LMFDB - Embedded newform 5577.2.a.bj.1.12

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:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - x^{17} - 29 x^{16} + 28 x^{15} + 341 x^{14} - 315 x^{13} - 2097 x^{12} + 1830 x^{11} + \cdots - 169 $ x^18 - x^17 - 29x^16 + 28x^15 + 341x^14 - 315x^13 - 2097x^12 + 1830x^11 + 7272x^10 - 5901x^9 - 14318x^8 + 10617x^7 + 15315x^6 - 10102x^5 - 8079x^4 + 4329x^3 + 1819x^2 - 533x - 169
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$0.933660$ 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)

$f(q)$	$=$	$q+0.933660 q^{2} -1.00000 q^{3} -1.12828 q^{4} -2.03159 q^{5} -0.933660 q^{6} -2.50464 q^{7} -2.92075 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.933660 q^{2} -1.00000 q^{3} -1.12828 q^{4} -2.03159 q^{5} -0.933660 q^{6} -2.50464 q^{7} -2.92075 q^{8} +1.00000 q^{9} -1.89682 q^{10} -1.00000 q^{11} +1.12828 q^{12} -2.33848 q^{14} +2.03159 q^{15} -0.470429 q^{16} -3.69616 q^{17} +0.933660 q^{18} +0.177845 q^{19} +2.29220 q^{20} +2.50464 q^{21} -0.933660 q^{22} -4.83581 q^{23} +2.92075 q^{24} -0.872636 q^{25} -1.00000 q^{27} +2.82593 q^{28} -7.73202 q^{29} +1.89682 q^{30} -8.83878 q^{31} +5.40228 q^{32} +1.00000 q^{33} -3.45095 q^{34} +5.08840 q^{35} -1.12828 q^{36} -10.6692 q^{37} +0.166047 q^{38} +5.93377 q^{40} -6.54853 q^{41} +2.33848 q^{42} -0.511245 q^{43} +1.12828 q^{44} -2.03159 q^{45} -4.51500 q^{46} -6.47604 q^{47} +0.470429 q^{48} -0.726802 q^{49} -0.814745 q^{50} +3.69616 q^{51} +11.2250 q^{53} -0.933660 q^{54} +2.03159 q^{55} +7.31541 q^{56} -0.177845 q^{57} -7.21908 q^{58} +0.823270 q^{59} -2.29220 q^{60} +5.60673 q^{61} -8.25242 q^{62} -2.50464 q^{63} +5.98475 q^{64} +0.933660 q^{66} +16.1893 q^{67} +4.17030 q^{68} +4.83581 q^{69} +4.75083 q^{70} -9.81162 q^{71} -2.92075 q^{72} -3.70987 q^{73} -9.96142 q^{74} +0.872636 q^{75} -0.200659 q^{76} +2.50464 q^{77} -9.36626 q^{79} +0.955719 q^{80} +1.00000 q^{81} -6.11410 q^{82} -7.01724 q^{83} -2.82593 q^{84} +7.50908 q^{85} -0.477329 q^{86} +7.73202 q^{87} +2.92075 q^{88} -10.0961 q^{89} -1.89682 q^{90} +5.45615 q^{92} +8.83878 q^{93} -6.04642 q^{94} -0.361309 q^{95} -5.40228 q^{96} -8.91674 q^{97} -0.678586 q^{98} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + q^{2} - 18 q^{3} + 23 q^{4} - 6 q^{5} - q^{6} + q^{7} + 18 q^{9}+O(q^{10}) $ 18 * q + q^2 - 18 * q^3 + 23 * q^4 - 6 * q^5 - q^6 + q^7 + 18 * q^9	$ 18 q + q^{2} - 18 q^{3} + 23 q^{4} - 6 q^{5} - q^{6} + q^{7} + 18 q^{9} + 11 q^{10} - 18 q^{11} - 23 q^{12} - 13 q^{14} + 6 q^{15} + 41 q^{16} - q^{17} + q^{18} - 12 q^{19} - 8 q^{20} - q^{21} - q^{22} + 44 q^{23} + 48 q^{25} - 18 q^{27} + 5 q^{28} - 17 q^{29} - 11 q^{30} + 18 q^{31} + q^{32} + 18 q^{33} - 8 q^{34} + 8 q^{35} + 23 q^{36} - 39 q^{37} + 4 q^{38} + 8 q^{40} - 6 q^{41} + 13 q^{42} + 36 q^{43} - 23 q^{44} - 6 q^{45} + 36 q^{46} - 19 q^{47} - 41 q^{48} + 41 q^{49} + 19 q^{50} + q^{51} + 7 q^{53} - q^{54} + 6 q^{55} - 73 q^{56} + 12 q^{57} - 69 q^{58} + 16 q^{59} + 8 q^{60} + 21 q^{61} + 8 q^{62} + q^{63} + 96 q^{64} + q^{66} + 40 q^{68} - 44 q^{69} + 107 q^{70} - 12 q^{71} + 7 q^{73} - 14 q^{74} - 48 q^{75} - 30 q^{76} - q^{77} + 62 q^{79} - 58 q^{80} + 18 q^{81} + 44 q^{82} + 11 q^{83} - 5 q^{84} + 46 q^{85} - 19 q^{86} + 17 q^{87} - 17 q^{89} + 11 q^{90} + 100 q^{92} - 18 q^{93} + 31 q^{94} + 42 q^{95} - q^{96} - 35 q^{97} + 94 q^{98} - 18 q^{99}+O(q^{100}) $ 18 * q + q^2 - 18 * q^3 + 23 * q^4 - 6 * q^5 - q^6 + q^7 + 18 * q^9 + 11 * q^10 - 18 * q^11 - 23 * q^12 - 13 * q^14 + 6 * q^15 + 41 * q^16 - q^17 + q^18 - 12 * q^19 - 8 * q^20 - q^21 - q^22 + 44 * q^23 + 48 * q^25 - 18 * q^27 + 5 * q^28 - 17 * q^29 - 11 * q^30 + 18 * q^31 + q^32 + 18 * q^33 - 8 * q^34 + 8 * q^35 + 23 * q^36 - 39 * q^37 + 4 * q^38 + 8 * q^40 - 6 * q^41 + 13 * q^42 + 36 * q^43 - 23 * q^44 - 6 * q^45 + 36 * q^46 - 19 * q^47 - 41 * q^48 + 41 * q^49 + 19 * q^50 + q^51 + 7 * q^53 - q^54 + 6 * q^55 - 73 * q^56 + 12 * q^57 - 69 * q^58 + 16 * q^59 + 8 * q^60 + 21 * q^61 + 8 * q^62 + q^63 + 96 * q^64 + q^66 + 40 * q^68 - 44 * q^69 + 107 * q^70 - 12 * q^71 + 7 * q^73 - 14 * q^74 - 48 * q^75 - 30 * q^76 - q^77 + 62 * q^79 - 58 * q^80 + 18 * q^81 + 44 * q^82 + 11 * q^83 - 5 * q^84 + 46 * q^85 - 19 * q^86 + 17 * q^87 - 17 * q^89 + 11 * q^90 + 100 * q^92 - 18 * q^93 + 31 * q^94 + 42 * q^95 - q^96 - 35 * q^97 + 94 * q^98 - 18 * 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 $

$2$	0.933660	0.660197	0.330099	−	0.943946i	$-0.392918\pi$
$2$	0.933660	0.660197	0.330099	+	0.943946i	$0.392918\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.12828	−0.564139
$5$	−2.03159	−0.908555	−0.454278	−	0.890860i	$-0.650103\pi$
$5$	−2.03159	−0.908555	−0.454278	+	0.890860i	$0.650103\pi$
$6$	−0.933660	−0.381165
$7$	−2.50464	−0.946663	−0.473332	−	0.880884i	$-0.656949\pi$
$7$	−2.50464	−0.946663	−0.473332	+	0.880884i	$0.656949\pi$
$8$	−2.92075	−1.03264
$9$	1.00000	0.333333
$10$	−1.89682	−0.599826
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	1.12828	0.325706
$13$	0	0
$13$	0	0
$14$	−2.33848	−0.624985
$15$	2.03159	0.524555
$16$	−0.470429	−0.117607
$17$	−3.69616	−0.896450	−0.448225	−	0.893921i	$-0.647944\pi$
$17$	−3.69616	−0.896450	−0.448225	+	0.893921i	$0.647944\pi$
$18$	0.933660	0.220066
$19$	0.177845	0.0408005	0.0204002	−	0.999792i	$-0.493506\pi$
$19$	0.177845	0.0408005	0.0204002	+	0.999792i	$0.493506\pi$
$20$	2.29220	0.512552
$21$	2.50464	0.546556
$22$	−0.933660	−0.199057
$23$	−4.83581	−1.00834	−0.504168	−	0.863605i	$-0.668201\pi$
$23$	−4.83581	−1.00834	−0.504168	+	0.863605i	$0.668201\pi$
$24$	2.92075	0.596195
$25$	−0.872636	−0.174527
$26$	0	0
$27$	−1.00000	−0.192450
$28$	2.82593	0.534050
$29$	−7.73202	−1.43580	−0.717900	−	0.696146i	$-0.754897\pi$
$29$	−7.73202	−1.43580	−0.717900	+	0.696146i	$0.754897\pi$
$30$	1.89682	0.346310
$31$	−8.83878	−1.58749	−0.793746	−	0.608250i	$-0.791872\pi$
$31$	−8.83878	−1.58749	−0.793746	+	0.608250i	$0.791872\pi$
$32$	5.40228	0.954997
$33$	1.00000	0.174078
$34$	−3.45095	−0.591834
$35$	5.08840	0.860096
$36$	−1.12828	−0.188046
$37$	−10.6692	−1.75401	−0.877004	−	0.480483i	$-0.840462\pi$
$37$	−10.6692	−1.75401	−0.877004	+	0.480483i	$0.840462\pi$
$38$	0.166047	0.0269364
$39$	0	0
$40$	5.93377	0.938211
$41$	−6.54853	−1.02271	−0.511354	−	0.859370i	$-0.670856\pi$
$41$	−6.54853	−1.02271	−0.511354	+	0.859370i	$0.670856\pi$
$42$	2.33848	0.360835
$43$	−0.511245	−0.0779641	−0.0389820	−	0.999240i	$-0.512412\pi$
$43$	−0.511245	−0.0779641	−0.0389820	+	0.999240i	$0.512412\pi$
$44$	1.12828	0.170094
$45$	−2.03159	−0.302852
$46$	−4.51500	−0.665701
$47$	−6.47604	−0.944628	−0.472314	−	0.881430i	$-0.656581\pi$
$47$	−6.47604	−0.944628	−0.472314	+	0.881430i	$0.656581\pi$
$48$	0.470429	0.0679006
$49$	−0.726802	−0.103829
$50$	−0.814745	−0.115222
$51$	3.69616	0.517566
$52$	0	0
$53$	11.2250	1.54188	0.770939	−	0.636909i	$-0.219787\pi$
$53$	11.2250	1.54188	0.770939	+	0.636909i	$0.219787\pi$
$54$	−0.933660	−0.127055
$55$	2.03159	0.273940
$56$	7.31541	0.977563
$57$	−0.177845	−0.0235562
$58$	−7.21908	−0.947912
$59$	0.823270	0.107181	0.0535903	−	0.998563i	$-0.482933\pi$
$59$	0.823270	0.107181	0.0535903	+	0.998563i	$0.482933\pi$
$60$	−2.29220	−0.295922
$61$	5.60673	0.717869	0.358934	−	0.933363i	$-0.383140\pi$
$61$	5.60673	0.717869	0.358934	+	0.933363i	$0.383140\pi$
$62$	−8.25242	−1.04806
$63$	−2.50464	−0.315554
$64$	5.98475	0.748094
$65$	0	0
$66$	0.933660	0.114926
$67$	16.1893	1.97784	0.988918	−	0.148464i	$-0.0474330\pi$
$67$	16.1893	1.97784	0.988918	+	0.148464i	$0.0474330\pi$
$68$	4.17030	0.505723
$69$	4.83581	0.582163
$70$	4.75083	0.567833
$71$	−9.81162	−1.16443	−0.582213	−	0.813036i	$-0.697813\pi$
$71$	−9.81162	−1.16443	−0.582213	+	0.813036i	$0.697813\pi$
$72$	−2.92075	−0.344214
$73$	−3.70987	−0.434208	−0.217104	−	0.976149i	$-0.569661\pi$
$73$	−3.70987	−0.434208	−0.217104	+	0.976149i	$0.569661\pi$
$74$	−9.96142	−1.15799
$75$	0.872636	0.100763
$76$	−0.200659	−0.0230171
$77$	2.50464	0.285430
$78$	0	0
$79$	−9.36626	−1.05379	−0.526893	−	0.849932i	$-0.676643\pi$
$79$	−9.36626	−1.05379	−0.526893	+	0.849932i	$0.676643\pi$
$80$	0.955719	0.106853
$81$	1.00000	0.111111
$82$	−6.11410	−0.675190
$83$	−7.01724	−0.770242	−0.385121	−	0.922866i	$-0.625840\pi$
$83$	−7.01724	−0.770242	−0.385121	+	0.922866i	$0.625840\pi$
$84$	−2.82593	−0.308334
$85$	7.50908	0.814474
$86$	−0.477329	−0.0514717
$87$	7.73202	0.828960
$88$	2.92075	0.311353
$89$	−10.0961	−1.07018	−0.535090	−	0.844795i	$-0.679722\pi$
$89$	−10.0961	−1.07018	−0.535090	+	0.844795i	$0.679722\pi$
$90$	−1.89682	−0.199942
$91$	0	0
$92$	5.45615	0.568842
$93$	8.83878	0.916539
$94$	−6.04642	−0.623641
$95$	−0.361309	−0.0370695
$96$	−5.40228	−0.551368
$97$	−8.91674	−0.905357	−0.452679	−	0.891674i	$-0.649532\pi$
$97$	−8.91674	−0.905357	−0.452679	+	0.891674i	$0.649532\pi$
$98$	−0.678586	−0.0685475
$99$	−1.00000	−0.100504
$100$	0.984576	0.0984576
$101$	−11.6608	−1.16029	−0.580145	−	0.814514i	$-0.697004\pi$
$101$	−11.6608	−1.16029	−0.580145	+	0.814514i	$0.697004\pi$
$102$	3.45095	0.341695
$103$	−0.907179	−0.0893870	−0.0446935	−	0.999001i	$-0.514231\pi$
$103$	−0.907179	−0.0893870	−0.0446935	+	0.999001i	$0.514231\pi$
$104$	0	0
$105$	−5.08840	−0.496577
$106$	10.4804	1.01794
$107$	−10.1756	−0.983711	−0.491856	−	0.870677i	$-0.663681\pi$
$107$	−10.1756	−0.983711	−0.491856	+	0.870677i	$0.663681\pi$
$108$	1.12828	0.108569
$109$	12.4405	1.19159	0.595794	−	0.803138i	$-0.296838\pi$
$109$	12.4405	1.19159	0.595794	+	0.803138i	$0.296838\pi$
$110$	1.89682	0.180854
$111$	10.6692	1.01268
$112$	1.17825	0.111334
$113$	−6.84823	−0.644227	−0.322113	−	0.946701i	$-0.604393\pi$
$113$	−6.84823	−0.644227	−0.322113	+	0.946701i	$0.604393\pi$
$114$	−0.166047	−0.0155517
$115$	9.82440	0.916130
$116$	8.72388	0.809992
$117$	0	0
$118$	0.768654	0.0707604
$119$	9.25753	0.848636
$120$	−5.93377	−0.541677
$121$	1.00000	0.0909091
$122$	5.23478	0.473935
$123$	6.54853	0.590461
$124$	9.97261	0.895567
$125$	11.9308	1.06712
$126$	−2.33848	−0.208328
$127$	10.0925	0.895561	0.447780	−	0.894144i	$-0.352215\pi$
$127$	10.0925	0.895561	0.447780	+	0.894144i	$0.352215\pi$
$128$	−5.21683	−0.461107
$129$	0.511245	0.0450126
$130$	0	0
$131$	15.1659	1.32505	0.662525	−	0.749040i	$-0.269485\pi$
$131$	15.1659	1.32505	0.662525	+	0.749040i	$0.269485\pi$
$132$	−1.12828	−0.0982041
$133$	−0.445437	−0.0386243
$134$	15.1153	1.30576
$135$	2.03159	0.174852
$136$	10.7955	0.925711
$137$	15.9212	1.36024	0.680119	−	0.733102i	$-0.261928\pi$
$137$	15.9212	1.36024	0.680119	+	0.733102i	$0.261928\pi$
$138$	4.51500	0.384343
$139$	−1.27724	−0.108334	−0.0541670	−	0.998532i	$-0.517250\pi$
$139$	−1.27724	−0.108334	−0.0541670	+	0.998532i	$0.517250\pi$
$140$	−5.74113	−0.485214
$141$	6.47604	0.545381
$142$	−9.16072	−0.768751
$143$	0	0
$144$	−0.470429	−0.0392024
$145$	15.7083	1.30450
$146$	−3.46376	−0.286663
$147$	0.726802	0.0599456
$148$	12.0379	0.989505
$149$	−0.507755	−0.0415969	−0.0207985	−	0.999784i	$-0.506621\pi$
$149$	−0.507755	−0.0415969	−0.0207985	+	0.999784i	$0.506621\pi$
$150$	0.814745	0.0665237
$151$	3.69825	0.300959	0.150480	−	0.988613i	$-0.451918\pi$
$151$	3.69825	0.300959	0.150480	+	0.988613i	$0.451918\pi$
$152$	−0.519441	−0.0421322
$153$	−3.69616	−0.298817
$154$	2.33848	0.188440
$155$	17.9568	1.44232
$156$	0	0
$157$	17.3393	1.38382	0.691912	−	0.721981i	$-0.256768\pi$
$157$	17.3393	1.38382	0.691912	+	0.721981i	$0.256768\pi$
$158$	−8.74490	−0.695707
$159$	−11.2250	−0.890204
$160$	−10.9752	−0.867667
$161$	12.1119	0.954555
$162$	0.933660	0.0733553
$163$	−18.1156	−1.41893	−0.709463	−	0.704742i	$-0.751063\pi$
$163$	−18.1156	−1.41893	−0.709463	+	0.704742i	$0.751063\pi$
$164$	7.38857	0.576950
$165$	−2.03159	−0.158159
$166$	−6.55172	−0.508512
$167$	−9.99386	−0.773348	−0.386674	−	0.922216i	$-0.626376\pi$
$167$	−9.99386	−0.773348	−0.386674	+	0.922216i	$0.626376\pi$
$168$	−7.31541	−0.564396
$169$	0	0
$170$	7.01093	0.537714
$171$	0.177845	0.0136002
$172$	0.576826	0.0439826
$173$	16.1831	1.23038	0.615188	−	0.788381i	$-0.289080\pi$
$173$	16.1831	1.23038	0.615188	+	0.788381i	$0.289080\pi$
$174$	7.21908	0.547277
$175$	2.18563	0.165218
$176$	0.470429	0.0354599
$177$	−0.823270	−0.0618808
$178$	−9.42628	−0.706530
$179$	0.397485	0.0297094	0.0148547	−	0.999890i	$-0.495271\pi$
$179$	0.397485	0.0297094	0.0148547	+	0.999890i	$0.495271\pi$
$180$	2.29220	0.170851
$181$	9.44541	0.702072	0.351036	−	0.936362i	$-0.385830\pi$
$181$	9.44541	0.702072	0.351036	+	0.936362i	$0.385830\pi$
$182$	0	0
$183$	−5.60673	−0.414462
$184$	14.1242	1.04125
$185$	21.6755	1.59361
$186$	8.25242	0.605096
$187$	3.69616	0.270290
$188$	7.30678	0.532902
$189$	2.50464	0.182185
$190$	−0.337339	−0.0244732
$191$	12.2524	0.886554	0.443277	−	0.896385i	$-0.353816\pi$
$191$	12.2524	0.886554	0.443277	+	0.896385i	$0.353816\pi$
$192$	−5.98475	−0.431912
$193$	−26.1205	−1.88019	−0.940096	−	0.340910i	$-0.889265\pi$
$193$	−26.1205	−1.88019	−0.940096	+	0.340910i	$0.889265\pi$
$194$	−8.32520	−0.597715
$195$	0	0
$196$	0.820035	0.0585739
$197$	3.94318	0.280940	0.140470	−	0.990085i	$-0.455139\pi$
$197$	3.94318	0.280940	0.140470	+	0.990085i	$0.455139\pi$
$198$	−0.933660	−0.0663523
$199$	8.34190	0.591342	0.295671	−	0.955290i	$-0.404457\pi$
$199$	8.34190	0.591342	0.295671	+	0.955290i	$0.404457\pi$
$200$	2.54875	0.180224
$201$	−16.1893	−1.14190
$202$	−10.8872	−0.766020
$203$	19.3659	1.35922
$204$	−4.17030	−0.291979
$205$	13.3039	0.929188
$206$	−0.846997	−0.0590131
$207$	−4.83581	−0.336112
$208$	0	0
$209$	−0.177845	−0.0123018
$210$	−4.75083	−0.327839
$211$	11.9162	0.820345	0.410173	−	0.912008i	$-0.365468\pi$
$211$	11.9162	0.820345	0.410173	+	0.912008i	$0.365468\pi$
$212$	−12.6650	−0.869834
$213$	9.81162	0.672281
$214$	−9.50054	−0.649444
$215$	1.03864	0.0708347
$216$	2.92075	0.198732
$217$	22.1379	1.50282
$218$	11.6152	0.786683
$219$	3.70987	0.250690
$220$	−2.29220	−0.154540
$221$	0	0
$222$	9.96142	0.668567
$223$	0.0455346	0.00304922	0.00152461	−	0.999999i	$-0.499515\pi$
$223$	0.0455346	0.00304922	0.00152461	+	0.999999i	$0.499515\pi$
$224$	−13.5307	−0.904060
$225$	−0.872636	−0.0581757
$226$	−6.39392	−0.425317
$227$	4.24286	0.281609	0.140804	−	0.990037i	$-0.455031\pi$
$227$	4.24286	0.281609	0.140804	+	0.990037i	$0.455031\pi$
$228$	0.200659	0.0132890
$229$	−3.89444	−0.257352	−0.128676	−	0.991687i	$-0.541073\pi$
$229$	−3.89444	−0.257352	−0.128676	+	0.991687i	$0.541073\pi$
$230$	9.17265	0.604826
$231$	−2.50464	−0.164793
$232$	22.5833	1.48267
$233$	−23.4650	−1.53724	−0.768621	−	0.639704i	$-0.779057\pi$
$233$	−23.4650	−1.53724	−0.768621	+	0.639704i	$0.779057\pi$
$234$	0	0
$235$	13.1567	0.858247
$236$	−0.928878	−0.0604648
$237$	9.36626	0.608404
$238$	8.64338	0.560267
$239$	8.60714	0.556750	0.278375	−	0.960473i	$-0.410204\pi$
$239$	8.60714	0.556750	0.278375	+	0.960473i	$0.410204\pi$
$240$	−0.955719	−0.0616914
$241$	−4.65244	−0.299690	−0.149845	−	0.988709i	$-0.547877\pi$
$241$	−4.65244	−0.299690	−0.149845	+	0.988709i	$0.547877\pi$
$242$	0.933660	0.0600179
$243$	−1.00000	−0.0641500
$244$	−6.32596	−0.404978
$245$	1.47656	0.0943342
$246$	6.11410	0.389821
$247$	0	0
$248$	25.8159	1.63931
$249$	7.01724	0.444700
$250$	11.1393	0.704512
$251$	−6.28034	−0.396411	−0.198206	−	0.980160i	$-0.563511\pi$
$251$	−6.28034	−0.396411	−0.198206	+	0.980160i	$0.563511\pi$
$252$	2.82593	0.178017
$253$	4.83581	0.304025
$254$	9.42292	0.591247
$255$	−7.50908	−0.470237
$256$	−16.8402	−1.05252
$257$	−9.63075	−0.600750	−0.300375	−	0.953821i	$-0.597112\pi$
$257$	−9.63075	−0.600750	−0.300375	+	0.953821i	$0.597112\pi$
$258$	0.477329	0.0297172
$259$	26.7225	1.66046
$260$	0	0
$261$	−7.73202	−0.478600
$262$	14.1598	0.874795
$263$	7.25654	0.447458	0.223729	−	0.974651i	$-0.428177\pi$
$263$	7.25654	0.447458	0.223729	+	0.974651i	$0.428177\pi$
$264$	−2.92075	−0.179760
$265$	−22.8047	−1.40088
$266$	−0.415887	−0.0254997
$267$	10.0961	0.617869
$268$	−18.2660	−1.11578
$269$	−5.47517	−0.333827	−0.166913	−	0.985972i	$-0.553380\pi$
$269$	−5.47517	−0.333827	−0.166913	+	0.985972i	$0.553380\pi$
$270$	1.89682	0.115437
$271$	−13.1132	−0.796568	−0.398284	−	0.917262i	$-0.630394\pi$
$271$	−13.1132	−0.796568	−0.398284	+	0.917262i	$0.630394\pi$
$272$	1.73878	0.105429
$273$	0	0
$274$	14.8650	0.898025
$275$	0.872636	0.0526219
$276$	−5.45615	−0.328421
$277$	−23.4734	−1.41038	−0.705189	−	0.709019i	$-0.749138\pi$
$277$	−23.4734	−1.41038	−0.705189	+	0.709019i	$0.749138\pi$
$278$	−1.19251	−0.0715219
$279$	−8.83878	−0.529164
$280$	−14.8619	−0.888170
$281$	−4.25275	−0.253698	−0.126849	−	0.991922i	$-0.540486\pi$
$281$	−4.25275	−0.253698	−0.126849	+	0.991922i	$0.540486\pi$
$282$	6.04642	0.360059
$283$	−8.93039	−0.530857	−0.265428	−	0.964131i	$-0.585513\pi$
$283$	−8.93039	−0.530857	−0.265428	+	0.964131i	$0.585513\pi$
$284$	11.0702	0.656898
$285$	0.361309	0.0214021
$286$	0	0
$287$	16.4017	0.968161
$288$	5.40228	0.318332
$289$	−3.33842	−0.196378
$290$	14.6662	0.861230
$291$	8.91674	0.522708
$292$	4.18577	0.244954
$293$	−3.35149	−0.195796	−0.0978980	−	0.995196i	$-0.531212\pi$
$293$	−3.35149	−0.195796	−0.0978980	+	0.995196i	$0.531212\pi$
$294$	0.678586	0.0395759
$295$	−1.67255	−0.0973795
$296$	31.1621	1.81126
$297$	1.00000	0.0580259
$298$	−0.474071	−0.0274622
$299$	0	0
$300$	−0.984576	−0.0568445
$301$	1.28048	0.0738057
$302$	3.45291	0.198693
$303$	11.6608	0.669893
$304$	−0.0836635	−0.00479843
$305$	−11.3906	−0.652223
$306$	−3.45095	−0.197278
$307$	−31.9777	−1.82506	−0.912532	−	0.409004i	$-0.865876\pi$
$307$	−31.9777	−1.82506	−0.912532	+	0.409004i	$0.865876\pi$
$308$	−2.82593	−0.161022
$309$	0.907179	0.0516076
$310$	16.7655	0.952219
$311$	−4.21584	−0.239059	−0.119529	−	0.992831i	$-0.538139\pi$
$311$	−4.21584	−0.239059	−0.119529	+	0.992831i	$0.538139\pi$
$312$	0	0
$313$	21.8252	1.23363	0.616817	−	0.787107i	$-0.288422\pi$
$313$	21.8252	1.23363	0.616817	+	0.787107i	$0.288422\pi$
$314$	16.1890	0.913598
$315$	5.08840	0.286699
$316$	10.5677	0.594482
$317$	8.75655	0.491817	0.245908	−	0.969293i	$-0.420914\pi$
$317$	8.75655	0.491817	0.245908	+	0.969293i	$0.420914\pi$
$318$	−10.4804	−0.587710
$319$	7.73202	0.432910
$320$	−12.1586	−0.679684
$321$	10.1756	0.567946
$322$	11.3084	0.630195
$323$	−0.657343	−0.0365756
$324$	−1.12828	−0.0626822
$325$	0	0
$326$	−16.9139	−0.936772
$327$	−12.4405	−0.687963
$328$	19.1266	1.05609
$329$	16.2201	0.894245
$330$	−1.89682	−0.104416
$331$	8.04496	0.442191	0.221096	−	0.975252i	$-0.429037\pi$
$331$	8.04496	0.442191	0.221096	+	0.975252i	$0.429037\pi$
$332$	7.91740	0.434524
$333$	−10.6692	−0.584670
$334$	−9.33087	−0.510562
$335$	−32.8900	−1.79697
$336$	−1.17825	−0.0642790
$337$	24.1310	1.31450	0.657250	−	0.753673i	$-0.271720\pi$
$337$	24.1310	1.31450	0.657250	+	0.753673i	$0.271720\pi$
$338$	0	0
$339$	6.84823	0.371945
$340$	−8.47234	−0.459477
$341$	8.83878	0.478647
$342$	0.166047	0.00897878
$343$	19.3528	1.04495
$344$	1.49322	0.0805089
$345$	−9.82440	−0.528928
$346$	15.1095	0.812291
$347$	−29.2386	−1.56961	−0.784804	−	0.619744i	$-0.787236\pi$
$347$	−29.2386	−1.56961	−0.784804	+	0.619744i	$0.787236\pi$
$348$	−8.72388	−0.467649
$349$	−16.3093	−0.873018	−0.436509	−	0.899700i	$-0.643785\pi$
$349$	−16.3093	−0.873018	−0.436509	+	0.899700i	$0.643785\pi$
$350$	2.04064	0.109077
$351$	0	0
$352$	−5.40228	−0.287942
$353$	−33.1018	−1.76183	−0.880917	−	0.473271i	$-0.843073\pi$
$353$	−33.1018	−1.76183	−0.880917	+	0.473271i	$0.843073\pi$
$354$	−0.768654	−0.0408535
$355$	19.9332	1.05795
$356$	11.3912	0.603731
$357$	−9.25753	−0.489960
$358$	0.371116	0.0196141
$359$	2.34626	0.123831	0.0619154	−	0.998081i	$-0.480279\pi$
$359$	2.34626	0.123831	0.0619154	+	0.998081i	$0.480279\pi$
$360$	5.93377	0.312737
$361$	−18.9684	−0.998335
$362$	8.81880	0.463506
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	7.53695	0.394502
$366$	−5.23478	−0.273627
$367$	−21.0725	−1.09998	−0.549989	−	0.835172i	$-0.685368\pi$
$367$	−21.0725	−1.09998	−0.549989	+	0.835172i	$0.685368\pi$
$368$	2.27491	0.118588
$369$	−6.54853	−0.340903
$370$	20.2375	1.05210
$371$	−28.1146	−1.45964
$372$	−9.97261	−0.517056
$373$	26.0992	1.35137	0.675683	−	0.737192i	$-0.263849\pi$
$373$	26.0992	1.35137	0.675683	+	0.737192i	$0.263849\pi$
$374$	3.45095	0.178445
$375$	−11.9308	−0.616104
$376$	18.9149	0.975461
$377$	0	0
$378$	2.33848	0.120278
$379$	−25.6871	−1.31946	−0.659729	−	0.751504i	$-0.729329\pi$
$379$	−25.6871	−1.31946	−0.659729	+	0.751504i	$0.729329\pi$
$380$	0.407657	0.0209124
$381$	−10.0925	−0.517052
$382$	11.4396	0.585301
$383$	14.7536	0.753874	0.376937	−	0.926239i	$-0.376977\pi$
$383$	14.7536	0.753874	0.376937	+	0.926239i	$0.376977\pi$
$384$	5.21683	0.266220
$385$	−5.08840	−0.259329
$386$	−24.3876	−1.24130
$387$	−0.511245	−0.0259880
$388$	10.0606	0.510748
$389$	−34.6262	−1.75562	−0.877808	−	0.479013i	$-0.840995\pi$
$389$	−34.6262	−1.75562	−0.877808	+	0.479013i	$0.840995\pi$
$390$	0	0
$391$	17.8739	0.903923
$392$	2.12281	0.107218
$393$	−15.1659	−0.765018
$394$	3.68159	0.185476
$395$	19.0284	0.957423
$396$	1.12828	0.0566981
$397$	−8.33430	−0.418287	−0.209143	−	0.977885i	$-0.567067\pi$
$397$	−8.33430	−0.418287	−0.209143	+	0.977885i	$0.567067\pi$
$398$	7.78850	0.390402
$399$	0.445437	0.0222997
$400$	0.410513	0.0205257
$401$	−15.2929	−0.763690	−0.381845	−	0.924226i	$-0.624711\pi$
$401$	−15.2929	−0.763690	−0.381845	+	0.924226i	$0.624711\pi$
$402$	−15.1153	−0.753882
$403$	0	0
$404$	13.1566	0.654565
$405$	−2.03159	−0.100951
$406$	18.0812	0.897353
$407$	10.6692	0.528853
$408$	−10.7955	−0.534459
$409$	7.14920	0.353505	0.176753	−	0.984255i	$-0.443441\pi$
$409$	7.14920	0.353505	0.176753	+	0.984255i	$0.443441\pi$
$410$	12.4214	0.613447
$411$	−15.9212	−0.785334
$412$	1.02355	0.0504267
$413$	−2.06199	−0.101464
$414$	−4.51500	−0.221900
$415$	14.2562	0.699808
$416$	0	0
$417$	1.27724	0.0625467
$418$	−0.166047	−0.00812162
$419$	8.51102	0.415790	0.207895	−	0.978151i	$-0.433339\pi$
$419$	8.51102	0.415790	0.207895	+	0.978151i	$0.433339\pi$
$420$	5.74113	0.280138
$421$	6.54048	0.318764	0.159382	−	0.987217i	$-0.449050\pi$
$421$	6.54048	0.318764	0.159382	+	0.987217i	$0.449050\pi$
$422$	11.1257	0.541590
$423$	−6.47604	−0.314876
$424$	−32.7855	−1.59221
$425$	3.22540	0.156455
$426$	9.16072	0.443838
$427$	−14.0428	−0.679580
$428$	11.4809	0.554950
$429$	0	0
$430$	0.969737	0.0467649
$431$	16.5683	0.798068	0.399034	−	0.916936i	$-0.369346\pi$
$431$	16.5683	0.798068	0.399034	+	0.916936i	$0.369346\pi$
$432$	0.470429	0.0226335
$433$	1.71264	0.0823040	0.0411520	−	0.999153i	$-0.486897\pi$
$433$	1.71264	0.0823040	0.0411520	+	0.999153i	$0.486897\pi$
$434$	20.6693	0.992158
$435$	−15.7083	−0.753156
$436$	−14.0364	−0.672221
$437$	−0.860025	−0.0411406
$438$	3.46376	0.165505
$439$	36.4385	1.73911	0.869557	−	0.493833i	$-0.164404\pi$
$439$	36.4385	1.73911	0.869557	+	0.493833i	$0.164404\pi$
$440$	−5.93377	−0.282881
$441$	−0.726802	−0.0346096
$442$	0	0
$443$	3.45652	0.164224	0.0821120	−	0.996623i	$-0.473833\pi$
$443$	3.45652	0.164224	0.0821120	+	0.996623i	$0.473833\pi$
$444$	−12.0379	−0.571291
$445$	20.5111	0.972318
$446$	0.0425138	0.00201309
$447$	0.507755	0.0240160
$448$	−14.9896	−0.708193
$449$	−24.1866	−1.14144	−0.570718	−	0.821146i	$-0.693335\pi$
$449$	−24.1866	−1.14144	−0.570718	+	0.821146i	$0.693335\pi$
$450$	−0.814745	−0.0384075
$451$	6.54853	0.308358
$452$	7.72671	0.363434
$453$	−3.69825	−0.173759
$454$	3.96139	0.185917
$455$	0	0
$456$	0.519441	0.0243250
$457$	22.3407	1.04506	0.522528	−	0.852622i	$-0.324989\pi$
$457$	22.3407	1.04506	0.522528	+	0.852622i	$0.324989\pi$
$458$	−3.63609	−0.169903
$459$	3.69616	0.172522
$460$	−11.0847	−0.516825
$461$	−29.4125	−1.36987	−0.684937	−	0.728602i	$-0.740170\pi$
$461$	−29.4125	−1.36987	−0.684937	+	0.728602i	$0.740170\pi$
$462$	−2.33848	−0.108796
$463$	22.9761	1.06779	0.533895	−	0.845551i	$-0.320728\pi$
$463$	22.9761	1.06779	0.533895	+	0.845551i	$0.320728\pi$
$464$	3.63737	0.168860
$465$	−17.9568	−0.832726
$466$	−21.9083	−1.01488
$467$	−1.35558	−0.0627286	−0.0313643	−	0.999508i	$-0.509985\pi$
$467$	−1.35558	−0.0627286	−0.0313643	+	0.999508i	$0.509985\pi$
$468$	0	0
$469$	−40.5482	−1.87234
$470$	12.2839	0.566612
$471$	−17.3393	−0.798952
$472$	−2.40457	−0.110679
$473$	0.511245	0.0235070
$474$	8.74490	0.401667
$475$	−0.155194	−0.00712079
$476$	−10.4451	−0.478749
$477$	11.2250	0.513959
$478$	8.03614	0.367565
$479$	−30.6882	−1.40218	−0.701090	−	0.713073i	$-0.747303\pi$
$479$	−30.6882	−1.40218	−0.701090	+	0.713073i	$0.747303\pi$
$480$	10.9752	0.500948
$481$	0	0
$482$	−4.34380	−0.197855
$483$	−12.1119	−0.551113
$484$	−1.12828	−0.0512854
$485$	18.1152	0.822567
$486$	−0.933660	−0.0423517
$487$	−13.8280	−0.626609	−0.313304	−	0.949653i	$-0.601436\pi$
$487$	−13.8280	−0.626609	−0.313304	+	0.949653i	$0.601436\pi$
$488$	−16.3759	−0.741300
$489$	18.1156	0.819218
$490$	1.37861	0.0622792
$491$	−8.69164	−0.392248	−0.196124	−	0.980579i	$-0.562836\pi$
$491$	−8.69164	−0.392248	−0.196124	+	0.980579i	$0.562836\pi$
$492$	−7.38857	−0.333102
$493$	28.5788	1.28712
$494$	0	0
$495$	2.03159	0.0913133
$496$	4.15802	0.186700
$497$	24.5745	1.10232
$498$	6.55172	0.293589
$499$	−17.4871	−0.782828	−0.391414	−	0.920215i	$-0.628014\pi$
$499$	−17.4871	−0.782828	−0.391414	+	0.920215i	$0.628014\pi$
$500$	−13.4613	−0.602006
$501$	9.99386	0.446493
$502$	−5.86370	−0.261710
$503$	23.0008	1.02556	0.512778	−	0.858521i	$-0.328616\pi$
$503$	23.0008	1.02556	0.512778	+	0.858521i	$0.328616\pi$
$504$	7.31541	0.325854
$505$	23.6899	1.05419
$506$	4.51500	0.200716
$507$	0	0
$508$	−11.3871	−0.505221
$509$	−9.67031	−0.428629	−0.214315	−	0.976765i	$-0.568752\pi$
$509$	−9.67031	−0.428629	−0.214315	+	0.976765i	$0.568752\pi$
$510$	−7.01093	−0.310449
$511$	9.29188	0.411049
$512$	−5.28940	−0.233761
$513$	−0.177845	−0.00785205
$514$	−8.99185	−0.396613
$515$	1.84302	0.0812130
$516$	−0.576826	−0.0253934
$517$	6.47604	0.284816
$518$	24.9497	1.09623
$519$	−16.1831	−0.710358
$520$	0	0
$521$	−21.7733	−0.953904	−0.476952	−	0.878929i	$-0.658258\pi$
$521$	−21.7733	−0.953904	−0.476952	+	0.878929i	$0.658258\pi$
$522$	−7.21908	−0.315971
$523$	12.7962	0.559537	0.279769	−	0.960067i	$-0.409742\pi$
$523$	12.7962	0.559537	0.279769	+	0.960067i	$0.409742\pi$
$524$	−17.1114	−0.747513
$525$	−2.18563	−0.0953889
$526$	6.77515	0.295410
$527$	32.6695	1.42311
$528$	−0.470429	−0.0204728
$529$	0.385083	0.0167427
$530$	−21.2918	−0.924859
$531$	0.823270	0.0357269
$532$	0.502577	0.0217895
$533$	0	0
$534$	9.42628	0.407915
$535$	20.6726	0.893756
$536$	−47.2848	−2.04239
$537$	−0.397485	−0.0171527
$538$	−5.11195	−0.220392
$539$	0.726802	0.0313056
$540$	−2.29220	−0.0986407
$541$	2.49681	0.107346	0.0536731	−	0.998559i	$-0.482907\pi$
$541$	2.49681	0.107346	0.0536731	+	0.998559i	$0.482907\pi$
$542$	−12.2432	−0.525892
$543$	−9.44541	−0.405341
$544$	−19.9677	−0.856107
$545$	−25.2741	−1.08262
$546$	0	0
$547$	−46.0277	−1.96800	−0.984001	−	0.178163i	$-0.942985\pi$
$547$	−46.0277	−1.96800	−0.984001	+	0.178163i	$0.942985\pi$
$548$	−17.9635	−0.767364
$549$	5.60673	0.239290
$550$	0.814745	0.0347408
$551$	−1.37510	−0.0585813
$552$	−14.1242	−0.601166
$553$	23.4591	0.997581
$554$	−21.9162	−0.931128
$555$	−21.6755	−0.920073
$556$	1.44108	0.0611155
$557$	39.4423	1.67122	0.835612	−	0.549320i	$-0.185113\pi$
$557$	39.4423	1.67122	0.835612	+	0.549320i	$0.185113\pi$
$558$	−8.25242	−0.349353
$559$	0	0
$560$	−2.39373	−0.101153
$561$	−3.69616	−0.156052
$562$	−3.97062	−0.167491
$563$	17.0130	0.717013	0.358506	−	0.933527i	$-0.383286\pi$
$563$	17.0130	0.717013	0.358506	+	0.933527i	$0.383286\pi$
$564$	−7.30678	−0.307671
$565$	13.9128	0.585316
$566$	−8.33795	−0.350470
$567$	−2.50464	−0.105185
$568$	28.6573	1.20243
$569$	−8.33821	−0.349556	−0.174778	−	0.984608i	$-0.555921\pi$
$569$	−8.33821	−0.349556	−0.174778	+	0.984608i	$0.555921\pi$
$570$	0.337339	0.0141296
$571$	29.1131	1.21835	0.609173	−	0.793038i	$-0.291502\pi$
$571$	29.1131	1.21835	0.609173	+	0.793038i	$0.291502\pi$
$572$	0	0
$573$	−12.2524	−0.511852
$574$	15.3136	0.639177
$575$	4.21990	0.175982
$576$	5.98475	0.249365
$577$	−39.6679	−1.65140	−0.825698	−	0.564113i	$-0.809218\pi$
$577$	−39.6679	−1.65140	−0.825698	+	0.564113i	$0.809218\pi$
$578$	−3.11695	−0.129648
$579$	26.1205	1.08553
$580$	−17.7234	−0.735922
$581$	17.5756	0.729160
$582$	8.32520	0.345091
$583$	−11.2250	−0.464894
$584$	10.8356	0.448381
$585$	0	0
$586$	−3.12915	−0.129264
$587$	−28.8349	−1.19015	−0.595073	−	0.803672i	$-0.702877\pi$
$587$	−28.8349	−1.19015	−0.595073	+	0.803672i	$0.702877\pi$
$588$	−0.820035	−0.0338177
$589$	−1.57193	−0.0647704
$590$	−1.56159	−0.0642897
$591$	−3.94318	−0.162201
$592$	5.01911	0.206284
$593$	29.8017	1.22381	0.611904	−	0.790932i	$-0.290404\pi$
$593$	29.8017	1.22381	0.611904	+	0.790932i	$0.290404\pi$
$594$	0.933660	0.0383085
$595$	−18.8075	−0.771033
$596$	0.572889	0.0234665
$597$	−8.34190	−0.341411
$598$	0	0
$599$	15.4759	0.632327	0.316163	−	0.948705i	$-0.397605\pi$
$599$	15.4759	0.632327	0.316163	+	0.948705i	$0.397605\pi$
$600$	−2.54875	−0.104052
$601$	−10.7692	−0.439283	−0.219642	−	0.975581i	$-0.570489\pi$
$601$	−10.7692	−0.439283	−0.219642	+	0.975581i	$0.570489\pi$
$602$	1.19553	0.0487263
$603$	16.1893	0.659279
$604$	−4.17266	−0.169783
$605$	−2.03159	−0.0825959
$606$	10.8872	0.442262
$607$	−9.75920	−0.396114	−0.198057	−	0.980191i	$-0.563463\pi$
$607$	−9.75920	−0.396114	−0.198057	+	0.980191i	$0.563463\pi$
$608$	0.960768	0.0389643
$609$	−19.3659	−0.784746
$610$	−10.6349	−0.430596
$611$	0	0
$612$	4.17030	0.168574
$613$	−32.5043	−1.31284	−0.656419	−	0.754396i	$-0.727930\pi$
$613$	−32.5043	−1.31284	−0.656419	+	0.754396i	$0.727930\pi$
$614$	−29.8563	−1.20490
$615$	−13.3039	−0.536467
$616$	−7.31541	−0.294746
$617$	−41.8641	−1.68539	−0.842693	−	0.538395i	$-0.819031\pi$
$617$	−41.8641	−1.68539	−0.842693	+	0.538395i	$0.819031\pi$
$618$	0.846997	0.0340712
$619$	−25.8333	−1.03833	−0.519164	−	0.854675i	$-0.673757\pi$
$619$	−25.8333	−1.03833	−0.519164	+	0.854675i	$0.673757\pi$
$620$	−20.2603	−0.813672
$621$	4.83581	0.194054
$622$	−3.93616	−0.157826
$623$	25.2869	1.01310
$624$	0	0
$625$	−19.8753	−0.795013
$626$	20.3773	0.814442
$627$	0.177845	0.00710245
$628$	−19.5635	−0.780670
$629$	39.4351	1.57238
$630$	4.75083	0.189278
$631$	15.6061	0.621270	0.310635	−	0.950529i	$-0.399458\pi$
$631$	15.6061	0.621270	0.310635	+	0.950529i	$0.399458\pi$
$632$	27.3565	1.08818
$633$	−11.9162	−0.473627
$634$	8.17564	0.324696
$635$	−20.5037	−0.813666
$636$	12.6650	0.502199
$637$	0	0
$638$	7.21908	0.285806
$639$	−9.81162	−0.388142
$640$	10.5985	0.418942
$641$	−7.48540	−0.295656	−0.147828	−	0.989013i	$-0.547228\pi$
$641$	−7.48540	−0.295656	−0.147828	+	0.989013i	$0.547228\pi$
$642$	9.50054	0.374956
$643$	45.6695	1.80103	0.900514	−	0.434827i	$-0.143191\pi$
$643$	45.6695	1.80103	0.900514	+	0.434827i	$0.143191\pi$
$644$	−13.6657	−0.538502
$645$	−1.03864	−0.0408964
$646$	−0.613735	−0.0241471
$647$	12.9569	0.509390	0.254695	−	0.967021i	$-0.418025\pi$
$647$	12.9569	0.509390	0.254695	+	0.967021i	$0.418025\pi$
$648$	−2.92075	−0.114738
$649$	−0.823270	−0.0323162
$650$	0	0
$651$	−22.1379	−0.867654
$652$	20.4395	0.800472
$653$	−11.0979	−0.434295	−0.217147	−	0.976139i	$-0.569675\pi$
$653$	−11.0979	−0.434295	−0.217147	+	0.976139i	$0.569675\pi$
$654$	−11.6152	−0.454192
$655$	−30.8109	−1.20388
$656$	3.08062	0.120278
$657$	−3.70987	−0.144736
$658$	15.1441	0.590378
$659$	19.8114	0.771741	0.385870	−	0.922553i	$-0.373901\pi$
$659$	19.8114	0.771741	0.385870	+	0.922553i	$0.373901\pi$
$660$	2.29220	0.0892238
$661$	16.6094	0.646032	0.323016	−	0.946393i	$-0.395303\pi$
$661$	16.6094	0.646032	0.323016	+	0.946393i	$0.395303\pi$
$662$	7.51126	0.291933
$663$	0	0
$664$	20.4956	0.795383
$665$	0.904946	0.0350923
$666$	−9.96142	−0.385997
$667$	37.3906	1.44777
$668$	11.2759	0.436276
$669$	−0.0455346	−0.00176047
$670$	−30.7081	−1.18636
$671$	−5.60673	−0.216446
$672$	13.5307	0.521959
$673$	2.04359	0.0787747	0.0393874	−	0.999224i	$-0.487459\pi$
$673$	2.04359	0.0787747	0.0393874	+	0.999224i	$0.487459\pi$
$674$	22.5302	0.867829
$675$	0.872636	0.0335878
$676$	0	0
$677$	−43.2409	−1.66188	−0.830941	−	0.556360i	$-0.812198\pi$
$677$	−43.2409	−1.66188	−0.830941	+	0.556360i	$0.812198\pi$
$678$	6.39392	0.245557
$679$	22.3332	0.857068
$680$	−21.9321	−0.841059
$681$	−4.24286	−0.162587
$682$	8.25242	0.316001
$683$	−7.47018	−0.285838	−0.142919	−	0.989734i	$-0.545649\pi$
$683$	−7.47018	−0.285838	−0.142919	+	0.989734i	$0.545649\pi$
$684$	−0.200659	−0.00767238
$685$	−32.3453	−1.23585
$686$	18.0690	0.689876
$687$	3.89444	0.148582
$688$	0.240504	0.00916914
$689$	0	0
$690$	−9.17265	−0.349197
$691$	−0.939734	−0.0357492	−0.0178746	−	0.999840i	$-0.505690\pi$
$691$	−0.939734	−0.0357492	−0.0178746	+	0.999840i	$0.505690\pi$
$692$	−18.2590	−0.694103
$693$	2.50464	0.0951432
$694$	−27.2989	−1.03625
$695$	2.59483	0.0984275
$696$	−22.5833	−0.856018
$697$	24.2044	0.916807
$698$	−15.2274	−0.576364
$699$	23.4650	0.887527
$700$	−2.46600	−0.0932062
$701$	−41.8724	−1.58150	−0.790749	−	0.612141i	$-0.790309\pi$
$701$	−41.8724	−1.58150	−0.790749	+	0.612141i	$0.790309\pi$
$702$	0	0
$703$	−1.89747	−0.0715643
$704$	−5.98475	−0.225559
$705$	−13.1567	−0.495509
$706$	−30.9059	−1.16316
$707$	29.2060	1.09840
$708$	0.928878	0.0349094
$709$	9.79637	0.367910	0.183955	−	0.982935i	$-0.441110\pi$
$709$	9.79637	0.367910	0.183955	+	0.982935i	$0.441110\pi$
$710$	18.6108	0.698453
$711$	−9.36626	−0.351262
$712$	29.4880	1.10511
$713$	42.7427	1.60073
$714$	−8.64338	−0.323470
$715$	0	0
$716$	−0.448474	−0.0167603
$717$	−8.60714	−0.321440
$718$	2.19061	0.0817528
$719$	23.3105	0.869334	0.434667	−	0.900591i	$-0.356866\pi$
$719$	23.3105	0.869334	0.434667	+	0.900591i	$0.356866\pi$
$720$	0.955719	0.0356176
$721$	2.27215	0.0846194
$722$	−17.7100	−0.659098
$723$	4.65244	0.173026
$724$	−10.6571	−0.396066
$725$	6.74724	0.250586
$726$	−0.933660	−0.0346514
$727$	40.9042	1.51705	0.758526	−	0.651642i	$-0.225920\pi$
$727$	40.9042	1.51705	0.758526	+	0.651642i	$0.225920\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	7.03695	0.260449
$731$	1.88964	0.0698909
$732$	6.32596	0.233814
$733$	53.7254	1.98439	0.992197	−	0.124682i	$-0.0397910\pi$
$733$	53.7254	1.98439	0.992197	+	0.124682i	$0.0397910\pi$
$734$	−19.6746	−0.726202
$735$	−1.47656	−0.0544639
$736$	−26.1244	−0.962958
$737$	−16.1893	−0.596340
$738$	−6.11410	−0.225063
$739$	−33.8116	−1.24378	−0.621890	−	0.783104i	$-0.713635\pi$
$739$	−33.8116	−1.24378	−0.621890	+	0.783104i	$0.713635\pi$
$740$	−24.4560	−0.899020
$741$	0	0
$742$	−26.2495	−0.963650
$743$	−30.7507	−1.12813	−0.564067	−	0.825729i	$-0.690764\pi$
$743$	−30.7507	−1.12813	−0.564067	+	0.825729i	$0.690764\pi$
$744$	−25.8159	−0.946455
$745$	1.03155	0.0377931
$746$	24.3678	0.892168
$747$	−7.01724	−0.256747
$748$	−4.17030	−0.152481
$749$	25.4861	0.931243
$750$	−11.1393	−0.406750
$751$	36.5263	1.33286	0.666431	−	0.745567i	$-0.267821\pi$
$751$	36.5263	1.33286	0.666431	+	0.745567i	$0.267821\pi$
$752$	3.04652	0.111095
$753$	6.28034	0.228868
$754$	0	0
$755$	−7.51333	−0.273438
$756$	−2.82593	−0.102778
$757$	44.0240	1.60008	0.800041	−	0.599946i	$-0.204811\pi$
$757$	44.0240	1.60008	0.800041	+	0.599946i	$0.204811\pi$
$758$	−23.9830	−0.871103
$759$	−4.83581	−0.175529
$760$	1.05529	0.0382794
$761$	−9.52391	−0.345242	−0.172621	−	0.984988i	$-0.555224\pi$
$761$	−9.52391	−0.345242	−0.172621	+	0.984988i	$0.555224\pi$
$762$	−9.42292	−0.341356
$763$	−31.1590	−1.12803
$764$	−13.8241	−0.500140
$765$	7.50908	0.271491
$766$	13.7748	0.497705
$767$	0	0
$768$	16.8402	0.607670
$769$	−24.1052	−0.869254	−0.434627	−	0.900610i	$-0.643120\pi$
$769$	−24.1052	−0.869254	−0.434627	+	0.900610i	$0.643120\pi$
$770$	−4.75083	−0.171208
$771$	9.63075	0.346843
$772$	29.4712	1.06069
$773$	−29.2427	−1.05179	−0.525894	−	0.850550i	$-0.676269\pi$
$773$	−29.2427	−1.05179	−0.525894	+	0.850550i	$0.676269\pi$
$774$	−0.477329	−0.0171572
$775$	7.71303	0.277060
$776$	26.0435	0.934909
$777$	−26.7225	−0.958664
$778$	−32.3291	−1.15905
$779$	−1.16462	−0.0417270
$780$	0	0
$781$	9.81162	0.351088
$782$	16.6882	0.596768
$783$	7.73202	0.276320
$784$	0.341908	0.0122110
$785$	−35.2263	−1.25728
$786$	−14.1598	−0.505063
$787$	−47.7730	−1.70292	−0.851462	−	0.524416i	$-0.824284\pi$
$787$	−47.7730	−1.70292	−0.851462	+	0.524416i	$0.824284\pi$
$788$	−4.44901	−0.158489
$789$	−7.25654	−0.258340
$790$	17.7661	0.632088
$791$	17.1523	0.609866
$792$	2.92075	0.103784
$793$	0	0
$794$	−7.78140	−0.276152
$795$	22.8047	0.808800
$796$	−9.41199	−0.333599
$797$	−37.6351	−1.33310	−0.666552	−	0.745458i	$-0.732231\pi$
$797$	−37.6351	−1.33310	−0.666552	+	0.745458i	$0.732231\pi$
$798$	0.415887	0.0147222
$799$	23.9365	0.846812
$800$	−4.71422	−0.166673
$801$	−10.0961	−0.356727
$802$	−14.2783	−0.504186
$803$	3.70987	0.130919
$804$	18.2660	0.644193
$805$	−24.6065	−0.867266
$806$	0	0
$807$	5.47517	0.192735
$808$	34.0582	1.19816
$809$	−4.54177	−0.159680	−0.0798400	−	0.996808i	$-0.525441\pi$
$809$	−4.54177	−0.159680	−0.0798400	+	0.996808i	$0.525441\pi$
$810$	−1.89682	−0.0666473
$811$	−40.7580	−1.43121	−0.715603	−	0.698507i	$-0.753848\pi$
$811$	−40.7580	−1.43121	−0.715603	+	0.698507i	$0.753848\pi$
$812$	−21.8501	−0.766789
$813$	13.1132	0.459899
$814$	9.96142	0.349148
$815$	36.8036	1.28917
$816$	−1.73878	−0.0608694
$817$	−0.0909223	−0.00318097
$818$	6.67493	0.233383
$819$	0	0
$820$	−15.0106	−0.524191
$821$	−2.96276	−0.103401	−0.0517006	−	0.998663i	$-0.516464\pi$
$821$	−2.96276	−0.103401	−0.0517006	+	0.998663i	$0.516464\pi$
$822$	−14.8650	−0.518475
$823$	18.6930	0.651596	0.325798	−	0.945439i	$-0.394367\pi$
$823$	18.6930	0.651596	0.325798	+	0.945439i	$0.394367\pi$
$824$	2.64964	0.0923047
$825$	−0.872636	−0.0303813
$826$	−1.92520	−0.0669862
$827$	5.66041	0.196832	0.0984159	−	0.995145i	$-0.468622\pi$
$827$	5.66041	0.196832	0.0984159	+	0.995145i	$0.468622\pi$
$828$	5.45615	0.189614
$829$	−15.0712	−0.523446	−0.261723	−	0.965143i	$-0.584291\pi$
$829$	−15.0712	−0.523446	−0.261723	+	0.965143i	$0.584291\pi$
$830$	13.3104	0.462011
$831$	23.4734	0.814283
$832$	0	0
$833$	2.68637	0.0930773
$834$	1.19251	0.0412932
$835$	20.3034	0.702630
$836$	0.200659	0.00693993
$837$	8.83878	0.305513
$838$	7.94640	0.274504
$839$	41.5445	1.43428	0.717139	−	0.696931i	$-0.245451\pi$
$839$	41.5445	1.43428	0.717139	+	0.696931i	$0.245451\pi$
$840$	14.8619	0.512785
$841$	30.7841	1.06152
$842$	6.10659	0.210447
$843$	4.25275	0.146472
$844$	−13.4448	−0.462789
$845$	0	0
$846$	−6.04642	−0.207880
$847$	−2.50464	−0.0860603
$848$	−5.28059	−0.181336
$849$	8.93039	0.306490
$850$	3.01143	0.103291
$851$	51.5943	1.76863
$852$	−11.0702	−0.379260
$853$	36.9040	1.26357	0.631784	−	0.775145i	$-0.282323\pi$
$853$	36.9040	1.26357	0.631784	+	0.775145i	$0.282323\pi$
$854$	−13.1112	−0.448657
$855$	−0.361309	−0.0123565
$856$	29.7203	1.01582
$857$	−50.3162	−1.71877	−0.859385	−	0.511329i	$-0.829153\pi$
$857$	−50.3162	−1.71877	−0.859385	+	0.511329i	$0.829153\pi$
$858$	0	0
$859$	34.3448	1.17183	0.585915	−	0.810372i	$-0.300735\pi$
$859$	34.3448	1.17183	0.585915	+	0.810372i	$0.300735\pi$
$860$	−1.17188	−0.0399606
$861$	−16.4017	−0.558968
$862$	15.4692	0.526882
$863$	−25.7340	−0.875997	−0.437999	−	0.898976i	$-0.644313\pi$
$863$	−25.7340	−0.875997	−0.437999	+	0.898976i	$0.644313\pi$
$864$	−5.40228	−0.183789
$865$	−32.8774	−1.11786
$866$	1.59902	0.0543369
$867$	3.33842	0.113379
$868$	−24.9777	−0.847800
$869$	9.36626	0.317728
$870$	−14.6662	−0.497231
$871$	0	0
$872$	−36.3357	−1.23048
$873$	−8.91674	−0.301786
$874$	−0.802971	−0.0271609
$875$	−29.8823	−1.01021
$876$	−4.18577	−0.141424
$877$	−21.2205	−0.716566	−0.358283	−	0.933613i	$-0.616638\pi$
$877$	−21.2205	−0.716566	−0.358283	+	0.933613i	$0.616638\pi$
$878$	34.0212	1.14816
$879$	3.35149	0.113043
$880$	−0.955719	−0.0322173
$881$	−18.6417	−0.628055	−0.314028	−	0.949414i	$-0.601678\pi$
$881$	−18.6417	−0.628055	−0.314028	+	0.949414i	$0.601678\pi$
$882$	−0.678586	−0.0228492
$883$	25.8112	0.868618	0.434309	−	0.900764i	$-0.356993\pi$
$883$	25.8112	0.868618	0.434309	+	0.900764i	$0.356993\pi$
$884$	0	0
$885$	1.67255	0.0562221
$886$	3.22721	0.108420
$887$	50.0092	1.67914	0.839572	−	0.543249i	$-0.182806\pi$
$887$	50.0092	1.67914	0.839572	+	0.543249i	$0.182806\pi$
$888$	−31.1621	−1.04573
$889$	−25.2779	−0.847794
$890$	19.1504	0.641922
$891$	−1.00000	−0.0335013
$892$	−0.0513757	−0.00172019
$893$	−1.15173	−0.0385413
$894$	0.474071	0.0158553
$895$	−0.807527	−0.0269927
$896$	13.0663	0.436513
$897$	0	0
$898$	−22.5821	−0.753573
$899$	68.3416	2.27932
$900$	0.984576	0.0328192
$901$	−41.4895	−1.38222
$902$	6.11410	0.203577
$903$	−1.28048	−0.0426117
$904$	20.0019	0.665255
$905$	−19.1892	−0.637871
$906$	−3.45291	−0.114715
$907$	16.1013	0.534636	0.267318	−	0.963608i	$-0.413863\pi$
$907$	16.1013	0.534636	0.267318	+	0.963608i	$0.413863\pi$
$908$	−4.78713	−0.158867
$909$	−11.6608	−0.386763
$910$	0	0
$911$	−22.7850	−0.754902	−0.377451	−	0.926030i	$-0.623199\pi$
$911$	−22.7850	−0.754902	−0.377451	+	0.926030i	$0.623199\pi$
$912$	0.0836635	0.00277037
$913$	7.01724	0.232237
$914$	20.8587	0.689943
$915$	11.3906	0.376561
$916$	4.39402	0.145182
$917$	−37.9850	−1.25438
$918$	3.45095	0.113898
$919$	−52.1592	−1.72057	−0.860286	−	0.509811i	$-0.829715\pi$
$919$	−52.1592	−1.72057	−0.860286	+	0.509811i	$0.829715\pi$
$920$	−28.6946	−0.946033
$921$	31.9777	1.05370
$922$	−27.4612	−0.904388
$923$	0	0
$924$	2.82593	0.0929662
$925$	9.31034	0.306122
$926$	21.4519	0.704952
$927$	−0.907179	−0.0297957
$928$	−41.7705	−1.37118
$929$	33.3550	1.09434	0.547172	−	0.837020i	$-0.315704\pi$
$929$	33.3550	1.09434	0.547172	+	0.837020i	$0.315704\pi$
$930$	−16.7655	−0.549764
$931$	−0.129258	−0.00423626
$932$	26.4750	0.867219
$933$	4.21584	0.138021
$934$	−1.26565	−0.0414132
$935$	−7.50908	−0.245573
$936$	0	0
$937$	−41.6208	−1.35969	−0.679846	−	0.733355i	$-0.737953\pi$
$937$	−41.6208	−1.35969	−0.679846	+	0.733355i	$0.737953\pi$
$938$	−37.8583	−1.23612
$939$	−21.8252	−0.712239
$940$	−14.8444	−0.484171
$941$	−45.3397	−1.47803	−0.739016	−	0.673688i	$-0.764709\pi$
$941$	−45.3397	−1.47803	−0.739016	+	0.673688i	$0.764709\pi$
$942$	−16.1890	−0.527466
$943$	31.6675	1.03123
$944$	−0.387290	−0.0126052
$945$	−5.08840	−0.165526
$946$	0.477329	0.0155193
$947$	−37.3932	−1.21512	−0.607558	−	0.794275i	$-0.707851\pi$
$947$	−37.3932	−1.21512	−0.607558	+	0.794275i	$0.707851\pi$
$948$	−10.5677	−0.343225
$949$	0	0
$950$	−0.144898	−0.00470112
$951$	−8.75655	−0.283951
$952$	−27.0389	−0.876336
$953$	−3.92144	−0.127028	−0.0635139	−	0.997981i	$-0.520231\pi$
$953$	−3.92144	−0.127028	−0.0635139	+	0.997981i	$0.520231\pi$
$954$	10.4804	0.339315
$955$	−24.8919	−0.805483
$956$	−9.71126	−0.314085
$957$	−7.73202	−0.249941
$958$	−28.6523	−0.925715
$959$	−39.8767	−1.28769
$960$	12.1586	0.392416
$961$	47.1240	1.52013
$962$	0	0
$963$	−10.1756	−0.327904
$964$	5.24925	0.169067
$965$	53.0661	1.70826
$966$	−11.3084	−0.363843
$967$	−4.17433	−0.134237	−0.0671186	−	0.997745i	$-0.521381\pi$
$967$	−4.17433	−0.134237	−0.0671186	+	0.997745i	$0.521381\pi$
$968$	−2.92075	−0.0938764
$969$	0.657343	0.0211169
$970$	16.9134	0.543057
$971$	−7.64577	−0.245364	−0.122682	−	0.992446i	$-0.539150\pi$
$971$	−7.64577	−0.245364	−0.122682	+	0.992446i	$0.539150\pi$
$972$	1.12828	0.0361896
$973$	3.19902	0.102556
$974$	−12.9107	−0.413685
$975$	0	0
$976$	−2.63757	−0.0844265
$977$	−32.1326	−1.02801	−0.514007	−	0.857786i	$-0.671839\pi$
$977$	−32.1326	−1.02801	−0.514007	+	0.857786i	$0.671839\pi$
$978$	16.9139	0.540845
$979$	10.0961	0.322671
$980$	−1.66598	−0.0532177
$981$	12.4405	0.397196
$982$	−8.11504	−0.258961
$983$	38.4881	1.22758	0.613789	−	0.789470i	$-0.289644\pi$
$983$	38.4881	1.22758	0.613789	+	0.789470i	$0.289644\pi$
$984$	−19.1266	−0.609734
$985$	−8.01093	−0.255249
$986$	26.6829	0.849755
$987$	−16.2201	−0.516292
$988$	0	0
$989$	2.47228	0.0786140
$990$	1.89682	0.0602848
$991$	−24.1600	−0.767469	−0.383734	−	0.923444i	$-0.625362\pi$
$991$	−24.1600	−0.767469	−0.383734	+	0.923444i	$0.625362\pi$
$992$	−47.7495	−1.51605
$993$	−8.04496	−0.255299
$994$	22.9443	0.727748
$995$	−16.9473	−0.537267
$996$	−7.91740	−0.250873
$997$	52.1318	1.65103	0.825515	−	0.564380i	$-0.190885\pi$
$997$	52.1318	1.65103	0.825515	+	0.564380i	$0.190885\pi$
$998$	−16.3270	−0.516821
$999$	10.6692	0.337559

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5577.2.a.bj.1.12	yes	18
13.12	even	2		5577.2.a.bh.1.7	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5577.2.a.bh.1.7	✓	18	13.12	even	2
5577.2.a.bj.1.12	yes	18	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)

\(f(q)\)	\(=\)	\(q+0.933660 q^{2} -1.00000 q^{3} -1.12828 q^{4} -2.03159 q^{5} -0.933660 q^{6} -2.50464 q^{7} -2.92075 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.933660 q^{2} -1.00000 q^{3} -1.12828 q^{4} -2.03159 q^{5} -0.933660 q^{6} -2.50464 q^{7} -2.92075 q^{8} +1.00000 q^{9} -1.89682 q^{10} -1.00000 q^{11} +1.12828 q^{12} -2.33848 q^{14} +2.03159 q^{15} -0.470429 q^{16} -3.69616 q^{17} +0.933660 q^{18} +0.177845 q^{19} +2.29220 q^{20} +2.50464 q^{21} -0.933660 q^{22} -4.83581 q^{23} +2.92075 q^{24} -0.872636 q^{25} -1.00000 q^{27} +2.82593 q^{28} -7.73202 q^{29} +1.89682 q^{30} -8.83878 q^{31} +5.40228 q^{32} +1.00000 q^{33} -3.45095 q^{34} +5.08840 q^{35} -1.12828 q^{36} -10.6692 q^{37} +0.166047 q^{38} +5.93377 q^{40} -6.54853 q^{41} +2.33848 q^{42} -0.511245 q^{43} +1.12828 q^{44} -2.03159 q^{45} -4.51500 q^{46} -6.47604 q^{47} +0.470429 q^{48} -0.726802 q^{49} -0.814745 q^{50} +3.69616 q^{51} +11.2250 q^{53} -0.933660 q^{54} +2.03159 q^{55} +7.31541 q^{56} -0.177845 q^{57} -7.21908 q^{58} +0.823270 q^{59} -2.29220 q^{60} +5.60673 q^{61} -8.25242 q^{62} -2.50464 q^{63} +5.98475 q^{64} +0.933660 q^{66} +16.1893 q^{67} +4.17030 q^{68} +4.83581 q^{69} +4.75083 q^{70} -9.81162 q^{71} -2.92075 q^{72} -3.70987 q^{73} -9.96142 q^{74} +0.872636 q^{75} -0.200659 q^{76} +2.50464 q^{77} -9.36626 q^{79} +0.955719 q^{80} +1.00000 q^{81} -6.11410 q^{82} -7.01724 q^{83} -2.82593 q^{84} +7.50908 q^{85} -0.477329 q^{86} +7.73202 q^{87} +2.92075 q^{88} -10.0961 q^{89} -1.89682 q^{90} +5.45615 q^{92} +8.83878 q^{93} -6.04642 q^{94} -0.361309 q^{95} -5.40228 q^{96} -8.91674 q^{97} -0.678586 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + q^{2} - 18 q^{3} + 23 q^{4} - 6 q^{5} - q^{6} + q^{7} + 18 q^{9}+O(q^{10}) \) 18 * q + q^2 - 18 * q^3 + 23 * q^4 - 6 * q^5 - q^6 + q^7 + 18 * q^9	\( 18 q + q^{2} - 18 q^{3} + 23 q^{4} - 6 q^{5} - q^{6} + q^{7} + 18 q^{9} + 11 q^{10} - 18 q^{11} - 23 q^{12} - 13 q^{14} + 6 q^{15} + 41 q^{16} - q^{17} + q^{18} - 12 q^{19} - 8 q^{20} - q^{21} - q^{22} + 44 q^{23} + 48 q^{25} - 18 q^{27} + 5 q^{28} - 17 q^{29} - 11 q^{30} + 18 q^{31} + q^{32} + 18 q^{33} - 8 q^{34} + 8 q^{35} + 23 q^{36} - 39 q^{37} + 4 q^{38} + 8 q^{40} - 6 q^{41} + 13 q^{42} + 36 q^{43} - 23 q^{44} - 6 q^{45} + 36 q^{46} - 19 q^{47} - 41 q^{48} + 41 q^{49} + 19 q^{50} + q^{51} + 7 q^{53} - q^{54} + 6 q^{55} - 73 q^{56} + 12 q^{57} - 69 q^{58} + 16 q^{59} + 8 q^{60} + 21 q^{61} + 8 q^{62} + q^{63} + 96 q^{64} + q^{66} + 40 q^{68} - 44 q^{69} + 107 q^{70} - 12 q^{71} + 7 q^{73} - 14 q^{74} - 48 q^{75} - 30 q^{76} - q^{77} + 62 q^{79} - 58 q^{80} + 18 q^{81} + 44 q^{82} + 11 q^{83} - 5 q^{84} + 46 q^{85} - 19 q^{86} + 17 q^{87} - 17 q^{89} + 11 q^{90} + 100 q^{92} - 18 q^{93} + 31 q^{94} + 42 q^{95} - q^{96} - 35 q^{97} + 94 q^{98} - 18 q^{99}+O(q^{100}) \) 18 * q + q^2 - 18 * q^3 + 23 * q^4 - 6 * q^5 - q^6 + q^7 + 18 * q^9 + 11 * q^10 - 18 * q^11 - 23 * q^12 - 13 * q^14 + 6 * q^15 + 41 * q^16 - q^17 + q^18 - 12 * q^19 - 8 * q^20 - q^21 - q^22 + 44 * q^23 + 48 * q^25 - 18 * q^27 + 5 * q^28 - 17 * q^29 - 11 * q^30 + 18 * q^31 + q^32 + 18 * q^33 - 8 * q^34 + 8 * q^35 + 23 * q^36 - 39 * q^37 + 4 * q^38 + 8 * q^40 - 6 * q^41 + 13 * q^42 + 36 * q^43 - 23 * q^44 - 6 * q^45 + 36 * q^46 - 19 * q^47 - 41 * q^48 + 41 * q^49 + 19 * q^50 + q^51 + 7 * q^53 - q^54 + 6 * q^55 - 73 * q^56 + 12 * q^57 - 69 * q^58 + 16 * q^59 + 8 * q^60 + 21 * q^61 + 8 * q^62 + q^63 + 96 * q^64 + q^66 + 40 * q^68 - 44 * q^69 + 107 * q^70 - 12 * q^71 + 7 * q^73 - 14 * q^74 - 48 * q^75 - 30 * q^76 - q^77 + 62 * q^79 - 58 * q^80 + 18 * q^81 + 44 * q^82 + 11 * q^83 - 5 * q^84 + 46 * q^85 - 19 * q^86 + 17 * q^87 - 17 * q^89 + 11 * q^90 + 100 * q^92 - 18 * q^93 + 31 * q^94 + 42 * q^95 - q^96 - 35 * q^97 + 94 * q^98 - 18 * q^99

Introduction

L-functions

Modular forms

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Database

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