LMFDB - Embedded newform 6750.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6750,2,Mod(1,6750)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6750, 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("6750.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6750 = 2 \cdot 3^{3} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6750.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:	$53.8990213644$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	6750.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} -1.00000 q^{8} -2.23607 q^{11} -4.23607 q^{13} -1.00000 q^{14} +1.00000 q^{16} +0.381966 q^{17} -2.38197 q^{19} +2.23607 q^{22} +6.23607 q^{23} +4.23607 q^{26} +1.00000 q^{28} +3.47214 q^{29} -10.8541 q^{31} -1.00000 q^{32} -0.381966 q^{34} +6.47214 q^{37} +2.38197 q^{38} +10.0902 q^{41} +5.70820 q^{43} -2.23607 q^{44} -6.23607 q^{46} +4.38197 q^{47} -6.00000 q^{49} -4.23607 q^{52} -3.61803 q^{53} -1.00000 q^{56} -3.47214 q^{58} +6.23607 q^{59} -1.14590 q^{61} +10.8541 q^{62} +1.00000 q^{64} +3.47214 q^{67} +0.381966 q^{68} -6.76393 q^{71} -6.09017 q^{73} -6.47214 q^{74} -2.38197 q^{76} -2.23607 q^{77} -6.00000 q^{79} -10.0902 q^{82} +14.9443 q^{83} -5.70820 q^{86} +2.23607 q^{88} -4.61803 q^{89} -4.23607 q^{91} +6.23607 q^{92} -4.38197 q^{94} -13.9443 q^{97} +6.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^7 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 2 q^{8} - 4 q^{13} - 2 q^{14} + 2 q^{16} + 3 q^{17} - 7 q^{19} + 8 q^{23} + 4 q^{26} + 2 q^{28} - 2 q^{29} - 15 q^{31} - 2 q^{32} - 3 q^{34} + 4 q^{37} + 7 q^{38} + 9 q^{41} - 2 q^{43} - 8 q^{46} + 11 q^{47} - 12 q^{49} - 4 q^{52} - 5 q^{53} - 2 q^{56} + 2 q^{58} + 8 q^{59} - 9 q^{61} + 15 q^{62} + 2 q^{64} - 2 q^{67} + 3 q^{68} - 18 q^{71} - q^{73} - 4 q^{74} - 7 q^{76} - 12 q^{79} - 9 q^{82} + 12 q^{83} + 2 q^{86} - 7 q^{89} - 4 q^{91} + 8 q^{92} - 11 q^{94} - 10 q^{97} + 12 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^7 - 2 * q^8 - 4 * q^13 - 2 * q^14 + 2 * q^16 + 3 * q^17 - 7 * q^19 + 8 * q^23 + 4 * q^26 + 2 * q^28 - 2 * q^29 - 15 * q^31 - 2 * q^32 - 3 * q^34 + 4 * q^37 + 7 * q^38 + 9 * q^41 - 2 * q^43 - 8 * q^46 + 11 * q^47 - 12 * q^49 - 4 * q^52 - 5 * q^53 - 2 * q^56 + 2 * q^58 + 8 * q^59 - 9 * q^61 + 15 * q^62 + 2 * q^64 - 2 * q^67 + 3 * q^68 - 18 * q^71 - q^73 - 4 * q^74 - 7 * q^76 - 12 * q^79 - 9 * q^82 + 12 * q^83 + 2 * q^86 - 7 * q^89 - 4 * q^91 + 8 * q^92 - 11 * q^94 - 10 * q^97 + 12 * 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
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	0	0
$11$	−2.23607	−0.674200	−0.337100	−	0.941469i	$-0.609446\pi$
$11$	−2.23607	−0.674200	−0.337100	+	0.941469i	$0.609446\pi$
$12$	0	0
$13$	−4.23607	−1.17487	−0.587437	−	0.809270i	$-0.699863\pi$
$13$	−4.23607	−1.17487	−0.587437	+	0.809270i	$0.699863\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	0.381966	0.0926404	0.0463202	−	0.998927i	$-0.485251\pi$
$17$	0.381966	0.0926404	0.0463202	+	0.998927i	$0.485251\pi$
$18$	0	0
$19$	−2.38197	−0.546460	−0.273230	−	0.961949i	$-0.588092\pi$
$19$	−2.38197	−0.546460	−0.273230	+	0.961949i	$0.588092\pi$
$20$	0	0
$21$	0	0
$22$	2.23607	0.476731
$23$	6.23607	1.30031	0.650155	−	0.759802i	$-0.274704\pi$
$23$	6.23607	1.30031	0.650155	+	0.759802i	$0.274704\pi$
$24$	0	0
$25$	0	0
$26$	4.23607	0.830761
$27$	0	0
$28$	1.00000	0.188982
$29$	3.47214	0.644759	0.322380	−	0.946610i	$-0.395517\pi$
$29$	3.47214	0.644759	0.322380	+	0.946610i	$0.395517\pi$
$30$	0	0
$31$	−10.8541	−1.94945	−0.974727	−	0.223399i	$-0.928285\pi$
$31$	−10.8541	−1.94945	−0.974727	+	0.223399i	$0.928285\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−0.381966	−0.0655066
$35$	0	0
$36$	0	0
$37$	6.47214	1.06401	0.532006	−	0.846740i	$-0.321438\pi$
$37$	6.47214	1.06401	0.532006	+	0.846740i	$0.321438\pi$
$38$	2.38197	0.386406
$39$	0	0
$40$	0	0
$41$	10.0902	1.57582	0.787910	−	0.615791i	$-0.211163\pi$
$41$	10.0902	1.57582	0.787910	+	0.615791i	$0.211163\pi$
$42$	0	0
$43$	5.70820	0.870493	0.435246	−	0.900311i	$-0.356661\pi$
$43$	5.70820	0.870493	0.435246	+	0.900311i	$0.356661\pi$
$44$	−2.23607	−0.337100
$45$	0	0
$46$	−6.23607	−0.919458
$47$	4.38197	0.639175	0.319588	−	0.947557i	$-0.396456\pi$
$47$	4.38197	0.639175	0.319588	+	0.947557i	$0.396456\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	0	0
$52$	−4.23607	−0.587437
$53$	−3.61803	−0.496975	−0.248488	−	0.968635i	$-0.579934\pi$
$53$	−3.61803	−0.496975	−0.248488	+	0.968635i	$0.579934\pi$
$54$	0	0
$55$	0	0
$56$	−1.00000	−0.133631
$57$	0	0
$58$	−3.47214	−0.455914
$59$	6.23607	0.811867	0.405933	−	0.913903i	$-0.366946\pi$
$59$	6.23607	0.811867	0.405933	+	0.913903i	$0.366946\pi$
$60$	0	0
$61$	−1.14590	−0.146717	−0.0733586	−	0.997306i	$-0.523372\pi$
$61$	−1.14590	−0.146717	−0.0733586	+	0.997306i	$0.523372\pi$
$62$	10.8541	1.37847
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	3.47214	0.424189	0.212094	−	0.977249i	$-0.431972\pi$
$67$	3.47214	0.424189	0.212094	+	0.977249i	$0.431972\pi$
$68$	0.381966	0.0463202
$69$	0	0
$70$	0	0
$71$	−6.76393	−0.802731	−0.401366	−	0.915918i	$-0.631464\pi$
$71$	−6.76393	−0.802731	−0.401366	+	0.915918i	$0.631464\pi$
$72$	0	0
$73$	−6.09017	−0.712800	−0.356400	−	0.934333i	$-0.615996\pi$
$73$	−6.09017	−0.712800	−0.356400	+	0.934333i	$0.615996\pi$
$74$	−6.47214	−0.752371
$75$	0	0
$76$	−2.38197	−0.273230
$77$	−2.23607	−0.254824
$78$	0	0
$79$	−6.00000	−0.675053	−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000	−0.675053	−0.337526	+	0.941316i	$0.609590\pi$
$80$	0	0
$81$	0	0
$82$	−10.0902	−1.11427
$83$	14.9443	1.64035	0.820173	−	0.572115i	$-0.193877\pi$
$83$	14.9443	1.64035	0.820173	+	0.572115i	$0.193877\pi$
$84$	0	0
$85$	0	0
$86$	−5.70820	−0.615531
$87$	0	0
$88$	2.23607	0.238366
$89$	−4.61803	−0.489511	−0.244755	−	0.969585i	$-0.578708\pi$
$89$	−4.61803	−0.489511	−0.244755	+	0.969585i	$0.578708\pi$
$90$	0	0
$91$	−4.23607	−0.444061
$92$	6.23607	0.650155
$93$	0	0
$94$	−4.38197	−0.451965
$95$	0	0
$96$	0	0
$97$	−13.9443	−1.41583	−0.707913	−	0.706299i	$-0.750363\pi$
$97$	−13.9443	−1.41583	−0.707913	+	0.706299i	$0.750363\pi$
$98$	6.00000	0.606092
$99$	0	0
$100$	0	0
$101$	−7.32624	−0.728988	−0.364494	−	0.931206i	$-0.618758\pi$
$101$	−7.32624	−0.728988	−0.364494	+	0.931206i	$0.618758\pi$
$102$	0	0
$103$	5.00000	0.492665	0.246332	−	0.969185i	$-0.420775\pi$
$103$	5.00000	0.492665	0.246332	+	0.969185i	$0.420775\pi$
$104$	4.23607	0.415381
$105$	0	0
$106$	3.61803	0.351415
$107$	−18.8885	−1.82602	−0.913012	−	0.407932i	$-0.866250\pi$
$107$	−18.8885	−1.82602	−0.913012	+	0.407932i	$0.866250\pi$
$108$	0	0
$109$	5.09017	0.487550	0.243775	−	0.969832i	$-0.421614\pi$
$109$	5.09017	0.487550	0.243775	+	0.969832i	$0.421614\pi$
$110$	0	0
$111$	0	0
$112$	1.00000	0.0944911
$113$	5.56231	0.523258	0.261629	−	0.965169i	$-0.415740\pi$
$113$	5.56231	0.523258	0.261629	+	0.965169i	$0.415740\pi$
$114$	0	0
$115$	0	0
$116$	3.47214	0.322380
$117$	0	0
$118$	−6.23607	−0.574077
$119$	0.381966	0.0350148
$120$	0	0
$121$	−6.00000	−0.545455
$122$	1.14590	0.103745
$123$	0	0
$124$	−10.8541	−0.974727
$125$	0	0
$126$	0	0
$127$	−3.14590	−0.279153	−0.139577	−	0.990211i	$-0.544574\pi$
$127$	−3.14590	−0.279153	−0.139577	+	0.990211i	$0.544574\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	−10.9443	−0.956205	−0.478103	−	0.878304i	$-0.658675\pi$
$131$	−10.9443	−0.956205	−0.478103	+	0.878304i	$0.658675\pi$
$132$	0	0
$133$	−2.38197	−0.206543
$134$	−3.47214	−0.299947
$135$	0	0
$136$	−0.381966	−0.0327533
$137$	−15.9443	−1.36221	−0.681106	−	0.732185i	$-0.738501\pi$
$137$	−15.9443	−1.36221	−0.681106	+	0.732185i	$0.738501\pi$
$138$	0	0
$139$	3.70820	0.314526	0.157263	−	0.987557i	$-0.449733\pi$
$139$	3.70820	0.314526	0.157263	+	0.987557i	$0.449733\pi$
$140$	0	0
$141$	0	0
$142$	6.76393	0.567617
$143$	9.47214	0.792100
$144$	0	0
$145$	0	0
$146$	6.09017	0.504026
$147$	0	0
$148$	6.47214	0.532006
$149$	−5.94427	−0.486974	−0.243487	−	0.969904i	$-0.578291\pi$
$149$	−5.94427	−0.486974	−0.243487	+	0.969904i	$0.578291\pi$
$150$	0	0
$151$	−20.4164	−1.66146	−0.830732	−	0.556673i	$-0.812078\pi$
$151$	−20.4164	−1.66146	−0.830732	+	0.556673i	$0.812078\pi$
$152$	2.38197	0.193203
$153$	0	0
$154$	2.23607	0.180187
$155$	0	0
$156$	0	0
$157$	−0.944272	−0.0753611	−0.0376806	−	0.999290i	$-0.511997\pi$
$157$	−0.944272	−0.0753611	−0.0376806	+	0.999290i	$0.511997\pi$
$158$	6.00000	0.477334
$159$	0	0
$160$	0	0
$161$	6.23607	0.491471
$162$	0	0
$163$	−5.14590	−0.403058	−0.201529	−	0.979483i	$-0.564591\pi$
$163$	−5.14590	−0.403058	−0.201529	+	0.979483i	$0.564591\pi$
$164$	10.0902	0.787910
$165$	0	0
$166$	−14.9443	−1.15990
$167$	5.23607	0.405179	0.202590	−	0.979264i	$-0.435064\pi$
$167$	5.23607	0.405179	0.202590	+	0.979264i	$0.435064\pi$
$168$	0	0
$169$	4.94427	0.380329
$170$	0	0
$171$	0	0
$172$	5.70820	0.435246
$173$	−5.38197	−0.409183	−0.204592	−	0.978847i	$-0.565587\pi$
$173$	−5.38197	−0.409183	−0.204592	+	0.978847i	$0.565587\pi$
$174$	0	0
$175$	0	0
$176$	−2.23607	−0.168550
$177$	0	0
$178$	4.61803	0.346136
$179$	−21.1803	−1.58309	−0.791546	−	0.611109i	$-0.790724\pi$
$179$	−21.1803	−1.58309	−0.791546	+	0.611109i	$0.790724\pi$
$180$	0	0
$181$	−20.4164	−1.51754	−0.758770	−	0.651359i	$-0.774199\pi$
$181$	−20.4164	−1.51754	−0.758770	+	0.651359i	$0.774199\pi$
$182$	4.23607	0.313998
$183$	0	0
$184$	−6.23607	−0.459729
$185$	0	0
$186$	0	0
$187$	−0.854102	−0.0624581
$188$	4.38197	0.319588
$189$	0	0
$190$	0	0
$191$	−1.76393	−0.127634	−0.0638168	−	0.997962i	$-0.520327\pi$
$191$	−1.76393	−0.127634	−0.0638168	+	0.997962i	$0.520327\pi$
$192$	0	0
$193$	11.8541	0.853277	0.426638	−	0.904422i	$-0.359698\pi$
$193$	11.8541	0.853277	0.426638	+	0.904422i	$0.359698\pi$
$194$	13.9443	1.00114
$195$	0	0
$196$	−6.00000	−0.428571
$197$	−7.14590	−0.509124	−0.254562	−	0.967056i	$-0.581931\pi$
$197$	−7.14590	−0.509124	−0.254562	+	0.967056i	$0.581931\pi$
$198$	0	0
$199$	7.27051	0.515393	0.257696	−	0.966226i	$-0.417037\pi$
$199$	7.27051	0.515393	0.257696	+	0.966226i	$0.417037\pi$
$200$	0	0
$201$	0	0
$202$	7.32624	0.515472
$203$	3.47214	0.243696
$204$	0	0
$205$	0	0
$206$	−5.00000	−0.348367
$207$	0	0
$208$	−4.23607	−0.293718
$209$	5.32624	0.368424
$210$	0	0
$211$	−13.4164	−0.923624	−0.461812	−	0.886978i	$-0.652800\pi$
$211$	−13.4164	−0.923624	−0.461812	+	0.886978i	$0.652800\pi$
$212$	−3.61803	−0.248488
$213$	0	0
$214$	18.8885	1.29119
$215$	0	0
$216$	0	0
$217$	−10.8541	−0.736824
$218$	−5.09017	−0.344750
$219$	0	0
$220$	0	0
$221$	−1.61803	−0.108841
$222$	0	0
$223$	21.6180	1.44765	0.723825	−	0.689983i	$-0.242382\pi$
$223$	21.6180	1.44765	0.723825	+	0.689983i	$0.242382\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	−5.56231	−0.369999
$227$	−1.09017	−0.0723571	−0.0361786	−	0.999345i	$-0.511519\pi$
$227$	−1.09017	−0.0723571	−0.0361786	+	0.999345i	$0.511519\pi$
$228$	0	0
$229$	−7.14590	−0.472214	−0.236107	−	0.971727i	$-0.575872\pi$
$229$	−7.14590	−0.472214	−0.236107	+	0.971727i	$0.575872\pi$
$230$	0	0
$231$	0	0
$232$	−3.47214	−0.227957
$233$	5.79837	0.379864	0.189932	−	0.981797i	$-0.439173\pi$
$233$	5.79837	0.379864	0.189932	+	0.981797i	$0.439173\pi$
$234$	0	0
$235$	0	0
$236$	6.23607	0.405933
$237$	0	0
$238$	−0.381966	−0.0247592
$239$	18.2705	1.18182	0.590911	−	0.806737i	$-0.298769\pi$
$239$	18.2705	1.18182	0.590911	+	0.806737i	$0.298769\pi$
$240$	0	0
$241$	−24.1803	−1.55759	−0.778796	−	0.627277i	$-0.784169\pi$
$241$	−24.1803	−1.55759	−0.778796	+	0.627277i	$0.784169\pi$
$242$	6.00000	0.385695
$243$	0	0
$244$	−1.14590	−0.0733586
$245$	0	0
$246$	0	0
$247$	10.0902	0.642022
$248$	10.8541	0.689236
$249$	0	0
$250$	0	0
$251$	6.70820	0.423418	0.211709	−	0.977333i	$-0.432097\pi$
$251$	6.70820	0.423418	0.211709	+	0.977333i	$0.432097\pi$
$252$	0	0
$253$	−13.9443	−0.876669
$254$	3.14590	0.197391
$255$	0	0
$256$	1.00000	0.0625000
$257$	−4.09017	−0.255138	−0.127569	−	0.991830i	$-0.540717\pi$
$257$	−4.09017	−0.255138	−0.127569	+	0.991830i	$0.540717\pi$
$258$	0	0
$259$	6.47214	0.402159
$260$	0	0
$261$	0	0
$262$	10.9443	0.676139
$263$	7.47214	0.460752	0.230376	−	0.973102i	$-0.426004\pi$
$263$	7.47214	0.460752	0.230376	+	0.973102i	$0.426004\pi$
$264$	0	0
$265$	0	0
$266$	2.38197	0.146048
$267$	0	0
$268$	3.47214	0.212094
$269$	13.7639	0.839202	0.419601	−	0.907709i	$-0.362170\pi$
$269$	13.7639	0.839202	0.419601	+	0.907709i	$0.362170\pi$
$270$	0	0
$271$	−7.61803	−0.462763	−0.231381	−	0.972863i	$-0.574324\pi$
$271$	−7.61803	−0.462763	−0.231381	+	0.972863i	$0.574324\pi$
$272$	0.381966	0.0231601
$273$	0	0
$274$	15.9443	0.963229
$275$	0	0
$276$	0	0
$277$	23.7639	1.42784	0.713918	−	0.700229i	$-0.246919\pi$
$277$	23.7639	1.42784	0.713918	+	0.700229i	$0.246919\pi$
$278$	−3.70820	−0.222403
$279$	0	0
$280$	0	0
$281$	6.09017	0.363309	0.181655	−	0.983362i	$-0.441855\pi$
$281$	6.09017	0.363309	0.181655	+	0.983362i	$0.441855\pi$
$282$	0	0
$283$	−30.1246	−1.79072	−0.895361	−	0.445341i	$-0.853083\pi$
$283$	−30.1246	−1.79072	−0.895361	+	0.445341i	$0.853083\pi$
$284$	−6.76393	−0.401366
$285$	0	0
$286$	−9.47214	−0.560099
$287$	10.0902	0.595604
$288$	0	0
$289$	−16.8541	−0.991418
$290$	0	0
$291$	0	0
$292$	−6.09017	−0.356400
$293$	4.61803	0.269788	0.134894	−	0.990860i	$-0.456931\pi$
$293$	4.61803	0.269788	0.134894	+	0.990860i	$0.456931\pi$
$294$	0	0
$295$	0	0
$296$	−6.47214	−0.376185
$297$	0	0
$298$	5.94427	0.344342
$299$	−26.4164	−1.52770
$300$	0	0
$301$	5.70820	0.329015
$302$	20.4164	1.17483
$303$	0	0
$304$	−2.38197	−0.136615
$305$	0	0
$306$	0	0
$307$	0.708204	0.0404193	0.0202097	−	0.999796i	$-0.493567\pi$
$307$	0.708204	0.0404193	0.0202097	+	0.999796i	$0.493567\pi$
$308$	−2.23607	−0.127412
$309$	0	0
$310$	0	0
$311$	5.29180	0.300070	0.150035	−	0.988681i	$-0.452061\pi$
$311$	5.29180	0.300070	0.150035	+	0.988681i	$0.452061\pi$
$312$	0	0
$313$	24.1246	1.36360	0.681802	−	0.731537i	$-0.261197\pi$
$313$	24.1246	1.36360	0.681802	+	0.731537i	$0.261197\pi$
$314$	0.944272	0.0532883
$315$	0	0
$316$	−6.00000	−0.337526
$317$	−7.94427	−0.446195	−0.223097	−	0.974796i	$-0.571617\pi$
$317$	−7.94427	−0.446195	−0.223097	+	0.974796i	$0.571617\pi$
$318$	0	0
$319$	−7.76393	−0.434697
$320$	0	0
$321$	0	0
$322$	−6.23607	−0.347522
$323$	−0.909830	−0.0506243
$324$	0	0
$325$	0	0
$326$	5.14590	0.285005
$327$	0	0
$328$	−10.0902	−0.557136
$329$	4.38197	0.241586
$330$	0	0
$331$	−11.7082	−0.643541	−0.321771	−	0.946818i	$-0.604278\pi$
$331$	−11.7082	−0.643541	−0.321771	+	0.946818i	$0.604278\pi$
$332$	14.9443	0.820173
$333$	0	0
$334$	−5.23607	−0.286505
$335$	0	0
$336$	0	0
$337$	20.8328	1.13484	0.567418	−	0.823430i	$-0.307942\pi$
$337$	20.8328	1.13484	0.567418	+	0.823430i	$0.307942\pi$
$338$	−4.94427	−0.268933
$339$	0	0
$340$	0	0
$341$	24.2705	1.31432
$342$	0	0
$343$	−13.0000	−0.701934
$344$	−5.70820	−0.307766
$345$	0	0
$346$	5.38197	0.289336
$347$	−21.3607	−1.14670	−0.573351	−	0.819310i	$-0.694357\pi$
$347$	−21.3607	−1.14670	−0.573351	+	0.819310i	$0.694357\pi$
$348$	0	0
$349$	2.70820	0.144967	0.0724834	−	0.997370i	$-0.476908\pi$
$349$	2.70820	0.144967	0.0724834	+	0.997370i	$0.476908\pi$
$350$	0	0
$351$	0	0
$352$	2.23607	0.119183
$353$	7.94427	0.422831	0.211415	−	0.977396i	$-0.432193\pi$
$353$	7.94427	0.422831	0.211415	+	0.977396i	$0.432193\pi$
$354$	0	0
$355$	0	0
$356$	−4.61803	−0.244755
$357$	0	0
$358$	21.1803	1.11942
$359$	−34.0902	−1.79921	−0.899605	−	0.436704i	$-0.856146\pi$
$359$	−34.0902	−1.79921	−0.899605	+	0.436704i	$0.856146\pi$
$360$	0	0
$361$	−13.3262	−0.701381
$362$	20.4164	1.07306
$363$	0	0
$364$	−4.23607	−0.222030
$365$	0	0
$366$	0	0
$367$	−23.8328	−1.24406	−0.622031	−	0.782992i	$-0.713692\pi$
$367$	−23.8328	−1.24406	−0.622031	+	0.782992i	$0.713692\pi$
$368$	6.23607	0.325078
$369$	0	0
$370$	0	0
$371$	−3.61803	−0.187839
$372$	0	0
$373$	6.00000	0.310668	0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000	0.310668	0.155334	+	0.987862i	$0.450355\pi$
$374$	0.854102	0.0441646
$375$	0	0
$376$	−4.38197	−0.225983
$377$	−14.7082	−0.757511
$378$	0	0
$379$	28.5066	1.46428	0.732142	−	0.681152i	$-0.238521\pi$
$379$	28.5066	1.46428	0.732142	+	0.681152i	$0.238521\pi$
$380$	0	0
$381$	0	0
$382$	1.76393	0.0902506
$383$	−26.6525	−1.36188	−0.680939	−	0.732340i	$-0.738428\pi$
$383$	−26.6525	−1.36188	−0.680939	+	0.732340i	$0.738428\pi$
$384$	0	0
$385$	0	0
$386$	−11.8541	−0.603358
$387$	0	0
$388$	−13.9443	−0.707913
$389$	29.0689	1.47385	0.736925	−	0.675974i	$-0.236277\pi$
$389$	29.0689	1.47385	0.736925	+	0.675974i	$0.236277\pi$
$390$	0	0
$391$	2.38197	0.120461
$392$	6.00000	0.303046
$393$	0	0
$394$	7.14590	0.360005
$395$	0	0
$396$	0	0
$397$	−12.6738	−0.636078	−0.318039	−	0.948078i	$-0.603024\pi$
$397$	−12.6738	−0.636078	−0.318039	+	0.948078i	$0.603024\pi$
$398$	−7.27051	−0.364438
$399$	0	0
$400$	0	0
$401$	20.6738	1.03240	0.516199	−	0.856469i	$-0.327346\pi$
$401$	20.6738	1.03240	0.516199	+	0.856469i	$0.327346\pi$
$402$	0	0
$403$	45.9787	2.29036
$404$	−7.32624	−0.364494
$405$	0	0
$406$	−3.47214	−0.172319
$407$	−14.4721	−0.717357
$408$	0	0
$409$	27.3050	1.35014	0.675071	−	0.737752i	$-0.264113\pi$
$409$	27.3050	1.35014	0.675071	+	0.737752i	$0.264113\pi$
$410$	0	0
$411$	0	0
$412$	5.00000	0.246332
$413$	6.23607	0.306857
$414$	0	0
$415$	0	0
$416$	4.23607	0.207690
$417$	0	0
$418$	−5.32624	−0.260515
$419$	19.9098	0.972659	0.486329	−	0.873776i	$-0.338336\pi$
$419$	19.9098	0.972659	0.486329	+	0.873776i	$0.338336\pi$
$420$	0	0
$421$	−23.3607	−1.13853	−0.569265	−	0.822154i	$-0.692772\pi$
$421$	−23.3607	−1.13853	−0.569265	+	0.822154i	$0.692772\pi$
$422$	13.4164	0.653101
$423$	0	0
$424$	3.61803	0.175707
$425$	0	0
$426$	0	0
$427$	−1.14590	−0.0554539
$428$	−18.8885	−0.913012
$429$	0	0
$430$	0	0
$431$	−18.0557	−0.869714	−0.434857	−	0.900500i	$-0.643201\pi$
$431$	−18.0557	−0.869714	−0.434857	+	0.900500i	$0.643201\pi$
$432$	0	0
$433$	−37.1246	−1.78409	−0.892047	−	0.451942i	$-0.850732\pi$
$433$	−37.1246	−1.78409	−0.892047	+	0.451942i	$0.850732\pi$
$434$	10.8541	0.521014
$435$	0	0
$436$	5.09017	0.243775
$437$	−14.8541	−0.710568
$438$	0	0
$439$	−25.8541	−1.23395	−0.616974	−	0.786983i	$-0.711642\pi$
$439$	−25.8541	−1.23395	−0.616974	+	0.786983i	$0.711642\pi$
$440$	0	0
$441$	0	0
$442$	1.61803	0.0769620
$443$	23.2148	1.10297	0.551484	−	0.834186i	$-0.314062\pi$
$443$	23.2148	1.10297	0.551484	+	0.834186i	$0.314062\pi$
$444$	0	0
$445$	0	0
$446$	−21.6180	−1.02364
$447$	0	0
$448$	1.00000	0.0472456
$449$	−35.1246	−1.65763	−0.828816	−	0.559521i	$-0.810985\pi$
$449$	−35.1246	−1.65763	−0.828816	+	0.559521i	$0.810985\pi$
$450$	0	0
$451$	−22.5623	−1.06242
$452$	5.56231	0.261629
$453$	0	0
$454$	1.09017	0.0511642
$455$	0	0
$456$	0	0
$457$	−27.8328	−1.30196	−0.650982	−	0.759093i	$-0.725643\pi$
$457$	−27.8328	−1.30196	−0.650982	+	0.759093i	$0.725643\pi$
$458$	7.14590	0.333906
$459$	0	0
$460$	0	0
$461$	−34.6525	−1.61393	−0.806963	−	0.590602i	$-0.798891\pi$
$461$	−34.6525	−1.61393	−0.806963	+	0.590602i	$0.798891\pi$
$462$	0	0
$463$	−17.4164	−0.809409	−0.404705	−	0.914447i	$-0.632626\pi$
$463$	−17.4164	−0.809409	−0.404705	+	0.914447i	$0.632626\pi$
$464$	3.47214	0.161190
$465$	0	0
$466$	−5.79837	−0.268604
$467$	−34.4164	−1.59260	−0.796301	−	0.604901i	$-0.793213\pi$
$467$	−34.4164	−1.59260	−0.796301	+	0.604901i	$0.793213\pi$
$468$	0	0
$469$	3.47214	0.160328
$470$	0	0
$471$	0	0
$472$	−6.23607	−0.287038
$473$	−12.7639	−0.586886
$474$	0	0
$475$	0	0
$476$	0.381966	0.0175074
$477$	0	0
$478$	−18.2705	−0.835674
$479$	16.1459	0.737725	0.368862	−	0.929484i	$-0.379747\pi$
$479$	16.1459	0.737725	0.368862	+	0.929484i	$0.379747\pi$
$480$	0	0
$481$	−27.4164	−1.25008
$482$	24.1803	1.10138
$483$	0	0
$484$	−6.00000	−0.272727
$485$	0	0
$486$	0	0
$487$	−15.4377	−0.699549	−0.349774	−	0.936834i	$-0.613742\pi$
$487$	−15.4377	−0.699549	−0.349774	+	0.936834i	$0.613742\pi$
$488$	1.14590	0.0518724
$489$	0	0
$490$	0	0
$491$	38.0344	1.71647	0.858235	−	0.513257i	$-0.171561\pi$
$491$	38.0344	1.71647	0.858235	+	0.513257i	$0.171561\pi$
$492$	0	0
$493$	1.32624	0.0597308
$494$	−10.0902	−0.453978
$495$	0	0
$496$	−10.8541	−0.487364
$497$	−6.76393	−0.303404
$498$	0	0
$499$	8.00000	0.358129	0.179065	−	0.983837i	$-0.442693\pi$
$499$	8.00000	0.358129	0.179065	+	0.983837i	$0.442693\pi$
$500$	0	0
$501$	0	0
$502$	−6.70820	−0.299402
$503$	−6.43769	−0.287043	−0.143521	−	0.989647i	$-0.545843\pi$
$503$	−6.43769	−0.287043	−0.143521	+	0.989647i	$0.545843\pi$
$504$	0	0
$505$	0	0
$506$	13.9443	0.619898
$507$	0	0
$508$	−3.14590	−0.139577
$509$	34.4164	1.52548	0.762740	−	0.646705i	$-0.223854\pi$
$509$	34.4164	1.52548	0.762740	+	0.646705i	$0.223854\pi$
$510$	0	0
$511$	−6.09017	−0.269413
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	4.09017	0.180410
$515$	0	0
$516$	0	0
$517$	−9.79837	−0.430932
$518$	−6.47214	−0.284369
$519$	0	0
$520$	0	0
$521$	−42.4721	−1.86074	−0.930369	−	0.366624i	$-0.880513\pi$
$521$	−42.4721	−1.86074	−0.930369	+	0.366624i	$0.880513\pi$
$522$	0	0
$523$	−10.8885	−0.476123	−0.238061	−	0.971250i	$-0.576512\pi$
$523$	−10.8885	−0.476123	−0.238061	+	0.971250i	$0.576512\pi$
$524$	−10.9443	−0.478103
$525$	0	0
$526$	−7.47214	−0.325801
$527$	−4.14590	−0.180598
$528$	0	0
$529$	15.8885	0.690806
$530$	0	0
$531$	0	0
$532$	−2.38197	−0.103271
$533$	−42.7426	−1.85139
$534$	0	0
$535$	0	0
$536$	−3.47214	−0.149973
$537$	0	0
$538$	−13.7639	−0.593405
$539$	13.4164	0.577886
$540$	0	0
$541$	−6.70820	−0.288408	−0.144204	−	0.989548i	$-0.546062\pi$
$541$	−6.70820	−0.288408	−0.144204	+	0.989548i	$0.546062\pi$
$542$	7.61803	0.327223
$543$	0	0
$544$	−0.381966	−0.0163767
$545$	0	0
$546$	0	0
$547$	8.29180	0.354532	0.177266	−	0.984163i	$-0.443275\pi$
$547$	8.29180	0.354532	0.177266	+	0.984163i	$0.443275\pi$
$548$	−15.9443	−0.681106
$549$	0	0
$550$	0	0
$551$	−8.27051	−0.352336
$552$	0	0
$553$	−6.00000	−0.255146
$554$	−23.7639	−1.00963
$555$	0	0
$556$	3.70820	0.157263
$557$	29.8328	1.26406	0.632028	−	0.774945i	$-0.282223\pi$
$557$	29.8328	1.26406	0.632028	+	0.774945i	$0.282223\pi$
$558$	0	0
$559$	−24.1803	−1.02272
$560$	0	0
$561$	0	0
$562$	−6.09017	−0.256898
$563$	−5.67376	−0.239121	−0.119560	−	0.992827i	$-0.538148\pi$
$563$	−5.67376	−0.239121	−0.119560	+	0.992827i	$0.538148\pi$
$564$	0	0
$565$	0	0
$566$	30.1246	1.26623
$567$	0	0
$568$	6.76393	0.283808
$569$	−27.8885	−1.16915	−0.584574	−	0.811340i	$-0.698738\pi$
$569$	−27.8885	−1.16915	−0.584574	+	0.811340i	$0.698738\pi$
$570$	0	0
$571$	−0.944272	−0.0395165	−0.0197583	−	0.999805i	$-0.506290\pi$
$571$	−0.944272	−0.0395165	−0.0197583	+	0.999805i	$0.506290\pi$
$572$	9.47214	0.396050
$573$	0	0
$574$	−10.0902	−0.421156
$575$	0	0
$576$	0	0
$577$	32.9787	1.37292	0.686461	−	0.727167i	$-0.259163\pi$
$577$	32.9787	1.37292	0.686461	+	0.727167i	$0.259163\pi$
$578$	16.8541	0.701038
$579$	0	0
$580$	0	0
$581$	14.9443	0.619993
$582$	0	0
$583$	8.09017	0.335061
$584$	6.09017	0.252013
$585$	0	0
$586$	−4.61803	−0.190769
$587$	−17.1803	−0.709109	−0.354554	−	0.935035i	$-0.615367\pi$
$587$	−17.1803	−0.709109	−0.354554	+	0.935035i	$0.615367\pi$
$588$	0	0
$589$	25.8541	1.06530
$590$	0	0
$591$	0	0
$592$	6.47214	0.266003
$593$	−33.1591	−1.36168	−0.680840	−	0.732433i	$-0.738385\pi$
$593$	−33.1591	−1.36168	−0.680840	+	0.732433i	$0.738385\pi$
$594$	0	0
$595$	0	0
$596$	−5.94427	−0.243487
$597$	0	0
$598$	26.4164	1.08025
$599$	−4.50658	−0.184134	−0.0920669	−	0.995753i	$-0.529347\pi$
$599$	−4.50658	−0.184134	−0.0920669	+	0.995753i	$0.529347\pi$
$600$	0	0
$601$	11.0000	0.448699	0.224350	−	0.974509i	$-0.427974\pi$
$601$	11.0000	0.448699	0.224350	+	0.974509i	$0.427974\pi$
$602$	−5.70820	−0.232649
$603$	0	0
$604$	−20.4164	−0.830732
$605$	0	0
$606$	0	0
$607$	−15.0000	−0.608831	−0.304416	−	0.952539i	$-0.598461\pi$
$607$	−15.0000	−0.608831	−0.304416	+	0.952539i	$0.598461\pi$
$608$	2.38197	0.0966015
$609$	0	0
$610$	0	0
$611$	−18.5623	−0.750951
$612$	0	0
$613$	−23.7082	−0.957565	−0.478783	−	0.877933i	$-0.658922\pi$
$613$	−23.7082	−0.957565	−0.478783	+	0.877933i	$0.658922\pi$
$614$	−0.708204	−0.0285808
$615$	0	0
$616$	2.23607	0.0900937
$617$	−8.11146	−0.326555	−0.163277	−	0.986580i	$-0.552207\pi$
$617$	−8.11146	−0.326555	−0.163277	+	0.986580i	$0.552207\pi$
$618$	0	0
$619$	27.5066	1.10558	0.552791	−	0.833320i	$-0.313563\pi$
$619$	27.5066	1.10558	0.552791	+	0.833320i	$0.313563\pi$
$620$	0	0
$621$	0	0
$622$	−5.29180	−0.212182
$623$	−4.61803	−0.185018
$624$	0	0
$625$	0	0
$626$	−24.1246	−0.964213
$627$	0	0
$628$	−0.944272	−0.0376806
$629$	2.47214	0.0985705
$630$	0	0
$631$	37.8328	1.50610	0.753050	−	0.657963i	$-0.228582\pi$
$631$	37.8328	1.50610	0.753050	+	0.657963i	$0.228582\pi$
$632$	6.00000	0.238667
$633$	0	0
$634$	7.94427	0.315507
$635$	0	0
$636$	0	0
$637$	25.4164	1.00703
$638$	7.76393	0.307377
$639$	0	0
$640$	0	0
$641$	26.1803	1.03406	0.517031	−	0.855967i	$-0.327037\pi$
$641$	26.1803	1.03406	0.517031	+	0.855967i	$0.327037\pi$
$642$	0	0
$643$	26.1591	1.03161	0.515806	−	0.856705i	$-0.327492\pi$
$643$	26.1591	1.03161	0.515806	+	0.856705i	$0.327492\pi$
$644$	6.23607	0.245736
$645$	0	0
$646$	0.909830	0.0357968
$647$	10.6525	0.418792	0.209396	−	0.977831i	$-0.432850\pi$
$647$	10.6525	0.418792	0.209396	+	0.977831i	$0.432850\pi$
$648$	0	0
$649$	−13.9443	−0.547361
$650$	0	0
$651$	0	0
$652$	−5.14590	−0.201529
$653$	19.9443	0.780480	0.390240	−	0.920713i	$-0.372392\pi$
$653$	19.9443	0.780480	0.390240	+	0.920713i	$0.372392\pi$
$654$	0	0
$655$	0	0
$656$	10.0902	0.393955
$657$	0	0
$658$	−4.38197	−0.170827
$659$	22.3607	0.871048	0.435524	−	0.900177i	$-0.356563\pi$
$659$	22.3607	0.871048	0.435524	+	0.900177i	$0.356563\pi$
$660$	0	0
$661$	−22.5066	−0.875404	−0.437702	−	0.899120i	$-0.644208\pi$
$661$	−22.5066	−0.875404	−0.437702	+	0.899120i	$0.644208\pi$
$662$	11.7082	0.455052
$663$	0	0
$664$	−14.9443	−0.579950
$665$	0	0
$666$	0	0
$667$	21.6525	0.838387
$668$	5.23607	0.202590
$669$	0	0
$670$	0	0
$671$	2.56231	0.0989167
$672$	0	0
$673$	21.7984	0.840266	0.420133	−	0.907463i	$-0.361983\pi$
$673$	21.7984	0.840266	0.420133	+	0.907463i	$0.361983\pi$
$674$	−20.8328	−0.802450
$675$	0	0
$676$	4.94427	0.190164
$677$	33.1591	1.27441	0.637203	−	0.770696i	$-0.280091\pi$
$677$	33.1591	1.27441	0.637203	+	0.770696i	$0.280091\pi$
$678$	0	0
$679$	−13.9443	−0.535132
$680$	0	0
$681$	0	0
$682$	−24.2705	−0.929366
$683$	20.5066	0.784662	0.392331	−	0.919824i	$-0.371669\pi$
$683$	20.5066	0.784662	0.392331	+	0.919824i	$0.371669\pi$
$684$	0	0
$685$	0	0
$686$	13.0000	0.496342
$687$	0	0
$688$	5.70820	0.217623
$689$	15.3262	0.583883
$690$	0	0
$691$	37.2492	1.41703	0.708514	−	0.705697i	$-0.249366\pi$
$691$	37.2492	1.41703	0.708514	+	0.705697i	$0.249366\pi$
$692$	−5.38197	−0.204592
$693$	0	0
$694$	21.3607	0.810840
$695$	0	0
$696$	0	0
$697$	3.85410	0.145985
$698$	−2.70820	−0.102507
$699$	0	0
$700$	0	0
$701$	18.6738	0.705298	0.352649	−	0.935756i	$-0.385281\pi$
$701$	18.6738	0.705298	0.352649	+	0.935756i	$0.385281\pi$
$702$	0	0
$703$	−15.4164	−0.581441
$704$	−2.23607	−0.0842750
$705$	0	0
$706$	−7.94427	−0.298987
$707$	−7.32624	−0.275532
$708$	0	0
$709$	−21.3607	−0.802217	−0.401109	−	0.916031i	$-0.631375\pi$
$709$	−21.3607	−0.802217	−0.401109	+	0.916031i	$0.631375\pi$
$710$	0	0
$711$	0	0
$712$	4.61803	0.173068
$713$	−67.6869	−2.53489
$714$	0	0
$715$	0	0
$716$	−21.1803	−0.791546
$717$	0	0
$718$	34.0902	1.27223
$719$	2.23607	0.0833913	0.0416956	−	0.999130i	$-0.486724\pi$
$719$	2.23607	0.0833913	0.0416956	+	0.999130i	$0.486724\pi$
$720$	0	0
$721$	5.00000	0.186210
$722$	13.3262	0.495951
$723$	0	0
$724$	−20.4164	−0.758770
$725$	0	0
$726$	0	0
$727$	23.2918	0.863845	0.431922	−	0.901911i	$-0.357835\pi$
$727$	23.2918	0.863845	0.431922	+	0.901911i	$0.357835\pi$
$728$	4.23607	0.156999
$729$	0	0
$730$	0	0
$731$	2.18034	0.0806428
$732$	0	0
$733$	2.90983	0.107477	0.0537385	−	0.998555i	$-0.482886\pi$
$733$	2.90983	0.107477	0.0537385	+	0.998555i	$0.482886\pi$
$734$	23.8328	0.879685
$735$	0	0
$736$	−6.23607	−0.229865
$737$	−7.76393	−0.285988
$738$	0	0
$739$	5.00000	0.183928	0.0919640	−	0.995762i	$-0.470686\pi$
$739$	5.00000	0.183928	0.0919640	+	0.995762i	$0.470686\pi$
$740$	0	0
$741$	0	0
$742$	3.61803	0.132822
$743$	−11.2016	−0.410948	−0.205474	−	0.978663i	$-0.565874\pi$
$743$	−11.2016	−0.410948	−0.205474	+	0.978663i	$0.565874\pi$
$744$	0	0
$745$	0	0
$746$	−6.00000	−0.219676
$747$	0	0
$748$	−0.854102	−0.0312291
$749$	−18.8885	−0.690172
$750$	0	0
$751$	−2.72949	−0.0996005	−0.0498003	−	0.998759i	$-0.515858\pi$
$751$	−2.72949	−0.0996005	−0.0498003	+	0.998759i	$0.515858\pi$
$752$	4.38197	0.159794
$753$	0	0
$754$	14.7082	0.535641
$755$	0	0
$756$	0	0
$757$	−9.70820	−0.352851	−0.176425	−	0.984314i	$-0.556453\pi$
$757$	−9.70820	−0.352851	−0.176425	+	0.984314i	$0.556453\pi$
$758$	−28.5066	−1.03541
$759$	0	0
$760$	0	0
$761$	−37.2492	−1.35028	−0.675142	−	0.737688i	$-0.735918\pi$
$761$	−37.2492	−1.35028	−0.675142	+	0.737688i	$0.735918\pi$
$762$	0	0
$763$	5.09017	0.184277
$764$	−1.76393	−0.0638168
$765$	0	0
$766$	26.6525	0.962993
$767$	−26.4164	−0.953841
$768$	0	0
$769$	−27.1246	−0.978139	−0.489069	−	0.872245i	$-0.662663\pi$
$769$	−27.1246	−0.978139	−0.489069	+	0.872245i	$0.662663\pi$
$770$	0	0
$771$	0	0
$772$	11.8541	0.426638
$773$	−47.6525	−1.71394	−0.856970	−	0.515366i	$-0.827656\pi$
$773$	−47.6525	−1.71394	−0.856970	+	0.515366i	$0.827656\pi$
$774$	0	0
$775$	0	0
$776$	13.9443	0.500570
$777$	0	0
$778$	−29.0689	−1.04217
$779$	−24.0344	−0.861123
$780$	0	0
$781$	15.1246	0.541201
$782$	−2.38197	−0.0851789
$783$	0	0
$784$	−6.00000	−0.214286
$785$	0	0
$786$	0	0
$787$	29.3607	1.04660	0.523298	−	0.852150i	$-0.324702\pi$
$787$	29.3607	1.04660	0.523298	+	0.852150i	$0.324702\pi$
$788$	−7.14590	−0.254562
$789$	0	0
$790$	0	0
$791$	5.56231	0.197773
$792$	0	0
$793$	4.85410	0.172374
$794$	12.6738	0.449775
$795$	0	0
$796$	7.27051	0.257696
$797$	−44.9443	−1.59201	−0.796004	−	0.605291i	$-0.793057\pi$
$797$	−44.9443	−1.59201	−0.796004	+	0.605291i	$0.793057\pi$
$798$	0	0
$799$	1.67376	0.0592134
$800$	0	0
$801$	0	0
$802$	−20.6738	−0.730016
$803$	13.6180	0.480570
$804$	0	0
$805$	0	0
$806$	−45.9787	−1.61953
$807$	0	0
$808$	7.32624	0.257736
$809$	−3.05573	−0.107434	−0.0537168	−	0.998556i	$-0.517107\pi$
$809$	−3.05573	−0.107434	−0.0537168	+	0.998556i	$0.517107\pi$
$810$	0	0
$811$	−26.0000	−0.912983	−0.456492	−	0.889728i	$-0.650894\pi$
$811$	−26.0000	−0.912983	−0.456492	+	0.889728i	$0.650894\pi$
$812$	3.47214	0.121848
$813$	0	0
$814$	14.4721	0.507248
$815$	0	0
$816$	0	0
$817$	−13.5967	−0.475690
$818$	−27.3050	−0.954695
$819$	0	0
$820$	0	0
$821$	−43.7984	−1.52857	−0.764287	−	0.644876i	$-0.776909\pi$
$821$	−43.7984	−1.52857	−0.764287	+	0.644876i	$0.776909\pi$
$822$	0	0
$823$	−19.3262	−0.673670	−0.336835	−	0.941564i	$-0.609357\pi$
$823$	−19.3262	−0.673670	−0.336835	+	0.941564i	$0.609357\pi$
$824$	−5.00000	−0.174183
$825$	0	0
$826$	−6.23607	−0.216981
$827$	−28.6312	−0.995604	−0.497802	−	0.867291i	$-0.665859\pi$
$827$	−28.6312	−0.995604	−0.497802	+	0.867291i	$0.665859\pi$
$828$	0	0
$829$	−49.5967	−1.72257	−0.861283	−	0.508125i	$-0.830339\pi$
$829$	−49.5967	−1.72257	−0.861283	+	0.508125i	$0.830339\pi$
$830$	0	0
$831$	0	0
$832$	−4.23607	−0.146859
$833$	−2.29180	−0.0794060
$834$	0	0
$835$	0	0
$836$	5.32624	0.184212
$837$	0	0
$838$	−19.9098	−0.687774
$839$	32.8673	1.13470	0.567352	−	0.823475i	$-0.307968\pi$
$839$	32.8673	1.13470	0.567352	+	0.823475i	$0.307968\pi$
$840$	0	0
$841$	−16.9443	−0.584285
$842$	23.3607	0.805062
$843$	0	0
$844$	−13.4164	−0.461812
$845$	0	0
$846$	0	0
$847$	−6.00000	−0.206162
$848$	−3.61803	−0.124244
$849$	0	0
$850$	0	0
$851$	40.3607	1.38355
$852$	0	0
$853$	48.4721	1.65965	0.829827	−	0.558021i	$-0.188439\pi$
$853$	48.4721	1.65965	0.829827	+	0.558021i	$0.188439\pi$
$854$	1.14590	0.0392118
$855$	0	0
$856$	18.8885	0.645597
$857$	−52.0902	−1.77937	−0.889683	−	0.456578i	$-0.849075\pi$
$857$	−52.0902	−1.77937	−0.889683	+	0.456578i	$0.849075\pi$
$858$	0	0
$859$	−41.4164	−1.41311	−0.706555	−	0.707658i	$-0.749752\pi$
$859$	−41.4164	−1.41311	−0.706555	+	0.707658i	$0.749752\pi$
$860$	0	0
$861$	0	0
$862$	18.0557	0.614981
$863$	49.0132	1.66843	0.834214	−	0.551441i	$-0.185922\pi$
$863$	49.0132	1.66843	0.834214	+	0.551441i	$0.185922\pi$
$864$	0	0
$865$	0	0
$866$	37.1246	1.26155
$867$	0	0
$868$	−10.8541	−0.368412
$869$	13.4164	0.455120
$870$	0	0
$871$	−14.7082	−0.498368
$872$	−5.09017	−0.172375
$873$	0	0
$874$	14.8541	0.502447
$875$	0	0
$876$	0	0
$877$	−19.8541	−0.670425	−0.335213	−	0.942142i	$-0.608808\pi$
$877$	−19.8541	−0.670425	−0.335213	+	0.942142i	$0.608808\pi$
$878$	25.8541	0.872534
$879$	0	0
$880$	0	0
$881$	−27.0000	−0.909653	−0.454827	−	0.890580i	$-0.650299\pi$
$881$	−27.0000	−0.909653	−0.454827	+	0.890580i	$0.650299\pi$
$882$	0	0
$883$	−49.4164	−1.66299	−0.831497	−	0.555529i	$-0.812516\pi$
$883$	−49.4164	−1.66299	−0.831497	+	0.555529i	$0.812516\pi$
$884$	−1.61803	−0.0544204
$885$	0	0
$886$	−23.2148	−0.779916
$887$	−21.9787	−0.737973	−0.368986	−	0.929435i	$-0.620295\pi$
$887$	−21.9787	−0.737973	−0.368986	+	0.929435i	$0.620295\pi$
$888$	0	0
$889$	−3.14590	−0.105510
$890$	0	0
$891$	0	0
$892$	21.6180	0.723825
$893$	−10.4377	−0.349284
$894$	0	0
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	35.1246	1.17212
$899$	−37.6869	−1.25693
$900$	0	0
$901$	−1.38197	−0.0460400
$902$	22.5623	0.751243
$903$	0	0
$904$	−5.56231	−0.185000
$905$	0	0
$906$	0	0
$907$	37.6738	1.25094	0.625468	−	0.780250i	$-0.284908\pi$
$907$	37.6738	1.25094	0.625468	+	0.780250i	$0.284908\pi$
$908$	−1.09017	−0.0361786
$909$	0	0
$910$	0	0
$911$	15.8754	0.525975	0.262988	−	0.964799i	$-0.415292\pi$
$911$	15.8754	0.525975	0.262988	+	0.964799i	$0.415292\pi$
$912$	0	0
$913$	−33.4164	−1.10592
$914$	27.8328	0.920628
$915$	0	0
$916$	−7.14590	−0.236107
$917$	−10.9443	−0.361412
$918$	0	0
$919$	−1.06888	−0.0352592	−0.0176296	−	0.999845i	$-0.505612\pi$
$919$	−1.06888	−0.0352592	−0.0176296	+	0.999845i	$0.505612\pi$
$920$	0	0
$921$	0	0
$922$	34.6525	1.14122
$923$	28.6525	0.943108
$924$	0	0
$925$	0	0
$926$	17.4164	0.572339
$927$	0	0
$928$	−3.47214	−0.113978
$929$	−56.8885	−1.86645	−0.933226	−	0.359289i	$-0.883019\pi$
$929$	−56.8885	−1.86645	−0.933226	+	0.359289i	$0.883019\pi$
$930$	0	0
$931$	14.2918	0.468395
$932$	5.79837	0.189932
$933$	0	0
$934$	34.4164	1.12614
$935$	0	0
$936$	0	0
$937$	20.2492	0.661513	0.330757	−	0.943716i	$-0.392696\pi$
$937$	20.2492	0.661513	0.330757	+	0.943716i	$0.392696\pi$
$938$	−3.47214	−0.113369
$939$	0	0
$940$	0	0
$941$	42.5410	1.38680	0.693399	−	0.720554i	$-0.256112\pi$
$941$	42.5410	1.38680	0.693399	+	0.720554i	$0.256112\pi$
$942$	0	0
$943$	62.9230	2.04905
$944$	6.23607	0.202967
$945$	0	0
$946$	12.7639	0.414991
$947$	13.5836	0.441407	0.220704	−	0.975341i	$-0.429165\pi$
$947$	13.5836	0.441407	0.220704	+	0.975341i	$0.429165\pi$
$948$	0	0
$949$	25.7984	0.837451
$950$	0	0
$951$	0	0
$952$	−0.381966	−0.0123796
$953$	−57.9443	−1.87700	−0.938500	−	0.345280i	$-0.887784\pi$
$953$	−57.9443	−1.87700	−0.938500	+	0.345280i	$0.887784\pi$
$954$	0	0
$955$	0	0
$956$	18.2705	0.590911
$957$	0	0
$958$	−16.1459	−0.521650
$959$	−15.9443	−0.514867
$960$	0	0
$961$	86.8115	2.80037
$962$	27.4164	0.883940
$963$	0	0
$964$	−24.1803	−0.778796
$965$	0	0
$966$	0	0
$967$	−49.1591	−1.58085	−0.790424	−	0.612560i	$-0.790140\pi$
$967$	−49.1591	−1.58085	−0.790424	+	0.612560i	$0.790140\pi$
$968$	6.00000	0.192847
$969$	0	0
$970$	0	0
$971$	21.3607	0.685497	0.342748	−	0.939427i	$-0.388642\pi$
$971$	21.3607	0.685497	0.342748	+	0.939427i	$0.388642\pi$
$972$	0	0
$973$	3.70820	0.118880
$974$	15.4377	0.494656
$975$	0	0
$976$	−1.14590	−0.0366793
$977$	−20.6738	−0.661412	−0.330706	−	0.943734i	$-0.607287\pi$
$977$	−20.6738	−0.661412	−0.330706	+	0.943734i	$0.607287\pi$
$978$	0	0
$979$	10.3262	0.330028
$980$	0	0
$981$	0	0
$982$	−38.0344	−1.21373
$983$	36.9787	1.17944	0.589719	−	0.807609i	$-0.299239\pi$
$983$	36.9787	1.17944	0.589719	+	0.807609i	$0.299239\pi$
$984$	0	0
$985$	0	0
$986$	−1.32624	−0.0422360
$987$	0	0
$988$	10.0902	0.321011
$989$	35.5967	1.13191
$990$	0	0
$991$	−0.944272	−0.0299958	−0.0149979	−	0.999888i	$-0.504774\pi$
$991$	−0.944272	−0.0299958	−0.0149979	+	0.999888i	$0.504774\pi$
$992$	10.8541	0.344618
$993$	0	0
$994$	6.76393	0.214539
$995$	0	0
$996$	0	0
$997$	−25.1246	−0.795704	−0.397852	−	0.917450i	$-0.630244\pi$
$997$	−25.1246	−0.795704	−0.397852	+	0.917450i	$0.630244\pi$
$998$	−8.00000	−0.253236
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6750.2.a.f.1.1	yes	2
3.2	odd	2		6750.2.a.n.1.2	yes	2
5.4	even	2		6750.2.a.k.1.1	yes	2
15.14	odd	2		6750.2.a.c.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6750.2.a.c.1.2	✓	2	15.14	odd	2
6750.2.a.f.1.1	yes	2	1.1	even	1	trivial
6750.2.a.k.1.1	yes	2	5.4	even	2
6750.2.a.n.1.2	yes	2	3.2	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:	\( N \)	\(=\)	\( 6750 = 2 \cdot 3^{3} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6750.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} -1.00000 q^{8} -2.23607 q^{11} -4.23607 q^{13} -1.00000 q^{14} +1.00000 q^{16} +0.381966 q^{17} -2.38197 q^{19} +2.23607 q^{22} +6.23607 q^{23} +4.23607 q^{26} +1.00000 q^{28} +3.47214 q^{29} -10.8541 q^{31} -1.00000 q^{32} -0.381966 q^{34} +6.47214 q^{37} +2.38197 q^{38} +10.0902 q^{41} +5.70820 q^{43} -2.23607 q^{44} -6.23607 q^{46} +4.38197 q^{47} -6.00000 q^{49} -4.23607 q^{52} -3.61803 q^{53} -1.00000 q^{56} -3.47214 q^{58} +6.23607 q^{59} -1.14590 q^{61} +10.8541 q^{62} +1.00000 q^{64} +3.47214 q^{67} +0.381966 q^{68} -6.76393 q^{71} -6.09017 q^{73} -6.47214 q^{74} -2.38197 q^{76} -2.23607 q^{77} -6.00000 q^{79} -10.0902 q^{82} +14.9443 q^{83} -5.70820 q^{86} +2.23607 q^{88} -4.61803 q^{89} -4.23607 q^{91} +6.23607 q^{92} -4.38197 q^{94} -13.9443 q^{97} +6.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^7 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 2 q^{8} - 4 q^{13} - 2 q^{14} + 2 q^{16} + 3 q^{17} - 7 q^{19} + 8 q^{23} + 4 q^{26} + 2 q^{28} - 2 q^{29} - 15 q^{31} - 2 q^{32} - 3 q^{34} + 4 q^{37} + 7 q^{38} + 9 q^{41} - 2 q^{43} - 8 q^{46} + 11 q^{47} - 12 q^{49} - 4 q^{52} - 5 q^{53} - 2 q^{56} + 2 q^{58} + 8 q^{59} - 9 q^{61} + 15 q^{62} + 2 q^{64} - 2 q^{67} + 3 q^{68} - 18 q^{71} - q^{73} - 4 q^{74} - 7 q^{76} - 12 q^{79} - 9 q^{82} + 12 q^{83} + 2 q^{86} - 7 q^{89} - 4 q^{91} + 8 q^{92} - 11 q^{94} - 10 q^{97} + 12 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^7 - 2 * q^8 - 4 * q^13 - 2 * q^14 + 2 * q^16 + 3 * q^17 - 7 * q^19 + 8 * q^23 + 4 * q^26 + 2 * q^28 - 2 * q^29 - 15 * q^31 - 2 * q^32 - 3 * q^34 + 4 * q^37 + 7 * q^38 + 9 * q^41 - 2 * q^43 - 8 * q^46 + 11 * q^47 - 12 * q^49 - 4 * q^52 - 5 * q^53 - 2 * q^56 + 2 * q^58 + 8 * q^59 - 9 * q^61 + 15 * q^62 + 2 * q^64 - 2 * q^67 + 3 * q^68 - 18 * q^71 - q^73 - 4 * q^74 - 7 * q^76 - 12 * q^79 - 9 * q^82 + 12 * q^83 + 2 * q^86 - 7 * q^89 - 4 * q^91 + 8 * q^92 - 11 * q^94 - 10 * q^97 + 12 * q^98

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