LMFDB - Embedded newform 5225.2.a.w.1.1

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 = [15,1,-4,13,0,-1,-11] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$41.7218350561$
Analytic rank:	$1$
Dimension:	$15$
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} - x^{14} - 21 x^{13} + 19 x^{12} + 170 x^{11} - 137 x^{10} - 669 x^{9} + 458 x^{8} + 1327 x^{7} + \cdots - 9 $ x^15 - x^14 - 21x^13 + 19x^12 + 170x^11 - 137x^10 - 669x^9 + 458x^8 + 1327x^7 - 687x^6 - 1256x^5 + 353x^4 + 519x^3 - 37x^2 - 75*x - 9
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			1.1
Root			$-2.47407$ 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)

$f(q)$	$=$	\(q-2.47407 q^{2} -3.11629 q^{3} +4.12105 q^{4} +7.70992 q^{6} -0.568753 q^{7} -5.24763 q^{8} +6.71124 q^{9} -1.00000 q^{11} -12.8424 q^{12} -1.23613 q^{13} +1.40714 q^{14} +4.74093 q^{16} -1.06253 q^{17} -16.6041 q^{18} -1.00000 q^{19} +1.77240 q^{21} +2.47407 q^{22} +5.72002 q^{23} +16.3531 q^{24} +3.05828 q^{26} -11.5653 q^{27} -2.34386 q^{28} +1.16291 q^{29} -0.252612 q^{31} -1.23416 q^{32} +3.11629 q^{33} +2.62878 q^{34} +27.6573 q^{36} +0.974602 q^{37} +2.47407 q^{38} +3.85214 q^{39} -1.90322 q^{41} -4.38504 q^{42} +3.31381 q^{43} -4.12105 q^{44} -14.1518 q^{46} +2.74967 q^{47} -14.7741 q^{48} -6.67652 q^{49} +3.31114 q^{51} -5.09416 q^{52} -4.08290 q^{53} +28.6134 q^{54} +2.98460 q^{56} +3.11629 q^{57} -2.87712 q^{58} -2.54912 q^{59} +7.49069 q^{61} +0.624981 q^{62} -3.81704 q^{63} -6.42846 q^{64} -7.70992 q^{66} -13.5077 q^{67} -4.37873 q^{68} -17.8252 q^{69} -1.42963 q^{71} -35.2181 q^{72} -4.15670 q^{73} -2.41124 q^{74} -4.12105 q^{76} +0.568753 q^{77} -9.53049 q^{78} -9.12259 q^{79} +15.9070 q^{81} +4.70870 q^{82} +10.1749 q^{83} +7.30413 q^{84} -8.19862 q^{86} -3.62396 q^{87} +5.24763 q^{88} -6.03138 q^{89} +0.703054 q^{91} +23.5725 q^{92} +0.787212 q^{93} -6.80289 q^{94} +3.84599 q^{96} +16.9507 q^{97} +16.5182 q^{98} -6.71124 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + q^{2} - 4 q^{3} + 13 q^{4} - q^{6} - 11 q^{7} + 3 q^{8} + 19 q^{9} - 15 q^{11} - 11 q^{12} - 3 q^{13} - 11 q^{14} + 13 q^{16} + 5 q^{17} - 12 q^{18} - 15 q^{19} + 10 q^{21} - q^{22} - 26 q^{23}+ \cdots - 19 q^{99}+O(q^{100}) $ 15 * q + q^2 - 4 * q^3 + 13 * q^4 - q^6 - 11 * q^7 + 3 * q^8 + 19 * q^9 - 15 * q^11 - 11 * q^12 - 3 * q^13 - 11 * q^14 + 13 * q^16 + 5 * q^17 - 12 * q^18 - 15 * q^19 + 10 * q^21 - q^22 - 26 * q^23 - 11 * q^24 - 5 * q^26 - 19 * q^27 - 18 * q^28 + 7 * q^29 - 10 * q^31 + 12 * q^32 + 4 * q^33 + 17 * q^34 + 24 * q^36 - 31 * q^37 - q^38 + 4 * q^39 + 2 * q^41 - 22 * q^42 - 26 * q^43 - 13 * q^44 - 23 * q^46 - 26 * q^47 - 46 * q^48 + 12 * q^49 + 12 * q^51 - 16 * q^52 - 21 * q^53 + 5 * q^54 - 10 * q^56 + 4 * q^57 - 34 * q^58 - 11 * q^59 + 20 * q^61 - 25 * q^62 - 27 * q^63 - 3 * q^64 + q^66 - 41 * q^67 + 6 * q^68 + q^69 + 25 * q^71 - 54 * q^72 + 6 * q^73 - 9 * q^74 - 13 * q^76 + 11 * q^77 - 28 * q^78 - 6 * q^79 + 43 * q^81 - 18 * q^82 + 20 * q^83 - 14 * q^84 + 35 * q^86 - 29 * q^87 - 3 * q^88 - 3 * q^89 + 30 * q^91 - 54 * q^92 - 2 * q^93 - 28 * q^94 - 61 * q^96 - 28 * q^97 + 2 * 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 $

$2$	−2.47407	−1.74944	−0.874718	−	0.484633i	$-0.838953\pi$
$2$	−2.47407	−1.74944	−0.874718	+	0.484633i	$0.838953\pi$
$3$	−3.11629	−1.79919	−0.899594	−	0.436727i	$-0.856138\pi$
$3$	−3.11629	−1.79919	−0.899594	+	0.436727i	$0.856138\pi$
$4$	4.12105	2.06052
$5$	0	0
$5$	0	0
$6$	7.70992	3.14756
$7$	−0.568753	−0.214968	−0.107484	−	0.994207i	$-0.534280\pi$
$7$	−0.568753	−0.214968	−0.107484	+	0.994207i	$0.534280\pi$
$8$	−5.24763	−1.85532
$9$	6.71124	2.23708
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−12.8424	−3.70727
$13$	−1.23613	−0.342841	−0.171421	−	0.985198i	$-0.554836\pi$
$13$	−1.23613	−0.342841	−0.171421	+	0.985198i	$0.554836\pi$
$14$	1.40714	0.376073
$15$	0	0
$16$	4.74093	1.18523
$17$	−1.06253	−0.257701	−0.128851	−	0.991664i	$-0.541129\pi$
$17$	−1.06253	−0.257701	−0.128851	+	0.991664i	$0.541129\pi$
$18$	−16.6041	−3.91362
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	1.77240	0.386769
$22$	2.47407	0.527475
$23$	5.72002	1.19271	0.596354	−	0.802722i	$-0.296616\pi$
$23$	5.72002	1.19271	0.596354	+	0.802722i	$0.296616\pi$
$24$	16.3531	3.33806
$25$	0	0
$26$	3.05828	0.599779
$27$	−11.5653	−2.22574
$28$	−2.34386	−0.442947
$29$	1.16291	0.215947	0.107973	−	0.994154i	$-0.465564\pi$
$29$	1.16291	0.215947	0.107973	+	0.994154i	$0.465564\pi$
$30$	0	0
$31$	−0.252612	−0.0453705	−0.0226852	−	0.999743i	$-0.507222\pi$
$31$	−0.252612	−0.0453705	−0.0226852	+	0.999743i	$0.507222\pi$
$32$	−1.23416	−0.218170
$33$	3.11629	0.542476
$34$	2.62878	0.450831
$35$	0	0
$36$	27.6573	4.60955
$37$	0.974602	0.160224	0.0801118	−	0.996786i	$-0.474472\pi$
$37$	0.974602	0.160224	0.0801118	+	0.996786i	$0.474472\pi$
$38$	2.47407	0.401348
$39$	3.85214	0.616836
$40$	0	0
$41$	−1.90322	−0.297232	−0.148616	−	0.988895i	$-0.547482\pi$
$41$	−1.90322	−0.297232	−0.148616	+	0.988895i	$0.547482\pi$
$42$	−4.38504	−0.676627
$43$	3.31381	0.505352	0.252676	−	0.967551i	$-0.418689\pi$
$43$	3.31381	0.505352	0.252676	+	0.967551i	$0.418689\pi$
$44$	−4.12105	−0.621271
$45$	0	0
$46$	−14.1518	−2.08656
$47$	2.74967	0.401081	0.200540	−	0.979685i	$-0.435730\pi$
$47$	2.74967	0.401081	0.200540	+	0.979685i	$0.435730\pi$
$48$	−14.7741	−2.13246
$49$	−6.67652	−0.953789
$50$	0	0
$51$	3.31114	0.463653
$52$	−5.09416	−0.706433
$53$	−4.08290	−0.560829	−0.280415	−	0.959879i	$-0.590472\pi$
$53$	−4.08290	−0.560829	−0.280415	+	0.959879i	$0.590472\pi$
$54$	28.6134	3.89378
$55$	0	0
$56$	2.98460	0.398834
$57$	3.11629	0.412762
$58$	−2.87712	−0.377785
$59$	−2.54912	−0.331868	−0.165934	−	0.986137i	$-0.553064\pi$
$59$	−2.54912	−0.331868	−0.165934	+	0.986137i	$0.553064\pi$
$60$	0	0
$61$	7.49069	0.959085	0.479542	−	0.877519i	$-0.340803\pi$
$61$	7.49069	0.959085	0.479542	+	0.877519i	$0.340803\pi$
$62$	0.624981	0.0793727
$63$	−3.81704	−0.480901
$64$	−6.42846	−0.803557
$65$	0	0
$66$	−7.70992	−0.949026
$67$	−13.5077	−1.65022	−0.825112	−	0.564969i	$-0.808888\pi$
$67$	−13.5077	−1.65022	−0.825112	+	0.564969i	$0.808888\pi$
$68$	−4.37873	−0.530999
$69$	−17.8252	−2.14591
$70$	0	0
$71$	−1.42963	−0.169665	−0.0848327	−	0.996395i	$-0.527036\pi$
$71$	−1.42963	−0.169665	−0.0848327	+	0.996395i	$0.527036\pi$
$72$	−35.2181	−4.15049
$73$	−4.15670	−0.486505	−0.243253	−	0.969963i	$-0.578214\pi$
$73$	−4.15670	−0.486505	−0.243253	+	0.969963i	$0.578214\pi$
$74$	−2.41124	−0.280301
$75$	0	0
$76$	−4.12105	−0.472716
$77$	0.568753	0.0648154
$78$	−9.53049	−1.07912
$79$	−9.12259	−1.02637	−0.513186	−	0.858278i	$-0.671535\pi$
$79$	−9.12259	−1.02637	−0.513186	+	0.858278i	$0.671535\pi$
$80$	0	0
$81$	15.9070	1.76744
$82$	4.70870	0.519989
$83$	10.1749	1.11684	0.558420	−	0.829558i	$-0.311408\pi$
$83$	10.1749	1.11684	0.558420	+	0.829558i	$0.311408\pi$
$84$	7.30413	0.796946
$85$	0	0
$86$	−8.19862	−0.884080
$87$	−3.62396	−0.388529
$88$	5.24763	0.559399
$89$	−6.03138	−0.639325	−0.319662	−	0.947531i	$-0.603570\pi$
$89$	−6.03138	−0.639325	−0.319662	+	0.947531i	$0.603570\pi$
$90$	0	0
$91$	0.703054	0.0737001
$92$	23.5725	2.45760
$93$	0.787212	0.0816301
$94$	−6.80289	−0.701665
$95$	0	0
$96$	3.84599	0.392529
$97$	16.9507	1.72108	0.860542	−	0.509380i	$-0.170125\pi$
$97$	16.9507	1.72108	0.860542	+	0.509380i	$0.170125\pi$
$98$	16.5182	1.66859
$99$	−6.71124	−0.674505
$100$	0	0
$101$	−3.69240	−0.367407	−0.183704	−	0.982982i	$-0.558809\pi$
$101$	−3.69240	−0.367407	−0.183704	+	0.982982i	$0.558809\pi$
$102$	−8.19202	−0.811130
$103$	−5.19662	−0.512038	−0.256019	−	0.966672i	$-0.582411\pi$
$103$	−5.19662	−0.512038	−0.256019	+	0.966672i	$0.582411\pi$
$104$	6.48676	0.636079
$105$	0	0
$106$	10.1014	0.981135
$107$	−1.32425	−0.128020	−0.0640101	−	0.997949i	$-0.520389\pi$
$107$	−1.32425	−0.128020	−0.0640101	+	0.997949i	$0.520389\pi$
$108$	−47.6610	−4.58619
$109$	15.6513	1.49912	0.749560	−	0.661936i	$-0.230265\pi$
$109$	15.6513	1.49912	0.749560	+	0.661936i	$0.230265\pi$
$110$	0	0
$111$	−3.03714	−0.288272
$112$	−2.69642	−0.254787
$113$	−16.1223	−1.51666	−0.758329	−	0.651873i	$-0.773984\pi$
$113$	−16.1223	−1.51666	−0.758329	+	0.651873i	$0.773984\pi$
$114$	−7.70992	−0.722101
$115$	0	0
$116$	4.79240	0.444963
$117$	−8.29598	−0.766964
$118$	6.30672	0.580581
$119$	0.604316	0.0553976
$120$	0	0
$121$	1.00000	0.0909091
$122$	−18.5325	−1.67786
$123$	5.93096	0.534777
$124$	−1.04103	−0.0934869
$125$	0	0
$126$	9.44363	0.841306
$127$	−2.42378	−0.215075	−0.107538	−	0.994201i	$-0.534297\pi$
$127$	−2.42378	−0.215075	−0.107538	+	0.994201i	$0.534297\pi$
$128$	18.3728	1.62394
$129$	−10.3268	−0.909223
$130$	0	0
$131$	17.0055	1.48577	0.742887	−	0.669417i	$-0.233456\pi$
$131$	17.0055	1.48577	0.742887	+	0.669417i	$0.233456\pi$
$132$	12.8424	1.11778
$133$	0.568753	0.0493171
$134$	33.4190	2.88696
$135$	0	0
$136$	5.57575	0.478117
$137$	13.9041	1.18791	0.593955	−	0.804498i	$-0.297566\pi$
$137$	13.9041	1.18791	0.593955	+	0.804498i	$0.297566\pi$
$138$	44.1009	3.75412
$139$	−5.48558	−0.465281	−0.232641	−	0.972563i	$-0.574737\pi$
$139$	−5.48558	−0.465281	−0.232641	+	0.972563i	$0.574737\pi$
$140$	0	0
$141$	−8.56876	−0.721620
$142$	3.53700	0.296819
$143$	1.23613	0.103371
$144$	31.8175	2.65146
$145$	0	0
$146$	10.2840	0.851109
$147$	20.8059	1.71605
$148$	4.01638	0.330144
$149$	12.5126	1.02508	0.512538	−	0.858665i	$-0.328705\pi$
$149$	12.5126	1.02508	0.512538	+	0.858665i	$0.328705\pi$
$150$	0	0
$151$	−18.1370	−1.47596	−0.737982	−	0.674820i	$-0.764221\pi$
$151$	−18.1370	−1.47596	−0.737982	+	0.674820i	$0.764221\pi$
$152$	5.24763	0.425639
$153$	−7.13088	−0.576498
$154$	−1.40714	−0.113390
$155$	0	0
$156$	15.8749	1.27101
$157$	−0.430792	−0.0343809	−0.0171905	−	0.999852i	$-0.505472\pi$
$157$	−0.430792	−0.0343809	−0.0171905	+	0.999852i	$0.505472\pi$
$158$	22.5700	1.79557
$159$	12.7235	1.00904
$160$	0	0
$161$	−3.25328	−0.256394
$162$	−39.3551	−3.09203
$163$	−19.8052	−1.55127	−0.775633	−	0.631184i	$-0.782569\pi$
$163$	−19.8052	−1.55127	−0.775633	+	0.631184i	$0.782569\pi$
$164$	−7.84324	−0.612454
$165$	0	0
$166$	−25.1735	−1.95384
$167$	4.73947	0.366751	0.183376	−	0.983043i	$-0.441298\pi$
$167$	4.73947	0.366751	0.183376	+	0.983043i	$0.441298\pi$
$168$	−9.30088	−0.717578
$169$	−11.4720	−0.882460
$170$	0	0
$171$	−6.71124	−0.513221
$172$	13.6564	1.04129
$173$	13.1702	1.00131	0.500654	−	0.865647i	$-0.333093\pi$
$173$	13.1702	1.00131	0.500654	+	0.865647i	$0.333093\pi$
$174$	8.96594	0.679706
$175$	0	0
$176$	−4.74093	−0.357361
$177$	7.94380	0.597093
$178$	14.9221	1.11846
$179$	−2.43120	−0.181717	−0.0908583	−	0.995864i	$-0.528961\pi$
$179$	−2.43120	−0.181717	−0.0908583	+	0.995864i	$0.528961\pi$
$180$	0	0
$181$	25.8909	1.92446	0.962229	−	0.272243i	$-0.0877653\pi$
$181$	25.8909	1.92446	0.962229	+	0.272243i	$0.0877653\pi$
$182$	−1.73941	−0.128934
$183$	−23.3431	−1.72557
$184$	−30.0165	−2.21285
$185$	0	0
$186$	−1.94762	−0.142806
$187$	1.06253	0.0776998
$188$	11.3315	0.826436
$189$	6.57779	0.478463
$190$	0	0
$191$	18.7924	1.35977	0.679886	−	0.733318i	$-0.262029\pi$
$191$	18.7924	1.35977	0.679886	+	0.733318i	$0.262029\pi$
$192$	20.0329	1.44575
$193$	16.0283	1.15374	0.576870	−	0.816836i	$-0.304274\pi$
$193$	16.0283	1.15374	0.576870	+	0.816836i	$0.304274\pi$
$194$	−41.9373	−3.01092
$195$	0	0
$196$	−27.5142	−1.96530
$197$	19.5771	1.39481	0.697405	−	0.716678i	$-0.254338\pi$
$197$	19.5771	1.39481	0.697405	+	0.716678i	$0.254338\pi$
$198$	16.6041	1.18000
$199$	−24.9130	−1.76604	−0.883019	−	0.469338i	$-0.844493\pi$
$199$	−24.9130	−1.76604	−0.883019	+	0.469338i	$0.844493\pi$
$200$	0	0
$201$	42.0937	2.96906
$202$	9.13526	0.642755
$203$	−0.661408	−0.0464217
$204$	13.6454	0.955367
$205$	0	0
$206$	12.8568	0.895778
$207$	38.3884	2.66818
$208$	−5.86042	−0.406347
$209$	1.00000	0.0691714
$210$	0	0
$211$	15.2896	1.05258	0.526290	−	0.850305i	$-0.323583\pi$
$211$	15.2896	1.05258	0.526290	+	0.850305i	$0.323583\pi$
$212$	−16.8258	−1.15560
$213$	4.45512	0.305260
$214$	3.27629	0.223963
$215$	0	0
$216$	60.6903	4.12945
$217$	0.143674	0.00975322
$218$	−38.7224	−2.62261
$219$	12.9535	0.875314
$220$	0	0
$221$	1.31343	0.0883506
$222$	7.51411	0.504314
$223$	−26.2223	−1.75598	−0.877989	−	0.478680i	$-0.841115\pi$
$223$	−26.2223	−1.75598	−0.877989	+	0.478680i	$0.841115\pi$
$224$	0.701931	0.0468997
$225$	0	0
$226$	39.8877	2.65329
$227$	21.0571	1.39761	0.698806	−	0.715311i	$-0.253715\pi$
$227$	21.0571	1.39761	0.698806	+	0.715311i	$0.253715\pi$
$228$	12.8424	0.850506
$229$	3.64009	0.240544	0.120272	−	0.992741i	$-0.461623\pi$
$229$	3.64009	0.240544	0.120272	+	0.992741i	$0.461623\pi$
$230$	0	0
$231$	−1.77240	−0.116615
$232$	−6.10251	−0.400650
$233$	17.0553	1.11733	0.558666	−	0.829393i	$-0.311313\pi$
$233$	17.0553	1.11733	0.558666	+	0.829393i	$0.311313\pi$
$234$	20.5249	1.34175
$235$	0	0
$236$	−10.5051	−0.683821
$237$	28.4286	1.84664
$238$	−1.49512	−0.0969145
$239$	−17.5075	−1.13247	−0.566233	−	0.824245i	$-0.691600\pi$
$239$	−17.5075	−1.13247	−0.566233	+	0.824245i	$0.691600\pi$
$240$	0	0
$241$	10.6563	0.686431	0.343215	−	0.939257i	$-0.388484\pi$
$241$	10.6563	0.686431	0.343215	+	0.939257i	$0.388484\pi$
$242$	−2.47407	−0.159040
$243$	−14.8749	−0.954226
$244$	30.8695	1.97622
$245$	0	0
$246$	−14.6736	−0.935558
$247$	1.23613	0.0786532
$248$	1.32561	0.0841766
$249$	−31.7079	−2.00941
$250$	0	0
$251$	25.5266	1.61123	0.805613	−	0.592442i	$-0.201836\pi$
$251$	25.5266	1.61123	0.805613	+	0.592442i	$0.201836\pi$
$252$	−15.7302	−0.990908
$253$	−5.72002	−0.359615
$254$	5.99660	0.376260
$255$	0	0
$256$	−32.5988	−2.03742
$257$	−0.270665	−0.0168836	−0.00844181	−	0.999964i	$-0.502687\pi$
$257$	−0.270665	−0.0168836	−0.00844181	+	0.999964i	$0.502687\pi$
$258$	25.5493	1.59063
$259$	−0.554308	−0.0344430
$260$	0	0
$261$	7.80456	0.483090
$262$	−42.0728	−2.59926
$263$	−24.6926	−1.52261	−0.761305	−	0.648394i	$-0.775441\pi$
$263$	−24.6926	−1.52261	−0.761305	+	0.648394i	$0.775441\pi$
$264$	−16.3531	−1.00646
$265$	0	0
$266$	−1.40714	−0.0862771
$267$	18.7955	1.15027
$268$	−55.6657	−3.40032
$269$	21.3173	1.29974	0.649869	−	0.760046i	$-0.274824\pi$
$269$	21.3173	1.29974	0.649869	+	0.760046i	$0.274824\pi$
$270$	0	0
$271$	12.1163	0.736013	0.368006	−	0.929823i	$-0.380040\pi$
$271$	12.1163	0.736013	0.368006	+	0.929823i	$0.380040\pi$
$272$	−5.03737	−0.305436
$273$	−2.19092	−0.132600
$274$	−34.3998	−2.07817
$275$	0	0
$276$	−73.4586	−4.42169
$277$	6.37705	0.383160	0.191580	−	0.981477i	$-0.438639\pi$
$277$	6.37705	0.383160	0.191580	+	0.981477i	$0.438639\pi$
$278$	13.5717	0.813979
$279$	−1.69534	−0.101497
$280$	0	0
$281$	−14.1056	−0.841469	−0.420735	−	0.907184i	$-0.638228\pi$
$281$	−14.1056	−0.841469	−0.420735	+	0.907184i	$0.638228\pi$
$282$	21.1998	1.26243
$283$	−13.9610	−0.829894	−0.414947	−	0.909845i	$-0.636200\pi$
$283$	−13.9610	−0.829894	−0.414947	+	0.909845i	$0.636200\pi$
$284$	−5.89156	−0.349600
$285$	0	0
$286$	−3.05828	−0.180840
$287$	1.08246	0.0638956
$288$	−8.28272	−0.488064
$289$	−15.8710	−0.933590
$290$	0	0
$291$	−52.8232	−3.09655
$292$	−17.1300	−1.00246
$293$	−0.0735511	−0.00429690	−0.00214845	−	0.999998i	$-0.500684\pi$
$293$	−0.0735511	−0.00429690	−0.00214845	+	0.999998i	$0.500684\pi$
$294$	−51.4755	−3.00211
$295$	0	0
$296$	−5.11435	−0.297265
$297$	11.5653	0.671085
$298$	−30.9572	−1.79330
$299$	−7.07071	−0.408910
$300$	0	0
$301$	−1.88474	−0.108635
$302$	44.8722	2.58210
$303$	11.5066	0.661035
$304$	−4.74093	−0.271911
$305$	0	0
$306$	17.6423	1.00855
$307$	−17.1790	−0.980458	−0.490229	−	0.871594i	$-0.663087\pi$
$307$	−17.1790	−0.980458	−0.490229	+	0.871594i	$0.663087\pi$
$308$	2.34386	0.133554
$309$	16.1942	0.921254
$310$	0	0
$311$	24.8461	1.40890	0.704448	−	0.709756i	$-0.251195\pi$
$311$	24.8461	1.40890	0.704448	+	0.709756i	$0.251195\pi$
$312$	−20.2146	−1.14443
$313$	20.2896	1.14684	0.573418	−	0.819263i	$-0.305617\pi$
$313$	20.2896	1.14684	0.573418	+	0.819263i	$0.305617\pi$
$314$	1.06581	0.0601472
$315$	0	0
$316$	−37.5946	−2.11486
$317$	0.475340	0.0266977	0.0133489	−	0.999911i	$-0.495751\pi$
$317$	0.475340	0.0266977	0.0133489	+	0.999911i	$0.495751\pi$
$318$	−31.4789	−1.76525
$319$	−1.16291	−0.0651104
$320$	0	0
$321$	4.12674	0.230332
$322$	8.04886	0.448545
$323$	1.06253	0.0591207
$324$	65.5535	3.64186
$325$	0	0
$326$	48.9996	2.71384
$327$	−48.7738	−2.69720
$328$	9.98736	0.551460
$329$	−1.56388	−0.0862197
$330$	0	0
$331$	−3.55187	−0.195228	−0.0976141	−	0.995224i	$-0.531121\pi$
$331$	−3.55187	−0.195228	−0.0976141	+	0.995224i	$0.531121\pi$
$332$	41.9312	2.30128
$333$	6.54078	0.358433
$334$	−11.7258	−0.641607
$335$	0	0
$336$	8.40281	0.458411
$337$	−13.0883	−0.712963	−0.356482	−	0.934302i	$-0.616024\pi$
$337$	−13.0883	−0.712963	−0.356482	+	0.934302i	$0.616024\pi$
$338$	28.3825	1.54381
$339$	50.2416	2.72875
$340$	0	0
$341$	0.252612	0.0136797
$342$	16.6041	0.897847
$343$	7.77856	0.420003
$344$	−17.3897	−0.937588
$345$	0	0
$346$	−32.5840	−1.75172
$347$	2.64724	0.142111	0.0710557	−	0.997472i	$-0.477363\pi$
$347$	2.64724	0.142111	0.0710557	+	0.997472i	$0.477363\pi$
$348$	−14.9345	−0.800573
$349$	−1.45049	−0.0776427	−0.0388214	−	0.999246i	$-0.512360\pi$
$349$	−1.45049	−0.0776427	−0.0388214	+	0.999246i	$0.512360\pi$
$350$	0	0
$351$	14.2962	0.763076
$352$	1.23416	0.0657808
$353$	−0.415398	−0.0221094	−0.0110547	−	0.999939i	$-0.503519\pi$
$353$	−0.415398	−0.0221094	−0.0110547	+	0.999939i	$0.503519\pi$
$354$	−19.6536	−1.04457
$355$	0	0
$356$	−24.8556	−1.31734
$357$	−1.88322	−0.0996707
$358$	6.01498	0.317901
$359$	8.01993	0.423276	0.211638	−	0.977348i	$-0.432120\pi$
$359$	8.01993	0.423276	0.211638	+	0.977348i	$0.432120\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−64.0561	−3.36671
$363$	−3.11629	−0.163563
$364$	2.89732	0.151861
$365$	0	0
$366$	57.7527	3.01878
$367$	−12.1728	−0.635415	−0.317707	−	0.948189i	$-0.602913\pi$
$367$	−12.1728	−0.635415	−0.317707	+	0.948189i	$0.602913\pi$
$368$	27.1182	1.41363
$369$	−12.7729	−0.664932
$370$	0	0
$371$	2.32216	0.120561
$372$	3.24414	0.168201
$373$	15.3210	0.793291	0.396646	−	0.917972i	$-0.370174\pi$
$373$	15.3210	0.793291	0.396646	+	0.917972i	$0.370174\pi$
$374$	−2.62878	−0.135931
$375$	0	0
$376$	−14.4292	−0.744132
$377$	−1.43751	−0.0740355
$378$	−16.2739	−0.837041
$379$	−9.78947	−0.502851	−0.251426	−	0.967877i	$-0.580899\pi$
$379$	−9.78947	−0.502851	−0.251426	+	0.967877i	$0.580899\pi$
$380$	0	0
$381$	7.55318	0.386961
$382$	−46.4939	−2.37883
$383$	−1.71875	−0.0878238	−0.0439119	−	0.999035i	$-0.513982\pi$
$383$	−1.71875	−0.0878238	−0.0439119	+	0.999035i	$0.513982\pi$
$384$	−57.2549	−2.92178
$385$	0	0
$386$	−39.6551	−2.01839
$387$	22.2398	1.13051
$388$	69.8546	3.54633
$389$	−24.8643	−1.26067	−0.630336	−	0.776323i	$-0.717083\pi$
$389$	−24.8643	−1.26067	−0.630336	+	0.776323i	$0.717083\pi$
$390$	0	0
$391$	−6.07769	−0.307362
$392$	35.0359	1.76958
$393$	−52.9939	−2.67319
$394$	−48.4352	−2.44013
$395$	0	0
$396$	−27.6573	−1.38983
$397$	−12.8294	−0.643887	−0.321943	−	0.946759i	$-0.604336\pi$
$397$	−12.8294	−0.643887	−0.321943	+	0.946759i	$0.604336\pi$
$398$	61.6367	3.08957
$399$	−1.77240	−0.0887308
$400$	0	0
$401$	−14.2796	−0.713089	−0.356544	−	0.934278i	$-0.616045\pi$
$401$	−14.2796	−0.713089	−0.356544	+	0.934278i	$0.616045\pi$
$402$	−104.143	−5.19418
$403$	0.312262	0.0155549
$404$	−15.2165	−0.757051
$405$	0	0
$406$	1.63637	0.0812118
$407$	−0.974602	−0.0483092
$408$	−17.3756	−0.860222
$409$	−14.7884	−0.731237	−0.365619	−	0.930765i	$-0.619143\pi$
$409$	−14.7884	−0.731237	−0.365619	+	0.930765i	$0.619143\pi$
$410$	0	0
$411$	−43.3292	−2.13727
$412$	−21.4155	−1.05507
$413$	1.44982	0.0713411
$414$	−94.9758	−4.66781
$415$	0	0
$416$	1.52558	0.0747978
$417$	17.0946	0.837128
$418$	−2.47407	−0.121011
$419$	11.1996	0.547135	0.273568	−	0.961853i	$-0.411796\pi$
$419$	11.1996	0.547135	0.273568	+	0.961853i	$0.411796\pi$
$420$	0	0
$421$	35.4914	1.72974	0.864872	−	0.501992i	$-0.167399\pi$
$421$	35.4914	1.72974	0.864872	+	0.501992i	$0.167399\pi$
$422$	−37.8277	−1.84142
$423$	18.4537	0.897250
$424$	21.4255	1.04052
$425$	0	0
$426$	−11.0223	−0.534033
$427$	−4.26035	−0.206173
$428$	−5.45730	−0.263788
$429$	−3.85214	−0.185983
$430$	0	0
$431$	−37.2681	−1.79514	−0.897571	−	0.440870i	$-0.854670\pi$
$431$	−37.2681	−1.79514	−0.897571	+	0.440870i	$0.854670\pi$
$432$	−54.8301	−2.63802
$433$	34.5772	1.66168	0.830838	−	0.556515i	$-0.187862\pi$
$433$	34.5772	1.66168	0.830838	+	0.556515i	$0.187862\pi$
$434$	−0.355460	−0.0170626
$435$	0	0
$436$	64.4996	3.08897
$437$	−5.72002	−0.273626
$438$	−32.0479	−1.53131
$439$	−34.7598	−1.65900	−0.829498	−	0.558510i	$-0.811373\pi$
$439$	−34.7598	−1.65900	−0.829498	+	0.558510i	$0.811373\pi$
$440$	0	0
$441$	−44.8077	−2.13370
$442$	−3.24951	−0.154564
$443$	−37.8832	−1.79988	−0.899942	−	0.436010i	$-0.856391\pi$
$443$	−37.8832	−1.79988	−0.899942	+	0.436010i	$0.856391\pi$
$444$	−12.5162	−0.593992
$445$	0	0
$446$	64.8761	3.07197
$447$	−38.9930	−1.84430
$448$	3.65620	0.172739
$449$	10.6044	0.500451	0.250226	−	0.968188i	$-0.419495\pi$
$449$	10.6044	0.500451	0.250226	+	0.968188i	$0.419495\pi$
$450$	0	0
$451$	1.90322	0.0896189
$452$	−66.4407	−3.12511
$453$	56.5200	2.65554
$454$	−52.0970	−2.44503
$455$	0	0
$456$	−16.3531	−0.765804
$457$	−4.25836	−0.199198	−0.0995988	−	0.995028i	$-0.531756\pi$
$457$	−4.25836	−0.199198	−0.0995988	+	0.995028i	$0.531756\pi$
$458$	−9.00585	−0.420816
$459$	12.2884	0.573575
$460$	0	0
$461$	36.0411	1.67860	0.839300	−	0.543669i	$-0.182965\pi$
$461$	36.0411	1.67860	0.839300	+	0.543669i	$0.182965\pi$
$462$	4.38504	0.204011
$463$	−11.9837	−0.556930	−0.278465	−	0.960446i	$-0.589826\pi$
$463$	−11.9837	−0.556930	−0.278465	+	0.960446i	$0.589826\pi$
$464$	5.51327	0.255947
$465$	0	0
$466$	−42.1962	−1.95470
$467$	−11.4583	−0.530228	−0.265114	−	0.964217i	$-0.585410\pi$
$467$	−11.4583	−0.530228	−0.265114	+	0.964217i	$0.585410\pi$
$468$	−34.1881	−1.58035
$469$	7.68252	0.354746
$470$	0	0
$471$	1.34247	0.0618577
$472$	13.3769	0.615719
$473$	−3.31381	−0.152369
$474$	−70.3345	−3.23057
$475$	0	0
$476$	2.49042	0.114148
$477$	−27.4013	−1.25462
$478$	43.3149	1.98118
$479$	11.8394	0.540956	0.270478	−	0.962726i	$-0.412818\pi$
$479$	11.8394	0.540956	0.270478	+	0.962726i	$0.412818\pi$
$480$	0	0
$481$	−1.20474	−0.0549313
$482$	−26.3644	−1.20087
$483$	10.1381	0.461302
$484$	4.12105	0.187320
$485$	0	0
$486$	36.8017	1.66936
$487$	−21.3119	−0.965733	−0.482866	−	0.875694i	$-0.660404\pi$
$487$	−21.3119	−0.965733	−0.482866	+	0.875694i	$0.660404\pi$
$488$	−39.3083	−1.77941
$489$	61.7188	2.79102
$490$	0	0
$491$	−30.6885	−1.38495	−0.692477	−	0.721440i	$-0.743481\pi$
$491$	−30.6885	−1.38495	−0.692477	+	0.721440i	$0.743481\pi$
$492$	24.4418	1.10192
$493$	−1.23562	−0.0556497
$494$	−3.05828	−0.137599
$495$	0	0
$496$	−1.19762	−0.0537745
$497$	0.813104	0.0364727
$498$	78.4477	3.51533
$499$	22.7442	1.01817	0.509086	−	0.860716i	$-0.329984\pi$
$499$	22.7442	1.01817	0.509086	+	0.860716i	$0.329984\pi$
$500$	0	0
$501$	−14.7695	−0.659854
$502$	−63.1548	−2.81874
$503$	−1.25443	−0.0559324	−0.0279662	−	0.999609i	$-0.508903\pi$
$503$	−1.25443	−0.0559324	−0.0279662	+	0.999609i	$0.508903\pi$
$504$	20.0304	0.892224
$505$	0	0
$506$	14.1518	0.629123
$507$	35.7500	1.58771
$508$	−9.98849	−0.443168
$509$	−31.1519	−1.38078	−0.690392	−	0.723435i	$-0.742562\pi$
$509$	−31.1519	−1.38078	−0.690392	+	0.723435i	$0.742562\pi$
$510$	0	0
$511$	2.36414	0.104583
$512$	43.9062	1.94040
$513$	11.5653	0.510619
$514$	0.669646	0.0295368
$515$	0	0
$516$	−42.5572	−1.87348
$517$	−2.74967	−0.120930
$518$	1.37140	0.0602558
$519$	−41.0420	−1.80154
$520$	0	0
$521$	−31.6591	−1.38701	−0.693506	−	0.720451i	$-0.743935\pi$
$521$	−31.6591	−1.38701	−0.693506	+	0.720451i	$0.743935\pi$
$522$	−19.3091	−0.845135
$523$	−12.9670	−0.567007	−0.283504	−	0.958971i	$-0.591497\pi$
$523$	−12.9670	−0.567007	−0.283504	+	0.958971i	$0.591497\pi$
$524$	70.0803	3.06147
$525$	0	0
$526$	61.0913	2.66371
$527$	0.268408	0.0116920
$528$	14.7741	0.642960
$529$	9.71866	0.422550
$530$	0	0
$531$	−17.1078	−0.742414
$532$	2.34386	0.101619
$533$	2.35263	0.101904
$534$	−46.5015	−2.01232
$535$	0	0
$536$	70.8832	3.06169
$537$	7.57632	0.326942
$538$	−52.7406	−2.27381
$539$	6.67652	0.287578
$540$	0	0
$541$	−2.93440	−0.126160	−0.0630799	−	0.998008i	$-0.520092\pi$
$541$	−2.93440	−0.126160	−0.0630799	+	0.998008i	$0.520092\pi$
$542$	−29.9766	−1.28761
$543$	−80.6835	−3.46246
$544$	1.31133	0.0562227
$545$	0	0
$546$	5.42049	0.231976
$547$	−15.5813	−0.666210	−0.333105	−	0.942890i	$-0.608096\pi$
$547$	−15.5813	−0.666210	−0.333105	+	0.942890i	$0.608096\pi$
$548$	57.2995	2.44771
$549$	50.2718	2.14555
$550$	0	0
$551$	−1.16291	−0.0495416
$552$	93.5401	3.98133
$553$	5.18850	0.220637
$554$	−15.7773	−0.670313
$555$	0	0
$556$	−22.6063	−0.958722
$557$	28.8338	1.22173	0.610864	−	0.791735i	$-0.290822\pi$
$557$	28.8338	1.22173	0.610864	+	0.791735i	$0.290822\pi$
$558$	4.19440	0.177563
$559$	−4.09631	−0.173256
$560$	0	0
$561$	−3.31114	−0.139797
$562$	34.8983	1.47210
$563$	−3.71838	−0.156711	−0.0783556	−	0.996925i	$-0.524967\pi$
$563$	−3.71838	−0.156711	−0.0783556	+	0.996925i	$0.524967\pi$
$564$	−35.3123	−1.48691
$565$	0	0
$566$	34.5405	1.45185
$567$	−9.04715	−0.379945
$568$	7.50215	0.314783
$569$	−12.9927	−0.544683	−0.272341	−	0.962201i	$-0.587798\pi$
$569$	−12.9927	−0.544683	−0.272341	+	0.962201i	$0.587798\pi$
$570$	0	0
$571$	34.2334	1.43262	0.716312	−	0.697780i	$-0.245829\pi$
$571$	34.2334	1.43262	0.716312	+	0.697780i	$0.245829\pi$
$572$	5.09416	0.212997
$573$	−58.5626	−2.44649
$574$	−2.67809	−0.111781
$575$	0	0
$576$	−43.1429	−1.79762
$577$	−5.29472	−0.220422	−0.110211	−	0.993908i	$-0.535153\pi$
$577$	−5.29472	−0.220422	−0.110211	+	0.993908i	$0.535153\pi$
$578$	39.2661	1.63326
$579$	−49.9487	−2.07580
$580$	0	0
$581$	−5.78700	−0.240085
$582$	130.689	5.41722
$583$	4.08290	0.169096
$584$	21.8128	0.902621
$585$	0	0
$586$	0.181971	0.00751715
$587$	15.9044	0.656447	0.328223	−	0.944600i	$-0.393550\pi$
$587$	15.9044	0.656447	0.328223	+	0.944600i	$0.393550\pi$
$588$	85.7423	3.53595
$589$	0.252612	0.0104087
$590$	0	0
$591$	−61.0078	−2.50952
$592$	4.62052	0.189902
$593$	15.8487	0.650828	0.325414	−	0.945572i	$-0.394496\pi$
$593$	15.8487	0.650828	0.325414	+	0.945572i	$0.394496\pi$
$594$	−28.6134	−1.17402
$595$	0	0
$596$	51.5652	2.11219
$597$	77.6361	3.17743
$598$	17.4935	0.715361
$599$	29.6191	1.21020	0.605102	−	0.796148i	$-0.293132\pi$
$599$	29.6191	1.21020	0.605102	+	0.796148i	$0.293132\pi$
$600$	0	0
$601$	−35.1273	−1.43287	−0.716435	−	0.697654i	$-0.754228\pi$
$601$	−35.1273	−1.43287	−0.716435	+	0.697654i	$0.754228\pi$
$602$	4.66299	0.190049
$603$	−90.6531	−3.69168
$604$	−74.7432	−3.04126
$605$	0	0
$606$	−28.4681	−1.15644
$607$	46.0823	1.87042	0.935211	−	0.354090i	$-0.115209\pi$
$607$	46.0823	1.87042	0.935211	+	0.354090i	$0.115209\pi$
$608$	1.23416	0.0500517
$609$	2.06114	0.0835214
$610$	0	0
$611$	−3.39896	−0.137507
$612$	−29.3867	−1.18789
$613$	−8.75866	−0.353759	−0.176880	−	0.984233i	$-0.556600\pi$
$613$	−8.75866	−0.353759	−0.176880	+	0.984233i	$0.556600\pi$
$614$	42.5022	1.71525
$615$	0	0
$616$	−2.98460	−0.120253
$617$	−23.4750	−0.945069	−0.472534	−	0.881312i	$-0.656661\pi$
$617$	−23.4750	−0.945069	−0.472534	+	0.881312i	$0.656661\pi$
$618$	−40.0656	−1.61167
$619$	−18.9663	−0.762319	−0.381160	−	0.924509i	$-0.624475\pi$
$619$	−18.9663	−0.762319	−0.381160	+	0.924509i	$0.624475\pi$
$620$	0	0
$621$	−66.1536	−2.65465
$622$	−61.4712	−2.46477
$623$	3.43036	0.137435
$624$	18.2627	0.731094
$625$	0	0
$626$	−50.1980	−2.00632
$627$	−3.11629	−0.124452
$628$	−1.77531	−0.0708426
$629$	−1.03554	−0.0412898
$630$	0	0
$631$	2.89078	0.115080	0.0575401	−	0.998343i	$-0.481674\pi$
$631$	2.89078	0.115080	0.0575401	+	0.998343i	$0.481674\pi$
$632$	47.8719	1.90424
$633$	−47.6468	−1.89379
$634$	−1.17603	−0.0467060
$635$	0	0
$636$	52.4341	2.07915
$637$	8.25306	0.326998
$638$	2.87712	0.113906
$639$	−9.59456	−0.379555
$640$	0	0
$641$	−36.0171	−1.42259	−0.711295	−	0.702894i	$-0.751891\pi$
$641$	−36.0171	−1.42259	−0.711295	+	0.702894i	$0.751891\pi$
$642$	−10.2099	−0.402951
$643$	−37.4426	−1.47659	−0.738296	−	0.674476i	$-0.764369\pi$
$643$	−37.4426	−1.47659	−0.738296	+	0.674476i	$0.764369\pi$
$644$	−13.4069	−0.528306
$645$	0	0
$646$	−2.62878	−0.103428
$647$	−10.6945	−0.420445	−0.210222	−	0.977654i	$-0.567419\pi$
$647$	−10.6945	−0.420445	−0.210222	+	0.977654i	$0.567419\pi$
$648$	−83.4740	−3.27917
$649$	2.54912	0.100062
$650$	0	0
$651$	−0.447729	−0.0175479
$652$	−81.6183	−3.19642
$653$	17.5656	0.687397	0.343698	−	0.939080i	$-0.388320\pi$
$653$	17.5656	0.687397	0.343698	+	0.939080i	$0.388320\pi$
$654$	120.670	4.71857
$655$	0	0
$656$	−9.02301	−0.352289
$657$	−27.8966	−1.08835
$658$	3.86917	0.150836
$659$	−21.1923	−0.825536	−0.412768	−	0.910836i	$-0.635438\pi$
$659$	−21.1923	−0.825536	−0.412768	+	0.910836i	$0.635438\pi$
$660$	0	0
$661$	37.8506	1.47222	0.736108	−	0.676864i	$-0.236661\pi$
$661$	37.8506	1.47222	0.736108	+	0.676864i	$0.236661\pi$
$662$	8.78759	0.341539
$663$	−4.09301	−0.158959
$664$	−53.3941	−2.07209
$665$	0	0
$666$	−16.1824	−0.627055
$667$	6.65187	0.257561
$668$	19.5316	0.755699
$669$	81.7163	3.15934
$670$	0	0
$671$	−7.49069	−0.289175
$672$	−2.18742	−0.0843814
$673$	−8.22334	−0.316986	−0.158493	−	0.987360i	$-0.550664\pi$
$673$	−8.22334	−0.316986	−0.158493	+	0.987360i	$0.550664\pi$
$674$	32.3813	1.24728
$675$	0	0
$676$	−47.2765	−1.81833
$677$	−38.5319	−1.48090	−0.740450	−	0.672111i	$-0.765388\pi$
$677$	−38.5319	−1.48090	−0.740450	+	0.672111i	$0.765388\pi$
$678$	−124.302	−4.77377
$679$	−9.64076	−0.369979
$680$	0	0
$681$	−65.6201	−2.51457
$682$	−0.624981	−0.0239318
$683$	−8.28374	−0.316968	−0.158484	−	0.987362i	$-0.550661\pi$
$683$	−8.28374	−0.316968	−0.158484	+	0.987362i	$0.550661\pi$
$684$	−27.6573	−1.05750
$685$	0	0
$686$	−19.2447	−0.734768
$687$	−11.3436	−0.432783
$688$	15.7106	0.598959
$689$	5.04701	0.192276
$690$	0	0
$691$	−28.9853	−1.10265	−0.551327	−	0.834289i	$-0.685878\pi$
$691$	−28.9853	−1.10265	−0.551327	+	0.834289i	$0.685878\pi$
$692$	54.2748	2.06322
$693$	3.81704	0.144997
$694$	−6.54947	−0.248615
$695$	0	0
$696$	19.0172	0.720844
$697$	2.02222	0.0765971
$698$	3.58861	0.135831
$699$	−53.1493	−2.01029
$700$	0	0
$701$	1.63265	0.0616643	0.0308321	−	0.999525i	$-0.490184\pi$
$701$	1.63265	0.0616643	0.0308321	+	0.999525i	$0.490184\pi$
$702$	−35.3699	−1.33495
$703$	−0.974602	−0.0367578
$704$	6.42846	0.242282
$705$	0	0
$706$	1.02773	0.0386790
$707$	2.10006	0.0789809
$708$	32.7368	1.23032
$709$	−35.8643	−1.34691	−0.673455	−	0.739228i	$-0.735191\pi$
$709$	−35.8643	−1.34691	−0.673455	+	0.739228i	$0.735191\pi$
$710$	0	0
$711$	−61.2239	−2.29607
$712$	31.6504	1.18615
$713$	−1.44495	−0.0541137
$714$	4.65923	0.174367
$715$	0	0
$716$	−10.0191	−0.374431
$717$	54.5584	2.03752
$718$	−19.8419	−0.740493
$719$	15.7314	0.586684	0.293342	−	0.956008i	$-0.405233\pi$
$719$	15.7314	0.586684	0.293342	+	0.956008i	$0.405233\pi$
$720$	0	0
$721$	2.95559	0.110072
$722$	−2.47407	−0.0920755
$723$	−33.2080	−1.23502
$724$	106.698	3.96539
$725$	0	0
$726$	7.70992	0.286142
$727$	−2.62799	−0.0974669	−0.0487334	−	0.998812i	$-0.515518\pi$
$727$	−2.62799	−0.0974669	−0.0487334	+	0.998812i	$0.515518\pi$
$728$	−3.68936	−0.136737
$729$	−1.36650	−0.0506111
$730$	0	0
$731$	−3.52102	−0.130230
$732$	−96.1981	−3.55558
$733$	−23.1636	−0.855567	−0.427783	−	0.903881i	$-0.640705\pi$
$733$	−23.1636	−0.855567	−0.427783	+	0.903881i	$0.640705\pi$
$734$	30.1164	1.11162
$735$	0	0
$736$	−7.05941	−0.260213
$737$	13.5077	0.497561
$738$	31.6012	1.16326
$739$	−15.0948	−0.555272	−0.277636	−	0.960686i	$-0.589551\pi$
$739$	−15.0948	−0.555272	−0.277636	+	0.960686i	$0.589551\pi$
$740$	0	0
$741$	−3.85214	−0.141512
$742$	−5.74520	−0.210913
$743$	−5.03276	−0.184634	−0.0923170	−	0.995730i	$-0.529427\pi$
$743$	−5.03276	−0.184634	−0.0923170	+	0.995730i	$0.529427\pi$
$744$	−4.13099	−0.151450
$745$	0	0
$746$	−37.9053	−1.38781
$747$	68.2862	2.49846
$748$	4.37873	0.160102
$749$	0.753171	0.0275203
$750$	0	0
$751$	28.6187	1.04431	0.522155	−	0.852850i	$-0.325128\pi$
$751$	28.6187	1.04431	0.522155	+	0.852850i	$0.325128\pi$
$752$	13.0360	0.475374
$753$	−79.5483	−2.89890
$754$	3.55651	0.129520
$755$	0	0
$756$	27.1074	0.985885
$757$	−47.5711	−1.72900	−0.864500	−	0.502633i	$-0.832365\pi$
$757$	−47.5711	−1.72900	−0.864500	+	0.502633i	$0.832365\pi$
$758$	24.2199	0.879706
$759$	17.8252	0.647015
$760$	0	0
$761$	40.8932	1.48238	0.741189	−	0.671296i	$-0.234262\pi$
$761$	40.8932	1.48238	0.741189	+	0.671296i	$0.234262\pi$
$762$	−18.6871	−0.676963
$763$	−8.90171	−0.322263
$764$	77.4444	2.80184
$765$	0	0
$766$	4.25230	0.153642
$767$	3.15106	0.113778
$768$	101.587	3.66571
$769$	−4.20621	−0.151680	−0.0758399	−	0.997120i	$-0.524164\pi$
$769$	−4.20621	−0.151680	−0.0758399	+	0.997120i	$0.524164\pi$
$770$	0	0
$771$	0.843470	0.0303768
$772$	66.0532	2.37731
$773$	43.8358	1.57666	0.788331	−	0.615251i	$-0.210945\pi$
$773$	43.8358	1.57666	0.788331	+	0.615251i	$0.210945\pi$
$774$	−55.0229	−1.97776
$775$	0	0
$776$	−88.9510	−3.19315
$777$	1.72738	0.0619695
$778$	61.5162	2.20546
$779$	1.90322	0.0681898
$780$	0	0
$781$	1.42963	0.0511561
$782$	15.0367	0.537710
$783$	−13.4494	−0.480641
$784$	−31.6529	−1.13046
$785$	0	0
$786$	131.111	4.67657
$787$	16.5893	0.591344	0.295672	−	0.955290i	$-0.404457\pi$
$787$	16.5893	0.591344	0.295672	+	0.955290i	$0.404457\pi$
$788$	80.6781	2.87404
$789$	76.9492	2.73946
$790$	0	0
$791$	9.16960	0.326033
$792$	35.2181	1.25142
$793$	−9.25949	−0.328814
$794$	31.7408	1.12644
$795$	0	0
$796$	−102.668	−3.63896
$797$	−50.2487	−1.77990	−0.889951	−	0.456056i	$-0.849262\pi$
$797$	−50.2487	−1.77990	−0.889951	+	0.456056i	$0.849262\pi$
$798$	4.38504	0.155229
$799$	−2.92160	−0.103359
$800$	0	0
$801$	−40.4780	−1.43022
$802$	35.3288	1.24750
$803$	4.15670	0.146687
$804$	173.470	6.11782
$805$	0	0
$806$	−0.772560	−0.0272123
$807$	−66.4308	−2.33847
$808$	19.3763	0.681656
$809$	−24.3490	−0.856066	−0.428033	−	0.903763i	$-0.640793\pi$
$809$	−24.3490	−0.856066	−0.428033	+	0.903763i	$0.640793\pi$
$810$	0	0
$811$	−19.9061	−0.698996	−0.349498	−	0.936937i	$-0.613648\pi$
$811$	−19.9061	−0.698996	−0.349498	+	0.936937i	$0.613648\pi$
$812$	−2.72569	−0.0956530
$813$	−37.7578	−1.32423
$814$	2.41124	0.0845138
$815$	0	0
$816$	15.6979	0.549536
$817$	−3.31381	−0.115936
$818$	36.5875	1.27925
$819$	4.71836	0.164873
$820$	0	0
$821$	−25.2915	−0.882679	−0.441339	−	0.897340i	$-0.645497\pi$
$821$	−25.2915	−0.882679	−0.441339	+	0.897340i	$0.645497\pi$
$822$	107.200	3.73902
$823$	8.61762	0.300391	0.150196	−	0.988656i	$-0.452010\pi$
$823$	8.61762	0.300391	0.150196	+	0.988656i	$0.452010\pi$
$824$	27.2699	0.949993
$825$	0	0
$826$	−3.58697	−0.124807
$827$	−12.5285	−0.435658	−0.217829	−	0.975987i	$-0.569898\pi$
$827$	−12.5285	−0.435658	−0.217829	+	0.975987i	$0.569898\pi$
$828$	158.200	5.49785
$829$	41.9775	1.45794	0.728970	−	0.684546i	$-0.239999\pi$
$829$	41.9775	1.45794	0.728970	+	0.684546i	$0.239999\pi$
$830$	0	0
$831$	−19.8727	−0.689376
$832$	7.94643	0.275493
$833$	7.09399	0.245792
$834$	−42.2934	−1.46450
$835$	0	0
$836$	4.12105	0.142529
$837$	2.92153	0.100983
$838$	−27.7086	−0.957177
$839$	13.8093	0.476750	0.238375	−	0.971173i	$-0.423385\pi$
$839$	13.8093	0.476750	0.238375	+	0.971173i	$0.423385\pi$
$840$	0	0
$841$	−27.6476	−0.953367
$842$	−87.8084	−3.02608
$843$	43.9571	1.51396
$844$	63.0092	2.16887
$845$	0	0
$846$	−45.6558	−1.56968
$847$	−0.568753	−0.0195426
$848$	−19.3567	−0.664713
$849$	43.5064	1.49314
$850$	0	0
$851$	5.57474	0.191100
$852$	18.3598	0.628995
$853$	35.7647	1.22456	0.612280	−	0.790641i	$-0.290253\pi$
$853$	35.7647	1.22456	0.612280	+	0.790641i	$0.290253\pi$
$854$	10.5404	0.360686
$855$	0	0
$856$	6.94917	0.237518
$857$	−6.18572	−0.211300	−0.105650	−	0.994403i	$-0.533692\pi$
$857$	−6.18572	−0.211300	−0.105650	+	0.994403i	$0.533692\pi$
$858$	9.53049	0.325366
$859$	40.5430	1.38331	0.691655	−	0.722228i	$-0.256882\pi$
$859$	40.5430	1.38331	0.691655	+	0.722228i	$0.256882\pi$
$860$	0	0
$861$	−3.37325	−0.114960
$862$	92.2041	3.14048
$863$	−7.58679	−0.258257	−0.129129	−	0.991628i	$-0.541218\pi$
$863$	−7.58679	−0.258257	−0.129129	+	0.991628i	$0.541218\pi$
$864$	14.2734	0.485590
$865$	0	0
$866$	−85.5466	−2.90699
$867$	49.4587	1.67970
$868$	0.592087	0.0200967
$869$	9.12259	0.309463
$870$	0	0
$871$	16.6973	0.565765
$872$	−82.1320	−2.78134
$873$	113.760	3.85020
$874$	14.1518	0.478691
$875$	0	0
$876$	53.3819	1.80361
$877$	−16.2379	−0.548316	−0.274158	−	0.961685i	$-0.588399\pi$
$877$	−16.2379	−0.548316	−0.274158	+	0.961685i	$0.588399\pi$
$878$	85.9984	2.90231
$879$	0.229206	0.00773094
$880$	0	0
$881$	0.927391	0.0312446	0.0156223	−	0.999878i	$-0.495027\pi$
$881$	0.927391	0.0312446	0.0156223	+	0.999878i	$0.495027\pi$
$882$	110.858	3.73277
$883$	−52.0058	−1.75013	−0.875067	−	0.484002i	$-0.839183\pi$
$883$	−52.0058	−1.75013	−0.875067	+	0.484002i	$0.839183\pi$
$884$	5.41269	0.182048
$885$	0	0
$886$	93.7258	3.14878
$887$	−0.333406	−0.0111947	−0.00559734	−	0.999984i	$-0.501782\pi$
$887$	−0.333406	−0.0111947	−0.00559734	+	0.999984i	$0.501782\pi$
$888$	15.9378	0.534836
$889$	1.37853	0.0462344
$890$	0	0
$891$	−15.9070	−0.532904
$892$	−108.064	−3.61823
$893$	−2.74967	−0.0920143
$894$	96.4715	3.22649
$895$	0	0
$896$	−10.4496	−0.349096
$897$	22.0343	0.735705
$898$	−26.2360	−0.875507
$899$	−0.293765	−0.00979761
$900$	0	0
$901$	4.33820	0.144526
$902$	−4.70870	−0.156782
$903$	5.87339	0.195454
$904$	84.6037	2.81388
$905$	0	0
$906$	−139.835	−4.64569
$907$	1.23050	0.0408580	0.0204290	−	0.999791i	$-0.493497\pi$
$907$	1.23050	0.0408580	0.0204290	+	0.999791i	$0.493497\pi$
$908$	86.7775	2.87981
$909$	−24.7805	−0.821919
$910$	0	0
$911$	23.8271	0.789428	0.394714	−	0.918804i	$-0.370844\pi$
$911$	23.8271	0.789428	0.394714	+	0.918804i	$0.370844\pi$
$912$	14.7741	0.489219
$913$	−10.1749	−0.336740
$914$	10.5355	0.348483
$915$	0	0
$916$	15.0010	0.495646
$917$	−9.67190	−0.319394
$918$	−30.4025	−1.00343
$919$	−23.4692	−0.774176	−0.387088	−	0.922043i	$-0.626519\pi$
$919$	−23.4692	−0.774176	−0.387088	+	0.922043i	$0.626519\pi$
$920$	0	0
$921$	53.5347	1.76403
$922$	−89.1683	−2.93660
$923$	1.76721	0.0581684
$924$	−7.30413	−0.240288
$925$	0	0
$926$	29.6486	0.974313
$927$	−34.8758	−1.14547
$928$	−1.43521	−0.0471132
$929$	−14.1081	−0.462872	−0.231436	−	0.972850i	$-0.574342\pi$
$929$	−14.1081	−0.462872	−0.231436	+	0.972850i	$0.574342\pi$
$930$	0	0
$931$	6.67652	0.218814
$932$	70.2858	2.30229
$933$	−77.4277	−2.53487
$934$	28.3488	0.927600
$935$	0	0
$936$	43.5342	1.42296
$937$	−32.2732	−1.05432	−0.527161	−	0.849766i	$-0.676743\pi$
$937$	−32.2732	−1.05432	−0.527161	+	0.849766i	$0.676743\pi$
$938$	−19.0071	−0.620605
$939$	−63.2282	−2.06337
$940$	0	0
$941$	4.62299	0.150705	0.0753526	−	0.997157i	$-0.475992\pi$
$941$	4.62299	0.150705	0.0753526	+	0.997157i	$0.475992\pi$
$942$	−3.32137	−0.108216
$943$	−10.8864	−0.354511
$944$	−12.0852	−0.393340
$945$	0	0
$946$	8.19862	0.266560
$947$	5.86027	0.190433	0.0952166	−	0.995457i	$-0.469646\pi$
$947$	5.86027	0.190433	0.0952166	+	0.995457i	$0.469646\pi$
$948$	117.156	3.80504
$949$	5.13823	0.166794
$950$	0	0
$951$	−1.48129	−0.0480343
$952$	−3.17123	−0.102780
$953$	43.9764	1.42454	0.712268	−	0.701908i	$-0.247668\pi$
$953$	43.9764	1.42454	0.712268	+	0.701908i	$0.247668\pi$
$954$	67.7929	2.19488
$955$	0	0
$956$	−72.1492	−2.33347
$957$	3.62396	0.117146
$958$	−29.2916	−0.946367
$959$	−7.90801	−0.255363
$960$	0	0
$961$	−30.9362	−0.997942
$962$	2.98061	0.0960987
$963$	−8.88736	−0.286391
$964$	43.9150	1.41441
$965$	0	0
$966$	−25.0825	−0.807018
$967$	21.7275	0.698710	0.349355	−	0.936990i	$-0.386401\pi$
$967$	21.7275	0.698710	0.349355	+	0.936990i	$0.386401\pi$
$968$	−5.24763	−0.168665
$969$	−3.31114	−0.106369
$970$	0	0
$971$	−10.5833	−0.339634	−0.169817	−	0.985476i	$-0.554318\pi$
$971$	−10.5833	−0.339634	−0.169817	+	0.985476i	$0.554318\pi$
$972$	−61.3002	−1.96621
$973$	3.11994	0.100021
$974$	52.7271	1.68949
$975$	0	0
$976$	35.5128	1.13674
$977$	−35.5223	−1.13646	−0.568230	−	0.822870i	$-0.692372\pi$
$977$	−35.5223	−1.13646	−0.568230	+	0.822870i	$0.692372\pi$
$978$	−152.697	−4.88271
$979$	6.03138	0.192764
$980$	0	0
$981$	105.039	3.35365
$982$	75.9257	2.42289
$983$	20.4638	0.652695	0.326348	−	0.945250i	$-0.394182\pi$
$983$	20.4638	0.652695	0.326348	+	0.945250i	$0.394182\pi$
$984$	−31.1235	−0.992180
$985$	0	0
$986$	3.05703	0.0973555
$987$	4.87351	0.155126
$988$	5.09416	0.162067
$989$	18.9551	0.602737
$990$	0	0
$991$	−26.0437	−0.827305	−0.413653	−	0.910435i	$-0.635747\pi$
$991$	−26.0437	−0.827305	−0.413653	+	0.910435i	$0.635747\pi$
$992$	0.311763	0.00989849
$993$	11.0686	0.351252
$994$	−2.01168	−0.0638066
$995$	0	0
$996$	−130.670	−4.14043
$997$	4.10364	0.129964	0.0649819	−	0.997886i	$-0.479301\pi$
$997$	4.10364	0.129964	0.0649819	+	0.997886i	$0.479301\pi$
$998$	−56.2709	−1.78123
$999$	−11.2715	−0.356616

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5225.2.a.w.1.1	yes	15
5.4	even	2		5225.2.a.t.1.15	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5225.2.a.t.1.15	✓	15	5.4	even	2
5225.2.a.w.1.1	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)

\(f(q)\)	\(=\)	\(q-2.47407 q^{2} -3.11629 q^{3} +4.12105 q^{4} +7.70992 q^{6} -0.568753 q^{7} -5.24763 q^{8} +6.71124 q^{9} -1.00000 q^{11} -12.8424 q^{12} -1.23613 q^{13} +1.40714 q^{14} +4.74093 q^{16} -1.06253 q^{17} -16.6041 q^{18} -1.00000 q^{19} +1.77240 q^{21} +2.47407 q^{22} +5.72002 q^{23} +16.3531 q^{24} +3.05828 q^{26} -11.5653 q^{27} -2.34386 q^{28} +1.16291 q^{29} -0.252612 q^{31} -1.23416 q^{32} +3.11629 q^{33} +2.62878 q^{34} +27.6573 q^{36} +0.974602 q^{37} +2.47407 q^{38} +3.85214 q^{39} -1.90322 q^{41} -4.38504 q^{42} +3.31381 q^{43} -4.12105 q^{44} -14.1518 q^{46} +2.74967 q^{47} -14.7741 q^{48} -6.67652 q^{49} +3.31114 q^{51} -5.09416 q^{52} -4.08290 q^{53} +28.6134 q^{54} +2.98460 q^{56} +3.11629 q^{57} -2.87712 q^{58} -2.54912 q^{59} +7.49069 q^{61} +0.624981 q^{62} -3.81704 q^{63} -6.42846 q^{64} -7.70992 q^{66} -13.5077 q^{67} -4.37873 q^{68} -17.8252 q^{69} -1.42963 q^{71} -35.2181 q^{72} -4.15670 q^{73} -2.41124 q^{74} -4.12105 q^{76} +0.568753 q^{77} -9.53049 q^{78} -9.12259 q^{79} +15.9070 q^{81} +4.70870 q^{82} +10.1749 q^{83} +7.30413 q^{84} -8.19862 q^{86} -3.62396 q^{87} +5.24763 q^{88} -6.03138 q^{89} +0.703054 q^{91} +23.5725 q^{92} +0.787212 q^{93} -6.80289 q^{94} +3.84599 q^{96} +16.9507 q^{97} +16.5182 q^{98} -6.71124 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + q^{2} - 4 q^{3} + 13 q^{4} - q^{6} - 11 q^{7} + 3 q^{8} + 19 q^{9} - 15 q^{11} - 11 q^{12} - 3 q^{13} - 11 q^{14} + 13 q^{16} + 5 q^{17} - 12 q^{18} - 15 q^{19} + 10 q^{21} - q^{22} - 26 q^{23}+ \cdots - 19 q^{99}+O(q^{100}) \) 15 * q + q^2 - 4 * q^3 + 13 * q^4 - q^6 - 11 * q^7 + 3 * q^8 + 19 * q^9 - 15 * q^11 - 11 * q^12 - 3 * q^13 - 11 * q^14 + 13 * q^16 + 5 * q^17 - 12 * q^18 - 15 * q^19 + 10 * q^21 - q^22 - 26 * q^23 - 11 * q^24 - 5 * q^26 - 19 * q^27 - 18 * q^28 + 7 * q^29 - 10 * q^31 + 12 * q^32 + 4 * q^33 + 17 * q^34 + 24 * q^36 - 31 * q^37 - q^38 + 4 * q^39 + 2 * q^41 - 22 * q^42 - 26 * q^43 - 13 * q^44 - 23 * q^46 - 26 * q^47 - 46 * q^48 + 12 * q^49 + 12 * q^51 - 16 * q^52 - 21 * q^53 + 5 * q^54 - 10 * q^56 + 4 * q^57 - 34 * q^58 - 11 * q^59 + 20 * q^61 - 25 * q^62 - 27 * q^63 - 3 * q^64 + q^66 - 41 * q^67 + 6 * q^68 + q^69 + 25 * q^71 - 54 * q^72 + 6 * q^73 - 9 * q^74 - 13 * q^76 + 11 * q^77 - 28 * q^78 - 6 * q^79 + 43 * q^81 - 18 * q^82 + 20 * q^83 - 14 * q^84 + 35 * q^86 - 29 * q^87 - 3 * q^88 - 3 * q^89 + 30 * q^91 - 54 * q^92 - 2 * q^93 - 28 * q^94 - 61 * q^96 - 28 * q^97 + 2 * q^98 - 19 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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