LMFDB - Embedded newform 2254.4.a.y.1.6

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:	$0$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 4 x^{10} - 212 x^{9} + 487 x^{8} + 16315 x^{7} - 9025 x^{6} - 516068 x^{5} - 504693 x^{4} + \cdots - 11394027 $ x^11 - 4x^10 - 212x^9 + 487x^8 + 16315x^7 - 9025x^6 - 516068x^5 - 504693x^4 + 5020566x^3 + 8901846x^2 - 4796415x - 11394027
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 322)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-1.55871$ 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)

$f(q)$	$=$	\(q+2.00000 q^{2} +3.55871 q^{3} +4.00000 q^{4} -17.9984 q^{5} +7.11742 q^{6} +8.00000 q^{8} -14.3356 q^{9} -35.9968 q^{10} -43.1362 q^{11} +14.2348 q^{12} +67.9392 q^{13} -64.0511 q^{15} +16.0000 q^{16} +5.75059 q^{17} -28.6712 q^{18} -102.206 q^{19} -71.9936 q^{20} -86.2723 q^{22} +23.0000 q^{23} +28.4697 q^{24} +198.943 q^{25} +135.878 q^{26} -147.101 q^{27} +23.0752 q^{29} -128.102 q^{30} -222.171 q^{31} +32.0000 q^{32} -153.509 q^{33} +11.5012 q^{34} -57.3423 q^{36} +39.2972 q^{37} -204.412 q^{38} +241.776 q^{39} -143.987 q^{40} +373.067 q^{41} +5.34174 q^{43} -172.545 q^{44} +258.018 q^{45} +46.0000 q^{46} +299.732 q^{47} +56.9394 q^{48} +397.885 q^{50} +20.4647 q^{51} +271.757 q^{52} -6.53059 q^{53} -294.203 q^{54} +776.382 q^{55} -363.721 q^{57} +46.1504 q^{58} +620.866 q^{59} -256.204 q^{60} -106.016 q^{61} -444.343 q^{62} +64.0000 q^{64} -1222.80 q^{65} -307.018 q^{66} -382.646 q^{67} +23.0024 q^{68} +81.8503 q^{69} -765.836 q^{71} -114.685 q^{72} +69.0434 q^{73} +78.5943 q^{74} +707.979 q^{75} -408.824 q^{76} +483.552 q^{78} +598.857 q^{79} -287.974 q^{80} -136.431 q^{81} +746.134 q^{82} +927.174 q^{83} -103.501 q^{85} +10.6835 q^{86} +82.1180 q^{87} -345.089 q^{88} +1549.42 q^{89} +516.035 q^{90} +92.0000 q^{92} -790.644 q^{93} +599.463 q^{94} +1839.54 q^{95} +113.879 q^{96} -314.236 q^{97} +618.382 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 22 q^{2} + 18 q^{3} + 44 q^{4} + 33 q^{5} + 36 q^{6} + 88 q^{8} + 171 q^{9} + 66 q^{10} + 8 q^{11} + 72 q^{12} + 185 q^{13} - 186 q^{15} + 176 q^{16} + 107 q^{17} + 342 q^{18} + 114 q^{19} + 132 q^{20}+ \cdots - 1729 q^{99}+O(q^{100}) $ 11 * q + 22 * q^2 + 18 * q^3 + 44 * q^4 + 33 * q^5 + 36 * q^6 + 88 * q^8 + 171 * q^9 + 66 * q^10 + 8 * q^11 + 72 * q^12 + 185 * q^13 - 186 * q^15 + 176 * q^16 + 107 * q^17 + 342 * q^18 + 114 * q^19 + 132 * q^20 + 16 * q^22 + 253 * q^23 + 144 * q^24 + 494 * q^25 + 370 * q^26 + 561 * q^27 - 79 * q^29 - 372 * q^30 + 488 * q^31 + 352 * q^32 + 304 * q^33 + 214 * q^34 + 684 * q^36 - 41 * q^37 + 228 * q^38 + 425 * q^39 + 264 * q^40 + 812 * q^41 + 219 * q^43 + 32 * q^44 + 619 * q^45 + 506 * q^46 + 1141 * q^47 + 288 * q^48 + 988 * q^50 + 83 * q^51 + 740 * q^52 + 579 * q^53 + 1122 * q^54 + 951 * q^55 + 383 * q^57 - 158 * q^58 + 1695 * q^59 - 744 * q^60 + 1105 * q^61 + 976 * q^62 + 704 * q^64 - 347 * q^65 + 608 * q^66 + 695 * q^67 + 428 * q^68 + 414 * q^69 + 65 * q^71 + 1368 * q^72 + 1935 * q^73 - 82 * q^74 + 1712 * q^75 + 456 * q^76 + 850 * q^78 + 447 * q^79 + 528 * q^80 + 2203 * q^81 + 1624 * q^82 + 1389 * q^83 + 153 * q^85 + 438 * q^86 + 3354 * q^87 + 64 * q^88 + 2698 * q^89 + 1238 * q^90 + 1012 * q^92 - 686 * q^93 + 2282 * q^94 + 827 * q^95 + 576 * q^96 + 1589 * q^97 - 1729 * 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
$3$	3.55871	0.684874	0.342437	−	0.939541i	$-0.388748\pi$
$3$	3.55871	0.684874	0.342437	+	0.939541i	$0.388748\pi$
$4$	4.00000	0.500000
$5$	−17.9984	−1.60983	−0.804913	−	0.593393i	$-0.797788\pi$
$5$	−17.9984	−1.60983	−0.804913	+	0.593393i	$0.797788\pi$
$6$	7.11742	0.484279
$7$	0	0
$7$	0	0
$8$	8.00000	0.353553
$9$	−14.3356	−0.530947
$10$	−35.9968	−1.13832
$11$	−43.1362	−1.18237	−0.591184	−	0.806537i	$-0.701339\pi$
$11$	−43.1362	−1.18237	−0.591184	+	0.806537i	$0.701339\pi$
$12$	14.2348	0.342437
$13$	67.9392	1.44946	0.724729	−	0.689034i	$-0.241965\pi$
$13$	67.9392	1.44946	0.724729	+	0.689034i	$0.241965\pi$
$14$	0	0
$15$	−64.0511	−1.10253
$16$	16.0000	0.250000
$17$	5.75059	0.0820425	0.0410213	−	0.999158i	$-0.486939\pi$
$17$	5.75059	0.0820425	0.0410213	+	0.999158i	$0.486939\pi$
$18$	−28.6712	−0.375436
$19$	−102.206	−1.23409	−0.617043	−	0.786929i	$-0.711670\pi$
$19$	−102.206	−1.23409	−0.617043	+	0.786929i	$0.711670\pi$
$20$	−71.9936	−0.804913
$21$	0	0
$22$	−86.2723	−0.836060
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	28.4697	0.242140
$25$	198.943	1.59154
$26$	135.878	1.02492
$27$	−147.101	−1.04851
$28$	0	0
$29$	23.0752	0.147757	0.0738786	−	0.997267i	$-0.476462\pi$
$29$	23.0752	0.147757	0.0738786	+	0.997267i	$0.476462\pi$
$30$	−128.102	−0.779605
$31$	−222.171	−1.28720	−0.643600	−	0.765362i	$-0.722560\pi$
$31$	−222.171	−1.28720	−0.643600	+	0.765362i	$0.722560\pi$
$32$	32.0000	0.176777
$33$	−153.509	−0.809773
$34$	11.5012	0.0580128
$35$	0	0
$36$	−57.3423	−0.265474
$37$	39.2972	0.174606	0.0873029	−	0.996182i	$-0.472175\pi$
$37$	39.2972	0.174606	0.0873029	+	0.996182i	$0.472175\pi$
$38$	−204.412	−0.872631
$39$	241.776	0.992696
$40$	−143.987	−0.569160
$41$	373.067	1.42105	0.710527	−	0.703670i	$-0.248456\pi$
$41$	373.067	1.42105	0.710527	+	0.703670i	$0.248456\pi$
$42$	0	0
$43$	5.34174	0.0189444	0.00947219	−	0.999955i	$-0.496985\pi$
$43$	5.34174	0.0189444	0.00947219	+	0.999955i	$0.496985\pi$
$44$	−172.545	−0.591184
$45$	258.018	0.854733
$46$	46.0000	0.147442
$47$	299.732	0.930220	0.465110	−	0.885253i	$-0.346015\pi$
$47$	299.732	0.930220	0.465110	+	0.885253i	$0.346015\pi$
$48$	56.9394	0.171219
$49$	0	0
$50$	397.885	1.12539
$51$	20.4647	0.0561888
$52$	271.757	0.724729
$53$	−6.53059	−0.0169254	−0.00846269	−	0.999964i	$-0.502694\pi$
$53$	−6.53059	−0.0169254	−0.00846269	+	0.999964i	$0.502694\pi$
$54$	−294.203	−0.741406
$55$	776.382	1.90341
$56$	0	0
$57$	−363.721	−0.845194
$58$	46.1504	0.104480
$59$	620.866	1.37000	0.684999	−	0.728544i	$-0.259802\pi$
$59$	620.866	1.37000	0.684999	+	0.728544i	$0.259802\pi$
$60$	−256.204	−0.551264
$61$	−106.016	−0.222524	−0.111262	−	0.993791i	$-0.535489\pi$
$61$	−106.016	−0.222524	−0.111262	+	0.993791i	$0.535489\pi$
$62$	−444.343	−0.910187
$63$	0	0
$64$	64.0000	0.125000
$65$	−1222.80	−2.33338
$66$	−307.018	−0.572596
$67$	−382.646	−0.697727	−0.348863	−	0.937174i	$-0.613432\pi$
$67$	−382.646	−0.697727	−0.348863	+	0.937174i	$0.613432\pi$
$68$	23.0024	0.0410213
$69$	81.8503	0.142806
$70$	0	0
$71$	−765.836	−1.28011	−0.640056	−	0.768328i	$-0.721089\pi$
$71$	−765.836	−1.28011	−0.640056	+	0.768328i	$0.721089\pi$
$72$	−114.685	−0.187718
$73$	69.0434	0.110697	0.0553487	−	0.998467i	$-0.482373\pi$
$73$	69.0434	0.110697	0.0553487	+	0.998467i	$0.482373\pi$
$74$	78.5943	0.123465
$75$	707.979	1.09001
$76$	−408.824	−0.617043
$77$	0	0
$78$	483.552	0.701942
$79$	598.857	0.852869	0.426434	−	0.904519i	$-0.359770\pi$
$79$	598.857	0.852869	0.426434	+	0.904519i	$0.359770\pi$
$80$	−287.974	−0.402457
$81$	−136.431	−0.187147
$82$	746.134	1.00484
$83$	927.174	1.22615	0.613076	−	0.790024i	$-0.289932\pi$
$83$	927.174	1.22615	0.613076	+	0.790024i	$0.289932\pi$
$84$	0	0
$85$	−103.501	−0.132074
$86$	10.6835	0.0133957
$87$	82.1180	0.101195
$88$	−345.089	−0.418030
$89$	1549.42	1.84537	0.922685	−	0.385554i	$-0.125990\pi$
$89$	1549.42	1.84537	0.922685	+	0.385554i	$0.125990\pi$
$90$	516.035	0.604388
$91$	0	0
$92$	92.0000	0.104257
$93$	−790.644	−0.881569
$94$	599.463	0.657765
$95$	1839.54	1.98666
$96$	113.879	0.121070
$97$	−314.236	−0.328926	−0.164463	−	0.986383i	$-0.552589\pi$
$97$	−314.236	−0.328926	−0.164463	+	0.986383i	$0.552589\pi$
$98$	0	0
$99$	618.382	0.627775
$100$	795.770	0.795770
$101$	−364.705	−0.359302	−0.179651	−	0.983730i	$-0.557497\pi$
$101$	−364.705	−0.359302	−0.179651	+	0.983730i	$0.557497\pi$
$102$	40.9294	0.0397315
$103$	1135.92	1.08665	0.543326	−	0.839522i	$-0.317165\pi$
$103$	1135.92	1.08665	0.543326	+	0.839522i	$0.317165\pi$
$104$	543.514	0.512461
$105$	0	0
$106$	−13.0612	−0.0119680
$107$	111.698	0.100919	0.0504593	−	0.998726i	$-0.483931\pi$
$107$	111.698	0.100919	0.0504593	+	0.998726i	$0.483931\pi$
$108$	−588.405	−0.524253
$109$	1242.59	1.09191	0.545957	−	0.837813i	$-0.316166\pi$
$109$	1242.59	1.09191	0.545957	+	0.837813i	$0.316166\pi$
$110$	1552.76	1.34591
$111$	139.847	0.119583
$112$	0	0
$113$	−1772.81	−1.47586	−0.737929	−	0.674879i	$-0.764196\pi$
$113$	−1772.81	−1.47586	−0.737929	+	0.674879i	$0.764196\pi$
$114$	−727.442	−0.597642
$115$	−413.963	−0.335672
$116$	92.3008	0.0738786
$117$	−973.948	−0.769586
$118$	1241.73	0.968735
$119$	0	0
$120$	−512.409	−0.389803
$121$	529.728	0.397992
$122$	−212.032	−0.157348
$123$	1327.64	0.973244
$124$	−888.686	−0.643600
$125$	−1330.85	−0.952278
$126$	0	0
$127$	−408.399	−0.285351	−0.142675	−	0.989770i	$-0.545570\pi$
$127$	−408.399	−0.285351	−0.142675	+	0.989770i	$0.545570\pi$
$128$	128.000	0.0883883
$129$	19.0097	0.0129745
$130$	−2445.60	−1.64995
$131$	2900.00	1.93415	0.967076	−	0.254489i	$-0.0819074\pi$
$131$	2900.00	1.93415	0.967076	+	0.254489i	$0.0819074\pi$
$132$	−614.036	−0.404886
$133$	0	0
$134$	−765.293	−0.493367
$135$	2647.59	1.68791
$136$	46.0047	0.0290064
$137$	529.278	0.330068	0.165034	−	0.986288i	$-0.447227\pi$
$137$	529.278	0.330068	0.165034	+	0.986288i	$0.447227\pi$
$138$	163.701	0.100979
$139$	1389.89	0.848121	0.424061	−	0.905634i	$-0.360604\pi$
$139$	1389.89	0.848121	0.424061	+	0.905634i	$0.360604\pi$
$140$	0	0
$141$	1066.66	0.637084
$142$	−1531.67	−0.905176
$143$	−2930.64	−1.71379
$144$	−229.369	−0.132737
$145$	−415.317	−0.237863
$146$	138.087	0.0782749
$147$	0	0
$148$	157.189	0.0873029
$149$	−2380.71	−1.30896	−0.654480	−	0.756079i	$-0.727112\pi$
$149$	−2380.71	−1.30896	−0.654480	+	0.756079i	$0.727112\pi$
$150$	1415.96	0.770750
$151$	420.845	0.226807	0.113404	−	0.993549i	$-0.463825\pi$
$151$	420.845	0.226807	0.113404	+	0.993549i	$0.463825\pi$
$152$	−817.647	−0.436315
$153$	−82.4380	−0.0435603
$154$	0	0
$155$	3998.73	2.07217
$156$	967.104	0.496348
$157$	189.463	0.0963107	0.0481553	−	0.998840i	$-0.484666\pi$
$157$	189.463	0.0963107	0.0481553	+	0.998840i	$0.484666\pi$
$158$	1197.71	0.603069
$159$	−23.2405	−0.0115918
$160$	−575.949	−0.284580
$161$	0	0
$162$	−272.861	−0.132333
$163$	3461.69	1.66344	0.831720	−	0.555196i	$-0.187357\pi$
$163$	3461.69	1.66344	0.831720	+	0.555196i	$0.187357\pi$
$164$	1492.27	0.710527
$165$	2762.92	1.30359
$166$	1854.35	0.867020
$167$	1480.75	0.686133	0.343067	−	0.939311i	$-0.388534\pi$
$167$	1480.75	0.686133	0.343067	+	0.939311i	$0.388534\pi$
$168$	0	0
$169$	2418.74	1.10093
$170$	−207.003	−0.0933905
$171$	1465.18	0.655235
$172$	21.3670	0.00947219
$173$	2966.34	1.30362	0.651811	−	0.758381i	$-0.274009\pi$
$173$	2966.34	1.30362	0.651811	+	0.758381i	$0.274009\pi$
$174$	164.236	0.0715557
$175$	0	0
$176$	−690.178	−0.295592
$177$	2209.48	0.938277
$178$	3098.84	1.30487
$179$	−4773.67	−1.99330	−0.996650	−	0.0817848i	$-0.973938\pi$
$179$	−4773.67	−1.99330	−0.996650	+	0.0817848i	$0.973938\pi$
$180$	1032.07	0.427367
$181$	−3285.16	−1.34908	−0.674541	−	0.738238i	$-0.735658\pi$
$181$	−3285.16	−1.34908	−0.674541	+	0.738238i	$0.735658\pi$
$182$	0	0
$183$	−377.281	−0.152401
$184$	184.000	0.0737210
$185$	−707.286	−0.281085
$186$	−1581.29	−0.623364
$187$	−248.058	−0.0970044
$188$	1198.93	0.465110
$189$	0	0
$190$	3679.09	1.40478
$191$	825.286	0.312647	0.156324	−	0.987706i	$-0.450036\pi$
$191$	825.286	0.312647	0.156324	+	0.987706i	$0.450036\pi$
$192$	227.757	0.0856093
$193$	−126.167	−0.0470553	−0.0235276	−	0.999723i	$-0.507490\pi$
$193$	−126.167	−0.0470553	−0.0235276	+	0.999723i	$0.507490\pi$
$194$	−628.472	−0.232586
$195$	−4351.58	−1.59807
$196$	0	0
$197$	3084.18	1.11542	0.557712	−	0.830034i	$-0.311679\pi$
$197$	3084.18	1.11542	0.557712	+	0.830034i	$0.311679\pi$
$198$	1236.76	0.443904
$199$	5119.28	1.82360	0.911798	−	0.410638i	$-0.134694\pi$
$199$	5119.28	1.82360	0.911798	+	0.410638i	$0.134694\pi$
$200$	1591.54	0.562695
$201$	−1361.73	−0.477855
$202$	−729.411	−0.254065
$203$	0	0
$204$	81.8587	0.0280944
$205$	−6714.61	−2.28765
$206$	2271.83	0.768379
$207$	−329.718	−0.110710
$208$	1087.03	0.362364
$209$	4408.77	1.45914
$210$	0	0
$211$	1904.78	0.621471	0.310735	−	0.950496i	$-0.399425\pi$
$211$	1904.78	0.621471	0.310735	+	0.950496i	$0.399425\pi$
$212$	−26.1223	−0.00846269
$213$	−2725.39	−0.876716
$214$	223.397	0.0713602
$215$	−96.1428	−0.0304971
$216$	−1176.81	−0.370703
$217$	0	0
$218$	2485.18	0.772100
$219$	245.705	0.0758138
$220$	3105.53	0.951703
$221$	390.691	0.118917
$222$	279.694	0.0845579
$223$	2576.01	0.773552	0.386776	−	0.922174i	$-0.373589\pi$
$223$	2576.01	0.773552	0.386776	+	0.922174i	$0.373589\pi$
$224$	0	0
$225$	−2851.96	−0.845025
$226$	−3545.62	−1.04359
$227$	−388.747	−0.113665	−0.0568327	−	0.998384i	$-0.518100\pi$
$227$	−388.747	−0.113665	−0.0568327	+	0.998384i	$0.518100\pi$
$228$	−1454.88	−0.422597
$229$	−3526.32	−1.01758	−0.508790	−	0.860891i	$-0.669907\pi$
$229$	−3526.32	−1.01758	−0.508790	+	0.860891i	$0.669907\pi$
$230$	−827.927	−0.237356
$231$	0	0
$232$	184.602	0.0522401
$233$	−1007.46	−0.283267	−0.141633	−	0.989919i	$-0.545235\pi$
$233$	−1007.46	−0.283267	−0.141633	+	0.989919i	$0.545235\pi$
$234$	−1947.90	−0.544179
$235$	−5394.69	−1.49749
$236$	2483.47	0.684999
$237$	2131.16	0.584108
$238$	0	0
$239$	1439.80	0.389676	0.194838	−	0.980835i	$-0.437582\pi$
$239$	1439.80	0.389676	0.194838	+	0.980835i	$0.437582\pi$
$240$	−1024.82	−0.275632
$241$	−2047.32	−0.547217	−0.273608	−	0.961841i	$-0.588217\pi$
$241$	−2047.32	−0.547217	−0.273608	+	0.961841i	$0.588217\pi$
$242$	1059.46	0.281423
$243$	3486.22	0.920334
$244$	−424.065	−0.111262
$245$	0	0
$246$	2655.27	0.688187
$247$	−6943.79	−1.78876
$248$	−1777.37	−0.455094
$249$	3299.54	0.839759
$250$	−2661.70	−0.673362
$251$	6266.17	1.57577	0.787883	−	0.615825i	$-0.211177\pi$
$251$	6266.17	1.57577	0.787883	+	0.615825i	$0.211177\pi$
$252$	0	0
$253$	−992.132	−0.246541
$254$	−816.798	−0.201773
$255$	−368.332	−0.0904542
$256$	256.000	0.0625000
$257$	5012.78	1.21669	0.608344	−	0.793674i	$-0.291834\pi$
$257$	5012.78	1.21669	0.608344	+	0.793674i	$0.291834\pi$
$258$	38.0194	0.00917436
$259$	0	0
$260$	−4891.19	−1.16669
$261$	−330.797	−0.0784513
$262$	5799.99	1.36765
$263$	−5343.47	−1.25282	−0.626411	−	0.779493i	$-0.715477\pi$
$263$	−5343.47	−1.25282	−0.626411	+	0.779493i	$0.715477\pi$
$264$	−1228.07	−0.286298
$265$	117.540	0.0272469
$266$	0	0
$267$	5513.93	1.26385
$268$	−1530.59	−0.348863
$269$	3353.50	0.760099	0.380049	−	0.924966i	$-0.375907\pi$
$269$	3353.50	0.760099	0.380049	+	0.924966i	$0.375907\pi$
$270$	5295.18	1.19353
$271$	−5898.14	−1.32209	−0.661046	−	0.750346i	$-0.729887\pi$
$271$	−5898.14	−1.32209	−0.661046	+	0.750346i	$0.729887\pi$
$272$	92.0094	0.0205106
$273$	0	0
$274$	1058.56	0.233393
$275$	−8581.62	−1.88179
$276$	327.401	0.0714031
$277$	375.574	0.0814659	0.0407329	−	0.999170i	$-0.487031\pi$
$277$	375.574	0.0814659	0.0407329	+	0.999170i	$0.487031\pi$
$278$	2779.78	0.599712
$279$	3184.96	0.683435
$280$	0	0
$281$	8138.46	1.72776	0.863879	−	0.503700i	$-0.168028\pi$
$281$	8138.46	1.72776	0.863879	+	0.503700i	$0.168028\pi$
$282$	2133.32	0.450486
$283$	5677.96	1.19265	0.596324	−	0.802744i	$-0.296627\pi$
$283$	5677.96	1.19265	0.596324	+	0.802744i	$0.296627\pi$
$284$	−3063.34	−0.640056
$285$	6546.40	1.36062
$286$	−5861.27	−1.21183
$287$	0	0
$288$	−458.739	−0.0938591
$289$	−4879.93	−0.993269
$290$	−830.634	−0.168195
$291$	−1118.28	−0.225273
$292$	276.174	0.0553487
$293$	−6359.25	−1.26796	−0.633978	−	0.773351i	$-0.718579\pi$
$293$	−6359.25	−1.26796	−0.633978	+	0.773351i	$0.718579\pi$
$294$	0	0
$295$	−11174.6	−2.20546
$296$	314.377	0.0617325
$297$	6345.39	1.23972
$298$	−4761.41	−0.925574
$299$	1562.60	0.302233
$300$	2831.92	0.545003
$301$	0	0
$302$	841.691	0.160377
$303$	−1297.88	−0.246077
$304$	−1635.29	−0.308522
$305$	1908.12	0.358225
$306$	−164.876	−0.0308017
$307$	−5457.19	−1.01452	−0.507261	−	0.861793i	$-0.669342\pi$
$307$	−5457.19	−1.01452	−0.507261	+	0.861793i	$0.669342\pi$
$308$	0	0
$309$	4042.40	0.744220
$310$	7997.47	1.46524
$311$	−10561.6	−1.92571	−0.962854	−	0.270022i	$-0.912969\pi$
$311$	−10561.6	−1.92571	−0.962854	+	0.270022i	$0.912969\pi$
$312$	1934.21	0.350971
$313$	−9944.05	−1.79575	−0.897877	−	0.440246i	$-0.854891\pi$
$313$	−9944.05	−1.79575	−0.897877	+	0.440246i	$0.854891\pi$
$314$	378.926	0.0681019
$315$	0	0
$316$	2395.43	0.426434
$317$	10308.7	1.82647	0.913237	−	0.407428i	$-0.133574\pi$
$317$	10308.7	1.82647	0.913237	+	0.407428i	$0.133574\pi$
$318$	−46.4809	−0.00819661
$319$	−995.376	−0.174703
$320$	−1151.90	−0.201228
$321$	397.502	0.0691165
$322$	0	0
$323$	−587.744	−0.101248
$324$	−545.722	−0.0935737
$325$	13516.0	2.30687
$326$	6923.38	1.17623
$327$	4422.02	0.747824
$328$	2984.53	0.502419
$329$	0	0
$330$	5525.84	0.921780
$331$	2570.62	0.426870	0.213435	−	0.976957i	$-0.431535\pi$
$331$	2570.62	0.426870	0.213435	+	0.976957i	$0.431535\pi$
$332$	3708.70	0.613076
$333$	−563.348	−0.0927065
$334$	2961.51	0.485169
$335$	6887.02	1.12322
$336$	0	0
$337$	74.1859	0.0119916	0.00599580	−	0.999982i	$-0.498091\pi$
$337$	74.1859	0.0119916	0.00599580	+	0.999982i	$0.498091\pi$
$338$	4837.48	0.778474
$339$	−6308.91	−1.01078
$340$	−414.006	−0.0660371
$341$	9583.62	1.52194
$342$	2930.36	0.463321
$343$	0	0
$344$	42.7339	0.00669785
$345$	−1473.18	−0.229893
$346$	5932.68	0.921800
$347$	6288.18	0.972816	0.486408	−	0.873732i	$-0.338307\pi$
$347$	6288.18	0.972816	0.486408	+	0.873732i	$0.338307\pi$
$348$	328.472	0.0505975
$349$	1527.11	0.234224	0.117112	−	0.993119i	$-0.462636\pi$
$349$	1527.11	0.234224	0.117112	+	0.993119i	$0.462636\pi$
$350$	0	0
$351$	−9993.95	−1.51977
$352$	−1380.36	−0.209015
$353$	−1304.25	−0.196652	−0.0983260	−	0.995154i	$-0.531349\pi$
$353$	−1304.25	−0.196652	−0.0983260	+	0.995154i	$0.531349\pi$
$354$	4418.97	0.663462
$355$	13783.8	2.06076
$356$	6197.67	0.922685
$357$	0	0
$358$	−9547.34	−1.40948
$359$	2620.95	0.385316	0.192658	−	0.981266i	$-0.438289\pi$
$359$	2620.95	0.385316	0.192658	+	0.981266i	$0.438289\pi$
$360$	2064.14	0.302194
$361$	3587.04	0.522969
$362$	−6570.31	−0.953945
$363$	1885.15	0.272575
$364$	0	0
$365$	−1242.67	−0.178204
$366$	−754.562	−0.107764
$367$	−1416.43	−0.201464	−0.100732	−	0.994914i	$-0.532118\pi$
$367$	−1416.43	−0.201464	−0.100732	+	0.994914i	$0.532118\pi$
$368$	368.000	0.0521286
$369$	−5348.13	−0.754505
$370$	−1414.57	−0.198757
$371$	0	0
$372$	−3162.58	−0.440785
$373$	−6774.40	−0.940389	−0.470194	−	0.882563i	$-0.655816\pi$
$373$	−6774.40	−0.940389	−0.470194	+	0.882563i	$0.655816\pi$
$374$	−496.117	−0.0685924
$375$	−4736.11	−0.652191
$376$	2397.85	0.328882
$377$	1567.71	0.214168
$378$	0	0
$379$	4197.46	0.568889	0.284445	−	0.958692i	$-0.408191\pi$
$379$	4197.46	0.568889	0.284445	+	0.958692i	$0.408191\pi$
$380$	7358.17	0.993332
$381$	−1453.37	−0.195429
$382$	1650.57	0.221075
$383$	1455.35	0.194164	0.0970819	−	0.995276i	$-0.469049\pi$
$383$	1455.35	0.194164	0.0970819	+	0.995276i	$0.469049\pi$
$384$	455.515	0.0605349
$385$	0	0
$386$	−252.333	−0.0332731
$387$	−76.5770	−0.0100585
$388$	−1256.94	−0.164463
$389$	−10330.7	−1.34649	−0.673246	−	0.739419i	$-0.735100\pi$
$389$	−10330.7	−1.34649	−0.673246	+	0.739419i	$0.735100\pi$
$390$	−8703.17	−1.13001
$391$	132.264	0.0171070
$392$	0	0
$393$	10320.2	1.32465
$394$	6168.36	0.788724
$395$	−10778.5	−1.37297
$396$	2473.53	0.313887
$397$	−3534.07	−0.446775	−0.223388	−	0.974730i	$-0.571712\pi$
$397$	−3534.07	−0.446775	−0.223388	+	0.974730i	$0.571712\pi$
$398$	10238.6	1.28948
$399$	0	0
$400$	3183.08	0.397885
$401$	−9140.78	−1.13833	−0.569163	−	0.822225i	$-0.692733\pi$
$401$	−9140.78	−1.13833	−0.569163	+	0.822225i	$0.692733\pi$
$402$	−2723.45	−0.337894
$403$	−15094.2	−1.86574
$404$	−1458.82	−0.179651
$405$	2455.53	0.301275
$406$	0	0
$407$	−1695.13	−0.206448
$408$	163.717	0.0198657
$409$	−7171.03	−0.866955	−0.433477	−	0.901164i	$-0.642714\pi$
$409$	−7171.03	−0.866955	−0.433477	+	0.901164i	$0.642714\pi$
$410$	−13429.2	−1.61761
$411$	1883.55	0.226055
$412$	4543.67	0.543326
$413$	0	0
$414$	−659.437	−0.0782839
$415$	−16687.6	−1.97389
$416$	2174.06	0.256230
$417$	4946.21	0.580856
$418$	8817.54	1.03177
$419$	−6995.48	−0.815636	−0.407818	−	0.913063i	$-0.633710\pi$
$419$	−6995.48	−0.815636	−0.407818	+	0.913063i	$0.633710\pi$
$420$	0	0
$421$	6600.97	0.764161	0.382080	−	0.924129i	$-0.375208\pi$
$421$	6600.97	0.764161	0.382080	+	0.924129i	$0.375208\pi$
$422$	3809.56	0.439446
$423$	−4296.82	−0.493898
$424$	−52.2447	−0.00598402
$425$	1144.04	0.130574
$426$	−5450.78	−0.619932
$427$	0	0
$428$	446.793	0.0504593
$429$	−10429.3	−1.17373
$430$	−192.286	−0.0215647
$431$	−6733.50	−0.752532	−0.376266	−	0.926512i	$-0.622792\pi$
$431$	−6733.50	−0.752532	−0.376266	+	0.926512i	$0.622792\pi$
$432$	−2353.62	−0.262127
$433$	−16434.7	−1.82402	−0.912009	−	0.410170i	$-0.865469\pi$
$433$	−16434.7	−1.82402	−0.912009	+	0.410170i	$0.865469\pi$
$434$	0	0
$435$	−1477.99	−0.162907
$436$	4970.37	0.545957
$437$	−2350.74	−0.257325
$438$	491.411	0.0536085
$439$	15748.8	1.71219	0.856093	−	0.516822i	$-0.172885\pi$
$439$	15748.8	1.71219	0.856093	+	0.516822i	$0.172885\pi$
$440$	6211.06	0.672956
$441$	0	0
$442$	781.381	0.0840871
$443$	−3594.34	−0.385490	−0.192745	−	0.981249i	$-0.561739\pi$
$443$	−3594.34	−0.385490	−0.192745	+	0.981249i	$0.561739\pi$
$444$	559.389	0.0597915
$445$	−27887.1	−2.97073
$446$	5152.01	0.546984
$447$	−8472.24	−0.896473
$448$	0	0
$449$	4998.01	0.525324	0.262662	−	0.964888i	$-0.415400\pi$
$449$	4998.01	0.525324	0.262662	+	0.964888i	$0.415400\pi$
$450$	−5703.92	−0.597523
$451$	−16092.7	−1.68021
$452$	−7091.24	−0.737929
$453$	1497.67	0.155334
$454$	−777.494	−0.0803736
$455$	0	0
$456$	−2909.77	−0.298821
$457$	−7277.12	−0.744878	−0.372439	−	0.928057i	$-0.621478\pi$
$457$	−7277.12	−0.744878	−0.372439	+	0.928057i	$0.621478\pi$
$458$	−7052.64	−0.719538
$459$	−845.919	−0.0860221
$460$	−1655.85	−0.167836
$461$	12878.9	1.30115	0.650575	−	0.759442i	$-0.274528\pi$
$461$	12878.9	1.30115	0.650575	+	0.759442i	$0.274528\pi$
$462$	0	0
$463$	−6951.02	−0.697713	−0.348857	−	0.937176i	$-0.613430\pi$
$463$	−6951.02	−0.697713	−0.348857	+	0.937176i	$0.613430\pi$
$464$	369.203	0.0369393
$465$	14230.3	1.41917
$466$	−2014.93	−0.200300
$467$	2405.13	0.238321	0.119161	−	0.992875i	$-0.461980\pi$
$467$	2405.13	0.238321	0.119161	+	0.992875i	$0.461980\pi$
$468$	−3895.79	−0.384793
$469$	0	0
$470$	−10789.4	−1.05889
$471$	674.243	0.0659607
$472$	4966.93	0.484368
$473$	−230.422	−0.0223992
$474$	4262.31	0.413027
$475$	−20333.1	−1.96410
$476$	0	0
$477$	93.6197	0.00898649
$478$	2879.59	0.275543
$479$	−806.577	−0.0769383	−0.0384692	−	0.999260i	$-0.512248\pi$
$479$	−806.577	−0.0769383	−0.0384692	+	0.999260i	$0.512248\pi$
$480$	−2049.64	−0.194901
$481$	2669.82	0.253084
$482$	−4094.63	−0.386941
$483$	0	0
$484$	2118.91	0.198996
$485$	5655.75	0.529514
$486$	6972.44	0.650774
$487$	18680.6	1.73819	0.869094	−	0.494647i	$-0.164703\pi$
$487$	18680.6	1.73819	0.869094	+	0.494647i	$0.164703\pi$
$488$	−848.129	−0.0786742
$489$	12319.2	1.13925
$490$	0	0
$491$	−17605.1	−1.61814	−0.809070	−	0.587712i	$-0.800029\pi$
$491$	−17605.1	−1.61814	−0.809070	+	0.587712i	$0.800029\pi$
$492$	5310.55	0.486622
$493$	132.696	0.0121224
$494$	−13887.6	−1.26484
$495$	−11129.9	−1.01061
$496$	−3554.74	−0.321800
$497$	0	0
$498$	6599.09	0.593799
$499$	10.8592	0.000974201 0	0.000487101	−	1.00000i	$-0.499845\pi$
$499$	10.8592	0.000974201 0	0.000487101		1.00000i	$0.499845\pi$
$500$	−5323.40	−0.476139
$501$	5269.58	0.469915
$502$	12532.3	1.11423
$503$	5870.47	0.520380	0.260190	−	0.965557i	$-0.416215\pi$
$503$	5870.47	0.520380	0.260190	+	0.965557i	$0.416215\pi$
$504$	0	0
$505$	6564.12	0.578414
$506$	−1984.26	−0.174331
$507$	8607.59	0.753997
$508$	−1633.60	−0.142675
$509$	−19054.4	−1.65928	−0.829640	−	0.558299i	$-0.811454\pi$
$509$	−19054.4	−1.65928	−0.829640	+	0.558299i	$0.811454\pi$
$510$	−736.663	−0.0639608
$511$	0	0
$512$	512.000	0.0441942
$513$	15034.6	1.29395
$514$	10025.6	0.860328
$515$	−20444.7	−1.74932
$516$	76.0389	0.00648726
$517$	−12929.3	−1.09986
$518$	0	0
$519$	10556.3	0.892818
$520$	−9782.38	−0.824973
$521$	13849.9	1.16463	0.582317	−	0.812962i	$-0.302146\pi$
$521$	13849.9	1.16463	0.582317	+	0.812962i	$0.302146\pi$
$522$	−661.593	−0.0554734
$523$	22916.8	1.91602	0.958012	−	0.286726i	$-0.0925671\pi$
$523$	22916.8	1.91602	0.958012	+	0.286726i	$0.0925671\pi$
$524$	11600.0	0.967076
$525$	0	0
$526$	−10686.9	−0.885879
$527$	−1277.62	−0.105605
$528$	−2456.15	−0.202443
$529$	529.000	0.0434783
$530$	235.080	0.0192665
$531$	−8900.48	−0.727397
$532$	0	0
$533$	25345.9	2.05976
$534$	11027.9	0.893674
$535$	−2010.39	−0.162461
$536$	−3061.17	−0.246684
$537$	−16988.1	−1.36516
$538$	6707.00	0.537471
$539$	0	0
$540$	10590.4	0.843957
$541$	−10238.4	−0.813648	−0.406824	−	0.913507i	$-0.633364\pi$
$541$	−10238.4	−0.813648	−0.406824	+	0.913507i	$0.633364\pi$
$542$	−11796.3	−0.934860
$543$	−11690.9	−0.923951
$544$	184.019	0.0145032
$545$	−22364.7	−1.75779
$546$	0	0
$547$	−1093.08	−0.0854420	−0.0427210	−	0.999087i	$-0.513603\pi$
$547$	−1093.08	−0.0854420	−0.0427210	+	0.999087i	$0.513603\pi$
$548$	2117.11	0.165034
$549$	1519.80	0.118149
$550$	−17163.2	−1.33062
$551$	−2358.42	−0.182345
$552$	654.803	0.0504896
$553$	0	0
$554$	751.148	0.0576051
$555$	−2517.03	−0.192508
$556$	5559.56	0.424061
$557$	21834.7	1.66098	0.830490	−	0.557033i	$-0.188060\pi$
$557$	21834.7	1.66098	0.830490	+	0.557033i	$0.188060\pi$
$558$	6369.91	0.483262
$559$	362.914	0.0274591
$560$	0	0
$561$	−882.768	−0.0664358
$562$	16276.9	1.22171
$563$	23343.3	1.74743	0.873713	−	0.486442i	$-0.161705\pi$
$563$	23343.3	1.74743	0.873713	+	0.486442i	$0.161705\pi$
$564$	4266.63	0.318542
$565$	31907.7	2.37587
$566$	11355.9	0.843330
$567$	0	0
$568$	−6126.69	−0.452588
$569$	7918.28	0.583394	0.291697	−	0.956511i	$-0.405780\pi$
$569$	7918.28	0.583394	0.291697	+	0.956511i	$0.405780\pi$
$570$	13092.8	0.962100
$571$	−5129.71	−0.375957	−0.187979	−	0.982173i	$-0.560194\pi$
$571$	−5129.71	−0.375957	−0.187979	+	0.982173i	$0.560194\pi$
$572$	−11722.5	−0.856896
$573$	2936.95	0.214124
$574$	0	0
$575$	4575.68	0.331859
$576$	−917.477	−0.0663684
$577$	12030.7	0.868018	0.434009	−	0.900909i	$-0.357099\pi$
$577$	12030.7	0.868018	0.434009	+	0.900909i	$0.357099\pi$
$578$	−9759.86	−0.702347
$579$	−448.990	−0.0322269
$580$	−1661.27	−0.118932
$581$	0	0
$582$	−2236.55	−0.159292
$583$	281.704	0.0200120
$584$	552.347	0.0391375
$585$	17529.5	1.23890
$586$	−12718.5	−0.896581
$587$	6880.38	0.483788	0.241894	−	0.970303i	$-0.422231\pi$
$587$	6880.38	0.483788	0.241894	+	0.970303i	$0.422231\pi$
$588$	0	0
$589$	22707.2	1.58851
$590$	−22349.2	−1.55950
$591$	10975.7	0.763926
$592$	628.755	0.0436514
$593$	−9730.88	−0.673860	−0.336930	−	0.941530i	$-0.609389\pi$
$593$	−9730.88	−0.673860	−0.336930	+	0.941530i	$0.609389\pi$
$594$	12690.8	0.876614
$595$	0	0
$596$	−9522.82	−0.654480
$597$	18218.0	1.24893
$598$	3125.20	0.213711
$599$	21409.4	1.46037	0.730187	−	0.683247i	$-0.239433\pi$
$599$	21409.4	1.46037	0.730187	+	0.683247i	$0.239433\pi$
$600$	5663.83	0.385375
$601$	−24045.1	−1.63198	−0.815991	−	0.578064i	$-0.803808\pi$
$601$	−24045.1	−1.63198	−0.815991	+	0.578064i	$0.803808\pi$
$602$	0	0
$603$	5485.46	0.370456
$604$	1683.38	0.113404
$605$	−9534.26	−0.640699
$606$	−2595.76	−0.174003
$607$	−14233.3	−0.951751	−0.475875	−	0.879513i	$-0.657869\pi$
$607$	−14233.3	−0.951751	−0.475875	+	0.879513i	$0.657869\pi$
$608$	−3270.59	−0.218158
$609$	0	0
$610$	3816.24	0.253304
$611$	20363.5	1.34831
$612$	−329.752	−0.0217801
$613$	23636.2	1.55736	0.778678	−	0.627424i	$-0.215891\pi$
$613$	23636.2	1.55736	0.778678	+	0.627424i	$0.215891\pi$
$614$	−10914.4	−0.717375
$615$	−23895.3	−1.56675
$616$	0	0
$617$	11758.3	0.767212	0.383606	−	0.923497i	$-0.374682\pi$
$617$	11758.3	0.767212	0.383606	+	0.923497i	$0.374682\pi$
$618$	8084.80	0.526243
$619$	−12821.0	−0.832501	−0.416251	−	0.909250i	$-0.636656\pi$
$619$	−12821.0	−0.832501	−0.416251	+	0.909250i	$0.636656\pi$
$620$	15994.9	1.03608
$621$	−3383.33	−0.218629
$622$	−21123.3	−1.36168
$623$	0	0
$624$	3868.42	0.248174
$625$	−914.661	−0.0585383
$626$	−19888.1	−1.26979
$627$	15689.5	0.999329
$628$	757.851	0.0481553
$629$	225.982	0.0143251
$630$	0	0
$631$	−12742.3	−0.803904	−0.401952	−	0.915661i	$-0.631668\pi$
$631$	−12742.3	−0.803904	−0.401952	+	0.915661i	$0.631668\pi$
$632$	4790.85	0.301535
$633$	6778.55	0.425629
$634$	20617.3	1.29151
$635$	7350.53	0.459365
$636$	−92.9619	−0.00579588
$637$	0	0
$638$	−1990.75	−0.123534
$639$	10978.7	0.679672
$640$	−2303.80	−0.142290
$641$	3519.47	0.216865	0.108433	−	0.994104i	$-0.465417\pi$
$641$	3519.47	0.216865	0.108433	+	0.994104i	$0.465417\pi$
$642$	795.004	0.0488728
$643$	3771.11	0.231288	0.115644	−	0.993291i	$-0.463107\pi$
$643$	3771.11	0.231288	0.115644	+	0.993291i	$0.463107\pi$
$644$	0	0
$645$	−342.145	−0.0208867
$646$	−1175.49	−0.0715928
$647$	9294.83	0.564787	0.282394	−	0.959299i	$-0.408872\pi$
$647$	9294.83	0.564787	0.282394	+	0.959299i	$0.408872\pi$
$648$	−1091.44	−0.0661666
$649$	−26781.8	−1.61984
$650$	27032.0	1.63120
$651$	0	0
$652$	13846.8	0.831720
$653$	24527.2	1.46987	0.734934	−	0.678138i	$-0.237213\pi$
$653$	24527.2	1.46987	0.734934	+	0.678138i	$0.237213\pi$
$654$	8844.05	0.528792
$655$	−52195.3	−3.11365
$656$	5969.07	0.355264
$657$	−989.777	−0.0587745
$658$	0	0
$659$	−26513.2	−1.56724	−0.783618	−	0.621244i	$-0.786628\pi$
$659$	−26513.2	−1.56724	−0.783618	+	0.621244i	$0.786628\pi$
$660$	11051.7	0.651797
$661$	−24583.9	−1.44660	−0.723301	−	0.690532i	$-0.757376\pi$
$661$	−24583.9	−1.44660	−0.723301	+	0.690532i	$0.757376\pi$
$662$	5141.24	0.301843
$663$	1390.35	0.0814433
$664$	7417.39	0.433510
$665$	0	0
$666$	−1126.70	−0.0655534
$667$	530.730	0.0308095
$668$	5923.02	0.343067
$669$	9167.26	0.529786
$670$	13774.0	0.794236
$671$	4573.13	0.263105
$672$	0	0
$673$	6229.66	0.356814	0.178407	−	0.983957i	$-0.442906\pi$
$673$	6229.66	0.356814	0.178407	+	0.983957i	$0.442906\pi$
$674$	148.372	0.00847933
$675$	−29264.7	−1.66874
$676$	9674.96	0.550464
$677$	19209.1	1.09050	0.545249	−	0.838274i	$-0.316435\pi$
$677$	19209.1	1.09050	0.545249	+	0.838274i	$0.316435\pi$
$678$	−12617.8	−0.714727
$679$	0	0
$680$	−828.011	−0.0466953
$681$	−1383.44	−0.0778465
$682$	19167.2	1.07618
$683$	22247.1	1.24636	0.623178	−	0.782080i	$-0.285841\pi$
$683$	22247.1	1.24636	0.623178	+	0.782080i	$0.285841\pi$
$684$	5860.72	0.327617
$685$	−9526.17	−0.531352
$686$	0	0
$687$	−12549.2	−0.696914
$688$	85.4679	0.00473609
$689$	−443.683	−0.0245326
$690$	−2946.35	−0.162559
$691$	−28905.7	−1.59135	−0.795675	−	0.605724i	$-0.792884\pi$
$691$	−28905.7	−1.59135	−0.795675	+	0.605724i	$0.792884\pi$
$692$	11865.4	0.651811
$693$	0	0
$694$	12576.4	0.687885
$695$	−25015.8	−1.36533
$696$	656.944	0.0357779
$697$	2145.35	0.116587
$698$	3054.22	0.165622
$699$	−3585.27	−0.194002
$700$	0	0
$701$	6015.44	0.324108	0.162054	−	0.986782i	$-0.448188\pi$
$701$	6015.44	0.324108	0.162054	+	0.986782i	$0.448188\pi$
$702$	−19987.9	−1.07464
$703$	−4016.40	−0.215479
$704$	−2760.71	−0.147796
$705$	−19198.1	−1.02559
$706$	−2608.50	−0.139054
$707$	0	0
$708$	8837.94	0.469138
$709$	−12028.4	−0.637147	−0.318574	−	0.947898i	$-0.603204\pi$
$709$	−12028.4	−0.637147	−0.318574	+	0.947898i	$0.603204\pi$
$710$	27567.6	1.45718
$711$	−8584.96	−0.452828
$712$	12395.3	0.652437
$713$	−5109.94	−0.268400
$714$	0	0
$715$	52746.8	2.75891
$716$	−19094.7	−0.996650
$717$	5123.82	0.266879
$718$	5241.90	0.272459
$719$	−8712.38	−0.451901	−0.225950	−	0.974139i	$-0.572549\pi$
$719$	−8712.38	−0.451901	−0.225950	+	0.974139i	$0.572549\pi$
$720$	4128.28	0.213683
$721$	0	0
$722$	7174.09	0.369795
$723$	−7285.81	−0.374775
$724$	−13140.6	−0.674541
$725$	4590.64	0.235162
$726$	3770.30	0.192739
$727$	6603.87	0.336897	0.168448	−	0.985710i	$-0.446124\pi$
$727$	6603.87	0.336897	0.168448	+	0.985710i	$0.446124\pi$
$728$	0	0
$729$	16090.1	0.817460
$730$	−2485.34	−0.126009
$731$	30.7182	0.00155424
$732$	−1509.12	−0.0762005
$733$	27608.7	1.39120	0.695601	−	0.718428i	$-0.255138\pi$
$733$	27608.7	1.39120	0.695601	+	0.718428i	$0.255138\pi$
$734$	−2832.87	−0.142456
$735$	0	0
$736$	736.000	0.0368605
$737$	16505.9	0.824969
$738$	−10696.3	−0.533516
$739$	−19825.8	−0.986880	−0.493440	−	0.869780i	$-0.664261\pi$
$739$	−19825.8	−0.986880	−0.493440	+	0.869780i	$0.664261\pi$
$740$	−2829.15	−0.140542
$741$	−24710.9	−1.22507
$742$	0	0
$743$	36526.2	1.80352	0.901760	−	0.432237i	$-0.142276\pi$
$743$	36526.2	1.80352	0.901760	+	0.432237i	$0.142276\pi$
$744$	−6325.15	−0.311682
$745$	42848.9	2.10720
$746$	−13548.8	−0.664955
$747$	−13291.6	−0.651022
$748$	−992.233	−0.0485022
$749$	0	0
$750$	−9472.21	−0.461169
$751$	−20082.8	−0.975809	−0.487905	−	0.872897i	$-0.662239\pi$
$751$	−20082.8	−0.975809	−0.487905	+	0.872897i	$0.662239\pi$
$752$	4795.70	0.232555
$753$	22299.5	1.07920
$754$	3135.42	0.151440
$755$	−7574.54	−0.365120
$756$	0	0
$757$	−33844.9	−1.62498	−0.812492	−	0.582973i	$-0.801889\pi$
$757$	−33844.9	−1.62498	−0.812492	+	0.582973i	$0.801889\pi$
$758$	8394.92	0.402266
$759$	−3530.71	−0.168849
$760$	14716.3	0.702392
$761$	782.869	0.0372917	0.0186458	−	0.999826i	$-0.494065\pi$
$761$	782.869	0.0372917	0.0186458	+	0.999826i	$0.494065\pi$
$762$	−2906.75	−0.138189
$763$	0	0
$764$	3301.14	0.156324
$765$	1483.75	0.0701244
$766$	2910.69	0.137295
$767$	42181.2	1.98576
$768$	911.030	0.0428046
$769$	34731.4	1.62867	0.814334	−	0.580396i	$-0.197102\pi$
$769$	34731.4	1.62867	0.814334	+	0.580396i	$0.197102\pi$
$770$	0	0
$771$	17839.0	0.833278
$772$	−504.666	−0.0235276
$773$	−28891.2	−1.34430	−0.672151	−	0.740414i	$-0.734629\pi$
$773$	−28891.2	−1.34430	−0.672151	+	0.740414i	$0.734629\pi$
$774$	−153.154	−0.00711241
$775$	−44199.4	−2.04863
$776$	−2513.89	−0.116293
$777$	0	0
$778$	−20661.3	−0.952113
$779$	−38129.6	−1.75370
$780$	−17406.3	−0.799034
$781$	33035.2	1.51356
$782$	264.527	0.0120965
$783$	−3394.39	−0.154924
$784$	0	0
$785$	−3410.03	−0.155043
$786$	20640.5	0.936669
$787$	6235.78	0.282442	0.141221	−	0.989978i	$-0.454897\pi$
$787$	6235.78	0.282442	0.141221	+	0.989978i	$0.454897\pi$
$788$	12336.7	0.557712
$789$	−19015.9	−0.858026
$790$	−21556.9	−0.970837
$791$	0	0
$792$	4947.05	0.221952
$793$	−7202.66	−0.322539
$794$	−7068.14	−0.315918
$795$	418.291	0.0186607
$796$	20477.1	0.911798
$797$	33498.5	1.48881	0.744404	−	0.667730i	$-0.232734\pi$
$797$	33498.5	1.48881	0.744404	+	0.667730i	$0.232734\pi$
$798$	0	0
$799$	1723.63	0.0763176
$800$	6366.16	0.281347
$801$	−22211.8	−0.979795
$802$	−18281.6	−0.804918
$803$	−2978.27	−0.130885
$804$	−5446.91	−0.238927
$805$	0	0
$806$	−30188.3	−1.31928
$807$	11934.1	0.520572
$808$	−2917.64	−0.127033
$809$	22047.6	0.958161	0.479080	−	0.877771i	$-0.340970\pi$
$809$	22047.6	0.958161	0.479080	+	0.877771i	$0.340970\pi$
$810$	4911.06	0.213034
$811$	45996.9	1.99158	0.995790	−	0.0916673i	$-0.0292196\pi$
$811$	45996.9	1.99158	0.995790	+	0.0916673i	$0.0292196\pi$
$812$	0	0
$813$	−20989.8	−0.905466
$814$	−3390.26	−0.145981
$815$	−62304.9	−2.67785
$816$	327.435	0.0140472
$817$	−545.957	−0.0233790
$818$	−14342.1	−0.613029
$819$	0	0
$820$	−26858.4	−1.14383
$821$	21679.0	0.921561	0.460780	−	0.887514i	$-0.347570\pi$
$821$	21679.0	0.921561	0.460780	+	0.887514i	$0.347570\pi$
$822$	3767.10	0.159845
$823$	18192.0	0.770514	0.385257	−	0.922809i	$-0.374113\pi$
$823$	18192.0	0.770514	0.385257	+	0.922809i	$0.374113\pi$
$824$	9087.33	0.384190
$825$	−30539.5	−1.28879
$826$	0	0
$827$	12583.1	0.529088	0.264544	−	0.964374i	$-0.414778\pi$
$827$	12583.1	0.529088	0.264544	+	0.964374i	$0.414778\pi$
$828$	−1318.87	−0.0553551
$829$	−1082.03	−0.0453321	−0.0226660	−	0.999743i	$-0.507215\pi$
$829$	−1082.03	−0.0453321	−0.0226660	+	0.999743i	$0.507215\pi$
$830$	−33375.3	−1.39575
$831$	1336.56	0.0557939
$832$	4348.11	0.181182
$833$	0	0
$834$	9892.43	0.410727
$835$	−26651.2	−1.10456
$836$	17635.1	0.729572
$837$	32681.7	1.34964
$838$	−13991.0	−0.576742
$839$	24091.4	0.991331	0.495666	−	0.868513i	$-0.334924\pi$
$839$	24091.4	0.991331	0.495666	+	0.868513i	$0.334924\pi$
$840$	0	0
$841$	−23856.5	−0.978168
$842$	13201.9	0.540343
$843$	28962.4	1.18330
$844$	7619.11	0.310735
$845$	−43533.4	−1.77230
$846$	−8593.65	−0.349238
$847$	0	0
$848$	−104.489	−0.00423134
$849$	20206.2	0.816814
$850$	2288.07	0.0923298
$851$	903.835	0.0364078
$852$	−10901.6	−0.438358
$853$	−6595.07	−0.264726	−0.132363	−	0.991201i	$-0.542256\pi$
$853$	−6595.07	−0.264726	−0.132363	+	0.991201i	$0.542256\pi$
$854$	0	0
$855$	−26370.9	−1.05481
$856$	893.587	0.0356801
$857$	−27321.0	−1.08900	−0.544498	−	0.838762i	$-0.683280\pi$
$857$	−27321.0	−1.08900	−0.544498	+	0.838762i	$0.683280\pi$
$858$	−20858.6	−0.829954
$859$	15291.6	0.607384	0.303692	−	0.952770i	$-0.401781\pi$
$859$	15291.6	0.607384	0.303692	+	0.952770i	$0.401781\pi$
$860$	−384.571	−0.0152486
$861$	0	0
$862$	−13467.0	−0.532121
$863$	−31878.5	−1.25742	−0.628712	−	0.777638i	$-0.716418\pi$
$863$	−31878.5	−1.25742	−0.628712	+	0.777638i	$0.716418\pi$
$864$	−4707.24	−0.185351
$865$	−53389.4	−2.09861
$866$	−32869.3	−1.28978
$867$	−17366.3	−0.680264
$868$	0	0
$869$	−25832.4	−1.00840
$870$	−2955.99	−0.115192
$871$	−25996.7	−1.01133
$872$	9940.74	0.386050
$873$	4504.76	0.174642
$874$	−4701.47	−0.181956
$875$	0	0
$876$	982.822	0.0379069
$877$	−237.037	−0.00912675	−0.00456338	−	0.999990i	$-0.501453\pi$
$877$	−237.037	−0.00912675	−0.00456338	+	0.999990i	$0.501453\pi$
$878$	31497.6	1.21070
$879$	−22630.7	−0.868391
$880$	12422.1	0.475851
$881$	−6739.60	−0.257733	−0.128867	−	0.991662i	$-0.541134\pi$
$881$	−6739.60	−0.257733	−0.128867	+	0.991662i	$0.541134\pi$
$882$	0	0
$883$	−19750.6	−0.752732	−0.376366	−	0.926471i	$-0.622826\pi$
$883$	−19750.6	−0.752732	−0.376366	+	0.926471i	$0.622826\pi$
$884$	1562.76	0.0594586
$885$	−39767.2	−1.51046
$886$	−7188.67	−0.272582
$887$	18982.8	0.718581	0.359290	−	0.933226i	$-0.383019\pi$
$887$	18982.8	0.718581	0.359290	+	0.933226i	$0.383019\pi$
$888$	1118.78	0.0422790
$889$	0	0
$890$	−55774.1	−2.10062
$891$	5885.09	0.221277
$892$	10304.0	0.386776
$893$	−30634.3	−1.14797
$894$	−16944.5	−0.633902
$895$	85918.4	3.20887
$896$	0	0
$897$	5560.85	0.206991
$898$	9996.01	0.371460
$899$	−5126.65	−0.190193
$900$	−11407.8	−0.422512
$901$	−37.5547	−0.00138860
$902$	−32185.3	−1.18809
$903$	0	0
$904$	−14182.5	−0.521794
$905$	59127.6	2.17179
$906$	2995.33	0.109838
$907$	40744.2	1.49161	0.745803	−	0.666166i	$-0.232066\pi$
$907$	40744.2	1.49161	0.745803	+	0.666166i	$0.232066\pi$
$908$	−1554.99	−0.0568327
$909$	5228.26	0.190771
$910$	0	0
$911$	6430.56	0.233868	0.116934	−	0.993140i	$-0.462693\pi$
$911$	6430.56	0.233868	0.116934	+	0.993140i	$0.462693\pi$
$912$	−5819.54	−0.211298
$913$	−39994.7	−1.44976
$914$	−14554.2	−0.526708
$915$	6790.45	0.245339
$916$	−14105.3	−0.508790
$917$	0	0
$918$	−1691.84	−0.0608268
$919$	−33090.0	−1.18774	−0.593872	−	0.804559i	$-0.702402\pi$
$919$	−33090.0	−1.18774	−0.593872	+	0.804559i	$0.702402\pi$
$920$	−3311.71	−0.118678
$921$	−19420.5	−0.694819
$922$	25757.8	0.920053
$923$	−52030.3	−1.85547
$924$	0	0
$925$	7817.88	0.277892
$926$	−13902.0	−0.493358
$927$	−16284.0	−0.576955
$928$	738.407	0.0261200
$929$	31170.1	1.10082	0.550408	−	0.834896i	$-0.314472\pi$
$929$	31170.1	1.10082	0.550408	+	0.834896i	$0.314472\pi$
$930$	28460.7	1.00351
$931$	0	0
$932$	−4029.86	−0.141633
$933$	−37585.8	−1.31887
$934$	4810.25	0.168518
$935$	4464.65	0.156160
$936$	−7791.59	−0.272090
$937$	9257.07	0.322749	0.161374	−	0.986893i	$-0.448407\pi$
$937$	9257.07	0.322749	0.161374	+	0.986893i	$0.448407\pi$
$938$	0	0
$939$	−35388.0	−1.22987
$940$	−21578.8	−0.748746
$941$	−54839.7	−1.89981	−0.949905	−	0.312538i	$-0.898821\pi$
$941$	−54839.7	−1.89981	−0.949905	+	0.312538i	$0.898821\pi$
$942$	1348.49	0.0466413
$943$	8580.54	0.296310
$944$	9933.86	0.342500
$945$	0	0
$946$	−460.844	−0.0158386
$947$	49151.7	1.68661	0.843303	−	0.537438i	$-0.180608\pi$
$947$	49151.7	1.68661	0.843303	+	0.537438i	$0.180608\pi$
$948$	8524.63	0.292054
$949$	4690.75	0.160451
$950$	−40666.2	−1.38883
$951$	36685.6	1.25091
$952$	0	0
$953$	−39430.5	−1.34027	−0.670136	−	0.742238i	$-0.733764\pi$
$953$	−39430.5	−1.34027	−0.670136	+	0.742238i	$0.733764\pi$
$954$	187.239	0.00635440
$955$	−14853.8	−0.503308
$956$	5759.18	0.194838
$957$	−3542.25	−0.119650
$958$	−1613.15	−0.0544036
$959$	0	0
$960$	−4099.27	−0.137816
$961$	19569.2	0.656882
$962$	5339.64	0.178957
$963$	−1601.26	−0.0535825
$964$	−8189.27	−0.273608
$965$	2270.80	0.0757508
$966$	0	0
$967$	−4299.88	−0.142993	−0.0714967	−	0.997441i	$-0.522778\pi$
$967$	−4299.88	−0.142993	−0.0714967	+	0.997441i	$0.522778\pi$
$968$	4237.82	0.140712
$969$	−2091.61	−0.0693418
$970$	11311.5	0.374423
$971$	−9069.64	−0.299751	−0.149876	−	0.988705i	$-0.547887\pi$
$971$	−9069.64	−0.299751	−0.149876	+	0.988705i	$0.547887\pi$
$972$	13944.9	0.460167
$973$	0	0
$974$	37361.1	1.22908
$975$	48099.6	1.57992
$976$	−1696.26	−0.0556311
$977$	−44846.3	−1.46854	−0.734268	−	0.678860i	$-0.762474\pi$
$977$	−44846.3	−1.46854	−0.734268	+	0.678860i	$0.762474\pi$
$978$	24638.3	0.805569
$979$	−66835.9	−2.18191
$980$	0	0
$981$	−17813.3	−0.579749
$982$	−35210.2	−1.14420
$983$	3678.82	0.119365	0.0596827	−	0.998217i	$-0.480991\pi$
$983$	3678.82	0.119365	0.0596827	+	0.998217i	$0.480991\pi$
$984$	10621.1	0.344094
$985$	−55510.3	−1.79564
$986$	265.392	0.00857181
$987$	0	0
$988$	−27775.2	−0.894378
$989$	122.860	0.00395017
$990$	−22259.8	−0.714608
$991$	5141.15	0.164797	0.0823986	−	0.996599i	$-0.473742\pi$
$991$	5141.15	0.164797	0.0823986	+	0.996599i	$0.473742\pi$
$992$	−7109.49	−0.227547
$993$	9148.10	0.292353
$994$	0	0
$995$	−92138.8	−2.93567
$996$	13198.2	0.419880
$997$	−38278.3	−1.21594	−0.607968	−	0.793962i	$-0.708015\pi$
$997$	−38278.3	−1.21594	−0.607968	+	0.793962i	$0.708015\pi$
$998$	21.7185	0.000688864 0
$999$	−5780.67	−0.183075

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2254.4.a.y.1.6		11
7.2	even	3		322.4.e.a.277.6	yes	22
7.4	even	3		322.4.e.a.93.6	✓	22
7.6	odd	2		2254.4.a.v.1.6		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.4.e.a.93.6	✓	22	7.4	even	3
322.4.e.a.277.6	yes	22	7.2	even	3
2254.4.a.v.1.6		11	7.6	odd	2
2254.4.a.y.1.6		11	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)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +3.55871 q^{3} +4.00000 q^{4} -17.9984 q^{5} +7.11742 q^{6} +8.00000 q^{8} -14.3356 q^{9} -35.9968 q^{10} -43.1362 q^{11} +14.2348 q^{12} +67.9392 q^{13} -64.0511 q^{15} +16.0000 q^{16} +5.75059 q^{17} -28.6712 q^{18} -102.206 q^{19} -71.9936 q^{20} -86.2723 q^{22} +23.0000 q^{23} +28.4697 q^{24} +198.943 q^{25} +135.878 q^{26} -147.101 q^{27} +23.0752 q^{29} -128.102 q^{30} -222.171 q^{31} +32.0000 q^{32} -153.509 q^{33} +11.5012 q^{34} -57.3423 q^{36} +39.2972 q^{37} -204.412 q^{38} +241.776 q^{39} -143.987 q^{40} +373.067 q^{41} +5.34174 q^{43} -172.545 q^{44} +258.018 q^{45} +46.0000 q^{46} +299.732 q^{47} +56.9394 q^{48} +397.885 q^{50} +20.4647 q^{51} +271.757 q^{52} -6.53059 q^{53} -294.203 q^{54} +776.382 q^{55} -363.721 q^{57} +46.1504 q^{58} +620.866 q^{59} -256.204 q^{60} -106.016 q^{61} -444.343 q^{62} +64.0000 q^{64} -1222.80 q^{65} -307.018 q^{66} -382.646 q^{67} +23.0024 q^{68} +81.8503 q^{69} -765.836 q^{71} -114.685 q^{72} +69.0434 q^{73} +78.5943 q^{74} +707.979 q^{75} -408.824 q^{76} +483.552 q^{78} +598.857 q^{79} -287.974 q^{80} -136.431 q^{81} +746.134 q^{82} +927.174 q^{83} -103.501 q^{85} +10.6835 q^{86} +82.1180 q^{87} -345.089 q^{88} +1549.42 q^{89} +516.035 q^{90} +92.0000 q^{92} -790.644 q^{93} +599.463 q^{94} +1839.54 q^{95} +113.879 q^{96} -314.236 q^{97} +618.382 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 22 q^{2} + 18 q^{3} + 44 q^{4} + 33 q^{5} + 36 q^{6} + 88 q^{8} + 171 q^{9} + 66 q^{10} + 8 q^{11} + 72 q^{12} + 185 q^{13} - 186 q^{15} + 176 q^{16} + 107 q^{17} + 342 q^{18} + 114 q^{19} + 132 q^{20}+ \cdots - 1729 q^{99}+O(q^{100}) \) 11 * q + 22 * q^2 + 18 * q^3 + 44 * q^4 + 33 * q^5 + 36 * q^6 + 88 * q^8 + 171 * q^9 + 66 * q^10 + 8 * q^11 + 72 * q^12 + 185 * q^13 - 186 * q^15 + 176 * q^16 + 107 * q^17 + 342 * q^18 + 114 * q^19 + 132 * q^20 + 16 * q^22 + 253 * q^23 + 144 * q^24 + 494 * q^25 + 370 * q^26 + 561 * q^27 - 79 * q^29 - 372 * q^30 + 488 * q^31 + 352 * q^32 + 304 * q^33 + 214 * q^34 + 684 * q^36 - 41 * q^37 + 228 * q^38 + 425 * q^39 + 264 * q^40 + 812 * q^41 + 219 * q^43 + 32 * q^44 + 619 * q^45 + 506 * q^46 + 1141 * q^47 + 288 * q^48 + 988 * q^50 + 83 * q^51 + 740 * q^52 + 579 * q^53 + 1122 * q^54 + 951 * q^55 + 383 * q^57 - 158 * q^58 + 1695 * q^59 - 744 * q^60 + 1105 * q^61 + 976 * q^62 + 704 * q^64 - 347 * q^65 + 608 * q^66 + 695 * q^67 + 428 * q^68 + 414 * q^69 + 65 * q^71 + 1368 * q^72 + 1935 * q^73 - 82 * q^74 + 1712 * q^75 + 456 * q^76 + 850 * q^78 + 447 * q^79 + 528 * q^80 + 2203 * q^81 + 1624 * q^82 + 1389 * q^83 + 153 * q^85 + 438 * q^86 + 3354 * q^87 + 64 * q^88 + 2698 * q^89 + 1238 * q^90 + 1012 * q^92 - 686 * q^93 + 2282 * q^94 + 827 * q^95 + 576 * q^96 + 1589 * q^97 - 1729 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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