LMFDB - Embedded newform 2275.2.a.y.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2275,2,Mod(1,2275)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2275.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2275, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2275 = 5^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2275.a (trivial)

Newform invariants

comment:select newform

sage:traces = [7,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$18.1659664598$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 8x^{5} - 2x^{4} + 16x^{3} + 5x^{2} - 6x - 2 $ x^7 - 8x^5 - 2x^4 + 16x^3 + 5x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 455)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.687115$ of defining polynomial
Character	$\chi$	$=$	2275.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.687115 q^{2} -1.04121 q^{3} -1.52787 q^{4} +0.715431 q^{6} -1.00000 q^{7} +2.42406 q^{8} -1.91588 q^{9} -1.53649 q^{11} +1.59083 q^{12} +1.00000 q^{13} +0.687115 q^{14} +1.39014 q^{16} -0.369953 q^{17} +1.31643 q^{18} +7.34406 q^{19} +1.04121 q^{21} +1.05574 q^{22} -1.57820 q^{23} -2.52395 q^{24} -0.687115 q^{26} +5.11846 q^{27} +1.52787 q^{28} -4.79474 q^{29} -4.60218 q^{31} -5.80330 q^{32} +1.59981 q^{33} +0.254200 q^{34} +2.92723 q^{36} +11.4924 q^{37} -5.04622 q^{38} -1.04121 q^{39} -1.14385 q^{41} -0.715431 q^{42} +9.28949 q^{43} +2.34756 q^{44} +1.08440 q^{46} +0.940075 q^{47} -1.44742 q^{48} +1.00000 q^{49} +0.385198 q^{51} -1.52787 q^{52} +10.4728 q^{53} -3.51698 q^{54} -2.42406 q^{56} -7.64671 q^{57} +3.29454 q^{58} -10.4249 q^{59} -9.72998 q^{61} +3.16223 q^{62} +1.91588 q^{63} +1.20726 q^{64} -1.09925 q^{66} +6.10516 q^{67} +0.565241 q^{68} +1.64323 q^{69} -14.5388 q^{71} -4.64421 q^{72} -15.3532 q^{73} -7.89661 q^{74} -11.2208 q^{76} +1.53649 q^{77} +0.715431 q^{78} -3.24752 q^{79} +0.418259 q^{81} +0.785958 q^{82} +5.16103 q^{83} -1.59083 q^{84} -6.38295 q^{86} +4.99233 q^{87} -3.72453 q^{88} +12.5955 q^{89} -1.00000 q^{91} +2.41129 q^{92} +4.79183 q^{93} -0.645940 q^{94} +6.04245 q^{96} -8.37280 q^{97} -0.687115 q^{98} +2.94373 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 2 q^{3} + 2 q^{4} - 8 q^{6} - 7 q^{7} + 6 q^{8} + q^{9} - 6 q^{11} - 3 q^{12} + 7 q^{13} - 4 q^{16} - 2 q^{17} + q^{18} - 10 q^{19} - 2 q^{21} - 18 q^{22} - 10 q^{23} - 5 q^{24} + 8 q^{27} - 2 q^{28}+ \cdots - 18 q^{99}+O(q^{100}) $ 7 * q + 2 * q^3 + 2 * q^4 - 8 * q^6 - 7 * q^7 + 6 * q^8 + q^9 - 6 * q^11 - 3 * q^12 + 7 * q^13 - 4 * q^16 - 2 * q^17 + q^18 - 10 * q^19 - 2 * q^21 - 18 * q^22 - 10 * q^23 - 5 * q^24 + 8 * q^27 - 2 * q^28 - 26 * q^29 - 6 * q^31 + 7 * q^32 - 4 * q^33 - q^34 - 6 * q^36 + 4 * q^37 + q^38 + 2 * q^39 - 28 * q^41 + 8 * q^42 - 2 * q^43 - 18 * q^46 + 20 * q^47 - 20 * q^48 + 7 * q^49 - 32 * q^51 + 2 * q^52 - 8 * q^53 + 11 * q^54 - 6 * q^56 - 14 * q^57 + 12 * q^58 - 28 * q^59 - 26 * q^61 + 22 * q^62 - q^63 - 8 * q^64 + 2 * q^66 - 2 * q^67 + 2 * q^68 - 22 * q^69 - 14 * q^71 - 4 * q^72 + 2 * q^73 - 12 * q^74 - 36 * q^76 + 6 * q^77 - 8 * q^78 + 14 * q^79 - 33 * q^81 - 19 * q^82 + 28 * q^83 + 3 * q^84 - 4 * q^86 - 22 * q^87 - 24 * q^88 - 30 * q^89 - 7 * q^91 - 42 * q^92 - 2 * q^93 + 14 * q^94 + 18 * q^97 - 18 * 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 $

$2$	−0.687115	−0.485864	−0.242932	−	0.970043i	$-0.578109\pi$
$2$	−0.687115	−0.485864	−0.242932	+	0.970043i	$0.578109\pi$
$3$	−1.04121	−0.601142	−0.300571	−	0.953759i	$-0.597177\pi$
$3$	−1.04121	−0.601142	−0.300571	+	0.953759i	$0.597177\pi$
$4$	−1.52787	−0.763936
$5$	0	0
$5$	0	0
$6$	0.715431	0.292073
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	2.42406	0.857033
$9$	−1.91588	−0.638628
$10$	0	0
$11$	−1.53649	−0.463269	−0.231634	−	0.972803i	$-0.574407\pi$
$11$	−1.53649	−0.463269	−0.231634	+	0.972803i	$0.574407\pi$
$12$	1.59083	0.459234
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0.687115	0.183639
$15$	0	0
$16$	1.39014	0.347535
$17$	−0.369953	−0.0897267	−0.0448634	−	0.998993i	$-0.514285\pi$
$17$	−0.369953	−0.0897267	−0.0448634	+	0.998993i	$0.514285\pi$
$18$	1.31643	0.310286
$19$	7.34406	1.68484	0.842422	−	0.538819i	$-0.181129\pi$
$19$	7.34406	1.68484	0.842422	+	0.538819i	$0.181129\pi$
$20$	0	0
$21$	1.04121	0.227210
$22$	1.05574	0.225086
$23$	−1.57820	−0.329077	−0.164539	−	0.986371i	$-0.552613\pi$
$23$	−1.57820	−0.329077	−0.164539	+	0.986371i	$0.552613\pi$
$24$	−2.52395	−0.515199
$25$	0	0
$26$	−0.687115	−0.134754
$27$	5.11846	0.985049
$28$	1.52787	0.288741
$29$	−4.79474	−0.890361	−0.445180	−	0.895441i	$-0.646860\pi$
$29$	−4.79474	−0.890361	−0.445180	+	0.895441i	$0.646860\pi$
$30$	0	0
$31$	−4.60218	−0.826575	−0.413288	−	0.910601i	$-0.635620\pi$
$31$	−4.60218	−0.826575	−0.413288	+	0.910601i	$0.635620\pi$
$32$	−5.80330	−1.02589
$33$	1.59981	0.278490
$34$	0.254200	0.0435950
$35$	0	0
$36$	2.92723	0.487871
$37$	11.4924	1.88934	0.944670	−	0.328022i	$-0.106382\pi$
$37$	11.4924	1.88934	0.944670	+	0.328022i	$0.106382\pi$
$38$	−5.04622	−0.818605
$39$	−1.04121	−0.166727
$40$	0	0
$41$	−1.14385	−0.178640	−0.0893198	−	0.996003i	$-0.528469\pi$
$41$	−1.14385	−0.178640	−0.0893198	+	0.996003i	$0.528469\pi$
$42$	−0.715431	−0.110393
$43$	9.28949	1.41663	0.708317	−	0.705894i	$-0.249455\pi$
$43$	9.28949	1.41663	0.708317	+	0.705894i	$0.249455\pi$
$44$	2.34756	0.353908
$45$	0	0
$46$	1.08440	0.159887
$47$	0.940075	0.137124	0.0685620	−	0.997647i	$-0.478159\pi$
$47$	0.940075	0.137124	0.0685620	+	0.997647i	$0.478159\pi$
$48$	−1.44742	−0.208918
$49$	1.00000	0.142857
$50$	0	0
$51$	0.385198	0.0539385
$52$	−1.52787	−0.211878
$53$	10.4728	1.43855	0.719275	−	0.694725i	$-0.244474\pi$
$53$	10.4728	1.43855	0.719275	+	0.694725i	$0.244474\pi$
$54$	−3.51698	−0.478600
$55$	0	0
$56$	−2.42406	−0.323928
$57$	−7.64671	−1.01283
$58$	3.29454	0.432594
$59$	−10.4249	−1.35721	−0.678605	−	0.734504i	$-0.737415\pi$
$59$	−10.4249	−1.35721	−0.678605	+	0.734504i	$0.737415\pi$
$60$	0	0
$61$	−9.72998	−1.24580	−0.622898	−	0.782303i	$-0.714045\pi$
$61$	−9.72998	−1.24580	−0.622898	+	0.782303i	$0.714045\pi$
$62$	3.16223	0.401603
$63$	1.91588	0.241379
$64$	1.20726	0.150907
$65$	0	0
$66$	−1.09925	−0.135308
$67$	6.10516	0.745865	0.372932	−	0.927859i	$-0.378352\pi$
$67$	6.10516	0.745865	0.372932	+	0.927859i	$0.378352\pi$
$68$	0.565241	0.0685455
$69$	1.64323	0.197822
$70$	0	0
$71$	−14.5388	−1.72544	−0.862722	−	0.505679i	$-0.831242\pi$
$71$	−14.5388	−1.72544	−0.862722	+	0.505679i	$0.831242\pi$
$72$	−4.64421	−0.547325
$73$	−15.3532	−1.79696	−0.898479	−	0.439016i	$-0.855327\pi$
$73$	−15.3532	−1.79696	−0.898479	+	0.439016i	$0.855327\pi$
$74$	−7.89661	−0.917963
$75$	0	0
$76$	−11.2208	−1.28711
$77$	1.53649	0.175099
$78$	0.715431	0.0810066
$79$	−3.24752	−0.365374	−0.182687	−	0.983171i	$-0.558480\pi$
$79$	−3.24752	−0.365374	−0.182687	+	0.983171i	$0.558480\pi$
$80$	0	0
$81$	0.418259	0.0464733
$82$	0.785958	0.0867946
$83$	5.16103	0.566497	0.283249	−	0.959047i	$-0.408588\pi$
$83$	5.16103	0.566497	0.283249	+	0.959047i	$0.408588\pi$
$84$	−1.59083	−0.173574
$85$	0	0
$86$	−6.38295	−0.688292
$87$	4.99233	0.535233
$88$	−3.72453	−0.397036
$89$	12.5955	1.33513	0.667563	−	0.744554i	$-0.267338\pi$
$89$	12.5955	1.33513	0.667563	+	0.744554i	$0.267338\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	2.41129	0.251394
$93$	4.79183	0.496889
$94$	−0.645940	−0.0666236
$95$	0	0
$96$	6.04245	0.616705
$97$	−8.37280	−0.850129	−0.425064	−	0.905163i	$-0.639749\pi$
$97$	−8.37280	−0.850129	−0.425064	+	0.905163i	$0.639749\pi$
$98$	−0.687115	−0.0694091
$99$	2.94373	0.295856
$100$	0	0
$101$	1.04015	0.103499	0.0517494	−	0.998660i	$-0.483520\pi$
$101$	1.04015	0.103499	0.0517494	+	0.998660i	$0.483520\pi$
$102$	−0.264676	−0.0262068
$103$	−11.6647	−1.14936	−0.574681	−	0.818378i	$-0.694874\pi$
$103$	−11.6647	−1.14936	−0.574681	+	0.818378i	$0.694874\pi$
$104$	2.42406	0.237698
$105$	0	0
$106$	−7.19603	−0.698940
$107$	10.0358	0.970198	0.485099	−	0.874459i	$-0.338784\pi$
$107$	10.0358	0.970198	0.485099	+	0.874459i	$0.338784\pi$
$108$	−7.82036	−0.752514
$109$	12.8186	1.22780	0.613898	−	0.789385i	$-0.289601\pi$
$109$	12.8186	1.22780	0.613898	+	0.789385i	$0.289601\pi$
$110$	0	0
$111$	−11.9660	−1.13576
$112$	−1.39014	−0.131356
$113$	−4.18984	−0.394147	−0.197074	−	0.980389i	$-0.563144\pi$
$113$	−4.18984	−0.394147	−0.197074	+	0.980389i	$0.563144\pi$
$114$	5.25417	0.492098
$115$	0	0
$116$	7.32575	0.680179
$117$	−1.91588	−0.177123
$118$	7.16313	0.659419
$119$	0.369953	0.0339135
$120$	0	0
$121$	−8.63921	−0.785382
$122$	6.68562	0.605288
$123$	1.19099	0.107388
$124$	7.03154	0.631451
$125$	0	0
$126$	−1.31643	−0.117277
$127$	−17.9910	−1.59644	−0.798222	−	0.602363i	$-0.794226\pi$
$127$	−17.9910	−1.59644	−0.798222	+	0.602363i	$0.794226\pi$
$128$	10.7771	0.952567
$129$	−9.67231	−0.851599
$130$	0	0
$131$	5.55142	0.485030	0.242515	−	0.970148i	$-0.422028\pi$
$131$	5.55142	0.485030	0.242515	+	0.970148i	$0.422028\pi$
$132$	−2.44430	−0.212749
$133$	−7.34406	−0.636811
$134$	−4.19495	−0.362389
$135$	0	0
$136$	−0.896786	−0.0768988
$137$	−20.8035	−1.77736	−0.888682	−	0.458524i	$-0.848378\pi$
$137$	−20.8035	−1.77736	−0.888682	+	0.458524i	$0.848378\pi$
$138$	−1.12909	−0.0961147
$139$	5.76606	0.489070	0.244535	−	0.969640i	$-0.421365\pi$
$139$	5.76606	0.489070	0.244535	+	0.969640i	$0.421365\pi$
$140$	0	0
$141$	−0.978815	−0.0824311
$142$	9.98987	0.838331
$143$	−1.53649	−0.128488
$144$	−2.66334	−0.221945
$145$	0	0
$146$	10.5494	0.873078
$147$	−1.04121	−0.0858775
$148$	−17.5589	−1.44334
$149$	−23.7177	−1.94303	−0.971514	−	0.236983i	$-0.923841\pi$
$149$	−23.7177	−1.94303	−0.971514	+	0.236983i	$0.923841\pi$
$150$	0	0
$151$	−8.70695	−0.708562	−0.354281	−	0.935139i	$-0.615274\pi$
$151$	−8.70695	−0.708562	−0.354281	+	0.935139i	$0.615274\pi$
$152$	17.8024	1.44397
$153$	0.708786	0.0573020
$154$	−1.05574	−0.0850743
$155$	0	0
$156$	1.59083	0.127369
$157$	4.41234	0.352143	0.176071	−	0.984377i	$-0.443661\pi$
$157$	4.41234	0.352143	0.176071	+	0.984377i	$0.443661\pi$
$158$	2.23142	0.177522
$159$	−10.9044	−0.864774
$160$	0	0
$161$	1.57820	0.124379
$162$	−0.287392	−0.0225797
$163$	2.34902	0.183989	0.0919947	−	0.995759i	$-0.470676\pi$
$163$	2.34902	0.183989	0.0919947	+	0.995759i	$0.470676\pi$
$164$	1.74766	0.136469
$165$	0	0
$166$	−3.54623	−0.275241
$167$	−12.9857	−1.00486	−0.502432	−	0.864616i	$-0.667561\pi$
$167$	−12.9857	−1.00486	−0.502432	+	0.864616i	$0.667561\pi$
$168$	2.52395	0.194727
$169$	1.00000	0.0769231
$170$	0	0
$171$	−14.0704	−1.07599
$172$	−14.1932	−1.08222
$173$	3.92016	0.298045	0.149022	−	0.988834i	$-0.452387\pi$
$173$	3.92016	0.298045	0.149022	+	0.988834i	$0.452387\pi$
$174$	−3.43030	−0.260051
$175$	0	0
$176$	−2.13593	−0.161002
$177$	10.8545	0.815876
$178$	−8.65460	−0.648689
$179$	−15.4509	−1.15486	−0.577429	−	0.816441i	$-0.695944\pi$
$179$	−15.4509	−1.15486	−0.577429	+	0.816441i	$0.695944\pi$
$180$	0	0
$181$	−7.00617	−0.520764	−0.260382	−	0.965506i	$-0.583849\pi$
$181$	−7.00617	−0.520764	−0.260382	+	0.965506i	$0.583849\pi$
$182$	0.687115	0.0509324
$183$	10.1309	0.748901
$184$	−3.82564	−0.282030
$185$	0	0
$186$	−3.29254	−0.241421
$187$	0.568428	0.0415676
$188$	−1.43631	−0.104754
$189$	−5.11846	−0.372313
$190$	0	0
$191$	−2.08008	−0.150509	−0.0752547	−	0.997164i	$-0.523977\pi$
$191$	−2.08008	−0.150509	−0.0752547	+	0.997164i	$0.523977\pi$
$192$	−1.25701	−0.0907168
$193$	−12.8703	−0.926428	−0.463214	−	0.886247i	$-0.653304\pi$
$193$	−12.8703	−0.926428	−0.463214	+	0.886247i	$0.653304\pi$
$194$	5.75308	0.413047
$195$	0	0
$196$	−1.52787	−0.109134
$197$	−17.0074	−1.21173	−0.605863	−	0.795569i	$-0.707172\pi$
$197$	−17.0074	−1.21173	−0.605863	+	0.795569i	$0.707172\pi$
$198$	−2.02268	−0.143746
$199$	6.90438	0.489439	0.244719	−	0.969594i	$-0.421304\pi$
$199$	6.90438	0.489439	0.244719	+	0.969594i	$0.421304\pi$
$200$	0	0
$201$	−6.35675	−0.448371
$202$	−0.714703	−0.0502863
$203$	4.79474	0.336525
$204$	−0.588534	−0.0412056
$205$	0	0
$206$	8.01503	0.558433
$207$	3.02364	0.210158
$208$	1.39014	0.0963888
$209$	−11.2841	−0.780535
$210$	0	0
$211$	−0.881563	−0.0606893	−0.0303446	−	0.999539i	$-0.509660\pi$
$211$	−0.881563	−0.0606893	−0.0303446	+	0.999539i	$0.509660\pi$
$212$	−16.0011	−1.09896
$213$	15.1380	1.03724
$214$	−6.89576	−0.471384
$215$	0	0
$216$	12.4074	0.844219
$217$	4.60218	0.312416
$218$	−8.80784	−0.596542
$219$	15.9859	1.08023
$220$	0	0
$221$	−0.369953	−0.0248857
$222$	8.22203	0.551826
$223$	22.7501	1.52346	0.761731	−	0.647894i	$-0.224350\pi$
$223$	22.7501	1.52346	0.761731	+	0.647894i	$0.224350\pi$
$224$	5.80330	0.387749
$225$	0	0
$226$	2.87890	0.191502
$227$	−7.15482	−0.474882	−0.237441	−	0.971402i	$-0.576309\pi$
$227$	−7.15482	−0.474882	−0.237441	+	0.971402i	$0.576309\pi$
$228$	11.6832	0.773738
$229$	−9.78522	−0.646625	−0.323313	−	0.946292i	$-0.604797\pi$
$229$	−9.78522	−0.646625	−0.323313	+	0.946292i	$0.604797\pi$
$230$	0	0
$231$	−1.59981	−0.105259
$232$	−11.6227	−0.763068
$233$	6.64600	0.435394	0.217697	−	0.976016i	$-0.430146\pi$
$233$	6.64600	0.435394	0.217697	+	0.976016i	$0.430146\pi$
$234$	1.31643	0.0860579
$235$	0	0
$236$	15.9279	1.03682
$237$	3.38135	0.219642
$238$	−0.254200	−0.0164774
$239$	−8.65984	−0.560158	−0.280079	−	0.959977i	$-0.590361\pi$
$239$	−8.65984	−0.560158	−0.280079	+	0.959977i	$0.590361\pi$
$240$	0	0
$241$	−3.60492	−0.232213	−0.116107	−	0.993237i	$-0.537041\pi$
$241$	−3.60492	−0.232213	−0.116107	+	0.993237i	$0.537041\pi$
$242$	5.93613	0.381589
$243$	−15.7909	−1.01299
$244$	14.8662	0.951709
$245$	0	0
$246$	−0.818347	−0.0521759
$247$	7.34406	0.467292
$248$	−11.1559	−0.708402
$249$	−5.37372	−0.340545
$250$	0	0
$251$	10.0906	0.636911	0.318456	−	0.947938i	$-0.396836\pi$
$251$	10.0906	0.636911	0.318456	+	0.947938i	$0.396836\pi$
$252$	−2.92723	−0.184398
$253$	2.42488	0.152451
$254$	12.3619	0.775655
$255$	0	0
$256$	−9.81961	−0.613725
$257$	−4.36140	−0.272057	−0.136028	−	0.990705i	$-0.543434\pi$
$257$	−4.36140	−0.272057	−0.136028	+	0.990705i	$0.543434\pi$
$258$	6.64599	0.413761
$259$	−11.4924	−0.714104
$260$	0	0
$261$	9.18616	0.568609
$262$	−3.81447	−0.235659
$263$	−18.7555	−1.15651	−0.578257	−	0.815855i	$-0.696267\pi$
$263$	−18.7555	−1.15651	−0.578257	+	0.815855i	$0.696267\pi$
$264$	3.87802	0.238675
$265$	0	0
$266$	5.04622	0.309404
$267$	−13.1146	−0.802601
$268$	−9.32791	−0.569793
$269$	−21.8350	−1.33130	−0.665652	−	0.746262i	$-0.731846\pi$
$269$	−21.8350	−1.33130	−0.665652	+	0.746262i	$0.731846\pi$
$270$	0	0
$271$	20.6330	1.25336	0.626681	−	0.779276i	$-0.284413\pi$
$271$	20.6330	1.25336	0.626681	+	0.779276i	$0.284413\pi$
$272$	−0.514286	−0.0311831
$273$	1.04121	0.0630168
$274$	14.2944	0.863557
$275$	0	0
$276$	−2.51065	−0.151124
$277$	−4.03668	−0.242541	−0.121270	−	0.992620i	$-0.538697\pi$
$277$	−4.03668	−0.242541	−0.121270	+	0.992620i	$0.538697\pi$
$278$	−3.96195	−0.237622
$279$	8.81723	0.527874
$280$	0	0
$281$	5.39373	0.321763	0.160882	−	0.986974i	$-0.448566\pi$
$281$	5.39373	0.321763	0.160882	+	0.986974i	$0.448566\pi$
$282$	0.672559	0.0400503
$283$	−19.5775	−1.16376	−0.581881	−	0.813274i	$-0.697683\pi$
$283$	−19.5775	−1.16376	−0.581881	+	0.813274i	$0.697683\pi$
$284$	22.2135	1.31813
$285$	0	0
$286$	1.05574	0.0624275
$287$	1.14385	0.0675194
$288$	11.1184	0.655160
$289$	−16.8631	−0.991949
$290$	0	0
$291$	8.71783	0.511048
$292$	23.4578	1.37276
$293$	−25.7857	−1.50642	−0.753208	−	0.657783i	$-0.771494\pi$
$293$	−25.7857	−1.50642	−0.753208	+	0.657783i	$0.771494\pi$
$294$	0.715431	0.0417248
$295$	0	0
$296$	27.8582	1.61923
$297$	−7.86446	−0.456342
$298$	16.2968	0.944047
$299$	−1.57820	−0.0912696
$300$	0	0
$301$	−9.28949	−0.535437
$302$	5.98268	0.344265
$303$	−1.08301	−0.0622175
$304$	10.2093	0.585541
$305$	0	0
$306$	−0.487018	−0.0278410
$307$	31.2935	1.78601	0.893006	−	0.450045i	$-0.148592\pi$
$307$	31.2935	1.78601	0.893006	+	0.450045i	$0.148592\pi$
$308$	−2.34756	−0.133764
$309$	12.1454	0.690930
$310$	0	0
$311$	0.604694	0.0342891	0.0171445	−	0.999853i	$-0.494542\pi$
$311$	0.604694	0.0342891	0.0171445	+	0.999853i	$0.494542\pi$
$312$	−2.52395	−0.142890
$313$	16.0469	0.907022	0.453511	−	0.891251i	$-0.350171\pi$
$313$	16.0469	0.907022	0.453511	+	0.891251i	$0.350171\pi$
$314$	−3.03178	−0.171094
$315$	0	0
$316$	4.96179	0.279123
$317$	22.2365	1.24893	0.624463	−	0.781054i	$-0.285318\pi$
$317$	22.2365	1.24893	0.624463	+	0.781054i	$0.285318\pi$
$318$	7.49257	0.420162
$319$	7.36706	0.412476
$320$	0	0
$321$	−10.4494	−0.583227
$322$	−1.08440	−0.0604315
$323$	−2.71696	−0.151176
$324$	−0.639047	−0.0355026
$325$	0	0
$326$	−1.61405	−0.0893938
$327$	−13.3468	−0.738080
$328$	−2.77276	−0.153100
$329$	−0.940075	−0.0518280
$330$	0	0
$331$	7.24423	0.398179	0.199089	−	0.979981i	$-0.436202\pi$
$331$	7.24423	0.398179	0.199089	+	0.979981i	$0.436202\pi$
$332$	−7.88540	−0.432768
$333$	−22.0181	−1.20659
$334$	8.92269	0.488228
$335$	0	0
$336$	1.44742	0.0789635
$337$	5.32218	0.289918	0.144959	−	0.989438i	$-0.453695\pi$
$337$	5.32218	0.289918	0.144959	+	0.989438i	$0.453695\pi$
$338$	−0.687115	−0.0373742
$339$	4.36250	0.236938
$340$	0	0
$341$	7.07119	0.382926
$342$	9.66797	0.522784
$343$	−1.00000	−0.0539949
$344$	22.5182	1.21410
$345$	0	0
$346$	−2.69361	−0.144809
$347$	4.58574	0.246175	0.123088	−	0.992396i	$-0.460720\pi$
$347$	4.58574	0.246175	0.123088	+	0.992396i	$0.460720\pi$
$348$	−7.62764	−0.408884
$349$	−34.4743	−1.84537	−0.922684	−	0.385558i	$-0.874009\pi$
$349$	−34.4743	−1.84537	−0.922684	+	0.385558i	$0.874009\pi$
$350$	0	0
$351$	5.11846	0.273203
$352$	8.91670	0.475261
$353$	−7.47105	−0.397644	−0.198822	−	0.980036i	$-0.563712\pi$
$353$	−7.47105	−0.397644	−0.198822	+	0.980036i	$0.563712\pi$
$354$	−7.45831	−0.396405
$355$	0	0
$356$	−19.2444	−1.01995
$357$	−0.385198	−0.0203869
$358$	10.6166	0.561104
$359$	−8.97801	−0.473841	−0.236921	−	0.971529i	$-0.576138\pi$
$359$	−8.97801	−0.473841	−0.236921	+	0.971529i	$0.576138\pi$
$360$	0	0
$361$	34.9353	1.83870
$362$	4.81404	0.253021
$363$	8.99522	0.472127
$364$	1.52787	0.0800823
$365$	0	0
$366$	−6.96113	−0.363864
$367$	11.3444	0.592173	0.296086	−	0.955161i	$-0.404318\pi$
$367$	11.3444	0.592173	0.296086	+	0.955161i	$0.404318\pi$
$368$	−2.19391	−0.114366
$369$	2.19149	0.114084
$370$	0	0
$371$	−10.4728	−0.543721
$372$	−7.32130	−0.379592
$373$	−21.5328	−1.11493	−0.557464	−	0.830201i	$-0.688225\pi$
$373$	−21.5328	−1.11493	−0.557464	+	0.830201i	$0.688225\pi$
$374$	−0.390576	−0.0201962
$375$	0	0
$376$	2.27879	0.117520
$377$	−4.79474	−0.246942
$378$	3.51698	0.180894
$379$	12.1253	0.622833	0.311417	−	0.950273i	$-0.399197\pi$
$379$	12.1253	0.622833	0.311417	+	0.950273i	$0.399197\pi$
$380$	0	0
$381$	18.7324	0.959691
$382$	1.42926	0.0731271
$383$	33.3086	1.70199	0.850996	−	0.525173i	$-0.175999\pi$
$383$	33.3086	1.70199	0.850996	+	0.525173i	$0.175999\pi$
$384$	−11.2212	−0.572629
$385$	0	0
$386$	8.84341	0.450118
$387$	−17.7976	−0.904702
$388$	12.7926	0.649444
$389$	28.5767	1.44889	0.724447	−	0.689330i	$-0.242095\pi$
$389$	28.5767	1.44889	0.724447	+	0.689330i	$0.242095\pi$
$390$	0	0
$391$	0.583859	0.0295270
$392$	2.42406	0.122433
$393$	−5.78019	−0.291572
$394$	11.6860	0.588734
$395$	0	0
$396$	−4.49765	−0.226015
$397$	−19.5788	−0.982630	−0.491315	−	0.870982i	$-0.663484\pi$
$397$	−19.5788	−0.982630	−0.491315	+	0.870982i	$0.663484\pi$
$398$	−4.74411	−0.237801
$399$	7.64671	0.382814
$400$	0	0
$401$	−5.29993	−0.264666	−0.132333	−	0.991205i	$-0.542247\pi$
$401$	−5.29993	−0.264666	−0.132333	+	0.991205i	$0.542247\pi$
$402$	4.36782	0.217847
$403$	−4.60218	−0.229251
$404$	−1.58922	−0.0790664
$405$	0	0
$406$	−3.29454	−0.163505
$407$	−17.6579	−0.875272
$408$	0.933742	0.0462271
$409$	24.3391	1.20349	0.601745	−	0.798688i	$-0.294472\pi$
$409$	24.3391	1.20349	0.601745	+	0.798688i	$0.294472\pi$
$410$	0	0
$411$	21.6608	1.06845
$412$	17.8222	0.878039
$413$	10.4249	0.512977
$414$	−2.07759	−0.102108
$415$	0	0
$416$	−5.80330	−0.284530
$417$	−6.00367	−0.294001
$418$	7.75345	0.379234
$419$	16.4092	0.801643	0.400821	−	0.916156i	$-0.368725\pi$
$419$	16.4092	0.801643	0.400821	+	0.916156i	$0.368725\pi$
$420$	0	0
$421$	8.95277	0.436331	0.218166	−	0.975912i	$-0.429993\pi$
$421$	8.95277	0.436331	0.218166	+	0.975912i	$0.429993\pi$
$422$	0.605735	0.0294867
$423$	−1.80107	−0.0875712
$424$	25.3867	1.23289
$425$	0	0
$426$	−10.4015	−0.503956
$427$	9.72998	0.470867
$428$	−15.3334	−0.741169
$429$	1.59981	0.0772393
$430$	0	0
$431$	4.29377	0.206824	0.103412	−	0.994639i	$-0.467024\pi$
$431$	4.29377	0.206824	0.103412	+	0.994639i	$0.467024\pi$
$432$	7.11537	0.342338
$433$	18.5366	0.890813	0.445407	−	0.895328i	$-0.353059\pi$
$433$	18.5366	0.890813	0.445407	+	0.895328i	$0.353059\pi$
$434$	−3.16223	−0.151792
$435$	0	0
$436$	−19.5851	−0.937958
$437$	−11.5904	−0.554443
$438$	−10.9842	−0.524844
$439$	19.9307	0.951238	0.475619	−	0.879651i	$-0.342224\pi$
$439$	19.9307	0.951238	0.475619	+	0.879651i	$0.342224\pi$
$440$	0	0
$441$	−1.91588	−0.0912325
$442$	0.254200	0.0120911
$443$	−4.27927	−0.203314	−0.101657	−	0.994819i	$-0.532414\pi$
$443$	−4.27927	−0.203314	−0.101657	+	0.994819i	$0.532414\pi$
$444$	18.2825	0.867650
$445$	0	0
$446$	−15.6320	−0.740195
$447$	24.6951	1.16804
$448$	−1.20726	−0.0570376
$449$	−33.9618	−1.60276	−0.801379	−	0.598157i	$-0.795900\pi$
$449$	−33.9618	−1.60276	−0.801379	+	0.598157i	$0.795900\pi$
$450$	0	0
$451$	1.75751	0.0827581
$452$	6.40154	0.301103
$453$	9.06576	0.425947
$454$	4.91619	0.230728
$455$	0	0
$456$	−18.5360	−0.868030
$457$	−12.8689	−0.601981	−0.300991	−	0.953627i	$-0.597317\pi$
$457$	−12.8689	−0.601981	−0.300991	+	0.953627i	$0.597317\pi$
$458$	6.72358	0.314172
$459$	−1.89359	−0.0883852
$460$	0	0
$461$	9.36566	0.436202	0.218101	−	0.975926i	$-0.430014\pi$
$461$	9.36566	0.436202	0.218101	+	0.975926i	$0.430014\pi$
$462$	1.09925	0.0511418
$463$	1.32674	0.0616590	0.0308295	−	0.999525i	$-0.490185\pi$
$463$	1.32674	0.0616590	0.0308295	+	0.999525i	$0.490185\pi$
$464$	−6.66535	−0.309431
$465$	0	0
$466$	−4.56657	−0.211542
$467$	29.5035	1.36526	0.682630	−	0.730764i	$-0.260836\pi$
$467$	29.5035	1.36526	0.682630	+	0.730764i	$0.260836\pi$
$468$	2.92723	0.135311
$469$	−6.10516	−0.281910
$470$	0	0
$471$	−4.59417	−0.211688
$472$	−25.2706	−1.16317
$473$	−14.2732	−0.656282
$474$	−2.32337	−0.106716
$475$	0	0
$476$	−0.565241	−0.0259078
$477$	−20.0647	−0.918698
$478$	5.95031	0.272161
$479$	−3.54307	−0.161887	−0.0809436	−	0.996719i	$-0.525793\pi$
$479$	−3.54307	−0.161887	−0.0809436	+	0.996719i	$0.525793\pi$
$480$	0	0
$481$	11.4924	0.524009
$482$	2.47699	0.112824
$483$	−1.64323	−0.0747698
$484$	13.1996	0.599982
$485$	0	0
$486$	10.8502	0.492173
$487$	7.37366	0.334132	0.167066	−	0.985946i	$-0.446571\pi$
$487$	7.37366	0.334132	0.167066	+	0.985946i	$0.446571\pi$
$488$	−23.5860	−1.06769
$489$	−2.44582	−0.110604
$490$	0	0
$491$	−5.42980	−0.245044	−0.122522	−	0.992466i	$-0.539098\pi$
$491$	−5.42980	−0.245044	−0.122522	+	0.992466i	$0.539098\pi$
$492$	−1.81968	−0.0820374
$493$	1.77383	0.0798891
$494$	−5.04622	−0.227040
$495$	0	0
$496$	−6.39766	−0.287264
$497$	14.5388	0.652156
$498$	3.69236	0.165459
$499$	−2.34930	−0.105169	−0.0525846	−	0.998616i	$-0.516746\pi$
$499$	−2.34930	−0.105169	−0.0525846	+	0.998616i	$0.516746\pi$
$500$	0	0
$501$	13.5208	0.604067
$502$	−6.93339	−0.309452
$503$	5.24148	0.233706	0.116853	−	0.993149i	$-0.462719\pi$
$503$	5.24148	0.233706	0.116853	+	0.993149i	$0.462719\pi$
$504$	4.64421	0.206869
$505$	0	0
$506$	−1.66617	−0.0740705
$507$	−1.04121	−0.0462417
$508$	27.4880	1.21958
$509$	−21.1506	−0.937485	−0.468743	−	0.883335i	$-0.655293\pi$
$509$	−21.1506	−0.937485	−0.468743	+	0.883335i	$0.655293\pi$
$510$	0	0
$511$	15.3532	0.679187
$512$	−14.8069	−0.654380
$513$	37.5903	1.65965
$514$	2.99679	0.132183
$515$	0	0
$516$	14.7780	0.650567
$517$	−1.44441	−0.0635252
$518$	7.89661	0.346957
$519$	−4.08171	−0.179167
$520$	0	0
$521$	−23.5219	−1.03051	−0.515256	−	0.857036i	$-0.672303\pi$
$521$	−23.5219	−1.03051	−0.515256	+	0.857036i	$0.672303\pi$
$522$	−6.31195	−0.276267
$523$	0.132078	0.00577535	0.00288768	−	0.999996i	$-0.499081\pi$
$523$	0.132078	0.00577535	0.00288768	+	0.999996i	$0.499081\pi$
$524$	−8.48186	−0.370532
$525$	0	0
$526$	12.8872	0.561908
$527$	1.70259	0.0741659
$528$	2.22395	0.0967850
$529$	−20.5093	−0.891708
$530$	0	0
$531$	19.9729	0.866752
$532$	11.2208	0.486483
$533$	−1.14385	−0.0495457
$534$	9.01125	0.389955
$535$	0	0
$536$	14.7993	0.639231
$537$	16.0877	0.694234
$538$	15.0032	0.646833
$539$	−1.53649	−0.0661812
$540$	0	0
$541$	−7.65572	−0.329145	−0.164572	−	0.986365i	$-0.552624\pi$
$541$	−7.65572	−0.329145	−0.164572	+	0.986365i	$0.552624\pi$
$542$	−14.1772	−0.608964
$543$	7.29488	0.313053
$544$	2.14695	0.0920495
$545$	0	0
$546$	−0.715431	−0.0306176
$547$	4.00033	0.171042	0.0855208	−	0.996336i	$-0.472745\pi$
$547$	4.00033	0.171042	0.0855208	+	0.996336i	$0.472745\pi$
$548$	31.7851	1.35779
$549$	18.6415	0.795601
$550$	0	0
$551$	−35.2129	−1.50012
$552$	3.98329	0.169540
$553$	3.24752	0.138098
$554$	2.77366	0.117842
$555$	0	0
$556$	−8.80980	−0.373619
$557$	−0.819142	−0.0347082	−0.0173541	−	0.999849i	$-0.505524\pi$
$557$	−0.819142	−0.0347082	−0.0173541	+	0.999849i	$0.505524\pi$
$558$	−6.05846	−0.256475
$559$	9.28949	0.392904
$560$	0	0
$561$	−0.591852	−0.0249880
$562$	−3.70612	−0.156333
$563$	−7.47385	−0.314985	−0.157493	−	0.987520i	$-0.550341\pi$
$563$	−7.47385	−0.314985	−0.157493	+	0.987520i	$0.550341\pi$
$564$	1.49550	0.0629721
$565$	0	0
$566$	13.4520	0.565430
$567$	−0.418259	−0.0175652
$568$	−35.2430	−1.47876
$569$	−5.51897	−0.231367	−0.115684	−	0.993286i	$-0.536906\pi$
$569$	−5.51897	−0.231367	−0.115684	+	0.993286i	$0.536906\pi$
$570$	0	0
$571$	43.2888	1.81158	0.905791	−	0.423725i	$-0.139278\pi$
$571$	43.2888	1.81158	0.905791	+	0.423725i	$0.139278\pi$
$572$	2.34756	0.0981563
$573$	2.16580	0.0904776
$574$	−0.785958	−0.0328053
$575$	0	0
$576$	−2.31297	−0.0963736
$577$	−17.8090	−0.741399	−0.370699	−	0.928753i	$-0.620882\pi$
$577$	−17.8090	−0.741399	−0.370699	+	0.928753i	$0.620882\pi$
$578$	11.5869	0.481952
$579$	13.4007	0.556915
$580$	0	0
$581$	−5.16103	−0.214116
$582$	−5.99016	−0.248300
$583$	−16.0913	−0.666435
$584$	−37.2171	−1.54005
$585$	0	0
$586$	17.7177	0.731913
$587$	46.9122	1.93627	0.968136	−	0.250423i	$-0.0805698\pi$
$587$	46.9122	1.93627	0.968136	+	0.250423i	$0.0805698\pi$
$588$	1.59083	0.0656049
$589$	−33.7987	−1.39265
$590$	0	0
$591$	17.7082	0.728420
$592$	15.9760	0.656611
$593$	5.76853	0.236885	0.118443	−	0.992961i	$-0.462210\pi$
$593$	5.76853	0.236885	0.118443	+	0.992961i	$0.462210\pi$
$594$	5.40379	0.221720
$595$	0	0
$596$	36.2376	1.48435
$597$	−7.18891	−0.294222
$598$	1.08440	0.0443446
$599$	−26.0972	−1.06630	−0.533150	−	0.846020i	$-0.678992\pi$
$599$	−26.0972	−1.06630	−0.533150	+	0.846020i	$0.678992\pi$
$600$	0	0
$601$	−8.22672	−0.335575	−0.167788	−	0.985823i	$-0.553662\pi$
$601$	−8.22672	−0.335575	−0.167788	+	0.985823i	$0.553662\pi$
$602$	6.38295	0.260150
$603$	−11.6968	−0.476330
$604$	13.3031	0.541296
$605$	0	0
$606$	0.744155	0.0302292
$607$	−12.6960	−0.515314	−0.257657	−	0.966236i	$-0.582951\pi$
$607$	−12.6960	−0.515314	−0.257657	+	0.966236i	$0.582951\pi$
$608$	−42.6198	−1.72846
$609$	−4.99233	−0.202299
$610$	0	0
$611$	0.940075	0.0380314
$612$	−1.08294	−0.0437751
$613$	−13.6414	−0.550972	−0.275486	−	0.961305i	$-0.588839\pi$
$613$	−13.6414	−0.550972	−0.275486	+	0.961305i	$0.588839\pi$
$614$	−21.5022	−0.867759
$615$	0	0
$616$	3.72453	0.150066
$617$	−10.3293	−0.415841	−0.207920	−	0.978146i	$-0.566669\pi$
$617$	−10.3293	−0.415841	−0.207920	+	0.978146i	$0.566669\pi$
$618$	−8.34532	−0.335698
$619$	−46.0719	−1.85178	−0.925892	−	0.377788i	$-0.876685\pi$
$619$	−46.0719	−1.85178	−0.925892	+	0.377788i	$0.876685\pi$
$620$	0	0
$621$	−8.07795	−0.324157
$622$	−0.415495	−0.0166598
$623$	−12.5955	−0.504630
$624$	−1.44742	−0.0579434
$625$	0	0
$626$	−11.0260	−0.440689
$627$	11.7491	0.469213
$628$	−6.74149	−0.269015
$629$	−4.25165	−0.169524
$630$	0	0
$631$	0.885927	0.0352682	0.0176341	−	0.999845i	$-0.494387\pi$
$631$	0.885927	0.0352682	0.0176341	+	0.999845i	$0.494387\pi$
$632$	−7.87216	−0.313138
$633$	0.917891	0.0364829
$634$	−15.2790	−0.606808
$635$	0	0
$636$	16.6605	0.660632
$637$	1.00000	0.0396214
$638$	−5.06202	−0.200407
$639$	27.8547	1.10192
$640$	0	0
$641$	−40.0387	−1.58143	−0.790716	−	0.612183i	$-0.790292\pi$
$641$	−40.0387	−1.58143	−0.790716	+	0.612183i	$0.790292\pi$
$642$	7.17993	0.283369
$643$	22.4584	0.885673	0.442837	−	0.896602i	$-0.353972\pi$
$643$	22.4584	0.885673	0.442837	+	0.896602i	$0.353972\pi$
$644$	−2.41129	−0.0950180
$645$	0	0
$646$	1.86686	0.0734507
$647$	−30.0514	−1.18144	−0.590722	−	0.806875i	$-0.701157\pi$
$647$	−30.0514	−1.18144	−0.590722	+	0.806875i	$0.701157\pi$
$648$	1.01388	0.0398291
$649$	16.0178	0.628752
$650$	0	0
$651$	−4.79183	−0.187807
$652$	−3.58900	−0.140556
$653$	3.48859	0.136519	0.0682595	−	0.997668i	$-0.478255\pi$
$653$	3.48859	0.136519	0.0682595	+	0.997668i	$0.478255\pi$
$654$	9.17080	0.358607
$655$	0	0
$656$	−1.59011	−0.0620834
$657$	29.4150	1.14759
$658$	0.645940	0.0251814
$659$	19.6470	0.765338	0.382669	−	0.923885i	$-0.375005\pi$
$659$	19.6470	0.765338	0.382669	+	0.923885i	$0.375005\pi$
$660$	0	0
$661$	27.7809	1.08055	0.540276	−	0.841488i	$-0.318320\pi$
$661$	27.7809	1.08055	0.540276	+	0.841488i	$0.318320\pi$
$662$	−4.97762	−0.193461
$663$	0.385198	0.0149599
$664$	12.5106	0.485507
$665$	0	0
$666$	15.1290	0.586236
$667$	7.56705	0.292997
$668$	19.8405	0.767653
$669$	−23.6877	−0.915817
$670$	0	0
$671$	14.9500	0.577138
$672$	−6.04245	−0.233092
$673$	−10.3875	−0.400408	−0.200204	−	0.979754i	$-0.564161\pi$
$673$	−10.3875	−0.400408	−0.200204	+	0.979754i	$0.564161\pi$
$674$	−3.65695	−0.140861
$675$	0	0
$676$	−1.52787	−0.0587643
$677$	−2.88769	−0.110983	−0.0554914	−	0.998459i	$-0.517673\pi$
$677$	−2.88769	−0.110983	−0.0554914	+	0.998459i	$0.517673\pi$
$678$	−2.99754	−0.115120
$679$	8.37280	0.321318
$680$	0	0
$681$	7.44966	0.285472
$682$	−4.85872	−0.186050
$683$	−33.7134	−1.29001	−0.645004	−	0.764179i	$-0.723144\pi$
$683$	−33.7134	−1.29001	−0.645004	+	0.764179i	$0.723144\pi$
$684$	21.4977	0.821986
$685$	0	0
$686$	0.687115	0.0262342
$687$	10.1885	0.388714
$688$	12.9137	0.492329
$689$	10.4728	0.398982
$690$	0	0
$691$	11.7338	0.446374	0.223187	−	0.974776i	$-0.428354\pi$
$691$	11.7338	0.446374	0.223187	+	0.974776i	$0.428354\pi$
$692$	−5.98951	−0.227687
$693$	−2.94373	−0.111823
$694$	−3.15093	−0.119608
$695$	0	0
$696$	12.1017	0.458713
$697$	0.423171	0.0160287
$698$	23.6878	0.896598
$699$	−6.91988	−0.261734
$700$	0	0
$701$	4.86500	0.183749	0.0918743	−	0.995771i	$-0.470714\pi$
$701$	4.86500	0.183749	0.0918743	+	0.995771i	$0.470714\pi$
$702$	−3.51698	−0.132740
$703$	84.4010	3.18324
$704$	−1.85494	−0.0699106
$705$	0	0
$706$	5.13347	0.193201
$707$	−1.04015	−0.0391188
$708$	−16.5843	−0.623277
$709$	−32.8091	−1.23217	−0.616085	−	0.787680i	$-0.711282\pi$
$709$	−32.8091	−1.23217	−0.616085	+	0.787680i	$0.711282\pi$
$710$	0	0
$711$	6.22186	0.233338
$712$	30.5323	1.14425
$713$	7.26315	0.272007
$714$	0.264676	0.00990524
$715$	0	0
$716$	23.6071	0.882237
$717$	9.01670	0.336735
$718$	6.16893	0.230222
$719$	−7.11273	−0.265260	−0.132630	−	0.991166i	$-0.542342\pi$
$719$	−7.11273	−0.265260	−0.132630	+	0.991166i	$0.542342\pi$
$720$	0	0
$721$	11.6647	0.434418
$722$	−24.0046	−0.893357
$723$	3.75347	0.139593
$724$	10.7045	0.397831
$725$	0	0
$726$	−6.18076	−0.229389
$727$	−21.8129	−0.808994	−0.404497	−	0.914539i	$-0.632553\pi$
$727$	−21.8129	−0.808994	−0.404497	+	0.914539i	$0.632553\pi$
$728$	−2.42406	−0.0898415
$729$	15.1868	0.562475
$730$	0	0
$731$	−3.43667	−0.127110
$732$	−15.4788	−0.572113
$733$	23.1573	0.855336	0.427668	−	0.903936i	$-0.359335\pi$
$733$	23.1573	0.855336	0.427668	+	0.903936i	$0.359335\pi$
$734$	−7.79491	−0.287715
$735$	0	0
$736$	9.15875	0.337596
$737$	−9.38051	−0.345536
$738$	−1.50580	−0.0554294
$739$	43.2337	1.59038	0.795188	−	0.606362i	$-0.207372\pi$
$739$	43.2337	1.59038	0.795188	+	0.606362i	$0.207372\pi$
$740$	0	0
$741$	−7.64671	−0.280909
$742$	7.19603	0.264174
$743$	−13.5435	−0.496864	−0.248432	−	0.968649i	$-0.579915\pi$
$743$	−13.5435	−0.496864	−0.248432	+	0.968649i	$0.579915\pi$
$744$	11.6157	0.425851
$745$	0	0
$746$	14.7955	0.541703
$747$	−9.88794	−0.361781
$748$	−0.868485	−0.0317550
$749$	−10.0358	−0.366700
$750$	0	0
$751$	−12.2824	−0.448193	−0.224096	−	0.974567i	$-0.571943\pi$
$751$	−12.2824	−0.448193	−0.224096	+	0.974567i	$0.571943\pi$
$752$	1.30683	0.0476553
$753$	−10.5064	−0.382874
$754$	3.29454	0.119980
$755$	0	0
$756$	7.82036	0.284424
$757$	−18.3523	−0.667025	−0.333513	−	0.942746i	$-0.608234\pi$
$757$	−18.3523	−0.667025	−0.333513	+	0.942746i	$0.608234\pi$
$758$	−8.33146	−0.302612
$759$	−2.52481	−0.0916448
$760$	0	0
$761$	−30.1167	−1.09173	−0.545865	−	0.837873i	$-0.683799\pi$
$761$	−30.1167	−1.09173	−0.545865	+	0.837873i	$0.683799\pi$
$762$	−12.8713	−0.466279
$763$	−12.8186	−0.464063
$764$	3.17810	0.114980
$765$	0	0
$766$	−22.8869	−0.826936
$767$	−10.4249	−0.376422
$768$	10.2243	0.368936
$769$	15.3983	0.555278	0.277639	−	0.960685i	$-0.410448\pi$
$769$	15.3983	0.555278	0.277639	+	0.960685i	$0.410448\pi$
$770$	0	0
$771$	4.54113	0.163545
$772$	19.6642	0.707732
$773$	23.9803	0.862512	0.431256	−	0.902230i	$-0.358071\pi$
$773$	23.9803	0.862512	0.431256	+	0.902230i	$0.358071\pi$
$774$	12.2290	0.439562
$775$	0	0
$776$	−20.2961	−0.728589
$777$	11.9660	0.429278
$778$	−19.6355	−0.703966
$779$	−8.40052	−0.300980
$780$	0	0
$781$	22.3388	0.799344
$782$	−0.401178	−0.0143461
$783$	−24.5417	−0.877048
$784$	1.39014	0.0496478
$785$	0	0
$786$	3.97166	0.141664
$787$	−24.9869	−0.890687	−0.445344	−	0.895360i	$-0.646918\pi$
$787$	−24.9869	−0.890687	−0.445344	+	0.895360i	$0.646918\pi$
$788$	25.9851	0.925681
$789$	19.5284	0.695229
$790$	0	0
$791$	4.18984	0.148974
$792$	7.13577	0.253559
$793$	−9.72998	−0.345522
$794$	13.4529	0.477425
$795$	0	0
$796$	−10.5490	−0.373900
$797$	−29.8255	−1.05647	−0.528237	−	0.849097i	$-0.677147\pi$
$797$	−29.8255	−1.05647	−0.528237	+	0.849097i	$0.677147\pi$
$798$	−5.25417	−0.185996
$799$	−0.347783	−0.0123037
$800$	0	0
$801$	−24.1316	−0.852648
$802$	3.64166	0.128592
$803$	23.5900	0.832474
$804$	9.71231	0.342527
$805$	0	0
$806$	3.16223	0.111385
$807$	22.7348	0.800303
$808$	2.52138	0.0887018
$809$	−20.6180	−0.724891	−0.362445	−	0.932005i	$-0.618058\pi$
$809$	−20.6180	−0.724891	−0.362445	+	0.932005i	$0.618058\pi$
$810$	0	0
$811$	−1.78309	−0.0626128	−0.0313064	−	0.999510i	$-0.509967\pi$
$811$	−1.78309	−0.0626128	−0.0313064	+	0.999510i	$0.509967\pi$
$812$	−7.32575	−0.257083
$813$	−21.4832	−0.753449
$814$	12.1330	0.425263
$815$	0	0
$816$	0.535479	0.0187455
$817$	68.2226	2.38681
$818$	−16.7238	−0.584733
$819$	1.91588	0.0669464
$820$	0	0
$821$	31.9774	1.11602	0.558010	−	0.829834i	$-0.311565\pi$
$821$	31.9774	1.11602	0.558010	+	0.829834i	$0.311565\pi$
$822$	−14.8835	−0.519121
$823$	−35.7006	−1.24445	−0.622223	−	0.782840i	$-0.713770\pi$
$823$	−35.7006	−1.24445	−0.622223	+	0.782840i	$0.713770\pi$
$824$	−28.2760	−0.985041
$825$	0	0
$826$	−7.16313	−0.249237
$827$	−29.6753	−1.03191	−0.515956	−	0.856615i	$-0.672563\pi$
$827$	−29.6753	−1.03191	−0.515956	+	0.856615i	$0.672563\pi$
$828$	−4.61974	−0.160547
$829$	35.5159	1.23352	0.616759	−	0.787152i	$-0.288445\pi$
$829$	35.5159	1.23352	0.616759	+	0.787152i	$0.288445\pi$
$830$	0	0
$831$	4.20303	0.145801
$832$	1.20726	0.0418542
$833$	−0.369953	−0.0128181
$834$	4.12521	0.142845
$835$	0	0
$836$	17.2406	0.596279
$837$	−23.5561	−0.814217
$838$	−11.2750	−0.389489
$839$	−19.6010	−0.676703	−0.338351	−	0.941020i	$-0.609869\pi$
$839$	−19.6010	−0.676703	−0.338351	+	0.941020i	$0.609869\pi$
$840$	0	0
$841$	−6.01049	−0.207258
$842$	−6.15158	−0.211998
$843$	−5.61601	−0.193425
$844$	1.34692	0.0463627
$845$	0	0
$846$	1.23755	0.0425477
$847$	8.63921	0.296847
$848$	14.5587	0.499946
$849$	20.3843	0.699587
$850$	0	0
$851$	−18.1373	−0.621739
$852$	−23.1289	−0.792383
$853$	27.1626	0.930031	0.465015	−	0.885303i	$-0.346049\pi$
$853$	27.1626	0.930031	0.465015	+	0.885303i	$0.346049\pi$
$854$	−6.68562	−0.228777
$855$	0	0
$856$	24.3274	0.831492
$857$	−41.7729	−1.42693	−0.713467	−	0.700689i	$-0.752876\pi$
$857$	−41.7729	−1.42693	−0.713467	+	0.700689i	$0.752876\pi$
$858$	−1.09925	−0.0375278
$859$	−56.2414	−1.91893	−0.959466	−	0.281824i	$-0.909061\pi$
$859$	−56.2414	−1.91893	−0.959466	+	0.281824i	$0.909061\pi$
$860$	0	0
$861$	−1.19099	−0.0405888
$862$	−2.95032	−0.100488
$863$	−33.7510	−1.14890	−0.574449	−	0.818540i	$-0.694784\pi$
$863$	−33.7510	−1.14890	−0.574449	+	0.818540i	$0.694784\pi$
$864$	−29.7040	−1.01055
$865$	0	0
$866$	−12.7368	−0.432814
$867$	17.5581	0.596303
$868$	−7.03154	−0.238666
$869$	4.98977	0.169266
$870$	0	0
$871$	6.10516	0.206866
$872$	31.0729	1.05226
$873$	16.0413	0.542916
$874$	7.96394	0.269384
$875$	0	0
$876$	−24.4244	−0.825225
$877$	30.3864	1.02608	0.513038	−	0.858366i	$-0.328520\pi$
$877$	30.3864	1.02608	0.513038	+	0.858366i	$0.328520\pi$
$878$	−13.6947	−0.462173
$879$	26.8483	0.905570
$880$	0	0
$881$	−39.3433	−1.32551	−0.662755	−	0.748836i	$-0.730613\pi$
$881$	−39.3433	−1.32551	−0.662755	+	0.748836i	$0.730613\pi$
$882$	1.31643	0.0443266
$883$	−11.9111	−0.400839	−0.200420	−	0.979710i	$-0.564231\pi$
$883$	−11.9111	−0.400839	−0.200420	+	0.979710i	$0.564231\pi$
$884$	0.565241	0.0190111
$885$	0	0
$886$	2.94035	0.0987831
$887$	4.07716	0.136898	0.0684489	−	0.997655i	$-0.478195\pi$
$887$	4.07716	0.136898	0.0684489	+	0.997655i	$0.478195\pi$
$888$	−29.0063	−0.973386
$889$	17.9910	0.603399
$890$	0	0
$891$	−0.642650	−0.0215296
$892$	−34.7593	−1.16383
$893$	6.90397	0.231033
$894$	−16.9684	−0.567507
$895$	0	0
$896$	−10.7771	−0.360037
$897$	1.64323	0.0548660
$898$	23.3357	0.778723
$899$	22.0662	0.735950
$900$	0	0
$901$	−3.87444	−0.129076
$902$	−1.20762	−0.0402092
$903$	9.67231	0.321874
$904$	−10.1564	−0.337797
$905$	0	0
$906$	−6.22922	−0.206952
$907$	−39.0382	−1.29624	−0.648122	−	0.761537i	$-0.724445\pi$
$907$	−39.0382	−1.29624	−0.648122	+	0.761537i	$0.724445\pi$
$908$	10.9317	0.362780
$909$	−1.99280	−0.0660972
$910$	0	0
$911$	−43.7851	−1.45066	−0.725332	−	0.688399i	$-0.758314\pi$
$911$	−43.7851	−1.45066	−0.725332	+	0.688399i	$0.758314\pi$
$912$	−10.6300	−0.351994
$913$	−7.92987	−0.262440
$914$	8.84241	0.292481
$915$	0	0
$916$	14.9506	0.493981
$917$	−5.55142	−0.183324
$918$	1.30111	0.0429432
$919$	−25.2107	−0.831624	−0.415812	−	0.909450i	$-0.636503\pi$
$919$	−25.2107	−0.831624	−0.415812	+	0.909450i	$0.636503\pi$
$920$	0	0
$921$	−32.5830	−1.07365
$922$	−6.43529	−0.211935
$923$	−14.5388	−0.478552
$924$	2.44430	0.0804115
$925$	0	0
$926$	−0.911626	−0.0299579
$927$	22.3483	0.734014
$928$	27.8253	0.913410
$929$	24.2947	0.797082	0.398541	−	0.917150i	$-0.369517\pi$
$929$	24.2947	0.797082	0.398541	+	0.917150i	$0.369517\pi$
$930$	0	0
$931$	7.34406	0.240692
$932$	−10.1542	−0.332613
$933$	−0.629613	−0.0206126
$934$	−20.2723	−0.663331
$935$	0	0
$936$	−4.64421	−0.151801
$937$	−30.5138	−0.996843	−0.498421	−	0.866935i	$-0.666087\pi$
$937$	−30.5138	−0.996843	−0.498421	+	0.866935i	$0.666087\pi$
$938$	4.19495	0.136970
$939$	−16.7081	−0.545249
$940$	0	0
$941$	−37.0335	−1.20726	−0.603629	−	0.797265i	$-0.706279\pi$
$941$	−37.0335	−1.20726	−0.603629	+	0.797265i	$0.706279\pi$
$942$	3.15672	0.102852
$943$	1.80522	0.0587862
$944$	−14.4921	−0.471677
$945$	0	0
$946$	9.80733	0.318864
$947$	−41.1054	−1.33575	−0.667874	−	0.744275i	$-0.732795\pi$
$947$	−41.1054	−1.33575	−0.667874	+	0.744275i	$0.732795\pi$
$948$	−5.16626	−0.167792
$949$	−15.3532	−0.498387
$950$	0	0
$951$	−23.1528	−0.750783
$952$	0.896786	0.0290650
$953$	−50.1540	−1.62465	−0.812324	−	0.583206i	$-0.801798\pi$
$953$	−50.1540	−1.62465	−0.812324	+	0.583206i	$0.801798\pi$
$954$	13.7868	0.446363
$955$	0	0
$956$	13.2311	0.427925
$957$	−7.67065	−0.247957
$958$	2.43450	0.0786552
$959$	20.8035	0.671780
$960$	0	0
$961$	−9.81997	−0.316773
$962$	−7.89661	−0.254597
$963$	−19.2274	−0.619596
$964$	5.50785	0.177396
$965$	0	0
$966$	1.12909	0.0363279
$967$	23.6062	0.759124	0.379562	−	0.925166i	$-0.376075\pi$
$967$	23.6062	0.759124	0.379562	+	0.925166i	$0.376075\pi$
$968$	−20.9419	−0.673099
$969$	2.82892	0.0908780
$970$	0	0
$971$	3.12343	0.100236	0.0501179	−	0.998743i	$-0.484040\pi$
$971$	3.12343	0.100236	0.0501179	+	0.998743i	$0.484040\pi$
$972$	24.1265	0.773856
$973$	−5.76606	−0.184851
$974$	−5.06656	−0.162343
$975$	0	0
$976$	−13.5260	−0.432957
$977$	44.9745	1.43886	0.719431	−	0.694563i	$-0.244402\pi$
$977$	44.9745	1.43886	0.719431	+	0.694563i	$0.244402\pi$
$978$	1.68056	0.0537384
$979$	−19.3529	−0.618522
$980$	0	0
$981$	−24.5589	−0.784105
$982$	3.73090	0.119058
$983$	42.7726	1.36423	0.682117	−	0.731243i	$-0.261060\pi$
$983$	42.7726	1.36423	0.682117	+	0.731243i	$0.261060\pi$
$984$	2.88702	0.0920349
$985$	0	0
$986$	−1.21882	−0.0388153
$987$	0.978815	0.0311560
$988$	−11.2208	−0.356981
$989$	−14.6607	−0.466182
$990$	0	0
$991$	57.3319	1.82121	0.910604	−	0.413279i	$-0.135617\pi$
$991$	57.3319	1.82121	0.910604	+	0.413279i	$0.135617\pi$
$992$	26.7078	0.847973
$993$	−7.54276	−0.239362
$994$	−9.98987	−0.316859
$995$	0	0
$996$	8.21035	0.260155
$997$	15.9308	0.504533	0.252267	−	0.967658i	$-0.418824\pi$
$997$	15.9308	0.504533	0.252267	+	0.967658i	$0.418824\pi$
$998$	1.61424	0.0510979
$999$	58.8235	1.86109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2275.2.a.y.1.3		7
5.2	odd	4		455.2.c.b.274.6	✓	14
5.3	odd	4		455.2.c.b.274.9	yes	14
5.4	even	2		2275.2.a.w.1.5		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
455.2.c.b.274.6	✓	14	5.2	odd	4
455.2.c.b.274.9	yes	14	5.3	odd	4
2275.2.a.w.1.5		7	5.4	even	2
2275.2.a.y.1.3		7	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2275 = 5^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2275.a (trivial)

\(f(q)\)	\(=\)	\(q-0.687115 q^{2} -1.04121 q^{3} -1.52787 q^{4} +0.715431 q^{6} -1.00000 q^{7} +2.42406 q^{8} -1.91588 q^{9} -1.53649 q^{11} +1.59083 q^{12} +1.00000 q^{13} +0.687115 q^{14} +1.39014 q^{16} -0.369953 q^{17} +1.31643 q^{18} +7.34406 q^{19} +1.04121 q^{21} +1.05574 q^{22} -1.57820 q^{23} -2.52395 q^{24} -0.687115 q^{26} +5.11846 q^{27} +1.52787 q^{28} -4.79474 q^{29} -4.60218 q^{31} -5.80330 q^{32} +1.59981 q^{33} +0.254200 q^{34} +2.92723 q^{36} +11.4924 q^{37} -5.04622 q^{38} -1.04121 q^{39} -1.14385 q^{41} -0.715431 q^{42} +9.28949 q^{43} +2.34756 q^{44} +1.08440 q^{46} +0.940075 q^{47} -1.44742 q^{48} +1.00000 q^{49} +0.385198 q^{51} -1.52787 q^{52} +10.4728 q^{53} -3.51698 q^{54} -2.42406 q^{56} -7.64671 q^{57} +3.29454 q^{58} -10.4249 q^{59} -9.72998 q^{61} +3.16223 q^{62} +1.91588 q^{63} +1.20726 q^{64} -1.09925 q^{66} +6.10516 q^{67} +0.565241 q^{68} +1.64323 q^{69} -14.5388 q^{71} -4.64421 q^{72} -15.3532 q^{73} -7.89661 q^{74} -11.2208 q^{76} +1.53649 q^{77} +0.715431 q^{78} -3.24752 q^{79} +0.418259 q^{81} +0.785958 q^{82} +5.16103 q^{83} -1.59083 q^{84} -6.38295 q^{86} +4.99233 q^{87} -3.72453 q^{88} +12.5955 q^{89} -1.00000 q^{91} +2.41129 q^{92} +4.79183 q^{93} -0.645940 q^{94} +6.04245 q^{96} -8.37280 q^{97} -0.687115 q^{98} +2.94373 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 2 q^{3} + 2 q^{4} - 8 q^{6} - 7 q^{7} + 6 q^{8} + q^{9} - 6 q^{11} - 3 q^{12} + 7 q^{13} - 4 q^{16} - 2 q^{17} + q^{18} - 10 q^{19} - 2 q^{21} - 18 q^{22} - 10 q^{23} - 5 q^{24} + 8 q^{27} - 2 q^{28}+ \cdots - 18 q^{99}+O(q^{100}) \) 7 * q + 2 * q^3 + 2 * q^4 - 8 * q^6 - 7 * q^7 + 6 * q^8 + q^9 - 6 * q^11 - 3 * q^12 + 7 * q^13 - 4 * q^16 - 2 * q^17 + q^18 - 10 * q^19 - 2 * q^21 - 18 * q^22 - 10 * q^23 - 5 * q^24 + 8 * q^27 - 2 * q^28 - 26 * q^29 - 6 * q^31 + 7 * q^32 - 4 * q^33 - q^34 - 6 * q^36 + 4 * q^37 + q^38 + 2 * q^39 - 28 * q^41 + 8 * q^42 - 2 * q^43 - 18 * q^46 + 20 * q^47 - 20 * q^48 + 7 * q^49 - 32 * q^51 + 2 * q^52 - 8 * q^53 + 11 * q^54 - 6 * q^56 - 14 * q^57 + 12 * q^58 - 28 * q^59 - 26 * q^61 + 22 * q^62 - q^63 - 8 * q^64 + 2 * q^66 - 2 * q^67 + 2 * q^68 - 22 * q^69 - 14 * q^71 - 4 * q^72 + 2 * q^73 - 12 * q^74 - 36 * q^76 + 6 * q^77 - 8 * q^78 + 14 * q^79 - 33 * q^81 - 19 * q^82 + 28 * q^83 + 3 * q^84 - 4 * q^86 - 22 * q^87 - 24 * q^88 - 30 * q^89 - 7 * q^91 - 42 * q^92 - 2 * q^93 + 14 * q^94 + 18 * q^97 - 18 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more