LMFDB - Embedded newform 5775.2.a.bt.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5775,2,Mod(1,5775)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5775, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5775.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5775 = 3 \cdot 5^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5775.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:	$46.1136071673$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1155)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	5775.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.21432 q^{2} -1.00000 q^{3} +2.90321 q^{4} -2.21432 q^{6} +1.00000 q^{7} +2.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.21432 q^{2} -1.00000 q^{3} +2.90321 q^{4} -2.21432 q^{6} +1.00000 q^{7} +2.00000 q^{8} +1.00000 q^{9} -1.00000 q^{11} -2.90321 q^{12} -1.31111 q^{13} +2.21432 q^{14} -1.37778 q^{16} -1.90321 q^{17} +2.21432 q^{18} +0.377784 q^{19} -1.00000 q^{21} -2.21432 q^{22} -2.52543 q^{23} -2.00000 q^{24} -2.90321 q^{26} -1.00000 q^{27} +2.90321 q^{28} -6.11753 q^{29} -4.96989 q^{31} -7.05086 q^{32} +1.00000 q^{33} -4.21432 q^{34} +2.90321 q^{36} +1.31111 q^{37} +0.836535 q^{38} +1.31111 q^{39} -2.68889 q^{41} -2.21432 q^{42} -6.11753 q^{43} -2.90321 q^{44} -5.59210 q^{46} +1.59210 q^{47} +1.37778 q^{48} +1.00000 q^{49} +1.90321 q^{51} -3.80642 q^{52} +11.4286 q^{53} -2.21432 q^{54} +2.00000 q^{56} -0.377784 q^{57} -13.5462 q^{58} -7.21432 q^{59} -1.04593 q^{61} -11.0049 q^{62} +1.00000 q^{63} -12.8573 q^{64} +2.21432 q^{66} +2.00000 q^{67} -5.52543 q^{68} +2.52543 q^{69} +4.92396 q^{71} +2.00000 q^{72} -12.4286 q^{73} +2.90321 q^{74} +1.09679 q^{76} -1.00000 q^{77} +2.90321 q^{78} -0.969888 q^{79} +1.00000 q^{81} -5.95407 q^{82} +6.71900 q^{83} -2.90321 q^{84} -13.5462 q^{86} +6.11753 q^{87} -2.00000 q^{88} -11.3620 q^{89} -1.31111 q^{91} -7.33185 q^{92} +4.96989 q^{93} +3.52543 q^{94} +7.05086 q^{96} -8.54617 q^{97} +2.21432 q^{98} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 2 q^{4} + 3 q^{7} + 6 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + 2 * q^4 + 3 * q^7 + 6 * q^8 + 3 * q^9	$ 3 q - 3 q^{3} + 2 q^{4} + 3 q^{7} + 6 q^{8} + 3 q^{9} - 3 q^{11} - 2 q^{12} - 4 q^{13} - 4 q^{16} + q^{17} + q^{19} - 3 q^{21} - q^{23} - 6 q^{24} - 2 q^{26} - 3 q^{27} + 2 q^{28} - 5 q^{29} - 8 q^{31} - 8 q^{32} + 3 q^{33} - 6 q^{34} + 2 q^{36} + 4 q^{37} - 4 q^{38} + 4 q^{39} - 8 q^{41} - 5 q^{43} - 2 q^{44} - 10 q^{46} - 2 q^{47} + 4 q^{48} + 3 q^{49} - q^{51} + 2 q^{52} + 21 q^{53} + 6 q^{56} - q^{57} - 14 q^{58} - 15 q^{59} - 23 q^{61} + 3 q^{63} - 12 q^{64} + 6 q^{67} - 10 q^{68} + q^{69} - 12 q^{71} + 6 q^{72} - 24 q^{73} + 2 q^{74} + 10 q^{76} - 3 q^{77} + 2 q^{78} + 4 q^{79} + 3 q^{81} + 2 q^{82} + 27 q^{83} - 2 q^{84} - 14 q^{86} + 5 q^{87} - 6 q^{88} - 21 q^{89} - 4 q^{91} - 2 q^{92} + 8 q^{93} + 4 q^{94} + 8 q^{96} + q^{97} - 3 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 2 * q^4 + 3 * q^7 + 6 * q^8 + 3 * q^9 - 3 * q^11 - 2 * q^12 - 4 * q^13 - 4 * q^16 + q^17 + q^19 - 3 * q^21 - q^23 - 6 * q^24 - 2 * q^26 - 3 * q^27 + 2 * q^28 - 5 * q^29 - 8 * q^31 - 8 * q^32 + 3 * q^33 - 6 * q^34 + 2 * q^36 + 4 * q^37 - 4 * q^38 + 4 * q^39 - 8 * q^41 - 5 * q^43 - 2 * q^44 - 10 * q^46 - 2 * q^47 + 4 * q^48 + 3 * q^49 - q^51 + 2 * q^52 + 21 * q^53 + 6 * q^56 - q^57 - 14 * q^58 - 15 * q^59 - 23 * q^61 + 3 * q^63 - 12 * q^64 + 6 * q^67 - 10 * q^68 + q^69 - 12 * q^71 + 6 * q^72 - 24 * q^73 + 2 * q^74 + 10 * q^76 - 3 * q^77 + 2 * q^78 + 4 * q^79 + 3 * q^81 + 2 * q^82 + 27 * q^83 - 2 * q^84 - 14 * q^86 + 5 * q^87 - 6 * q^88 - 21 * q^89 - 4 * q^91 - 2 * q^92 + 8 * q^93 + 4 * q^94 + 8 * q^96 + q^97 - 3 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	2.21432	1.56576	0.782880	−	0.622172i	$-0.213750\pi$
$2$	2.21432	1.56576	0.782880	+	0.622172i	$0.213750\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	2.90321	1.45161
$5$	0	0
$5$	0	0
$6$	−2.21432	−0.903992
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.00000	0.707107
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−2.90321	−0.838085
$13$	−1.31111	−0.363636	−0.181818	−	0.983332i	$-0.558198\pi$
$13$	−1.31111	−0.363636	−0.181818	+	0.983332i	$0.558198\pi$
$14$	2.21432	0.591802
$15$	0	0
$16$	−1.37778	−0.344446
$17$	−1.90321	−0.461597	−0.230798	−	0.973002i	$-0.574134\pi$
$17$	−1.90321	−0.461597	−0.230798	+	0.973002i	$0.574134\pi$
$18$	2.21432	0.521920
$19$	0.377784	0.0866697	0.0433348	−	0.999061i	$-0.486202\pi$
$19$	0.377784	0.0866697	0.0433348	+	0.999061i	$0.486202\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	−2.21432	−0.472095
$23$	−2.52543	−0.526588	−0.263294	−	0.964716i	$-0.584809\pi$
$23$	−2.52543	−0.526588	−0.263294	+	0.964716i	$0.584809\pi$
$24$	−2.00000	−0.408248
$25$	0	0
$26$	−2.90321	−0.569367
$27$	−1.00000	−0.192450
$28$	2.90321	0.548655
$29$	−6.11753	−1.13600	−0.567999	−	0.823030i	$-0.692282\pi$
$29$	−6.11753	−1.13600	−0.567999	+	0.823030i	$0.692282\pi$
$30$	0	0
$31$	−4.96989	−0.892618	−0.446309	−	0.894879i	$-0.647262\pi$
$31$	−4.96989	−0.892618	−0.446309	+	0.894879i	$0.647262\pi$
$32$	−7.05086	−1.24643
$33$	1.00000	0.174078
$34$	−4.21432	−0.722750
$35$	0	0
$36$	2.90321	0.483869
$37$	1.31111	0.215545	0.107772	−	0.994176i	$-0.465628\pi$
$37$	1.31111	0.215545	0.107772	+	0.994176i	$0.465628\pi$
$38$	0.836535	0.135704
$39$	1.31111	0.209945
$40$	0	0
$41$	−2.68889	−0.419934	−0.209967	−	0.977708i	$-0.567336\pi$
$41$	−2.68889	−0.419934	−0.209967	+	0.977708i	$0.567336\pi$
$42$	−2.21432	−0.341677
$43$	−6.11753	−0.932915	−0.466457	−	0.884544i	$-0.654470\pi$
$43$	−6.11753	−0.932915	−0.466457	+	0.884544i	$0.654470\pi$
$44$	−2.90321	−0.437676
$45$	0	0
$46$	−5.59210	−0.824511
$47$	1.59210	0.232232	0.116116	−	0.993236i	$-0.462956\pi$
$47$	1.59210	0.232232	0.116116	+	0.993236i	$0.462956\pi$
$48$	1.37778	0.198866
$49$	1.00000	0.142857
$50$	0	0
$51$	1.90321	0.266503
$52$	−3.80642	−0.527856
$53$	11.4286	1.56984	0.784922	−	0.619594i	$-0.212703\pi$
$53$	11.4286	1.56984	0.784922	+	0.619594i	$0.212703\pi$
$54$	−2.21432	−0.301331
$55$	0	0
$56$	2.00000	0.267261
$57$	−0.377784	−0.0500388
$58$	−13.5462	−1.77870
$59$	−7.21432	−0.939224	−0.469612	−	0.882873i	$-0.655606\pi$
$59$	−7.21432	−0.939224	−0.469612	+	0.882873i	$0.655606\pi$
$60$	0	0
$61$	−1.04593	−0.133918	−0.0669590	−	0.997756i	$-0.521330\pi$
$61$	−1.04593	−0.133918	−0.0669590	+	0.997756i	$0.521330\pi$
$62$	−11.0049	−1.39763
$63$	1.00000	0.125988
$64$	−12.8573	−1.60716
$65$	0	0
$66$	2.21432	0.272564
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	−5.52543	−0.670057
$69$	2.52543	0.304026
$70$	0	0
$71$	4.92396	0.584366	0.292183	−	0.956362i	$-0.405618\pi$
$71$	4.92396	0.584366	0.292183	+	0.956362i	$0.405618\pi$
$72$	2.00000	0.235702
$73$	−12.4286	−1.45466	−0.727331	−	0.686287i	$-0.759240\pi$
$73$	−12.4286	−1.45466	−0.727331	+	0.686287i	$0.759240\pi$
$74$	2.90321	0.337492
$75$	0	0
$76$	1.09679	0.125810
$77$	−1.00000	−0.113961
$78$	2.90321	0.328724
$79$	−0.969888	−0.109121	−0.0545605	−	0.998510i	$-0.517376\pi$
$79$	−0.969888	−0.109121	−0.0545605	+	0.998510i	$0.517376\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−5.95407	−0.657517
$83$	6.71900	0.737506	0.368753	−	0.929527i	$-0.379785\pi$
$83$	6.71900	0.737506	0.368753	+	0.929527i	$0.379785\pi$
$84$	−2.90321	−0.316766
$85$	0	0
$86$	−13.5462	−1.46072
$87$	6.11753	0.655868
$88$	−2.00000	−0.213201
$89$	−11.3620	−1.20437	−0.602183	−	0.798358i	$-0.705702\pi$
$89$	−11.3620	−1.20437	−0.602183	+	0.798358i	$0.705702\pi$
$90$	0	0
$91$	−1.31111	−0.137441
$92$	−7.33185	−0.764398
$93$	4.96989	0.515353
$94$	3.52543	0.363620
$95$	0	0
$96$	7.05086	0.719625
$97$	−8.54617	−0.867732	−0.433866	−	0.900977i	$-0.642851\pi$
$97$	−8.54617	−0.867732	−0.433866	+	0.900977i	$0.642851\pi$
$98$	2.21432	0.223680
$99$	−1.00000	−0.100504
$100$	0	0
$101$	−4.75557	−0.473197	−0.236598	−	0.971608i	$-0.576033\pi$
$101$	−4.75557	−0.473197	−0.236598	+	0.971608i	$0.576033\pi$
$102$	4.21432	0.417280
$103$	−11.3017	−1.11359	−0.556797	−	0.830649i	$-0.687970\pi$
$103$	−11.3017	−1.11359	−0.556797	+	0.830649i	$0.687970\pi$
$104$	−2.62222	−0.257129
$105$	0	0
$106$	25.3067	2.45800
$107$	8.83654	0.854260	0.427130	−	0.904190i	$-0.359525\pi$
$107$	8.83654	0.854260	0.427130	+	0.904190i	$0.359525\pi$
$108$	−2.90321	−0.279362
$109$	3.39853	0.325520	0.162760	−	0.986666i	$-0.447960\pi$
$109$	3.39853	0.325520	0.162760	+	0.986666i	$0.447960\pi$
$110$	0	0
$111$	−1.31111	−0.124445
$112$	−1.37778	−0.130188
$113$	−2.76049	−0.259685	−0.129843	−	0.991535i	$-0.541447\pi$
$113$	−2.76049	−0.259685	−0.129843	+	0.991535i	$0.541447\pi$
$114$	−0.836535	−0.0783487
$115$	0	0
$116$	−17.7605	−1.64902
$117$	−1.31111	−0.121212
$118$	−15.9748	−1.47060
$119$	−1.90321	−0.174467
$120$	0	0
$121$	1.00000	0.0909091
$122$	−2.31603	−0.209684
$123$	2.68889	0.242449
$124$	−14.4286	−1.29573
$125$	0	0
$126$	2.21432	0.197267
$127$	2.45875	0.218179	0.109089	−	0.994032i	$-0.465207\pi$
$127$	2.45875	0.218179	0.109089	+	0.994032i	$0.465207\pi$
$128$	−14.3684	−1.27000
$129$	6.11753	0.538619
$130$	0	0
$131$	−5.73975	−0.501484	−0.250742	−	0.968054i	$-0.580675\pi$
$131$	−5.73975	−0.501484	−0.250742	+	0.968054i	$0.580675\pi$
$132$	2.90321	0.252692
$133$	0.377784	0.0327581
$134$	4.42864	0.382576
$135$	0	0
$136$	−3.80642	−0.326398
$137$	13.5669	1.15910	0.579550	−	0.814937i	$-0.303228\pi$
$137$	13.5669	1.15910	0.579550	+	0.814937i	$0.303228\pi$
$138$	5.59210	0.476032
$139$	6.01429	0.510125	0.255063	−	0.966925i	$-0.417904\pi$
$139$	6.01429	0.510125	0.255063	+	0.966925i	$0.417904\pi$
$140$	0	0
$141$	−1.59210	−0.134079
$142$	10.9032	0.914977
$143$	1.31111	0.109640
$144$	−1.37778	−0.114815
$145$	0	0
$146$	−27.5210	−2.27765
$147$	−1.00000	−0.0824786
$148$	3.80642	0.312886
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	0	0
$151$	−5.05086	−0.411033	−0.205516	−	0.978654i	$-0.565887\pi$
$151$	−5.05086	−0.411033	−0.205516	+	0.978654i	$0.565887\pi$
$152$	0.755569	0.0612847
$153$	−1.90321	−0.153866
$154$	−2.21432	−0.178435
$155$	0	0
$156$	3.80642	0.304758
$157$	8.88739	0.709291	0.354645	−	0.935001i	$-0.384602\pi$
$157$	8.88739	0.709291	0.354645	+	0.935001i	$0.384602\pi$
$158$	−2.14764	−0.170857
$159$	−11.4286	−0.906350
$160$	0	0
$161$	−2.52543	−0.199032
$162$	2.21432	0.173973
$163$	−1.31111	−0.102694	−0.0513469	−	0.998681i	$-0.516351\pi$
$163$	−1.31111	−0.102694	−0.0513469	+	0.998681i	$0.516351\pi$
$164$	−7.80642	−0.609579
$165$	0	0
$166$	14.8780	1.15476
$167$	15.1383	1.17143	0.585717	−	0.810515i	$-0.300813\pi$
$167$	15.1383	1.17143	0.585717	+	0.810515i	$0.300813\pi$
$168$	−2.00000	−0.154303
$169$	−11.2810	−0.867769
$170$	0	0
$171$	0.377784	0.0288899
$172$	−17.7605	−1.35422
$173$	−21.8622	−1.66215	−0.831076	−	0.556159i	$-0.812275\pi$
$173$	−21.8622	−1.66215	−0.831076	+	0.556159i	$0.812275\pi$
$174$	13.5462	1.02693
$175$	0	0
$176$	1.37778	0.103854
$177$	7.21432	0.542261
$178$	−25.1590	−1.88575
$179$	−15.8129	−1.18191	−0.590955	−	0.806705i	$-0.701249\pi$
$179$	−15.8129	−1.18191	−0.590955	+	0.806705i	$0.701249\pi$
$180$	0	0
$181$	−21.5812	−1.60412	−0.802059	−	0.597245i	$-0.796262\pi$
$181$	−21.5812	−1.60412	−0.802059	+	0.597245i	$0.796262\pi$
$182$	−2.90321	−0.215200
$183$	1.04593	0.0773176
$184$	−5.05086	−0.372354
$185$	0	0
$186$	11.0049	0.806920
$187$	1.90321	0.139177
$188$	4.62222	0.337110
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−5.53972	−0.400840	−0.200420	−	0.979710i	$-0.564231\pi$
$191$	−5.53972	−0.400840	−0.200420	+	0.979710i	$0.564231\pi$
$192$	12.8573	0.927894
$193$	9.31756	0.670693	0.335346	−	0.942095i	$-0.391147\pi$
$193$	9.31756	0.670693	0.335346	+	0.942095i	$0.391147\pi$
$194$	−18.9240	−1.35866
$195$	0	0
$196$	2.90321	0.207372
$197$	−27.0321	−1.92596	−0.962979	−	0.269575i	$-0.913117\pi$
$197$	−27.0321	−1.92596	−0.962979	+	0.269575i	$0.913117\pi$
$198$	−2.21432	−0.157365
$199$	−4.06022	−0.287822	−0.143911	−	0.989591i	$-0.545968\pi$
$199$	−4.06022	−0.287822	−0.143911	+	0.989591i	$0.545968\pi$
$200$	0	0
$201$	−2.00000	−0.141069
$202$	−10.5303	−0.740913
$203$	−6.11753	−0.429367
$204$	5.52543	0.386857
$205$	0	0
$206$	−25.0257	−1.74362
$207$	−2.52543	−0.175529
$208$	1.80642	0.125253
$209$	−0.377784	−0.0261319
$210$	0	0
$211$	5.41927	0.373078	0.186539	−	0.982448i	$-0.440273\pi$
$211$	5.41927	0.373078	0.186539	+	0.982448i	$0.440273\pi$
$212$	33.1798	2.27880
$213$	−4.92396	−0.337384
$214$	19.5669	1.33757
$215$	0	0
$216$	−2.00000	−0.136083
$217$	−4.96989	−0.337378
$218$	7.52543	0.509686
$219$	12.4286	0.839850
$220$	0	0
$221$	2.49532	0.167853
$222$	−2.90321	−0.194851
$223$	−18.2924	−1.22495	−0.612474	−	0.790491i	$-0.709826\pi$
$223$	−18.2924	−1.22495	−0.612474	+	0.790491i	$0.709826\pi$
$224$	−7.05086	−0.471105
$225$	0	0
$226$	−6.11261	−0.406605
$227$	9.17484	0.608956	0.304478	−	0.952519i	$-0.401518\pi$
$227$	9.17484	0.608956	0.304478	+	0.952519i	$0.401518\pi$
$228$	−1.09679	−0.0726366
$229$	0.112610	0.00744145	0.00372072	−	0.999993i	$-0.498816\pi$
$229$	0.112610	0.00744145	0.00372072	+	0.999993i	$0.498816\pi$
$230$	0	0
$231$	1.00000	0.0657952
$232$	−12.2351	−0.803271
$233$	20.3892	1.33574	0.667869	−	0.744279i	$-0.267207\pi$
$233$	20.3892	1.33574	0.667869	+	0.744279i	$0.267207\pi$
$234$	−2.90321	−0.189789
$235$	0	0
$236$	−20.9447	−1.36338
$237$	0.969888	0.0630010
$238$	−4.21432	−0.273174
$239$	25.0207	1.61846	0.809229	−	0.587494i	$-0.199885\pi$
$239$	25.0207	1.61846	0.809229	+	0.587494i	$0.199885\pi$
$240$	0	0
$241$	−1.18865	−0.0765679	−0.0382840	−	0.999267i	$-0.512189\pi$
$241$	−1.18865	−0.0765679	−0.0382840	+	0.999267i	$0.512189\pi$
$242$	2.21432	0.142342
$243$	−1.00000	−0.0641500
$244$	−3.03657	−0.194396
$245$	0	0
$246$	5.95407	0.379617
$247$	−0.495316	−0.0315162
$248$	−9.93978	−0.631176
$249$	−6.71900	−0.425800
$250$	0	0
$251$	−27.8163	−1.75575	−0.877874	−	0.478892i	$-0.841038\pi$
$251$	−27.8163	−1.75575	−0.877874	+	0.478892i	$0.841038\pi$
$252$	2.90321	0.182885
$253$	2.52543	0.158772
$254$	5.44446	0.341616
$255$	0	0
$256$	−6.10171	−0.381357
$257$	31.6336	1.97325	0.986625	−	0.163009i	$-0.0521199\pi$
$257$	31.6336	1.97325	0.986625	+	0.163009i	$0.0521199\pi$
$258$	13.5462	0.843348
$259$	1.31111	0.0814683
$260$	0	0
$261$	−6.11753	−0.378666
$262$	−12.7096	−0.785204
$263$	22.8573	1.40944	0.704720	−	0.709485i	$-0.251073\pi$
$263$	22.8573	1.40944	0.704720	+	0.709485i	$0.251073\pi$
$264$	2.00000	0.123091
$265$	0	0
$266$	0.836535	0.0512913
$267$	11.3620	0.695341
$268$	5.80642	0.354684
$269$	1.48103	0.0902997	0.0451499	−	0.998980i	$-0.485623\pi$
$269$	1.48103	0.0902997	0.0451499	+	0.998980i	$0.485623\pi$
$270$	0	0
$271$	−25.4844	−1.54807	−0.774034	−	0.633144i	$-0.781764\pi$
$271$	−25.4844	−1.54807	−0.774034	+	0.633144i	$0.781764\pi$
$272$	2.62222	0.158995
$273$	1.31111	0.0793519
$274$	30.0415	1.81487
$275$	0	0
$276$	7.33185	0.441326
$277$	−14.2395	−0.855569	−0.427785	−	0.903881i	$-0.640706\pi$
$277$	−14.2395	−0.855569	−0.427785	+	0.903881i	$0.640706\pi$
$278$	13.3176	0.798734
$279$	−4.96989	−0.297539
$280$	0	0
$281$	−2.23951	−0.133598	−0.0667990	−	0.997766i	$-0.521279\pi$
$281$	−2.23951	−0.133598	−0.0667990	+	0.997766i	$0.521279\pi$
$282$	−3.52543	−0.209936
$283$	−24.8321	−1.47611	−0.738057	−	0.674738i	$-0.764257\pi$
$283$	−24.8321	−1.47611	−0.738057	+	0.674738i	$0.764257\pi$
$284$	14.2953	0.848269
$285$	0	0
$286$	2.90321	0.171671
$287$	−2.68889	−0.158720
$288$	−7.05086	−0.415476
$289$	−13.3778	−0.786928
$290$	0	0
$291$	8.54617	0.500985
$292$	−36.0830	−2.11160
$293$	−25.6035	−1.49577	−0.747886	−	0.663828i	$-0.768931\pi$
$293$	−25.6035	−1.49577	−0.747886	+	0.663828i	$0.768931\pi$
$294$	−2.21432	−0.129142
$295$	0	0
$296$	2.62222	0.152413
$297$	1.00000	0.0580259
$298$	22.1432	1.28272
$299$	3.31111	0.191486
$300$	0	0
$301$	−6.11753	−0.352609
$302$	−11.1842	−0.643579
$303$	4.75557	0.273200
$304$	−0.520505	−0.0298530
$305$	0	0
$306$	−4.21432	−0.240917
$307$	5.09234	0.290635	0.145318	−	0.989385i	$-0.453580\pi$
$307$	5.09234	0.290635	0.145318	+	0.989385i	$0.453580\pi$
$308$	−2.90321	−0.165426
$309$	11.3017	0.642934
$310$	0	0
$311$	17.3274	0.982547	0.491274	−	0.871005i	$-0.336532\pi$
$311$	17.3274	0.982547	0.491274	+	0.871005i	$0.336532\pi$
$312$	2.62222	0.148454
$313$	15.4909	0.875596	0.437798	−	0.899073i	$-0.355759\pi$
$313$	15.4909	0.875596	0.437798	+	0.899073i	$0.355759\pi$
$314$	19.6795	1.11058
$315$	0	0
$316$	−2.81579	−0.158401
$317$	−12.4143	−0.697259	−0.348630	−	0.937261i	$-0.613353\pi$
$317$	−12.4143	−0.697259	−0.348630	+	0.937261i	$0.613353\pi$
$318$	−25.3067	−1.41913
$319$	6.11753	0.342516
$320$	0	0
$321$	−8.83654	−0.493207
$322$	−5.59210	−0.311636
$323$	−0.719004	−0.0400064
$324$	2.90321	0.161290
$325$	0	0
$326$	−2.90321	−0.160794
$327$	−3.39853	−0.187939
$328$	−5.37778	−0.296938
$329$	1.59210	0.0877755
$330$	0	0
$331$	32.5210	1.78751	0.893757	−	0.448551i	$-0.148060\pi$
$331$	32.5210	1.78751	0.893757	+	0.448551i	$0.148060\pi$
$332$	19.5067	1.07057
$333$	1.31111	0.0718483
$334$	33.5210	1.83419
$335$	0	0
$336$	1.37778	0.0751643
$337$	−3.06668	−0.167053	−0.0835263	−	0.996506i	$-0.526618\pi$
$337$	−3.06668	−0.167053	−0.0835263	+	0.996506i	$0.526618\pi$
$338$	−24.9797	−1.35872
$339$	2.76049	0.149929
$340$	0	0
$341$	4.96989	0.269135
$342$	0.836535	0.0452347
$343$	1.00000	0.0539949
$344$	−12.2351	−0.659670
$345$	0	0
$346$	−48.4099	−2.60253
$347$	23.7146	1.27306	0.636532	−	0.771250i	$-0.280368\pi$
$347$	23.7146	1.27306	0.636532	+	0.771250i	$0.280368\pi$
$348$	17.7605	0.952062
$349$	21.8385	1.16899	0.584495	−	0.811397i	$-0.301293\pi$
$349$	21.8385	1.16899	0.584495	+	0.811397i	$0.301293\pi$
$350$	0	0
$351$	1.31111	0.0699818
$352$	7.05086	0.375812
$353$	31.1941	1.66029	0.830146	−	0.557546i	$-0.188257\pi$
$353$	31.1941	1.66029	0.830146	+	0.557546i	$0.188257\pi$
$354$	15.9748	0.849052
$355$	0	0
$356$	−32.9862	−1.74826
$357$	1.90321	0.100729
$358$	−35.0148	−1.85059
$359$	−7.16839	−0.378333	−0.189166	−	0.981945i	$-0.560579\pi$
$359$	−7.16839	−0.378333	−0.189166	+	0.981945i	$0.560579\pi$
$360$	0	0
$361$	−18.8573	−0.992488
$362$	−47.7877	−2.51167
$363$	−1.00000	−0.0524864
$364$	−3.80642	−0.199511
$365$	0	0
$366$	2.31603	0.121061
$367$	17.9195	0.935391	0.467695	−	0.883890i	$-0.345084\pi$
$367$	17.9195	0.935391	0.467695	+	0.883890i	$0.345084\pi$
$368$	3.47949	0.181381
$369$	−2.68889	−0.139978
$370$	0	0
$371$	11.4286	0.593345
$372$	14.4286	0.748090
$373$	−2.22369	−0.115138	−0.0575691	−	0.998342i	$-0.518335\pi$
$373$	−2.22369	−0.115138	−0.0575691	+	0.998342i	$0.518335\pi$
$374$	4.21432	0.217917
$375$	0	0
$376$	3.18421	0.164213
$377$	8.02074	0.413089
$378$	−2.21432	−0.113892
$379$	17.6780	0.908058	0.454029	−	0.890987i	$-0.349986\pi$
$379$	17.6780	0.908058	0.454029	+	0.890987i	$0.349986\pi$
$380$	0	0
$381$	−2.45875	−0.125966
$382$	−12.2667	−0.627619
$383$	−16.8365	−0.860307	−0.430153	−	0.902756i	$-0.641541\pi$
$383$	−16.8365	−0.860307	−0.430153	+	0.902756i	$0.641541\pi$
$384$	14.3684	0.733235
$385$	0	0
$386$	20.6321	1.05014
$387$	−6.11753	−0.310972
$388$	−24.8113	−1.25961
$389$	−15.4859	−0.785169	−0.392584	−	0.919716i	$-0.628419\pi$
$389$	−15.4859	−0.785169	−0.392584	+	0.919716i	$0.628419\pi$
$390$	0	0
$391$	4.80642	0.243071
$392$	2.00000	0.101015
$393$	5.73975	0.289532
$394$	−59.8578	−3.01559
$395$	0	0
$396$	−2.90321	−0.145892
$397$	16.9032	0.848348	0.424174	−	0.905581i	$-0.360565\pi$
$397$	16.9032	0.848348	0.424174	+	0.905581i	$0.360565\pi$
$398$	−8.99063	−0.450660
$399$	−0.377784	−0.0189129
$400$	0	0
$401$	23.3210	1.16459	0.582296	−	0.812977i	$-0.302154\pi$
$401$	23.3210	1.16459	0.582296	+	0.812977i	$0.302154\pi$
$402$	−4.42864	−0.220880
$403$	6.51606	0.324588
$404$	−13.8064	−0.686895
$405$	0	0
$406$	−13.5462	−0.672285
$407$	−1.31111	−0.0649892
$408$	3.80642	0.188446
$409$	−2.60793	−0.128954	−0.0644768	−	0.997919i	$-0.520538\pi$
$409$	−2.60793	−0.128954	−0.0644768	+	0.997919i	$0.520538\pi$
$410$	0	0
$411$	−13.5669	−0.669207
$412$	−32.8113	−1.61650
$413$	−7.21432	−0.354993
$414$	−5.59210	−0.274837
$415$	0	0
$416$	9.24443	0.453246
$417$	−6.01429	−0.294521
$418$	−0.836535	−0.0409163
$419$	−15.4222	−0.753423	−0.376712	−	0.926331i	$-0.622945\pi$
$419$	−15.4222	−0.753423	−0.376712	+	0.926331i	$0.622945\pi$
$420$	0	0
$421$	6.28592	0.306357	0.153178	−	0.988199i	$-0.451049\pi$
$421$	6.28592	0.306357	0.153178	+	0.988199i	$0.451049\pi$
$422$	12.0000	0.584151
$423$	1.59210	0.0774108
$424$	22.8573	1.11005
$425$	0	0
$426$	−10.9032	−0.528262
$427$	−1.04593	−0.0506162
$428$	25.6543	1.24005
$429$	−1.31111	−0.0633009
$430$	0	0
$431$	16.3082	0.785538	0.392769	−	0.919637i	$-0.371517\pi$
$431$	16.3082	0.785538	0.392769	+	0.919637i	$0.371517\pi$
$432$	1.37778	0.0662887
$433$	14.8859	0.715369	0.357684	−	0.933843i	$-0.383566\pi$
$433$	14.8859	0.715369	0.357684	+	0.933843i	$0.383566\pi$
$434$	−11.0049	−0.528253
$435$	0	0
$436$	9.86665	0.472527
$437$	−0.954067	−0.0456392
$438$	27.5210	1.31500
$439$	10.3176	0.492430	0.246215	−	0.969215i	$-0.420813\pi$
$439$	10.3176	0.492430	0.246215	+	0.969215i	$0.420813\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	5.52543	0.262818
$443$	4.17929	0.198564	0.0992819	−	0.995059i	$-0.468345\pi$
$443$	4.17929	0.198564	0.0992819	+	0.995059i	$0.468345\pi$
$444$	−3.80642	−0.180645
$445$	0	0
$446$	−40.5052	−1.91797
$447$	−10.0000	−0.472984
$448$	−12.8573	−0.607449
$449$	−25.2192	−1.19017	−0.595085	−	0.803663i	$-0.702882\pi$
$449$	−25.2192	−1.19017	−0.595085	+	0.803663i	$0.702882\pi$
$450$	0	0
$451$	2.68889	0.126615
$452$	−8.01429	−0.376960
$453$	5.05086	0.237310
$454$	20.3160	0.953479
$455$	0	0
$456$	−0.755569	−0.0353827
$457$	27.5970	1.29093	0.645467	−	0.763788i	$-0.276663\pi$
$457$	27.5970	1.29093	0.645467	+	0.763788i	$0.276663\pi$
$458$	0.249353	0.0116515
$459$	1.90321	0.0888343
$460$	0	0
$461$	33.5812	1.56403	0.782016	−	0.623258i	$-0.214191\pi$
$461$	33.5812	1.56403	0.782016	+	0.623258i	$0.214191\pi$
$462$	2.21432	0.103019
$463$	−22.8889	−1.06374	−0.531869	−	0.846827i	$-0.678510\pi$
$463$	−22.8889	−1.06374	−0.531869	+	0.846827i	$0.678510\pi$
$464$	8.42864	0.391290
$465$	0	0
$466$	45.1481	2.09145
$467$	−6.38715	−0.295562	−0.147781	−	0.989020i	$-0.547213\pi$
$467$	−6.38715	−0.295562	−0.147781	+	0.989020i	$0.547213\pi$
$468$	−3.80642	−0.175952
$469$	2.00000	0.0923514
$470$	0	0
$471$	−8.88739	−0.409509
$472$	−14.4286	−0.664132
$473$	6.11753	0.281284
$474$	2.14764	0.0986445
$475$	0	0
$476$	−5.52543	−0.253258
$477$	11.4286	0.523281
$478$	55.4039	2.53412
$479$	−34.5970	−1.58078	−0.790389	−	0.612605i	$-0.790122\pi$
$479$	−34.5970	−1.58078	−0.790389	+	0.612605i	$0.790122\pi$
$480$	0	0
$481$	−1.71900	−0.0783798
$482$	−2.63206	−0.119887
$483$	2.52543	0.114911
$484$	2.90321	0.131964
$485$	0	0
$486$	−2.21432	−0.100444
$487$	−12.4572	−0.564491	−0.282245	−	0.959342i	$-0.591079\pi$
$487$	−12.4572	−0.564491	−0.282245	+	0.959342i	$0.591079\pi$
$488$	−2.09187	−0.0946943
$489$	1.31111	0.0592903
$490$	0	0
$491$	16.4271	0.741345	0.370673	−	0.928764i	$-0.379127\pi$
$491$	16.4271	0.741345	0.370673	+	0.928764i	$0.379127\pi$
$492$	7.80642	0.351941
$493$	11.6430	0.524373
$494$	−1.09679	−0.0493468
$495$	0	0
$496$	6.84743	0.307459
$497$	4.92396	0.220870
$498$	−14.8780	−0.666700
$499$	−37.2623	−1.66809	−0.834044	−	0.551698i	$-0.813980\pi$
$499$	−37.2623	−1.66809	−0.834044	+	0.551698i	$0.813980\pi$
$500$	0	0
$501$	−15.1383	−0.676328
$502$	−61.5941	−2.74908
$503$	9.23506	0.411771	0.205886	−	0.978576i	$-0.433993\pi$
$503$	9.23506	0.411771	0.205886	+	0.978576i	$0.433993\pi$
$504$	2.00000	0.0890871
$505$	0	0
$506$	5.59210	0.248599
$507$	11.2810	0.501007
$508$	7.13828	0.316710
$509$	−43.3847	−1.92299	−0.961497	−	0.274816i	$-0.911383\pi$
$509$	−43.3847	−1.92299	−0.961497	+	0.274816i	$0.911383\pi$
$510$	0	0
$511$	−12.4286	−0.549811
$512$	15.2257	0.672887
$513$	−0.377784	−0.0166796
$514$	70.0469	3.08964
$515$	0	0
$516$	17.7605	0.781862
$517$	−1.59210	−0.0700207
$518$	2.90321	0.127560
$519$	21.8622	0.959644
$520$	0	0
$521$	33.9195	1.48604	0.743020	−	0.669269i	$-0.233393\pi$
$521$	33.9195	1.48604	0.743020	+	0.669269i	$0.233393\pi$
$522$	−13.5462	−0.592900
$523$	−19.6731	−0.860243	−0.430122	−	0.902771i	$-0.641529\pi$
$523$	−19.6731	−0.860243	−0.430122	+	0.902771i	$0.641529\pi$
$524$	−16.6637	−0.727957
$525$	0	0
$526$	50.6133	2.20685
$527$	9.45875	0.412030
$528$	−1.37778	−0.0599604
$529$	−16.6222	−0.722705
$530$	0	0
$531$	−7.21432	−0.313075
$532$	1.09679	0.0475518
$533$	3.52543	0.152703
$534$	25.1590	1.08874
$535$	0	0
$536$	4.00000	0.172774
$537$	15.8129	0.682376
$538$	3.27946	0.141388
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−26.2844	−1.13005	−0.565027	−	0.825072i	$-0.691134\pi$
$541$	−26.2844	−1.13005	−0.565027	+	0.825072i	$0.691134\pi$
$542$	−56.4306	−2.42390
$543$	21.5812	0.926138
$544$	13.4193	0.575347
$545$	0	0
$546$	2.90321	0.124246
$547$	−3.43648	−0.146933	−0.0734666	−	0.997298i	$-0.523406\pi$
$547$	−3.43648	−0.146933	−0.0734666	+	0.997298i	$0.523406\pi$
$548$	39.3876	1.68256
$549$	−1.04593	−0.0446393
$550$	0	0
$551$	−2.31111	−0.0984565
$552$	5.05086	0.214979
$553$	−0.969888	−0.0412439
$554$	−31.5308	−1.33962
$555$	0	0
$556$	17.4608	0.740501
$557$	35.1941	1.49122	0.745610	−	0.666383i	$-0.232158\pi$
$557$	35.1941	1.49122	0.745610	+	0.666383i	$0.232158\pi$
$558$	−11.0049	−0.465876
$559$	8.02074	0.339241
$560$	0	0
$561$	−1.90321	−0.0803537
$562$	−4.95899	−0.209182
$563$	16.6780	0.702894	0.351447	−	0.936208i	$-0.385690\pi$
$563$	16.6780	0.702894	0.351447	+	0.936208i	$0.385690\pi$
$564$	−4.62222	−0.194630
$565$	0	0
$566$	−54.9862	−2.31124
$567$	1.00000	0.0419961
$568$	9.84791	0.413209
$569$	−41.9481	−1.75856	−0.879278	−	0.476309i	$-0.841974\pi$
$569$	−41.9481	−1.75856	−0.879278	+	0.476309i	$0.841974\pi$
$570$	0	0
$571$	−3.76694	−0.157642	−0.0788209	−	0.996889i	$-0.525116\pi$
$571$	−3.76694	−0.157642	−0.0788209	+	0.996889i	$0.525116\pi$
$572$	3.80642	0.159155
$573$	5.53972	0.231425
$574$	−5.95407	−0.248518
$575$	0	0
$576$	−12.8573	−0.535720
$577$	−43.5812	−1.81431	−0.907155	−	0.420797i	$-0.861750\pi$
$577$	−43.5812	−1.81431	−0.907155	+	0.420797i	$0.861750\pi$
$578$	−29.6227	−1.23214
$579$	−9.31756	−0.387225
$580$	0	0
$581$	6.71900	0.278751
$582$	18.9240	0.784423
$583$	−11.4286	−0.473326
$584$	−24.8573	−1.02860
$585$	0	0
$586$	−56.6943	−2.34202
$587$	20.6015	0.850314	0.425157	−	0.905120i	$-0.360219\pi$
$587$	20.6015	0.850314	0.425157	+	0.905120i	$0.360219\pi$
$588$	−2.90321	−0.119726
$589$	−1.87755	−0.0773629
$590$	0	0
$591$	27.0321	1.11195
$592$	−1.80642	−0.0742436
$593$	7.33185	0.301083	0.150542	−	0.988604i	$-0.451898\pi$
$593$	7.33185	0.301083	0.150542	+	0.988604i	$0.451898\pi$
$594$	2.21432	0.0908546
$595$	0	0
$596$	29.0321	1.18920
$597$	4.06022	0.166174
$598$	7.33185	0.299822
$599$	−2.16193	−0.0883342	−0.0441671	−	0.999024i	$-0.514063\pi$
$599$	−2.16193	−0.0883342	−0.0441671	+	0.999024i	$0.514063\pi$
$600$	0	0
$601$	−3.51606	−0.143423	−0.0717115	−	0.997425i	$-0.522846\pi$
$601$	−3.51606	−0.143423	−0.0717115	+	0.997425i	$0.522846\pi$
$602$	−13.5462	−0.552101
$603$	2.00000	0.0814463
$604$	−14.6637	−0.596658
$605$	0	0
$606$	10.5303	0.427766
$607$	43.2958	1.75732	0.878660	−	0.477447i	$-0.158438\pi$
$607$	43.2958	1.75732	0.878660	+	0.477447i	$0.158438\pi$
$608$	−2.66370	−0.108027
$609$	6.11753	0.247895
$610$	0	0
$611$	−2.08742	−0.0844480
$612$	−5.52543	−0.223352
$613$	36.3466	1.46803	0.734013	−	0.679135i	$-0.237645\pi$
$613$	36.3466	1.46803	0.734013	+	0.679135i	$0.237645\pi$
$614$	11.2761	0.455065
$615$	0	0
$616$	−2.00000	−0.0805823
$617$	−31.1896	−1.25565	−0.627823	−	0.778356i	$-0.716054\pi$
$617$	−31.1896	−1.25565	−0.627823	+	0.778356i	$0.716054\pi$
$618$	25.0257	1.00668
$619$	14.5906	0.586445	0.293222	−	0.956044i	$-0.405272\pi$
$619$	14.5906	0.586445	0.293222	+	0.956044i	$0.405272\pi$
$620$	0	0
$621$	2.52543	0.101342
$622$	38.3684	1.53843
$623$	−11.3620	−0.455207
$624$	−1.80642	−0.0723148
$625$	0	0
$626$	34.3017	1.37097
$627$	0.377784	0.0150873
$628$	25.8020	1.02961
$629$	−2.49532	−0.0994948
$630$	0	0
$631$	21.6365	0.861336	0.430668	−	0.902511i	$-0.358278\pi$
$631$	21.6365	0.861336	0.430668	+	0.902511i	$0.358278\pi$
$632$	−1.93978	−0.0771602
$633$	−5.41927	−0.215397
$634$	−27.4893	−1.09174
$635$	0	0
$636$	−33.1798	−1.31566
$637$	−1.31111	−0.0519480
$638$	13.5462	0.536298
$639$	4.92396	0.194789
$640$	0	0
$641$	−4.78415	−0.188963	−0.0944813	−	0.995527i	$-0.530119\pi$
$641$	−4.78415	−0.188963	−0.0944813	+	0.995527i	$0.530119\pi$
$642$	−19.5669	−0.772245
$643$	24.5832	0.969467	0.484734	−	0.874662i	$-0.338917\pi$
$643$	24.5832	0.969467	0.484734	+	0.874662i	$0.338917\pi$
$644$	−7.33185	−0.288915
$645$	0	0
$646$	−1.59210	−0.0626405
$647$	−19.0638	−0.749474	−0.374737	−	0.927131i	$-0.622267\pi$
$647$	−19.0638	−0.749474	−0.374737	+	0.927131i	$0.622267\pi$
$648$	2.00000	0.0785674
$649$	7.21432	0.283187
$650$	0	0
$651$	4.96989	0.194785
$652$	−3.80642	−0.149071
$653$	25.7654	1.00828	0.504139	−	0.863622i	$-0.331810\pi$
$653$	25.7654	1.00828	0.504139	+	0.863622i	$0.331810\pi$
$654$	−7.52543	−0.294268
$655$	0	0
$656$	3.70471	0.144645
$657$	−12.4286	−0.484887
$658$	3.52543	0.137435
$659$	50.0212	1.94855	0.974275	−	0.225362i	$-0.0723566\pi$
$659$	50.0212	1.94855	0.974275	+	0.225362i	$0.0723566\pi$
$660$	0	0
$661$	36.5195	1.42044	0.710221	−	0.703979i	$-0.248595\pi$
$661$	36.5195	1.42044	0.710221	+	0.703979i	$0.248595\pi$
$662$	72.0119	2.79882
$663$	−2.49532	−0.0969100
$664$	13.4380	0.521496
$665$	0	0
$666$	2.90321	0.112497
$667$	15.4494	0.598203
$668$	43.9496	1.70046
$669$	18.2924	0.707224
$670$	0	0
$671$	1.04593	0.0403778
$672$	7.05086	0.271993
$673$	27.5926	1.06362	0.531808	−	0.846865i	$-0.321513\pi$
$673$	27.5926	1.06362	0.531808	+	0.846865i	$0.321513\pi$
$674$	−6.79060	−0.261564
$675$	0	0
$676$	−32.7511	−1.25966
$677$	17.3511	0.666856	0.333428	−	0.942776i	$-0.391795\pi$
$677$	17.3511	0.666856	0.333428	+	0.942776i	$0.391795\pi$
$678$	6.11261	0.234753
$679$	−8.54617	−0.327972
$680$	0	0
$681$	−9.17484	−0.351581
$682$	11.0049	0.421400
$683$	21.8983	0.837915	0.418957	−	0.908006i	$-0.362396\pi$
$683$	21.8983	0.837915	0.418957	+	0.908006i	$0.362396\pi$
$684$	1.09679	0.0419367
$685$	0	0
$686$	2.21432	0.0845431
$687$	−0.112610	−0.00429632
$688$	8.42864	0.321339
$689$	−14.9842	−0.570852
$690$	0	0
$691$	−0.726989	−0.0276559	−0.0138280	−	0.999904i	$-0.504402\pi$
$691$	−0.726989	−0.0276559	−0.0138280	+	0.999904i	$0.504402\pi$
$692$	−63.4706	−2.41279
$693$	−1.00000	−0.0379869
$694$	52.5116	1.99331
$695$	0	0
$696$	12.2351	0.463769
$697$	5.11753	0.193840
$698$	48.3575	1.83036
$699$	−20.3892	−0.771189
$700$	0	0
$701$	27.9940	1.05732	0.528660	−	0.848834i	$-0.322695\pi$
$701$	27.9940	1.05732	0.528660	+	0.848834i	$0.322695\pi$
$702$	2.90321	0.109575
$703$	0.495316	0.0186812
$704$	12.8573	0.484577
$705$	0	0
$706$	69.0736	2.59962
$707$	−4.75557	−0.178852
$708$	20.9447	0.787150
$709$	−40.3319	−1.51469	−0.757347	−	0.653012i	$-0.773505\pi$
$709$	−40.3319	−1.51469	−0.757347	+	0.653012i	$0.773505\pi$
$710$	0	0
$711$	−0.969888	−0.0363737
$712$	−22.7239	−0.851615
$713$	12.5511	0.470042
$714$	4.21432	0.157717
$715$	0	0
$716$	−45.9081	−1.71567
$717$	−25.0207	−0.934417
$718$	−15.8731	−0.592379
$719$	−40.9418	−1.52687	−0.763435	−	0.645884i	$-0.776489\pi$
$719$	−40.9418	−1.52687	−0.763435	+	0.645884i	$0.776489\pi$
$720$	0	0
$721$	−11.3017	−0.420899
$722$	−41.7560	−1.55400
$723$	1.18865	0.0442065
$724$	−62.6548	−2.32855
$725$	0	0
$726$	−2.21432	−0.0821811
$727$	26.2464	0.973427	0.486713	−	0.873562i	$-0.338196\pi$
$727$	26.2464	0.973427	0.486713	+	0.873562i	$0.338196\pi$
$728$	−2.62222	−0.0971858
$729$	1.00000	0.0370370
$730$	0	0
$731$	11.6430	0.430630
$732$	3.03657	0.112235
$733$	24.0129	0.886937	0.443468	−	0.896290i	$-0.353748\pi$
$733$	24.0129	0.886937	0.443468	+	0.896290i	$0.353748\pi$
$734$	39.6795	1.46460
$735$	0	0
$736$	17.8064	0.656353
$737$	−2.00000	−0.0736709
$738$	−5.95407	−0.219172
$739$	−29.2958	−1.07766	−0.538831	−	0.842414i	$-0.681134\pi$
$739$	−29.2958	−1.07766	−0.538831	+	0.842414i	$0.681134\pi$
$740$	0	0
$741$	0.495316	0.0181959
$742$	25.3067	0.929037
$743$	−40.0830	−1.47050	−0.735251	−	0.677795i	$-0.762936\pi$
$743$	−40.0830	−1.47050	−0.735251	+	0.677795i	$0.762936\pi$
$744$	9.93978	0.364410
$745$	0	0
$746$	−4.92396	−0.180279
$747$	6.71900	0.245835
$748$	5.52543	0.202030
$749$	8.83654	0.322880
$750$	0	0
$751$	−12.2444	−0.446806	−0.223403	−	0.974726i	$-0.571717\pi$
$751$	−12.2444	−0.446806	−0.223403	+	0.974726i	$0.571717\pi$
$752$	−2.19358	−0.0799915
$753$	27.8163	1.01368
$754$	17.7605	0.646799
$755$	0	0
$756$	−2.90321	−0.105589
$757$	49.1655	1.78695	0.893475	−	0.449113i	$-0.148260\pi$
$757$	49.1655	1.78695	0.893475	+	0.449113i	$0.148260\pi$
$758$	39.1447	1.42180
$759$	−2.52543	−0.0916672
$760$	0	0
$761$	−37.6860	−1.36612	−0.683058	−	0.730364i	$-0.739350\pi$
$761$	−37.6860	−1.36612	−0.683058	+	0.730364i	$0.739350\pi$
$762$	−5.44446	−0.197232
$763$	3.39853	0.123035
$764$	−16.0830	−0.581862
$765$	0	0
$766$	−37.2815	−1.34703
$767$	9.45875	0.341536
$768$	6.10171	0.220177
$769$	−7.78415	−0.280704	−0.140352	−	0.990102i	$-0.544823\pi$
$769$	−7.78415	−0.280704	−0.140352	+	0.990102i	$0.544823\pi$
$770$	0	0
$771$	−31.6336	−1.13926
$772$	27.0509	0.973582
$773$	43.0450	1.54822	0.774111	−	0.633050i	$-0.218197\pi$
$773$	43.0450	1.54822	0.774111	+	0.633050i	$0.218197\pi$
$774$	−13.5462	−0.486907
$775$	0	0
$776$	−17.0923	−0.613579
$777$	−1.31111	−0.0470357
$778$	−34.2908	−1.22939
$779$	−1.01582	−0.0363956
$780$	0	0
$781$	−4.92396	−0.176193
$782$	10.6430	0.380591
$783$	6.11753	0.218623
$784$	−1.37778	−0.0492066
$785$	0	0
$786$	12.7096	0.453338
$787$	−9.21585	−0.328510	−0.164255	−	0.986418i	$-0.552522\pi$
$787$	−9.21585	−0.328510	−0.164255	+	0.986418i	$0.552522\pi$
$788$	−78.4800	−2.79573
$789$	−22.8573	−0.813741
$790$	0	0
$791$	−2.76049	−0.0981518
$792$	−2.00000	−0.0710669
$793$	1.37133	0.0486974
$794$	37.4291	1.32831
$795$	0	0
$796$	−11.7877	−0.417804
$797$	15.4982	0.548975	0.274488	−	0.961591i	$-0.411492\pi$
$797$	15.4982	0.548975	0.274488	+	0.961591i	$0.411492\pi$
$798$	−0.836535	−0.0296130
$799$	−3.03011	−0.107198
$800$	0	0
$801$	−11.3620	−0.401455
$802$	51.6400	1.82347
$803$	12.4286	0.438597
$804$	−5.80642	−0.204777
$805$	0	0
$806$	14.4286	0.508227
$807$	−1.48103	−0.0521346
$808$	−9.51114	−0.334601
$809$	−20.9304	−0.735874	−0.367937	−	0.929851i	$-0.619936\pi$
$809$	−20.9304	−0.735874	−0.367937	+	0.929851i	$0.619936\pi$
$810$	0	0
$811$	−49.2770	−1.73035	−0.865175	−	0.501470i	$-0.832793\pi$
$811$	−49.2770	−1.73035	−0.865175	+	0.501470i	$0.832793\pi$
$812$	−17.7605	−0.623271
$813$	25.4844	0.893778
$814$	−2.90321	−0.101758
$815$	0	0
$816$	−2.62222	−0.0917959
$817$	−2.31111	−0.0808554
$818$	−5.77478	−0.201910
$819$	−1.31111	−0.0458138
$820$	0	0
$821$	−44.1862	−1.54211	−0.771055	−	0.636769i	$-0.780271\pi$
$821$	−44.1862	−1.54211	−0.771055	+	0.636769i	$0.780271\pi$
$822$	−30.0415	−1.04782
$823$	−7.10462	−0.247652	−0.123826	−	0.992304i	$-0.539516\pi$
$823$	−7.10462	−0.247652	−0.123826	+	0.992304i	$0.539516\pi$
$824$	−22.6035	−0.787430
$825$	0	0
$826$	−15.9748	−0.555835
$827$	41.3461	1.43775	0.718873	−	0.695141i	$-0.244658\pi$
$827$	41.3461	1.43775	0.718873	+	0.695141i	$0.244658\pi$
$828$	−7.33185	−0.254799
$829$	−3.40943	−0.118414	−0.0592072	−	0.998246i	$-0.518857\pi$
$829$	−3.40943	−0.118414	−0.0592072	+	0.998246i	$0.518857\pi$
$830$	0	0
$831$	14.2395	0.493963
$832$	16.8573	0.584421
$833$	−1.90321	−0.0659424
$834$	−13.3176	−0.461149
$835$	0	0
$836$	−1.09679	−0.0379332
$837$	4.96989	0.171784
$838$	−34.1497	−1.17968
$839$	5.09526	0.175908	0.0879539	−	0.996125i	$-0.471967\pi$
$839$	5.09526	0.175908	0.0879539	+	0.996125i	$0.471967\pi$
$840$	0	0
$841$	8.42419	0.290489
$842$	13.9190	0.479682
$843$	2.23951	0.0771328
$844$	15.7333	0.541562
$845$	0	0
$846$	3.52543	0.121207
$847$	1.00000	0.0343604
$848$	−15.7462	−0.540727
$849$	24.8321	0.852235
$850$	0	0
$851$	−3.31111	−0.113503
$852$	−14.2953	−0.489748
$853$	−23.0257	−0.788384	−0.394192	−	0.919028i	$-0.628975\pi$
$853$	−23.0257	−0.788384	−0.394192	+	0.919028i	$0.628975\pi$
$854$	−2.31603	−0.0792529
$855$	0	0
$856$	17.6731	0.604053
$857$	12.6923	0.433560	0.216780	−	0.976220i	$-0.430445\pi$
$857$	12.6923	0.433560	0.216780	+	0.976220i	$0.430445\pi$
$858$	−2.90321	−0.0991140
$859$	41.9418	1.43104	0.715518	−	0.698595i	$-0.246191\pi$
$859$	41.9418	1.43104	0.715518	+	0.698595i	$0.246191\pi$
$860$	0	0
$861$	2.68889	0.0916372
$862$	36.1116	1.22996
$863$	45.1659	1.53747	0.768733	−	0.639569i	$-0.220887\pi$
$863$	45.1659	1.53747	0.768733	+	0.639569i	$0.220887\pi$
$864$	7.05086	0.239875
$865$	0	0
$866$	32.9621	1.12010
$867$	13.3778	0.454333
$868$	−14.4286	−0.489740
$869$	0.969888	0.0329012
$870$	0	0
$871$	−2.62222	−0.0888504
$872$	6.79706	0.230177
$873$	−8.54617	−0.289244
$874$	−2.11261	−0.0714601
$875$	0	0
$876$	36.0830	1.21913
$877$	−47.6671	−1.60960	−0.804802	−	0.593544i	$-0.797728\pi$
$877$	−47.6671	−1.60960	−0.804802	+	0.593544i	$0.797728\pi$
$878$	22.8464	0.771028
$879$	25.6035	0.863584
$880$	0	0
$881$	−8.53633	−0.287596	−0.143798	−	0.989607i	$-0.545932\pi$
$881$	−8.53633	−0.287596	−0.143798	+	0.989607i	$0.545932\pi$
$882$	2.21432	0.0745600
$883$	−17.6258	−0.593154	−0.296577	−	0.955009i	$-0.595845\pi$
$883$	−17.6258	−0.593154	−0.296577	+	0.955009i	$0.595845\pi$
$884$	7.24443	0.243657
$885$	0	0
$886$	9.25428	0.310903
$887$	−14.2761	−0.479344	−0.239672	−	0.970854i	$-0.577040\pi$
$887$	−14.2761	−0.479344	−0.239672	+	0.970854i	$0.577040\pi$
$888$	−2.62222	−0.0879958
$889$	2.45875	0.0824639
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	−53.1066	−1.77814
$893$	0.601472	0.0201275
$894$	−22.1432	−0.740579
$895$	0	0
$896$	−14.3684	−0.480015
$897$	−3.31111	−0.110555
$898$	−55.8435	−1.86352
$899$	30.4035	1.01401
$900$	0	0
$901$	−21.7511	−0.724635
$902$	5.95407	0.198249
$903$	6.11753	0.203579
$904$	−5.52098	−0.183625
$905$	0	0
$906$	11.1842	0.371570
$907$	46.8736	1.55641	0.778206	−	0.628009i	$-0.216130\pi$
$907$	46.8736	1.55641	0.778206	+	0.628009i	$0.216130\pi$
$908$	26.6365	0.883963
$909$	−4.75557	−0.157732
$910$	0	0
$911$	−14.5018	−0.480465	−0.240233	−	0.970715i	$-0.577224\pi$
$911$	−14.5018	−0.480465	−0.240233	+	0.970715i	$0.577224\pi$
$912$	0.520505	0.0172357
$913$	−6.71900	−0.222367
$914$	61.1086	2.02129
$915$	0	0
$916$	0.326929	0.0108020
$917$	−5.73975	−0.189543
$918$	4.21432	0.139093
$919$	0.990632	0.0326779	0.0163390	−	0.999867i	$-0.494799\pi$
$919$	0.990632	0.0326779	0.0163390	+	0.999867i	$0.494799\pi$
$920$	0	0
$921$	−5.09234	−0.167798
$922$	74.3595	2.44890
$923$	−6.45584	−0.212496
$924$	2.90321	0.0955087
$925$	0	0
$926$	−50.6834	−1.66556
$927$	−11.3017	−0.371198
$928$	43.1338	1.41594
$929$	−40.8943	−1.34170	−0.670850	−	0.741593i	$-0.734070\pi$
$929$	−40.8943	−1.34170	−0.670850	+	0.741593i	$0.734070\pi$
$930$	0	0
$931$	0.377784	0.0123814
$932$	59.1941	1.93897
$933$	−17.3274	−0.567274
$934$	−14.1432	−0.462780
$935$	0	0
$936$	−2.62222	−0.0857098
$937$	−16.5872	−0.541880	−0.270940	−	0.962596i	$-0.587334\pi$
$937$	−16.5872	−0.541880	−0.270940	+	0.962596i	$0.587334\pi$
$938$	4.42864	0.144600
$939$	−15.4909	−0.505525
$940$	0	0
$941$	35.9561	1.17213	0.586067	−	0.810262i	$-0.300675\pi$
$941$	35.9561	1.17213	0.586067	+	0.810262i	$0.300675\pi$
$942$	−19.6795	−0.641194
$943$	6.79060	0.221132
$944$	9.93978	0.323512
$945$	0	0
$946$	13.5462	0.440424
$947$	−41.9675	−1.36376	−0.681879	−	0.731465i	$-0.738837\pi$
$947$	−41.9675	−1.36376	−0.681879	+	0.731465i	$0.738837\pi$
$948$	2.81579	0.0914527
$949$	16.2953	0.528967
$950$	0	0
$951$	12.4143	0.402563
$952$	−3.80642	−0.123367
$953$	−7.83500	−0.253801	−0.126900	−	0.991915i	$-0.540503\pi$
$953$	−7.83500	−0.253801	−0.126900	+	0.991915i	$0.540503\pi$
$954$	25.3067	0.819333
$955$	0	0
$956$	72.6405	2.34936
$957$	−6.11753	−0.197752
$958$	−76.6089	−2.47512
$959$	13.5669	0.438099
$960$	0	0
$961$	−6.30021	−0.203233
$962$	−3.80642	−0.122724
$963$	8.83654	0.284753
$964$	−3.45091	−0.111146
$965$	0	0
$966$	5.59210	0.179923
$967$	29.5783	0.951174	0.475587	−	0.879669i	$-0.342236\pi$
$967$	29.5783	0.951174	0.475587	+	0.879669i	$0.342236\pi$
$968$	2.00000	0.0642824
$969$	0.719004	0.0230977
$970$	0	0
$971$	13.3749	0.429220	0.214610	−	0.976700i	$-0.431152\pi$
$971$	13.3749	0.429220	0.214610	+	0.976700i	$0.431152\pi$
$972$	−2.90321	−0.0931206
$973$	6.01429	0.192809
$974$	−27.5843	−0.883857
$975$	0	0
$976$	1.44107	0.0461275
$977$	−57.1245	−1.82757	−0.913787	−	0.406194i	$-0.866856\pi$
$977$	−57.1245	−1.82757	−0.913787	+	0.406194i	$0.866856\pi$
$978$	2.90321	0.0928345
$979$	11.3620	0.363130
$980$	0	0
$981$	3.39853	0.108507
$982$	36.3749	1.16077
$983$	11.4924	0.366551	0.183275	−	0.983062i	$-0.441330\pi$
$983$	11.4924	0.366551	0.183275	+	0.983062i	$0.441330\pi$
$984$	5.37778	0.171438
$985$	0	0
$986$	25.7812	0.821042
$987$	−1.59210	−0.0506772
$988$	−1.43801	−0.0457491
$989$	15.4494	0.491262
$990$	0	0
$991$	23.4987	0.746461	0.373231	−	0.927739i	$-0.378250\pi$
$991$	23.4987	0.746461	0.373231	+	0.927739i	$0.378250\pi$
$992$	35.0420	1.11258
$993$	−32.5210	−1.03202
$994$	10.9032	0.345829
$995$	0	0
$996$	−19.5067	−0.618093
$997$	−24.3970	−0.772661	−0.386330	−	0.922360i	$-0.626258\pi$
$997$	−24.3970	−0.772661	−0.386330	+	0.922360i	$0.626258\pi$
$998$	−82.5106	−2.61183
$999$	−1.31111	−0.0414816

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5775.2.a.bt.1.3		3
5.2	odd	4		1155.2.c.c.694.6	yes	6
5.3	odd	4		1155.2.c.c.694.1	✓	6
5.4	even	2		5775.2.a.bu.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1155.2.c.c.694.1	✓	6	5.3	odd	4
1155.2.c.c.694.6	yes	6	5.2	odd	4
5775.2.a.bt.1.3		3	1.1	even	1	trivial
5775.2.a.bu.1.1		3	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:	\( N \)	\(=\)	\( 5775 = 3 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5775.a (trivial)

\(f(q)\)	\(=\)	\(q+2.21432 q^{2} -1.00000 q^{3} +2.90321 q^{4} -2.21432 q^{6} +1.00000 q^{7} +2.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.21432 q^{2} -1.00000 q^{3} +2.90321 q^{4} -2.21432 q^{6} +1.00000 q^{7} +2.00000 q^{8} +1.00000 q^{9} -1.00000 q^{11} -2.90321 q^{12} -1.31111 q^{13} +2.21432 q^{14} -1.37778 q^{16} -1.90321 q^{17} +2.21432 q^{18} +0.377784 q^{19} -1.00000 q^{21} -2.21432 q^{22} -2.52543 q^{23} -2.00000 q^{24} -2.90321 q^{26} -1.00000 q^{27} +2.90321 q^{28} -6.11753 q^{29} -4.96989 q^{31} -7.05086 q^{32} +1.00000 q^{33} -4.21432 q^{34} +2.90321 q^{36} +1.31111 q^{37} +0.836535 q^{38} +1.31111 q^{39} -2.68889 q^{41} -2.21432 q^{42} -6.11753 q^{43} -2.90321 q^{44} -5.59210 q^{46} +1.59210 q^{47} +1.37778 q^{48} +1.00000 q^{49} +1.90321 q^{51} -3.80642 q^{52} +11.4286 q^{53} -2.21432 q^{54} +2.00000 q^{56} -0.377784 q^{57} -13.5462 q^{58} -7.21432 q^{59} -1.04593 q^{61} -11.0049 q^{62} +1.00000 q^{63} -12.8573 q^{64} +2.21432 q^{66} +2.00000 q^{67} -5.52543 q^{68} +2.52543 q^{69} +4.92396 q^{71} +2.00000 q^{72} -12.4286 q^{73} +2.90321 q^{74} +1.09679 q^{76} -1.00000 q^{77} +2.90321 q^{78} -0.969888 q^{79} +1.00000 q^{81} -5.95407 q^{82} +6.71900 q^{83} -2.90321 q^{84} -13.5462 q^{86} +6.11753 q^{87} -2.00000 q^{88} -11.3620 q^{89} -1.31111 q^{91} -7.33185 q^{92} +4.96989 q^{93} +3.52543 q^{94} +7.05086 q^{96} -8.54617 q^{97} +2.21432 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 2 q^{4} + 3 q^{7} + 6 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + 2 * q^4 + 3 * q^7 + 6 * q^8 + 3 * q^9	\( 3 q - 3 q^{3} + 2 q^{4} + 3 q^{7} + 6 q^{8} + 3 q^{9} - 3 q^{11} - 2 q^{12} - 4 q^{13} - 4 q^{16} + q^{17} + q^{19} - 3 q^{21} - q^{23} - 6 q^{24} - 2 q^{26} - 3 q^{27} + 2 q^{28} - 5 q^{29} - 8 q^{31} - 8 q^{32} + 3 q^{33} - 6 q^{34} + 2 q^{36} + 4 q^{37} - 4 q^{38} + 4 q^{39} - 8 q^{41} - 5 q^{43} - 2 q^{44} - 10 q^{46} - 2 q^{47} + 4 q^{48} + 3 q^{49} - q^{51} + 2 q^{52} + 21 q^{53} + 6 q^{56} - q^{57} - 14 q^{58} - 15 q^{59} - 23 q^{61} + 3 q^{63} - 12 q^{64} + 6 q^{67} - 10 q^{68} + q^{69} - 12 q^{71} + 6 q^{72} - 24 q^{73} + 2 q^{74} + 10 q^{76} - 3 q^{77} + 2 q^{78} + 4 q^{79} + 3 q^{81} + 2 q^{82} + 27 q^{83} - 2 q^{84} - 14 q^{86} + 5 q^{87} - 6 q^{88} - 21 q^{89} - 4 q^{91} - 2 q^{92} + 8 q^{93} + 4 q^{94} + 8 q^{96} + q^{97} - 3 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 2 * q^4 + 3 * q^7 + 6 * q^8 + 3 * q^9 - 3 * q^11 - 2 * q^12 - 4 * q^13 - 4 * q^16 + q^17 + q^19 - 3 * q^21 - q^23 - 6 * q^24 - 2 * q^26 - 3 * q^27 + 2 * q^28 - 5 * q^29 - 8 * q^31 - 8 * q^32 + 3 * q^33 - 6 * q^34 + 2 * q^36 + 4 * q^37 - 4 * q^38 + 4 * q^39 - 8 * q^41 - 5 * q^43 - 2 * q^44 - 10 * q^46 - 2 * q^47 + 4 * q^48 + 3 * q^49 - q^51 + 2 * q^52 + 21 * q^53 + 6 * q^56 - q^57 - 14 * q^58 - 15 * q^59 - 23 * q^61 + 3 * q^63 - 12 * q^64 + 6 * q^67 - 10 * q^68 + q^69 - 12 * q^71 + 6 * q^72 - 24 * q^73 + 2 * q^74 + 10 * q^76 - 3 * q^77 + 2 * q^78 + 4 * q^79 + 3 * q^81 + 2 * q^82 + 27 * q^83 - 2 * q^84 - 14 * q^86 + 5 * q^87 - 6 * q^88 - 21 * q^89 - 4 * q^91 - 2 * q^92 + 8 * q^93 + 4 * q^94 + 8 * q^96 + q^97 - 3 * q^99

Introduction

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