0	0
$256$	−4362.48	−1.06506
$257$	−4009.50	−0.973174	−0.486587	−	0.873632i	$-0.661758\pi$
$257$	−4009.50	−0.973174	−0.486587	+	0.873632i	$0.661758\pi$
$258$	0	0
$259$	0	0
$260$	−47.6948	−0.0113766
$261$	0	0
$262$	−4145.60	−0.977542
$263$	−3585.19	−0.840579	−0.420290	−	0.907390i	$-0.638072\pi$
$263$	−3585.19	−0.840579	−0.420290	+	0.907390i	$0.638072\pi$
$264$	0	0
$265$	−2985.28	−0.692016
$266$	0	0
$267$	0	0
$268$	−2105.94	−0.480004
$269$	−5509.19	−1.24870	−0.624352	−	0.781143i	$-0.714637\pi$
$269$	−5509.19	−1.24870	−0.624352	+	0.781143i	$0.714637\pi$
$270$	0	0
$271$	5819.89	1.30455	0.652276	−	0.757982i	$-0.273814\pi$
$271$	5819.89	1.30455	0.652276	+	0.757982i	$0.273814\pi$
$272$	−1319.60	−0.294165
$273$	0	0
$274$	347.563	0.0766316
$275$	595.864	0.130662
$276$	0	0
$277$	−1150.45	−0.249545	−0.124772	−	0.992185i	$-0.539820\pi$
$277$	−1150.45	−0.249545	−0.124772	+	0.992185i	$0.539820\pi$
$278$	−4643.24	−1.00174
$279$	0	0
$280$	0	0
$281$	−895.631	−0.190138	−0.0950692	−	0.995471i	$-0.530307\pi$
$281$	−895.631	−0.190138	−0.0950692	+	0.995471i	$0.530307\pi$
$282$	0	0
$283$	−122.450	−0.0257205	−0.0128603	−	0.999917i	$-0.504094\pi$
$283$	−122.450	−0.0257205	−0.0128603	+	0.999917i	$0.504094\pi$
$284$	2178.07	0.455087
$285$	0	0
$286$	68.8324	0.0142313
$287$	0	0
$288$	0	0
$289$	4740.33	0.964854
$290$	5353.03	1.08393
$291$	0	0
$292$	−2620.01	−0.525084
$293$	7601.77	1.51570	0.757850	−	0.652429i	$-0.226250\pi$
$293$	7601.77	1.51570	0.757850	+	0.652429i	$0.226250\pi$
$294$	0	0
$295$	4851.33	0.957475
$296$	−7430.50	−1.45909
$297$	0	0
$298$	2922.20	0.568050
$299$	52.4347	0.0101417
$300$	0	0
$301$	0	0
$302$	804.441	0.153279
$303$	0	0
$304$	1423.69	0.268599
$305$	−10231.6	−1.92085
$306$	0	0
$307$	3539.05	0.657929	0.328964	−	0.944342i	$-0.393300\pi$
$307$	3539.05	0.657929	0.328964	+	0.944342i	$0.393300\pi$
$308$	0	0
$309$	0	0
$310$	2976.33	0.545304
$311$	3894.93	0.710165	0.355083	−	0.934835i	$-0.384453\pi$
$311$	3894.93	0.710165	0.355083	+	0.934835i	$0.384453\pi$
$312$	0	0
$313$	−8682.58	−1.56795	−0.783975	−	0.620793i	$-0.786811\pi$
$313$	−8682.58	−1.56795	−0.783975	+	0.620793i	$0.786811\pi$
$314$	−888.792	−0.159737
$315$	0	0
$316$	−1329.00	−0.236589
$317$	8495.52	1.50522	0.752612	−	0.658464i	$-0.228794\pi$
$317$	8495.52	1.50522	0.752612	+	0.658464i	$0.228794\pi$
$318$	0	0
$319$	8364.97	1.46818
$320$	5106.39	0.892049
$321$	0	0
$322$	0	0
$323$	−10414.7	−1.79409
$324$	0	0
$325$	15.7835	0.00269389
$326$	1466.29	0.249112
$327$	0	0
$328$	10006.6	1.68451
$329$	0	0
$330$	0	0
$331$	1270.71	0.211011	0.105505	−	0.994419i	$-0.466354\pi$
$331$	1270.71	0.211011	0.105505	+	0.994419i	$0.466354\pi$
$332$	−5444.76	−0.900061
$333$	0	0
$334$	−1015.73	−0.166402
$335$	6020.44	0.981886
$336$	0	0
$337$	4695.47	0.758986	0.379493	−	0.925195i	$-0.376098\pi$
$337$	4695.47	0.758986	0.379493	+	0.925195i	$0.376098\pi$
$338$	−4303.97	−0.692618
$339$	0	0
$340$	−4858.43	−0.774957
$341$	4651.00	0.738609
$342$	0	0
$343$	0	0
$344$	−12467.3	−1.95404
$345$	0	0
$346$	−2574.23	−0.399976
$347$	6513.97	1.00775	0.503874	−	0.863777i	$-0.331908\pi$
$347$	6513.97	1.00775	0.503874	+	0.863777i	$0.331908\pi$
$348$	0	0
$349$	−7184.75	−1.10198	−0.550990	−	0.834512i	$-0.685750\pi$
$349$	−7184.75	−1.10198	−0.550990	+	0.834512i	$0.685750\pi$
$350$	0	0
$351$	0	0
$352$	5983.23	0.905987
$353$	−6068.55	−0.915004	−0.457502	−	0.889209i	$-0.651256\pi$
$353$	−6068.55	−0.915004	−0.457502	+	0.889209i	$0.651256\pi$
$354$	0	0
$355$	−6226.64	−0.930918
$356$	1449.74	0.215832
$357$	0	0
$358$	8100.34	1.19586
$359$	1743.71	0.256350	0.128175	−	0.991752i	$-0.459088\pi$
$359$	1743.71	0.256350	0.128175	+	0.991752i	$0.459088\pi$
$360$	0	0
$361$	4377.18	0.638167
$362$	−7278.96	−1.05683
$363$	0	0
$364$	0	0
$365$	7490.06	1.07410
$366$	0	0
$367$	9000.12	1.28012	0.640058	−	0.768327i	$-0.278910\pi$
$367$	9000.12	1.28012	0.640058	+	0.768327i	$0.278910\pi$
$368$	−730.148	−0.103428
$369$	0	0
$370$	7265.91	1.02091
$371$	0	0
$372$	0	0
$373$	5271.75	0.731799	0.365899	−	0.930654i	$-0.380761\pi$
$373$	5271.75	0.731799	0.365899	+	0.930654i	$0.380761\pi$
$374$	7011.61	0.969417
$375$	0	0
$376$	−6441.86	−0.883547
$377$	221.575	0.0302698
$378$	0	0
$379$	10480.0	1.42037	0.710185	−	0.704015i	$-0.248611\pi$
$379$	10480.0	1.42037	0.710185	+	0.704015i	$0.248611\pi$
$380$	5241.63	0.707606
$381$	0	0
$382$	−6013.55	−0.805445
$383$	−1374.03	−0.183315	−0.0916574	−	0.995791i	$-0.529216\pi$
$383$	−1374.03	−0.183315	−0.0916574	+	0.995791i	$0.529216\pi$
$384$	0	0
$385$	0	0
$386$	4095.17	0.539996
$387$	0	0
$388$	673.739	0.0881544
$389$	1898.27	0.247419	0.123710	−	0.992318i	$-0.460521\pi$
$389$	1898.27	0.247419	0.123710	+	0.992318i	$0.460521\pi$
$390$	0	0
$391$	5341.26	0.690842
$392$	0	0
$393$	0	0
$394$	−7310.02	−0.934704
$395$	3799.32	0.483961
$396$	0	0
$397$	−5572.30	−0.704448	−0.352224	−	0.935916i	$-0.614574\pi$
$397$	−5572.30	−0.704448	−0.352224	+	0.935916i	$0.614574\pi$
$398$	−4133.02	−0.520527
$399$	0	0
$400$	−219.784	−0.0274730
$401$	−5716.38	−0.711877	−0.355938	−	0.934509i	$-0.615839\pi$
$401$	−5716.38	−0.711877	−0.355938	+	0.934509i	$0.615839\pi$
$402$	0	0
$403$	123.198	0.0152281
$404$	908.083	0.111829
$405$	0	0
$406$	0	0
$407$	11354.2	1.38281
$408$	0	0
$409$	10348.5	1.25111	0.625553	−	0.780182i	$-0.284874\pi$
$409$	10348.5	1.25111	0.625553	+	0.780182i	$0.284874\pi$
$410$	−9784.93	−1.17864
$411$	0	0
$412$	961.103	0.114927
$413$	0	0
$414$	0	0
$415$	15565.4	1.84115
$416$	158.487	0.0186790
$417$	0	0
$418$	−7564.65	−0.885165
$419$	−3251.80	−0.379142	−0.189571	−	0.981867i	$-0.560710\pi$
$419$	−3251.80	−0.379142	−0.189571	+	0.981867i	$0.560710\pi$
$420$	0	0
$421$	3708.63	0.429329	0.214664	−	0.976688i	$-0.431134\pi$
$421$	3708.63	0.429329	0.214664	+	0.976688i	$0.431134\pi$
$422$	−3030.96	−0.349632
$423$	0	0
$424$	5983.23	0.685310
$425$	1607.79	0.183504
$426$	0	0
$427$	0	0
$428$	7757.03	0.876052
$429$	0	0
$430$	12191.1	1.36723
$431$	4148.74	0.463661	0.231831	−	0.972756i	$-0.425529\pi$
$431$	4148.74	0.463661	0.231831	+	0.972756i	$0.425529\pi$
$432$	0	0
$433$	1985.64	0.220378	0.110189	−	0.993911i	$-0.464854\pi$
$433$	1985.64	0.220378	0.110189	+	0.993911i	$0.464854\pi$
$434$	0	0
$435$	0	0
$436$	−2499.57	−0.274559
$437$	−5762.55	−0.630802
$438$	0	0
$439$	−6914.42	−0.751725	−0.375863	−	0.926675i	$-0.622654\pi$
$439$	−6914.42	−0.751725	−0.375863	+	0.926675i	$0.622654\pi$
$440$	−10316.8	−1.11781
$441$	0	0
$442$	185.727	0.0199867
$443$	4237.47	0.454466	0.227233	−	0.973840i	$-0.427032\pi$
$443$	4237.47	0.454466	0.227233	+	0.973840i	$0.427032\pi$
$444$	0	0
$445$	−4144.51	−0.441502
$446$	−9525.65	−1.01133
$447$	0	0
$448$	0	0
$449$	−2097.02	−0.220411	−0.110206	−	0.993909i	$-0.535151\pi$
$449$	−2097.02	−0.220411	−0.110206	+	0.993909i	$0.535151\pi$
$450$	0	0
$451$	−15290.5	−1.59646
$452$	1976.97	0.205728
$453$	0	0
$454$	−1373.74	−0.142011
$455$	0	0
$456$	0	0
$457$	−18676.3	−1.91168	−0.955842	−	0.293880i	$-0.905053\pi$
$457$	−18676.3	−1.91168	−0.955842	+	0.293880i	$0.905053\pi$
$458$	7240.15	0.738668
$459$	0	0
$460$	−2688.21	−0.272475
$461$	−13879.5	−1.40224	−0.701121	−	0.713043i	$-0.747317\pi$
$461$	−13879.5	−1.40224	−0.701121	+	0.713043i	$0.747317\pi$
$462$	0	0
$463$	4780.89	0.479885	0.239942	−	0.970787i	$-0.422871\pi$
$463$	4780.89	0.479885	0.239942	+	0.970787i	$0.422871\pi$
$464$	−3085.41	−0.308699
$465$	0	0
$466$	674.819	0.0670824
$467$	8376.19	0.829986	0.414993	−	0.909824i	$-0.363784\pi$
$467$	8376.19	0.829986	0.414993	+	0.909824i	$0.363784\pi$
$468$	0	0
$469$	0	0
$470$	6299.17	0.618211
$471$	0	0
$472$	−9723.25	−0.948197
$473$	19050.6	1.85189
$474$	0	0
$475$	−1734.60	−0.167556
$476$	0	0
$477$	0	0
$478$	12041.7	1.15225
$479$	−17011.4	−1.62269	−0.811346	−	0.584567i	$-0.801265\pi$
$479$	−17011.4	−1.62269	−0.811346	+	0.584567i	$0.801265\pi$
$480$	0	0
$481$	300.754	0.0285098
$482$	11026.3	1.04198
$483$	0	0
$484$	21.1781	0.00198893
$485$	−1926.07	−0.180327
$486$	0	0
$487$	6017.65	0.559930	0.279965	−	0.960010i	$-0.409677\pi$
$487$	6017.65	0.559930	0.279965	+	0.960010i	$0.409677\pi$
$488$	20506.6	1.90224
$489$	0	0
$490$	0	0
$491$	−12306.6	−1.13113	−0.565567	−	0.824702i	$-0.691343\pi$
$491$	−12306.6	−1.13113	−0.565567	+	0.824702i	$0.691343\pi$
$492$	0	0
$493$	22570.8	2.06194
$494$	−200.376	−0.0182497
$495$	0	0
$496$	−1715.51	−0.155300
$497$	0	0
$498$	0	0
$499$	8565.93	0.768464	0.384232	−	0.923237i	$-0.374466\pi$
$499$	8565.93	0.768464	0.384232	+	0.923237i	$0.374466\pi$
$500$	5371.94	0.480481
$501$	0	0
$502$	1020.02	0.0906890
$503$	−10520.4	−0.932568	−0.466284	−	0.884635i	$-0.654408\pi$
$503$	−10520.4	−0.932568	−0.466284	+	0.884635i	$0.654408\pi$
$504$	0	0
$505$	−2596.02	−0.228755
$506$	3879.58	0.340847
$507$	0	0
$508$	120.975	0.0105658
$509$	5445.41	0.474192	0.237096	−	0.971486i	$-0.423804\pi$
$509$	5445.41	0.474192	0.237096	+	0.971486i	$0.423804\pi$
$510$	0	0
$511$	0	0
$512$	−4767.35	−0.411502
$513$	0	0
$514$	−7858.02	−0.674324
$515$	−2747.59	−0.235094
$516$	0	0
$517$	9843.46	0.837360
$518$	0	0
$519$	0	0
$520$	−273.277	−0.0230461
$521$	22217.9	1.86830	0.934149	−	0.356883i	$-0.116160\pi$
$521$	22217.9	1.86830	0.934149	+	0.356883i	$0.116160\pi$
$522$	0	0
$523$	22109.9	1.84856	0.924281	−	0.381712i	$-0.124665\pi$
$523$	22109.9	1.84856	0.924281	+	0.381712i	$0.124665\pi$
$524$	8797.36	0.733425
$525$	0	0
$526$	−7026.44	−0.582447
$527$	12549.5	1.03732
$528$	0	0
$529$	−9211.64	−0.757100
$530$	−5850.70	−0.479506
$531$	0	0
$532$	0	0
$533$	−405.022	−0.0329146
$534$	0	0
$535$	−22175.7	−1.79204
$536$	−12066.4	−0.972371
$537$	0	0
$538$	−10797.2	−0.865241
$539$	0	0
$540$	0	0
$541$	−8631.34	−0.685934	−0.342967	−	0.939347i	$-0.611432\pi$
$541$	−8631.34	−0.685934	−0.342967	+	0.939347i	$0.611432\pi$
$542$	11406.1	0.903939
$543$	0	0
$544$	16144.2	1.27239
$545$	7145.74	0.561633
$546$	0	0
$547$	−17846.7	−1.39501	−0.697505	−	0.716580i	$-0.745707\pi$
$547$	−17846.7	−1.39501	−0.697505	+	0.716580i	$0.745707\pi$
$548$	−737.562	−0.0574947
$549$	0	0
$550$	1167.80	0.0905370
$551$	−24351.0	−1.88274
$552$	0	0
$553$	0	0
$554$	−2254.71	−0.172913
$555$	0	0
$556$	9853.39	0.751577
$557$	15703.3	1.19456	0.597279	−	0.802033i	$-0.296248\pi$
$557$	15703.3	1.19456	0.597279	+	0.802033i	$0.296248\pi$
$558$	0	0
$559$	504.620	0.0381810
$560$	0	0
$561$	0	0
$562$	−1755.30	−0.131749
$563$	−673.679	−0.0504302	−0.0252151	−	0.999682i	$-0.508027\pi$
$563$	−673.679	−0.0504302	−0.0252151	+	0.999682i	$0.508027\pi$
$564$	0	0
$565$	−5651.74	−0.420833
$566$	−239.984	−0.0178221
$567$	0	0
$568$	12479.7	0.921896
$569$	−15186.2	−1.11887	−0.559436	−	0.828873i	$-0.688982\pi$
$569$	−15186.2	−1.11887	−0.559436	+	0.828873i	$0.688982\pi$
$570$	0	0
$571$	2526.66	0.185180	0.0925898	−	0.995704i	$-0.470485\pi$
$571$	2526.66	0.185180	0.0925898	+	0.995704i	$0.470485\pi$
$572$	−146.069	−0.0106774
$573$	0	0
$574$	0	0
$575$	889.603	0.0645200
$576$	0	0
$577$	−21680.4	−1.56424	−0.782119	−	0.623129i	$-0.785861\pi$
$577$	−21680.4	−1.56424	−0.782119	+	0.623129i	$0.785861\pi$
$578$	9290.33	0.668559
$579$	0	0
$580$	−11359.6	−0.813248
$581$	0	0
$582$	0	0
$583$	−9142.66	−0.649486
$584$	−15011.9	−1.06369
$585$	0	0
$586$	14898.3	1.05025
$587$	17458.1	1.22755	0.613776	−	0.789481i	$-0.289650\pi$
$587$	17458.1	1.22755	0.613776	+	0.789481i	$0.289650\pi$
$588$	0	0
$589$	−13539.4	−0.947165
$590$	9507.87	0.663446
$591$	0	0
$592$	−4187.97	−0.290751
$593$	−5487.00	−0.379973	−0.189987	−	0.981787i	$-0.560844\pi$
$593$	−5487.00	−0.379973	−0.189987	+	0.981787i	$0.560844\pi$
$594$	0	0
$595$	0	0
$596$	−6201.20	−0.426193
$597$	0	0
$598$	102.764	0.00702732
$599$	−22997.3	−1.56869	−0.784343	−	0.620327i	$-0.787000\pi$
$599$	−22997.3	−1.56869	−0.784343	+	0.620327i	$0.787000\pi$
$600$	0	0
$601$	−18027.1	−1.22353	−0.611764	−	0.791040i	$-0.709540\pi$
$601$	−18027.1	−1.22353	−0.611764	+	0.791040i	$0.709540\pi$
$602$	0	0
$603$	0	0
$604$	−1707.10	−0.115002
$605$	−60.5437	−0.00406852
$606$	0	0
$607$	10285.1	0.687744	0.343872	−	0.939017i	$-0.388261\pi$
$607$	10285.1	0.687744	0.343872	+	0.939017i	$0.388261\pi$
$608$	−17417.6	−1.16181
$609$	0	0
$610$	−20052.4	−1.33098
$611$	260.738	0.0172641
$612$	0	0
$613$	−1596.93	−0.105219	−0.0526096	−	0.998615i	$-0.516754\pi$
$613$	−1596.93	−0.105219	−0.0526096	+	0.998615i	$0.516754\pi$
$614$	6936.00	0.455886
$615$	0	0
$616$	0	0
$617$	5850.74	0.381753	0.190877	−	0.981614i	$-0.438867\pi$
$617$	5850.74	0.381753	0.190877	+	0.981614i	$0.438867\pi$
$618$	0	0
$619$	1721.62	0.111789	0.0558947	−	0.998437i	$-0.482199\pi$
$619$	1721.62	0.111789	0.0558947	+	0.998437i	$0.482199\pi$
$620$	−6316.06	−0.409128
$621$	0	0
$622$	7633.48	0.492082
$623$	0	0
$624$	0	0
$625$	−17402.7	−1.11377
$626$	−17016.5	−1.08645
$627$	0	0
$628$	1886.10	0.119847
$629$	30636.3	1.94205
$630$	0	0
$631$	−19533.5	−1.23235	−0.616177	−	0.787608i	$-0.711319\pi$
$631$	−19533.5	−1.23235	−0.616177	+	0.787608i	$0.711319\pi$
$632$	−7614.77	−0.479271
$633$	0	0
$634$	16649.9	1.04299
$635$	−345.842	−0.0216131
$636$	0	0
$637$	0	0
$638$	16394.1	1.01732
$639$	0	0
$640$	−5621.52	−0.347203
$641$	−23496.2	−1.44781	−0.723903	−	0.689902i	$-0.757654\pi$
$641$	−23496.2	−1.44781	−0.723903	+	0.689902i	$0.757654\pi$
$642$	0	0
$643$	1537.40	0.0942913	0.0471456	−	0.998888i	$-0.484988\pi$
$643$	1537.40	0.0942913	0.0471456	+	0.998888i	$0.484988\pi$
$644$	0	0
$645$	0	0
$646$	−20411.3	−1.24315
$647$	−28273.0	−1.71797	−0.858985	−	0.512000i	$-0.828905\pi$
$647$	−28273.0	−1.71797	−0.858985	+	0.512000i	$0.828905\pi$
$648$	0	0
$649$	14857.6	0.898630
$650$	30.9333	0.00186662
$651$	0	0
$652$	−3111.62	−0.186902
$653$	−2668.97	−0.159946	−0.0799730	−	0.996797i	$-0.525483\pi$
$653$	−2668.97	−0.159946	−0.0799730	+	0.996797i	$0.525483\pi$
$654$	0	0
$655$	−25149.8	−1.50028
$656$	5639.89	0.335672
$657$	0	0
$658$	0	0
$659$	−6343.74	−0.374988	−0.187494	−	0.982266i	$-0.560036\pi$
$659$	−6343.74	−0.374988	−0.187494	+	0.982266i	$0.560036\pi$
$660$	0	0
$661$	8409.15	0.494823	0.247412	−	0.968910i	$-0.420420\pi$
$661$	8409.15	0.494823	0.247412	+	0.968910i	$0.420420\pi$
$662$	2490.40	0.146212
$663$	0	0
$664$	−31196.9	−1.82331
$665$	0	0
$666$	0	0
$667$	12488.6	0.724977
$668$	2155.48	0.124847
$669$	0	0
$670$	11799.2	0.680360
$671$	−31335.1	−1.80280
$672$	0	0
$673$	−15326.7	−0.877862	−0.438931	−	0.898521i	$-0.644643\pi$
$673$	−15326.7	−0.877862	−0.438931	+	0.898521i	$0.644643\pi$
$674$	9202.41	0.525910
$675$	0	0
$676$	9133.43	0.519653
$677$	13011.6	0.738666	0.369333	−	0.929297i	$-0.379586\pi$
$677$	13011.6	0.738666	0.369333	+	0.929297i	$0.379586\pi$
$678$	0	0
$679$	0	0
$680$	−27837.4	−1.56987
$681$	0	0
$682$	9115.25	0.511790
$683$	−17472.0	−0.978840	−0.489420	−	0.872048i	$-0.662791\pi$
$683$	−17472.0	−0.978840	−0.489420	+	0.872048i	$0.662791\pi$
$684$	0	0
$685$	2108.53	0.117610
$686$	0	0
$687$	0	0
$688$	−7026.78	−0.389380
$689$	−242.175	−0.0133906
$690$	0	0
$691$	7317.10	0.402830	0.201415	−	0.979506i	$-0.435446\pi$
$691$	7317.10	0.402830	0.201415	+	0.979506i	$0.435446\pi$
$692$	5462.77	0.300091
$693$	0	0
$694$	12766.4	0.698280
$695$	−28168.7	−1.53741
$696$	0	0
$697$	−41257.6	−2.24210
$698$	−14081.0	−0.763575
$699$	0	0
$700$	0	0
$701$	7874.65	0.424282	0.212141	−	0.977239i	$-0.431956\pi$
$701$	7874.65	0.424282	0.212141	+	0.977239i	$0.431956\pi$
$702$	0	0
$703$	−33052.7	−1.77327
$704$	15638.7	0.837225
$705$	0	0
$706$	−11893.4	−0.634017
$707$	0	0
$708$	0	0
$709$	−25236.8	−1.33680	−0.668398	−	0.743804i	$-0.733020\pi$
$709$	−25236.8	−1.33680	−0.668398	+	0.743804i	$0.733020\pi$
$710$	−12203.3	−0.645044
$711$	0	0
$712$	8306.61	0.437224
$713$	6943.76	0.364721
$714$	0	0
$715$	417.580	0.0218414
$716$	−17189.7	−0.897220
$717$	0	0
$718$	3417.42	0.177628
$719$	−19673.0	−1.02041	−0.510207	−	0.860052i	$-0.670431\pi$
$719$	−19673.0	−1.02041	−0.510207	+	0.860052i	$0.670431\pi$
$720$	0	0
$721$	0	0
$722$	8578.62	0.442193
$723$	0	0
$724$	15446.6	0.792914
$725$	3759.22	0.192571
$726$	0	0
$727$	31921.7	1.62849	0.814243	−	0.580525i	$-0.197152\pi$
$727$	31921.7	1.62849	0.814243	+	0.580525i	$0.197152\pi$
$728$	0	0
$729$	0	0
$730$	14679.4	0.744258
$731$	51403.1	2.60084
$732$	0	0
$733$	−13545.3	−0.682547	−0.341274	−	0.939964i	$-0.610858\pi$
$733$	−13545.3	−0.682547	−0.341274	+	0.939964i	$0.610858\pi$
$734$	17638.9	0.887007
$735$	0	0
$736$	8932.74	0.447371
$737$	18438.1	0.921541
$738$	0	0
$739$	25497.6	1.26921	0.634604	−	0.772837i	$-0.281163\pi$
$739$	25497.6	1.26921	0.634604	+	0.772837i	$0.281163\pi$
$740$	−15419.0	−0.765963
$741$	0	0
$742$	0	0
$743$	−11146.9	−0.550390	−0.275195	−	0.961388i	$-0.588742\pi$
$743$	−11146.9	−0.550390	−0.275195	+	0.961388i	$0.588742\pi$
$744$	0	0
$745$	17727.9	0.871812
$746$	10331.8	0.507072
$747$	0	0
$748$	−14879.3	−0.727328
$749$	0	0
$750$	0	0
$751$	21412.5	1.04042	0.520208	−	0.854040i	$-0.325854\pi$
$751$	21412.5	1.04042	0.520208	+	0.854040i	$0.325854\pi$
$752$	−3630.75	−0.176064
$753$	0	0
$754$	434.254	0.0209743
$755$	4880.24	0.235245
$756$	0	0
$757$	−33963.9	−1.63070	−0.815350	−	0.578968i	$-0.803456\pi$
$757$	−33963.9	−1.63070	−0.815350	+	0.578968i	$0.803456\pi$
$758$	20539.2	0.984191
$759$	0	0
$760$	30033.0	1.43344
$761$	19920.0	0.948880	0.474440	−	0.880288i	$-0.342651\pi$
$761$	19920.0	0.948880	0.474440	+	0.880288i	$0.342651\pi$
$762$	0	0
$763$	0	0
$764$	12761.3	0.604305
$765$	0	0
$766$	−2692.89	−0.127021
$767$	393.554	0.0185273
$768$	0	0
$769$	−10039.2	−0.470769	−0.235385	−	0.971902i	$-0.575635\pi$
$769$	−10039.2	−0.470769	−0.235385	+	0.971902i	$0.575635\pi$
$770$	0	0
$771$	0	0
$772$	−8690.33	−0.405145
$773$	21385.3	0.995054	0.497527	−	0.867449i	$-0.334242\pi$
$773$	21385.3	0.995054	0.497527	+	0.867449i	$0.334242\pi$
$774$	0	0
$775$	2090.16	0.0968785
$776$	3860.33	0.178579
$777$	0	0
$778$	3720.32	0.171440
$779$	44511.8	2.04724
$780$	0	0
$781$	−19069.6	−0.873704
$782$	10468.1	0.478693
$783$	0	0
$784$	0	0
$785$	−5391.96	−0.245156
$786$	0	0
$787$	−37421.3	−1.69495	−0.847475	−	0.530835i	$-0.821878\pi$
$787$	−37421.3	−1.69495	−0.847475	+	0.530835i	$0.821878\pi$
$788$	15512.6	0.701284
$789$	0	0
$790$	7446.10	0.335342
$791$	0	0
$792$	0	0
$793$	−830.018	−0.0371687
$794$	−10920.9	−0.488120
$795$	0	0
$796$	8770.67	0.390538
$797$	−30888.7	−1.37281	−0.686407	−	0.727217i	$-0.740813\pi$
$797$	−30888.7	−1.37281	−0.686407	+	0.727217i	$0.740813\pi$
$798$	0	0
$799$	26560.1	1.17601
$800$	2688.87	0.118832
$801$	0	0
$802$	−11203.3	−0.493268
$803$	22938.9	1.00809
$804$	0	0
$805$	0	0
$806$	241.449	0.0105517
$807$	0	0
$808$	5203.05	0.226538
$809$	7238.50	0.314576	0.157288	−	0.987553i	$-0.449725\pi$
$809$	7238.50	0.314576	0.157288	+	0.987553i	$0.449725\pi$
$810$	0	0
$811$	35466.4	1.53563	0.767814	−	0.640673i	$-0.221344\pi$
$811$	35466.4	1.53563	0.767814	+	0.640673i	$0.221344\pi$
$812$	0	0
$813$	0	0
$814$	22252.4	0.958166
$815$	8895.44	0.382324
$816$	0	0
$817$	−55457.5	−2.37480
$818$	20281.6	0.866905
$819$	0	0
$820$	20764.6	0.884304
$821$	−37947.3	−1.61312	−0.806558	−	0.591154i	$-0.798672\pi$
$821$	−37947.3	−1.61312	−0.806558	+	0.591154i	$0.798672\pi$
$822$	0	0
$823$	26932.2	1.14070	0.570351	−	0.821401i	$-0.306807\pi$
$823$	26932.2	1.14070	0.570351	+	0.821401i	$0.306807\pi$
$824$	5506.84	0.232815
$825$	0	0
$826$	0	0
$827$	−1913.98	−0.0804783	−0.0402391	−	0.999190i	$-0.512812\pi$
$827$	−1913.98	−0.0804783	−0.0402391	+	0.999190i	$0.512812\pi$
$828$	0	0
$829$	15281.9	0.640245	0.320122	−	0.947376i	$-0.396276\pi$
$829$	15281.9	0.640245	0.320122	+	0.947376i	$0.396276\pi$
$830$	30505.9	1.27575
$831$	0	0
$832$	414.246	0.0172613
$833$	0	0
$834$	0	0
$835$	−6162.04	−0.255385
$836$	16052.9	0.664117
$837$	0	0
$838$	−6373.03	−0.262712
$839$	−25779.5	−1.06079	−0.530397	−	0.847749i	$-0.677957\pi$
$839$	−25779.5	−1.06079	−0.530397	+	0.847749i	$0.677957\pi$
$840$	0	0
$841$	28384.4	1.16382
$842$	7268.35	0.297487
$843$	0	0
$844$	6431.98	0.262320
$845$	−26110.5	−1.06299
$846$	0	0
$847$	0	0
$848$	3372.26	0.136561
$849$	0	0
$850$	3151.03	0.127152
$851$	16951.3	0.682825
$852$	0	0
$853$	6452.21	0.258991	0.129496	−	0.991580i	$-0.458664\pi$
$853$	6452.21	0.258991	0.129496	+	0.991580i	$0.458664\pi$
$854$	0	0
$855$	0	0
$856$	44445.5	1.77467
$857$	−32061.8	−1.27796	−0.638980	−	0.769224i	$-0.720643\pi$
$857$	−32061.8	−1.27796	−0.638980	+	0.769224i	$0.720643\pi$
$858$	0	0
$859$	3934.64	0.156284	0.0781421	−	0.996942i	$-0.475101\pi$
$859$	3934.64	0.156284	0.0781421	+	0.996942i	$0.475101\pi$
$860$	−25870.7	−1.02579
$861$	0	0
$862$	8130.91	0.321276
$863$	−14043.1	−0.553921	−0.276961	−	0.960881i	$-0.589327\pi$
$863$	−14043.1	−0.553921	−0.276961	+	0.960881i	$0.589327\pi$
$864$	0	0
$865$	−15616.9	−0.613861
$866$	3891.56	0.152703
$867$	0	0
$868$	0	0
$869$	11635.7	0.454217
$870$	0	0
$871$	488.396	0.0189996
$872$	−14321.8	−0.556190
$873$	0	0
$874$	−11293.7	−0.437090
$875$	0	0
$876$	0	0
$877$	2091.31	0.0805228	0.0402614	−	0.999189i	$-0.487181\pi$
$877$	2091.31	0.0805228	0.0402614	+	0.999189i	$0.487181\pi$
$878$	−13551.2	−0.520879
$879$	0	0
$880$	−5814.75	−0.222745
$881$	−11548.6	−0.441639	−0.220819	−	0.975315i	$-0.570873\pi$
$881$	−11548.6	−0.441639	−0.220819	+	0.975315i	$0.570873\pi$
$882$	0	0
$883$	7531.24	0.287029	0.143514	−	0.989648i	$-0.454160\pi$
$883$	7531.24	0.287029	0.143514	+	0.989648i	$0.454160\pi$
$884$	−394.130	−0.0149955
$885$	0	0
$886$	8304.80	0.314904
$887$	38502.7	1.45749	0.728745	−	0.684785i	$-0.240104\pi$
$887$	38502.7	1.45749	0.728745	+	0.684785i	$0.240104\pi$
$888$	0	0
$889$	0	0
$890$	−8122.61	−0.305922
$891$	0	0
$892$	20214.3	0.758774
$893$	−28655.0	−1.07380
$894$	0	0
$895$	49141.7	1.83534
$896$	0	0
$897$	0	0
$898$	−4109.85	−0.152725
$899$	29342.5	1.08857
$900$	0	0
$901$	−24669.2	−0.912152
$902$	−29967.1	−1.10620
$903$	0	0
$904$	11327.5	0.416755
$905$	−44158.7	−1.62197
$906$	0	0
$907$	−31306.7	−1.14611	−0.573054	−	0.819517i	$-0.694242\pi$
$907$	−31306.7	−1.14611	−0.573054	+	0.819517i	$0.694242\pi$
$908$	2915.21	0.106547
$909$	0	0
$910$	0	0
$911$	34154.3	1.24213	0.621067	−	0.783758i	$-0.286700\pi$
$911$	34154.3	1.24213	0.621067	+	0.783758i	$0.286700\pi$
$912$	0	0
$913$	47670.3	1.72799
$914$	−36602.7	−1.32463
$915$	0	0
$916$	−15364.3	−0.554204
$917$	0	0
$918$	0	0
$919$	35988.9	1.29180	0.645899	−	0.763423i	$-0.276483\pi$
$919$	35988.9	1.29180	0.645899	+	0.763423i	$0.276483\pi$
$920$	−15402.6	−0.551967
$921$	0	0
$922$	−27201.8	−0.971629
$923$	−505.124	−0.0180134
$924$	0	0
$925$	5102.57	0.181374
$926$	9369.82	0.332518
$927$	0	0
$928$	37747.4	1.33526
$929$	−13906.0	−0.491109	−0.245554	−	0.969383i	$-0.578970\pi$
$929$	−13906.0	−0.491109	−0.245554	+	0.969383i	$0.578970\pi$
$930$	0	0
$931$	0	0
$932$	−1432.03	−0.0503302
$933$	0	0
$934$	16416.1	0.575107
$935$	42536.8	1.48781
$936$	0	0
$937$	40905.3	1.42617	0.713083	−	0.701079i	$-0.247298\pi$
$937$	40905.3	1.42617	0.713083	+	0.701079i	$0.247298\pi$
$938$	0	0
$939$	0	0
$940$	−13367.4	−0.463828
$941$	8067.50	0.279482	0.139741	−	0.990188i	$-0.455373\pi$
$941$	8067.50	0.279482	0.139741	+	0.990188i	$0.455373\pi$
$942$	0	0
$943$	−22828.2	−0.788321
$944$	−5480.20	−0.188946
$945$	0	0
$946$	37336.2	1.28320
$947$	39781.2	1.36506	0.682531	−	0.730856i	$-0.260879\pi$
$947$	39781.2	1.36506	0.682531	+	0.730856i	$0.260879\pi$
$948$	0	0
$949$	607.616	0.0207840
$950$	−3399.56	−0.116101
$951$	0	0
$952$	0	0
$953$	32354.0	1.09974	0.549868	−	0.835252i	$-0.314678\pi$
$953$	32354.0	1.09974	0.549868	+	0.835252i	$0.314678\pi$
$954$	0	0
$955$	−36481.9	−1.23615
$956$	−25553.7	−0.864504
$957$	0	0
$958$	−33339.7	−1.12438
$959$	0	0
$960$	0	0
$961$	−13476.3	−0.452362
$962$	589.433	0.0197548
$963$	0	0
$964$	−23398.9	−0.781773
$965$	24843.8	0.828757
$966$	0	0
$967$	17389.2	0.578282	0.289141	−	0.957287i	$-0.406630\pi$
$967$	17389.2	0.578282	0.289141	+	0.957287i	$0.406630\pi$
$968$	121.344	0.00402909
$969$	0	0
$970$	−3774.82	−0.124951
$971$	−52406.2	−1.73202	−0.866012	−	0.500023i	$-0.833325\pi$
$971$	−52406.2	−1.73202	−0.866012	+	0.500023i	$0.833325\pi$
$972$	0	0
$973$	0	0
$974$	11793.7	0.387982
$975$	0	0
$976$	11557.9	0.379057
$977$	6846.70	0.224202	0.112101	−	0.993697i	$-0.464242\pi$
$977$	6846.70	0.224202	0.112101	+	0.993697i	$0.464242\pi$
$978$	0	0
$979$	−12692.9	−0.414368
$980$	0	0
$981$	0	0
$982$	−24119.0	−0.783776
$983$	−37647.9	−1.22155	−0.610774	−	0.791805i	$-0.709142\pi$
$983$	−37647.9	−1.22155	−0.610774	+	0.791805i	$0.709142\pi$
$984$	0	0
$985$	−44347.1	−1.43453
$986$	44235.3	1.42874
$987$	0	0
$988$	425.217	0.0136923
$989$	28441.8	0.914454
$990$	0	0
$991$	2036.17	0.0652686	0.0326343	−	0.999467i	$-0.489610\pi$
$991$	2036.17	0.0652686	0.0326343	+	0.999467i	$0.489610\pi$
$992$	20987.9	0.671740
$993$	0	0
$994$	0	0
$995$	−25073.5	−0.798877
$996$	0	0
$997$	13540.6	0.430127	0.215064	−	0.976600i	$-0.431004\pi$
$997$	13540.6	0.430127	0.215064	+	0.976600i	$0.431004\pi$
$998$	16787.9	0.532478
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.4.a.x.1.6	yes	8
3.2	odd	2	inner	441.4.a.x.1.3	✓	8
7.2	even	3		441.4.e.z.361.3		16
7.3	odd	6		441.4.e.z.226.4		16
7.4	even	3		441.4.e.z.226.3		16
7.5	odd	6		441.4.e.z.361.4		16
7.6	odd	2	inner	441.4.a.x.1.5	yes	8
21.2	odd	6		441.4.e.z.361.6		16
21.5	even	6		441.4.e.z.361.5		16
21.11	odd	6		441.4.e.z.226.6		16
21.17	even	6		441.4.e.z.226.5		16
21.20	even	2	inner	441.4.a.x.1.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.4.a.x.1.3	✓	8	3.2	odd	2	inner
441.4.a.x.1.4	yes	8	21.20	even	2	inner
441.4.a.x.1.5	yes	8	7.6	odd	2	inner
441.4.a.x.1.6	yes	8	1.1	even	1	trivial
441.4.e.z.226.3		16	7.4	even	3
441.4.e.z.226.4		16	7.3	odd	6
441.4.e.z.226.5		16	21.17	even	6
441.4.e.z.226.6		16	21.11	odd	6
441.4.e.z.361.3		16	7.2	even	3
441.4.e.z.361.4		16	7.5	odd	6
441.4.e.z.361.5		16	21.5	even	6
441.4.e.z.361.6		16	21.2	odd	6

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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	441.a (trivial)

\(f(q)\)	\(=\)	\(q+1.95985 q^{2} -4.15899 q^{4} +11.8897 q^{5} -23.8298 q^{8} +O(q^{10})\)	\(q+1.95985 q^{2} -4.15899 q^{4} +11.8897 q^{5} -23.8298 q^{8} +23.3019 q^{10} +36.4130 q^{11} +0.964525 q^{13} -13.4309 q^{16} +98.2514 q^{17} -106.001 q^{19} -49.4490 q^{20} +71.3640 q^{22} +54.3633 q^{23} +16.3640 q^{25} +1.89032 q^{26} +229.725 q^{29} +127.729 q^{31} +164.316 q^{32} +192.558 q^{34} +311.816 q^{37} -207.746 q^{38} -283.328 q^{40} -419.919 q^{41} +523.180 q^{43} -151.441 q^{44} +106.544 q^{46} +270.328 q^{47} +32.0711 q^{50} -4.01145 q^{52} -251.082 q^{53} +432.938 q^{55} +450.226 q^{58} +408.029 q^{59} -860.546 q^{61} +250.329 q^{62} +429.481 q^{64} +11.4679 q^{65} +506.360 q^{67} -408.626 q^{68} -523.702 q^{71} +629.964 q^{73} +611.112 q^{74} +440.856 q^{76} +319.548 q^{79} -159.689 q^{80} -822.978 q^{82} +1309.16 q^{83} +1168.18 q^{85} +1025.35 q^{86} -867.715 q^{88} -348.581 q^{89} -226.096 q^{92} +529.803 q^{94} -1260.31 q^{95} -161.996 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 68 q^{4}+O(q^{10}) \) 8 * q + 68 * q^4	\( 8 q + 68 q^{4} + 804 q^{16} + 976 q^{22} + 536 q^{25} + 64 q^{37} + 2160 q^{43} - 768 q^{46} + 2184 q^{58} + 7588 q^{64} + 5392 q^{79} + 2864 q^{85} + 5616 q^{88}+O(q^{100}) \) 8 * q + 68 * q^4 + 804 * q^16 + 976 * q^22 + 536 * q^25 + 64 * q^37 + 2160 * q^43 - 768 * q^46 + 2184 * q^58 + 7588 * q^64 + 5392 * q^79 + 2864 * q^85 + 5616 * q^88

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