LMFDB - Embedded newform 8092.2.a.v.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8092,2,Mod(1,8092)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8092, 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("8092.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8092 = 2^{2} \cdot 7 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8092.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:	$64.6149453156$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 24 x^{10} - 2 x^{9} + 216 x^{8} + 15 x^{7} - 924 x^{6} + 15 x^{5} + 1947 x^{4} - 251 x^{3} + \cdots + 456 $ x^12 - 24x^10 - 2x^9 + 216x^8 + 15x^7 - 924x^6 + 15x^5 + 1947x^4 - 251x^3 - 1806x^2 + 360x + 456
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$-2.41089$ of defining polynomial
Character	$\chi$	$=$	8092.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.41089 q^{3} +0.568538 q^{5} -1.00000 q^{7} +2.81238 q^{9} +O(q^{10})$	$q+2.41089 q^{3} +0.568538 q^{5} -1.00000 q^{7} +2.81238 q^{9} -5.65717 q^{11} -2.69720 q^{13} +1.37068 q^{15} +7.35893 q^{19} -2.41089 q^{21} -2.33235 q^{23} -4.67677 q^{25} -0.452327 q^{27} +8.38860 q^{29} +3.89756 q^{31} -13.6388 q^{33} -0.568538 q^{35} +7.68633 q^{37} -6.50264 q^{39} -7.27530 q^{41} -12.0872 q^{43} +1.59894 q^{45} -9.00152 q^{47} +1.00000 q^{49} -6.10982 q^{53} -3.21632 q^{55} +17.7415 q^{57} +1.55671 q^{59} -1.71488 q^{61} -2.81238 q^{63} -1.53346 q^{65} +13.3442 q^{67} -5.62304 q^{69} -7.99168 q^{71} -12.0752 q^{73} -11.2752 q^{75} +5.65717 q^{77} +1.65340 q^{79} -9.52765 q^{81} -7.40791 q^{83} +20.2240 q^{87} -8.97278 q^{89} +2.69720 q^{91} +9.39659 q^{93} +4.18383 q^{95} +1.10846 q^{97} -15.9101 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{7} + 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^7 + 12 * q^9	$ 12 q - 12 q^{7} + 12 q^{9} + 6 q^{11} - 12 q^{13} - 9 q^{15} - 12 q^{19} + 9 q^{23} - 6 q^{25} - 6 q^{27} + 6 q^{29} + 9 q^{31} - 24 q^{33} + 12 q^{37} + 3 q^{39} - 21 q^{41} - 3 q^{43} + 33 q^{45} - 30 q^{47} + 12 q^{49} - 27 q^{53} - 15 q^{55} + 30 q^{57} - 30 q^{59} + 3 q^{61} - 12 q^{63} + 24 q^{65} - 3 q^{67} - 3 q^{69} + 27 q^{71} - 15 q^{73} - 63 q^{75} - 6 q^{77} + 6 q^{79} - 36 q^{81} - 54 q^{83} - 36 q^{87} - 21 q^{89} + 12 q^{91} - 51 q^{93} + 18 q^{95} - 45 q^{97} + 30 q^{99}+O(q^{100}) $ 12 * q - 12 * q^7 + 12 * q^9 + 6 * q^11 - 12 * q^13 - 9 * q^15 - 12 * q^19 + 9 * q^23 - 6 * q^25 - 6 * q^27 + 6 * q^29 + 9 * q^31 - 24 * q^33 + 12 * q^37 + 3 * q^39 - 21 * q^41 - 3 * q^43 + 33 * q^45 - 30 * q^47 + 12 * q^49 - 27 * q^53 - 15 * q^55 + 30 * q^57 - 30 * q^59 + 3 * q^61 - 12 * q^63 + 24 * q^65 - 3 * q^67 - 3 * q^69 + 27 * q^71 - 15 * q^73 - 63 * q^75 - 6 * q^77 + 6 * q^79 - 36 * q^81 - 54 * q^83 - 36 * q^87 - 21 * q^89 + 12 * q^91 - 51 * q^93 + 18 * q^95 - 45 * q^97 + 30 * 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
$3$	2.41089	1.39193	0.695963	−	0.718077i	$-0.254978\pi$
$3$	2.41089	1.39193	0.695963	+	0.718077i	$0.254978\pi$
$4$	0	0
$5$	0.568538	0.254258	0.127129	−	0.991886i	$-0.459424\pi$
$5$	0.568538	0.254258	0.127129	+	0.991886i	$0.459424\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	2.81238	0.937461
$10$	0	0
$11$	−5.65717	−1.70570	−0.852851	−	0.522154i	$-0.825128\pi$
$11$	−5.65717	−1.70570	−0.852851	+	0.522154i	$0.825128\pi$
$12$	0	0
$13$	−2.69720	−0.748068	−0.374034	−	0.927415i	$-0.622026\pi$
$13$	−2.69720	−0.748068	−0.374034	+	0.927415i	$0.622026\pi$
$14$	0	0
$15$	1.37068	0.353908
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	7.35893	1.68825	0.844127	−	0.536144i	$-0.180119\pi$
$19$	7.35893	1.68825	0.844127	+	0.536144i	$0.180119\pi$
$20$	0	0
$21$	−2.41089	−0.526099
$22$	0	0
$23$	−2.33235	−0.486329	−0.243164	−	0.969985i	$-0.578185\pi$
$23$	−2.33235	−0.486329	−0.243164	+	0.969985i	$0.578185\pi$
$24$	0	0
$25$	−4.67677	−0.935353
$26$	0	0
$27$	−0.452327	−0.0870503
$28$	0	0
$29$	8.38860	1.55772	0.778862	−	0.627195i	$-0.215797\pi$
$29$	8.38860	1.55772	0.778862	+	0.627195i	$0.215797\pi$
$30$	0	0
$31$	3.89756	0.700023	0.350012	−	0.936745i	$-0.386178\pi$
$31$	3.89756	0.700023	0.350012	+	0.936745i	$0.386178\pi$
$32$	0	0
$33$	−13.6388	−2.37421
$34$	0	0
$35$	−0.568538	−0.0961004
$36$	0	0
$37$	7.68633	1.26362	0.631812	−	0.775122i	$-0.282311\pi$
$37$	7.68633	1.26362	0.631812	+	0.775122i	$0.282311\pi$
$38$	0	0
$39$	−6.50264	−1.04126
$40$	0	0
$41$	−7.27530	−1.13621	−0.568105	−	0.822956i	$-0.692323\pi$
$41$	−7.27530	−1.13621	−0.568105	+	0.822956i	$0.692323\pi$
$42$	0	0
$43$	−12.0872	−1.84327	−0.921637	−	0.388054i	$-0.873148\pi$
$43$	−12.0872	−1.84327	−0.921637	+	0.388054i	$0.873148\pi$
$44$	0	0
$45$	1.59894	0.238357
$46$	0	0
$47$	−9.00152	−1.31301	−0.656503	−	0.754323i	$-0.727965\pi$
$47$	−9.00152	−1.31301	−0.656503	+	0.754323i	$0.727965\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−6.10982	−0.839248	−0.419624	−	0.907698i	$-0.637838\pi$
$53$	−6.10982	−0.839248	−0.419624	+	0.907698i	$0.637838\pi$
$54$	0	0
$55$	−3.21632	−0.433688
$56$	0	0
$57$	17.7415	2.34993
$58$	0	0
$59$	1.55671	0.202667	0.101333	−	0.994853i	$-0.467689\pi$
$59$	1.55671	0.202667	0.101333	+	0.994853i	$0.467689\pi$
$60$	0	0
$61$	−1.71488	−0.219568	−0.109784	−	0.993955i	$-0.535016\pi$
$61$	−1.71488	−0.219568	−0.109784	+	0.993955i	$0.535016\pi$
$62$	0	0
$63$	−2.81238	−0.354327
$64$	0	0
$65$	−1.53346	−0.190202
$66$	0	0
$67$	13.3442	1.63026	0.815130	−	0.579279i	$-0.196666\pi$
$67$	13.3442	1.63026	0.815130	+	0.579279i	$0.196666\pi$
$68$	0	0
$69$	−5.62304	−0.676934
$70$	0	0
$71$	−7.99168	−0.948438	−0.474219	−	0.880407i	$-0.657269\pi$
$71$	−7.99168	−0.948438	−0.474219	+	0.880407i	$0.657269\pi$
$72$	0	0
$73$	−12.0752	−1.41329	−0.706645	−	0.707568i	$-0.749792\pi$
$73$	−12.0752	−1.41329	−0.706645	+	0.707568i	$0.749792\pi$
$74$	0	0
$75$	−11.2752	−1.30194
$76$	0	0
$77$	5.65717	0.644695
$78$	0	0
$79$	1.65340	0.186022	0.0930108	−	0.995665i	$-0.470351\pi$
$79$	1.65340	0.186022	0.0930108	+	0.995665i	$0.470351\pi$
$80$	0	0
$81$	−9.52765	−1.05863
$82$	0	0
$83$	−7.40791	−0.813124	−0.406562	−	0.913623i	$-0.633272\pi$
$83$	−7.40791	−0.813124	−0.406562	+	0.913623i	$0.633272\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	20.2240	2.16824
$88$	0	0
$89$	−8.97278	−0.951112	−0.475556	−	0.879685i	$-0.657753\pi$
$89$	−8.97278	−0.951112	−0.475556	+	0.879685i	$0.657753\pi$
$90$	0	0
$91$	2.69720	0.282743
$92$	0	0
$93$	9.39659	0.974381
$94$	0	0
$95$	4.18383	0.429251
$96$	0	0
$97$	1.10846	0.112547	0.0562736	−	0.998415i	$-0.482078\pi$
$97$	1.10846	0.112547	0.0562736	+	0.998415i	$0.482078\pi$
$98$	0	0
$99$	−15.9101	−1.59903
$100$	0	0
$101$	12.3238	1.22626	0.613130	−	0.789982i	$-0.289910\pi$
$101$	12.3238	1.22626	0.613130	+	0.789982i	$0.289910\pi$
$102$	0	0
$103$	−1.31926	−0.129990	−0.0649951	−	0.997886i	$-0.520703\pi$
$103$	−1.31926	−0.129990	−0.0649951	+	0.997886i	$0.520703\pi$
$104$	0	0
$105$	−1.37068	−0.133765
$106$	0	0
$107$	−2.94798	−0.284992	−0.142496	−	0.989795i	$-0.545513\pi$
$107$	−2.94798	−0.284992	−0.142496	+	0.989795i	$0.545513\pi$
$108$	0	0
$109$	2.18034	0.208838	0.104419	−	0.994533i	$-0.466702\pi$
$109$	2.18034	0.208838	0.104419	+	0.994533i	$0.466702\pi$
$110$	0	0
$111$	18.5309	1.75887
$112$	0	0
$113$	−13.1626	−1.23823	−0.619115	−	0.785300i	$-0.712509\pi$
$113$	−13.1626	−1.23823	−0.619115	+	0.785300i	$0.712509\pi$
$114$	0	0
$115$	−1.32603	−0.123653
$116$	0	0
$117$	−7.58555	−0.701284
$118$	0	0
$119$	0	0
$120$	0	0
$121$	21.0036	1.90942
$122$	0	0
$123$	−17.5399	−1.58152
$124$	0	0
$125$	−5.50160	−0.492078
$126$	0	0
$127$	−17.7029	−1.57088	−0.785439	−	0.618939i	$-0.787563\pi$
$127$	−17.7029	−1.57088	−0.785439	+	0.618939i	$0.787563\pi$
$128$	0	0
$129$	−29.1408	−2.56570
$130$	0	0
$131$	0.811228	0.0708773	0.0354386	−	0.999372i	$-0.488717\pi$
$131$	0.811228	0.0708773	0.0354386	+	0.999372i	$0.488717\pi$
$132$	0	0
$133$	−7.35893	−0.638100
$134$	0	0
$135$	−0.257165	−0.0221332
$136$	0	0
$137$	−15.2387	−1.30193	−0.650967	−	0.759106i	$-0.725636\pi$
$137$	−15.2387	−1.30193	−0.650967	+	0.759106i	$0.725636\pi$
$138$	0	0
$139$	−0.699928	−0.0593671	−0.0296836	−	0.999559i	$-0.509450\pi$
$139$	−0.699928	−0.0593671	−0.0296836	+	0.999559i	$0.509450\pi$
$140$	0	0
$141$	−21.7016	−1.82761
$142$	0	0
$143$	15.2585	1.27598
$144$	0	0
$145$	4.76923	0.396063
$146$	0	0
$147$	2.41089	0.198847
$148$	0	0
$149$	0.115394	0.00945346	0.00472673	−	0.999989i	$-0.498495\pi$
$149$	0.115394	0.00945346	0.00472673	+	0.999989i	$0.498495\pi$
$150$	0	0
$151$	−17.3722	−1.41373	−0.706866	−	0.707348i	$-0.749891\pi$
$151$	−17.3722	−1.41373	−0.706866	+	0.707348i	$0.749891\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.21591	0.177986
$156$	0	0
$157$	−8.96992	−0.715877	−0.357939	−	0.933745i	$-0.616520\pi$
$157$	−8.96992	−0.715877	−0.357939	+	0.933745i	$0.616520\pi$
$158$	0	0
$159$	−14.7301	−1.16817
$160$	0	0
$161$	2.33235	0.183815
$162$	0	0
$163$	8.29783	0.649936	0.324968	−	0.945725i	$-0.394646\pi$
$163$	8.29783	0.649936	0.324968	+	0.945725i	$0.394646\pi$
$164$	0	0
$165$	−7.75418	−0.603662
$166$	0	0
$167$	−15.3437	−1.18733	−0.593664	−	0.804713i	$-0.702319\pi$
$167$	−15.3437	−1.18733	−0.593664	+	0.804713i	$0.702319\pi$
$168$	0	0
$169$	−5.72513	−0.440394
$170$	0	0
$171$	20.6961	1.58267
$172$	0	0
$173$	2.41676	0.183743	0.0918716	−	0.995771i	$-0.470715\pi$
$173$	2.41676	0.183743	0.0918716	+	0.995771i	$0.470715\pi$
$174$	0	0
$175$	4.67677	0.353530
$176$	0	0
$177$	3.75306	0.282097
$178$	0	0
$179$	4.96414	0.371037	0.185519	−	0.982641i	$-0.440603\pi$
$179$	4.96414	0.371037	0.185519	+	0.982641i	$0.440603\pi$
$180$	0	0
$181$	15.5483	1.15569	0.577847	−	0.816145i	$-0.303893\pi$
$181$	15.5483	1.15569	0.577847	+	0.816145i	$0.303893\pi$
$182$	0	0
$183$	−4.13438	−0.305623
$184$	0	0
$185$	4.36996	0.321286
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0.452327	0.0329019
$190$	0	0
$191$	−9.35478	−0.676888	−0.338444	−	0.940986i	$-0.609901\pi$
$191$	−9.35478	−0.676888	−0.338444	+	0.940986i	$0.609901\pi$
$192$	0	0
$193$	−5.41662	−0.389897	−0.194948	−	0.980813i	$-0.562454\pi$
$193$	−5.41662	−0.389897	−0.194948	+	0.980813i	$0.562454\pi$
$194$	0	0
$195$	−3.69700	−0.264747
$196$	0	0
$197$	−0.575238	−0.0409840	−0.0204920	−	0.999790i	$-0.506523\pi$
$197$	−0.575238	−0.0409840	−0.0204920	+	0.999790i	$0.506523\pi$
$198$	0	0
$199$	−22.7235	−1.61083	−0.805415	−	0.592712i	$-0.798057\pi$
$199$	−22.7235	−1.61083	−0.805415	+	0.592712i	$0.798057\pi$
$200$	0	0
$201$	32.1715	2.26920
$202$	0	0
$203$	−8.38860	−0.588764
$204$	0	0
$205$	−4.13628	−0.288890
$206$	0	0
$207$	−6.55946	−0.455914
$208$	0	0
$209$	−41.6307	−2.87966
$210$	0	0
$211$	−6.16118	−0.424153	−0.212077	−	0.977253i	$-0.568023\pi$
$211$	−6.16118	−0.424153	−0.212077	+	0.977253i	$0.568023\pi$
$212$	0	0
$213$	−19.2670	−1.32016
$214$	0	0
$215$	−6.87200	−0.468667
$216$	0	0
$217$	−3.89756	−0.264584
$218$	0	0
$219$	−29.1118	−1.96720
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−7.21427	−0.483103	−0.241552	−	0.970388i	$-0.577656\pi$
$223$	−7.21427	−0.483103	−0.241552	+	0.970388i	$0.577656\pi$
$224$	0	0
$225$	−13.1528	−0.876857
$226$	0	0
$227$	15.1440	1.00514	0.502571	−	0.864536i	$-0.332387\pi$
$227$	15.1440	1.00514	0.502571	+	0.864536i	$0.332387\pi$
$228$	0	0
$229$	−0.443837	−0.0293296	−0.0146648	−	0.999892i	$-0.504668\pi$
$229$	−0.443837	−0.0293296	−0.0146648	+	0.999892i	$0.504668\pi$
$230$	0	0
$231$	13.6388	0.897368
$232$	0	0
$233$	−1.82426	−0.119511	−0.0597555	−	0.998213i	$-0.519032\pi$
$233$	−1.82426	−0.119511	−0.0597555	+	0.998213i	$0.519032\pi$
$234$	0	0
$235$	−5.11770	−0.333842
$236$	0	0
$237$	3.98615	0.258928
$238$	0	0
$239$	−21.2627	−1.37537	−0.687684	−	0.726010i	$-0.741373\pi$
$239$	−21.2627	−1.37537	−0.687684	+	0.726010i	$0.741373\pi$
$240$	0	0
$241$	13.3876	0.862372	0.431186	−	0.902263i	$-0.358095\pi$
$241$	13.3876	0.862372	0.431186	+	0.902263i	$0.358095\pi$
$242$	0	0
$243$	−21.6131	−1.38648
$244$	0	0
$245$	0.568538	0.0363225
$246$	0	0
$247$	−19.8485	−1.26293
$248$	0	0
$249$	−17.8596	−1.13181
$250$	0	0
$251$	23.4979	1.48317	0.741587	−	0.670857i	$-0.234074\pi$
$251$	23.4979	1.48317	0.741587	+	0.670857i	$0.234074\pi$
$252$	0	0
$253$	13.1945	0.829532
$254$	0	0
$255$	0	0
$256$	0	0
$257$	11.2174	0.699724	0.349862	−	0.936801i	$-0.386228\pi$
$257$	11.2174	0.699724	0.349862	+	0.936801i	$0.386228\pi$
$258$	0	0
$259$	−7.68633	−0.477605
$260$	0	0
$261$	23.5919	1.46030
$262$	0	0
$263$	−31.0027	−1.91171	−0.955856	−	0.293837i	$-0.905068\pi$
$263$	−31.0027	−1.91171	−0.955856	+	0.293837i	$0.905068\pi$
$264$	0	0
$265$	−3.47366	−0.213385
$266$	0	0
$267$	−21.6324	−1.32388
$268$	0	0
$269$	−0.614893	−0.0374907	−0.0187453	−	0.999824i	$-0.505967\pi$
$269$	−0.614893	−0.0374907	−0.0187453	+	0.999824i	$0.505967\pi$
$270$	0	0
$271$	28.3262	1.72070	0.860348	−	0.509708i	$-0.170246\pi$
$271$	28.3262	1.72070	0.860348	+	0.509708i	$0.170246\pi$
$272$	0	0
$273$	6.50264	0.393558
$274$	0	0
$275$	26.4573	1.59543
$276$	0	0
$277$	13.4763	0.809714	0.404857	−	0.914380i	$-0.367321\pi$
$277$	13.4763	0.809714	0.404857	+	0.914380i	$0.367321\pi$
$278$	0	0
$279$	10.9614	0.656244
$280$	0	0
$281$	31.7130	1.89184	0.945920	−	0.324400i	$-0.105163\pi$
$281$	31.7130	1.89184	0.945920	+	0.324400i	$0.105163\pi$
$282$	0	0
$283$	6.94253	0.412690	0.206345	−	0.978479i	$-0.433843\pi$
$283$	6.94253	0.412690	0.206345	+	0.978479i	$0.433843\pi$
$284$	0	0
$285$	10.0867	0.597487
$286$	0	0
$287$	7.27530	0.429447
$288$	0	0
$289$	0	0
$290$	0	0
$291$	2.67238	0.156657
$292$	0	0
$293$	−33.1190	−1.93483	−0.967417	−	0.253190i	$-0.918520\pi$
$293$	−33.1190	−1.93483	−0.967417	+	0.253190i	$0.918520\pi$
$294$	0	0
$295$	0.885050	0.0515296
$296$	0	0
$297$	2.55889	0.148482
$298$	0	0
$299$	6.29081	0.363807
$300$	0	0
$301$	12.0872	0.696692
$302$	0	0
$303$	29.7112	1.70686
$304$	0	0
$305$	−0.974974	−0.0558268
$306$	0	0
$307$	19.4377	1.10937	0.554685	−	0.832061i	$-0.312839\pi$
$307$	19.4377	1.10937	0.554685	+	0.832061i	$0.312839\pi$
$308$	0	0
$309$	−3.18058	−0.180937
$310$	0	0
$311$	19.6306	1.11315	0.556575	−	0.830798i	$-0.312115\pi$
$311$	19.6306	1.11315	0.556575	+	0.830798i	$0.312115\pi$
$312$	0	0
$313$	−0.289182	−0.0163455	−0.00817276	−	0.999967i	$-0.502602\pi$
$313$	−0.289182	−0.0163455	−0.00817276	+	0.999967i	$0.502602\pi$
$314$	0	0
$315$	−1.59894	−0.0900903
$316$	0	0
$317$	−9.06811	−0.509316	−0.254658	−	0.967031i	$-0.581963\pi$
$317$	−9.06811	−0.509316	−0.254658	+	0.967031i	$0.581963\pi$
$318$	0	0
$319$	−47.4558	−2.65701
$320$	0	0
$321$	−7.10726	−0.396689
$322$	0	0
$323$	0	0
$324$	0	0
$325$	12.6142	0.699708
$326$	0	0
$327$	5.25655	0.290688
$328$	0	0
$329$	9.00152	0.496270
$330$	0	0
$331$	−34.9745	−1.92237	−0.961185	−	0.275904i	$-0.911023\pi$
$331$	−34.9745	−1.92237	−0.961185	+	0.275904i	$0.911023\pi$
$332$	0	0
$333$	21.6169	1.18460
$334$	0	0
$335$	7.58670	0.414506
$336$	0	0
$337$	−0.462674	−0.0252034	−0.0126017	−	0.999921i	$-0.504011\pi$
$337$	−0.462674	−0.0252034	−0.0126017	+	0.999921i	$0.504011\pi$
$338$	0	0
$339$	−31.7335	−1.72353
$340$	0	0
$341$	−22.0492	−1.19403
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−3.19691	−0.172116
$346$	0	0
$347$	6.26754	0.336459	0.168230	−	0.985748i	$-0.446195\pi$
$347$	6.26754	0.336459	0.168230	+	0.985748i	$0.446195\pi$
$348$	0	0
$349$	23.9591	1.28250	0.641250	−	0.767332i	$-0.278416\pi$
$349$	23.9591	1.28250	0.641250	+	0.767332i	$0.278416\pi$
$350$	0	0
$351$	1.22001	0.0651196
$352$	0	0
$353$	28.6998	1.52753	0.763767	−	0.645492i	$-0.223348\pi$
$353$	28.6998	1.52753	0.763767	+	0.645492i	$0.223348\pi$
$354$	0	0
$355$	−4.54357	−0.241148
$356$	0	0
$357$	0	0
$358$	0	0
$359$	22.2307	1.17329	0.586645	−	0.809845i	$-0.300449\pi$
$359$	22.2307	1.17329	0.586645	+	0.809845i	$0.300449\pi$
$360$	0	0
$361$	35.1538	1.85020
$362$	0	0
$363$	50.6374	2.65777
$364$	0	0
$365$	−6.86518	−0.359340
$366$	0	0
$367$	−24.8200	−1.29559	−0.647796	−	0.761814i	$-0.724309\pi$
$367$	−24.8200	−1.29559	−0.647796	+	0.761814i	$0.724309\pi$
$368$	0	0
$369$	−20.4609	−1.06515
$370$	0	0
$371$	6.10982	0.317206
$372$	0	0
$373$	−9.54940	−0.494449	−0.247225	−	0.968958i	$-0.579519\pi$
$373$	−9.54940	−0.494449	−0.247225	+	0.968958i	$0.579519\pi$
$374$	0	0
$375$	−13.2638	−0.684937
$376$	0	0
$377$	−22.6257	−1.16528
$378$	0	0
$379$	29.9899	1.54048	0.770238	−	0.637756i	$-0.220137\pi$
$379$	29.9899	1.54048	0.770238	+	0.637756i	$0.220137\pi$
$380$	0	0
$381$	−42.6797	−2.18655
$382$	0	0
$383$	−22.6937	−1.15959	−0.579796	−	0.814761i	$-0.696868\pi$
$383$	−22.6937	−1.15959	−0.579796	+	0.814761i	$0.696868\pi$
$384$	0	0
$385$	3.21632	0.163919
$386$	0	0
$387$	−33.9937	−1.72800
$388$	0	0
$389$	−2.53996	−0.128781	−0.0643904	−	0.997925i	$-0.520510\pi$
$389$	−2.53996	−0.128781	−0.0643904	+	0.997925i	$0.520510\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	1.95578	0.0986560
$394$	0	0
$395$	0.940017	0.0472974
$396$	0	0
$397$	22.1188	1.11011	0.555056	−	0.831813i	$-0.312697\pi$
$397$	22.1188	1.11011	0.555056	+	0.831813i	$0.312697\pi$
$398$	0	0
$399$	−17.7415	−0.888188
$400$	0	0
$401$	16.5121	0.824576	0.412288	−	0.911054i	$-0.364730\pi$
$401$	16.5121	0.824576	0.412288	+	0.911054i	$0.364730\pi$
$402$	0	0
$403$	−10.5125	−0.523665
$404$	0	0
$405$	−5.41683	−0.269164
$406$	0	0
$407$	−43.4829	−2.15537
$408$	0	0
$409$	10.0076	0.494844	0.247422	−	0.968908i	$-0.420417\pi$
$409$	10.0076	0.494844	0.247422	+	0.968908i	$0.420417\pi$
$410$	0	0
$411$	−36.7389	−1.81220
$412$	0	0
$413$	−1.55671	−0.0766009
$414$	0	0
$415$	−4.21168	−0.206743
$416$	0	0
$417$	−1.68745	−0.0826347
$418$	0	0
$419$	32.3557	1.58068	0.790341	−	0.612668i	$-0.209904\pi$
$419$	32.3557	1.58068	0.790341	+	0.612668i	$0.209904\pi$
$420$	0	0
$421$	25.6663	1.25090	0.625450	−	0.780264i	$-0.284915\pi$
$421$	25.6663	1.25090	0.625450	+	0.780264i	$0.284915\pi$
$422$	0	0
$423$	−25.3157	−1.23089
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1.71488	0.0829889
$428$	0	0
$429$	36.7866	1.77607
$430$	0	0
$431$	−14.6239	−0.704407	−0.352203	−	0.935923i	$-0.614567\pi$
$431$	−14.6239	−0.704407	−0.352203	+	0.935923i	$0.614567\pi$
$432$	0	0
$433$	−10.9350	−0.525501	−0.262751	−	0.964864i	$-0.584630\pi$
$433$	−10.9350	−0.525501	−0.262751	+	0.964864i	$0.584630\pi$
$434$	0	0
$435$	11.4981	0.551291
$436$	0	0
$437$	−17.1636	−0.821046
$438$	0	0
$439$	17.6890	0.844251	0.422126	−	0.906537i	$-0.361284\pi$
$439$	17.6890	0.844251	0.422126	+	0.906537i	$0.361284\pi$
$440$	0	0
$441$	2.81238	0.133923
$442$	0	0
$443$	9.31621	0.442626	0.221313	−	0.975203i	$-0.428966\pi$
$443$	9.31621	0.442626	0.221313	+	0.975203i	$0.428966\pi$
$444$	0	0
$445$	−5.10136	−0.241828
$446$	0	0
$447$	0.278203	0.0131585
$448$	0	0
$449$	2.16217	0.102039	0.0510196	−	0.998698i	$-0.483753\pi$
$449$	2.16217	0.102039	0.0510196	+	0.998698i	$0.483753\pi$
$450$	0	0
$451$	41.1576	1.93804
$452$	0	0
$453$	−41.8825	−1.96781
$454$	0	0
$455$	1.53346	0.0718896
$456$	0	0
$457$	2.45984	0.115066	0.0575331	−	0.998344i	$-0.481677\pi$
$457$	2.45984	0.115066	0.0575331	+	0.998344i	$0.481677\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0.117028	0.00545054	0.00272527	−	0.999996i	$-0.499133\pi$
$461$	0.117028	0.00545054	0.00272527	+	0.999996i	$0.499133\pi$
$462$	0	0
$463$	−10.0103	−0.465220	−0.232610	−	0.972570i	$-0.574727\pi$
$463$	−10.0103	−0.465220	−0.232610	+	0.972570i	$0.574727\pi$
$464$	0	0
$465$	5.34232	0.247744
$466$	0	0
$467$	22.8895	1.05920	0.529599	−	0.848248i	$-0.322342\pi$
$467$	22.8895	1.05920	0.529599	+	0.848248i	$0.322342\pi$
$468$	0	0
$469$	−13.3442	−0.616180
$470$	0	0
$471$	−21.6255	−0.996449
$472$	0	0
$473$	68.3791	3.14408
$474$	0	0
$475$	−34.4160	−1.57911
$476$	0	0
$477$	−17.1831	−0.786762
$478$	0	0
$479$	28.0599	1.28209	0.641046	−	0.767503i	$-0.278501\pi$
$479$	28.0599	1.28209	0.641046	+	0.767503i	$0.278501\pi$
$480$	0	0
$481$	−20.7315	−0.945277
$482$	0	0
$483$	5.62304	0.255857
$484$	0	0
$485$	0.630202	0.0286160
$486$	0	0
$487$	31.6111	1.43244	0.716218	−	0.697877i	$-0.245872\pi$
$487$	31.6111	1.43244	0.716218	+	0.697877i	$0.245872\pi$
$488$	0	0
$489$	20.0051	0.904664
$490$	0	0
$491$	−2.98626	−0.134768	−0.0673841	−	0.997727i	$-0.521465\pi$
$491$	−2.98626	−0.134768	−0.0673841	+	0.997727i	$0.521465\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−9.04551	−0.406565
$496$	0	0
$497$	7.99168	0.358476
$498$	0	0
$499$	23.1244	1.03519	0.517596	−	0.855625i	$-0.326827\pi$
$499$	23.1244	1.03519	0.517596	+	0.855625i	$0.326827\pi$
$500$	0	0
$501$	−36.9918	−1.65267
$502$	0	0
$503$	−27.9544	−1.24643	−0.623213	−	0.782052i	$-0.714173\pi$
$503$	−27.9544	−1.24643	−0.623213	+	0.782052i	$0.714173\pi$
$504$	0	0
$505$	7.00652	0.311786
$506$	0	0
$507$	−13.8026	−0.612997
$508$	0	0
$509$	−43.3280	−1.92048	−0.960239	−	0.279178i	$-0.909938\pi$
$509$	−43.3280	−1.92048	−0.960239	+	0.279178i	$0.909938\pi$
$510$	0	0
$511$	12.0752	0.534173
$512$	0	0
$513$	−3.32864	−0.146963
$514$	0	0
$515$	−0.750047	−0.0330510
$516$	0	0
$517$	50.9231	2.23960
$518$	0	0
$519$	5.82655	0.255757
$520$	0	0
$521$	34.8491	1.52677	0.763384	−	0.645945i	$-0.223537\pi$
$521$	34.8491	1.52677	0.763384	+	0.645945i	$0.223537\pi$
$522$	0	0
$523$	−13.2675	−0.580146	−0.290073	−	0.957005i	$-0.593680\pi$
$523$	−13.2675	−0.580146	−0.290073	+	0.957005i	$0.593680\pi$
$524$	0	0
$525$	11.2752	0.492088
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−17.5601	−0.763484
$530$	0	0
$531$	4.37807	0.189992
$532$	0	0
$533$	19.6229	0.849963
$534$	0	0
$535$	−1.67604	−0.0724615
$536$	0	0
$537$	11.9680	0.516457
$538$	0	0
$539$	−5.65717	−0.243672
$540$	0	0
$541$	23.2078	0.997782	0.498891	−	0.866665i	$-0.333741\pi$
$541$	23.2078	0.997782	0.498891	+	0.866665i	$0.333741\pi$
$542$	0	0
$543$	37.4852	1.60864
$544$	0	0
$545$	1.23960	0.0530988
$546$	0	0
$547$	14.4698	0.618685	0.309343	−	0.950951i	$-0.399891\pi$
$547$	14.4698	0.618685	0.309343	+	0.950951i	$0.399891\pi$
$548$	0	0
$549$	−4.82290	−0.205836
$550$	0	0
$551$	61.7311	2.62983
$552$	0	0
$553$	−1.65340	−0.0703095
$554$	0	0
$555$	10.5355	0.447207
$556$	0	0
$557$	−13.6049	−0.576458	−0.288229	−	0.957562i	$-0.593066\pi$
$557$	−13.6049	−0.576458	−0.288229	+	0.957562i	$0.593066\pi$
$558$	0	0
$559$	32.6014	1.37889
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−20.8321	−0.877967	−0.438984	−	0.898495i	$-0.644661\pi$
$563$	−20.8321	−0.877967	−0.438984	+	0.898495i	$0.644661\pi$
$564$	0	0
$565$	−7.48342	−0.314830
$566$	0	0
$567$	9.52765	0.400124
$568$	0	0
$569$	−24.1604	−1.01285	−0.506427	−	0.862283i	$-0.669034\pi$
$569$	−24.1604	−1.01285	−0.506427	+	0.862283i	$0.669034\pi$
$570$	0	0
$571$	3.58514	0.150033	0.0750167	−	0.997182i	$-0.476099\pi$
$571$	3.58514	0.150033	0.0750167	+	0.997182i	$0.476099\pi$
$572$	0	0
$573$	−22.5533	−0.942179
$574$	0	0
$575$	10.9079	0.454889
$576$	0	0
$577$	−5.25905	−0.218937	−0.109469	−	0.993990i	$-0.534915\pi$
$577$	−5.25905	−0.218937	−0.109469	+	0.993990i	$0.534915\pi$
$578$	0	0
$579$	−13.0589	−0.542708
$580$	0	0
$581$	7.40791	0.307332
$582$	0	0
$583$	34.5643	1.43151
$584$	0	0
$585$	−4.31267	−0.178307
$586$	0	0
$587$	−46.1592	−1.90519	−0.952596	−	0.304238i	$-0.901598\pi$
$587$	−46.1592	−1.90519	−0.952596	+	0.304238i	$0.901598\pi$
$588$	0	0
$589$	28.6819	1.18182
$590$	0	0
$591$	−1.38683	−0.0570468
$592$	0	0
$593$	−12.9389	−0.531335	−0.265668	−	0.964065i	$-0.585592\pi$
$593$	−12.9389	−0.531335	−0.265668	+	0.964065i	$0.585592\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−54.7839	−2.24216
$598$	0	0
$599$	−10.1670	−0.415414	−0.207707	−	0.978191i	$-0.566600\pi$
$599$	−10.1670	−0.415414	−0.207707	+	0.978191i	$0.566600\pi$
$600$	0	0
$601$	−15.0848	−0.615322	−0.307661	−	0.951496i	$-0.599546\pi$
$601$	−15.0848	−0.615322	−0.307661	+	0.951496i	$0.599546\pi$
$602$	0	0
$603$	37.5291	1.52830
$604$	0	0
$605$	11.9413	0.485485
$606$	0	0
$607$	1.73230	0.0703121	0.0351560	−	0.999382i	$-0.488807\pi$
$607$	1.73230	0.0703121	0.0351560	+	0.999382i	$0.488807\pi$
$608$	0	0
$609$	−20.2240	−0.819517
$610$	0	0
$611$	24.2789	0.982218
$612$	0	0
$613$	−31.7087	−1.28070	−0.640351	−	0.768083i	$-0.721211\pi$
$613$	−31.7087	−1.28070	−0.640351	+	0.768083i	$0.721211\pi$
$614$	0	0
$615$	−9.97211	−0.402114
$616$	0	0
$617$	−31.9478	−1.28617	−0.643085	−	0.765795i	$-0.722346\pi$
$617$	−31.9478	−1.28617	−0.643085	+	0.765795i	$0.722346\pi$
$618$	0	0
$619$	40.1301	1.61297	0.806483	−	0.591257i	$-0.201368\pi$
$619$	40.1301	1.61297	0.806483	+	0.591257i	$0.201368\pi$
$620$	0	0
$621$	1.05498	0.0423351
$622$	0	0
$623$	8.97278	0.359487
$624$	0	0
$625$	20.2560	0.810238
$626$	0	0
$627$	−100.367	−4.00827
$628$	0	0
$629$	0	0
$630$	0	0
$631$	5.65656	0.225184	0.112592	−	0.993641i	$-0.464085\pi$
$631$	5.65656	0.225184	0.112592	+	0.993641i	$0.464085\pi$
$632$	0	0
$633$	−14.8539	−0.590390
$634$	0	0
$635$	−10.0648	−0.399408
$636$	0	0
$637$	−2.69720	−0.106867
$638$	0	0
$639$	−22.4757	−0.889123
$640$	0	0
$641$	6.39021	0.252398	0.126199	−	0.992005i	$-0.459722\pi$
$641$	6.39021	0.252398	0.126199	+	0.992005i	$0.459722\pi$
$642$	0	0
$643$	−19.1196	−0.754004	−0.377002	−	0.926212i	$-0.623045\pi$
$643$	−19.1196	−0.754004	−0.377002	+	0.926212i	$0.623045\pi$
$644$	0	0
$645$	−16.5676	−0.652350
$646$	0	0
$647$	32.3455	1.27163	0.635817	−	0.771840i	$-0.280663\pi$
$647$	32.3455	1.27163	0.635817	+	0.771840i	$0.280663\pi$
$648$	0	0
$649$	−8.80660	−0.345689
$650$	0	0
$651$	−9.39659	−0.368281
$652$	0	0
$653$	3.03022	0.118582	0.0592909	−	0.998241i	$-0.481116\pi$
$653$	3.03022	0.118582	0.0592909	+	0.998241i	$0.481116\pi$
$654$	0	0
$655$	0.461213	0.0180211
$656$	0	0
$657$	−33.9599	−1.32490
$658$	0	0
$659$	39.0093	1.51959	0.759793	−	0.650166i	$-0.225300\pi$
$659$	39.0093	1.51959	0.759793	+	0.650166i	$0.225300\pi$
$660$	0	0
$661$	−6.96595	−0.270944	−0.135472	−	0.990781i	$-0.543255\pi$
$661$	−6.96595	−0.270944	−0.135472	+	0.990781i	$0.543255\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−4.18383	−0.162242
$666$	0	0
$667$	−19.5652	−0.757566
$668$	0	0
$669$	−17.3928	−0.672445
$670$	0	0
$671$	9.70138	0.374517
$672$	0	0
$673$	−13.7759	−0.531020	−0.265510	−	0.964108i	$-0.585540\pi$
$673$	−13.7759	−0.531020	−0.265510	+	0.964108i	$0.585540\pi$
$674$	0	0
$675$	2.11543	0.0814228
$676$	0	0
$677$	47.6413	1.83100	0.915502	−	0.402313i	$-0.131794\pi$
$677$	47.6413	1.83100	0.915502	+	0.402313i	$0.131794\pi$
$678$	0	0
$679$	−1.10846	−0.0425388
$680$	0	0
$681$	36.5105	1.39909
$682$	0	0
$683$	24.6741	0.944130	0.472065	−	0.881564i	$-0.343509\pi$
$683$	24.6741	0.944130	0.472065	+	0.881564i	$0.343509\pi$
$684$	0	0
$685$	−8.66380	−0.331027
$686$	0	0
$687$	−1.07004	−0.0408246
$688$	0	0
$689$	16.4794	0.627814
$690$	0	0
$691$	−43.0484	−1.63764	−0.818820	−	0.574051i	$-0.805371\pi$
$691$	−43.0484	−1.63764	−0.818820	+	0.574051i	$0.805371\pi$
$692$	0	0
$693$	15.9101	0.604376
$694$	0	0
$695$	−0.397935	−0.0150946
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−4.39808	−0.166351
$700$	0	0
$701$	−36.7893	−1.38951	−0.694757	−	0.719244i	$-0.744488\pi$
$701$	−36.7893	−1.38951	−0.694757	+	0.719244i	$0.744488\pi$
$702$	0	0
$703$	56.5631	2.13332
$704$	0	0
$705$	−12.3382	−0.464684
$706$	0	0
$707$	−12.3238	−0.463483
$708$	0	0
$709$	−23.8175	−0.894484	−0.447242	−	0.894413i	$-0.647594\pi$
$709$	−23.8175	−0.894484	−0.447242	+	0.894413i	$0.647594\pi$
$710$	0	0
$711$	4.64998	0.174388
$712$	0	0
$713$	−9.09049	−0.340441
$714$	0	0
$715$	8.67504	0.324428
$716$	0	0
$717$	−51.2619	−1.91441
$718$	0	0
$719$	26.9059	1.00342	0.501710	−	0.865036i	$-0.332704\pi$
$719$	26.9059	1.00342	0.501710	+	0.865036i	$0.332704\pi$
$720$	0	0
$721$	1.31926	0.0491317
$722$	0	0
$723$	32.2760	1.20036
$724$	0	0
$725$	−39.2315	−1.45702
$726$	0	0
$727$	11.9924	0.444773	0.222387	−	0.974959i	$-0.428615\pi$
$727$	11.9924	0.444773	0.222387	+	0.974959i	$0.428615\pi$
$728$	0	0
$729$	−23.5239	−0.871255
$730$	0	0
$731$	0	0
$732$	0	0
$733$	2.17406	0.0803009	0.0401505	−	0.999194i	$-0.487216\pi$
$733$	2.17406	0.0803009	0.0401505	+	0.999194i	$0.487216\pi$
$734$	0	0
$735$	1.37068	0.0505583
$736$	0	0
$737$	−75.4907	−2.78074
$738$	0	0
$739$	−17.3808	−0.639363	−0.319682	−	0.947525i	$-0.603576\pi$
$739$	−17.3808	−0.639363	−0.319682	+	0.947525i	$0.603576\pi$
$740$	0	0
$741$	−47.8525	−1.75790
$742$	0	0
$743$	23.0790	0.846686	0.423343	−	0.905970i	$-0.360857\pi$
$743$	23.0790	0.846686	0.423343	+	0.905970i	$0.360857\pi$
$744$	0	0
$745$	0.0656059	0.00240362
$746$	0	0
$747$	−20.8339	−0.762271
$748$	0	0
$749$	2.94798	0.107717
$750$	0	0
$751$	−46.9497	−1.71322	−0.856609	−	0.515966i	$-0.827433\pi$
$751$	−46.9497	−1.71322	−0.856609	+	0.515966i	$0.827433\pi$
$752$	0	0
$753$	56.6508	2.06447
$754$	0	0
$755$	−9.87676	−0.359452
$756$	0	0
$757$	2.52126	0.0916366	0.0458183	−	0.998950i	$-0.485410\pi$
$757$	2.52126	0.0916366	0.0458183	+	0.998950i	$0.485410\pi$
$758$	0	0
$759$	31.8105	1.15465
$760$	0	0
$761$	5.47670	0.198530	0.0992650	−	0.995061i	$-0.468351\pi$
$761$	5.47670	0.198530	0.0992650	+	0.995061i	$0.468351\pi$
$762$	0	0
$763$	−2.18034	−0.0789335
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−4.19876	−0.151609
$768$	0	0
$769$	54.0302	1.94838	0.974189	−	0.225732i	$-0.0724774\pi$
$769$	54.0302	1.94838	0.974189	+	0.225732i	$0.0724774\pi$
$770$	0	0
$771$	27.0440	0.973965
$772$	0	0
$773$	−30.1384	−1.08400	−0.542002	−	0.840377i	$-0.682333\pi$
$773$	−30.1384	−1.08400	−0.542002	+	0.840377i	$0.682333\pi$
$774$	0	0
$775$	−18.2280	−0.654769
$776$	0	0
$777$	−18.5309	−0.664791
$778$	0	0
$779$	−53.5384	−1.91821
$780$	0	0
$781$	45.2103	1.61775
$782$	0	0
$783$	−3.79439	−0.135600
$784$	0	0
$785$	−5.09973	−0.182017
$786$	0	0
$787$	−34.9551	−1.24602	−0.623008	−	0.782216i	$-0.714090\pi$
$787$	−34.9551	−1.24602	−0.623008	+	0.782216i	$0.714090\pi$
$788$	0	0
$789$	−74.7442	−2.66096
$790$	0	0
$791$	13.1626	0.468007
$792$	0	0
$793$	4.62537	0.164252
$794$	0	0
$795$	−8.37460	−0.297017
$796$	0	0
$797$	18.2322	0.645819	0.322909	−	0.946430i	$-0.395339\pi$
$797$	18.2322	0.645819	0.322909	+	0.946430i	$0.395339\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−25.2349	−0.891630
$802$	0	0
$803$	68.3112	2.41065
$804$	0	0
$805$	1.32603	0.0467364
$806$	0	0
$807$	−1.48244	−0.0521843
$808$	0	0
$809$	28.4383	0.999839	0.499919	−	0.866072i	$-0.333363\pi$
$809$	28.4383	0.999839	0.499919	+	0.866072i	$0.333363\pi$
$810$	0	0
$811$	−21.6351	−0.759712	−0.379856	−	0.925046i	$-0.624026\pi$
$811$	−21.6351	−0.759712	−0.379856	+	0.925046i	$0.624026\pi$
$812$	0	0
$813$	68.2913	2.39508
$814$	0	0
$815$	4.71763	0.165251
$816$	0	0
$817$	−88.9485	−3.11191
$818$	0	0
$819$	7.58555	0.265060
$820$	0	0
$821$	−25.6565	−0.895419	−0.447710	−	0.894179i	$-0.647760\pi$
$821$	−25.6565	−0.895419	−0.447710	+	0.894179i	$0.647760\pi$
$822$	0	0
$823$	36.5612	1.27444	0.637222	−	0.770680i	$-0.280083\pi$
$823$	36.5612	1.27444	0.637222	+	0.770680i	$0.280083\pi$
$824$	0	0
$825$	63.7855	2.22073
$826$	0	0
$827$	30.5146	1.06110	0.530548	−	0.847655i	$-0.321986\pi$
$827$	30.5146	1.06110	0.530548	+	0.847655i	$0.321986\pi$
$828$	0	0
$829$	25.5316	0.886750	0.443375	−	0.896336i	$-0.353781\pi$
$829$	25.5316	0.886750	0.443375	+	0.896336i	$0.353781\pi$
$830$	0	0
$831$	32.4899	1.12706
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−8.72345	−0.301887
$836$	0	0
$837$	−1.76297	−0.0609373
$838$	0	0
$839$	2.41584	0.0834041	0.0417020	−	0.999130i	$-0.486722\pi$
$839$	2.41584	0.0834041	0.0417020	+	0.999130i	$0.486722\pi$
$840$	0	0
$841$	41.3686	1.42650
$842$	0	0
$843$	76.4565	2.63330
$844$	0	0
$845$	−3.25495	−0.111974
$846$	0	0
$847$	−21.0036	−0.721693
$848$	0	0
$849$	16.7377	0.574435
$850$	0	0
$851$	−17.9272	−0.614537
$852$	0	0
$853$	3.01848	0.103351	0.0516754	−	0.998664i	$-0.483544\pi$
$853$	3.01848	0.103351	0.0516754	+	0.998664i	$0.483544\pi$
$854$	0	0
$855$	11.7665	0.402406
$856$	0	0
$857$	26.8230	0.916256	0.458128	−	0.888886i	$-0.348520\pi$
$857$	26.8230	0.916256	0.458128	+	0.888886i	$0.348520\pi$
$858$	0	0
$859$	14.9316	0.509460	0.254730	−	0.967012i	$-0.418013\pi$
$859$	14.9316	0.509460	0.254730	+	0.967012i	$0.418013\pi$
$860$	0	0
$861$	17.5399	0.597759
$862$	0	0
$863$	−18.9280	−0.644318	−0.322159	−	0.946686i	$-0.604408\pi$
$863$	−18.9280	−0.644318	−0.322159	+	0.946686i	$0.604408\pi$
$864$	0	0
$865$	1.37402	0.0467181
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−9.35354	−0.317297
$870$	0	0
$871$	−35.9921	−1.21954
$872$	0	0
$873$	3.11741	0.105509
$874$	0	0
$875$	5.50160	0.185988
$876$	0	0
$877$	28.8803	0.975218	0.487609	−	0.873062i	$-0.337869\pi$
$877$	28.8803	0.975218	0.487609	+	0.873062i	$0.337869\pi$
$878$	0	0
$879$	−79.8462	−2.69315
$880$	0	0
$881$	21.9080	0.738099	0.369050	−	0.929410i	$-0.379683\pi$
$881$	21.9080	0.738099	0.369050	+	0.929410i	$0.379683\pi$
$882$	0	0
$883$	23.8295	0.801927	0.400963	−	0.916094i	$-0.368675\pi$
$883$	23.8295	0.801927	0.400963	+	0.916094i	$0.368675\pi$
$884$	0	0
$885$	2.13376	0.0717254
$886$	0	0
$887$	6.84186	0.229727	0.114864	−	0.993381i	$-0.463357\pi$
$887$	6.84186	0.229727	0.114864	+	0.993381i	$0.463357\pi$
$888$	0	0
$889$	17.7029	0.593736
$890$	0	0
$891$	53.8996	1.80570
$892$	0	0
$893$	−66.2415	−2.21669
$894$	0	0
$895$	2.82230	0.0943391
$896$	0	0
$897$	15.1664	0.506393
$898$	0	0
$899$	32.6951	1.09044
$900$	0	0
$901$	0	0
$902$	0	0
$903$	29.1408	0.969744
$904$	0	0
$905$	8.83978	0.293844
$906$	0	0
$907$	−55.3689	−1.83849	−0.919246	−	0.393683i	$-0.871201\pi$
$907$	−55.3689	−1.83849	−0.919246	+	0.393683i	$0.871201\pi$
$908$	0	0
$909$	34.6591	1.14957
$910$	0	0
$911$	−33.9987	−1.12643	−0.563213	−	0.826312i	$-0.690435\pi$
$911$	−33.9987	−1.12643	−0.563213	+	0.826312i	$0.690435\pi$
$912$	0	0
$913$	41.9078	1.38695
$914$	0	0
$915$	−2.35055	−0.0777069
$916$	0	0
$917$	−0.811228	−0.0267891
$918$	0	0
$919$	−54.2903	−1.79087	−0.895437	−	0.445189i	$-0.853136\pi$
$919$	−54.2903	−1.79087	−0.895437	+	0.445189i	$0.853136\pi$
$920$	0	0
$921$	46.8622	1.54416
$922$	0	0
$923$	21.5551	0.709496
$924$	0	0
$925$	−35.9471	−1.18193
$926$	0	0
$927$	−3.71026	−0.121861
$928$	0	0
$929$	−29.6824	−0.973849	−0.486925	−	0.873444i	$-0.661881\pi$
$929$	−29.6824	−0.973849	−0.486925	+	0.873444i	$0.661881\pi$
$930$	0	0
$931$	7.35893	0.241179
$932$	0	0
$933$	47.3272	1.54942
$934$	0	0
$935$	0	0
$936$	0	0
$937$	37.1014	1.21205	0.606025	−	0.795446i	$-0.292763\pi$
$937$	37.1014	1.21205	0.606025	+	0.795446i	$0.292763\pi$
$938$	0	0
$939$	−0.697185	−0.0227518
$940$	0	0
$941$	6.79710	0.221579	0.110790	−	0.993844i	$-0.464662\pi$
$941$	6.79710	0.221579	0.110790	+	0.993844i	$0.464662\pi$
$942$	0	0
$943$	16.9686	0.552572
$944$	0	0
$945$	0.257165	0.00836557
$946$	0	0
$947$	−2.95594	−0.0960551	−0.0480275	−	0.998846i	$-0.515294\pi$
$947$	−2.95594	−0.0960551	−0.0480275	+	0.998846i	$0.515294\pi$
$948$	0	0
$949$	32.5691	1.05724
$950$	0	0
$951$	−21.8622	−0.708930
$952$	0	0
$953$	49.5333	1.60454	0.802270	−	0.596961i	$-0.203625\pi$
$953$	49.5333	1.60454	0.802270	+	0.596961i	$0.203625\pi$
$954$	0	0
$955$	−5.31854	−0.172104
$956$	0	0
$957$	−114.411	−3.69837
$958$	0	0
$959$	15.2387	0.492085
$960$	0	0
$961$	−15.8090	−0.509967
$962$	0	0
$963$	−8.29086	−0.267169
$964$	0	0
$965$	−3.07955	−0.0991343
$966$	0	0
$967$	51.3202	1.65035	0.825173	−	0.564880i	$-0.191077\pi$
$967$	51.3202	1.65035	0.825173	+	0.564880i	$0.191077\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−43.6740	−1.40157	−0.700783	−	0.713374i	$-0.747166\pi$
$971$	−43.6740	−1.40157	−0.700783	+	0.713374i	$0.747166\pi$
$972$	0	0
$973$	0.699928	0.0224387
$974$	0	0
$975$	30.4113	0.973942
$976$	0	0
$977$	14.5093	0.464193	0.232096	−	0.972693i	$-0.425442\pi$
$977$	14.5093	0.464193	0.232096	+	0.972693i	$0.425442\pi$
$978$	0	0
$979$	50.7605	1.62231
$980$	0	0
$981$	6.13194	0.195778
$982$	0	0
$983$	−55.2463	−1.76208	−0.881041	−	0.473039i	$-0.843157\pi$
$983$	−55.2463	−1.76208	−0.881041	+	0.473039i	$0.843157\pi$
$984$	0	0
$985$	−0.327044	−0.0104205
$986$	0	0
$987$	21.7016	0.690771
$988$	0	0
$989$	28.1915	0.896437
$990$	0	0
$991$	−2.28047	−0.0724415	−0.0362207	−	0.999344i	$-0.511532\pi$
$991$	−2.28047	−0.0724415	−0.0362207	+	0.999344i	$0.511532\pi$
$992$	0	0
$993$	−84.3195	−2.67580
$994$	0	0
$995$	−12.9192	−0.409566
$996$	0	0
$997$	16.9944	0.538217	0.269109	−	0.963110i	$-0.413271\pi$
$997$	16.9944	0.538217	0.269109	+	0.963110i	$0.413271\pi$
$998$	0	0
$999$	−3.47673	−0.109999

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8092.2.a.v.1.11	✓	12
17.16	even	2		8092.2.a.w.1.2	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8092.2.a.v.1.11	✓	12	1.1	even	1	trivial
8092.2.a.w.1.2	yes	12	17.16	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8092 = 2^{2} \cdot 7 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8092.a (trivial)

\(f(q)\)	\(=\)	\(q+2.41089 q^{3} +0.568538 q^{5} -1.00000 q^{7} +2.81238 q^{9} +O(q^{10})\)	\(q+2.41089 q^{3} +0.568538 q^{5} -1.00000 q^{7} +2.81238 q^{9} -5.65717 q^{11} -2.69720 q^{13} +1.37068 q^{15} +7.35893 q^{19} -2.41089 q^{21} -2.33235 q^{23} -4.67677 q^{25} -0.452327 q^{27} +8.38860 q^{29} +3.89756 q^{31} -13.6388 q^{33} -0.568538 q^{35} +7.68633 q^{37} -6.50264 q^{39} -7.27530 q^{41} -12.0872 q^{43} +1.59894 q^{45} -9.00152 q^{47} +1.00000 q^{49} -6.10982 q^{53} -3.21632 q^{55} +17.7415 q^{57} +1.55671 q^{59} -1.71488 q^{61} -2.81238 q^{63} -1.53346 q^{65} +13.3442 q^{67} -5.62304 q^{69} -7.99168 q^{71} -12.0752 q^{73} -11.2752 q^{75} +5.65717 q^{77} +1.65340 q^{79} -9.52765 q^{81} -7.40791 q^{83} +20.2240 q^{87} -8.97278 q^{89} +2.69720 q^{91} +9.39659 q^{93} +4.18383 q^{95} +1.10846 q^{97} -15.9101 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{7} + 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^7 + 12 * q^9	\( 12 q - 12 q^{7} + 12 q^{9} + 6 q^{11} - 12 q^{13} - 9 q^{15} - 12 q^{19} + 9 q^{23} - 6 q^{25} - 6 q^{27} + 6 q^{29} + 9 q^{31} - 24 q^{33} + 12 q^{37} + 3 q^{39} - 21 q^{41} - 3 q^{43} + 33 q^{45} - 30 q^{47} + 12 q^{49} - 27 q^{53} - 15 q^{55} + 30 q^{57} - 30 q^{59} + 3 q^{61} - 12 q^{63} + 24 q^{65} - 3 q^{67} - 3 q^{69} + 27 q^{71} - 15 q^{73} - 63 q^{75} - 6 q^{77} + 6 q^{79} - 36 q^{81} - 54 q^{83} - 36 q^{87} - 21 q^{89} + 12 q^{91} - 51 q^{93} + 18 q^{95} - 45 q^{97} + 30 q^{99}+O(q^{100}) \) 12 * q - 12 * q^7 + 12 * q^9 + 6 * q^11 - 12 * q^13 - 9 * q^15 - 12 * q^19 + 9 * q^23 - 6 * q^25 - 6 * q^27 + 6 * q^29 + 9 * q^31 - 24 * q^33 + 12 * q^37 + 3 * q^39 - 21 * q^41 - 3 * q^43 + 33 * q^45 - 30 * q^47 + 12 * q^49 - 27 * q^53 - 15 * q^55 + 30 * q^57 - 30 * q^59 + 3 * q^61 - 12 * q^63 + 24 * q^65 - 3 * q^67 - 3 * q^69 + 27 * q^71 - 15 * q^73 - 63 * q^75 - 6 * q^77 + 6 * q^79 - 36 * q^81 - 54 * q^83 - 36 * q^87 - 21 * q^89 + 12 * q^91 - 51 * q^93 + 18 * q^95 - 45 * q^97 + 30 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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