LMFDB - Embedded newform 6348.2.a.s.1.7

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:	$1$
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			1.7
Root			$2.46393$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +1.04710 q^{5} +1.46307 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.04710 q^{5} +1.46307 q^{7} +1.00000 q^{9} -2.04776 q^{11} -6.89348 q^{13} +1.04710 q^{15} -7.77763 q^{17} +6.37212 q^{19} +1.46307 q^{21} -3.90357 q^{25} +1.00000 q^{27} +4.14709 q^{29} +0.909997 q^{31} -2.04776 q^{33} +1.53199 q^{35} +4.07919 q^{37} -6.89348 q^{39} +4.89363 q^{41} -3.50314 q^{43} +1.04710 q^{45} -1.92699 q^{47} -4.85943 q^{49} -7.77763 q^{51} -7.82919 q^{53} -2.14422 q^{55} +6.37212 q^{57} +14.4629 q^{59} -7.53528 q^{61} +1.46307 q^{63} -7.21819 q^{65} -10.0950 q^{67} -0.197482 q^{71} -7.96038 q^{73} -3.90357 q^{75} -2.99601 q^{77} -4.62746 q^{79} +1.00000 q^{81} +8.28322 q^{83} -8.14399 q^{85} +4.14709 q^{87} -0.275247 q^{89} -10.0856 q^{91} +0.909997 q^{93} +6.67228 q^{95} -7.35665 q^{97} -2.04776 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	1.04710	0.468279	0.234140	−	0.972203i	$-0.424773\pi$
$5$	1.04710	0.468279	0.234140	+	0.972203i	$0.424773\pi$
$6$	0	0
$7$	1.46307	0.552988	0.276494	−	0.961016i	$-0.410827\pi$
$7$	1.46307	0.552988	0.276494	+	0.961016i	$0.410827\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.04776	−0.617423	−0.308712	−	0.951156i	$-0.599898\pi$
$11$	−2.04776	−0.617423	−0.308712	+	0.951156i	$0.599898\pi$
$12$	0	0
$13$	−6.89348	−1.91191	−0.955953	−	0.293519i	$-0.905173\pi$
$13$	−6.89348	−1.91191	−0.955953	+	0.293519i	$0.905173\pi$
$14$	0	0
$15$	1.04710	0.270361
$16$	0	0
$17$	−7.77763	−1.88635	−0.943176	−	0.332294i	$-0.892177\pi$
$17$	−7.77763	−1.88635	−0.943176	+	0.332294i	$0.892177\pi$
$18$	0	0
$19$	6.37212	1.46187	0.730933	−	0.682450i	$-0.239085\pi$
$19$	6.37212	1.46187	0.730933	+	0.682450i	$0.239085\pi$
$20$	0	0
$21$	1.46307	0.319268
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−3.90357	−0.780714
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	4.14709	0.770095	0.385048	−	0.922897i	$-0.374185\pi$
$29$	4.14709	0.770095	0.385048	+	0.922897i	$0.374185\pi$
$30$	0	0
$31$	0.909997	0.163440	0.0817201	−	0.996655i	$-0.473959\pi$
$31$	0.909997	0.163440	0.0817201	+	0.996655i	$0.473959\pi$
$32$	0	0
$33$	−2.04776	−0.356469
$34$	0	0
$35$	1.53199	0.258953
$36$	0	0
$37$	4.07919	0.670614	0.335307	−	0.942109i	$-0.391160\pi$
$37$	4.07919	0.670614	0.335307	+	0.942109i	$0.391160\pi$
$38$	0	0
$39$	−6.89348	−1.10384
$40$	0	0
$41$	4.89363	0.764257	0.382129	−	0.924109i	$-0.375191\pi$
$41$	4.89363	0.764257	0.382129	+	0.924109i	$0.375191\pi$
$42$	0	0
$43$	−3.50314	−0.534224	−0.267112	−	0.963665i	$-0.586069\pi$
$43$	−3.50314	−0.534224	−0.267112	+	0.963665i	$0.586069\pi$
$44$	0	0
$45$	1.04710	0.156093
$46$	0	0
$47$	−1.92699	−0.281080	−0.140540	−	0.990075i	$-0.544884\pi$
$47$	−1.92699	−0.281080	−0.140540	+	0.990075i	$0.544884\pi$
$48$	0	0
$49$	−4.85943	−0.694205
$50$	0	0
$51$	−7.77763	−1.08909
$52$	0	0
$53$	−7.82919	−1.07542	−0.537711	−	0.843129i	$-0.680711\pi$
$53$	−7.82919	−1.07542	−0.537711	+	0.843129i	$0.680711\pi$
$54$	0	0
$55$	−2.14422	−0.289127
$56$	0	0
$57$	6.37212	0.844008
$58$	0	0
$59$	14.4629	1.88291	0.941453	−	0.337145i	$-0.109461\pi$
$59$	14.4629	1.88291	0.941453	+	0.337145i	$0.109461\pi$
$60$	0	0
$61$	−7.53528	−0.964794	−0.482397	−	0.875953i	$-0.660234\pi$
$61$	−7.53528	−0.964794	−0.482397	+	0.875953i	$0.660234\pi$
$62$	0	0
$63$	1.46307	0.184329
$64$	0	0
$65$	−7.21819	−0.895306
$66$	0	0
$67$	−10.0950	−1.23330	−0.616650	−	0.787237i	$-0.711511\pi$
$67$	−10.0950	−1.23330	−0.616650	+	0.787237i	$0.711511\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−0.197482	−0.0234368	−0.0117184	−	0.999931i	$-0.503730\pi$
$71$	−0.197482	−0.0234368	−0.0117184	+	0.999931i	$0.503730\pi$
$72$	0	0
$73$	−7.96038	−0.931692	−0.465846	−	0.884866i	$-0.654250\pi$
$73$	−7.96038	−0.931692	−0.465846	+	0.884866i	$0.654250\pi$
$74$	0	0
$75$	−3.90357	−0.450746
$76$	0	0
$77$	−2.99601	−0.341427
$78$	0	0
$79$	−4.62746	−0.520630	−0.260315	−	0.965524i	$-0.583826\pi$
$79$	−4.62746	−0.520630	−0.260315	+	0.965524i	$0.583826\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	8.28322	0.909201	0.454601	−	0.890695i	$-0.349782\pi$
$83$	8.28322	0.909201	0.454601	+	0.890695i	$0.349782\pi$
$84$	0	0
$85$	−8.14399	−0.883340
$86$	0	0
$87$	4.14709	0.444615
$88$	0	0
$89$	−0.275247	−0.0291762	−0.0145881	−	0.999894i	$-0.504644\pi$
$89$	−0.275247	−0.0291762	−0.0145881	+	0.999894i	$0.504644\pi$
$90$	0	0
$91$	−10.0856	−1.05726
$92$	0	0
$93$	0.909997	0.0943623
$94$	0	0
$95$	6.67228	0.684561
$96$	0	0
$97$	−7.35665	−0.746954	−0.373477	−	0.927639i	$-0.621835\pi$
$97$	−7.35665	−0.746954	−0.373477	+	0.927639i	$0.621835\pi$
$98$	0	0
$99$	−2.04776	−0.205808
$100$	0	0
$101$	−14.3167	−1.42456	−0.712282	−	0.701893i	$-0.752338\pi$
$101$	−14.3167	−1.42456	−0.712282	+	0.701893i	$0.752338\pi$
$102$	0	0
$103$	−4.01467	−0.395577	−0.197788	−	0.980245i	$-0.563376\pi$
$103$	−4.01467	−0.395577	−0.197788	+	0.980245i	$0.563376\pi$
$104$	0	0
$105$	1.53199	0.149506
$106$	0	0
$107$	−13.0252	−1.25919	−0.629597	−	0.776922i	$-0.716780\pi$
$107$	−13.0252	−1.25919	−0.629597	+	0.776922i	$0.716780\pi$
$108$	0	0
$109$	−3.54533	−0.339581	−0.169791	−	0.985480i	$-0.554309\pi$
$109$	−3.54533	−0.339581	−0.169791	+	0.985480i	$0.554309\pi$
$110$	0	0
$111$	4.07919	0.387179
$112$	0	0
$113$	−15.3074	−1.43999	−0.719997	−	0.693977i	$-0.755857\pi$
$113$	−15.3074	−1.43999	−0.719997	+	0.693977i	$0.755857\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−6.89348	−0.637302
$118$	0	0
$119$	−11.3792	−1.04313
$120$	0	0
$121$	−6.80668	−0.618789
$122$	0	0
$123$	4.89363	0.441244
$124$	0	0
$125$	−9.32297	−0.833872
$126$	0	0
$127$	13.7788	1.22267	0.611335	−	0.791372i	$-0.290633\pi$
$127$	13.7788	1.22267	0.611335	+	0.791372i	$0.290633\pi$
$128$	0	0
$129$	−3.50314	−0.308434
$130$	0	0
$131$	−9.61471	−0.840041	−0.420020	−	0.907515i	$-0.637977\pi$
$131$	−9.61471	−0.840041	−0.420020	+	0.907515i	$0.637977\pi$
$132$	0	0
$133$	9.32285	0.808393
$134$	0	0
$135$	1.04710	0.0901204
$136$	0	0
$137$	3.53079	0.301656	0.150828	−	0.988560i	$-0.451806\pi$
$137$	3.53079	0.301656	0.150828	+	0.988560i	$0.451806\pi$
$138$	0	0
$139$	2.28333	0.193670	0.0968349	−	0.995300i	$-0.469128\pi$
$139$	2.28333	0.193670	0.0968349	+	0.995300i	$0.469128\pi$
$140$	0	0
$141$	−1.92699	−0.162282
$142$	0	0
$143$	14.1162	1.18046
$144$	0	0
$145$	4.34244	0.360620
$146$	0	0
$147$	−4.85943	−0.400799
$148$	0	0
$149$	−13.5130	−1.10703	−0.553515	−	0.832839i	$-0.686714\pi$
$149$	−13.5130	−1.10703	−0.553515	+	0.832839i	$0.686714\pi$
$150$	0	0
$151$	−2.31041	−0.188019	−0.0940093	−	0.995571i	$-0.529968\pi$
$151$	−2.31041	−0.188019	−0.0940093	+	0.995571i	$0.529968\pi$
$152$	0	0
$153$	−7.77763	−0.628784
$154$	0	0
$155$	0.952862	0.0765357
$156$	0	0
$157$	−12.0945	−0.965245	−0.482622	−	0.875829i	$-0.660316\pi$
$157$	−12.0945	−0.965245	−0.482622	+	0.875829i	$0.660316\pi$
$158$	0	0
$159$	−7.82919	−0.620895
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−21.6370	−1.69474	−0.847371	−	0.531001i	$-0.821816\pi$
$163$	−21.6370	−1.69474	−0.847371	+	0.531001i	$0.821816\pi$
$164$	0	0
$165$	−2.14422	−0.166927
$166$	0	0
$167$	−5.53960	−0.428667	−0.214333	−	0.976761i	$-0.568758\pi$
$167$	−5.53960	−0.428667	−0.214333	+	0.976761i	$0.568758\pi$
$168$	0	0
$169$	34.5200	2.65539
$170$	0	0
$171$	6.37212	0.487288
$172$	0	0
$173$	10.9773	0.834590	0.417295	−	0.908771i	$-0.362978\pi$
$173$	10.9773	0.834590	0.417295	+	0.908771i	$0.362978\pi$
$174$	0	0
$175$	−5.71119	−0.431725
$176$	0	0
$177$	14.4629	1.08710
$178$	0	0
$179$	18.9602	1.41715	0.708576	−	0.705634i	$-0.249338\pi$
$179$	18.9602	1.41715	0.708576	+	0.705634i	$0.249338\pi$
$180$	0	0
$181$	−16.7037	−1.24158	−0.620789	−	0.783978i	$-0.713188\pi$
$181$	−16.7037	−1.24158	−0.620789	+	0.783978i	$0.713188\pi$
$182$	0	0
$183$	−7.53528	−0.557024
$184$	0	0
$185$	4.27134	0.314035
$186$	0	0
$187$	15.9267	1.16468
$188$	0	0
$189$	1.46307	0.106423
$190$	0	0
$191$	10.1256	0.732663	0.366332	−	0.930484i	$-0.380614\pi$
$191$	10.1256	0.732663	0.366332	+	0.930484i	$0.380614\pi$
$192$	0	0
$193$	14.1573	1.01906	0.509532	−	0.860452i	$-0.329819\pi$
$193$	14.1573	1.01906	0.509532	+	0.860452i	$0.329819\pi$
$194$	0	0
$195$	−7.21819	−0.516905
$196$	0	0
$197$	−25.3524	−1.80628	−0.903140	−	0.429346i	$-0.858744\pi$
$197$	−25.3524	−1.80628	−0.903140	+	0.429346i	$0.858744\pi$
$198$	0	0
$199$	13.2342	0.938148	0.469074	−	0.883159i	$-0.344588\pi$
$199$	13.2342	0.938148	0.469074	+	0.883159i	$0.344588\pi$
$200$	0	0
$201$	−10.0950	−0.712047
$202$	0	0
$203$	6.06748	0.425853
$204$	0	0
$205$	5.12415	0.357886
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−13.0486	−0.902589
$210$	0	0
$211$	−1.29644	−0.0892509	−0.0446254	−	0.999004i	$-0.514209\pi$
$211$	−1.29644	−0.0892509	−0.0446254	+	0.999004i	$0.514209\pi$
$212$	0	0
$213$	−0.197482	−0.0135313
$214$	0	0
$215$	−3.66816	−0.250166
$216$	0	0
$217$	1.33139	0.0903804
$218$	0	0
$219$	−7.96038	−0.537913
$220$	0	0
$221$	53.6149	3.60653
$222$	0	0
$223$	−12.1561	−0.814030	−0.407015	−	0.913422i	$-0.633430\pi$
$223$	−12.1561	−0.814030	−0.407015	+	0.913422i	$0.633430\pi$
$224$	0	0
$225$	−3.90357	−0.260238
$226$	0	0
$227$	−23.5950	−1.56606	−0.783029	−	0.621985i	$-0.786326\pi$
$227$	−23.5950	−1.56606	−0.783029	+	0.621985i	$0.786326\pi$
$228$	0	0
$229$	15.4803	1.02296	0.511482	−	0.859294i	$-0.329097\pi$
$229$	15.4803	1.02296	0.511482	+	0.859294i	$0.329097\pi$
$230$	0	0
$231$	−2.99601	−0.197123
$232$	0	0
$233$	12.4984	0.818800	0.409400	−	0.912355i	$-0.365738\pi$
$233$	12.4984	0.818800	0.409400	+	0.912355i	$0.365738\pi$
$234$	0	0
$235$	−2.01776	−0.131624
$236$	0	0
$237$	−4.62746	−0.300586
$238$	0	0
$239$	20.8825	1.35078	0.675390	−	0.737461i	$-0.263975\pi$
$239$	20.8825	1.35078	0.675390	+	0.737461i	$0.263975\pi$
$240$	0	0
$241$	2.57940	0.166154	0.0830768	−	0.996543i	$-0.473525\pi$
$241$	2.57940	0.166154	0.0830768	+	0.996543i	$0.473525\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−5.08833	−0.325082
$246$	0	0
$247$	−43.9261	−2.79495
$248$	0	0
$249$	8.28322	0.524928
$250$	0	0
$251$	6.31385	0.398527	0.199263	−	0.979946i	$-0.436145\pi$
$251$	6.31385	0.398527	0.199263	+	0.979946i	$0.436145\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−8.14399	−0.509996
$256$	0	0
$257$	12.3028	0.767425	0.383713	−	0.923453i	$-0.374645\pi$
$257$	12.3028	0.767425	0.383713	+	0.923453i	$0.374645\pi$
$258$	0	0
$259$	5.96813	0.370842
$260$	0	0
$261$	4.14709	0.256698
$262$	0	0
$263$	7.09090	0.437244	0.218622	−	0.975810i	$-0.429844\pi$
$263$	7.09090	0.437244	0.218622	+	0.975810i	$0.429844\pi$
$264$	0	0
$265$	−8.19798	−0.503598
$266$	0	0
$267$	−0.275247	−0.0168449
$268$	0	0
$269$	−14.9741	−0.912989	−0.456494	−	0.889726i	$-0.650895\pi$
$269$	−14.9741	−0.912989	−0.456494	+	0.889726i	$0.650895\pi$
$270$	0	0
$271$	18.3737	1.11612	0.558061	−	0.829800i	$-0.311545\pi$
$271$	18.3737	1.11612	0.558061	+	0.829800i	$0.311545\pi$
$272$	0	0
$273$	−10.0856	−0.610410
$274$	0	0
$275$	7.99358	0.482031
$276$	0	0
$277$	13.9414	0.837657	0.418829	−	0.908065i	$-0.362441\pi$
$277$	13.9414	0.837657	0.418829	+	0.908065i	$0.362441\pi$
$278$	0	0
$279$	0.909997	0.0544801
$280$	0	0
$281$	−7.75769	−0.462785	−0.231392	−	0.972861i	$-0.574328\pi$
$281$	−7.75769	−0.462785	−0.231392	+	0.972861i	$0.574328\pi$
$282$	0	0
$283$	−32.5795	−1.93665	−0.968325	−	0.249693i	$-0.919670\pi$
$283$	−32.5795	−1.93665	−0.968325	+	0.249693i	$0.919670\pi$
$284$	0	0
$285$	6.67228	0.395232
$286$	0	0
$287$	7.15972	0.422625
$288$	0	0
$289$	43.4915	2.55832
$290$	0	0
$291$	−7.35665	−0.431254
$292$	0	0
$293$	21.4612	1.25378	0.626889	−	0.779108i	$-0.284328\pi$
$293$	21.4612	1.25378	0.626889	+	0.779108i	$0.284328\pi$
$294$	0	0
$295$	15.1441	0.881726
$296$	0	0
$297$	−2.04776	−0.118823
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−5.12533	−0.295419
$302$	0	0
$303$	−14.3167	−0.822473
$304$	0	0
$305$	−7.89023	−0.451793
$306$	0	0
$307$	−4.31387	−0.246205	−0.123103	−	0.992394i	$-0.539284\pi$
$307$	−4.31387	−0.246205	−0.123103	+	0.992394i	$0.539284\pi$
$308$	0	0
$309$	−4.01467	−0.228386
$310$	0	0
$311$	12.9425	0.733899	0.366950	−	0.930241i	$-0.380402\pi$
$311$	12.9425	0.733899	0.366950	+	0.930241i	$0.380402\pi$
$312$	0	0
$313$	3.69404	0.208800	0.104400	−	0.994535i	$-0.466708\pi$
$313$	3.69404	0.208800	0.104400	+	0.994535i	$0.466708\pi$
$314$	0	0
$315$	1.53199	0.0863176
$316$	0	0
$317$	18.7068	1.05068	0.525339	−	0.850893i	$-0.323938\pi$
$317$	18.7068	1.05068	0.525339	+	0.850893i	$0.323938\pi$
$318$	0	0
$319$	−8.49225	−0.475475
$320$	0	0
$321$	−13.0252	−0.726996
$322$	0	0
$323$	−49.5600	−2.75759
$324$	0	0
$325$	26.9092	1.49265
$326$	0	0
$327$	−3.54533	−0.196057
$328$	0	0
$329$	−2.81931	−0.155434
$330$	0	0
$331$	−4.54026	−0.249555	−0.124778	−	0.992185i	$-0.539822\pi$
$331$	−4.54026	−0.249555	−0.124778	+	0.992185i	$0.539822\pi$
$332$	0	0
$333$	4.07919	0.223538
$334$	0	0
$335$	−10.5705	−0.577530
$336$	0	0
$337$	−22.6085	−1.23156	−0.615782	−	0.787917i	$-0.711160\pi$
$337$	−22.6085	−1.23156	−0.615782	+	0.787917i	$0.711160\pi$
$338$	0	0
$339$	−15.3074	−0.831381
$340$	0	0
$341$	−1.86346	−0.100912
$342$	0	0
$343$	−17.3512	−0.936874
$344$	0	0
$345$	0	0
$346$	0	0
$347$	18.0403	0.968452	0.484226	−	0.874943i	$-0.339101\pi$
$347$	18.0403	0.968452	0.484226	+	0.874943i	$0.339101\pi$
$348$	0	0
$349$	−5.02311	−0.268881	−0.134441	−	0.990922i	$-0.542924\pi$
$349$	−5.02311	−0.268881	−0.134441	+	0.990922i	$0.542924\pi$
$350$	0	0
$351$	−6.89348	−0.367947
$352$	0	0
$353$	12.7057	0.676254	0.338127	−	0.941100i	$-0.390207\pi$
$353$	12.7057	0.676254	0.338127	+	0.941100i	$0.390207\pi$
$354$	0	0
$355$	−0.206785	−0.0109750
$356$	0	0
$357$	−11.3792	−0.602251
$358$	0	0
$359$	5.33498	0.281569	0.140785	−	0.990040i	$-0.455037\pi$
$359$	5.33498	0.281569	0.140785	+	0.990040i	$0.455037\pi$
$360$	0	0
$361$	21.6039	1.13705
$362$	0	0
$363$	−6.80668	−0.357258
$364$	0	0
$365$	−8.33535	−0.436292
$366$	0	0
$367$	−1.47822	−0.0771624	−0.0385812	−	0.999255i	$-0.512284\pi$
$367$	−1.47822	−0.0771624	−0.0385812	+	0.999255i	$0.512284\pi$
$368$	0	0
$369$	4.89363	0.254752
$370$	0	0
$371$	−11.4546	−0.594695
$372$	0	0
$373$	17.5700	0.909740	0.454870	−	0.890558i	$-0.349686\pi$
$373$	17.5700	0.909740	0.454870	+	0.890558i	$0.349686\pi$
$374$	0	0
$375$	−9.32297	−0.481436
$376$	0	0
$377$	−28.5879	−1.47235
$378$	0	0
$379$	−5.17831	−0.265992	−0.132996	−	0.991117i	$-0.542460\pi$
$379$	−5.17831	−0.265992	−0.132996	+	0.991117i	$0.542460\pi$
$380$	0	0
$381$	13.7788	0.705909
$382$	0	0
$383$	−27.7003	−1.41542	−0.707708	−	0.706505i	$-0.750271\pi$
$383$	−27.7003	−1.41542	−0.707708	+	0.706505i	$0.750271\pi$
$384$	0	0
$385$	−3.13714	−0.159883
$386$	0	0
$387$	−3.50314	−0.178075
$388$	0	0
$389$	5.37281	0.272412	0.136206	−	0.990681i	$-0.456509\pi$
$389$	5.37281	0.272412	0.136206	+	0.990681i	$0.456509\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−9.61471	−0.484998
$394$	0	0
$395$	−4.84543	−0.243800
$396$	0	0
$397$	−36.1923	−1.81644	−0.908219	−	0.418496i	$-0.862557\pi$
$397$	−36.1923	−1.81644	−0.908219	+	0.418496i	$0.862557\pi$
$398$	0	0
$399$	9.32285	0.466726
$400$	0	0
$401$	−9.19341	−0.459097	−0.229549	−	0.973297i	$-0.573725\pi$
$401$	−9.19341	−0.459097	−0.229549	+	0.973297i	$0.573725\pi$
$402$	0	0
$403$	−6.27304	−0.312482
$404$	0	0
$405$	1.04710	0.0520311
$406$	0	0
$407$	−8.35320	−0.414053
$408$	0	0
$409$	−0.848516	−0.0419564	−0.0209782	−	0.999780i	$-0.506678\pi$
$409$	−0.848516	−0.0419564	−0.0209782	+	0.999780i	$0.506678\pi$
$410$	0	0
$411$	3.53079	0.174161
$412$	0	0
$413$	21.1602	1.04122
$414$	0	0
$415$	8.67340	0.425760
$416$	0	0
$417$	2.28333	0.111815
$418$	0	0
$419$	24.7145	1.20738	0.603691	−	0.797219i	$-0.293696\pi$
$419$	24.7145	1.20738	0.603691	+	0.797219i	$0.293696\pi$
$420$	0	0
$421$	15.6587	0.763157	0.381578	−	0.924336i	$-0.375381\pi$
$421$	15.6587	0.763157	0.381578	+	0.924336i	$0.375381\pi$
$422$	0	0
$423$	−1.92699	−0.0936933
$424$	0	0
$425$	30.3605	1.47270
$426$	0	0
$427$	−11.0246	−0.533519
$428$	0	0
$429$	14.1162	0.681536
$430$	0	0
$431$	−6.65212	−0.320421	−0.160211	−	0.987083i	$-0.551217\pi$
$431$	−6.65212	−0.320421	−0.160211	+	0.987083i	$0.551217\pi$
$432$	0	0
$433$	20.2388	0.972613	0.486306	−	0.873788i	$-0.338344\pi$
$433$	20.2388	0.972613	0.486306	+	0.873788i	$0.338344\pi$
$434$	0	0
$435$	4.34244	0.208204
$436$	0	0
$437$	0	0
$438$	0	0
$439$	4.12378	0.196817	0.0984087	−	0.995146i	$-0.468625\pi$
$439$	4.12378	0.196817	0.0984087	+	0.995146i	$0.468625\pi$
$440$	0	0
$441$	−4.85943	−0.231402
$442$	0	0
$443$	30.6761	1.45747	0.728733	−	0.684798i	$-0.240110\pi$
$443$	30.6761	1.45747	0.728733	+	0.684798i	$0.240110\pi$
$444$	0	0
$445$	−0.288213	−0.0136626
$446$	0	0
$447$	−13.5130	−0.639144
$448$	0	0
$449$	22.6823	1.07045	0.535223	−	0.844711i	$-0.320228\pi$
$449$	22.6823	1.07045	0.535223	+	0.844711i	$0.320228\pi$
$450$	0	0
$451$	−10.0210	−0.471870
$452$	0	0
$453$	−2.31041	−0.108553
$454$	0	0
$455$	−10.5607	−0.495093
$456$	0	0
$457$	−28.1141	−1.31512	−0.657561	−	0.753401i	$-0.728412\pi$
$457$	−28.1141	−1.31512	−0.657561	+	0.753401i	$0.728412\pi$
$458$	0	0
$459$	−7.77763	−0.363029
$460$	0	0
$461$	−4.98643	−0.232241	−0.116121	−	0.993235i	$-0.537046\pi$
$461$	−4.98643	−0.232241	−0.116121	+	0.993235i	$0.537046\pi$
$462$	0	0
$463$	−26.2974	−1.22214	−0.611072	−	0.791575i	$-0.709261\pi$
$463$	−26.2974	−1.22214	−0.611072	+	0.791575i	$0.709261\pi$
$464$	0	0
$465$	0.952862	0.0441879
$466$	0	0
$467$	35.6620	1.65024	0.825119	−	0.564959i	$-0.191108\pi$
$467$	35.6620	1.65024	0.825119	+	0.564959i	$0.191108\pi$
$468$	0	0
$469$	−14.7697	−0.682000
$470$	0	0
$471$	−12.0945	−0.557284
$472$	0	0
$473$	7.17360	0.329842
$474$	0	0
$475$	−24.8740	−1.14130
$476$	0	0
$477$	−7.82919	−0.358474
$478$	0	0
$479$	−42.3745	−1.93614	−0.968069	−	0.250682i	$-0.919345\pi$
$479$	−42.3745	−1.93614	−0.968069	+	0.250682i	$0.919345\pi$
$480$	0	0
$481$	−28.1198	−1.28215
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−7.70318	−0.349783
$486$	0	0
$487$	−29.8140	−1.35100	−0.675501	−	0.737359i	$-0.736073\pi$
$487$	−29.8140	−1.35100	−0.675501	+	0.737359i	$0.736073\pi$
$488$	0	0
$489$	−21.6370	−0.978460
$490$	0	0
$491$	13.0685	0.589773	0.294886	−	0.955532i	$-0.404718\pi$
$491$	13.0685	0.589773	0.294886	+	0.955532i	$0.404718\pi$
$492$	0	0
$493$	−32.2545	−1.45267
$494$	0	0
$495$	−2.14422	−0.0963755
$496$	0	0
$497$	−0.288930	−0.0129603
$498$	0	0
$499$	−21.7301	−0.972773	−0.486386	−	0.873744i	$-0.661685\pi$
$499$	−21.7301	−0.972773	−0.486386	+	0.873744i	$0.661685\pi$
$500$	0	0
$501$	−5.53960	−0.247491
$502$	0	0
$503$	−18.2149	−0.812161	−0.406080	−	0.913837i	$-0.633105\pi$
$503$	−18.2149	−0.812161	−0.406080	+	0.913837i	$0.633105\pi$
$504$	0	0
$505$	−14.9911	−0.667094
$506$	0	0
$507$	34.5200	1.53309
$508$	0	0
$509$	−1.51886	−0.0673224	−0.0336612	−	0.999433i	$-0.510717\pi$
$509$	−1.51886	−0.0673224	−0.0336612	+	0.999433i	$0.510717\pi$
$510$	0	0
$511$	−11.6466	−0.515214
$512$	0	0
$513$	6.37212	0.281336
$514$	0	0
$515$	−4.20378	−0.185240
$516$	0	0
$517$	3.94601	0.173545
$518$	0	0
$519$	10.9773	0.481851
$520$	0	0
$521$	14.7934	0.648112	0.324056	−	0.946038i	$-0.394953\pi$
$521$	14.7934	0.648112	0.324056	+	0.946038i	$0.394953\pi$
$522$	0	0
$523$	19.5761	0.856001	0.428001	−	0.903778i	$-0.359218\pi$
$523$	19.5761	0.856001	0.428001	+	0.903778i	$0.359218\pi$
$524$	0	0
$525$	−5.71119	−0.249257
$526$	0	0
$527$	−7.07761	−0.308306
$528$	0	0
$529$	0	0
$530$	0	0
$531$	14.4629	0.627635
$532$	0	0
$533$	−33.7341	−1.46119
$534$	0	0
$535$	−13.6387	−0.589654
$536$	0	0
$537$	18.9602	0.818193
$538$	0	0
$539$	9.95095	0.428618
$540$	0	0
$541$	−1.41169	−0.0606934	−0.0303467	−	0.999539i	$-0.509661\pi$
$541$	−1.41169	−0.0606934	−0.0303467	+	0.999539i	$0.509661\pi$
$542$	0	0
$543$	−16.7037	−0.716825
$544$	0	0
$545$	−3.71233	−0.159019
$546$	0	0
$547$	−22.4315	−0.959102	−0.479551	−	0.877514i	$-0.659200\pi$
$547$	−22.4315	−0.959102	−0.479551	+	0.877514i	$0.659200\pi$
$548$	0	0
$549$	−7.53528	−0.321598
$550$	0	0
$551$	26.4258	1.12578
$552$	0	0
$553$	−6.77028	−0.287902
$554$	0	0
$555$	4.27134	0.181308
$556$	0	0
$557$	34.8082	1.47487	0.737436	−	0.675417i	$-0.236036\pi$
$557$	34.8082	1.47487	0.737436	+	0.675417i	$0.236036\pi$
$558$	0	0
$559$	24.1488	1.02139
$560$	0	0
$561$	15.9267	0.672427
$562$	0	0
$563$	−46.7994	−1.97236	−0.986179	−	0.165685i	$-0.947017\pi$
$563$	−46.7994	−1.97236	−0.986179	+	0.165685i	$0.947017\pi$
$564$	0	0
$565$	−16.0284	−0.674320
$566$	0	0
$567$	1.46307	0.0614431
$568$	0	0
$569$	−8.73896	−0.366356	−0.183178	−	0.983080i	$-0.558639\pi$
$569$	−8.73896	−0.366356	−0.183178	+	0.983080i	$0.558639\pi$
$570$	0	0
$571$	19.7017	0.824491	0.412246	−	0.911073i	$-0.364745\pi$
$571$	19.7017	0.824491	0.412246	+	0.911073i	$0.364745\pi$
$572$	0	0
$573$	10.1256	0.423003
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−17.2532	−0.718260	−0.359130	−	0.933288i	$-0.616927\pi$
$577$	−17.2532	−0.718260	−0.359130	+	0.933288i	$0.616927\pi$
$578$	0	0
$579$	14.1573	0.588357
$580$	0	0
$581$	12.1189	0.502777
$582$	0	0
$583$	16.0323	0.663990
$584$	0	0
$585$	−7.21819	−0.298435
$586$	0	0
$587$	23.3777	0.964900	0.482450	−	0.875923i	$-0.339747\pi$
$587$	23.3777	0.964900	0.482450	+	0.875923i	$0.339747\pi$
$588$	0	0
$589$	5.79861	0.238928
$590$	0	0
$591$	−25.3524	−1.04286
$592$	0	0
$593$	14.6390	0.601152	0.300576	−	0.953758i	$-0.402821\pi$
$593$	14.6390	0.601152	0.300576	+	0.953758i	$0.402821\pi$
$594$	0	0
$595$	−11.9152	−0.488476
$596$	0	0
$597$	13.2342	0.541640
$598$	0	0
$599$	−18.2295	−0.744838	−0.372419	−	0.928065i	$-0.621472\pi$
$599$	−18.2295	−0.744838	−0.372419	+	0.928065i	$0.621472\pi$
$600$	0	0
$601$	−33.9613	−1.38531	−0.692655	−	0.721269i	$-0.743559\pi$
$601$	−33.9613	−1.38531	−0.692655	+	0.721269i	$0.743559\pi$
$602$	0	0
$603$	−10.0950	−0.411100
$604$	0	0
$605$	−7.12730	−0.289766
$606$	0	0
$607$	48.9794	1.98801	0.994007	−	0.109314i	$-0.0348655\pi$
$607$	48.9794	1.98801	0.994007	+	0.109314i	$0.0348655\pi$
$608$	0	0
$609$	6.06748	0.245866
$610$	0	0
$611$	13.2836	0.537398
$612$	0	0
$613$	20.6235	0.832974	0.416487	−	0.909142i	$-0.363261\pi$
$613$	20.6235	0.832974	0.416487	+	0.909142i	$0.363261\pi$
$614$	0	0
$615$	5.12415	0.206626
$616$	0	0
$617$	−27.0051	−1.08718	−0.543592	−	0.839350i	$-0.682936\pi$
$617$	−27.0051	−1.08718	−0.543592	+	0.839350i	$0.682936\pi$
$618$	0	0
$619$	27.3293	1.09846	0.549228	−	0.835673i	$-0.314922\pi$
$619$	27.3293	1.09846	0.549228	+	0.835673i	$0.314922\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−0.402706	−0.0161341
$624$	0	0
$625$	9.75573	0.390229
$626$	0	0
$627$	−13.0486	−0.521110
$628$	0	0
$629$	−31.7264	−1.26501
$630$	0	0
$631$	40.2664	1.60298	0.801491	−	0.598007i	$-0.204041\pi$
$631$	40.2664	1.60298	0.801491	+	0.598007i	$0.204041\pi$
$632$	0	0
$633$	−1.29644	−0.0515290
$634$	0	0
$635$	14.4278	0.572551
$636$	0	0
$637$	33.4984	1.32725
$638$	0	0
$639$	−0.197482	−0.00781227
$640$	0	0
$641$	−3.44875	−0.136217	−0.0681087	−	0.997678i	$-0.521696\pi$
$641$	−3.44875	−0.136217	−0.0681087	+	0.997678i	$0.521696\pi$
$642$	0	0
$643$	−23.4827	−0.926065	−0.463033	−	0.886341i	$-0.653239\pi$
$643$	−23.4827	−0.926065	−0.463033	+	0.886341i	$0.653239\pi$
$644$	0	0
$645$	−3.66816	−0.144434
$646$	0	0
$647$	39.1906	1.54074	0.770371	−	0.637595i	$-0.220071\pi$
$647$	39.1906	1.54074	0.770371	+	0.637595i	$0.220071\pi$
$648$	0	0
$649$	−29.6165	−1.16255
$650$	0	0
$651$	1.33139	0.0521812
$652$	0	0
$653$	−4.39915	−0.172152	−0.0860760	−	0.996289i	$-0.527433\pi$
$653$	−4.39915	−0.172152	−0.0860760	+	0.996289i	$0.527433\pi$
$654$	0	0
$655$	−10.0676	−0.393374
$656$	0	0
$657$	−7.96038	−0.310564
$658$	0	0
$659$	15.4695	0.602606	0.301303	−	0.953528i	$-0.402578\pi$
$659$	15.4695	0.602606	0.301303	+	0.953528i	$0.402578\pi$
$660$	0	0
$661$	−25.5574	−0.994069	−0.497035	−	0.867731i	$-0.665578\pi$
$661$	−25.5574	−0.994069	−0.497035	+	0.867731i	$0.665578\pi$
$662$	0	0
$663$	53.6149	2.08223
$664$	0	0
$665$	9.76200	0.378554
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−12.1561	−0.469980
$670$	0	0
$671$	15.4305	0.595686
$672$	0	0
$673$	−16.4470	−0.633986	−0.316993	−	0.948428i	$-0.602673\pi$
$673$	−16.4470	−0.633986	−0.316993	+	0.948428i	$0.602673\pi$
$674$	0	0
$675$	−3.90357	−0.150249
$676$	0	0
$677$	44.9835	1.72886	0.864429	−	0.502755i	$-0.167680\pi$
$677$	44.9835	1.72886	0.864429	+	0.502755i	$0.167680\pi$
$678$	0	0
$679$	−10.7633	−0.413057
$680$	0	0
$681$	−23.5950	−0.904164
$682$	0	0
$683$	5.31043	0.203198	0.101599	−	0.994825i	$-0.467604\pi$
$683$	5.31043	0.203198	0.101599	+	0.994825i	$0.467604\pi$
$684$	0	0
$685$	3.69711	0.141259
$686$	0	0
$687$	15.4803	0.590609
$688$	0	0
$689$	53.9703	2.05611
$690$	0	0
$691$	−22.2506	−0.846454	−0.423227	−	0.906024i	$-0.639103\pi$
$691$	−22.2506	−0.846454	−0.423227	+	0.906024i	$0.639103\pi$
$692$	0	0
$693$	−2.99601	−0.113809
$694$	0	0
$695$	2.39089	0.0906916
$696$	0	0
$697$	−38.0609	−1.44166
$698$	0	0
$699$	12.4984	0.472734
$700$	0	0
$701$	31.1228	1.17549	0.587745	−	0.809046i	$-0.300016\pi$
$701$	31.1228	1.17549	0.587745	+	0.809046i	$0.300016\pi$
$702$	0	0
$703$	25.9931	0.980348
$704$	0	0
$705$	−2.01776	−0.0759931
$706$	0	0
$707$	−20.9463	−0.787767
$708$	0	0
$709$	22.9277	0.861067	0.430534	−	0.902575i	$-0.358325\pi$
$709$	22.9277	0.861067	0.430534	+	0.902575i	$0.358325\pi$
$710$	0	0
$711$	−4.62746	−0.173543
$712$	0	0
$713$	0	0
$714$	0	0
$715$	14.7811	0.552783
$716$	0	0
$717$	20.8825	0.779873
$718$	0	0
$719$	45.9238	1.71267	0.856335	−	0.516421i	$-0.172736\pi$
$719$	45.9238	1.71267	0.856335	+	0.516421i	$0.172736\pi$
$720$	0	0
$721$	−5.87373	−0.218749
$722$	0	0
$723$	2.57940	0.0959288
$724$	0	0
$725$	−16.1885	−0.601224
$726$	0	0
$727$	20.6813	0.767028	0.383514	−	0.923535i	$-0.374714\pi$
$727$	20.6813	0.767028	0.383514	+	0.923535i	$0.374714\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	27.2461	1.00773
$732$	0	0
$733$	38.9570	1.43891	0.719454	−	0.694540i	$-0.244392\pi$
$733$	38.9570	1.43891	0.719454	+	0.694540i	$0.244392\pi$
$734$	0	0
$735$	−5.08833	−0.187686
$736$	0	0
$737$	20.6722	0.761469
$738$	0	0
$739$	29.1549	1.07248	0.536239	−	0.844066i	$-0.319844\pi$
$739$	29.1549	1.07248	0.536239	+	0.844066i	$0.319844\pi$
$740$	0	0
$741$	−43.9261	−1.61366
$742$	0	0
$743$	−2.48734	−0.0912516	−0.0456258	−	0.998959i	$-0.514528\pi$
$743$	−2.48734	−0.0912516	−0.0456258	+	0.998959i	$0.514528\pi$
$744$	0	0
$745$	−14.1495	−0.518399
$746$	0	0
$747$	8.28322	0.303067
$748$	0	0
$749$	−19.0567	−0.696318
$750$	0	0
$751$	−26.9151	−0.982146	−0.491073	−	0.871118i	$-0.663395\pi$
$751$	−26.9151	−0.982146	−0.491073	+	0.871118i	$0.663395\pi$
$752$	0	0
$753$	6.31385	0.230089
$754$	0	0
$755$	−2.41924	−0.0880453
$756$	0	0
$757$	15.0755	0.547928	0.273964	−	0.961740i	$-0.411665\pi$
$757$	15.0755	0.547928	0.273964	+	0.961740i	$0.411665\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	20.1657	0.731005	0.365502	−	0.930810i	$-0.380897\pi$
$761$	20.1657	0.731005	0.365502	+	0.930810i	$0.380897\pi$
$762$	0	0
$763$	−5.18706	−0.187784
$764$	0	0
$765$	−8.14399	−0.294447
$766$	0	0
$767$	−99.6994	−3.59994
$768$	0	0
$769$	−31.0151	−1.11843	−0.559216	−	0.829022i	$-0.688898\pi$
$769$	−31.0151	−1.11843	−0.559216	+	0.829022i	$0.688898\pi$
$770$	0	0
$771$	12.3028	0.443073
$772$	0	0
$773$	14.8542	0.534269	0.267135	−	0.963659i	$-0.413923\pi$
$773$	14.8542	0.534269	0.267135	+	0.963659i	$0.413923\pi$
$774$	0	0
$775$	−3.55224	−0.127600
$776$	0	0
$777$	5.96813	0.214105
$778$	0	0
$779$	31.1828	1.11724
$780$	0	0
$781$	0.404396	0.0144704
$782$	0	0
$783$	4.14709	0.148205
$784$	0	0
$785$	−12.6642	−0.452004
$786$	0	0
$787$	−23.8432	−0.849919	−0.424959	−	0.905212i	$-0.639712\pi$
$787$	−23.8432	−0.849919	−0.424959	+	0.905212i	$0.639712\pi$
$788$	0	0
$789$	7.09090	0.252443
$790$	0	0
$791$	−22.3957	−0.796299
$792$	0	0
$793$	51.9443	1.84460
$794$	0	0
$795$	−8.19798	−0.290752
$796$	0	0
$797$	−0.0152420	−0.000539900 0	−0.000269950	−	1.00000i	$-0.500086\pi$
$797$	−0.0152420	−0.000539900 0	−0.000269950		1.00000i	$0.500086\pi$
$798$	0	0
$799$	14.9874	0.530215
$800$	0	0
$801$	−0.275247	−0.00972539
$802$	0	0
$803$	16.3010	0.575248
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−14.9741	−0.527114
$808$	0	0
$809$	−34.3498	−1.20767	−0.603836	−	0.797108i	$-0.706362\pi$
$809$	−34.3498	−1.20767	−0.603836	+	0.797108i	$0.706362\pi$
$810$	0	0
$811$	−35.6554	−1.25203	−0.626016	−	0.779810i	$-0.715315\pi$
$811$	−35.6554	−1.25203	−0.626016	+	0.779810i	$0.715315\pi$
$812$	0	0
$813$	18.3737	0.644394
$814$	0	0
$815$	−22.6562	−0.793613
$816$	0	0
$817$	−22.3224	−0.780964
$818$	0	0
$819$	−10.0856	−0.352420
$820$	0	0
$821$	34.7214	1.21178	0.605892	−	0.795547i	$-0.292816\pi$
$821$	34.7214	1.21178	0.605892	+	0.795547i	$0.292816\pi$
$822$	0	0
$823$	−26.4728	−0.922783	−0.461391	−	0.887197i	$-0.652650\pi$
$823$	−26.4728	−0.922783	−0.461391	+	0.887197i	$0.652650\pi$
$824$	0	0
$825$	7.99358	0.278301
$826$	0	0
$827$	−25.7194	−0.894351	−0.447176	−	0.894446i	$-0.647570\pi$
$827$	−25.7194	−0.894351	−0.447176	+	0.894446i	$0.647570\pi$
$828$	0	0
$829$	2.78348	0.0966741	0.0483371	−	0.998831i	$-0.484608\pi$
$829$	2.78348	0.0966741	0.0483371	+	0.998831i	$0.484608\pi$
$830$	0	0
$831$	13.9414	0.483622
$832$	0	0
$833$	37.7948	1.30951
$834$	0	0
$835$	−5.80054	−0.200736
$836$	0	0
$837$	0.909997	0.0314541
$838$	0	0
$839$	−41.7432	−1.44113	−0.720567	−	0.693385i	$-0.756118\pi$
$839$	−41.7432	−1.44113	−0.720567	+	0.693385i	$0.756118\pi$
$840$	0	0
$841$	−11.8016	−0.406953
$842$	0	0
$843$	−7.75769	−0.267189
$844$	0	0
$845$	36.1461	1.24346
$846$	0	0
$847$	−9.95863	−0.342183
$848$	0	0
$849$	−32.5795	−1.11813
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−27.0756	−0.927050	−0.463525	−	0.886084i	$-0.653416\pi$
$853$	−27.0756	−0.927050	−0.463525	+	0.886084i	$0.653416\pi$
$854$	0	0
$855$	6.67228	0.228187
$856$	0	0
$857$	31.7827	1.08568	0.542838	−	0.839837i	$-0.317350\pi$
$857$	31.7827	1.08568	0.542838	+	0.839837i	$0.317350\pi$
$858$	0	0
$859$	15.5728	0.531338	0.265669	−	0.964064i	$-0.414407\pi$
$859$	15.5728	0.531338	0.265669	+	0.964064i	$0.414407\pi$
$860$	0	0
$861$	7.15972	0.244003
$862$	0	0
$863$	3.13566	0.106739	0.0533695	−	0.998575i	$-0.483004\pi$
$863$	3.13566	0.106739	0.0533695	+	0.998575i	$0.483004\pi$
$864$	0	0
$865$	11.4944	0.390821
$866$	0	0
$867$	43.4915	1.47705
$868$	0	0
$869$	9.47593	0.321449
$870$	0	0
$871$	69.5897	2.35796
$872$	0	0
$873$	−7.35665	−0.248985
$874$	0	0
$875$	−13.6401	−0.461121
$876$	0	0
$877$	−7.55790	−0.255212	−0.127606	−	0.991825i	$-0.540729\pi$
$877$	−7.55790	−0.255212	−0.127606	+	0.991825i	$0.540729\pi$
$878$	0	0
$879$	21.4612	0.723869
$880$	0	0
$881$	−21.8132	−0.734905	−0.367453	−	0.930042i	$-0.619770\pi$
$881$	−21.8132	−0.734905	−0.367453	+	0.930042i	$0.619770\pi$
$882$	0	0
$883$	1.98696	0.0668664	0.0334332	−	0.999441i	$-0.489356\pi$
$883$	1.98696	0.0668664	0.0334332	+	0.999441i	$0.489356\pi$
$884$	0	0
$885$	15.1441	0.509065
$886$	0	0
$887$	−14.7683	−0.495870	−0.247935	−	0.968777i	$-0.579752\pi$
$887$	−14.7683	−0.495870	−0.247935	+	0.968777i	$0.579752\pi$
$888$	0	0
$889$	20.1593	0.676121
$890$	0	0
$891$	−2.04776	−0.0686026
$892$	0	0
$893$	−12.2790	−0.410901
$894$	0	0
$895$	19.8533	0.663623
$896$	0	0
$897$	0	0
$898$	0	0
$899$	3.77384	0.125865
$900$	0	0
$901$	60.8925	2.02862
$902$	0	0
$903$	−5.12533	−0.170560
$904$	0	0
$905$	−17.4905	−0.581405
$906$	0	0
$907$	7.95757	0.264227	0.132113	−	0.991235i	$-0.457824\pi$
$907$	7.95757	0.264227	0.132113	+	0.991235i	$0.457824\pi$
$908$	0	0
$909$	−14.3167	−0.474855
$910$	0	0
$911$	28.6491	0.949186	0.474593	−	0.880205i	$-0.342595\pi$
$911$	28.6491	0.949186	0.474593	+	0.880205i	$0.342595\pi$
$912$	0	0
$913$	−16.9620	−0.561362
$914$	0	0
$915$	−7.89023	−0.260843
$916$	0	0
$917$	−14.0670	−0.464532
$918$	0	0
$919$	−33.7887	−1.11458	−0.557292	−	0.830316i	$-0.688160\pi$
$919$	−33.7887	−1.11458	−0.557292	+	0.830316i	$0.688160\pi$
$920$	0	0
$921$	−4.31387	−0.142147
$922$	0	0
$923$	1.36134	0.0448090
$924$	0	0
$925$	−15.9234	−0.523558
$926$	0	0
$927$	−4.01467	−0.131859
$928$	0	0
$929$	−33.5555	−1.10092	−0.550460	−	0.834862i	$-0.685547\pi$
$929$	−33.5555	−1.10092	−0.550460	+	0.834862i	$0.685547\pi$
$930$	0	0
$931$	−30.9649	−1.01483
$932$	0	0
$933$	12.9425	0.423717
$934$	0	0
$935$	16.6769	0.545394
$936$	0	0
$937$	21.9489	0.717039	0.358520	−	0.933522i	$-0.383282\pi$
$937$	21.9489	0.717039	0.358520	+	0.933522i	$0.383282\pi$
$938$	0	0
$939$	3.69404	0.120551
$940$	0	0
$941$	32.7362	1.06717	0.533585	−	0.845746i	$-0.320844\pi$
$941$	32.7362	1.06717	0.533585	+	0.845746i	$0.320844\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	1.53199	0.0498355
$946$	0	0
$947$	32.0690	1.04210	0.521051	−	0.853526i	$-0.325540\pi$
$947$	32.0690	1.04210	0.521051	+	0.853526i	$0.325540\pi$
$948$	0	0
$949$	54.8747	1.78131
$950$	0	0
$951$	18.7068	0.606609
$952$	0	0
$953$	−24.4717	−0.792717	−0.396358	−	0.918096i	$-0.629726\pi$
$953$	−24.4717	−0.792717	−0.396358	+	0.918096i	$0.629726\pi$
$954$	0	0
$955$	10.6026	0.343091
$956$	0	0
$957$	−8.49225	−0.274515
$958$	0	0
$959$	5.16579	0.166812
$960$	0	0
$961$	−30.1719	−0.973287
$962$	0	0
$963$	−13.0252	−0.419731
$964$	0	0
$965$	14.8242	0.477207
$966$	0	0
$967$	17.6787	0.568508	0.284254	−	0.958749i	$-0.408254\pi$
$967$	17.6787	0.568508	0.284254	+	0.958749i	$0.408254\pi$
$968$	0	0
$969$	−49.5600	−1.59210
$970$	0	0
$971$	5.19762	0.166799	0.0833997	−	0.996516i	$-0.473422\pi$
$971$	5.19762	0.166799	0.0833997	+	0.996516i	$0.473422\pi$
$972$	0	0
$973$	3.34067	0.107097
$974$	0	0
$975$	26.9092	0.861783
$976$	0	0
$977$	−23.3734	−0.747783	−0.373891	−	0.927473i	$-0.621977\pi$
$977$	−23.3734	−0.747783	−0.373891	+	0.927473i	$0.621977\pi$
$978$	0	0
$979$	0.563641	0.0180140
$980$	0	0
$981$	−3.54533	−0.113194
$982$	0	0
$983$	33.4288	1.06621	0.533107	−	0.846048i	$-0.321024\pi$
$983$	33.4288	1.06621	0.533107	+	0.846048i	$0.321024\pi$
$984$	0	0
$985$	−26.5466	−0.845844
$986$	0	0
$987$	−2.81931	−0.0897397
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0.897557	0.0285118	0.0142559	−	0.999898i	$-0.495462\pi$
$991$	0.897557	0.0285118	0.0142559	+	0.999898i	$0.495462\pi$
$992$	0	0
$993$	−4.54026	−0.144081
$994$	0	0
$995$	13.8576	0.439316
$996$	0	0
$997$	50.0344	1.58460	0.792302	−	0.610129i	$-0.208882\pi$
$997$	50.0344	1.58460	0.792302	+	0.610129i	$0.208882\pi$
$998$	0	0
$999$	4.07919	0.129060

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6348.2.a.s.1.7		10
23.3	even	11		276.2.i.a.193.1	yes	20
23.8	even	11		276.2.i.a.133.1	✓	20
23.22	odd	2		6348.2.a.t.1.4		10
69.8	odd	22		828.2.q.c.685.2		20
69.26	odd	22		828.2.q.c.469.2		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
276.2.i.a.133.1	✓	20	23.8	even	11
276.2.i.a.193.1	yes	20	23.3	even	11
828.2.q.c.469.2		20	69.26	odd	22
828.2.q.c.685.2		20	69.8	odd	22
6348.2.a.s.1.7		10	1.1	even	1	trivial
6348.2.a.t.1.4		10	23.22	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:	\( N \)	\(=\)	\( 6348 = 2^{2} \cdot 3 \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6348.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.04710 q^{5} +1.46307 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.04710 q^{5} +1.46307 q^{7} +1.00000 q^{9} -2.04776 q^{11} -6.89348 q^{13} +1.04710 q^{15} -7.77763 q^{17} +6.37212 q^{19} +1.46307 q^{21} -3.90357 q^{25} +1.00000 q^{27} +4.14709 q^{29} +0.909997 q^{31} -2.04776 q^{33} +1.53199 q^{35} +4.07919 q^{37} -6.89348 q^{39} +4.89363 q^{41} -3.50314 q^{43} +1.04710 q^{45} -1.92699 q^{47} -4.85943 q^{49} -7.77763 q^{51} -7.82919 q^{53} -2.14422 q^{55} +6.37212 q^{57} +14.4629 q^{59} -7.53528 q^{61} +1.46307 q^{63} -7.21819 q^{65} -10.0950 q^{67} -0.197482 q^{71} -7.96038 q^{73} -3.90357 q^{75} -2.99601 q^{77} -4.62746 q^{79} +1.00000 q^{81} +8.28322 q^{83} -8.14399 q^{85} +4.14709 q^{87} -0.275247 q^{89} -10.0856 q^{91} +0.909997 q^{93} +6.67228 q^{95} -7.35665 q^{97} -2.04776 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 10 q^{3} - 2 q^{5} - 11 q^{7} + 10 q^{9}+O(q^{10}) \) 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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