LMFDB - Embedded newform 5577.2.a.bk.1.15

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} + 26 x^{15} + 347 x^{14} - 277 x^{13} - 2215 x^{12} + 1560 x^{11} + \cdots - 71 $ x^18 - x^17 - 29x^16 + 26x^15 + 347x^14 - 277x^13 - 2215x^12 + 1560x^11 + 8156x^10 - 4977x^9 - 17520x^8 + 8853x^7 + 21169x^6 - 7818x^5 - 13097x^4 + 2237x^3 + 3379x^2 + 285x - 71
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			1.15
Root			$1.88763$ 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)

$f(q)$	$=$	$q+1.88763 q^{2} +1.00000 q^{3} +1.56314 q^{4} +2.17541 q^{5} +1.88763 q^{6} -3.72459 q^{7} -0.824626 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.88763 q^{2} +1.00000 q^{3} +1.56314 q^{4} +2.17541 q^{5} +1.88763 q^{6} -3.72459 q^{7} -0.824626 q^{8} +1.00000 q^{9} +4.10637 q^{10} -1.00000 q^{11} +1.56314 q^{12} -7.03065 q^{14} +2.17541 q^{15} -4.68287 q^{16} +2.48835 q^{17} +1.88763 q^{18} +4.08363 q^{19} +3.40047 q^{20} -3.72459 q^{21} -1.88763 q^{22} +6.28232 q^{23} -0.824626 q^{24} -0.267594 q^{25} +1.00000 q^{27} -5.82207 q^{28} +9.95915 q^{29} +4.10637 q^{30} +1.11531 q^{31} -7.19027 q^{32} -1.00000 q^{33} +4.69707 q^{34} -8.10252 q^{35} +1.56314 q^{36} +4.55187 q^{37} +7.70838 q^{38} -1.79390 q^{40} +1.89825 q^{41} -7.03065 q^{42} +8.75103 q^{43} -1.56314 q^{44} +2.17541 q^{45} +11.8587 q^{46} -0.202980 q^{47} -4.68287 q^{48} +6.87260 q^{49} -0.505118 q^{50} +2.48835 q^{51} +3.06306 q^{53} +1.88763 q^{54} -2.17541 q^{55} +3.07140 q^{56} +4.08363 q^{57} +18.7992 q^{58} +3.39937 q^{59} +3.40047 q^{60} -2.93389 q^{61} +2.10528 q^{62} -3.72459 q^{63} -4.20682 q^{64} -1.88763 q^{66} +4.67187 q^{67} +3.88964 q^{68} +6.28232 q^{69} -15.2945 q^{70} +3.32844 q^{71} -0.824626 q^{72} +3.80477 q^{73} +8.59224 q^{74} -0.267594 q^{75} +6.38330 q^{76} +3.72459 q^{77} +12.0191 q^{79} -10.1872 q^{80} +1.00000 q^{81} +3.58320 q^{82} -13.2116 q^{83} -5.82207 q^{84} +5.41317 q^{85} +16.5187 q^{86} +9.95915 q^{87} +0.824626 q^{88} -16.0591 q^{89} +4.10637 q^{90} +9.82015 q^{92} +1.11531 q^{93} -0.383151 q^{94} +8.88357 q^{95} -7.19027 q^{96} +8.11178 q^{97} +12.9729 q^{98} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + q^{2} + 18 q^{3} + 23 q^{4} - 6 q^{5} + q^{6} + q^{7} + 6 q^{8} + 18 q^{9}+O(q^{10}) $ 18 * q + q^2 + 18 * q^3 + 23 * q^4 - 6 * q^5 + q^6 + q^7 + 6 * q^8 + 18 * q^9	$ 18 q + q^{2} + 18 q^{3} + 23 q^{4} - 6 q^{5} + q^{6} + q^{7} + 6 q^{8} + 18 q^{9} - 7 q^{10} - 18 q^{11} + 23 q^{12} + 7 q^{14} - 6 q^{15} + 25 q^{16} + 19 q^{17} + q^{18} - 2 q^{20} + q^{21} - q^{22} + 38 q^{23} + 6 q^{24} + 36 q^{25} + 18 q^{27} + 33 q^{28} + 47 q^{29} - 7 q^{30} - 22 q^{31} + 33 q^{32} - 18 q^{33} + 4 q^{34} + 20 q^{35} + 23 q^{36} - 41 q^{37} + 14 q^{38} - 28 q^{40} + 8 q^{41} + 7 q^{42} + 28 q^{43} - 23 q^{44} - 6 q^{45} + 6 q^{46} + 11 q^{47} + 25 q^{48} + 9 q^{49} - q^{50} + 19 q^{51} + 49 q^{53} + q^{54} + 6 q^{55} + 35 q^{56} - 27 q^{58} - 2 q^{59} - 2 q^{60} - 13 q^{61} + 46 q^{62} + q^{63} + 40 q^{64} - q^{66} + 4 q^{67} + 30 q^{68} + 38 q^{69} - 83 q^{70} + 2 q^{71} + 6 q^{72} + 39 q^{73} + 42 q^{74} + 36 q^{75} + 20 q^{76} - q^{77} + 18 q^{79} + 18 q^{80} + 18 q^{81} - 12 q^{82} + 5 q^{83} + 33 q^{84} + 2 q^{85} + 57 q^{86} + 47 q^{87} - 6 q^{88} - 9 q^{89} - 7 q^{90} + 86 q^{92} - 22 q^{93} - 27 q^{94} + 74 q^{95} + 33 q^{96} - 17 q^{97} + 4 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 + 6 * q^8 + 18 * q^9 - 7 * q^10 - 18 * q^11 + 23 * q^12 + 7 * q^14 - 6 * q^15 + 25 * q^16 + 19 * q^17 + q^18 - 2 * q^20 + q^21 - q^22 + 38 * q^23 + 6 * q^24 + 36 * q^25 + 18 * q^27 + 33 * q^28 + 47 * q^29 - 7 * q^30 - 22 * q^31 + 33 * q^32 - 18 * q^33 + 4 * q^34 + 20 * q^35 + 23 * q^36 - 41 * q^37 + 14 * q^38 - 28 * q^40 + 8 * q^41 + 7 * q^42 + 28 * q^43 - 23 * q^44 - 6 * q^45 + 6 * q^46 + 11 * q^47 + 25 * q^48 + 9 * q^49 - q^50 + 19 * q^51 + 49 * q^53 + q^54 + 6 * q^55 + 35 * q^56 - 27 * q^58 - 2 * q^59 - 2 * q^60 - 13 * q^61 + 46 * q^62 + q^63 + 40 * q^64 - q^66 + 4 * q^67 + 30 * q^68 + 38 * q^69 - 83 * q^70 + 2 * q^71 + 6 * q^72 + 39 * q^73 + 42 * q^74 + 36 * q^75 + 20 * q^76 - q^77 + 18 * q^79 + 18 * q^80 + 18 * q^81 - 12 * q^82 + 5 * q^83 + 33 * q^84 + 2 * q^85 + 57 * q^86 + 47 * q^87 - 6 * q^88 - 9 * q^89 - 7 * q^90 + 86 * q^92 - 22 * q^93 - 27 * q^94 + 74 * q^95 + 33 * q^96 - 17 * q^97 + 4 * 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 $

$2$	1.88763	1.33476	0.667378	−	0.744720i	$-0.267417\pi$
$2$	1.88763	1.33476	0.667378	+	0.744720i	$0.267417\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.56314	0.781571
$5$	2.17541	0.972873	0.486436	−	0.873716i	$-0.338297\pi$
$5$	2.17541	0.972873	0.486436	+	0.873716i	$0.338297\pi$
$6$	1.88763	0.770621
$7$	−3.72459	−1.40776	−0.703882	−	0.710317i	$-0.748552\pi$
$7$	−3.72459	−1.40776	−0.703882	+	0.710317i	$0.748552\pi$
$8$	−0.824626	−0.291549
$9$	1.00000	0.333333
$10$	4.10637	1.29855
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	1.56314	0.451240
$13$	0	0
$13$	0	0
$14$	−7.03065	−1.87902
$15$	2.17541	0.561688
$16$	−4.68287	−1.17072
$17$	2.48835	0.603513	0.301756	−	0.953385i	$-0.402427\pi$
$17$	2.48835	0.603513	0.301756	+	0.953385i	$0.402427\pi$
$18$	1.88763	0.444918
$19$	4.08363	0.936849	0.468425	−	0.883503i	$-0.344822\pi$
$19$	4.08363	0.936849	0.468425	+	0.883503i	$0.344822\pi$
$20$	3.40047	0.760369
$21$	−3.72459	−0.812773
$22$	−1.88763	−0.402444
$23$	6.28232	1.30995	0.654977	−	0.755649i	$-0.272678\pi$
$23$	6.28232	1.30995	0.654977	+	0.755649i	$0.272678\pi$
$24$	−0.824626	−0.168326
$25$	−0.267594	−0.0535188
$26$	0	0
$27$	1.00000	0.192450
$28$	−5.82207	−1.10027
$29$	9.95915	1.84937	0.924684	−	0.380736i	$-0.124329\pi$
$29$	9.95915	1.84937	0.924684	+	0.380736i	$0.124329\pi$
$30$	4.10637	0.749716
$31$	1.11531	0.200315	0.100157	−	0.994972i	$-0.468065\pi$
$31$	1.11531	0.200315	0.100157	+	0.994972i	$0.468065\pi$
$32$	−7.19027	−1.27107
$33$	−1.00000	−0.174078
$34$	4.69707	0.805542
$35$	−8.10252	−1.36958
$36$	1.56314	0.260524
$37$	4.55187	0.748323	0.374161	−	0.927364i	$-0.377931\pi$
$37$	4.55187	0.748323	0.374161	+	0.927364i	$0.377931\pi$
$38$	7.70838	1.25046
$39$	0	0
$40$	−1.79390	−0.283640
$41$	1.89825	0.296457	0.148229	−	0.988953i	$-0.452643\pi$
$41$	1.89825	0.296457	0.148229	+	0.988953i	$0.452643\pi$
$42$	−7.03065	−1.08485
$43$	8.75103	1.33452	0.667259	−	0.744825i	$-0.267467\pi$
$43$	8.75103	1.33452	0.667259	+	0.744825i	$0.267467\pi$
$44$	−1.56314	−0.235653
$45$	2.17541	0.324291
$46$	11.8587	1.74847
$47$	−0.202980	−0.0296077	−0.0148039	−	0.999890i	$-0.504712\pi$
$47$	−0.202980	−0.0296077	−0.0148039	+	0.999890i	$0.504712\pi$
$48$	−4.68287	−0.675914
$49$	6.87260	0.981800
$50$	−0.505118	−0.0714344
$51$	2.48835	0.348438
$52$	0	0
$53$	3.06306	0.420743	0.210372	−	0.977621i	$-0.432533\pi$
$53$	3.06306	0.420743	0.210372	+	0.977621i	$0.432533\pi$
$54$	1.88763	0.256874
$55$	−2.17541	−0.293332
$56$	3.07140	0.410433
$57$	4.08363	0.540890
$58$	18.7992	2.46845
$59$	3.39937	0.442560	0.221280	−	0.975210i	$-0.428976\pi$
$59$	3.39937	0.442560	0.221280	+	0.975210i	$0.428976\pi$
$60$	3.40047	0.438999
$61$	−2.93389	−0.375646	−0.187823	−	0.982203i	$-0.560143\pi$
$61$	−2.93389	−0.375646	−0.187823	+	0.982203i	$0.560143\pi$
$62$	2.10528	0.267371
$63$	−3.72459	−0.469255
$64$	−4.20682	−0.525852
$65$	0	0
$66$	−1.88763	−0.232351
$67$	4.67187	0.570760	0.285380	−	0.958415i	$-0.407880\pi$
$67$	4.67187	0.570760	0.285380	+	0.958415i	$0.407880\pi$
$68$	3.88964	0.471688
$69$	6.28232	0.756302
$70$	−15.2945	−1.82805
$71$	3.32844	0.395013	0.197506	−	0.980302i	$-0.436716\pi$
$71$	3.32844	0.395013	0.197506	+	0.980302i	$0.436716\pi$
$72$	−0.824626	−0.0971831
$73$	3.80477	0.445315	0.222657	−	0.974897i	$-0.428527\pi$
$73$	3.80477	0.445315	0.222657	+	0.974897i	$0.428527\pi$
$74$	8.59224	0.998827
$75$	−0.267594	−0.0308991
$76$	6.38330	0.732214
$77$	3.72459	0.424457
$78$	0	0
$79$	12.0191	1.35225	0.676127	−	0.736785i	$-0.263657\pi$
$79$	12.0191	1.35225	0.676127	+	0.736785i	$0.263657\pi$
$80$	−10.1872	−1.13896
$81$	1.00000	0.111111
$82$	3.58320	0.395698
$83$	−13.2116	−1.45016	−0.725080	−	0.688665i	$-0.758197\pi$
$83$	−13.2116	−1.45016	−0.725080	+	0.688665i	$0.758197\pi$
$84$	−5.82207	−0.635240
$85$	5.41317	0.587141
$86$	16.5187	1.78126
$87$	9.95915	1.06773
$88$	0.824626	0.0879054
$89$	−16.0591	−1.70226	−0.851132	−	0.524952i	$-0.824083\pi$
$89$	−16.0591	−1.70226	−0.851132	+	0.524952i	$0.824083\pi$
$90$	4.10637	0.432849
$91$	0	0
$92$	9.82015	1.02382
$93$	1.11531	0.115652
$94$	−0.383151	−0.0395191
$95$	8.88357	0.911435
$96$	−7.19027	−0.733854
$97$	8.11178	0.823626	0.411813	−	0.911268i	$-0.364896\pi$
$97$	8.11178	0.823626	0.411813	+	0.911268i	$0.364896\pi$
$98$	12.9729	1.31046
$99$	−1.00000	−0.100504
$100$	−0.418287	−0.0418287
$101$	−5.29685	−0.527056	−0.263528	−	0.964652i	$-0.584886\pi$
$101$	−5.29685	−0.527056	−0.263528	+	0.964652i	$0.584886\pi$
$102$	4.69707	0.465080
$103$	−16.4097	−1.61689	−0.808446	−	0.588571i	$-0.799691\pi$
$103$	−16.4097	−1.61689	−0.808446	+	0.588571i	$0.799691\pi$
$104$	0	0
$105$	−8.10252	−0.790725
$106$	5.78191	0.561589
$107$	14.2903	1.38150	0.690749	−	0.723095i	$-0.257281\pi$
$107$	14.2903	1.38150	0.690749	+	0.723095i	$0.257281\pi$
$108$	1.56314	0.150413
$109$	−16.3539	−1.56642	−0.783208	−	0.621760i	$-0.786418\pi$
$109$	−16.3539	−1.56642	−0.783208	+	0.621760i	$0.786418\pi$
$110$	−4.10637	−0.391527
$111$	4.55187	0.432044
$112$	17.4418	1.64809
$113$	−3.99905	−0.376199	−0.188099	−	0.982150i	$-0.560233\pi$
$113$	−3.99905	−0.376199	−0.188099	+	0.982150i	$0.560233\pi$
$114$	7.70838	0.721956
$115$	13.6666	1.27442
$116$	15.5676	1.44541
$117$	0	0
$118$	6.41675	0.590709
$119$	−9.26808	−0.849604
$120$	−1.79390	−0.163760
$121$	1.00000	0.0909091
$122$	−5.53809	−0.501395
$123$	1.89825	0.171160
$124$	1.74338	0.156560
$125$	−11.4592	−1.02494
$126$	−7.03065	−0.626340
$127$	5.50159	0.488187	0.244094	−	0.969752i	$-0.421510\pi$
$127$	5.50159	0.488187	0.244094	+	0.969752i	$0.421510\pi$
$128$	6.43963	0.569188
$129$	8.75103	0.770485
$130$	0	0
$131$	10.8722	0.949913	0.474956	−	0.880009i	$-0.342464\pi$
$131$	10.8722	0.949913	0.474956	+	0.880009i	$0.342464\pi$
$132$	−1.56314	−0.136054
$133$	−15.2099	−1.31886
$134$	8.81875	0.761824
$135$	2.17541	0.187229
$136$	−2.05195	−0.175954
$137$	−14.4347	−1.23324	−0.616620	−	0.787261i	$-0.711499\pi$
$137$	−14.4347	−1.23324	−0.616620	+	0.787261i	$0.711499\pi$
$138$	11.8587	1.00948
$139$	−7.08593	−0.601021	−0.300510	−	0.953779i	$-0.597157\pi$
$139$	−7.08593	−0.601021	−0.300510	+	0.953779i	$0.597157\pi$
$140$	−12.6654	−1.07042
$141$	−0.202980	−0.0170940
$142$	6.28285	0.527245
$143$	0	0
$144$	−4.68287	−0.390239
$145$	21.6652	1.79920
$146$	7.18199	0.594386
$147$	6.87260	0.566842
$148$	7.11522	0.584867
$149$	21.1037	1.72888	0.864442	−	0.502732i	$-0.167672\pi$
$149$	21.1037	1.72888	0.864442	+	0.502732i	$0.167672\pi$
$150$	−0.505118	−0.0412427
$151$	1.96608	0.159997	0.0799987	−	0.996795i	$-0.474508\pi$
$151$	1.96608	0.159997	0.0799987	+	0.996795i	$0.474508\pi$
$152$	−3.36747	−0.273138
$153$	2.48835	0.201171
$154$	7.03065	0.566546
$155$	2.42625	0.194881
$156$	0	0
$157$	−22.6164	−1.80499	−0.902493	−	0.430705i	$-0.858265\pi$
$157$	−22.6164	−1.80499	−0.902493	+	0.430705i	$0.858265\pi$
$158$	22.6876	1.80493
$159$	3.06306	0.242916
$160$	−15.6418	−1.23659
$161$	−23.3991	−1.84411
$162$	1.88763	0.148306
$163$	16.6713	1.30580	0.652898	−	0.757446i	$-0.273553\pi$
$163$	16.6713	1.30580	0.652898	+	0.757446i	$0.273553\pi$
$164$	2.96724	0.231703
$165$	−2.17541	−0.169355
$166$	−24.9386	−1.93561
$167$	21.9953	1.70205	0.851024	−	0.525127i	$-0.175982\pi$
$167$	21.9953	1.70205	0.851024	+	0.525127i	$0.175982\pi$
$168$	3.07140	0.236963
$169$	0	0
$170$	10.2181	0.783689
$171$	4.08363	0.312283
$172$	13.6791	1.04302
$173$	−19.3912	−1.47428	−0.737142	−	0.675738i	$-0.763825\pi$
$173$	−19.3912	−1.47428	−0.737142	+	0.675738i	$0.763825\pi$
$174$	18.7992	1.42516
$175$	0.996678	0.0753418
$176$	4.68287	0.352985
$177$	3.39937	0.255512
$178$	−30.3137	−2.27210
$179$	−8.44338	−0.631088	−0.315544	−	0.948911i	$-0.602187\pi$
$179$	−8.44338	−0.631088	−0.315544	+	0.948911i	$0.602187\pi$
$180$	3.40047	0.253456
$181$	19.7268	1.46628	0.733141	−	0.680076i	$-0.238053\pi$
$181$	19.7268	1.46628	0.733141	+	0.680076i	$0.238053\pi$
$182$	0	0
$183$	−2.93389	−0.216879
$184$	−5.18056	−0.381916
$185$	9.90218	0.728023
$186$	2.10528	0.154367
$187$	−2.48835	−0.181966
$188$	−0.317287	−0.0231405
$189$	−3.72459	−0.270924
$190$	16.7689	1.21654
$191$	1.08306	0.0783672	0.0391836	−	0.999232i	$-0.487524\pi$
$191$	1.08306	0.0783672	0.0391836	+	0.999232i	$0.487524\pi$
$192$	−4.20682	−0.303601
$193$	−21.6348	−1.55731	−0.778654	−	0.627454i	$-0.784097\pi$
$193$	−21.6348	−1.55731	−0.778654	+	0.627454i	$0.784097\pi$
$194$	15.3120	1.09934
$195$	0	0
$196$	10.7428	0.767346
$197$	13.4237	0.956401	0.478201	−	0.878251i	$-0.341289\pi$
$197$	13.4237	0.956401	0.478201	+	0.878251i	$0.341289\pi$
$198$	−1.88763	−0.134148
$199$	−1.29548	−0.0918343	−0.0459172	−	0.998945i	$-0.514621\pi$
$199$	−1.29548	−0.0918343	−0.0459172	+	0.998945i	$0.514621\pi$
$200$	0.220665	0.0156034
$201$	4.67187	0.329528
$202$	−9.99848	−0.703490
$203$	−37.0938	−2.60347
$204$	3.88964	0.272329
$205$	4.12948	0.288415
$206$	−30.9753	−2.15815
$207$	6.28232	0.436651
$208$	0	0
$209$	−4.08363	−0.282471
$210$	−15.2945	−1.05542
$211$	−0.305491	−0.0210309	−0.0105154	−	0.999945i	$-0.503347\pi$
$211$	−0.305491	−0.0210309	−0.0105154	+	0.999945i	$0.503347\pi$
$212$	4.78799	0.328841
$213$	3.32844	0.228061
$214$	26.9748	1.84396
$215$	19.0371	1.29832
$216$	−0.824626	−0.0561087
$217$	−4.15406	−0.281996
$218$	−30.8700	−2.09078
$219$	3.80477	0.257103
$220$	−3.40047	−0.229260
$221$	0	0
$222$	8.59224	0.576673
$223$	−12.7781	−0.855683	−0.427842	−	0.903854i	$-0.640726\pi$
$223$	−12.7781	−0.855683	−0.427842	+	0.903854i	$0.640726\pi$
$224$	26.7808	1.78937
$225$	−0.267594	−0.0178396
$226$	−7.54872	−0.502133
$227$	−0.951199	−0.0631333	−0.0315666	−	0.999502i	$-0.510050\pi$
$227$	−0.951199	−0.0631333	−0.0315666	+	0.999502i	$0.510050\pi$
$228$	6.38330	0.422744
$229$	−21.6950	−1.43364	−0.716822	−	0.697256i	$-0.754404\pi$
$229$	−21.6950	−1.43364	−0.716822	+	0.697256i	$0.754404\pi$
$230$	25.7975	1.70104
$231$	3.72459	0.245060
$232$	−8.21257	−0.539182
$233$	−28.1585	−1.84472	−0.922362	−	0.386327i	$-0.873744\pi$
$233$	−28.1585	−1.84472	−0.922362	+	0.386327i	$0.873744\pi$
$234$	0	0
$235$	−0.441565	−0.0288045
$236$	5.31370	0.345892
$237$	12.0191	0.780724
$238$	−17.4947	−1.13401
$239$	−26.7292	−1.72897	−0.864486	−	0.502657i	$-0.832356\pi$
$239$	−26.7292	−1.72897	−0.864486	+	0.502657i	$0.832356\pi$
$240$	−10.1872	−0.657578
$241$	24.4629	1.57579	0.787897	−	0.615807i	$-0.211170\pi$
$241$	24.4629	1.57579	0.787897	+	0.615807i	$0.211170\pi$
$242$	1.88763	0.121341
$243$	1.00000	0.0641500
$244$	−4.58608	−0.293594
$245$	14.9507	0.955166
$246$	3.58320	0.228456
$247$	0	0
$248$	−0.919710	−0.0584017
$249$	−13.2116	−0.837250
$250$	−21.6307	−1.36804
$251$	6.85130	0.432450	0.216225	−	0.976344i	$-0.430625\pi$
$251$	6.85130	0.432450	0.216225	+	0.976344i	$0.430625\pi$
$252$	−5.82207	−0.366756
$253$	−6.28232	−0.394966
$254$	10.3850	0.651610
$255$	5.41317	0.338986
$256$	20.5693	1.28558
$257$	−2.08107	−0.129813	−0.0649067	−	0.997891i	$-0.520675\pi$
$257$	−2.08107	−0.129813	−0.0649067	+	0.997891i	$0.520675\pi$
$258$	16.5187	1.02841
$259$	−16.9539	−1.05346
$260$	0	0
$261$	9.95915	0.616456
$262$	20.5228	1.26790
$263$	6.76875	0.417379	0.208689	−	0.977982i	$-0.433080\pi$
$263$	6.76875	0.417379	0.208689	+	0.977982i	$0.433080\pi$
$264$	0.824626	0.0507522
$265$	6.66340	0.409330
$266$	−28.7106	−1.76036
$267$	−16.0591	−0.982802
$268$	7.30279	0.446089
$269$	−26.0731	−1.58971	−0.794854	−	0.606801i	$-0.792452\pi$
$269$	−26.0731	−1.58971	−0.794854	+	0.606801i	$0.792452\pi$
$270$	4.10637	0.249905
$271$	−14.0343	−0.852521	−0.426260	−	0.904600i	$-0.640169\pi$
$271$	−14.0343	−0.852521	−0.426260	+	0.904600i	$0.640169\pi$
$272$	−11.6526	−0.706543
$273$	0	0
$274$	−27.2474	−1.64607
$275$	0.267594	0.0161365
$276$	9.82015	0.591104
$277$	30.6267	1.84018	0.920089	−	0.391709i	$-0.128116\pi$
$277$	30.6267	1.84018	0.920089	+	0.391709i	$0.128116\pi$
$278$	−13.3756	−0.802216
$279$	1.11531	0.0667716
$280$	6.68154	0.399299
$281$	−19.6249	−1.17072	−0.585360	−	0.810773i	$-0.699047\pi$
$281$	−19.6249	−1.17072	−0.585360	+	0.810773i	$0.699047\pi$
$282$	−0.383151	−0.0228163
$283$	18.9839	1.12848	0.564238	−	0.825612i	$-0.309170\pi$
$283$	18.9839	1.12848	0.564238	+	0.825612i	$0.309170\pi$
$284$	5.20282	0.308730
$285$	8.88357	0.526217
$286$	0	0
$287$	−7.07022	−0.417342
$288$	−7.19027	−0.423691
$289$	−10.8081	−0.635772
$290$	40.8959	2.40149
$291$	8.11178	0.475521
$292$	5.94740	0.348045
$293$	−17.1572	−1.00233	−0.501167	−	0.865351i	$-0.667096\pi$
$293$	−17.1572	−1.00233	−0.501167	+	0.865351i	$0.667096\pi$
$294$	12.9729	0.756596
$295$	7.39502	0.430555
$296$	−3.75359	−0.218173
$297$	−1.00000	−0.0580259
$298$	39.8360	2.30764
$299$	0	0
$300$	−0.418287	−0.0241498
$301$	−32.5940	−1.87869
$302$	3.71123	0.213557
$303$	−5.29685	−0.304296
$304$	−19.1231	−1.09679
$305$	−6.38241	−0.365456
$306$	4.69707	0.268514
$307$	7.69678	0.439278	0.219639	−	0.975581i	$-0.429512\pi$
$307$	7.69678	0.439278	0.219639	+	0.975581i	$0.429512\pi$
$308$	5.82207	0.331743
$309$	−16.4097	−0.933513
$310$	4.57985	0.260118
$311$	−10.8021	−0.612530	−0.306265	−	0.951946i	$-0.599079\pi$
$311$	−10.8021	−0.612530	−0.306265	+	0.951946i	$0.599079\pi$
$312$	0	0
$313$	32.8078	1.85441	0.927204	−	0.374557i	$-0.122205\pi$
$313$	32.8078	1.85441	0.927204	+	0.374557i	$0.122205\pi$
$314$	−42.6914	−2.40921
$315$	−8.10252	−0.456525
$316$	18.7875	1.05688
$317$	2.68357	0.150725	0.0753623	−	0.997156i	$-0.475989\pi$
$317$	2.68357	0.150725	0.0753623	+	0.997156i	$0.475989\pi$
$318$	5.78191	0.324234
$319$	−9.95915	−0.557605
$320$	−9.15155	−0.511587
$321$	14.2903	0.797608
$322$	−44.1688	−2.46143
$323$	10.1615	0.565400
$324$	1.56314	0.0868412
$325$	0	0
$326$	31.4692	1.74292
$327$	−16.3539	−0.904371
$328$	−1.56535	−0.0864319
$329$	0.756019	0.0416807
$330$	−4.10637	−0.226048
$331$	−2.81774	−0.154877	−0.0774384	−	0.996997i	$-0.524674\pi$
$331$	−2.81774	−0.154877	−0.0774384	+	0.996997i	$0.524674\pi$
$332$	−20.6516	−1.13340
$333$	4.55187	0.249441
$334$	41.5190	2.27182
$335$	10.1632	0.555276
$336$	17.4418	0.951528
$337$	13.9799	0.761535	0.380768	−	0.924671i	$-0.375660\pi$
$337$	13.9799	0.761535	0.380768	+	0.924671i	$0.375660\pi$
$338$	0	0
$339$	−3.99905	−0.217198
$340$	8.46156	0.458892
$341$	−1.11531	−0.0603972
$342$	7.70838	0.416821
$343$	0.474517	0.0256215
$344$	−7.21632	−0.389078
$345$	13.6666	0.735786
$346$	−36.6033	−1.96781
$347$	−12.5502	−0.673728	−0.336864	−	0.941553i	$-0.609366\pi$
$347$	−12.5502	−0.673728	−0.336864	+	0.941553i	$0.609366\pi$
$348$	15.5676	0.834509
$349$	16.3234	0.873771	0.436886	−	0.899517i	$-0.356081\pi$
$349$	16.3234	0.873771	0.436886	+	0.899517i	$0.356081\pi$
$350$	1.88136	0.100563
$351$	0	0
$352$	7.19027	0.383243
$353$	−7.62792	−0.405993	−0.202997	−	0.979179i	$-0.565068\pi$
$353$	−7.62792	−0.405993	−0.202997	+	0.979179i	$0.565068\pi$
$354$	6.41675	0.341046
$355$	7.24071	0.384297
$356$	−25.1027	−1.33044
$357$	−9.26808	−0.490519
$358$	−15.9380	−0.842348
$359$	−16.0231	−0.845668	−0.422834	−	0.906207i	$-0.638965\pi$
$359$	−16.0231	−0.845668	−0.422834	+	0.906207i	$0.638965\pi$
$360$	−1.79390	−0.0945468
$361$	−2.32395	−0.122313
$362$	37.2369	1.95713
$363$	1.00000	0.0524864
$364$	0	0
$365$	8.27693	0.433235
$366$	−5.53809	−0.289481
$367$	0.0239305	0.00124916	0.000624581	−	1.00000i	$-0.499801\pi$
$367$	0.0239305	0.00124916	0.000624581		1.00000i	$0.499801\pi$
$368$	−29.4193	−1.53359
$369$	1.89825	0.0988191
$370$	18.6916	0.971732
$371$	−11.4086	−0.592307
$372$	1.74338	0.0903901
$373$	3.33442	0.172650	0.0863248	−	0.996267i	$-0.472488\pi$
$373$	3.33442	0.172650	0.0863248	+	0.996267i	$0.472488\pi$
$374$	−4.69707	−0.242880
$375$	−11.4592	−0.591749
$376$	0.167383	0.00863211
$377$	0	0
$378$	−7.03065	−0.361618
$379$	−5.91352	−0.303757	−0.151879	−	0.988399i	$-0.548532\pi$
$379$	−5.91352	−0.303757	−0.151879	+	0.988399i	$0.548532\pi$
$380$	13.8863	0.712351
$381$	5.50159	0.281855
$382$	2.04441	0.104601
$383$	2.50155	0.127823	0.0639117	−	0.997956i	$-0.479642\pi$
$383$	2.50155	0.127823	0.0639117	+	0.997956i	$0.479642\pi$
$384$	6.43963	0.328621
$385$	8.10252	0.412942
$386$	−40.8385	−2.07862
$387$	8.75103	0.444840
$388$	12.6799	0.643722
$389$	7.28946	0.369590	0.184795	−	0.982777i	$-0.440838\pi$
$389$	7.28946	0.369590	0.184795	+	0.982777i	$0.440838\pi$
$390$	0	0
$391$	15.6326	0.790574
$392$	−5.66732	−0.286243
$393$	10.8722	0.548432
$394$	25.3390	1.27656
$395$	26.1464	1.31557
$396$	−1.56314	−0.0785508
$397$	−2.63302	−0.132148	−0.0660738	−	0.997815i	$-0.521047\pi$
$397$	−2.63302	−0.132148	−0.0660738	+	0.997815i	$0.521047\pi$
$398$	−2.44539	−0.122576
$399$	−15.2099	−0.761446
$400$	1.25311	0.0626554
$401$	−19.6879	−0.983165	−0.491583	−	0.870831i	$-0.663581\pi$
$401$	−19.6879	−0.983165	−0.491583	+	0.870831i	$0.663581\pi$
$402$	8.81875	0.439839
$403$	0	0
$404$	−8.27972	−0.411932
$405$	2.17541	0.108097
$406$	−70.0193	−3.47500
$407$	−4.55187	−0.225628
$408$	−2.05195	−0.101587
$409$	7.32108	0.362004	0.181002	−	0.983483i	$-0.442066\pi$
$409$	7.32108	0.362004	0.181002	+	0.983483i	$0.442066\pi$
$410$	7.79492	0.384964
$411$	−14.4347	−0.712012
$412$	−25.6506	−1.26372
$413$	−12.6613	−0.623020
$414$	11.8587	0.582822
$415$	−28.7406	−1.41082
$416$	0	0
$417$	−7.08593	−0.347000
$418$	−7.70838	−0.377029
$419$	21.4197	1.04642	0.523210	−	0.852204i	$-0.324734\pi$
$419$	21.4197	1.04642	0.523210	+	0.852204i	$0.324734\pi$
$420$	−12.6654	−0.618007
$421$	5.35484	0.260979	0.130489	−	0.991450i	$-0.458345\pi$
$421$	5.35484	0.260979	0.130489	+	0.991450i	$0.458345\pi$
$422$	−0.576654	−0.0280711
$423$	−0.202980	−0.00986924
$424$	−2.52588	−0.122667
$425$	−0.665866	−0.0322993
$426$	6.28285	0.304405
$427$	10.9275	0.528821
$428$	22.3378	1.07974
$429$	0	0
$430$	35.9349	1.73294
$431$	−16.6627	−0.802612	−0.401306	−	0.915944i	$-0.631444\pi$
$431$	−16.6627	−0.802612	−0.401306	+	0.915944i	$0.631444\pi$
$432$	−4.68287	−0.225305
$433$	18.9631	0.911310	0.455655	−	0.890156i	$-0.349405\pi$
$433$	18.9631	0.911310	0.455655	+	0.890156i	$0.349405\pi$
$434$	−7.84133	−0.376396
$435$	21.6652	1.03877
$436$	−25.5634	−1.22427
$437$	25.6547	1.22723
$438$	7.18199	0.343169
$439$	19.7565	0.942926	0.471463	−	0.881886i	$-0.343726\pi$
$439$	19.7565	0.942926	0.471463	+	0.881886i	$0.343726\pi$
$440$	1.79390	0.0855208
$441$	6.87260	0.327267
$442$	0	0
$443$	−3.77886	−0.179539	−0.0897696	−	0.995963i	$-0.528613\pi$
$443$	−3.77886	−0.179539	−0.0897696	+	0.995963i	$0.528613\pi$
$444$	7.11522	0.337673
$445$	−34.9352	−1.65609
$446$	−24.1203	−1.14213
$447$	21.1037	0.998172
$448$	15.6687	0.740276
$449$	−0.246336	−0.0116253	−0.00581266	−	0.999983i	$-0.501850\pi$
$449$	−0.246336	−0.0116253	−0.00581266	+	0.999983i	$0.501850\pi$
$450$	−0.505118	−0.0238115
$451$	−1.89825	−0.0893853
$452$	−6.25108	−0.294026
$453$	1.96608	0.0923745
$454$	−1.79551	−0.0842675
$455$	0	0
$456$	−3.36747	−0.157696
$457$	−22.7491	−1.06416	−0.532078	−	0.846695i	$-0.678589\pi$
$457$	−22.7491	−1.06416	−0.532078	+	0.846695i	$0.678589\pi$
$458$	−40.9521	−1.91356
$459$	2.48835	0.116146
$460$	21.3629	0.996048
$461$	−0.862073	−0.0401507	−0.0200754	−	0.999798i	$-0.506391\pi$
$461$	−0.862073	−0.0401507	−0.0200754	+	0.999798i	$0.506391\pi$
$462$	7.03065	0.327095
$463$	16.5483	0.769065	0.384532	−	0.923111i	$-0.374363\pi$
$463$	16.5483	0.769065	0.384532	+	0.923111i	$0.374363\pi$
$464$	−46.6374	−2.16509
$465$	2.42625	0.112515
$466$	−53.1528	−2.46225
$467$	−2.65641	−0.122924	−0.0614619	−	0.998109i	$-0.519576\pi$
$467$	−2.65641	−0.122924	−0.0614619	+	0.998109i	$0.519576\pi$
$468$	0	0
$469$	−17.4008	−0.803495
$470$	−0.833511	−0.0384470
$471$	−22.6164	−1.04211
$472$	−2.80321	−0.129028
$473$	−8.75103	−0.402373
$474$	22.6876	1.04208
$475$	−1.09275	−0.0501390
$476$	−14.4873	−0.664025
$477$	3.06306	0.140248
$478$	−50.4549	−2.30775
$479$	40.9236	1.86985	0.934923	−	0.354850i	$-0.115468\pi$
$479$	40.9236	1.86985	0.934923	+	0.354850i	$0.115468\pi$
$480$	−15.6418	−0.713946
$481$	0	0
$482$	46.1769	2.10330
$483$	−23.3991	−1.06469
$484$	1.56314	0.0710519
$485$	17.6464	0.801284
$486$	1.88763	0.0856246
$487$	−19.8472	−0.899362	−0.449681	−	0.893189i	$-0.648462\pi$
$487$	−19.8472	−0.899362	−0.449681	+	0.893189i	$0.648462\pi$
$488$	2.41936	0.109519
$489$	16.6713	0.753902
$490$	28.2214	1.27491
$491$	32.1096	1.44909	0.724543	−	0.689229i	$-0.242051\pi$
$491$	32.1096	1.44909	0.724543	+	0.689229i	$0.242051\pi$
$492$	2.96724	0.133774
$493$	24.7818	1.11612
$494$	0	0
$495$	−2.17541	−0.0977774
$496$	−5.22284	−0.234512
$497$	−12.3971	−0.556085
$498$	−24.9386	−1.11752
$499$	38.0257	1.70226	0.851132	−	0.524951i	$-0.175916\pi$
$499$	38.0257	1.70226	0.851132	+	0.524951i	$0.175916\pi$
$500$	−17.9123	−0.801063
$501$	21.9953	0.982678
$502$	12.9327	0.577215
$503$	12.1082	0.539880	0.269940	−	0.962877i	$-0.412996\pi$
$503$	12.1082	0.539880	0.269940	+	0.962877i	$0.412996\pi$
$504$	3.07140	0.136811
$505$	−11.5228	−0.512758
$506$	−11.8587	−0.527183
$507$	0	0
$508$	8.59977	0.381553
$509$	−35.5620	−1.57626	−0.788129	−	0.615510i	$-0.788950\pi$
$509$	−35.5620	−1.57626	−0.788129	+	0.615510i	$0.788950\pi$
$510$	10.2181	0.452463
$511$	−14.1712	−0.626898
$512$	25.9479	1.14675
$513$	4.08363	0.180297
$514$	−3.92828	−0.173269
$515$	−35.6977	−1.57303
$516$	13.6791	0.602189
$517$	0.202980	0.00892706
$518$	−32.0026	−1.40611
$519$	−19.3912	−0.851178
$520$	0	0
$521$	11.4222	0.500415	0.250207	−	0.968192i	$-0.419501\pi$
$521$	11.4222	0.500415	0.250207	+	0.968192i	$0.419501\pi$
$522$	18.7992	0.822817
$523$	9.93549	0.434449	0.217224	−	0.976122i	$-0.430300\pi$
$523$	9.93549	0.434449	0.217224	+	0.976122i	$0.430300\pi$
$524$	16.9949	0.742424
$525$	0.996678	0.0434986
$526$	12.7769	0.557098
$527$	2.77527	0.120893
$528$	4.68287	0.203796
$529$	16.4675	0.715978
$530$	12.5780	0.546355
$531$	3.39937	0.147520
$532$	−23.7752	−1.03078
$533$	0	0
$534$	−30.3137	−1.31180
$535$	31.0873	1.34402
$536$	−3.85254	−0.166405
$537$	−8.44338	−0.364359
$538$	−49.2164	−2.12187
$539$	−6.87260	−0.296024
$540$	3.40047	0.146333
$541$	24.8916	1.07017	0.535087	−	0.844797i	$-0.320279\pi$
$541$	24.8916	1.07017	0.535087	+	0.844797i	$0.320279\pi$
$542$	−26.4915	−1.13791
$543$	19.7268	0.846559
$544$	−17.8919	−0.767108
$545$	−35.5763	−1.52392
$546$	0	0
$547$	−10.3838	−0.443980	−0.221990	−	0.975049i	$-0.571255\pi$
$547$	−10.3838	−0.443980	−0.221990	+	0.975049i	$0.571255\pi$
$548$	−22.5635	−0.963865
$549$	−2.93389	−0.125215
$550$	0.505118	0.0215383
$551$	40.6695	1.73258
$552$	−5.18056	−0.220499
$553$	−44.7662	−1.90365
$554$	57.8118	2.45619
$555$	9.90218	0.420324
$556$	−11.0763	−0.469741
$557$	−27.0308	−1.14533	−0.572666	−	0.819789i	$-0.694091\pi$
$557$	−27.0308	−1.14533	−0.572666	+	0.819789i	$0.694091\pi$
$558$	2.10528	0.0891238
$559$	0	0
$560$	37.9430	1.60339
$561$	−2.48835	−0.105058
$562$	−37.0444	−1.56263
$563$	−40.6960	−1.71513	−0.857566	−	0.514374i	$-0.828024\pi$
$563$	−40.6960	−1.71513	−0.857566	+	0.514374i	$0.828024\pi$
$564$	−0.317287	−0.0133602
$565$	−8.69957	−0.365994
$566$	35.8346	1.50624
$567$	−3.72459	−0.156418
$568$	−2.74471	−0.115166
$569$	37.0846	1.55467	0.777334	−	0.629088i	$-0.216571\pi$
$569$	37.0846	1.55467	0.777334	+	0.629088i	$0.216571\pi$
$570$	16.7689	0.702371
$571$	−13.0144	−0.544636	−0.272318	−	0.962207i	$-0.587790\pi$
$571$	−13.0144	−0.544636	−0.272318	+	0.962207i	$0.587790\pi$
$572$	0	0
$573$	1.08306	0.0452453
$574$	−13.3460	−0.557050
$575$	−1.68111	−0.0701071
$576$	−4.20682	−0.175284
$577$	−31.0425	−1.29232	−0.646158	−	0.763204i	$-0.723625\pi$
$577$	−31.0425	−1.29232	−0.646158	+	0.763204i	$0.723625\pi$
$578$	−20.4017	−0.848600
$579$	−21.6348	−0.899112
$580$	33.8658	1.40620
$581$	49.2078	2.04148
$582$	15.3120	0.634704
$583$	−3.06306	−0.126859
$584$	−3.13751	−0.129831
$585$	0	0
$586$	−32.3864	−1.33787
$587$	−19.1803	−0.791654	−0.395827	−	0.918325i	$-0.629542\pi$
$587$	−19.1803	−0.791654	−0.395827	+	0.918325i	$0.629542\pi$
$588$	10.7428	0.443028
$589$	4.55450	0.187665
$590$	13.9591	0.574685
$591$	13.4237	0.552178
$592$	−21.3158	−0.876074
$593$	−10.3002	−0.422978	−0.211489	−	0.977380i	$-0.567831\pi$
$593$	−10.3002	−0.422978	−0.211489	+	0.977380i	$0.567831\pi$
$594$	−1.88763	−0.0774503
$595$	−20.1619	−0.826556
$596$	32.9881	1.35125
$597$	−1.29548	−0.0530206
$598$	0	0
$599$	40.3713	1.64953	0.824764	−	0.565477i	$-0.191308\pi$
$599$	40.3713	1.64953	0.824764	+	0.565477i	$0.191308\pi$
$600$	0.220665	0.00900860
$601$	0.739156	0.0301508	0.0150754	−	0.999886i	$-0.495201\pi$
$601$	0.739156	0.0301508	0.0150754	+	0.999886i	$0.495201\pi$
$602$	−61.5254	−2.50759
$603$	4.67187	0.190253
$604$	3.07326	0.125049
$605$	2.17541	0.0884430
$606$	−9.99848	−0.406160
$607$	−44.2097	−1.79442	−0.897209	−	0.441606i	$-0.854409\pi$
$607$	−44.2097	−1.79442	−0.897209	+	0.441606i	$0.854409\pi$
$608$	−29.3624	−1.19080
$609$	−37.0938	−1.50312
$610$	−12.0476	−0.487794
$611$	0	0
$612$	3.88964	0.157229
$613$	1.29495	0.0523024	0.0261512	−	0.999658i	$-0.491675\pi$
$613$	1.29495	0.0523024	0.0261512	+	0.999658i	$0.491675\pi$
$614$	14.5287	0.586329
$615$	4.12948	0.166517
$616$	−3.07140	−0.123750
$617$	−30.8274	−1.24106	−0.620532	−	0.784181i	$-0.713083\pi$
$617$	−30.8274	−1.24106	−0.620532	+	0.784181i	$0.713083\pi$
$618$	−30.9753	−1.24601
$619$	−21.4325	−0.861445	−0.430723	−	0.902484i	$-0.641741\pi$
$619$	−21.4325	−0.861445	−0.430723	+	0.902484i	$0.641741\pi$
$620$	3.79257	0.152313
$621$	6.28232	0.252101
$622$	−20.3903	−0.817577
$623$	59.8137	2.39639
$624$	0	0
$625$	−23.5904	−0.943617
$626$	61.9290	2.47518
$627$	−4.08363	−0.163085
$628$	−35.3527	−1.41072
$629$	11.3266	0.451622
$630$	−15.2945	−0.609349
$631$	−33.7655	−1.34418	−0.672092	−	0.740468i	$-0.734604\pi$
$631$	−33.7655	−1.34418	−0.672092	+	0.740468i	$0.734604\pi$
$632$	−9.91125	−0.394248
$633$	−0.305491	−0.0121422
$634$	5.06559	0.201180
$635$	11.9682	0.474944
$636$	4.78799	0.189856
$637$	0	0
$638$	−18.7992	−0.744266
$639$	3.32844	0.131671
$640$	14.0088	0.553748
$641$	47.4137	1.87273	0.936363	−	0.351032i	$-0.114169\pi$
$641$	47.4137	1.87273	0.936363	+	0.351032i	$0.114169\pi$
$642$	26.9748	1.06461
$643$	−1.46549	−0.0577934	−0.0288967	−	0.999582i	$-0.509199\pi$
$643$	−1.46549	−0.0577934	−0.0288967	+	0.999582i	$0.509199\pi$
$644$	−36.5761	−1.44130
$645$	19.0371	0.749584
$646$	19.1811	0.754671
$647$	30.7427	1.20862	0.604309	−	0.796750i	$-0.293449\pi$
$647$	30.7427	1.20862	0.604309	+	0.796750i	$0.293449\pi$
$648$	−0.824626	−0.0323944
$649$	−3.39937	−0.133437
$650$	0	0
$651$	−4.15406	−0.162811
$652$	26.0596	1.02057
$653$	34.1637	1.33693	0.668464	−	0.743744i	$-0.266952\pi$
$653$	34.1637	1.33693	0.668464	+	0.743744i	$0.266952\pi$
$654$	−30.8700	−1.20711
$655$	23.6516	0.924144
$656$	−8.88928	−0.347068
$657$	3.80477	0.148438
$658$	1.42708	0.0556335
$659$	38.9925	1.51893	0.759465	−	0.650548i	$-0.225461\pi$
$659$	38.9925	1.51893	0.759465	+	0.650548i	$0.225461\pi$
$660$	−3.40047	−0.132363
$661$	−33.2315	−1.29255	−0.646277	−	0.763103i	$-0.723675\pi$
$661$	−33.2315	−1.29255	−0.646277	+	0.763103i	$0.723675\pi$
$662$	−5.31884	−0.206723
$663$	0	0
$664$	10.8946	0.422793
$665$	−33.0877	−1.28309
$666$	8.59224	0.332942
$667$	62.5665	2.42259
$668$	34.3818	1.33027
$669$	−12.7781	−0.494029
$670$	19.1844	0.741158
$671$	2.93389	0.113262
$672$	26.7808	1.03309
$673$	−20.4272	−0.787409	−0.393705	−	0.919237i	$-0.628807\pi$
$673$	−20.4272	−0.787409	−0.393705	+	0.919237i	$0.628807\pi$
$674$	26.3889	1.01646
$675$	−0.267594	−0.0102997
$676$	0	0
$677$	−6.43652	−0.247375	−0.123688	−	0.992321i	$-0.539472\pi$
$677$	−6.43652	−0.247375	−0.123688	+	0.992321i	$0.539472\pi$
$678$	−7.54872	−0.289907
$679$	−30.2131	−1.15947
$680$	−4.46384	−0.171180
$681$	−0.951199	−0.0364500
$682$	−2.10528	−0.0806155
$683$	36.8208	1.40891	0.704455	−	0.709749i	$-0.251191\pi$
$683$	36.8208	1.40891	0.704455	+	0.709749i	$0.251191\pi$
$684$	6.38330	0.244071
$685$	−31.4014	−1.19979
$686$	0.895713	0.0341985
$687$	−21.6950	−0.827715
$688$	−40.9799	−1.56235
$689$	0	0
$690$	25.7975	0.982094
$691$	32.5317	1.23756	0.618782	−	0.785563i	$-0.287626\pi$
$691$	32.5317	1.23756	0.618782	+	0.785563i	$0.287626\pi$
$692$	−30.3112	−1.15226
$693$	3.72459	0.141486
$694$	−23.6901	−0.899262
$695$	−15.4148	−0.584717
$696$	−8.21257	−0.311297
$697$	4.72351	0.178916
$698$	30.8125	1.16627
$699$	−28.1585	−1.06505
$700$	1.55795	0.0588850
$701$	−5.10437	−0.192789	−0.0963946	−	0.995343i	$-0.530731\pi$
$701$	−5.10437	−0.192789	−0.0963946	+	0.995343i	$0.530731\pi$
$702$	0	0
$703$	18.5882	0.701065
$704$	4.20682	0.158550
$705$	−0.441565	−0.0166303
$706$	−14.3987	−0.541902
$707$	19.7286	0.741970
$708$	5.31370	0.199701
$709$	−50.2974	−1.88896	−0.944479	−	0.328572i	$-0.893433\pi$
$709$	−50.2974	−1.88896	−0.944479	+	0.328572i	$0.893433\pi$
$710$	13.6678	0.512942
$711$	12.0191	0.450751
$712$	13.2428	0.496294
$713$	7.00671	0.262403
$714$	−17.4947	−0.654722
$715$	0	0
$716$	−13.1982	−0.493240
$717$	−26.7292	−0.998222
$718$	−30.2457	−1.12876
$719$	−46.1034	−1.71937	−0.859683	−	0.510827i	$-0.829339\pi$
$719$	−46.1034	−1.71937	−0.859683	+	0.510827i	$0.829339\pi$
$720$	−10.1872	−0.379653
$721$	61.1193	2.27620
$722$	−4.38676	−0.163258
$723$	24.4629	0.909785
$724$	30.8358	1.14600
$725$	−2.66501	−0.0989758
$726$	1.88763	0.0700565
$727$	22.0396	0.817403	0.408702	−	0.912668i	$-0.365982\pi$
$727$	22.0396	0.817403	0.408702	+	0.912668i	$0.365982\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	15.6238	0.578262
$731$	21.7756	0.805399
$732$	−4.58608	−0.169507
$733$	48.3895	1.78731	0.893653	−	0.448758i	$-0.148133\pi$
$733$	48.3895	1.78731	0.893653	+	0.448758i	$0.148133\pi$
$734$	0.0451719	0.00166732
$735$	14.9507	0.551465
$736$	−45.1716	−1.66505
$737$	−4.67187	−0.172090
$738$	3.58320	0.131899
$739$	28.3045	1.04120	0.520599	−	0.853801i	$-0.325709\pi$
$739$	28.3045	1.04120	0.520599	+	0.853801i	$0.325709\pi$
$740$	15.4785	0.569001
$741$	0	0
$742$	−21.5353	−0.790585
$743$	13.6972	0.502502	0.251251	−	0.967922i	$-0.419158\pi$
$743$	13.6972	0.502502	0.251251	+	0.967922i	$0.419158\pi$
$744$	−0.919710	−0.0337182
$745$	45.9092	1.68198
$746$	6.29414	0.230445
$747$	−13.2116	−0.483386
$748$	−3.88964	−0.142219
$749$	−53.2256	−1.94482
$750$	−21.6307	−0.789840
$751$	−11.5640	−0.421975	−0.210987	−	0.977489i	$-0.567668\pi$
$751$	−11.5640	−0.421975	−0.210987	+	0.977489i	$0.567668\pi$
$752$	0.950531	0.0346623
$753$	6.85130	0.249675
$754$	0	0
$755$	4.27703	0.155657
$756$	−5.82207	−0.211747
$757$	14.2908	0.519407	0.259704	−	0.965688i	$-0.416375\pi$
$757$	14.2908	0.519407	0.259704	+	0.965688i	$0.416375\pi$
$758$	−11.1625	−0.405441
$759$	−6.28232	−0.228034
$760$	−7.32562	−0.265728
$761$	−27.3098	−0.989980	−0.494990	−	0.868899i	$-0.664828\pi$
$761$	−27.3098	−0.989980	−0.494990	+	0.868899i	$0.664828\pi$
$762$	10.3850	0.376207
$763$	60.9115	2.20514
$764$	1.69297	0.0612495
$765$	5.41317	0.195714
$766$	4.72200	0.170613
$767$	0	0
$768$	20.5693	0.742229
$769$	−24.4384	−0.881273	−0.440636	−	0.897686i	$-0.645247\pi$
$769$	−24.4384	−0.881273	−0.440636	+	0.897686i	$0.645247\pi$
$770$	15.2945	0.551177
$771$	−2.08107	−0.0749478
$772$	−33.8183	−1.21715
$773$	−0.731344	−0.0263046	−0.0131523	−	0.999914i	$-0.504187\pi$
$773$	−0.731344	−0.0263046	−0.0131523	+	0.999914i	$0.504187\pi$
$774$	16.5187	0.593752
$775$	−0.298449	−0.0107206
$776$	−6.68918	−0.240128
$777$	−16.9539	−0.608216
$778$	13.7598	0.493312
$779$	7.75177	0.277736
$780$	0	0
$781$	−3.32844	−0.119101
$782$	29.5085	1.05522
$783$	9.95915	0.355911
$784$	−32.1835	−1.14941
$785$	−49.1999	−1.75602
$786$	20.5228	0.732023
$787$	−37.2133	−1.32651	−0.663256	−	0.748392i	$-0.730826\pi$
$787$	−37.2133	−1.32651	−0.663256	+	0.748392i	$0.730826\pi$
$788$	20.9832	0.747495
$789$	6.76875	0.240974
$790$	49.3548	1.75596
$791$	14.8948	0.529599
$792$	0.824626	0.0293018
$793$	0	0
$794$	−4.97017	−0.176385
$795$	6.66340	0.236327
$796$	−2.02502	−0.0717751
$797$	15.7534	0.558013	0.279006	−	0.960289i	$-0.409995\pi$
$797$	15.7534	0.558013	0.279006	+	0.960289i	$0.409995\pi$
$798$	−28.7106	−1.01634
$799$	−0.505085	−0.0178686
$800$	1.92407	0.0680262
$801$	−16.0591	−0.567421
$802$	−37.1634	−1.31228
$803$	−3.80477	−0.134267
$804$	7.30279	0.257550
$805$	−50.9026	−1.79408
$806$	0	0
$807$	−26.0731	−0.917818
$808$	4.36791	0.153663
$809$	18.0582	0.634892	0.317446	−	0.948276i	$-0.397175\pi$
$809$	18.0582	0.634892	0.317446	+	0.948276i	$0.397175\pi$
$810$	4.10637	0.144283
$811$	17.3700	0.609944	0.304972	−	0.952361i	$-0.401353\pi$
$811$	17.3700	0.609944	0.304972	+	0.952361i	$0.401353\pi$
$812$	−57.9828	−2.03480
$813$	−14.0343	−0.492203
$814$	−8.59224	−0.301158
$815$	36.2669	1.27037
$816$	−11.6526	−0.407923
$817$	35.7360	1.25024
$818$	13.8195	0.483187
$819$	0	0
$820$	6.45496	0.225417
$821$	41.4490	1.44658	0.723291	−	0.690544i	$-0.242629\pi$
$821$	41.4490	1.44658	0.723291	+	0.690544i	$0.242629\pi$
$822$	−27.2474	−0.950361
$823$	30.2849	1.05566	0.527832	−	0.849349i	$-0.323005\pi$
$823$	30.2849	1.05566	0.527832	+	0.849349i	$0.323005\pi$
$824$	13.5318	0.471403
$825$	0.267594	0.00931642
$826$	−23.8998	−0.831580
$827$	−9.75249	−0.339127	−0.169564	−	0.985519i	$-0.554236\pi$
$827$	−9.75249	−0.339127	−0.169564	+	0.985519i	$0.554236\pi$
$828$	9.82015	0.341274
$829$	−32.2526	−1.12018	−0.560090	−	0.828432i	$-0.689234\pi$
$829$	−32.2526	−1.12018	−0.560090	+	0.828432i	$0.689234\pi$
$830$	−54.2516	−1.88310
$831$	30.6267	1.06243
$832$	0	0
$833$	17.1014	0.592529
$834$	−13.3756	−0.463159
$835$	47.8488	1.65588
$836$	−6.38330	−0.220771
$837$	1.11531	0.0385506
$838$	40.4324	1.39671
$839$	8.38113	0.289349	0.144674	−	0.989479i	$-0.453787\pi$
$839$	8.38113	0.289349	0.144674	+	0.989479i	$0.453787\pi$
$840$	6.68154	0.230535
$841$	70.1846	2.42016
$842$	10.1079	0.348343
$843$	−19.6249	−0.675916
$844$	−0.477526	−0.0164371
$845$	0	0
$846$	−0.383151	−0.0131730
$847$	−3.72459	−0.127979
$848$	−14.3439	−0.492572
$849$	18.9839	0.651526
$850$	−1.25691	−0.0431116
$851$	28.5963	0.980268
$852$	5.20282	0.178246
$853$	41.2839	1.41354	0.706768	−	0.707446i	$-0.250153\pi$
$853$	41.2839	1.41354	0.706768	+	0.707446i	$0.250153\pi$
$854$	20.6271	0.705846
$855$	8.88357	0.303812
$856$	−11.7842	−0.402774
$857$	−7.62243	−0.260377	−0.130189	−	0.991489i	$-0.541558\pi$
$857$	−7.62243	−0.260377	−0.130189	+	0.991489i	$0.541558\pi$
$858$	0	0
$859$	5.31091	0.181206	0.0906030	−	0.995887i	$-0.471121\pi$
$859$	5.31091	0.181206	0.0906030	+	0.995887i	$0.471121\pi$
$860$	29.7576	1.01473
$861$	−7.07022	−0.240953
$862$	−31.4529	−1.07129
$863$	1.95334	0.0664924	0.0332462	−	0.999447i	$-0.489415\pi$
$863$	1.95334	0.0664924	0.0332462	+	0.999447i	$0.489415\pi$
$864$	−7.19027	−0.244618
$865$	−42.1837	−1.43429
$866$	35.7953	1.21638
$867$	−10.8081	−0.367063
$868$	−6.49339	−0.220400
$869$	−12.0191	−0.407720
$870$	40.8959	1.38650
$871$	0	0
$872$	13.4858	0.456687
$873$	8.11178	0.274542
$874$	48.4265	1.63805
$875$	42.6808	1.44287
$876$	5.94740	0.200944
$877$	0.191918	0.00648062	0.00324031	−	0.999995i	$-0.498969\pi$
$877$	0.191918	0.00648062	0.00324031	+	0.999995i	$0.498969\pi$
$878$	37.2929	1.25858
$879$	−17.1572	−0.578698
$880$	10.1872	0.343409
$881$	−9.11658	−0.307145	−0.153573	−	0.988137i	$-0.549078\pi$
$881$	−9.11658	−0.307145	−0.153573	+	0.988137i	$0.549078\pi$
$882$	12.9729	0.436821
$883$	−53.8461	−1.81207	−0.906033	−	0.423206i	$-0.860905\pi$
$883$	−53.8461	−1.81207	−0.906033	+	0.423206i	$0.860905\pi$
$884$	0	0
$885$	7.39502	0.248581
$886$	−7.13309	−0.239641
$887$	13.3872	0.449498	0.224749	−	0.974417i	$-0.427844\pi$
$887$	13.3872	0.449498	0.224749	+	0.974417i	$0.427844\pi$
$888$	−3.75359	−0.125962
$889$	−20.4912	−0.687252
$890$	−65.9446	−2.21047
$891$	−1.00000	−0.0335013
$892$	−19.9739	−0.668777
$893$	−0.828897	−0.0277380
$894$	39.8360	1.33231
$895$	−18.3678	−0.613968
$896$	−23.9850	−0.801283
$897$	0	0
$898$	−0.464991	−0.0155169
$899$	11.1075	0.370456
$900$	−0.418287	−0.0139429
$901$	7.62195	0.253924
$902$	−3.58320	−0.119307
$903$	−32.5940	−1.08466
$904$	3.29772	0.109680
$905$	42.9139	1.42651
$906$	3.71123	0.123297
$907$	−17.9755	−0.596867	−0.298434	−	0.954430i	$-0.596464\pi$
$907$	−17.9755	−0.596867	−0.298434	+	0.954430i	$0.596464\pi$
$908$	−1.48686	−0.0493431
$909$	−5.29685	−0.175685
$910$	0	0
$911$	−25.3591	−0.840185	−0.420093	−	0.907481i	$-0.638002\pi$
$911$	−25.3591	−0.840185	−0.420093	+	0.907481i	$0.638002\pi$
$912$	−19.1231	−0.633230
$913$	13.2116	0.437239
$914$	−42.9418	−1.42039
$915$	−6.38241	−0.210996
$916$	−33.9123	−1.12049
$917$	−40.4947	−1.33725
$918$	4.69707	0.155027
$919$	−37.2892	−1.23006	−0.615028	−	0.788505i	$-0.710855\pi$
$919$	−37.2892	−1.23006	−0.615028	+	0.788505i	$0.710855\pi$
$920$	−11.2698	−0.371556
$921$	7.69678	0.253618
$922$	−1.62727	−0.0535914
$923$	0	0
$924$	5.82207	0.191532
$925$	−1.21805	−0.0400493
$926$	31.2371	1.02651
$927$	−16.4097	−0.538964
$928$	−71.6090	−2.35068
$929$	54.9615	1.80323	0.901614	−	0.432542i	$-0.142383\pi$
$929$	54.9615	1.80323	0.901614	+	0.432542i	$0.142383\pi$
$930$	4.57985	0.150179
$931$	28.0652	0.919798
$932$	−44.0157	−1.44178
$933$	−10.8021	−0.353644
$934$	−5.01431	−0.164073
$935$	−5.41317	−0.177030
$936$	0	0
$937$	−54.1109	−1.76773	−0.883864	−	0.467744i	$-0.845067\pi$
$937$	−54.1109	−1.76773	−0.883864	+	0.467744i	$0.845067\pi$
$938$	−32.8463	−1.07247
$939$	32.8078	1.07064
$940$	−0.690229	−0.0225128
$941$	−2.14684	−0.0699849	−0.0349925	−	0.999388i	$-0.511141\pi$
$941$	−2.14684	−0.0699849	−0.0349925	+	0.999388i	$0.511141\pi$
$942$	−42.6914	−1.39096
$943$	11.9254	0.388345
$944$	−15.9188	−0.518113
$945$	−8.10252	−0.263575
$946$	−16.5187	−0.537069
$947$	−26.0510	−0.846542	−0.423271	−	0.906003i	$-0.639118\pi$
$947$	−26.0510	−0.846542	−0.423271	+	0.906003i	$0.639118\pi$
$948$	18.7875	0.610191
$949$	0	0
$950$	−2.06271	−0.0669233
$951$	2.68357	0.0870209
$952$	7.64270	0.247701
$953$	3.49931	0.113354	0.0566769	−	0.998393i	$-0.481950\pi$
$953$	3.49931	0.113354	0.0566769	+	0.998393i	$0.481950\pi$
$954$	5.78191	0.187196
$955$	2.35609	0.0762413
$956$	−41.7816	−1.35131
$957$	−9.95915	−0.321934
$958$	77.2485	2.49579
$959$	53.7634	1.73611
$960$	−9.15155	−0.295365
$961$	−29.7561	−0.959874
$962$	0	0
$963$	14.2903	0.460499
$964$	38.2390	1.23160
$965$	−47.0646	−1.51506
$966$	−44.1688	−1.42111
$967$	51.4221	1.65362	0.826812	−	0.562478i	$-0.190152\pi$
$967$	51.4221	1.65362	0.826812	+	0.562478i	$0.190152\pi$
$968$	−0.824626	−0.0265045
$969$	10.1615	0.326434
$970$	33.3099	1.06952
$971$	−39.1185	−1.25537	−0.627686	−	0.778467i	$-0.715998\pi$
$971$	−39.1185	−1.25537	−0.627686	+	0.778467i	$0.715998\pi$
$972$	1.56314	0.0501378
$973$	26.3922	0.846096
$974$	−37.4641	−1.20043
$975$	0	0
$976$	13.7390	0.439775
$977$	−4.03653	−0.129140	−0.0645700	−	0.997913i	$-0.520568\pi$
$977$	−4.03653	−0.129140	−0.0645700	+	0.997913i	$0.520568\pi$
$978$	31.4692	1.00627
$979$	16.0591	0.513252
$980$	23.3701	0.746530
$981$	−16.3539	−0.522139
$982$	60.6110	1.93418
$983$	2.21453	0.0706327	0.0353163	−	0.999376i	$-0.488756\pi$
$983$	2.21453	0.0706327	0.0353163	+	0.999376i	$0.488756\pi$
$984$	−1.56535	−0.0499015
$985$	29.2021	0.930456
$986$	46.7789	1.48974
$987$	0.756019	0.0240644
$988$	0	0
$989$	54.9767	1.74816
$990$	−4.10637	−0.130509
$991$	−21.2051	−0.673601	−0.336801	−	0.941576i	$-0.609345\pi$
$991$	−21.2051	−0.673601	−0.336801	+	0.941576i	$0.609345\pi$
$992$	−8.01935	−0.254615
$993$	−2.81774	−0.0894181
$994$	−23.4011	−0.742237
$995$	−2.81821	−0.0893431
$996$	−20.6516	−0.654370
$997$	−43.7454	−1.38543	−0.692715	−	0.721211i	$-0.743586\pi$
$997$	−43.7454	−1.38543	−0.692715	+	0.721211i	$0.743586\pi$
$998$	71.7784	2.27211
$999$	4.55187	0.144015

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5577.2.a.bk.1.15	yes	18
13.12	even	2		5577.2.a.bi.1.4	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5577.2.a.bi.1.4	✓	18	13.12	even	2
5577.2.a.bk.1.15	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)

\(f(q)\)	\(=\)	\(q+1.88763 q^{2} +1.00000 q^{3} +1.56314 q^{4} +2.17541 q^{5} +1.88763 q^{6} -3.72459 q^{7} -0.824626 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.88763 q^{2} +1.00000 q^{3} +1.56314 q^{4} +2.17541 q^{5} +1.88763 q^{6} -3.72459 q^{7} -0.824626 q^{8} +1.00000 q^{9} +4.10637 q^{10} -1.00000 q^{11} +1.56314 q^{12} -7.03065 q^{14} +2.17541 q^{15} -4.68287 q^{16} +2.48835 q^{17} +1.88763 q^{18} +4.08363 q^{19} +3.40047 q^{20} -3.72459 q^{21} -1.88763 q^{22} +6.28232 q^{23} -0.824626 q^{24} -0.267594 q^{25} +1.00000 q^{27} -5.82207 q^{28} +9.95915 q^{29} +4.10637 q^{30} +1.11531 q^{31} -7.19027 q^{32} -1.00000 q^{33} +4.69707 q^{34} -8.10252 q^{35} +1.56314 q^{36} +4.55187 q^{37} +7.70838 q^{38} -1.79390 q^{40} +1.89825 q^{41} -7.03065 q^{42} +8.75103 q^{43} -1.56314 q^{44} +2.17541 q^{45} +11.8587 q^{46} -0.202980 q^{47} -4.68287 q^{48} +6.87260 q^{49} -0.505118 q^{50} +2.48835 q^{51} +3.06306 q^{53} +1.88763 q^{54} -2.17541 q^{55} +3.07140 q^{56} +4.08363 q^{57} +18.7992 q^{58} +3.39937 q^{59} +3.40047 q^{60} -2.93389 q^{61} +2.10528 q^{62} -3.72459 q^{63} -4.20682 q^{64} -1.88763 q^{66} +4.67187 q^{67} +3.88964 q^{68} +6.28232 q^{69} -15.2945 q^{70} +3.32844 q^{71} -0.824626 q^{72} +3.80477 q^{73} +8.59224 q^{74} -0.267594 q^{75} +6.38330 q^{76} +3.72459 q^{77} +12.0191 q^{79} -10.1872 q^{80} +1.00000 q^{81} +3.58320 q^{82} -13.2116 q^{83} -5.82207 q^{84} +5.41317 q^{85} +16.5187 q^{86} +9.95915 q^{87} +0.824626 q^{88} -16.0591 q^{89} +4.10637 q^{90} +9.82015 q^{92} +1.11531 q^{93} -0.383151 q^{94} +8.88357 q^{95} -7.19027 q^{96} +8.11178 q^{97} +12.9729 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + q^{2} + 18 q^{3} + 23 q^{4} - 6 q^{5} + q^{6} + q^{7} + 6 q^{8} + 18 q^{9}+O(q^{10}) \) 18 * q + q^2 + 18 * q^3 + 23 * q^4 - 6 * q^5 + q^6 + q^7 + 6 * q^8 + 18 * q^9	\( 18 q + q^{2} + 18 q^{3} + 23 q^{4} - 6 q^{5} + q^{6} + q^{7} + 6 q^{8} + 18 q^{9} - 7 q^{10} - 18 q^{11} + 23 q^{12} + 7 q^{14} - 6 q^{15} + 25 q^{16} + 19 q^{17} + q^{18} - 2 q^{20} + q^{21} - q^{22} + 38 q^{23} + 6 q^{24} + 36 q^{25} + 18 q^{27} + 33 q^{28} + 47 q^{29} - 7 q^{30} - 22 q^{31} + 33 q^{32} - 18 q^{33} + 4 q^{34} + 20 q^{35} + 23 q^{36} - 41 q^{37} + 14 q^{38} - 28 q^{40} + 8 q^{41} + 7 q^{42} + 28 q^{43} - 23 q^{44} - 6 q^{45} + 6 q^{46} + 11 q^{47} + 25 q^{48} + 9 q^{49} - q^{50} + 19 q^{51} + 49 q^{53} + q^{54} + 6 q^{55} + 35 q^{56} - 27 q^{58} - 2 q^{59} - 2 q^{60} - 13 q^{61} + 46 q^{62} + q^{63} + 40 q^{64} - q^{66} + 4 q^{67} + 30 q^{68} + 38 q^{69} - 83 q^{70} + 2 q^{71} + 6 q^{72} + 39 q^{73} + 42 q^{74} + 36 q^{75} + 20 q^{76} - q^{77} + 18 q^{79} + 18 q^{80} + 18 q^{81} - 12 q^{82} + 5 q^{83} + 33 q^{84} + 2 q^{85} + 57 q^{86} + 47 q^{87} - 6 q^{88} - 9 q^{89} - 7 q^{90} + 86 q^{92} - 22 q^{93} - 27 q^{94} + 74 q^{95} + 33 q^{96} - 17 q^{97} + 4 q^{98} - 18 q^{99}+O(q^{100}) \) 18 * q + q^2 + 18 * q^3 + 23 * q^4 - 6 * q^5 + q^6 + q^7 + 6 * q^8 + 18 * q^9 - 7 * q^10 - 18 * q^11 + 23 * q^12 + 7 * q^14 - 6 * q^15 + 25 * q^16 + 19 * q^17 + q^18 - 2 * q^20 + q^21 - q^22 + 38 * q^23 + 6 * q^24 + 36 * q^25 + 18 * q^27 + 33 * q^28 + 47 * q^29 - 7 * q^30 - 22 * q^31 + 33 * q^32 - 18 * q^33 + 4 * q^34 + 20 * q^35 + 23 * q^36 - 41 * q^37 + 14 * q^38 - 28 * q^40 + 8 * q^41 + 7 * q^42 + 28 * q^43 - 23 * q^44 - 6 * q^45 + 6 * q^46 + 11 * q^47 + 25 * q^48 + 9 * q^49 - q^50 + 19 * q^51 + 49 * q^53 + q^54 + 6 * q^55 + 35 * q^56 - 27 * q^58 - 2 * q^59 - 2 * q^60 - 13 * q^61 + 46 * q^62 + q^63 + 40 * q^64 - q^66 + 4 * q^67 + 30 * q^68 + 38 * q^69 - 83 * q^70 + 2 * q^71 + 6 * q^72 + 39 * q^73 + 42 * q^74 + 36 * q^75 + 20 * q^76 - q^77 + 18 * q^79 + 18 * q^80 + 18 * q^81 - 12 * q^82 + 5 * q^83 + 33 * q^84 + 2 * q^85 + 57 * q^86 + 47 * q^87 - 6 * q^88 - 9 * q^89 - 7 * q^90 + 86 * q^92 - 22 * q^93 - 27 * q^94 + 74 * q^95 + 33 * q^96 - 17 * q^97 + 4 * q^98 - 18 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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