LMFDB - Embedded newform 3364.2.a.q.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3364,2,Mod(1,3364)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3364.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3364, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3364 = 2^{2} \cdot 29^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3364.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,-4,0,-1,0,-2,0,4,0,-5,0,4,0,17,0,-15,0,-11,0,-19,0,-3,0, 3,0,-16,0,0,0,9,0,-16,0,9,0,2,0,-9,0,-38,0,3,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(45)] == traces)

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

Self dual:	yes
Analytic conductor:	$26.8616752400$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.3266578125.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 10x^{6} + 6x^{5} + 29x^{4} - 9x^{3} - 25x^{2} - x + 1 $ x^8 - x^7 - 10x^6 + 6x^5 + 29x^4 - 9x^3 - 25*x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.178891$ of defining polynomial
Character	$\chi$	$=$	3364.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.498509 q^{3} +1.93765 q^{5} +0.267484 q^{7} -2.75149 q^{9} -6.18245 q^{11} +3.63066 q^{13} +0.965937 q^{15} -3.20847 q^{17} -0.562951 q^{19} +0.133343 q^{21} +3.58574 q^{23} -1.24551 q^{25} -2.86717 q^{27} +2.38700 q^{31} -3.08201 q^{33} +0.518291 q^{35} -8.08624 q^{37} +1.80992 q^{39} +5.86162 q^{41} +8.47514 q^{43} -5.33143 q^{45} -5.79919 q^{47} -6.92845 q^{49} -1.59945 q^{51} -7.74932 q^{53} -11.9794 q^{55} -0.280636 q^{57} -12.0464 q^{59} -14.3583 q^{61} -0.735980 q^{63} +7.03496 q^{65} +12.0926 q^{67} +1.78752 q^{69} -1.35301 q^{71} -9.77812 q^{73} -0.620896 q^{75} -1.65371 q^{77} +3.54940 q^{79} +6.82516 q^{81} -8.00413 q^{83} -6.21690 q^{85} -10.0204 q^{89} +0.971145 q^{91} +1.18994 q^{93} -1.09080 q^{95} +4.53082 q^{97} +17.0109 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{3} - q^{5} - 2 q^{7} + 4 q^{9} - 5 q^{11} + 4 q^{13} + 17 q^{15} - 15 q^{17} - 11 q^{19} - 19 q^{21} - 3 q^{23} + 3 q^{25} - 16 q^{27} + 9 q^{31} - 16 q^{33} + 9 q^{35} + 2 q^{37} - 9 q^{39}+ \cdots + 11 q^{99}+O(q^{100}) $ 8 * q - 4 * q^3 - q^5 - 2 * q^7 + 4 * q^9 - 5 * q^11 + 4 * q^13 + 17 * q^15 - 15 * q^17 - 11 * q^19 - 19 * q^21 - 3 * q^23 + 3 * q^25 - 16 * q^27 + 9 * q^31 - 16 * q^33 + 9 * q^35 + 2 * q^37 - 9 * q^39 - 38 * q^41 + 3 * q^43 + 3 * q^45 + 7 * q^47 + 12 * q^49 + 3 * q^51 - 5 * q^53 - 34 * q^55 + 34 * q^57 - 7 * q^59 - 39 * q^61 - 6 * q^63 - 21 * q^65 - 26 * q^67 - 34 * q^69 + 52 * q^71 + 34 * q^73 - 10 * q^77 - 31 * q^79 + 24 * q^81 - 17 * q^83 - 18 * q^85 - 24 * q^89 + 54 * q^91 + 6 * q^93 - 9 * q^95 - 22 * q^97 + 11 * q^99

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
$3$	0.498509	0.287814	0.143907	−	0.989591i	$-0.454033\pi$
$3$	0.498509	0.287814	0.143907	+	0.989591i	$0.454033\pi$
$4$	0	0
$5$	1.93765	0.866544	0.433272	−	0.901263i	$-0.357359\pi$
$5$	1.93765	0.866544	0.433272	+	0.901263i	$0.357359\pi$
$6$	0	0
$7$	0.267484	0.101100	0.0505498	−	0.998722i	$-0.483903\pi$
$7$	0.267484	0.101100	0.0505498	+	0.998722i	$0.483903\pi$
$8$	0	0
$9$	−2.75149	−0.917163
$10$	0	0
$11$	−6.18245	−1.86408	−0.932040	−	0.362356i	$-0.881972\pi$
$11$	−6.18245	−1.86408	−0.932040	+	0.362356i	$0.881972\pi$
$12$	0	0
$13$	3.63066	1.00696	0.503482	−	0.864005i	$-0.332052\pi$
$13$	3.63066	1.00696	0.503482	+	0.864005i	$0.332052\pi$
$14$	0	0
$15$	0.965937	0.249404
$16$	0	0
$17$	−3.20847	−0.778169	−0.389084	−	0.921202i	$-0.627209\pi$
$17$	−3.20847	−0.778169	−0.389084	+	0.921202i	$0.627209\pi$
$18$	0	0
$19$	−0.562951	−0.129150	−0.0645749	−	0.997913i	$-0.520569\pi$
$19$	−0.562951	−0.129150	−0.0645749	+	0.997913i	$0.520569\pi$
$20$	0	0
$21$	0.133343	0.0290979
$22$	0	0
$23$	3.58574	0.747678	0.373839	−	0.927494i	$-0.378041\pi$
$23$	3.58574	0.747678	0.373839	+	0.927494i	$0.378041\pi$
$24$	0	0
$25$	−1.24551	−0.249101
$26$	0	0
$27$	−2.86717	−0.551787
$28$	0	0
$29$	0	0
$29$	0	0
$30$	0	0
$31$	2.38700	0.428718	0.214359	−	0.976755i	$-0.431234\pi$
$31$	2.38700	0.428718	0.214359	+	0.976755i	$0.431234\pi$
$32$	0	0
$33$	−3.08201	−0.536509
$34$	0	0
$35$	0.518291	0.0876072
$36$	0	0
$37$	−8.08624	−1.32937	−0.664685	−	0.747124i	$-0.731434\pi$
$37$	−8.08624	−1.32937	−0.664685	+	0.747124i	$0.731434\pi$
$38$	0	0
$39$	1.80992	0.289819
$40$	0	0
$41$	5.86162	0.915431	0.457716	−	0.889099i	$-0.348668\pi$
$41$	5.86162	0.915431	0.457716	+	0.889099i	$0.348668\pi$
$42$	0	0
$43$	8.47514	1.29245	0.646223	−	0.763149i	$-0.276348\pi$
$43$	8.47514	1.29245	0.646223	+	0.763149i	$0.276348\pi$
$44$	0	0
$45$	−5.33143	−0.794762
$46$	0	0
$47$	−5.79919	−0.845899	−0.422949	−	0.906153i	$-0.639005\pi$
$47$	−5.79919	−0.845899	−0.422949	+	0.906153i	$0.639005\pi$
$48$	0	0
$49$	−6.92845	−0.989779
$50$	0	0
$51$	−1.59945	−0.223968
$52$	0	0
$53$	−7.74932	−1.06445	−0.532225	−	0.846603i	$-0.678644\pi$
$53$	−7.74932	−1.06445	−0.532225	+	0.846603i	$0.678644\pi$
$54$	0	0
$55$	−11.9794	−1.61531
$56$	0	0
$57$	−0.280636	−0.0371711
$58$	0	0
$59$	−12.0464	−1.56831	−0.784156	−	0.620564i	$-0.786904\pi$
$59$	−12.0464	−1.56831	−0.784156	+	0.620564i	$0.786904\pi$
$60$	0	0
$61$	−14.3583	−1.83839	−0.919196	−	0.393799i	$-0.871161\pi$
$61$	−14.3583	−1.83839	−0.919196	+	0.393799i	$0.871161\pi$
$62$	0	0
$63$	−0.735980	−0.0927248
$64$	0	0
$65$	7.03496	0.872579
$66$	0	0
$67$	12.0926	1.47734	0.738671	−	0.674066i	$-0.235454\pi$
$67$	12.0926	1.47734	0.738671	+	0.674066i	$0.235454\pi$
$68$	0	0
$69$	1.78752	0.215193
$70$	0	0
$71$	−1.35301	−0.160573	−0.0802865	−	0.996772i	$-0.525584\pi$
$71$	−1.35301	−0.160573	−0.0802865	+	0.996772i	$0.525584\pi$
$72$	0	0
$73$	−9.77812	−1.14444	−0.572221	−	0.820099i	$-0.693918\pi$
$73$	−9.77812	−1.14444	−0.572221	+	0.820099i	$0.693918\pi$
$74$	0	0
$75$	−0.620896	−0.0716949
$76$	0	0
$77$	−1.65371	−0.188458
$78$	0	0
$79$	3.54940	0.399338	0.199669	−	0.979863i	$-0.436013\pi$
$79$	3.54940	0.399338	0.199669	+	0.979863i	$0.436013\pi$
$80$	0	0
$81$	6.82516	0.758351
$82$	0	0
$83$	−8.00413	−0.878568	−0.439284	−	0.898348i	$-0.644768\pi$
$83$	−8.00413	−0.878568	−0.439284	+	0.898348i	$0.644768\pi$
$84$	0	0
$85$	−6.21690	−0.674317
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−10.0204	−1.06216	−0.531081	−	0.847321i	$-0.678214\pi$
$89$	−10.0204	−1.06216	−0.531081	+	0.847321i	$0.678214\pi$
$90$	0	0
$91$	0.971145	0.101804
$92$	0	0
$93$	1.18994	0.123391
$94$	0	0
$95$	−1.09080	−0.111914
$96$	0	0
$97$	4.53082	0.460035	0.230018	−	0.973186i	$-0.426122\pi$
$97$	4.53082	0.460035	0.230018	+	0.973186i	$0.426122\pi$
$98$	0	0
$99$	17.0109	1.70966
$100$	0	0
$101$	−1.23526	−0.122913	−0.0614565	−	0.998110i	$-0.519575\pi$
$101$	−1.23526	−0.122913	−0.0614565	+	0.998110i	$0.519575\pi$
$102$	0	0
$103$	10.8296	1.06707	0.533535	−	0.845778i	$-0.320863\pi$
$103$	10.8296	1.06707	0.533535	+	0.845778i	$0.320863\pi$
$104$	0	0
$105$	0.258373	0.0252146
$106$	0	0
$107$	−8.72039	−0.843032	−0.421516	−	0.906821i	$-0.638502\pi$
$107$	−8.72039	−0.843032	−0.421516	+	0.906821i	$0.638502\pi$
$108$	0	0
$109$	−16.1356	−1.54551	−0.772753	−	0.634707i	$-0.781121\pi$
$109$	−16.1356	−1.54551	−0.772753	+	0.634707i	$0.781121\pi$
$110$	0	0
$111$	−4.03106	−0.382612
$112$	0	0
$113$	−15.8835	−1.49420	−0.747098	−	0.664714i	$-0.768553\pi$
$113$	−15.8835	−1.49420	−0.747098	+	0.664714i	$0.768553\pi$
$114$	0	0
$115$	6.94792	0.647896
$116$	0	0
$117$	−9.98973	−0.923551
$118$	0	0
$119$	−0.858216	−0.0786725
$120$	0	0
$121$	27.2227	2.47479
$122$	0	0
$123$	2.92207	0.263474
$124$	0	0
$125$	−12.1016	−1.08240
$126$	0	0
$127$	−15.6634	−1.38990	−0.694951	−	0.719057i	$-0.744574\pi$
$127$	−15.6634	−1.38990	−0.694951	+	0.719057i	$0.744574\pi$
$128$	0	0
$129$	4.22493	0.371984
$130$	0	0
$131$	12.6657	1.10661	0.553303	−	0.832980i	$-0.313367\pi$
$131$	12.6657	1.10661	0.553303	+	0.832980i	$0.313367\pi$
$132$	0	0
$133$	−0.150580	−0.0130570
$134$	0	0
$135$	−5.55557	−0.478148
$136$	0	0
$137$	−4.45856	−0.380921	−0.190460	−	0.981695i	$-0.560998\pi$
$137$	−4.45856	−0.380921	−0.190460	+	0.981695i	$0.560998\pi$
$138$	0	0
$139$	3.22696	0.273707	0.136853	−	0.990591i	$-0.456301\pi$
$139$	3.22696	0.273707	0.136853	+	0.990591i	$0.456301\pi$
$140$	0	0
$141$	−2.89095	−0.243462
$142$	0	0
$143$	−22.4464	−1.87706
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−3.45390	−0.284872
$148$	0	0
$149$	4.31553	0.353542	0.176771	−	0.984252i	$-0.443435\pi$
$149$	4.31553	0.353542	0.176771	+	0.984252i	$0.443435\pi$
$150$	0	0
$151$	19.5819	1.59355	0.796775	−	0.604277i	$-0.206538\pi$
$151$	19.5819	1.59355	0.796775	+	0.604277i	$0.206538\pi$
$152$	0	0
$153$	8.82807	0.713707
$154$	0	0
$155$	4.62518	0.371503
$156$	0	0
$157$	0.568812	0.0453961	0.0226981	−	0.999742i	$-0.492774\pi$
$157$	0.568812	0.0453961	0.0226981	+	0.999742i	$0.492774\pi$
$158$	0	0
$159$	−3.86310	−0.306364
$160$	0	0
$161$	0.959129	0.0755900
$162$	0	0
$163$	18.8668	1.47776	0.738880	−	0.673837i	$-0.235355\pi$
$163$	18.8668	1.47776	0.738880	+	0.673837i	$0.235355\pi$
$164$	0	0
$165$	−5.97186	−0.464908
$166$	0	0
$167$	−12.0978	−0.936154	−0.468077	−	0.883688i	$-0.655053\pi$
$167$	−12.0978	−0.936154	−0.468077	+	0.883688i	$0.655053\pi$
$168$	0	0
$169$	0.181714	0.0139780
$170$	0	0
$171$	1.54895	0.118451
$172$	0	0
$173$	25.0399	1.90375	0.951875	−	0.306488i	$-0.0991539\pi$
$173$	25.0399	1.90375	0.951875	+	0.306488i	$0.0991539\pi$
$174$	0	0
$175$	−0.333153	−0.0251840
$176$	0	0
$177$	−6.00526	−0.451383
$178$	0	0
$179$	9.94203	0.743102	0.371551	−	0.928412i	$-0.378826\pi$
$179$	9.94203	0.743102	0.371551	+	0.928412i	$0.378826\pi$
$180$	0	0
$181$	−4.94428	−0.367505	−0.183753	−	0.982973i	$-0.558825\pi$
$181$	−4.94428	−0.367505	−0.183753	+	0.982973i	$0.558825\pi$
$182$	0	0
$183$	−7.15774	−0.529116
$184$	0	0
$185$	−15.6683	−1.15196
$186$	0	0
$187$	19.8362	1.45057
$188$	0	0
$189$	−0.766923	−0.0557854
$190$	0	0
$191$	7.50104	0.542757	0.271378	−	0.962473i	$-0.412521\pi$
$191$	7.50104	0.542757	0.271378	+	0.962473i	$0.412521\pi$
$192$	0	0
$193$	11.8818	0.855271	0.427635	−	0.903951i	$-0.359347\pi$
$193$	11.8818	0.855271	0.427635	+	0.903951i	$0.359347\pi$
$194$	0	0
$195$	3.50699	0.251141
$196$	0	0
$197$	22.3390	1.59159	0.795794	−	0.605568i	$-0.207054\pi$
$197$	22.3390	1.59159	0.795794	+	0.605568i	$0.207054\pi$
$198$	0	0
$199$	−11.1648	−0.791453	−0.395727	−	0.918368i	$-0.629507\pi$
$199$	−11.1648	−0.791453	−0.395727	+	0.918368i	$0.629507\pi$
$200$	0	0
$201$	6.02825	0.425200
$202$	0	0
$203$	0	0
$204$	0	0
$205$	11.3578	0.793262
$206$	0	0
$207$	−9.86612	−0.685743
$208$	0	0
$209$	3.48042	0.240745
$210$	0	0
$211$	−10.2812	−0.707784	−0.353892	−	0.935286i	$-0.615142\pi$
$211$	−10.2812	−0.707784	−0.353892	+	0.935286i	$0.615142\pi$
$212$	0	0
$213$	−0.674489	−0.0462152
$214$	0	0
$215$	16.4219	1.11996
$216$	0	0
$217$	0.638485	0.0433432
$218$	0	0
$219$	−4.87448	−0.329387
$220$	0	0
$221$	−11.6489	−0.783588
$222$	0	0
$223$	−24.1973	−1.62037	−0.810186	−	0.586173i	$-0.800634\pi$
$223$	−24.1973	−1.62037	−0.810186	+	0.586173i	$0.800634\pi$
$224$	0	0
$225$	3.42700	0.228466
$226$	0	0
$227$	−18.2464	−1.21106	−0.605530	−	0.795823i	$-0.707039\pi$
$227$	−18.2464	−1.21106	−0.605530	+	0.795823i	$0.707039\pi$
$228$	0	0
$229$	−16.3646	−1.08141	−0.540703	−	0.841214i	$-0.681842\pi$
$229$	−16.3646	−1.08141	−0.540703	+	0.841214i	$0.681842\pi$
$230$	0	0
$231$	−0.824389	−0.0542408
$232$	0	0
$233$	−13.8382	−0.906568	−0.453284	−	0.891366i	$-0.649748\pi$
$233$	−13.8382	−0.906568	−0.453284	+	0.891366i	$0.649748\pi$
$234$	0	0
$235$	−11.2368	−0.733009
$236$	0	0
$237$	1.76941	0.114935
$238$	0	0
$239$	0.783539	0.0506829	0.0253415	−	0.999679i	$-0.491933\pi$
$239$	0.783539	0.0506829	0.0253415	+	0.999679i	$0.491933\pi$
$240$	0	0
$241$	23.6055	1.52056	0.760281	−	0.649595i	$-0.225061\pi$
$241$	23.6055	1.52056	0.760281	+	0.649595i	$0.225061\pi$
$242$	0	0
$243$	12.0039	0.770051
$244$	0	0
$245$	−13.4249	−0.857687
$246$	0	0
$247$	−2.04388	−0.130049
$248$	0	0
$249$	−3.99013	−0.252864
$250$	0	0
$251$	−12.6595	−0.799058	−0.399529	−	0.916721i	$-0.630826\pi$
$251$	−12.6595	−0.799058	−0.399529	+	0.916721i	$0.630826\pi$
$252$	0	0
$253$	−22.1687	−1.39373
$254$	0	0
$255$	−3.09918	−0.194078
$256$	0	0
$257$	10.3460	0.645366	0.322683	−	0.946507i	$-0.395415\pi$
$257$	10.3460	0.645366	0.322683	+	0.946507i	$0.395415\pi$
$258$	0	0
$259$	−2.16294	−0.134399
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−16.0395	−0.989035	−0.494517	−	0.869168i	$-0.664655\pi$
$263$	−16.0395	−0.989035	−0.494517	+	0.869168i	$0.664655\pi$
$264$	0	0
$265$	−15.0155	−0.922393
$266$	0	0
$267$	−4.99527	−0.305705
$268$	0	0
$269$	−8.93282	−0.544643	−0.272322	−	0.962206i	$-0.587791\pi$
$269$	−8.93282	−0.544643	−0.272322	+	0.962206i	$0.587791\pi$
$270$	0	0
$271$	26.0649	1.58333	0.791666	−	0.610955i	$-0.209214\pi$
$271$	26.0649	1.58333	0.791666	+	0.610955i	$0.209214\pi$
$272$	0	0
$273$	0.484125	0.0293006
$274$	0	0
$275$	7.70028	0.464344
$276$	0	0
$277$	−18.8513	−1.13266	−0.566331	−	0.824178i	$-0.691638\pi$
$277$	−18.8513	−1.13266	−0.566331	+	0.824178i	$0.691638\pi$
$278$	0	0
$279$	−6.56780	−0.393204
$280$	0	0
$281$	20.4308	1.21880	0.609401	−	0.792862i	$-0.291410\pi$
$281$	20.4308	1.21880	0.609401	+	0.792862i	$0.291410\pi$
$282$	0	0
$283$	24.1511	1.43563	0.717816	−	0.696232i	$-0.245142\pi$
$283$	24.1511	1.43563	0.717816	+	0.696232i	$0.245142\pi$
$284$	0	0
$285$	−0.543775	−0.0322104
$286$	0	0
$287$	1.56789	0.0925497
$288$	0	0
$289$	−6.70571	−0.394454
$290$	0	0
$291$	2.25866	0.132405
$292$	0	0
$293$	−0.901085	−0.0526420	−0.0263210	−	0.999654i	$-0.508379\pi$
$293$	−0.901085	−0.0526420	−0.0263210	+	0.999654i	$0.508379\pi$
$294$	0	0
$295$	−23.3418	−1.35901
$296$	0	0
$297$	17.7261	1.02857
$298$	0	0
$299$	13.0186	0.752886
$300$	0	0
$301$	2.26697	0.130666
$302$	0	0
$303$	−0.615788	−0.0353761
$304$	0	0
$305$	−27.8214	−1.59305
$306$	0	0
$307$	−2.26687	−0.129377	−0.0646885	−	0.997906i	$-0.520605\pi$
$307$	−2.26687	−0.129377	−0.0646885	+	0.997906i	$0.520605\pi$
$308$	0	0
$309$	5.39864	0.307118
$310$	0	0
$311$	30.8593	1.74987	0.874936	−	0.484239i	$-0.160903\pi$
$311$	30.8593	1.74987	0.874936	+	0.484239i	$0.160903\pi$
$312$	0	0
$313$	6.37922	0.360575	0.180288	−	0.983614i	$-0.442297\pi$
$313$	6.37922	0.360575	0.180288	+	0.983614i	$0.442297\pi$
$314$	0	0
$315$	−1.42607	−0.0803501
$316$	0	0
$317$	−10.5309	−0.591473	−0.295737	−	0.955270i	$-0.595565\pi$
$317$	−10.5309	−0.591473	−0.295737	+	0.955270i	$0.595565\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−4.34719	−0.242637
$322$	0	0
$323$	1.80621	0.100500
$324$	0	0
$325$	−4.52201	−0.250836
$326$	0	0
$327$	−8.04372	−0.444819
$328$	0	0
$329$	−1.55119	−0.0855200
$330$	0	0
$331$	3.07669	0.169110	0.0845552	−	0.996419i	$-0.473053\pi$
$331$	3.07669	0.169110	0.0845552	+	0.996419i	$0.473053\pi$
$332$	0	0
$333$	22.2492	1.21925
$334$	0	0
$335$	23.4312	1.28018
$336$	0	0
$337$	−15.0517	−0.819919	−0.409960	−	0.912104i	$-0.634457\pi$
$337$	−15.0517	−0.819919	−0.409960	+	0.912104i	$0.634457\pi$
$338$	0	0
$339$	−7.91807	−0.430051
$340$	0	0
$341$	−14.7575	−0.799164
$342$	0	0
$343$	−3.72564	−0.201166
$344$	0	0
$345$	3.46360	0.186474
$346$	0	0
$347$	−22.9764	−1.23344	−0.616719	−	0.787184i	$-0.711538\pi$
$347$	−22.9764	−1.23344	−0.616719	+	0.787184i	$0.711538\pi$
$348$	0	0
$349$	−3.47138	−0.185819	−0.0929094	−	0.995675i	$-0.529617\pi$
$349$	−3.47138	−0.185819	−0.0929094	+	0.995675i	$0.529617\pi$
$350$	0	0
$351$	−10.4097	−0.555630
$352$	0	0
$353$	7.15857	0.381012	0.190506	−	0.981686i	$-0.438987\pi$
$353$	7.15857	0.381012	0.190506	+	0.981686i	$0.438987\pi$
$354$	0	0
$355$	−2.62167	−0.139144
$356$	0	0
$357$	−0.427828	−0.0226431
$358$	0	0
$359$	−16.9081	−0.892376	−0.446188	−	0.894939i	$-0.647219\pi$
$359$	−16.9081	−0.892376	−0.446188	+	0.894939i	$0.647219\pi$
$360$	0	0
$361$	−18.6831	−0.983320
$362$	0	0
$363$	13.5708	0.712280
$364$	0	0
$365$	−18.9466	−0.991710
$366$	0	0
$367$	−22.4984	−1.17441	−0.587204	−	0.809439i	$-0.699771\pi$
$367$	−22.4984	−1.17441	−0.587204	+	0.809439i	$0.699771\pi$
$368$	0	0
$369$	−16.1282	−0.839600
$370$	0	0
$371$	−2.07282	−0.107615
$372$	0	0
$373$	−7.53650	−0.390225	−0.195113	−	0.980781i	$-0.562507\pi$
$373$	−7.53650	−0.390225	−0.195113	+	0.980781i	$0.562507\pi$
$374$	0	0
$375$	−6.03276	−0.311531
$376$	0	0
$377$	0	0
$378$	0	0
$379$	11.5167	0.591571	0.295785	−	0.955254i	$-0.404419\pi$
$379$	11.5167	0.591571	0.295785	+	0.955254i	$0.404419\pi$
$380$	0	0
$381$	−7.80834	−0.400034
$382$	0	0
$383$	−38.3541	−1.95980	−0.979901	−	0.199484i	$-0.936073\pi$
$383$	−38.3541	−1.95980	−0.979901	+	0.199484i	$0.936073\pi$
$384$	0	0
$385$	−3.20431	−0.163307
$386$	0	0
$387$	−23.3192	−1.18538
$388$	0	0
$389$	−33.0783	−1.67714	−0.838569	−	0.544795i	$-0.816607\pi$
$389$	−33.0783	−1.67714	−0.838569	+	0.544795i	$0.816607\pi$
$390$	0	0
$391$	−11.5047	−0.581820
$392$	0	0
$393$	6.31395	0.318497
$394$	0	0
$395$	6.87750	0.346044
$396$	0	0
$397$	27.7994	1.39521	0.697606	−	0.716481i	$-0.254249\pi$
$397$	27.7994	1.39521	0.697606	+	0.716481i	$0.254249\pi$
$398$	0	0
$399$	−0.0750657	−0.00375799
$400$	0	0
$401$	8.67153	0.433036	0.216518	−	0.976279i	$-0.430530\pi$
$401$	8.67153	0.433036	0.216518	+	0.976279i	$0.430530\pi$
$402$	0	0
$403$	8.66639	0.431704
$404$	0	0
$405$	13.2248	0.657145
$406$	0	0
$407$	49.9928	2.47805
$408$	0	0
$409$	12.4891	0.617548	0.308774	−	0.951135i	$-0.400081\pi$
$409$	12.4891	0.617548	0.308774	+	0.951135i	$0.400081\pi$
$410$	0	0
$411$	−2.22263	−0.109634
$412$	0	0
$413$	−3.22223	−0.158556
$414$	0	0
$415$	−15.5092	−0.761318
$416$	0	0
$417$	1.60867	0.0787768
$418$	0	0
$419$	17.6374	0.861645	0.430823	−	0.902437i	$-0.358223\pi$
$419$	17.6374	0.861645	0.430823	+	0.902437i	$0.358223\pi$
$420$	0	0
$421$	16.8056	0.819054	0.409527	−	0.912298i	$-0.365694\pi$
$421$	16.8056	0.819054	0.409527	+	0.912298i	$0.365694\pi$
$422$	0	0
$423$	15.9564	0.775827
$424$	0	0
$425$	3.99617	0.193843
$426$	0	0
$427$	−3.84062	−0.185861
$428$	0	0
$429$	−11.1897	−0.540245
$430$	0	0
$431$	9.31283	0.448583	0.224292	−	0.974522i	$-0.427993\pi$
$431$	9.31283	0.448583	0.224292	+	0.974522i	$0.427993\pi$
$432$	0	0
$433$	2.38638	0.114682	0.0573410	−	0.998355i	$-0.481738\pi$
$433$	2.38638	0.114682	0.0573410	+	0.998355i	$0.481738\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.01859	−0.0965625
$438$	0	0
$439$	32.1760	1.53567	0.767837	−	0.640645i	$-0.221333\pi$
$439$	32.1760	1.53567	0.767837	+	0.640645i	$0.221333\pi$
$440$	0	0
$441$	19.0636	0.907789
$442$	0	0
$443$	29.2967	1.39193	0.695964	−	0.718077i	$-0.254977\pi$
$443$	29.2967	1.39193	0.695964	+	0.718077i	$0.254977\pi$
$444$	0	0
$445$	−19.4161	−0.920411
$446$	0	0
$447$	2.15133	0.101754
$448$	0	0
$449$	−6.83909	−0.322757	−0.161378	−	0.986893i	$-0.551594\pi$
$449$	−6.83909	−0.322757	−0.161378	+	0.986893i	$0.551594\pi$
$450$	0	0
$451$	−36.2392	−1.70644
$452$	0	0
$453$	9.76173	0.458646
$454$	0	0
$455$	1.88174	0.0882174
$456$	0	0
$457$	−2.37038	−0.110882	−0.0554408	−	0.998462i	$-0.517656\pi$
$457$	−2.37038	−0.110882	−0.0554408	+	0.998462i	$0.517656\pi$
$458$	0	0
$459$	9.19923	0.429383
$460$	0	0
$461$	20.6640	0.962417	0.481209	−	0.876606i	$-0.340198\pi$
$461$	20.6640	0.962417	0.481209	+	0.876606i	$0.340198\pi$
$462$	0	0
$463$	−5.94160	−0.276130	−0.138065	−	0.990423i	$-0.544088\pi$
$463$	−5.94160	−0.276130	−0.138065	+	0.990423i	$0.544088\pi$
$464$	0	0
$465$	2.30569	0.106924
$466$	0	0
$467$	21.4516	0.992664	0.496332	−	0.868133i	$-0.334680\pi$
$467$	21.4516	0.992664	0.496332	+	0.868133i	$0.334680\pi$
$468$	0	0
$469$	3.23457	0.149359
$470$	0	0
$471$	0.283558	0.0130657
$472$	0	0
$473$	−52.3971	−2.40922
$474$	0	0
$475$	0.701158	0.0321714
$476$	0	0
$477$	21.3222	0.976275
$478$	0	0
$479$	5.28551	0.241501	0.120751	−	0.992683i	$-0.461470\pi$
$479$	5.28551	0.241501	0.120751	+	0.992683i	$0.461470\pi$
$480$	0	0
$481$	−29.3584	−1.33863
$482$	0	0
$483$	0.478134	0.0217559
$484$	0	0
$485$	8.77916	0.398641
$486$	0	0
$487$	1.56145	0.0707561	0.0353781	−	0.999374i	$-0.488736\pi$
$487$	1.56145	0.0707561	0.0353781	+	0.999374i	$0.488736\pi$
$488$	0	0
$489$	9.40525	0.425320
$490$	0	0
$491$	16.4010	0.740165	0.370082	−	0.928999i	$-0.379329\pi$
$491$	16.4010	0.740165	0.370082	+	0.928999i	$0.379329\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	32.9613	1.48150
$496$	0	0
$497$	−0.361910	−0.0162339
$498$	0	0
$499$	19.4787	0.871985	0.435993	−	0.899950i	$-0.356397\pi$
$499$	19.4787	0.871985	0.435993	+	0.899950i	$0.356397\pi$
$500$	0	0
$501$	−6.03085	−0.269438
$502$	0	0
$503$	−6.53697	−0.291469	−0.145734	−	0.989324i	$-0.546555\pi$
$503$	−6.53697	−0.291469	−0.145734	+	0.989324i	$0.546555\pi$
$504$	0	0
$505$	−2.39350	−0.106510
$506$	0	0
$507$	0.0905862	0.00402307
$508$	0	0
$509$	28.6063	1.26795	0.633976	−	0.773353i	$-0.281422\pi$
$509$	28.6063	1.26795	0.633976	+	0.773353i	$0.281422\pi$
$510$	0	0
$511$	−2.61549	−0.115703
$512$	0	0
$513$	1.61407	0.0712631
$514$	0	0
$515$	20.9840	0.924664
$516$	0	0
$517$	35.8532	1.57682
$518$	0	0
$519$	12.4826	0.547926
$520$	0	0
$521$	−12.5478	−0.549731	−0.274866	−	0.961483i	$-0.588633\pi$
$521$	−12.5478	−0.549731	−0.274866	+	0.961483i	$0.588633\pi$
$522$	0	0
$523$	−16.8553	−0.737031	−0.368516	−	0.929622i	$-0.620134\pi$
$523$	−16.8553	−0.737031	−0.368516	+	0.929622i	$0.620134\pi$
$524$	0	0
$525$	−0.166080	−0.00724832
$526$	0	0
$527$	−7.65862	−0.333615
$528$	0	0
$529$	−10.1425	−0.440977
$530$	0	0
$531$	33.1456	1.43840
$532$	0	0
$533$	21.2816	0.921807
$534$	0	0
$535$	−16.8971	−0.730524
$536$	0	0
$537$	4.95619	0.213875
$538$	0	0
$539$	42.8348	1.84503
$540$	0	0
$541$	23.9231	1.02853	0.514267	−	0.857630i	$-0.328064\pi$
$541$	23.9231	1.02853	0.514267	+	0.857630i	$0.328064\pi$
$542$	0	0
$543$	−2.46477	−0.105773
$544$	0	0
$545$	−31.2651	−1.33925
$546$	0	0
$547$	−16.7482	−0.716100	−0.358050	−	0.933702i	$-0.616558\pi$
$547$	−16.7482	−0.716100	−0.358050	+	0.933702i	$0.616558\pi$
$548$	0	0
$549$	39.5067	1.68611
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0.949408	0.0403729
$554$	0	0
$555$	−7.81080	−0.331550
$556$	0	0
$557$	−5.21880	−0.221127	−0.110564	−	0.993869i	$-0.535266\pi$
$557$	−5.21880	−0.221127	−0.110564	+	0.993869i	$0.535266\pi$
$558$	0	0
$559$	30.7704	1.30145
$560$	0	0
$561$	9.88853	0.417494
$562$	0	0
$563$	−8.88005	−0.374250	−0.187125	−	0.982336i	$-0.559917\pi$
$563$	−8.88005	−0.374250	−0.187125	+	0.982336i	$0.559917\pi$
$564$	0	0
$565$	−30.7767	−1.29479
$566$	0	0
$567$	1.82562	0.0766689
$568$	0	0
$569$	21.1360	0.886067	0.443034	−	0.896505i	$-0.353902\pi$
$569$	21.1360	0.886067	0.443034	+	0.896505i	$0.353902\pi$
$570$	0	0
$571$	−4.85753	−0.203281	−0.101641	−	0.994821i	$-0.532409\pi$
$571$	−4.85753	−0.203281	−0.101641	+	0.994821i	$0.532409\pi$
$572$	0	0
$573$	3.73934	0.156213
$574$	0	0
$575$	−4.46606	−0.186248
$576$	0	0
$577$	29.0667	1.21006	0.605031	−	0.796202i	$-0.293161\pi$
$577$	29.0667	1.21006	0.605031	+	0.796202i	$0.293161\pi$
$578$	0	0
$579$	5.92318	0.246159
$580$	0	0
$581$	−2.14098	−0.0888228
$582$	0	0
$583$	47.9098	1.98422
$584$	0	0
$585$	−19.3566	−0.800298
$586$	0	0
$587$	−38.1573	−1.57492	−0.787461	−	0.616365i	$-0.788605\pi$
$587$	−38.1573	−1.57492	−0.787461	+	0.616365i	$0.788605\pi$
$588$	0	0
$589$	−1.34376	−0.0553688
$590$	0	0
$591$	11.1362	0.458082
$592$	0	0
$593$	−16.1688	−0.663973	−0.331986	−	0.943284i	$-0.607719\pi$
$593$	−16.1688	−0.663973	−0.331986	+	0.943284i	$0.607719\pi$
$594$	0	0
$595$	−1.66292	−0.0681732
$596$	0	0
$597$	−5.56577	−0.227792
$598$	0	0
$599$	28.2890	1.15586	0.577928	−	0.816088i	$-0.303861\pi$
$599$	28.2890	1.15586	0.577928	+	0.816088i	$0.303861\pi$
$600$	0	0
$601$	−10.2439	−0.417856	−0.208928	−	0.977931i	$-0.566997\pi$
$601$	−10.2439	−0.417856	−0.208928	+	0.977931i	$0.566997\pi$
$602$	0	0
$603$	−33.2726	−1.35496
$604$	0	0
$605$	52.7481	2.14452
$606$	0	0
$607$	45.3280	1.83981	0.919903	−	0.392147i	$-0.128267\pi$
$607$	45.3280	1.83981	0.919903	+	0.392147i	$0.128267\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−21.0549	−0.851790
$612$	0	0
$613$	−23.8059	−0.961512	−0.480756	−	0.876854i	$-0.659638\pi$
$613$	−23.8059	−0.961512	−0.480756	+	0.876854i	$0.659638\pi$
$614$	0	0
$615$	5.66195	0.228312
$616$	0	0
$617$	−21.6000	−0.869583	−0.434791	−	0.900531i	$-0.643178\pi$
$617$	−21.6000	−0.869583	−0.434791	+	0.900531i	$0.643178\pi$
$618$	0	0
$619$	−15.2773	−0.614046	−0.307023	−	0.951702i	$-0.599333\pi$
$619$	−15.2773	−0.614046	−0.307023	+	0.951702i	$0.599333\pi$
$620$	0	0
$621$	−10.2809	−0.412559
$622$	0	0
$623$	−2.68030	−0.107384
$624$	0	0
$625$	−17.2212	−0.688847
$626$	0	0
$627$	1.73502	0.0692900
$628$	0	0
$629$	25.9445	1.03447
$630$	0	0
$631$	29.2658	1.16505	0.582527	−	0.812811i	$-0.302064\pi$
$631$	29.2658	1.16505	0.582527	+	0.812811i	$0.302064\pi$
$632$	0	0
$633$	−5.12525	−0.203710
$634$	0	0
$635$	−30.3502	−1.20441
$636$	0	0
$637$	−25.1549	−0.996672
$638$	0	0
$639$	3.72280	0.147272
$640$	0	0
$641$	1.68377	0.0665048	0.0332524	−	0.999447i	$-0.489413\pi$
$641$	1.68377	0.0665048	0.0332524	+	0.999447i	$0.489413\pi$
$642$	0	0
$643$	−48.0543	−1.89508	−0.947538	−	0.319642i	$-0.896437\pi$
$643$	−48.0543	−1.89508	−0.947538	+	0.319642i	$0.896437\pi$
$644$	0	0
$645$	8.18644	0.322341
$646$	0	0
$647$	14.7176	0.578608	0.289304	−	0.957237i	$-0.406576\pi$
$647$	14.7176	0.578608	0.289304	+	0.957237i	$0.406576\pi$
$648$	0	0
$649$	74.4765	2.92346
$650$	0	0
$651$	0.318291	0.0124748
$652$	0	0
$653$	−21.4828	−0.840687	−0.420343	−	0.907365i	$-0.638090\pi$
$653$	−21.4828	−0.840687	−0.420343	+	0.907365i	$0.638090\pi$
$654$	0	0
$655$	24.5417	0.958922
$656$	0	0
$657$	26.9044	1.04964
$658$	0	0
$659$	34.3590	1.33844	0.669218	−	0.743066i	$-0.266629\pi$
$659$	34.3590	1.33844	0.669218	+	0.743066i	$0.266629\pi$
$660$	0	0
$661$	−17.4294	−0.677924	−0.338962	−	0.940800i	$-0.610076\pi$
$661$	−17.4294	−0.677924	−0.338962	+	0.940800i	$0.610076\pi$
$662$	0	0
$663$	−5.80707	−0.225528
$664$	0	0
$665$	−0.291773	−0.0113145
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−12.0626	−0.466366
$670$	0	0
$671$	88.7695	3.42691
$672$	0	0
$673$	−21.6549	−0.834735	−0.417367	−	0.908738i	$-0.637047\pi$
$673$	−21.6549	−0.834735	−0.417367	+	0.908738i	$0.637047\pi$
$674$	0	0
$675$	3.57107	0.137451
$676$	0	0
$677$	32.0534	1.23191	0.615956	−	0.787781i	$-0.288770\pi$
$677$	32.0534	1.23191	0.615956	+	0.787781i	$0.288770\pi$
$678$	0	0
$679$	1.21192	0.0465094
$680$	0	0
$681$	−9.09602	−0.348560
$682$	0	0
$683$	10.1276	0.387523	0.193762	−	0.981049i	$-0.437931\pi$
$683$	10.1276	0.387523	0.193762	+	0.981049i	$0.437931\pi$
$684$	0	0
$685$	−8.63914	−0.330085
$686$	0	0
$687$	−8.15792	−0.311244
$688$	0	0
$689$	−28.1352	−1.07186
$690$	0	0
$691$	25.9048	0.985465	0.492733	−	0.870181i	$-0.335998\pi$
$691$	25.9048	0.985465	0.492733	+	0.870181i	$0.335998\pi$
$692$	0	0
$693$	4.55016	0.172846
$694$	0	0
$695$	6.25272	0.237179
$696$	0	0
$697$	−18.8068	−0.712360
$698$	0	0
$699$	−6.89845	−0.260923
$700$	0	0
$701$	−3.55651	−0.134328	−0.0671638	−	0.997742i	$-0.521395\pi$
$701$	−3.55651	−0.134328	−0.0671638	+	0.997742i	$0.521395\pi$
$702$	0	0
$703$	4.55215	0.171688
$704$	0	0
$705$	−5.60165	−0.210970
$706$	0	0
$707$	−0.330413	−0.0124265
$708$	0	0
$709$	−34.8259	−1.30791	−0.653956	−	0.756532i	$-0.726892\pi$
$709$	−34.8259	−1.30791	−0.653956	+	0.756532i	$0.726892\pi$
$710$	0	0
$711$	−9.76613	−0.366258
$712$	0	0
$713$	8.55916	0.320543
$714$	0	0
$715$	−43.4933	−1.62656
$716$	0	0
$717$	0.390601	0.0145873
$718$	0	0
$719$	−7.83549	−0.292215	−0.146107	−	0.989269i	$-0.546674\pi$
$719$	−7.83549	−0.292215	−0.146107	+	0.989269i	$0.546674\pi$
$720$	0	0
$721$	2.89674	0.107880
$722$	0	0
$723$	11.7675	0.437639
$724$	0	0
$725$	0	0
$726$	0	0
$727$	7.51051	0.278549	0.139275	−	0.990254i	$-0.455523\pi$
$727$	7.51051	0.278549	0.139275	+	0.990254i	$0.455523\pi$
$728$	0	0
$729$	−14.4914	−0.536719
$730$	0	0
$731$	−27.1922	−1.00574
$732$	0	0
$733$	27.2379	1.00605	0.503027	−	0.864271i	$-0.332220\pi$
$733$	27.2379	1.00605	0.503027	+	0.864271i	$0.332220\pi$
$734$	0	0
$735$	−6.69245	−0.246855
$736$	0	0
$737$	−74.7617	−2.75388
$738$	0	0
$739$	−9.99385	−0.367630	−0.183815	−	0.982961i	$-0.558845\pi$
$739$	−9.99385	−0.367630	−0.183815	+	0.982961i	$0.558845\pi$
$740$	0	0
$741$	−1.01889	−0.0374300
$742$	0	0
$743$	−25.6412	−0.940686	−0.470343	−	0.882484i	$-0.655870\pi$
$743$	−25.6412	−0.940686	−0.470343	+	0.882484i	$0.655870\pi$
$744$	0	0
$745$	8.36198	0.306359
$746$	0	0
$747$	22.0233	0.805790
$748$	0	0
$749$	−2.33257	−0.0852301
$750$	0	0
$751$	40.1392	1.46470	0.732349	−	0.680929i	$-0.238424\pi$
$751$	40.1392	1.46470	0.732349	+	0.680929i	$0.238424\pi$
$752$	0	0
$753$	−6.31085	−0.229980
$754$	0	0
$755$	37.9428	1.38088
$756$	0	0
$757$	4.12630	0.149973	0.0749865	−	0.997185i	$-0.476109\pi$
$757$	4.12630	0.149973	0.0749865	+	0.997185i	$0.476109\pi$
$758$	0	0
$759$	−11.0513	−0.401136
$760$	0	0
$761$	−32.5074	−1.17839	−0.589196	−	0.807990i	$-0.700555\pi$
$761$	−32.5074	−1.17839	−0.589196	+	0.807990i	$0.700555\pi$
$762$	0	0
$763$	−4.31601	−0.156250
$764$	0	0
$765$	17.1057	0.618459
$766$	0	0
$767$	−43.7366	−1.57924
$768$	0	0
$769$	11.6275	0.419300	0.209650	−	0.977777i	$-0.432768\pi$
$769$	11.6275	0.419300	0.209650	+	0.977777i	$0.432768\pi$
$770$	0	0
$771$	5.15757	0.185745
$772$	0	0
$773$	7.96103	0.286338	0.143169	−	0.989698i	$-0.454271\pi$
$773$	7.96103	0.286338	0.143169	+	0.989698i	$0.454271\pi$
$774$	0	0
$775$	−2.97302	−0.106794
$776$	0	0
$777$	−1.07825	−0.0386819
$778$	0	0
$779$	−3.29980	−0.118228
$780$	0	0
$781$	8.36493	0.299321
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1.10216	0.0393378
$786$	0	0
$787$	2.35358	0.0838962	0.0419481	−	0.999120i	$-0.486644\pi$
$787$	2.35358	0.0838962	0.0419481	+	0.999120i	$0.486644\pi$
$788$	0	0
$789$	−7.99581	−0.284658
$790$	0	0
$791$	−4.24859	−0.151062
$792$	0	0
$793$	−52.1302	−1.85120
$794$	0	0
$795$	−7.48535	−0.265478
$796$	0	0
$797$	−19.0106	−0.673391	−0.336695	−	0.941614i	$-0.609309\pi$
$797$	−19.0106	−0.673391	−0.336695	+	0.941614i	$0.609309\pi$
$798$	0	0
$799$	18.6065	0.658252
$800$	0	0
$801$	27.5711	0.974176
$802$	0	0
$803$	60.4528	2.13333
$804$	0	0
$805$	1.85846	0.0655021
$806$	0	0
$807$	−4.45309	−0.156756
$808$	0	0
$809$	35.6176	1.25225	0.626124	−	0.779724i	$-0.284641\pi$
$809$	35.6176	1.25225	0.626124	+	0.779724i	$0.284641\pi$
$810$	0	0
$811$	−15.6295	−0.548827	−0.274414	−	0.961612i	$-0.588484\pi$
$811$	−15.6295	−0.548827	−0.274414	+	0.961612i	$0.588484\pi$
$812$	0	0
$813$	12.9936	0.455705
$814$	0	0
$815$	36.5572	1.28054
$816$	0	0
$817$	−4.77108	−0.166919
$818$	0	0
$819$	−2.67210	−0.0933706
$820$	0	0
$821$	−14.8071	−0.516773	−0.258386	−	0.966042i	$-0.583191\pi$
$821$	−14.8071	−0.516773	−0.258386	+	0.966042i	$0.583191\pi$
$822$	0	0
$823$	14.8341	0.517083	0.258542	−	0.966000i	$-0.416758\pi$
$823$	14.8341	0.517083	0.258542	+	0.966000i	$0.416758\pi$
$824$	0	0
$825$	3.83866	0.133645
$826$	0	0
$827$	34.0092	1.18262	0.591308	−	0.806446i	$-0.298612\pi$
$827$	34.0092	1.18262	0.591308	+	0.806446i	$0.298612\pi$
$828$	0	0
$829$	−25.0496	−0.870010	−0.435005	−	0.900428i	$-0.643253\pi$
$829$	−25.0496	−0.870010	−0.435005	+	0.900428i	$0.643253\pi$
$830$	0	0
$831$	−9.39752	−0.325996
$832$	0	0
$833$	22.2297	0.770215
$834$	0	0
$835$	−23.4413	−0.811219
$836$	0	0
$837$	−6.84393	−0.236561
$838$	0	0
$839$	−40.5261	−1.39912	−0.699558	−	0.714576i	$-0.746620\pi$
$839$	−40.5261	−1.39912	−0.699558	+	0.714576i	$0.746620\pi$
$840$	0	0
$841$	0	0
$842$	0	0
$843$	10.1850	0.350788
$844$	0	0
$845$	0.352099	0.0121126
$846$	0	0
$847$	7.28165	0.250200
$848$	0	0
$849$	12.0395	0.413196
$850$	0	0
$851$	−28.9952	−0.993941
$852$	0	0
$853$	24.9274	0.853497	0.426749	−	0.904370i	$-0.359659\pi$
$853$	24.9274	0.853497	0.426749	+	0.904370i	$0.359659\pi$
$854$	0	0
$855$	3.00133	0.102643
$856$	0	0
$857$	6.91827	0.236324	0.118162	−	0.992994i	$-0.462300\pi$
$857$	6.91827	0.236324	0.118162	+	0.992994i	$0.462300\pi$
$858$	0	0
$859$	11.7507	0.400928	0.200464	−	0.979701i	$-0.435755\pi$
$859$	11.7507	0.400928	0.200464	+	0.979701i	$0.435755\pi$
$860$	0	0
$861$	0.781608	0.0266371
$862$	0	0
$863$	−17.1630	−0.584235	−0.292117	−	0.956382i	$-0.594360\pi$
$863$	−17.1630	−0.584235	−0.292117	+	0.956382i	$0.594360\pi$
$864$	0	0
$865$	48.5186	1.64968
$866$	0	0
$867$	−3.34286	−0.113529
$868$	0	0
$869$	−21.9440	−0.744398
$870$	0	0
$871$	43.9041	1.48763
$872$	0	0
$873$	−12.4665	−0.421927
$874$	0	0
$875$	−3.23699	−0.109430
$876$	0	0
$877$	14.9090	0.503440	0.251720	−	0.967800i	$-0.419004\pi$
$877$	14.9090	0.503440	0.251720	+	0.967800i	$0.419004\pi$
$878$	0	0
$879$	−0.449199	−0.0151511
$880$	0	0
$881$	−35.7490	−1.20441	−0.602207	−	0.798340i	$-0.705712\pi$
$881$	−35.7490	−1.20441	−0.602207	+	0.798340i	$0.705712\pi$
$882$	0	0
$883$	49.6477	1.67078	0.835390	−	0.549658i	$-0.185242\pi$
$883$	49.6477	1.67078	0.835390	+	0.549658i	$0.185242\pi$
$884$	0	0
$885$	−11.6361	−0.391143
$886$	0	0
$887$	−30.5983	−1.02739	−0.513695	−	0.857973i	$-0.671724\pi$
$887$	−30.5983	−1.02739	−0.513695	+	0.857973i	$0.671724\pi$
$888$	0	0
$889$	−4.18971	−0.140518
$890$	0	0
$891$	−42.1962	−1.41363
$892$	0	0
$893$	3.26466	0.109248
$894$	0	0
$895$	19.2642	0.643931
$896$	0	0
$897$	6.48990	0.216691
$898$	0	0
$899$	0	0
$900$	0	0
$901$	24.8635	0.828322
$902$	0	0
$903$	1.13010	0.0376075
$904$	0	0
$905$	−9.58029	−0.318460
$906$	0	0
$907$	12.3531	0.410177	0.205088	−	0.978743i	$-0.434252\pi$
$907$	12.3531	0.410177	0.205088	+	0.978743i	$0.434252\pi$
$908$	0	0
$909$	3.39881	0.112731
$910$	0	0
$911$	−42.2608	−1.40016	−0.700081	−	0.714064i	$-0.746853\pi$
$911$	−42.2608	−1.40016	−0.700081	+	0.714064i	$0.746853\pi$
$912$	0	0
$913$	49.4852	1.63772
$914$	0	0
$915$	−13.8692	−0.458502
$916$	0	0
$917$	3.38787	0.111877
$918$	0	0
$919$	−6.72007	−0.221675	−0.110837	−	0.993839i	$-0.535353\pi$
$919$	−6.72007	−0.221675	−0.110837	+	0.993839i	$0.535353\pi$
$920$	0	0
$921$	−1.13005	−0.0372366
$922$	0	0
$923$	−4.91233	−0.161691
$924$	0	0
$925$	10.0715	0.331148
$926$	0	0
$927$	−29.7975	−0.978678
$928$	0	0
$929$	37.2715	1.22284	0.611419	−	0.791307i	$-0.290599\pi$
$929$	37.2715	1.22284	0.611419	+	0.791307i	$0.290599\pi$
$930$	0	0
$931$	3.90038	0.127830
$932$	0	0
$933$	15.3837	0.503638
$934$	0	0
$935$	38.4357	1.25698
$936$	0	0
$937$	5.20730	0.170115	0.0850576	−	0.996376i	$-0.472893\pi$
$937$	5.20730	0.170115	0.0850576	+	0.996376i	$0.472893\pi$
$938$	0	0
$939$	3.18010	0.103779
$940$	0	0
$941$	0.339552	0.0110691	0.00553454	−	0.999985i	$-0.498238\pi$
$941$	0.339552	0.0110691	0.00553454	+	0.999985i	$0.498238\pi$
$942$	0	0
$943$	21.0182	0.684448
$944$	0	0
$945$	−1.48603	−0.0483405
$946$	0	0
$947$	−31.9120	−1.03700	−0.518500	−	0.855077i	$-0.673510\pi$
$947$	−31.9120	−1.03700	−0.518500	+	0.855077i	$0.673510\pi$
$948$	0	0
$949$	−35.5011	−1.15241
$950$	0	0
$951$	−5.24974	−0.170234
$952$	0	0
$953$	14.9302	0.483637	0.241819	−	0.970321i	$-0.422256\pi$
$953$	14.9302	0.483637	0.241819	+	0.970321i	$0.422256\pi$
$954$	0	0
$955$	14.5344	0.470323
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−1.19260	−0.0385109
$960$	0	0
$961$	−25.3022	−0.816201
$962$	0	0
$963$	23.9940	0.773197
$964$	0	0
$965$	23.0228	0.741130
$966$	0	0
$967$	25.5534	0.821741	0.410871	−	0.911694i	$-0.365225\pi$
$967$	25.5534	0.821741	0.410871	+	0.911694i	$0.365225\pi$
$968$	0	0
$969$	0.900412	0.0289254
$970$	0	0
$971$	−33.0036	−1.05914	−0.529568	−	0.848268i	$-0.677646\pi$
$971$	−33.0036	−1.05914	−0.529568	+	0.848268i	$0.677646\pi$
$972$	0	0
$973$	0.863161	0.0276717
$974$	0	0
$975$	−2.25426	−0.0721942
$976$	0	0
$977$	27.5051	0.879966	0.439983	−	0.898006i	$-0.354984\pi$
$977$	27.5051	0.879966	0.439983	+	0.898006i	$0.354984\pi$
$978$	0	0
$979$	61.9507	1.97995
$980$	0	0
$981$	44.3968	1.41748
$982$	0	0
$983$	54.4728	1.73741	0.868707	−	0.495327i	$-0.164952\pi$
$983$	54.4728	1.73741	0.868707	+	0.495327i	$0.164952\pi$
$984$	0	0
$985$	43.2852	1.37918
$986$	0	0
$987$	−0.773283	−0.0246139
$988$	0	0
$989$	30.3896	0.966334
$990$	0	0
$991$	−3.84175	−0.122037	−0.0610186	−	0.998137i	$-0.519435\pi$
$991$	−3.84175	−0.122037	−0.0610186	+	0.998137i	$0.519435\pi$
$992$	0	0
$993$	1.53376	0.0486724
$994$	0	0
$995$	−21.6335	−0.685829
$996$	0	0
$997$	−3.05652	−0.0968011	−0.0484005	−	0.998828i	$-0.515412\pi$
$997$	−3.05652	−0.0968011	−0.0484005	+	0.998828i	$0.515412\pi$
$998$	0	0
$999$	23.1846	0.733529

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3364.2.a.q.1.6	✓	8
29.12	odd	4		3364.2.c.k.1681.11		16
29.17	odd	4		3364.2.c.k.1681.6		16
29.28	even	2		3364.2.a.r.1.3	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3364.2.a.q.1.6	✓	8	1.1	even	1	trivial
3364.2.a.r.1.3	yes	8	29.28	even	2
3364.2.c.k.1681.6		16	29.17	odd	4
3364.2.c.k.1681.11		16	29.12	odd	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3364 = 2^{2} \cdot 29^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3364.a (trivial)

\(f(q)\)	\(=\)	\(q+0.498509 q^{3} +1.93765 q^{5} +0.267484 q^{7} -2.75149 q^{9} -6.18245 q^{11} +3.63066 q^{13} +0.965937 q^{15} -3.20847 q^{17} -0.562951 q^{19} +0.133343 q^{21} +3.58574 q^{23} -1.24551 q^{25} -2.86717 q^{27} +2.38700 q^{31} -3.08201 q^{33} +0.518291 q^{35} -8.08624 q^{37} +1.80992 q^{39} +5.86162 q^{41} +8.47514 q^{43} -5.33143 q^{45} -5.79919 q^{47} -6.92845 q^{49} -1.59945 q^{51} -7.74932 q^{53} -11.9794 q^{55} -0.280636 q^{57} -12.0464 q^{59} -14.3583 q^{61} -0.735980 q^{63} +7.03496 q^{65} +12.0926 q^{67} +1.78752 q^{69} -1.35301 q^{71} -9.77812 q^{73} -0.620896 q^{75} -1.65371 q^{77} +3.54940 q^{79} +6.82516 q^{81} -8.00413 q^{83} -6.21690 q^{85} -10.0204 q^{89} +0.971145 q^{91} +1.18994 q^{93} -1.09080 q^{95} +4.53082 q^{97} +17.0109 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{3} - q^{5} - 2 q^{7} + 4 q^{9} - 5 q^{11} + 4 q^{13} + 17 q^{15} - 15 q^{17} - 11 q^{19} - 19 q^{21} - 3 q^{23} + 3 q^{25} - 16 q^{27} + 9 q^{31} - 16 q^{33} + 9 q^{35} + 2 q^{37} - 9 q^{39}+ \cdots + 11 q^{99}+O(q^{100}) \) 8 * q - 4 * q^3 - q^5 - 2 * q^7 + 4 * q^9 - 5 * q^11 + 4 * q^13 + 17 * q^15 - 15 * q^17 - 11 * q^19 - 19 * q^21 - 3 * q^23 + 3 * q^25 - 16 * q^27 + 9 * q^31 - 16 * q^33 + 9 * q^35 + 2 * q^37 - 9 * q^39 - 38 * q^41 + 3 * q^43 + 3 * q^45 + 7 * q^47 + 12 * q^49 + 3 * q^51 - 5 * q^53 - 34 * q^55 + 34 * q^57 - 7 * q^59 - 39 * q^61 - 6 * q^63 - 21 * q^65 - 26 * q^67 - 34 * q^69 + 52 * q^71 + 34 * q^73 - 10 * q^77 - 31 * q^79 + 24 * q^81 - 17 * q^83 - 18 * q^85 - 24 * q^89 + 54 * q^91 + 6 * q^93 - 9 * q^95 - 22 * q^97 + 11 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more