LMFDB - Embedded newform 6292.2.a.z.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6292,2,Mod(1,6292)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6292, 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("6292.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6292 = 2^{2} \cdot 11^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6292.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:	$50.2418729518$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 3 x^{13} - 31 x^{12} + 97 x^{11} + 339 x^{10} - 1140 x^{9} - 1495 x^{8} + 5888 x^{7} + \cdots + 80 $ x^14 - 3x^13 - 31x^12 + 97x^11 + 339x^10 - 1140x^9 - 1495x^8 + 5888x^7 + 1859x^6 - 12750x^5 + 2100x^4 + 9280x^3 - 3480x^2 - 800*x + 80
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 572)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.40915$ of defining polynomial
Character	$\chi$	$=$	6292.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.40915 q^{3} +2.99160 q^{5} +3.46369 q^{7} +8.62230 q^{9} +O(q^{10})$	$q-3.40915 q^{3} +2.99160 q^{5} +3.46369 q^{7} +8.62230 q^{9} -1.00000 q^{13} -10.1988 q^{15} -6.00545 q^{17} -4.10518 q^{19} -11.8082 q^{21} +5.25830 q^{23} +3.94968 q^{25} -19.1673 q^{27} -1.94363 q^{29} -2.23592 q^{31} +10.3620 q^{35} +7.55090 q^{37} +3.40915 q^{39} +3.29848 q^{41} -8.24966 q^{43} +25.7945 q^{45} +6.40804 q^{47} +4.99714 q^{49} +20.4735 q^{51} +6.62939 q^{53} +13.9952 q^{57} -4.62867 q^{59} +4.97344 q^{61} +29.8650 q^{63} -2.99160 q^{65} +1.08509 q^{67} -17.9263 q^{69} +2.79559 q^{71} +14.6849 q^{73} -13.4651 q^{75} -0.381142 q^{79} +39.4772 q^{81} -3.52753 q^{83} -17.9659 q^{85} +6.62611 q^{87} +4.38192 q^{89} -3.46369 q^{91} +7.62260 q^{93} -12.2810 q^{95} +5.92091 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 3 q^{3} + 9 q^{5} + 3 q^{7} + 29 q^{9}+O(q^{10}) $ 14 * q + 3 * q^3 + 9 * q^5 + 3 * q^7 + 29 * q^9	$ 14 q + 3 q^{3} + 9 q^{5} + 3 q^{7} + 29 q^{9} - 14 q^{13} + 13 q^{15} - 4 q^{17} - q^{19} - 6 q^{21} + 17 q^{23} + 37 q^{25} - 3 q^{27} + 4 q^{29} + 14 q^{31} + 22 q^{35} + 25 q^{37} - 3 q^{39} - 13 q^{41} - 4 q^{43} + 31 q^{45} + 24 q^{47} + 13 q^{49} + 12 q^{51} + 44 q^{53} - 3 q^{57} + 19 q^{59} - 7 q^{61} + 3 q^{63} - 9 q^{65} + 5 q^{67} + 9 q^{69} + 24 q^{71} + 20 q^{73} + 13 q^{75} + 33 q^{79} + 86 q^{81} - 34 q^{83} + 36 q^{85} + 28 q^{87} + 41 q^{89} - 3 q^{91} + 36 q^{93} - 17 q^{95} + 16 q^{97}+O(q^{100}) $ 14 * q + 3 * q^3 + 9 * q^5 + 3 * q^7 + 29 * q^9 - 14 * q^13 + 13 * q^15 - 4 * q^17 - q^19 - 6 * q^21 + 17 * q^23 + 37 * q^25 - 3 * q^27 + 4 * q^29 + 14 * q^31 + 22 * q^35 + 25 * q^37 - 3 * q^39 - 13 * q^41 - 4 * q^43 + 31 * q^45 + 24 * q^47 + 13 * q^49 + 12 * q^51 + 44 * q^53 - 3 * q^57 + 19 * q^59 - 7 * q^61 + 3 * q^63 - 9 * q^65 + 5 * q^67 + 9 * q^69 + 24 * q^71 + 20 * q^73 + 13 * q^75 + 33 * q^79 + 86 * q^81 - 34 * q^83 + 36 * q^85 + 28 * q^87 + 41 * q^89 - 3 * q^91 + 36 * q^93 - 17 * q^95 + 16 * q^97

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
$3$	−3.40915	−1.96827	−0.984137	−	0.177411i	$-0.943228\pi$
$3$	−3.40915	−1.96827	−0.984137	+	0.177411i	$0.943228\pi$
$4$	0	0
$5$	2.99160	1.33788	0.668942	−	0.743314i	$-0.266747\pi$
$5$	2.99160	1.33788	0.668942	+	0.743314i	$0.266747\pi$
$6$	0	0
$7$	3.46369	1.30915	0.654576	−	0.755996i	$-0.272847\pi$
$7$	3.46369	1.30915	0.654576	+	0.755996i	$0.272847\pi$
$8$	0	0
$9$	8.62230	2.87410
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−10.1988	−2.63332
$16$	0	0
$17$	−6.00545	−1.45653	−0.728267	−	0.685293i	$-0.759674\pi$
$17$	−6.00545	−1.45653	−0.728267	+	0.685293i	$0.759674\pi$
$18$	0	0
$19$	−4.10518	−0.941792	−0.470896	−	0.882189i	$-0.656069\pi$
$19$	−4.10518	−0.941792	−0.470896	+	0.882189i	$0.656069\pi$
$20$	0	0
$21$	−11.8082	−2.57677
$22$	0	0
$23$	5.25830	1.09643	0.548216	−	0.836337i	$-0.315307\pi$
$23$	5.25830	1.09643	0.548216	+	0.836337i	$0.315307\pi$
$24$	0	0
$25$	3.94968	0.789936
$26$	0	0
$27$	−19.1673	−3.68874
$28$	0	0
$29$	−1.94363	−0.360922	−0.180461	−	0.983582i	$-0.557759\pi$
$29$	−1.94363	−0.360922	−0.180461	+	0.983582i	$0.557759\pi$
$30$	0	0
$31$	−2.23592	−0.401584	−0.200792	−	0.979634i	$-0.564352\pi$
$31$	−2.23592	−0.401584	−0.200792	+	0.979634i	$0.564352\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	10.3620	1.75149
$36$	0	0
$37$	7.55090	1.24136	0.620680	−	0.784064i	$-0.286857\pi$
$37$	7.55090	1.24136	0.620680	+	0.784064i	$0.286857\pi$
$38$	0	0
$39$	3.40915	0.545901
$40$	0	0
$41$	3.29848	0.515135	0.257568	−	0.966260i	$-0.417079\pi$
$41$	3.29848	0.515135	0.257568	+	0.966260i	$0.417079\pi$
$42$	0	0
$43$	−8.24966	−1.25806	−0.629031	−	0.777380i	$-0.716548\pi$
$43$	−8.24966	−1.25806	−0.629031	+	0.777380i	$0.716548\pi$
$44$	0	0
$45$	25.7945	3.84522
$46$	0	0
$47$	6.40804	0.934708	0.467354	−	0.884070i	$-0.345207\pi$
$47$	6.40804	0.934708	0.467354	+	0.884070i	$0.345207\pi$
$48$	0	0
$49$	4.99714	0.713878
$50$	0	0
$51$	20.4735	2.86686
$52$	0	0
$53$	6.62939	0.910617	0.455309	−	0.890334i	$-0.349529\pi$
$53$	6.62939	0.910617	0.455309	+	0.890334i	$0.349529\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	13.9952	1.85370
$58$	0	0
$59$	−4.62867	−0.602602	−0.301301	−	0.953529i	$-0.597421\pi$
$59$	−4.62867	−0.602602	−0.301301	+	0.953529i	$0.597421\pi$
$60$	0	0
$61$	4.97344	0.636783	0.318392	−	0.947959i	$-0.396857\pi$
$61$	4.97344	0.636783	0.318392	+	0.947959i	$0.396857\pi$
$62$	0	0
$63$	29.8650	3.76263
$64$	0	0
$65$	−2.99160	−0.371063
$66$	0	0
$67$	1.08509	0.132565	0.0662824	−	0.997801i	$-0.478886\pi$
$67$	1.08509	0.132565	0.0662824	+	0.997801i	$0.478886\pi$
$68$	0	0
$69$	−17.9263	−2.15808
$70$	0	0
$71$	2.79559	0.331775	0.165888	−	0.986145i	$-0.446951\pi$
$71$	2.79559	0.331775	0.165888	+	0.986145i	$0.446951\pi$
$72$	0	0
$73$	14.6849	1.71874	0.859370	−	0.511354i	$-0.170856\pi$
$73$	14.6849	1.71874	0.859370	+	0.511354i	$0.170856\pi$
$74$	0	0
$75$	−13.4651	−1.55481
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−0.381142	−0.0428818	−0.0214409	−	0.999770i	$-0.506825\pi$
$79$	−0.381142	−0.0428818	−0.0214409	+	0.999770i	$0.506825\pi$
$80$	0	0
$81$	39.4772	4.38636
$82$	0	0
$83$	−3.52753	−0.387196	−0.193598	−	0.981081i	$-0.562016\pi$
$83$	−3.52753	−0.387196	−0.193598	+	0.981081i	$0.562016\pi$
$84$	0	0
$85$	−17.9659	−1.94868
$86$	0	0
$87$	6.62611	0.710394
$88$	0	0
$89$	4.38192	0.464482	0.232241	−	0.972658i	$-0.425394\pi$
$89$	4.38192	0.464482	0.232241	+	0.972658i	$0.425394\pi$
$90$	0	0
$91$	−3.46369	−0.363093
$92$	0	0
$93$	7.62260	0.790427
$94$	0	0
$95$	−12.2810	−1.26001
$96$	0	0
$97$	5.92091	0.601177	0.300588	−	0.953754i	$-0.402817\pi$
$97$	5.92091	0.601177	0.300588	+	0.953754i	$0.402817\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	8.55461	0.851216	0.425608	−	0.904908i	$-0.360060\pi$
$101$	8.55461	0.851216	0.425608	+	0.904908i	$0.360060\pi$
$102$	0	0
$103$	4.21260	0.415079	0.207540	−	0.978227i	$-0.433454\pi$
$103$	4.21260	0.415079	0.207540	+	0.978227i	$0.433454\pi$
$104$	0	0
$105$	−35.3255	−3.44742
$106$	0	0
$107$	−13.4631	−1.30152	−0.650761	−	0.759282i	$-0.725550\pi$
$107$	−13.4631	−1.30152	−0.650761	+	0.759282i	$0.725550\pi$
$108$	0	0
$109$	14.0855	1.34915	0.674574	−	0.738207i	$-0.264327\pi$
$109$	14.0855	1.34915	0.674574	+	0.738207i	$0.264327\pi$
$110$	0	0
$111$	−25.7421	−2.44334
$112$	0	0
$113$	−14.7250	−1.38522	−0.692608	−	0.721314i	$-0.743538\pi$
$113$	−14.7250	−1.38522	−0.692608	+	0.721314i	$0.743538\pi$
$114$	0	0
$115$	15.7307	1.46690
$116$	0	0
$117$	−8.62230	−0.797132
$118$	0	0
$119$	−20.8010	−1.90682
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−11.2450	−1.01393
$124$	0	0
$125$	−3.14214	−0.281041
$126$	0	0
$127$	1.51971	0.134852	0.0674262	−	0.997724i	$-0.478521\pi$
$127$	1.51971	0.134852	0.0674262	+	0.997724i	$0.478521\pi$
$128$	0	0
$129$	28.1243	2.47621
$130$	0	0
$131$	5.42306	0.473815	0.236907	−	0.971532i	$-0.423866\pi$
$131$	5.42306	0.473815	0.236907	+	0.971532i	$0.423866\pi$
$132$	0	0
$133$	−14.2191	−1.23295
$134$	0	0
$135$	−57.3409	−4.93512
$136$	0	0
$137$	7.32321	0.625664	0.312832	−	0.949808i	$-0.398722\pi$
$137$	7.32321	0.625664	0.312832	+	0.949808i	$0.398722\pi$
$138$	0	0
$139$	2.92145	0.247795	0.123897	−	0.992295i	$-0.460461\pi$
$139$	2.92145	0.247795	0.123897	+	0.992295i	$0.460461\pi$
$140$	0	0
$141$	−21.8460	−1.83976
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−5.81455	−0.482872
$146$	0	0
$147$	−17.0360	−1.40511
$148$	0	0
$149$	1.00255	0.0821320	0.0410660	−	0.999156i	$-0.486925\pi$
$149$	1.00255	0.0821320	0.0410660	+	0.999156i	$0.486925\pi$
$150$	0	0
$151$	0.685816	0.0558109	0.0279055	−	0.999611i	$-0.491116\pi$
$151$	0.685816	0.0558109	0.0279055	+	0.999611i	$0.491116\pi$
$152$	0	0
$153$	−51.7808	−4.18623
$154$	0	0
$155$	−6.68900	−0.537273
$156$	0	0
$157$	16.9709	1.35443	0.677213	−	0.735787i	$-0.263188\pi$
$157$	16.9709	1.35443	0.677213	+	0.735787i	$0.263188\pi$
$158$	0	0
$159$	−22.6006	−1.79234
$160$	0	0
$161$	18.2131	1.43540
$162$	0	0
$163$	10.4460	0.818196	0.409098	−	0.912490i	$-0.365843\pi$
$163$	10.4460	0.818196	0.409098	+	0.912490i	$0.365843\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2.53015	0.195789	0.0978945	−	0.995197i	$-0.468789\pi$
$167$	2.53015	0.195789	0.0978945	+	0.995197i	$0.468789\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−35.3961	−2.70680
$172$	0	0
$173$	14.9309	1.13518	0.567588	−	0.823312i	$-0.307877\pi$
$173$	14.9309	1.13518	0.567588	+	0.823312i	$0.307877\pi$
$174$	0	0
$175$	13.6805	1.03415
$176$	0	0
$177$	15.7798	1.18608
$178$	0	0
$179$	−0.0619254	−0.00462852	−0.00231426	−	0.999997i	$-0.500737\pi$
$179$	−0.0619254	−0.00462852	−0.00231426	+	0.999997i	$0.500737\pi$
$180$	0	0
$181$	−21.5579	−1.60239	−0.801193	−	0.598406i	$-0.795801\pi$
$181$	−21.5579	−1.60239	−0.801193	+	0.598406i	$0.795801\pi$
$182$	0	0
$183$	−16.9552	−1.25336
$184$	0	0
$185$	22.5893	1.66080
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−66.3895	−4.82913
$190$	0	0
$191$	23.5524	1.70419	0.852096	−	0.523385i	$-0.175331\pi$
$191$	23.5524	1.70419	0.852096	+	0.523385i	$0.175331\pi$
$192$	0	0
$193$	−4.71986	−0.339743	−0.169871	−	0.985466i	$-0.554335\pi$
$193$	−4.71986	−0.339743	−0.169871	+	0.985466i	$0.554335\pi$
$194$	0	0
$195$	10.1988	0.730353
$196$	0	0
$197$	15.1947	1.08258	0.541289	−	0.840837i	$-0.317937\pi$
$197$	15.1947	1.08258	0.541289	+	0.840837i	$0.317937\pi$
$198$	0	0
$199$	3.14455	0.222911	0.111456	−	0.993769i	$-0.464449\pi$
$199$	3.14455	0.222911	0.111456	+	0.993769i	$0.464449\pi$
$200$	0	0
$201$	−3.69923	−0.260924
$202$	0	0
$203$	−6.73212	−0.472502
$204$	0	0
$205$	9.86773	0.689192
$206$	0	0
$207$	45.3387	3.15126
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−1.39332	−0.0959201	−0.0479601	−	0.998849i	$-0.515272\pi$
$211$	−1.39332	−0.0959201	−0.0479601	+	0.998849i	$0.515272\pi$
$212$	0	0
$213$	−9.53057	−0.653024
$214$	0	0
$215$	−24.6797	−1.68314
$216$	0	0
$217$	−7.74455	−0.525734
$218$	0	0
$219$	−50.0631	−3.38295
$220$	0	0
$221$	6.00545	0.403970
$222$	0	0
$223$	2.13267	0.142814	0.0714070	−	0.997447i	$-0.477251\pi$
$223$	2.13267	0.142814	0.0714070	+	0.997447i	$0.477251\pi$
$224$	0	0
$225$	34.0554	2.27036
$226$	0	0
$227$	1.33285	0.0884641	0.0442320	−	0.999021i	$-0.485916\pi$
$227$	1.33285	0.0884641	0.0442320	+	0.999021i	$0.485916\pi$
$228$	0	0
$229$	−4.47194	−0.295514	−0.147757	−	0.989024i	$-0.547205\pi$
$229$	−4.47194	−0.295514	−0.147757	+	0.989024i	$0.547205\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	28.9252	1.89495	0.947476	−	0.319826i	$-0.103625\pi$
$233$	28.9252	1.89495	0.947476	+	0.319826i	$0.103625\pi$
$234$	0	0
$235$	19.1703	1.25053
$236$	0	0
$237$	1.29937	0.0844031
$238$	0	0
$239$	18.5113	1.19740	0.598699	−	0.800974i	$-0.295685\pi$
$239$	18.5113	1.19740	0.598699	+	0.800974i	$0.295685\pi$
$240$	0	0
$241$	−19.6851	−1.26803	−0.634014	−	0.773321i	$-0.718594\pi$
$241$	−19.6851	−1.26803	−0.634014	+	0.773321i	$0.718594\pi$
$242$	0	0
$243$	−77.0819	−4.94481
$244$	0	0
$245$	14.9495	0.955086
$246$	0	0
$247$	4.10518	0.261206
$248$	0	0
$249$	12.0259	0.762109
$250$	0	0
$251$	−17.2042	−1.08592	−0.542961	−	0.839758i	$-0.682697\pi$
$251$	−17.2042	−1.08592	−0.542961	+	0.839758i	$0.682697\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	61.2485	3.83553
$256$	0	0
$257$	0.901238	0.0562177	0.0281088	−	0.999605i	$-0.491051\pi$
$257$	0.901238	0.0562177	0.0281088	+	0.999605i	$0.491051\pi$
$258$	0	0
$259$	26.1540	1.62513
$260$	0	0
$261$	−16.7585	−1.03733
$262$	0	0
$263$	−1.15980	−0.0715166	−0.0357583	−	0.999360i	$-0.511385\pi$
$263$	−1.15980	−0.0715166	−0.0357583	+	0.999360i	$0.511385\pi$
$264$	0	0
$265$	19.8325	1.21830
$266$	0	0
$267$	−14.9386	−0.914228
$268$	0	0
$269$	3.56399	0.217300	0.108650	−	0.994080i	$-0.465347\pi$
$269$	3.56399	0.217300	0.108650	+	0.994080i	$0.465347\pi$
$270$	0	0
$271$	−15.6354	−0.949780	−0.474890	−	0.880045i	$-0.657512\pi$
$271$	−15.6354	−0.949780	−0.474890	+	0.880045i	$0.657512\pi$
$272$	0	0
$273$	11.8082	0.714667
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−5.01644	−0.301409	−0.150704	−	0.988579i	$-0.548154\pi$
$277$	−5.01644	−0.301409	−0.150704	+	0.988579i	$0.548154\pi$
$278$	0	0
$279$	−19.2788	−1.15419
$280$	0	0
$281$	23.5802	1.40668	0.703340	−	0.710854i	$-0.251691\pi$
$281$	23.5802	1.40668	0.703340	+	0.710854i	$0.251691\pi$
$282$	0	0
$283$	−10.6638	−0.633899	−0.316950	−	0.948442i	$-0.602659\pi$
$283$	−10.6638	−0.633899	−0.316950	+	0.948442i	$0.602659\pi$
$284$	0	0
$285$	41.8679	2.48004
$286$	0	0
$287$	11.4249	0.674390
$288$	0	0
$289$	19.0654	1.12149
$290$	0	0
$291$	−20.1853	−1.18328
$292$	0	0
$293$	−20.6610	−1.20703	−0.603515	−	0.797352i	$-0.706234\pi$
$293$	−20.6610	−1.20703	−0.603515	+	0.797352i	$0.706234\pi$
$294$	0	0
$295$	−13.8471	−0.806212
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−5.25830	−0.304095
$300$	0	0
$301$	−28.5743	−1.64699
$302$	0	0
$303$	−29.1640	−1.67543
$304$	0	0
$305$	14.8785	0.851943
$306$	0	0
$307$	4.28073	0.244314	0.122157	−	0.992511i	$-0.461019\pi$
$307$	4.28073	0.244314	0.122157	+	0.992511i	$0.461019\pi$
$308$	0	0
$309$	−14.3614	−0.816990
$310$	0	0
$311$	−16.4974	−0.935483	−0.467741	−	0.883865i	$-0.654932\pi$
$311$	−16.4974	−0.935483	−0.467741	+	0.883865i	$0.654932\pi$
$312$	0	0
$313$	23.7720	1.34367	0.671835	−	0.740701i	$-0.265506\pi$
$313$	23.7720	1.34367	0.671835	+	0.740701i	$0.265506\pi$
$314$	0	0
$315$	89.3441	5.03397
$316$	0	0
$317$	19.3939	1.08927	0.544635	−	0.838673i	$-0.316668\pi$
$317$	19.3939	1.08927	0.544635	+	0.838673i	$0.316668\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	45.8976	2.56175
$322$	0	0
$323$	24.6534	1.37175
$324$	0	0
$325$	−3.94968	−0.219089
$326$	0	0
$327$	−48.0197	−2.65549
$328$	0	0
$329$	22.1954	1.22367
$330$	0	0
$331$	1.19772	0.0658327	0.0329164	−	0.999458i	$-0.489521\pi$
$331$	1.19772	0.0658327	0.0329164	+	0.999458i	$0.489521\pi$
$332$	0	0
$333$	65.1061	3.56779
$334$	0	0
$335$	3.24615	0.177356
$336$	0	0
$337$	−21.7020	−1.18218	−0.591090	−	0.806605i	$-0.701302\pi$
$337$	−21.7020	−1.18218	−0.591090	+	0.806605i	$0.701302\pi$
$338$	0	0
$339$	50.1999	2.72648
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−6.93727	−0.374577
$344$	0	0
$345$	−53.6285	−2.88726
$346$	0	0
$347$	−7.15467	−0.384083	−0.192041	−	0.981387i	$-0.561511\pi$
$347$	−7.15467	−0.384083	−0.192041	+	0.981387i	$0.561511\pi$
$348$	0	0
$349$	−31.6067	−1.69187	−0.845933	−	0.533289i	$-0.820956\pi$
$349$	−31.6067	−1.69187	−0.845933	+	0.533289i	$0.820956\pi$
$350$	0	0
$351$	19.1673	1.02307
$352$	0	0
$353$	10.2675	0.546483	0.273242	−	0.961945i	$-0.411904\pi$
$353$	10.2675	0.546483	0.273242	+	0.961945i	$0.411904\pi$
$354$	0	0
$355$	8.36328	0.443877
$356$	0	0
$357$	70.9137	3.75315
$358$	0	0
$359$	14.7931	0.780751	0.390375	−	0.920656i	$-0.372345\pi$
$359$	14.7931	0.780751	0.390375	+	0.920656i	$0.372345\pi$
$360$	0	0
$361$	−2.14754	−0.113028
$362$	0	0
$363$	0	0
$364$	0	0
$365$	43.9314	2.29948
$366$	0	0
$367$	16.7733	0.875559	0.437779	−	0.899082i	$-0.355765\pi$
$367$	16.7733	0.875559	0.437779	+	0.899082i	$0.355765\pi$
$368$	0	0
$369$	28.4405	1.48055
$370$	0	0
$371$	22.9622	1.19214
$372$	0	0
$373$	37.2795	1.93026	0.965130	−	0.261769i	$-0.0843060\pi$
$373$	37.2795	1.93026	0.965130	+	0.261769i	$0.0843060\pi$
$374$	0	0
$375$	10.7120	0.553166
$376$	0	0
$377$	1.94363	0.100102
$378$	0	0
$379$	32.7763	1.68360	0.841802	−	0.539786i	$-0.181495\pi$
$379$	32.7763	1.68360	0.841802	+	0.539786i	$0.181495\pi$
$380$	0	0
$381$	−5.18092	−0.265427
$382$	0	0
$383$	19.9667	1.02025	0.510125	−	0.860100i	$-0.329599\pi$
$383$	19.9667	1.02025	0.510125	+	0.860100i	$0.329599\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−71.1311	−3.61580
$388$	0	0
$389$	14.7333	0.747007	0.373504	−	0.927629i	$-0.378156\pi$
$389$	14.7333	0.747007	0.373504	+	0.927629i	$0.378156\pi$
$390$	0	0
$391$	−31.5785	−1.59699
$392$	0	0
$393$	−18.4880	−0.932597
$394$	0	0
$395$	−1.14022	−0.0573709
$396$	0	0
$397$	27.3620	1.37326	0.686629	−	0.727008i	$-0.259089\pi$
$397$	27.3620	1.37326	0.686629	+	0.727008i	$0.259089\pi$
$398$	0	0
$399$	48.4749	2.42678
$400$	0	0
$401$	7.26936	0.363015	0.181507	−	0.983390i	$-0.441902\pi$
$401$	7.26936	0.363015	0.181507	+	0.983390i	$0.441902\pi$
$402$	0	0
$403$	2.23592	0.111379
$404$	0	0
$405$	118.100	5.86844
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−33.2629	−1.64474	−0.822371	−	0.568952i	$-0.807349\pi$
$409$	−33.2629	−1.64474	−0.822371	+	0.568952i	$0.807349\pi$
$410$	0	0
$411$	−24.9659	−1.23148
$412$	0	0
$413$	−16.0323	−0.788897
$414$	0	0
$415$	−10.5530	−0.518024
$416$	0	0
$417$	−9.95968	−0.487727
$418$	0	0
$419$	5.67635	0.277308	0.138654	−	0.990341i	$-0.455722\pi$
$419$	5.67635	0.277308	0.138654	+	0.990341i	$0.455722\pi$
$420$	0	0
$421$	4.36832	0.212899	0.106449	−	0.994318i	$-0.466052\pi$
$421$	4.36832	0.212899	0.106449	+	0.994318i	$0.466052\pi$
$422$	0	0
$423$	55.2520	2.68645
$424$	0	0
$425$	−23.7196	−1.15057
$426$	0	0
$427$	17.2264	0.833646
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−30.2691	−1.45801	−0.729005	−	0.684509i	$-0.760017\pi$
$431$	−30.2691	−1.45801	−0.729005	+	0.684509i	$0.760017\pi$
$432$	0	0
$433$	−12.9207	−0.620932	−0.310466	−	0.950585i	$-0.600485\pi$
$433$	−12.9207	−0.620932	−0.310466	+	0.950585i	$0.600485\pi$
$434$	0	0
$435$	19.8227	0.950425
$436$	0	0
$437$	−21.5863	−1.03261
$438$	0	0
$439$	29.5168	1.40876	0.704380	−	0.709824i	$-0.251225\pi$
$439$	29.5168	1.40876	0.704380	+	0.709824i	$0.251225\pi$
$440$	0	0
$441$	43.0869	2.05176
$442$	0	0
$443$	−17.6372	−0.837968	−0.418984	−	0.907994i	$-0.637614\pi$
$443$	−17.6372	−0.837968	−0.418984	+	0.907994i	$0.637614\pi$
$444$	0	0
$445$	13.1090	0.621424
$446$	0	0
$447$	−3.41784	−0.161658
$448$	0	0
$449$	−16.3031	−0.769389	−0.384694	−	0.923044i	$-0.625693\pi$
$449$	−16.3031	−0.769389	−0.384694	+	0.923044i	$0.625693\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−2.33805	−0.109851
$454$	0	0
$455$	−10.3620	−0.485777
$456$	0	0
$457$	−20.6471	−0.965830	−0.482915	−	0.875667i	$-0.660422\pi$
$457$	−20.6471	−0.965830	−0.482915	+	0.875667i	$0.660422\pi$
$458$	0	0
$459$	115.108	5.37278
$460$	0	0
$461$	−4.54374	−0.211623	−0.105812	−	0.994386i	$-0.533744\pi$
$461$	−4.54374	−0.211623	−0.105812	+	0.994386i	$0.533744\pi$
$462$	0	0
$463$	25.0385	1.16364	0.581820	−	0.813318i	$-0.302341\pi$
$463$	25.0385	1.16364	0.581820	+	0.813318i	$0.302341\pi$
$464$	0	0
$465$	22.8038	1.05750
$466$	0	0
$467$	−39.1890	−1.81345	−0.906725	−	0.421723i	$-0.861426\pi$
$467$	−39.1890	−1.81345	−0.906725	+	0.421723i	$0.861426\pi$
$468$	0	0
$469$	3.75841	0.173547
$470$	0	0
$471$	−57.8564	−2.66588
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−16.2141	−0.743955
$476$	0	0
$477$	57.1606	2.61721
$478$	0	0
$479$	5.37437	0.245561	0.122781	−	0.992434i	$-0.460819\pi$
$479$	5.37437	0.245561	0.122781	+	0.992434i	$0.460819\pi$
$480$	0	0
$481$	−7.55090	−0.344291
$482$	0	0
$483$	−62.0913	−2.82525
$484$	0	0
$485$	17.7130	0.804305
$486$	0	0
$487$	−0.0824319	−0.00373534	−0.00186767	−	0.999998i	$-0.500594\pi$
$487$	−0.0824319	−0.00373534	−0.00186767	+	0.999998i	$0.500594\pi$
$488$	0	0
$489$	−35.6121	−1.61043
$490$	0	0
$491$	−29.2846	−1.32159	−0.660797	−	0.750564i	$-0.729782\pi$
$491$	−29.2846	−1.32159	−0.660797	+	0.750564i	$0.729782\pi$
$492$	0	0
$493$	11.6723	0.525696
$494$	0	0
$495$	0	0
$496$	0	0
$497$	9.68304	0.434344
$498$	0	0
$499$	27.3915	1.22621	0.613107	−	0.790000i	$-0.289919\pi$
$499$	27.3915	1.22621	0.613107	+	0.790000i	$0.289919\pi$
$500$	0	0
$501$	−8.62566	−0.385366
$502$	0	0
$503$	33.8029	1.50720	0.753599	−	0.657335i	$-0.228316\pi$
$503$	33.8029	1.50720	0.753599	+	0.657335i	$0.228316\pi$
$504$	0	0
$505$	25.5920	1.13883
$506$	0	0
$507$	−3.40915	−0.151406
$508$	0	0
$509$	10.2629	0.454896	0.227448	−	0.973790i	$-0.426962\pi$
$509$	10.2629	0.454896	0.227448	+	0.973790i	$0.426962\pi$
$510$	0	0
$511$	50.8640	2.25009
$512$	0	0
$513$	78.6850	3.47403
$514$	0	0
$515$	12.6024	0.555329
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−50.9017	−2.23434
$520$	0	0
$521$	14.6712	0.642758	0.321379	−	0.946951i	$-0.395854\pi$
$521$	14.6712	0.642758	0.321379	+	0.946951i	$0.395854\pi$
$522$	0	0
$523$	41.0494	1.79496	0.897482	−	0.441051i	$-0.145394\pi$
$523$	41.0494	1.79496	0.897482	+	0.441051i	$0.145394\pi$
$524$	0	0
$525$	−46.6388	−2.03548
$526$	0	0
$527$	13.4277	0.584921
$528$	0	0
$529$	4.64974	0.202163
$530$	0	0
$531$	−39.9098	−1.73194
$532$	0	0
$533$	−3.29848	−0.142873
$534$	0	0
$535$	−40.2761	−1.74129
$536$	0	0
$537$	0.211113	0.00911020
$538$	0	0
$539$	0	0
$540$	0	0
$541$	9.79836	0.421264	0.210632	−	0.977565i	$-0.432448\pi$
$541$	9.79836	0.421264	0.210632	+	0.977565i	$0.432448\pi$
$542$	0	0
$543$	73.4941	3.15393
$544$	0	0
$545$	42.1383	1.80501
$546$	0	0
$547$	27.9908	1.19680	0.598400	−	0.801197i	$-0.295803\pi$
$547$	27.9908	1.19680	0.598400	+	0.801197i	$0.295803\pi$
$548$	0	0
$549$	42.8825	1.83018
$550$	0	0
$551$	7.97892	0.339914
$552$	0	0
$553$	−1.32016	−0.0561388
$554$	0	0
$555$	−77.0102	−3.26890
$556$	0	0
$557$	−32.7914	−1.38942	−0.694709	−	0.719291i	$-0.744467\pi$
$557$	−32.7914	−1.38942	−0.694709	+	0.719291i	$0.744467\pi$
$558$	0	0
$559$	8.24966	0.348924
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−36.1272	−1.52258	−0.761290	−	0.648412i	$-0.775433\pi$
$563$	−36.1272	−1.52258	−0.761290	+	0.648412i	$0.775433\pi$
$564$	0	0
$565$	−44.0515	−1.85326
$566$	0	0
$567$	136.737	5.74241
$568$	0	0
$569$	−25.2569	−1.05882	−0.529412	−	0.848365i	$-0.677587\pi$
$569$	−25.2569	−1.05882	−0.529412	+	0.848365i	$0.677587\pi$
$570$	0	0
$571$	−20.2018	−0.845419	−0.422709	−	0.906265i	$-0.638921\pi$
$571$	−20.2018	−0.845419	−0.422709	+	0.906265i	$0.638921\pi$
$572$	0	0
$573$	−80.2937	−3.35432
$574$	0	0
$575$	20.7686	0.866111
$576$	0	0
$577$	27.0253	1.12508	0.562538	−	0.826771i	$-0.309825\pi$
$577$	27.0253	1.12508	0.562538	+	0.826771i	$0.309825\pi$
$578$	0	0
$579$	16.0907	0.668707
$580$	0	0
$581$	−12.2183	−0.506899
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−25.7945	−1.06647
$586$	0	0
$587$	24.7520	1.02162	0.510812	−	0.859692i	$-0.329345\pi$
$587$	24.7520	1.02162	0.510812	+	0.859692i	$0.329345\pi$
$588$	0	0
$589$	9.17886	0.378208
$590$	0	0
$591$	−51.8010	−2.13081
$592$	0	0
$593$	−9.38104	−0.385233	−0.192617	−	0.981274i	$-0.561697\pi$
$593$	−9.38104	−0.385233	−0.192617	+	0.981274i	$0.561697\pi$
$594$	0	0
$595$	−62.2283	−2.55111
$596$	0	0
$597$	−10.7202	−0.438750
$598$	0	0
$599$	4.75731	0.194378	0.0971891	−	0.995266i	$-0.469015\pi$
$599$	4.75731	0.194378	0.0971891	+	0.995266i	$0.469015\pi$
$600$	0	0
$601$	−41.2264	−1.68166	−0.840830	−	0.541299i	$-0.817933\pi$
$601$	−41.2264	−1.68166	−0.840830	+	0.541299i	$0.817933\pi$
$602$	0	0
$603$	9.35597	0.381004
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−14.6972	−0.596542	−0.298271	−	0.954481i	$-0.596410\pi$
$607$	−14.6972	−0.596542	−0.298271	+	0.954481i	$0.596410\pi$
$608$	0	0
$609$	22.9508	0.930013
$610$	0	0
$611$	−6.40804	−0.259241
$612$	0	0
$613$	−29.7265	−1.20064	−0.600321	−	0.799759i	$-0.704961\pi$
$613$	−29.7265	−1.20064	−0.600321	+	0.799759i	$0.704961\pi$
$614$	0	0
$615$	−33.6406	−1.35652
$616$	0	0
$617$	32.3194	1.30113	0.650565	−	0.759450i	$-0.274532\pi$
$617$	32.3194	1.30113	0.650565	+	0.759450i	$0.274532\pi$
$618$	0	0
$619$	−23.0698	−0.927252	−0.463626	−	0.886031i	$-0.653452\pi$
$619$	−23.0698	−0.927252	−0.463626	+	0.886031i	$0.653452\pi$
$620$	0	0
$621$	−100.787	−4.04446
$622$	0	0
$623$	15.1776	0.608078
$624$	0	0
$625$	−29.1484	−1.16594
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−45.3465	−1.80808
$630$	0	0
$631$	27.7927	1.10641	0.553204	−	0.833046i	$-0.313405\pi$
$631$	27.7927	1.10641	0.553204	+	0.833046i	$0.313405\pi$
$632$	0	0
$633$	4.75004	0.188797
$634$	0	0
$635$	4.54637	0.180417
$636$	0	0
$637$	−4.99714	−0.197994
$638$	0	0
$639$	24.1044	0.953555
$640$	0	0
$641$	−33.6198	−1.32790	−0.663951	−	0.747776i	$-0.731122\pi$
$641$	−33.6198	−1.32790	−0.663951	+	0.747776i	$0.731122\pi$
$642$	0	0
$643$	−6.83964	−0.269729	−0.134865	−	0.990864i	$-0.543060\pi$
$643$	−6.83964	−0.269729	−0.134865	+	0.990864i	$0.543060\pi$
$644$	0	0
$645$	84.1368	3.31288
$646$	0	0
$647$	29.7187	1.16836	0.584181	−	0.811624i	$-0.301416\pi$
$647$	29.7187	1.16836	0.584181	+	0.811624i	$0.301416\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	26.4023	1.03479
$652$	0	0
$653$	−3.23253	−0.126499	−0.0632493	−	0.997998i	$-0.520146\pi$
$653$	−3.23253	−0.126499	−0.0632493	+	0.997998i	$0.520146\pi$
$654$	0	0
$655$	16.2236	0.633909
$656$	0	0
$657$	126.618	4.93983
$658$	0	0
$659$	−11.5331	−0.449264	−0.224632	−	0.974444i	$-0.572118\pi$
$659$	−11.5331	−0.449264	−0.224632	+	0.974444i	$0.572118\pi$
$660$	0	0
$661$	−41.0038	−1.59486	−0.797431	−	0.603410i	$-0.793808\pi$
$661$	−41.0038	−1.59486	−0.797431	+	0.603410i	$0.793808\pi$
$662$	0	0
$663$	−20.4735	−0.795124
$664$	0	0
$665$	−42.5377	−1.64954
$666$	0	0
$667$	−10.2202	−0.395727
$668$	0	0
$669$	−7.27058	−0.281097
$670$	0	0
$671$	0	0
$672$	0	0
$673$	9.10588	0.351006	0.175503	−	0.984479i	$-0.443845\pi$
$673$	9.10588	0.351006	0.175503	+	0.984479i	$0.443845\pi$
$674$	0	0
$675$	−75.7046	−2.91387
$676$	0	0
$677$	−11.1405	−0.428166	−0.214083	−	0.976816i	$-0.568676\pi$
$677$	−11.1405	−0.428166	−0.214083	+	0.976816i	$0.568676\pi$
$678$	0	0
$679$	20.5082	0.787032
$680$	0	0
$681$	−4.54387	−0.174122
$682$	0	0
$683$	22.0121	0.842271	0.421136	−	0.906998i	$-0.361632\pi$
$683$	22.0121	0.842271	0.421136	+	0.906998i	$0.361632\pi$
$684$	0	0
$685$	21.9081	0.837067
$686$	0	0
$687$	15.2455	0.581653
$688$	0	0
$689$	−6.62939	−0.252560
$690$	0	0
$691$	−37.7529	−1.43619	−0.718093	−	0.695947i	$-0.754985\pi$
$691$	−37.7529	−1.43619	−0.718093	+	0.695947i	$0.754985\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	8.73983	0.331521
$696$	0	0
$697$	−19.8088	−0.750313
$698$	0	0
$699$	−98.6103	−3.72979
$700$	0	0
$701$	2.71657	0.102604	0.0513018	−	0.998683i	$-0.483663\pi$
$701$	2.71657	0.102604	0.0513018	+	0.998683i	$0.483663\pi$
$702$	0	0
$703$	−30.9978	−1.16910
$704$	0	0
$705$	−65.3544	−2.46139
$706$	0	0
$707$	29.6305	1.11437
$708$	0	0
$709$	21.3118	0.800382	0.400191	−	0.916432i	$-0.368944\pi$
$709$	21.3118	0.800382	0.400191	+	0.916432i	$0.368944\pi$
$710$	0	0
$711$	−3.28632	−0.123247
$712$	0	0
$713$	−11.7572	−0.440309
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−63.1079	−2.35681
$718$	0	0
$719$	−27.4720	−1.02453	−0.512266	−	0.858827i	$-0.671194\pi$
$719$	−27.4720	−1.02453	−0.512266	+	0.858827i	$0.671194\pi$
$720$	0	0
$721$	14.5911	0.543402
$722$	0	0
$723$	67.1095	2.49583
$724$	0	0
$725$	−7.67670	−0.285106
$726$	0	0
$727$	−32.8448	−1.21815	−0.609073	−	0.793114i	$-0.708458\pi$
$727$	−32.8448	−1.21815	−0.609073	+	0.793114i	$0.708458\pi$
$728$	0	0
$729$	144.352	5.34638
$730$	0	0
$731$	49.5429	1.83241
$732$	0	0
$733$	1.90739	0.0704509	0.0352255	−	0.999379i	$-0.488785\pi$
$733$	1.90739	0.0704509	0.0352255	+	0.999379i	$0.488785\pi$
$734$	0	0
$735$	−50.9650	−1.87987
$736$	0	0
$737$	0	0
$738$	0	0
$739$	49.2448	1.81150	0.905749	−	0.423815i	$-0.139309\pi$
$739$	49.2448	1.81150	0.905749	+	0.423815i	$0.139309\pi$
$740$	0	0
$741$	−13.9952	−0.514125
$742$	0	0
$743$	27.8455	1.02155	0.510776	−	0.859714i	$-0.329358\pi$
$743$	27.8455	1.02155	0.510776	+	0.859714i	$0.329358\pi$
$744$	0	0
$745$	2.99923	0.109883
$746$	0	0
$747$	−30.4154	−1.11284
$748$	0	0
$749$	−46.6318	−1.70389
$750$	0	0
$751$	−21.2523	−0.775509	−0.387754	−	0.921763i	$-0.626749\pi$
$751$	−21.2523	−0.775509	−0.387754	+	0.921763i	$0.626749\pi$
$752$	0	0
$753$	58.6518	2.13739
$754$	0	0
$755$	2.05169	0.0746686
$756$	0	0
$757$	4.24052	0.154124	0.0770621	−	0.997026i	$-0.475446\pi$
$757$	4.24052	0.154124	0.0770621	+	0.997026i	$0.475446\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−13.2136	−0.478992	−0.239496	−	0.970897i	$-0.576982\pi$
$761$	−13.2136	−0.478992	−0.239496	+	0.970897i	$0.576982\pi$
$762$	0	0
$763$	48.7879	1.76624
$764$	0	0
$765$	−154.907	−5.60069
$766$	0	0
$767$	4.62867	0.167132
$768$	0	0
$769$	33.1577	1.19570	0.597849	−	0.801609i	$-0.296022\pi$
$769$	33.1577	1.19570	0.597849	+	0.801609i	$0.296022\pi$
$770$	0	0
$771$	−3.07246	−0.110652
$772$	0	0
$773$	−35.5706	−1.27939	−0.639693	−	0.768631i	$-0.720938\pi$
$773$	−35.5706	−1.27939	−0.639693	+	0.768631i	$0.720938\pi$
$774$	0	0
$775$	−8.83119	−0.317226
$776$	0	0
$777$	−89.1628	−3.19870
$778$	0	0
$779$	−13.5408	−0.485150
$780$	0	0
$781$	0	0
$782$	0	0
$783$	37.2540	1.33135
$784$	0	0
$785$	50.7702	1.81207
$786$	0	0
$787$	37.6862	1.34337	0.671684	−	0.740837i	$-0.265571\pi$
$787$	37.6862	1.34337	0.671684	+	0.740837i	$0.265571\pi$
$788$	0	0
$789$	3.95395	0.140764
$790$	0	0
$791$	−51.0030	−1.81346
$792$	0	0
$793$	−4.97344	−0.176612
$794$	0	0
$795$	−67.6120	−2.39795
$796$	0	0
$797$	−15.2948	−0.541770	−0.270885	−	0.962612i	$-0.587316\pi$
$797$	−15.2948	−0.541770	−0.270885	+	0.962612i	$0.587316\pi$
$798$	0	0
$799$	−38.4831	−1.36143
$800$	0	0
$801$	37.7822	1.33497
$802$	0	0
$803$	0	0
$804$	0	0
$805$	54.4864	1.92039
$806$	0	0
$807$	−12.1502	−0.427707
$808$	0	0
$809$	49.0261	1.72367	0.861833	−	0.507192i	$-0.169317\pi$
$809$	49.0261	1.72367	0.861833	+	0.507192i	$0.169317\pi$
$810$	0	0
$811$	9.39472	0.329893	0.164947	−	0.986302i	$-0.447255\pi$
$811$	9.39472	0.329893	0.164947	+	0.986302i	$0.447255\pi$
$812$	0	0
$813$	53.3033	1.86943
$814$	0	0
$815$	31.2503	1.09465
$816$	0	0
$817$	33.8663	1.18483
$818$	0	0
$819$	−29.8650	−1.04357
$820$	0	0
$821$	3.48261	0.121544	0.0607720	−	0.998152i	$-0.480644\pi$
$821$	3.48261	0.121544	0.0607720	+	0.998152i	$0.480644\pi$
$822$	0	0
$823$	−34.5447	−1.20415	−0.602076	−	0.798439i	$-0.705660\pi$
$823$	−34.5447	−1.20415	−0.602076	+	0.798439i	$0.705660\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	44.0530	1.53187	0.765936	−	0.642917i	$-0.222276\pi$
$827$	44.0530	1.53187	0.765936	+	0.642917i	$0.222276\pi$
$828$	0	0
$829$	−40.3457	−1.40127	−0.700633	−	0.713522i	$-0.747099\pi$
$829$	−40.3457	−1.40127	−0.700633	+	0.713522i	$0.747099\pi$
$830$	0	0
$831$	17.1018	0.593255
$832$	0	0
$833$	−30.0101	−1.03979
$834$	0	0
$835$	7.56920	0.261943
$836$	0	0
$837$	42.8566	1.48134
$838$	0	0
$839$	12.8094	0.442229	0.221115	−	0.975248i	$-0.429031\pi$
$839$	12.8094	0.442229	0.221115	+	0.975248i	$0.429031\pi$
$840$	0	0
$841$	−25.2223	−0.869735
$842$	0	0
$843$	−80.3886	−2.76873
$844$	0	0
$845$	2.99160	0.102914
$846$	0	0
$847$	0	0
$848$	0	0
$849$	36.3546	1.24769
$850$	0	0
$851$	39.7049	1.36107
$852$	0	0
$853$	−10.6822	−0.365753	−0.182876	−	0.983136i	$-0.558541\pi$
$853$	−10.6822	−0.365753	−0.182876	+	0.983136i	$0.558541\pi$
$854$	0	0
$855$	−105.891	−3.62139
$856$	0	0
$857$	48.6806	1.66290	0.831449	−	0.555602i	$-0.187512\pi$
$857$	48.6806	1.66290	0.831449	+	0.555602i	$0.187512\pi$
$858$	0	0
$859$	17.7081	0.604193	0.302096	−	0.953277i	$-0.402314\pi$
$859$	17.7081	0.604193	0.302096	+	0.953277i	$0.402314\pi$
$860$	0	0
$861$	−38.9492	−1.32738
$862$	0	0
$863$	11.4421	0.389494	0.194747	−	0.980853i	$-0.437611\pi$
$863$	11.4421	0.389494	0.194747	+	0.980853i	$0.437611\pi$
$864$	0	0
$865$	44.6674	1.51874
$866$	0	0
$867$	−64.9968	−2.20741
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−1.08509	−0.0367668
$872$	0	0
$873$	51.0518	1.72784
$874$	0	0
$875$	−10.8834	−0.367925
$876$	0	0
$877$	38.4880	1.29965	0.649824	−	0.760085i	$-0.274843\pi$
$877$	38.4880	1.29965	0.649824	+	0.760085i	$0.274843\pi$
$878$	0	0
$879$	70.4365	2.37576
$880$	0	0
$881$	17.7509	0.598044	0.299022	−	0.954246i	$-0.403340\pi$
$881$	17.7509	0.598044	0.299022	+	0.954246i	$0.403340\pi$
$882$	0	0
$883$	20.5733	0.692347	0.346173	−	0.938171i	$-0.387481\pi$
$883$	20.5733	0.692347	0.346173	+	0.938171i	$0.387481\pi$
$884$	0	0
$885$	47.2070	1.58684
$886$	0	0
$887$	−7.98395	−0.268075	−0.134037	−	0.990976i	$-0.542794\pi$
$887$	−7.98395	−0.268075	−0.134037	+	0.990976i	$0.542794\pi$
$888$	0	0
$889$	5.26380	0.176542
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−26.3061	−0.880300
$894$	0	0
$895$	−0.185256	−0.00619243
$896$	0	0
$897$	17.9263	0.598543
$898$	0	0
$899$	4.34580	0.144941
$900$	0	0
$901$	−39.8125	−1.32635
$902$	0	0
$903$	97.4140	3.24173
$904$	0	0
$905$	−64.4926	−2.14381
$906$	0	0
$907$	16.9221	0.561890	0.280945	−	0.959724i	$-0.409352\pi$
$907$	16.9221	0.561890	0.280945	+	0.959724i	$0.409352\pi$
$908$	0	0
$909$	73.7605	2.44648
$910$	0	0
$911$	−49.1440	−1.62821	−0.814106	−	0.580716i	$-0.802773\pi$
$911$	−49.1440	−1.62821	−0.814106	+	0.580716i	$0.802773\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−50.7232	−1.67686
$916$	0	0
$917$	18.7838	0.620295
$918$	0	0
$919$	−32.2180	−1.06278	−0.531388	−	0.847129i	$-0.678329\pi$
$919$	−32.2180	−1.06278	−0.531388	+	0.847129i	$0.678329\pi$
$920$	0	0
$921$	−14.5937	−0.480878
$922$	0	0
$923$	−2.79559	−0.0920178
$924$	0	0
$925$	29.8236	0.980595
$926$	0	0
$927$	36.3223	1.19298
$928$	0	0
$929$	23.0489	0.756209	0.378105	−	0.925763i	$-0.376576\pi$
$929$	23.0489	0.756209	0.378105	+	0.925763i	$0.376576\pi$
$930$	0	0
$931$	−20.5142	−0.672324
$932$	0	0
$933$	56.2422	1.84129
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−12.7219	−0.415605	−0.207803	−	0.978171i	$-0.566631\pi$
$937$	−12.7219	−0.415605	−0.207803	+	0.978171i	$0.566631\pi$
$938$	0	0
$939$	−81.0422	−2.64471
$940$	0	0
$941$	29.5231	0.962425	0.481212	−	0.876604i	$-0.340197\pi$
$941$	29.5231	0.962425	0.481212	+	0.876604i	$0.340197\pi$
$942$	0	0
$943$	17.3444	0.564811
$944$	0	0
$945$	−198.611	−6.46081
$946$	0	0
$947$	−56.4744	−1.83517	−0.917585	−	0.397539i	$-0.869865\pi$
$947$	−56.4744	−1.83517	−0.917585	+	0.397539i	$0.869865\pi$
$948$	0	0
$949$	−14.6849	−0.476693
$950$	0	0
$951$	−66.1167	−2.14398
$952$	0	0
$953$	−29.2081	−0.946143	−0.473072	−	0.881024i	$-0.656855\pi$
$953$	−29.2081	−0.946143	−0.473072	+	0.881024i	$0.656855\pi$
$954$	0	0
$955$	70.4594	2.28001
$956$	0	0
$957$	0	0
$958$	0	0
$959$	25.3653	0.819090
$960$	0	0
$961$	−26.0006	−0.838730
$962$	0	0
$963$	−116.083	−3.74071
$964$	0	0
$965$	−14.1199	−0.454537
$966$	0	0
$967$	26.2505	0.844160	0.422080	−	0.906559i	$-0.361300\pi$
$967$	26.2505	0.844160	0.422080	+	0.906559i	$0.361300\pi$
$968$	0	0
$969$	−84.0472	−2.69998
$970$	0	0
$971$	−5.58097	−0.179102	−0.0895509	−	0.995982i	$-0.528543\pi$
$971$	−5.58097	−0.179102	−0.0895509	+	0.995982i	$0.528543\pi$
$972$	0	0
$973$	10.1190	0.324401
$974$	0	0
$975$	13.4651	0.431227
$976$	0	0
$977$	35.2641	1.12820	0.564100	−	0.825707i	$-0.309223\pi$
$977$	35.2641	1.12820	0.564100	+	0.825707i	$0.309223\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	121.450	3.87759
$982$	0	0
$983$	58.0029	1.85001	0.925003	−	0.379959i	$-0.124062\pi$
$983$	58.0029	1.85001	0.925003	+	0.379959i	$0.124062\pi$
$984$	0	0
$985$	45.4565	1.44836
$986$	0	0
$987$	−75.6676	−2.40853
$988$	0	0
$989$	−43.3792	−1.37938
$990$	0	0
$991$	27.6564	0.878534	0.439267	−	0.898357i	$-0.355238\pi$
$991$	27.6564	0.878534	0.439267	+	0.898357i	$0.355238\pi$
$992$	0	0
$993$	−4.08321	−0.129577
$994$	0	0
$995$	9.40724	0.298230
$996$	0	0
$997$	−48.6718	−1.54145	−0.770725	−	0.637168i	$-0.780106\pi$
$997$	−48.6718	−1.54145	−0.770725	+	0.637168i	$0.780106\pi$
$998$	0	0
$999$	−144.730	−4.57906

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6292.2.a.z.1.1		14
11.5	even	5		572.2.n.b.157.1	✓	28
11.9	even	5		572.2.n.b.521.1	yes	28
11.10	odd	2		6292.2.a.y.1.1		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
572.2.n.b.157.1	✓	28	11.5	even	5
572.2.n.b.521.1	yes	28	11.9	even	5
6292.2.a.y.1.1		14	11.10	odd	2
6292.2.a.z.1.1		14	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:	\( N \)	\(=\)	\( 6292 = 2^{2} \cdot 11^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6292.a (trivial)

\(f(q)\)	\(=\)	\(q-3.40915 q^{3} +2.99160 q^{5} +3.46369 q^{7} +8.62230 q^{9} +O(q^{10})\)	\(q-3.40915 q^{3} +2.99160 q^{5} +3.46369 q^{7} +8.62230 q^{9} -1.00000 q^{13} -10.1988 q^{15} -6.00545 q^{17} -4.10518 q^{19} -11.8082 q^{21} +5.25830 q^{23} +3.94968 q^{25} -19.1673 q^{27} -1.94363 q^{29} -2.23592 q^{31} +10.3620 q^{35} +7.55090 q^{37} +3.40915 q^{39} +3.29848 q^{41} -8.24966 q^{43} +25.7945 q^{45} +6.40804 q^{47} +4.99714 q^{49} +20.4735 q^{51} +6.62939 q^{53} +13.9952 q^{57} -4.62867 q^{59} +4.97344 q^{61} +29.8650 q^{63} -2.99160 q^{65} +1.08509 q^{67} -17.9263 q^{69} +2.79559 q^{71} +14.6849 q^{73} -13.4651 q^{75} -0.381142 q^{79} +39.4772 q^{81} -3.52753 q^{83} -17.9659 q^{85} +6.62611 q^{87} +4.38192 q^{89} -3.46369 q^{91} +7.62260 q^{93} -12.2810 q^{95} +5.92091 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 3 q^{3} + 9 q^{5} + 3 q^{7} + 29 q^{9}+O(q^{10}) \) 14 * q + 3 * q^3 + 9 * q^5 + 3 * q^7 + 29 * q^9	\( 14 q + 3 q^{3} + 9 q^{5} + 3 q^{7} + 29 q^{9} - 14 q^{13} + 13 q^{15} - 4 q^{17} - q^{19} - 6 q^{21} + 17 q^{23} + 37 q^{25} - 3 q^{27} + 4 q^{29} + 14 q^{31} + 22 q^{35} + 25 q^{37} - 3 q^{39} - 13 q^{41} - 4 q^{43} + 31 q^{45} + 24 q^{47} + 13 q^{49} + 12 q^{51} + 44 q^{53} - 3 q^{57} + 19 q^{59} - 7 q^{61} + 3 q^{63} - 9 q^{65} + 5 q^{67} + 9 q^{69} + 24 q^{71} + 20 q^{73} + 13 q^{75} + 33 q^{79} + 86 q^{81} - 34 q^{83} + 36 q^{85} + 28 q^{87} + 41 q^{89} - 3 q^{91} + 36 q^{93} - 17 q^{95} + 16 q^{97}+O(q^{100}) \) 14 * q + 3 * q^3 + 9 * q^5 + 3 * q^7 + 29 * q^9 - 14 * q^13 + 13 * q^15 - 4 * q^17 - q^19 - 6 * q^21 + 17 * q^23 + 37 * q^25 - 3 * q^27 + 4 * q^29 + 14 * q^31 + 22 * q^35 + 25 * q^37 - 3 * q^39 - 13 * q^41 - 4 * q^43 + 31 * q^45 + 24 * q^47 + 13 * q^49 + 12 * q^51 + 44 * q^53 - 3 * q^57 + 19 * q^59 - 7 * q^61 + 3 * q^63 - 9 * q^65 + 5 * q^67 + 9 * q^69 + 24 * q^71 + 20 * q^73 + 13 * q^75 + 33 * q^79 + 86 * q^81 - 34 * q^83 + 36 * q^85 + 28 * q^87 + 41 * q^89 - 3 * q^91 + 36 * q^93 - 17 * q^95 + 16 * q^97

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