LMFDB - Embedded newform 9264.2.a.bp.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9264,2,Mod(1,9264)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9264, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("9264.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9264 = 2^{4} \cdot 3 \cdot 193 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9264.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:	$73.9734124325$
Analytic rank:	$0$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 2 x^{12} - 20 x^{11} + 39 x^{10} + 148 x^{9} - 275 x^{8} - 508 x^{7} + 865 x^{6} + 823 x^{5} + \cdots - 48 $ x^13 - 2x^12 - 20x^11 + 39x^10 + 148x^9 - 275x^8 - 508x^7 + 865x^6 + 823x^5 - 1187x^4 - 556x^3 + 576x^2 + 64x - 48
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 579)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$0.294774$ of defining polynomial
Character	$\chi$	$=$	9264.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +0.386624 q^{5} -2.95689 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +0.386624 q^{5} -2.95689 q^{7} +1.00000 q^{9} +1.64913 q^{11} +2.41755 q^{13} -0.386624 q^{15} +1.61931 q^{17} -4.40371 q^{19} +2.95689 q^{21} -6.60080 q^{23} -4.85052 q^{25} -1.00000 q^{27} -1.93504 q^{29} +4.51636 q^{31} -1.64913 q^{33} -1.14321 q^{35} +5.95105 q^{37} -2.41755 q^{39} +5.29889 q^{41} -8.10944 q^{43} +0.386624 q^{45} -0.468667 q^{47} +1.74322 q^{49} -1.61931 q^{51} -13.5817 q^{53} +0.637594 q^{55} +4.40371 q^{57} +11.3947 q^{59} +8.10307 q^{61} -2.95689 q^{63} +0.934684 q^{65} +1.47560 q^{67} +6.60080 q^{69} +12.1544 q^{71} +4.78747 q^{73} +4.85052 q^{75} -4.87631 q^{77} +6.29813 q^{79} +1.00000 q^{81} +4.71520 q^{83} +0.626064 q^{85} +1.93504 q^{87} -3.18967 q^{89} -7.14845 q^{91} -4.51636 q^{93} -1.70258 q^{95} +9.16299 q^{97} +1.64913 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 13 q^{3} + 6 q^{5} - 15 q^{7} + 13 q^{9}+O(q^{10}) $ 13 * q - 13 * q^3 + 6 * q^5 - 15 * q^7 + 13 * q^9	$ 13 q - 13 q^{3} + 6 q^{5} - 15 q^{7} + 13 q^{9} - 3 q^{11} + 11 q^{13} - 6 q^{15} - 2 q^{17} - 9 q^{19} + 15 q^{21} + 8 q^{23} + 21 q^{25} - 13 q^{27} + 5 q^{29} - 25 q^{31} + 3 q^{33} + 10 q^{35} + 29 q^{37} - 11 q^{39} - 11 q^{41} - 8 q^{43} + 6 q^{45} + 12 q^{47} + 20 q^{49} + 2 q^{51} + 14 q^{53} - 12 q^{55} + 9 q^{57} - 10 q^{59} + 6 q^{61} - 15 q^{63} - 15 q^{65} - 25 q^{67} - 8 q^{69} + 8 q^{73} - 21 q^{75} - 25 q^{77} - 7 q^{79} + 13 q^{81} + 28 q^{83} - 3 q^{85} - 5 q^{87} + 7 q^{89} - 7 q^{91} + 25 q^{93} + 26 q^{95} + 26 q^{97} - 3 q^{99}+O(q^{100}) $ 13 * q - 13 * q^3 + 6 * q^5 - 15 * q^7 + 13 * q^9 - 3 * q^11 + 11 * q^13 - 6 * q^15 - 2 * q^17 - 9 * q^19 + 15 * q^21 + 8 * q^23 + 21 * q^25 - 13 * q^27 + 5 * q^29 - 25 * q^31 + 3 * q^33 + 10 * q^35 + 29 * q^37 - 11 * q^39 - 11 * q^41 - 8 * q^43 + 6 * q^45 + 12 * q^47 + 20 * q^49 + 2 * q^51 + 14 * q^53 - 12 * q^55 + 9 * q^57 - 10 * q^59 + 6 * q^61 - 15 * q^63 - 15 * q^65 - 25 * q^67 - 8 * q^69 + 8 * q^73 - 21 * q^75 - 25 * q^77 - 7 * q^79 + 13 * q^81 + 28 * q^83 - 3 * q^85 - 5 * q^87 + 7 * q^89 - 7 * q^91 + 25 * q^93 + 26 * q^95 + 26 * q^97 - 3 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	0.386624	0.172903	0.0864517	−	0.996256i	$-0.472447\pi$
$5$	0.386624	0.172903	0.0864517	+	0.996256i	$0.472447\pi$
$6$	0	0
$7$	−2.95689	−1.11760	−0.558800	−	0.829302i	$-0.688738\pi$
$7$	−2.95689	−1.11760	−0.558800	+	0.829302i	$0.688738\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.64913	0.497232	0.248616	−	0.968602i	$-0.420024\pi$
$11$	1.64913	0.497232	0.248616	+	0.968602i	$0.420024\pi$
$12$	0	0
$13$	2.41755	0.670509	0.335254	−	0.942128i	$-0.391178\pi$
$13$	2.41755	0.670509	0.335254	+	0.942128i	$0.391178\pi$
$14$	0	0
$15$	−0.386624	−0.0998258
$16$	0	0
$17$	1.61931	0.392740	0.196370	−	0.980530i	$-0.437085\pi$
$17$	1.61931	0.392740	0.196370	+	0.980530i	$0.437085\pi$
$18$	0	0
$19$	−4.40371	−1.01028	−0.505140	−	0.863037i	$-0.668559\pi$
$19$	−4.40371	−1.01028	−0.505140	+	0.863037i	$0.668559\pi$
$20$	0	0
$21$	2.95689	0.645247
$22$	0	0
$23$	−6.60080	−1.37636	−0.688181	−	0.725539i	$-0.741591\pi$
$23$	−6.60080	−1.37636	−0.688181	+	0.725539i	$0.741591\pi$
$24$	0	0
$25$	−4.85052	−0.970104
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−1.93504	−0.359328	−0.179664	−	0.983728i	$-0.557501\pi$
$29$	−1.93504	−0.359328	−0.179664	+	0.983728i	$0.557501\pi$
$30$	0	0
$31$	4.51636	0.811162	0.405581	−	0.914059i	$-0.367069\pi$
$31$	4.51636	0.811162	0.405581	+	0.914059i	$0.367069\pi$
$32$	0	0
$33$	−1.64913	−0.287077
$34$	0	0
$35$	−1.14321	−0.193237
$36$	0	0
$37$	5.95105	0.978346	0.489173	−	0.872187i	$-0.337299\pi$
$37$	5.95105	0.978346	0.489173	+	0.872187i	$0.337299\pi$
$38$	0	0
$39$	−2.41755	−0.387118
$40$	0	0
$41$	5.29889	0.827547	0.413774	−	0.910380i	$-0.364210\pi$
$41$	5.29889	0.827547	0.413774	+	0.910380i	$0.364210\pi$
$42$	0	0
$43$	−8.10944	−1.23668	−0.618339	−	0.785912i	$-0.712194\pi$
$43$	−8.10944	−1.23668	−0.618339	+	0.785912i	$0.712194\pi$
$44$	0	0
$45$	0.386624	0.0576345
$46$	0	0
$47$	−0.468667	−0.0683622	−0.0341811	−	0.999416i	$-0.510882\pi$
$47$	−0.468667	−0.0683622	−0.0341811	+	0.999416i	$0.510882\pi$
$48$	0	0
$49$	1.74322	0.249032
$50$	0	0
$51$	−1.61931	−0.226749
$52$	0	0
$53$	−13.5817	−1.86559	−0.932795	−	0.360408i	$-0.882638\pi$
$53$	−13.5817	−1.86559	−0.932795	+	0.360408i	$0.882638\pi$
$54$	0	0
$55$	0.637594	0.0859732
$56$	0	0
$57$	4.40371	0.583285
$58$	0	0
$59$	11.3947	1.48346	0.741730	−	0.670699i	$-0.234006\pi$
$59$	11.3947	1.48346	0.741730	+	0.670699i	$0.234006\pi$
$60$	0	0
$61$	8.10307	1.03749	0.518746	−	0.854929i	$-0.326399\pi$
$61$	8.10307	1.03749	0.518746	+	0.854929i	$0.326399\pi$
$62$	0	0
$63$	−2.95689	−0.372534
$64$	0	0
$65$	0.934684	0.115933
$66$	0	0
$67$	1.47560	0.180273	0.0901366	−	0.995929i	$-0.471270\pi$
$67$	1.47560	0.180273	0.0901366	+	0.995929i	$0.471270\pi$
$68$	0	0
$69$	6.60080	0.794643
$70$	0	0
$71$	12.1544	1.44246	0.721231	−	0.692695i	$-0.243577\pi$
$71$	12.1544	1.44246	0.721231	+	0.692695i	$0.243577\pi$
$72$	0	0
$73$	4.78747	0.560331	0.280166	−	0.959952i	$-0.409611\pi$
$73$	4.78747	0.560331	0.280166	+	0.959952i	$0.409611\pi$
$74$	0	0
$75$	4.85052	0.560090
$76$	0	0
$77$	−4.87631	−0.555707
$78$	0	0
$79$	6.29813	0.708595	0.354298	−	0.935133i	$-0.384720\pi$
$79$	6.29813	0.708595	0.354298	+	0.935133i	$0.384720\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.71520	0.517560	0.258780	−	0.965936i	$-0.416679\pi$
$83$	4.71520	0.517560	0.258780	+	0.965936i	$0.416679\pi$
$84$	0	0
$85$	0.626064	0.0679062
$86$	0	0
$87$	1.93504	0.207458
$88$	0	0
$89$	−3.18967	−0.338105	−0.169052	−	0.985607i	$-0.554071\pi$
$89$	−3.18967	−0.338105	−0.169052	+	0.985607i	$0.554071\pi$
$90$	0	0
$91$	−7.14845	−0.749361
$92$	0	0
$93$	−4.51636	−0.468325
$94$	0	0
$95$	−1.70258	−0.174681
$96$	0	0
$97$	9.16299	0.930360	0.465180	−	0.885216i	$-0.345990\pi$
$97$	9.16299	0.930360	0.465180	+	0.885216i	$0.345990\pi$
$98$	0	0
$99$	1.64913	0.165744
$100$	0	0
$101$	−4.65337	−0.463028	−0.231514	−	0.972832i	$-0.574368\pi$
$101$	−4.65337	−0.463028	−0.231514	+	0.972832i	$0.574368\pi$
$102$	0	0
$103$	10.5042	1.03501	0.517505	−	0.855680i	$-0.326861\pi$
$103$	10.5042	1.03501	0.517505	+	0.855680i	$0.326861\pi$
$104$	0	0
$105$	1.14321	0.111565
$106$	0	0
$107$	−11.1723	−1.08006	−0.540031	−	0.841645i	$-0.681588\pi$
$107$	−11.1723	−1.08006	−0.540031	+	0.841645i	$0.681588\pi$
$108$	0	0
$109$	−12.3271	−1.18072	−0.590360	−	0.807140i	$-0.701014\pi$
$109$	−12.3271	−1.18072	−0.590360	+	0.807140i	$0.701014\pi$
$110$	0	0
$111$	−5.95105	−0.564848
$112$	0	0
$113$	−10.5371	−0.991244	−0.495622	−	0.868538i	$-0.665060\pi$
$113$	−10.5371	−0.991244	−0.495622	+	0.868538i	$0.665060\pi$
$114$	0	0
$115$	−2.55203	−0.237978
$116$	0	0
$117$	2.41755	0.223503
$118$	0	0
$119$	−4.78813	−0.438927
$120$	0	0
$121$	−8.28036	−0.752760
$122$	0	0
$123$	−5.29889	−0.477785
$124$	0	0
$125$	−3.80845	−0.340638
$126$	0	0
$127$	−10.1860	−0.903860	−0.451930	−	0.892054i	$-0.649264\pi$
$127$	−10.1860	−0.903860	−0.451930	+	0.892054i	$0.649264\pi$
$128$	0	0
$129$	8.10944	0.713996
$130$	0	0
$131$	14.0131	1.22433	0.612166	−	0.790729i	$-0.290298\pi$
$131$	14.0131	1.22433	0.612166	+	0.790729i	$0.290298\pi$
$132$	0	0
$133$	13.0213	1.12909
$134$	0	0
$135$	−0.386624	−0.0332753
$136$	0	0
$137$	−2.28952	−0.195607	−0.0978034	−	0.995206i	$-0.531182\pi$
$137$	−2.28952	−0.195607	−0.0978034	+	0.995206i	$0.531182\pi$
$138$	0	0
$139$	−5.39959	−0.457987	−0.228994	−	0.973428i	$-0.573544\pi$
$139$	−5.39959	−0.457987	−0.228994	+	0.973428i	$0.573544\pi$
$140$	0	0
$141$	0.468667	0.0394689
$142$	0	0
$143$	3.98687	0.333399
$144$	0	0
$145$	−0.748133	−0.0621290
$146$	0	0
$147$	−1.74322	−0.143779
$148$	0	0
$149$	−18.9607	−1.55332	−0.776660	−	0.629920i	$-0.783088\pi$
$149$	−18.9607	−1.55332	−0.776660	+	0.629920i	$0.783088\pi$
$150$	0	0
$151$	−3.13560	−0.255172	−0.127586	−	0.991828i	$-0.540723\pi$
$151$	−3.13560	−0.255172	−0.127586	+	0.991828i	$0.540723\pi$
$152$	0	0
$153$	1.61931	0.130913
$154$	0	0
$155$	1.74613	0.140253
$156$	0	0
$157$	−0.982994	−0.0784514	−0.0392257	−	0.999230i	$-0.512489\pi$
$157$	−0.982994	−0.0784514	−0.0392257	+	0.999230i	$0.512489\pi$
$158$	0	0
$159$	13.5817	1.07710
$160$	0	0
$161$	19.5179	1.53822
$162$	0	0
$163$	−9.07562	−0.710858	−0.355429	−	0.934703i	$-0.615665\pi$
$163$	−9.07562	−0.710858	−0.355429	+	0.934703i	$0.615665\pi$
$164$	0	0
$165$	−0.637594	−0.0496366
$166$	0	0
$167$	−8.80527	−0.681372	−0.340686	−	0.940177i	$-0.610659\pi$
$167$	−8.80527	−0.681372	−0.340686	+	0.940177i	$0.610659\pi$
$168$	0	0
$169$	−7.15544	−0.550418
$170$	0	0
$171$	−4.40371	−0.336760
$172$	0	0
$173$	10.4539	0.794797	0.397399	−	0.917646i	$-0.369913\pi$
$173$	10.4539	0.794797	0.397399	+	0.917646i	$0.369913\pi$
$174$	0	0
$175$	14.3425	1.08419
$176$	0	0
$177$	−11.3947	−0.856476
$178$	0	0
$179$	−0.748038	−0.0559110	−0.0279555	−	0.999609i	$-0.508900\pi$
$179$	−0.748038	−0.0559110	−0.0279555	+	0.999609i	$0.508900\pi$
$180$	0	0
$181$	23.6329	1.75662	0.878312	−	0.478089i	$-0.158670\pi$
$181$	23.6329	1.75662	0.878312	+	0.478089i	$0.158670\pi$
$182$	0	0
$183$	−8.10307	−0.598996
$184$	0	0
$185$	2.30082	0.169159
$186$	0	0
$187$	2.67046	0.195283
$188$	0	0
$189$	2.95689	0.215082
$190$	0	0
$191$	2.29905	0.166354	0.0831768	−	0.996535i	$-0.473493\pi$
$191$	2.29905	0.166354	0.0831768	+	0.996535i	$0.473493\pi$
$192$	0	0
$193$	−1.00000	−0.0719816
$193$	−1.00000	−0.0719816
$194$	0	0
$195$	−0.934684	−0.0669341
$196$	0	0
$197$	10.4443	0.744126	0.372063	−	0.928208i	$-0.378651\pi$
$197$	10.4443	0.744126	0.372063	+	0.928208i	$0.378651\pi$
$198$	0	0
$199$	22.2811	1.57947	0.789734	−	0.613450i	$-0.210219\pi$
$199$	22.2811	1.57947	0.789734	+	0.613450i	$0.210219\pi$
$200$	0	0
$201$	−1.47560	−0.104081
$202$	0	0
$203$	5.72171	0.401585
$204$	0	0
$205$	2.04868	0.143086
$206$	0	0
$207$	−6.60080	−0.458788
$208$	0	0
$209$	−7.26230	−0.502344
$210$	0	0
$211$	15.0302	1.03473	0.517363	−	0.855766i	$-0.326914\pi$
$211$	15.0302	1.03473	0.517363	+	0.855766i	$0.326914\pi$
$212$	0	0
$213$	−12.1544	−0.832805
$214$	0	0
$215$	−3.13530	−0.213826
$216$	0	0
$217$	−13.3544	−0.906556
$218$	0	0
$219$	−4.78747	−0.323507
$220$	0	0
$221$	3.91477	0.263336
$222$	0	0
$223$	−1.92583	−0.128963	−0.0644815	−	0.997919i	$-0.520539\pi$
$223$	−1.92583	−0.128963	−0.0644815	+	0.997919i	$0.520539\pi$
$224$	0	0
$225$	−4.85052	−0.323368
$226$	0	0
$227$	−15.3169	−1.01662	−0.508311	−	0.861174i	$-0.669730\pi$
$227$	−15.3169	−1.01662	−0.508311	+	0.861174i	$0.669730\pi$
$228$	0	0
$229$	−8.99568	−0.594451	−0.297226	−	0.954807i	$-0.596061\pi$
$229$	−8.99568	−0.594451	−0.297226	+	0.954807i	$0.596061\pi$
$230$	0	0
$231$	4.87631	0.320838
$232$	0	0
$233$	−9.57715	−0.627420	−0.313710	−	0.949519i	$-0.601572\pi$
$233$	−9.57715	−0.627420	−0.313710	+	0.949519i	$0.601572\pi$
$234$	0	0
$235$	−0.181198	−0.0118200
$236$	0	0
$237$	−6.29813	−0.409108
$238$	0	0
$239$	9.43060	0.610015	0.305007	−	0.952350i	$-0.401341\pi$
$239$	9.43060	0.610015	0.305007	+	0.952350i	$0.401341\pi$
$240$	0	0
$241$	26.1794	1.68636	0.843181	−	0.537629i	$-0.180680\pi$
$241$	26.1794	1.68636	0.843181	+	0.537629i	$0.180680\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0.673971	0.0430584
$246$	0	0
$247$	−10.6462	−0.677401
$248$	0	0
$249$	−4.71520	−0.298814
$250$	0	0
$251$	19.3074	1.21868	0.609338	−	0.792911i	$-0.291435\pi$
$251$	19.3074	1.21868	0.609338	+	0.792911i	$0.291435\pi$
$252$	0	0
$253$	−10.8856	−0.684372
$254$	0	0
$255$	−0.626064	−0.0392056
$256$	0	0
$257$	26.4489	1.64984	0.824919	−	0.565251i	$-0.191221\pi$
$257$	26.4489	1.64984	0.824919	+	0.565251i	$0.191221\pi$
$258$	0	0
$259$	−17.5966	−1.09340
$260$	0	0
$261$	−1.93504	−0.119776
$262$	0	0
$263$	−0.0522246	−0.00322031	−0.00161015	−	0.999999i	$-0.500513\pi$
$263$	−0.0522246	−0.00322031	−0.00161015	+	0.999999i	$0.500513\pi$
$264$	0	0
$265$	−5.25101	−0.322567
$266$	0	0
$267$	3.18967	0.195205
$268$	0	0
$269$	28.9995	1.76813	0.884066	−	0.467363i	$-0.154796\pi$
$269$	28.9995	1.76813	0.884066	+	0.467363i	$0.154796\pi$
$270$	0	0
$271$	−9.97419	−0.605889	−0.302944	−	0.953008i	$-0.597970\pi$
$271$	−9.97419	−0.605889	−0.302944	+	0.953008i	$0.597970\pi$
$272$	0	0
$273$	7.14845	0.432644
$274$	0	0
$275$	−7.99916	−0.482367
$276$	0	0
$277$	30.8325	1.85254	0.926272	−	0.376856i	$-0.122995\pi$
$277$	30.8325	1.85254	0.926272	+	0.376856i	$0.122995\pi$
$278$	0	0
$279$	4.51636	0.270387
$280$	0	0
$281$	−24.7955	−1.47918	−0.739588	−	0.673060i	$-0.764980\pi$
$281$	−24.7955	−1.47918	−0.739588	+	0.673060i	$0.764980\pi$
$282$	0	0
$283$	11.5613	0.687245	0.343623	−	0.939108i	$-0.388346\pi$
$283$	11.5613	0.687245	0.343623	+	0.939108i	$0.388346\pi$
$284$	0	0
$285$	1.70258	0.100852
$286$	0	0
$287$	−15.6683	−0.924868
$288$	0	0
$289$	−14.3778	−0.845755
$290$	0	0
$291$	−9.16299	−0.537144
$292$	0	0
$293$	−0.0556420	−0.00325064	−0.00162532	−	0.999999i	$-0.500517\pi$
$293$	−0.0556420	−0.00325064	−0.00162532	+	0.999999i	$0.500517\pi$
$294$	0	0
$295$	4.40545	0.256495
$296$	0	0
$297$	−1.64913	−0.0956924
$298$	0	0
$299$	−15.9578	−0.922863
$300$	0	0
$301$	23.9788	1.38211
$302$	0	0
$303$	4.65337	0.267329
$304$	0	0
$305$	3.13284	0.179386
$306$	0	0
$307$	−9.31741	−0.531773	−0.265886	−	0.964004i	$-0.585665\pi$
$307$	−9.31741	−0.531773	−0.265886	+	0.964004i	$0.585665\pi$
$308$	0	0
$309$	−10.5042	−0.597563
$310$	0	0
$311$	10.4901	0.594842	0.297421	−	0.954746i	$-0.403874\pi$
$311$	10.4901	0.594842	0.297421	+	0.954746i	$0.403874\pi$
$312$	0	0
$313$	5.09321	0.287885	0.143943	−	0.989586i	$-0.454022\pi$
$313$	5.09321	0.287885	0.143943	+	0.989586i	$0.454022\pi$
$314$	0	0
$315$	−1.14321	−0.0644123
$316$	0	0
$317$	10.2012	0.572954	0.286477	−	0.958087i	$-0.407516\pi$
$317$	10.2012	0.572954	0.286477	+	0.958087i	$0.407516\pi$
$318$	0	0
$319$	−3.19114	−0.178669
$320$	0	0
$321$	11.1723	0.623574
$322$	0	0
$323$	−7.13097	−0.396778
$324$	0	0
$325$	−11.7264	−0.650463
$326$	0	0
$327$	12.3271	0.681689
$328$	0	0
$329$	1.38580	0.0764016
$330$	0	0
$331$	5.48496	0.301481	0.150740	−	0.988573i	$-0.451834\pi$
$331$	5.48496	0.301481	0.150740	+	0.988573i	$0.451834\pi$
$332$	0	0
$333$	5.95105	0.326115
$334$	0	0
$335$	0.570502	0.0311698
$336$	0	0
$337$	19.6957	1.07289	0.536446	−	0.843935i	$-0.319767\pi$
$337$	19.6957	1.07289	0.536446	+	0.843935i	$0.319767\pi$
$338$	0	0
$339$	10.5371	0.572295
$340$	0	0
$341$	7.44808	0.403336
$342$	0	0
$343$	15.5437	0.839283
$344$	0	0
$345$	2.55203	0.137397
$346$	0	0
$347$	19.5450	1.04923	0.524614	−	0.851340i	$-0.324210\pi$
$347$	19.5450	1.04923	0.524614	+	0.851340i	$0.324210\pi$
$348$	0	0
$349$	16.5795	0.887481	0.443741	−	0.896155i	$-0.353651\pi$
$349$	16.5795	0.887481	0.443741	+	0.896155i	$0.353651\pi$
$350$	0	0
$351$	−2.41755	−0.129039
$352$	0	0
$353$	10.8463	0.577293	0.288646	−	0.957436i	$-0.406795\pi$
$353$	10.8463	0.577293	0.288646	+	0.957436i	$0.406795\pi$
$354$	0	0
$355$	4.69918	0.249406
$356$	0	0
$357$	4.78813	0.253415
$358$	0	0
$359$	−10.9324	−0.576989	−0.288495	−	0.957482i	$-0.593155\pi$
$359$	−10.9324	−0.576989	−0.288495	+	0.957482i	$0.593155\pi$
$360$	0	0
$361$	0.392642	0.0206654
$362$	0	0
$363$	8.28036	0.434606
$364$	0	0
$365$	1.85095	0.0968832
$366$	0	0
$367$	23.1751	1.20973	0.604866	−	0.796327i	$-0.293227\pi$
$367$	23.1751	1.20973	0.604866	+	0.796327i	$0.293227\pi$
$368$	0	0
$369$	5.29889	0.275849
$370$	0	0
$371$	40.1596	2.08498
$372$	0	0
$373$	14.1209	0.731152	0.365576	−	0.930781i	$-0.380872\pi$
$373$	14.1209	0.731152	0.365576	+	0.930781i	$0.380872\pi$
$374$	0	0
$375$	3.80845	0.196667
$376$	0	0
$377$	−4.67806	−0.240933
$378$	0	0
$379$	4.66154	0.239447	0.119724	−	0.992807i	$-0.461799\pi$
$379$	4.66154	0.239447	0.119724	+	0.992807i	$0.461799\pi$
$380$	0	0
$381$	10.1860	0.521844
$382$	0	0
$383$	31.0020	1.58413	0.792064	−	0.610439i	$-0.209007\pi$
$383$	31.0020	1.58413	0.792064	+	0.610439i	$0.209007\pi$
$384$	0	0
$385$	−1.88530	−0.0960837
$386$	0	0
$387$	−8.10944	−0.412226
$388$	0	0
$389$	36.4992	1.85058	0.925292	−	0.379256i	$-0.123820\pi$
$389$	36.4992	1.85058	0.925292	+	0.379256i	$0.123820\pi$
$390$	0	0
$391$	−10.6888	−0.540553
$392$	0	0
$393$	−14.0131	−0.706869
$394$	0	0
$395$	2.43501	0.122519
$396$	0	0
$397$	−6.45524	−0.323979	−0.161990	−	0.986792i	$-0.551791\pi$
$397$	−6.45524	−0.323979	−0.161990	+	0.986792i	$0.551791\pi$
$398$	0	0
$399$	−13.0213	−0.651880
$400$	0	0
$401$	26.8631	1.34148	0.670739	−	0.741694i	$-0.265977\pi$
$401$	26.8631	1.34148	0.670739	+	0.741694i	$0.265977\pi$
$402$	0	0
$403$	10.9185	0.543891
$404$	0	0
$405$	0.386624	0.0192115
$406$	0	0
$407$	9.81407	0.486465
$408$	0	0
$409$	−26.6585	−1.31818	−0.659090	−	0.752064i	$-0.729058\pi$
$409$	−26.6585	−1.31818	−0.659090	+	0.752064i	$0.729058\pi$
$410$	0	0
$411$	2.28952	0.112934
$412$	0	0
$413$	−33.6928	−1.65792
$414$	0	0
$415$	1.82301	0.0894880
$416$	0	0
$417$	5.39959	0.264419
$418$	0	0
$419$	15.4930	0.756881	0.378440	−	0.925626i	$-0.376461\pi$
$419$	15.4930	0.756881	0.378440	+	0.925626i	$0.376461\pi$
$420$	0	0
$421$	12.3456	0.601688	0.300844	−	0.953673i	$-0.402732\pi$
$421$	12.3456	0.601688	0.300844	+	0.953673i	$0.402732\pi$
$422$	0	0
$423$	−0.468667	−0.0227874
$424$	0	0
$425$	−7.85450	−0.380999
$426$	0	0
$427$	−23.9599	−1.15950
$428$	0	0
$429$	−3.98687	−0.192488
$430$	0	0
$431$	33.5269	1.61494	0.807468	−	0.589912i	$-0.200837\pi$
$431$	33.5269	1.61494	0.807468	+	0.589912i	$0.200837\pi$
$432$	0	0
$433$	−37.4726	−1.80082	−0.900409	−	0.435045i	$-0.856732\pi$
$433$	−37.4726	−1.80082	−0.900409	+	0.435045i	$0.856732\pi$
$434$	0	0
$435$	0.748133	0.0358702
$436$	0	0
$437$	29.0680	1.39051
$438$	0	0
$439$	−25.1610	−1.20087	−0.600433	−	0.799675i	$-0.705005\pi$
$439$	−25.1610	−1.20087	−0.600433	+	0.799675i	$0.705005\pi$
$440$	0	0
$441$	1.74322	0.0830106
$442$	0	0
$443$	5.63898	0.267916	0.133958	−	0.990987i	$-0.457231\pi$
$443$	5.63898	0.267916	0.133958	+	0.990987i	$0.457231\pi$
$444$	0	0
$445$	−1.23320	−0.0584595
$446$	0	0
$447$	18.9607	0.896810
$448$	0	0
$449$	28.6396	1.35158	0.675792	−	0.737092i	$-0.263802\pi$
$449$	28.6396	1.35158	0.675792	+	0.737092i	$0.263802\pi$
$450$	0	0
$451$	8.73857	0.411483
$452$	0	0
$453$	3.13560	0.147323
$454$	0	0
$455$	−2.76376	−0.129567
$456$	0	0
$457$	−2.45744	−0.114954	−0.0574771	−	0.998347i	$-0.518306\pi$
$457$	−2.45744	−0.114954	−0.0574771	+	0.998347i	$0.518306\pi$
$458$	0	0
$459$	−1.61931	−0.0755829
$460$	0	0
$461$	29.5424	1.37593	0.687964	−	0.725744i	$-0.258505\pi$
$461$	29.5424	1.37593	0.687964	+	0.725744i	$0.258505\pi$
$462$	0	0
$463$	17.9373	0.833616	0.416808	−	0.908995i	$-0.363149\pi$
$463$	17.9373	0.833616	0.416808	+	0.908995i	$0.363149\pi$
$464$	0	0
$465$	−1.74613	−0.0809749
$466$	0	0
$467$	1.62765	0.0753189	0.0376594	−	0.999291i	$-0.488010\pi$
$467$	1.62765	0.0753189	0.0376594	+	0.999291i	$0.488010\pi$
$468$	0	0
$469$	−4.36319	−0.201473
$470$	0	0
$471$	0.982994	0.0452939
$472$	0	0
$473$	−13.3735	−0.614916
$474$	0	0
$475$	21.3603	0.980077
$476$	0	0
$477$	−13.5817	−0.621863
$478$	0	0
$479$	−1.13852	−0.0520205	−0.0260103	−	0.999662i	$-0.508280\pi$
$479$	−1.13852	−0.0520205	−0.0260103	+	0.999662i	$0.508280\pi$
$480$	0	0
$481$	14.3870	0.655990
$482$	0	0
$483$	−19.5179	−0.888094
$484$	0	0
$485$	3.54263	0.160862
$486$	0	0
$487$	22.2904	1.01007	0.505037	−	0.863098i	$-0.331479\pi$
$487$	22.2904	1.01007	0.505037	+	0.863098i	$0.331479\pi$
$488$	0	0
$489$	9.07562	0.410414
$490$	0	0
$491$	−0.713701	−0.0322089	−0.0161044	−	0.999870i	$-0.505126\pi$
$491$	−0.713701	−0.0322089	−0.0161044	+	0.999870i	$0.505126\pi$
$492$	0	0
$493$	−3.13343	−0.141123
$494$	0	0
$495$	0.637594	0.0286577
$496$	0	0
$497$	−35.9393	−1.61210
$498$	0	0
$499$	22.5329	1.00871	0.504355	−	0.863496i	$-0.331730\pi$
$499$	22.5329	1.00871	0.504355	+	0.863496i	$0.331730\pi$
$500$	0	0
$501$	8.80527	0.393390
$502$	0	0
$503$	−9.57917	−0.427114	−0.213557	−	0.976931i	$-0.568505\pi$
$503$	−9.57917	−0.427114	−0.213557	+	0.976931i	$0.568505\pi$
$504$	0	0
$505$	−1.79910	−0.0800591
$506$	0	0
$507$	7.15544	0.317784
$508$	0	0
$509$	−11.3671	−0.503837	−0.251918	−	0.967748i	$-0.581061\pi$
$509$	−11.3671	−0.503837	−0.251918	+	0.967748i	$0.581061\pi$
$510$	0	0
$511$	−14.1560	−0.626227
$512$	0	0
$513$	4.40371	0.194428
$514$	0	0
$515$	4.06117	0.178957
$516$	0	0
$517$	−0.772895	−0.0339919
$518$	0	0
$519$	−10.4539	−0.458877
$520$	0	0
$521$	−32.2592	−1.41330	−0.706650	−	0.707564i	$-0.749794\pi$
$521$	−32.2592	−1.41330	−0.706650	+	0.707564i	$0.749794\pi$
$522$	0	0
$523$	19.6194	0.857898	0.428949	−	0.903329i	$-0.358884\pi$
$523$	19.6194	0.857898	0.428949	+	0.903329i	$0.358884\pi$
$524$	0	0
$525$	−14.3425	−0.625957
$526$	0	0
$527$	7.31339	0.318576
$528$	0	0
$529$	20.5706	0.894374
$530$	0	0
$531$	11.3947	0.494487
$532$	0	0
$533$	12.8103	0.554878
$534$	0	0
$535$	−4.31946	−0.186747
$536$	0	0
$537$	0.748038	0.0322802
$538$	0	0
$539$	2.87481	0.123827
$540$	0	0
$541$	−35.0884	−1.50857	−0.754283	−	0.656549i	$-0.772015\pi$
$541$	−35.0884	−1.50857	−0.754283	+	0.656549i	$0.772015\pi$
$542$	0	0
$543$	−23.6329	−1.01419
$544$	0	0
$545$	−4.76594	−0.204150
$546$	0	0
$547$	22.1329	0.946335	0.473167	−	0.880972i	$-0.343111\pi$
$547$	22.1329	0.946335	0.473167	+	0.880972i	$0.343111\pi$
$548$	0	0
$549$	8.10307	0.345830
$550$	0	0
$551$	8.52135	0.363022
$552$	0	0
$553$	−18.6229	−0.791927
$554$	0	0
$555$	−2.30082	−0.0976642
$556$	0	0
$557$	24.9672	1.05789	0.528947	−	0.848655i	$-0.322587\pi$
$557$	24.9672	1.05789	0.528947	+	0.848655i	$0.322587\pi$
$558$	0	0
$559$	−19.6050	−0.829203
$560$	0	0
$561$	−2.67046	−0.112747
$562$	0	0
$563$	−35.1548	−1.48160	−0.740800	−	0.671726i	$-0.765553\pi$
$563$	−35.1548	−1.48160	−0.740800	+	0.671726i	$0.765553\pi$
$564$	0	0
$565$	−4.07388	−0.171390
$566$	0	0
$567$	−2.95689	−0.124178
$568$	0	0
$569$	10.2789	0.430914	0.215457	−	0.976513i	$-0.430876\pi$
$569$	10.2789	0.430914	0.215457	+	0.976513i	$0.430876\pi$
$570$	0	0
$571$	3.31684	0.138805	0.0694027	−	0.997589i	$-0.477891\pi$
$571$	3.31684	0.138805	0.0694027	+	0.997589i	$0.477891\pi$
$572$	0	0
$573$	−2.29905	−0.0960443
$574$	0	0
$575$	32.0173	1.33522
$576$	0	0
$577$	−40.5837	−1.68952	−0.844761	−	0.535143i	$-0.820258\pi$
$577$	−40.5837	−1.68952	−0.844761	+	0.535143i	$0.820258\pi$
$578$	0	0
$579$	1.00000	0.0415586
$580$	0	0
$581$	−13.9423	−0.578426
$582$	0	0
$583$	−22.3980	−0.927632
$584$	0	0
$585$	0.934684	0.0386444
$586$	0	0
$587$	−2.51455	−0.103786	−0.0518932	−	0.998653i	$-0.516526\pi$
$587$	−2.51455	−0.103786	−0.0518932	+	0.998653i	$0.516526\pi$
$588$	0	0
$589$	−19.8887	−0.819501
$590$	0	0
$591$	−10.4443	−0.429621
$592$	0	0
$593$	15.1643	0.622725	0.311362	−	0.950291i	$-0.399215\pi$
$593$	15.1643	0.622725	0.311362	+	0.950291i	$0.399215\pi$
$594$	0	0
$595$	−1.85120	−0.0758920
$596$	0	0
$597$	−22.2811	−0.911906
$598$	0	0
$599$	−25.4874	−1.04139	−0.520693	−	0.853744i	$-0.674326\pi$
$599$	−25.4874	−1.04139	−0.520693	+	0.853744i	$0.674326\pi$
$600$	0	0
$601$	8.83051	0.360204	0.180102	−	0.983648i	$-0.442357\pi$
$601$	8.83051	0.360204	0.180102	+	0.983648i	$0.442357\pi$
$602$	0	0
$603$	1.47560	0.0600910
$604$	0	0
$605$	−3.20138	−0.130155
$606$	0	0
$607$	−34.5555	−1.40256	−0.701282	−	0.712884i	$-0.747389\pi$
$607$	−34.5555	−1.40256	−0.701282	+	0.712884i	$0.747389\pi$
$608$	0	0
$609$	−5.72171	−0.231855
$610$	0	0
$611$	−1.13303	−0.0458374
$612$	0	0
$613$	−10.9389	−0.441816	−0.220908	−	0.975295i	$-0.570902\pi$
$613$	−10.9389	−0.441816	−0.220908	+	0.975295i	$0.570902\pi$
$614$	0	0
$615$	−2.04868	−0.0826106
$616$	0	0
$617$	−10.4100	−0.419092	−0.209546	−	0.977799i	$-0.567199\pi$
$617$	−10.4100	−0.419092	−0.209546	+	0.977799i	$0.567199\pi$
$618$	0	0
$619$	17.3083	0.695679	0.347840	−	0.937554i	$-0.386915\pi$
$619$	17.3083	0.695679	0.347840	+	0.937554i	$0.386915\pi$
$620$	0	0
$621$	6.60080	0.264881
$622$	0	0
$623$	9.43153	0.377866
$624$	0	0
$625$	22.7802	0.911207
$626$	0	0
$627$	7.26230	0.290028
$628$	0	0
$629$	9.63659	0.384236
$630$	0	0
$631$	−18.8163	−0.749065	−0.374533	−	0.927214i	$-0.622197\pi$
$631$	−18.8163	−0.749065	−0.374533	+	0.927214i	$0.622197\pi$
$632$	0	0
$633$	−15.0302	−0.597399
$634$	0	0
$635$	−3.93814	−0.156280
$636$	0	0
$637$	4.21433	0.166978
$638$	0	0
$639$	12.1544	0.480820
$640$	0	0
$641$	−1.48842	−0.0587889	−0.0293945	−	0.999568i	$-0.509358\pi$
$641$	−1.48842	−0.0587889	−0.0293945	+	0.999568i	$0.509358\pi$
$642$	0	0
$643$	12.0794	0.476364	0.238182	−	0.971221i	$-0.423449\pi$
$643$	12.0794	0.476364	0.238182	+	0.971221i	$0.423449\pi$
$644$	0	0
$645$	3.13530	0.123452
$646$	0	0
$647$	14.9732	0.588659	0.294329	−	0.955704i	$-0.404904\pi$
$647$	14.9732	0.588659	0.294329	+	0.955704i	$0.404904\pi$
$648$	0	0
$649$	18.7913	0.737624
$650$	0	0
$651$	13.3544	0.523400
$652$	0	0
$653$	−42.5269	−1.66421	−0.832103	−	0.554621i	$-0.812863\pi$
$653$	−42.5269	−1.66421	−0.832103	+	0.554621i	$0.812863\pi$
$654$	0	0
$655$	5.41781	0.211691
$656$	0	0
$657$	4.78747	0.186777
$658$	0	0
$659$	43.1340	1.68026	0.840130	−	0.542384i	$-0.182478\pi$
$659$	43.1340	1.68026	0.840130	+	0.542384i	$0.182478\pi$
$660$	0	0
$661$	−8.21019	−0.319339	−0.159670	−	0.987170i	$-0.551043\pi$
$661$	−8.21019	−0.319339	−0.159670	+	0.987170i	$0.551043\pi$
$662$	0	0
$663$	−3.91477	−0.152037
$664$	0	0
$665$	5.03434	0.195223
$666$	0	0
$667$	12.7728	0.494566
$668$	0	0
$669$	1.92583	0.0744568
$670$	0	0
$671$	13.3630	0.515874
$672$	0	0
$673$	2.96230	0.114188	0.0570941	−	0.998369i	$-0.481816\pi$
$673$	2.96230	0.114188	0.0570941	+	0.998369i	$0.481816\pi$
$674$	0	0
$675$	4.85052	0.186697
$676$	0	0
$677$	25.5130	0.980546	0.490273	−	0.871569i	$-0.336897\pi$
$677$	25.5130	0.980546	0.490273	+	0.871569i	$0.336897\pi$
$678$	0	0
$679$	−27.0940	−1.03977
$680$	0	0
$681$	15.3169	0.586947
$682$	0	0
$683$	13.1682	0.503868	0.251934	−	0.967744i	$-0.418933\pi$
$683$	13.1682	0.503868	0.251934	+	0.967744i	$0.418933\pi$
$684$	0	0
$685$	−0.885183	−0.0338211
$686$	0	0
$687$	8.99568	0.343207
$688$	0	0
$689$	−32.8345	−1.25089
$690$	0	0
$691$	13.9444	0.530469	0.265235	−	0.964184i	$-0.414551\pi$
$691$	13.9444	0.530469	0.265235	+	0.964184i	$0.414551\pi$
$692$	0	0
$693$	−4.87631	−0.185236
$694$	0	0
$695$	−2.08761	−0.0791876
$696$	0	0
$697$	8.58055	0.325011
$698$	0	0
$699$	9.57715	0.362241
$700$	0	0
$701$	18.4080	0.695261	0.347630	−	0.937632i	$-0.386986\pi$
$701$	18.4080	0.695261	0.347630	+	0.937632i	$0.386986\pi$
$702$	0	0
$703$	−26.2067	−0.988404
$704$	0	0
$705$	0.181198	0.00682431
$706$	0	0
$707$	13.7595	0.517480
$708$	0	0
$709$	−19.5435	−0.733973	−0.366987	−	0.930226i	$-0.619611\pi$
$709$	−19.5435	−0.733973	−0.366987	+	0.930226i	$0.619611\pi$
$710$	0	0
$711$	6.29813	0.236198
$712$	0	0
$713$	−29.8116	−1.11645
$714$	0	0
$715$	1.54142	0.0576457
$716$	0	0
$717$	−9.43060	−0.352192
$718$	0	0
$719$	41.7189	1.55585	0.777926	−	0.628356i	$-0.216272\pi$
$719$	41.7189	1.55585	0.777926	+	0.628356i	$0.216272\pi$
$720$	0	0
$721$	−31.0598	−1.15673
$722$	0	0
$723$	−26.1794	−0.973622
$724$	0	0
$725$	9.38596	0.348586
$726$	0	0
$727$	36.7985	1.36478	0.682391	−	0.730988i	$-0.260940\pi$
$727$	36.7985	1.36478	0.682391	+	0.730988i	$0.260940\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−13.1317	−0.485693
$732$	0	0
$733$	14.2467	0.526214	0.263107	−	0.964767i	$-0.415253\pi$
$733$	14.2467	0.526214	0.263107	+	0.964767i	$0.415253\pi$
$734$	0	0
$735$	−0.673971	−0.0248598
$736$	0	0
$737$	2.43346	0.0896376
$738$	0	0
$739$	5.54835	0.204099	0.102050	−	0.994779i	$-0.467460\pi$
$739$	5.54835	0.204099	0.102050	+	0.994779i	$0.467460\pi$
$740$	0	0
$741$	10.6462	0.391098
$742$	0	0
$743$	25.0698	0.919721	0.459860	−	0.887991i	$-0.347899\pi$
$743$	25.0698	0.919721	0.459860	+	0.887991i	$0.347899\pi$
$744$	0	0
$745$	−7.33065	−0.268574
$746$	0	0
$747$	4.71520	0.172520
$748$	0	0
$749$	33.0352	1.20708
$750$	0	0
$751$	−8.14507	−0.297218	−0.148609	−	0.988896i	$-0.547480\pi$
$751$	−8.14507	−0.297218	−0.148609	+	0.988896i	$0.547480\pi$
$752$	0	0
$753$	−19.3074	−0.703602
$754$	0	0
$755$	−1.21230	−0.0441200
$756$	0	0
$757$	−46.9170	−1.70523	−0.852614	−	0.522541i	$-0.824984\pi$
$757$	−46.9170	−1.70523	−0.852614	+	0.522541i	$0.824984\pi$
$758$	0	0
$759$	10.8856	0.395122
$760$	0	0
$761$	8.37311	0.303525	0.151763	−	0.988417i	$-0.451505\pi$
$761$	8.37311	0.303525	0.151763	+	0.988417i	$0.451505\pi$
$762$	0	0
$763$	36.4499	1.31957
$764$	0	0
$765$	0.626064	0.0226354
$766$	0	0
$767$	27.5472	0.994673
$768$	0	0
$769$	−2.79625	−0.100836	−0.0504178	−	0.998728i	$-0.516055\pi$
$769$	−2.79625	−0.100836	−0.0504178	+	0.998728i	$0.516055\pi$
$770$	0	0
$771$	−26.4489	−0.952534
$772$	0	0
$773$	−24.9660	−0.897964	−0.448982	−	0.893541i	$-0.648213\pi$
$773$	−24.9660	−0.897964	−0.448982	+	0.893541i	$0.648213\pi$
$774$	0	0
$775$	−21.9067	−0.786912
$776$	0	0
$777$	17.5966	0.631275
$778$	0	0
$779$	−23.3348	−0.836054
$780$	0	0
$781$	20.0442	0.717238
$782$	0	0
$783$	1.93504	0.0691527
$784$	0	0
$785$	−0.380049	−0.0135645
$786$	0	0
$787$	−31.1004	−1.10861	−0.554305	−	0.832314i	$-0.687016\pi$
$787$	−31.1004	−1.10861	−0.554305	+	0.832314i	$0.687016\pi$
$788$	0	0
$789$	0.0522246	0.00185925
$790$	0	0
$791$	31.1570	1.10782
$792$	0	0
$793$	19.5896	0.695647
$794$	0	0
$795$	5.25101	0.186234
$796$	0	0
$797$	32.9709	1.16789	0.583945	−	0.811793i	$-0.301509\pi$
$797$	32.9709	1.16789	0.583945	+	0.811793i	$0.301509\pi$
$798$	0	0
$799$	−0.758918	−0.0268486
$800$	0	0
$801$	−3.18967	−0.112702
$802$	0	0
$803$	7.89518	0.278615
$804$	0	0
$805$	7.54608	0.265964
$806$	0	0
$807$	−28.9995	−1.02083
$808$	0	0
$809$	11.3949	0.400624	0.200312	−	0.979732i	$-0.435804\pi$
$809$	11.3949	0.400624	0.200312	+	0.979732i	$0.435804\pi$
$810$	0	0
$811$	−38.4944	−1.35172	−0.675861	−	0.737029i	$-0.736228\pi$
$811$	−38.4944	−1.35172	−0.675861	+	0.737029i	$0.736228\pi$
$812$	0	0
$813$	9.97419	0.349810
$814$	0	0
$815$	−3.50885	−0.122910
$816$	0	0
$817$	35.7116	1.24939
$818$	0	0
$819$	−7.14845	−0.249787
$820$	0	0
$821$	33.2875	1.16174	0.580871	−	0.813995i	$-0.302712\pi$
$821$	33.2875	1.16174	0.580871	+	0.813995i	$0.302712\pi$
$822$	0	0
$823$	−36.1651	−1.26064	−0.630318	−	0.776337i	$-0.717075\pi$
$823$	−36.1651	−1.26064	−0.630318	+	0.776337i	$0.717075\pi$
$824$	0	0
$825$	7.99916	0.278495
$826$	0	0
$827$	15.3233	0.532844	0.266422	−	0.963856i	$-0.414159\pi$
$827$	15.3233	0.532844	0.266422	+	0.963856i	$0.414159\pi$
$828$	0	0
$829$	38.9987	1.35448	0.677241	−	0.735761i	$-0.263175\pi$
$829$	38.9987	1.35448	0.677241	+	0.735761i	$0.263175\pi$
$830$	0	0
$831$	−30.8325	−1.06957
$832$	0	0
$833$	2.82282	0.0978049
$834$	0	0
$835$	−3.40433	−0.117812
$836$	0	0
$837$	−4.51636	−0.156108
$838$	0	0
$839$	−41.0475	−1.41712	−0.708558	−	0.705652i	$-0.750654\pi$
$839$	−41.0475	−1.41712	−0.708558	+	0.705652i	$0.750654\pi$
$840$	0	0
$841$	−25.2556	−0.870883
$842$	0	0
$843$	24.7955	0.854003
$844$	0	0
$845$	−2.76646	−0.0951692
$846$	0	0
$847$	24.4841	0.841285
$848$	0	0
$849$	−11.5613	−0.396781
$850$	0	0
$851$	−39.2817	−1.34656
$852$	0	0
$853$	−6.76197	−0.231525	−0.115763	−	0.993277i	$-0.536931\pi$
$853$	−6.76197	−0.231525	−0.115763	+	0.993277i	$0.536931\pi$
$854$	0	0
$855$	−1.70258	−0.0582269
$856$	0	0
$857$	−54.8218	−1.87268	−0.936338	−	0.351101i	$-0.885808\pi$
$857$	−54.8218	−1.87268	−0.936338	+	0.351101i	$0.885808\pi$
$858$	0	0
$859$	−23.1196	−0.788832	−0.394416	−	0.918932i	$-0.629053\pi$
$859$	−23.1196	−0.788832	−0.394416	+	0.918932i	$0.629053\pi$
$860$	0	0
$861$	15.6683	0.533973
$862$	0	0
$863$	−56.7469	−1.93169	−0.965843	−	0.259129i	$-0.916565\pi$
$863$	−56.7469	−1.93169	−0.965843	+	0.259129i	$0.916565\pi$
$864$	0	0
$865$	4.04174	0.137423
$866$	0	0
$867$	14.3778	0.488297
$868$	0	0
$869$	10.3865	0.352337
$870$	0	0
$871$	3.56734	0.120875
$872$	0	0
$873$	9.16299	0.310120
$874$	0	0
$875$	11.2612	0.380697
$876$	0	0
$877$	49.6547	1.67672	0.838361	−	0.545116i	$-0.183514\pi$
$877$	49.6547	1.67672	0.838361	+	0.545116i	$0.183514\pi$
$878$	0	0
$879$	0.0556420	0.00187676
$880$	0	0
$881$	49.7113	1.67481	0.837407	−	0.546579i	$-0.184070\pi$
$881$	49.7113	1.67481	0.837407	+	0.546579i	$0.184070\pi$
$882$	0	0
$883$	−25.4170	−0.855350	−0.427675	−	0.903933i	$-0.640667\pi$
$883$	−25.4170	−0.855350	−0.427675	+	0.903933i	$0.640667\pi$
$884$	0	0
$885$	−4.40545	−0.148088
$886$	0	0
$887$	23.7582	0.797722	0.398861	−	0.917011i	$-0.369406\pi$
$887$	23.7582	0.797722	0.398861	+	0.917011i	$0.369406\pi$
$888$	0	0
$889$	30.1189	1.01015
$890$	0	0
$891$	1.64913	0.0552480
$892$	0	0
$893$	2.06387	0.0690649
$894$	0	0
$895$	−0.289209	−0.00966720
$896$	0	0
$897$	15.9578	0.532815
$898$	0	0
$899$	−8.73934	−0.291473
$900$	0	0
$901$	−21.9930	−0.732693
$902$	0	0
$903$	−23.9788	−0.797963
$904$	0	0
$905$	9.13706	0.303726
$906$	0	0
$907$	−28.6444	−0.951122	−0.475561	−	0.879683i	$-0.657755\pi$
$907$	−28.6444	−0.951122	−0.475561	+	0.879683i	$0.657755\pi$
$908$	0	0
$909$	−4.65337	−0.154343
$910$	0	0
$911$	−10.5213	−0.348587	−0.174294	−	0.984694i	$-0.555764\pi$
$911$	−10.5213	−0.348587	−0.174294	+	0.984694i	$0.555764\pi$
$912$	0	0
$913$	7.77599	0.257348
$914$	0	0
$915$	−3.13284	−0.103568
$916$	0	0
$917$	−41.4353	−1.36831
$918$	0	0
$919$	25.6911	0.847472	0.423736	−	0.905786i	$-0.360718\pi$
$919$	25.6911	0.847472	0.423736	+	0.905786i	$0.360718\pi$
$920$	0	0
$921$	9.31741	0.307019
$922$	0	0
$923$	29.3839	0.967183
$924$	0	0
$925$	−28.8657	−0.949098
$926$	0	0
$927$	10.5042	0.345003
$928$	0	0
$929$	−0.526368	−0.0172696	−0.00863479	−	0.999963i	$-0.502749\pi$
$929$	−0.526368	−0.0172696	−0.00863479	+	0.999963i	$0.502749\pi$
$930$	0	0
$931$	−7.67664	−0.251592
$932$	0	0
$933$	−10.4901	−0.343432
$934$	0	0
$935$	1.03246	0.0337651
$936$	0	0
$937$	13.3213	0.435189	0.217595	−	0.976039i	$-0.430179\pi$
$937$	13.3213	0.435189	0.217595	+	0.976039i	$0.430179\pi$
$938$	0	0
$939$	−5.09321	−0.166211
$940$	0	0
$941$	10.8935	0.355118	0.177559	−	0.984110i	$-0.443180\pi$
$941$	10.8935	0.355118	0.177559	+	0.984110i	$0.443180\pi$
$942$	0	0
$943$	−34.9769	−1.13901
$944$	0	0
$945$	1.14321	0.0371885
$946$	0	0
$947$	14.8853	0.483708	0.241854	−	0.970313i	$-0.422244\pi$
$947$	14.8853	0.483708	0.241854	+	0.970313i	$0.422244\pi$
$948$	0	0
$949$	11.5740	0.375707
$950$	0	0
$951$	−10.2012	−0.330795
$952$	0	0
$953$	50.4649	1.63472	0.817359	−	0.576129i	$-0.195437\pi$
$953$	50.4649	1.63472	0.817359	+	0.576129i	$0.195437\pi$
$954$	0	0
$955$	0.888869	0.0287631
$956$	0	0
$957$	3.19114	0.103155
$958$	0	0
$959$	6.76987	0.218610
$960$	0	0
$961$	−10.6025	−0.342016
$962$	0	0
$963$	−11.1723	−0.360021
$964$	0	0
$965$	−0.386624	−0.0124459
$966$	0	0
$967$	12.1726	0.391444	0.195722	−	0.980659i	$-0.437295\pi$
$967$	12.1726	0.391444	0.195722	+	0.980659i	$0.437295\pi$
$968$	0	0
$969$	7.13097	0.229080
$970$	0	0
$971$	−8.88819	−0.285235	−0.142618	−	0.989778i	$-0.545552\pi$
$971$	−8.88819	−0.285235	−0.142618	+	0.989778i	$0.545552\pi$
$972$	0	0
$973$	15.9660	0.511847
$974$	0	0
$975$	11.7264	0.375545
$976$	0	0
$977$	31.0236	0.992533	0.496266	−	0.868170i	$-0.334704\pi$
$977$	31.0236	0.992533	0.496266	+	0.868170i	$0.334704\pi$
$978$	0	0
$979$	−5.26020	−0.168117
$980$	0	0
$981$	−12.3271	−0.393573
$982$	0	0
$983$	−10.8807	−0.347039	−0.173520	−	0.984830i	$-0.555514\pi$
$983$	−10.8807	−0.347039	−0.173520	+	0.984830i	$0.555514\pi$
$984$	0	0
$985$	4.03802	0.128662
$986$	0	0
$987$	−1.38580	−0.0441105
$988$	0	0
$989$	53.5288	1.70212
$990$	0	0
$991$	−30.8501	−0.979987	−0.489994	−	0.871726i	$-0.663001\pi$
$991$	−30.8501	−0.979987	−0.489994	+	0.871726i	$0.663001\pi$
$992$	0	0
$993$	−5.48496	−0.174060
$994$	0	0
$995$	8.61441	0.273095
$996$	0	0
$997$	25.6836	0.813407	0.406704	−	0.913560i	$-0.366678\pi$
$997$	25.6836	0.813407	0.406704	+	0.913560i	$0.366678\pi$
$998$	0	0
$999$	−5.95105	−0.188283

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9264.2.a.bp.1.7		13
4.3	odd	2		579.2.a.g.1.7	✓	13
12.11	even	2		1737.2.a.j.1.7		13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
579.2.a.g.1.7	✓	13	4.3	odd	2
1737.2.a.j.1.7		13	12.11	even	2
9264.2.a.bp.1.7		13	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:	\( N \)	\(=\)	\( 9264 = 2^{4} \cdot 3 \cdot 193 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9264.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +0.386624 q^{5} -2.95689 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +0.386624 q^{5} -2.95689 q^{7} +1.00000 q^{9} +1.64913 q^{11} +2.41755 q^{13} -0.386624 q^{15} +1.61931 q^{17} -4.40371 q^{19} +2.95689 q^{21} -6.60080 q^{23} -4.85052 q^{25} -1.00000 q^{27} -1.93504 q^{29} +4.51636 q^{31} -1.64913 q^{33} -1.14321 q^{35} +5.95105 q^{37} -2.41755 q^{39} +5.29889 q^{41} -8.10944 q^{43} +0.386624 q^{45} -0.468667 q^{47} +1.74322 q^{49} -1.61931 q^{51} -13.5817 q^{53} +0.637594 q^{55} +4.40371 q^{57} +11.3947 q^{59} +8.10307 q^{61} -2.95689 q^{63} +0.934684 q^{65} +1.47560 q^{67} +6.60080 q^{69} +12.1544 q^{71} +4.78747 q^{73} +4.85052 q^{75} -4.87631 q^{77} +6.29813 q^{79} +1.00000 q^{81} +4.71520 q^{83} +0.626064 q^{85} +1.93504 q^{87} -3.18967 q^{89} -7.14845 q^{91} -4.51636 q^{93} -1.70258 q^{95} +9.16299 q^{97} +1.64913 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 13 q^{3} + 6 q^{5} - 15 q^{7} + 13 q^{9}+O(q^{10}) \) 13 * q - 13 * q^3 + 6 * q^5 - 15 * q^7 + 13 * q^9	\( 13 q - 13 q^{3} + 6 q^{5} - 15 q^{7} + 13 q^{9} - 3 q^{11} + 11 q^{13} - 6 q^{15} - 2 q^{17} - 9 q^{19} + 15 q^{21} + 8 q^{23} + 21 q^{25} - 13 q^{27} + 5 q^{29} - 25 q^{31} + 3 q^{33} + 10 q^{35} + 29 q^{37} - 11 q^{39} - 11 q^{41} - 8 q^{43} + 6 q^{45} + 12 q^{47} + 20 q^{49} + 2 q^{51} + 14 q^{53} - 12 q^{55} + 9 q^{57} - 10 q^{59} + 6 q^{61} - 15 q^{63} - 15 q^{65} - 25 q^{67} - 8 q^{69} + 8 q^{73} - 21 q^{75} - 25 q^{77} - 7 q^{79} + 13 q^{81} + 28 q^{83} - 3 q^{85} - 5 q^{87} + 7 q^{89} - 7 q^{91} + 25 q^{93} + 26 q^{95} + 26 q^{97} - 3 q^{99}+O(q^{100}) \) 13 * q - 13 * q^3 + 6 * q^5 - 15 * q^7 + 13 * q^9 - 3 * q^11 + 11 * q^13 - 6 * q^15 - 2 * q^17 - 9 * q^19 + 15 * q^21 + 8 * q^23 + 21 * q^25 - 13 * q^27 + 5 * q^29 - 25 * q^31 + 3 * q^33 + 10 * q^35 + 29 * q^37 - 11 * q^39 - 11 * q^41 - 8 * q^43 + 6 * q^45 + 12 * q^47 + 20 * q^49 + 2 * q^51 + 14 * q^53 - 12 * q^55 + 9 * q^57 - 10 * q^59 + 6 * q^61 - 15 * q^63 - 15 * q^65 - 25 * q^67 - 8 * q^69 + 8 * q^73 - 21 * q^75 - 25 * q^77 - 7 * q^79 + 13 * q^81 + 28 * q^83 - 3 * q^85 - 5 * q^87 + 7 * q^89 - 7 * q^91 + 25 * q^93 + 26 * q^95 + 26 * q^97 - 3 * q^99

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