LMFDB - Embedded newform 363.3.c.e.241.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [363,3,Mod(241,363)]

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, 1]))

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

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

chi := DirichletCharacter("363.241");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 363 = 3 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	363.c (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:	$9.89103359628$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} + 3 x^{14} - 4 x^{13} + 77 x^{12} + 88 x^{11} - 577 x^{10} + 578 x^{9} + 1520 x^{8} + \cdots + 83521 $ x^16 - 2x^15 + 3x^14 - 4x^13 + 77x^12 + 88x^11 - 577x^10 + 578x^9 + 1520x^8 + 1868x^7 - 1619x^6 - 16804x^5 + 32427x^4 + 43316x^3 - 71672x^2 + 83521
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8}\cdot 11^{4} $
Twist minimal:	no (minimal twist has level 33)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			241.14
Root			$0.988132 + 0.846795i$ of defining polynomial
Character	$\chi$	$=$	363.241
Dual form			363.3.c.e.241.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.53176i q^{2} -1.73205 q^{3} -2.40981 q^{4} -8.44690 q^{5} -4.38514i q^{6} -2.44043i q^{7} +4.02597i q^{8} +3.00000 q^{9} +O(q^{10})$	\(q+2.53176i q^{2} -1.73205 q^{3} -2.40981 q^{4} -8.44690 q^{5} -4.38514i q^{6} -2.44043i q^{7} +4.02597i q^{8} +3.00000 q^{9} -21.3855i q^{10} +4.17392 q^{12} -11.7756i q^{13} +6.17858 q^{14} +14.6305 q^{15} -19.8321 q^{16} -2.02988i q^{17} +7.59528i q^{18} -8.47011i q^{19} +20.3555 q^{20} +4.22695i q^{21} +41.9571 q^{23} -6.97318i q^{24} +46.3502 q^{25} +29.8131 q^{26} -5.19615 q^{27} +5.88098i q^{28} +24.6142i q^{29} +37.0409i q^{30} +21.8135 q^{31} -34.1061i q^{32} +5.13918 q^{34} +20.6141i q^{35} -7.22944 q^{36} -16.0044 q^{37} +21.4443 q^{38} +20.3960i q^{39} -34.0070i q^{40} -52.6166i q^{41} -10.7016 q^{42} -42.3507i q^{43} -25.3407 q^{45} +106.225i q^{46} -16.1814 q^{47} +34.3501 q^{48} +43.0443 q^{49} +117.348i q^{50} +3.51586i q^{51} +28.3771i q^{52} -49.5329 q^{53} -13.1554i q^{54} +9.82509 q^{56} +14.6707i q^{57} -62.3174 q^{58} +19.9324 q^{59} -35.2567 q^{60} -118.906i q^{61} +55.2266i q^{62} -7.32129i q^{63} +7.02040 q^{64} +99.4678i q^{65} -4.41442 q^{67} +4.89164i q^{68} -72.6718 q^{69} -52.1899 q^{70} +6.03591 q^{71} +12.0779i q^{72} +6.08043i q^{73} -40.5193i q^{74} -80.2809 q^{75} +20.4114i q^{76} -51.6379 q^{78} -103.463i q^{79} +167.519 q^{80} +9.00000 q^{81} +133.213 q^{82} +30.1663i q^{83} -10.1862i q^{84} +17.1462i q^{85} +107.222 q^{86} -42.6331i q^{87} -60.4650 q^{89} -64.1566i q^{90} -28.7376 q^{91} -101.109 q^{92} -37.7821 q^{93} -40.9674i q^{94} +71.5462i q^{95} +59.0736i q^{96} +36.6892 q^{97} +108.978i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 20 q^{4} - 4 q^{5} + 48 q^{9}+O(q^{10}) $ 16 * q - 20 * q^4 - 4 * q^5 + 48 * q^9	$ 16 q - 20 q^{4} - 4 q^{5} + 48 q^{9} - 24 q^{12} - 52 q^{14} + 36 q^{15} - 44 q^{16} - 108 q^{20} + 132 q^{23} + 88 q^{25} - 4 q^{26} + 40 q^{31} - 368 q^{34} - 60 q^{36} - 16 q^{37} + 280 q^{38} + 36 q^{42} - 12 q^{45} + 80 q^{47} + 144 q^{48} - 140 q^{49} - 128 q^{53} + 524 q^{56} + 140 q^{58} - 220 q^{59} - 384 q^{60} - 8 q^{64} + 36 q^{67} - 180 q^{69} - 100 q^{70} + 644 q^{71} + 312 q^{75} - 312 q^{78} + 264 q^{80} + 144 q^{81} - 476 q^{82} + 76 q^{86} + 76 q^{89} - 624 q^{91} + 120 q^{92} - 336 q^{93} + 216 q^{97}+O(q^{100}) $ 16 * q - 20 * q^4 - 4 * q^5 + 48 * q^9 - 24 * q^12 - 52 * q^14 + 36 * q^15 - 44 * q^16 - 108 * q^20 + 132 * q^23 + 88 * q^25 - 4 * q^26 + 40 * q^31 - 368 * q^34 - 60 * q^36 - 16 * q^37 + 280 * q^38 + 36 * q^42 - 12 * q^45 + 80 * q^47 + 144 * q^48 - 140 * q^49 - 128 * q^53 + 524 * q^56 + 140 * q^58 - 220 * q^59 - 384 * q^60 - 8 * q^64 + 36 * q^67 - 180 * q^69 - 100 * q^70 + 644 * q^71 + 312 * q^75 - 312 * q^78 + 264 * q^80 + 144 * q^81 - 476 * q^82 + 76 * q^86 + 76 * q^89 - 624 * q^91 + 120 * q^92 - 336 * q^93 + 216 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/363\mathbb{Z}\right)^\times$.

$n$	$122$	$244$
$\chi(n)$	$1$	$-1$

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 $

$2$	2.53176i	1.26588i	0.774201	+	0.632940i	$0.218152\pi$
$2$	2.53176i	1.26588i	−0.774201	+	0.632940i	$0.781848\pi$
$3$	−1.73205	−0.577350
$3$	−1.73205	−0.577350
$4$	−2.40981	−0.602454
$5$	−8.44690	−1.68938	−0.844690	−	0.535255i	$-0.820215\pi$
$5$	−8.44690	−1.68938	−0.844690	+	0.535255i	$0.820215\pi$
$6$	− 4.38514i	− 0.730856i
$7$	− 2.44043i	− 0.348633i	−0.984690	−	0.174316i	$-0.944228\pi$
$7$	− 2.44043i	− 0.348633i	0.984690	−	0.174316i	$-0.0557715\pi$
$8$	4.02597i	0.503246i
$9$	3.00000	0.333333
$10$	− 21.3855i	− 2.13855i
$11$	0	0
$11$	0	0
$12$	4.17392	0.347827
$13$	− 11.7756i	− 0.905819i	−0.891556	−	0.452910i	$-0.850386\pi$
$13$	− 11.7756i	− 0.905819i	0.891556	−	0.452910i	$-0.149614\pi$
$14$	6.17858	0.441327
$15$	14.6305	0.975365
$16$	−19.8321	−1.23950
$17$	− 2.02988i	− 0.119405i	−0.998216	−	0.0597024i	$-0.980985\pi$
$17$	− 2.02988i	− 0.119405i	0.998216	−	0.0597024i	$-0.0190152\pi$
$18$	7.59528i	0.421960i
$19$	− 8.47011i	− 0.445795i	−0.974842	−	0.222898i	$-0.928448\pi$
$19$	− 8.47011i	− 0.445795i	0.974842	−	0.222898i	$-0.0715516\pi$
$20$	20.3555	1.01777
$21$	4.22695i	0.201283i
$22$	0	0
$23$	41.9571	1.82422	0.912110	−	0.409946i	$-0.134452\pi$
$23$	41.9571	1.82422	0.912110	+	0.409946i	$0.134452\pi$
$24$	− 6.97318i	− 0.290549i
$25$	46.3502	1.85401
$26$	29.8131	1.14666
$27$	−5.19615	−0.192450
$28$	5.88098i	0.210035i
$29$	24.6142i	0.848767i	0.905483	+	0.424383i	$0.139509\pi$
$29$	24.6142i	0.848767i	−0.905483	+	0.424383i	$0.860491\pi$
$30$	37.0409i	1.23470i
$31$	21.8135	0.703661	0.351831	−	0.936064i	$-0.385559\pi$
$31$	21.8135	0.703661	0.351831	+	0.936064i	$0.385559\pi$
$32$	− 34.1061i	− 1.06582i
$33$	0	0
$34$	5.13918	0.151152
$35$	20.6141i	0.588973i
$36$	−7.22944	−0.200818
$37$	−16.0044	−0.432551	−0.216276	−	0.976332i	$-0.569391\pi$
$37$	−16.0044	−0.432551	−0.216276	+	0.976332i	$0.569391\pi$
$38$	21.4443	0.564324
$39$	20.3960i	0.522975i
$40$	− 34.0070i	− 0.850174i
$41$	− 52.6166i	− 1.28333i	−0.766985	−	0.641665i	$-0.778244\pi$
$41$	− 52.6166i	− 1.28333i	0.766985	−	0.641665i	$-0.221756\pi$
$42$	−10.7016	−0.254800
$43$	− 42.3507i	− 0.984900i	−0.870341	−	0.492450i	$-0.836101\pi$
$43$	− 42.3507i	− 0.984900i	0.870341	−	0.492450i	$-0.163899\pi$
$44$	0	0
$45$	−25.3407	−0.563127
$46$	106.225i	2.30924i
$47$	−16.1814	−0.344284	−0.172142	−	0.985072i	$-0.555069\pi$
$47$	−16.1814	−0.344284	−0.172142	+	0.985072i	$0.555069\pi$
$48$	34.3501	0.715628
$49$	43.0443	0.878455
$50$	117.348i	2.34695i
$51$	3.51586i	0.0689384i
$52$	28.3771i	0.545714i
$53$	−49.5329	−0.934582	−0.467291	−	0.884104i	$-0.654770\pi$
$53$	−49.5329	−0.934582	−0.467291	+	0.884104i	$0.654770\pi$
$54$	− 13.1554i	− 0.243619i
$55$	0	0
$56$	9.82509	0.175448
$57$	14.6707i	0.257380i
$58$	−62.3174	−1.07444
$59$	19.9324	0.337838	0.168919	−	0.985630i	$-0.445972\pi$
$59$	19.9324	0.337838	0.168919	+	0.985630i	$0.445972\pi$
$60$	−35.2567	−0.587612
$61$	− 118.906i	− 1.94928i	−0.223784	−	0.974639i	$-0.571841\pi$
$61$	− 118.906i	− 1.94928i	0.223784	−	0.974639i	$-0.428159\pi$
$62$	55.2266i	0.890751i
$63$	− 7.32129i	− 0.116211i
$64$	7.02040	0.109694
$65$	99.4678i	1.53027i
$66$	0	0
$67$	−4.41442	−0.0658869	−0.0329434	−	0.999457i	$-0.510488\pi$
$67$	−4.41442	−0.0658869	−0.0329434	+	0.999457i	$0.510488\pi$
$68$	4.89164i	0.0719359i
$69$	−72.6718	−1.05321
$70$	−52.1899	−0.745570
$71$	6.03591	0.0850128	0.0425064	−	0.999096i	$-0.486466\pi$
$71$	6.03591	0.0850128	0.0425064	+	0.999096i	$0.486466\pi$
$72$	12.0779i	0.167749i
$73$	6.08043i	0.0832936i	0.999132	+	0.0416468i	$0.0132604\pi$
$73$	6.08043i	0.0832936i	−0.999132	+	0.0416468i	$0.986740\pi$
$74$	− 40.5193i	− 0.547558i
$75$	−80.2809	−1.07041
$76$	20.4114i	0.268571i
$77$	0	0
$78$	−51.6379	−0.662024
$79$	− 103.463i	− 1.30965i	−0.755779	−	0.654827i	$-0.772742\pi$
$79$	− 103.463i	− 1.30965i	0.755779	−	0.654827i	$-0.227258\pi$
$80$	167.519	2.09399
$81$	9.00000	0.111111
$82$	133.213	1.62454
$83$	30.1663i	0.363449i	0.983349	+	0.181724i	$0.0581679\pi$
$83$	30.1663i	0.363449i	−0.983349	+	0.181724i	$0.941832\pi$
$84$	− 10.1862i	− 0.121264i
$85$	17.1462i	0.201720i
$86$	107.222	1.24677
$87$	− 42.6331i	− 0.490036i
$88$	0	0
$89$	−60.4650	−0.679382	−0.339691	−	0.940537i	$-0.610323\pi$
$89$	−60.4650	−0.679382	−0.339691	+	0.940537i	$0.610323\pi$
$90$	− 64.1566i	− 0.712851i
$91$	−28.7376	−0.315798
$92$	−101.109	−1.09901
$93$	−37.7821	−0.406259
$94$	− 40.9674i	− 0.435823i
$95$	71.5462i	0.753118i
$96$	59.0736i	0.615350i
$97$	36.6892	0.378239	0.189120	−	0.981954i	$-0.439437\pi$
$97$	36.6892	0.378239	0.189120	+	0.981954i	$0.439437\pi$
$98$	108.978i	1.11202i
$99$	0	0
$100$	−111.695	−1.11695
$101$	− 141.754i	− 1.40350i	−0.712422	−	0.701752i	$-0.752402\pi$
$101$	− 141.754i	− 1.40350i	0.712422	−	0.701752i	$-0.247598\pi$
$102$	−8.90131	−0.0872678
$103$	39.0213	0.378847	0.189424	−	0.981895i	$-0.439338\pi$
$103$	39.0213	0.378847	0.189424	+	0.981895i	$0.439338\pi$
$104$	47.4084	0.455850
$105$	− 35.7046i	− 0.340044i
$106$	− 125.405i	− 1.18307i
$107$	− 116.623i	− 1.08994i	−0.838456	−	0.544969i	$-0.816541\pi$
$107$	− 116.623i	− 1.08994i	0.838456	−	0.544969i	$-0.183459\pi$
$108$	12.5218	0.115942
$109$	81.2670i	0.745569i	0.927918	+	0.372784i	$0.121597\pi$
$109$	81.2670i	0.745569i	−0.927918	+	0.372784i	$0.878403\pi$
$110$	0	0
$111$	27.7204	0.249733
$112$	48.3987i	0.432131i
$113$	−76.4454	−0.676508	−0.338254	−	0.941055i	$-0.609836\pi$
$113$	−76.4454	−0.676508	−0.338254	+	0.941055i	$0.609836\pi$
$114$	−37.1426	−0.325812
$115$	−354.407	−3.08180
$116$	− 59.3157i	− 0.511343i
$117$	− 35.3269i	− 0.301940i
$118$	50.4642i	0.427663i
$119$	−4.95378	−0.0416284
$120$	58.9018i	0.490848i
$121$	0	0
$122$	301.041	2.46755
$123$	91.1345i	0.740931i
$124$	−52.5665	−0.423923
$125$	−180.343	−1.44274
$126$	18.5357	0.147109
$127$	− 21.0623i	− 0.165845i	−0.996556	−	0.0829224i	$-0.973575\pi$
$127$	− 21.0623i	− 0.165845i	0.996556	−	0.0829224i	$-0.0264254\pi$
$128$	− 118.651i	− 0.926958i
$129$	73.3536i	0.568632i
$130$	−251.829	−1.93714
$131$	− 106.847i	− 0.815628i	−0.913065	−	0.407814i	$-0.866291\pi$
$131$	− 106.847i	− 0.815628i	0.913065	−	0.407814i	$-0.133709\pi$
$132$	0	0
$133$	−20.6707	−0.155419
$134$	− 11.1763i	− 0.0834049i
$135$	43.8914	0.325122
$136$	8.17224	0.0600900
$137$	119.183	0.869946	0.434973	−	0.900443i	$-0.356758\pi$
$137$	119.183	0.869946	0.434973	+	0.900443i	$0.356758\pi$
$138$	− 183.988i	− 1.33324i
$139$	7.42431i	0.0534123i	0.999643	+	0.0267061i	$0.00850184\pi$
$139$	7.42431i	0.0534123i	−0.999643	+	0.0267061i	$0.991498\pi$
$140$	− 49.6761i	− 0.354829i
$141$	28.0270	0.198773
$142$	15.2815i	0.107616i
$143$	0	0
$144$	−59.4962	−0.413168
$145$	− 207.914i	− 1.43389i
$146$	−15.3942	−0.105440
$147$	−74.5549	−0.507176
$148$	38.5676	0.260592
$149$	− 141.263i	− 0.948075i	−0.880505	−	0.474038i	$-0.842796\pi$
$149$	− 141.263i	− 0.948075i	0.880505	−	0.474038i	$-0.157204\pi$
$150$	− 203.252i	− 1.35501i
$151$	63.7331i	0.422074i	0.977478	+	0.211037i	$0.0676840\pi$
$151$	63.7331i	0.422074i	−0.977478	+	0.211037i	$0.932316\pi$
$152$	34.1004	0.224345
$153$	− 6.08965i	− 0.0398016i
$154$	0	0
$155$	−184.257	−1.18875
$156$	− 49.1506i	− 0.315068i
$157$	36.8901	0.234969	0.117485	−	0.993075i	$-0.462517\pi$
$157$	36.8901	0.234969	0.117485	+	0.993075i	$0.462517\pi$
$158$	261.943	1.65787
$159$	85.7934	0.539581
$160$	288.091i	1.80057i
$161$	− 102.393i	− 0.635983i
$162$	22.7859i	0.140653i
$163$	−53.6360	−0.329055	−0.164528	−	0.986372i	$-0.552610\pi$
$163$	−53.6360	−0.329055	−0.164528	+	0.986372i	$0.552610\pi$
$164$	126.796i	0.773147i
$165$	0	0
$166$	−76.3738	−0.460083
$167$	131.995i	0.790387i	0.918598	+	0.395194i	$0.129323\pi$
$167$	131.995i	0.790387i	−0.918598	+	0.395194i	$0.870677\pi$
$168$	−17.0176	−0.101295
$169$	30.3341	0.179492
$170$	−43.4101	−0.255354
$171$	− 25.4103i	− 0.148598i
$172$	102.057i	0.593357i
$173$	− 66.5688i	− 0.384791i	−0.981317	−	0.192395i	$-0.938374\pi$
$173$	− 66.5688i	− 0.384791i	0.981317	−	0.192395i	$-0.0616256\pi$
$174$	107.937	0.620327
$175$	− 113.114i	− 0.646368i
$176$	0	0
$177$	−34.5240	−0.195051
$178$	− 153.083i	− 0.860017i
$179$	178.511	0.997268	0.498634	−	0.866813i	$-0.333835\pi$
$179$	178.511	0.997268	0.498634	+	0.866813i	$0.333835\pi$
$180$	61.0664	0.339258
$181$	−52.1144	−0.287925	−0.143962	−	0.989583i	$-0.545984\pi$
$181$	−52.1144	−0.287925	−0.143962	+	0.989583i	$0.545984\pi$
$182$	− 72.7568i	− 0.399763i
$183$	205.951i	1.12542i
$184$	168.918i	0.918032i
$185$	135.188	0.730744
$186$	− 95.6553i	− 0.514276i
$187$	0	0
$188$	38.9941	0.207415
$189$	12.6808i	0.0670944i
$190$	−181.138	−0.953358
$191$	−358.333	−1.87609	−0.938046	−	0.346512i	$-0.887366\pi$
$191$	−358.333	−1.87609	−0.938046	+	0.346512i	$0.887366\pi$
$192$	−12.1597	−0.0633317
$193$	213.899i	1.10829i	0.832422	+	0.554143i	$0.186954\pi$
$193$	213.899i	1.10829i	−0.832422	+	0.554143i	$0.813046\pi$
$194$	92.8883i	0.478806i
$195$	− 172.283i	− 0.883504i
$196$	−103.729	−0.529229
$197$	276.568i	1.40390i	0.712227	+	0.701949i	$0.247687\pi$
$197$	276.568i	1.40390i	−0.712227	+	0.701949i	$0.752313\pi$
$198$	0	0
$199$	78.8085	0.396022	0.198011	−	0.980200i	$-0.436552\pi$
$199$	78.8085	0.396022	0.198011	+	0.980200i	$0.436552\pi$
$200$	186.604i	0.933022i
$201$	7.64600	0.0380398
$202$	358.887	1.77667
$203$	60.0693	0.295908
$204$	− 8.47257i	− 0.0415322i
$205$	444.447i	2.16803i
$206$	98.7926i	0.479576i
$207$	125.871	0.608073
$208$	233.535i	1.12277i
$209$	0	0
$210$	90.3955	0.430455
$211$	218.966i	1.03776i	0.854849	+	0.518878i	$0.173650\pi$
$211$	218.966i	1.03776i	−0.854849	+	0.518878i	$0.826350\pi$
$212$	119.365	0.563042
$213$	−10.4545	−0.0490821
$214$	295.263	1.37973
$215$	357.732i	1.66387i
$216$	− 20.9196i	− 0.0968498i
$217$	− 53.2343i	− 0.245319i
$218$	−205.749	−0.943801
$219$	− 10.5316i	− 0.0480896i
$220$	0	0
$221$	−23.9032	−0.108159
$222$	70.1815i	0.316133i
$223$	−209.548	−0.939679	−0.469839	−	0.882752i	$-0.655688\pi$
$223$	−209.548	−0.939679	−0.469839	+	0.882752i	$0.655688\pi$
$224$	−83.2336	−0.371579
$225$	139.051	0.618003
$226$	− 193.542i	− 0.856379i
$227$	83.3414i	0.367143i	0.983006	+	0.183571i	$0.0587658\pi$
$227$	83.3414i	0.367143i	−0.983006	+	0.183571i	$0.941234\pi$
$228$	− 35.3536i	− 0.155060i
$229$	330.662	1.44394	0.721969	−	0.691925i	$-0.243237\pi$
$229$	330.662	1.44394	0.721969	+	0.691925i	$0.243237\pi$
$230$	− 897.274i	− 3.90119i
$231$	0	0
$232$	−99.0962	−0.427139
$233$	165.410i	0.709915i	0.934882	+	0.354958i	$0.115505\pi$
$233$	165.410i	0.709915i	−0.934882	+	0.354958i	$0.884495\pi$
$234$	89.4394	0.382220
$235$	136.682	0.581628
$236$	−48.0335	−0.203532
$237$	179.203i	0.756129i
$238$	− 12.5418i	− 0.0526966i
$239$	− 179.946i	− 0.752911i	−0.926435	−	0.376455i	$-0.877143\pi$
$239$	− 179.946i	− 0.752911i	0.926435	−	0.376455i	$-0.122857\pi$
$240$	−290.152	−1.20897
$241$	− 226.632i	− 0.940380i	−0.882565	−	0.470190i	$-0.844185\pi$
$241$	− 226.632i	− 0.940380i	0.882565	−	0.470190i	$-0.155815\pi$
$242$	0	0
$243$	−15.5885	−0.0641500
$244$	286.541i	1.17435i
$245$	−363.591	−1.48405
$246$	−230.731	−0.937930
$247$	−99.7410	−0.403810
$248$	87.8205i	0.354115i
$249$	− 52.2495i	− 0.209837i
$250$	− 456.586i	− 1.82634i
$251$	−79.1282	−0.315252	−0.157626	−	0.987499i	$-0.550384\pi$
$251$	−79.1282	−0.315252	−0.157626	+	0.987499i	$0.550384\pi$
$252$	17.6429i	0.0700117i
$253$	0	0
$254$	53.3247	0.209940
$255$	− 29.6981i	− 0.116463i
$256$	328.477	1.28311
$257$	298.533	1.16161	0.580803	−	0.814044i	$-0.302739\pi$
$257$	298.533	1.16161	0.580803	+	0.814044i	$0.302739\pi$
$258$	−185.714	−0.719821
$259$	39.0576i	0.150801i
$260$	− 239.699i	− 0.921919i
$261$	73.8427i	0.282922i
$262$	270.512	1.03249
$263$	60.3285i	0.229386i	0.993401	+	0.114693i	$0.0365884\pi$
$263$	60.3285i	0.229386i	−0.993401	+	0.114693i	$0.963412\pi$
$264$	0	0
$265$	418.399	1.57887
$266$	− 52.3333i	− 0.196742i
$267$	104.728	0.392242
$268$	10.6379	0.0396938
$269$	−195.972	−0.728522	−0.364261	−	0.931297i	$-0.618678\pi$
$269$	−195.972	−0.728522	−0.364261	+	0.931297i	$0.618678\pi$
$270$	111.123i	0.411565i
$271$	− 340.500i	− 1.25646i	−0.778029	−	0.628229i	$-0.783780\pi$
$271$	− 340.500i	− 1.25646i	0.778029	−	0.628229i	$-0.216220\pi$
$272$	40.2567i	0.148003i
$273$	49.7750	0.182326
$274$	301.742i	1.10125i
$275$	0	0
$276$	175.125	0.634513
$277$	350.746i	1.26623i	0.774057	+	0.633116i	$0.218224\pi$
$277$	350.746i	1.26623i	−0.774057	+	0.633116i	$0.781776\pi$
$278$	−18.7966	−0.0676136
$279$	65.4405	0.234554
$280$	−82.9916	−0.296399
$281$	− 408.440i	− 1.45352i	−0.686890	−	0.726762i	$-0.741024\pi$
$281$	− 408.440i	− 1.45352i	0.686890	−	0.726762i	$-0.258976\pi$
$282$	70.9576i	0.251623i
$283$	321.158i	1.13483i	0.823431	+	0.567417i	$0.192057\pi$
$283$	321.158i	1.13483i	−0.823431	+	0.567417i	$0.807943\pi$
$284$	−14.5454	−0.0512163
$285$	− 123.922i	− 0.434813i
$286$	0	0
$287$	−128.407	−0.447411
$288$	− 102.318i	− 0.355272i
$289$	284.880	0.985742
$290$	526.389	1.81513
$291$	−63.5475	−0.218376
$292$	− 14.6527i	− 0.0501805i
$293$	377.069i	1.28692i	0.765478	+	0.643462i	$0.222503\pi$
$293$	377.069i	1.28692i	−0.765478	+	0.643462i	$0.777497\pi$
$294$	− 188.755i	− 0.642025i
$295$	−168.367	−0.570737
$296$	− 64.4332i	− 0.217680i
$297$	0	0
$298$	357.645	1.20015
$299$	− 494.072i	− 1.65241i
$300$	193.462	0.644874
$301$	−103.354	−0.343368
$302$	−161.357	−0.534295
$303$	245.525i	0.810313i
$304$	167.980i	0.552565i
$305$	1004.39i	3.29307i
$306$	15.4175	0.0503841
$307$	− 505.973i	− 1.64812i	−0.566502	−	0.824060i	$-0.691704\pi$
$307$	− 505.973i	− 1.64812i	0.566502	−	0.824060i	$-0.308296\pi$
$308$	0	0
$309$	−67.5869	−0.218728
$310$	− 466.494i	− 1.50482i
$311$	−166.073	−0.533997	−0.266998	−	0.963697i	$-0.586032\pi$
$311$	−166.073	−0.533997	−0.266998	+	0.963697i	$0.586032\pi$
$312$	−82.1138	−0.263185
$313$	93.6449	0.299185	0.149593	−	0.988748i	$-0.452204\pi$
$313$	93.6449	0.299185	0.149593	+	0.988748i	$0.452204\pi$
$314$	93.3970i	0.297443i
$315$	61.8422i	0.196324i
$316$	249.326i	0.789006i
$317$	149.549	0.471764	0.235882	−	0.971782i	$-0.424202\pi$
$317$	149.549	0.471764	0.235882	+	0.971782i	$0.424202\pi$
$318$	217.208i	0.683045i
$319$	0	0
$320$	−59.3006	−0.185314
$321$	201.998i	0.629276i
$322$	259.235	0.805078
$323$	−17.1933	−0.0532301
$324$	−21.6883	−0.0669393
$325$	− 545.804i	− 1.67940i
$326$	− 135.794i	− 0.416545i
$327$	− 140.759i	− 0.430454i
$328$	211.833	0.645831
$329$	39.4895i	0.120029i
$330$	0	0
$331$	−271.711	−0.820880	−0.410440	−	0.911888i	$-0.634625\pi$
$331$	−271.711	−0.820880	−0.410440	+	0.911888i	$0.634625\pi$
$332$	− 72.6951i	− 0.218961i
$333$	−48.0132	−0.144184
$334$	−334.179	−1.00054
$335$	37.2882	0.111308
$336$	− 83.8290i	− 0.249491i
$337$	− 500.414i	− 1.48491i	−0.669898	−	0.742454i	$-0.733662\pi$
$337$	− 500.414i	− 1.48491i	0.669898	−	0.742454i	$-0.266338\pi$
$338$	76.7987i	0.227215i
$339$	132.407	0.390582
$340$	− 41.3192i	− 0.121527i
$341$	0	0
$342$	64.3329	0.188108
$343$	− 224.628i	− 0.654891i
$344$	170.503	0.495647
$345$	613.851	1.77928
$346$	168.536	0.487099
$347$	− 242.729i	− 0.699509i	−0.936841	−	0.349754i	$-0.886265\pi$
$347$	− 242.729i	− 0.699509i	0.936841	−	0.349754i	$-0.113735\pi$
$348$	102.738i	0.295224i
$349$	− 447.324i	− 1.28173i	−0.767653	−	0.640866i	$-0.778576\pi$
$349$	− 447.324i	− 1.28173i	0.767653	−	0.640866i	$-0.221424\pi$
$350$	286.378	0.818224
$351$	61.1881i	0.174325i
$352$	0	0
$353$	−226.910	−0.642805	−0.321403	−	0.946943i	$-0.604154\pi$
$353$	−226.910	−0.642805	−0.321403	+	0.946943i	$0.604154\pi$
$354$	− 87.4066i	− 0.246911i
$355$	−50.9847	−0.143619
$356$	145.710	0.409296
$357$	8.58020	0.0240342
$358$	451.947i	1.26242i
$359$	− 642.653i	− 1.79012i	−0.445947	−	0.895059i	$-0.647133\pi$
$359$	− 642.653i	− 1.79012i	0.445947	−	0.895059i	$-0.352867\pi$
$360$	− 102.021i	− 0.283391i
$361$	289.257	0.801267
$362$	− 131.941i	− 0.364478i
$363$	0	0
$364$	69.2524	0.190254
$365$	− 51.3608i	− 0.140715i
$366$	−521.419	−1.42464
$367$	−717.882	−1.95608	−0.978041	−	0.208415i	$-0.933170\pi$
$367$	−717.882	−1.95608	−0.978041	+	0.208415i	$0.933170\pi$
$368$	−832.094	−2.26113
$369$	− 157.850i	− 0.427777i
$370$	342.263i	0.925034i
$371$	120.881i	0.325826i
$372$	91.0479	0.244752
$373$	− 18.1857i	− 0.0487552i	−0.999703	−	0.0243776i	$-0.992240\pi$
$373$	− 18.1857i	− 0.0487552i	0.999703	−	0.0243776i	$-0.00776040\pi$
$374$	0	0
$375$	312.363	0.832969
$376$	− 65.1457i	− 0.173260i
$377$	289.849	0.768829
$378$	−32.1049	−0.0849335
$379$	−155.717	−0.410864	−0.205432	−	0.978671i	$-0.565860\pi$
$379$	−155.717	−0.410864	−0.205432	+	0.978671i	$0.565860\pi$
$380$	− 172.413i	− 0.453719i
$381$	36.4810i	0.0957505i
$382$	− 907.215i	− 2.37491i
$383$	−720.395	−1.88093	−0.940464	−	0.339894i	$-0.889609\pi$
$383$	−720.395	−1.88093	−0.940464	+	0.339894i	$0.889609\pi$
$384$	205.509i	0.535179i
$385$	0	0
$386$	−541.542	−1.40296
$387$	− 127.052i	− 0.328300i
$388$	−88.4142	−0.227872
$389$	8.88848	0.0228496	0.0114248	−	0.999935i	$-0.496363\pi$
$389$	8.88848	0.0228496	0.0114248	+	0.999935i	$0.496363\pi$
$390$	436.180	1.11841
$391$	− 85.1679i	− 0.217821i
$392$	173.295i	0.442079i
$393$	185.065i	0.470903i
$394$	−700.204	−1.77717
$395$	873.939i	2.21250i
$396$	0	0
$397$	513.254	1.29283	0.646416	−	0.762985i	$-0.276267\pi$
$397$	513.254	1.29283	0.646416	+	0.762985i	$0.276267\pi$
$398$	199.524i	0.501317i
$399$	35.8027	0.0897311
$400$	−919.219	−2.29805
$401$	548.823	1.36863	0.684317	−	0.729184i	$-0.260100\pi$
$401$	548.823	1.36863	0.684317	+	0.729184i	$0.260100\pi$
$402$	19.3578i	0.0481539i
$403$	− 256.868i	− 0.637390i
$404$	341.600i	0.845546i
$405$	−76.0221	−0.187709
$406$	152.081i	0.374584i
$407$	0	0
$408$	−14.1547	−0.0346930
$409$	506.077i	1.23735i	0.785646	+	0.618676i	$0.212331\pi$
$409$	506.077i	1.23735i	−0.785646	+	0.618676i	$0.787669\pi$
$410$	−1125.23	−2.74447
$411$	−206.430	−0.502264
$412$	−94.0341	−0.228238
$413$	− 48.6437i	− 0.117781i
$414$	318.676i	0.769748i
$415$	− 254.812i	− 0.614004i
$416$	−401.622	−0.965437
$417$	− 12.8593i	− 0.0308376i
$418$	0	0
$419$	−111.580	−0.266300	−0.133150	−	0.991096i	$-0.542509\pi$
$419$	−111.580	−0.266300	−0.133150	+	0.991096i	$0.542509\pi$
$420$	86.0415i	0.204861i
$421$	373.299	0.886696	0.443348	−	0.896350i	$-0.353791\pi$
$421$	373.299	0.886696	0.443348	+	0.896350i	$0.353791\pi$
$422$	−554.370	−1.31367
$423$	−48.5441	−0.114761
$424$	− 199.418i	− 0.470325i
$425$	− 94.0854i	− 0.221377i
$426$	− 26.4683i	− 0.0621321i
$427$	−290.181	−0.679582
$428$	281.041i	0.656637i
$429$	0	0
$430$	−905.693	−2.10626
$431$	78.0726i	0.181143i	0.995890	+	0.0905714i	$0.0288693\pi$
$431$	78.0726i	0.181143i	−0.995890	+	0.0905714i	$0.971131\pi$
$432$	103.050	0.238543
$433$	711.481	1.64314	0.821572	−	0.570105i	$-0.193097\pi$
$433$	711.481	1.64314	0.821572	+	0.570105i	$0.193097\pi$
$434$	134.777	0.310545
$435$	360.118i	0.827857i
$436$	− 195.838i	− 0.449170i
$437$	− 355.381i	− 0.813229i
$438$	26.6635	0.0608757
$439$	262.674i	0.598347i	0.954199	+	0.299173i	$0.0967109\pi$
$439$	262.674i	0.598347i	−0.954199	+	0.299173i	$0.903289\pi$
$440$	0	0
$441$	129.133	0.292818
$442$	− 60.5171i	− 0.136917i
$443$	−528.705	−1.19346	−0.596732	−	0.802441i	$-0.703534\pi$
$443$	−528.705	−1.19346	−0.596732	+	0.802441i	$0.703534\pi$
$444$	−66.8011	−0.150453
$445$	510.742	1.14774
$446$	− 530.526i	− 1.18952i
$447$	244.675i	0.547371i
$448$	− 17.1328i	− 0.0382428i
$449$	−287.807	−0.640995	−0.320498	−	0.947249i	$-0.603850\pi$
$449$	−287.807	−0.640995	−0.320498	+	0.947249i	$0.603850\pi$
$450$	352.043i	0.782318i
$451$	0	0
$452$	184.219	0.407565
$453$	− 110.389i	− 0.243684i
$454$	−211.000	−0.464759
$455$	242.744	0.533503
$456$	−59.0636	−0.129526
$457$	826.946i	1.80951i	0.425932	+	0.904755i	$0.359946\pi$
$457$	826.946i	1.80951i	−0.425932	+	0.904755i	$0.640054\pi$
$458$	837.157i	1.82785i
$459$	10.5476i	0.0229795i
$460$	854.056	1.85664
$461$	− 325.354i	− 0.705756i	−0.935669	−	0.352878i	$-0.885203\pi$
$461$	− 325.354i	− 0.705756i	0.935669	−	0.352878i	$-0.114797\pi$
$462$	0	0
$463$	−711.656	−1.53705	−0.768527	−	0.639818i	$-0.779010\pi$
$463$	−711.656	−1.53705	−0.768527	+	0.639818i	$0.779010\pi$
$464$	− 488.151i	− 1.05205i
$465$	319.142	0.686326
$466$	−418.779	−0.898668
$467$	458.119	0.980982	0.490491	−	0.871446i	$-0.336817\pi$
$467$	458.119	0.980982	0.490491	+	0.871446i	$0.336817\pi$
$468$	85.1314i	0.181905i
$469$	10.7731i	0.0229703i
$470$	346.047i	0.736271i
$471$	−63.8956	−0.135659
$472$	80.2474i	0.170016i
$473$	0	0
$474$	−453.698	−0.957169
$475$	− 392.591i	− 0.826508i
$476$	11.9377	0.0250792
$477$	−148.599	−0.311527
$478$	455.579	0.953095
$479$	592.433i	1.23681i	0.785859	+	0.618406i	$0.212221\pi$
$479$	592.433i	1.23681i	−0.785859	+	0.618406i	$0.787779\pi$
$480$	− 498.989i	− 1.03956i
$481$	188.462i	0.391813i
$482$	573.777	1.19041
$483$	177.350i	0.367185i
$484$	0	0
$485$	−309.910	−0.638990
$486$	− 39.4663i	− 0.0812063i
$487$	437.813	0.899001	0.449500	−	0.893280i	$-0.351602\pi$
$487$	437.813	0.899001	0.449500	+	0.893280i	$0.351602\pi$
$488$	478.712	0.980967
$489$	92.9003	0.189980
$490$	− 920.526i	− 1.87862i
$491$	− 774.734i	− 1.57787i	−0.614477	−	0.788935i	$-0.710633\pi$
$491$	− 774.734i	− 1.57787i	0.614477	−	0.788935i	$-0.289367\pi$
$492$	− 219.617i	− 0.446377i
$493$	49.9640	0.101347
$494$	− 252.521i	− 0.511175i
$495$	0	0
$496$	−432.607	−0.872191
$497$	− 14.7302i	− 0.0296382i
$498$	132.283	0.265629
$499$	262.110	0.525271	0.262636	−	0.964895i	$-0.415408\pi$
$499$	262.110	0.525271	0.262636	+	0.964895i	$0.415408\pi$
$500$	434.593	0.869187
$501$	− 228.622i	− 0.456330i
$502$	− 200.334i	− 0.399071i
$503$	− 153.502i	− 0.305174i	−0.988290	−	0.152587i	$-0.951240\pi$
$503$	− 153.502i	− 0.305174i	0.988290	−	0.152587i	$-0.0487604\pi$
$504$	29.4753	0.0584827
$505$	1197.38i	2.37105i
$506$	0	0
$507$	−52.5402	−0.103630
$508$	50.7562i	0.0999138i
$509$	−782.781	−1.53788	−0.768940	−	0.639321i	$-0.779215\pi$
$509$	−782.781	−1.53788	−0.768940	+	0.639321i	$0.779215\pi$
$510$	75.1886	0.147429
$511$	14.8389	0.0290389
$512$	357.022i	0.697308i
$513$	44.0120i	0.0857933i
$514$	755.813i	1.47045i
$515$	−329.609	−0.640018
$516$	− 176.769i	− 0.342575i
$517$	0	0
$518$	−98.8844	−0.190897
$519$	115.301i	0.222159i
$520$	−400.454	−0.770104
$521$	−180.526	−0.346498	−0.173249	−	0.984878i	$-0.555427\pi$
$521$	−180.526	−0.346498	−0.173249	+	0.984878i	$0.555427\pi$
$522$	−186.952	−0.358146
$523$	151.833i	0.290311i	0.989409	+	0.145156i	$0.0463683\pi$
$523$	151.833i	0.290311i	−0.989409	+	0.145156i	$0.953632\pi$
$524$	257.482i	0.491378i
$525$	195.920i	0.373181i
$526$	−152.737	−0.290375
$527$	− 44.2788i	− 0.0840206i
$528$	0	0
$529$	1231.39	2.32778
$530$	1059.29i	1.99865i
$531$	59.7973	0.112613
$532$	49.8126	0.0936326
$533$	−619.594	−1.16247
$534$	265.148i	0.496531i
$535$	985.107i	1.84132i
$536$	− 17.7723i	− 0.0331573i
$537$	−309.190	−0.575773
$538$	− 496.155i	− 0.922222i
$539$	0	0
$540$	−105.770	−0.195871
$541$	287.853i	0.532075i	0.963963	+	0.266038i	$0.0857146\pi$
$541$	287.853i	0.532075i	−0.963963	+	0.266038i	$0.914285\pi$
$542$	862.065	1.59052
$543$	90.2647	0.166233
$544$	−69.2314	−0.127264
$545$	− 686.454i	− 1.25955i
$546$	126.018i	0.230803i
$547$	122.440i	0.223839i	0.993717	+	0.111920i	$0.0356999\pi$
$547$	122.440i	0.223839i	−0.993717	+	0.111920i	$0.964300\pi$
$548$	−287.208	−0.524102
$549$	− 356.718i	− 0.649759i
$550$	0	0
$551$	208.485	0.378376
$552$	− 292.574i	− 0.530026i
$553$	−252.493	−0.456588
$554$	−888.005	−1.60290
$555$	−234.152	−0.421895
$556$	− 17.8912i	− 0.0321784i
$557$	− 560.374i	− 1.00606i	−0.864270	−	0.503028i	$-0.832219\pi$
$557$	− 560.374i	− 1.00606i	0.864270	−	0.503028i	$-0.167781\pi$
$558$	165.680i	0.296917i
$559$	−498.707	−0.892141
$560$	− 408.819i	− 0.730034i
$561$	0	0
$562$	1034.07	1.83999
$563$	− 461.422i	− 0.819577i	−0.912181	−	0.409788i	$-0.865603\pi$
$563$	− 461.422i	− 0.819577i	0.912181	−	0.409788i	$-0.134397\pi$
$564$	−67.5398	−0.119751
$565$	645.727	1.14288
$566$	−813.095	−1.43656
$567$	− 21.9639i	− 0.0387370i
$568$	24.3004i	0.0427824i
$569$	− 850.975i	− 1.49556i	−0.663945	−	0.747781i	$-0.731119\pi$
$569$	− 850.975i	− 1.49556i	0.663945	−	0.747781i	$-0.268881\pi$
$570$	313.740	0.550421
$571$	− 558.153i	− 0.977500i	−0.872424	−	0.488750i	$-0.837453\pi$
$571$	− 558.153i	− 0.977500i	0.872424	−	0.488750i	$-0.162547\pi$
$572$	0	0
$573$	620.652	1.08316
$574$	− 325.096i	− 0.566369i
$575$	1944.72	3.38212
$576$	21.0612	0.0365646
$577$	1063.74	1.84358	0.921789	−	0.387692i	$-0.126728\pi$
$577$	1063.74	1.84358	0.921789	+	0.387692i	$0.126728\pi$
$578$	721.247i	1.24783i
$579$	− 370.484i	− 0.639869i
$580$	501.034i	0.863852i
$581$	73.6186	0.126710
$582$	− 160.887i	− 0.276439i
$583$	0	0
$584$	−24.4796	−0.0419172
$585$	298.403i	0.510091i
$586$	−954.648	−1.62909
$587$	−137.730	−0.234633	−0.117317	−	0.993095i	$-0.537429\pi$
$587$	−137.730	−0.234633	−0.117317	+	0.993095i	$0.537429\pi$
$588$	179.664	0.305550
$589$	− 184.763i	− 0.313689i
$590$	− 426.266i	− 0.722485i
$591$	− 479.030i	− 0.810541i
$592$	317.400	0.536148
$593$	− 42.1576i	− 0.0710921i	−0.999368	−	0.0355461i	$-0.988683\pi$
$593$	− 42.1576i	− 0.0710921i	0.999368	−	0.0355461i	$-0.0113170\pi$
$594$	0	0
$595$	41.8441	0.0703263
$596$	340.418i	0.571171i
$597$	−136.500	−0.228644
$598$	1250.87	2.09176
$599$	−323.005	−0.539240	−0.269620	−	0.962967i	$-0.586898\pi$
$599$	−323.005	−0.539240	−0.269620	+	0.962967i	$0.586898\pi$
$600$	− 323.208i	− 0.538681i
$601$	− 682.717i	− 1.13597i	−0.823039	−	0.567985i	$-0.807723\pi$
$601$	− 682.717i	− 1.13597i	0.823039	−	0.567985i	$-0.192277\pi$
$602$	− 261.667i	− 0.434663i
$603$	−13.2433	−0.0219623
$604$	− 153.585i	− 0.254280i
$605$	0	0
$606$	−621.610	−1.02576
$607$	− 219.873i	− 0.362229i	−0.983462	−	0.181114i	$-0.942030\pi$
$607$	− 219.873i	− 0.362229i	0.983462	−	0.181114i	$-0.0579705\pi$
$608$	−288.883	−0.475136
$609$	−104.043	−0.170842
$610$	−2542.87	−4.16864
$611$	190.546i	0.311859i
$612$	14.6749i	0.0239786i
$613$	198.057i	0.323094i	0.986865	+	0.161547i	$0.0516483\pi$
$613$	198.057i	0.323094i	−0.986865	+	0.161547i	$0.948352\pi$
$614$	1281.00	2.08632
$615$	− 769.805i	− 1.25172i
$616$	0	0
$617$	666.299	1.07990	0.539951	−	0.841697i	$-0.318443\pi$
$617$	666.299	1.07990	0.539951	+	0.841697i	$0.318443\pi$
$618$	− 171.114i	− 0.276883i
$619$	506.706	0.818589	0.409294	−	0.912402i	$-0.365775\pi$
$619$	506.706	0.818589	0.409294	+	0.912402i	$0.365775\pi$
$620$	444.024	0.716168
$621$	−218.015	−0.351071
$622$	− 420.457i	− 0.675976i
$623$	147.561i	0.236855i
$624$	− 404.495i	− 0.648229i
$625$	364.586	0.583337
$626$	237.087i	0.378733i
$627$	0	0
$628$	−88.8984	−0.141558
$629$	32.4870i	0.0516487i
$630$	−156.570	−0.248523
$631$	761.035	1.20608	0.603039	−	0.797712i	$-0.293956\pi$
$631$	761.035	1.20608	0.603039	+	0.797712i	$0.293956\pi$
$632$	416.537	0.659078
$633$	− 379.261i	− 0.599148i
$634$	378.623i	0.597197i
$635$	177.911i	0.280175i
$636$	−206.746	−0.325073
$637$	− 506.875i	− 0.795722i
$638$	0	0
$639$	18.1077	0.0283376
$640$	1002.23i	1.56598i
$641$	203.013	0.316714	0.158357	−	0.987382i	$-0.449380\pi$
$641$	203.013	0.316714	0.158357	+	0.987382i	$0.449380\pi$
$642$	−511.410	−0.796589
$643$	−455.154	−0.707860	−0.353930	−	0.935272i	$-0.615155\pi$
$643$	−455.154	−0.707860	−0.353930	+	0.935272i	$0.615155\pi$
$644$	246.749i	0.383150i
$645$	− 619.611i	− 0.960637i
$646$	− 43.5294i	− 0.0673830i
$647$	535.496	0.827659	0.413830	−	0.910354i	$-0.364191\pi$
$647$	535.496	0.827659	0.413830	+	0.910354i	$0.364191\pi$
$648$	36.2337i	0.0559162i
$649$	0	0
$650$	1381.84	2.12591
$651$	92.2045i	0.141635i
$652$	129.253	0.198241
$653$	−829.067	−1.26963	−0.634814	−	0.772665i	$-0.718923\pi$
$653$	−829.067	−1.26963	−0.634814	+	0.772665i	$0.718923\pi$
$654$	356.367	0.544904
$655$	902.529i	1.37791i
$656$	1043.49i	1.59069i
$657$	18.2413i	0.0277645i
$658$	−99.9779	−0.151942
$659$	− 1089.67i	− 1.65353i	−0.562551	−	0.826763i	$-0.690180\pi$
$659$	− 1089.67i	− 1.65353i	0.562551	−	0.826763i	$-0.309820\pi$
$660$	0	0
$661$	−920.151	−1.39206	−0.696029	−	0.718013i	$-0.745052\pi$
$661$	−920.151	−1.39206	−0.696029	+	0.718013i	$0.745052\pi$
$662$	− 687.908i	− 1.03914i
$663$	41.4015	0.0624457
$664$	−121.448	−0.182904
$665$	174.603	0.262562
$666$	− 121.558i	− 0.182519i
$667$	1032.74i	1.54834i
$668$	− 318.083i	− 0.476172i
$669$	362.948	0.542524
$670$	94.4048i	0.140903i
$671$	0	0
$672$	144.165	0.214531
$673$	− 645.812i	− 0.959602i	−0.877377	−	0.479801i	$-0.840709\pi$
$673$	− 645.812i	− 0.959602i	0.877377	−	0.479801i	$-0.159291\pi$
$674$	1266.93	1.87972
$675$	−240.843	−0.356804
$676$	−73.0996	−0.108135
$677$	− 737.620i	− 1.08954i	−0.838585	−	0.544771i	$-0.816617\pi$
$677$	− 737.620i	− 1.08954i	0.838585	−	0.544771i	$-0.183383\pi$
$678$	335.224i	0.494430i
$679$	− 89.5373i	− 0.131866i
$680$	−69.0302	−0.101515
$681$	− 144.352i	− 0.211970i
$682$	0	0
$683$	527.576	0.772440	0.386220	−	0.922407i	$-0.373781\pi$
$683$	527.576	0.772440	0.386220	+	0.922407i	$0.373781\pi$
$684$	61.2342i	0.0895237i
$685$	−1006.72	−1.46967
$686$	568.703	0.829014
$687$	−572.723	−0.833658
$688$	839.901i	1.22079i
$689$	583.281i	0.846562i
$690$	1554.13i	2.25236i
$691$	−516.694	−0.747748	−0.373874	−	0.927479i	$-0.621971\pi$
$691$	−516.694	−0.747748	−0.373874	+	0.927479i	$0.621971\pi$
$692$	160.419i	0.231819i
$693$	0	0
$694$	614.533	0.885494
$695$	− 62.7124i	− 0.0902337i
$696$	171.640	0.246609
$697$	−106.805	−0.153236
$698$	1132.52	1.62252
$699$	− 286.499i	− 0.409870i
$700$	272.585i	0.389407i
$701$	699.987i	0.998555i	0.866442	+	0.499278i	$0.166401\pi$
$701$	699.987i	0.998555i	−0.866442	+	0.499278i	$0.833599\pi$
$702$	−154.914	−0.220675
$703$	135.559i	0.192829i
$704$	0	0
$705$	−236.741	−0.335803
$706$	− 574.482i	− 0.813714i
$707$	−345.940	−0.489307
$708$	83.1965	0.117509
$709$	−649.664	−0.916310	−0.458155	−	0.888872i	$-0.651490\pi$
$709$	−649.664	−0.916310	−0.458155	+	0.888872i	$0.651490\pi$
$710$	− 129.081i	− 0.181804i
$711$	− 310.388i	− 0.436551i
$712$	− 243.430i	− 0.341897i
$713$	915.230	1.28363
$714$	21.7230i	0.0304244i
$715$	0	0
$716$	−430.178	−0.600808
$717$	311.675i	0.434693i
$718$	1627.04	2.26608
$719$	694.522	0.965955	0.482978	−	0.875633i	$-0.339555\pi$
$719$	694.522	0.965955	0.482978	+	0.875633i	$0.339555\pi$
$720$	502.558	0.697998
$721$	− 95.2287i	− 0.132079i
$722$	732.330i	1.01431i
$723$	392.537i	0.542929i
$724$	125.586	0.173461
$725$	1140.87i	1.57362i
$726$	0	0
$727$	−970.108	−1.33440	−0.667200	−	0.744879i	$-0.732507\pi$
$727$	−970.108	−1.33440	−0.667200	+	0.744879i	$0.732507\pi$
$728$	− 115.697i	− 0.158924i
$729$	27.0000	0.0370370
$730$	130.033	0.178128
$731$	−85.9669	−0.117602
$732$	− 496.304i	− 0.678011i
$733$	157.602i	0.215009i	0.994205	+	0.107504i	$0.0342860\pi$
$733$	157.602i	0.215009i	−0.994205	+	0.107504i	$0.965714\pi$
$734$	− 1817.51i	− 2.47616i
$735$	629.758	0.856814
$736$	− 1430.99i	− 1.94428i
$737$	0	0
$738$	399.638	0.541514
$739$	685.531i	0.927647i	0.885928	+	0.463823i	$0.153523\pi$
$739$	685.531i	0.927647i	−0.885928	+	0.463823i	$0.846477\pi$
$740$	−325.777	−0.440239
$741$	172.757	0.233140
$742$	−306.043	−0.412457
$743$	516.079i	0.694588i	0.937756	+	0.347294i	$0.112899\pi$
$743$	516.079i	0.694588i	−0.937756	+	0.347294i	$0.887101\pi$
$744$	− 152.110i	− 0.204448i
$745$	1193.24i	1.60166i
$746$	46.0418	0.0617183
$747$	90.4988i	0.121150i
$748$	0	0
$749$	−284.611	−0.379988
$750$	790.829i	1.05444i
$751$	−853.690	−1.13674	−0.568369	−	0.822774i	$-0.692425\pi$
$751$	−853.690	−1.13674	−0.568369	+	0.822774i	$0.692425\pi$
$752$	320.910	0.426742
$753$	137.054	0.182011
$754$	733.827i	0.973246i
$755$	− 538.347i	− 0.713043i
$756$	− 30.5585i	− 0.0404213i
$757$	−496.049	−0.655282	−0.327641	−	0.944802i	$-0.606254\pi$
$757$	−496.049	−0.655282	−0.327641	+	0.944802i	$0.606254\pi$
$758$	− 394.239i	− 0.520104i
$759$	0	0
$760$	−288.043	−0.379004
$761$	1.29340i	0.00169961i	1.00000		0.000849804i	$0.000270501\pi$
$761$	1.29340i	0.00169961i	−1.00000		0.000849804i	$0.999729\pi$
$762$	−92.3611	−0.121209
$763$	198.326	0.259930
$764$	863.517	1.13026
$765$	51.4387i	0.0672401i
$766$	− 1823.87i	− 2.38103i
$767$	− 234.717i	− 0.306020i
$768$	−568.938	−0.740805
$769$	398.342i	0.518000i	0.965877	+	0.259000i	$0.0833929\pi$
$769$	398.342i	0.518000i	−0.965877	+	0.259000i	$0.916607\pi$
$770$	0	0
$771$	−517.073	−0.670653
$772$	− 515.457i	− 0.667691i
$773$	−637.493	−0.824700	−0.412350	−	0.911025i	$-0.635292\pi$
$773$	−637.493	−0.824700	−0.412350	+	0.911025i	$0.635292\pi$
$774$	321.666	0.415589
$775$	1011.06	1.30459
$776$	147.710i	0.190347i
$777$	− 67.6497i	− 0.0870652i
$778$	22.5035i	0.0289248i
$779$	−445.668	−0.572103
$780$	415.171i	0.532270i
$781$	0	0
$782$	215.625	0.275735
$783$	− 127.899i	− 0.163345i
$784$	−853.657	−1.08885
$785$	−311.608	−0.396952
$786$	−468.540	−0.596107
$787$	1070.60i	1.36036i	0.733045	+	0.680180i	$0.238099\pi$
$787$	1070.60i	1.36036i	−0.733045	+	0.680180i	$0.761901\pi$
$788$	− 666.477i	− 0.845783i
$789$	− 104.492i	− 0.132436i
$790$	−2212.60	−2.80077
$791$	186.560i	0.235853i
$792$	0	0
$793$	−1400.19	−1.76569
$794$	1299.44i	1.63657i
$795$	−724.689	−0.911558
$796$	−189.914	−0.238585
$797$	365.858	0.459044	0.229522	−	0.973303i	$-0.426284\pi$
$797$	365.858	0.459044	0.229522	+	0.973303i	$0.426284\pi$
$798$	90.6439i	0.113589i
$799$	32.8463i	0.0411092i
$800$	− 1580.83i	− 1.97603i
$801$	−181.395	−0.226461
$802$	1389.49i	1.73253i
$803$	0	0
$804$	−18.4254	−0.0229172
$805$	864.906i	1.07442i
$806$	650.329	0.806860
$807$	339.434	0.420612
$808$	570.697	0.706308
$809$	176.666i	0.218375i	0.994021	+	0.109188i	$0.0348249\pi$
$809$	176.666i	0.218375i	−0.994021	+	0.109188i	$0.965175\pi$
$810$	− 192.470i	− 0.237617i
$811$	256.402i	0.316156i	0.987427	+	0.158078i	$0.0505297\pi$
$811$	256.402i	0.316156i	−0.987427	+	0.158078i	$0.949470\pi$
$812$	−144.756	−0.178271
$813$	589.763i	0.725416i
$814$	0	0
$815$	453.058	0.555900
$816$	− 69.7267i	− 0.0854494i
$817$	−358.715	−0.439064
$818$	−1281.27	−1.56634
$819$	−86.2129	−0.105266
$820$	− 1071.03i	− 1.30614i
$821$	938.646i	1.14330i	0.820499	+	0.571648i	$0.193696\pi$
$821$	938.646i	1.14330i	−0.820499	+	0.571648i	$0.806304\pi$
$822$	− 522.632i	− 0.635806i
$823$	558.391	0.678483	0.339241	−	0.940699i	$-0.389830\pi$
$823$	558.391	0.678483	0.339241	+	0.940699i	$0.389830\pi$
$824$	157.099i	0.190654i
$825$	0	0
$826$	123.154	0.149097
$827$	1326.63i	1.60414i	0.597228	+	0.802072i	$0.296269\pi$
$827$	1326.63i	1.60414i	−0.597228	+	0.802072i	$0.703731\pi$
$828$	−303.326	−0.366336
$829$	−899.388	−1.08491	−0.542454	−	0.840086i	$-0.682505\pi$
$829$	−899.388	−1.08491	−0.542454	+	0.840086i	$0.682505\pi$
$830$	645.122	0.777255
$831$	− 607.510i	− 0.731059i
$832$	− 82.6697i	− 0.0993626i
$833$	− 87.3749i	− 0.104892i
$834$	32.5566	0.0390367
$835$	− 1114.95i	− 1.33527i
$836$	0	0
$837$	−113.346	−0.135420
$838$	− 282.494i	− 0.337104i
$839$	283.608	0.338032	0.169016	−	0.985613i	$-0.445941\pi$
$839$	283.608	0.338032	0.169016	+	0.985613i	$0.445941\pi$
$840$	143.746	0.171126
$841$	235.139	0.279595
$842$	945.104i	1.12245i
$843$	707.439i	0.839192i
$844$	− 527.668i	− 0.625199i
$845$	−256.229	−0.303230
$846$	− 122.902i	− 0.145274i
$847$	0	0
$848$	982.338	1.15842
$849$	− 556.262i	− 0.655197i
$850$	238.202	0.280237
$851$	−671.497	−0.789068
$852$	25.1934	0.0295697
$853$	− 440.959i	− 0.516951i	−0.966018	−	0.258476i	$-0.916780\pi$
$853$	− 440.959i	− 0.516951i	0.966018	−	0.258476i	$-0.0832202\pi$
$854$	− 734.670i	− 0.860269i
$855$	214.639i	0.251039i
$856$	469.522	0.548507
$857$	1378.52i	1.60854i	0.594264	+	0.804270i	$0.297443\pi$
$857$	1378.52i	1.60854i	−0.594264	+	0.804270i	$0.702557\pi$
$858$	0	0
$859$	1387.60	1.61537	0.807685	−	0.589614i	$-0.200720\pi$
$859$	1387.60	1.61537	0.807685	+	0.589614i	$0.200720\pi$
$860$	− 862.069i	− 1.00241i
$861$	222.407	0.258313
$862$	−197.661	−0.229305
$863$	−514.959	−0.596708	−0.298354	−	0.954455i	$-0.596438\pi$
$863$	−514.959	−0.596708	−0.298354	+	0.954455i	$0.596438\pi$
$864$	177.221i	0.205117i
$865$	562.301i	0.650058i
$866$	1801.30i	2.08002i
$867$	−493.426	−0.569119
$868$	128.285i	0.147794i
$869$	0	0
$870$	−911.732	−1.04797
$871$	51.9827i	0.0596816i
$872$	−327.178	−0.375205
$873$	110.068	0.126080
$874$	899.740	1.02945
$875$	440.114i	0.502988i
$876$	25.3793i	0.0289717i
$877$	− 742.107i	− 0.846188i	−0.906086	−	0.423094i	$-0.860944\pi$
$877$	− 742.107i	− 0.846188i	0.906086	−	0.423094i	$-0.139056\pi$
$878$	−665.028	−0.757435
$879$	− 653.102i	− 0.743006i
$880$	0	0
$881$	−620.478	−0.704288	−0.352144	−	0.935946i	$-0.614547\pi$
$881$	−620.478	−0.704288	−0.352144	+	0.935946i	$0.614547\pi$
$882$	326.934i	0.370673i
$883$	674.168	0.763498	0.381749	−	0.924266i	$-0.375322\pi$
$883$	674.168	0.763498	0.381749	+	0.924266i	$0.375322\pi$
$884$	57.6022	0.0651609
$885$	291.621	0.329515
$886$	− 1338.55i	− 1.51078i
$887$	1098.43i	1.23837i	0.785245	+	0.619185i	$0.212537\pi$
$887$	1098.43i	1.23837i	−0.785245	+	0.619185i	$0.787463\pi$
$888$	111.602i	0.125677i
$889$	−51.4010	−0.0578189
$890$	1293.08i	1.45290i
$891$	0	0
$892$	504.973	0.566113
$893$	137.058i	0.153480i
$894$	−619.459	−0.692907
$895$	−1507.86	−1.68477
$896$	−289.558	−0.323168
$897$	855.757i	0.954021i
$898$	− 728.658i	− 0.811424i
$899$	536.923i	0.597244i
$900$	−335.086	−0.372318
$901$	100.546i	0.111594i
$902$	0	0
$903$	179.014	0.198244
$904$	− 307.767i	− 0.340450i
$905$	440.205	0.486415
$906$	279.479	0.308475
$907$	247.155	0.272498	0.136249	−	0.990675i	$-0.456495\pi$
$907$	247.155	0.272498	0.136249	+	0.990675i	$0.456495\pi$
$908$	− 200.837i	− 0.221186i
$909$	− 425.262i	− 0.467834i
$910$	614.570i	0.675351i
$911$	686.752	0.753844	0.376922	−	0.926245i	$-0.376982\pi$
$911$	686.752	0.753844	0.376922	+	0.926245i	$0.376982\pi$
$912$	− 290.949i	− 0.319023i
$913$	0	0
$914$	−2093.63	−2.29062
$915$	− 1739.65i	− 1.90126i
$916$	−796.834	−0.869906
$917$	−260.753	−0.284355
$918$	−26.7039	−0.0290893
$919$	− 1485.25i	− 1.61615i	−0.589077	−	0.808077i	$-0.700509\pi$
$919$	− 1485.25i	− 1.61615i	0.589077	−	0.808077i	$-0.299491\pi$
$920$	− 1426.83i	− 1.55091i
$921$	876.371i	0.951543i
$922$	823.718	0.893403
$923$	− 71.0767i	− 0.0770062i
$924$	0	0
$925$	−741.807	−0.801953
$926$	− 1801.74i	− 1.94573i
$927$	117.064	0.126282
$928$	839.497	0.904630
$929$	−681.402	−0.733479	−0.366740	−	0.930324i	$-0.619526\pi$
$929$	−681.402	−0.733479	−0.366740	+	0.930324i	$0.619526\pi$
$930$	807.991i	0.868807i
$931$	− 364.590i	− 0.391611i
$932$	− 398.608i	− 0.427691i
$933$	287.647	0.308303
$934$	1159.85i	1.24181i
$935$	0	0
$936$	142.225	0.151950
$937$	− 88.5349i	− 0.0944876i	−0.998883	−	0.0472438i	$-0.984956\pi$
$937$	− 88.5349i	− 0.0944876i	0.998883	−	0.0472438i	$-0.0150438\pi$
$938$	−27.2749	−0.0290777
$939$	−162.198	−0.172735
$940$	−329.379	−0.350404
$941$	− 582.065i	− 0.618560i	−0.950971	−	0.309280i	$-0.899912\pi$
$941$	− 582.065i	− 0.618560i	0.950971	−	0.309280i	$-0.100088\pi$
$942$	− 161.768i	− 0.171729i
$943$	− 2207.64i	− 2.34108i
$944$	−395.301	−0.418751
$945$	− 107.114i	− 0.113348i
$946$	0	0
$947$	−912.836	−0.963924	−0.481962	−	0.876192i	$-0.660076\pi$
$947$	−912.836	−0.963924	−0.481962	+	0.876192i	$0.660076\pi$
$948$	− 431.845i	− 0.455533i
$949$	71.6011	0.0754489
$950$	993.947	1.04626
$951$	−259.027	−0.272373
$952$	− 19.9438i	− 0.0209493i
$953$	554.670i	0.582026i	0.956719	+	0.291013i	$0.0939922\pi$
$953$	554.670i	0.582026i	−0.956719	+	0.291013i	$0.906008\pi$
$954$	− 376.216i	− 0.394356i
$955$	3026.81	3.16943
$956$	433.636i	0.453594i
$957$	0	0
$958$	−1499.90	−1.56566
$959$	− 290.857i	− 0.303292i
$960$	102.712	0.106991
$961$	−485.171	−0.504861
$962$	−477.141	−0.495989
$963$	− 349.870i	− 0.363313i
$964$	546.140i	0.566535i
$965$	− 1806.79i	− 1.87232i
$966$	−449.008	−0.464812
$967$	1781.13i	1.84191i	0.389667	+	0.920956i	$0.372590\pi$
$967$	1781.13i	1.84191i	−0.389667	+	0.920956i	$0.627410\pi$
$968$	0	0
$969$	29.7797	0.0307324
$970$	− 784.618i	− 0.808885i
$971$	−990.964	−1.02056	−0.510280	−	0.860008i	$-0.670458\pi$
$971$	−990.964	−1.02056	−0.510280	+	0.860008i	$0.670458\pi$
$972$	37.5653	0.0386474
$973$	18.1185	0.0186213
$974$	1108.44i	1.13803i
$975$	945.360i	0.969600i
$976$	2358.15i	2.41614i
$977$	754.917	0.772689	0.386344	−	0.922355i	$-0.373738\pi$
$977$	754.917	0.772689	0.386344	+	0.922355i	$0.373738\pi$
$978$	235.201i	0.240492i
$979$	0	0
$980$	876.187	0.894069
$981$	243.801i	0.248523i
$982$	1961.44	1.99740
$983$	−206.512	−0.210084	−0.105042	−	0.994468i	$-0.533498\pi$
$983$	−206.512	−0.210084	−0.105042	+	0.994468i	$0.533498\pi$
$984$	−366.905	−0.372871
$985$	− 2336.14i	− 2.37172i
$986$	126.497i	0.128293i
$987$	− 68.3978i	− 0.0692987i
$988$	240.357	0.243277
$989$	− 1776.91i	− 1.79667i
$990$	0	0
$991$	852.133	0.859872	0.429936	−	0.902859i	$-0.358536\pi$
$991$	852.133	0.859872	0.429936	+	0.902859i	$0.358536\pi$
$992$	− 743.975i	− 0.749974i
$993$	470.618	0.473935
$994$	37.2933	0.0375185
$995$	−665.687	−0.669033
$996$	125.912i	0.126417i
$997$	− 1168.11i	− 1.17162i	−0.810448	−	0.585810i	$-0.800776\pi$
$997$	− 1168.11i	− 1.17162i	0.810448	−	0.585810i	$-0.199224\pi$
$998$	663.601i	0.664931i
$999$	83.1613	0.0832445

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	363.3.c.e.241.14		16
3.2	odd	2		1089.3.c.m.604.3		16
11.2	odd	10		363.3.g.a.40.2		16
11.3	even	5		363.3.g.f.112.3		16
11.4	even	5		33.3.g.a.28.2	yes	16
11.5	even	5		363.3.g.a.118.2		16
11.6	odd	10		363.3.g.g.118.3		16
11.7	odd	10		363.3.g.f.94.3		16
11.8	odd	10		33.3.g.a.13.2	✓	16
11.9	even	5		363.3.g.g.40.3		16
11.10	odd	2	inner	363.3.c.e.241.3		16
33.8	even	10		99.3.k.c.46.3		16
33.26	odd	10		99.3.k.c.28.3		16
33.32	even	2		1089.3.c.m.604.14		16
44.15	odd	10		528.3.bf.b.193.1		16
44.19	even	10		528.3.bf.b.145.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.3.g.a.13.2	✓	16	11.8	odd	10
33.3.g.a.28.2	yes	16	11.4	even	5
99.3.k.c.28.3		16	33.26	odd	10
99.3.k.c.46.3		16	33.8	even	10
363.3.c.e.241.3		16	11.10	odd	2	inner
363.3.c.e.241.14		16	1.1	even	1	trivial
363.3.g.a.40.2		16	11.2	odd	10
363.3.g.a.118.2		16	11.5	even	5
363.3.g.f.94.3		16	11.7	odd	10
363.3.g.f.112.3		16	11.3	even	5
363.3.g.g.40.3		16	11.9	even	5
363.3.g.g.118.3		16	11.6	odd	10
528.3.bf.b.145.1		16	44.19	even	10
528.3.bf.b.193.1		16	44.15	odd	10
1089.3.c.m.604.3		16	3.2	odd	2
1089.3.c.m.604.14		16	33.32	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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	363.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.53176i q^{2} -1.73205 q^{3} -2.40981 q^{4} -8.44690 q^{5} -4.38514i q^{6} -2.44043i q^{7} +4.02597i q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q+2.53176i q^{2} -1.73205 q^{3} -2.40981 q^{4} -8.44690 q^{5} -4.38514i q^{6} -2.44043i q^{7} +4.02597i q^{8} +3.00000 q^{9} -21.3855i q^{10} +4.17392 q^{12} -11.7756i q^{13} +6.17858 q^{14} +14.6305 q^{15} -19.8321 q^{16} -2.02988i q^{17} +7.59528i q^{18} -8.47011i q^{19} +20.3555 q^{20} +4.22695i q^{21} +41.9571 q^{23} -6.97318i q^{24} +46.3502 q^{25} +29.8131 q^{26} -5.19615 q^{27} +5.88098i q^{28} +24.6142i q^{29} +37.0409i q^{30} +21.8135 q^{31} -34.1061i q^{32} +5.13918 q^{34} +20.6141i q^{35} -7.22944 q^{36} -16.0044 q^{37} +21.4443 q^{38} +20.3960i q^{39} -34.0070i q^{40} -52.6166i q^{41} -10.7016 q^{42} -42.3507i q^{43} -25.3407 q^{45} +106.225i q^{46} -16.1814 q^{47} +34.3501 q^{48} +43.0443 q^{49} +117.348i q^{50} +3.51586i q^{51} +28.3771i q^{52} -49.5329 q^{53} -13.1554i q^{54} +9.82509 q^{56} +14.6707i q^{57} -62.3174 q^{58} +19.9324 q^{59} -35.2567 q^{60} -118.906i q^{61} +55.2266i q^{62} -7.32129i q^{63} +7.02040 q^{64} +99.4678i q^{65} -4.41442 q^{67} +4.89164i q^{68} -72.6718 q^{69} -52.1899 q^{70} +6.03591 q^{71} +12.0779i q^{72} +6.08043i q^{73} -40.5193i q^{74} -80.2809 q^{75} +20.4114i q^{76} -51.6379 q^{78} -103.463i q^{79} +167.519 q^{80} +9.00000 q^{81} +133.213 q^{82} +30.1663i q^{83} -10.1862i q^{84} +17.1462i q^{85} +107.222 q^{86} -42.6331i q^{87} -60.4650 q^{89} -64.1566i q^{90} -28.7376 q^{91} -101.109 q^{92} -37.7821 q^{93} -40.9674i q^{94} +71.5462i q^{95} +59.0736i q^{96} +36.6892 q^{97} +108.978i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 20 q^{4} - 4 q^{5} + 48 q^{9}+O(q^{10}) \) 16 * q - 20 * q^4 - 4 * q^5 + 48 * q^9	\( 16 q - 20 q^{4} - 4 q^{5} + 48 q^{9} - 24 q^{12} - 52 q^{14} + 36 q^{15} - 44 q^{16} - 108 q^{20} + 132 q^{23} + 88 q^{25} - 4 q^{26} + 40 q^{31} - 368 q^{34} - 60 q^{36} - 16 q^{37} + 280 q^{38} + 36 q^{42} - 12 q^{45} + 80 q^{47} + 144 q^{48} - 140 q^{49} - 128 q^{53} + 524 q^{56} + 140 q^{58} - 220 q^{59} - 384 q^{60} - 8 q^{64} + 36 q^{67} - 180 q^{69} - 100 q^{70} + 644 q^{71} + 312 q^{75} - 312 q^{78} + 264 q^{80} + 144 q^{81} - 476 q^{82} + 76 q^{86} + 76 q^{89} - 624 q^{91} + 120 q^{92} - 336 q^{93} + 216 q^{97}+O(q^{100}) \) 16 * q - 20 * q^4 - 4 * q^5 + 48 * q^9 - 24 * q^12 - 52 * q^14 + 36 * q^15 - 44 * q^16 - 108 * q^20 + 132 * q^23 + 88 * q^25 - 4 * q^26 + 40 * q^31 - 368 * q^34 - 60 * q^36 - 16 * q^37 + 280 * q^38 + 36 * q^42 - 12 * q^45 + 80 * q^47 + 144 * q^48 - 140 * q^49 - 128 * q^53 + 524 * q^56 + 140 * q^58 - 220 * q^59 - 384 * q^60 - 8 * q^64 + 36 * q^67 - 180 * q^69 - 100 * q^70 + 644 * q^71 + 312 * q^75 - 312 * q^78 + 264 * q^80 + 144 * q^81 - 476 * q^82 + 76 * q^86 + 76 * q^89 - 624 * q^91 + 120 * q^92 - 336 * q^93 + 216 * q^97

Introduction

L-functions

Modular forms

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