LMFDB - Embedded newform 2254.4.a.n.1.3

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:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 38x^{4} + 203x^{2} - 100 $ x^6 - 38x^4 + 203x^2 - 100
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.39944$ 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)

$f(q)$	$=$	\(q+2.00000 q^{2} -3.34803 q^{3} +4.00000 q^{4} -2.71681 q^{5} -6.69606 q^{6} +8.00000 q^{8} -15.7907 q^{9} -5.43361 q^{10} +14.0668 q^{11} -13.3921 q^{12} -44.4558 q^{13} +9.09595 q^{15} +16.0000 q^{16} +107.929 q^{17} -31.5814 q^{18} +34.7498 q^{19} -10.8672 q^{20} +28.1336 q^{22} +23.0000 q^{23} -26.7843 q^{24} -117.619 q^{25} -88.9116 q^{26} +143.265 q^{27} -237.901 q^{29} +18.1919 q^{30} +132.575 q^{31} +32.0000 q^{32} -47.0960 q^{33} +215.859 q^{34} -63.1627 q^{36} +58.0639 q^{37} +69.4996 q^{38} +148.839 q^{39} -21.7344 q^{40} +72.2508 q^{41} -97.4243 q^{43} +56.2671 q^{44} +42.9002 q^{45} +46.0000 q^{46} +258.518 q^{47} -53.5685 q^{48} -235.238 q^{50} -361.351 q^{51} -177.823 q^{52} +46.6039 q^{53} +286.529 q^{54} -38.2167 q^{55} -116.343 q^{57} -475.801 q^{58} +140.566 q^{59} +36.3838 q^{60} +17.3939 q^{61} +265.150 q^{62} +64.0000 q^{64} +120.778 q^{65} -94.1920 q^{66} -685.931 q^{67} +431.717 q^{68} -77.0047 q^{69} +331.606 q^{71} -126.325 q^{72} -943.497 q^{73} +116.128 q^{74} +393.792 q^{75} +138.999 q^{76} +297.679 q^{78} +827.511 q^{79} -43.4689 q^{80} -53.3059 q^{81} +144.502 q^{82} -53.7529 q^{83} -293.223 q^{85} -194.849 q^{86} +796.499 q^{87} +112.534 q^{88} +276.921 q^{89} +85.8004 q^{90} +92.0000 q^{92} -443.866 q^{93} +517.037 q^{94} -94.4084 q^{95} -107.137 q^{96} +971.792 q^{97} -222.124 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 12 q^{2} + 24 q^{4} + 48 q^{8} - 52 q^{9} - 84 q^{11} + 52 q^{15} + 96 q^{16} - 104 q^{18} - 168 q^{22} + 138 q^{23} - 286 q^{25} - 22 q^{29} + 104 q^{30} + 192 q^{32} - 208 q^{36} - 180 q^{37} - 414 q^{39}+ \cdots + 600 q^{99}+O(q^{100}) $ 6 * q + 12 * q^2 + 24 * q^4 + 48 * q^8 - 52 * q^9 - 84 * q^11 + 52 * q^15 + 96 * q^16 - 104 * q^18 - 168 * q^22 + 138 * q^23 - 286 * q^25 - 22 * q^29 + 104 * q^30 + 192 * q^32 - 208 * q^36 - 180 * q^37 - 414 * q^39 - 600 * q^43 - 336 * q^44 + 276 * q^46 - 572 * q^50 + 464 * q^51 - 876 * q^53 - 1452 * q^57 - 44 * q^58 + 208 * q^60 + 384 * q^64 - 1532 * q^65 - 92 * q^67 - 1434 * q^71 - 416 * q^72 - 360 * q^74 - 828 * q^78 - 2820 * q^79 - 2198 * q^81 - 184 * q^85 - 1200 * q^86 - 672 * q^88 + 552 * q^92 - 2742 * q^93 - 2888 * q^95 + 600 * 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
$3$	−3.34803	−0.644329	−0.322165	−	0.946684i	$-0.604410\pi$
$3$	−3.34803	−0.644329	−0.322165	+	0.946684i	$0.604410\pi$
$4$	4.00000	0.500000
$5$	−2.71681	−0.242998	−0.121499	−	0.992592i	$-0.538770\pi$
$5$	−2.71681	−0.242998	−0.121499	+	0.992592i	$0.538770\pi$
$6$	−6.69606	−0.455609
$7$	0	0
$7$	0	0
$8$	8.00000	0.353553
$9$	−15.7907	−0.584840
$10$	−5.43361	−0.171826
$11$	14.0668	0.385572	0.192786	−	0.981241i	$-0.438248\pi$
$11$	14.0668	0.385572	0.192786	+	0.981241i	$0.438248\pi$
$12$	−13.3921	−0.322165
$13$	−44.4558	−0.948448	−0.474224	−	0.880404i	$-0.657271\pi$
$13$	−44.4558	−0.948448	−0.474224	+	0.880404i	$0.657271\pi$
$14$	0	0
$15$	9.09595	0.156571
$16$	16.0000	0.250000
$17$	107.929	1.53981	0.769903	−	0.638161i	$-0.220305\pi$
$17$	107.929	1.53981	0.769903	+	0.638161i	$0.220305\pi$
$18$	−31.5814	−0.413544
$19$	34.7498	0.419587	0.209793	−	0.977746i	$-0.432721\pi$
$19$	34.7498	0.419587	0.209793	+	0.977746i	$0.432721\pi$
$20$	−10.8672	−0.121499
$21$	0	0
$22$	28.1336	0.272641
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	−26.7843	−0.227805
$25$	−117.619	−0.940952
$26$	−88.9116	−0.670654
$27$	143.265	1.02116
$28$	0	0
$29$	−237.901	−1.52335	−0.761673	−	0.647962i	$-0.775622\pi$
$29$	−237.901	−1.52335	−0.761673	+	0.647962i	$0.775622\pi$
$30$	18.1919	0.110712
$31$	132.575	0.768103	0.384052	−	0.923312i	$-0.374528\pi$
$31$	132.575	0.768103	0.384052	+	0.923312i	$0.374528\pi$
$32$	32.0000	0.176777
$33$	−47.0960	−0.248435
$34$	215.859	1.08881
$35$	0	0
$36$	−63.1627	−0.292420
$37$	58.0639	0.257990	0.128995	−	0.991645i	$-0.458825\pi$
$37$	58.0639	0.257990	0.128995	+	0.991645i	$0.458825\pi$
$38$	69.4996	0.296693
$39$	148.839	0.611113
$40$	−21.7344	−0.0859129
$41$	72.2508	0.275212	0.137606	−	0.990487i	$-0.456059\pi$
$41$	72.2508	0.275212	0.137606	+	0.990487i	$0.456059\pi$
$42$	0	0
$43$	−97.4243	−0.345513	−0.172757	−	0.984965i	$-0.555267\pi$
$43$	−97.4243	−0.345513	−0.172757	+	0.984965i	$0.555267\pi$
$44$	56.2671	0.192786
$45$	42.9002	0.142115
$46$	46.0000	0.147442
$47$	258.518	0.802314	0.401157	−	0.916009i	$-0.368608\pi$
$47$	258.518	0.802314	0.401157	+	0.916009i	$0.368608\pi$
$48$	−53.5685	−0.161082
$49$	0	0
$50$	−235.238	−0.665353
$51$	−361.351	−0.992141
$52$	−177.823	−0.474224
$53$	46.6039	0.120784	0.0603918	−	0.998175i	$-0.480765\pi$
$53$	46.6039	0.120784	0.0603918	+	0.998175i	$0.480765\pi$
$54$	286.529	0.722068
$55$	−38.2167	−0.0936934
$56$	0	0
$57$	−116.343	−0.270352
$58$	−475.801	−1.07717
$59$	140.566	0.310172	0.155086	−	0.987901i	$-0.450435\pi$
$59$	140.566	0.310172	0.155086	+	0.987901i	$0.450435\pi$
$60$	36.3838	0.0782855
$61$	17.3939	0.0365093	0.0182546	−	0.999833i	$-0.494189\pi$
$61$	17.3939	0.0365093	0.0182546	+	0.999833i	$0.494189\pi$
$62$	265.150	0.543131
$63$	0	0
$64$	64.0000	0.125000
$65$	120.778	0.230471
$66$	−94.1920	−0.175670
$67$	−685.931	−1.25074	−0.625372	−	0.780327i	$-0.715053\pi$
$67$	−685.931	−1.25074	−0.625372	+	0.780327i	$0.715053\pi$
$68$	431.717	0.769903
$69$	−77.0047	−0.134352
$70$	0	0
$71$	331.606	0.554287	0.277144	−	0.960829i	$-0.410612\pi$
$71$	331.606	0.554287	0.277144	+	0.960829i	$0.410612\pi$
$72$	−126.325	−0.206772
$73$	−943.497	−1.51271	−0.756356	−	0.654160i	$-0.773022\pi$
$73$	−943.497	−1.51271	−0.756356	+	0.654160i	$0.773022\pi$
$74$	116.128	0.182427
$75$	393.792	0.606283
$76$	138.999	0.209793
$77$	0	0
$78$	297.679	0.432122
$79$	827.511	1.17851	0.589255	−	0.807947i	$-0.299421\pi$
$79$	827.511	1.17851	0.589255	+	0.807947i	$0.299421\pi$
$80$	−43.4689	−0.0607496
$81$	−53.3059	−0.0731220
$82$	144.502	0.194604
$83$	−53.7529	−0.0710861	−0.0355430	−	0.999368i	$-0.511316\pi$
$83$	−53.7529	−0.0710861	−0.0355430	+	0.999368i	$0.511316\pi$
$84$	0	0
$85$	−293.223	−0.374170
$86$	−194.849	−0.244315
$87$	796.499	0.981536
$88$	112.534	0.136320
$89$	276.921	0.329815	0.164908	−	0.986309i	$-0.447267\pi$
$89$	276.921	0.329815	0.164908	+	0.986309i	$0.447267\pi$
$90$	85.8004	0.100491
$91$	0	0
$92$	92.0000	0.104257
$93$	−443.866	−0.494911
$94$	517.037	0.567322
$95$	−94.4084	−0.101959
$96$	−107.137	−0.113902
$97$	971.792	1.01722	0.508611	−	0.860996i	$-0.330159\pi$
$97$	971.792	1.01722	0.508611	+	0.860996i	$0.330159\pi$
$98$	0	0
$99$	−222.124	−0.225498
$100$	−470.476	−0.470476
$101$	−1626.51	−1.60241	−0.801205	−	0.598390i	$-0.795807\pi$
$101$	−1626.51	−1.60241	−0.801205	+	0.598390i	$0.795807\pi$
$102$	−722.701	−0.701550
$103$	−1919.35	−1.83611	−0.918055	−	0.396453i	$-0.870241\pi$
$103$	−1919.35	−1.83611	−0.918055	+	0.396453i	$0.870241\pi$
$104$	−355.647	−0.335327
$105$	0	0
$106$	93.2077	0.0854070
$107$	−672.883	−0.607944	−0.303972	−	0.952681i	$-0.598313\pi$
$107$	−672.883	−0.607944	−0.303972	+	0.952681i	$0.598313\pi$
$108$	573.058	0.510579
$109$	−1857.07	−1.63188	−0.815942	−	0.578134i	$-0.803781\pi$
$109$	−1857.07	−1.63188	−0.815942	+	0.578134i	$0.803781\pi$
$110$	−76.4334	−0.0662513
$111$	−194.400	−0.166231
$112$	0	0
$113$	−1327.98	−1.10554	−0.552769	−	0.833335i	$-0.686429\pi$
$113$	−1327.98	−1.10554	−0.552769	+	0.833335i	$0.686429\pi$
$114$	−232.687	−0.191168
$115$	−62.4865	−0.0506687
$116$	−951.602	−0.761673
$117$	701.988	0.554690
$118$	281.132	0.219325
$119$	0	0
$120$	72.7676	0.0553562
$121$	−1133.13	−0.851334
$122$	34.7879	0.0258159
$123$	−241.898	−0.177327
$124$	530.301	0.384052
$125$	659.149	0.471648
$126$	0	0
$127$	1979.97	1.38341	0.691707	−	0.722178i	$-0.256859\pi$
$127$	1979.97	1.38341	0.691707	+	0.722178i	$0.256859\pi$
$128$	128.000	0.0883883
$129$	326.180	0.222624
$130$	241.556	0.162968
$131$	−263.738	−0.175900	−0.0879501	−	0.996125i	$-0.528032\pi$
$131$	−263.738	−0.175900	−0.0879501	+	0.996125i	$0.528032\pi$
$132$	−188.384	−0.124218
$133$	0	0
$134$	−1371.86	−0.884409
$135$	−389.222	−0.248140
$136$	863.434	0.544403
$137$	−1040.20	−0.648687	−0.324344	−	0.945939i	$-0.605143\pi$
$137$	−1040.20	−0.648687	−0.324344	+	0.945939i	$0.605143\pi$
$138$	−154.009	−0.0950011
$139$	394.748	0.240878	0.120439	−	0.992721i	$-0.461570\pi$
$139$	394.748	0.240878	0.120439	+	0.992721i	$0.461570\pi$
$140$	0	0
$141$	−865.528	−0.516954
$142$	663.212	0.391940
$143$	−625.350	−0.365695
$144$	−252.651	−0.146210
$145$	646.330	0.370171
$146$	−1886.99	−1.06965
$147$	0	0
$148$	232.256	0.128995
$149$	−2239.11	−1.23111	−0.615553	−	0.788095i	$-0.711067\pi$
$149$	−2239.11	−1.23111	−0.615553	+	0.788095i	$0.711067\pi$
$150$	787.584	0.428706
$151$	−532.290	−0.286868	−0.143434	−	0.989660i	$-0.545815\pi$
$151$	−532.290	−0.286868	−0.143434	+	0.989660i	$0.545815\pi$
$152$	277.998	0.148346
$153$	−1704.28	−0.900540
$154$	0	0
$155$	−360.181	−0.186648
$156$	595.358	0.305556
$157$	−1936.90	−0.984594	−0.492297	−	0.870427i	$-0.663843\pi$
$157$	−1936.90	−0.984594	−0.492297	+	0.870427i	$0.663843\pi$
$158$	1655.02	0.833332
$159$	−156.031	−0.0778244
$160$	−86.9378	−0.0429565
$161$	0	0
$162$	−106.612	−0.0517051
$163$	−117.333	−0.0563819	−0.0281910	−	0.999603i	$-0.508975\pi$
$163$	−117.333	−0.0563819	−0.0281910	+	0.999603i	$0.508975\pi$
$164$	289.003	0.137606
$165$	127.951	0.0603694
$166$	−107.506	−0.0502654
$167$	2193.72	1.01650	0.508250	−	0.861210i	$-0.330293\pi$
$167$	2193.72	1.01650	0.508250	+	0.861210i	$0.330293\pi$
$168$	0	0
$169$	−220.680	−0.100446
$170$	−586.446	−0.264578
$171$	−548.723	−0.245391
$172$	−389.697	−0.172757
$173$	1423.67	0.625663	0.312832	−	0.949809i	$-0.398722\pi$
$173$	1423.67	0.625663	0.312832	+	0.949809i	$0.398722\pi$
$174$	1593.00	0.694051
$175$	0	0
$176$	225.068	0.0963930
$177$	−470.620	−0.199853
$178$	553.842	0.233215
$179$	−2374.20	−0.991374	−0.495687	−	0.868501i	$-0.665084\pi$
$179$	−2374.20	−0.991374	−0.495687	+	0.868501i	$0.665084\pi$
$180$	171.601	0.0710576
$181$	3991.73	1.63924	0.819622	−	0.572905i	$-0.194184\pi$
$181$	3991.73	1.63924	0.819622	+	0.572905i	$0.194184\pi$
$182$	0	0
$183$	−58.2354	−0.0235240
$184$	184.000	0.0737210
$185$	−157.748	−0.0626913
$186$	−887.732	−0.349955
$187$	1518.22	0.593706
$188$	1034.07	0.401157
$189$	0	0
$190$	−188.817	−0.0720959
$191$	3287.70	1.24550	0.622748	−	0.782422i	$-0.286016\pi$
$191$	3287.70	1.24550	0.622748	+	0.782422i	$0.286016\pi$
$192$	−214.274	−0.0805411
$193$	−3520.56	−1.31303	−0.656517	−	0.754311i	$-0.727971\pi$
$193$	−3520.56	−1.31303	−0.656517	+	0.754311i	$0.727971\pi$
$194$	1943.58	0.719284
$195$	−404.368	−0.148499
$196$	0	0
$197$	3002.85	1.08601	0.543005	−	0.839730i	$-0.317286\pi$
$197$	3002.85	1.08601	0.543005	+	0.839730i	$0.317286\pi$
$198$	−444.248	−0.159451
$199$	220.682	0.0786116	0.0393058	−	0.999227i	$-0.487485\pi$
$199$	220.682	0.0786116	0.0393058	+	0.999227i	$0.487485\pi$
$200$	−940.952	−0.332677
$201$	2296.52	0.805890
$202$	−3253.01	−1.13307
$203$	0	0
$204$	−1445.40	−0.496071
$205$	−196.292	−0.0668761
$206$	−3838.70	−1.29833
$207$	−363.186	−0.121948
$208$	−711.293	−0.237112
$209$	488.818	0.161781
$210$	0	0
$211$	−4453.76	−1.45313	−0.726564	−	0.687099i	$-0.758884\pi$
$211$	−4453.76	−1.45313	−0.726564	+	0.687099i	$0.758884\pi$
$212$	186.415	0.0603918
$213$	−1110.23	−0.357143
$214$	−1345.77	−0.429882
$215$	264.683	0.0839592
$216$	1146.12	0.361034
$217$	0	0
$218$	−3714.15	−1.15392
$219$	3158.86	0.974684
$220$	−152.867	−0.0468467
$221$	−4798.08	−1.46043
$222$	−388.800	−0.117543
$223$	−429.190	−0.128882	−0.0644411	−	0.997922i	$-0.520526\pi$
$223$	−429.190	−0.128882	−0.0644411	+	0.997922i	$0.520526\pi$
$224$	0	0
$225$	1857.28	0.550306
$226$	−2655.96	−0.781733
$227$	2729.41	0.798049	0.399025	−	0.916940i	$-0.369349\pi$
$227$	2729.41	0.798049	0.399025	+	0.916940i	$0.369349\pi$
$228$	−465.374	−0.135176
$229$	−6497.18	−1.87487	−0.937436	−	0.348157i	$-0.886808\pi$
$229$	−6497.18	−1.87487	−0.937436	+	0.348157i	$0.886808\pi$
$230$	−124.973	−0.0358282
$231$	0	0
$232$	−1903.20	−0.538584
$233$	3125.41	0.878767	0.439383	−	0.898300i	$-0.355197\pi$
$233$	3125.41	0.878767	0.439383	+	0.898300i	$0.355197\pi$
$234$	1403.98	0.392225
$235$	−702.344	−0.194961
$236$	562.264	0.155086
$237$	−2770.53	−0.759348
$238$	0	0
$239$	65.9000	0.0178356	0.00891782	−	0.999960i	$-0.497161\pi$
$239$	65.9000	0.0178356	0.00891782	+	0.999960i	$0.497161\pi$
$240$	145.535	0.0391427
$241$	−3954.42	−1.05696	−0.528479	−	0.848947i	$-0.677237\pi$
$241$	−3954.42	−1.05696	−0.528479	+	0.848947i	$0.677237\pi$
$242$	−2266.25	−0.601984
$243$	−3689.67	−0.974044
$244$	69.5757	0.0182546
$245$	0	0
$246$	−483.796	−0.125389
$247$	−1544.83	−0.397956
$248$	1060.60	0.271566
$249$	179.966	0.0458028
$250$	1318.30	0.333506
$251$	−3446.26	−0.866639	−0.433319	−	0.901240i	$-0.642658\pi$
$251$	−3446.26	−0.866639	−0.433319	+	0.901240i	$0.642658\pi$
$252$	0	0
$253$	323.536	0.0803973
$254$	3959.93	0.978221
$255$	981.719	0.241089
$256$	256.000	0.0625000
$257$	3530.64	0.856946	0.428473	−	0.903554i	$-0.359052\pi$
$257$	3530.64	0.856946	0.428473	+	0.903554i	$0.359052\pi$
$258$	652.359	0.157419
$259$	0	0
$260$	483.111	0.115236
$261$	3756.61	0.890914
$262$	−527.477	−0.124380
$263$	−4541.28	−1.06474	−0.532371	−	0.846511i	$-0.678699\pi$
$263$	−4541.28	−1.06474	−0.532371	+	0.846511i	$0.678699\pi$
$264$	−376.768	−0.0878351
$265$	−126.614	−0.0293502
$266$	0	0
$267$	−927.140	−0.212510
$268$	−2743.72	−0.625372
$269$	−1751.64	−0.397024	−0.198512	−	0.980098i	$-0.563611\pi$
$269$	−1751.64	−0.397024	−0.198512	+	0.980098i	$0.563611\pi$
$270$	−778.444	−0.175461
$271$	−7466.78	−1.67371	−0.836854	−	0.547427i	$-0.815608\pi$
$271$	−7466.78	−1.67371	−0.836854	+	0.547427i	$0.815608\pi$
$272$	1726.87	0.384951
$273$	0	0
$274$	−2080.40	−0.458691
$275$	−1654.52	−0.362805
$276$	−308.019	−0.0671759
$277$	−2196.36	−0.476413	−0.238207	−	0.971215i	$-0.576560\pi$
$277$	−2196.36	−0.476413	−0.238207	+	0.971215i	$0.576560\pi$
$278$	789.496	0.170327
$279$	−2093.45	−0.449218
$280$	0	0
$281$	−1306.50	−0.277363	−0.138681	−	0.990337i	$-0.544286\pi$
$281$	−1306.50	−0.277363	−0.138681	+	0.990337i	$0.544286\pi$
$282$	−1731.06	−0.365542
$283$	−89.8988	−0.0188831	−0.00944157	−	0.999955i	$-0.503005\pi$
$283$	−89.8988	−0.0188831	−0.00944157	+	0.999955i	$0.503005\pi$
$284$	1326.42	0.277144
$285$	316.082	0.0656951
$286$	−1250.70	−0.258585
$287$	0	0
$288$	−505.302	−0.103386
$289$	6735.73	1.37100
$290$	1292.66	0.261750
$291$	−3253.59	−0.655426
$292$	−3773.99	−0.756356
$293$	−2114.68	−0.421641	−0.210821	−	0.977525i	$-0.567614\pi$
$293$	−2114.68	−0.421641	−0.210821	+	0.977525i	$0.567614\pi$
$294$	0	0
$295$	−381.891	−0.0753713
$296$	464.511	0.0912134
$297$	2015.27	0.393730
$298$	−4478.22	−0.870524
$299$	−1022.48	−0.197765
$300$	1575.17	0.303141
$301$	0	0
$302$	−1064.58	−0.202847
$303$	5445.59	1.03248
$304$	555.997	0.104897
$305$	−47.2559	−0.00887169
$306$	−3408.55	−0.636778
$307$	−9228.48	−1.71563	−0.857813	−	0.513962i	$-0.828177\pi$
$307$	−9228.48	−1.71563	−0.857813	+	0.513962i	$0.828177\pi$
$308$	0	0
$309$	6426.05	1.18306
$310$	−720.362	−0.131980
$311$	3531.94	0.643980	0.321990	−	0.946743i	$-0.395648\pi$
$311$	3531.94	0.643980	0.321990	+	0.946743i	$0.395648\pi$
$312$	1190.72	0.216061
$313$	−7131.97	−1.28793	−0.643966	−	0.765054i	$-0.722712\pi$
$313$	−7131.97	−1.28793	−0.643966	+	0.765054i	$0.722712\pi$
$314$	−3873.80	−0.696213
$315$	0	0
$316$	3310.04	0.589255
$317$	−7702.64	−1.36474	−0.682372	−	0.731005i	$-0.739051\pi$
$317$	−7702.64	−1.36474	−0.682372	+	0.731005i	$0.739051\pi$
$318$	−312.063	−0.0550302
$319$	−3346.49	−0.587360
$320$	−173.876	−0.0303748
$321$	2252.83	0.391716
$322$	0	0
$323$	3750.52	0.646082
$324$	−213.224	−0.0365610
$325$	5228.85	0.892444
$326$	−234.667	−0.0398680
$327$	6217.54	1.05147
$328$	578.007	0.0973021
$329$	0	0
$330$	255.901	0.0426876
$331$	5114.80	0.849350	0.424675	−	0.905346i	$-0.360388\pi$
$331$	5114.80	0.849350	0.424675	+	0.905346i	$0.360388\pi$
$332$	−215.011	−0.0355430
$333$	−916.869	−0.150883
$334$	4387.45	0.718773
$335$	1863.54	0.303929
$336$	0	0
$337$	−12351.9	−1.99659	−0.998297	−	0.0583313i	$-0.981422\pi$
$337$	−12351.9	−1.99659	−0.998297	+	0.0583313i	$0.981422\pi$
$338$	−441.361	−0.0710263
$339$	4446.12	0.712330
$340$	−1172.89	−0.187085
$341$	1864.91	0.296159
$342$	−1097.45	−0.173518
$343$	0	0
$344$	−779.394	−0.122157
$345$	209.207	0.0326473
$346$	2847.34	0.442411
$347$	−9580.04	−1.48209	−0.741043	−	0.671458i	$-0.765668\pi$
$347$	−9580.04	−1.48209	−0.741043	+	0.671458i	$0.765668\pi$
$348$	3186.00	0.490768
$349$	2084.47	0.319711	0.159856	−	0.987140i	$-0.448897\pi$
$349$	2084.47	0.319711	0.159856	+	0.987140i	$0.448897\pi$
$350$	0	0
$351$	−6368.94	−0.968516
$352$	450.137	0.0681602
$353$	5195.91	0.783430	0.391715	−	0.920087i	$-0.371882\pi$
$353$	5195.91	0.783430	0.391715	+	0.920087i	$0.371882\pi$
$354$	−941.239	−0.141317
$355$	−900.909	−0.134691
$356$	1107.68	0.164908
$357$	0	0
$358$	−4748.40	−0.701007
$359$	2768.61	0.407024	0.203512	−	0.979072i	$-0.434764\pi$
$359$	2768.61	0.407024	0.203512	+	0.979072i	$0.434764\pi$
$360$	343.202	0.0502453
$361$	−5651.45	−0.823947
$362$	7983.46	1.15912
$363$	3793.74	0.548539
$364$	0	0
$365$	2563.30	0.367587
$366$	−116.471	−0.0166340
$367$	−8763.87	−1.24651	−0.623257	−	0.782017i	$-0.714191\pi$
$367$	−8763.87	−1.24651	−0.623257	+	0.782017i	$0.714191\pi$
$368$	368.000	0.0521286
$369$	−1140.89	−0.160955
$370$	−315.497	−0.0443294
$371$	0	0
$372$	−1775.46	−0.247456
$373$	2136.84	0.296626	0.148313	−	0.988940i	$-0.452616\pi$
$373$	2136.84	0.296626	0.148313	+	0.988940i	$0.452616\pi$
$374$	3036.43	0.419814
$375$	−2206.85	−0.303897
$376$	2068.15	0.283661
$377$	10576.1	1.44481
$378$	0	0
$379$	−5065.66	−0.686558	−0.343279	−	0.939233i	$-0.611538\pi$
$379$	−5065.66	−0.686558	−0.343279	+	0.939233i	$0.611538\pi$
$380$	−377.634	−0.0509795
$381$	−6628.99	−0.891374
$382$	6575.41	0.880699
$383$	−10590.5	−1.41292	−0.706458	−	0.707755i	$-0.749708\pi$
$383$	−10590.5	−1.41292	−0.706458	+	0.707755i	$0.749708\pi$
$384$	−428.548	−0.0569512
$385$	0	0
$386$	−7041.12	−0.928455
$387$	1538.40	0.202070
$388$	3887.17	0.508611
$389$	3352.08	0.436909	0.218454	−	0.975847i	$-0.429899\pi$
$389$	3352.08	0.436909	0.218454	+	0.975847i	$0.429899\pi$
$390$	−808.736	−0.105005
$391$	2482.37	0.321072
$392$	0	0
$393$	883.005	0.113338
$394$	6005.69	0.767925
$395$	−2248.19	−0.286376
$396$	−888.496	−0.112749
$397$	5136.45	0.649348	0.324674	−	0.945826i	$-0.394745\pi$
$397$	5136.45	0.649348	0.324674	+	0.945826i	$0.394745\pi$
$398$	441.363	0.0555868
$399$	0	0
$400$	−1881.90	−0.235238
$401$	6489.15	0.808111	0.404056	−	0.914734i	$-0.367600\pi$
$401$	6489.15	0.808111	0.404056	+	0.914734i	$0.367600\pi$
$402$	4593.04	0.569850
$403$	−5893.74	−0.728506
$404$	−6506.02	−0.801205
$405$	144.822	0.0177685
$406$	0	0
$407$	816.772	0.0994739
$408$	−2890.81	−0.350775
$409$	12092.3	1.46192	0.730958	−	0.682422i	$-0.239073\pi$
$409$	12092.3	1.46192	0.730958	+	0.682422i	$0.239073\pi$
$410$	−392.583	−0.0472885
$411$	3482.62	0.417968
$412$	−7677.41	−0.918055
$413$	0	0
$414$	−726.371	−0.0862300
$415$	146.036	0.0172738
$416$	−1422.59	−0.167664
$417$	−1321.63	−0.155205
$418$	977.635	0.114396
$419$	4134.54	0.482066	0.241033	−	0.970517i	$-0.422514\pi$
$419$	4134.54	0.482066	0.241033	+	0.970517i	$0.422514\pi$
$420$	0	0
$421$	1768.64	0.204746	0.102373	−	0.994746i	$-0.467356\pi$
$421$	1768.64	0.204746	0.102373	+	0.994746i	$0.467356\pi$
$422$	−8907.53	−1.02752
$423$	−4082.18	−0.469226
$424$	372.831	0.0427035
$425$	−12694.5	−1.44888
$426$	−2220.45	−0.252538
$427$	0	0
$428$	−2691.53	−0.303972
$429$	2093.69	0.235628
$430$	529.366	0.0593681
$431$	−3528.60	−0.394355	−0.197177	−	0.980368i	$-0.563177\pi$
$431$	−3528.60	−0.394355	−0.197177	+	0.980368i	$0.563177\pi$
$432$	2292.23	0.255290
$433$	−10991.6	−1.21991	−0.609955	−	0.792436i	$-0.708813\pi$
$433$	−10991.6	−1.21991	−0.609955	+	0.792436i	$0.708813\pi$
$434$	0	0
$435$	−2163.93	−0.238512
$436$	−7428.29	−0.815942
$437$	799.245	0.0874899
$438$	6317.72	0.689206
$439$	9488.77	1.03160	0.515802	−	0.856708i	$-0.327494\pi$
$439$	9488.77	1.03160	0.515802	+	0.856708i	$0.327494\pi$
$440$	−305.734	−0.0331256
$441$	0	0
$442$	−9596.17	−1.03268
$443$	9092.32	0.975145	0.487573	−	0.873082i	$-0.337882\pi$
$443$	9092.32	0.975145	0.487573	+	0.873082i	$0.337882\pi$
$444$	−777.599	−0.0831154
$445$	−752.340	−0.0801446
$446$	−858.381	−0.0911334
$447$	7496.61	0.793238
$448$	0	0
$449$	−7127.43	−0.749141	−0.374570	−	0.927198i	$-0.622210\pi$
$449$	−7127.43	−0.749141	−0.374570	+	0.927198i	$0.622210\pi$
$450$	3714.57	0.389125
$451$	1016.34	0.106114
$452$	−5311.92	−0.552769
$453$	1782.12	0.184838
$454$	5458.82	0.564306
$455$	0	0
$456$	−930.747	−0.0955839
$457$	1286.52	0.131687	0.0658436	−	0.997830i	$-0.479026\pi$
$457$	1286.52	0.131687	0.0658436	+	0.997830i	$0.479026\pi$
$458$	−12994.4	−1.32573
$459$	15462.4	1.57239
$460$	−249.946	−0.0253343
$461$	−14471.7	−1.46207	−0.731037	−	0.682338i	$-0.760963\pi$
$461$	−14471.7	−1.46207	−0.731037	+	0.682338i	$0.760963\pi$
$462$	0	0
$463$	−5418.87	−0.543923	−0.271961	−	0.962308i	$-0.587672\pi$
$463$	−5418.87	−0.543923	−0.271961	+	0.962308i	$0.587672\pi$
$464$	−3806.41	−0.380837
$465$	1205.90	0.120263
$466$	6250.83	0.621382
$467$	1421.79	0.140884	0.0704418	−	0.997516i	$-0.477559\pi$
$467$	1421.79	0.140884	0.0704418	+	0.997516i	$0.477559\pi$
$468$	2807.95	0.277345
$469$	0	0
$470$	−1404.69	−0.137858
$471$	6484.80	0.634403
$472$	1124.53	0.109662
$473$	−1370.45	−0.133220
$474$	−5541.07	−0.536940
$475$	−4087.24	−0.394811
$476$	0	0
$477$	−735.907	−0.0706391
$478$	131.800	0.0126117
$479$	12046.2	1.14907	0.574535	−	0.818480i	$-0.305183\pi$
$479$	12046.2	1.14907	0.574535	+	0.818480i	$0.305183\pi$
$480$	291.070	0.0276781
$481$	−2581.28	−0.244691
$482$	−7908.84	−0.747382
$483$	0	0
$484$	−4532.50	−0.425667
$485$	−2640.17	−0.247183
$486$	−7379.35	−0.688753
$487$	6788.00	0.631609	0.315804	−	0.948824i	$-0.397726\pi$
$487$	6788.00	0.631609	0.315804	+	0.948824i	$0.397726\pi$
$488$	139.151	0.0129080
$489$	392.836	0.0363285
$490$	0	0
$491$	2710.93	0.249170	0.124585	−	0.992209i	$-0.460240\pi$
$491$	2710.93	0.249170	0.124585	+	0.992209i	$0.460240\pi$
$492$	−967.593	−0.0886635
$493$	−25676.4	−2.34566
$494$	−3089.66	−0.281398
$495$	603.468	0.0547957
$496$	2121.20	0.192026
$497$	0	0
$498$	359.933	0.0323875
$499$	9408.15	0.844021	0.422011	−	0.906591i	$-0.361324\pi$
$499$	9408.15	0.844021	0.422011	+	0.906591i	$0.361324\pi$
$500$	2636.59	0.235824
$501$	−7344.65	−0.654960
$502$	−6892.53	−0.612806
$503$	−11819.6	−1.04773	−0.523865	−	0.851801i	$-0.675510\pi$
$503$	−11819.6	−1.04773	−0.523865	+	0.851801i	$0.675510\pi$
$504$	0	0
$505$	4418.90	0.389383
$506$	647.072	0.0568495
$507$	738.845	0.0647205
$508$	7919.86	0.691707
$509$	−5776.74	−0.503044	−0.251522	−	0.967852i	$-0.580931\pi$
$509$	−5776.74	−0.503044	−0.251522	+	0.967852i	$0.580931\pi$
$510$	1963.44	0.170476
$511$	0	0
$512$	512.000	0.0441942
$513$	4978.41	0.428465
$514$	7061.28	0.605953
$515$	5214.51	0.446172
$516$	1304.72	0.111312
$517$	3636.52	0.309350
$518$	0	0
$519$	−4766.50	−0.403133
$520$	966.222	0.0814840
$521$	7824.93	0.657997	0.328999	−	0.944330i	$-0.393289\pi$
$521$	7824.93	0.657997	0.328999	+	0.944330i	$0.393289\pi$
$522$	7513.23	0.629971
$523$	20822.5	1.74093	0.870465	−	0.492231i	$-0.163818\pi$
$523$	20822.5	1.74093	0.870465	+	0.492231i	$0.163818\pi$
$524$	−1054.95	−0.0879501
$525$	0	0
$526$	−9082.56	−0.752886
$527$	14308.7	1.18273
$528$	−753.536	−0.0621088
$529$	529.000	0.0434783
$530$	−253.227	−0.0207538
$531$	−2219.63	−0.181401
$532$	0	0
$533$	−3211.97	−0.261024
$534$	−1854.28	−0.150267
$535$	1828.09	0.147730
$536$	−5487.45	−0.442205
$537$	7948.89	0.638771
$538$	−3503.28	−0.280738
$539$	0	0
$540$	−1556.89	−0.124070
$541$	18074.4	1.43637	0.718186	−	0.695851i	$-0.244973\pi$
$541$	18074.4	1.43637	0.718186	+	0.695851i	$0.244973\pi$
$542$	−14933.6	−1.18349
$543$	−13364.4	−1.05621
$544$	3453.74	0.272202
$545$	5045.31	0.396545
$546$	0	0
$547$	6046.24	0.472612	0.236306	−	0.971679i	$-0.424063\pi$
$547$	6046.24	0.472612	0.236306	+	0.971679i	$0.424063\pi$
$548$	−4160.79	−0.324344
$549$	−274.662	−0.0213521
$550$	−3309.04	−0.256542
$551$	−8267.00	−0.639176
$552$	−616.038	−0.0475006
$553$	0	0
$554$	−4392.72	−0.336875
$555$	528.146	0.0403938
$556$	1578.99	0.120439
$557$	−11786.4	−0.896598	−0.448299	−	0.893884i	$-0.647970\pi$
$557$	−11786.4	−0.896598	−0.448299	+	0.893884i	$0.647970\pi$
$558$	−4186.91	−0.317645
$559$	4331.08	0.327701
$560$	0	0
$561$	−5083.04	−0.382542
$562$	−2612.99	−0.196125
$563$	−21680.7	−1.62297	−0.811486	−	0.584372i	$-0.801341\pi$
$563$	−21680.7	−1.62297	−0.811486	+	0.584372i	$0.801341\pi$
$564$	−3462.11	−0.258477
$565$	3607.86	0.268644
$566$	−179.798	−0.0133524
$567$	0	0
$568$	2652.85	0.195970
$569$	3846.99	0.283435	0.141717	−	0.989907i	$-0.454738\pi$
$569$	3846.99	0.283435	0.141717	+	0.989907i	$0.454738\pi$
$570$	632.165	0.0464535
$571$	−3404.68	−0.249529	−0.124765	−	0.992186i	$-0.539818\pi$
$571$	−3404.68	−0.249529	−0.124765	+	0.992186i	$0.539818\pi$
$572$	−2501.40	−0.182848
$573$	−11007.3	−0.802510
$574$	0	0
$575$	−2705.24	−0.196202
$576$	−1010.60	−0.0731050
$577$	13520.4	0.975495	0.487747	−	0.872985i	$-0.337819\pi$
$577$	13520.4	0.975495	0.487747	+	0.872985i	$0.337819\pi$
$578$	13471.5	0.969444
$579$	11787.0	0.846026
$580$	2585.32	0.185085
$581$	0	0
$582$	−6507.18	−0.463456
$583$	655.566	0.0465708
$584$	−7547.98	−0.534825
$585$	−1907.16	−0.134789
$586$	−4229.36	−0.298145
$587$	3743.91	0.263250	0.131625	−	0.991300i	$-0.457980\pi$
$587$	3743.91	0.263250	0.131625	+	0.991300i	$0.457980\pi$
$588$	0	0
$589$	4606.96	0.322286
$590$	−763.781	−0.0532956
$591$	−10053.6	−0.699748
$592$	929.022	0.0644976
$593$	22462.0	1.55549	0.777744	−	0.628582i	$-0.216364\pi$
$593$	22462.0	1.55549	0.777744	+	0.628582i	$0.216364\pi$
$594$	4030.54	0.278409
$595$	0	0
$596$	−8956.44	−0.615553
$597$	−738.849	−0.0506517
$598$	−2044.97	−0.139841
$599$	2804.54	0.191303	0.0956514	−	0.995415i	$-0.469507\pi$
$599$	2804.54	0.191303	0.0956514	+	0.995415i	$0.469507\pi$
$600$	3150.34	0.214353
$601$	11993.5	0.814020	0.407010	−	0.913424i	$-0.366571\pi$
$601$	11993.5	0.814020	0.407010	+	0.913424i	$0.366571\pi$
$602$	0	0
$603$	10831.3	0.731485
$604$	−2129.16	−0.143434
$605$	3078.48	0.206873
$606$	10891.2	0.730073
$607$	−9757.57	−0.652467	−0.326234	−	0.945289i	$-0.605780\pi$
$607$	−9757.57	−0.652467	−0.326234	+	0.945289i	$0.605780\pi$
$608$	1111.99	0.0741732
$609$	0	0
$610$	−94.5119	−0.00627324
$611$	−11492.6	−0.760954
$612$	−6817.11	−0.450270
$613$	−10140.1	−0.668118	−0.334059	−	0.942552i	$-0.608418\pi$
$613$	−10140.1	−0.668118	−0.334059	+	0.942552i	$0.608418\pi$
$614$	−18457.0	−1.21313
$615$	657.190	0.0430902
$616$	0	0
$617$	−14606.8	−0.953076	−0.476538	−	0.879154i	$-0.658109\pi$
$617$	−14606.8	−0.953076	−0.476538	+	0.879154i	$0.658109\pi$
$618$	12852.1	0.836549
$619$	22544.3	1.46386	0.731931	−	0.681379i	$-0.238619\pi$
$619$	22544.3	1.46386	0.731931	+	0.681379i	$0.238619\pi$
$620$	−1440.72	−0.0933240
$621$	3295.09	0.212926
$622$	7063.88	0.455363
$623$	0	0
$624$	2381.43	0.152778
$625$	12911.6	0.826342
$626$	−14263.9	−0.910706
$627$	−1636.58	−0.104240
$628$	−7747.59	−0.492297
$629$	6266.79	0.397255
$630$	0	0
$631$	−17047.0	−1.07549	−0.537743	−	0.843109i	$-0.680723\pi$
$631$	−17047.0	−1.07549	−0.537743	+	0.843109i	$0.680723\pi$
$632$	6620.09	0.416666
$633$	14911.3	0.936292
$634$	−15405.3	−0.965019
$635$	−5379.18	−0.336167
$636$	−624.125	−0.0389122
$637$	0	0
$638$	−6692.99	−0.415326
$639$	−5236.28	−0.324169
$640$	−347.751	−0.0214782
$641$	20950.9	1.29097	0.645484	−	0.763774i	$-0.276656\pi$
$641$	20950.9	1.29097	0.645484	+	0.763774i	$0.276656\pi$
$642$	4505.67	0.276985
$643$	17140.2	1.05124	0.525618	−	0.850721i	$-0.323834\pi$
$643$	17140.2	1.05124	0.525618	+	0.850721i	$0.323834\pi$
$644$	0	0
$645$	−886.167	−0.0540973
$646$	7501.04	0.456849
$647$	15648.3	0.950849	0.475425	−	0.879757i	$-0.342294\pi$
$647$	15648.3	0.950849	0.475425	+	0.879757i	$0.342294\pi$
$648$	−426.447	−0.0258525
$649$	1977.31	0.119594
$650$	10457.7	0.631053
$651$	0	0
$652$	−469.333	−0.0281910
$653$	1529.72	0.0916732	0.0458366	−	0.998949i	$-0.485405\pi$
$653$	1529.72	0.0916732	0.0458366	+	0.998949i	$0.485405\pi$
$654$	12435.1	0.743502
$655$	716.526	0.0427435
$656$	1156.01	0.0688030
$657$	14898.5	0.884695
$658$	0	0
$659$	19059.9	1.12666	0.563329	−	0.826233i	$-0.309520\pi$
$659$	19059.9	1.12666	0.563329	+	0.826233i	$0.309520\pi$
$660$	511.803	0.0301847
$661$	−28402.8	−1.67132	−0.835658	−	0.549251i	$-0.814913\pi$
$661$	−28402.8	−1.67132	−0.835658	+	0.549251i	$0.814913\pi$
$662$	10229.6	0.600581
$663$	16064.1	0.940995
$664$	−430.023	−0.0251327
$665$	0	0
$666$	−1833.74	−0.106690
$667$	−5471.71	−0.317640
$668$	8774.89	0.508250
$669$	1436.94	0.0830425
$670$	3727.08	0.214910
$671$	244.677	0.0140770
$672$	0	0
$673$	26105.0	1.49521	0.747603	−	0.664146i	$-0.231205\pi$
$673$	26105.0	1.49521	0.747603	+	0.664146i	$0.231205\pi$
$674$	−24703.9	−1.41181
$675$	−16850.6	−0.960861
$676$	−882.722	−0.0502231
$677$	−7378.78	−0.418892	−0.209446	−	0.977820i	$-0.567166\pi$
$677$	−7378.78	−0.418892	−0.209446	+	0.977820i	$0.567166\pi$
$678$	8892.23	0.503694
$679$	0	0
$680$	−2345.78	−0.132289
$681$	−9138.15	−0.514206
$682$	3729.81	0.209416
$683$	17170.3	0.961936	0.480968	−	0.876738i	$-0.340285\pi$
$683$	17170.3	0.961936	0.480968	+	0.876738i	$0.340285\pi$
$684$	−2194.89	−0.122696
$685$	2826.02	0.157630
$686$	0	0
$687$	21752.8	1.20803
$688$	−1558.79	−0.0863783
$689$	−2071.81	−0.114557
$690$	418.414	0.0230851
$691$	12082.1	0.665158	0.332579	−	0.943075i	$-0.392081\pi$
$691$	12082.1	0.665158	0.332579	+	0.943075i	$0.392081\pi$
$692$	5694.69	0.312832
$693$	0	0
$694$	−19160.1	−1.04799
$695$	−1072.45	−0.0585331
$696$	6371.99	0.347025
$697$	7797.98	0.423773
$698$	4168.94	0.226070
$699$	−10464.0	−0.566215
$700$	0	0
$701$	24855.8	1.33922	0.669608	−	0.742715i	$-0.266462\pi$
$701$	24855.8	1.33922	0.669608	+	0.742715i	$0.266462\pi$
$702$	−12737.9	−0.684844
$703$	2017.71	0.108249
$704$	900.274	0.0481965
$705$	2351.47	0.125619
$706$	10391.8	0.553968
$707$	0	0
$708$	−1882.48	−0.0999264
$709$	5750.35	0.304596	0.152298	−	0.988335i	$-0.451333\pi$
$709$	5750.35	0.304596	0.152298	+	0.988335i	$0.451333\pi$
$710$	−1801.82	−0.0952409
$711$	−13067.0	−0.689240
$712$	2215.37	0.116607
$713$	3049.23	0.160161
$714$	0	0
$715$	1698.95	0.0888634
$716$	−9496.79	−0.495687
$717$	−220.635	−0.0114920
$718$	5537.22	0.287809
$719$	−28485.9	−1.47753	−0.738767	−	0.673961i	$-0.764591\pi$
$719$	−28485.9	−1.47753	−0.738767	+	0.673961i	$0.764591\pi$
$720$	686.403	0.0355288
$721$	0	0
$722$	−11302.9	−0.582618
$723$	13239.5	0.681028
$724$	15966.9	0.819622
$725$	27981.6	1.43340
$726$	7587.48	0.387876
$727$	−7730.71	−0.394383	−0.197191	−	0.980365i	$-0.563182\pi$
$727$	−7730.71	−0.394383	−0.197191	+	0.980365i	$0.563182\pi$
$728$	0	0
$729$	13792.4	0.700727
$730$	5126.60	0.259923
$731$	−10514.9	−0.532023
$732$	−232.942	−0.0117620
$733$	−20633.5	−1.03972	−0.519862	−	0.854250i	$-0.674017\pi$
$733$	−20633.5	−1.03972	−0.519862	+	0.854250i	$0.674017\pi$
$734$	−17527.7	−0.881418
$735$	0	0
$736$	736.000	0.0368605
$737$	−9648.84	−0.482252
$738$	−2281.78	−0.113812
$739$	−33798.1	−1.68239	−0.841194	−	0.540734i	$-0.818147\pi$
$739$	−33798.1	−1.68239	−0.841194	+	0.540734i	$0.818147\pi$
$740$	−630.993	−0.0313456
$741$	5172.14	0.256415
$742$	0	0
$743$	10083.2	0.497869	0.248934	−	0.968520i	$-0.419920\pi$
$743$	10083.2	0.497869	0.248934	+	0.968520i	$0.419920\pi$
$744$	−3550.93	−0.174978
$745$	6083.22	0.299157
$746$	4273.68	0.209746
$747$	848.795	0.0415740
$748$	6072.87	0.296853
$749$	0	0
$750$	−4413.70	−0.214887
$751$	15448.5	0.750629	0.375315	−	0.926897i	$-0.377535\pi$
$751$	15448.5	0.750629	0.375315	+	0.926897i	$0.377535\pi$
$752$	4136.29	0.200579
$753$	11538.2	0.558400
$754$	21152.1	1.02164
$755$	1446.13	0.0697086
$756$	0	0
$757$	15907.3	0.763752	0.381876	−	0.924213i	$-0.375278\pi$
$757$	15907.3	0.763752	0.381876	+	0.924213i	$0.375278\pi$
$758$	−10131.3	−0.485470
$759$	−1083.21	−0.0518023
$760$	−755.268	−0.0360479
$761$	23874.7	1.13726	0.568632	−	0.822592i	$-0.307473\pi$
$761$	23874.7	1.13726	0.568632	+	0.822592i	$0.307473\pi$
$762$	−13258.0	−0.630296
$763$	0	0
$764$	13150.8	0.622748
$765$	4630.19	0.218830
$766$	−21180.9	−0.999082
$767$	−6248.98	−0.294182
$768$	−857.096	−0.0402706
$769$	13225.1	0.620170	0.310085	−	0.950709i	$-0.399642\pi$
$769$	13225.1	0.620170	0.310085	+	0.950709i	$0.399642\pi$
$770$	0	0
$771$	−11820.7	−0.552155
$772$	−14082.2	−0.656517
$773$	−5278.62	−0.245613	−0.122806	−	0.992431i	$-0.539189\pi$
$773$	−5278.62	−0.245613	−0.122806	+	0.992431i	$0.539189\pi$
$774$	3076.79	0.142885
$775$	−15593.4	−0.722748
$776$	7774.33	0.359642
$777$	0	0
$778$	6704.17	0.308941
$779$	2510.70	0.115475
$780$	−1617.47	−0.0742497
$781$	4664.63	0.213718
$782$	4964.75	0.227032
$783$	−34082.7	−1.55558
$784$	0	0
$785$	5262.18	0.239255
$786$	1766.01	0.0801418
$787$	24986.0	1.13171	0.565854	−	0.824505i	$-0.308547\pi$
$787$	24986.0	1.13171	0.565854	+	0.824505i	$0.308547\pi$
$788$	12011.4	0.543005
$789$	15204.3	0.686044
$790$	−4496.37	−0.202498
$791$	0	0
$792$	−1776.99	−0.0797256
$793$	−773.261	−0.0346271
$794$	10272.9	0.459158
$795$	423.907	0.0189112
$796$	882.727	0.0393058
$797$	3366.67	0.149628	0.0748141	−	0.997198i	$-0.476164\pi$
$797$	3366.67	0.149628	0.0748141	+	0.997198i	$0.476164\pi$
$798$	0	0
$799$	27901.7	1.23541
$800$	−3763.81	−0.166338
$801$	−4372.77	−0.192889
$802$	12978.3	0.571421
$803$	−13272.0	−0.583260
$804$	9186.08	0.402945
$805$	0	0
$806$	−11787.5	−0.515132
$807$	5864.55	0.255814
$808$	−13012.0	−0.566537
$809$	−25761.2	−1.11955	−0.559776	−	0.828644i	$-0.689113\pi$
$809$	−25761.2	−1.11955	−0.559776	+	0.828644i	$0.689113\pi$
$810$	289.644	0.0125642
$811$	−16654.4	−0.721104	−0.360552	−	0.932739i	$-0.617412\pi$
$811$	−16654.4	−0.721104	−0.360552	+	0.932739i	$0.617412\pi$
$812$	0	0
$813$	24999.0	1.07842
$814$	1633.54	0.0703387
$815$	318.772	0.0137007
$816$	−5781.61	−0.248035
$817$	−3385.47	−0.144973
$818$	24184.5	1.03373
$819$	0	0
$820$	−785.166	−0.0334380
$821$	−9226.83	−0.392227	−0.196114	−	0.980581i	$-0.562832\pi$
$821$	−9226.83	−0.392227	−0.196114	+	0.980581i	$0.562832\pi$
$822$	6965.24	0.295548
$823$	−5194.26	−0.220001	−0.110000	−	0.993932i	$-0.535085\pi$
$823$	−5194.26	−0.220001	−0.110000	+	0.993932i	$0.535085\pi$
$824$	−15354.8	−0.649163
$825$	5539.39	0.233766
$826$	0	0
$827$	24987.6	1.05067	0.525334	−	0.850896i	$-0.323940\pi$
$827$	24987.6	1.05067	0.525334	+	0.850896i	$0.323940\pi$
$828$	−1452.74	−0.0609738
$829$	−551.268	−0.0230957	−0.0115479	−	0.999933i	$-0.503676\pi$
$829$	−551.268	−0.0230957	−0.0115479	+	0.999933i	$0.503676\pi$
$830$	292.072	0.0122144
$831$	7353.48	0.306967
$832$	−2845.17	−0.118556
$833$	0	0
$834$	−2643.26	−0.109747
$835$	−5959.92	−0.247008
$836$	1955.27	0.0808905
$837$	18993.3	0.784355
$838$	8269.08	0.340872
$839$	−45472.8	−1.87115	−0.935575	−	0.353128i	$-0.885118\pi$
$839$	−45472.8	−1.87115	−0.935575	+	0.353128i	$0.885118\pi$
$840$	0	0
$841$	32207.7	1.32058
$842$	3537.28	0.144778
$843$	4374.19	0.178713
$844$	−17815.1	−0.726564
$845$	599.546	0.0244083
$846$	−8164.36	−0.331793
$847$	0	0
$848$	745.662	0.0301959
$849$	300.984	0.0121670
$850$	−25389.1	−1.02451
$851$	1335.47	0.0537947
$852$	−4440.91	−0.178572
$853$	15598.0	0.626103	0.313052	−	0.949736i	$-0.398649\pi$
$853$	15598.0	0.626103	0.313052	+	0.949736i	$0.398649\pi$
$854$	0	0
$855$	1490.77	0.0596297
$856$	−5383.06	−0.214941
$857$	−5443.13	−0.216959	−0.108479	−	0.994099i	$-0.534598\pi$
$857$	−5443.13	−0.216959	−0.108479	+	0.994099i	$0.534598\pi$
$858$	4187.38	0.166614
$859$	−6233.26	−0.247586	−0.123793	−	0.992308i	$-0.539506\pi$
$859$	−6233.26	−0.247586	−0.123793	+	0.992308i	$0.539506\pi$
$860$	1058.73	0.0419796
$861$	0	0
$862$	−7057.21	−0.278851
$863$	−37857.2	−1.49325	−0.746624	−	0.665246i	$-0.768327\pi$
$863$	−37857.2	−1.49325	−0.746624	+	0.665246i	$0.768327\pi$
$864$	4584.47	0.180517
$865$	−3867.84	−0.152035
$866$	−21983.1	−0.862607
$867$	−22551.4	−0.883376
$868$	0	0
$869$	11640.4	0.454400
$870$	−4327.87	−0.168653
$871$	30493.6	1.18627
$872$	−14856.6	−0.576958
$873$	−15345.3	−0.594912
$874$	1598.49	0.0618647
$875$	0	0
$876$	12635.4	0.487342
$877$	5772.86	0.222275	0.111138	−	0.993805i	$-0.464551\pi$
$877$	5772.86	0.222275	0.111138	+	0.993805i	$0.464551\pi$
$878$	18977.5	0.729454
$879$	7080.01	0.271676
$880$	−611.467	−0.0234234
$881$	15933.8	0.609335	0.304668	−	0.952459i	$-0.401455\pi$
$881$	15933.8	0.609335	0.304668	+	0.952459i	$0.401455\pi$
$882$	0	0
$883$	−38496.6	−1.46717	−0.733587	−	0.679595i	$-0.762155\pi$
$883$	−38496.6	−1.46717	−0.733587	+	0.679595i	$0.762155\pi$
$884$	−19192.3	−0.730213
$885$	1278.58	0.0485639
$886$	18184.6	0.689532
$887$	22380.5	0.847198	0.423599	−	0.905850i	$-0.360767\pi$
$887$	22380.5	0.847198	0.423599	+	0.905850i	$0.360767\pi$
$888$	−1555.20	−0.0587714
$889$	0	0
$890$	−1504.68	−0.0566708
$891$	−749.843	−0.0281938
$892$	−1716.76	−0.0644411
$893$	8983.46	0.336641
$894$	14993.2	0.560904
$895$	6450.23	0.240902
$896$	0	0
$897$	3423.31	0.127426
$898$	−14254.9	−0.529723
$899$	−31539.7	−1.17009
$900$	7429.13	0.275153
$901$	5029.92	0.185983
$902$	2032.67	0.0750339
$903$	0	0
$904$	−10623.8	−0.390867
$905$	−10844.8	−0.398334
$906$	3564.25	0.130700
$907$	34170.5	1.25095	0.625475	−	0.780244i	$-0.284905\pi$
$907$	34170.5	1.25095	0.625475	+	0.780244i	$0.284905\pi$
$908$	10917.6	0.399025
$909$	25683.6	0.937153
$910$	0	0
$911$	2638.77	0.0959674	0.0479837	−	0.998848i	$-0.484720\pi$
$911$	2638.77	0.0959674	0.0479837	+	0.998848i	$0.484720\pi$
$912$	−1861.49	−0.0675880
$913$	−756.130	−0.0274088
$914$	2573.05	0.0931170
$915$	158.214	0.00571629
$916$	−25988.7	−0.937436
$917$	0	0
$918$	30924.9	1.11184
$919$	33287.2	1.19482	0.597412	−	0.801935i	$-0.296196\pi$
$919$	33287.2	1.19482	0.597412	+	0.801935i	$0.296196\pi$
$920$	−499.892	−0.0179141
$921$	30897.2	1.10543
$922$	−28943.5	−1.03384
$923$	−14741.8	−0.525713
$924$	0	0
$925$	−6829.42	−0.242757
$926$	−10837.7	−0.384611
$927$	30307.9	1.07383
$928$	−7612.82	−0.269292
$929$	15350.2	0.542112	0.271056	−	0.962564i	$-0.412627\pi$
$929$	15350.2	0.542112	0.271056	+	0.962564i	$0.412627\pi$
$930$	2411.80	0.0850386
$931$	0	0
$932$	12501.7	0.439383
$933$	−11825.0	−0.414935
$934$	2843.58	0.0996198
$935$	−4124.70	−0.144270
$936$	5615.90	0.196113
$937$	17483.9	0.609578	0.304789	−	0.952420i	$-0.401414\pi$
$937$	17483.9	0.609578	0.304789	+	0.952420i	$0.401414\pi$
$938$	0	0
$939$	23878.1	0.829852
$940$	−2809.38	−0.0974806
$941$	−1203.31	−0.0416864	−0.0208432	−	0.999783i	$-0.506635\pi$
$941$	−1203.31	−0.0416864	−0.0208432	+	0.999783i	$0.506635\pi$
$942$	12969.6	0.448590
$943$	1661.77	0.0573856
$944$	2249.06	0.0775430
$945$	0	0
$946$	−2740.89	−0.0942009
$947$	−24703.3	−0.847675	−0.423838	−	0.905738i	$-0.639317\pi$
$947$	−24703.3	−0.847675	−0.423838	+	0.905738i	$0.639317\pi$
$948$	−11082.1	−0.379674
$949$	41943.9	1.43473
$950$	−8174.47	−0.279174
$951$	25788.7	0.879344
$952$	0	0
$953$	1027.78	0.0349349	0.0174674	−	0.999847i	$-0.494440\pi$
$953$	1027.78	0.0349349	0.0174674	+	0.999847i	$0.494440\pi$
$954$	−1471.81	−0.0499494
$955$	−8932.05	−0.302654
$956$	263.600	0.00891782
$957$	11204.2	0.378453
$958$	24092.4	0.812516
$959$	0	0
$960$	582.141	0.0195714
$961$	−12214.8	−0.410017
$962$	−5162.56	−0.173022
$963$	10625.3	0.355550
$964$	−15817.7	−0.528479
$965$	9564.68	0.319065
$966$	0	0
$967$	−1549.75	−0.0515373	−0.0257687	−	0.999668i	$-0.508203\pi$
$967$	−1549.75	−0.0515373	−0.0257687	+	0.999668i	$0.508203\pi$
$968$	−9065.01	−0.300992
$969$	−12556.9	−0.416289
$970$	−5280.34	−0.174785
$971$	−17142.9	−0.566574	−0.283287	−	0.959035i	$-0.591425\pi$
$971$	−17142.9	−0.566574	−0.283287	+	0.959035i	$0.591425\pi$
$972$	−14758.7	−0.487022
$973$	0	0
$974$	13576.0	0.446615
$975$	−17506.3	−0.575027
$976$	278.303	0.00912731
$977$	16269.5	0.532760	0.266380	−	0.963868i	$-0.414172\pi$
$977$	16269.5	0.532760	0.266380	+	0.963868i	$0.414172\pi$
$978$	785.671	0.0256881
$979$	3895.38	0.127168
$980$	0	0
$981$	29324.4	0.954391
$982$	5421.86	0.176190
$983$	12235.2	0.396991	0.198495	−	0.980102i	$-0.436395\pi$
$983$	12235.2	0.396991	0.198495	+	0.980102i	$0.436395\pi$
$984$	−1935.19	−0.0626946
$985$	−8158.15	−0.263899
$986$	−51352.9	−1.65863
$987$	0	0
$988$	−6179.32	−0.198978
$989$	−2240.76	−0.0720445
$990$	1206.94	0.0387464
$991$	−31854.0	−1.02106	−0.510532	−	0.859858i	$-0.670552\pi$
$991$	−31854.0	−1.02106	−0.510532	+	0.859858i	$0.670552\pi$
$992$	4242.41	0.135783
$993$	−17124.5	−0.547261
$994$	0	0
$995$	−599.549	−0.0191025
$996$	719.865	0.0229014
$997$	−15670.5	−0.497782	−0.248891	−	0.968532i	$-0.580066\pi$
$997$	−15670.5	−0.497782	−0.248891	+	0.968532i	$0.580066\pi$
$998$	18816.3	0.596813
$999$	8318.50	0.263449

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2254.4.a.n.1.3	✓	6
7.6	odd	2	inner	2254.4.a.n.1.4	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2254.4.a.n.1.3	✓	6	1.1	even	1	trivial
2254.4.a.n.1.4	yes	6	7.6	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} -3.34803 q^{3} +4.00000 q^{4} -2.71681 q^{5} -6.69606 q^{6} +8.00000 q^{8} -15.7907 q^{9} -5.43361 q^{10} +14.0668 q^{11} -13.3921 q^{12} -44.4558 q^{13} +9.09595 q^{15} +16.0000 q^{16} +107.929 q^{17} -31.5814 q^{18} +34.7498 q^{19} -10.8672 q^{20} +28.1336 q^{22} +23.0000 q^{23} -26.7843 q^{24} -117.619 q^{25} -88.9116 q^{26} +143.265 q^{27} -237.901 q^{29} +18.1919 q^{30} +132.575 q^{31} +32.0000 q^{32} -47.0960 q^{33} +215.859 q^{34} -63.1627 q^{36} +58.0639 q^{37} +69.4996 q^{38} +148.839 q^{39} -21.7344 q^{40} +72.2508 q^{41} -97.4243 q^{43} +56.2671 q^{44} +42.9002 q^{45} +46.0000 q^{46} +258.518 q^{47} -53.5685 q^{48} -235.238 q^{50} -361.351 q^{51} -177.823 q^{52} +46.6039 q^{53} +286.529 q^{54} -38.2167 q^{55} -116.343 q^{57} -475.801 q^{58} +140.566 q^{59} +36.3838 q^{60} +17.3939 q^{61} +265.150 q^{62} +64.0000 q^{64} +120.778 q^{65} -94.1920 q^{66} -685.931 q^{67} +431.717 q^{68} -77.0047 q^{69} +331.606 q^{71} -126.325 q^{72} -943.497 q^{73} +116.128 q^{74} +393.792 q^{75} +138.999 q^{76} +297.679 q^{78} +827.511 q^{79} -43.4689 q^{80} -53.3059 q^{81} +144.502 q^{82} -53.7529 q^{83} -293.223 q^{85} -194.849 q^{86} +796.499 q^{87} +112.534 q^{88} +276.921 q^{89} +85.8004 q^{90} +92.0000 q^{92} -443.866 q^{93} +517.037 q^{94} -94.4084 q^{95} -107.137 q^{96} +971.792 q^{97} -222.124 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 12 q^{2} + 24 q^{4} + 48 q^{8} - 52 q^{9} - 84 q^{11} + 52 q^{15} + 96 q^{16} - 104 q^{18} - 168 q^{22} + 138 q^{23} - 286 q^{25} - 22 q^{29} + 104 q^{30} + 192 q^{32} - 208 q^{36} - 180 q^{37} - 414 q^{39}+ \cdots + 600 q^{99}+O(q^{100}) \) 6 * q + 12 * q^2 + 24 * q^4 + 48 * q^8 - 52 * q^9 - 84 * q^11 + 52 * q^15 + 96 * q^16 - 104 * q^18 - 168 * q^22 + 138 * q^23 - 286 * q^25 - 22 * q^29 + 104 * q^30 + 192 * q^32 - 208 * q^36 - 180 * q^37 - 414 * q^39 - 600 * q^43 - 336 * q^44 + 276 * q^46 - 572 * q^50 + 464 * q^51 - 876 * q^53 - 1452 * q^57 - 44 * q^58 + 208 * q^60 + 384 * q^64 - 1532 * q^65 - 92 * q^67 - 1434 * q^71 - 416 * q^72 - 360 * q^74 - 828 * q^78 - 2820 * q^79 - 2198 * q^81 - 184 * q^85 - 1200 * q^86 - 672 * q^88 + 552 * q^92 - 2742 * q^93 - 2888 * q^95 + 600 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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