LMFDB - Embedded newform 8464.2.a.bu.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8464,2,Mod(1,8464)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8464, 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("8464.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8464 = 2^{4} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8464.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:	$67.5853802708$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\Q(\zeta_{22})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1 $ x^5 - x^4 - 4x^3 + 3x^2 + 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 46)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.91899$ of defining polynomial
Character	$\chi$	$=$	8464.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.99223 q^{3} +1.39788 q^{5} +4.29177 q^{7} +5.95343 q^{9} +O(q^{10})$	$q+2.99223 q^{3} +1.39788 q^{5} +4.29177 q^{7} +5.95343 q^{9} -0.221566 q^{11} +1.98798 q^{13} +4.18277 q^{15} +0.111215 q^{17} -3.23585 q^{19} +12.8420 q^{21} -3.04594 q^{25} +8.83734 q^{27} -3.56863 q^{29} -6.15787 q^{31} -0.662975 q^{33} +5.99937 q^{35} +4.22808 q^{37} +5.94850 q^{39} +3.41316 q^{41} +3.24797 q^{43} +8.32217 q^{45} -7.73852 q^{47} +11.4193 q^{49} +0.332780 q^{51} +0.146111 q^{53} -0.309721 q^{55} -9.68240 q^{57} +13.1693 q^{59} -5.68965 q^{61} +25.5508 q^{63} +2.77896 q^{65} +14.3384 q^{67} +5.84511 q^{71} -3.79241 q^{73} -9.11415 q^{75} -0.950909 q^{77} +8.72882 q^{79} +8.58305 q^{81} +15.6494 q^{83} +0.155465 q^{85} -10.6782 q^{87} +1.01934 q^{89} +8.53197 q^{91} -18.4258 q^{93} -4.52332 q^{95} -18.2132 q^{97} -1.31908 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{5} + 9 q^{7} + 7 q^{9}+O(q^{10}) $ 5 * q - 2 * q^5 + 9 * q^7 + 7 * q^9	$ 5 q - 2 q^{5} + 9 q^{7} + 7 q^{9} + q^{11} + q^{13} + 11 q^{15} + q^{17} + 10 q^{19} - 11 q^{25} + 10 q^{29} - 11 q^{33} + 3 q^{35} - 20 q^{37} + 22 q^{39} - 3 q^{41} + 27 q^{43} - 5 q^{45} + 11 q^{47} + 12 q^{49} + 2 q^{53} + 4 q^{55} - 11 q^{57} + 17 q^{59} - 7 q^{61} + 39 q^{63} + 15 q^{65} + 28 q^{67} + 8 q^{73} - 22 q^{75} + 26 q^{77} + 16 q^{79} + q^{81} + 18 q^{83} - 7 q^{85} - 22 q^{87} + 14 q^{89} + 26 q^{91} - 44 q^{93} - 26 q^{95} - 47 q^{97} + 8 q^{99}+O(q^{100}) $ 5 * q - 2 * q^5 + 9 * q^7 + 7 * q^9 + q^11 + q^13 + 11 * q^15 + q^17 + 10 * q^19 - 11 * q^25 + 10 * q^29 - 11 * q^33 + 3 * q^35 - 20 * q^37 + 22 * q^39 - 3 * q^41 + 27 * q^43 - 5 * q^45 + 11 * q^47 + 12 * q^49 + 2 * q^53 + 4 * q^55 - 11 * q^57 + 17 * q^59 - 7 * q^61 + 39 * q^63 + 15 * q^65 + 28 * q^67 + 8 * q^73 - 22 * q^75 + 26 * q^77 + 16 * q^79 + q^81 + 18 * q^83 - 7 * q^85 - 22 * q^87 + 14 * q^89 + 26 * q^91 - 44 * q^93 - 26 * q^95 - 47 * q^97 + 8 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	2.99223	1.72756	0.863782	−	0.503866i	$-0.168089\pi$
$3$	2.99223	1.72756	0.863782	+	0.503866i	$0.168089\pi$
$4$	0	0
$5$	1.39788	0.625150	0.312575	−	0.949893i	$-0.398808\pi$
$5$	1.39788	0.625150	0.312575	+	0.949893i	$0.398808\pi$
$6$	0	0
$7$	4.29177	1.62214	0.811069	−	0.584951i	$-0.198886\pi$
$7$	4.29177	1.62214	0.811069	+	0.584951i	$0.198886\pi$
$8$	0	0
$9$	5.95343	1.98448
$10$	0	0
$11$	−0.221566	−0.0668045	−0.0334023	−	0.999442i	$-0.510634\pi$
$11$	−0.221566	−0.0668045	−0.0334023	+	0.999442i	$0.510634\pi$
$12$	0	0
$13$	1.98798	0.551367	0.275684	−	0.961248i	$-0.411096\pi$
$13$	1.98798	0.551367	0.275684	+	0.961248i	$0.411096\pi$
$14$	0	0
$15$	4.18277	1.07999
$16$	0	0
$17$	0.111215	0.0269736	0.0134868	−	0.999909i	$-0.495707\pi$
$17$	0.111215	0.0269736	0.0134868	+	0.999909i	$0.495707\pi$
$18$	0	0
$19$	−3.23585	−0.742355	−0.371177	−	0.928562i	$-0.621046\pi$
$19$	−3.23585	−0.742355	−0.371177	+	0.928562i	$0.621046\pi$
$20$	0	0
$21$	12.8420	2.80235
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−3.04594	−0.609188
$26$	0	0
$27$	8.83734	1.70075
$28$	0	0
$29$	−3.56863	−0.662678	−0.331339	−	0.943512i	$-0.607500\pi$
$29$	−3.56863	−0.662678	−0.331339	+	0.943512i	$0.607500\pi$
$30$	0	0
$31$	−6.15787	−1.10599	−0.552993	−	0.833186i	$-0.686514\pi$
$31$	−6.15787	−1.10599	−0.552993	+	0.833186i	$0.686514\pi$
$32$	0	0
$33$	−0.662975	−0.115409
$34$	0	0
$35$	5.99937	1.01408
$36$	0	0
$37$	4.22808	0.695092	0.347546	−	0.937663i	$-0.387015\pi$
$37$	4.22808	0.695092	0.347546	+	0.937663i	$0.387015\pi$
$38$	0	0
$39$	5.94850	0.952523
$40$	0	0
$41$	3.41316	0.533047	0.266523	−	0.963828i	$-0.414125\pi$
$41$	3.41316	0.533047	0.266523	+	0.963828i	$0.414125\pi$
$42$	0	0
$43$	3.24797	0.495311	0.247656	−	0.968848i	$-0.420340\pi$
$43$	3.24797	0.495311	0.247656	+	0.968848i	$0.420340\pi$
$44$	0	0
$45$	8.32217	1.24060
$46$	0	0
$47$	−7.73852	−1.12878	−0.564389	−	0.825509i	$-0.690888\pi$
$47$	−7.73852	−1.12878	−0.564389	+	0.825509i	$0.690888\pi$
$48$	0	0
$49$	11.4193	1.63133
$50$	0	0
$51$	0.332780	0.0465986
$52$	0	0
$53$	0.146111	0.0200699	0.0100349	−	0.999950i	$-0.496806\pi$
$53$	0.146111	0.0200699	0.0100349	+	0.999950i	$0.496806\pi$
$54$	0	0
$55$	−0.309721	−0.0417628
$56$	0	0
$57$	−9.68240	−1.28247
$58$	0	0
$59$	13.1693	1.71449	0.857246	−	0.514908i	$-0.172174\pi$
$59$	13.1693	1.71449	0.857246	+	0.514908i	$0.172174\pi$
$60$	0	0
$61$	−5.68965	−0.728485	−0.364242	−	0.931304i	$-0.618672\pi$
$61$	−5.68965	−0.728485	−0.364242	+	0.931304i	$0.618672\pi$
$62$	0	0
$63$	25.5508	3.21909
$64$	0	0
$65$	2.77896	0.344687
$66$	0	0
$67$	14.3384	1.75172	0.875859	−	0.482567i	$-0.160296\pi$
$67$	14.3384	1.75172	0.875859	+	0.482567i	$0.160296\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	5.84511	0.693687	0.346844	−	0.937923i	$-0.387253\pi$
$71$	5.84511	0.693687	0.346844	+	0.937923i	$0.387253\pi$
$72$	0	0
$73$	−3.79241	−0.443868	−0.221934	−	0.975062i	$-0.571237\pi$
$73$	−3.79241	−0.443868	−0.221934	+	0.975062i	$0.571237\pi$
$74$	0	0
$75$	−9.11415	−1.05241
$76$	0	0
$77$	−0.950909	−0.108366
$78$	0	0
$79$	8.72882	0.982069	0.491034	−	0.871140i	$-0.336619\pi$
$79$	8.72882	0.982069	0.491034	+	0.871140i	$0.336619\pi$
$80$	0	0
$81$	8.58305	0.953673
$82$	0	0
$83$	15.6494	1.71774	0.858872	−	0.512191i	$-0.171166\pi$
$83$	15.6494	1.71774	0.858872	+	0.512191i	$0.171166\pi$
$84$	0	0
$85$	0.155465	0.0168625
$86$	0	0
$87$	−10.6782	−1.14482
$88$	0	0
$89$	1.01934	0.108050	0.0540248	−	0.998540i	$-0.482795\pi$
$89$	1.01934	0.108050	0.0540248	+	0.998540i	$0.482795\pi$
$90$	0	0
$91$	8.53197	0.894394
$92$	0	0
$93$	−18.4258	−1.91066
$94$	0	0
$95$	−4.52332	−0.464083
$96$	0	0
$97$	−18.2132	−1.84927	−0.924633	−	0.380858i	$-0.875628\pi$
$97$	−18.2132	−1.84927	−0.924633	+	0.380858i	$0.875628\pi$
$98$	0	0
$99$	−1.31908	−0.132572
$100$	0	0
$101$	3.25891	0.324273	0.162137	−	0.986768i	$-0.448161\pi$
$101$	3.25891	0.324273	0.162137	+	0.986768i	$0.448161\pi$
$102$	0	0
$103$	10.1126	0.996420	0.498210	−	0.867056i	$-0.333991\pi$
$103$	10.1126	0.996420	0.498210	+	0.867056i	$0.333991\pi$
$104$	0	0
$105$	17.9515	1.75189
$106$	0	0
$107$	0.475845	0.0460017	0.0230009	−	0.999735i	$-0.492678\pi$
$107$	0.475845	0.0460017	0.0230009	+	0.999735i	$0.492678\pi$
$108$	0	0
$109$	9.80872	0.939505	0.469753	−	0.882798i	$-0.344343\pi$
$109$	9.80872	0.939505	0.469753	+	0.882798i	$0.344343\pi$
$110$	0	0
$111$	12.6514	1.20082
$112$	0	0
$113$	8.86799	0.834231	0.417115	−	0.908854i	$-0.363041\pi$
$113$	8.86799	0.834231	0.417115	+	0.908854i	$0.363041\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	11.8353	1.09418
$118$	0	0
$119$	0.477309	0.0437548
$120$	0	0
$121$	−10.9509	−0.995537
$122$	0	0
$123$	10.2130	0.920872
$124$	0	0
$125$	−11.2472	−1.00598
$126$	0	0
$127$	−2.61962	−0.232454	−0.116227	−	0.993223i	$-0.537080\pi$
$127$	−2.61962	−0.232454	−0.116227	+	0.993223i	$0.537080\pi$
$128$	0	0
$129$	9.71867	0.855682
$130$	0	0
$131$	−15.3207	−1.33857	−0.669286	−	0.743005i	$-0.733400\pi$
$131$	−15.3207	−1.33857	−0.669286	+	0.743005i	$0.733400\pi$
$132$	0	0
$133$	−13.8875	−1.20420
$134$	0	0
$135$	12.3535	1.06322
$136$	0	0
$137$	−10.3092	−0.880771	−0.440386	−	0.897809i	$-0.645158\pi$
$137$	−10.3092	−0.880771	−0.440386	+	0.897809i	$0.645158\pi$
$138$	0	0
$139$	0.569454	0.0483005	0.0241502	−	0.999708i	$-0.492312\pi$
$139$	0.569454	0.0483005	0.0241502	+	0.999708i	$0.492312\pi$
$140$	0	0
$141$	−23.1554	−1.95004
$142$	0	0
$143$	−0.440469	−0.0368338
$144$	0	0
$145$	−4.98851	−0.414273
$146$	0	0
$147$	34.1692	2.81823
$148$	0	0
$149$	13.1652	1.07853	0.539267	−	0.842135i	$-0.318701\pi$
$149$	13.1652	1.07853	0.539267	+	0.842135i	$0.318701\pi$
$150$	0	0
$151$	9.08797	0.739569	0.369784	−	0.929118i	$-0.379432\pi$
$151$	9.08797	0.739569	0.369784	+	0.929118i	$0.379432\pi$
$152$	0	0
$153$	0.662110	0.0535285
$154$	0	0
$155$	−8.60795	−0.691407
$156$	0	0
$157$	−8.71215	−0.695306	−0.347653	−	0.937623i	$-0.613021\pi$
$157$	−8.71215	−0.695306	−0.347653	+	0.937623i	$0.613021\pi$
$158$	0	0
$159$	0.437197	0.0346720
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−10.6812	−0.836616	−0.418308	−	0.908305i	$-0.637377\pi$
$163$	−10.6812	−0.836616	−0.418308	+	0.908305i	$0.637377\pi$
$164$	0	0
$165$	−0.926757	−0.0721480
$166$	0	0
$167$	−7.93820	−0.614276	−0.307138	−	0.951665i	$-0.599371\pi$
$167$	−7.93820	−0.614276	−0.307138	+	0.951665i	$0.599371\pi$
$168$	0	0
$169$	−9.04792	−0.695994
$170$	0	0
$171$	−19.2644	−1.47319
$172$	0	0
$173$	−7.45918	−0.567111	−0.283555	−	0.958956i	$-0.591514\pi$
$173$	−7.45918	−0.567111	−0.283555	+	0.958956i	$0.591514\pi$
$174$	0	0
$175$	−13.0725	−0.988186
$176$	0	0
$177$	39.4054	2.96189
$178$	0	0
$179$	−20.8381	−1.55751	−0.778757	−	0.627326i	$-0.784149\pi$
$179$	−20.8381	−1.55751	−0.778757	+	0.627326i	$0.784149\pi$
$180$	0	0
$181$	−4.22222	−0.313835	−0.156917	−	0.987612i	$-0.550156\pi$
$181$	−4.22222	−0.313835	−0.156917	+	0.987612i	$0.550156\pi$
$182$	0	0
$183$	−17.0247	−1.25850
$184$	0	0
$185$	5.91033	0.434536
$186$	0	0
$187$	−0.0246414	−0.00180196
$188$	0	0
$189$	37.9279	2.75885
$190$	0	0
$191$	14.1891	1.02669	0.513344	−	0.858183i	$-0.328407\pi$
$191$	14.1891	1.02669	0.513344	+	0.858183i	$0.328407\pi$
$192$	0	0
$193$	2.00602	0.144397	0.0721983	−	0.997390i	$-0.476999\pi$
$193$	2.00602	0.144397	0.0721983	+	0.997390i	$0.476999\pi$
$194$	0	0
$195$	8.31528	0.595469
$196$	0	0
$197$	12.9688	0.923985	0.461993	−	0.886884i	$-0.347135\pi$
$197$	12.9688	0.923985	0.461993	+	0.886884i	$0.347135\pi$
$198$	0	0
$199$	11.8687	0.841347	0.420673	−	0.907212i	$-0.361794\pi$
$199$	11.8687	0.841347	0.420673	+	0.907212i	$0.361794\pi$
$200$	0	0
$201$	42.9039	3.02620
$202$	0	0
$203$	−15.3157	−1.07495
$204$	0	0
$205$	4.77119	0.333234
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0.716953	0.0495927
$210$	0	0
$211$	−19.6017	−1.34943	−0.674717	−	0.738077i	$-0.735734\pi$
$211$	−19.6017	−1.34943	−0.674717	+	0.738077i	$0.735734\pi$
$212$	0	0
$213$	17.4899	1.19839
$214$	0	0
$215$	4.54027	0.309644
$216$	0	0
$217$	−26.4282	−1.79406
$218$	0	0
$219$	−11.3477	−0.766810
$220$	0	0
$221$	0.221093	0.0148724
$222$	0	0
$223$	16.7226	1.11983	0.559913	−	0.828551i	$-0.310835\pi$
$223$	16.7226	1.11983	0.559913	+	0.828551i	$0.310835\pi$
$224$	0	0
$225$	−18.1338	−1.20892
$226$	0	0
$227$	−14.1479	−0.939029	−0.469514	−	0.882925i	$-0.655571\pi$
$227$	−14.1479	−0.939029	−0.469514	+	0.882925i	$0.655571\pi$
$228$	0	0
$229$	−23.3847	−1.54531	−0.772653	−	0.634828i	$-0.781071\pi$
$229$	−23.3847	−1.54531	−0.772653	+	0.634828i	$0.781071\pi$
$230$	0	0
$231$	−2.84534	−0.187209
$232$	0	0
$233$	−8.64601	−0.566419	−0.283209	−	0.959058i	$-0.591399\pi$
$233$	−8.64601	−0.566419	−0.283209	+	0.959058i	$0.591399\pi$
$234$	0	0
$235$	−10.8175	−0.705656
$236$	0	0
$237$	26.1186	1.69659
$238$	0	0
$239$	−8.84860	−0.572368	−0.286184	−	0.958175i	$-0.592387\pi$
$239$	−8.84860	−0.572368	−0.286184	+	0.958175i	$0.592387\pi$
$240$	0	0
$241$	5.58741	0.359917	0.179958	−	0.983674i	$-0.442404\pi$
$241$	5.58741	0.359917	0.179958	+	0.983674i	$0.442404\pi$
$242$	0	0
$243$	−0.829571	−0.0532170
$244$	0	0
$245$	15.9628	1.01982
$246$	0	0
$247$	−6.43282	−0.409310
$248$	0	0
$249$	46.8265	2.96751
$250$	0	0
$251$	19.1964	1.21167	0.605834	−	0.795591i	$-0.292840\pi$
$251$	19.1964	1.21167	0.605834	+	0.795591i	$0.292840\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0.465186	0.0291311
$256$	0	0
$257$	8.43545	0.526189	0.263095	−	0.964770i	$-0.415257\pi$
$257$	8.43545	0.526189	0.263095	+	0.964770i	$0.415257\pi$
$258$	0	0
$259$	18.1459	1.12753
$260$	0	0
$261$	−21.2456	−1.31507
$262$	0	0
$263$	−21.4538	−1.32290	−0.661448	−	0.749991i	$-0.730058\pi$
$263$	−21.4538	−1.32290	−0.661448	+	0.749991i	$0.730058\pi$
$264$	0	0
$265$	0.204245	0.0125467
$266$	0	0
$267$	3.05009	0.186662
$268$	0	0
$269$	−21.5844	−1.31602	−0.658012	−	0.753008i	$-0.728602\pi$
$269$	−21.5844	−1.31602	−0.658012	+	0.753008i	$0.728602\pi$
$270$	0	0
$271$	12.8063	0.777930	0.388965	−	0.921252i	$-0.372833\pi$
$271$	12.8063	0.777930	0.388965	+	0.921252i	$0.372833\pi$
$272$	0	0
$273$	25.5296	1.54512
$274$	0	0
$275$	0.674875	0.0406965
$276$	0	0
$277$	−15.6781	−0.942007	−0.471004	−	0.882131i	$-0.656108\pi$
$277$	−15.6781	−0.942007	−0.471004	+	0.882131i	$0.656108\pi$
$278$	0	0
$279$	−36.6605	−2.19480
$280$	0	0
$281$	−9.41093	−0.561409	−0.280705	−	0.959794i	$-0.590568\pi$
$281$	−9.41093	−0.561409	−0.280705	+	0.959794i	$0.590568\pi$
$282$	0	0
$283$	−7.08348	−0.421069	−0.210534	−	0.977586i	$-0.567520\pi$
$283$	−7.08348	−0.421069	−0.210534	+	0.977586i	$0.567520\pi$
$284$	0	0
$285$	−13.5348	−0.801733
$286$	0	0
$287$	14.6485	0.864675
$288$	0	0
$289$	−16.9876	−0.999272
$290$	0	0
$291$	−54.4980	−3.19473
$292$	0	0
$293$	−26.4324	−1.54419	−0.772097	−	0.635504i	$-0.780792\pi$
$293$	−26.4324	−1.54419	−0.772097	+	0.635504i	$0.780792\pi$
$294$	0	0
$295$	18.4090	1.07181
$296$	0	0
$297$	−1.95805	−0.113618
$298$	0	0
$299$	0	0
$300$	0	0
$301$	13.9396	0.803463
$302$	0	0
$303$	9.75140	0.560203
$304$	0	0
$305$	−7.95343	−0.455412
$306$	0	0
$307$	−11.2935	−0.644554	−0.322277	−	0.946645i	$-0.604448\pi$
$307$	−11.2935	−0.644554	−0.322277	+	0.946645i	$0.604448\pi$
$308$	0	0
$309$	30.2591	1.72138
$310$	0	0
$311$	8.16799	0.463164	0.231582	−	0.972815i	$-0.425610\pi$
$311$	8.16799	0.463164	0.231582	+	0.972815i	$0.425610\pi$
$312$	0	0
$313$	−7.85449	−0.443962	−0.221981	−	0.975051i	$-0.571252\pi$
$313$	−7.85449	−0.443962	−0.221981	+	0.975051i	$0.571252\pi$
$314$	0	0
$315$	35.7168	2.01242
$316$	0	0
$317$	13.2797	0.745861	0.372931	−	0.927859i	$-0.378353\pi$
$317$	13.2797	0.745861	0.372931	+	0.927859i	$0.378353\pi$
$318$	0	0
$319$	0.790685	0.0442699
$320$	0	0
$321$	1.42384	0.0794709
$322$	0	0
$323$	−0.359875	−0.0200240
$324$	0	0
$325$	−6.05528	−0.335886
$326$	0	0
$327$	29.3499	1.62306
$328$	0	0
$329$	−33.2120	−1.83103
$330$	0	0
$331$	−1.31086	−0.0720516	−0.0360258	−	0.999351i	$-0.511470\pi$
$331$	−1.31086	−0.0720516	−0.0360258	+	0.999351i	$0.511470\pi$
$332$	0	0
$333$	25.1716	1.37939
$334$	0	0
$335$	20.0434	1.09509
$336$	0	0
$337$	−34.2683	−1.86672	−0.933358	−	0.358947i	$-0.883136\pi$
$337$	−34.2683	−1.86672	−0.933358	+	0.358947i	$0.883136\pi$
$338$	0	0
$339$	26.5351	1.44119
$340$	0	0
$341$	1.36437	0.0738849
$342$	0	0
$343$	18.9666	1.02410
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−24.6201	−1.32168	−0.660838	−	0.750528i	$-0.729799\pi$
$347$	−24.6201	−1.32168	−0.660838	+	0.750528i	$0.729799\pi$
$348$	0	0
$349$	−20.5915	−1.10224	−0.551119	−	0.834427i	$-0.685799\pi$
$349$	−20.5915	−1.10224	−0.551119	+	0.834427i	$0.685799\pi$
$350$	0	0
$351$	17.5685	0.937737
$352$	0	0
$353$	27.3130	1.45372	0.726862	−	0.686784i	$-0.240978\pi$
$353$	27.3130	1.45372	0.726862	+	0.686784i	$0.240978\pi$
$354$	0	0
$355$	8.17075	0.433659
$356$	0	0
$357$	1.42822	0.0755893
$358$	0	0
$359$	37.5903	1.98394	0.991971	−	0.126468i	$-0.0403641\pi$
$359$	37.5903	1.98394	0.991971	+	0.126468i	$0.0403641\pi$
$360$	0	0
$361$	−8.52928	−0.448909
$362$	0	0
$363$	−32.7676	−1.71985
$364$	0	0
$365$	−5.30132	−0.277484
$366$	0	0
$367$	22.3474	1.16653	0.583263	−	0.812283i	$-0.301776\pi$
$367$	22.3474	1.16653	0.583263	+	0.812283i	$0.301776\pi$
$368$	0	0
$369$	20.3200	1.05782
$370$	0	0
$371$	0.627075	0.0325561
$372$	0	0
$373$	29.7918	1.54256	0.771279	−	0.636497i	$-0.219617\pi$
$373$	29.7918	1.54256	0.771279	+	0.636497i	$0.219617\pi$
$374$	0	0
$375$	−33.6543	−1.73790
$376$	0	0
$377$	−7.09438	−0.365379
$378$	0	0
$379$	−21.5505	−1.10698	−0.553488	−	0.832857i	$-0.686703\pi$
$379$	−21.5505	−1.10698	−0.553488	+	0.832857i	$0.686703\pi$
$380$	0	0
$381$	−7.83851	−0.401579
$382$	0	0
$383$	−12.8513	−0.656668	−0.328334	−	0.944562i	$-0.606487\pi$
$383$	−12.8513	−0.656668	−0.328334	+	0.944562i	$0.606487\pi$
$384$	0	0
$385$	−1.32925	−0.0677450
$386$	0	0
$387$	19.3366	0.982934
$388$	0	0
$389$	−0.489944	−0.0248411	−0.0124206	−	0.999923i	$-0.503954\pi$
$389$	−0.489944	−0.0248411	−0.0124206	+	0.999923i	$0.503954\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−45.8429	−2.31247
$394$	0	0
$395$	12.2018	0.613940
$396$	0	0
$397$	14.9274	0.749183	0.374592	−	0.927190i	$-0.377783\pi$
$397$	14.9274	0.749183	0.374592	+	0.927190i	$0.377783\pi$
$398$	0	0
$399$	−41.5547	−2.08033
$400$	0	0
$401$	36.8123	1.83832	0.919159	−	0.393886i	$-0.128870\pi$
$401$	36.8123	1.83832	0.919159	+	0.393886i	$0.128870\pi$
$402$	0	0
$403$	−12.2417	−0.609805
$404$	0	0
$405$	11.9981	0.596188
$406$	0	0
$407$	−0.936796	−0.0464353
$408$	0	0
$409$	−27.3001	−1.34990	−0.674951	−	0.737862i	$-0.735835\pi$
$409$	−27.3001	−1.34990	−0.674951	+	0.737862i	$0.735835\pi$
$410$	0	0
$411$	−30.8474	−1.52159
$412$	0	0
$413$	56.5194	2.78114
$414$	0	0
$415$	21.8759	1.07385
$416$	0	0
$417$	1.70394	0.0834422
$418$	0	0
$419$	16.9066	0.825943	0.412972	−	0.910744i	$-0.364491\pi$
$419$	16.9066	0.825943	0.412972	+	0.910744i	$0.364491\pi$
$420$	0	0
$421$	4.43267	0.216035	0.108017	−	0.994149i	$-0.465550\pi$
$421$	4.43267	0.216035	0.108017	+	0.994149i	$0.465550\pi$
$422$	0	0
$423$	−46.0707	−2.24004
$424$	0	0
$425$	−0.338754	−0.0164320
$426$	0	0
$427$	−24.4187	−1.18170
$428$	0	0
$429$	−1.31798	−0.0636328
$430$	0	0
$431$	−27.9986	−1.34865	−0.674324	−	0.738436i	$-0.735565\pi$
$431$	−27.9986	−1.34865	−0.674324	+	0.738436i	$0.735565\pi$
$432$	0	0
$433$	−12.2280	−0.587641	−0.293820	−	0.955861i	$-0.594927\pi$
$433$	−12.2280	−0.587641	−0.293820	+	0.955861i	$0.594927\pi$
$434$	0	0
$435$	−14.9268	−0.715683
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−15.3054	−0.730487	−0.365244	−	0.930912i	$-0.619014\pi$
$439$	−15.3054	−0.730487	−0.365244	+	0.930912i	$0.619014\pi$
$440$	0	0
$441$	67.9840	3.23734
$442$	0	0
$443$	−7.54063	−0.358266	−0.179133	−	0.983825i	$-0.557329\pi$
$443$	−7.54063	−0.358266	−0.179133	+	0.983825i	$0.557329\pi$
$444$	0	0
$445$	1.42491	0.0675471
$446$	0	0
$447$	39.3933	1.86324
$448$	0	0
$449$	−12.4935	−0.589606	−0.294803	−	0.955558i	$-0.595254\pi$
$449$	−12.4935	−0.589606	−0.294803	+	0.955558i	$0.595254\pi$
$450$	0	0
$451$	−0.756240	−0.0356099
$452$	0	0
$453$	27.1933	1.27765
$454$	0	0
$455$	11.9267	0.559130
$456$	0	0
$457$	−27.6630	−1.29402	−0.647011	−	0.762481i	$-0.723981\pi$
$457$	−27.6630	−1.29402	−0.647011	+	0.762481i	$0.723981\pi$
$458$	0	0
$459$	0.982844	0.0458752
$460$	0	0
$461$	11.2174	0.522445	0.261223	−	0.965279i	$-0.415874\pi$
$461$	11.2174	0.522445	0.261223	+	0.965279i	$0.415874\pi$
$462$	0	0
$463$	−9.47539	−0.440359	−0.220179	−	0.975459i	$-0.570664\pi$
$463$	−9.47539	−0.440359	−0.220179	+	0.975459i	$0.570664\pi$
$464$	0	0
$465$	−25.7570	−1.19445
$466$	0	0
$467$	−9.33895	−0.432155	−0.216077	−	0.976376i	$-0.569326\pi$
$467$	−9.33895	−0.432155	−0.216077	+	0.976376i	$0.569326\pi$
$468$	0	0
$469$	61.5373	2.84153
$470$	0	0
$471$	−26.0688	−1.20118
$472$	0	0
$473$	−0.719639	−0.0330890
$474$	0	0
$475$	9.85620	0.452233
$476$	0	0
$477$	0.869862	0.0398282
$478$	0	0
$479$	−1.20307	−0.0549699	−0.0274849	−	0.999622i	$-0.508750\pi$
$479$	−1.20307	−0.0549699	−0.0274849	+	0.999622i	$0.508750\pi$
$480$	0	0
$481$	8.40535	0.383251
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−25.4598	−1.15607
$486$	0	0
$487$	4.47248	0.202667	0.101334	−	0.994852i	$-0.467689\pi$
$487$	4.47248	0.202667	0.101334	+	0.994852i	$0.467689\pi$
$488$	0	0
$489$	−31.9606	−1.44531
$490$	0	0
$491$	−12.0574	−0.544143	−0.272071	−	0.962277i	$-0.587709\pi$
$491$	−12.0574	−0.544143	−0.272071	+	0.962277i	$0.587709\pi$
$492$	0	0
$493$	−0.396885	−0.0178748
$494$	0	0
$495$	−1.84391	−0.0828774
$496$	0	0
$497$	25.0859	1.12526
$498$	0	0
$499$	−3.64574	−0.163206	−0.0816029	−	0.996665i	$-0.526004\pi$
$499$	−3.64574	−0.163206	−0.0816029	+	0.996665i	$0.526004\pi$
$500$	0	0
$501$	−23.7529	−1.06120
$502$	0	0
$503$	37.0763	1.65315	0.826575	−	0.562826i	$-0.190286\pi$
$503$	37.0763	1.65315	0.826575	+	0.562826i	$0.190286\pi$
$504$	0	0
$505$	4.55555	0.202720
$506$	0	0
$507$	−27.0734	−1.20237
$508$	0	0
$509$	−35.5360	−1.57511	−0.787554	−	0.616246i	$-0.788653\pi$
$509$	−35.5360	−1.57511	−0.787554	+	0.616246i	$0.788653\pi$
$510$	0	0
$511$	−16.2761	−0.720014
$512$	0	0
$513$	−28.5963	−1.26256
$514$	0	0
$515$	14.1361	0.622912
$516$	0	0
$517$	1.71459	0.0754075
$518$	0	0
$519$	−22.3196	−0.979720
$520$	0	0
$521$	−5.66152	−0.248036	−0.124018	−	0.992280i	$-0.539578\pi$
$521$	−5.66152	−0.248036	−0.124018	+	0.992280i	$0.539578\pi$
$522$	0	0
$523$	−3.79404	−0.165902	−0.0829508	−	0.996554i	$-0.526434\pi$
$523$	−3.79404	−0.165902	−0.0829508	+	0.996554i	$0.526434\pi$
$524$	0	0
$525$	−39.1158	−1.70715
$526$	0	0
$527$	−0.684847	−0.0298324
$528$	0	0
$529$	0	0
$530$	0	0
$531$	78.4023	3.40237
$532$	0	0
$533$	6.78532	0.293905
$534$	0	0
$535$	0.665174	0.0287580
$536$	0	0
$537$	−62.3524	−2.69070
$538$	0	0
$539$	−2.53012	−0.108980
$540$	0	0
$541$	10.4793	0.450538	0.225269	−	0.974297i	$-0.427674\pi$
$541$	10.4793	0.450538	0.225269	+	0.974297i	$0.427674\pi$
$542$	0	0
$543$	−12.6338	−0.542170
$544$	0	0
$545$	13.7114	0.587332
$546$	0	0
$547$	28.6896	1.22668	0.613339	−	0.789820i	$-0.289826\pi$
$547$	28.6896	1.22668	0.613339	+	0.789820i	$0.289826\pi$
$548$	0	0
$549$	−33.8729	−1.44566
$550$	0	0
$551$	11.5475	0.491942
$552$	0	0
$553$	37.4621	1.59305
$554$	0	0
$555$	17.6851	0.750689
$556$	0	0
$557$	40.9955	1.73704	0.868518	−	0.495658i	$-0.165073\pi$
$557$	40.9955	1.73704	0.868518	+	0.495658i	$0.165073\pi$
$558$	0	0
$559$	6.45692	0.273098
$560$	0	0
$561$	−0.0737327	−0.00311300
$562$	0	0
$563$	−11.3662	−0.479030	−0.239515	−	0.970893i	$-0.576988\pi$
$563$	−11.3662	−0.479030	−0.239515	+	0.970893i	$0.576988\pi$
$564$	0	0
$565$	12.3964	0.521519
$566$	0	0
$567$	36.8365	1.54699
$568$	0	0
$569$	−35.3055	−1.48008	−0.740042	−	0.672561i	$-0.765194\pi$
$569$	−35.3055	−1.48008	−0.740042	+	0.672561i	$0.765194\pi$
$570$	0	0
$571$	−14.3742	−0.601542	−0.300771	−	0.953696i	$-0.597244\pi$
$571$	−14.3742	−0.601542	−0.300771	+	0.953696i	$0.597244\pi$
$572$	0	0
$573$	42.4570	1.77367
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−19.1095	−0.795540	−0.397770	−	0.917485i	$-0.630216\pi$
$577$	−19.1095	−0.795540	−0.397770	+	0.917485i	$0.630216\pi$
$578$	0	0
$579$	6.00248	0.249454
$580$	0	0
$581$	67.1636	2.78642
$582$	0	0
$583$	−0.0323732	−0.00134076
$584$	0	0
$585$	16.5443	0.684024
$586$	0	0
$587$	42.3710	1.74884	0.874420	−	0.485170i	$-0.161242\pi$
$587$	42.3710	1.74884	0.874420	+	0.485170i	$0.161242\pi$
$588$	0	0
$589$	19.9259	0.821034
$590$	0	0
$591$	38.8055	1.59624
$592$	0	0
$593$	32.5539	1.33683	0.668414	−	0.743789i	$-0.266973\pi$
$593$	32.5539	1.33683	0.668414	+	0.743789i	$0.266973\pi$
$594$	0	0
$595$	0.667220	0.0273533
$596$	0	0
$597$	35.5137	1.45348
$598$	0	0
$599$	35.4803	1.44969	0.724843	−	0.688914i	$-0.241912\pi$
$599$	35.4803	1.44969	0.724843	+	0.688914i	$0.241912\pi$
$600$	0	0
$601$	13.9894	0.570641	0.285320	−	0.958432i	$-0.407900\pi$
$601$	13.9894	0.570641	0.285320	+	0.958432i	$0.407900\pi$
$602$	0	0
$603$	85.3629	3.47624
$604$	0	0
$605$	−15.3080	−0.622360
$606$	0	0
$607$	30.3298	1.23105	0.615525	−	0.788118i	$-0.288944\pi$
$607$	30.3298	1.23105	0.615525	+	0.788118i	$0.288944\pi$
$608$	0	0
$609$	−45.8282	−1.85705
$610$	0	0
$611$	−15.3840	−0.622372
$612$	0	0
$613$	13.8504	0.559413	0.279706	−	0.960086i	$-0.409763\pi$
$613$	13.8504	0.559413	0.279706	+	0.960086i	$0.409763\pi$
$614$	0	0
$615$	14.2765	0.575683
$616$	0	0
$617$	−46.8999	−1.88812	−0.944060	−	0.329774i	$-0.893027\pi$
$617$	−46.8999	−1.88812	−0.944060	+	0.329774i	$0.893027\pi$
$618$	0	0
$619$	−15.3922	−0.618666	−0.309333	−	0.950954i	$-0.600106\pi$
$619$	−15.3922	−0.618666	−0.309333	+	0.950954i	$0.600106\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	4.37476	0.175271
$624$	0	0
$625$	−0.492563	−0.0197025
$626$	0	0
$627$	2.14529	0.0856745
$628$	0	0
$629$	0.470225	0.0187491
$630$	0	0
$631$	26.2229	1.04392	0.521959	−	0.852971i	$-0.325201\pi$
$631$	26.2229	1.04392	0.521959	+	0.852971i	$0.325201\pi$
$632$	0	0
$633$	−58.6526	−2.33123
$634$	0	0
$635$	−3.66191	−0.145318
$636$	0	0
$637$	22.7014	0.899462
$638$	0	0
$639$	34.7985	1.37661
$640$	0	0
$641$	21.3508	0.843305	0.421653	−	0.906757i	$-0.361450\pi$
$641$	21.3508	0.843305	0.421653	+	0.906757i	$0.361450\pi$
$642$	0	0
$643$	−21.3545	−0.842139	−0.421069	−	0.907028i	$-0.638345\pi$
$643$	−21.3545	−0.842139	−0.421069	+	0.907028i	$0.638345\pi$
$644$	0	0
$645$	13.5855	0.534929
$646$	0	0
$647$	10.7365	0.422095	0.211047	−	0.977476i	$-0.432313\pi$
$647$	10.7365	0.422095	0.211047	+	0.977476i	$0.432313\pi$
$648$	0	0
$649$	−2.91785	−0.114536
$650$	0	0
$651$	−79.0792	−3.09936
$652$	0	0
$653$	−21.6445	−0.847015	−0.423507	−	0.905893i	$-0.639201\pi$
$653$	−21.6445	−0.847015	−0.423507	+	0.905893i	$0.639201\pi$
$654$	0	0
$655$	−21.4164	−0.836808
$656$	0	0
$657$	−22.5778	−0.880845
$658$	0	0
$659$	8.50264	0.331216	0.165608	−	0.986192i	$-0.447041\pi$
$659$	8.50264	0.331216	0.165608	+	0.986192i	$0.447041\pi$
$660$	0	0
$661$	−26.7131	−1.03902	−0.519510	−	0.854464i	$-0.673886\pi$
$661$	−26.7131	−1.03902	−0.519510	+	0.854464i	$0.673886\pi$
$662$	0	0
$663$	0.661562	0.0256929
$664$	0	0
$665$	−19.4131	−0.752806
$666$	0	0
$667$	0	0
$668$	0	0
$669$	50.0377	1.93457
$670$	0	0
$671$	1.26063	0.0486661
$672$	0	0
$673$	−1.60196	−0.0617509	−0.0308754	−	0.999523i	$-0.509830\pi$
$673$	−1.60196	−0.0617509	−0.0308754	+	0.999523i	$0.509830\pi$
$674$	0	0
$675$	−26.9180	−1.03607
$676$	0	0
$677$	−17.2239	−0.661969	−0.330984	−	0.943636i	$-0.607381\pi$
$677$	−17.2239	−0.661969	−0.330984	+	0.943636i	$0.607381\pi$
$678$	0	0
$679$	−78.1667	−2.99976
$680$	0	0
$681$	−42.3337	−1.62223
$682$	0	0
$683$	9.47633	0.362602	0.181301	−	0.983428i	$-0.441969\pi$
$683$	9.47633	0.362602	0.181301	+	0.983428i	$0.441969\pi$
$684$	0	0
$685$	−14.4109	−0.550614
$686$	0	0
$687$	−69.9725	−2.66962
$688$	0	0
$689$	0.290466	0.0110659
$690$	0	0
$691$	21.3629	0.812685	0.406342	−	0.913721i	$-0.366804\pi$
$691$	21.3629	0.812685	0.406342	+	0.913721i	$0.366804\pi$
$692$	0	0
$693$	−5.66117	−0.215050
$694$	0	0
$695$	0.796027	0.0301950
$696$	0	0
$697$	0.379595	0.0143782
$698$	0	0
$699$	−25.8708	−0.978525
$700$	0	0
$701$	5.54455	0.209415	0.104707	−	0.994503i	$-0.466609\pi$
$701$	5.54455	0.209415	0.104707	+	0.994503i	$0.466609\pi$
$702$	0	0
$703$	−13.6814	−0.516005
$704$	0	0
$705$	−32.3684	−1.21907
$706$	0	0
$707$	13.9865	0.526016
$708$	0	0
$709$	4.00516	0.150417	0.0752086	−	0.997168i	$-0.476038\pi$
$709$	4.00516	0.150417	0.0752086	+	0.997168i	$0.476038\pi$
$710$	0	0
$711$	51.9664	1.94889
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−0.615721	−0.0230267
$716$	0	0
$717$	−26.4770	−0.988803
$718$	0	0
$719$	22.6271	0.843849	0.421925	−	0.906631i	$-0.361355\pi$
$719$	22.6271	0.843849	0.421925	+	0.906631i	$0.361355\pi$
$720$	0	0
$721$	43.4008	1.61633
$722$	0	0
$723$	16.7188	0.621780
$724$	0	0
$725$	10.8698	0.403695
$726$	0	0
$727$	−37.3558	−1.38545	−0.692725	−	0.721201i	$-0.743590\pi$
$727$	−37.3558	−1.38545	−0.692725	+	0.721201i	$0.743590\pi$
$728$	0	0
$729$	−28.2314	−1.04561
$730$	0	0
$731$	0.361223	0.0133603
$732$	0	0
$733$	2.09461	0.0773661	0.0386831	−	0.999252i	$-0.487684\pi$
$733$	2.09461	0.0773661	0.0386831	+	0.999252i	$0.487684\pi$
$734$	0	0
$735$	47.7643	1.76181
$736$	0	0
$737$	−3.17690	−0.117023
$738$	0	0
$739$	−18.7148	−0.688436	−0.344218	−	0.938890i	$-0.611856\pi$
$739$	−18.7148	−0.688436	−0.344218	+	0.938890i	$0.611856\pi$
$740$	0	0
$741$	−19.2485	−0.707110
$742$	0	0
$743$	−31.2984	−1.14823	−0.574113	−	0.818776i	$-0.694653\pi$
$743$	−31.2984	−1.14823	−0.574113	+	0.818776i	$0.694653\pi$
$744$	0	0
$745$	18.4033	0.674246
$746$	0	0
$747$	93.1676	3.40882
$748$	0	0
$749$	2.04222	0.0746211
$750$	0	0
$751$	−48.2216	−1.75963	−0.879816	−	0.475314i	$-0.842334\pi$
$751$	−48.2216	−1.75963	−0.879816	+	0.475314i	$0.842334\pi$
$752$	0	0
$753$	57.4401	2.09323
$754$	0	0
$755$	12.7039	0.462341
$756$	0	0
$757$	−18.4533	−0.670696	−0.335348	−	0.942094i	$-0.608854\pi$
$757$	−18.4533	−0.670696	−0.335348	+	0.942094i	$0.608854\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	18.4322	0.668166	0.334083	−	0.942544i	$-0.391573\pi$
$761$	18.4322	0.668166	0.334083	+	0.942544i	$0.391573\pi$
$762$	0	0
$763$	42.0968	1.52401
$764$	0	0
$765$	0.925549	0.0334633
$766$	0	0
$767$	26.1803	0.945315
$768$	0	0
$769$	−34.4332	−1.24169	−0.620846	−	0.783933i	$-0.713211\pi$
$769$	−34.4332	−1.24169	−0.620846	+	0.783933i	$0.713211\pi$
$770$	0	0
$771$	25.2408	0.909026
$772$	0	0
$773$	5.84600	0.210266	0.105133	−	0.994458i	$-0.466473\pi$
$773$	5.84600	0.210266	0.105133	+	0.994458i	$0.466473\pi$
$774$	0	0
$775$	18.7565	0.673753
$776$	0	0
$777$	54.2968	1.94789
$778$	0	0
$779$	−11.0445	−0.395710
$780$	0	0
$781$	−1.29508	−0.0463415
$782$	0	0
$783$	−31.5372	−1.12705
$784$	0	0
$785$	−12.1785	−0.434670
$786$	0	0
$787$	−5.47501	−0.195163	−0.0975816	−	0.995228i	$-0.531111\pi$
$787$	−5.47501	−0.195163	−0.0975816	+	0.995228i	$0.531111\pi$
$788$	0	0
$789$	−64.1946	−2.28539
$790$	0	0
$791$	38.0594	1.35324
$792$	0	0
$793$	−11.3109	−0.401663
$794$	0	0
$795$	0.611148	0.0216752
$796$	0	0
$797$	2.24779	0.0796207	0.0398103	−	0.999207i	$-0.487325\pi$
$797$	2.24779	0.0796207	0.0398103	+	0.999207i	$0.487325\pi$
$798$	0	0
$799$	−0.860639	−0.0304472
$800$	0	0
$801$	6.06855	0.214422
$802$	0	0
$803$	0.840267	0.0296524
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−64.5854	−2.27351
$808$	0	0
$809$	12.4644	0.438225	0.219112	−	0.975700i	$-0.429684\pi$
$809$	12.4644	0.438225	0.219112	+	0.975700i	$0.429684\pi$
$810$	0	0
$811$	−23.9769	−0.841944	−0.420972	−	0.907074i	$-0.638311\pi$
$811$	−23.9769	−0.841944	−0.420972	+	0.907074i	$0.638311\pi$
$812$	0	0
$813$	38.3195	1.34392
$814$	0	0
$815$	−14.9310	−0.523010
$816$	0	0
$817$	−10.5099	−0.367697
$818$	0	0
$819$	50.7945	1.77490
$820$	0	0
$821$	−29.8885	−1.04311	−0.521557	−	0.853216i	$-0.674649\pi$
$821$	−29.8885	−1.04311	−0.521557	+	0.853216i	$0.674649\pi$
$822$	0	0
$823$	−10.7506	−0.374741	−0.187370	−	0.982289i	$-0.559996\pi$
$823$	−10.7506	−0.374741	−0.187370	+	0.982289i	$0.559996\pi$
$824$	0	0
$825$	2.01938	0.0703058
$826$	0	0
$827$	−4.42943	−0.154026	−0.0770132	−	0.997030i	$-0.524538\pi$
$827$	−4.42943	−0.154026	−0.0770132	+	0.997030i	$0.524538\pi$
$828$	0	0
$829$	−7.87479	−0.273503	−0.136751	−	0.990605i	$-0.543666\pi$
$829$	−7.87479	−0.273503	−0.136751	+	0.990605i	$0.543666\pi$
$830$	0	0
$831$	−46.9125	−1.62738
$832$	0	0
$833$	1.27000	0.0440028
$834$	0	0
$835$	−11.0966	−0.384015
$836$	0	0
$837$	−54.4192	−1.88100
$838$	0	0
$839$	11.8973	0.410740	0.205370	−	0.978684i	$-0.434160\pi$
$839$	11.8973	0.410740	0.205370	+	0.978684i	$0.434160\pi$
$840$	0	0
$841$	−16.2649	−0.560858
$842$	0	0
$843$	−28.1597	−0.969870
$844$	0	0
$845$	−12.6479	−0.435100
$846$	0	0
$847$	−46.9988	−1.61490
$848$	0	0
$849$	−21.1954	−0.727424
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−34.0150	−1.16465	−0.582326	−	0.812955i	$-0.697857\pi$
$853$	−34.0150	−1.16465	−0.582326	+	0.812955i	$0.697857\pi$
$854$	0	0
$855$	−26.9293	−0.920962
$856$	0	0
$857$	37.2645	1.27293	0.636466	−	0.771305i	$-0.280396\pi$
$857$	37.2645	1.27293	0.636466	+	0.771305i	$0.280396\pi$
$858$	0	0
$859$	37.7361	1.28754	0.643769	−	0.765220i	$-0.277370\pi$
$859$	37.7361	1.28754	0.643769	+	0.765220i	$0.277370\pi$
$860$	0	0
$861$	43.8317	1.49378
$862$	0	0
$863$	−13.5482	−0.461188	−0.230594	−	0.973050i	$-0.574067\pi$
$863$	−13.5482	−0.461188	−0.230594	+	0.973050i	$0.574067\pi$
$864$	0	0
$865$	−10.4270	−0.354529
$866$	0	0
$867$	−50.8309	−1.72631
$868$	0	0
$869$	−1.93401	−0.0656067
$870$	0	0
$871$	28.5046	0.965840
$872$	0	0
$873$	−108.431	−3.66983
$874$	0	0
$875$	−48.2706	−1.63184
$876$	0	0
$877$	7.07074	0.238762	0.119381	−	0.992849i	$-0.461909\pi$
$877$	7.07074	0.238762	0.119381	+	0.992849i	$0.461909\pi$
$878$	0	0
$879$	−79.0917	−2.66770
$880$	0	0
$881$	8.19673	0.276155	0.138077	−	0.990421i	$-0.455908\pi$
$881$	8.19673	0.276155	0.138077	+	0.990421i	$0.455908\pi$
$882$	0	0
$883$	−7.75231	−0.260886	−0.130443	−	0.991456i	$-0.541640\pi$
$883$	−7.75231	−0.260886	−0.130443	+	0.991456i	$0.541640\pi$
$884$	0	0
$885$	55.0840	1.85163
$886$	0	0
$887$	45.2584	1.51963	0.759815	−	0.650140i	$-0.225290\pi$
$887$	45.2584	1.51963	0.759815	+	0.650140i	$0.225290\pi$
$888$	0	0
$889$	−11.2428	−0.377072
$890$	0	0
$891$	−1.90171	−0.0637096
$892$	0	0
$893$	25.0407	0.837954
$894$	0	0
$895$	−29.1291	−0.973679
$896$	0	0
$897$	0	0
$898$	0	0
$899$	21.9752	0.732913
$900$	0	0
$901$	0.0162497	0.000541357 0
$902$	0	0
$903$	41.7103	1.38803
$904$	0	0
$905$	−5.90214	−0.196194
$906$	0	0
$907$	−54.3258	−1.80386	−0.901929	−	0.431885i	$-0.857849\pi$
$907$	−54.3258	−1.80386	−0.901929	+	0.431885i	$0.857849\pi$
$908$	0	0
$909$	19.4017	0.643513
$910$	0	0
$911$	−25.0359	−0.829477	−0.414739	−	0.909941i	$-0.636127\pi$
$911$	−25.0359	−0.829477	−0.414739	+	0.909941i	$0.636127\pi$
$912$	0	0
$913$	−3.46737	−0.114753
$914$	0	0
$915$	−23.7985	−0.786754
$916$	0	0
$917$	−65.7528	−2.17135
$918$	0	0
$919$	23.8024	0.785170	0.392585	−	0.919716i	$-0.371581\pi$
$919$	23.8024	0.785170	0.392585	+	0.919716i	$0.371581\pi$
$920$	0	0
$921$	−33.7927	−1.11351
$922$	0	0
$923$	11.6200	0.382477
$924$	0	0
$925$	−12.8785	−0.423441
$926$	0	0
$927$	60.2044	1.97737
$928$	0	0
$929$	−14.3204	−0.469838	−0.234919	−	0.972015i	$-0.575483\pi$
$929$	−14.3204	−0.469838	−0.234919	+	0.972015i	$0.575483\pi$
$930$	0	0
$931$	−36.9511	−1.21102
$932$	0	0
$933$	24.4405	0.800146
$934$	0	0
$935$	−0.0344456	−0.00112649
$936$	0	0
$937$	37.8951	1.23798	0.618989	−	0.785400i	$-0.287543\pi$
$937$	37.8951	1.23798	0.618989	+	0.785400i	$0.287543\pi$
$938$	0	0
$939$	−23.5024	−0.766973
$940$	0	0
$941$	−26.3514	−0.859031	−0.429516	−	0.903059i	$-0.641316\pi$
$941$	−26.3514	−0.859031	−0.429516	+	0.903059i	$0.641316\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	53.0185	1.72469
$946$	0	0
$947$	12.9255	0.420023	0.210011	−	0.977699i	$-0.432650\pi$
$947$	12.9255	0.420023	0.210011	+	0.977699i	$0.432650\pi$
$948$	0	0
$949$	−7.53924	−0.244734
$950$	0	0
$951$	39.7358	1.28852
$952$	0	0
$953$	9.84574	0.318935	0.159467	−	0.987203i	$-0.449022\pi$
$953$	9.84574	0.318935	0.159467	+	0.987203i	$0.449022\pi$
$954$	0	0
$955$	19.8346	0.641833
$956$	0	0
$957$	2.36591	0.0764791
$958$	0	0
$959$	−44.2446	−1.42873
$960$	0	0
$961$	6.91939	0.223206
$962$	0	0
$963$	2.83291	0.0912894
$964$	0	0
$965$	2.80417	0.0902695
$966$	0	0
$967$	59.2387	1.90499	0.952495	−	0.304555i	$-0.0985077\pi$
$967$	59.2387	1.90499	0.952495	+	0.304555i	$0.0985077\pi$
$968$	0	0
$969$	−1.07683	−0.0345927
$970$	0	0
$971$	6.72775	0.215904	0.107952	−	0.994156i	$-0.465571\pi$
$971$	6.72775	0.215904	0.107952	+	0.994156i	$0.465571\pi$
$972$	0	0
$973$	2.44397	0.0783500
$974$	0	0
$975$	−18.1188	−0.580265
$976$	0	0
$977$	32.7644	1.04823	0.524113	−	0.851649i	$-0.324397\pi$
$977$	32.7644	1.04823	0.524113	+	0.851649i	$0.324397\pi$
$978$	0	0
$979$	−0.225850	−0.00721820
$980$	0	0
$981$	58.3956	1.86443
$982$	0	0
$983$	−25.1809	−0.803146	−0.401573	−	0.915827i	$-0.631536\pi$
$983$	−25.1809	−0.803146	−0.401573	+	0.915827i	$0.631536\pi$
$984$	0	0
$985$	18.1287	0.577629
$986$	0	0
$987$	−99.3777	−3.16323
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−35.7334	−1.13511	−0.567554	−	0.823336i	$-0.692110\pi$
$991$	−35.7334	−1.13511	−0.567554	+	0.823336i	$0.692110\pi$
$992$	0	0
$993$	−3.92241	−0.124474
$994$	0	0
$995$	16.5909	0.525968
$996$	0	0
$997$	5.86890	0.185870	0.0929350	−	0.995672i	$-0.470375\pi$
$997$	5.86890	0.185870	0.0929350	+	0.995672i	$0.470375\pi$
$998$	0	0
$999$	37.3650	1.18218

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8464.2.a.bu.1.5		5
4.3	odd	2		1058.2.a.j.1.1		5
12.11	even	2		9522.2.a.bz.1.1		5
23.9	even	11		368.2.m.a.81.1		10
23.18	even	11		368.2.m.a.209.1		10
23.22	odd	2		8464.2.a.bv.1.5		5
92.55	odd	22		46.2.c.b.35.1	yes	10
92.87	odd	22		46.2.c.b.25.1	✓	10
92.91	even	2		1058.2.a.k.1.1		5
276.179	even	22		414.2.i.c.163.1		10
276.239	even	22		414.2.i.c.127.1		10
276.275	odd	2		9522.2.a.bw.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
46.2.c.b.25.1	✓	10	92.87	odd	22
46.2.c.b.35.1	yes	10	92.55	odd	22
368.2.m.a.81.1		10	23.9	even	11
368.2.m.a.209.1		10	23.18	even	11
414.2.i.c.127.1		10	276.239	even	22
414.2.i.c.163.1		10	276.179	even	22
1058.2.a.j.1.1		5	4.3	odd	2
1058.2.a.k.1.1		5	92.91	even	2
8464.2.a.bu.1.5		5	1.1	even	1	trivial
8464.2.a.bv.1.5		5	23.22	odd	2
9522.2.a.bw.1.5		5	276.275	odd	2
9522.2.a.bz.1.1		5	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8464 = 2^{4} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8464.a (trivial)

\(f(q)\)	\(=\)	\(q+2.99223 q^{3} +1.39788 q^{5} +4.29177 q^{7} +5.95343 q^{9} +O(q^{10})\)	\(q+2.99223 q^{3} +1.39788 q^{5} +4.29177 q^{7} +5.95343 q^{9} -0.221566 q^{11} +1.98798 q^{13} +4.18277 q^{15} +0.111215 q^{17} -3.23585 q^{19} +12.8420 q^{21} -3.04594 q^{25} +8.83734 q^{27} -3.56863 q^{29} -6.15787 q^{31} -0.662975 q^{33} +5.99937 q^{35} +4.22808 q^{37} +5.94850 q^{39} +3.41316 q^{41} +3.24797 q^{43} +8.32217 q^{45} -7.73852 q^{47} +11.4193 q^{49} +0.332780 q^{51} +0.146111 q^{53} -0.309721 q^{55} -9.68240 q^{57} +13.1693 q^{59} -5.68965 q^{61} +25.5508 q^{63} +2.77896 q^{65} +14.3384 q^{67} +5.84511 q^{71} -3.79241 q^{73} -9.11415 q^{75} -0.950909 q^{77} +8.72882 q^{79} +8.58305 q^{81} +15.6494 q^{83} +0.155465 q^{85} -10.6782 q^{87} +1.01934 q^{89} +8.53197 q^{91} -18.4258 q^{93} -4.52332 q^{95} -18.2132 q^{97} -1.31908 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{5} + 9 q^{7} + 7 q^{9}+O(q^{10}) \) 5 * q - 2 * q^5 + 9 * q^7 + 7 * q^9	\( 5 q - 2 q^{5} + 9 q^{7} + 7 q^{9} + q^{11} + q^{13} + 11 q^{15} + q^{17} + 10 q^{19} - 11 q^{25} + 10 q^{29} - 11 q^{33} + 3 q^{35} - 20 q^{37} + 22 q^{39} - 3 q^{41} + 27 q^{43} - 5 q^{45} + 11 q^{47} + 12 q^{49} + 2 q^{53} + 4 q^{55} - 11 q^{57} + 17 q^{59} - 7 q^{61} + 39 q^{63} + 15 q^{65} + 28 q^{67} + 8 q^{73} - 22 q^{75} + 26 q^{77} + 16 q^{79} + q^{81} + 18 q^{83} - 7 q^{85} - 22 q^{87} + 14 q^{89} + 26 q^{91} - 44 q^{93} - 26 q^{95} - 47 q^{97} + 8 q^{99}+O(q^{100}) \) 5 * q - 2 * q^5 + 9 * q^7 + 7 * q^9 + q^11 + q^13 + 11 * q^15 + q^17 + 10 * q^19 - 11 * q^25 + 10 * q^29 - 11 * q^33 + 3 * q^35 - 20 * q^37 + 22 * q^39 - 3 * q^41 + 27 * q^43 - 5 * q^45 + 11 * q^47 + 12 * q^49 + 2 * q^53 + 4 * q^55 - 11 * q^57 + 17 * q^59 - 7 * q^61 + 39 * q^63 + 15 * q^65 + 28 * q^67 + 8 * q^73 - 22 * q^75 + 26 * q^77 + 16 * q^79 + q^81 + 18 * q^83 - 7 * q^85 - 22 * q^87 + 14 * q^89 + 26 * q^91 - 44 * q^93 - 26 * q^95 - 47 * q^97 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more