LMFDB - Embedded newform 8092.2.a.w.1.8

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			1.8
Root			$1.15276$ 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)

$f(q)$	$=$	$q+1.15276 q^{3} -2.31274 q^{5} +1.00000 q^{7} -1.67113 q^{9} +O(q^{10})$	$q+1.15276 q^{3} -2.31274 q^{5} +1.00000 q^{7} -1.67113 q^{9} +2.26723 q^{11} +1.33004 q^{13} -2.66604 q^{15} +1.19393 q^{19} +1.15276 q^{21} +0.651235 q^{23} +0.348764 q^{25} -5.38472 q^{27} +1.30761 q^{29} -0.161788 q^{31} +2.61358 q^{33} -2.31274 q^{35} -5.85195 q^{37} +1.53322 q^{39} -2.77666 q^{41} -12.0852 q^{43} +3.86490 q^{45} +8.78849 q^{47} +1.00000 q^{49} +0.105878 q^{53} -5.24351 q^{55} +1.37632 q^{57} +0.416213 q^{59} -12.0311 q^{61} -1.67113 q^{63} -3.07603 q^{65} +1.79590 q^{67} +0.750721 q^{69} +1.25386 q^{71} +5.88184 q^{73} +0.402043 q^{75} +2.26723 q^{77} +13.9211 q^{79} -1.19391 q^{81} -10.8518 q^{83} +1.50737 q^{87} +0.333536 q^{89} +1.33004 q^{91} -0.186504 q^{93} -2.76125 q^{95} -8.85840 q^{97} -3.78885 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.15276	0.665549	0.332774	−	0.943006i	$-0.392015\pi$
$3$	1.15276	0.665549	0.332774	+	0.943006i	$0.392015\pi$
$4$	0	0
$5$	−2.31274	−1.03429	−0.517144	−	0.855898i	$-0.673005\pi$
$5$	−2.31274	−1.03429	−0.517144	+	0.855898i	$0.673005\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−1.67113	−0.557045
$10$	0	0
$11$	2.26723	0.683596	0.341798	−	0.939774i	$-0.388964\pi$
$11$	2.26723	0.683596	0.341798	+	0.939774i	$0.388964\pi$
$12$	0	0
$13$	1.33004	0.368886	0.184443	−	0.982843i	$-0.440952\pi$
$13$	1.33004	0.368886	0.184443	+	0.982843i	$0.440952\pi$
$14$	0	0
$15$	−2.66604	−0.688369
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	1.19393	0.273906	0.136953	−	0.990578i	$-0.456269\pi$
$19$	1.19393	0.273906	0.136953	+	0.990578i	$0.456269\pi$
$20$	0	0
$21$	1.15276	0.251554
$22$	0	0
$23$	0.651235	0.135792	0.0678960	−	0.997692i	$-0.478371\pi$
$23$	0.651235	0.135792	0.0678960	+	0.997692i	$0.478371\pi$
$24$	0	0
$25$	0.348764	0.0697528
$26$	0	0
$27$	−5.38472	−1.03629
$28$	0	0
$29$	1.30761	0.242818	0.121409	−	0.992603i	$-0.461259\pi$
$29$	1.30761	0.242818	0.121409	+	0.992603i	$0.461259\pi$
$30$	0	0
$31$	−0.161788	−0.0290580	−0.0145290	−	0.999894i	$-0.504625\pi$
$31$	−0.161788	−0.0290580	−0.0145290	+	0.999894i	$0.504625\pi$
$32$	0	0
$33$	2.61358	0.454966
$34$	0	0
$35$	−2.31274	−0.390924
$36$	0	0
$37$	−5.85195	−0.962055	−0.481028	−	0.876705i	$-0.659736\pi$
$37$	−5.85195	−0.962055	−0.481028	+	0.876705i	$0.659736\pi$
$38$	0	0
$39$	1.53322	0.245512
$40$	0	0
$41$	−2.77666	−0.433641	−0.216821	−	0.976211i	$-0.569569\pi$
$41$	−2.77666	−0.433641	−0.216821	+	0.976211i	$0.569569\pi$
$42$	0	0
$43$	−12.0852	−1.84298	−0.921489	−	0.388404i	$-0.873027\pi$
$43$	−12.0852	−1.84298	−0.921489	+	0.388404i	$0.873027\pi$
$44$	0	0
$45$	3.86490	0.576145
$46$	0	0
$47$	8.78849	1.28193	0.640966	−	0.767569i	$-0.278534\pi$
$47$	8.78849	1.28193	0.640966	+	0.767569i	$0.278534\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0.105878	0.0145435	0.00727173	−	0.999974i	$-0.497685\pi$
$53$	0.105878	0.0145435	0.00727173	+	0.999974i	$0.497685\pi$
$54$	0	0
$55$	−5.24351	−0.707035
$56$	0	0
$57$	1.37632	0.182298
$58$	0	0
$59$	0.416213	0.0541863	0.0270932	−	0.999633i	$-0.491375\pi$
$59$	0.416213	0.0541863	0.0270932	+	0.999633i	$0.491375\pi$
$60$	0	0
$61$	−12.0311	−1.54042	−0.770212	−	0.637788i	$-0.779850\pi$
$61$	−12.0311	−1.54042	−0.770212	+	0.637788i	$0.779850\pi$
$62$	0	0
$63$	−1.67113	−0.210543
$64$	0	0
$65$	−3.07603	−0.381535
$66$	0	0
$67$	1.79590	0.219405	0.109702	−	0.993964i	$-0.465010\pi$
$67$	1.79590	0.219405	0.109702	+	0.993964i	$0.465010\pi$
$68$	0	0
$69$	0.750721	0.0903762
$70$	0	0
$71$	1.25386	0.148805	0.0744026	−	0.997228i	$-0.476295\pi$
$71$	1.25386	0.148805	0.0744026	+	0.997228i	$0.476295\pi$
$72$	0	0
$73$	5.88184	0.688418	0.344209	−	0.938893i	$-0.388147\pi$
$73$	5.88184	0.688418	0.344209	+	0.938893i	$0.388147\pi$
$74$	0	0
$75$	0.402043	0.0464239
$76$	0	0
$77$	2.26723	0.258375
$78$	0	0
$79$	13.9211	1.56624	0.783120	−	0.621870i	$-0.213627\pi$
$79$	13.9211	1.56624	0.783120	+	0.621870i	$0.213627\pi$
$80$	0	0
$81$	−1.19391	−0.132656
$82$	0	0
$83$	−10.8518	−1.19113	−0.595567	−	0.803305i	$-0.703073\pi$
$83$	−10.8518	−1.19113	−0.595567	+	0.803305i	$0.703073\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.50737	0.161607
$88$	0	0
$89$	0.333536	0.0353548	0.0176774	−	0.999844i	$-0.494373\pi$
$89$	0.333536	0.0353548	0.0176774	+	0.999844i	$0.494373\pi$
$90$	0	0
$91$	1.33004	0.139426
$92$	0	0
$93$	−0.186504	−0.0193395
$94$	0	0
$95$	−2.76125	−0.283298
$96$	0	0
$97$	−8.85840	−0.899435	−0.449717	−	0.893171i	$-0.648475\pi$
$97$	−8.85840	−0.899435	−0.449717	+	0.893171i	$0.648475\pi$
$98$	0	0
$99$	−3.78885	−0.380793
$100$	0	0
$101$	−1.44796	−0.144078	−0.0720388	−	0.997402i	$-0.522951\pi$
$101$	−1.44796	−0.144078	−0.0720388	+	0.997402i	$0.522951\pi$
$102$	0	0
$103$	2.47908	0.244271	0.122136	−	0.992513i	$-0.461026\pi$
$103$	2.47908	0.244271	0.122136	+	0.992513i	$0.461026\pi$
$104$	0	0
$105$	−2.66604	−0.260179
$106$	0	0
$107$	4.84546	0.468428	0.234214	−	0.972185i	$-0.424748\pi$
$107$	4.84546	0.468428	0.234214	+	0.972185i	$0.424748\pi$
$108$	0	0
$109$	5.26885	0.504664	0.252332	−	0.967641i	$-0.418802\pi$
$109$	5.26885	0.504664	0.252332	+	0.967641i	$0.418802\pi$
$110$	0	0
$111$	−6.74592	−0.640295
$112$	0	0
$113$	4.32117	0.406501	0.203251	−	0.979127i	$-0.434849\pi$
$113$	4.32117	0.406501	0.203251	+	0.979127i	$0.434849\pi$
$114$	0	0
$115$	−1.50614	−0.140448
$116$	0	0
$117$	−2.22267	−0.205486
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−5.85967	−0.532697
$122$	0	0
$123$	−3.20083	−0.288609
$124$	0	0
$125$	10.7571	0.962144
$126$	0	0
$127$	−0.878404	−0.0779457	−0.0389729	−	0.999240i	$-0.512409\pi$
$127$	−0.878404	−0.0779457	−0.0389729	+	0.999240i	$0.512409\pi$
$128$	0	0
$129$	−13.9314	−1.22659
$130$	0	0
$131$	4.48806	0.392124	0.196062	−	0.980592i	$-0.437185\pi$
$131$	4.48806	0.392124	0.196062	+	0.980592i	$0.437185\pi$
$132$	0	0
$133$	1.19393	0.103527
$134$	0	0
$135$	12.4534	1.07182
$136$	0	0
$137$	−10.6185	−0.907197	−0.453599	−	0.891206i	$-0.649860\pi$
$137$	−10.6185	−0.907197	−0.453599	+	0.891206i	$0.649860\pi$
$138$	0	0
$139$	−7.09684	−0.601946	−0.300973	−	0.953633i	$-0.597311\pi$
$139$	−7.09684	−0.601946	−0.300973	+	0.953633i	$0.597311\pi$
$140$	0	0
$141$	10.1311	0.853189
$142$	0	0
$143$	3.01550	0.252169
$144$	0	0
$145$	−3.02417	−0.251144
$146$	0	0
$147$	1.15276	0.0950784
$148$	0	0
$149$	−8.00104	−0.655471	−0.327735	−	0.944770i	$-0.606285\pi$
$149$	−8.00104	−0.655471	−0.327735	+	0.944770i	$0.606285\pi$
$150$	0	0
$151$	10.9842	0.893879	0.446940	−	0.894564i	$-0.352514\pi$
$151$	10.9842	0.893879	0.446940	+	0.894564i	$0.352514\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0.374174	0.0300544
$156$	0	0
$157$	−17.3607	−1.38553	−0.692766	−	0.721163i	$-0.743608\pi$
$157$	−17.3607	−1.38553	−0.692766	+	0.721163i	$0.743608\pi$
$158$	0	0
$159$	0.122052	0.00967938
$160$	0	0
$161$	0.651235	0.0513245
$162$	0	0
$163$	11.2974	0.884878	0.442439	−	0.896799i	$-0.354113\pi$
$163$	11.2974	0.884878	0.442439	+	0.896799i	$0.354113\pi$
$164$	0	0
$165$	−6.04453	−0.470566
$166$	0	0
$167$	5.91986	0.458092	0.229046	−	0.973416i	$-0.426439\pi$
$167$	5.91986	0.458092	0.229046	+	0.973416i	$0.426439\pi$
$168$	0	0
$169$	−11.2310	−0.863923
$170$	0	0
$171$	−1.99522	−0.152578
$172$	0	0
$173$	−24.5073	−1.86325	−0.931626	−	0.363418i	$-0.881610\pi$
$173$	−24.5073	−1.86325	−0.931626	+	0.363418i	$0.881610\pi$
$174$	0	0
$175$	0.348764	0.0263641
$176$	0	0
$177$	0.479796	0.0360637
$178$	0	0
$179$	−21.5119	−1.60787	−0.803936	−	0.594716i	$-0.797265\pi$
$179$	−21.5119	−1.60787	−0.803936	+	0.594716i	$0.797265\pi$
$180$	0	0
$181$	−1.14746	−0.0852898	−0.0426449	−	0.999090i	$-0.513578\pi$
$181$	−1.14746	−0.0852898	−0.0426449	+	0.999090i	$0.513578\pi$
$182$	0	0
$183$	−13.8690	−1.02523
$184$	0	0
$185$	13.5340	0.995043
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−5.38472	−0.391681
$190$	0	0
$191$	−17.1107	−1.23809	−0.619044	−	0.785356i	$-0.712480\pi$
$191$	−17.1107	−1.23809	−0.619044	+	0.785356i	$0.712480\pi$
$192$	0	0
$193$	7.43945	0.535503	0.267752	−	0.963488i	$-0.413719\pi$
$193$	7.43945	0.535503	0.267752	+	0.963488i	$0.413719\pi$
$194$	0	0
$195$	−3.54594	−0.253930
$196$	0	0
$197$	−20.9482	−1.49250	−0.746249	−	0.665667i	$-0.768147\pi$
$197$	−20.9482	−1.49250	−0.746249	+	0.665667i	$0.768147\pi$
$198$	0	0
$199$	−7.67426	−0.544014	−0.272007	−	0.962295i	$-0.587687\pi$
$199$	−7.67426	−0.544014	−0.272007	+	0.962295i	$0.587687\pi$
$200$	0	0
$201$	2.07025	0.146024
$202$	0	0
$203$	1.30761	0.0917765
$204$	0	0
$205$	6.42169	0.448510
$206$	0	0
$207$	−1.08830	−0.0756422
$208$	0	0
$209$	2.70692	0.187241
$210$	0	0
$211$	21.3551	1.47014	0.735072	−	0.677989i	$-0.237148\pi$
$211$	21.3551	1.47014	0.735072	+	0.677989i	$0.237148\pi$
$212$	0	0
$213$	1.44540	0.0990372
$214$	0	0
$215$	27.9500	1.90617
$216$	0	0
$217$	−0.161788	−0.0109829
$218$	0	0
$219$	6.78038	0.458176
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−18.0088	−1.20596	−0.602980	−	0.797756i	$-0.706020\pi$
$223$	−18.0088	−1.20596	−0.602980	+	0.797756i	$0.706020\pi$
$224$	0	0
$225$	−0.582831	−0.0388554
$226$	0	0
$227$	−5.43867	−0.360977	−0.180488	−	0.983577i	$-0.557768\pi$
$227$	−5.43867	−0.360977	−0.180488	+	0.983577i	$0.557768\pi$
$228$	0	0
$229$	1.32501	0.0875588	0.0437794	−	0.999041i	$-0.486060\pi$
$229$	1.32501	0.0875588	0.0437794	+	0.999041i	$0.486060\pi$
$230$	0	0
$231$	2.61358	0.171961
$232$	0	0
$233$	−7.20241	−0.471846	−0.235923	−	0.971772i	$-0.575811\pi$
$233$	−7.20241	−0.471846	−0.235923	+	0.971772i	$0.575811\pi$
$234$	0	0
$235$	−20.3255	−1.32589
$236$	0	0
$237$	16.0477	1.04241
$238$	0	0
$239$	−21.4894	−1.39003	−0.695017	−	0.718993i	$-0.744603\pi$
$239$	−21.4894	−1.39003	−0.695017	+	0.718993i	$0.744603\pi$
$240$	0	0
$241$	9.33934	0.601600	0.300800	−	0.953687i	$-0.402746\pi$
$241$	9.33934	0.601600	0.300800	+	0.953687i	$0.402746\pi$
$242$	0	0
$243$	14.7779	0.948000
$244$	0	0
$245$	−2.31274	−0.147756
$246$	0	0
$247$	1.58797	0.101040
$248$	0	0
$249$	−12.5095	−0.792758
$250$	0	0
$251$	13.8611	0.874905	0.437453	−	0.899241i	$-0.355881\pi$
$251$	13.8611	0.874905	0.437453	+	0.899241i	$0.355881\pi$
$252$	0	0
$253$	1.47650	0.0928268
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−21.0447	−1.31273	−0.656366	−	0.754442i	$-0.727907\pi$
$257$	−21.0447	−1.31273	−0.656366	+	0.754442i	$0.727907\pi$
$258$	0	0
$259$	−5.85195	−0.363623
$260$	0	0
$261$	−2.18520	−0.135260
$262$	0	0
$263$	−11.1208	−0.685740	−0.342870	−	0.939383i	$-0.611399\pi$
$263$	−11.1208	−0.685740	−0.342870	+	0.939383i	$0.611399\pi$
$264$	0	0
$265$	−0.244868	−0.0150421
$266$	0	0
$267$	0.384488	0.0235303
$268$	0	0
$269$	−29.5859	−1.80388	−0.901942	−	0.431858i	$-0.857858\pi$
$269$	−29.5859	−1.80388	−0.901942	+	0.431858i	$0.857858\pi$
$270$	0	0
$271$	9.42606	0.572592	0.286296	−	0.958141i	$-0.407576\pi$
$271$	9.42606	0.572592	0.286296	+	0.958141i	$0.407576\pi$
$272$	0	0
$273$	1.53322	0.0927947
$274$	0	0
$275$	0.790728	0.0476827
$276$	0	0
$277$	−31.1219	−1.86993	−0.934966	−	0.354738i	$-0.884570\pi$
$277$	−31.1219	−1.86993	−0.934966	+	0.354738i	$0.884570\pi$
$278$	0	0
$279$	0.270370	0.0161866
$280$	0	0
$281$	−10.1855	−0.607614	−0.303807	−	0.952734i	$-0.598258\pi$
$281$	−10.1855	−0.607614	−0.303807	+	0.952734i	$0.598258\pi$
$282$	0	0
$283$	3.33585	0.198296	0.0991479	−	0.995073i	$-0.468388\pi$
$283$	3.33585	0.198296	0.0991479	+	0.995073i	$0.468388\pi$
$284$	0	0
$285$	−3.18307	−0.188549
$286$	0	0
$287$	−2.77666	−0.163901
$288$	0	0
$289$	0	0
$290$	0	0
$291$	−10.2117	−0.598618
$292$	0	0
$293$	−20.0139	−1.16923	−0.584613	−	0.811313i	$-0.698753\pi$
$293$	−20.0139	−1.16923	−0.584613	+	0.811313i	$0.698753\pi$
$294$	0	0
$295$	−0.962593	−0.0560443
$296$	0	0
$297$	−12.2084	−0.708403
$298$	0	0
$299$	0.866168	0.0500918
$300$	0	0
$301$	−12.0852	−0.696580
$302$	0	0
$303$	−1.66916	−0.0958907
$304$	0	0
$305$	27.8248	1.59324
$306$	0	0
$307$	14.0693	0.802975	0.401487	−	0.915865i	$-0.368493\pi$
$307$	14.0693	0.802975	0.401487	+	0.915865i	$0.368493\pi$
$308$	0	0
$309$	2.85780	0.162574
$310$	0	0
$311$	−3.74344	−0.212271	−0.106135	−	0.994352i	$-0.533848\pi$
$311$	−3.74344	−0.212271	−0.106135	+	0.994352i	$0.533848\pi$
$312$	0	0
$313$	14.2299	0.804322	0.402161	−	0.915569i	$-0.368259\pi$
$313$	14.2299	0.804322	0.402161	+	0.915569i	$0.368259\pi$
$314$	0	0
$315$	3.86490	0.217762
$316$	0	0
$317$	−16.6723	−0.936411	−0.468205	−	0.883620i	$-0.655099\pi$
$317$	−16.6723	−0.936411	−0.468205	+	0.883620i	$0.655099\pi$
$318$	0	0
$319$	2.96466	0.165989
$320$	0	0
$321$	5.58567	0.311762
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0.463869	0.0257308
$326$	0	0
$327$	6.07374	0.335879
$328$	0	0
$329$	8.78849	0.484525
$330$	0	0
$331$	5.41643	0.297714	0.148857	−	0.988859i	$-0.452441\pi$
$331$	5.41643	0.297714	0.148857	+	0.988859i	$0.452441\pi$
$332$	0	0
$333$	9.77940	0.535908
$334$	0	0
$335$	−4.15346	−0.226928
$336$	0	0
$337$	−21.9920	−1.19798	−0.598990	−	0.800756i	$-0.704431\pi$
$337$	−21.9920	−1.19798	−0.598990	+	0.800756i	$0.704431\pi$
$338$	0	0
$339$	4.98129	0.270546
$340$	0	0
$341$	−0.366811	−0.0198640
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−1.73622	−0.0934750
$346$	0	0
$347$	18.2684	0.980699	0.490350	−	0.871526i	$-0.336869\pi$
$347$	18.2684	0.980699	0.490350	+	0.871526i	$0.336869\pi$
$348$	0	0
$349$	3.22879	0.172833	0.0864165	−	0.996259i	$-0.472458\pi$
$349$	3.22879	0.172833	0.0864165	+	0.996259i	$0.472458\pi$
$350$	0	0
$351$	−7.16188	−0.382273
$352$	0	0
$353$	−27.3584	−1.45614	−0.728070	−	0.685503i	$-0.759583\pi$
$353$	−27.3584	−1.45614	−0.728070	+	0.685503i	$0.759583\pi$
$354$	0	0
$355$	−2.89984	−0.153908
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−33.1587	−1.75005	−0.875024	−	0.484080i	$-0.839155\pi$
$359$	−33.1587	−1.75005	−0.875024	+	0.484080i	$0.839155\pi$
$360$	0	0
$361$	−17.5745	−0.924975
$362$	0	0
$363$	−6.75482	−0.354536
$364$	0	0
$365$	−13.6032	−0.712023
$366$	0	0
$367$	−9.09286	−0.474643	−0.237322	−	0.971431i	$-0.576270\pi$
$367$	−9.09286	−0.474643	−0.237322	+	0.971431i	$0.576270\pi$
$368$	0	0
$369$	4.64017	0.241558
$370$	0	0
$371$	0.105878	0.00549691
$372$	0	0
$373$	3.16381	0.163816	0.0819079	−	0.996640i	$-0.473899\pi$
$373$	3.16381	0.163816	0.0819079	+	0.996640i	$0.473899\pi$
$374$	0	0
$375$	12.4004	0.640354
$376$	0	0
$377$	1.73918	0.0895721
$378$	0	0
$379$	18.1769	0.933684	0.466842	−	0.884341i	$-0.345392\pi$
$379$	18.1769	0.933684	0.466842	+	0.884341i	$0.345392\pi$
$380$	0	0
$381$	−1.01259	−0.0518767
$382$	0	0
$383$	0.712746	0.0364196	0.0182098	−	0.999834i	$-0.494203\pi$
$383$	0.712746	0.0364196	0.0182098	+	0.999834i	$0.494203\pi$
$384$	0	0
$385$	−5.24351	−0.267234
$386$	0	0
$387$	20.1960	1.02662
$388$	0	0
$389$	19.6481	0.996197	0.498099	−	0.867120i	$-0.334032\pi$
$389$	19.6481	0.996197	0.498099	+	0.867120i	$0.334032\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	5.17368	0.260977
$394$	0	0
$395$	−32.1958	−1.61994
$396$	0	0
$397$	24.0961	1.20935	0.604676	−	0.796472i	$-0.293303\pi$
$397$	24.0961	1.20935	0.604676	+	0.796472i	$0.293303\pi$
$398$	0	0
$399$	1.37632	0.0689022
$400$	0	0
$401$	−14.4764	−0.722915	−0.361458	−	0.932389i	$-0.617721\pi$
$401$	−14.4764	−0.722915	−0.361458	+	0.932389i	$0.617721\pi$
$402$	0	0
$403$	−0.215185	−0.0107191
$404$	0	0
$405$	2.76119	0.137205
$406$	0	0
$407$	−13.2677	−0.657657
$408$	0	0
$409$	21.9045	1.08311	0.541554	−	0.840666i	$-0.317836\pi$
$409$	21.9045	1.08311	0.541554	+	0.840666i	$0.317836\pi$
$410$	0	0
$411$	−12.2406	−0.603784
$412$	0	0
$413$	0.416213	0.0204805
$414$	0	0
$415$	25.0973	1.23198
$416$	0	0
$417$	−8.18098	−0.400624
$418$	0	0
$419$	5.32259	0.260025	0.130013	−	0.991512i	$-0.458498\pi$
$419$	5.32259	0.260025	0.130013	+	0.991512i	$0.458498\pi$
$420$	0	0
$421$	−37.4407	−1.82475	−0.912375	−	0.409355i	$-0.865754\pi$
$421$	−37.4407	−1.82475	−0.912375	+	0.409355i	$0.865754\pi$
$422$	0	0
$423$	−14.6867	−0.714094
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−12.0311	−0.582225
$428$	0	0
$429$	3.47616	0.167831
$430$	0	0
$431$	−8.00641	−0.385655	−0.192828	−	0.981233i	$-0.561766\pi$
$431$	−8.00641	−0.385655	−0.192828	+	0.981233i	$0.561766\pi$
$432$	0	0
$433$	3.72258	0.178896	0.0894478	−	0.995992i	$-0.471490\pi$
$433$	3.72258	0.178896	0.0894478	+	0.995992i	$0.471490\pi$
$434$	0	0
$435$	−3.48616	−0.167148
$436$	0	0
$437$	0.777530	0.0371943
$438$	0	0
$439$	21.4772	1.02505	0.512526	−	0.858672i	$-0.328710\pi$
$439$	21.4772	1.02505	0.512526	+	0.858672i	$0.328710\pi$
$440$	0	0
$441$	−1.67113	−0.0795778
$442$	0	0
$443$	5.13834	0.244130	0.122065	−	0.992522i	$-0.461048\pi$
$443$	5.13834	0.244130	0.122065	+	0.992522i	$0.461048\pi$
$444$	0	0
$445$	−0.771382	−0.0365670
$446$	0	0
$447$	−9.22331	−0.436248
$448$	0	0
$449$	10.9292	0.515779	0.257889	−	0.966174i	$-0.416973\pi$
$449$	10.9292	0.515779	0.257889	+	0.966174i	$0.416973\pi$
$450$	0	0
$451$	−6.29532	−0.296435
$452$	0	0
$453$	12.6622	0.594920
$454$	0	0
$455$	−3.07603	−0.144207
$456$	0	0
$457$	25.4486	1.19044	0.595219	−	0.803564i	$-0.297065\pi$
$457$	25.4486	1.19044	0.595219	+	0.803564i	$0.297065\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−11.4142	−0.531611	−0.265805	−	0.964027i	$-0.585638\pi$
$461$	−11.4142	−0.531611	−0.265805	+	0.964027i	$0.585638\pi$
$462$	0	0
$463$	−9.80318	−0.455593	−0.227796	−	0.973709i	$-0.573152\pi$
$463$	−9.80318	−0.455593	−0.227796	+	0.973709i	$0.573152\pi$
$464$	0	0
$465$	0.431335	0.0200027
$466$	0	0
$467$	−21.3322	−0.987137	−0.493569	−	0.869707i	$-0.664308\pi$
$467$	−21.3322	−0.987137	−0.493569	+	0.869707i	$0.664308\pi$
$468$	0	0
$469$	1.79590	0.0829271
$470$	0	0
$471$	−20.0127	−0.922139
$472$	0	0
$473$	−27.4000	−1.25985
$474$	0	0
$475$	0.416400	0.0191057
$476$	0	0
$477$	−0.176936	−0.00810136
$478$	0	0
$479$	15.8687	0.725058	0.362529	−	0.931972i	$-0.381913\pi$
$479$	15.8687	0.725058	0.362529	+	0.931972i	$0.381913\pi$
$480$	0	0
$481$	−7.78332	−0.354889
$482$	0	0
$483$	0.750721	0.0341590
$484$	0	0
$485$	20.4872	0.930275
$486$	0	0
$487$	−5.95142	−0.269684	−0.134842	−	0.990867i	$-0.543053\pi$
$487$	−5.95142	−0.269684	−0.134842	+	0.990867i	$0.543053\pi$
$488$	0	0
$489$	13.0232	0.588929
$490$	0	0
$491$	−41.8478	−1.88856	−0.944282	−	0.329137i	$-0.893242\pi$
$491$	−41.8478	−1.88856	−0.944282	+	0.329137i	$0.893242\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	8.76261	0.393850
$496$	0	0
$497$	1.25386	0.0562431
$498$	0	0
$499$	3.90302	0.174723	0.0873615	−	0.996177i	$-0.472156\pi$
$499$	3.90302	0.174723	0.0873615	+	0.996177i	$0.472156\pi$
$500$	0	0
$501$	6.82420	0.304883
$502$	0	0
$503$	−37.0603	−1.65244	−0.826219	−	0.563349i	$-0.809513\pi$
$503$	−37.0603	−1.65244	−0.826219	+	0.563349i	$0.809513\pi$
$504$	0	0
$505$	3.34876	0.149018
$506$	0	0
$507$	−12.9467	−0.574983
$508$	0	0
$509$	40.8876	1.81231	0.906154	−	0.422947i	$-0.139004\pi$
$509$	40.8876	1.81231	0.906154	+	0.422947i	$0.139004\pi$
$510$	0	0
$511$	5.88184	0.260197
$512$	0	0
$513$	−6.42898	−0.283846
$514$	0	0
$515$	−5.73347	−0.252647
$516$	0	0
$517$	19.9255	0.876324
$518$	0	0
$519$	−28.2511	−1.24009
$520$	0	0
$521$	−26.9804	−1.18203	−0.591017	−	0.806659i	$-0.701273\pi$
$521$	−26.9804	−1.18203	−0.591017	+	0.806659i	$0.701273\pi$
$522$	0	0
$523$	32.6840	1.42917	0.714586	−	0.699548i	$-0.246615\pi$
$523$	32.6840	1.42917	0.714586	+	0.699548i	$0.246615\pi$
$524$	0	0
$525$	0.402043	0.0175466
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−22.5759	−0.981561
$530$	0	0
$531$	−0.695548	−0.0301842
$532$	0	0
$533$	−3.69306	−0.159964
$534$	0	0
$535$	−11.2063	−0.484490
$536$	0	0
$537$	−24.7981	−1.07012
$538$	0	0
$539$	2.26723	0.0976565
$540$	0	0
$541$	−16.7541	−0.720314	−0.360157	−	0.932892i	$-0.617277\pi$
$541$	−16.7541	−0.720314	−0.360157	+	0.932892i	$0.617277\pi$
$542$	0	0
$543$	−1.32275	−0.0567645
$544$	0	0
$545$	−12.1855	−0.521968
$546$	0	0
$547$	23.6901	1.01292	0.506459	−	0.862264i	$-0.330954\pi$
$547$	23.6901	1.01292	0.506459	+	0.862264i	$0.330954\pi$
$548$	0	0
$549$	20.1056	0.858085
$550$	0	0
$551$	1.56120	0.0665094
$552$	0	0
$553$	13.9211	0.591983
$554$	0	0
$555$	15.6016	0.662250
$556$	0	0
$557$	−22.8321	−0.967429	−0.483715	−	0.875226i	$-0.660713\pi$
$557$	−22.8321	−0.967429	−0.483715	+	0.875226i	$0.660713\pi$
$558$	0	0
$559$	−16.0738	−0.679849
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−34.0404	−1.43463	−0.717315	−	0.696749i	$-0.754629\pi$
$563$	−34.0404	−1.43463	−0.717315	+	0.696749i	$0.754629\pi$
$564$	0	0
$565$	−9.99374	−0.420440
$566$	0	0
$567$	−1.19391	−0.0501394
$568$	0	0
$569$	18.2631	0.765630	0.382815	−	0.923825i	$-0.374955\pi$
$569$	18.2631	0.765630	0.382815	+	0.923825i	$0.374955\pi$
$570$	0	0
$571$	11.8596	0.496309	0.248155	−	0.968720i	$-0.420176\pi$
$571$	11.8596	0.496309	0.248155	+	0.968720i	$0.420176\pi$
$572$	0	0
$573$	−19.7246	−0.824008
$574$	0	0
$575$	0.227127	0.00947187
$576$	0	0
$577$	−14.3386	−0.596925	−0.298462	−	0.954421i	$-0.596474\pi$
$577$	−14.3386	−0.596925	−0.298462	+	0.954421i	$0.596474\pi$
$578$	0	0
$579$	8.57593	0.356403
$580$	0	0
$581$	−10.8518	−0.450207
$582$	0	0
$583$	0.240050	0.00994185
$584$	0	0
$585$	5.14046	0.212532
$586$	0	0
$587$	24.7830	1.02291	0.511453	−	0.859311i	$-0.329108\pi$
$587$	24.7830	1.02291	0.511453	+	0.859311i	$0.329108\pi$
$588$	0	0
$589$	−0.193164	−0.00795919
$590$	0	0
$591$	−24.1483	−0.993330
$592$	0	0
$593$	−13.2495	−0.544092	−0.272046	−	0.962284i	$-0.587700\pi$
$593$	−13.2495	−0.544092	−0.272046	+	0.962284i	$0.587700\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−8.84661	−0.362068
$598$	0	0
$599$	24.0574	0.982957	0.491479	−	0.870890i	$-0.336457\pi$
$599$	24.0574	0.982957	0.491479	+	0.870890i	$0.336457\pi$
$600$	0	0
$601$	42.9683	1.75271	0.876357	−	0.481661i	$-0.159966\pi$
$601$	42.9683	1.75271	0.876357	+	0.481661i	$0.159966\pi$
$602$	0	0
$603$	−3.00120	−0.122218
$604$	0	0
$605$	13.5519	0.550962
$606$	0	0
$607$	20.5585	0.834443	0.417221	−	0.908805i	$-0.363004\pi$
$607$	20.5585	0.834443	0.417221	+	0.908805i	$0.363004\pi$
$608$	0	0
$609$	1.50737	0.0610818
$610$	0	0
$611$	11.6890	0.472887
$612$	0	0
$613$	12.3794	0.499999	0.250000	−	0.968246i	$-0.419570\pi$
$613$	12.3794	0.499999	0.250000	+	0.968246i	$0.419570\pi$
$614$	0	0
$615$	7.40269	0.298505
$616$	0	0
$617$	28.5340	1.14874	0.574369	−	0.818597i	$-0.305248\pi$
$617$	28.5340	1.14874	0.574369	+	0.818597i	$0.305248\pi$
$618$	0	0
$619$	21.6669	0.870867	0.435434	−	0.900221i	$-0.356595\pi$
$619$	21.6669	0.870867	0.435434	+	0.900221i	$0.356595\pi$
$620$	0	0
$621$	−3.50672	−0.140720
$622$	0	0
$623$	0.333536	0.0133628
$624$	0	0
$625$	−26.6222	−1.06489
$626$	0	0
$627$	3.12044	0.124618
$628$	0	0
$629$	0	0
$630$	0	0
$631$	18.6790	0.743600	0.371800	−	0.928313i	$-0.378741\pi$
$631$	18.6790	0.743600	0.371800	+	0.928313i	$0.378741\pi$
$632$	0	0
$633$	24.6174	0.978452
$634$	0	0
$635$	2.03152	0.0806184
$636$	0	0
$637$	1.33004	0.0526980
$638$	0	0
$639$	−2.09536	−0.0828912
$640$	0	0
$641$	49.3506	1.94923	0.974616	−	0.223885i	$-0.0718740\pi$
$641$	49.3506	1.94923	0.974616	+	0.223885i	$0.0718740\pi$
$642$	0	0
$643$	32.9146	1.29803	0.649013	−	0.760777i	$-0.275182\pi$
$643$	32.9146	1.29803	0.649013	+	0.760777i	$0.275182\pi$
$644$	0	0
$645$	32.2197	1.26865
$646$	0	0
$647$	−15.9166	−0.625747	−0.312873	−	0.949795i	$-0.601292\pi$
$647$	−15.9166	−0.625747	−0.312873	+	0.949795i	$0.601292\pi$
$648$	0	0
$649$	0.943651	0.0370415
$650$	0	0
$651$	−0.186504	−0.00730966
$652$	0	0
$653$	10.8620	0.425062	0.212531	−	0.977154i	$-0.431829\pi$
$653$	10.8620	0.425062	0.212531	+	0.977154i	$0.431829\pi$
$654$	0	0
$655$	−10.3797	−0.405569
$656$	0	0
$657$	−9.82935	−0.383480
$658$	0	0
$659$	19.1978	0.747842	0.373921	−	0.927461i	$-0.378013\pi$
$659$	19.1978	0.747842	0.373921	+	0.927461i	$0.378013\pi$
$660$	0	0
$661$	−14.9864	−0.582903	−0.291452	−	0.956586i	$-0.594138\pi$
$661$	−14.9864	−0.582903	−0.291452	+	0.956586i	$0.594138\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2.76125	−0.107077
$666$	0	0
$667$	0.851565	0.0329727
$668$	0	0
$669$	−20.7599	−0.802625
$670$	0	0
$671$	−27.2773	−1.05303
$672$	0	0
$673$	−22.9478	−0.884572	−0.442286	−	0.896874i	$-0.645832\pi$
$673$	−22.9478	−0.884572	−0.442286	+	0.896874i	$0.645832\pi$
$674$	0	0
$675$	−1.87800	−0.0722841
$676$	0	0
$677$	36.0140	1.38413	0.692065	−	0.721835i	$-0.256701\pi$
$677$	36.0140	1.38413	0.692065	+	0.721835i	$0.256701\pi$
$678$	0	0
$679$	−8.85840	−0.339954
$680$	0	0
$681$	−6.26950	−0.240248
$682$	0	0
$683$	−24.5836	−0.940667	−0.470333	−	0.882489i	$-0.655866\pi$
$683$	−24.5836	−0.940667	−0.470333	+	0.882489i	$0.655866\pi$
$684$	0	0
$685$	24.5578	0.938304
$686$	0	0
$687$	1.52742	0.0582747
$688$	0	0
$689$	0.140822	0.00536488
$690$	0	0
$691$	28.1139	1.06950	0.534752	−	0.845009i	$-0.320405\pi$
$691$	28.1139	1.06950	0.534752	+	0.845009i	$0.320405\pi$
$692$	0	0
$693$	−3.78885	−0.143926
$694$	0	0
$695$	16.4131	0.622586
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−8.30268	−0.314036
$700$	0	0
$701$	−11.4774	−0.433497	−0.216749	−	0.976227i	$-0.569545\pi$
$701$	−11.4774	−0.433497	−0.216749	+	0.976227i	$0.569545\pi$
$702$	0	0
$703$	−6.98683	−0.263513
$704$	0	0
$705$	−23.4305	−0.882443
$706$	0	0
$707$	−1.44796	−0.0544562
$708$	0	0
$709$	24.0442	0.902998	0.451499	−	0.892272i	$-0.350889\pi$
$709$	24.0442	0.902998	0.451499	+	0.892272i	$0.350889\pi$
$710$	0	0
$711$	−23.2639	−0.872466
$712$	0	0
$713$	−0.105362	−0.00394585
$714$	0	0
$715$	−6.97407	−0.260815
$716$	0	0
$717$	−24.7722	−0.925136
$718$	0	0
$719$	−17.9377	−0.668963	−0.334481	−	0.942402i	$-0.608561\pi$
$719$	−17.9377	−0.668963	−0.334481	+	0.942402i	$0.608561\pi$
$720$	0	0
$721$	2.47908	0.0923258
$722$	0	0
$723$	10.7661	0.400394
$724$	0	0
$725$	0.456049	0.0169372
$726$	0	0
$727$	−4.37698	−0.162333	−0.0811666	−	0.996701i	$-0.525865\pi$
$727$	−4.37698	−0.162333	−0.0811666	+	0.996701i	$0.525865\pi$
$728$	0	0
$729$	20.6171	0.763597
$730$	0	0
$731$	0	0
$732$	0	0
$733$	40.5661	1.49834	0.749172	−	0.662375i	$-0.230451\pi$
$733$	40.5661	1.49834	0.749172	+	0.662375i	$0.230451\pi$
$734$	0	0
$735$	−2.66604	−0.0983385
$736$	0	0
$737$	4.07173	0.149984
$738$	0	0
$739$	−14.6497	−0.538898	−0.269449	−	0.963015i	$-0.586842\pi$
$739$	−14.6497	−0.538898	−0.269449	+	0.963015i	$0.586842\pi$
$740$	0	0
$741$	1.83056	0.0672472
$742$	0	0
$743$	37.1647	1.36344	0.681721	−	0.731613i	$-0.261232\pi$
$743$	37.1647	1.36344	0.681721	+	0.731613i	$0.261232\pi$
$744$	0	0
$745$	18.5043	0.677946
$746$	0	0
$747$	18.1347	0.663515
$748$	0	0
$749$	4.84546	0.177049
$750$	0	0
$751$	−38.7945	−1.41563	−0.707815	−	0.706398i	$-0.750319\pi$
$751$	−38.7945	−1.41563	−0.707815	+	0.706398i	$0.750319\pi$
$752$	0	0
$753$	15.9786	0.582292
$754$	0	0
$755$	−25.4035	−0.924529
$756$	0	0
$757$	−18.4944	−0.672192	−0.336096	−	0.941828i	$-0.609107\pi$
$757$	−18.4944	−0.672192	−0.336096	+	0.941828i	$0.609107\pi$
$758$	0	0
$759$	1.70206	0.0617808
$760$	0	0
$761$	21.9658	0.796260	0.398130	−	0.917329i	$-0.369659\pi$
$761$	21.9658	0.796260	0.398130	+	0.917329i	$0.369659\pi$
$762$	0	0
$763$	5.26885	0.190745
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0.553579	0.0199886
$768$	0	0
$769$	−14.9545	−0.539273	−0.269636	−	0.962962i	$-0.586903\pi$
$769$	−14.9545	−0.539273	−0.269636	+	0.962962i	$0.586903\pi$
$770$	0	0
$771$	−24.2596	−0.873688
$772$	0	0
$773$	35.1315	1.26359	0.631796	−	0.775134i	$-0.282318\pi$
$773$	35.1315	1.26359	0.631796	+	0.775134i	$0.282318\pi$
$774$	0	0
$775$	−0.0564259	−0.00202688
$776$	0	0
$777$	−6.74592	−0.242009
$778$	0	0
$779$	−3.31514	−0.118777
$780$	0	0
$781$	2.84278	0.101723
$782$	0	0
$783$	−7.04113	−0.251630
$784$	0	0
$785$	40.1507	1.43304
$786$	0	0
$787$	19.3311	0.689079	0.344540	−	0.938772i	$-0.388035\pi$
$787$	19.3311	0.689079	0.344540	+	0.938772i	$0.388035\pi$
$788$	0	0
$789$	−12.8197	−0.456393
$790$	0	0
$791$	4.32117	0.153643
$792$	0	0
$793$	−16.0018	−0.568241
$794$	0	0
$795$	−0.282275	−0.0100113
$796$	0	0
$797$	−46.7781	−1.65697	−0.828483	−	0.560014i	$-0.810796\pi$
$797$	−46.7781	−1.65697	−0.828483	+	0.560014i	$0.810796\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−0.557384	−0.0196942
$802$	0	0
$803$	13.3355	0.470599
$804$	0	0
$805$	−1.50614	−0.0530844
$806$	0	0
$807$	−34.1056	−1.20057
$808$	0	0
$809$	10.9520	0.385051	0.192525	−	0.981292i	$-0.438332\pi$
$809$	10.9520	0.385051	0.192525	+	0.981292i	$0.438332\pi$
$810$	0	0
$811$	−14.4530	−0.507513	−0.253756	−	0.967268i	$-0.581666\pi$
$811$	−14.4530	−0.507513	−0.253756	+	0.967268i	$0.581666\pi$
$812$	0	0
$813$	10.8660	0.381088
$814$	0	0
$815$	−26.1279	−0.915219
$816$	0	0
$817$	−14.4289	−0.504804
$818$	0	0
$819$	−2.22267	−0.0776665
$820$	0	0
$821$	−35.0751	−1.22413	−0.612065	−	0.790808i	$-0.709661\pi$
$821$	−35.0751	−1.22413	−0.612065	+	0.790808i	$0.709661\pi$
$822$	0	0
$823$	40.8014	1.42225	0.711123	−	0.703067i	$-0.248187\pi$
$823$	40.8014	1.42225	0.711123	+	0.703067i	$0.248187\pi$
$824$	0	0
$825$	0.911523	0.0317352
$826$	0	0
$827$	27.2258	0.946734	0.473367	−	0.880865i	$-0.343038\pi$
$827$	27.2258	0.946734	0.473367	+	0.880865i	$0.343038\pi$
$828$	0	0
$829$	24.1511	0.838803	0.419401	−	0.907801i	$-0.362240\pi$
$829$	24.1511	0.838803	0.419401	+	0.907801i	$0.362240\pi$
$830$	0	0
$831$	−35.8762	−1.24453
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−13.6911	−0.473800
$836$	0	0
$837$	0.871185	0.0301125
$838$	0	0
$839$	4.05374	0.139950	0.0699752	−	0.997549i	$-0.477708\pi$
$839$	4.05374	0.139950	0.0699752	+	0.997549i	$0.477708\pi$
$840$	0	0
$841$	−27.2901	−0.941039
$842$	0	0
$843$	−11.7414	−0.404397
$844$	0	0
$845$	25.9744	0.893546
$846$	0	0
$847$	−5.85967	−0.201341
$848$	0	0
$849$	3.84545	0.131975
$850$	0	0
$851$	−3.81100	−0.130639
$852$	0	0
$853$	−29.3344	−1.00439	−0.502195	−	0.864755i	$-0.667474\pi$
$853$	−29.3344	−1.00439	−0.502195	+	0.864755i	$0.667474\pi$
$854$	0	0
$855$	4.61442	0.157810
$856$	0	0
$857$	−0.769593	−0.0262888	−0.0131444	−	0.999914i	$-0.504184\pi$
$857$	−0.769593	−0.0262888	−0.0131444	+	0.999914i	$0.504184\pi$
$858$	0	0
$859$	−33.9254	−1.15752	−0.578760	−	0.815498i	$-0.696463\pi$
$859$	−33.9254	−1.15752	−0.578760	+	0.815498i	$0.696463\pi$
$860$	0	0
$861$	−3.20083	−0.109084
$862$	0	0
$863$	11.7791	0.400966	0.200483	−	0.979697i	$-0.435749\pi$
$863$	11.7791	0.400966	0.200483	+	0.979697i	$0.435749\pi$
$864$	0	0
$865$	56.6789	1.92714
$866$	0	0
$867$	0	0
$868$	0	0
$869$	31.5622	1.07068
$870$	0	0
$871$	2.38862	0.0809353
$872$	0	0
$873$	14.8036	0.501025
$874$	0	0
$875$	10.7571	0.363656
$876$	0	0
$877$	23.3570	0.788712	0.394356	−	0.918958i	$-0.370968\pi$
$877$	23.3570	0.788712	0.394356	+	0.918958i	$0.370968\pi$
$878$	0	0
$879$	−23.0713	−0.778176
$880$	0	0
$881$	11.8916	0.400639	0.200319	−	0.979731i	$-0.435802\pi$
$881$	11.8916	0.400639	0.200319	+	0.979731i	$0.435802\pi$
$882$	0	0
$883$	−27.6831	−0.931612	−0.465806	−	0.884887i	$-0.654236\pi$
$883$	−27.6831	−0.931612	−0.465806	+	0.884887i	$0.654236\pi$
$884$	0	0
$885$	−1.10964	−0.0373002
$886$	0	0
$887$	18.9155	0.635120	0.317560	−	0.948238i	$-0.397137\pi$
$887$	18.9155	0.635120	0.317560	+	0.948238i	$0.397137\pi$
$888$	0	0
$889$	−0.878404	−0.0294607
$890$	0	0
$891$	−2.70686	−0.0906832
$892$	0	0
$893$	10.4928	0.351130
$894$	0	0
$895$	49.7513	1.66300
$896$	0	0
$897$	0.998487	0.0333385
$898$	0	0
$899$	−0.211557	−0.00705581
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−13.9314	−0.463608
$904$	0	0
$905$	2.65377	0.0882142
$906$	0	0
$907$	−24.2771	−0.806109	−0.403055	−	0.915176i	$-0.632052\pi$
$907$	−24.2771	−0.806109	−0.403055	+	0.915176i	$0.632052\pi$
$908$	0	0
$909$	2.41974	0.0802577
$910$	0	0
$911$	−30.4468	−1.00875	−0.504374	−	0.863485i	$-0.668277\pi$
$911$	−30.4468	−1.00875	−0.504374	+	0.863485i	$0.668277\pi$
$912$	0	0
$913$	−24.6034	−0.814254
$914$	0	0
$915$	32.0754	1.06038
$916$	0	0
$917$	4.48806	0.148209
$918$	0	0
$919$	28.3915	0.936550	0.468275	−	0.883583i	$-0.344876\pi$
$919$	28.3915	0.936550	0.468275	+	0.883583i	$0.344876\pi$
$920$	0	0
$921$	16.2185	0.534419
$922$	0	0
$923$	1.66768	0.0548922
$924$	0	0
$925$	−2.04095	−0.0671060
$926$	0	0
$927$	−4.14288	−0.136070
$928$	0	0
$929$	52.0241	1.70685	0.853427	−	0.521212i	$-0.174520\pi$
$929$	52.0241	1.70685	0.853427	+	0.521212i	$0.174520\pi$
$930$	0	0
$931$	1.19393	0.0391295
$932$	0	0
$933$	−4.31530	−0.141277
$934$	0	0
$935$	0	0
$936$	0	0
$937$	35.5380	1.16098	0.580488	−	0.814269i	$-0.302862\pi$
$937$	35.5380	1.16098	0.580488	+	0.814269i	$0.302862\pi$
$938$	0	0
$939$	16.4037	0.535316
$940$	0	0
$941$	18.8315	0.613889	0.306944	−	0.951727i	$-0.400693\pi$
$941$	18.8315	0.613889	0.306944	+	0.951727i	$0.400693\pi$
$942$	0	0
$943$	−1.80826	−0.0588850
$944$	0	0
$945$	12.4534	0.405111
$946$	0	0
$947$	8.93847	0.290461	0.145231	−	0.989398i	$-0.453608\pi$
$947$	8.93847	0.290461	0.145231	+	0.989398i	$0.453608\pi$
$948$	0	0
$949$	7.82308	0.253948
$950$	0	0
$951$	−19.2193	−0.623227
$952$	0	0
$953$	−35.0196	−1.13440	−0.567199	−	0.823581i	$-0.691973\pi$
$953$	−35.0196	−1.13440	−0.567199	+	0.823581i	$0.691973\pi$
$954$	0	0
$955$	39.5726	1.28054
$956$	0	0
$957$	3.41756	0.110474
$958$	0	0
$959$	−10.6185	−0.342888
$960$	0	0
$961$	−30.9738	−0.999156
$962$	0	0
$963$	−8.09741	−0.260936
$964$	0	0
$965$	−17.2055	−0.553865
$966$	0	0
$967$	−22.2637	−0.715954	−0.357977	−	0.933730i	$-0.616533\pi$
$967$	−22.2637	−0.715954	−0.357977	+	0.933730i	$0.616533\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	44.0288	1.41295	0.706476	−	0.707737i	$-0.250284\pi$
$971$	44.0288	1.41295	0.706476	+	0.707737i	$0.250284\pi$
$972$	0	0
$973$	−7.09684	−0.227514
$974$	0	0
$975$	0.534732	0.0171251
$976$	0	0
$977$	54.5658	1.74572	0.872858	−	0.487975i	$-0.162264\pi$
$977$	54.5658	1.74572	0.872858	+	0.487975i	$0.162264\pi$
$978$	0	0
$979$	0.756203	0.0241684
$980$	0	0
$981$	−8.80495	−0.281120
$982$	0	0
$983$	−1.53489	−0.0489554	−0.0244777	−	0.999700i	$-0.507792\pi$
$983$	−1.53489	−0.0489554	−0.0244777	+	0.999700i	$0.507792\pi$
$984$	0	0
$985$	48.4477	1.54367
$986$	0	0
$987$	10.1311	0.322475
$988$	0	0
$989$	−7.87032	−0.250262
$990$	0	0
$991$	−23.9416	−0.760531	−0.380266	−	0.924877i	$-0.624167\pi$
$991$	−23.9416	−0.760531	−0.380266	+	0.924877i	$0.624167\pi$
$992$	0	0
$993$	6.24386	0.198143
$994$	0	0
$995$	17.7486	0.562667
$996$	0	0
$997$	−46.8138	−1.48261	−0.741305	−	0.671169i	$-0.765793\pi$
$997$	−46.8138	−1.48261	−0.741305	+	0.671169i	$0.765793\pi$
$998$	0	0
$999$	31.5111	0.996968

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8092.2.a.w.1.8	yes	12
17.16	even	2		8092.2.a.v.1.5	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8092.2.a.v.1.5	✓	12	17.16	even	2
8092.2.a.w.1.8	yes	12	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:	\( N \)	\(=\)	\( 8092 = 2^{2} \cdot 7 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8092.a (trivial)

\(f(q)\)	\(=\)	\(q+1.15276 q^{3} -2.31274 q^{5} +1.00000 q^{7} -1.67113 q^{9} +O(q^{10})\)	\(q+1.15276 q^{3} -2.31274 q^{5} +1.00000 q^{7} -1.67113 q^{9} +2.26723 q^{11} +1.33004 q^{13} -2.66604 q^{15} +1.19393 q^{19} +1.15276 q^{21} +0.651235 q^{23} +0.348764 q^{25} -5.38472 q^{27} +1.30761 q^{29} -0.161788 q^{31} +2.61358 q^{33} -2.31274 q^{35} -5.85195 q^{37} +1.53322 q^{39} -2.77666 q^{41} -12.0852 q^{43} +3.86490 q^{45} +8.78849 q^{47} +1.00000 q^{49} +0.105878 q^{53} -5.24351 q^{55} +1.37632 q^{57} +0.416213 q^{59} -12.0311 q^{61} -1.67113 q^{63} -3.07603 q^{65} +1.79590 q^{67} +0.750721 q^{69} +1.25386 q^{71} +5.88184 q^{73} +0.402043 q^{75} +2.26723 q^{77} +13.9211 q^{79} -1.19391 q^{81} -10.8518 q^{83} +1.50737 q^{87} +0.333536 q^{89} +1.33004 q^{91} -0.186504 q^{93} -2.76125 q^{95} -8.85840 q^{97} -3.78885 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{7} + 12 q^{9}+O(q^{10}) \) 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

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Groups

Database

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