LMFDB - Embedded newform 2254.4.a.n.1.4

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			1.4
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	$-0.395590\pi$
$3$	3.34803	0.644329	0.322165	+	0.946684i	$0.395590\pi$
$4$	4.00000	0.500000
$5$	2.71681	0.242998	0.121499	−	0.992592i	$-0.461230\pi$
$5$	2.71681	0.242998	0.121499	+	0.992592i	$0.461230\pi$
$6$	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	$-0.342729\pi$
$13$	44.4558	0.948448	0.474224	+	0.880404i	$0.342729\pi$
$14$	0	0
$15$	9.09595	0.156571
$16$	16.0000	0.250000
$17$	−107.929	−1.53981	−0.769903	−	0.638161i	$-0.779695\pi$
$17$	−107.929	−1.53981	−0.769903	+	0.638161i	$0.779695\pi$
$18$	−31.5814	−0.413544
$19$	−34.7498	−0.419587	−0.209793	−	0.977746i	$-0.567279\pi$
$19$	−34.7498	−0.419587	−0.209793	+	0.977746i	$0.567279\pi$
$20$	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	$-0.625472\pi$
$31$	−132.575	−0.768103	−0.384052	+	0.923312i	$0.625472\pi$
$32$	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	$-0.543941\pi$
$41$	−72.2508	−0.275212	−0.137606	+	0.990487i	$0.543941\pi$
$42$	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	$-0.631392\pi$
$47$	−258.518	−0.802314	−0.401157	+	0.916009i	$0.631392\pi$
$48$	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	$-0.549565\pi$
$59$	−140.566	−0.310172	−0.155086	+	0.987901i	$0.549565\pi$
$60$	36.3838	0.0782855
$61$	−17.3939	−0.0365093	−0.0182546	−	0.999833i	$-0.505811\pi$
$61$	−17.3939	−0.0365093	−0.0182546	+	0.999833i	$0.505811\pi$
$62$	−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	$-0.226978\pi$
$73$	943.497	1.51271	0.756356	+	0.654160i	$0.226978\pi$
$74$	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	$-0.488684\pi$
$83$	53.7529	0.0710861	0.0355430	+	0.999368i	$0.488684\pi$
$84$	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	$-0.552733\pi$
$89$	−276.921	−0.329815	−0.164908	+	0.986309i	$0.552733\pi$
$90$	−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	$-0.669841\pi$
$97$	−971.792	−1.01722	−0.508611	+	0.860996i	$0.669841\pi$
$98$	0	0
$99$	−222.124	−0.225498
$100$	−470.476	−0.470476
$101$	1626.51	1.60241	0.801205	−	0.598390i	$-0.204193\pi$
$101$	1626.51	1.60241	0.801205	+	0.598390i	$0.204193\pi$
$102$	−722.701	−0.701550
$103$	1919.35	1.83611	0.918055	−	0.396453i	$-0.129759\pi$
$103$	1919.35	1.83611	0.918055	+	0.396453i	$0.129759\pi$
$104$	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	$-0.471968\pi$
$131$	263.738	0.175900	0.0879501	+	0.996125i	$0.471968\pi$
$132$	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	$-0.538430\pi$
$139$	−394.748	−0.240878	−0.120439	+	0.992721i	$0.538430\pi$
$140$	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	$-0.336157\pi$
$157$	1936.90	0.984594	0.492297	+	0.870427i	$0.336157\pi$
$158$	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	$-0.669707\pi$
$167$	−2193.72	−1.01650	−0.508250	+	0.861210i	$0.669707\pi$
$168$	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	$-0.601278\pi$
$173$	−1423.67	−0.625663	−0.312832	+	0.949809i	$0.601278\pi$
$174$	−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	$-0.805816\pi$
$181$	−3991.73	−1.63924	−0.819622	+	0.572905i	$0.805816\pi$
$182$	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	$-0.512515\pi$
$199$	−220.682	−0.0786116	−0.0393058	+	0.999227i	$0.512515\pi$
$200$	−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	$-0.479474\pi$
$223$	429.190	0.128882	0.0644411	+	0.997922i	$0.479474\pi$
$224$	0	0
$225$	1857.28	0.550306
$226$	−2655.96	−0.781733
$227$	−2729.41	−0.798049	−0.399025	−	0.916940i	$-0.630651\pi$
$227$	−2729.41	−0.798049	−0.399025	+	0.916940i	$0.630651\pi$
$228$	−465.374	−0.135176
$229$	6497.18	1.87487	0.937436	−	0.348157i	$-0.113192\pi$
$229$	6497.18	1.87487	0.937436	+	0.348157i	$0.113192\pi$
$230$	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	$-0.322763\pi$
$241$	3954.42	1.05696	0.528479	+	0.848947i	$0.322763\pi$
$242$	−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	$-0.357342\pi$
$251$	3446.26	0.866639	0.433319	+	0.901240i	$0.357342\pi$
$252$	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	$-0.640948\pi$
$257$	−3530.64	−0.856946	−0.428473	+	0.903554i	$0.640948\pi$
$258$	−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	$-0.436389\pi$
$269$	1751.64	0.397024	0.198512	+	0.980098i	$0.436389\pi$
$270$	−778.444	−0.175461
$271$	7466.78	1.67371	0.836854	−	0.547427i	$-0.184392\pi$
$271$	7466.78	1.67371	0.836854	+	0.547427i	$0.184392\pi$
$272$	−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	$-0.496995\pi$
$283$	89.8988	0.0188831	0.00944157	+	0.999955i	$0.496995\pi$
$284$	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	$-0.432386\pi$
$293$	2114.68	0.421641	0.210821	+	0.977525i	$0.432386\pi$
$294$	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	$-0.171823\pi$
$307$	9228.48	1.71563	0.857813	+	0.513962i	$0.171823\pi$
$308$	0	0
$309$	6426.05	1.18306
$310$	−720.362	−0.131980
$311$	−3531.94	−0.643980	−0.321990	−	0.946743i	$-0.604352\pi$
$311$	−3531.94	−0.643980	−0.321990	+	0.946743i	$0.604352\pi$
$312$	1190.72	0.216061
$313$	7131.97	1.28793	0.643966	−	0.765054i	$-0.277288\pi$
$313$	7131.97	1.28793	0.643966	+	0.765054i	$0.277288\pi$
$314$	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	$-0.551103\pi$
$349$	−2084.47	−0.319711	−0.159856	+	0.987140i	$0.551103\pi$
$350$	0	0
$351$	−6368.94	−0.968516
$352$	450.137	0.0681602
$353$	−5195.91	−0.783430	−0.391715	−	0.920087i	$-0.628118\pi$
$353$	−5195.91	−0.783430	−0.391715	+	0.920087i	$0.628118\pi$
$354$	−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	$-0.285809\pi$
$367$	8763.87	1.24651	0.623257	+	0.782017i	$0.285809\pi$
$368$	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	$-0.250292\pi$
$383$	10590.5	1.41292	0.706458	+	0.707755i	$0.250292\pi$
$384$	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	$-0.605255\pi$
$397$	−5136.45	−0.649348	−0.324674	+	0.945826i	$0.605255\pi$
$398$	−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	$-0.760927\pi$
$409$	−12092.3	−1.46192	−0.730958	+	0.682422i	$0.760927\pi$
$410$	−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	$-0.577486\pi$
$419$	−4134.54	−0.482066	−0.241033	+	0.970517i	$0.577486\pi$
$420$	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	$-0.291187\pi$
$433$	10991.6	1.21991	0.609955	+	0.792436i	$0.291187\pi$
$434$	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	$-0.672506\pi$
$439$	−9488.77	−1.03160	−0.515802	+	0.856708i	$0.672506\pi$
$440$	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	$-0.239037\pi$
$461$	14471.7	1.46207	0.731037	+	0.682338i	$0.239037\pi$
$462$	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	$-0.522441\pi$
$467$	−1421.79	−0.140884	−0.0704418	+	0.997516i	$0.522441\pi$
$468$	−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	$-0.694817\pi$
$479$	−12046.2	−1.14907	−0.574535	+	0.818480i	$0.694817\pi$
$480$	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	$-0.324490\pi$
$503$	11819.6	1.04773	0.523865	+	0.851801i	$0.324490\pi$
$504$	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	$-0.419069\pi$
$509$	5776.74	0.503044	0.251522	+	0.967852i	$0.419069\pi$
$510$	−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	$-0.606711\pi$
$521$	−7824.93	−0.657997	−0.328999	+	0.944330i	$0.606711\pi$
$522$	7513.23	0.629971
$523$	−20822.5	−1.74093	−0.870465	−	0.492231i	$-0.836182\pi$
$523$	−20822.5	−1.74093	−0.870465	+	0.492231i	$0.836182\pi$
$524$	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	$-0.198659\pi$
$563$	21680.7	1.62297	0.811486	+	0.584372i	$0.198659\pi$
$564$	−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	$-0.662181\pi$
$577$	−13520.4	−0.975495	−0.487747	+	0.872985i	$0.662181\pi$
$578$	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	$-0.542020\pi$
$587$	−3743.91	−0.263250	−0.131625	+	0.991300i	$0.542020\pi$
$588$	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	$-0.783636\pi$
$593$	−22462.0	−1.55549	−0.777744	+	0.628582i	$0.783636\pi$
$594$	−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	$-0.633429\pi$
$601$	−11993.5	−0.814020	−0.407010	+	0.913424i	$0.633429\pi$
$602$	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	$-0.394220\pi$
$607$	9757.57	0.652467	0.326234	+	0.945289i	$0.394220\pi$
$608$	−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	$-0.761381\pi$
$619$	−22544.3	−1.46386	−0.731931	+	0.681379i	$0.761381\pi$
$620$	−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	$-0.676166\pi$
$643$	−17140.2	−1.05124	−0.525618	+	0.850721i	$0.676166\pi$
$644$	0	0
$645$	−886.167	−0.0540973
$646$	7501.04	0.456849
$647$	−15648.3	−0.950849	−0.475425	−	0.879757i	$-0.657706\pi$
$647$	−15648.3	−0.950849	−0.475425	+	0.879757i	$0.657706\pi$
$648$	−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	$-0.185087\pi$
$661$	28402.8	1.67132	0.835658	+	0.549251i	$0.185087\pi$
$662$	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	$-0.432834\pi$
$677$	7378.78	0.418892	0.209446	+	0.977820i	$0.432834\pi$
$678$	−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	$-0.607919\pi$
$691$	−12082.1	−0.665158	−0.332579	+	0.943075i	$0.607919\pi$
$692$	−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	$-0.235409\pi$
$719$	28485.9	1.47753	0.738767	+	0.673961i	$0.235409\pi$
$720$	−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	$-0.436818\pi$
$727$	7730.71	0.394383	0.197191	+	0.980365i	$0.436818\pi$
$728$	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	$-0.325983\pi$
$733$	20633.5	1.03972	0.519862	+	0.854250i	$0.325983\pi$
$734$	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	$-0.692527\pi$
$761$	−23874.7	−1.13726	−0.568632	+	0.822592i	$0.692527\pi$
$762$	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	$-0.600358\pi$
$769$	−13225.1	−0.620170	−0.310085	+	0.950709i	$0.600358\pi$
$770$	0	0
$771$	−11820.7	−0.552155
$772$	−14082.2	−0.656517
$773$	5278.62	0.245613	0.122806	−	0.992431i	$-0.460811\pi$
$773$	5278.62	0.245613	0.122806	+	0.992431i	$0.460811\pi$
$774$	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	$-0.691453\pi$
$787$	−24986.0	−1.13171	−0.565854	+	0.824505i	$0.691453\pi$
$788$	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	$-0.523836\pi$
$797$	−3366.67	−0.149628	−0.0748141	+	0.997198i	$0.523836\pi$
$798$	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	$-0.382588\pi$
$811$	16654.4	0.721104	0.360552	+	0.932739i	$0.382588\pi$
$812$	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	$-0.496324\pi$
$829$	551.268	0.0230957	0.0115479	+	0.999933i	$0.496324\pi$
$830$	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	$-0.114882\pi$
$839$	45472.8	1.87115	0.935575	+	0.353128i	$0.114882\pi$
$840$	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	$-0.601351\pi$
$853$	−15598.0	−0.626103	−0.313052	+	0.949736i	$0.601351\pi$
$854$	0	0
$855$	1490.77	0.0596297
$856$	−5383.06	−0.214941
$857$	5443.13	0.216959	0.108479	−	0.994099i	$-0.465402\pi$
$857$	5443.13	0.216959	0.108479	+	0.994099i	$0.465402\pi$
$858$	4187.38	0.166614
$859$	6233.26	0.247586	0.123793	−	0.992308i	$-0.460494\pi$
$859$	6233.26	0.247586	0.123793	+	0.992308i	$0.460494\pi$
$860$	−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	$-0.598545\pi$
$881$	−15933.8	−0.609335	−0.304668	+	0.952459i	$0.598545\pi$
$882$	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	$-0.639233\pi$
$887$	−22380.5	−0.847198	−0.423599	+	0.905850i	$0.639233\pi$
$888$	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	$-0.587373\pi$
$929$	−15350.2	−0.542112	−0.271056	+	0.962564i	$0.587373\pi$
$930$	−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	$-0.598586\pi$
$937$	−17483.9	−0.609578	−0.304789	+	0.952420i	$0.598586\pi$
$938$	0	0
$939$	23878.1	0.829852
$940$	−2809.38	−0.0974806
$941$	1203.31	0.0416864	0.0208432	−	0.999783i	$-0.493365\pi$
$941$	1203.31	0.0416864	0.0208432	+	0.999783i	$0.493365\pi$
$942$	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	$-0.408575\pi$
$971$	17142.9	0.566574	0.283287	+	0.959035i	$0.408575\pi$
$972$	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	$-0.563605\pi$
$983$	−12235.2	−0.396991	−0.198495	+	0.980102i	$0.563605\pi$
$984$	−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	$-0.419934\pi$
$997$	15670.5	0.497782	0.248891	+	0.968532i	$0.419934\pi$
$998$	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.1.4	yes	6
7.6	odd	2	inner	2254.4.a.n.1.3	✓	6

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2254 = 2 \cdot 7^{2} \cdot 23 \)
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

Related objects

Downloads

Learn more