LMFDB - Embedded newform 5225.2.a.u.1.3

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,-1,17] 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} - x^{14} - 21 x^{13} + 21 x^{12} + 168 x^{11} - 165 x^{10} - 645 x^{9} + 606 x^{8} + 1239 x^{7} + \cdots - 5 $ x^15 - x^14 - 21x^13 + 21x^12 + 168x^11 - 165x^10 - 645x^9 + 606x^8 + 1239x^7 - 1055x^6 - 1160x^5 + 793x^4 + 485x^3 - 171x^2 - 93*x - 5
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			1.3
Root			$1.98766$ 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)

$f(q)$	$=$	\(q-1.98766 q^{2} -0.104882 q^{3} +1.95080 q^{4} +0.208469 q^{6} +1.24528 q^{7} +0.0977875 q^{8} -2.98900 q^{9} -1.00000 q^{11} -0.204603 q^{12} -2.45314 q^{13} -2.47520 q^{14} -4.09597 q^{16} -8.07996 q^{17} +5.94112 q^{18} +1.00000 q^{19} -0.130607 q^{21} +1.98766 q^{22} -0.595102 q^{23} -0.0102561 q^{24} +4.87602 q^{26} +0.628136 q^{27} +2.42930 q^{28} -6.20473 q^{29} -9.51628 q^{31} +7.94584 q^{32} +0.104882 q^{33} +16.0602 q^{34} -5.83095 q^{36} +6.49334 q^{37} -1.98766 q^{38} +0.257290 q^{39} -8.06971 q^{41} +0.259603 q^{42} -2.33313 q^{43} -1.95080 q^{44} +1.18286 q^{46} -8.72660 q^{47} +0.429593 q^{48} -5.44927 q^{49} +0.847439 q^{51} -4.78560 q^{52} +10.8618 q^{53} -1.24852 q^{54} +0.121773 q^{56} -0.104882 q^{57} +12.3329 q^{58} +0.601714 q^{59} -1.23483 q^{61} +18.9152 q^{62} -3.72215 q^{63} -7.60170 q^{64} -0.208469 q^{66} -5.79923 q^{67} -15.7624 q^{68} +0.0624153 q^{69} +7.74139 q^{71} -0.292287 q^{72} +13.3744 q^{73} -12.9066 q^{74} +1.95080 q^{76} -1.24528 q^{77} -0.511405 q^{78} +1.79880 q^{79} +8.90112 q^{81} +16.0399 q^{82} +15.9421 q^{83} -0.254789 q^{84} +4.63748 q^{86} +0.650763 q^{87} -0.0977875 q^{88} -7.25863 q^{89} -3.05486 q^{91} -1.16093 q^{92} +0.998084 q^{93} +17.3455 q^{94} -0.833373 q^{96} +8.72541 q^{97} +10.8313 q^{98} +2.98900 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - q^{2} + 4 q^{3} + 13 q^{4} - q^{6} + 17 q^{7} + 3 q^{8} + 15 q^{9} - 15 q^{11} + 9 q^{12} + 3 q^{13} - 11 q^{14} + 13 q^{16} + q^{17} - 10 q^{18} + 15 q^{19} + 16 q^{21} + q^{22} + 18 q^{23}+ \cdots - 15 q^{99}+O(q^{100}) $ 15 * q - q^2 + 4 * q^3 + 13 * q^4 - q^6 + 17 * q^7 + 3 * q^8 + 15 * q^9 - 15 * q^11 + 9 * q^12 + 3 * q^13 - 11 * q^14 + 13 * q^16 + q^17 - 10 * q^18 + 15 * q^19 + 16 * q^21 + q^22 + 18 * q^23 - 27 * q^24 + 15 * q^26 + 31 * q^27 + 34 * q^28 + 11 * q^29 - 2 * q^31 + 10 * q^32 - 4 * q^33 - 13 * q^34 + 8 * q^36 + 27 * q^37 - q^38 + 4 * q^39 + 6 * q^41 + 2 * q^42 + 38 * q^43 - 13 * q^44 - 9 * q^46 + 14 * q^47 + 32 * q^48 + 28 * q^49 - 32 * q^51 + 16 * q^52 + 11 * q^53 - 11 * q^54 + 2 * q^56 + 4 * q^57 - 6 * q^58 + 13 * q^59 + 8 * q^61 + 7 * q^62 + 49 * q^63 + 9 * q^64 + q^66 + 31 * q^67 - 26 * q^68 + 39 * q^69 - 3 * q^71 - 18 * q^72 + 18 * q^73 + 7 * q^74 + 13 * q^76 - 17 * q^77 + 18 * q^78 + 10 * q^79 + 31 * q^81 + 58 * q^82 + 16 * q^83 + 112 * q^84 - 63 * q^86 + 67 * q^87 - 3 * q^88 - 7 * q^89 - 50 * q^91 + 98 * q^92 + 26 * q^93 + 22 * q^94 - 37 * q^96 + 24 * q^97 - 46 * q^98 - 15 * 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 $

$2$	−1.98766	−1.40549	−0.702745	−	0.711442i	$-0.748042\pi$
$2$	−1.98766	−1.40549	−0.702745	+	0.711442i	$0.748042\pi$
$3$	−0.104882	−0.0605534	−0.0302767	−	0.999542i	$-0.509639\pi$
$3$	−0.104882	−0.0605534	−0.0302767	+	0.999542i	$0.509639\pi$
$4$	1.95080	0.975401
$5$	0	0
$5$	0	0
$6$	0.208469	0.0851073
$7$	1.24528	0.470673	0.235336	−	0.971914i	$-0.424381\pi$
$7$	1.24528	0.470673	0.235336	+	0.971914i	$0.424381\pi$
$8$	0.0977875	0.0345731
$9$	−2.98900	−0.996333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−0.204603	−0.0590639
$13$	−2.45314	−0.680379	−0.340190	−	0.940357i	$-0.610491\pi$
$13$	−2.45314	−0.680379	−0.340190	+	0.940357i	$0.610491\pi$
$14$	−2.47520	−0.661526
$15$	0	0
$16$	−4.09597	−1.02399
$17$	−8.07996	−1.95968	−0.979839	−	0.199789i	$-0.935974\pi$
$17$	−8.07996	−1.95968	−0.979839	+	0.199789i	$0.935974\pi$
$18$	5.94112	1.40034
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−0.130607	−0.0285009
$22$	1.98766	0.423771
$23$	−0.595102	−0.124087	−0.0620437	−	0.998073i	$-0.519762\pi$
$23$	−0.595102	−0.124087	−0.0620437	+	0.998073i	$0.519762\pi$
$24$	−0.0102561	−0.00209352
$25$	0	0
$26$	4.87602	0.956266
$27$	0.628136	0.120885
$28$	2.42930	0.459095
$29$	−6.20473	−1.15219	−0.576095	−	0.817383i	$-0.695424\pi$
$29$	−6.20473	−1.15219	−0.576095	+	0.817383i	$0.695424\pi$
$30$	0	0
$31$	−9.51628	−1.70918	−0.854588	−	0.519307i	$-0.826190\pi$
$31$	−9.51628	−1.70918	−0.854588	+	0.519307i	$0.826190\pi$
$32$	7.94584	1.40464
$33$	0.104882	0.0182576
$34$	16.0602	2.75431
$35$	0	0
$36$	−5.83095	−0.971825
$37$	6.49334	1.06750	0.533749	−	0.845643i	$-0.320783\pi$
$37$	6.49334	1.06750	0.533749	+	0.845643i	$0.320783\pi$
$38$	−1.98766	−0.322441
$39$	0.257290	0.0411993
$40$	0	0
$41$	−8.06971	−1.26028	−0.630139	−	0.776483i	$-0.717002\pi$
$41$	−8.06971	−1.26028	−0.630139	+	0.776483i	$0.717002\pi$
$42$	0.259603	0.0400577
$43$	−2.33313	−0.355800	−0.177900	−	0.984049i	$-0.556930\pi$
$43$	−2.33313	−0.355800	−0.177900	+	0.984049i	$0.556930\pi$
$44$	−1.95080	−0.294095
$45$	0	0
$46$	1.18286	0.174403
$47$	−8.72660	−1.27291	−0.636453	−	0.771316i	$-0.719599\pi$
$47$	−8.72660	−1.27291	−0.636453	+	0.771316i	$0.719599\pi$
$48$	0.429593	0.0620063
$49$	−5.44927	−0.778467
$50$	0	0
$51$	0.847439	0.118665
$52$	−4.78560	−0.663643
$53$	10.8618	1.49198	0.745990	−	0.665957i	$-0.231977\pi$
$53$	10.8618	1.49198	0.745990	+	0.665957i	$0.231977\pi$
$54$	−1.24852	−0.169902
$55$	0	0
$56$	0.121773	0.0162726
$57$	−0.104882	−0.0138919
$58$	12.3329	1.61939
$59$	0.601714	0.0783365	0.0391683	−	0.999233i	$-0.487529\pi$
$59$	0.601714	0.0783365	0.0391683	+	0.999233i	$0.487529\pi$
$60$	0	0
$61$	−1.23483	−0.158104	−0.0790522	−	0.996870i	$-0.525189\pi$
$61$	−1.23483	−0.158104	−0.0790522	+	0.996870i	$0.525189\pi$
$62$	18.9152	2.40223
$63$	−3.72215	−0.468947
$64$	−7.60170	−0.950213
$65$	0	0
$66$	−0.208469	−0.0256608
$67$	−5.79923	−0.708488	−0.354244	−	0.935153i	$-0.615262\pi$
$67$	−5.79923	−0.708488	−0.354244	+	0.935153i	$0.615262\pi$
$68$	−15.7624	−1.91147
$69$	0.0624153	0.00751392
$70$	0	0
$71$	7.74139	0.918734	0.459367	−	0.888247i	$-0.348076\pi$
$71$	7.74139	0.918734	0.459367	+	0.888247i	$0.348076\pi$
$72$	−0.292287	−0.0344463
$73$	13.3744	1.56536	0.782680	−	0.622424i	$-0.213852\pi$
$73$	13.3744	1.56536	0.782680	+	0.622424i	$0.213852\pi$
$74$	−12.9066	−1.50036
$75$	0	0
$76$	1.95080	0.223772
$77$	−1.24528	−0.141913
$78$	−0.511405	−0.0579052
$79$	1.79880	0.202381	0.101190	−	0.994867i	$-0.467735\pi$
$79$	1.79880	0.202381	0.101190	+	0.994867i	$0.467735\pi$
$80$	0	0
$81$	8.90112	0.989013
$82$	16.0399	1.77131
$83$	15.9421	1.74987	0.874934	−	0.484242i	$-0.160905\pi$
$83$	15.9421	1.74987	0.874934	+	0.484242i	$0.160905\pi$
$84$	−0.254789	−0.0277998
$85$	0	0
$86$	4.63748	0.500073
$87$	0.650763	0.0697691
$88$	−0.0977875	−0.0104242
$89$	−7.25863	−0.769414	−0.384707	−	0.923039i	$-0.625697\pi$
$89$	−7.25863	−0.769414	−0.384707	+	0.923039i	$0.625697\pi$
$90$	0	0
$91$	−3.05486	−0.320236
$92$	−1.16093	−0.121035
$93$	0.998084	0.103496
$94$	17.3455	1.78906
$95$	0	0
$96$	−0.833373	−0.0850558
$97$	8.72541	0.885931	0.442966	−	0.896539i	$-0.353926\pi$
$97$	8.72541	0.885931	0.442966	+	0.896539i	$0.353926\pi$
$98$	10.8313	1.09413
$99$	2.98900	0.300406
$100$	0	0
$101$	−5.79671	−0.576794	−0.288397	−	0.957511i	$-0.593122\pi$
$101$	−5.79671	−0.576794	−0.288397	+	0.957511i	$0.593122\pi$
$102$	−1.68442	−0.166783
$103$	7.85458	0.773935	0.386967	−	0.922093i	$-0.373523\pi$
$103$	7.85458	0.773935	0.386967	+	0.922093i	$0.373523\pi$
$104$	−0.239887	−0.0235228
$105$	0	0
$106$	−21.5895	−2.09696
$107$	−0.0994255	−0.00961183	−0.00480591	−	0.999988i	$-0.501530\pi$
$107$	−0.0994255	−0.00961183	−0.00480591	+	0.999988i	$0.501530\pi$
$108$	1.22537	0.117911
$109$	16.9318	1.62177	0.810885	−	0.585206i	$-0.198986\pi$
$109$	16.9318	1.62177	0.810885	+	0.585206i	$0.198986\pi$
$110$	0	0
$111$	−0.681032	−0.0646407
$112$	−5.10065	−0.481966
$113$	3.54408	0.333399	0.166699	−	0.986008i	$-0.446689\pi$
$113$	3.54408	0.333399	0.166699	+	0.986008i	$0.446689\pi$
$114$	0.208469	0.0195249
$115$	0	0
$116$	−12.1042	−1.12385
$117$	7.33244	0.677885
$118$	−1.19601	−0.110101
$119$	−10.0618	−0.922367
$120$	0	0
$121$	1.00000	0.0909091
$122$	2.45443	0.222214
$123$	0.846365	0.0763142
$124$	−18.5644	−1.66713
$125$	0	0
$126$	7.39838	0.659100
$127$	8.94688	0.793907	0.396954	−	0.917839i	$-0.370067\pi$
$127$	8.94688	0.793907	0.396954	+	0.917839i	$0.370067\pi$
$128$	−0.782063	−0.0691253
$129$	0.244703	0.0215449
$130$	0	0
$131$	−16.5671	−1.44747	−0.723735	−	0.690078i	$-0.757576\pi$
$131$	−16.5671	−1.44747	−0.723735	+	0.690078i	$0.757576\pi$
$132$	0.204603	0.0178084
$133$	1.24528	0.107980
$134$	11.5269	0.995773
$135$	0	0
$136$	−0.790119	−0.0677521
$137$	−2.55929	−0.218655	−0.109327	−	0.994006i	$-0.534870\pi$
$137$	−2.55929	−0.218655	−0.109327	+	0.994006i	$0.534870\pi$
$138$	−0.124061	−0.0105607
$139$	17.7342	1.50419	0.752096	−	0.659053i	$-0.229043\pi$
$139$	17.7342	1.50419	0.752096	+	0.659053i	$0.229043\pi$
$140$	0	0
$141$	0.915260	0.0770788
$142$	−15.3873	−1.29127
$143$	2.45314	0.205142
$144$	12.2429	1.02024
$145$	0	0
$146$	−26.5839	−2.20010
$147$	0.571529	0.0471389
$148$	12.6672	1.04124
$149$	−9.38089	−0.768513	−0.384256	−	0.923226i	$-0.625542\pi$
$149$	−9.38089	−0.768513	−0.384256	+	0.923226i	$0.625542\pi$
$150$	0	0
$151$	−12.7995	−1.04161	−0.520804	−	0.853676i	$-0.674368\pi$
$151$	−12.7995	−1.04161	−0.520804	+	0.853676i	$0.674368\pi$
$152$	0.0977875	0.00793161
$153$	24.1510	1.95249
$154$	2.47520	0.199457
$155$	0	0
$156$	0.501921	0.0401859
$157$	−20.1863	−1.61104	−0.805520	−	0.592569i	$-0.798114\pi$
$157$	−20.1863	−1.61104	−0.805520	+	0.592569i	$0.798114\pi$
$158$	−3.57540	−0.284444
$159$	−1.13920	−0.0903445
$160$	0	0
$161$	−0.741070	−0.0584045
$162$	−17.6924	−1.39005
$163$	17.1940	1.34674	0.673368	−	0.739307i	$-0.264847\pi$
$163$	17.1940	1.34674	0.673368	+	0.739307i	$0.264847\pi$
$164$	−15.7424	−1.22928
$165$	0	0
$166$	−31.6874	−2.45942
$167$	−10.6651	−0.825286	−0.412643	−	0.910893i	$-0.635394\pi$
$167$	−10.6651	−0.825286	−0.412643	+	0.910893i	$0.635394\pi$
$168$	−0.0127718	−0.000985363 0
$169$	−6.98209	−0.537084
$170$	0	0
$171$	−2.98900	−0.228575
$172$	−4.55148	−0.347047
$173$	9.39794	0.714512	0.357256	−	0.934007i	$-0.383712\pi$
$173$	9.39794	0.714512	0.357256	+	0.934007i	$0.383712\pi$
$174$	−1.29350	−0.0980597
$175$	0	0
$176$	4.09597	0.308746
$177$	−0.0631088	−0.00474355
$178$	14.4277	1.08140
$179$	12.3622	0.923997	0.461999	−	0.886881i	$-0.347132\pi$
$179$	12.3622	0.923997	0.461999	+	0.886881i	$0.347132\pi$
$180$	0	0
$181$	10.5514	0.784279	0.392139	−	0.919906i	$-0.371735\pi$
$181$	10.5514	0.784279	0.392139	+	0.919906i	$0.371735\pi$
$182$	6.07202	0.450088
$183$	0.129511	0.00957376
$184$	−0.0581935	−0.00429008
$185$	0	0
$186$	−1.98385	−0.145463
$187$	8.07996	0.590865
$188$	−17.0239	−1.24159
$189$	0.782207	0.0568972
$190$	0	0
$191$	7.75577	0.561188	0.280594	−	0.959827i	$-0.409469\pi$
$191$	7.75577	0.561188	0.280594	+	0.959827i	$0.409469\pi$
$192$	0.797279	0.0575386
$193$	−18.0868	−1.30192	−0.650958	−	0.759114i	$-0.725632\pi$
$193$	−18.0868	−1.30192	−0.650958	+	0.759114i	$0.725632\pi$
$194$	−17.3432	−1.24517
$195$	0	0
$196$	−10.6305	−0.759318
$197$	18.6144	1.32622	0.663110	−	0.748522i	$-0.269236\pi$
$197$	18.6144	1.32622	0.663110	+	0.748522i	$0.269236\pi$
$198$	−5.94112	−0.422217
$199$	−6.39608	−0.453406	−0.226703	−	0.973964i	$-0.572795\pi$
$199$	−6.39608	−0.453406	−0.226703	+	0.973964i	$0.572795\pi$
$200$	0	0
$201$	0.608232	0.0429014
$202$	11.5219	0.810678
$203$	−7.72665	−0.542304
$204$	1.65319	0.115746
$205$	0	0
$206$	−15.6123	−1.08776
$207$	1.77876	0.123632
$208$	10.0480	0.696704
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	−18.2797	−1.25842	−0.629212	−	0.777234i	$-0.716622\pi$
$211$	−18.2797	−1.25842	−0.629212	+	0.777234i	$0.716622\pi$
$212$	21.1892	1.45528
$213$	−0.811930	−0.0556325
$214$	0.197624	0.0135093
$215$	0	0
$216$	0.0614239	0.00417936
$217$	−11.8505	−0.804462
$218$	−33.6546	−2.27938
$219$	−1.40273	−0.0947879
$220$	0	0
$221$	19.8213	1.33332
$222$	1.35366	0.0908519
$223$	−10.9820	−0.735411	−0.367705	−	0.929942i	$-0.619856\pi$
$223$	−10.9820	−0.735411	−0.367705	+	0.929942i	$0.619856\pi$
$224$	9.89482	0.661125
$225$	0	0
$226$	−7.04443	−0.468589
$227$	−16.6800	−1.10709	−0.553545	−	0.832819i	$-0.686725\pi$
$227$	−16.6800	−1.10709	−0.553545	+	0.832819i	$0.686725\pi$
$228$	−0.204603	−0.0135502
$229$	20.2154	1.33587	0.667935	−	0.744220i	$-0.267179\pi$
$229$	20.2154	1.33587	0.667935	+	0.744220i	$0.267179\pi$
$230$	0	0
$231$	0.130607	0.00859333
$232$	−0.606745	−0.0398348
$233$	−12.3865	−0.811468	−0.405734	−	0.913991i	$-0.632984\pi$
$233$	−12.3865	−0.811468	−0.405734	+	0.913991i	$0.632984\pi$
$234$	−14.5744	−0.952760
$235$	0	0
$236$	1.17383	0.0764096
$237$	−0.188661	−0.0122548
$238$	19.9995	1.29638
$239$	−5.13989	−0.332472	−0.166236	−	0.986086i	$-0.553161\pi$
$239$	−5.13989	−0.332472	−0.166236	+	0.986086i	$0.553161\pi$
$240$	0	0
$241$	20.3113	1.30837	0.654183	−	0.756336i	$-0.273012\pi$
$241$	20.3113	1.30837	0.654183	+	0.756336i	$0.273012\pi$
$242$	−1.98766	−0.127772
$243$	−2.81797	−0.180773
$244$	−2.40892	−0.154215
$245$	0	0
$246$	−1.68229	−0.107259
$247$	−2.45314	−0.156090
$248$	−0.930573	−0.0590915
$249$	−1.67203	−0.105961
$250$	0	0
$251$	−20.0586	−1.26609	−0.633043	−	0.774117i	$-0.718194\pi$
$251$	−20.0586	−1.26609	−0.633043	+	0.774117i	$0.718194\pi$
$252$	−7.26118	−0.457411
$253$	0.595102	0.0374137
$254$	−17.7834	−1.11583
$255$	0	0
$256$	16.7579	1.04737
$257$	−8.63146	−0.538416	−0.269208	−	0.963082i	$-0.586762\pi$
$257$	−8.63146	−0.538416	−0.269208	+	0.963082i	$0.586762\pi$
$258$	−0.486387	−0.0302811
$259$	8.08605	0.502442
$260$	0	0
$261$	18.5460	1.14797
$262$	32.9297	2.03441
$263$	4.81636	0.296990	0.148495	−	0.988913i	$-0.452557\pi$
$263$	4.81636	0.296990	0.148495	+	0.988913i	$0.452557\pi$
$264$	0.0102561	0.000631220 0
$265$	0	0
$266$	−2.47520	−0.151764
$267$	0.761297	0.0465906
$268$	−11.3131	−0.691060
$269$	3.30933	0.201774	0.100887	−	0.994898i	$-0.467832\pi$
$269$	3.30933	0.201774	0.100887	+	0.994898i	$0.467832\pi$
$270$	0	0
$271$	0.697805	0.0423886	0.0211943	−	0.999775i	$-0.493253\pi$
$271$	0.697805	0.0423886	0.0211943	+	0.999775i	$0.493253\pi$
$272$	33.0953	2.00670
$273$	0.320398	0.0193914
$274$	5.08700	0.307317
$275$	0	0
$276$	0.121760	0.00732908
$277$	−4.85656	−0.291803	−0.145901	−	0.989299i	$-0.546608\pi$
$277$	−4.85656	−0.291803	−0.145901	+	0.989299i	$0.546608\pi$
$278$	−35.2495	−2.11413
$279$	28.4442	1.70291
$280$	0	0
$281$	12.0878	0.721095	0.360548	−	0.932741i	$-0.382590\pi$
$281$	12.0878	0.721095	0.360548	+	0.932741i	$0.382590\pi$
$282$	−1.81923	−0.108333
$283$	16.2537	0.966182	0.483091	−	0.875570i	$-0.339514\pi$
$283$	16.2537	0.966182	0.483091	+	0.875570i	$0.339514\pi$
$284$	15.1019	0.896134
$285$	0	0
$286$	−4.87602	−0.288325
$287$	−10.0491	−0.593178
$288$	−23.7501	−1.39949
$289$	48.2857	2.84034
$290$	0	0
$291$	−0.915136	−0.0536462
$292$	26.0909	1.52685
$293$	10.8945	0.636462	0.318231	−	0.948013i	$-0.396911\pi$
$293$	10.8945	0.636462	0.318231	+	0.948013i	$0.396911\pi$
$294$	−1.13601	−0.0662532
$295$	0	0
$296$	0.634967	0.0369067
$297$	−0.628136	−0.0364482
$298$	18.6460	1.08014
$299$	1.45987	0.0844265
$300$	0	0
$301$	−2.90541	−0.167465
$302$	25.4411	1.46397
$303$	0.607968	0.0349269
$304$	−4.09597	−0.234920
$305$	0	0
$306$	−48.0040	−2.74421
$307$	−1.11684	−0.0637414	−0.0318707	−	0.999492i	$-0.510146\pi$
$307$	−1.11684	−0.0637414	−0.0318707	+	0.999492i	$0.510146\pi$
$308$	−2.42930	−0.138422
$309$	−0.823801	−0.0468644
$310$	0	0
$311$	−31.2395	−1.77143	−0.885714	−	0.464231i	$-0.846331\pi$
$311$	−31.2395	−1.77143	−0.885714	+	0.464231i	$0.846331\pi$
$312$	0.0251597	0.00142439
$313$	−15.9427	−0.901133	−0.450566	−	0.892743i	$-0.648778\pi$
$313$	−15.9427	−0.901133	−0.450566	+	0.892743i	$0.648778\pi$
$314$	40.1235	2.26430
$315$	0	0
$316$	3.50910	0.197402
$317$	−10.2653	−0.576558	−0.288279	−	0.957546i	$-0.593083\pi$
$317$	−10.2653	−0.576558	−0.288279	+	0.957546i	$0.593083\pi$
$318$	2.26435	0.126978
$319$	6.20473	0.347398
$320$	0	0
$321$	0.0104279	0.000582029 0
$322$	1.47300	0.0820869
$323$	−8.07996	−0.449581
$324$	17.3643	0.964685
$325$	0	0
$326$	−34.1758	−1.89282
$327$	−1.77583	−0.0982037
$328$	−0.789117	−0.0435717
$329$	−10.8671	−0.599122
$330$	0	0
$331$	22.2045	1.22047	0.610234	−	0.792221i	$-0.291075\pi$
$331$	22.2045	1.22047	0.610234	+	0.792221i	$0.291075\pi$
$332$	31.0998	1.70682
$333$	−19.4086	−1.06358
$334$	21.1985	1.15993
$335$	0	0
$336$	0.534964	0.0291847
$337$	−21.7331	−1.18388	−0.591938	−	0.805983i	$-0.701637\pi$
$337$	−21.7331	−1.18388	−0.591938	+	0.805983i	$0.701637\pi$
$338$	13.8780	0.754866
$339$	−0.371709	−0.0201884
$340$	0	0
$341$	9.51628	0.515336
$342$	5.94112	0.321259
$343$	−15.5029	−0.837076
$344$	−0.228151	−0.0123011
$345$	0	0
$346$	−18.6799	−1.00424
$347$	6.96031	0.373649	0.186824	−	0.982393i	$-0.440180\pi$
$347$	6.96031	0.373649	0.186824	+	0.982393i	$0.440180\pi$
$348$	1.26951	0.0680529
$349$	−8.22249	−0.440140	−0.220070	−	0.975484i	$-0.570629\pi$
$349$	−8.22249	−0.440140	−0.220070	+	0.975484i	$0.570629\pi$
$350$	0	0
$351$	−1.54091	−0.0822476
$352$	−7.94584	−0.423515
$353$	22.9432	1.22114	0.610571	−	0.791962i	$-0.290940\pi$
$353$	22.9432	1.22114	0.610571	+	0.791962i	$0.290940\pi$
$354$	0.125439	0.00666701
$355$	0	0
$356$	−14.1602	−0.750487
$357$	1.05530	0.0558525
$358$	−24.5720	−1.29867
$359$	−0.321493	−0.0169678	−0.00848388	−	0.999964i	$-0.502701\pi$
$359$	−0.321493	−0.0169678	−0.00848388	+	0.999964i	$0.502701\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−20.9726	−1.10230
$363$	−0.104882	−0.00550486
$364$	−5.95942	−0.312359
$365$	0	0
$366$	−0.257425	−0.0134558
$367$	19.0521	0.994510	0.497255	−	0.867604i	$-0.334341\pi$
$367$	19.0521	0.994510	0.497255	+	0.867604i	$0.334341\pi$
$368$	2.43752	0.127065
$369$	24.1204	1.25566
$370$	0	0
$371$	13.5260	0.702234
$372$	1.94706	0.100951
$373$	−5.23572	−0.271096	−0.135548	−	0.990771i	$-0.543279\pi$
$373$	−5.23572	−0.271096	−0.135548	+	0.990771i	$0.543279\pi$
$374$	−16.0602	−0.830455
$375$	0	0
$376$	−0.853352	−0.0440083
$377$	15.2211	0.783926
$378$	−1.55476	−0.0799684
$379$	−21.2726	−1.09270	−0.546349	−	0.837558i	$-0.683983\pi$
$379$	−21.2726	−1.09270	−0.546349	+	0.837558i	$0.683983\pi$
$380$	0	0
$381$	−0.938364	−0.0480738
$382$	−15.4159	−0.788744
$383$	−17.1531	−0.876482	−0.438241	−	0.898857i	$-0.644398\pi$
$383$	−17.1531	−0.876482	−0.438241	+	0.898857i	$0.644398\pi$
$384$	0.0820241	0.00418577
$385$	0	0
$386$	35.9504	1.82983
$387$	6.97374	0.354495
$388$	17.0216	0.864139
$389$	16.3320	0.828065	0.414033	−	0.910262i	$-0.364120\pi$
$389$	16.3320	0.828065	0.414033	+	0.910262i	$0.364120\pi$
$390$	0	0
$391$	4.80840	0.243171
$392$	−0.532870	−0.0269140
$393$	1.73758	0.0876494
$394$	−36.9991	−1.86399
$395$	0	0
$396$	5.83095	0.293016
$397$	35.5061	1.78200	0.891000	−	0.454003i	$-0.150004\pi$
$397$	35.5061	1.78200	0.891000	+	0.454003i	$0.150004\pi$
$398$	12.7133	0.637258
$399$	−0.130607	−0.00653854
$400$	0	0
$401$	30.9517	1.54565	0.772827	−	0.634617i	$-0.218842\pi$
$401$	30.9517	1.54565	0.772827	+	0.634617i	$0.218842\pi$
$402$	−1.20896	−0.0602975
$403$	23.3448	1.16289
$404$	−11.3082	−0.562606
$405$	0	0
$406$	15.3580	0.762203
$407$	−6.49334	−0.321863
$408$	0.0828690	0.00410262
$409$	−3.57490	−0.176767	−0.0883837	−	0.996087i	$-0.528170\pi$
$409$	−3.57490	−0.176767	−0.0883837	+	0.996087i	$0.528170\pi$
$410$	0	0
$411$	0.268423	0.0132403
$412$	15.3227	0.754897
$413$	0.749304	0.0368709
$414$	−3.53557	−0.173764
$415$	0	0
$416$	−19.4923	−0.955687
$417$	−1.85999	−0.0910840
$418$	1.98766	0.0972198
$419$	21.8193	1.06594	0.532972	−	0.846133i	$-0.321075\pi$
$419$	21.8193	1.06594	0.532972	+	0.846133i	$0.321075\pi$
$420$	0	0
$421$	36.1289	1.76081	0.880407	−	0.474220i	$-0.157270\pi$
$421$	36.1289	1.76081	0.880407	+	0.474220i	$0.157270\pi$
$422$	36.3338	1.76870
$423$	26.0838	1.26824
$424$	1.06215	0.0515824
$425$	0	0
$426$	1.61384	0.0781909
$427$	−1.53772	−0.0744154
$428$	−0.193960	−0.00937539
$429$	−0.257290	−0.0124221
$430$	0	0
$431$	−26.0067	−1.25270	−0.626349	−	0.779542i	$-0.715452\pi$
$431$	−26.0067	−1.25270	−0.626349	+	0.779542i	$0.715452\pi$
$432$	−2.57283	−0.123785
$433$	−32.2109	−1.54796	−0.773978	−	0.633212i	$-0.781736\pi$
$433$	−32.2109	−1.54796	−0.773978	+	0.633212i	$0.781736\pi$
$434$	23.5547	1.13066
$435$	0	0
$436$	33.0305	1.58188
$437$	−0.595102	−0.0284676
$438$	2.78816	0.133223
$439$	−36.1699	−1.72630	−0.863149	−	0.504950i	$-0.831511\pi$
$439$	−36.1699	−1.72630	−0.863149	+	0.504950i	$0.831511\pi$
$440$	0	0
$441$	16.2879	0.775613
$442$	−39.3980	−1.87397
$443$	−6.35992	−0.302169	−0.151084	−	0.988521i	$-0.548277\pi$
$443$	−6.35992	−0.302169	−0.151084	+	0.988521i	$0.548277\pi$
$444$	−1.32856	−0.0630507
$445$	0	0
$446$	21.8286	1.03361
$447$	0.983883	0.0465361
$448$	−9.46627	−0.447239
$449$	29.0551	1.37120	0.685598	−	0.727981i	$-0.259541\pi$
$449$	29.0551	1.37120	0.685598	+	0.727981i	$0.259541\pi$
$450$	0	0
$451$	8.06971	0.379988
$452$	6.91380	0.325198
$453$	1.34243	0.0630729
$454$	33.1542	1.55600
$455$	0	0
$456$	−0.0102561	−0.000480286 0
$457$	18.2529	0.853832	0.426916	−	0.904291i	$-0.359600\pi$
$457$	18.2529	0.853832	0.426916	+	0.904291i	$0.359600\pi$
$458$	−40.1813	−1.87755
$459$	−5.07531	−0.236895
$460$	0	0
$461$	22.6967	1.05709	0.528545	−	0.848905i	$-0.322738\pi$
$461$	22.6967	1.05709	0.528545	+	0.848905i	$0.322738\pi$
$462$	−0.259603	−0.0120778
$463$	25.3667	1.17889	0.589446	−	0.807808i	$-0.299346\pi$
$463$	25.3667	1.17889	0.589446	+	0.807808i	$0.299346\pi$
$464$	25.4144	1.17984
$465$	0	0
$466$	24.6202	1.14051
$467$	−35.1641	−1.62720	−0.813601	−	0.581423i	$-0.802496\pi$
$467$	−35.1641	−1.62720	−0.813601	+	0.581423i	$0.802496\pi$
$468$	14.3041	0.661210
$469$	−7.22168	−0.333466
$470$	0	0
$471$	2.11717	0.0975540
$472$	0.0588401	0.00270834
$473$	2.33313	0.107278
$474$	0.374994	0.0172241
$475$	0	0
$476$	−19.6287	−0.899678
$477$	−32.4658	−1.48651
$478$	10.2164	0.467286
$479$	−19.4028	−0.886536	−0.443268	−	0.896389i	$-0.646181\pi$
$479$	−19.4028	−0.886536	−0.443268	+	0.896389i	$0.646181\pi$
$480$	0	0
$481$	−15.9291	−0.726304
$482$	−40.3720	−1.83890
$483$	0.0777247	0.00353659
$484$	1.95080	0.0886729
$485$	0	0
$486$	5.60118	0.254075
$487$	25.6957	1.16438	0.582191	−	0.813052i	$-0.302196\pi$
$487$	25.6957	1.16438	0.582191	+	0.813052i	$0.302196\pi$
$488$	−0.120751	−0.00546616
$489$	−1.80333	−0.0815496
$490$	0	0
$491$	−4.76339	−0.214969	−0.107484	−	0.994207i	$-0.534280\pi$
$491$	−4.76339	−0.214969	−0.107484	+	0.994207i	$0.534280\pi$
$492$	1.65109	0.0744369
$493$	50.1340	2.25792
$494$	4.87602	0.219382
$495$	0	0
$496$	38.9785	1.75018
$497$	9.64022	0.432423
$498$	3.32343	0.148926
$499$	26.8553	1.20221	0.601104	−	0.799171i	$-0.294728\pi$
$499$	26.8553	1.20221	0.601104	+	0.799171i	$0.294728\pi$
$500$	0	0
$501$	1.11857	0.0499739
$502$	39.8697	1.77947
$503$	−2.37364	−0.105836	−0.0529178	−	0.998599i	$-0.516852\pi$
$503$	−2.37364	−0.105836	−0.0529178	+	0.998599i	$0.516852\pi$
$504$	−0.363980	−0.0162129
$505$	0	0
$506$	−1.18286	−0.0525846
$507$	0.732293	0.0325223
$508$	17.4536	0.774378
$509$	−13.3350	−0.591063	−0.295531	−	0.955333i	$-0.595497\pi$
$509$	−13.3350	−0.591063	−0.295531	+	0.955333i	$0.595497\pi$
$510$	0	0
$511$	16.6550	0.736772
$512$	−31.7449	−1.40294
$513$	0.628136	0.0277329
$514$	17.1564	0.756738
$515$	0	0
$516$	0.477367	0.0210149
$517$	8.72660	0.383795
$518$	−16.0723	−0.706178
$519$	−0.985671	−0.0432662
$520$	0	0
$521$	−30.9479	−1.35585	−0.677926	−	0.735131i	$-0.737121\pi$
$521$	−30.9479	−1.35585	−0.677926	+	0.735131i	$0.737121\pi$
$522$	−36.8631	−1.61345
$523$	17.2379	0.753762	0.376881	−	0.926262i	$-0.376996\pi$
$523$	17.2379	0.753762	0.376881	+	0.926262i	$0.376996\pi$
$524$	−32.3191	−1.41187
$525$	0	0
$526$	−9.57330	−0.417416
$527$	76.8912	3.34943
$528$	−0.429593	−0.0186956
$529$	−22.6459	−0.984602
$530$	0	0
$531$	−1.79852	−0.0780493
$532$	2.42930	0.105324
$533$	19.7962	0.857467
$534$	−1.51320	−0.0654827
$535$	0	0
$536$	−0.567092	−0.0244946
$537$	−1.29657	−0.0559512
$538$	−6.57784	−0.283591
$539$	5.44927	0.234717
$540$	0	0
$541$	−28.2470	−1.21443	−0.607217	−	0.794536i	$-0.707714\pi$
$541$	−28.2470	−1.21443	−0.607217	+	0.794536i	$0.707714\pi$
$542$	−1.38700	−0.0595768
$543$	−1.10665	−0.0474908
$544$	−64.2021	−2.75264
$545$	0	0
$546$	−0.636844	−0.0272544
$547$	−7.04078	−0.301042	−0.150521	−	0.988607i	$-0.548095\pi$
$547$	−7.04078	−0.301042	−0.150521	+	0.988607i	$0.548095\pi$
$548$	−4.99267	−0.213276
$549$	3.69092	0.157525
$550$	0	0
$551$	−6.20473	−0.264331
$552$	0.00610343	0.000259779 0
$553$	2.24001	0.0952550
$554$	9.65321	0.410126
$555$	0	0
$556$	34.5959	1.46719
$557$	−26.5954	−1.12688	−0.563441	−	0.826156i	$-0.690523\pi$
$557$	−26.5954	−1.12688	−0.563441	+	0.826156i	$0.690523\pi$
$558$	−56.5374	−2.39342
$559$	5.72351	0.242079
$560$	0	0
$561$	−0.847439	−0.0357789
$562$	−24.0264	−1.01349
$563$	3.77560	0.159122	0.0795612	−	0.996830i	$-0.474648\pi$
$563$	3.77560	0.159122	0.0795612	+	0.996830i	$0.474648\pi$
$564$	1.78549	0.0751828
$565$	0	0
$566$	−32.3069	−1.35796
$567$	11.0844	0.465501
$568$	0.757011	0.0317635
$569$	33.4474	1.40219	0.701094	−	0.713069i	$-0.252695\pi$
$569$	33.4474	1.40219	0.701094	+	0.713069i	$0.252695\pi$
$570$	0	0
$571$	−12.1608	−0.508914	−0.254457	−	0.967084i	$-0.581897\pi$
$571$	−12.1608	−0.508914	−0.254457	+	0.967084i	$0.581897\pi$
$572$	4.78560	0.200096
$573$	−0.813438	−0.0339819
$574$	19.9742	0.833706
$575$	0	0
$576$	22.7215	0.946728
$577$	8.99197	0.374341	0.187170	−	0.982327i	$-0.440068\pi$
$577$	8.99197	0.374341	0.187170	+	0.982327i	$0.440068\pi$
$578$	−95.9757	−3.99206
$579$	1.89697	0.0788355
$580$	0	0
$581$	19.8524	0.823615
$582$	1.81898	0.0753992
$583$	−10.8618	−0.449849
$584$	1.30785	0.0541193
$585$	0	0
$586$	−21.6545	−0.894541
$587$	34.4915	1.42362	0.711809	−	0.702373i	$-0.247876\pi$
$587$	34.4915	1.42362	0.711809	+	0.702373i	$0.247876\pi$
$588$	1.11494	0.0459793
$589$	−9.51628	−0.392112
$590$	0	0
$591$	−1.95231	−0.0803072
$592$	−26.5966	−1.09311
$593$	−13.9176	−0.571527	−0.285763	−	0.958300i	$-0.592247\pi$
$593$	−13.9176	−0.571527	−0.285763	+	0.958300i	$0.592247\pi$
$594$	1.24852	0.0512275
$595$	0	0
$596$	−18.3003	−0.749608
$597$	0.670832	0.0274553
$598$	−2.90173	−0.118661
$599$	−19.8981	−0.813014	−0.406507	−	0.913648i	$-0.633253\pi$
$599$	−19.8981	−0.813014	−0.406507	+	0.913648i	$0.633253\pi$
$600$	0	0
$601$	−35.9397	−1.46601	−0.733005	−	0.680224i	$-0.761883\pi$
$601$	−35.9397	−1.46601	−0.733005	+	0.680224i	$0.761883\pi$
$602$	5.77498	0.235371
$603$	17.3339	0.705890
$604$	−24.9693	−1.01599
$605$	0	0
$606$	−1.20844	−0.0490894
$607$	3.78275	0.153537	0.0767686	−	0.997049i	$-0.475540\pi$
$607$	3.78275	0.153537	0.0767686	+	0.997049i	$0.475540\pi$
$608$	7.94584	0.322246
$609$	0.810384	0.0328384
$610$	0	0
$611$	21.4076	0.866058
$612$	47.1138	1.90446
$613$	−1.87040	−0.0755446	−0.0377723	−	0.999286i	$-0.512026\pi$
$613$	−1.87040	−0.0755446	−0.0377723	+	0.999286i	$0.512026\pi$
$614$	2.21990	0.0895879
$615$	0	0
$616$	−0.121773	−0.00490638
$617$	48.3584	1.94683	0.973417	−	0.229040i	$-0.0735585\pi$
$617$	48.3584	1.94683	0.973417	+	0.229040i	$0.0735585\pi$
$618$	1.63744	0.0658674
$619$	24.5105	0.985162	0.492581	−	0.870267i	$-0.336054\pi$
$619$	24.5105	0.985162	0.492581	+	0.870267i	$0.336054\pi$
$620$	0	0
$621$	−0.373805	−0.0150003
$622$	62.0935	2.48972
$623$	−9.03905	−0.362142
$624$	−1.05385	−0.0421878
$625$	0	0
$626$	31.6886	1.26653
$627$	0.104882	0.00418857
$628$	−39.3794	−1.57141
$629$	−52.4659	−2.09195
$630$	0	0
$631$	30.7812	1.22538	0.612691	−	0.790323i	$-0.290087\pi$
$631$	30.7812	1.22538	0.612691	+	0.790323i	$0.290087\pi$
$632$	0.175900	0.00699692
$633$	1.91720	0.0762019
$634$	20.4040	0.810346
$635$	0	0
$636$	−2.22236	−0.0881222
$637$	13.3678	0.529653
$638$	−12.3329	−0.488265
$639$	−23.1390	−0.915365
$640$	0	0
$641$	18.2583	0.721159	0.360579	−	0.932729i	$-0.382579\pi$
$641$	18.2583	0.721159	0.360579	+	0.932729i	$0.382579\pi$
$642$	−0.0207272	−0.000818036 0
$643$	24.9299	0.983140	0.491570	−	0.870838i	$-0.336423\pi$
$643$	24.9299	0.983140	0.491570	+	0.870838i	$0.336423\pi$
$644$	−1.44568	−0.0569678
$645$	0	0
$646$	16.0602	0.631881
$647$	−10.0526	−0.395209	−0.197604	−	0.980282i	$-0.563316\pi$
$647$	−10.0526	−0.395209	−0.197604	+	0.980282i	$0.563316\pi$
$648$	0.870418	0.0341932
$649$	−0.601714	−0.0236194
$650$	0	0
$651$	1.24290	0.0487129
$652$	33.5421	1.31361
$653$	3.99482	0.156330	0.0781648	−	0.996940i	$-0.475094\pi$
$653$	3.99482	0.156330	0.0781648	+	0.996940i	$0.475094\pi$
$654$	3.52975	0.138024
$655$	0	0
$656$	33.0533	1.29052
$657$	−39.9762	−1.55962
$658$	21.6001	0.842059
$659$	−16.1838	−0.630430	−0.315215	−	0.949020i	$-0.602077\pi$
$659$	−16.1838	−0.630430	−0.315215	+	0.949020i	$0.602077\pi$
$660$	0	0
$661$	27.2908	1.06149	0.530744	−	0.847532i	$-0.321912\pi$
$661$	27.2908	1.06149	0.530744	+	0.847532i	$0.321912\pi$
$662$	−44.1350	−1.71536
$663$	−2.07889	−0.0807374
$664$	1.55893	0.0604983
$665$	0	0
$666$	38.5777	1.49486
$667$	3.69245	0.142972
$668$	−20.8054	−0.804986
$669$	1.15181	0.0445317
$670$	0	0
$671$	1.23483	0.0476703
$672$	−1.03778	−0.0400334
$673$	23.9335	0.922568	0.461284	−	0.887252i	$-0.347389\pi$
$673$	23.9335	0.922568	0.461284	+	0.887252i	$0.347389\pi$
$674$	43.1980	1.66393
$675$	0	0
$676$	−13.6207	−0.523873
$677$	−4.23402	−0.162727	−0.0813633	−	0.996685i	$-0.525927\pi$
$677$	−4.23402	−0.162727	−0.0813633	+	0.996685i	$0.525927\pi$
$678$	0.738832	0.0283747
$679$	10.8656	0.416984
$680$	0	0
$681$	1.74942	0.0670381
$682$	−18.9152	−0.724299
$683$	14.4350	0.552341	0.276170	−	0.961109i	$-0.410935\pi$
$683$	14.4350	0.552341	0.276170	+	0.961109i	$0.410935\pi$
$684$	−5.83095	−0.222952
$685$	0	0
$686$	30.8145	1.17650
$687$	−2.12022	−0.0808915
$688$	9.55646	0.364336
$689$	−26.6455	−1.01511
$690$	0	0
$691$	48.4203	1.84200	0.920998	−	0.389568i	$-0.127376\pi$
$691$	48.4203	1.84200	0.920998	+	0.389568i	$0.127376\pi$
$692$	18.3335	0.696936
$693$	3.72215	0.141393
$694$	−13.8347	−0.525159
$695$	0	0
$696$	0.0636365	0.00241213
$697$	65.2030	2.46974
$698$	16.3435	0.618612
$699$	1.29912	0.0491372
$700$	0	0
$701$	22.7724	0.860103	0.430052	−	0.902804i	$-0.358495\pi$
$701$	22.7724	0.860103	0.430052	+	0.902804i	$0.358495\pi$
$702$	3.06280	0.115598
$703$	6.49334	0.244901
$704$	7.60170	0.286500
$705$	0	0
$706$	−45.6033	−1.71630
$707$	−7.21854	−0.271481
$708$	−0.123113	−0.00462686
$709$	32.9979	1.23926	0.619631	−	0.784893i	$-0.287282\pi$
$709$	32.9979	1.23926	0.619631	+	0.784893i	$0.287282\pi$
$710$	0	0
$711$	−5.37661	−0.201639
$712$	−0.709803	−0.0266010
$713$	5.66316	0.212087
$714$	−2.09758	−0.0785001
$715$	0	0
$716$	24.1163	0.901268
$717$	0.539080	0.0201323
$718$	0.639020	0.0238480
$719$	−25.3585	−0.945712	−0.472856	−	0.881140i	$-0.656777\pi$
$719$	−25.3585	−0.945712	−0.472856	+	0.881140i	$0.656777\pi$
$720$	0	0
$721$	9.78117	0.364270
$722$	−1.98766	−0.0739731
$723$	−2.13028	−0.0792261
$724$	20.5837	0.764986
$725$	0	0
$726$	0.208469	0.00773702
$727$	−37.2296	−1.38077	−0.690385	−	0.723442i	$-0.742559\pi$
$727$	−37.2296	−1.38077	−0.690385	+	0.723442i	$0.742559\pi$
$728$	−0.298727	−0.0110715
$729$	−26.4078	−0.978067
$730$	0	0
$731$	18.8516	0.697253
$732$	0.252651	0.00933826
$733$	48.0618	1.77520	0.887601	−	0.460614i	$-0.152371\pi$
$733$	48.0618	1.77520	0.887601	+	0.460614i	$0.152371\pi$
$734$	−37.8691	−1.39777
$735$	0	0
$736$	−4.72859	−0.174298
$737$	5.79923	0.213617
$738$	−47.9432	−1.76481
$739$	39.7257	1.46133	0.730667	−	0.682734i	$-0.239209\pi$
$739$	39.7257	1.46133	0.730667	+	0.682734i	$0.239209\pi$
$740$	0	0
$741$	0.257290	0.00945177
$742$	−26.8851	−0.986983
$743$	−35.6645	−1.30840	−0.654202	−	0.756320i	$-0.726995\pi$
$743$	−35.6645	−1.30840	−0.654202	+	0.756320i	$0.726995\pi$
$744$	0.0976001	0.00357819
$745$	0	0
$746$	10.4069	0.381022
$747$	−47.6508	−1.74345
$748$	15.7624	0.576331
$749$	−0.123813	−0.00452402
$750$	0	0
$751$	−35.0581	−1.27929	−0.639645	−	0.768671i	$-0.720918\pi$
$751$	−35.0581	−1.27929	−0.639645	+	0.768671i	$0.720918\pi$
$752$	35.7439	1.30345
$753$	2.10378	0.0766658
$754$	−30.2544	−1.10180
$755$	0	0
$756$	1.52593	0.0554976
$757$	42.8016	1.55565	0.777826	−	0.628480i	$-0.216323\pi$
$757$	42.8016	1.55565	0.777826	+	0.628480i	$0.216323\pi$
$758$	42.2827	1.53578
$759$	−0.0624153	−0.00226553
$760$	0	0
$761$	−12.6482	−0.458497	−0.229249	−	0.973368i	$-0.573627\pi$
$761$	−12.6482	−0.458497	−0.229249	+	0.973368i	$0.573627\pi$
$762$	1.86515	0.0675673
$763$	21.0848	0.763322
$764$	15.1300	0.547384
$765$	0	0
$766$	34.0946	1.23189
$767$	−1.47609	−0.0532986
$768$	−1.75759	−0.0634217
$769$	−9.67462	−0.348876	−0.174438	−	0.984668i	$-0.555811\pi$
$769$	−9.67462	−0.348876	−0.174438	+	0.984668i	$0.555811\pi$
$770$	0	0
$771$	0.905282	0.0326029
$772$	−35.2838	−1.26989
$773$	−31.1687	−1.12106	−0.560531	−	0.828133i	$-0.689403\pi$
$773$	−31.1687	−1.12106	−0.560531	+	0.828133i	$0.689403\pi$
$774$	−13.8614	−0.498239
$775$	0	0
$776$	0.853236	0.0306294
$777$	−0.848078	−0.0304246
$778$	−32.4625	−1.16384
$779$	−8.06971	−0.289128
$780$	0	0
$781$	−7.74139	−0.277009
$782$	−9.55748	−0.341775
$783$	−3.89742	−0.139282
$784$	22.3201	0.797145
$785$	0	0
$786$	−3.45372	−0.123190
$787$	44.4253	1.58359	0.791795	−	0.610787i	$-0.209147\pi$
$787$	44.4253	1.58359	0.791795	+	0.610787i	$0.209147\pi$
$788$	36.3130	1.29360
$789$	−0.505148	−0.0179837
$790$	0	0
$791$	4.41338	0.156922
$792$	0.292287	0.0103860
$793$	3.02923	0.107571
$794$	−70.5742	−2.50458
$795$	0	0
$796$	−12.4775	−0.442253
$797$	20.1652	0.714288	0.357144	−	0.934049i	$-0.383750\pi$
$797$	20.1652	0.714288	0.357144	+	0.934049i	$0.383750\pi$
$798$	0.259603	0.00918986
$799$	70.5106	2.49448
$800$	0	0
$801$	21.6961	0.766592
$802$	−61.5215	−2.17240
$803$	−13.3744	−0.471974
$804$	1.18654	0.0418461
$805$	0	0
$806$	−46.4016	−1.63443
$807$	−0.347088	−0.0122181
$808$	−0.566846	−0.0199416
$809$	−32.6594	−1.14824	−0.574121	−	0.818770i	$-0.694656\pi$
$809$	−32.6594	−1.14824	−0.574121	+	0.818770i	$0.694656\pi$
$810$	0	0
$811$	45.7373	1.60606	0.803028	−	0.595942i	$-0.203221\pi$
$811$	45.7373	1.60606	0.803028	+	0.595942i	$0.203221\pi$
$812$	−15.0732	−0.528964
$813$	−0.0731870	−0.00256678
$814$	12.9066	0.452375
$815$	0	0
$816$	−3.47109	−0.121512
$817$	−2.33313	−0.0816260
$818$	7.10569	0.248445
$819$	9.13096	0.319062
$820$	0	0
$821$	13.5002	0.471159	0.235580	−	0.971855i	$-0.424301\pi$
$821$	13.5002	0.471159	0.235580	+	0.971855i	$0.424301\pi$
$822$	−0.533533	−0.0186091
$823$	45.8337	1.59766	0.798831	−	0.601556i	$-0.205452\pi$
$823$	45.8337	1.59766	0.798831	+	0.601556i	$0.205452\pi$
$824$	0.768079	0.0267573
$825$	0	0
$826$	−1.48936	−0.0518216
$827$	−48.5796	−1.68928	−0.844639	−	0.535337i	$-0.820185\pi$
$827$	−48.5796	−1.68928	−0.844639	+	0.535337i	$0.820185\pi$
$828$	3.47001	0.120591
$829$	−23.5476	−0.817842	−0.408921	−	0.912570i	$-0.634095\pi$
$829$	−23.5476	−0.817842	−0.408921	+	0.912570i	$0.634095\pi$
$830$	0	0
$831$	0.509364	0.0176697
$832$	18.6481	0.646505
$833$	44.0299	1.52555
$834$	3.69703	0.128018
$835$	0	0
$836$	−1.95080	−0.0674699
$837$	−5.97752	−0.206613
$838$	−43.3695	−1.49817
$839$	−30.7980	−1.06327	−0.531633	−	0.846975i	$-0.678421\pi$
$839$	−30.7980	−1.06327	−0.531633	+	0.846975i	$0.678421\pi$
$840$	0	0
$841$	9.49873	0.327543
$842$	−71.8120	−2.47480
$843$	−1.26778	−0.0436648
$844$	−35.6600	−1.22747
$845$	0	0
$846$	−51.8458	−1.78250
$847$	1.24528	0.0427884
$848$	−44.4896	−1.52778
$849$	−1.70471	−0.0585056
$850$	0	0
$851$	−3.86420	−0.132463
$852$	−1.58391	−0.0542640
$853$	−8.62823	−0.295425	−0.147712	−	0.989030i	$-0.547191\pi$
$853$	−8.62823	−0.295425	−0.147712	+	0.989030i	$0.547191\pi$
$854$	3.05646	0.104590
$855$	0	0
$856$	−0.00972257	−0.000332311 0
$857$	−47.7829	−1.63223	−0.816117	−	0.577887i	$-0.803877\pi$
$857$	−47.7829	−1.63223	−0.816117	+	0.577887i	$0.803877\pi$
$858$	0.511405	0.0174591
$859$	−38.9923	−1.33040	−0.665200	−	0.746665i	$-0.731654\pi$
$859$	−38.9923	−1.33040	−0.665200	+	0.746665i	$0.731654\pi$
$860$	0	0
$861$	1.05396	0.0359190
$862$	51.6926	1.76066
$863$	−34.7056	−1.18139	−0.590697	−	0.806894i	$-0.701147\pi$
$863$	−34.7056	−1.18139	−0.590697	+	0.806894i	$0.701147\pi$
$864$	4.99107	0.169800
$865$	0	0
$866$	64.0244	2.17564
$867$	−5.06429	−0.171992
$868$	−23.1179	−0.784673
$869$	−1.79880	−0.0610201
$870$	0	0
$871$	14.2263	0.482041
$872$	1.65571	0.0560696
$873$	−26.0803	−0.882683
$874$	1.18286	0.0400109
$875$	0	0
$876$	−2.73646	−0.0924563
$877$	−14.3162	−0.483422	−0.241711	−	0.970348i	$-0.577709\pi$
$877$	−14.3162	−0.483422	−0.241711	+	0.970348i	$0.577709\pi$
$878$	71.8936	2.42629
$879$	−1.14263	−0.0385400
$880$	0	0
$881$	0.516699	0.0174080	0.00870402	−	0.999962i	$-0.497229\pi$
$881$	0.516699	0.0174080	0.00870402	+	0.999962i	$0.497229\pi$
$882$	−32.3748	−1.09012
$883$	51.8049	1.74337	0.871687	−	0.490063i	$-0.163026\pi$
$883$	51.8049	1.74337	0.871687	+	0.490063i	$0.163026\pi$
$884$	38.6674	1.30053
$885$	0	0
$886$	12.6414	0.424695
$887$	14.6407	0.491585	0.245793	−	0.969322i	$-0.420952\pi$
$887$	14.6407	0.491585	0.245793	+	0.969322i	$0.420952\pi$
$888$	−0.0665964	−0.00223483
$889$	11.1414	0.373670
$890$	0	0
$891$	−8.90112	−0.298199
$892$	−21.4238	−0.717321
$893$	−8.72660	−0.292025
$894$	−1.95563	−0.0654060
$895$	0	0
$896$	−0.973890	−0.0325354
$897$	−0.153114	−0.00511231
$898$	−57.7518	−1.92720
$899$	59.0460	1.96930
$900$	0	0
$901$	−87.7627	−2.92380
$902$	−16.0399	−0.534069
$903$	0.304724	0.0101406
$904$	0.346566	0.0115266
$905$	0	0
$906$	−2.66830	−0.0886484
$907$	13.4655	0.447114	0.223557	−	0.974691i	$-0.428233\pi$
$907$	13.4655	0.447114	0.223557	+	0.974691i	$0.428233\pi$
$908$	−32.5394	−1.07986
$909$	17.3264	0.574679
$910$	0	0
$911$	−3.43920	−0.113946	−0.0569729	−	0.998376i	$-0.518145\pi$
$911$	−3.43920	−0.113946	−0.0569729	+	0.998376i	$0.518145\pi$
$912$	0.429593	0.0142252
$913$	−15.9421	−0.527605
$914$	−36.2805	−1.20005
$915$	0	0
$916$	39.4362	1.30301
$917$	−20.6307	−0.681285
$918$	10.0880	0.332954
$919$	46.3892	1.53024	0.765120	−	0.643888i	$-0.222680\pi$
$919$	46.3892	1.53024	0.765120	+	0.643888i	$0.222680\pi$
$920$	0	0
$921$	0.117136	0.00385976
$922$	−45.1134	−1.48573
$923$	−18.9907	−0.625087
$924$	0.254789	0.00838195
$925$	0	0
$926$	−50.4205	−1.65692
$927$	−23.4773	−0.771097
$928$	−49.3018	−1.61841
$929$	−29.7987	−0.977665	−0.488832	−	0.872378i	$-0.662577\pi$
$929$	−29.7987	−0.977665	−0.488832	+	0.872378i	$0.662577\pi$
$930$	0	0
$931$	−5.44927	−0.178593
$932$	−24.1636	−0.791507
$933$	3.27645	0.107266
$934$	69.8945	2.28702
$935$	0	0
$936$	0.717021	0.0234366
$937$	−43.2661	−1.41344	−0.706720	−	0.707493i	$-0.749826\pi$
$937$	−43.2661	−1.41344	−0.706720	+	0.707493i	$0.749826\pi$
$938$	14.3543	0.468683
$939$	1.67209	0.0545667
$940$	0	0
$941$	25.8405	0.842377	0.421189	−	0.906973i	$-0.361613\pi$
$941$	25.8405	0.842377	0.421189	+	0.906973i	$0.361613\pi$
$942$	−4.20822	−0.137111
$943$	4.80230	0.156385
$944$	−2.46461	−0.0802161
$945$	0	0
$946$	−4.63748	−0.150778
$947$	42.8107	1.39116	0.695580	−	0.718449i	$-0.255147\pi$
$947$	42.8107	1.39116	0.695580	+	0.718449i	$0.255147\pi$
$948$	−0.368040	−0.0119534
$949$	−32.8094	−1.06504
$950$	0	0
$951$	1.07664	0.0349126
$952$	−0.983921	−0.0318891
$953$	35.0067	1.13398	0.566990	−	0.823725i	$-0.308108\pi$
$953$	35.0067	1.13398	0.566990	+	0.823725i	$0.308108\pi$
$954$	64.5312	2.08927
$955$	0	0
$956$	−10.0269	−0.324293
$957$	−0.650763	−0.0210362
$958$	38.5662	1.24602
$959$	−3.18704	−0.102915
$960$	0	0
$961$	59.5597	1.92128
$962$	31.6617	1.02081
$963$	0.297183	0.00957658
$964$	39.6234	1.27618
$965$	0	0
$966$	−0.154490	−0.00497065
$967$	−2.63095	−0.0846057	−0.0423028	−	0.999105i	$-0.513469\pi$
$967$	−2.63095	−0.0846057	−0.0423028	+	0.999105i	$0.513469\pi$
$968$	0.0977875	0.00314301
$969$	0.847439	0.0272237
$970$	0	0
$971$	26.6425	0.854999	0.427499	−	0.904016i	$-0.359395\pi$
$971$	26.6425	0.854999	0.427499	+	0.904016i	$0.359395\pi$
$972$	−5.49731	−0.176326
$973$	22.0840	0.707982
$974$	−51.0743	−1.63653
$975$	0	0
$976$	5.05785	0.161898
$977$	−33.7219	−1.07886	−0.539429	−	0.842031i	$-0.681360\pi$
$977$	−33.7219	−1.07886	−0.539429	+	0.842031i	$0.681360\pi$
$978$	3.58442	0.114617
$979$	7.25863	0.231987
$980$	0	0
$981$	−50.6090	−1.61582
$982$	9.46802	0.302137
$983$	−6.93604	−0.221225	−0.110613	−	0.993864i	$-0.535281\pi$
$983$	−6.93604	−0.221225	−0.110613	+	0.993864i	$0.535281\pi$
$984$	0.0827639	0.00263842
$985$	0	0
$986$	−99.6495	−3.17349
$987$	1.13976	0.0362789
$988$	−4.78560	−0.152250
$989$	1.38845	0.0441502
$990$	0	0
$991$	15.8403	0.503183	0.251591	−	0.967834i	$-0.419046\pi$
$991$	15.8403	0.503183	0.251591	+	0.967834i	$0.419046\pi$
$992$	−75.6149	−2.40077
$993$	−2.32884	−0.0739036
$994$	−19.1615	−0.607766
$995$	0	0
$996$	−3.26180	−0.103354
$997$	−53.3663	−1.69013	−0.845064	−	0.534665i	$-0.820438\pi$
$997$	−53.3663	−1.69013	−0.845064	+	0.534665i	$0.820438\pi$
$998$	−53.3792	−1.68969
$999$	4.07870	0.129044

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5225.2.a.u.1.3	✓	15
5.4	even	2		5225.2.a.v.1.13	yes	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5225.2.a.u.1.3	✓	15	1.1	even	1	trivial
5225.2.a.v.1.13	yes	15	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.98766 q^{2} -0.104882 q^{3} +1.95080 q^{4} +0.208469 q^{6} +1.24528 q^{7} +0.0977875 q^{8} -2.98900 q^{9} -1.00000 q^{11} -0.204603 q^{12} -2.45314 q^{13} -2.47520 q^{14} -4.09597 q^{16} -8.07996 q^{17} +5.94112 q^{18} +1.00000 q^{19} -0.130607 q^{21} +1.98766 q^{22} -0.595102 q^{23} -0.0102561 q^{24} +4.87602 q^{26} +0.628136 q^{27} +2.42930 q^{28} -6.20473 q^{29} -9.51628 q^{31} +7.94584 q^{32} +0.104882 q^{33} +16.0602 q^{34} -5.83095 q^{36} +6.49334 q^{37} -1.98766 q^{38} +0.257290 q^{39} -8.06971 q^{41} +0.259603 q^{42} -2.33313 q^{43} -1.95080 q^{44} +1.18286 q^{46} -8.72660 q^{47} +0.429593 q^{48} -5.44927 q^{49} +0.847439 q^{51} -4.78560 q^{52} +10.8618 q^{53} -1.24852 q^{54} +0.121773 q^{56} -0.104882 q^{57} +12.3329 q^{58} +0.601714 q^{59} -1.23483 q^{61} +18.9152 q^{62} -3.72215 q^{63} -7.60170 q^{64} -0.208469 q^{66} -5.79923 q^{67} -15.7624 q^{68} +0.0624153 q^{69} +7.74139 q^{71} -0.292287 q^{72} +13.3744 q^{73} -12.9066 q^{74} +1.95080 q^{76} -1.24528 q^{77} -0.511405 q^{78} +1.79880 q^{79} +8.90112 q^{81} +16.0399 q^{82} +15.9421 q^{83} -0.254789 q^{84} +4.63748 q^{86} +0.650763 q^{87} -0.0977875 q^{88} -7.25863 q^{89} -3.05486 q^{91} -1.16093 q^{92} +0.998084 q^{93} +17.3455 q^{94} -0.833373 q^{96} +8.72541 q^{97} +10.8313 q^{98} +2.98900 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - q^{2} + 4 q^{3} + 13 q^{4} - q^{6} + 17 q^{7} + 3 q^{8} + 15 q^{9} - 15 q^{11} + 9 q^{12} + 3 q^{13} - 11 q^{14} + 13 q^{16} + q^{17} - 10 q^{18} + 15 q^{19} + 16 q^{21} + q^{22} + 18 q^{23}+ \cdots - 15 q^{99}+O(q^{100}) \) 15 * q - q^2 + 4 * q^3 + 13 * q^4 - q^6 + 17 * q^7 + 3 * q^8 + 15 * q^9 - 15 * q^11 + 9 * q^12 + 3 * q^13 - 11 * q^14 + 13 * q^16 + q^17 - 10 * q^18 + 15 * q^19 + 16 * q^21 + q^22 + 18 * q^23 - 27 * q^24 + 15 * q^26 + 31 * q^27 + 34 * q^28 + 11 * q^29 - 2 * q^31 + 10 * q^32 - 4 * q^33 - 13 * q^34 + 8 * q^36 + 27 * q^37 - q^38 + 4 * q^39 + 6 * q^41 + 2 * q^42 + 38 * q^43 - 13 * q^44 - 9 * q^46 + 14 * q^47 + 32 * q^48 + 28 * q^49 - 32 * q^51 + 16 * q^52 + 11 * q^53 - 11 * q^54 + 2 * q^56 + 4 * q^57 - 6 * q^58 + 13 * q^59 + 8 * q^61 + 7 * q^62 + 49 * q^63 + 9 * q^64 + q^66 + 31 * q^67 - 26 * q^68 + 39 * q^69 - 3 * q^71 - 18 * q^72 + 18 * q^73 + 7 * q^74 + 13 * q^76 - 17 * q^77 + 18 * q^78 + 10 * q^79 + 31 * q^81 + 58 * q^82 + 16 * q^83 + 112 * q^84 - 63 * q^86 + 67 * q^87 - 3 * q^88 - 7 * q^89 - 50 * q^91 + 98 * q^92 + 26 * q^93 + 22 * q^94 - 37 * q^96 + 24 * q^97 - 46 * q^98 - 15 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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