LMFDB - Embedded newform 5577.2.a.bc.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5577,2,Mod(1,5577)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5577, 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("5577.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5577 = 3 \cdot 11 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5577.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:	$44.5325692073$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} - 14 x^{10} + 13 x^{9} + 70 x^{8} - 61 x^{7} - 152 x^{6} + 127 x^{5} + 138 x^{4} + \cdots + 1 $ x^12 - x^11 - 14x^10 + 13x^9 + 70x^8 - 61x^7 - 152x^6 + 127x^5 + 138x^4 - 110x^3 - 40x^2 + 29x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$0.645300$ of defining polynomial
Character	$\chi$	$=$	5577.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.645300 q^{2} -1.00000 q^{3} -1.58359 q^{4} -1.52961 q^{5} -0.645300 q^{6} -1.04662 q^{7} -2.31249 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.645300 q^{2} -1.00000 q^{3} -1.58359 q^{4} -1.52961 q^{5} -0.645300 q^{6} -1.04662 q^{7} -2.31249 q^{8} +1.00000 q^{9} -0.987060 q^{10} +1.00000 q^{11} +1.58359 q^{12} -0.675382 q^{14} +1.52961 q^{15} +1.67492 q^{16} +4.81161 q^{17} +0.645300 q^{18} -1.94773 q^{19} +2.42228 q^{20} +1.04662 q^{21} +0.645300 q^{22} -4.04134 q^{23} +2.31249 q^{24} -2.66028 q^{25} -1.00000 q^{27} +1.65741 q^{28} -3.70073 q^{29} +0.987060 q^{30} +1.80069 q^{31} +5.70581 q^{32} -1.00000 q^{33} +3.10493 q^{34} +1.60092 q^{35} -1.58359 q^{36} +7.65554 q^{37} -1.25687 q^{38} +3.53721 q^{40} -1.30748 q^{41} +0.675382 q^{42} +3.96306 q^{43} -1.58359 q^{44} -1.52961 q^{45} -2.60788 q^{46} +9.48276 q^{47} -1.67492 q^{48} -5.90460 q^{49} -1.71668 q^{50} -4.81161 q^{51} +5.64847 q^{53} -0.645300 q^{54} -1.52961 q^{55} +2.42029 q^{56} +1.94773 q^{57} -2.38809 q^{58} -2.62254 q^{59} -2.42228 q^{60} +5.34894 q^{61} +1.16198 q^{62} -1.04662 q^{63} +0.332114 q^{64} -0.645300 q^{66} +1.26790 q^{67} -7.61960 q^{68} +4.04134 q^{69} +1.03307 q^{70} +8.20926 q^{71} -2.31249 q^{72} +8.10366 q^{73} +4.94012 q^{74} +2.66028 q^{75} +3.08440 q^{76} -1.04662 q^{77} +13.5719 q^{79} -2.56198 q^{80} +1.00000 q^{81} -0.843720 q^{82} -3.84882 q^{83} -1.65741 q^{84} -7.35990 q^{85} +2.55737 q^{86} +3.70073 q^{87} -2.31249 q^{88} -2.89507 q^{89} -0.987060 q^{90} +6.39982 q^{92} -1.80069 q^{93} +6.11923 q^{94} +2.97927 q^{95} -5.70581 q^{96} -8.29891 q^{97} -3.81024 q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + q^{2} - 12 q^{3} + 5 q^{4} - 6 q^{5} - q^{6} - q^{7} + 12 q^{9}+O(q^{10}) $ 12 * q + q^2 - 12 * q^3 + 5 * q^4 - 6 * q^5 - q^6 - q^7 + 12 * q^9	$ 12 q + q^{2} - 12 q^{3} + 5 q^{4} - 6 q^{5} - q^{6} - q^{7} + 12 q^{9} - 11 q^{10} + 12 q^{11} - 5 q^{12} + 9 q^{14} + 6 q^{15} - 9 q^{16} + 3 q^{17} + q^{18} + 6 q^{19} - 20 q^{20} + q^{21} + q^{22} - 22 q^{23} - 2 q^{25} - 12 q^{27} - 11 q^{28} + 13 q^{29} + 11 q^{30} + 2 q^{31} + 11 q^{32} - 12 q^{33} + 10 q^{34} - 14 q^{35} + 5 q^{36} + 3 q^{37} - 18 q^{38} - 18 q^{40} + 4 q^{41} - 9 q^{42} - 26 q^{43} + 5 q^{44} - 6 q^{45} - 18 q^{46} - 9 q^{47} + 9 q^{48} - 3 q^{49} + 29 q^{50} - 3 q^{51} - 5 q^{53} - q^{54} - 6 q^{55} + 5 q^{56} - 6 q^{57} + 37 q^{58} - 22 q^{59} + 20 q^{60} - 11 q^{61} - 18 q^{62} - q^{63} - 10 q^{64} - q^{66} - 28 q^{67} + 12 q^{68} + 22 q^{69} - 29 q^{70} + 10 q^{71} - 7 q^{73} - 6 q^{74} + 2 q^{75} + 32 q^{76} - q^{77} - 40 q^{79} + 30 q^{80} + 12 q^{81} + 26 q^{82} + q^{83} + 11 q^{84} - 32 q^{85} - 9 q^{86} - 13 q^{87} + 11 q^{89} - 11 q^{90} - 38 q^{92} - 2 q^{93} - 25 q^{94} - 10 q^{95} - 11 q^{96} + 7 q^{97} + 28 q^{98} + 12 q^{99}+O(q^{100}) $ 12 * q + q^2 - 12 * q^3 + 5 * q^4 - 6 * q^5 - q^6 - q^7 + 12 * q^9 - 11 * q^10 + 12 * q^11 - 5 * q^12 + 9 * q^14 + 6 * q^15 - 9 * q^16 + 3 * q^17 + q^18 + 6 * q^19 - 20 * q^20 + q^21 + q^22 - 22 * q^23 - 2 * q^25 - 12 * q^27 - 11 * q^28 + 13 * q^29 + 11 * q^30 + 2 * q^31 + 11 * q^32 - 12 * q^33 + 10 * q^34 - 14 * q^35 + 5 * q^36 + 3 * q^37 - 18 * q^38 - 18 * q^40 + 4 * q^41 - 9 * q^42 - 26 * q^43 + 5 * q^44 - 6 * q^45 - 18 * q^46 - 9 * q^47 + 9 * q^48 - 3 * q^49 + 29 * q^50 - 3 * q^51 - 5 * q^53 - q^54 - 6 * q^55 + 5 * q^56 - 6 * q^57 + 37 * q^58 - 22 * q^59 + 20 * q^60 - 11 * q^61 - 18 * q^62 - q^63 - 10 * q^64 - q^66 - 28 * q^67 + 12 * q^68 + 22 * q^69 - 29 * q^70 + 10 * q^71 - 7 * q^73 - 6 * q^74 + 2 * q^75 + 32 * q^76 - q^77 - 40 * q^79 + 30 * q^80 + 12 * q^81 + 26 * q^82 + q^83 + 11 * q^84 - 32 * q^85 - 9 * q^86 - 13 * q^87 + 11 * q^89 - 11 * q^90 - 38 * q^92 - 2 * q^93 - 25 * q^94 - 10 * q^95 - 11 * q^96 + 7 * q^97 + 28 * q^98 + 12 * 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 $

$2$	0.645300	0.456296	0.228148	−	0.973626i	$-0.426733\pi$
$2$	0.645300	0.456296	0.228148	+	0.973626i	$0.426733\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.58359	−0.791794
$5$	−1.52961	−0.684064	−0.342032	−	0.939688i	$-0.611115\pi$
$5$	−1.52961	−0.684064	−0.342032	+	0.939688i	$0.611115\pi$
$6$	−0.645300	−0.263443
$7$	−1.04662	−0.395584	−0.197792	−	0.980244i	$-0.563377\pi$
$7$	−1.04662	−0.395584	−0.197792	+	0.980244i	$0.563377\pi$
$8$	−2.31249	−0.817589
$9$	1.00000	0.333333
$10$	−0.987060	−0.312136
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	1.58359	0.457142
$13$	0	0
$13$	0	0
$14$	−0.675382	−0.180503
$15$	1.52961	0.394944
$16$	1.67492	0.418731
$17$	4.81161	1.16699	0.583493	−	0.812118i	$-0.301685\pi$
$17$	4.81161	1.16699	0.583493	+	0.812118i	$0.301685\pi$
$18$	0.645300	0.152099
$19$	−1.94773	−0.446839	−0.223420	−	0.974722i	$-0.571722\pi$
$19$	−1.94773	−0.446839	−0.223420	+	0.974722i	$0.571722\pi$
$20$	2.42228	0.541637
$21$	1.04662	0.228390
$22$	0.645300	0.137579
$23$	−4.04134	−0.842679	−0.421339	−	0.906903i	$-0.638440\pi$
$23$	−4.04134	−0.842679	−0.421339	+	0.906903i	$0.638440\pi$
$24$	2.31249	0.472035
$25$	−2.66028	−0.532057
$26$	0	0
$27$	−1.00000	−0.192450
$28$	1.65741	0.313221
$29$	−3.70073	−0.687209	−0.343605	−	0.939114i	$-0.611648\pi$
$29$	−3.70073	−0.687209	−0.343605	+	0.939114i	$0.611648\pi$
$30$	0.987060	0.180212
$31$	1.80069	0.323413	0.161706	−	0.986839i	$-0.448300\pi$
$31$	1.80069	0.323413	0.161706	+	0.986839i	$0.448300\pi$
$32$	5.70581	1.00865
$33$	−1.00000	−0.174078
$34$	3.10493	0.532492
$35$	1.60092	0.270604
$36$	−1.58359	−0.263931
$37$	7.65554	1.25856	0.629281	−	0.777178i	$-0.283349\pi$
$37$	7.65554	1.25856	0.629281	+	0.777178i	$0.283349\pi$
$38$	−1.25687	−0.203891
$39$	0	0
$40$	3.53721	0.559283
$41$	−1.30748	−0.204195	−0.102097	−	0.994774i	$-0.532555\pi$
$41$	−1.30748	−0.204195	−0.102097	+	0.994774i	$0.532555\pi$
$42$	0.675382	0.104214
$43$	3.96306	0.604361	0.302181	−	0.953251i	$-0.402285\pi$
$43$	3.96306	0.604361	0.302181	+	0.953251i	$0.402285\pi$
$44$	−1.58359	−0.238735
$45$	−1.52961	−0.228021
$46$	−2.60788	−0.384511
$47$	9.48276	1.38320	0.691602	−	0.722279i	$-0.256905\pi$
$47$	9.48276	1.38320	0.691602	+	0.722279i	$0.256905\pi$
$48$	−1.67492	−0.241754
$49$	−5.90460	−0.843514
$50$	−1.71668	−0.242776
$51$	−4.81161	−0.673760
$52$	0	0
$53$	5.64847	0.775878	0.387939	−	0.921685i	$-0.373187\pi$
$53$	5.64847	0.775878	0.387939	+	0.921685i	$0.373187\pi$
$54$	−0.645300	−0.0878143
$55$	−1.52961	−0.206253
$56$	2.42029	0.323425
$57$	1.94773	0.257983
$58$	−2.38809	−0.313571
$59$	−2.62254	−0.341426	−0.170713	−	0.985321i	$-0.554607\pi$
$59$	−2.62254	−0.341426	−0.170713	+	0.985321i	$0.554607\pi$
$60$	−2.42228	−0.312714
$61$	5.34894	0.684862	0.342431	−	0.939543i	$-0.388750\pi$
$61$	5.34894	0.684862	0.342431	+	0.939543i	$0.388750\pi$
$62$	1.16198	0.147572
$63$	−1.04662	−0.131861
$64$	0.332114	0.0415143
$65$	0	0
$66$	−0.645300	−0.0794310
$67$	1.26790	0.154898	0.0774490	−	0.996996i	$-0.475322\pi$
$67$	1.26790	0.154898	0.0774490	+	0.996996i	$0.475322\pi$
$68$	−7.61960	−0.924013
$69$	4.04134	0.486521
$70$	1.03307	0.123476
$71$	8.20926	0.974260	0.487130	−	0.873330i	$-0.338044\pi$
$71$	8.20926	0.974260	0.487130	+	0.873330i	$0.338044\pi$
$72$	−2.31249	−0.272530
$73$	8.10366	0.948462	0.474231	−	0.880401i	$-0.342726\pi$
$73$	8.10366	0.948462	0.474231	+	0.880401i	$0.342726\pi$
$74$	4.94012	0.574278
$75$	2.66028	0.307183
$76$	3.08440	0.353805
$77$	−1.04662	−0.119273
$78$	0	0
$79$	13.5719	1.52696	0.763478	−	0.645834i	$-0.223490\pi$
$79$	13.5719	1.52696	0.763478	+	0.645834i	$0.223490\pi$
$80$	−2.56198	−0.286439
$81$	1.00000	0.111111
$82$	−0.843720	−0.0931733
$83$	−3.84882	−0.422463	−0.211232	−	0.977436i	$-0.567747\pi$
$83$	−3.84882	−0.422463	−0.211232	+	0.977436i	$0.567747\pi$
$84$	−1.65741	−0.180838
$85$	−7.35990	−0.798293
$86$	2.55737	0.275768
$87$	3.70073	0.396760
$88$	−2.31249	−0.246512
$89$	−2.89507	−0.306876	−0.153438	−	0.988158i	$-0.549035\pi$
$89$	−2.89507	−0.306876	−0.153438	+	0.988158i	$0.549035\pi$
$90$	−0.987060	−0.104045
$91$	0	0
$92$	6.39982	0.667228
$93$	−1.80069	−0.186722
$94$	6.11923	0.631150
$95$	2.97927	0.305667
$96$	−5.70581	−0.582347
$97$	−8.29891	−0.842626	−0.421313	−	0.906915i	$-0.638431\pi$
$97$	−8.29891	−0.842626	−0.421313	+	0.906915i	$0.638431\pi$
$98$	−3.81024	−0.384892
$99$	1.00000	0.100504
$100$	4.21279	0.421279
$101$	−5.30237	−0.527606	−0.263803	−	0.964577i	$-0.584977\pi$
$101$	−5.30237	−0.527606	−0.263803	+	0.964577i	$0.584977\pi$
$102$	−3.10493	−0.307434
$103$	−14.4233	−1.42117	−0.710584	−	0.703612i	$-0.751569\pi$
$103$	−14.4233	−1.42117	−0.710584	+	0.703612i	$0.751569\pi$
$104$	0	0
$105$	−1.60092	−0.156233
$106$	3.64496	0.354030
$107$	−9.89563	−0.956647	−0.478323	−	0.878184i	$-0.658755\pi$
$107$	−9.89563	−0.956647	−0.478323	+	0.878184i	$0.658755\pi$
$108$	1.58359	0.152381
$109$	−5.62995	−0.539251	−0.269626	−	0.962965i	$-0.586900\pi$
$109$	−5.62995	−0.539251	−0.269626	+	0.962965i	$0.586900\pi$
$110$	−0.987060	−0.0941125
$111$	−7.65554	−0.726632
$112$	−1.75300	−0.165643
$113$	−10.5981	−0.996984	−0.498492	−	0.866894i	$-0.666113\pi$
$113$	−10.5981	−0.996984	−0.498492	+	0.866894i	$0.666113\pi$
$114$	1.25687	0.117717
$115$	6.18169	0.576446
$116$	5.86044	0.544128
$117$	0	0
$118$	−1.69233	−0.155791
$119$	−5.03591	−0.461641
$120$	−3.53721	−0.322902
$121$	1.00000	0.0909091
$122$	3.45167	0.312500
$123$	1.30748	0.117892
$124$	−2.85154	−0.256076
$125$	11.7173	1.04802
$126$	−0.675382	−0.0601678
$127$	−1.37203	−0.121748	−0.0608738	−	0.998145i	$-0.519389\pi$
$127$	−1.37203	−0.121748	−0.0608738	+	0.998145i	$0.519389\pi$
$128$	−11.1973	−0.989711
$129$	−3.96306	−0.348928
$130$	0	0
$131$	10.5736	0.923823	0.461912	−	0.886926i	$-0.347164\pi$
$131$	10.5736	0.923823	0.461912	+	0.886926i	$0.347164\pi$
$132$	1.58359	0.137834
$133$	2.03852	0.176762
$134$	0.818173	0.0706794
$135$	1.52961	0.131648
$136$	−11.1268	−0.954115
$137$	3.21030	0.274275	0.137137	−	0.990552i	$-0.456210\pi$
$137$	3.21030	0.274275	0.137137	+	0.990552i	$0.456210\pi$
$138$	2.60788	0.221998
$139$	−13.5273	−1.14737	−0.573686	−	0.819075i	$-0.694487\pi$
$139$	−13.5273	−1.14737	−0.573686	+	0.819075i	$0.694487\pi$
$140$	−2.53519	−0.214263
$141$	−9.48276	−0.798593
$142$	5.29744	0.444551
$143$	0	0
$144$	1.67492	0.139577
$145$	5.66069	0.470095
$146$	5.22929	0.432780
$147$	5.90460	0.487003
$148$	−12.1232	−0.996522
$149$	−10.4277	−0.854267	−0.427134	−	0.904189i	$-0.640477\pi$
$149$	−10.4277	−0.854267	−0.427134	+	0.904189i	$0.640477\pi$
$150$	1.71668	0.140167
$151$	−10.5698	−0.860157	−0.430078	−	0.902792i	$-0.641514\pi$
$151$	−10.5698	−0.860157	−0.430078	+	0.902792i	$0.641514\pi$
$152$	4.50410	0.365331
$153$	4.81161	0.388995
$154$	−0.675382	−0.0544238
$155$	−2.75435	−0.221235
$156$	0	0
$157$	12.2812	0.980147	0.490073	−	0.871681i	$-0.336970\pi$
$157$	12.2812	0.980147	0.490073	+	0.871681i	$0.336970\pi$
$158$	8.75794	0.696744
$159$	−5.64847	−0.447953
$160$	−8.72768	−0.689984
$161$	4.22973	0.333350
$162$	0.645300	0.0506996
$163$	10.0543	0.787516	0.393758	−	0.919214i	$-0.371175\pi$
$163$	10.0543	0.787516	0.393758	+	0.919214i	$0.371175\pi$
$164$	2.07051	0.161680
$165$	1.52961	0.119080
$166$	−2.48365	−0.192768
$167$	−10.4117	−0.805682	−0.402841	−	0.915270i	$-0.631977\pi$
$167$	−10.4117	−0.805682	−0.402841	+	0.915270i	$0.631977\pi$
$168$	−2.42029	−0.186729
$169$	0	0
$170$	−4.74934	−0.364258
$171$	−1.94773	−0.148946
$172$	−6.27586	−0.478530
$173$	−6.60938	−0.502502	−0.251251	−	0.967922i	$-0.580842\pi$
$173$	−6.60938	−0.502502	−0.251251	+	0.967922i	$0.580842\pi$
$174$	2.38809	0.181040
$175$	2.78430	0.210473
$176$	1.67492	0.126252
$177$	2.62254	0.197122
$178$	−1.86819	−0.140027
$179$	−25.1584	−1.88042	−0.940212	−	0.340590i	$-0.889373\pi$
$179$	−25.1584	−1.88042	−0.940212	+	0.340590i	$0.889373\pi$
$180$	2.42228	0.180546
$181$	−15.6416	−1.16263	−0.581314	−	0.813680i	$-0.697461\pi$
$181$	−15.6416	−1.16263	−0.581314	+	0.813680i	$0.697461\pi$
$182$	0	0
$183$	−5.34894	−0.395405
$184$	9.34557	0.688965
$185$	−11.7100	−0.860937
$186$	−1.16198	−0.0852008
$187$	4.81161	0.351860
$188$	−15.0168	−1.09521
$189$	1.04662	0.0761301
$190$	1.92252	0.139475
$191$	−15.5248	−1.12333	−0.561666	−	0.827364i	$-0.689839\pi$
$191$	−15.5248	−1.12333	−0.561666	+	0.827364i	$0.689839\pi$
$192$	−0.332114	−0.0239683
$193$	−14.7089	−1.05877	−0.529385	−	0.848381i	$-0.677577\pi$
$193$	−14.7089	−1.05877	−0.529385	+	0.848381i	$0.677577\pi$
$194$	−5.35529	−0.384487
$195$	0	0
$196$	9.35044	0.667889
$197$	2.45124	0.174644	0.0873220	−	0.996180i	$-0.472169\pi$
$197$	2.45124	0.174644	0.0873220	+	0.996180i	$0.472169\pi$
$198$	0.645300	0.0458595
$199$	−12.8327	−0.909688	−0.454844	−	0.890571i	$-0.650305\pi$
$199$	−12.8327	−0.909688	−0.454844	+	0.890571i	$0.650305\pi$
$200$	6.15188	0.435004
$201$	−1.26790	−0.0894304
$202$	−3.42162	−0.240745
$203$	3.87325	0.271849
$204$	7.61960	0.533479
$205$	1.99994	0.139682
$206$	−9.30735	−0.648474
$207$	−4.04134	−0.280893
$208$	0	0
$209$	−1.94773	−0.134727
$210$	−1.03307	−0.0712888
$211$	−19.6202	−1.35071	−0.675356	−	0.737491i	$-0.736010\pi$
$211$	−19.6202	−1.35071	−0.675356	+	0.737491i	$0.736010\pi$
$212$	−8.94485	−0.614335
$213$	−8.20926	−0.562489
$214$	−6.38565	−0.436514
$215$	−6.06195	−0.413422
$216$	2.31249	0.157345
$217$	−1.88463	−0.127937
$218$	−3.63301	−0.246058
$219$	−8.10366	−0.547595
$220$	2.42228	0.163310
$221$	0	0
$222$	−4.94012	−0.331559
$223$	−26.6417	−1.78406	−0.892029	−	0.451979i	$-0.850718\pi$
$223$	−26.6417	−1.78406	−0.892029	+	0.451979i	$0.850718\pi$
$224$	−5.97179	−0.399007
$225$	−2.66028	−0.177352
$226$	−6.83895	−0.454920
$227$	−10.6290	−0.705468	−0.352734	−	0.935724i	$-0.614748\pi$
$227$	−10.6290	−0.705468	−0.352734	+	0.935724i	$0.614748\pi$
$228$	−3.08440	−0.204269
$229$	0.164187	0.0108498	0.00542488	−	0.999985i	$-0.498273\pi$
$229$	0.164187	0.0108498	0.00542488	+	0.999985i	$0.498273\pi$
$230$	3.98905	0.263030
$231$	1.04662	0.0688623
$232$	8.55791	0.561854
$233$	11.3669	0.744673	0.372337	−	0.928098i	$-0.378557\pi$
$233$	11.3669	0.744673	0.372337	+	0.928098i	$0.378557\pi$
$234$	0	0
$235$	−14.5050	−0.946199
$236$	4.15303	0.270339
$237$	−13.5719	−0.881588
$238$	−3.24967	−0.210645
$239$	−21.9048	−1.41690	−0.708451	−	0.705760i	$-0.750606\pi$
$239$	−21.9048	−1.41690	−0.708451	+	0.705760i	$0.750606\pi$
$240$	2.56198	0.165375
$241$	−20.3921	−1.31357	−0.656787	−	0.754076i	$-0.728085\pi$
$241$	−20.3921	−1.31357	−0.656787	+	0.754076i	$0.728085\pi$
$242$	0.645300	0.0414815
$243$	−1.00000	−0.0641500
$244$	−8.47051	−0.542269
$245$	9.03174	0.577017
$246$	0.843720	0.0537936
$247$	0	0
$248$	−4.16407	−0.264419
$249$	3.84882	0.243909
$250$	7.56116	0.478210
$251$	7.55764	0.477034	0.238517	−	0.971138i	$-0.423339\pi$
$251$	7.55764	0.477034	0.238517	+	0.971138i	$0.423339\pi$
$252$	1.65741	0.104407
$253$	−4.04134	−0.254077
$254$	−0.885369	−0.0555530
$255$	7.35990	0.460895
$256$	−7.88985	−0.493116
$257$	15.6870	0.978531	0.489265	−	0.872135i	$-0.337265\pi$
$257$	15.6870	0.978531	0.489265	+	0.872135i	$0.337265\pi$
$258$	−2.55737	−0.159215
$259$	−8.01241	−0.497867
$260$	0	0
$261$	−3.70073	−0.229070
$262$	6.82317	0.421537
$263$	6.88124	0.424316	0.212158	−	0.977235i	$-0.431951\pi$
$263$	6.88124	0.424316	0.212158	+	0.977235i	$0.431951\pi$
$264$	2.31249	0.142324
$265$	−8.63998	−0.530750
$266$	1.31546	0.0806560
$267$	2.89507	0.177175
$268$	−2.00782	−0.122647
$269$	9.32248	0.568401	0.284201	−	0.958765i	$-0.408272\pi$
$269$	9.32248	0.568401	0.284201	+	0.958765i	$0.408272\pi$
$270$	0.987060	0.0600705
$271$	27.9034	1.69501	0.847505	−	0.530787i	$-0.178103\pi$
$271$	27.9034	1.69501	0.847505	+	0.530787i	$0.178103\pi$
$272$	8.05908	0.488653
$273$	0	0
$274$	2.07161	0.125151
$275$	−2.66028	−0.160421
$276$	−6.39982	−0.385224
$277$	−12.9095	−0.775654	−0.387827	−	0.921732i	$-0.626774\pi$
$277$	−12.9095	−0.775654	−0.387827	+	0.921732i	$0.626774\pi$
$278$	−8.72919	−0.523542
$279$	1.80069	0.107804
$280$	−3.70210	−0.221243
$281$	19.3205	1.15257	0.576283	−	0.817250i	$-0.304503\pi$
$281$	19.3205	1.15257	0.576283	+	0.817250i	$0.304503\pi$
$282$	−6.11923	−0.364395
$283$	−30.0065	−1.78370	−0.891852	−	0.452328i	$-0.850594\pi$
$283$	−30.0065	−1.78370	−0.891852	+	0.452328i	$0.850594\pi$
$284$	−13.0001	−0.771413
$285$	−2.97927	−0.176477
$286$	0	0
$287$	1.36843	0.0807761
$288$	5.70581	0.336218
$289$	6.15158	0.361857
$290$	3.65285	0.214502
$291$	8.29891	0.486491
$292$	−12.8329	−0.750986
$293$	29.2614	1.70947	0.854735	−	0.519065i	$-0.173720\pi$
$293$	29.2614	1.70947	0.854735	+	0.519065i	$0.173720\pi$
$294$	3.81024	0.222218
$295$	4.01148	0.233557
$296$	−17.7034	−1.02899
$297$	−1.00000	−0.0580259
$298$	−6.72897	−0.389799
$299$	0	0
$300$	−4.21279	−0.243226
$301$	−4.14780	−0.239075
$302$	−6.82069	−0.392486
$303$	5.30237	0.304613
$304$	−3.26230	−0.187105
$305$	−8.18181	−0.468489
$306$	3.10493	0.177497
$307$	13.1223	0.748927	0.374464	−	0.927242i	$-0.377827\pi$
$307$	13.1223	0.748927	0.374464	+	0.927242i	$0.377827\pi$
$308$	1.65741	0.0944396
$309$	14.4233	0.820512
$310$	−1.77738	−0.100949
$311$	−9.14166	−0.518376	−0.259188	−	0.965827i	$-0.583455\pi$
$311$	−9.14166	−0.518376	−0.259188	+	0.965827i	$0.583455\pi$
$312$	0	0
$313$	13.0032	0.734982	0.367491	−	0.930027i	$-0.380217\pi$
$313$	13.0032	0.734982	0.367491	+	0.930027i	$0.380217\pi$
$314$	7.92507	0.447237
$315$	1.60092	0.0902014
$316$	−21.4923	−1.20903
$317$	1.66130	0.0933081	0.0466540	−	0.998911i	$-0.485144\pi$
$317$	1.66130	0.0933081	0.0466540	+	0.998911i	$0.485144\pi$
$318$	−3.64496	−0.204399
$319$	−3.70073	−0.207201
$320$	−0.508006	−0.0283984
$321$	9.89563	0.552320
$322$	2.72945	0.152106
$323$	−9.37170	−0.521456
$324$	−1.58359	−0.0879771
$325$	0	0
$326$	6.48807	0.359341
$327$	5.62995	0.311337
$328$	3.02354	0.166947
$329$	−9.92481	−0.547172
$330$	0.987060	0.0543358
$331$	8.91786	0.490170	0.245085	−	0.969502i	$-0.421184\pi$
$331$	8.91786	0.490170	0.245085	+	0.969502i	$0.421184\pi$
$332$	6.09495	0.334504
$333$	7.65554	0.419521
$334$	−6.71868	−0.367630
$335$	−1.93939	−0.105960
$336$	1.75300	0.0956341
$337$	16.1182	0.878014	0.439007	−	0.898484i	$-0.355330\pi$
$337$	16.1182	0.878014	0.439007	+	0.898484i	$0.355330\pi$
$338$	0	0
$339$	10.5981	0.575609
$340$	11.6550	0.632083
$341$	1.80069	0.0975126
$342$	−1.25687	−0.0679637
$343$	13.5062	0.729264
$344$	−9.16454	−0.494119
$345$	−6.18169	−0.332811
$346$	−4.26504	−0.229290
$347$	18.8024	1.00937	0.504683	−	0.863305i	$-0.331609\pi$
$347$	18.8024	1.00937	0.504683	+	0.863305i	$0.331609\pi$
$348$	−5.86044	−0.314152
$349$	10.5308	0.563699	0.281850	−	0.959459i	$-0.409052\pi$
$349$	10.5308	0.563699	0.281850	+	0.959459i	$0.409052\pi$
$350$	1.79671	0.0960381
$351$	0	0
$352$	5.70581	0.304121
$353$	−12.5042	−0.665533	−0.332766	−	0.943009i	$-0.607982\pi$
$353$	−12.5042	−0.665533	−0.332766	+	0.943009i	$0.607982\pi$
$354$	1.69233	0.0899462
$355$	−12.5570	−0.666456
$356$	4.58459	0.242983
$357$	5.03591	0.266528
$358$	−16.2347	−0.858030
$359$	13.4100	0.707755	0.353877	−	0.935292i	$-0.384863\pi$
$359$	13.4100	0.707755	0.353877	+	0.935292i	$0.384863\pi$
$360$	3.53721	0.186428
$361$	−15.2064	−0.800335
$362$	−10.0935	−0.530503
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	−12.3955	−0.648808
$366$	−3.45167	−0.180422
$367$	−8.51561	−0.444511	−0.222255	−	0.974988i	$-0.571342\pi$
$367$	−8.51561	−0.444511	−0.222255	+	0.974988i	$0.571342\pi$
$368$	−6.76894	−0.352856
$369$	−1.30748	−0.0680649
$370$	−7.55647	−0.392842
$371$	−5.91178	−0.306924
$372$	2.85154	0.147846
$373$	−1.08791	−0.0563298	−0.0281649	−	0.999603i	$-0.508966\pi$
$373$	−1.08791	−0.0563298	−0.0281649	+	0.999603i	$0.508966\pi$
$374$	3.10493	0.160552
$375$	−11.7173	−0.605077
$376$	−21.9288	−1.13089
$377$	0	0
$378$	0.675382	0.0347379
$379$	30.1686	1.54966	0.774829	−	0.632170i	$-0.217836\pi$
$379$	30.1686	1.54966	0.774829	+	0.632170i	$0.217836\pi$
$380$	−4.71793	−0.242025
$381$	1.37203	0.0702910
$382$	−10.0181	−0.512572
$383$	14.8323	0.757895	0.378948	−	0.925418i	$-0.376286\pi$
$383$	14.8323	0.757895	0.378948	+	0.925418i	$0.376286\pi$
$384$	11.1973	0.571410
$385$	1.60092	0.0815903
$386$	−9.49167	−0.483113
$387$	3.96306	0.201454
$388$	13.1420	0.667186
$389$	−14.5775	−0.739111	−0.369556	−	0.929209i	$-0.620490\pi$
$389$	−14.5775	−0.739111	−0.369556	+	0.929209i	$0.620490\pi$
$390$	0	0
$391$	−19.4454	−0.983395
$392$	13.6543	0.689647
$393$	−10.5736	−0.533369
$394$	1.58179	0.0796894
$395$	−20.7597	−1.04453
$396$	−1.58359	−0.0795783
$397$	26.3452	1.32223	0.661115	−	0.750285i	$-0.270084\pi$
$397$	26.3452	1.32223	0.661115	+	0.750285i	$0.270084\pi$
$398$	−8.28097	−0.415087
$399$	−2.03852	−0.102054
$400$	−4.45577	−0.222789
$401$	−10.8946	−0.544050	−0.272025	−	0.962290i	$-0.587693\pi$
$401$	−10.8946	−0.544050	−0.272025	+	0.962290i	$0.587693\pi$
$402$	−0.818173	−0.0408068
$403$	0	0
$404$	8.39677	0.417755
$405$	−1.52961	−0.0760071
$406$	2.49941	0.124044
$407$	7.65554	0.379471
$408$	11.1268	0.550859
$409$	15.7220	0.777405	0.388702	−	0.921363i	$-0.372923\pi$
$409$	15.7220	0.777405	0.388702	+	0.921363i	$0.372923\pi$
$410$	1.29056	0.0637364
$411$	−3.21030	−0.158353
$412$	22.8405	1.12527
$413$	2.74480	0.135063
$414$	−2.60788	−0.128170
$415$	5.88721	0.288992
$416$	0	0
$417$	13.5273	0.662436
$418$	−1.25687	−0.0614755
$419$	16.9576	0.828432	0.414216	−	0.910179i	$-0.364056\pi$
$419$	16.9576	0.828432	0.414216	+	0.910179i	$0.364056\pi$
$420$	2.53519	0.123705
$421$	21.4776	1.04676	0.523378	−	0.852101i	$-0.324672\pi$
$421$	21.4776	1.04676	0.523378	+	0.852101i	$0.324672\pi$
$422$	−12.6609	−0.616325
$423$	9.48276	0.461068
$424$	−13.0620	−0.634349
$425$	−12.8002	−0.620903
$426$	−5.29744	−0.256662
$427$	−5.59829	−0.270920
$428$	15.6706	0.757467
$429$	0	0
$430$	−3.91178	−0.188643
$431$	28.7860	1.38657	0.693287	−	0.720662i	$-0.256162\pi$
$431$	28.7860	1.38657	0.693287	+	0.720662i	$0.256162\pi$
$432$	−1.67492	−0.0805848
$433$	−32.8717	−1.57971	−0.789856	−	0.613293i	$-0.789845\pi$
$433$	−32.8717	−1.57971	−0.789856	+	0.613293i	$0.789845\pi$
$434$	−1.21615	−0.0583771
$435$	−5.66069	−0.271409
$436$	8.91552	0.426976
$437$	7.87144	0.376542
$438$	−5.22929	−0.249865
$439$	−31.0552	−1.48219	−0.741093	−	0.671402i	$-0.765692\pi$
$439$	−31.0552	−1.48219	−0.741093	+	0.671402i	$0.765692\pi$
$440$	3.53721	0.168630
$441$	−5.90460	−0.281171
$442$	0	0
$443$	−25.8318	−1.22731	−0.613653	−	0.789576i	$-0.710301\pi$
$443$	−25.8318	−1.22731	−0.613653	+	0.789576i	$0.710301\pi$
$444$	12.1232	0.575342
$445$	4.42833	0.209923
$446$	−17.1919	−0.814059
$447$	10.4277	0.493211
$448$	−0.347596	−0.0164224
$449$	−13.4596	−0.635199	−0.317599	−	0.948225i	$-0.602877\pi$
$449$	−13.4596	−0.635199	−0.317599	+	0.948225i	$0.602877\pi$
$450$	−1.71668	−0.0809252
$451$	−1.30748	−0.0615670
$452$	16.7830	0.789406
$453$	10.5698	0.496612
$454$	−6.85887	−0.321903
$455$	0	0
$456$	−4.50410	−0.210924
$457$	−19.4344	−0.909104	−0.454552	−	0.890720i	$-0.650201\pi$
$457$	−19.4344	−0.909104	−0.454552	+	0.890720i	$0.650201\pi$
$458$	0.105950	0.00495071
$459$	−4.81161	−0.224587
$460$	−9.78925	−0.456426
$461$	20.0618	0.934371	0.467186	−	0.884159i	$-0.345268\pi$
$461$	20.0618	0.934371	0.467186	+	0.884159i	$0.345268\pi$
$462$	0.675382	0.0314216
$463$	24.3316	1.13078	0.565392	−	0.824822i	$-0.308725\pi$
$463$	24.3316	1.13078	0.565392	+	0.824822i	$0.308725\pi$
$464$	−6.19845	−0.287756
$465$	2.75435	0.127730
$466$	7.33509	0.339792
$467$	−28.4233	−1.31527	−0.657637	−	0.753335i	$-0.728444\pi$
$467$	−28.4233	−1.31527	−0.657637	+	0.753335i	$0.728444\pi$
$468$	0	0
$469$	−1.32700	−0.0612751
$470$	−9.36005	−0.431747
$471$	−12.2812	−0.565888
$472$	6.06461	0.279146
$473$	3.96306	0.182222
$474$	−8.75794	−0.402265
$475$	5.18151	0.237744
$476$	7.97480	0.365524
$477$	5.64847	0.258626
$478$	−14.1352	−0.646527
$479$	−25.9477	−1.18558	−0.592791	−	0.805356i	$-0.701974\pi$
$479$	−25.9477	−1.18558	−0.592791	+	0.805356i	$0.701974\pi$
$480$	8.72768	0.398362
$481$	0	0
$482$	−13.1591	−0.599379
$483$	−4.22973	−0.192460
$484$	−1.58359	−0.0719812
$485$	12.6941	0.576410
$486$	−0.645300	−0.0292714
$487$	29.3703	1.33090	0.665448	−	0.746445i	$-0.268241\pi$
$487$	29.3703	1.33090	0.665448	+	0.746445i	$0.268241\pi$
$488$	−12.3694	−0.559935
$489$	−10.0543	−0.454673
$490$	5.82819	0.263291
$491$	0.444397	0.0200553	0.0100277	−	0.999950i	$-0.496808\pi$
$491$	0.444397	0.0200553	0.0100277	+	0.999950i	$0.496808\pi$
$492$	−2.07051	−0.0933460
$493$	−17.8065	−0.801964
$494$	0	0
$495$	−1.52961	−0.0687510
$496$	3.01601	0.135423
$497$	−8.59194	−0.385401
$498$	2.48365	0.111295
$499$	−9.73446	−0.435774	−0.217887	−	0.975974i	$-0.569916\pi$
$499$	−9.73446	−0.435774	−0.217887	+	0.975974i	$0.569916\pi$
$500$	−18.5553	−0.829819
$501$	10.4117	0.465161
$502$	4.87695	0.217669
$503$	−11.5182	−0.513570	−0.256785	−	0.966469i	$-0.582663\pi$
$503$	−11.5182	−0.513570	−0.256785	+	0.966469i	$0.582663\pi$
$504$	2.42029	0.107808
$505$	8.11058	0.360916
$506$	−2.60788	−0.115934
$507$	0	0
$508$	2.17272	0.0963990
$509$	−1.01969	−0.0451968	−0.0225984	−	0.999745i	$-0.507194\pi$
$509$	−1.01969	−0.0451968	−0.0225984	+	0.999745i	$0.507194\pi$
$510$	4.74934	0.210305
$511$	−8.48142	−0.375196
$512$	17.3033	0.764704
$513$	1.94773	0.0859943
$514$	10.1229	0.446500
$515$	22.0620	0.972169
$516$	6.27586	0.276279
$517$	9.48276	0.417051
$518$	−5.17041	−0.227175
$519$	6.60938	0.290120
$520$	0	0
$521$	−4.73095	−0.207267	−0.103633	−	0.994616i	$-0.533047\pi$
$521$	−4.73095	−0.207267	−0.103633	+	0.994616i	$0.533047\pi$
$522$	−2.38809	−0.104524
$523$	−29.5966	−1.29417	−0.647085	−	0.762418i	$-0.724012\pi$
$523$	−29.5966	−1.29417	−0.647085	+	0.762418i	$0.724012\pi$
$524$	−16.7443	−0.731477
$525$	−2.78430	−0.121517
$526$	4.44047	0.193614
$527$	8.66420	0.377418
$528$	−1.67492	−0.0728917
$529$	−6.66753	−0.289893
$530$	−5.57538	−0.242179
$531$	−2.62254	−0.113809
$532$	−3.22818	−0.139959
$533$	0	0
$534$	1.86819	0.0808444
$535$	15.1365	0.654407
$536$	−2.93200	−0.126643
$537$	25.1584	1.08566
$538$	6.01580	0.259359
$539$	−5.90460	−0.254329
$540$	−2.42228	−0.104238
$541$	−20.4684	−0.880006	−0.440003	−	0.897996i	$-0.645023\pi$
$541$	−20.4684	−0.880006	−0.440003	+	0.897996i	$0.645023\pi$
$542$	18.0061	0.773427
$543$	15.6416	0.671243
$544$	27.4541	1.17709
$545$	8.61164	0.368882
$546$	0	0
$547$	16.6789	0.713139	0.356570	−	0.934269i	$-0.383946\pi$
$547$	16.6789	0.713139	0.356570	+	0.934269i	$0.383946\pi$
$548$	−5.08380	−0.217169
$549$	5.34894	0.228287
$550$	−1.71668	−0.0731996
$551$	7.20802	0.307072
$552$	−9.34557	−0.397774
$553$	−14.2045	−0.604039
$554$	−8.33047	−0.353928
$555$	11.7100	0.497062
$556$	21.4217	0.908482
$557$	−39.3844	−1.66877	−0.834384	−	0.551183i	$-0.814177\pi$
$557$	−39.3844	−1.66877	−0.834384	+	0.551183i	$0.814177\pi$
$558$	1.16198	0.0491907
$559$	0	0
$560$	2.68141	0.113310
$561$	−4.81161	−0.203146
$562$	12.4675	0.525912
$563$	15.7641	0.664378	0.332189	−	0.943213i	$-0.392213\pi$
$563$	15.7641	0.664378	0.332189	+	0.943213i	$0.392213\pi$
$564$	15.0168	0.632321
$565$	16.2110	0.682001
$566$	−19.3632	−0.813897
$567$	−1.04662	−0.0439537
$568$	−18.9838	−0.796544
$569$	28.7679	1.20601	0.603006	−	0.797737i	$-0.293970\pi$
$569$	28.7679	1.20601	0.603006	+	0.797737i	$0.293970\pi$
$570$	−1.92252	−0.0805257
$571$	−14.1836	−0.593565	−0.296783	−	0.954945i	$-0.595914\pi$
$571$	−14.1836	−0.593565	−0.296783	+	0.954945i	$0.595914\pi$
$572$	0	0
$573$	15.5248	0.648556
$574$	0.883051	0.0368578
$575$	10.7511	0.448353
$576$	0.332114	0.0138381
$577$	−0.663739	−0.0276318	−0.0138159	−	0.999905i	$-0.504398\pi$
$577$	−0.663739	−0.0276318	−0.0138159	+	0.999905i	$0.504398\pi$
$578$	3.96961	0.165114
$579$	14.7089	0.611282
$580$	−8.96420	−0.372218
$581$	4.02824	0.167120
$582$	5.35529	0.221984
$583$	5.64847	0.233936
$584$	−18.7396	−0.775452
$585$	0	0
$586$	18.8824	0.780024
$587$	−33.1801	−1.36949	−0.684745	−	0.728783i	$-0.740086\pi$
$587$	−33.1801	−1.36949	−0.684745	+	0.728783i	$0.740086\pi$
$588$	−9.35044	−0.385606
$589$	−3.50725	−0.144514
$590$	2.58861	0.106571
$591$	−2.45124	−0.100831
$592$	12.8224	0.526999
$593$	23.7220	0.974146	0.487073	−	0.873361i	$-0.338065\pi$
$593$	23.7220	0.974146	0.487073	+	0.873361i	$0.338065\pi$
$594$	−0.645300	−0.0264770
$595$	7.70298	0.315792
$596$	16.5131	0.676403
$597$	12.8327	0.525209
$598$	0	0
$599$	1.14126	0.0466307	0.0233153	−	0.999728i	$-0.492578\pi$
$599$	1.14126	0.0466307	0.0233153	+	0.999728i	$0.492578\pi$
$600$	−6.15188	−0.251150
$601$	4.69350	0.191452	0.0957259	−	0.995408i	$-0.469483\pi$
$601$	4.69350	0.191452	0.0957259	+	0.995408i	$0.469483\pi$
$602$	−2.67658	−0.109089
$603$	1.26790	0.0516327
$604$	16.7382	0.681067
$605$	−1.52961	−0.0621876
$606$	3.42162	0.138994
$607$	−36.3906	−1.47705	−0.738525	−	0.674226i	$-0.764477\pi$
$607$	−36.3906	−1.47705	−0.738525	+	0.674226i	$0.764477\pi$
$608$	−11.1134	−0.450706
$609$	−3.87325	−0.156952
$610$	−5.27972	−0.213770
$611$	0	0
$612$	−7.61960	−0.308004
$613$	−11.2702	−0.455198	−0.227599	−	0.973755i	$-0.573088\pi$
$613$	−11.2702	−0.455198	−0.227599	+	0.973755i	$0.573088\pi$
$614$	8.46780	0.341733
$615$	−1.99994	−0.0806455
$616$	2.42029	0.0975162
$617$	−20.1227	−0.810109	−0.405054	−	0.914293i	$-0.632747\pi$
$617$	−20.1227	−0.810109	−0.405054	+	0.914293i	$0.632747\pi$
$618$	9.30735	0.374396
$619$	−7.70638	−0.309746	−0.154873	−	0.987934i	$-0.549497\pi$
$619$	−7.70638	−0.309746	−0.154873	+	0.987934i	$0.549497\pi$
$620$	4.36176	0.175172
$621$	4.04134	0.162174
$622$	−5.89912	−0.236533
$623$	3.03002	0.121395
$624$	0	0
$625$	−4.62146	−0.184858
$626$	8.39095	0.335370
$627$	1.94773	0.0777848
$628$	−19.4484	−0.776074
$629$	36.8355	1.46873
$630$	1.03307	0.0411586
$631$	−43.7723	−1.74255	−0.871274	−	0.490797i	$-0.836706\pi$
$631$	−43.7723	−1.74255	−0.871274	+	0.490797i	$0.836706\pi$
$632$	−31.3848	−1.24842
$633$	19.6202	0.779835
$634$	1.07204	0.0425761
$635$	2.09867	0.0832831
$636$	8.94485	0.354686
$637$	0	0
$638$	−2.38809	−0.0945452
$639$	8.20926	0.324753
$640$	17.1275	0.677026
$641$	42.3607	1.67315	0.836574	−	0.547853i	$-0.184555\pi$
$641$	42.3607	1.67315	0.836574	+	0.547853i	$0.184555\pi$
$642$	6.38565	0.252022
$643$	26.3239	1.03812	0.519058	−	0.854739i	$-0.326283\pi$
$643$	26.3239	1.03812	0.519058	+	0.854739i	$0.326283\pi$
$644$	−6.69815	−0.263944
$645$	6.06195	0.238689
$646$	−6.04756	−0.237938
$647$	3.77540	0.148426	0.0742131	−	0.997242i	$-0.476355\pi$
$647$	3.77540	0.148426	0.0742131	+	0.997242i	$0.476355\pi$
$648$	−2.31249	−0.0908432
$649$	−2.62254	−0.102944
$650$	0	0
$651$	1.88463	0.0738643
$652$	−15.9219	−0.623551
$653$	−31.7180	−1.24122	−0.620611	−	0.784118i	$-0.713116\pi$
$653$	−31.7180	−1.24122	−0.620611	+	0.784118i	$0.713116\pi$
$654$	3.63301	0.142062
$655$	−16.1736	−0.631954
$656$	−2.18994	−0.0855026
$657$	8.10366	0.316154
$658$	−6.40448	−0.249673
$659$	49.5482	1.93012	0.965061	−	0.262025i	$-0.0843902\pi$
$659$	49.5482	1.93012	0.965061	+	0.262025i	$0.0843902\pi$
$660$	−2.42228	−0.0942869
$661$	23.6910	0.921474	0.460737	−	0.887537i	$-0.347585\pi$
$661$	23.6910	0.921474	0.460737	+	0.887537i	$0.347585\pi$
$662$	5.75470	0.223663
$663$	0	0
$664$	8.90037	0.345401
$665$	−3.11815	−0.120917
$666$	4.94012	0.191426
$667$	14.9559	0.579096
$668$	16.4878	0.637934
$669$	26.6417	1.03003
$670$	−1.25149	−0.0483492
$671$	5.34894	0.206494
$672$	5.97179	0.230367
$673$	−36.5156	−1.40757	−0.703787	−	0.710411i	$-0.748509\pi$
$673$	−36.5156	−1.40757	−0.703787	+	0.710411i	$0.748509\pi$
$674$	10.4011	0.400634
$675$	2.66028	0.102394
$676$	0	0
$677$	−30.8562	−1.18590	−0.592950	−	0.805240i	$-0.702037\pi$
$677$	−30.8562	−1.18590	−0.592950	+	0.805240i	$0.702037\pi$
$678$	6.83895	0.262648
$679$	8.68577	0.333329
$680$	17.0197	0.652675
$681$	10.6290	0.407302
$682$	1.16198	0.0444947
$683$	18.4889	0.707458	0.353729	−	0.935348i	$-0.384914\pi$
$683$	18.4889	0.707458	0.353729	+	0.935348i	$0.384914\pi$
$684$	3.08440	0.117935
$685$	−4.91052	−0.187621
$686$	8.71553	0.332760
$687$	−0.164187	−0.00626411
$688$	6.63783	0.253065
$689$	0	0
$690$	−3.98905	−0.151860
$691$	9.13545	0.347529	0.173764	−	0.984787i	$-0.444407\pi$
$691$	9.13545	0.347529	0.173764	+	0.984787i	$0.444407\pi$
$692$	10.4665	0.397878
$693$	−1.04662	−0.0397576
$694$	12.1332	0.460570
$695$	20.6916	0.784876
$696$	−8.55791	−0.324387
$697$	−6.29110	−0.238292
$698$	6.79551	0.257214
$699$	−11.3669	−0.429937
$700$	−4.40918	−0.166651
$701$	33.4414	1.26306	0.631532	−	0.775350i	$-0.282426\pi$
$701$	33.4414	1.26306	0.631532	+	0.775350i	$0.282426\pi$
$702$	0	0
$703$	−14.9109	−0.562375
$704$	0.332114	0.0125170
$705$	14.5050	0.546288
$706$	−8.06898	−0.303680
$707$	5.54955	0.208712
$708$	−4.15303	−0.156080
$709$	−39.4674	−1.48223	−0.741114	−	0.671379i	$-0.765702\pi$
$709$	−39.4674	−1.48223	−0.741114	+	0.671379i	$0.765702\pi$
$710$	−8.10303	−0.304101
$711$	13.5719	0.508985
$712$	6.69481	0.250899
$713$	−7.27719	−0.272533
$714$	3.24967	0.121616
$715$	0	0
$716$	39.8404	1.48891
$717$	21.9048	0.818049
$718$	8.65350	0.322946
$719$	−27.2536	−1.01639	−0.508194	−	0.861242i	$-0.669687\pi$
$719$	−27.2536	−1.01639	−0.508194	+	0.861242i	$0.669687\pi$
$720$	−2.56198	−0.0954795
$721$	15.0956	0.562191
$722$	−9.81267	−0.365190
$723$	20.3921	0.758392
$724$	24.7698	0.920561
$725$	9.84501	0.365634
$726$	−0.645300	−0.0239493
$727$	−39.2488	−1.45566	−0.727829	−	0.685758i	$-0.759471\pi$
$727$	−39.2488	−1.45566	−0.727829	+	0.685758i	$0.759471\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−7.99880	−0.296049
$731$	19.0687	0.705282
$732$	8.47051	0.313079
$733$	−17.4146	−0.643223	−0.321612	−	0.946872i	$-0.604225\pi$
$733$	−17.4146	−0.643223	−0.321612	+	0.946872i	$0.604225\pi$
$734$	−5.49512	−0.202829
$735$	−9.03174	−0.333141
$736$	−23.0591	−0.849971
$737$	1.26790	0.0467035
$738$	−0.843720	−0.0310578
$739$	−10.4114	−0.382990	−0.191495	−	0.981494i	$-0.561334\pi$
$739$	−10.4114	−0.382990	−0.191495	+	0.981494i	$0.561334\pi$
$740$	18.5438	0.681684
$741$	0	0
$742$	−3.81488	−0.140048
$743$	−27.1166	−0.994813	−0.497406	−	0.867518i	$-0.665714\pi$
$743$	−27.1166	−0.994813	−0.497406	+	0.867518i	$0.665714\pi$
$744$	4.16407	0.152662
$745$	15.9503	0.584373
$746$	−0.702029	−0.0257031
$747$	−3.84882	−0.140821
$748$	−7.61960	−0.278600
$749$	10.3569	0.378434
$750$	−7.56116	−0.276094
$751$	−43.2313	−1.57753	−0.788766	−	0.614694i	$-0.789280\pi$
$751$	−43.2313	−1.57753	−0.788766	+	0.614694i	$0.789280\pi$
$752$	15.8829	0.579190
$753$	−7.55764	−0.275416
$754$	0	0
$755$	16.1677	0.588402
$756$	−1.65741	−0.0602793
$757$	24.1815	0.878893	0.439447	−	0.898269i	$-0.355175\pi$
$757$	24.1815	0.878893	0.439447	+	0.898269i	$0.355175\pi$
$758$	19.4678	0.707104
$759$	4.04134	0.146692
$760$	−6.88953	−0.249910
$761$	1.62749	0.0589965	0.0294983	−	0.999565i	$-0.490609\pi$
$761$	1.62749	0.0589965	0.0294983	+	0.999565i	$0.490609\pi$
$762$	0.885369	0.0320735
$763$	5.89239	0.213319
$764$	24.5848	0.889447
$765$	−7.35990	−0.266098
$766$	9.57129	0.345825
$767$	0	0
$768$	7.88985	0.284701
$769$	−20.8745	−0.752752	−0.376376	−	0.926467i	$-0.622830\pi$
$769$	−20.8745	−0.752752	−0.376376	+	0.926467i	$0.622830\pi$
$770$	1.03307	0.0372293
$771$	−15.6870	−0.564955
$772$	23.2928	0.838328
$773$	−22.2132	−0.798955	−0.399477	−	0.916743i	$-0.630808\pi$
$773$	−22.2132	−0.798955	−0.399477	+	0.916743i	$0.630808\pi$
$774$	2.55737	0.0919226
$775$	−4.79034	−0.172074
$776$	19.1911	0.688922
$777$	8.01241	0.287443
$778$	−9.40690	−0.337254
$779$	2.54662	0.0912422
$780$	0	0
$781$	8.20926	0.293750
$782$	−12.5481	−0.448719
$783$	3.70073	0.132253
$784$	−9.88975	−0.353205
$785$	−18.7855	−0.670483
$786$	−6.82317	−0.243375
$787$	−37.1502	−1.32426	−0.662130	−	0.749389i	$-0.730347\pi$
$787$	−37.1502	−1.32426	−0.662130	+	0.749389i	$0.730347\pi$
$788$	−3.88176	−0.138282
$789$	−6.88124	−0.244979
$790$	−13.3963	−0.476617
$791$	11.0921	0.394391
$792$	−2.31249	−0.0821708
$793$	0	0
$794$	17.0006	0.603329
$795$	8.63998	0.306428
$796$	20.3218	0.720286
$797$	−11.4702	−0.406297	−0.203148	−	0.979148i	$-0.565117\pi$
$797$	−11.4702	−0.406297	−0.203148	+	0.979148i	$0.565117\pi$
$798$	−1.31546	−0.0465668
$799$	45.6273	1.61418
$800$	−15.1791	−0.536662
$801$	−2.89507	−0.102292
$802$	−7.03029	−0.248248
$803$	8.10366	0.285972
$804$	2.00782	0.0708105
$805$	−6.46986	−0.228032
$806$	0	0
$807$	−9.32248	−0.328167
$808$	12.2617	0.431365
$809$	−3.04605	−0.107093	−0.0535467	−	0.998565i	$-0.517053\pi$
$809$	−3.04605	−0.107093	−0.0535467	+	0.998565i	$0.517053\pi$
$810$	−0.987060	−0.0346817
$811$	−15.2259	−0.534653	−0.267326	−	0.963606i	$-0.586140\pi$
$811$	−15.2259	−0.534653	−0.267326	+	0.963606i	$0.586140\pi$
$812$	−6.13362	−0.215248
$813$	−27.9034	−0.978615
$814$	4.94012	0.173151
$815$	−15.3792	−0.538711
$816$	−8.05908	−0.282124
$817$	−7.71897	−0.270053
$818$	10.1454	0.354727
$819$	0	0
$820$	−3.16709	−0.110599
$821$	−33.7250	−1.17701	−0.588505	−	0.808494i	$-0.700283\pi$
$821$	−33.7250	−1.17701	−0.588505	+	0.808494i	$0.700283\pi$
$822$	−2.07161	−0.0722557
$823$	−25.5176	−0.889489	−0.444744	−	0.895658i	$-0.646705\pi$
$823$	−25.5176	−0.889489	−0.444744	+	0.895658i	$0.646705\pi$
$824$	33.3537	1.16193
$825$	2.66028	0.0926192
$826$	1.77122	0.0616285
$827$	−11.9311	−0.414884	−0.207442	−	0.978247i	$-0.566514\pi$
$827$	−11.9311	−0.414884	−0.207442	+	0.978247i	$0.566514\pi$
$828$	6.39982	0.222409
$829$	44.0821	1.53104	0.765518	−	0.643414i	$-0.222483\pi$
$829$	44.0821	1.53104	0.765518	+	0.643414i	$0.222483\pi$
$830$	3.79902	0.131866
$831$	12.9095	0.447824
$832$	0	0
$833$	−28.4106	−0.984369
$834$	8.72919	0.302267
$835$	15.9259	0.551138
$836$	3.08440	0.106676
$837$	−1.80069	−0.0622408
$838$	10.9427	0.378011
$839$	15.6959	0.541884	0.270942	−	0.962596i	$-0.412665\pi$
$839$	15.6959	0.541884	0.270942	+	0.962596i	$0.412665\pi$
$840$	3.70210	0.127735
$841$	−15.3046	−0.527744
$842$	13.8595	0.477631
$843$	−19.3205	−0.665434
$844$	31.0704	1.06949
$845$	0	0
$846$	6.11923	0.210383
$847$	−1.04662	−0.0359621
$848$	9.46076	0.324884
$849$	30.0065	1.02982
$850$	−8.26001	−0.283316
$851$	−30.9387	−1.06056
$852$	13.0001	0.445375
$853$	−33.8572	−1.15925	−0.579624	−	0.814884i	$-0.696800\pi$
$853$	−33.8572	−1.15925	−0.579624	+	0.814884i	$0.696800\pi$
$854$	−3.61258	−0.123620
$855$	2.97927	0.101889
$856$	22.8835	0.782144
$857$	24.2665	0.828927	0.414464	−	0.910066i	$-0.363969\pi$
$857$	24.2665	0.828927	0.414464	+	0.910066i	$0.363969\pi$
$858$	0	0
$859$	−13.7121	−0.467852	−0.233926	−	0.972254i	$-0.575157\pi$
$859$	−13.7121	−0.467852	−0.233926	+	0.972254i	$0.575157\pi$
$860$	9.59963	0.327345
$861$	−1.36843	−0.0466361
$862$	18.5756	0.632688
$863$	6.30850	0.214744	0.107372	−	0.994219i	$-0.465756\pi$
$863$	6.30850	0.214744	0.107372	+	0.994219i	$0.465756\pi$
$864$	−5.70581	−0.194116
$865$	10.1098	0.343743
$866$	−21.2121	−0.720816
$867$	−6.15158	−0.208918
$868$	2.98447	0.101300
$869$	13.5719	0.460394
$870$	−3.65285	−0.123843
$871$	0	0
$872$	13.0192	0.440886
$873$	−8.29891	−0.280875
$874$	5.07944	0.171815
$875$	−12.2635	−0.414581
$876$	12.8329	0.433582
$877$	−25.1286	−0.848534	−0.424267	−	0.905537i	$-0.639468\pi$
$877$	−25.1286	−0.848534	−0.424267	+	0.905537i	$0.639468\pi$
$878$	−20.0400	−0.676316
$879$	−29.2614	−0.986963
$880$	−2.56198	−0.0863645
$881$	39.7946	1.34071	0.670357	−	0.742039i	$-0.266141\pi$
$881$	39.7946	1.34071	0.670357	+	0.742039i	$0.266141\pi$
$882$	−3.81024	−0.128297
$883$	22.2676	0.749365	0.374682	−	0.927153i	$-0.377752\pi$
$883$	22.2676	0.749365	0.374682	+	0.927153i	$0.377752\pi$
$884$	0	0
$885$	−4.01148	−0.134844
$886$	−16.6693	−0.560016
$887$	−11.9141	−0.400036	−0.200018	−	0.979792i	$-0.564100\pi$
$887$	−11.9141	−0.400036	−0.200018	+	0.979792i	$0.564100\pi$
$888$	17.7034	0.594086
$889$	1.43598	0.0481614
$890$	2.85760	0.0957871
$891$	1.00000	0.0335013
$892$	42.1894	1.41261
$893$	−18.4698	−0.618070
$894$	6.72897	0.225050
$895$	38.4825	1.28633
$896$	11.7193	0.391514
$897$	0	0
$898$	−8.68550	−0.289839
$899$	−6.66386	−0.222252
$900$	4.21279	0.140426
$901$	27.1782	0.905439
$902$	−0.843720	−0.0280928
$903$	4.14780	0.138030
$904$	24.5080	0.815123
$905$	23.9255	0.795311
$906$	6.82069	0.226602
$907$	9.74823	0.323685	0.161842	−	0.986817i	$-0.448256\pi$
$907$	9.74823	0.323685	0.161842	+	0.986817i	$0.448256\pi$
$908$	16.8319	0.558585
$909$	−5.30237	−0.175869
$910$	0	0
$911$	17.6700	0.585434	0.292717	−	0.956199i	$-0.405441\pi$
$911$	17.6700	0.585434	0.292717	+	0.956199i	$0.405441\pi$
$912$	3.26230	0.108025
$913$	−3.84882	−0.127377
$914$	−12.5410	−0.414821
$915$	8.18181	0.270482
$916$	−0.260004	−0.00859078
$917$	−11.0665	−0.365449
$918$	−3.10493	−0.102478
$919$	−38.9801	−1.28583	−0.642917	−	0.765936i	$-0.722276\pi$
$919$	−38.9801	−1.28583	−0.642917	+	0.765936i	$0.722276\pi$
$920$	−14.2951	−0.471296
$921$	−13.1223	−0.432393
$922$	12.9459	0.426350
$923$	0	0
$924$	−1.65741	−0.0545247
$925$	−20.3659	−0.669627
$926$	15.7012	0.515973
$927$	−14.4233	−0.473723
$928$	−21.1157	−0.693156
$929$	40.3636	1.32429	0.662144	−	0.749377i	$-0.269647\pi$
$929$	40.3636	1.32429	0.662144	+	0.749377i	$0.269647\pi$
$930$	1.77738	0.0582827
$931$	11.5005	0.376915
$932$	−18.0005	−0.589627
$933$	9.14166	0.299285
$934$	−18.3416	−0.600154
$935$	−7.35990	−0.240694
$936$	0	0
$937$	−47.0901	−1.53837	−0.769183	−	0.639029i	$-0.779337\pi$
$937$	−47.0901	−1.53837	−0.769183	+	0.639029i	$0.779337\pi$
$938$	−0.856313	−0.0279596
$939$	−13.0032	−0.424342
$940$	22.9699	0.749194
$941$	−16.5420	−0.539254	−0.269627	−	0.962965i	$-0.586900\pi$
$941$	−16.5420	−0.539254	−0.269627	+	0.962965i	$0.586900\pi$
$942$	−7.92507	−0.258213
$943$	5.28399	0.172070
$944$	−4.39256	−0.142966
$945$	−1.60092	−0.0520778
$946$	2.55737	0.0831471
$947$	15.6163	0.507461	0.253730	−	0.967275i	$-0.418342\pi$
$947$	15.6163	0.507461	0.253730	+	0.967275i	$0.418342\pi$
$948$	21.4923	0.698036
$949$	0	0
$950$	3.34363	0.108482
$951$	−1.66130	−0.0538715
$952$	11.6455	0.377432
$953$	19.9075	0.644867	0.322433	−	0.946592i	$-0.395499\pi$
$953$	19.9075	0.644867	0.322433	+	0.946592i	$0.395499\pi$
$954$	3.64496	0.118010
$955$	23.7469	0.768431
$956$	34.6881	1.12189
$957$	3.70073	0.119628
$958$	−16.7441	−0.540977
$959$	−3.35996	−0.108499
$960$	0.508006	0.0163958
$961$	−27.7575	−0.895404
$962$	0	0
$963$	−9.89563	−0.318882
$964$	32.2927	1.04008
$965$	22.4989	0.724266
$966$	−2.72945	−0.0878186
$967$	−40.1352	−1.29066	−0.645330	−	0.763904i	$-0.723280\pi$
$967$	−40.1352	−1.29066	−0.645330	+	0.763904i	$0.723280\pi$
$968$	−2.31249	−0.0743263
$969$	9.37170	0.301063
$970$	8.19152	0.263014
$971$	0.545093	0.0174929	0.00874643	−	0.999962i	$-0.497216\pi$
$971$	0.545093	0.0174929	0.00874643	+	0.999962i	$0.497216\pi$
$972$	1.58359	0.0507936
$973$	14.1579	0.453882
$974$	18.9527	0.607282
$975$	0	0
$976$	8.95907	0.286773
$977$	50.8185	1.62583	0.812915	−	0.582383i	$-0.197880\pi$
$977$	50.8185	1.62583	0.812915	+	0.582383i	$0.197880\pi$
$978$	−6.48807	−0.207466
$979$	−2.89507	−0.0925267
$980$	−14.3026	−0.456878
$981$	−5.62995	−0.179750
$982$	0.286769	0.00915118
$983$	−17.6666	−0.563476	−0.281738	−	0.959491i	$-0.590911\pi$
$983$	−17.6666	−0.563476	−0.281738	+	0.959491i	$0.590911\pi$
$984$	−3.02354	−0.0963871
$985$	−3.74945	−0.119468
$986$	−11.4905	−0.365933
$987$	9.92481	0.315910
$988$	0	0
$989$	−16.0161	−0.509282
$990$	−0.987060	−0.0313708
$991$	23.8658	0.758123	0.379062	−	0.925371i	$-0.376247\pi$
$991$	23.8658	0.758123	0.379062	+	0.925371i	$0.376247\pi$
$992$	10.2744	0.326212
$993$	−8.91786	−0.283000
$994$	−5.54438	−0.175857
$995$	19.6291	0.622285
$996$	−6.09495	−0.193126
$997$	−50.5447	−1.60077	−0.800383	−	0.599489i	$-0.795371\pi$
$997$	−50.5447	−1.60077	−0.800383	+	0.599489i	$0.795371\pi$
$998$	−6.28165	−0.198842
$999$	−7.65554	−0.242211

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5577.2.a.bc.1.7	yes	12
13.12	even	2		5577.2.a.ba.1.6	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5577.2.a.ba.1.6	✓	12	13.12	even	2
5577.2.a.bc.1.7	yes	12	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:	\( N \)	\(=\)	\( 5577 = 3 \cdot 11 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5577.a (trivial)

\(f(q)\)	\(=\)	\(q+0.645300 q^{2} -1.00000 q^{3} -1.58359 q^{4} -1.52961 q^{5} -0.645300 q^{6} -1.04662 q^{7} -2.31249 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.645300 q^{2} -1.00000 q^{3} -1.58359 q^{4} -1.52961 q^{5} -0.645300 q^{6} -1.04662 q^{7} -2.31249 q^{8} +1.00000 q^{9} -0.987060 q^{10} +1.00000 q^{11} +1.58359 q^{12} -0.675382 q^{14} +1.52961 q^{15} +1.67492 q^{16} +4.81161 q^{17} +0.645300 q^{18} -1.94773 q^{19} +2.42228 q^{20} +1.04662 q^{21} +0.645300 q^{22} -4.04134 q^{23} +2.31249 q^{24} -2.66028 q^{25} -1.00000 q^{27} +1.65741 q^{28} -3.70073 q^{29} +0.987060 q^{30} +1.80069 q^{31} +5.70581 q^{32} -1.00000 q^{33} +3.10493 q^{34} +1.60092 q^{35} -1.58359 q^{36} +7.65554 q^{37} -1.25687 q^{38} +3.53721 q^{40} -1.30748 q^{41} +0.675382 q^{42} +3.96306 q^{43} -1.58359 q^{44} -1.52961 q^{45} -2.60788 q^{46} +9.48276 q^{47} -1.67492 q^{48} -5.90460 q^{49} -1.71668 q^{50} -4.81161 q^{51} +5.64847 q^{53} -0.645300 q^{54} -1.52961 q^{55} +2.42029 q^{56} +1.94773 q^{57} -2.38809 q^{58} -2.62254 q^{59} -2.42228 q^{60} +5.34894 q^{61} +1.16198 q^{62} -1.04662 q^{63} +0.332114 q^{64} -0.645300 q^{66} +1.26790 q^{67} -7.61960 q^{68} +4.04134 q^{69} +1.03307 q^{70} +8.20926 q^{71} -2.31249 q^{72} +8.10366 q^{73} +4.94012 q^{74} +2.66028 q^{75} +3.08440 q^{76} -1.04662 q^{77} +13.5719 q^{79} -2.56198 q^{80} +1.00000 q^{81} -0.843720 q^{82} -3.84882 q^{83} -1.65741 q^{84} -7.35990 q^{85} +2.55737 q^{86} +3.70073 q^{87} -2.31249 q^{88} -2.89507 q^{89} -0.987060 q^{90} +6.39982 q^{92} -1.80069 q^{93} +6.11923 q^{94} +2.97927 q^{95} -5.70581 q^{96} -8.29891 q^{97} -3.81024 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + q^{2} - 12 q^{3} + 5 q^{4} - 6 q^{5} - q^{6} - q^{7} + 12 q^{9}+O(q^{10}) \) 12 * q + q^2 - 12 * q^3 + 5 * q^4 - 6 * q^5 - q^6 - q^7 + 12 * q^9	\( 12 q + q^{2} - 12 q^{3} + 5 q^{4} - 6 q^{5} - q^{6} - q^{7} + 12 q^{9} - 11 q^{10} + 12 q^{11} - 5 q^{12} + 9 q^{14} + 6 q^{15} - 9 q^{16} + 3 q^{17} + q^{18} + 6 q^{19} - 20 q^{20} + q^{21} + q^{22} - 22 q^{23} - 2 q^{25} - 12 q^{27} - 11 q^{28} + 13 q^{29} + 11 q^{30} + 2 q^{31} + 11 q^{32} - 12 q^{33} + 10 q^{34} - 14 q^{35} + 5 q^{36} + 3 q^{37} - 18 q^{38} - 18 q^{40} + 4 q^{41} - 9 q^{42} - 26 q^{43} + 5 q^{44} - 6 q^{45} - 18 q^{46} - 9 q^{47} + 9 q^{48} - 3 q^{49} + 29 q^{50} - 3 q^{51} - 5 q^{53} - q^{54} - 6 q^{55} + 5 q^{56} - 6 q^{57} + 37 q^{58} - 22 q^{59} + 20 q^{60} - 11 q^{61} - 18 q^{62} - q^{63} - 10 q^{64} - q^{66} - 28 q^{67} + 12 q^{68} + 22 q^{69} - 29 q^{70} + 10 q^{71} - 7 q^{73} - 6 q^{74} + 2 q^{75} + 32 q^{76} - q^{77} - 40 q^{79} + 30 q^{80} + 12 q^{81} + 26 q^{82} + q^{83} + 11 q^{84} - 32 q^{85} - 9 q^{86} - 13 q^{87} + 11 q^{89} - 11 q^{90} - 38 q^{92} - 2 q^{93} - 25 q^{94} - 10 q^{95} - 11 q^{96} + 7 q^{97} + 28 q^{98} + 12 q^{99}+O(q^{100}) \) 12 * q + q^2 - 12 * q^3 + 5 * q^4 - 6 * q^5 - q^6 - q^7 + 12 * q^9 - 11 * q^10 + 12 * q^11 - 5 * q^12 + 9 * q^14 + 6 * q^15 - 9 * q^16 + 3 * q^17 + q^18 + 6 * q^19 - 20 * q^20 + q^21 + q^22 - 22 * q^23 - 2 * q^25 - 12 * q^27 - 11 * q^28 + 13 * q^29 + 11 * q^30 + 2 * q^31 + 11 * q^32 - 12 * q^33 + 10 * q^34 - 14 * q^35 + 5 * q^36 + 3 * q^37 - 18 * q^38 - 18 * q^40 + 4 * q^41 - 9 * q^42 - 26 * q^43 + 5 * q^44 - 6 * q^45 - 18 * q^46 - 9 * q^47 + 9 * q^48 - 3 * q^49 + 29 * q^50 - 3 * q^51 - 5 * q^53 - q^54 - 6 * q^55 + 5 * q^56 - 6 * q^57 + 37 * q^58 - 22 * q^59 + 20 * q^60 - 11 * q^61 - 18 * q^62 - q^63 - 10 * q^64 - q^66 - 28 * q^67 + 12 * q^68 + 22 * q^69 - 29 * q^70 + 10 * q^71 - 7 * q^73 - 6 * q^74 + 2 * q^75 + 32 * q^76 - q^77 - 40 * q^79 + 30 * q^80 + 12 * q^81 + 26 * q^82 + q^83 + 11 * q^84 - 32 * q^85 - 9 * q^86 - 13 * q^87 + 11 * q^89 - 11 * q^90 - 38 * q^92 - 2 * q^93 - 25 * q^94 - 10 * q^95 - 11 * q^96 + 7 * q^97 + 28 * q^98 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more