LMFDB - Embedded newform 5225.2.a.s.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5225,2,Mod(1,5225)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

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")

//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);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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} - 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			1.5
Root			$1.38376$ 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)

$f(q)$	$=$	\(q-1.38376 q^{2} +0.179641 q^{3} -0.0852182 q^{4} -0.248579 q^{6} -4.32482 q^{7} +2.88543 q^{8} -2.96773 q^{9} +1.00000 q^{11} -0.0153087 q^{12} -0.663216 q^{13} +5.98449 q^{14} -3.82230 q^{16} +5.90319 q^{17} +4.10661 q^{18} -1.00000 q^{19} -0.776914 q^{21} -1.38376 q^{22} -0.697968 q^{23} +0.518342 q^{24} +0.917729 q^{26} -1.07205 q^{27} +0.368553 q^{28} -3.32909 q^{29} +10.4922 q^{31} -0.481734 q^{32} +0.179641 q^{33} -8.16858 q^{34} +0.252904 q^{36} -10.8094 q^{37} +1.38376 q^{38} -0.119141 q^{39} +6.31607 q^{41} +1.07506 q^{42} -2.76329 q^{43} -0.0852182 q^{44} +0.965818 q^{46} -10.4716 q^{47} -0.686641 q^{48} +11.7040 q^{49} +1.06045 q^{51} +0.0565180 q^{52} +8.10587 q^{53} +1.48345 q^{54} -12.4790 q^{56} -0.179641 q^{57} +4.60665 q^{58} -0.264035 q^{59} +14.2963 q^{61} -14.5186 q^{62} +12.8349 q^{63} +8.31121 q^{64} -0.248579 q^{66} +14.6511 q^{67} -0.503059 q^{68} -0.125384 q^{69} -9.64601 q^{71} -8.56319 q^{72} +0.803845 q^{73} +14.9576 q^{74} +0.0852182 q^{76} -4.32482 q^{77} +0.164862 q^{78} +8.25623 q^{79} +8.71060 q^{81} -8.73990 q^{82} -1.39198 q^{83} +0.0662072 q^{84} +3.82372 q^{86} -0.598040 q^{87} +2.88543 q^{88} +9.69068 q^{89} +2.86829 q^{91} +0.0594796 q^{92} +1.88482 q^{93} +14.4901 q^{94} -0.0865391 q^{96} -6.48020 q^{97} -16.1955 q^{98} -2.96773 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.38376	−0.978464	−0.489232	−	0.872154i	$-0.662723\pi$
$2$	−1.38376	−0.978464	−0.489232	+	0.872154i	$0.662723\pi$
$3$	0.179641	0.103716	0.0518578	−	0.998654i	$-0.483486\pi$
$3$	0.179641	0.103716	0.0518578	+	0.998654i	$0.483486\pi$
$4$	−0.0852182	−0.0426091
$5$	0	0
$5$	0	0
$6$	−0.248579	−0.101482
$7$	−4.32482	−1.63463	−0.817313	−	0.576193i	$-0.804538\pi$
$7$	−4.32482	−1.63463	−0.817313	+	0.576193i	$0.804538\pi$
$8$	2.88543	1.02015
$9$	−2.96773	−0.989243
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−0.0153087	−0.00441923
$13$	−0.663216	−0.183943	−0.0919715	−	0.995762i	$-0.529317\pi$
$13$	−0.663216	−0.183943	−0.0919715	+	0.995762i	$0.529317\pi$
$14$	5.98449	1.59942
$15$	0	0
$16$	−3.82230	−0.955575
$17$	5.90319	1.43173	0.715867	−	0.698237i	$-0.246032\pi$
$17$	5.90319	1.43173	0.715867	+	0.698237i	$0.246032\pi$
$18$	4.10661	0.967938
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−0.776914	−0.169536
$22$	−1.38376	−0.295018
$23$	−0.697968	−0.145536	−0.0727682	−	0.997349i	$-0.523183\pi$
$23$	−0.697968	−0.145536	−0.0727682	+	0.997349i	$0.523183\pi$
$24$	0.518342	0.105806
$25$	0	0
$26$	0.917729	0.179982
$27$	−1.07205	−0.206316
$28$	0.368553	0.0696500
$29$	−3.32909	−0.618196	−0.309098	−	0.951030i	$-0.600027\pi$
$29$	−3.32909	−0.618196	−0.309098	+	0.951030i	$0.600027\pi$
$30$	0	0
$31$	10.4922	1.88445	0.942224	−	0.334984i	$-0.108731\pi$
$31$	10.4922	1.88445	0.942224	+	0.334984i	$0.108731\pi$
$32$	−0.481734	−0.0851593
$33$	0.179641	0.0312715
$34$	−8.16858	−1.40090
$35$	0	0
$36$	0.252904	0.0421507
$37$	−10.8094	−1.77706	−0.888530	−	0.458819i	$-0.848273\pi$
$37$	−10.8094	−1.77706	−0.888530	+	0.458819i	$0.848273\pi$
$38$	1.38376	0.224475
$39$	−0.119141	−0.0190778
$40$	0	0
$41$	6.31607	0.986405	0.493202	−	0.869915i	$-0.335826\pi$
$41$	6.31607	0.986405	0.493202	+	0.869915i	$0.335826\pi$
$42$	1.07506	0.165885
$43$	−2.76329	−0.421397	−0.210699	−	0.977551i	$-0.567574\pi$
$43$	−2.76329	−0.421397	−0.210699	+	0.977551i	$0.567574\pi$
$44$	−0.0852182	−0.0128471
$45$	0	0
$46$	0.965818	0.142402
$47$	−10.4716	−1.52744	−0.763720	−	0.645548i	$-0.776629\pi$
$47$	−10.4716	−1.52744	−0.763720	+	0.645548i	$0.776629\pi$
$48$	−0.686641	−0.0991082
$49$	11.7040	1.67201
$50$	0	0
$51$	1.06045	0.148493
$52$	0.0565180	0.00783764
$53$	8.10587	1.11343	0.556713	−	0.830705i	$-0.312062\pi$
$53$	8.10587	1.11343	0.556713	+	0.830705i	$0.312062\pi$
$54$	1.48345	0.201872
$55$	0	0
$56$	−12.4790	−1.66757
$57$	−0.179641	−0.0237940
$58$	4.60665	0.604883
$59$	−0.264035	−0.0343744	−0.0171872	−	0.999852i	$-0.505471\pi$
$59$	−0.264035	−0.0343744	−0.0171872	+	0.999852i	$0.505471\pi$
$60$	0	0
$61$	14.2963	1.83045	0.915227	−	0.402939i	$-0.132011\pi$
$61$	14.2963	1.83045	0.915227	+	0.402939i	$0.132011\pi$
$62$	−14.5186	−1.84386
$63$	12.8349	1.61704
$64$	8.31121	1.03890
$65$	0	0
$66$	−0.248579	−0.0305980
$67$	14.6511	1.78992	0.894959	−	0.446149i	$-0.147205\pi$
$67$	14.6511	1.78992	0.894959	+	0.446149i	$0.147205\pi$
$68$	−0.503059	−0.0610049
$69$	−0.125384	−0.0150944
$70$	0	0
$71$	−9.64601	−1.14477	−0.572385	−	0.819985i	$-0.693982\pi$
$71$	−9.64601	−1.14477	−0.572385	+	0.819985i	$0.693982\pi$
$72$	−8.56319	−1.00918
$73$	0.803845	0.0940830	0.0470415	−	0.998893i	$-0.485021\pi$
$73$	0.803845	0.0940830	0.0470415	+	0.998893i	$0.485021\pi$
$74$	14.9576	1.73879
$75$	0	0
$76$	0.0852182	0.00977520
$77$	−4.32482	−0.492859
$78$	0.164862	0.0186669
$79$	8.25623	0.928899	0.464449	−	0.885600i	$-0.346252\pi$
$79$	8.25623	0.928899	0.464449	+	0.885600i	$0.346252\pi$
$80$	0	0
$81$	8.71060	0.967845
$82$	−8.73990	−0.965161
$83$	−1.39198	−0.152790	−0.0763951	−	0.997078i	$-0.524341\pi$
$83$	−1.39198	−0.152790	−0.0763951	+	0.997078i	$0.524341\pi$
$84$	0.0662072	0.00722379
$85$	0	0
$86$	3.82372	0.412322
$87$	−0.598040	−0.0641167
$88$	2.88543	0.307588
$89$	9.69068	1.02721	0.513605	−	0.858027i	$-0.328310\pi$
$89$	9.69068	1.02721	0.513605	+	0.858027i	$0.328310\pi$
$90$	0	0
$91$	2.86829	0.300678
$92$	0.0594796	0.00620117
$93$	1.88482	0.195447
$94$	14.4901	1.49454
$95$	0	0
$96$	−0.0865391	−0.00883236
$97$	−6.48020	−0.657964	−0.328982	−	0.944336i	$-0.606706\pi$
$97$	−6.48020	−0.657964	−0.328982	+	0.944336i	$0.606706\pi$
$98$	−16.1955	−1.63600
$99$	−2.96773	−0.298268
$100$	0	0
$101$	−8.25408	−0.821312	−0.410656	−	0.911790i	$-0.634700\pi$
$101$	−8.25408	−0.821312	−0.410656	+	0.911790i	$0.634700\pi$
$102$	−1.46741	−0.145295
$103$	−11.2648	−1.10996	−0.554979	−	0.831864i	$-0.687274\pi$
$103$	−11.2648	−1.10996	−0.554979	+	0.831864i	$0.687274\pi$
$104$	−1.91367	−0.187650
$105$	0	0
$106$	−11.2165	−1.08945
$107$	−11.9862	−1.15875	−0.579377	−	0.815060i	$-0.696704\pi$
$107$	−11.9862	−1.15875	−0.579377	+	0.815060i	$0.696704\pi$
$108$	0.0913580	0.00879092
$109$	0.188775	0.0180813	0.00904067	−	0.999959i	$-0.497122\pi$
$109$	0.188775	0.0180813	0.00904067	+	0.999959i	$0.497122\pi$
$110$	0	0
$111$	−1.94181	−0.184309
$112$	16.5308	1.56201
$113$	12.2684	1.15411	0.577056	−	0.816704i	$-0.304201\pi$
$113$	12.2684	1.15411	0.577056	+	0.816704i	$0.304201\pi$
$114$	0.248579	0.0232816
$115$	0	0
$116$	0.283699	0.0263408
$117$	1.96825	0.181964
$118$	0.365360	0.0336341
$119$	−25.5302	−2.34035
$120$	0	0
$121$	1.00000	0.0909091
$122$	−19.7826	−1.79103
$123$	1.13462	0.102306
$124$	−0.894123	−0.0802946
$125$	0	0
$126$	−17.7604	−1.58222
$127$	7.42133	0.658537	0.329268	−	0.944236i	$-0.393198\pi$
$127$	7.42133	0.658537	0.329268	+	0.944236i	$0.393198\pi$
$128$	−10.5372	−0.931367
$129$	−0.496399	−0.0437055
$130$	0	0
$131$	−1.15539	−0.100947	−0.0504734	−	0.998725i	$-0.516073\pi$
$131$	−1.15539	−0.100947	−0.0504734	+	0.998725i	$0.516073\pi$
$132$	−0.0153087	−0.00133245
$133$	4.32482	0.375009
$134$	−20.2736	−1.75137
$135$	0	0
$136$	17.0333	1.46059
$137$	14.7417	1.25947	0.629735	−	0.776810i	$-0.283163\pi$
$137$	14.7417	1.25947	0.629735	+	0.776810i	$0.283163\pi$
$138$	0.173500	0.0147693
$139$	6.38501	0.541570	0.270785	−	0.962640i	$-0.412717\pi$
$139$	6.38501	0.541570	0.270785	+	0.962640i	$0.412717\pi$
$140$	0	0
$141$	−1.88113	−0.158419
$142$	13.3477	1.12012
$143$	−0.663216	−0.0554609
$144$	11.3436	0.945296
$145$	0	0
$146$	−1.11233	−0.0920568
$147$	2.10252	0.173413
$148$	0.921160	0.0757189
$149$	−7.66886	−0.628257	−0.314129	−	0.949380i	$-0.601712\pi$
$149$	−7.66886	−0.628257	−0.314129	+	0.949380i	$0.601712\pi$
$150$	0	0
$151$	7.24054	0.589227	0.294613	−	0.955617i	$-0.404809\pi$
$151$	7.24054	0.589227	0.294613	+	0.955617i	$0.404809\pi$
$152$	−2.88543	−0.234040
$153$	−17.5191	−1.41633
$154$	5.98449	0.482244
$155$	0	0
$156$	0.0101529	0.000812887 0
$157$	−6.79146	−0.542018	−0.271009	−	0.962577i	$-0.587357\pi$
$157$	−6.79146	−0.542018	−0.271009	+	0.962577i	$0.587357\pi$
$158$	−11.4246	−0.908894
$159$	1.45614	0.115480
$160$	0	0
$161$	3.01858	0.237898
$162$	−12.0534	−0.947001
$163$	−11.6881	−0.915481	−0.457741	−	0.889086i	$-0.651341\pi$
$163$	−11.6881	−0.915481	−0.457741	+	0.889086i	$0.651341\pi$
$164$	−0.538244	−0.0420298
$165$	0	0
$166$	1.92617	0.149500
$167$	−4.48188	−0.346818	−0.173409	−	0.984850i	$-0.555478\pi$
$167$	−4.48188	−0.346818	−0.173409	+	0.984850i	$0.555478\pi$
$168$	−2.24173	−0.172953
$169$	−12.5601	−0.966165
$170$	0	0
$171$	2.96773	0.226948
$172$	0.235482	0.0179554
$173$	−18.6634	−1.41895	−0.709475	−	0.704731i	$-0.751068\pi$
$173$	−18.6634	−1.41895	−0.709475	+	0.704731i	$0.751068\pi$
$174$	0.827542	0.0627358
$175$	0	0
$176$	−3.82230	−0.288117
$177$	−0.0474315	−0.00356517
$178$	−13.4095	−1.00509
$179$	3.78670	0.283031	0.141516	−	0.989936i	$-0.454802\pi$
$179$	3.78670	0.283031	0.141516	+	0.989936i	$0.454802\pi$
$180$	0	0
$181$	7.77095	0.577610	0.288805	−	0.957388i	$-0.406742\pi$
$181$	7.77095	0.577610	0.288805	+	0.957388i	$0.406742\pi$
$182$	−3.96901	−0.294203
$183$	2.56820	0.189847
$184$	−2.01394	−0.148470
$185$	0	0
$186$	−2.60813	−0.191238
$187$	5.90319	0.431684
$188$	0.892370	0.0650828
$189$	4.63641	0.337249
$190$	0	0
$191$	−19.2865	−1.39552	−0.697760	−	0.716331i	$-0.745820\pi$
$191$	−19.2865	−1.39552	−0.697760	+	0.716331i	$0.745820\pi$
$192$	1.49303	0.107750
$193$	−7.76965	−0.559272	−0.279636	−	0.960106i	$-0.590214\pi$
$193$	−7.76965	−0.559272	−0.279636	+	0.960106i	$0.590214\pi$
$194$	8.96701	0.643794
$195$	0	0
$196$	−0.997397	−0.0712426
$197$	−19.8618	−1.41510	−0.707548	−	0.706665i	$-0.750199\pi$
$197$	−19.8618	−1.41510	−0.707548	+	0.706665i	$0.750199\pi$
$198$	4.10661	0.291844
$199$	3.55464	0.251982	0.125991	−	0.992031i	$-0.459789\pi$
$199$	3.55464	0.251982	0.125991	+	0.992031i	$0.459789\pi$
$200$	0	0
$201$	2.63194	0.185642
$202$	11.4216	0.803624
$203$	14.3977	1.01052
$204$	−0.0903700	−0.00632716
$205$	0	0
$206$	15.5878	1.08605
$207$	2.07138	0.143971
$208$	2.53501	0.175771
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	−20.0441	−1.37990	−0.689948	−	0.723859i	$-0.742367\pi$
$211$	−20.0441	−1.37990	−0.689948	+	0.723859i	$0.742367\pi$
$212$	−0.690767	−0.0474421
$213$	−1.73282	−0.118731
$214$	16.5860	1.13380
$215$	0	0
$216$	−3.09332	−0.210474
$217$	−45.3767	−3.08037
$218$	−0.261218	−0.0176919
$219$	0.144403	0.00975788
$220$	0	0
$221$	−3.91509	−0.263357
$222$	2.68700	0.180340
$223$	−6.29774	−0.421728	−0.210864	−	0.977515i	$-0.567628\pi$
$223$	−6.29774	−0.421728	−0.210864	+	0.977515i	$0.567628\pi$
$224$	2.08341	0.139204
$225$	0	0
$226$	−16.9765	−1.12926
$227$	23.7789	1.57826	0.789131	−	0.614225i	$-0.210531\pi$
$227$	23.7789	1.57826	0.789131	+	0.614225i	$0.210531\pi$
$228$	0.0153087	0.00101384
$229$	−4.01952	−0.265618	−0.132809	−	0.991142i	$-0.542400\pi$
$229$	−4.01952	−0.265618	−0.132809	+	0.991142i	$0.542400\pi$
$230$	0	0
$231$	−0.776914	−0.0511172
$232$	−9.60587	−0.630656
$233$	−26.6054	−1.74298	−0.871488	−	0.490416i	$-0.836845\pi$
$233$	−26.6054	−1.74298	−0.871488	+	0.490416i	$0.836845\pi$
$234$	−2.72357	−0.178045
$235$	0	0
$236$	0.0225006	0.00146466
$237$	1.48316	0.0963414
$238$	35.3276	2.28995
$239$	24.8559	1.60780	0.803899	−	0.594766i	$-0.202755\pi$
$239$	24.8559	1.60780	0.803899	+	0.594766i	$0.202755\pi$
$240$	0	0
$241$	−1.33695	−0.0861208	−0.0430604	−	0.999072i	$-0.513711\pi$
$241$	−1.33695	−0.0861208	−0.0430604	+	0.999072i	$0.513711\pi$
$242$	−1.38376	−0.0889512
$243$	4.78092	0.306696
$244$	−1.21830	−0.0779940
$245$	0	0
$246$	−1.57004	−0.100102
$247$	0.663216	0.0421994
$248$	30.2744	1.92243
$249$	−0.250057	−0.0158467
$250$	0	0
$251$	6.30211	0.397786	0.198893	−	0.980021i	$-0.436265\pi$
$251$	6.30211	0.397786	0.198893	+	0.980021i	$0.436265\pi$
$252$	−1.09377	−0.0689007
$253$	−0.697968	−0.0438809
$254$	−10.2693	−0.644354
$255$	0	0
$256$	−2.04147	−0.127592
$257$	−10.4318	−0.650718	−0.325359	−	0.945591i	$-0.605485\pi$
$257$	−10.4318	−0.650718	−0.325359	+	0.945591i	$0.605485\pi$
$258$	0.686896	0.0427643
$259$	46.7488	2.90483
$260$	0	0
$261$	9.87984	0.611547
$262$	1.59878	0.0987728
$263$	−21.5818	−1.33079	−0.665396	−	0.746490i	$-0.731737\pi$
$263$	−21.5818	−1.33079	−0.665396	+	0.746490i	$0.731737\pi$
$264$	0.518342	0.0319017
$265$	0	0
$266$	−5.98449	−0.366933
$267$	1.74084	0.106538
$268$	−1.24854	−0.0762667
$269$	17.5582	1.07054	0.535272	−	0.844680i	$-0.320209\pi$
$269$	17.5582	1.07054	0.535272	+	0.844680i	$0.320209\pi$
$270$	0	0
$271$	−17.3550	−1.05424	−0.527122	−	0.849790i	$-0.676729\pi$
$271$	−17.3550	−1.05424	−0.527122	+	0.849790i	$0.676729\pi$
$272$	−22.5638	−1.36813
$273$	0.515261	0.0311850
$274$	−20.3989	−1.23235
$275$	0	0
$276$	0.0106850	0.000643159 0
$277$	−19.6803	−1.18247	−0.591237	−	0.806498i	$-0.701360\pi$
$277$	−19.6803	−1.18247	−0.591237	+	0.806498i	$0.701360\pi$
$278$	−8.83530	−0.529906
$279$	−31.1379	−1.86418
$280$	0	0
$281$	−3.35786	−0.200313	−0.100157	−	0.994972i	$-0.531934\pi$
$281$	−3.35786	−0.200313	−0.100157	+	0.994972i	$0.531934\pi$
$282$	2.60302	0.155008
$283$	−17.1783	−1.02114	−0.510571	−	0.859836i	$-0.670566\pi$
$283$	−17.1783	−1.02114	−0.510571	+	0.859836i	$0.670566\pi$
$284$	0.822015	0.0487776
$285$	0	0
$286$	0.917729	0.0542665
$287$	−27.3158	−1.61240
$288$	1.42966	0.0842433
$289$	17.8477	1.04986
$290$	0	0
$291$	−1.16411	−0.0682412
$292$	−0.0685022	−0.00400879
$293$	23.9141	1.39707	0.698537	−	0.715574i	$-0.253835\pi$
$293$	23.9141	1.39707	0.698537	+	0.715574i	$0.253835\pi$
$294$	−2.90938	−0.169678
$295$	0	0
$296$	−31.1899	−1.81288
$297$	−1.07205	−0.0622065
$298$	10.6118	0.614727
$299$	0.462903	0.0267704
$300$	0	0
$301$	11.9507	0.688828
$302$	−10.0191	−0.576537
$303$	−1.48277	−0.0851829
$304$	3.82230	0.219224
$305$	0	0
$306$	24.2421	1.38583
$307$	29.5003	1.68367	0.841835	−	0.539735i	$-0.181476\pi$
$307$	29.5003	1.68367	0.841835	+	0.539735i	$0.181476\pi$
$308$	0.368553	0.0210003
$309$	−2.02363	−0.115120
$310$	0	0
$311$	−24.1117	−1.36725	−0.683625	−	0.729834i	$-0.739598\pi$
$311$	−24.1117	−1.36725	−0.683625	+	0.729834i	$0.739598\pi$
$312$	−0.343773	−0.0194623
$313$	8.25794	0.466766	0.233383	−	0.972385i	$-0.425020\pi$
$313$	8.25794	0.466766	0.233383	+	0.972385i	$0.425020\pi$
$314$	9.39773	0.530344
$315$	0	0
$316$	−0.703581	−0.0395795
$317$	−7.65440	−0.429914	−0.214957	−	0.976624i	$-0.568961\pi$
$317$	−7.65440	−0.429914	−0.214957	+	0.976624i	$0.568961\pi$
$318$	−2.01495	−0.112993
$319$	−3.32909	−0.186393
$320$	0	0
$321$	−2.15322	−0.120181
$322$	−4.17698	−0.232774
$323$	−5.90319	−0.328462
$324$	−0.742302	−0.0412390
$325$	0	0
$326$	16.1735	0.895765
$327$	0.0339116	0.00187532
$328$	18.2246	1.00629
$329$	45.2877	2.49679
$330$	0	0
$331$	11.6128	0.638299	0.319149	−	0.947704i	$-0.396603\pi$
$331$	11.6128	0.638299	0.319149	+	0.947704i	$0.396603\pi$
$332$	0.118622	0.00651025
$333$	32.0795	1.75794
$334$	6.20182	0.339349
$335$	0	0
$336$	2.96960	0.162005
$337$	−20.4093	−1.11177	−0.555883	−	0.831261i	$-0.687620\pi$
$337$	−20.4093	−1.11177	−0.555883	+	0.831261i	$0.687620\pi$
$338$	17.3802	0.945357
$339$	2.20390	0.119700
$340$	0	0
$341$	10.4922	0.568182
$342$	−4.10661	−0.222060
$343$	−20.3441	−1.09848
$344$	−7.97328	−0.429891
$345$	0	0
$346$	25.8256	1.38839
$347$	26.5715	1.42643	0.713216	−	0.700945i	$-0.247238\pi$
$347$	26.5715	1.42643	0.713216	+	0.700945i	$0.247238\pi$
$348$	0.0509639	0.00273195
$349$	1.10501	0.0591499	0.0295749	−	0.999563i	$-0.490585\pi$
$349$	1.10501	0.0591499	0.0295749	+	0.999563i	$0.490585\pi$
$350$	0	0
$351$	0.710999	0.0379503
$352$	−0.481734	−0.0256765
$353$	−29.9830	−1.59584	−0.797918	−	0.602766i	$-0.794065\pi$
$353$	−29.9830	−1.59584	−0.797918	+	0.602766i	$0.794065\pi$
$354$	0.0656336	0.00348839
$355$	0	0
$356$	−0.825822	−0.0437685
$357$	−4.58627	−0.242731
$358$	−5.23987	−0.276936
$359$	−14.7854	−0.780343	−0.390172	−	0.920742i	$-0.627584\pi$
$359$	−14.7854	−0.780343	−0.390172	+	0.920742i	$0.627584\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−10.7531	−0.565170
$363$	0.179641	0.00942870
$364$	−0.244430	−0.0128116
$365$	0	0
$366$	−3.55376	−0.185758
$367$	−27.7019	−1.44602	−0.723012	−	0.690835i	$-0.757243\pi$
$367$	−27.7019	−1.44602	−0.723012	+	0.690835i	$0.757243\pi$
$368$	2.66784	0.139071
$369$	−18.7444	−0.975794
$370$	0	0
$371$	−35.0564	−1.82004
$372$	−0.160621	−0.00832781
$373$	7.75317	0.401444	0.200722	−	0.979648i	$-0.435671\pi$
$373$	7.75317	0.401444	0.200722	+	0.979648i	$0.435671\pi$
$374$	−8.16858	−0.422387
$375$	0	0
$376$	−30.2151	−1.55822
$377$	2.20791	0.113713
$378$	−6.41566	−0.329986
$379$	−30.5335	−1.56840	−0.784200	−	0.620508i	$-0.786926\pi$
$379$	−30.5335	−1.56840	−0.784200	+	0.620508i	$0.786926\pi$
$380$	0	0
$381$	1.33317	0.0683006
$382$	26.6878	1.36547
$383$	24.7369	1.26400	0.631999	−	0.774969i	$-0.282235\pi$
$383$	24.7369	1.26400	0.631999	+	0.774969i	$0.282235\pi$
$384$	−1.89291	−0.0965974
$385$	0	0
$386$	10.7513	0.547227
$387$	8.20069	0.416864
$388$	0.552230	0.0280353
$389$	26.9954	1.36872	0.684360	−	0.729144i	$-0.260082\pi$
$389$	26.9954	1.36872	0.684360	+	0.729144i	$0.260082\pi$
$390$	0	0
$391$	−4.12024	−0.208369
$392$	33.7712	1.70570
$393$	−0.207555	−0.0104698
$394$	27.4839	1.38462
$395$	0	0
$396$	0.252904	0.0127089
$397$	−6.77500	−0.340028	−0.170014	−	0.985442i	$-0.554381\pi$
$397$	−6.77500	−0.340028	−0.170014	+	0.985442i	$0.554381\pi$
$398$	−4.91876	−0.246555
$399$	0.776914	0.0388943
$400$	0	0
$401$	20.9804	1.04771	0.523856	−	0.851807i	$-0.324493\pi$
$401$	20.9804	1.04771	0.523856	+	0.851807i	$0.324493\pi$
$402$	−3.64196	−0.181644
$403$	−6.95857	−0.346631
$404$	0.703398	0.0349953
$405$	0	0
$406$	−19.9229	−0.988758
$407$	−10.8094	−0.535804
$408$	3.05987	0.151486
$409$	−22.0672	−1.09115	−0.545575	−	0.838062i	$-0.683689\pi$
$409$	−22.0672	−1.09115	−0.545575	+	0.838062i	$0.683689\pi$
$410$	0	0
$411$	2.64821	0.130627
$412$	0.959969	0.0472943
$413$	1.14190	0.0561894
$414$	−2.86629	−0.140870
$415$	0	0
$416$	0.319494	0.0156645
$417$	1.14701	0.0561693
$418$	1.38376	0.0676817
$419$	−26.1712	−1.27855	−0.639273	−	0.768980i	$-0.720764\pi$
$419$	−26.1712	−1.27855	−0.639273	+	0.768980i	$0.720764\pi$
$420$	0	0
$421$	16.2156	0.790300	0.395150	−	0.918617i	$-0.370693\pi$
$421$	16.2156	0.790300	0.395150	+	0.918617i	$0.370693\pi$
$422$	27.7362	1.35018
$423$	31.0769	1.51101
$424$	23.3889	1.13587
$425$	0	0
$426$	2.39780	0.116174
$427$	−61.8289	−2.99211
$428$	1.02145	0.0493734
$429$	−0.119141	−0.00575217
$430$	0	0
$431$	−39.4232	−1.89895	−0.949474	−	0.313847i	$-0.898382\pi$
$431$	−39.4232	−1.89895	−0.949474	+	0.313847i	$0.898382\pi$
$432$	4.09769	0.197150
$433$	21.8969	1.05230	0.526148	−	0.850393i	$-0.323636\pi$
$433$	21.8969	1.05230	0.526148	+	0.850393i	$0.323636\pi$
$434$	62.7903	3.01403
$435$	0	0
$436$	−0.0160870	−0.000770429 0
$437$	0.697968	0.0333883
$438$	−0.199819	−0.00954773
$439$	−23.8693	−1.13922	−0.569609	−	0.821916i	$-0.692905\pi$
$439$	−23.8693	−1.13922	−0.569609	+	0.821916i	$0.692905\pi$
$440$	0	0
$441$	−34.7344	−1.65402
$442$	5.41753	0.257686
$443$	5.09632	0.242134	0.121067	−	0.992644i	$-0.461368\pi$
$443$	5.09632	0.242134	0.121067	+	0.992644i	$0.461368\pi$
$444$	0.165478	0.00785323
$445$	0	0
$446$	8.71454	0.412645
$447$	−1.37764	−0.0651601
$448$	−35.9444	−1.69821
$449$	5.00982	0.236428	0.118214	−	0.992988i	$-0.462283\pi$
$449$	5.00982	0.236428	0.118214	+	0.992988i	$0.462283\pi$
$450$	0	0
$451$	6.31607	0.297412
$452$	−1.04549	−0.0491757
$453$	1.30070	0.0611120
$454$	−32.9042	−1.54427
$455$	0	0
$456$	−0.518342	−0.0242736
$457$	−1.49477	−0.0699222	−0.0349611	−	0.999389i	$-0.511131\pi$
$457$	−1.49477	−0.0699222	−0.0349611	+	0.999389i	$0.511131\pi$
$458$	5.56204	0.259897
$459$	−6.32850	−0.295389
$460$	0	0
$461$	−13.3892	−0.623598	−0.311799	−	0.950148i	$-0.600932\pi$
$461$	−13.3892	−0.623598	−0.311799	+	0.950148i	$0.600932\pi$
$462$	1.07506	0.0500163
$463$	22.0666	1.02552	0.512761	−	0.858531i	$-0.328623\pi$
$463$	22.0666	1.02552	0.512761	+	0.858531i	$0.328623\pi$
$464$	12.7248	0.590733
$465$	0	0
$466$	36.8154	1.70544
$467$	6.60761	0.305764	0.152882	−	0.988244i	$-0.451145\pi$
$467$	6.60761	0.305764	0.152882	+	0.988244i	$0.451145\pi$
$468$	−0.167730	−0.00775333
$469$	−63.3633	−2.92585
$470$	0	0
$471$	−1.22002	−0.0562157
$472$	−0.761856	−0.0350672
$473$	−2.76329	−0.127056
$474$	−2.05233	−0.0942665
$475$	0	0
$476$	2.17564	0.0997202
$477$	−24.0560	−1.10145
$478$	−34.3946	−1.57317
$479$	−21.8942	−1.00037	−0.500185	−	0.865919i	$-0.666735\pi$
$479$	−21.8942	−1.00037	−0.500185	+	0.865919i	$0.666735\pi$
$480$	0	0
$481$	7.16899	0.326878
$482$	1.85002	0.0842661
$483$	0.542261	0.0246737
$484$	−0.0852182	−0.00387355
$485$	0	0
$486$	−6.61563	−0.300091
$487$	3.48029	0.157707	0.0788535	−	0.996886i	$-0.474874\pi$
$487$	3.48029	0.157707	0.0788535	+	0.996886i	$0.474874\pi$
$488$	41.2510	1.86735
$489$	−2.09966	−0.0949498
$490$	0	0
$491$	−25.4084	−1.14666	−0.573332	−	0.819323i	$-0.694349\pi$
$491$	−25.4084	−1.14666	−0.573332	+	0.819323i	$0.694349\pi$
$492$	−0.0966906	−0.00435915
$493$	−19.6523	−0.885093
$494$	−0.917729	−0.0412906
$495$	0	0
$496$	−40.1042	−1.80073
$497$	41.7172	1.87127
$498$	0.346018	0.0155054
$499$	8.27283	0.370343	0.185171	−	0.982706i	$-0.440716\pi$
$499$	8.27283	0.370343	0.185171	+	0.982706i	$0.440716\pi$
$500$	0	0
$501$	−0.805128	−0.0359705
$502$	−8.72059	−0.389219
$503$	−31.4555	−1.40253	−0.701265	−	0.712901i	$-0.747381\pi$
$503$	−31.4555	−1.40253	−0.701265	+	0.712901i	$0.747381\pi$
$504$	37.0342	1.64963
$505$	0	0
$506$	0.965818	0.0429358
$507$	−2.25631	−0.100206
$508$	−0.632433	−0.0280597
$509$	−28.6481	−1.26981	−0.634903	−	0.772592i	$-0.718960\pi$
$509$	−28.6481	−1.26981	−0.634903	+	0.772592i	$0.718960\pi$
$510$	0	0
$511$	−3.47648	−0.153791
$512$	23.8993	1.05621
$513$	1.07205	0.0473321
$514$	14.4351	0.636704
$515$	0	0
$516$	0.0423022	0.00186225
$517$	−10.4716	−0.460540
$518$	−64.6889	−2.84227
$519$	−3.35270	−0.147167
$520$	0	0
$521$	14.2218	0.623067	0.311534	−	0.950235i	$-0.399157\pi$
$521$	14.2218	0.623067	0.311534	+	0.950235i	$0.399157\pi$
$522$	−13.6713	−0.598376
$523$	−32.8102	−1.43469	−0.717345	−	0.696718i	$-0.754643\pi$
$523$	−32.8102	−1.43469	−0.717345	+	0.696718i	$0.754643\pi$
$524$	0.0984602	0.00430125
$525$	0	0
$526$	29.8640	1.30213
$527$	61.9372	2.69803
$528$	−0.686641	−0.0298822
$529$	−22.5128	−0.978819
$530$	0	0
$531$	0.783584	0.0340047
$532$	−0.368553	−0.0159788
$533$	−4.18892	−0.181442
$534$	−2.40890	−0.104243
$535$	0	0
$536$	42.2748	1.82599
$537$	0.680246	0.0293548
$538$	−24.2963	−1.04749
$539$	11.7040	0.504128
$540$	0	0
$541$	−12.2505	−0.526689	−0.263345	−	0.964702i	$-0.584826\pi$
$541$	−12.2505	−0.526689	−0.263345	+	0.964702i	$0.584826\pi$
$542$	24.0151	1.03154
$543$	1.39598	0.0599072
$544$	−2.84377	−0.121926
$545$	0	0
$546$	−0.712996	−0.0305134
$547$	3.35144	0.143297	0.0716487	−	0.997430i	$-0.477174\pi$
$547$	3.35144	0.143297	0.0716487	+	0.997430i	$0.477174\pi$
$548$	−1.25626	−0.0536649
$549$	−42.4275	−1.81076
$550$	0	0
$551$	3.32909	0.141824
$552$	−0.361786	−0.0153986
$553$	−35.7067	−1.51840
$554$	27.2327	1.15701
$555$	0	0
$556$	−0.544119	−0.0230758
$557$	32.8902	1.39360	0.696801	−	0.717264i	$-0.254606\pi$
$557$	32.8902	1.39360	0.696801	+	0.717264i	$0.254606\pi$
$558$	43.0873	1.82403
$559$	1.83266	0.0775131
$560$	0	0
$561$	1.06045	0.0447724
$562$	4.64647	0.195999
$563$	−17.9736	−0.757495	−0.378747	−	0.925500i	$-0.623645\pi$
$563$	−17.9736	−0.757495	−0.378747	+	0.925500i	$0.623645\pi$
$564$	0.160306	0.00675011
$565$	0	0
$566$	23.7705	0.999150
$567$	−37.6718	−1.58207
$568$	−27.8329	−1.16784
$569$	0.968772	0.0406130	0.0203065	−	0.999794i	$-0.493536\pi$
$569$	0.968772	0.0406130	0.0203065	+	0.999794i	$0.493536\pi$
$570$	0	0
$571$	−18.5167	−0.774901	−0.387450	−	0.921891i	$-0.626644\pi$
$571$	−18.5167	−0.774901	−0.387450	+	0.921891i	$0.626644\pi$
$572$	0.0565180	0.00236314
$573$	−3.46464	−0.144737
$574$	37.7985	1.57768
$575$	0	0
$576$	−24.6654	−1.02773
$577$	−13.3376	−0.555252	−0.277626	−	0.960689i	$-0.589548\pi$
$577$	−13.3376	−0.555252	−0.277626	+	0.960689i	$0.589548\pi$
$578$	−24.6968	−1.02725
$579$	−1.39575	−0.0580052
$580$	0	0
$581$	6.02008	0.249755
$582$	1.61084	0.0667715
$583$	8.10587	0.335711
$584$	2.31944	0.0959792
$585$	0	0
$586$	−33.0912	−1.36699
$587$	−30.6611	−1.26552	−0.632760	−	0.774348i	$-0.718078\pi$
$587$	−30.6611	−1.26552	−0.632760	+	0.774348i	$0.718078\pi$
$588$	−0.179173	−0.00738898
$589$	−10.4922	−0.432322
$590$	0	0
$591$	−3.56799	−0.146768
$592$	41.3169	1.69811
$593$	−14.1679	−0.581806	−0.290903	−	0.956753i	$-0.593956\pi$
$593$	−14.1679	−0.581806	−0.290903	+	0.956753i	$0.593956\pi$
$594$	1.48345	0.0608668
$595$	0	0
$596$	0.653526	0.0267695
$597$	0.638559	0.0261345
$598$	−0.640546	−0.0261939
$599$	3.17933	0.129904	0.0649519	−	0.997888i	$-0.479311\pi$
$599$	3.17933	0.129904	0.0649519	+	0.997888i	$0.479311\pi$
$600$	0	0
$601$	26.0743	1.06359	0.531796	−	0.846872i	$-0.321517\pi$
$601$	26.0743	1.06359	0.531796	+	0.846872i	$0.321517\pi$
$602$	−16.5369	−0.673993
$603$	−43.4805	−1.77066
$604$	−0.617025	−0.0251064
$605$	0	0
$606$	2.05179	0.0833484
$607$	−30.1201	−1.22254	−0.611269	−	0.791423i	$-0.709341\pi$
$607$	−30.1201	−1.22254	−0.611269	+	0.791423i	$0.709341\pi$
$608$	0.481734	0.0195369
$609$	2.58642	0.104807
$610$	0	0
$611$	6.94493	0.280962
$612$	1.49294	0.0603487
$613$	26.6754	1.07741	0.538704	−	0.842495i	$-0.318914\pi$
$613$	26.6754	1.07741	0.538704	+	0.842495i	$0.318914\pi$
$614$	−40.8212	−1.64741
$615$	0	0
$616$	−12.4790	−0.502792
$617$	5.41171	0.217867	0.108934	−	0.994049i	$-0.465256\pi$
$617$	5.41171	0.217867	0.108934	+	0.994049i	$0.465256\pi$
$618$	2.80020	0.112641
$619$	35.8748	1.44193	0.720965	−	0.692971i	$-0.243699\pi$
$619$	35.8748	1.44193	0.720965	+	0.692971i	$0.243699\pi$
$620$	0	0
$621$	0.748255	0.0300264
$622$	33.3647	1.33780
$623$	−41.9104	−1.67910
$624$	0.455392	0.0182303
$625$	0	0
$626$	−11.4270	−0.456714
$627$	−0.179641	−0.00717416
$628$	0.578756	0.0230949
$629$	−63.8101	−2.54428
$630$	0	0
$631$	10.3550	0.412226	0.206113	−	0.978528i	$-0.433919\pi$
$631$	10.3550	0.412226	0.206113	+	0.978528i	$0.433919\pi$
$632$	23.8228	0.947621
$633$	−3.60075	−0.143117
$634$	10.5918	0.420655
$635$	0	0
$636$	−0.124090	−0.00492049
$637$	−7.76230	−0.307554
$638$	4.60665	0.182379
$639$	28.6267	1.13246
$640$	0	0
$641$	6.49289	0.256454	0.128227	−	0.991745i	$-0.459071\pi$
$641$	6.49289	0.256454	0.128227	+	0.991745i	$0.459071\pi$
$642$	2.97953	0.117593
$643$	12.9459	0.510538	0.255269	−	0.966870i	$-0.417836\pi$
$643$	12.9459	0.510538	0.255269	+	0.966870i	$0.417836\pi$
$644$	−0.257238	−0.0101366
$645$	0	0
$646$	8.16858	0.321388
$647$	8.43170	0.331484	0.165742	−	0.986169i	$-0.446998\pi$
$647$	8.43170	0.331484	0.165742	+	0.986169i	$0.446998\pi$
$648$	25.1339	0.987352
$649$	−0.264035	−0.0103643
$650$	0	0
$651$	−8.15150	−0.319483
$652$	0.996037	0.0390078
$653$	37.3640	1.46217	0.731083	−	0.682288i	$-0.239015\pi$
$653$	37.3640	1.46217	0.731083	+	0.682288i	$0.239015\pi$
$654$	−0.0469255	−0.00183493
$655$	0	0
$656$	−24.1419	−0.942584
$657$	−2.38560	−0.0930709
$658$	−62.6672	−2.44302
$659$	14.3895	0.560535	0.280267	−	0.959922i	$-0.409577\pi$
$659$	14.3895	0.560535	0.280267	+	0.959922i	$0.409577\pi$
$660$	0	0
$661$	−18.1075	−0.704300	−0.352150	−	0.935944i	$-0.614549\pi$
$661$	−18.1075	−0.704300	−0.352150	+	0.935944i	$0.614549\pi$
$662$	−16.0693	−0.624552
$663$	−0.703310	−0.0273143
$664$	−4.01648	−0.155870
$665$	0	0
$666$	−44.3902	−1.72008
$667$	2.32360	0.0899701
$668$	0.381937	0.0147776
$669$	−1.13133	−0.0437398
$670$	0	0
$671$	14.2963	0.551903
$672$	0.374266	0.0144376
$673$	−24.7942	−0.955747	−0.477873	−	0.878429i	$-0.658592\pi$
$673$	−24.7942	−0.955747	−0.477873	+	0.878429i	$0.658592\pi$
$674$	28.2415	1.08782
$675$	0	0
$676$	1.07035	0.0411674
$677$	−5.73973	−0.220596	−0.110298	−	0.993899i	$-0.535180\pi$
$677$	−5.73973	−0.220596	−0.110298	+	0.993899i	$0.535180\pi$
$678$	−3.04966	−0.117122
$679$	28.0257	1.07553
$680$	0	0
$681$	4.27166	0.163690
$682$	−14.5186	−0.555946
$683$	16.8301	0.643986	0.321993	−	0.946742i	$-0.395647\pi$
$683$	16.8301	0.643986	0.321993	+	0.946742i	$0.395647\pi$
$684$	−0.252904	−0.00967004
$685$	0	0
$686$	28.1513	1.07482
$687$	−0.722071	−0.0275487
$688$	10.5621	0.402677
$689$	−5.37594	−0.204807
$690$	0	0
$691$	−11.7122	−0.445553	−0.222776	−	0.974870i	$-0.571512\pi$
$691$	−11.7122	−0.445553	−0.222776	+	0.974870i	$0.571512\pi$
$692$	1.59046	0.0604602
$693$	12.8349	0.487557
$694$	−36.7684	−1.39571
$695$	0	0
$696$	−1.72561	−0.0654089
$697$	37.2850	1.41227
$698$	−1.52907	−0.0578760
$699$	−4.77941	−0.180774
$700$	0	0
$701$	−21.2822	−0.803818	−0.401909	−	0.915680i	$-0.631653\pi$
$701$	−21.2822	−0.803818	−0.401909	+	0.915680i	$0.631653\pi$
$702$	−0.983850	−0.0371330
$703$	10.8094	0.407685
$704$	8.31121	0.313240
$705$	0	0
$706$	41.4892	1.56147
$707$	35.6974	1.34254
$708$	0.00404202	0.000151909 0
$709$	−33.7749	−1.26844	−0.634222	−	0.773151i	$-0.718679\pi$
$709$	−33.7749	−1.26844	−0.634222	+	0.773151i	$0.718679\pi$
$710$	0	0
$711$	−24.5023	−0.918907
$712$	27.9618	1.04791
$713$	−7.32319	−0.274256
$714$	6.34628	0.237504
$715$	0	0
$716$	−0.322696	−0.0120597
$717$	4.46514	0.166754
$718$	20.4594	0.763537
$719$	−17.4677	−0.651434	−0.325717	−	0.945467i	$-0.605606\pi$
$719$	−17.4677	−0.651434	−0.325717	+	0.945467i	$0.605606\pi$
$720$	0	0
$721$	48.7184	1.81437
$722$	−1.38376	−0.0514981
$723$	−0.240172	−0.00893208
$724$	−0.662226	−0.0246114
$725$	0	0
$726$	−0.248579	−0.00922564
$727$	−43.0612	−1.59705	−0.798525	−	0.601961i	$-0.794386\pi$
$727$	−43.0612	−1.59705	−0.798525	+	0.601961i	$0.794386\pi$
$728$	8.27625	0.306738
$729$	−25.2730	−0.936036
$730$	0	0
$731$	−16.3122	−0.603329
$732$	−0.218857	−0.00808920
$733$	32.1119	1.18608	0.593039	−	0.805173i	$-0.297928\pi$
$733$	32.1119	1.18608	0.593039	+	0.805173i	$0.297928\pi$
$734$	38.3326	1.41488
$735$	0	0
$736$	0.336235	0.0123938
$737$	14.6511	0.539680
$738$	25.9377	0.954779
$739$	8.07571	0.297070	0.148535	−	0.988907i	$-0.452544\pi$
$739$	8.07571	0.297070	0.148535	+	0.988907i	$0.452544\pi$
$740$	0	0
$741$	0.119141	0.00437674
$742$	48.5095	1.78084
$743$	−14.9512	−0.548507	−0.274253	−	0.961657i	$-0.588431\pi$
$743$	−14.9512	−0.548507	−0.274253	+	0.961657i	$0.588431\pi$
$744$	5.43853	0.199386
$745$	0	0
$746$	−10.7285	−0.392798
$747$	4.13103	0.151147
$748$	−0.503059	−0.0183937
$749$	51.8383	1.89413
$750$	0	0
$751$	48.7426	1.77864	0.889322	−	0.457282i	$-0.151177\pi$
$751$	48.7426	1.77864	0.889322	+	0.457282i	$0.151177\pi$
$752$	40.0256	1.45958
$753$	1.13212	0.0412566
$754$	−3.05520	−0.111264
$755$	0	0
$756$	−0.395106	−0.0143699
$757$	47.6174	1.73068	0.865342	−	0.501182i	$-0.167101\pi$
$757$	47.6174	1.73068	0.865342	+	0.501182i	$0.167101\pi$
$758$	42.2509	1.53462
$759$	−0.125384	−0.00455113
$760$	0	0
$761$	30.9317	1.12127	0.560637	−	0.828062i	$-0.310556\pi$
$761$	30.9317	1.12127	0.560637	+	0.828062i	$0.310556\pi$
$762$	−1.84479	−0.0668297
$763$	−0.816416	−0.0295562
$764$	1.64356	0.0594619
$765$	0	0
$766$	−34.2299	−1.23678
$767$	0.175112	0.00632294
$768$	−0.366731	−0.0132333
$769$	27.4711	0.990633	0.495316	−	0.868713i	$-0.335052\pi$
$769$	27.4711	0.990633	0.495316	+	0.868713i	$0.335052\pi$
$770$	0	0
$771$	−1.87398	−0.0674897
$772$	0.662115	0.0238301
$773$	−32.0937	−1.15433	−0.577166	−	0.816627i	$-0.695841\pi$
$773$	−32.0937	−1.15433	−0.577166	+	0.816627i	$0.695841\pi$
$774$	−11.3478	−0.407887
$775$	0	0
$776$	−18.6982	−0.671225
$777$	8.39799	0.301276
$778$	−37.3550	−1.33924
$779$	−6.31607	−0.226297
$780$	0	0
$781$	−9.64601	−0.345161
$782$	5.70141	0.203882
$783$	3.56894	0.127544
$784$	−44.7364	−1.59773
$785$	0	0
$786$	0.287206	0.0102443
$787$	−21.1724	−0.754715	−0.377357	−	0.926068i	$-0.623167\pi$
$787$	−21.1724	−0.754715	−0.377357	+	0.926068i	$0.623167\pi$
$788$	1.69259	0.0602959
$789$	−3.87698	−0.138024
$790$	0	0
$791$	−53.0585	−1.88654
$792$	−8.56319	−0.304280
$793$	−9.48153	−0.336699
$794$	9.37495	0.332705
$795$	0	0
$796$	−0.302920	−0.0107367
$797$	−0.325852	−0.0115423	−0.00577114	−	0.999983i	$-0.501837\pi$
$797$	−0.325852	−0.0115423	−0.00577114	+	0.999983i	$0.501837\pi$
$798$	−1.07506	−0.0380567
$799$	−61.8158	−2.18689
$800$	0	0
$801$	−28.7593	−1.01616
$802$	−29.0318	−1.02515
$803$	0.803845	0.0283671
$804$	−0.224289	−0.00791006
$805$	0	0
$806$	9.62896	0.339166
$807$	3.15418	0.111032
$808$	−23.8166	−0.837865
$809$	−5.48167	−0.192725	−0.0963626	−	0.995346i	$-0.530721\pi$
$809$	−5.48167	−0.192725	−0.0963626	+	0.995346i	$0.530721\pi$
$810$	0	0
$811$	−46.4009	−1.62935	−0.814677	−	0.579915i	$-0.803086\pi$
$811$	−46.4009	−1.62935	−0.814677	+	0.579915i	$0.803086\pi$
$812$	−1.22695	−0.0430574
$813$	−3.11767	−0.109342
$814$	14.9576	0.524264
$815$	0	0
$816$	−4.05338	−0.141897
$817$	2.76329	0.0966752
$818$	30.5356	1.06765
$819$	−8.51230	−0.297444
$820$	0	0
$821$	−29.0800	−1.01490	−0.507449	−	0.861682i	$-0.669411\pi$
$821$	−29.0800	−1.01490	−0.507449	+	0.861682i	$0.669411\pi$
$822$	−3.66448	−0.127814
$823$	29.3196	1.02202	0.511008	−	0.859576i	$-0.329272\pi$
$823$	29.3196	1.02202	0.511008	+	0.859576i	$0.329272\pi$
$824$	−32.5040	−1.13233
$825$	0	0
$826$	−1.58012	−0.0549793
$827$	−11.4469	−0.398048	−0.199024	−	0.979995i	$-0.563777\pi$
$827$	−11.4469	−0.398048	−0.199024	+	0.979995i	$0.563777\pi$
$828$	−0.176519	−0.00613447
$829$	−50.5560	−1.75588	−0.877941	−	0.478770i	$-0.841083\pi$
$829$	−50.5560	−1.75588	−0.877941	+	0.478770i	$0.841083\pi$
$830$	0	0
$831$	−3.53538	−0.122641
$832$	−5.51212	−0.191098
$833$	69.0912	2.39387
$834$	−1.58718	−0.0549596
$835$	0	0
$836$	0.0852182	0.00294733
$837$	−11.2481	−0.388791
$838$	36.2145	1.25101
$839$	1.09625	0.0378466	0.0189233	−	0.999821i	$-0.493976\pi$
$839$	1.09625	0.0378466	0.0189233	+	0.999821i	$0.493976\pi$
$840$	0	0
$841$	−17.9172	−0.617833
$842$	−22.4384	−0.773280
$843$	−0.603210	−0.0207756
$844$	1.70813	0.0587961
$845$	0	0
$846$	−43.0028	−1.47847
$847$	−4.32482	−0.148602
$848$	−30.9831	−1.06396
$849$	−3.08592	−0.105908
$850$	0	0
$851$	7.54464	0.258627
$852$	0.147668	0.00505901
$853$	13.1321	0.449633	0.224816	−	0.974401i	$-0.427822\pi$
$853$	13.1321	0.449633	0.224816	+	0.974401i	$0.427822\pi$
$854$	85.5561	2.92767
$855$	0	0
$856$	−34.5855	−1.18211
$857$	−18.5275	−0.632888	−0.316444	−	0.948611i	$-0.602489\pi$
$857$	−18.5275	−0.632888	−0.316444	+	0.948611i	$0.602489\pi$
$858$	0.164862	0.00562828
$859$	24.1651	0.824502	0.412251	−	0.911070i	$-0.364743\pi$
$859$	24.1651	0.824502	0.412251	+	0.911070i	$0.364743\pi$
$860$	0	0
$861$	−4.90704	−0.167232
$862$	54.5521	1.85805
$863$	−8.57175	−0.291786	−0.145893	−	0.989300i	$-0.546605\pi$
$863$	−8.57175	−0.291786	−0.145893	+	0.989300i	$0.546605\pi$
$864$	0.516442	0.0175697
$865$	0	0
$866$	−30.2999	−1.02963
$867$	3.20617	0.108887
$868$	3.86692	0.131252
$869$	8.25623	0.280074
$870$	0	0
$871$	−9.71684	−0.329243
$872$	0.544697	0.0184458
$873$	19.2315	0.650886
$874$	−0.965818	−0.0326693
$875$	0	0
$876$	−0.0123058	−0.000415774 0
$877$	−10.4764	−0.353763	−0.176881	−	0.984232i	$-0.556601\pi$
$877$	−10.4764	−0.353763	−0.176881	+	0.984232i	$0.556601\pi$
$878$	33.0293	1.11468
$879$	4.29594	0.144899
$880$	0	0
$881$	−0.125726	−0.00423583	−0.00211792	−	0.999998i	$-0.500674\pi$
$881$	−0.125726	−0.00423583	−0.00211792	+	0.999998i	$0.500674\pi$
$882$	48.0640	1.61840
$883$	−53.1047	−1.78712	−0.893558	−	0.448947i	$-0.851799\pi$
$883$	−53.1047	−1.78712	−0.893558	+	0.448947i	$0.851799\pi$
$884$	0.333637	0.0112214
$885$	0	0
$886$	−7.05207	−0.236919
$887$	−1.80239	−0.0605183	−0.0302591	−	0.999542i	$-0.509633\pi$
$887$	−1.80239	−0.0605183	−0.0302591	+	0.999542i	$0.509633\pi$
$888$	−5.60298	−0.188024
$889$	−32.0959	−1.07646
$890$	0	0
$891$	8.71060	0.291816
$892$	0.536682	0.0179694
$893$	10.4716	0.350419
$894$	1.90632	0.0637568
$895$	0	0
$896$	45.5715	1.52244
$897$	0.0831564	0.00277651
$898$	−6.93237	−0.231336
$899$	−34.9294	−1.16496
$900$	0	0
$901$	47.8505	1.59413
$902$	−8.73990	−0.291007
$903$	2.14684	0.0714422
$904$	35.3996	1.17737
$905$	0	0
$906$	−1.79985	−0.0597959
$907$	1.11693	0.0370869	0.0185435	−	0.999828i	$-0.494097\pi$
$907$	1.11693	0.0370869	0.0185435	+	0.999828i	$0.494097\pi$
$908$	−2.02639	−0.0672483
$909$	24.4959	0.812477
$910$	0	0
$911$	46.5149	1.54111	0.770553	−	0.637376i	$-0.219980\pi$
$911$	46.5149	1.54111	0.770553	+	0.637376i	$0.219980\pi$
$912$	0.686641	0.0227370
$913$	−1.39198	−0.0460680
$914$	2.06839	0.0684163
$915$	0	0
$916$	0.342537	0.0113177
$917$	4.99685	0.165010
$918$	8.75711	0.289028
$919$	−40.3852	−1.33218	−0.666092	−	0.745870i	$-0.732034\pi$
$919$	−40.3852	−1.33218	−0.666092	+	0.745870i	$0.732034\pi$
$920$	0	0
$921$	5.29945	0.174623
$922$	18.5274	0.610168
$923$	6.39739	0.210573
$924$	0.0662072	0.00217806
$925$	0	0
$926$	−30.5348	−1.00344
$927$	33.4310	1.09802
$928$	1.60374	0.0526452
$929$	−23.3618	−0.766476	−0.383238	−	0.923650i	$-0.625191\pi$
$929$	−23.3618	−0.766476	−0.383238	+	0.923650i	$0.625191\pi$
$930$	0	0
$931$	−11.7040	−0.383584
$932$	2.26726	0.0742666
$933$	−4.33145	−0.141805
$934$	−9.14332	−0.299179
$935$	0	0
$936$	5.67924	0.185632
$937$	−13.6780	−0.446842	−0.223421	−	0.974722i	$-0.571722\pi$
$937$	−13.6780	−0.446842	−0.223421	+	0.974722i	$0.571722\pi$
$938$	87.6794	2.86283
$939$	1.48346	0.0484110
$940$	0	0
$941$	41.2212	1.34377	0.671887	−	0.740654i	$-0.265484\pi$
$941$	41.2212	1.34377	0.671887	+	0.740654i	$0.265484\pi$
$942$	1.68822	0.0550050
$943$	−4.40842	−0.143558
$944$	1.00922	0.0328474
$945$	0	0
$946$	3.82372	0.124320
$947$	−7.43620	−0.241644	−0.120822	−	0.992674i	$-0.538553\pi$
$947$	−7.43620	−0.241644	−0.120822	+	0.992674i	$0.538553\pi$
$948$	−0.126392	−0.00410502
$949$	−0.533123	−0.0173059
$950$	0	0
$951$	−1.37504	−0.0445888
$952$	−73.6658	−2.38752
$953$	3.32135	0.107589	0.0537945	−	0.998552i	$-0.482868\pi$
$953$	3.32135	0.107589	0.0537945	+	0.998552i	$0.482868\pi$
$954$	33.2877	1.07773
$955$	0	0
$956$	−2.11818	−0.0685068
$957$	−0.598040	−0.0193319
$958$	30.2962	0.978825
$959$	−63.7552	−2.05876
$960$	0	0
$961$	79.0855	2.55114
$962$	−9.92013	−0.319838
$963$	35.5719	1.14629
$964$	0.113933	0.00366953
$965$	0	0
$966$	−0.750357	−0.0241423
$967$	40.1346	1.29064	0.645321	−	0.763911i	$-0.276723\pi$
$967$	40.1346	1.29064	0.645321	+	0.763911i	$0.276723\pi$
$968$	2.88543	0.0927414
$969$	−1.06045	−0.0340667
$970$	0	0
$971$	27.7956	0.892005	0.446002	−	0.895032i	$-0.352847\pi$
$971$	27.7956	0.892005	0.446002	+	0.895032i	$0.352847\pi$
$972$	−0.407422	−0.0130681
$973$	−27.6140	−0.885265
$974$	−4.81588	−0.154311
$975$	0	0
$976$	−54.6448	−1.74914
$977$	55.0675	1.76177	0.880883	−	0.473335i	$-0.156950\pi$
$977$	55.0675	1.76177	0.880883	+	0.473335i	$0.156950\pi$
$978$	2.90541	0.0929049
$979$	9.69068	0.309715
$980$	0	0
$981$	−0.560232	−0.0178868
$982$	35.1590	1.12197
$983$	22.0581	0.703545	0.351773	−	0.936085i	$-0.385579\pi$
$983$	22.0581	0.703545	0.351773	+	0.936085i	$0.385579\pi$
$984$	3.27388	0.104368
$985$	0	0
$986$	27.1939	0.866031
$987$	8.13552	0.258957
$988$	−0.0565180	−0.00179808
$989$	1.92869	0.0613287
$990$	0	0
$991$	13.3501	0.424080	0.212040	−	0.977261i	$-0.431989\pi$
$991$	13.3501	0.424080	0.212040	+	0.977261i	$0.431989\pi$
$992$	−5.05443	−0.160478
$993$	2.08614	0.0662016
$994$	−57.7265	−1.83097
$995$	0	0
$996$	0.0213094	0.000675215 0
$997$	40.0293	1.26774	0.633870	−	0.773439i	$-0.281465\pi$
$997$	40.0293	1.26774	0.633870	+	0.773439i	$0.281465\pi$
$998$	−11.4476	−0.362367
$999$	11.5882	0.366635

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5225.2.a.s.1.5	✓	15
5.4	even	2		5225.2.a.x.1.11	yes	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5225.2.a.s.1.5	✓	15	1.1	even	1	trivial
5225.2.a.x.1.11	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-1.38376 q^{2} +0.179641 q^{3} -0.0852182 q^{4} -0.248579 q^{6} -4.32482 q^{7} +2.88543 q^{8} -2.96773 q^{9} +1.00000 q^{11} -0.0153087 q^{12} -0.663216 q^{13} +5.98449 q^{14} -3.82230 q^{16} +5.90319 q^{17} +4.10661 q^{18} -1.00000 q^{19} -0.776914 q^{21} -1.38376 q^{22} -0.697968 q^{23} +0.518342 q^{24} +0.917729 q^{26} -1.07205 q^{27} +0.368553 q^{28} -3.32909 q^{29} +10.4922 q^{31} -0.481734 q^{32} +0.179641 q^{33} -8.16858 q^{34} +0.252904 q^{36} -10.8094 q^{37} +1.38376 q^{38} -0.119141 q^{39} +6.31607 q^{41} +1.07506 q^{42} -2.76329 q^{43} -0.0852182 q^{44} +0.965818 q^{46} -10.4716 q^{47} -0.686641 q^{48} +11.7040 q^{49} +1.06045 q^{51} +0.0565180 q^{52} +8.10587 q^{53} +1.48345 q^{54} -12.4790 q^{56} -0.179641 q^{57} +4.60665 q^{58} -0.264035 q^{59} +14.2963 q^{61} -14.5186 q^{62} +12.8349 q^{63} +8.31121 q^{64} -0.248579 q^{66} +14.6511 q^{67} -0.503059 q^{68} -0.125384 q^{69} -9.64601 q^{71} -8.56319 q^{72} +0.803845 q^{73} +14.9576 q^{74} +0.0852182 q^{76} -4.32482 q^{77} +0.164862 q^{78} +8.25623 q^{79} +8.71060 q^{81} -8.73990 q^{82} -1.39198 q^{83} +0.0662072 q^{84} +3.82372 q^{86} -0.598040 q^{87} +2.88543 q^{88} +9.69068 q^{89} +2.86829 q^{91} +0.0594796 q^{92} +1.88482 q^{93} +14.4901 q^{94} -0.0865391 q^{96} -6.48020 q^{97} -16.1955 q^{98} -2.96773 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 5 q^{2} - 4 q^{3} + 17 q^{4} - q^{6} - 15 q^{7} - 15 q^{8} + 19 q^{9} + 15 q^{11} - 9 q^{12} - 13 q^{13} + 9 q^{14} + 21 q^{16} - 11 q^{17} - 16 q^{18} - 15 q^{19} - 5 q^{22} - 10 q^{23} + 17 q^{24}+ \cdots + 19 q^{99}+O(q^{100}) \) 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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