LMFDB - Embedded newform 6348.2.a.t.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6348,2,Mod(1,6348)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6348, 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("6348.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6348 = 2^{2} \cdot 3 \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6348.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:	$50.6890352031$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 4x^{9} - 14x^{8} + 65x^{7} + 57x^{6} - 354x^{5} - 46x^{4} + 714x^{3} - 74x^{2} - 323x - 23 $ x^10 - 4x^9 - 14x^8 + 65x^7 + 57x^6 - 354x^5 - 46x^4 + 714x^3 - 74x^2 - 323*x - 23
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 276)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$2.84010$ of defining polynomial
Character	$\chi$	$=$	6348.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.80838 q^{5} +3.41002 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.80838 q^{5} +3.41002 q^{7} +1.00000 q^{9} +1.92751 q^{11} +5.29803 q^{13} +1.80838 q^{15} +0.758011 q^{17} -4.73895 q^{19} +3.41002 q^{21} -1.72977 q^{25} +1.00000 q^{27} +4.26626 q^{29} -3.42219 q^{31} +1.92751 q^{33} +6.16660 q^{35} +3.67687 q^{37} +5.29803 q^{39} -7.03987 q^{41} +3.67323 q^{43} +1.80838 q^{45} +10.3519 q^{47} +4.62823 q^{49} +0.758011 q^{51} -7.86365 q^{53} +3.48566 q^{55} -4.73895 q^{57} +10.8055 q^{59} +12.8972 q^{61} +3.41002 q^{63} +9.58083 q^{65} +6.20464 q^{67} -14.7922 q^{71} -7.59888 q^{73} -1.72977 q^{75} +6.57283 q^{77} -6.36068 q^{79} +1.00000 q^{81} +9.77918 q^{83} +1.37077 q^{85} +4.26626 q^{87} -11.1469 q^{89} +18.0664 q^{91} -3.42219 q^{93} -8.56981 q^{95} -7.62022 q^{97} +1.92751 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 10 q^{3} + 2 q^{5} + 11 q^{7} + 10 q^{9}+O(q^{10}) $ 10 * q + 10 * q^3 + 2 * q^5 + 11 * q^7 + 10 * q^9	$ 10 q + 10 q^{3} + 2 q^{5} + 11 q^{7} + 10 q^{9} + 11 q^{11} + 2 q^{15} + 13 q^{17} + 18 q^{19} + 11 q^{21} + 10 q^{25} + 10 q^{27} + 5 q^{29} + 15 q^{31} + 11 q^{33} - 13 q^{35} + 5 q^{37} + 24 q^{41} + 40 q^{43} + 2 q^{45} - 9 q^{47} + 5 q^{49} + 13 q^{51} + 6 q^{53} - 14 q^{55} + 18 q^{57} + 28 q^{59} + 39 q^{61} + 11 q^{63} + 14 q^{65} + 32 q^{67} - 33 q^{71} - 50 q^{73} + 10 q^{75} + 19 q^{77} + 33 q^{79} + 10 q^{81} + 29 q^{83} - 21 q^{85} + 5 q^{87} + 17 q^{89} + 36 q^{91} + 15 q^{93} - 42 q^{95} + 46 q^{97} + 11 q^{99}+O(q^{100}) $ 10 * q + 10 * q^3 + 2 * q^5 + 11 * q^7 + 10 * q^9 + 11 * q^11 + 2 * q^15 + 13 * q^17 + 18 * q^19 + 11 * q^21 + 10 * q^25 + 10 * q^27 + 5 * q^29 + 15 * q^31 + 11 * q^33 - 13 * q^35 + 5 * q^37 + 24 * q^41 + 40 * q^43 + 2 * q^45 - 9 * q^47 + 5 * q^49 + 13 * q^51 + 6 * q^53 - 14 * q^55 + 18 * q^57 + 28 * q^59 + 39 * q^61 + 11 * q^63 + 14 * q^65 + 32 * q^67 - 33 * q^71 - 50 * q^73 + 10 * q^75 + 19 * q^77 + 33 * q^79 + 10 * q^81 + 29 * q^83 - 21 * q^85 + 5 * q^87 + 17 * q^89 + 36 * q^91 + 15 * q^93 - 42 * q^95 + 46 * q^97 + 11 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	1.80838	0.808731	0.404365	−	0.914598i	$-0.367492\pi$
$5$	1.80838	0.808731	0.404365	+	0.914598i	$0.367492\pi$
$6$	0	0
$7$	3.41002	1.28887	0.644433	−	0.764661i	$-0.277093\pi$
$7$	3.41002	1.28887	0.644433	+	0.764661i	$0.277093\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.92751	0.581165	0.290582	−	0.956850i	$-0.406151\pi$
$11$	1.92751	0.581165	0.290582	+	0.956850i	$0.406151\pi$
$12$	0	0
$13$	5.29803	1.46941	0.734704	−	0.678387i	$-0.237321\pi$
$13$	5.29803	1.46941	0.734704	+	0.678387i	$0.237321\pi$
$14$	0	0
$15$	1.80838	0.466921
$16$	0	0
$17$	0.758011	0.183845	0.0919224	−	0.995766i	$-0.470699\pi$
$17$	0.758011	0.183845	0.0919224	+	0.995766i	$0.470699\pi$
$18$	0	0
$19$	−4.73895	−1.08719	−0.543595	−	0.839348i	$-0.682937\pi$
$19$	−4.73895	−1.08719	−0.543595	+	0.839348i	$0.682937\pi$
$20$	0	0
$21$	3.41002	0.744127
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−1.72977	−0.345955
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	4.26626	0.792225	0.396112	−	0.918202i	$-0.370359\pi$
$29$	4.26626	0.792225	0.396112	+	0.918202i	$0.370359\pi$
$30$	0	0
$31$	−3.42219	−0.614643	−0.307321	−	0.951606i	$-0.599433\pi$
$31$	−3.42219	−0.614643	−0.307321	+	0.951606i	$0.599433\pi$
$32$	0	0
$33$	1.92751	0.335536
$34$	0	0
$35$	6.16660	1.04235
$36$	0	0
$37$	3.67687	0.604474	0.302237	−	0.953233i	$-0.402267\pi$
$37$	3.67687	0.604474	0.302237	+	0.953233i	$0.402267\pi$
$38$	0	0
$39$	5.29803	0.848364
$40$	0	0
$41$	−7.03987	−1.09944	−0.549722	−	0.835348i	$-0.685266\pi$
$41$	−7.03987	−1.09944	−0.549722	+	0.835348i	$0.685266\pi$
$42$	0	0
$43$	3.67323	0.560162	0.280081	−	0.959976i	$-0.409639\pi$
$43$	3.67323	0.560162	0.280081	+	0.959976i	$0.409639\pi$
$44$	0	0
$45$	1.80838	0.269577
$46$	0	0
$47$	10.3519	1.50998	0.754989	−	0.655738i	$-0.227642\pi$
$47$	10.3519	1.50998	0.754989	+	0.655738i	$0.227642\pi$
$48$	0	0
$49$	4.62823	0.661175
$50$	0	0
$51$	0.758011	0.106143
$52$	0	0
$53$	−7.86365	−1.08016	−0.540078	−	0.841615i	$-0.681605\pi$
$53$	−7.86365	−1.08016	−0.540078	+	0.841615i	$0.681605\pi$
$54$	0	0
$55$	3.48566	0.470006
$56$	0	0
$57$	−4.73895	−0.627689
$58$	0	0
$59$	10.8055	1.40676	0.703378	−	0.710816i	$-0.251674\pi$
$59$	10.8055	1.40676	0.703378	+	0.710816i	$0.251674\pi$
$60$	0	0
$61$	12.8972	1.65132	0.825658	−	0.564171i	$-0.190804\pi$
$61$	12.8972	1.65132	0.825658	+	0.564171i	$0.190804\pi$
$62$	0	0
$63$	3.41002	0.429622
$64$	0	0
$65$	9.58083	1.18836
$66$	0	0
$67$	6.20464	0.758018	0.379009	−	0.925393i	$-0.376265\pi$
$67$	6.20464	0.758018	0.379009	+	0.925393i	$0.376265\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−14.7922	−1.75551	−0.877756	−	0.479108i	$-0.840960\pi$
$71$	−14.7922	−1.75551	−0.877756	+	0.479108i	$0.840960\pi$
$72$	0	0
$73$	−7.59888	−0.889382	−0.444691	−	0.895684i	$-0.646686\pi$
$73$	−7.59888	−0.889382	−0.444691	+	0.895684i	$0.646686\pi$
$74$	0	0
$75$	−1.72977	−0.199737
$76$	0	0
$77$	6.57283	0.749043
$78$	0	0
$79$	−6.36068	−0.715633	−0.357816	−	0.933792i	$-0.616479\pi$
$79$	−6.36068	−0.715633	−0.357816	+	0.933792i	$0.616479\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	9.77918	1.07340	0.536702	−	0.843772i	$-0.319670\pi$
$83$	9.77918	1.07340	0.536702	+	0.843772i	$0.319670\pi$
$84$	0	0
$85$	1.37077	0.148681
$86$	0	0
$87$	4.26626	0.457391
$88$	0	0
$89$	−11.1469	−1.18157	−0.590784	−	0.806830i	$-0.701181\pi$
$89$	−11.1469	−1.18157	−0.590784	+	0.806830i	$0.701181\pi$
$90$	0	0
$91$	18.0664	1.89387
$92$	0	0
$93$	−3.42219	−0.354864
$94$	0	0
$95$	−8.56981	−0.879243
$96$	0	0
$97$	−7.62022	−0.773717	−0.386858	−	0.922139i	$-0.626440\pi$
$97$	−7.62022	−0.773717	−0.386858	+	0.922139i	$0.626440\pi$
$98$	0	0
$99$	1.92751	0.193722
$100$	0	0
$101$	8.91174	0.886752	0.443376	−	0.896336i	$-0.353781\pi$
$101$	8.91174	0.886752	0.443376	+	0.896336i	$0.353781\pi$
$102$	0	0
$103$	0.885137	0.0872152	0.0436076	−	0.999049i	$-0.486115\pi$
$103$	0.885137	0.0872152	0.0436076	+	0.999049i	$0.486115\pi$
$104$	0	0
$105$	6.16660	0.601798
$106$	0	0
$107$	7.91236	0.764917	0.382458	−	0.923973i	$-0.375078\pi$
$107$	7.91236	0.764917	0.382458	+	0.923973i	$0.375078\pi$
$108$	0	0
$109$	1.75994	0.168571	0.0842856	−	0.996442i	$-0.473139\pi$
$109$	1.75994	0.168571	0.0842856	+	0.996442i	$0.473139\pi$
$110$	0	0
$111$	3.67687	0.348993
$112$	0	0
$113$	−16.7242	−1.57328	−0.786642	−	0.617409i	$-0.788182\pi$
$113$	−16.7242	−1.57328	−0.786642	+	0.617409i	$0.788182\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	5.29803	0.489803
$118$	0	0
$119$	2.58483	0.236951
$120$	0	0
$121$	−7.28472	−0.662248
$122$	0	0
$123$	−7.03987	−0.634764
$124$	0	0
$125$	−12.1700	−1.08851
$126$	0	0
$127$	1.27902	0.113495	0.0567473	−	0.998389i	$-0.481927\pi$
$127$	1.27902	0.113495	0.0567473	+	0.998389i	$0.481927\pi$
$128$	0	0
$129$	3.67323	0.323410
$130$	0	0
$131$	−10.0849	−0.881121	−0.440560	−	0.897723i	$-0.645220\pi$
$131$	−10.0849	−0.881121	−0.440560	+	0.897723i	$0.645220\pi$
$132$	0	0
$133$	−16.1599	−1.40124
$134$	0	0
$135$	1.80838	0.155640
$136$	0	0
$137$	−6.53140	−0.558015	−0.279007	−	0.960289i	$-0.590005\pi$
$137$	−6.53140	−0.558015	−0.279007	+	0.960289i	$0.590005\pi$
$138$	0	0
$139$	−8.55305	−0.725461	−0.362730	−	0.931894i	$-0.618155\pi$
$139$	−8.55305	−0.725461	−0.362730	+	0.931894i	$0.618155\pi$
$140$	0	0
$141$	10.3519	0.871786
$142$	0	0
$143$	10.2120	0.853968
$144$	0	0
$145$	7.71501	0.640696
$146$	0	0
$147$	4.62823	0.381730
$148$	0	0
$149$	−3.23686	−0.265174	−0.132587	−	0.991171i	$-0.542328\pi$
$149$	−3.23686	−0.265174	−0.132587	+	0.991171i	$0.542328\pi$
$150$	0	0
$151$	7.02936	0.572041	0.286021	−	0.958223i	$-0.407667\pi$
$151$	7.02936	0.572041	0.286021	+	0.958223i	$0.407667\pi$
$152$	0	0
$153$	0.758011	0.0612816
$154$	0	0
$155$	−6.18860	−0.497081
$156$	0	0
$157$	20.0537	1.60046	0.800230	−	0.599693i	$-0.204711\pi$
$157$	20.0537	1.60046	0.800230	+	0.599693i	$0.204711\pi$
$158$	0	0
$159$	−7.86365	−0.623628
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−14.8460	−1.16283	−0.581414	−	0.813608i	$-0.697500\pi$
$163$	−14.8460	−1.16283	−0.581414	+	0.813608i	$0.697500\pi$
$164$	0	0
$165$	3.48566	0.271358
$166$	0	0
$167$	−24.6323	−1.90611	−0.953053	−	0.302805i	$-0.902077\pi$
$167$	−24.6323	−1.90611	−0.953053	+	0.302805i	$0.902077\pi$
$168$	0	0
$169$	15.0691	1.15916
$170$	0	0
$171$	−4.73895	−0.362396
$172$	0	0
$173$	−15.6072	−1.18659	−0.593296	−	0.804985i	$-0.702173\pi$
$173$	−15.6072	−1.18659	−0.593296	+	0.804985i	$0.702173\pi$
$174$	0	0
$175$	−5.89856	−0.445889
$176$	0	0
$177$	10.8055	0.812191
$178$	0	0
$179$	−18.1421	−1.35601	−0.678003	−	0.735059i	$-0.737154\pi$
$179$	−18.1421	−1.35601	−0.678003	+	0.735059i	$0.737154\pi$
$180$	0	0
$181$	26.6007	1.97721	0.988607	−	0.150516i	$-0.0480937\pi$
$181$	26.6007	1.97721	0.988607	+	0.150516i	$0.0480937\pi$
$182$	0	0
$183$	12.8972	0.953388
$184$	0	0
$185$	6.64917	0.488856
$186$	0	0
$187$	1.46107	0.106844
$188$	0	0
$189$	3.41002	0.248042
$190$	0	0
$191$	24.7928	1.79394	0.896972	−	0.442087i	$-0.145762\pi$
$191$	24.7928	1.79394	0.896972	+	0.442087i	$0.145762\pi$
$192$	0	0
$193$	−4.71905	−0.339684	−0.169842	−	0.985471i	$-0.554326\pi$
$193$	−4.71905	−0.339684	−0.169842	+	0.985471i	$0.554326\pi$
$194$	0	0
$195$	9.58083	0.686098
$196$	0	0
$197$	−11.1321	−0.793128	−0.396564	−	0.918007i	$-0.629798\pi$
$197$	−11.1321	−0.793128	−0.396564	+	0.918007i	$0.629798\pi$
$198$	0	0
$199$	7.88804	0.559168	0.279584	−	0.960121i	$-0.409803\pi$
$199$	7.88804	0.559168	0.279584	+	0.960121i	$0.409803\pi$
$200$	0	0
$201$	6.20464	0.437642
$202$	0	0
$203$	14.5480	1.02107
$204$	0	0
$205$	−12.7307	−0.889154
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−9.13435	−0.631836
$210$	0	0
$211$	−18.6582	−1.28448	−0.642242	−	0.766502i	$-0.721996\pi$
$211$	−18.6582	−1.28448	−0.642242	+	0.766502i	$0.721996\pi$
$212$	0	0
$213$	−14.7922	−1.01355
$214$	0	0
$215$	6.64258	0.453020
$216$	0	0
$217$	−11.6697	−0.792192
$218$	0	0
$219$	−7.59888	−0.513485
$220$	0	0
$221$	4.01597	0.270143
$222$	0	0
$223$	23.1363	1.54932	0.774661	−	0.632377i	$-0.217921\pi$
$223$	23.1363	1.54932	0.774661	+	0.632377i	$0.217921\pi$
$224$	0	0
$225$	−1.72977	−0.115318
$226$	0	0
$227$	−12.3327	−0.818551	−0.409276	−	0.912411i	$-0.634219\pi$
$227$	−12.3327	−0.818551	−0.409276	+	0.912411i	$0.634219\pi$
$228$	0	0
$229$	11.4857	0.758998	0.379499	−	0.925192i	$-0.376096\pi$
$229$	11.4857	0.758998	0.379499	+	0.925192i	$0.376096\pi$
$230$	0	0
$231$	6.57283	0.432460
$232$	0	0
$233$	−5.35110	−0.350562	−0.175281	−	0.984518i	$-0.556083\pi$
$233$	−5.35110	−0.350562	−0.175281	+	0.984518i	$0.556083\pi$
$234$	0	0
$235$	18.7201	1.22117
$236$	0	0
$237$	−6.36068	−0.413171
$238$	0	0
$239$	11.4413	0.740079	0.370039	−	0.929016i	$-0.379344\pi$
$239$	11.4413	0.740079	0.370039	+	0.929016i	$0.379344\pi$
$240$	0	0
$241$	−3.50702	−0.225907	−0.112954	−	0.993600i	$-0.536031\pi$
$241$	−3.50702	−0.225907	−0.112954	+	0.993600i	$0.536031\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	8.36958	0.534713
$246$	0	0
$247$	−25.1071	−1.59753
$248$	0	0
$249$	9.77918	0.619730
$250$	0	0
$251$	11.8422	0.747473	0.373737	−	0.927535i	$-0.378076\pi$
$251$	11.8422	0.747473	0.373737	+	0.927535i	$0.378076\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	1.37077	0.0858410
$256$	0	0
$257$	−25.5236	−1.59212	−0.796059	−	0.605219i	$-0.793086\pi$
$257$	−25.5236	−1.59212	−0.796059	+	0.605219i	$0.793086\pi$
$258$	0	0
$259$	12.5382	0.779085
$260$	0	0
$261$	4.26626	0.264075
$262$	0	0
$263$	3.59494	0.221674	0.110837	−	0.993839i	$-0.464647\pi$
$263$	3.59494	0.221674	0.110837	+	0.993839i	$0.464647\pi$
$264$	0	0
$265$	−14.2204	−0.873555
$266$	0	0
$267$	−11.1469	−0.682179
$268$	0	0
$269$	16.2119	0.988454	0.494227	−	0.869333i	$-0.335451\pi$
$269$	16.2119	0.988454	0.494227	+	0.869333i	$0.335451\pi$
$270$	0	0
$271$	−4.34094	−0.263693	−0.131847	−	0.991270i	$-0.542091\pi$
$271$	−4.34094	−0.263693	−0.131847	+	0.991270i	$0.542091\pi$
$272$	0	0
$273$	18.0664	1.09343
$274$	0	0
$275$	−3.33415	−0.201057
$276$	0	0
$277$	28.6009	1.71846	0.859230	−	0.511589i	$-0.170943\pi$
$277$	28.6009	1.71846	0.859230	+	0.511589i	$0.170943\pi$
$278$	0	0
$279$	−3.42219	−0.204881
$280$	0	0
$281$	11.3291	0.675836	0.337918	−	0.941176i	$-0.390277\pi$
$281$	11.3291	0.675836	0.337918	+	0.941176i	$0.390277\pi$
$282$	0	0
$283$	5.33421	0.317086	0.158543	−	0.987352i	$-0.449320\pi$
$283$	5.33421	0.317086	0.158543	+	0.987352i	$0.449320\pi$
$284$	0	0
$285$	−8.56981	−0.507631
$286$	0	0
$287$	−24.0061	−1.41704
$288$	0	0
$289$	−16.4254	−0.966201
$290$	0	0
$291$	−7.62022	−0.446705
$292$	0	0
$293$	23.6546	1.38192	0.690959	−	0.722894i	$-0.257189\pi$
$293$	23.6546	1.38192	0.690959	+	0.722894i	$0.257189\pi$
$294$	0	0
$295$	19.5404	1.13769
$296$	0	0
$297$	1.92751	0.111845
$298$	0	0
$299$	0	0
$300$	0	0
$301$	12.5258	0.721974
$302$	0	0
$303$	8.91174	0.511966
$304$	0	0
$305$	23.3230	1.33547
$306$	0	0
$307$	−19.5823	−1.11762	−0.558809	−	0.829296i	$-0.688742\pi$
$307$	−19.5823	−1.11762	−0.558809	+	0.829296i	$0.688742\pi$
$308$	0	0
$309$	0.885137	0.0503537
$310$	0	0
$311$	−1.91968	−0.108855	−0.0544275	−	0.998518i	$-0.517333\pi$
$311$	−1.91968	−0.108855	−0.0544275	+	0.998518i	$0.517333\pi$
$312$	0	0
$313$	12.6859	0.717047	0.358524	−	0.933521i	$-0.383280\pi$
$313$	12.6859	0.717047	0.358524	+	0.933521i	$0.383280\pi$
$314$	0	0
$315$	6.16660	0.347448
$316$	0	0
$317$	3.12473	0.175503	0.0877513	−	0.996142i	$-0.472032\pi$
$317$	3.12473	0.175503	0.0877513	+	0.996142i	$0.472032\pi$
$318$	0	0
$319$	8.22324	0.460413
$320$	0	0
$321$	7.91236	0.441625
$322$	0	0
$323$	−3.59218	−0.199874
$324$	0	0
$325$	−9.16439	−0.508349
$326$	0	0
$327$	1.75994	0.0973247
$328$	0	0
$329$	35.3001	1.94616
$330$	0	0
$331$	−12.8689	−0.707337	−0.353669	−	0.935371i	$-0.615066\pi$
$331$	−12.8689	−0.707337	−0.353669	+	0.935371i	$0.615066\pi$
$332$	0	0
$333$	3.67687	0.201491
$334$	0	0
$335$	11.2203	0.613032
$336$	0	0
$337$	15.4541	0.841840	0.420920	−	0.907098i	$-0.361707\pi$
$337$	15.4541	0.841840	0.420920	+	0.907098i	$0.361707\pi$
$338$	0	0
$339$	−16.7242	−0.908336
$340$	0	0
$341$	−6.59628	−0.357209
$342$	0	0
$343$	−8.08779	−0.436700
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−18.1546	−0.974587	−0.487294	−	0.873238i	$-0.662016\pi$
$347$	−18.1546	−0.974587	−0.487294	+	0.873238i	$0.662016\pi$
$348$	0	0
$349$	21.8805	1.17123	0.585617	−	0.810588i	$-0.300852\pi$
$349$	21.8805	1.17123	0.585617	+	0.810588i	$0.300852\pi$
$350$	0	0
$351$	5.29803	0.282788
$352$	0	0
$353$	12.0997	0.644002	0.322001	−	0.946739i	$-0.395645\pi$
$353$	12.0997	0.644002	0.322001	+	0.946739i	$0.395645\pi$
$354$	0	0
$355$	−26.7499	−1.41974
$356$	0	0
$357$	2.58483	0.136804
$358$	0	0
$359$	3.54542	0.187120	0.0935601	−	0.995614i	$-0.470175\pi$
$359$	3.54542	0.187120	0.0935601	+	0.995614i	$0.470175\pi$
$360$	0	0
$361$	3.45764	0.181981
$362$	0	0
$363$	−7.28472	−0.382349
$364$	0	0
$365$	−13.7416	−0.719270
$366$	0	0
$367$	26.1379	1.36439	0.682194	−	0.731171i	$-0.261026\pi$
$367$	26.1379	1.36439	0.682194	+	0.731171i	$0.261026\pi$
$368$	0	0
$369$	−7.03987	−0.366481
$370$	0	0
$371$	−26.8152	−1.39218
$372$	0	0
$373$	−14.4617	−0.748798	−0.374399	−	0.927268i	$-0.622151\pi$
$373$	−14.4617	−0.748798	−0.374399	+	0.927268i	$0.622151\pi$
$374$	0	0
$375$	−12.1700	−0.628454
$376$	0	0
$377$	22.6028	1.16410
$378$	0	0
$379$	27.9789	1.43718	0.718590	−	0.695434i	$-0.244788\pi$
$379$	27.9789	1.43718	0.718590	+	0.695434i	$0.244788\pi$
$380$	0	0
$381$	1.27902	0.0655261
$382$	0	0
$383$	18.4801	0.944291	0.472145	−	0.881521i	$-0.343480\pi$
$383$	18.4801	0.944291	0.472145	+	0.881521i	$0.343480\pi$
$384$	0	0
$385$	11.8862	0.605774
$386$	0	0
$387$	3.67323	0.186721
$388$	0	0
$389$	−25.7151	−1.30381	−0.651903	−	0.758302i	$-0.726029\pi$
$389$	−25.7151	−1.30381	−0.651903	+	0.758302i	$0.726029\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−10.0849	−0.508715
$394$	0	0
$395$	−11.5025	−0.578754
$396$	0	0
$397$	−27.9380	−1.40217	−0.701085	−	0.713078i	$-0.747300\pi$
$397$	−27.9380	−1.40217	−0.701085	+	0.713078i	$0.747300\pi$
$398$	0	0
$399$	−16.1599	−0.809007
$400$	0	0
$401$	−24.4261	−1.21978	−0.609890	−	0.792486i	$-0.708787\pi$
$401$	−24.4261	−1.21978	−0.609890	+	0.792486i	$0.708787\pi$
$402$	0	0
$403$	−18.1308	−0.903162
$404$	0	0
$405$	1.80838	0.0898590
$406$	0	0
$407$	7.08718	0.351299
$408$	0	0
$409$	−36.1790	−1.78894	−0.894469	−	0.447130i	$-0.852446\pi$
$409$	−36.1790	−1.78894	−0.894469	+	0.447130i	$0.852446\pi$
$410$	0	0
$411$	−6.53140	−0.322170
$412$	0	0
$413$	36.8470	1.81312
$414$	0	0
$415$	17.6844	0.868095
$416$	0	0
$417$	−8.55305	−0.418845
$418$	0	0
$419$	−8.28020	−0.404514	−0.202257	−	0.979332i	$-0.564828\pi$
$419$	−8.28020	−0.404514	−0.202257	+	0.979332i	$0.564828\pi$
$420$	0	0
$421$	−35.7531	−1.74250	−0.871250	−	0.490840i	$-0.836690\pi$
$421$	−35.7531	−1.74250	−0.871250	+	0.490840i	$0.836690\pi$
$422$	0	0
$423$	10.3519	0.503326
$424$	0	0
$425$	−1.31119	−0.0636020
$426$	0	0
$427$	43.9797	2.12832
$428$	0	0
$429$	10.2120	0.493039
$430$	0	0
$431$	−21.8226	−1.05116	−0.525578	−	0.850745i	$-0.676151\pi$
$431$	−21.8226	−1.05116	−0.525578	+	0.850745i	$0.676151\pi$
$432$	0	0
$433$	−30.3982	−1.46085	−0.730423	−	0.682995i	$-0.760677\pi$
$433$	−30.3982	−1.46085	−0.730423	+	0.682995i	$0.760677\pi$
$434$	0	0
$435$	7.71501	0.369906
$436$	0	0
$437$	0	0
$438$	0	0
$439$	20.7108	0.988471	0.494235	−	0.869328i	$-0.335448\pi$
$439$	20.7108	0.988471	0.494235	+	0.869328i	$0.335448\pi$
$440$	0	0
$441$	4.62823	0.220392
$442$	0	0
$443$	24.6057	1.16905	0.584527	−	0.811374i	$-0.301280\pi$
$443$	24.6057	1.16905	0.584527	+	0.811374i	$0.301280\pi$
$444$	0	0
$445$	−20.1578	−0.955570
$446$	0	0
$447$	−3.23686	−0.153098
$448$	0	0
$449$	39.3310	1.85614	0.928072	−	0.372402i	$-0.121466\pi$
$449$	39.3310	1.85614	0.928072	+	0.372402i	$0.121466\pi$
$450$	0	0
$451$	−13.5694	−0.638958
$452$	0	0
$453$	7.02936	0.330268
$454$	0	0
$455$	32.6708	1.53163
$456$	0	0
$457$	−17.1738	−0.803355	−0.401678	−	0.915781i	$-0.631573\pi$
$457$	−17.1738	−0.803355	−0.401678	+	0.915781i	$0.631573\pi$
$458$	0	0
$459$	0.758011	0.0353809
$460$	0	0
$461$	1.59212	0.0741524	0.0370762	−	0.999312i	$-0.488196\pi$
$461$	1.59212	0.0741524	0.0370762	+	0.999312i	$0.488196\pi$
$462$	0	0
$463$	−14.5914	−0.678122	−0.339061	−	0.940764i	$-0.610109\pi$
$463$	−14.5914	−0.678122	−0.339061	+	0.940764i	$0.610109\pi$
$464$	0	0
$465$	−6.18860	−0.286990
$466$	0	0
$467$	29.5473	1.36728	0.683642	−	0.729817i	$-0.260395\pi$
$467$	29.5473	1.36728	0.683642	+	0.729817i	$0.260395\pi$
$468$	0	0
$469$	21.1579	0.976983
$470$	0	0
$471$	20.0537	0.924026
$472$	0	0
$473$	7.08017	0.325547
$474$	0	0
$475$	8.19731	0.376118
$476$	0	0
$477$	−7.86365	−0.360052
$478$	0	0
$479$	−3.16929	−0.144808	−0.0724042	−	0.997375i	$-0.523067\pi$
$479$	−3.16929	−0.144808	−0.0724042	+	0.997375i	$0.523067\pi$
$480$	0	0
$481$	19.4802	0.888219
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−13.7802	−0.625728
$486$	0	0
$487$	31.8315	1.44242	0.721212	−	0.692715i	$-0.243585\pi$
$487$	31.8315	1.44242	0.721212	+	0.692715i	$0.243585\pi$
$488$	0	0
$489$	−14.8460	−0.671359
$490$	0	0
$491$	11.1533	0.503342	0.251671	−	0.967813i	$-0.419020\pi$
$491$	11.1533	0.503342	0.251671	+	0.967813i	$0.419020\pi$
$492$	0	0
$493$	3.23387	0.145646
$494$	0	0
$495$	3.48566	0.156669
$496$	0	0
$497$	−50.4417	−2.26262
$498$	0	0
$499$	3.88983	0.174133	0.0870664	−	0.996203i	$-0.472251\pi$
$499$	3.88983	0.174133	0.0870664	+	0.996203i	$0.472251\pi$
$500$	0	0
$501$	−24.6323	−1.10049
$502$	0	0
$503$	−37.6158	−1.67720	−0.838602	−	0.544745i	$-0.816626\pi$
$503$	−37.6158	−1.67720	−0.838602	+	0.544745i	$0.816626\pi$
$504$	0	0
$505$	16.1158	0.717143
$506$	0	0
$507$	15.0691	0.669243
$508$	0	0
$509$	19.7613	0.875904	0.437952	−	0.898998i	$-0.355704\pi$
$509$	19.7613	0.875904	0.437952	+	0.898998i	$0.355704\pi$
$510$	0	0
$511$	−25.9123	−1.14629
$512$	0	0
$513$	−4.73895	−0.209230
$514$	0	0
$515$	1.60066	0.0705336
$516$	0	0
$517$	19.9533	0.877546
$518$	0	0
$519$	−15.6072	−0.685079
$520$	0	0
$521$	−27.4915	−1.20442	−0.602212	−	0.798336i	$-0.705714\pi$
$521$	−27.4915	−1.20442	−0.602212	+	0.798336i	$0.705714\pi$
$522$	0	0
$523$	8.12792	0.355409	0.177705	−	0.984084i	$-0.443133\pi$
$523$	8.12792	0.355409	0.177705	+	0.984084i	$0.443133\pi$
$524$	0	0
$525$	−5.89856	−0.257434
$526$	0	0
$527$	−2.59406	−0.112999
$528$	0	0
$529$	0	0
$530$	0	0
$531$	10.8055	0.468919
$532$	0	0
$533$	−37.2975	−1.61553
$534$	0	0
$535$	14.3085	0.618612
$536$	0	0
$537$	−18.1421	−0.782891
$538$	0	0
$539$	8.92093	0.384252
$540$	0	0
$541$	23.4068	1.00634	0.503168	−	0.864189i	$-0.332168\pi$
$541$	23.4068	1.00634	0.503168	+	0.864189i	$0.332168\pi$
$542$	0	0
$543$	26.6007	1.14155
$544$	0	0
$545$	3.18263	0.136329
$546$	0	0
$547$	16.3796	0.700339	0.350170	−	0.936686i	$-0.386124\pi$
$547$	16.3796	0.700339	0.350170	+	0.936686i	$0.386124\pi$
$548$	0	0
$549$	12.8972	0.550439
$550$	0	0
$551$	−20.2176	−0.861298
$552$	0	0
$553$	−21.6900	−0.922354
$554$	0	0
$555$	6.64917	0.282241
$556$	0	0
$557$	−19.1105	−0.809739	−0.404870	−	0.914374i	$-0.632683\pi$
$557$	−19.1105	−0.809739	−0.404870	+	0.914374i	$0.632683\pi$
$558$	0	0
$559$	19.4609	0.823107
$560$	0	0
$561$	1.46107	0.0616865
$562$	0	0
$563$	−20.2704	−0.854295	−0.427147	−	0.904182i	$-0.640481\pi$
$563$	−20.2704	−0.854295	−0.427147	+	0.904182i	$0.640481\pi$
$564$	0	0
$565$	−30.2437	−1.27236
$566$	0	0
$567$	3.41002	0.143207
$568$	0	0
$569$	18.5611	0.778122	0.389061	−	0.921212i	$-0.372799\pi$
$569$	18.5611	0.778122	0.389061	+	0.921212i	$0.372799\pi$
$570$	0	0
$571$	46.7442	1.95619	0.978093	−	0.208169i	$-0.0667504\pi$
$571$	46.7442	1.95619	0.978093	+	0.208169i	$0.0667504\pi$
$572$	0	0
$573$	24.7928	1.03573
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−35.8223	−1.49130	−0.745650	−	0.666338i	$-0.767861\pi$
$577$	−35.8223	−1.49130	−0.745650	+	0.666338i	$0.767861\pi$
$578$	0	0
$579$	−4.71905	−0.196117
$580$	0	0
$581$	33.3472	1.38347
$582$	0	0
$583$	−15.1572	−0.627748
$584$	0	0
$585$	9.58083	0.396119
$586$	0	0
$587$	0.0185507	0.000765668 0	0.000382834	−	1.00000i	$-0.499878\pi$
$587$	0.0185507	0.000765668 0	0.000382834		1.00000i	$0.499878\pi$
$588$	0	0
$589$	16.2176	0.668233
$590$	0	0
$591$	−11.1321	−0.457913
$592$	0	0
$593$	−6.49894	−0.266880	−0.133440	−	0.991057i	$-0.542602\pi$
$593$	−6.49894	−0.266880	−0.133440	+	0.991057i	$0.542602\pi$
$594$	0	0
$595$	4.67435	0.191630
$596$	0	0
$597$	7.88804	0.322836
$598$	0	0
$599$	−7.49989	−0.306437	−0.153219	−	0.988192i	$-0.548964\pi$
$599$	−7.49989	−0.306437	−0.153219	+	0.988192i	$0.548964\pi$
$600$	0	0
$601$	25.7294	1.04953	0.524763	−	0.851248i	$-0.324154\pi$
$601$	25.7294	1.04953	0.524763	+	0.851248i	$0.324154\pi$
$602$	0	0
$603$	6.20464	0.252673
$604$	0	0
$605$	−13.1735	−0.535580
$606$	0	0
$607$	28.9734	1.17599	0.587997	−	0.808863i	$-0.299917\pi$
$607$	28.9734	1.17599	0.587997	+	0.808863i	$0.299917\pi$
$608$	0	0
$609$	14.5480	0.589516
$610$	0	0
$611$	54.8446	2.21877
$612$	0	0
$613$	−2.35394	−0.0950747	−0.0475374	−	0.998869i	$-0.515137\pi$
$613$	−2.35394	−0.0950747	−0.0475374	+	0.998869i	$0.515137\pi$
$614$	0	0
$615$	−12.7307	−0.513353
$616$	0	0
$617$	−15.2703	−0.614757	−0.307379	−	0.951587i	$-0.599452\pi$
$617$	−15.2703	−0.614757	−0.307379	+	0.951587i	$0.599452\pi$
$618$	0	0
$619$	11.4977	0.462131	0.231066	−	0.972938i	$-0.425779\pi$
$619$	11.4977	0.462131	0.231066	+	0.972938i	$0.425779\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−38.0111	−1.52288
$624$	0	0
$625$	−13.3590	−0.534361
$626$	0	0
$627$	−9.13435	−0.364791
$628$	0	0
$629$	2.78711	0.111129
$630$	0	0
$631$	14.1283	0.562440	0.281220	−	0.959643i	$-0.409261\pi$
$631$	14.1283	0.562440	0.281220	+	0.959643i	$0.409261\pi$
$632$	0	0
$633$	−18.6582	−0.741598
$634$	0	0
$635$	2.31295	0.0917865
$636$	0	0
$637$	24.5205	0.971537
$638$	0	0
$639$	−14.7922	−0.585171
$640$	0	0
$641$	24.8912	0.983144	0.491572	−	0.870837i	$-0.336422\pi$
$641$	24.8912	0.983144	0.491572	+	0.870837i	$0.336422\pi$
$642$	0	0
$643$	−23.5235	−0.927678	−0.463839	−	0.885919i	$-0.653528\pi$
$643$	−23.5235	−0.927678	−0.463839	+	0.885919i	$0.653528\pi$
$644$	0	0
$645$	6.64258	0.261551
$646$	0	0
$647$	−20.0221	−0.787149	−0.393575	−	0.919293i	$-0.628762\pi$
$647$	−20.0221	−0.787149	−0.393575	+	0.919293i	$0.628762\pi$
$648$	0	0
$649$	20.8277	0.817557
$650$	0	0
$651$	−11.6697	−0.457373
$652$	0	0
$653$	−7.46797	−0.292244	−0.146122	−	0.989267i	$-0.546679\pi$
$653$	−7.46797	−0.292244	−0.146122	+	0.989267i	$0.546679\pi$
$654$	0	0
$655$	−18.2373	−0.712589
$656$	0	0
$657$	−7.59888	−0.296461
$658$	0	0
$659$	−10.9051	−0.424804	−0.212402	−	0.977182i	$-0.568129\pi$
$659$	−10.9051	−0.424804	−0.212402	+	0.977182i	$0.568129\pi$
$660$	0	0
$661$	−17.2616	−0.671398	−0.335699	−	0.941969i	$-0.608972\pi$
$661$	−17.2616	−0.671398	−0.335699	+	0.941969i	$0.608972\pi$
$662$	0	0
$663$	4.01597	0.155967
$664$	0	0
$665$	−29.2232	−1.13323
$666$	0	0
$667$	0	0
$668$	0	0
$669$	23.1363	0.894501
$670$	0	0
$671$	24.8594	0.959686
$672$	0	0
$673$	−7.05328	−0.271884	−0.135942	−	0.990717i	$-0.543406\pi$
$673$	−7.05328	−0.271884	−0.135942	+	0.990717i	$0.543406\pi$
$674$	0	0
$675$	−1.72977	−0.0665790
$676$	0	0
$677$	−26.1331	−1.00437	−0.502187	−	0.864759i	$-0.667471\pi$
$677$	−26.1331	−1.00437	−0.502187	+	0.864759i	$0.667471\pi$
$678$	0	0
$679$	−25.9851	−0.997217
$680$	0	0
$681$	−12.3327	−0.472591
$682$	0	0
$683$	−4.22451	−0.161647	−0.0808233	−	0.996728i	$-0.525755\pi$
$683$	−4.22451	−0.161647	−0.0808233	+	0.996728i	$0.525755\pi$
$684$	0	0
$685$	−11.8112	−0.451284
$686$	0	0
$687$	11.4857	0.438208
$688$	0	0
$689$	−41.6619	−1.58719
$690$	0	0
$691$	35.2154	1.33966	0.669829	−	0.742516i	$-0.266367\pi$
$691$	35.2154	1.33966	0.669829	+	0.742516i	$0.266367\pi$
$692$	0	0
$693$	6.57283	0.249681
$694$	0	0
$695$	−15.4671	−0.586702
$696$	0	0
$697$	−5.33630	−0.202127
$698$	0	0
$699$	−5.35110	−0.202397
$700$	0	0
$701$	−38.7455	−1.46340	−0.731698	−	0.681629i	$-0.761272\pi$
$701$	−38.7455	−1.46340	−0.731698	+	0.681629i	$0.761272\pi$
$702$	0	0
$703$	−17.4245	−0.657177
$704$	0	0
$705$	18.7201	0.705040
$706$	0	0
$707$	30.3892	1.14290
$708$	0	0
$709$	14.1192	0.530258	0.265129	−	0.964213i	$-0.414585\pi$
$709$	14.1192	0.530258	0.265129	+	0.964213i	$0.414585\pi$
$710$	0	0
$711$	−6.36068	−0.238544
$712$	0	0
$713$	0	0
$714$	0	0
$715$	18.4671	0.690631
$716$	0	0
$717$	11.4413	0.427285
$718$	0	0
$719$	47.7130	1.77940	0.889698	−	0.456550i	$-0.150915\pi$
$719$	47.7130	1.77940	0.889698	+	0.456550i	$0.150915\pi$
$720$	0	0
$721$	3.01833	0.112409
$722$	0	0
$723$	−3.50702	−0.130427
$724$	0	0
$725$	−7.37966	−0.274074
$726$	0	0
$727$	36.7935	1.36460	0.682298	−	0.731075i	$-0.260981\pi$
$727$	36.7935	1.36460	0.682298	+	0.731075i	$0.260981\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	2.78435	0.102983
$732$	0	0
$733$	−24.7615	−0.914588	−0.457294	−	0.889316i	$-0.651181\pi$
$733$	−24.7615	−0.914588	−0.457294	+	0.889316i	$0.651181\pi$
$734$	0	0
$735$	8.36958	0.308717
$736$	0	0
$737$	11.9595	0.440533
$738$	0	0
$739$	18.7732	0.690585	0.345292	−	0.938495i	$-0.387780\pi$
$739$	18.7732	0.690585	0.345292	+	0.938495i	$0.387780\pi$
$740$	0	0
$741$	−25.1071	−0.922332
$742$	0	0
$743$	30.4315	1.11642	0.558212	−	0.829698i	$-0.311487\pi$
$743$	30.4315	1.11642	0.558212	+	0.829698i	$0.311487\pi$
$744$	0	0
$745$	−5.85347	−0.214454
$746$	0	0
$747$	9.77918	0.357801
$748$	0	0
$749$	26.9813	0.985875
$750$	0	0
$751$	−36.2616	−1.32321	−0.661603	−	0.749854i	$-0.730124\pi$
$751$	−36.2616	−1.32321	−0.661603	+	0.749854i	$0.730124\pi$
$752$	0	0
$753$	11.8422	0.431554
$754$	0	0
$755$	12.7117	0.462628
$756$	0	0
$757$	43.6441	1.58627	0.793136	−	0.609045i	$-0.208447\pi$
$757$	43.6441	1.58627	0.793136	+	0.609045i	$0.208447\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−3.10891	−0.112698	−0.0563490	−	0.998411i	$-0.517946\pi$
$761$	−3.10891	−0.112698	−0.0563490	+	0.998411i	$0.517946\pi$
$762$	0	0
$763$	6.00141	0.217266
$764$	0	0
$765$	1.37077	0.0495603
$766$	0	0
$767$	57.2479	2.06710
$768$	0	0
$769$	44.6155	1.60887	0.804437	−	0.594038i	$-0.202467\pi$
$769$	44.6155	1.60887	0.804437	+	0.594038i	$0.202467\pi$
$770$	0	0
$771$	−25.5236	−0.919210
$772$	0	0
$773$	−17.1132	−0.615519	−0.307759	−	0.951464i	$-0.599579\pi$
$773$	−17.1132	−0.615519	−0.307759	+	0.951464i	$0.599579\pi$
$774$	0	0
$775$	5.91961	0.212639
$776$	0	0
$777$	12.5382	0.449805
$778$	0	0
$779$	33.3616	1.19530
$780$	0	0
$781$	−28.5121	−1.02024
$782$	0	0
$783$	4.26626	0.152464
$784$	0	0
$785$	36.2647	1.29434
$786$	0	0
$787$	43.7737	1.56036	0.780182	−	0.625553i	$-0.215127\pi$
$787$	43.7737	1.56036	0.780182	+	0.625553i	$0.215127\pi$
$788$	0	0
$789$	3.59494	0.127983
$790$	0	0
$791$	−57.0300	−2.02775
$792$	0	0
$793$	68.3297	2.42646
$794$	0	0
$795$	−14.2204	−0.504347
$796$	0	0
$797$	−28.7708	−1.01911	−0.509557	−	0.860437i	$-0.670191\pi$
$797$	−28.7708	−1.01911	−0.509557	+	0.860437i	$0.670191\pi$
$798$	0	0
$799$	7.84685	0.277601
$800$	0	0
$801$	−11.1469	−0.393856
$802$	0	0
$803$	−14.6469	−0.516877
$804$	0	0
$805$	0	0
$806$	0	0
$807$	16.2119	0.570684
$808$	0	0
$809$	−13.3028	−0.467703	−0.233851	−	0.972272i	$-0.575133\pi$
$809$	−13.3028	−0.467703	−0.233851	+	0.972272i	$0.575133\pi$
$810$	0	0
$811$	−45.9548	−1.61369	−0.806846	−	0.590762i	$-0.798827\pi$
$811$	−45.9548	−1.61369	−0.806846	+	0.590762i	$0.798827\pi$
$812$	0	0
$813$	−4.34094	−0.152243
$814$	0	0
$815$	−26.8471	−0.940414
$816$	0	0
$817$	−17.4072	−0.609003
$818$	0	0
$819$	18.0664	0.631290
$820$	0	0
$821$	−48.1024	−1.67879	−0.839393	−	0.543525i	$-0.817089\pi$
$821$	−48.1024	−1.67879	−0.839393	+	0.543525i	$0.817089\pi$
$822$	0	0
$823$	28.7767	1.00309	0.501547	−	0.865130i	$-0.332764\pi$
$823$	28.7767	1.00309	0.501547	+	0.865130i	$0.332764\pi$
$824$	0	0
$825$	−3.33415	−0.116080
$826$	0	0
$827$	−47.0915	−1.63753	−0.818766	−	0.574128i	$-0.805341\pi$
$827$	−47.0915	−1.63753	−0.818766	+	0.574128i	$0.805341\pi$
$828$	0	0
$829$	−27.9669	−0.971330	−0.485665	−	0.874145i	$-0.661423\pi$
$829$	−27.9669	−0.971330	−0.485665	+	0.874145i	$0.661423\pi$
$830$	0	0
$831$	28.6009	0.992153
$832$	0	0
$833$	3.50825	0.121554
$834$	0	0
$835$	−44.5445	−1.54153
$836$	0	0
$837$	−3.42219	−0.118288
$838$	0	0
$839$	−18.5797	−0.641443	−0.320722	−	0.947174i	$-0.603925\pi$
$839$	−18.5797	−0.641443	−0.320722	+	0.947174i	$0.603925\pi$
$840$	0	0
$841$	−10.7990	−0.372380
$842$	0	0
$843$	11.3291	0.390194
$844$	0	0
$845$	27.2506	0.937450
$846$	0	0
$847$	−24.8410	−0.853548
$848$	0	0
$849$	5.33421	0.183070
$850$	0	0
$851$	0	0
$852$	0	0
$853$	43.6988	1.49622	0.748109	−	0.663576i	$-0.230962\pi$
$853$	43.6988	1.49622	0.748109	+	0.663576i	$0.230962\pi$
$854$	0	0
$855$	−8.56981	−0.293081
$856$	0	0
$857$	−35.9455	−1.22787	−0.613937	−	0.789355i	$-0.710415\pi$
$857$	−35.9455	−1.22787	−0.613937	+	0.789355i	$0.710415\pi$
$858$	0	0
$859$	−27.7971	−0.948426	−0.474213	−	0.880410i	$-0.657267\pi$
$859$	−27.7971	−0.948426	−0.474213	+	0.880410i	$0.657267\pi$
$860$	0	0
$861$	−24.0061	−0.818126
$862$	0	0
$863$	−6.73210	−0.229163	−0.114582	−	0.993414i	$-0.536553\pi$
$863$	−6.73210	−0.229163	−0.114582	+	0.993414i	$0.536553\pi$
$864$	0	0
$865$	−28.2236	−0.959633
$866$	0	0
$867$	−16.4254	−0.557836
$868$	0	0
$869$	−12.2602	−0.415900
$870$	0	0
$871$	32.8724	1.11384
$872$	0	0
$873$	−7.62022	−0.257906
$874$	0	0
$875$	−41.4998	−1.40295
$876$	0	0
$877$	12.5250	0.422940	0.211470	−	0.977384i	$-0.432175\pi$
$877$	12.5250	0.422940	0.211470	+	0.977384i	$0.432175\pi$
$878$	0	0
$879$	23.6546	0.797850
$880$	0	0
$881$	29.8983	1.00730	0.503649	−	0.863908i	$-0.331991\pi$
$881$	29.8983	1.00730	0.503649	+	0.863908i	$0.331991\pi$
$882$	0	0
$883$	19.3188	0.650129	0.325064	−	0.945692i	$-0.394614\pi$
$883$	19.3188	0.650129	0.325064	+	0.945692i	$0.394614\pi$
$884$	0	0
$885$	19.5404	0.656844
$886$	0	0
$887$	−52.7571	−1.77141	−0.885705	−	0.464248i	$-0.846324\pi$
$887$	−52.7571	−1.77141	−0.885705	+	0.464248i	$0.846324\pi$
$888$	0	0
$889$	4.36148	0.146279
$890$	0	0
$891$	1.92751	0.0645739
$892$	0	0
$893$	−49.0571	−1.64163
$894$	0	0
$895$	−32.8078	−1.09664
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−14.5999	−0.486935
$900$	0	0
$901$	−5.96074	−0.198581
$902$	0	0
$903$	12.5258	0.416832
$904$	0	0
$905$	48.1041	1.59903
$906$	0	0
$907$	−5.82536	−0.193428	−0.0967140	−	0.995312i	$-0.530833\pi$
$907$	−5.82536	−0.193428	−0.0967140	+	0.995312i	$0.530833\pi$
$908$	0	0
$909$	8.91174	0.295584
$910$	0	0
$911$	35.9253	1.19026	0.595129	−	0.803630i	$-0.297101\pi$
$911$	35.9253	1.19026	0.595129	+	0.803630i	$0.297101\pi$
$912$	0	0
$913$	18.8494	0.623825
$914$	0	0
$915$	23.3230	0.771034
$916$	0	0
$917$	−34.3897	−1.13565
$918$	0	0
$919$	5.76019	0.190011	0.0950055	−	0.995477i	$-0.469713\pi$
$919$	5.76019	0.190011	0.0950055	+	0.995477i	$0.469713\pi$
$920$	0	0
$921$	−19.5823	−0.645257
$922$	0	0
$923$	−78.3695	−2.57956
$924$	0	0
$925$	−6.36015	−0.209120
$926$	0	0
$927$	0.885137	0.0290717
$928$	0	0
$929$	19.7791	0.648932	0.324466	−	0.945897i	$-0.394815\pi$
$929$	19.7791	0.648932	0.324466	+	0.945897i	$0.394815\pi$
$930$	0	0
$931$	−21.9329	−0.718823
$932$	0	0
$933$	−1.91968	−0.0628474
$934$	0	0
$935$	2.64217	0.0864081
$936$	0	0
$937$	−0.548515	−0.0179192	−0.00895961	−	0.999960i	$-0.502852\pi$
$937$	−0.548515	−0.0179192	−0.00895961	+	0.999960i	$0.502852\pi$
$938$	0	0
$939$	12.6859	0.413987
$940$	0	0
$941$	−45.1280	−1.47113	−0.735565	−	0.677454i	$-0.763083\pi$
$941$	−45.1280	−1.47113	−0.735565	+	0.677454i	$0.763083\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	6.16660	0.200599
$946$	0	0
$947$	3.89487	0.126566	0.0632831	−	0.997996i	$-0.479843\pi$
$947$	3.89487	0.126566	0.0632831	+	0.997996i	$0.479843\pi$
$948$	0	0
$949$	−40.2591	−1.30687
$950$	0	0
$951$	3.12473	0.101326
$952$	0	0
$953$	48.2589	1.56326	0.781629	−	0.623743i	$-0.214389\pi$
$953$	48.2589	1.56326	0.781629	+	0.623743i	$0.214389\pi$
$954$	0	0
$955$	44.8347	1.45082
$956$	0	0
$957$	8.22324	0.265820
$958$	0	0
$959$	−22.2722	−0.719206
$960$	0	0
$961$	−19.2886	−0.622214
$962$	0	0
$963$	7.91236	0.254972
$964$	0	0
$965$	−8.53381	−0.274713
$966$	0	0
$967$	−52.3950	−1.68491	−0.842456	−	0.538766i	$-0.818891\pi$
$967$	−52.3950	−1.68491	−0.842456	+	0.538766i	$0.818891\pi$
$968$	0	0
$969$	−3.59218	−0.115397
$970$	0	0
$971$	−0.443074	−0.0142189	−0.00710946	−	0.999975i	$-0.502263\pi$
$971$	−0.443074	−0.0142189	−0.00710946	+	0.999975i	$0.502263\pi$
$972$	0	0
$973$	−29.1661	−0.935021
$974$	0	0
$975$	−9.16439	−0.293495
$976$	0	0
$977$	−52.6871	−1.68561	−0.842805	−	0.538219i	$-0.819097\pi$
$977$	−52.6871	−1.68561	−0.842805	+	0.538219i	$0.819097\pi$
$978$	0	0
$979$	−21.4857	−0.686686
$980$	0	0
$981$	1.75994	0.0561904
$982$	0	0
$983$	−15.9658	−0.509231	−0.254615	−	0.967042i	$-0.581949\pi$
$983$	−15.9658	−0.509231	−0.254615	+	0.967042i	$0.581949\pi$
$984$	0	0
$985$	−20.1310	−0.641427
$986$	0	0
$987$	35.3001	1.12362
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−21.7399	−0.690590	−0.345295	−	0.938494i	$-0.612221\pi$
$991$	−21.7399	−0.690590	−0.345295	+	0.938494i	$0.612221\pi$
$992$	0	0
$993$	−12.8689	−0.408381
$994$	0	0
$995$	14.2645	0.452216
$996$	0	0
$997$	32.2016	1.01984	0.509918	−	0.860223i	$-0.329676\pi$
$997$	32.2016	1.01984	0.509918	+	0.860223i	$0.329676\pi$
$998$	0	0
$999$	3.67687	0.116331

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6348.2.a.t.1.8		10
23.11	odd	22		276.2.i.a.121.1	yes	20
23.21	odd	22		276.2.i.a.73.1	✓	20
23.22	odd	2		6348.2.a.s.1.3		10
69.11	even	22		828.2.q.c.397.2		20
69.44	even	22		828.2.q.c.73.2		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
276.2.i.a.73.1	✓	20	23.21	odd	22
276.2.i.a.121.1	yes	20	23.11	odd	22
828.2.q.c.73.2		20	69.44	even	22
828.2.q.c.397.2		20	69.11	even	22
6348.2.a.s.1.3		10	23.22	odd	2
6348.2.a.t.1.8		10	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:	\( N \)	\(=\)	\( 6348 = 2^{2} \cdot 3 \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6348.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.80838 q^{5} +3.41002 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.80838 q^{5} +3.41002 q^{7} +1.00000 q^{9} +1.92751 q^{11} +5.29803 q^{13} +1.80838 q^{15} +0.758011 q^{17} -4.73895 q^{19} +3.41002 q^{21} -1.72977 q^{25} +1.00000 q^{27} +4.26626 q^{29} -3.42219 q^{31} +1.92751 q^{33} +6.16660 q^{35} +3.67687 q^{37} +5.29803 q^{39} -7.03987 q^{41} +3.67323 q^{43} +1.80838 q^{45} +10.3519 q^{47} +4.62823 q^{49} +0.758011 q^{51} -7.86365 q^{53} +3.48566 q^{55} -4.73895 q^{57} +10.8055 q^{59} +12.8972 q^{61} +3.41002 q^{63} +9.58083 q^{65} +6.20464 q^{67} -14.7922 q^{71} -7.59888 q^{73} -1.72977 q^{75} +6.57283 q^{77} -6.36068 q^{79} +1.00000 q^{81} +9.77918 q^{83} +1.37077 q^{85} +4.26626 q^{87} -11.1469 q^{89} +18.0664 q^{91} -3.42219 q^{93} -8.56981 q^{95} -7.62022 q^{97} +1.92751 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 10 q^{3} + 2 q^{5} + 11 q^{7} + 10 q^{9}+O(q^{10}) \) 10 * q + 10 * q^3 + 2 * q^5 + 11 * q^7 + 10 * q^9	\( 10 q + 10 q^{3} + 2 q^{5} + 11 q^{7} + 10 q^{9} + 11 q^{11} + 2 q^{15} + 13 q^{17} + 18 q^{19} + 11 q^{21} + 10 q^{25} + 10 q^{27} + 5 q^{29} + 15 q^{31} + 11 q^{33} - 13 q^{35} + 5 q^{37} + 24 q^{41} + 40 q^{43} + 2 q^{45} - 9 q^{47} + 5 q^{49} + 13 q^{51} + 6 q^{53} - 14 q^{55} + 18 q^{57} + 28 q^{59} + 39 q^{61} + 11 q^{63} + 14 q^{65} + 32 q^{67} - 33 q^{71} - 50 q^{73} + 10 q^{75} + 19 q^{77} + 33 q^{79} + 10 q^{81} + 29 q^{83} - 21 q^{85} + 5 q^{87} + 17 q^{89} + 36 q^{91} + 15 q^{93} - 42 q^{95} + 46 q^{97} + 11 q^{99}+O(q^{100}) \) 10 * q + 10 * q^3 + 2 * q^5 + 11 * q^7 + 10 * q^9 + 11 * q^11 + 2 * q^15 + 13 * q^17 + 18 * q^19 + 11 * q^21 + 10 * q^25 + 10 * q^27 + 5 * q^29 + 15 * q^31 + 11 * q^33 - 13 * q^35 + 5 * q^37 + 24 * q^41 + 40 * q^43 + 2 * q^45 - 9 * q^47 + 5 * q^49 + 13 * q^51 + 6 * q^53 - 14 * q^55 + 18 * q^57 + 28 * q^59 + 39 * q^61 + 11 * q^63 + 14 * q^65 + 32 * q^67 - 33 * q^71 - 50 * q^73 + 10 * q^75 + 19 * q^77 + 33 * q^79 + 10 * q^81 + 29 * q^83 - 21 * q^85 + 5 * q^87 + 17 * q^89 + 36 * q^91 + 15 * q^93 - 42 * q^95 + 46 * q^97 + 11 * q^99

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