LMFDB - Embedded newform 1143.2.a.k.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1143,2,Mod(1,1143)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1143, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1143.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1143 = 3^{2} \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1143.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$9.12690095103$
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} - 26x^{14} + 269x^{12} - 1408x^{10} + 3924x^{8} - 5655x^{6} + 3886x^{4} - 1107x^{2} + 108 $ x^16 - 26x^14 + 269x^12 - 1408x^10 + 3924x^8 - 5655x^6 + 3886x^4 - 1107*x^2 + 108
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Root			$2.59858$ of defining polynomial
Character	$\chi$	$=$	1143.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.59858 q^{2} +4.75262 q^{4} +0.560562 q^{5} +4.70134 q^{7} +7.15292 q^{8} +O(q^{10})$	\(q+2.59858 q^{2} +4.75262 q^{4} +0.560562 q^{5} +4.70134 q^{7} +7.15292 q^{8} +1.45667 q^{10} -3.80859 q^{11} -1.05307 q^{13} +12.2168 q^{14} +9.08219 q^{16} -5.70697 q^{17} +2.99026 q^{19} +2.66414 q^{20} -9.89692 q^{22} -6.22047 q^{23} -4.68577 q^{25} -2.73649 q^{26} +22.3437 q^{28} -1.50607 q^{29} -1.47216 q^{31} +9.29498 q^{32} -14.8300 q^{34} +2.63540 q^{35} -4.96775 q^{37} +7.77044 q^{38} +4.00966 q^{40} +1.85767 q^{41} +10.9035 q^{43} -18.1008 q^{44} -16.1644 q^{46} +4.18532 q^{47} +15.1026 q^{49} -12.1764 q^{50} -5.00486 q^{52} +13.2401 q^{53} -2.13495 q^{55} +33.6283 q^{56} -3.91363 q^{58} +7.57386 q^{59} -14.8716 q^{61} -3.82552 q^{62} +5.98937 q^{64} -0.590313 q^{65} -2.00966 q^{67} -27.1231 q^{68} +6.84829 q^{70} -4.49409 q^{71} -1.65793 q^{73} -12.9091 q^{74} +14.2116 q^{76} -17.9055 q^{77} +3.97574 q^{79} +5.09113 q^{80} +4.82731 q^{82} -15.1233 q^{83} -3.19911 q^{85} +28.3337 q^{86} -27.2425 q^{88} +6.12112 q^{89} -4.95085 q^{91} -29.5635 q^{92} +10.8759 q^{94} +1.67623 q^{95} +10.2087 q^{97} +39.2454 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 20 q^{4} + 10 q^{7}+O(q^{10}) $ 16 * q + 20 * q^4 + 10 * q^7	$ 16 q + 20 q^{4} + 10 q^{7} + 14 q^{10} + 20 q^{13} + 28 q^{16} + 12 q^{19} + 18 q^{22} + 52 q^{25} + 42 q^{28} + 18 q^{31} + 10 q^{34} + 16 q^{37} + 6 q^{40} + 26 q^{43} - 24 q^{46} + 54 q^{49} + 52 q^{52} + 20 q^{55} - 14 q^{58} + 36 q^{61} - 4 q^{64} + 26 q^{67} + 36 q^{70} + 60 q^{73} - 20 q^{76} + 12 q^{79} - 20 q^{82} - 12 q^{85} + 8 q^{88} - 24 q^{91} - 26 q^{94} + 108 q^{97}+O(q^{100}) $ 16 * q + 20 * q^4 + 10 * q^7 + 14 * q^10 + 20 * q^13 + 28 * q^16 + 12 * q^19 + 18 * q^22 + 52 * q^25 + 42 * q^28 + 18 * q^31 + 10 * q^34 + 16 * q^37 + 6 * q^40 + 26 * q^43 - 24 * q^46 + 54 * q^49 + 52 * q^52 + 20 * q^55 - 14 * q^58 + 36 * q^61 - 4 * q^64 + 26 * q^67 + 36 * q^70 + 60 * q^73 - 20 * q^76 + 12 * q^79 - 20 * q^82 - 12 * q^85 + 8 * q^88 - 24 * q^91 - 26 * q^94 + 108 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.59858	1.83747	0.918737	−	0.394869i	$-0.129210\pi$
$2$	2.59858	1.83747	0.918737	+	0.394869i	$0.129210\pi$
$3$	0	0
$3$	0	0
$4$	4.75262	2.37631
$5$	0.560562	0.250691	0.125346	−	0.992113i	$-0.459996\pi$
$5$	0.560562	0.250691	0.125346	+	0.992113i	$0.459996\pi$
$6$	0	0
$7$	4.70134	1.77694	0.888470	−	0.458934i	$-0.151769\pi$
$7$	4.70134	1.77694	0.888470	+	0.458934i	$0.151769\pi$
$8$	7.15292	2.52894
$9$	0	0
$10$	1.45667	0.460638
$11$	−3.80859	−1.14833	−0.574166	−	0.818739i	$-0.694674\pi$
$11$	−3.80859	−1.14833	−0.574166	+	0.818739i	$0.694674\pi$
$12$	0	0
$13$	−1.05307	−0.292070	−0.146035	−	0.989279i	$-0.546651\pi$
$13$	−1.05307	−0.292070	−0.146035	+	0.989279i	$0.546651\pi$
$14$	12.2168	3.26508
$15$	0	0
$16$	9.08219	2.27055
$17$	−5.70697	−1.38414	−0.692071	−	0.721829i	$-0.743302\pi$
$17$	−5.70697	−1.38414	−0.692071	+	0.721829i	$0.743302\pi$
$18$	0	0
$19$	2.99026	0.686013	0.343006	−	0.939333i	$-0.388555\pi$
$19$	2.99026	0.686013	0.343006	+	0.939333i	$0.388555\pi$
$20$	2.66414	0.595720
$21$	0	0
$22$	−9.89692	−2.11003
$23$	−6.22047	−1.29706	−0.648528	−	0.761191i	$-0.724615\pi$
$23$	−6.22047	−1.29706	−0.648528	+	0.761191i	$0.724615\pi$
$24$	0	0
$25$	−4.68577	−0.937154
$26$	−2.73649	−0.536671
$27$	0	0
$28$	22.3437	4.22257
$29$	−1.50607	−0.279669	−0.139835	−	0.990175i	$-0.544657\pi$
$29$	−1.50607	−0.279669	−0.139835	+	0.990175i	$0.544657\pi$
$30$	0	0
$31$	−1.47216	−0.264407	−0.132204	−	0.991223i	$-0.542205\pi$
$31$	−1.47216	−0.264407	−0.132204	+	0.991223i	$0.542205\pi$
$32$	9.29498	1.64314
$33$	0	0
$34$	−14.8300	−2.54333
$35$	2.63540	0.445463
$36$	0	0
$37$	−4.96775	−0.816693	−0.408346	−	0.912827i	$-0.633894\pi$
$37$	−4.96775	−0.816693	−0.408346	+	0.912827i	$0.633894\pi$
$38$	7.77044	1.26053
$39$	0	0
$40$	4.00966	0.633982
$41$	1.85767	0.290120	0.145060	−	0.989423i	$-0.453663\pi$
$41$	1.85767	0.290120	0.145060	+	0.989423i	$0.453663\pi$
$42$	0	0
$43$	10.9035	1.66277	0.831385	−	0.555697i	$-0.187548\pi$
$43$	10.9035	1.66277	0.831385	+	0.555697i	$0.187548\pi$
$44$	−18.1008	−2.72880
$45$	0	0
$46$	−16.1644	−2.38331
$47$	4.18532	0.610492	0.305246	−	0.952274i	$-0.401261\pi$
$47$	4.18532	0.610492	0.305246	+	0.952274i	$0.401261\pi$
$48$	0	0
$49$	15.1026	2.15752
$50$	−12.1764	−1.72200
$51$	0	0
$52$	−5.00486	−0.694049
$53$	13.2401	1.81867	0.909335	−	0.416064i	$-0.136591\pi$
$53$	13.2401	1.81867	0.909335	+	0.416064i	$0.136591\pi$
$54$	0	0
$55$	−2.13495	−0.287877
$56$	33.6283	4.49377
$57$	0	0
$58$	−3.91363	−0.513885
$59$	7.57386	0.986033	0.493017	−	0.870020i	$-0.335894\pi$
$59$	7.57386	0.986033	0.493017	+	0.870020i	$0.335894\pi$
$60$	0	0
$61$	−14.8716	−1.90411	−0.952054	−	0.305930i	$-0.901033\pi$
$61$	−14.8716	−1.90411	−0.952054	+	0.305930i	$0.901033\pi$
$62$	−3.82552	−0.485841
$63$	0	0
$64$	5.98937	0.748671
$65$	−0.590313	−0.0732193
$66$	0	0
$67$	−2.00966	−0.245519	−0.122759	−	0.992436i	$-0.539174\pi$
$67$	−2.00966	−0.245519	−0.122759	+	0.992436i	$0.539174\pi$
$68$	−27.1231	−3.28916
$69$	0	0
$70$	6.84829	0.818527
$71$	−4.49409	−0.533350	−0.266675	−	0.963787i	$-0.585925\pi$
$71$	−4.49409	−0.533350	−0.266675	+	0.963787i	$0.585925\pi$
$72$	0	0
$73$	−1.65793	−0.194046	−0.0970228	−	0.995282i	$-0.530932\pi$
$73$	−1.65793	−0.194046	−0.0970228	+	0.995282i	$0.530932\pi$
$74$	−12.9091	−1.50065
$75$	0	0
$76$	14.2116	1.63018
$77$	−17.9055	−2.04052
$78$	0	0
$79$	3.97574	0.447305	0.223653	−	0.974669i	$-0.428202\pi$
$79$	3.97574	0.447305	0.223653	+	0.974669i	$0.428202\pi$
$80$	5.09113	0.569206
$81$	0	0
$82$	4.82731	0.533088
$83$	−15.1233	−1.66000	−0.829998	−	0.557766i	$-0.811659\pi$
$83$	−15.1233	−1.66000	−0.829998	+	0.557766i	$0.811659\pi$
$84$	0	0
$85$	−3.19911	−0.346992
$86$	28.3337	3.05530
$87$	0	0
$88$	−27.2425	−2.90406
$89$	6.12112	0.648838	0.324419	−	0.945914i	$-0.394831\pi$
$89$	6.12112	0.648838	0.324419	+	0.945914i	$0.394831\pi$
$90$	0	0
$91$	−4.95085	−0.518990
$92$	−29.5635	−3.08221
$93$	0	0
$94$	10.8759	1.12176
$95$	1.67623	0.171977
$96$	0	0
$97$	10.2087	1.03654	0.518269	−	0.855218i	$-0.326577\pi$
$97$	10.2087	1.03654	0.518269	+	0.855218i	$0.326577\pi$
$98$	39.2454	3.96438
$99$	0	0
$100$	−22.2697	−2.22697
$101$	11.1150	1.10598	0.552992	−	0.833187i	$-0.313486\pi$
$101$	11.1150	1.10598	0.552992	+	0.833187i	$0.313486\pi$
$102$	0	0
$103$	10.6702	1.05137	0.525684	−	0.850680i	$-0.323809\pi$
$103$	10.6702	1.05137	0.525684	+	0.850680i	$0.323809\pi$
$104$	−7.53254	−0.738626
$105$	0	0
$106$	34.4055	3.34176
$107$	3.36295	0.325109	0.162554	−	0.986700i	$-0.448027\pi$
$107$	3.36295	0.325109	0.162554	+	0.986700i	$0.448027\pi$
$108$	0	0
$109$	−11.6114	−1.11217	−0.556085	−	0.831125i	$-0.687697\pi$
$109$	−11.6114	−1.11217	−0.556085	+	0.831125i	$0.687697\pi$
$110$	−5.54784	−0.528966
$111$	0	0
$112$	42.6985	4.03463
$113$	4.14656	0.390076	0.195038	−	0.980796i	$-0.437517\pi$
$113$	4.14656	0.390076	0.195038	+	0.980796i	$0.437517\pi$
$114$	0	0
$115$	−3.48696	−0.325161
$116$	−7.15776	−0.664582
$117$	0	0
$118$	19.6813	1.81181
$119$	−26.8304	−2.45954
$120$	0	0
$121$	3.50533	0.318667
$122$	−38.6450	−3.49875
$123$	0	0
$124$	−6.99661	−0.628314
$125$	−5.42948	−0.485627
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	−3.02609	−0.267472
$129$	0	0
$130$	−1.53398	−0.134539
$131$	−5.13867	−0.448967	−0.224484	−	0.974478i	$-0.572070\pi$
$131$	−5.13867	−0.448967	−0.224484	+	0.974478i	$0.572070\pi$
$132$	0	0
$133$	14.0582	1.21900
$134$	−5.22226	−0.451134
$135$	0	0
$136$	−40.8215	−3.50041
$137$	−19.4152	−1.65875	−0.829376	−	0.558691i	$-0.811304\pi$
$137$	−19.4152	−1.65875	−0.829376	+	0.558691i	$0.811304\pi$
$138$	0	0
$139$	19.1087	1.62078	0.810389	−	0.585892i	$-0.199256\pi$
$139$	19.1087	1.62078	0.810389	+	0.585892i	$0.199256\pi$
$140$	12.5250	1.05856
$141$	0	0
$142$	−11.6782	−0.980017
$143$	4.01072	0.335393
$144$	0	0
$145$	−0.844243	−0.0701106
$146$	−4.30826	−0.356554
$147$	0	0
$148$	−23.6098	−1.94072
$149$	−5.16462	−0.423102	−0.211551	−	0.977367i	$-0.567851\pi$
$149$	−5.16462	−0.423102	−0.211551	+	0.977367i	$0.567851\pi$
$150$	0	0
$151$	9.01378	0.733531	0.366766	−	0.930313i	$-0.380465\pi$
$151$	9.01378	0.733531	0.366766	+	0.930313i	$0.380465\pi$
$152$	21.3891	1.73488
$153$	0	0
$154$	−46.5288	−3.74940
$155$	−0.825236	−0.0662845
$156$	0	0
$157$	19.0273	1.51854	0.759271	−	0.650774i	$-0.225556\pi$
$157$	19.0273	1.51854	0.759271	+	0.650774i	$0.225556\pi$
$158$	10.3313	0.821912
$159$	0	0
$160$	5.21041	0.411919
$161$	−29.2445	−2.30479
$162$	0	0
$163$	−17.2873	−1.35404	−0.677022	−	0.735963i	$-0.736730\pi$
$163$	−17.2873	−1.35404	−0.677022	+	0.735963i	$0.736730\pi$
$164$	8.82882	0.689415
$165$	0	0
$166$	−39.2991	−3.05020
$167$	−8.87640	−0.686877	−0.343438	−	0.939175i	$-0.611592\pi$
$167$	−8.87640	−0.686877	−0.343438	+	0.939175i	$0.611592\pi$
$168$	0	0
$169$	−11.8910	−0.914695
$170$	−8.31315	−0.637589
$171$	0	0
$172$	51.8203	3.95126
$173$	−2.73839	−0.208196	−0.104098	−	0.994567i	$-0.533196\pi$
$173$	−2.73839	−0.208196	−0.104098	+	0.994567i	$0.533196\pi$
$174$	0	0
$175$	−22.0294	−1.66527
$176$	−34.5903	−2.60734
$177$	0	0
$178$	15.9062	1.19222
$179$	−22.6566	−1.69344	−0.846719	−	0.532041i	$-0.821425\pi$
$179$	−22.6566	−1.69344	−0.846719	+	0.532041i	$0.821425\pi$
$180$	0	0
$181$	17.4609	1.29786	0.648930	−	0.760848i	$-0.275217\pi$
$181$	17.4609	1.29786	0.648930	+	0.760848i	$0.275217\pi$
$182$	−12.8652	−0.953632
$183$	0	0
$184$	−44.4945	−3.28018
$185$	−2.78473	−0.204738
$186$	0	0
$187$	21.7355	1.58946
$188$	19.8913	1.45072
$189$	0	0
$190$	4.35581	0.316004
$191$	21.2967	1.54098	0.770489	−	0.637453i	$-0.220012\pi$
$191$	21.2967	1.54098	0.770489	+	0.637453i	$0.220012\pi$
$192$	0	0
$193$	−0.181450	−0.0130610	−0.00653051	−	0.999979i	$-0.502079\pi$
$193$	−0.181450	−0.0130610	−0.00653051	+	0.999979i	$0.502079\pi$
$194$	26.5282	1.90461
$195$	0	0
$196$	71.7771	5.12694
$197$	20.9903	1.49550	0.747748	−	0.663982i	$-0.231135\pi$
$197$	20.9903	1.49550	0.747748	+	0.663982i	$0.231135\pi$
$198$	0	0
$199$	−13.2078	−0.936276	−0.468138	−	0.883655i	$-0.655075\pi$
$199$	−13.2078	−0.936276	−0.468138	+	0.883655i	$0.655075\pi$
$200$	−33.5169	−2.37001
$201$	0	0
$202$	28.8832	2.03222
$203$	−7.08053	−0.496956
$204$	0	0
$205$	1.04134	0.0727305
$206$	27.7274	1.93186
$207$	0	0
$208$	−9.56420	−0.663158
$209$	−11.3887	−0.787771
$210$	0	0
$211$	−12.5494	−0.863934	−0.431967	−	0.901889i	$-0.642180\pi$
$211$	−12.5494	−0.863934	−0.431967	+	0.901889i	$0.642180\pi$
$212$	62.9253	4.32173
$213$	0	0
$214$	8.73890	0.597379
$215$	6.11210	0.416842
$216$	0	0
$217$	−6.92111	−0.469836
$218$	−30.1732	−2.04358
$219$	0	0
$220$	−10.1466	−0.684085
$221$	6.00985	0.404266
$222$	0	0
$223$	13.5149	0.905025	0.452513	−	0.891758i	$-0.350528\pi$
$223$	13.5149	0.905025	0.452513	+	0.891758i	$0.350528\pi$
$224$	43.6989	2.91975
$225$	0	0
$226$	10.7752	0.716754
$227$	−2.50327	−0.166148	−0.0830739	−	0.996543i	$-0.526474\pi$
$227$	−2.50327	−0.166148	−0.0830739	+	0.996543i	$0.526474\pi$
$228$	0	0
$229$	−1.58055	−0.104446	−0.0522230	−	0.998635i	$-0.516631\pi$
$229$	−1.58055	−0.104446	−0.0522230	+	0.998635i	$0.516631\pi$
$230$	−9.06114	−0.597474
$231$	0	0
$232$	−10.7728	−0.707266
$233$	14.5221	0.951376	0.475688	−	0.879614i	$-0.342199\pi$
$233$	14.5221	0.951376	0.475688	+	0.879614i	$0.342199\pi$
$234$	0	0
$235$	2.34613	0.153045
$236$	35.9957	2.34312
$237$	0	0
$238$	−69.7210	−4.51934
$239$	8.44877	0.546505	0.273253	−	0.961942i	$-0.411900\pi$
$239$	8.44877	0.546505	0.273253	+	0.961942i	$0.411900\pi$
$240$	0	0
$241$	−24.6256	−1.58628	−0.793138	−	0.609042i	$-0.791554\pi$
$241$	−24.6256	−1.58628	−0.793138	+	0.609042i	$0.791554\pi$
$242$	9.10889	0.585542
$243$	0	0
$244$	−70.6789	−4.52476
$245$	8.46596	0.540870
$246$	0	0
$247$	−3.14896	−0.200364
$248$	−10.5302	−0.668670
$249$	0	0
$250$	−14.1089	−0.892328
$251$	9.05087	0.571286	0.285643	−	0.958336i	$-0.407793\pi$
$251$	9.05087	0.571286	0.285643	+	0.958336i	$0.407793\pi$
$252$	0	0
$253$	23.6912	1.48945
$254$	2.59858	0.163049
$255$	0	0
$256$	−19.8423	−1.24014
$257$	12.7480	0.795200	0.397600	−	0.917559i	$-0.369843\pi$
$257$	12.7480	0.795200	0.397600	+	0.917559i	$0.369843\pi$
$258$	0	0
$259$	−23.3551	−1.45121
$260$	−2.80553	−0.173992
$261$	0	0
$262$	−13.3532	−0.824966
$263$	−24.6964	−1.52284	−0.761422	−	0.648257i	$-0.775498\pi$
$263$	−24.6964	−1.52284	−0.761422	+	0.648257i	$0.775498\pi$
$264$	0	0
$265$	7.42191	0.455925
$266$	36.5315	2.23989
$267$	0	0
$268$	−9.55114	−0.583429
$269$	27.4338	1.67267	0.836335	−	0.548219i	$-0.184694\pi$
$269$	27.4338	1.67267	0.836335	+	0.548219i	$0.184694\pi$
$270$	0	0
$271$	20.3668	1.23719	0.618597	−	0.785709i	$-0.287702\pi$
$271$	20.3668	1.23719	0.618597	+	0.785709i	$0.287702\pi$
$272$	−51.8318	−3.14276
$273$	0	0
$274$	−50.4519	−3.04791
$275$	17.8462	1.07616
$276$	0	0
$277$	−24.8624	−1.49384	−0.746920	−	0.664914i	$-0.768468\pi$
$277$	−24.8624	−1.49384	−0.746920	+	0.664914i	$0.768468\pi$
$278$	49.6555	2.97814
$279$	0	0
$280$	18.8508	1.12655
$281$	−3.85302	−0.229852	−0.114926	−	0.993374i	$-0.536663\pi$
$281$	−3.85302	−0.229852	−0.114926	+	0.993374i	$0.536663\pi$
$282$	0	0
$283$	−0.991584	−0.0589436	−0.0294718	−	0.999566i	$-0.509383\pi$
$283$	−0.991584	−0.0589436	−0.0294718	+	0.999566i	$0.509383\pi$
$284$	−21.3587	−1.26741
$285$	0	0
$286$	10.4222	0.616276
$287$	8.73356	0.515526
$288$	0	0
$289$	15.5695	0.915851
$290$	−2.19383	−0.128826
$291$	0	0
$292$	−7.87951	−0.461113
$293$	−1.79802	−0.105041	−0.0525207	−	0.998620i	$-0.516726\pi$
$293$	−1.79802	−0.105041	−0.0525207	+	0.998620i	$0.516726\pi$
$294$	0	0
$295$	4.24562	0.247190
$296$	−35.5339	−2.06537
$297$	0	0
$298$	−13.4207	−0.777439
$299$	6.55060	0.378831
$300$	0	0
$301$	51.2611	2.95464
$302$	23.4230	1.34784
$303$	0	0
$304$	27.1581	1.55763
$305$	−8.33644	−0.477343
$306$	0	0
$307$	−24.9003	−1.42113	−0.710567	−	0.703629i	$-0.751562\pi$
$307$	−24.9003	−1.42113	−0.710567	+	0.703629i	$0.751562\pi$
$308$	−85.0980	−4.84891
$309$	0	0
$310$	−2.14444	−0.121796
$311$	24.1880	1.37157	0.685787	−	0.727802i	$-0.259458\pi$
$311$	24.1880	1.37157	0.685787	+	0.727802i	$0.259458\pi$
$312$	0	0
$313$	24.7956	1.40153	0.700764	−	0.713393i	$-0.252843\pi$
$313$	24.7956	1.40153	0.700764	+	0.713393i	$0.252843\pi$
$314$	49.4439	2.79028
$315$	0	0
$316$	18.8952	1.06294
$317$	−26.2674	−1.47533	−0.737663	−	0.675169i	$-0.764071\pi$
$317$	−26.2674	−1.47533	−0.737663	+	0.675169i	$0.764071\pi$
$318$	0	0
$319$	5.73598	0.321153
$320$	3.35741	0.187685
$321$	0	0
$322$	−75.9943	−4.23500
$323$	−17.0653	−0.949540
$324$	0	0
$325$	4.93445	0.273714
$326$	−44.9224	−2.48802
$327$	0	0
$328$	13.2878	0.733695
$329$	19.6766	1.08481
$330$	0	0
$331$	−6.92323	−0.380535	−0.190268	−	0.981732i	$-0.560936\pi$
$331$	−6.92323	−0.380535	−0.190268	+	0.981732i	$0.560936\pi$
$332$	−71.8753	−3.94467
$333$	0	0
$334$	−23.0661	−1.26212
$335$	−1.12654	−0.0615493
$336$	0	0
$337$	28.8849	1.57346	0.786731	−	0.617297i	$-0.211772\pi$
$337$	28.8849	1.57346	0.786731	+	0.617297i	$0.211772\pi$
$338$	−30.8998	−1.68073
$339$	0	0
$340$	−15.2042	−0.824562
$341$	5.60684	0.303627
$342$	0	0
$343$	38.0932	2.05684
$344$	77.9919	4.20504
$345$	0	0
$346$	−7.11594	−0.382555
$347$	−6.89735	−0.370269	−0.185135	−	0.982713i	$-0.559272\pi$
$347$	−6.89735	−0.370269	−0.185135	+	0.982713i	$0.559272\pi$
$348$	0	0
$349$	−14.2502	−0.762795	−0.381398	−	0.924411i	$-0.624557\pi$
$349$	−14.2502	−0.762795	−0.381398	+	0.924411i	$0.624557\pi$
$350$	−57.2452	−3.05989
$351$	0	0
$352$	−35.4007	−1.88686
$353$	8.76388	0.466454	0.233227	−	0.972422i	$-0.425071\pi$
$353$	8.76388	0.466454	0.233227	+	0.972422i	$0.425071\pi$
$354$	0	0
$355$	−2.51922	−0.133706
$356$	29.0914	1.54184
$357$	0	0
$358$	−58.8751	−3.11165
$359$	13.0809	0.690381	0.345191	−	0.938533i	$-0.387814\pi$
$359$	13.0809	0.690381	0.345191	+	0.938533i	$0.387814\pi$
$360$	0	0
$361$	−10.0583	−0.529386
$362$	45.3736	2.38479
$363$	0	0
$364$	−23.5295	−1.23328
$365$	−0.929371	−0.0486455
$366$	0	0
$367$	−7.38367	−0.385425	−0.192712	−	0.981255i	$-0.561728\pi$
$367$	−7.38367	−0.385425	−0.192712	+	0.981255i	$0.561728\pi$
$368$	−56.4955	−2.94503
$369$	0	0
$370$	−7.23636	−0.376200
$371$	62.2464	3.23167
$372$	0	0
$373$	−5.84837	−0.302817	−0.151408	−	0.988471i	$-0.548381\pi$
$373$	−5.84837	−0.302817	−0.151408	+	0.988471i	$0.548381\pi$
$374$	56.4814	2.92058
$375$	0	0
$376$	29.9373	1.54390
$377$	1.58600	0.0816829
$378$	0	0
$379$	0.850266	0.0436752	0.0218376	−	0.999762i	$-0.493048\pi$
$379$	0.850266	0.0436752	0.0218376	+	0.999762i	$0.493048\pi$
$380$	7.96648	0.408672
$381$	0	0
$382$	55.3413	2.83151
$383$	−13.7492	−0.702550	−0.351275	−	0.936272i	$-0.614252\pi$
$383$	−13.7492	−0.702550	−0.351275	+	0.936272i	$0.614252\pi$
$384$	0	0
$385$	−10.0371	−0.511540
$386$	−0.471512	−0.0239993
$387$	0	0
$388$	48.5182	2.46314
$389$	−33.2501	−1.68584	−0.842922	−	0.538035i	$-0.819167\pi$
$389$	−33.2501	−1.68584	−0.842922	+	0.538035i	$0.819167\pi$
$390$	0	0
$391$	35.5000	1.79531
$392$	108.028	5.45623
$393$	0	0
$394$	54.5450	2.74794
$395$	2.22865	0.112135
$396$	0	0
$397$	5.81644	0.291919	0.145959	−	0.989291i	$-0.453373\pi$
$397$	5.81644	0.291919	0.145959	+	0.989291i	$0.453373\pi$
$398$	−34.3215	−1.72038
$399$	0	0
$400$	−42.5571	−2.12785
$401$	−12.0669	−0.602590	−0.301295	−	0.953531i	$-0.597419\pi$
$401$	−12.0669	−0.602590	−0.301295	+	0.953531i	$0.597419\pi$
$402$	0	0
$403$	1.55029	0.0772253
$404$	52.8254	2.62816
$405$	0	0
$406$	−18.3993	−0.913143
$407$	18.9201	0.937835
$408$	0	0
$409$	27.9245	1.38078	0.690390	−	0.723438i	$-0.257439\pi$
$409$	27.9245	1.38078	0.690390	+	0.723438i	$0.257439\pi$
$410$	2.70601	0.133640
$411$	0	0
$412$	50.7115	2.49838
$413$	35.6073	1.75212
$414$	0	0
$415$	−8.47755	−0.416146
$416$	−9.78828	−0.479910
$417$	0	0
$418$	−29.5944	−1.44751
$419$	−20.6915	−1.01085	−0.505424	−	0.862871i	$-0.668664\pi$
$419$	−20.6915	−1.01085	−0.505424	+	0.862871i	$0.668664\pi$
$420$	0	0
$421$	8.19195	0.399251	0.199626	−	0.979872i	$-0.436027\pi$
$421$	8.19195	0.399251	0.199626	+	0.979872i	$0.436027\pi$
$422$	−32.6105	−1.58746
$423$	0	0
$424$	94.7055	4.59931
$425$	26.7415	1.29715
$426$	0	0
$427$	−69.9163	−3.38349
$428$	15.9828	0.772559
$429$	0	0
$430$	15.8828	0.765936
$431$	35.5258	1.71122	0.855608	−	0.517625i	$-0.173184\pi$
$431$	35.5258	1.71122	0.855608	+	0.517625i	$0.173184\pi$
$432$	0	0
$433$	35.4508	1.70366	0.851828	−	0.523822i	$-0.175494\pi$
$433$	35.4508	1.70366	0.851828	+	0.523822i	$0.175494\pi$
$434$	−17.9851	−0.863311
$435$	0	0
$436$	−55.1846	−2.64286
$437$	−18.6008	−0.889797
$438$	0	0
$439$	25.0505	1.19560	0.597798	−	0.801647i	$-0.296042\pi$
$439$	25.0505	1.19560	0.597798	+	0.801647i	$0.296042\pi$
$440$	−15.2711	−0.728022
$441$	0	0
$442$	15.6171	0.742829
$443$	−3.76881	−0.179062	−0.0895308	−	0.995984i	$-0.528537\pi$
$443$	−3.76881	−0.179062	−0.0895308	+	0.995984i	$0.528537\pi$
$444$	0	0
$445$	3.43127	0.162658
$446$	35.1196	1.66296
$447$	0	0
$448$	28.1581	1.33034
$449$	33.5667	1.58411	0.792056	−	0.610449i	$-0.209011\pi$
$449$	33.5667	1.58411	0.792056	+	0.610449i	$0.209011\pi$
$450$	0	0
$451$	−7.07511	−0.333154
$452$	19.7071	0.926942
$453$	0	0
$454$	−6.50495	−0.305292
$455$	−2.77526	−0.130106
$456$	0	0
$457$	2.78962	0.130493	0.0652464	−	0.997869i	$-0.479217\pi$
$457$	2.78962	0.130493	0.0652464	+	0.997869i	$0.479217\pi$
$458$	−4.10720	−0.191917
$459$	0	0
$460$	−16.5722	−0.772683
$461$	21.9615	1.02285	0.511425	−	0.859328i	$-0.329118\pi$
$461$	21.9615	1.02285	0.511425	+	0.859328i	$0.329118\pi$
$462$	0	0
$463$	−10.9677	−0.509711	−0.254855	−	0.966979i	$-0.582028\pi$
$463$	−10.9677	−0.509711	−0.254855	+	0.966979i	$0.582028\pi$
$464$	−13.6784	−0.635002
$465$	0	0
$466$	37.7369	1.74813
$467$	−10.2697	−0.475225	−0.237613	−	0.971360i	$-0.576365\pi$
$467$	−10.2697	−0.475225	−0.237613	+	0.971360i	$0.576365\pi$
$468$	0	0
$469$	−9.44809	−0.436272
$470$	6.09662	0.281216
$471$	0	0
$472$	54.1752	2.49362
$473$	−41.5270	−1.90941
$474$	0	0
$475$	−14.0117	−0.642900
$476$	−127.515	−5.84463
$477$	0	0
$478$	21.9548	1.00419
$479$	−1.80620	−0.0825274	−0.0412637	−	0.999148i	$-0.513138\pi$
$479$	−1.80620	−0.0825274	−0.0412637	+	0.999148i	$0.513138\pi$
$480$	0	0
$481$	5.23140	0.238531
$482$	−63.9917	−2.91474
$483$	0	0
$484$	16.6595	0.757251
$485$	5.72262	0.259851
$486$	0	0
$487$	−6.40098	−0.290056	−0.145028	−	0.989428i	$-0.546327\pi$
$487$	−6.40098	−0.290056	−0.145028	+	0.989428i	$0.546327\pi$
$488$	−106.375	−4.81537
$489$	0	0
$490$	21.9995	0.993836
$491$	−8.08703	−0.364963	−0.182481	−	0.983209i	$-0.558413\pi$
$491$	−8.08703	−0.364963	−0.182481	+	0.983209i	$0.558413\pi$
$492$	0	0
$493$	8.59506	0.387102
$494$	−8.18283	−0.368163
$495$	0	0
$496$	−13.3704	−0.600349
$497$	−21.1282	−0.947731
$498$	0	0
$499$	−14.2866	−0.639558	−0.319779	−	0.947492i	$-0.603609\pi$
$499$	−14.2866	−0.639558	−0.319779	+	0.947492i	$0.603609\pi$
$500$	−25.8043	−1.15400
$501$	0	0
$502$	23.5194	1.04972
$503$	2.23856	0.0998127	0.0499063	−	0.998754i	$-0.484108\pi$
$503$	2.23856	0.0998127	0.0499063	+	0.998754i	$0.484108\pi$
$504$	0	0
$505$	6.23065	0.277260
$506$	61.5635	2.73683
$507$	0	0
$508$	4.75262	0.210864
$509$	−11.7376	−0.520258	−0.260129	−	0.965574i	$-0.583765\pi$
$509$	−11.7376	−0.520258	−0.260129	+	0.965574i	$0.583765\pi$
$510$	0	0
$511$	−7.79448	−0.344808
$512$	−45.5096	−2.01126
$513$	0	0
$514$	33.1268	1.46116
$515$	5.98132	0.263569
$516$	0	0
$517$	−15.9402	−0.701047
$518$	−60.6901	−2.66657
$519$	0	0
$520$	−4.22246	−0.185167
$521$	30.0540	1.31669	0.658345	−	0.752717i	$-0.271257\pi$
$521$	30.0540	1.31669	0.658345	+	0.752717i	$0.271257\pi$
$522$	0	0
$523$	−11.7788	−0.515049	−0.257525	−	0.966272i	$-0.582907\pi$
$523$	−11.7788	−0.515049	−0.257525	+	0.966272i	$0.582907\pi$
$524$	−24.4221	−1.06689
$525$	0	0
$526$	−64.1755	−2.79819
$527$	8.40155	0.365977
$528$	0	0
$529$	15.6942	0.682356
$530$	19.2864	0.837750
$531$	0	0
$532$	66.8135	2.89673
$533$	−1.95626	−0.0847352
$534$	0	0
$535$	1.88514	0.0815018
$536$	−14.3749	−0.620902
$537$	0	0
$538$	71.2890	3.07349
$539$	−57.5196	−2.47755
$540$	0	0
$541$	−6.16617	−0.265104	−0.132552	−	0.991176i	$-0.542317\pi$
$541$	−6.16617	−0.265104	−0.132552	+	0.991176i	$0.542317\pi$
$542$	52.9247	2.27331
$543$	0	0
$544$	−53.0461	−2.27433
$545$	−6.50891	−0.278811
$546$	0	0
$547$	−29.5976	−1.26550	−0.632751	−	0.774355i	$-0.718074\pi$
$547$	−29.5976	−1.26550	−0.632751	+	0.774355i	$0.718074\pi$
$548$	−92.2731	−3.94171
$549$	0	0
$550$	46.3747	1.97742
$551$	−4.50353	−0.191857
$552$	0	0
$553$	18.6913	0.794835
$554$	−64.6071	−2.74489
$555$	0	0
$556$	90.8165	3.85148
$557$	28.7389	1.21771	0.608853	−	0.793283i	$-0.291630\pi$
$557$	28.7389	1.21771	0.608853	+	0.793283i	$0.291630\pi$
$558$	0	0
$559$	−11.4822	−0.485645
$560$	23.9352	1.01145
$561$	0	0
$562$	−10.0124	−0.422347
$563$	−29.9843	−1.26369	−0.631844	−	0.775096i	$-0.717701\pi$
$563$	−29.9843	−1.26369	−0.631844	+	0.775096i	$0.717701\pi$
$564$	0	0
$565$	2.32441	0.0977885
$566$	−2.57671	−0.108307
$567$	0	0
$568$	−32.1458	−1.34881
$569$	0.555374	0.0232825	0.0116412	−	0.999932i	$-0.496294\pi$
$569$	0.555374	0.0232825	0.0116412	+	0.999932i	$0.496294\pi$
$570$	0	0
$571$	−27.5161	−1.15151	−0.575756	−	0.817622i	$-0.695292\pi$
$571$	−27.5161	−1.15151	−0.575756	+	0.817622i	$0.695292\pi$
$572$	19.0614	0.796999
$573$	0	0
$574$	22.6949	0.947265
$575$	29.1477	1.21554
$576$	0	0
$577$	−33.3797	−1.38961	−0.694807	−	0.719197i	$-0.744510\pi$
$577$	−33.3797	−1.38961	−0.694807	+	0.719197i	$0.744510\pi$
$578$	40.4585	1.68285
$579$	0	0
$580$	−4.01237	−0.166605
$581$	−71.0998	−2.94971
$582$	0	0
$583$	−50.4262	−2.08844
$584$	−11.8590	−0.490730
$585$	0	0
$586$	−4.67230	−0.193011
$587$	36.6967	1.51463	0.757317	−	0.653048i	$-0.226510\pi$
$587$	36.6967	1.51463	0.757317	+	0.653048i	$0.226510\pi$
$588$	0	0
$589$	−4.40213	−0.181387
$590$	11.0326	0.454205
$591$	0	0
$592$	−45.1181	−1.85434
$593$	−10.3388	−0.424564	−0.212282	−	0.977208i	$-0.568090\pi$
$593$	−10.3388	−0.424564	−0.212282	+	0.977208i	$0.568090\pi$
$594$	0	0
$595$	−15.0401	−0.616585
$596$	−24.5455	−1.00542
$597$	0	0
$598$	17.0223	0.696092
$599$	−11.3524	−0.463846	−0.231923	−	0.972734i	$-0.574502\pi$
$599$	−11.3524	−0.463846	−0.231923	+	0.972734i	$0.574502\pi$
$600$	0	0
$601$	32.5042	1.32587	0.662937	−	0.748675i	$-0.269310\pi$
$601$	32.5042	1.32587	0.662937	+	0.748675i	$0.269310\pi$
$602$	133.206	5.42908
$603$	0	0
$604$	42.8391	1.74310
$605$	1.96496	0.0798869
$606$	0	0
$607$	−43.0815	−1.74862	−0.874312	−	0.485365i	$-0.838687\pi$
$607$	−43.0815	−1.74862	−0.874312	+	0.485365i	$0.838687\pi$
$608$	27.7944	1.12721
$609$	0	0
$610$	−21.6629	−0.877105
$611$	−4.40745	−0.178306
$612$	0	0
$613$	−4.07621	−0.164636	−0.0823182	−	0.996606i	$-0.526232\pi$
$613$	−4.07621	−0.164636	−0.0823182	+	0.996606i	$0.526232\pi$
$614$	−64.7054	−2.61130
$615$	0	0
$616$	−128.076	−5.16034
$617$	20.2709	0.816075	0.408037	−	0.912965i	$-0.366213\pi$
$617$	20.2709	0.816075	0.408037	+	0.912965i	$0.366213\pi$
$618$	0	0
$619$	7.04298	0.283081	0.141541	−	0.989932i	$-0.454794\pi$
$619$	7.04298	0.283081	0.141541	+	0.989932i	$0.454794\pi$
$620$	−3.92204	−0.157513
$621$	0	0
$622$	62.8545	2.52023
$623$	28.7775	1.15295
$624$	0	0
$625$	20.3853	0.815412
$626$	64.4333	2.57527
$627$	0	0
$628$	90.4295	3.60853
$629$	28.3508	1.13042
$630$	0	0
$631$	−17.4805	−0.695887	−0.347944	−	0.937515i	$-0.613120\pi$
$631$	−17.4805	−0.695887	−0.347944	+	0.937515i	$0.613120\pi$
$632$	28.4381	1.13121
$633$	0	0
$634$	−68.2581	−2.71087
$635$	0.560562	0.0222452
$636$	0	0
$637$	−15.9042	−0.630145
$638$	14.9054	0.590111
$639$	0	0
$640$	−1.69631	−0.0670527
$641$	−35.6340	−1.40746	−0.703730	−	0.710468i	$-0.748484\pi$
$641$	−35.6340	−1.40746	−0.703730	+	0.710468i	$0.748484\pi$
$642$	0	0
$643$	3.91349	0.154333	0.0771664	−	0.997018i	$-0.475413\pi$
$643$	3.91349	0.154333	0.0771664	+	0.997018i	$0.475413\pi$
$644$	−138.988	−5.47691
$645$	0	0
$646$	−44.3456	−1.74475
$647$	14.1447	0.556087	0.278043	−	0.960569i	$-0.410314\pi$
$647$	14.1447	0.556087	0.278043	+	0.960569i	$0.410314\pi$
$648$	0	0
$649$	−28.8457	−1.13229
$650$	12.8226	0.502943
$651$	0	0
$652$	−82.1599	−3.21763
$653$	−6.80559	−0.266323	−0.133162	−	0.991094i	$-0.542513\pi$
$653$	−6.80559	−0.266323	−0.133162	+	0.991094i	$0.542513\pi$
$654$	0	0
$655$	−2.88054	−0.112552
$656$	16.8717	0.658731
$657$	0	0
$658$	51.1313	1.99331
$659$	32.6649	1.27244	0.636222	−	0.771506i	$-0.280496\pi$
$659$	32.6649	1.27244	0.636222	+	0.771506i	$0.280496\pi$
$660$	0	0
$661$	16.7002	0.649561	0.324781	−	0.945789i	$-0.394710\pi$
$661$	16.7002	0.649561	0.324781	+	0.945789i	$0.394710\pi$
$662$	−17.9906	−0.699224
$663$	0	0
$664$	−108.176	−4.19803
$665$	7.88052	0.305593
$666$	0	0
$667$	9.36843	0.362747
$668$	−42.1862	−1.63223
$669$	0	0
$670$	−2.92740	−0.113095
$671$	56.6396	2.18655
$672$	0	0
$673$	39.7605	1.53265	0.766327	−	0.642451i	$-0.222082\pi$
$673$	39.7605	1.53265	0.766327	+	0.642451i	$0.222082\pi$
$674$	75.0598	2.89119
$675$	0	0
$676$	−56.5136	−2.17360
$677$	43.1065	1.65672	0.828359	−	0.560198i	$-0.189275\pi$
$677$	43.1065	1.65672	0.828359	+	0.560198i	$0.189275\pi$
$678$	0	0
$679$	47.9946	1.84187
$680$	−22.8830	−0.877522
$681$	0	0
$682$	14.5698	0.557907
$683$	−3.98149	−0.152347	−0.0761737	−	0.997095i	$-0.524270\pi$
$683$	−3.98149	−0.152347	−0.0761737	+	0.997095i	$0.524270\pi$
$684$	0	0
$685$	−10.8834	−0.415834
$686$	98.9883	3.77939
$687$	0	0
$688$	99.0278	3.77540
$689$	−13.9428	−0.531179
$690$	0	0
$691$	−35.0059	−1.33169	−0.665843	−	0.746092i	$-0.731928\pi$
$691$	−35.0059	−1.33169	−0.665843	+	0.746092i	$0.731928\pi$
$692$	−13.0146	−0.494739
$693$	0	0
$694$	−17.9233	−0.680360
$695$	10.7116	0.406315
$696$	0	0
$697$	−10.6017	−0.401567
$698$	−37.0303	−1.40162
$699$	0	0
$700$	−104.698	−3.95719
$701$	−5.39784	−0.203874	−0.101937	−	0.994791i	$-0.532504\pi$
$701$	−5.39784	−0.203874	−0.101937	+	0.994791i	$0.532504\pi$
$702$	0	0
$703$	−14.8549	−0.560262
$704$	−22.8110	−0.859723
$705$	0	0
$706$	22.7736	0.857097
$707$	52.2554	1.96527
$708$	0	0
$709$	−20.2547	−0.760680	−0.380340	−	0.924847i	$-0.624193\pi$
$709$	−20.2547	−0.760680	−0.380340	+	0.924847i	$0.624193\pi$
$710$	−6.54639	−0.245681
$711$	0	0
$712$	43.7839	1.64087
$713$	9.15750	0.342951
$714$	0	0
$715$	2.24826	0.0840800
$716$	−107.679	−4.02414
$717$	0	0
$718$	33.9917	1.26856
$719$	−36.1262	−1.34728	−0.673640	−	0.739060i	$-0.735270\pi$
$719$	−36.1262	−1.34728	−0.673640	+	0.739060i	$0.735270\pi$
$720$	0	0
$721$	50.1643	1.86822
$722$	−26.1374	−0.972734
$723$	0	0
$724$	82.9852	3.08412
$725$	7.05707	0.262093
$726$	0	0
$727$	−26.5788	−0.985751	−0.492876	−	0.870100i	$-0.664054\pi$
$727$	−26.5788	−0.985751	−0.492876	+	0.870100i	$0.664054\pi$
$728$	−35.4131	−1.31250
$729$	0	0
$730$	−2.41505	−0.0893849
$731$	−62.2260	−2.30151
$732$	0	0
$733$	−41.4745	−1.53189	−0.765947	−	0.642904i	$-0.777730\pi$
$733$	−41.4745	−1.53189	−0.765947	+	0.642904i	$0.777730\pi$
$734$	−19.1871	−0.708208
$735$	0	0
$736$	−57.8191	−2.13124
$737$	7.65395	0.281937
$738$	0	0
$739$	32.6849	1.20233	0.601167	−	0.799123i	$-0.294703\pi$
$739$	32.6849	1.20233	0.601167	+	0.799123i	$0.294703\pi$
$740$	−13.2348	−0.486521
$741$	0	0
$742$	161.752	5.93811
$743$	−22.3876	−0.821320	−0.410660	−	0.911789i	$-0.634702\pi$
$743$	−22.3876	−0.821320	−0.410660	+	0.911789i	$0.634702\pi$
$744$	0	0
$745$	−2.89509	−0.106068
$746$	−15.1975	−0.556418
$747$	0	0
$748$	103.301	3.77704
$749$	15.8104	0.577699
$750$	0	0
$751$	27.5093	1.00383	0.501915	−	0.864917i	$-0.332629\pi$
$751$	27.5093	1.00383	0.501915	+	0.864917i	$0.332629\pi$
$752$	38.0119	1.38615
$753$	0	0
$754$	4.12134	0.150090
$755$	5.05279	0.183890
$756$	0	0
$757$	47.2192	1.71621	0.858106	−	0.513473i	$-0.171641\pi$
$757$	47.2192	1.71621	0.858106	+	0.513473i	$0.171641\pi$
$758$	2.20948	0.0802521
$759$	0	0
$760$	11.9899	0.434920
$761$	7.21112	0.261403	0.130702	−	0.991422i	$-0.458277\pi$
$761$	7.21112	0.261403	0.130702	+	0.991422i	$0.458277\pi$
$762$	0	0
$763$	−54.5891	−1.97626
$764$	101.215	3.66185
$765$	0	0
$766$	−35.7283	−1.29092
$767$	−7.97583	−0.287990
$768$	0	0
$769$	−8.30677	−0.299550	−0.149775	−	0.988720i	$-0.547855\pi$
$769$	−8.30677	−0.299550	−0.149775	+	0.988720i	$0.547855\pi$
$770$	−26.0823	−0.939941
$771$	0	0
$772$	−0.862362	−0.0310371
$773$	22.8973	0.823559	0.411780	−	0.911283i	$-0.364907\pi$
$773$	22.8973	0.823559	0.411780	+	0.911283i	$0.364907\pi$
$774$	0	0
$775$	6.89819	0.247790
$776$	73.0221	2.62134
$777$	0	0
$778$	−86.4030	−3.09770
$779$	5.55493	0.199026
$780$	0	0
$781$	17.1161	0.612463
$782$	92.2496	3.29884
$783$	0	0
$784$	137.165	4.89875
$785$	10.6660	0.380685
$786$	0	0
$787$	−14.6424	−0.521946	−0.260973	−	0.965346i	$-0.584043\pi$
$787$	−14.6424	−0.521946	−0.260973	+	0.965346i	$0.584043\pi$
$788$	99.7590	3.55377
$789$	0	0
$790$	5.79132	0.206046
$791$	19.4944	0.693141
$792$	0	0
$793$	15.6608	0.556132
$794$	15.1145	0.536393
$795$	0	0
$796$	−62.7717	−2.22488
$797$	−29.8636	−1.05782	−0.528912	−	0.848676i	$-0.677400\pi$
$797$	−29.8636	−1.05782	−0.528912	+	0.848676i	$0.677400\pi$
$798$	0	0
$799$	−23.8855	−0.845008
$800$	−43.5541	−1.53987
$801$	0	0
$802$	−31.3567	−1.10724
$803$	6.31436	0.222829
$804$	0	0
$805$	−16.3934	−0.577791
$806$	4.02855	0.141900
$807$	0	0
$808$	79.5047	2.79697
$809$	42.6554	1.49969	0.749843	−	0.661616i	$-0.230129\pi$
$809$	42.6554	1.49969	0.749843	+	0.661616i	$0.230129\pi$
$810$	0	0
$811$	6.64944	0.233493	0.116747	−	0.993162i	$-0.462753\pi$
$811$	6.64944	0.233493	0.116747	+	0.993162i	$0.462753\pi$
$812$	−33.6511	−1.18092
$813$	0	0
$814$	49.1654	1.72325
$815$	−9.69059	−0.339447
$816$	0	0
$817$	32.6043	1.14068
$818$	72.5642	2.53715
$819$	0	0
$820$	4.94911	0.172830
$821$	11.4870	0.400901	0.200450	−	0.979704i	$-0.435759\pi$
$821$	11.4870	0.400901	0.200450	+	0.979704i	$0.435759\pi$
$822$	0	0
$823$	−15.3682	−0.535703	−0.267852	−	0.963460i	$-0.586314\pi$
$823$	−15.3682	−0.535703	−0.267852	+	0.963460i	$0.586314\pi$
$824$	76.3232	2.65884
$825$	0	0
$826$	92.5285	3.21948
$827$	45.6096	1.58600	0.793001	−	0.609220i	$-0.208517\pi$
$827$	45.6096	1.58600	0.793001	+	0.609220i	$0.208517\pi$
$828$	0	0
$829$	−14.7166	−0.511128	−0.255564	−	0.966792i	$-0.582261\pi$
$829$	−14.7166	−0.511128	−0.255564	+	0.966792i	$0.582261\pi$
$830$	−22.0296	−0.764658
$831$	0	0
$832$	−6.30724	−0.218664
$833$	−86.1902	−2.98631
$834$	0	0
$835$	−4.97578	−0.172194
$836$	−54.1261	−1.87199
$837$	0	0
$838$	−53.7686	−1.85741
$839$	−18.0951	−0.624711	−0.312355	−	0.949965i	$-0.601118\pi$
$839$	−18.0951	−0.624711	−0.312355	+	0.949965i	$0.601118\pi$
$840$	0	0
$841$	−26.7318	−0.921785
$842$	21.2874	0.733614
$843$	0	0
$844$	−59.6424	−2.05298
$845$	−6.66567	−0.229306
$846$	0	0
$847$	16.4798	0.566251
$848$	120.249	4.12938
$849$	0	0
$850$	69.4900	2.38349
$851$	30.9017	1.05930
$852$	0	0
$853$	−23.8148	−0.815403	−0.407702	−	0.913115i	$-0.633670\pi$
$853$	−23.8148	−0.815403	−0.407702	+	0.913115i	$0.633670\pi$
$854$	−181.683	−6.21707
$855$	0	0
$856$	24.0549	0.822180
$857$	−33.6816	−1.15054	−0.575271	−	0.817963i	$-0.695103\pi$
$857$	−33.6816	−1.15054	−0.575271	+	0.817963i	$0.695103\pi$
$858$	0	0
$859$	−40.7180	−1.38928	−0.694641	−	0.719357i	$-0.744437\pi$
$859$	−40.7180	−1.38928	−0.694641	+	0.719357i	$0.744437\pi$
$860$	29.0485	0.990546
$861$	0	0
$862$	92.3166	3.14431
$863$	−47.1366	−1.60455	−0.802275	−	0.596955i	$-0.796377\pi$
$863$	−47.1366	−1.60455	−0.802275	+	0.596955i	$0.796377\pi$
$864$	0	0
$865$	−1.53504	−0.0521929
$866$	92.1218	3.13042
$867$	0	0
$868$	−32.8935	−1.11648
$869$	−15.1419	−0.513655
$870$	0	0
$871$	2.11631	0.0717086
$872$	−83.0554	−2.81261
$873$	0	0
$874$	−48.3357	−1.63498
$875$	−25.5258	−0.862931
$876$	0	0
$877$	−11.9832	−0.404643	−0.202321	−	0.979319i	$-0.564849\pi$
$877$	−11.9832	−0.404643	−0.202321	+	0.979319i	$0.564849\pi$
$878$	65.0958	2.19688
$879$	0	0
$880$	−19.3900	−0.653638
$881$	−22.0033	−0.741311	−0.370655	−	0.928770i	$-0.620867\pi$
$881$	−22.0033	−0.741311	−0.370655	+	0.928770i	$0.620867\pi$
$882$	0	0
$883$	2.29701	0.0773004	0.0386502	−	0.999253i	$-0.487694\pi$
$883$	2.29701	0.0773004	0.0386502	+	0.999253i	$0.487694\pi$
$884$	28.5626	0.960663
$885$	0	0
$886$	−9.79356	−0.329021
$887$	23.7072	0.796010	0.398005	−	0.917383i	$-0.369703\pi$
$887$	23.7072	0.796010	0.398005	+	0.917383i	$0.369703\pi$
$888$	0	0
$889$	4.70134	0.157678
$890$	8.91643	0.298880
$891$	0	0
$892$	64.2313	2.15062
$893$	12.5152	0.418805
$894$	0	0
$895$	−12.7005	−0.424530
$896$	−14.2267	−0.475281
$897$	0	0
$898$	87.2259	2.91076
$899$	2.21716	0.0739466
$900$	0	0
$901$	−75.5609	−2.51730
$902$	−18.3852	−0.612162
$903$	0	0
$904$	29.6600	0.986478
$905$	9.78794	0.325362
$906$	0	0
$907$	23.8707	0.792613	0.396307	−	0.918118i	$-0.370292\pi$
$907$	23.8707	0.792613	0.396307	+	0.918118i	$0.370292\pi$
$908$	−11.8971	−0.394819
$909$	0	0
$910$	−7.21174	−0.239067
$911$	−36.5020	−1.20937	−0.604683	−	0.796466i	$-0.706700\pi$
$911$	−36.5020	−1.20937	−0.604683	+	0.796466i	$0.706700\pi$
$912$	0	0
$913$	57.5984	1.90623
$914$	7.24904	0.239777
$915$	0	0
$916$	−7.51178	−0.248196
$917$	−24.1586	−0.797788
$918$	0	0
$919$	−27.8968	−0.920232	−0.460116	−	0.887859i	$-0.652192\pi$
$919$	−27.8968	−0.920232	−0.460116	+	0.887859i	$0.652192\pi$
$920$	−24.9419	−0.822311
$921$	0	0
$922$	57.0688	1.87946
$923$	4.73260	0.155775
$924$	0	0
$925$	23.2777	0.765367
$926$	−28.5004	−0.936580
$927$	0	0
$928$	−13.9988	−0.459534
$929$	36.0576	1.18301	0.591505	−	0.806301i	$-0.298534\pi$
$929$	36.0576	1.18301	0.591505	+	0.806301i	$0.298534\pi$
$930$	0	0
$931$	45.1608	1.48008
$932$	69.0182	2.26077
$933$	0	0
$934$	−26.6866	−0.873214
$935$	12.1841	0.398462
$936$	0	0
$937$	31.9487	1.04372	0.521860	−	0.853031i	$-0.325238\pi$
$937$	31.9487	1.04372	0.521860	+	0.853031i	$0.325238\pi$
$938$	−24.5516	−0.801639
$939$	0	0
$940$	11.1503	0.363682
$941$	29.7734	0.970584	0.485292	−	0.874352i	$-0.338713\pi$
$941$	29.7734	0.970584	0.485292	+	0.874352i	$0.338713\pi$
$942$	0	0
$943$	−11.5556	−0.376302
$944$	68.7873	2.23884
$945$	0	0
$946$	−107.911	−3.50850
$947$	2.96288	0.0962808	0.0481404	−	0.998841i	$-0.484671\pi$
$947$	2.96288	0.0962808	0.0481404	+	0.998841i	$0.484671\pi$
$948$	0	0
$949$	1.74592	0.0566749
$950$	−36.4105	−1.18131
$951$	0	0
$952$	−191.916	−6.22002
$953$	15.9303	0.516035	0.258017	−	0.966140i	$-0.416931\pi$
$953$	15.9303	0.516035	0.258017	+	0.966140i	$0.416931\pi$
$954$	0	0
$955$	11.9382	0.386310
$956$	40.1538	1.29867
$957$	0	0
$958$	−4.69356	−0.151642
$959$	−91.2775	−2.94750
$960$	0	0
$961$	−28.8328	−0.930089
$962$	13.5942	0.438295
$963$	0	0
$964$	−117.036	−3.76949
$965$	−0.101714	−0.00327428
$966$	0	0
$967$	−2.73034	−0.0878019	−0.0439010	−	0.999036i	$-0.513979\pi$
$967$	−2.73034	−0.0878019	−0.0439010	+	0.999036i	$0.513979\pi$
$968$	25.0734	0.805888
$969$	0	0
$970$	14.8707	0.477469
$971$	−38.9478	−1.24989	−0.624947	−	0.780667i	$-0.714880\pi$
$971$	−38.9478	−1.24989	−0.624947	+	0.780667i	$0.714880\pi$
$972$	0	0
$973$	89.8365	2.88003
$974$	−16.6335	−0.532971
$975$	0	0
$976$	−135.066	−4.32337
$977$	57.8449	1.85062	0.925311	−	0.379208i	$-0.123804\pi$
$977$	57.8449	1.85062	0.925311	+	0.379208i	$0.123804\pi$
$978$	0	0
$979$	−23.3128	−0.745081
$980$	40.2355	1.28528
$981$	0	0
$982$	−21.0148	−0.670609
$983$	−55.4740	−1.76935	−0.884673	−	0.466211i	$-0.845619\pi$
$983$	−55.4740	−1.76935	−0.884673	+	0.466211i	$0.845619\pi$
$984$	0	0
$985$	11.7664	0.374908
$986$	22.3350	0.711290
$987$	0	0
$988$	−14.9658	−0.476126
$989$	−67.8249	−2.15671
$990$	0	0
$991$	59.6622	1.89523	0.947615	−	0.319413i	$-0.103486\pi$
$991$	59.6622	1.89523	0.947615	+	0.319413i	$0.103486\pi$
$992$	−13.6837	−0.434457
$993$	0	0
$994$	−54.9034	−1.74143
$995$	−7.40379	−0.234716
$996$	0	0
$997$	−27.1018	−0.858322	−0.429161	−	0.903228i	$-0.641191\pi$
$997$	−27.1018	−0.858322	−0.429161	+	0.903228i	$0.641191\pi$
$998$	−37.1250	−1.17517
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1143.2.a.k.1.16	yes	16
3.2	odd	2	inner	1143.2.a.k.1.1	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1143.2.a.k.1.1	✓	16	3.2	odd	2	inner
1143.2.a.k.1.16	yes	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:	\( N \)	\(=\)	\( 1143 = 3^{2} \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1143.a (trivial)

\(f(q)\)	\(=\)	\(q+2.59858 q^{2} +4.75262 q^{4} +0.560562 q^{5} +4.70134 q^{7} +7.15292 q^{8} +O(q^{10})\)	\(q+2.59858 q^{2} +4.75262 q^{4} +0.560562 q^{5} +4.70134 q^{7} +7.15292 q^{8} +1.45667 q^{10} -3.80859 q^{11} -1.05307 q^{13} +12.2168 q^{14} +9.08219 q^{16} -5.70697 q^{17} +2.99026 q^{19} +2.66414 q^{20} -9.89692 q^{22} -6.22047 q^{23} -4.68577 q^{25} -2.73649 q^{26} +22.3437 q^{28} -1.50607 q^{29} -1.47216 q^{31} +9.29498 q^{32} -14.8300 q^{34} +2.63540 q^{35} -4.96775 q^{37} +7.77044 q^{38} +4.00966 q^{40} +1.85767 q^{41} +10.9035 q^{43} -18.1008 q^{44} -16.1644 q^{46} +4.18532 q^{47} +15.1026 q^{49} -12.1764 q^{50} -5.00486 q^{52} +13.2401 q^{53} -2.13495 q^{55} +33.6283 q^{56} -3.91363 q^{58} +7.57386 q^{59} -14.8716 q^{61} -3.82552 q^{62} +5.98937 q^{64} -0.590313 q^{65} -2.00966 q^{67} -27.1231 q^{68} +6.84829 q^{70} -4.49409 q^{71} -1.65793 q^{73} -12.9091 q^{74} +14.2116 q^{76} -17.9055 q^{77} +3.97574 q^{79} +5.09113 q^{80} +4.82731 q^{82} -15.1233 q^{83} -3.19911 q^{85} +28.3337 q^{86} -27.2425 q^{88} +6.12112 q^{89} -4.95085 q^{91} -29.5635 q^{92} +10.8759 q^{94} +1.67623 q^{95} +10.2087 q^{97} +39.2454 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 20 q^{4} + 10 q^{7}+O(q^{10}) \) 16 * q + 20 * q^4 + 10 * q^7	\( 16 q + 20 q^{4} + 10 q^{7} + 14 q^{10} + 20 q^{13} + 28 q^{16} + 12 q^{19} + 18 q^{22} + 52 q^{25} + 42 q^{28} + 18 q^{31} + 10 q^{34} + 16 q^{37} + 6 q^{40} + 26 q^{43} - 24 q^{46} + 54 q^{49} + 52 q^{52} + 20 q^{55} - 14 q^{58} + 36 q^{61} - 4 q^{64} + 26 q^{67} + 36 q^{70} + 60 q^{73} - 20 q^{76} + 12 q^{79} - 20 q^{82} - 12 q^{85} + 8 q^{88} - 24 q^{91} - 26 q^{94} + 108 q^{97}+O(q^{100}) \) 16 * q + 20 * q^4 + 10 * q^7 + 14 * q^10 + 20 * q^13 + 28 * q^16 + 12 * q^19 + 18 * q^22 + 52 * q^25 + 42 * q^28 + 18 * q^31 + 10 * q^34 + 16 * q^37 + 6 * q^40 + 26 * q^43 - 24 * q^46 + 54 * q^49 + 52 * q^52 + 20 * q^55 - 14 * q^58 + 36 * q^61 - 4 * q^64 + 26 * q^67 + 36 * q^70 + 60 * q^73 - 20 * q^76 + 12 * q^79 - 20 * q^82 - 12 * q^85 + 8 * q^88 - 24 * q^91 - 26 * q^94 + 108 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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