LMFDB - Embedded newform 1275.4.a.bf.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1275,4,Mod(1,1275)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1275, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1275.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1275 = 3 \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1275.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:	$75.2274352573$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 2 x^{13} - 88 x^{12} + 151 x^{11} + 2969 x^{10} - 4260 x^{9} - 48218 x^{8} + 56779 x^{7} + \cdots - 40000 $ x^14 - 2x^13 - 88x^12 + 151x^11 + 2969x^10 - 4260x^9 - 48218x^8 + 56779x^7 + 385032x^6 - 364360x^5 - 1363392x^4 + 986348x^3 + 1567872x^2 - 1068240*x - 40000
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8}\cdot 5^{3} $
Twist minimal:	no (minimal twist has level 255)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1.25969$ of defining polynomial
Character	$\chi$	$=$	1275.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.25969 q^{2} -3.00000 q^{3} -6.41317 q^{4} +3.77908 q^{6} -12.7244 q^{7} +18.1562 q^{8} +9.00000 q^{9} -62.4899 q^{11} +19.2395 q^{12} +82.6391 q^{13} +16.0289 q^{14} +28.4341 q^{16} +17.0000 q^{17} -11.3373 q^{18} +80.2399 q^{19} +38.1733 q^{21} +78.7182 q^{22} -200.609 q^{23} -54.4686 q^{24} -104.100 q^{26} -27.0000 q^{27} +81.6040 q^{28} -188.476 q^{29} -11.0660 q^{31} -181.068 q^{32} +187.470 q^{33} -21.4148 q^{34} -57.7185 q^{36} -313.895 q^{37} -101.078 q^{38} -247.917 q^{39} +49.9679 q^{41} -48.0868 q^{42} +15.2173 q^{43} +400.758 q^{44} +252.706 q^{46} -311.467 q^{47} -85.3022 q^{48} -181.088 q^{49} -51.0000 q^{51} -529.979 q^{52} -244.318 q^{53} +34.0118 q^{54} -231.028 q^{56} -240.720 q^{57} +237.423 q^{58} +471.083 q^{59} +539.301 q^{61} +13.9397 q^{62} -114.520 q^{63} +0.617610 q^{64} -236.155 q^{66} -185.229 q^{67} -109.024 q^{68} +601.827 q^{69} +1013.09 q^{71} +163.406 q^{72} +167.726 q^{73} +395.411 q^{74} -514.592 q^{76} +795.149 q^{77} +312.300 q^{78} -883.216 q^{79} +81.0000 q^{81} -62.9444 q^{82} -760.655 q^{83} -244.812 q^{84} -19.1691 q^{86} +565.429 q^{87} -1134.58 q^{88} -1114.45 q^{89} -1051.54 q^{91} +1286.54 q^{92} +33.1979 q^{93} +392.354 q^{94} +543.204 q^{96} +21.6086 q^{97} +228.116 q^{98} -562.409 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 2 q^{2} - 42 q^{3} + 68 q^{4} + 6 q^{6} - 8 q^{7} - 51 q^{8} + 126 q^{9} + 58 q^{11} - 204 q^{12} + 72 q^{13} + 257 q^{14} + 404 q^{16} + 238 q^{17} - 18 q^{18} + 282 q^{19} + 24 q^{21} - 361 q^{22}+ \cdots + 522 q^{99}+O(q^{100}) $ 14 * q - 2 * q^2 - 42 * q^3 + 68 * q^4 + 6 * q^6 - 8 * q^7 - 51 * q^8 + 126 * q^9 + 58 * q^11 - 204 * q^12 + 72 * q^13 + 257 * q^14 + 404 * q^16 + 238 * q^17 - 18 * q^18 + 282 * q^19 + 24 * q^21 - 361 * q^22 - 344 * q^23 + 153 * q^24 + 174 * q^26 - 378 * q^27 + q^28 + 434 * q^29 + 226 * q^31 - 870 * q^32 - 174 * q^33 - 34 * q^34 + 612 * q^36 - 166 * q^37 - 637 * q^38 - 216 * q^39 + 618 * q^41 - 771 * q^42 + 590 * q^43 + 1443 * q^44 + 130 * q^46 - 944 * q^47 - 1212 * q^48 + 1718 * q^49 - 714 * q^51 + 246 * q^52 - 628 * q^53 + 54 * q^54 + 1902 * q^56 - 846 * q^57 - 89 * q^58 + 564 * q^59 + 1680 * q^61 - 1556 * q^62 - 72 * q^63 + 2855 * q^64 + 1083 * q^66 + 658 * q^67 + 1156 * q^68 + 1032 * q^69 + 2582 * q^71 - 459 * q^72 + 1114 * q^73 + 61 * q^74 + 5993 * q^76 - 780 * q^77 - 522 * q^78 + 902 * q^79 + 1134 * q^81 - 849 * q^82 - 2082 * q^83 - 3 * q^84 + 1608 * q^86 - 1302 * q^87 - 2324 * q^88 + 992 * q^89 + 2416 * q^91 - 4410 * q^92 - 678 * q^93 - 541 * q^94 + 2610 * q^96 + 980 * q^97 + 1237 * q^98 + 522 * 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 $

$2$	−1.25969	−0.445369	−0.222685	−	0.974891i	$-0.571482\pi$
$2$	−1.25969	−0.445369	−0.222685	+	0.974891i	$0.571482\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−6.41317	−0.801646
$5$	0	0
$5$	0	0
$6$	3.77908	0.257134
$7$	−12.7244	−0.687056	−0.343528	−	0.939143i	$-0.611622\pi$
$7$	−12.7244	−0.687056	−0.343528	+	0.939143i	$0.611622\pi$
$8$	18.1562	0.802398
$9$	9.00000	0.333333
$10$	0	0
$11$	−62.4899	−1.71285	−0.856427	−	0.516267i	$-0.827321\pi$
$11$	−62.4899	−1.71285	−0.856427	+	0.516267i	$0.827321\pi$
$12$	19.2395	0.462831
$13$	82.6391	1.76307	0.881537	−	0.472114i	$-0.156509\pi$
$13$	82.6391	1.76307	0.881537	+	0.472114i	$0.156509\pi$
$14$	16.0289	0.305994
$15$	0	0
$16$	28.4341	0.444282
$17$	17.0000	0.242536
$17$	17.0000	0.242536
$18$	−11.3373	−0.148456
$19$	80.2399	0.968858	0.484429	−	0.874831i	$-0.339027\pi$
$19$	80.2399	0.968858	0.484429	+	0.874831i	$0.339027\pi$
$20$	0	0
$21$	38.1733	0.396672
$22$	78.7182	0.762853
$23$	−200.609	−1.81869	−0.909345	−	0.416042i	$-0.863417\pi$
$23$	−200.609	−1.81869	−0.909345	+	0.416042i	$0.863417\pi$
$24$	−54.4686	−0.463265
$25$	0	0
$26$	−104.100	−0.785220
$27$	−27.0000	−0.192450
$28$	81.6040	0.550775
$29$	−188.476	−1.20687	−0.603434	−	0.797413i	$-0.706201\pi$
$29$	−188.476	−1.20687	−0.603434	+	0.797413i	$0.706201\pi$
$30$	0	0
$31$	−11.0660	−0.0641130	−0.0320565	−	0.999486i	$-0.510206\pi$
$31$	−11.0660	−0.0641130	−0.0320565	+	0.999486i	$0.510206\pi$
$32$	−181.068	−1.00027
$33$	187.470	0.988917
$34$	−21.4148	−0.108018
$35$	0	0
$36$	−57.7185	−0.267215
$37$	−313.895	−1.39470	−0.697351	−	0.716730i	$-0.745638\pi$
$37$	−313.895	−1.39470	−0.697351	+	0.716730i	$0.745638\pi$
$38$	−101.078	−0.431500
$39$	−247.917	−1.01791
$40$	0	0
$41$	49.9679	0.190334	0.0951668	−	0.995461i	$-0.469662\pi$
$41$	49.9679	0.190334	0.0951668	+	0.995461i	$0.469662\pi$
$42$	−48.0868	−0.176665
$43$	15.2173	0.0539678	0.0269839	−	0.999636i	$-0.491410\pi$
$43$	15.2173	0.0539678	0.0269839	+	0.999636i	$0.491410\pi$
$44$	400.758	1.37310
$45$	0	0
$46$	252.706	0.809989
$47$	−311.467	−0.966642	−0.483321	−	0.875443i	$-0.660570\pi$
$47$	−311.467	−0.966642	−0.483321	+	0.875443i	$0.660570\pi$
$48$	−85.3022	−0.256507
$49$	−181.088	−0.527955
$50$	0	0
$51$	−51.0000	−0.140028
$52$	−529.979	−1.41336
$53$	−244.318	−0.633202	−0.316601	−	0.948559i	$-0.602542\pi$
$53$	−244.318	−0.633202	−0.316601	+	0.948559i	$0.602542\pi$
$54$	34.0118	0.0857114
$55$	0	0
$56$	−231.028	−0.551292
$57$	−240.720	−0.559370
$58$	237.423	0.537502
$59$	471.083	1.03949	0.519744	−	0.854322i	$-0.326027\pi$
$59$	471.083	1.03949	0.519744	+	0.854322i	$0.326027\pi$
$60$	0	0
$61$	539.301	1.13197	0.565987	−	0.824414i	$-0.308495\pi$
$61$	539.301	1.13197	0.565987	+	0.824414i	$0.308495\pi$
$62$	13.9397	0.0285540
$63$	−114.520	−0.229019
$64$	0.617610	0.00120627
$65$	0	0
$66$	−236.155	−0.440434
$67$	−185.229	−0.337751	−0.168875	−	0.985637i	$-0.554014\pi$
$67$	−185.229	−0.337751	−0.168875	+	0.985637i	$0.554014\pi$
$68$	−109.024	−0.194428
$69$	601.827	1.05002
$70$	0	0
$71$	1013.09	1.69340	0.846698	−	0.532074i	$-0.178587\pi$
$71$	1013.09	1.69340	0.846698	+	0.532074i	$0.178587\pi$
$72$	163.406	0.267466
$73$	167.726	0.268915	0.134458	−	0.990919i	$-0.457071\pi$
$73$	167.726	0.268915	0.134458	+	0.990919i	$0.457071\pi$
$74$	395.411	0.621157
$75$	0	0
$76$	−514.592	−0.776681
$77$	795.149	1.17683
$78$	312.300	0.453347
$79$	−883.216	−1.25784	−0.628921	−	0.777469i	$-0.716503\pi$
$79$	−883.216	−1.25784	−0.628921	+	0.777469i	$0.716503\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	−62.9444	−0.0847688
$83$	−760.655	−1.00594	−0.502968	−	0.864305i	$-0.667759\pi$
$83$	−760.655	−1.00594	−0.502968	+	0.864305i	$0.667759\pi$
$84$	−244.812	−0.317990
$85$	0	0
$86$	−19.1691	−0.0240356
$87$	565.429	0.696785
$88$	−1134.58	−1.37439
$89$	−1114.45	−1.32732	−0.663659	−	0.748036i	$-0.730997\pi$
$89$	−1114.45	−1.32732	−0.663659	+	0.748036i	$0.730997\pi$
$90$	0	0
$91$	−1051.54	−1.21133
$92$	1286.54	1.45795
$93$	33.1979	0.0370157
$94$	392.354	0.430513
$95$	0	0
$96$	543.204	0.577505
$97$	21.6086	0.0226187	0.0113094	−	0.999936i	$-0.496400\pi$
$97$	21.6086	0.0226187	0.0113094	+	0.999936i	$0.496400\pi$
$98$	228.116	0.235135
$99$	−562.409	−0.570952
$100$	0	0
$101$	−368.472	−0.363013	−0.181507	−	0.983390i	$-0.558097\pi$
$101$	−368.472	−0.363013	−0.181507	+	0.983390i	$0.558097\pi$
$102$	64.2444	0.0623642
$103$	271.410	0.259639	0.129820	−	0.991538i	$-0.458560\pi$
$103$	271.410	0.259639	0.129820	+	0.991538i	$0.458560\pi$
$104$	1500.41	1.41469
$105$	0	0
$106$	307.766	0.282009
$107$	−1486.02	−1.34260	−0.671302	−	0.741184i	$-0.734265\pi$
$107$	−1486.02	−1.34260	−0.671302	+	0.741184i	$0.734265\pi$
$108$	173.156	0.154277
$109$	−1530.13	−1.34459	−0.672295	−	0.740284i	$-0.734691\pi$
$109$	−1530.13	−1.34459	−0.672295	+	0.740284i	$0.734691\pi$
$110$	0	0
$111$	941.684	0.805231
$112$	−361.808	−0.305247
$113$	−22.1792	−0.0184641	−0.00923206	−	0.999957i	$-0.502939\pi$
$113$	−22.1792	−0.0184641	−0.00923206	+	0.999957i	$0.502939\pi$
$114$	303.233	0.249126
$115$	0	0
$116$	1208.73	0.967480
$117$	743.752	0.587692
$118$	−593.421	−0.462956
$119$	−216.316	−0.166635
$120$	0	0
$121$	2573.98	1.93387
$122$	−679.355	−0.504147
$123$	−149.904	−0.109889
$124$	70.9678	0.0513960
$125$	0	0
$126$	144.260	0.101998
$127$	−416.040	−0.290690	−0.145345	−	0.989381i	$-0.546429\pi$
$127$	−416.040	−0.290690	−0.145345	+	0.989381i	$0.546429\pi$
$128$	1447.76	0.999731
$129$	−45.6518	−0.0311583
$130$	0	0
$131$	−668.770	−0.446036	−0.223018	−	0.974814i	$-0.571591\pi$
$131$	−668.770	−0.446036	−0.223018	+	0.974814i	$0.571591\pi$
$132$	−1202.27	−0.792762
$133$	−1021.01	−0.665659
$134$	233.332	0.150424
$135$	0	0
$136$	308.655	0.194610
$137$	1744.43	1.08786	0.543931	−	0.839130i	$-0.316935\pi$
$137$	1744.43	1.08786	0.543931	+	0.839130i	$0.316935\pi$
$138$	−758.119	−0.467648
$139$	2952.05	1.80136	0.900682	−	0.434479i	$-0.143067\pi$
$139$	2952.05	1.80136	0.900682	+	0.434479i	$0.143067\pi$
$140$	0	0
$141$	934.402	0.558091
$142$	−1276.18	−0.754187
$143$	−5164.11	−3.01989
$144$	255.907	0.148094
$145$	0	0
$146$	−211.283	−0.119767
$147$	543.265	0.304815
$148$	2013.06	1.11806
$149$	1973.80	1.08524	0.542618	−	0.839979i	$-0.317433\pi$
$149$	1973.80	1.08524	0.542618	+	0.839979i	$0.317433\pi$
$150$	0	0
$151$	1959.35	1.05596	0.527980	−	0.849257i	$-0.322950\pi$
$151$	1959.35	1.05596	0.527980	+	0.849257i	$0.322950\pi$
$152$	1456.85	0.777410
$153$	153.000	0.0808452
$154$	−1001.65	−0.524123
$155$	0	0
$156$	1589.94	0.816005
$157$	2470.75	1.25597	0.627986	−	0.778224i	$-0.283879\pi$
$157$	2470.75	1.25597	0.627986	+	0.778224i	$0.283879\pi$
$158$	1112.58	0.560204
$159$	732.955	0.365579
$160$	0	0
$161$	2552.64	1.24954
$162$	−102.035	−0.0494855
$163$	−2422.36	−1.16401	−0.582005	−	0.813185i	$-0.697731\pi$
$163$	−2422.36	−1.16401	−0.582005	+	0.813185i	$0.697731\pi$
$164$	−320.453	−0.152580
$165$	0	0
$166$	958.193	0.448013
$167$	870.144	0.403196	0.201598	−	0.979468i	$-0.435386\pi$
$167$	870.144	0.403196	0.201598	+	0.979468i	$0.435386\pi$
$168$	693.083	0.318289
$169$	4632.23	2.10843
$170$	0	0
$171$	722.159	0.322953
$172$	−97.5910	−0.0432630
$173$	739.243	0.324876	0.162438	−	0.986719i	$-0.448064\pi$
$173$	739.243	0.324876	0.162438	+	0.986719i	$0.448064\pi$
$174$	−712.268	−0.310327
$175$	0	0
$176$	−1776.84	−0.760991
$177$	−1413.25	−0.600149
$178$	1403.86	0.591147
$179$	3780.32	1.57852	0.789258	−	0.614062i	$-0.210465\pi$
$179$	3780.32	1.57852	0.789258	+	0.614062i	$0.210465\pi$
$180$	0	0
$181$	1240.89	0.509583	0.254792	−	0.966996i	$-0.417993\pi$
$181$	1240.89	0.509583	0.254792	+	0.966996i	$0.417993\pi$
$182$	1324.62	0.539489
$183$	−1617.90	−0.653545
$184$	−3642.30	−1.45931
$185$	0	0
$186$	−41.8192	−0.0164857
$187$	−1062.33	−0.415428
$188$	1997.49	0.774905
$189$	343.560	0.132224
$190$	0	0
$191$	−343.172	−0.130006	−0.0650028	−	0.997885i	$-0.520706\pi$
$191$	−343.172	−0.130006	−0.0650028	+	0.997885i	$0.520706\pi$
$192$	−1.85283	−0.000696440 0
$193$	−252.881	−0.0943148	−0.0471574	−	0.998887i	$-0.515016\pi$
$193$	−252.881	−0.0943148	−0.0471574	+	0.998887i	$0.515016\pi$
$194$	−27.2202	−0.0100737
$195$	0	0
$196$	1161.35	0.423233
$197$	2036.95	0.736685	0.368343	−	0.929690i	$-0.379925\pi$
$197$	2036.95	0.736685	0.368343	+	0.929690i	$0.379925\pi$
$198$	708.464	0.254284
$199$	329.405	0.117341	0.0586707	−	0.998277i	$-0.481314\pi$
$199$	329.405	0.117341	0.0586707	+	0.998277i	$0.481314\pi$
$200$	0	0
$201$	555.687	0.195001
$202$	464.163	0.161675
$203$	2398.26	0.829185
$204$	327.072	0.112253
$205$	0	0
$206$	−341.894	−0.115635
$207$	−1805.48	−0.606230
$208$	2349.77	0.783303
$209$	−5014.18	−1.65951
$210$	0	0
$211$	233.272	0.0761096	0.0380548	−	0.999276i	$-0.487884\pi$
$211$	233.272	0.0761096	0.0380548	+	0.999276i	$0.487884\pi$
$212$	1566.85	0.507604
$213$	−3039.26	−0.977683
$214$	1871.93	0.597955
$215$	0	0
$216$	−490.217	−0.154422
$217$	140.808	0.0440492
$218$	1927.50	0.598839
$219$	−503.177	−0.155258
$220$	0	0
$221$	1404.87	0.427608
$222$	−1186.23	−0.358625
$223$	−311.137	−0.0934317	−0.0467158	−	0.998908i	$-0.514876\pi$
$223$	−311.137	−0.0934317	−0.0467158	+	0.998908i	$0.514876\pi$
$224$	2303.99	0.687240
$225$	0	0
$226$	27.9391	0.00822335
$227$	−5854.93	−1.71192	−0.855959	−	0.517044i	$-0.827032\pi$
$227$	−5854.93	−1.71192	−0.855959	+	0.517044i	$0.827032\pi$
$228$	1543.78	0.448417
$229$	3167.85	0.914136	0.457068	−	0.889432i	$-0.348900\pi$
$229$	3167.85	0.914136	0.457068	+	0.889432i	$0.348900\pi$
$230$	0	0
$231$	−2385.45	−0.679441
$232$	−3422.01	−0.968388
$233$	3726.78	1.04785	0.523926	−	0.851764i	$-0.324467\pi$
$233$	3726.78	1.04785	0.523926	+	0.851764i	$0.324467\pi$
$234$	−936.901	−0.261740
$235$	0	0
$236$	−3021.14	−0.833302
$237$	2649.65	0.726215
$238$	272.492	0.0742143
$239$	−2335.34	−0.632054	−0.316027	−	0.948750i	$-0.602349\pi$
$239$	−2335.34	−0.632054	−0.316027	+	0.948750i	$0.602349\pi$
$240$	0	0
$241$	5424.08	1.44977	0.724887	−	0.688867i	$-0.241892\pi$
$241$	5424.08	1.44977	0.724887	+	0.688867i	$0.241892\pi$
$242$	−3242.43	−0.861287
$243$	−243.000	−0.0641500
$244$	−3458.63	−0.907442
$245$	0	0
$246$	188.833	0.0489413
$247$	6630.96	1.70817
$248$	−200.916	−0.0514442
$249$	2281.96	0.580778
$250$	0	0
$251$	1283.75	0.322828	0.161414	−	0.986887i	$-0.448395\pi$
$251$	1283.75	0.322828	0.161414	+	0.986887i	$0.448395\pi$
$252$	734.436	0.183592
$253$	12536.0	3.11515
$254$	524.083	0.129464
$255$	0	0
$256$	−1828.68	−0.446456
$257$	−4082.27	−0.990838	−0.495419	−	0.868654i	$-0.664985\pi$
$257$	−4082.27	−0.990838	−0.495419	+	0.868654i	$0.664985\pi$
$258$	57.5074	0.0138770
$259$	3994.13	0.958237
$260$	0	0
$261$	−1696.29	−0.402289
$262$	842.447	0.198651
$263$	−3450.41	−0.808979	−0.404490	−	0.914543i	$-0.632551\pi$
$263$	−3450.41	−0.808979	−0.404490	+	0.914543i	$0.632551\pi$
$264$	3403.73	0.793505
$265$	0	0
$266$	1286.16	0.296464
$267$	3343.34	0.766327
$268$	1187.90	0.270757
$269$	269.468	0.0610771	0.0305386	−	0.999534i	$-0.490278\pi$
$269$	269.468	0.0610771	0.0305386	+	0.999534i	$0.490278\pi$
$270$	0	0
$271$	−2713.52	−0.608246	−0.304123	−	0.952633i	$-0.598363\pi$
$271$	−2713.52	−0.608246	−0.304123	+	0.952633i	$0.598363\pi$
$272$	483.379	0.107754
$273$	3154.61	0.699362
$274$	−2197.45	−0.484500
$275$	0	0
$276$	−3859.62	−0.841746
$277$	−749.213	−0.162512	−0.0812560	−	0.996693i	$-0.525893\pi$
$277$	−749.213	−0.162512	−0.0812560	+	0.996693i	$0.525893\pi$
$278$	−3718.68	−0.802272
$279$	−99.5936	−0.0213710
$280$	0	0
$281$	3890.04	0.825837	0.412918	−	0.910768i	$-0.364509\pi$
$281$	3890.04	0.825837	0.412918	+	0.910768i	$0.364509\pi$
$282$	−1177.06	−0.248557
$283$	−2355.55	−0.494780	−0.247390	−	0.968916i	$-0.579573\pi$
$283$	−2355.55	−0.494780	−0.247390	+	0.968916i	$0.579573\pi$
$284$	−6497.09	−1.35750
$285$	0	0
$286$	6505.20	1.34497
$287$	−635.814	−0.130770
$288$	−1629.61	−0.333423
$289$	289.000	0.0588235
$290$	0	0
$291$	−64.8257	−0.0130589
$292$	−1075.65	−0.215575
$293$	−1280.74	−0.255365	−0.127682	−	0.991815i	$-0.540754\pi$
$293$	−1280.74	−0.255365	−0.127682	+	0.991815i	$0.540754\pi$
$294$	−684.349	−0.135755
$295$	0	0
$296$	−5699.13	−1.11911
$297$	1687.23	0.329639
$298$	−2486.39	−0.483331
$299$	−16578.2	−3.20649
$300$	0	0
$301$	−193.631	−0.0370788
$302$	−2468.19	−0.470292
$303$	1105.42	0.209586
$304$	2281.55	0.430447
$305$	0	0
$306$	−192.733	−0.0360060
$307$	−5433.85	−1.01018	−0.505092	−	0.863066i	$-0.668541\pi$
$307$	−5433.85	−1.01018	−0.505092	+	0.863066i	$0.668541\pi$
$308$	−5099.42	−0.943398
$309$	−814.231	−0.149903
$310$	0	0
$311$	6485.11	1.18243	0.591217	−	0.806512i	$-0.298648\pi$
$311$	6485.11	1.18243	0.591217	+	0.806512i	$0.298648\pi$
$312$	−4501.24	−0.816770
$313$	4842.72	0.874526	0.437263	−	0.899334i	$-0.355948\pi$
$313$	4842.72	0.874526	0.437263	+	0.899334i	$0.355948\pi$
$314$	−3112.40	−0.559372
$315$	0	0
$316$	5664.21	1.00834
$317$	413.344	0.0732357	0.0366179	−	0.999329i	$-0.488342\pi$
$317$	413.344	0.0732357	0.0366179	+	0.999329i	$0.488342\pi$
$318$	−923.299	−0.162818
$319$	11777.9	2.06719
$320$	0	0
$321$	4458.05	0.775153
$322$	−3215.55	−0.556508
$323$	1364.08	0.234983
$324$	−519.467	−0.0890718
$325$	0	0
$326$	3051.43	0.518414
$327$	4590.40	0.776299
$328$	907.228	0.152723
$329$	3963.25	0.664137
$330$	0	0
$331$	−8712.38	−1.44675	−0.723377	−	0.690453i	$-0.757411\pi$
$331$	−8712.38	−1.44675	−0.723377	+	0.690453i	$0.757411\pi$
$332$	4878.21	0.806405
$333$	−2825.05	−0.464900
$334$	−1096.12	−0.179571
$335$	0	0
$336$	1085.42	0.176234
$337$	9425.37	1.52354	0.761769	−	0.647848i	$-0.224331\pi$
$337$	9425.37	1.52354	0.761769	+	0.647848i	$0.224331\pi$
$338$	−5835.19	−0.939031
$339$	66.5377	0.0106603
$340$	0	0
$341$	691.510	0.109816
$342$	−909.700	−0.143833
$343$	6668.74	1.04979
$344$	276.288	0.0433036
$345$	0	0
$346$	−931.221	−0.144690
$347$	6135.13	0.949139	0.474570	−	0.880218i	$-0.342604\pi$
$347$	6135.13	0.949139	0.474570	+	0.880218i	$0.342604\pi$
$348$	−3626.19	−0.558575
$349$	7271.83	1.11533	0.557667	−	0.830064i	$-0.311696\pi$
$349$	7271.83	1.11533	0.557667	+	0.830064i	$0.311696\pi$
$350$	0	0
$351$	−2231.26	−0.339304
$352$	11314.9	1.71331
$353$	433.170	0.0653126	0.0326563	−	0.999467i	$-0.489603\pi$
$353$	433.170	0.0653126	0.0326563	+	0.999467i	$0.489603\pi$
$354$	1780.26	0.267288
$355$	0	0
$356$	7147.14	1.06404
$357$	648.947	0.0962070
$358$	−4762.05	−0.703023
$359$	2063.42	0.303352	0.151676	−	0.988430i	$-0.451533\pi$
$359$	2063.42	0.303352	0.151676	+	0.988430i	$0.451533\pi$
$360$	0	0
$361$	−420.554	−0.0613142
$362$	−1563.14	−0.226953
$363$	−7721.95	−1.11652
$364$	6743.68	0.971058
$365$	0	0
$366$	2038.06	0.291069
$367$	5283.59	0.751501	0.375751	−	0.926721i	$-0.377385\pi$
$367$	5283.59	0.751501	0.375751	+	0.926721i	$0.377385\pi$
$368$	−5704.14	−0.808013
$369$	449.711	0.0634446
$370$	0	0
$371$	3108.81	0.435045
$372$	−212.903	−0.0296735
$373$	9240.42	1.28271	0.641355	−	0.767244i	$-0.278372\pi$
$373$	9240.42	1.28271	0.641355	+	0.767244i	$0.278372\pi$
$374$	1338.21	0.185019
$375$	0	0
$376$	−5655.06	−0.775631
$377$	−15575.5	−2.12780
$378$	−432.781	−0.0588885
$379$	−14148.2	−1.91753	−0.958765	−	0.284200i	$-0.908272\pi$
$379$	−14148.2	−1.91753	−0.958765	+	0.284200i	$0.908272\pi$
$380$	0	0
$381$	1248.12	0.167830
$382$	432.292	0.0579005
$383$	55.5041	0.00740503	0.00370252	−	0.999993i	$-0.498821\pi$
$383$	55.5041	0.00740503	0.00370252	+	0.999993i	$0.498821\pi$
$384$	−4343.29	−0.577195
$385$	0	0
$386$	318.553	0.0420049
$387$	136.956	0.0179893
$388$	−138.579	−0.0181322
$389$	−5020.91	−0.654422	−0.327211	−	0.944951i	$-0.606109\pi$
$389$	−5020.91	−0.654422	−0.327211	+	0.944951i	$0.606109\pi$
$390$	0	0
$391$	−3410.35	−0.441097
$392$	−3287.88	−0.423630
$393$	2006.31	0.257519
$394$	−2565.94	−0.328097
$395$	0	0
$396$	3606.82	0.457701
$397$	7924.10	1.00176	0.500881	−	0.865516i	$-0.333010\pi$
$397$	7924.10	1.00176	0.500881	+	0.865516i	$0.333010\pi$
$398$	−414.950	−0.0522602
$399$	3063.03	0.384319
$400$	0	0
$401$	−4752.42	−0.591832	−0.295916	−	0.955214i	$-0.595625\pi$
$401$	−4752.42	−0.591832	−0.295916	+	0.955214i	$0.595625\pi$
$402$	−699.996	−0.0868473
$403$	−914.481	−0.113036
$404$	2363.07	0.291008
$405$	0	0
$406$	−3021.07	−0.369294
$407$	19615.2	2.38892
$408$	−925.966	−0.112358
$409$	−9089.47	−1.09889	−0.549444	−	0.835530i	$-0.685161\pi$
$409$	−9089.47	−1.09889	−0.549444	+	0.835530i	$0.685161\pi$
$410$	0	0
$411$	−5233.30	−0.628077
$412$	−1740.60	−0.208139
$413$	−5994.27	−0.714186
$414$	2274.36	0.269996
$415$	0	0
$416$	−14963.3	−1.76355
$417$	−8856.15	−1.04002
$418$	6316.34	0.739096
$419$	1867.23	0.217710	0.108855	−	0.994058i	$-0.465282\pi$
$419$	1867.23	0.217710	0.108855	+	0.994058i	$0.465282\pi$
$420$	0	0
$421$	−7820.00	−0.905282	−0.452641	−	0.891693i	$-0.649518\pi$
$421$	−7820.00	−0.905282	−0.452641	+	0.891693i	$0.649518\pi$
$422$	−293.852	−0.0338969
$423$	−2803.20	−0.322214
$424$	−4435.89	−0.508080
$425$	0	0
$426$	3828.54	0.435430
$427$	−6862.31	−0.777729
$428$	9530.08	1.07629
$429$	15492.3	1.74353
$430$	0	0
$431$	7389.33	0.825827	0.412913	−	0.910770i	$-0.364511\pi$
$431$	7389.33	0.825827	0.412913	+	0.910770i	$0.364511\pi$
$432$	−767.720	−0.0855022
$433$	−6231.69	−0.691630	−0.345815	−	0.938303i	$-0.612398\pi$
$433$	−6231.69	−0.691630	−0.345815	+	0.938303i	$0.612398\pi$
$434$	−177.375	−0.0196182
$435$	0	0
$436$	9813.01	1.07788
$437$	−16096.9	−1.76205
$438$	633.849	0.0691473
$439$	−10096.9	−1.09772	−0.548859	−	0.835915i	$-0.684937\pi$
$439$	−10096.9	−1.09772	−0.548859	+	0.835915i	$0.684937\pi$
$440$	0	0
$441$	−1629.80	−0.175985
$442$	−1769.70	−0.190444
$443$	−5125.14	−0.549668	−0.274834	−	0.961492i	$-0.588623\pi$
$443$	−5125.14	−0.549668	−0.274834	+	0.961492i	$0.588623\pi$
$444$	−6039.18	−0.645510
$445$	0	0
$446$	391.938	0.0416116
$447$	−5921.41	−0.626562
$448$	−7.85875	−0.000828775 0
$449$	15122.0	1.58942	0.794711	−	0.606987i	$-0.207622\pi$
$449$	15122.0	1.58942	0.794711	+	0.606987i	$0.207622\pi$
$450$	0	0
$451$	−3122.49	−0.326014
$452$	142.239	0.0148017
$453$	−5878.06	−0.609659
$454$	7375.43	0.762436
$455$	0	0
$456$	−4370.56	−0.448838
$457$	10562.0	1.08112	0.540560	−	0.841306i	$-0.318213\pi$
$457$	10562.0	1.08112	0.540560	+	0.841306i	$0.318213\pi$
$458$	−3990.52	−0.407128
$459$	−459.000	−0.0466760
$460$	0	0
$461$	11262.7	1.13787	0.568936	−	0.822382i	$-0.307355\pi$
$461$	11262.7	1.13787	0.568936	+	0.822382i	$0.307355\pi$
$462$	3004.94	0.302602
$463$	4669.94	0.468749	0.234374	−	0.972146i	$-0.424696\pi$
$463$	4669.94	0.468749	0.234374	+	0.972146i	$0.424696\pi$
$464$	−5359.15	−0.536190
$465$	0	0
$466$	−4694.61	−0.466681
$467$	−11972.4	−1.18633	−0.593166	−	0.805080i	$-0.702122\pi$
$467$	−11972.4	−1.18633	−0.593166	+	0.805080i	$0.702122\pi$
$468$	−4769.81	−0.471121
$469$	2356.94	0.232054
$470$	0	0
$471$	−7412.26	−0.725136
$472$	8553.08	0.834083
$473$	−950.926	−0.0924389
$474$	−3337.75	−0.323434
$475$	0	0
$476$	1387.27	0.133583
$477$	−2198.86	−0.211067
$478$	2941.82	0.281498
$479$	−9807.66	−0.935540	−0.467770	−	0.883850i	$-0.654942\pi$
$479$	−9807.66	−0.935540	−0.467770	+	0.883850i	$0.654942\pi$
$480$	0	0
$481$	−25940.0	−2.45896
$482$	−6832.69	−0.645685
$483$	−7657.92	−0.721423
$484$	−16507.4	−1.55028
$485$	0	0
$486$	306.106	0.0285705
$487$	−19150.5	−1.78192	−0.890959	−	0.454084i	$-0.849967\pi$
$487$	−19150.5	−1.78192	−0.890959	+	0.454084i	$0.849967\pi$
$488$	9791.65	0.908294
$489$	7267.07	0.672041
$490$	0	0
$491$	18563.3	1.70621	0.853107	−	0.521736i	$-0.174716\pi$
$491$	18563.3	1.70621	0.853107	+	0.521736i	$0.174716\pi$
$492$	961.358	0.0880923
$493$	−3204.10	−0.292708
$494$	−8352.98	−0.760766
$495$	0	0
$496$	−314.650	−0.0284843
$497$	−12891.0	−1.16346
$498$	−2874.58	−0.258661
$499$	−4574.89	−0.410421	−0.205211	−	0.978718i	$-0.565788\pi$
$499$	−4574.89	−0.410421	−0.205211	+	0.978718i	$0.565788\pi$
$500$	0	0
$501$	−2610.43	−0.232786
$502$	−1617.14	−0.143778
$503$	16704.8	1.48077	0.740387	−	0.672181i	$-0.234642\pi$
$503$	16704.8	1.48077	0.740387	+	0.672181i	$0.234642\pi$
$504$	−2079.25	−0.183764
$505$	0	0
$506$	−15791.6	−1.38739
$507$	−13896.7	−1.21730
$508$	2668.13	0.233030
$509$	12612.8	1.09833	0.549166	−	0.835713i	$-0.314945\pi$
$509$	12612.8	1.09833	0.549166	+	0.835713i	$0.314945\pi$
$510$	0	0
$511$	−2134.22	−0.184760
$512$	−9278.53	−0.800893
$513$	−2166.48	−0.186457
$514$	5142.42	0.441289
$515$	0	0
$516$	292.773	0.0249779
$517$	19463.5	1.65572
$518$	−5031.39	−0.426770
$519$	−2217.73	−0.187568
$520$	0	0
$521$	−14198.7	−1.19397	−0.596983	−	0.802254i	$-0.703634\pi$
$521$	−14198.7	−1.19397	−0.596983	+	0.802254i	$0.703634\pi$
$522$	2136.80	0.179167
$523$	6550.31	0.547658	0.273829	−	0.961778i	$-0.411710\pi$
$523$	6550.31	0.547658	0.273829	+	0.961778i	$0.411710\pi$
$524$	4288.94	0.357563
$525$	0	0
$526$	4346.47	0.360295
$527$	−188.121	−0.0155497
$528$	5330.53	0.439359
$529$	28077.0	2.30764
$530$	0	0
$531$	4239.75	0.346496
$532$	6547.90	0.533623
$533$	4129.31	0.335572
$534$	−4211.59	−0.341299
$535$	0	0
$536$	−3363.05	−0.271011
$537$	−11341.0	−0.911357
$538$	−339.447	−0.0272019
$539$	11316.2	0.904310
$540$	0	0
$541$	12735.0	1.01205	0.506027	−	0.862518i	$-0.331114\pi$
$541$	12735.0	1.01205	0.506027	+	0.862518i	$0.331114\pi$
$542$	3418.21	0.270894
$543$	−3722.67	−0.294208
$544$	−3078.15	−0.242601
$545$	0	0
$546$	−3973.85	−0.311474
$547$	6526.99	0.510190	0.255095	−	0.966916i	$-0.417893\pi$
$547$	6526.99	0.510190	0.255095	+	0.966916i	$0.417893\pi$
$548$	−11187.3	−0.872080
$549$	4853.71	0.377325
$550$	0	0
$551$	−15123.3	−1.16928
$552$	10926.9	0.842535
$553$	11238.4	0.864207
$554$	943.780	0.0723779
$555$	0	0
$556$	−18932.0	−1.44406
$557$	12318.8	0.937099	0.468550	−	0.883437i	$-0.344777\pi$
$557$	12318.8	0.937099	0.468550	+	0.883437i	$0.344777\pi$
$558$	125.458	0.00951799
$559$	1257.54	0.0951492
$560$	0	0
$561$	3186.98	0.239848
$562$	−4900.26	−0.367802
$563$	−23933.5	−1.79161	−0.895805	−	0.444448i	$-0.853400\pi$
$563$	−23933.5	−1.79161	−0.895805	+	0.444448i	$0.853400\pi$
$564$	−5992.48	−0.447391
$565$	0	0
$566$	2967.27	0.220360
$567$	−1030.68	−0.0763395
$568$	18393.8	1.35878
$569$	−7904.74	−0.582397	−0.291199	−	0.956663i	$-0.594054\pi$
$569$	−7904.74	−0.582397	−0.291199	+	0.956663i	$0.594054\pi$
$570$	0	0
$571$	15930.7	1.16756	0.583782	−	0.811911i	$-0.301572\pi$
$571$	15930.7	1.16756	0.583782	+	0.811911i	$0.301572\pi$
$572$	33118.3	2.42088
$573$	1029.52	0.0750588
$574$	800.932	0.0582409
$575$	0	0
$576$	5.55849	0.000402090 0
$577$	−12078.8	−0.871483	−0.435742	−	0.900072i	$-0.643514\pi$
$577$	−12078.8	−0.871483	−0.435742	+	0.900072i	$0.643514\pi$
$578$	−364.052	−0.0261982
$579$	758.642	0.0544527
$580$	0	0
$581$	9678.91	0.691134
$582$	81.6606	0.00581605
$583$	15267.4	1.08458
$584$	3045.26	0.215777
$585$	0	0
$586$	1613.35	0.113732
$587$	6424.82	0.451756	0.225878	−	0.974156i	$-0.427475\pi$
$587$	6424.82	0.451756	0.225878	+	0.974156i	$0.427475\pi$
$588$	−3484.05	−0.244354
$589$	−887.931	−0.0621164
$590$	0	0
$591$	−6110.86	−0.425325
$592$	−8925.30	−0.619641
$593$	24448.4	1.69305	0.846523	−	0.532352i	$-0.178692\pi$
$593$	24448.4	1.69305	0.846523	+	0.532352i	$0.178692\pi$
$594$	−2125.39	−0.146811
$595$	0	0
$596$	−12658.3	−0.869976
$597$	−988.216	−0.0677470
$598$	20883.4	1.42807
$599$	5099.60	0.347853	0.173926	−	0.984759i	$-0.444354\pi$
$599$	5099.60	0.347853	0.173926	+	0.984759i	$0.444354\pi$
$600$	0	0
$601$	−25127.3	−1.70543	−0.852715	−	0.522377i	$-0.825045\pi$
$601$	−25127.3	−1.70543	−0.852715	+	0.522377i	$0.825045\pi$
$602$	243.917	0.0165138
$603$	−1667.06	−0.112584
$604$	−12565.7	−0.846507
$605$	0	0
$606$	−1392.49	−0.0933432
$607$	12725.4	0.850921	0.425460	−	0.904977i	$-0.360112\pi$
$607$	12725.4	0.850921	0.425460	+	0.904977i	$0.360112\pi$
$608$	−14528.9	−0.969118
$609$	−7194.77	−0.478730
$610$	0	0
$611$	−25739.4	−1.70426
$612$	−981.215	−0.0648092
$613$	4997.30	0.329264	0.164632	−	0.986355i	$-0.447356\pi$
$613$	4997.30	0.329264	0.164632	+	0.986355i	$0.447356\pi$
$614$	6844.99	0.449905
$615$	0	0
$616$	14436.9	0.944283
$617$	23828.5	1.55478	0.777390	−	0.629019i	$-0.216543\pi$
$617$	23828.5	1.55478	0.777390	+	0.629019i	$0.216543\pi$
$618$	1025.68	0.0667622
$619$	3551.58	0.230614	0.115307	−	0.993330i	$-0.463215\pi$
$619$	3551.58	0.230614	0.115307	+	0.993330i	$0.463215\pi$
$620$	0	0
$621$	5416.45	0.350007
$622$	−8169.26	−0.526620
$623$	14180.7	0.911941
$624$	−7049.30	−0.452240
$625$	0	0
$626$	−6100.35	−0.389487
$627$	15042.5	0.958120
$628$	−15845.4	−1.00685
$629$	−5336.21	−0.338265
$630$	0	0
$631$	29200.1	1.84222	0.921109	−	0.389305i	$-0.127285\pi$
$631$	29200.1	1.84222	0.921109	+	0.389305i	$0.127285\pi$
$632$	−16035.8	−1.00929
$633$	−699.817	−0.0439419
$634$	−520.688	−0.0326170
$635$	0	0
$636$	−4700.56	−0.293065
$637$	−14965.0	−0.930824
$638$	−14836.5	−0.920663
$639$	9117.77	0.564465
$640$	0	0
$641$	−26828.1	−1.65311	−0.826557	−	0.562853i	$-0.809704\pi$
$641$	−26828.1	−1.65311	−0.826557	+	0.562853i	$0.809704\pi$
$642$	−5615.78	−0.345229
$643$	−9467.48	−0.580655	−0.290327	−	0.956927i	$-0.593764\pi$
$643$	−9467.48	−0.580655	−0.290327	+	0.956927i	$0.593764\pi$
$644$	−16370.5	−1.00169
$645$	0	0
$646$	−1718.32	−0.104654
$647$	−14497.3	−0.880906	−0.440453	−	0.897776i	$-0.645182\pi$
$647$	−14497.3	−0.880906	−0.440453	+	0.897776i	$0.645182\pi$
$648$	1470.65	0.0891553
$649$	−29437.9	−1.78049
$650$	0	0
$651$	−422.424	−0.0254318
$652$	15535.0	0.933124
$653$	−11678.8	−0.699889	−0.349944	−	0.936770i	$-0.613800\pi$
$653$	−11678.8	−0.699889	−0.349944	+	0.936770i	$0.613800\pi$
$654$	−5782.51	−0.345740
$655$	0	0
$656$	1420.79	0.0845619
$657$	1509.53	0.0896384
$658$	−4992.48	−0.295786
$659$	5738.86	0.339233	0.169616	−	0.985510i	$-0.445747\pi$
$659$	5738.86	0.339233	0.169616	+	0.985510i	$0.445747\pi$
$660$	0	0
$661$	7047.04	0.414672	0.207336	−	0.978270i	$-0.433521\pi$
$661$	7047.04	0.414672	0.207336	+	0.978270i	$0.433521\pi$
$662$	10974.9	0.644340
$663$	−4214.60	−0.246880
$664$	−13810.6	−0.807162
$665$	0	0
$666$	3558.70	0.207052
$667$	37810.0	2.19492
$668$	−5580.38	−0.323221
$669$	933.411	0.0539428
$670$	0	0
$671$	−33700.8	−1.93891
$672$	−6911.96	−0.396778
$673$	3737.64	0.214080	0.107040	−	0.994255i	$-0.465863\pi$
$673$	3737.64	0.214080	0.107040	+	0.994255i	$0.465863\pi$
$674$	−11873.1	−0.678538
$675$	0	0
$676$	−29707.2	−1.69022
$677$	−21346.3	−1.21182	−0.605911	−	0.795532i	$-0.707191\pi$
$677$	−21346.3	−1.21182	−0.605911	+	0.795532i	$0.707191\pi$
$678$	−83.8172	−0.00474776
$679$	−274.957	−0.0155403
$680$	0	0
$681$	17564.8	0.988376
$682$	−871.092	−0.0489088
$683$	−15238.4	−0.853707	−0.426854	−	0.904321i	$-0.640378\pi$
$683$	−15238.4	−0.853707	−0.426854	+	0.904321i	$0.640378\pi$
$684$	−4631.33	−0.258894
$685$	0	0
$686$	−8400.57	−0.467544
$687$	−9503.54	−0.527777
$688$	432.689	0.0239769
$689$	−20190.2	−1.11638
$690$	0	0
$691$	−9114.57	−0.501787	−0.250893	−	0.968015i	$-0.580724\pi$
$691$	−9114.57	−0.501787	−0.250893	+	0.968015i	$0.580724\pi$
$692$	−4740.89	−0.260436
$693$	7156.34	0.392275
$694$	−7728.40	−0.422718
$695$	0	0
$696$	10266.0	0.559099
$697$	849.455	0.0461627
$698$	−9160.28	−0.496736
$699$	−11180.3	−0.604978
$700$	0	0
$701$	−7625.96	−0.410882	−0.205441	−	0.978669i	$-0.565863\pi$
$701$	−7625.96	−0.410882	−0.205441	+	0.978669i	$0.565863\pi$
$702$	2810.70	0.151116
$703$	−25186.9	−1.35127
$704$	−38.5944	−0.00206617
$705$	0	0
$706$	−545.662	−0.0290882
$707$	4688.61	0.249410
$708$	9063.41	0.481107
$709$	−11352.5	−0.601343	−0.300672	−	0.953728i	$-0.597211\pi$
$709$	−11352.5	−0.601343	−0.300672	+	0.953728i	$0.597211\pi$
$710$	0	0
$711$	−7948.94	−0.419281
$712$	−20234.1	−1.06504
$713$	2219.93	0.116602
$714$	−817.475	−0.0428477
$715$	0	0
$716$	−24243.8	−1.26541
$717$	7006.03	0.364917
$718$	−2599.29	−0.135104
$719$	10470.9	0.543111	0.271556	−	0.962423i	$-0.412462\pi$
$719$	10470.9	0.543111	0.271556	+	0.962423i	$0.412462\pi$
$720$	0	0
$721$	−3453.55	−0.178387
$722$	529.770	0.0273075
$723$	−16272.2	−0.837028
$724$	−7958.03	−0.408505
$725$	0	0
$726$	9727.30	0.497265
$727$	9894.05	0.504746	0.252373	−	0.967630i	$-0.418789\pi$
$727$	9894.05	0.504746	0.252373	+	0.967630i	$0.418789\pi$
$728$	−19091.9	−0.971969
$729$	729.000	0.0370370
$730$	0	0
$731$	258.694	0.0130891
$732$	10375.9	0.523912
$733$	20969.4	1.05665	0.528323	−	0.849044i	$-0.322821\pi$
$733$	20969.4	1.05665	0.528323	+	0.849044i	$0.322821\pi$
$734$	−6655.71	−0.334696
$735$	0	0
$736$	36323.9	1.81918
$737$	11574.9	0.578518
$738$	−566.499	−0.0282563
$739$	34627.8	1.72369	0.861843	−	0.507175i	$-0.169310\pi$
$739$	34627.8	1.72369	0.861843	+	0.507175i	$0.169310\pi$
$740$	0	0
$741$	−19892.9	−0.986212
$742$	−3916.16	−0.193756
$743$	−29823.9	−1.47259	−0.736295	−	0.676661i	$-0.763426\pi$
$743$	−29823.9	−1.47259	−0.736295	+	0.676661i	$0.763426\pi$
$744$	602.747	0.0297013
$745$	0	0
$746$	−11640.1	−0.571280
$747$	−6845.89	−0.335312
$748$	6812.89	0.333026
$749$	18908.7	0.922444
$750$	0	0
$751$	28027.3	1.36183	0.680913	−	0.732365i	$-0.261583\pi$
$751$	28027.3	1.36183	0.680913	+	0.732365i	$0.261583\pi$
$752$	−8856.28	−0.429462
$753$	−3851.26	−0.186385
$754$	19620.4	0.947656
$755$	0	0
$756$	−2203.31	−0.105997
$757$	−19086.0	−0.916369	−0.458185	−	0.888857i	$-0.651500\pi$
$757$	−19086.0	−0.916369	−0.458185	+	0.888857i	$0.651500\pi$
$758$	17822.4	0.854009
$759$	−37608.1	−1.79853
$760$	0	0
$761$	41604.6	1.98182	0.990910	−	0.134524i	$-0.0429504\pi$
$761$	41604.6	1.98182	0.990910	+	0.134524i	$0.0429504\pi$
$762$	−1572.25	−0.0747462
$763$	19470.1	0.923808
$764$	2200.82	0.104218
$765$	0	0
$766$	−69.9182	−0.00329797
$767$	38929.9	1.83270
$768$	5486.05	0.257761
$769$	10647.1	0.499278	0.249639	−	0.968339i	$-0.419688\pi$
$769$	10647.1	0.499278	0.249639	+	0.968339i	$0.419688\pi$
$770$	0	0
$771$	12246.8	0.572060
$772$	1621.77	0.0756071
$773$	−33674.9	−1.56688	−0.783442	−	0.621465i	$-0.786538\pi$
$773$	−33674.9	−1.56688	−0.783442	+	0.621465i	$0.786538\pi$
$774$	−172.522	−0.00801186
$775$	0	0
$776$	392.329	0.0181492
$777$	−11982.4	−0.553238
$778$	6324.81	0.291459
$779$	4009.42	0.184406
$780$	0	0
$781$	−63307.6	−2.90054
$782$	4296.01	0.196451
$783$	5088.86	0.232262
$784$	−5149.08	−0.234561
$785$	0	0
$786$	−2527.34	−0.114691
$787$	9870.65	0.447079	0.223539	−	0.974695i	$-0.428239\pi$
$787$	9870.65	0.447079	0.223539	+	0.974695i	$0.428239\pi$
$788$	−13063.3	−0.590561
$789$	10351.2	0.467064
$790$	0	0
$791$	282.218	0.0126859
$792$	−10211.2	−0.458131
$793$	44567.4	1.99575
$794$	−9981.95	−0.446154
$795$	0	0
$796$	−2112.53	−0.0940662
$797$	−33355.5	−1.48245	−0.741225	−	0.671256i	$-0.765755\pi$
$797$	−33355.5	−1.48245	−0.741225	+	0.671256i	$0.765755\pi$
$798$	−3858.48	−0.171164
$799$	−5294.94	−0.234445
$800$	0	0
$801$	−10030.0	−0.442439
$802$	5986.60	0.263584
$803$	−10481.2	−0.460613
$804$	−3563.71	−0.156321
$805$	0	0
$806$	1151.97	0.0503428
$807$	−808.403	−0.0352629
$808$	−6690.05	−0.291281
$809$	11930.1	0.518467	0.259233	−	0.965815i	$-0.416530\pi$
$809$	11930.1	0.518467	0.259233	+	0.965815i	$0.416530\pi$
$810$	0	0
$811$	40278.5	1.74398	0.871991	−	0.489522i	$-0.162829\pi$
$811$	40278.5	1.74398	0.871991	+	0.489522i	$0.162829\pi$
$812$	−15380.4	−0.664713
$813$	8140.56	0.351171
$814$	−24709.2	−1.06395
$815$	0	0
$816$	−1450.14	−0.0622120
$817$	1221.03	0.0522871
$818$	11450.0	0.489412
$819$	−9463.83	−0.403777
$820$	0	0
$821$	14099.1	0.599345	0.299672	−	0.954042i	$-0.403123\pi$
$821$	14099.1	0.599345	0.299672	+	0.954042i	$0.403123\pi$
$822$	6592.36	0.279726
$823$	−42568.6	−1.80297	−0.901487	−	0.432805i	$-0.857524\pi$
$823$	−42568.6	−1.80297	−0.901487	+	0.432805i	$0.857524\pi$
$824$	4927.78	0.208334
$825$	0	0
$826$	7550.95	0.318077
$827$	−333.622	−0.0140280	−0.00701401	−	0.999975i	$-0.502233\pi$
$827$	−333.622	−0.0140280	−0.00701401	+	0.999975i	$0.502233\pi$
$828$	11578.9	0.485982
$829$	−39151.8	−1.64029	−0.820143	−	0.572159i	$-0.806106\pi$
$829$	−39151.8	−1.64029	−0.820143	+	0.572159i	$0.806106\pi$
$830$	0	0
$831$	2247.64	0.0938264
$832$	51.0388	0.00212674
$833$	−3078.50	−0.128048
$834$	11156.0	0.463192
$835$	0	0
$836$	32156.8	1.33034
$837$	298.781	0.0123386
$838$	−2352.14	−0.0969612
$839$	32928.1	1.35495	0.677477	−	0.735544i	$-0.263074\pi$
$839$	32928.1	1.35495	0.677477	+	0.735544i	$0.263074\pi$
$840$	0	0
$841$	11134.3	0.456529
$842$	9850.82	0.403185
$843$	−11670.1	−0.476797
$844$	−1496.01	−0.0610130
$845$	0	0
$846$	3531.18	0.143504
$847$	−32752.5	−1.32868
$848$	−6946.96	−0.281320
$849$	7066.64	0.285661
$850$	0	0
$851$	62970.1	2.53653
$852$	19491.3	0.783756
$853$	40373.8	1.62060	0.810300	−	0.586015i	$-0.199304\pi$
$853$	40373.8	1.62060	0.810300	+	0.586015i	$0.199304\pi$
$854$	8644.41	0.346377
$855$	0	0
$856$	−26980.4	−1.07730
$857$	18125.7	0.722477	0.361238	−	0.932473i	$-0.382354\pi$
$857$	18125.7	0.722477	0.361238	+	0.932473i	$0.382354\pi$
$858$	−19515.6	−0.776517
$859$	22072.2	0.876710	0.438355	−	0.898802i	$-0.355561\pi$
$859$	22072.2	0.876710	0.438355	+	0.898802i	$0.355561\pi$
$860$	0	0
$861$	1907.44	0.0755000
$862$	−9308.30	−0.367798
$863$	7038.85	0.277642	0.138821	−	0.990317i	$-0.455669\pi$
$863$	7038.85	0.277642	0.138821	+	0.990317i	$0.455669\pi$
$864$	4888.83	0.192502
$865$	0	0
$866$	7850.03	0.308031
$867$	−867.000	−0.0339618
$868$	−903.026	−0.0353119
$869$	55192.0	2.15450
$870$	0	0
$871$	−15307.2	−0.595480
$872$	−27781.4	−1.07890
$873$	194.477	0.00753957
$874$	20277.1	0.784765
$875$	0	0
$876$	3226.96	0.124462
$877$	−23593.1	−0.908420	−0.454210	−	0.890895i	$-0.650078\pi$
$877$	−23593.1	−0.908420	−0.454210	+	0.890895i	$0.650078\pi$
$878$	12719.0	0.488890
$879$	3842.23	0.147435
$880$	0	0
$881$	20302.0	0.776380	0.388190	−	0.921579i	$-0.373100\pi$
$881$	20302.0	0.776380	0.388190	+	0.921579i	$0.373100\pi$
$882$	2053.05	0.0783783
$883$	17644.1	0.672448	0.336224	−	0.941782i	$-0.390850\pi$
$883$	17644.1	0.672448	0.336224	+	0.941782i	$0.390850\pi$
$884$	−9009.64	−0.342791
$885$	0	0
$886$	6456.12	0.244805
$887$	21960.5	0.831298	0.415649	−	0.909525i	$-0.363555\pi$
$887$	21960.5	0.831298	0.415649	+	0.909525i	$0.363555\pi$
$888$	17097.4	0.646116
$889$	5293.88	0.199720
$890$	0	0
$891$	−5061.68	−0.190317
$892$	1995.37	0.0748991
$893$	−24992.1	−0.936539
$894$	7459.17	0.279051
$895$	0	0
$896$	−18422.0	−0.686870
$897$	49734.5	1.85127
$898$	−19049.1	−0.707880
$899$	2085.67	0.0773759
$900$	0	0
$901$	−4153.41	−0.153574
$902$	3933.39	0.145197
$903$	580.894	0.0214075
$904$	−402.690	−0.0148156
$905$	0	0
$906$	7404.57	0.271523
$907$	32773.5	1.19981	0.599904	−	0.800072i	$-0.295205\pi$
$907$	32773.5	1.19981	0.599904	+	0.800072i	$0.295205\pi$
$908$	37548.7	1.37235
$909$	−3316.25	−0.121004
$910$	0	0
$911$	17733.6	0.644942	0.322471	−	0.946579i	$-0.395487\pi$
$911$	17733.6	0.644942	0.322471	+	0.946579i	$0.395487\pi$
$912$	−6844.65	−0.248518
$913$	47533.2	1.72302
$914$	−13305.0	−0.481497
$915$	0	0
$916$	−20315.9	−0.732814
$917$	8509.73	0.306452
$918$	578.200	0.0207881
$919$	−6048.55	−0.217109	−0.108555	−	0.994090i	$-0.534622\pi$
$919$	−6048.55	−0.217109	−0.108555	+	0.994090i	$0.534622\pi$
$920$	0	0
$921$	16301.6	0.583230
$922$	−14187.6	−0.506773
$923$	83720.5	2.98558
$924$	15298.3	0.544671
$925$	0	0
$926$	−5882.71	−0.208766
$927$	2442.69	0.0865465
$928$	34127.0	1.20719
$929$	−31468.8	−1.11137	−0.555683	−	0.831394i	$-0.687543\pi$
$929$	−31468.8	−1.11137	−0.555683	+	0.831394i	$0.687543\pi$
$930$	0	0
$931$	−14530.5	−0.511513
$932$	−23900.5	−0.840007
$933$	−19455.3	−0.682679
$934$	15081.6	0.528356
$935$	0	0
$936$	13503.7	0.471563
$937$	29463.7	1.02725	0.513627	−	0.858014i	$-0.328302\pi$
$937$	29463.7	1.02725	0.513627	+	0.858014i	$0.328302\pi$
$938$	−2969.02	−0.103350
$939$	−14528.2	−0.504908
$940$	0	0
$941$	4529.95	0.156931	0.0784655	−	0.996917i	$-0.474998\pi$
$941$	4529.95	0.156931	0.0784655	+	0.996917i	$0.474998\pi$
$942$	9337.19	0.322953
$943$	−10024.0	−0.346158
$944$	13394.8	0.461826
$945$	0	0
$946$	1197.88	0.0411695
$947$	−38985.6	−1.33776	−0.668882	−	0.743369i	$-0.733227\pi$
$947$	−38985.6	−1.33776	−0.668882	+	0.743369i	$0.733227\pi$
$948$	−16992.6	−0.582168
$949$	13860.7	0.474117
$950$	0	0
$951$	−1240.03	−0.0422827
$952$	−3927.47	−0.133708
$953$	−18133.6	−0.616376	−0.308188	−	0.951325i	$-0.599723\pi$
$953$	−18133.6	−0.616376	−0.308188	+	0.951325i	$0.599723\pi$
$954$	2769.90	0.0940029
$955$	0	0
$956$	14977.0	0.506684
$957$	−35333.6	−1.19349
$958$	12354.7	0.416661
$959$	−22196.9	−0.747421
$960$	0	0
$961$	−29668.5	−0.995890
$962$	32676.4	1.09515
$963$	−13374.2	−0.447535
$964$	−34785.5	−1.16221
$965$	0	0
$966$	9646.64	0.321300
$967$	18377.4	0.611145	0.305572	−	0.952169i	$-0.401152\pi$
$967$	18377.4	0.611145	0.305572	+	0.952169i	$0.401152\pi$
$968$	46733.8	1.55174
$969$	−4092.24	−0.135667
$970$	0	0
$971$	−25113.2	−0.829992	−0.414996	−	0.909823i	$-0.636217\pi$
$971$	−25113.2	−0.829992	−0.414996	+	0.909823i	$0.636217\pi$
$972$	1558.40	0.0514256
$973$	−37563.2	−1.23764
$974$	24123.8	0.793612
$975$	0	0
$976$	15334.5	0.502916
$977$	2528.49	0.0827978	0.0413989	−	0.999143i	$-0.486819\pi$
$977$	2528.49	0.0827978	0.0413989	+	0.999143i	$0.486819\pi$
$978$	−9154.29	−0.299307
$979$	69641.7	2.27350
$980$	0	0
$981$	−13771.2	−0.448197
$982$	−23384.1	−0.759895
$983$	28913.4	0.938143	0.469072	−	0.883160i	$-0.344589\pi$
$983$	28913.4	0.938143	0.469072	+	0.883160i	$0.344589\pi$
$984$	−2721.68	−0.0881749
$985$	0	0
$986$	4036.18	0.130363
$987$	−11889.7	−0.383439
$988$	−42525.4	−1.36935
$989$	−3052.73	−0.0981507
$990$	0	0
$991$	13310.7	0.426670	0.213335	−	0.976979i	$-0.431567\pi$
$991$	13310.7	0.426670	0.213335	+	0.976979i	$0.431567\pi$
$992$	2003.69	0.0641302
$993$	26137.1	0.835284
$994$	16238.7	0.518168
$995$	0	0
$996$	−14634.6	−0.465578
$997$	32942.2	1.04643	0.523215	−	0.852201i	$-0.324732\pi$
$997$	32942.2	1.04643	0.523215	+	0.852201i	$0.324732\pi$
$998$	5762.97	0.182789
$999$	8475.15	0.268410

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1275.4.a.bf.1.6		14
5.2	odd	4		255.4.b.b.154.12	✓	28
5.3	odd	4		255.4.b.b.154.17	yes	28
5.4	even	2		1275.4.a.bg.1.9		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
255.4.b.b.154.12	✓	28	5.2	odd	4
255.4.b.b.154.17	yes	28	5.3	odd	4
1275.4.a.bf.1.6		14	1.1	even	1	trivial
1275.4.a.bg.1.9		14	5.4	even	2

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:	\( N \)	\(=\)	\( 1275 = 3 \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1275.a (trivial)

\(f(q)\)	\(=\)	\(q-1.25969 q^{2} -3.00000 q^{3} -6.41317 q^{4} +3.77908 q^{6} -12.7244 q^{7} +18.1562 q^{8} +9.00000 q^{9} -62.4899 q^{11} +19.2395 q^{12} +82.6391 q^{13} +16.0289 q^{14} +28.4341 q^{16} +17.0000 q^{17} -11.3373 q^{18} +80.2399 q^{19} +38.1733 q^{21} +78.7182 q^{22} -200.609 q^{23} -54.4686 q^{24} -104.100 q^{26} -27.0000 q^{27} +81.6040 q^{28} -188.476 q^{29} -11.0660 q^{31} -181.068 q^{32} +187.470 q^{33} -21.4148 q^{34} -57.7185 q^{36} -313.895 q^{37} -101.078 q^{38} -247.917 q^{39} +49.9679 q^{41} -48.0868 q^{42} +15.2173 q^{43} +400.758 q^{44} +252.706 q^{46} -311.467 q^{47} -85.3022 q^{48} -181.088 q^{49} -51.0000 q^{51} -529.979 q^{52} -244.318 q^{53} +34.0118 q^{54} -231.028 q^{56} -240.720 q^{57} +237.423 q^{58} +471.083 q^{59} +539.301 q^{61} +13.9397 q^{62} -114.520 q^{63} +0.617610 q^{64} -236.155 q^{66} -185.229 q^{67} -109.024 q^{68} +601.827 q^{69} +1013.09 q^{71} +163.406 q^{72} +167.726 q^{73} +395.411 q^{74} -514.592 q^{76} +795.149 q^{77} +312.300 q^{78} -883.216 q^{79} +81.0000 q^{81} -62.9444 q^{82} -760.655 q^{83} -244.812 q^{84} -19.1691 q^{86} +565.429 q^{87} -1134.58 q^{88} -1114.45 q^{89} -1051.54 q^{91} +1286.54 q^{92} +33.1979 q^{93} +392.354 q^{94} +543.204 q^{96} +21.6086 q^{97} +228.116 q^{98} -562.409 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 2 q^{2} - 42 q^{3} + 68 q^{4} + 6 q^{6} - 8 q^{7} - 51 q^{8} + 126 q^{9} + 58 q^{11} - 204 q^{12} + 72 q^{13} + 257 q^{14} + 404 q^{16} + 238 q^{17} - 18 q^{18} + 282 q^{19} + 24 q^{21} - 361 q^{22}+ \cdots + 522 q^{99}+O(q^{100}) \) 14 * q - 2 * q^2 - 42 * q^3 + 68 * q^4 + 6 * q^6 - 8 * q^7 - 51 * q^8 + 126 * q^9 + 58 * q^11 - 204 * q^12 + 72 * q^13 + 257 * q^14 + 404 * q^16 + 238 * q^17 - 18 * q^18 + 282 * q^19 + 24 * q^21 - 361 * q^22 - 344 * q^23 + 153 * q^24 + 174 * q^26 - 378 * q^27 + q^28 + 434 * q^29 + 226 * q^31 - 870 * q^32 - 174 * q^33 - 34 * q^34 + 612 * q^36 - 166 * q^37 - 637 * q^38 - 216 * q^39 + 618 * q^41 - 771 * q^42 + 590 * q^43 + 1443 * q^44 + 130 * q^46 - 944 * q^47 - 1212 * q^48 + 1718 * q^49 - 714 * q^51 + 246 * q^52 - 628 * q^53 + 54 * q^54 + 1902 * q^56 - 846 * q^57 - 89 * q^58 + 564 * q^59 + 1680 * q^61 - 1556 * q^62 - 72 * q^63 + 2855 * q^64 + 1083 * q^66 + 658 * q^67 + 1156 * q^68 + 1032 * q^69 + 2582 * q^71 - 459 * q^72 + 1114 * q^73 + 61 * q^74 + 5993 * q^76 - 780 * q^77 - 522 * q^78 + 902 * q^79 + 1134 * q^81 - 849 * q^82 - 2082 * q^83 - 3 * q^84 + 1608 * q^86 - 1302 * q^87 - 2324 * q^88 + 992 * q^89 + 2416 * q^91 - 4410 * q^92 - 678 * q^93 - 541 * q^94 + 2610 * q^96 + 980 * q^97 + 1237 * q^98 + 522 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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