LMFDB - Embedded newform 5075.2.a.ba.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5075,2,Mod(1,5075)] 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("5075.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5075, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5075 = 5^{2} \cdot 7 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5075.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,-1,-4,5,0,-6,-8] 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:	$40.5240790258$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 10x^{6} + 11x^{5} + 25x^{4} - 25x^{3} - 16x^{2} + 9x + 4 $ x^8 - x^7 - 10x^6 + 11x^5 + 25x^4 - 25x^3 - 16x^2 + 9x + 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1015)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.742598$ of defining polynomial
Character	$\chi$	$=$	5075.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.742598 q^{2} -1.24299 q^{3} -1.44855 q^{4} +0.923043 q^{6} -1.00000 q^{7} +2.56088 q^{8} -1.45497 q^{9} +2.72358 q^{11} +1.80053 q^{12} -2.45958 q^{13} +0.742598 q^{14} +0.995389 q^{16} -1.07696 q^{17} +1.08046 q^{18} +3.42397 q^{19} +1.24299 q^{21} -2.02252 q^{22} -8.67622 q^{23} -3.18316 q^{24} +1.82648 q^{26} +5.53749 q^{27} +1.44855 q^{28} +1.00000 q^{29} +5.27754 q^{31} -5.86094 q^{32} -3.38538 q^{33} +0.799746 q^{34} +2.10760 q^{36} +8.24831 q^{37} -2.54263 q^{38} +3.05724 q^{39} -5.65526 q^{41} -0.923043 q^{42} -9.12362 q^{43} -3.94523 q^{44} +6.44294 q^{46} -2.71814 q^{47} -1.23726 q^{48} +1.00000 q^{49} +1.33865 q^{51} +3.56282 q^{52} +1.79592 q^{53} -4.11213 q^{54} -2.56088 q^{56} -4.25596 q^{57} -0.742598 q^{58} -12.2626 q^{59} -5.46390 q^{61} -3.91909 q^{62} +1.45497 q^{63} +2.36155 q^{64} +2.51398 q^{66} +6.14811 q^{67} +1.56002 q^{68} +10.7845 q^{69} +7.91130 q^{71} -3.72601 q^{72} -2.94337 q^{73} -6.12518 q^{74} -4.95978 q^{76} -2.72358 q^{77} -2.27030 q^{78} -10.6271 q^{79} -2.51815 q^{81} +4.19958 q^{82} +4.85777 q^{83} -1.80053 q^{84} +6.77518 q^{86} -1.24299 q^{87} +6.97477 q^{88} +1.52378 q^{89} +2.45958 q^{91} +12.5679 q^{92} -6.55993 q^{93} +2.01849 q^{94} +7.28510 q^{96} -2.85316 q^{97} -0.742598 q^{98} -3.96273 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{2} - 4 q^{3} + 5 q^{4} - 6 q^{6} - 8 q^{7} + 6 q^{8} + 14 q^{9} + q^{11} + 7 q^{12} + q^{13} + q^{14} + 3 q^{16} - 22 q^{17} - 13 q^{18} + 8 q^{19} + 4 q^{21} + 3 q^{22} - 9 q^{23} + 16 q^{24}+ \cdots - 63 q^{99}+O(q^{100}) $ 8 * q - q^2 - 4 * q^3 + 5 * q^4 - 6 * q^6 - 8 * q^7 + 6 * q^8 + 14 * q^9 + q^11 + 7 * q^12 + q^13 + q^14 + 3 * q^16 - 22 * q^17 - 13 * q^18 + 8 * q^19 + 4 * q^21 + 3 * q^22 - 9 * q^23 + 16 * q^24 + 27 * q^26 - q^27 - 5 * q^28 + 8 * q^29 + 3 * q^31 + 26 * q^32 - 14 * q^33 + q^34 - 13 * q^36 - 9 * q^37 + 5 * q^38 + 3 * q^39 + 24 * q^41 + 6 * q^42 - 7 * q^43 + 10 * q^44 - 27 * q^46 - 21 * q^47 + 8 * q^48 + 8 * q^49 - 8 * q^51 + 7 * q^52 + 2 * q^53 - 33 * q^54 - 6 * q^56 + 10 * q^57 - q^58 + 18 * q^59 + 26 * q^61 - 15 * q^62 - 14 * q^63 + 4 * q^64 + 55 * q^66 + 8 * q^67 - 6 * q^68 - 2 * q^69 - 12 * q^71 - 9 * q^72 + q^73 + 14 * q^74 + 15 * q^76 - q^77 - 8 * q^78 - 14 * q^79 + 40 * q^81 + 42 * q^82 + 10 * q^83 - 7 * q^84 + 19 * q^86 - 4 * q^87 + 29 * q^88 - 4 * q^89 - q^91 - 47 * q^92 + 24 * q^93 - 30 * q^94 - q^96 + 11 * q^97 - q^98 - 63 * 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 $

$2$	−0.742598	−0.525096	−0.262548	−	0.964919i	$-0.584563\pi$
$2$	−0.742598	−0.525096	−0.262548	+	0.964919i	$0.584563\pi$
$3$	−1.24299	−0.717642	−0.358821	−	0.933406i	$-0.616821\pi$
$3$	−1.24299	−0.717642	−0.358821	+	0.933406i	$0.616821\pi$
$4$	−1.44855	−0.724274
$5$	0	0
$5$	0	0
$6$	0.923043	0.376831
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	2.56088	0.905409
$9$	−1.45497	−0.484990
$10$	0	0
$11$	2.72358	0.821189	0.410595	−	0.911818i	$-0.365321\pi$
$11$	2.72358	0.821189	0.410595	+	0.911818i	$0.365321\pi$
$12$	1.80053	0.519769
$13$	−2.45958	−0.682165	−0.341083	−	0.940033i	$-0.610794\pi$
$13$	−2.45958	−0.682165	−0.341083	+	0.940033i	$0.610794\pi$
$14$	0.742598	0.198468
$15$	0	0
$16$	0.995389	0.248847
$17$	−1.07696	−0.261200	−0.130600	−	0.991435i	$-0.541690\pi$
$17$	−1.07696	−0.261200	−0.130600	+	0.991435i	$0.541690\pi$
$18$	1.08046	0.254667
$19$	3.42397	0.785511	0.392756	−	0.919643i	$-0.371522\pi$
$19$	3.42397	0.785511	0.392756	+	0.919643i	$0.371522\pi$
$20$	0	0
$21$	1.24299	0.271243
$22$	−2.02252	−0.431203
$23$	−8.67622	−1.80912	−0.904558	−	0.426350i	$-0.859799\pi$
$23$	−8.67622	−1.80912	−0.904558	+	0.426350i	$0.859799\pi$
$24$	−3.18316	−0.649760
$25$	0	0
$26$	1.82648	0.358202
$27$	5.53749	1.06569
$28$	1.44855	0.273750
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	5.27754	0.947873	0.473937	−	0.880559i	$-0.342833\pi$
$31$	5.27754	0.947873	0.473937	+	0.880559i	$0.342833\pi$
$32$	−5.86094	−1.03608
$33$	−3.38538	−0.589320
$34$	0.799746	0.137155
$35$	0	0
$36$	2.10760	0.351266
$37$	8.24831	1.35601	0.678007	−	0.735056i	$-0.262844\pi$
$37$	8.24831	1.35601	0.678007	+	0.735056i	$0.262844\pi$
$38$	−2.54263	−0.412469
$39$	3.05724	0.489550
$40$	0	0
$41$	−5.65526	−0.883203	−0.441602	−	0.897211i	$-0.645590\pi$
$41$	−5.65526	−0.883203	−0.441602	+	0.897211i	$0.645590\pi$
$42$	−0.923043	−0.142429
$43$	−9.12362	−1.39134	−0.695670	−	0.718362i	$-0.744892\pi$
$43$	−9.12362	−1.39134	−0.695670	+	0.718362i	$0.744892\pi$
$44$	−3.94523	−0.594766
$45$	0	0
$46$	6.44294	0.949960
$47$	−2.71814	−0.396482	−0.198241	−	0.980153i	$-0.563523\pi$
$47$	−2.71814	−0.396482	−0.198241	+	0.980153i	$0.563523\pi$
$48$	−1.23726	−0.178583
$49$	1.00000	0.142857
$50$	0	0
$51$	1.33865	0.187448
$52$	3.56282	0.494075
$53$	1.79592	0.246689	0.123345	−	0.992364i	$-0.460638\pi$
$53$	1.79592	0.246689	0.123345	+	0.992364i	$0.460638\pi$
$54$	−4.11213	−0.559590
$55$	0	0
$56$	−2.56088	−0.342213
$57$	−4.25596	−0.563716
$58$	−0.742598	−0.0975079
$59$	−12.2626	−1.59646	−0.798230	−	0.602353i	$-0.794230\pi$
$59$	−12.2626	−1.59646	−0.798230	+	0.602353i	$0.794230\pi$
$60$	0	0
$61$	−5.46390	−0.699581	−0.349790	−	0.936828i	$-0.613747\pi$
$61$	−5.46390	−0.699581	−0.349790	+	0.936828i	$0.613747\pi$
$62$	−3.91909	−0.497725
$63$	1.45497	0.183309
$64$	2.36155	0.295193
$65$	0	0
$66$	2.51398	0.309449
$67$	6.14811	0.751112	0.375556	−	0.926800i	$-0.377452\pi$
$67$	6.14811	0.751112	0.375556	+	0.926800i	$0.377452\pi$
$68$	1.56002	0.189181
$69$	10.7845	1.29830
$70$	0	0
$71$	7.91130	0.938899	0.469450	−	0.882959i	$-0.344452\pi$
$71$	7.91130	0.938899	0.469450	+	0.882959i	$0.344452\pi$
$72$	−3.72601	−0.439115
$73$	−2.94337	−0.344496	−0.172248	−	0.985054i	$-0.555103\pi$
$73$	−2.94337	−0.344496	−0.172248	+	0.985054i	$0.555103\pi$
$74$	−6.12518	−0.712037
$75$	0	0
$76$	−4.95978	−0.568926
$77$	−2.72358	−0.310380
$78$	−2.27030	−0.257061
$79$	−10.6271	−1.19564	−0.597819	−	0.801631i	$-0.703966\pi$
$79$	−10.6271	−1.19564	−0.597819	+	0.801631i	$0.703966\pi$
$80$	0	0
$81$	−2.51815	−0.279794
$82$	4.19958	0.463766
$83$	4.85777	0.533210	0.266605	−	0.963806i	$-0.414098\pi$
$83$	4.85777	0.533210	0.266605	+	0.963806i	$0.414098\pi$
$84$	−1.80053	−0.196454
$85$	0	0
$86$	6.77518	0.730587
$87$	−1.24299	−0.133263
$88$	6.97477	0.743513
$89$	1.52378	0.161520	0.0807600	−	0.996734i	$-0.474265\pi$
$89$	1.52378	0.161520	0.0807600	+	0.996734i	$0.474265\pi$
$90$	0	0
$91$	2.45958	0.257834
$92$	12.5679	1.31030
$93$	−6.55993	−0.680234
$94$	2.01849	0.208191
$95$	0	0
$96$	7.28510	0.743533
$97$	−2.85316	−0.289695	−0.144847	−	0.989454i	$-0.546269\pi$
$97$	−2.85316	−0.289695	−0.144847	+	0.989454i	$0.546269\pi$
$98$	−0.742598	−0.0750137
$99$	−3.96273	−0.398269
$100$	0	0
$101$	2.81363	0.279966	0.139983	−	0.990154i	$-0.455295\pi$
$101$	2.81363	0.279966	0.139983	+	0.990154i	$0.455295\pi$
$102$	−0.994078	−0.0984284
$103$	3.95445	0.389643	0.194822	−	0.980839i	$-0.437587\pi$
$103$	3.95445	0.389643	0.194822	+	0.980839i	$0.437587\pi$
$104$	−6.29871	−0.617639
$105$	0	0
$106$	−1.33365	−0.129535
$107$	−11.2230	−1.08497	−0.542484	−	0.840066i	$-0.682516\pi$
$107$	−11.2230	−1.08497	−0.542484	+	0.840066i	$0.682516\pi$
$108$	−8.02133	−0.771853
$109$	−3.55434	−0.340444	−0.170222	−	0.985406i	$-0.554448\pi$
$109$	−3.55434	−0.340444	−0.170222	+	0.985406i	$0.554448\pi$
$110$	0	0
$111$	−10.2526	−0.973132
$112$	−0.995389	−0.0940555
$113$	−6.35561	−0.597886	−0.298943	−	0.954271i	$-0.596634\pi$
$113$	−6.35561	−0.597886	−0.298943	+	0.954271i	$0.596634\pi$
$114$	3.16047	0.296005
$115$	0	0
$116$	−1.44855	−0.134494
$117$	3.57862	0.330844
$118$	9.10621	0.838294
$119$	1.07696	0.0987245
$120$	0	0
$121$	−3.58213	−0.325648
$122$	4.05748	0.367347
$123$	7.02944	0.633823
$124$	−7.64477	−0.686520
$125$	0	0
$126$	−1.08046	−0.0962549
$127$	−3.91918	−0.347771	−0.173885	−	0.984766i	$-0.555632\pi$
$127$	−3.91918	−0.347771	−0.173885	+	0.984766i	$0.555632\pi$
$128$	9.96821	0.881073
$129$	11.3406	0.998483
$130$	0	0
$131$	−9.59543	−0.838356	−0.419178	−	0.907904i	$-0.637682\pi$
$131$	−9.59543	−0.838356	−0.419178	+	0.907904i	$0.637682\pi$
$132$	4.90389	0.426829
$133$	−3.42397	−0.296895
$134$	−4.56558	−0.394406
$135$	0	0
$136$	−2.75796	−0.236493
$137$	12.9022	1.10231	0.551155	−	0.834403i	$-0.314187\pi$
$137$	12.9022	1.10231	0.551155	+	0.834403i	$0.314187\pi$
$138$	−8.00852	−0.681731
$139$	−1.46037	−0.123867	−0.0619335	−	0.998080i	$-0.519727\pi$
$139$	−1.46037	−0.123867	−0.0619335	+	0.998080i	$0.519727\pi$
$140$	0	0
$141$	3.37863	0.284532
$142$	−5.87492	−0.493012
$143$	−6.69886	−0.560187
$144$	−1.44826	−0.120689
$145$	0	0
$146$	2.18574	0.180893
$147$	−1.24299	−0.102520
$148$	−11.9481	−0.982126
$149$	−17.7888	−1.45731	−0.728657	−	0.684879i	$-0.759855\pi$
$149$	−17.7888	−1.45731	−0.728657	+	0.684879i	$0.759855\pi$
$150$	0	0
$151$	8.73963	0.711221	0.355610	−	0.934634i	$-0.384273\pi$
$151$	8.73963	0.711221	0.355610	+	0.934634i	$0.384273\pi$
$152$	8.76838	0.711210
$153$	1.56694	0.126680
$154$	2.02252	0.162980
$155$	0	0
$156$	−4.42856	−0.354569
$157$	−24.9441	−1.99076	−0.995379	−	0.0960239i	$-0.969387\pi$
$157$	−24.9441	−1.99076	−0.995379	+	0.0960239i	$0.969387\pi$
$158$	7.89164	0.627825
$159$	−2.23232	−0.177034
$160$	0	0
$161$	8.67622	0.683782
$162$	1.86997	0.146919
$163$	20.2931	1.58948	0.794740	−	0.606950i	$-0.207607\pi$
$163$	20.2931	1.58948	0.794740	+	0.606950i	$0.207607\pi$
$164$	8.19192	0.639681
$165$	0	0
$166$	−3.60737	−0.279986
$167$	11.1652	0.863992	0.431996	−	0.901876i	$-0.357809\pi$
$167$	11.1652	0.863992	0.431996	+	0.901876i	$0.357809\pi$
$168$	3.18316	0.245586
$169$	−6.95046	−0.534651
$170$	0	0
$171$	−4.98177	−0.380966
$172$	13.2160	1.00771
$173$	12.1695	0.925229	0.462614	−	0.886560i	$-0.346911\pi$
$173$	12.1695	0.925229	0.462614	+	0.886560i	$0.346911\pi$
$174$	0.923043	0.0699757
$175$	0	0
$176$	2.71102	0.204351
$177$	15.2424	1.14569
$178$	−1.13155	−0.0848135
$179$	0.143504	0.0107260	0.00536298	−	0.999986i	$-0.498293\pi$
$179$	0.143504	0.0107260	0.00536298	+	0.999986i	$0.498293\pi$
$180$	0	0
$181$	−0.703601	−0.0522983	−0.0261491	−	0.999658i	$-0.508324\pi$
$181$	−0.703601	−0.0522983	−0.0261491	+	0.999658i	$0.508324\pi$
$182$	−1.82648	−0.135388
$183$	6.79159	0.502048
$184$	−22.2188	−1.63799
$185$	0	0
$186$	4.87139	0.357188
$187$	−2.93318	−0.214495
$188$	3.93736	0.287161
$189$	−5.53749	−0.402793
$190$	0	0
$191$	−3.62970	−0.262636	−0.131318	−	0.991340i	$-0.541921\pi$
$191$	−3.62970	−0.262636	−0.131318	+	0.991340i	$0.541921\pi$
$192$	−2.93538	−0.211843
$193$	13.4971	0.971539	0.485770	−	0.874087i	$-0.338539\pi$
$193$	13.4971	0.971539	0.485770	+	0.874087i	$0.338539\pi$
$194$	2.11875	0.152118
$195$	0	0
$196$	−1.44855	−0.103468
$197$	2.37107	0.168931	0.0844657	−	0.996426i	$-0.473082\pi$
$197$	2.37107	0.168931	0.0844657	+	0.996426i	$0.473082\pi$
$198$	2.94271	0.209129
$199$	23.5634	1.67036	0.835182	−	0.549973i	$-0.185362\pi$
$199$	23.5634	1.67036	0.835182	+	0.549973i	$0.185362\pi$
$200$	0	0
$201$	−7.64206	−0.539029
$202$	−2.08939	−0.147009
$203$	−1.00000	−0.0701862
$204$	−1.93910	−0.135764
$205$	0	0
$206$	−2.93656	−0.204600
$207$	12.6237	0.877404
$208$	−2.44824	−0.169755
$209$	9.32543	0.645054
$210$	0	0
$211$	4.36903	0.300777	0.150388	−	0.988627i	$-0.451948\pi$
$211$	4.36903	0.300777	0.150388	+	0.988627i	$0.451948\pi$
$212$	−2.60148	−0.178671
$213$	−9.83369	−0.673793
$214$	8.33417	0.569712
$215$	0	0
$216$	14.1809	0.964887
$217$	−5.27754	−0.358263
$218$	2.63944	0.178766
$219$	3.65859	0.247225
$220$	0	0
$221$	2.64886	0.178182
$222$	7.61354	0.510988
$223$	2.90285	0.194390	0.0971948	−	0.995265i	$-0.469013\pi$
$223$	2.90285	0.194390	0.0971948	+	0.995265i	$0.469013\pi$
$224$	5.86094	0.391601
$225$	0	0
$226$	4.71966	0.313947
$227$	3.91031	0.259536	0.129768	−	0.991544i	$-0.458577\pi$
$227$	3.91031	0.259536	0.129768	+	0.991544i	$0.458577\pi$
$228$	6.16497	0.408285
$229$	−6.63824	−0.438667	−0.219334	−	0.975650i	$-0.570388\pi$
$229$	−6.63824	−0.438667	−0.219334	+	0.975650i	$0.570388\pi$
$230$	0	0
$231$	3.38538	0.222742
$232$	2.56088	0.168130
$233$	−8.58359	−0.562330	−0.281165	−	0.959659i	$-0.590721\pi$
$233$	−8.58359	−0.562330	−0.281165	+	0.959659i	$0.590721\pi$
$234$	−2.65748	−0.173725
$235$	0	0
$236$	17.7630	1.15627
$237$	13.2094	0.858040
$238$	−0.799746	−0.0518398
$239$	7.06424	0.456948	0.228474	−	0.973550i	$-0.426626\pi$
$239$	7.06424	0.456948	0.228474	+	0.973550i	$0.426626\pi$
$240$	0	0
$241$	12.1441	0.782272	0.391136	−	0.920333i	$-0.372082\pi$
$241$	12.1441	0.782272	0.391136	+	0.920333i	$0.372082\pi$
$242$	2.66008	0.170996
$243$	−13.4824	−0.864899
$244$	7.91473	0.506688
$245$	0	0
$246$	−5.22005	−0.332818
$247$	−8.42152	−0.535849
$248$	13.5152	0.858214
$249$	−6.03817	−0.382654
$250$	0	0
$251$	23.9247	1.51011	0.755057	−	0.655659i	$-0.227609\pi$
$251$	23.9247	1.51011	0.755057	+	0.655659i	$0.227609\pi$
$252$	−2.10760	−0.132766
$253$	−23.6304	−1.48563
$254$	2.91037	0.182613
$255$	0	0
$256$	−12.1255	−0.757841
$257$	16.8350	1.05014	0.525069	−	0.851059i	$-0.324039\pi$
$257$	16.8350	1.05014	0.525069	+	0.851059i	$0.324039\pi$
$258$	−8.42150	−0.524299
$259$	−8.24831	−0.512525
$260$	0	0
$261$	−1.45497	−0.0900605
$262$	7.12554	0.440218
$263$	24.6760	1.52159	0.760793	−	0.648994i	$-0.224810\pi$
$263$	24.6760	1.52159	0.760793	+	0.648994i	$0.224810\pi$
$264$	−8.66958	−0.533576
$265$	0	0
$266$	2.54263	0.155899
$267$	−1.89404	−0.115914
$268$	−8.90584	−0.544011
$269$	18.2635	1.11355	0.556774	−	0.830664i	$-0.312039\pi$
$269$	18.2635	1.11355	0.556774	+	0.830664i	$0.312039\pi$
$270$	0	0
$271$	−8.75928	−0.532088	−0.266044	−	0.963961i	$-0.585717\pi$
$271$	−8.75928	−0.532088	−0.266044	+	0.963961i	$0.585717\pi$
$272$	−1.07199	−0.0649990
$273$	−3.05724	−0.185033
$274$	−9.58115	−0.578818
$275$	0	0
$276$	−15.6218	−0.940324
$277$	16.0346	0.963428	0.481714	−	0.876328i	$-0.340014\pi$
$277$	16.0346	0.963428	0.481714	+	0.876328i	$0.340014\pi$
$278$	1.08447	0.0650420
$279$	−7.67866	−0.459710
$280$	0	0
$281$	29.0005	1.73002	0.865011	−	0.501753i	$-0.167311\pi$
$281$	29.0005	1.73002	0.865011	+	0.501753i	$0.167311\pi$
$282$	−2.50896	−0.149406
$283$	12.8625	0.764599	0.382299	−	0.924039i	$-0.375132\pi$
$283$	12.8625	0.764599	0.382299	+	0.924039i	$0.375132\pi$
$284$	−11.4599	−0.680020
$285$	0	0
$286$	4.97456	0.294152
$287$	5.65526	0.333819
$288$	8.52750	0.502488
$289$	−15.8402	−0.931774
$290$	0	0
$291$	3.54646	0.207897
$292$	4.26362	0.249509
$293$	−17.1854	−1.00398	−0.501992	−	0.864872i	$-0.667399\pi$
$293$	−17.1854	−1.00398	−0.501992	+	0.864872i	$0.667399\pi$
$294$	0.923043	0.0538330
$295$	0	0
$296$	21.1230	1.22775
$297$	15.0818	0.875134
$298$	13.2099	0.765229
$299$	21.3399	1.23412
$300$	0	0
$301$	9.12362	0.525877
$302$	−6.49003	−0.373459
$303$	−3.49732	−0.200916
$304$	3.40818	0.195472
$305$	0	0
$306$	−1.16361	−0.0665190
$307$	20.6312	1.17748	0.588741	−	0.808321i	$-0.299624\pi$
$307$	20.6312	1.17748	0.588741	+	0.808321i	$0.299624\pi$
$308$	3.94523	0.224801
$309$	−4.91535	−0.279624
$310$	0	0
$311$	−12.9489	−0.734263	−0.367131	−	0.930169i	$-0.619660\pi$
$311$	−12.9489	−0.734263	−0.367131	+	0.930169i	$0.619660\pi$
$312$	7.82924	0.443243
$313$	−13.0052	−0.735097	−0.367548	−	0.930004i	$-0.619803\pi$
$313$	−13.0052	−0.735097	−0.367548	+	0.930004i	$0.619803\pi$
$314$	18.5235	1.04534
$315$	0	0
$316$	15.3938	0.865970
$317$	−2.92051	−0.164032	−0.0820162	−	0.996631i	$-0.526136\pi$
$317$	−2.92051	−0.164032	−0.0820162	+	0.996631i	$0.526136\pi$
$318$	1.65771	0.0929600
$319$	2.72358	0.152491
$320$	0	0
$321$	13.9501	0.778618
$322$	−6.44294	−0.359051
$323$	−3.68746	−0.205176
$324$	3.64766	0.202648
$325$	0	0
$326$	−15.0696	−0.834630
$327$	4.41801	0.244317
$328$	−14.4825	−0.799661
$329$	2.71814	0.149856
$330$	0	0
$331$	−29.9566	−1.64656	−0.823282	−	0.567633i	$-0.807859\pi$
$331$	−29.9566	−1.64656	−0.823282	+	0.567633i	$0.807859\pi$
$332$	−7.03672	−0.386190
$333$	−12.0011	−0.657654
$334$	−8.29128	−0.453679
$335$	0	0
$336$	1.23726	0.0674981
$337$	15.0887	0.821932	0.410966	−	0.911651i	$-0.365192\pi$
$337$	15.0887	0.821932	0.410966	+	0.911651i	$0.365192\pi$
$338$	5.16139	0.280743
$339$	7.89998	0.429068
$340$	0	0
$341$	14.3738	0.778384
$342$	3.69945	0.200043
$343$	−1.00000	−0.0539949
$344$	−23.3645	−1.25973
$345$	0	0
$346$	−9.03703	−0.485834
$347$	−10.8847	−0.584323	−0.292161	−	0.956369i	$-0.594374\pi$
$347$	−10.8847	−0.584323	−0.292161	+	0.956369i	$0.594374\pi$
$348$	1.80053	0.0965187
$349$	9.08964	0.486557	0.243279	−	0.969956i	$-0.421777\pi$
$349$	9.08964	0.486557	0.243279	+	0.969956i	$0.421777\pi$
$350$	0	0
$351$	−13.6199	−0.726977
$352$	−15.9627	−0.850816
$353$	19.8419	1.05608	0.528039	−	0.849220i	$-0.322927\pi$
$353$	19.8419	1.05608	0.528039	+	0.849220i	$0.322927\pi$
$354$	−11.3189	−0.601595
$355$	0	0
$356$	−2.20726	−0.116985
$357$	−1.33865	−0.0708488
$358$	−0.106566	−0.00563216
$359$	−15.8550	−0.836792	−0.418396	−	0.908265i	$-0.637408\pi$
$359$	−15.8550	−0.836792	−0.418396	+	0.908265i	$0.637408\pi$
$360$	0	0
$361$	−7.27646	−0.382972
$362$	0.522493	0.0274616
$363$	4.45256	0.233699
$364$	−3.56282	−0.186743
$365$	0	0
$366$	−5.04342	−0.263624
$367$	28.2994	1.47722	0.738609	−	0.674134i	$-0.235483\pi$
$367$	28.2994	1.47722	0.738609	+	0.674134i	$0.235483\pi$
$368$	−8.63622	−0.450194
$369$	8.22824	0.428345
$370$	0	0
$371$	−1.79592	−0.0932397
$372$	9.50238	0.492676
$373$	27.3265	1.41491	0.707456	−	0.706757i	$-0.249843\pi$
$373$	27.3265	1.41491	0.707456	+	0.706757i	$0.249843\pi$
$374$	2.17817	0.112630
$375$	0	0
$376$	−6.96085	−0.358978
$377$	−2.45958	−0.126675
$378$	4.11213	0.211505
$379$	9.67982	0.497219	0.248609	−	0.968604i	$-0.420026\pi$
$379$	9.67982	0.497219	0.248609	+	0.968604i	$0.420026\pi$
$380$	0	0
$381$	4.87150	0.249575
$382$	2.69541	0.137909
$383$	9.45520	0.483138	0.241569	−	0.970384i	$-0.422338\pi$
$383$	9.45520	0.483138	0.241569	+	0.970384i	$0.422338\pi$
$384$	−12.3904	−0.632295
$385$	0	0
$386$	−10.0229	−0.510151
$387$	13.2746	0.674786
$388$	4.13295	0.209819
$389$	1.42855	0.0724302	0.0362151	−	0.999344i	$-0.488470\pi$
$389$	1.42855	0.0724302	0.0362151	+	0.999344i	$0.488470\pi$
$390$	0	0
$391$	9.34391	0.472542
$392$	2.56088	0.129344
$393$	11.9270	0.601639
$394$	−1.76075	−0.0887052
$395$	0	0
$396$	5.74020	0.288456
$397$	−2.24309	−0.112577	−0.0562887	−	0.998415i	$-0.517927\pi$
$397$	−2.24309	−0.112577	−0.0562887	+	0.998415i	$0.517927\pi$
$398$	−17.4981	−0.877102
$399$	4.25596	0.213065
$400$	0	0
$401$	13.6739	0.682840	0.341420	−	0.939911i	$-0.389092\pi$
$401$	13.6739	0.682840	0.341420	+	0.939911i	$0.389092\pi$
$402$	5.67497	0.283042
$403$	−12.9805	−0.646606
$404$	−4.07568	−0.202772
$405$	0	0
$406$	0.742598	0.0368545
$407$	22.4649	1.11354
$408$	3.42812	0.169717
$409$	31.2533	1.54538	0.772688	−	0.634786i	$-0.218912\pi$
$409$	31.2533	1.54538	0.772688	+	0.634786i	$0.218912\pi$
$410$	0	0
$411$	−16.0373	−0.791063
$412$	−5.72821	−0.282209
$413$	12.2626	0.603405
$414$	−9.37430	−0.460721
$415$	0	0
$416$	14.4155	0.706777
$417$	1.81523	0.0888921
$418$	−6.92505	−0.338715
$419$	29.3917	1.43588	0.717940	−	0.696105i	$-0.245085\pi$
$419$	29.3917	1.43588	0.717940	+	0.696105i	$0.245085\pi$
$420$	0	0
$421$	−9.85922	−0.480509	−0.240254	−	0.970710i	$-0.577231\pi$
$421$	−9.85922	−0.480509	−0.240254	+	0.970710i	$0.577231\pi$
$422$	−3.24443	−0.157937
$423$	3.95482	0.192290
$424$	4.59915	0.223355
$425$	0	0
$426$	7.30247	0.353806
$427$	5.46390	0.264417
$428$	16.2570	0.785814
$429$	8.32663	0.402013
$430$	0	0
$431$	−8.84559	−0.426077	−0.213038	−	0.977044i	$-0.568336\pi$
$431$	−8.84559	−0.426077	−0.213038	+	0.977044i	$0.568336\pi$
$432$	5.51196	0.265194
$433$	−7.92806	−0.380998	−0.190499	−	0.981687i	$-0.561011\pi$
$433$	−7.92806	−0.380998	−0.190499	+	0.981687i	$0.561011\pi$
$434$	3.91909	0.188122
$435$	0	0
$436$	5.14863	0.246575
$437$	−29.7071	−1.42108
$438$	−2.71686	−0.129817
$439$	31.9269	1.52379	0.761893	−	0.647703i	$-0.224270\pi$
$439$	31.9269	1.52379	0.761893	+	0.647703i	$0.224270\pi$
$440$	0	0
$441$	−1.45497	−0.0692843
$442$	−1.96704	−0.0935626
$443$	−22.2449	−1.05689	−0.528443	−	0.848969i	$-0.677224\pi$
$443$	−22.2449	−1.05689	−0.528443	+	0.848969i	$0.677224\pi$
$444$	14.8514	0.704814
$445$	0	0
$446$	−2.15565	−0.102073
$447$	22.1113	1.04583
$448$	−2.36155	−0.111573
$449$	17.6495	0.832931	0.416466	−	0.909151i	$-0.363269\pi$
$449$	17.6495	0.832931	0.416466	+	0.909151i	$0.363269\pi$
$450$	0	0
$451$	−15.4025	−0.725277
$452$	9.20641	0.433033
$453$	−10.8633	−0.510402
$454$	−2.90379	−0.136281
$455$	0	0
$456$	−10.8990	−0.510394
$457$	31.3744	1.46763	0.733815	−	0.679349i	$-0.237738\pi$
$457$	31.3744	1.46763	0.733815	+	0.679349i	$0.237738\pi$
$458$	4.92954	0.230343
$459$	−5.96364	−0.278359
$460$	0	0
$461$	39.8145	1.85435	0.927173	−	0.374633i	$-0.122231\pi$
$461$	39.8145	1.85435	0.927173	+	0.374633i	$0.122231\pi$
$462$	−2.51398	−0.116961
$463$	−29.2989	−1.36164	−0.680818	−	0.732452i	$-0.738376\pi$
$463$	−29.2989	−1.36164	−0.680818	+	0.732452i	$0.738376\pi$
$464$	0.995389	0.0462098
$465$	0	0
$466$	6.37416	0.295277
$467$	−26.7553	−1.23809	−0.619044	−	0.785356i	$-0.712480\pi$
$467$	−26.7553	−1.23809	−0.619044	+	0.785356i	$0.712480\pi$
$468$	−5.18381	−0.239621
$469$	−6.14811	−0.283894
$470$	0	0
$471$	31.0053	1.42865
$472$	−31.4032	−1.44545
$473$	−24.8489	−1.14255
$474$	−9.80924	−0.450553
$475$	0	0
$476$	−1.56002	−0.0715036
$477$	−2.61302	−0.119642
$478$	−5.24589	−0.239942
$479$	1.57462	0.0719464	0.0359732	−	0.999353i	$-0.488547\pi$
$479$	1.57462	0.0719464	0.0359732	+	0.999353i	$0.488547\pi$
$480$	0	0
$481$	−20.2874	−0.925025
$482$	−9.01820	−0.410768
$483$	−10.7845	−0.490710
$484$	5.18889	0.235858
$485$	0	0
$486$	10.0120	0.454155
$487$	37.1271	1.68239	0.841195	−	0.540732i	$-0.181853\pi$
$487$	37.1271	1.68239	0.841195	+	0.540732i	$0.181853\pi$
$488$	−13.9924	−0.633407
$489$	−25.2242	−1.14068
$490$	0	0
$491$	23.0418	1.03986	0.519931	−	0.854208i	$-0.325958\pi$
$491$	23.0418	1.03986	0.519931	+	0.854208i	$0.325958\pi$
$492$	−10.1825	−0.459062
$493$	−1.07696	−0.0485037
$494$	6.25380	0.281372
$495$	0	0
$496$	5.25320	0.235876
$497$	−7.91130	−0.354870
$498$	4.48393	0.200930
$499$	−5.14747	−0.230432	−0.115216	−	0.993340i	$-0.536756\pi$
$499$	−5.14747	−0.230432	−0.115216	+	0.993340i	$0.536756\pi$
$500$	0	0
$501$	−13.8783	−0.620037
$502$	−17.7664	−0.792955
$503$	−9.65742	−0.430603	−0.215301	−	0.976548i	$-0.569073\pi$
$503$	−9.65742	−0.430603	−0.215301	+	0.976548i	$0.569073\pi$
$504$	3.72601	0.165970
$505$	0	0
$506$	17.5478	0.780097
$507$	8.63936	0.383688
$508$	5.67712	0.251881
$509$	−17.5804	−0.779238	−0.389619	−	0.920976i	$-0.627393\pi$
$509$	−17.5804	−0.779238	−0.389619	+	0.920976i	$0.627393\pi$
$510$	0	0
$511$	2.94337	0.130207
$512$	−10.9321	−0.483134
$513$	18.9602	0.837113
$514$	−12.5016	−0.551424
$515$	0	0
$516$	−16.4274	−0.723176
$517$	−7.40307	−0.325586
$518$	6.12518	0.269125
$519$	−15.1266	−0.663983
$520$	0	0
$521$	31.5139	1.38065	0.690324	−	0.723500i	$-0.257468\pi$
$521$	31.5139	1.38065	0.690324	+	0.723500i	$0.257468\pi$
$522$	1.08046	0.0472904
$523$	43.2499	1.89118	0.945592	−	0.325355i	$-0.105484\pi$
$523$	43.2499	1.89118	0.945592	+	0.325355i	$0.105484\pi$
$524$	13.8994	0.607200
$525$	0	0
$526$	−18.3243	−0.798979
$527$	−5.68368	−0.247585
$528$	−3.36978	−0.146651
$529$	52.2768	2.27290
$530$	0	0
$531$	17.8418	0.774267
$532$	4.95978	0.215034
$533$	13.9096	0.602491
$534$	1.40651	0.0608657
$535$	0	0
$536$	15.7446	0.680064
$537$	−0.178374	−0.00769740
$538$	−13.5625	−0.584720
$539$	2.72358	0.117313
$540$	0	0
$541$	−0.871762	−0.0374800	−0.0187400	−	0.999824i	$-0.505965\pi$
$541$	−0.871762	−0.0374800	−0.0187400	+	0.999824i	$0.505965\pi$
$542$	6.50462	0.279397
$543$	0.874571	0.0375314
$544$	6.31198	0.270624
$545$	0	0
$546$	2.27030	0.0971599
$547$	−31.0585	−1.32797	−0.663983	−	0.747747i	$-0.731135\pi$
$547$	−31.0585	−1.32797	−0.663983	+	0.747747i	$0.731135\pi$
$548$	−18.6895	−0.798375
$549$	7.94982	0.339290
$550$	0	0
$551$	3.42397	0.145866
$552$	27.6178	1.17549
$553$	10.6271	0.451909
$554$	−11.9073	−0.505892
$555$	0	0
$556$	2.11542	0.0897136
$557$	−27.0230	−1.14500	−0.572501	−	0.819904i	$-0.694027\pi$
$557$	−27.0230	−1.14500	−0.572501	+	0.819904i	$0.694027\pi$
$558$	5.70216	0.241392
$559$	22.4403	0.949123
$560$	0	0
$561$	3.64591	0.153931
$562$	−21.5357	−0.908427
$563$	12.8101	0.539881	0.269941	−	0.962877i	$-0.412996\pi$
$563$	12.8101	0.539881	0.269941	+	0.962877i	$0.412996\pi$
$564$	−4.89411	−0.206079
$565$	0	0
$566$	−9.55170	−0.401488
$567$	2.51815	0.105752
$568$	20.2599	0.850088
$569$	−9.73685	−0.408190	−0.204095	−	0.978951i	$-0.565425\pi$
$569$	−9.73685	−0.408190	−0.204095	+	0.978951i	$0.565425\pi$
$570$	0	0
$571$	29.6366	1.24025	0.620126	−	0.784502i	$-0.287081\pi$
$571$	29.6366	1.24025	0.620126	+	0.784502i	$0.287081\pi$
$572$	9.70362	0.405729
$573$	4.51169	0.188479
$574$	−4.19958	−0.175287
$575$	0	0
$576$	−3.43598	−0.143166
$577$	33.5601	1.39713	0.698563	−	0.715549i	$-0.253823\pi$
$577$	33.5601	1.39713	0.698563	+	0.715549i	$0.253823\pi$
$578$	11.7629	0.489271
$579$	−16.7767	−0.697217
$580$	0	0
$581$	−4.85777	−0.201534
$582$	−2.63359	−0.109166
$583$	4.89134	0.202578
$584$	−7.53764	−0.311910
$585$	0	0
$586$	12.7619	0.527188
$587$	−37.6485	−1.55392	−0.776959	−	0.629551i	$-0.783239\pi$
$587$	−37.6485	−1.55392	−0.776959	+	0.629551i	$0.783239\pi$
$588$	1.80053	0.0742528
$589$	18.0701	0.744566
$590$	0	0
$591$	−2.94722	−0.121232
$592$	8.21028	0.337440
$593$	−22.8430	−0.938049	−0.469025	−	0.883185i	$-0.655394\pi$
$593$	−22.8430	−0.938049	−0.469025	+	0.883185i	$0.655394\pi$
$594$	−11.1997	−0.459529
$595$	0	0
$596$	25.7679	1.05549
$597$	−29.2891	−1.19872
$598$	−15.8469	−0.648030
$599$	−3.81859	−0.156023	−0.0780116	−	0.996952i	$-0.524857\pi$
$599$	−3.81859	−0.156023	−0.0780116	+	0.996952i	$0.524857\pi$
$600$	0	0
$601$	3.94457	0.160902	0.0804512	−	0.996759i	$-0.474364\pi$
$601$	3.94457	0.160902	0.0804512	+	0.996759i	$0.474364\pi$
$602$	−6.77518	−0.276136
$603$	−8.94533	−0.364282
$604$	−12.6598	−0.515119
$605$	0	0
$606$	2.59710	0.105500
$607$	−34.6570	−1.40668	−0.703342	−	0.710852i	$-0.748310\pi$
$607$	−34.6570	−1.40668	−0.703342	+	0.710852i	$0.748310\pi$
$608$	−20.0677	−0.813851
$609$	1.24299	0.0503686
$610$	0	0
$611$	6.68549	0.270466
$612$	−2.26979	−0.0917508
$613$	21.8403	0.882123	0.441061	−	0.897477i	$-0.354602\pi$
$613$	21.8403	0.882123	0.441061	+	0.897477i	$0.354602\pi$
$614$	−15.3207	−0.618292
$615$	0	0
$616$	−6.97477	−0.281021
$617$	−30.4898	−1.22747	−0.613737	−	0.789511i	$-0.710334\pi$
$617$	−30.4898	−1.22747	−0.613737	+	0.789511i	$0.710334\pi$
$618$	3.65013	0.146830
$619$	−25.7318	−1.03425	−0.517124	−	0.855911i	$-0.672997\pi$
$619$	−25.7318	−1.03425	−0.517124	+	0.855911i	$0.672997\pi$
$620$	0	0
$621$	−48.0445	−1.92796
$622$	9.61580	0.385559
$623$	−1.52378	−0.0610488
$624$	3.04314	0.121823
$625$	0	0
$626$	9.65763	0.385996
$627$	−11.5914	−0.462917
$628$	36.1328	1.44185
$629$	−8.88307	−0.354191
$630$	0	0
$631$	−22.5172	−0.896394	−0.448197	−	0.893935i	$-0.647934\pi$
$631$	−22.5172	−0.896394	−0.448197	+	0.893935i	$0.647934\pi$
$632$	−27.2147	−1.08254
$633$	−5.43067	−0.215850
$634$	2.16877	0.0861327
$635$	0	0
$636$	3.23362	0.128221
$637$	−2.45958	−0.0974522
$638$	−2.02252	−0.0800724
$639$	−11.5107	−0.455357
$640$	0	0
$641$	13.1318	0.518674	0.259337	−	0.965787i	$-0.416496\pi$
$641$	13.1318	0.518674	0.259337	+	0.965787i	$0.416496\pi$
$642$	−10.3593	−0.408849
$643$	25.8524	1.01952	0.509760	−	0.860317i	$-0.329734\pi$
$643$	25.8524	1.01952	0.509760	+	0.860317i	$0.329734\pi$
$644$	−12.5679	−0.495246
$645$	0	0
$646$	2.73830	0.107737
$647$	−27.2681	−1.07202	−0.536010	−	0.844212i	$-0.680069\pi$
$647$	−27.2681	−1.07202	−0.536010	+	0.844212i	$0.680069\pi$
$648$	−6.44868	−0.253328
$649$	−33.3982	−1.31100
$650$	0	0
$651$	6.55993	0.257104
$652$	−29.3956	−1.15122
$653$	−6.86403	−0.268610	−0.134305	−	0.990940i	$-0.542880\pi$
$653$	−6.86403	−0.268610	−0.134305	+	0.990940i	$0.542880\pi$
$654$	−3.28080	−0.128290
$655$	0	0
$656$	−5.62919	−0.219783
$657$	4.28252	0.167077
$658$	−2.01849	−0.0786888
$659$	−44.3372	−1.72713	−0.863566	−	0.504236i	$-0.831774\pi$
$659$	−44.3372	−1.72713	−0.863566	+	0.504236i	$0.831774\pi$
$660$	0	0
$661$	−9.42036	−0.366410	−0.183205	−	0.983075i	$-0.558647\pi$
$661$	−9.42036	−0.366410	−0.183205	+	0.983075i	$0.558647\pi$
$662$	22.2457	0.864604
$663$	−3.29252	−0.127871
$664$	12.4402	0.482773
$665$	0	0
$666$	8.91195	0.345331
$667$	−8.67622	−0.335945
$668$	−16.1734	−0.625767
$669$	−3.60822	−0.139502
$670$	0	0
$671$	−14.8814	−0.574488
$672$	−7.28510	−0.281029
$673$	25.5578	0.985180	0.492590	−	0.870262i	$-0.336050\pi$
$673$	25.5578	0.985180	0.492590	+	0.870262i	$0.336050\pi$
$674$	−11.2048	−0.431593
$675$	0	0
$676$	10.0681	0.387234
$677$	−18.8886	−0.725949	−0.362974	−	0.931799i	$-0.618239\pi$
$677$	−18.8886	−0.725949	−0.362974	+	0.931799i	$0.618239\pi$
$678$	−5.86651	−0.225302
$679$	2.85316	0.109494
$680$	0	0
$681$	−4.86048	−0.186254
$682$	−10.6739	−0.408726
$683$	13.3522	0.510907	0.255454	−	0.966821i	$-0.417775\pi$
$683$	13.3522	0.510907	0.255454	+	0.966821i	$0.417775\pi$
$684$	7.21634	0.275924
$685$	0	0
$686$	0.742598	0.0283525
$687$	8.25128	0.314806
$688$	−9.08156	−0.346231
$689$	−4.41722	−0.168283
$690$	0	0
$691$	−18.5029	−0.703882	−0.351941	−	0.936022i	$-0.614478\pi$
$691$	−18.5029	−0.703882	−0.351941	+	0.936022i	$0.614478\pi$
$692$	−17.6281	−0.670119
$693$	3.96273	0.150532
$694$	8.08297	0.306825
$695$	0	0
$696$	−3.18316	−0.120657
$697$	6.09047	0.230693
$698$	−6.74994	−0.255489
$699$	10.6693	0.403551
$700$	0	0
$701$	−12.3177	−0.465232	−0.232616	−	0.972569i	$-0.574729\pi$
$701$	−12.3177	−0.465232	−0.232616	+	0.972569i	$0.574729\pi$
$702$	10.1141	0.381733
$703$	28.2419	1.06516
$704$	6.43185	0.242409
$705$	0	0
$706$	−14.7346	−0.554542
$707$	−2.81363	−0.105817
$708$	−22.0793	−0.829791
$709$	38.0330	1.42836	0.714179	−	0.699963i	$-0.246800\pi$
$709$	38.0330	1.42836	0.714179	+	0.699963i	$0.246800\pi$
$710$	0	0
$711$	15.4621	0.579873
$712$	3.90222	0.146242
$713$	−45.7891	−1.71481
$714$	0.994078	0.0372024
$715$	0	0
$716$	−0.207872	−0.00776854
$717$	−8.78080	−0.327925
$718$	11.7739	0.439396
$719$	−43.6760	−1.62884	−0.814419	−	0.580277i	$-0.802944\pi$
$719$	−43.6760	−1.62884	−0.814419	+	0.580277i	$0.802944\pi$
$720$	0	0
$721$	−3.95445	−0.147271
$722$	5.40349	0.201097
$723$	−15.0951	−0.561391
$724$	1.01920	0.0378783
$725$	0	0
$726$	−3.30646	−0.122714
$727$	22.7092	0.842236	0.421118	−	0.907006i	$-0.361638\pi$
$727$	22.7092	0.842236	0.421118	+	0.907006i	$0.361638\pi$
$728$	6.29871	0.233446
$729$	24.3130	0.900482
$730$	0	0
$731$	9.82575	0.363418
$732$	−9.83794	−0.363621
$733$	14.3053	0.528377	0.264189	−	0.964471i	$-0.414896\pi$
$733$	14.3053	0.528377	0.264189	+	0.964471i	$0.414896\pi$
$734$	−21.0151	−0.775681
$735$	0	0
$736$	50.8508	1.87439
$737$	16.7449	0.616805
$738$	−6.11027	−0.224922
$739$	9.21757	0.339074	0.169537	−	0.985524i	$-0.445773\pi$
$739$	9.21757	0.339074	0.169537	+	0.985524i	$0.445773\pi$
$740$	0	0
$741$	10.4679	0.384547
$742$	1.33365	0.0489598
$743$	21.7153	0.796656	0.398328	−	0.917243i	$-0.369591\pi$
$743$	21.7153	0.796656	0.398328	+	0.917243i	$0.369591\pi$
$744$	−16.7992	−0.615890
$745$	0	0
$746$	−20.2926	−0.742965
$747$	−7.06792	−0.258602
$748$	4.24885	0.155353
$749$	11.2230	0.410079
$750$	0	0
$751$	−43.9777	−1.60477	−0.802384	−	0.596808i	$-0.796436\pi$
$751$	−43.9777	−1.60477	−0.802384	+	0.596808i	$0.796436\pi$
$752$	−2.70561	−0.0986634
$753$	−29.7382	−1.08372
$754$	1.82648	0.0665165
$755$	0	0
$756$	8.02133	0.291733
$757$	29.2006	1.06131	0.530657	−	0.847586i	$-0.321945\pi$
$757$	29.2006	1.06131	0.530657	+	0.847586i	$0.321945\pi$
$758$	−7.18821	−0.261088
$759$	29.3723	1.06615
$760$	0	0
$761$	42.0780	1.52532	0.762662	−	0.646797i	$-0.223892\pi$
$761$	42.0780	1.52532	0.762662	+	0.646797i	$0.223892\pi$
$762$	−3.61757	−0.131051
$763$	3.55434	0.128676
$764$	5.25780	0.190221
$765$	0	0
$766$	−7.02141	−0.253694
$767$	30.1610	1.08905
$768$	15.0718	0.543858
$769$	−0.619891	−0.0223538	−0.0111769	−	0.999938i	$-0.503558\pi$
$769$	−0.619891	−0.0223538	−0.0111769	+	0.999938i	$0.503558\pi$
$770$	0	0
$771$	−20.9258	−0.753623
$772$	−19.5511	−0.703661
$773$	46.9574	1.68894	0.844470	−	0.535603i	$-0.179916\pi$
$773$	46.9574	1.68894	0.844470	+	0.535603i	$0.179916\pi$
$774$	−9.85770	−0.354328
$775$	0	0
$776$	−7.30662	−0.262292
$777$	10.2526	0.367809
$778$	−1.06084	−0.0380328
$779$	−19.3634	−0.693766
$780$	0	0
$781$	21.5470	0.771014
$782$	−6.93877	−0.248130
$783$	5.53749	0.197894
$784$	0.995389	0.0355496
$785$	0	0
$786$	−8.85699	−0.315918
$787$	3.51055	0.125137	0.0625687	−	0.998041i	$-0.480071\pi$
$787$	3.51055	0.125137	0.0625687	+	0.998041i	$0.480071\pi$
$788$	−3.43460	−0.122353
$789$	−30.6720	−1.09195
$790$	0	0
$791$	6.35561	0.225980
$792$	−10.1481	−0.360596
$793$	13.4389	0.477230
$794$	1.66571	0.0591139
$795$	0	0
$796$	−34.1327	−1.20980
$797$	32.0308	1.13459	0.567295	−	0.823515i	$-0.307990\pi$
$797$	32.0308	1.13459	0.567295	+	0.823515i	$0.307990\pi$
$798$	−3.16047	−0.111879
$799$	2.92732	0.103561
$800$	0	0
$801$	−2.21705	−0.0783357
$802$	−10.1542	−0.358557
$803$	−8.01650	−0.282896
$804$	11.0699	0.390405
$805$	0	0
$806$	9.63932	0.339530
$807$	−22.7014	−0.799128
$808$	7.20538	0.253484
$809$	38.5580	1.35563	0.677813	−	0.735234i	$-0.262928\pi$
$809$	38.5580	1.35563	0.677813	+	0.735234i	$0.262928\pi$
$810$	0	0
$811$	−14.5869	−0.512215	−0.256107	−	0.966648i	$-0.582440\pi$
$811$	−14.5869	−0.512215	−0.256107	+	0.966648i	$0.582440\pi$
$812$	1.44855	0.0508341
$813$	10.8877	0.381849
$814$	−16.6824	−0.584717
$815$	0	0
$816$	1.33248	0.0466460
$817$	−31.2390	−1.09291
$818$	−23.2086	−0.811471
$819$	−3.57862	−0.125047
$820$	0	0
$821$	9.77420	0.341122	0.170561	−	0.985347i	$-0.445442\pi$
$821$	9.77420	0.341122	0.170561	+	0.985347i	$0.445442\pi$
$822$	11.9093	0.415384
$823$	44.5331	1.55233	0.776163	−	0.630532i	$-0.217163\pi$
$823$	44.5331	1.55233	0.776163	+	0.630532i	$0.217163\pi$
$824$	10.1269	0.352787
$825$	0	0
$826$	−9.10621	−0.316845
$827$	31.4132	1.09234	0.546172	−	0.837673i	$-0.316084\pi$
$827$	31.4132	1.09234	0.546172	+	0.837673i	$0.316084\pi$
$828$	−18.2860	−0.635481
$829$	37.9866	1.31933	0.659665	−	0.751560i	$-0.270698\pi$
$829$	37.9866	1.31933	0.659665	+	0.751560i	$0.270698\pi$
$830$	0	0
$831$	−19.9309	−0.691396
$832$	−5.80841	−0.201370
$833$	−1.07696	−0.0373143
$834$	−1.34798	−0.0466769
$835$	0	0
$836$	−13.5083	−0.467196
$837$	29.2243	1.01014
$838$	−21.8262	−0.753975
$839$	39.6890	1.37021	0.685107	−	0.728442i	$-0.259755\pi$
$839$	39.6890	1.37021	0.685107	+	0.728442i	$0.259755\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	7.32143	0.252313
$843$	−36.0473	−1.24154
$844$	−6.32876	−0.217845
$845$	0	0
$846$	−2.93684	−0.100971
$847$	3.58213	0.123083
$848$	1.78764	0.0613879
$849$	−15.9880	−0.548708
$850$	0	0
$851$	−71.5641	−2.45319
$852$	14.2446	0.488011
$853$	−5.58178	−0.191116	−0.0955582	−	0.995424i	$-0.530464\pi$
$853$	−5.58178	−0.191116	−0.0955582	+	0.995424i	$0.530464\pi$
$854$	−4.05748	−0.138844
$855$	0	0
$856$	−28.7408	−0.982340
$857$	−45.2887	−1.54703	−0.773516	−	0.633776i	$-0.781504\pi$
$857$	−45.2887	−1.54703	−0.773516	+	0.633776i	$0.781504\pi$
$858$	−6.18334	−0.211096
$859$	5.37261	0.183311	0.0916555	−	0.995791i	$-0.470784\pi$
$859$	5.37261	0.183311	0.0916555	+	0.995791i	$0.470784\pi$
$860$	0	0
$861$	−7.02944	−0.239563
$862$	6.56871	0.223731
$863$	12.6972	0.432219	0.216109	−	0.976369i	$-0.430663\pi$
$863$	12.6972	0.432219	0.216109	+	0.976369i	$0.430663\pi$
$864$	−32.4549	−1.10414
$865$	0	0
$866$	5.88736	0.200061
$867$	19.6892	0.668680
$868$	7.64477	0.259480
$869$	−28.9436	−0.981846
$870$	0	0
$871$	−15.1218	−0.512382
$872$	−9.10224	−0.308241
$873$	4.15127	0.140499
$874$	22.0604	0.746204
$875$	0	0
$876$	−5.29964	−0.179058
$877$	22.6456	0.764687	0.382343	−	0.924020i	$-0.375117\pi$
$877$	22.6456	0.764687	0.382343	+	0.924020i	$0.375117\pi$
$878$	−23.7088	−0.800134
$879$	21.3614	0.720501
$880$	0	0
$881$	−35.6474	−1.20099	−0.600495	−	0.799629i	$-0.705030\pi$
$881$	−35.6474	−1.20099	−0.600495	+	0.799629i	$0.705030\pi$
$882$	1.08046	0.0363809
$883$	−40.3777	−1.35882	−0.679409	−	0.733760i	$-0.737764\pi$
$883$	−40.3777	−1.35882	−0.679409	+	0.733760i	$0.737764\pi$
$884$	−3.83701	−0.129053
$885$	0	0
$886$	16.5190	0.554967
$887$	24.5494	0.824287	0.412143	−	0.911119i	$-0.364780\pi$
$887$	24.5494	0.824287	0.412143	+	0.911119i	$0.364780\pi$
$888$	−26.2557	−0.881083
$889$	3.91918	0.131445
$890$	0	0
$891$	−6.85836	−0.229764
$892$	−4.20493	−0.140791
$893$	−9.30682	−0.311441
$894$	−16.4198	−0.549160
$895$	0	0
$896$	−9.96821	−0.333014
$897$	−26.5253	−0.885654
$898$	−13.1065	−0.437369
$899$	5.27754	0.176016
$900$	0	0
$901$	−1.93413	−0.0644353
$902$	11.4379	0.380840
$903$	−11.3406	−0.377391
$904$	−16.2760	−0.541331
$905$	0	0
$906$	8.06705	0.268010
$907$	−2.75276	−0.0914039	−0.0457020	−	0.998955i	$-0.514552\pi$
$907$	−2.75276	−0.0914039	−0.0457020	+	0.998955i	$0.514552\pi$
$908$	−5.66427	−0.187975
$909$	−4.09375	−0.135781
$910$	0	0
$911$	−33.4937	−1.10970	−0.554848	−	0.831952i	$-0.687223\pi$
$911$	−33.4937	−1.10970	−0.554848	+	0.831952i	$0.687223\pi$
$912$	−4.23634	−0.140279
$913$	13.2305	0.437866
$914$	−23.2985	−0.770647
$915$	0	0
$916$	9.61582	0.317716
$917$	9.59543	0.316869
$918$	4.42859	0.146165
$919$	−52.4710	−1.73086	−0.865429	−	0.501031i	$-0.832954\pi$
$919$	−52.4710	−1.73086	−0.865429	+	0.501031i	$0.832954\pi$
$920$	0	0
$921$	−25.6444	−0.845011
$922$	−29.5662	−0.973710
$923$	−19.4585	−0.640484
$924$	−4.90389	−0.161326
$925$	0	0
$926$	21.7573	0.714990
$927$	−5.75361	−0.188973
$928$	−5.86094	−0.192395
$929$	47.5181	1.55902	0.779509	−	0.626391i	$-0.215469\pi$
$929$	47.5181	1.55902	0.779509	+	0.626391i	$0.215469\pi$
$930$	0	0
$931$	3.42397	0.112216
$932$	12.4337	0.407281
$933$	16.0953	0.526938
$934$	19.8684	0.650115
$935$	0	0
$936$	9.16443	0.299549
$937$	13.4354	0.438914	0.219457	−	0.975622i	$-0.429571\pi$
$937$	13.4354	0.438914	0.219457	+	0.975622i	$0.429571\pi$
$938$	4.56558	0.149071
$939$	16.1653	0.527536
$940$	0	0
$941$	−33.5211	−1.09276	−0.546379	−	0.837538i	$-0.683994\pi$
$941$	−33.5211	−1.09276	−0.546379	+	0.837538i	$0.683994\pi$
$942$	−23.0245	−0.750179
$943$	49.0663	1.59782
$944$	−12.2061	−0.397275
$945$	0	0
$946$	18.4527	0.599950
$947$	−40.8903	−1.32876	−0.664378	−	0.747397i	$-0.731303\pi$
$947$	−40.8903	−1.32876	−0.664378	+	0.747397i	$0.731303\pi$
$948$	−19.1344	−0.621456
$949$	7.23947	0.235003
$950$	0	0
$951$	3.63017	0.117716
$952$	2.75796	0.0893861
$953$	8.77147	0.284136	0.142068	−	0.989857i	$-0.454625\pi$
$953$	8.77147	0.284136	0.142068	+	0.989857i	$0.454625\pi$
$954$	1.94042	0.0628234
$955$	0	0
$956$	−10.2329	−0.330956
$957$	−3.38538	−0.109434
$958$	−1.16931	−0.0377788
$959$	−12.9022	−0.416634
$960$	0	0
$961$	−3.14761	−0.101536
$962$	15.0654	0.485727
$963$	16.3291	0.526199
$964$	−17.5914	−0.566579
$965$	0	0
$966$	8.00852	0.257670
$967$	−49.6974	−1.59816	−0.799080	−	0.601224i	$-0.794680\pi$
$967$	−49.6974	−1.59816	−0.799080	+	0.601224i	$0.794680\pi$
$968$	−9.17342	−0.294845
$969$	4.58349	0.147243
$970$	0	0
$971$	−1.64292	−0.0527239	−0.0263620	−	0.999652i	$-0.508392\pi$
$971$	−1.64292	−0.0527239	−0.0263620	+	0.999652i	$0.508392\pi$
$972$	19.5300	0.626424
$973$	1.46037	0.0468173
$974$	−27.5705	−0.883416
$975$	0	0
$976$	−5.43871	−0.174089
$977$	−44.7190	−1.43069	−0.715344	−	0.698773i	$-0.753730\pi$
$977$	−44.7190	−1.43069	−0.715344	+	0.698773i	$0.753730\pi$
$978$	18.7314	0.598965
$979$	4.15012	0.132639
$980$	0	0
$981$	5.17146	0.165112
$982$	−17.1108	−0.546028
$983$	4.88404	0.155777	0.0778883	−	0.996962i	$-0.475182\pi$
$983$	4.88404	0.155777	0.0778883	+	0.996962i	$0.475182\pi$
$984$	18.0016	0.573870
$985$	0	0
$986$	0.799746	0.0254691
$987$	−3.37863	−0.107543
$988$	12.1990	0.388101
$989$	79.1586	2.51710
$990$	0	0
$991$	−22.9522	−0.729101	−0.364550	−	0.931184i	$-0.618777\pi$
$991$	−22.9522	−0.729101	−0.364550	+	0.931184i	$0.618777\pi$
$992$	−30.9313	−0.982071
$993$	37.2358	1.18164
$994$	5.87492	0.186341
$995$	0	0
$996$	8.74659	0.277146
$997$	−33.3087	−1.05490	−0.527448	−	0.849587i	$-0.676851\pi$
$997$	−33.3087	−1.05490	−0.527448	+	0.849587i	$0.676851\pi$
$998$	3.82250	0.120999
$999$	45.6749	1.44509

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5075.2.a.ba.1.4		8
5.4	even	2		1015.2.a.l.1.5	✓	8
15.14	odd	2		9135.2.a.bh.1.4		8
35.34	odd	2		7105.2.a.t.1.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1015.2.a.l.1.5	✓	8	5.4	even	2
5075.2.a.ba.1.4		8	1.1	even	1	trivial
7105.2.a.t.1.5		8	35.34	odd	2
9135.2.a.bh.1.4		8	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:	\( N \)	\(=\)	\( 5075 = 5^{2} \cdot 7 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5075.a (trivial)

\(f(q)\)	\(=\)	\(q-0.742598 q^{2} -1.24299 q^{3} -1.44855 q^{4} +0.923043 q^{6} -1.00000 q^{7} +2.56088 q^{8} -1.45497 q^{9} +2.72358 q^{11} +1.80053 q^{12} -2.45958 q^{13} +0.742598 q^{14} +0.995389 q^{16} -1.07696 q^{17} +1.08046 q^{18} +3.42397 q^{19} +1.24299 q^{21} -2.02252 q^{22} -8.67622 q^{23} -3.18316 q^{24} +1.82648 q^{26} +5.53749 q^{27} +1.44855 q^{28} +1.00000 q^{29} +5.27754 q^{31} -5.86094 q^{32} -3.38538 q^{33} +0.799746 q^{34} +2.10760 q^{36} +8.24831 q^{37} -2.54263 q^{38} +3.05724 q^{39} -5.65526 q^{41} -0.923043 q^{42} -9.12362 q^{43} -3.94523 q^{44} +6.44294 q^{46} -2.71814 q^{47} -1.23726 q^{48} +1.00000 q^{49} +1.33865 q^{51} +3.56282 q^{52} +1.79592 q^{53} -4.11213 q^{54} -2.56088 q^{56} -4.25596 q^{57} -0.742598 q^{58} -12.2626 q^{59} -5.46390 q^{61} -3.91909 q^{62} +1.45497 q^{63} +2.36155 q^{64} +2.51398 q^{66} +6.14811 q^{67} +1.56002 q^{68} +10.7845 q^{69} +7.91130 q^{71} -3.72601 q^{72} -2.94337 q^{73} -6.12518 q^{74} -4.95978 q^{76} -2.72358 q^{77} -2.27030 q^{78} -10.6271 q^{79} -2.51815 q^{81} +4.19958 q^{82} +4.85777 q^{83} -1.80053 q^{84} +6.77518 q^{86} -1.24299 q^{87} +6.97477 q^{88} +1.52378 q^{89} +2.45958 q^{91} +12.5679 q^{92} -6.55993 q^{93} +2.01849 q^{94} +7.28510 q^{96} -2.85316 q^{97} -0.742598 q^{98} -3.96273 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{2} - 4 q^{3} + 5 q^{4} - 6 q^{6} - 8 q^{7} + 6 q^{8} + 14 q^{9} + q^{11} + 7 q^{12} + q^{13} + q^{14} + 3 q^{16} - 22 q^{17} - 13 q^{18} + 8 q^{19} + 4 q^{21} + 3 q^{22} - 9 q^{23} + 16 q^{24}+ \cdots - 63 q^{99}+O(q^{100}) \) 8 * q - q^2 - 4 * q^3 + 5 * q^4 - 6 * q^6 - 8 * q^7 + 6 * q^8 + 14 * q^9 + q^11 + 7 * q^12 + q^13 + q^14 + 3 * q^16 - 22 * q^17 - 13 * q^18 + 8 * q^19 + 4 * q^21 + 3 * q^22 - 9 * q^23 + 16 * q^24 + 27 * q^26 - q^27 - 5 * q^28 + 8 * q^29 + 3 * q^31 + 26 * q^32 - 14 * q^33 + q^34 - 13 * q^36 - 9 * q^37 + 5 * q^38 + 3 * q^39 + 24 * q^41 + 6 * q^42 - 7 * q^43 + 10 * q^44 - 27 * q^46 - 21 * q^47 + 8 * q^48 + 8 * q^49 - 8 * q^51 + 7 * q^52 + 2 * q^53 - 33 * q^54 - 6 * q^56 + 10 * q^57 - q^58 + 18 * q^59 + 26 * q^61 - 15 * q^62 - 14 * q^63 + 4 * q^64 + 55 * q^66 + 8 * q^67 - 6 * q^68 - 2 * q^69 - 12 * q^71 - 9 * q^72 + q^73 + 14 * q^74 + 15 * q^76 - q^77 - 8 * q^78 - 14 * q^79 + 40 * q^81 + 42 * q^82 + 10 * q^83 - 7 * q^84 + 19 * q^86 - 4 * q^87 + 29 * q^88 - 4 * q^89 - q^91 - 47 * q^92 + 24 * q^93 - 30 * q^94 - q^96 + 11 * q^97 - q^98 - 63 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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