LMFDB - Embedded newform 2254.4.a.l.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2254,4,Mod(1,2254)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2254, 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("2254.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2254 = 2 \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2254.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:	$132.990305153$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 92x^{3} - 28x^{2} + 1593x - 1782 $ x^5 - 92x^3 - 28x^2 + 1593*x - 1782
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 322)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-6.17693$ of defining polynomial
Character	$\chi$	$=$	2254.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.00000 q^{2} -7.17693 q^{3} +4.00000 q^{4} -18.7050 q^{5} -14.3539 q^{6} +8.00000 q^{8} +24.5083 q^{9} -37.4099 q^{10} -36.8949 q^{11} -28.7077 q^{12} -22.7900 q^{13} +134.244 q^{15} +16.0000 q^{16} -66.6941 q^{17} +49.0166 q^{18} +106.301 q^{19} -74.8199 q^{20} -73.7897 q^{22} +23.0000 q^{23} -57.4154 q^{24} +224.876 q^{25} -45.5800 q^{26} +17.8829 q^{27} -199.773 q^{29} +268.488 q^{30} +262.034 q^{31} +32.0000 q^{32} +264.792 q^{33} -133.388 q^{34} +98.0331 q^{36} +202.174 q^{37} +212.601 q^{38} +163.562 q^{39} -149.640 q^{40} -44.4679 q^{41} -18.2027 q^{43} -147.579 q^{44} -458.426 q^{45} +46.0000 q^{46} +306.525 q^{47} -114.831 q^{48} +449.751 q^{50} +478.659 q^{51} -91.1600 q^{52} +300.565 q^{53} +35.7658 q^{54} +690.117 q^{55} -762.913 q^{57} -399.547 q^{58} -255.485 q^{59} +536.977 q^{60} -598.006 q^{61} +524.068 q^{62} +64.0000 q^{64} +426.286 q^{65} +529.584 q^{66} -636.773 q^{67} -266.776 q^{68} -165.069 q^{69} +320.916 q^{71} +196.066 q^{72} +24.5948 q^{73} +404.348 q^{74} -1613.92 q^{75} +425.203 q^{76} +327.124 q^{78} +1121.21 q^{79} -299.279 q^{80} -790.068 q^{81} -88.9358 q^{82} +767.685 q^{83} +1247.51 q^{85} -36.4055 q^{86} +1433.76 q^{87} -295.159 q^{88} +807.439 q^{89} -916.853 q^{90} +92.0000 q^{92} -1880.60 q^{93} +613.051 q^{94} -1988.35 q^{95} -229.662 q^{96} -1170.29 q^{97} -904.230 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 10 q^{2} - 5 q^{3} + 20 q^{4} - 22 q^{5} - 10 q^{6} + 40 q^{8} + 54 q^{9} - 44 q^{10} + 42 q^{11} - 20 q^{12} - 107 q^{13} + 122 q^{15} + 80 q^{16} - 218 q^{17} + 108 q^{18} - 194 q^{19} - 88 q^{20}+ \cdots - 2616 q^{99}+O(q^{100}) $ 5 * q + 10 * q^2 - 5 * q^3 + 20 * q^4 - 22 * q^5 - 10 * q^6 + 40 * q^8 + 54 * q^9 - 44 * q^10 + 42 * q^11 - 20 * q^12 - 107 * q^13 + 122 * q^15 + 80 * q^16 - 218 * q^17 + 108 * q^18 - 194 * q^19 - 88 * q^20 + 84 * q^22 + 115 * q^23 - 40 * q^24 + 219 * q^25 - 214 * q^26 - 203 * q^27 + 33 * q^29 + 244 * q^30 - 19 * q^31 + 160 * q^32 - 240 * q^33 - 436 * q^34 + 216 * q^36 + 214 * q^37 - 388 * q^38 - 243 * q^39 - 176 * q^40 - 395 * q^41 - 284 * q^43 + 168 * q^44 - 470 * q^45 + 230 * q^46 + 493 * q^47 - 80 * q^48 + 438 * q^50 - 244 * q^51 - 428 * q^52 - 70 * q^53 - 406 * q^54 + 186 * q^55 + 22 * q^57 + 66 * q^58 - 226 * q^59 + 488 * q^60 - 1606 * q^61 - 38 * q^62 + 320 * q^64 + 362 * q^65 - 480 * q^66 + 298 * q^67 - 872 * q^68 - 115 * q^69 - 969 * q^71 + 432 * q^72 - 697 * q^73 + 428 * q^74 - 1065 * q^75 - 776 * q^76 - 486 * q^78 + 186 * q^79 - 352 * q^80 - 335 * q^81 - 790 * q^82 + 1104 * q^83 - 1344 * q^85 - 568 * q^86 + 1603 * q^87 + 336 * q^88 + 814 * q^89 - 940 * q^90 + 460 * q^92 - 3423 * q^93 + 986 * q^94 - 2812 * q^95 - 160 * q^96 + 256 * q^97 - 2616 * q^99

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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	−7.17693	−1.38120	−0.690600	−	0.723237i	$-0.742654\pi$
$3$	−7.17693	−1.38120	−0.690600	+	0.723237i	$0.742654\pi$
$4$	4.00000	0.500000
$5$	−18.7050	−1.67302	−0.836511	−	0.547949i	$-0.815409\pi$
$5$	−18.7050	−1.67302	−0.836511	+	0.547949i	$0.815409\pi$
$6$	−14.3539	−0.976656
$7$	0	0
$7$	0	0
$8$	8.00000	0.353553
$9$	24.5083	0.907714
$10$	−37.4099	−1.18301
$11$	−36.8949	−1.01129	−0.505646	−	0.862741i	$-0.668746\pi$
$11$	−36.8949	−1.01129	−0.505646	+	0.862741i	$0.668746\pi$
$12$	−28.7077	−0.690600
$13$	−22.7900	−0.486216	−0.243108	−	0.969999i	$-0.578167\pi$
$13$	−22.7900	−0.486216	−0.243108	+	0.969999i	$0.578167\pi$
$14$	0	0
$15$	134.244	2.31078
$16$	16.0000	0.250000
$17$	−66.6941	−0.951511	−0.475756	−	0.879577i	$-0.657825\pi$
$17$	−66.6941	−0.951511	−0.475756	+	0.879577i	$0.657825\pi$
$18$	49.0166	0.641851
$19$	106.301	1.28353	0.641765	−	0.766901i	$-0.278203\pi$
$19$	106.301	1.28353	0.641765	+	0.766901i	$0.278203\pi$
$20$	−74.8199	−0.836511
$21$	0	0
$22$	−73.7897	−0.715092
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	−57.4154	−0.488328
$25$	224.876	1.79901
$26$	−45.5800	−0.343807
$27$	17.8829	0.127465
$28$	0	0
$29$	−199.773	−1.27921	−0.639603	−	0.768705i	$-0.720901\pi$
$29$	−199.773	−1.27921	−0.639603	+	0.768705i	$0.720901\pi$
$30$	268.488	1.63397
$31$	262.034	1.51815	0.759075	−	0.651003i	$-0.225651\pi$
$31$	262.034	1.51815	0.759075	+	0.651003i	$0.225651\pi$
$32$	32.0000	0.176777
$33$	264.792	1.39680
$34$	−133.388	−0.672820
$35$	0	0
$36$	98.0331	0.453857
$37$	202.174	0.898302	0.449151	−	0.893456i	$-0.351726\pi$
$37$	202.174	0.898302	0.449151	+	0.893456i	$0.351726\pi$
$38$	212.601	0.907593
$39$	163.562	0.671562
$40$	−149.640	−0.591503
$41$	−44.4679	−0.169383	−0.0846917	−	0.996407i	$-0.526991\pi$
$41$	−44.4679	−0.169383	−0.0846917	+	0.996407i	$0.526991\pi$
$42$	0	0
$43$	−18.2027	−0.0645556	−0.0322778	−	0.999479i	$-0.510276\pi$
$43$	−18.2027	−0.0645556	−0.0322778	+	0.999479i	$0.510276\pi$
$44$	−147.579	−0.505646
$45$	−458.426	−1.51863
$46$	46.0000	0.147442
$47$	306.525	0.951305	0.475652	−	0.879633i	$-0.342212\pi$
$47$	306.525	0.951305	0.475652	+	0.879633i	$0.342212\pi$
$48$	−114.831	−0.345300
$49$	0	0
$50$	449.751	1.27209
$51$	478.659	1.31423
$52$	−91.1600	−0.243108
$53$	300.565	0.778977	0.389488	−	0.921031i	$-0.372652\pi$
$53$	300.565	0.778977	0.389488	+	0.921031i	$0.372652\pi$
$54$	35.7658	0.0901317
$55$	690.117	1.69192
$56$	0	0
$57$	−762.913	−1.77281
$58$	−399.547	−0.904535
$59$	−255.485	−0.563750	−0.281875	−	0.959451i	$-0.590956\pi$
$59$	−255.485	−0.563750	−0.281875	+	0.959451i	$0.590956\pi$
$60$	536.977	1.15539
$61$	−598.006	−1.25519	−0.627597	−	0.778538i	$-0.715961\pi$
$61$	−598.006	−1.25519	−0.627597	+	0.778538i	$0.715961\pi$
$62$	524.068	1.07349
$63$	0	0
$64$	64.0000	0.125000
$65$	426.286	0.813451
$66$	529.584	0.987685
$67$	−636.773	−1.16111	−0.580553	−	0.814222i	$-0.697164\pi$
$67$	−636.773	−1.16111	−0.580553	+	0.814222i	$0.697164\pi$
$68$	−266.776	−0.475756
$69$	−165.069	−0.288000
$70$	0	0
$71$	320.916	0.536418	0.268209	−	0.963361i	$-0.413568\pi$
$71$	320.916	0.536418	0.268209	+	0.963361i	$0.413568\pi$
$72$	196.066	0.320925
$73$	24.5948	0.0394330	0.0197165	−	0.999806i	$-0.493724\pi$
$73$	24.5948	0.0394330	0.0197165	+	0.999806i	$0.493724\pi$
$74$	404.348	0.635196
$75$	−1613.92	−2.48479
$76$	425.203	0.641765
$77$	0	0
$78$	327.124	0.474866
$79$	1121.21	1.59678	0.798390	−	0.602141i	$-0.205685\pi$
$79$	1121.21	1.59678	0.798390	+	0.602141i	$0.205685\pi$
$80$	−299.279	−0.418256
$81$	−790.068	−1.08377
$82$	−88.9358	−0.119772
$83$	767.685	1.01523	0.507617	−	0.861583i	$-0.330527\pi$
$83$	767.685	1.01523	0.507617	+	0.861583i	$0.330527\pi$
$84$	0	0
$85$	1247.51	1.59190
$86$	−36.4055	−0.0456477
$87$	1433.76	1.76684
$88$	−295.159	−0.357546
$89$	807.439	0.961667	0.480833	−	0.876812i	$-0.340334\pi$
$89$	807.439	0.961667	0.480833	+	0.876812i	$0.340334\pi$
$90$	−916.853	−1.07383
$91$	0	0
$92$	92.0000	0.104257
$93$	−1880.60	−2.09687
$94$	613.051	0.672674
$95$	−1988.35	−2.14737
$96$	−229.662	−0.244164
$97$	−1170.29	−1.22500	−0.612500	−	0.790471i	$-0.709836\pi$
$97$	−1170.29	−1.22500	−0.612500	+	0.790471i	$0.709836\pi$
$98$	0	0
$99$	−904.230	−0.917965
$100$	899.503	0.899503
$101$	554.631	0.546414	0.273207	−	0.961955i	$-0.411916\pi$
$101$	554.631	0.546414	0.273207	+	0.961955i	$0.411916\pi$
$102$	957.317	0.929299
$103$	1018.33	0.974166	0.487083	−	0.873356i	$-0.338061\pi$
$103$	1018.33	0.974166	0.487083	+	0.873356i	$0.338061\pi$
$104$	−182.320	−0.171903
$105$	0	0
$106$	601.130	0.550820
$107$	1451.75	1.31164	0.655821	−	0.754916i	$-0.272322\pi$
$107$	1451.75	1.31164	0.655821	+	0.754916i	$0.272322\pi$
$108$	71.5317	0.0637327
$109$	1719.61	1.51109	0.755544	−	0.655098i	$-0.227372\pi$
$109$	1719.61	1.51109	0.755544	+	0.655098i	$0.227372\pi$
$110$	1380.23	1.19637
$111$	−1450.99	−1.24074
$112$	0	0
$113$	−2162.47	−1.80025	−0.900126	−	0.435630i	$-0.856525\pi$
$113$	−2162.47	−1.80025	−0.900126	+	0.435630i	$0.856525\pi$
$114$	−1525.83	−1.25357
$115$	−430.214	−0.348849
$116$	−799.093	−0.639603
$117$	−558.544	−0.441345
$118$	−510.969	−0.398631
$119$	0	0
$120$	1073.95	0.816984
$121$	30.2313	0.0227132
$122$	−1196.01	−0.887556
$123$	319.143	0.233952
$124$	1048.14	0.759075
$125$	−1868.17	−1.33675
$126$	0	0
$127$	1462.69	1.02199	0.510994	−	0.859584i	$-0.329278\pi$
$127$	1462.69	1.02199	0.510994	+	0.859584i	$0.329278\pi$
$128$	128.000	0.0883883
$129$	130.640	0.0891642
$130$	852.573	0.575197
$131$	−1970.02	−1.31390	−0.656952	−	0.753932i	$-0.728155\pi$
$131$	−1970.02	−1.31390	−0.656952	+	0.753932i	$0.728155\pi$
$132$	1059.17	0.698399
$133$	0	0
$134$	−1273.55	−0.821027
$135$	−334.499	−0.213253
$136$	−533.553	−0.336410
$137$	−2064.12	−1.28722	−0.643611	−	0.765353i	$-0.722565\pi$
$137$	−2064.12	−1.28722	−0.643611	+	0.765353i	$0.722565\pi$
$138$	−330.139	−0.203647
$139$	−9.19537	−0.00561108	−0.00280554	−	0.999996i	$-0.500893\pi$
$139$	−9.19537	−0.00561108	−0.00280554	+	0.999996i	$0.500893\pi$
$140$	0	0
$141$	−2199.91	−1.31394
$142$	641.831	0.379305
$143$	840.834	0.491707
$144$	392.132	0.226928
$145$	3736.75	2.14014
$146$	49.1897	0.0278833
$147$	0	0
$148$	808.696	0.449151
$149$	−619.044	−0.340363	−0.170181	−	0.985413i	$-0.554435\pi$
$149$	−619.044	−0.340363	−0.170181	+	0.985413i	$0.554435\pi$
$150$	−3227.83	−1.75701
$151$	162.870	0.0877758	0.0438879	−	0.999036i	$-0.486026\pi$
$151$	162.870	0.0877758	0.0438879	+	0.999036i	$0.486026\pi$
$152$	850.406	0.453796
$153$	−1634.56	−0.863700
$154$	0	0
$155$	−4901.33	−2.53990
$156$	654.249	0.335781
$157$	1320.45	0.671234	0.335617	−	0.941999i	$-0.391055\pi$
$157$	1320.45	0.671234	0.335617	+	0.941999i	$0.391055\pi$
$158$	2242.41	1.12909
$159$	−2157.13	−1.07592
$160$	−598.559	−0.295751
$161$	0	0
$162$	−1580.14	−0.766341
$163$	−2193.25	−1.05392	−0.526960	−	0.849890i	$-0.676668\pi$
$163$	−2193.25	−1.05392	−0.526960	+	0.849890i	$0.676668\pi$
$164$	−177.872	−0.0846917
$165$	−4952.92	−2.33687
$166$	1535.37	0.717878
$167$	−2117.18	−0.981030	−0.490515	−	0.871433i	$-0.663191\pi$
$167$	−2117.18	−0.981030	−0.490515	+	0.871433i	$0.663191\pi$
$168$	0	0
$169$	−1677.62	−0.763594
$170$	2495.02	1.12564
$171$	2605.25	1.16508
$172$	−72.8109	−0.0322778
$173$	2576.00	1.13208	0.566040	−	0.824378i	$-0.308475\pi$
$173$	2576.00	1.13208	0.566040	+	0.824378i	$0.308475\pi$
$174$	2867.52	1.24934
$175$	0	0
$176$	−590.318	−0.252823
$177$	1833.59	0.778652
$178$	1614.88	0.680001
$179$	−3008.14	−1.25608	−0.628042	−	0.778179i	$-0.716144\pi$
$179$	−3008.14	−1.25608	−0.628042	+	0.778179i	$0.716144\pi$
$180$	−1833.71	−0.759313
$181$	−202.379	−0.0831090	−0.0415545	−	0.999136i	$-0.513231\pi$
$181$	−202.379	−0.0831090	−0.0415545	+	0.999136i	$0.513231\pi$
$182$	0	0
$183$	4291.85	1.73367
$184$	184.000	0.0737210
$185$	−3781.66	−1.50288
$186$	−3761.19	−1.48271
$187$	2460.67	0.962257
$188$	1226.10	0.475652
$189$	0	0
$190$	−3976.70	−1.51842
$191$	4260.13	1.61389	0.806943	−	0.590629i	$-0.201120\pi$
$191$	4260.13	1.61389	0.806943	+	0.590629i	$0.201120\pi$
$192$	−459.323	−0.172650
$193$	2071.29	0.772510	0.386255	−	0.922392i	$-0.373768\pi$
$193$	2071.29	0.772510	0.386255	+	0.922392i	$0.373768\pi$
$194$	−2340.58	−0.866206
$195$	−3059.43	−1.12354
$196$	0	0
$197$	−4143.04	−1.49837	−0.749187	−	0.662359i	$-0.769555\pi$
$197$	−4143.04	−1.49837	−0.749187	+	0.662359i	$0.769555\pi$
$198$	−1808.46	−0.649099
$199$	−1113.48	−0.396647	−0.198323	−	0.980137i	$-0.563550\pi$
$199$	−1113.48	−0.396647	−0.198323	+	0.980137i	$0.563550\pi$
$200$	1799.01	0.636045
$201$	4570.07	1.60372
$202$	1109.26	0.386373
$203$	0	0
$204$	1914.63	0.657114
$205$	831.770	0.283382
$206$	2036.66	0.688840
$207$	563.690	0.189271
$208$	−364.640	−0.121554
$209$	−3921.95	−1.29802
$210$	0	0
$211$	1213.02	0.395772	0.197886	−	0.980225i	$-0.436592\pi$
$211$	1213.02	0.395772	0.197886	+	0.980225i	$0.436592\pi$
$212$	1202.26	0.389488
$213$	−2303.19	−0.740901
$214$	2903.50	0.927471
$215$	340.481	0.108003
$216$	143.063	0.0450659
$217$	0	0
$218$	3439.22	1.06850
$219$	−176.515	−0.0544649
$220$	2760.47	0.845958
$221$	1519.96	0.462640
$222$	−2901.97	−0.877332
$223$	3263.55	0.980016	0.490008	−	0.871718i	$-0.336994\pi$
$223$	3263.55	0.980016	0.490008	+	0.871718i	$0.336994\pi$
$224$	0	0
$225$	5511.32	1.63298
$226$	−4324.95	−1.27297
$227$	−6282.29	−1.83687	−0.918436	−	0.395569i	$-0.870547\pi$
$227$	−6282.29	−1.83687	−0.918436	+	0.395569i	$0.870547\pi$
$228$	−3051.65	−0.886406
$229$	2034.66	0.587136	0.293568	−	0.955938i	$-0.405157\pi$
$229$	2034.66	0.587136	0.293568	+	0.955938i	$0.405157\pi$
$230$	−860.428	−0.246674
$231$	0	0
$232$	−1598.19	−0.452268
$233$	1687.13	0.474368	0.237184	−	0.971465i	$-0.423776\pi$
$233$	1687.13	0.474368	0.237184	+	0.971465i	$0.423776\pi$
$234$	−1117.09	−0.312078
$235$	−5733.55	−1.59155
$236$	−1021.94	−0.281875
$237$	−8046.82	−2.20547
$238$	0	0
$239$	−6511.95	−1.76244	−0.881220	−	0.472707i	$-0.843277\pi$
$239$	−6511.95	−1.76244	−0.881220	+	0.472707i	$0.843277\pi$
$240$	2147.91	0.577695
$241$	−3163.26	−0.845492	−0.422746	−	0.906248i	$-0.638934\pi$
$241$	−3163.26	−0.845492	−0.422746	+	0.906248i	$0.638934\pi$
$242$	60.4625	0.0160607
$243$	5187.42	1.36944
$244$	−2392.02	−0.627597
$245$	0	0
$246$	638.286	0.165429
$247$	−2422.60	−0.624073
$248$	2096.27	0.536747
$249$	−5509.62	−1.40224
$250$	−3736.34	−0.945228
$251$	−518.424	−0.130369	−0.0651845	−	0.997873i	$-0.520764\pi$
$251$	−518.424	−0.130369	−0.0651845	+	0.997873i	$0.520764\pi$
$252$	0	0
$253$	−848.582	−0.210869
$254$	2925.37	0.722654
$255$	−8953.29	−2.19873
$256$	256.000	0.0625000
$257$	−5445.71	−1.32177	−0.660884	−	0.750488i	$-0.729818\pi$
$257$	−5445.71	−1.32177	−0.660884	+	0.750488i	$0.729818\pi$
$258$	261.279	0.0630486
$259$	0	0
$260$	1705.15	0.406726
$261$	−4896.10	−1.16115
$262$	−3940.04	−0.929071
$263$	5094.98	1.19456	0.597281	−	0.802032i	$-0.296248\pi$
$263$	5094.98	1.19456	0.597281	+	0.802032i	$0.296248\pi$
$264$	2118.33	0.493843
$265$	−5622.05	−1.30325
$266$	0	0
$267$	−5794.93	−1.32825
$268$	−2547.09	−0.580553
$269$	−7694.70	−1.74407	−0.872034	−	0.489446i	$-0.837199\pi$
$269$	−7694.70	−1.74407	−0.872034	+	0.489446i	$0.837199\pi$
$270$	−668.999	−0.150792
$271$	898.571	0.201418	0.100709	−	0.994916i	$-0.467889\pi$
$271$	898.571	0.201418	0.100709	+	0.994916i	$0.467889\pi$
$272$	−1067.11	−0.237878
$273$	0	0
$274$	−4128.24	−0.910204
$275$	−8296.76	−1.81932
$276$	−660.277	−0.144000
$277$	−3660.31	−0.793959	−0.396979	−	0.917828i	$-0.629942\pi$
$277$	−3660.31	−0.793959	−0.396979	+	0.917828i	$0.629942\pi$
$278$	−18.3907	−0.00396764
$279$	6422.00	1.37805
$280$	0	0
$281$	3392.36	0.720183	0.360092	−	0.932917i	$-0.382745\pi$
$281$	3392.36	0.720183	0.360092	+	0.932917i	$0.382745\pi$
$282$	−4399.82	−0.929097
$283$	−8150.68	−1.71204	−0.856021	−	0.516942i	$-0.827070\pi$
$283$	−8150.68	−1.71204	−0.856021	+	0.516942i	$0.827070\pi$
$284$	1283.66	0.268209
$285$	14270.3	2.96595
$286$	1681.67	0.347689
$287$	0	0
$288$	784.265	0.160463
$289$	−464.898	−0.0946261
$290$	7473.51	1.51331
$291$	8399.09	1.69197
$292$	98.3794	0.0197165
$293$	−6020.89	−1.20049	−0.600246	−	0.799816i	$-0.704931\pi$
$293$	−6020.89	−1.20049	−0.600246	+	0.799816i	$0.704931\pi$
$294$	0	0
$295$	4778.83	0.943167
$296$	1617.39	0.317598
$297$	−659.788	−0.128905
$298$	−1238.09	−0.240673
$299$	−524.170	−0.101383
$300$	−6455.67	−1.24239
$301$	0	0
$302$	325.739	0.0620669
$303$	−3980.54	−0.754707
$304$	1700.81	0.320882
$305$	11185.7	2.09997
$306$	−3269.11	−0.610728
$307$	−2860.72	−0.531824	−0.265912	−	0.963997i	$-0.585673\pi$
$307$	−2860.72	−0.531824	−0.265912	+	0.963997i	$0.585673\pi$
$308$	0	0
$309$	−7308.49	−1.34552
$310$	−9802.67	−1.79598
$311$	−1380.64	−0.251732	−0.125866	−	0.992047i	$-0.540171\pi$
$311$	−1380.64	−0.251732	−0.125866	+	0.992047i	$0.540171\pi$
$312$	1308.50	0.237433
$313$	−298.973	−0.0539902	−0.0269951	−	0.999636i	$-0.508594\pi$
$313$	−298.973	−0.0539902	−0.0269951	+	0.999636i	$0.508594\pi$
$314$	2640.91	0.474634
$315$	0	0
$316$	4484.83	0.798390
$317$	−2343.58	−0.415232	−0.207616	−	0.978210i	$-0.566570\pi$
$317$	−2343.58	−0.415232	−0.207616	+	0.978210i	$0.566570\pi$
$318$	−4314.26	−0.760792
$319$	7370.61	1.29365
$320$	−1197.12	−0.209128
$321$	−10419.1	−1.81164
$322$	0	0
$323$	−7089.63	−1.22129
$324$	−3160.27	−0.541885
$325$	−5124.92	−0.874706
$326$	−4386.51	−0.745234
$327$	−12341.5	−2.08712
$328$	−355.743	−0.0598861
$329$	0	0
$330$	−9905.84	−1.65242
$331$	1886.83	0.313322	0.156661	−	0.987652i	$-0.449927\pi$
$331$	1886.83	0.313322	0.156661	+	0.987652i	$0.449927\pi$
$332$	3070.74	0.507617
$333$	4954.93	0.815402
$334$	−4234.35	−0.693693
$335$	11910.8	1.94256
$336$	0	0
$337$	10943.8	1.76898	0.884488	−	0.466564i	$-0.154508\pi$
$337$	10943.8	1.76898	0.884488	+	0.466564i	$0.154508\pi$
$338$	−3355.23	−0.539942
$339$	15519.9	2.48651
$340$	4990.04	0.795950
$341$	−9667.70	−1.53529
$342$	5210.50	0.823834
$343$	0	0
$344$	−145.622	−0.0228238
$345$	3087.62	0.481831
$346$	5152.00	0.800501
$347$	10567.0	1.63477	0.817385	−	0.576091i	$-0.195423\pi$
$347$	10567.0	1.63477	0.817385	+	0.576091i	$0.195423\pi$
$348$	5735.03	0.883420
$349$	−1332.92	−0.204440	−0.102220	−	0.994762i	$-0.532595\pi$
$349$	−1332.92	−0.204440	−0.102220	+	0.994762i	$0.532595\pi$
$350$	0	0
$351$	−407.552	−0.0619758
$352$	−1180.64	−0.178773
$353$	3880.43	0.585083	0.292541	−	0.956253i	$-0.405499\pi$
$353$	3880.43	0.585083	0.292541	+	0.956253i	$0.405499\pi$
$354$	3667.19	0.550590
$355$	−6002.72	−0.897440
$356$	3229.76	0.480833
$357$	0	0
$358$	−6016.29	−0.888186
$359$	−12612.1	−1.85415	−0.927073	−	0.374880i	$-0.877684\pi$
$359$	−12612.1	−1.85415	−0.927073	+	0.374880i	$0.877684\pi$
$360$	−3667.41	−0.536915
$361$	4440.85	0.647448
$362$	−404.758	−0.0587669
$363$	−216.968	−0.0313715
$364$	0	0
$365$	−460.046	−0.0659723
$366$	8583.69	1.22589
$367$	5092.93	0.724384	0.362192	−	0.932104i	$-0.382029\pi$
$367$	5092.93	0.724384	0.362192	+	0.932104i	$0.382029\pi$
$368$	368.000	0.0521286
$369$	−1089.83	−0.153752
$370$	−7563.31	−1.06270
$371$	0	0
$372$	−7522.39	−1.04843
$373$	−12578.8	−1.74612	−0.873062	−	0.487609i	$-0.837869\pi$
$373$	−12578.8	−1.74612	−0.873062	+	0.487609i	$0.837869\pi$
$374$	4921.34	0.680418
$375$	13407.7	1.84633
$376$	2452.20	0.336337
$377$	4552.84	0.621971
$378$	0	0
$379$	3831.85	0.519338	0.259669	−	0.965698i	$-0.416387\pi$
$379$	3831.85	0.519338	0.259669	+	0.965698i	$0.416387\pi$
$380$	−7953.41	−1.07369
$381$	−10497.6	−1.41157
$382$	8520.27	1.14119
$383$	7165.67	0.956002	0.478001	−	0.878359i	$-0.341362\pi$
$383$	7165.67	0.956002	0.478001	+	0.878359i	$0.341362\pi$
$384$	−918.647	−0.122082
$385$	0	0
$386$	4142.57	0.546247
$387$	−446.118	−0.0585980
$388$	−4681.16	−0.612500
$389$	10808.2	1.40873	0.704364	−	0.709839i	$-0.251232\pi$
$389$	10808.2	1.40873	0.704364	+	0.709839i	$0.251232\pi$
$390$	−6118.85	−0.794462
$391$	−1533.96	−0.198404
$392$	0	0
$393$	14138.7	1.81477
$394$	−8286.08	−1.05951
$395$	−20972.1	−2.67145
$396$	−3616.92	−0.458982
$397$	12829.0	1.62183	0.810917	−	0.585162i	$-0.198969\pi$
$397$	12829.0	1.62183	0.810917	+	0.585162i	$0.198969\pi$
$398$	−2226.97	−0.280472
$399$	0	0
$400$	3598.01	0.449751
$401$	10168.2	1.26628	0.633139	−	0.774038i	$-0.281766\pi$
$401$	10168.2	1.26628	0.633139	+	0.774038i	$0.281766\pi$
$402$	9140.14	1.13400
$403$	−5971.75	−0.738149
$404$	2218.52	0.273207
$405$	14778.2	1.81317
$406$	0	0
$407$	−7459.18	−0.908447
$408$	3829.27	0.464650
$409$	−1594.36	−0.192753	−0.0963766	−	0.995345i	$-0.530725\pi$
$409$	−1594.36	−0.192753	−0.0963766	+	0.995345i	$0.530725\pi$
$410$	1663.54	0.200382
$411$	14814.0	1.77791
$412$	4073.32	0.487083
$413$	0	0
$414$	1127.38	0.133835
$415$	−14359.5	−1.69851
$416$	−729.280	−0.0859517
$417$	65.9945	0.00775003
$418$	−7843.90	−0.917842
$419$	−8355.45	−0.974201	−0.487101	−	0.873346i	$-0.661945\pi$
$419$	−8355.45	−0.974201	−0.487101	+	0.873346i	$0.661945\pi$
$420$	0	0
$421$	−1727.23	−0.199952	−0.0999762	−	0.994990i	$-0.531877\pi$
$421$	−1727.23	−0.199952	−0.0999762	+	0.994990i	$0.531877\pi$
$422$	2426.04	0.279853
$423$	7512.41	0.863513
$424$	2404.52	0.275410
$425$	−14997.9	−1.71177
$426$	−4606.38	−0.523896
$427$	0	0
$428$	5806.99	0.655821
$429$	−6034.61	−0.679146
$430$	680.963	0.0763696
$431$	5240.54	0.585680	0.292840	−	0.956161i	$-0.405400\pi$
$431$	5240.54	0.585680	0.292840	+	0.956161i	$0.405400\pi$
$432$	286.127	0.0318664
$433$	6074.43	0.674177	0.337088	−	0.941473i	$-0.390558\pi$
$433$	6074.43	0.674177	0.337088	+	0.941473i	$0.390558\pi$
$434$	0	0
$435$	−26818.4	−2.95596
$436$	6878.44	0.755544
$437$	2444.92	0.267634
$438$	−353.031	−0.0385125
$439$	10981.3	1.19388	0.596938	−	0.802288i	$-0.296384\pi$
$439$	10981.3	1.19388	0.596938	+	0.802288i	$0.296384\pi$
$440$	5520.94	0.598183
$441$	0	0
$442$	3039.92	0.327136
$443$	4102.85	0.440028	0.220014	−	0.975497i	$-0.429390\pi$
$443$	4102.85	0.440028	0.220014	+	0.975497i	$0.429390\pi$
$444$	−5803.95	−0.620368
$445$	−15103.1	−1.60889
$446$	6527.10	0.692976
$447$	4442.83	0.470109
$448$	0	0
$449$	16448.6	1.72886	0.864429	−	0.502754i	$-0.167680\pi$
$449$	16448.6	1.72886	0.864429	+	0.502754i	$0.167680\pi$
$450$	11022.6	1.15469
$451$	1640.64	0.171296
$452$	−8649.89	−0.900126
$453$	−1168.90	−0.121236
$454$	−12564.6	−1.29886
$455$	0	0
$456$	−6103.30	−0.626783
$457$	14213.7	1.45490	0.727452	−	0.686159i	$-0.240704\pi$
$457$	14213.7	1.45490	0.727452	+	0.686159i	$0.240704\pi$
$458$	4069.32	0.415168
$459$	−1192.68	−0.121285
$460$	−1720.86	−0.174425
$461$	6087.14	0.614982	0.307491	−	0.951551i	$-0.400511\pi$
$461$	6087.14	0.614982	0.307491	+	0.951551i	$0.400511\pi$
$462$	0	0
$463$	5359.06	0.537919	0.268959	−	0.963151i	$-0.413320\pi$
$463$	5359.06	0.537919	0.268959	+	0.963151i	$0.413320\pi$
$464$	−3196.37	−0.319801
$465$	35176.5	3.50811
$466$	3374.27	0.335429
$467$	17780.9	1.76188	0.880942	−	0.473224i	$-0.156910\pi$
$467$	17780.9	1.76188	0.880942	+	0.473224i	$0.156910\pi$
$468$	−2234.18	−0.220673
$469$	0	0
$470$	−11467.1	−1.12540
$471$	−9476.80	−0.927108
$472$	−2043.88	−0.199316
$473$	671.587	0.0652846
$474$	−16093.6	−1.55950
$475$	23904.5	2.30908
$476$	0	0
$477$	7366.33	0.707088
$478$	−13023.9	−1.24623
$479$	12340.2	1.17711	0.588555	−	0.808457i	$-0.299697\pi$
$479$	12340.2	1.17711	0.588555	+	0.808457i	$0.299697\pi$
$480$	4295.81	0.408492
$481$	−4607.55	−0.436769
$482$	−6326.52	−0.597853
$483$	0	0
$484$	120.925	0.0113566
$485$	21890.2	2.04945
$486$	10374.8	0.968338
$487$	−14345.9	−1.33486	−0.667428	−	0.744674i	$-0.732605\pi$
$487$	−14345.9	−1.33486	−0.667428	+	0.744674i	$0.732605\pi$
$488$	−4784.05	−0.443778
$489$	15740.8	1.45567
$490$	0	0
$491$	−839.846	−0.0771929	−0.0385964	−	0.999255i	$-0.512289\pi$
$491$	−839.846	−0.0771929	−0.0385964	+	0.999255i	$0.512289\pi$
$492$	1276.57	0.116976
$493$	13323.7	1.21718
$494$	−4845.19	−0.441286
$495$	16913.6	1.53578
$496$	4192.54	0.379538
$497$	0	0
$498$	−11019.2	−0.991534
$499$	−10473.4	−0.939587	−0.469794	−	0.882776i	$-0.655672\pi$
$499$	−10473.4	−0.939587	−0.469794	+	0.882776i	$0.655672\pi$
$500$	−7472.69	−0.668377
$501$	15194.8	1.35500
$502$	−1036.85	−0.0921849
$503$	−21797.3	−1.93219	−0.966097	−	0.258180i	$-0.916877\pi$
$503$	−21797.3	−1.93219	−0.966097	+	0.258180i	$0.916877\pi$
$504$	0	0
$505$	−10374.3	−0.914163
$506$	−1697.16	−0.149107
$507$	12040.1	1.05468
$508$	5850.74	0.510994
$509$	−18466.3	−1.60806	−0.804031	−	0.594587i	$-0.797316\pi$
$509$	−18466.3	−1.60806	−0.804031	+	0.594587i	$0.797316\pi$
$510$	−17906.6	−1.55474
$511$	0	0
$512$	512.000	0.0441942
$513$	1900.97	0.163606
$514$	−10891.4	−0.934631
$515$	−19047.8	−1.62980
$516$	522.559	0.0445821
$517$	−11309.2	−0.962048
$518$	0	0
$519$	−18487.8	−1.56363
$520$	3410.29	0.287598
$521$	−970.557	−0.0816139	−0.0408070	−	0.999167i	$-0.512993\pi$
$521$	−970.557	−0.0816139	−0.0408070	+	0.999167i	$0.512993\pi$
$522$	−9792.20	−0.821059
$523$	−2501.73	−0.209164	−0.104582	−	0.994516i	$-0.533350\pi$
$523$	−2501.73	−0.209164	−0.104582	+	0.994516i	$0.533350\pi$
$524$	−7880.08	−0.656952
$525$	0	0
$526$	10190.0	0.844683
$527$	−17476.1	−1.44454
$528$	4236.67	0.349199
$529$	529.000	0.0434783
$530$	−11244.1	−0.921534
$531$	−6261.49	−0.511724
$532$	0	0
$533$	1013.42	0.0823570
$534$	−11589.9	−0.939218
$535$	−27154.9	−2.19441
$536$	−5094.18	−0.410513
$537$	21589.2	1.73490
$538$	−15389.4	−1.23324
$539$	0	0
$540$	−1338.00	−0.106626
$541$	−5519.57	−0.438641	−0.219321	−	0.975653i	$-0.570384\pi$
$541$	−5519.57	−0.438641	−0.219321	+	0.975653i	$0.570384\pi$
$542$	1797.14	0.142424
$543$	1452.46	0.114790
$544$	−2134.21	−0.168205
$545$	−32165.2	−2.52809
$546$	0	0
$547$	−15172.9	−1.18601	−0.593004	−	0.805199i	$-0.702058\pi$
$547$	−15172.9	−1.18601	−0.593004	+	0.805199i	$0.702058\pi$
$548$	−8256.47	−0.643611
$549$	−14656.1	−1.13936
$550$	−16593.5	−1.28645
$551$	−21236.1	−1.64190
$552$	−1320.55	−0.101823
$553$	0	0
$554$	−7320.61	−0.561413
$555$	27140.7	2.07578
$556$	−36.7815	−0.00280554
$557$	−2907.15	−0.221149	−0.110574	−	0.993868i	$-0.535269\pi$
$557$	−2907.15	−0.221149	−0.110574	+	0.993868i	$0.535269\pi$
$558$	12844.0	0.974426
$559$	414.840	0.0313880
$560$	0	0
$561$	−17660.0	−1.32907
$562$	6784.73	0.509246
$563$	−1318.69	−0.0987141	−0.0493570	−	0.998781i	$-0.515717\pi$
$563$	−1318.69	−0.0987141	−0.0493570	+	0.998781i	$0.515717\pi$
$564$	−8799.64	−0.656971
$565$	40449.0	3.01186
$566$	−16301.4	−1.21060
$567$	0	0
$568$	2567.33	0.189652
$569$	−2920.60	−0.215181	−0.107591	−	0.994195i	$-0.534314\pi$
$569$	−2920.60	−0.215181	−0.107591	+	0.994195i	$0.534314\pi$
$570$	28540.5	2.09725
$571$	11200.3	0.820873	0.410436	−	0.911889i	$-0.365376\pi$
$571$	11200.3	0.820873	0.410436	+	0.911889i	$0.365376\pi$
$572$	3363.34	0.245854
$573$	−30574.7	−2.22910
$574$	0	0
$575$	5172.14	0.375119
$576$	1568.53	0.113464
$577$	3416.96	0.246534	0.123267	−	0.992374i	$-0.460663\pi$
$577$	3416.96	0.246534	0.123267	+	0.992374i	$0.460663\pi$
$578$	−929.796	−0.0669108
$579$	−14865.5	−1.06699
$580$	14947.0	1.07007
$581$	0	0
$582$	16798.2	1.19640
$583$	−11089.3	−0.787773
$584$	196.759	0.0139417
$585$	10447.5	0.738381
$586$	−12041.8	−0.848876
$587$	16614.4	1.16823	0.584114	−	0.811671i	$-0.301442\pi$
$587$	16614.4	1.16823	0.584114	+	0.811671i	$0.301442\pi$
$588$	0	0
$589$	27854.4	1.94859
$590$	9557.66	0.666920
$591$	29734.3	2.06955
$592$	3234.78	0.224576
$593$	16559.0	1.14671	0.573354	−	0.819308i	$-0.305642\pi$
$593$	16559.0	1.14671	0.573354	+	0.819308i	$0.305642\pi$
$594$	−1319.58	−0.0911496
$595$	0	0
$596$	−2476.18	−0.170181
$597$	7991.39	0.547849
$598$	−1048.34	−0.0716887
$599$	10995.8	0.750043	0.375021	−	0.927016i	$-0.377635\pi$
$599$	10995.8	0.750043	0.375021	+	0.927016i	$0.377635\pi$
$600$	−12911.3	−0.878505
$601$	−9311.94	−0.632016	−0.316008	−	0.948756i	$-0.602343\pi$
$601$	−9311.94	−0.632016	−0.316008	+	0.948756i	$0.602343\pi$
$602$	0	0
$603$	−15606.2	−1.05395
$604$	651.479	0.0438879
$605$	−565.475	−0.0379997
$606$	−7961.09	−0.533659
$607$	19247.2	1.28702	0.643510	−	0.765438i	$-0.277477\pi$
$607$	19247.2	1.28702	0.643510	+	0.765438i	$0.277477\pi$
$608$	3401.62	0.226898
$609$	0	0
$610$	22371.4	1.48490
$611$	−6985.72	−0.462540
$612$	−6538.23	−0.431850
$613$	−10419.4	−0.686517	−0.343259	−	0.939241i	$-0.611531\pi$
$613$	−10419.4	−0.686517	−0.343259	+	0.939241i	$0.611531\pi$
$614$	−5721.44	−0.376057
$615$	−5969.56	−0.391408
$616$	0	0
$617$	15890.0	1.03681	0.518403	−	0.855137i	$-0.326527\pi$
$617$	15890.0	1.03681	0.518403	+	0.855137i	$0.326527\pi$
$618$	−14617.0	−0.951425
$619$	2325.59	0.151007	0.0755035	−	0.997146i	$-0.475944\pi$
$619$	2325.59	0.151007	0.0755035	+	0.997146i	$0.475944\pi$
$620$	−19605.3	−1.26995
$621$	411.307	0.0265784
$622$	−2761.28	−0.178002
$623$	0	0
$624$	2617.00	0.167891
$625$	6834.62	0.437416
$626$	−597.945	−0.0381768
$627$	28147.6	1.79283
$628$	5281.82	0.335617
$629$	−13483.8	−0.854745
$630$	0	0
$631$	25002.0	1.57736	0.788678	−	0.614806i	$-0.210766\pi$
$631$	25002.0	1.57736	0.788678	+	0.614806i	$0.210766\pi$
$632$	8969.65	0.564547
$633$	−8705.77	−0.546640
$634$	−4687.15	−0.293613
$635$	−27359.5	−1.70981
$636$	−8628.53	−0.537961
$637$	0	0
$638$	14741.2	0.914750
$639$	7865.09	0.486914
$640$	−2394.24	−0.147876
$641$	−23855.1	−1.46992	−0.734960	−	0.678110i	$-0.762799\pi$
$641$	−23855.1	−1.46992	−0.734960	+	0.678110i	$0.762799\pi$
$642$	−20838.2	−1.28102
$643$	−28805.3	−1.76667	−0.883337	−	0.468738i	$-0.844709\pi$
$643$	−28805.3	−1.76667	−0.883337	+	0.468738i	$0.844709\pi$
$644$	0	0
$645$	−2443.61	−0.149174
$646$	−14179.3	−0.863585
$647$	−32498.4	−1.97472	−0.987360	−	0.158492i	$-0.949337\pi$
$647$	−32498.4	−1.97472	−0.987360	+	0.158492i	$0.949337\pi$
$648$	−6320.54	−0.383170
$649$	9426.07	0.570116
$650$	−10249.8	−0.618510
$651$	0	0
$652$	−8773.02	−0.526960
$653$	16341.1	0.979290	0.489645	−	0.871922i	$-0.337126\pi$
$653$	16341.1	0.979290	0.489645	+	0.871922i	$0.337126\pi$
$654$	−24683.0	−1.47581
$655$	36849.2	2.19819
$656$	−711.486	−0.0423458
$657$	602.777	0.0357939
$658$	0	0
$659$	5445.43	0.321887	0.160944	−	0.986964i	$-0.448546\pi$
$659$	5445.43	0.321887	0.160944	+	0.986964i	$0.448546\pi$
$660$	−19811.7	−1.16844
$661$	−17006.6	−1.00073	−0.500363	−	0.865816i	$-0.666800\pi$
$661$	−17006.6	−1.00073	−0.500363	+	0.865816i	$0.666800\pi$
$662$	3773.66	0.221552
$663$	−10908.6	−0.638999
$664$	6141.48	0.358939
$665$	0	0
$666$	9909.87	0.576576
$667$	−4594.79	−0.266733
$668$	−8468.70	−0.490515
$669$	−23422.3	−1.35360
$670$	23821.6	1.37360
$671$	22063.4	1.26937
$672$	0	0
$673$	−20763.5	−1.18926	−0.594631	−	0.803999i	$-0.702702\pi$
$673$	−20763.5	−1.18926	−0.594631	+	0.803999i	$0.702702\pi$
$674$	21887.5	1.25085
$675$	4021.43	0.229311
$676$	−6710.46	−0.381797
$677$	18958.9	1.07629	0.538146	−	0.842851i	$-0.319125\pi$
$677$	18958.9	1.07629	0.538146	+	0.842851i	$0.319125\pi$
$678$	31039.8	1.75823
$679$	0	0
$680$	9980.08	0.562822
$681$	45087.5	2.53709
$682$	−19335.4	−1.08562
$683$	22546.4	1.26312	0.631562	−	0.775325i	$-0.282414\pi$
$683$	22546.4	1.26312	0.631562	+	0.775325i	$0.282414\pi$
$684$	10421.0	0.582539
$685$	38609.3	2.15355
$686$	0	0
$687$	−14602.6	−0.810952
$688$	−291.244	−0.0161389
$689$	−6849.88	−0.378751
$690$	6175.23	0.340706
$691$	−27690.6	−1.52446	−0.762228	−	0.647309i	$-0.775894\pi$
$691$	−27690.6	−1.52446	−0.762228	+	0.647309i	$0.775894\pi$
$692$	10304.0	0.566040
$693$	0	0
$694$	21134.0	1.15596
$695$	171.999	0.00938747
$696$	11470.1	0.624672
$697$	2965.75	0.161170
$698$	−2665.84	−0.144561
$699$	−12108.4	−0.655198
$700$	0	0
$701$	2135.07	0.115036	0.0575182	−	0.998344i	$-0.481681\pi$
$701$	2135.07	0.115036	0.0575182	+	0.998344i	$0.481681\pi$
$702$	−815.104	−0.0438235
$703$	21491.2	1.15300
$704$	−2361.27	−0.126412
$705$	41149.2	2.19826
$706$	7760.85	0.413716
$707$	0	0
$708$	7334.37	0.389326
$709$	−18765.6	−0.994016	−0.497008	−	0.867746i	$-0.665568\pi$
$709$	−18765.6	−0.994016	−0.497008	+	0.867746i	$0.665568\pi$
$710$	−12005.4	−0.634586
$711$	27478.8	1.44942
$712$	6459.51	0.340001
$713$	6026.78	0.316556
$714$	0	0
$715$	−15727.8	−0.822637
$716$	−12032.6	−0.628042
$717$	46735.8	2.43428
$718$	−25224.1	−1.31108
$719$	−24563.0	−1.27405	−0.637027	−	0.770842i	$-0.719836\pi$
$719$	−24563.0	−1.27405	−0.637027	+	0.770842i	$0.719836\pi$
$720$	−7334.82	−0.379657
$721$	0	0
$722$	8881.70	0.457815
$723$	22702.5	1.16779
$724$	−809.517	−0.0415545
$725$	−44924.2	−2.30130
$726$	−433.935	−0.0221830
$727$	−1895.61	−0.0967045	−0.0483522	−	0.998830i	$-0.515397\pi$
$727$	−1895.61	−0.0967045	−0.0483522	+	0.998830i	$0.515397\pi$
$728$	0	0
$729$	−15897.9	−0.807697
$730$	−920.091	−0.0466495
$731$	1214.01	0.0614254
$732$	17167.4	0.866837
$733$	8040.15	0.405143	0.202571	−	0.979268i	$-0.435070\pi$
$733$	8040.15	0.405143	0.202571	+	0.979268i	$0.435070\pi$
$734$	10185.9	0.512217
$735$	0	0
$736$	736.000	0.0368605
$737$	23493.6	1.17422
$738$	−2179.66	−0.108719
$739$	7025.86	0.349730	0.174865	−	0.984592i	$-0.444051\pi$
$739$	7025.86	0.349730	0.174865	+	0.984592i	$0.444051\pi$
$740$	−15126.6	−0.751440
$741$	17386.8	0.861970
$742$	0	0
$743$	35267.7	1.74138	0.870691	−	0.491830i	$-0.163672\pi$
$743$	35267.7	1.74138	0.870691	+	0.491830i	$0.163672\pi$
$744$	−15044.8	−0.741355
$745$	11579.2	0.569435
$746$	−25157.5	−1.23470
$747$	18814.6	0.921542
$748$	9842.68	0.481128
$749$	0	0
$750$	26815.5	1.30555
$751$	−26262.0	−1.27605	−0.638026	−	0.770015i	$-0.720249\pi$
$751$	−26262.0	−1.27605	−0.638026	+	0.770015i	$0.720249\pi$
$752$	4904.41	0.237826
$753$	3720.69	0.180066
$754$	9105.67	0.439800
$755$	−3046.47	−0.146851
$756$	0	0
$757$	20312.0	0.975234	0.487617	−	0.873058i	$-0.337866\pi$
$757$	20312.0	0.975234	0.487617	+	0.873058i	$0.337866\pi$
$758$	7663.70	0.367227
$759$	6090.21	0.291252
$760$	−15906.8	−0.759212
$761$	−5976.37	−0.284682	−0.142341	−	0.989818i	$-0.545463\pi$
$761$	−5976.37	−0.284682	−0.142341	+	0.989818i	$0.545463\pi$
$762$	−20995.2	−0.998130
$763$	0	0
$764$	17040.5	0.806943
$765$	30574.3	1.44499
$766$	14331.3	0.675995
$767$	5822.50	0.274104
$768$	−1837.29	−0.0863250
$769$	−20927.5	−0.981360	−0.490680	−	0.871340i	$-0.663252\pi$
$769$	−20927.5	−0.981360	−0.490680	+	0.871340i	$0.663252\pi$
$770$	0	0
$771$	39083.5	1.82563
$772$	8285.15	0.386255
$773$	−15071.3	−0.701262	−0.350631	−	0.936514i	$-0.614033\pi$
$773$	−15071.3	−0.701262	−0.350631	+	0.936514i	$0.614033\pi$
$774$	−892.235	−0.0414350
$775$	58925.0	2.73116
$776$	−9362.33	−0.433103
$777$	0	0
$778$	21616.3	0.996121
$779$	−4726.97	−0.217409
$780$	−12237.7	−0.561769
$781$	−11840.1	−0.542476
$782$	−3067.93	−0.140293
$783$	−3572.53	−0.163055
$784$	0	0
$785$	−24699.0	−1.12299
$786$	28277.4	1.28323
$787$	9818.43	0.444713	0.222357	−	0.974965i	$-0.428625\pi$
$787$	9818.43	0.444713	0.222357	+	0.974965i	$0.428625\pi$
$788$	−16572.2	−0.749187
$789$	−36566.3	−1.64993
$790$	−41944.3	−1.88900
$791$	0	0
$792$	−7233.84	−0.324550
$793$	13628.6	0.610296
$794$	25658.0	1.14681
$795$	40349.1	1.80004
$796$	−4453.93	−0.198323
$797$	25841.3	1.14849	0.574244	−	0.818684i	$-0.305296\pi$
$797$	25841.3	1.14849	0.574244	+	0.818684i	$0.305296\pi$
$798$	0	0
$799$	−20443.4	−0.905177
$800$	7196.02	0.318022
$801$	19788.9	0.872918
$802$	20336.5	0.895393
$803$	−907.423	−0.0398783
$804$	18280.3	0.801861
$805$	0	0
$806$	−11943.5	−0.521950
$807$	55224.3	2.40891
$808$	4437.05	0.193187
$809$	−44965.3	−1.95414	−0.977068	−	0.212926i	$-0.931701\pi$
$809$	−44965.3	−1.95414	−0.977068	+	0.212926i	$0.931701\pi$
$810$	29556.4	1.28211
$811$	23927.0	1.03599	0.517997	−	0.855382i	$-0.326678\pi$
$811$	23927.0	1.03599	0.517997	+	0.855382i	$0.326678\pi$
$812$	0	0
$813$	−6448.98	−0.278199
$814$	−14918.4	−0.642369
$815$	41024.8	1.76323
$816$	7658.54	0.328557
$817$	−1934.96	−0.0828590
$818$	−3188.72	−0.136297
$819$	0	0
$820$	3327.08	0.141691
$821$	−45775.0	−1.94587	−0.972934	−	0.231082i	$-0.925773\pi$
$821$	−45775.0	−1.94587	−0.972934	+	0.231082i	$0.925773\pi$
$822$	29628.0	1.25717
$823$	187.951	0.00796060	0.00398030	−	0.999992i	$-0.498733\pi$
$823$	187.951	0.00796060	0.00398030	+	0.999992i	$0.498733\pi$
$824$	8146.65	0.344420
$825$	59545.2	2.51285
$826$	0	0
$827$	−11598.2	−0.487676	−0.243838	−	0.969816i	$-0.578406\pi$
$827$	−11598.2	−0.487676	−0.243838	+	0.969816i	$0.578406\pi$
$828$	2254.76	0.0946357
$829$	4238.86	0.177590	0.0887948	−	0.996050i	$-0.471698\pi$
$829$	4238.86	0.177590	0.0887948	+	0.996050i	$0.471698\pi$
$830$	−28719.0	−1.20103
$831$	26269.8	1.09662
$832$	−1458.56	−0.0607770
$833$	0	0
$834$	131.989	0.00548010
$835$	39601.7	1.64129
$836$	−15687.8	−0.649012
$837$	4685.93	0.193512
$838$	−16710.9	−0.688864
$839$	30548.4	1.25703	0.628514	−	0.777798i	$-0.283663\pi$
$839$	30548.4	1.25703	0.628514	+	0.777798i	$0.283663\pi$
$840$	0	0
$841$	15520.4	0.636368
$842$	−3454.46	−0.141388
$843$	−24346.7	−0.994717
$844$	4852.09	0.197886
$845$	31379.7	1.27751
$846$	15024.8	0.610596
$847$	0	0
$848$	4809.04	0.194744
$849$	58496.8	2.36467
$850$	−29995.8	−1.21041
$851$	4650.00	0.187309
$852$	−9212.75	−0.370450
$853$	−7956.01	−0.319354	−0.159677	−	0.987169i	$-0.551045\pi$
$853$	−7956.01	−0.319354	−0.159677	+	0.987169i	$0.551045\pi$
$854$	0	0
$855$	−48731.1	−1.94920
$856$	11614.0	0.463736
$857$	−32534.3	−1.29679	−0.648396	−	0.761303i	$-0.724560\pi$
$857$	−32534.3	−1.29679	−0.648396	+	0.761303i	$0.724560\pi$
$858$	−12069.2	−0.480229
$859$	818.271	0.0325018	0.0162509	−	0.999868i	$-0.494827\pi$
$859$	818.271	0.0325018	0.0162509	+	0.999868i	$0.494827\pi$
$860$	1361.93	0.0540015
$861$	0	0
$862$	10481.1	0.414138
$863$	−20161.0	−0.795238	−0.397619	−	0.917551i	$-0.630163\pi$
$863$	−20161.0	−0.795238	−0.397619	+	0.917551i	$0.630163\pi$
$864$	572.253	0.0225329
$865$	−48184.0	−1.89400
$866$	12148.9	0.476715
$867$	3336.54	0.130698
$868$	0	0
$869$	−41366.8	−1.61481
$870$	−53636.8	−2.09018
$871$	14512.1	0.564549
$872$	13756.9	0.534250
$873$	−28681.8	−1.11195
$874$	4889.83	0.189246
$875$	0	0
$876$	−706.061	−0.0272324
$877$	−33048.9	−1.27250	−0.636250	−	0.771483i	$-0.719515\pi$
$877$	−33048.9	−1.27250	−0.636250	+	0.771483i	$0.719515\pi$
$878$	21962.7	0.844197
$879$	43211.5	1.65812
$880$	11041.9	0.422979
$881$	−15531.2	−0.593938	−0.296969	−	0.954887i	$-0.595976\pi$
$881$	−15531.2	−0.593938	−0.296969	+	0.954887i	$0.595976\pi$
$882$	0	0
$883$	−22079.1	−0.841474	−0.420737	−	0.907183i	$-0.638228\pi$
$883$	−22079.1	−0.841474	−0.420737	+	0.907183i	$0.638228\pi$
$884$	6079.84	0.231320
$885$	−34297.3	−1.30270
$886$	8205.70	0.311147
$887$	−8427.81	−0.319028	−0.159514	−	0.987196i	$-0.550993\pi$
$887$	−8427.81	−0.319028	−0.159514	+	0.987196i	$0.550993\pi$
$888$	−11607.9	−0.438666
$889$	0	0
$890$	−30206.2	−1.13766
$891$	29149.4	1.09601
$892$	13054.2	0.490008
$893$	32583.9	1.22103
$894$	8885.67	0.332417
$895$	56267.2	2.10146
$896$	0	0
$897$	3761.93	0.140030
$898$	32897.2	1.22249
$899$	−52347.4	−1.94203
$900$	22045.3	0.816491
$901$	−20045.9	−0.741205
$902$	3281.27	0.121125
$903$	0	0
$904$	−17299.8	−0.636485
$905$	3785.50	0.139043
$906$	−2337.81	−0.0857268
$907$	−28182.9	−1.03175	−0.515876	−	0.856663i	$-0.672533\pi$
$907$	−28182.9	−1.03175	−0.515876	+	0.856663i	$0.672533\pi$
$908$	−25129.1	−0.918436
$909$	13593.0	0.495988
$910$	0	0
$911$	−48586.1	−1.76699	−0.883496	−	0.468439i	$-0.844816\pi$
$911$	−48586.1	−1.76699	−0.883496	+	0.468439i	$0.844816\pi$
$912$	−12206.6	−0.443203
$913$	−28323.6	−1.02670
$914$	28427.5	1.02877
$915$	−80278.8	−2.90048
$916$	8138.64	0.293568
$917$	0	0
$918$	−2385.37	−0.0857614
$919$	−48873.3	−1.75428	−0.877139	−	0.480236i	$-0.840551\pi$
$919$	−48873.3	−1.75428	−0.877139	+	0.480236i	$0.840551\pi$
$920$	−3441.71	−0.123337
$921$	20531.2	0.734556
$922$	12174.3	0.434858
$923$	−7313.67	−0.260815
$924$	0	0
$925$	45464.0	1.61605
$926$	10718.1	0.380366
$927$	24957.5	0.884264
$928$	−6392.75	−0.226134
$929$	−36236.5	−1.27974	−0.639872	−	0.768482i	$-0.721012\pi$
$929$	−36236.5	−1.27974	−0.639872	+	0.768482i	$0.721012\pi$
$930$	70353.0	2.48061
$931$	0	0
$932$	6748.53	0.237184
$933$	9908.74	0.347693
$934$	35561.7	1.24584
$935$	−46026.7	−1.60988
$936$	−4468.35	−0.156039
$937$	37238.1	1.29831	0.649154	−	0.760657i	$-0.275123\pi$
$937$	37238.1	1.29831	0.649154	+	0.760657i	$0.275123\pi$
$938$	0	0
$939$	2145.70	0.0745713
$940$	−22934.2	−0.795777
$941$	7961.47	0.275809	0.137905	−	0.990446i	$-0.455963\pi$
$941$	7961.47	0.275809	0.137905	+	0.990446i	$0.455963\pi$
$942$	−18953.6	−0.655565
$943$	−1022.76	−0.0353189
$944$	−4087.75	−0.140938
$945$	0	0
$946$	1343.17	0.0461632
$947$	5376.57	0.184493	0.0922465	−	0.995736i	$-0.470595\pi$
$947$	5376.57	0.184493	0.0922465	+	0.995736i	$0.470595\pi$
$948$	−32187.3	−1.10274
$949$	−560.517	−0.0191730
$950$	47808.9	1.63276
$951$	16819.7	0.573518
$952$	0	0
$953$	−19459.6	−0.661448	−0.330724	−	0.943728i	$-0.607293\pi$
$953$	−19459.6	−0.661448	−0.330724	+	0.943728i	$0.607293\pi$
$954$	14732.7	0.499987
$955$	−79685.6	−2.70007
$956$	−26047.8	−0.881220
$957$	−52898.3	−1.78679
$958$	24680.3	0.832343
$959$	0	0
$960$	8591.63	0.288847
$961$	38870.7	1.30478
$962$	−9215.09	−0.308842
$963$	35579.8	1.19060
$964$	−12653.0	−0.422746
$965$	−38743.4	−1.29243
$966$	0	0
$967$	−7286.78	−0.242324	−0.121162	−	0.992633i	$-0.538662\pi$
$967$	−7286.78	−0.242324	−0.121162	+	0.992633i	$0.538662\pi$
$968$	241.850	0.00803033
$969$	50881.8	1.68685
$970$	43780.5	1.44918
$971$	1401.06	0.0463050	0.0231525	−	0.999732i	$-0.492630\pi$
$971$	1401.06	0.0463050	0.0231525	+	0.999732i	$0.492630\pi$
$972$	20749.7	0.684718
$973$	0	0
$974$	−28691.8	−0.943886
$975$	36781.2	1.20814
$976$	−9568.10	−0.313799
$977$	28822.9	0.943833	0.471917	−	0.881643i	$-0.343562\pi$
$977$	28822.9	0.943833	0.471917	+	0.881643i	$0.343562\pi$
$978$	31481.7	1.02932
$979$	−29790.4	−0.972527
$980$	0	0
$981$	42144.6	1.37164
$982$	−1679.69	−0.0545836
$983$	−8699.13	−0.282257	−0.141129	−	0.989991i	$-0.545073\pi$
$983$	−8699.13	−0.282257	−0.141129	+	0.989991i	$0.545073\pi$
$984$	2553.14	0.0827146
$985$	77495.5	2.50681
$986$	26647.4	0.860676
$987$	0	0
$988$	−9690.38	−0.312037
$989$	−418.663	−0.0134608
$990$	33827.2	1.08596
$991$	40870.4	1.31008	0.655041	−	0.755593i	$-0.272651\pi$
$991$	40870.4	1.31008	0.655041	+	0.755593i	$0.272651\pi$
$992$	8385.08	0.268374
$993$	−13541.6	−0.432760
$994$	0	0
$995$	20827.7	0.663599
$996$	−22038.5	−0.701120
$997$	36867.5	1.17112	0.585560	−	0.810629i	$-0.300875\pi$
$997$	36867.5	1.17112	0.585560	+	0.810629i	$0.300875\pi$
$998$	−20946.8	−0.664389
$999$	3615.46	0.114503

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2254.4.a.l.1.2		5
7.6	odd	2		322.4.a.h.1.4	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.4.a.h.1.4	✓	5	7.6	odd	2
2254.4.a.l.1.2		5	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:	\( N \)	\(=\)	\( 2254 = 2 \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2254.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} -7.17693 q^{3} +4.00000 q^{4} -18.7050 q^{5} -14.3539 q^{6} +8.00000 q^{8} +24.5083 q^{9} -37.4099 q^{10} -36.8949 q^{11} -28.7077 q^{12} -22.7900 q^{13} +134.244 q^{15} +16.0000 q^{16} -66.6941 q^{17} +49.0166 q^{18} +106.301 q^{19} -74.8199 q^{20} -73.7897 q^{22} +23.0000 q^{23} -57.4154 q^{24} +224.876 q^{25} -45.5800 q^{26} +17.8829 q^{27} -199.773 q^{29} +268.488 q^{30} +262.034 q^{31} +32.0000 q^{32} +264.792 q^{33} -133.388 q^{34} +98.0331 q^{36} +202.174 q^{37} +212.601 q^{38} +163.562 q^{39} -149.640 q^{40} -44.4679 q^{41} -18.2027 q^{43} -147.579 q^{44} -458.426 q^{45} +46.0000 q^{46} +306.525 q^{47} -114.831 q^{48} +449.751 q^{50} +478.659 q^{51} -91.1600 q^{52} +300.565 q^{53} +35.7658 q^{54} +690.117 q^{55} -762.913 q^{57} -399.547 q^{58} -255.485 q^{59} +536.977 q^{60} -598.006 q^{61} +524.068 q^{62} +64.0000 q^{64} +426.286 q^{65} +529.584 q^{66} -636.773 q^{67} -266.776 q^{68} -165.069 q^{69} +320.916 q^{71} +196.066 q^{72} +24.5948 q^{73} +404.348 q^{74} -1613.92 q^{75} +425.203 q^{76} +327.124 q^{78} +1121.21 q^{79} -299.279 q^{80} -790.068 q^{81} -88.9358 q^{82} +767.685 q^{83} +1247.51 q^{85} -36.4055 q^{86} +1433.76 q^{87} -295.159 q^{88} +807.439 q^{89} -916.853 q^{90} +92.0000 q^{92} -1880.60 q^{93} +613.051 q^{94} -1988.35 q^{95} -229.662 q^{96} -1170.29 q^{97} -904.230 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 10 q^{2} - 5 q^{3} + 20 q^{4} - 22 q^{5} - 10 q^{6} + 40 q^{8} + 54 q^{9} - 44 q^{10} + 42 q^{11} - 20 q^{12} - 107 q^{13} + 122 q^{15} + 80 q^{16} - 218 q^{17} + 108 q^{18} - 194 q^{19} - 88 q^{20}+ \cdots - 2616 q^{99}+O(q^{100}) \) 5 * q + 10 * q^2 - 5 * q^3 + 20 * q^4 - 22 * q^5 - 10 * q^6 + 40 * q^8 + 54 * q^9 - 44 * q^10 + 42 * q^11 - 20 * q^12 - 107 * q^13 + 122 * q^15 + 80 * q^16 - 218 * q^17 + 108 * q^18 - 194 * q^19 - 88 * q^20 + 84 * q^22 + 115 * q^23 - 40 * q^24 + 219 * q^25 - 214 * q^26 - 203 * q^27 + 33 * q^29 + 244 * q^30 - 19 * q^31 + 160 * q^32 - 240 * q^33 - 436 * q^34 + 216 * q^36 + 214 * q^37 - 388 * q^38 - 243 * q^39 - 176 * q^40 - 395 * q^41 - 284 * q^43 + 168 * q^44 - 470 * q^45 + 230 * q^46 + 493 * q^47 - 80 * q^48 + 438 * q^50 - 244 * q^51 - 428 * q^52 - 70 * q^53 - 406 * q^54 + 186 * q^55 + 22 * q^57 + 66 * q^58 - 226 * q^59 + 488 * q^60 - 1606 * q^61 - 38 * q^62 + 320 * q^64 + 362 * q^65 - 480 * q^66 + 298 * q^67 - 872 * q^68 - 115 * q^69 - 969 * q^71 + 432 * q^72 - 697 * q^73 + 428 * q^74 - 1065 * q^75 - 776 * q^76 - 486 * q^78 + 186 * q^79 - 352 * q^80 - 335 * q^81 - 790 * q^82 + 1104 * q^83 - 1344 * q^85 - 568 * q^86 + 1603 * q^87 + 336 * q^88 + 814 * q^89 - 940 * q^90 + 460 * q^92 - 3423 * q^93 + 986 * q^94 - 2812 * q^95 - 160 * q^96 + 256 * q^97 - 2616 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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