LMFDB - Embedded newform 98.3.b.b.97.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [98,3,Mod(97,98)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(98, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("98.97");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 98 = 2 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	98.b (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$2.67030659073$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 14)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			97.1
Root			$0.707107 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	98.97
Dual form			98.3.b.b.97.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} -4.18154i q^{3} +2.00000 q^{4} -3.16693i q^{5} +5.91359i q^{6} -2.82843 q^{8} -8.48528 q^{9} +O(q^{10})$	\(q-1.41421 q^{2} -4.18154i q^{3} +2.00000 q^{4} -3.16693i q^{5} +5.91359i q^{6} -2.82843 q^{8} -8.48528 q^{9} +4.47871i q^{10} -13.2426 q^{11} -8.36308i q^{12} -5.49333i q^{13} -13.2426 q^{15} +4.00000 q^{16} -13.5592i q^{17} +12.0000 q^{18} +0.717439i q^{19} -6.33386i q^{20} +18.7279 q^{22} -2.27208 q^{23} +11.8272i q^{24} +14.9706 q^{25} +7.76874i q^{26} -2.15232i q^{27} +20.4853 q^{29} +18.7279 q^{30} +24.6180i q^{31} -5.65685 q^{32} +55.3746i q^{33} +19.1757i q^{34} -16.9706 q^{36} +64.9411 q^{37} -1.01461i q^{38} -22.9706 q^{39} +8.95743i q^{40} -21.0308i q^{41} +6.48528 q^{43} -26.4853 q^{44} +26.8723i q^{45} +3.21320 q^{46} -47.7800i q^{47} -16.7262i q^{48} -21.1716 q^{50} -56.6985 q^{51} -10.9867i q^{52} +22.0294 q^{53} +3.04384i q^{54} +41.9385i q^{55} +3.00000 q^{57} -28.9706 q^{58} -83.7539i q^{59} -26.4853 q^{60} -66.2593i q^{61} -34.8151i q^{62} +8.00000 q^{64} -17.3970 q^{65} -78.3116i q^{66} +92.6396 q^{67} -27.1185i q^{68} +9.50079i q^{69} -48.4264 q^{71} +24.0000 q^{72} +130.991i q^{73} -91.8406 q^{74} -62.6000i q^{75} +1.43488i q^{76} +32.4853 q^{78} -76.2132 q^{79} -12.6677i q^{80} -85.3675 q^{81} +29.7420i q^{82} +107.981i q^{83} -42.9411 q^{85} -9.17157 q^{86} -85.6600i q^{87} +37.4558 q^{88} +167.907i q^{89} -38.0031i q^{90} -4.54416 q^{92} +102.941 q^{93} +67.5711i q^{94} +2.27208 q^{95} +23.6544i q^{96} +25.5816i q^{97} +112.368 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{4}+O(q^{10}) $ 4 * q + 8 * q^4	$ 4 q + 8 q^{4} - 36 q^{11} - 36 q^{15} + 16 q^{16} + 48 q^{18} + 24 q^{22} - 60 q^{23} - 8 q^{25} + 48 q^{29} + 24 q^{30} + 124 q^{37} - 24 q^{39} - 8 q^{43} - 72 q^{44} - 72 q^{46} - 96 q^{50} - 108 q^{51} + 156 q^{53} + 12 q^{57} - 48 q^{58} - 72 q^{60} + 32 q^{64} + 168 q^{65} + 116 q^{67} - 24 q^{71} + 96 q^{72} - 192 q^{74} + 96 q^{78} - 220 q^{79} - 36 q^{81} - 36 q^{85} - 48 q^{86} + 48 q^{88} - 120 q^{92} + 276 q^{93} + 60 q^{95} + 144 q^{99}+O(q^{100}) $ 4 * q + 8 * q^4 - 36 * q^11 - 36 * q^15 + 16 * q^16 + 48 * q^18 + 24 * q^22 - 60 * q^23 - 8 * q^25 + 48 * q^29 + 24 * q^30 + 124 * q^37 - 24 * q^39 - 8 * q^43 - 72 * q^44 - 72 * q^46 - 96 * q^50 - 108 * q^51 + 156 * q^53 + 12 * q^57 - 48 * q^58 - 72 * q^60 + 32 * q^64 + 168 * q^65 + 116 * q^67 - 24 * q^71 + 96 * q^72 - 192 * q^74 + 96 * q^78 - 220 * q^79 - 36 * q^81 - 36 * q^85 - 48 * q^86 + 48 * q^88 - 120 * q^92 + 276 * q^93 + 60 * q^95 + 144 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/98\mathbb{Z}\right)^\times$.

$n$	$3$
$\chi(n)$	$-1$

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 $

$2$	−1.41421	−0.707107
$2$	−1.41421	−0.707107
$3$	− 4.18154i	− 1.39385i	−0.717146	−	0.696923i	$-0.754552\pi$
$3$	− 4.18154i	− 1.39385i	0.717146	−	0.696923i	$-0.245448\pi$
$4$	2.00000	0.500000
$5$	− 3.16693i	− 0.633386i	−0.948528	−	0.316693i	$-0.897428\pi$
$5$	− 3.16693i	− 0.633386i	0.948528	−	0.316693i	$-0.102572\pi$
$6$	5.91359i	0.985599i
$7$	0	0
$7$	0	0
$8$	−2.82843	−0.353553
$9$	−8.48528	−0.942809
$10$	4.47871i	0.447871i
$11$	−13.2426	−1.20388	−0.601938	−	0.798543i	$-0.705605\pi$
$11$	−13.2426	−1.20388	−0.601938	+	0.798543i	$0.705605\pi$
$12$	− 8.36308i	− 0.696923i
$13$	− 5.49333i	− 0.422563i	−0.977425	−	0.211282i	$-0.932236\pi$
$13$	− 5.49333i	− 0.422563i	0.977425	−	0.211282i	$-0.0677638\pi$
$14$	0	0
$15$	−13.2426	−0.882843
$16$	4.00000	0.250000
$17$	− 13.5592i	− 0.797602i	−0.917037	−	0.398801i	$-0.869426\pi$
$17$	− 13.5592i	− 0.797602i	0.917037	−	0.398801i	$-0.130574\pi$
$18$	12.0000	0.666667
$19$	0.717439i	0.0377599i	0.999822	+	0.0188800i	$0.00601004\pi$
$19$	0.717439i	0.0377599i	−0.999822	+	0.0188800i	$0.993990\pi$
$20$	− 6.33386i	− 0.316693i
$21$	0	0
$22$	18.7279	0.851269
$23$	−2.27208	−0.0987860	−0.0493930	−	0.998779i	$-0.515729\pi$
$23$	−2.27208	−0.0987860	−0.0493930	+	0.998779i	$0.515729\pi$
$24$	11.8272i	0.492799i
$25$	14.9706	0.598823
$26$	7.76874i	0.298798i
$27$	− 2.15232i	− 0.0797154i
$28$	0	0
$29$	20.4853	0.706389	0.353195	−	0.935550i	$-0.385095\pi$
$29$	20.4853	0.706389	0.353195	+	0.935550i	$0.385095\pi$
$30$	18.7279	0.624264
$31$	24.6180i	0.794129i	0.917791	+	0.397064i	$0.129971\pi$
$31$	24.6180i	0.794129i	−0.917791	+	0.397064i	$0.870029\pi$
$32$	−5.65685	−0.176777
$33$	55.3746i	1.67802i
$34$	19.1757i	0.563990i
$35$	0	0
$36$	−16.9706	−0.471405
$37$	64.9411	1.75517	0.877583	−	0.479425i	$-0.159155\pi$
$37$	64.9411	1.75517	0.877583	+	0.479425i	$0.159155\pi$
$38$	− 1.01461i	− 0.0267003i
$39$	−22.9706	−0.588989
$40$	8.95743i	0.223936i
$41$	− 21.0308i	− 0.512946i	−0.966551	−	0.256473i	$-0.917439\pi$
$41$	− 21.0308i	− 0.512946i	0.966551	−	0.256473i	$-0.0825605\pi$
$42$	0	0
$43$	6.48528	0.150820	0.0754102	−	0.997153i	$-0.475973\pi$
$43$	6.48528	0.150820	0.0754102	+	0.997153i	$0.475973\pi$
$44$	−26.4853	−0.601938
$45$	26.8723i	0.597162i
$46$	3.21320	0.0698522
$47$	− 47.7800i	− 1.01660i	−0.861181	−	0.508298i	$-0.830275\pi$
$47$	− 47.7800i	− 1.01660i	0.861181	−	0.508298i	$-0.169725\pi$
$48$	− 16.7262i	− 0.348462i
$49$	0	0
$50$	−21.1716	−0.423431
$51$	−56.6985	−1.11173
$52$	− 10.9867i	− 0.211282i
$53$	22.0294	0.415650	0.207825	−	0.978166i	$-0.433362\pi$
$53$	22.0294	0.415650	0.207825	+	0.978166i	$0.433362\pi$
$54$	3.04384i	0.0563673i
$55$	41.9385i	0.762518i
$56$	0	0
$57$	3.00000	0.0526316
$58$	−28.9706	−0.499492
$59$	− 83.7539i	− 1.41956i	−0.704425	−	0.709779i	$-0.748795\pi$
$59$	− 83.7539i	− 1.41956i	0.704425	−	0.709779i	$-0.251205\pi$
$60$	−26.4853	−0.441421
$61$	− 66.2593i	− 1.08622i	−0.839662	−	0.543109i	$-0.817247\pi$
$61$	− 66.2593i	− 1.08622i	0.839662	−	0.543109i	$-0.182753\pi$
$62$	− 34.8151i	− 0.561534i
$63$	0	0
$64$	8.00000	0.125000
$65$	−17.3970	−0.267646
$66$	− 78.3116i	− 1.18654i
$67$	92.6396	1.38268	0.691340	−	0.722529i	$-0.257021\pi$
$67$	92.6396	1.38268	0.691340	+	0.722529i	$0.257021\pi$
$68$	− 27.1185i	− 0.398801i
$69$	9.50079i	0.137693i
$70$	0	0
$71$	−48.4264	−0.682062	−0.341031	−	0.940052i	$-0.610776\pi$
$71$	−48.4264	−0.682062	−0.341031	+	0.940052i	$0.610776\pi$
$72$	24.0000	0.333333
$73$	130.991i	1.79439i	0.441634	+	0.897195i	$0.354399\pi$
$73$	130.991i	1.79439i	−0.441634	+	0.897195i	$0.645601\pi$
$74$	−91.8406	−1.24109
$75$	− 62.6000i	− 0.834667i
$76$	1.43488i	0.0188800i
$77$	0	0
$78$	32.4853	0.416478
$79$	−76.2132	−0.964724	−0.482362	−	0.875972i	$-0.660221\pi$
$79$	−76.2132	−0.964724	−0.482362	+	0.875972i	$0.660221\pi$
$80$	− 12.6677i	− 0.158346i
$81$	−85.3675	−1.05392
$82$	29.7420i	0.362708i
$83$	107.981i	1.30098i	0.759514	+	0.650491i	$0.225437\pi$
$83$	107.981i	1.30098i	−0.759514	+	0.650491i	$0.774563\pi$
$84$	0	0
$85$	−42.9411	−0.505190
$86$	−9.17157	−0.106646
$87$	− 85.6600i	− 0.984598i
$88$	37.4558	0.425635
$89$	167.907i	1.88659i	0.331949	+	0.943297i	$0.392294\pi$
$89$	167.907i	1.88659i	−0.331949	+	0.943297i	$0.607706\pi$
$90$	− 38.0031i	− 0.422257i
$91$	0	0
$92$	−4.54416	−0.0493930
$93$	102.941	1.10689
$94$	67.5711i	0.718841i
$95$	2.27208	0.0239166
$96$	23.6544i	0.246400i
$97$	25.5816i	0.263728i	0.991268	+	0.131864i	$0.0420962\pi$
$97$	25.5816i	0.263728i	−0.991268	+	0.131864i	$0.957904\pi$
$98$	0	0
$99$	112.368	1.13503
$100$	29.9411	0.299411
$101$	28.5024i	0.282202i	0.989995	+	0.141101i	$0.0450642\pi$
$101$	28.5024i	0.282202i	−0.989995	+	0.141101i	$0.954936\pi$
$102$	80.1838	0.786115
$103$	− 56.4912i	− 0.548458i	−0.961664	−	0.274229i	$-0.911577\pi$
$103$	− 56.4912i	− 0.548458i	0.961664	−	0.274229i	$-0.0884227\pi$
$104$	15.5375i	0.149399i
$105$	0	0
$106$	−31.1543	−0.293909
$107$	47.6102	0.444955	0.222477	−	0.974938i	$-0.428586\pi$
$107$	47.6102	0.444955	0.222477	+	0.974938i	$0.428586\pi$
$108$	− 4.30463i	− 0.0398577i
$109$	75.3087	0.690905	0.345453	−	0.938436i	$-0.387725\pi$
$109$	75.3087	0.690905	0.345453	+	0.938436i	$0.387725\pi$
$110$	− 59.3100i	− 0.539182i
$111$	− 271.554i	− 2.44643i
$112$	0	0
$113$	85.4558	0.756246	0.378123	−	0.925755i	$-0.376570\pi$
$113$	85.4558	0.756246	0.378123	+	0.925755i	$0.376570\pi$
$114$	−4.24264	−0.0372161
$115$	7.19551i	0.0625696i
$116$	40.9706	0.353195
$117$	46.6124i	0.398397i
$118$	118.446i	1.00378i
$119$	0	0
$120$	37.4558	0.312132
$121$	54.3675	0.449318
$122$	93.7048i	0.768072i
$123$	−87.9411	−0.714968
$124$	49.2360i	0.397064i
$125$	− 126.584i	− 1.01267i
$126$	0	0
$127$	−60.6619	−0.477653	−0.238826	−	0.971062i	$-0.576763\pi$
$127$	−60.6619	−0.477653	−0.238826	+	0.971062i	$0.576763\pi$
$128$	−11.3137	−0.0883883
$129$	− 27.1185i	− 0.210221i
$130$	24.6030	0.189254
$131$	132.948i	1.01487i	0.861691	+	0.507434i	$0.169406\pi$
$131$	132.948i	1.01487i	−0.861691	+	0.507434i	$0.830594\pi$
$132$	110.749i	0.839010i
$133$	0	0
$134$	−131.012	−0.977703
$135$	−6.81623	−0.0504906
$136$	38.3513i	0.281995i
$137$	−117.426	−0.857127	−0.428564	−	0.903512i	$-0.640980\pi$
$137$	−117.426	−0.857127	−0.428564	+	0.903512i	$0.640980\pi$
$138$	− 13.4361i	− 0.0973633i
$139$	68.5857i	0.493422i	0.969089	+	0.246711i	$0.0793499\pi$
$139$	68.5857i	0.493422i	−0.969089	+	0.246711i	$0.920650\pi$
$140$	0	0
$141$	−199.794	−1.41698
$142$	68.4853	0.482291
$143$	72.7461i	0.508714i
$144$	−33.9411	−0.235702
$145$	− 64.8754i	− 0.447417i
$146$	− 185.249i	− 1.26883i
$147$	0	0
$148$	129.882	0.877583
$149$	−26.3970	−0.177161	−0.0885804	−	0.996069i	$-0.528233\pi$
$149$	−26.3970	−0.177161	−0.0885804	+	0.996069i	$0.528233\pi$
$150$	88.5298i	0.590199i
$151$	−134.213	−0.888829	−0.444415	−	0.895821i	$-0.646588\pi$
$151$	−134.213	−0.888829	−0.444415	+	0.895821i	$0.646588\pi$
$152$	− 2.02922i	− 0.0133502i
$153$	115.054i	0.751986i
$154$	0	0
$155$	77.9634	0.502990
$156$	−45.9411	−0.294494
$157$	− 226.695i	− 1.44392i	−0.691937	−	0.721958i	$-0.743243\pi$
$157$	− 226.695i	− 1.44392i	0.691937	−	0.721958i	$-0.256757\pi$
$158$	107.782	0.682163
$159$	− 92.1170i	− 0.579352i
$160$	17.9149i	0.111968i
$161$	0	0
$162$	120.728	0.745234
$163$	−91.9777	−0.564280	−0.282140	−	0.959373i	$-0.591044\pi$
$163$	−91.9777	−0.564280	−0.282140	+	0.959373i	$0.591044\pi$
$164$	− 42.0616i	− 0.256473i
$165$	175.368	1.06283
$166$	− 152.709i	− 0.919933i
$167$	203.482i	1.21845i	0.792996	+	0.609227i	$0.208520\pi$
$167$	203.482i	1.21845i	−0.792996	+	0.609227i	$0.791480\pi$
$168$	0	0
$169$	138.823	0.821440
$170$	60.7279	0.357223
$171$	− 6.08767i	− 0.0356004i
$172$	12.9706	0.0754102
$173$	− 70.8101i	− 0.409307i	−0.978834	−	0.204654i	$-0.934393\pi$
$173$	− 70.8101i	− 0.409307i	0.978834	−	0.204654i	$-0.0656068\pi$
$174$	121.142i	0.696216i
$175$	0	0
$176$	−52.9706	−0.300969
$177$	−350.220	−1.97865
$178$	− 237.456i	− 1.33402i
$179$	108.816	0.607912	0.303956	−	0.952686i	$-0.401692\pi$
$179$	108.816	0.607912	0.303956	+	0.952686i	$0.401692\pi$
$180$	53.7446i	0.298581i
$181$	− 99.6607i	− 0.550611i	−0.961357	−	0.275306i	$-0.911221\pi$
$181$	− 99.6607i	− 0.550611i	0.961357	−	0.275306i	$-0.0887791\pi$
$182$	0	0
$183$	−277.066	−1.51402
$184$	6.42641	0.0349261
$185$	− 205.664i	− 1.11170i
$186$	−145.581	−0.782692
$187$	179.560i	0.960214i
$188$	− 95.5600i	− 0.508298i
$189$	0	0
$190$	−3.21320	−0.0169116
$191$	69.9045	0.365992	0.182996	−	0.983114i	$-0.441420\pi$
$191$	69.9045	0.365992	0.182996	+	0.983114i	$0.441420\pi$
$192$	− 33.4523i	− 0.174231i
$193$	−32.3381	−0.167555	−0.0837774	−	0.996484i	$-0.526698\pi$
$193$	−32.3381	−0.167555	−0.0837774	+	0.996484i	$0.526698\pi$
$194$	− 36.1779i	− 0.186484i
$195$	72.7461i	0.373057i
$196$	0	0
$197$	277.103	1.40661	0.703306	−	0.710887i	$-0.251706\pi$
$197$	277.103	1.40661	0.703306	+	0.710887i	$0.251706\pi$
$198$	−158.912	−0.802584
$199$	167.444i	0.841429i	0.907193	+	0.420715i	$0.138221\pi$
$199$	167.444i	0.841429i	−0.907193	+	0.420715i	$0.861779\pi$
$200$	−42.3431	−0.211716
$201$	− 387.376i	− 1.92725i
$202$	− 40.3084i	− 0.199547i
$203$	0	0
$204$	−113.397	−0.555867
$205$	−66.6030	−0.324893
$206$	79.8907i	0.387819i
$207$	19.2792	0.0931363
$208$	− 21.9733i	− 0.105641i
$209$	− 9.50079i	− 0.0454583i
$210$	0	0
$211$	−128.073	−0.606982	−0.303491	−	0.952834i	$-0.598152\pi$
$211$	−128.073	−0.606982	−0.303491	+	0.952834i	$0.598152\pi$
$212$	44.0589	0.207825
$213$	202.497i	0.950690i
$214$	−67.3310	−0.314631
$215$	− 20.5384i	− 0.0955276i
$216$	6.08767i	0.0281837i
$217$	0	0
$218$	−106.503	−0.488544
$219$	547.742	2.50111
$220$	83.8770i	0.381259i
$221$	−74.4853	−0.337037
$222$	384.035i	1.72989i
$223$	− 417.169i	− 1.87071i	−0.353705	−	0.935357i	$-0.615078\pi$
$223$	− 417.169i	− 1.87071i	0.353705	−	0.935357i	$-0.384922\pi$
$224$	0	0
$225$	−127.029	−0.564575
$226$	−120.853	−0.534747
$227$	232.260i	1.02317i	0.859232	+	0.511586i	$0.170942\pi$
$227$	232.260i	1.02317i	−0.859232	+	0.511586i	$0.829058\pi$
$228$	6.00000	0.0263158
$229$	83.6221i	0.365162i	0.983191	+	0.182581i	$0.0584452\pi$
$229$	83.6221i	0.365162i	−0.983191	+	0.182581i	$0.941555\pi$
$230$	− 10.1760i	− 0.0442434i
$231$	0	0
$232$	−57.9411	−0.249746
$233$	219.073	0.940228	0.470114	−	0.882606i	$-0.344213\pi$
$233$	219.073	0.940228	0.470114	+	0.882606i	$0.344213\pi$
$234$	− 65.9199i	− 0.281709i
$235$	−151.316	−0.643897
$236$	− 167.508i	− 0.709779i
$237$	318.689i	1.34468i
$238$	0	0
$239$	193.103	0.807961	0.403980	−	0.914768i	$-0.367626\pi$
$239$	193.103	0.807961	0.403980	+	0.914768i	$0.367626\pi$
$240$	−52.9706	−0.220711
$241$	49.5332i	0.205532i	0.994706	+	0.102766i	$0.0327692\pi$
$241$	49.5332i	0.205532i	−0.994706	+	0.102766i	$0.967231\pi$
$242$	−76.8873	−0.317716
$243$	337.597i	1.38929i
$244$	− 132.519i	− 0.543109i
$245$	0	0
$246$	124.368	0.505559
$247$	3.94113	0.0159560
$248$	− 69.6302i	− 0.280767i
$249$	451.529	1.81337
$250$	179.017i	0.716067i
$251$	162.524i	0.647507i	0.946141	+	0.323754i	$0.104945\pi$
$251$	162.524i	0.647507i	−0.946141	+	0.323754i	$0.895055\pi$
$252$	0	0
$253$	30.0883	0.118926
$254$	85.7889	0.337752
$255$	179.560i	0.704157i
$256$	16.0000	0.0625000
$257$	99.1595i	0.385835i	0.981215	+	0.192917i	$0.0617949\pi$
$257$	99.1595i	0.385835i	−0.981215	+	0.192917i	$0.938205\pi$
$258$	38.3513i	0.148648i
$259$	0	0
$260$	−34.7939	−0.133823
$261$	−173.823	−0.665990
$262$	− 188.016i	− 0.717620i
$263$	434.345	1.65150	0.825751	−	0.564034i	$-0.190751\pi$
$263$	434.345	1.65150	0.825751	+	0.564034i	$0.190751\pi$
$264$	− 156.623i	− 0.593269i
$265$	− 69.7657i	− 0.263267i
$266$	0	0
$267$	702.110	2.62962
$268$	185.279	0.691340
$269$	− 91.4083i	− 0.339808i	−0.985461	−	0.169904i	$-0.945654\pi$
$269$	− 91.4083i	− 0.339808i	0.985461	−	0.169904i	$-0.0543457\pi$
$270$	9.63961	0.0357023
$271$	− 17.0954i	− 0.0630828i	−0.999502	−	0.0315414i	$-0.989958\pi$
$271$	− 17.0954i	− 0.0630828i	0.999502	−	0.0315414i	$-0.0100416\pi$
$272$	− 54.2369i	− 0.199400i
$273$	0	0
$274$	166.066	0.606080
$275$	−198.250	−0.720908
$276$	19.0016i	0.0688463i
$277$	−400.411	−1.44553	−0.722764	−	0.691095i	$-0.757129\pi$
$277$	−400.411	−1.44553	−0.722764	+	0.691095i	$0.757129\pi$
$278$	− 96.9948i	− 0.348902i
$279$	− 208.891i	− 0.748712i
$280$	0	0
$281$	−538.690	−1.91705	−0.958524	−	0.285012i	$-0.908002\pi$
$281$	−538.690	−1.91705	−0.958524	+	0.285012i	$0.908002\pi$
$282$	282.551	1.00195
$283$	− 309.209i	− 1.09261i	−0.837586	−	0.546306i	$-0.816034\pi$
$283$	− 309.209i	− 1.09261i	0.837586	−	0.546306i	$-0.183966\pi$
$284$	−96.8528	−0.341031
$285$	− 9.50079i	− 0.0333361i
$286$	− 102.879i	− 0.359715i
$287$	0	0
$288$	48.0000	0.166667
$289$	105.147	0.363831
$290$	91.7477i	0.316371i
$291$	106.971	0.367596
$292$	261.981i	0.897195i
$293$	327.391i	1.11738i	0.829378	+	0.558688i	$0.188695\pi$
$293$	327.391i	1.11738i	−0.829378	+	0.558688i	$0.811305\pi$
$294$	0	0
$295$	−265.243	−0.899128
$296$	−183.681	−0.620545
$297$	28.5024i	0.0959675i
$298$	37.3310	0.125272
$299$	12.4813i	0.0417434i
$300$	− 125.200i	− 0.417333i
$301$	0	0
$302$	189.806	0.628497
$303$	119.184	0.393346
$304$	2.86976i	0.00943999i
$305$	−209.839	−0.687995
$306$	− 162.711i	− 0.531735i
$307$	256.140i	0.834331i	0.908831	+	0.417165i	$0.136976\pi$
$307$	256.140i	0.834331i	−0.908831	+	0.417165i	$0.863024\pi$
$308$	0	0
$309$	−236.220	−0.764467
$310$	−110.257	−0.355668
$311$	216.332i	0.695602i	0.937568	+	0.347801i	$0.113072\pi$
$311$	216.332i	0.695602i	−0.937568	+	0.347801i	$0.886928\pi$
$312$	64.9706	0.208239
$313$	− 156.818i	− 0.501017i	−0.968114	−	0.250509i	$-0.919402\pi$
$313$	− 156.818i	− 0.501017i	0.968114	−	0.250509i	$-0.0805978\pi$
$314$	320.595i	1.02100i
$315$	0	0
$316$	−152.426	−0.482362
$317$	−448.029	−1.41334	−0.706671	−	0.707542i	$-0.749804\pi$
$317$	−448.029	−1.41334	−0.706671	+	0.707542i	$0.749804\pi$
$318$	130.273i	0.409664i
$319$	−271.279	−0.850405
$320$	− 25.3354i	− 0.0791732i
$321$	− 199.084i	− 0.620199i
$322$	0	0
$323$	9.72792	0.0301174
$324$	−170.735	−0.526960
$325$	− 82.2382i	− 0.253041i
$326$	130.076	0.399006
$327$	− 314.906i	− 0.963016i
$328$	59.4841i	0.181354i
$329$	0	0
$330$	−248.007	−0.751537
$331$	−55.0071	−0.166185	−0.0830924	−	0.996542i	$-0.526480\pi$
$331$	−55.0071	−0.166185	−0.0830924	+	0.996542i	$0.526480\pi$
$332$	215.963i	0.650491i
$333$	−551.044	−1.65479
$334$	− 287.767i	− 0.861577i
$335$	− 293.383i	− 0.875770i
$336$	0	0
$337$	−111.632	−0.331254	−0.165627	−	0.986189i	$-0.552965\pi$
$337$	−111.632	−0.331254	−0.165627	+	0.986189i	$0.552965\pi$
$338$	−196.326	−0.580846
$339$	− 357.337i	− 1.05409i
$340$	−85.8823	−0.252595
$341$	− 326.007i	− 0.956033i
$342$	8.60927i	0.0251733i
$343$	0	0
$344$	−18.3431	−0.0533231
$345$	30.0883	0.0872125
$346$	100.141i	0.289424i
$347$	377.257	1.08720	0.543598	−	0.839346i	$-0.317062\pi$
$347$	377.257	1.08720	0.543598	+	0.839346i	$0.317062\pi$
$348$	− 171.320i	− 0.492299i
$349$	− 204.034i	− 0.584624i	−0.956323	−	0.292312i	$-0.905575\pi$
$349$	− 204.034i	− 0.584624i	0.956323	−	0.292312i	$-0.0944246\pi$
$350$	0	0
$351$	−11.8234	−0.0336848
$352$	74.9117	0.212817
$353$	− 417.076i	− 1.18152i	−0.806848	−	0.590759i	$-0.798828\pi$
$353$	− 417.076i	− 1.18152i	0.806848	−	0.590759i	$-0.201172\pi$
$354$	495.286	1.39911
$355$	153.363i	0.432008i
$356$	335.814i	0.943297i
$357$	0	0
$358$	−153.889	−0.429859
$359$	−178.831	−0.498135	−0.249068	−	0.968486i	$-0.580124\pi$
$359$	−178.831	−0.498135	−0.249068	+	0.968486i	$0.580124\pi$
$360$	− 76.0063i	− 0.211129i
$361$	360.485	0.998574
$362$	140.941i	0.389341i
$363$	− 227.340i	− 0.626281i
$364$	0	0
$365$	414.838	1.13654
$366$	391.831	1.07058
$367$	− 628.993i	− 1.71388i	−0.515418	−	0.856939i	$-0.672363\pi$
$367$	− 628.993i	− 1.71388i	0.515418	−	0.856939i	$-0.327637\pi$
$368$	−9.08831	−0.0246965
$369$	178.452i	0.483610i
$370$	290.853i	0.786088i
$371$	0	0
$372$	205.882	0.553447
$373$	−255.558	−0.685143	−0.342572	−	0.939492i	$-0.611298\pi$
$373$	−255.558	−0.685143	−0.342572	+	0.939492i	$0.611298\pi$
$374$	− 253.936i	− 0.678974i
$375$	−529.316	−1.41151
$376$	135.142i	0.359421i
$377$	− 112.532i	− 0.298494i
$378$	0	0
$379$	219.750	0.579816	0.289908	−	0.957055i	$-0.406375\pi$
$379$	219.750	0.579816	0.289908	+	0.957055i	$0.406375\pi$
$380$	4.54416	0.0119583
$381$	253.660i	0.665775i
$382$	−98.8600	−0.258796
$383$	17.0357i	0.0444797i	0.999753	+	0.0222398i	$0.00707974\pi$
$383$	17.0357i	0.0444797i	−0.999753	+	0.0222398i	$0.992920\pi$
$384$	47.3087i	0.123200i
$385$	0	0
$386$	45.7330	0.118479
$387$	−55.0294	−0.142195
$388$	51.1632i	0.131864i
$389$	−152.220	−0.391312	−0.195656	−	0.980673i	$-0.562684\pi$
$389$	−152.220	−0.391312	−0.195656	+	0.980673i	$0.562684\pi$
$390$	− 102.879i	− 0.263791i
$391$	30.8076i	0.0787919i
$392$	0	0
$393$	555.926	1.41457
$394$	−391.882	−0.994625
$395$	241.362i	0.611042i
$396$	224.735	0.567513
$397$	372.722i	0.938845i	0.882974	+	0.469423i	$0.155538\pi$
$397$	372.722i	0.938845i	−0.882974	+	0.469423i	$0.844462\pi$
$398$	− 236.802i	− 0.594980i
$399$	0	0
$400$	59.8823	0.149706
$401$	651.573	1.62487	0.812435	−	0.583052i	$-0.198142\pi$
$401$	651.573	1.62487	0.812435	+	0.583052i	$0.198142\pi$
$402$	547.833i	1.36277i
$403$	135.235	0.335570
$404$	57.0047i	0.141101i
$405$	270.353i	0.667538i
$406$	0	0
$407$	−859.992	−2.11300
$408$	160.368	0.393058
$409$	533.565i	1.30456i	0.757978	+	0.652280i	$0.226187\pi$
$409$	533.565i	1.30456i	−0.757978	+	0.652280i	$0.773813\pi$
$410$	94.1909	0.229734
$411$	491.023i	1.19470i
$412$	− 112.982i	− 0.274229i
$413$	0	0
$414$	−27.2649	−0.0658573
$415$	341.970	0.824023
$416$	31.0749i	0.0746994i
$417$	286.794	0.687755
$418$	13.4361i	0.0321439i
$419$	534.252i	1.27507i	0.770423	+	0.637533i	$0.220045\pi$
$419$	534.252i	1.27507i	−0.770423	+	0.637533i	$0.779955\pi$
$420$	0	0
$421$	157.220	0.373445	0.186723	−	0.982413i	$-0.440213\pi$
$421$	157.220	0.373445	0.186723	+	0.982413i	$0.440213\pi$
$422$	181.123	0.429201
$423$	405.427i	0.958455i
$424$	−62.3087	−0.146954
$425$	− 202.989i	− 0.477622i
$426$	− 286.374i	− 0.672239i
$427$	0	0
$428$	95.2203	0.222477
$429$	304.191	0.709070
$430$	29.0457i	0.0675482i
$431$	−228.536	−0.530246	−0.265123	−	0.964215i	$-0.585413\pi$
$431$	−228.536	−0.530246	−0.265123	+	0.964215i	$0.585413\pi$
$432$	− 8.60927i	− 0.0199289i
$433$	− 47.5549i	− 0.109827i	−0.998491	−	0.0549133i	$-0.982512\pi$
$433$	− 47.5549i	− 0.109827i	0.998491	−	0.0549133i	$-0.0174882\pi$
$434$	0	0
$435$	−271.279	−0.623630
$436$	150.617	0.345453
$437$	− 1.63008i	− 0.00373015i
$438$	−774.624	−1.76855
$439$	73.8540i	0.168232i	0.996456	+	0.0841161i	$0.0268067\pi$
$439$	73.8540i	0.168232i	−0.996456	+	0.0841161i	$0.973193\pi$
$440$	− 118.620i	− 0.269591i
$441$	0	0
$442$	105.338	0.238321
$443$	234.640	0.529661	0.264830	−	0.964295i	$-0.414684\pi$
$443$	234.640	0.529661	0.264830	+	0.964295i	$0.414684\pi$
$444$	− 543.108i	− 1.22322i
$445$	531.749	1.19494
$446$	589.966i	1.32279i
$447$	110.380i	0.246935i
$448$	0	0
$449$	−255.161	−0.568288	−0.284144	−	0.958782i	$-0.591709\pi$
$449$	−255.161	−0.568288	−0.284144	+	0.958782i	$0.591709\pi$
$450$	179.647	0.399215
$451$	278.503i	0.617524i
$452$	170.912	0.378123
$453$	561.218i	1.23889i
$454$	− 328.465i	− 0.723492i
$455$	0	0
$456$	−8.48528	−0.0186081
$457$	−145.735	−0.318895	−0.159448	−	0.987206i	$-0.550971\pi$
$457$	−145.735	−0.318895	−0.159448	+	0.987206i	$0.550971\pi$
$458$	− 118.259i	− 0.258208i
$459$	−29.1838	−0.0635812
$460$	14.3910i	0.0312848i
$461$	− 888.329i	− 1.92696i	−0.267777	−	0.963481i	$-0.586289\pi$
$461$	− 888.329i	− 1.92696i	0.267777	−	0.963481i	$-0.413711\pi$
$462$	0	0
$463$	234.014	0.505430	0.252715	−	0.967541i	$-0.418676\pi$
$463$	234.014	0.505430	0.252715	+	0.967541i	$0.418676\pi$
$464$	81.9411	0.176597
$465$	− 326.007i	− 0.701091i
$466$	−309.816	−0.664842
$467$	− 786.618i	− 1.68441i	−0.539159	−	0.842204i	$-0.681258\pi$
$467$	− 786.618i	− 1.68441i	0.539159	−	0.842204i	$-0.318742\pi$
$468$	93.2248i	0.199198i
$469$	0	0
$470$	213.993	0.455304
$471$	−947.933	−2.01260
$472$	236.892i	0.501889i
$473$	−85.8823	−0.181569
$474$	− 450.694i	− 0.950831i
$475$	10.7405i	0.0226115i
$476$	0	0
$477$	−186.926	−0.391878
$478$	−273.088	−0.571314
$479$	736.932i	1.53848i	0.638960	+	0.769240i	$0.279365\pi$
$479$	736.932i	1.53848i	−0.638960	+	0.769240i	$0.720635\pi$
$480$	74.9117	0.156066
$481$	− 356.743i	− 0.741669i
$482$	− 70.0505i	− 0.145333i
$483$	0	0
$484$	108.735	0.224659
$485$	81.0152	0.167042
$486$	− 477.434i	− 0.982375i
$487$	270.698	0.555849	0.277925	−	0.960603i	$-0.410353\pi$
$487$	270.698	0.555849	0.277925	+	0.960603i	$0.410353\pi$
$488$	187.410i	0.384036i
$489$	384.609i	0.786520i
$490$	0	0
$491$	760.161	1.54819	0.774094	−	0.633070i	$-0.218206\pi$
$491$	760.161	1.54819	0.774094	+	0.633070i	$0.218206\pi$
$492$	−175.882	−0.357484
$493$	− 277.765i	− 0.563417i
$494$	−5.57359	−0.0112826
$495$	− 355.860i	− 0.718909i
$496$	98.4720i	0.198532i
$497$	0	0
$498$	−638.558	−1.28225
$499$	125.492	0.251488	0.125744	−	0.992063i	$-0.459868\pi$
$499$	125.492	0.251488	0.125744	+	0.992063i	$0.459868\pi$
$500$	− 253.168i	− 0.506336i
$501$	850.867	1.69834
$502$	− 229.844i	− 0.457857i
$503$	− 117.083i	− 0.232770i	−0.993204	−	0.116385i	$-0.962869\pi$
$503$	− 117.083i	− 0.232770i	0.993204	−	0.116385i	$-0.0371306\pi$
$504$	0	0
$505$	90.2649	0.178742
$506$	−42.5513	−0.0840935
$507$	− 580.496i	− 1.14496i
$508$	−121.324	−0.238826
$509$	662.925i	1.30241i	0.758903	+	0.651204i	$0.225736\pi$
$509$	662.925i	1.30241i	−0.758903	+	0.651204i	$0.774264\pi$
$510$	− 253.936i	− 0.497914i
$511$	0	0
$512$	−22.6274	−0.0441942
$513$	1.54416	0.00301005
$514$	− 140.233i	− 0.272826i
$515$	−178.904	−0.347386
$516$	− 54.2369i	− 0.105110i
$517$	632.733i	1.22386i
$518$	0	0
$519$	−296.095	−0.570511
$520$	49.2061	0.0946270
$521$	− 47.1383i	− 0.0904765i	−0.998976	−	0.0452383i	$-0.985595\pi$
$521$	− 47.1383i	− 0.0904765i	0.998976	−	0.0452383i	$-0.0144047\pi$
$522$	245.823	0.470926
$523$	− 499.471i	− 0.955011i	−0.878629	−	0.477506i	$-0.841541\pi$
$523$	− 499.471i	− 0.955011i	0.878629	−	0.477506i	$-0.158459\pi$
$524$	265.895i	0.507434i
$525$	0	0
$526$	−614.257	−1.16779
$527$	333.801	0.633399
$528$	221.499i	0.419505i
$529$	−523.838	−0.990241
$530$	98.6635i	0.186158i
$531$	710.675i	1.33837i
$532$	0	0
$533$	−115.529	−0.216752
$534$	−992.933	−1.85943
$535$	− 150.778i	− 0.281828i
$536$	−262.024	−0.488851
$537$	− 455.019i	− 0.847336i
$538$	129.271i	0.240280i
$539$	0	0
$540$	−13.6325	−0.0252453
$541$	498.809	0.922013	0.461007	−	0.887397i	$-0.347488\pi$
$541$	498.809	0.922013	0.461007	+	0.887397i	$0.347488\pi$
$542$	24.1766i	0.0446063i
$543$	−416.735	−0.767468
$544$	76.7026i	0.140997i
$545$	− 238.497i	− 0.437609i
$546$	0	0
$547$	−279.897	−0.511694	−0.255847	−	0.966717i	$-0.582354\pi$
$547$	−279.897	−0.511694	−0.255847	+	0.966717i	$0.582354\pi$
$548$	−234.853	−0.428564
$549$	562.229i	1.02410i
$550$	280.368	0.509759
$551$	14.6969i	0.0266732i
$552$	− 26.8723i	− 0.0486817i
$553$	0	0
$554$	566.267	1.02214
$555$	−859.992	−1.54954
$556$	137.171i	0.246711i
$557$	−261.780	−0.469981	−0.234991	−	0.971998i	$-0.575506\pi$
$557$	−261.780	−0.469981	−0.234991	+	0.971998i	$0.575506\pi$
$558$	295.416i	0.529419i
$559$	− 35.6258i	− 0.0637312i
$560$	0	0
$561$	750.838	1.33839
$562$	761.823	1.35556
$563$	− 485.062i	− 0.861567i	−0.902455	−	0.430784i	$-0.858237\pi$
$563$	− 485.062i	− 0.861567i	0.902455	−	0.430784i	$-0.141763\pi$
$564$	−399.588	−0.708489
$565$	− 270.633i	− 0.478996i
$566$	437.287i	0.772593i
$567$	0	0
$568$	136.971	0.241145
$569$	−453.999	−0.797890	−0.398945	−	0.916975i	$-0.630623\pi$
$569$	−453.999	−0.797890	−0.398945	+	0.916975i	$0.630623\pi$
$570$	13.4361i	0.0235722i
$571$	−231.537	−0.405494	−0.202747	−	0.979231i	$-0.564987\pi$
$571$	−231.537	−0.405494	−0.202747	+	0.979231i	$0.564987\pi$
$572$	145.492i	0.254357i
$573$	− 292.309i	− 0.510137i
$574$	0	0
$575$	−34.0143	−0.0591553
$576$	−67.8823	−0.117851
$577$	651.267i	1.12871i	0.825531	+	0.564356i	$0.190876\pi$
$577$	651.267i	1.12871i	−0.825531	+	0.564356i	$0.809124\pi$
$578$	−148.701	−0.257267
$579$	135.223i	0.233546i
$580$	− 129.751i	− 0.223708i
$581$	0	0
$582$	−151.279	−0.259930
$583$	−291.728	−0.500391
$584$	− 370.497i	− 0.634413i
$585$	147.618	0.252339
$586$	− 463.001i	− 0.790104i
$587$	− 823.029i	− 1.40209i	−0.713116	−	0.701046i	$-0.752717\pi$
$587$	− 823.029i	− 1.40209i	0.713116	−	0.701046i	$-0.247283\pi$
$588$	0	0
$589$	−17.6619	−0.0299863
$590$	375.110	0.635779
$591$	− 1158.72i	− 1.96060i
$592$	259.765	0.438791
$593$	808.418i	1.36327i	0.731694	+	0.681634i	$0.238730\pi$
$593$	808.418i	1.36327i	−0.731694	+	0.681634i	$0.761270\pi$
$594$	− 40.3084i	− 0.0678593i
$595$	0	0
$596$	−52.7939	−0.0885804
$597$	700.176	1.17282
$598$	− 17.6512i	− 0.0295170i
$599$	530.845	0.886218	0.443109	−	0.896468i	$-0.353875\pi$
$599$	530.845	0.886218	0.443109	+	0.896468i	$0.353875\pi$
$600$	177.060i	0.295099i
$601$	936.503i	1.55824i	0.626874	+	0.779121i	$0.284334\pi$
$601$	936.503i	1.55824i	−0.626874	+	0.779121i	$0.715666\pi$
$602$	0	0
$603$	−786.073	−1.30360
$604$	−268.426	−0.444415
$605$	− 172.178i	− 0.284592i
$606$	−168.551	−0.278137
$607$	− 602.121i	− 0.991962i	−0.868333	−	0.495981i	$-0.834809\pi$
$607$	− 602.121i	− 0.991962i	0.868333	−	0.495981i	$-0.165191\pi$
$608$	− 4.05845i	− 0.00667508i
$609$	0	0
$610$	296.756	0.486486
$611$	−262.471	−0.429576
$612$	230.108i	0.375993i
$613$	1096.90	1.78939	0.894695	−	0.446677i	$-0.147393\pi$
$613$	1096.90	1.78939	0.894695	+	0.446677i	$0.147393\pi$
$614$	− 362.236i	− 0.589961i
$615$	278.503i	0.452851i
$616$	0	0
$617$	−432.956	−0.701712	−0.350856	−	0.936429i	$-0.614109\pi$
$617$	−432.956	−0.701712	−0.350856	+	0.936429i	$0.614109\pi$
$618$	334.066	0.540560
$619$	225.110i	0.363668i	0.983329	+	0.181834i	$0.0582034\pi$
$619$	225.110i	0.363668i	−0.983329	+	0.181834i	$0.941797\pi$
$620$	155.927	0.251495
$621$	4.89023i	0.00787477i
$622$	− 305.940i	− 0.491865i
$623$	0	0
$624$	−91.8823	−0.147247
$625$	−26.6182	−0.0425891
$626$	221.775i	0.354273i
$627$	−39.7279	−0.0633619
$628$	− 453.389i	− 0.721958i
$629$	− 880.552i	− 1.39992i
$630$	0	0
$631$	750.514	1.18940	0.594702	−	0.803946i	$-0.297270\pi$
$631$	750.514	1.18940	0.594702	+	0.803946i	$0.297270\pi$
$632$	215.563	0.341081
$633$	535.543i	0.846040i
$634$	633.609	0.999384
$635$	192.112i	0.302538i
$636$	− 184.234i	− 0.289676i
$637$	0	0
$638$	383.647	0.601327
$639$	410.912	0.643054
$640$	35.8297i	0.0559839i
$641$	−1161.85	−1.81256	−0.906281	−	0.422675i	$-0.861091\pi$
$641$	−1161.85	−1.81256	−0.906281	+	0.422675i	$0.861091\pi$
$642$	281.547i	0.438547i
$643$	− 121.957i	− 0.189669i	−0.995493	−	0.0948347i	$-0.969768\pi$
$643$	− 121.957i	− 0.189669i	0.995493	−	0.0948347i	$-0.0302322\pi$
$644$	0	0
$645$	−85.8823	−0.133151
$646$	−13.7574	−0.0212962
$647$	− 158.775i	− 0.245403i	−0.992444	−	0.122701i	$-0.960844\pi$
$647$	− 158.775i	− 0.245403i	0.992444	−	0.122701i	$-0.0391557\pi$
$648$	241.456	0.372617
$649$	1109.12i	1.70897i
$650$	116.302i	0.178927i
$651$	0	0
$652$	−183.955	−0.282140
$653$	−390.941	−0.598685	−0.299342	−	0.954146i	$-0.596767\pi$
$653$	−390.941	−0.598685	−0.299342	+	0.954146i	$0.596767\pi$
$654$	445.345i	0.680955i
$655$	421.036	0.642803
$656$	− 84.1232i	− 0.128237i
$657$	− 1111.49i	− 1.69177i
$658$	0	0
$659$	−331.955	−0.503726	−0.251863	−	0.967763i	$-0.581043\pi$
$659$	−331.955	−0.503726	−0.251863	+	0.967763i	$0.581043\pi$
$660$	350.735	0.531417
$661$	647.820i	0.980061i	0.871705	+	0.490031i	$0.163014\pi$
$661$	647.820i	0.980061i	−0.871705	+	0.490031i	$0.836986\pi$
$662$	77.7918	0.117510
$663$	311.463i	0.469779i
$664$	− 305.418i	− 0.459967i
$665$	0	0
$666$	779.294	1.17011
$667$	−46.5442	−0.0697813
$668$	406.963i	0.609227i
$669$	−1744.41	−2.60749
$670$	414.906i	0.619263i
$671$	877.448i	1.30767i
$672$	0	0
$673$	100.956	0.150009	0.0750047	−	0.997183i	$-0.476103\pi$
$673$	100.956	0.150009	0.0750047	+	0.997183i	$0.476103\pi$
$674$	157.872	0.234232
$675$	− 32.2214i	− 0.0477354i
$676$	277.647	0.410720
$677$	− 743.177i	− 1.09775i	−0.835905	−	0.548875i	$-0.815056\pi$
$677$	− 743.177i	− 1.09775i	0.835905	−	0.548875i	$-0.184944\pi$
$678$	505.351i	0.745355i
$679$	0	0
$680$	121.456	0.178612
$681$	971.205	1.42615
$682$	461.044i	0.676017i
$683$	4.43442	0.00649256	0.00324628	−	0.999995i	$-0.498967\pi$
$683$	4.43442	0.00649256	0.00324628	+	0.999995i	$0.498967\pi$
$684$	− 12.1753i	− 0.0178002i
$685$	371.881i	0.542892i
$686$	0	0
$687$	349.669	0.508980
$688$	25.9411	0.0377051
$689$	− 121.015i	− 0.175638i
$690$	−42.5513	−0.0616685
$691$	977.169i	1.41414i	0.707145	+	0.707069i	$0.249983\pi$
$691$	977.169i	1.41414i	−0.707145	+	0.707069i	$0.750017\pi$
$692$	− 141.620i	− 0.204654i
$693$	0	0
$694$	−533.522	−0.768763
$695$	217.206	0.312527
$696$	242.283i	0.348108i
$697$	−285.161	−0.409127
$698$	288.547i	0.413392i
$699$	− 916.063i	− 1.31053i
$700$	0	0
$701$	−840.177	−1.19854	−0.599270	−	0.800547i	$-0.704542\pi$
$701$	−840.177	−1.19854	−0.599270	+	0.800547i	$0.704542\pi$
$702$	16.7208	0.0238188
$703$	46.5913i	0.0662750i
$704$	−105.941	−0.150485
$705$	632.733i	0.897494i
$706$	589.835i	0.835460i
$707$	0	0
$708$	−700.441	−0.989323
$709$	682.558	0.962705	0.481352	−	0.876527i	$-0.340146\pi$
$709$	682.558	0.962705	0.481352	+	0.876527i	$0.340146\pi$
$710$	− 216.888i	− 0.305476i
$711$	646.690	0.909551
$712$	− 474.913i	− 0.667012i
$713$	− 55.9340i	− 0.0784488i
$714$	0	0
$715$	230.382	0.322212
$716$	217.632	0.303956
$717$	− 807.466i	− 1.12617i
$718$	252.905	0.352235
$719$	− 137.625i	− 0.191412i	−0.995410	−	0.0957060i	$-0.969489\pi$
$719$	− 137.625i	− 0.191412i	0.995410	−	0.0957060i	$-0.0305109\pi$
$720$	107.489i	0.149290i
$721$	0	0
$722$	−509.803	−0.706099
$723$	207.125	0.286480
$724$	− 199.321i	− 0.275306i
$725$	306.676	0.423002
$726$	321.507i	0.442848i
$727$	− 264.137i	− 0.363325i	−0.983361	−	0.181662i	$-0.941852\pi$
$727$	− 264.137i	− 0.363325i	0.983361	−	0.181662i	$-0.0581478\pi$
$728$	0	0
$729$	643.368	0.882534
$730$	−586.669	−0.803656
$731$	− 87.9354i	− 0.120295i
$732$	−554.132	−0.757011
$733$	− 579.319i	− 0.790340i	−0.918608	−	0.395170i	$-0.870686\pi$
$733$	− 579.319i	− 0.790340i	0.918608	−	0.395170i	$-0.129314\pi$
$734$	889.530i	1.21189i
$735$	0	0
$736$	12.8528	0.0174631
$737$	−1226.79	−1.66458
$738$	− 252.370i	− 0.341964i
$739$	−198.095	−0.268059	−0.134029	−	0.990977i	$-0.542792\pi$
$739$	−198.095	−0.268059	−0.134029	+	0.990977i	$0.542792\pi$
$740$	− 411.328i	− 0.555848i
$741$	− 16.4800i	− 0.0222402i
$742$	0	0
$743$	976.690	1.31452	0.657261	−	0.753663i	$-0.271715\pi$
$743$	976.690	1.31452	0.657261	+	0.753663i	$0.271715\pi$
$744$	−291.161	−0.391346
$745$	83.5973i	0.112211i
$746$	361.414	0.484469
$747$	− 916.253i	− 1.22658i
$748$	359.120i	0.480107i
$749$	0	0
$750$	748.566	0.998087
$751$	−835.330	−1.11229	−0.556145	−	0.831085i	$-0.687720\pi$
$751$	−835.330	−1.11229	−0.556145	+	0.831085i	$0.687720\pi$
$752$	− 191.120i	− 0.254149i
$753$	679.602	0.902526
$754$	159.145i	0.211067i
$755$	425.044i	0.562972i
$756$	0	0
$757$	104.221	0.137677	0.0688383	−	0.997628i	$-0.478071\pi$
$757$	104.221	0.137677	0.0688383	+	0.997628i	$0.478071\pi$
$758$	−310.774	−0.409992
$759$	− 125.815i	− 0.165765i
$760$	−6.42641	−0.00845580
$761$	547.080i	0.718897i	0.933165	+	0.359448i	$0.117035\pi$
$761$	547.080i	0.718897i	−0.933165	+	0.359448i	$0.882965\pi$
$762$	− 358.730i	− 0.470774i
$763$	0	0
$764$	139.809	0.182996
$765$	364.368	0.476297
$766$	− 24.0921i	− 0.0314519i
$767$	−460.087	−0.599853
$768$	− 66.9046i	− 0.0871154i
$769$	341.205i	0.443700i	0.975081	+	0.221850i	$0.0712095\pi$
$769$	341.205i	0.443700i	−0.975081	+	0.221850i	$0.928790\pi$
$770$	0	0
$771$	414.640	0.537795
$772$	−64.6762	−0.0837774
$773$	− 490.993i	− 0.635179i	−0.948228	−	0.317590i	$-0.897127\pi$
$773$	− 490.993i	− 0.635179i	0.948228	−	0.317590i	$-0.102873\pi$
$774$	77.8234	0.100547
$775$	368.545i	0.475542i
$776$	− 72.3557i	− 0.0932419i
$777$	0	0
$778$	215.272	0.276699
$779$	15.0883	0.0193688
$780$	145.492i	0.186529i
$781$	641.294	0.821118
$782$	− 43.5686i	− 0.0557143i
$783$	− 44.0908i	− 0.0563101i
$784$	0	0
$785$	−717.926	−0.914555
$786$	−786.198	−1.00025
$787$	300.455i	0.381773i	0.981612	+	0.190887i	$0.0611363\pi$
$787$	300.455i	0.381773i	−0.981612	+	0.190887i	$0.938864\pi$
$788$	554.205	0.703306
$789$	− 1816.23i	− 2.30194i
$790$	− 341.337i	− 0.432072i
$791$	0	0
$792$	−317.823	−0.401292
$793$	−363.984	−0.458996
$794$	− 527.108i	− 0.663864i
$795$	−291.728	−0.366953
$796$	334.889i	0.420715i
$797$	370.072i	0.464331i	0.972676	+	0.232165i	$0.0745811\pi$
$797$	370.072i	0.464331i	−0.972676	+	0.232165i	$0.925419\pi$
$798$	0	0
$799$	−647.860	−0.810838
$800$	−84.6863	−0.105858
$801$	− 1424.74i	− 1.77870i
$802$	−921.463	−1.14896
$803$	− 1734.66i	− 2.16022i
$804$	− 774.753i	− 0.963623i
$805$	0	0
$806$	−191.251	−0.237284
$807$	−382.227	−0.473640
$808$	− 80.6168i	− 0.0997733i
$809$	491.235	0.607213	0.303607	−	0.952797i	$-0.401809\pi$
$809$	491.235	0.607213	0.303607	+	0.952797i	$0.401809\pi$
$810$	− 382.337i	− 0.472021i
$811$	− 156.802i	− 0.193344i	−0.995316	−	0.0966722i	$-0.969180\pi$
$811$	− 156.802i	− 0.193344i	0.995316	−	0.0966722i	$-0.0308199\pi$
$812$	0	0
$813$	−71.4853	−0.0879278
$814$	1216.21	1.49412
$815$	291.287i	0.357407i
$816$	−226.794	−0.277934
$817$	4.65279i	0.00569497i
$818$	− 754.575i	− 0.922463i
$819$	0	0
$820$	−133.206	−0.162446
$821$	−430.632	−0.524522	−0.262261	−	0.964997i	$-0.584468\pi$
$821$	−430.632	−0.524522	−0.262261	+	0.964997i	$0.584468\pi$
$822$	− 694.412i	− 0.844783i
$823$	−708.741	−0.861168	−0.430584	−	0.902550i	$-0.641692\pi$
$823$	−708.741	−0.861168	−0.430584	+	0.902550i	$0.641692\pi$
$824$	159.781i	0.193909i
$825$	828.990i	1.00484i
$826$	0	0
$827$	−1460.10	−1.76554	−0.882770	−	0.469805i	$-0.844324\pi$
$827$	−1460.10	−1.76554	−0.882770	+	0.469805i	$0.844324\pi$
$828$	38.5584	0.0465682
$829$	− 257.608i	− 0.310745i	−0.987856	−	0.155373i	$-0.950342\pi$
$829$	− 257.608i	− 0.310745i	0.987856	−	0.155373i	$-0.0496578\pi$
$830$	−483.618	−0.582673
$831$	1674.34i	2.01484i
$832$	− 43.9466i	− 0.0528204i
$833$	0	0
$834$	−405.588	−0.486316
$835$	644.412	0.771751
$836$	− 19.0016i	− 0.0227292i
$837$	52.9857	0.0633043
$838$	− 755.547i	− 0.901607i
$839$	213.621i	0.254613i	0.991863	+	0.127307i	$0.0406332\pi$
$839$	213.621i	0.254613i	−0.991863	+	0.127307i	$0.959367\pi$
$840$	0	0
$841$	−421.353	−0.501015
$842$	−222.343	−0.264065
$843$	2252.56i	2.67207i
$844$	−256.146	−0.303491
$845$	− 439.644i	− 0.520288i
$846$	− 573.360i	− 0.677730i
$847$	0	0
$848$	88.1177	0.103912
$849$	−1292.97	−1.52293
$850$	287.070i	0.337730i
$851$	−147.551	−0.173386
$852$	404.994i	0.475345i
$853$	1127.37i	1.32165i	0.750539	+	0.660826i	$0.229794\pi$
$853$	1127.37i	1.32165i	−0.750539	+	0.660826i	$0.770206\pi$
$854$	0	0
$855$	−19.2792	−0.0225488
$856$	−134.662	−0.157315
$857$	1270.42i	1.48241i	0.671280	+	0.741204i	$0.265745\pi$
$857$	1270.42i	1.48241i	−0.671280	+	0.741204i	$0.734255\pi$
$858$	−430.191	−0.501388
$859$	255.753i	0.297733i	0.988857	+	0.148867i	$0.0475625\pi$
$859$	255.753i	0.297733i	−0.988857	+	0.148867i	$0.952438\pi$
$860$	− 41.0768i	− 0.0477638i
$861$	0	0
$862$	323.199	0.374941
$863$	1114.73	1.29169	0.645844	−	0.763469i	$-0.276506\pi$
$863$	1114.73	1.29169	0.645844	+	0.763469i	$0.276506\pi$
$864$	12.1753i	0.0140918i
$865$	−224.251	−0.259249
$866$	67.2528i	0.0776591i
$867$	− 439.677i	− 0.507125i
$868$	0	0
$869$	1009.26	1.16141
$870$	383.647	0.440973
$871$	− 508.900i	− 0.584270i
$872$	−213.005	−0.244272
$873$	− 217.067i	− 0.248645i
$874$	2.30528i	0.00263762i
$875$	0	0
$876$	1095.48	1.25055
$877$	−1101.81	−1.25634	−0.628169	−	0.778077i	$-0.716195\pi$
$877$	−1101.81	−1.25634	−0.628169	+	0.778077i	$0.716195\pi$
$878$	− 104.445i	− 0.118958i
$879$	1369.00	1.55745
$880$	167.754i	0.190630i
$881$	− 217.067i	− 0.246387i	−0.992383	−	0.123194i	$-0.960686\pi$
$881$	− 217.067i	− 0.246387i	0.992383	−	0.123194i	$-0.0393136\pi$
$882$	0	0
$883$	−516.544	−0.584988	−0.292494	−	0.956267i	$-0.594485\pi$
$883$	−516.544	−0.584988	−0.292494	+	0.956267i	$0.594485\pi$
$884$	−148.971	−0.168519
$885$	1109.12i	1.25325i
$886$	−331.831	−0.374527
$887$	1129.81i	1.27374i	0.770970	+	0.636872i	$0.219772\pi$
$887$	1129.81i	1.27374i	−0.770970	+	0.636872i	$0.780228\pi$
$888$	768.071i	0.864944i
$889$	0	0
$890$	−752.007	−0.844952
$891$	1130.49	1.26879
$892$	− 834.339i	− 0.935357i
$893$	34.2792	0.0383866
$894$	− 156.101i	− 0.174609i
$895$	− 344.613i	− 0.385043i
$896$	0	0
$897$	52.1909	0.0581838
$898$	360.853	0.401841
$899$	504.306i	0.560964i
$900$	−254.059	−0.282288
$901$	− 298.702i	− 0.331523i
$902$	− 393.863i	− 0.436655i
$903$	0	0
$904$	−241.706	−0.267373
$905$	−315.618	−0.348749
$906$	− 793.682i	− 0.876029i
$907$	60.0223	0.0661767	0.0330884	−	0.999452i	$-0.489466\pi$
$907$	60.0223	0.0661767	0.0330884	+	0.999452i	$0.489466\pi$
$908$	464.520i	0.511586i
$909$	− 241.851i	− 0.266062i
$910$	0	0
$911$	1422.25	1.56120	0.780598	−	0.625033i	$-0.214915\pi$
$911$	1422.25	1.56120	0.780598	+	0.625033i	$0.214915\pi$
$912$	12.0000	0.0131579
$913$	− 1429.96i	− 1.56622i
$914$	206.101	0.225493
$915$	877.448i	0.958960i
$916$	167.244i	0.182581i
$917$	0	0
$918$	41.2721	0.0449587
$919$	1669.70	1.81686	0.908432	−	0.418033i	$-0.137280\pi$
$919$	1669.70	1.81686	0.908432	+	0.418033i	$0.137280\pi$
$920$	− 20.3520i	− 0.0221217i
$921$	1071.06	1.16293
$922$	1256.29i	1.36257i
$923$	266.022i	0.288215i
$924$	0	0
$925$	972.205	1.05103
$926$	−330.946	−0.357393
$927$	479.344i	0.517092i
$928$	−115.882	−0.124873
$929$	− 968.860i	− 1.04291i	−0.853280	−	0.521453i	$-0.825390\pi$
$929$	− 968.860i	− 1.04291i	0.853280	−	0.521453i	$-0.174610\pi$
$930$	461.044i	0.495746i
$931$	0	0
$932$	438.146	0.470114
$933$	904.602	0.969563
$934$	1112.45i	1.19106i
$935$	568.654	0.608186
$936$	− 131.840i	− 0.140854i
$937$	− 1212.57i	− 1.29410i	−0.762449	−	0.647049i	$-0.776003\pi$
$937$	− 1212.57i	− 1.29410i	0.762449	−	0.647049i	$-0.223997\pi$
$938$	0	0
$939$	−655.742	−0.698341
$940$	−302.632	−0.321949
$941$	− 1494.06i	− 1.58774i	−0.608087	−	0.793870i	$-0.708063\pi$
$941$	− 1494.06i	− 1.58774i	0.608087	−	0.793870i	$-0.291937\pi$
$942$	1340.58	1.42312
$943$	47.7836i	0.0506719i
$944$	− 335.016i	− 0.354889i
$945$	0	0
$946$	121.456	0.128389
$947$	775.462	0.818862	0.409431	−	0.912341i	$-0.365727\pi$
$947$	775.462	0.818862	0.409431	+	0.912341i	$0.365727\pi$
$948$	637.377i	0.672339i
$949$	719.574	0.758244
$950$	− 15.1893i	− 0.0159887i
$951$	1873.45i	1.96998i
$952$	0	0
$953$	1055.40	1.10745	0.553723	−	0.832701i	$-0.313206\pi$
$953$	1055.40	1.10745	0.553723	+	0.832701i	$0.313206\pi$
$954$	264.353	0.277100
$955$	− 221.383i	− 0.231814i
$956$	386.205	0.403980
$957$	1134.37i	1.18533i
$958$	− 1042.18i	− 1.08787i
$959$	0	0
$960$	−105.941	−0.110355
$961$	354.955	0.369360
$962$	504.510i	0.524439i
$963$	−403.986	−0.419507
$964$	99.0663i	0.102766i
$965$	102.412i	0.106127i
$966$	0	0
$967$	1221.63	1.26332	0.631661	−	0.775245i	$-0.282373\pi$
$967$	1221.63	1.26332	0.631661	+	0.775245i	$0.282373\pi$
$968$	−153.775	−0.158858
$969$	− 40.6777i	− 0.0419791i
$970$	−114.573	−0.118116
$971$	526.259i	0.541976i	0.962583	+	0.270988i	$0.0873504\pi$
$971$	526.259i	0.541976i	−0.962583	+	0.270988i	$0.912650\pi$
$972$	675.194i	0.694644i
$973$	0	0
$974$	−382.825	−0.393045
$975$	−343.882	−0.352700
$976$	− 265.037i	− 0.271555i
$977$	−1000.10	−1.02365	−0.511823	−	0.859091i	$-0.671030\pi$
$977$	−1000.10	−1.02365	−0.511823	+	0.859091i	$0.671030\pi$
$978$	− 543.919i	− 0.556154i
$979$	− 2223.53i	− 2.27123i
$980$	0	0
$981$	−639.015	−0.651392
$982$	−1075.03	−1.09473
$983$	1075.70i	1.09430i	0.837033	+	0.547152i	$0.184288\pi$
$983$	1075.70i	1.09430i	−0.837033	+	0.547152i	$0.815712\pi$
$984$	248.735	0.252780
$985$	− 877.564i	− 0.890928i
$986$	392.819i	0.398396i
$987$	0	0
$988$	7.88225	0.00797799
$989$	−14.7351	−0.0148990
$990$	503.262i	0.508345i
$991$	−1876.03	−1.89307	−0.946536	−	0.322597i	$-0.895444\pi$
$991$	−1876.03	−1.89307	−0.946536	+	0.322597i	$0.895444\pi$
$992$	− 139.260i	− 0.140383i
$993$	230.015i	0.231636i
$994$	0	0
$995$	530.285	0.532949
$996$	903.058	0.906685
$997$	582.224i	0.583976i	0.956422	+	0.291988i	$0.0943167\pi$
$997$	582.224i	0.583976i	−0.956422	+	0.291988i	$0.905683\pi$
$998$	−177.473	−0.177829
$999$	− 139.774i	− 0.139914i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	98.3.b.b.97.1		4
3.2	odd	2		882.3.c.f.685.4		4
4.3	odd	2		784.3.c.e.97.4		4
7.2	even	3		98.3.d.a.31.2		4
7.3	odd	6		98.3.d.a.19.2		4
7.4	even	3		14.3.d.a.5.2	yes	4
7.5	odd	6		14.3.d.a.3.2	✓	4
7.6	odd	2	inner	98.3.b.b.97.2		4
21.2	odd	6		882.3.n.b.325.1		4
21.5	even	6		126.3.n.c.73.1		4
21.11	odd	6		126.3.n.c.19.1		4
21.17	even	6		882.3.n.b.19.1		4
21.20	even	2		882.3.c.f.685.3		4
28.3	even	6		784.3.s.c.705.1		4
28.11	odd	6		112.3.s.b.33.2		4
28.19	even	6		112.3.s.b.17.2		4
28.23	odd	6		784.3.s.c.129.1		4
28.27	even	2		784.3.c.e.97.1		4
35.4	even	6		350.3.k.a.201.1		4
35.12	even	12		350.3.i.a.199.1		8
35.18	odd	12		350.3.i.a.299.1		8
35.19	odd	6		350.3.k.a.101.1		4
35.32	odd	12		350.3.i.a.299.4		8
35.33	even	12		350.3.i.a.199.4		8
56.5	odd	6		448.3.s.d.129.2		4
56.11	odd	6		448.3.s.c.257.1		4
56.19	even	6		448.3.s.c.129.1		4
56.53	even	6		448.3.s.d.257.2		4
84.11	even	6		1008.3.cg.l.145.1		4
84.47	odd	6		1008.3.cg.l.577.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.3.d.a.3.2	✓	4	7.5	odd	6
14.3.d.a.5.2	yes	4	7.4	even	3
98.3.b.b.97.1		4	1.1	even	1	trivial
98.3.b.b.97.2		4	7.6	odd	2	inner
98.3.d.a.19.2		4	7.3	odd	6
98.3.d.a.31.2		4	7.2	even	3
112.3.s.b.17.2		4	28.19	even	6
112.3.s.b.33.2		4	28.11	odd	6
126.3.n.c.19.1		4	21.11	odd	6
126.3.n.c.73.1		4	21.5	even	6
350.3.i.a.199.1		8	35.12	even	12
350.3.i.a.199.4		8	35.33	even	12
350.3.i.a.299.1		8	35.18	odd	12
350.3.i.a.299.4		8	35.32	odd	12
350.3.k.a.101.1		4	35.19	odd	6
350.3.k.a.201.1		4	35.4	even	6
448.3.s.c.129.1		4	56.19	even	6
448.3.s.c.257.1		4	56.11	odd	6
448.3.s.d.129.2		4	56.5	odd	6
448.3.s.d.257.2		4	56.53	even	6
784.3.c.e.97.1		4	28.27	even	2
784.3.c.e.97.4		4	4.3	odd	2
784.3.s.c.129.1		4	28.23	odd	6
784.3.s.c.705.1		4	28.3	even	6
882.3.c.f.685.3		4	21.20	even	2
882.3.c.f.685.4		4	3.2	odd	2
882.3.n.b.19.1		4	21.17	even	6
882.3.n.b.325.1		4	21.2	odd	6
1008.3.cg.l.145.1		4	84.11	even	6
1008.3.cg.l.577.1		4	84.47	odd	6

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:	\( N \)	\(=\)	\( 98 = 2 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	98.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} -4.18154i q^{3} +2.00000 q^{4} -3.16693i q^{5} +5.91359i q^{6} -2.82843 q^{8} -8.48528 q^{9} +O(q^{10})\)	\(q-1.41421 q^{2} -4.18154i q^{3} +2.00000 q^{4} -3.16693i q^{5} +5.91359i q^{6} -2.82843 q^{8} -8.48528 q^{9} +4.47871i q^{10} -13.2426 q^{11} -8.36308i q^{12} -5.49333i q^{13} -13.2426 q^{15} +4.00000 q^{16} -13.5592i q^{17} +12.0000 q^{18} +0.717439i q^{19} -6.33386i q^{20} +18.7279 q^{22} -2.27208 q^{23} +11.8272i q^{24} +14.9706 q^{25} +7.76874i q^{26} -2.15232i q^{27} +20.4853 q^{29} +18.7279 q^{30} +24.6180i q^{31} -5.65685 q^{32} +55.3746i q^{33} +19.1757i q^{34} -16.9706 q^{36} +64.9411 q^{37} -1.01461i q^{38} -22.9706 q^{39} +8.95743i q^{40} -21.0308i q^{41} +6.48528 q^{43} -26.4853 q^{44} +26.8723i q^{45} +3.21320 q^{46} -47.7800i q^{47} -16.7262i q^{48} -21.1716 q^{50} -56.6985 q^{51} -10.9867i q^{52} +22.0294 q^{53} +3.04384i q^{54} +41.9385i q^{55} +3.00000 q^{57} -28.9706 q^{58} -83.7539i q^{59} -26.4853 q^{60} -66.2593i q^{61} -34.8151i q^{62} +8.00000 q^{64} -17.3970 q^{65} -78.3116i q^{66} +92.6396 q^{67} -27.1185i q^{68} +9.50079i q^{69} -48.4264 q^{71} +24.0000 q^{72} +130.991i q^{73} -91.8406 q^{74} -62.6000i q^{75} +1.43488i q^{76} +32.4853 q^{78} -76.2132 q^{79} -12.6677i q^{80} -85.3675 q^{81} +29.7420i q^{82} +107.981i q^{83} -42.9411 q^{85} -9.17157 q^{86} -85.6600i q^{87} +37.4558 q^{88} +167.907i q^{89} -38.0031i q^{90} -4.54416 q^{92} +102.941 q^{93} +67.5711i q^{94} +2.27208 q^{95} +23.6544i q^{96} +25.5816i q^{97} +112.368 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{4}+O(q^{10}) \) 4 * q + 8 * q^4	\( 4 q + 8 q^{4} - 36 q^{11} - 36 q^{15} + 16 q^{16} + 48 q^{18} + 24 q^{22} - 60 q^{23} - 8 q^{25} + 48 q^{29} + 24 q^{30} + 124 q^{37} - 24 q^{39} - 8 q^{43} - 72 q^{44} - 72 q^{46} - 96 q^{50} - 108 q^{51} + 156 q^{53} + 12 q^{57} - 48 q^{58} - 72 q^{60} + 32 q^{64} + 168 q^{65} + 116 q^{67} - 24 q^{71} + 96 q^{72} - 192 q^{74} + 96 q^{78} - 220 q^{79} - 36 q^{81} - 36 q^{85} - 48 q^{86} + 48 q^{88} - 120 q^{92} + 276 q^{93} + 60 q^{95} + 144 q^{99}+O(q^{100}) \) 4 * q + 8 * q^4 - 36 * q^11 - 36 * q^15 + 16 * q^16 + 48 * q^18 + 24 * q^22 - 60 * q^23 - 8 * q^25 + 48 * q^29 + 24 * q^30 + 124 * q^37 - 24 * q^39 - 8 * q^43 - 72 * q^44 - 72 * q^46 - 96 * q^50 - 108 * q^51 + 156 * q^53 + 12 * q^57 - 48 * q^58 - 72 * q^60 + 32 * q^64 + 168 * q^65 + 116 * q^67 - 24 * q^71 + 96 * q^72 - 192 * q^74 + 96 * q^78 - 220 * q^79 - 36 * q^81 - 36 * q^85 - 48 * q^86 + 48 * q^88 - 120 * q^92 + 276 * q^93 + 60 * q^95 + 144 * q^99

Introduction

L-functions

Modular forms

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