LMFDB - Embedded newform 1336.2.a.e.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1336,2,Mod(1,1336)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1336, 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("1336.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1336 = 2^{3} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1336.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:	$10.6680137100$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5 x^{11} - 15 x^{10} + 100 x^{9} + 36 x^{8} - 641 x^{7} + 129 x^{6} + 1804 x^{5} - 433 x^{4} + \cdots + 224 $ x^12 - 5x^11 - 15x^10 + 100x^9 + 36x^8 - 641x^7 + 129x^6 + 1804x^5 - 433x^4 - 2399x^3 + 148x^2 + 1224*x + 224
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			1.10
Root			$2.51502$ of defining polynomial
Character	$\chi$	$=$	1336.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.51502 q^{3} +3.24915 q^{5} +3.64134 q^{7} +3.32531 q^{9} +O(q^{10})$	$q+2.51502 q^{3} +3.24915 q^{5} +3.64134 q^{7} +3.32531 q^{9} -4.55087 q^{11} -1.90798 q^{13} +8.17165 q^{15} -2.76209 q^{17} -0.000966172 q^{19} +9.15804 q^{21} +3.53395 q^{23} +5.55695 q^{25} +0.818149 q^{27} +3.21352 q^{29} +6.76881 q^{31} -11.4455 q^{33} +11.8313 q^{35} -10.7480 q^{37} -4.79861 q^{39} +6.22849 q^{41} -12.1891 q^{43} +10.8044 q^{45} -1.34868 q^{47} +6.25939 q^{49} -6.94671 q^{51} -9.40289 q^{53} -14.7865 q^{55} -0.00242994 q^{57} -1.70673 q^{59} -2.75807 q^{61} +12.1086 q^{63} -6.19931 q^{65} -10.8739 q^{67} +8.88795 q^{69} +5.42558 q^{71} -7.26127 q^{73} +13.9758 q^{75} -16.5713 q^{77} +12.1864 q^{79} -7.91826 q^{81} +14.7034 q^{83} -8.97444 q^{85} +8.08206 q^{87} +7.89532 q^{89} -6.94762 q^{91} +17.0237 q^{93} -0.00313923 q^{95} +14.7647 q^{97} -15.1330 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 5 q^{3} - 2 q^{5} + 10 q^{7} + 19 q^{9}+O(q^{10}) $ 12 * q + 5 * q^3 - 2 * q^5 + 10 * q^7 + 19 * q^9	$ 12 q + 5 q^{3} - 2 q^{5} + 10 q^{7} + 19 q^{9} + 14 q^{11} - 9 q^{13} + 8 q^{15} + 4 q^{17} + 9 q^{19} + 7 q^{21} + 15 q^{23} + 18 q^{25} + 20 q^{27} + 17 q^{29} + 11 q^{31} + 8 q^{33} + 19 q^{35} - 29 q^{37} + 26 q^{39} + 14 q^{41} + 15 q^{43} - 26 q^{45} + 17 q^{47} + 20 q^{49} + 36 q^{51} - 7 q^{53} + 13 q^{55} + 3 q^{57} + 32 q^{59} - 8 q^{61} + 50 q^{63} + 37 q^{65} + 39 q^{67} + q^{69} + 35 q^{71} - 12 q^{73} + 29 q^{75} + 13 q^{77} + 36 q^{79} + 44 q^{81} + 25 q^{83} - 38 q^{85} + 14 q^{87} + 23 q^{89} - 2 q^{91} - 25 q^{93} + 38 q^{95} - 2 q^{97} + 43 q^{99}+O(q^{100}) $ 12 * q + 5 * q^3 - 2 * q^5 + 10 * q^7 + 19 * q^9 + 14 * q^11 - 9 * q^13 + 8 * q^15 + 4 * q^17 + 9 * q^19 + 7 * q^21 + 15 * q^23 + 18 * q^25 + 20 * q^27 + 17 * q^29 + 11 * q^31 + 8 * q^33 + 19 * q^35 - 29 * q^37 + 26 * q^39 + 14 * q^41 + 15 * q^43 - 26 * q^45 + 17 * q^47 + 20 * q^49 + 36 * q^51 - 7 * q^53 + 13 * q^55 + 3 * q^57 + 32 * q^59 - 8 * q^61 + 50 * q^63 + 37 * q^65 + 39 * q^67 + q^69 + 35 * q^71 - 12 * q^73 + 29 * q^75 + 13 * q^77 + 36 * q^79 + 44 * q^81 + 25 * q^83 - 38 * q^85 + 14 * q^87 + 23 * q^89 - 2 * q^91 - 25 * q^93 + 38 * q^95 - 2 * q^97 + 43 * 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
$3$	2.51502	1.45205	0.726023	−	0.687671i	$-0.241367\pi$
$3$	2.51502	1.45205	0.726023	+	0.687671i	$0.241367\pi$
$4$	0	0
$5$	3.24915	1.45306	0.726531	−	0.687134i	$-0.241131\pi$
$5$	3.24915	1.45306	0.726531	+	0.687134i	$0.241131\pi$
$6$	0	0
$7$	3.64134	1.37630	0.688149	−	0.725569i	$-0.258423\pi$
$7$	3.64134	1.37630	0.688149	+	0.725569i	$0.258423\pi$
$8$	0	0
$9$	3.32531	1.10844
$10$	0	0
$11$	−4.55087	−1.37214	−0.686070	−	0.727536i	$-0.740666\pi$
$11$	−4.55087	−1.37214	−0.686070	+	0.727536i	$0.740666\pi$
$12$	0	0
$13$	−1.90798	−0.529179	−0.264590	−	0.964361i	$-0.585236\pi$
$13$	−1.90798	−0.529179	−0.264590	+	0.964361i	$0.585236\pi$
$14$	0	0
$15$	8.17165	2.10991
$16$	0	0
$17$	−2.76209	−0.669906	−0.334953	−	0.942235i	$-0.608720\pi$
$17$	−2.76209	−0.669906	−0.334953	+	0.942235i	$0.608720\pi$
$18$	0	0
$19$	−0.000966172 0	−0.000221655 0	−0.000110828	−	1.00000i	$-0.500035\pi$
$19$	−0.000966172 0	−0.000221655 0	−0.000110828		1.00000i	$0.500035\pi$
$20$	0	0
$21$	9.15804	1.99845
$22$	0	0
$23$	3.53395	0.736880	0.368440	−	0.929651i	$-0.379892\pi$
$23$	3.53395	0.736880	0.368440	+	0.929651i	$0.379892\pi$
$24$	0	0
$25$	5.55695	1.11139
$26$	0	0
$27$	0.818149	0.157453
$28$	0	0
$29$	3.21352	0.596736	0.298368	−	0.954451i	$-0.403558\pi$
$29$	3.21352	0.596736	0.298368	+	0.954451i	$0.403558\pi$
$30$	0	0
$31$	6.76881	1.21571	0.607857	−	0.794047i	$-0.292029\pi$
$31$	6.76881	1.21571	0.607857	+	0.794047i	$0.292029\pi$
$32$	0	0
$33$	−11.4455	−1.99241
$34$	0	0
$35$	11.8313	1.99985
$36$	0	0
$37$	−10.7480	−1.76696	−0.883478	−	0.468473i	$-0.844804\pi$
$37$	−10.7480	−1.76696	−0.883478	+	0.468473i	$0.844804\pi$
$38$	0	0
$39$	−4.79861	−0.768392
$40$	0	0
$41$	6.22849	0.972727	0.486364	−	0.873756i	$-0.338323\pi$
$41$	6.22849	0.972727	0.486364	+	0.873756i	$0.338323\pi$
$42$	0	0
$43$	−12.1891	−1.85883	−0.929413	−	0.369041i	$-0.879686\pi$
$43$	−12.1891	−1.85883	−0.929413	+	0.369041i	$0.879686\pi$
$44$	0	0
$45$	10.8044	1.61063
$46$	0	0
$47$	−1.34868	−0.196726	−0.0983629	−	0.995151i	$-0.531361\pi$
$47$	−1.34868	−0.196726	−0.0983629	+	0.995151i	$0.531361\pi$
$48$	0	0
$49$	6.25939	0.894198
$50$	0	0
$51$	−6.94671	−0.972734
$52$	0	0
$53$	−9.40289	−1.29159	−0.645793	−	0.763512i	$-0.723473\pi$
$53$	−9.40289	−1.29159	−0.645793	+	0.763512i	$0.723473\pi$
$54$	0	0
$55$	−14.7865	−1.99380
$56$	0	0
$57$	−0.00242994	−0.000321853 0
$58$	0	0
$59$	−1.70673	−0.222198	−0.111099	−	0.993809i	$-0.535437\pi$
$59$	−1.70673	−0.222198	−0.111099	+	0.993809i	$0.535437\pi$
$60$	0	0
$61$	−2.75807	−0.353134	−0.176567	−	0.984289i	$-0.556499\pi$
$61$	−2.75807	−0.353134	−0.176567	+	0.984289i	$0.556499\pi$
$62$	0	0
$63$	12.1086	1.52554
$64$	0	0
$65$	−6.19931	−0.768930
$66$	0	0
$67$	−10.8739	−1.32845	−0.664227	−	0.747531i	$-0.731239\pi$
$67$	−10.8739	−1.32845	−0.664227	+	0.747531i	$0.731239\pi$
$68$	0	0
$69$	8.88795	1.06998
$70$	0	0
$71$	5.42558	0.643898	0.321949	−	0.946757i	$-0.395662\pi$
$71$	5.42558	0.643898	0.321949	+	0.946757i	$0.395662\pi$
$72$	0	0
$73$	−7.26127	−0.849868	−0.424934	−	0.905224i	$-0.639703\pi$
$73$	−7.26127	−0.849868	−0.424934	+	0.905224i	$0.639703\pi$
$74$	0	0
$75$	13.9758	1.61379
$76$	0	0
$77$	−16.5713	−1.88847
$78$	0	0
$79$	12.1864	1.37108	0.685540	−	0.728035i	$-0.259566\pi$
$79$	12.1864	1.37108	0.685540	+	0.728035i	$0.259566\pi$
$80$	0	0
$81$	−7.91826	−0.879807
$82$	0	0
$83$	14.7034	1.61391	0.806956	−	0.590611i	$-0.201113\pi$
$83$	14.7034	1.61391	0.806956	+	0.590611i	$0.201113\pi$
$84$	0	0
$85$	−8.97444	−0.973415
$86$	0	0
$87$	8.08206	0.866488
$88$	0	0
$89$	7.89532	0.836903	0.418451	−	0.908239i	$-0.362573\pi$
$89$	7.89532	0.836903	0.418451	+	0.908239i	$0.362573\pi$
$90$	0	0
$91$	−6.94762	−0.728308
$92$	0	0
$93$	17.0237	1.76527
$94$	0	0
$95$	−0.00313923	−0.000322079 0
$96$	0	0
$97$	14.7647	1.49913	0.749565	−	0.661931i	$-0.230263\pi$
$97$	14.7647	1.49913	0.749565	+	0.661931i	$0.230263\pi$
$98$	0	0
$99$	−15.1330	−1.52093
$100$	0	0
$101$	−0.720775	−0.0717198	−0.0358599	−	0.999357i	$-0.511417\pi$
$101$	−0.720775	−0.0717198	−0.0358599	+	0.999357i	$0.511417\pi$
$102$	0	0
$103$	−12.9281	−1.27384	−0.636922	−	0.770929i	$-0.719793\pi$
$103$	−12.9281	−1.27384	−0.636922	+	0.770929i	$0.719793\pi$
$104$	0	0
$105$	29.7558	2.90387
$106$	0	0
$107$	10.1744	0.983592	0.491796	−	0.870711i	$-0.336341\pi$
$107$	10.1744	0.983592	0.491796	+	0.870711i	$0.336341\pi$
$108$	0	0
$109$	3.33822	0.319744	0.159872	−	0.987138i	$-0.448892\pi$
$109$	3.33822	0.319744	0.159872	+	0.987138i	$0.448892\pi$
$110$	0	0
$111$	−27.0313	−2.56570
$112$	0	0
$113$	−13.1540	−1.23743	−0.618713	−	0.785617i	$-0.712346\pi$
$113$	−13.1540	−1.23743	−0.618713	+	0.785617i	$0.712346\pi$
$114$	0	0
$115$	11.4823	1.07073
$116$	0	0
$117$	−6.34462	−0.586561
$118$	0	0
$119$	−10.0577	−0.921991
$120$	0	0
$121$	9.71045	0.882768
$122$	0	0
$123$	15.6648	1.41244
$124$	0	0
$125$	1.80961	0.161857
$126$	0	0
$127$	9.83499	0.872715	0.436357	−	0.899773i	$-0.356268\pi$
$127$	9.83499	0.872715	0.436357	+	0.899773i	$0.356268\pi$
$128$	0	0
$129$	−30.6559	−2.69910
$130$	0	0
$131$	4.54870	0.397422	0.198711	−	0.980058i	$-0.436325\pi$
$131$	4.54870	0.397422	0.198711	+	0.980058i	$0.436325\pi$
$132$	0	0
$133$	−0.00351817	−0.000305064 0
$134$	0	0
$135$	2.65828	0.228789
$136$	0	0
$137$	−6.11611	−0.522534	−0.261267	−	0.965267i	$-0.584140\pi$
$137$	−6.11611	−0.522534	−0.261267	+	0.965267i	$0.584140\pi$
$138$	0	0
$139$	11.2357	0.952999	0.476500	−	0.879175i	$-0.341905\pi$
$139$	11.2357	0.952999	0.476500	+	0.879175i	$0.341905\pi$
$140$	0	0
$141$	−3.39196	−0.285655
$142$	0	0
$143$	8.68299	0.726108
$144$	0	0
$145$	10.4412	0.867095
$146$	0	0
$147$	15.7425	1.29842
$148$	0	0
$149$	−2.89920	−0.237512	−0.118756	−	0.992923i	$-0.537891\pi$
$149$	−2.89920	−0.237512	−0.118756	+	0.992923i	$0.537891\pi$
$150$	0	0
$151$	−12.0721	−0.982417	−0.491208	−	0.871042i	$-0.663445\pi$
$151$	−12.0721	−0.982417	−0.491208	+	0.871042i	$0.663445\pi$
$152$	0	0
$153$	−9.18480	−0.742547
$154$	0	0
$155$	21.9928	1.76651
$156$	0	0
$157$	−2.15189	−0.171740	−0.0858698	−	0.996306i	$-0.527367\pi$
$157$	−2.15189	−0.171740	−0.0858698	+	0.996306i	$0.527367\pi$
$158$	0	0
$159$	−23.6484	−1.87544
$160$	0	0
$161$	12.8683	1.01417
$162$	0	0
$163$	14.1711	1.10996	0.554981	−	0.831863i	$-0.312725\pi$
$163$	14.1711	1.10996	0.554981	+	0.831863i	$0.312725\pi$
$164$	0	0
$165$	−37.1882	−2.89509
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−9.35960	−0.719970
$170$	0	0
$171$	−0.00321282	−0.000245690 0
$172$	0	0
$173$	−7.16876	−0.545031	−0.272515	−	0.962151i	$-0.587856\pi$
$173$	−7.16876	−0.545031	−0.272515	+	0.962151i	$0.587856\pi$
$174$	0	0
$175$	20.2348	1.52960
$176$	0	0
$177$	−4.29246	−0.322641
$178$	0	0
$179$	−2.71339	−0.202808	−0.101404	−	0.994845i	$-0.532334\pi$
$179$	−2.71339	−0.202808	−0.101404	+	0.994845i	$0.532334\pi$
$180$	0	0
$181$	−22.7247	−1.68912	−0.844558	−	0.535465i	$-0.820137\pi$
$181$	−22.7247	−1.68912	−0.844558	+	0.535465i	$0.820137\pi$
$182$	0	0
$183$	−6.93658	−0.512767
$184$	0	0
$185$	−34.9217	−2.56750
$186$	0	0
$187$	12.5699	0.919205
$188$	0	0
$189$	2.97916	0.216702
$190$	0	0
$191$	25.5292	1.84723	0.923613	−	0.383326i	$-0.125221\pi$
$191$	25.5292	1.84723	0.923613	+	0.383326i	$0.125221\pi$
$192$	0	0
$193$	−2.88828	−0.207903	−0.103951	−	0.994582i	$-0.533149\pi$
$193$	−2.88828	−0.207903	−0.103951	+	0.994582i	$0.533149\pi$
$194$	0	0
$195$	−15.5914	−1.11652
$196$	0	0
$197$	1.67383	0.119256	0.0596279	−	0.998221i	$-0.481009\pi$
$197$	1.67383	0.119256	0.0596279	+	0.998221i	$0.481009\pi$
$198$	0	0
$199$	−13.5815	−0.962768	−0.481384	−	0.876510i	$-0.659866\pi$
$199$	−13.5815	−0.962768	−0.481384	+	0.876510i	$0.659866\pi$
$200$	0	0
$201$	−27.3480	−1.92898
$202$	0	0
$203$	11.7015	0.821287
$204$	0	0
$205$	20.2373	1.41343
$206$	0	0
$207$	11.7515	0.816784
$208$	0	0
$209$	0.00439693	0.000304142 0
$210$	0	0
$211$	23.8316	1.64063	0.820316	−	0.571911i	$-0.193798\pi$
$211$	23.8316	1.64063	0.820316	+	0.571911i	$0.193798\pi$
$212$	0	0
$213$	13.6454	0.934969
$214$	0	0
$215$	−39.6043	−2.70099
$216$	0	0
$217$	24.6476	1.67319
$218$	0	0
$219$	−18.2622	−1.23405
$220$	0	0
$221$	5.27002	0.354500
$222$	0	0
$223$	6.86645	0.459812	0.229906	−	0.973213i	$-0.426158\pi$
$223$	6.86645	0.459812	0.229906	+	0.973213i	$0.426158\pi$
$224$	0	0
$225$	18.4786	1.23190
$226$	0	0
$227$	9.52258	0.632036	0.316018	−	0.948753i	$-0.397654\pi$
$227$	9.52258	0.632036	0.316018	+	0.948753i	$0.397654\pi$
$228$	0	0
$229$	−16.4364	−1.08615	−0.543074	−	0.839685i	$-0.682740\pi$
$229$	−16.4364	−1.08615	−0.543074	+	0.839685i	$0.682740\pi$
$230$	0	0
$231$	−41.6771	−2.74215
$232$	0	0
$233$	−21.2957	−1.39513	−0.697564	−	0.716523i	$-0.745733\pi$
$233$	−21.2957	−1.39513	−0.697564	+	0.716523i	$0.745733\pi$
$234$	0	0
$235$	−4.38207	−0.285855
$236$	0	0
$237$	30.6491	1.99087
$238$	0	0
$239$	−3.85882	−0.249606	−0.124803	−	0.992182i	$-0.539830\pi$
$239$	−3.85882	−0.249606	−0.124803	+	0.992182i	$0.539830\pi$
$240$	0	0
$241$	24.8126	1.59832	0.799160	−	0.601118i	$-0.205278\pi$
$241$	24.8126	1.59832	0.799160	+	0.601118i	$0.205278\pi$
$242$	0	0
$243$	−22.3690	−1.43497
$244$	0	0
$245$	20.3377	1.29933
$246$	0	0
$247$	0.00184344	0.000117295 0
$248$	0	0
$249$	36.9794	2.34347
$250$	0	0
$251$	26.1842	1.65273	0.826367	−	0.563131i	$-0.190404\pi$
$251$	26.1842	1.65273	0.826367	+	0.563131i	$0.190404\pi$
$252$	0	0
$253$	−16.0826	−1.01110
$254$	0	0
$255$	−22.5709	−1.41344
$256$	0	0
$257$	−22.5995	−1.40972	−0.704858	−	0.709349i	$-0.748989\pi$
$257$	−22.5995	−1.40972	−0.704858	+	0.709349i	$0.748989\pi$
$258$	0	0
$259$	−39.1371	−2.43186
$260$	0	0
$261$	10.6859	0.661443
$262$	0	0
$263$	21.0875	1.30031	0.650157	−	0.759800i	$-0.274703\pi$
$263$	21.0875	1.30031	0.650157	+	0.759800i	$0.274703\pi$
$264$	0	0
$265$	−30.5514	−1.87676
$266$	0	0
$267$	19.8569	1.21522
$268$	0	0
$269$	17.7594	1.08281	0.541405	−	0.840762i	$-0.317893\pi$
$269$	17.7594	1.08281	0.541405	+	0.840762i	$0.317893\pi$
$270$	0	0
$271$	−8.48230	−0.515263	−0.257631	−	0.966243i	$-0.582942\pi$
$271$	−8.48230	−0.515263	−0.257631	+	0.966243i	$0.582942\pi$
$272$	0	0
$273$	−17.4734	−1.05754
$274$	0	0
$275$	−25.2890	−1.52498
$276$	0	0
$277$	−6.61144	−0.397243	−0.198621	−	0.980076i	$-0.563646\pi$
$277$	−6.61144	−0.397243	−0.198621	+	0.980076i	$0.563646\pi$
$278$	0	0
$279$	22.5084	1.34754
$280$	0	0
$281$	21.7564	1.29788	0.648938	−	0.760841i	$-0.275213\pi$
$281$	21.7564	1.29788	0.648938	+	0.760841i	$0.275213\pi$
$282$	0	0
$283$	26.1381	1.55375	0.776873	−	0.629657i	$-0.216805\pi$
$283$	26.1381	1.55375	0.776873	+	0.629657i	$0.216805\pi$
$284$	0	0
$285$	−0.00789523	−0.000467673 0
$286$	0	0
$287$	22.6801	1.33876
$288$	0	0
$289$	−9.37084	−0.551226
$290$	0	0
$291$	37.1335	2.17680
$292$	0	0
$293$	17.4554	1.01975	0.509877	−	0.860248i	$-0.329691\pi$
$293$	17.4554	1.01975	0.509877	+	0.860248i	$0.329691\pi$
$294$	0	0
$295$	−5.54542	−0.322867
$296$	0	0
$297$	−3.72329	−0.216047
$298$	0	0
$299$	−6.74272	−0.389942
$300$	0	0
$301$	−44.3849	−2.55830
$302$	0	0
$303$	−1.81276	−0.104140
$304$	0	0
$305$	−8.96136	−0.513126
$306$	0	0
$307$	−10.0662	−0.574506	−0.287253	−	0.957855i	$-0.592742\pi$
$307$	−10.0662	−0.574506	−0.287253	+	0.957855i	$0.592742\pi$
$308$	0	0
$309$	−32.5144	−1.84968
$310$	0	0
$311$	16.5331	0.937508	0.468754	−	0.883329i	$-0.344703\pi$
$311$	16.5331	0.937508	0.468754	+	0.883329i	$0.344703\pi$
$312$	0	0
$313$	22.4385	1.26830	0.634149	−	0.773211i	$-0.281351\pi$
$313$	22.4385	1.26830	0.634149	+	0.773211i	$0.281351\pi$
$314$	0	0
$315$	39.3426	2.21670
$316$	0	0
$317$	−19.6568	−1.10404	−0.552018	−	0.833832i	$-0.686142\pi$
$317$	−19.6568	−1.10404	−0.552018	+	0.833832i	$0.686142\pi$
$318$	0	0
$319$	−14.6243	−0.818805
$320$	0	0
$321$	25.5887	1.42822
$322$	0	0
$323$	0.00266866	0.000148488 0
$324$	0	0
$325$	−10.6026	−0.588124
$326$	0	0
$327$	8.39568	0.464282
$328$	0	0
$329$	−4.91102	−0.270753
$330$	0	0
$331$	−31.9910	−1.75838	−0.879192	−	0.476468i	$-0.841917\pi$
$331$	−31.9910	−1.75838	−0.879192	+	0.476468i	$0.841917\pi$
$332$	0	0
$333$	−35.7403	−1.95856
$334$	0	0
$335$	−35.3308	−1.93033
$336$	0	0
$337$	−5.97813	−0.325650	−0.162825	−	0.986655i	$-0.552061\pi$
$337$	−5.97813	−0.325650	−0.162825	+	0.986655i	$0.552061\pi$
$338$	0	0
$339$	−33.0826	−1.79680
$340$	0	0
$341$	−30.8040	−1.66813
$342$	0	0
$343$	−2.69682	−0.145615
$344$	0	0
$345$	28.8783	1.55475
$346$	0	0
$347$	−26.8363	−1.44065	−0.720325	−	0.693637i	$-0.756007\pi$
$347$	−26.8363	−1.44065	−0.720325	+	0.693637i	$0.756007\pi$
$348$	0	0
$349$	10.6736	0.571347	0.285674	−	0.958327i	$-0.407783\pi$
$349$	10.6736	0.571347	0.285674	+	0.958327i	$0.407783\pi$
$350$	0	0
$351$	−1.56101	−0.0833207
$352$	0	0
$353$	6.20762	0.330398	0.165199	−	0.986260i	$-0.447173\pi$
$353$	6.20762	0.330398	0.165199	+	0.986260i	$0.447173\pi$
$354$	0	0
$355$	17.6285	0.935624
$356$	0	0
$357$	−25.2954	−1.33877
$358$	0	0
$359$	11.5474	0.609448	0.304724	−	0.952441i	$-0.401436\pi$
$359$	11.5474	0.609448	0.304724	+	0.952441i	$0.401436\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	24.4219	1.28182
$364$	0	0
$365$	−23.5929	−1.23491
$366$	0	0
$367$	−31.4525	−1.64181	−0.820903	−	0.571067i	$-0.806530\pi$
$367$	−31.4525	−1.64181	−0.820903	+	0.571067i	$0.806530\pi$
$368$	0	0
$369$	20.7116	1.07821
$370$	0	0
$371$	−34.2392	−1.77761
$372$	0	0
$373$	−20.4841	−1.06063	−0.530314	−	0.847801i	$-0.677926\pi$
$373$	−20.4841	−1.06063	−0.530314	+	0.847801i	$0.677926\pi$
$374$	0	0
$375$	4.55121	0.235023
$376$	0	0
$377$	−6.13134	−0.315780
$378$	0	0
$379$	10.0226	0.514828	0.257414	−	0.966301i	$-0.417130\pi$
$379$	10.0226	0.514828	0.257414	+	0.966301i	$0.417130\pi$
$380$	0	0
$381$	24.7352	1.26722
$382$	0	0
$383$	−20.5190	−1.04847	−0.524237	−	0.851572i	$-0.675649\pi$
$383$	−20.5190	−1.04847	−0.524237	+	0.851572i	$0.675649\pi$
$384$	0	0
$385$	−53.8426	−2.74407
$386$	0	0
$387$	−40.5326	−2.06039
$388$	0	0
$389$	39.1984	1.98744	0.993720	−	0.111899i	$-0.0356934\pi$
$389$	39.1984	1.98744	0.993720	+	0.111899i	$0.0356934\pi$
$390$	0	0
$391$	−9.76111	−0.493641
$392$	0	0
$393$	11.4400	0.577074
$394$	0	0
$395$	39.5955	1.99227
$396$	0	0
$397$	21.6589	1.08703	0.543515	−	0.839399i	$-0.317093\pi$
$397$	21.6589	1.08703	0.543515	+	0.839399i	$0.317093\pi$
$398$	0	0
$399$	−0.00884824	−0.000442966 0
$400$	0	0
$401$	−9.58930	−0.478867	−0.239433	−	0.970913i	$-0.576962\pi$
$401$	−9.58930	−0.478867	−0.239433	+	0.970913i	$0.576962\pi$
$402$	0	0
$403$	−12.9148	−0.643330
$404$	0	0
$405$	−25.7276	−1.27841
$406$	0	0
$407$	48.9127	2.42451
$408$	0	0
$409$	11.5275	0.569997	0.284998	−	0.958528i	$-0.408007\pi$
$409$	11.5275	0.569997	0.284998	+	0.958528i	$0.408007\pi$
$410$	0	0
$411$	−15.3821	−0.758744
$412$	0	0
$413$	−6.21480	−0.305810
$414$	0	0
$415$	47.7736	2.34512
$416$	0	0
$417$	28.2580	1.38380
$418$	0	0
$419$	−12.3931	−0.605444	−0.302722	−	0.953079i	$-0.597895\pi$
$419$	−12.3931	−0.605444	−0.302722	+	0.953079i	$0.597895\pi$
$420$	0	0
$421$	−14.8893	−0.725658	−0.362829	−	0.931856i	$-0.618189\pi$
$421$	−14.8893	−0.725658	−0.362829	+	0.931856i	$0.618189\pi$
$422$	0	0
$423$	−4.48479	−0.218058
$424$	0	0
$425$	−15.3488	−0.744527
$426$	0	0
$427$	−10.0431	−0.486018
$428$	0	0
$429$	21.8378	1.05434
$430$	0	0
$431$	−21.8253	−1.05129	−0.525644	−	0.850705i	$-0.676175\pi$
$431$	−21.8253	−1.05129	−0.525644	+	0.850705i	$0.676175\pi$
$432$	0	0
$433$	35.3162	1.69719	0.848594	−	0.529045i	$-0.177450\pi$
$433$	35.3162	1.69719	0.848594	+	0.529045i	$0.177450\pi$
$434$	0	0
$435$	26.2598	1.25906
$436$	0	0
$437$	−0.00341441	−0.000163333 0
$438$	0	0
$439$	−5.00817	−0.239027	−0.119513	−	0.992833i	$-0.538133\pi$
$439$	−5.00817	−0.239027	−0.119513	+	0.992833i	$0.538133\pi$
$440$	0	0
$441$	20.8144	0.991161
$442$	0	0
$443$	−34.1728	−1.62360	−0.811799	−	0.583936i	$-0.801512\pi$
$443$	−34.1728	−1.62360	−0.811799	+	0.583936i	$0.801512\pi$
$444$	0	0
$445$	25.6531	1.21607
$446$	0	0
$447$	−7.29154	−0.344878
$448$	0	0
$449$	−14.8065	−0.698764	−0.349382	−	0.936980i	$-0.613608\pi$
$449$	−14.8065	−0.698764	−0.349382	+	0.936980i	$0.613608\pi$
$450$	0	0
$451$	−28.3451	−1.33472
$452$	0	0
$453$	−30.3616	−1.42651
$454$	0	0
$455$	−22.5738	−1.05828
$456$	0	0
$457$	−4.60631	−0.215474	−0.107737	−	0.994179i	$-0.534360\pi$
$457$	−4.60631	−0.215474	−0.107737	+	0.994179i	$0.534360\pi$
$458$	0	0
$459$	−2.25980	−0.105479
$460$	0	0
$461$	14.3061	0.666300	0.333150	−	0.942874i	$-0.391888\pi$
$461$	14.3061	0.666300	0.333150	+	0.942874i	$0.391888\pi$
$462$	0	0
$463$	24.9928	1.16151	0.580757	−	0.814077i	$-0.302757\pi$
$463$	24.9928	1.16151	0.580757	+	0.814077i	$0.302757\pi$
$464$	0	0
$465$	55.3124	2.56505
$466$	0	0
$467$	16.3297	0.755649	0.377824	−	0.925877i	$-0.376672\pi$
$467$	16.3297	0.755649	0.377824	+	0.925877i	$0.376672\pi$
$468$	0	0
$469$	−39.5955	−1.82835
$470$	0	0
$471$	−5.41204	−0.249374
$472$	0	0
$473$	55.4712	2.55057
$474$	0	0
$475$	−0.00536897	−0.000246345 0
$476$	0	0
$477$	−31.2675	−1.43164
$478$	0	0
$479$	16.7781	0.766611	0.383306	−	0.923622i	$-0.374786\pi$
$479$	16.7781	0.766611	0.383306	+	0.923622i	$0.374786\pi$
$480$	0	0
$481$	20.5069	0.935036
$482$	0	0
$483$	32.3641	1.47262
$484$	0	0
$485$	47.9727	2.17833
$486$	0	0
$487$	−35.3206	−1.60053	−0.800264	−	0.599647i	$-0.795308\pi$
$487$	−35.3206	−1.60053	−0.800264	+	0.599647i	$0.795308\pi$
$488$	0	0
$489$	35.6404	1.61172
$490$	0	0
$491$	−6.36939	−0.287446	−0.143723	−	0.989618i	$-0.545907\pi$
$491$	−6.36939	−0.287446	−0.143723	+	0.989618i	$0.545907\pi$
$492$	0	0
$493$	−8.87605	−0.399757
$494$	0	0
$495$	−49.1695	−2.21000
$496$	0	0
$497$	19.7564	0.886196
$498$	0	0
$499$	42.8357	1.91759	0.958796	−	0.284097i	$-0.0916938\pi$
$499$	42.8357	1.91759	0.958796	+	0.284097i	$0.0916938\pi$
$500$	0	0
$501$	2.51502	0.112363
$502$	0	0
$503$	36.1170	1.61038	0.805189	−	0.593018i	$-0.202064\pi$
$503$	36.1170	1.61038	0.805189	+	0.593018i	$0.202064\pi$
$504$	0	0
$505$	−2.34190	−0.104213
$506$	0	0
$507$	−23.5396	−1.04543
$508$	0	0
$509$	−18.4304	−0.816912	−0.408456	−	0.912778i	$-0.633933\pi$
$509$	−18.4304	−0.816912	−0.408456	+	0.912778i	$0.633933\pi$
$510$	0	0
$511$	−26.4408	−1.16967
$512$	0	0
$513$	−0.000790473 0	−3.49002e−5 0
$514$	0	0
$515$	−42.0053	−1.85097
$516$	0	0
$517$	6.13769	0.269935
$518$	0	0
$519$	−18.0295	−0.791409
$520$	0	0
$521$	28.7858	1.26113	0.630565	−	0.776137i	$-0.282823\pi$
$521$	28.7858	1.26113	0.630565	+	0.776137i	$0.282823\pi$
$522$	0	0
$523$	−12.5265	−0.547744	−0.273872	−	0.961766i	$-0.588304\pi$
$523$	−12.5265	−0.547744	−0.273872	+	0.961766i	$0.588304\pi$
$524$	0	0
$525$	50.8908	2.22106
$526$	0	0
$527$	−18.6961	−0.814414
$528$	0	0
$529$	−10.5112	−0.457007
$530$	0	0
$531$	−5.67541	−0.246292
$532$	0	0
$533$	−11.8839	−0.514747
$534$	0	0
$535$	33.0580	1.42922
$536$	0	0
$537$	−6.82422	−0.294487
$538$	0	0
$539$	−28.4857	−1.22697
$540$	0	0
$541$	−41.3549	−1.77799	−0.888994	−	0.457920i	$-0.848595\pi$
$541$	−41.3549	−1.77799	−0.888994	+	0.457920i	$0.848595\pi$
$542$	0	0
$543$	−57.1530	−2.45267
$544$	0	0
$545$	10.8464	0.464608
$546$	0	0
$547$	−27.5776	−1.17913	−0.589566	−	0.807720i	$-0.700701\pi$
$547$	−27.5776	−1.17913	−0.589566	+	0.807720i	$0.700701\pi$
$548$	0	0
$549$	−9.17141	−0.391426
$550$	0	0
$551$	−0.00310482	−0.000132270 0
$552$	0	0
$553$	44.3750	1.88702
$554$	0	0
$555$	−87.8287	−3.72812
$556$	0	0
$557$	17.2923	0.732698	0.366349	−	0.930478i	$-0.380608\pi$
$557$	17.2923	0.732698	0.366349	+	0.930478i	$0.380608\pi$
$558$	0	0
$559$	23.2567	0.983652
$560$	0	0
$561$	31.6136	1.33473
$562$	0	0
$563$	−4.83039	−0.203577	−0.101788	−	0.994806i	$-0.532456\pi$
$563$	−4.83039	−0.203577	−0.101788	+	0.994806i	$0.532456\pi$
$564$	0	0
$565$	−42.7393	−1.79806
$566$	0	0
$567$	−28.8331	−1.21088
$568$	0	0
$569$	14.7588	0.618720	0.309360	−	0.950945i	$-0.399885\pi$
$569$	14.7588	0.618720	0.309360	+	0.950945i	$0.399885\pi$
$570$	0	0
$571$	−11.9854	−0.501572	−0.250786	−	0.968043i	$-0.580689\pi$
$571$	−11.9854	−0.501572	−0.250786	+	0.968043i	$0.580689\pi$
$572$	0	0
$573$	64.2063	2.68226
$574$	0	0
$575$	19.6380	0.818962
$576$	0	0
$577$	2.61426	0.108833	0.0544165	−	0.998518i	$-0.482670\pi$
$577$	2.61426	0.108833	0.0544165	+	0.998518i	$0.482670\pi$
$578$	0	0
$579$	−7.26406	−0.301884
$580$	0	0
$581$	53.5403	2.22123
$582$	0	0
$583$	42.7914	1.77224
$584$	0	0
$585$	−20.6146	−0.852309
$586$	0	0
$587$	−7.14254	−0.294804	−0.147402	−	0.989077i	$-0.547091\pi$
$587$	−7.14254	−0.294804	−0.147402	+	0.989077i	$0.547091\pi$
$588$	0	0
$589$	−0.00653983	−0.000269469 0
$590$	0	0
$591$	4.20972	0.173165
$592$	0	0
$593$	1.41303	0.0580261	0.0290131	−	0.999579i	$-0.490764\pi$
$593$	1.41303	0.0580261	0.0290131	+	0.999579i	$0.490764\pi$
$594$	0	0
$595$	−32.6790	−1.33971
$596$	0	0
$597$	−34.1577	−1.39798
$598$	0	0
$599$	−43.0839	−1.76036	−0.880180	−	0.474640i	$-0.842578\pi$
$599$	−43.0839	−1.76036	−0.880180	+	0.474640i	$0.842578\pi$
$600$	0	0
$601$	12.5275	0.511009	0.255504	−	0.966808i	$-0.417758\pi$
$601$	12.5275	0.511009	0.255504	+	0.966808i	$0.417758\pi$
$602$	0	0
$603$	−36.1589	−1.47251
$604$	0	0
$605$	31.5507	1.28272
$606$	0	0
$607$	−8.40187	−0.341021	−0.170511	−	0.985356i	$-0.554542\pi$
$607$	−8.40187	−0.341021	−0.170511	+	0.985356i	$0.554542\pi$
$608$	0	0
$609$	29.4296	1.19255
$610$	0	0
$611$	2.57327	0.104103
$612$	0	0
$613$	−16.9667	−0.685279	−0.342639	−	0.939467i	$-0.611321\pi$
$613$	−16.9667	−0.685279	−0.342639	+	0.939467i	$0.611321\pi$
$614$	0	0
$615$	50.8971	2.05237
$616$	0	0
$617$	29.7146	1.19627	0.598133	−	0.801397i	$-0.295909\pi$
$617$	29.7146	1.19627	0.598133	+	0.801397i	$0.295909\pi$
$618$	0	0
$619$	−3.78344	−0.152069	−0.0760347	−	0.997105i	$-0.524226\pi$
$619$	−3.78344	−0.152069	−0.0760347	+	0.997105i	$0.524226\pi$
$620$	0	0
$621$	2.89130	0.116024
$622$	0	0
$623$	28.7496	1.15183
$624$	0	0
$625$	−21.9051	−0.876202
$626$	0	0
$627$	0.0110583	0.000441628 0
$628$	0	0
$629$	29.6869	1.18369
$630$	0	0
$631$	2.93933	0.117013	0.0585064	−	0.998287i	$-0.481366\pi$
$631$	2.93933	0.117013	0.0585064	+	0.998287i	$0.481366\pi$
$632$	0	0
$633$	59.9367	2.38227
$634$	0	0
$635$	31.9553	1.26811
$636$	0	0
$637$	−11.9428	−0.473191
$638$	0	0
$639$	18.0417	0.713719
$640$	0	0
$641$	33.4963	1.32302	0.661512	−	0.749935i	$-0.269915\pi$
$641$	33.4963	1.32302	0.661512	+	0.749935i	$0.269915\pi$
$642$	0	0
$643$	42.7269	1.68498	0.842492	−	0.538709i	$-0.181088\pi$
$643$	42.7269	1.68498	0.842492	+	0.538709i	$0.181088\pi$
$644$	0	0
$645$	−99.6055	−3.92196
$646$	0	0
$647$	2.13701	0.0840147	0.0420074	−	0.999117i	$-0.486625\pi$
$647$	2.13701	0.0840147	0.0420074	+	0.999117i	$0.486625\pi$
$648$	0	0
$649$	7.76712	0.304886
$650$	0	0
$651$	61.9890	2.42954
$652$	0	0
$653$	−1.50047	−0.0587178	−0.0293589	−	0.999569i	$-0.509347\pi$
$653$	−1.50047	−0.0587178	−0.0293589	+	0.999569i	$0.509347\pi$
$654$	0	0
$655$	14.7794	0.577478
$656$	0	0
$657$	−24.1460	−0.942023
$658$	0	0
$659$	27.4880	1.07078	0.535390	−	0.844605i	$-0.320165\pi$
$659$	27.4880	1.07078	0.535390	+	0.844605i	$0.320165\pi$
$660$	0	0
$661$	34.4182	1.33871	0.669357	−	0.742941i	$-0.266570\pi$
$661$	34.4182	1.33871	0.669357	+	0.742941i	$0.266570\pi$
$662$	0	0
$663$	13.2542	0.514750
$664$	0	0
$665$	−0.0114310	−0.000443277 0
$666$	0	0
$667$	11.3564	0.439723
$668$	0	0
$669$	17.2692	0.667667
$670$	0	0
$671$	12.5516	0.484550
$672$	0	0
$673$	−35.4485	−1.36644	−0.683219	−	0.730214i	$-0.739421\pi$
$673$	−35.4485	−1.36644	−0.683219	+	0.730214i	$0.739421\pi$
$674$	0	0
$675$	4.54641	0.174991
$676$	0	0
$677$	0.190869	0.00733571	0.00366785	−	0.999993i	$-0.498832\pi$
$677$	0.190869	0.00733571	0.00366785	+	0.999993i	$0.498832\pi$
$678$	0	0
$679$	53.7634	2.06325
$680$	0	0
$681$	23.9495	0.917745
$682$	0	0
$683$	21.2202	0.811970	0.405985	−	0.913880i	$-0.366929\pi$
$683$	21.2202	0.811970	0.405985	+	0.913880i	$0.366929\pi$
$684$	0	0
$685$	−19.8721	−0.759275
$686$	0	0
$687$	−41.3378	−1.57714
$688$	0	0
$689$	17.9405	0.683481
$690$	0	0
$691$	28.5479	1.08601	0.543007	−	0.839728i	$-0.317286\pi$
$691$	28.5479	1.08601	0.543007	+	0.839728i	$0.317286\pi$
$692$	0	0
$693$	−55.1046	−2.09325
$694$	0	0
$695$	36.5064	1.38477
$696$	0	0
$697$	−17.2037	−0.651636
$698$	0	0
$699$	−53.5590	−2.02579
$700$	0	0
$701$	22.0285	0.832005	0.416002	−	0.909364i	$-0.363431\pi$
$701$	22.0285	0.832005	0.416002	+	0.909364i	$0.363431\pi$
$702$	0	0
$703$	0.0103844	0.000391655 0
$704$	0	0
$705$	−11.0210	−0.415074
$706$	0	0
$707$	−2.62459	−0.0987079
$708$	0	0
$709$	9.26142	0.347820	0.173910	−	0.984762i	$-0.444360\pi$
$709$	9.26142	0.347820	0.173910	+	0.984762i	$0.444360\pi$
$710$	0	0
$711$	40.5236	1.51975
$712$	0	0
$713$	23.9207	0.895835
$714$	0	0
$715$	28.2123	1.05508
$716$	0	0
$717$	−9.70499	−0.362439
$718$	0	0
$719$	−3.19471	−0.119142	−0.0595712	−	0.998224i	$-0.518973\pi$
$719$	−3.19471	−0.119142	−0.0595712	+	0.998224i	$0.518973\pi$
$720$	0	0
$721$	−47.0756	−1.75319
$722$	0	0
$723$	62.4041	2.32083
$724$	0	0
$725$	17.8574	0.663207
$726$	0	0
$727$	−32.4166	−1.20226	−0.601132	−	0.799150i	$-0.705284\pi$
$727$	−32.4166	−1.20226	−0.601132	+	0.799150i	$0.705284\pi$
$728$	0	0
$729$	−32.5036	−1.20384
$730$	0	0
$731$	33.6675	1.24524
$732$	0	0
$733$	18.5529	0.685267	0.342634	−	0.939469i	$-0.388681\pi$
$733$	18.5529	0.685267	0.342634	+	0.939469i	$0.388681\pi$
$734$	0	0
$735$	51.1496	1.88668
$736$	0	0
$737$	49.4856	1.82283
$738$	0	0
$739$	−31.2754	−1.15049	−0.575243	−	0.817983i	$-0.695092\pi$
$739$	−31.2754	−1.15049	−0.575243	+	0.817983i	$0.695092\pi$
$740$	0	0
$741$	0.00463628	0.000170318 0
$742$	0	0
$743$	39.7250	1.45737	0.728685	−	0.684849i	$-0.240132\pi$
$743$	39.7250	1.45737	0.728685	+	0.684849i	$0.240132\pi$
$744$	0	0
$745$	−9.41993	−0.345119
$746$	0	0
$747$	48.8934	1.78892
$748$	0	0
$749$	37.0483	1.35372
$750$	0	0
$751$	4.49920	0.164178	0.0820891	−	0.996625i	$-0.473841\pi$
$751$	4.49920	0.164178	0.0820891	+	0.996625i	$0.473841\pi$
$752$	0	0
$753$	65.8538	2.39985
$754$	0	0
$755$	−39.2241	−1.42751
$756$	0	0
$757$	−43.1446	−1.56812	−0.784059	−	0.620687i	$-0.786854\pi$
$757$	−43.1446	−1.56812	−0.784059	+	0.620687i	$0.786854\pi$
$758$	0	0
$759$	−40.4479	−1.46817
$760$	0	0
$761$	35.3114	1.28004	0.640018	−	0.768360i	$-0.278927\pi$
$761$	35.3114	1.28004	0.640018	+	0.768360i	$0.278927\pi$
$762$	0	0
$763$	12.1556	0.440063
$764$	0	0
$765$	−29.8428	−1.07897
$766$	0	0
$767$	3.25641	0.117582
$768$	0	0
$769$	−39.3217	−1.41798	−0.708988	−	0.705220i	$-0.750848\pi$
$769$	−39.3217	−1.41798	−0.708988	+	0.705220i	$0.750848\pi$
$770$	0	0
$771$	−56.8380	−2.04697
$772$	0	0
$773$	28.3996	1.02146	0.510731	−	0.859740i	$-0.329375\pi$
$773$	28.3996	1.02146	0.510731	+	0.859740i	$0.329375\pi$
$774$	0	0
$775$	37.6139	1.35113
$776$	0	0
$777$	−98.4303	−3.53117
$778$	0	0
$779$	−0.00601780	−0.000215610 0
$780$	0	0
$781$	−24.6911	−0.883519
$782$	0	0
$783$	2.62914	0.0939578
$784$	0	0
$785$	−6.99181	−0.249548
$786$	0	0
$787$	−42.7067	−1.52233	−0.761165	−	0.648558i	$-0.775372\pi$
$787$	−42.7067	−1.52233	−0.761165	+	0.648558i	$0.775372\pi$
$788$	0	0
$789$	53.0355	1.88811
$790$	0	0
$791$	−47.8983	−1.70307
$792$	0	0
$793$	5.26234	0.186871
$794$	0	0
$795$	−76.8372	−2.72513
$796$	0	0
$797$	14.9261	0.528708	0.264354	−	0.964426i	$-0.414841\pi$
$797$	14.9261	0.528708	0.264354	+	0.964426i	$0.414841\pi$
$798$	0	0
$799$	3.72519	0.131788
$800$	0	0
$801$	26.2544	0.927652
$802$	0	0
$803$	33.0451	1.16614
$804$	0	0
$805$	41.8111	1.47365
$806$	0	0
$807$	44.6652	1.57229
$808$	0	0
$809$	−26.3823	−0.927553	−0.463776	−	0.885952i	$-0.653506\pi$
$809$	−26.3823	−0.927553	−0.463776	+	0.885952i	$0.653506\pi$
$810$	0	0
$811$	52.6197	1.84773	0.923863	−	0.382722i	$-0.125013\pi$
$811$	52.6197	1.84773	0.923863	+	0.382722i	$0.125013\pi$
$812$	0	0
$813$	−21.3331	−0.748185
$814$	0	0
$815$	46.0438	1.61284
$816$	0	0
$817$	0.0117768	0.000412018 0
$818$	0	0
$819$	−23.1030	−0.807283
$820$	0	0
$821$	−30.4637	−1.06319	−0.531596	−	0.846998i	$-0.678408\pi$
$821$	−30.4637	−1.06319	−0.531596	+	0.846998i	$0.678408\pi$
$822$	0	0
$823$	42.7600	1.49052	0.745260	−	0.666774i	$-0.232325\pi$
$823$	42.7600	1.49052	0.745260	+	0.666774i	$0.232325\pi$
$824$	0	0
$825$	−63.6022	−2.21434
$826$	0	0
$827$	−29.7870	−1.03580	−0.517898	−	0.855443i	$-0.673285\pi$
$827$	−29.7870	−1.03580	−0.517898	+	0.855443i	$0.673285\pi$
$828$	0	0
$829$	0.107481	0.00373298	0.00186649	−	0.999998i	$-0.499406\pi$
$829$	0.107481	0.00373298	0.00186649	+	0.999998i	$0.499406\pi$
$830$	0	0
$831$	−16.6279	−0.576815
$832$	0	0
$833$	−17.2890	−0.599029
$834$	0	0
$835$	3.24915	0.112441
$836$	0	0
$837$	5.53789	0.191418
$838$	0	0
$839$	17.5630	0.606341	0.303170	−	0.952936i	$-0.401955\pi$
$839$	17.5630	0.606341	0.303170	+	0.952936i	$0.401955\pi$
$840$	0	0
$841$	−18.6733	−0.643906
$842$	0	0
$843$	54.7176	1.88458
$844$	0	0
$845$	−30.4107	−1.04616
$846$	0	0
$847$	35.3591	1.21495
$848$	0	0
$849$	65.7376	2.25611
$850$	0	0
$851$	−37.9828	−1.30203
$852$	0	0
$853$	−25.9714	−0.889243	−0.444622	−	0.895718i	$-0.646662\pi$
$853$	−25.9714	−0.889243	−0.444622	+	0.895718i	$0.646662\pi$
$854$	0	0
$855$	−0.0104389	−0.000357003 0
$856$	0	0
$857$	−24.8894	−0.850205	−0.425103	−	0.905145i	$-0.639762\pi$
$857$	−24.8894	−0.850205	−0.425103	+	0.905145i	$0.639762\pi$
$858$	0	0
$859$	2.96781	0.101260	0.0506302	−	0.998717i	$-0.483877\pi$
$859$	2.96781	0.101260	0.0506302	+	0.998717i	$0.483877\pi$
$860$	0	0
$861$	57.0408	1.94394
$862$	0	0
$863$	−3.22785	−0.109877	−0.0549386	−	0.998490i	$-0.517496\pi$
$863$	−3.22785	−0.109877	−0.0549386	+	0.998490i	$0.517496\pi$
$864$	0	0
$865$	−23.2923	−0.791963
$866$	0	0
$867$	−23.5678	−0.800405
$868$	0	0
$869$	−55.4589	−1.88131
$870$	0	0
$871$	20.7471	0.702990
$872$	0	0
$873$	49.0972	1.66169
$874$	0	0
$875$	6.58943	0.222763
$876$	0	0
$877$	27.1001	0.915106	0.457553	−	0.889182i	$-0.348726\pi$
$877$	27.1001	0.915106	0.457553	+	0.889182i	$0.348726\pi$
$878$	0	0
$879$	43.9005	1.48073
$880$	0	0
$881$	−30.9784	−1.04369	−0.521845	−	0.853041i	$-0.674756\pi$
$881$	−30.9784	−1.04369	−0.521845	+	0.853041i	$0.674756\pi$
$882$	0	0
$883$	11.8647	0.399280	0.199640	−	0.979869i	$-0.436023\pi$
$883$	11.8647	0.399280	0.199640	+	0.979869i	$0.436023\pi$
$884$	0	0
$885$	−13.9468	−0.468817
$886$	0	0
$887$	30.0242	1.00811	0.504057	−	0.863670i	$-0.331840\pi$
$887$	30.0242	1.00811	0.504057	+	0.863670i	$0.331840\pi$
$888$	0	0
$889$	35.8126	1.20112
$890$	0	0
$891$	36.0350	1.20722
$892$	0	0
$893$	0.00130306	4.36053e−5 0
$894$	0	0
$895$	−8.81620	−0.294693
$896$	0	0
$897$	−16.9581	−0.566213
$898$	0	0
$899$	21.7517	0.725460
$900$	0	0
$901$	25.9717	0.865242
$902$	0	0
$903$	−111.629	−3.71477
$904$	0	0
$905$	−73.8359	−2.45439
$906$	0	0
$907$	−11.9441	−0.396598	−0.198299	−	0.980142i	$-0.563542\pi$
$907$	−11.9441	−0.396598	−0.198299	+	0.980142i	$0.563542\pi$
$908$	0	0
$909$	−2.39680	−0.0794968
$910$	0	0
$911$	43.0780	1.42724	0.713619	−	0.700534i	$-0.247055\pi$
$911$	43.0780	1.42724	0.713619	+	0.700534i	$0.247055\pi$
$912$	0	0
$913$	−66.9135	−2.21451
$914$	0	0
$915$	−22.5380	−0.745082
$916$	0	0
$917$	16.5634	0.546971
$918$	0	0
$919$	−51.4174	−1.69610	−0.848052	−	0.529913i	$-0.822225\pi$
$919$	−51.4174	−1.69610	−0.848052	+	0.529913i	$0.822225\pi$
$920$	0	0
$921$	−25.3166	−0.834209
$922$	0	0
$923$	−10.3519	−0.340737
$924$	0	0
$925$	−59.7259	−1.96378
$926$	0	0
$927$	−42.9899	−1.41197
$928$	0	0
$929$	−18.3255	−0.601240	−0.300620	−	0.953744i	$-0.597194\pi$
$929$	−18.3255	−0.601240	−0.300620	+	0.953744i	$0.597194\pi$
$930$	0	0
$931$	−0.00604765	−0.000198204 0
$932$	0	0
$933$	41.5811	1.36130
$934$	0	0
$935$	40.8416	1.33566
$936$	0	0
$937$	14.3623	0.469196	0.234598	−	0.972092i	$-0.424623\pi$
$937$	14.3623	0.469196	0.234598	+	0.972092i	$0.424623\pi$
$938$	0	0
$939$	56.4332	1.84163
$940$	0	0
$941$	25.3328	0.825827	0.412913	−	0.910770i	$-0.364511\pi$
$941$	25.3328	0.825827	0.412913	+	0.910770i	$0.364511\pi$
$942$	0	0
$943$	22.0112	0.716784
$944$	0	0
$945$	9.67973	0.314882
$946$	0	0
$947$	−36.9577	−1.20096	−0.600482	−	0.799639i	$-0.705024\pi$
$947$	−36.9577	−1.20096	−0.600482	+	0.799639i	$0.705024\pi$
$948$	0	0
$949$	13.8544	0.449732
$950$	0	0
$951$	−49.4372	−1.60311
$952$	0	0
$953$	42.7684	1.38540	0.692701	−	0.721224i	$-0.256420\pi$
$953$	42.7684	1.38540	0.692701	+	0.721224i	$0.256420\pi$
$954$	0	0
$955$	82.9480	2.68413
$956$	0	0
$957$	−36.7804	−1.18894
$958$	0	0
$959$	−22.2709	−0.719164
$960$	0	0
$961$	14.8167	0.477960
$962$	0	0
$963$	33.8328	1.09025
$964$	0	0
$965$	−9.38443	−0.302096
$966$	0	0
$967$	5.62959	0.181035	0.0905177	−	0.995895i	$-0.471148\pi$
$967$	5.62959	0.181035	0.0905177	+	0.995895i	$0.471148\pi$
$968$	0	0
$969$	0.00671172	0.000215611 0
$970$	0	0
$971$	27.5265	0.883369	0.441684	−	0.897171i	$-0.354381\pi$
$971$	27.5265	0.883369	0.441684	+	0.897171i	$0.354381\pi$
$972$	0	0
$973$	40.9130	1.31161
$974$	0	0
$975$	−26.6656	−0.853983
$976$	0	0
$977$	−13.1282	−0.420007	−0.210004	−	0.977701i	$-0.567348\pi$
$977$	−13.1282	−0.420007	−0.210004	+	0.977701i	$0.567348\pi$
$978$	0	0
$979$	−35.9306	−1.14835
$980$	0	0
$981$	11.1006	0.354415
$982$	0	0
$983$	−37.4193	−1.19349	−0.596745	−	0.802431i	$-0.703540\pi$
$983$	−37.4193	−1.19349	−0.596745	+	0.802431i	$0.703540\pi$
$984$	0	0
$985$	5.43853	0.173286
$986$	0	0
$987$	−12.3513	−0.393146
$988$	0	0
$989$	−43.0759	−1.36973
$990$	0	0
$991$	−35.8113	−1.13758	−0.568792	−	0.822482i	$-0.692589\pi$
$991$	−35.8113	−1.13758	−0.568792	+	0.822482i	$0.692589\pi$
$992$	0	0
$993$	−80.4578	−2.55325
$994$	0	0
$995$	−44.1283	−1.39896
$996$	0	0
$997$	53.2909	1.68774	0.843870	−	0.536548i	$-0.180272\pi$
$997$	53.2909	1.68774	0.843870	+	0.536548i	$0.180272\pi$
$998$	0	0
$999$	−8.79344	−0.278212

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1336.2.a.e.1.10	✓	12
4.3	odd	2		2672.2.a.n.1.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1336.2.a.e.1.10	✓	12	1.1	even	1	trivial
2672.2.a.n.1.3		12	4.3	odd	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:	\( N \)	\(=\)	\( 1336 = 2^{3} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1336.a (trivial)

\(f(q)\)	\(=\)	\(q+2.51502 q^{3} +3.24915 q^{5} +3.64134 q^{7} +3.32531 q^{9} +O(q^{10})\)	\(q+2.51502 q^{3} +3.24915 q^{5} +3.64134 q^{7} +3.32531 q^{9} -4.55087 q^{11} -1.90798 q^{13} +8.17165 q^{15} -2.76209 q^{17} -0.000966172 q^{19} +9.15804 q^{21} +3.53395 q^{23} +5.55695 q^{25} +0.818149 q^{27} +3.21352 q^{29} +6.76881 q^{31} -11.4455 q^{33} +11.8313 q^{35} -10.7480 q^{37} -4.79861 q^{39} +6.22849 q^{41} -12.1891 q^{43} +10.8044 q^{45} -1.34868 q^{47} +6.25939 q^{49} -6.94671 q^{51} -9.40289 q^{53} -14.7865 q^{55} -0.00242994 q^{57} -1.70673 q^{59} -2.75807 q^{61} +12.1086 q^{63} -6.19931 q^{65} -10.8739 q^{67} +8.88795 q^{69} +5.42558 q^{71} -7.26127 q^{73} +13.9758 q^{75} -16.5713 q^{77} +12.1864 q^{79} -7.91826 q^{81} +14.7034 q^{83} -8.97444 q^{85} +8.08206 q^{87} +7.89532 q^{89} -6.94762 q^{91} +17.0237 q^{93} -0.00313923 q^{95} +14.7647 q^{97} -15.1330 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 5 q^{3} - 2 q^{5} + 10 q^{7} + 19 q^{9}+O(q^{10}) \) 12 * q + 5 * q^3 - 2 * q^5 + 10 * q^7 + 19 * q^9	\( 12 q + 5 q^{3} - 2 q^{5} + 10 q^{7} + 19 q^{9} + 14 q^{11} - 9 q^{13} + 8 q^{15} + 4 q^{17} + 9 q^{19} + 7 q^{21} + 15 q^{23} + 18 q^{25} + 20 q^{27} + 17 q^{29} + 11 q^{31} + 8 q^{33} + 19 q^{35} - 29 q^{37} + 26 q^{39} + 14 q^{41} + 15 q^{43} - 26 q^{45} + 17 q^{47} + 20 q^{49} + 36 q^{51} - 7 q^{53} + 13 q^{55} + 3 q^{57} + 32 q^{59} - 8 q^{61} + 50 q^{63} + 37 q^{65} + 39 q^{67} + q^{69} + 35 q^{71} - 12 q^{73} + 29 q^{75} + 13 q^{77} + 36 q^{79} + 44 q^{81} + 25 q^{83} - 38 q^{85} + 14 q^{87} + 23 q^{89} - 2 q^{91} - 25 q^{93} + 38 q^{95} - 2 q^{97} + 43 q^{99}+O(q^{100}) \) 12 * q + 5 * q^3 - 2 * q^5 + 10 * q^7 + 19 * q^9 + 14 * q^11 - 9 * q^13 + 8 * q^15 + 4 * q^17 + 9 * q^19 + 7 * q^21 + 15 * q^23 + 18 * q^25 + 20 * q^27 + 17 * q^29 + 11 * q^31 + 8 * q^33 + 19 * q^35 - 29 * q^37 + 26 * q^39 + 14 * q^41 + 15 * q^43 - 26 * q^45 + 17 * q^47 + 20 * q^49 + 36 * q^51 - 7 * q^53 + 13 * q^55 + 3 * q^57 + 32 * q^59 - 8 * q^61 + 50 * q^63 + 37 * q^65 + 39 * q^67 + q^69 + 35 * q^71 - 12 * q^73 + 29 * q^75 + 13 * q^77 + 36 * q^79 + 44 * q^81 + 25 * q^83 - 38 * q^85 + 14 * q^87 + 23 * q^89 - 2 * q^91 - 25 * q^93 + 38 * q^95 - 2 * q^97 + 43 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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