LMFDB - Embedded newform 1336.2.a.e.1.6

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			1.6
Root			$-0.203398$ 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)

$f(q)$	$=$	$q-0.203398 q^{3} +1.32800 q^{5} +0.559903 q^{7} -2.95863 q^{9} +O(q^{10})$	$q-0.203398 q^{3} +1.32800 q^{5} +0.559903 q^{7} -2.95863 q^{9} +1.10814 q^{11} +6.62673 q^{13} -0.270112 q^{15} -6.57694 q^{17} +1.20962 q^{19} -0.113883 q^{21} +9.45529 q^{23} -3.23643 q^{25} +1.21197 q^{27} +6.95229 q^{29} +0.00258064 q^{31} -0.225394 q^{33} +0.743549 q^{35} -4.42235 q^{37} -1.34786 q^{39} +2.82131 q^{41} +8.21365 q^{43} -3.92905 q^{45} +2.86600 q^{47} -6.68651 q^{49} +1.33774 q^{51} +9.96231 q^{53} +1.47161 q^{55} -0.246035 q^{57} -12.1824 q^{59} +4.00000 q^{61} -1.65654 q^{63} +8.80028 q^{65} +13.6873 q^{67} -1.92319 q^{69} +14.0119 q^{71} -14.5327 q^{73} +0.658282 q^{75} +0.620452 q^{77} -3.34668 q^{79} +8.62938 q^{81} +5.00331 q^{83} -8.73416 q^{85} -1.41408 q^{87} -4.41222 q^{89} +3.71033 q^{91} -0.000524897 q^{93} +1.60637 q^{95} +8.32498 q^{97} -3.27858 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.203398	−0.117432	−0.0587159	−	0.998275i	$-0.518701\pi$
$3$	−0.203398	−0.117432	−0.0587159	+	0.998275i	$0.518701\pi$
$4$	0	0
$5$	1.32800	0.593898	0.296949	−	0.954893i	$-0.404031\pi$
$5$	1.32800	0.593898	0.296949	+	0.954893i	$0.404031\pi$
$6$	0	0
$7$	0.559903	0.211623	0.105812	−	0.994386i	$-0.466256\pi$
$7$	0.559903	0.211623	0.105812	+	0.994386i	$0.466256\pi$
$8$	0	0
$9$	−2.95863	−0.986210
$10$	0	0
$11$	1.10814	0.334118	0.167059	−	0.985947i	$-0.446573\pi$
$11$	1.10814	0.334118	0.167059	+	0.985947i	$0.446573\pi$
$12$	0	0
$13$	6.62673	1.83793	0.918963	−	0.394345i	$-0.129028\pi$
$13$	6.62673	1.83793	0.918963	+	0.394345i	$0.129028\pi$
$14$	0	0
$15$	−0.270112	−0.0697425
$16$	0	0
$17$	−6.57694	−1.59514	−0.797572	−	0.603224i	$-0.793882\pi$
$17$	−6.57694	−1.59514	−0.797572	+	0.603224i	$0.793882\pi$
$18$	0	0
$19$	1.20962	0.277507	0.138753	−	0.990327i	$-0.455690\pi$
$19$	1.20962	0.277507	0.138753	+	0.990327i	$0.455690\pi$
$20$	0	0
$21$	−0.113883	−0.0248513
$22$	0	0
$23$	9.45529	1.97156	0.985782	−	0.168029i	$-0.0537402\pi$
$23$	9.45529	1.97156	0.985782	+	0.168029i	$0.0537402\pi$
$24$	0	0
$25$	−3.23643	−0.647285
$26$	0	0
$27$	1.21197	0.233244
$28$	0	0
$29$	6.95229	1.29101	0.645504	−	0.763757i	$-0.276648\pi$
$29$	6.95229	1.29101	0.645504	+	0.763757i	$0.276648\pi$
$30$	0	0
$31$	0.00258064	0.000463497 0	0.000231748	−	1.00000i	$-0.499926\pi$
$31$	0.00258064	0.000463497 0	0.000231748		1.00000i	$0.499926\pi$
$32$	0	0
$33$	−0.225394	−0.0392361
$34$	0	0
$35$	0.743549	0.125683
$36$	0	0
$37$	−4.42235	−0.727030	−0.363515	−	0.931588i	$-0.618424\pi$
$37$	−4.42235	−0.727030	−0.363515	+	0.931588i	$0.618424\pi$
$38$	0	0
$39$	−1.34786	−0.215831
$40$	0	0
$41$	2.82131	0.440614	0.220307	−	0.975431i	$-0.429294\pi$
$41$	2.82131	0.440614	0.220307	+	0.975431i	$0.429294\pi$
$42$	0	0
$43$	8.21365	1.25257	0.626285	−	0.779594i	$-0.284575\pi$
$43$	8.21365	1.25257	0.626285	+	0.779594i	$0.284575\pi$
$44$	0	0
$45$	−3.92905	−0.585708
$46$	0	0
$47$	2.86600	0.418049	0.209024	−	0.977910i	$-0.432971\pi$
$47$	2.86600	0.418049	0.209024	+	0.977910i	$0.432971\pi$
$48$	0	0
$49$	−6.68651	−0.955216
$50$	0	0
$51$	1.33774	0.187321
$52$	0	0
$53$	9.96231	1.36843	0.684214	−	0.729281i	$-0.260145\pi$
$53$	9.96231	1.36843	0.684214	+	0.729281i	$0.260145\pi$
$54$	0	0
$55$	1.47161	0.198432
$56$	0	0
$57$	−0.246035	−0.0325881
$58$	0	0
$59$	−12.1824	−1.58602	−0.793008	−	0.609211i	$-0.791486\pi$
$59$	−12.1824	−1.58602	−0.793008	+	0.609211i	$0.791486\pi$
$60$	0	0
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	0	0
$63$	−1.65654	−0.208705
$64$	0	0
$65$	8.80028	1.09154
$66$	0	0
$67$	13.6873	1.67216	0.836082	−	0.548604i	$-0.184841\pi$
$67$	13.6873	1.67216	0.836082	+	0.548604i	$0.184841\pi$
$68$	0	0
$69$	−1.92319	−0.231524
$70$	0	0
$71$	14.0119	1.66290	0.831452	−	0.555596i	$-0.187510\pi$
$71$	14.0119	1.66290	0.831452	+	0.555596i	$0.187510\pi$
$72$	0	0
$73$	−14.5327	−1.70093	−0.850463	−	0.526035i	$-0.823678\pi$
$73$	−14.5327	−1.70093	−0.850463	+	0.526035i	$0.823678\pi$
$74$	0	0
$75$	0.658282	0.0760119
$76$	0	0
$77$	0.620452	0.0707071
$78$	0	0
$79$	−3.34668	−0.376531	−0.188265	−	0.982118i	$-0.560286\pi$
$79$	−3.34668	−0.376531	−0.188265	+	0.982118i	$0.560286\pi$
$80$	0	0
$81$	8.62938	0.958819
$82$	0	0
$83$	5.00331	0.549184	0.274592	−	0.961561i	$-0.411457\pi$
$83$	5.00331	0.549184	0.274592	+	0.961561i	$0.411457\pi$
$84$	0	0
$85$	−8.73416	−0.947352
$86$	0	0
$87$	−1.41408	−0.151605
$88$	0	0
$89$	−4.41222	−0.467695	−0.233847	−	0.972273i	$-0.575132\pi$
$89$	−4.41222	−0.467695	−0.233847	+	0.972273i	$0.575132\pi$
$90$	0	0
$91$	3.71033	0.388948
$92$	0	0
$93$	−0.000524897 0	−5.44293e−5 0
$94$	0	0
$95$	1.60637	0.164811
$96$	0	0
$97$	8.32498	0.845274	0.422637	−	0.906299i	$-0.361105\pi$
$97$	8.32498	0.845274	0.422637	+	0.906299i	$0.361105\pi$
$98$	0	0
$99$	−3.27858	−0.329510
$100$	0	0
$101$	6.49027	0.645806	0.322903	−	0.946432i	$-0.395341\pi$
$101$	6.49027	0.645806	0.322903	+	0.946432i	$0.395341\pi$
$102$	0	0
$103$	−1.18810	−0.117067	−0.0585334	−	0.998285i	$-0.518642\pi$
$103$	−1.18810	−0.117067	−0.0585334	+	0.998285i	$0.518642\pi$
$104$	0	0
$105$	−0.151236	−0.0147591
$106$	0	0
$107$	−11.4481	−1.10673	−0.553365	−	0.832939i	$-0.686657\pi$
$107$	−11.4481	−1.10673	−0.553365	+	0.832939i	$0.686657\pi$
$108$	0	0
$109$	−9.79662	−0.938346	−0.469173	−	0.883106i	$-0.655448\pi$
$109$	−9.79662	−0.938346	−0.469173	+	0.883106i	$0.655448\pi$
$110$	0	0
$111$	0.899497	0.0853765
$112$	0	0
$113$	6.62331	0.623068	0.311534	−	0.950235i	$-0.399157\pi$
$113$	6.62331	0.623068	0.311534	+	0.950235i	$0.399157\pi$
$114$	0	0
$115$	12.5566	1.17091
$116$	0	0
$117$	−19.6060	−1.81258
$118$	0	0
$119$	−3.68245	−0.337570
$120$	0	0
$121$	−9.77202	−0.888365
$122$	0	0
$123$	−0.573848	−0.0517421
$124$	0	0
$125$	−10.9379	−0.978319
$126$	0	0
$127$	12.4938	1.10865	0.554324	−	0.832301i	$-0.312977\pi$
$127$	12.4938	1.10865	0.554324	+	0.832301i	$0.312977\pi$
$128$	0	0
$129$	−1.67064	−0.147092
$130$	0	0
$131$	17.9395	1.56738	0.783690	−	0.621153i	$-0.213335\pi$
$131$	17.9395	1.56738	0.783690	+	0.621153i	$0.213335\pi$
$132$	0	0
$133$	0.677271	0.0587269
$134$	0	0
$135$	1.60950	0.138523
$136$	0	0
$137$	5.08448	0.434397	0.217198	−	0.976127i	$-0.430308\pi$
$137$	5.08448	0.434397	0.217198	+	0.976127i	$0.430308\pi$
$138$	0	0
$139$	14.7422	1.25042	0.625209	−	0.780457i	$-0.285014\pi$
$139$	14.7422	1.25042	0.625209	+	0.780457i	$0.285014\pi$
$140$	0	0
$141$	−0.582938	−0.0490923
$142$	0	0
$143$	7.34337	0.614083
$144$	0	0
$145$	9.23261	0.766727
$146$	0	0
$147$	1.36002	0.112173
$148$	0	0
$149$	5.95852	0.488141	0.244071	−	0.969757i	$-0.421517\pi$
$149$	5.95852	0.488141	0.244071	+	0.969757i	$0.421517\pi$
$150$	0	0
$151$	−5.58757	−0.454710	−0.227355	−	0.973812i	$-0.573008\pi$
$151$	−5.58757	−0.454710	−0.227355	+	0.973812i	$0.573008\pi$
$152$	0	0
$153$	19.4587	1.57315
$154$	0	0
$155$	0.00342708	0.000275270 0
$156$	0	0
$157$	−15.7830	−1.25962	−0.629810	−	0.776749i	$-0.716867\pi$
$157$	−15.7830	−1.25962	−0.629810	+	0.776749i	$0.716867\pi$
$158$	0	0
$159$	−2.02631	−0.160697
$160$	0	0
$161$	5.29404	0.417229
$162$	0	0
$163$	−10.7295	−0.840397	−0.420199	−	0.907432i	$-0.638040\pi$
$163$	−10.7295	−0.840397	−0.420199	+	0.907432i	$0.638040\pi$
$164$	0	0
$165$	−0.299322	−0.0233022
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	30.9136	2.37797
$170$	0	0
$171$	−3.57883	−0.273680
$172$	0	0
$173$	0.805366	0.0612309	0.0306154	−	0.999531i	$-0.490253\pi$
$173$	0.805366	0.0612309	0.0306154	+	0.999531i	$0.490253\pi$
$174$	0	0
$175$	−1.81208	−0.136981
$176$	0	0
$177$	2.47788	0.186249
$178$	0	0
$179$	−22.8581	−1.70849	−0.854247	−	0.519867i	$-0.825981\pi$
$179$	−22.8581	−1.70849	−0.854247	+	0.519867i	$0.825981\pi$
$180$	0	0
$181$	7.57419	0.562985	0.281493	−	0.959563i	$-0.409170\pi$
$181$	7.57419	0.562985	0.281493	+	0.959563i	$0.409170\pi$
$182$	0	0
$183$	−0.813592	−0.0601424
$184$	0	0
$185$	−5.87287	−0.431782
$186$	0	0
$187$	−7.28820	−0.532966
$188$	0	0
$189$	0.678587	0.0493599
$190$	0	0
$191$	−12.6614	−0.916146	−0.458073	−	0.888915i	$-0.651460\pi$
$191$	−12.6614	−0.916146	−0.458073	+	0.888915i	$0.651460\pi$
$192$	0	0
$193$	−5.12510	−0.368913	−0.184457	−	0.982841i	$-0.559052\pi$
$193$	−5.12510	−0.368913	−0.184457	+	0.982841i	$0.559052\pi$
$194$	0	0
$195$	−1.78996	−0.128182
$196$	0	0
$197$	−13.9442	−0.993486	−0.496743	−	0.867898i	$-0.665471\pi$
$197$	−13.9442	−0.993486	−0.496743	+	0.867898i	$0.665471\pi$
$198$	0	0
$199$	16.6320	1.17901	0.589506	−	0.807764i	$-0.299322\pi$
$199$	16.6320	1.17901	0.589506	+	0.807764i	$0.299322\pi$
$200$	0	0
$201$	−2.78396	−0.196365
$202$	0	0
$203$	3.89260	0.273207
$204$	0	0
$205$	3.74668	0.261680
$206$	0	0
$207$	−27.9747	−1.94438
$208$	0	0
$209$	1.34044	0.0927199
$210$	0	0
$211$	−23.5864	−1.62375	−0.811876	−	0.583830i	$-0.801553\pi$
$211$	−23.5864	−1.62375	−0.811876	+	0.583830i	$0.801553\pi$
$212$	0	0
$213$	−2.84999	−0.195278
$214$	0	0
$215$	10.9077	0.743899
$216$	0	0
$217$	0.00144491	9.80867e−5 0
$218$	0	0
$219$	2.95593	0.199743
$220$	0	0
$221$	−43.5837	−2.93175
$222$	0	0
$223$	−3.71967	−0.249088	−0.124544	−	0.992214i	$-0.539747\pi$
$223$	−3.71967	−0.249088	−0.124544	+	0.992214i	$0.539747\pi$
$224$	0	0
$225$	9.57539	0.638359
$226$	0	0
$227$	4.61311	0.306183	0.153092	−	0.988212i	$-0.451077\pi$
$227$	4.61311	0.306183	0.153092	+	0.988212i	$0.451077\pi$
$228$	0	0
$229$	−19.6522	−1.29865	−0.649327	−	0.760509i	$-0.724949\pi$
$229$	−19.6522	−1.29865	−0.649327	+	0.760509i	$0.724949\pi$
$230$	0	0
$231$	−0.126199	−0.00830327
$232$	0	0
$233$	−14.5438	−0.952799	−0.476399	−	0.879229i	$-0.658058\pi$
$233$	−14.5438	−0.952799	−0.476399	+	0.879229i	$0.658058\pi$
$234$	0	0
$235$	3.80604	0.248278
$236$	0	0
$237$	0.680707	0.0442167
$238$	0	0
$239$	−20.3322	−1.31518	−0.657589	−	0.753376i	$-0.728424\pi$
$239$	−20.3322	−1.31518	−0.657589	+	0.753376i	$0.728424\pi$
$240$	0	0
$241$	−30.3419	−1.95449	−0.977247	−	0.212103i	$-0.931969\pi$
$241$	−30.3419	−1.95449	−0.977247	+	0.212103i	$0.931969\pi$
$242$	0	0
$243$	−5.39112	−0.345840
$244$	0	0
$245$	−8.87966	−0.567301
$246$	0	0
$247$	8.01585	0.510036
$248$	0	0
$249$	−1.01766	−0.0644917
$250$	0	0
$251$	−12.1509	−0.766955	−0.383478	−	0.923550i	$-0.625274\pi$
$251$	−12.1509	−0.766955	−0.383478	+	0.923550i	$0.625274\pi$
$252$	0	0
$253$	10.4778	0.658735
$254$	0	0
$255$	1.77651	0.111249
$256$	0	0
$257$	−18.2314	−1.13724	−0.568622	−	0.822599i	$-0.692523\pi$
$257$	−18.2314	−1.13724	−0.568622	+	0.822599i	$0.692523\pi$
$258$	0	0
$259$	−2.47609	−0.153857
$260$	0	0
$261$	−20.5692	−1.27320
$262$	0	0
$263$	23.8289	1.46935	0.734677	−	0.678417i	$-0.237334\pi$
$263$	23.8289	1.46935	0.734677	+	0.678417i	$0.237334\pi$
$264$	0	0
$265$	13.2299	0.812707
$266$	0	0
$267$	0.897437	0.0549222
$268$	0	0
$269$	13.9025	0.847651	0.423826	−	0.905744i	$-0.360687\pi$
$269$	13.9025	0.847651	0.423826	+	0.905744i	$0.360687\pi$
$270$	0	0
$271$	17.3663	1.05493	0.527463	−	0.849578i	$-0.323143\pi$
$271$	17.3663	1.05493	0.527463	+	0.849578i	$0.323143\pi$
$272$	0	0
$273$	−0.754673	−0.0456749
$274$	0	0
$275$	−3.58642	−0.216269
$276$	0	0
$277$	−14.2760	−0.857763	−0.428882	−	0.903361i	$-0.641092\pi$
$277$	−14.2760	−0.857763	−0.428882	+	0.903361i	$0.641092\pi$
$278$	0	0
$279$	−0.00763516	−0.000457105 0
$280$	0	0
$281$	−23.9693	−1.42989	−0.714945	−	0.699180i	$-0.753548\pi$
$281$	−23.9693	−1.42989	−0.714945	+	0.699180i	$0.753548\pi$
$282$	0	0
$283$	−22.3185	−1.32670	−0.663349	−	0.748310i	$-0.730865\pi$
$283$	−22.3185	−1.32670	−0.663349	+	0.748310i	$0.730865\pi$
$284$	0	0
$285$	−0.326733	−0.0193540
$286$	0	0
$287$	1.57966	0.0932442
$288$	0	0
$289$	26.2562	1.54448
$290$	0	0
$291$	−1.69328	−0.0992621
$292$	0	0
$293$	11.3845	0.665088	0.332544	−	0.943088i	$-0.392093\pi$
$293$	11.3845	0.665088	0.332544	+	0.943088i	$0.392093\pi$
$294$	0	0
$295$	−16.1782	−0.941932
$296$	0	0
$297$	1.34304	0.0779311
$298$	0	0
$299$	62.6577	3.62359
$300$	0	0
$301$	4.59884	0.265073
$302$	0	0
$303$	−1.32011	−0.0758382
$304$	0	0
$305$	5.31199	0.304163
$306$	0	0
$307$	−31.1629	−1.77856	−0.889281	−	0.457360i	$-0.848795\pi$
$307$	−31.1629	−1.77856	−0.889281	+	0.457360i	$0.848795\pi$
$308$	0	0
$309$	0.241657	0.0137474
$310$	0	0
$311$	22.7439	1.28969	0.644843	−	0.764315i	$-0.276923\pi$
$311$	22.7439	1.28969	0.644843	+	0.764315i	$0.276923\pi$
$312$	0	0
$313$	1.31118	0.0741125	0.0370562	−	0.999313i	$-0.488202\pi$
$313$	1.31118	0.0741125	0.0370562	+	0.999313i	$0.488202\pi$
$314$	0	0
$315$	−2.19988	−0.123949
$316$	0	0
$317$	−0.942950	−0.0529614	−0.0264807	−	0.999649i	$-0.508430\pi$
$317$	−0.942950	−0.0529614	−0.0264807	+	0.999649i	$0.508430\pi$
$318$	0	0
$319$	7.70413	0.431348
$320$	0	0
$321$	2.32852	0.129965
$322$	0	0
$323$	−7.95562	−0.442663
$324$	0	0
$325$	−21.4469	−1.18966
$326$	0	0
$327$	1.99261	0.110192
$328$	0	0
$329$	1.60468	0.0884689
$330$	0	0
$331$	8.24777	0.453339	0.226669	−	0.973972i	$-0.427216\pi$
$331$	8.24777	0.453339	0.226669	+	0.973972i	$0.427216\pi$
$332$	0	0
$333$	13.0841	0.717004
$334$	0	0
$335$	18.1766	0.993095
$336$	0	0
$337$	2.52300	0.137436	0.0687182	−	0.997636i	$-0.478109\pi$
$337$	2.52300	0.137436	0.0687182	+	0.997636i	$0.478109\pi$
$338$	0	0
$339$	−1.34717	−0.0731681
$340$	0	0
$341$	0.00285972	0.000154862 0
$342$	0	0
$343$	−7.66311	−0.413769
$344$	0	0
$345$	−2.55398	−0.137502
$346$	0	0
$347$	23.8408	1.27984	0.639920	−	0.768442i	$-0.278967\pi$
$347$	23.8408	1.27984	0.639920	+	0.768442i	$0.278967\pi$
$348$	0	0
$349$	−19.7860	−1.05912	−0.529559	−	0.848273i	$-0.677643\pi$
$349$	−19.7860	−1.05912	−0.529559	+	0.848273i	$0.677643\pi$
$350$	0	0
$351$	8.03142	0.428686
$352$	0	0
$353$	23.3507	1.24283	0.621417	−	0.783480i	$-0.286557\pi$
$353$	23.3507	1.24283	0.621417	+	0.783480i	$0.286557\pi$
$354$	0	0
$355$	18.6077	0.987596
$356$	0	0
$357$	0.749003	0.0396414
$358$	0	0
$359$	−24.3122	−1.28315	−0.641574	−	0.767061i	$-0.721718\pi$
$359$	−24.3122	−1.28315	−0.641574	+	0.767061i	$0.721718\pi$
$360$	0	0
$361$	−17.5368	−0.922990
$362$	0	0
$363$	1.98761	0.104322
$364$	0	0
$365$	−19.2994	−1.01018
$366$	0	0
$367$	−30.9690	−1.61657	−0.808283	−	0.588794i	$-0.799603\pi$
$367$	−30.9690	−1.61657	−0.808283	+	0.588794i	$0.799603\pi$
$368$	0	0
$369$	−8.34720	−0.434538
$370$	0	0
$371$	5.57792	0.289591
$372$	0	0
$373$	−5.62327	−0.291162	−0.145581	−	0.989346i	$-0.546505\pi$
$373$	−5.62327	−0.291162	−0.145581	+	0.989346i	$0.546505\pi$
$374$	0	0
$375$	2.22475	0.114886
$376$	0	0
$377$	46.0710	2.37277
$378$	0	0
$379$	−12.7554	−0.655200	−0.327600	−	0.944817i	$-0.606240\pi$
$379$	−12.7554	−0.655200	−0.327600	+	0.944817i	$0.606240\pi$
$380$	0	0
$381$	−2.54122	−0.130191
$382$	0	0
$383$	−36.0721	−1.84320	−0.921600	−	0.388141i	$-0.873117\pi$
$383$	−36.0721	−1.84320	−0.921600	+	0.388141i	$0.873117\pi$
$384$	0	0
$385$	0.823958	0.0419928
$386$	0	0
$387$	−24.3011	−1.23530
$388$	0	0
$389$	20.1954	1.02395	0.511974	−	0.859001i	$-0.328915\pi$
$389$	20.1954	1.02395	0.511974	+	0.859001i	$0.328915\pi$
$390$	0	0
$391$	−62.1869	−3.14493
$392$	0	0
$393$	−3.64885	−0.184060
$394$	0	0
$395$	−4.44438	−0.223621
$396$	0	0
$397$	−11.6658	−0.585489	−0.292745	−	0.956191i	$-0.594569\pi$
$397$	−11.6658	−0.585489	−0.292745	+	0.956191i	$0.594569\pi$
$398$	0	0
$399$	−0.137756	−0.00689640
$400$	0	0
$401$	−20.5468	−1.02606	−0.513028	−	0.858372i	$-0.671476\pi$
$401$	−20.5468	−1.02606	−0.513028	+	0.858372i	$0.671476\pi$
$402$	0	0
$403$	0.0171012	0.000851872 0
$404$	0	0
$405$	11.4598	0.569441
$406$	0	0
$407$	−4.90060	−0.242914
$408$	0	0
$409$	−7.30802	−0.361358	−0.180679	−	0.983542i	$-0.557830\pi$
$409$	−7.30802	−0.361358	−0.180679	+	0.983542i	$0.557830\pi$
$410$	0	0
$411$	−1.03417	−0.0510120
$412$	0	0
$413$	−6.82097	−0.335638
$414$	0	0
$415$	6.64437	0.326159
$416$	0	0
$417$	−2.99854	−0.146839
$418$	0	0
$419$	−26.9436	−1.31628	−0.658141	−	0.752895i	$-0.728657\pi$
$419$	−26.9436	−1.31628	−0.658141	+	0.752895i	$0.728657\pi$
$420$	0	0
$421$	6.47841	0.315738	0.157869	−	0.987460i	$-0.449538\pi$
$421$	6.47841	0.315738	0.157869	+	0.987460i	$0.449538\pi$
$422$	0	0
$423$	−8.47943	−0.412284
$424$	0	0
$425$	21.2858	1.03251
$426$	0	0
$427$	2.23961	0.108382
$428$	0	0
$429$	−1.49363	−0.0721129
$430$	0	0
$431$	−10.7273	−0.516717	−0.258358	−	0.966049i	$-0.583182\pi$
$431$	−10.7273	−0.516717	−0.258358	+	0.966049i	$0.583182\pi$
$432$	0	0
$433$	6.93748	0.333394	0.166697	−	0.986008i	$-0.446690\pi$
$433$	6.93748	0.333394	0.166697	+	0.986008i	$0.446690\pi$
$434$	0	0
$435$	−1.87789	−0.0900381
$436$	0	0
$437$	11.4373	0.547122
$438$	0	0
$439$	16.3600	0.780819	0.390410	−	0.920641i	$-0.372333\pi$
$439$	16.3600	0.780819	0.390410	+	0.920641i	$0.372333\pi$
$440$	0	0
$441$	19.7829	0.942043
$442$	0	0
$443$	34.8020	1.65349	0.826746	−	0.562575i	$-0.190189\pi$
$443$	34.8020	1.65349	0.826746	+	0.562575i	$0.190189\pi$
$444$	0	0
$445$	−5.85941	−0.277763
$446$	0	0
$447$	−1.21195	−0.0573233
$448$	0	0
$449$	22.5517	1.06428	0.532140	−	0.846656i	$-0.321388\pi$
$449$	22.5517	1.06428	0.532140	+	0.846656i	$0.321388\pi$
$450$	0	0
$451$	3.12641	0.147217
$452$	0	0
$453$	1.13650	0.0533974
$454$	0	0
$455$	4.92730	0.230995
$456$	0	0
$457$	25.9646	1.21457	0.607286	−	0.794483i	$-0.292258\pi$
$457$	25.9646	1.21457	0.607286	+	0.794483i	$0.292258\pi$
$458$	0	0
$459$	−7.97108	−0.372058
$460$	0	0
$461$	32.3272	1.50563	0.752813	−	0.658234i	$-0.228696\pi$
$461$	32.3272	1.50563	0.752813	+	0.658234i	$0.228696\pi$
$462$	0	0
$463$	9.69473	0.450552	0.225276	−	0.974295i	$-0.427672\pi$
$463$	9.69473	0.450552	0.225276	+	0.974295i	$0.427672\pi$
$464$	0	0
$465$	−0.000697061 0	−3.23254e−5 0
$466$	0	0
$467$	14.5436	0.672998	0.336499	−	0.941684i	$-0.390757\pi$
$467$	14.5436	0.672998	0.336499	+	0.941684i	$0.390757\pi$
$468$	0	0
$469$	7.66353	0.353869
$470$	0	0
$471$	3.21023	0.147920
$472$	0	0
$473$	9.10190	0.418506
$474$	0	0
$475$	−3.91486	−0.179626
$476$	0	0
$477$	−29.4748	−1.34956
$478$	0	0
$479$	24.6974	1.12845	0.564226	−	0.825621i	$-0.309175\pi$
$479$	24.6974	1.12845	0.564226	+	0.825621i	$0.309175\pi$
$480$	0	0
$481$	−29.3058	−1.33623
$482$	0	0
$483$	−1.07680	−0.0489960
$484$	0	0
$485$	11.0555	0.502006
$486$	0	0
$487$	−21.1865	−0.960052	−0.480026	−	0.877254i	$-0.659373\pi$
$487$	−21.1865	−0.960052	−0.480026	+	0.877254i	$0.659373\pi$
$488$	0	0
$489$	2.18235	0.0986894
$490$	0	0
$491$	9.16362	0.413548	0.206774	−	0.978389i	$-0.433703\pi$
$491$	9.16362	0.413548	0.206774	+	0.978389i	$0.433703\pi$
$492$	0	0
$493$	−45.7248	−2.05934
$494$	0	0
$495$	−4.35395	−0.195695
$496$	0	0
$497$	7.84529	0.351909
$498$	0	0
$499$	−37.2494	−1.66751	−0.833755	−	0.552134i	$-0.813814\pi$
$499$	−37.2494	−1.66751	−0.833755	+	0.552134i	$0.813814\pi$
$500$	0	0
$501$	−0.203398	−0.00908715
$502$	0	0
$503$	−24.7861	−1.10516	−0.552580	−	0.833460i	$-0.686357\pi$
$503$	−24.7861	−1.10516	−0.552580	+	0.833460i	$0.686357\pi$
$504$	0	0
$505$	8.61905	0.383543
$506$	0	0
$507$	−6.28776	−0.279249
$508$	0	0
$509$	27.3483	1.21219	0.606095	−	0.795392i	$-0.292735\pi$
$509$	27.3483	1.21219	0.606095	+	0.795392i	$0.292735\pi$
$510$	0	0
$511$	−8.13691	−0.359956
$512$	0	0
$513$	1.46603	0.0647268
$514$	0	0
$515$	−1.57779	−0.0695257
$516$	0	0
$517$	3.17594	0.139678
$518$	0	0
$519$	−0.163810	−0.00719045
$520$	0	0
$521$	24.6770	1.08112	0.540559	−	0.841306i	$-0.318213\pi$
$521$	24.6770	1.08112	0.540559	+	0.841306i	$0.318213\pi$
$522$	0	0
$523$	−5.52099	−0.241416	−0.120708	−	0.992688i	$-0.538516\pi$
$523$	−5.52099	−0.241416	−0.120708	+	0.992688i	$0.538516\pi$
$524$	0	0
$525$	0.368574	0.0160859
$526$	0	0
$527$	−0.0169727	−0.000739344 0
$528$	0	0
$529$	66.4025	2.88706
$530$	0	0
$531$	36.0433	1.56414
$532$	0	0
$533$	18.6960	0.809816
$534$	0	0
$535$	−15.2030	−0.657285
$536$	0	0
$537$	4.64929	0.200632
$538$	0	0
$539$	−7.40961	−0.319154
$540$	0	0
$541$	23.5678	1.01326	0.506630	−	0.862164i	$-0.330891\pi$
$541$	23.5678	1.01326	0.506630	+	0.862164i	$0.330891\pi$
$542$	0	0
$543$	−1.54058	−0.0661124
$544$	0	0
$545$	−13.0099	−0.557282
$546$	0	0
$547$	−26.6836	−1.14091	−0.570455	−	0.821329i	$-0.693233\pi$
$547$	−26.6836	−1.14091	−0.570455	+	0.821329i	$0.693233\pi$
$548$	0	0
$549$	−11.8345	−0.505085
$550$	0	0
$551$	8.40965	0.358263
$552$	0	0
$553$	−1.87381	−0.0796827
$554$	0	0
$555$	1.19453	0.0507049
$556$	0	0
$557$	29.9546	1.26922	0.634609	−	0.772834i	$-0.281161\pi$
$557$	29.9546	1.26922	0.634609	+	0.772834i	$0.281161\pi$
$558$	0	0
$559$	54.4297	2.30213
$560$	0	0
$561$	1.48240	0.0625872
$562$	0	0
$563$	−38.6698	−1.62974	−0.814869	−	0.579645i	$-0.803191\pi$
$563$	−38.6698	−1.62974	−0.814869	+	0.579645i	$0.803191\pi$
$564$	0	0
$565$	8.79573	0.370039
$566$	0	0
$567$	4.83161	0.202909
$568$	0	0
$569$	5.79012	0.242734	0.121367	−	0.992608i	$-0.461272\pi$
$569$	5.79012	0.242734	0.121367	+	0.992608i	$0.461272\pi$
$570$	0	0
$571$	26.6663	1.11595	0.557975	−	0.829858i	$-0.311579\pi$
$571$	26.6663	1.11595	0.557975	+	0.829858i	$0.311579\pi$
$572$	0	0
$573$	2.57530	0.107585
$574$	0	0
$575$	−30.6013	−1.27616
$576$	0	0
$577$	−38.5907	−1.60655	−0.803276	−	0.595608i	$-0.796911\pi$
$577$	−38.5907	−1.60655	−0.803276	+	0.595608i	$0.796911\pi$
$578$	0	0
$579$	1.04244	0.0433221
$580$	0	0
$581$	2.80136	0.116220
$582$	0	0
$583$	11.0397	0.457216
$584$	0	0
$585$	−26.0368	−1.07649
$586$	0	0
$587$	16.5467	0.682954	0.341477	−	0.939890i	$-0.389073\pi$
$587$	16.5467	0.682954	0.341477	+	0.939890i	$0.389073\pi$
$588$	0	0
$589$	0.00312160	0.000128623 0
$590$	0	0
$591$	2.83623	0.116667
$592$	0	0
$593$	−17.0512	−0.700210	−0.350105	−	0.936710i	$-0.613854\pi$
$593$	−17.0512	−0.700210	−0.350105	+	0.936710i	$0.613854\pi$
$594$	0	0
$595$	−4.89028	−0.200482
$596$	0	0
$597$	−3.38292	−0.138454
$598$	0	0
$599$	44.3288	1.81123	0.905613	−	0.424105i	$-0.139411\pi$
$599$	44.3288	1.81123	0.905613	+	0.424105i	$0.139411\pi$
$600$	0	0
$601$	−28.2608	−1.15278	−0.576392	−	0.817174i	$-0.695540\pi$
$601$	−28.2608	−1.15278	−0.576392	+	0.817174i	$0.695540\pi$
$602$	0	0
$603$	−40.4955	−1.64911
$604$	0	0
$605$	−12.9772	−0.527598
$606$	0	0
$607$	−6.75766	−0.274285	−0.137142	−	0.990551i	$-0.543792\pi$
$607$	−6.75766	−0.274285	−0.137142	+	0.990551i	$0.543792\pi$
$608$	0	0
$609$	−0.791748	−0.0320832
$610$	0	0
$611$	18.9922	0.768343
$612$	0	0
$613$	7.85318	0.317187	0.158593	−	0.987344i	$-0.449304\pi$
$613$	7.85318	0.317187	0.158593	+	0.987344i	$0.449304\pi$
$614$	0	0
$615$	−0.762068	−0.0307295
$616$	0	0
$617$	34.6498	1.39495	0.697475	−	0.716609i	$-0.254307\pi$
$617$	34.6498	1.39495	0.697475	+	0.716609i	$0.254307\pi$
$618$	0	0
$619$	21.0331	0.845394	0.422697	−	0.906271i	$-0.361084\pi$
$619$	21.0331	0.845394	0.422697	+	0.906271i	$0.361084\pi$
$620$	0	0
$621$	11.4596	0.459856
$622$	0	0
$623$	−2.47041	−0.0989751
$624$	0	0
$625$	1.65659	0.0662635
$626$	0	0
$627$	−0.272642	−0.0108883
$628$	0	0
$629$	29.0856	1.15972
$630$	0	0
$631$	8.95385	0.356447	0.178224	−	0.983990i	$-0.442965\pi$
$631$	8.95385	0.356447	0.178224	+	0.983990i	$0.442965\pi$
$632$	0	0
$633$	4.79742	0.190680
$634$	0	0
$635$	16.5918	0.658424
$636$	0	0
$637$	−44.3097	−1.75561
$638$	0	0
$639$	−41.4560	−1.63997
$640$	0	0
$641$	−4.49943	−0.177717	−0.0888584	−	0.996044i	$-0.528322\pi$
$641$	−4.49943	−0.177717	−0.0888584	+	0.996044i	$0.528322\pi$
$642$	0	0
$643$	−3.11132	−0.122699	−0.0613493	−	0.998116i	$-0.519540\pi$
$643$	−3.11132	−0.122699	−0.0613493	+	0.998116i	$0.519540\pi$
$644$	0	0
$645$	−2.21860	−0.0873574
$646$	0	0
$647$	−0.0891930	−0.00350654	−0.00175327	−	0.999998i	$-0.500558\pi$
$647$	−0.0891930	−0.00350654	−0.00175327	+	0.999998i	$0.500558\pi$
$648$	0	0
$649$	−13.4999	−0.529916
$650$	0	0
$651$	−0.000293891 0	−1.15185e−5 0
$652$	0	0
$653$	−39.7493	−1.55551	−0.777756	−	0.628566i	$-0.783642\pi$
$653$	−39.7493	−1.55551	−0.777756	+	0.628566i	$0.783642\pi$
$654$	0	0
$655$	23.8236	0.930863
$656$	0	0
$657$	42.9969	1.67747
$658$	0	0
$659$	−13.1683	−0.512965	−0.256483	−	0.966549i	$-0.582564\pi$
$659$	−13.1683	−0.512965	−0.256483	+	0.966549i	$0.582564\pi$
$660$	0	0
$661$	1.57719	0.0613456	0.0306728	−	0.999529i	$-0.490235\pi$
$661$	1.57719	0.0613456	0.0306728	+	0.999529i	$0.490235\pi$
$662$	0	0
$663$	8.86483	0.344281
$664$	0	0
$665$	0.899413	0.0348778
$666$	0	0
$667$	65.7359	2.54530
$668$	0	0
$669$	0.756574	0.0292508
$670$	0	0
$671$	4.43257	0.171118
$672$	0	0
$673$	37.4362	1.44306	0.721530	−	0.692383i	$-0.243439\pi$
$673$	37.4362	1.44306	0.721530	+	0.692383i	$0.243439\pi$
$674$	0	0
$675$	−3.92246	−0.150976
$676$	0	0
$677$	−32.9327	−1.26571	−0.632853	−	0.774272i	$-0.718116\pi$
$677$	−32.9327	−1.26571	−0.632853	+	0.774272i	$0.718116\pi$
$678$	0	0
$679$	4.66118	0.178880
$680$	0	0
$681$	−0.938298	−0.0359556
$682$	0	0
$683$	1.79081	0.0685234	0.0342617	−	0.999413i	$-0.489092\pi$
$683$	1.79081	0.0685234	0.0342617	+	0.999413i	$0.489092\pi$
$684$	0	0
$685$	6.75217	0.257987
$686$	0	0
$687$	3.99722	0.152503
$688$	0	0
$689$	66.0176	2.51507
$690$	0	0
$691$	−47.6194	−1.81153	−0.905765	−	0.423781i	$-0.860703\pi$
$691$	−47.6194	−1.81153	−0.905765	+	0.423781i	$0.860703\pi$
$692$	0	0
$693$	−1.83569	−0.0697320
$694$	0	0
$695$	19.5776	0.742621
$696$	0	0
$697$	−18.5556	−0.702843
$698$	0	0
$699$	2.95819	0.111889
$700$	0	0
$701$	−29.1728	−1.10184	−0.550920	−	0.834558i	$-0.685723\pi$
$701$	−29.1728	−1.10184	−0.550920	+	0.834558i	$0.685723\pi$
$702$	0	0
$703$	−5.34938	−0.201756
$704$	0	0
$705$	−0.774140	−0.0291558
$706$	0	0
$707$	3.63392	0.136668
$708$	0	0
$709$	27.5136	1.03329	0.516647	−	0.856198i	$-0.327180\pi$
$709$	27.5136	1.03329	0.516647	+	0.856198i	$0.327180\pi$
$710$	0	0
$711$	9.90158	0.371338
$712$	0	0
$713$	0.0244007	0.000913813 0
$714$	0	0
$715$	9.75197	0.364703
$716$	0	0
$717$	4.13552	0.154444
$718$	0	0
$719$	4.08370	0.152296	0.0761482	−	0.997097i	$-0.475738\pi$
$719$	4.08370	0.152296	0.0761482	+	0.997097i	$0.475738\pi$
$720$	0	0
$721$	−0.665219	−0.0247740
$722$	0	0
$723$	6.17148	0.229520
$724$	0	0
$725$	−22.5006	−0.835650
$726$	0	0
$727$	6.35141	0.235561	0.117780	−	0.993040i	$-0.462422\pi$
$727$	6.35141	0.235561	0.117780	+	0.993040i	$0.462422\pi$
$728$	0	0
$729$	−24.7916	−0.918207
$730$	0	0
$731$	−54.0207	−1.99803
$732$	0	0
$733$	4.55052	0.168077	0.0840387	−	0.996462i	$-0.473218\pi$
$733$	4.55052	0.168077	0.0840387	+	0.996462i	$0.473218\pi$
$734$	0	0
$735$	1.80610	0.0666192
$736$	0	0
$737$	15.1674	0.558700
$738$	0	0
$739$	22.1490	0.814766	0.407383	−	0.913257i	$-0.366441\pi$
$739$	22.1490	0.814766	0.407383	+	0.913257i	$0.366441\pi$
$740$	0	0
$741$	−1.63041	−0.0598945
$742$	0	0
$743$	3.45278	0.126670	0.0633352	−	0.997992i	$-0.479826\pi$
$743$	3.45278	0.126670	0.0633352	+	0.997992i	$0.479826\pi$
$744$	0	0
$745$	7.91289	0.289906
$746$	0	0
$747$	−14.8029	−0.541611
$748$	0	0
$749$	−6.40983	−0.234210
$750$	0	0
$751$	48.9929	1.78778	0.893889	−	0.448289i	$-0.147966\pi$
$751$	48.9929	1.78778	0.893889	+	0.448289i	$0.147966\pi$
$752$	0	0
$753$	2.47146	0.0900650
$754$	0	0
$755$	−7.42027	−0.270051
$756$	0	0
$757$	−12.3628	−0.449333	−0.224667	−	0.974436i	$-0.572129\pi$
$757$	−12.3628	−0.449333	−0.224667	+	0.974436i	$0.572129\pi$
$758$	0	0
$759$	−2.13117	−0.0773564
$760$	0	0
$761$	40.9080	1.48291	0.741456	−	0.671001i	$-0.234135\pi$
$761$	40.9080	1.48291	0.741456	+	0.671001i	$0.234135\pi$
$762$	0	0
$763$	−5.48515	−0.198576
$764$	0	0
$765$	25.8411	0.934288
$766$	0	0
$767$	−80.7296	−2.91498
$768$	0	0
$769$	10.2747	0.370515	0.185258	−	0.982690i	$-0.440688\pi$
$769$	10.2747	0.370515	0.185258	+	0.982690i	$0.440688\pi$
$770$	0	0
$771$	3.70823	0.133549
$772$	0	0
$773$	1.55020	0.0557568	0.0278784	−	0.999611i	$-0.491125\pi$
$773$	1.55020	0.0557568	0.0278784	+	0.999611i	$0.491125\pi$
$774$	0	0
$775$	−0.00835205	−0.000300015 0
$776$	0	0
$777$	0.503631	0.0180677
$778$	0	0
$779$	3.41272	0.122273
$780$	0	0
$781$	15.5272	0.555606
$782$	0	0
$783$	8.42598	0.301120
$784$	0	0
$785$	−20.9598	−0.748086
$786$	0	0
$787$	44.0821	1.57136	0.785679	−	0.618634i	$-0.212314\pi$
$787$	44.0821	1.57136	0.785679	+	0.618634i	$0.212314\pi$
$788$	0	0
$789$	−4.84675	−0.172549
$790$	0	0
$791$	3.70841	0.131856
$792$	0	0
$793$	26.5069	0.941289
$794$	0	0
$795$	−2.69094	−0.0954377
$796$	0	0
$797$	30.1417	1.06768	0.533838	−	0.845587i	$-0.320749\pi$
$797$	30.1417	1.06768	0.533838	+	0.845587i	$0.320749\pi$
$798$	0	0
$799$	−18.8495	−0.666848
$800$	0	0
$801$	13.0541	0.461245
$802$	0	0
$803$	−16.1043	−0.568310
$804$	0	0
$805$	7.03047	0.247791
$806$	0	0
$807$	−2.82774	−0.0995413
$808$	0	0
$809$	48.4411	1.70310	0.851550	−	0.524274i	$-0.175663\pi$
$809$	48.4411	1.70310	0.851550	+	0.524274i	$0.175663\pi$
$810$	0	0
$811$	24.6395	0.865209	0.432605	−	0.901584i	$-0.357595\pi$
$811$	24.6395	0.865209	0.432605	+	0.901584i	$0.357595\pi$
$812$	0	0
$813$	−3.53227	−0.123882
$814$	0	0
$815$	−14.2487	−0.499110
$816$	0	0
$817$	9.93542	0.347596
$818$	0	0
$819$	−10.9775	−0.383584
$820$	0	0
$821$	37.9949	1.32603	0.663016	−	0.748605i	$-0.269276\pi$
$821$	37.9949	1.32603	0.663016	+	0.748605i	$0.269276\pi$
$822$	0	0
$823$	−15.6752	−0.546402	−0.273201	−	0.961957i	$-0.588082\pi$
$823$	−15.6752	−0.546402	−0.273201	+	0.961957i	$0.588082\pi$
$824$	0	0
$825$	0.729471	0.0253969
$826$	0	0
$827$	−28.5096	−0.991377	−0.495688	−	0.868500i	$-0.665084\pi$
$827$	−28.5096	−0.991377	−0.495688	+	0.868500i	$0.665084\pi$
$828$	0	0
$829$	−15.8088	−0.549061	−0.274530	−	0.961578i	$-0.588522\pi$
$829$	−15.8088	−0.549061	−0.274530	+	0.961578i	$0.588522\pi$
$830$	0	0
$831$	2.90371	0.100729
$832$	0	0
$833$	43.9768	1.52371
$834$	0	0
$835$	1.32800	0.0459572
$836$	0	0
$837$	0.00312767	0.000108108 0
$838$	0	0
$839$	−23.7311	−0.819287	−0.409644	−	0.912246i	$-0.634347\pi$
$839$	−23.7311	−0.819287	−0.409644	+	0.912246i	$0.634347\pi$
$840$	0	0
$841$	19.3343	0.666700
$842$	0	0
$843$	4.87531	0.167915
$844$	0	0
$845$	41.0531	1.41227
$846$	0	0
$847$	−5.47138	−0.187999
$848$	0	0
$849$	4.53954	0.155797
$850$	0	0
$851$	−41.8146	−1.43339
$852$	0	0
$853$	−50.8766	−1.74198	−0.870990	−	0.491301i	$-0.836522\pi$
$853$	−50.8766	−1.74198	−0.870990	+	0.491301i	$0.836522\pi$
$854$	0	0
$855$	−4.75267	−0.162538
$856$	0	0
$857$	−0.410381	−0.0140183	−0.00700917	−	0.999975i	$-0.502231\pi$
$857$	−0.410381	−0.0140183	−0.00700917	+	0.999975i	$0.502231\pi$
$858$	0	0
$859$	35.6543	1.21651	0.608255	−	0.793742i	$-0.291870\pi$
$859$	35.6543	1.21651	0.608255	+	0.793742i	$0.291870\pi$
$860$	0	0
$861$	−0.321299	−0.0109498
$862$	0	0
$863$	7.07940	0.240986	0.120493	−	0.992714i	$-0.461553\pi$
$863$	7.07940	0.240986	0.120493	+	0.992714i	$0.461553\pi$
$864$	0	0
$865$	1.06952	0.0363649
$866$	0	0
$867$	−5.34046	−0.181371
$868$	0	0
$869$	−3.70860	−0.125806
$870$	0	0
$871$	90.7018	3.07331
$872$	0	0
$873$	−24.6305	−0.833617
$874$	0	0
$875$	−6.12418	−0.207035
$876$	0	0
$877$	−0.959215	−0.0323904	−0.0161952	−	0.999869i	$-0.505155\pi$
$877$	−0.959215	−0.0323904	−0.0161952	+	0.999869i	$0.505155\pi$
$878$	0	0
$879$	−2.31558	−0.0781026
$880$	0	0
$881$	8.89142	0.299560	0.149780	−	0.988719i	$-0.452144\pi$
$881$	8.89142	0.299560	0.149780	+	0.988719i	$0.452144\pi$
$882$	0	0
$883$	−13.5910	−0.457372	−0.228686	−	0.973500i	$-0.573443\pi$
$883$	−13.5910	−0.457372	−0.228686	+	0.973500i	$0.573443\pi$
$884$	0	0
$885$	3.29061	0.110613
$886$	0	0
$887$	−8.01973	−0.269276	−0.134638	−	0.990895i	$-0.542987\pi$
$887$	−8.01973	−0.269276	−0.134638	+	0.990895i	$0.542987\pi$
$888$	0	0
$889$	6.99533	0.234616
$890$	0	0
$891$	9.56258	0.320359
$892$	0	0
$893$	3.46678	0.116011
$894$	0	0
$895$	−30.3555	−1.01467
$896$	0	0
$897$	−12.7444	−0.425525
$898$	0	0
$899$	0.0179414	0.000598378 0
$900$	0	0
$901$	−65.5216	−2.18284
$902$	0	0
$903$	−0.935395	−0.0311280
$904$	0	0
$905$	10.0585	0.334356
$906$	0	0
$907$	−47.2123	−1.56766	−0.783829	−	0.620977i	$-0.786736\pi$
$907$	−47.2123	−1.56766	−0.783829	+	0.620977i	$0.786736\pi$
$908$	0	0
$909$	−19.2023	−0.636900
$910$	0	0
$911$	16.6156	0.550498	0.275249	−	0.961373i	$-0.411240\pi$
$911$	16.6156	0.550498	0.275249	+	0.961373i	$0.411240\pi$
$912$	0	0
$913$	5.54438	0.183492
$914$	0	0
$915$	−1.08045	−0.0357185
$916$	0	0
$917$	10.0444	0.331694
$918$	0	0
$919$	1.78083	0.0587442	0.0293721	−	0.999569i	$-0.490649\pi$
$919$	1.78083	0.0587442	0.0293721	+	0.999569i	$0.490649\pi$
$920$	0	0
$921$	6.33848	0.208860
$922$	0	0
$923$	92.8530	3.05629
$924$	0	0
$925$	14.3126	0.470596
$926$	0	0
$927$	3.51514	0.115452
$928$	0	0
$929$	−39.0211	−1.28024	−0.640120	−	0.768275i	$-0.721115\pi$
$929$	−39.0211	−1.28024	−0.640120	+	0.768275i	$0.721115\pi$
$930$	0	0
$931$	−8.08816	−0.265079
$932$	0	0
$933$	−4.62605	−0.151450
$934$	0	0
$935$	−9.67870	−0.316527
$936$	0	0
$937$	−24.3769	−0.796360	−0.398180	−	0.917307i	$-0.630358\pi$
$937$	−24.3769	−0.796360	−0.398180	+	0.917307i	$0.630358\pi$
$938$	0	0
$939$	−0.266692	−0.00870316
$940$	0	0
$941$	−15.3205	−0.499434	−0.249717	−	0.968319i	$-0.580338\pi$
$941$	−15.3205	−0.499434	−0.249717	+	0.968319i	$0.580338\pi$
$942$	0	0
$943$	26.6763	0.868699
$944$	0	0
$945$	0.901161	0.0293148
$946$	0	0
$947$	5.72492	0.186035	0.0930175	−	0.995664i	$-0.470349\pi$
$947$	5.72492	0.186035	0.0930175	+	0.995664i	$0.470349\pi$
$948$	0	0
$949$	−96.3045	−3.12618
$950$	0	0
$951$	0.191794	0.00621935
$952$	0	0
$953$	−30.6641	−0.993308	−0.496654	−	0.867949i	$-0.665438\pi$
$953$	−30.6641	−0.993308	−0.496654	+	0.867949i	$0.665438\pi$
$954$	0	0
$955$	−16.8143	−0.544097
$956$	0	0
$957$	−1.56700	−0.0506540
$958$	0	0
$959$	2.84682	0.0919285
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	33.8707	1.09147
$964$	0	0
$965$	−6.80612	−0.219097
$966$	0	0
$967$	−32.9044	−1.05813	−0.529067	−	0.848580i	$-0.677458\pi$
$967$	−32.9044	−1.05813	−0.529067	+	0.848580i	$0.677458\pi$
$968$	0	0
$969$	1.61816	0.0519827
$970$	0	0
$971$	24.2187	0.777215	0.388608	−	0.921403i	$-0.372956\pi$
$971$	24.2187	0.777215	0.388608	+	0.921403i	$0.372956\pi$
$972$	0	0
$973$	8.25420	0.264618
$974$	0	0
$975$	4.36226	0.139704
$976$	0	0
$977$	−12.4876	−0.399514	−0.199757	−	0.979845i	$-0.564015\pi$
$977$	−12.4876	−0.399514	−0.199757	+	0.979845i	$0.564015\pi$
$978$	0	0
$979$	−4.88937	−0.156265
$980$	0	0
$981$	28.9846	0.925406
$982$	0	0
$983$	55.9056	1.78311	0.891556	−	0.452911i	$-0.149614\pi$
$983$	55.9056	1.78311	0.891556	+	0.452911i	$0.149614\pi$
$984$	0	0
$985$	−18.5179	−0.590029
$986$	0	0
$987$	−0.326389	−0.0103891
$988$	0	0
$989$	77.6624	2.46952
$990$	0	0
$991$	−29.1786	−0.926887	−0.463444	−	0.886126i	$-0.653386\pi$
$991$	−29.1786	−0.926887	−0.463444	+	0.886126i	$0.653386\pi$
$992$	0	0
$993$	−1.67758	−0.0532364
$994$	0	0
$995$	22.0873	0.700213
$996$	0	0
$997$	24.5460	0.777379	0.388690	−	0.921369i	$-0.372928\pi$
$997$	24.5460	0.777379	0.388690	+	0.921369i	$0.372928\pi$
$998$	0	0
$999$	−5.35977	−0.169576

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1336.2.a.e.1.6	✓	12
4.3	odd	2		2672.2.a.n.1.7		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1336.2.a.e.1.6	✓	12	1.1	even	1	trivial
2672.2.a.n.1.7		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)

\(f(q)\)	\(=\)	\(q-0.203398 q^{3} +1.32800 q^{5} +0.559903 q^{7} -2.95863 q^{9} +O(q^{10})\)	\(q-0.203398 q^{3} +1.32800 q^{5} +0.559903 q^{7} -2.95863 q^{9} +1.10814 q^{11} +6.62673 q^{13} -0.270112 q^{15} -6.57694 q^{17} +1.20962 q^{19} -0.113883 q^{21} +9.45529 q^{23} -3.23643 q^{25} +1.21197 q^{27} +6.95229 q^{29} +0.00258064 q^{31} -0.225394 q^{33} +0.743549 q^{35} -4.42235 q^{37} -1.34786 q^{39} +2.82131 q^{41} +8.21365 q^{43} -3.92905 q^{45} +2.86600 q^{47} -6.68651 q^{49} +1.33774 q^{51} +9.96231 q^{53} +1.47161 q^{55} -0.246035 q^{57} -12.1824 q^{59} +4.00000 q^{61} -1.65654 q^{63} +8.80028 q^{65} +13.6873 q^{67} -1.92319 q^{69} +14.0119 q^{71} -14.5327 q^{73} +0.658282 q^{75} +0.620452 q^{77} -3.34668 q^{79} +8.62938 q^{81} +5.00331 q^{83} -8.73416 q^{85} -1.41408 q^{87} -4.41222 q^{89} +3.71033 q^{91} -0.000524897 q^{93} +1.60637 q^{95} +8.32498 q^{97} -3.27858 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 5 q^{3} - 2 q^{5} + 10 q^{7} + 19 q^{9}+O(q^{10}) \) 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

Fields

Representations

Groups

Database

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