LMFDB - Embedded newform 5225.2.a.x.1.2

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 = [15,5,4,17,0,-1,15] 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:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 5 x^{14} - 11 x^{13} + 85 x^{12} + 6 x^{11} - 537 x^{10} + 327 x^{9} + 1556 x^{8} - 1451 x^{7} + \cdots - 13 $ x^15 - 5x^14 - 11x^13 + 85x^12 + 6x^11 - 537x^10 + 327x^9 + 1556x^8 - 1451x^7 - 2033x^6 + 2316x^5 + 927x^4 - 1295x^3 + 17x^2 + 103x - 13
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.15791$ 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)

$f(q)$	$=$	\(q-2.15791 q^{2} -2.27846 q^{3} +2.65657 q^{4} +4.91670 q^{6} +4.96679 q^{7} -1.41681 q^{8} +2.19136 q^{9} +1.00000 q^{11} -6.05287 q^{12} +7.07849 q^{13} -10.7179 q^{14} -2.25579 q^{16} -0.882524 q^{17} -4.72875 q^{18} -1.00000 q^{19} -11.3166 q^{21} -2.15791 q^{22} -0.450956 q^{23} +3.22814 q^{24} -15.2747 q^{26} +1.84245 q^{27} +13.1946 q^{28} +4.90646 q^{29} -4.59098 q^{31} +7.70140 q^{32} -2.27846 q^{33} +1.90440 q^{34} +5.82149 q^{36} +9.57939 q^{37} +2.15791 q^{38} -16.1280 q^{39} -3.73151 q^{41} +24.4202 q^{42} +6.29514 q^{43} +2.65657 q^{44} +0.973122 q^{46} -5.17054 q^{47} +5.13971 q^{48} +17.6690 q^{49} +2.01079 q^{51} +18.8045 q^{52} -3.00676 q^{53} -3.97584 q^{54} -7.03699 q^{56} +2.27846 q^{57} -10.5877 q^{58} +7.98874 q^{59} +2.63405 q^{61} +9.90692 q^{62} +10.8840 q^{63} -12.1073 q^{64} +4.91670 q^{66} +14.4306 q^{67} -2.34448 q^{68} +1.02748 q^{69} +11.8877 q^{71} -3.10474 q^{72} -11.3797 q^{73} -20.6714 q^{74} -2.65657 q^{76} +4.96679 q^{77} +34.8028 q^{78} +6.34343 q^{79} -10.7720 q^{81} +8.05225 q^{82} +17.2366 q^{83} -30.0633 q^{84} -13.5843 q^{86} -11.1791 q^{87} -1.41681 q^{88} +5.27278 q^{89} +35.1573 q^{91} -1.19799 q^{92} +10.4604 q^{93} +11.1575 q^{94} -17.5473 q^{96} -19.4258 q^{97} -38.1280 q^{98} +2.19136 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 5 q^{2} + 4 q^{3} + 17 q^{4} - q^{6} + 15 q^{7} + 15 q^{8} + 19 q^{9} + 15 q^{11} + 9 q^{12} + 13 q^{13} + 9 q^{14} + 21 q^{16} + 11 q^{17} + 16 q^{18} - 15 q^{19} + 5 q^{22} + 10 q^{23} + 17 q^{24}+ \cdots + 19 q^{99}+O(q^{100}) $ 15 * q + 5 * q^2 + 4 * q^3 + 17 * q^4 - q^6 + 15 * q^7 + 15 * q^8 + 19 * q^9 + 15 * q^11 + 9 * q^12 + 13 * q^13 + 9 * q^14 + 21 * q^16 + 11 * q^17 + 16 * q^18 - 15 * q^19 + 5 * q^22 + 10 * q^23 + 17 * q^24 - 17 * q^26 + 13 * q^27 + 30 * q^28 + 5 * q^29 + 6 * q^31 + 40 * q^32 + 4 * q^33 + 17 * q^34 + 28 * q^36 + 13 * q^37 - 5 * q^38 - 22 * q^39 + 30 * q^42 + 36 * q^43 + 17 * q^44 + 13 * q^46 - 6 * q^47 + 14 * q^48 + 16 * q^49 + 4 * q^51 + 50 * q^52 + 9 * q^53 + 9 * q^54 - 18 * q^56 - 4 * q^57 + 2 * q^58 - 7 * q^59 - 2 * q^61 + 11 * q^62 + 39 * q^63 + 17 * q^64 - q^66 + 35 * q^67 - 18 * q^68 - 9 * q^69 + 13 * q^71 + 68 * q^72 + 2 * q^73 + 13 * q^74 - 17 * q^76 + 15 * q^77 - 10 * q^78 + 6 * q^79 + 11 * q^81 + 14 * q^82 + 30 * q^83 - 6 * q^84 - 25 * q^86 + 19 * q^87 + 15 * q^88 + 55 * q^89 + 26 * q^91 - 18 * q^92 + 14 * q^93 - 22 * q^94 - 17 * q^96 + 28 * q^97 - 22 * q^98 + 19 * 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 $

$2$	−2.15791	−1.52587	−0.762936	−	0.646474i	$-0.776243\pi$
$2$	−2.15791	−1.52587	−0.762936	+	0.646474i	$0.776243\pi$
$3$	−2.27846	−1.31547	−0.657733	−	0.753251i	$-0.728485\pi$
$3$	−2.27846	−1.31547	−0.657733	+	0.753251i	$0.728485\pi$
$4$	2.65657	1.32828
$5$	0	0
$5$	0	0
$6$	4.91670	2.00723
$7$	4.96679	1.87727	0.938634	−	0.344914i	$-0.112092\pi$
$7$	4.96679	1.87727	0.938634	+	0.344914i	$0.112092\pi$
$8$	−1.41681	−0.500918
$9$	2.19136	0.730453
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−6.05287	−1.74731
$13$	7.07849	1.96322	0.981610	−	0.190898i	$-0.0611400\pi$
$13$	7.07849	1.96322	0.981610	+	0.190898i	$0.0611400\pi$
$14$	−10.7179	−2.86447
$15$	0	0
$16$	−2.25579	−0.563947
$17$	−0.882524	−0.214043	−0.107022	−	0.994257i	$-0.534131\pi$
$17$	−0.882524	−0.214043	−0.107022	+	0.994257i	$0.534131\pi$
$18$	−4.72875	−1.11458
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−11.3166	−2.46948
$22$	−2.15791	−0.460068
$23$	−0.450956	−0.0940308	−0.0470154	−	0.998894i	$-0.514971\pi$
$23$	−0.450956	−0.0940308	−0.0470154	+	0.998894i	$0.514971\pi$
$24$	3.22814	0.658941
$25$	0	0
$26$	−15.2747	−2.99562
$27$	1.84245	0.354580
$28$	13.1946	2.49354
$29$	4.90646	0.911106	0.455553	−	0.890209i	$-0.349441\pi$
$29$	4.90646	0.911106	0.455553	+	0.890209i	$0.349441\pi$
$30$	0	0
$31$	−4.59098	−0.824565	−0.412283	−	0.911056i	$-0.635268\pi$
$31$	−4.59098	−0.824565	−0.412283	+	0.911056i	$0.635268\pi$
$32$	7.70140	1.36143
$33$	−2.27846	−0.396628
$34$	1.90440	0.326603
$35$	0	0
$36$	5.82149	0.970249
$37$	9.57939	1.57484	0.787421	−	0.616416i	$-0.211416\pi$
$37$	9.57939	1.57484	0.787421	+	0.616416i	$0.211416\pi$
$38$	2.15791	0.350059
$39$	−16.1280	−2.58255
$40$	0	0
$41$	−3.73151	−0.582764	−0.291382	−	0.956607i	$-0.594115\pi$
$41$	−3.73151	−0.582764	−0.291382	+	0.956607i	$0.594115\pi$
$42$	24.4202	3.76812
$43$	6.29514	0.960000	0.480000	−	0.877269i	$-0.340637\pi$
$43$	6.29514	0.960000	0.480000	+	0.877269i	$0.340637\pi$
$44$	2.65657	0.400493
$45$	0	0
$46$	0.973122	0.143479
$47$	−5.17054	−0.754200	−0.377100	−	0.926173i	$-0.623079\pi$
$47$	−5.17054	−0.754200	−0.377100	+	0.926173i	$0.623079\pi$
$48$	5.13971	0.741853
$49$	17.6690	2.52414
$50$	0	0
$51$	2.01079	0.281567
$52$	18.8045	2.60771
$53$	−3.00676	−0.413010	−0.206505	−	0.978446i	$-0.566209\pi$
$53$	−3.00676	−0.413010	−0.206505	+	0.978446i	$0.566209\pi$
$54$	−3.97584	−0.541043
$55$	0	0
$56$	−7.03699	−0.940358
$57$	2.27846	0.301789
$58$	−10.5877	−1.39023
$59$	7.98874	1.04005	0.520023	−	0.854152i	$-0.325923\pi$
$59$	7.98874	1.04005	0.520023	+	0.854152i	$0.325923\pi$
$60$	0	0
$61$	2.63405	0.337256	0.168628	−	0.985680i	$-0.446066\pi$
$61$	2.63405	0.337256	0.168628	+	0.985680i	$0.446066\pi$
$62$	9.90692	1.25818
$63$	10.8840	1.37126
$64$	−12.1073	−1.51342
$65$	0	0
$66$	4.91670	0.605204
$67$	14.4306	1.76298	0.881491	−	0.472200i	$-0.156540\pi$
$67$	14.4306	1.76298	0.881491	+	0.472200i	$0.156540\pi$
$68$	−2.34448	−0.284310
$69$	1.02748	0.123694
$70$	0	0
$71$	11.8877	1.41081	0.705404	−	0.708805i	$-0.250765\pi$
$71$	11.8877	1.41081	0.705404	+	0.708805i	$0.250765\pi$
$72$	−3.10474	−0.365897
$73$	−11.3797	−1.33190	−0.665949	−	0.745997i	$-0.731973\pi$
$73$	−11.3797	−1.33190	−0.665949	+	0.745997i	$0.731973\pi$
$74$	−20.6714	−2.40301
$75$	0	0
$76$	−2.65657	−0.304729
$77$	4.96679	0.566018
$78$	34.8028	3.94064
$79$	6.34343	0.713691	0.356846	−	0.934163i	$-0.383852\pi$
$79$	6.34343	0.713691	0.356846	+	0.934163i	$0.383852\pi$
$80$	0	0
$81$	−10.7720	−1.19689
$82$	8.05225	0.889223
$83$	17.2366	1.89196	0.945982	−	0.324221i	$-0.105102\pi$
$83$	17.2366	1.89196	0.945982	+	0.324221i	$0.105102\pi$
$84$	−30.0633	−3.28018
$85$	0	0
$86$	−13.5843	−1.46484
$87$	−11.1791	−1.19853
$88$	−1.41681	−0.151033
$89$	5.27278	0.558914	0.279457	−	0.960158i	$-0.409846\pi$
$89$	5.27278	0.558914	0.279457	+	0.960158i	$0.409846\pi$
$90$	0	0
$91$	35.1573	3.68549
$92$	−1.19799	−0.124900
$93$	10.4604	1.08469
$94$	11.1575	1.15081
$95$	0	0
$96$	−17.5473	−1.79091
$97$	−19.4258	−1.97239	−0.986195	−	0.165587i	$-0.947048\pi$
$97$	−19.4258	−1.97239	−0.986195	+	0.165587i	$0.947048\pi$
$98$	−38.1280	−3.85151
$99$	2.19136	0.220240
$100$	0	0
$101$	−8.82167	−0.877789	−0.438895	−	0.898539i	$-0.644630\pi$
$101$	−8.82167	−0.877789	−0.438895	+	0.898539i	$0.644630\pi$
$102$	−4.33910	−0.429635
$103$	5.46174	0.538161	0.269080	−	0.963118i	$-0.413280\pi$
$103$	5.46174	0.538161	0.269080	+	0.963118i	$0.413280\pi$
$104$	−10.0289	−0.983413
$105$	0	0
$106$	6.48830	0.630200
$107$	−17.9262	−1.73299	−0.866497	−	0.499182i	$-0.833634\pi$
$107$	−17.9262	−1.73299	−0.866497	+	0.499182i	$0.833634\pi$
$108$	4.89459	0.470983
$109$	17.0978	1.63767	0.818835	−	0.574029i	$-0.194620\pi$
$109$	17.0978	1.63767	0.818835	+	0.574029i	$0.194620\pi$
$110$	0	0
$111$	−21.8262	−2.07165
$112$	−11.2040	−1.05868
$113$	3.10231	0.291841	0.145920	−	0.989296i	$-0.453386\pi$
$113$	3.10231	0.291841	0.145920	+	0.989296i	$0.453386\pi$
$114$	−4.91670	−0.460491
$115$	0	0
$116$	13.0343	1.21021
$117$	15.5115	1.43404
$118$	−17.2390	−1.58698
$119$	−4.38331	−0.401817
$120$	0	0
$121$	1.00000	0.0909091
$122$	−5.68405	−0.514609
$123$	8.50208	0.766606
$124$	−12.1963	−1.09526
$125$	0	0
$126$	−23.4867	−2.09236
$127$	16.3093	1.44722	0.723609	−	0.690210i	$-0.242482\pi$
$127$	16.3093	1.44722	0.723609	+	0.690210i	$0.242482\pi$
$128$	10.7237	0.947852
$129$	−14.3432	−1.26285
$130$	0	0
$131$	−19.3238	−1.68833	−0.844165	−	0.536083i	$-0.819904\pi$
$131$	−19.3238	−1.68833	−0.844165	+	0.536083i	$0.819904\pi$
$132$	−6.05287	−0.526835
$133$	−4.96679	−0.430675
$134$	−31.1400	−2.69008
$135$	0	0
$136$	1.25037	0.107218
$137$	10.4835	0.895663	0.447832	−	0.894118i	$-0.352196\pi$
$137$	10.4835	0.895663	0.447832	+	0.894118i	$0.352196\pi$
$138$	−2.21721	−0.188742
$139$	−6.68861	−0.567321	−0.283660	−	0.958925i	$-0.591549\pi$
$139$	−6.68861	−0.567321	−0.283660	+	0.958925i	$0.591549\pi$
$140$	0	0
$141$	11.7808	0.992125
$142$	−25.6525	−2.15271
$143$	7.07849	0.591933
$144$	−4.94324	−0.411937
$145$	0	0
$146$	24.5564	2.03231
$147$	−40.2579	−3.32042
$148$	25.4483	2.09184
$149$	−18.7751	−1.53811	−0.769057	−	0.639180i	$-0.779274\pi$
$149$	−18.7751	−1.53811	−0.769057	+	0.639180i	$0.779274\pi$
$150$	0	0
$151$	−6.60661	−0.537638	−0.268819	−	0.963191i	$-0.586633\pi$
$151$	−6.60661	−0.537638	−0.268819	+	0.963191i	$0.586633\pi$
$152$	1.41681	0.114919
$153$	−1.93393	−0.156349
$154$	−10.7179	−0.863670
$155$	0	0
$156$	−42.8452	−3.43036
$157$	4.79891	0.382995	0.191497	−	0.981493i	$-0.438666\pi$
$157$	4.79891	0.382995	0.191497	+	0.981493i	$0.438666\pi$
$158$	−13.6885	−1.08900
$159$	6.85076	0.543301
$160$	0	0
$161$	−2.23980	−0.176521
$162$	23.2450	1.82630
$163$	−13.1690	−1.03148	−0.515739	−	0.856746i	$-0.672483\pi$
$163$	−13.1690	−1.03148	−0.515739	+	0.856746i	$0.672483\pi$
$164$	−9.91300	−0.774075
$165$	0	0
$166$	−37.1950	−2.88689
$167$	3.07374	0.237853	0.118926	−	0.992903i	$-0.462055\pi$
$167$	3.07374	0.237853	0.118926	+	0.992903i	$0.462055\pi$
$168$	16.0335	1.23701
$169$	37.1050	2.85423
$170$	0	0
$171$	−2.19136	−0.167577
$172$	16.7235	1.27515
$173$	5.63958	0.428769	0.214385	−	0.976749i	$-0.431225\pi$
$173$	5.63958	0.428769	0.214385	+	0.976749i	$0.431225\pi$
$174$	24.1236	1.82880
$175$	0	0
$176$	−2.25579	−0.170036
$177$	−18.2020	−1.36815
$178$	−11.3782	−0.852831
$179$	1.40812	0.105248	0.0526239	−	0.998614i	$-0.483242\pi$
$179$	1.40812	0.105248	0.0526239	+	0.998614i	$0.483242\pi$
$180$	0	0
$181$	−13.3227	−0.990270	−0.495135	−	0.868816i	$-0.664881\pi$
$181$	−13.3227	−0.990270	−0.495135	+	0.868816i	$0.664881\pi$
$182$	−75.8663	−5.62358
$183$	−6.00157	−0.443649
$184$	0.638919	0.0471018
$185$	0	0
$186$	−22.5725	−1.65509
$187$	−0.882524	−0.0645365
$188$	−13.7359	−1.00179
$189$	9.15106	0.665642
$190$	0	0
$191$	−10.0000	−0.723576	−0.361788	−	0.932260i	$-0.617834\pi$
$191$	−10.0000	−0.723576	−0.361788	+	0.932260i	$0.617834\pi$
$192$	27.5860	1.99085
$193$	9.04980	0.651419	0.325710	−	0.945470i	$-0.394397\pi$
$193$	9.04980	0.651419	0.325710	+	0.945470i	$0.394397\pi$
$194$	41.9191	3.00961
$195$	0	0
$196$	46.9388	3.35277
$197$	−4.60794	−0.328302	−0.164151	−	0.986435i	$-0.552488\pi$
$197$	−4.60794	−0.328302	−0.164151	+	0.986435i	$0.552488\pi$
$198$	−4.72875	−0.336058
$199$	6.28722	0.445689	0.222845	−	0.974854i	$-0.428466\pi$
$199$	6.28722	0.445689	0.222845	+	0.974854i	$0.428466\pi$
$200$	0	0
$201$	−32.8796	−2.31915
$202$	19.0364	1.33939
$203$	24.3693	1.71039
$204$	5.34180	0.374001
$205$	0	0
$206$	−11.7859	−0.821164
$207$	−0.988207	−0.0686851
$208$	−15.9676	−1.10715
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	−8.43652	−0.580794	−0.290397	−	0.956906i	$-0.593787\pi$
$211$	−8.43652	−0.580794	−0.290397	+	0.956906i	$0.593787\pi$
$212$	−7.98765	−0.548594
$213$	−27.0856	−1.85587
$214$	38.6832	2.64433
$215$	0	0
$216$	−2.61040	−0.177616
$217$	−22.8024	−1.54793
$218$	−36.8954	−2.49887
$219$	25.9282	1.75207
$220$	0	0
$221$	−6.24693	−0.420214
$222$	47.0989	3.16107
$223$	7.41729	0.496699	0.248349	−	0.968671i	$-0.420112\pi$
$223$	7.41729	0.496699	0.248349	+	0.968671i	$0.420112\pi$
$224$	38.2512	2.55577
$225$	0	0
$226$	−6.69450	−0.445312
$227$	−26.0253	−1.72736	−0.863680	−	0.504040i	$-0.831846\pi$
$227$	−26.0253	−1.72736	−0.863680	+	0.504040i	$0.831846\pi$
$228$	6.05287	0.400861
$229$	−4.45087	−0.294122	−0.147061	−	0.989127i	$-0.546981\pi$
$229$	−4.45087	−0.294122	−0.147061	+	0.989127i	$0.546981\pi$
$230$	0	0
$231$	−11.3166	−0.744578
$232$	−6.95152	−0.456390
$233$	6.22176	0.407601	0.203801	−	0.979012i	$-0.434671\pi$
$233$	6.22176	0.407601	0.203801	+	0.979012i	$0.434671\pi$
$234$	−33.4724	−2.18816
$235$	0	0
$236$	21.2226	1.38148
$237$	−14.4532	−0.938837
$238$	9.45877	0.613121
$239$	−3.20257	−0.207157	−0.103578	−	0.994621i	$-0.533029\pi$
$239$	−3.20257	−0.207157	−0.103578	+	0.994621i	$0.533029\pi$
$240$	0	0
$241$	6.27683	0.404326	0.202163	−	0.979352i	$-0.435203\pi$
$241$	6.27683	0.404326	0.202163	+	0.979352i	$0.435203\pi$
$242$	−2.15791	−0.138716
$243$	19.0162	1.21989
$244$	6.99754	0.447972
$245$	0	0
$246$	−18.3467	−1.16974
$247$	−7.07849	−0.450393
$248$	6.50456	0.413040
$249$	−39.2728	−2.48881
$250$	0	0
$251$	−2.74929	−0.173534	−0.0867668	−	0.996229i	$-0.527654\pi$
$251$	−2.74929	−0.173534	−0.0867668	+	0.996229i	$0.527654\pi$
$252$	28.9141	1.82142
$253$	−0.450956	−0.0283514
$254$	−35.1940	−2.20827
$255$	0	0
$256$	1.07387	0.0671167
$257$	10.7158	0.668433	0.334216	−	0.942496i	$-0.391528\pi$
$257$	10.7158	0.668433	0.334216	+	0.942496i	$0.391528\pi$
$258$	30.9513	1.92694
$259$	47.5788	2.95640
$260$	0	0
$261$	10.7518	0.665521
$262$	41.6990	2.57618
$263$	23.5337	1.45115	0.725577	−	0.688141i	$-0.241573\pi$
$263$	23.5337	1.45115	0.725577	+	0.688141i	$0.241573\pi$
$264$	3.22814	0.198678
$265$	0	0
$266$	10.7179	0.657155
$267$	−12.0138	−0.735233
$268$	38.3359	2.34174
$269$	−9.95531	−0.606986	−0.303493	−	0.952834i	$-0.598153\pi$
$269$	−9.95531	−0.606986	−0.303493	+	0.952834i	$0.598153\pi$
$270$	0	0
$271$	−23.8531	−1.44897	−0.724485	−	0.689291i	$-0.757922\pi$
$271$	−23.8531	−1.44897	−0.724485	+	0.689291i	$0.757922\pi$
$272$	1.99078	0.120709
$273$	−80.1044	−4.84814
$274$	−22.6224	−1.36667
$275$	0	0
$276$	2.72958	0.164301
$277$	3.64629	0.219084	0.109542	−	0.993982i	$-0.465062\pi$
$277$	3.64629	0.219084	0.109542	+	0.993982i	$0.465062\pi$
$278$	14.4334	0.865658
$279$	−10.0605	−0.602306
$280$	0	0
$281$	−17.4125	−1.03875	−0.519373	−	0.854548i	$-0.673834\pi$
$281$	−17.4125	−1.03875	−0.519373	+	0.854548i	$0.673834\pi$
$282$	−25.4220	−1.51386
$283$	−4.12829	−0.245401	−0.122701	−	0.992444i	$-0.539155\pi$
$283$	−4.12829	−0.245401	−0.122701	+	0.992444i	$0.539155\pi$
$284$	31.5804	1.87395
$285$	0	0
$286$	−15.2747	−0.903214
$287$	−18.5336	−1.09400
$288$	16.8765	0.994460
$289$	−16.2212	−0.954185
$290$	0	0
$291$	44.2608	2.59461
$292$	−30.2310	−1.76914
$293$	−15.3602	−0.897354	−0.448677	−	0.893694i	$-0.648105\pi$
$293$	−15.3602	−0.897354	−0.448677	+	0.893694i	$0.648105\pi$
$294$	86.8729	5.06653
$295$	0	0
$296$	−13.5722	−0.788867
$297$	1.84245	0.106910
$298$	40.5149	2.34696
$299$	−3.19209	−0.184603
$300$	0	0
$301$	31.2666	1.80218
$302$	14.2565	0.820367
$303$	20.0998	1.15470
$304$	2.25579	0.129378
$305$	0	0
$306$	4.17324	0.238568
$307$	−15.3305	−0.874955	−0.437478	−	0.899229i	$-0.644128\pi$
$307$	−15.3305	−0.874955	−0.437478	+	0.899229i	$0.644128\pi$
$308$	13.1946	0.751832
$309$	−12.4443	−0.707933
$310$	0	0
$311$	18.7433	1.06284	0.531418	−	0.847110i	$-0.321659\pi$
$311$	18.7433	1.06284	0.531418	+	0.847110i	$0.321659\pi$
$312$	22.8504	1.29365
$313$	15.9384	0.900894	0.450447	−	0.892803i	$-0.351265\pi$
$313$	15.9384	0.900894	0.450447	+	0.892803i	$0.351265\pi$
$314$	−10.3556	−0.584401
$315$	0	0
$316$	16.8517	0.947984
$317$	9.04934	0.508261	0.254131	−	0.967170i	$-0.418211\pi$
$317$	9.04934	0.508261	0.254131	+	0.967170i	$0.418211\pi$
$318$	−14.7833	−0.829007
$319$	4.90646	0.274709
$320$	0	0
$321$	40.8441	2.27970
$322$	4.83329	0.269348
$323$	0.882524	0.0491049
$324$	−28.6166	−1.58981
$325$	0	0
$326$	28.4175	1.57390
$327$	−38.9565	−2.15430
$328$	5.28684	0.291917
$329$	−25.6809	−1.41584
$330$	0	0
$331$	−5.01790	−0.275809	−0.137904	−	0.990446i	$-0.544037\pi$
$331$	−5.01790	−0.275809	−0.137904	+	0.990446i	$0.544037\pi$
$332$	45.7902	2.51306
$333$	20.9919	1.15035
$334$	−6.63284	−0.362933
$335$	0	0
$336$	25.5278	1.39266
$337$	−10.0176	−0.545696	−0.272848	−	0.962057i	$-0.587966\pi$
$337$	−10.0176	−0.545696	−0.272848	+	0.962057i	$0.587966\pi$
$338$	−80.0692	−4.35519
$339$	−7.06848	−0.383907
$340$	0	0
$341$	−4.59098	−0.248616
$342$	4.72875	0.255702
$343$	52.9904	2.86121
$344$	−8.91902	−0.480881
$345$	0	0
$346$	−12.1697	−0.654246
$347$	8.03717	0.431458	0.215729	−	0.976453i	$-0.430787\pi$
$347$	8.03717	0.431458	0.215729	+	0.976453i	$0.430787\pi$
$348$	−29.6981	−1.59199
$349$	12.1481	0.650271	0.325135	−	0.945668i	$-0.394590\pi$
$349$	12.1481	0.650271	0.325135	+	0.945668i	$0.394590\pi$
$350$	0	0
$351$	13.0418	0.696118
$352$	7.70140	0.410486
$353$	−23.7465	−1.26390	−0.631949	−	0.775010i	$-0.717745\pi$
$353$	−23.7465	−1.26390	−0.631949	+	0.775010i	$0.717745\pi$
$354$	39.2782	2.08761
$355$	0	0
$356$	14.0075	0.742396
$357$	9.98717	0.528577
$358$	−3.03859	−0.160595
$359$	−3.60073	−0.190040	−0.0950198	−	0.995475i	$-0.530291\pi$
$359$	−3.60073	−0.190040	−0.0950198	+	0.995475i	$0.530291\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	28.7492	1.51102
$363$	−2.27846	−0.119588
$364$	93.3978	4.89538
$365$	0	0
$366$	12.9508	0.676952
$367$	−17.7138	−0.924651	−0.462325	−	0.886710i	$-0.652985\pi$
$367$	−17.7138	−0.924651	−0.462325	+	0.886710i	$0.652985\pi$
$368$	1.01726	0.0530284
$369$	−8.17708	−0.425682
$370$	0	0
$371$	−14.9339	−0.775330
$372$	27.7886	1.44077
$373$	23.9253	1.23881	0.619403	−	0.785073i	$-0.287375\pi$
$373$	23.9253	1.23881	0.619403	+	0.785073i	$0.287375\pi$
$374$	1.90440	0.0984744
$375$	0	0
$376$	7.32567	0.377793
$377$	34.7303	1.78870
$378$	−19.7471	−1.01568
$379$	15.5016	0.796261	0.398131	−	0.917329i	$-0.369659\pi$
$379$	15.5016	0.796261	0.398131	+	0.917329i	$0.369659\pi$
$380$	0	0
$381$	−37.1601	−1.90377
$382$	21.5791	1.10408
$383$	−5.45842	−0.278912	−0.139456	−	0.990228i	$-0.544535\pi$
$383$	−5.45842	−0.278912	−0.139456	+	0.990228i	$0.544535\pi$
$384$	−24.4335	−1.24687
$385$	0	0
$386$	−19.5286	−0.993982
$387$	13.7949	0.701235
$388$	−51.6059	−2.61989
$389$	−33.2696	−1.68683	−0.843417	−	0.537260i	$-0.819459\pi$
$389$	−33.2696	−1.68683	−0.843417	+	0.537260i	$0.819459\pi$
$390$	0	0
$391$	0.397979	0.0201267
$392$	−25.0336	−1.26439
$393$	44.0285	2.22094
$394$	9.94351	0.500947
$395$	0	0
$396$	5.82149	0.292541
$397$	−5.90789	−0.296509	−0.148254	−	0.988949i	$-0.547365\pi$
$397$	−5.90789	−0.296509	−0.148254	+	0.988949i	$0.547365\pi$
$398$	−13.5672	−0.680064
$399$	11.3166	0.566539
$400$	0	0
$401$	−23.6503	−1.18104	−0.590520	−	0.807023i	$-0.701077\pi$
$401$	−23.6503	−1.18104	−0.590520	+	0.807023i	$0.701077\pi$
$402$	70.9511	3.53872
$403$	−32.4972	−1.61880
$404$	−23.4354	−1.16595
$405$	0	0
$406$	−52.5868	−2.60984
$407$	9.57939	0.474833
$408$	−2.84891	−0.141042
$409$	−18.7712	−0.928177	−0.464089	−	0.885789i	$-0.653618\pi$
$409$	−18.7712	−0.928177	−0.464089	+	0.885789i	$0.653618\pi$
$410$	0	0
$411$	−23.8861	−1.17822
$412$	14.5095	0.714830
$413$	39.6784	1.95244
$414$	2.13246	0.104805
$415$	0	0
$416$	54.5143	2.67278
$417$	15.2397	0.746292
$418$	2.15791	0.105547
$419$	18.8197	0.919403	0.459701	−	0.888074i	$-0.347956\pi$
$419$	18.8197	0.919403	0.459701	+	0.888074i	$0.347956\pi$
$420$	0	0
$421$	−7.61118	−0.370946	−0.185473	−	0.982649i	$-0.559382\pi$
$421$	−7.61118	−0.370946	−0.185473	+	0.982649i	$0.559382\pi$
$422$	18.2052	0.886217
$423$	−11.3305	−0.550908
$424$	4.26000	0.206884
$425$	0	0
$426$	58.4481	2.83182
$427$	13.0828	0.633120
$428$	−47.6222	−2.30191
$429$	−16.1280	−0.778668
$430$	0	0
$431$	10.7318	0.516931	0.258465	−	0.966021i	$-0.416783\pi$
$431$	10.7318	0.516931	0.258465	+	0.966021i	$0.416783\pi$
$432$	−4.15618	−0.199964
$433$	−30.3651	−1.45925	−0.729626	−	0.683846i	$-0.760306\pi$
$433$	−30.3651	−1.45925	−0.729626	+	0.683846i	$0.760306\pi$
$434$	49.2056	2.36194
$435$	0	0
$436$	45.4214	2.17529
$437$	0.450956	0.0215722
$438$	−55.9508	−2.67343
$439$	12.6527	0.603880	0.301940	−	0.953327i	$-0.402366\pi$
$439$	12.6527	0.603880	0.301940	+	0.953327i	$0.402366\pi$
$440$	0	0
$441$	38.7190	1.84376
$442$	13.4803	0.641193
$443$	−23.6862	−1.12537	−0.562683	−	0.826673i	$-0.690231\pi$
$443$	−23.6862	−1.12537	−0.562683	+	0.826673i	$0.690231\pi$
$444$	−57.9828	−2.75174
$445$	0	0
$446$	−16.0058	−0.757898
$447$	42.7782	2.02334
$448$	−60.1346	−2.84109
$449$	20.2496	0.955638	0.477819	−	0.878458i	$-0.341428\pi$
$449$	20.2496	0.955638	0.477819	+	0.878458i	$0.341428\pi$
$450$	0	0
$451$	−3.73151	−0.175710
$452$	8.24150	0.387647
$453$	15.0529	0.707245
$454$	56.1602	2.63573
$455$	0	0
$456$	−3.22814	−0.151172
$457$	−13.1720	−0.616159	−0.308080	−	0.951361i	$-0.599686\pi$
$457$	−13.1720	−0.616159	−0.308080	+	0.951361i	$0.599686\pi$
$458$	9.60457	0.448792
$459$	−1.62601	−0.0758955
$460$	0	0
$461$	22.4846	1.04721	0.523605	−	0.851961i	$-0.324587\pi$
$461$	22.4846	1.04721	0.523605	+	0.851961i	$0.324587\pi$
$462$	24.4202	1.13613
$463$	19.4962	0.906064	0.453032	−	0.891494i	$-0.350342\pi$
$463$	19.4962	0.906064	0.453032	+	0.891494i	$0.350342\pi$
$464$	−11.0679	−0.513815
$465$	0	0
$466$	−13.4260	−0.621947
$467$	13.5551	0.627254	0.313627	−	0.949546i	$-0.398456\pi$
$467$	13.5551	0.627254	0.313627	+	0.949546i	$0.398456\pi$
$468$	41.2074	1.90481
$469$	71.6739	3.30959
$470$	0	0
$471$	−10.9341	−0.503817
$472$	−11.3185	−0.520978
$473$	6.29514	0.289451
$474$	31.1887	1.43254
$475$	0	0
$476$	−11.6445	−0.533727
$477$	−6.58888	−0.301684
$478$	6.91084	0.316095
$479$	−23.0340	−1.05245	−0.526224	−	0.850346i	$-0.676393\pi$
$479$	−23.0340	−1.05245	−0.526224	+	0.850346i	$0.676393\pi$
$480$	0	0
$481$	67.8076	3.09176
$482$	−13.5448	−0.616950
$483$	5.10329	0.232208
$484$	2.65657	0.120753
$485$	0	0
$486$	−41.0353	−1.86140
$487$	−29.5256	−1.33793	−0.668966	−	0.743293i	$-0.733263\pi$
$487$	−29.5256	−1.33793	−0.668966	+	0.743293i	$0.733263\pi$
$488$	−3.73196	−0.168938
$489$	30.0050	1.35687
$490$	0	0
$491$	4.66909	0.210713	0.105357	−	0.994435i	$-0.466402\pi$
$491$	4.66909	0.210713	0.105357	+	0.994435i	$0.466402\pi$
$492$	22.5863	1.01827
$493$	−4.33006	−0.195016
$494$	15.2747	0.687243
$495$	0	0
$496$	10.3563	0.465011
$497$	59.0436	2.64847
$498$	84.7472	3.79761
$499$	6.17644	0.276495	0.138248	−	0.990398i	$-0.455853\pi$
$499$	6.17644	0.276495	0.138248	+	0.990398i	$0.455853\pi$
$500$	0	0
$501$	−7.00337	−0.312888
$502$	5.93272	0.264790
$503$	30.2556	1.34903	0.674514	−	0.738262i	$-0.264353\pi$
$503$	30.2556	1.34903	0.674514	+	0.738262i	$0.264353\pi$
$504$	−15.4206	−0.686887
$505$	0	0
$506$	0.973122	0.0432605
$507$	−84.5421	−3.75465
$508$	43.3268	1.92232
$509$	40.9940	1.81703	0.908515	−	0.417853i	$-0.137217\pi$
$509$	40.9940	1.81703	0.908515	+	0.417853i	$0.137217\pi$
$510$	0	0
$511$	−56.5207	−2.50033
$512$	−23.7648	−1.05026
$513$	−1.84245	−0.0813462
$514$	−23.1237	−1.01994
$515$	0	0
$516$	−38.1037	−1.67742
$517$	−5.17054	−0.227400
$518$	−102.671	−4.51109
$519$	−12.8495	−0.564031
$520$	0	0
$521$	−1.91886	−0.0840666	−0.0420333	−	0.999116i	$-0.513384\pi$
$521$	−1.91886	−0.0840666	−0.0420333	+	0.999116i	$0.513384\pi$
$522$	−23.2014	−1.01550
$523$	−7.45333	−0.325911	−0.162956	−	0.986633i	$-0.552103\pi$
$523$	−7.45333	−0.325911	−0.162956	+	0.986633i	$0.552103\pi$
$524$	−51.3350	−2.24258
$525$	0	0
$526$	−50.7837	−2.21427
$527$	4.05165	0.176493
$528$	5.13971	0.223677
$529$	−22.7966	−0.991158
$530$	0	0
$531$	17.5062	0.759704
$532$	−13.1946	−0.572058
$533$	−26.4134	−1.14409
$534$	25.9247	1.12187
$535$	0	0
$536$	−20.4455	−0.883110
$537$	−3.20834	−0.138450
$538$	21.4826	0.926182
$539$	17.6690	0.761056
$540$	0	0
$541$	9.31376	0.400430	0.200215	−	0.979752i	$-0.435836\pi$
$541$	9.31376	0.400430	0.200215	+	0.979752i	$0.435836\pi$
$542$	51.4727	2.21094
$543$	30.3552	1.30267
$544$	−6.79667	−0.291405
$545$	0	0
$546$	172.858	7.39764
$547$	24.8678	1.06327	0.531636	−	0.846973i	$-0.321578\pi$
$547$	24.8678	1.06327	0.531636	+	0.846973i	$0.321578\pi$
$548$	27.8500	1.18969
$549$	5.77216	0.246350
$550$	0	0
$551$	−4.90646	−0.209022
$552$	−1.45575	−0.0619608
$553$	31.5064	1.33979
$554$	−7.86835	−0.334294
$555$	0	0
$556$	−17.7687	−0.753563
$557$	40.6011	1.72032	0.860161	−	0.510022i	$-0.170363\pi$
$557$	40.6011	1.72032	0.860161	+	0.510022i	$0.170363\pi$
$558$	21.7096	0.919042
$559$	44.5601	1.88469
$560$	0	0
$561$	2.01079	0.0848957
$562$	37.5747	1.58499
$563$	−12.6733	−0.534114	−0.267057	−	0.963681i	$-0.586051\pi$
$563$	−12.6733	−0.534114	−0.267057	+	0.963681i	$0.586051\pi$
$564$	31.2966	1.31782
$565$	0	0
$566$	8.90846	0.374451
$567$	−53.5023	−2.24689
$568$	−16.8426	−0.706700
$569$	2.72793	0.114361	0.0571803	−	0.998364i	$-0.481789\pi$
$569$	2.72793	0.114361	0.0571803	+	0.998364i	$0.481789\pi$
$570$	0	0
$571$	19.8177	0.829346	0.414673	−	0.909971i	$-0.363896\pi$
$571$	19.8177	0.829346	0.414673	+	0.909971i	$0.363896\pi$
$572$	18.8045	0.786255
$573$	22.7846	0.951840
$574$	39.9938	1.66931
$575$	0	0
$576$	−26.5315	−1.10548
$577$	4.34646	0.180946	0.0904728	−	0.995899i	$-0.471162\pi$
$577$	4.34646	0.180946	0.0904728	+	0.995899i	$0.471162\pi$
$578$	35.0038	1.45596
$579$	−20.6196	−0.856920
$580$	0	0
$581$	85.6105	3.55172
$582$	−95.5107	−3.95905
$583$	−3.00676	−0.124527
$584$	16.1229	0.667172
$585$	0	0
$586$	33.1460	1.36925
$587$	15.4236	0.636601	0.318301	−	0.947990i	$-0.396888\pi$
$587$	15.4236	0.636601	0.318301	+	0.947990i	$0.396888\pi$
$588$	−106.948	−4.41046
$589$	4.59098	0.189168
$590$	0	0
$591$	10.4990	0.431870
$592$	−21.6090	−0.888126
$593$	−41.6736	−1.71133	−0.855666	−	0.517529i	$-0.826852\pi$
$593$	−41.6736	−1.71133	−0.855666	+	0.517529i	$0.826852\pi$
$594$	−3.97584	−0.163131
$595$	0	0
$596$	−49.8772	−2.04305
$597$	−14.3251	−0.586289
$598$	6.88823	0.281681
$599$	7.51792	0.307174	0.153587	−	0.988135i	$-0.450917\pi$
$599$	7.51792	0.307174	0.153587	+	0.988135i	$0.450917\pi$
$600$	0	0
$601$	−13.1820	−0.537705	−0.268853	−	0.963181i	$-0.586645\pi$
$601$	−13.1820	−0.537705	−0.268853	+	0.963181i	$0.586645\pi$
$602$	−67.4705	−2.74989
$603$	31.6227	1.28778
$604$	−17.5509	−0.714136
$605$	0	0
$606$	−43.3735	−1.76193
$607$	18.0725	0.733542	0.366771	−	0.930311i	$-0.380463\pi$
$607$	18.0725	0.733542	0.366771	+	0.930311i	$0.380463\pi$
$608$	−7.70140	−0.312333
$609$	−55.5244	−2.24996
$610$	0	0
$611$	−36.5996	−1.48066
$612$	−5.13761	−0.207675
$613$	−7.35354	−0.297007	−0.148503	−	0.988912i	$-0.547446\pi$
$613$	−7.35354	−0.297007	−0.148503	+	0.988912i	$0.547446\pi$
$614$	33.0817	1.33507
$615$	0	0
$616$	−7.03699	−0.283529
$617$	36.9153	1.48615	0.743077	−	0.669206i	$-0.233366\pi$
$617$	36.9153	1.48615	0.743077	+	0.669206i	$0.233366\pi$
$618$	26.8537	1.08021
$619$	35.6415	1.43255	0.716276	−	0.697817i	$-0.245845\pi$
$619$	35.6415	1.43255	0.716276	+	0.697817i	$0.245845\pi$
$620$	0	0
$621$	−0.830864	−0.0333414
$622$	−40.4463	−1.62175
$623$	26.1888	1.04923
$624$	36.3814	1.45642
$625$	0	0
$626$	−34.3937	−1.37465
$627$	2.27846	0.0909927
$628$	12.7486	0.508726
$629$	−8.45403	−0.337084
$630$	0	0
$631$	−0.205084	−0.00816426	−0.00408213	−	0.999992i	$-0.501299\pi$
$631$	−0.205084	−0.00816426	−0.00408213	+	0.999992i	$0.501299\pi$
$632$	−8.98743	−0.357501
$633$	19.2222	0.764015
$634$	−19.5276	−0.775541
$635$	0	0
$636$	18.1995	0.721657
$637$	125.070	4.95543
$638$	−10.5877	−0.419170
$639$	26.0502	1.03053
$640$	0	0
$641$	10.3863	0.410234	0.205117	−	0.978737i	$-0.434242\pi$
$641$	10.3863	0.410234	0.205117	+	0.978737i	$0.434242\pi$
$642$	−88.1379	−3.47852
$643$	−4.09055	−0.161315	−0.0806577	−	0.996742i	$-0.525702\pi$
$643$	−4.09055	−0.161315	−0.0806577	+	0.996742i	$0.525702\pi$
$644$	−5.95018	−0.234470
$645$	0	0
$646$	−1.90440	−0.0749278
$647$	−31.7398	−1.24782	−0.623910	−	0.781496i	$-0.714457\pi$
$647$	−31.7398	−1.24782	−0.623910	+	0.781496i	$0.714457\pi$
$648$	15.2619	0.599545
$649$	7.98874	0.313585
$650$	0	0
$651$	51.9543	2.03625
$652$	−34.9844	−1.37009
$653$	−27.0609	−1.05898	−0.529488	−	0.848317i	$-0.677616\pi$
$653$	−27.0609	−1.05898	−0.529488	+	0.848317i	$0.677616\pi$
$654$	84.0646	3.28718
$655$	0	0
$656$	8.41749	0.328648
$657$	−24.9371	−0.972889
$658$	55.4171	2.16038
$659$	−40.6494	−1.58348	−0.791738	−	0.610861i	$-0.790823\pi$
$659$	−40.6494	−1.58348	−0.791738	+	0.610861i	$0.790823\pi$
$660$	0	0
$661$	−26.1629	−1.01762	−0.508810	−	0.860879i	$-0.669914\pi$
$661$	−26.1629	−1.01762	−0.508810	+	0.860879i	$0.669914\pi$
$662$	10.8282	0.420849
$663$	14.2334	0.552778
$664$	−24.4210	−0.947719
$665$	0	0
$666$	−45.2985	−1.75528
$667$	−2.21260	−0.0856721
$668$	8.16559	0.315936
$669$	−16.9000	−0.653391
$670$	0	0
$671$	2.63405	0.101687
$672$	−87.1537	−3.36203
$673$	−12.1030	−0.466537	−0.233269	−	0.972412i	$-0.574942\pi$
$673$	−12.1030	−0.466537	−0.233269	+	0.972412i	$0.574942\pi$
$674$	21.6171	0.832661
$675$	0	0
$676$	98.5719	3.79123
$677$	3.66596	0.140894	0.0704471	−	0.997516i	$-0.477557\pi$
$677$	3.66596	0.140894	0.0704471	+	0.997516i	$0.477557\pi$
$678$	15.2531	0.585793
$679$	−96.4837	−3.70271
$680$	0	0
$681$	59.2975	2.27229
$682$	9.90692	0.379356
$683$	−39.3905	−1.50724	−0.753618	−	0.657313i	$-0.771693\pi$
$683$	−39.3905	−1.50724	−0.753618	+	0.657313i	$0.771693\pi$
$684$	−5.82149	−0.222590
$685$	0	0
$686$	−114.348	−4.36584
$687$	10.1411	0.386907
$688$	−14.2005	−0.541389
$689$	−21.2833	−0.810829
$690$	0	0
$691$	15.8240	0.601972	0.300986	−	0.953629i	$-0.402684\pi$
$691$	15.8240	0.601972	0.300986	+	0.953629i	$0.402684\pi$
$692$	14.9819	0.569527
$693$	10.8840	0.413449
$694$	−17.3435	−0.658349
$695$	0	0
$696$	15.8387	0.600366
$697$	3.29314	0.124737
$698$	−26.2144	−0.992229
$699$	−14.1760	−0.536186
$700$	0	0
$701$	−48.1331	−1.81796	−0.908980	−	0.416839i	$-0.863138\pi$
$701$	−48.1331	−1.81796	−0.908980	+	0.416839i	$0.863138\pi$
$702$	−28.1429	−1.06219
$703$	−9.57939	−0.361293
$704$	−12.1073	−0.456313
$705$	0	0
$706$	51.2427	1.92854
$707$	−43.8153	−1.64785
$708$	−48.3548	−1.81728
$709$	26.1687	0.982785	0.491392	−	0.870938i	$-0.336488\pi$
$709$	26.1687	0.982785	0.491392	+	0.870938i	$0.336488\pi$
$710$	0	0
$711$	13.9007	0.521318
$712$	−7.47053	−0.279970
$713$	2.07033	0.0775345
$714$	−21.5514	−0.806540
$715$	0	0
$716$	3.74077	0.139799
$717$	7.29690	0.272508
$718$	7.77005	0.289976
$719$	−29.4161	−1.09703	−0.548517	−	0.836140i	$-0.684807\pi$
$719$	−29.4161	−1.09703	−0.548517	+	0.836140i	$0.684807\pi$
$720$	0	0
$721$	27.1273	1.01027
$722$	−2.15791	−0.0803090
$723$	−14.3015	−0.531878
$724$	−35.3927	−1.31536
$725$	0	0
$726$	4.91670	0.182476
$727$	42.1640	1.56378	0.781888	−	0.623419i	$-0.214257\pi$
$727$	42.1640	1.56378	0.781888	+	0.623419i	$0.214257\pi$
$728$	−49.8113	−1.84613
$729$	−11.0115	−0.407835
$730$	0	0
$731$	−5.55561	−0.205482
$732$	−15.9436	−0.589292
$733$	−36.0591	−1.33187	−0.665937	−	0.746008i	$-0.731968\pi$
$733$	−36.0591	−1.33187	−0.665937	+	0.746008i	$0.731968\pi$
$734$	38.2247	1.41090
$735$	0	0
$736$	−3.47299	−0.128016
$737$	14.4306	0.531559
$738$	17.6454	0.649535
$739$	−17.3701	−0.638969	−0.319484	−	0.947592i	$-0.603510\pi$
$739$	−17.3701	−0.638969	−0.319484	+	0.947592i	$0.603510\pi$
$740$	0	0
$741$	16.1280	0.592478
$742$	32.2260	1.18305
$743$	31.8035	1.16676	0.583378	−	0.812201i	$-0.301731\pi$
$743$	31.8035	1.16676	0.583378	+	0.812201i	$0.301731\pi$
$744$	−14.8203	−0.543340
$745$	0	0
$746$	−51.6286	−1.89026
$747$	37.7716	1.38199
$748$	−2.34448	−0.0857228
$749$	−89.0358	−3.25330
$750$	0	0
$751$	−23.6143	−0.861699	−0.430850	−	0.902424i	$-0.641786\pi$
$751$	−23.6143	−0.861699	−0.430850	+	0.902424i	$0.641786\pi$
$752$	11.6636	0.425329
$753$	6.26414	0.228278
$754$	−74.9448	−2.72933
$755$	0	0
$756$	24.3104	0.884161
$757$	−38.5407	−1.40079	−0.700393	−	0.713757i	$-0.746992\pi$
$757$	−38.5407	−1.40079	−0.700393	+	0.713757i	$0.746992\pi$
$758$	−33.4509	−1.21499
$759$	1.02748	0.0372953
$760$	0	0
$761$	−2.30664	−0.0836156	−0.0418078	−	0.999126i	$-0.513312\pi$
$761$	−2.30664	−0.0836156	−0.0418078	+	0.999126i	$0.513312\pi$
$762$	80.1880	2.90490
$763$	84.9210	3.07435
$764$	−26.5657	−0.961114
$765$	0	0
$766$	11.7788	0.425584
$767$	56.5482	2.04184
$768$	−2.44676	−0.0882899
$769$	−25.9610	−0.936178	−0.468089	−	0.883681i	$-0.655057\pi$
$769$	−25.9610	−0.936178	−0.468089	+	0.883681i	$0.655057\pi$
$770$	0	0
$771$	−24.4155	−0.879301
$772$	24.0414	0.865269
$773$	21.2402	0.763958	0.381979	−	0.924171i	$-0.375243\pi$
$773$	21.2402	0.763958	0.381979	+	0.924171i	$0.375243\pi$
$774$	−29.7682	−1.06999
$775$	0	0
$776$	27.5227	0.988006
$777$	−108.406	−3.88905
$778$	71.7926	2.57389
$779$	3.73151	0.133695
$780$	0	0
$781$	11.8877	0.425375
$782$	−0.858803	−0.0307107
$783$	9.03991	0.323060
$784$	−39.8574	−1.42348
$785$	0	0
$786$	−95.0094	−3.38887
$787$	3.82340	0.136290	0.0681448	−	0.997675i	$-0.478292\pi$
$787$	3.82340	0.136290	0.0681448	+	0.997675i	$0.478292\pi$
$788$	−12.2413	−0.436078
$789$	−53.6206	−1.90894
$790$	0	0
$791$	15.4085	0.547864
$792$	−3.10474	−0.110322
$793$	18.6451	0.662108
$794$	12.7487	0.452434
$795$	0	0
$796$	16.7024	0.592001
$797$	−0.761886	−0.0269874	−0.0134937	−	0.999909i	$-0.504295\pi$
$797$	−0.761886	−0.0269874	−0.0134937	+	0.999909i	$0.504295\pi$
$798$	−24.4202	−0.864465
$799$	4.56312	0.161432
$800$	0	0
$801$	11.5546	0.408260
$802$	51.0352	1.80211
$803$	−11.3797	−0.401582
$804$	−87.3467	−3.08048
$805$	0	0
$806$	70.1260	2.47008
$807$	22.6827	0.798470
$808$	12.4986	0.439701
$809$	31.0757	1.09256	0.546282	−	0.837601i	$-0.316043\pi$
$809$	31.0757	1.09256	0.546282	+	0.837601i	$0.316043\pi$
$810$	0	0
$811$	−9.80439	−0.344279	−0.172139	−	0.985073i	$-0.555068\pi$
$811$	−9.80439	−0.344279	−0.172139	+	0.985073i	$0.555068\pi$
$812$	64.7387	2.27188
$813$	54.3481	1.90607
$814$	−20.6714	−0.724533
$815$	0	0
$816$	−4.53591	−0.158789
$817$	−6.29514	−0.220239
$818$	40.5066	1.41628
$819$	77.0424	2.69208
$820$	0	0
$821$	−29.5439	−1.03109	−0.515544	−	0.856863i	$-0.672410\pi$
$821$	−29.5439	−1.03109	−0.515544	+	0.856863i	$0.672410\pi$
$822$	51.5441	1.79781
$823$	35.9452	1.25297	0.626486	−	0.779433i	$-0.284493\pi$
$823$	35.9452	1.25297	0.626486	+	0.779433i	$0.284493\pi$
$824$	−7.73825	−0.269575
$825$	0	0
$826$	−85.6222	−2.97918
$827$	37.2264	1.29449	0.647244	−	0.762283i	$-0.275922\pi$
$827$	37.2264	1.29449	0.647244	+	0.762283i	$0.275922\pi$
$828$	−2.62524	−0.0912333
$829$	−20.3779	−0.707755	−0.353878	−	0.935292i	$-0.615137\pi$
$829$	−20.3779	−0.707755	−0.353878	+	0.935292i	$0.615137\pi$
$830$	0	0
$831$	−8.30790	−0.288198
$832$	−85.7017	−2.97117
$833$	−15.5933	−0.540275
$834$	−32.8859	−1.13874
$835$	0	0
$836$	−2.65657	−0.0918793
$837$	−8.45866	−0.292374
$838$	−40.6112	−1.40289
$839$	−22.0874	−0.762543	−0.381271	−	0.924463i	$-0.624514\pi$
$839$	−22.0874	−0.762543	−0.381271	+	0.924463i	$0.624514\pi$
$840$	0	0
$841$	−4.92667	−0.169885
$842$	16.4242	0.566017
$843$	39.6737	1.36643
$844$	−22.4122	−0.771459
$845$	0	0
$846$	24.4502	0.840615
$847$	4.96679	0.170661
$848$	6.78260	0.232915
$849$	9.40612	0.322817
$850$	0	0
$851$	−4.31988	−0.148084
$852$	−71.9546	−2.46512
$853$	−36.5266	−1.25065	−0.625323	−	0.780366i	$-0.715033\pi$
$853$	−36.5266	−1.25065	−0.625323	+	0.780366i	$0.715033\pi$
$854$	−28.2314	−0.966060
$855$	0	0
$856$	25.3981	0.868089
$857$	−35.3847	−1.20872	−0.604359	−	0.796712i	$-0.706571\pi$
$857$	−35.3847	−1.20872	−0.604359	+	0.796712i	$0.706571\pi$
$858$	34.8028	1.18815
$859$	14.4130	0.491765	0.245883	−	0.969300i	$-0.420922\pi$
$859$	14.4130	0.491765	0.245883	+	0.969300i	$0.420922\pi$
$860$	0	0
$861$	42.2280	1.43913
$862$	−23.1581	−0.788770
$863$	16.8842	0.574745	0.287372	−	0.957819i	$-0.407218\pi$
$863$	16.8842	0.574745	0.287372	+	0.957819i	$0.407218\pi$
$864$	14.1895	0.482735
$865$	0	0
$866$	65.5250	2.22663
$867$	36.9592	1.25520
$868$	−60.5762	−2.05609
$869$	6.34343	0.215186
$870$	0	0
$871$	102.147	3.46112
$872$	−24.2243	−0.820338
$873$	−42.5689	−1.44074
$874$	−0.973122	−0.0329163
$875$	0	0
$876$	68.8801	2.32724
$877$	5.83820	0.197142	0.0985711	−	0.995130i	$-0.468573\pi$
$877$	5.83820	0.197142	0.0985711	+	0.995130i	$0.468573\pi$
$878$	−27.3034	−0.921444
$879$	34.9976	1.18044
$880$	0	0
$881$	17.6524	0.594723	0.297361	−	0.954765i	$-0.403893\pi$
$881$	17.6524	0.594723	0.297361	+	0.954765i	$0.403893\pi$
$882$	−83.5521	−2.81335
$883$	−22.4787	−0.756468	−0.378234	−	0.925710i	$-0.623469\pi$
$883$	−22.4787	−0.756468	−0.378234	+	0.925710i	$0.623469\pi$
$884$	−16.5954	−0.558164
$885$	0	0
$886$	51.1127	1.71716
$887$	34.9174	1.17241	0.586205	−	0.810163i	$-0.300621\pi$
$887$	34.9174	1.17241	0.586205	+	0.810163i	$0.300621\pi$
$888$	30.9236	1.03773
$889$	81.0049	2.71682
$890$	0	0
$891$	−10.7720	−0.360876
$892$	19.7045	0.659757
$893$	5.17054	0.173025
$894$	−92.3113	−3.08735
$895$	0	0
$896$	53.2624	1.77937
$897$	7.27303	0.242839
$898$	−43.6968	−1.45818
$899$	−22.5255	−0.751267
$900$	0	0
$901$	2.65353	0.0884020
$902$	8.05225	0.268111
$903$	−71.2396	−2.37070
$904$	−4.39539	−0.146188
$905$	0	0
$906$	−32.4827	−1.07917
$907$	−2.00996	−0.0667395	−0.0333698	−	0.999443i	$-0.510624\pi$
$907$	−2.00996	−0.0667395	−0.0333698	+	0.999443i	$0.510624\pi$
$908$	−69.1380	−2.29442
$909$	−19.3315	−0.641184
$910$	0	0
$911$	21.7282	0.719887	0.359944	−	0.932974i	$-0.382796\pi$
$911$	21.7282	0.719887	0.359944	+	0.932974i	$0.382796\pi$
$912$	−5.13971	−0.170193
$913$	17.2366	0.570448
$914$	28.4239	0.940179
$915$	0	0
$916$	−11.8240	−0.390677
$917$	−95.9773	−3.16945
$918$	3.50877	0.115807
$919$	−32.8035	−1.08209	−0.541044	−	0.840995i	$-0.681971\pi$
$919$	−32.8035	−1.08209	−0.541044	+	0.840995i	$0.681971\pi$
$920$	0	0
$921$	34.9298	1.15097
$922$	−48.5196	−1.59791
$923$	84.1468	2.76973
$924$	−30.0633	−0.989010
$925$	0	0
$926$	−42.0709	−1.38254
$927$	11.9686	0.393101
$928$	37.7866	1.24041
$929$	−8.32036	−0.272982	−0.136491	−	0.990641i	$-0.543582\pi$
$929$	−8.32036	−0.272982	−0.136491	+	0.990641i	$0.543582\pi$
$930$	0	0
$931$	−17.6690	−0.579077
$932$	16.5285	0.541410
$933$	−42.7058	−1.39813
$934$	−29.2506	−0.957109
$935$	0	0
$936$	−21.9769	−0.718337
$937$	−11.2558	−0.367709	−0.183855	−	0.982953i	$-0.558858\pi$
$937$	−11.2558	−0.367709	−0.183855	+	0.982953i	$0.558858\pi$
$938$	−154.666	−5.05001
$939$	−36.3150	−1.18510
$940$	0	0
$941$	30.8893	1.00696	0.503480	−	0.864007i	$-0.332053\pi$
$941$	30.8893	1.00696	0.503480	+	0.864007i	$0.332053\pi$
$942$	23.5948	0.768760
$943$	1.68275	0.0547978
$944$	−18.0209	−0.586530
$945$	0	0
$946$	−13.5843	−0.441665
$947$	8.80666	0.286178	0.143089	−	0.989710i	$-0.454296\pi$
$947$	8.80666	0.286178	0.143089	+	0.989710i	$0.454296\pi$
$948$	−38.3959	−1.24704
$949$	−80.5514	−2.61481
$950$	0	0
$951$	−20.6185	−0.668601
$952$	6.21031	0.201277
$953$	34.0563	1.10319	0.551595	−	0.834112i	$-0.314019\pi$
$953$	34.0563	1.10319	0.551595	+	0.834112i	$0.314019\pi$
$954$	14.2182	0.460331
$955$	0	0
$956$	−8.50783	−0.275163
$957$	−11.1791	−0.361370
$958$	49.7052	1.60590
$959$	52.0692	1.68140
$960$	0	0
$961$	−9.92286	−0.320092
$962$	−146.323	−4.71763
$963$	−39.2828	−1.26587
$964$	16.6748	0.537060
$965$	0	0
$966$	−11.0124	−0.354319
$967$	36.4838	1.17324	0.586619	−	0.809863i	$-0.300459\pi$
$967$	36.4838	1.17324	0.586619	+	0.809863i	$0.300459\pi$
$968$	−1.41681	−0.0455380
$969$	−2.01079	−0.0645959
$970$	0	0
$971$	12.7147	0.408035	0.204017	−	0.978967i	$-0.434600\pi$
$971$	12.7147	0.408035	0.204017	+	0.978967i	$0.434600\pi$
$972$	50.5179	1.62036
$973$	−33.2209	−1.06501
$974$	63.7135	2.04151
$975$	0	0
$976$	−5.94186	−0.190194
$977$	−7.45252	−0.238427	−0.119214	−	0.992869i	$-0.538037\pi$
$977$	−7.45252	−0.238427	−0.119214	+	0.992869i	$0.538037\pi$
$978$	−64.7481	−2.07042
$979$	5.27278	0.168519
$980$	0	0
$981$	37.4674	1.19624
$982$	−10.0755	−0.321521
$983$	56.6122	1.80565	0.902824	−	0.430011i	$-0.141490\pi$
$983$	56.6122	1.80565	0.902824	+	0.430011i	$0.141490\pi$
$984$	−12.0458	−0.384007
$985$	0	0
$986$	9.34388	0.297570
$987$	58.5129	1.86249
$988$	−18.8045	−0.598250
$989$	−2.83883	−0.0902696
$990$	0	0
$991$	−10.6165	−0.337244	−0.168622	−	0.985681i	$-0.553932\pi$
$991$	−10.6165	−0.337244	−0.168622	+	0.985681i	$0.553932\pi$
$992$	−35.3570	−1.12259
$993$	11.4331	0.362817
$994$	−127.411	−4.04122
$995$	0	0
$996$	−104.331	−3.30585
$997$	25.1438	0.796311	0.398155	−	0.917318i	$-0.369650\pi$
$997$	25.1438	0.796311	0.398155	+	0.917318i	$0.369650\pi$
$998$	−13.3282	−0.421896
$999$	17.6495	0.558407

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5225.2.a.x.1.2	yes	15
5.4	even	2		5225.2.a.s.1.14	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5225.2.a.s.1.14	✓	15	5.4	even	2
5225.2.a.x.1.2	yes	15	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-2.15791 q^{2} -2.27846 q^{3} +2.65657 q^{4} +4.91670 q^{6} +4.96679 q^{7} -1.41681 q^{8} +2.19136 q^{9} +1.00000 q^{11} -6.05287 q^{12} +7.07849 q^{13} -10.7179 q^{14} -2.25579 q^{16} -0.882524 q^{17} -4.72875 q^{18} -1.00000 q^{19} -11.3166 q^{21} -2.15791 q^{22} -0.450956 q^{23} +3.22814 q^{24} -15.2747 q^{26} +1.84245 q^{27} +13.1946 q^{28} +4.90646 q^{29} -4.59098 q^{31} +7.70140 q^{32} -2.27846 q^{33} +1.90440 q^{34} +5.82149 q^{36} +9.57939 q^{37} +2.15791 q^{38} -16.1280 q^{39} -3.73151 q^{41} +24.4202 q^{42} +6.29514 q^{43} +2.65657 q^{44} +0.973122 q^{46} -5.17054 q^{47} +5.13971 q^{48} +17.6690 q^{49} +2.01079 q^{51} +18.8045 q^{52} -3.00676 q^{53} -3.97584 q^{54} -7.03699 q^{56} +2.27846 q^{57} -10.5877 q^{58} +7.98874 q^{59} +2.63405 q^{61} +9.90692 q^{62} +10.8840 q^{63} -12.1073 q^{64} +4.91670 q^{66} +14.4306 q^{67} -2.34448 q^{68} +1.02748 q^{69} +11.8877 q^{71} -3.10474 q^{72} -11.3797 q^{73} -20.6714 q^{74} -2.65657 q^{76} +4.96679 q^{77} +34.8028 q^{78} +6.34343 q^{79} -10.7720 q^{81} +8.05225 q^{82} +17.2366 q^{83} -30.0633 q^{84} -13.5843 q^{86} -11.1791 q^{87} -1.41681 q^{88} +5.27278 q^{89} +35.1573 q^{91} -1.19799 q^{92} +10.4604 q^{93} +11.1575 q^{94} -17.5473 q^{96} -19.4258 q^{97} -38.1280 q^{98} +2.19136 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 5 q^{2} + 4 q^{3} + 17 q^{4} - q^{6} + 15 q^{7} + 15 q^{8} + 19 q^{9} + 15 q^{11} + 9 q^{12} + 13 q^{13} + 9 q^{14} + 21 q^{16} + 11 q^{17} + 16 q^{18} - 15 q^{19} + 5 q^{22} + 10 q^{23} + 17 q^{24}+ \cdots + 19 q^{99}+O(q^{100}) \) 15 * q + 5 * q^2 + 4 * q^3 + 17 * q^4 - q^6 + 15 * q^7 + 15 * q^8 + 19 * q^9 + 15 * q^11 + 9 * q^12 + 13 * q^13 + 9 * q^14 + 21 * q^16 + 11 * q^17 + 16 * q^18 - 15 * q^19 + 5 * q^22 + 10 * q^23 + 17 * q^24 - 17 * q^26 + 13 * q^27 + 30 * q^28 + 5 * q^29 + 6 * q^31 + 40 * q^32 + 4 * q^33 + 17 * q^34 + 28 * q^36 + 13 * q^37 - 5 * q^38 - 22 * q^39 + 30 * q^42 + 36 * q^43 + 17 * q^44 + 13 * q^46 - 6 * q^47 + 14 * q^48 + 16 * q^49 + 4 * q^51 + 50 * q^52 + 9 * q^53 + 9 * q^54 - 18 * q^56 - 4 * q^57 + 2 * q^58 - 7 * q^59 - 2 * q^61 + 11 * q^62 + 39 * q^63 + 17 * q^64 - q^66 + 35 * q^67 - 18 * q^68 - 9 * q^69 + 13 * q^71 + 68 * q^72 + 2 * q^73 + 13 * q^74 - 17 * q^76 + 15 * q^77 - 10 * q^78 + 6 * q^79 + 11 * q^81 + 14 * q^82 + 30 * q^83 - 6 * q^84 - 25 * q^86 + 19 * q^87 + 15 * q^88 + 55 * q^89 + 26 * q^91 - 18 * q^92 + 14 * q^93 - 22 * q^94 - 17 * q^96 + 28 * q^97 - 22 * q^98 + 19 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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