LMFDB - Embedded newform 1336.2.a.a.1.2

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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ 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)

$f(q)$	$=$	$q+2.61803 q^{3} -2.23607 q^{5} -2.38197 q^{7} +3.85410 q^{9} +O(q^{10})$	$q+2.61803 q^{3} -2.23607 q^{5} -2.38197 q^{7} +3.85410 q^{9} -4.00000 q^{11} +3.38197 q^{13} -5.85410 q^{15} -8.09017 q^{17} -8.47214 q^{19} -6.23607 q^{21} +0.854102 q^{23} +2.23607 q^{27} -1.76393 q^{29} +5.23607 q^{31} -10.4721 q^{33} +5.32624 q^{35} +9.94427 q^{37} +8.85410 q^{39} +2.23607 q^{41} +1.47214 q^{43} -8.61803 q^{45} -4.23607 q^{47} -1.32624 q^{49} -21.1803 q^{51} -4.76393 q^{53} +8.94427 q^{55} -22.1803 q^{57} +11.7082 q^{59} -7.76393 q^{61} -9.18034 q^{63} -7.56231 q^{65} +1.76393 q^{67} +2.23607 q^{69} +2.14590 q^{71} +6.09017 q^{73} +9.52786 q^{77} -3.00000 q^{79} -5.70820 q^{81} -15.9443 q^{83} +18.0902 q^{85} -4.61803 q^{87} -3.23607 q^{89} -8.05573 q^{91} +13.7082 q^{93} +18.9443 q^{95} +16.0344 q^{97} -15.4164 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{3} - 7 q^{7} + q^{9}+O(q^{10}) $ 2 * q + 3 * q^3 - 7 * q^7 + q^9	$ 2 q + 3 q^{3} - 7 q^{7} + q^{9} - 8 q^{11} + 9 q^{13} - 5 q^{15} - 5 q^{17} - 8 q^{19} - 8 q^{21} - 5 q^{23} - 8 q^{29} + 6 q^{31} - 12 q^{33} - 5 q^{35} + 2 q^{37} + 11 q^{39} - 6 q^{43} - 15 q^{45} - 4 q^{47} + 13 q^{49} - 20 q^{51} - 14 q^{53} - 22 q^{57} + 10 q^{59} - 20 q^{61} + 4 q^{63} + 5 q^{65} + 8 q^{67} + 11 q^{71} + q^{73} + 28 q^{77} - 6 q^{79} + 2 q^{81} - 14 q^{83} + 25 q^{85} - 7 q^{87} - 2 q^{89} - 34 q^{91} + 14 q^{93} + 20 q^{95} + 3 q^{97} - 4 q^{99}+O(q^{100}) $ 2 * q + 3 * q^3 - 7 * q^7 + q^9 - 8 * q^11 + 9 * q^13 - 5 * q^15 - 5 * q^17 - 8 * q^19 - 8 * q^21 - 5 * q^23 - 8 * q^29 + 6 * q^31 - 12 * q^33 - 5 * q^35 + 2 * q^37 + 11 * q^39 - 6 * q^43 - 15 * q^45 - 4 * q^47 + 13 * q^49 - 20 * q^51 - 14 * q^53 - 22 * q^57 + 10 * q^59 - 20 * q^61 + 4 * q^63 + 5 * q^65 + 8 * q^67 + 11 * q^71 + q^73 + 28 * q^77 - 6 * q^79 + 2 * q^81 - 14 * q^83 + 25 * q^85 - 7 * q^87 - 2 * q^89 - 34 * q^91 + 14 * q^93 + 20 * q^95 + 3 * q^97 - 4 * 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
$3$	2.61803	1.51152	0.755761	−	0.654847i	$-0.227267\pi$
$3$	2.61803	1.51152	0.755761	+	0.654847i	$0.227267\pi$
$4$	0	0
$5$	−2.23607	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$5$	−2.23607	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$6$	0	0
$7$	−2.38197	−0.900299	−0.450149	−	0.892953i	$-0.648629\pi$
$7$	−2.38197	−0.900299	−0.450149	+	0.892953i	$0.648629\pi$
$8$	0	0
$9$	3.85410	1.28470
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	3.38197	0.937989	0.468994	−	0.883201i	$-0.344616\pi$
$13$	3.38197	0.937989	0.468994	+	0.883201i	$0.344616\pi$
$14$	0	0
$15$	−5.85410	−1.51152
$16$	0	0
$17$	−8.09017	−1.96215	−0.981077	−	0.193617i	$-0.937978\pi$
$17$	−8.09017	−1.96215	−0.981077	+	0.193617i	$0.937978\pi$
$18$	0	0
$19$	−8.47214	−1.94364	−0.971821	−	0.235722i	$-0.924255\pi$
$19$	−8.47214	−1.94364	−0.971821	+	0.235722i	$0.924255\pi$
$20$	0	0
$21$	−6.23607	−1.36082
$22$	0	0
$23$	0.854102	0.178093	0.0890463	−	0.996027i	$-0.471618\pi$
$23$	0.854102	0.178093	0.0890463	+	0.996027i	$0.471618\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	2.23607	0.430331
$28$	0	0
$29$	−1.76393	−0.327554	−0.163777	−	0.986497i	$-0.552368\pi$
$29$	−1.76393	−0.327554	−0.163777	+	0.986497i	$0.552368\pi$
$30$	0	0
$31$	5.23607	0.940426	0.470213	−	0.882553i	$-0.344177\pi$
$31$	5.23607	0.940426	0.470213	+	0.882553i	$0.344177\pi$
$32$	0	0
$33$	−10.4721	−1.82296
$34$	0	0
$35$	5.32624	0.900299
$36$	0	0
$37$	9.94427	1.63483	0.817414	−	0.576050i	$-0.195407\pi$
$37$	9.94427	1.63483	0.817414	+	0.576050i	$0.195407\pi$
$38$	0	0
$39$	8.85410	1.41779
$40$	0	0
$41$	2.23607	0.349215	0.174608	−	0.984638i	$-0.444134\pi$
$41$	2.23607	0.349215	0.174608	+	0.984638i	$0.444134\pi$
$42$	0	0
$43$	1.47214	0.224499	0.112249	−	0.993680i	$-0.464194\pi$
$43$	1.47214	0.224499	0.112249	+	0.993680i	$0.464194\pi$
$44$	0	0
$45$	−8.61803	−1.28470
$46$	0	0
$47$	−4.23607	−0.617894	−0.308947	−	0.951079i	$-0.599977\pi$
$47$	−4.23607	−0.617894	−0.308947	+	0.951079i	$0.599977\pi$
$48$	0	0
$49$	−1.32624	−0.189463
$50$	0	0
$51$	−21.1803	−2.96584
$52$	0	0
$53$	−4.76393	−0.654376	−0.327188	−	0.944959i	$-0.606101\pi$
$53$	−4.76393	−0.654376	−0.327188	+	0.944959i	$0.606101\pi$
$54$	0	0
$55$	8.94427	1.20605
$56$	0	0
$57$	−22.1803	−2.93786
$58$	0	0
$59$	11.7082	1.52428	0.762139	−	0.647413i	$-0.224149\pi$
$59$	11.7082	1.52428	0.762139	+	0.647413i	$0.224149\pi$
$60$	0	0
$61$	−7.76393	−0.994070	−0.497035	−	0.867731i	$-0.665578\pi$
$61$	−7.76393	−0.994070	−0.497035	+	0.867731i	$0.665578\pi$
$62$	0	0
$63$	−9.18034	−1.15661
$64$	0	0
$65$	−7.56231	−0.937989
$66$	0	0
$67$	1.76393	0.215499	0.107749	−	0.994178i	$-0.465636\pi$
$67$	1.76393	0.215499	0.107749	+	0.994178i	$0.465636\pi$
$68$	0	0
$69$	2.23607	0.269191
$70$	0	0
$71$	2.14590	0.254671	0.127336	−	0.991860i	$-0.459357\pi$
$71$	2.14590	0.254671	0.127336	+	0.991860i	$0.459357\pi$
$72$	0	0
$73$	6.09017	0.712800	0.356400	−	0.934333i	$-0.384004\pi$
$73$	6.09017	0.712800	0.356400	+	0.934333i	$0.384004\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	9.52786	1.08580
$78$	0	0
$79$	−3.00000	−0.337526	−0.168763	−	0.985657i	$-0.553977\pi$
$79$	−3.00000	−0.337526	−0.168763	+	0.985657i	$0.553977\pi$
$80$	0	0
$81$	−5.70820	−0.634245
$82$	0	0
$83$	−15.9443	−1.75011	−0.875056	−	0.484022i	$-0.839175\pi$
$83$	−15.9443	−1.75011	−0.875056	+	0.484022i	$0.839175\pi$
$84$	0	0
$85$	18.0902	1.96215
$86$	0	0
$87$	−4.61803	−0.495105
$88$	0	0
$89$	−3.23607	−0.343023	−0.171511	−	0.985182i	$-0.554865\pi$
$89$	−3.23607	−0.343023	−0.171511	+	0.985182i	$0.554865\pi$
$90$	0	0
$91$	−8.05573	−0.844470
$92$	0	0
$93$	13.7082	1.42147
$94$	0	0
$95$	18.9443	1.94364
$96$	0	0
$97$	16.0344	1.62805	0.814025	−	0.580829i	$-0.197272\pi$
$97$	16.0344	1.62805	0.814025	+	0.580829i	$0.197272\pi$
$98$	0	0
$99$	−15.4164	−1.54941
$100$	0	0
$101$	−5.94427	−0.591477	−0.295739	−	0.955269i	$-0.595566\pi$
$101$	−5.94427	−0.591477	−0.295739	+	0.955269i	$0.595566\pi$
$102$	0	0
$103$	−10.3262	−1.01747	−0.508737	−	0.860922i	$-0.669888\pi$
$103$	−10.3262	−1.01747	−0.508737	+	0.860922i	$0.669888\pi$
$104$	0	0
$105$	13.9443	1.36082
$106$	0	0
$107$	−13.6525	−1.31983	−0.659917	−	0.751338i	$-0.729409\pi$
$107$	−13.6525	−1.31983	−0.659917	+	0.751338i	$0.729409\pi$
$108$	0	0
$109$	7.52786	0.721039	0.360519	−	0.932752i	$-0.382599\pi$
$109$	7.52786	0.721039	0.360519	+	0.932752i	$0.382599\pi$
$110$	0	0
$111$	26.0344	2.47108
$112$	0	0
$113$	−1.76393	−0.165937	−0.0829684	−	0.996552i	$-0.526440\pi$
$113$	−1.76393	−0.165937	−0.0829684	+	0.996552i	$0.526440\pi$
$114$	0	0
$115$	−1.90983	−0.178093
$116$	0	0
$117$	13.0344	1.20503
$118$	0	0
$119$	19.2705	1.76652
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	5.85410	0.527847
$124$	0	0
$125$	11.1803	1.00000
$126$	0	0
$127$	−18.8885	−1.67609	−0.838044	−	0.545603i	$-0.816301\pi$
$127$	−18.8885	−1.67609	−0.838044	+	0.545603i	$0.816301\pi$
$128$	0	0
$129$	3.85410	0.339335
$130$	0	0
$131$	1.94427	0.169872	0.0849359	−	0.996386i	$-0.472931\pi$
$131$	1.94427	0.169872	0.0849359	+	0.996386i	$0.472931\pi$
$132$	0	0
$133$	20.1803	1.74986
$134$	0	0
$135$	−5.00000	−0.430331
$136$	0	0
$137$	0.944272	0.0806746	0.0403373	−	0.999186i	$-0.487157\pi$
$137$	0.944272	0.0806746	0.0403373	+	0.999186i	$0.487157\pi$
$138$	0	0
$139$	22.0344	1.86894	0.934468	−	0.356046i	$-0.115875\pi$
$139$	22.0344	1.86894	0.934468	+	0.356046i	$0.115875\pi$
$140$	0	0
$141$	−11.0902	−0.933961
$142$	0	0
$143$	−13.5279	−1.13126
$144$	0	0
$145$	3.94427	0.327554
$146$	0	0
$147$	−3.47214	−0.286377
$148$	0	0
$149$	14.8541	1.21690	0.608448	−	0.793594i	$-0.291793\pi$
$149$	14.8541	1.21690	0.608448	+	0.793594i	$0.291793\pi$
$150$	0	0
$151$	−8.85410	−0.720537	−0.360268	−	0.932849i	$-0.617315\pi$
$151$	−8.85410	−0.720537	−0.360268	+	0.932849i	$0.617315\pi$
$152$	0	0
$153$	−31.1803	−2.52078
$154$	0	0
$155$	−11.7082	−0.940426
$156$	0	0
$157$	15.7639	1.25810	0.629049	−	0.777365i	$-0.283444\pi$
$157$	15.7639	1.25810	0.629049	+	0.777365i	$0.283444\pi$
$158$	0	0
$159$	−12.4721	−0.989105
$160$	0	0
$161$	−2.03444	−0.160336
$162$	0	0
$163$	−21.2361	−1.66334	−0.831669	−	0.555272i	$-0.812614\pi$
$163$	−21.2361	−1.66334	−0.831669	+	0.555272i	$0.812614\pi$
$164$	0	0
$165$	23.4164	1.82296
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−1.56231	−0.120177
$170$	0	0
$171$	−32.6525	−2.49700
$172$	0	0
$173$	5.47214	0.416039	0.208019	−	0.978125i	$-0.433298\pi$
$173$	5.47214	0.416039	0.208019	+	0.978125i	$0.433298\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	30.6525	2.30398
$178$	0	0
$179$	−5.79837	−0.433391	−0.216695	−	0.976239i	$-0.569528\pi$
$179$	−5.79837	−0.433391	−0.216695	+	0.976239i	$0.569528\pi$
$180$	0	0
$181$	−22.9443	−1.70543	−0.852717	−	0.522373i	$-0.825047\pi$
$181$	−22.9443	−1.70543	−0.852717	+	0.522373i	$0.825047\pi$
$182$	0	0
$183$	−20.3262	−1.50256
$184$	0	0
$185$	−22.2361	−1.63483
$186$	0	0
$187$	32.3607	2.36645
$188$	0	0
$189$	−5.32624	−0.387427
$190$	0	0
$191$	15.0902	1.09189	0.545943	−	0.837822i	$-0.316171\pi$
$191$	15.0902	1.09189	0.545943	+	0.837822i	$0.316171\pi$
$192$	0	0
$193$	5.38197	0.387402	0.193701	−	0.981061i	$-0.437951\pi$
$193$	5.38197	0.387402	0.193701	+	0.981061i	$0.437951\pi$
$194$	0	0
$195$	−19.7984	−1.41779
$196$	0	0
$197$	1.14590	0.0816419	0.0408209	−	0.999166i	$-0.487003\pi$
$197$	1.14590	0.0816419	0.0408209	+	0.999166i	$0.487003\pi$
$198$	0	0
$199$	10.9098	0.773377	0.386689	−	0.922210i	$-0.373619\pi$
$199$	10.9098	0.773377	0.386689	+	0.922210i	$0.373619\pi$
$200$	0	0
$201$	4.61803	0.325731
$202$	0	0
$203$	4.20163	0.294896
$204$	0	0
$205$	−5.00000	−0.349215
$206$	0	0
$207$	3.29180	0.228796
$208$	0	0
$209$	33.8885	2.34412
$210$	0	0
$211$	−15.1459	−1.04269	−0.521343	−	0.853347i	$-0.674569\pi$
$211$	−15.1459	−1.04269	−0.521343	+	0.853347i	$0.674569\pi$
$212$	0	0
$213$	5.61803	0.384941
$214$	0	0
$215$	−3.29180	−0.224499
$216$	0	0
$217$	−12.4721	−0.846664
$218$	0	0
$219$	15.9443	1.07741
$220$	0	0
$221$	−27.3607	−1.84048
$222$	0	0
$223$	26.5066	1.77501	0.887506	−	0.460796i	$-0.152436\pi$
$223$	26.5066	1.77501	0.887506	+	0.460796i	$0.152436\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−11.0000	−0.730096	−0.365048	−	0.930989i	$-0.618947\pi$
$227$	−11.0000	−0.730096	−0.365048	+	0.930989i	$0.618947\pi$
$228$	0	0
$229$	15.3820	1.01647	0.508234	−	0.861219i	$-0.330298\pi$
$229$	15.3820	1.01647	0.508234	+	0.861219i	$0.330298\pi$
$230$	0	0
$231$	24.9443	1.64121
$232$	0	0
$233$	7.79837	0.510888	0.255444	−	0.966824i	$-0.417778\pi$
$233$	7.79837	0.510888	0.255444	+	0.966824i	$0.417778\pi$
$234$	0	0
$235$	9.47214	0.617894
$236$	0	0
$237$	−7.85410	−0.510179
$238$	0	0
$239$	−10.5623	−0.683219	−0.341609	−	0.939842i	$-0.610972\pi$
$239$	−10.5623	−0.683219	−0.341609	+	0.939842i	$0.610972\pi$
$240$	0	0
$241$	−12.0344	−0.775207	−0.387603	−	0.921826i	$-0.626697\pi$
$241$	−12.0344	−0.775207	−0.387603	+	0.921826i	$0.626697\pi$
$242$	0	0
$243$	−21.6525	−1.38901
$244$	0	0
$245$	2.96556	0.189463
$246$	0	0
$247$	−28.6525	−1.82311
$248$	0	0
$249$	−41.7426	−2.64533
$250$	0	0
$251$	−7.29180	−0.460254	−0.230127	−	0.973161i	$-0.573914\pi$
$251$	−7.29180	−0.460254	−0.230127	+	0.973161i	$0.573914\pi$
$252$	0	0
$253$	−3.41641	−0.214788
$254$	0	0
$255$	47.3607	2.96584
$256$	0	0
$257$	−22.9787	−1.43337	−0.716686	−	0.697396i	$-0.754342\pi$
$257$	−22.9787	−1.43337	−0.716686	+	0.697396i	$0.754342\pi$
$258$	0	0
$259$	−23.6869	−1.47183
$260$	0	0
$261$	−6.79837	−0.420809
$262$	0	0
$263$	−14.4164	−0.888954	−0.444477	−	0.895790i	$-0.646610\pi$
$263$	−14.4164	−0.888954	−0.444477	+	0.895790i	$0.646610\pi$
$264$	0	0
$265$	10.6525	0.654376
$266$	0	0
$267$	−8.47214	−0.518486
$268$	0	0
$269$	22.5066	1.37225	0.686125	−	0.727484i	$-0.259310\pi$
$269$	22.5066	1.37225	0.686125	+	0.727484i	$0.259310\pi$
$270$	0	0
$271$	−12.0000	−0.728948	−0.364474	−	0.931214i	$-0.618751\pi$
$271$	−12.0000	−0.728948	−0.364474	+	0.931214i	$0.618751\pi$
$272$	0	0
$273$	−21.0902	−1.27644
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−19.9787	−1.20040	−0.600202	−	0.799848i	$-0.704913\pi$
$277$	−19.9787	−1.20040	−0.600202	+	0.799848i	$0.704913\pi$
$278$	0	0
$279$	20.1803	1.20817
$280$	0	0
$281$	4.88854	0.291626	0.145813	−	0.989312i	$-0.453420\pi$
$281$	4.88854	0.291626	0.145813	+	0.989312i	$0.453420\pi$
$282$	0	0
$283$	4.94427	0.293906	0.146953	−	0.989143i	$-0.453053\pi$
$283$	4.94427	0.293906	0.146953	+	0.989143i	$0.453053\pi$
$284$	0	0
$285$	49.5967	2.93786
$286$	0	0
$287$	−5.32624	−0.314398
$288$	0	0
$289$	48.4508	2.85005
$290$	0	0
$291$	41.9787	2.46084
$292$	0	0
$293$	14.9443	0.873054	0.436527	−	0.899691i	$-0.356208\pi$
$293$	14.9443	0.873054	0.436527	+	0.899691i	$0.356208\pi$
$294$	0	0
$295$	−26.1803	−1.52428
$296$	0	0
$297$	−8.94427	−0.518999
$298$	0	0
$299$	2.88854	0.167049
$300$	0	0
$301$	−3.50658	−0.202116
$302$	0	0
$303$	−15.5623	−0.894031
$304$	0	0
$305$	17.3607	0.994070
$306$	0	0
$307$	28.0902	1.60319	0.801595	−	0.597867i	$-0.203985\pi$
$307$	28.0902	1.60319	0.801595	+	0.597867i	$0.203985\pi$
$308$	0	0
$309$	−27.0344	−1.53794
$310$	0	0
$311$	−24.4164	−1.38453	−0.692264	−	0.721645i	$-0.743387\pi$
$311$	−24.4164	−1.38453	−0.692264	+	0.721645i	$0.743387\pi$
$312$	0	0
$313$	−22.2148	−1.25565	−0.627827	−	0.778353i	$-0.716055\pi$
$313$	−22.2148	−1.25565	−0.627827	+	0.778353i	$0.716055\pi$
$314$	0	0
$315$	20.5279	1.15661
$316$	0	0
$317$	−21.0344	−1.18141	−0.590706	−	0.806887i	$-0.701151\pi$
$317$	−21.0344	−1.18141	−0.590706	+	0.806887i	$0.701151\pi$
$318$	0	0
$319$	7.05573	0.395045
$320$	0	0
$321$	−35.7426	−1.99496
$322$	0	0
$323$	68.5410	3.81372
$324$	0	0
$325$	0	0
$326$	0	0
$327$	19.7082	1.08987
$328$	0	0
$329$	10.0902	0.556289
$330$	0	0
$331$	−7.00000	−0.384755	−0.192377	−	0.981321i	$-0.561620\pi$
$331$	−7.00000	−0.384755	−0.192377	+	0.981321i	$0.561620\pi$
$332$	0	0
$333$	38.3262	2.10026
$334$	0	0
$335$	−3.94427	−0.215499
$336$	0	0
$337$	25.6525	1.39738	0.698690	−	0.715425i	$-0.253767\pi$
$337$	25.6525	1.39738	0.698690	+	0.715425i	$0.253767\pi$
$338$	0	0
$339$	−4.61803	−0.250817
$340$	0	0
$341$	−20.9443	−1.13420
$342$	0	0
$343$	19.8328	1.07087
$344$	0	0
$345$	−5.00000	−0.269191
$346$	0	0
$347$	−9.38197	−0.503650	−0.251825	−	0.967773i	$-0.581031\pi$
$347$	−9.38197	−0.503650	−0.251825	+	0.967773i	$0.581031\pi$
$348$	0	0
$349$	−31.0000	−1.65939	−0.829696	−	0.558216i	$-0.811486\pi$
$349$	−31.0000	−1.65939	−0.829696	+	0.558216i	$0.811486\pi$
$350$	0	0
$351$	7.56231	0.403646
$352$	0	0
$353$	6.76393	0.360008	0.180004	−	0.983666i	$-0.442389\pi$
$353$	6.76393	0.360008	0.180004	+	0.983666i	$0.442389\pi$
$354$	0	0
$355$	−4.79837	−0.254671
$356$	0	0
$357$	50.4508	2.67014
$358$	0	0
$359$	−6.81966	−0.359928	−0.179964	−	0.983673i	$-0.557598\pi$
$359$	−6.81966	−0.359928	−0.179964	+	0.983673i	$0.557598\pi$
$360$	0	0
$361$	52.7771	2.77774
$362$	0	0
$363$	13.0902	0.687056
$364$	0	0
$365$	−13.6180	−0.712800
$366$	0	0
$367$	20.7082	1.08096	0.540480	−	0.841357i	$-0.318243\pi$
$367$	20.7082	1.08096	0.540480	+	0.841357i	$0.318243\pi$
$368$	0	0
$369$	8.61803	0.448637
$370$	0	0
$371$	11.3475	0.589134
$372$	0	0
$373$	25.7984	1.33579	0.667895	−	0.744256i	$-0.267196\pi$
$373$	25.7984	1.33579	0.667895	+	0.744256i	$0.267196\pi$
$374$	0	0
$375$	29.2705	1.51152
$376$	0	0
$377$	−5.96556	−0.307242
$378$	0	0
$379$	−31.3607	−1.61089	−0.805445	−	0.592671i	$-0.798074\pi$
$379$	−31.3607	−1.61089	−0.805445	+	0.592671i	$0.798074\pi$
$380$	0	0
$381$	−49.4508	−2.53344
$382$	0	0
$383$	7.94427	0.405933	0.202967	−	0.979186i	$-0.434942\pi$
$383$	7.94427	0.405933	0.202967	+	0.979186i	$0.434942\pi$
$384$	0	0
$385$	−21.3050	−1.08580
$386$	0	0
$387$	5.67376	0.288414
$388$	0	0
$389$	6.88854	0.349263	0.174632	−	0.984634i	$-0.444127\pi$
$389$	6.88854	0.349263	0.174632	+	0.984634i	$0.444127\pi$
$390$	0	0
$391$	−6.90983	−0.349445
$392$	0	0
$393$	5.09017	0.256765
$394$	0	0
$395$	6.70820	0.337526
$396$	0	0
$397$	10.7984	0.541955	0.270977	−	0.962586i	$-0.412653\pi$
$397$	10.7984	0.541955	0.270977	+	0.962586i	$0.412653\pi$
$398$	0	0
$399$	52.8328	2.64495
$400$	0	0
$401$	−29.6180	−1.47905	−0.739527	−	0.673127i	$-0.764951\pi$
$401$	−29.6180	−1.47905	−0.739527	+	0.673127i	$0.764951\pi$
$402$	0	0
$403$	17.7082	0.882108
$404$	0	0
$405$	12.7639	0.634245
$406$	0	0
$407$	−39.7771	−1.97168
$408$	0	0
$409$	5.61803	0.277794	0.138897	−	0.990307i	$-0.455644\pi$
$409$	5.61803	0.277794	0.138897	+	0.990307i	$0.455644\pi$
$410$	0	0
$411$	2.47214	0.121941
$412$	0	0
$413$	−27.8885	−1.37231
$414$	0	0
$415$	35.6525	1.75011
$416$	0	0
$417$	57.6869	2.82494
$418$	0	0
$419$	13.4721	0.658157	0.329078	−	0.944303i	$-0.393262\pi$
$419$	13.4721	0.658157	0.329078	+	0.944303i	$0.393262\pi$
$420$	0	0
$421$	30.6525	1.49391	0.746955	−	0.664874i	$-0.231515\pi$
$421$	30.6525	1.49391	0.746955	+	0.664874i	$0.231515\pi$
$422$	0	0
$423$	−16.3262	−0.793809
$424$	0	0
$425$	0	0
$426$	0	0
$427$	18.4934	0.894959
$428$	0	0
$429$	−35.4164	−1.70992
$430$	0	0
$431$	−33.2361	−1.60093	−0.800463	−	0.599383i	$-0.795413\pi$
$431$	−33.2361	−1.60093	−0.800463	+	0.599383i	$0.795413\pi$
$432$	0	0
$433$	−27.7984	−1.33590	−0.667952	−	0.744204i	$-0.732829\pi$
$433$	−27.7984	−1.33590	−0.667952	+	0.744204i	$0.732829\pi$
$434$	0	0
$435$	10.3262	0.495105
$436$	0	0
$437$	−7.23607	−0.346148
$438$	0	0
$439$	−0.124612	−0.00594740	−0.00297370	−	0.999996i	$-0.500947\pi$
$439$	−0.124612	−0.00594740	−0.00297370	+	0.999996i	$0.500947\pi$
$440$	0	0
$441$	−5.11146	−0.243403
$442$	0	0
$443$	−13.2705	−0.630501	−0.315250	−	0.949009i	$-0.602089\pi$
$443$	−13.2705	−0.630501	−0.315250	+	0.949009i	$0.602089\pi$
$444$	0	0
$445$	7.23607	0.343023
$446$	0	0
$447$	38.8885	1.83937
$448$	0	0
$449$	29.0689	1.37185	0.685923	−	0.727674i	$-0.259399\pi$
$449$	29.0689	1.37185	0.685923	+	0.727674i	$0.259399\pi$
$450$	0	0
$451$	−8.94427	−0.421169
$452$	0	0
$453$	−23.1803	−1.08911
$454$	0	0
$455$	18.0132	0.844470
$456$	0	0
$457$	−11.3820	−0.532426	−0.266213	−	0.963914i	$-0.585772\pi$
$457$	−11.3820	−0.532426	−0.266213	+	0.963914i	$0.585772\pi$
$458$	0	0
$459$	−18.0902	−0.844377
$460$	0	0
$461$	−9.90983	−0.461547	−0.230773	−	0.973008i	$-0.574126\pi$
$461$	−9.90983	−0.461547	−0.230773	+	0.973008i	$0.574126\pi$
$462$	0	0
$463$	−20.7082	−0.962392	−0.481196	−	0.876613i	$-0.659797\pi$
$463$	−20.7082	−0.962392	−0.481196	+	0.876613i	$0.659797\pi$
$464$	0	0
$465$	−30.6525	−1.42147
$466$	0	0
$467$	16.4377	0.760646	0.380323	−	0.924854i	$-0.375813\pi$
$467$	16.4377	0.760646	0.380323	+	0.924854i	$0.375813\pi$
$468$	0	0
$469$	−4.20163	−0.194013
$470$	0	0
$471$	41.2705	1.90164
$472$	0	0
$473$	−5.88854	−0.270756
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−18.3607	−0.840678
$478$	0	0
$479$	−28.0000	−1.27935	−0.639676	−	0.768644i	$-0.720932\pi$
$479$	−28.0000	−1.27935	−0.639676	+	0.768644i	$0.720932\pi$
$480$	0	0
$481$	33.6312	1.53345
$482$	0	0
$483$	−5.32624	−0.242352
$484$	0	0
$485$	−35.8541	−1.62805
$486$	0	0
$487$	−23.8328	−1.07997	−0.539984	−	0.841675i	$-0.681570\pi$
$487$	−23.8328	−1.07997	−0.539984	+	0.841675i	$0.681570\pi$
$488$	0	0
$489$	−55.5967	−2.51417
$490$	0	0
$491$	12.8885	0.581652	0.290826	−	0.956776i	$-0.406070\pi$
$491$	12.8885	0.581652	0.290826	+	0.956776i	$0.406070\pi$
$492$	0	0
$493$	14.2705	0.642711
$494$	0	0
$495$	34.4721	1.54941
$496$	0	0
$497$	−5.11146	−0.229280
$498$	0	0
$499$	−3.79837	−0.170039	−0.0850193	−	0.996379i	$-0.527095\pi$
$499$	−3.79837	−0.170039	−0.0850193	+	0.996379i	$0.527095\pi$
$500$	0	0
$501$	−2.61803	−0.116965
$502$	0	0
$503$	−34.0902	−1.52001	−0.760003	−	0.649920i	$-0.774802\pi$
$503$	−34.0902	−1.52001	−0.760003	+	0.649920i	$0.774802\pi$
$504$	0	0
$505$	13.2918	0.591477
$506$	0	0
$507$	−4.09017	−0.181651
$508$	0	0
$509$	1.43769	0.0637247	0.0318623	−	0.999492i	$-0.489856\pi$
$509$	1.43769	0.0637247	0.0318623	+	0.999492i	$0.489856\pi$
$510$	0	0
$511$	−14.5066	−0.641733
$512$	0	0
$513$	−18.9443	−0.836410
$514$	0	0
$515$	23.0902	1.01747
$516$	0	0
$517$	16.9443	0.745208
$518$	0	0
$519$	14.3262	0.628852
$520$	0	0
$521$	2.76393	0.121090	0.0605450	−	0.998165i	$-0.480716\pi$
$521$	2.76393	0.121090	0.0605450	+	0.998165i	$0.480716\pi$
$522$	0	0
$523$	−28.2361	−1.23468	−0.617339	−	0.786698i	$-0.711789\pi$
$523$	−28.2361	−1.23468	−0.617339	+	0.786698i	$0.711789\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−42.3607	−1.84526
$528$	0	0
$529$	−22.2705	−0.968283
$530$	0	0
$531$	45.1246	1.95824
$532$	0	0
$533$	7.56231	0.327560
$534$	0	0
$535$	30.5279	1.31983
$536$	0	0
$537$	−15.1803	−0.655080
$538$	0	0
$539$	5.30495	0.228500
$540$	0	0
$541$	21.9443	0.943458	0.471729	−	0.881744i	$-0.343630\pi$
$541$	21.9443	0.943458	0.471729	+	0.881744i	$0.343630\pi$
$542$	0	0
$543$	−60.0689	−2.57780
$544$	0	0
$545$	−16.8328	−0.721039
$546$	0	0
$547$	4.38197	0.187359	0.0936797	−	0.995602i	$-0.470137\pi$
$547$	4.38197	0.187359	0.0936797	+	0.995602i	$0.470137\pi$
$548$	0	0
$549$	−29.9230	−1.27708
$550$	0	0
$551$	14.9443	0.636647
$552$	0	0
$553$	7.14590	0.303874
$554$	0	0
$555$	−58.2148	−2.47108
$556$	0	0
$557$	18.7082	0.792692	0.396346	−	0.918101i	$-0.370278\pi$
$557$	18.7082	0.792692	0.396346	+	0.918101i	$0.370278\pi$
$558$	0	0
$559$	4.97871	0.210577
$560$	0	0
$561$	84.7214	3.57694
$562$	0	0
$563$	−2.09017	−0.0880902	−0.0440451	−	0.999030i	$-0.514025\pi$
$563$	−2.09017	−0.0880902	−0.0440451	+	0.999030i	$0.514025\pi$
$564$	0	0
$565$	3.94427	0.165937
$566$	0	0
$567$	13.5967	0.571010
$568$	0	0
$569$	−28.2361	−1.18372	−0.591859	−	0.806042i	$-0.701606\pi$
$569$	−28.2361	−1.18372	−0.591859	+	0.806042i	$0.701606\pi$
$570$	0	0
$571$	20.6738	0.865170	0.432585	−	0.901593i	$-0.357602\pi$
$571$	20.6738	0.865170	0.432585	+	0.901593i	$0.357602\pi$
$572$	0	0
$573$	39.5066	1.65041
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−31.0902	−1.29430	−0.647150	−	0.762362i	$-0.724039\pi$
$577$	−31.0902	−1.29430	−0.647150	+	0.762362i	$0.724039\pi$
$578$	0	0
$579$	14.0902	0.585567
$580$	0	0
$581$	37.9787	1.57562
$582$	0	0
$583$	19.0557	0.789208
$584$	0	0
$585$	−29.1459	−1.20503
$586$	0	0
$587$	−5.79837	−0.239325	−0.119662	−	0.992815i	$-0.538181\pi$
$587$	−5.79837	−0.239325	−0.119662	+	0.992815i	$0.538181\pi$
$588$	0	0
$589$	−44.3607	−1.82785
$590$	0	0
$591$	3.00000	0.123404
$592$	0	0
$593$	−1.00000	−0.0410651	−0.0205325	−	0.999789i	$-0.506536\pi$
$593$	−1.00000	−0.0410651	−0.0205325	+	0.999789i	$0.506536\pi$
$594$	0	0
$595$	−43.0902	−1.76652
$596$	0	0
$597$	28.5623	1.16898
$598$	0	0
$599$	−38.2148	−1.56141	−0.780707	−	0.624897i	$-0.785141\pi$
$599$	−38.2148	−1.56141	−0.780707	+	0.624897i	$0.785141\pi$
$600$	0	0
$601$	2.96556	0.120968	0.0604838	−	0.998169i	$-0.480736\pi$
$601$	2.96556	0.120968	0.0604838	+	0.998169i	$0.480736\pi$
$602$	0	0
$603$	6.79837	0.276851
$604$	0	0
$605$	−11.1803	−0.454545
$606$	0	0
$607$	−5.09017	−0.206604	−0.103302	−	0.994650i	$-0.532941\pi$
$607$	−5.09017	−0.206604	−0.103302	+	0.994650i	$0.532941\pi$
$608$	0	0
$609$	11.0000	0.445742
$610$	0	0
$611$	−14.3262	−0.579578
$612$	0	0
$613$	9.67376	0.390720	0.195360	−	0.980732i	$-0.437413\pi$
$613$	9.67376	0.390720	0.195360	+	0.980732i	$0.437413\pi$
$614$	0	0
$615$	−13.0902	−0.527847
$616$	0	0
$617$	−42.1246	−1.69587	−0.847937	−	0.530098i	$-0.822155\pi$
$617$	−42.1246	−1.69587	−0.847937	+	0.530098i	$0.822155\pi$
$618$	0	0
$619$	18.5410	0.745227	0.372613	−	0.927987i	$-0.378462\pi$
$619$	18.5410	0.745227	0.372613	+	0.927987i	$0.378462\pi$
$620$	0	0
$621$	1.90983	0.0766388
$622$	0	0
$623$	7.70820	0.308823
$624$	0	0
$625$	−25.0000	−1.00000
$626$	0	0
$627$	88.7214	3.54319
$628$	0	0
$629$	−80.4508	−3.20779
$630$	0	0
$631$	13.1459	0.523330	0.261665	−	0.965159i	$-0.415728\pi$
$631$	13.1459	0.523330	0.261665	+	0.965159i	$0.415728\pi$
$632$	0	0
$633$	−39.6525	−1.57604
$634$	0	0
$635$	42.2361	1.67609
$636$	0	0
$637$	−4.48529	−0.177714
$638$	0	0
$639$	8.27051	0.327176
$640$	0	0
$641$	−47.5623	−1.87860	−0.939299	−	0.343099i	$-0.888523\pi$
$641$	−47.5623	−1.87860	−0.939299	+	0.343099i	$0.888523\pi$
$642$	0	0
$643$	−26.9098	−1.06122	−0.530610	−	0.847616i	$-0.678037\pi$
$643$	−26.9098	−1.06122	−0.530610	+	0.847616i	$0.678037\pi$
$644$	0	0
$645$	−8.61803	−0.339335
$646$	0	0
$647$	−15.0344	−0.591065	−0.295532	−	0.955333i	$-0.595497\pi$
$647$	−15.0344	−0.591065	−0.295532	+	0.955333i	$0.595497\pi$
$648$	0	0
$649$	−46.8328	−1.83835
$650$	0	0
$651$	−32.6525	−1.27975
$652$	0	0
$653$	−26.3262	−1.03023	−0.515113	−	0.857122i	$-0.672250\pi$
$653$	−26.3262	−1.03023	−0.515113	+	0.857122i	$0.672250\pi$
$654$	0	0
$655$	−4.34752	−0.169872
$656$	0	0
$657$	23.4721	0.915735
$658$	0	0
$659$	33.0132	1.28601	0.643005	−	0.765862i	$-0.277687\pi$
$659$	33.0132	1.28601	0.643005	+	0.765862i	$0.277687\pi$
$660$	0	0
$661$	27.4377	1.06720	0.533601	−	0.845736i	$-0.320838\pi$
$661$	27.4377	1.06720	0.533601	+	0.845736i	$0.320838\pi$
$662$	0	0
$663$	−71.6312	−2.78192
$664$	0	0
$665$	−45.1246	−1.74986
$666$	0	0
$667$	−1.50658	−0.0583349
$668$	0	0
$669$	69.3951	2.68297
$670$	0	0
$671$	31.0557	1.19889
$672$	0	0
$673$	18.0000	0.693849	0.346925	−	0.937893i	$-0.387226\pi$
$673$	18.0000	0.693849	0.346925	+	0.937893i	$0.387226\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	18.6525	0.716873	0.358436	−	0.933554i	$-0.383310\pi$
$677$	18.6525	0.716873	0.358436	+	0.933554i	$0.383310\pi$
$678$	0	0
$679$	−38.1935	−1.46573
$680$	0	0
$681$	−28.7984	−1.10356
$682$	0	0
$683$	−7.56231	−0.289364	−0.144682	−	0.989478i	$-0.546216\pi$
$683$	−7.56231	−0.289364	−0.144682	+	0.989478i	$0.546216\pi$
$684$	0	0
$685$	−2.11146	−0.0806746
$686$	0	0
$687$	40.2705	1.53642
$688$	0	0
$689$	−16.1115	−0.613798
$690$	0	0
$691$	−20.8885	−0.794638	−0.397319	−	0.917681i	$-0.630059\pi$
$691$	−20.8885	−0.794638	−0.397319	+	0.917681i	$0.630059\pi$
$692$	0	0
$693$	36.7214	1.39493
$694$	0	0
$695$	−49.2705	−1.86894
$696$	0	0
$697$	−18.0902	−0.685214
$698$	0	0
$699$	20.4164	0.772219
$700$	0	0
$701$	14.1246	0.533479	0.266740	−	0.963769i	$-0.414054\pi$
$701$	14.1246	0.533479	0.266740	+	0.963769i	$0.414054\pi$
$702$	0	0
$703$	−84.2492	−3.17752
$704$	0	0
$705$	24.7984	0.933961
$706$	0	0
$707$	14.1591	0.532506
$708$	0	0
$709$	13.2016	0.495797	0.247899	−	0.968786i	$-0.420260\pi$
$709$	13.2016	0.495797	0.247899	+	0.968786i	$0.420260\pi$
$710$	0	0
$711$	−11.5623	−0.433620
$712$	0	0
$713$	4.47214	0.167483
$714$	0	0
$715$	30.2492	1.13126
$716$	0	0
$717$	−27.6525	−1.03270
$718$	0	0
$719$	−10.4377	−0.389260	−0.194630	−	0.980877i	$-0.562351\pi$
$719$	−10.4377	−0.389260	−0.194630	+	0.980877i	$0.562351\pi$
$720$	0	0
$721$	24.5967	0.916031
$722$	0	0
$723$	−31.5066	−1.17174
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−11.9098	−0.441711	−0.220856	−	0.975307i	$-0.570885\pi$
$727$	−11.9098	−0.441711	−0.220856	+	0.975307i	$0.570885\pi$
$728$	0	0
$729$	−39.5623	−1.46527
$730$	0	0
$731$	−11.9098	−0.440501
$732$	0	0
$733$	−18.9443	−0.699723	−0.349861	−	0.936802i	$-0.613771\pi$
$733$	−18.9443	−0.699723	−0.349861	+	0.936802i	$0.613771\pi$
$734$	0	0
$735$	7.76393	0.286377
$736$	0	0
$737$	−7.05573	−0.259901
$738$	0	0
$739$	25.6869	0.944909	0.472454	−	0.881355i	$-0.343368\pi$
$739$	25.6869	0.944909	0.472454	+	0.881355i	$0.343368\pi$
$740$	0	0
$741$	−75.0132	−2.75568
$742$	0	0
$743$	−32.3820	−1.18798	−0.593990	−	0.804473i	$-0.702448\pi$
$743$	−32.3820	−1.18798	−0.593990	+	0.804473i	$0.702448\pi$
$744$	0	0
$745$	−33.2148	−1.21690
$746$	0	0
$747$	−61.4508	−2.24837
$748$	0	0
$749$	32.5197	1.18825
$750$	0	0
$751$	−20.4164	−0.745005	−0.372503	−	0.928031i	$-0.621500\pi$
$751$	−20.4164	−0.745005	−0.372503	+	0.928031i	$0.621500\pi$
$752$	0	0
$753$	−19.0902	−0.695684
$754$	0	0
$755$	19.7984	0.720537
$756$	0	0
$757$	31.5066	1.14513	0.572563	−	0.819861i	$-0.305949\pi$
$757$	31.5066	1.14513	0.572563	+	0.819861i	$0.305949\pi$
$758$	0	0
$759$	−8.94427	−0.324657
$760$	0	0
$761$	−15.6869	−0.568650	−0.284325	−	0.958728i	$-0.591770\pi$
$761$	−15.6869	−0.568650	−0.284325	+	0.958728i	$0.591770\pi$
$762$	0	0
$763$	−17.9311	−0.649150
$764$	0	0
$765$	69.7214	2.52078
$766$	0	0
$767$	39.5967	1.42976
$768$	0	0
$769$	1.83282	0.0660930	0.0330465	−	0.999454i	$-0.489479\pi$
$769$	1.83282	0.0660930	0.0330465	+	0.999454i	$0.489479\pi$
$770$	0	0
$771$	−60.1591	−2.16658
$772$	0	0
$773$	52.0689	1.87279	0.936394	−	0.350951i	$-0.114142\pi$
$773$	52.0689	1.87279	0.936394	+	0.350951i	$0.114142\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−62.0132	−2.22471
$778$	0	0
$779$	−18.9443	−0.678749
$780$	0	0
$781$	−8.58359	−0.307145
$782$	0	0
$783$	−3.94427	−0.140957
$784$	0	0
$785$	−35.2492	−1.25810
$786$	0	0
$787$	14.1803	0.505475	0.252737	−	0.967535i	$-0.418669\pi$
$787$	14.1803	0.505475	0.252737	+	0.967535i	$0.418669\pi$
$788$	0	0
$789$	−37.7426	−1.34367
$790$	0	0
$791$	4.20163	0.149393
$792$	0	0
$793$	−26.2574	−0.932426
$794$	0	0
$795$	27.8885	0.989105
$796$	0	0
$797$	−22.2361	−0.787642	−0.393821	−	0.919187i	$-0.628847\pi$
$797$	−22.2361	−0.787642	−0.393821	+	0.919187i	$0.628847\pi$
$798$	0	0
$799$	34.2705	1.21240
$800$	0	0
$801$	−12.4721	−0.440681
$802$	0	0
$803$	−24.3607	−0.859670
$804$	0	0
$805$	4.54915	0.160336
$806$	0	0
$807$	58.9230	2.07419
$808$	0	0
$809$	44.5066	1.56477	0.782384	−	0.622796i	$-0.214003\pi$
$809$	44.5066	1.56477	0.782384	+	0.622796i	$0.214003\pi$
$810$	0	0
$811$	−10.8197	−0.379930	−0.189965	−	0.981791i	$-0.560837\pi$
$811$	−10.8197	−0.379930	−0.189965	+	0.981791i	$0.560837\pi$
$812$	0	0
$813$	−31.4164	−1.10182
$814$	0	0
$815$	47.4853	1.66334
$816$	0	0
$817$	−12.4721	−0.436345
$818$	0	0
$819$	−31.0476	−1.08489
$820$	0	0
$821$	−9.14590	−0.319194	−0.159597	−	0.987182i	$-0.551019\pi$
$821$	−9.14590	−0.319194	−0.159597	+	0.987182i	$0.551019\pi$
$822$	0	0
$823$	−23.9656	−0.835387	−0.417693	−	0.908588i	$-0.637161\pi$
$823$	−23.9656	−0.835387	−0.417693	+	0.908588i	$0.637161\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	34.5623	1.20185	0.600925	−	0.799306i	$-0.294799\pi$
$827$	34.5623	1.20185	0.600925	+	0.799306i	$0.294799\pi$
$828$	0	0
$829$	6.76393	0.234921	0.117461	−	0.993078i	$-0.462525\pi$
$829$	6.76393	0.234921	0.117461	+	0.993078i	$0.462525\pi$
$830$	0	0
$831$	−52.3050	−1.81444
$832$	0	0
$833$	10.7295	0.371755
$834$	0	0
$835$	2.23607	0.0773823
$836$	0	0
$837$	11.7082	0.404695
$838$	0	0
$839$	41.2918	1.42555	0.712776	−	0.701392i	$-0.247438\pi$
$839$	41.2918	1.42555	0.712776	+	0.701392i	$0.247438\pi$
$840$	0	0
$841$	−25.8885	−0.892708
$842$	0	0
$843$	12.7984	0.440799
$844$	0	0
$845$	3.49342	0.120177
$846$	0	0
$847$	−11.9098	−0.409227
$848$	0	0
$849$	12.9443	0.444246
$850$	0	0
$851$	8.49342	0.291151
$852$	0	0
$853$	54.4508	1.86436	0.932181	−	0.361993i	$-0.117904\pi$
$853$	54.4508	1.86436	0.932181	+	0.361993i	$0.117904\pi$
$854$	0	0
$855$	73.0132	2.49700
$856$	0	0
$857$	6.21478	0.212293	0.106146	−	0.994351i	$-0.466149\pi$
$857$	6.21478	0.212293	0.106146	+	0.994351i	$0.466149\pi$
$858$	0	0
$859$	−23.7082	−0.808913	−0.404457	−	0.914557i	$-0.632539\pi$
$859$	−23.7082	−0.808913	−0.404457	+	0.914557i	$0.632539\pi$
$860$	0	0
$861$	−13.9443	−0.475220
$862$	0	0
$863$	−51.5755	−1.75565	−0.877825	−	0.478982i	$-0.841006\pi$
$863$	−51.5755	−1.75565	−0.877825	+	0.478982i	$0.841006\pi$
$864$	0	0
$865$	−12.2361	−0.416039
$866$	0	0
$867$	126.846	4.30792
$868$	0	0
$869$	12.0000	0.407072
$870$	0	0
$871$	5.96556	0.202135
$872$	0	0
$873$	61.7984	2.09156
$874$	0	0
$875$	−26.6312	−0.900299
$876$	0	0
$877$	−22.6312	−0.764201	−0.382100	−	0.924121i	$-0.624799\pi$
$877$	−22.6312	−0.764201	−0.382100	+	0.924121i	$0.624799\pi$
$878$	0	0
$879$	39.1246	1.31964
$880$	0	0
$881$	22.8328	0.769257	0.384629	−	0.923071i	$-0.374330\pi$
$881$	22.8328	0.769257	0.384629	+	0.923071i	$0.374330\pi$
$882$	0	0
$883$	10.2016	0.343312	0.171656	−	0.985157i	$-0.445088\pi$
$883$	10.2016	0.343312	0.171656	+	0.985157i	$0.445088\pi$
$884$	0	0
$885$	−68.5410	−2.30398
$886$	0	0
$887$	22.7771	0.764780	0.382390	−	0.924001i	$-0.375101\pi$
$887$	22.7771	0.764780	0.382390	+	0.924001i	$0.375101\pi$
$888$	0	0
$889$	44.9919	1.50898
$890$	0	0
$891$	22.8328	0.764928
$892$	0	0
$893$	35.8885	1.20096
$894$	0	0
$895$	12.9656	0.433391
$896$	0	0
$897$	7.56231	0.252498
$898$	0	0
$899$	−9.23607	−0.308040
$900$	0	0
$901$	38.5410	1.28399
$902$	0	0
$903$	−9.18034	−0.305503
$904$	0	0
$905$	51.3050	1.70543
$906$	0	0
$907$	38.3050	1.27190	0.635948	−	0.771732i	$-0.280609\pi$
$907$	38.3050	1.27190	0.635948	+	0.771732i	$0.280609\pi$
$908$	0	0
$909$	−22.9098	−0.759871
$910$	0	0
$911$	46.5967	1.54382	0.771910	−	0.635732i	$-0.219302\pi$
$911$	46.5967	1.54382	0.771910	+	0.635732i	$0.219302\pi$
$912$	0	0
$913$	63.7771	2.11071
$914$	0	0
$915$	45.4508	1.50256
$916$	0	0
$917$	−4.63119	−0.152935
$918$	0	0
$919$	−33.3820	−1.10117	−0.550585	−	0.834779i	$-0.685595\pi$
$919$	−33.3820	−1.10117	−0.550585	+	0.834779i	$0.685595\pi$
$920$	0	0
$921$	73.5410	2.42326
$922$	0	0
$923$	7.25735	0.238879
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−39.7984	−1.30715
$928$	0	0
$929$	27.1591	0.891060	0.445530	−	0.895267i	$-0.353015\pi$
$929$	27.1591	0.891060	0.445530	+	0.895267i	$0.353015\pi$
$930$	0	0
$931$	11.2361	0.368247
$932$	0	0
$933$	−63.9230	−2.09274
$934$	0	0
$935$	−72.3607	−2.36645
$936$	0	0
$937$	−0.291796	−0.00953256	−0.00476628	−	0.999989i	$-0.501517\pi$
$937$	−0.291796	−0.00953256	−0.00476628	+	0.999989i	$0.501517\pi$
$938$	0	0
$939$	−58.1591	−1.89795
$940$	0	0
$941$	−2.88854	−0.0941638	−0.0470819	−	0.998891i	$-0.514992\pi$
$941$	−2.88854	−0.0941638	−0.0470819	+	0.998891i	$0.514992\pi$
$942$	0	0
$943$	1.90983	0.0621926
$944$	0	0
$945$	11.9098	0.387427
$946$	0	0
$947$	−11.0902	−0.360382	−0.180191	−	0.983632i	$-0.557672\pi$
$947$	−11.0902	−0.360382	−0.180191	+	0.983632i	$0.557672\pi$
$948$	0	0
$949$	20.5967	0.668599
$950$	0	0
$951$	−55.0689	−1.78573
$952$	0	0
$953$	18.4934	0.599061	0.299530	−	0.954087i	$-0.403170\pi$
$953$	18.4934	0.599061	0.299530	+	0.954087i	$0.403170\pi$
$954$	0	0
$955$	−33.7426	−1.09189
$956$	0	0
$957$	18.4721	0.597119
$958$	0	0
$959$	−2.24922	−0.0726312
$960$	0	0
$961$	−3.58359	−0.115600
$962$	0	0
$963$	−52.6180	−1.69559
$964$	0	0
$965$	−12.0344	−0.387402
$966$	0	0
$967$	55.8541	1.79615	0.898073	−	0.439846i	$-0.144967\pi$
$967$	55.8541	1.79615	0.898073	+	0.439846i	$0.144967\pi$
$968$	0	0
$969$	179.443	5.76453
$970$	0	0
$971$	50.5967	1.62373	0.811863	−	0.583847i	$-0.198453\pi$
$971$	50.5967	1.62373	0.811863	+	0.583847i	$0.198453\pi$
$972$	0	0
$973$	−52.4853	−1.68260
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−30.9787	−0.991097	−0.495548	−	0.868580i	$-0.665033\pi$
$977$	−30.9787	−0.991097	−0.495548	+	0.868580i	$0.665033\pi$
$978$	0	0
$979$	12.9443	0.413701
$980$	0	0
$981$	29.0132	0.926319
$982$	0	0
$983$	5.16718	0.164808	0.0824038	−	0.996599i	$-0.473740\pi$
$983$	5.16718	0.164808	0.0824038	+	0.996599i	$0.473740\pi$
$984$	0	0
$985$	−2.56231	−0.0816419
$986$	0	0
$987$	26.4164	0.840844
$988$	0	0
$989$	1.25735	0.0399815
$990$	0	0
$991$	−16.9443	−0.538253	−0.269126	−	0.963105i	$-0.586735\pi$
$991$	−16.9443	−0.538253	−0.269126	+	0.963105i	$0.586735\pi$
$992$	0	0
$993$	−18.3262	−0.581566
$994$	0	0
$995$	−24.3951	−0.773377
$996$	0	0
$997$	51.6656	1.63627	0.818133	−	0.575028i	$-0.195009\pi$
$997$	51.6656	1.63627	0.818133	+	0.575028i	$0.195009\pi$
$998$	0	0
$999$	22.2361	0.703518

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1336.2.a.a.1.2	✓	2
4.3	odd	2		2672.2.a.b.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1336.2.a.a.1.2	✓	2	1.1	even	1	trivial
2672.2.a.b.1.1		2	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)

\(f(q)\)	\(=\)	\(q+2.61803 q^{3} -2.23607 q^{5} -2.38197 q^{7} +3.85410 q^{9} +O(q^{10})\)	\(q+2.61803 q^{3} -2.23607 q^{5} -2.38197 q^{7} +3.85410 q^{9} -4.00000 q^{11} +3.38197 q^{13} -5.85410 q^{15} -8.09017 q^{17} -8.47214 q^{19} -6.23607 q^{21} +0.854102 q^{23} +2.23607 q^{27} -1.76393 q^{29} +5.23607 q^{31} -10.4721 q^{33} +5.32624 q^{35} +9.94427 q^{37} +8.85410 q^{39} +2.23607 q^{41} +1.47214 q^{43} -8.61803 q^{45} -4.23607 q^{47} -1.32624 q^{49} -21.1803 q^{51} -4.76393 q^{53} +8.94427 q^{55} -22.1803 q^{57} +11.7082 q^{59} -7.76393 q^{61} -9.18034 q^{63} -7.56231 q^{65} +1.76393 q^{67} +2.23607 q^{69} +2.14590 q^{71} +6.09017 q^{73} +9.52786 q^{77} -3.00000 q^{79} -5.70820 q^{81} -15.9443 q^{83} +18.0902 q^{85} -4.61803 q^{87} -3.23607 q^{89} -8.05573 q^{91} +13.7082 q^{93} +18.9443 q^{95} +16.0344 q^{97} -15.4164 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{3} - 7 q^{7} + q^{9}+O(q^{10}) \) 2 * q + 3 * q^3 - 7 * q^7 + q^9	\( 2 q + 3 q^{3} - 7 q^{7} + q^{9} - 8 q^{11} + 9 q^{13} - 5 q^{15} - 5 q^{17} - 8 q^{19} - 8 q^{21} - 5 q^{23} - 8 q^{29} + 6 q^{31} - 12 q^{33} - 5 q^{35} + 2 q^{37} + 11 q^{39} - 6 q^{43} - 15 q^{45} - 4 q^{47} + 13 q^{49} - 20 q^{51} - 14 q^{53} - 22 q^{57} + 10 q^{59} - 20 q^{61} + 4 q^{63} + 5 q^{65} + 8 q^{67} + 11 q^{71} + q^{73} + 28 q^{77} - 6 q^{79} + 2 q^{81} - 14 q^{83} + 25 q^{85} - 7 q^{87} - 2 q^{89} - 34 q^{91} + 14 q^{93} + 20 q^{95} + 3 q^{97} - 4 q^{99}+O(q^{100}) \) 2 * q + 3 * q^3 - 7 * q^7 + q^9 - 8 * q^11 + 9 * q^13 - 5 * q^15 - 5 * q^17 - 8 * q^19 - 8 * q^21 - 5 * q^23 - 8 * q^29 + 6 * q^31 - 12 * q^33 - 5 * q^35 + 2 * q^37 + 11 * q^39 - 6 * q^43 - 15 * q^45 - 4 * q^47 + 13 * q^49 - 20 * q^51 - 14 * q^53 - 22 * q^57 + 10 * q^59 - 20 * q^61 + 4 * q^63 + 5 * q^65 + 8 * q^67 + 11 * q^71 + q^73 + 28 * q^77 - 6 * q^79 + 2 * q^81 - 14 * q^83 + 25 * q^85 - 7 * q^87 - 2 * q^89 - 34 * q^91 + 14 * q^93 + 20 * q^95 + 3 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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