LMFDB - Embedded newform 6975.2.a.bf.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6975,2,Mod(1,6975)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6975.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6975, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6975 = 3^{2} \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6975.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,1,0,5,0,0,-8,3,0,0,-2,0,-4,8,0,5,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$55.6956554098$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.564.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 3 $ x^3 - x^2 - 5*x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 465)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.571993$ of defining polynomial
Character	$\chi$	$=$	6975.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.571993 q^{2} -1.67282 q^{4} -2.42801 q^{7} -2.10083 q^{8} -1.14399 q^{11} -1.57199 q^{13} -1.38880 q^{14} +2.14399 q^{16} +4.67282 q^{17} +5.34565 q^{19} -0.654353 q^{22} -1.81681 q^{23} -0.899170 q^{26} +4.06163 q^{28} -1.95684 q^{29} +1.00000 q^{31} +5.42801 q^{32} +2.67282 q^{34} -3.57199 q^{37} +3.05767 q^{38} +3.14399 q^{41} -2.85601 q^{43} +1.91369 q^{44} -1.03920 q^{46} +7.16246 q^{47} -1.10478 q^{49} +2.62967 q^{52} -3.81681 q^{53} +5.10083 q^{56} -1.11930 q^{58} +6.24482 q^{59} -2.00000 q^{61} +0.571993 q^{62} -1.18319 q^{64} +0.917641 q^{67} -7.81681 q^{68} +6.44648 q^{71} -6.71598 q^{73} -2.04316 q^{74} -8.94233 q^{76} +2.77761 q^{77} +9.52884 q^{79} +1.79834 q^{82} -3.16246 q^{83} -1.63362 q^{86} +2.40332 q^{88} +15.2241 q^{89} +3.81681 q^{91} +3.03920 q^{92} +4.09688 q^{94} -10.7776 q^{97} -0.631929 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} + 5 q^{4} - 8 q^{7} + 3 q^{8} - 2 q^{11} - 4 q^{13} + 8 q^{14} + 5 q^{16} + 4 q^{17} - 4 q^{19} - 22 q^{22} + 6 q^{23} - 12 q^{26} - 10 q^{28} + 2 q^{29} + 3 q^{31} + 17 q^{32} - 2 q^{34} - 10 q^{37}+ \cdots - 57 q^{98}+O(q^{100}) $ 3 * q + q^2 + 5 * q^4 - 8 * q^7 + 3 * q^8 - 2 * q^11 - 4 * q^13 + 8 * q^14 + 5 * q^16 + 4 * q^17 - 4 * q^19 - 22 * q^22 + 6 * q^23 - 12 * q^26 - 10 * q^28 + 2 * q^29 + 3 * q^31 + 17 * q^32 - 2 * q^34 - 10 * q^37 - 8 * q^38 + 8 * q^41 - 10 * q^43 - 10 * q^44 - 16 * q^46 - 10 * q^47 + 11 * q^49 - 10 * q^52 + 6 * q^56 + 36 * q^58 + 8 * q^59 - 6 * q^61 + q^62 - 15 * q^64 - 18 * q^67 - 12 * q^68 - 10 * q^71 - 18 * q^73 - 14 * q^74 - 44 * q^76 - 16 * q^77 + 20 * q^79 + 24 * q^82 + 22 * q^83 + 18 * q^86 - 30 * q^88 - 8 * q^89 + 22 * q^92 + 8 * q^94 - 8 * q^97 - 57 * q^98

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 $

$2$	0.571993	0.404460	0.202230	−	0.979338i	$-0.435181\pi$
$2$	0.571993	0.404460	0.202230	+	0.979338i	$0.435181\pi$
$3$	0	0
$3$	0	0
$4$	−1.67282	−0.836412
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.42801	−0.917700	−0.458850	−	0.888514i	$-0.651739\pi$
$7$	−2.42801	−0.917700	−0.458850	+	0.888514i	$0.651739\pi$
$8$	−2.10083	−0.742756
$9$	0	0
$10$	0	0
$11$	−1.14399	−0.344925	−0.172462	−	0.985016i	$-0.555172\pi$
$11$	−1.14399	−0.344925	−0.172462	+	0.985016i	$0.555172\pi$
$12$	0	0
$13$	−1.57199	−0.435992	−0.217996	−	0.975950i	$-0.569952\pi$
$13$	−1.57199	−0.435992	−0.217996	+	0.975950i	$0.569952\pi$
$14$	−1.38880	−0.371173
$15$	0	0
$16$	2.14399	0.535997
$17$	4.67282	1.13333	0.566663	−	0.823950i	$-0.308234\pi$
$17$	4.67282	1.13333	0.566663	+	0.823950i	$0.308234\pi$
$18$	0	0
$19$	5.34565	1.22638	0.613188	−	0.789937i	$-0.289887\pi$
$19$	5.34565	1.22638	0.613188	+	0.789937i	$0.289887\pi$
$20$	0	0
$21$	0	0
$22$	−0.654353	−0.139508
$23$	−1.81681	−0.378831	−0.189416	−	0.981897i	$-0.560659\pi$
$23$	−1.81681	−0.378831	−0.189416	+	0.981897i	$0.560659\pi$
$24$	0	0
$25$	0	0
$26$	−0.899170	−0.176342
$27$	0	0
$28$	4.06163	0.767575
$29$	−1.95684	−0.363377	−0.181688	−	0.983356i	$-0.558156\pi$
$29$	−1.95684	−0.363377	−0.181688	+	0.983356i	$0.558156\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	5.42801	0.959545
$33$	0	0
$34$	2.67282	0.458385
$35$	0	0
$36$	0	0
$37$	−3.57199	−0.587232	−0.293616	−	0.955923i	$-0.594859\pi$
$37$	−3.57199	−0.587232	−0.293616	+	0.955923i	$0.594859\pi$
$38$	3.05767	0.496020
$39$	0	0
$40$	0	0
$41$	3.14399	0.491008	0.245504	−	0.969396i	$-0.421047\pi$
$41$	3.14399	0.491008	0.245504	+	0.969396i	$0.421047\pi$
$42$	0	0
$43$	−2.85601	−0.435538	−0.217769	−	0.976000i	$-0.569878\pi$
$43$	−2.85601	−0.435538	−0.217769	+	0.976000i	$0.569878\pi$
$44$	1.91369	0.288499
$45$	0	0
$46$	−1.03920	−0.153222
$47$	7.16246	1.04475	0.522376	−	0.852715i	$-0.325046\pi$
$47$	7.16246	1.04475	0.522376	+	0.852715i	$0.325046\pi$
$48$	0	0
$49$	−1.10478	−0.157826
$50$	0	0
$51$	0	0
$52$	2.62967	0.364669
$53$	−3.81681	−0.524279	−0.262140	−	0.965030i	$-0.584428\pi$
$53$	−3.81681	−0.524279	−0.262140	+	0.965030i	$0.584428\pi$
$54$	0	0
$55$	0	0
$56$	5.10083	0.681627
$57$	0	0
$58$	−1.11930	−0.146971
$59$	6.24482	0.813006	0.406503	−	0.913649i	$-0.366748\pi$
$59$	6.24482	0.813006	0.406503	+	0.913649i	$0.366748\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0.571993	0.0726432
$63$	0	0
$64$	−1.18319	−0.147899
$65$	0	0
$66$	0	0
$67$	0.917641	0.112108	0.0560538	−	0.998428i	$-0.482148\pi$
$67$	0.917641	0.112108	0.0560538	+	0.998428i	$0.482148\pi$
$68$	−7.81681	−0.947927
$69$	0	0
$70$	0	0
$71$	6.44648	0.765056	0.382528	−	0.923944i	$-0.375054\pi$
$71$	6.44648	0.765056	0.382528	+	0.923944i	$0.375054\pi$
$72$	0	0
$73$	−6.71598	−0.786046	−0.393023	−	0.919529i	$-0.628571\pi$
$73$	−6.71598	−0.786046	−0.393023	+	0.919529i	$0.628571\pi$
$74$	−2.04316	−0.237512
$75$	0	0
$76$	−8.94233	−1.02576
$77$	2.77761	0.316538
$78$	0	0
$79$	9.52884	1.07208	0.536039	−	0.844193i	$-0.319920\pi$
$79$	9.52884	1.07208	0.536039	+	0.844193i	$0.319920\pi$
$80$	0	0
$81$	0	0
$82$	1.79834	0.198593
$83$	−3.16246	−0.347125	−0.173562	−	0.984823i	$-0.555528\pi$
$83$	−3.16246	−0.347125	−0.173562	+	0.984823i	$0.555528\pi$
$84$	0	0
$85$	0	0
$86$	−1.63362	−0.176158
$87$	0	0
$88$	2.40332	0.256195
$89$	15.2241	1.61375	0.806875	−	0.590722i	$-0.201157\pi$
$89$	15.2241	1.61375	0.806875	+	0.590722i	$0.201157\pi$
$90$	0	0
$91$	3.81681	0.400110
$92$	3.03920	0.316859
$93$	0	0
$94$	4.09688	0.422561
$95$	0	0
$96$	0	0
$97$	−10.7776	−1.09430	−0.547150	−	0.837035i	$-0.684287\pi$
$97$	−10.7776	−1.09430	−0.547150	+	0.837035i	$0.684287\pi$
$98$	−0.631929	−0.0638344
$99$	0	0
$100$	0	0
$101$	3.51037	0.349294	0.174647	−	0.984631i	$-0.444121\pi$
$101$	3.51037	0.349294	0.174647	+	0.984631i	$0.444121\pi$
$102$	0	0
$103$	−18.2633	−1.79954	−0.899768	−	0.436369i	$-0.856264\pi$
$103$	−18.2633	−1.79954	−0.899768	+	0.436369i	$0.856264\pi$
$104$	3.30249	0.323836
$105$	0	0
$106$	−2.18319	−0.212050
$107$	−4.96080	−0.479578	−0.239789	−	0.970825i	$-0.577078\pi$
$107$	−4.96080	−0.479578	−0.239789	+	0.970825i	$0.577078\pi$
$108$	0	0
$109$	6.30644	0.604048	0.302024	−	0.953300i	$-0.402338\pi$
$109$	6.30644	0.604048	0.302024	+	0.953300i	$0.402338\pi$
$110$	0	0
$111$	0	0
$112$	−5.20561	−0.491884
$113$	−3.43196	−0.322852	−0.161426	−	0.986885i	$-0.551609\pi$
$113$	−3.43196	−0.322852	−0.161426	+	0.986885i	$0.551609\pi$
$114$	0	0
$115$	0	0
$116$	3.27345	0.303933
$117$	0	0
$118$	3.57199	0.328829
$119$	−11.3456	−1.04005
$120$	0	0
$121$	−9.69129	−0.881027
$122$	−1.14399	−0.103572
$123$	0	0
$124$	−1.67282	−0.150224
$125$	0	0
$126$	0	0
$127$	−13.8353	−1.22768	−0.613841	−	0.789429i	$-0.710377\pi$
$127$	−13.8353	−1.22768	−0.613841	+	0.789429i	$0.710377\pi$
$128$	−11.5328	−1.01936
$129$	0	0
$130$	0	0
$131$	−9.79213	−0.855542	−0.427771	−	0.903887i	$-0.640701\pi$
$131$	−9.79213	−0.855542	−0.427771	+	0.903887i	$0.640701\pi$
$132$	0	0
$133$	−12.9793	−1.12545
$134$	0.524884	0.0453431
$135$	0	0
$136$	−9.81681	−0.841785
$137$	15.3641	1.31265	0.656323	−	0.754480i	$-0.272111\pi$
$137$	15.3641	1.31265	0.656323	+	0.754480i	$0.272111\pi$
$138$	0	0
$139$	−19.1809	−1.62691	−0.813453	−	0.581631i	$-0.802415\pi$
$139$	−19.1809	−1.62691	−0.813453	+	0.581631i	$0.802415\pi$
$140$	0	0
$141$	0	0
$142$	3.68734	0.309435
$143$	1.79834	0.150385
$144$	0	0
$145$	0	0
$146$	−3.84150	−0.317924
$147$	0	0
$148$	5.97531	0.491168
$149$	11.5473	0.945992	0.472996	−	0.881064i	$-0.343172\pi$
$149$	11.5473	0.945992	0.472996	+	0.881064i	$0.343172\pi$
$150$	0	0
$151$	5.61515	0.456954	0.228477	−	0.973549i	$-0.426625\pi$
$151$	5.61515	0.456954	0.228477	+	0.973549i	$0.426625\pi$
$152$	−11.2303	−0.910898
$153$	0	0
$154$	1.58877	0.128027
$155$	0	0
$156$	0	0
$157$	16.1233	1.28678	0.643388	−	0.765540i	$-0.277528\pi$
$157$	16.1233	1.28678	0.643388	+	0.765540i	$0.277528\pi$
$158$	5.45043	0.433613
$159$	0	0
$160$	0	0
$161$	4.41123	0.347653
$162$	0	0
$163$	−6.71598	−0.526036	−0.263018	−	0.964791i	$-0.584718\pi$
$163$	−6.71598	−0.526036	−0.263018	+	0.964791i	$0.584718\pi$
$164$	−5.25934	−0.410685
$165$	0	0
$166$	−1.80890	−0.140398
$167$	2.16472	0.167511	0.0837555	−	0.996486i	$-0.473309\pi$
$167$	2.16472	0.167511	0.0837555	+	0.996486i	$0.473309\pi$
$168$	0	0
$169$	−10.5288	−0.809911
$170$	0	0
$171$	0	0
$172$	4.77761	0.364289
$173$	−14.8560	−1.12948	−0.564741	−	0.825268i	$-0.691024\pi$
$173$	−14.8560	−1.12948	−0.564741	+	0.825268i	$0.691024\pi$
$174$	0	0
$175$	0	0
$176$	−2.45269	−0.184879
$177$	0	0
$178$	8.70807	0.652698
$179$	−20.0369	−1.49763	−0.748816	−	0.662778i	$-0.769377\pi$
$179$	−20.0369	−1.49763	−0.748816	+	0.662778i	$0.769377\pi$
$180$	0	0
$181$	−0.287973	−0.0214049	−0.0107024	−	0.999943i	$-0.503407\pi$
$181$	−0.287973	−0.0214049	−0.0107024	+	0.999943i	$0.503407\pi$
$182$	2.18319	0.161829
$183$	0	0
$184$	3.81681	0.281379
$185$	0	0
$186$	0	0
$187$	−5.34565	−0.390912
$188$	−11.9815	−0.873843
$189$	0	0
$190$	0	0
$191$	−10.2448	−0.741289	−0.370644	−	0.928775i	$-0.620863\pi$
$191$	−10.2448	−0.741289	−0.370644	+	0.928775i	$0.620863\pi$
$192$	0	0
$193$	−17.8353	−1.28381	−0.641906	−	0.766783i	$-0.721856\pi$
$193$	−17.8353	−1.28381	−0.641906	+	0.766783i	$0.721856\pi$
$194$	−6.16472	−0.442601
$195$	0	0
$196$	1.84811	0.132008
$197$	3.32718	0.237051	0.118526	−	0.992951i	$-0.462183\pi$
$197$	3.32718	0.237051	0.118526	+	0.992951i	$0.462183\pi$
$198$	0	0
$199$	−5.65209	−0.400666	−0.200333	−	0.979728i	$-0.564202\pi$
$199$	−5.65209	−0.400666	−0.200333	+	0.979728i	$0.564202\pi$
$200$	0	0
$201$	0	0
$202$	2.00791	0.141276
$203$	4.75123	0.333471
$204$	0	0
$205$	0	0
$206$	−10.4465	−0.727841
$207$	0	0
$208$	−3.37033	−0.233691
$209$	−6.11535	−0.423008
$210$	0	0
$211$	−22.6050	−1.55619	−0.778096	−	0.628146i	$-0.783814\pi$
$211$	−22.6050	−1.55619	−0.778096	+	0.628146i	$0.783814\pi$
$212$	6.38485	0.438513
$213$	0	0
$214$	−2.83754	−0.193970
$215$	0	0
$216$	0	0
$217$	−2.42801	−0.164824
$218$	3.60724	0.244313
$219$	0	0
$220$	0	0
$221$	−7.34565	−0.494122
$222$	0	0
$223$	4.69129	0.314152	0.157076	−	0.987586i	$-0.449793\pi$
$223$	4.69129	0.314152	0.157076	+	0.987586i	$0.449793\pi$
$224$	−13.1792	−0.880575
$225$	0	0
$226$	−1.96306	−0.130581
$227$	1.90312	0.126315	0.0631573	−	0.998004i	$-0.479883\pi$
$227$	1.90312	0.126315	0.0631573	+	0.998004i	$0.479883\pi$
$228$	0	0
$229$	−21.1440	−1.39723	−0.698617	−	0.715496i	$-0.746201\pi$
$229$	−21.1440	−1.39723	−0.698617	+	0.715496i	$0.746201\pi$
$230$	0	0
$231$	0	0
$232$	4.11100	0.269900
$233$	−26.7882	−1.75495	−0.877476	−	0.479621i	$-0.840774\pi$
$233$	−26.7882	−1.75495	−0.877476	+	0.479621i	$0.840774\pi$
$234$	0	0
$235$	0	0
$236$	−10.4465	−0.680008
$237$	0	0
$238$	−6.48963	−0.420660
$239$	−24.3249	−1.57345	−0.786724	−	0.617305i	$-0.788224\pi$
$239$	−24.3249	−1.57345	−0.786724	+	0.617305i	$0.788224\pi$
$240$	0	0
$241$	6.48963	0.418034	0.209017	−	0.977912i	$-0.432974\pi$
$241$	6.48963	0.418034	0.209017	+	0.977912i	$0.432974\pi$
$242$	−5.54336	−0.356340
$243$	0	0
$244$	3.34565	0.214183
$245$	0	0
$246$	0	0
$247$	−8.40332	−0.534691
$248$	−2.10083	−0.133403
$249$	0	0
$250$	0	0
$251$	21.9216	1.38368	0.691839	−	0.722051i	$-0.256801\pi$
$251$	21.9216	1.38368	0.691839	+	0.722051i	$0.256801\pi$
$252$	0	0
$253$	2.07841	0.130668
$254$	−7.91369	−0.496549
$255$	0	0
$256$	−4.23030	−0.264394
$257$	−7.73050	−0.482215	−0.241108	−	0.970498i	$-0.577511\pi$
$257$	−7.73050	−0.482215	−0.241108	+	0.970498i	$0.577511\pi$
$258$	0	0
$259$	8.67282	0.538903
$260$	0	0
$261$	0	0
$262$	−5.60103	−0.346033
$263$	25.4689	1.57048	0.785240	−	0.619192i	$-0.212540\pi$
$263$	25.4689	1.57048	0.785240	+	0.619192i	$0.212540\pi$
$264$	0	0
$265$	0	0
$266$	−7.42405	−0.455198
$267$	0	0
$268$	−1.53505	−0.0937682
$269$	3.75518	0.228958	0.114479	−	0.993426i	$-0.463480\pi$
$269$	3.75518	0.228958	0.114479	+	0.993426i	$0.463480\pi$
$270$	0	0
$271$	−13.6257	−0.827703	−0.413852	−	0.910344i	$-0.635817\pi$
$271$	−13.6257	−0.827703	−0.413852	+	0.910344i	$0.635817\pi$
$272$	10.0185	0.607459
$273$	0	0
$274$	8.78817	0.530913
$275$	0	0
$276$	0	0
$277$	19.3994	1.16560	0.582798	−	0.812617i	$-0.301958\pi$
$277$	19.3994	1.16560	0.582798	+	0.812617i	$0.301958\pi$
$278$	−10.9714	−0.658019
$279$	0	0
$280$	0	0
$281$	9.14399	0.545485	0.272742	−	0.962087i	$-0.412069\pi$
$281$	9.14399	0.545485	0.272742	+	0.962087i	$0.412069\pi$
$282$	0	0
$283$	−22.0616	−1.31143	−0.655714	−	0.755010i	$-0.727632\pi$
$283$	−22.0616	−1.31143	−0.655714	+	0.755010i	$0.727632\pi$
$284$	−10.7838	−0.639902
$285$	0	0
$286$	1.02864	0.0608246
$287$	−7.63362	−0.450598
$288$	0	0
$289$	4.83528	0.284428
$290$	0	0
$291$	0	0
$292$	11.2347	0.657458
$293$	−7.90312	−0.461705	−0.230853	−	0.972989i	$-0.574152\pi$
$293$	−7.90312	−0.461705	−0.230853	+	0.972989i	$0.574152\pi$
$294$	0	0
$295$	0	0
$296$	7.50415	0.436170
$297$	0	0
$298$	6.60498	0.382616
$299$	2.85601	0.165168
$300$	0	0
$301$	6.93442	0.399693
$302$	3.21183	0.184820
$303$	0	0
$304$	11.4610	0.657333
$305$	0	0
$306$	0	0
$307$	24.3865	1.39181	0.695907	−	0.718132i	$-0.255003\pi$
$307$	24.3865	1.39181	0.695907	+	0.718132i	$0.255003\pi$
$308$	−4.64645	−0.264756
$309$	0	0
$310$	0	0
$311$	−32.3681	−1.83542	−0.917712	−	0.397245i	$-0.869966\pi$
$311$	−32.3681	−1.83542	−0.917712	+	0.397245i	$0.869966\pi$
$312$	0	0
$313$	−10.8313	−0.612223	−0.306111	−	0.951996i	$-0.599028\pi$
$313$	−10.8313	−0.612223	−0.306111	+	0.951996i	$0.599028\pi$
$314$	9.22239	0.520450
$315$	0	0
$316$	−15.9401	−0.896699
$317$	−15.3641	−0.862935	−0.431467	−	0.902129i	$-0.642004\pi$
$317$	−15.3641	−0.862935	−0.431467	+	0.902129i	$0.642004\pi$
$318$	0	0
$319$	2.23860	0.125338
$320$	0	0
$321$	0	0
$322$	2.52319	0.140612
$323$	24.9793	1.38988
$324$	0	0
$325$	0	0
$326$	−3.84150	−0.212761
$327$	0	0
$328$	−6.60498	−0.364699
$329$	−17.3905	−0.958769
$330$	0	0
$331$	22.1417	1.21702	0.608510	−	0.793546i	$-0.291768\pi$
$331$	22.1417	1.21702	0.608510	+	0.793546i	$0.291768\pi$
$332$	5.29023	0.290339
$333$	0	0
$334$	1.23820	0.0677515
$335$	0	0
$336$	0	0
$337$	−25.6873	−1.39928	−0.699639	−	0.714496i	$-0.746656\pi$
$337$	−25.6873	−1.39928	−0.699639	+	0.714496i	$0.746656\pi$
$338$	−6.02242	−0.327577
$339$	0	0
$340$	0	0
$341$	−1.14399	−0.0619503
$342$	0	0
$343$	19.6785	1.06254
$344$	6.00000	0.323498
$345$	0	0
$346$	−8.49754	−0.456831
$347$	17.2593	0.926530	0.463265	−	0.886220i	$-0.346678\pi$
$347$	17.2593	0.926530	0.463265	+	0.886220i	$0.346678\pi$
$348$	0	0
$349$	−30.7098	−1.64386	−0.821928	−	0.569591i	$-0.807101\pi$
$349$	−30.7098	−1.64386	−0.821928	+	0.569591i	$0.807101\pi$
$350$	0	0
$351$	0	0
$352$	−6.20957	−0.330971
$353$	−32.1338	−1.71031	−0.855155	−	0.518372i	$-0.826538\pi$
$353$	−32.1338	−1.71031	−0.855155	+	0.518372i	$0.826538\pi$
$354$	0	0
$355$	0	0
$356$	−25.4672	−1.34976
$357$	0	0
$358$	−11.4610	−0.605732
$359$	25.0594	1.32258	0.661291	−	0.750129i	$-0.270009\pi$
$359$	25.0594	1.32258	0.661291	+	0.750129i	$0.270009\pi$
$360$	0	0
$361$	9.57595	0.503997
$362$	−0.164719	−0.00865742
$363$	0	0
$364$	−6.38485	−0.334657
$365$	0	0
$366$	0	0
$367$	1.71203	0.0893671	0.0446835	−	0.999001i	$-0.485772\pi$
$367$	1.71203	0.0893671	0.0446835	+	0.999001i	$0.485772\pi$
$368$	−3.89522	−0.203052
$369$	0	0
$370$	0	0
$371$	9.26724	0.481131
$372$	0	0
$373$	32.4482	1.68010	0.840051	−	0.542507i	$-0.182525\pi$
$373$	32.4482	1.68010	0.840051	+	0.542507i	$0.182525\pi$
$374$	−3.05767	−0.158109
$375$	0	0
$376$	−15.0471	−0.775995
$377$	3.07615	0.158430
$378$	0	0
$379$	−1.51827	−0.0779884	−0.0389942	−	0.999239i	$-0.512415\pi$
$379$	−1.51827	−0.0779884	−0.0389942	+	0.999239i	$0.512415\pi$
$380$	0	0
$381$	0	0
$382$	−5.85997	−0.299822
$383$	30.2201	1.54418	0.772088	−	0.635515i	$-0.219212\pi$
$383$	30.2201	1.54418	0.772088	+	0.635515i	$0.219212\pi$
$384$	0	0
$385$	0	0
$386$	−10.2017	−0.519251
$387$	0	0
$388$	18.0290	0.915286
$389$	−7.67678	−0.389228	−0.194614	−	0.980880i	$-0.562345\pi$
$389$	−7.67678	−0.389228	−0.194614	+	0.980880i	$0.562345\pi$
$390$	0	0
$391$	−8.48963	−0.429339
$392$	2.32096	0.117226
$393$	0	0
$394$	1.90312	0.0958779
$395$	0	0
$396$	0	0
$397$	14.3664	0.721028	0.360514	−	0.932754i	$-0.382601\pi$
$397$	14.3664	0.721028	0.360514	+	0.932754i	$0.382601\pi$
$398$	−3.23296	−0.162054
$399$	0	0
$400$	0	0
$401$	−10.2448	−0.511602	−0.255801	−	0.966729i	$-0.582339\pi$
$401$	−10.2448	−0.511602	−0.255801	+	0.966729i	$0.582339\pi$
$402$	0	0
$403$	−1.57199	−0.0783066
$404$	−5.87222	−0.292154
$405$	0	0
$406$	2.71767	0.134876
$407$	4.08631	0.202551
$408$	0	0
$409$	−24.2880	−1.20096	−0.600481	−	0.799639i	$-0.705024\pi$
$409$	−24.2880	−1.20096	−0.600481	+	0.799639i	$0.705024\pi$
$410$	0	0
$411$	0	0
$412$	30.5513	1.50515
$413$	−15.1625	−0.746096
$414$	0	0
$415$	0	0
$416$	−8.53279	−0.418354
$417$	0	0
$418$	−3.49794	−0.171090
$419$	37.9938	1.85612	0.928059	−	0.372433i	$-0.121476\pi$
$419$	37.9938	1.85612	0.928059	+	0.372433i	$0.121476\pi$
$420$	0	0
$421$	31.5658	1.53842	0.769211	−	0.638995i	$-0.220650\pi$
$421$	31.5658	1.53842	0.769211	+	0.638995i	$0.220650\pi$
$422$	−12.9299	−0.629418
$423$	0	0
$424$	8.01847	0.389411
$425$	0	0
$426$	0	0
$427$	4.85601	0.234999
$428$	8.29854	0.401125
$429$	0	0
$430$	0	0
$431$	14.8992	0.717668	0.358834	−	0.933401i	$-0.383174\pi$
$431$	14.8992	0.717668	0.358834	+	0.933401i	$0.383174\pi$
$432$	0	0
$433$	15.9753	0.767725	0.383862	−	0.923390i	$-0.374594\pi$
$433$	15.9753	0.767725	0.383862	+	0.923390i	$0.374594\pi$
$434$	−1.38880	−0.0666647
$435$	0	0
$436$	−10.5496	−0.505233
$437$	−9.71203	−0.464589
$438$	0	0
$439$	−32.0739	−1.53080	−0.765401	−	0.643553i	$-0.777460\pi$
$439$	−32.0739	−1.53080	−0.765401	+	0.643553i	$0.777460\pi$
$440$	0	0
$441$	0	0
$442$	−4.20166	−0.199853
$443$	−12.5944	−0.598379	−0.299189	−	0.954194i	$-0.596716\pi$
$443$	−12.5944	−0.598379	−0.299189	+	0.954194i	$0.596716\pi$
$444$	0	0
$445$	0	0
$446$	2.68339	0.127062
$447$	0	0
$448$	2.87279	0.135727
$449$	7.46721	0.352399	0.176200	−	0.984354i	$-0.443620\pi$
$449$	7.46721	0.352399	0.176200	+	0.984354i	$0.443620\pi$
$450$	0	0
$451$	−3.59668	−0.169361
$452$	5.74106	0.270037
$453$	0	0
$454$	1.08857	0.0510893
$455$	0	0
$456$	0	0
$457$	20.1849	0.944209	0.472104	−	0.881543i	$-0.343495\pi$
$457$	20.1849	0.944209	0.472104	+	0.881543i	$0.343495\pi$
$458$	−12.0942	−0.565126
$459$	0	0
$460$	0	0
$461$	−28.0017	−1.30417	−0.652084	−	0.758146i	$-0.726105\pi$
$461$	−28.0017	−1.30417	−0.652084	+	0.758146i	$0.726105\pi$
$462$	0	0
$463$	38.4896	1.78876	0.894382	−	0.447303i	$-0.147615\pi$
$463$	38.4896	1.78876	0.894382	+	0.447303i	$0.147615\pi$
$464$	−4.19545	−0.194769
$465$	0	0
$466$	−15.3227	−0.709808
$467$	−20.2985	−0.939304	−0.469652	−	0.882852i	$-0.655621\pi$
$467$	−20.2985	−0.939304	−0.469652	+	0.882852i	$0.655621\pi$
$468$	0	0
$469$	−2.22804	−0.102881
$470$	0	0
$471$	0	0
$472$	−13.1193	−0.603865
$473$	3.26724	0.150228
$474$	0	0
$475$	0	0
$476$	18.9793	0.869913
$477$	0	0
$478$	−13.9137	−0.636397
$479$	19.0594	0.870845	0.435422	−	0.900226i	$-0.356599\pi$
$479$	19.0594	0.870845	0.435422	+	0.900226i	$0.356599\pi$
$480$	0	0
$481$	5.61515	0.256029
$482$	3.71203	0.169078
$483$	0	0
$484$	16.2118	0.736901
$485$	0	0
$486$	0	0
$487$	−13.0656	−0.592058	−0.296029	−	0.955179i	$-0.595662\pi$
$487$	−13.0656	−0.592058	−0.296029	+	0.955179i	$0.595662\pi$
$488$	4.20166	0.190200
$489$	0	0
$490$	0	0
$491$	−24.3170	−1.09741	−0.548706	−	0.836016i	$-0.684879\pi$
$491$	−24.3170	−1.09741	−0.548706	+	0.836016i	$0.684879\pi$
$492$	0	0
$493$	−9.14399	−0.411824
$494$	−4.80664	−0.216261
$495$	0	0
$496$	2.14399	0.0962678
$497$	−15.6521	−0.702092
$498$	0	0
$499$	−16.4033	−0.734314	−0.367157	−	0.930159i	$-0.619669\pi$
$499$	−16.4033	−0.734314	−0.367157	+	0.930159i	$0.619669\pi$
$500$	0	0
$501$	0	0
$502$	12.5390	0.559643
$503$	−0.837542	−0.0373442	−0.0186721	−	0.999826i	$-0.505944\pi$
$503$	−0.837542	−0.0373442	−0.0186721	+	0.999826i	$0.505944\pi$
$504$	0	0
$505$	0	0
$506$	1.18883	0.0528501
$507$	0	0
$508$	23.1440	1.02685
$509$	1.38880	0.0615576	0.0307788	−	0.999526i	$-0.490201\pi$
$509$	1.38880	0.0615576	0.0307788	+	0.999526i	$0.490201\pi$
$510$	0	0
$511$	16.3064	0.721355
$512$	20.6459	0.912428
$513$	0	0
$514$	−4.42179	−0.195037
$515$	0	0
$516$	0	0
$517$	−8.19376	−0.360361
$518$	4.96080	0.217965
$519$	0	0
$520$	0	0
$521$	−33.1025	−1.45025	−0.725124	−	0.688618i	$-0.758218\pi$
$521$	−33.1025	−1.45025	−0.725124	+	0.688618i	$0.758218\pi$
$522$	0	0
$523$	−1.62571	−0.0710875	−0.0355438	−	0.999368i	$-0.511316\pi$
$523$	−1.62571	−0.0710875	−0.0355438	+	0.999368i	$0.511316\pi$
$524$	16.3805	0.715585
$525$	0	0
$526$	14.5680	0.635197
$527$	4.67282	0.203551
$528$	0	0
$529$	−19.6992	−0.856487
$530$	0	0
$531$	0	0
$532$	21.7120	0.941336
$533$	−4.94233	−0.214076
$534$	0	0
$535$	0	0
$536$	−1.92781	−0.0832686
$537$	0	0
$538$	2.14794	0.0926042
$539$	1.26386	0.0544382
$540$	0	0
$541$	26.5865	1.14304	0.571522	−	0.820587i	$-0.306353\pi$
$541$	26.5865	1.14304	0.571522	+	0.820587i	$0.306353\pi$
$542$	−7.79382	−0.334773
$543$	0	0
$544$	25.3641	1.08748
$545$	0	0
$546$	0	0
$547$	19.9753	0.854083	0.427041	−	0.904232i	$-0.359556\pi$
$547$	19.9753	0.854083	0.427041	+	0.904232i	$0.359556\pi$
$548$	−25.7015	−1.09791
$549$	0	0
$550$	0	0
$551$	−10.4606	−0.445636
$552$	0	0
$553$	−23.1361	−0.983846
$554$	11.0963	0.471437
$555$	0	0
$556$	32.0863	1.36076
$557$	−29.2488	−1.23931	−0.619655	−	0.784874i	$-0.712728\pi$
$557$	−29.2488	−1.23931	−0.619655	+	0.784874i	$0.712728\pi$
$558$	0	0
$559$	4.48963	0.189891
$560$	0	0
$561$	0	0
$562$	5.23030	0.220627
$563$	0.960797	0.0404928	0.0202464	−	0.999795i	$-0.493555\pi$
$563$	0.960797	0.0404928	0.0202464	+	0.999795i	$0.493555\pi$
$564$	0	0
$565$	0	0
$566$	−12.6191	−0.530420
$567$	0	0
$568$	−13.5430	−0.568250
$569$	3.87844	0.162593	0.0812963	−	0.996690i	$-0.474094\pi$
$569$	3.87844	0.162593	0.0812963	+	0.996690i	$0.474094\pi$
$570$	0	0
$571$	−6.20166	−0.259531	−0.129766	−	0.991545i	$-0.541423\pi$
$571$	−6.20166	−0.259531	−0.129766	+	0.991545i	$0.541423\pi$
$572$	−3.00830	−0.125784
$573$	0	0
$574$	−4.36638	−0.182249
$575$	0	0
$576$	0	0
$577$	−35.6336	−1.48345	−0.741724	−	0.670706i	$-0.765991\pi$
$577$	−35.6336	−1.48345	−0.741724	+	0.670706i	$0.765991\pi$
$578$	2.76575	0.115040
$579$	0	0
$580$	0	0
$581$	7.67847	0.318557
$582$	0	0
$583$	4.36638	0.180837
$584$	14.1091	0.583840
$585$	0	0
$586$	−4.52053	−0.186741
$587$	−43.7569	−1.80604	−0.903020	−	0.429599i	$-0.858655\pi$
$587$	−43.7569	−1.80604	−0.903020	+	0.429599i	$0.858655\pi$
$588$	0	0
$589$	5.34565	0.220264
$590$	0	0
$591$	0	0
$592$	−7.65831	−0.314754
$593$	17.3536	0.712625	0.356313	−	0.934367i	$-0.384034\pi$
$593$	17.3536	0.712625	0.356313	+	0.934367i	$0.384034\pi$
$594$	0	0
$595$	0	0
$596$	−19.3166	−0.791239
$597$	0	0
$598$	1.63362	0.0668037
$599$	−23.7552	−0.970610	−0.485305	−	0.874345i	$-0.661292\pi$
$599$	−23.7552	−0.970610	−0.485305	+	0.874345i	$0.661292\pi$
$600$	0	0
$601$	−41.9506	−1.71120	−0.855601	−	0.517636i	$-0.826812\pi$
$601$	−41.9506	−1.71120	−0.855601	+	0.517636i	$0.826812\pi$
$602$	3.96644	0.161660
$603$	0	0
$604$	−9.39315	−0.382202
$605$	0	0
$606$	0	0
$607$	4.25538	0.172721	0.0863603	−	0.996264i	$-0.472476\pi$
$607$	4.25538	0.172721	0.0863603	+	0.996264i	$0.472476\pi$
$608$	29.0162	1.17676
$609$	0	0
$610$	0	0
$611$	−11.2593	−0.455504
$612$	0	0
$613$	−11.7367	−0.474041	−0.237021	−	0.971505i	$-0.576171\pi$
$613$	−11.7367	−0.474041	−0.237021	+	0.971505i	$0.576171\pi$
$614$	13.9489	0.562933
$615$	0	0
$616$	−5.83528	−0.235110
$617$	−10.5759	−0.425772	−0.212886	−	0.977077i	$-0.568286\pi$
$617$	−10.5759	−0.425772	−0.212886	+	0.977077i	$0.568286\pi$
$618$	0	0
$619$	29.1994	1.17362	0.586811	−	0.809724i	$-0.300383\pi$
$619$	29.1994	1.17362	0.586811	+	0.809724i	$0.300383\pi$
$620$	0	0
$621$	0	0
$622$	−18.5143	−0.742357
$623$	−36.9642	−1.48094
$624$	0	0
$625$	0	0
$626$	−6.19545	−0.247620
$627$	0	0
$628$	−26.9714	−1.07627
$629$	−16.6913	−0.665526
$630$	0	0
$631$	19.6706	0.783073	0.391536	−	0.920163i	$-0.371944\pi$
$631$	19.6706	0.783073	0.391536	+	0.920163i	$0.371944\pi$
$632$	−20.0185	−0.796292
$633$	0	0
$634$	−8.78817	−0.349023
$635$	0	0
$636$	0	0
$637$	1.73671	0.0688110
$638$	1.28047	0.0506941
$639$	0	0
$640$	0	0
$641$	−40.3681	−1.59444	−0.797221	−	0.603687i	$-0.793698\pi$
$641$	−40.3681	−1.59444	−0.797221	+	0.603687i	$0.793698\pi$
$642$	0	0
$643$	−2.20166	−0.0868250	−0.0434125	−	0.999057i	$-0.513823\pi$
$643$	−2.20166	−0.0868250	−0.0434125	+	0.999057i	$0.513823\pi$
$644$	−7.37921	−0.290781
$645$	0	0
$646$	14.2880	0.562153
$647$	41.0162	1.61251	0.806257	−	0.591566i	$-0.201490\pi$
$647$	41.0162	1.61251	0.806257	+	0.591566i	$0.201490\pi$
$648$	0	0
$649$	−7.14399	−0.280426
$650$	0	0
$651$	0	0
$652$	11.2347	0.439983
$653$	−5.00226	−0.195754	−0.0978768	−	0.995199i	$-0.531205\pi$
$653$	−5.00226	−0.195754	−0.0978768	+	0.995199i	$0.531205\pi$
$654$	0	0
$655$	0	0
$656$	6.74066	0.263179
$657$	0	0
$658$	−9.94725	−0.387784
$659$	−11.5826	−0.451192	−0.225596	−	0.974221i	$-0.572433\pi$
$659$	−11.5826	−0.451192	−0.225596	+	0.974221i	$0.572433\pi$
$660$	0	0
$661$	48.2386	1.87626	0.938132	−	0.346278i	$-0.112554\pi$
$661$	48.2386	1.87626	0.938132	+	0.346278i	$0.112554\pi$
$662$	12.6649	0.492236
$663$	0	0
$664$	6.64379	0.257829
$665$	0	0
$666$	0	0
$667$	3.55521	0.137658
$668$	−3.62119	−0.140108
$669$	0	0
$670$	0	0
$671$	2.28797	0.0883262
$672$	0	0
$673$	−18.3417	−0.707020	−0.353510	−	0.935431i	$-0.615012\pi$
$673$	−18.3417	−0.707020	−0.353510	+	0.935431i	$0.615012\pi$
$674$	−14.6930	−0.565953
$675$	0	0
$676$	17.6129	0.677419
$677$	5.33508	0.205044	0.102522	−	0.994731i	$-0.467309\pi$
$677$	5.33508	0.205044	0.102522	+	0.994731i	$0.467309\pi$
$678$	0	0
$679$	26.1681	1.00424
$680$	0	0
$681$	0	0
$682$	−0.654353	−0.0250565
$683$	14.2695	0.546007	0.273004	−	0.962013i	$-0.411983\pi$
$683$	14.2695	0.546007	0.273004	+	0.962013i	$0.411983\pi$
$684$	0	0
$685$	0	0
$686$	11.2560	0.429754
$687$	0	0
$688$	−6.12325	−0.233447
$689$	6.00000	0.228582
$690$	0	0
$691$	−24.1233	−0.917692	−0.458846	−	0.888516i	$-0.651737\pi$
$691$	−24.1233	−0.917692	−0.458846	+	0.888516i	$0.651737\pi$
$692$	24.8515	0.944712
$693$	0	0
$694$	9.87222	0.374744
$695$	0	0
$696$	0	0
$697$	14.6913	0.556472
$698$	−17.5658	−0.664875
$699$	0	0
$700$	0	0
$701$	3.26724	0.123402	0.0617010	−	0.998095i	$-0.480347\pi$
$701$	3.26724	0.123402	0.0617010	+	0.998095i	$0.480347\pi$
$702$	0	0
$703$	−19.0946	−0.720167
$704$	1.35355	0.0510140
$705$	0	0
$706$	−18.3803	−0.691753
$707$	−8.52319	−0.320548
$708$	0	0
$709$	49.1104	1.84438	0.922190	−	0.386736i	$-0.126398\pi$
$709$	49.1104	1.84438	0.922190	+	0.386736i	$0.126398\pi$
$710$	0	0
$711$	0	0
$712$	−31.9832	−1.19862
$713$	−1.81681	−0.0680401
$714$	0	0
$715$	0	0
$716$	33.5183	1.25264
$717$	0	0
$718$	14.3338	0.534932
$719$	−11.4320	−0.426340	−0.213170	−	0.977015i	$-0.568379\pi$
$719$	−11.4320	−0.426340	−0.213170	+	0.977015i	$0.568379\pi$
$720$	0	0
$721$	44.3434	1.65143
$722$	5.47738	0.203847
$723$	0	0
$724$	0.481728	0.0179033
$725$	0	0
$726$	0	0
$727$	0.514319	0.0190750	0.00953752	−	0.999955i	$-0.496964\pi$
$727$	0.514319	0.0190750	0.00953752	+	0.999955i	$0.496964\pi$
$728$	−8.01847	−0.297184
$729$	0	0
$730$	0	0
$731$	−13.3456	−0.493607
$732$	0	0
$733$	−35.7859	−1.32178	−0.660891	−	0.750482i	$-0.729822\pi$
$733$	−35.7859	−1.32178	−0.660891	+	0.750482i	$0.729822\pi$
$734$	0.979268	0.0361454
$735$	0	0
$736$	−9.86166	−0.363506
$737$	−1.04977	−0.0386687
$738$	0	0
$739$	33.2857	1.22443	0.612217	−	0.790690i	$-0.290278\pi$
$739$	33.2857	1.22443	0.612217	+	0.790690i	$0.290278\pi$
$740$	0	0
$741$	0	0
$742$	5.30080	0.194598
$743$	38.0448	1.39573	0.697865	−	0.716229i	$-0.254134\pi$
$743$	38.0448	1.39573	0.697865	+	0.716229i	$0.254134\pi$
$744$	0	0
$745$	0	0
$746$	18.5601	0.679535
$747$	0	0
$748$	8.94233	0.326964
$749$	12.0448	0.440109
$750$	0	0
$751$	−6.20166	−0.226302	−0.113151	−	0.993578i	$-0.536094\pi$
$751$	−6.20166	−0.226302	−0.113151	+	0.993578i	$0.536094\pi$
$752$	15.3562	0.559983
$753$	0	0
$754$	1.75953	0.0640785
$755$	0	0
$756$	0	0
$757$	0.522225	0.0189806	0.00949029	−	0.999955i	$-0.496979\pi$
$757$	0.522225	0.0189806	0.00949029	+	0.999955i	$0.496979\pi$
$758$	−0.868441	−0.0315432
$759$	0	0
$760$	0	0
$761$	4.07219	0.147617	0.0738084	−	0.997272i	$-0.476485\pi$
$761$	4.07219	0.147617	0.0738084	+	0.997272i	$0.476485\pi$
$762$	0	0
$763$	−15.3121	−0.554335
$764$	17.1378	0.620023
$765$	0	0
$766$	17.2857	0.624558
$767$	−9.81681	−0.354464
$768$	0	0
$769$	−19.8089	−0.714327	−0.357164	−	0.934042i	$-0.616256\pi$
$769$	−19.8089	−0.714327	−0.357164	+	0.934042i	$0.616256\pi$
$770$	0	0
$771$	0	0
$772$	29.8353	1.07380
$773$	−40.3803	−1.45238	−0.726190	−	0.687494i	$-0.758711\pi$
$773$	−40.3803	−1.45238	−0.726190	+	0.687494i	$0.758711\pi$
$774$	0	0
$775$	0	0
$776$	22.6419	0.812798
$777$	0	0
$778$	−4.39106	−0.157427
$779$	16.8066	0.602160
$780$	0	0
$781$	−7.37468	−0.263887
$782$	−4.85601	−0.173651
$783$	0	0
$784$	−2.36864	−0.0845943
$785$	0	0
$786$	0	0
$787$	−19.1730	−0.683444	−0.341722	−	0.939801i	$-0.611010\pi$
$787$	−19.1730	−0.683444	−0.341722	+	0.939801i	$0.611010\pi$
$788$	−5.56578	−0.198273
$789$	0	0
$790$	0	0
$791$	8.33282	0.296281
$792$	0	0
$793$	3.14399	0.111646
$794$	8.21747	0.291627
$795$	0	0
$796$	9.45495	0.335122
$797$	39.0347	1.38268	0.691340	−	0.722530i	$-0.257021\pi$
$797$	39.0347	1.38268	0.691340	+	0.722530i	$0.257021\pi$
$798$	0	0
$799$	33.4689	1.18404
$800$	0	0
$801$	0	0
$802$	−5.85997	−0.206923
$803$	7.68299	0.271127
$804$	0	0
$805$	0	0
$806$	−0.899170	−0.0316719
$807$	0	0
$808$	−7.37468	−0.259440
$809$	25.4627	0.895220	0.447610	−	0.894229i	$-0.352275\pi$
$809$	25.4627	0.895220	0.447610	+	0.894229i	$0.352275\pi$
$810$	0	0
$811$	−43.1440	−1.51499	−0.757495	−	0.652841i	$-0.773577\pi$
$811$	−43.1440	−1.51499	−0.757495	+	0.652841i	$0.773577\pi$
$812$	−7.94797	−0.278919
$813$	0	0
$814$	2.33734	0.0819238
$815$	0	0
$816$	0	0
$817$	−15.2672	−0.534133
$818$	−13.8926	−0.485742
$819$	0	0
$820$	0	0
$821$	−12.2527	−0.427623	−0.213811	−	0.976875i	$-0.568588\pi$
$821$	−12.2527	−0.427623	−0.213811	+	0.976875i	$0.568588\pi$
$822$	0	0
$823$	33.7569	1.17669	0.588345	−	0.808610i	$-0.299780\pi$
$823$	33.7569	1.17669	0.588345	+	0.808610i	$0.299780\pi$
$824$	38.3681	1.33662
$825$	0	0
$826$	−8.67282	−0.301766
$827$	−37.2778	−1.29628	−0.648138	−	0.761523i	$-0.724452\pi$
$827$	−37.2778	−1.29628	−0.648138	+	0.761523i	$0.724452\pi$
$828$	0	0
$829$	26.3170	0.914028	0.457014	−	0.889460i	$-0.348919\pi$
$829$	26.3170	0.914028	0.457014	+	0.889460i	$0.348919\pi$
$830$	0	0
$831$	0	0
$832$	1.85997	0.0644827
$833$	−5.16246	−0.178869
$834$	0	0
$835$	0	0
$836$	10.2299	0.353808
$837$	0	0
$838$	21.7322	0.750726
$839$	−28.1664	−0.972412	−0.486206	−	0.873844i	$-0.661620\pi$
$839$	−28.1664	−0.972412	−0.486206	+	0.873844i	$0.661620\pi$
$840$	0	0
$841$	−25.1708	−0.867957
$842$	18.0554	0.622231
$843$	0	0
$844$	37.8142	1.30162
$845$	0	0
$846$	0	0
$847$	23.5305	0.808519
$848$	−8.18319	−0.281012
$849$	0	0
$850$	0	0
$851$	6.48963	0.222462
$852$	0	0
$853$	7.67056	0.262635	0.131318	−	0.991340i	$-0.458079\pi$
$853$	7.67056	0.262635	0.131318	+	0.991340i	$0.458079\pi$
$854$	2.77761	0.0950478
$855$	0	0
$856$	10.4218	0.356210
$857$	−42.6992	−1.45858	−0.729288	−	0.684206i	$-0.760149\pi$
$857$	−42.6992	−1.45858	−0.729288	+	0.684206i	$0.760149\pi$
$858$	0	0
$859$	−44.2465	−1.50967	−0.754836	−	0.655914i	$-0.772283\pi$
$859$	−44.2465	−1.50967	−0.754836	+	0.655914i	$0.772283\pi$
$860$	0	0
$861$	0	0
$862$	8.52222	0.290268
$863$	22.8639	0.778297	0.389148	−	0.921175i	$-0.372769\pi$
$863$	22.8639	0.778297	0.389148	+	0.921175i	$0.372769\pi$
$864$	0	0
$865$	0	0
$866$	9.13777	0.310514
$867$	0	0
$868$	4.06163	0.137861
$869$	−10.9009	−0.369786
$870$	0	0
$871$	−1.44252	−0.0488781
$872$	−13.2488	−0.448660
$873$	0	0
$874$	−5.55521	−0.187908
$875$	0	0
$876$	0	0
$877$	−31.0577	−1.04874	−0.524372	−	0.851490i	$-0.675700\pi$
$877$	−31.0577	−1.04874	−0.524372	+	0.851490i	$0.675700\pi$
$878$	−18.3460	−0.619149
$879$	0	0
$880$	0	0
$881$	17.8784	0.602340	0.301170	−	0.953570i	$-0.402623\pi$
$881$	17.8784	0.602340	0.301170	+	0.953570i	$0.402623\pi$
$882$	0	0
$883$	22.2096	0.747411	0.373706	−	0.927547i	$-0.378087\pi$
$883$	22.2096	0.747411	0.373706	+	0.927547i	$0.378087\pi$
$884$	12.2880	0.413289
$885$	0	0
$886$	−7.20392	−0.242020
$887$	−4.10478	−0.137825	−0.0689126	−	0.997623i	$-0.521953\pi$
$887$	−4.10478	−0.137825	−0.0689126	+	0.997623i	$0.521953\pi$
$888$	0	0
$889$	33.5922	1.12664
$890$	0	0
$891$	0	0
$892$	−7.84771	−0.262761
$893$	38.2880	1.28126
$894$	0	0
$895$	0	0
$896$	28.0017	0.935471
$897$	0	0
$898$	4.27119	0.142532
$899$	−1.95684	−0.0652644
$900$	0	0
$901$	−17.8353	−0.594179
$902$	−2.05728	−0.0684998
$903$	0	0
$904$	7.20997	0.239800
$905$	0	0
$906$	0	0
$907$	−8.62967	−0.286543	−0.143272	−	0.989683i	$-0.545762\pi$
$907$	−8.62967	−0.286543	−0.143272	+	0.989683i	$0.545762\pi$
$908$	−3.18359	−0.105651
$909$	0	0
$910$	0	0
$911$	−7.83528	−0.259594	−0.129797	−	0.991541i	$-0.541433\pi$
$911$	−7.83528	−0.259594	−0.129797	+	0.991541i	$0.541433\pi$
$912$	0	0
$913$	3.61781	0.119732
$914$	11.5456	0.381895
$915$	0	0
$916$	35.3702	1.16866
$917$	23.7753	0.785131
$918$	0	0
$919$	15.2514	0.503098	0.251549	−	0.967845i	$-0.419060\pi$
$919$	15.2514	0.503098	0.251549	+	0.967845i	$0.419060\pi$
$920$	0	0
$921$	0	0
$922$	−16.0168	−0.527485
$923$	−10.1338	−0.333559
$924$	0	0
$925$	0	0
$926$	22.0158	0.723484
$927$	0	0
$928$	−10.6218	−0.348676
$929$	4.15850	0.136436	0.0682181	−	0.997670i	$-0.478269\pi$
$929$	4.15850	0.136436	0.0682181	+	0.997670i	$0.478269\pi$
$930$	0	0
$931$	−5.90578	−0.193554
$932$	44.8119	1.46786
$933$	0	0
$934$	−11.6106	−0.379911
$935$	0	0
$936$	0	0
$937$	18.2175	0.595139	0.297569	−	0.954700i	$-0.403824\pi$
$937$	18.2175	0.595139	0.297569	+	0.954700i	$0.403824\pi$
$938$	−1.27442	−0.0416114
$939$	0	0
$940$	0	0
$941$	5.84150	0.190427	0.0952137	−	0.995457i	$-0.469647\pi$
$941$	5.84150	0.190427	0.0952137	+	0.995457i	$0.469647\pi$
$942$	0	0
$943$	−5.71203	−0.186009
$944$	13.3888	0.435768
$945$	0	0
$946$	1.86884	0.0607612
$947$	57.9276	1.88240	0.941198	−	0.337856i	$-0.109702\pi$
$947$	57.9276	1.88240	0.941198	+	0.337856i	$0.109702\pi$
$948$	0	0
$949$	10.5575	0.342710
$950$	0	0
$951$	0	0
$952$	23.8353	0.772506
$953$	22.9977	0.744970	0.372485	−	0.928038i	$-0.378506\pi$
$953$	22.9977	0.744970	0.372485	+	0.928038i	$0.378506\pi$
$954$	0	0
$955$	0	0
$956$	40.6913	1.31605
$957$	0	0
$958$	10.9018	0.352222
$959$	−37.3042	−1.20461
$960$	0	0
$961$	1.00000	0.0322581
$962$	3.21183	0.103553
$963$	0	0
$964$	−10.8560	−0.349649
$965$	0	0
$966$	0	0
$967$	−45.2672	−1.45570	−0.727848	−	0.685738i	$-0.759479\pi$
$967$	−45.2672	−1.45570	−0.727848	+	0.685738i	$0.759479\pi$
$968$	20.3598	0.654388
$969$	0	0
$970$	0	0
$971$	−7.59837	−0.243843	−0.121922	−	0.992540i	$-0.538906\pi$
$971$	−7.59837	−0.243843	−0.121922	+	0.992540i	$0.538906\pi$
$972$	0	0
$973$	46.5714	1.49301
$974$	−7.47342	−0.239464
$975$	0	0
$976$	−4.28797	−0.137255
$977$	−19.1519	−0.612723	−0.306362	−	0.951915i	$-0.599112\pi$
$977$	−19.1519	−0.612723	−0.306362	+	0.951915i	$0.599112\pi$
$978$	0	0
$979$	−17.4161	−0.556623
$980$	0	0
$981$	0	0
$982$	−13.9092	−0.443859
$983$	4.20957	0.134264	0.0671322	−	0.997744i	$-0.478615\pi$
$983$	4.20957	0.134264	0.0671322	+	0.997744i	$0.478615\pi$
$984$	0	0
$985$	0	0
$986$	−5.23030	−0.166567
$987$	0	0
$988$	14.0573	0.447222
$989$	5.18883	0.164995
$990$	0	0
$991$	−37.3121	−1.18526	−0.592629	−	0.805476i	$-0.701910\pi$
$991$	−37.3121	−1.18526	−0.592629	+	0.805476i	$0.701910\pi$
$992$	5.42801	0.172339
$993$	0	0
$994$	−8.95289	−0.283968
$995$	0	0
$996$	0	0
$997$	62.0034	1.96367	0.981833	−	0.189745i	$-0.0607661\pi$
$997$	62.0034	1.96367	0.981833	+	0.189745i	$0.0607661\pi$
$998$	−9.38259	−0.297001
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6975.2.a.bf.1.2		3
3.2	odd	2		2325.2.a.r.1.2		3
5.4	even	2		1395.2.a.j.1.2		3
15.2	even	4		2325.2.c.k.1024.3		6
15.8	even	4		2325.2.c.k.1024.4		6
15.14	odd	2		465.2.a.e.1.2	✓	3
60.59	even	2		7440.2.a.bs.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
465.2.a.e.1.2	✓	3	15.14	odd	2
1395.2.a.j.1.2		3	5.4	even	2
2325.2.a.r.1.2		3	3.2	odd	2
2325.2.c.k.1024.3		6	15.2	even	4
2325.2.c.k.1024.4		6	15.8	even	4
6975.2.a.bf.1.2		3	1.1	even	1	trivial
7440.2.a.bs.1.2		3	60.59	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6975 = 3^{2} \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6975.a (trivial)

\(f(q)\)	\(=\)	\(q+0.571993 q^{2} -1.67282 q^{4} -2.42801 q^{7} -2.10083 q^{8} -1.14399 q^{11} -1.57199 q^{13} -1.38880 q^{14} +2.14399 q^{16} +4.67282 q^{17} +5.34565 q^{19} -0.654353 q^{22} -1.81681 q^{23} -0.899170 q^{26} +4.06163 q^{28} -1.95684 q^{29} +1.00000 q^{31} +5.42801 q^{32} +2.67282 q^{34} -3.57199 q^{37} +3.05767 q^{38} +3.14399 q^{41} -2.85601 q^{43} +1.91369 q^{44} -1.03920 q^{46} +7.16246 q^{47} -1.10478 q^{49} +2.62967 q^{52} -3.81681 q^{53} +5.10083 q^{56} -1.11930 q^{58} +6.24482 q^{59} -2.00000 q^{61} +0.571993 q^{62} -1.18319 q^{64} +0.917641 q^{67} -7.81681 q^{68} +6.44648 q^{71} -6.71598 q^{73} -2.04316 q^{74} -8.94233 q^{76} +2.77761 q^{77} +9.52884 q^{79} +1.79834 q^{82} -3.16246 q^{83} -1.63362 q^{86} +2.40332 q^{88} +15.2241 q^{89} +3.81681 q^{91} +3.03920 q^{92} +4.09688 q^{94} -10.7776 q^{97} -0.631929 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + 5 q^{4} - 8 q^{7} + 3 q^{8} - 2 q^{11} - 4 q^{13} + 8 q^{14} + 5 q^{16} + 4 q^{17} - 4 q^{19} - 22 q^{22} + 6 q^{23} - 12 q^{26} - 10 q^{28} + 2 q^{29} + 3 q^{31} + 17 q^{32} - 2 q^{34} - 10 q^{37}+ \cdots - 57 q^{98}+O(q^{100}) \) 3 * q + q^2 + 5 * q^4 - 8 * q^7 + 3 * q^8 - 2 * q^11 - 4 * q^13 + 8 * q^14 + 5 * q^16 + 4 * q^17 - 4 * q^19 - 22 * q^22 + 6 * q^23 - 12 * q^26 - 10 * q^28 + 2 * q^29 + 3 * q^31 + 17 * q^32 - 2 * q^34 - 10 * q^37 - 8 * q^38 + 8 * q^41 - 10 * q^43 - 10 * q^44 - 16 * q^46 - 10 * q^47 + 11 * q^49 - 10 * q^52 + 6 * q^56 + 36 * q^58 + 8 * q^59 - 6 * q^61 + q^62 - 15 * q^64 - 18 * q^67 - 12 * q^68 - 10 * q^71 - 18 * q^73 - 14 * q^74 - 44 * q^76 - 16 * q^77 + 20 * q^79 + 24 * q^82 + 22 * q^83 + 18 * q^86 - 30 * q^88 - 8 * q^89 + 22 * q^92 + 8 * q^94 - 8 * q^97 - 57 * q^98

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