LMFDB - Embedded newform 363.2.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [363,2,Mod(1,363)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(363, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("363.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 363 = 3 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	363.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:	$2.89856959337$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 33)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	363.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} -2.61803 q^{5} +0.618034 q^{6} -3.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} -2.61803 q^{5} +0.618034 q^{6} -3.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} -1.61803 q^{10} -1.61803 q^{12} -1.76393 q^{13} -1.85410 q^{14} -2.61803 q^{15} +1.85410 q^{16} -1.61803 q^{17} +0.618034 q^{18} -5.85410 q^{19} +4.23607 q^{20} -3.00000 q^{21} +3.47214 q^{23} -2.23607 q^{24} +1.85410 q^{25} -1.09017 q^{26} +1.00000 q^{27} +4.85410 q^{28} -4.47214 q^{29} -1.61803 q^{30} +2.85410 q^{31} +5.61803 q^{32} -1.00000 q^{34} +7.85410 q^{35} -1.61803 q^{36} +0.236068 q^{37} -3.61803 q^{38} -1.76393 q^{39} +5.85410 q^{40} +11.9443 q^{41} -1.85410 q^{42} -6.23607 q^{43} -2.61803 q^{45} +2.14590 q^{46} +1.61803 q^{47} +1.85410 q^{48} +2.00000 q^{49} +1.14590 q^{50} -1.61803 q^{51} +2.85410 q^{52} -9.61803 q^{53} +0.618034 q^{54} +6.70820 q^{56} -5.85410 q^{57} -2.76393 q^{58} +10.3262 q^{59} +4.23607 q^{60} -7.85410 q^{61} +1.76393 q^{62} -3.00000 q^{63} -0.236068 q^{64} +4.61803 q^{65} -9.56231 q^{67} +2.61803 q^{68} +3.47214 q^{69} +4.85410 q^{70} -5.56231 q^{71} -2.23607 q^{72} +3.23607 q^{73} +0.145898 q^{74} +1.85410 q^{75} +9.47214 q^{76} -1.09017 q^{78} -9.47214 q^{79} -4.85410 q^{80} +1.00000 q^{81} +7.38197 q^{82} -0.708204 q^{83} +4.85410 q^{84} +4.23607 q^{85} -3.85410 q^{86} -4.47214 q^{87} +0.527864 q^{89} -1.61803 q^{90} +5.29180 q^{91} -5.61803 q^{92} +2.85410 q^{93} +1.00000 q^{94} +15.3262 q^{95} +5.61803 q^{96} -14.0344 q^{97} +1.23607 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} + 2 q^{3} - q^{4} - 3 q^{5} - q^{6} - 6 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - q^2 + 2 * q^3 - q^4 - 3 * q^5 - q^6 - 6 * q^7 + 2 * q^9	$ 2 q - q^{2} + 2 q^{3} - q^{4} - 3 q^{5} - q^{6} - 6 q^{7} + 2 q^{9} - q^{10} - q^{12} - 8 q^{13} + 3 q^{14} - 3 q^{15} - 3 q^{16} - q^{17} - q^{18} - 5 q^{19} + 4 q^{20} - 6 q^{21} - 2 q^{23} - 3 q^{25} + 9 q^{26} + 2 q^{27} + 3 q^{28} - q^{30} - q^{31} + 9 q^{32} - 2 q^{34} + 9 q^{35} - q^{36} - 4 q^{37} - 5 q^{38} - 8 q^{39} + 5 q^{40} + 6 q^{41} + 3 q^{42} - 8 q^{43} - 3 q^{45} + 11 q^{46} + q^{47} - 3 q^{48} + 4 q^{49} + 9 q^{50} - q^{51} - q^{52} - 17 q^{53} - q^{54} - 5 q^{57} - 10 q^{58} + 5 q^{59} + 4 q^{60} - 9 q^{61} + 8 q^{62} - 6 q^{63} + 4 q^{64} + 7 q^{65} + q^{67} + 3 q^{68} - 2 q^{69} + 3 q^{70} + 9 q^{71} + 2 q^{73} + 7 q^{74} - 3 q^{75} + 10 q^{76} + 9 q^{78} - 10 q^{79} - 3 q^{80} + 2 q^{81} + 17 q^{82} + 12 q^{83} + 3 q^{84} + 4 q^{85} - q^{86} + 10 q^{89} - q^{90} + 24 q^{91} - 9 q^{92} - q^{93} + 2 q^{94} + 15 q^{95} + 9 q^{96} + q^{97} - 2 q^{98}+O(q^{100}) $ 2 * q - q^2 + 2 * q^3 - q^4 - 3 * q^5 - q^6 - 6 * q^7 + 2 * q^9 - q^10 - q^12 - 8 * q^13 + 3 * q^14 - 3 * q^15 - 3 * q^16 - q^17 - q^18 - 5 * q^19 + 4 * q^20 - 6 * q^21 - 2 * q^23 - 3 * q^25 + 9 * q^26 + 2 * q^27 + 3 * q^28 - q^30 - q^31 + 9 * q^32 - 2 * q^34 + 9 * q^35 - q^36 - 4 * q^37 - 5 * q^38 - 8 * q^39 + 5 * q^40 + 6 * q^41 + 3 * q^42 - 8 * q^43 - 3 * q^45 + 11 * q^46 + q^47 - 3 * q^48 + 4 * q^49 + 9 * q^50 - q^51 - q^52 - 17 * q^53 - q^54 - 5 * q^57 - 10 * q^58 + 5 * q^59 + 4 * q^60 - 9 * q^61 + 8 * q^62 - 6 * q^63 + 4 * q^64 + 7 * q^65 + q^67 + 3 * q^68 - 2 * q^69 + 3 * q^70 + 9 * q^71 + 2 * q^73 + 7 * q^74 - 3 * q^75 + 10 * q^76 + 9 * q^78 - 10 * q^79 - 3 * q^80 + 2 * q^81 + 17 * q^82 + 12 * q^83 + 3 * q^84 + 4 * q^85 - q^86 + 10 * q^89 - q^90 + 24 * q^91 - 9 * q^92 - q^93 + 2 * q^94 + 15 * q^95 + 9 * q^96 + q^97 - 2 * q^98

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 $

$2$	0.618034	0.437016	0.218508	−	0.975835i	$-0.429881\pi$
$2$	0.618034	0.437016	0.218508	+	0.975835i	$0.429881\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.61803	−0.809017
$5$	−2.61803	−1.17082	−0.585410	−	0.810737i	$-0.699067\pi$
$5$	−2.61803	−1.17082	−0.585410	+	0.810737i	$0.699067\pi$
$6$	0.618034	0.252311
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	−2.23607	−0.790569
$9$	1.00000	0.333333
$10$	−1.61803	−0.511667
$11$	0	0
$11$	0	0
$12$	−1.61803	−0.467086
$13$	−1.76393	−0.489227	−0.244613	−	0.969621i	$-0.578661\pi$
$13$	−1.76393	−0.489227	−0.244613	+	0.969621i	$0.578661\pi$
$14$	−1.85410	−0.495530
$15$	−2.61803	−0.675973
$16$	1.85410	0.463525
$17$	−1.61803	−0.392431	−0.196215	−	0.980561i	$-0.562865\pi$
$17$	−1.61803	−0.392431	−0.196215	+	0.980561i	$0.562865\pi$
$18$	0.618034	0.145672
$19$	−5.85410	−1.34302	−0.671512	−	0.740994i	$-0.734355\pi$
$19$	−5.85410	−1.34302	−0.671512	+	0.740994i	$0.734355\pi$
$20$	4.23607	0.947214
$21$	−3.00000	−0.654654
$22$	0	0
$23$	3.47214	0.723990	0.361995	−	0.932180i	$-0.382096\pi$
$23$	3.47214	0.723990	0.361995	+	0.932180i	$0.382096\pi$
$24$	−2.23607	−0.456435
$25$	1.85410	0.370820
$26$	−1.09017	−0.213800
$27$	1.00000	0.192450
$28$	4.85410	0.917339
$29$	−4.47214	−0.830455	−0.415227	−	0.909718i	$-0.636298\pi$
$29$	−4.47214	−0.830455	−0.415227	+	0.909718i	$0.636298\pi$
$30$	−1.61803	−0.295411
$31$	2.85410	0.512612	0.256306	−	0.966596i	$-0.417495\pi$
$31$	2.85410	0.512612	0.256306	+	0.966596i	$0.417495\pi$
$32$	5.61803	0.993137
$33$	0	0
$34$	−1.00000	−0.171499
$35$	7.85410	1.32759
$36$	−1.61803	−0.269672
$37$	0.236068	0.0388093	0.0194047	−	0.999812i	$-0.493823\pi$
$37$	0.236068	0.0388093	0.0194047	+	0.999812i	$0.493823\pi$
$38$	−3.61803	−0.586923
$39$	−1.76393	−0.282455
$40$	5.85410	0.925615
$41$	11.9443	1.86538	0.932691	−	0.360677i	$-0.117454\pi$
$41$	11.9443	1.86538	0.932691	+	0.360677i	$0.117454\pi$
$42$	−1.85410	−0.286094
$43$	−6.23607	−0.950991	−0.475496	−	0.879718i	$-0.657731\pi$
$43$	−6.23607	−0.950991	−0.475496	+	0.879718i	$0.657731\pi$
$44$	0	0
$45$	−2.61803	−0.390273
$46$	2.14590	0.316395
$47$	1.61803	0.236015	0.118007	−	0.993013i	$-0.462349\pi$
$47$	1.61803	0.236015	0.118007	+	0.993013i	$0.462349\pi$
$48$	1.85410	0.267617
$49$	2.00000	0.285714
$50$	1.14590	0.162054
$51$	−1.61803	−0.226570
$52$	2.85410	0.395793
$53$	−9.61803	−1.32114	−0.660569	−	0.750765i	$-0.729685\pi$
$53$	−9.61803	−1.32114	−0.660569	+	0.750765i	$0.729685\pi$
$54$	0.618034	0.0841038
$55$	0	0
$56$	6.70820	0.896421
$57$	−5.85410	−0.775395
$58$	−2.76393	−0.362922
$59$	10.3262	1.34436	0.672181	−	0.740387i	$-0.265358\pi$
$59$	10.3262	1.34436	0.672181	+	0.740387i	$0.265358\pi$
$60$	4.23607	0.546874
$61$	−7.85410	−1.00561	−0.502807	−	0.864398i	$-0.667699\pi$
$61$	−7.85410	−1.00561	−0.502807	+	0.864398i	$0.667699\pi$
$62$	1.76393	0.224020
$63$	−3.00000	−0.377964
$64$	−0.236068	−0.0295085
$65$	4.61803	0.572797
$66$	0	0
$67$	−9.56231	−1.16822	−0.584111	−	0.811674i	$-0.698557\pi$
$67$	−9.56231	−1.16822	−0.584111	+	0.811674i	$0.698557\pi$
$68$	2.61803	0.317483
$69$	3.47214	0.417996
$70$	4.85410	0.580176
$71$	−5.56231	−0.660124	−0.330062	−	0.943959i	$-0.607070\pi$
$71$	−5.56231	−0.660124	−0.330062	+	0.943959i	$0.607070\pi$
$72$	−2.23607	−0.263523
$73$	3.23607	0.378753	0.189377	−	0.981905i	$-0.439353\pi$
$73$	3.23607	0.378753	0.189377	+	0.981905i	$0.439353\pi$
$74$	0.145898	0.0169603
$75$	1.85410	0.214093
$76$	9.47214	1.08653
$77$	0	0
$78$	−1.09017	−0.123437
$79$	−9.47214	−1.06570	−0.532849	−	0.846210i	$-0.678879\pi$
$79$	−9.47214	−1.06570	−0.532849	+	0.846210i	$0.678879\pi$
$80$	−4.85410	−0.542705
$81$	1.00000	0.111111
$82$	7.38197	0.815202
$83$	−0.708204	−0.0777355	−0.0388677	−	0.999244i	$-0.512375\pi$
$83$	−0.708204	−0.0777355	−0.0388677	+	0.999244i	$0.512375\pi$
$84$	4.85410	0.529626
$85$	4.23607	0.459466
$86$	−3.85410	−0.415599
$87$	−4.47214	−0.479463
$88$	0	0
$89$	0.527864	0.0559535	0.0279767	−	0.999609i	$-0.491094\pi$
$89$	0.527864	0.0559535	0.0279767	+	0.999609i	$0.491094\pi$
$90$	−1.61803	−0.170556
$91$	5.29180	0.554731
$92$	−5.61803	−0.585721
$93$	2.85410	0.295957
$94$	1.00000	0.103142
$95$	15.3262	1.57244
$96$	5.61803	0.573388
$97$	−14.0344	−1.42498	−0.712491	−	0.701681i	$-0.752433\pi$
$97$	−14.0344	−1.42498	−0.712491	+	0.701681i	$0.752433\pi$
$98$	1.23607	0.124862
$99$	0	0
$100$	−3.00000	−0.300000
$101$	3.00000	0.298511	0.149256	−	0.988799i	$-0.452312\pi$
$101$	3.00000	0.298511	0.149256	+	0.988799i	$0.452312\pi$
$102$	−1.00000	−0.0990148
$103$	−6.00000	−0.591198	−0.295599	−	0.955312i	$-0.595519\pi$
$103$	−6.00000	−0.591198	−0.295599	+	0.955312i	$0.595519\pi$
$104$	3.94427	0.386768
$105$	7.85410	0.766482
$106$	−5.94427	−0.577359
$107$	4.23607	0.409516	0.204758	−	0.978813i	$-0.434359\pi$
$107$	4.23607	0.409516	0.204758	+	0.978813i	$0.434359\pi$
$108$	−1.61803	−0.155695
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	0	0
$111$	0.236068	0.0224066
$112$	−5.56231	−0.525589
$113$	0.708204	0.0666222	0.0333111	−	0.999445i	$-0.489395\pi$
$113$	0.708204	0.0666222	0.0333111	+	0.999445i	$0.489395\pi$
$114$	−3.61803	−0.338860
$115$	−9.09017	−0.847663
$116$	7.23607	0.671852
$117$	−1.76393	−0.163076
$118$	6.38197	0.587508
$119$	4.85410	0.444975
$120$	5.85410	0.534404
$121$	0	0
$122$	−4.85410	−0.439470
$123$	11.9443	1.07698
$124$	−4.61803	−0.414712
$125$	8.23607	0.736656
$126$	−1.85410	−0.165177
$127$	3.70820	0.329050	0.164525	−	0.986373i	$-0.447391\pi$
$127$	3.70820	0.329050	0.164525	+	0.986373i	$0.447391\pi$
$128$	−11.3820	−1.00603
$129$	−6.23607	−0.549055
$130$	2.85410	0.250321
$131$	7.14590	0.624340	0.312170	−	0.950026i	$-0.398944\pi$
$131$	7.14590	0.624340	0.312170	+	0.950026i	$0.398944\pi$
$132$	0	0
$133$	17.5623	1.52285
$134$	−5.90983	−0.510532
$135$	−2.61803	−0.225324
$136$	3.61803	0.310244
$137$	7.47214	0.638388	0.319194	−	0.947689i	$-0.396588\pi$
$137$	7.47214	0.638388	0.319194	+	0.947689i	$0.396588\pi$
$138$	2.14590	0.182671
$139$	0.854102	0.0724440	0.0362220	−	0.999344i	$-0.488468\pi$
$139$	0.854102	0.0724440	0.0362220	+	0.999344i	$0.488468\pi$
$140$	−12.7082	−1.07404
$141$	1.61803	0.136263
$142$	−3.43769	−0.288485
$143$	0	0
$144$	1.85410	0.154508
$145$	11.7082	0.972313
$146$	2.00000	0.165521
$147$	2.00000	0.164957
$148$	−0.381966	−0.0313974
$149$	−15.0000	−1.22885	−0.614424	−	0.788976i	$-0.710612\pi$
$149$	−15.0000	−1.22885	−0.614424	+	0.788976i	$0.710612\pi$
$150$	1.14590	0.0935622
$151$	−2.00000	−0.162758	−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000	−0.162758	−0.0813788	+	0.996683i	$0.525932\pi$
$152$	13.0902	1.06175
$153$	−1.61803	−0.130810
$154$	0	0
$155$	−7.47214	−0.600176
$156$	2.85410	0.228511
$157$	−3.70820	−0.295947	−0.147973	−	0.988991i	$-0.547275\pi$
$157$	−3.70820	−0.295947	−0.147973	+	0.988991i	$0.547275\pi$
$158$	−5.85410	−0.465727
$159$	−9.61803	−0.762760
$160$	−14.7082	−1.16279
$161$	−10.4164	−0.820928
$162$	0.618034	0.0485573
$163$	18.2705	1.43106	0.715528	−	0.698584i	$-0.246186\pi$
$163$	18.2705	1.43106	0.715528	+	0.698584i	$0.246186\pi$
$164$	−19.3262	−1.50913
$165$	0	0
$166$	−0.437694	−0.0339717
$167$	−10.0344	−0.776488	−0.388244	−	0.921557i	$-0.626918\pi$
$167$	−10.0344	−0.776488	−0.388244	+	0.921557i	$0.626918\pi$
$168$	6.70820	0.517549
$169$	−9.88854	−0.760657
$170$	2.61803	0.200794
$171$	−5.85410	−0.447674
$172$	10.0902	0.769368
$173$	−15.3820	−1.16947	−0.584735	−	0.811225i	$-0.698801\pi$
$173$	−15.3820	−1.16947	−0.584735	+	0.811225i	$0.698801\pi$
$174$	−2.76393	−0.209533
$175$	−5.56231	−0.420471
$176$	0	0
$177$	10.3262	0.776168
$178$	0.326238	0.0244526
$179$	−2.23607	−0.167132	−0.0835658	−	0.996502i	$-0.526631\pi$
$179$	−2.23607	−0.167132	−0.0835658	+	0.996502i	$0.526631\pi$
$180$	4.23607	0.315738
$181$	−17.4721	−1.29869	−0.649347	−	0.760492i	$-0.724958\pi$
$181$	−17.4721	−1.29869	−0.649347	+	0.760492i	$0.724958\pi$
$182$	3.27051	0.242426
$183$	−7.85410	−0.580592
$184$	−7.76393	−0.572365
$185$	−0.618034	−0.0454388
$186$	1.76393	0.129338
$187$	0	0
$188$	−2.61803	−0.190940
$189$	−3.00000	−0.218218
$190$	9.47214	0.687181
$191$	−7.47214	−0.540665	−0.270332	−	0.962767i	$-0.587134\pi$
$191$	−7.47214	−0.540665	−0.270332	+	0.962767i	$0.587134\pi$
$192$	−0.236068	−0.0170367
$193$	18.5623	1.33614	0.668072	−	0.744097i	$-0.267120\pi$
$193$	18.5623	1.33614	0.668072	+	0.744097i	$0.267120\pi$
$194$	−8.67376	−0.622740
$195$	4.61803	0.330704
$196$	−3.23607	−0.231148
$197$	−24.3820	−1.73714	−0.868572	−	0.495564i	$-0.834961\pi$
$197$	−24.3820	−1.73714	−0.868572	+	0.495564i	$0.834961\pi$
$198$	0	0
$199$	−16.7082	−1.18441	−0.592207	−	0.805786i	$-0.701743\pi$
$199$	−16.7082	−1.18441	−0.592207	+	0.805786i	$0.701743\pi$
$200$	−4.14590	−0.293159
$201$	−9.56231	−0.674473
$202$	1.85410	0.130454
$203$	13.4164	0.941647
$204$	2.61803	0.183299
$205$	−31.2705	−2.18403
$206$	−3.70820	−0.258363
$207$	3.47214	0.241330
$208$	−3.27051	−0.226769
$209$	0	0
$210$	4.85410	0.334965
$211$	22.2705	1.53317	0.766583	−	0.642146i	$-0.221956\pi$
$211$	22.2705	1.53317	0.766583	+	0.642146i	$0.221956\pi$
$212$	15.5623	1.06882
$213$	−5.56231	−0.381123
$214$	2.61803	0.178965
$215$	16.3262	1.11344
$216$	−2.23607	−0.152145
$217$	−8.56231	−0.581247
$218$	0	0
$219$	3.23607	0.218673
$220$	0	0
$221$	2.85410	0.191988
$222$	0.145898	0.00979203
$223$	0.708204	0.0474248	0.0237124	−	0.999719i	$-0.492451\pi$
$223$	0.708204	0.0474248	0.0237124	+	0.999719i	$0.492451\pi$
$224$	−16.8541	−1.12611
$225$	1.85410	0.123607
$226$	0.437694	0.0291150
$227$	24.8885	1.65191	0.825955	−	0.563736i	$-0.190636\pi$
$227$	24.8885	1.65191	0.825955	+	0.563736i	$0.190636\pi$
$228$	9.47214	0.627308
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	−5.61803	−0.370442
$231$	0	0
$232$	10.0000	0.656532
$233$	−24.3262	−1.59366	−0.796832	−	0.604200i	$-0.793493\pi$
$233$	−24.3262	−1.59366	−0.796832	+	0.604200i	$0.793493\pi$
$234$	−1.09017	−0.0712666
$235$	−4.23607	−0.276331
$236$	−16.7082	−1.08761
$237$	−9.47214	−0.615281
$238$	3.00000	0.194461
$239$	−2.56231	−0.165742	−0.0828709	−	0.996560i	$-0.526409\pi$
$239$	−2.56231	−0.165742	−0.0828709	+	0.996560i	$0.526409\pi$
$240$	−4.85410	−0.313331
$241$	23.1246	1.48959	0.744794	−	0.667295i	$-0.232548\pi$
$241$	23.1246	1.48959	0.744794	+	0.667295i	$0.232548\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	12.7082	0.813559
$245$	−5.23607	−0.334520
$246$	7.38197	0.470657
$247$	10.3262	0.657043
$248$	−6.38197	−0.405255
$249$	−0.708204	−0.0448806
$250$	5.09017	0.321931
$251$	−7.79837	−0.492229	−0.246114	−	0.969241i	$-0.579154\pi$
$251$	−7.79837	−0.492229	−0.246114	+	0.969241i	$0.579154\pi$
$252$	4.85410	0.305780
$253$	0	0
$254$	2.29180	0.143800
$255$	4.23607	0.265273
$256$	−6.56231	−0.410144
$257$	−11.6738	−0.728189	−0.364095	−	0.931362i	$-0.618622\pi$
$257$	−11.6738	−0.728189	−0.364095	+	0.931362i	$0.618622\pi$
$258$	−3.85410	−0.239946
$259$	−0.708204	−0.0440057
$260$	−7.47214	−0.463402
$261$	−4.47214	−0.276818
$262$	4.41641	0.272847
$263$	16.3262	1.00672	0.503359	−	0.864077i	$-0.332097\pi$
$263$	16.3262	1.00672	0.503359	+	0.864077i	$0.332097\pi$
$264$	0	0
$265$	25.1803	1.54682
$266$	10.8541	0.665508
$267$	0.527864	0.0323048
$268$	15.4721	0.945111
$269$	15.5279	0.946751	0.473375	−	0.880861i	$-0.343035\pi$
$269$	15.5279	0.946751	0.473375	+	0.880861i	$0.343035\pi$
$270$	−1.61803	−0.0984704
$271$	27.2705	1.65657	0.828283	−	0.560310i	$-0.189318\pi$
$271$	27.2705	1.65657	0.828283	+	0.560310i	$0.189318\pi$
$272$	−3.00000	−0.181902
$273$	5.29180	0.320274
$274$	4.61803	0.278986
$275$	0	0
$276$	−5.61803	−0.338166
$277$	−30.5623	−1.83631	−0.918155	−	0.396220i	$-0.870322\pi$
$277$	−30.5623	−1.83631	−0.918155	+	0.396220i	$0.870322\pi$
$278$	0.527864	0.0316592
$279$	2.85410	0.170871
$280$	−17.5623	−1.04955
$281$	0.763932	0.0455724	0.0227862	−	0.999740i	$-0.492746\pi$
$281$	0.763932	0.0455724	0.0227862	+	0.999740i	$0.492746\pi$
$282$	1.00000	0.0595491
$283$	−0.180340	−0.0107201	−0.00536005	−	0.999986i	$-0.501706\pi$
$283$	−0.180340	−0.0107201	−0.00536005	+	0.999986i	$0.501706\pi$
$284$	9.00000	0.534052
$285$	15.3262	0.907848
$286$	0	0
$287$	−35.8328	−2.11514
$288$	5.61803	0.331046
$289$	−14.3820	−0.845998
$290$	7.23607	0.424917
$291$	−14.0344	−0.822714
$292$	−5.23607	−0.306418
$293$	−0.0557281	−0.00325567	−0.00162783	−	0.999999i	$-0.500518\pi$
$293$	−0.0557281	−0.00325567	−0.00162783	+	0.999999i	$0.500518\pi$
$294$	1.23607	0.0720889
$295$	−27.0344	−1.57401
$296$	−0.527864	−0.0306815
$297$	0	0
$298$	−9.27051	−0.537026
$299$	−6.12461	−0.354195
$300$	−3.00000	−0.173205
$301$	18.7082	1.07832
$302$	−1.23607	−0.0711277
$303$	3.00000	0.172345
$304$	−10.8541	−0.622525
$305$	20.5623	1.17739
$306$	−1.00000	−0.0571662
$307$	−0.562306	−0.0320925	−0.0160462	−	0.999871i	$-0.505108\pi$
$307$	−0.562306	−0.0320925	−0.0160462	+	0.999871i	$0.505108\pi$
$308$	0	0
$309$	−6.00000	−0.341328
$310$	−4.61803	−0.262287
$311$	2.52786	0.143342	0.0716710	−	0.997428i	$-0.477167\pi$
$311$	2.52786	0.143342	0.0716710	+	0.997428i	$0.477167\pi$
$312$	3.94427	0.223300
$313$	25.8328	1.46016	0.730079	−	0.683363i	$-0.239483\pi$
$313$	25.8328	1.46016	0.730079	+	0.683363i	$0.239483\pi$
$314$	−2.29180	−0.129334
$315$	7.85410	0.442529
$316$	15.3262	0.862168
$317$	−19.3607	−1.08740	−0.543702	−	0.839278i	$-0.682978\pi$
$317$	−19.3607	−1.08740	−0.543702	+	0.839278i	$0.682978\pi$
$318$	−5.94427	−0.333338
$319$	0	0
$320$	0.618034	0.0345492
$321$	4.23607	0.236434
$322$	−6.43769	−0.358759
$323$	9.47214	0.527044
$324$	−1.61803	−0.0898908
$325$	−3.27051	−0.181415
$326$	11.2918	0.625395
$327$	0	0
$328$	−26.7082	−1.47471
$329$	−4.85410	−0.267615
$330$	0	0
$331$	26.5967	1.46189	0.730945	−	0.682437i	$-0.239080\pi$
$331$	26.5967	1.46189	0.730945	+	0.682437i	$0.239080\pi$
$332$	1.14590	0.0628893
$333$	0.236068	0.0129364
$334$	−6.20163	−0.339338
$335$	25.0344	1.36778
$336$	−5.56231	−0.303449
$337$	0.291796	0.0158951	0.00794757	−	0.999968i	$-0.497470\pi$
$337$	0.291796	0.0158951	0.00794757	+	0.999968i	$0.497470\pi$
$338$	−6.11146	−0.332419
$339$	0.708204	0.0384644
$340$	−6.85410	−0.371716
$341$	0	0
$342$	−3.61803	−0.195641
$343$	15.0000	0.809924
$344$	13.9443	0.751825
$345$	−9.09017	−0.489398
$346$	−9.50658	−0.511077
$347$	20.9443	1.12435	0.562174	−	0.827019i	$-0.309965\pi$
$347$	20.9443	1.12435	0.562174	+	0.827019i	$0.309965\pi$
$348$	7.23607	0.387894
$349$	10.1246	0.541958	0.270979	−	0.962585i	$-0.412653\pi$
$349$	10.1246	0.541958	0.270979	+	0.962585i	$0.412653\pi$
$350$	−3.43769	−0.183752
$351$	−1.76393	−0.0941517
$352$	0	0
$353$	−10.4721	−0.557376	−0.278688	−	0.960382i	$-0.589899\pi$
$353$	−10.4721	−0.557376	−0.278688	+	0.960382i	$0.589899\pi$
$354$	6.38197	0.339198
$355$	14.5623	0.772887
$356$	−0.854102	−0.0452673
$357$	4.85410	0.256906
$358$	−1.38197	−0.0730392
$359$	12.7639	0.673655	0.336827	−	0.941566i	$-0.390646\pi$
$359$	12.7639	0.673655	0.336827	+	0.941566i	$0.390646\pi$
$360$	5.85410	0.308538
$361$	15.2705	0.803711
$362$	−10.7984	−0.567550
$363$	0	0
$364$	−8.56231	−0.448787
$365$	−8.47214	−0.443452
$366$	−4.85410	−0.253728
$367$	5.56231	0.290350	0.145175	−	0.989406i	$-0.453625\pi$
$367$	5.56231	0.290350	0.145175	+	0.989406i	$0.453625\pi$
$368$	6.43769	0.335588
$369$	11.9443	0.621794
$370$	−0.381966	−0.0198575
$371$	28.8541	1.49803
$372$	−4.61803	−0.239434
$373$	4.41641	0.228673	0.114336	−	0.993442i	$-0.463526\pi$
$373$	4.41641	0.228673	0.114336	+	0.993442i	$0.463526\pi$
$374$	0	0
$375$	8.23607	0.425309
$376$	−3.61803	−0.186586
$377$	7.88854	0.406281
$378$	−1.85410	−0.0953647
$379$	1.58359	0.0813437	0.0406718	−	0.999173i	$-0.487050\pi$
$379$	1.58359	0.0813437	0.0406718	+	0.999173i	$0.487050\pi$
$380$	−24.7984	−1.27213
$381$	3.70820	0.189977
$382$	−4.61803	−0.236279
$383$	26.8885	1.37394	0.686970	−	0.726686i	$-0.258940\pi$
$383$	26.8885	1.37394	0.686970	+	0.726686i	$0.258940\pi$
$384$	−11.3820	−0.580834
$385$	0	0
$386$	11.4721	0.583916
$387$	−6.23607	−0.316997
$388$	22.7082	1.15283
$389$	24.2705	1.23056	0.615282	−	0.788307i	$-0.289042\pi$
$389$	24.2705	1.23056	0.615282	+	0.788307i	$0.289042\pi$
$390$	2.85410	0.144523
$391$	−5.61803	−0.284116
$392$	−4.47214	−0.225877
$393$	7.14590	0.360463
$394$	−15.0689	−0.759159
$395$	24.7984	1.24774
$396$	0	0
$397$	−38.7082	−1.94271	−0.971355	−	0.237635i	$-0.923628\pi$
$397$	−38.7082	−1.94271	−0.971355	+	0.237635i	$0.923628\pi$
$398$	−10.3262	−0.517608
$399$	17.5623	0.879215
$400$	3.43769	0.171885
$401$	−26.0902	−1.30288	−0.651440	−	0.758700i	$-0.725835\pi$
$401$	−26.0902	−1.30288	−0.651440	+	0.758700i	$0.725835\pi$
$402$	−5.90983	−0.294756
$403$	−5.03444	−0.250783
$404$	−4.85410	−0.241501
$405$	−2.61803	−0.130091
$406$	8.29180	0.411515
$407$	0	0
$408$	3.61803	0.179119
$409$	11.0557	0.546671	0.273335	−	0.961919i	$-0.411873\pi$
$409$	11.0557	0.546671	0.273335	+	0.961919i	$0.411873\pi$
$410$	−19.3262	−0.954455
$411$	7.47214	0.368573
$412$	9.70820	0.478289
$413$	−30.9787	−1.52436
$414$	2.14590	0.105465
$415$	1.85410	0.0910143
$416$	−9.90983	−0.485869
$417$	0.854102	0.0418256
$418$	0	0
$419$	16.5066	0.806399	0.403200	−	0.915112i	$-0.367898\pi$
$419$	16.5066	0.806399	0.403200	+	0.915112i	$0.367898\pi$
$420$	−12.7082	−0.620097
$421$	−37.2705	−1.81645	−0.908227	−	0.418478i	$-0.862564\pi$
$421$	−37.2705	−1.81645	−0.908227	+	0.418478i	$0.862564\pi$
$422$	13.7639	0.670018
$423$	1.61803	0.0786715
$424$	21.5066	1.04445
$425$	−3.00000	−0.145521
$426$	−3.43769	−0.166557
$427$	23.5623	1.14026
$428$	−6.85410	−0.331306
$429$	0	0
$430$	10.0902	0.486591
$431$	39.5066	1.90296	0.951482	−	0.307703i	$-0.0995603\pi$
$431$	39.5066	1.90296	0.951482	+	0.307703i	$0.0995603\pi$
$432$	1.85410	0.0892055
$433$	−6.00000	−0.288342	−0.144171	−	0.989553i	$-0.546051\pi$
$433$	−6.00000	−0.288342	−0.144171	+	0.989553i	$0.546051\pi$
$434$	−5.29180	−0.254014
$435$	11.7082	0.561365
$436$	0	0
$437$	−20.3262	−0.972336
$438$	2.00000	0.0955637
$439$	−3.29180	−0.157109	−0.0785544	−	0.996910i	$-0.525030\pi$
$439$	−3.29180	−0.157109	−0.0785544	+	0.996910i	$0.525030\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	1.76393	0.0839017
$443$	−41.1246	−1.95389	−0.976945	−	0.213493i	$-0.931516\pi$
$443$	−41.1246	−1.95389	−0.976945	+	0.213493i	$0.931516\pi$
$444$	−0.381966	−0.0181273
$445$	−1.38197	−0.0655115
$446$	0.437694	0.0207254
$447$	−15.0000	−0.709476
$448$	0.708204	0.0334595
$449$	−24.4721	−1.15491	−0.577456	−	0.816422i	$-0.695954\pi$
$449$	−24.4721	−1.15491	−0.577456	+	0.816422i	$0.695954\pi$
$450$	1.14590	0.0540182
$451$	0	0
$452$	−1.14590	−0.0538985
$453$	−2.00000	−0.0939682
$454$	15.3820	0.721911
$455$	−13.8541	−0.649490
$456$	13.0902	0.613003
$457$	−8.20163	−0.383656	−0.191828	−	0.981429i	$-0.561442\pi$
$457$	−8.20163	−0.383656	−0.191828	+	0.981429i	$0.561442\pi$
$458$	6.18034	0.288788
$459$	−1.61803	−0.0755234
$460$	14.7082	0.685774
$461$	21.0902	0.982267	0.491134	−	0.871084i	$-0.336583\pi$
$461$	21.0902	0.982267	0.491134	+	0.871084i	$0.336583\pi$
$462$	0	0
$463$	−15.7984	−0.734213	−0.367106	−	0.930179i	$-0.619651\pi$
$463$	−15.7984	−0.734213	−0.367106	+	0.930179i	$0.619651\pi$
$464$	−8.29180	−0.384937
$465$	−7.47214	−0.346512
$466$	−15.0344	−0.696457
$467$	−9.76393	−0.451821	−0.225910	−	0.974148i	$-0.572536\pi$
$467$	−9.76393	−0.451821	−0.225910	+	0.974148i	$0.572536\pi$
$468$	2.85410	0.131931
$469$	28.6869	1.32464
$470$	−2.61803	−0.120761
$471$	−3.70820	−0.170865
$472$	−23.0902	−1.06281
$473$	0	0
$474$	−5.85410	−0.268888
$475$	−10.8541	−0.498020
$476$	−7.85410	−0.359992
$477$	−9.61803	−0.440380
$478$	−1.58359	−0.0724318
$479$	28.0902	1.28347	0.641736	−	0.766925i	$-0.278214\pi$
$479$	28.0902	1.28347	0.641736	+	0.766925i	$0.278214\pi$
$480$	−14.7082	−0.671335
$481$	−0.416408	−0.0189866
$482$	14.2918	0.650973
$483$	−10.4164	−0.473963
$484$	0	0
$485$	36.7426	1.66840
$486$	0.618034	0.0280346
$487$	16.8197	0.762172	0.381086	−	0.924540i	$-0.375550\pi$
$487$	16.8197	0.762172	0.381086	+	0.924540i	$0.375550\pi$
$488$	17.5623	0.795008
$489$	18.2705	0.826221
$490$	−3.23607	−0.146191
$491$	−25.2148	−1.13793	−0.568964	−	0.822363i	$-0.692655\pi$
$491$	−25.2148	−1.13793	−0.568964	+	0.822363i	$0.692655\pi$
$492$	−19.3262	−0.871294
$493$	7.23607	0.325896
$494$	6.38197	0.287138
$495$	0	0
$496$	5.29180	0.237609
$497$	16.6869	0.748511
$498$	−0.437694	−0.0196135
$499$	−17.5623	−0.786197	−0.393098	−	0.919496i	$-0.628597\pi$
$499$	−17.5623	−0.786197	−0.393098	+	0.919496i	$0.628597\pi$
$500$	−13.3262	−0.595967
$501$	−10.0344	−0.448306
$502$	−4.81966	−0.215112
$503$	−28.0689	−1.25153	−0.625765	−	0.780012i	$-0.715213\pi$
$503$	−28.0689	−1.25153	−0.625765	+	0.780012i	$0.715213\pi$
$504$	6.70820	0.298807
$505$	−7.85410	−0.349503
$506$	0	0
$507$	−9.88854	−0.439166
$508$	−6.00000	−0.266207
$509$	23.6180	1.04685	0.523425	−	0.852071i	$-0.324654\pi$
$509$	23.6180	1.04685	0.523425	+	0.852071i	$0.324654\pi$
$510$	2.61803	0.115928
$511$	−9.70820	−0.429466
$512$	18.7082	0.826794
$513$	−5.85410	−0.258465
$514$	−7.21478	−0.318230
$515$	15.7082	0.692186
$516$	10.0902	0.444195
$517$	0	0
$518$	−0.437694	−0.0192312
$519$	−15.3820	−0.675193
$520$	−10.3262	−0.452835
$521$	−14.8328	−0.649837	−0.324919	−	0.945742i	$-0.605337\pi$
$521$	−14.8328	−0.649837	−0.324919	+	0.945742i	$0.605337\pi$
$522$	−2.76393	−0.120974
$523$	11.9787	0.523793	0.261896	−	0.965096i	$-0.415652\pi$
$523$	11.9787	0.523793	0.261896	+	0.965096i	$0.415652\pi$
$524$	−11.5623	−0.505102
$525$	−5.56231	−0.242759
$526$	10.0902	0.439952
$527$	−4.61803	−0.201165
$528$	0	0
$529$	−10.9443	−0.475838
$530$	15.5623	0.675983
$531$	10.3262	0.448121
$532$	−28.4164	−1.23201
$533$	−21.0689	−0.912595
$534$	0.326238	0.0141177
$535$	−11.0902	−0.479470
$536$	21.3820	0.923560
$537$	−2.23607	−0.0964935
$538$	9.59675	0.413745
$539$	0	0
$540$	4.23607	0.182291
$541$	−19.5623	−0.841049	−0.420525	−	0.907281i	$-0.638154\pi$
$541$	−19.5623	−0.841049	−0.420525	+	0.907281i	$0.638154\pi$
$542$	16.8541	0.723946
$543$	−17.4721	−0.749801
$544$	−9.09017	−0.389738
$545$	0	0
$546$	3.27051	0.139965
$547$	−21.6180	−0.924320	−0.462160	−	0.886796i	$-0.652925\pi$
$547$	−21.6180	−0.924320	−0.462160	+	0.886796i	$0.652925\pi$
$548$	−12.0902	−0.516466
$549$	−7.85410	−0.335205
$550$	0	0
$551$	26.1803	1.11532
$552$	−7.76393	−0.330455
$553$	28.4164	1.20839
$554$	−18.8885	−0.802497
$555$	−0.618034	−0.0262341
$556$	−1.38197	−0.0586084
$557$	−14.9098	−0.631750	−0.315875	−	0.948801i	$-0.602298\pi$
$557$	−14.9098	−0.631750	−0.315875	+	0.948801i	$0.602298\pi$
$558$	1.76393	0.0746732
$559$	11.0000	0.465250
$560$	14.5623	0.615370
$561$	0	0
$562$	0.472136	0.0199159
$563$	8.88854	0.374607	0.187304	−	0.982302i	$-0.440025\pi$
$563$	8.88854	0.374607	0.187304	+	0.982302i	$0.440025\pi$
$564$	−2.61803	−0.110239
$565$	−1.85410	−0.0780027
$566$	−0.111456	−0.00468485
$567$	−3.00000	−0.125988
$568$	12.4377	0.521874
$569$	24.0689	1.00902	0.504510	−	0.863406i	$-0.331673\pi$
$569$	24.0689	1.00902	0.504510	+	0.863406i	$0.331673\pi$
$570$	9.47214	0.396744
$571$	−34.6869	−1.45160	−0.725801	−	0.687905i	$-0.758531\pi$
$571$	−34.6869	−1.45160	−0.725801	+	0.687905i	$0.758531\pi$
$572$	0	0
$573$	−7.47214	−0.312153
$574$	−22.1459	−0.924352
$575$	6.43769	0.268470
$576$	−0.236068	−0.00983617
$577$	10.7639	0.448108	0.224054	−	0.974577i	$-0.428071\pi$
$577$	10.7639	0.448108	0.224054	+	0.974577i	$0.428071\pi$
$578$	−8.88854	−0.369715
$579$	18.5623	0.771423
$580$	−18.9443	−0.786618
$581$	2.12461	0.0881437
$582$	−8.67376	−0.359539
$583$	0	0
$584$	−7.23607	−0.299431
$585$	4.61803	0.190932
$586$	−0.0344419	−0.00142278
$587$	−38.3050	−1.58101	−0.790507	−	0.612453i	$-0.790183\pi$
$587$	−38.3050	−1.58101	−0.790507	+	0.612453i	$0.790183\pi$
$588$	−3.23607	−0.133453
$589$	−16.7082	−0.688450
$590$	−16.7082	−0.687866
$591$	−24.3820	−1.00294
$592$	0.437694	0.0179891
$593$	−22.2148	−0.912252	−0.456126	−	0.889915i	$-0.650763\pi$
$593$	−22.2148	−0.912252	−0.456126	+	0.889915i	$0.650763\pi$
$594$	0	0
$595$	−12.7082	−0.520986
$596$	24.2705	0.994159
$597$	−16.7082	−0.683821
$598$	−3.78522	−0.154789
$599$	−8.29180	−0.338794	−0.169397	−	0.985548i	$-0.554182\pi$
$599$	−8.29180	−0.338794	−0.169397	+	0.985548i	$0.554182\pi$
$600$	−4.14590	−0.169256
$601$	−33.8328	−1.38007	−0.690035	−	0.723776i	$-0.742405\pi$
$601$	−33.8328	−1.38007	−0.690035	+	0.723776i	$0.742405\pi$
$602$	11.5623	0.471244
$603$	−9.56231	−0.389407
$604$	3.23607	0.131674
$605$	0	0
$606$	1.85410	0.0753177
$607$	−13.3262	−0.540895	−0.270448	−	0.962735i	$-0.587172\pi$
$607$	−13.3262	−0.540895	−0.270448	+	0.962735i	$0.587172\pi$
$608$	−32.8885	−1.33381
$609$	13.4164	0.543660
$610$	12.7082	0.514540
$611$	−2.85410	−0.115465
$612$	2.61803	0.105828
$613$	−34.6525	−1.39960	−0.699800	−	0.714339i	$-0.746728\pi$
$613$	−34.6525	−1.39960	−0.699800	+	0.714339i	$0.746728\pi$
$614$	−0.347524	−0.0140249
$615$	−31.2705	−1.26095
$616$	0	0
$617$	19.5836	0.788406	0.394203	−	0.919023i	$-0.371021\pi$
$617$	19.5836	0.788406	0.394203	+	0.919023i	$0.371021\pi$
$618$	−3.70820	−0.149166
$619$	−8.81966	−0.354492	−0.177246	−	0.984167i	$-0.556719\pi$
$619$	−8.81966	−0.354492	−0.177246	+	0.984167i	$0.556719\pi$
$620$	12.0902	0.485553
$621$	3.47214	0.139332
$622$	1.56231	0.0626428
$623$	−1.58359	−0.0634453
$624$	−3.27051	−0.130925
$625$	−30.8328	−1.23331
$626$	15.9656	0.638112
$627$	0	0
$628$	6.00000	0.239426
$629$	−0.381966	−0.0152300
$630$	4.85410	0.193392
$631$	46.2705	1.84200	0.921000	−	0.389563i	$-0.127374\pi$
$631$	46.2705	1.84200	0.921000	+	0.389563i	$0.127374\pi$
$632$	21.1803	0.842509
$633$	22.2705	0.885173
$634$	−11.9656	−0.475213
$635$	−9.70820	−0.385258
$636$	15.5623	0.617086
$637$	−3.52786	−0.139779
$638$	0	0
$639$	−5.56231	−0.220041
$640$	29.7984	1.17788
$641$	35.7426	1.41175	0.705875	−	0.708337i	$-0.250554\pi$
$641$	35.7426	1.41175	0.705875	+	0.708337i	$0.250554\pi$
$642$	2.61803	0.103326
$643$	−38.5623	−1.52075	−0.760374	−	0.649485i	$-0.774985\pi$
$643$	−38.5623	−1.52075	−0.760374	+	0.649485i	$0.774985\pi$
$644$	16.8541	0.664145
$645$	16.3262	0.642845
$646$	5.85410	0.230327
$647$	27.7984	1.09287	0.546433	−	0.837503i	$-0.315985\pi$
$647$	27.7984	1.09287	0.546433	+	0.837503i	$0.315985\pi$
$648$	−2.23607	−0.0878410
$649$	0	0
$650$	−2.02129	−0.0792814
$651$	−8.56231	−0.335583
$652$	−29.5623	−1.15775
$653$	51.0344	1.99713	0.998566	−	0.0535342i	$-0.0170486\pi$
$653$	51.0344	1.99713	0.998566	+	0.0535342i	$0.0170486\pi$
$654$	0	0
$655$	−18.7082	−0.730990
$656$	22.1459	0.864652
$657$	3.23607	0.126251
$658$	−3.00000	−0.116952
$659$	10.6525	0.414962	0.207481	−	0.978239i	$-0.433474\pi$
$659$	10.6525	0.414962	0.207481	+	0.978239i	$0.433474\pi$
$660$	0	0
$661$	−9.90983	−0.385448	−0.192724	−	0.981253i	$-0.561732\pi$
$661$	−9.90983	−0.385448	−0.192724	+	0.981253i	$0.561732\pi$
$662$	16.4377	0.638869
$663$	2.85410	0.110844
$664$	1.58359	0.0614553
$665$	−45.9787	−1.78298
$666$	0.145898	0.00565343
$667$	−15.5279	−0.601241
$668$	16.2361	0.628192
$669$	0.708204	0.0273807
$670$	15.4721	0.597741
$671$	0	0
$672$	−16.8541	−0.650161
$673$	−12.4164	−0.478617	−0.239309	−	0.970944i	$-0.576921\pi$
$673$	−12.4164	−0.478617	−0.239309	+	0.970944i	$0.576921\pi$
$674$	0.180340	0.00694643
$675$	1.85410	0.0713644
$676$	16.0000	0.615385
$677$	−13.5279	−0.519918	−0.259959	−	0.965620i	$-0.583709\pi$
$677$	−13.5279	−0.519918	−0.259959	+	0.965620i	$0.583709\pi$
$678$	0.437694	0.0168095
$679$	42.1033	1.61578
$680$	−9.47214	−0.363240
$681$	24.8885	0.953731
$682$	0	0
$683$	−3.11146	−0.119057	−0.0595283	−	0.998227i	$-0.518960\pi$
$683$	−3.11146	−0.119057	−0.0595283	+	0.998227i	$0.518960\pi$
$684$	9.47214	0.362176
$685$	−19.5623	−0.747437
$686$	9.27051	0.353950
$687$	10.0000	0.381524
$688$	−11.5623	−0.440809
$689$	16.9656	0.646336
$690$	−5.61803	−0.213875
$691$	−26.2918	−1.00019	−0.500094	−	0.865971i	$-0.666701\pi$
$691$	−26.2918	−1.00019	−0.500094	+	0.865971i	$0.666701\pi$
$692$	24.8885	0.946120
$693$	0	0
$694$	12.9443	0.491358
$695$	−2.23607	−0.0848189
$696$	10.0000	0.379049
$697$	−19.3262	−0.732033
$698$	6.25735	0.236844
$699$	−24.3262	−0.920103
$700$	9.00000	0.340168
$701$	10.6869	0.403639	0.201820	−	0.979423i	$-0.435315\pi$
$701$	10.6869	0.403639	0.201820	+	0.979423i	$0.435315\pi$
$702$	−1.09017	−0.0411458
$703$	−1.38197	−0.0521218
$704$	0	0
$705$	−4.23607	−0.159540
$706$	−6.47214	−0.243582
$707$	−9.00000	−0.338480
$708$	−16.7082	−0.627933
$709$	48.7426	1.83057	0.915284	−	0.402809i	$-0.131966\pi$
$709$	48.7426	1.83057	0.915284	+	0.402809i	$0.131966\pi$
$710$	9.00000	0.337764
$711$	−9.47214	−0.355233
$712$	−1.18034	−0.0442351
$713$	9.90983	0.371126
$714$	3.00000	0.112272
$715$	0	0
$716$	3.61803	0.135212
$717$	−2.56231	−0.0956911
$718$	7.88854	0.294398
$719$	−1.58359	−0.0590580	−0.0295290	−	0.999564i	$-0.509401\pi$
$719$	−1.58359	−0.0590580	−0.0295290	+	0.999564i	$0.509401\pi$
$720$	−4.85410	−0.180902
$721$	18.0000	0.670355
$722$	9.43769	0.351235
$723$	23.1246	0.860014
$724$	28.2705	1.05067
$725$	−8.29180	−0.307950
$726$	0	0
$727$	38.8541	1.44102	0.720509	−	0.693445i	$-0.243908\pi$
$727$	38.8541	1.44102	0.720509	+	0.693445i	$0.243908\pi$
$728$	−11.8328	−0.438553
$729$	1.00000	0.0370370
$730$	−5.23607	−0.193796
$731$	10.0902	0.373198
$732$	12.7082	0.469709
$733$	37.7082	1.39278	0.696392	−	0.717661i	$-0.254787\pi$
$733$	37.7082	1.39278	0.696392	+	0.717661i	$0.254787\pi$
$734$	3.43769	0.126888
$735$	−5.23607	−0.193135
$736$	19.5066	0.719022
$737$	0	0
$738$	7.38197	0.271734
$739$	25.0000	0.919640	0.459820	−	0.888012i	$-0.347914\pi$
$739$	25.0000	0.919640	0.459820	+	0.888012i	$0.347914\pi$
$740$	1.00000	0.0367607
$741$	10.3262	0.379344
$742$	17.8328	0.654663
$743$	−35.1803	−1.29064	−0.645321	−	0.763912i	$-0.723276\pi$
$743$	−35.1803	−1.29064	−0.645321	+	0.763912i	$0.723276\pi$
$744$	−6.38197	−0.233974
$745$	39.2705	1.43876
$746$	2.72949	0.0999337
$747$	−0.708204	−0.0259118
$748$	0	0
$749$	−12.7082	−0.464348
$750$	5.09017	0.185867
$751$	11.9230	0.435076	0.217538	−	0.976052i	$-0.430197\pi$
$751$	11.9230	0.435076	0.217538	+	0.976052i	$0.430197\pi$
$752$	3.00000	0.109399
$753$	−7.79837	−0.284189
$754$	4.87539	0.177551
$755$	5.23607	0.190560
$756$	4.85410	0.176542
$757$	1.94427	0.0706658	0.0353329	−	0.999376i	$-0.488751\pi$
$757$	1.94427	0.0706658	0.0353329	+	0.999376i	$0.488751\pi$
$758$	0.978714	0.0355485
$759$	0	0
$760$	−34.2705	−1.24312
$761$	30.8885	1.11971	0.559854	−	0.828591i	$-0.310857\pi$
$761$	30.8885	1.11971	0.559854	+	0.828591i	$0.310857\pi$
$762$	2.29180	0.0830230
$763$	0	0
$764$	12.0902	0.437407
$765$	4.23607	0.153155
$766$	16.6180	0.600434
$767$	−18.2148	−0.657698
$768$	−6.56231	−0.236797
$769$	−12.6869	−0.457502	−0.228751	−	0.973485i	$-0.573464\pi$
$769$	−12.6869	−0.457502	−0.228751	+	0.973485i	$0.573464\pi$
$770$	0	0
$771$	−11.6738	−0.420420
$772$	−30.0344	−1.08096
$773$	−31.2492	−1.12396	−0.561978	−	0.827152i	$-0.689960\pi$
$773$	−31.2492	−1.12396	−0.561978	+	0.827152i	$0.689960\pi$
$774$	−3.85410	−0.138533
$775$	5.29180	0.190087
$776$	31.3820	1.12655
$777$	−0.708204	−0.0254067
$778$	15.0000	0.537776
$779$	−69.9230	−2.50525
$780$	−7.47214	−0.267545
$781$	0	0
$782$	−3.47214	−0.124163
$783$	−4.47214	−0.159821
$784$	3.70820	0.132436
$785$	9.70820	0.346501
$786$	4.41641	0.157528
$787$	10.2918	0.366863	0.183431	−	0.983033i	$-0.441279\pi$
$787$	10.2918	0.366863	0.183431	+	0.983033i	$0.441279\pi$
$788$	39.4508	1.40538
$789$	16.3262	0.581229
$790$	15.3262	0.545283
$791$	−2.12461	−0.0755425
$792$	0	0
$793$	13.8541	0.491974
$794$	−23.9230	−0.848995
$795$	25.1803	0.893055
$796$	27.0344	0.958210
$797$	10.7639	0.381278	0.190639	−	0.981660i	$-0.438944\pi$
$797$	10.7639	0.381278	0.190639	+	0.981660i	$0.438944\pi$
$798$	10.8541	0.384231
$799$	−2.61803	−0.0926194
$800$	10.4164	0.368276
$801$	0.527864	0.0186512
$802$	−16.1246	−0.569380
$803$	0	0
$804$	15.4721	0.545660
$805$	27.2705	0.961159
$806$	−3.11146	−0.109596
$807$	15.5279	0.546607
$808$	−6.70820	−0.235994
$809$	−9.67376	−0.340111	−0.170056	−	0.985434i	$-0.554395\pi$
$809$	−9.67376	−0.340111	−0.170056	+	0.985434i	$0.554395\pi$
$810$	−1.61803	−0.0568519
$811$	−3.25735	−0.114381	−0.0571906	−	0.998363i	$-0.518214\pi$
$811$	−3.25735	−0.114381	−0.0571906	+	0.998363i	$0.518214\pi$
$812$	−21.7082	−0.761809
$813$	27.2705	0.956419
$814$	0	0
$815$	−47.8328	−1.67551
$816$	−3.00000	−0.105021
$817$	36.5066	1.27720
$818$	6.83282	0.238904
$819$	5.29180	0.184910
$820$	50.5967	1.76692
$821$	−40.4164	−1.41054	−0.705271	−	0.708938i	$-0.749175\pi$
$821$	−40.4164	−1.41054	−0.705271	+	0.708938i	$0.749175\pi$
$822$	4.61803	0.161072
$823$	−35.4721	−1.23648	−0.618240	−	0.785989i	$-0.712154\pi$
$823$	−35.4721	−1.23648	−0.618240	+	0.785989i	$0.712154\pi$
$824$	13.4164	0.467383
$825$	0	0
$826$	−19.1459	−0.666171
$827$	−53.1246	−1.84732	−0.923662	−	0.383208i	$-0.874819\pi$
$827$	−53.1246	−1.84732	−0.923662	+	0.383208i	$0.874819\pi$
$828$	−5.61803	−0.195240
$829$	17.6869	0.614292	0.307146	−	0.951662i	$-0.400626\pi$
$829$	17.6869	0.614292	0.307146	+	0.951662i	$0.400626\pi$
$830$	1.14590	0.0397747
$831$	−30.5623	−1.06019
$832$	0.416408	0.0144363
$833$	−3.23607	−0.112123
$834$	0.527864	0.0182784
$835$	26.2705	0.909128
$836$	0	0
$837$	2.85410	0.0986522
$838$	10.2016	0.352409
$839$	−36.7082	−1.26731	−0.633654	−	0.773617i	$-0.718446\pi$
$839$	−36.7082	−1.26731	−0.633654	+	0.773617i	$0.718446\pi$
$840$	−17.5623	−0.605957
$841$	−9.00000	−0.310345
$842$	−23.0344	−0.793819
$843$	0.763932	0.0263112
$844$	−36.0344	−1.24036
$845$	25.8885	0.890593
$846$	1.00000	0.0343807
$847$	0	0
$848$	−17.8328	−0.612381
$849$	−0.180340	−0.00618925
$850$	−1.85410	−0.0635952
$851$	0.819660	0.0280976
$852$	9.00000	0.308335
$853$	9.94427	0.340485	0.170243	−	0.985402i	$-0.445545\pi$
$853$	9.94427	0.340485	0.170243	+	0.985402i	$0.445545\pi$
$854$	14.5623	0.498312
$855$	15.3262	0.524146
$856$	−9.47214	−0.323751
$857$	−47.7214	−1.63013	−0.815065	−	0.579369i	$-0.803299\pi$
$857$	−47.7214	−1.63013	−0.815065	+	0.579369i	$0.803299\pi$
$858$	0	0
$859$	7.11146	0.242640	0.121320	−	0.992613i	$-0.461287\pi$
$859$	7.11146	0.242640	0.121320	+	0.992613i	$0.461287\pi$
$860$	−26.4164	−0.900792
$861$	−35.8328	−1.22118
$862$	24.4164	0.831626
$863$	11.8885	0.404691	0.202345	−	0.979314i	$-0.435144\pi$
$863$	11.8885	0.404691	0.202345	+	0.979314i	$0.435144\pi$
$864$	5.61803	0.191129
$865$	40.2705	1.36924
$866$	−3.70820	−0.126010
$867$	−14.3820	−0.488437
$868$	13.8541	0.470239
$869$	0	0
$870$	7.23607	0.245326
$871$	16.8673	0.571525
$872$	0	0
$873$	−14.0344	−0.474994
$874$	−12.5623	−0.424926
$875$	−24.7082	−0.835290
$876$	−5.23607	−0.176910
$877$	−6.41641	−0.216667	−0.108333	−	0.994115i	$-0.534551\pi$
$877$	−6.41641	−0.216667	−0.108333	+	0.994115i	$0.534551\pi$
$878$	−2.03444	−0.0686591
$879$	−0.0557281	−0.00187966
$880$	0	0
$881$	13.9098	0.468634	0.234317	−	0.972160i	$-0.424715\pi$
$881$	13.9098	0.468634	0.234317	+	0.972160i	$0.424715\pi$
$882$	1.23607	0.0416206
$883$	10.5836	0.356166	0.178083	−	0.984015i	$-0.443010\pi$
$883$	10.5836	0.356166	0.178083	+	0.984015i	$0.443010\pi$
$884$	−4.61803	−0.155321
$885$	−27.0344	−0.908753
$886$	−25.4164	−0.853881
$887$	−3.00000	−0.100730	−0.0503651	−	0.998731i	$-0.516038\pi$
$887$	−3.00000	−0.100730	−0.0503651	+	0.998731i	$0.516038\pi$
$888$	−0.527864	−0.0177140
$889$	−11.1246	−0.373108
$890$	−0.854102	−0.0286296
$891$	0	0
$892$	−1.14590	−0.0383675
$893$	−9.47214	−0.316973
$894$	−9.27051	−0.310052
$895$	5.85410	0.195681
$896$	34.1459	1.14073
$897$	−6.12461	−0.204495
$898$	−15.1246	−0.504715
$899$	−12.7639	−0.425701
$900$	−3.00000	−0.100000
$901$	15.5623	0.518456
$902$	0	0
$903$	18.7082	0.622570
$904$	−1.58359	−0.0526695
$905$	45.7426	1.52054
$906$	−1.23607	−0.0410656
$907$	−42.9787	−1.42708	−0.713542	−	0.700612i	$-0.752910\pi$
$907$	−42.9787	−1.42708	−0.713542	+	0.700612i	$0.752910\pi$
$908$	−40.2705	−1.33642
$909$	3.00000	0.0995037
$910$	−8.56231	−0.283838
$911$	18.0557	0.598213	0.299106	−	0.954220i	$-0.403311\pi$
$911$	18.0557	0.598213	0.299106	+	0.954220i	$0.403311\pi$
$912$	−10.8541	−0.359415
$913$	0	0
$914$	−5.06888	−0.167664
$915$	20.5623	0.679769
$916$	−16.1803	−0.534613
$917$	−21.4377	−0.707935
$918$	−1.00000	−0.0330049
$919$	−46.9574	−1.54898	−0.774491	−	0.632585i	$-0.781994\pi$
$919$	−46.9574	−1.54898	−0.774491	+	0.632585i	$0.781994\pi$
$920$	20.3262	0.670136
$921$	−0.562306	−0.0185286
$922$	13.0344	0.429266
$923$	9.81153	0.322950
$924$	0	0
$925$	0.437694	0.0143913
$926$	−9.76393	−0.320863
$927$	−6.00000	−0.197066
$928$	−25.1246	−0.824756
$929$	−32.8885	−1.07904	−0.539519	−	0.841973i	$-0.681394\pi$
$929$	−32.8885	−1.07904	−0.539519	+	0.841973i	$0.681394\pi$
$930$	−4.61803	−0.151431
$931$	−11.7082	−0.383721
$932$	39.3607	1.28930
$933$	2.52786	0.0827586
$934$	−6.03444	−0.197453
$935$	0	0
$936$	3.94427	0.128923
$937$	16.5967	0.542192	0.271096	−	0.962552i	$-0.412614\pi$
$937$	16.5967	0.542192	0.271096	+	0.962552i	$0.412614\pi$
$938$	17.7295	0.578888
$939$	25.8328	0.843022
$940$	6.85410	0.223556
$941$	44.6312	1.45494	0.727468	−	0.686142i	$-0.240697\pi$
$941$	44.6312	1.45494	0.727468	+	0.686142i	$0.240697\pi$
$942$	−2.29180	−0.0746708
$943$	41.4721	1.35052
$944$	19.1459	0.623146
$945$	7.85410	0.255494
$946$	0	0
$947$	18.3262	0.595523	0.297761	−	0.954640i	$-0.403760\pi$
$947$	18.3262	0.595523	0.297761	+	0.954640i	$0.403760\pi$
$948$	15.3262	0.497773
$949$	−5.70820	−0.185296
$950$	−6.70820	−0.217643
$951$	−19.3607	−0.627813
$952$	−10.8541	−0.351783
$953$	−37.8197	−1.22510	−0.612549	−	0.790432i	$-0.709856\pi$
$953$	−37.8197	−1.22510	−0.612549	+	0.790432i	$0.709856\pi$
$954$	−5.94427	−0.192453
$955$	19.5623	0.633021
$956$	4.14590	0.134088
$957$	0	0
$958$	17.3607	0.560898
$959$	−22.4164	−0.723864
$960$	0.618034	0.0199470
$961$	−22.8541	−0.737229
$962$	−0.257354	−0.00829743
$963$	4.23607	0.136505
$964$	−37.4164	−1.20510
$965$	−48.5967	−1.56438
$966$	−6.43769	−0.207129
$967$	34.6869	1.11546	0.557728	−	0.830024i	$-0.311673\pi$
$967$	34.6869	1.11546	0.557728	+	0.830024i	$0.311673\pi$
$968$	0	0
$969$	9.47214	0.304289
$970$	22.7082	0.729116
$971$	37.7771	1.21232	0.606162	−	0.795341i	$-0.292708\pi$
$971$	37.7771	1.21232	0.606162	+	0.795341i	$0.292708\pi$
$972$	−1.61803	−0.0518985
$973$	−2.56231	−0.0821438
$974$	10.3951	0.333081
$975$	−3.27051	−0.104740
$976$	−14.5623	−0.466128
$977$	−4.63932	−0.148425	−0.0742125	−	0.997242i	$-0.523644\pi$
$977$	−4.63932	−0.148425	−0.0742125	+	0.997242i	$0.523644\pi$
$978$	11.2918	0.361072
$979$	0	0
$980$	8.47214	0.270632
$981$	0	0
$982$	−15.5836	−0.497292
$983$	−27.3050	−0.870893	−0.435446	−	0.900215i	$-0.643409\pi$
$983$	−27.3050	−0.870893	−0.435446	+	0.900215i	$0.643409\pi$
$984$	−26.7082	−0.851426
$985$	63.8328	2.03388
$986$	4.47214	0.142422
$987$	−4.85410	−0.154508
$988$	−16.7082	−0.531559
$989$	−21.6525	−0.688509
$990$	0	0
$991$	21.2705	0.675680	0.337840	−	0.941204i	$-0.390304\pi$
$991$	21.2705	0.675680	0.337840	+	0.941204i	$0.390304\pi$
$992$	16.0344	0.509094
$993$	26.5967	0.844022
$994$	10.3131	0.327111
$995$	43.7426	1.38674
$996$	1.14590	0.0363092
$997$	12.9787	0.411040	0.205520	−	0.978653i	$-0.434111\pi$
$997$	12.9787	0.411040	0.205520	+	0.978653i	$0.434111\pi$
$998$	−10.8541	−0.343581
$999$	0.236068	0.00746886

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	363.2.a.e.1.2		2
3.2	odd	2		1089.2.a.s.1.1		2
4.3	odd	2		5808.2.a.bm.1.1		2
5.4	even	2		9075.2.a.bv.1.1		2
11.2	odd	10		363.2.e.h.202.1		4
11.3	even	5		363.2.e.j.130.1		4
11.4	even	5		363.2.e.j.148.1		4
11.5	even	5		363.2.e.c.124.1		4
11.6	odd	10		363.2.e.h.124.1		4
11.7	odd	10		33.2.e.a.16.1	✓	4
11.8	odd	10		33.2.e.a.31.1	yes	4
11.9	even	5		363.2.e.c.202.1		4
11.10	odd	2		363.2.a.h.1.1		2
33.8	even	10		99.2.f.b.64.1		4
33.29	even	10		99.2.f.b.82.1		4
33.32	even	2		1089.2.a.m.1.2		2
44.7	even	10		528.2.y.f.49.1		4
44.19	even	10		528.2.y.f.97.1		4
44.43	even	2		5808.2.a.bl.1.1		2
55.7	even	20		825.2.bx.b.49.2		8
55.8	even	20		825.2.bx.b.724.2		8
55.18	even	20		825.2.bx.b.49.1		8
55.19	odd	10		825.2.n.f.526.1		4
55.29	odd	10		825.2.n.f.676.1		4
55.52	even	20		825.2.bx.b.724.1		8
55.54	odd	2		9075.2.a.x.1.2		2
99.7	odd	30		891.2.n.d.676.1		8
99.29	even	30		891.2.n.a.676.1		8
99.40	odd	30		891.2.n.d.379.1		8
99.41	even	30		891.2.n.a.460.1		8
99.52	odd	30		891.2.n.d.757.1		8
99.74	even	30		891.2.n.a.757.1		8
99.85	odd	30		891.2.n.d.460.1		8
99.95	even	30		891.2.n.a.379.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.2.e.a.16.1	✓	4	11.7	odd	10
33.2.e.a.31.1	yes	4	11.8	odd	10
99.2.f.b.64.1		4	33.8	even	10
99.2.f.b.82.1		4	33.29	even	10
363.2.a.e.1.2		2	1.1	even	1	trivial
363.2.a.h.1.1		2	11.10	odd	2
363.2.e.c.124.1		4	11.5	even	5
363.2.e.c.202.1		4	11.9	even	5
363.2.e.h.124.1		4	11.6	odd	10
363.2.e.h.202.1		4	11.2	odd	10
363.2.e.j.130.1		4	11.3	even	5
363.2.e.j.148.1		4	11.4	even	5
528.2.y.f.49.1		4	44.7	even	10
528.2.y.f.97.1		4	44.19	even	10
825.2.n.f.526.1		4	55.19	odd	10
825.2.n.f.676.1		4	55.29	odd	10
825.2.bx.b.49.1		8	55.18	even	20
825.2.bx.b.49.2		8	55.7	even	20
825.2.bx.b.724.1		8	55.52	even	20
825.2.bx.b.724.2		8	55.8	even	20
891.2.n.a.379.1		8	99.95	even	30
891.2.n.a.460.1		8	99.41	even	30
891.2.n.a.676.1		8	99.29	even	30
891.2.n.a.757.1		8	99.74	even	30
891.2.n.d.379.1		8	99.40	odd	30
891.2.n.d.460.1		8	99.85	odd	30
891.2.n.d.676.1		8	99.7	odd	30
891.2.n.d.757.1		8	99.52	odd	30
1089.2.a.m.1.2		2	33.32	even	2
1089.2.a.s.1.1		2	3.2	odd	2
5808.2.a.bl.1.1		2	44.43	even	2
5808.2.a.bm.1.1		2	4.3	odd	2
9075.2.a.x.1.2		2	55.54	odd	2
9075.2.a.bv.1.1		2	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 363 = 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	363.a (trivial)

\(f(q)\)	\(=\)	\(q+0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} -2.61803 q^{5} +0.618034 q^{6} -3.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} -2.61803 q^{5} +0.618034 q^{6} -3.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} -1.61803 q^{10} -1.61803 q^{12} -1.76393 q^{13} -1.85410 q^{14} -2.61803 q^{15} +1.85410 q^{16} -1.61803 q^{17} +0.618034 q^{18} -5.85410 q^{19} +4.23607 q^{20} -3.00000 q^{21} +3.47214 q^{23} -2.23607 q^{24} +1.85410 q^{25} -1.09017 q^{26} +1.00000 q^{27} +4.85410 q^{28} -4.47214 q^{29} -1.61803 q^{30} +2.85410 q^{31} +5.61803 q^{32} -1.00000 q^{34} +7.85410 q^{35} -1.61803 q^{36} +0.236068 q^{37} -3.61803 q^{38} -1.76393 q^{39} +5.85410 q^{40} +11.9443 q^{41} -1.85410 q^{42} -6.23607 q^{43} -2.61803 q^{45} +2.14590 q^{46} +1.61803 q^{47} +1.85410 q^{48} +2.00000 q^{49} +1.14590 q^{50} -1.61803 q^{51} +2.85410 q^{52} -9.61803 q^{53} +0.618034 q^{54} +6.70820 q^{56} -5.85410 q^{57} -2.76393 q^{58} +10.3262 q^{59} +4.23607 q^{60} -7.85410 q^{61} +1.76393 q^{62} -3.00000 q^{63} -0.236068 q^{64} +4.61803 q^{65} -9.56231 q^{67} +2.61803 q^{68} +3.47214 q^{69} +4.85410 q^{70} -5.56231 q^{71} -2.23607 q^{72} +3.23607 q^{73} +0.145898 q^{74} +1.85410 q^{75} +9.47214 q^{76} -1.09017 q^{78} -9.47214 q^{79} -4.85410 q^{80} +1.00000 q^{81} +7.38197 q^{82} -0.708204 q^{83} +4.85410 q^{84} +4.23607 q^{85} -3.85410 q^{86} -4.47214 q^{87} +0.527864 q^{89} -1.61803 q^{90} +5.29180 q^{91} -5.61803 q^{92} +2.85410 q^{93} +1.00000 q^{94} +15.3262 q^{95} +5.61803 q^{96} -14.0344 q^{97} +1.23607 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 2 q^{3} - q^{4} - 3 q^{5} - q^{6} - 6 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - q^2 + 2 * q^3 - q^4 - 3 * q^5 - q^6 - 6 * q^7 + 2 * q^9	\( 2 q - q^{2} + 2 q^{3} - q^{4} - 3 q^{5} - q^{6} - 6 q^{7} + 2 q^{9} - q^{10} - q^{12} - 8 q^{13} + 3 q^{14} - 3 q^{15} - 3 q^{16} - q^{17} - q^{18} - 5 q^{19} + 4 q^{20} - 6 q^{21} - 2 q^{23} - 3 q^{25} + 9 q^{26} + 2 q^{27} + 3 q^{28} - q^{30} - q^{31} + 9 q^{32} - 2 q^{34} + 9 q^{35} - q^{36} - 4 q^{37} - 5 q^{38} - 8 q^{39} + 5 q^{40} + 6 q^{41} + 3 q^{42} - 8 q^{43} - 3 q^{45} + 11 q^{46} + q^{47} - 3 q^{48} + 4 q^{49} + 9 q^{50} - q^{51} - q^{52} - 17 q^{53} - q^{54} - 5 q^{57} - 10 q^{58} + 5 q^{59} + 4 q^{60} - 9 q^{61} + 8 q^{62} - 6 q^{63} + 4 q^{64} + 7 q^{65} + q^{67} + 3 q^{68} - 2 q^{69} + 3 q^{70} + 9 q^{71} + 2 q^{73} + 7 q^{74} - 3 q^{75} + 10 q^{76} + 9 q^{78} - 10 q^{79} - 3 q^{80} + 2 q^{81} + 17 q^{82} + 12 q^{83} + 3 q^{84} + 4 q^{85} - q^{86} + 10 q^{89} - q^{90} + 24 q^{91} - 9 q^{92} - q^{93} + 2 q^{94} + 15 q^{95} + 9 q^{96} + q^{97} - 2 q^{98}+O(q^{100}) \) 2 * q - q^2 + 2 * q^3 - q^4 - 3 * q^5 - q^6 - 6 * q^7 + 2 * q^9 - q^10 - q^12 - 8 * q^13 + 3 * q^14 - 3 * q^15 - 3 * q^16 - q^17 - q^18 - 5 * q^19 + 4 * q^20 - 6 * q^21 - 2 * q^23 - 3 * q^25 + 9 * q^26 + 2 * q^27 + 3 * q^28 - q^30 - q^31 + 9 * q^32 - 2 * q^34 + 9 * q^35 - q^36 - 4 * q^37 - 5 * q^38 - 8 * q^39 + 5 * q^40 + 6 * q^41 + 3 * q^42 - 8 * q^43 - 3 * q^45 + 11 * q^46 + q^47 - 3 * q^48 + 4 * q^49 + 9 * q^50 - q^51 - q^52 - 17 * q^53 - q^54 - 5 * q^57 - 10 * q^58 + 5 * q^59 + 4 * q^60 - 9 * q^61 + 8 * q^62 - 6 * q^63 + 4 * q^64 + 7 * q^65 + q^67 + 3 * q^68 - 2 * q^69 + 3 * q^70 + 9 * q^71 + 2 * q^73 + 7 * q^74 - 3 * q^75 + 10 * q^76 + 9 * q^78 - 10 * q^79 - 3 * q^80 + 2 * q^81 + 17 * q^82 + 12 * q^83 + 3 * q^84 + 4 * q^85 - q^86 + 10 * q^89 - q^90 + 24 * q^91 - 9 * q^92 - q^93 + 2 * q^94 + 15 * q^95 + 9 * q^96 + q^97 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more