LMFDB - Embedded newform 1225.4.a.p.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1225,4,Mod(1,1225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1225, 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("1225.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1225 = 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1225.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:	$72.2773397570$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{11}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 11 $ x^2 - 11
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 245)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.31662$ of defining polynomial
Character	$\chi$	$=$	1225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.31662 q^{2} -5.00000 q^{3} -2.63325 q^{4} -11.5831 q^{6} -24.6332 q^{8} -2.00000 q^{9} +O(q^{10})$	\(q+2.31662 q^{2} -5.00000 q^{3} -2.63325 q^{4} -11.5831 q^{6} -24.6332 q^{8} -2.00000 q^{9} +46.2665 q^{11} +13.1662 q^{12} +61.3325 q^{13} -36.0000 q^{16} -101.332 q^{17} -4.63325 q^{18} +3.66750 q^{19} +107.182 q^{22} -84.8655 q^{23} +123.166 q^{24} +142.084 q^{26} +145.000 q^{27} +30.1980 q^{29} +188.997 q^{31} +113.668 q^{32} -231.332 q^{33} -234.749 q^{34} +5.26650 q^{36} -18.0685 q^{37} +8.49623 q^{38} -306.662 q^{39} +481.662 q^{41} +97.7995 q^{43} -121.831 q^{44} -196.602 q^{46} -117.665 q^{47} +180.000 q^{48} +506.662 q^{51} -161.504 q^{52} -667.995 q^{53} +335.911 q^{54} -18.3375 q^{57} +69.9574 q^{58} +57.3350 q^{59} +738.997 q^{61} +437.836 q^{62} +551.325 q^{64} -535.911 q^{66} -552.396 q^{67} +266.834 q^{68} +424.327 q^{69} -740.264 q^{71} +49.2665 q^{72} -233.325 q^{73} -41.8580 q^{74} -9.65745 q^{76} -710.422 q^{78} -1075.19 q^{79} -671.000 q^{81} +1115.83 q^{82} -683.325 q^{83} +226.565 q^{86} -150.990 q^{87} -1139.69 q^{88} -1380.32 q^{89} +223.472 q^{92} -944.987 q^{93} -272.586 q^{94} -568.338 q^{96} -218.008 q^{97} -92.5330 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 10 q^{3} + 8 q^{4} + 10 q^{6} - 36 q^{8} - 4 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 - 10 * q^3 + 8 * q^4 + 10 * q^6 - 36 * q^8 - 4 * q^9	$ 2 q - 2 q^{2} - 10 q^{3} + 8 q^{4} + 10 q^{6} - 36 q^{8} - 4 q^{9} + 66 q^{11} - 40 q^{12} - 10 q^{13} - 72 q^{16} - 70 q^{17} + 4 q^{18} + 140 q^{19} + 22 q^{22} + 16 q^{23} + 180 q^{24} + 450 q^{26} + 290 q^{27} - 258 q^{29} - 20 q^{31} + 360 q^{32} - 330 q^{33} - 370 q^{34} - 16 q^{36} - 328 q^{37} - 580 q^{38} + 50 q^{39} + 300 q^{41} + 116 q^{43} + 88 q^{44} - 632 q^{46} + 30 q^{47} + 360 q^{48} + 350 q^{51} - 920 q^{52} - 540 q^{53} - 290 q^{54} - 700 q^{57} + 1314 q^{58} + 380 q^{59} + 1080 q^{61} + 1340 q^{62} - 224 q^{64} - 110 q^{66} - 468 q^{67} + 600 q^{68} - 80 q^{69} - 1056 q^{71} + 72 q^{72} + 860 q^{73} + 1296 q^{74} + 1440 q^{76} - 2250 q^{78} + 158 q^{79} - 1342 q^{81} + 1900 q^{82} - 40 q^{83} + 148 q^{86} + 1290 q^{87} - 1364 q^{88} - 240 q^{89} + 1296 q^{92} + 100 q^{93} - 910 q^{94} - 1800 q^{96} - 1630 q^{97} - 132 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 - 10 * q^3 + 8 * q^4 + 10 * q^6 - 36 * q^8 - 4 * q^9 + 66 * q^11 - 40 * q^12 - 10 * q^13 - 72 * q^16 - 70 * q^17 + 4 * q^18 + 140 * q^19 + 22 * q^22 + 16 * q^23 + 180 * q^24 + 450 * q^26 + 290 * q^27 - 258 * q^29 - 20 * q^31 + 360 * q^32 - 330 * q^33 - 370 * q^34 - 16 * q^36 - 328 * q^37 - 580 * q^38 + 50 * q^39 + 300 * q^41 + 116 * q^43 + 88 * q^44 - 632 * q^46 + 30 * q^47 + 360 * q^48 + 350 * q^51 - 920 * q^52 - 540 * q^53 - 290 * q^54 - 700 * q^57 + 1314 * q^58 + 380 * q^59 + 1080 * q^61 + 1340 * q^62 - 224 * q^64 - 110 * q^66 - 468 * q^67 + 600 * q^68 - 80 * q^69 - 1056 * q^71 + 72 * q^72 + 860 * q^73 + 1296 * q^74 + 1440 * q^76 - 2250 * q^78 + 158 * q^79 - 1342 * q^81 + 1900 * q^82 - 40 * q^83 + 148 * q^86 + 1290 * q^87 - 1364 * q^88 - 240 * q^89 + 1296 * q^92 + 100 * q^93 - 910 * q^94 - 1800 * q^96 - 1630 * q^97 - 132 * q^99

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 $

$2$	2.31662	0.819051	0.409525	−	0.912299i	$-0.365694\pi$
$2$	2.31662	0.819051	0.409525	+	0.912299i	$0.365694\pi$
$3$	−5.00000	−0.962250	−0.481125	−	0.876652i	$-0.659772\pi$
$3$	−5.00000	−0.962250	−0.481125	+	0.876652i	$0.659772\pi$
$4$	−2.63325	−0.329156
$5$	0	0
$5$	0	0
$6$	−11.5831	−0.788132
$7$	0	0
$7$	0	0
$8$	−24.6332	−1.08865
$9$	−2.00000	−0.0740741
$10$	0	0
$11$	46.2665	1.26817	0.634085	−	0.773263i	$-0.281377\pi$
$11$	46.2665	1.26817	0.634085	+	0.773263i	$0.281377\pi$
$12$	13.1662	0.316731
$13$	61.3325	1.30851	0.654253	−	0.756276i	$-0.272983\pi$
$13$	61.3325	1.30851	0.654253	+	0.756276i	$0.272983\pi$
$14$	0	0
$15$	0	0
$16$	−36.0000	−0.562500
$17$	−101.332	−1.44569	−0.722845	−	0.691010i	$-0.757166\pi$
$17$	−101.332	−1.44569	−0.722845	+	0.691010i	$0.757166\pi$
$18$	−4.63325	−0.0606704
$19$	3.66750	0.0442833	0.0221417	−	0.999755i	$-0.492952\pi$
$19$	3.66750	0.0442833	0.0221417	+	0.999755i	$0.492952\pi$
$20$	0	0
$21$	0	0
$22$	107.182	1.03870
$23$	−84.8655	−0.769377	−0.384689	−	0.923046i	$-0.625691\pi$
$23$	−84.8655	−0.769377	−0.384689	+	0.923046i	$0.625691\pi$
$24$	123.166	1.04755
$25$	0	0
$26$	142.084	1.07173
$27$	145.000	1.03353
$28$	0	0
$29$	30.1980	0.193366	0.0966832	−	0.995315i	$-0.469177\pi$
$29$	30.1980	0.193366	0.0966832	+	0.995315i	$0.469177\pi$
$30$	0	0
$31$	188.997	1.09500	0.547499	−	0.836806i	$-0.315580\pi$
$31$	188.997	1.09500	0.547499	+	0.836806i	$0.315580\pi$
$32$	113.668	0.627930
$33$	−231.332	−1.22030
$34$	−234.749	−1.18409
$35$	0	0
$36$	5.26650	0.0243819
$37$	−18.0685	−0.0802823	−0.0401411	−	0.999194i	$-0.512781\pi$
$37$	−18.0685	−0.0802823	−0.0401411	+	0.999194i	$0.512781\pi$
$38$	8.49623	0.0362703
$39$	−306.662	−1.25911
$40$	0	0
$41$	481.662	1.83471	0.917354	−	0.398072i	$-0.130321\pi$
$41$	481.662	1.83471	0.917354	+	0.398072i	$0.130321\pi$
$42$	0	0
$43$	97.7995	0.346844	0.173422	−	0.984848i	$-0.444518\pi$
$43$	97.7995	0.346844	0.173422	+	0.984848i	$0.444518\pi$
$44$	−121.831	−0.417426
$45$	0	0
$46$	−196.602	−0.630159
$47$	−117.665	−0.365175	−0.182587	−	0.983190i	$-0.558447\pi$
$47$	−117.665	−0.365175	−0.182587	+	0.983190i	$0.558447\pi$
$48$	180.000	0.541266
$49$	0	0
$50$	0	0
$51$	506.662	1.39112
$52$	−161.504	−0.430703
$53$	−667.995	−1.73125	−0.865624	−	0.500694i	$-0.833078\pi$
$53$	−667.995	−1.73125	−0.865624	+	0.500694i	$0.833078\pi$
$54$	335.911	0.846512
$55$	0	0
$56$	0	0
$57$	−18.3375	−0.0426116
$58$	69.9574	0.158377
$59$	57.3350	0.126515	0.0632575	−	0.997997i	$-0.479851\pi$
$59$	57.3350	0.126515	0.0632575	+	0.997997i	$0.479851\pi$
$60$	0	0
$61$	738.997	1.55113	0.775565	−	0.631268i	$-0.217465\pi$
$61$	738.997	1.55113	0.775565	+	0.631268i	$0.217465\pi$
$62$	437.836	0.896859
$63$	0	0
$64$	551.325	1.07681
$65$	0	0
$66$	−535.911	−0.999485
$67$	−552.396	−1.00725	−0.503626	−	0.863922i	$-0.668001\pi$
$67$	−552.396	−1.00725	−0.503626	+	0.863922i	$0.668001\pi$
$68$	266.834	0.475858
$69$	424.327	0.740334
$70$	0	0
$71$	−740.264	−1.23737	−0.618684	−	0.785640i	$-0.712334\pi$
$71$	−740.264	−1.23737	−0.618684	+	0.785640i	$0.712334\pi$
$72$	49.2665	0.0806405
$73$	−233.325	−0.374091	−0.187045	−	0.982351i	$-0.559891\pi$
$73$	−233.325	−0.374091	−0.187045	+	0.982351i	$0.559891\pi$
$74$	−41.8580	−0.0657553
$75$	0	0
$76$	−9.65745	−0.0145761
$77$	0	0
$78$	−710.422	−1.03127
$79$	−1075.19	−1.53124	−0.765619	−	0.643294i	$-0.777567\pi$
$79$	−1075.19	−1.53124	−0.765619	+	0.643294i	$0.777567\pi$
$80$	0	0
$81$	−671.000	−0.920439
$82$	1115.83	1.50272
$83$	−683.325	−0.903671	−0.451835	−	0.892101i	$-0.649231\pi$
$83$	−683.325	−0.903671	−0.451835	+	0.892101i	$0.649231\pi$
$84$	0	0
$85$	0	0
$86$	226.565	0.284083
$87$	−150.990	−0.186067
$88$	−1139.69	−1.38059
$89$	−1380.32	−1.64397	−0.821985	−	0.569509i	$-0.807133\pi$
$89$	−1380.32	−1.64397	−0.821985	+	0.569509i	$0.807133\pi$
$90$	0	0
$91$	0	0
$92$	223.472	0.253245
$93$	−944.987	−1.05366
$94$	−272.586	−0.299096
$95$	0	0
$96$	−568.338	−0.604226
$97$	−218.008	−0.228199	−0.114100	−	0.993469i	$-0.536398\pi$
$97$	−218.008	−0.228199	−0.114100	+	0.993469i	$0.536398\pi$
$98$	0	0
$99$	−92.5330	−0.0939385
$100$	0	0
$101$	−1474.33	−1.45249	−0.726243	−	0.687438i	$-0.758735\pi$
$101$	−1474.33	−1.45249	−0.726243	+	0.687438i	$0.758735\pi$
$102$	1173.75	1.13939
$103$	810.990	0.775818	0.387909	−	0.921698i	$-0.373198\pi$
$103$	810.990	0.775818	0.387909	+	0.921698i	$0.373198\pi$
$104$	−1510.82	−1.42450
$105$	0	0
$106$	−1547.49	−1.41798
$107$	−440.660	−0.398133	−0.199066	−	0.979986i	$-0.563791\pi$
$107$	−440.660	−0.398133	−0.199066	+	0.979986i	$0.563791\pi$
$108$	−381.821	−0.340192
$109$	−1906.19	−1.67504	−0.837522	−	0.546404i	$-0.815996\pi$
$109$	−1906.19	−1.67504	−0.837522	+	0.546404i	$0.815996\pi$
$110$	0	0
$111$	90.3425	0.0772517
$112$	0	0
$113$	−962.470	−0.801252	−0.400626	−	0.916242i	$-0.631207\pi$
$113$	−962.470	−0.801252	−0.400626	+	0.916242i	$0.631207\pi$
$114$	−42.4812	−0.0349011
$115$	0	0
$116$	−79.5188	−0.0636478
$117$	−122.665	−0.0969263
$118$	132.824	0.103622
$119$	0	0
$120$	0	0
$121$	809.589	0.608256
$122$	1711.98	1.27045
$123$	−2408.31	−1.76545
$124$	−497.678	−0.360426
$125$	0	0
$126$	0	0
$127$	1621.74	1.13312	0.566558	−	0.824022i	$-0.308275\pi$
$127$	1621.74	1.13312	0.566558	+	0.824022i	$0.308275\pi$
$128$	367.873	0.254029
$129$	−488.997	−0.333751
$130$	0	0
$131$	−1380.32	−0.920602	−0.460301	−	0.887763i	$-0.652259\pi$
$131$	−1380.32	−0.920602	−0.460301	+	0.887763i	$0.652259\pi$
$132$	609.156	0.401668
$133$	0	0
$134$	−1279.69	−0.824991
$135$	0	0
$136$	2496.15	1.57385
$137$	1949.66	1.21584	0.607921	−	0.793997i	$-0.292004\pi$
$137$	1949.66	1.21584	0.607921	+	0.793997i	$0.292004\pi$
$138$	983.008	0.606371
$139$	2800.00	1.70858	0.854291	−	0.519795i	$-0.173992\pi$
$139$	2800.00	1.70858	0.854291	+	0.519795i	$0.173992\pi$
$140$	0	0
$141$	588.325	0.351389
$142$	−1714.91	−1.01347
$143$	2837.64	1.65941
$144$	72.0000	0.0416667
$145$	0	0
$146$	−540.526	−0.306399
$147$	0	0
$148$	47.5789	0.0264254
$149$	−1434.12	−0.788506	−0.394253	−	0.919002i	$-0.628997\pi$
$149$	−1434.12	−0.788506	−0.394253	+	0.919002i	$0.628997\pi$
$150$	0	0
$151$	−1985.58	−1.07009	−0.535047	−	0.844822i	$-0.679706\pi$
$151$	−1985.58	−1.07009	−0.535047	+	0.844822i	$0.679706\pi$
$152$	−90.3425	−0.0482089
$153$	202.665	0.107088
$154$	0	0
$155$	0	0
$156$	807.519	0.414444
$157$	−40.6600	−0.0206689	−0.0103345	−	0.999947i	$-0.503290\pi$
$157$	−40.6600	−0.0206689	−0.0103345	+	0.999947i	$0.503290\pi$
$158$	−2490.80	−1.25416
$159$	3339.97	1.66589
$160$	0	0
$161$	0	0
$162$	−1554.46	−0.753886
$163$	3953.98	1.90000	0.950000	−	0.312250i	$-0.101083\pi$
$163$	3953.98	1.90000	0.950000	+	0.312250i	$0.101083\pi$
$164$	−1268.34	−0.603906
$165$	0	0
$166$	−1583.01	−0.740152
$167$	−3380.30	−1.56632	−0.783161	−	0.621819i	$-0.786394\pi$
$167$	−3380.30	−1.56632	−0.783161	+	0.621819i	$0.786394\pi$
$168$	0	0
$169$	1564.68	0.712187
$170$	0	0
$171$	−7.33501	−0.00328025
$172$	−257.530	−0.114166
$173$	−3206.66	−1.40924	−0.704619	−	0.709586i	$-0.748882\pi$
$173$	−3206.66	−1.40924	−0.704619	+	0.709586i	$0.748882\pi$
$174$	−349.787	−0.152398
$175$	0	0
$176$	−1665.59	−0.713346
$177$	−286.675	−0.121739
$178$	−3197.68	−1.34649
$179$	1442.65	0.602395	0.301198	−	0.953562i	$-0.402614\pi$
$179$	1442.65	0.602395	0.301198	+	0.953562i	$0.402614\pi$
$180$	0	0
$181$	−908.680	−0.373158	−0.186579	−	0.982440i	$-0.559740\pi$
$181$	−908.680	−0.373158	−0.186579	+	0.982440i	$0.559740\pi$
$182$	0	0
$183$	−3694.99	−1.49258
$184$	2090.51	0.837580
$185$	0	0
$186$	−2189.18	−0.863003
$187$	−4688.30	−1.83338
$188$	309.841	0.120199
$189$	0	0
$190$	0	0
$191$	2474.64	0.937479	0.468739	−	0.883336i	$-0.344708\pi$
$191$	2474.64	0.937479	0.468739	+	0.883336i	$0.344708\pi$
$192$	−2756.62	−1.03616
$193$	−3533.52	−1.31787	−0.658934	−	0.752201i	$-0.728992\pi$
$193$	−3533.52	−1.31787	−0.658934	+	0.752201i	$0.728992\pi$
$194$	−505.042	−0.186907
$195$	0	0
$196$	0	0
$197$	1952.57	0.706165	0.353083	−	0.935592i	$-0.385133\pi$
$197$	1952.57	0.706165	0.353083	+	0.935592i	$0.385133\pi$
$198$	−214.364	−0.0769404
$199$	4064.67	1.44792	0.723962	−	0.689840i	$-0.242319\pi$
$199$	4064.67	1.44792	0.723962	+	0.689840i	$0.242319\pi$
$200$	0	0
$201$	2761.98	0.969229
$202$	−3415.46	−1.18966
$203$	0	0
$204$	−1334.17	−0.457895
$205$	0	0
$206$	1878.76	0.635434
$207$	169.731	0.0569909
$208$	−2207.97	−0.736034
$209$	169.683	0.0561588
$210$	0	0
$211$	−4325.34	−1.41123	−0.705613	−	0.708598i	$-0.749328\pi$
$211$	−4325.34	−1.41123	−0.705613	+	0.708598i	$0.749328\pi$
$212$	1759.00	0.569851
$213$	3701.32	1.19066
$214$	−1020.84	−0.326091
$215$	0	0
$216$	−3571.82	−1.12515
$217$	0	0
$218$	−4415.92	−1.37194
$219$	1166.62	0.359969
$220$	0	0
$221$	−6214.97	−1.89169
$222$	209.290	0.0632730
$223$	982.970	0.295177	0.147589	−	0.989049i	$-0.452849\pi$
$223$	982.970	0.295177	0.147589	+	0.989049i	$0.452849\pi$
$224$	0	0
$225$	0	0
$226$	−2229.68	−0.656266
$227$	1660.96	0.485648	0.242824	−	0.970070i	$-0.421926\pi$
$227$	1660.96	0.485648	0.242824	+	0.970070i	$0.421926\pi$
$228$	48.2873	0.0140259
$229$	−574.327	−0.165732	−0.0828660	−	0.996561i	$-0.526407\pi$
$229$	−574.327	−0.165732	−0.0828660	+	0.996561i	$0.526407\pi$
$230$	0	0
$231$	0	0
$232$	−743.875	−0.210508
$233$	2316.48	0.651320	0.325660	−	0.945487i	$-0.394414\pi$
$233$	2316.48	0.651320	0.325660	+	0.945487i	$0.394414\pi$
$234$	−284.169	−0.0793876
$235$	0	0
$236$	−150.977	−0.0416432
$237$	5375.93	1.47343
$238$	0	0
$239$	−3659.31	−0.990382	−0.495191	−	0.868784i	$-0.664902\pi$
$239$	−3659.31	−0.990382	−0.495191	+	0.868784i	$0.664902\pi$
$240$	0	0
$241$	2446.33	0.653868	0.326934	−	0.945047i	$-0.393985\pi$
$241$	2446.33	0.653868	0.326934	+	0.945047i	$0.393985\pi$
$242$	1875.51	0.498193
$243$	−560.000	−0.147835
$244$	−1945.96	−0.510564
$245$	0	0
$246$	−5579.16	−1.44599
$247$	224.937	0.0579450
$248$	−4655.62	−1.19207
$249$	3416.62	0.869557
$250$	0	0
$251$	−2909.29	−0.731605	−0.365802	−	0.930693i	$-0.619205\pi$
$251$	−2909.29	−0.731605	−0.365802	+	0.930693i	$0.619205\pi$
$252$	0	0
$253$	−3926.43	−0.975702
$254$	3756.95	0.928080
$255$	0	0
$256$	−3558.38	−0.868744
$257$	−168.680	−0.0409415	−0.0204708	−	0.999790i	$-0.506517\pi$
$257$	−168.680	−0.0409415	−0.0204708	+	0.999790i	$0.506517\pi$
$258$	−1132.82	−0.273359
$259$	0	0
$260$	0	0
$261$	−60.3960	−0.0143234
$262$	−3197.68	−0.754020
$263$	3244.47	0.760695	0.380347	−	0.924844i	$-0.375804\pi$
$263$	3244.47	0.760695	0.380347	+	0.924844i	$0.375804\pi$
$264$	5698.47	1.32847
$265$	0	0
$266$	0	0
$267$	6901.59	1.58191
$268$	1454.60	0.331543
$269$	−2848.65	−0.645671	−0.322836	−	0.946455i	$-0.604636\pi$
$269$	−2848.65	−0.645671	−0.322836	+	0.946455i	$0.604636\pi$
$270$	0	0
$271$	−2850.98	−0.639057	−0.319529	−	0.947577i	$-0.603525\pi$
$271$	−2850.98	−0.639057	−0.319529	+	0.947577i	$0.603525\pi$
$272$	3647.97	0.813201
$273$	0	0
$274$	4516.62	0.995837
$275$	0	0
$276$	−1117.36	−0.243685
$277$	−2298.63	−0.498597	−0.249298	−	0.968427i	$-0.580200\pi$
$277$	−2298.63	−0.498597	−0.249298	+	0.968427i	$0.580200\pi$
$278$	6486.55	1.39942
$279$	−377.995	−0.0811110
$280$	0	0
$281$	6109.20	1.29695	0.648477	−	0.761234i	$-0.275406\pi$
$281$	6109.20	1.29695	0.648477	+	0.761234i	$0.275406\pi$
$282$	1362.93	0.287806
$283$	−5854.95	−1.22983	−0.614913	−	0.788595i	$-0.710809\pi$
$283$	−5854.95	−1.22983	−0.614913	+	0.788595i	$0.710809\pi$
$284$	1949.30	0.407288
$285$	0	0
$286$	6573.75	1.35914
$287$	0	0
$288$	−227.335	−0.0465133
$289$	5355.27	1.09002
$290$	0	0
$291$	1090.04	0.219585
$292$	614.403	0.123134
$293$	5135.34	1.02392	0.511962	−	0.859008i	$-0.328919\pi$
$293$	5135.34	1.02392	0.511962	+	0.859008i	$0.328919\pi$
$294$	0	0
$295$	0	0
$296$	445.086	0.0873990
$297$	6708.64	1.31069
$298$	−3322.31	−0.645827
$299$	−5205.01	−1.00673
$300$	0	0
$301$	0	0
$302$	−4599.85	−0.876462
$303$	7371.64	1.39766
$304$	−132.030	−0.0249094
$305$	0	0
$306$	469.499	0.0877106
$307$	−2102.97	−0.390954	−0.195477	−	0.980708i	$-0.562625\pi$
$307$	−2102.97	−0.390954	−0.195477	+	0.980708i	$0.562625\pi$
$308$	0	0
$309$	−4054.95	−0.746531
$310$	0	0
$311$	−5764.30	−1.05101	−0.525504	−	0.850791i	$-0.676123\pi$
$311$	−5764.30	−1.05101	−0.525504	+	0.850791i	$0.676123\pi$
$312$	7554.09	1.37073
$313$	1360.01	0.245599	0.122799	−	0.992432i	$-0.460813\pi$
$313$	1360.01	0.245599	0.122799	+	0.992432i	$0.460813\pi$
$314$	−94.1939	−0.0169289
$315$	0	0
$316$	2831.23	0.504017
$317$	−5138.95	−0.910511	−0.455256	−	0.890361i	$-0.650452\pi$
$317$	−5138.95	−0.910511	−0.455256	+	0.890361i	$0.650452\pi$
$318$	7737.47	1.36445
$319$	1397.16	0.245222
$320$	0	0
$321$	2203.30	0.383103
$322$	0	0
$323$	−371.637	−0.0640200
$324$	1766.91	0.302968
$325$	0	0
$326$	9159.90	1.55620
$327$	9530.94	1.61181
$328$	−11864.9	−1.99735
$329$	0	0
$330$	0	0
$331$	1825.70	0.303171	0.151585	−	0.988444i	$-0.451562\pi$
$331$	1825.70	0.303171	0.151585	+	0.988444i	$0.451562\pi$
$332$	1799.37	0.297449
$333$	36.1370	0.00594684
$334$	−7830.90	−1.28290
$335$	0	0
$336$	0	0
$337$	−153.985	−0.0248905	−0.0124452	−	0.999923i	$-0.503962\pi$
$337$	−153.985	−0.0248905	−0.0124452	+	0.999923i	$0.503962\pi$
$338$	3624.76	0.583317
$339$	4812.35	0.771005
$340$	0	0
$341$	8744.25	1.38864
$342$	−16.9925	−0.00268669
$343$	0	0
$344$	−2409.12	−0.377590
$345$	0	0
$346$	−7428.63	−1.15424
$347$	−4359.39	−0.674421	−0.337211	−	0.941429i	$-0.609483\pi$
$347$	−4359.39	−0.674421	−0.337211	+	0.941429i	$0.609483\pi$
$348$	397.594	0.0612451
$349$	1689.00	0.259054	0.129527	−	0.991576i	$-0.458654\pi$
$349$	1689.00	0.259054	0.129527	+	0.991576i	$0.458654\pi$
$350$	0	0
$351$	8893.21	1.35238
$352$	5259.00	0.796322
$353$	3921.36	0.591254	0.295627	−	0.955303i	$-0.404471\pi$
$353$	3921.36	0.591254	0.295627	+	0.955303i	$0.404471\pi$
$354$	−664.119	−0.0997105
$355$	0	0
$356$	3634.72	0.541123
$357$	0	0
$358$	3342.08	0.493392
$359$	−2867.86	−0.421616	−0.210808	−	0.977528i	$-0.567609\pi$
$359$	−2867.86	−0.421616	−0.210808	+	0.977528i	$0.567609\pi$
$360$	0	0
$361$	−6845.55	−0.998039
$362$	−2105.07	−0.305636
$363$	−4047.94	−0.585295
$364$	0	0
$365$	0	0
$366$	−8559.90	−1.22249
$367$	−11503.0	−1.63611	−0.818054	−	0.575142i	$-0.804947\pi$
$367$	−11503.0	−1.63611	−0.818054	+	0.575142i	$0.804947\pi$
$368$	3055.16	0.432775
$369$	−963.325	−0.135904
$370$	0	0
$371$	0	0
$372$	2488.39	0.346820
$373$	−5086.43	−0.706073	−0.353037	−	0.935610i	$-0.614851\pi$
$373$	−5086.43	−0.706073	−0.353037	+	0.935610i	$0.614851\pi$
$374$	−10861.0	−1.50163
$375$	0	0
$376$	2898.47	0.397546
$377$	1852.12	0.253021
$378$	0	0
$379$	954.827	0.129409	0.0647047	−	0.997904i	$-0.479389\pi$
$379$	954.827	0.129409	0.0647047	+	0.997904i	$0.479389\pi$
$380$	0	0
$381$	−8108.68	−1.09034
$382$	5732.81	0.767843
$383$	−3083.91	−0.411437	−0.205719	−	0.978611i	$-0.565953\pi$
$383$	−3083.91	−0.411437	−0.205719	+	0.978611i	$0.565953\pi$
$384$	−1839.37	−0.244439
$385$	0	0
$386$	−8185.84	−1.07940
$387$	−195.599	−0.0256921
$388$	574.068	0.0751131
$389$	−6331.15	−0.825198	−0.412599	−	0.910913i	$-0.635379\pi$
$389$	−6331.15	−0.825198	−0.412599	+	0.910913i	$0.635379\pi$
$390$	0	0
$391$	8599.63	1.11228
$392$	0	0
$393$	6901.59	0.885850
$394$	4523.36	0.578385
$395$	0	0
$396$	243.662	0.0309205
$397$	12133.2	1.53388	0.766939	−	0.641720i	$-0.221779\pi$
$397$	12133.2	1.53388	0.766939	+	0.641720i	$0.221779\pi$
$398$	9416.32	1.18592
$399$	0	0
$400$	0	0
$401$	−270.669	−0.0337072	−0.0168536	−	0.999858i	$-0.505365\pi$
$401$	−270.669	−0.0337072	−0.0168536	+	0.999858i	$0.505365\pi$
$402$	6398.47	0.793848
$403$	11591.7	1.43281
$404$	3882.27	0.478095
$405$	0	0
$406$	0	0
$407$	−835.967	−0.101812
$408$	−12480.7	−1.51443
$409$	4019.92	0.485996	0.242998	−	0.970027i	$-0.421869\pi$
$409$	4019.92	0.485996	0.242998	+	0.970027i	$0.421869\pi$
$410$	0	0
$411$	−9748.29	−1.16995
$412$	−2135.54	−0.255365
$413$	0	0
$414$	393.203	0.0466784
$415$	0	0
$416$	6971.51	0.821650
$417$	−14000.0	−1.64408
$418$	393.091	0.0459969
$419$	2437.28	0.284175	0.142087	−	0.989854i	$-0.454619\pi$
$419$	2437.28	0.284175	0.142087	+	0.989854i	$0.454619\pi$
$420$	0	0
$421$	−4751.36	−0.550041	−0.275020	−	0.961438i	$-0.588685\pi$
$421$	−4751.36	−0.550041	−0.275020	+	0.961438i	$0.588685\pi$
$422$	−10020.2	−1.15586
$423$	235.330	0.0270500
$424$	16454.9	1.88472
$425$	0	0
$426$	8574.57	0.975210
$427$	0	0
$428$	1160.37	0.131048
$429$	−14188.2	−1.59677
$430$	0	0
$431$	7925.19	0.885714	0.442857	−	0.896592i	$-0.353965\pi$
$431$	7925.19	0.885714	0.442857	+	0.896592i	$0.353965\pi$
$432$	−5220.00	−0.581360
$433$	11487.3	1.27492	0.637462	−	0.770481i	$-0.279984\pi$
$433$	11487.3	1.27492	0.637462	+	0.770481i	$0.279984\pi$
$434$	0	0
$435$	0	0
$436$	5019.47	0.551351
$437$	−311.245	−0.0340706
$438$	2702.63	0.294833
$439$	−9147.92	−0.994548	−0.497274	−	0.867594i	$-0.665666\pi$
$439$	−9147.92	−0.994548	−0.497274	+	0.867594i	$0.665666\pi$
$440$	0	0
$441$	0	0
$442$	−14397.8	−1.54939
$443$	−1864.35	−0.199950	−0.0999752	−	0.994990i	$-0.531876\pi$
$443$	−1864.35	−0.199950	−0.0999752	+	0.994990i	$0.531876\pi$
$444$	−237.894	−0.0254279
$445$	0	0
$446$	2277.17	0.241765
$447$	7170.58	0.758741
$448$	0	0
$449$	4490.88	0.472022	0.236011	−	0.971750i	$-0.424160\pi$
$449$	4490.88	0.472022	0.236011	+	0.971750i	$0.424160\pi$
$450$	0	0
$451$	22284.8	2.32672
$452$	2534.42	0.263737
$453$	9927.91	1.02970
$454$	3847.83	0.397770
$455$	0	0
$456$	451.713	0.0463890
$457$	14343.8	1.46822	0.734109	−	0.679032i	$-0.237600\pi$
$457$	14343.8	1.46822	0.734109	+	0.679032i	$0.237600\pi$
$458$	−1330.50	−0.135743
$459$	−14693.2	−1.49416
$460$	0	0
$461$	−14558.7	−1.47086	−0.735429	−	0.677602i	$-0.763019\pi$
$461$	−14558.7	−1.47086	−0.735429	+	0.677602i	$0.763019\pi$
$462$	0	0
$463$	1809.56	0.181636	0.0908178	−	0.995868i	$-0.471052\pi$
$463$	1809.56	0.181636	0.0908178	+	0.995868i	$0.471052\pi$
$464$	−1087.13	−0.108769
$465$	0	0
$466$	5366.41	0.533464
$467$	−5981.65	−0.592715	−0.296357	−	0.955077i	$-0.595772\pi$
$467$	−5981.65	−0.592715	−0.296357	+	0.955077i	$0.595772\pi$
$468$	323.008	0.0319039
$469$	0	0
$470$	0	0
$471$	203.300	0.0198887
$472$	−1412.35	−0.137730
$473$	4524.84	0.439857
$474$	12454.0	1.20682
$475$	0	0
$476$	0	0
$477$	1335.99	0.128241
$478$	−8477.25	−0.811173
$479$	11527.5	1.09959	0.549796	−	0.835299i	$-0.314705\pi$
$479$	11527.5	1.09959	0.549796	+	0.835299i	$0.314705\pi$
$480$	0	0
$481$	−1108.19	−0.105050
$482$	5667.23	0.535551
$483$	0	0
$484$	−2131.85	−0.200211
$485$	0	0
$486$	−1297.31	−0.121085
$487$	−15791.5	−1.46936	−0.734682	−	0.678411i	$-0.762669\pi$
$487$	−15791.5	−1.46936	−0.734682	+	0.678411i	$0.762669\pi$
$488$	−18203.9	−1.68863
$489$	−19769.9	−1.82828
$490$	0	0
$491$	13064.9	1.20083	0.600417	−	0.799687i	$-0.295001\pi$
$491$	13064.9	1.20083	0.600417	+	0.799687i	$0.295001\pi$
$492$	6341.69	0.581108
$493$	−3060.04	−0.279548
$494$	521.095	0.0474599
$495$	0	0
$496$	−6803.91	−0.615937
$497$	0	0
$498$	7915.04	0.712211
$499$	−20135.8	−1.80642	−0.903209	−	0.429201i	$-0.858795\pi$
$499$	−20135.8	−1.80642	−0.903209	+	0.429201i	$0.858795\pi$
$500$	0	0
$501$	16901.5	1.50719
$502$	−6739.73	−0.599221
$503$	−751.675	−0.0666313	−0.0333156	−	0.999445i	$-0.510607\pi$
$503$	−751.675	−0.0666313	−0.0333156	+	0.999445i	$0.510607\pi$
$504$	0	0
$505$	0	0
$506$	−9096.06	−0.799149
$507$	−7823.38	−0.685302
$508$	−4270.44	−0.372972
$509$	12334.5	1.07410	0.537049	−	0.843551i	$-0.319539\pi$
$509$	12334.5	1.07410	0.537049	+	0.843551i	$0.319539\pi$
$510$	0	0
$511$	0	0
$512$	−11186.4	−0.965574
$513$	531.788	0.0457681
$514$	−390.768	−0.0335332
$515$	0	0
$516$	1287.65	0.109856
$517$	−5443.95	−0.463104
$518$	0	0
$519$	16033.3	1.35604
$520$	0	0
$521$	1736.43	0.146016	0.0730082	−	0.997331i	$-0.476740\pi$
$521$	1736.43	0.146016	0.0730082	+	0.997331i	$0.476740\pi$
$522$	−139.915	−0.0117316
$523$	−1421.42	−0.118842	−0.0594210	−	0.998233i	$-0.518925\pi$
$523$	−1421.42	−0.118842	−0.0594210	+	0.998233i	$0.518925\pi$
$524$	3634.72	0.303022
$525$	0	0
$526$	7516.23	0.623048
$527$	−19151.6	−1.58303
$528$	8327.97	0.686417
$529$	−4964.85	−0.408059
$530$	0	0
$531$	−114.670	−0.00937148
$532$	0	0
$533$	29541.6	2.40073
$534$	15988.4	1.29567
$535$	0	0
$536$	13607.3	1.09654
$537$	−7213.25	−0.579655
$538$	−6599.26	−0.528837
$539$	0	0
$540$	0	0
$541$	5773.27	0.458802	0.229401	−	0.973332i	$-0.426323\pi$
$541$	5773.27	0.458802	0.229401	+	0.973332i	$0.426323\pi$
$542$	−6604.64	−0.523420
$543$	4543.40	0.359072
$544$	−11518.2	−0.907793
$545$	0	0
$546$	0	0
$547$	3941.30	0.308076	0.154038	−	0.988065i	$-0.450772\pi$
$547$	3941.30	0.308076	0.154038	+	0.988065i	$0.450772\pi$
$548$	−5133.93	−0.400202
$549$	−1477.99	−0.114899
$550$	0	0
$551$	110.751	0.00856291
$552$	−10452.6	−0.805961
$553$	0	0
$554$	−5325.06	−0.408376
$555$	0	0
$556$	−7373.10	−0.562390
$557$	6951.74	0.528823	0.264412	−	0.964410i	$-0.414822\pi$
$557$	6951.74	0.528823	0.264412	+	0.964410i	$0.414822\pi$
$558$	−875.673	−0.0664340
$559$	5998.29	0.453847
$560$	0	0
$561$	23441.5	1.76417
$562$	14152.7	1.06227
$563$	−24284.6	−1.81789	−0.908946	−	0.416913i	$-0.863112\pi$
$563$	−24284.6	−1.81789	−0.908946	+	0.416913i	$0.863112\pi$
$564$	−1549.21	−0.115662
$565$	0	0
$566$	−13563.7	−1.00729
$567$	0	0
$568$	18235.1	1.34706
$569$	−21563.4	−1.58873	−0.794363	−	0.607443i	$-0.792195\pi$
$569$	−21563.4	−1.58873	−0.794363	+	0.607443i	$0.792195\pi$
$570$	0	0
$571$	−3689.56	−0.270409	−0.135204	−	0.990818i	$-0.543169\pi$
$571$	−3689.56	−0.270409	−0.135204	+	0.990818i	$0.543169\pi$
$572$	−7472.21	−0.546204
$573$	−12373.2	−0.902090
$574$	0	0
$575$	0	0
$576$	−1102.65	−0.0797634
$577$	22183.9	1.60057	0.800285	−	0.599620i	$-0.204682\pi$
$577$	22183.9	1.60057	0.800285	+	0.599620i	$0.204682\pi$
$578$	12406.2	0.892783
$579$	17667.6	1.26812
$580$	0	0
$581$	0	0
$582$	2525.21	0.179851
$583$	−30905.8	−2.19552
$584$	5747.55	0.407252
$585$	0	0
$586$	11896.7	0.838646
$587$	−10605.3	−0.745705	−0.372852	−	0.927891i	$-0.621620\pi$
$587$	−10605.3	−0.745705	−0.372852	+	0.927891i	$0.621620\pi$
$588$	0	0
$589$	693.149	0.0484902
$590$	0	0
$591$	−9762.83	−0.679508
$592$	650.466	0.0451588
$593$	−6277.25	−0.434698	−0.217349	−	0.976094i	$-0.569741\pi$
$593$	−6277.25	−0.434698	−0.217349	+	0.976094i	$0.569741\pi$
$594$	15541.4	1.07352
$595$	0	0
$596$	3776.39	0.259542
$597$	−20323.4	−1.39327
$598$	−12058.1	−0.824567
$599$	−9970.73	−0.680122	−0.340061	−	0.940403i	$-0.610448\pi$
$599$	−9970.73	−0.680122	−0.340061	+	0.940403i	$0.610448\pi$
$600$	0	0
$601$	−24619.2	−1.67094	−0.835472	−	0.549533i	$-0.814806\pi$
$601$	−24619.2	−1.67094	−0.835472	+	0.549533i	$0.814806\pi$
$602$	0	0
$603$	1104.79	0.0746113
$604$	5228.53	0.352228
$605$	0	0
$606$	17077.3	1.14475
$607$	−11252.9	−0.752460	−0.376230	−	0.926526i	$-0.622780\pi$
$607$	−11252.9	−0.752460	−0.376230	+	0.926526i	$0.622780\pi$
$608$	416.876	0.0278068
$609$	0	0
$610$	0	0
$611$	−7216.69	−0.477833
$612$	−533.668	−0.0352487
$613$	15293.2	1.00765	0.503824	−	0.863807i	$-0.331926\pi$
$613$	15293.2	1.00765	0.503824	+	0.863807i	$0.331926\pi$
$614$	−4871.79	−0.320211
$615$	0	0
$616$	0	0
$617$	17589.4	1.14769	0.573843	−	0.818966i	$-0.305452\pi$
$617$	17589.4	1.14769	0.573843	+	0.818966i	$0.305452\pi$
$618$	−9393.80	−0.611446
$619$	−23467.4	−1.52380	−0.761900	−	0.647694i	$-0.775733\pi$
$619$	−23467.4	−1.52380	−0.761900	+	0.647694i	$0.775733\pi$
$620$	0	0
$621$	−12305.5	−0.795173
$622$	−13353.7	−0.860829
$623$	0	0
$624$	11039.8	0.708249
$625$	0	0
$626$	3150.64	0.201158
$627$	−848.413	−0.0540388
$628$	107.068	0.00680330
$629$	1830.93	0.116063
$630$	0	0
$631$	−6040.86	−0.381114	−0.190557	−	0.981676i	$-0.561029\pi$
$631$	−6040.86	−0.381114	−0.190557	+	0.981676i	$0.561029\pi$
$632$	26485.3	1.66698
$633$	21626.7	1.35795
$634$	−11905.0	−0.745755
$635$	0	0
$636$	−8794.99	−0.548340
$637$	0	0
$638$	3236.68	0.200849
$639$	1480.53	0.0916569
$640$	0	0
$641$	25111.6	1.54735	0.773673	−	0.633586i	$-0.218418\pi$
$641$	25111.6	1.54735	0.773673	+	0.633586i	$0.218418\pi$
$642$	5104.22	0.313781
$643$	−3095.03	−0.189823	−0.0949113	−	0.995486i	$-0.530257\pi$
$643$	−3095.03	−0.189823	−0.0949113	+	0.995486i	$0.530257\pi$
$644$	0	0
$645$	0	0
$646$	−860.944	−0.0524356
$647$	−9178.63	−0.557727	−0.278863	−	0.960331i	$-0.589958\pi$
$647$	−9178.63	−0.557727	−0.278863	+	0.960331i	$0.589958\pi$
$648$	16528.9	1.00203
$649$	2652.69	0.160443
$650$	0	0
$651$	0	0
$652$	−10411.8	−0.625397
$653$	14438.4	0.865265	0.432632	−	0.901570i	$-0.357585\pi$
$653$	14438.4	0.865265	0.432632	+	0.901570i	$0.357585\pi$
$654$	22079.6	1.32015
$655$	0	0
$656$	−17339.8	−1.03202
$657$	466.650	0.0277104
$658$	0	0
$659$	−2900.64	−0.171461	−0.0857305	−	0.996318i	$-0.527322\pi$
$659$	−2900.64	−0.171461	−0.0857305	+	0.996318i	$0.527322\pi$
$660$	0	0
$661$	9976.52	0.587053	0.293526	−	0.955951i	$-0.405171\pi$
$661$	9976.52	0.587053	0.293526	+	0.955951i	$0.405171\pi$
$662$	4229.46	0.248312
$663$	31074.9	1.82028
$664$	16832.5	0.983777
$665$	0	0
$666$	83.7159	0.00487076
$667$	−2562.77	−0.148772
$668$	8901.19	0.515565
$669$	−4914.85	−0.284034
$670$	0	0
$671$	34190.8	1.96710
$672$	0	0
$673$	−20760.8	−1.18911	−0.594554	−	0.804055i	$-0.702671\pi$
$673$	−20760.8	−1.18911	−0.594554	+	0.804055i	$0.702671\pi$
$674$	−356.725	−0.0203866
$675$	0	0
$676$	−4120.18	−0.234421
$677$	−3209.13	−0.182181	−0.0910907	−	0.995843i	$-0.529035\pi$
$677$	−3209.13	−0.182181	−0.0910907	+	0.995843i	$0.529035\pi$
$678$	11148.4	0.631492
$679$	0	0
$680$	0	0
$681$	−8304.82	−0.467315
$682$	20257.2	1.13737
$683$	−4333.57	−0.242781	−0.121391	−	0.992605i	$-0.538735\pi$
$683$	−4333.57	−0.242781	−0.121391	+	0.992605i	$0.538735\pi$
$684$	19.3149	0.00107971
$685$	0	0
$686$	0	0
$687$	2871.64	0.159476
$688$	−3520.78	−0.195100
$689$	−40969.8	−2.26535
$690$	0	0
$691$	−14446.0	−0.795297	−0.397649	−	0.917538i	$-0.630174\pi$
$691$	−14446.0	−0.795297	−0.397649	+	0.917538i	$0.630174\pi$
$692$	8443.94	0.463859
$693$	0	0
$694$	−10099.1	−0.552385
$695$	0	0
$696$	3719.37	0.202561
$697$	−48808.1	−2.65242
$698$	3912.77	0.212179
$699$	−11582.4	−0.626733
$700$	0	0
$701$	−859.801	−0.0463256	−0.0231628	−	0.999732i	$-0.507374\pi$
$701$	−859.801	−0.0463256	−0.0231628	+	0.999732i	$0.507374\pi$
$702$	20602.2	1.10767
$703$	−66.2663	−0.00355517
$704$	25507.9	1.36557
$705$	0	0
$706$	9084.31	0.484267
$707$	0	0
$708$	754.887	0.0400712
$709$	−7979.13	−0.422655	−0.211327	−	0.977415i	$-0.567779\pi$
$709$	−7979.13	−0.422655	−0.211327	+	0.977415i	$0.567779\pi$
$710$	0	0
$711$	2150.37	0.113425
$712$	34001.7	1.78970
$713$	−16039.4	−0.842467
$714$	0	0
$715$	0	0
$716$	−3798.86	−0.198282
$717$	18296.6	0.952995
$718$	−6643.76	−0.345325
$719$	−33703.3	−1.74815	−0.874076	−	0.485790i	$-0.838532\pi$
$719$	−33703.3	−1.74815	−0.874076	+	0.485790i	$0.838532\pi$
$720$	0	0
$721$	0	0
$722$	−15858.6	−0.817444
$723$	−12231.7	−0.629185
$724$	2392.78	0.122827
$725$	0	0
$726$	−9377.57	−0.479386
$727$	30277.0	1.54458	0.772290	−	0.635270i	$-0.219111\pi$
$727$	30277.0	1.54458	0.772290	+	0.635270i	$0.219111\pi$
$728$	0	0
$729$	20917.0	1.06269
$730$	0	0
$731$	−9910.27	−0.501429
$732$	9729.82	0.491291
$733$	−19363.9	−0.975749	−0.487874	−	0.872914i	$-0.662228\pi$
$733$	−19363.9	−0.975749	−0.487874	+	0.872914i	$0.662228\pi$
$734$	−26648.1	−1.34005
$735$	0	0
$736$	−9646.45	−0.483115
$737$	−25557.4	−1.27737
$738$	−2231.66	−0.111313
$739$	24952.4	1.24207	0.621035	−	0.783783i	$-0.286713\pi$
$739$	24952.4	1.24207	0.621035	+	0.783783i	$0.286713\pi$
$740$	0	0
$741$	−1124.69	−0.0557576
$742$	0	0
$743$	8154.54	0.402640	0.201320	−	0.979526i	$-0.435477\pi$
$743$	8154.54	0.402640	0.201320	+	0.979526i	$0.435477\pi$
$744$	23278.1	1.14707
$745$	0	0
$746$	−11783.3	−0.578310
$747$	1366.65	0.0669386
$748$	12345.5	0.603469
$749$	0	0
$750$	0	0
$751$	−4311.26	−0.209481	−0.104740	−	0.994500i	$-0.533401\pi$
$751$	−4311.26	−0.209481	−0.104740	+	0.994500i	$0.533401\pi$
$752$	4235.94	0.205411
$753$	14546.4	0.703987
$754$	4290.66	0.207237
$755$	0	0
$756$	0	0
$757$	−3624.79	−0.174036	−0.0870179	−	0.996207i	$-0.527734\pi$
$757$	−3624.79	−0.174036	−0.0870179	+	0.996207i	$0.527734\pi$
$758$	2211.98	0.105993
$759$	19632.1	0.938869
$760$	0	0
$761$	20576.4	0.980150	0.490075	−	0.871680i	$-0.336969\pi$
$761$	20576.4	0.980150	0.490075	+	0.871680i	$0.336969\pi$
$762$	−18784.8	−0.893045
$763$	0	0
$764$	−6516.34	−0.308577
$765$	0	0
$766$	−7144.26	−0.336988
$767$	3516.50	0.165546
$768$	17791.9	0.835949
$769$	−3066.14	−0.143781	−0.0718907	−	0.997413i	$-0.522903\pi$
$769$	−3066.14	−0.143781	−0.0718907	+	0.997413i	$0.522903\pi$
$770$	0	0
$771$	843.400	0.0393960
$772$	9304.64	0.433784
$773$	19387.0	0.902074	0.451037	−	0.892505i	$-0.351054\pi$
$773$	19387.0	0.902074	0.451037	+	0.892505i	$0.351054\pi$
$774$	−453.129	−0.0210432
$775$	0	0
$776$	5370.23	0.248428
$777$	0	0
$778$	−14666.9	−0.675879
$779$	1766.50	0.0812470
$780$	0	0
$781$	−34249.4	−1.56919
$782$	19922.1	0.911015
$783$	4378.71	0.199850
$784$	0	0
$785$	0	0
$786$	15988.4	0.725556
$787$	−43363.4	−1.96409	−0.982044	−	0.188651i	$-0.939588\pi$
$787$	−43363.4	−1.96409	−0.982044	+	0.188651i	$0.939588\pi$
$788$	−5141.59	−0.232439
$789$	−16222.4	−0.731979
$790$	0	0
$791$	0	0
$792$	2279.39	0.102266
$793$	45324.6	2.02966
$794$	28108.2	1.25632
$795$	0	0
$796$	−10703.3	−0.476593
$797$	−17132.6	−0.761439	−0.380720	−	0.924691i	$-0.624324\pi$
$797$	−17132.6	−0.761439	−0.380720	+	0.924691i	$0.624324\pi$
$798$	0	0
$799$	11923.3	0.527929
$800$	0	0
$801$	2760.63	0.121776
$802$	−627.039	−0.0276079
$803$	−10795.1	−0.474411
$804$	−7272.98	−0.319028
$805$	0	0
$806$	26853.6	1.17355
$807$	14243.3	0.621297
$808$	36317.5	1.58124
$809$	−1080.49	−0.0469566	−0.0234783	−	0.999724i	$-0.507474\pi$
$809$	−1080.49	−0.0469566	−0.0234783	+	0.999724i	$0.507474\pi$
$810$	0	0
$811$	19593.9	0.848378	0.424189	−	0.905574i	$-0.360559\pi$
$811$	19593.9	0.848378	0.424189	+	0.905574i	$0.360559\pi$
$812$	0	0
$813$	14254.9	0.614933
$814$	−1936.62	−0.0833889
$815$	0	0
$816$	−18239.8	−0.782503
$817$	358.680	0.0153594
$818$	9312.66	0.398056
$819$	0	0
$820$	0	0
$821$	5123.80	0.217810	0.108905	−	0.994052i	$-0.465266\pi$
$821$	5123.80	0.217810	0.108905	+	0.994052i	$0.465266\pi$
$822$	−22583.1	−0.958244
$823$	13184.1	0.558405	0.279202	−	0.960232i	$-0.409930\pi$
$823$	13184.1	0.558405	0.279202	+	0.960232i	$0.409930\pi$
$824$	−19977.3	−0.844591
$825$	0	0
$826$	0	0
$827$	−24658.7	−1.03684	−0.518421	−	0.855126i	$-0.673480\pi$
$827$	−24658.7	−1.03684	−0.518421	+	0.855126i	$0.673480\pi$
$828$	−446.944	−0.0187589
$829$	−28562.3	−1.19664	−0.598318	−	0.801259i	$-0.704164\pi$
$829$	−28562.3	−1.19664	−0.598318	+	0.801259i	$0.704164\pi$
$830$	0	0
$831$	11493.1	0.479775
$832$	33814.1	1.40901
$833$	0	0
$834$	−32432.7	−1.34659
$835$	0	0
$836$	−446.817	−0.0184850
$837$	27404.6	1.13171
$838$	5646.27	0.232753
$839$	31106.0	1.27997	0.639987	−	0.768386i	$-0.278940\pi$
$839$	31106.0	1.27997	0.639987	+	0.768386i	$0.278940\pi$
$840$	0	0
$841$	−23477.1	−0.962609
$842$	−11007.1	−0.450511
$843$	−30546.0	−1.24800
$844$	11389.7	0.464514
$845$	0	0
$846$	545.171	0.0221553
$847$	0	0
$848$	24047.8	0.973827
$849$	29274.7	1.18340
$850$	0	0
$851$	1533.39	0.0617674
$852$	−9746.50	−0.391913
$853$	20567.9	0.825596	0.412798	−	0.910823i	$-0.364552\pi$
$853$	20567.9	0.825596	0.412798	+	0.910823i	$0.364552\pi$
$854$	0	0
$855$	0	0
$856$	10854.9	0.433426
$857$	6459.44	0.257468	0.128734	−	0.991679i	$-0.458909\pi$
$857$	6459.44	0.257468	0.128734	+	0.991679i	$0.458909\pi$
$858$	−32868.7	−1.30783
$859$	48214.4	1.91508	0.957541	−	0.288298i	$-0.0930895\pi$
$859$	48214.4	1.91508	0.957541	+	0.288298i	$0.0930895\pi$
$860$	0	0
$861$	0	0
$862$	18359.7	0.725445
$863$	31709.5	1.25076	0.625378	−	0.780322i	$-0.284945\pi$
$863$	31709.5	1.25076	0.625378	+	0.780322i	$0.284945\pi$
$864$	16481.8	0.648984
$865$	0	0
$866$	26611.7	1.04423
$867$	−26776.4	−1.04887
$868$	0	0
$869$	−49745.1	−1.94187
$870$	0	0
$871$	−33879.8	−1.31800
$872$	46955.6	1.82353
$873$	436.015	0.0169036
$874$	−721.037	−0.0279055
$875$	0	0
$876$	−3072.01	−0.118486
$877$	−25654.8	−0.987799	−0.493900	−	0.869519i	$-0.664429\pi$
$877$	−25654.8	−0.987799	−0.493900	+	0.869519i	$0.664429\pi$
$878$	−21192.3	−0.814585
$879$	−25676.7	−0.985272
$880$	0	0
$881$	11470.4	0.438647	0.219323	−	0.975652i	$-0.429615\pi$
$881$	11470.4	0.438647	0.219323	+	0.975652i	$0.429615\pi$
$882$	0	0
$883$	−39124.0	−1.49108	−0.745542	−	0.666459i	$-0.767809\pi$
$883$	−39124.0	−1.49108	−0.745542	+	0.666459i	$0.767809\pi$
$884$	16365.6	0.622663
$885$	0	0
$886$	−4319.01	−0.163770
$887$	15585.8	0.589987	0.294994	−	0.955499i	$-0.404682\pi$
$887$	15585.8	0.589987	0.294994	+	0.955499i	$0.404682\pi$
$888$	−2225.43	−0.0840997
$889$	0	0
$890$	0	0
$891$	−31044.8	−1.16727
$892$	−2588.40	−0.0971594
$893$	−431.537	−0.0161711
$894$	16611.6	0.621447
$895$	0	0
$896$	0	0
$897$	26025.1	0.968731
$898$	10403.7	0.386610
$899$	5707.34	0.211736
$900$	0	0
$901$	67689.6	2.50285
$902$	51625.6	1.90570
$903$	0	0
$904$	23708.8	0.872280
$905$	0	0
$906$	22999.2	0.843376
$907$	−27596.1	−1.01027	−0.505134	−	0.863041i	$-0.668557\pi$
$907$	−27596.1	−1.01027	−0.505134	+	0.863041i	$0.668557\pi$
$908$	−4373.73	−0.159854
$909$	2948.65	0.107592
$910$	0	0
$911$	−14396.2	−0.523565	−0.261782	−	0.965127i	$-0.584310\pi$
$911$	−14396.2	−0.523565	−0.261782	+	0.965127i	$0.584310\pi$
$912$	660.151	0.0239691
$913$	−31615.1	−1.14601
$914$	33229.2	1.20254
$915$	0	0
$916$	1512.35	0.0545517
$917$	0	0
$918$	−34038.7	−1.22379
$919$	10279.6	0.368978	0.184489	−	0.982835i	$-0.440937\pi$
$919$	10279.6	0.368978	0.184489	+	0.982835i	$0.440937\pi$
$920$	0	0
$921$	10514.8	0.376196
$922$	−33727.0	−1.20471
$923$	−45402.2	−1.61910
$924$	0	0
$925$	0	0
$926$	4192.06	0.148769
$927$	−1621.98	−0.0574680
$928$	3432.53	0.121421
$929$	6499.87	0.229552	0.114776	−	0.993391i	$-0.463385\pi$
$929$	6499.87	0.229552	0.114776	+	0.993391i	$0.463385\pi$
$930$	0	0
$931$	0	0
$932$	−6099.86	−0.214386
$933$	28821.5	1.01133
$934$	−13857.2	−0.485463
$935$	0	0
$936$	3021.64	0.105518
$937$	10269.8	0.358056	0.179028	−	0.983844i	$-0.442705\pi$
$937$	10269.8	0.358056	0.179028	+	0.983844i	$0.442705\pi$
$938$	0	0
$939$	−6800.06	−0.236328
$940$	0	0
$941$	34396.2	1.19159	0.595794	−	0.803137i	$-0.296837\pi$
$941$	34396.2	1.19159	0.595794	+	0.803137i	$0.296837\pi$
$942$	470.969	0.0162898
$943$	−40876.5	−1.41158
$944$	−2064.06	−0.0711647
$945$	0	0
$946$	10482.4	0.360265
$947$	−27192.1	−0.933078	−0.466539	−	0.884501i	$-0.654499\pi$
$947$	−27192.1	−0.933078	−0.466539	+	0.884501i	$0.654499\pi$
$948$	−14156.2	−0.484990
$949$	−14310.4	−0.489500
$950$	0	0
$951$	25694.7	0.876140
$952$	0	0
$953$	−49965.2	−1.69836	−0.849178	−	0.528107i	$-0.822902\pi$
$953$	−49965.2	−1.69836	−0.849178	+	0.528107i	$0.822902\pi$
$954$	3094.99	0.105036
$955$	0	0
$956$	9635.88	0.325990
$957$	−6985.78	−0.235965
$958$	26704.9	0.900622
$959$	0	0
$960$	0	0
$961$	5929.05	0.199022
$962$	−2567.25	−0.0860411
$963$	881.320	0.0294913
$964$	−6441.80	−0.215225
$965$	0	0
$966$	0	0
$967$	16755.5	0.557208	0.278604	−	0.960406i	$-0.410128\pi$
$967$	16755.5	0.557208	0.278604	+	0.960406i	$0.410128\pi$
$968$	−19942.8	−0.662176
$969$	1858.19	0.0616033
$970$	0	0
$971$	−37617.4	−1.24325	−0.621626	−	0.783314i	$-0.713528\pi$
$971$	−37617.4	−1.24325	−0.621626	+	0.783314i	$0.713528\pi$
$972$	1474.62	0.0486610
$973$	0	0
$974$	−36583.0	−1.20348
$975$	0	0
$976$	−26603.9	−0.872511
$977$	27690.9	0.906767	0.453384	−	0.891315i	$-0.350217\pi$
$977$	27690.9	0.906767	0.453384	+	0.891315i	$0.350217\pi$
$978$	−45799.5	−1.49745
$979$	−63862.5	−2.08483
$980$	0	0
$981$	3812.38	0.124077
$982$	30266.4	0.983544
$983$	22754.2	0.738299	0.369149	−	0.929370i	$-0.379649\pi$
$983$	22754.2	0.738299	0.369149	+	0.929370i	$0.379649\pi$
$984$	59324.6	1.92195
$985$	0	0
$986$	−7088.96	−0.228964
$987$	0	0
$988$	−592.316	−0.0190729
$989$	−8299.80	−0.266854
$990$	0	0
$991$	−55470.9	−1.77809	−0.889047	−	0.457816i	$-0.848632\pi$
$991$	−55470.9	−1.77809	−0.889047	+	0.457816i	$0.848632\pi$
$992$	21482.9	0.687583
$993$	−9128.50	−0.291726
$994$	0	0
$995$	0	0
$996$	−8996.83	−0.286220
$997$	15181.9	0.482264	0.241132	−	0.970492i	$-0.422481\pi$
$997$	15181.9	0.482264	0.241132	+	0.970492i	$0.422481\pi$
$998$	−46647.1	−1.47955
$999$	−2619.93	−0.0829740

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1225.4.a.p.1.2		2
5.4	even	2		245.4.a.j.1.1	yes	2
7.6	odd	2		1225.4.a.q.1.2		2
15.14	odd	2		2205.4.a.w.1.2		2
35.4	even	6		245.4.e.j.226.2		4
35.9	even	6		245.4.e.j.116.2		4
35.19	odd	6		245.4.e.k.116.2		4
35.24	odd	6		245.4.e.k.226.2		4
35.34	odd	2		245.4.a.i.1.1	✓	2
105.104	even	2		2205.4.a.x.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.4.a.i.1.1	✓	2	35.34	odd	2
245.4.a.j.1.1	yes	2	5.4	even	2
245.4.e.j.116.2		4	35.9	even	6
245.4.e.j.226.2		4	35.4	even	6
245.4.e.k.116.2		4	35.19	odd	6
245.4.e.k.226.2		4	35.24	odd	6
1225.4.a.p.1.2		2	1.1	even	1	trivial
1225.4.a.q.1.2		2	7.6	odd	2
2205.4.a.w.1.2		2	15.14	odd	2
2205.4.a.x.1.2		2	105.104	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:	\( N \)	\(=\)	\( 1225 = 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1225.a (trivial)

\(f(q)\)	\(=\)	\(q+2.31662 q^{2} -5.00000 q^{3} -2.63325 q^{4} -11.5831 q^{6} -24.6332 q^{8} -2.00000 q^{9} +O(q^{10})\)	\(q+2.31662 q^{2} -5.00000 q^{3} -2.63325 q^{4} -11.5831 q^{6} -24.6332 q^{8} -2.00000 q^{9} +46.2665 q^{11} +13.1662 q^{12} +61.3325 q^{13} -36.0000 q^{16} -101.332 q^{17} -4.63325 q^{18} +3.66750 q^{19} +107.182 q^{22} -84.8655 q^{23} +123.166 q^{24} +142.084 q^{26} +145.000 q^{27} +30.1980 q^{29} +188.997 q^{31} +113.668 q^{32} -231.332 q^{33} -234.749 q^{34} +5.26650 q^{36} -18.0685 q^{37} +8.49623 q^{38} -306.662 q^{39} +481.662 q^{41} +97.7995 q^{43} -121.831 q^{44} -196.602 q^{46} -117.665 q^{47} +180.000 q^{48} +506.662 q^{51} -161.504 q^{52} -667.995 q^{53} +335.911 q^{54} -18.3375 q^{57} +69.9574 q^{58} +57.3350 q^{59} +738.997 q^{61} +437.836 q^{62} +551.325 q^{64} -535.911 q^{66} -552.396 q^{67} +266.834 q^{68} +424.327 q^{69} -740.264 q^{71} +49.2665 q^{72} -233.325 q^{73} -41.8580 q^{74} -9.65745 q^{76} -710.422 q^{78} -1075.19 q^{79} -671.000 q^{81} +1115.83 q^{82} -683.325 q^{83} +226.565 q^{86} -150.990 q^{87} -1139.69 q^{88} -1380.32 q^{89} +223.472 q^{92} -944.987 q^{93} -272.586 q^{94} -568.338 q^{96} -218.008 q^{97} -92.5330 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 10 q^{3} + 8 q^{4} + 10 q^{6} - 36 q^{8} - 4 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 - 10 * q^3 + 8 * q^4 + 10 * q^6 - 36 * q^8 - 4 * q^9	\( 2 q - 2 q^{2} - 10 q^{3} + 8 q^{4} + 10 q^{6} - 36 q^{8} - 4 q^{9} + 66 q^{11} - 40 q^{12} - 10 q^{13} - 72 q^{16} - 70 q^{17} + 4 q^{18} + 140 q^{19} + 22 q^{22} + 16 q^{23} + 180 q^{24} + 450 q^{26} + 290 q^{27} - 258 q^{29} - 20 q^{31} + 360 q^{32} - 330 q^{33} - 370 q^{34} - 16 q^{36} - 328 q^{37} - 580 q^{38} + 50 q^{39} + 300 q^{41} + 116 q^{43} + 88 q^{44} - 632 q^{46} + 30 q^{47} + 360 q^{48} + 350 q^{51} - 920 q^{52} - 540 q^{53} - 290 q^{54} - 700 q^{57} + 1314 q^{58} + 380 q^{59} + 1080 q^{61} + 1340 q^{62} - 224 q^{64} - 110 q^{66} - 468 q^{67} + 600 q^{68} - 80 q^{69} - 1056 q^{71} + 72 q^{72} + 860 q^{73} + 1296 q^{74} + 1440 q^{76} - 2250 q^{78} + 158 q^{79} - 1342 q^{81} + 1900 q^{82} - 40 q^{83} + 148 q^{86} + 1290 q^{87} - 1364 q^{88} - 240 q^{89} + 1296 q^{92} + 100 q^{93} - 910 q^{94} - 1800 q^{96} - 1630 q^{97} - 132 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 10 * q^3 + 8 * q^4 + 10 * q^6 - 36 * q^8 - 4 * q^9 + 66 * q^11 - 40 * q^12 - 10 * q^13 - 72 * q^16 - 70 * q^17 + 4 * q^18 + 140 * q^19 + 22 * q^22 + 16 * q^23 + 180 * q^24 + 450 * q^26 + 290 * q^27 - 258 * q^29 - 20 * q^31 + 360 * q^32 - 330 * q^33 - 370 * q^34 - 16 * q^36 - 328 * q^37 - 580 * q^38 + 50 * q^39 + 300 * q^41 + 116 * q^43 + 88 * q^44 - 632 * q^46 + 30 * q^47 + 360 * q^48 + 350 * q^51 - 920 * q^52 - 540 * q^53 - 290 * q^54 - 700 * q^57 + 1314 * q^58 + 380 * q^59 + 1080 * q^61 + 1340 * q^62 - 224 * q^64 - 110 * q^66 - 468 * q^67 + 600 * q^68 - 80 * q^69 - 1056 * q^71 + 72 * q^72 + 860 * q^73 + 1296 * q^74 + 1440 * q^76 - 2250 * q^78 + 158 * q^79 - 1342 * q^81 + 1900 * q^82 - 40 * q^83 + 148 * q^86 + 1290 * q^87 - 1364 * q^88 - 240 * q^89 + 1296 * q^92 + 100 * q^93 - 910 * q^94 - 1800 * q^96 - 1630 * q^97 - 132 * q^99

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