LMFDB - Embedded newform 189.3.b.c.134.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [189,3,Mod(134,189)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(189, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

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

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

chi := DirichletCharacter("189.134");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 189 = 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	189.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$5.14987699641$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.1997017344.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 14x^{6} + 53x^{4} + 56x^{2} + 4 $ x^8 + 14x^6 + 53x^4 + 56*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			134.2
Root			$0.277334i$ of defining polynomial
Character	$\chi$	$=$	189.134
Dual form			189.3.b.c.134.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.27733i q^{2} -1.18625 q^{4} +4.94527i q^{5} +2.64575 q^{7} -6.40785i q^{8} +O(q^{10})$	\(q-2.27733i q^{2} -1.18625 q^{4} +4.94527i q^{5} +2.64575 q^{7} -6.40785i q^{8} +11.2620 q^{10} -20.1892i q^{11} +25.8541 q^{13} -6.02526i q^{14} -19.3378 q^{16} -6.60531i q^{17} +5.53147 q^{19} -5.86633i q^{20} -45.9776 q^{22} +14.5333i q^{23} +0.544307 q^{25} -58.8783i q^{26} -3.13853 q^{28} +17.4877i q^{29} -45.3926 q^{31} +18.4073i q^{32} -15.0425 q^{34} +13.0840i q^{35} +40.8908 q^{37} -12.5970i q^{38} +31.6885 q^{40} +30.4091i q^{41} -53.5305 q^{43} +23.9495i q^{44} +33.0973 q^{46} +27.8541i q^{47} +7.00000 q^{49} -1.23957i q^{50} -30.6694 q^{52} -6.45594i q^{53} +99.8411 q^{55} -16.9536i q^{56} +39.8254 q^{58} -42.8050i q^{59} +35.5980 q^{61} +103.374i q^{62} -35.4317 q^{64} +127.855i q^{65} -83.7309 q^{67} +7.83556i q^{68} +29.7965 q^{70} +113.821i q^{71} -25.5762 q^{73} -93.1220i q^{74} -6.56172 q^{76} -53.4156i q^{77} +81.8944 q^{79} -95.6307i q^{80} +69.2516 q^{82} +28.7852i q^{83} +32.6651 q^{85} +121.907i q^{86} -129.369 q^{88} +5.92931i q^{89} +68.4034 q^{91} -17.2402i q^{92} +63.4332 q^{94} +27.3546i q^{95} -94.4881 q^{97} -15.9413i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 12 q^{4}+O(q^{10}) $ 8 * q - 12 * q^4	$ 8 q - 12 q^{4} - 52 q^{10} + 36 q^{13} + 132 q^{16} + 12 q^{19} - 136 q^{22} - 108 q^{25} + 56 q^{28} - 28 q^{31} - 12 q^{34} - 4 q^{37} + 336 q^{40} - 152 q^{43} + 108 q^{46} + 56 q^{49} - 272 q^{52} + 196 q^{55} - 220 q^{58} + 180 q^{61} - 700 q^{64} - 132 q^{67} + 196 q^{70} + 272 q^{73} + 544 q^{76} + 316 q^{79} + 28 q^{82} - 228 q^{85} - 56 q^{91} - 348 q^{94} - 364 q^{97}+O(q^{100}) $ 8 * q - 12 * q^4 - 52 * q^10 + 36 * q^13 + 132 * q^16 + 12 * q^19 - 136 * q^22 - 108 * q^25 + 56 * q^28 - 28 * q^31 - 12 * q^34 - 4 * q^37 + 336 * q^40 - 152 * q^43 + 108 * q^46 + 56 * q^49 - 272 * q^52 + 196 * q^55 - 220 * q^58 + 180 * q^61 - 700 * q^64 - 132 * q^67 + 196 * q^70 + 272 * q^73 + 544 * q^76 + 316 * q^79 + 28 * q^82 - 228 * q^85 - 56 * q^91 - 348 * q^94 - 364 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/189\mathbb{Z}\right)^\times$.

$n$	$29$	$136$
$\chi(n)$	$-1$	$1$

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 $

$2$	− 2.27733i	− 1.13867i	−0.822107	−	0.569334i	$-0.807201\pi$
$2$	− 2.27733i	− 1.13867i	0.822107	−	0.569334i	$-0.192799\pi$
$3$	0	0
$3$	0	0
$4$	−1.18625	−0.296563
$5$	4.94527i	0.989054i	0.869162	+	0.494527i	$0.164659\pi$
$5$	4.94527i	0.989054i	−0.869162	+	0.494527i	$0.835341\pi$
$6$	0	0
$7$	2.64575	0.377964
$7$	2.64575	0.377964
$8$	− 6.40785i	− 0.800981i
$9$	0	0
$10$	11.2620	1.12620
$11$	− 20.1892i	− 1.83538i	−0.397295	−	0.917691i	$-0.630051\pi$
$11$	− 20.1892i	− 1.83538i	0.397295	−	0.917691i	$-0.369949\pi$
$12$	0	0
$13$	25.8541	1.98877	0.994387	−	0.105803i	$-0.0337414\pi$
$13$	25.8541	1.98877	0.994387	+	0.105803i	$0.0337414\pi$
$14$	− 6.02526i	− 0.430376i
$15$	0	0
$16$	−19.3378	−1.20861
$17$	− 6.60531i	− 0.388548i	−0.980947	−	0.194274i	$-0.937765\pi$
$17$	− 6.60531i	− 0.388548i	0.980947	−	0.194274i	$-0.0622351\pi$
$18$	0	0
$19$	5.53147	0.291130	0.145565	−	0.989349i	$-0.453500\pi$
$19$	5.53147	0.291130	0.145565	+	0.989349i	$0.453500\pi$
$20$	− 5.86633i	− 0.293317i
$21$	0	0
$22$	−45.9776	−2.08989
$23$	14.5333i	0.631884i	0.948778	+	0.315942i	$0.102321\pi$
$23$	14.5333i	0.631884i	−0.948778	+	0.315942i	$0.897679\pi$
$24$	0	0
$25$	0.544307	0.0217723
$26$	− 58.8783i	− 2.26455i
$27$	0	0
$28$	−3.13853	−0.112090
$29$	17.4877i	0.603025i	0.953462	+	0.301512i	$0.0974914\pi$
$29$	17.4877i	0.603025i	−0.953462	+	0.301512i	$0.902509\pi$
$30$	0	0
$31$	−45.3926	−1.46428	−0.732138	−	0.681156i	$-0.761478\pi$
$31$	−45.3926	−1.46428	−0.732138	+	0.681156i	$0.761478\pi$
$32$	18.4073i	0.575228i
$33$	0	0
$34$	−15.0425	−0.442427
$35$	13.0840i	0.373827i
$36$	0	0
$37$	40.8908	1.10516	0.552578	−	0.833461i	$-0.313644\pi$
$37$	40.8908	1.10516	0.552578	+	0.833461i	$0.313644\pi$
$38$	− 12.5970i	− 0.331500i
$39$	0	0
$40$	31.6885	0.792213
$41$	30.4091i	0.741684i	0.928696	+	0.370842i	$0.120931\pi$
$41$	30.4091i	0.741684i	−0.928696	+	0.370842i	$0.879069\pi$
$42$	0	0
$43$	−53.5305	−1.24490	−0.622448	−	0.782661i	$-0.713862\pi$
$43$	−53.5305	−1.24490	−0.622448	+	0.782661i	$0.713862\pi$
$44$	23.9495i	0.544306i
$45$	0	0
$46$	33.0973	0.719506
$47$	27.8541i	0.592641i	0.955089	+	0.296321i	$0.0957597\pi$
$47$	27.8541i	0.592641i	−0.955089	+	0.296321i	$0.904240\pi$
$48$	0	0
$49$	7.00000	0.142857
$50$	− 1.23957i	− 0.0247914i
$51$	0	0
$52$	−30.6694	−0.589797
$53$	− 6.45594i	− 0.121810i	−0.998144	−	0.0609051i	$-0.980601\pi$
$53$	− 6.45594i	− 0.121810i	0.998144	−	0.0609051i	$-0.0193987\pi$
$54$	0	0
$55$	99.8411	1.81529
$56$	− 16.9536i	− 0.302742i
$57$	0	0
$58$	39.8254	0.686644
$59$	− 42.8050i	− 0.725508i	−0.931885	−	0.362754i	$-0.881837\pi$
$59$	− 42.8050i	− 0.725508i	0.931885	−	0.362754i	$-0.118163\pi$
$60$	0	0
$61$	35.5980	0.583574	0.291787	−	0.956483i	$-0.405750\pi$
$61$	35.5980	0.583574	0.291787	+	0.956483i	$0.405750\pi$
$62$	103.374i	1.66732i
$63$	0	0
$64$	−35.4317	−0.553621
$65$	127.855i	1.96700i
$66$	0	0
$67$	−83.7309	−1.24971	−0.624857	−	0.780739i	$-0.714843\pi$
$67$	−83.7309	−1.24971	−0.624857	+	0.780739i	$0.714843\pi$
$68$	7.83556i	0.115229i
$69$	0	0
$70$	29.7965	0.425665
$71$	113.821i	1.60311i	0.597923	+	0.801554i	$0.295993\pi$
$71$	113.821i	1.60311i	−0.597923	+	0.801554i	$0.704007\pi$
$72$	0	0
$73$	−25.5762	−0.350358	−0.175179	−	0.984537i	$-0.556050\pi$
$73$	−25.5762	−0.350358	−0.175179	+	0.984537i	$0.556050\pi$
$74$	− 93.1220i	− 1.25840i
$75$	0	0
$76$	−6.56172	−0.0863384
$77$	− 53.4156i	− 0.693709i
$78$	0	0
$79$	81.8944	1.03664	0.518319	−	0.855187i	$-0.326558\pi$
$79$	81.8944	1.03664	0.518319	+	0.855187i	$0.326558\pi$
$80$	− 95.6307i	− 1.19538i
$81$	0	0
$82$	69.2516	0.844532
$83$	28.7852i	0.346810i	0.984851	+	0.173405i	$0.0554769\pi$
$83$	28.7852i	0.346810i	−0.984851	+	0.173405i	$0.944523\pi$
$84$	0	0
$85$	32.6651	0.384295
$86$	121.907i	1.41752i
$87$	0	0
$88$	−129.369	−1.47011
$89$	5.92931i	0.0666214i	0.999445	+	0.0333107i	$0.0106051\pi$
$89$	5.92931i	0.0666214i	−0.999445	+	0.0333107i	$0.989395\pi$
$90$	0	0
$91$	68.4034	0.751686
$92$	− 17.2402i	− 0.187393i
$93$	0	0
$94$	63.4332	0.674821
$95$	27.3546i	0.287944i
$96$	0	0
$97$	−94.4881	−0.974104	−0.487052	−	0.873373i	$-0.661928\pi$
$97$	−94.4881	−0.974104	−0.487052	+	0.873373i	$0.661928\pi$
$98$	− 15.9413i	− 0.162667i
$99$	0	0
$100$	−0.645685	−0.00645685
$101$	123.052i	1.21834i	0.793042	+	0.609168i	$0.208496\pi$
$101$	123.052i	1.21834i	−0.793042	+	0.609168i	$0.791504\pi$
$102$	0	0
$103$	−41.5755	−0.403646	−0.201823	−	0.979422i	$-0.564687\pi$
$103$	−41.5755	−0.403646	−0.201823	+	0.979422i	$0.564687\pi$
$104$	− 165.669i	− 1.59297i
$105$	0	0
$106$	−14.7023	−0.138701
$107$	− 2.53060i	− 0.0236505i	−0.999930	−	0.0118252i	$-0.996236\pi$
$107$	− 2.53060i	− 0.0236505i	0.999930	−	0.0118252i	$-0.00376418\pi$
$108$	0	0
$109$	−27.9757	−0.256658	−0.128329	−	0.991732i	$-0.540961\pi$
$109$	−27.9757	−0.256658	−0.128329	+	0.991732i	$0.540961\pi$
$110$	− 227.371i	− 2.06701i
$111$	0	0
$112$	−51.1630	−0.456813
$113$	− 183.509i	− 1.62397i	−0.583678	−	0.811985i	$-0.698387\pi$
$113$	− 183.509i	− 1.62397i	0.583678	−	0.811985i	$-0.301613\pi$
$114$	0	0
$115$	−71.8713	−0.624968
$116$	− 20.7448i	− 0.178835i
$117$	0	0
$118$	−97.4812	−0.826112
$119$	− 17.4760i	− 0.146857i
$120$	0	0
$121$	−286.604	−2.36863
$122$	− 81.0686i	− 0.664497i
$123$	0	0
$124$	53.8470	0.434250
$125$	126.323i	1.01059i
$126$	0	0
$127$	−117.574	−0.925780	−0.462890	−	0.886416i	$-0.653187\pi$
$127$	−117.574	−0.925780	−0.462890	+	0.886416i	$0.653187\pi$
$128$	154.319i	1.20562i
$129$	0	0
$130$	291.169	2.23976
$131$	− 175.442i	− 1.33926i	−0.742697	−	0.669628i	$-0.766454\pi$
$131$	− 175.442i	− 1.33926i	0.742697	−	0.669628i	$-0.233546\pi$
$132$	0	0
$133$	14.6349	0.110037
$134$	190.683i	1.42301i
$135$	0	0
$136$	−42.3258	−0.311219
$137$	3.72691i	0.0272037i	0.999907	+	0.0136019i	$0.00432974\pi$
$137$	3.72691i	0.0272037i	−0.999907	+	0.0136019i	$0.995670\pi$
$138$	0	0
$139$	118.450	0.852159	0.426079	−	0.904686i	$-0.359894\pi$
$139$	118.450	0.852159	0.426079	+	0.904686i	$0.359894\pi$
$140$	− 15.5209i	− 0.110863i
$141$	0	0
$142$	259.208	1.82541
$143$	− 521.973i	− 3.65016i
$144$	0	0
$145$	−86.4815	−0.596424
$146$	58.2455i	0.398942i
$147$	0	0
$148$	−48.5068	−0.327748
$149$	208.169i	1.39711i	0.715556	+	0.698555i	$0.246173\pi$
$149$	208.169i	1.39711i	−0.715556	+	0.698555i	$0.753827\pi$
$150$	0	0
$151$	77.7758	0.515071	0.257536	−	0.966269i	$-0.417090\pi$
$151$	77.7758	0.515071	0.257536	+	0.966269i	$0.417090\pi$
$152$	− 35.4448i	− 0.233190i
$153$	0	0
$154$	−121.645	−0.789904
$155$	− 224.479i	− 1.44825i
$156$	0	0
$157$	113.687	0.724118	0.362059	−	0.932155i	$-0.382074\pi$
$157$	113.687	0.724118	0.362059	+	0.932155i	$0.382074\pi$
$158$	− 186.501i	− 1.18039i
$159$	0	0
$160$	−91.0290	−0.568931
$161$	38.4516i	0.238830i
$162$	0	0
$163$	−143.154	−0.878243	−0.439122	−	0.898428i	$-0.644710\pi$
$163$	−143.154	−0.878243	−0.439122	+	0.898428i	$0.644710\pi$
$164$	− 36.0728i	− 0.219956i
$165$	0	0
$166$	65.5536	0.394901
$167$	208.697i	1.24968i	0.780752	+	0.624841i	$0.214836\pi$
$167$	208.697i	1.24968i	−0.780752	+	0.624841i	$0.785164\pi$
$168$	0	0
$169$	499.433	2.95522
$170$	− 74.3892i	− 0.437584i
$171$	0	0
$172$	63.5007	0.369190
$173$	165.376i	0.955930i	0.878379	+	0.477965i	$0.158625\pi$
$173$	165.376i	0.955930i	−0.878379	+	0.477965i	$0.841375\pi$
$174$	0	0
$175$	1.44010	0.00822914
$176$	390.415i	2.21827i
$177$	0	0
$178$	13.5030	0.0758596
$179$	− 6.53386i	− 0.0365020i	−0.999833	−	0.0182510i	$-0.994190\pi$
$179$	− 6.53386i	− 0.0365020i	0.999833	−	0.0182510i	$-0.00580980\pi$
$180$	0	0
$181$	47.6268	0.263132	0.131566	−	0.991307i	$-0.458000\pi$
$181$	47.6268	0.263132	0.131566	+	0.991307i	$0.458000\pi$
$182$	− 155.777i	− 0.855920i
$183$	0	0
$184$	93.1274	0.506127
$185$	202.216i	1.09306i
$186$	0	0
$187$	−133.356	−0.713134
$188$	− 33.0420i	− 0.175755i
$189$	0	0
$190$	62.2956	0.327872
$191$	− 59.7719i	− 0.312942i	−0.987683	−	0.156471i	$-0.949988\pi$
$191$	− 59.7719i	− 0.312942i	0.987683	−	0.156471i	$-0.0500118\pi$
$192$	0	0
$193$	−178.540	−0.925079	−0.462540	−	0.886599i	$-0.653062\pi$
$193$	−178.540	−0.925079	−0.462540	+	0.886599i	$0.653062\pi$
$194$	215.181i	1.10918i
$195$	0	0
$196$	−8.30376	−0.0423661
$197$	69.0446i	0.350480i	0.984526	+	0.175240i	$0.0560702\pi$
$197$	69.0446i	0.350480i	−0.984526	+	0.175240i	$0.943930\pi$
$198$	0	0
$199$	360.982	1.81398	0.906989	−	0.421155i	$-0.138375\pi$
$199$	360.982	1.81398	0.906989	+	0.421155i	$0.138375\pi$
$200$	− 3.48783i	− 0.0174392i
$201$	0	0
$202$	280.230	1.38728
$203$	46.2681i	0.227922i
$204$	0	0
$205$	−150.381	−0.733566
$206$	94.6813i	0.459618i
$207$	0	0
$208$	−499.961	−2.40366
$209$	− 111.676i	− 0.534335i
$210$	0	0
$211$	−45.0507	−0.213510	−0.106755	−	0.994285i	$-0.534046\pi$
$211$	−45.0507	−0.213510	−0.106755	+	0.994285i	$0.534046\pi$
$212$	7.65837i	0.0361244i
$213$	0	0
$214$	−5.76302	−0.0269300
$215$	− 264.723i	− 1.23127i
$216$	0	0
$217$	−120.097	−0.553445
$218$	63.7101i	0.292248i
$219$	0	0
$220$	−118.437	−0.538348
$221$	− 170.774i	− 0.772734i
$222$	0	0
$223$	−185.805	−0.833209	−0.416604	−	0.909088i	$-0.636780\pi$
$223$	−185.805	−0.833209	−0.416604	+	0.909088i	$0.636780\pi$
$224$	48.7011i	0.217416i
$225$	0	0
$226$	−417.910	−1.84916
$227$	199.040i	0.876827i	0.898773	+	0.438414i	$0.144460\pi$
$227$	199.040i	0.876827i	−0.898773	+	0.438414i	$0.855540\pi$
$228$	0	0
$229$	−321.236	−1.40278	−0.701389	−	0.712779i	$-0.747436\pi$
$229$	−321.236	−1.40278	−0.701389	+	0.712779i	$0.747436\pi$
$230$	163.675i	0.711630i
$231$	0	0
$232$	112.059	0.483011
$233$	215.361i	0.924295i	0.886803	+	0.462147i	$0.152921\pi$
$233$	215.361i	0.924295i	−0.886803	+	0.462147i	$0.847079\pi$
$234$	0	0
$235$	−137.746	−0.586154
$236$	50.7775i	0.215159i
$237$	0	0
$238$	−39.7987	−0.167222
$239$	− 165.314i	− 0.691691i	−0.938291	−	0.345846i	$-0.887592\pi$
$239$	− 165.314i	− 0.691691i	0.938291	−	0.345846i	$-0.112408\pi$
$240$	0	0
$241$	251.336	1.04289	0.521443	−	0.853286i	$-0.325394\pi$
$241$	251.336	1.04289	0.521443	+	0.853286i	$0.325394\pi$
$242$	652.693i	2.69708i
$243$	0	0
$244$	−42.2282	−0.173066
$245$	34.6169i	0.141293i
$246$	0	0
$247$	143.011	0.578992
$248$	290.869i	1.17286i
$249$	0	0
$250$	287.681	1.15072
$251$	10.7301i	0.0427496i	0.999772	+	0.0213748i	$0.00680432\pi$
$251$	10.7301i	0.0427496i	−0.999772	+	0.0213748i	$0.993196\pi$
$252$	0	0
$253$	293.417	1.15975
$254$	267.755i	1.05415i
$255$	0	0
$256$	209.709	0.819176
$257$	− 34.3713i	− 0.133741i	−0.997762	−	0.0668703i	$-0.978699\pi$
$257$	− 34.3713i	− 0.133741i	0.997762	−	0.0668703i	$-0.0213014\pi$
$258$	0	0
$259$	108.187	0.417710
$260$	− 151.669i	− 0.583341i
$261$	0	0
$262$	−399.541	−1.52497
$263$	− 438.760i	− 1.66829i	−0.551546	−	0.834144i	$-0.685962\pi$
$263$	− 438.760i	− 1.66829i	0.551546	−	0.834144i	$-0.314038\pi$
$264$	0	0
$265$	31.9264	0.120477
$266$	− 33.3286i	− 0.125295i
$267$	0	0
$268$	99.3259	0.370619
$269$	− 244.971i	− 0.910673i	−0.890320	−	0.455336i	$-0.849519\pi$
$269$	− 244.971i	− 0.910673i	0.890320	−	0.455336i	$-0.150481\pi$
$270$	0	0
$271$	162.048	0.597962	0.298981	−	0.954259i	$-0.403353\pi$
$271$	162.048	0.597962	0.298981	+	0.954259i	$0.403353\pi$
$272$	127.732i	0.469604i
$273$	0	0
$274$	8.48743	0.0309760
$275$	− 10.9891i	− 0.0399604i
$276$	0	0
$277$	−282.791	−1.02091	−0.510453	−	0.859906i	$-0.670522\pi$
$277$	−282.791	−1.02091	−0.510453	+	0.859906i	$0.670522\pi$
$278$	− 269.750i	− 0.970325i
$279$	0	0
$280$	83.8400	0.299428
$281$	− 68.2322i	− 0.242819i	−0.992602	−	0.121410i	$-0.961259\pi$
$281$	− 68.2322i	− 0.242819i	0.992602	−	0.121410i	$-0.0387415\pi$
$282$	0	0
$283$	194.706	0.688005	0.344003	−	0.938969i	$-0.388217\pi$
$283$	194.706	0.688005	0.344003	+	0.938969i	$0.388217\pi$
$284$	− 135.020i	− 0.475422i
$285$	0	0
$286$	−1188.71	−4.15632
$287$	80.4548i	0.280330i
$288$	0	0
$289$	245.370	0.849031
$290$	196.947i	0.679128i
$291$	0	0
$292$	30.3398	0.103903
$293$	− 233.076i	− 0.795482i	−0.917498	−	0.397741i	$-0.869794\pi$
$293$	− 233.076i	− 0.795482i	0.917498	−	0.397741i	$-0.130206\pi$
$294$	0	0
$295$	211.682	0.717567
$296$	− 262.022i	− 0.885209i
$297$	0	0
$298$	474.071	1.59084
$299$	375.746i	1.25668i
$300$	0	0
$301$	−141.628	−0.470526
$302$	− 177.121i	− 0.586495i
$303$	0	0
$304$	−106.967	−0.351864
$305$	176.042i	0.577186i
$306$	0	0
$307$	167.051	0.544140	0.272070	−	0.962277i	$-0.412292\pi$
$307$	167.051	0.544140	0.272070	+	0.962277i	$0.412292\pi$
$308$	63.3644i	0.205728i
$309$	0	0
$310$	−511.213	−1.64907
$311$	− 194.748i	− 0.626199i	−0.949720	−	0.313099i	$-0.898633\pi$
$311$	− 194.748i	− 0.626199i	0.949720	−	0.313099i	$-0.101367\pi$
$312$	0	0
$313$	328.297	1.04887	0.524436	−	0.851450i	$-0.324276\pi$
$313$	328.297	1.04887	0.524436	+	0.851450i	$0.324276\pi$
$314$	− 258.902i	− 0.824529i
$315$	0	0
$316$	−97.1474	−0.307428
$317$	− 435.769i	− 1.37467i	−0.726342	−	0.687333i	$-0.758781\pi$
$317$	− 435.769i	− 1.37467i	0.726342	−	0.687333i	$-0.241219\pi$
$318$	0	0
$319$	353.063	1.10678
$320$	− 175.219i	− 0.547561i
$321$	0	0
$322$	87.5671	0.271948
$323$	− 36.5371i	− 0.113118i
$324$	0	0
$325$	14.0725	0.0433001
$326$	326.009i	1.00003i
$327$	0	0
$328$	194.857	0.594075
$329$	73.6951i	0.223997i
$330$	0	0
$331$	441.180	1.33287	0.666436	−	0.745563i	$-0.267819\pi$
$331$	441.180	1.33287	0.666436	+	0.745563i	$0.267819\pi$
$332$	− 34.1465i	− 0.102851i
$333$	0	0
$334$	475.273	1.42297
$335$	− 414.072i	− 1.23603i
$336$	0	0
$337$	−226.991	−0.673564	−0.336782	−	0.941583i	$-0.609339\pi$
$337$	−226.991	−0.673564	−0.336782	+	0.941583i	$0.609339\pi$
$338$	− 1137.38i	− 3.36502i
$339$	0	0
$340$	−38.7490	−0.113968
$341$	916.440i	2.68751i
$342$	0	0
$343$	18.5203	0.0539949
$344$	343.015i	0.997138i
$345$	0	0
$346$	376.616	1.08849
$347$	− 127.698i	− 0.368005i	−0.982926	−	0.184003i	$-0.941095\pi$
$347$	− 127.698i	− 0.368005i	0.982926	−	0.184003i	$-0.0589055\pi$
$348$	0	0
$349$	68.5010	0.196278	0.0981390	−	0.995173i	$-0.468711\pi$
$349$	68.5010	0.196278	0.0981390	+	0.995173i	$0.468711\pi$
$350$	− 3.27959i	− 0.00937026i
$351$	0	0
$352$	371.628	1.05576
$353$	− 436.129i	− 1.23549i	−0.786377	−	0.617747i	$-0.788046\pi$
$353$	− 436.129i	− 1.23549i	0.786377	−	0.617747i	$-0.211954\pi$
$354$	0	0
$355$	−562.874	−1.58556
$356$	− 7.03365i	− 0.0197574i
$357$	0	0
$358$	−14.8798	−0.0415637
$359$	− 557.516i	− 1.55297i	−0.630136	−	0.776485i	$-0.717001\pi$
$359$	− 557.516i	− 1.55297i	0.630136	−	0.776485i	$-0.282999\pi$
$360$	0	0
$361$	−330.403	−0.915243
$362$	− 108.462i	− 0.299619i
$363$	0	0
$364$	−81.1437	−0.222922
$365$	− 126.481i	− 0.346523i
$366$	0	0
$367$	−540.446	−1.47260	−0.736302	−	0.676653i	$-0.763430\pi$
$367$	−540.446	−1.47260	−0.736302	+	0.676653i	$0.763430\pi$
$368$	− 281.043i	− 0.763704i
$369$	0	0
$370$	460.513	1.24463
$371$	− 17.0808i	− 0.0460399i
$372$	0	0
$373$	−130.478	−0.349807	−0.174903	−	0.984586i	$-0.555961\pi$
$373$	−130.478	−0.349807	−0.174903	+	0.984586i	$0.555961\pi$
$374$	303.696i	0.812022i
$375$	0	0
$376$	178.485	0.474694
$377$	452.128i	1.19928i
$378$	0	0
$379$	−345.177	−0.910757	−0.455379	−	0.890298i	$-0.650496\pi$
$379$	−345.177	−0.910757	−0.455379	+	0.890298i	$0.650496\pi$
$380$	− 32.4495i	− 0.0853934i
$381$	0	0
$382$	−136.121	−0.356337
$383$	− 105.528i	− 0.275529i	−0.990465	−	0.137764i	$-0.956008\pi$
$383$	− 105.528i	− 0.275529i	0.990465	−	0.137764i	$-0.0439917\pi$
$384$	0	0
$385$	264.155	0.686116
$386$	406.596i	1.05336i
$387$	0	0
$388$	112.087	0.288883
$389$	498.611i	1.28178i	0.767634	+	0.640889i	$0.221434\pi$
$389$	498.611i	1.28178i	−0.767634	+	0.640889i	$0.778566\pi$
$390$	0	0
$391$	95.9972	0.245517
$392$	− 44.8549i	− 0.114426i
$393$	0	0
$394$	157.238	0.399080
$395$	404.990i	1.02529i
$396$	0	0
$397$	119.497	0.301000	0.150500	−	0.988610i	$-0.451912\pi$
$397$	119.497	0.301000	0.150500	+	0.988610i	$0.451912\pi$
$398$	− 822.076i	− 2.06552i
$399$	0	0
$400$	−10.5257	−0.0263143
$401$	602.317i	1.50204i	0.660280	+	0.751019i	$0.270437\pi$
$401$	602.317i	1.50204i	−0.660280	+	0.751019i	$0.729563\pi$
$402$	0	0
$403$	−1173.58	−2.91212
$404$	− 145.970i	− 0.361313i
$405$	0	0
$406$	105.368	0.259527
$407$	− 825.552i	− 2.02838i
$408$	0	0
$409$	−382.189	−0.934447	−0.467224	−	0.884139i	$-0.654746\pi$
$409$	−382.189	−0.934447	−0.467224	+	0.884139i	$0.654746\pi$
$410$	342.468i	0.835287i
$411$	0	0
$412$	49.3190	0.119706
$413$	− 113.251i	− 0.274216i
$414$	0	0
$415$	−142.351	−0.343014
$416$	475.903i	1.14400i
$417$	0	0
$418$	−254.324	−0.608430
$419$	101.655i	0.242612i	0.992615	+	0.121306i	$0.0387083\pi$
$419$	101.655i	0.242612i	−0.992615	+	0.121306i	$0.961292\pi$
$420$	0	0
$421$	−425.972	−1.01181	−0.505905	−	0.862589i	$-0.668842\pi$
$421$	−425.972	−1.01181	−0.505905	+	0.862589i	$0.668842\pi$
$422$	102.595i	0.243117i
$423$	0	0
$424$	−41.3687	−0.0975676
$425$	− 3.59532i	− 0.00845957i
$426$	0	0
$427$	94.1835	0.220570
$428$	3.00193i	0.00701385i
$429$	0	0
$430$	−602.863	−1.40201
$431$	− 256.315i	− 0.594698i	−0.954769	−	0.297349i	$-0.903898\pi$
$431$	− 256.315i	− 0.594698i	0.954769	−	0.297349i	$-0.0961025\pi$
$432$	0	0
$433$	487.980	1.12697	0.563487	−	0.826125i	$-0.309460\pi$
$433$	487.980	1.12697	0.563487	+	0.826125i	$0.309460\pi$
$434$	273.502i	0.630189i
$435$	0	0
$436$	33.1863	0.0761153
$437$	80.3908i	0.183961i
$438$	0	0
$439$	−353.322	−0.804833	−0.402416	−	0.915457i	$-0.631830\pi$
$439$	−353.322	−0.804833	−0.402416	+	0.915457i	$0.631830\pi$
$440$	− 639.766i	− 1.45401i
$441$	0	0
$442$	−388.910	−0.879887
$443$	− 14.5063i	− 0.0327455i	−0.999866	−	0.0163728i	$-0.994788\pi$
$443$	− 14.5063i	− 0.0327455i	0.999866	−	0.0163728i	$-0.00521184\pi$
$444$	0	0
$445$	−29.3220	−0.0658922
$446$	423.141i	0.948747i
$447$	0	0
$448$	−93.7435	−0.209249
$449$	− 848.396i	− 1.88952i	−0.327759	−	0.944761i	$-0.606293\pi$
$449$	− 848.396i	− 1.88952i	0.327759	−	0.944761i	$-0.393707\pi$
$450$	0	0
$451$	613.935	1.36127
$452$	217.687i	0.481609i
$453$	0	0
$454$	453.280	0.998415
$455$	338.273i	0.743458i
$456$	0	0
$457$	165.684	0.362546	0.181273	−	0.983433i	$-0.441978\pi$
$457$	165.684	0.362546	0.181273	+	0.983433i	$0.441978\pi$
$458$	731.562i	1.59730i
$459$	0	0
$460$	85.2574	0.185342
$461$	785.218i	1.70329i	0.524117	+	0.851646i	$0.324395\pi$
$461$	785.218i	1.70329i	−0.524117	+	0.851646i	$0.675605\pi$
$462$	0	0
$463$	−575.009	−1.24192	−0.620960	−	0.783842i	$-0.713257\pi$
$463$	−575.009	−1.24192	−0.620960	+	0.783842i	$0.713257\pi$
$464$	− 338.174i	− 0.728824i
$465$	0	0
$466$	490.448	1.05246
$467$	773.867i	1.65710i	0.559913	+	0.828551i	$0.310834\pi$
$467$	773.867i	1.65710i	−0.559913	+	0.828551i	$0.689166\pi$
$468$	0	0
$469$	−221.531	−0.472348
$470$	313.694i	0.667435i
$471$	0	0
$472$	−274.288	−0.581118
$473$	1080.74i	2.28486i
$474$	0	0
$475$	3.01082	0.00633857
$476$	20.7310i	0.0435524i
$477$	0	0
$478$	−376.476	−0.787606
$479$	489.868i	1.02269i	0.859376	+	0.511345i	$0.170852\pi$
$479$	489.868i	1.02269i	−0.859376	+	0.511345i	$0.829148\pi$
$480$	0	0
$481$	1057.19	2.19791
$482$	− 572.376i	− 1.18750i
$483$	0	0
$484$	339.984	0.702447
$485$	− 467.269i	− 0.963442i
$486$	0	0
$487$	750.641	1.54136	0.770678	−	0.637224i	$-0.219918\pi$
$487$	750.641	1.54136	0.770678	+	0.637224i	$0.219918\pi$
$488$	− 228.107i	− 0.467432i
$489$	0	0
$490$	78.8342	0.160886
$491$	− 154.588i	− 0.314843i	−0.987532	−	0.157421i	$-0.949682\pi$
$491$	− 154.588i	− 0.314843i	0.987532	−	0.157421i	$-0.0503181\pi$
$492$	0	0
$493$	115.512	0.234304
$494$	− 325.684i	− 0.659279i
$495$	0	0
$496$	877.793	1.76974
$497$	301.141i	0.605918i
$498$	0	0
$499$	−144.447	−0.289473	−0.144736	−	0.989470i	$-0.546233\pi$
$499$	−144.447	−0.289473	−0.144736	+	0.989470i	$0.546233\pi$
$500$	− 149.851i	− 0.299703i
$501$	0	0
$502$	24.4361	0.0486775
$503$	110.317i	0.219318i	0.993969	+	0.109659i	$0.0349760\pi$
$503$	110.317i	0.219318i	−0.993969	+	0.109659i	$0.965024\pi$
$504$	0	0
$505$	−608.525	−1.20500
$506$	− 668.208i	− 1.32057i
$507$	0	0
$508$	139.472	0.274552
$509$	469.475i	0.922347i	0.887310	+	0.461173i	$0.152571\pi$
$509$	469.475i	0.922347i	−0.887310	+	0.461173i	$0.847429\pi$
$510$	0	0
$511$	−67.6682	−0.132423
$512$	139.698i	0.272848i
$513$	0	0
$514$	−78.2750	−0.152286
$515$	− 205.602i	− 0.399227i
$516$	0	0
$517$	562.353	1.08772
$518$	− 246.378i	− 0.475632i
$519$	0	0
$520$	819.277	1.57553
$521$	470.639i	0.903338i	0.892186	+	0.451669i	$0.149171\pi$
$521$	470.639i	0.903338i	−0.892186	+	0.451669i	$0.850829\pi$
$522$	0	0
$523$	13.5759	0.0259577	0.0129788	−	0.999916i	$-0.495869\pi$
$523$	13.5759	0.0259577	0.0129788	+	0.999916i	$0.495869\pi$
$524$	208.119i	0.397174i
$525$	0	0
$526$	−999.203	−1.89963
$527$	299.832i	0.568941i
$528$	0	0
$529$	317.782	0.600722
$530$	− 72.7070i	− 0.137183i
$531$	0	0
$532$	−17.3607	−0.0326329
$533$	786.198i	1.47504i
$534$	0	0
$535$	12.5145	0.0233916
$536$	536.534i	1.00100i
$537$	0	0
$538$	−557.881	−1.03695
$539$	− 141.324i	− 0.262197i
$540$	0	0
$541$	15.7625	0.0291358	0.0145679	−	0.999894i	$-0.495363\pi$
$541$	15.7625	0.0291358	0.0145679	+	0.999894i	$0.495363\pi$
$542$	− 369.037i	− 0.680880i
$543$	0	0
$544$	121.586	0.223503
$545$	− 138.348i	− 0.253849i
$546$	0	0
$547$	59.2428	0.108305	0.0541525	−	0.998533i	$-0.482754\pi$
$547$	59.2428	0.108305	0.0541525	+	0.998533i	$0.482754\pi$
$548$	− 4.42106i	− 0.00806762i
$549$	0	0
$550$	−25.0259	−0.0455016
$551$	96.7328i	0.175559i
$552$	0	0
$553$	216.672	0.391812
$554$	644.010i	1.16247i
$555$	0	0
$556$	−140.512	−0.252719
$557$	− 736.312i	− 1.32192i	−0.750419	−	0.660962i	$-0.770148\pi$
$557$	− 736.312i	− 1.32192i	0.750419	−	0.660962i	$-0.229852\pi$
$558$	0	0
$559$	−1383.98	−2.47582
$560$	− 253.015i	− 0.451813i
$561$	0	0
$562$	−155.388	−0.276490
$563$	− 188.768i	− 0.335290i	−0.985847	−	0.167645i	$-0.946384\pi$
$563$	− 188.768i	− 0.335290i	0.985847	−	0.167645i	$-0.0536162\pi$
$564$	0	0
$565$	907.500	1.60619
$566$	− 443.410i	− 0.783409i
$567$	0	0
$568$	729.345	1.28406
$569$	− 804.768i	− 1.41435i	−0.707036	−	0.707177i	$-0.749968\pi$
$569$	− 804.768i	− 1.41435i	0.707036	−	0.707177i	$-0.250032\pi$
$570$	0	0
$571$	563.341	0.986586	0.493293	−	0.869863i	$-0.335793\pi$
$571$	563.341	0.986586	0.493293	+	0.869863i	$0.335793\pi$
$572$	619.191i	1.08250i
$573$	0	0
$574$	183.222	0.319203
$575$	7.91059i	0.0137576i
$576$	0	0
$577$	−389.601	−0.675218	−0.337609	−	0.941286i	$-0.609618\pi$
$577$	−389.601	−0.675218	−0.337609	+	0.941286i	$0.609618\pi$
$578$	− 558.789i	− 0.966763i
$579$	0	0
$580$	102.589	0.176877
$581$	76.1585i	0.131082i
$582$	0	0
$583$	−130.340	−0.223568
$584$	163.888i	0.280630i
$585$	0	0
$586$	−530.793	−0.905789
$587$	894.933i	1.52459i	0.647231	+	0.762294i	$0.275927\pi$
$587$	894.933i	1.52459i	−0.647231	+	0.762294i	$0.724073\pi$
$588$	0	0
$589$	−251.088	−0.426295
$590$	− 482.071i	− 0.817069i
$591$	0	0
$592$	−790.738	−1.33571
$593$	− 684.579i	− 1.15443i	−0.816591	−	0.577217i	$-0.804139\pi$
$593$	− 684.579i	− 1.15443i	0.816591	−	0.577217i	$-0.195861\pi$
$594$	0	0
$595$	86.4236	0.145250
$596$	− 246.941i	− 0.414331i
$597$	0	0
$598$	855.699	1.43093
$599$	− 159.478i	− 0.266241i	−0.991100	−	0.133121i	$-0.957500\pi$
$599$	− 159.478i	− 0.266241i	0.991100	−	0.133121i	$-0.0424998\pi$
$600$	0	0
$601$	−434.985	−0.723769	−0.361884	−	0.932223i	$-0.617867\pi$
$601$	−434.985	−0.723769	−0.361884	+	0.932223i	$0.617867\pi$
$602$	322.535i	0.535773i
$603$	0	0
$604$	−92.2616	−0.152751
$605$	− 1417.33i	− 2.34270i
$606$	0	0
$607$	224.469	0.369800	0.184900	−	0.982757i	$-0.440804\pi$
$607$	224.469	0.369800	0.184900	+	0.982757i	$0.440804\pi$
$608$	101.819i	0.167466i
$609$	0	0
$610$	400.906	0.657223
$611$	720.143i	1.17863i
$612$	0	0
$613$	−550.312	−0.897735	−0.448868	−	0.893598i	$-0.648173\pi$
$613$	−550.312	−0.897735	−0.448868	+	0.893598i	$0.648173\pi$
$614$	− 380.431i	− 0.619595i
$615$	0	0
$616$	−342.279	−0.555648
$617$	− 69.7803i	− 0.113096i	−0.998400	−	0.0565481i	$-0.981991\pi$
$617$	− 69.7803i	− 0.113096i	0.998400	−	0.0565481i	$-0.0180094\pi$
$618$	0	0
$619$	74.8169	0.120867	0.0604337	−	0.998172i	$-0.480752\pi$
$619$	74.8169	0.120867	0.0604337	+	0.998172i	$0.480752\pi$
$620$	266.288i	0.429497i
$621$	0	0
$622$	−443.506	−0.713032
$623$	15.6875i	0.0251805i
$624$	0	0
$625$	−611.096	−0.977754
$626$	− 747.641i	− 1.19432i
$627$	0	0
$628$	−134.861	−0.214747
$629$	− 270.096i	− 0.429406i
$630$	0	0
$631$	576.881	0.914233	0.457116	−	0.889407i	$-0.348882\pi$
$631$	576.881	0.914233	0.457116	+	0.889407i	$0.348882\pi$
$632$	− 524.767i	− 0.830327i
$633$	0	0
$634$	−992.392	−1.56529
$635$	− 581.435i	− 0.915646i
$636$	0	0
$637$	180.978	0.284111
$638$	− 804.043i	− 1.26025i
$639$	0	0
$640$	−763.149	−1.19242
$641$	808.585i	1.26144i	0.776009	+	0.630722i	$0.217241\pi$
$641$	808.585i	1.26144i	−0.776009	+	0.630722i	$0.782759\pi$
$642$	0	0
$643$	911.869	1.41815	0.709074	−	0.705135i	$-0.249113\pi$
$643$	911.869	1.41815	0.709074	+	0.705135i	$0.249113\pi$
$644$	− 45.6133i	− 0.0708281i
$645$	0	0
$646$	−83.2072	−0.128804
$647$	1259.24i	1.94628i	0.230210	+	0.973141i	$0.426059\pi$
$647$	1259.24i	1.94628i	−0.230210	+	0.973141i	$0.573941\pi$
$648$	0	0
$649$	−864.198	−1.33158
$650$	− 32.0479i	− 0.0493044i
$651$	0	0
$652$	169.816	0.260454
$653$	− 159.535i	− 0.244310i	−0.992511	−	0.122155i	$-0.961019\pi$
$653$	− 159.535i	− 0.244310i	0.992511	−	0.122155i	$-0.0389806\pi$
$654$	0	0
$655$	867.610	1.32460
$656$	− 588.045i	− 0.896410i
$657$	0	0
$658$	167.828	0.255058
$659$	− 123.079i	− 0.186767i	−0.995630	−	0.0933834i	$-0.970232\pi$
$659$	− 123.079i	− 0.186767i	0.995630	−	0.0933834i	$-0.0297682\pi$
$660$	0	0
$661$	−717.352	−1.08525	−0.542626	−	0.839974i	$-0.682570\pi$
$661$	−717.352	−1.08525	−0.542626	+	0.839974i	$0.682570\pi$
$662$	− 1004.72i	− 1.51770i
$663$	0	0
$664$	184.451	0.277788
$665$	72.3736i	0.108832i
$666$	0	0
$667$	−254.155	−0.381042
$668$	− 247.567i	− 0.370609i
$669$	0	0
$670$	−942.980	−1.40743
$671$	− 718.696i	− 1.07108i
$672$	0	0
$673$	664.319	0.987101	0.493550	−	0.869717i	$-0.335699\pi$
$673$	664.319	0.987101	0.493550	+	0.869717i	$0.335699\pi$
$674$	516.935i	0.766966i
$675$	0	0
$676$	−592.453	−0.876410
$677$	468.775i	0.692429i	0.938155	+	0.346215i	$0.112533\pi$
$677$	468.775i	0.692429i	−0.938155	+	0.346215i	$0.887467\pi$
$678$	0	0
$679$	−249.992	−0.368177
$680$	− 209.313i	− 0.307813i
$681$	0	0
$682$	2087.04	3.06018
$683$	− 633.346i	− 0.927300i	−0.886019	−	0.463650i	$-0.846540\pi$
$683$	− 633.346i	− 0.927300i	0.886019	−	0.463650i	$-0.153460\pi$
$684$	0	0
$685$	−18.4306	−0.0269060
$686$	− 42.1768i	− 0.0614822i
$687$	0	0
$688$	1035.16	1.50460
$689$	− 166.912i	− 0.242253i
$690$	0	0
$691$	−640.635	−0.927113	−0.463557	−	0.886067i	$-0.653427\pi$
$691$	−640.635	−0.927113	−0.463557	+	0.886067i	$0.653427\pi$
$692$	− 196.177i	− 0.283493i
$693$	0	0
$694$	−290.811	−0.419035
$695$	585.768i	0.842831i
$696$	0	0
$697$	200.861	0.288180
$698$	− 156.000i	− 0.223495i
$699$	0	0
$700$	−1.70832	−0.00244046
$701$	− 963.347i	− 1.37425i	−0.726541	−	0.687123i	$-0.758873\pi$
$701$	− 963.347i	− 1.37425i	0.726541	−	0.687123i	$-0.241127\pi$
$702$	0	0
$703$	226.186	0.321744
$704$	715.338i	1.01611i
$705$	0	0
$706$	−993.212	−1.40682
$707$	325.565i	0.460487i
$708$	0	0
$709$	−235.904	−0.332728	−0.166364	−	0.986064i	$-0.553203\pi$
$709$	−235.904	−0.332728	−0.166364	+	0.986064i	$0.553203\pi$
$710$	1281.85i	1.80542i
$711$	0	0
$712$	37.9941	0.0533625
$713$	− 659.706i	− 0.925253i
$714$	0	0
$715$	2581.30	3.61021
$716$	7.75081i	0.0108252i
$717$	0	0
$718$	−1269.65	−1.76832
$719$	87.8999i	0.122253i	0.998130	+	0.0611265i	$0.0194693\pi$
$719$	87.8999i	0.122253i	−0.998130	+	0.0611265i	$0.980531\pi$
$720$	0	0
$721$	−109.998	−0.152564
$722$	752.438i	1.04216i
$723$	0	0
$724$	−56.4974	−0.0780351
$725$	9.51868i	0.0131292i
$726$	0	0
$727$	−878.296	−1.20811	−0.604055	−	0.796942i	$-0.706449\pi$
$727$	−878.296	−1.20811	−0.604055	+	0.796942i	$0.706449\pi$
$728$	− 438.319i	− 0.602086i
$729$	0	0
$730$	−288.040	−0.394575
$731$	353.586i	0.483702i
$732$	0	0
$733$	162.061	0.221093	0.110547	−	0.993871i	$-0.464740\pi$
$733$	162.061	0.221093	0.110547	+	0.993871i	$0.464740\pi$
$734$	1230.78i	1.67681i
$735$	0	0
$736$	−267.519	−0.363477
$737$	1690.46i	2.29370i
$738$	0	0
$739$	42.1893	0.0570897	0.0285449	−	0.999593i	$-0.490913\pi$
$739$	42.1893	0.0570897	0.0285449	+	0.999593i	$0.490913\pi$
$740$	− 239.879i	− 0.324161i
$741$	0	0
$742$	−38.8987	−0.0524242
$743$	− 720.211i	− 0.969328i	−0.874700	−	0.484664i	$-0.838942\pi$
$743$	− 720.211i	− 0.969328i	0.874700	−	0.484664i	$-0.161058\pi$
$744$	0	0
$745$	−1029.45	−1.38182
$746$	297.142i	0.398313i
$747$	0	0
$748$	158.194	0.211489
$749$	− 6.69534i	− 0.00893904i
$750$	0	0
$751$	1309.52	1.74370	0.871850	−	0.489773i	$-0.162920\pi$
$751$	1309.52	1.74370	0.871850	+	0.489773i	$0.162920\pi$
$752$	− 538.638i	− 0.716274i
$753$	0	0
$754$	1029.65	1.36558
$755$	384.622i	0.509433i
$756$	0	0
$757$	−695.235	−0.918409	−0.459204	−	0.888331i	$-0.651865\pi$
$757$	−695.235	−0.918409	−0.459204	+	0.888331i	$0.651865\pi$
$758$	786.083i	1.03705i
$759$	0	0
$760$	175.284	0.230637
$761$	− 1060.27i	− 1.39325i	−0.717434	−	0.696626i	$-0.754684\pi$
$761$	− 1060.27i	− 1.39325i	0.717434	−	0.696626i	$-0.245316\pi$
$762$	0	0
$763$	−74.0169	−0.0970077
$764$	70.9045i	0.0928070i
$765$	0	0
$766$	−240.322	−0.313736
$767$	− 1106.68i	− 1.44287i
$768$	0	0
$769$	−1255.65	−1.63283	−0.816415	−	0.577466i	$-0.804042\pi$
$769$	−1255.65	−1.63283	−0.816415	+	0.577466i	$0.804042\pi$
$770$	− 601.568i	− 0.781258i
$771$	0	0
$772$	211.794	0.274344
$773$	− 1182.69i	− 1.53000i	−0.644033	−	0.764998i	$-0.722740\pi$
$773$	− 1182.69i	− 1.53000i	0.644033	−	0.764998i	$-0.277260\pi$
$774$	0	0
$775$	−24.7075	−0.0318806
$776$	605.465i	0.780239i
$777$	0	0
$778$	1135.50	1.45952
$779$	168.207i	0.215927i
$780$	0	0
$781$	2297.95	2.94231
$782$	− 218.618i	− 0.279562i
$783$	0	0
$784$	−135.365	−0.172659
$785$	562.211i	0.716192i
$786$	0	0
$787$	630.329	0.800927	0.400463	−	0.916313i	$-0.368849\pi$
$787$	630.329	0.800927	0.400463	+	0.916313i	$0.368849\pi$
$788$	− 81.9043i	− 0.103939i
$789$	0	0
$790$	922.297	1.16746
$791$	− 485.518i	− 0.613803i
$792$	0	0
$793$	920.354	1.16060
$794$	− 272.135i	− 0.342739i
$795$	0	0
$796$	−428.215	−0.537958
$797$	− 273.631i	− 0.343326i	−0.985156	−	0.171663i	$-0.945086\pi$
$797$	− 273.631i	− 0.343326i	0.985156	−	0.171663i	$-0.0549140\pi$
$798$	0	0
$799$	183.985	0.230269
$800$	10.0192i	0.0125240i
$801$	0	0
$802$	1371.68	1.71032
$803$	516.363i	0.643042i
$804$	0	0
$805$	−190.154	−0.236216
$806$	2672.64i	3.31593i
$807$	0	0
$808$	788.497	0.975863
$809$	− 182.061i	− 0.225044i	−0.993649	−	0.112522i	$-0.964107\pi$
$809$	− 182.061i	− 0.225044i	0.993649	−	0.112522i	$-0.0358929\pi$
$810$	0	0
$811$	98.8163	0.121845	0.0609225	−	0.998142i	$-0.480596\pi$
$811$	98.8163	0.121845	0.0609225	+	0.998142i	$0.480596\pi$
$812$	− 54.8857i	− 0.0675932i
$813$	0	0
$814$	−1880.06	−2.30965
$815$	− 707.933i	− 0.868630i
$816$	0	0
$817$	−296.103	−0.362427
$818$	870.372i	1.06402i
$819$	0	0
$820$	178.390	0.217548
$821$	566.107i	0.689534i	0.938688	+	0.344767i	$0.112042\pi$
$821$	566.107i	0.689534i	−0.938688	+	0.344767i	$0.887958\pi$
$822$	0	0
$823$	452.893	0.550296	0.275148	−	0.961402i	$-0.411273\pi$
$823$	452.893	0.550296	0.275148	+	0.961402i	$0.411273\pi$
$824$	266.409i	0.323312i
$825$	0	0
$826$	−257.911	−0.312241
$827$	1268.79i	1.53421i	0.641521	+	0.767105i	$0.278304\pi$
$827$	1268.79i	1.53421i	−0.641521	+	0.767105i	$0.721696\pi$
$828$	0	0
$829$	1017.98	1.22797	0.613983	−	0.789319i	$-0.289566\pi$
$829$	1017.98	1.22797	0.613983	+	0.789319i	$0.289566\pi$
$830$	324.180i	0.390578i
$831$	0	0
$832$	−916.054	−1.10103
$833$	− 46.2372i	− 0.0555068i
$834$	0	0
$835$	−1032.06	−1.23600
$836$	132.476i	0.158464i
$837$	0	0
$838$	231.501	0.276255
$839$	1175.45i	1.40101i	0.713646	+	0.700507i	$0.247043\pi$
$839$	1175.45i	1.40101i	−0.713646	+	0.700507i	$0.752957\pi$
$840$	0	0
$841$	535.180	0.636361
$842$	970.082i	1.15212i
$843$	0	0
$844$	53.4414	0.0633192
$845$	2469.83i	2.92287i
$846$	0	0
$847$	−758.283	−0.895257
$848$	124.844i	0.147221i
$849$	0	0
$850$	−8.18774	−0.00963263
$851$	594.279i	0.698331i
$852$	0	0
$853$	1058.27	1.24065	0.620323	−	0.784346i	$-0.287001\pi$
$853$	1058.27	1.24065	0.620323	+	0.784346i	$0.287001\pi$
$854$	− 214.487i	− 0.251156i
$855$	0	0
$856$	−16.2157	−0.0189436
$857$	− 1392.71i	− 1.62510i	−0.582888	−	0.812552i	$-0.698077\pi$
$857$	− 1392.71i	− 1.62510i	0.582888	−	0.812552i	$-0.301923\pi$
$858$	0	0
$859$	−748.263	−0.871086	−0.435543	−	0.900168i	$-0.643444\pi$
$859$	−748.263	−0.871086	−0.435543	+	0.900168i	$0.643444\pi$
$860$	314.028i	0.365149i
$861$	0	0
$862$	−583.714	−0.677163
$863$	− 908.810i	− 1.05308i	−0.850150	−	0.526541i	$-0.823489\pi$
$863$	− 908.810i	− 1.05308i	0.850150	−	0.526541i	$-0.176511\pi$
$864$	0	0
$865$	−817.829	−0.945467
$866$	− 1111.29i	− 1.28325i
$867$	0	0
$868$	142.466	0.164131
$869$	− 1653.38i	− 1.90263i
$870$	0	0
$871$	−2164.78	−2.48540
$872$	179.264i	0.205578i
$873$	0	0
$874$	183.077	0.209470
$875$	334.221i	0.381966i
$876$	0	0
$877$	−587.685	−0.670108	−0.335054	−	0.942199i	$-0.608755\pi$
$877$	−587.685	−0.670108	−0.335054	+	0.942199i	$0.608755\pi$
$878$	804.631i	0.916437i
$879$	0	0
$880$	−1930.71	−2.19399
$881$	− 1218.08i	− 1.38261i	−0.722565	−	0.691303i	$-0.757037\pi$
$881$	− 1218.08i	− 1.38261i	0.722565	−	0.691303i	$-0.242963\pi$
$882$	0	0
$883$	−635.067	−0.719215	−0.359607	−	0.933104i	$-0.617089\pi$
$883$	−635.067	−0.719215	−0.359607	+	0.933104i	$0.617089\pi$
$884$	202.581i	0.229164i
$885$	0	0
$886$	−33.0356	−0.0372862
$887$	− 603.102i	− 0.679935i	−0.940437	−	0.339967i	$-0.889584\pi$
$887$	− 603.102i	− 0.679935i	0.940437	−	0.339967i	$-0.110416\pi$
$888$	0	0
$889$	−311.072	−0.349912
$890$	66.7760i	0.0750293i
$891$	0	0
$892$	220.412	0.247099
$893$	154.074i	0.172536i
$894$	0	0
$895$	32.3117	0.0361025
$896$	408.290i	0.455680i
$897$	0	0
$898$	−1932.08	−2.15154
$899$	− 793.812i	− 0.882995i
$900$	0	0
$901$	−42.6435	−0.0473291
$902$	− 1398.13i	− 1.55004i
$903$	0	0
$904$	−1175.89	−1.30077
$905$	235.527i	0.260251i
$906$	0	0
$907$	275.919	0.304210	0.152105	−	0.988364i	$-0.451395\pi$
$907$	275.919	0.304210	0.152105	+	0.988364i	$0.451395\pi$
$908$	− 236.111i	− 0.260035i
$909$	0	0
$910$	770.362	0.846551
$911$	515.110i	0.565434i	0.959203	+	0.282717i	$0.0912357\pi$
$911$	515.110i	0.565434i	−0.959203	+	0.282717i	$0.908764\pi$
$912$	0	0
$913$	581.151	0.636529
$914$	− 377.317i	− 0.412819i
$915$	0	0
$916$	381.067	0.416012
$917$	− 464.177i	− 0.506191i
$918$	0	0
$919$	−245.344	−0.266968	−0.133484	−	0.991051i	$-0.542616\pi$
$919$	−245.344	−0.266968	−0.133484	+	0.991051i	$0.542616\pi$
$920$	460.540i	0.500587i
$921$	0	0
$922$	1788.20	1.93948
$923$	2942.73i	3.18822i
$924$	0	0
$925$	22.2571	0.0240618
$926$	1309.49i	1.41413i
$927$	0	0
$928$	−321.901	−0.346876
$929$	− 903.534i	− 0.972587i	−0.873795	−	0.486294i	$-0.838349\pi$
$929$	− 903.534i	− 0.972587i	0.873795	−	0.486294i	$-0.161651\pi$
$930$	0	0
$931$	38.7203	0.0415900
$932$	− 255.472i	− 0.274112i
$933$	0	0
$934$	1762.35	1.88689
$935$	− 659.481i	− 0.705328i
$936$	0	0
$937$	−1413.35	−1.50837	−0.754187	−	0.656660i	$-0.771969\pi$
$937$	−1413.35	−1.50837	−0.754187	+	0.656660i	$0.771969\pi$
$938$	504.500i	0.537847i
$939$	0	0
$940$	163.402	0.173832
$941$	581.423i	0.617878i	0.951082	+	0.308939i	$0.0999739\pi$
$941$	581.423i	0.617878i	−0.951082	+	0.308939i	$0.900026\pi$
$942$	0	0
$943$	−441.945	−0.468659
$944$	827.755i	0.876859i
$945$	0	0
$946$	2461.20	2.60170
$947$	957.202i	1.01077i	0.862893	+	0.505386i	$0.168650\pi$
$947$	957.202i	1.01077i	−0.862893	+	0.505386i	$0.831350\pi$
$948$	0	0
$949$	−661.248	−0.696784
$950$	− 6.85664i	− 0.00721752i
$951$	0	0
$952$	−111.984	−0.117630
$953$	− 1060.55i	− 1.11285i	−0.830898	−	0.556425i	$-0.812173\pi$
$953$	− 1060.55i	− 1.11285i	0.830898	−	0.556425i	$-0.187827\pi$
$954$	0	0
$955$	295.588	0.309517
$956$	196.104i	0.205130i
$957$	0	0
$958$	1115.59	1.16450
$959$	9.86048i	0.0102820i
$960$	0	0
$961$	1099.49	1.14411
$962$	− 2407.58i	− 2.50268i
$963$	0	0
$964$	−298.147	−0.309282
$965$	− 882.930i	− 0.914953i
$966$	0	0
$967$	−1651.29	−1.70764	−0.853821	−	0.520567i	$-0.825720\pi$
$967$	−1651.29	−1.70764	−0.853821	+	0.520567i	$0.825720\pi$
$968$	1836.51i	1.89723i
$969$	0	0
$970$	−1064.13	−1.09704
$971$	− 1166.15i	− 1.20098i	−0.799633	−	0.600490i	$-0.794972\pi$
$971$	− 1166.15i	− 1.20098i	0.799633	−	0.600490i	$-0.205028\pi$
$972$	0	0
$973$	313.389	0.322086
$974$	− 1709.46i	− 1.75509i
$975$	0	0
$976$	−688.388	−0.705316
$977$	− 11.4573i	− 0.0117271i	−0.999983	−	0.00586353i	$-0.998134\pi$
$977$	− 11.4573i	− 0.0117271i	0.999983	−	0.00586353i	$-0.00186643\pi$
$978$	0	0
$979$	119.708	0.122276
$980$	− 41.0643i	− 0.0419024i
$981$	0	0
$982$	−352.048	−0.358501
$983$	− 1270.31i	− 1.29228i	−0.763221	−	0.646138i	$-0.776383\pi$
$983$	− 1270.31i	− 1.29228i	0.763221	−	0.646138i	$-0.223617\pi$
$984$	0	0
$985$	−341.444	−0.346644
$986$	− 263.059i	− 0.266794i
$987$	0	0
$988$	−169.647	−0.171708
$989$	− 777.977i	− 0.786630i
$990$	0	0
$991$	107.860	0.108839	0.0544196	−	0.998518i	$-0.482669\pi$
$991$	107.860	0.108839	0.0544196	+	0.998518i	$0.482669\pi$
$992$	− 835.554i	− 0.842292i
$993$	0	0
$994$	685.799	0.689939
$995$	1785.15i	1.79412i
$996$	0	0
$997$	−208.773	−0.209402	−0.104701	−	0.994504i	$-0.533388\pi$
$997$	−208.773	−0.209402	−0.104701	+	0.994504i	$0.533388\pi$
$998$	328.954i	0.329613i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.3.b.c.134.2	✓	8
3.2	odd	2	inner	189.3.b.c.134.7	yes	8
4.3	odd	2		3024.3.d.j.1457.6		8
9.2	odd	6		567.3.r.e.134.7		16
9.4	even	3		567.3.r.e.512.7		16
9.5	odd	6		567.3.r.e.512.2		16
9.7	even	3		567.3.r.e.134.2		16
12.11	even	2		3024.3.d.j.1457.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.3.b.c.134.2	✓	8	1.1	even	1	trivial
189.3.b.c.134.7	yes	8	3.2	odd	2	inner
567.3.r.e.134.2		16	9.7	even	3
567.3.r.e.134.7		16	9.2	odd	6
567.3.r.e.512.2		16	9.5	odd	6
567.3.r.e.512.7		16	9.4	even	3
3024.3.d.j.1457.3		8	12.11	even	2
3024.3.d.j.1457.6		8	4.3	odd	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:	\( N \)	\(=\)	\( 189 = 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	189.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.27733i q^{2} -1.18625 q^{4} +4.94527i q^{5} +2.64575 q^{7} -6.40785i q^{8} +O(q^{10})\)	\(q-2.27733i q^{2} -1.18625 q^{4} +4.94527i q^{5} +2.64575 q^{7} -6.40785i q^{8} +11.2620 q^{10} -20.1892i q^{11} +25.8541 q^{13} -6.02526i q^{14} -19.3378 q^{16} -6.60531i q^{17} +5.53147 q^{19} -5.86633i q^{20} -45.9776 q^{22} +14.5333i q^{23} +0.544307 q^{25} -58.8783i q^{26} -3.13853 q^{28} +17.4877i q^{29} -45.3926 q^{31} +18.4073i q^{32} -15.0425 q^{34} +13.0840i q^{35} +40.8908 q^{37} -12.5970i q^{38} +31.6885 q^{40} +30.4091i q^{41} -53.5305 q^{43} +23.9495i q^{44} +33.0973 q^{46} +27.8541i q^{47} +7.00000 q^{49} -1.23957i q^{50} -30.6694 q^{52} -6.45594i q^{53} +99.8411 q^{55} -16.9536i q^{56} +39.8254 q^{58} -42.8050i q^{59} +35.5980 q^{61} +103.374i q^{62} -35.4317 q^{64} +127.855i q^{65} -83.7309 q^{67} +7.83556i q^{68} +29.7965 q^{70} +113.821i q^{71} -25.5762 q^{73} -93.1220i q^{74} -6.56172 q^{76} -53.4156i q^{77} +81.8944 q^{79} -95.6307i q^{80} +69.2516 q^{82} +28.7852i q^{83} +32.6651 q^{85} +121.907i q^{86} -129.369 q^{88} +5.92931i q^{89} +68.4034 q^{91} -17.2402i q^{92} +63.4332 q^{94} +27.3546i q^{95} -94.4881 q^{97} -15.9413i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 12 q^{4}+O(q^{10}) \) 8 * q - 12 * q^4	\( 8 q - 12 q^{4} - 52 q^{10} + 36 q^{13} + 132 q^{16} + 12 q^{19} - 136 q^{22} - 108 q^{25} + 56 q^{28} - 28 q^{31} - 12 q^{34} - 4 q^{37} + 336 q^{40} - 152 q^{43} + 108 q^{46} + 56 q^{49} - 272 q^{52} + 196 q^{55} - 220 q^{58} + 180 q^{61} - 700 q^{64} - 132 q^{67} + 196 q^{70} + 272 q^{73} + 544 q^{76} + 316 q^{79} + 28 q^{82} - 228 q^{85} - 56 q^{91} - 348 q^{94} - 364 q^{97}+O(q^{100}) \) 8 * q - 12 * q^4 - 52 * q^10 + 36 * q^13 + 132 * q^16 + 12 * q^19 - 136 * q^22 - 108 * q^25 + 56 * q^28 - 28 * q^31 - 12 * q^34 - 4 * q^37 + 336 * q^40 - 152 * q^43 + 108 * q^46 + 56 * q^49 - 272 * q^52 + 196 * q^55 - 220 * q^58 + 180 * q^61 - 700 * q^64 - 132 * q^67 + 196 * q^70 + 272 * q^73 + 544 * q^76 + 316 * q^79 + 28 * q^82 - 228 * q^85 - 56 * q^91 - 348 * q^94 - 364 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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