LMFDB - Embedded newform 1089.3.c.m.604.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,3,Mod(604,1089)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1089, 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("1089.604");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1089 = 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1089.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:	$29.6731007888$
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			604.14
Root			$0.988132 + 0.846795i$ of defining polynomial
Character	$\chi$	$=$	1089.604
Dual form			1089.3.c.m.604.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.53176i q^{2} -2.40981 q^{4} +8.44690 q^{5} +2.44043i q^{7} +4.02597i q^{8} +O(q^{10})$	$q+2.53176i q^{2} -2.40981 q^{4} +8.44690 q^{5} +2.44043i q^{7} +4.02597i q^{8} +21.3855i q^{10} +11.7756i q^{13} -6.17858 q^{14} -19.8321 q^{16} -2.02988i q^{17} +8.47011i q^{19} -20.3555 q^{20} -41.9571 q^{23} +46.3502 q^{25} -29.8131 q^{26} -5.88098i q^{28} +24.6142i q^{29} +21.8135 q^{31} -34.1061i q^{32} +5.13918 q^{34} +20.6141i q^{35} -16.0044 q^{37} -21.4443 q^{38} +34.0070i q^{40} -52.6166i q^{41} +42.3507i q^{43} -106.225i q^{46} +16.1814 q^{47} +43.0443 q^{49} +117.348i q^{50} -28.3771i q^{52} +49.5329 q^{53} -9.82509 q^{56} -62.3174 q^{58} -19.9324 q^{59} +118.906i q^{61} +55.2266i q^{62} +7.02040 q^{64} +99.4678i q^{65} -4.41442 q^{67} +4.89164i q^{68} -52.1899 q^{70} -6.03591 q^{71} -6.08043i q^{73} -40.5193i q^{74} -20.4114i q^{76} +103.463i q^{79} -167.519 q^{80} +133.213 q^{82} +30.1663i q^{83} -17.1462i q^{85} -107.222 q^{86} +60.4650 q^{89} -28.7376 q^{91} +101.109 q^{92} +40.9674i q^{94} +71.5462i q^{95} +36.6892 q^{97} +108.978i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 20 q^{4} + 4 q^{5}+O(q^{10}) $ 16 * q - 20 * q^4 + 4 * q^5	$ 16 q - 20 q^{4} + 4 q^{5} + 52 q^{14} - 44 q^{16} + 108 q^{20} - 132 q^{23} + 88 q^{25} + 4 q^{26} + 40 q^{31} - 368 q^{34} - 16 q^{37} - 280 q^{38} - 80 q^{47} - 140 q^{49} + 128 q^{53} - 524 q^{56} + 140 q^{58} + 220 q^{59} - 8 q^{64} + 36 q^{67} - 100 q^{70} - 644 q^{71} - 264 q^{80} - 476 q^{82} - 76 q^{86} - 76 q^{89} - 624 q^{91} - 120 q^{92} + 216 q^{97}+O(q^{100}) $ 16 * q - 20 * q^4 + 4 * q^5 + 52 * q^14 - 44 * q^16 + 108 * q^20 - 132 * q^23 + 88 * q^25 + 4 * q^26 + 40 * q^31 - 368 * q^34 - 16 * q^37 - 280 * q^38 - 80 * q^47 - 140 * q^49 + 128 * q^53 - 524 * q^56 + 140 * q^58 + 220 * q^59 - 8 * q^64 + 36 * q^67 - 100 * q^70 - 644 * q^71 - 264 * q^80 - 476 * q^82 - 76 * q^86 - 76 * q^89 - 624 * q^91 - 120 * q^92 + 216 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times$.

$n$	$244$	$848$
$\chi(n)$	$-1$	$1$

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	$0.781848\pi$
$3$	0	0
$3$	0	0
$4$	−2.40981	−0.602454
$5$	8.44690	1.68938	0.844690	−	0.535255i	$-0.179785\pi$
$5$	8.44690	1.68938	0.844690	+	0.535255i	$0.179785\pi$
$6$	0	0
$7$	2.44043i	0.348633i	0.984690	+	0.174316i	$0.0557715\pi$
$7$	2.44043i	0.348633i	−0.984690	+	0.174316i	$0.944228\pi$
$8$	4.02597i	0.503246i
$9$	0	0
$10$	21.3855i	2.13855i
$11$	0	0
$11$	0	0
$12$	0	0
$13$	11.7756i	0.905819i	0.891556	+	0.452910i	$0.149614\pi$
$13$	11.7756i	0.905819i	−0.891556	+	0.452910i	$0.850386\pi$
$14$	−6.17858	−0.441327
$15$	0	0
$16$	−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	$-0.0190152\pi$
$18$	0	0
$19$	8.47011i	0.445795i	0.974842	+	0.222898i	$0.0715516\pi$
$19$	8.47011i	0.445795i	−0.974842	+	0.222898i	$0.928448\pi$
$20$	−20.3555	−1.01777
$21$	0	0
$22$	0	0
$23$	−41.9571	−1.82422	−0.912110	−	0.409946i	$-0.865548\pi$
$23$	−41.9571	−1.82422	−0.912110	+	0.409946i	$0.865548\pi$
$24$	0	0
$25$	46.3502	1.85401
$26$	−29.8131	−1.14666
$27$	0	0
$28$	− 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	$0.860491\pi$
$30$	0	0
$31$	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
$36$	0	0
$37$	−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
$39$	0	0
$40$	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	$-0.221756\pi$
$42$	0	0
$43$	42.3507i	0.984900i	0.870341	+	0.492450i	$0.163899\pi$
$43$	42.3507i	0.984900i	−0.870341	+	0.492450i	$0.836101\pi$
$44$	0	0
$45$	0	0
$46$	− 106.225i	− 2.30924i
$47$	16.1814	0.344284	0.172142	−	0.985072i	$-0.444931\pi$
$47$	16.1814	0.344284	0.172142	+	0.985072i	$0.444931\pi$
$48$	0	0
$49$	43.0443	0.878455
$50$	117.348i	2.34695i
$51$	0	0
$52$	− 28.3771i	− 0.545714i
$53$	49.5329	0.934582	0.467291	−	0.884104i	$-0.345230\pi$
$53$	49.5329	0.934582	0.467291	+	0.884104i	$0.345230\pi$
$54$	0	0
$55$	0	0
$56$	−9.82509	−0.175448
$57$	0	0
$58$	−62.3174	−1.07444
$59$	−19.9324	−0.337838	−0.168919	−	0.985630i	$-0.554028\pi$
$59$	−19.9324	−0.337838	−0.168919	+	0.985630i	$0.554028\pi$
$60$	0	0
$61$	118.906i	1.94928i	0.223784	+	0.974639i	$0.428159\pi$
$61$	118.906i	1.94928i	−0.223784	+	0.974639i	$0.571841\pi$
$62$	55.2266i	0.890751i
$63$	0	0
$64$	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
$69$	0	0
$70$	−52.1899	−0.745570
$71$	−6.03591	−0.0850128	−0.0425064	−	0.999096i	$-0.513534\pi$
$71$	−6.03591	−0.0850128	−0.0425064	+	0.999096i	$0.513534\pi$
$72$	0	0
$73$	− 6.08043i	− 0.0832936i	−0.999132	−	0.0416468i	$-0.986740\pi$
$73$	− 6.08043i	− 0.0832936i	0.999132	−	0.0416468i	$-0.0132604\pi$
$74$	− 40.5193i	− 0.547558i
$75$	0	0
$76$	− 20.4114i	− 0.268571i
$77$	0	0
$78$	0	0
$79$	103.463i	1.30965i	0.755779	+	0.654827i	$0.227258\pi$
$79$	103.463i	1.30965i	−0.755779	+	0.654827i	$0.772742\pi$
$80$	−167.519	−2.09399
$81$	0	0
$82$	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	$0.941832\pi$
$84$	0	0
$85$	− 17.1462i	− 0.201720i
$86$	−107.222	−1.24677
$87$	0	0
$88$	0	0
$89$	60.4650	0.679382	0.339691	−	0.940537i	$-0.389677\pi$
$89$	60.4650	0.679382	0.339691	+	0.940537i	$0.389677\pi$
$90$	0	0
$91$	−28.7376	−0.315798
$92$	101.109	1.09901
$93$	0	0
$94$	40.9674i	0.435823i
$95$	71.5462i	0.753118i
$96$	0	0
$97$	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	$-0.247598\pi$
$102$	0	0
$103$	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
$105$	0	0
$106$	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	$-0.183459\pi$
$108$	0	0
$109$	− 81.2670i	− 0.745569i	−0.927918	−	0.372784i	$-0.878403\pi$
$109$	− 81.2670i	− 0.745569i	0.927918	−	0.372784i	$-0.121597\pi$
$110$	0	0
$111$	0	0
$112$	− 48.3987i	− 0.432131i
$113$	76.4454	0.676508	0.338254	−	0.941055i	$-0.390164\pi$
$113$	76.4454	0.676508	0.338254	+	0.941055i	$0.390164\pi$
$114$	0	0
$115$	−354.407	−3.08180
$116$	− 59.3157i	− 0.511343i
$117$	0	0
$118$	− 50.4642i	− 0.427663i
$119$	4.95378	0.0416284
$120$	0	0
$121$	0	0
$122$	−301.041	−2.46755
$123$	0	0
$124$	−52.5665	−0.423923
$125$	180.343	1.44274
$126$	0	0
$127$	21.0623i	0.165845i	0.996556	+	0.0829224i	$0.0264254\pi$
$127$	21.0623i	0.165845i	−0.996556	+	0.0829224i	$0.973575\pi$
$128$	− 118.651i	− 0.926958i
$129$	0	0
$130$	−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
$135$	0	0
$136$	8.17224	0.0600900
$137$	−119.183	−0.869946	−0.434973	−	0.900443i	$-0.643242\pi$
$137$	−119.183	−0.869946	−0.434973	+	0.900443i	$0.643242\pi$
$138$	0	0
$139$	− 7.42431i	− 0.0534123i	−0.999643	−	0.0267061i	$-0.991498\pi$
$139$	− 7.42431i	− 0.0534123i	0.999643	−	0.0267061i	$-0.00850184\pi$
$140$	− 49.6761i	− 0.354829i
$141$	0	0
$142$	− 15.2815i	− 0.107616i
$143$	0	0
$144$	0	0
$145$	207.914i	1.43389i
$146$	15.3942	0.105440
$147$	0	0
$148$	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	$-0.157204\pi$
$150$	0	0
$151$	− 63.7331i	− 0.422074i	−0.977478	−	0.211037i	$-0.932316\pi$
$151$	− 63.7331i	− 0.422074i	0.977478	−	0.211037i	$-0.0676840\pi$
$152$	−34.1004	−0.224345
$153$	0	0
$154$	0	0
$155$	184.257	1.18875
$156$	0	0
$157$	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
$159$	0	0
$160$	− 288.091i	− 1.80057i
$161$	− 102.393i	− 0.635983i
$162$	0	0
$163$	−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	$0.870677\pi$
$168$	0	0
$169$	30.3341	0.179492
$170$	43.4101	0.255354
$171$	0	0
$172$	− 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	$-0.0616256\pi$
$174$	0	0
$175$	113.114i	0.646368i
$176$	0	0
$177$	0	0
$178$	153.083i	0.860017i
$179$	−178.511	−0.997268	−0.498634	−	0.866813i	$-0.666165\pi$
$179$	−178.511	−0.997268	−0.498634	+	0.866813i	$0.666165\pi$
$180$	0	0
$181$	−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
$183$	0	0
$184$	− 168.918i	− 0.918032i
$185$	−135.188	−0.730744
$186$	0	0
$187$	0	0
$188$	−38.9941	−0.207415
$189$	0	0
$190$	−181.138	−0.953358
$191$	358.333	1.87609	0.938046	−	0.346512i	$-0.112634\pi$
$191$	358.333	1.87609	0.938046	+	0.346512i	$0.112634\pi$
$192$	0	0
$193$	− 213.899i	− 1.10829i	−0.832422	−	0.554143i	$-0.813046\pi$
$193$	− 213.899i	− 1.10829i	0.832422	−	0.554143i	$-0.186954\pi$
$194$	92.8883i	0.478806i
$195$	0	0
$196$	−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
$201$	0	0
$202$	358.887	1.77667
$203$	−60.0693	−0.295908
$204$	0	0
$205$	− 444.447i	− 2.16803i
$206$	98.7926i	0.479576i
$207$	0	0
$208$	− 233.535i	− 1.12277i
$209$	0	0
$210$	0	0
$211$	− 218.966i	− 1.03776i	−0.854849	−	0.518878i	$-0.826350\pi$
$211$	− 218.966i	− 1.03776i	0.854849	−	0.518878i	$-0.173650\pi$
$212$	−119.365	−0.563042
$213$	0	0
$214$	295.263	1.37973
$215$	357.732i	1.66387i
$216$	0	0
$217$	53.2343i	0.245319i
$218$	205.749	0.943801
$219$	0	0
$220$	0	0
$221$	23.9032	0.108159
$222$	0	0
$223$	−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
$225$	0	0
$226$	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	$0.941234\pi$
$228$	0	0
$229$	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	$0.884495\pi$
$234$	0	0
$235$	136.682	0.581628
$236$	48.0335	0.203532
$237$	0	0
$238$	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	$-0.122857\pi$
$240$	0	0
$241$	226.632i	0.940380i	0.882565	+	0.470190i	$0.155815\pi$
$241$	226.632i	0.940380i	−0.882565	+	0.470190i	$0.844185\pi$
$242$	0	0
$243$	0	0
$244$	− 286.541i	− 1.17435i
$245$	363.591	1.48405
$246$	0	0
$247$	−99.7410	−0.403810
$248$	87.8205i	0.354115i
$249$	0	0
$250$	456.586i	1.82634i
$251$	79.1282	0.315252	0.157626	−	0.987499i	$-0.449616\pi$
$251$	79.1282	0.315252	0.157626	+	0.987499i	$0.449616\pi$
$252$	0	0
$253$	0	0
$254$	−53.3247	−0.209940
$255$	0	0
$256$	328.477	1.28311
$257$	−298.533	−1.16161	−0.580803	−	0.814044i	$-0.697261\pi$
$257$	−298.533	−1.16161	−0.580803	+	0.814044i	$0.697261\pi$
$258$	0	0
$259$	− 39.0576i	− 0.150801i
$260$	− 239.699i	− 0.921919i
$261$	0	0
$262$	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
$267$	0	0
$268$	10.6379	0.0396938
$269$	195.972	0.728522	0.364261	−	0.931297i	$-0.381322\pi$
$269$	195.972	0.728522	0.364261	+	0.931297i	$0.381322\pi$
$270$	0	0
$271$	340.500i	1.25646i	0.778029	+	0.628229i	$0.216220\pi$
$271$	340.500i	1.25646i	−0.778029	+	0.628229i	$0.783780\pi$
$272$	40.2567i	0.148003i
$273$	0	0
$274$	− 301.742i	− 1.10125i
$275$	0	0
$276$	0	0
$277$	− 350.746i	− 1.26623i	−0.774057	−	0.633116i	$-0.781776\pi$
$277$	− 350.746i	− 1.26623i	0.774057	−	0.633116i	$-0.218224\pi$
$278$	18.7966	0.0676136
$279$	0	0
$280$	−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	$-0.258976\pi$
$282$	0	0
$283$	− 321.158i	− 1.13483i	−0.823431	−	0.567417i	$-0.807943\pi$
$283$	− 321.158i	− 1.13483i	0.823431	−	0.567417i	$-0.192057\pi$
$284$	14.5454	0.0512163
$285$	0	0
$286$	0	0
$287$	128.407	0.447411
$288$	0	0
$289$	284.880	0.985742
$290$	−526.389	−1.81513
$291$	0	0
$292$	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	$0.777497\pi$
$294$	0	0
$295$	−168.367	−0.570737
$296$	− 64.4332i	− 0.217680i
$297$	0	0
$298$	357.645	1.20015
$299$	− 494.072i	− 1.65241i
$300$	0	0
$301$	−103.354	−0.343368
$302$	161.357	0.534295
$303$	0	0
$304$	− 167.980i	− 0.552565i
$305$	1004.39i	3.29307i
$306$	0	0
$307$	505.973i	1.64812i	0.566502	+	0.824060i	$0.308296\pi$
$307$	505.973i	1.64812i	−0.566502	+	0.824060i	$0.691704\pi$
$308$	0	0
$309$	0	0
$310$	466.494i	1.50482i
$311$	166.073	0.533997	0.266998	−	0.963697i	$-0.413968\pi$
$311$	166.073	0.533997	0.266998	+	0.963697i	$0.413968\pi$
$312$	0	0
$313$	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
$315$	0	0
$316$	− 249.326i	− 0.789006i
$317$	−149.549	−0.471764	−0.235882	−	0.971782i	$-0.575798\pi$
$317$	−149.549	−0.471764	−0.235882	+	0.971782i	$0.575798\pi$
$318$	0	0
$319$	0	0
$320$	59.3006	0.185314
$321$	0	0
$322$	259.235	0.805078
$323$	17.1933	0.0532301
$324$	0	0
$325$	545.804i	1.67940i
$326$	− 135.794i	− 0.416545i
$327$	0	0
$328$	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
$333$	0	0
$334$	−334.179	−1.00054
$335$	−37.2882	−0.111308
$336$	0	0
$337$	500.414i	1.48491i	0.669898	+	0.742454i	$0.266338\pi$
$337$	500.414i	1.48491i	−0.669898	+	0.742454i	$0.733662\pi$
$338$	76.7987i	0.227215i
$339$	0	0
$340$	41.3192i	0.121527i
$341$	0	0
$342$	0	0
$343$	224.628i	0.654891i
$344$	−170.503	−0.495647
$345$	0	0
$346$	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	$-0.113735\pi$
$348$	0	0
$349$	447.324i	1.28173i	0.767653	+	0.640866i	$0.221424\pi$
$349$	447.324i	1.28173i	−0.767653	+	0.640866i	$0.778576\pi$
$350$	−286.378	−0.818224
$351$	0	0
$352$	0	0
$353$	226.910	0.642805	0.321403	−	0.946943i	$-0.395846\pi$
$353$	226.910	0.642805	0.321403	+	0.946943i	$0.395846\pi$
$354$	0	0
$355$	−50.9847	−0.143619
$356$	−145.710	−0.409296
$357$	0	0
$358$	− 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	$-0.352867\pi$
$360$	0	0
$361$	289.257	0.801267
$362$	− 131.941i	− 0.364478i
$363$	0	0
$364$	69.2524	0.190254
$365$	− 51.3608i	− 0.140715i
$366$	0	0
$367$	−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
$369$	0	0
$370$	− 342.263i	− 0.925034i
$371$	120.881i	0.325826i
$372$	0	0
$373$	18.1857i	0.0487552i	0.999703	+	0.0243776i	$0.00776040\pi$
$373$	18.1857i	0.0487552i	−0.999703	+	0.0243776i	$0.992240\pi$
$374$	0	0
$375$	0	0
$376$	65.1457i	0.173260i
$377$	−289.849	−0.768829
$378$	0	0
$379$	−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
$381$	0	0
$382$	907.215i	2.37491i
$383$	720.395	1.88093	0.940464	−	0.339894i	$-0.110391\pi$
$383$	720.395	1.88093	0.940464	+	0.339894i	$0.110391\pi$
$384$	0	0
$385$	0	0
$386$	541.542	1.40296
$387$	0	0
$388$	−88.4142	−0.227872
$389$	−8.88848	−0.0228496	−0.0114248	−	0.999935i	$-0.503637\pi$
$389$	−8.88848	−0.0228496	−0.0114248	+	0.999935i	$0.503637\pi$
$390$	0	0
$391$	85.1679i	0.217821i
$392$	173.295i	0.442079i
$393$	0	0
$394$	−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
$399$	0	0
$400$	−919.219	−2.29805
$401$	−548.823	−1.36863	−0.684317	−	0.729184i	$-0.739900\pi$
$401$	−548.823	−1.36863	−0.684317	+	0.729184i	$0.739900\pi$
$402$	0	0
$403$	256.868i	0.637390i
$404$	341.600i	0.845546i
$405$	0	0
$406$	− 152.081i	− 0.374584i
$407$	0	0
$408$	0	0
$409$	− 506.077i	− 1.23735i	−0.785646	−	0.618676i	$-0.787669\pi$
$409$	− 506.077i	− 1.23735i	0.785646	−	0.618676i	$-0.212331\pi$
$410$	1125.23	2.74447
$411$	0	0
$412$	−94.0341	−0.228238
$413$	− 48.6437i	− 0.117781i
$414$	0	0
$415$	254.812i	0.614004i
$416$	401.622	0.965437
$417$	0	0
$418$	0	0
$419$	111.580	0.266300	0.133150	−	0.991096i	$-0.457491\pi$
$419$	111.580	0.266300	0.133150	+	0.991096i	$0.457491\pi$
$420$	0	0
$421$	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
$423$	0	0
$424$	199.418i	0.470325i
$425$	− 94.0854i	− 0.221377i
$426$	0	0
$427$	−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	$0.971131\pi$
$432$	0	0
$433$	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
$435$	0	0
$436$	195.838i	0.449170i
$437$	− 355.381i	− 0.813229i
$438$	0	0
$439$	− 262.674i	− 0.598347i	−0.954199	−	0.299173i	$-0.903289\pi$
$439$	− 262.674i	− 0.598347i	0.954199	−	0.299173i	$-0.0967109\pi$
$440$	0	0
$441$	0	0
$442$	60.5171i	0.136917i
$443$	528.705	1.19346	0.596732	−	0.802441i	$-0.296466\pi$
$443$	528.705	1.19346	0.596732	+	0.802441i	$0.296466\pi$
$444$	0	0
$445$	510.742	1.14774
$446$	− 530.526i	− 1.18952i
$447$	0	0
$448$	17.1328i	0.0382428i
$449$	287.807	0.640995	0.320498	−	0.947249i	$-0.396150\pi$
$449$	287.807	0.640995	0.320498	+	0.947249i	$0.396150\pi$
$450$	0	0
$451$	0	0
$452$	−184.219	−0.407565
$453$	0	0
$454$	−211.000	−0.464759
$455$	−242.744	−0.533503
$456$	0	0
$457$	− 826.946i	− 1.80951i	−0.425932	−	0.904755i	$-0.640054\pi$
$457$	− 826.946i	− 1.80951i	0.425932	−	0.904755i	$-0.359946\pi$
$458$	837.157i	1.82785i
$459$	0	0
$460$	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
$465$	0	0
$466$	−418.779	−0.898668
$467$	−458.119	−0.980982	−0.490491	−	0.871446i	$-0.663183\pi$
$467$	−458.119	−0.980982	−0.490491	+	0.871446i	$0.663183\pi$
$468$	0	0
$469$	− 10.7731i	− 0.0229703i
$470$	346.047i	0.736271i
$471$	0	0
$472$	− 80.2474i	− 0.170016i
$473$	0	0
$474$	0	0
$475$	392.591i	0.826508i
$476$	−11.9377	−0.0250792
$477$	0	0
$478$	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	$0.787779\pi$
$480$	0	0
$481$	− 188.462i	− 0.391813i
$482$	−573.777	−1.19041
$483$	0	0
$484$	0	0
$485$	309.910	0.638990
$486$	0	0
$487$	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
$489$	0	0
$490$	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	$-0.289367\pi$
$492$	0	0
$493$	49.9640	0.101347
$494$	− 252.521i	− 0.511175i
$495$	0	0
$496$	−432.607	−0.872191
$497$	− 14.7302i	− 0.0296382i
$498$	0	0
$499$	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
$501$	0	0
$502$	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	$-0.0487604\pi$
$504$	0	0
$505$	− 1197.38i	− 2.37105i
$506$	0	0
$507$	0	0
$508$	− 50.7562i	− 0.0999138i
$509$	782.781	1.53788	0.768940	−	0.639321i	$-0.220785\pi$
$509$	782.781	1.53788	0.768940	+	0.639321i	$0.220785\pi$
$510$	0	0
$511$	14.8389	0.0290389
$512$	357.022i	0.697308i
$513$	0	0
$514$	− 755.813i	− 1.47045i
$515$	329.609	0.640018
$516$	0	0
$517$	0	0
$518$	98.8844	0.190897
$519$	0	0
$520$	−400.454	−0.770104
$521$	180.526	0.346498	0.173249	−	0.984878i	$-0.444573\pi$
$521$	180.526	0.346498	0.173249	+	0.984878i	$0.444573\pi$
$522$	0	0
$523$	− 151.833i	− 0.290311i	−0.989409	−	0.145156i	$-0.953632\pi$
$523$	− 151.833i	− 0.290311i	0.989409	−	0.145156i	$-0.0463683\pi$
$524$	257.482i	0.491378i
$525$	0	0
$526$	−152.737	−0.290375
$527$	− 44.2788i	− 0.0840206i
$528$	0	0
$529$	1231.39	2.32778
$530$	1059.29i	1.99865i
$531$	0	0
$532$	49.8126	0.0936326
$533$	619.594	1.16247
$534$	0	0
$535$	− 985.107i	− 1.84132i
$536$	− 17.7723i	− 0.0331573i
$537$	0	0
$538$	496.155i	0.922222i
$539$	0	0
$540$	0	0
$541$	− 287.853i	− 0.532075i	−0.963963	−	0.266038i	$-0.914285\pi$
$541$	− 287.853i	− 0.532075i	0.963963	−	0.266038i	$-0.0857146\pi$
$542$	−862.065	−1.59052
$543$	0	0
$544$	−69.2314	−0.127264
$545$	− 686.454i	− 1.25955i
$546$	0	0
$547$	− 122.440i	− 0.223839i	−0.993717	−	0.111920i	$-0.964300\pi$
$547$	− 122.440i	− 0.223839i	0.993717	−	0.111920i	$-0.0356999\pi$
$548$	287.208	0.524102
$549$	0	0
$550$	0	0
$551$	−208.485	−0.378376
$552$	0	0
$553$	−252.493	−0.456588
$554$	888.005	1.60290
$555$	0	0
$556$	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	$-0.167781\pi$
$558$	0	0
$559$	−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	$-0.134397\pi$
$564$	0	0
$565$	645.727	1.14288
$566$	813.095	1.43656
$567$	0	0
$568$	− 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	$-0.268881\pi$
$570$	0	0
$571$	558.153i	0.977500i	0.872424	+	0.488750i	$0.162547\pi$
$571$	558.153i	0.977500i	−0.872424	+	0.488750i	$0.837453\pi$
$572$	0	0
$573$	0	0
$574$	325.096i	0.566369i
$575$	−1944.72	−3.38212
$576$	0	0
$577$	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
$579$	0	0
$580$	− 501.034i	− 0.863852i
$581$	−73.6186	−0.126710
$582$	0	0
$583$	0	0
$584$	24.4796	0.0419172
$585$	0	0
$586$	−954.648	−1.62909
$587$	137.730	0.234633	0.117317	−	0.993095i	$-0.462571\pi$
$587$	137.730	0.234633	0.117317	+	0.993095i	$0.462571\pi$
$588$	0	0
$589$	184.763i	0.313689i
$590$	− 426.266i	− 0.722485i
$591$	0	0
$592$	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
$597$	0	0
$598$	1250.87	2.09176
$599$	323.005	0.539240	0.269620	−	0.962967i	$-0.413102\pi$
$599$	323.005	0.539240	0.269620	+	0.962967i	$0.413102\pi$
$600$	0	0
$601$	682.717i	1.13597i	0.823039	+	0.567985i	$0.192277\pi$
$601$	682.717i	1.13597i	−0.823039	+	0.567985i	$0.807723\pi$
$602$	− 261.667i	− 0.434663i
$603$	0	0
$604$	153.585i	0.254280i
$605$	0	0
$606$	0	0
$607$	219.873i	0.362229i	0.983462	+	0.181114i	$0.0579705\pi$
$607$	219.873i	0.362229i	−0.983462	+	0.181114i	$0.942030\pi$
$608$	288.883	0.475136
$609$	0	0
$610$	−2542.87	−4.16864
$611$	190.546i	0.311859i
$612$	0	0
$613$	− 198.057i	− 0.323094i	−0.986865	−	0.161547i	$-0.948352\pi$
$613$	− 198.057i	− 0.323094i	0.986865	−	0.161547i	$-0.0516483\pi$
$614$	−1281.00	−2.08632
$615$	0	0
$616$	0	0
$617$	−666.299	−1.07990	−0.539951	−	0.841697i	$-0.681557\pi$
$617$	−666.299	−1.07990	−0.539951	+	0.841697i	$0.681557\pi$
$618$	0	0
$619$	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
$621$	0	0
$622$	420.457i	0.675976i
$623$	147.561i	0.236855i
$624$	0	0
$625$	364.586	0.583337
$626$	237.087i	0.378733i
$627$	0	0
$628$	−88.8984	−0.141558
$629$	32.4870i	0.0516487i
$630$	0	0
$631$	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
$633$	0	0
$634$	− 378.623i	− 0.597197i
$635$	177.911i	0.280175i
$636$	0	0
$637$	506.875i	0.795722i
$638$	0	0
$639$	0	0
$640$	− 1002.23i	− 1.56598i
$641$	−203.013	−0.316714	−0.158357	−	0.987382i	$-0.550620\pi$
$641$	−203.013	−0.316714	−0.158357	+	0.987382i	$0.550620\pi$
$642$	0	0
$643$	−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
$645$	0	0
$646$	43.5294i	0.0673830i
$647$	−535.496	−0.827659	−0.413830	−	0.910354i	$-0.635809\pi$
$647$	−535.496	−0.827659	−0.413830	+	0.910354i	$0.635809\pi$
$648$	0	0
$649$	0	0
$650$	−1381.84	−2.12591
$651$	0	0
$652$	129.253	0.198241
$653$	829.067	1.26963	0.634814	−	0.772665i	$-0.281077\pi$
$653$	829.067	1.26963	0.634814	+	0.772665i	$0.281077\pi$
$654$	0	0
$655$	− 902.529i	− 1.37791i
$656$	1043.49i	1.59069i
$657$	0	0
$658$	−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
$663$	0	0
$664$	−121.448	−0.182904
$665$	−174.603	−0.262562
$666$	0	0
$667$	− 1032.74i	− 1.54834i
$668$	− 318.083i	− 0.476172i
$669$	0	0
$670$	− 94.4048i	− 0.140903i
$671$	0	0
$672$	0	0
$673$	645.812i	0.959602i	0.877377	+	0.479801i	$0.159291\pi$
$673$	645.812i	0.959602i	−0.877377	+	0.479801i	$0.840709\pi$
$674$	−1266.93	−1.87972
$675$	0	0
$676$	−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	$-0.183383\pi$
$678$	0	0
$679$	89.5373i	0.131866i
$680$	69.0302	0.101515
$681$	0	0
$682$	0	0
$683$	−527.576	−0.772440	−0.386220	−	0.922407i	$-0.626219\pi$
$683$	−527.576	−0.772440	−0.386220	+	0.922407i	$0.626219\pi$
$684$	0	0
$685$	−1006.72	−1.46967
$686$	−568.703	−0.829014
$687$	0	0
$688$	− 839.901i	− 1.22079i
$689$	583.281i	0.846562i
$690$	0	0
$691$	−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
$696$	0	0
$697$	−106.805	−0.153236
$698$	−1132.52	−1.62252
$699$	0	0
$700$	− 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	$0.833599\pi$
$702$	0	0
$703$	− 135.559i	− 0.192829i
$704$	0	0
$705$	0	0
$706$	574.482i	0.813714i
$707$	345.940	0.489307
$708$	0	0
$709$	−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
$711$	0	0
$712$	243.430i	0.341897i
$713$	−915.230	−1.28363
$714$	0	0
$715$	0	0
$716$	430.178	0.600808
$717$	0	0
$718$	1627.04	2.26608
$719$	−694.522	−0.965955	−0.482978	−	0.875633i	$-0.660445\pi$
$719$	−694.522	−0.965955	−0.482978	+	0.875633i	$0.660445\pi$
$720$	0	0
$721$	95.2287i	0.132079i
$722$	732.330i	1.01431i
$723$	0	0
$724$	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
$729$	0	0
$730$	130.033	0.178128
$731$	85.9669	0.117602
$732$	0	0
$733$	− 157.602i	− 0.215009i	−0.994205	−	0.107504i	$-0.965714\pi$
$733$	− 157.602i	− 0.215009i	0.994205	−	0.107504i	$-0.0342860\pi$
$734$	− 1817.51i	− 2.47616i
$735$	0	0
$736$	1430.99i	1.94428i
$737$	0	0
$738$	0	0
$739$	− 685.531i	− 0.927647i	−0.885928	−	0.463823i	$-0.846477\pi$
$739$	− 685.531i	− 0.927647i	0.885928	−	0.463823i	$-0.153523\pi$
$740$	325.777	0.440239
$741$	0	0
$742$	−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	$0.887101\pi$
$744$	0	0
$745$	− 1193.24i	− 1.60166i
$746$	−46.0418	−0.0617183
$747$	0	0
$748$	0	0
$749$	284.611	0.379988
$750$	0	0
$751$	−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
$753$	0	0
$754$	− 733.827i	− 0.973246i
$755$	− 538.347i	− 0.713043i
$756$	0	0
$757$	−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	$0.999729\pi$
$762$	0	0
$763$	198.326	0.259930
$764$	−863.517	−1.13026
$765$	0	0
$766$	1823.87i	2.38103i
$767$	− 234.717i	− 0.306020i
$768$	0	0
$769$	− 398.342i	− 0.518000i	−0.965877	−	0.259000i	$-0.916607\pi$
$769$	− 398.342i	− 0.518000i	0.965877	−	0.259000i	$-0.0833929\pi$
$770$	0	0
$771$	0	0
$772$	515.457i	0.667691i
$773$	637.493	0.824700	0.412350	−	0.911025i	$-0.364708\pi$
$773$	637.493	0.824700	0.412350	+	0.911025i	$0.364708\pi$
$774$	0	0
$775$	1011.06	1.30459
$776$	147.710i	0.190347i
$777$	0	0
$778$	− 22.5035i	− 0.0289248i
$779$	445.668	0.572103
$780$	0	0
$781$	0	0
$782$	−215.625	−0.275735
$783$	0	0
$784$	−853.657	−1.08885
$785$	311.608	0.396952
$786$	0	0
$787$	− 1070.60i	− 1.36036i	−0.733045	−	0.680180i	$-0.761901\pi$
$787$	− 1070.60i	− 1.36036i	0.733045	−	0.680180i	$-0.238099\pi$
$788$	− 666.477i	− 0.845783i
$789$	0	0
$790$	−2212.60	−2.80077
$791$	186.560i	0.235853i
$792$	0	0
$793$	−1400.19	−1.76569
$794$	1299.44i	1.63657i
$795$	0	0
$796$	−189.914	−0.238585
$797$	−365.858	−0.459044	−0.229522	−	0.973303i	$-0.573716\pi$
$797$	−365.858	−0.459044	−0.229522	+	0.973303i	$0.573716\pi$
$798$	0	0
$799$	− 32.8463i	− 0.0411092i
$800$	− 1580.83i	− 1.97603i
$801$	0	0
$802$	− 1389.49i	− 1.73253i
$803$	0	0
$804$	0	0
$805$	− 864.906i	− 1.07442i
$806$	−650.329	−0.806860
$807$	0	0
$808$	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	$0.965175\pi$
$810$	0	0
$811$	− 256.402i	− 0.316156i	−0.987427	−	0.158078i	$-0.949470\pi$
$811$	− 256.402i	− 0.316156i	0.987427	−	0.158078i	$-0.0505297\pi$
$812$	144.756	0.178271
$813$	0	0
$814$	0	0
$815$	−453.058	−0.555900
$816$	0	0
$817$	−358.715	−0.439064
$818$	1281.27	1.56634
$819$	0	0
$820$	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	$0.806304\pi$
$822$	0	0
$823$	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	$0.703731\pi$
$828$	0	0
$829$	−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
$831$	0	0
$832$	82.6697i	0.0993626i
$833$	− 87.3749i	− 0.104892i
$834$	0	0
$835$	1114.95i	1.33527i
$836$	0	0
$837$	0	0
$838$	282.494i	0.337104i
$839$	−283.608	−0.338032	−0.169016	−	0.985613i	$-0.554059\pi$
$839$	−283.608	−0.338032	−0.169016	+	0.985613i	$0.554059\pi$
$840$	0	0
$841$	235.139	0.279595
$842$	945.104i	1.12245i
$843$	0	0
$844$	527.668i	0.625199i
$845$	256.229	0.303230
$846$	0	0
$847$	0	0
$848$	−982.338	−1.15842
$849$	0	0
$850$	238.202	0.280237
$851$	671.497	0.789068
$852$	0	0
$853$	440.959i	0.516951i	0.966018	+	0.258476i	$0.0832202\pi$
$853$	440.959i	0.516951i	−0.966018	+	0.258476i	$0.916780\pi$
$854$	− 734.670i	− 0.860269i
$855$	0	0
$856$	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
$861$	0	0
$862$	−197.661	−0.229305
$863$	514.959	0.596708	0.298354	−	0.954455i	$-0.403562\pi$
$863$	514.959	0.596708	0.298354	+	0.954455i	$0.403562\pi$
$864$	0	0
$865$	− 562.301i	− 0.650058i
$866$	1801.30i	2.08002i
$867$	0	0
$868$	− 128.285i	− 0.147794i
$869$	0	0
$870$	0	0
$871$	− 51.9827i	− 0.0596816i
$872$	327.178	0.375205
$873$	0	0
$874$	899.740	1.02945
$875$	440.114i	0.502988i
$876$	0	0
$877$	742.107i	0.846188i	0.906086	+	0.423094i	$0.139056\pi$
$877$	742.107i	0.846188i	−0.906086	+	0.423094i	$0.860944\pi$
$878$	665.028	0.757435
$879$	0	0
$880$	0	0
$881$	620.478	0.704288	0.352144	−	0.935946i	$-0.385453\pi$
$881$	620.478	0.704288	0.352144	+	0.935946i	$0.385453\pi$
$882$	0	0
$883$	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
$885$	0	0
$886$	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	$0.787463\pi$
$888$	0	0
$889$	−51.4010	−0.0578189
$890$	1293.08i	1.45290i
$891$	0	0
$892$	504.973	0.566113
$893$	137.058i	0.153480i
$894$	0	0
$895$	−1507.86	−1.68477
$896$	289.558	0.323168
$897$	0	0
$898$	728.658i	0.811424i
$899$	536.923i	0.597244i
$900$	0	0
$901$	− 100.546i	− 0.111594i
$902$	0	0
$903$	0	0
$904$	307.767i	0.340450i
$905$	−440.205	−0.486415
$906$	0	0
$907$	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
$909$	0	0
$910$	− 614.570i	− 0.675351i
$911$	−686.752	−0.753844	−0.376922	−	0.926245i	$-0.623018\pi$
$911$	−686.752	−0.753844	−0.376922	+	0.926245i	$0.623018\pi$
$912$	0	0
$913$	0	0
$914$	2093.63	2.29062
$915$	0	0
$916$	−796.834	−0.869906
$917$	260.753	0.284355
$918$	0	0
$919$	1485.25i	1.61615i	0.589077	+	0.808077i	$0.299491\pi$
$919$	1485.25i	1.61615i	−0.589077	+	0.808077i	$0.700509\pi$
$920$	− 1426.83i	− 1.55091i
$921$	0	0
$922$	823.718	0.893403
$923$	− 71.0767i	− 0.0770062i
$924$	0	0
$925$	−741.807	−0.801953
$926$	− 1801.74i	− 1.94573i
$927$	0	0
$928$	839.497	0.904630
$929$	681.402	0.733479	0.366740	−	0.930324i	$-0.380474\pi$
$929$	681.402	0.733479	0.366740	+	0.930324i	$0.380474\pi$
$930$	0	0
$931$	364.590i	0.391611i
$932$	− 398.608i	− 0.427691i
$933$	0	0
$934$	− 1159.85i	− 1.24181i
$935$	0	0
$936$	0	0
$937$	88.5349i	0.0944876i	0.998883	+	0.0472438i	$0.0150438\pi$
$937$	88.5349i	0.0944876i	−0.998883	+	0.0472438i	$0.984956\pi$
$938$	27.2749	0.0290777
$939$	0	0
$940$	−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	$-0.100088\pi$
$942$	0	0
$943$	2207.64i	2.34108i
$944$	395.301	0.418751
$945$	0	0
$946$	0	0
$947$	912.836	0.963924	0.481962	−	0.876192i	$-0.339924\pi$
$947$	912.836	0.963924	0.481962	+	0.876192i	$0.339924\pi$
$948$	0	0
$949$	71.6011	0.0754489
$950$	−993.947	−1.04626
$951$	0	0
$952$	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	$0.906008\pi$
$954$	0	0
$955$	3026.81	3.16943
$956$	433.636i	0.453594i
$957$	0	0
$958$	−1499.90	−1.56566
$959$	− 290.857i	− 0.303292i
$960$	0	0
$961$	−485.171	−0.504861
$962$	477.141	0.495989
$963$	0	0
$964$	− 546.140i	− 0.566535i
$965$	− 1806.79i	− 1.87232i
$966$	0	0
$967$	− 1781.13i	− 1.84191i	−0.389667	−	0.920956i	$-0.627410\pi$
$967$	− 1781.13i	− 1.84191i	0.389667	−	0.920956i	$-0.372590\pi$
$968$	0	0
$969$	0	0
$970$	784.618i	0.808885i
$971$	990.964	1.02056	0.510280	−	0.860008i	$-0.329542\pi$
$971$	990.964	1.02056	0.510280	+	0.860008i	$0.329542\pi$
$972$	0	0
$973$	18.1185	0.0186213
$974$	1108.44i	1.13803i
$975$	0	0
$976$	− 2358.15i	− 2.41614i
$977$	−754.917	−0.772689	−0.386344	−	0.922355i	$-0.626262\pi$
$977$	−754.917	−0.772689	−0.386344	+	0.922355i	$0.626262\pi$
$978$	0	0
$979$	0	0
$980$	−876.187	−0.894069
$981$	0	0
$982$	1961.44	1.99740
$983$	206.512	0.210084	0.105042	−	0.994468i	$-0.466502\pi$
$983$	206.512	0.210084	0.105042	+	0.994468i	$0.466502\pi$
$984$	0	0
$985$	2336.14i	2.37172i
$986$	126.497i	0.128293i
$987$	0	0
$988$	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
$993$	0	0
$994$	37.2933	0.0375185
$995$	665.687	0.669033
$996$	0	0
$997$	1168.11i	1.17162i	0.810448	+	0.585810i	$0.199224\pi$
$997$	1168.11i	1.17162i	−0.810448	+	0.585810i	$0.800776\pi$
$998$	663.601i	0.664931i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.3.c.m.604.14		16
3.2	odd	2		363.3.c.e.241.3		16
11.3	even	5		99.3.k.c.46.3		16
11.7	odd	10		99.3.k.c.28.3		16
11.10	odd	2	inner	1089.3.c.m.604.3		16
33.2	even	10		363.3.g.g.40.3		16
33.5	odd	10		363.3.g.g.118.3		16
33.8	even	10		363.3.g.f.112.3		16
33.14	odd	10		33.3.g.a.13.2	✓	16
33.17	even	10		363.3.g.a.118.2		16
33.20	odd	10		363.3.g.a.40.2		16
33.26	odd	10		363.3.g.f.94.3		16
33.29	even	10		33.3.g.a.28.2	yes	16
33.32	even	2		363.3.c.e.241.14		16
132.47	even	10		528.3.bf.b.145.1		16
132.95	odd	10		528.3.bf.b.193.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.3.g.a.13.2	✓	16	33.14	odd	10
33.3.g.a.28.2	yes	16	33.29	even	10
99.3.k.c.28.3		16	11.7	odd	10
99.3.k.c.46.3		16	11.3	even	5
363.3.c.e.241.3		16	3.2	odd	2
363.3.c.e.241.14		16	33.32	even	2
363.3.g.a.40.2		16	33.20	odd	10
363.3.g.a.118.2		16	33.17	even	10
363.3.g.f.94.3		16	33.26	odd	10
363.3.g.f.112.3		16	33.8	even	10
363.3.g.g.40.3		16	33.2	even	10
363.3.g.g.118.3		16	33.5	odd	10
528.3.bf.b.145.1		16	132.47	even	10
528.3.bf.b.193.1		16	132.95	odd	10
1089.3.c.m.604.3		16	11.10	odd	2	inner
1089.3.c.m.604.14		16	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1089 = 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1089.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.53176i q^{2} -2.40981 q^{4} +8.44690 q^{5} +2.44043i q^{7} +4.02597i q^{8} +O(q^{10})\)	\(q+2.53176i q^{2} -2.40981 q^{4} +8.44690 q^{5} +2.44043i q^{7} +4.02597i q^{8} +21.3855i q^{10} +11.7756i q^{13} -6.17858 q^{14} -19.8321 q^{16} -2.02988i q^{17} +8.47011i q^{19} -20.3555 q^{20} -41.9571 q^{23} +46.3502 q^{25} -29.8131 q^{26} -5.88098i q^{28} +24.6142i q^{29} +21.8135 q^{31} -34.1061i q^{32} +5.13918 q^{34} +20.6141i q^{35} -16.0044 q^{37} -21.4443 q^{38} +34.0070i q^{40} -52.6166i q^{41} +42.3507i q^{43} -106.225i q^{46} +16.1814 q^{47} +43.0443 q^{49} +117.348i q^{50} -28.3771i q^{52} +49.5329 q^{53} -9.82509 q^{56} -62.3174 q^{58} -19.9324 q^{59} +118.906i q^{61} +55.2266i q^{62} +7.02040 q^{64} +99.4678i q^{65} -4.41442 q^{67} +4.89164i q^{68} -52.1899 q^{70} -6.03591 q^{71} -6.08043i q^{73} -40.5193i q^{74} -20.4114i q^{76} +103.463i q^{79} -167.519 q^{80} +133.213 q^{82} +30.1663i q^{83} -17.1462i q^{85} -107.222 q^{86} +60.4650 q^{89} -28.7376 q^{91} +101.109 q^{92} +40.9674i q^{94} +71.5462i q^{95} +36.6892 q^{97} +108.978i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 20 q^{4} + 4 q^{5}+O(q^{10}) \) 16 * q - 20 * q^4 + 4 * q^5	\( 16 q - 20 q^{4} + 4 q^{5} + 52 q^{14} - 44 q^{16} + 108 q^{20} - 132 q^{23} + 88 q^{25} + 4 q^{26} + 40 q^{31} - 368 q^{34} - 16 q^{37} - 280 q^{38} - 80 q^{47} - 140 q^{49} + 128 q^{53} - 524 q^{56} + 140 q^{58} + 220 q^{59} - 8 q^{64} + 36 q^{67} - 100 q^{70} - 644 q^{71} - 264 q^{80} - 476 q^{82} - 76 q^{86} - 76 q^{89} - 624 q^{91} - 120 q^{92} + 216 q^{97}+O(q^{100}) \) 16 * q - 20 * q^4 + 4 * q^5 + 52 * q^14 - 44 * q^16 + 108 * q^20 - 132 * q^23 + 88 * q^25 + 4 * q^26 + 40 * q^31 - 368 * q^34 - 16 * q^37 - 280 * q^38 - 80 * q^47 - 140 * q^49 + 128 * q^53 - 524 * q^56 + 140 * q^58 + 220 * q^59 - 8 * q^64 + 36 * q^67 - 100 * q^70 - 644 * q^71 - 264 * q^80 - 476 * q^82 - 76 * q^86 - 76 * q^89 - 624 * q^91 - 120 * q^92 + 216 * q^97

Introduction

L-functions

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Database

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