LMFDB - Embedded newform 5775.2.a.bu.1.1

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			1.1
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	$-0.786250\pi$
$2$	−2.21432	−1.56576	−0.782880	+	0.622172i	$0.786250\pi$
$3$	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	$-0.441802\pi$
$13$	1.31111	0.363636	0.181818	+	0.983332i	$0.441802\pi$
$14$	2.21432	0.591802
$15$	0	0
$16$	−1.37778	−0.344446
$17$	1.90321	0.461597	0.230798	−	0.973002i	$-0.425866\pi$
$17$	1.90321	0.461597	0.230798	+	0.973002i	$0.425866\pi$
$18$	−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	$-0.415191\pi$
$23$	2.52543	0.526588	0.263294	+	0.964716i	$0.415191\pi$
$24$	−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	$-0.534372\pi$
$37$	−1.31111	−0.215545	−0.107772	+	0.994176i	$0.534372\pi$
$38$	−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	$-0.345530\pi$
$43$	6.11753	0.932915	0.466457	+	0.884544i	$0.345530\pi$
$44$	−2.90321	−0.437676
$45$	0	0
$46$	−5.59210	−0.824511
$47$	−1.59210	−0.232232	−0.116116	−	0.993236i	$-0.537044\pi$
$47$	−1.59210	−0.232232	−0.116116	+	0.993236i	$0.537044\pi$
$48$	−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	$-0.787297\pi$
$53$	−11.4286	−1.56984	−0.784922	+	0.619594i	$0.787297\pi$
$54$	−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	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	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	$-0.240760\pi$
$73$	12.4286	1.45466	0.727331	+	0.686287i	$0.240760\pi$
$74$	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	$-0.620215\pi$
$83$	−6.71900	−0.737506	−0.368753	+	0.929527i	$0.620215\pi$
$84$	−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	$-0.357149\pi$
$97$	8.54617	0.867732	0.433866	+	0.900977i	$0.357149\pi$
$98$	−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	$-0.312030\pi$
$103$	11.3017	1.11359	0.556797	+	0.830649i	$0.312030\pi$
$104$	−2.62222	−0.257129
$105$	0	0
$106$	25.3067	2.45800
$107$	−8.83654	−0.854260	−0.427130	−	0.904190i	$-0.640475\pi$
$107$	−8.83654	−0.854260	−0.427130	+	0.904190i	$0.640475\pi$
$108$	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	$-0.458553\pi$
$113$	2.76049	0.259685	0.129843	+	0.991535i	$0.458553\pi$
$114$	−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	$-0.534793\pi$
$127$	−2.45875	−0.218179	−0.109089	+	0.994032i	$0.534793\pi$
$128$	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	$-0.696772\pi$
$137$	−13.5669	−1.15910	−0.579550	+	0.814937i	$0.696772\pi$
$138$	−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	$-0.615398\pi$
$157$	−8.88739	−0.709291	−0.354645	+	0.935001i	$0.615398\pi$
$158$	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	$-0.483649\pi$
$163$	1.31111	0.102694	0.0513469	+	0.998681i	$0.483649\pi$
$164$	−7.80642	−0.609579
$165$	0	0
$166$	14.8780	1.15476
$167$	−15.1383	−1.17143	−0.585717	−	0.810515i	$-0.699187\pi$
$167$	−15.1383	−1.17143	−0.585717	+	0.810515i	$0.699187\pi$
$168$	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	$-0.187725\pi$
$173$	21.8622	1.66215	0.831076	+	0.556159i	$0.187725\pi$
$174$	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	$-0.608853\pi$
$193$	−9.31756	−0.670693	−0.335346	+	0.942095i	$0.608853\pi$
$194$	−18.9240	−1.35866
$195$	0	0
$196$	2.90321	0.207372
$197$	27.0321	1.92596	0.962979	−	0.269575i	$-0.0868832\pi$
$197$	27.0321	1.92596	0.962979	+	0.269575i	$0.0868832\pi$
$198$	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	$-0.290174\pi$
$223$	18.2924	1.22495	0.612474	+	0.790491i	$0.290174\pi$
$224$	−7.05086	−0.471105
$225$	0	0
$226$	−6.11261	−0.406605
$227$	−9.17484	−0.608956	−0.304478	−	0.952519i	$-0.598482\pi$
$227$	−9.17484	−0.608956	−0.304478	+	0.952519i	$0.598482\pi$
$228$	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	$-0.732793\pi$
$233$	−20.3892	−1.33574	−0.667869	+	0.744279i	$0.732793\pi$
$234$	−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	$-0.947880\pi$
$257$	−31.6336	−1.97325	−0.986625	+	0.163009i	$0.947880\pi$
$258$	−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	$-0.748927\pi$
$263$	−22.8573	−1.40944	−0.704720	+	0.709485i	$0.748927\pi$
$264$	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	$-0.359294\pi$
$277$	14.2395	0.855569	0.427785	+	0.903881i	$0.359294\pi$
$278$	−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	$-0.235743\pi$
$283$	24.8321	1.47611	0.738057	+	0.674738i	$0.235743\pi$
$284$	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	$-0.231069\pi$
$293$	25.6035	1.49577	0.747886	+	0.663828i	$0.231069\pi$
$294$	−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	$-0.546420\pi$
$307$	−5.09234	−0.290635	−0.145318	+	0.989385i	$0.546420\pi$
$308$	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	$-0.644241\pi$
$313$	−15.4909	−0.875596	−0.437798	+	0.899073i	$0.644241\pi$
$314$	19.6795	1.11058
$315$	0	0
$316$	−2.81579	−0.158401
$317$	12.4143	0.697259	0.348630	−	0.937261i	$-0.386647\pi$
$317$	12.4143	0.697259	0.348630	+	0.937261i	$0.386647\pi$
$318$	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	$-0.473382\pi$
$337$	3.06668	0.167053	0.0835263	+	0.996506i	$0.473382\pi$
$338$	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	$-0.719632\pi$
$347$	−23.7146	−1.27306	−0.636532	+	0.771250i	$0.719632\pi$
$348$	−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	$-0.811743\pi$
$353$	−31.1941	−1.66029	−0.830146	+	0.557546i	$0.811743\pi$
$354$	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	$-0.654916\pi$
$367$	−17.9195	−0.935391	−0.467695	+	0.883890i	$0.654916\pi$
$368$	−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	$-0.481665\pi$
$373$	2.22369	0.115138	0.0575691	+	0.998342i	$0.481665\pi$
$374$	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	$-0.358459\pi$
$383$	16.8365	0.860307	0.430153	+	0.902756i	$0.358459\pi$
$384$	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	$-0.639435\pi$
$397$	−16.9032	−0.848348	−0.424174	+	0.905581i	$0.639435\pi$
$398$	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	$-0.616434\pi$
$433$	−14.8859	−0.715369	−0.357684	+	0.933843i	$0.616434\pi$
$434$	−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	$-0.531655\pi$
$443$	−4.17929	−0.198564	−0.0992819	+	0.995059i	$0.531655\pi$
$444$	−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	$-0.723337\pi$
$457$	−27.5970	−1.29093	−0.645467	+	0.763788i	$0.723337\pi$
$458$	−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	$-0.321490\pi$
$463$	22.8889	1.06374	0.531869	+	0.846827i	$0.321490\pi$
$464$	8.42864	0.391290
$465$	0	0
$466$	45.1481	2.09145
$467$	6.38715	0.295562	0.147781	−	0.989020i	$-0.452787\pi$
$467$	6.38715	0.295562	0.147781	+	0.989020i	$0.452787\pi$
$468$	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	$-0.408921\pi$
$487$	12.4572	0.564491	0.282245	+	0.959342i	$0.408921\pi$
$488$	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	$-0.566007\pi$
$503$	−9.23506	−0.411771	−0.205886	+	0.978576i	$0.566007\pi$
$504$	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	$-0.358471\pi$
$523$	19.6731	0.860243	0.430122	+	0.902771i	$0.358471\pi$
$524$	−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	$-0.476594\pi$
$547$	3.43648	0.146933	0.0734666	+	0.997298i	$0.476594\pi$
$548$	−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	$-0.767842\pi$
$557$	−35.1941	−1.49122	−0.745610	+	0.666383i	$0.767842\pi$
$558$	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	$-0.614310\pi$
$563$	−16.6780	−0.702894	−0.351447	+	0.936208i	$0.614310\pi$
$564$	−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	$-0.138250\pi$
$577$	43.5812	1.81431	0.907155	+	0.420797i	$0.138250\pi$
$578$	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	$-0.639781\pi$
$587$	−20.6015	−0.850314	−0.425157	+	0.905120i	$0.639781\pi$
$588$	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	$-0.548102\pi$
$593$	−7.33185	−0.301083	−0.150542	+	0.988604i	$0.548102\pi$
$594$	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	$-0.841562\pi$
$607$	−43.2958	−1.75732	−0.878660	+	0.477447i	$0.841562\pi$
$608$	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	$-0.762355\pi$
$613$	−36.3466	−1.46803	−0.734013	+	0.679135i	$0.762355\pi$
$614$	11.2761	0.455065
$615$	0	0
$616$	−2.00000	−0.0805823
$617$	31.1896	1.25565	0.627823	−	0.778356i	$-0.283946\pi$
$617$	31.1896	1.25565	0.627823	+	0.778356i	$0.283946\pi$
$618$	−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	$-0.661083\pi$
$643$	−24.5832	−0.969467	−0.484734	+	0.874662i	$0.661083\pi$
$644$	−7.33185	−0.288915
$645$	0	0
$646$	−1.59210	−0.0626405
$647$	19.0638	0.749474	0.374737	−	0.927131i	$-0.377733\pi$
$647$	19.0638	0.749474	0.374737	+	0.927131i	$0.377733\pi$
$648$	−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	$-0.668190\pi$
$653$	−25.7654	−1.00828	−0.504139	+	0.863622i	$0.668190\pi$
$654$	−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	$-0.678487\pi$
$673$	−27.5926	−1.06362	−0.531808	+	0.846865i	$0.678487\pi$
$674$	−6.79060	−0.261564
$675$	0	0
$676$	−32.7511	−1.25966
$677$	−17.3511	−0.666856	−0.333428	−	0.942776i	$-0.608205\pi$
$677$	−17.3511	−0.666856	−0.333428	+	0.942776i	$0.608205\pi$
$678$	−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	$-0.637604\pi$
$683$	−21.8983	−0.837915	−0.418957	+	0.908006i	$0.637604\pi$
$684$	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	$-0.661804\pi$
$727$	−26.2464	−0.973427	−0.486713	+	0.873562i	$0.661804\pi$
$728$	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	$-0.646252\pi$
$733$	−24.0129	−0.886937	−0.443468	+	0.896290i	$0.646252\pi$
$734$	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	$-0.237064\pi$
$743$	40.0830	1.47050	0.735251	+	0.677795i	$0.237064\pi$
$744$	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	$-0.851740\pi$
$757$	−49.1655	−1.78695	−0.893475	+	0.449113i	$0.851740\pi$
$758$	−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	$-0.781803\pi$
$773$	−43.0450	−1.54822	−0.774111	+	0.633050i	$0.781803\pi$
$774$	−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	$-0.447478\pi$
$787$	9.21585	0.328510	0.164255	+	0.986418i	$0.447478\pi$
$788$	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	$-0.588508\pi$
$797$	−15.4982	−0.548975	−0.274488	+	0.961591i	$0.588508\pi$
$798$	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	$-0.460484\pi$
$823$	7.10462	0.247652	0.123826	+	0.992304i	$0.460484\pi$
$824$	−22.6035	−0.787430
$825$	0	0
$826$	−15.9748	−0.555835
$827$	−41.3461	−1.43775	−0.718873	−	0.695141i	$-0.755342\pi$
$827$	−41.3461	−1.43775	−0.718873	+	0.695141i	$0.755342\pi$
$828$	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	$-0.371025\pi$
$853$	23.0257	0.788384	0.394192	+	0.919028i	$0.371025\pi$
$854$	−2.31603	−0.0792529
$855$	0	0
$856$	17.6731	0.604053
$857$	−12.6923	−0.433560	−0.216780	−	0.976220i	$-0.569555\pi$
$857$	−12.6923	−0.433560	−0.216780	+	0.976220i	$0.569555\pi$
$858$	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	$-0.779113\pi$
$863$	−45.1659	−1.53747	−0.768733	+	0.639569i	$0.779113\pi$
$864$	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	$-0.202272\pi$
$877$	47.6671	1.60960	0.804802	+	0.593544i	$0.202272\pi$
$878$	−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	$-0.404155\pi$
$883$	17.6258	0.593154	0.296577	+	0.955009i	$0.404155\pi$
$884$	7.24443	0.243657
$885$	0	0
$886$	9.25428	0.310903
$887$	14.2761	0.479344	0.239672	−	0.970854i	$-0.422960\pi$
$887$	14.2761	0.479344	0.239672	+	0.970854i	$0.422960\pi$
$888$	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	$-0.783870\pi$
$907$	−46.8736	−1.55641	−0.778206	+	0.628009i	$0.783870\pi$
$908$	−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	$-0.412666\pi$
$937$	16.5872	0.541880	0.270940	+	0.962596i	$0.412666\pi$
$938$	−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	$-0.261163\pi$
$947$	41.9675	1.36376	0.681879	+	0.731465i	$0.261163\pi$
$948$	−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	$-0.459497\pi$
$953$	7.83500	0.253801	0.126900	+	0.991915i	$0.459497\pi$
$954$	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	$-0.657764\pi$
$967$	−29.5783	−0.951174	−0.475587	+	0.879669i	$0.657764\pi$
$968$	−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	$-0.133144\pi$
$977$	57.1245	1.82757	0.913787	+	0.406194i	$0.133144\pi$
$978$	−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	$-0.558670\pi$
$983$	−11.4924	−0.366551	−0.183275	+	0.983062i	$0.558670\pi$
$984$	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	$-0.373742\pi$
$997$	24.3970	0.772661	0.386330	+	0.922360i	$0.373742\pi$
$998$	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.bu.1.1		3
5.2	odd	4		1155.2.c.c.694.1	✓	6
5.3	odd	4		1155.2.c.c.694.6	yes	6
5.4	even	2		5775.2.a.bt.1.3		3

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

L-functions

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