LMFDB - Embedded newform 2008.2.a.c.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2008,2,Mod(1,2008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2008, 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("2008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2008 = 2^{3} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2008.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:	$16.0339607259$
Analytic rank:	$1$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 6 x^{18} - 21 x^{17} + 179 x^{16} + 90 x^{15} - 2109 x^{14} + 926 x^{13} + 12681 x^{12} + \cdots + 4 $ x^19 - 6x^18 - 21x^17 + 179x^16 + 90x^15 - 2109x^14 + 926x^13 + 12681x^12 - 10845x^11 - 41921x^10 + 43551x^9 + 76260x^8 - 80907x^7 - 72526x^6 + 64793x^5 + 33209x^4 - 13777x^3 - 7607x^2 - 747x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Root			$-1.53958$ of defining polynomial
Character	$\chi$	$=$	2008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.53958 q^{3} -0.551255 q^{5} +1.16112 q^{7} -0.629704 q^{9} +O(q^{10})$	$q+1.53958 q^{3} -0.551255 q^{5} +1.16112 q^{7} -0.629704 q^{9} -4.36036 q^{11} +4.01578 q^{13} -0.848700 q^{15} -4.62573 q^{17} -6.90960 q^{19} +1.78764 q^{21} -9.20684 q^{23} -4.69612 q^{25} -5.58821 q^{27} -8.36816 q^{29} +7.71697 q^{31} -6.71310 q^{33} -0.640075 q^{35} +3.55197 q^{37} +6.18260 q^{39} +7.65470 q^{41} +10.3914 q^{43} +0.347128 q^{45} +5.43471 q^{47} -5.65179 q^{49} -7.12167 q^{51} -4.47955 q^{53} +2.40367 q^{55} -10.6379 q^{57} -4.30455 q^{59} +5.04626 q^{61} -0.731164 q^{63} -2.21372 q^{65} +7.56274 q^{67} -14.1746 q^{69} -3.20862 q^{71} +4.28354 q^{73} -7.23003 q^{75} -5.06291 q^{77} -14.7379 q^{79} -6.71436 q^{81} -5.61737 q^{83} +2.54996 q^{85} -12.8834 q^{87} +6.23159 q^{89} +4.66281 q^{91} +11.8809 q^{93} +3.80895 q^{95} +2.25605 q^{97} +2.74574 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 6 q^{3} - 8 q^{5} - 11 q^{7} + 21 q^{9}+O(q^{10}) $ 19 * q - 6 * q^3 - 8 * q^5 - 11 * q^7 + 21 * q^9	$ 19 q - 6 q^{3} - 8 q^{5} - 11 q^{7} + 21 q^{9} - 15 q^{11} - 8 q^{13} - 17 q^{15} - 4 q^{17} - 14 q^{19} - 9 q^{21} - 28 q^{23} + 25 q^{25} - 21 q^{27} - 13 q^{29} - 20 q^{31} - 6 q^{33} - 32 q^{35} - 16 q^{37} - 27 q^{39} + 2 q^{41} - 28 q^{43} - 29 q^{45} - 37 q^{47} + 36 q^{49} - 35 q^{51} - 37 q^{53} - 24 q^{55} - 11 q^{57} - 32 q^{59} - 7 q^{61} - 45 q^{63} + q^{65} - 45 q^{67} - 12 q^{69} - 49 q^{71} + 16 q^{73} - 35 q^{75} - 40 q^{77} - 33 q^{79} + 15 q^{81} - 43 q^{83} - 28 q^{85} - 48 q^{87} + 3 q^{89} - 56 q^{91} - 48 q^{93} - 43 q^{95} + 8 q^{97} - 74 q^{99}+O(q^{100}) $ 19 * q - 6 * q^3 - 8 * q^5 - 11 * q^7 + 21 * q^9 - 15 * q^11 - 8 * q^13 - 17 * q^15 - 4 * q^17 - 14 * q^19 - 9 * q^21 - 28 * q^23 + 25 * q^25 - 21 * q^27 - 13 * q^29 - 20 * q^31 - 6 * q^33 - 32 * q^35 - 16 * q^37 - 27 * q^39 + 2 * q^41 - 28 * q^43 - 29 * q^45 - 37 * q^47 + 36 * q^49 - 35 * q^51 - 37 * q^53 - 24 * q^55 - 11 * q^57 - 32 * q^59 - 7 * q^61 - 45 * q^63 + q^65 - 45 * q^67 - 12 * q^69 - 49 * q^71 + 16 * q^73 - 35 * q^75 - 40 * q^77 - 33 * q^79 + 15 * q^81 - 43 * q^83 - 28 * q^85 - 48 * q^87 + 3 * q^89 - 56 * q^91 - 48 * q^93 - 43 * q^95 + 8 * q^97 - 74 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.53958	0.888875	0.444437	−	0.895810i	$-0.353404\pi$
$3$	1.53958	0.888875	0.444437	+	0.895810i	$0.353404\pi$
$4$	0	0
$5$	−0.551255	−0.246529	−0.123264	−	0.992374i	$-0.539336\pi$
$5$	−0.551255	−0.246529	−0.123264	+	0.992374i	$0.539336\pi$
$6$	0	0
$7$	1.16112	0.438863	0.219432	−	0.975628i	$-0.429580\pi$
$7$	1.16112	0.438863	0.219432	+	0.975628i	$0.429580\pi$
$8$	0	0
$9$	−0.629704	−0.209901
$10$	0	0
$11$	−4.36036	−1.31470	−0.657348	−	0.753587i	$-0.728322\pi$
$11$	−4.36036	−1.31470	−0.657348	+	0.753587i	$0.728322\pi$
$12$	0	0
$13$	4.01578	1.11378	0.556888	−	0.830588i	$-0.311995\pi$
$13$	4.01578	1.11378	0.556888	+	0.830588i	$0.311995\pi$
$14$	0	0
$15$	−0.848700	−0.219133
$16$	0	0
$17$	−4.62573	−1.12191	−0.560953	−	0.827848i	$-0.689565\pi$
$17$	−4.62573	−1.12191	−0.560953	+	0.827848i	$0.689565\pi$
$18$	0	0
$19$	−6.90960	−1.58517	−0.792585	−	0.609761i	$-0.791265\pi$
$19$	−6.90960	−1.58517	−0.792585	+	0.609761i	$0.791265\pi$
$20$	0	0
$21$	1.78764	0.390094
$22$	0	0
$23$	−9.20684	−1.91976	−0.959879	−	0.280414i	$-0.909528\pi$
$23$	−9.20684	−1.91976	−0.959879	+	0.280414i	$0.909528\pi$
$24$	0	0
$25$	−4.69612	−0.939224
$26$	0	0
$27$	−5.58821	−1.07545
$28$	0	0
$29$	−8.36816	−1.55393	−0.776964	−	0.629545i	$-0.783241\pi$
$29$	−8.36816	−1.55393	−0.776964	+	0.629545i	$0.783241\pi$
$30$	0	0
$31$	7.71697	1.38601	0.693004	−	0.720933i	$-0.256287\pi$
$31$	7.71697	1.38601	0.693004	+	0.720933i	$0.256287\pi$
$32$	0	0
$33$	−6.71310	−1.16860
$34$	0	0
$35$	−0.640075	−0.108192
$36$	0	0
$37$	3.55197	0.583940	0.291970	−	0.956428i	$-0.405689\pi$
$37$	3.55197	0.583940	0.291970	+	0.956428i	$0.405689\pi$
$38$	0	0
$39$	6.18260	0.990008
$40$	0	0
$41$	7.65470	1.19546	0.597732	−	0.801696i	$-0.296069\pi$
$41$	7.65470	1.19546	0.597732	+	0.801696i	$0.296069\pi$
$42$	0	0
$43$	10.3914	1.58468	0.792338	−	0.610083i	$-0.208864\pi$
$43$	10.3914	1.58468	0.792338	+	0.610083i	$0.208864\pi$
$44$	0	0
$45$	0.347128	0.0517468
$46$	0	0
$47$	5.43471	0.792733	0.396367	−	0.918092i	$-0.370271\pi$
$47$	5.43471	0.792733	0.396367	+	0.918092i	$0.370271\pi$
$48$	0	0
$49$	−5.65179	−0.807399
$50$	0	0
$51$	−7.12167	−0.997233
$52$	0	0
$53$	−4.47955	−0.615314	−0.307657	−	0.951497i	$-0.599545\pi$
$53$	−4.47955	−0.615314	−0.307657	+	0.951497i	$0.599545\pi$
$54$	0	0
$55$	2.40367	0.324111
$56$	0	0
$57$	−10.6379	−1.40902
$58$	0	0
$59$	−4.30455	−0.560405	−0.280203	−	0.959941i	$-0.590402\pi$
$59$	−4.30455	−0.560405	−0.280203	+	0.959941i	$0.590402\pi$
$60$	0	0
$61$	5.04626	0.646107	0.323054	−	0.946381i	$-0.395291\pi$
$61$	5.04626	0.646107	0.323054	+	0.946381i	$0.395291\pi$
$62$	0	0
$63$	−0.731164	−0.0921180
$64$	0	0
$65$	−2.21372	−0.274578
$66$	0	0
$67$	7.56274	0.923936	0.461968	−	0.886897i	$-0.347143\pi$
$67$	7.56274	0.923936	0.461968	+	0.886897i	$0.347143\pi$
$68$	0	0
$69$	−14.1746	−1.70643
$70$	0	0
$71$	−3.20862	−0.380793	−0.190396	−	0.981707i	$-0.560977\pi$
$71$	−3.20862	−0.380793	−0.190396	+	0.981707i	$0.560977\pi$
$72$	0	0
$73$	4.28354	0.501351	0.250675	−	0.968071i	$-0.419347\pi$
$73$	4.28354	0.501351	0.250675	+	0.968071i	$0.419347\pi$
$74$	0	0
$75$	−7.23003	−0.834852
$76$	0	0
$77$	−5.06291	−0.576972
$78$	0	0
$79$	−14.7379	−1.65815	−0.829073	−	0.559140i	$-0.811132\pi$
$79$	−14.7379	−1.65815	−0.829073	+	0.559140i	$0.811132\pi$
$80$	0	0
$81$	−6.71436	−0.746040
$82$	0	0
$83$	−5.61737	−0.616587	−0.308293	−	0.951291i	$-0.599758\pi$
$83$	−5.61737	−0.616587	−0.308293	+	0.951291i	$0.599758\pi$
$84$	0	0
$85$	2.54996	0.276582
$86$	0	0
$87$	−12.8834	−1.38125
$88$	0	0
$89$	6.23159	0.660548	0.330274	−	0.943885i	$-0.392859\pi$
$89$	6.23159	0.660548	0.330274	+	0.943885i	$0.392859\pi$
$90$	0	0
$91$	4.66281	0.488795
$92$	0	0
$93$	11.8809	1.23199
$94$	0	0
$95$	3.80895	0.390790
$96$	0	0
$97$	2.25605	0.229067	0.114533	−	0.993419i	$-0.463463\pi$
$97$	2.25605	0.229067	0.114533	+	0.993419i	$0.463463\pi$
$98$	0	0
$99$	2.74574	0.275957
$100$	0	0
$101$	−9.64660	−0.959873	−0.479936	−	0.877303i	$-0.659340\pi$
$101$	−9.64660	−0.959873	−0.479936	+	0.877303i	$0.659340\pi$
$102$	0	0
$103$	−12.2916	−1.21113	−0.605564	−	0.795797i	$-0.707052\pi$
$103$	−12.2916	−1.21113	−0.605564	+	0.795797i	$0.707052\pi$
$104$	0	0
$105$	−0.985444	−0.0961695
$106$	0	0
$107$	9.81737	0.949081	0.474541	−	0.880234i	$-0.342614\pi$
$107$	9.81737	0.949081	0.474541	+	0.880234i	$0.342614\pi$
$108$	0	0
$109$	6.32547	0.605871	0.302935	−	0.953011i	$-0.402033\pi$
$109$	6.32547	0.605871	0.302935	+	0.953011i	$0.402033\pi$
$110$	0	0
$111$	5.46852	0.519049
$112$	0	0
$113$	14.1214	1.32843	0.664217	−	0.747540i	$-0.268765\pi$
$113$	14.1214	1.32843	0.664217	+	0.747540i	$0.268765\pi$
$114$	0	0
$115$	5.07532	0.473276
$116$	0	0
$117$	−2.52875	−0.233783
$118$	0	0
$119$	−5.37104	−0.492363
$120$	0	0
$121$	8.01270	0.728427
$122$	0	0
$123$	11.7850	1.06262
$124$	0	0
$125$	5.34504	0.478074
$126$	0	0
$127$	−5.00649	−0.444254	−0.222127	−	0.975018i	$-0.571300\pi$
$127$	−5.00649	−0.444254	−0.222127	+	0.975018i	$0.571300\pi$
$128$	0	0
$129$	15.9984	1.40858
$130$	0	0
$131$	−17.9307	−1.56662	−0.783308	−	0.621633i	$-0.786469\pi$
$131$	−17.9307	−1.56662	−0.783308	+	0.621633i	$0.786469\pi$
$132$	0	0
$133$	−8.02289	−0.695673
$134$	0	0
$135$	3.08053	0.265130
$136$	0	0
$137$	−0.845966	−0.0722757	−0.0361379	−	0.999347i	$-0.511506\pi$
$137$	−0.845966	−0.0722757	−0.0361379	+	0.999347i	$0.511506\pi$
$138$	0	0
$139$	−23.5553	−1.99794	−0.998968	−	0.0454155i	$-0.985539\pi$
$139$	−23.5553	−1.99794	−0.998968	+	0.0454155i	$0.985539\pi$
$140$	0	0
$141$	8.36714	0.704641
$142$	0	0
$143$	−17.5102	−1.46428
$144$	0	0
$145$	4.61299	0.383088
$146$	0	0
$147$	−8.70137	−0.717677
$148$	0	0
$149$	3.16938	0.259645	0.129823	−	0.991537i	$-0.458559\pi$
$149$	3.16938	0.259645	0.129823	+	0.991537i	$0.458559\pi$
$150$	0	0
$151$	18.4543	1.50179	0.750895	−	0.660421i	$-0.229622\pi$
$151$	18.4543	1.50179	0.750895	+	0.660421i	$0.229622\pi$
$152$	0	0
$153$	2.91284	0.235490
$154$	0	0
$155$	−4.25402	−0.341691
$156$	0	0
$157$	1.07412	0.0857237	0.0428619	−	0.999081i	$-0.486352\pi$
$157$	1.07412	0.0857237	0.0428619	+	0.999081i	$0.486352\pi$
$158$	0	0
$159$	−6.89662	−0.546937
$160$	0	0
$161$	−10.6903	−0.842511
$162$	0	0
$163$	−11.8887	−0.931195	−0.465598	−	0.884997i	$-0.654161\pi$
$163$	−11.8887	−0.931195	−0.465598	+	0.884997i	$0.654161\pi$
$164$	0	0
$165$	3.70063	0.288094
$166$	0	0
$167$	3.73459	0.288992	0.144496	−	0.989505i	$-0.453844\pi$
$167$	3.73459	0.288992	0.144496	+	0.989505i	$0.453844\pi$
$168$	0	0
$169$	3.12647	0.240497
$170$	0	0
$171$	4.35100	0.332730
$172$	0	0
$173$	−20.7247	−1.57567	−0.787833	−	0.615889i	$-0.788797\pi$
$173$	−20.7247	−1.57567	−0.787833	+	0.615889i	$0.788797\pi$
$174$	0	0
$175$	−5.45277	−0.412191
$176$	0	0
$177$	−6.62719	−0.498130
$178$	0	0
$179$	17.9210	1.33948	0.669740	−	0.742595i	$-0.266405\pi$
$179$	17.9210	1.33948	0.669740	+	0.742595i	$0.266405\pi$
$180$	0	0
$181$	15.4407	1.14770	0.573850	−	0.818960i	$-0.305449\pi$
$181$	15.4407	1.14770	0.573850	+	0.818960i	$0.305449\pi$
$182$	0	0
$183$	7.76910	0.574309
$184$	0	0
$185$	−1.95804	−0.143958
$186$	0	0
$187$	20.1698	1.47496
$188$	0	0
$189$	−6.48859	−0.471976
$190$	0	0
$191$	−2.23516	−0.161730	−0.0808651	−	0.996725i	$-0.525768\pi$
$191$	−2.23516	−0.161730	−0.0808651	+	0.996725i	$0.525768\pi$
$192$	0	0
$193$	17.5006	1.25972	0.629859	−	0.776710i	$-0.283113\pi$
$193$	17.5006	1.25972	0.629859	+	0.776710i	$0.283113\pi$
$194$	0	0
$195$	−3.40819	−0.244065
$196$	0	0
$197$	2.28548	0.162834	0.0814169	−	0.996680i	$-0.474055\pi$
$197$	2.28548	0.162834	0.0814169	+	0.996680i	$0.474055\pi$
$198$	0	0
$199$	−1.14571	−0.0812170	−0.0406085	−	0.999175i	$-0.512930\pi$
$199$	−1.14571	−0.0812170	−0.0406085	+	0.999175i	$0.512930\pi$
$200$	0	0
$201$	11.6434	0.821263
$202$	0	0
$203$	−9.71646	−0.681962
$204$	0	0
$205$	−4.21969	−0.294716
$206$	0	0
$207$	5.79759	0.402960
$208$	0	0
$209$	30.1283	2.08402
$210$	0	0
$211$	−25.8214	−1.77762	−0.888809	−	0.458279i	$-0.848466\pi$
$211$	−25.8214	−1.77762	−0.888809	+	0.458279i	$0.848466\pi$
$212$	0	0
$213$	−4.93991	−0.338477
$214$	0	0
$215$	−5.72832	−0.390668
$216$	0	0
$217$	8.96035	0.608268
$218$	0	0
$219$	6.59484	0.445638
$220$	0	0
$221$	−18.5759	−1.24955
$222$	0	0
$223$	−24.2487	−1.62382	−0.811908	−	0.583786i	$-0.801571\pi$
$223$	−24.2487	−1.62382	−0.811908	+	0.583786i	$0.801571\pi$
$224$	0	0
$225$	2.95717	0.197144
$226$	0	0
$227$	17.7240	1.17638	0.588191	−	0.808722i	$-0.299840\pi$
$227$	17.7240	1.17638	0.588191	+	0.808722i	$0.299840\pi$
$228$	0	0
$229$	−20.5048	−1.35500	−0.677498	−	0.735525i	$-0.736936\pi$
$229$	−20.5048	−1.35500	−0.677498	+	0.735525i	$0.736936\pi$
$230$	0	0
$231$	−7.79473	−0.512856
$232$	0	0
$233$	16.2167	1.06239	0.531194	−	0.847250i	$-0.321743\pi$
$233$	16.2167	1.06239	0.531194	+	0.847250i	$0.321743\pi$
$234$	0	0
$235$	−2.99591	−0.195432
$236$	0	0
$237$	−22.6902	−1.47389
$238$	0	0
$239$	−22.7083	−1.46888	−0.734440	−	0.678673i	$-0.762555\pi$
$239$	−22.7083	−1.46888	−0.734440	+	0.678673i	$0.762555\pi$
$240$	0	0
$241$	−5.65543	−0.364298	−0.182149	−	0.983271i	$-0.558305\pi$
$241$	−5.65543	−0.364298	−0.182149	+	0.983271i	$0.558305\pi$
$242$	0	0
$243$	6.42735	0.412315
$244$	0	0
$245$	3.11558	0.199047
$246$	0	0
$247$	−27.7474	−1.76553
$248$	0	0
$249$	−8.64838	−0.548069
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	40.1451	2.52390
$254$	0	0
$255$	3.92586	0.245847
$256$	0	0
$257$	24.0694	1.50141	0.750703	−	0.660639i	$-0.229715\pi$
$257$	24.0694	1.50141	0.750703	+	0.660639i	$0.229715\pi$
$258$	0	0
$259$	4.12427	0.256270
$260$	0	0
$261$	5.26947	0.326172
$262$	0	0
$263$	7.94125	0.489679	0.244839	−	0.969564i	$-0.421265\pi$
$263$	7.94125	0.489679	0.244839	+	0.969564i	$0.421265\pi$
$264$	0	0
$265$	2.46938	0.151693
$266$	0	0
$267$	9.59401	0.587144
$268$	0	0
$269$	−3.70661	−0.225996	−0.112998	−	0.993595i	$-0.536045\pi$
$269$	−3.70661	−0.225996	−0.112998	+	0.993595i	$0.536045\pi$
$270$	0	0
$271$	16.6020	1.00850	0.504250	−	0.863558i	$-0.331769\pi$
$271$	16.6020	1.00850	0.504250	+	0.863558i	$0.331769\pi$
$272$	0	0
$273$	7.17875	0.434478
$274$	0	0
$275$	20.4767	1.23479
$276$	0	0
$277$	20.2259	1.21525	0.607627	−	0.794222i	$-0.292122\pi$
$277$	20.2259	1.21525	0.607627	+	0.794222i	$0.292122\pi$
$278$	0	0
$279$	−4.85941	−0.290925
$280$	0	0
$281$	−10.4710	−0.624647	−0.312324	−	0.949976i	$-0.601107\pi$
$281$	−10.4710	−0.624647	−0.312324	+	0.949976i	$0.601107\pi$
$282$	0	0
$283$	−17.2832	−1.02738	−0.513688	−	0.857977i	$-0.671721\pi$
$283$	−17.2832	−1.02738	−0.513688	+	0.857977i	$0.671721\pi$
$284$	0	0
$285$	5.86417	0.347364
$286$	0	0
$287$	8.88805	0.524645
$288$	0	0
$289$	4.39740	0.258671
$290$	0	0
$291$	3.47336	0.203612
$292$	0	0
$293$	−15.2888	−0.893184	−0.446592	−	0.894738i	$-0.647362\pi$
$293$	−15.2888	−0.893184	−0.446592	+	0.894738i	$0.647362\pi$
$294$	0	0
$295$	2.37291	0.138156
$296$	0	0
$297$	24.3666	1.41389
$298$	0	0
$299$	−36.9726	−2.13818
$300$	0	0
$301$	12.0657	0.695455
$302$	0	0
$303$	−14.8517	−0.853207
$304$	0	0
$305$	−2.78178	−0.159284
$306$	0	0
$307$	28.2703	1.61347	0.806735	−	0.590914i	$-0.201233\pi$
$307$	28.2703	1.61347	0.806735	+	0.590914i	$0.201233\pi$
$308$	0	0
$309$	−18.9239	−1.07654
$310$	0	0
$311$	−26.4029	−1.49717	−0.748585	−	0.663038i	$-0.769267\pi$
$311$	−26.4029	−1.49717	−0.748585	+	0.663038i	$0.769267\pi$
$312$	0	0
$313$	−29.5834	−1.67215	−0.836077	−	0.548612i	$-0.815157\pi$
$313$	−29.5834	−1.67215	−0.836077	+	0.548612i	$0.815157\pi$
$314$	0	0
$315$	0.403058	0.0227097
$316$	0	0
$317$	30.7841	1.72901	0.864503	−	0.502627i	$-0.167633\pi$
$317$	30.7841	1.72901	0.864503	+	0.502627i	$0.167633\pi$
$318$	0	0
$319$	36.4881	2.04294
$320$	0	0
$321$	15.1146	0.843614
$322$	0	0
$323$	31.9620	1.77841
$324$	0	0
$325$	−18.8586	−1.04608
$326$	0	0
$327$	9.73855	0.538543
$328$	0	0
$329$	6.31036	0.347901
$330$	0	0
$331$	−25.6660	−1.41073	−0.705367	−	0.708843i	$-0.749218\pi$
$331$	−25.6660	−1.41073	−0.705367	+	0.708843i	$0.749218\pi$
$332$	0	0
$333$	−2.23669	−0.122570
$334$	0	0
$335$	−4.16900	−0.227777
$336$	0	0
$337$	14.8134	0.806938	0.403469	−	0.914993i	$-0.367804\pi$
$337$	14.8134	0.806938	0.403469	+	0.914993i	$0.367804\pi$
$338$	0	0
$339$	21.7410	1.18081
$340$	0	0
$341$	−33.6487	−1.82218
$342$	0	0
$343$	−14.6903	−0.793201
$344$	0	0
$345$	7.81384	0.420683
$346$	0	0
$347$	13.3204	0.715076	0.357538	−	0.933899i	$-0.383616\pi$
$347$	13.3204	0.715076	0.357538	+	0.933899i	$0.383616\pi$
$348$	0	0
$349$	12.2810	0.657389	0.328695	−	0.944436i	$-0.393391\pi$
$349$	12.2810	0.657389	0.328695	+	0.944436i	$0.393391\pi$
$350$	0	0
$351$	−22.4410	−1.19781
$352$	0	0
$353$	12.7374	0.677943	0.338972	−	0.940797i	$-0.389921\pi$
$353$	12.7374	0.677943	0.338972	+	0.940797i	$0.389921\pi$
$354$	0	0
$355$	1.76877	0.0938764
$356$	0	0
$357$	−8.26913	−0.437649
$358$	0	0
$359$	−21.7466	−1.14774	−0.573871	−	0.818945i	$-0.694559\pi$
$359$	−21.7466	−1.14774	−0.573871	+	0.818945i	$0.694559\pi$
$360$	0	0
$361$	28.7425	1.51276
$362$	0	0
$363$	12.3362	0.647481
$364$	0	0
$365$	−2.36132	−0.123597
$366$	0	0
$367$	14.8465	0.774983	0.387491	−	0.921873i	$-0.373342\pi$
$367$	14.8465	0.774983	0.387491	+	0.921873i	$0.373342\pi$
$368$	0	0
$369$	−4.82020	−0.250930
$370$	0	0
$371$	−5.20131	−0.270039
$372$	0	0
$373$	−18.4432	−0.954952	−0.477476	−	0.878645i	$-0.658448\pi$
$373$	−18.4432	−0.954952	−0.477476	+	0.878645i	$0.658448\pi$
$374$	0	0
$375$	8.22909	0.424948
$376$	0	0
$377$	−33.6047	−1.73073
$378$	0	0
$379$	−19.8723	−1.02077	−0.510386	−	0.859946i	$-0.670497\pi$
$379$	−19.8723	−1.02077	−0.510386	+	0.859946i	$0.670497\pi$
$380$	0	0
$381$	−7.70788	−0.394887
$382$	0	0
$383$	23.4774	1.19964	0.599819	−	0.800135i	$-0.295239\pi$
$383$	23.4774	1.19964	0.599819	+	0.800135i	$0.295239\pi$
$384$	0	0
$385$	2.79095	0.142240
$386$	0	0
$387$	−6.54352	−0.332626
$388$	0	0
$389$	3.20920	0.162713	0.0813565	−	0.996685i	$-0.474075\pi$
$389$	3.20920	0.162713	0.0813565	+	0.996685i	$0.474075\pi$
$390$	0	0
$391$	42.5884	2.15379
$392$	0	0
$393$	−27.6058	−1.39253
$394$	0	0
$395$	8.12436	0.408781
$396$	0	0
$397$	−12.6928	−0.637032	−0.318516	−	0.947917i	$-0.603185\pi$
$397$	−12.6928	−0.637032	−0.318516	+	0.947917i	$0.603185\pi$
$398$	0	0
$399$	−12.3519	−0.618366
$400$	0	0
$401$	28.6740	1.43191	0.715954	−	0.698147i	$-0.245992\pi$
$401$	28.6740	1.43191	0.715954	+	0.698147i	$0.245992\pi$
$402$	0	0
$403$	30.9896	1.54370
$404$	0	0
$405$	3.70133	0.183920
$406$	0	0
$407$	−15.4878	−0.767703
$408$	0	0
$409$	−4.42281	−0.218694	−0.109347	−	0.994004i	$-0.534876\pi$
$409$	−4.42281	−0.218694	−0.109347	+	0.994004i	$0.534876\pi$
$410$	0	0
$411$	−1.30243	−0.0642441
$412$	0	0
$413$	−4.99812	−0.245941
$414$	0	0
$415$	3.09661	0.152006
$416$	0	0
$417$	−36.2652	−1.77592
$418$	0	0
$419$	14.9349	0.729619	0.364809	−	0.931082i	$-0.381134\pi$
$419$	14.9349	0.729619	0.364809	+	0.931082i	$0.381134\pi$
$420$	0	0
$421$	−20.3245	−0.990557	−0.495279	−	0.868734i	$-0.664934\pi$
$421$	−20.3245	−0.990557	−0.495279	+	0.868734i	$0.664934\pi$
$422$	0	0
$423$	−3.42226	−0.166396
$424$	0	0
$425$	21.7230	1.05372
$426$	0	0
$427$	5.85933	0.283553
$428$	0	0
$429$	−26.9583	−1.30156
$430$	0	0
$431$	−23.8347	−1.14808	−0.574040	−	0.818828i	$-0.694624\pi$
$431$	−23.8347	−1.14808	−0.574040	+	0.818828i	$0.694624\pi$
$432$	0	0
$433$	−0.273200	−0.0131292	−0.00656459	−	0.999978i	$-0.502090\pi$
$433$	−0.273200	−0.0131292	−0.00656459	+	0.999978i	$0.502090\pi$
$434$	0	0
$435$	7.10205	0.340517
$436$	0	0
$437$	63.6155	3.04314
$438$	0	0
$439$	−2.28322	−0.108972	−0.0544862	−	0.998515i	$-0.517352\pi$
$439$	−2.28322	−0.108972	−0.0544862	+	0.998515i	$0.517352\pi$
$440$	0	0
$441$	3.55896	0.169474
$442$	0	0
$443$	−10.5646	−0.501939	−0.250970	−	0.967995i	$-0.580749\pi$
$443$	−10.5646	−0.501939	−0.250970	+	0.967995i	$0.580749\pi$
$444$	0	0
$445$	−3.43520	−0.162844
$446$	0	0
$447$	4.87950	0.230792
$448$	0	0
$449$	−0.0287827	−0.00135834	−0.000679169	−	1.00000i	$-0.500216\pi$
$449$	−0.0287827	−0.00135834	−0.000679169		1.00000i	$0.500216\pi$
$450$	0	0
$451$	−33.3772	−1.57167
$452$	0	0
$453$	28.4118	1.33490
$454$	0	0
$455$	−2.57040	−0.120502
$456$	0	0
$457$	−4.90927	−0.229646	−0.114823	−	0.993386i	$-0.536630\pi$
$457$	−4.90927	−0.229646	−0.114823	+	0.993386i	$0.536630\pi$
$458$	0	0
$459$	25.8496	1.20655
$460$	0	0
$461$	−19.6413	−0.914786	−0.457393	−	0.889265i	$-0.651217\pi$
$461$	−19.6413	−0.914786	−0.457393	+	0.889265i	$0.651217\pi$
$462$	0	0
$463$	−11.8949	−0.552805	−0.276402	−	0.961042i	$-0.589142\pi$
$463$	−11.8949	−0.552805	−0.276402	+	0.961042i	$0.589142\pi$
$464$	0	0
$465$	−6.54939	−0.303721
$466$	0	0
$467$	−5.14416	−0.238044	−0.119022	−	0.992892i	$-0.537976\pi$
$467$	−5.14416	−0.238044	−0.119022	+	0.992892i	$0.537976\pi$
$468$	0	0
$469$	8.78127	0.405481
$470$	0	0
$471$	1.65368	0.0761977
$472$	0	0
$473$	−45.3102	−2.08337
$474$	0	0
$475$	32.4483	1.48883
$476$	0	0
$477$	2.82080	0.129155
$478$	0	0
$479$	−19.3109	−0.882338	−0.441169	−	0.897424i	$-0.645436\pi$
$479$	−19.3109	−0.882338	−0.441169	+	0.897424i	$0.645436\pi$
$480$	0	0
$481$	14.2639	0.650378
$482$	0	0
$483$	−16.4585	−0.748887
$484$	0	0
$485$	−1.24366	−0.0564716
$486$	0	0
$487$	−16.5470	−0.749815	−0.374908	−	0.927062i	$-0.622326\pi$
$487$	−16.5470	−0.749815	−0.374908	+	0.927062i	$0.622326\pi$
$488$	0	0
$489$	−18.3036	−0.827716
$490$	0	0
$491$	−18.8050	−0.848656	−0.424328	−	0.905509i	$-0.639490\pi$
$491$	−18.8050	−0.848656	−0.424328	+	0.905509i	$0.639490\pi$
$492$	0	0
$493$	38.7089	1.74336
$494$	0	0
$495$	−1.51360	−0.0680313
$496$	0	0
$497$	−3.72560	−0.167116
$498$	0	0
$499$	−4.03940	−0.180828	−0.0904142	−	0.995904i	$-0.528819\pi$
$499$	−4.03940	−0.180828	−0.0904142	+	0.995904i	$0.528819\pi$
$500$	0	0
$501$	5.74969	0.256877
$502$	0	0
$503$	16.3967	0.731094	0.365547	−	0.930793i	$-0.380882\pi$
$503$	16.3967	0.731094	0.365547	+	0.930793i	$0.380882\pi$
$504$	0	0
$505$	5.31774	0.236636
$506$	0	0
$507$	4.81343	0.213772
$508$	0	0
$509$	16.4297	0.728232	0.364116	−	0.931354i	$-0.381371\pi$
$509$	16.4297	0.728232	0.364116	+	0.931354i	$0.381371\pi$
$510$	0	0
$511$	4.97372	0.220024
$512$	0	0
$513$	38.6123	1.70477
$514$	0	0
$515$	6.77581	0.298578
$516$	0	0
$517$	−23.6972	−1.04220
$518$	0	0
$519$	−31.9072	−1.40057
$520$	0	0
$521$	12.0099	0.526163	0.263082	−	0.964774i	$-0.415261\pi$
$521$	12.0099	0.526163	0.263082	+	0.964774i	$0.415261\pi$
$522$	0	0
$523$	−30.5873	−1.33749	−0.668745	−	0.743492i	$-0.733168\pi$
$523$	−30.5873	−1.33749	−0.668745	+	0.743492i	$0.733168\pi$
$524$	0	0
$525$	−8.39495	−0.366386
$526$	0	0
$527$	−35.6966	−1.55497
$528$	0	0
$529$	61.7659	2.68547
$530$	0	0
$531$	2.71060	0.117630
$532$	0	0
$533$	30.7396	1.33148
$534$	0	0
$535$	−5.41188	−0.233976
$536$	0	0
$537$	27.5908	1.19063
$538$	0	0
$539$	24.6438	1.06148
$540$	0	0
$541$	19.3820	0.833298	0.416649	−	0.909067i	$-0.363204\pi$
$541$	19.3820	0.833298	0.416649	+	0.909067i	$0.363204\pi$
$542$	0	0
$543$	23.7722	1.02016
$544$	0	0
$545$	−3.48695	−0.149365
$546$	0	0
$547$	15.2984	0.654113	0.327056	−	0.945005i	$-0.393943\pi$
$547$	15.2984	0.654113	0.327056	+	0.945005i	$0.393943\pi$
$548$	0	0
$549$	−3.17765	−0.135619
$550$	0	0
$551$	57.8206	2.46324
$552$	0	0
$553$	−17.1125	−0.727699
$554$	0	0
$555$	−3.01455	−0.127961
$556$	0	0
$557$	38.3198	1.62366	0.811831	−	0.583893i	$-0.198471\pi$
$557$	38.3198	1.62366	0.811831	+	0.583893i	$0.198471\pi$
$558$	0	0
$559$	41.7296	1.76497
$560$	0	0
$561$	31.0530	1.31106
$562$	0	0
$563$	26.5408	1.11856	0.559282	−	0.828978i	$-0.311077\pi$
$563$	26.5408	1.11856	0.559282	+	0.828978i	$0.311077\pi$
$564$	0	0
$565$	−7.78452	−0.327497
$566$	0	0
$567$	−7.79619	−0.327409
$568$	0	0
$569$	−21.7320	−0.911052	−0.455526	−	0.890222i	$-0.650549\pi$
$569$	−21.7320	−0.911052	−0.455526	+	0.890222i	$0.650549\pi$
$570$	0	0
$571$	−33.7128	−1.41084	−0.705418	−	0.708792i	$-0.749241\pi$
$571$	−33.7128	−1.41084	−0.705418	+	0.708792i	$0.749241\pi$
$572$	0	0
$573$	−3.44119	−0.143758
$574$	0	0
$575$	43.2364	1.80308
$576$	0	0
$577$	17.2501	0.718132	0.359066	−	0.933312i	$-0.383095\pi$
$577$	17.2501	0.718132	0.359066	+	0.933312i	$0.383095\pi$
$578$	0	0
$579$	26.9434	1.11973
$580$	0	0
$581$	−6.52246	−0.270597
$582$	0	0
$583$	19.5325	0.808951
$584$	0	0
$585$	1.39399	0.0576343
$586$	0	0
$587$	−21.7942	−0.899544	−0.449772	−	0.893144i	$-0.648495\pi$
$587$	−21.7942	−0.899544	−0.449772	+	0.893144i	$0.648495\pi$
$588$	0	0
$589$	−53.3212	−2.19706
$590$	0	0
$591$	3.51867	0.144739
$592$	0	0
$593$	22.6214	0.928950	0.464475	−	0.885586i	$-0.346243\pi$
$593$	22.6214	0.928950	0.464475	+	0.885586i	$0.346243\pi$
$594$	0	0
$595$	2.96082	0.121382
$596$	0	0
$597$	−1.76390	−0.0721918
$598$	0	0
$599$	−0.931155	−0.0380459	−0.0190230	−	0.999819i	$-0.506056\pi$
$599$	−0.931155	−0.0380459	−0.0190230	+	0.999819i	$0.506056\pi$
$600$	0	0
$601$	−20.1131	−0.820429	−0.410214	−	0.911989i	$-0.634546\pi$
$601$	−20.1131	−0.820429	−0.410214	+	0.911989i	$0.634546\pi$
$602$	0	0
$603$	−4.76229	−0.193936
$604$	0	0
$605$	−4.41704	−0.179578
$606$	0	0
$607$	28.0630	1.13904	0.569521	−	0.821976i	$-0.307129\pi$
$607$	28.0630	1.13904	0.569521	+	0.821976i	$0.307129\pi$
$608$	0	0
$609$	−14.9592	−0.606179
$610$	0	0
$611$	21.8246	0.882927
$612$	0	0
$613$	25.5465	1.03181	0.515907	−	0.856645i	$-0.327455\pi$
$613$	25.5465	1.03181	0.515907	+	0.856645i	$0.327455\pi$
$614$	0	0
$615$	−6.49654	−0.261966
$616$	0	0
$617$	6.80975	0.274150	0.137075	−	0.990561i	$-0.456230\pi$
$617$	6.80975	0.274150	0.137075	+	0.990561i	$0.456230\pi$
$618$	0	0
$619$	−23.0610	−0.926902	−0.463451	−	0.886123i	$-0.653389\pi$
$619$	−23.0610	−0.926902	−0.463451	+	0.886123i	$0.653389\pi$
$620$	0	0
$621$	51.4497	2.06461
$622$	0	0
$623$	7.23564	0.289890
$624$	0	0
$625$	20.5341	0.821364
$626$	0	0
$627$	46.3848	1.85243
$628$	0	0
$629$	−16.4304	−0.655125
$630$	0	0
$631$	34.3457	1.36728	0.683641	−	0.729819i	$-0.260395\pi$
$631$	34.3457	1.36728	0.683641	+	0.729819i	$0.260395\pi$
$632$	0	0
$633$	−39.7540	−1.58008
$634$	0	0
$635$	2.75986	0.109522
$636$	0	0
$637$	−22.6963	−0.899262
$638$	0	0
$639$	2.02048	0.0799290
$640$	0	0
$641$	−0.998757	−0.0394485	−0.0197243	−	0.999805i	$-0.506279\pi$
$641$	−0.998757	−0.0394485	−0.0197243	+	0.999805i	$0.506279\pi$
$642$	0	0
$643$	−25.2016	−0.993854	−0.496927	−	0.867792i	$-0.665538\pi$
$643$	−25.2016	−0.993854	−0.496927	+	0.867792i	$0.665538\pi$
$644$	0	0
$645$	−8.81918	−0.347255
$646$	0	0
$647$	5.63817	0.221659	0.110830	−	0.993839i	$-0.464649\pi$
$647$	5.63817	0.221659	0.110830	+	0.993839i	$0.464649\pi$
$648$	0	0
$649$	18.7694	0.736763
$650$	0	0
$651$	13.7951	0.540674
$652$	0	0
$653$	−44.9850	−1.76040	−0.880200	−	0.474602i	$-0.842592\pi$
$653$	−44.9850	−1.76040	−0.880200	+	0.474602i	$0.842592\pi$
$654$	0	0
$655$	9.88442	0.386216
$656$	0	0
$657$	−2.69737	−0.105234
$658$	0	0
$659$	20.1850	0.786297	0.393148	−	0.919475i	$-0.371386\pi$
$659$	20.1850	0.786297	0.393148	+	0.919475i	$0.371386\pi$
$660$	0	0
$661$	2.55309	0.0993035	0.0496518	−	0.998767i	$-0.484189\pi$
$661$	2.55309	0.0993035	0.0496518	+	0.998767i	$0.484189\pi$
$662$	0	0
$663$	−28.5990	−1.11069
$664$	0	0
$665$	4.42266	0.171503
$666$	0	0
$667$	77.0443	2.98317
$668$	0	0
$669$	−37.3328	−1.44337
$670$	0	0
$671$	−22.0035	−0.849435
$672$	0	0
$673$	6.35139	0.244828	0.122414	−	0.992479i	$-0.460936\pi$
$673$	6.35139	0.244828	0.122414	+	0.992479i	$0.460936\pi$
$674$	0	0
$675$	26.2429	1.01009
$676$	0	0
$677$	−49.5563	−1.90460	−0.952302	−	0.305157i	$-0.901291\pi$
$677$	−49.5563	−1.90460	−0.952302	+	0.305157i	$0.901291\pi$
$678$	0	0
$679$	2.61955	0.100529
$680$	0	0
$681$	27.2874	1.04566
$682$	0	0
$683$	−4.77352	−0.182653	−0.0913267	−	0.995821i	$-0.529111\pi$
$683$	−4.77352	−0.182653	−0.0913267	+	0.995821i	$0.529111\pi$
$684$	0	0
$685$	0.466343	0.0178180
$686$	0	0
$687$	−31.5687	−1.20442
$688$	0	0
$689$	−17.9889	−0.685322
$690$	0	0
$691$	−39.6672	−1.50901	−0.754506	−	0.656293i	$-0.772123\pi$
$691$	−39.6672	−1.50901	−0.754506	+	0.656293i	$0.772123\pi$
$692$	0	0
$693$	3.18814	0.121107
$694$	0	0
$695$	12.9850	0.492549
$696$	0	0
$697$	−35.4086	−1.34120
$698$	0	0
$699$	24.9668	0.944331
$700$	0	0
$701$	−44.1101	−1.66601	−0.833007	−	0.553263i	$-0.813383\pi$
$701$	−44.1101	−1.66601	−0.833007	+	0.553263i	$0.813383\pi$
$702$	0	0
$703$	−24.5427	−0.925644
$704$	0	0
$705$	−4.61243	−0.173714
$706$	0	0
$707$	−11.2009	−0.421253
$708$	0	0
$709$	14.8323	0.557038	0.278519	−	0.960431i	$-0.410157\pi$
$709$	14.8323	0.557038	0.278519	+	0.960431i	$0.410157\pi$
$710$	0	0
$711$	9.28054	0.348047
$712$	0	0
$713$	−71.0489	−2.66080
$714$	0	0
$715$	9.65260	0.360987
$716$	0	0
$717$	−34.9612	−1.30565
$718$	0	0
$719$	25.3580	0.945695	0.472848	−	0.881144i	$-0.343226\pi$
$719$	25.3580	0.945695	0.472848	+	0.881144i	$0.343226\pi$
$720$	0	0
$721$	−14.2721	−0.531519
$722$	0	0
$723$	−8.70696	−0.323815
$724$	0	0
$725$	39.2979	1.45949
$726$	0	0
$727$	24.7816	0.919099	0.459549	−	0.888152i	$-0.348011\pi$
$727$	24.7816	0.919099	0.459549	+	0.888152i	$0.348011\pi$
$728$	0	0
$729$	30.0385	1.11254
$730$	0	0
$731$	−48.0679	−1.77785
$732$	0	0
$733$	−7.68344	−0.283794	−0.141897	−	0.989881i	$-0.545320\pi$
$733$	−7.68344	−0.283794	−0.141897	+	0.989881i	$0.545320\pi$
$734$	0	0
$735$	4.79667	0.176928
$736$	0	0
$737$	−32.9762	−1.21470
$738$	0	0
$739$	−35.9063	−1.32084	−0.660418	−	0.750899i	$-0.729621\pi$
$739$	−35.9063	−1.32084	−0.660418	+	0.750899i	$0.729621\pi$
$740$	0	0
$741$	−42.7192	−1.56933
$742$	0	0
$743$	30.6140	1.12312	0.561559	−	0.827437i	$-0.310202\pi$
$743$	30.6140	1.12312	0.561559	+	0.827437i	$0.310202\pi$
$744$	0	0
$745$	−1.74713	−0.0640101
$746$	0	0
$747$	3.53729	0.129423
$748$	0	0
$749$	11.3992	0.416517
$750$	0	0
$751$	8.22282	0.300055	0.150028	−	0.988682i	$-0.452064\pi$
$751$	8.22282	0.300055	0.150028	+	0.988682i	$0.452064\pi$
$752$	0	0
$753$	−1.53958	−0.0561053
$754$	0	0
$755$	−10.1730	−0.370235
$756$	0	0
$757$	−22.0360	−0.800911	−0.400456	−	0.916316i	$-0.631148\pi$
$757$	−22.0360	−0.800911	−0.400456	+	0.916316i	$0.631148\pi$
$758$	0	0
$759$	61.8064	2.24343
$760$	0	0
$761$	1.84922	0.0670343	0.0335172	−	0.999438i	$-0.489329\pi$
$761$	1.84922	0.0670343	0.0335172	+	0.999438i	$0.489329\pi$
$762$	0	0
$763$	7.34465	0.265894
$764$	0	0
$765$	−1.60572	−0.0580550
$766$	0	0
$767$	−17.2861	−0.624166
$768$	0	0
$769$	37.4558	1.35069	0.675345	−	0.737502i	$-0.263995\pi$
$769$	37.4558	1.35069	0.675345	+	0.737502i	$0.263995\pi$
$770$	0	0
$771$	37.0567	1.33456
$772$	0	0
$773$	−7.46672	−0.268559	−0.134280	−	0.990943i	$-0.542872\pi$
$773$	−7.46672	−0.268559	−0.134280	+	0.990943i	$0.542872\pi$
$774$	0	0
$775$	−36.2398	−1.30177
$776$	0	0
$777$	6.34963	0.227792
$778$	0	0
$779$	−52.8909	−1.89501
$780$	0	0
$781$	13.9907	0.500627
$782$	0	0
$783$	46.7630	1.67117
$784$	0	0
$785$	−0.592112	−0.0211334
$786$	0	0
$787$	19.9721	0.711930	0.355965	−	0.934499i	$-0.384152\pi$
$787$	19.9721	0.711930	0.355965	+	0.934499i	$0.384152\pi$
$788$	0	0
$789$	12.2262	0.435263
$790$	0	0
$791$	16.3967	0.583000
$792$	0	0
$793$	20.2647	0.719619
$794$	0	0
$795$	3.80180	0.134836
$796$	0	0
$797$	−48.3269	−1.71183	−0.855913	−	0.517120i	$-0.827004\pi$
$797$	−48.3269	−1.71183	−0.855913	+	0.517120i	$0.827004\pi$
$798$	0	0
$799$	−25.1395	−0.889371
$800$	0	0
$801$	−3.92406	−0.138650
$802$	0	0
$803$	−18.6778	−0.659124
$804$	0	0
$805$	5.89307	0.207703
$806$	0	0
$807$	−5.70661	−0.200882
$808$	0	0
$809$	−37.6955	−1.32530	−0.662652	−	0.748927i	$-0.730569\pi$
$809$	−37.6955	−1.32530	−0.662652	+	0.748927i	$0.730569\pi$
$810$	0	0
$811$	−22.3182	−0.783699	−0.391849	−	0.920029i	$-0.628165\pi$
$811$	−22.3182	−0.783699	−0.391849	+	0.920029i	$0.628165\pi$
$812$	0	0
$813$	25.5600	0.896430
$814$	0	0
$815$	6.55371	0.229566
$816$	0	0
$817$	−71.8005	−2.51198
$818$	0	0
$819$	−2.93619	−0.102599
$820$	0	0
$821$	−34.9672	−1.22036	−0.610182	−	0.792262i	$-0.708904\pi$
$821$	−34.9672	−1.22036	−0.610182	+	0.792262i	$0.708904\pi$
$822$	0	0
$823$	36.6074	1.27605	0.638026	−	0.770015i	$-0.279751\pi$
$823$	36.6074	1.27605	0.638026	+	0.770015i	$0.279751\pi$
$824$	0	0
$825$	31.5255	1.09758
$826$	0	0
$827$	−37.2631	−1.29576	−0.647882	−	0.761741i	$-0.724345\pi$
$827$	−37.2631	−1.29576	−0.647882	+	0.761741i	$0.724345\pi$
$828$	0	0
$829$	27.0980	0.941153	0.470576	−	0.882359i	$-0.344046\pi$
$829$	27.0980	0.941153	0.470576	+	0.882359i	$0.344046\pi$
$830$	0	0
$831$	31.1393	1.08021
$832$	0	0
$833$	26.1437	0.905825
$834$	0	0
$835$	−2.05871	−0.0712447
$836$	0	0
$837$	−43.1240	−1.49058
$838$	0	0
$839$	−14.3862	−0.496666	−0.248333	−	0.968675i	$-0.579883\pi$
$839$	−14.3862	−0.496666	−0.248333	+	0.968675i	$0.579883\pi$
$840$	0	0
$841$	41.0261	1.41469
$842$	0	0
$843$	−16.1209	−0.555233
$844$	0	0
$845$	−1.72348	−0.0592896
$846$	0	0
$847$	9.30372	0.319680
$848$	0	0
$849$	−26.6087	−0.913209
$850$	0	0
$851$	−32.7024	−1.12102
$852$	0	0
$853$	−46.3738	−1.58781	−0.793904	−	0.608043i	$-0.791955\pi$
$853$	−46.3738	−1.58781	−0.793904	+	0.608043i	$0.791955\pi$
$854$	0	0
$855$	−2.39851	−0.0820274
$856$	0	0
$857$	−37.5807	−1.28373	−0.641866	−	0.766817i	$-0.721840\pi$
$857$	−37.5807	−1.28373	−0.641866	+	0.766817i	$0.721840\pi$
$858$	0	0
$859$	−0.794126	−0.0270952	−0.0135476	−	0.999908i	$-0.504312\pi$
$859$	−0.794126	−0.0270952	−0.0135476	+	0.999908i	$0.504312\pi$
$860$	0	0
$861$	13.6838	0.466344
$862$	0	0
$863$	−13.0699	−0.444904	−0.222452	−	0.974944i	$-0.571406\pi$
$863$	−13.0699	−0.444904	−0.222452	+	0.974944i	$0.571406\pi$
$864$	0	0
$865$	11.4246	0.388447
$866$	0	0
$867$	6.77014	0.229926
$868$	0	0
$869$	64.2626	2.17996
$870$	0	0
$871$	30.3703	1.02906
$872$	0	0
$873$	−1.42064	−0.0480815
$874$	0	0
$875$	6.20624	0.209809
$876$	0	0
$877$	14.6738	0.495498	0.247749	−	0.968824i	$-0.420309\pi$
$877$	14.6738	0.495498	0.247749	+	0.968824i	$0.420309\pi$
$878$	0	0
$879$	−23.5383	−0.793929
$880$	0	0
$881$	2.96655	0.0999457	0.0499729	−	0.998751i	$-0.484087\pi$
$881$	2.96655	0.0999457	0.0499729	+	0.998751i	$0.484087\pi$
$882$	0	0
$883$	41.3908	1.39291	0.696456	−	0.717600i	$-0.254759\pi$
$883$	41.3908	1.39291	0.696456	+	0.717600i	$0.254759\pi$
$884$	0	0
$885$	3.65327	0.122803
$886$	0	0
$887$	5.28066	0.177307	0.0886535	−	0.996063i	$-0.471744\pi$
$887$	5.28066	0.177307	0.0886535	+	0.996063i	$0.471744\pi$
$888$	0	0
$889$	−5.81315	−0.194967
$890$	0	0
$891$	29.2770	0.980816
$892$	0	0
$893$	−37.5516	−1.25662
$894$	0	0
$895$	−9.87906	−0.330221
$896$	0	0
$897$	−56.9222	−1.90058
$898$	0	0
$899$	−64.5768	−2.15376
$900$	0	0
$901$	20.7212	0.690324
$902$	0	0
$903$	18.5761	0.618173
$904$	0	0
$905$	−8.51179	−0.282941
$906$	0	0
$907$	5.22874	0.173617	0.0868087	−	0.996225i	$-0.472333\pi$
$907$	5.22874	0.173617	0.0868087	+	0.996225i	$0.472333\pi$
$908$	0	0
$909$	6.07451	0.201479
$910$	0	0
$911$	17.3347	0.574324	0.287162	−	0.957882i	$-0.407288\pi$
$911$	17.3347	0.574324	0.287162	+	0.957882i	$0.407288\pi$
$912$	0	0
$913$	24.4937	0.810625
$914$	0	0
$915$	−4.28276	−0.141584
$916$	0	0
$917$	−20.8198	−0.687530
$918$	0	0
$919$	−37.9498	−1.25185	−0.625924	−	0.779884i	$-0.715278\pi$
$919$	−37.9498	−1.25185	−0.625924	+	0.779884i	$0.715278\pi$
$920$	0	0
$921$	43.5242	1.43417
$922$	0	0
$923$	−12.8851	−0.424118
$924$	0	0
$925$	−16.6804	−0.548450
$926$	0	0
$927$	7.74007	0.254217
$928$	0	0
$929$	−38.1618	−1.25205	−0.626024	−	0.779804i	$-0.715319\pi$
$929$	−38.1618	−1.25205	−0.626024	+	0.779804i	$0.715319\pi$
$930$	0	0
$931$	39.0516	1.27987
$932$	0	0
$933$	−40.6493	−1.33080
$934$	0	0
$935$	−11.1187	−0.363621
$936$	0	0
$937$	−24.7076	−0.807163	−0.403582	−	0.914944i	$-0.632235\pi$
$937$	−24.7076	−0.807163	−0.403582	+	0.914944i	$0.632235\pi$
$938$	0	0
$939$	−45.5460	−1.48634
$940$	0	0
$941$	1.63156	0.0531873	0.0265937	−	0.999646i	$-0.491534\pi$
$941$	1.63156	0.0531873	0.0265937	+	0.999646i	$0.491534\pi$
$942$	0	0
$943$	−70.4756	−2.29500
$944$	0	0
$945$	3.57687	0.116356
$946$	0	0
$947$	−0.325926	−0.0105912	−0.00529559	−	0.999986i	$-0.501686\pi$
$947$	−0.325926	−0.0105912	−0.00529559	+	0.999986i	$0.501686\pi$
$948$	0	0
$949$	17.2018	0.558392
$950$	0	0
$951$	47.3944	1.53687
$952$	0	0
$953$	26.3575	0.853804	0.426902	−	0.904298i	$-0.359605\pi$
$953$	26.3575	0.853804	0.426902	+	0.904298i	$0.359605\pi$
$954$	0	0
$955$	1.23214	0.0398712
$956$	0	0
$957$	56.1763	1.81592
$958$	0	0
$959$	−0.982270	−0.0317191
$960$	0	0
$961$	28.5516	0.921021
$962$	0	0
$963$	−6.18204	−0.199214
$964$	0	0
$965$	−9.64727	−0.310557
$966$	0	0
$967$	−56.2795	−1.80983	−0.904914	−	0.425595i	$-0.860065\pi$
$967$	−56.2795	−1.80983	−0.904914	+	0.425595i	$0.860065\pi$
$968$	0	0
$969$	49.2079	1.58078
$970$	0	0
$971$	14.7867	0.474528	0.237264	−	0.971445i	$-0.423749\pi$
$971$	14.7867	0.474528	0.237264	+	0.971445i	$0.423749\pi$
$972$	0	0
$973$	−27.3506	−0.876821
$974$	0	0
$975$	−29.0342	−0.929839
$976$	0	0
$977$	−21.8132	−0.697866	−0.348933	−	0.937148i	$-0.613456\pi$
$977$	−21.8132	−0.697866	−0.348933	+	0.937148i	$0.613456\pi$
$978$	0	0
$979$	−27.1720	−0.868420
$980$	0	0
$981$	−3.98318	−0.127173
$982$	0	0
$983$	48.5738	1.54926	0.774632	−	0.632412i	$-0.217935\pi$
$983$	48.5738	1.54926	0.774632	+	0.632412i	$0.217935\pi$
$984$	0	0
$985$	−1.25988	−0.0401432
$986$	0	0
$987$	9.71528	0.309241
$988$	0	0
$989$	−95.6720	−3.04219
$990$	0	0
$991$	22.9683	0.729611	0.364806	−	0.931084i	$-0.381135\pi$
$991$	22.9683	0.729611	0.364806	+	0.931084i	$0.381135\pi$
$992$	0	0
$993$	−39.5148	−1.25397
$994$	0	0
$995$	0.631577	0.0200223
$996$	0	0
$997$	−15.7685	−0.499393	−0.249697	−	0.968324i	$-0.580331\pi$
$997$	−15.7685	−0.499393	−0.249697	+	0.968324i	$0.580331\pi$
$998$	0	0
$999$	−19.8491	−0.627998

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2008.2.a.c.1.15	✓	19
4.3	odd	2		4016.2.a.l.1.5		19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.2.a.c.1.15	✓	19	1.1	even	1	trivial
4016.2.a.l.1.5		19	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:	\( N \)	\(=\)	\( 2008 = 2^{3} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2008.a (trivial)

\(f(q)\)	\(=\)	\(q+1.53958 q^{3} -0.551255 q^{5} +1.16112 q^{7} -0.629704 q^{9} +O(q^{10})\)	\(q+1.53958 q^{3} -0.551255 q^{5} +1.16112 q^{7} -0.629704 q^{9} -4.36036 q^{11} +4.01578 q^{13} -0.848700 q^{15} -4.62573 q^{17} -6.90960 q^{19} +1.78764 q^{21} -9.20684 q^{23} -4.69612 q^{25} -5.58821 q^{27} -8.36816 q^{29} +7.71697 q^{31} -6.71310 q^{33} -0.640075 q^{35} +3.55197 q^{37} +6.18260 q^{39} +7.65470 q^{41} +10.3914 q^{43} +0.347128 q^{45} +5.43471 q^{47} -5.65179 q^{49} -7.12167 q^{51} -4.47955 q^{53} +2.40367 q^{55} -10.6379 q^{57} -4.30455 q^{59} +5.04626 q^{61} -0.731164 q^{63} -2.21372 q^{65} +7.56274 q^{67} -14.1746 q^{69} -3.20862 q^{71} +4.28354 q^{73} -7.23003 q^{75} -5.06291 q^{77} -14.7379 q^{79} -6.71436 q^{81} -5.61737 q^{83} +2.54996 q^{85} -12.8834 q^{87} +6.23159 q^{89} +4.66281 q^{91} +11.8809 q^{93} +3.80895 q^{95} +2.25605 q^{97} +2.74574 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 6 q^{3} - 8 q^{5} - 11 q^{7} + 21 q^{9}+O(q^{10}) \) 19 * q - 6 * q^3 - 8 * q^5 - 11 * q^7 + 21 * q^9	\( 19 q - 6 q^{3} - 8 q^{5} - 11 q^{7} + 21 q^{9} - 15 q^{11} - 8 q^{13} - 17 q^{15} - 4 q^{17} - 14 q^{19} - 9 q^{21} - 28 q^{23} + 25 q^{25} - 21 q^{27} - 13 q^{29} - 20 q^{31} - 6 q^{33} - 32 q^{35} - 16 q^{37} - 27 q^{39} + 2 q^{41} - 28 q^{43} - 29 q^{45} - 37 q^{47} + 36 q^{49} - 35 q^{51} - 37 q^{53} - 24 q^{55} - 11 q^{57} - 32 q^{59} - 7 q^{61} - 45 q^{63} + q^{65} - 45 q^{67} - 12 q^{69} - 49 q^{71} + 16 q^{73} - 35 q^{75} - 40 q^{77} - 33 q^{79} + 15 q^{81} - 43 q^{83} - 28 q^{85} - 48 q^{87} + 3 q^{89} - 56 q^{91} - 48 q^{93} - 43 q^{95} + 8 q^{97} - 74 q^{99}+O(q^{100}) \) 19 * q - 6 * q^3 - 8 * q^5 - 11 * q^7 + 21 * q^9 - 15 * q^11 - 8 * q^13 - 17 * q^15 - 4 * q^17 - 14 * q^19 - 9 * q^21 - 28 * q^23 + 25 * q^25 - 21 * q^27 - 13 * q^29 - 20 * q^31 - 6 * q^33 - 32 * q^35 - 16 * q^37 - 27 * q^39 + 2 * q^41 - 28 * q^43 - 29 * q^45 - 37 * q^47 + 36 * q^49 - 35 * q^51 - 37 * q^53 - 24 * q^55 - 11 * q^57 - 32 * q^59 - 7 * q^61 - 45 * q^63 + q^65 - 45 * q^67 - 12 * q^69 - 49 * q^71 + 16 * q^73 - 35 * q^75 - 40 * q^77 - 33 * q^79 + 15 * q^81 - 43 * q^83 - 28 * q^85 - 48 * q^87 + 3 * q^89 - 56 * q^91 - 48 * q^93 - 43 * q^95 + 8 * q^97 - 74 * q^99

Introduction

L-functions

Modular forms

Varieties

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