LMFDB - Embedded newform 5225.2.a.k.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5225,2,Mod(1,5225)] 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("5225.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5225 = 5^{2} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5225.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,-3,8,0,2,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$41.7218350561$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.131947641.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 10x^{4} + 25x^{2} - 3x - 9 $ x^6 - 10x^4 + 25x^2 - 3*x - 9
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1045)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-2.36323$ of defining polynomial
Character	$\chi$	$=$	5225.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.36323 q^{2} -2.87321 q^{3} +3.58485 q^{4} -6.79006 q^{6} +0.269449 q^{7} +3.74537 q^{8} +5.25535 q^{9} -1.00000 q^{11} -10.3000 q^{12} +1.50998 q^{13} +0.636771 q^{14} +1.68146 q^{16} -3.65972 q^{17} +12.4196 q^{18} -1.00000 q^{19} -0.774186 q^{21} -2.36323 q^{22} -2.55781 q^{23} -10.7612 q^{24} +3.56844 q^{26} -6.48011 q^{27} +0.965936 q^{28} +0.572530 q^{29} +5.82024 q^{31} -3.51706 q^{32} +2.87321 q^{33} -8.64876 q^{34} +18.8397 q^{36} -7.49463 q^{37} -2.36323 q^{38} -4.33851 q^{39} +11.2021 q^{41} -1.82958 q^{42} +3.61858 q^{43} -3.58485 q^{44} -6.04469 q^{46} -13.6320 q^{47} -4.83120 q^{48} -6.92740 q^{49} +10.5152 q^{51} +5.41307 q^{52} +3.41181 q^{53} -15.3140 q^{54} +1.00919 q^{56} +2.87321 q^{57} +1.35302 q^{58} +1.76356 q^{59} -11.4676 q^{61} +13.7546 q^{62} +1.41605 q^{63} -11.6745 q^{64} +6.79006 q^{66} -3.50098 q^{67} -13.1196 q^{68} +7.34913 q^{69} -10.1638 q^{71} +19.6832 q^{72} +0.482272 q^{73} -17.7115 q^{74} -3.58485 q^{76} -0.269449 q^{77} -10.2529 q^{78} -1.93576 q^{79} +2.85268 q^{81} +26.4731 q^{82} +4.76907 q^{83} -2.77534 q^{84} +8.55154 q^{86} -1.64500 q^{87} -3.74537 q^{88} +0.150946 q^{89} +0.406864 q^{91} -9.16937 q^{92} -16.7228 q^{93} -32.2156 q^{94} +10.1053 q^{96} +1.43635 q^{97} -16.3710 q^{98} -5.25535 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{3} + 8 q^{4} + 2 q^{6} - 5 q^{7} + 9 q^{9} - 6 q^{11} - 19 q^{12} + 9 q^{13} + 18 q^{14} + 4 q^{16} + 5 q^{17} - 2 q^{18} - 6 q^{19} + 3 q^{21} - 8 q^{23} - 7 q^{24} - 22 q^{26} - 30 q^{27} - 10 q^{28}+ \cdots - 9 q^{99}+O(q^{100}) $ 6 * q - 3 * q^3 + 8 * q^4 + 2 * q^6 - 5 * q^7 + 9 * q^9 - 6 * q^11 - 19 * q^12 + 9 * q^13 + 18 * q^14 + 4 * q^16 + 5 * q^17 - 2 * q^18 - 6 * q^19 + 3 * q^21 - 8 * q^23 - 7 * q^24 - 22 * q^26 - 30 * q^27 - 10 * q^28 - 5 * q^29 - q^31 - 15 * q^32 + 3 * q^33 - 22 * q^34 + 12 * q^36 - 9 * q^37 - 32 * q^39 + 25 * q^41 - 11 * q^42 - 15 * q^43 - 8 * q^44 - 16 * q^46 - 24 * q^47 + 4 * q^48 + 13 * q^49 + 27 * q^52 - 5 * q^53 - 11 * q^54 - 12 * q^56 + 3 * q^57 - 13 * q^58 + 39 * q^59 - 11 * q^61 + 42 * q^62 - 38 * q^63 - 14 * q^64 - 2 * q^66 - 24 * q^67 - 45 * q^68 + 14 * q^69 - 24 * q^71 + 61 * q^72 + 26 * q^73 + q^74 - 8 * q^76 + 5 * q^77 + 29 * q^78 + 11 * q^79 + 30 * q^81 - 8 * q^82 - 39 * q^83 + 25 * q^84 + 18 * q^86 + 16 * q^87 + 22 * q^89 - 26 * q^91 + 11 * q^92 + 6 * q^93 - 30 * q^94 - 15 * q^96 - 22 * q^97 - 33 * q^98 - 9 * 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 $

$2$	2.36323	1.67106	0.835528	−	0.549448i	$-0.185162\pi$
$2$	2.36323	1.67106	0.835528	+	0.549448i	$0.185162\pi$
$3$	−2.87321	−1.65885	−0.829425	−	0.558618i	$-0.811332\pi$
$3$	−2.87321	−1.65885	−0.829425	+	0.558618i	$0.811332\pi$
$4$	3.58485	1.79243
$5$	0	0
$5$	0	0
$6$	−6.79006	−2.77203
$7$	0.269449	0.101842	0.0509212	−	0.998703i	$-0.483784\pi$
$7$	0.269449	0.101842	0.0509212	+	0.998703i	$0.483784\pi$
$8$	3.74537	1.32419
$9$	5.25535	1.75178
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−10.3000	−2.97337
$13$	1.50998	0.418794	0.209397	−	0.977831i	$-0.432850\pi$
$13$	1.50998	0.418794	0.209397	+	0.977831i	$0.432850\pi$
$14$	0.636771	0.170184
$15$	0	0
$16$	1.68146	0.420366
$17$	−3.65972	−0.887613	−0.443806	−	0.896123i	$-0.646372\pi$
$17$	−3.65972	−0.887613	−0.443806	+	0.896123i	$0.646372\pi$
$18$	12.4196	2.92733
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−0.774186	−0.168941
$22$	−2.36323	−0.503842
$23$	−2.55781	−0.533340	−0.266670	−	0.963788i	$-0.585923\pi$
$23$	−2.55781	−0.533340	−0.266670	+	0.963788i	$0.585923\pi$
$24$	−10.7612	−2.19663
$25$	0	0
$26$	3.56844	0.699828
$27$	−6.48011	−1.24710
$28$	0.965936	0.182545
$29$	0.572530	0.106316	0.0531581	−	0.998586i	$-0.483071\pi$
$29$	0.572530	0.106316	0.0531581	+	0.998586i	$0.483071\pi$
$30$	0	0
$31$	5.82024	1.04535	0.522673	−	0.852533i	$-0.324935\pi$
$31$	5.82024	1.04535	0.522673	+	0.852533i	$0.324935\pi$
$32$	−3.51706	−0.621734
$33$	2.87321	0.500162
$34$	−8.64876	−1.48325
$35$	0	0
$36$	18.8397	3.13994
$37$	−7.49463	−1.23211	−0.616054	−	0.787704i	$-0.711270\pi$
$37$	−7.49463	−1.23211	−0.616054	+	0.787704i	$0.711270\pi$
$38$	−2.36323	−0.383366
$39$	−4.33851	−0.694717
$40$	0	0
$41$	11.2021	1.74947	0.874735	−	0.484602i	$-0.161035\pi$
$41$	11.2021	1.74947	0.874735	+	0.484602i	$0.161035\pi$
$42$	−1.82958	−0.282310
$43$	3.61858	0.551829	0.275914	−	0.961182i	$-0.411019\pi$
$43$	3.61858	0.551829	0.275914	+	0.961182i	$0.411019\pi$
$44$	−3.58485	−0.540437
$45$	0	0
$46$	−6.04469	−0.891241
$47$	−13.6320	−1.98844	−0.994219	−	0.107373i	$-0.965756\pi$
$47$	−13.6320	−1.98844	−0.994219	+	0.107373i	$0.965756\pi$
$48$	−4.83120	−0.697324
$49$	−6.92740	−0.989628
$50$	0	0
$51$	10.5152	1.47242
$52$	5.41307	0.750658
$53$	3.41181	0.468648	0.234324	−	0.972159i	$-0.424712\pi$
$53$	3.41181	0.468648	0.234324	+	0.972159i	$0.424712\pi$
$54$	−15.3140	−2.08397
$55$	0	0
$56$	1.00919	0.134858
$57$	2.87321	0.380566
$58$	1.35302	0.177660
$59$	1.76356	0.229596	0.114798	−	0.993389i	$-0.463378\pi$
$59$	1.76356	0.229596	0.114798	+	0.993389i	$0.463378\pi$
$60$	0	0
$61$	−11.4676	−1.46828	−0.734140	−	0.678998i	$-0.762415\pi$
$61$	−11.4676	−1.46828	−0.734140	+	0.678998i	$0.762415\pi$
$62$	13.7546	1.74683
$63$	1.41605	0.178406
$64$	−11.6745	−1.45932
$65$	0	0
$66$	6.79006	0.835799
$67$	−3.50098	−0.427712	−0.213856	−	0.976865i	$-0.568602\pi$
$67$	−3.50098	−0.427712	−0.213856	+	0.976865i	$0.568602\pi$
$68$	−13.1196	−1.59098
$69$	7.34913	0.884732
$70$	0	0
$71$	−10.1638	−1.20622	−0.603109	−	0.797659i	$-0.706071\pi$
$71$	−10.1638	−1.20622	−0.603109	+	0.797659i	$0.706071\pi$
$72$	19.6832	2.31969
$73$	0.482272	0.0564456	0.0282228	−	0.999602i	$-0.491015\pi$
$73$	0.482272	0.0564456	0.0282228	+	0.999602i	$0.491015\pi$
$74$	−17.7115	−2.05892
$75$	0	0
$76$	−3.58485	−0.411211
$77$	−0.269449	−0.0307066
$78$	−10.2529	−1.16091
$79$	−1.93576	−0.217790	−0.108895	−	0.994053i	$-0.534731\pi$
$79$	−1.93576	−0.217790	−0.108895	+	0.994053i	$0.534731\pi$
$80$	0	0
$81$	2.85268	0.316964
$82$	26.4731	2.92346
$83$	4.76907	0.523474	0.261737	−	0.965139i	$-0.415705\pi$
$83$	4.76907	0.523474	0.261737	+	0.965139i	$0.415705\pi$
$84$	−2.77534	−0.302815
$85$	0	0
$86$	8.55154	0.922136
$87$	−1.64500	−0.176363
$88$	−3.74537	−0.399258
$89$	0.150946	0.0160002	0.00800011	−	0.999968i	$-0.497453\pi$
$89$	0.150946	0.0160002	0.00800011	+	0.999968i	$0.497453\pi$
$90$	0	0
$91$	0.406864	0.0426510
$92$	−9.16937	−0.955973
$93$	−16.7228	−1.73407
$94$	−32.2156	−3.32279
$95$	0	0
$96$	10.1053	1.03136
$97$	1.43635	0.145839	0.0729196	−	0.997338i	$-0.476768\pi$
$97$	1.43635	0.145839	0.0729196	+	0.997338i	$0.476768\pi$
$98$	−16.3710	−1.65372
$99$	−5.25535	−0.528183
$100$	0	0
$101$	−5.48472	−0.545750	−0.272875	−	0.962049i	$-0.587975\pi$
$101$	−5.48472	−0.545750	−0.272875	+	0.962049i	$0.587975\pi$
$102$	24.8497	2.46049
$103$	10.2028	1.00531	0.502656	−	0.864487i	$-0.332356\pi$
$103$	10.2028	1.00531	0.502656	+	0.864487i	$0.332356\pi$
$104$	5.65545	0.554562
$105$	0	0
$106$	8.06288	0.783136
$107$	−12.3928	−1.19805	−0.599027	−	0.800729i	$-0.704446\pi$
$107$	−12.3928	−1.19805	−0.599027	+	0.800729i	$0.704446\pi$
$108$	−23.2302	−2.23533
$109$	−3.96699	−0.379969	−0.189985	−	0.981787i	$-0.560844\pi$
$109$	−3.96699	−0.379969	−0.189985	+	0.981787i	$0.560844\pi$
$110$	0	0
$111$	21.5337	2.04388
$112$	0.453069	0.0428110
$113$	−2.08887	−0.196504	−0.0982520	−	0.995162i	$-0.531325\pi$
$113$	−2.08887	−0.196504	−0.0982520	+	0.995162i	$0.531325\pi$
$114$	6.79006	0.635948
$115$	0	0
$116$	2.05244	0.190564
$117$	7.93550	0.733637
$118$	4.16769	0.383667
$119$	−0.986110	−0.0903965
$120$	0	0
$121$	1.00000	0.0909091
$122$	−27.1007	−2.45358
$123$	−32.1859	−2.90211
$124$	20.8647	1.87370
$125$	0	0
$126$	3.34646	0.298126
$127$	−11.6605	−1.03470	−0.517350	−	0.855774i	$-0.673081\pi$
$127$	−11.6605	−1.03470	−0.517350	+	0.855774i	$0.673081\pi$
$128$	−20.5555	−1.81687
$129$	−10.3970	−0.915401
$130$	0	0
$131$	2.52155	0.220309	0.110155	−	0.993914i	$-0.464865\pi$
$131$	2.52155	0.220309	0.110155	+	0.993914i	$0.464865\pi$
$132$	10.3000	0.896504
$133$	−0.269449	−0.0233642
$134$	−8.27361	−0.714731
$135$	0	0
$136$	−13.7070	−1.17537
$137$	9.53639	0.814749	0.407374	−	0.913261i	$-0.366444\pi$
$137$	9.53639	0.814749	0.407374	+	0.913261i	$0.366444\pi$
$138$	17.3677	1.47844
$139$	17.0241	1.44396	0.721982	−	0.691911i	$-0.243231\pi$
$139$	17.0241	1.44396	0.721982	+	0.691911i	$0.243231\pi$
$140$	0	0
$141$	39.1678	3.29852
$142$	−24.0193	−2.01566
$143$	−1.50998	−0.126271
$144$	8.83668	0.736390
$145$	0	0
$146$	1.13972	0.0943238
$147$	19.9039	1.64165
$148$	−26.8671	−2.20846
$149$	5.02421	0.411600	0.205800	−	0.978594i	$-0.434020\pi$
$149$	5.02421	0.411600	0.205800	+	0.978594i	$0.434020\pi$
$150$	0	0
$151$	−22.1631	−1.80361	−0.901806	−	0.432142i	$-0.857758\pi$
$151$	−22.1631	−1.80361	−0.901806	+	0.432142i	$0.857758\pi$
$152$	−3.74537	−0.303790
$153$	−19.2331	−1.55491
$154$	−0.636771	−0.0513124
$155$	0	0
$156$	−15.5529	−1.24523
$157$	13.2234	1.05534	0.527671	−	0.849449i	$-0.323065\pi$
$157$	13.2234	1.05534	0.527671	+	0.849449i	$0.323065\pi$
$158$	−4.57464	−0.363939
$159$	−9.80285	−0.777417
$160$	0	0
$161$	−0.689200	−0.0543166
$162$	6.74154	0.529665
$163$	−17.0772	−1.33759	−0.668795	−	0.743447i	$-0.733190\pi$
$163$	−17.0772	−1.33759	−0.668795	+	0.743447i	$0.733190\pi$
$164$	40.1578	3.13580
$165$	0	0
$166$	11.2704	0.874753
$167$	−8.08784	−0.625856	−0.312928	−	0.949777i	$-0.601310\pi$
$167$	−8.08784	−0.625856	−0.312928	+	0.949777i	$0.601310\pi$
$168$	−2.89961	−0.223710
$169$	−10.7199	−0.824611
$170$	0	0
$171$	−5.25535	−0.401887
$172$	12.9721	0.989112
$173$	−1.93735	−0.147294	−0.0736472	−	0.997284i	$-0.523464\pi$
$173$	−1.93735	−0.147294	−0.0736472	+	0.997284i	$0.523464\pi$
$174$	−3.88751	−0.294712
$175$	0	0
$176$	−1.68146	−0.126745
$177$	−5.06708	−0.380865
$178$	0.356719	0.0267373
$179$	4.24521	0.317302	0.158651	−	0.987335i	$-0.449286\pi$
$179$	4.24521	0.317302	0.158651	+	0.987335i	$0.449286\pi$
$180$	0	0
$181$	−15.7971	−1.17419	−0.587096	−	0.809517i	$-0.699729\pi$
$181$	−15.7971	−1.17419	−0.587096	+	0.809517i	$0.699729\pi$
$182$	0.961514	0.0712721
$183$	32.9490	2.43566
$184$	−9.57994	−0.706243
$185$	0	0
$186$	−39.5198	−2.89773
$187$	3.65972	0.267625
$188$	−48.8689	−3.56413
$189$	−1.74606	−0.127007
$190$	0	0
$191$	−17.6702	−1.27857	−0.639285	−	0.768969i	$-0.720770\pi$
$191$	−17.6702	−1.27857	−0.639285	+	0.768969i	$0.720770\pi$
$192$	33.5434	2.42079
$193$	2.61056	0.187912	0.0939561	−	0.995576i	$-0.470049\pi$
$193$	2.61056	0.187912	0.0939561	+	0.995576i	$0.470049\pi$
$194$	3.39442	0.243705
$195$	0	0
$196$	−24.8337	−1.77384
$197$	−8.58931	−0.611963	−0.305982	−	0.952037i	$-0.598985\pi$
$197$	−8.58931	−0.611963	−0.305982	+	0.952037i	$0.598985\pi$
$198$	−12.4196	−0.882623
$199$	−26.1761	−1.85557	−0.927786	−	0.373113i	$-0.878290\pi$
$199$	−26.1761	−1.85557	−0.927786	+	0.373113i	$0.878290\pi$
$200$	0	0
$201$	10.0591	0.709511
$202$	−12.9617	−0.911979
$203$	0.154268	0.0108275
$204$	37.6953	2.63920
$205$	0	0
$206$	24.1115	1.67993
$207$	−13.4422	−0.934297
$208$	2.53898	0.176047
$209$	1.00000	0.0691714
$210$	0	0
$211$	−17.4084	−1.19845	−0.599223	−	0.800582i	$-0.704524\pi$
$211$	−17.4084	−1.19845	−0.599223	+	0.800582i	$0.704524\pi$
$212$	12.2308	0.840017
$213$	29.2027	2.00093
$214$	−29.2870	−2.00201
$215$	0	0
$216$	−24.2704	−1.65139
$217$	1.56826	0.106460
$218$	−9.37491	−0.634949
$219$	−1.38567	−0.0936349
$220$	0	0
$221$	−5.52612	−0.371727
$222$	50.8890	3.41544
$223$	−13.6764	−0.915840	−0.457920	−	0.888993i	$-0.651405\pi$
$223$	−13.6764	−0.915840	−0.457920	+	0.888993i	$0.651405\pi$
$224$	−0.947669	−0.0633188
$225$	0	0
$226$	−4.93647	−0.328369
$227$	−11.8910	−0.789232	−0.394616	−	0.918846i	$-0.629122\pi$
$227$	−11.8910	−0.789232	−0.394616	+	0.918846i	$0.629122\pi$
$228$	10.3000	0.682137
$229$	9.56459	0.632046	0.316023	−	0.948752i	$-0.397652\pi$
$229$	9.56459	0.632046	0.316023	+	0.948752i	$0.397652\pi$
$230$	0	0
$231$	0.774186	0.0509377
$232$	2.14434	0.140783
$233$	−2.75254	−0.180325	−0.0901626	−	0.995927i	$-0.528739\pi$
$233$	−2.75254	−0.180325	−0.0901626	+	0.995927i	$0.528739\pi$
$234$	18.7534	1.22595
$235$	0	0
$236$	6.32209	0.411533
$237$	5.56185	0.361281
$238$	−2.33040	−0.151058
$239$	−12.8426	−0.830716	−0.415358	−	0.909658i	$-0.636344\pi$
$239$	−12.8426	−0.830716	−0.415358	+	0.909658i	$0.636344\pi$
$240$	0	0
$241$	2.50671	0.161471	0.0807356	−	0.996736i	$-0.474273\pi$
$241$	2.50671	0.161471	0.0807356	+	0.996736i	$0.474273\pi$
$242$	2.36323	0.151914
$243$	11.2440	0.721301
$244$	−41.1098	−2.63178
$245$	0	0
$246$	−76.0628	−4.84959
$247$	−1.50998	−0.0960780
$248$	21.7989	1.38423
$249$	−13.7026	−0.868364
$250$	0	0
$251$	−10.0939	−0.637123	−0.318562	−	0.947902i	$-0.603200\pi$
$251$	−10.0939	−0.637123	−0.318562	+	0.947902i	$0.603200\pi$
$252$	5.07634	0.319779
$253$	2.55781	0.160808
$254$	−27.5564	−1.72904
$255$	0	0
$256$	−25.2283	−1.57677
$257$	17.4000	1.08538	0.542691	−	0.839932i	$-0.317405\pi$
$257$	17.4000	1.08538	0.542691	+	0.839932i	$0.317405\pi$
$258$	−24.5704	−1.52969
$259$	−2.01942	−0.125481
$260$	0	0
$261$	3.00885	0.186243
$262$	5.95901	0.368149
$263$	−4.18923	−0.258319	−0.129159	−	0.991624i	$-0.541228\pi$
$263$	−4.18923	−0.258319	−0.129159	+	0.991624i	$0.541228\pi$
$264$	10.7612	0.662309
$265$	0	0
$266$	−0.636771	−0.0390429
$267$	−0.433699	−0.0265420
$268$	−12.5505	−0.766643
$269$	−3.64105	−0.221999	−0.110999	−	0.993820i	$-0.535405\pi$
$269$	−3.64105	−0.221999	−0.110999	+	0.993820i	$0.535405\pi$
$270$	0	0
$271$	−22.1472	−1.34535	−0.672674	−	0.739939i	$-0.734854\pi$
$271$	−22.1472	−1.34535	−0.672674	+	0.739939i	$0.734854\pi$
$272$	−6.15368	−0.373122
$273$	−1.16901	−0.0707516
$274$	22.5367	1.36149
$275$	0	0
$276$	26.3456	1.58582
$277$	−3.32919	−0.200032	−0.100016	−	0.994986i	$-0.531889\pi$
$277$	−3.32919	−0.200032	−0.100016	+	0.994986i	$0.531889\pi$
$278$	40.2318	2.41294
$279$	30.5874	1.83122
$280$	0	0
$281$	26.8500	1.60174	0.800868	−	0.598841i	$-0.204372\pi$
$281$	26.8500	1.60174	0.800868	+	0.598841i	$0.204372\pi$
$282$	92.5624	5.51201
$283$	10.7579	0.639489	0.319745	−	0.947504i	$-0.396403\pi$
$283$	10.7579	0.639489	0.319745	+	0.947504i	$0.396403\pi$
$284$	−36.4356	−2.16206
$285$	0	0
$286$	−3.56844	−0.211006
$287$	3.01839	0.178170
$288$	−18.4834	−1.08914
$289$	−3.60644	−0.212144
$290$	0	0
$291$	−4.12694	−0.241925
$292$	1.72887	0.101175
$293$	19.3755	1.13193	0.565964	−	0.824430i	$-0.308504\pi$
$293$	19.3755	1.13193	0.565964	+	0.824430i	$0.308504\pi$
$294$	47.0375	2.74328
$295$	0	0
$296$	−28.0701	−1.63154
$297$	6.48011	0.376014
$298$	11.8734	0.687806
$299$	−3.86225	−0.223360
$300$	0	0
$301$	0.975025	0.0561995
$302$	−52.3766	−3.01393
$303$	15.7588	0.905318
$304$	−1.68146	−0.0964385
$305$	0	0
$306$	−45.4523	−2.59833
$307$	−26.8951	−1.53499	−0.767493	−	0.641058i	$-0.778496\pi$
$307$	−26.8951	−1.53499	−0.767493	+	0.641058i	$0.778496\pi$
$308$	−0.965936	−0.0550393
$309$	−29.3148	−1.66766
$310$	0	0
$311$	7.36426	0.417589	0.208794	−	0.977960i	$-0.433046\pi$
$311$	7.36426	0.417589	0.208794	+	0.977960i	$0.433046\pi$
$312$	−16.2493	−0.919936
$313$	17.3778	0.982254	0.491127	−	0.871088i	$-0.336585\pi$
$313$	17.3778	0.982254	0.491127	+	0.871088i	$0.336585\pi$
$314$	31.2499	1.76354
$315$	0	0
$316$	−6.93941	−0.390372
$317$	15.4600	0.868321	0.434161	−	0.900835i	$-0.357045\pi$
$317$	15.4600	0.868321	0.434161	+	0.900835i	$0.357045\pi$
$318$	−23.1664	−1.29911
$319$	−0.572530	−0.0320555
$320$	0	0
$321$	35.6071	1.98739
$322$	−1.62874	−0.0907661
$323$	3.65972	0.203632
$324$	10.2264	0.568135
$325$	0	0
$326$	−40.3574	−2.23519
$327$	11.3980	0.630312
$328$	41.9559	2.31663
$329$	−3.67315	−0.202507
$330$	0	0
$331$	35.6299	1.95840	0.979198	−	0.202905i	$-0.0650382\pi$
$331$	35.6299	1.95840	0.979198	+	0.202905i	$0.0650382\pi$
$332$	17.0964	0.938288
$333$	−39.3869	−2.15839
$334$	−19.1134	−1.04584
$335$	0	0
$336$	−1.30176	−0.0710170
$337$	0.207036	0.0112780	0.00563898	−	0.999984i	$-0.498205\pi$
$337$	0.207036	0.0112780	0.00563898	+	0.999984i	$0.498205\pi$
$338$	−25.3337	−1.37797
$339$	6.00176	0.325971
$340$	0	0
$341$	−5.82024	−0.315184
$342$	−12.4196	−0.671575
$343$	−3.75273	−0.202628
$344$	13.5529	0.730725
$345$	0	0
$346$	−4.57841	−0.246137
$347$	−7.01279	−0.376467	−0.188233	−	0.982124i	$-0.560276\pi$
$347$	−7.01279	−0.376467	−0.188233	+	0.982124i	$0.560276\pi$
$348$	−5.89709	−0.316117
$349$	−18.5946	−0.995348	−0.497674	−	0.867364i	$-0.665812\pi$
$349$	−18.5946	−0.995348	−0.497674	+	0.867364i	$0.665812\pi$
$350$	0	0
$351$	−9.78486	−0.522277
$352$	3.51706	0.187460
$353$	28.2404	1.50308	0.751541	−	0.659686i	$-0.229311\pi$
$353$	28.2404	1.50308	0.751541	+	0.659686i	$0.229311\pi$
$354$	−11.9747	−0.636446
$355$	0	0
$356$	0.541118	0.0286792
$357$	2.83330	0.149954
$358$	10.0324	0.530229
$359$	−29.2222	−1.54229	−0.771143	−	0.636661i	$-0.780315\pi$
$359$	−29.2222	−1.54229	−0.771143	+	0.636661i	$0.780315\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−37.3323	−1.96214
$363$	−2.87321	−0.150805
$364$	1.45855	0.0764487
$365$	0	0
$366$	77.8660	4.07012
$367$	20.0779	1.04806	0.524028	−	0.851701i	$-0.324429\pi$
$367$	20.0779	1.04806	0.524028	+	0.851701i	$0.324429\pi$
$368$	−4.30086	−0.224198
$369$	58.8709	3.06469
$370$	0	0
$371$	0.919310	0.0477282
$372$	−59.9487	−3.10820
$373$	−32.7276	−1.69457	−0.847284	−	0.531139i	$-0.821764\pi$
$373$	−32.7276	−1.69457	−0.847284	+	0.531139i	$0.821764\pi$
$374$	8.64876	0.447217
$375$	0	0
$376$	−51.0570	−2.63307
$377$	0.864511	0.0445246
$378$	−4.12635	−0.212236
$379$	20.4080	1.04829	0.524144	−	0.851629i	$-0.324385\pi$
$379$	20.4080	1.04829	0.524144	+	0.851629i	$0.324385\pi$
$380$	0	0
$381$	33.5030	1.71641
$382$	−41.7587	−2.13656
$383$	32.7540	1.67365	0.836824	−	0.547471i	$-0.184410\pi$
$383$	32.7540	1.67365	0.836824	+	0.547471i	$0.184410\pi$
$384$	59.0603	3.01391
$385$	0	0
$386$	6.16935	0.314012
$387$	19.0169	0.966685
$388$	5.14910	0.261406
$389$	30.7107	1.55710	0.778548	−	0.627585i	$-0.215956\pi$
$389$	30.7107	1.55710	0.778548	+	0.627585i	$0.215956\pi$
$390$	0	0
$391$	9.36087	0.473400
$392$	−25.9457	−1.31045
$393$	−7.24496	−0.365460
$394$	−20.2985	−1.02262
$395$	0	0
$396$	−18.8397	−0.946729
$397$	−14.5208	−0.728779	−0.364390	−	0.931247i	$-0.618722\pi$
$397$	−14.5208	−0.728779	−0.364390	+	0.931247i	$0.618722\pi$
$398$	−61.8600	−3.10076
$399$	0.774186	0.0387578
$400$	0	0
$401$	−9.20455	−0.459653	−0.229827	−	0.973232i	$-0.573816\pi$
$401$	−9.20455	−0.459653	−0.229827	+	0.973232i	$0.573816\pi$
$402$	23.7719	1.18563
$403$	8.78847	0.437785
$404$	−19.6619	−0.978217
$405$	0	0
$406$	0.364570	0.0180933
$407$	7.49463	0.371495
$408$	39.3832	1.94976
$409$	23.8193	1.17779	0.588894	−	0.808210i	$-0.299563\pi$
$409$	23.8193	1.17779	0.588894	+	0.808210i	$0.299563\pi$
$410$	0	0
$411$	−27.4001	−1.35155
$412$	36.5755	1.80195
$413$	0.475190	0.0233826
$414$	−31.7670	−1.56126
$415$	0	0
$416$	−5.31070	−0.260379
$417$	−48.9138	−2.39532
$418$	2.36323	0.115589
$419$	6.91381	0.337761	0.168881	−	0.985636i	$-0.445985\pi$
$419$	6.91381	0.337761	0.168881	+	0.985636i	$0.445985\pi$
$420$	0	0
$421$	4.96199	0.241833	0.120916	−	0.992663i	$-0.461417\pi$
$421$	4.96199	0.241833	0.120916	+	0.992663i	$0.461417\pi$
$422$	−41.1401	−2.00267
$423$	−71.6412	−3.48331
$424$	12.7785	0.620578
$425$	0	0
$426$	69.0126	3.34367
$427$	−3.08995	−0.149533
$428$	−44.4262	−2.14742
$429$	4.33851	0.209465
$430$	0	0
$431$	−29.4702	−1.41953	−0.709764	−	0.704439i	$-0.751198\pi$
$431$	−29.4702	−1.41953	−0.709764	+	0.704439i	$0.751198\pi$
$432$	−10.8961	−0.524237
$433$	−15.1549	−0.728299	−0.364149	−	0.931341i	$-0.618640\pi$
$433$	−15.1549	−0.728299	−0.364149	+	0.931341i	$0.618640\pi$
$434$	3.70616	0.177901
$435$	0	0
$436$	−14.2211	−0.681066
$437$	2.55781	0.122357
$438$	−3.27465	−0.156469
$439$	−26.0950	−1.24545	−0.622724	−	0.782442i	$-0.713974\pi$
$439$	−26.0950	−1.24545	−0.622724	+	0.782442i	$0.713974\pi$
$440$	0	0
$441$	−36.4059	−1.73362
$442$	−13.0595	−0.621176
$443$	39.2407	1.86438	0.932191	−	0.361966i	$-0.117894\pi$
$443$	39.2407	1.86438	0.932191	+	0.361966i	$0.117894\pi$
$444$	77.1950	3.66351
$445$	0	0
$446$	−32.3205	−1.53042
$447$	−14.4356	−0.682782
$448$	−3.14570	−0.148620
$449$	−15.0258	−0.709112	−0.354556	−	0.935035i	$-0.615368\pi$
$449$	−15.0258	−0.709112	−0.354556	+	0.935035i	$0.615368\pi$
$450$	0	0
$451$	−11.2021	−0.527485
$452$	−7.48827	−0.352219
$453$	63.6794	2.99192
$454$	−28.1011	−1.31885
$455$	0	0
$456$	10.7612	0.503941
$457$	38.5344	1.80256	0.901282	−	0.433233i	$-0.142627\pi$
$457$	38.5344	1.80256	0.901282	+	0.433233i	$0.142627\pi$
$458$	22.6033	1.05618
$459$	23.7154	1.10694
$460$	0	0
$461$	33.5770	1.56384	0.781918	−	0.623381i	$-0.214242\pi$
$461$	33.5770	1.56384	0.781918	+	0.623381i	$0.214242\pi$
$462$	1.82958	0.0851197
$463$	22.5403	1.04753	0.523767	−	0.851861i	$-0.324526\pi$
$463$	22.5403	1.04753	0.523767	+	0.851861i	$0.324526\pi$
$464$	0.962688	0.0446917
$465$	0	0
$466$	−6.50489	−0.301333
$467$	−20.5656	−0.951664	−0.475832	−	0.879536i	$-0.657853\pi$
$467$	−20.5656	−0.951664	−0.475832	+	0.879536i	$0.657853\pi$
$468$	28.4476	1.31499
$469$	−0.943336	−0.0435592
$470$	0	0
$471$	−37.9937	−1.75066
$472$	6.60518	0.304028
$473$	−3.61858	−0.166383
$474$	13.1439	0.603720
$475$	0	0
$476$	−3.53506	−0.162029
$477$	17.9303	0.820970
$478$	−30.3499	−1.38817
$479$	−12.6051	−0.575942	−0.287971	−	0.957639i	$-0.592981\pi$
$479$	−12.6051	−0.575942	−0.287971	+	0.957639i	$0.592981\pi$
$480$	0	0
$481$	−11.3168	−0.516000
$482$	5.92393	0.269827
$483$	1.98022	0.0901031
$484$	3.58485	0.162948
$485$	0	0
$486$	26.5721	1.20533
$487$	−3.20539	−0.145250	−0.0726250	−	0.997359i	$-0.523138\pi$
$487$	−3.20539	−0.145250	−0.0726250	+	0.997359i	$0.523138\pi$
$488$	−42.9505	−1.94428
$489$	49.0665	2.21886
$490$	0	0
$491$	44.0578	1.98830	0.994151	−	0.107999i	$-0.0344444\pi$
$491$	44.0578	1.98830	0.994151	+	0.107999i	$0.0344444\pi$
$492$	−115.382	−5.20182
$493$	−2.09530	−0.0943676
$494$	−3.56844	−0.160552
$495$	0	0
$496$	9.78651	0.439427
$497$	−2.73862	−0.122844
$498$	−32.3823	−1.45109
$499$	2.95628	0.132341	0.0661706	−	0.997808i	$-0.478922\pi$
$499$	2.95628	0.132341	0.0661706	+	0.997808i	$0.478922\pi$
$500$	0	0
$501$	23.2381	1.03820
$502$	−23.8543	−1.06467
$503$	22.3611	0.997034	0.498517	−	0.866880i	$-0.333878\pi$
$503$	22.3611	0.997034	0.498517	+	0.866880i	$0.333878\pi$
$504$	5.30364	0.236243
$505$	0	0
$506$	6.04469	0.268719
$507$	30.8007	1.36791
$508$	−41.8011	−1.85462
$509$	−22.6466	−1.00379	−0.501897	−	0.864927i	$-0.667364\pi$
$509$	−22.6466	−1.00379	−0.501897	+	0.864927i	$0.667364\pi$
$510$	0	0
$511$	0.129948	0.00574855
$512$	−18.5092	−0.817999
$513$	6.48011	0.286104
$514$	41.1202	1.81373
$515$	0	0
$516$	−37.2716	−1.64079
$517$	13.6320	0.599536
$518$	−4.77236	−0.209685
$519$	5.56643	0.244339
$520$	0	0
$521$	8.55935	0.374992	0.187496	−	0.982265i	$-0.439963\pi$
$521$	8.55935	0.374992	0.187496	+	0.982265i	$0.439963\pi$
$522$	7.11060	0.311222
$523$	24.2895	1.06210	0.531052	−	0.847339i	$-0.321797\pi$
$523$	24.2895	1.06210	0.531052	+	0.847339i	$0.321797\pi$
$524$	9.03940	0.394888
$525$	0	0
$526$	−9.90010	−0.431665
$527$	−21.3004	−0.927862
$528$	4.83120	0.210251
$529$	−16.4576	−0.715548
$530$	0	0
$531$	9.26812	0.402202
$532$	−0.965936	−0.0418787
$533$	16.9150	0.732668
$534$	−1.02493	−0.0443531
$535$	0	0
$536$	−13.1125	−0.566372
$537$	−12.1974	−0.526357
$538$	−8.60463	−0.370972
$539$	6.92740	0.298384
$540$	0	0
$541$	−16.3282	−0.702003	−0.351001	−	0.936375i	$-0.614159\pi$
$541$	−16.3282	−0.702003	−0.351001	+	0.936375i	$0.614159\pi$
$542$	−52.3390	−2.24815
$543$	45.3885	1.94781
$544$	12.8715	0.551859
$545$	0	0
$546$	−2.76263	−0.118230
$547$	30.9668	1.32404	0.662022	−	0.749485i	$-0.269699\pi$
$547$	30.9668	1.32404	0.662022	+	0.749485i	$0.269699\pi$
$548$	34.1865	1.46038
$549$	−60.2665	−2.57211
$550$	0	0
$551$	−0.572530	−0.0243906
$552$	27.5252	1.17155
$553$	−0.521589	−0.0221802
$554$	−7.86764	−0.334264
$555$	0	0
$556$	61.0289	2.58820
$557$	43.5226	1.84411	0.922055	−	0.387058i	$-0.126509\pi$
$557$	43.5226	1.84411	0.922055	+	0.387058i	$0.126509\pi$
$558$	72.2851	3.06007
$559$	5.46400	0.231103
$560$	0	0
$561$	−10.5152	−0.443950
$562$	63.4527	2.67659
$563$	16.5384	0.697010	0.348505	−	0.937307i	$-0.386689\pi$
$563$	16.5384	0.697010	0.348505	+	0.937307i	$0.386689\pi$
$564$	140.411	5.91235
$565$	0	0
$566$	25.4233	1.06862
$567$	0.768653	0.0322804
$568$	−38.0671	−1.59726
$569$	44.4882	1.86504	0.932522	−	0.361113i	$-0.117603\pi$
$569$	44.4882	1.86504	0.932522	+	0.361113i	$0.117603\pi$
$570$	0	0
$571$	40.2470	1.68429	0.842143	−	0.539255i	$-0.181294\pi$
$571$	40.2470	1.68429	0.842143	+	0.539255i	$0.181294\pi$
$572$	−5.41307	−0.226332
$573$	50.7703	2.12096
$574$	7.13315	0.297732
$575$	0	0
$576$	−61.3538	−2.55641
$577$	7.74126	0.322273	0.161137	−	0.986932i	$-0.448484\pi$
$577$	7.74126	0.322273	0.161137	+	0.986932i	$0.448484\pi$
$578$	−8.52285	−0.354504
$579$	−7.50069	−0.311718
$580$	0	0
$581$	1.28502	0.0533118
$582$	−9.75290	−0.404271
$583$	−3.41181	−0.141303
$584$	1.80629	0.0747446
$585$	0	0
$586$	45.7887	1.89151
$587$	−14.7960	−0.610697	−0.305349	−	0.952241i	$-0.598773\pi$
$587$	−14.7960	−0.610697	−0.305349	+	0.952241i	$0.598773\pi$
$588$	71.3525	2.94253
$589$	−5.82024	−0.239819
$590$	0	0
$591$	24.6789	1.01516
$592$	−12.6019	−0.517936
$593$	27.4949	1.12908	0.564540	−	0.825406i	$-0.309054\pi$
$593$	27.4949	1.12908	0.564540	+	0.825406i	$0.309054\pi$
$594$	15.3140	0.628341
$595$	0	0
$596$	18.0111	0.737762
$597$	75.2094	3.07812
$598$	−9.12739	−0.373247
$599$	22.3573	0.913496	0.456748	−	0.889596i	$-0.349014\pi$
$599$	22.3573	0.913496	0.456748	+	0.889596i	$0.349014\pi$
$600$	0	0
$601$	4.91993	0.200688	0.100344	−	0.994953i	$-0.468006\pi$
$601$	4.91993	0.200688	0.100344	+	0.994953i	$0.468006\pi$
$602$	2.30421	0.0939125
$603$	−18.3989	−0.749260
$604$	−79.4516	−3.23284
$605$	0	0
$606$	37.2416	1.51284
$607$	−25.5414	−1.03669	−0.518346	−	0.855171i	$-0.673452\pi$
$607$	−25.5414	−1.03669	−0.518346	+	0.855171i	$0.673452\pi$
$608$	3.51706	0.142636
$609$	−0.443245	−0.0179612
$610$	0	0
$611$	−20.5842	−0.832746
$612$	−68.9479	−2.78705
$613$	34.4326	1.39072	0.695359	−	0.718663i	$-0.255245\pi$
$613$	34.4326	1.39072	0.695359	+	0.718663i	$0.255245\pi$
$614$	−63.5593	−2.56505
$615$	0	0
$616$	−1.00919	−0.0406613
$617$	20.3399	0.818855	0.409427	−	0.912343i	$-0.365729\pi$
$617$	20.3399	0.818855	0.409427	+	0.912343i	$0.365729\pi$
$618$	−69.2776	−2.78675
$619$	36.8315	1.48038	0.740192	−	0.672395i	$-0.234734\pi$
$619$	36.8315	1.48038	0.740192	+	0.672395i	$0.234734\pi$
$620$	0	0
$621$	16.5749	0.665128
$622$	17.4034	0.697814
$623$	0.0406723	0.00162950
$624$	−7.29503	−0.292035
$625$	0	0
$626$	41.0678	1.64140
$627$	−2.87321	−0.114745
$628$	47.4039	1.89162
$629$	27.4282	1.09364
$630$	0	0
$631$	−29.8615	−1.18877	−0.594384	−	0.804181i	$-0.702604\pi$
$631$	−29.8615	−1.18877	−0.594384	+	0.804181i	$0.702604\pi$
$632$	−7.25013	−0.288395
$633$	50.0182	1.98804
$634$	36.5356	1.45101
$635$	0	0
$636$	−35.1418	−1.39346
$637$	−10.4603	−0.414450
$638$	−1.35302	−0.0535666
$639$	−53.4142	−2.11303
$640$	0	0
$641$	8.93034	0.352727	0.176364	−	0.984325i	$-0.443567\pi$
$641$	8.93034	0.352727	0.176364	+	0.984325i	$0.443567\pi$
$642$	84.1477	3.32104
$643$	−35.6606	−1.40632	−0.703158	−	0.711034i	$-0.748227\pi$
$643$	−35.6606	−1.40632	−0.703158	+	0.711034i	$0.748227\pi$
$644$	−2.47068	−0.0973585
$645$	0	0
$646$	8.64876	0.340281
$647$	−44.1953	−1.73750	−0.868749	−	0.495253i	$-0.835075\pi$
$647$	−44.1953	−1.73750	−0.868749	+	0.495253i	$0.835075\pi$
$648$	10.6843	0.419721
$649$	−1.76356	−0.0692257
$650$	0	0
$651$	−4.50594	−0.176602
$652$	−61.2193	−2.39753
$653$	−34.4397	−1.34773	−0.673864	−	0.738855i	$-0.735367\pi$
$653$	−34.4397	−1.34773	−0.673864	+	0.738855i	$0.735367\pi$
$654$	26.9361	1.05329
$655$	0	0
$656$	18.8359	0.735417
$657$	2.53451	0.0988806
$658$	−8.68048	−0.338401
$659$	16.4152	0.639445	0.319722	−	0.947511i	$-0.396410\pi$
$659$	16.4152	0.639445	0.319722	+	0.947511i	$0.396410\pi$
$660$	0	0
$661$	16.6165	0.646307	0.323153	−	0.946347i	$-0.395257\pi$
$661$	16.6165	0.646307	0.323153	+	0.946347i	$0.395257\pi$
$662$	84.2016	3.27259
$663$	15.8777	0.616640
$664$	17.8619	0.693178
$665$	0	0
$666$	−93.0803	−3.60679
$667$	−1.46442	−0.0567027
$668$	−28.9937	−1.12180
$669$	39.2952	1.51924
$670$	0	0
$671$	11.4676	0.442703
$672$	2.72286	0.105036
$673$	−24.7365	−0.953522	−0.476761	−	0.879033i	$-0.658189\pi$
$673$	−24.7365	−0.953522	−0.476761	+	0.879033i	$0.658189\pi$
$674$	0.489273	0.0188461
$675$	0	0
$676$	−38.4294	−1.47806
$677$	2.57708	0.0990452	0.0495226	−	0.998773i	$-0.484230\pi$
$677$	2.57708	0.0990452	0.0495226	+	0.998773i	$0.484230\pi$
$678$	14.1835	0.544715
$679$	0.387024	0.0148526
$680$	0	0
$681$	34.1653	1.30922
$682$	−13.7546	−0.526689
$683$	23.7289	0.907961	0.453980	−	0.891012i	$-0.350004\pi$
$683$	23.7289	0.907961	0.453980	+	0.891012i	$0.350004\pi$
$684$	−18.8397	−0.720353
$685$	0	0
$686$	−8.86856	−0.338603
$687$	−27.4811	−1.04847
$688$	6.08451	0.231970
$689$	5.15177	0.196267
$690$	0	0
$691$	49.0740	1.86686	0.933432	−	0.358755i	$-0.116799\pi$
$691$	49.0740	1.86686	0.933432	+	0.358755i	$0.116799\pi$
$692$	−6.94513	−0.264014
$693$	−1.41605	−0.0537914
$694$	−16.5728	−0.629096
$695$	0	0
$696$	−6.16114	−0.233537
$697$	−40.9965	−1.55285
$698$	−43.9434	−1.66328
$699$	7.90864	0.299132
$700$	0	0
$701$	−38.2763	−1.44568	−0.722838	−	0.691017i	$-0.757163\pi$
$701$	−38.2763	−1.44568	−0.722838	+	0.691017i	$0.757163\pi$
$702$	−23.1239	−0.872754
$703$	7.49463	0.282665
$704$	11.6745	0.440001
$705$	0	0
$706$	66.7384	2.51173
$707$	−1.47785	−0.0555805
$708$	−18.1647	−0.682672
$709$	−35.1393	−1.31969	−0.659843	−	0.751404i	$-0.729377\pi$
$709$	−35.1393	−1.31969	−0.659843	+	0.751404i	$0.729377\pi$
$710$	0	0
$711$	−10.1731	−0.381521
$712$	0.565348	0.0211873
$713$	−14.8871	−0.557525
$714$	6.69575	0.250582
$715$	0	0
$716$	15.2185	0.568740
$717$	36.8994	1.37803
$718$	−69.0587	−2.57725
$719$	50.4256	1.88056	0.940279	−	0.340405i	$-0.110564\pi$
$719$	50.4256	1.88056	0.940279	+	0.340405i	$0.110564\pi$
$720$	0	0
$721$	2.74914	0.102383
$722$	2.36323	0.0879503
$723$	−7.20231	−0.267857
$724$	−56.6304	−2.10465
$725$	0	0
$726$	−6.79006	−0.252003
$727$	29.9437	1.11055	0.555275	−	0.831667i	$-0.312613\pi$
$727$	29.9437	1.11055	0.555275	+	0.831667i	$0.312613\pi$
$728$	1.52386	0.0564779
$729$	−40.8644	−1.51350
$730$	0	0
$731$	−13.2430	−0.489810
$732$	118.117	4.36574
$733$	−10.5029	−0.387934	−0.193967	−	0.981008i	$-0.562135\pi$
$733$	−10.5029	−0.387934	−0.193967	+	0.981008i	$0.562135\pi$
$734$	47.4486	1.75136
$735$	0	0
$736$	8.99597	0.331596
$737$	3.50098	0.128960
$738$	139.125	5.12127
$739$	−26.3721	−0.970114	−0.485057	−	0.874482i	$-0.661201\pi$
$739$	−26.3721	−0.970114	−0.485057	+	0.874482i	$0.661201\pi$
$740$	0	0
$741$	4.33851	0.159379
$742$	2.17254	0.0797564
$743$	−19.2435	−0.705975	−0.352988	−	0.935628i	$-0.614834\pi$
$743$	−19.2435	−0.705975	−0.352988	+	0.935628i	$0.614834\pi$
$744$	−62.6330	−2.29624
$745$	0	0
$746$	−77.3427	−2.83172
$747$	25.0632	0.917013
$748$	13.1196	0.479699
$749$	−3.33922	−0.122013
$750$	0	0
$751$	24.9991	0.912229	0.456114	−	0.889921i	$-0.349241\pi$
$751$	24.9991	0.912229	0.456114	+	0.889921i	$0.349241\pi$
$752$	−22.9218	−0.835871
$753$	29.0020	1.05689
$754$	2.04304	0.0744031
$755$	0	0
$756$	−6.25938	−0.227651
$757$	−8.23644	−0.299359	−0.149679	−	0.988735i	$-0.547824\pi$
$757$	−8.23644	−0.299359	−0.149679	+	0.988735i	$0.547824\pi$
$758$	48.2288	1.75175
$759$	−7.34913	−0.266757
$760$	0	0
$761$	9.52256	0.345192	0.172596	−	0.984993i	$-0.444784\pi$
$761$	9.52256	0.345192	0.172596	+	0.984993i	$0.444784\pi$
$762$	79.1753	2.86822
$763$	−1.06890	−0.0386969
$764$	−63.3451	−2.29174
$765$	0	0
$766$	77.4051	2.79676
$767$	2.66294	0.0961533
$768$	72.4862	2.61562
$769$	−20.5017	−0.739309	−0.369654	−	0.929169i	$-0.620524\pi$
$769$	−20.5017	−0.739309	−0.369654	+	0.929169i	$0.620524\pi$
$770$	0	0
$771$	−49.9939	−1.80049
$772$	9.35847	0.336819
$773$	−14.0912	−0.506824	−0.253412	−	0.967359i	$-0.581553\pi$
$773$	−14.0912	−0.506824	−0.253412	+	0.967359i	$0.581553\pi$
$774$	44.9414	1.61538
$775$	0	0
$776$	5.37966	0.193119
$777$	5.80223	0.208154
$778$	72.5765	2.60200
$779$	−11.2021	−0.401356
$780$	0	0
$781$	10.1638	0.363688
$782$	22.1219	0.791077
$783$	−3.71006	−0.132587
$784$	−11.6482	−0.416006
$785$	0	0
$786$	−17.1215	−0.610704
$787$	−11.7044	−0.417218	−0.208609	−	0.977999i	$-0.566894\pi$
$787$	−11.7044	−0.417218	−0.208609	+	0.977999i	$0.566894\pi$
$788$	−30.7914	−1.09690
$789$	12.0365	0.428512
$790$	0	0
$791$	−0.562844	−0.0200124
$792$	−19.6832	−0.699414
$793$	−17.3159	−0.614907
$794$	−34.3160	−1.21783
$795$	0	0
$796$	−93.8373	−3.32598
$797$	−10.7297	−0.380066	−0.190033	−	0.981778i	$-0.560859\pi$
$797$	−10.7297	−0.380066	−0.190033	+	0.981778i	$0.560859\pi$
$798$	1.82958	0.0647664
$799$	49.8895	1.76496
$800$	0	0
$801$	0.793273	0.0280289
$802$	−21.7525	−0.768106
$803$	−0.482272	−0.0170190
$804$	36.0602	1.27175
$805$	0	0
$806$	20.7692	0.731562
$807$	10.4615	0.368262
$808$	−20.5423	−0.722676
$809$	−29.9973	−1.05465	−0.527324	−	0.849664i	$-0.676805\pi$
$809$	−29.9973	−1.05465	−0.527324	+	0.849664i	$0.676805\pi$
$810$	0	0
$811$	−34.0557	−1.19586	−0.597929	−	0.801549i	$-0.704009\pi$
$811$	−34.0557	−1.19586	−0.597929	+	0.801549i	$0.704009\pi$
$812$	0.553028	0.0194075
$813$	63.6337	2.23173
$814$	17.7115	0.620788
$815$	0	0
$816$	17.6808	0.618953
$817$	−3.61858	−0.126598
$818$	56.2905	1.96815
$819$	2.13822	0.0747153
$820$	0	0
$821$	−16.9488	−0.591518	−0.295759	−	0.955263i	$-0.595573\pi$
$821$	−16.9488	−0.591518	−0.295759	+	0.955263i	$0.595573\pi$
$822$	−64.7527	−2.25851
$823$	−46.6538	−1.62625	−0.813124	−	0.582090i	$-0.802235\pi$
$823$	−46.6538	−1.62625	−0.813124	+	0.582090i	$0.802235\pi$
$824$	38.2132	1.33122
$825$	0	0
$826$	1.12298	0.0390735
$827$	−10.0896	−0.350849	−0.175425	−	0.984493i	$-0.556130\pi$
$827$	−10.0896	−0.350849	−0.175425	+	0.984493i	$0.556130\pi$
$828$	−48.1883	−1.67466
$829$	−4.61277	−0.160208	−0.0801040	−	0.996787i	$-0.525525\pi$
$829$	−4.61277	−0.160208	−0.0801040	+	0.996787i	$0.525525\pi$
$830$	0	0
$831$	9.56548	0.331823
$832$	−17.6284	−0.611154
$833$	25.3523	0.878407
$834$	−115.595	−4.00271
$835$	0	0
$836$	3.58485	0.123985
$837$	−37.7158	−1.30365
$838$	16.3389	0.564418
$839$	−36.5758	−1.26274	−0.631368	−	0.775484i	$-0.717506\pi$
$839$	−36.5758	−1.26274	−0.631368	+	0.775484i	$0.717506\pi$
$840$	0	0
$841$	−28.6722	−0.988697
$842$	11.7263	0.404116
$843$	−77.1458	−2.65704
$844$	−62.4067	−2.14813
$845$	0	0
$846$	−169.305	−5.82081
$847$	0.269449	0.00925839
$848$	5.73683	0.197003
$849$	−30.9097	−1.06082
$850$	0	0
$851$	19.1698	0.657133
$852$	104.687	3.58653
$853$	24.3932	0.835206	0.417603	−	0.908630i	$-0.362870\pi$
$853$	24.3932	0.835206	0.417603	+	0.908630i	$0.362870\pi$
$854$	−7.30226	−0.249878
$855$	0	0
$856$	−46.4155	−1.58645
$857$	−31.4495	−1.07429	−0.537147	−	0.843489i	$-0.680498\pi$
$857$	−31.4495	−1.07429	−0.537147	+	0.843489i	$0.680498\pi$
$858$	10.2529	0.350028
$859$	1.06992	0.0365051	0.0182526	−	0.999833i	$-0.494190\pi$
$859$	1.06992	0.0365051	0.0182526	+	0.999833i	$0.494190\pi$
$860$	0	0
$861$	−8.67248	−0.295557
$862$	−69.6448	−2.37211
$863$	−35.8665	−1.22091	−0.610455	−	0.792051i	$-0.709014\pi$
$863$	−35.8665	−1.22091	−0.610455	+	0.792051i	$0.709014\pi$
$864$	22.7909	0.775363
$865$	0	0
$866$	−35.8145	−1.21703
$867$	10.3621	0.351915
$868$	5.62198	0.190822
$869$	1.93576	0.0656661
$870$	0	0
$871$	−5.28642	−0.179123
$872$	−14.8579	−0.503150
$873$	7.54853	0.255479
$874$	6.04469	0.204465
$875$	0	0
$876$	−4.96742	−0.167834
$877$	−18.2115	−0.614959	−0.307479	−	0.951555i	$-0.599486\pi$
$877$	−18.2115	−0.614959	−0.307479	+	0.951555i	$0.599486\pi$
$878$	−61.6685	−2.08121
$879$	−55.6699	−1.87770
$880$	0	0
$881$	21.4148	0.721482	0.360741	−	0.932666i	$-0.382524\pi$
$881$	21.4148	0.721482	0.360741	+	0.932666i	$0.382524\pi$
$882$	−86.0355	−2.89697
$883$	26.6128	0.895593	0.447797	−	0.894135i	$-0.352209\pi$
$883$	26.6128	0.895593	0.447797	+	0.894135i	$0.352209\pi$
$884$	−19.8103	−0.666293
$885$	0	0
$886$	92.7348	3.11549
$887$	−11.9257	−0.400424	−0.200212	−	0.979753i	$-0.564163\pi$
$887$	−11.9257	−0.400424	−0.200212	+	0.979753i	$0.564163\pi$
$888$	80.6515	2.70649
$889$	−3.14191	−0.105376
$890$	0	0
$891$	−2.85268	−0.0955684
$892$	−49.0279	−1.64158
$893$	13.6320	0.456179
$894$	−34.1147	−1.14097
$895$	0	0
$896$	−5.53867	−0.185034
$897$	11.0971	0.370520
$898$	−35.5094	−1.18497
$899$	3.33226	0.111137
$900$	0	0
$901$	−12.4863	−0.415978
$902$	−26.4731	−0.881457
$903$	−2.80145	−0.0932266
$904$	−7.82357	−0.260208
$905$	0	0
$906$	150.489	4.99967
$907$	−34.7821	−1.15492	−0.577460	−	0.816419i	$-0.695956\pi$
$907$	−34.7821	−1.15492	−0.577460	+	0.816419i	$0.695956\pi$
$908$	−42.6274	−1.41464
$909$	−28.8241	−0.956037
$910$	0	0
$911$	2.18025	0.0722351	0.0361175	−	0.999348i	$-0.488501\pi$
$911$	2.18025	0.0722351	0.0361175	+	0.999348i	$0.488501\pi$
$912$	4.83120	0.159977
$913$	−4.76907	−0.157833
$914$	91.0657	3.01218
$915$	0	0
$916$	34.2876	1.13290
$917$	0.679431	0.0224368
$918$	56.0449	1.84976
$919$	34.9180	1.15184	0.575920	−	0.817506i	$-0.304644\pi$
$919$	34.9180	1.15184	0.575920	+	0.817506i	$0.304644\pi$
$920$	0	0
$921$	77.2754	2.54631
$922$	79.3501	2.61326
$923$	−15.3471	−0.505157
$924$	2.77534	0.0913020
$925$	0	0
$926$	53.2678	1.75049
$927$	53.6193	1.76109
$928$	−2.01362	−0.0661004
$929$	−26.9768	−0.885080	−0.442540	−	0.896749i	$-0.645922\pi$
$929$	−26.9768	−0.885080	−0.442540	+	0.896749i	$0.645922\pi$
$930$	0	0
$931$	6.92740	0.227036
$932$	−9.86746	−0.323219
$933$	−21.1591	−0.692717
$934$	−48.6013	−1.59028
$935$	0	0
$936$	29.7214	0.971474
$937$	19.3062	0.630707	0.315354	−	0.948974i	$-0.397877\pi$
$937$	19.3062	0.630707	0.315354	+	0.948974i	$0.397877\pi$
$938$	−2.22932	−0.0727899
$939$	−49.9302	−1.62941
$940$	0	0
$941$	−0.207907	−0.00677756	−0.00338878	−	0.999994i	$-0.501079\pi$
$941$	−0.207907	−0.00677756	−0.00338878	+	0.999994i	$0.501079\pi$
$942$	−89.7877	−2.92544
$943$	−28.6528	−0.933063
$944$	2.96536	0.0965141
$945$	0	0
$946$	−8.55154	−0.278035
$947$	−51.9687	−1.68876	−0.844378	−	0.535748i	$-0.820030\pi$
$947$	−51.9687	−1.68876	−0.844378	+	0.535748i	$0.820030\pi$
$948$	19.9384	0.647569
$949$	0.728222	0.0236391
$950$	0	0
$951$	−44.4199	−1.44042
$952$	−3.69335	−0.119702
$953$	16.0553	0.520081	0.260040	−	0.965598i	$-0.416264\pi$
$953$	16.0553	0.520081	0.260040	+	0.965598i	$0.416264\pi$
$954$	42.3733	1.37189
$955$	0	0
$956$	−46.0386	−1.48900
$957$	1.64500	0.0531753
$958$	−29.7888	−0.962431
$959$	2.56957	0.0829759
$960$	0	0
$961$	2.87517	0.0927476
$962$	−26.7441	−0.862265
$963$	−65.1284	−2.09873
$964$	8.98618	0.289425
$965$	0	0
$966$	4.67971	0.150567
$967$	10.4378	0.335656	0.167828	−	0.985816i	$-0.446325\pi$
$967$	10.4378	0.335656	0.167828	+	0.985816i	$0.446325\pi$
$968$	3.74537	0.120381
$969$	−10.5152	−0.337796
$970$	0	0
$971$	−55.0457	−1.76650	−0.883250	−	0.468903i	$-0.844649\pi$
$971$	−55.0457	−1.76650	−0.883250	+	0.468903i	$0.844649\pi$
$972$	40.3080	1.29288
$973$	4.58713	0.147057
$974$	−7.57507	−0.242721
$975$	0	0
$976$	−19.2824	−0.617215
$977$	55.8927	1.78817	0.894083	−	0.447902i	$-0.147828\pi$
$977$	55.8927	1.78817	0.894083	+	0.447902i	$0.147828\pi$
$978$	115.955	3.70784
$979$	−0.150946	−0.00482425
$980$	0	0
$981$	−20.8480	−0.665624
$982$	104.119	3.32256
$983$	−35.4032	−1.12919	−0.564593	−	0.825370i	$-0.690967\pi$
$983$	−35.4032	−1.12919	−0.564593	+	0.825370i	$0.690967\pi$
$984$	−120.548	−3.84294
$985$	0	0
$986$	−4.95168	−0.157693
$987$	10.5537	0.335929
$988$	−5.41307	−0.172213
$989$	−9.25565	−0.294312
$990$	0	0
$991$	−53.6641	−1.70470	−0.852349	−	0.522974i	$-0.824823\pi$
$991$	−53.6641	−1.70470	−0.852349	+	0.522974i	$0.824823\pi$
$992$	−20.4701	−0.649927
$993$	−102.372	−3.24869
$994$	−6.47199	−0.205279
$995$	0	0
$996$	−49.1217	−1.55648
$997$	46.1890	1.46282	0.731410	−	0.681938i	$-0.238863\pi$
$997$	46.1890	1.46282	0.731410	+	0.681938i	$0.238863\pi$
$998$	6.98636	0.221149
$999$	48.5660	1.53656

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5225.2.a.k.1.6		6
5.4	even	2		1045.2.a.g.1.1	✓	6
15.14	odd	2		9405.2.a.w.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.a.g.1.1	✓	6	5.4	even	2
5225.2.a.k.1.6		6	1.1	even	1	trivial
9405.2.a.w.1.6		6	15.14	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5225.a (trivial)

\(f(q)\)	\(=\)	\(q+2.36323 q^{2} -2.87321 q^{3} +3.58485 q^{4} -6.79006 q^{6} +0.269449 q^{7} +3.74537 q^{8} +5.25535 q^{9} -1.00000 q^{11} -10.3000 q^{12} +1.50998 q^{13} +0.636771 q^{14} +1.68146 q^{16} -3.65972 q^{17} +12.4196 q^{18} -1.00000 q^{19} -0.774186 q^{21} -2.36323 q^{22} -2.55781 q^{23} -10.7612 q^{24} +3.56844 q^{26} -6.48011 q^{27} +0.965936 q^{28} +0.572530 q^{29} +5.82024 q^{31} -3.51706 q^{32} +2.87321 q^{33} -8.64876 q^{34} +18.8397 q^{36} -7.49463 q^{37} -2.36323 q^{38} -4.33851 q^{39} +11.2021 q^{41} -1.82958 q^{42} +3.61858 q^{43} -3.58485 q^{44} -6.04469 q^{46} -13.6320 q^{47} -4.83120 q^{48} -6.92740 q^{49} +10.5152 q^{51} +5.41307 q^{52} +3.41181 q^{53} -15.3140 q^{54} +1.00919 q^{56} +2.87321 q^{57} +1.35302 q^{58} +1.76356 q^{59} -11.4676 q^{61} +13.7546 q^{62} +1.41605 q^{63} -11.6745 q^{64} +6.79006 q^{66} -3.50098 q^{67} -13.1196 q^{68} +7.34913 q^{69} -10.1638 q^{71} +19.6832 q^{72} +0.482272 q^{73} -17.7115 q^{74} -3.58485 q^{76} -0.269449 q^{77} -10.2529 q^{78} -1.93576 q^{79} +2.85268 q^{81} +26.4731 q^{82} +4.76907 q^{83} -2.77534 q^{84} +8.55154 q^{86} -1.64500 q^{87} -3.74537 q^{88} +0.150946 q^{89} +0.406864 q^{91} -9.16937 q^{92} -16.7228 q^{93} -32.2156 q^{94} +10.1053 q^{96} +1.43635 q^{97} -16.3710 q^{98} -5.25535 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{3} + 8 q^{4} + 2 q^{6} - 5 q^{7} + 9 q^{9} - 6 q^{11} - 19 q^{12} + 9 q^{13} + 18 q^{14} + 4 q^{16} + 5 q^{17} - 2 q^{18} - 6 q^{19} + 3 q^{21} - 8 q^{23} - 7 q^{24} - 22 q^{26} - 30 q^{27} - 10 q^{28}+ \cdots - 9 q^{99}+O(q^{100}) \) 6 * q - 3 * q^3 + 8 * q^4 + 2 * q^6 - 5 * q^7 + 9 * q^9 - 6 * q^11 - 19 * q^12 + 9 * q^13 + 18 * q^14 + 4 * q^16 + 5 * q^17 - 2 * q^18 - 6 * q^19 + 3 * q^21 - 8 * q^23 - 7 * q^24 - 22 * q^26 - 30 * q^27 - 10 * q^28 - 5 * q^29 - q^31 - 15 * q^32 + 3 * q^33 - 22 * q^34 + 12 * q^36 - 9 * q^37 - 32 * q^39 + 25 * q^41 - 11 * q^42 - 15 * q^43 - 8 * q^44 - 16 * q^46 - 24 * q^47 + 4 * q^48 + 13 * q^49 + 27 * q^52 - 5 * q^53 - 11 * q^54 - 12 * q^56 + 3 * q^57 - 13 * q^58 + 39 * q^59 - 11 * q^61 + 42 * q^62 - 38 * q^63 - 14 * q^64 - 2 * q^66 - 24 * q^67 - 45 * q^68 + 14 * q^69 - 24 * q^71 + 61 * q^72 + 26 * q^73 + q^74 - 8 * q^76 + 5 * q^77 + 29 * q^78 + 11 * q^79 + 30 * q^81 - 8 * q^82 - 39 * q^83 + 25 * q^84 + 18 * q^86 + 16 * q^87 + 22 * q^89 - 26 * q^91 + 11 * q^92 + 6 * q^93 - 30 * q^94 - 15 * q^96 - 22 * q^97 - 33 * q^98 - 9 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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