LMFDB - Embedded newform 525.4.a.k.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [525,4,Mod(1,525)] 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("525.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(525, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 525 = 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	525.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-1,-6,17,0,3,14,-33,18,0,-22] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$30.9760027530$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{65}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 16 $ x^2 - x - 16
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.53113$ of defining polynomial
Character	$\chi$	$=$	525.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+3.53113 q^{2} -3.00000 q^{3} +4.46887 q^{4} -10.5934 q^{6} +7.00000 q^{7} -12.4689 q^{8} +9.00000 q^{9} -2.93774 q^{11} -13.4066 q^{12} +19.0623 q^{13} +24.7179 q^{14} -79.7802 q^{16} -122.498 q^{17} +31.7802 q^{18} +107.436 q^{19} -21.0000 q^{21} -10.3735 q^{22} -210.623 q^{23} +37.4066 q^{24} +67.3113 q^{26} -27.0000 q^{27} +31.2821 q^{28} +95.4942 q^{29} -94.3074 q^{31} -181.963 q^{32} +8.81323 q^{33} -432.556 q^{34} +40.2198 q^{36} -97.1206 q^{37} +379.370 q^{38} -57.1868 q^{39} -491.113 q^{41} -74.1537 q^{42} +43.0039 q^{43} -13.1284 q^{44} -743.735 q^{46} -473.494 q^{47} +239.340 q^{48} +49.0000 q^{49} +367.494 q^{51} +85.1868 q^{52} +183.677 q^{53} -95.3405 q^{54} -87.2821 q^{56} -322.307 q^{57} +337.202 q^{58} -760.615 q^{59} -198.747 q^{61} -333.012 q^{62} +63.0000 q^{63} -4.29373 q^{64} +31.1206 q^{66} +309.992 q^{67} -547.428 q^{68} +631.868 q^{69} +665.693 q^{71} -112.220 q^{72} -621.288 q^{73} -342.945 q^{74} +480.117 q^{76} -20.5642 q^{77} -201.934 q^{78} -24.7626 q^{79} +81.0000 q^{81} -1734.18 q^{82} +406.724 q^{83} -93.8463 q^{84} +151.852 q^{86} -286.483 q^{87} +36.6303 q^{88} +261.751 q^{89} +133.436 q^{91} -941.245 q^{92} +282.922 q^{93} -1671.97 q^{94} +545.889 q^{96} +1004.77 q^{97} +173.025 q^{98} -26.4397 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - 6 q^{3} + 17 q^{4} + 3 q^{6} + 14 q^{7} - 33 q^{8} + 18 q^{9} - 22 q^{11} - 51 q^{12} + 22 q^{13} - 7 q^{14} - 87 q^{16} - 116 q^{17} - 9 q^{18} + 102 q^{19} - 42 q^{21} + 76 q^{22} - 260 q^{23}+ \cdots - 198 q^{99}+O(q^{100}) $ 2 * q - q^2 - 6 * q^3 + 17 * q^4 + 3 * q^6 + 14 * q^7 - 33 * q^8 + 18 * q^9 - 22 * q^11 - 51 * q^12 + 22 * q^13 - 7 * q^14 - 87 * q^16 - 116 * q^17 - 9 * q^18 + 102 * q^19 - 42 * q^21 + 76 * q^22 - 260 * q^23 + 99 * q^24 + 54 * q^26 - 54 * q^27 + 119 * q^28 - 196 * q^29 + 150 * q^31 + 15 * q^32 + 66 * q^33 - 462 * q^34 + 153 * q^36 + 96 * q^37 + 404 * q^38 - 66 * q^39 - 176 * q^41 + 21 * q^42 + 344 * q^43 - 252 * q^44 - 520 * q^46 - 560 * q^47 + 261 * q^48 + 98 * q^49 + 348 * q^51 + 122 * q^52 - 326 * q^53 + 27 * q^54 - 231 * q^56 - 306 * q^57 + 1658 * q^58 - 844 * q^59 - 204 * q^61 - 1440 * q^62 + 126 * q^63 - 839 * q^64 - 228 * q^66 + 104 * q^67 - 466 * q^68 + 780 * q^69 + 1670 * q^71 - 297 * q^72 + 386 * q^73 - 1218 * q^74 + 412 * q^76 - 154 * q^77 - 162 * q^78 - 888 * q^79 + 162 * q^81 - 3162 * q^82 - 928 * q^83 - 357 * q^84 - 1212 * q^86 + 588 * q^87 + 428 * q^88 + 588 * q^89 + 154 * q^91 - 1560 * q^92 - 450 * q^93 - 1280 * q^94 - 45 * q^96 - 522 * q^97 - 49 * q^98 - 198 * 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 $

$2$	3.53113	1.24844	0.624221	−	0.781248i	$-0.285416\pi$
$2$	3.53113	1.24844	0.624221	+	0.781248i	$0.285416\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	4.46887	0.558609
$5$	0	0
$5$	0	0
$6$	−10.5934	−0.720789
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−12.4689	−0.551051
$9$	9.00000	0.333333
$10$	0	0
$11$	−2.93774	−0.0805239	−0.0402619	−	0.999189i	$-0.512819\pi$
$11$	−2.93774	−0.0805239	−0.0402619	+	0.999189i	$0.512819\pi$
$12$	−13.4066	−0.322513
$13$	19.0623	0.406686	0.203343	−	0.979108i	$-0.434819\pi$
$13$	19.0623	0.406686	0.203343	+	0.979108i	$0.434819\pi$
$14$	24.7179	0.471867
$15$	0	0
$16$	−79.7802	−1.24656
$17$	−122.498	−1.74766	−0.873828	−	0.486236i	$-0.838370\pi$
$17$	−122.498	−1.74766	−0.873828	+	0.486236i	$0.838370\pi$
$18$	31.7802	0.416148
$19$	107.436	1.29723	0.648617	−	0.761115i	$-0.275348\pi$
$19$	107.436	1.29723	0.648617	+	0.761115i	$0.275348\pi$
$20$	0	0
$21$	−21.0000	−0.218218
$22$	−10.3735	−0.100529
$23$	−210.623	−1.90947	−0.954736	−	0.297455i	$-0.903862\pi$
$23$	−210.623	−1.90947	−0.954736	+	0.297455i	$0.903862\pi$
$24$	37.4066	0.318150
$25$	0	0
$26$	67.3113	0.507724
$27$	−27.0000	−0.192450
$28$	31.2821	0.211134
$29$	95.4942	0.611477	0.305738	−	0.952116i	$-0.401097\pi$
$29$	95.4942	0.611477	0.305738	+	0.952116i	$0.401097\pi$
$30$	0	0
$31$	−94.3074	−0.546391	−0.273195	−	0.961959i	$-0.588081\pi$
$31$	−94.3074	−0.546391	−0.273195	+	0.961959i	$0.588081\pi$
$32$	−181.963	−1.00521
$33$	8.81323	0.0464905
$34$	−432.556	−2.18185
$35$	0	0
$36$	40.2198	0.186203
$37$	−97.1206	−0.431528	−0.215764	−	0.976446i	$-0.569224\pi$
$37$	−97.1206	−0.431528	−0.215764	+	0.976446i	$0.569224\pi$
$38$	379.370	1.61952
$39$	−57.1868	−0.234800
$40$	0	0
$41$	−491.113	−1.87071	−0.935353	−	0.353716i	$-0.884918\pi$
$41$	−491.113	−1.87071	−0.935353	+	0.353716i	$0.884918\pi$
$42$	−74.1537	−0.272433
$43$	43.0039	0.152512	0.0762562	−	0.997088i	$-0.475703\pi$
$43$	43.0039	0.152512	0.0762562	+	0.997088i	$0.475703\pi$
$44$	−13.1284	−0.0449814
$45$	0	0
$46$	−743.735	−2.38387
$47$	−473.494	−1.46949	−0.734747	−	0.678341i	$-0.762699\pi$
$47$	−473.494	−1.46949	−0.734747	+	0.678341i	$0.762699\pi$
$48$	239.340	0.719705
$49$	49.0000	0.142857
$50$	0	0
$51$	367.494	1.00901
$52$	85.1868	0.227178
$53$	183.677	0.476038	0.238019	−	0.971261i	$-0.423502\pi$
$53$	183.677	0.476038	0.238019	+	0.971261i	$0.423502\pi$
$54$	−95.3405	−0.240263
$55$	0	0
$56$	−87.2821	−0.208278
$57$	−322.307	−0.748959
$58$	337.202	0.763394
$59$	−760.615	−1.67837	−0.839183	−	0.543849i	$-0.816966\pi$
$59$	−760.615	−1.67837	−0.839183	+	0.543849i	$0.816966\pi$
$60$	0	0
$61$	−198.747	−0.417163	−0.208582	−	0.978005i	$-0.566885\pi$
$61$	−198.747	−0.417163	−0.208582	+	0.978005i	$0.566885\pi$
$62$	−333.012	−0.682137
$63$	63.0000	0.125988
$64$	−4.29373	−0.00838618
$65$	0	0
$66$	31.1206	0.0580407
$67$	309.992	0.565247	0.282624	−	0.959231i	$-0.408795\pi$
$67$	309.992	0.565247	0.282624	+	0.959231i	$0.408795\pi$
$68$	−547.428	−0.976256
$69$	631.868	1.10243
$70$	0	0
$71$	665.693	1.11272	0.556360	−	0.830941i	$-0.312197\pi$
$71$	665.693	1.11272	0.556360	+	0.830941i	$0.312197\pi$
$72$	−112.220	−0.183684
$73$	−621.288	−0.996113	−0.498057	−	0.867145i	$-0.665953\pi$
$73$	−621.288	−0.996113	−0.498057	+	0.867145i	$0.665953\pi$
$74$	−342.945	−0.538738
$75$	0	0
$76$	480.117	0.724647
$77$	−20.5642	−0.0304352
$78$	−201.934	−0.293135
$79$	−24.7626	−0.0352659	−0.0176330	−	0.999845i	$-0.505613\pi$
$79$	−24.7626	−0.0352659	−0.0176330	+	0.999845i	$0.505613\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	−1734.18	−2.33547
$83$	406.724	0.537876	0.268938	−	0.963157i	$-0.413327\pi$
$83$	406.724	0.537876	0.268938	+	0.963157i	$0.413327\pi$
$84$	−93.8463	−0.121898
$85$	0	0
$86$	151.852	0.190403
$87$	−286.483	−0.353036
$88$	36.6303	0.0443728
$89$	261.751	0.311748	0.155874	−	0.987777i	$-0.450181\pi$
$89$	261.751	0.311748	0.155874	+	0.987777i	$0.450181\pi$
$90$	0	0
$91$	133.436	0.153713
$92$	−941.245	−1.06665
$93$	282.922	0.315459
$94$	−1671.97	−1.83458
$95$	0	0
$96$	545.889	0.580360
$97$	1004.77	1.05175	0.525873	−	0.850563i	$-0.323739\pi$
$97$	1004.77	1.05175	0.525873	+	0.850563i	$0.323739\pi$
$98$	173.025	0.178349
$99$	−26.4397	−0.0268413
$100$	0	0
$101$	−128.872	−0.126962	−0.0634812	−	0.997983i	$-0.520220\pi$
$101$	−128.872	−0.126962	−0.0634812	+	0.997983i	$0.520220\pi$
$102$	1297.67	1.25969
$103$	−806.008	−0.771051	−0.385526	−	0.922697i	$-0.625980\pi$
$103$	−806.008	−0.771051	−0.385526	+	0.922697i	$0.625980\pi$
$104$	−237.685	−0.224105
$105$	0	0
$106$	648.587	0.594305
$107$	769.712	0.695429	0.347714	−	0.937600i	$-0.386958\pi$
$107$	769.712	0.695429	0.347714	+	0.937600i	$0.386958\pi$
$108$	−120.660	−0.107504
$109$	−780.856	−0.686169	−0.343085	−	0.939304i	$-0.611472\pi$
$109$	−780.856	−0.686169	−0.343085	+	0.939304i	$0.611472\pi$
$110$	0	0
$111$	291.362	0.249143
$112$	−558.461	−0.471157
$113$	1115.65	0.928771	0.464386	−	0.885633i	$-0.346275\pi$
$113$	1115.65	0.928771	0.464386	+	0.885633i	$0.346275\pi$
$114$	−1138.11	−0.935032
$115$	0	0
$116$	426.751	0.341576
$117$	171.560	0.135562
$118$	−2685.83	−2.09534
$119$	−857.486	−0.660552
$120$	0	0
$121$	−1322.37	−0.993516
$122$	−701.802	−0.520804
$123$	1473.34	1.08005
$124$	−421.448	−0.305219
$125$	0	0
$126$	222.461	0.157289
$127$	1875.98	1.31076	0.655381	−	0.755299i	$-0.272508\pi$
$127$	1875.98	1.31076	0.655381	+	0.755299i	$0.272508\pi$
$128$	1440.54	0.994744
$129$	−129.012	−0.0880530
$130$	0	0
$131$	364.203	0.242905	0.121452	−	0.992597i	$-0.461245\pi$
$131$	364.203	0.242905	0.121452	+	0.992597i	$0.461245\pi$
$132$	39.3852	0.0259700
$133$	752.051	0.490309
$134$	1094.62	0.705679
$135$	0	0
$136$	1527.41	0.963048
$137$	−1603.13	−0.999743	−0.499872	−	0.866099i	$-0.666620\pi$
$137$	−1603.13	−0.999743	−0.499872	+	0.866099i	$0.666620\pi$
$138$	2231.21	1.37633
$139$	2431.12	1.48349	0.741746	−	0.670681i	$-0.233998\pi$
$139$	2431.12	1.48349	0.741746	+	0.670681i	$0.233998\pi$
$140$	0	0
$141$	1420.48	0.848413
$142$	2350.65	1.38917
$143$	−56.0000	−0.0327479
$144$	−718.021	−0.415522
$145$	0	0
$146$	−2193.85	−1.24359
$147$	−147.000	−0.0824786
$148$	−434.020	−0.241055
$149$	2341.57	1.28744	0.643722	−	0.765260i	$-0.277389\pi$
$149$	2341.57	1.28744	0.643722	+	0.765260i	$0.277389\pi$
$150$	0	0
$151$	−2104.07	−1.13395	−0.566976	−	0.823734i	$-0.691887\pi$
$151$	−2104.07	−1.13395	−0.566976	+	0.823734i	$0.691887\pi$
$152$	−1339.60	−0.714843
$153$	−1102.48	−0.582552
$154$	−72.6148	−0.0379966
$155$	0	0
$156$	−255.560	−0.131162
$157$	593.467	0.301680	0.150840	−	0.988558i	$-0.451802\pi$
$157$	593.467	0.301680	0.150840	+	0.988558i	$0.451802\pi$
$158$	−87.4399	−0.0440275
$159$	−551.031	−0.274840
$160$	0	0
$161$	−1474.36	−0.721712
$162$	286.021	0.138716
$163$	−2178.71	−1.04693	−0.523465	−	0.852047i	$-0.675361\pi$
$163$	−2178.71	−1.04693	−0.523465	+	0.852047i	$0.675361\pi$
$164$	−2194.72	−1.04499
$165$	0	0
$166$	1436.19	0.671508
$167$	−799.502	−0.370463	−0.185231	−	0.982695i	$-0.559303\pi$
$167$	−799.502	−0.370463	−0.185231	+	0.982695i	$0.559303\pi$
$168$	261.846	0.120249
$169$	−1833.63	−0.834606
$170$	0	0
$171$	966.922	0.432412
$172$	192.179	0.0851947
$173$	1444.36	0.634754	0.317377	−	0.948299i	$-0.397198\pi$
$173$	1444.36	0.634754	0.317377	+	0.948299i	$0.397198\pi$
$174$	−1011.61	−0.440745
$175$	0	0
$176$	234.374	0.100378
$177$	2281.84	0.969005
$178$	924.276	0.389199
$179$	3343.49	1.39611	0.698056	−	0.716043i	$-0.254048\pi$
$179$	3343.49	1.39611	0.698056	+	0.716043i	$0.254048\pi$
$180$	0	0
$181$	2251.81	0.924729	0.462365	−	0.886690i	$-0.347001\pi$
$181$	2251.81	0.924729	0.462365	+	0.886690i	$0.347001\pi$
$182$	471.179	0.191902
$183$	596.241	0.240849
$184$	2626.23	1.05222
$185$	0	0
$186$	999.035	0.393832
$187$	359.868	0.140728
$188$	−2115.98	−0.820873
$189$	−189.000	−0.0727393
$190$	0	0
$191$	−1001.93	−0.379565	−0.189782	−	0.981826i	$-0.560778\pi$
$191$	−1001.93	−0.379565	−0.189782	+	0.981826i	$0.560778\pi$
$192$	12.8812	0.00484177
$193$	4054.97	1.51235	0.756173	−	0.654372i	$-0.227067\pi$
$193$	4054.97	1.51235	0.756173	+	0.654372i	$0.227067\pi$
$194$	3547.99	1.31304
$195$	0	0
$196$	218.975	0.0798013
$197$	5140.23	1.85902	0.929508	−	0.368802i	$-0.120232\pi$
$197$	5140.23	1.85902	0.929508	+	0.368802i	$0.120232\pi$
$198$	−93.3619	−0.0335098
$199$	585.631	0.208614	0.104307	−	0.994545i	$-0.466737\pi$
$199$	585.631	0.208614	0.104307	+	0.994545i	$0.466737\pi$
$200$	0	0
$201$	−929.977	−0.326346
$202$	−455.062	−0.158505
$203$	668.459	0.231116
$204$	1642.28	0.563642
$205$	0	0
$206$	−2846.12	−0.962614
$207$	−1895.60	−0.636490
$208$	−1520.79	−0.506961
$209$	−315.619	−0.104458
$210$	0	0
$211$	−1055.16	−0.344266	−0.172133	−	0.985074i	$-0.555066\pi$
$211$	−1055.16	−0.344266	−0.172133	+	0.985074i	$0.555066\pi$
$212$	820.829	0.265919
$213$	−1997.08	−0.642430
$214$	2717.95	0.868203
$215$	0	0
$216$	336.660	0.106050
$217$	−660.152	−0.206516
$218$	−2757.30	−0.856643
$219$	1863.86	0.575106
$220$	0	0
$221$	−2335.09	−0.710747
$222$	1028.84	0.311040
$223$	−4675.85	−1.40412	−0.702059	−	0.712119i	$-0.747736\pi$
$223$	−4675.85	−1.40412	−0.702059	+	0.712119i	$0.747736\pi$
$224$	−1273.74	−0.379935
$225$	0	0
$226$	3939.49	1.15952
$227$	−5443.11	−1.59151	−0.795754	−	0.605621i	$-0.792925\pi$
$227$	−5443.11	−1.59151	−0.795754	+	0.605621i	$0.792925\pi$
$228$	−1440.35	−0.418375
$229$	−536.303	−0.154759	−0.0773797	−	0.997002i	$-0.524655\pi$
$229$	−536.303	−0.154759	−0.0773797	+	0.997002i	$0.524655\pi$
$230$	0	0
$231$	61.6926	0.0175717
$232$	−1190.70	−0.336955
$233$	183.490	0.0515916	0.0257958	−	0.999667i	$-0.491788\pi$
$233$	183.490	0.0515916	0.0257958	+	0.999667i	$0.491788\pi$
$234$	605.802	0.169241
$235$	0	0
$236$	−3399.09	−0.937550
$237$	74.2878	0.0203608
$238$	−3027.90	−0.824661
$239$	643.218	0.174085	0.0870425	−	0.996205i	$-0.472258\pi$
$239$	643.218	0.174085	0.0870425	+	0.996205i	$0.472258\pi$
$240$	0	0
$241$	−5755.61	−1.53839	−0.769194	−	0.639015i	$-0.779342\pi$
$241$	−5755.61	−1.53839	−0.769194	+	0.639015i	$0.779342\pi$
$242$	−4669.46	−1.24035
$243$	−243.000	−0.0641500
$244$	−888.175	−0.233031
$245$	0	0
$246$	5202.55	1.34838
$247$	2047.97	0.527567
$248$	1175.91	0.301089
$249$	−1220.17	−0.310543
$250$	0	0
$251$	−5132.27	−1.29062	−0.645311	−	0.763920i	$-0.723272\pi$
$251$	−5132.27	−1.29062	−0.645311	+	0.763920i	$0.723272\pi$
$252$	281.539	0.0703781
$253$	618.755	0.153758
$254$	6624.34	1.63641
$255$	0	0
$256$	5121.09	1.25027
$257$	−5041.74	−1.22372	−0.611859	−	0.790967i	$-0.709578\pi$
$257$	−5041.74	−1.22372	−0.611859	+	0.790967i	$0.709578\pi$
$258$	−455.557	−0.109929
$259$	−679.844	−0.163102
$260$	0	0
$261$	859.448	0.203826
$262$	1286.05	0.303253
$263$	−7577.00	−1.77649	−0.888246	−	0.459367i	$-0.848076\pi$
$263$	−7577.00	−1.77649	−0.888246	+	0.459367i	$0.848076\pi$
$264$	−109.891	−0.0256186
$265$	0	0
$266$	2655.59	0.612122
$267$	−785.253	−0.179988
$268$	1385.32	0.315752
$269$	1023.10	0.231893	0.115947	−	0.993255i	$-0.463010\pi$
$269$	1023.10	0.231893	0.115947	+	0.993255i	$0.463010\pi$
$270$	0	0
$271$	−2251.98	−0.504790	−0.252395	−	0.967624i	$-0.581218\pi$
$271$	−2251.98	−0.504790	−0.252395	+	0.967624i	$0.581218\pi$
$272$	9772.91	2.17857
$273$	−400.307	−0.0887462
$274$	−5660.87	−1.24812
$275$	0	0
$276$	2823.74	0.615829
$277$	8630.72	1.87209	0.936047	−	0.351875i	$-0.114456\pi$
$277$	8630.72	1.87209	0.936047	+	0.351875i	$0.114456\pi$
$278$	8584.61	1.85205
$279$	−848.767	−0.182130
$280$	0	0
$281$	−7521.62	−1.59680	−0.798402	−	0.602124i	$-0.794321\pi$
$281$	−7521.62	−1.59680	−0.798402	+	0.602124i	$0.794321\pi$
$282$	5015.91	1.05919
$283$	−14.8169	−0.00311226	−0.00155613	−	0.999999i	$-0.500495\pi$
$283$	−14.8169	−0.00311226	−0.00155613	+	0.999999i	$0.500495\pi$
$284$	2974.89	0.621576
$285$	0	0
$286$	−197.743	−0.0408839
$287$	−3437.79	−0.707060
$288$	−1637.67	−0.335071
$289$	10092.8	2.05430
$290$	0	0
$291$	−3014.32	−0.607226
$292$	−2776.46	−0.556438
$293$	−6913.39	−1.37844	−0.689222	−	0.724550i	$-0.742048\pi$
$293$	−6913.39	−1.37844	−0.689222	+	0.724550i	$0.742048\pi$
$294$	−519.076	−0.102970
$295$	0	0
$296$	1210.98	0.237794
$297$	79.3190	0.0154968
$298$	8268.39	1.60730
$299$	−4014.94	−0.776555
$300$	0	0
$301$	301.027	0.0576442
$302$	−7429.74	−1.41567
$303$	386.615	0.0733018
$304$	−8571.25	−1.61709
$305$	0	0
$306$	−3893.01	−0.727283
$307$	7644.12	1.42108	0.710542	−	0.703655i	$-0.248450\pi$
$307$	7644.12	1.42108	0.710542	+	0.703655i	$0.248450\pi$
$308$	−91.8987	−0.0170014
$309$	2418.02	0.445167
$310$	0	0
$311$	7593.99	1.38462	0.692308	−	0.721602i	$-0.256594\pi$
$311$	7593.99	1.38462	0.692308	+	0.721602i	$0.256594\pi$
$312$	713.055	0.129387
$313$	9127.84	1.64836	0.824179	−	0.566329i	$-0.191637\pi$
$313$	9127.84	1.64836	0.824179	+	0.566329i	$0.191637\pi$
$314$	2095.61	0.376631
$315$	0	0
$316$	−110.661	−0.0196999
$317$	4929.81	0.873456	0.436728	−	0.899593i	$-0.356137\pi$
$317$	4929.81	0.873456	0.436728	+	0.899593i	$0.356137\pi$
$318$	−1945.76	−0.343122
$319$	−280.537	−0.0492385
$320$	0	0
$321$	−2309.14	−0.401506
$322$	−5206.15	−0.901016
$323$	−13160.7	−2.26712
$324$	361.979	0.0620677
$325$	0	0
$326$	−7693.30	−1.30703
$327$	2342.57	0.396160
$328$	6123.62	1.03086
$329$	−3314.46	−0.555417
$330$	0	0
$331$	1221.67	0.202867	0.101433	−	0.994842i	$-0.467657\pi$
$331$	1221.67	0.202867	0.101433	+	0.994842i	$0.467657\pi$
$332$	1817.60	0.300463
$333$	−874.086	−0.143843
$334$	−2823.14	−0.462502
$335$	0	0
$336$	1675.38	0.272023
$337$	8744.83	1.41354	0.706768	−	0.707446i	$-0.250153\pi$
$337$	8744.83	1.41354	0.706768	+	0.707446i	$0.250153\pi$
$338$	−6474.79	−1.04196
$339$	−3346.94	−0.536226
$340$	0	0
$341$	277.051	0.0439975
$342$	3414.33	0.539841
$343$	343.000	0.0539949
$344$	−536.210	−0.0840421
$345$	0	0
$346$	5100.21	0.792454
$347$	−4589.56	−0.710031	−0.355015	−	0.934860i	$-0.615524\pi$
$347$	−4589.56	−0.710031	−0.355015	+	0.934860i	$0.615524\pi$
$348$	−1280.25	−0.197209
$349$	−3989.89	−0.611960	−0.305980	−	0.952038i	$-0.598984\pi$
$349$	−3989.89	−0.611960	−0.305980	+	0.952038i	$0.598984\pi$
$350$	0	0
$351$	−514.681	−0.0782668
$352$	534.561	0.0809437
$353$	−2416.35	−0.364333	−0.182166	−	0.983268i	$-0.558311\pi$
$353$	−2416.35	−0.364333	−0.182166	+	0.983268i	$0.558311\pi$
$354$	8057.49	1.20975
$355$	0	0
$356$	1169.73	0.174145
$357$	2572.46	0.381370
$358$	11806.3	1.74297
$359$	−2756.24	−0.405206	−0.202603	−	0.979261i	$-0.564940\pi$
$359$	−2756.24	−0.405206	−0.202603	+	0.979261i	$0.564940\pi$
$360$	0	0
$361$	4683.45	0.682818
$362$	7951.44	1.15447
$363$	3967.11	0.573607
$364$	596.307	0.0858654
$365$	0	0
$366$	2105.40	0.300687
$367$	−11112.8	−1.58061	−0.790307	−	0.612711i	$-0.790079\pi$
$367$	−11112.8	−1.58061	−0.790307	+	0.612711i	$0.790079\pi$
$368$	16803.5	2.38028
$369$	−4420.02	−0.623569
$370$	0	0
$371$	1285.74	0.179925
$372$	1264.34	0.176218
$373$	−6091.09	−0.845535	−0.422768	−	0.906238i	$-0.638941\pi$
$373$	−6091.09	−0.845535	−0.422768	+	0.906238i	$0.638941\pi$
$374$	1270.74	0.175691
$375$	0	0
$376$	5903.94	0.809767
$377$	1820.33	0.248679
$378$	−667.383	−0.0908108
$379$	3984.29	0.539998	0.269999	−	0.962861i	$-0.412977\pi$
$379$	3984.29	0.539998	0.269999	+	0.962861i	$0.412977\pi$
$380$	0	0
$381$	−5627.95	−0.756768
$382$	−3537.93	−0.473865
$383$	−318.475	−0.0424890	−0.0212445	−	0.999774i	$-0.506763\pi$
$383$	−318.475	−0.0424890	−0.0212445	+	0.999774i	$0.506763\pi$
$384$	−4321.63	−0.574316
$385$	0	0
$386$	14318.6	1.88808
$387$	387.035	0.0508374
$388$	4490.21	0.587515
$389$	3885.46	0.506429	0.253214	−	0.967410i	$-0.418512\pi$
$389$	3885.46	0.506429	0.253214	+	0.967410i	$0.418512\pi$
$390$	0	0
$391$	25800.9	3.33710
$392$	−610.975	−0.0787216
$393$	−1092.61	−0.140241
$394$	18150.8	2.32088
$395$	0	0
$396$	−118.156	−0.0149938
$397$	−4806.04	−0.607578	−0.303789	−	0.952739i	$-0.598252\pi$
$397$	−4806.04	−0.607578	−0.303789	+	0.952739i	$0.598252\pi$
$398$	2067.94	0.260443
$399$	−2256.15	−0.283080
$400$	0	0
$401$	3618.59	0.450633	0.225316	−	0.974286i	$-0.427658\pi$
$401$	3618.59	0.450633	0.225316	+	0.974286i	$0.427658\pi$
$402$	−3283.87	−0.407424
$403$	−1797.71	−0.222209
$404$	−575.911	−0.0709223
$405$	0	0
$406$	2360.42	0.288536
$407$	285.315	0.0347483
$408$	−4582.24	−0.556016
$409$	−2109.05	−0.254978	−0.127489	−	0.991840i	$-0.540692\pi$
$409$	−2109.05	−0.254978	−0.127489	+	0.991840i	$0.540692\pi$
$410$	0	0
$411$	4809.40	0.577202
$412$	−3601.94	−0.430716
$413$	−5324.30	−0.634363
$414$	−6693.62	−0.794622
$415$	0	0
$416$	−3468.63	−0.408806
$417$	−7293.37	−0.856494
$418$	−1114.49	−0.130410
$419$	−6905.91	−0.805193	−0.402597	−	0.915377i	$-0.631892\pi$
$419$	−6905.91	−0.805193	−0.402597	+	0.915377i	$0.631892\pi$
$420$	0	0
$421$	−9647.54	−1.11685	−0.558423	−	0.829556i	$-0.688593\pi$
$421$	−9647.54	−1.11685	−0.558423	+	0.829556i	$0.688593\pi$
$422$	−3725.91	−0.429797
$423$	−4261.45	−0.489831
$424$	−2290.25	−0.262321
$425$	0	0
$426$	−7051.94	−0.802037
$427$	−1391.23	−0.157673
$428$	3439.74	0.388473
$429$	168.000	0.0189070
$430$	0	0
$431$	−13002.7	−1.45318	−0.726589	−	0.687073i	$-0.758895\pi$
$431$	−13002.7	−1.45318	−0.726589	+	0.687073i	$0.758895\pi$
$432$	2154.06	0.239902
$433$	−7356.07	−0.816420	−0.408210	−	0.912888i	$-0.633847\pi$
$433$	−7356.07	−0.816420	−0.408210	+	0.912888i	$0.633847\pi$
$434$	−2331.08	−0.257824
$435$	0	0
$436$	−3489.55	−0.383300
$437$	−22628.4	−2.47703
$438$	6581.54	0.717987
$439$	6909.21	0.751159	0.375579	−	0.926790i	$-0.377444\pi$
$439$	6909.21	0.751159	0.375579	+	0.926790i	$0.377444\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	−8245.50	−0.887327
$443$	14812.6	1.58864	0.794318	−	0.607502i	$-0.207828\pi$
$443$	14812.6	1.58864	0.794318	+	0.607502i	$0.207828\pi$
$444$	1302.06	0.139173
$445$	0	0
$446$	−16511.0	−1.75296
$447$	−7024.72	−0.743306
$448$	−30.0561	−0.00316968
$449$	−10654.5	−1.11986	−0.559932	−	0.828538i	$-0.689173\pi$
$449$	−10654.5	−1.11986	−0.559932	+	0.828538i	$0.689173\pi$
$450$	0	0
$451$	1442.76	0.150636
$452$	4985.68	0.518820
$453$	6312.21	0.654688
$454$	−19220.3	−1.98691
$455$	0	0
$456$	4018.81	0.412715
$457$	5855.16	0.599328	0.299664	−	0.954045i	$-0.403125\pi$
$457$	5855.16	0.599328	0.299664	+	0.954045i	$0.403125\pi$
$458$	−1893.76	−0.193208
$459$	3307.45	0.336336
$460$	0	0
$461$	3204.74	0.323774	0.161887	−	0.986809i	$-0.448242\pi$
$461$	3204.74	0.323774	0.161887	+	0.986809i	$0.448242\pi$
$462$	217.844	0.0219373
$463$	−371.658	−0.0373054	−0.0186527	−	0.999826i	$-0.505938\pi$
$463$	−371.658	−0.0373054	−0.0186527	+	0.999826i	$0.505938\pi$
$464$	−7618.54	−0.762245
$465$	0	0
$466$	647.927	0.0644091
$467$	19752.3	1.95723	0.978614	−	0.205703i	$-0.0659482\pi$
$467$	19752.3	1.95723	0.978614	+	0.205703i	$0.0659482\pi$
$468$	766.681	0.0757262
$469$	2169.95	0.213643
$470$	0	0
$471$	−1780.40	−0.174175
$472$	9484.01	0.924866
$473$	−126.334	−0.0122809
$474$	262.320	0.0254193
$475$	0	0
$476$	−3832.00	−0.368990
$477$	1653.09	0.158679
$478$	2271.28	0.217335
$479$	−20762.0	−1.98046	−0.990232	−	0.139433i	$-0.955472\pi$
$479$	−20762.0	−1.98046	−0.990232	+	0.139433i	$0.955472\pi$
$480$	0	0
$481$	−1851.34	−0.175496
$482$	−20323.8	−1.92059
$483$	4423.07	0.416681
$484$	−5909.50	−0.554987
$485$	0	0
$486$	−858.064	−0.0800876
$487$	−17647.6	−1.64207	−0.821035	−	0.570878i	$-0.806603\pi$
$487$	−17647.6	−1.64207	−0.821035	+	0.570878i	$0.806603\pi$
$488$	2478.15	0.229878
$489$	6536.12	0.604445
$490$	0	0
$491$	−5637.46	−0.518157	−0.259078	−	0.965856i	$-0.583419\pi$
$491$	−5637.46	−0.518157	−0.259078	+	0.965856i	$0.583419\pi$
$492$	6584.16	0.603327
$493$	−11697.9	−1.06865
$494$	7231.64	0.658638
$495$	0	0
$496$	7523.86	0.681112
$497$	4659.85	0.420569
$498$	−4308.58	−0.387695
$499$	−17474.1	−1.56764	−0.783818	−	0.620991i	$-0.786730\pi$
$499$	−17474.1	−1.56764	−0.783818	+	0.620991i	$0.786730\pi$
$500$	0	0
$501$	2398.51	0.213887
$502$	−18122.7	−1.61127
$503$	−7444.81	−0.659936	−0.329968	−	0.943992i	$-0.607038\pi$
$503$	−7444.81	−0.659936	−0.329968	+	0.943992i	$0.607038\pi$
$504$	−785.539	−0.0694260
$505$	0	0
$506$	2184.90	0.191958
$507$	5500.89	0.481860
$508$	8383.53	0.732203
$509$	−3384.48	−0.294724	−0.147362	−	0.989083i	$-0.547078\pi$
$509$	−3384.48	−0.294724	−0.147362	+	0.989083i	$0.547078\pi$
$510$	0	0
$511$	−4349.02	−0.376495
$512$	6558.89	0.566142
$513$	−2900.77	−0.249653
$514$	−17803.0	−1.52774
$515$	0	0
$516$	−576.536	−0.0491872
$517$	1391.00	0.118329
$518$	−2400.62	−0.203624
$519$	−4333.07	−0.366476
$520$	0	0
$521$	2973.12	0.250009	0.125005	−	0.992156i	$-0.460105\pi$
$521$	2973.12	0.250009	0.125005	+	0.992156i	$0.460105\pi$
$522$	3034.82	0.254465
$523$	−2689.02	−0.224823	−0.112412	−	0.993662i	$-0.535858\pi$
$523$	−2689.02	−0.224823	−0.112412	+	0.993662i	$0.535858\pi$
$524$	1627.57	0.135689
$525$	0	0
$526$	−26755.3	−2.21785
$527$	11552.5	0.954903
$528$	−703.121	−0.0579534
$529$	32194.9	2.64608
$530$	0	0
$531$	−6845.53	−0.559455
$532$	3360.82	0.273891
$533$	−9361.72	−0.760790
$534$	−2772.83	−0.224704
$535$	0	0
$536$	−3865.25	−0.311480
$537$	−10030.5	−0.806046
$538$	3612.69	0.289506
$539$	−143.949	−0.0115034
$540$	0	0
$541$	−14429.5	−1.14671	−0.573356	−	0.819306i	$-0.694359\pi$
$541$	−14429.5	−1.14671	−0.573356	+	0.819306i	$0.694359\pi$
$542$	−7952.03	−0.630201
$543$	−6755.44	−0.533893
$544$	22290.1	1.75677
$545$	0	0
$546$	−1413.54	−0.110795
$547$	−13811.2	−1.07957	−0.539784	−	0.841804i	$-0.681494\pi$
$547$	−13811.2	−1.07957	−0.539784	+	0.841804i	$0.681494\pi$
$548$	−7164.19	−0.558466
$549$	−1788.72	−0.139054
$550$	0	0
$551$	10259.5	0.793229
$552$	−7878.68	−0.607498
$553$	−173.338	−0.0133293
$554$	30476.2	2.33720
$555$	0	0
$556$	10864.4	0.828692
$557$	6033.26	0.458954	0.229477	−	0.973314i	$-0.426298\pi$
$557$	6033.26	0.458954	0.229477	+	0.973314i	$0.426298\pi$
$558$	−2997.10	−0.227379
$559$	819.751	0.0620246
$560$	0	0
$561$	−1079.60	−0.0812493
$562$	−26559.8	−1.99352
$563$	6958.47	0.520896	0.260448	−	0.965488i	$-0.416130\pi$
$563$	6958.47	0.520896	0.260448	+	0.965488i	$0.416130\pi$
$564$	6347.95	0.473931
$565$	0	0
$566$	−52.3202	−0.00388548
$567$	567.000	0.0419961
$568$	−8300.44	−0.613166
$569$	−13396.4	−0.987009	−0.493505	−	0.869743i	$-0.664284\pi$
$569$	−13396.4	−0.987009	−0.493505	+	0.869743i	$0.664284\pi$
$570$	0	0
$571$	−8055.84	−0.590414	−0.295207	−	0.955433i	$-0.595389\pi$
$571$	−8055.84	−0.590414	−0.295207	+	0.955433i	$0.595389\pi$
$572$	−250.257	−0.0182933
$573$	3005.78	0.219142
$574$	−12139.3	−0.882724
$575$	0	0
$576$	−38.6435	−0.00279539
$577$	21456.9	1.54812	0.774059	−	0.633114i	$-0.218223\pi$
$577$	21456.9	1.54812	0.774059	+	0.633114i	$0.218223\pi$
$578$	35638.9	2.56468
$579$	−12164.9	−0.873153
$580$	0	0
$581$	2847.07	0.203298
$582$	−10644.0	−0.758087
$583$	−539.596	−0.0383324
$584$	7746.76	0.548910
$585$	0	0
$586$	−24412.1	−1.72091
$587$	−20156.3	−1.41728	−0.708638	−	0.705572i	$-0.750690\pi$
$587$	−20156.3	−1.41728	−0.708638	+	0.705572i	$0.750690\pi$
$588$	−656.924	−0.0460733
$589$	−10132.0	−0.708797
$590$	0	0
$591$	−15420.7	−1.07330
$592$	7748.30	0.537928
$593$	−599.307	−0.0415018	−0.0207509	−	0.999785i	$-0.506606\pi$
$593$	−599.307	−0.0415018	−0.0207509	+	0.999785i	$0.506606\pi$
$594$	280.086	0.0193469
$595$	0	0
$596$	10464.2	0.719177
$597$	−1756.89	−0.120444
$598$	−14177.3	−0.969485
$599$	−5493.05	−0.374691	−0.187346	−	0.982294i	$-0.559988\pi$
$599$	−5493.05	−0.374691	−0.187346	+	0.982294i	$0.559988\pi$
$600$	0	0
$601$	24292.8	1.64879	0.824396	−	0.566014i	$-0.191515\pi$
$601$	24292.8	1.64879	0.824396	+	0.566014i	$0.191515\pi$
$602$	1062.97	0.0719655
$603$	2789.93	0.188416
$604$	−9402.82	−0.633436
$605$	0	0
$606$	1365.19	0.0915131
$607$	−3029.50	−0.202576	−0.101288	−	0.994857i	$-0.532296\pi$
$607$	−3029.50	−0.202576	−0.101288	+	0.994857i	$0.532296\pi$
$608$	−19549.3	−1.30400
$609$	−2005.38	−0.133435
$610$	0	0
$611$	−9025.87	−0.597623
$612$	−4926.85	−0.325419
$613$	19339.6	1.27426	0.637129	−	0.770757i	$-0.280122\pi$
$613$	19339.6	1.27426	0.637129	+	0.770757i	$0.280122\pi$
$614$	26992.4	1.77414
$615$	0	0
$616$	256.412	0.0167713
$617$	5743.91	0.374783	0.187391	−	0.982285i	$-0.439997\pi$
$617$	5743.91	0.374783	0.187391	+	0.982285i	$0.439997\pi$
$618$	8538.35	0.555765
$619$	−8243.35	−0.535264	−0.267632	−	0.963521i	$-0.586241\pi$
$619$	−8243.35	−0.535264	−0.267632	+	0.963521i	$0.586241\pi$
$620$	0	0
$621$	5686.81	0.367478
$622$	26815.4	1.72861
$623$	1832.26	0.117830
$624$	4562.37	0.292694
$625$	0	0
$626$	32231.6	2.05788
$627$	946.856	0.0603091
$628$	2652.13	0.168521
$629$	11897.1	0.754162
$630$	0	0
$631$	−4376.56	−0.276114	−0.138057	−	0.990424i	$-0.544086\pi$
$631$	−4376.56	−0.276114	−0.138057	+	0.990424i	$0.544086\pi$
$632$	308.762	0.0194334
$633$	3165.48	0.198762
$634$	17407.8	1.09046
$635$	0	0
$636$	−2462.49	−0.153528
$637$	934.051	0.0580980
$638$	−990.613	−0.0614714
$639$	5991.23	0.370907
$640$	0	0
$641$	11836.6	0.729357	0.364678	−	0.931133i	$-0.381179\pi$
$641$	11836.6	0.729357	0.364678	+	0.931133i	$0.381179\pi$
$642$	−8153.86	−0.501257
$643$	−1448.21	−0.0888209	−0.0444104	−	0.999013i	$-0.514141\pi$
$643$	−1448.21	−0.0888209	−0.0444104	+	0.999013i	$0.514141\pi$
$644$	−6588.72	−0.403155
$645$	0	0
$646$	−46472.0	−2.83037
$647$	8732.95	0.530646	0.265323	−	0.964160i	$-0.414521\pi$
$647$	8732.95	0.530646	0.265323	+	0.964160i	$0.414521\pi$
$648$	−1009.98	−0.0612279
$649$	2234.49	0.135149
$650$	0	0
$651$	1980.46	0.119232
$652$	−9736.37	−0.584824
$653$	21978.4	1.31712	0.658562	−	0.752527i	$-0.271165\pi$
$653$	21978.4	1.31712	0.658562	+	0.752527i	$0.271165\pi$
$654$	8271.91	0.494583
$655$	0	0
$656$	39181.1	2.33196
$657$	−5591.59	−0.332038
$658$	−11703.8	−0.693406
$659$	−27761.7	−1.64103	−0.820516	−	0.571623i	$-0.806314\pi$
$659$	−27761.7	−1.64103	−0.820516	+	0.571623i	$0.806314\pi$
$660$	0	0
$661$	−8573.72	−0.504507	−0.252254	−	0.967661i	$-0.581172\pi$
$661$	−8573.72	−0.504507	−0.252254	+	0.967661i	$0.581172\pi$
$662$	4313.86	0.253267
$663$	7005.27	0.410350
$664$	−5071.39	−0.296398
$665$	0	0
$666$	−3086.51	−0.179579
$667$	−20113.2	−1.16760
$668$	−3572.87	−0.206944
$669$	14027.6	0.810668
$670$	0	0
$671$	583.868	0.0335916
$672$	3821.22	0.219356
$673$	−27159.2	−1.55559	−0.777795	−	0.628518i	$-0.783662\pi$
$673$	−27159.2	−1.55559	−0.777795	+	0.628518i	$0.783662\pi$
$674$	30879.1	1.76472
$675$	0	0
$676$	−8194.26	−0.466219
$677$	1392.30	0.0790404	0.0395202	−	0.999219i	$-0.487417\pi$
$677$	1392.30	0.0790404	0.0395202	+	0.999219i	$0.487417\pi$
$678$	−11818.5	−0.669448
$679$	7033.42	0.397523
$680$	0	0
$681$	16329.3	0.918857
$682$	978.302	0.0549283
$683$	8675.09	0.486007	0.243004	−	0.970025i	$-0.421867\pi$
$683$	8675.09	0.486007	0.243004	+	0.970025i	$0.421867\pi$
$684$	4321.05	0.241549
$685$	0	0
$686$	1211.18	0.0674096
$687$	1608.91	0.0893504
$688$	−3430.86	−0.190117
$689$	3501.30	0.193598
$690$	0	0
$691$	−21426.0	−1.17957	−0.589785	−	0.807561i	$-0.700787\pi$
$691$	−21426.0	−1.17957	−0.589785	+	0.807561i	$0.700787\pi$
$692$	6454.65	0.354579
$693$	−185.078	−0.0101451
$694$	−16206.3	−0.886433
$695$	0	0
$696$	3572.11	0.194541
$697$	60160.4	3.26935
$698$	−14088.8	−0.763997
$699$	−550.470	−0.0297864
$700$	0	0
$701$	24840.5	1.33839	0.669197	−	0.743085i	$-0.266638\pi$
$701$	24840.5	1.33839	0.669197	+	0.743085i	$0.266638\pi$
$702$	−1817.40	−0.0977116
$703$	−10434.2	−0.559793
$704$	12.6139	0.000675288 0
$705$	0	0
$706$	−8532.45	−0.454848
$707$	−902.101	−0.0479873
$708$	10197.3	0.541295
$709$	12525.0	0.663450	0.331725	−	0.943376i	$-0.392369\pi$
$709$	12525.0	0.663450	0.331725	+	0.943376i	$0.392369\pi$
$710$	0	0
$711$	−222.863	−0.0117553
$712$	−3263.74	−0.171789
$713$	19863.3	1.04332
$714$	9083.69	0.476118
$715$	0	0
$716$	14941.6	0.779881
$717$	−1929.65	−0.100508
$718$	−9732.66	−0.505877
$719$	28085.0	1.45674	0.728369	−	0.685185i	$-0.240279\pi$
$719$	28085.0	1.45674	0.728369	+	0.685185i	$0.240279\pi$
$720$	0	0
$721$	−5642.05	−0.291430
$722$	16537.9	0.852460
$723$	17266.8	0.888189
$724$	10063.1	0.516562
$725$	0	0
$726$	14008.4	0.716115
$727$	14326.2	0.730851	0.365426	−	0.930841i	$-0.380923\pi$
$727$	14326.2	0.730851	0.365426	+	0.930841i	$0.380923\pi$
$728$	−1663.79	−0.0847037
$729$	729.000	0.0370370
$730$	0	0
$731$	−5267.89	−0.266539
$732$	2664.53	0.134541
$733$	−6727.85	−0.339016	−0.169508	−	0.985529i	$-0.554218\pi$
$733$	−6727.85	−0.339016	−0.169508	+	0.985529i	$0.554218\pi$
$734$	−39240.8	−1.97331
$735$	0	0
$736$	38325.5	1.91943
$737$	−910.677	−0.0455159
$738$	−15607.6	−0.778490
$739$	−3418.51	−0.170165	−0.0850826	−	0.996374i	$-0.527115\pi$
$739$	−3418.51	−0.170165	−0.0850826	+	0.996374i	$0.527115\pi$
$740$	0	0
$741$	−6143.91	−0.304591
$742$	4540.11	0.224626
$743$	−8095.50	−0.399724	−0.199862	−	0.979824i	$-0.564049\pi$
$743$	−8095.50	−0.399724	−0.199862	+	0.979824i	$0.564049\pi$
$744$	−3527.72	−0.173834
$745$	0	0
$746$	−21508.4	−1.05560
$747$	3660.51	0.179292
$748$	1608.20	0.0786119
$749$	5387.99	0.262847
$750$	0	0
$751$	13446.8	0.653371	0.326686	−	0.945133i	$-0.394068\pi$
$751$	13446.8	0.653371	0.326686	+	0.945133i	$0.394068\pi$
$752$	37775.4	1.83182
$753$	15396.8	0.745141
$754$	6427.84	0.310462
$755$	0	0
$756$	−844.617	−0.0406328
$757$	2593.24	0.124508	0.0622541	−	0.998060i	$-0.480171\pi$
$757$	2593.24	0.124508	0.0622541	+	0.998060i	$0.480171\pi$
$758$	14069.0	0.674156
$759$	−1856.26	−0.0887722
$760$	0	0
$761$	27079.4	1.28992	0.644959	−	0.764217i	$-0.276875\pi$
$761$	27079.4	1.28992	0.644959	+	0.764217i	$0.276875\pi$
$762$	−19873.0	−0.944782
$763$	−5465.99	−0.259348
$764$	−4477.48	−0.212028
$765$	0	0
$766$	−1124.58	−0.0530451
$767$	−14499.0	−0.682568
$768$	−15363.3	−0.721842
$769$	2138.72	0.100292	0.0501458	−	0.998742i	$-0.484031\pi$
$769$	2138.72	0.100292	0.0501458	+	0.998742i	$0.484031\pi$
$770$	0	0
$771$	15125.2	0.706513
$772$	18121.1	0.844810
$773$	−25864.0	−1.20345	−0.601724	−	0.798704i	$-0.705519\pi$
$773$	−25864.0	−1.20345	−0.601724	+	0.798704i	$0.705519\pi$
$774$	1366.67	0.0634676
$775$	0	0
$776$	−12528.4	−0.579566
$777$	2039.53	0.0941671
$778$	13720.1	0.632247
$779$	−52763.1	−2.42675
$780$	0	0
$781$	−1955.63	−0.0896006
$782$	91106.2	4.16618
$783$	−2578.34	−0.117679
$784$	−3909.23	−0.178081
$785$	0	0
$786$	−3858.14	−0.175083
$787$	32371.3	1.46621	0.733107	−	0.680113i	$-0.238069\pi$
$787$	32371.3	1.46621	0.733107	+	0.680113i	$0.238069\pi$
$788$	22971.0	1.03846
$789$	22731.0	1.02566
$790$	0	0
$791$	7809.52	0.351043
$792$	329.673	0.0147909
$793$	−3788.57	−0.169654
$794$	−16970.8	−0.758526
$795$	0	0
$796$	2617.11	0.116534
$797$	−2024.33	−0.0899691	−0.0449845	−	0.998988i	$-0.514324\pi$
$797$	−2024.33	−0.0899691	−0.0449845	+	0.998988i	$0.514324\pi$
$798$	−7966.76	−0.353409
$799$	58002.1	2.56817
$800$	0	0
$801$	2355.76	0.103916
$802$	12777.7	0.562589
$803$	1825.18	0.0802109
$804$	−4155.95	−0.182300
$805$	0	0
$806$	−6347.95	−0.277416
$807$	−3069.29	−0.133884
$808$	1606.88	0.0699628
$809$	12391.7	0.538526	0.269263	−	0.963067i	$-0.413220\pi$
$809$	12391.7	0.538526	0.269263	+	0.963067i	$0.413220\pi$
$810$	0	0
$811$	14654.5	0.634511	0.317256	−	0.948340i	$-0.397239\pi$
$811$	14654.5	0.634511	0.317256	+	0.948340i	$0.397239\pi$
$812$	2987.26	0.129104
$813$	6755.94	0.291441
$814$	1007.49	0.0433813
$815$	0	0
$816$	−29318.7	−1.25780
$817$	4620.16	0.197844
$818$	−7447.33	−0.318325
$819$	1200.92	0.0512376
$820$	0	0
$821$	23887.9	1.01546	0.507731	−	0.861516i	$-0.330485\pi$
$821$	23887.9	1.01546	0.507731	+	0.861516i	$0.330485\pi$
$822$	16982.6	0.720604
$823$	4008.41	0.169774	0.0848871	−	0.996391i	$-0.472947\pi$
$823$	4008.41	0.169774	0.0848871	+	0.996391i	$0.472947\pi$
$824$	10050.0	0.424889
$825$	0	0
$826$	−18800.8	−0.791966
$827$	45110.4	1.89679	0.948394	−	0.317096i	$-0.102708\pi$
$827$	45110.4	1.89679	0.948394	+	0.317096i	$0.102708\pi$
$828$	−8471.21	−0.355549
$829$	16165.4	0.677260	0.338630	−	0.940920i	$-0.390036\pi$
$829$	16165.4	0.677260	0.338630	+	0.940920i	$0.390036\pi$
$830$	0	0
$831$	−25892.2	−1.08085
$832$	−81.8481	−0.00341054
$833$	−6002.41	−0.249665
$834$	−25753.8	−1.06928
$835$	0	0
$836$	−1410.46	−0.0583514
$837$	2546.30	0.105153
$838$	−24385.7	−1.00524
$839$	−25244.4	−1.03878	−0.519388	−	0.854538i	$-0.673840\pi$
$839$	−25244.4	−1.03878	−0.519388	+	0.854538i	$0.673840\pi$
$840$	0	0
$841$	−15269.9	−0.626096
$842$	−34066.7	−1.39432
$843$	22564.9	0.921916
$844$	−4715.37	−0.192310
$845$	0	0
$846$	−15047.7	−0.611526
$847$	−9256.59	−0.375514
$848$	−14653.8	−0.593412
$849$	44.4506	0.00179687
$850$	0	0
$851$	20455.8	0.823990
$852$	−8924.68	−0.358867
$853$	30168.1	1.21094	0.605472	−	0.795867i	$-0.292984\pi$
$853$	30168.1	1.21094	0.605472	+	0.795867i	$0.292984\pi$
$854$	−4912.61	−0.196846
$855$	0	0
$856$	−9597.44	−0.383217
$857$	13393.6	0.533857	0.266929	−	0.963716i	$-0.413991\pi$
$857$	13393.6	0.533857	0.266929	+	0.963716i	$0.413991\pi$
$858$	593.230	0.0236043
$859$	19060.4	0.757081	0.378541	−	0.925585i	$-0.376426\pi$
$859$	19060.4	0.757081	0.378541	+	0.925585i	$0.376426\pi$
$860$	0	0
$861$	10313.4	0.408222
$862$	−45914.3	−1.81421
$863$	9466.86	0.373413	0.186707	−	0.982416i	$-0.440219\pi$
$863$	9466.86	0.373413	0.186707	+	0.982416i	$0.440219\pi$
$864$	4913.00	0.193453
$865$	0	0
$866$	−25975.2	−1.01925
$867$	−30278.3	−1.18605
$868$	−2950.13	−0.115362
$869$	72.7461	0.00283975
$870$	0	0
$871$	5909.15	0.229878
$872$	9736.39	0.378115
$873$	9042.97	0.350582
$874$	−79903.8	−3.09243
$875$	0	0
$876$	8329.37	0.321259
$877$	−37740.6	−1.45315	−0.726573	−	0.687090i	$-0.758888\pi$
$877$	−37740.6	−1.45315	−0.726573	+	0.687090i	$0.758888\pi$
$878$	24397.3	0.937778
$879$	20740.2	0.795845
$880$	0	0
$881$	25991.5	0.993957	0.496979	−	0.867763i	$-0.334443\pi$
$881$	25991.5	0.993957	0.496979	+	0.867763i	$0.334443\pi$
$882$	1557.23	0.0594496
$883$	−39420.3	−1.50238	−0.751189	−	0.660087i	$-0.770519\pi$
$883$	−39420.3	−1.50238	−0.751189	+	0.660087i	$0.770519\pi$
$884$	−10435.2	−0.397030
$885$	0	0
$886$	52305.0	1.98332
$887$	−46005.2	−1.74149	−0.870745	−	0.491735i	$-0.836363\pi$
$887$	−46005.2	−1.74149	−0.870745	+	0.491735i	$0.836363\pi$
$888$	−3632.95	−0.137290
$889$	13131.9	0.495421
$890$	0	0
$891$	−237.957	−0.00894710
$892$	−20895.8	−0.784353
$893$	−50870.2	−1.90628
$894$	−24805.2	−0.927975
$895$	0	0
$896$	10083.8	0.375978
$897$	12044.8	0.448345
$898$	−37622.6	−1.39809
$899$	−9005.81	−0.334105
$900$	0	0
$901$	−22500.1	−0.831950
$902$	5094.58	0.188061
$903$	−903.081	−0.0332809
$904$	−13910.8	−0.511801
$905$	0	0
$906$	22289.2	0.817340
$907$	2838.97	0.103932	0.0519661	−	0.998649i	$-0.483451\pi$
$907$	2838.97	0.103932	0.0519661	+	0.998649i	$0.483451\pi$
$908$	−24324.6	−0.889030
$909$	−1159.84	−0.0423208
$910$	0	0
$911$	39890.9	1.45076	0.725382	−	0.688347i	$-0.241663\pi$
$911$	39890.9	1.45076	0.725382	+	0.688347i	$0.241663\pi$
$912$	25713.7	0.933626
$913$	−1194.85	−0.0433119
$914$	20675.3	0.748226
$915$	0	0
$916$	−2396.67	−0.0864500
$917$	2549.42	0.0918094
$918$	11679.0	0.419897
$919$	−646.475	−0.0232048	−0.0116024	−	0.999933i	$-0.503693\pi$
$919$	−646.475	−0.0232048	−0.0116024	+	0.999933i	$0.503693\pi$
$920$	0	0
$921$	−22932.4	−0.820463
$922$	11316.3	0.404213
$923$	12689.6	0.452528
$924$	275.696	0.00981574
$925$	0	0
$926$	−1312.37	−0.0465737
$927$	−7254.07	−0.257017
$928$	−17376.4	−0.614665
$929$	51188.2	1.80778	0.903892	−	0.427760i	$-0.140697\pi$
$929$	51188.2	1.80778	0.903892	+	0.427760i	$0.140697\pi$
$930$	0	0
$931$	5264.35	0.185319
$932$	819.993	0.0288195
$933$	−22782.0	−0.799409
$934$	69747.8	2.44349
$935$	0	0
$936$	−2139.16	−0.0747017
$937$	−29786.1	−1.03849	−0.519247	−	0.854624i	$-0.673788\pi$
$937$	−29786.1	−1.03849	−0.519247	+	0.854624i	$0.673788\pi$
$938$	7662.36	0.266722
$939$	−27383.5	−0.951680
$940$	0	0
$941$	−44817.4	−1.55261	−0.776304	−	0.630358i	$-0.782908\pi$
$941$	−44817.4	−1.55261	−0.776304	+	0.630358i	$0.782908\pi$
$942$	−6286.82	−0.217448
$943$	103439.	3.57206
$944$	60682.0	2.09219
$945$	0	0
$946$	−446.103	−0.0153320
$947$	−54697.1	−1.87689	−0.938446	−	0.345425i	$-0.887735\pi$
$947$	−54697.1	−1.87689	−0.938446	+	0.345425i	$0.887735\pi$
$948$	331.983	0.0113737
$949$	−11843.2	−0.405105
$950$	0	0
$951$	−14789.4	−0.504290
$952$	10691.9	0.363998
$953$	7577.51	0.257565	0.128783	−	0.991673i	$-0.458893\pi$
$953$	7577.51	0.257565	0.128783	+	0.991673i	$0.458893\pi$
$954$	5837.29	0.198102
$955$	0	0
$956$	2874.46	0.0972454
$957$	841.612	0.0284278
$958$	−73313.4	−2.47249
$959$	−11221.9	−0.377868
$960$	0	0
$961$	−20897.1	−0.701457
$962$	−6537.32	−0.219097
$963$	6927.41	0.231810
$964$	−25721.1	−0.859357
$965$	0	0
$966$	15618.4	0.520202
$967$	−50779.0	−1.68867	−0.844334	−	0.535817i	$-0.820004\pi$
$967$	−50779.0	−1.68867	−0.844334	+	0.535817i	$0.820004\pi$
$968$	16488.5	0.547478
$969$	39482.0	1.30892
$970$	0	0
$971$	−15313.2	−0.506102	−0.253051	−	0.967453i	$-0.581434\pi$
$971$	−15313.2	−0.506102	−0.253051	+	0.967453i	$0.581434\pi$
$972$	−1085.94	−0.0358348
$973$	17017.9	0.560707
$974$	−62315.9	−2.05003
$975$	0	0
$976$	15856.1	0.520021
$977$	−46620.4	−1.52663	−0.763316	−	0.646025i	$-0.776430\pi$
$977$	−46620.4	−1.52663	−0.763316	+	0.646025i	$0.776430\pi$
$978$	23079.9	0.754615
$979$	−768.957	−0.0251031
$980$	0	0
$981$	−7027.70	−0.228723
$982$	−19906.6	−0.646889
$983$	2824.37	0.0916414	0.0458207	−	0.998950i	$-0.485410\pi$
$983$	2824.37	0.0916414	0.0458207	+	0.998950i	$0.485410\pi$
$984$	−18370.9	−0.595165
$985$	0	0
$986$	−41306.6	−1.33415
$987$	9943.38	0.320670
$988$	9152.11	0.294704
$989$	−9057.59	−0.291218
$990$	0	0
$991$	16951.4	0.543370	0.271685	−	0.962386i	$-0.412419\pi$
$991$	16951.4	0.543370	0.271685	+	0.962386i	$0.412419\pi$
$992$	17160.5	0.549239
$993$	−3665.00	−0.117125
$994$	16454.5	0.525056
$995$	0	0
$996$	−5452.79	−0.173472
$997$	−23847.8	−0.757540	−0.378770	−	0.925491i	$-0.623653\pi$
$997$	−23847.8	−0.757540	−0.378770	+	0.925491i	$0.623653\pi$
$998$	−61703.4	−1.95710
$999$	2622.26	0.0830476

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.4.a.k.1.2		2
3.2	odd	2		1575.4.a.w.1.1		2
5.2	odd	4		525.4.d.h.274.3		4
5.3	odd	4		525.4.d.h.274.2		4
5.4	even	2		105.4.a.f.1.1	✓	2
15.14	odd	2		315.4.a.i.1.2		2
20.19	odd	2		1680.4.a.bg.1.1		2
35.34	odd	2		735.4.a.p.1.1		2
105.104	even	2		2205.4.a.z.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.a.f.1.1	✓	2	5.4	even	2
315.4.a.i.1.2		2	15.14	odd	2
525.4.a.k.1.2		2	1.1	even	1	trivial
525.4.d.h.274.2		4	5.3	odd	4
525.4.d.h.274.3		4	5.2	odd	4
735.4.a.p.1.1		2	35.34	odd	2
1575.4.a.w.1.1		2	3.2	odd	2
1680.4.a.bg.1.1		2	20.19	odd	2
2205.4.a.z.1.2		2	105.104	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	525.a (trivial)

\(f(q)\)	\(=\)	\(q+3.53113 q^{2} -3.00000 q^{3} +4.46887 q^{4} -10.5934 q^{6} +7.00000 q^{7} -12.4689 q^{8} +9.00000 q^{9} -2.93774 q^{11} -13.4066 q^{12} +19.0623 q^{13} +24.7179 q^{14} -79.7802 q^{16} -122.498 q^{17} +31.7802 q^{18} +107.436 q^{19} -21.0000 q^{21} -10.3735 q^{22} -210.623 q^{23} +37.4066 q^{24} +67.3113 q^{26} -27.0000 q^{27} +31.2821 q^{28} +95.4942 q^{29} -94.3074 q^{31} -181.963 q^{32} +8.81323 q^{33} -432.556 q^{34} +40.2198 q^{36} -97.1206 q^{37} +379.370 q^{38} -57.1868 q^{39} -491.113 q^{41} -74.1537 q^{42} +43.0039 q^{43} -13.1284 q^{44} -743.735 q^{46} -473.494 q^{47} +239.340 q^{48} +49.0000 q^{49} +367.494 q^{51} +85.1868 q^{52} +183.677 q^{53} -95.3405 q^{54} -87.2821 q^{56} -322.307 q^{57} +337.202 q^{58} -760.615 q^{59} -198.747 q^{61} -333.012 q^{62} +63.0000 q^{63} -4.29373 q^{64} +31.1206 q^{66} +309.992 q^{67} -547.428 q^{68} +631.868 q^{69} +665.693 q^{71} -112.220 q^{72} -621.288 q^{73} -342.945 q^{74} +480.117 q^{76} -20.5642 q^{77} -201.934 q^{78} -24.7626 q^{79} +81.0000 q^{81} -1734.18 q^{82} +406.724 q^{83} -93.8463 q^{84} +151.852 q^{86} -286.483 q^{87} +36.6303 q^{88} +261.751 q^{89} +133.436 q^{91} -941.245 q^{92} +282.922 q^{93} -1671.97 q^{94} +545.889 q^{96} +1004.77 q^{97} +173.025 q^{98} -26.4397 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - 6 q^{3} + 17 q^{4} + 3 q^{6} + 14 q^{7} - 33 q^{8} + 18 q^{9} - 22 q^{11} - 51 q^{12} + 22 q^{13} - 7 q^{14} - 87 q^{16} - 116 q^{17} - 9 q^{18} + 102 q^{19} - 42 q^{21} + 76 q^{22} - 260 q^{23}+ \cdots - 198 q^{99}+O(q^{100}) \) 2 * q - q^2 - 6 * q^3 + 17 * q^4 + 3 * q^6 + 14 * q^7 - 33 * q^8 + 18 * q^9 - 22 * q^11 - 51 * q^12 + 22 * q^13 - 7 * q^14 - 87 * q^16 - 116 * q^17 - 9 * q^18 + 102 * q^19 - 42 * q^21 + 76 * q^22 - 260 * q^23 + 99 * q^24 + 54 * q^26 - 54 * q^27 + 119 * q^28 - 196 * q^29 + 150 * q^31 + 15 * q^32 + 66 * q^33 - 462 * q^34 + 153 * q^36 + 96 * q^37 + 404 * q^38 - 66 * q^39 - 176 * q^41 + 21 * q^42 + 344 * q^43 - 252 * q^44 - 520 * q^46 - 560 * q^47 + 261 * q^48 + 98 * q^49 + 348 * q^51 + 122 * q^52 - 326 * q^53 + 27 * q^54 - 231 * q^56 - 306 * q^57 + 1658 * q^58 - 844 * q^59 - 204 * q^61 - 1440 * q^62 + 126 * q^63 - 839 * q^64 - 228 * q^66 + 104 * q^67 - 466 * q^68 + 780 * q^69 + 1670 * q^71 - 297 * q^72 + 386 * q^73 - 1218 * q^74 + 412 * q^76 - 154 * q^77 - 162 * q^78 - 888 * q^79 + 162 * q^81 - 3162 * q^82 - 928 * q^83 - 357 * q^84 - 1212 * q^86 + 588 * q^87 + 428 * q^88 + 588 * q^89 + 154 * q^91 - 1560 * q^92 - 450 * q^93 - 1280 * q^94 - 45 * q^96 - 522 * q^97 - 49 * q^98 - 198 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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