LMFDB - Embedded newform 2254.4.a.y.1.7

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			1.7
Root			$-1.83354$ 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)

$f(q)$	$=$	\(q+2.00000 q^{2} +3.83354 q^{3} +4.00000 q^{4} +17.7022 q^{5} +7.66708 q^{6} +8.00000 q^{8} -12.3040 q^{9} +35.4044 q^{10} -12.8553 q^{11} +15.3342 q^{12} +1.76114 q^{13} +67.8620 q^{15} +16.0000 q^{16} -35.3413 q^{17} -24.6079 q^{18} +23.6430 q^{19} +70.8087 q^{20} -25.7106 q^{22} +23.0000 q^{23} +30.6683 q^{24} +188.367 q^{25} +3.52229 q^{26} -150.673 q^{27} +308.379 q^{29} +135.724 q^{30} +120.796 q^{31} +32.0000 q^{32} -49.2812 q^{33} -70.6827 q^{34} -49.2159 q^{36} +437.043 q^{37} +47.2860 q^{38} +6.75141 q^{39} +141.617 q^{40} -468.412 q^{41} -374.758 q^{43} -51.4211 q^{44} -217.807 q^{45} +46.0000 q^{46} +409.343 q^{47} +61.3366 q^{48} +376.734 q^{50} -135.482 q^{51} +7.04457 q^{52} +25.6072 q^{53} -301.347 q^{54} -227.567 q^{55} +90.6364 q^{57} +616.759 q^{58} +637.206 q^{59} +271.448 q^{60} +624.507 q^{61} +241.592 q^{62} +64.0000 q^{64} +31.1761 q^{65} -98.5625 q^{66} +376.758 q^{67} -141.365 q^{68} +88.1714 q^{69} +435.818 q^{71} -98.4318 q^{72} +988.465 q^{73} +874.086 q^{74} +722.113 q^{75} +94.5721 q^{76} +13.5028 q^{78} +524.171 q^{79} +283.235 q^{80} -245.405 q^{81} -936.823 q^{82} +1087.48 q^{83} -625.619 q^{85} -749.517 q^{86} +1182.18 q^{87} -102.842 q^{88} +133.439 q^{89} -435.614 q^{90} +92.0000 q^{92} +463.077 q^{93} +818.685 q^{94} +418.533 q^{95} +122.673 q^{96} -1160.72 q^{97} +158.171 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	3.83354	0.737765	0.368883	−	0.929476i	$-0.379740\pi$
$3$	3.83354	0.737765	0.368883	+	0.929476i	$0.379740\pi$
$4$	4.00000	0.500000
$5$	17.7022	1.58333	0.791665	−	0.610955i	$-0.209214\pi$
$5$	17.7022	1.58333	0.791665	+	0.610955i	$0.209214\pi$
$6$	7.66708	0.521679
$7$	0	0
$7$	0	0
$8$	8.00000	0.353553
$9$	−12.3040	−0.455703
$10$	35.4044	1.11958
$11$	−12.8553	−0.352365	−0.176182	−	0.984358i	$-0.556375\pi$
$11$	−12.8553	−0.352365	−0.176182	+	0.984358i	$0.556375\pi$
$12$	15.3342	0.368883
$13$	1.76114	0.0375733	0.0187867	−	0.999824i	$-0.494020\pi$
$13$	1.76114	0.0375733	0.0187867	+	0.999824i	$0.494020\pi$
$14$	0	0
$15$	67.8620	1.16813
$16$	16.0000	0.250000
$17$	−35.3413	−0.504208	−0.252104	−	0.967700i	$-0.581123\pi$
$17$	−35.3413	−0.504208	−0.252104	+	0.967700i	$0.581123\pi$
$18$	−24.6079	−0.322230
$19$	23.6430	0.285478	0.142739	−	0.989760i	$-0.454409\pi$
$19$	23.6430	0.285478	0.142739	+	0.989760i	$0.454409\pi$
$20$	70.8087	0.791665
$21$	0	0
$22$	−25.7106	−0.249160
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	30.6683	0.260839
$25$	188.367	1.50694
$26$	3.52229	0.0265684
$27$	−150.673	−1.07397
$28$	0	0
$29$	308.379	1.97464	0.987321	−	0.158739i	$-0.0507428\pi$
$29$	308.379	1.97464	0.987321	+	0.158739i	$0.0507428\pi$
$30$	135.724	0.825990
$31$	120.796	0.699859	0.349929	−	0.936776i	$-0.386206\pi$
$31$	120.796	0.699859	0.349929	+	0.936776i	$0.386206\pi$
$32$	32.0000	0.176777
$33$	−49.2812	−0.259963
$34$	−70.6827	−0.356529
$35$	0	0
$36$	−49.2159	−0.227851
$37$	437.043	1.94188	0.970938	−	0.239331i	$-0.0769282\pi$
$37$	437.043	1.94188	0.970938	+	0.239331i	$0.0769282\pi$
$38$	47.2860	0.201863
$39$	6.75141	0.0277203
$40$	141.617	0.559792
$41$	−468.412	−1.78423	−0.892117	−	0.451804i	$-0.850781\pi$
$41$	−468.412	−1.78423	−0.892117	+	0.451804i	$0.850781\pi$
$42$	0	0
$43$	−374.758	−1.32907	−0.664536	−	0.747256i	$-0.731371\pi$
$43$	−374.758	−1.32907	−0.664536	+	0.747256i	$0.731371\pi$
$44$	−51.4211	−0.176182
$45$	−217.807	−0.721528
$46$	46.0000	0.147442
$47$	409.343	1.27040	0.635200	−	0.772348i	$-0.280918\pi$
$47$	409.343	1.27040	0.635200	+	0.772348i	$0.280918\pi$
$48$	61.3366	0.184441
$49$	0	0
$50$	376.734	1.06556
$51$	−135.482	−0.371987
$52$	7.04457	0.0187867
$53$	25.6072	0.0663663	0.0331832	−	0.999449i	$-0.489436\pi$
$53$	25.6072	0.0663663	0.0331832	+	0.999449i	$0.489436\pi$
$54$	−301.347	−0.759409
$55$	−227.567	−0.557910
$56$	0	0
$57$	90.6364	0.210616
$58$	616.759	1.39628
$59$	637.206	1.40605	0.703027	−	0.711163i	$-0.251831\pi$
$59$	637.206	1.40605	0.703027	+	0.711163i	$0.251831\pi$
$60$	271.448	0.584063
$61$	624.507	1.31082	0.655409	−	0.755274i	$-0.272496\pi$
$61$	624.507	1.31082	0.655409	+	0.755274i	$0.272496\pi$
$62$	241.592	0.494875
$63$	0	0
$64$	64.0000	0.125000
$65$	31.1761	0.0594910
$66$	−98.5625	−0.183821
$67$	376.758	0.686990	0.343495	−	0.939155i	$-0.388389\pi$
$67$	376.758	0.686990	0.343495	+	0.939155i	$0.388389\pi$
$68$	−141.365	−0.252104
$69$	88.1714	0.153835
$70$	0	0
$71$	435.818	0.728480	0.364240	−	0.931305i	$-0.381329\pi$
$71$	435.818	0.728480	0.364240	+	0.931305i	$0.381329\pi$
$72$	−98.4318	−0.161115
$73$	988.465	1.58481	0.792405	−	0.609995i	$-0.208829\pi$
$73$	988.465	1.58481	0.792405	+	0.609995i	$0.208829\pi$
$74$	874.086	1.37311
$75$	722.113	1.11177
$76$	94.5721	0.142739
$77$	0	0
$78$	13.5028	0.0196012
$79$	524.171	0.746504	0.373252	−	0.927730i	$-0.378243\pi$
$79$	524.171	0.746504	0.373252	+	0.927730i	$0.378243\pi$
$80$	283.235	0.395833
$81$	−245.405	−0.336632
$82$	−936.823	−1.26164
$83$	1087.48	1.43814	0.719072	−	0.694935i	$-0.244567\pi$
$83$	1087.48	1.43814	0.719072	+	0.694935i	$0.244567\pi$
$84$	0	0
$85$	−625.619	−0.798328
$86$	−749.517	−0.939796
$87$	1182.18	1.45682
$88$	−102.842	−0.124580
$89$	133.439	0.158927	0.0794637	−	0.996838i	$-0.474679\pi$
$89$	133.439	0.158927	0.0794637	+	0.996838i	$0.474679\pi$
$90$	−435.614	−0.510197
$91$	0	0
$92$	92.0000	0.104257
$93$	463.077	0.516331
$94$	818.685	0.898308
$95$	418.533	0.452006
$96$	122.673	0.130420
$97$	−1160.72	−1.21498	−0.607492	−	0.794326i	$-0.707824\pi$
$97$	−1160.72	−1.21498	−0.607492	+	0.794326i	$0.707824\pi$
$98$	0	0
$99$	158.171	0.160574
$100$	753.468	0.753468
$101$	−1652.54	−1.62806	−0.814031	−	0.580822i	$-0.802731\pi$
$101$	−1652.54	−1.62806	−0.814031	+	0.580822i	$0.802731\pi$
$102$	−270.965	−0.263035
$103$	−683.529	−0.653885	−0.326942	−	0.945044i	$-0.606018\pi$
$103$	−683.529	−0.653885	−0.326942	+	0.945044i	$0.606018\pi$
$104$	14.0891	0.0132842
$105$	0	0
$106$	51.2143	0.0469281
$107$	113.864	0.102876	0.0514378	−	0.998676i	$-0.483620\pi$
$107$	113.864	0.102876	0.0514378	+	0.998676i	$0.483620\pi$
$108$	−602.693	−0.536983
$109$	99.4872	0.0874234	0.0437117	−	0.999044i	$-0.486082\pi$
$109$	99.4872	0.0874234	0.0437117	+	0.999044i	$0.486082\pi$
$110$	−455.133	−0.394502
$111$	1675.42	1.43265
$112$	0	0
$113$	802.538	0.668110	0.334055	−	0.942554i	$-0.391583\pi$
$113$	802.538	0.668110	0.334055	+	0.942554i	$0.391583\pi$
$114$	181.273	0.148928
$115$	407.150	0.330147
$116$	1233.52	0.987321
$117$	−21.6691	−0.0171223
$118$	1274.41	0.994230
$119$	0	0
$120$	542.896	0.412995
$121$	−1165.74	−0.875839
$122$	1249.01	0.926889
$123$	−1795.67	−1.31635
$124$	483.184	0.349929
$125$	1121.73	0.802648
$126$	0	0
$127$	−991.403	−0.692699	−0.346349	−	0.938106i	$-0.612579\pi$
$127$	−991.403	−0.692699	−0.346349	+	0.938106i	$0.612579\pi$
$128$	128.000	0.0883883
$129$	−1436.65	−0.980544
$130$	62.3521	0.0420665
$131$	1137.96	0.758960	0.379480	−	0.925200i	$-0.376103\pi$
$131$	1137.96	0.758960	0.379480	+	0.925200i	$0.376103\pi$
$132$	−197.125	−0.129981
$133$	0	0
$134$	753.516	0.485775
$135$	−2667.25	−1.70044
$136$	−282.731	−0.178264
$137$	−292.179	−0.182208	−0.0911042	−	0.995841i	$-0.529040\pi$
$137$	−292.179	−0.182208	−0.0911042	+	0.995841i	$0.529040\pi$
$138$	176.343	0.108778
$139$	1024.65	0.625247	0.312623	−	0.949877i	$-0.398792\pi$
$139$	1024.65	0.625247	0.312623	+	0.949877i	$0.398792\pi$
$140$	0	0
$141$	1569.23	0.937256
$142$	871.636	0.515113
$143$	−22.6400	−0.0132395
$144$	−196.864	−0.113926
$145$	5458.98	3.12651
$146$	1976.93	1.12063
$147$	0	0
$148$	1748.17	0.970938
$149$	−2851.69	−1.56791	−0.783957	−	0.620815i	$-0.786802\pi$
$149$	−2851.69	−1.56791	−0.783957	+	0.620815i	$0.786802\pi$
$150$	1444.23	0.786137
$151$	190.564	0.102701	0.0513505	−	0.998681i	$-0.483647\pi$
$151$	190.564	0.102701	0.0513505	+	0.998681i	$0.483647\pi$
$152$	189.144	0.100932
$153$	434.839	0.229769
$154$	0	0
$155$	2138.35	1.10811
$156$	27.0056	0.0138601
$157$	−3246.21	−1.65016	−0.825081	−	0.565015i	$-0.808871\pi$
$157$	−3246.21	−1.65016	−0.825081	+	0.565015i	$0.808871\pi$
$158$	1048.34	0.527858
$159$	98.1661	0.0489628
$160$	566.470	0.279896
$161$	0	0
$162$	−490.810	−0.238035
$163$	−1102.67	−0.529866	−0.264933	−	0.964267i	$-0.585350\pi$
$163$	−1102.67	−0.529866	−0.264933	+	0.964267i	$0.585350\pi$
$164$	−1873.65	−0.892117
$165$	−872.385	−0.411607
$166$	2174.95	1.01692
$167$	−3145.55	−1.45755	−0.728773	−	0.684755i	$-0.759909\pi$
$167$	−3145.55	−1.45755	−0.728773	+	0.684755i	$0.759909\pi$
$168$	0	0
$169$	−2193.90	−0.998588
$170$	−1251.24	−0.564503
$171$	−290.903	−0.130093
$172$	−1499.03	−0.664536
$173$	3639.65	1.59952	0.799761	−	0.600319i	$-0.204960\pi$
$173$	3639.65	1.59952	0.799761	+	0.600319i	$0.204960\pi$
$174$	2364.37	1.03013
$175$	0	0
$176$	−205.685	−0.0880912
$177$	2442.75	1.03734
$178$	266.879	0.112379
$179$	−1453.90	−0.607094	−0.303547	−	0.952816i	$-0.598171\pi$
$179$	−1453.90	−0.607094	−0.303547	+	0.952816i	$0.598171\pi$
$180$	−871.228	−0.360764
$181$	−1659.81	−0.681618	−0.340809	−	0.940133i	$-0.610701\pi$
$181$	−1659.81	−0.681618	−0.340809	+	0.940133i	$0.610701\pi$
$182$	0	0
$183$	2394.07	0.967076
$184$	184.000	0.0737210
$185$	7736.61	3.07463
$186$	926.153	0.365101
$187$	454.323	0.177665
$188$	1637.37	0.635200
$189$	0	0
$190$	837.066	0.319616
$191$	−2117.44	−0.802161	−0.401081	−	0.916043i	$-0.631365\pi$
$191$	−2117.44	−0.802161	−0.401081	+	0.916043i	$0.631365\pi$
$192$	245.347	0.0922206
$193$	2182.27	0.813904	0.406952	−	0.913450i	$-0.366592\pi$
$193$	2182.27	0.813904	0.406952	+	0.913450i	$0.366592\pi$
$194$	−2321.44	−0.859123
$195$	119.515	0.0438904
$196$	0	0
$197$	−2360.71	−0.853776	−0.426888	−	0.904304i	$-0.640390\pi$
$197$	−2360.71	−0.853776	−0.426888	+	0.904304i	$0.640390\pi$
$198$	316.342	0.113543
$199$	−1637.87	−0.583445	−0.291723	−	0.956503i	$-0.594228\pi$
$199$	−1637.87	−0.583445	−0.291723	+	0.956503i	$0.594228\pi$
$200$	1506.94	0.532782
$201$	1444.32	0.506837
$202$	−3305.09	−1.15121
$203$	0	0
$204$	−541.930	−0.185994
$205$	−8291.91	−2.82503
$206$	−1367.06	−0.462366
$207$	−282.991	−0.0950206
$208$	28.1783	0.00939333
$209$	−303.938	−0.100592
$210$	0	0
$211$	−3272.61	−1.06775	−0.533877	−	0.845562i	$-0.679265\pi$
$211$	−3272.61	−1.06775	−0.533877	+	0.845562i	$0.679265\pi$
$212$	102.429	0.0331832
$213$	1670.73	0.537447
$214$	227.729	0.0727441
$215$	−6634.04	−2.10436
$216$	−1205.39	−0.379705
$217$	0	0
$218$	198.974	0.0618177
$219$	3789.32	1.16922
$220$	−910.266	−0.278955
$221$	−62.2412	−0.0189448
$222$	3350.84	1.01304
$223$	958.723	0.287896	0.143948	−	0.989585i	$-0.454020\pi$
$223$	958.723	0.287896	0.143948	+	0.989585i	$0.454020\pi$
$224$	0	0
$225$	−2317.66	−0.686715
$226$	1605.08	0.472425
$227$	−5186.74	−1.51655	−0.758273	−	0.651937i	$-0.773957\pi$
$227$	−5186.74	−1.51655	−0.758273	+	0.651937i	$0.773957\pi$
$228$	362.546	0.105308
$229$	2383.61	0.687833	0.343916	−	0.939000i	$-0.388246\pi$
$229$	2383.61	0.687833	0.343916	+	0.939000i	$0.388246\pi$
$230$	814.300	0.233449
$231$	0	0
$232$	2467.03	0.698141
$233$	−2433.77	−0.684298	−0.342149	−	0.939646i	$-0.611155\pi$
$233$	−2433.77	−0.684298	−0.342149	+	0.939646i	$0.611155\pi$
$234$	−43.3381	−0.0121073
$235$	7246.26	2.01146
$236$	2548.82	0.703027
$237$	2009.43	0.550745
$238$	0	0
$239$	2088.03	0.565118	0.282559	−	0.959250i	$-0.408817\pi$
$239$	2088.03	0.565118	0.282559	+	0.959250i	$0.408817\pi$
$240$	1085.79	0.292032
$241$	−2757.56	−0.737055	−0.368527	−	0.929617i	$-0.620138\pi$
$241$	−2757.56	−0.737055	−0.368527	+	0.929617i	$0.620138\pi$
$242$	−2331.48	−0.619312
$243$	3127.41	0.825611
$244$	2498.03	0.655409
$245$	0	0
$246$	−3591.35	−0.930797
$247$	41.6387	0.0107264
$248$	966.369	0.247437
$249$	4168.88	1.06101
$250$	2243.47	0.567558
$251$	3013.45	0.757799	0.378899	−	0.925438i	$-0.376303\pi$
$251$	3013.45	0.757799	0.378899	+	0.925438i	$0.376303\pi$
$252$	0	0
$253$	−295.672	−0.0734732
$254$	−1982.81	−0.489812
$255$	−2398.33	−0.588979
$256$	256.000	0.0625000
$257$	3800.48	0.922441	0.461220	−	0.887286i	$-0.347412\pi$
$257$	3800.48	0.922441	0.461220	+	0.887286i	$0.347412\pi$
$258$	−2873.30	−0.693349
$259$	0	0
$260$	124.704	0.0297455
$261$	−3794.29	−0.899849
$262$	2275.91	0.536666
$263$	−6668.63	−1.56352	−0.781759	−	0.623581i	$-0.785677\pi$
$263$	−6668.63	−1.56352	−0.781759	+	0.623581i	$0.785677\pi$
$264$	−394.250	−0.0919106
$265$	453.303	0.105080
$266$	0	0
$267$	511.545	0.117251
$268$	1507.03	0.343495
$269$	1576.33	0.357287	0.178644	−	0.983914i	$-0.442829\pi$
$269$	1576.33	0.357287	0.178644	+	0.983914i	$0.442829\pi$
$270$	−5334.49	−1.20240
$271$	−2209.64	−0.495298	−0.247649	−	0.968850i	$-0.579658\pi$
$271$	−2209.64	−0.495298	−0.247649	+	0.968850i	$0.579658\pi$
$272$	−565.462	−0.126052
$273$	0	0
$274$	−584.359	−0.128841
$275$	−2421.51	−0.530992
$276$	352.686	0.0769173
$277$	−5837.67	−1.26625	−0.633126	−	0.774049i	$-0.718228\pi$
$277$	−5837.67	−1.26625	−0.633126	+	0.774049i	$0.718228\pi$
$278$	2049.29	0.442116
$279$	−1486.27	−0.318927
$280$	0	0
$281$	−1211.54	−0.257205	−0.128602	−	0.991696i	$-0.541049\pi$
$281$	−1211.54	−0.257205	−0.128602	+	0.991696i	$0.541049\pi$
$282$	3138.46	0.662740
$283$	405.766	0.0852306	0.0426153	−	0.999092i	$-0.486431\pi$
$283$	405.766	0.0852306	0.0426153	+	0.999092i	$0.486431\pi$
$284$	1743.27	0.364240
$285$	1604.46	0.333474
$286$	−45.2800	−0.00936175
$287$	0	0
$288$	−393.727	−0.0805576
$289$	−3663.99	−0.745774
$290$	10918.0	2.21078
$291$	−4449.67	−0.896373
$292$	3953.86	0.792405
$293$	−207.808	−0.0414343	−0.0207172	−	0.999785i	$-0.506595\pi$
$293$	−207.808	−0.0414343	−0.0207172	+	0.999785i	$0.506595\pi$
$294$	0	0
$295$	11279.9	2.22625
$296$	3496.34	0.686557
$297$	1936.95	0.378428
$298$	−5703.37	−1.10868
$299$	40.5063	0.00783458
$300$	2888.45	0.555883
$301$	0	0
$302$	381.127	0.0726205
$303$	−6335.09	−1.20113
$304$	378.288	0.0713695
$305$	11055.1	2.07546
$306$	869.678	0.162471
$307$	176.655	0.0328411	0.0164206	−	0.999865i	$-0.494773\pi$
$307$	176.655	0.0328411	0.0164206	+	0.999865i	$0.494773\pi$
$308$	0	0
$309$	−2620.34	−0.482413
$310$	4276.71	0.783551
$311$	5548.20	1.01161	0.505804	−	0.862649i	$-0.331196\pi$
$311$	5548.20	1.01161	0.505804	+	0.862649i	$0.331196\pi$
$312$	54.0113	0.00980060
$313$	604.396	0.109145	0.0545727	−	0.998510i	$-0.482620\pi$
$313$	604.396	0.109145	0.0545727	+	0.998510i	$0.482620\pi$
$314$	−6492.41	−1.16684
$315$	0	0
$316$	2096.68	0.373252
$317$	3411.08	0.604370	0.302185	−	0.953249i	$-0.402284\pi$
$317$	3411.08	0.604370	0.302185	+	0.953249i	$0.402284\pi$
$318$	196.332	0.0346219
$319$	−3964.30	−0.695794
$320$	1132.94	0.197916
$321$	436.504	0.0758980
$322$	0	0
$323$	−835.576	−0.143940
$324$	−981.620	−0.168316
$325$	331.741	0.0566206
$326$	−2205.35	−0.374672
$327$	381.388	0.0644979
$328$	−3747.29	−0.630822
$329$	0	0
$330$	−1744.77	−0.291050
$331$	4188.89	0.695596	0.347798	−	0.937570i	$-0.386930\pi$
$331$	4188.89	0.695596	0.347798	+	0.937570i	$0.386930\pi$
$332$	4349.91	0.719072
$333$	−5377.36	−0.884918
$334$	−6291.11	−1.03064
$335$	6669.44	1.08773
$336$	0	0
$337$	−6636.34	−1.07271	−0.536357	−	0.843991i	$-0.680200\pi$
$337$	−6636.34	−1.07271	−0.536357	+	0.843991i	$0.680200\pi$
$338$	−4387.80	−0.706109
$339$	3076.56	0.492908
$340$	−2502.48	−0.399164
$341$	−1552.87	−0.246606
$342$	−581.806	−0.0919897
$343$	0	0
$344$	−2998.07	−0.469898
$345$	1560.83	0.243571
$346$	7279.30	1.13103
$347$	−6131.61	−0.948594	−0.474297	−	0.880365i	$-0.657298\pi$
$347$	−6131.61	−0.948594	−0.474297	+	0.880365i	$0.657298\pi$
$348$	4728.74	0.728411
$349$	3262.19	0.500346	0.250173	−	0.968201i	$-0.419512\pi$
$349$	3262.19	0.500346	0.250173	+	0.968201i	$0.419512\pi$
$350$	0	0
$351$	−265.357	−0.0403525
$352$	−411.369	−0.0622899
$353$	−3571.08	−0.538440	−0.269220	−	0.963079i	$-0.586766\pi$
$353$	−3571.08	−0.538440	−0.269220	+	0.963079i	$0.586766\pi$
$354$	4885.51	0.733508
$355$	7714.93	1.15342
$356$	533.757	0.0794637
$357$	0	0
$358$	−2907.81	−0.429280
$359$	−9336.26	−1.37256	−0.686280	−	0.727337i	$-0.740758\pi$
$359$	−9336.26	−1.37256	−0.686280	+	0.727337i	$0.740758\pi$
$360$	−1742.46	−0.255099
$361$	−6300.01	−0.918502
$362$	−3319.62	−0.481976
$363$	−4468.92	−0.646163
$364$	0	0
$365$	17498.0	2.50928
$366$	4788.15	0.683826
$367$	3146.68	0.447562	0.223781	−	0.974639i	$-0.428160\pi$
$367$	3146.68	0.447562	0.223781	+	0.974639i	$0.428160\pi$
$368$	368.000	0.0521286
$369$	5763.32	0.813080
$370$	15473.2	2.17409
$371$	0	0
$372$	1852.31	0.258166
$373$	6076.92	0.843568	0.421784	−	0.906696i	$-0.361404\pi$
$373$	6076.92	0.843568	0.421784	+	0.906696i	$0.361404\pi$
$374$	908.646	0.125628
$375$	4300.21	0.592166
$376$	3274.74	0.449154
$377$	543.100	0.0741938
$378$	0	0
$379$	11561.3	1.56692	0.783462	−	0.621439i	$-0.213452\pi$
$379$	11561.3	1.56692	0.783462	+	0.621439i	$0.213452\pi$
$380$	1674.13	0.226003
$381$	−3800.58	−0.511049
$382$	−4234.88	−0.567214
$383$	−5732.41	−0.764784	−0.382392	−	0.924000i	$-0.624900\pi$
$383$	−5732.41	−0.764784	−0.382392	+	0.924000i	$0.624900\pi$
$384$	490.693	0.0652098
$385$	0	0
$386$	4364.54	0.575517
$387$	4611.02	0.605662
$388$	−4642.89	−0.607492
$389$	5054.23	0.658766	0.329383	−	0.944196i	$-0.393159\pi$
$389$	5054.23	0.658766	0.329383	+	0.944196i	$0.393159\pi$
$390$	239.029	0.0310352
$391$	−812.851	−0.105135
$392$	0	0
$393$	4362.40	0.559934
$394$	−4721.43	−0.603711
$395$	9278.96	1.18196
$396$	632.684	0.0802868
$397$	−11883.1	−1.50226	−0.751128	−	0.660157i	$-0.770490\pi$
$397$	−11883.1	−1.50226	−0.751128	+	0.660157i	$0.770490\pi$
$398$	−3275.74	−0.412558
$399$	0	0
$400$	3013.87	0.376734
$401$	9048.50	1.12683	0.563417	−	0.826172i	$-0.309486\pi$
$401$	9048.50	1.12683	0.563417	+	0.826172i	$0.309486\pi$
$402$	2888.63	0.358388
$403$	212.739	0.0262960
$404$	−6610.17	−0.814031
$405$	−4344.20	−0.533001
$406$	0	0
$407$	−5618.31	−0.684249
$408$	−1083.86	−0.131517
$409$	−4185.35	−0.505995	−0.252998	−	0.967467i	$-0.581416\pi$
$409$	−4185.35	−0.505995	−0.252998	+	0.967467i	$0.581416\pi$
$410$	−16583.8	−1.99760
$411$	−1120.08	−0.134427
$412$	−2734.12	−0.326942
$413$	0	0
$414$	−565.983	−0.0671897
$415$	19250.7	2.27706
$416$	56.3566	0.00664209
$417$	3928.02	0.461285
$418$	−607.875	−0.0711296
$419$	2669.40	0.311238	0.155619	−	0.987817i	$-0.450263\pi$
$419$	2669.40	0.311238	0.155619	+	0.987817i	$0.450263\pi$
$420$	0	0
$421$	2415.44	0.279623	0.139811	−	0.990178i	$-0.455350\pi$
$421$	2415.44	0.279623	0.139811	+	0.990178i	$0.455350\pi$
$422$	−6545.22	−0.755016
$423$	−5036.54	−0.578924
$424$	204.857	0.0234640
$425$	−6657.15	−0.759809
$426$	3341.45	0.380032
$427$	0	0
$428$	455.458	0.0514378
$429$	−86.7913	−0.00976766
$430$	−13268.1	−1.48801
$431$	1589.04	0.177590	0.0887951	−	0.996050i	$-0.471698\pi$
$431$	1589.04	0.177590	0.0887951	+	0.996050i	$0.471698\pi$
$432$	−2410.77	−0.268492
$433$	15266.3	1.69435	0.847174	−	0.531316i	$-0.178302\pi$
$433$	15266.3	1.69435	0.847174	+	0.531316i	$0.178302\pi$
$434$	0	0
$435$	20927.2	2.30663
$436$	397.949	0.0437117
$437$	543.789	0.0595263
$438$	7578.64	0.826762
$439$	−6933.07	−0.753753	−0.376876	−	0.926264i	$-0.623002\pi$
$439$	−6933.07	−0.753753	−0.376876	+	0.926264i	$0.623002\pi$
$440$	−1820.53	−0.197251
$441$	0	0
$442$	−124.482	−0.0133960
$443$	−16808.6	−1.80271	−0.901356	−	0.433078i	$-0.857427\pi$
$443$	−16808.6	−1.80271	−0.901356	+	0.433078i	$0.857427\pi$
$444$	6701.69	0.716324
$445$	2362.17	0.251635
$446$	1917.45	0.203573
$447$	−10932.1	−1.15675
$448$	0	0
$449$	859.080	0.0902951	0.0451476	−	0.998980i	$-0.485624\pi$
$449$	859.080	0.0902951	0.0451476	+	0.998980i	$0.485624\pi$
$450$	−4635.33	−0.485581
$451$	6021.56	0.628702
$452$	3210.15	0.334055
$453$	730.533	0.0757692
$454$	−10373.5	−1.07236
$455$	0	0
$456$	725.092	0.0744639
$457$	16138.1	1.65188	0.825939	−	0.563759i	$-0.190645\pi$
$457$	16138.1	1.65188	0.825939	+	0.563759i	$0.190645\pi$
$458$	4767.23	0.486371
$459$	5325.00	0.541503
$460$	1628.60	0.165074
$461$	−15143.7	−1.52996	−0.764982	−	0.644052i	$-0.777252\pi$
$461$	−15143.7	−1.52996	−0.764982	+	0.644052i	$0.777252\pi$
$462$	0	0
$463$	−1664.21	−0.167046	−0.0835230	−	0.996506i	$-0.526617\pi$
$463$	−1664.21	−0.167046	−0.0835230	+	0.996506i	$0.526617\pi$
$464$	4934.07	0.493660
$465$	8197.46	0.817523
$466$	−4867.53	−0.483872
$467$	−5224.77	−0.517716	−0.258858	−	0.965915i	$-0.583346\pi$
$467$	−5224.77	−0.517716	−0.258858	+	0.965915i	$0.583346\pi$
$468$	−86.6762	−0.00856113
$469$	0	0
$470$	14492.5	1.42232
$471$	−12444.5	−1.21743
$472$	5097.65	0.497115
$473$	4817.63	0.468319
$474$	4018.86	0.389435
$475$	4453.56	0.430197
$476$	0	0
$477$	−315.070	−0.0302433
$478$	4176.05	0.399599
$479$	17412.2	1.66092	0.830461	−	0.557077i	$-0.188077\pi$
$479$	17412.2	1.66092	0.830461	+	0.557077i	$0.188077\pi$
$480$	2171.58	0.206497
$481$	769.695	0.0729627
$482$	−5515.12	−0.521176
$483$	0	0
$484$	−4662.97	−0.437919
$485$	−20547.3	−1.92372
$486$	6254.82	0.583795
$487$	−4595.93	−0.427642	−0.213821	−	0.976873i	$-0.568591\pi$
$487$	−4595.93	−0.427642	−0.213821	+	0.976873i	$0.568591\pi$
$488$	4996.06	0.463444
$489$	−4227.15	−0.390917
$490$	0	0
$491$	19635.7	1.80478	0.902390	−	0.430921i	$-0.141811\pi$
$491$	19635.7	1.80478	0.902390	+	0.430921i	$0.141811\pi$
$492$	−7182.70	−0.658173
$493$	−10898.5	−0.995630
$494$	83.2775	0.00758468
$495$	2799.97	0.254241
$496$	1932.74	0.174965
$497$	0	0
$498$	8337.77	0.750250
$499$	560.550	0.0502879	0.0251439	−	0.999684i	$-0.491996\pi$
$499$	560.550	0.0502879	0.0251439	+	0.999684i	$0.491996\pi$
$500$	4486.94	0.401324
$501$	−12058.6	−1.07533
$502$	6026.91	0.535845
$503$	10650.7	0.944114	0.472057	−	0.881568i	$-0.343512\pi$
$503$	10650.7	0.944114	0.472057	+	0.881568i	$0.343512\pi$
$504$	0	0
$505$	−29253.6	−2.57776
$506$	−591.343	−0.0519534
$507$	−8410.40	−0.736724
$508$	−3965.61	−0.346349
$509$	16579.1	1.44372	0.721862	−	0.692037i	$-0.243287\pi$
$509$	16579.1	1.44372	0.721862	+	0.692037i	$0.243287\pi$
$510$	−4796.67	−0.416471
$511$	0	0
$512$	512.000	0.0441942
$513$	−3562.37	−0.306594
$514$	7600.95	0.652264
$515$	−12100.0	−1.03532
$516$	−5746.61	−0.490272
$517$	−5262.22	−0.447644
$518$	0	0
$519$	13952.7	1.18007
$520$	249.409	0.0210332
$521$	−9210.95	−0.774547	−0.387274	−	0.921965i	$-0.626583\pi$
$521$	−9210.95	−0.774547	−0.387274	+	0.921965i	$0.626583\pi$
$522$	−7588.58	−0.636289
$523$	4593.65	0.384066	0.192033	−	0.981388i	$-0.438492\pi$
$523$	4593.65	0.384066	0.192033	+	0.981388i	$0.438492\pi$
$524$	4551.83	0.379480
$525$	0	0
$526$	−13337.3	−1.10557
$527$	−4269.10	−0.352874
$528$	−788.500	−0.0649906
$529$	529.000	0.0434783
$530$	906.605	0.0743027
$531$	−7840.16	−0.640742
$532$	0	0
$533$	−824.940	−0.0670396
$534$	1023.09	0.0829091
$535$	2015.65	0.162886
$536$	3014.07	0.242888
$537$	−5573.60	−0.447893
$538$	3152.65	0.252640
$539$	0	0
$540$	−10669.0	−0.850222
$541$	−809.793	−0.0643543	−0.0321772	−	0.999482i	$-0.510244\pi$
$541$	−809.793	−0.0643543	−0.0321772	+	0.999482i	$0.510244\pi$
$542$	−4419.27	−0.350229
$543$	−6362.95	−0.502874
$544$	−1130.92	−0.0891322
$545$	1761.14	0.138420
$546$	0	0
$547$	19890.4	1.55476	0.777378	−	0.629034i	$-0.216549\pi$
$547$	19890.4	1.55476	0.777378	+	0.629034i	$0.216549\pi$
$548$	−1168.72	−0.0911042
$549$	−7683.92	−0.597343
$550$	−4843.02	−0.375468
$551$	7291.02	0.563716
$552$	705.371	0.0543888
$553$	0	0
$554$	−11675.3	−0.895375
$555$	29658.6	2.26836
$556$	4098.58	0.312623
$557$	−19742.2	−1.50180	−0.750899	−	0.660417i	$-0.770380\pi$
$557$	−19742.2	−1.50180	−0.750899	+	0.660417i	$0.770380\pi$
$558$	−2972.54	−0.225516
$559$	−660.003	−0.0499377
$560$	0	0
$561$	1741.67	0.131075
$562$	−2423.08	−0.181871
$563$	−2565.33	−0.192035	−0.0960175	−	0.995380i	$-0.530610\pi$
$563$	−2565.33	−0.192035	−0.0960175	+	0.995380i	$0.530610\pi$
$564$	6276.93	0.468628
$565$	14206.7	1.05784
$566$	811.532	0.0602672
$567$	0	0
$568$	3486.54	0.257557
$569$	3759.53	0.276990	0.138495	−	0.990363i	$-0.455773\pi$
$569$	3759.53	0.276990	0.138495	+	0.990363i	$0.455773\pi$
$570$	3208.92	0.235802
$571$	−3599.36	−0.263798	−0.131899	−	0.991263i	$-0.542107\pi$
$571$	−3599.36	−0.263798	−0.131899	+	0.991263i	$0.542107\pi$
$572$	−90.5600	−0.00661976
$573$	−8117.30	−0.591806
$574$	0	0
$575$	4332.44	0.314218
$576$	−787.454	−0.0569628
$577$	−10241.5	−0.738926	−0.369463	−	0.929245i	$-0.620458\pi$
$577$	−10241.5	−0.738926	−0.369463	+	0.929245i	$0.620458\pi$
$578$	−7327.98	−0.527342
$579$	8365.83	0.600470
$580$	21835.9	1.56326
$581$	0	0
$582$	−8899.35	−0.633831
$583$	−329.187	−0.0233852
$584$	7907.72	0.560315
$585$	−383.589	−0.0271102
$586$	−415.615	−0.0292985
$587$	26021.6	1.82969	0.914845	−	0.403805i	$-0.132313\pi$
$587$	26021.6	1.82969	0.914845	+	0.403805i	$0.132313\pi$
$588$	0	0
$589$	2855.98	0.199794
$590$	22559.9	1.57419
$591$	−9049.89	−0.629886
$592$	6992.69	0.485469
$593$	9600.90	0.664859	0.332430	−	0.943128i	$-0.392132\pi$
$593$	9600.90	0.664859	0.332430	+	0.943128i	$0.392132\pi$
$594$	3873.90	0.267589
$595$	0	0
$596$	−11406.7	−0.783957
$597$	−6278.84	−0.430445
$598$	81.0126	0.00553988
$599$	4093.24	0.279208	0.139604	−	0.990207i	$-0.455417\pi$
$599$	4093.24	0.279208	0.139604	+	0.990207i	$0.455417\pi$
$600$	5776.90	0.393068
$601$	−23177.0	−1.57306	−0.786531	−	0.617551i	$-0.788125\pi$
$601$	−23177.0	−1.57306	−0.786531	+	0.617551i	$0.788125\pi$
$602$	0	0
$603$	−4635.62	−0.313063
$604$	762.254	0.0513505
$605$	−20636.2	−1.38674
$606$	−12670.2	−0.849325
$607$	16351.4	1.09338	0.546690	−	0.837335i	$-0.315888\pi$
$607$	16351.4	1.09338	0.546690	+	0.837335i	$0.315888\pi$
$608$	756.576	0.0504658
$609$	0	0
$610$	22110.3	1.46757
$611$	720.911	0.0477331
$612$	1739.36	0.114884
$613$	1213.17	0.0799337	0.0399669	−	0.999201i	$-0.487275\pi$
$613$	1213.17	0.0799337	0.0399669	+	0.999201i	$0.487275\pi$
$614$	353.310	0.0232222
$615$	−31787.4	−2.08421
$616$	0	0
$617$	−15954.8	−1.04103	−0.520516	−	0.853852i	$-0.674261\pi$
$617$	−15954.8	−1.04103	−0.520516	+	0.853852i	$0.674261\pi$
$618$	−5240.67	−0.341118
$619$	25491.1	1.65521	0.827606	−	0.561310i	$-0.189702\pi$
$619$	25491.1	1.65521	0.827606	+	0.561310i	$0.189702\pi$
$620$	8553.42	0.554054
$621$	−3465.49	−0.223938
$622$	11096.4	0.715314
$623$	0	0
$624$	108.023	0.00693007
$625$	−3688.74	−0.236079
$626$	1208.79	0.0771774
$627$	−1165.16	−0.0742136
$628$	−12984.8	−0.825081
$629$	−15445.7	−0.979109
$630$	0	0
$631$	749.075	0.0472587	0.0236293	−	0.999721i	$-0.492478\pi$
$631$	749.075	0.0472587	0.0236293	+	0.999721i	$0.492478\pi$
$632$	4193.37	0.263929
$633$	−12545.7	−0.787751
$634$	6822.16	0.427354
$635$	−17550.0	−1.09677
$636$	392.664	0.0244814
$637$	0	0
$638$	−7928.61	−0.492001
$639$	−5362.29	−0.331970
$640$	2265.88	0.139948
$641$	13102.8	0.807378	0.403689	−	0.914896i	$-0.367728\pi$
$641$	13102.8	0.807378	0.403689	+	0.914896i	$0.367728\pi$
$642$	873.008	0.0536680
$643$	−2023.67	−0.124115	−0.0620575	−	0.998073i	$-0.519766\pi$
$643$	−2023.67	−0.124115	−0.0620575	+	0.998073i	$0.519766\pi$
$644$	0	0
$645$	−25431.9	−1.55252
$646$	−1671.15	−0.101781
$647$	−10793.3	−0.655842	−0.327921	−	0.944705i	$-0.606348\pi$
$647$	−10793.3	−0.655842	−0.327921	+	0.944705i	$0.606348\pi$
$648$	−1963.24	−0.119018
$649$	−8191.46	−0.495444
$650$	663.483	0.0400368
$651$	0	0
$652$	−4410.70	−0.264933
$653$	−28123.9	−1.68541	−0.842705	−	0.538376i	$-0.819038\pi$
$653$	−28123.9	−1.68541	−0.842705	+	0.538376i	$0.819038\pi$
$654$	762.776	0.0456069
$655$	20144.3	1.20168
$656$	−7494.59	−0.446059
$657$	−12162.0	−0.722202
$658$	0	0
$659$	−8247.28	−0.487509	−0.243755	−	0.969837i	$-0.578379\pi$
$659$	−8247.28	−0.487509	−0.243755	+	0.969837i	$0.578379\pi$
$660$	−3489.54	−0.205803
$661$	1576.85	0.0927870	0.0463935	−	0.998923i	$-0.485227\pi$
$661$	1576.85	0.0927870	0.0463935	+	0.998923i	$0.485227\pi$
$662$	8377.78	0.491860
$663$	−238.604	−0.0139768
$664$	8699.81	0.508461
$665$	0	0
$666$	−10754.7	−0.625731
$667$	7092.72	0.411741
$668$	−12582.2	−0.728773
$669$	3675.30	0.212400
$670$	13338.9	0.769143
$671$	−8028.22	−0.461886
$672$	0	0
$673$	13689.8	0.784108	0.392054	−	0.919942i	$-0.371765\pi$
$673$	13689.8	0.784108	0.392054	+	0.919942i	$0.371765\pi$
$674$	−13272.7	−0.758523
$675$	−28381.9	−1.61840
$676$	−8775.59	−0.499294
$677$	24316.7	1.38045	0.690226	−	0.723594i	$-0.257511\pi$
$677$	24316.7	1.38045	0.690226	+	0.723594i	$0.257511\pi$
$678$	6153.12	0.348539
$679$	0	0
$680$	−5004.95	−0.282252
$681$	−19883.6	−1.11885
$682$	−3105.74	−0.174377
$683$	−30643.9	−1.71677	−0.858386	−	0.513004i	$-0.828532\pi$
$683$	−30643.9	−1.71677	−0.858386	+	0.513004i	$0.828532\pi$
$684$	−1163.61	−0.0650465
$685$	−5172.21	−0.288496
$686$	0	0
$687$	9137.68	0.507459
$688$	−5996.14	−0.332268
$689$	45.0979	0.00249360
$690$	3121.65	0.172231
$691$	−10971.3	−0.604005	−0.302003	−	0.953307i	$-0.597655\pi$
$691$	−10971.3	−0.604005	−0.302003	+	0.953307i	$0.597655\pi$
$692$	14558.6	0.799761
$693$	0	0
$694$	−12263.2	−0.670757
$695$	18138.4	0.989972
$696$	9457.47	0.515064
$697$	16554.3	0.899625
$698$	6524.38	0.353798
$699$	−9329.94	−0.504851
$700$	0	0
$701$	−19013.4	−1.02443	−0.512215	−	0.858857i	$-0.671175\pi$
$701$	−19013.4	−1.02443	−0.512215	+	0.858857i	$0.671175\pi$
$702$	−530.715	−0.0285335
$703$	10333.0	0.554363
$704$	−822.738	−0.0440456
$705$	27778.8	1.48399
$706$	−7142.16	−0.380735
$707$	0	0
$708$	9771.02	0.518669
$709$	17276.9	0.915158	0.457579	−	0.889169i	$-0.348717\pi$
$709$	17276.9	0.915158	0.457579	+	0.889169i	$0.348717\pi$
$710$	15429.9	0.815594
$711$	−6449.38	−0.340184
$712$	1067.51	0.0561893
$713$	2778.31	0.145931
$714$	0	0
$715$	−400.777	−0.0209625
$716$	−5815.62	−0.303547
$717$	8004.54	0.416925
$718$	−18672.5	−0.970547
$719$	−33295.7	−1.72701	−0.863504	−	0.504342i	$-0.831735\pi$
$719$	−33295.7	−1.72701	−0.863504	+	0.504342i	$0.831735\pi$
$720$	−3484.91	−0.180382
$721$	0	0
$722$	−12600.0	−0.649479
$723$	−10571.2	−0.543773
$724$	−6639.24	−0.340809
$725$	58088.5	2.97566
$726$	−8937.83	−0.456907
$727$	8288.99	0.422863	0.211432	−	0.977393i	$-0.432187\pi$
$727$	8288.99	0.422863	0.211432	+	0.977393i	$0.432187\pi$
$728$	0	0
$729$	18615.0	0.945739
$730$	34996.0	1.77433
$731$	13244.5	0.670129
$732$	9576.29	0.483538
$733$	−18902.0	−0.952473	−0.476236	−	0.879317i	$-0.657999\pi$
$733$	−18902.0	−0.952473	−0.476236	+	0.879317i	$0.657999\pi$
$734$	6293.36	0.316474
$735$	0	0
$736$	736.000	0.0368605
$737$	−4843.33	−0.242071
$738$	11526.6	0.574935
$739$	7399.62	0.368335	0.184167	−	0.982895i	$-0.441041\pi$
$739$	7399.62	0.368335	0.184167	+	0.982895i	$0.441041\pi$
$740$	30946.4	1.53732
$741$	159.624	0.00791353
$742$	0	0
$743$	−1405.27	−0.0693866	−0.0346933	−	0.999398i	$-0.511045\pi$
$743$	−1405.27	−0.0693866	−0.0346933	+	0.999398i	$0.511045\pi$
$744$	3704.61	0.182551
$745$	−50481.0	−2.48253
$746$	12153.8	0.596493
$747$	−13380.3	−0.655366
$748$	1817.29	0.0888326
$749$	0	0
$750$	8600.43	0.418724
$751$	2729.49	0.132624	0.0663118	−	0.997799i	$-0.478877\pi$
$751$	2729.49	0.132624	0.0663118	+	0.997799i	$0.478877\pi$
$752$	6549.48	0.317600
$753$	11552.2	0.559078
$754$	1086.20	0.0524630
$755$	3373.39	0.162610
$756$	0	0
$757$	5187.94	0.249087	0.124544	−	0.992214i	$-0.460253\pi$
$757$	5187.94	0.249087	0.124544	+	0.992214i	$0.460253\pi$
$758$	23122.6	1.10798
$759$	−1133.47	−0.0542059
$760$	3348.26	0.159808
$761$	−8820.39	−0.420156	−0.210078	−	0.977685i	$-0.567372\pi$
$761$	−8820.39	−0.420156	−0.210078	+	0.977685i	$0.567372\pi$
$762$	−7601.16	−0.361366
$763$	0	0
$764$	−8469.77	−0.401081
$765$	7697.60	0.363800
$766$	−11464.8	−0.540784
$767$	1122.21	0.0528301
$768$	981.386	0.0461103
$769$	−16564.8	−0.776778	−0.388389	−	0.921495i	$-0.626968\pi$
$769$	−16564.8	−0.776778	−0.388389	+	0.921495i	$0.626968\pi$
$770$	0	0
$771$	14569.3	0.680545
$772$	8729.09	0.406952
$773$	−2034.36	−0.0946585	−0.0473292	−	0.998879i	$-0.515071\pi$
$773$	−2034.36	−0.0946585	−0.0473292	+	0.998879i	$0.515071\pi$
$774$	9222.03	0.428268
$775$	22754.0	1.05464
$776$	−9285.77	−0.429562
$777$	0	0
$778$	10108.5	0.465818
$779$	−11074.7	−0.509359
$780$	478.059	0.0219452
$781$	−5602.56	−0.256691
$782$	−1625.70	−0.0743414
$783$	−46464.5	−2.12070
$784$	0	0
$785$	−57464.9	−2.61275
$786$	8724.80	0.395933
$787$	−4732.04	−0.214332	−0.107166	−	0.994241i	$-0.534178\pi$
$787$	−4732.04	−0.214332	−0.107166	+	0.994241i	$0.534178\pi$
$788$	−9442.86	−0.426888
$789$	−25564.5	−1.15351
$790$	18557.9	0.835774
$791$	0	0
$792$	1265.37	0.0567713
$793$	1099.85	0.0492518
$794$	−23766.2	−1.06226
$795$	1737.75	0.0775243
$796$	−6551.48	−0.291723
$797$	25545.5	1.13534	0.567671	−	0.823256i	$-0.307845\pi$
$797$	25545.5	1.13534	0.567671	+	0.823256i	$0.307845\pi$
$798$	0	0
$799$	−14466.7	−0.640546
$800$	6027.75	0.266391
$801$	−1641.83	−0.0724237
$802$	18097.0	0.796792
$803$	−12707.0	−0.558431
$804$	5777.27	0.253419
$805$	0	0
$806$	425.478	0.0185941
$807$	6042.91	0.263594
$808$	−13220.3	−0.575607
$809$	−26674.6	−1.15924	−0.579622	−	0.814885i	$-0.696800\pi$
$809$	−26674.6	−1.15924	−0.579622	+	0.814885i	$0.696800\pi$
$810$	−8688.41	−0.376888
$811$	34427.4	1.49064	0.745320	−	0.666706i	$-0.232297\pi$
$811$	34427.4	1.49064	0.745320	+	0.666706i	$0.232297\pi$
$812$	0	0
$813$	−8470.73	−0.365414
$814$	−11236.6	−0.483837
$815$	−19519.7	−0.838953
$816$	−2167.72	−0.0929968
$817$	−8860.42	−0.379421
$818$	−8370.69	−0.357793
$819$	0	0
$820$	−33167.6	−1.41252
$821$	−7283.95	−0.309637	−0.154818	−	0.987943i	$-0.549479\pi$
$821$	−7283.95	−0.309637	−0.154818	+	0.987943i	$0.549479\pi$
$822$	−2240.16	−0.0950543
$823$	−5862.49	−0.248303	−0.124152	−	0.992263i	$-0.539621\pi$
$823$	−5862.49	−0.248303	−0.124152	+	0.992263i	$0.539621\pi$
$824$	−5468.23	−0.231183
$825$	−9282.96	−0.391747
$826$	0	0
$827$	−47141.6	−1.98219	−0.991097	−	0.133143i	$-0.957493\pi$
$827$	−47141.6	−1.98219	−0.991097	+	0.133143i	$0.957493\pi$
$828$	−1131.97	−0.0475103
$829$	5182.32	0.217116	0.108558	−	0.994090i	$-0.465377\pi$
$829$	5182.32	0.217116	0.108558	+	0.994090i	$0.465377\pi$
$830$	38501.4	1.61012
$831$	−22379.0	−0.934197
$832$	112.713	0.00469667
$833$	0	0
$834$	7856.04	0.326178
$835$	−55683.1	−2.30778
$836$	−1215.75	−0.0502962
$837$	−18200.8	−0.751625
$838$	5338.80	0.220079
$839$	23541.5	0.968704	0.484352	−	0.874873i	$-0.339055\pi$
$839$	23541.5	0.968704	0.484352	+	0.874873i	$0.339055\pi$
$840$	0	0
$841$	70708.8	2.89921
$842$	4830.87	0.197723
$843$	−4644.50	−0.189757
$844$	−13090.4	−0.533877
$845$	−38836.8	−1.58110
$846$	−10073.1	−0.409361
$847$	0	0
$848$	409.715	0.0165916
$849$	1555.52	0.0628802
$850$	−13314.3	−0.537266
$851$	10052.0	0.404909
$852$	6682.90	0.268724
$853$	5075.40	0.203726	0.101863	−	0.994798i	$-0.467520\pi$
$853$	5075.40	0.203726	0.101863	+	0.994798i	$0.467520\pi$
$854$	0	0
$855$	−5149.62	−0.205980
$856$	910.916	0.0363720
$857$	8940.83	0.356374	0.178187	−	0.983997i	$-0.442977\pi$
$857$	8940.83	0.356374	0.178187	+	0.983997i	$0.442977\pi$
$858$	−173.583	−0.00690678
$859$	3592.54	0.142696	0.0713480	−	0.997451i	$-0.477270\pi$
$859$	3592.54	0.142696	0.0713480	+	0.997451i	$0.477270\pi$
$860$	−26536.2	−1.05218
$861$	0	0
$862$	3178.08	0.125575
$863$	−3297.35	−0.130062	−0.0650308	−	0.997883i	$-0.520715\pi$
$863$	−3297.35	−0.130062	−0.0650308	+	0.997883i	$0.520715\pi$
$864$	−4821.55	−0.189852
$865$	64429.7	2.53257
$866$	30532.6	1.19808
$867$	−14046.0	−0.550206
$868$	0	0
$869$	−6738.36	−0.263042
$870$	41854.5	1.63103
$871$	663.525	0.0258125
$872$	795.898	0.0309088
$873$	14281.5	0.553671
$874$	1087.58	0.0420914
$875$	0	0
$876$	15157.3	0.584609
$877$	27099.1	1.04341	0.521706	−	0.853125i	$-0.325296\pi$
$877$	27099.1	1.04341	0.521706	+	0.853125i	$0.325296\pi$
$878$	−13866.1	−0.532984
$879$	−796.639	−0.0305688
$880$	−3641.06	−0.139478
$881$	−36788.0	−1.40683	−0.703417	−	0.710777i	$-0.748343\pi$
$881$	−36788.0	−1.40683	−0.703417	+	0.710777i	$0.748343\pi$
$882$	0	0
$883$	5402.83	0.205911	0.102956	−	0.994686i	$-0.467170\pi$
$883$	5402.83	0.205911	0.102956	+	0.994686i	$0.467170\pi$
$884$	−248.965	−0.00947239
$885$	43242.1	1.64245
$886$	−33617.2	−1.27471
$887$	16458.0	0.623004	0.311502	−	0.950246i	$-0.399168\pi$
$887$	16458.0	0.623004	0.311502	+	0.950246i	$0.399168\pi$
$888$	13403.4	0.506518
$889$	0	0
$890$	4724.33	0.177933
$891$	3154.75	0.118617
$892$	3834.89	0.143948
$893$	9678.10	0.362671
$894$	−21864.1	−0.817947
$895$	−25737.3	−0.961231
$896$	0	0
$897$	155.282	0.00578008
$898$	1718.16	0.0638483
$899$	37251.0	1.38197
$900$	−9270.65	−0.343357
$901$	−904.992	−0.0334624
$902$	12043.1	0.444559
$903$	0	0
$904$	6420.30	0.236213
$905$	−29382.3	−1.07923
$906$	1461.07	0.0535769
$907$	51745.1	1.89434	0.947170	−	0.320731i	$-0.103929\pi$
$907$	51745.1	1.89434	0.947170	+	0.320731i	$0.103929\pi$
$908$	−20747.0	−0.758273
$909$	20332.8	0.741912
$910$	0	0
$911$	9591.20	0.348815	0.174408	−	0.984674i	$-0.444199\pi$
$911$	9591.20	0.348815	0.174408	+	0.984674i	$0.444199\pi$
$912$	1450.18	0.0526539
$913$	−13979.8	−0.506752
$914$	32276.2	1.16805
$915$	42380.3	1.53120
$916$	9534.46	0.343916
$917$	0	0
$918$	10650.0	0.382900
$919$	−34675.0	−1.24464	−0.622319	−	0.782763i	$-0.713809\pi$
$919$	−34675.0	−1.24464	−0.622319	+	0.782763i	$0.713809\pi$
$920$	3257.20	0.116725
$921$	677.214	0.0242290
$922$	−30287.4	−1.08185
$923$	767.538	0.0273714
$924$	0	0
$925$	82324.5	2.92628
$926$	−3328.42	−0.118119
$927$	8410.12	0.297977
$928$	9868.14	0.349071
$929$	10925.9	0.385864	0.192932	−	0.981212i	$-0.438200\pi$
$929$	10925.9	0.385864	0.192932	+	0.981212i	$0.438200\pi$
$930$	16394.9	0.578076
$931$	0	0
$932$	−9735.07	−0.342149
$933$	21269.3	0.746328
$934$	−10449.5	−0.366081
$935$	8042.51	0.281303
$936$	−173.352	−0.00605363
$937$	30674.2	1.06946	0.534729	−	0.845024i	$-0.320414\pi$
$937$	30674.2	1.06946	0.534729	+	0.845024i	$0.320414\pi$
$938$	0	0
$939$	2316.98	0.0805236
$940$	28985.0	1.00573
$941$	10554.9	0.365653	0.182826	−	0.983145i	$-0.441475\pi$
$941$	10554.9	0.365653	0.182826	+	0.983145i	$0.441475\pi$
$942$	−24888.9	−0.860854
$943$	−10773.5	−0.372039
$944$	10195.3	0.351513
$945$	0	0
$946$	9635.25	0.331151
$947$	−23938.4	−0.821431	−0.410715	−	0.911764i	$-0.634721\pi$
$947$	−23938.4	−0.821431	−0.410715	+	0.911764i	$0.634721\pi$
$948$	8037.72	0.275372
$949$	1740.83	0.0595466
$950$	8907.13	0.304195
$951$	13076.5	0.445883
$952$	0	0
$953$	36298.6	1.23382	0.616908	−	0.787035i	$-0.288385\pi$
$953$	36298.6	1.23382	0.616908	+	0.787035i	$0.288385\pi$
$954$	−630.140	−0.0213853
$955$	−37483.3	−1.27009
$956$	8352.11	0.282559
$957$	−15197.3	−0.513333
$958$	34824.3	1.17445
$959$	0	0
$960$	4343.17	0.146016
$961$	−15199.3	−0.510198
$962$	1539.39	0.0515924
$963$	−1400.98	−0.0468807
$964$	−11030.2	−0.368527
$965$	38631.0	1.28868
$966$	0	0
$967$	35482.9	1.17999	0.589996	−	0.807406i	$-0.299129\pi$
$967$	35482.9	1.17999	0.589996	+	0.807406i	$0.299129\pi$
$968$	−9325.93	−0.309656
$969$	−3203.21	−0.106194
$970$	−41094.6	−1.36028
$971$	−15292.1	−0.505405	−0.252703	−	0.967544i	$-0.581319\pi$
$971$	−15292.1	−0.505405	−0.252703	+	0.967544i	$0.581319\pi$
$972$	12509.6	0.412805
$973$	0	0
$974$	−9191.87	−0.302389
$975$	1271.74	0.0417727
$976$	9992.11	0.327705
$977$	−41684.6	−1.36501	−0.682503	−	0.730883i	$-0.739108\pi$
$977$	−41684.6	−1.36501	−0.682503	+	0.730883i	$0.739108\pi$
$978$	−8454.29	−0.276420
$979$	−1715.40	−0.0560005
$980$	0	0
$981$	−1224.09	−0.0398391
$982$	39271.4	1.27617
$983$	−10056.7	−0.326305	−0.163153	−	0.986601i	$-0.552166\pi$
$983$	−10056.7	−0.326305	−0.163153	+	0.986601i	$0.552166\pi$
$984$	−14365.4	−0.465399
$985$	−41789.8	−1.35181
$986$	−21797.1	−0.704017
$987$	0	0
$988$	166.555	0.00536318
$989$	−8619.44	−0.277131
$990$	5599.94	0.179776
$991$	56618.5	1.81488	0.907440	−	0.420181i	$-0.138034\pi$
$991$	56618.5	1.81488	0.907440	+	0.420181i	$0.138034\pi$
$992$	3865.47	0.123719
$993$	16058.3	0.513186
$994$	0	0
$995$	−28993.9	−0.923786
$996$	16675.5	0.530507
$997$	−1853.84	−0.0588885	−0.0294443	−	0.999566i	$-0.509374\pi$
$997$	−1853.84	−0.0588885	−0.0294443	+	0.999566i	$0.509374\pi$
$998$	1121.10	0.0355589
$999$	−65850.7	−2.08551

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2254.4.a.y.1.7		11
7.2	even	3		322.4.e.a.277.5	yes	22
7.4	even	3		322.4.e.a.93.5	✓	22
7.6	odd	2		2254.4.a.v.1.5		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.4.e.a.93.5	✓	22	7.4	even	3
322.4.e.a.277.5	yes	22	7.2	even	3
2254.4.a.v.1.5		11	7.6	odd	2
2254.4.a.y.1.7		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)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +3.83354 q^{3} +4.00000 q^{4} +17.7022 q^{5} +7.66708 q^{6} +8.00000 q^{8} -12.3040 q^{9} +35.4044 q^{10} -12.8553 q^{11} +15.3342 q^{12} +1.76114 q^{13} +67.8620 q^{15} +16.0000 q^{16} -35.3413 q^{17} -24.6079 q^{18} +23.6430 q^{19} +70.8087 q^{20} -25.7106 q^{22} +23.0000 q^{23} +30.6683 q^{24} +188.367 q^{25} +3.52229 q^{26} -150.673 q^{27} +308.379 q^{29} +135.724 q^{30} +120.796 q^{31} +32.0000 q^{32} -49.2812 q^{33} -70.6827 q^{34} -49.2159 q^{36} +437.043 q^{37} +47.2860 q^{38} +6.75141 q^{39} +141.617 q^{40} -468.412 q^{41} -374.758 q^{43} -51.4211 q^{44} -217.807 q^{45} +46.0000 q^{46} +409.343 q^{47} +61.3366 q^{48} +376.734 q^{50} -135.482 q^{51} +7.04457 q^{52} +25.6072 q^{53} -301.347 q^{54} -227.567 q^{55} +90.6364 q^{57} +616.759 q^{58} +637.206 q^{59} +271.448 q^{60} +624.507 q^{61} +241.592 q^{62} +64.0000 q^{64} +31.1761 q^{65} -98.5625 q^{66} +376.758 q^{67} -141.365 q^{68} +88.1714 q^{69} +435.818 q^{71} -98.4318 q^{72} +988.465 q^{73} +874.086 q^{74} +722.113 q^{75} +94.5721 q^{76} +13.5028 q^{78} +524.171 q^{79} +283.235 q^{80} -245.405 q^{81} -936.823 q^{82} +1087.48 q^{83} -625.619 q^{85} -749.517 q^{86} +1182.18 q^{87} -102.842 q^{88} +133.439 q^{89} -435.614 q^{90} +92.0000 q^{92} +463.077 q^{93} +818.685 q^{94} +418.533 q^{95} +122.673 q^{96} -1160.72 q^{97} +158.171 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 22 q^{2} + 18 q^{3} + 44 q^{4} + 33 q^{5} + 36 q^{6} + 88 q^{8} + 171 q^{9} + 66 q^{10} + 8 q^{11} + 72 q^{12} + 185 q^{13} - 186 q^{15} + 176 q^{16} + 107 q^{17} + 342 q^{18} + 114 q^{19} + 132 q^{20}+ \cdots - 1729 q^{99}+O(q^{100}) \) 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

Related objects

Downloads

Learn more