LMFDB - Embedded newform 6724.2.a.j.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6724,2,Mod(1,6724)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6724, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6724.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6724 = 2^{2} \cdot 41^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6724.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$53.6914103191$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 39x^{14} + 594x^{12} - 4428x^{10} + 16529x^{8} - 28236x^{6} + 17856x^{4} - 4032x^{2} + 256 $ x^16 - 39x^14 + 594x^12 - 4428x^10 + 16529x^8 - 28236x^6 + 17856x^4 - 4032*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 164)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Root			$3.12433$ of defining polynomial
Character	$\chi$	$=$	6724.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.12433 q^{3} +4.24501 q^{5} +1.30863 q^{7} +6.76146 q^{9} +O(q^{10})$	$q+3.12433 q^{3} +4.24501 q^{5} +1.30863 q^{7} +6.76146 q^{9} -1.62925 q^{11} +3.29297 q^{13} +13.2628 q^{15} -1.71565 q^{17} +4.45022 q^{19} +4.08859 q^{21} -6.76699 q^{23} +13.0201 q^{25} +11.7521 q^{27} -6.13704 q^{29} -3.16021 q^{31} -5.09032 q^{33} +5.55513 q^{35} -2.83232 q^{37} +10.2883 q^{39} +0.937313 q^{43} +28.7025 q^{45} -1.88404 q^{47} -5.28750 q^{49} -5.36028 q^{51} +0.389398 q^{53} -6.91617 q^{55} +13.9040 q^{57} +7.04699 q^{59} -6.07590 q^{61} +8.84823 q^{63} +13.9787 q^{65} -4.83541 q^{67} -21.1423 q^{69} -9.65102 q^{71} +8.76246 q^{73} +40.6791 q^{75} -2.13208 q^{77} -14.3387 q^{79} +16.4330 q^{81} +13.9185 q^{83} -7.28296 q^{85} -19.1742 q^{87} -2.53391 q^{89} +4.30927 q^{91} -9.87356 q^{93} +18.8912 q^{95} -1.36564 q^{97} -11.0161 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 6 q^{5} + 30 q^{9}+O(q^{10}) $ 16 * q + 6 * q^5 + 30 * q^9	$ 16 q + 6 q^{5} + 30 q^{9} + 46 q^{21} - 8 q^{23} + 38 q^{25} - 2 q^{31} + 20 q^{33} + 12 q^{37} + 8 q^{39} - 48 q^{43} + 92 q^{45} - 12 q^{49} + 42 q^{51} + 22 q^{57} - 32 q^{59} + 6 q^{61} - 34 q^{73} + 92 q^{77} + 108 q^{81} + 12 q^{83} - 34 q^{87}+O(q^{100}) $ 16 * q + 6 * q^5 + 30 * q^9 + 46 * q^21 - 8 * q^23 + 38 * q^25 - 2 * q^31 + 20 * q^33 + 12 * q^37 + 8 * q^39 - 48 * q^43 + 92 * q^45 - 12 * q^49 + 42 * q^51 + 22 * q^57 - 32 * q^59 + 6 * q^61 - 34 * q^73 + 92 * q^77 + 108 * q^81 + 12 * q^83 - 34 * q^87

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
$3$	3.12433	1.80384	0.901918	−	0.431908i	$-0.142160\pi$
$3$	3.12433	1.80384	0.901918	+	0.431908i	$0.142160\pi$
$4$	0	0
$5$	4.24501	1.89842	0.949212	−	0.314637i	$-0.101883\pi$
$5$	4.24501	1.89842	0.949212	+	0.314637i	$0.101883\pi$
$6$	0	0
$7$	1.30863	0.494615	0.247307	−	0.968937i	$-0.420454\pi$
$7$	1.30863	0.494615	0.247307	+	0.968937i	$0.420454\pi$
$8$	0	0
$9$	6.76146	2.25382
$10$	0	0
$11$	−1.62925	−0.491237	−0.245618	−	0.969367i	$-0.578991\pi$
$11$	−1.62925	−0.491237	−0.245618	+	0.969367i	$0.578991\pi$
$12$	0	0
$13$	3.29297	0.913307	0.456653	−	0.889645i	$-0.349048\pi$
$13$	3.29297	0.913307	0.456653	+	0.889645i	$0.349048\pi$
$14$	0	0
$15$	13.2628	3.42444
$16$	0	0
$17$	−1.71565	−0.416107	−0.208054	−	0.978117i	$-0.566713\pi$
$17$	−1.71565	−0.416107	−0.208054	+	0.978117i	$0.566713\pi$
$18$	0	0
$19$	4.45022	1.02095	0.510475	−	0.859892i	$-0.329470\pi$
$19$	4.45022	1.02095	0.510475	+	0.859892i	$0.329470\pi$
$20$	0	0
$21$	4.08859	0.892203
$22$	0	0
$23$	−6.76699	−1.41101	−0.705507	−	0.708703i	$-0.749281\pi$
$23$	−6.76699	−1.41101	−0.705507	+	0.708703i	$0.749281\pi$
$24$	0	0
$25$	13.0201	2.60401
$26$	0	0
$27$	11.7521	2.26169
$28$	0	0
$29$	−6.13704	−1.13962	−0.569810	−	0.821776i	$-0.692983\pi$
$29$	−6.13704	−1.13962	−0.569810	+	0.821776i	$0.692983\pi$
$30$	0	0
$31$	−3.16021	−0.567591	−0.283795	−	0.958885i	$-0.591594\pi$
$31$	−3.16021	−0.567591	−0.283795	+	0.958885i	$0.591594\pi$
$32$	0	0
$33$	−5.09032	−0.886110
$34$	0	0
$35$	5.55513	0.938988
$36$	0	0
$37$	−2.83232	−0.465631	−0.232815	−	0.972521i	$-0.574794\pi$
$37$	−2.83232	−0.465631	−0.232815	+	0.972521i	$0.574794\pi$
$38$	0	0
$39$	10.2883	1.64745
$40$	0	0
$41$	0	0
$41$	0	0
$42$	0	0
$43$	0.937313	0.142939	0.0714694	−	0.997443i	$-0.477231\pi$
$43$	0.937313	0.142939	0.0714694	+	0.997443i	$0.477231\pi$
$44$	0	0
$45$	28.7025	4.27871
$46$	0	0
$47$	−1.88404	−0.274815	−0.137407	−	0.990515i	$-0.543877\pi$
$47$	−1.88404	−0.274815	−0.137407	+	0.990515i	$0.543877\pi$
$48$	0	0
$49$	−5.28750	−0.755356
$50$	0	0
$51$	−5.36028	−0.750589
$52$	0	0
$53$	0.389398	0.0534879	0.0267440	−	0.999642i	$-0.491486\pi$
$53$	0.389398	0.0534879	0.0267440	+	0.999642i	$0.491486\pi$
$54$	0	0
$55$	−6.91617	−0.932576
$56$	0	0
$57$	13.9040	1.84163
$58$	0	0
$59$	7.04699	0.917441	0.458720	−	0.888581i	$-0.348308\pi$
$59$	7.04699	0.917441	0.458720	+	0.888581i	$0.348308\pi$
$60$	0	0
$61$	−6.07590	−0.777939	−0.388969	−	0.921251i	$-0.627169\pi$
$61$	−6.07590	−0.777939	−0.388969	+	0.921251i	$0.627169\pi$
$62$	0	0
$63$	8.84823	1.11477
$64$	0	0
$65$	13.9787	1.73384
$66$	0	0
$67$	−4.83541	−0.590740	−0.295370	−	0.955383i	$-0.595443\pi$
$67$	−4.83541	−0.590740	−0.295370	+	0.955383i	$0.595443\pi$
$68$	0	0
$69$	−21.1423	−2.54524
$70$	0	0
$71$	−9.65102	−1.14537	−0.572683	−	0.819777i	$-0.694097\pi$
$71$	−9.65102	−1.14537	−0.572683	+	0.819777i	$0.694097\pi$
$72$	0	0
$73$	8.76246	1.02557	0.512784	−	0.858517i	$-0.328614\pi$
$73$	8.76246	1.02557	0.512784	+	0.858517i	$0.328614\pi$
$74$	0	0
$75$	40.6791	4.69721
$76$	0	0
$77$	−2.13208	−0.242973
$78$	0	0
$79$	−14.3387	−1.61323	−0.806617	−	0.591075i	$-0.798704\pi$
$79$	−14.3387	−1.61323	−0.806617	+	0.591075i	$0.798704\pi$
$80$	0	0
$81$	16.4330	1.82589
$82$	0	0
$83$	13.9185	1.52775	0.763877	−	0.645362i	$-0.223293\pi$
$83$	13.9185	1.52775	0.763877	+	0.645362i	$0.223293\pi$
$84$	0	0
$85$	−7.28296	−0.789948
$86$	0	0
$87$	−19.1742	−2.05569
$88$	0	0
$89$	−2.53391	−0.268594	−0.134297	−	0.990941i	$-0.542878\pi$
$89$	−2.53391	−0.268594	−0.134297	+	0.990941i	$0.542878\pi$
$90$	0	0
$91$	4.30927	0.451735
$92$	0	0
$93$	−9.87356	−1.02384
$94$	0	0
$95$	18.8912	1.93820
$96$	0	0
$97$	−1.36564	−0.138660	−0.0693299	−	0.997594i	$-0.522086\pi$
$97$	−1.36564	−0.138660	−0.0693299	+	0.997594i	$0.522086\pi$
$98$	0	0
$99$	−11.0161	−1.10716
$100$	0	0
$101$	−8.41286	−0.837111	−0.418555	−	0.908191i	$-0.637463\pi$
$101$	−8.41286	−0.837111	−0.418555	+	0.908191i	$0.637463\pi$
$102$	0	0
$103$	2.98458	0.294079	0.147040	−	0.989131i	$-0.453026\pi$
$103$	2.98458	0.294079	0.147040	+	0.989131i	$0.453026\pi$
$104$	0	0
$105$	17.3561	1.69378
$106$	0	0
$107$	−9.98689	−0.965469	−0.482735	−	0.875767i	$-0.660356\pi$
$107$	−9.98689	−0.965469	−0.482735	+	0.875767i	$0.660356\pi$
$108$	0	0
$109$	−13.8644	−1.32797	−0.663983	−	0.747748i	$-0.731135\pi$
$109$	−13.8644	−1.32797	−0.663983	+	0.747748i	$0.731135\pi$
$110$	0	0
$111$	−8.84912	−0.839921
$112$	0	0
$113$	−3.61828	−0.340379	−0.170189	−	0.985411i	$-0.554438\pi$
$113$	−3.61828	−0.340379	−0.170189	+	0.985411i	$0.554438\pi$
$114$	0	0
$115$	−28.7259	−2.67870
$116$	0	0
$117$	22.2653	2.05843
$118$	0	0
$119$	−2.24515	−0.205813
$120$	0	0
$121$	−8.34555	−0.758686
$122$	0	0
$123$	0	0
$124$	0	0
$125$	34.0453	3.04510
$126$	0	0
$127$	8.22026	0.729430	0.364715	−	0.931119i	$-0.381166\pi$
$127$	8.22026	0.729430	0.364715	+	0.931119i	$0.381166\pi$
$128$	0	0
$129$	2.92848	0.257838
$130$	0	0
$131$	1.28689	0.112436	0.0562180	−	0.998419i	$-0.482096\pi$
$131$	1.28689	0.112436	0.0562180	+	0.998419i	$0.482096\pi$
$132$	0	0
$133$	5.82368	0.504977
$134$	0	0
$135$	49.8876	4.29364
$136$	0	0
$137$	15.1788	1.29682	0.648408	−	0.761293i	$-0.275435\pi$
$137$	15.1788	1.29682	0.648408	+	0.761293i	$0.275435\pi$
$138$	0	0
$139$	−16.4397	−1.39440	−0.697199	−	0.716877i	$-0.745571\pi$
$139$	−16.4397	−1.39440	−0.697199	+	0.716877i	$0.745571\pi$
$140$	0	0
$141$	−5.88636	−0.495721
$142$	0	0
$143$	−5.36507	−0.448650
$144$	0	0
$145$	−26.0518	−2.16348
$146$	0	0
$147$	−16.5199	−1.36254
$148$	0	0
$149$	21.7804	1.78432	0.892159	−	0.451721i	$-0.149190\pi$
$149$	21.7804	1.78432	0.892159	+	0.451721i	$0.149190\pi$
$150$	0	0
$151$	−2.86932	−0.233502	−0.116751	−	0.993161i	$-0.537248\pi$
$151$	−2.86932	−0.233502	−0.116751	+	0.993161i	$0.537248\pi$
$152$	0	0
$153$	−11.6003	−0.937831
$154$	0	0
$155$	−13.4151	−1.07753
$156$	0	0
$157$	6.50191	0.518909	0.259454	−	0.965755i	$-0.416457\pi$
$157$	6.50191	0.518909	0.259454	+	0.965755i	$0.416457\pi$
$158$	0	0
$159$	1.21661	0.0964834
$160$	0	0
$161$	−8.85546	−0.697908
$162$	0	0
$163$	1.99726	0.156437	0.0782186	−	0.996936i	$-0.475077\pi$
$163$	1.99726	0.156437	0.0782186	+	0.996936i	$0.475077\pi$
$164$	0	0
$165$	−21.6084	−1.68221
$166$	0	0
$167$	2.25420	0.174435	0.0872177	−	0.996189i	$-0.472202\pi$
$167$	2.25420	0.174435	0.0872177	+	0.996189i	$0.472202\pi$
$168$	0	0
$169$	−2.15632	−0.165871
$170$	0	0
$171$	30.0900	2.30104
$172$	0	0
$173$	22.1020	1.68038	0.840191	−	0.542290i	$-0.182443\pi$
$173$	22.1020	1.68038	0.840191	+	0.542290i	$0.182443\pi$
$174$	0	0
$175$	17.0384	1.28798
$176$	0	0
$177$	22.0172	1.65491
$178$	0	0
$179$	−6.40445	−0.478691	−0.239346	−	0.970934i	$-0.576933\pi$
$179$	−6.40445	−0.478691	−0.239346	+	0.970934i	$0.576933\pi$
$180$	0	0
$181$	−3.70348	−0.275278	−0.137639	−	0.990482i	$-0.543951\pi$
$181$	−3.70348	−0.275278	−0.137639	+	0.990482i	$0.543951\pi$
$182$	0	0
$183$	−18.9831	−1.40327
$184$	0	0
$185$	−12.0232	−0.883965
$186$	0	0
$187$	2.79523	0.204407
$188$	0	0
$189$	15.3791	1.11866
$190$	0	0
$191$	10.1282	0.732848	0.366424	−	0.930448i	$-0.380582\pi$
$191$	10.1282	0.732848	0.366424	+	0.930448i	$0.380582\pi$
$192$	0	0
$193$	3.07064	0.221029	0.110515	−	0.993874i	$-0.464750\pi$
$193$	3.07064	0.221029	0.110515	+	0.993874i	$0.464750\pi$
$194$	0	0
$195$	43.6741	3.12757
$196$	0	0
$197$	3.79845	0.270628	0.135314	−	0.990803i	$-0.456796\pi$
$197$	3.79845	0.270628	0.135314	+	0.990803i	$0.456796\pi$
$198$	0	0
$199$	−24.5111	−1.73754	−0.868772	−	0.495212i	$-0.835090\pi$
$199$	−24.5111	−1.73754	−0.868772	+	0.495212i	$0.835090\pi$
$200$	0	0
$201$	−15.1075	−1.06560
$202$	0	0
$203$	−8.03110	−0.563673
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−45.7547	−3.18017
$208$	0	0
$209$	−7.25051	−0.501528
$210$	0	0
$211$	−14.0403	−0.966575	−0.483287	−	0.875462i	$-0.660557\pi$
$211$	−14.0403	−0.966575	−0.483287	+	0.875462i	$0.660557\pi$
$212$	0	0
$213$	−30.1530	−2.06605
$214$	0	0
$215$	3.97890	0.271359
$216$	0	0
$217$	−4.13554	−0.280739
$218$	0	0
$219$	27.3769	1.84996
$220$	0	0
$221$	−5.64961	−0.380034
$222$	0	0
$223$	3.80771	0.254983	0.127492	−	0.991840i	$-0.459307\pi$
$223$	3.80771	0.254983	0.127492	+	0.991840i	$0.459307\pi$
$224$	0	0
$225$	88.0348	5.86898
$226$	0	0
$227$	−6.71185	−0.445481	−0.222741	−	0.974878i	$-0.571500\pi$
$227$	−6.71185	−0.445481	−0.222741	+	0.974878i	$0.571500\pi$
$228$	0	0
$229$	−4.00230	−0.264479	−0.132240	−	0.991218i	$-0.542217\pi$
$229$	−4.00230	−0.264479	−0.132240	+	0.991218i	$0.542217\pi$
$230$	0	0
$231$	−6.66133	−0.438283
$232$	0	0
$233$	13.3476	0.874427	0.437214	−	0.899358i	$-0.355965\pi$
$233$	13.3476	0.874427	0.437214	+	0.899358i	$0.355965\pi$
$234$	0	0
$235$	−7.99775	−0.521715
$236$	0	0
$237$	−44.7990	−2.91001
$238$	0	0
$239$	2.77816	0.179704	0.0898520	−	0.995955i	$-0.471361\pi$
$239$	2.77816	0.179704	0.0898520	+	0.995955i	$0.471361\pi$
$240$	0	0
$241$	22.0721	1.42179	0.710895	−	0.703298i	$-0.248290\pi$
$241$	22.0721	1.42179	0.710895	+	0.703298i	$0.248290\pi$
$242$	0	0
$243$	16.0860	1.03192
$244$	0	0
$245$	−22.4454	−1.43399
$246$	0	0
$247$	14.6545	0.932441
$248$	0	0
$249$	43.4860	2.75582
$250$	0	0
$251$	21.6252	1.36497	0.682484	−	0.730900i	$-0.260900\pi$
$251$	21.6252	1.36497	0.682484	+	0.730900i	$0.260900\pi$
$252$	0	0
$253$	11.0251	0.693142
$254$	0	0
$255$	−22.7544	−1.42494
$256$	0	0
$257$	8.55514	0.533655	0.266827	−	0.963744i	$-0.414025\pi$
$257$	8.55514	0.533655	0.266827	+	0.963744i	$0.414025\pi$
$258$	0	0
$259$	−3.70645	−0.230308
$260$	0	0
$261$	−41.4954	−2.56850
$262$	0	0
$263$	15.4713	0.954000	0.477000	−	0.878903i	$-0.341724\pi$
$263$	15.4713	0.954000	0.477000	+	0.878903i	$0.341724\pi$
$264$	0	0
$265$	1.65300	0.101543
$266$	0	0
$267$	−7.91677	−0.484499
$268$	0	0
$269$	13.3063	0.811302	0.405651	−	0.914028i	$-0.367045\pi$
$269$	13.3063	0.811302	0.405651	+	0.914028i	$0.367045\pi$
$270$	0	0
$271$	18.1167	1.10051	0.550255	−	0.834997i	$-0.314531\pi$
$271$	18.1167	1.10051	0.550255	+	0.834997i	$0.314531\pi$
$272$	0	0
$273$	13.4636	0.814855
$274$	0	0
$275$	−21.2129	−1.27919
$276$	0	0
$277$	−9.95584	−0.598188	−0.299094	−	0.954224i	$-0.596684\pi$
$277$	−9.95584	−0.598188	−0.299094	+	0.954224i	$0.596684\pi$
$278$	0	0
$279$	−21.3677	−1.27925
$280$	0	0
$281$	−3.16811	−0.188994	−0.0944968	−	0.995525i	$-0.530124\pi$
$281$	−3.16811	−0.188994	−0.0944968	+	0.995525i	$0.530124\pi$
$282$	0	0
$283$	−9.61475	−0.571538	−0.285769	−	0.958299i	$-0.592249\pi$
$283$	−9.61475	−0.571538	−0.285769	+	0.958299i	$0.592249\pi$
$284$	0	0
$285$	59.0224	3.49619
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−14.0565	−0.826855
$290$	0	0
$291$	−4.26672	−0.250120
$292$	0	0
$293$	28.0976	1.64148	0.820738	−	0.571304i	$-0.193562\pi$
$293$	28.0976	1.64148	0.820738	+	0.571304i	$0.193562\pi$
$294$	0	0
$295$	29.9145	1.74169
$296$	0	0
$297$	−19.1470	−1.11102
$298$	0	0
$299$	−22.2835	−1.28869
$300$	0	0
$301$	1.22659	0.0706996
$302$	0	0
$303$	−26.2846	−1.51001
$304$	0	0
$305$	−25.7922	−1.47686
$306$	0	0
$307$	−18.6327	−1.06343	−0.531713	−	0.846925i	$-0.678451\pi$
$307$	−18.6327	−1.06343	−0.531713	+	0.846925i	$0.678451\pi$
$308$	0	0
$309$	9.32482	0.530471
$310$	0	0
$311$	−17.3361	−0.983041	−0.491520	−	0.870866i	$-0.663559\pi$
$311$	−17.3361	−0.983041	−0.491520	+	0.870866i	$0.663559\pi$
$312$	0	0
$313$	−24.8463	−1.40440	−0.702199	−	0.711981i	$-0.747798\pi$
$313$	−24.8463	−1.40440	−0.702199	+	0.711981i	$0.747798\pi$
$314$	0	0
$315$	37.5608	2.11631
$316$	0	0
$317$	−4.75841	−0.267259	−0.133629	−	0.991031i	$-0.542663\pi$
$317$	−4.75841	−0.267259	−0.133629	+	0.991031i	$0.542663\pi$
$318$	0	0
$319$	9.99877	0.559823
$320$	0	0
$321$	−31.2024	−1.74155
$322$	0	0
$323$	−7.63504	−0.424825
$324$	0	0
$325$	42.8748	2.37826
$326$	0	0
$327$	−43.3169	−2.39543
$328$	0	0
$329$	−2.46550	−0.135927
$330$	0	0
$331$	−28.8815	−1.58747	−0.793737	−	0.608262i	$-0.791867\pi$
$331$	−28.8815	−1.58747	−0.793737	+	0.608262i	$0.791867\pi$
$332$	0	0
$333$	−19.1506	−1.04945
$334$	0	0
$335$	−20.5264	−1.12148
$336$	0	0
$337$	2.54693	0.138740	0.0693702	−	0.997591i	$-0.477901\pi$
$337$	2.54693	0.138740	0.0693702	+	0.997591i	$0.477901\pi$
$338$	0	0
$339$	−11.3047	−0.613987
$340$	0	0
$341$	5.14877	0.278822
$342$	0	0
$343$	−16.0797	−0.868225
$344$	0	0
$345$	−89.7493	−4.83194
$346$	0	0
$347$	−9.81004	−0.526630	−0.263315	−	0.964710i	$-0.584816\pi$
$347$	−9.81004	−0.526630	−0.263315	+	0.964710i	$0.584816\pi$
$348$	0	0
$349$	37.2561	1.99427	0.997137	−	0.0756154i	$-0.0240921\pi$
$349$	37.2561	1.99427	0.997137	+	0.0756154i	$0.0240921\pi$
$350$	0	0
$351$	38.6993	2.06561
$352$	0	0
$353$	26.8768	1.43051	0.715254	−	0.698864i	$-0.246311\pi$
$353$	26.8768	1.43051	0.715254	+	0.698864i	$0.246311\pi$
$354$	0	0
$355$	−40.9686	−2.17439
$356$	0	0
$357$	−7.01460	−0.371252
$358$	0	0
$359$	−4.38380	−0.231368	−0.115684	−	0.993286i	$-0.536906\pi$
$359$	−4.38380	−0.231368	−0.115684	+	0.993286i	$0.536906\pi$
$360$	0	0
$361$	0.804455	0.0423397
$362$	0	0
$363$	−26.0743	−1.36855
$364$	0	0
$365$	37.1967	1.94696
$366$	0	0
$367$	−27.3710	−1.42875	−0.714377	−	0.699761i	$-0.753290\pi$
$367$	−27.3710	−1.42875	−0.714377	+	0.699761i	$0.753290\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0.509577	0.0264559
$372$	0	0
$373$	−8.78477	−0.454858	−0.227429	−	0.973795i	$-0.573032\pi$
$373$	−8.78477	−0.454858	−0.227429	+	0.973795i	$0.573032\pi$
$374$	0	0
$375$	106.369	5.49286
$376$	0	0
$377$	−20.2091	−1.04082
$378$	0	0
$379$	24.4047	1.25358	0.626792	−	0.779187i	$-0.284368\pi$
$379$	24.4047	1.25358	0.626792	+	0.779187i	$0.284368\pi$
$380$	0	0
$381$	25.6828	1.31577
$382$	0	0
$383$	7.25535	0.370731	0.185365	−	0.982670i	$-0.440653\pi$
$383$	7.25535	0.370731	0.185365	+	0.982670i	$0.440653\pi$
$384$	0	0
$385$	−9.05069	−0.461266
$386$	0	0
$387$	6.33761	0.322159
$388$	0	0
$389$	−5.17764	−0.262517	−0.131258	−	0.991348i	$-0.541902\pi$
$389$	−5.17764	−0.262517	−0.131258	+	0.991348i	$0.541902\pi$
$390$	0	0
$391$	11.6098	0.587133
$392$	0	0
$393$	4.02067	0.202816
$394$	0	0
$395$	−60.8680	−3.06260
$396$	0	0
$397$	18.2317	0.915022	0.457511	−	0.889204i	$-0.348741\pi$
$397$	18.2317	0.915022	0.457511	+	0.889204i	$0.348741\pi$
$398$	0	0
$399$	18.1951	0.910895
$400$	0	0
$401$	12.9649	0.647436	0.323718	−	0.946154i	$-0.395067\pi$
$401$	12.9649	0.647436	0.323718	+	0.946154i	$0.395067\pi$
$402$	0	0
$403$	−10.4065	−0.518384
$404$	0	0
$405$	69.7582	3.46631
$406$	0	0
$407$	4.61456	0.228735
$408$	0	0
$409$	35.0515	1.73319	0.866594	−	0.499015i	$-0.166305\pi$
$409$	35.0515	1.73319	0.866594	+	0.499015i	$0.166305\pi$
$410$	0	0
$411$	47.4237	2.33924
$412$	0	0
$413$	9.22189	0.453779
$414$	0	0
$415$	59.0841	2.90033
$416$	0	0
$417$	−51.3632	−2.51527
$418$	0	0
$419$	−19.4258	−0.949010	−0.474505	−	0.880253i	$-0.657373\pi$
$419$	−19.4258	−0.949010	−0.474505	+	0.880253i	$0.657373\pi$
$420$	0	0
$421$	−31.5171	−1.53605	−0.768024	−	0.640421i	$-0.778760\pi$
$421$	−31.5171	−1.53605	−0.768024	+	0.640421i	$0.778760\pi$
$422$	0	0
$423$	−12.7388	−0.619384
$424$	0	0
$425$	−22.3379	−1.08355
$426$	0	0
$427$	−7.95108	−0.384780
$428$	0	0
$429$	−16.7623	−0.809290
$430$	0	0
$431$	−11.7872	−0.567771	−0.283886	−	0.958858i	$-0.591624\pi$
$431$	−11.7872	−0.567771	−0.283886	+	0.958858i	$0.591624\pi$
$432$	0	0
$433$	16.3127	0.783940	0.391970	−	0.919978i	$-0.371794\pi$
$433$	16.3127	0.783940	0.391970	+	0.919978i	$0.371794\pi$
$434$	0	0
$435$	−81.3945	−3.90257
$436$	0	0
$437$	−30.1146	−1.44058
$438$	0	0
$439$	−31.5933	−1.50786	−0.753932	−	0.656952i	$-0.771845\pi$
$439$	−31.5933	−1.50786	−0.753932	+	0.656952i	$0.771845\pi$
$440$	0	0
$441$	−35.7512	−1.70244
$442$	0	0
$443$	29.7833	1.41505	0.707525	−	0.706689i	$-0.249812\pi$
$443$	29.7833	1.41505	0.707525	+	0.706689i	$0.249812\pi$
$444$	0	0
$445$	−10.7565	−0.509905
$446$	0	0
$447$	68.0492	3.21862
$448$	0	0
$449$	13.0591	0.616298	0.308149	−	0.951338i	$-0.400290\pi$
$449$	13.0591	0.616298	0.308149	+	0.951338i	$0.400290\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−8.96470	−0.421199
$454$	0	0
$455$	18.2929	0.857584
$456$	0	0
$457$	−23.8824	−1.11717	−0.558587	−	0.829446i	$-0.688656\pi$
$457$	−23.8824	−1.11717	−0.558587	+	0.829446i	$0.688656\pi$
$458$	0	0
$459$	−20.1625	−0.941104
$460$	0	0
$461$	−14.8073	−0.689647	−0.344823	−	0.938668i	$-0.612061\pi$
$461$	−14.8073	−0.689647	−0.344823	+	0.938668i	$0.612061\pi$
$462$	0	0
$463$	22.9536	1.06675	0.533373	−	0.845880i	$-0.320924\pi$
$463$	22.9536	1.06675	0.533373	+	0.845880i	$0.320924\pi$
$464$	0	0
$465$	−41.9133	−1.94368
$466$	0	0
$467$	37.5761	1.73881	0.869407	−	0.494096i	$-0.164501\pi$
$467$	37.5761	1.73881	0.869407	+	0.494096i	$0.164501\pi$
$468$	0	0
$469$	−6.32775	−0.292189
$470$	0	0
$471$	20.3141	0.936026
$472$	0	0
$473$	−1.52712	−0.0702168
$474$	0	0
$475$	57.9422	2.65857
$476$	0	0
$477$	2.63290	0.120552
$478$	0	0
$479$	−28.3618	−1.29588	−0.647942	−	0.761689i	$-0.724370\pi$
$479$	−28.3618	−1.29588	−0.647942	+	0.761689i	$0.724370\pi$
$480$	0	0
$481$	−9.32676	−0.425264
$482$	0	0
$483$	−27.6674	−1.25891
$484$	0	0
$485$	−5.79715	−0.263235
$486$	0	0
$487$	5.72348	0.259356	0.129678	−	0.991556i	$-0.458606\pi$
$487$	5.72348	0.259356	0.129678	+	0.991556i	$0.458606\pi$
$488$	0	0
$489$	6.24010	0.282187
$490$	0	0
$491$	−22.0836	−0.996621	−0.498310	−	0.866999i	$-0.666046\pi$
$491$	−22.0836	−0.996621	−0.498310	+	0.866999i	$0.666046\pi$
$492$	0	0
$493$	10.5290	0.474204
$494$	0	0
$495$	−46.7634	−2.10186
$496$	0	0
$497$	−12.6296	−0.566515
$498$	0	0
$499$	3.89011	0.174145	0.0870727	−	0.996202i	$-0.472249\pi$
$499$	3.89011	0.174145	0.0870727	+	0.996202i	$0.472249\pi$
$500$	0	0
$501$	7.04288	0.314653
$502$	0	0
$503$	18.0406	0.804392	0.402196	−	0.915554i	$-0.368247\pi$
$503$	18.0406	0.804392	0.402196	+	0.915554i	$0.368247\pi$
$504$	0	0
$505$	−35.7126	−1.58919
$506$	0	0
$507$	−6.73708	−0.299204
$508$	0	0
$509$	24.8748	1.10255	0.551277	−	0.834322i	$-0.314141\pi$
$509$	24.8748	1.10255	0.551277	+	0.834322i	$0.314141\pi$
$510$	0	0
$511$	11.4668	0.507261
$512$	0	0
$513$	52.2993	2.30907
$514$	0	0
$515$	12.6696	0.558287
$516$	0	0
$517$	3.06956	0.134999
$518$	0	0
$519$	69.0540	3.03113
$520$	0	0
$521$	−9.36774	−0.410408	−0.205204	−	0.978719i	$-0.565786\pi$
$521$	−9.36774	−0.410408	−0.205204	+	0.978719i	$0.565786\pi$
$522$	0	0
$523$	39.7438	1.73787	0.868937	−	0.494923i	$-0.164804\pi$
$523$	39.7438	1.73787	0.868937	+	0.494923i	$0.164804\pi$
$524$	0	0
$525$	53.2337	2.32331
$526$	0	0
$527$	5.42183	0.236179
$528$	0	0
$529$	22.7921	0.990962
$530$	0	0
$531$	47.6480	2.06775
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−42.3944	−1.83287
$536$	0	0
$537$	−20.0096	−0.863480
$538$	0	0
$539$	8.61464	0.371059
$540$	0	0
$541$	21.3819	0.919278	0.459639	−	0.888106i	$-0.347979\pi$
$541$	21.3819	0.919278	0.459639	+	0.888106i	$0.347979\pi$
$542$	0	0
$543$	−11.5709	−0.496555
$544$	0	0
$545$	−58.8543	−2.52104
$546$	0	0
$547$	22.8905	0.978725	0.489363	−	0.872080i	$-0.337229\pi$
$547$	22.8905	0.978725	0.489363	+	0.872080i	$0.337229\pi$
$548$	0	0
$549$	−41.0819	−1.75333
$550$	0	0
$551$	−27.3112	−1.16350
$552$	0	0
$553$	−18.7641	−0.797929
$554$	0	0
$555$	−37.5646	−1.59453
$556$	0	0
$557$	−25.8288	−1.09440	−0.547200	−	0.837002i	$-0.684306\pi$
$557$	−25.8288	−1.09440	−0.547200	+	0.837002i	$0.684306\pi$
$558$	0	0
$559$	3.08655	0.130547
$560$	0	0
$561$	8.73322	0.368717
$562$	0	0
$563$	1.28723	0.0542505	0.0271252	−	0.999632i	$-0.491365\pi$
$563$	1.28723	0.0542505	0.0271252	+	0.999632i	$0.491365\pi$
$564$	0	0
$565$	−15.3596	−0.646183
$566$	0	0
$567$	21.5047	0.903111
$568$	0	0
$569$	−15.9180	−0.667319	−0.333659	−	0.942694i	$-0.608284\pi$
$569$	−15.9180	−0.667319	−0.333659	+	0.942694i	$0.608284\pi$
$570$	0	0
$571$	37.0527	1.55061	0.775303	−	0.631589i	$-0.217597\pi$
$571$	37.0527	1.55061	0.775303	+	0.631589i	$0.217597\pi$
$572$	0	0
$573$	31.6437	1.32194
$574$	0	0
$575$	−88.1067	−3.67430
$576$	0	0
$577$	−39.1427	−1.62953	−0.814766	−	0.579790i	$-0.803135\pi$
$577$	−39.1427	−1.62953	−0.814766	+	0.579790i	$0.803135\pi$
$578$	0	0
$579$	9.59370	0.398701
$580$	0	0
$581$	18.2141	0.755649
$582$	0	0
$583$	−0.634426	−0.0262752
$584$	0	0
$585$	94.5164	3.90777
$586$	0	0
$587$	32.0841	1.32425	0.662127	−	0.749392i	$-0.269654\pi$
$587$	32.0841	1.32425	0.662127	+	0.749392i	$0.269654\pi$
$588$	0	0
$589$	−14.0636	−0.579482
$590$	0	0
$591$	11.8676	0.488168
$592$	0	0
$593$	−11.7512	−0.482565	−0.241283	−	0.970455i	$-0.577568\pi$
$593$	−11.7512	−0.482565	−0.241283	+	0.970455i	$0.577568\pi$
$594$	0	0
$595$	−9.53068	−0.390720
$596$	0	0
$597$	−76.5808	−3.13424
$598$	0	0
$599$	−7.68620	−0.314050	−0.157025	−	0.987595i	$-0.550190\pi$
$599$	−7.68620	−0.314050	−0.157025	+	0.987595i	$0.550190\pi$
$600$	0	0
$601$	32.3357	1.31900	0.659501	−	0.751704i	$-0.270768\pi$
$601$	32.3357	1.31900	0.659501	+	0.751704i	$0.270768\pi$
$602$	0	0
$603$	−32.6945	−1.33142
$604$	0	0
$605$	−35.4269	−1.44031
$606$	0	0
$607$	−24.9324	−1.01197	−0.505987	−	0.862541i	$-0.668872\pi$
$607$	−24.9324	−1.01197	−0.505987	+	0.862541i	$0.668872\pi$
$608$	0	0
$609$	−25.0918	−1.01677
$610$	0	0
$611$	−6.20408	−0.250990
$612$	0	0
$613$	16.9106	0.683012	0.341506	−	0.939880i	$-0.389063\pi$
$613$	16.9106	0.683012	0.341506	+	0.939880i	$0.389063\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−21.1687	−0.852220	−0.426110	−	0.904671i	$-0.640116\pi$
$617$	−21.1687	−0.852220	−0.426110	+	0.904671i	$0.640116\pi$
$618$	0	0
$619$	8.47067	0.340465	0.170233	−	0.985404i	$-0.445548\pi$
$619$	8.47067	0.340465	0.170233	+	0.985404i	$0.445548\pi$
$620$	0	0
$621$	−79.5261	−3.19127
$622$	0	0
$623$	−3.31594	−0.132850
$624$	0	0
$625$	79.4220	3.17688
$626$	0	0
$627$	−22.6530	−0.904675
$628$	0	0
$629$	4.85929	0.193752
$630$	0	0
$631$	8.62095	0.343194	0.171597	−	0.985167i	$-0.445107\pi$
$631$	8.62095	0.343194	0.171597	+	0.985167i	$0.445107\pi$
$632$	0	0
$633$	−43.8666	−1.74354
$634$	0	0
$635$	34.8951	1.38477
$636$	0	0
$637$	−17.4116	−0.689872
$638$	0	0
$639$	−65.2550	−2.58145
$640$	0	0
$641$	35.2251	1.39131	0.695654	−	0.718377i	$-0.255114\pi$
$641$	35.2251	1.39131	0.695654	+	0.718377i	$0.255114\pi$
$642$	0	0
$643$	−37.7016	−1.48681	−0.743403	−	0.668843i	$-0.766790\pi$
$643$	−37.7016	−1.48681	−0.743403	+	0.668843i	$0.766790\pi$
$644$	0	0
$645$	12.4314	0.489486
$646$	0	0
$647$	−21.1744	−0.832453	−0.416226	−	0.909261i	$-0.636648\pi$
$647$	−21.1744	−0.832453	−0.416226	+	0.909261i	$0.636648\pi$
$648$	0	0
$649$	−11.4813	−0.450681
$650$	0	0
$651$	−12.9208	−0.506406
$652$	0	0
$653$	−3.47444	−0.135965	−0.0679827	−	0.997686i	$-0.521656\pi$
$653$	−3.47444	−0.135965	−0.0679827	+	0.997686i	$0.521656\pi$
$654$	0	0
$655$	5.46285	0.213451
$656$	0	0
$657$	59.2471	2.31145
$658$	0	0
$659$	16.9041	0.658491	0.329245	−	0.944244i	$-0.393206\pi$
$659$	16.9041	0.658491	0.329245	+	0.944244i	$0.393206\pi$
$660$	0	0
$661$	26.6890	1.03808	0.519040	−	0.854750i	$-0.326289\pi$
$661$	26.6890	1.03808	0.519040	+	0.854750i	$0.326289\pi$
$662$	0	0
$663$	−17.6513	−0.685518
$664$	0	0
$665$	24.7215	0.958660
$666$	0	0
$667$	41.5293	1.60802
$668$	0	0
$669$	11.8966	0.459947
$670$	0	0
$671$	9.89914	0.382152
$672$	0	0
$673$	35.4618	1.36695	0.683476	−	0.729973i	$-0.260467\pi$
$673$	35.4618	1.36695	0.683476	+	0.729973i	$0.260467\pi$
$674$	0	0
$675$	153.013	5.88947
$676$	0	0
$677$	−4.26217	−0.163809	−0.0819043	−	0.996640i	$-0.526100\pi$
$677$	−4.26217	−0.163809	−0.0819043	+	0.996640i	$0.526100\pi$
$678$	0	0
$679$	−1.78712	−0.0685832
$680$	0	0
$681$	−20.9701	−0.803575
$682$	0	0
$683$	32.1424	1.22989	0.614947	−	0.788568i	$-0.289177\pi$
$683$	32.1424	1.22989	0.614947	+	0.788568i	$0.289177\pi$
$684$	0	0
$685$	64.4342	2.46191
$686$	0	0
$687$	−12.5045	−0.477077
$688$	0	0
$689$	1.28228	0.0488509
$690$	0	0
$691$	−28.0798	−1.06820	−0.534102	−	0.845420i	$-0.679350\pi$
$691$	−28.0798	−1.06820	−0.534102	+	0.845420i	$0.679350\pi$
$692$	0	0
$693$	−14.4160	−0.547617
$694$	0	0
$695$	−69.7867	−2.64716
$696$	0	0
$697$	0	0
$698$	0	0
$699$	41.7022	1.57732
$700$	0	0
$701$	13.6240	0.514573	0.257286	−	0.966335i	$-0.417172\pi$
$701$	13.6240	0.514573	0.257286	+	0.966335i	$0.417172\pi$
$702$	0	0
$703$	−12.6045	−0.475386
$704$	0	0
$705$	−24.9876	−0.941089
$706$	0	0
$707$	−11.0093	−0.414047
$708$	0	0
$709$	17.0211	0.639239	0.319620	−	0.947546i	$-0.396445\pi$
$709$	17.0211	0.639239	0.319620	+	0.947546i	$0.396445\pi$
$710$	0	0
$711$	−96.9508	−3.63594
$712$	0	0
$713$	21.3851	0.800879
$714$	0	0
$715$	−22.7748	−0.851728
$716$	0	0
$717$	8.67989	0.324156
$718$	0	0
$719$	−14.7030	−0.548329	−0.274165	−	0.961683i	$-0.588401\pi$
$719$	−14.7030	−0.548329	−0.274165	+	0.961683i	$0.588401\pi$
$720$	0	0
$721$	3.90570	0.145456
$722$	0	0
$723$	68.9607	2.56468
$724$	0	0
$725$	−79.9047	−2.96759
$726$	0	0
$727$	52.3149	1.94025	0.970126	−	0.242601i	$-0.0780006\pi$
$727$	52.3149	1.94025	0.970126	+	0.242601i	$0.0780006\pi$
$728$	0	0
$729$	0.958942	0.0355164
$730$	0	0
$731$	−1.60810	−0.0594779
$732$	0	0
$733$	−18.1753	−0.671321	−0.335660	−	0.941983i	$-0.608959\pi$
$733$	−18.1753	−0.671321	−0.335660	+	0.941983i	$0.608959\pi$
$734$	0	0
$735$	−70.1271	−2.58668
$736$	0	0
$737$	7.87809	0.290193
$738$	0	0
$739$	−3.34138	−0.122915	−0.0614573	−	0.998110i	$-0.519575\pi$
$739$	−3.34138	−0.122915	−0.0614573	+	0.998110i	$0.519575\pi$
$740$	0	0
$741$	45.7854	1.68197
$742$	0	0
$743$	20.2109	0.741467	0.370733	−	0.928739i	$-0.379106\pi$
$743$	20.2109	0.741467	0.370733	+	0.928739i	$0.379106\pi$
$744$	0	0
$745$	92.4579	3.38739
$746$	0	0
$747$	94.1094	3.44328
$748$	0	0
$749$	−13.0691	−0.477535
$750$	0	0
$751$	−8.16093	−0.297797	−0.148898	−	0.988853i	$-0.547573\pi$
$751$	−8.16093	−0.297797	−0.148898	+	0.988853i	$0.547573\pi$
$752$	0	0
$753$	67.5642	2.46218
$754$	0	0
$755$	−12.1803	−0.443285
$756$	0	0
$757$	−49.4463	−1.79716	−0.898578	−	0.438814i	$-0.855399\pi$
$757$	−49.4463	−1.79716	−0.898578	+	0.438814i	$0.855399\pi$
$758$	0	0
$759$	34.4461	1.25031
$760$	0	0
$761$	38.7161	1.40346	0.701728	−	0.712445i	$-0.252412\pi$
$761$	38.7161	1.40346	0.701728	+	0.712445i	$0.252412\pi$
$762$	0	0
$763$	−18.1433	−0.656831
$764$	0	0
$765$	−49.2435	−1.78040
$766$	0	0
$767$	23.2056	0.837904
$768$	0	0
$769$	30.7992	1.11065	0.555324	−	0.831634i	$-0.312594\pi$
$769$	30.7992	1.11065	0.555324	+	0.831634i	$0.312594\pi$
$770$	0	0
$771$	26.7291	0.962625
$772$	0	0
$773$	−16.2483	−0.584411	−0.292206	−	0.956355i	$-0.594389\pi$
$773$	−16.2483	−0.584411	−0.292206	+	0.956355i	$0.594389\pi$
$774$	0	0
$775$	−41.1462	−1.47801
$776$	0	0
$777$	−11.5802	−0.415437
$778$	0	0
$779$	0	0
$780$	0	0
$781$	15.7239	0.562646
$782$	0	0
$783$	−72.1229	−2.57746
$784$	0	0
$785$	27.6006	0.985109
$786$	0	0
$787$	32.3492	1.15312	0.576562	−	0.817053i	$-0.304394\pi$
$787$	32.3492	1.15312	0.576562	+	0.817053i	$0.304394\pi$
$788$	0	0
$789$	48.3375	1.72086
$790$	0	0
$791$	−4.73497	−0.168356
$792$	0	0
$793$	−20.0078	−0.710497
$794$	0	0
$795$	5.16451	0.183166
$796$	0	0
$797$	−37.5685	−1.33074	−0.665372	−	0.746512i	$-0.731727\pi$
$797$	−37.5685	−1.33074	−0.665372	+	0.746512i	$0.731727\pi$
$798$	0	0
$799$	3.23236	0.114353
$800$	0	0
$801$	−17.1329	−0.605362
$802$	0	0
$803$	−14.2762	−0.503797
$804$	0	0
$805$	−37.5915	−1.32493
$806$	0	0
$807$	41.5734	1.46345
$808$	0	0
$809$	−8.51107	−0.299233	−0.149617	−	0.988744i	$-0.547804\pi$
$809$	−8.51107	−0.299233	−0.149617	+	0.988744i	$0.547804\pi$
$810$	0	0
$811$	31.7221	1.11391	0.556957	−	0.830541i	$-0.311969\pi$
$811$	31.7221	1.11391	0.556957	+	0.830541i	$0.311969\pi$
$812$	0	0
$813$	56.6026	1.98514
$814$	0	0
$815$	8.47837	0.296984
$816$	0	0
$817$	4.17125	0.145933
$818$	0	0
$819$	29.1370	1.01813
$820$	0	0
$821$	−32.6476	−1.13941	−0.569705	−	0.821849i	$-0.692943\pi$
$821$	−32.6476	−1.13941	−0.569705	+	0.821849i	$0.692943\pi$
$822$	0	0
$823$	−24.4517	−0.852334	−0.426167	−	0.904644i	$-0.640136\pi$
$823$	−24.4517	−0.852334	−0.426167	+	0.904644i	$0.640136\pi$
$824$	0	0
$825$	−66.2763	−2.30744
$826$	0	0
$827$	−25.8431	−0.898652	−0.449326	−	0.893368i	$-0.648336\pi$
$827$	−25.8431	−0.898652	−0.449326	+	0.893368i	$0.648336\pi$
$828$	0	0
$829$	−11.8261	−0.410737	−0.205368	−	0.978685i	$-0.565839\pi$
$829$	−11.8261	−0.410737	−0.205368	+	0.978685i	$0.565839\pi$
$830$	0	0
$831$	−31.1054	−1.07903
$832$	0	0
$833$	9.07151	0.314309
$834$	0	0
$835$	9.56910	0.331153
$836$	0	0
$837$	−37.1390	−1.28371
$838$	0	0
$839$	−9.45468	−0.326412	−0.163206	−	0.986592i	$-0.552183\pi$
$839$	−9.45468	−0.326412	−0.163206	+	0.986592i	$0.552183\pi$
$840$	0	0
$841$	8.66329	0.298734
$842$	0	0
$843$	−9.89824	−0.340913
$844$	0	0
$845$	−9.15361	−0.314894
$846$	0	0
$847$	−10.9212	−0.375257
$848$	0	0
$849$	−30.0397	−1.03096
$850$	0	0
$851$	19.1663	0.657012
$852$	0	0
$853$	−37.8113	−1.29463	−0.647316	−	0.762221i	$-0.724109\pi$
$853$	−37.8113	−1.29463	−0.647316	+	0.762221i	$0.724109\pi$
$854$	0	0
$855$	127.732	4.36835
$856$	0	0
$857$	51.4857	1.75872	0.879359	−	0.476160i	$-0.157972\pi$
$857$	51.4857	1.75872	0.879359	+	0.476160i	$0.157972\pi$
$858$	0	0
$859$	−35.8804	−1.22422	−0.612112	−	0.790771i	$-0.709680\pi$
$859$	−35.8804	−1.22422	−0.612112	+	0.790771i	$0.709680\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	10.9319	0.372125	0.186062	−	0.982538i	$-0.440427\pi$
$863$	10.9319	0.372125	0.186062	+	0.982538i	$0.440427\pi$
$864$	0	0
$865$	93.8231	3.19008
$866$	0	0
$867$	−43.9173	−1.49151
$868$	0	0
$869$	23.3614	0.792480
$870$	0	0
$871$	−15.9229	−0.539527
$872$	0	0
$873$	−9.23373	−0.312514
$874$	0	0
$875$	44.5526	1.50615
$876$	0	0
$877$	32.8803	1.11029	0.555144	−	0.831754i	$-0.312663\pi$
$877$	32.8803	1.11029	0.555144	+	0.831754i	$0.312663\pi$
$878$	0	0
$879$	87.7861	2.96095
$880$	0	0
$881$	−53.7395	−1.81053	−0.905266	−	0.424846i	$-0.860328\pi$
$881$	−53.7395	−1.81053	−0.905266	+	0.424846i	$0.860328\pi$
$882$	0	0
$883$	13.6745	0.460183	0.230092	−	0.973169i	$-0.426097\pi$
$883$	13.6745	0.460183	0.230092	+	0.973169i	$0.426097\pi$
$884$	0	0
$885$	93.4630	3.14172
$886$	0	0
$887$	40.4909	1.35955	0.679776	−	0.733419i	$-0.262077\pi$
$887$	40.4909	1.35955	0.679776	+	0.733419i	$0.262077\pi$
$888$	0	0
$889$	10.7573	0.360787
$890$	0	0
$891$	−26.7734	−0.896944
$892$	0	0
$893$	−8.38438	−0.280572
$894$	0	0
$895$	−27.1869	−0.908759
$896$	0	0
$897$	−69.6211	−2.32458
$898$	0	0
$899$	19.3944	0.646838
$900$	0	0
$901$	−0.668072	−0.0222567
$902$	0	0
$903$	3.83229	0.127530
$904$	0	0
$905$	−15.7213	−0.522594
$906$	0	0
$907$	−45.2952	−1.50400	−0.752001	−	0.659162i	$-0.770911\pi$
$907$	−45.2952	−1.50400	−0.752001	+	0.659162i	$0.770911\pi$
$908$	0	0
$909$	−56.8832	−1.88670
$910$	0	0
$911$	−45.7103	−1.51445	−0.757225	−	0.653155i	$-0.773445\pi$
$911$	−45.7103	−1.51445	−0.757225	+	0.653155i	$0.773445\pi$
$912$	0	0
$913$	−22.6767	−0.750489
$914$	0	0
$915$	−80.5835	−2.66401
$916$	0	0
$917$	1.68406	0.0556125
$918$	0	0
$919$	44.2320	1.45908	0.729539	−	0.683939i	$-0.239734\pi$
$919$	44.2320	1.45908	0.729539	+	0.683939i	$0.239734\pi$
$920$	0	0
$921$	−58.2148	−1.91824
$922$	0	0
$923$	−31.7806	−1.04607
$924$	0	0
$925$	−36.8770	−1.21251
$926$	0	0
$927$	20.1801	0.662802
$928$	0	0
$929$	11.2971	0.370645	0.185322	−	0.982678i	$-0.440667\pi$
$929$	11.2971	0.370645	0.185322	+	0.982678i	$0.440667\pi$
$930$	0	0
$931$	−23.5305	−0.771181
$932$	0	0
$933$	−54.1638	−1.77324
$934$	0	0
$935$	11.8658	0.388052
$936$	0	0
$937$	41.7645	1.36439	0.682194	−	0.731171i	$-0.261026\pi$
$937$	41.7645	1.36439	0.682194	+	0.731171i	$0.261026\pi$
$938$	0	0
$939$	−77.6282	−2.53330
$940$	0	0
$941$	36.4765	1.18910	0.594550	−	0.804058i	$-0.297330\pi$
$941$	36.4765	1.18910	0.594550	+	0.804058i	$0.297330\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	65.2843	2.12370
$946$	0	0
$947$	46.9192	1.52467	0.762334	−	0.647183i	$-0.224053\pi$
$947$	46.9192	1.52467	0.762334	+	0.647183i	$0.224053\pi$
$948$	0	0
$949$	28.8546	0.936659
$950$	0	0
$951$	−14.8669	−0.482091
$952$	0	0
$953$	28.6742	0.928849	0.464425	−	0.885613i	$-0.346261\pi$
$953$	28.6742	0.928849	0.464425	+	0.885613i	$0.346261\pi$
$954$	0	0
$955$	42.9941	1.39126
$956$	0	0
$957$	31.2395	1.00983
$958$	0	0
$959$	19.8634	0.641424
$960$	0	0
$961$	−21.0131	−0.677841
$962$	0	0
$963$	−67.5260	−2.17599
$964$	0	0
$965$	13.0349	0.419608
$966$	0	0
$967$	0.658156	0.0211649	0.0105824	−	0.999944i	$-0.496631\pi$
$967$	0.658156	0.0211649	0.0105824	+	0.999944i	$0.496631\pi$
$968$	0	0
$969$	−23.8544	−0.766314
$970$	0	0
$971$	0.114974	0.00368971	0.00184485	−	0.999998i	$-0.499413\pi$
$971$	0.114974	0.00368971	0.00184485	+	0.999998i	$0.499413\pi$
$972$	0	0
$973$	−21.5135	−0.689690
$974$	0	0
$975$	133.955	4.29000
$976$	0	0
$977$	−17.2209	−0.550944	−0.275472	−	0.961309i	$-0.588834\pi$
$977$	−17.2209	−0.550944	−0.275472	+	0.961309i	$0.588834\pi$
$978$	0	0
$979$	4.12836	0.131943
$980$	0	0
$981$	−93.7434	−2.99300
$982$	0	0
$983$	58.6708	1.87131	0.935654	−	0.352920i	$-0.114811\pi$
$983$	58.6708	1.87131	0.935654	+	0.352920i	$0.114811\pi$
$984$	0	0
$985$	16.1244	0.513767
$986$	0	0
$987$	−7.70305	−0.245191
$988$	0	0
$989$	−6.34278	−0.201689
$990$	0	0
$991$	−16.2164	−0.515131	−0.257566	−	0.966261i	$-0.582920\pi$
$991$	−16.2164	−0.515131	−0.257566	+	0.966261i	$0.582920\pi$
$992$	0	0
$993$	−90.2356	−2.86354
$994$	0	0
$995$	−104.050	−3.29860
$996$	0	0
$997$	10.6265	0.336544	0.168272	−	0.985741i	$-0.446181\pi$
$997$	10.6265	0.336544	0.168272	+	0.985741i	$0.446181\pi$
$998$	0	0
$999$	−33.2856	−1.05311

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6724.2.a.j.1.15		16
41.8	even	20		164.2.k.a.105.4	yes	16
41.36	even	20		164.2.k.a.25.1	✓	16
41.40	even	2	inner	6724.2.a.j.1.2		16
123.8	odd	20		1476.2.bb.b.433.1		16
123.77	odd	20		1476.2.bb.b.1009.1		16
164.131	odd	20		656.2.be.e.433.1		16
164.159	odd	20		656.2.be.e.353.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
164.2.k.a.25.1	✓	16	41.36	even	20
164.2.k.a.105.4	yes	16	41.8	even	20
656.2.be.e.353.4		16	164.159	odd	20
656.2.be.e.433.1		16	164.131	odd	20
1476.2.bb.b.433.1		16	123.8	odd	20
1476.2.bb.b.1009.1		16	123.77	odd	20
6724.2.a.j.1.2		16	41.40	even	2	inner
6724.2.a.j.1.15		16	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:	\( N \)	\(=\)	\( 6724 = 2^{2} \cdot 41^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6724.a (trivial)

\(f(q)\)	\(=\)	\(q+3.12433 q^{3} +4.24501 q^{5} +1.30863 q^{7} +6.76146 q^{9} +O(q^{10})\)	\(q+3.12433 q^{3} +4.24501 q^{5} +1.30863 q^{7} +6.76146 q^{9} -1.62925 q^{11} +3.29297 q^{13} +13.2628 q^{15} -1.71565 q^{17} +4.45022 q^{19} +4.08859 q^{21} -6.76699 q^{23} +13.0201 q^{25} +11.7521 q^{27} -6.13704 q^{29} -3.16021 q^{31} -5.09032 q^{33} +5.55513 q^{35} -2.83232 q^{37} +10.2883 q^{39} +0.937313 q^{43} +28.7025 q^{45} -1.88404 q^{47} -5.28750 q^{49} -5.36028 q^{51} +0.389398 q^{53} -6.91617 q^{55} +13.9040 q^{57} +7.04699 q^{59} -6.07590 q^{61} +8.84823 q^{63} +13.9787 q^{65} -4.83541 q^{67} -21.1423 q^{69} -9.65102 q^{71} +8.76246 q^{73} +40.6791 q^{75} -2.13208 q^{77} -14.3387 q^{79} +16.4330 q^{81} +13.9185 q^{83} -7.28296 q^{85} -19.1742 q^{87} -2.53391 q^{89} +4.30927 q^{91} -9.87356 q^{93} +18.8912 q^{95} -1.36564 q^{97} -11.0161 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 6 q^{5} + 30 q^{9}+O(q^{10}) \) 16 * q + 6 * q^5 + 30 * q^9	\( 16 q + 6 q^{5} + 30 q^{9} + 46 q^{21} - 8 q^{23} + 38 q^{25} - 2 q^{31} + 20 q^{33} + 12 q^{37} + 8 q^{39} - 48 q^{43} + 92 q^{45} - 12 q^{49} + 42 q^{51} + 22 q^{57} - 32 q^{59} + 6 q^{61} - 34 q^{73} + 92 q^{77} + 108 q^{81} + 12 q^{83} - 34 q^{87}+O(q^{100}) \) 16 * q + 6 * q^5 + 30 * q^9 + 46 * q^21 - 8 * q^23 + 38 * q^25 - 2 * q^31 + 20 * q^33 + 12 * q^37 + 8 * q^39 - 48 * q^43 + 92 * q^45 - 12 * q^49 + 42 * q^51 + 22 * q^57 - 32 * q^59 + 6 * q^61 - 34 * q^73 + 92 * q^77 + 108 * q^81 + 12 * q^83 - 34 * q^87

Introduction

L-functions

Modular forms

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