LMFDB - Embedded newform 8550.2.a.co.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8550,2,Mod(1,8550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8550, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8550.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8550.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:	$68.2720937282$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.257.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 3 $ x^3 - x^2 - 4*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 950)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.713538$ of defining polynomial
Character	$\chi$	$=$	8550.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +4.69527 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} +4.69527 q^{7} +1.00000 q^{8} -6.40880 q^{11} -1.06379 q^{13} +4.69527 q^{14} +1.00000 q^{16} +1.91794 q^{17} -1.00000 q^{19} -6.40880 q^{22} +1.79560 q^{23} -1.06379 q^{26} +4.69527 q^{28} -2.93621 q^{29} -5.55465 q^{31} +1.00000 q^{32} +1.91794 q^{34} +11.4088 q^{37} -1.00000 q^{38} +1.14585 q^{41} +3.55465 q^{43} -6.40880 q^{44} +1.79560 q^{46} +10.8359 q^{47} +15.0455 q^{49} -1.06379 q^{52} -8.69527 q^{53} +4.69527 q^{56} -2.93621 q^{58} +5.63148 q^{59} -3.39053 q^{61} -5.55465 q^{62} +1.00000 q^{64} +8.82284 q^{67} +1.91794 q^{68} +1.42708 q^{71} +12.6132 q^{73} +11.4088 q^{74} -1.00000 q^{76} -30.0910 q^{77} -1.96345 q^{79} +1.14585 q^{82} +16.2447 q^{83} +3.55465 q^{86} -6.40880 q^{88} +10.0000 q^{89} -4.99477 q^{91} +1.79560 q^{92} +10.8359 q^{94} +14.9452 q^{97} +15.0455 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} - 2 q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q + 3 * q^2 + 3 * q^4 - 2 * q^7 + 3 * q^8	$ 3 q + 3 q^{2} + 3 q^{4} - 2 q^{7} + 3 q^{8} - 2 q^{11} + 2 q^{13} - 2 q^{14} + 3 q^{16} - 4 q^{17} - 3 q^{19} - 2 q^{22} + 14 q^{23} + 2 q^{26} - 2 q^{28} - 14 q^{29} - 4 q^{31} + 3 q^{32} - 4 q^{34} + 17 q^{37} - 3 q^{38} + 8 q^{41} - 2 q^{43} - 2 q^{44} + 14 q^{46} + 13 q^{47} + 25 q^{49} + 2 q^{52} - 10 q^{53} - 2 q^{56} - 14 q^{58} + 6 q^{59} + 22 q^{61} - 4 q^{62} + 3 q^{64} - 4 q^{68} + 2 q^{71} + 12 q^{73} + 17 q^{74} - 3 q^{76} - 50 q^{77} + 24 q^{79} + 8 q^{82} + 12 q^{83} - 2 q^{86} - 2 q^{88} + 30 q^{89} - 7 q^{91} + 14 q^{92} + 13 q^{94} + 25 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 - 2 * q^7 + 3 * q^8 - 2 * q^11 + 2 * q^13 - 2 * q^14 + 3 * q^16 - 4 * q^17 - 3 * q^19 - 2 * q^22 + 14 * q^23 + 2 * q^26 - 2 * q^28 - 14 * q^29 - 4 * q^31 + 3 * q^32 - 4 * q^34 + 17 * q^37 - 3 * q^38 + 8 * q^41 - 2 * q^43 - 2 * q^44 + 14 * q^46 + 13 * q^47 + 25 * q^49 + 2 * q^52 - 10 * q^53 - 2 * q^56 - 14 * q^58 + 6 * q^59 + 22 * q^61 - 4 * q^62 + 3 * q^64 - 4 * q^68 + 2 * q^71 + 12 * q^73 + 17 * q^74 - 3 * q^76 - 50 * q^77 + 24 * q^79 + 8 * q^82 + 12 * q^83 - 2 * q^86 - 2 * q^88 + 30 * q^89 - 7 * q^91 + 14 * q^92 + 13 * q^94 + 25 * q^98

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.69527	1.77464	0.887322	−	0.461151i	$-0.152563\pi$
$7$	4.69527	1.77464	0.887322	+	0.461151i	$0.152563\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	0	0
$11$	−6.40880	−1.93233	−0.966163	−	0.257931i	$-0.916959\pi$
$11$	−6.40880	−1.93233	−0.966163	+	0.257931i	$0.916959\pi$
$12$	0	0
$13$	−1.06379	−0.295042	−0.147521	−	0.989059i	$-0.547129\pi$
$13$	−1.06379	−0.295042	−0.147521	+	0.989059i	$0.547129\pi$
$14$	4.69527	1.25486
$15$	0	0
$16$	1.00000	0.250000
$17$	1.91794	0.465169	0.232584	−	0.972576i	$-0.425282\pi$
$17$	1.91794	0.465169	0.232584	+	0.972576i	$0.425282\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	−6.40880	−1.36636
$23$	1.79560	0.374408	0.187204	−	0.982321i	$-0.440057\pi$
$23$	1.79560	0.374408	0.187204	+	0.982321i	$0.440057\pi$
$24$	0	0
$25$	0	0
$26$	−1.06379	−0.208626
$27$	0	0
$28$	4.69527	0.887322
$29$	−2.93621	−0.545241	−0.272620	−	0.962122i	$-0.587890\pi$
$29$	−2.93621	−0.545241	−0.272620	+	0.962122i	$0.587890\pi$
$30$	0	0
$31$	−5.55465	−0.997645	−0.498822	−	0.866704i	$-0.666234\pi$
$31$	−5.55465	−0.997645	−0.498822	+	0.866704i	$0.666234\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	1.91794	0.328924
$35$	0	0
$36$	0	0
$37$	11.4088	1.87560	0.937798	−	0.347182i	$-0.112861\pi$
$37$	11.4088	1.87560	0.937798	+	0.347182i	$0.112861\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	0	0
$41$	1.14585	0.178951	0.0894757	−	0.995989i	$-0.471481\pi$
$41$	1.14585	0.178951	0.0894757	+	0.995989i	$0.471481\pi$
$42$	0	0
$43$	3.55465	0.542079	0.271040	−	0.962568i	$-0.412633\pi$
$43$	3.55465	0.542079	0.271040	+	0.962568i	$0.412633\pi$
$44$	−6.40880	−0.966163
$45$	0	0
$46$	1.79560	0.264747
$47$	10.8359	1.58058	0.790288	−	0.612736i	$-0.209931\pi$
$47$	10.8359	1.58058	0.790288	+	0.612736i	$0.209931\pi$
$48$	0	0
$49$	15.0455	2.14936
$50$	0	0
$51$	0	0
$52$	−1.06379	−0.147521
$53$	−8.69527	−1.19439	−0.597193	−	0.802097i	$-0.703717\pi$
$53$	−8.69527	−1.19439	−0.597193	+	0.802097i	$0.703717\pi$
$54$	0	0
$55$	0	0
$56$	4.69527	0.627431
$57$	0	0
$58$	−2.93621	−0.385544
$59$	5.63148	0.733156	0.366578	−	0.930387i	$-0.380529\pi$
$59$	5.63148	0.733156	0.366578	+	0.930387i	$0.380529\pi$
$60$	0	0
$61$	−3.39053	−0.434113	−0.217056	−	0.976159i	$-0.569646\pi$
$61$	−3.39053	−0.434113	−0.217056	+	0.976159i	$0.569646\pi$
$62$	−5.55465	−0.705441
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	8.82284	1.07788	0.538941	−	0.842344i	$-0.318825\pi$
$67$	8.82284	1.07788	0.538941	+	0.842344i	$0.318825\pi$
$68$	1.91794	0.232584
$69$	0	0
$70$	0	0
$71$	1.42708	0.169363	0.0846814	−	0.996408i	$-0.473013\pi$
$71$	1.42708	0.169363	0.0846814	+	0.996408i	$0.473013\pi$
$72$	0	0
$73$	12.6132	1.47626	0.738132	−	0.674656i	$-0.235708\pi$
$73$	12.6132	1.47626	0.738132	+	0.674656i	$0.235708\pi$
$74$	11.4088	1.32625
$75$	0	0
$76$	−1.00000	−0.114708
$77$	−30.0910	−3.42919
$78$	0	0
$79$	−1.96345	−0.220906	−0.110453	−	0.993881i	$-0.535230\pi$
$79$	−1.96345	−0.220906	−0.110453	+	0.993881i	$0.535230\pi$
$80$	0	0
$81$	0	0
$82$	1.14585	0.126538
$83$	16.2447	1.78309	0.891543	−	0.452937i	$-0.149624\pi$
$83$	16.2447	1.78309	0.891543	+	0.452937i	$0.149624\pi$
$84$	0	0
$85$	0	0
$86$	3.55465	0.383308
$87$	0	0
$88$	−6.40880	−0.683181
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	−4.99477	−0.523594
$92$	1.79560	0.187204
$93$	0	0
$94$	10.8359	1.11764
$95$	0	0
$96$	0	0
$97$	14.9452	1.51745	0.758727	−	0.651409i	$-0.225822\pi$
$97$	14.9452	1.51745	0.758727	+	0.651409i	$0.225822\pi$
$98$	15.0455	1.51983
$99$	0	0
$100$	0	0
$101$	−10.5364	−1.04841	−0.524204	−	0.851592i	$-0.675637\pi$
$101$	−10.5364	−1.04841	−0.524204	+	0.851592i	$0.675637\pi$
$102$	0	0
$103$	16.9817	1.67326	0.836630	−	0.547769i	$-0.184523\pi$
$103$	16.9817	1.67326	0.836630	+	0.547769i	$0.184523\pi$
$104$	−1.06379	−0.104313
$105$	0	0
$106$	−8.69527	−0.844559
$107$	−1.79036	−0.173081	−0.0865405	−	0.996248i	$-0.527581\pi$
$107$	−1.79036	−0.173081	−0.0865405	+	0.996248i	$0.527581\pi$
$108$	0	0
$109$	2.41404	0.231223	0.115611	−	0.993295i	$-0.463117\pi$
$109$	2.41404	0.231223	0.115611	+	0.993295i	$0.463117\pi$
$110$	0	0
$111$	0	0
$112$	4.69527	0.443661
$113$	7.14585	0.672225	0.336112	−	0.941822i	$-0.390888\pi$
$113$	7.14585	0.672225	0.336112	+	0.941822i	$0.390888\pi$
$114$	0	0
$115$	0	0
$116$	−2.93621	−0.272620
$117$	0	0
$118$	5.63148	0.518420
$119$	9.00523	0.825508
$120$	0	0
$121$	30.0728	2.73389
$122$	−3.39053	−0.306964
$123$	0	0
$124$	−5.55465	−0.498822
$125$	0	0
$126$	0	0
$127$	−9.26295	−0.821954	−0.410977	−	0.911646i	$-0.634812\pi$
$127$	−9.26295	−0.821954	−0.410977	+	0.911646i	$0.634812\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0	0
$131$	−11.5181	−1.00634	−0.503171	−	0.864187i	$-0.667833\pi$
$131$	−11.5181	−1.00634	−0.503171	+	0.864187i	$0.667833\pi$
$132$	0	0
$133$	−4.69527	−0.407131
$134$	8.82284	0.762177
$135$	0	0
$136$	1.91794	0.164462
$137$	15.1041	1.29043	0.645214	−	0.764002i	$-0.276768\pi$
$137$	15.1041	1.29043	0.645214	+	0.764002i	$0.276768\pi$
$138$	0	0
$139$	−0.700500	−0.0594156	−0.0297078	−	0.999559i	$-0.509458\pi$
$139$	−0.700500	−0.0594156	−0.0297078	+	0.999559i	$0.509458\pi$
$140$	0	0
$141$	0	0
$142$	1.42708	0.119758
$143$	6.81761	0.570117
$144$	0	0
$145$	0	0
$146$	12.6132	1.04388
$147$	0	0
$148$	11.4088	0.937798
$149$	−12.9452	−1.06051	−0.530255	−	0.847838i	$-0.677904\pi$
$149$	−12.9452	−1.06051	−0.530255	+	0.847838i	$0.677904\pi$
$150$	0	0
$151$	5.70830	0.464535	0.232268	−	0.972652i	$-0.425385\pi$
$151$	5.70830	0.464535	0.232268	+	0.972652i	$0.425385\pi$
$152$	−1.00000	−0.0811107
$153$	0	0
$154$	−30.0910	−2.42480
$155$	0	0
$156$	0	0
$157$	−9.26295	−0.739264	−0.369632	−	0.929178i	$-0.620516\pi$
$157$	−9.26295	−0.739264	−0.369632	+	0.929178i	$0.620516\pi$
$158$	−1.96345	−0.156204
$159$	0	0
$160$	0	0
$161$	8.43081	0.664441
$162$	0	0
$163$	−4.28123	−0.335332	−0.167666	−	0.985844i	$-0.553623\pi$
$163$	−4.28123	−0.335332	−0.167666	+	0.985844i	$0.553623\pi$
$164$	1.14585	0.0894757
$165$	0	0
$166$	16.2447	1.26083
$167$	−15.2264	−1.17825	−0.589127	−	0.808040i	$-0.700528\pi$
$167$	−15.2264	−1.17825	−0.589127	+	0.808040i	$0.700528\pi$
$168$	0	0
$169$	−11.8684	−0.912950
$170$	0	0
$171$	0	0
$172$	3.55465	0.271040
$173$	−12.2630	−0.932335	−0.466168	−	0.884696i	$-0.654366\pi$
$173$	−12.2630	−0.932335	−0.466168	+	0.884696i	$0.654366\pi$
$174$	0	0
$175$	0	0
$176$	−6.40880	−0.483082
$177$	0	0
$178$	10.0000	0.749532
$179$	1.57292	0.117566	0.0587829	−	0.998271i	$-0.481278\pi$
$179$	1.57292	0.117566	0.0587829	+	0.998271i	$0.481278\pi$
$180$	0	0
$181$	11.4088	0.848010	0.424005	−	0.905660i	$-0.360624\pi$
$181$	11.4088	0.848010	0.424005	+	0.905660i	$0.360624\pi$
$182$	−4.99477	−0.370237
$183$	0	0
$184$	1.79560	0.132373
$185$	0	0
$186$	0	0
$187$	−12.2917	−0.898858
$188$	10.8359	0.790288
$189$	0	0
$190$	0	0
$191$	6.35805	0.460053	0.230026	−	0.973184i	$-0.426119\pi$
$191$	6.35805	0.460053	0.230026	+	0.973184i	$0.426119\pi$
$192$	0	0
$193$	−14.4998	−1.04372	−0.521860	−	0.853031i	$-0.674762\pi$
$193$	−14.4998	−1.04372	−0.521860	+	0.853031i	$0.674762\pi$
$194$	14.9452	1.07300
$195$	0	0
$196$	15.0455	1.07468
$197$	−5.14585	−0.366627	−0.183313	−	0.983055i	$-0.558682\pi$
$197$	−5.14585	−0.366627	−0.183313	+	0.983055i	$0.558682\pi$
$198$	0	0
$199$	3.87766	0.274880	0.137440	−	0.990510i	$-0.456113\pi$
$199$	3.87766	0.274880	0.137440	+	0.990510i	$0.456113\pi$
$200$	0	0
$201$	0	0
$202$	−10.5364	−0.741337
$203$	−13.7863	−0.967608
$204$	0	0
$205$	0	0
$206$	16.9817	1.18317
$207$	0	0
$208$	−1.06379	−0.0737604
$209$	6.40880	0.443306
$210$	0	0
$211$	17.1496	1.18063	0.590313	−	0.807174i	$-0.299004\pi$
$211$	17.1496	1.18063	0.590313	+	0.807174i	$0.299004\pi$
$212$	−8.69527	−0.597193
$213$	0	0
$214$	−1.79036	−0.122387
$215$	0	0
$216$	0	0
$217$	−26.0806	−1.77046
$218$	2.41404	0.163499
$219$	0	0
$220$	0	0
$221$	−2.04028	−0.137244
$222$	0	0
$223$	−7.84635	−0.525430	−0.262715	−	0.964873i	$-0.584618\pi$
$223$	−7.84635	−0.525430	−0.262715	+	0.964873i	$0.584618\pi$
$224$	4.69527	0.313716
$225$	0	0
$226$	7.14585	0.475335
$227$	6.28646	0.417247	0.208624	−	0.977996i	$-0.433102\pi$
$227$	6.28646	0.417247	0.208624	+	0.977996i	$0.433102\pi$
$228$	0	0
$229$	−3.14585	−0.207884	−0.103942	−	0.994583i	$-0.533146\pi$
$229$	−3.14585	−0.207884	−0.103942	+	0.994583i	$0.533146\pi$
$230$	0	0
$231$	0	0
$232$	−2.93621	−0.192772
$233$	0.182394	0.0119490	0.00597451	−	0.999982i	$-0.498098\pi$
$233$	0.182394	0.0119490	0.00597451	+	0.999982i	$0.498098\pi$
$234$	0	0
$235$	0	0
$236$	5.63148	0.366578
$237$	0	0
$238$	9.00523	0.583723
$239$	11.5039	0.744126	0.372063	−	0.928208i	$-0.378651\pi$
$239$	11.5039	0.744126	0.372063	+	0.928208i	$0.378651\pi$
$240$	0	0
$241$	−0.445349	−0.0286874	−0.0143437	−	0.999897i	$-0.504566\pi$
$241$	−0.445349	−0.0286874	−0.0143437	+	0.999897i	$0.504566\pi$
$242$	30.0728	1.93315
$243$	0	0
$244$	−3.39053	−0.217056
$245$	0	0
$246$	0	0
$247$	1.06379	0.0676872
$248$	−5.55465	−0.352721
$249$	0	0
$250$	0	0
$251$	−13.2630	−0.837150	−0.418575	−	0.908182i	$-0.637470\pi$
$251$	−13.2630	−0.837150	−0.418575	+	0.908182i	$0.637470\pi$
$252$	0	0
$253$	−11.5076	−0.723479
$254$	−9.26295	−0.581209
$255$	0	0
$256$	1.00000	0.0625000
$257$	−23.4816	−1.46474	−0.732370	−	0.680907i	$-0.761586\pi$
$257$	−23.4816	−1.46474	−0.732370	+	0.680907i	$0.761586\pi$
$258$	0	0
$259$	53.5674	3.32851
$260$	0	0
$261$	0	0
$262$	−11.5181	−0.711591
$263$	27.9269	1.72205	0.861023	−	0.508565i	$-0.169824\pi$
$263$	27.9269	1.72205	0.861023	+	0.508565i	$0.169824\pi$
$264$	0	0
$265$	0	0
$266$	−4.69527	−0.287885
$267$	0	0
$268$	8.82284	0.538941
$269$	−16.4816	−1.00490	−0.502449	−	0.864607i	$-0.667568\pi$
$269$	−16.4816	−1.00490	−0.502449	+	0.864607i	$0.667568\pi$
$270$	0	0
$271$	−1.46736	−0.0891356	−0.0445678	−	0.999006i	$-0.514191\pi$
$271$	−1.46736	−0.0891356	−0.0445678	+	0.999006i	$0.514191\pi$
$272$	1.91794	0.116292
$273$	0	0
$274$	15.1041	0.912470
$275$	0	0
$276$	0	0
$277$	−15.8359	−0.951486	−0.475743	−	0.879584i	$-0.657821\pi$
$277$	−15.8359	−0.951486	−0.475743	+	0.879584i	$0.657821\pi$
$278$	−0.700500	−0.0420132
$279$	0	0
$280$	0	0
$281$	6.25515	0.373151	0.186576	−	0.982441i	$-0.440261\pi$
$281$	6.25515	0.373151	0.186576	+	0.982441i	$0.440261\pi$
$282$	0	0
$283$	16.7005	0.992742	0.496371	−	0.868111i	$-0.334666\pi$
$283$	16.7005	0.992742	0.496371	+	0.868111i	$0.334666\pi$
$284$	1.42708	0.0846814
$285$	0	0
$286$	6.81761	0.403134
$287$	5.38006	0.317575
$288$	0	0
$289$	−13.3215	−0.783618
$290$	0	0
$291$	0	0
$292$	12.6132	0.738132
$293$	0.644516	0.0376530	0.0188265	−	0.999823i	$-0.494007\pi$
$293$	0.644516	0.0376530	0.0188265	+	0.999823i	$0.494007\pi$
$294$	0	0
$295$	0	0
$296$	11.4088	0.663123
$297$	0	0
$298$	−12.9452	−0.749894
$299$	−1.91014	−0.110466
$300$	0	0
$301$	16.6900	0.961997
$302$	5.70830	0.328476
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	0	0
$307$	−26.6457	−1.52075	−0.760375	−	0.649485i	$-0.774985\pi$
$307$	−26.6457	−1.52075	−0.760375	+	0.649485i	$0.774985\pi$
$308$	−30.0910	−1.71460
$309$	0	0
$310$	0	0
$311$	−11.5494	−0.654907	−0.327454	−	0.944867i	$-0.606191\pi$
$311$	−11.5494	−0.654907	−0.327454	+	0.944867i	$0.606191\pi$
$312$	0	0
$313$	23.0638	1.30364	0.651821	−	0.758373i	$-0.274005\pi$
$313$	23.0638	1.30364	0.651821	+	0.758373i	$0.274005\pi$
$314$	−9.26295	−0.522739
$315$	0	0
$316$	−1.96345	−0.110453
$317$	−13.9232	−0.782003	−0.391002	−	0.920390i	$-0.627871\pi$
$317$	−13.9232	−0.782003	−0.391002	+	0.920390i	$0.627871\pi$
$318$	0	0
$319$	18.8176	1.05358
$320$	0	0
$321$	0	0
$322$	8.43081	0.469831
$323$	−1.91794	−0.106717
$324$	0	0
$325$	0	0
$326$	−4.28123	−0.237115
$327$	0	0
$328$	1.14585	0.0632689
$329$	50.8773	2.80496
$330$	0	0
$331$	−0.735546	−0.0404292	−0.0202146	−	0.999796i	$-0.506435\pi$
$331$	−0.735546	−0.0404292	−0.0202146	+	0.999796i	$0.506435\pi$
$332$	16.2447	0.891543
$333$	0	0
$334$	−15.2264	−0.833152
$335$	0	0
$336$	0	0
$337$	2.28123	0.124266	0.0621332	−	0.998068i	$-0.480210\pi$
$337$	2.28123	0.124266	0.0621332	+	0.998068i	$0.480210\pi$
$338$	−11.8684	−0.645553
$339$	0	0
$340$	0	0
$341$	35.5987	1.92778
$342$	0	0
$343$	37.7758	2.03970
$344$	3.55465	0.191654
$345$	0	0
$346$	−12.2630	−0.659261
$347$	−2.56246	−0.137560	−0.0687799	−	0.997632i	$-0.521911\pi$
$347$	−2.56246	−0.137560	−0.0687799	+	0.997632i	$0.521911\pi$
$348$	0	0
$349$	−5.67176	−0.303602	−0.151801	−	0.988411i	$-0.548507\pi$
$349$	−5.67176	−0.303602	−0.151801	+	0.988411i	$0.548507\pi$
$350$	0	0
$351$	0	0
$352$	−6.40880	−0.341590
$353$	−23.6665	−1.25964	−0.629821	−	0.776740i	$-0.716872\pi$
$353$	−23.6665	−1.25964	−0.629821	+	0.776740i	$0.716872\pi$
$354$	0	0
$355$	0	0
$356$	10.0000	0.529999
$357$	0	0
$358$	1.57292	0.0831316
$359$	−31.9724	−1.68744	−0.843720	−	0.536784i	$-0.819639\pi$
$359$	−31.9724	−1.68744	−0.843720	+	0.536784i	$0.819639\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	11.4088	0.599633
$363$	0	0
$364$	−4.99477	−0.261797
$365$	0	0
$366$	0	0
$367$	−1.14585	−0.0598128	−0.0299064	−	0.999553i	$-0.509521\pi$
$367$	−1.14585	−0.0598128	−0.0299064	+	0.999553i	$0.509521\pi$
$368$	1.79560	0.0936020
$369$	0	0
$370$	0	0
$371$	−40.8266	−2.11961
$372$	0	0
$373$	−32.8579	−1.70132	−0.850658	−	0.525719i	$-0.823796\pi$
$373$	−32.8579	−1.70132	−0.850658	+	0.525719i	$0.823796\pi$
$374$	−12.2917	−0.635588
$375$	0	0
$376$	10.8359	0.558818
$377$	3.12351	0.160869
$378$	0	0
$379$	−18.2865	−0.939312	−0.469656	−	0.882849i	$-0.655622\pi$
$379$	−18.2865	−0.939312	−0.469656	+	0.882849i	$0.655622\pi$
$380$	0	0
$381$	0	0
$382$	6.35805	0.325306
$383$	20.8542	1.06560	0.532799	−	0.846242i	$-0.321140\pi$
$383$	20.8542	1.06560	0.532799	+	0.846242i	$0.321140\pi$
$384$	0	0
$385$	0	0
$386$	−14.4998	−0.738022
$387$	0	0
$388$	14.9452	0.758727
$389$	24.9086	1.26292	0.631459	−	0.775409i	$-0.282456\pi$
$389$	24.9086	1.26292	0.631459	+	0.775409i	$0.282456\pi$
$390$	0	0
$391$	3.44385	0.174163
$392$	15.0455	0.759913
$393$	0	0
$394$	−5.14585	−0.259244
$395$	0	0
$396$	0	0
$397$	24.7445	1.24189	0.620946	−	0.783853i	$-0.286749\pi$
$397$	24.7445	1.24189	0.620946	+	0.783853i	$0.286749\pi$
$398$	3.87766	0.194369
$399$	0	0
$400$	0	0
$401$	−15.6457	−0.781308	−0.390654	−	0.920538i	$-0.627751\pi$
$401$	−15.6457	−0.781308	−0.390654	+	0.920538i	$0.627751\pi$
$402$	0	0
$403$	5.90897	0.294347
$404$	−10.5364	−0.524204
$405$	0	0
$406$	−13.7863	−0.684202
$407$	−73.1168	−3.62426
$408$	0	0
$409$	30.5804	1.51210	0.756052	−	0.654512i	$-0.227126\pi$
$409$	30.5804	1.51210	0.756052	+	0.654512i	$0.227126\pi$
$410$	0	0
$411$	0	0
$412$	16.9817	0.836630
$413$	26.4413	1.30109
$414$	0	0
$415$	0	0
$416$	−1.06379	−0.0521565
$417$	0	0
$418$	6.40880	0.313465
$419$	−1.96345	−0.0959210	−0.0479605	−	0.998849i	$-0.515272\pi$
$419$	−1.96345	−0.0959210	−0.0479605	+	0.998849i	$0.515272\pi$
$420$	0	0
$421$	−25.5949	−1.24742	−0.623710	−	0.781656i	$-0.714376\pi$
$421$	−25.5949	−1.24742	−0.623710	+	0.781656i	$0.714376\pi$
$422$	17.1496	0.834829
$423$	0	0
$424$	−8.69527	−0.422279
$425$	0	0
$426$	0	0
$427$	−15.9194	−0.770396
$428$	−1.79036	−0.0865405
$429$	0	0
$430$	0	0
$431$	15.3540	0.739575	0.369788	−	0.929116i	$-0.379430\pi$
$431$	15.3540	0.739575	0.369788	+	0.929116i	$0.379430\pi$
$432$	0	0
$433$	−16.6640	−0.800819	−0.400409	−	0.916336i	$-0.631132\pi$
$433$	−16.6640	−0.800819	−0.400409	+	0.916336i	$0.631132\pi$
$434$	−26.0806	−1.25191
$435$	0	0
$436$	2.41404	0.115611
$437$	−1.79560	−0.0858951
$438$	0	0
$439$	−19.5987	−0.935393	−0.467697	−	0.883889i	$-0.654916\pi$
$439$	−19.5987	−0.935393	−0.467697	+	0.883889i	$0.654916\pi$
$440$	0	0
$441$	0	0
$442$	−2.04028	−0.0970462
$443$	13.0183	0.618517	0.309258	−	0.950978i	$-0.399919\pi$
$443$	13.0183	0.618517	0.309258	+	0.950978i	$0.399919\pi$
$444$	0	0
$445$	0	0
$446$	−7.84635	−0.371535
$447$	0	0
$448$	4.69527	0.221830
$449$	−6.77359	−0.319666	−0.159833	−	0.987144i	$-0.551095\pi$
$449$	−6.77359	−0.319666	−0.159833	+	0.987144i	$0.551095\pi$
$450$	0	0
$451$	−7.34352	−0.345793
$452$	7.14585	0.336112
$453$	0	0
$454$	6.28646	0.295038
$455$	0	0
$456$	0	0
$457$	27.3189	1.27793	0.638963	−	0.769237i	$-0.279364\pi$
$457$	27.3189	1.27793	0.638963	+	0.769237i	$0.279364\pi$
$458$	−3.14585	−0.146996
$459$	0	0
$460$	0	0
$461$	27.7993	1.29474	0.647372	−	0.762174i	$-0.275868\pi$
$461$	27.7993	1.29474	0.647372	+	0.762174i	$0.275868\pi$
$462$	0	0
$463$	−38.2369	−1.77702	−0.888509	−	0.458859i	$-0.848258\pi$
$463$	−38.2369	−1.77702	−0.888509	+	0.458859i	$0.848258\pi$
$464$	−2.93621	−0.136310
$465$	0	0
$466$	0.182394	0.00844923
$467$	23.4711	1.08611	0.543056	−	0.839696i	$-0.317267\pi$
$467$	23.4711	1.08611	0.543056	+	0.839696i	$0.317267\pi$
$468$	0	0
$469$	41.4256	1.91286
$470$	0	0
$471$	0	0
$472$	5.63148	0.259210
$473$	−22.7811	−1.04747
$474$	0	0
$475$	0	0
$476$	9.00523	0.412754
$477$	0	0
$478$	11.5039	0.526176
$479$	19.6900	0.899660	0.449830	−	0.893114i	$-0.351484\pi$
$479$	19.6900	0.899660	0.449830	+	0.893114i	$0.351484\pi$
$480$	0	0
$481$	−12.1365	−0.553379
$482$	−0.445349	−0.0202851
$483$	0	0
$484$	30.0728	1.36694
$485$	0	0
$486$	0	0
$487$	−16.3357	−0.740242	−0.370121	−	0.928984i	$-0.620684\pi$
$487$	−16.3357	−0.740242	−0.370121	+	0.928984i	$0.620684\pi$
$488$	−3.39053	−0.153482
$489$	0	0
$490$	0	0
$491$	−27.1093	−1.22343	−0.611713	−	0.791080i	$-0.709519\pi$
$491$	−27.1093	−1.22343	−0.611713	+	0.791080i	$0.709519\pi$
$492$	0	0
$493$	−5.63148	−0.253629
$494$	1.06379	0.0478621
$495$	0	0
$496$	−5.55465	−0.249411
$497$	6.70050	0.300558
$498$	0	0
$499$	−4.69003	−0.209955	−0.104977	−	0.994475i	$-0.533477\pi$
$499$	−4.69003	−0.209955	−0.104977	+	0.994475i	$0.533477\pi$
$500$	0	0
$501$	0	0
$502$	−13.2630	−0.591955
$503$	5.19136	0.231471	0.115736	−	0.993280i	$-0.463077\pi$
$503$	5.19136	0.231471	0.115736	+	0.993280i	$0.463077\pi$
$504$	0	0
$505$	0	0
$506$	−11.5076	−0.511577
$507$	0	0
$508$	−9.26295	−0.410977
$509$	4.51811	0.200262	0.100131	−	0.994974i	$-0.468074\pi$
$509$	4.51811	0.200262	0.100131	+	0.994974i	$0.468074\pi$
$510$	0	0
$511$	59.2223	2.61984
$512$	1.00000	0.0441942
$513$	0	0
$514$	−23.4816	−1.03573
$515$	0	0
$516$	0	0
$517$	−69.4450	−3.05419
$518$	53.5674	2.35361
$519$	0	0
$520$	0	0
$521$	−14.4453	−0.632862	−0.316431	−	0.948615i	$-0.602485\pi$
$521$	−14.4453	−0.632862	−0.316431	+	0.948615i	$0.602485\pi$
$522$	0	0
$523$	18.2134	0.796415	0.398208	−	0.917295i	$-0.369632\pi$
$523$	18.2134	0.796415	0.398208	+	0.917295i	$0.369632\pi$
$524$	−11.5181	−0.503171
$525$	0	0
$526$	27.9269	1.21767
$527$	−10.6535	−0.464073
$528$	0	0
$529$	−19.7758	−0.859819
$530$	0	0
$531$	0	0
$532$	−4.69527	−0.203566
$533$	−1.21894	−0.0527981
$534$	0	0
$535$	0	0
$536$	8.82284	0.381089
$537$	0	0
$538$	−16.4816	−0.710571
$539$	−96.4237	−4.15326
$540$	0	0
$541$	37.3905	1.60754	0.803772	−	0.594937i	$-0.202823\pi$
$541$	37.3905	1.60754	0.803772	+	0.594937i	$0.202823\pi$
$542$	−1.46736	−0.0630284
$543$	0	0
$544$	1.91794	0.0822310
$545$	0	0
$546$	0	0
$547$	−29.5621	−1.26399	−0.631993	−	0.774974i	$-0.717763\pi$
$547$	−29.5621	−1.26399	−0.631993	+	0.774974i	$0.717763\pi$
$548$	15.1041	0.645214
$549$	0	0
$550$	0	0
$551$	2.93621	0.125087
$552$	0	0
$553$	−9.21894	−0.392029
$554$	−15.8359	−0.672802
$555$	0	0
$556$	−0.700500	−0.0297078
$557$	14.3723	0.608972	0.304486	−	0.952517i	$-0.401515\pi$
$557$	14.3723	0.608972	0.304486	+	0.952517i	$0.401515\pi$
$558$	0	0
$559$	−3.78139	−0.159936
$560$	0	0
$561$	0	0
$562$	6.25515	0.263858
$563$	−6.39053	−0.269329	−0.134664	−	0.990891i	$-0.542996\pi$
$563$	−6.39053	−0.269329	−0.134664	+	0.990891i	$0.542996\pi$
$564$	0	0
$565$	0	0
$566$	16.7005	0.701974
$567$	0	0
$568$	1.42708	0.0598788
$569$	44.9086	1.88267	0.941334	−	0.337476i	$-0.109573\pi$
$569$	44.9086	1.88267	0.941334	+	0.337476i	$0.109573\pi$
$570$	0	0
$571$	12.4193	0.519730	0.259865	−	0.965645i	$-0.416322\pi$
$571$	12.4193	0.519730	0.259865	+	0.965645i	$0.416322\pi$
$572$	6.81761	0.285058
$573$	0	0
$574$	5.38006	0.224559
$575$	0	0
$576$	0	0
$577$	4.20440	0.175032	0.0875158	−	0.996163i	$-0.472107\pi$
$577$	4.20440	0.175032	0.0875158	+	0.996163i	$0.472107\pi$
$578$	−13.3215	−0.554102
$579$	0	0
$580$	0	0
$581$	76.2731	3.16434
$582$	0	0
$583$	55.7262	2.30795
$584$	12.6132	0.521938
$585$	0	0
$586$	0.644516	0.0266247
$587$	24.7915	1.02326	0.511628	−	0.859207i	$-0.329043\pi$
$587$	24.7915	1.02326	0.511628	+	0.859207i	$0.329043\pi$
$588$	0	0
$589$	5.55465	0.228875
$590$	0	0
$591$	0	0
$592$	11.4088	0.468899
$593$	4.67176	0.191846	0.0959231	−	0.995389i	$-0.469420\pi$
$593$	4.67176	0.191846	0.0959231	+	0.995389i	$0.469420\pi$
$594$	0	0
$595$	0	0
$596$	−12.9452	−0.530255
$597$	0	0
$598$	−1.91014	−0.0781113
$599$	−39.2369	−1.60318	−0.801588	−	0.597877i	$-0.796011\pi$
$599$	−39.2369	−1.60318	−0.801588	+	0.597877i	$0.796011\pi$
$600$	0	0
$601$	−42.4267	−1.73062	−0.865311	−	0.501235i	$-0.832879\pi$
$601$	−42.4267	−1.73062	−0.865311	+	0.501235i	$0.832879\pi$
$602$	16.6900	0.680235
$603$	0	0
$604$	5.70830	0.232268
$605$	0	0
$606$	0	0
$607$	2.98173	0.121025	0.0605123	−	0.998167i	$-0.480727\pi$
$607$	2.98173	0.121025	0.0605123	+	0.998167i	$0.480727\pi$
$608$	−1.00000	−0.0405554
$609$	0	0
$610$	0	0
$611$	−11.5271	−0.466336
$612$	0	0
$613$	28.9452	1.16908	0.584542	−	0.811363i	$-0.301274\pi$
$613$	28.9452	1.16908	0.584542	+	0.811363i	$0.301274\pi$
$614$	−26.6457	−1.07533
$615$	0	0
$616$	−30.0910	−1.21240
$617$	−15.0365	−0.605349	−0.302674	−	0.953094i	$-0.597879\pi$
$617$	−15.0365	−0.605349	−0.302674	+	0.953094i	$0.597879\pi$
$618$	0	0
$619$	8.25515	0.331803	0.165901	−	0.986142i	$-0.446947\pi$
$619$	8.25515	0.331803	0.165901	+	0.986142i	$0.446947\pi$
$620$	0	0
$621$	0	0
$622$	−11.5494	−0.463089
$623$	46.9527	1.88112
$624$	0	0
$625$	0	0
$626$	23.0638	0.921814
$627$	0	0
$628$	−9.26295	−0.369632
$629$	21.8814	0.872468
$630$	0	0
$631$	−9.09103	−0.361908	−0.180954	−	0.983492i	$-0.557919\pi$
$631$	−9.09103	−0.361908	−0.180954	+	0.983492i	$0.557919\pi$
$632$	−1.96345	−0.0781020
$633$	0	0
$634$	−13.9232	−0.552960
$635$	0	0
$636$	0	0
$637$	−16.0052	−0.634150
$638$	18.8176	0.744996
$639$	0	0
$640$	0	0
$641$	−30.1171	−1.18955	−0.594777	−	0.803891i	$-0.702760\pi$
$641$	−30.1171	−1.18955	−0.594777	+	0.803891i	$0.702760\pi$
$642$	0	0
$643$	35.3174	1.39278	0.696392	−	0.717662i	$-0.254788\pi$
$643$	35.3174	1.39278	0.696392	+	0.717662i	$0.254788\pi$
$644$	8.43081	0.332220
$645$	0	0
$646$	−1.91794	−0.0754603
$647$	4.12234	0.162066	0.0810330	−	0.996711i	$-0.474178\pi$
$647$	4.12234	0.162066	0.0810330	+	0.996711i	$0.474178\pi$
$648$	0	0
$649$	−36.0910	−1.41670
$650$	0	0
$651$	0	0
$652$	−4.28123	−0.167666
$653$	8.62741	0.337617	0.168808	−	0.985649i	$-0.446008\pi$
$653$	8.62741	0.337617	0.168808	+	0.985649i	$0.446008\pi$
$654$	0	0
$655$	0	0
$656$	1.14585	0.0447379
$657$	0	0
$658$	50.8773	1.98340
$659$	37.3853	1.45632	0.728162	−	0.685405i	$-0.240375\pi$
$659$	37.3853	1.45632	0.728162	+	0.685405i	$0.240375\pi$
$660$	0	0
$661$	−12.8997	−0.501739	−0.250869	−	0.968021i	$-0.580716\pi$
$661$	−12.8997	−0.501739	−0.250869	+	0.968021i	$0.580716\pi$
$662$	−0.735546	−0.0285878
$663$	0	0
$664$	16.2447	0.630416
$665$	0	0
$666$	0	0
$667$	−5.27226	−0.204143
$668$	−15.2264	−0.589127
$669$	0	0
$670$	0	0
$671$	21.7292	0.838848
$672$	0	0
$673$	−10.4349	−0.402235	−0.201118	−	0.979567i	$-0.564457\pi$
$673$	−10.4349	−0.402235	−0.201118	+	0.979567i	$0.564457\pi$
$674$	2.28123	0.0878696
$675$	0	0
$676$	−11.8684	−0.456475
$677$	12.4401	0.478112	0.239056	−	0.971006i	$-0.423162\pi$
$677$	12.4401	0.478112	0.239056	+	0.971006i	$0.423162\pi$
$678$	0	0
$679$	70.1716	2.69294
$680$	0	0
$681$	0	0
$682$	35.5987	1.36314
$683$	17.5364	0.671011	0.335505	−	0.942038i	$-0.391093\pi$
$683$	17.5364	0.671011	0.335505	+	0.942038i	$0.391093\pi$
$684$	0	0
$685$	0	0
$686$	37.7758	1.44229
$687$	0	0
$688$	3.55465	0.135520
$689$	9.24992	0.352394
$690$	0	0
$691$	−25.2734	−0.961446	−0.480723	−	0.876872i	$-0.659626\pi$
$691$	−25.2734	−0.961446	−0.480723	+	0.876872i	$0.659626\pi$
$692$	−12.2630	−0.466168
$693$	0	0
$694$	−2.56246	−0.0972695
$695$	0	0
$696$	0	0
$697$	2.19767	0.0832426
$698$	−5.67176	−0.214679
$699$	0	0
$700$	0	0
$701$	16.0545	0.606370	0.303185	−	0.952932i	$-0.401950\pi$
$701$	16.0545	0.606370	0.303185	+	0.952932i	$0.401950\pi$
$702$	0	0
$703$	−11.4088	−0.430291
$704$	−6.40880	−0.241541
$705$	0	0
$706$	−23.6665	−0.890701
$707$	−49.4711	−1.86055
$708$	0	0
$709$	13.5076	0.507290	0.253645	−	0.967297i	$-0.418371\pi$
$709$	13.5076	0.507290	0.253645	+	0.967297i	$0.418371\pi$
$710$	0	0
$711$	0	0
$712$	10.0000	0.374766
$713$	−9.97392	−0.373526
$714$	0	0
$715$	0	0
$716$	1.57292	0.0587829
$717$	0	0
$718$	−31.9724	−1.19320
$719$	−0.803402	−0.0299619	−0.0149809	−	0.999888i	$-0.504769\pi$
$719$	−0.803402	−0.0299619	−0.0149809	+	0.999888i	$0.504769\pi$
$720$	0	0
$721$	79.7337	2.96944
$722$	1.00000	0.0372161
$723$	0	0
$724$	11.4088	0.424005
$725$	0	0
$726$	0	0
$727$	−51.6860	−1.91693	−0.958463	−	0.285217i	$-0.907934\pi$
$727$	−51.6860	−1.91693	−0.958463	+	0.285217i	$0.907934\pi$
$728$	−4.99477	−0.185118
$729$	0	0
$730$	0	0
$731$	6.81761	0.252158
$732$	0	0
$733$	22.3723	0.826338	0.413169	−	0.910654i	$-0.364422\pi$
$733$	22.3723	0.826338	0.413169	+	0.910654i	$0.364422\pi$
$734$	−1.14585	−0.0422940
$735$	0	0
$736$	1.79560	0.0661866
$737$	−56.5438	−2.08282
$738$	0	0
$739$	33.4271	1.22963	0.614817	−	0.788669i	$-0.289230\pi$
$739$	33.4271	1.22963	0.614817	+	0.788669i	$0.289230\pi$
$740$	0	0
$741$	0	0
$742$	−40.8266	−1.49879
$743$	14.4998	0.531947	0.265974	−	0.963980i	$-0.414307\pi$
$743$	14.4998	0.531947	0.265974	+	0.963980i	$0.414307\pi$
$744$	0	0
$745$	0	0
$746$	−32.8579	−1.20301
$747$	0	0
$748$	−12.2917	−0.449429
$749$	−8.40623	−0.307157
$750$	0	0
$751$	−26.6169	−0.971266	−0.485633	−	0.874163i	$-0.661411\pi$
$751$	−26.6169	−0.971266	−0.485633	+	0.874163i	$0.661411\pi$
$752$	10.8359	0.395144
$753$	0	0
$754$	3.12351	0.113751
$755$	0	0
$756$	0	0
$757$	31.0362	1.12803	0.564015	−	0.825764i	$-0.309256\pi$
$757$	31.0362	1.12803	0.564015	+	0.825764i	$0.309256\pi$
$758$	−18.2865	−0.664194
$759$	0	0
$760$	0	0
$761$	27.8083	1.00805	0.504025	−	0.863689i	$-0.331852\pi$
$761$	27.8083	1.00805	0.504025	+	0.863689i	$0.331852\pi$
$762$	0	0
$763$	11.3345	0.410338
$764$	6.35805	0.230026
$765$	0	0
$766$	20.8542	0.753491
$767$	−5.99070	−0.216312
$768$	0	0
$769$	−19.2279	−0.693376	−0.346688	−	0.937980i	$-0.612694\pi$
$769$	−19.2279	−0.693376	−0.346688	+	0.937980i	$0.612694\pi$
$770$	0	0
$771$	0	0
$772$	−14.4998	−0.521860
$773$	6.00373	0.215939	0.107970	−	0.994154i	$-0.465565\pi$
$773$	6.00373	0.215939	0.107970	+	0.994154i	$0.465565\pi$
$774$	0	0
$775$	0	0
$776$	14.9452	0.536501
$777$	0	0
$778$	24.9086	0.893018
$779$	−1.14585	−0.0410543
$780$	0	0
$781$	−9.14585	−0.327264
$782$	3.44385	0.123152
$783$	0	0
$784$	15.0455	0.537340
$785$	0	0
$786$	0	0
$787$	−22.2797	−0.794187	−0.397093	−	0.917778i	$-0.629981\pi$
$787$	−22.2797	−0.794187	−0.397093	+	0.917778i	$0.629981\pi$
$788$	−5.14585	−0.183313
$789$	0	0
$790$	0	0
$791$	33.5517	1.19296
$792$	0	0
$793$	3.60680	0.128081
$794$	24.7445	0.878150
$795$	0	0
$796$	3.87766	0.137440
$797$	26.6259	0.943138	0.471569	−	0.881829i	$-0.343688\pi$
$797$	26.6259	0.943138	0.471569	+	0.881829i	$0.343688\pi$
$798$	0	0
$799$	20.7826	0.735234
$800$	0	0
$801$	0	0
$802$	−15.6457	−0.552468
$803$	−80.8355	−2.85262
$804$	0	0
$805$	0	0
$806$	5.90897	0.208135
$807$	0	0
$808$	−10.5364	−0.370669
$809$	−0.651250	−0.0228967	−0.0114484	−	0.999934i	$-0.503644\pi$
$809$	−0.651250	−0.0228967	−0.0114484	+	0.999934i	$0.503644\pi$
$810$	0	0
$811$	−13.0780	−0.459230	−0.229615	−	0.973281i	$-0.573747\pi$
$811$	−13.0780	−0.459230	−0.229615	+	0.973281i	$0.573747\pi$
$812$	−13.7863	−0.483804
$813$	0	0
$814$	−73.1168	−2.56274
$815$	0	0
$816$	0	0
$817$	−3.55465	−0.124362
$818$	30.5804	1.06922
$819$	0	0
$820$	0	0
$821$	−14.6640	−0.511776	−0.255888	−	0.966706i	$-0.582368\pi$
$821$	−14.6640	−0.511776	−0.255888	+	0.966706i	$0.582368\pi$
$822$	0	0
$823$	18.2850	0.637374	0.318687	−	0.947860i	$-0.396758\pi$
$823$	18.2850	0.637374	0.318687	+	0.947860i	$0.396758\pi$
$824$	16.9817	0.591587
$825$	0	0
$826$	26.4413	0.920010
$827$	40.1910	1.39758	0.698790	−	0.715327i	$-0.253722\pi$
$827$	40.1910	1.39758	0.698790	+	0.715327i	$0.253722\pi$
$828$	0	0
$829$	28.3760	0.985539	0.492769	−	0.870160i	$-0.335985\pi$
$829$	28.3760	0.985539	0.492769	+	0.870160i	$0.335985\pi$
$830$	0	0
$831$	0	0
$832$	−1.06379	−0.0368802
$833$	28.8564	0.999815
$834$	0	0
$835$	0	0
$836$	6.40880	0.221653
$837$	0	0
$838$	−1.96345	−0.0678264
$839$	−6.93471	−0.239413	−0.119706	−	0.992809i	$-0.538195\pi$
$839$	−6.93471	−0.239413	−0.119706	+	0.992809i	$0.538195\pi$
$840$	0	0
$841$	−20.3787	−0.702712
$842$	−25.5949	−0.882060
$843$	0	0
$844$	17.1496	0.590313
$845$	0	0
$846$	0	0
$847$	141.200	4.85167
$848$	−8.69527	−0.298597
$849$	0	0
$850$	0	0
$851$	20.4856	0.702238
$852$	0	0
$853$	−21.9739	−0.752373	−0.376186	−	0.926544i	$-0.622765\pi$
$853$	−21.9739	−0.752373	−0.376186	+	0.926544i	$0.622765\pi$
$854$	−15.9194	−0.544752
$855$	0	0
$856$	−1.79036	−0.0611934
$857$	13.6091	0.464879	0.232440	−	0.972611i	$-0.425329\pi$
$857$	13.6091	0.464879	0.232440	+	0.972611i	$0.425329\pi$
$858$	0	0
$859$	5.17192	0.176464	0.0882319	−	0.996100i	$-0.471878\pi$
$859$	5.17192	0.176464	0.0882319	+	0.996100i	$0.471878\pi$
$860$	0	0
$861$	0	0
$862$	15.3540	0.522959
$863$	15.9635	0.543402	0.271701	−	0.962382i	$-0.412414\pi$
$863$	15.9635	0.543402	0.271701	+	0.962382i	$0.412414\pi$
$864$	0	0
$865$	0	0
$866$	−16.6640	−0.566264
$867$	0	0
$868$	−26.0806	−0.885232
$869$	12.5834	0.426862
$870$	0	0
$871$	−9.38563	−0.318020
$872$	2.41404	0.0817496
$873$	0	0
$874$	−1.79560	−0.0607370
$875$	0	0
$876$	0	0
$877$	15.8866	0.536453	0.268227	−	0.963356i	$-0.413562\pi$
$877$	15.8866	0.536453	0.268227	+	0.963356i	$0.413562\pi$
$878$	−19.5987	−0.661423
$879$	0	0
$880$	0	0
$881$	16.1458	0.543967	0.271984	−	0.962302i	$-0.412320\pi$
$881$	16.1458	0.543967	0.271984	+	0.962302i	$0.412320\pi$
$882$	0	0
$883$	−38.2887	−1.28852	−0.644259	−	0.764808i	$-0.722834\pi$
$883$	−38.2887	−1.28852	−0.644259	+	0.764808i	$0.722834\pi$
$884$	−2.04028	−0.0686221
$885$	0	0
$886$	13.0183	0.437357
$887$	19.2809	0.647389	0.323695	−	0.946162i	$-0.395075\pi$
$887$	19.2809	0.647389	0.323695	+	0.946162i	$0.395075\pi$
$888$	0	0
$889$	−43.4920	−1.45868
$890$	0	0
$891$	0	0
$892$	−7.84635	−0.262715
$893$	−10.8359	−0.362609
$894$	0	0
$895$	0	0
$896$	4.69527	0.156858
$897$	0	0
$898$	−6.77359	−0.226038
$899$	16.3096	0.543957
$900$	0	0
$901$	−16.6770	−0.555591
$902$	−7.34352	−0.244512
$903$	0	0
$904$	7.14585	0.237667
$905$	0	0
$906$	0	0
$907$	−43.0205	−1.42847	−0.714236	−	0.699905i	$-0.753226\pi$
$907$	−43.0205	−1.42847	−0.714236	+	0.699905i	$0.753226\pi$
$908$	6.28646	0.208624
$909$	0	0
$910$	0	0
$911$	25.7733	0.853906	0.426953	−	0.904274i	$-0.359587\pi$
$911$	25.7733	0.853906	0.426953	+	0.904274i	$0.359587\pi$
$912$	0	0
$913$	−104.109	−3.44550
$914$	27.3189	0.903630
$915$	0	0
$916$	−3.14585	−0.103942
$917$	−54.0806	−1.78590
$918$	0	0
$919$	−29.5897	−0.976074	−0.488037	−	0.872823i	$-0.662287\pi$
$919$	−29.5897	−0.976074	−0.488037	+	0.872823i	$0.662287\pi$
$920$	0	0
$921$	0	0
$922$	27.7993	0.915522
$923$	−1.51811	−0.0499691
$924$	0	0
$925$	0	0
$926$	−38.2369	−1.25654
$927$	0	0
$928$	−2.93621	−0.0963859
$929$	1.42334	0.0466983	0.0233491	−	0.999727i	$-0.492567\pi$
$929$	1.42334	0.0466983	0.0233491	+	0.999727i	$0.492567\pi$
$930$	0	0
$931$	−15.0455	−0.493097
$932$	0.182394	0.00597451
$933$	0	0
$934$	23.4711	0.767998
$935$	0	0
$936$	0	0
$937$	−28.2276	−0.922155	−0.461077	−	0.887360i	$-0.652537\pi$
$937$	−28.2276	−0.922155	−0.461077	+	0.887360i	$0.652537\pi$
$938$	41.4256	1.35259
$939$	0	0
$940$	0	0
$941$	15.8672	0.517256	0.258628	−	0.965977i	$-0.416730\pi$
$941$	15.8672	0.517256	0.258628	+	0.965977i	$0.416730\pi$
$942$	0	0
$943$	2.05748	0.0670009
$944$	5.63148	0.183289
$945$	0	0
$946$	−22.7811	−0.740676
$947$	43.6718	1.41914	0.709571	−	0.704634i	$-0.248889\pi$
$947$	43.6718	1.41914	0.709571	+	0.704634i	$0.248889\pi$
$948$	0	0
$949$	−13.4178	−0.435559
$950$	0	0
$951$	0	0
$952$	9.00523	0.291861
$953$	16.0261	0.519136	0.259568	−	0.965725i	$-0.416420\pi$
$953$	16.0261	0.519136	0.259568	+	0.965725i	$0.416420\pi$
$954$	0	0
$955$	0	0
$956$	11.5039	0.372063
$957$	0	0
$958$	19.6900	0.636156
$959$	70.9176	2.29005
$960$	0	0
$961$	−0.145848	−0.00470478
$962$	−12.1365	−0.391298
$963$	0	0
$964$	−0.445349	−0.0143437
$965$	0	0
$966$	0	0
$967$	11.5987	0.372988	0.186494	−	0.982456i	$-0.440288\pi$
$967$	11.5987	0.372988	0.186494	+	0.982456i	$0.440288\pi$
$968$	30.0728	0.966575
$969$	0	0
$970$	0	0
$971$	27.0362	0.867633	0.433817	−	0.901001i	$-0.357167\pi$
$971$	27.0362	0.867633	0.433817	+	0.901001i	$0.357167\pi$
$972$	0	0
$973$	−3.28903	−0.105442
$974$	−16.3357	−0.523430
$975$	0	0
$976$	−3.39053	−0.108528
$977$	14.1537	0.452815	0.226408	−	0.974033i	$-0.427302\pi$
$977$	14.1537	0.452815	0.226408	+	0.974033i	$0.427302\pi$
$978$	0	0
$979$	−64.0880	−2.04826
$980$	0	0
$981$	0	0
$982$	−27.1093	−0.865093
$983$	−32.8542	−1.04788	−0.523942	−	0.851754i	$-0.675539\pi$
$983$	−32.8542	−1.04788	−0.523942	+	0.851754i	$0.675539\pi$
$984$	0	0
$985$	0	0
$986$	−5.63148	−0.179343
$987$	0	0
$988$	1.06379	0.0338436
$989$	6.38273	0.202959
$990$	0	0
$991$	47.9709	1.52385	0.761923	−	0.647667i	$-0.224255\pi$
$991$	47.9709	1.52385	0.761923	+	0.647667i	$0.224255\pi$
$992$	−5.55465	−0.176360
$993$	0	0
$994$	6.70050	0.212527
$995$	0	0
$996$	0	0
$997$	−44.1716	−1.39893	−0.699464	−	0.714668i	$-0.746578\pi$
$997$	−44.1716	−1.39893	−0.699464	+	0.714668i	$0.746578\pi$
$998$	−4.69003	−0.148460
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8550.2.a.co.1.3		3
3.2	odd	2		950.2.a.k.1.1	✓	3
5.4	even	2		8550.2.a.cj.1.1		3
12.11	even	2		7600.2.a.cb.1.3		3
15.2	even	4		950.2.b.g.799.3		6
15.8	even	4		950.2.b.g.799.4		6
15.14	odd	2		950.2.a.m.1.3	yes	3
60.59	even	2		7600.2.a.bm.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.a.k.1.1	✓	3	3.2	odd	2
950.2.a.m.1.3	yes	3	15.14	odd	2
950.2.b.g.799.3		6	15.2	even	4
950.2.b.g.799.4		6	15.8	even	4
7600.2.a.bm.1.1		3	60.59	even	2
7600.2.a.cb.1.3		3	12.11	even	2
8550.2.a.cj.1.1		3	5.4	even	2
8550.2.a.co.1.3		3	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:	\( N \)	\(=\)	\( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8550.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +4.69527 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +4.69527 q^{7} +1.00000 q^{8} -6.40880 q^{11} -1.06379 q^{13} +4.69527 q^{14} +1.00000 q^{16} +1.91794 q^{17} -1.00000 q^{19} -6.40880 q^{22} +1.79560 q^{23} -1.06379 q^{26} +4.69527 q^{28} -2.93621 q^{29} -5.55465 q^{31} +1.00000 q^{32} +1.91794 q^{34} +11.4088 q^{37} -1.00000 q^{38} +1.14585 q^{41} +3.55465 q^{43} -6.40880 q^{44} +1.79560 q^{46} +10.8359 q^{47} +15.0455 q^{49} -1.06379 q^{52} -8.69527 q^{53} +4.69527 q^{56} -2.93621 q^{58} +5.63148 q^{59} -3.39053 q^{61} -5.55465 q^{62} +1.00000 q^{64} +8.82284 q^{67} +1.91794 q^{68} +1.42708 q^{71} +12.6132 q^{73} +11.4088 q^{74} -1.00000 q^{76} -30.0910 q^{77} -1.96345 q^{79} +1.14585 q^{82} +16.2447 q^{83} +3.55465 q^{86} -6.40880 q^{88} +10.0000 q^{89} -4.99477 q^{91} +1.79560 q^{92} +10.8359 q^{94} +14.9452 q^{97} +15.0455 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} - 2 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^4 - 2 * q^7 + 3 * q^8	\( 3 q + 3 q^{2} + 3 q^{4} - 2 q^{7} + 3 q^{8} - 2 q^{11} + 2 q^{13} - 2 q^{14} + 3 q^{16} - 4 q^{17} - 3 q^{19} - 2 q^{22} + 14 q^{23} + 2 q^{26} - 2 q^{28} - 14 q^{29} - 4 q^{31} + 3 q^{32} - 4 q^{34} + 17 q^{37} - 3 q^{38} + 8 q^{41} - 2 q^{43} - 2 q^{44} + 14 q^{46} + 13 q^{47} + 25 q^{49} + 2 q^{52} - 10 q^{53} - 2 q^{56} - 14 q^{58} + 6 q^{59} + 22 q^{61} - 4 q^{62} + 3 q^{64} - 4 q^{68} + 2 q^{71} + 12 q^{73} + 17 q^{74} - 3 q^{76} - 50 q^{77} + 24 q^{79} + 8 q^{82} + 12 q^{83} - 2 q^{86} - 2 q^{88} + 30 q^{89} - 7 q^{91} + 14 q^{92} + 13 q^{94} + 25 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 - 2 * q^7 + 3 * q^8 - 2 * q^11 + 2 * q^13 - 2 * q^14 + 3 * q^16 - 4 * q^17 - 3 * q^19 - 2 * q^22 + 14 * q^23 + 2 * q^26 - 2 * q^28 - 14 * q^29 - 4 * q^31 + 3 * q^32 - 4 * q^34 + 17 * q^37 - 3 * q^38 + 8 * q^41 - 2 * q^43 - 2 * q^44 + 14 * q^46 + 13 * q^47 + 25 * q^49 + 2 * q^52 - 10 * q^53 - 2 * q^56 - 14 * q^58 + 6 * q^59 + 22 * q^61 - 4 * q^62 + 3 * q^64 - 4 * q^68 + 2 * q^71 + 12 * q^73 + 17 * q^74 - 3 * q^76 - 50 * q^77 + 24 * q^79 + 8 * q^82 + 12 * q^83 - 2 * q^86 - 2 * q^88 + 30 * q^89 - 7 * q^91 + 14 * q^92 + 13 * q^94 + 25 * q^98

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