LMFDB - Embedded newform 2151.2.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2151,2,Mod(1,2151)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2151, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2151.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2151 = 3^{2} \cdot 239 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2151.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:	$17.1758214748$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 717)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	2151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.618034 q^{2} -1.61803 q^{4} +1.00000 q^{5} -2.23607 q^{8} +O(q^{10})$	$q+0.618034 q^{2} -1.61803 q^{4} +1.00000 q^{5} -2.23607 q^{8} +0.618034 q^{10} -4.23607 q^{11} -0.763932 q^{13} +1.85410 q^{16} +5.00000 q^{17} +5.23607 q^{19} -1.61803 q^{20} -2.61803 q^{22} +6.47214 q^{23} -4.00000 q^{25} -0.472136 q^{26} -7.47214 q^{29} -8.70820 q^{31} +5.61803 q^{32} +3.09017 q^{34} -6.00000 q^{37} +3.23607 q^{38} -2.23607 q^{40} +1.70820 q^{41} -10.0000 q^{43} +6.85410 q^{44} +4.00000 q^{46} -7.70820 q^{47} -7.00000 q^{49} -2.47214 q^{50} +1.23607 q^{52} -2.47214 q^{53} -4.23607 q^{55} -4.61803 q^{58} -3.23607 q^{59} -3.94427 q^{61} -5.38197 q^{62} -0.236068 q^{64} -0.763932 q^{65} -6.23607 q^{67} -8.09017 q^{68} +4.94427 q^{71} -2.94427 q^{73} -3.70820 q^{74} -8.47214 q^{76} +2.94427 q^{79} +1.85410 q^{80} +1.05573 q^{82} +4.23607 q^{83} +5.00000 q^{85} -6.18034 q^{86} +9.47214 q^{88} -12.9443 q^{89} -10.4721 q^{92} -4.76393 q^{94} +5.23607 q^{95} -8.18034 q^{97} -4.32624 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - q^{4} + 2 q^{5}+O(q^{10}) $ 2 * q - q^2 - q^4 + 2 * q^5	$ 2 q - q^{2} - q^{4} + 2 q^{5} - q^{10} - 4 q^{11} - 6 q^{13} - 3 q^{16} + 10 q^{17} + 6 q^{19} - q^{20} - 3 q^{22} + 4 q^{23} - 8 q^{25} + 8 q^{26} - 6 q^{29} - 4 q^{31} + 9 q^{32} - 5 q^{34} - 12 q^{37} + 2 q^{38} - 10 q^{41} - 20 q^{43} + 7 q^{44} + 8 q^{46} - 2 q^{47} - 14 q^{49} + 4 q^{50} - 2 q^{52} + 4 q^{53} - 4 q^{55} - 7 q^{58} - 2 q^{59} + 10 q^{61} - 13 q^{62} + 4 q^{64} - 6 q^{65} - 8 q^{67} - 5 q^{68} - 8 q^{71} + 12 q^{73} + 6 q^{74} - 8 q^{76} - 12 q^{79} - 3 q^{80} + 20 q^{82} + 4 q^{83} + 10 q^{85} + 10 q^{86} + 10 q^{88} - 8 q^{89} - 12 q^{92} - 14 q^{94} + 6 q^{95} + 6 q^{97} + 7 q^{98}+O(q^{100}) $ 2 * q - q^2 - q^4 + 2 * q^5 - q^10 - 4 * q^11 - 6 * q^13 - 3 * q^16 + 10 * q^17 + 6 * q^19 - q^20 - 3 * q^22 + 4 * q^23 - 8 * q^25 + 8 * q^26 - 6 * q^29 - 4 * q^31 + 9 * q^32 - 5 * q^34 - 12 * q^37 + 2 * q^38 - 10 * q^41 - 20 * q^43 + 7 * q^44 + 8 * q^46 - 2 * q^47 - 14 * q^49 + 4 * q^50 - 2 * q^52 + 4 * q^53 - 4 * q^55 - 7 * q^58 - 2 * q^59 + 10 * q^61 - 13 * q^62 + 4 * q^64 - 6 * q^65 - 8 * q^67 - 5 * q^68 - 8 * q^71 + 12 * q^73 + 6 * q^74 - 8 * q^76 - 12 * q^79 - 3 * q^80 + 20 * q^82 + 4 * q^83 + 10 * q^85 + 10 * q^86 + 10 * q^88 - 8 * q^89 - 12 * q^92 - 14 * q^94 + 6 * q^95 + 6 * q^97 + 7 * q^98

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 $

$2$	0.618034	0.437016	0.218508	−	0.975835i	$-0.429881\pi$
$2$	0.618034	0.437016	0.218508	+	0.975835i	$0.429881\pi$
$3$	0	0
$3$	0	0
$4$	−1.61803	−0.809017
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	−2.23607	−0.790569
$9$	0	0
$10$	0.618034	0.195440
$11$	−4.23607	−1.27722	−0.638611	−	0.769529i	$-0.720491\pi$
$11$	−4.23607	−1.27722	−0.638611	+	0.769529i	$0.720491\pi$
$12$	0	0
$13$	−0.763932	−0.211877	−0.105938	−	0.994373i	$-0.533785\pi$
$13$	−0.763932	−0.211877	−0.105938	+	0.994373i	$0.533785\pi$
$14$	0	0
$15$	0	0
$16$	1.85410	0.463525
$17$	5.00000	1.21268	0.606339	−	0.795206i	$-0.292637\pi$
$17$	5.00000	1.21268	0.606339	+	0.795206i	$0.292637\pi$
$18$	0	0
$19$	5.23607	1.20124	0.600618	−	0.799536i	$-0.294921\pi$
$19$	5.23607	1.20124	0.600618	+	0.799536i	$0.294921\pi$
$20$	−1.61803	−0.361803
$21$	0	0
$22$	−2.61803	−0.558167
$23$	6.47214	1.34953	0.674767	−	0.738031i	$-0.264244\pi$
$23$	6.47214	1.34953	0.674767	+	0.738031i	$0.264244\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	−0.472136	−0.0925935
$27$	0	0
$28$	0	0
$29$	−7.47214	−1.38754	−0.693770	−	0.720196i	$-0.744052\pi$
$29$	−7.47214	−1.38754	−0.693770	+	0.720196i	$0.744052\pi$
$30$	0	0
$31$	−8.70820	−1.56404	−0.782020	−	0.623254i	$-0.785810\pi$
$31$	−8.70820	−1.56404	−0.782020	+	0.623254i	$0.785810\pi$
$32$	5.61803	0.993137
$33$	0	0
$34$	3.09017	0.529960
$35$	0	0
$36$	0	0
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	3.23607	0.524960
$39$	0	0
$40$	−2.23607	−0.353553
$41$	1.70820	0.266777	0.133388	−	0.991064i	$-0.457414\pi$
$41$	1.70820	0.266777	0.133388	+	0.991064i	$0.457414\pi$
$42$	0	0
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	6.85410	1.03329
$45$	0	0
$46$	4.00000	0.589768
$47$	−7.70820	−1.12436	−0.562179	−	0.827016i	$-0.690037\pi$
$47$	−7.70820	−1.12436	−0.562179	+	0.827016i	$0.690037\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	−2.47214	−0.349613
$51$	0	0
$52$	1.23607	0.171412
$53$	−2.47214	−0.339574	−0.169787	−	0.985481i	$-0.554308\pi$
$53$	−2.47214	−0.339574	−0.169787	+	0.985481i	$0.554308\pi$
$54$	0	0
$55$	−4.23607	−0.571191
$56$	0	0
$57$	0	0
$58$	−4.61803	−0.606378
$59$	−3.23607	−0.421300	−0.210650	−	0.977562i	$-0.567558\pi$
$59$	−3.23607	−0.421300	−0.210650	+	0.977562i	$0.567558\pi$
$60$	0	0
$61$	−3.94427	−0.505012	−0.252506	−	0.967595i	$-0.581255\pi$
$61$	−3.94427	−0.505012	−0.252506	+	0.967595i	$0.581255\pi$
$62$	−5.38197	−0.683510
$63$	0	0
$64$	−0.236068	−0.0295085
$65$	−0.763932	−0.0947541
$66$	0	0
$67$	−6.23607	−0.761857	−0.380928	−	0.924605i	$-0.624396\pi$
$67$	−6.23607	−0.761857	−0.380928	+	0.924605i	$0.624396\pi$
$68$	−8.09017	−0.981077
$69$	0	0
$70$	0	0
$71$	4.94427	0.586777	0.293389	−	0.955993i	$-0.405217\pi$
$71$	4.94427	0.586777	0.293389	+	0.955993i	$0.405217\pi$
$72$	0	0
$73$	−2.94427	−0.344601	−0.172300	−	0.985044i	$-0.555120\pi$
$73$	−2.94427	−0.344601	−0.172300	+	0.985044i	$0.555120\pi$
$74$	−3.70820	−0.431070
$75$	0	0
$76$	−8.47214	−0.971821
$77$	0	0
$78$	0	0
$79$	2.94427	0.331256	0.165628	−	0.986188i	$-0.447035\pi$
$79$	2.94427	0.331256	0.165628	+	0.986188i	$0.447035\pi$
$80$	1.85410	0.207295
$81$	0	0
$82$	1.05573	0.116586
$83$	4.23607	0.464969	0.232484	−	0.972600i	$-0.425315\pi$
$83$	4.23607	0.464969	0.232484	+	0.972600i	$0.425315\pi$
$84$	0	0
$85$	5.00000	0.542326
$86$	−6.18034	−0.666443
$87$	0	0
$88$	9.47214	1.00973
$89$	−12.9443	−1.37209	−0.686045	−	0.727559i	$-0.740655\pi$
$89$	−12.9443	−1.37209	−0.686045	+	0.727559i	$0.740655\pi$
$90$	0	0
$91$	0	0
$92$	−10.4721	−1.09180
$93$	0	0
$94$	−4.76393	−0.491362
$95$	5.23607	0.537209
$96$	0	0
$97$	−8.18034	−0.830588	−0.415294	−	0.909687i	$-0.636321\pi$
$97$	−8.18034	−0.830588	−0.415294	+	0.909687i	$0.636321\pi$
$98$	−4.32624	−0.437016
$99$	0	0
$100$	6.47214	0.647214
$101$	13.4164	1.33498	0.667491	−	0.744618i	$-0.267368\pi$
$101$	13.4164	1.33498	0.667491	+	0.744618i	$0.267368\pi$
$102$	0	0
$103$	5.23607	0.515925	0.257963	−	0.966155i	$-0.416949\pi$
$103$	5.23607	0.515925	0.257963	+	0.966155i	$0.416949\pi$
$104$	1.70820	0.167503
$105$	0	0
$106$	−1.52786	−0.148399
$107$	6.47214	0.625685	0.312842	−	0.949805i	$-0.398719\pi$
$107$	6.47214	0.625685	0.312842	+	0.949805i	$0.398719\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	−2.61803	−0.249620
$111$	0	0
$112$	0	0
$113$	14.9443	1.40584	0.702919	−	0.711269i	$-0.251879\pi$
$113$	14.9443	1.40584	0.702919	+	0.711269i	$0.251879\pi$
$114$	0	0
$115$	6.47214	0.603530
$116$	12.0902	1.12254
$117$	0	0
$118$	−2.00000	−0.184115
$119$	0	0
$120$	0	0
$121$	6.94427	0.631297
$122$	−2.43769	−0.220698
$123$	0	0
$124$	14.0902	1.26533
$125$	−9.00000	−0.804984
$126$	0	0
$127$	−13.7639	−1.22135	−0.610676	−	0.791881i	$-0.709102\pi$
$127$	−13.7639	−1.22135	−0.610676	+	0.791881i	$0.709102\pi$
$128$	−11.3820	−1.00603
$129$	0	0
$130$	−0.472136	−0.0414091
$131$	11.5279	1.00719	0.503597	−	0.863939i	$-0.332010\pi$
$131$	11.5279	1.00719	0.503597	+	0.863939i	$0.332010\pi$
$132$	0	0
$133$	0	0
$134$	−3.85410	−0.332944
$135$	0	0
$136$	−11.1803	−0.958706
$137$	−18.4721	−1.57818	−0.789091	−	0.614277i	$-0.789448\pi$
$137$	−18.4721	−1.57818	−0.789091	+	0.614277i	$0.789448\pi$
$138$	0	0
$139$	8.94427	0.758643	0.379322	−	0.925265i	$-0.376157\pi$
$139$	8.94427	0.758643	0.379322	+	0.925265i	$0.376157\pi$
$140$	0	0
$141$	0	0
$142$	3.05573	0.256431
$143$	3.23607	0.270614
$144$	0	0
$145$	−7.47214	−0.620527
$146$	−1.81966	−0.150596
$147$	0	0
$148$	9.70820	0.798009
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	−16.1803	−1.31674	−0.658369	−	0.752696i	$-0.728753\pi$
$151$	−16.1803	−1.31674	−0.658369	+	0.752696i	$0.728753\pi$
$152$	−11.7082	−0.949661
$153$	0	0
$154$	0	0
$155$	−8.70820	−0.699460
$156$	0	0
$157$	23.3607	1.86439	0.932193	−	0.361963i	$-0.117893\pi$
$157$	23.3607	1.86439	0.932193	+	0.361963i	$0.117893\pi$
$158$	1.81966	0.144764
$159$	0	0
$160$	5.61803	0.444145
$161$	0	0
$162$	0	0
$163$	−20.2361	−1.58501	−0.792506	−	0.609865i	$-0.791224\pi$
$163$	−20.2361	−1.58501	−0.792506	+	0.609865i	$0.791224\pi$
$164$	−2.76393	−0.215827
$165$	0	0
$166$	2.61803	0.203199
$167$	−15.4164	−1.19296	−0.596479	−	0.802629i	$-0.703434\pi$
$167$	−15.4164	−1.19296	−0.596479	+	0.802629i	$0.703434\pi$
$168$	0	0
$169$	−12.4164	−0.955108
$170$	3.09017	0.237005
$171$	0	0
$172$	16.1803	1.23374
$173$	−6.47214	−0.492067	−0.246034	−	0.969261i	$-0.579127\pi$
$173$	−6.47214	−0.492067	−0.246034	+	0.969261i	$0.579127\pi$
$174$	0	0
$175$	0	0
$176$	−7.85410	−0.592025
$177$	0	0
$178$	−8.00000	−0.599625
$179$	19.1246	1.42944	0.714720	−	0.699410i	$-0.246554\pi$
$179$	19.1246	1.42944	0.714720	+	0.699410i	$0.246554\pi$
$180$	0	0
$181$	3.23607	0.240535	0.120268	−	0.992742i	$-0.461625\pi$
$181$	3.23607	0.240535	0.120268	+	0.992742i	$0.461625\pi$
$182$	0	0
$183$	0	0
$184$	−14.4721	−1.06690
$185$	−6.00000	−0.441129
$186$	0	0
$187$	−21.1803	−1.54886
$188$	12.4721	0.909624
$189$	0	0
$190$	3.23607	0.234769
$191$	15.2361	1.10244	0.551222	−	0.834359i	$-0.314162\pi$
$191$	15.2361	1.10244	0.551222	+	0.834359i	$0.314162\pi$
$192$	0	0
$193$	−14.5279	−1.04574	−0.522869	−	0.852413i	$-0.675138\pi$
$193$	−14.5279	−1.04574	−0.522869	+	0.852413i	$0.675138\pi$
$194$	−5.05573	−0.362980
$195$	0	0
$196$	11.3262	0.809017
$197$	−25.9443	−1.84845	−0.924226	−	0.381845i	$-0.875289\pi$
$197$	−25.9443	−1.84845	−0.924226	+	0.381845i	$0.875289\pi$
$198$	0	0
$199$	2.00000	0.141776	0.0708881	−	0.997484i	$-0.477417\pi$
$199$	2.00000	0.141776	0.0708881	+	0.997484i	$0.477417\pi$
$200$	8.94427	0.632456
$201$	0	0
$202$	8.29180	0.583409
$203$	0	0
$204$	0	0
$205$	1.70820	0.119306
$206$	3.23607	0.225468
$207$	0	0
$208$	−1.41641	−0.0982102
$209$	−22.1803	−1.53425
$210$	0	0
$211$	−20.2361	−1.39311	−0.696554	−	0.717504i	$-0.745284\pi$
$211$	−20.2361	−1.39311	−0.696554	+	0.717504i	$0.745284\pi$
$212$	4.00000	0.274721
$213$	0	0
$214$	4.00000	0.273434
$215$	−10.0000	−0.681994
$216$	0	0
$217$	0	0
$218$	−1.23607	−0.0837171
$219$	0	0
$220$	6.85410	0.462103
$221$	−3.81966	−0.256938
$222$	0	0
$223$	29.1246	1.95033	0.975164	−	0.221483i	$-0.0710899\pi$
$223$	29.1246	1.95033	0.975164	+	0.221483i	$0.0710899\pi$
$224$	0	0
$225$	0	0
$226$	9.23607	0.614374
$227$	25.7082	1.70631	0.853157	−	0.521655i	$-0.174685\pi$
$227$	25.7082	1.70631	0.853157	+	0.521655i	$0.174685\pi$
$228$	0	0
$229$	29.5967	1.95581	0.977904	−	0.209054i	$-0.0670385\pi$
$229$	29.5967	1.95581	0.977904	+	0.209054i	$0.0670385\pi$
$230$	4.00000	0.263752
$231$	0	0
$232$	16.7082	1.09695
$233$	−14.4721	−0.948101	−0.474051	−	0.880498i	$-0.657209\pi$
$233$	−14.4721	−0.948101	−0.474051	+	0.880498i	$0.657209\pi$
$234$	0	0
$235$	−7.70820	−0.502828
$236$	5.23607	0.340839
$237$	0	0
$238$	0	0
$239$	1.00000	0.0646846
$239$	1.00000	0.0646846
$240$	0	0
$241$	18.3607	1.18272	0.591358	−	0.806409i	$-0.298592\pi$
$241$	18.3607	1.18272	0.591358	+	0.806409i	$0.298592\pi$
$242$	4.29180	0.275887
$243$	0	0
$244$	6.38197	0.408564
$245$	−7.00000	−0.447214
$246$	0	0
$247$	−4.00000	−0.254514
$248$	19.4721	1.23648
$249$	0	0
$250$	−5.56231	−0.351791
$251$	3.29180	0.207776	0.103888	−	0.994589i	$-0.466872\pi$
$251$	3.29180	0.207776	0.103888	+	0.994589i	$0.466872\pi$
$252$	0	0
$253$	−27.4164	−1.72365
$254$	−8.50658	−0.533750
$255$	0	0
$256$	−6.56231	−0.410144
$257$	11.9443	0.745063	0.372532	−	0.928020i	$-0.378490\pi$
$257$	11.9443	0.745063	0.372532	+	0.928020i	$0.378490\pi$
$258$	0	0
$259$	0	0
$260$	1.23607	0.0766577
$261$	0	0
$262$	7.12461	0.440160
$263$	−7.18034	−0.442759	−0.221379	−	0.975188i	$-0.571056\pi$
$263$	−7.18034	−0.442759	−0.221379	+	0.975188i	$0.571056\pi$
$264$	0	0
$265$	−2.47214	−0.151862
$266$	0	0
$267$	0	0
$268$	10.0902	0.616355
$269$	17.4721	1.06529	0.532647	−	0.846337i	$-0.321197\pi$
$269$	17.4721	1.06529	0.532647	+	0.846337i	$0.321197\pi$
$270$	0	0
$271$	−4.70820	−0.286003	−0.143002	−	0.989722i	$-0.545675\pi$
$271$	−4.70820	−0.286003	−0.143002	+	0.989722i	$0.545675\pi$
$272$	9.27051	0.562107
$273$	0	0
$274$	−11.4164	−0.689690
$275$	16.9443	1.02178
$276$	0	0
$277$	1.23607	0.0742681	0.0371341	−	0.999310i	$-0.488177\pi$
$277$	1.23607	0.0742681	0.0371341	+	0.999310i	$0.488177\pi$
$278$	5.52786	0.331539
$279$	0	0
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	14.2361	0.846246	0.423123	−	0.906072i	$-0.360934\pi$
$283$	14.2361	0.846246	0.423123	+	0.906072i	$0.360934\pi$
$284$	−8.00000	−0.474713
$285$	0	0
$286$	2.00000	0.118262
$287$	0	0
$288$	0	0
$289$	8.00000	0.470588
$290$	−4.61803	−0.271180
$291$	0	0
$292$	4.76393	0.278788
$293$	10.9443	0.639371	0.319686	−	0.947524i	$-0.396423\pi$
$293$	10.9443	0.639371	0.319686	+	0.947524i	$0.396423\pi$
$294$	0	0
$295$	−3.23607	−0.188411
$296$	13.4164	0.779813
$297$	0	0
$298$	−3.70820	−0.214810
$299$	−4.94427	−0.285935
$300$	0	0
$301$	0	0
$302$	−10.0000	−0.575435
$303$	0	0
$304$	9.70820	0.556804
$305$	−3.94427	−0.225848
$306$	0	0
$307$	2.34752	0.133980	0.0669901	−	0.997754i	$-0.478660\pi$
$307$	2.34752	0.133980	0.0669901	+	0.997754i	$0.478660\pi$
$308$	0	0
$309$	0	0
$310$	−5.38197	−0.305675
$311$	10.1246	0.574114	0.287057	−	0.957913i	$-0.407323\pi$
$311$	10.1246	0.574114	0.287057	+	0.957913i	$0.407323\pi$
$312$	0	0
$313$	−19.2361	−1.08729	−0.543643	−	0.839316i	$-0.682956\pi$
$313$	−19.2361	−1.08729	−0.543643	+	0.839316i	$0.682956\pi$
$314$	14.4377	0.814766
$315$	0	0
$316$	−4.76393	−0.267992
$317$	−4.76393	−0.267569	−0.133785	−	0.991010i	$-0.542713\pi$
$317$	−4.76393	−0.267569	−0.133785	+	0.991010i	$0.542713\pi$
$318$	0	0
$319$	31.6525	1.77220
$320$	−0.236068	−0.0131966
$321$	0	0
$322$	0	0
$323$	26.1803	1.45671
$324$	0	0
$325$	3.05573	0.169501
$326$	−12.5066	−0.692675
$327$	0	0
$328$	−3.81966	−0.210905
$329$	0	0
$330$	0	0
$331$	31.5967	1.73671	0.868357	−	0.495939i	$-0.165176\pi$
$331$	31.5967	1.73671	0.868357	+	0.495939i	$0.165176\pi$
$332$	−6.85410	−0.376168
$333$	0	0
$334$	−9.52786	−0.521342
$335$	−6.23607	−0.340713
$336$	0	0
$337$	−10.5279	−0.573489	−0.286745	−	0.958007i	$-0.592573\pi$
$337$	−10.5279	−0.573489	−0.286745	+	0.958007i	$0.592573\pi$
$338$	−7.67376	−0.417398
$339$	0	0
$340$	−8.09017	−0.438751
$341$	36.8885	1.99763
$342$	0	0
$343$	0	0
$344$	22.3607	1.20561
$345$	0	0
$346$	−4.00000	−0.215041
$347$	31.6525	1.69919	0.849597	−	0.527432i	$-0.176845\pi$
$347$	31.6525	1.69919	0.849597	+	0.527432i	$0.176845\pi$
$348$	0	0
$349$	−31.8328	−1.70397	−0.851986	−	0.523565i	$-0.824602\pi$
$349$	−31.8328	−1.70397	−0.851986	+	0.523565i	$0.824602\pi$
$350$	0	0
$351$	0	0
$352$	−23.7984	−1.26846
$353$	−16.2918	−0.867125	−0.433562	−	0.901124i	$-0.642744\pi$
$353$	−16.2918	−0.867125	−0.433562	+	0.901124i	$0.642744\pi$
$354$	0	0
$355$	4.94427	0.262415
$356$	20.9443	1.11004
$357$	0	0
$358$	11.8197	0.624688
$359$	−9.52786	−0.502861	−0.251431	−	0.967875i	$-0.580901\pi$
$359$	−9.52786	−0.502861	−0.251431	+	0.967875i	$0.580901\pi$
$360$	0	0
$361$	8.41641	0.442969
$362$	2.00000	0.105118
$363$	0	0
$364$	0	0
$365$	−2.94427	−0.154110
$366$	0	0
$367$	−2.70820	−0.141367	−0.0706835	−	0.997499i	$-0.522518\pi$
$367$	−2.70820	−0.141367	−0.0706835	+	0.997499i	$0.522518\pi$
$368$	12.0000	0.625543
$369$	0	0
$370$	−3.70820	−0.192780
$371$	0	0
$372$	0	0
$373$	−31.8885	−1.65113	−0.825563	−	0.564310i	$-0.809142\pi$
$373$	−31.8885	−1.65113	−0.825563	+	0.564310i	$0.809142\pi$
$374$	−13.0902	−0.676877
$375$	0	0
$376$	17.2361	0.888882
$377$	5.70820	0.293987
$378$	0	0
$379$	−18.1803	−0.933861	−0.466931	−	0.884294i	$-0.654640\pi$
$379$	−18.1803	−0.933861	−0.466931	+	0.884294i	$0.654640\pi$
$380$	−8.47214	−0.434611
$381$	0	0
$382$	9.41641	0.481785
$383$	15.4164	0.787742	0.393871	−	0.919166i	$-0.371136\pi$
$383$	15.4164	0.787742	0.393871	+	0.919166i	$0.371136\pi$
$384$	0	0
$385$	0	0
$386$	−8.97871	−0.457004
$387$	0	0
$388$	13.2361	0.671960
$389$	26.5279	1.34502	0.672508	−	0.740090i	$-0.265217\pi$
$389$	26.5279	1.34502	0.672508	+	0.740090i	$0.265217\pi$
$390$	0	0
$391$	32.3607	1.63655
$392$	15.6525	0.790569
$393$	0	0
$394$	−16.0344	−0.807804
$395$	2.94427	0.148142
$396$	0	0
$397$	20.0000	1.00377	0.501886	−	0.864934i	$-0.332640\pi$
$397$	20.0000	1.00377	0.501886	+	0.864934i	$0.332640\pi$
$398$	1.23607	0.0619585
$399$	0	0
$400$	−7.41641	−0.370820
$401$	−31.8328	−1.58965	−0.794827	−	0.606835i	$-0.792439\pi$
$401$	−31.8328	−1.58965	−0.794827	+	0.606835i	$0.792439\pi$
$402$	0	0
$403$	6.65248	0.331383
$404$	−21.7082	−1.08002
$405$	0	0
$406$	0	0
$407$	25.4164	1.25984
$408$	0	0
$409$	−3.00000	−0.148340	−0.0741702	−	0.997246i	$-0.523631\pi$
$409$	−3.00000	−0.148340	−0.0741702	+	0.997246i	$0.523631\pi$
$410$	1.05573	0.0521387
$411$	0	0
$412$	−8.47214	−0.417392
$413$	0	0
$414$	0	0
$415$	4.23607	0.207940
$416$	−4.29180	−0.210423
$417$	0	0
$418$	−13.7082	−0.670490
$419$	−10.1246	−0.494620	−0.247310	−	0.968936i	$-0.579547\pi$
$419$	−10.1246	−0.494620	−0.247310	+	0.968936i	$0.579547\pi$
$420$	0	0
$421$	30.9443	1.50813	0.754066	−	0.656799i	$-0.228090\pi$
$421$	30.9443	1.50813	0.754066	+	0.656799i	$0.228090\pi$
$422$	−12.5066	−0.608811
$423$	0	0
$424$	5.52786	0.268457
$425$	−20.0000	−0.970143
$426$	0	0
$427$	0	0
$428$	−10.4721	−0.506190
$429$	0	0
$430$	−6.18034	−0.298042
$431$	9.18034	0.442201	0.221101	−	0.975251i	$-0.429035\pi$
$431$	9.18034	0.442201	0.221101	+	0.975251i	$0.429035\pi$
$432$	0	0
$433$	−3.23607	−0.155516	−0.0777578	−	0.996972i	$-0.524776\pi$
$433$	−3.23607	−0.155516	−0.0777578	+	0.996972i	$0.524776\pi$
$434$	0	0
$435$	0	0
$436$	3.23607	0.154980
$437$	33.8885	1.62111
$438$	0	0
$439$	7.41641	0.353966	0.176983	−	0.984214i	$-0.443366\pi$
$439$	7.41641	0.353966	0.176983	+	0.984214i	$0.443366\pi$
$440$	9.47214	0.451566
$441$	0	0
$442$	−2.36068	−0.112286
$443$	6.70820	0.318716	0.159358	−	0.987221i	$-0.449058\pi$
$443$	6.70820	0.318716	0.159358	+	0.987221i	$0.449058\pi$
$444$	0	0
$445$	−12.9443	−0.613617
$446$	18.0000	0.852325
$447$	0	0
$448$	0	0
$449$	−17.1246	−0.808160	−0.404080	−	0.914724i	$-0.632408\pi$
$449$	−17.1246	−0.808160	−0.404080	+	0.914724i	$0.632408\pi$
$450$	0	0
$451$	−7.23607	−0.340733
$452$	−24.1803	−1.13735
$453$	0	0
$454$	15.8885	0.745686
$455$	0	0
$456$	0	0
$457$	20.4721	0.957646	0.478823	−	0.877911i	$-0.341064\pi$
$457$	20.4721	0.957646	0.478823	+	0.877911i	$0.341064\pi$
$458$	18.2918	0.854719
$459$	0	0
$460$	−10.4721	−0.488266
$461$	−13.5279	−0.630055	−0.315028	−	0.949082i	$-0.602014\pi$
$461$	−13.5279	−0.630055	−0.315028	+	0.949082i	$0.602014\pi$
$462$	0	0
$463$	−13.1246	−0.609952	−0.304976	−	0.952360i	$-0.598649\pi$
$463$	−13.1246	−0.609952	−0.304976	+	0.952360i	$0.598649\pi$
$464$	−13.8541	−0.643161
$465$	0	0
$466$	−8.94427	−0.414335
$467$	11.3475	0.525101	0.262550	−	0.964918i	$-0.415436\pi$
$467$	11.3475	0.525101	0.262550	+	0.964918i	$0.415436\pi$
$468$	0	0
$469$	0	0
$470$	−4.76393	−0.219744
$471$	0	0
$472$	7.23607	0.333067
$473$	42.3607	1.94775
$474$	0	0
$475$	−20.9443	−0.960989
$476$	0	0
$477$	0	0
$478$	0.618034	0.0282682
$479$	−34.1246	−1.55919	−0.779597	−	0.626282i	$-0.784576\pi$
$479$	−34.1246	−1.55919	−0.779597	+	0.626282i	$0.784576\pi$
$480$	0	0
$481$	4.58359	0.208994
$482$	11.3475	0.516866
$483$	0	0
$484$	−11.2361	−0.510730
$485$	−8.18034	−0.371450
$486$	0	0
$487$	27.1803	1.23166	0.615829	−	0.787880i	$-0.288821\pi$
$487$	27.1803	1.23166	0.615829	+	0.787880i	$0.288821\pi$
$488$	8.81966	0.399247
$489$	0	0
$490$	−4.32624	−0.195440
$491$	−2.00000	−0.0902587	−0.0451294	−	0.998981i	$-0.514370\pi$
$491$	−2.00000	−0.0902587	−0.0451294	+	0.998981i	$0.514370\pi$
$492$	0	0
$493$	−37.3607	−1.68264
$494$	−2.47214	−0.111227
$495$	0	0
$496$	−16.1459	−0.724972
$497$	0	0
$498$	0	0
$499$	16.9443	0.758530	0.379265	−	0.925288i	$-0.376177\pi$
$499$	16.9443	0.758530	0.379265	+	0.925288i	$0.376177\pi$
$500$	14.5623	0.651246
$501$	0	0
$502$	2.03444	0.0908016
$503$	12.2361	0.545579	0.272790	−	0.962074i	$-0.412054\pi$
$503$	12.2361	0.545579	0.272790	+	0.962074i	$0.412054\pi$
$504$	0	0
$505$	13.4164	0.597022
$506$	−16.9443	−0.753265
$507$	0	0
$508$	22.2705	0.988094
$509$	−30.3050	−1.34324	−0.671622	−	0.740894i	$-0.734402\pi$
$509$	−30.3050	−1.34324	−0.671622	+	0.740894i	$0.734402\pi$
$510$	0	0
$511$	0	0
$512$	18.7082	0.826794
$513$	0	0
$514$	7.38197	0.325605
$515$	5.23607	0.230729
$516$	0	0
$517$	32.6525	1.43605
$518$	0	0
$519$	0	0
$520$	1.70820	0.0749097
$521$	26.3607	1.15488	0.577441	−	0.816432i	$-0.304051\pi$
$521$	26.3607	1.15488	0.577441	+	0.816432i	$0.304051\pi$
$522$	0	0
$523$	−4.23607	−0.185230	−0.0926152	−	0.995702i	$-0.529523\pi$
$523$	−4.23607	−0.185230	−0.0926152	+	0.995702i	$0.529523\pi$
$524$	−18.6525	−0.814837
$525$	0	0
$526$	−4.43769	−0.193493
$527$	−43.5410	−1.89668
$528$	0	0
$529$	18.8885	0.821241
$530$	−1.52786	−0.0663662
$531$	0	0
$532$	0	0
$533$	−1.30495	−0.0565237
$534$	0	0
$535$	6.47214	0.279815
$536$	13.9443	0.602301
$537$	0	0
$538$	10.7984	0.465551
$539$	29.6525	1.27722
$540$	0	0
$541$	4.47214	0.192272	0.0961361	−	0.995368i	$-0.469352\pi$
$541$	4.47214	0.192272	0.0961361	+	0.995368i	$0.469352\pi$
$542$	−2.90983	−0.124988
$543$	0	0
$544$	28.0902	1.20436
$545$	−2.00000	−0.0856706
$546$	0	0
$547$	−32.0689	−1.37117	−0.685583	−	0.727994i	$-0.740453\pi$
$547$	−32.0689	−1.37117	−0.685583	+	0.727994i	$0.740453\pi$
$548$	29.8885	1.27678
$549$	0	0
$550$	10.4721	0.446533
$551$	−39.1246	−1.66676
$552$	0	0
$553$	0	0
$554$	0.763932	0.0324564
$555$	0	0
$556$	−14.4721	−0.613755
$557$	2.76393	0.117112	0.0585558	−	0.998284i	$-0.481350\pi$
$557$	2.76393	0.117112	0.0585558	+	0.998284i	$0.481350\pi$
$558$	0	0
$559$	7.63932	0.323109
$560$	0	0
$561$	0	0
$562$	−6.18034	−0.260702
$563$	−14.8328	−0.625129	−0.312564	−	0.949897i	$-0.601188\pi$
$563$	−14.8328	−0.625129	−0.312564	+	0.949897i	$0.601188\pi$
$564$	0	0
$565$	14.9443	0.628710
$566$	8.79837	0.369823
$567$	0	0
$568$	−11.0557	−0.463888
$569$	−37.9443	−1.59071	−0.795353	−	0.606146i	$-0.792715\pi$
$569$	−37.9443	−1.59071	−0.795353	+	0.606146i	$0.792715\pi$
$570$	0	0
$571$	1.88854	0.0790331	0.0395165	−	0.999219i	$-0.487418\pi$
$571$	1.88854	0.0790331	0.0395165	+	0.999219i	$0.487418\pi$
$572$	−5.23607	−0.218931
$573$	0	0
$574$	0	0
$575$	−25.8885	−1.07963
$576$	0	0
$577$	4.11146	0.171162	0.0855811	−	0.996331i	$-0.472725\pi$
$577$	4.11146	0.171162	0.0855811	+	0.996331i	$0.472725\pi$
$578$	4.94427	0.205655
$579$	0	0
$580$	12.0902	0.502017
$581$	0	0
$582$	0	0
$583$	10.4721	0.433712
$584$	6.58359	0.272431
$585$	0	0
$586$	6.76393	0.279415
$587$	6.23607	0.257390	0.128695	−	0.991684i	$-0.458921\pi$
$587$	6.23607	0.257390	0.128695	+	0.991684i	$0.458921\pi$
$588$	0	0
$589$	−45.5967	−1.87878
$590$	−2.00000	−0.0823387
$591$	0	0
$592$	−11.1246	−0.457219
$593$	−30.1803	−1.23936	−0.619679	−	0.784855i	$-0.712737\pi$
$593$	−30.1803	−1.23936	−0.619679	+	0.784855i	$0.712737\pi$
$594$	0	0
$595$	0	0
$596$	9.70820	0.397664
$597$	0	0
$598$	−3.05573	−0.124958
$599$	−25.8885	−1.05778	−0.528889	−	0.848691i	$-0.677391\pi$
$599$	−25.8885	−1.05778	−0.528889	+	0.848691i	$0.677391\pi$
$600$	0	0
$601$	41.7771	1.70412	0.852061	−	0.523442i	$-0.175352\pi$
$601$	41.7771	1.70412	0.852061	+	0.523442i	$0.175352\pi$
$602$	0	0
$603$	0	0
$604$	26.1803	1.06526
$605$	6.94427	0.282325
$606$	0	0
$607$	23.5279	0.954967	0.477483	−	0.878641i	$-0.341549\pi$
$607$	23.5279	0.954967	0.477483	+	0.878641i	$0.341549\pi$
$608$	29.4164	1.19299
$609$	0	0
$610$	−2.43769	−0.0986993
$611$	5.88854	0.238225
$612$	0	0
$613$	−2.00000	−0.0807792	−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000	−0.0807792	−0.0403896	+	0.999184i	$0.512860\pi$
$614$	1.45085	0.0585515
$615$	0	0
$616$	0	0
$617$	−29.7082	−1.19601	−0.598004	−	0.801493i	$-0.704039\pi$
$617$	−29.7082	−1.19601	−0.598004	+	0.801493i	$0.704039\pi$
$618$	0	0
$619$	−49.3050	−1.98173	−0.990867	−	0.134845i	$-0.956946\pi$
$619$	−49.3050	−1.98173	−0.990867	+	0.134845i	$0.956946\pi$
$620$	14.0902	0.565875
$621$	0	0
$622$	6.25735	0.250897
$623$	0	0
$624$	0	0
$625$	11.0000	0.440000
$626$	−11.8885	−0.475162
$627$	0	0
$628$	−37.7984	−1.50832
$629$	−30.0000	−1.19618
$630$	0	0
$631$	−20.1246	−0.801148	−0.400574	−	0.916264i	$-0.631189\pi$
$631$	−20.1246	−0.801148	−0.400574	+	0.916264i	$0.631189\pi$
$632$	−6.58359	−0.261881
$633$	0	0
$634$	−2.94427	−0.116932
$635$	−13.7639	−0.546205
$636$	0	0
$637$	5.34752	0.211877
$638$	19.5623	0.774479
$639$	0	0
$640$	−11.3820	−0.449912
$641$	−11.1115	−0.438876	−0.219438	−	0.975626i	$-0.570422\pi$
$641$	−11.1115	−0.438876	−0.219438	+	0.975626i	$0.570422\pi$
$642$	0	0
$643$	33.1803	1.30850	0.654252	−	0.756276i	$-0.272983\pi$
$643$	33.1803	1.30850	0.654252	+	0.756276i	$0.272983\pi$
$644$	0	0
$645$	0	0
$646$	16.1803	0.636607
$647$	14.8197	0.582621	0.291310	−	0.956629i	$-0.405909\pi$
$647$	14.8197	0.582621	0.291310	+	0.956629i	$0.405909\pi$
$648$	0	0
$649$	13.7082	0.538094
$650$	1.88854	0.0740748
$651$	0	0
$652$	32.7426	1.28230
$653$	−32.1803	−1.25931	−0.629657	−	0.776873i	$-0.716805\pi$
$653$	−32.1803	−1.25931	−0.629657	+	0.776873i	$0.716805\pi$
$654$	0	0
$655$	11.5279	0.450431
$656$	3.16718	0.123658
$657$	0	0
$658$	0	0
$659$	−11.7082	−0.456087	−0.228043	−	0.973651i	$-0.573233\pi$
$659$	−11.7082	−0.456087	−0.228043	+	0.973651i	$0.573233\pi$
$660$	0	0
$661$	−18.5279	−0.720650	−0.360325	−	0.932827i	$-0.617334\pi$
$661$	−18.5279	−0.720650	−0.360325	+	0.932827i	$0.617334\pi$
$662$	19.5279	0.758972
$663$	0	0
$664$	−9.47214	−0.367590
$665$	0	0
$666$	0	0
$667$	−48.3607	−1.87253
$668$	24.9443	0.965123
$669$	0	0
$670$	−3.85410	−0.148897
$671$	16.7082	0.645013
$672$	0	0
$673$	−46.5410	−1.79402	−0.897012	−	0.442006i	$-0.854267\pi$
$673$	−46.5410	−1.79402	−0.897012	+	0.442006i	$0.854267\pi$
$674$	−6.50658	−0.250624
$675$	0	0
$676$	20.0902	0.772699
$677$	3.70820	0.142518	0.0712589	−	0.997458i	$-0.477298\pi$
$677$	3.70820	0.142518	0.0712589	+	0.997458i	$0.477298\pi$
$678$	0	0
$679$	0	0
$680$	−11.1803	−0.428746
$681$	0	0
$682$	22.7984	0.872995
$683$	−18.0689	−0.691387	−0.345693	−	0.938348i	$-0.612356\pi$
$683$	−18.0689	−0.691387	−0.345693	+	0.938348i	$0.612356\pi$
$684$	0	0
$685$	−18.4721	−0.705784
$686$	0	0
$687$	0	0
$688$	−18.5410	−0.706870
$689$	1.88854	0.0719478
$690$	0	0
$691$	−21.7639	−0.827939	−0.413969	−	0.910291i	$-0.635858\pi$
$691$	−21.7639	−0.827939	−0.413969	+	0.910291i	$0.635858\pi$
$692$	10.4721	0.398091
$693$	0	0
$694$	19.5623	0.742575
$695$	8.94427	0.339276
$696$	0	0
$697$	8.54102	0.323514
$698$	−19.6738	−0.744663
$699$	0	0
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	−31.4164	−1.18489
$704$	1.00000	0.0376889
$705$	0	0
$706$	−10.0689	−0.378947
$707$	0	0
$708$	0	0
$709$	36.6525	1.37651	0.688256	−	0.725468i	$-0.258376\pi$
$709$	36.6525	1.37651	0.688256	+	0.725468i	$0.258376\pi$
$710$	3.05573	0.114679
$711$	0	0
$712$	28.9443	1.08473
$713$	−56.3607	−2.11072
$714$	0	0
$715$	3.23607	0.121022
$716$	−30.9443	−1.15644
$717$	0	0
$718$	−5.88854	−0.219759
$719$	−33.2918	−1.24157	−0.620787	−	0.783979i	$-0.713187\pi$
$719$	−33.2918	−1.24157	−0.620787	+	0.783979i	$0.713187\pi$
$720$	0	0
$721$	0	0
$722$	5.20163	0.193584
$723$	0	0
$724$	−5.23607	−0.194597
$725$	29.8885	1.11003
$726$	0	0
$727$	−1.88854	−0.0700422	−0.0350211	−	0.999387i	$-0.511150\pi$
$727$	−1.88854	−0.0700422	−0.0350211	+	0.999387i	$0.511150\pi$
$728$	0	0
$729$	0	0
$730$	−1.81966	−0.0673486
$731$	−50.0000	−1.84932
$732$	0	0
$733$	27.9443	1.03215	0.516073	−	0.856545i	$-0.327393\pi$
$733$	27.9443	1.03215	0.516073	+	0.856545i	$0.327393\pi$
$734$	−1.67376	−0.0617797
$735$	0	0
$736$	36.3607	1.34027
$737$	26.4164	0.973061
$738$	0	0
$739$	42.8328	1.57563	0.787815	−	0.615912i	$-0.211212\pi$
$739$	42.8328	1.57563	0.787815	+	0.615912i	$0.211212\pi$
$740$	9.70820	0.356881
$741$	0	0
$742$	0	0
$743$	12.2918	0.450942	0.225471	−	0.974250i	$-0.427608\pi$
$743$	12.2918	0.450942	0.225471	+	0.974250i	$0.427608\pi$
$744$	0	0
$745$	−6.00000	−0.219823
$746$	−19.7082	−0.721569
$747$	0	0
$748$	34.2705	1.25305
$749$	0	0
$750$	0	0
$751$	30.4721	1.11194	0.555972	−	0.831201i	$-0.312346\pi$
$751$	30.4721	1.11194	0.555972	+	0.831201i	$0.312346\pi$
$752$	−14.2918	−0.521168
$753$	0	0
$754$	3.52786	0.128477
$755$	−16.1803	−0.588863
$756$	0	0
$757$	42.5279	1.54570	0.772851	−	0.634588i	$-0.218830\pi$
$757$	42.5279	1.54570	0.772851	+	0.634588i	$0.218830\pi$
$758$	−11.2361	−0.408112
$759$	0	0
$760$	−11.7082	−0.424701
$761$	45.7771	1.65942	0.829709	−	0.558196i	$-0.188506\pi$
$761$	45.7771	1.65942	0.829709	+	0.558196i	$0.188506\pi$
$762$	0	0
$763$	0	0
$764$	−24.6525	−0.891895
$765$	0	0
$766$	9.52786	0.344256
$767$	2.47214	0.0892637
$768$	0	0
$769$	10.8754	0.392177	0.196088	−	0.980586i	$-0.437176\pi$
$769$	10.8754	0.392177	0.196088	+	0.980586i	$0.437176\pi$
$770$	0	0
$771$	0	0
$772$	23.5066	0.846020
$773$	5.59675	0.201301	0.100651	−	0.994922i	$-0.467908\pi$
$773$	5.59675	0.201301	0.100651	+	0.994922i	$0.467908\pi$
$774$	0	0
$775$	34.8328	1.25123
$776$	18.2918	0.656637
$777$	0	0
$778$	16.3951	0.587794
$779$	8.94427	0.320462
$780$	0	0
$781$	−20.9443	−0.749445
$782$	20.0000	0.715199
$783$	0	0
$784$	−12.9787	−0.463525
$785$	23.3607	0.833778
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	41.9787	1.49543
$789$	0	0
$790$	1.81966	0.0647406
$791$	0	0
$792$	0	0
$793$	3.01316	0.107000
$794$	12.3607	0.438664
$795$	0	0
$796$	−3.23607	−0.114699
$797$	37.3607	1.32338	0.661692	−	0.749776i	$-0.269839\pi$
$797$	37.3607	1.32338	0.661692	+	0.749776i	$0.269839\pi$
$798$	0	0
$799$	−38.5410	−1.36348
$800$	−22.4721	−0.794510
$801$	0	0
$802$	−19.6738	−0.694705
$803$	12.4721	0.440132
$804$	0	0
$805$	0	0
$806$	4.11146	0.144820
$807$	0	0
$808$	−30.0000	−1.05540
$809$	24.6525	0.866735	0.433367	−	0.901217i	$-0.357325\pi$
$809$	24.6525	0.866735	0.433367	+	0.901217i	$0.357325\pi$
$810$	0	0
$811$	−6.00000	−0.210688	−0.105344	−	0.994436i	$-0.533594\pi$
$811$	−6.00000	−0.210688	−0.105344	+	0.994436i	$0.533594\pi$
$812$	0	0
$813$	0	0
$814$	15.7082	0.550572
$815$	−20.2361	−0.708839
$816$	0	0
$817$	−52.3607	−1.83187
$818$	−1.85410	−0.0648272
$819$	0	0
$820$	−2.76393	−0.0965207
$821$	8.00000	0.279202	0.139601	−	0.990208i	$-0.455418\pi$
$821$	8.00000	0.279202	0.139601	+	0.990208i	$0.455418\pi$
$822$	0	0
$823$	−50.3607	−1.75546	−0.877731	−	0.479153i	$-0.840944\pi$
$823$	−50.3607	−1.75546	−0.877731	+	0.479153i	$0.840944\pi$
$824$	−11.7082	−0.407875
$825$	0	0
$826$	0	0
$827$	34.1246	1.18663	0.593315	−	0.804971i	$-0.297819\pi$
$827$	34.1246	1.18663	0.593315	+	0.804971i	$0.297819\pi$
$828$	0	0
$829$	−35.7082	−1.24020	−0.620099	−	0.784524i	$-0.712907\pi$
$829$	−35.7082	−1.24020	−0.620099	+	0.784524i	$0.712907\pi$
$830$	2.61803	0.0908733
$831$	0	0
$832$	0.180340	0.00625216
$833$	−35.0000	−1.21268
$834$	0	0
$835$	−15.4164	−0.533507
$836$	35.8885	1.24123
$837$	0	0
$838$	−6.25735	−0.216157
$839$	22.5967	0.780126	0.390063	−	0.920788i	$-0.372453\pi$
$839$	22.5967	0.780126	0.390063	+	0.920788i	$0.372453\pi$
$840$	0	0
$841$	26.8328	0.925270
$842$	19.1246	0.659078
$843$	0	0
$844$	32.7426	1.12705
$845$	−12.4164	−0.427137
$846$	0	0
$847$	0	0
$848$	−4.58359	−0.157401
$849$	0	0
$850$	−12.3607	−0.423968
$851$	−38.8328	−1.33117
$852$	0	0
$853$	12.4164	0.425130	0.212565	−	0.977147i	$-0.431818\pi$
$853$	12.4164	0.425130	0.212565	+	0.977147i	$0.431818\pi$
$854$	0	0
$855$	0	0
$856$	−14.4721	−0.494647
$857$	−5.41641	−0.185021	−0.0925105	−	0.995712i	$-0.529489\pi$
$857$	−5.41641	−0.185021	−0.0925105	+	0.995712i	$0.529489\pi$
$858$	0	0
$859$	23.0689	0.787100	0.393550	−	0.919303i	$-0.371247\pi$
$859$	23.0689	0.787100	0.393550	+	0.919303i	$0.371247\pi$
$860$	16.1803	0.551745
$861$	0	0
$862$	5.67376	0.193249
$863$	−33.4853	−1.13985	−0.569926	−	0.821696i	$-0.693028\pi$
$863$	−33.4853	−1.13985	−0.569926	+	0.821696i	$0.693028\pi$
$864$	0	0
$865$	−6.47214	−0.220059
$866$	−2.00000	−0.0679628
$867$	0	0
$868$	0	0
$869$	−12.4721	−0.423088
$870$	0	0
$871$	4.76393	0.161420
$872$	4.47214	0.151446
$873$	0	0
$874$	20.9443	0.708451
$875$	0	0
$876$	0	0
$877$	16.5279	0.558106	0.279053	−	0.960276i	$-0.409979\pi$
$877$	16.5279	0.558106	0.279053	+	0.960276i	$0.409979\pi$
$878$	4.58359	0.154689
$879$	0	0
$880$	−7.85410	−0.264762
$881$	26.2492	0.884359	0.442179	−	0.896927i	$-0.354205\pi$
$881$	26.2492	0.884359	0.442179	+	0.896927i	$0.354205\pi$
$882$	0	0
$883$	38.0132	1.27924	0.639622	−	0.768689i	$-0.279091\pi$
$883$	38.0132	1.27924	0.639622	+	0.768689i	$0.279091\pi$
$884$	6.18034	0.207867
$885$	0	0
$886$	4.14590	0.139284
$887$	−16.3607	−0.549338	−0.274669	−	0.961539i	$-0.588568\pi$
$887$	−16.3607	−0.549338	−0.274669	+	0.961539i	$0.588568\pi$
$888$	0	0
$889$	0	0
$890$	−8.00000	−0.268161
$891$	0	0
$892$	−47.1246	−1.57785
$893$	−40.3607	−1.35062
$894$	0	0
$895$	19.1246	0.639265
$896$	0	0
$897$	0	0
$898$	−10.5836	−0.353179
$899$	65.0689	2.17017
$900$	0	0
$901$	−12.3607	−0.411794
$902$	−4.47214	−0.148906
$903$	0	0
$904$	−33.4164	−1.11141
$905$	3.23607	0.107571
$906$	0	0
$907$	32.0689	1.06483	0.532415	−	0.846484i	$-0.321285\pi$
$907$	32.0689	1.06483	0.532415	+	0.846484i	$0.321285\pi$
$908$	−41.5967	−1.38044
$909$	0	0
$910$	0	0
$911$	18.1115	0.600059	0.300030	−	0.953930i	$-0.403003\pi$
$911$	18.1115	0.600059	0.300030	+	0.953930i	$0.403003\pi$
$912$	0	0
$913$	−17.9443	−0.593869
$914$	12.6525	0.418507
$915$	0	0
$916$	−47.8885	−1.58228
$917$	0	0
$918$	0	0
$919$	−52.9574	−1.74690	−0.873452	−	0.486910i	$-0.838124\pi$
$919$	−52.9574	−1.74690	−0.873452	+	0.486910i	$0.838124\pi$
$920$	−14.4721	−0.477132
$921$	0	0
$922$	−8.36068	−0.275344
$923$	−3.77709	−0.124324
$924$	0	0
$925$	24.0000	0.789115
$926$	−8.11146	−0.266559
$927$	0	0
$928$	−41.9787	−1.37802
$929$	34.6525	1.13691	0.568455	−	0.822714i	$-0.307541\pi$
$929$	34.6525	1.13691	0.568455	+	0.822714i	$0.307541\pi$
$930$	0	0
$931$	−36.6525	−1.20124
$932$	23.4164	0.767030
$933$	0	0
$934$	7.01316	0.229477
$935$	−21.1803	−0.692671
$936$	0	0
$937$	0.0557281	0.00182056	0.000910279	−	1.00000i	$-0.499710\pi$
$937$	0.0557281	0.00182056	0.000910279		1.00000i	$0.499710\pi$
$938$	0	0
$939$	0	0
$940$	12.4721	0.406796
$941$	7.81966	0.254914	0.127457	−	0.991844i	$-0.459319\pi$
$941$	7.81966	0.254914	0.127457	+	0.991844i	$0.459319\pi$
$942$	0	0
$943$	11.0557	0.360024
$944$	−6.00000	−0.195283
$945$	0	0
$946$	26.1803	0.851196
$947$	35.3050	1.14726	0.573628	−	0.819116i	$-0.305535\pi$
$947$	35.3050	1.14726	0.573628	+	0.819116i	$0.305535\pi$
$948$	0	0
$949$	2.24922	0.0730129
$950$	−12.9443	−0.419968
$951$	0	0
$952$	0	0
$953$	34.3607	1.11305	0.556526	−	0.830830i	$-0.312134\pi$
$953$	34.3607	1.11305	0.556526	+	0.830830i	$0.312134\pi$
$954$	0	0
$955$	15.2361	0.493028
$956$	−1.61803	−0.0523310
$957$	0	0
$958$	−21.0902	−0.681392
$959$	0	0
$960$	0	0
$961$	44.8328	1.44622
$962$	2.83282	0.0913336
$963$	0	0
$964$	−29.7082	−0.956837
$965$	−14.5279	−0.467668
$966$	0	0
$967$	−44.9443	−1.44531	−0.722655	−	0.691209i	$-0.757079\pi$
$967$	−44.9443	−1.44531	−0.722655	+	0.691209i	$0.757079\pi$
$968$	−15.5279	−0.499084
$969$	0	0
$970$	−5.05573	−0.162330
$971$	51.4296	1.65045	0.825227	−	0.564802i	$-0.191047\pi$
$971$	51.4296	1.65045	0.825227	+	0.564802i	$0.191047\pi$
$972$	0	0
$973$	0	0
$974$	16.7984	0.538255
$975$	0	0
$976$	−7.31308	−0.234086
$977$	−18.8328	−0.602515	−0.301258	−	0.953543i	$-0.597406\pi$
$977$	−18.8328	−0.602515	−0.301258	+	0.953543i	$0.597406\pi$
$978$	0	0
$979$	54.8328	1.75246
$980$	11.3262	0.361803
$981$	0	0
$982$	−1.23607	−0.0394445
$983$	11.5410	0.368101	0.184051	−	0.982917i	$-0.441079\pi$
$983$	11.5410	0.368101	0.184051	+	0.982917i	$0.441079\pi$
$984$	0	0
$985$	−25.9443	−0.826653
$986$	−23.0902	−0.735341
$987$	0	0
$988$	6.47214	0.205906
$989$	−64.7214	−2.05802
$990$	0	0
$991$	33.1246	1.05224	0.526119	−	0.850411i	$-0.323647\pi$
$991$	33.1246	1.05224	0.526119	+	0.850411i	$0.323647\pi$
$992$	−48.9230	−1.55331
$993$	0	0
$994$	0	0
$995$	2.00000	0.0634043
$996$	0	0
$997$	−45.8885	−1.45330	−0.726652	−	0.687005i	$-0.758925\pi$
$997$	−45.8885	−1.45330	−0.726652	+	0.687005i	$0.758925\pi$
$998$	10.4721	0.331490
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.2.a.c.1.2		2
3.2	odd	2		717.2.a.a.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.2.a.a.1.1	✓	2	3.2	odd	2
2151.2.a.c.1.2		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2151.a (trivial)

\(f(q)\)	\(=\)	\(q+0.618034 q^{2} -1.61803 q^{4} +1.00000 q^{5} -2.23607 q^{8} +O(q^{10})\)	\(q+0.618034 q^{2} -1.61803 q^{4} +1.00000 q^{5} -2.23607 q^{8} +0.618034 q^{10} -4.23607 q^{11} -0.763932 q^{13} +1.85410 q^{16} +5.00000 q^{17} +5.23607 q^{19} -1.61803 q^{20} -2.61803 q^{22} +6.47214 q^{23} -4.00000 q^{25} -0.472136 q^{26} -7.47214 q^{29} -8.70820 q^{31} +5.61803 q^{32} +3.09017 q^{34} -6.00000 q^{37} +3.23607 q^{38} -2.23607 q^{40} +1.70820 q^{41} -10.0000 q^{43} +6.85410 q^{44} +4.00000 q^{46} -7.70820 q^{47} -7.00000 q^{49} -2.47214 q^{50} +1.23607 q^{52} -2.47214 q^{53} -4.23607 q^{55} -4.61803 q^{58} -3.23607 q^{59} -3.94427 q^{61} -5.38197 q^{62} -0.236068 q^{64} -0.763932 q^{65} -6.23607 q^{67} -8.09017 q^{68} +4.94427 q^{71} -2.94427 q^{73} -3.70820 q^{74} -8.47214 q^{76} +2.94427 q^{79} +1.85410 q^{80} +1.05573 q^{82} +4.23607 q^{83} +5.00000 q^{85} -6.18034 q^{86} +9.47214 q^{88} -12.9443 q^{89} -10.4721 q^{92} -4.76393 q^{94} +5.23607 q^{95} -8.18034 q^{97} -4.32624 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - q^{4} + 2 q^{5}+O(q^{10}) \) 2 * q - q^2 - q^4 + 2 * q^5	\( 2 q - q^{2} - q^{4} + 2 q^{5} - q^{10} - 4 q^{11} - 6 q^{13} - 3 q^{16} + 10 q^{17} + 6 q^{19} - q^{20} - 3 q^{22} + 4 q^{23} - 8 q^{25} + 8 q^{26} - 6 q^{29} - 4 q^{31} + 9 q^{32} - 5 q^{34} - 12 q^{37} + 2 q^{38} - 10 q^{41} - 20 q^{43} + 7 q^{44} + 8 q^{46} - 2 q^{47} - 14 q^{49} + 4 q^{50} - 2 q^{52} + 4 q^{53} - 4 q^{55} - 7 q^{58} - 2 q^{59} + 10 q^{61} - 13 q^{62} + 4 q^{64} - 6 q^{65} - 8 q^{67} - 5 q^{68} - 8 q^{71} + 12 q^{73} + 6 q^{74} - 8 q^{76} - 12 q^{79} - 3 q^{80} + 20 q^{82} + 4 q^{83} + 10 q^{85} + 10 q^{86} + 10 q^{88} - 8 q^{89} - 12 q^{92} - 14 q^{94} + 6 q^{95} + 6 q^{97} + 7 q^{98}+O(q^{100}) \) 2 * q - q^2 - q^4 + 2 * q^5 - q^10 - 4 * q^11 - 6 * q^13 - 3 * q^16 + 10 * q^17 + 6 * q^19 - q^20 - 3 * q^22 + 4 * q^23 - 8 * q^25 + 8 * q^26 - 6 * q^29 - 4 * q^31 + 9 * q^32 - 5 * q^34 - 12 * q^37 + 2 * q^38 - 10 * q^41 - 20 * q^43 + 7 * q^44 + 8 * q^46 - 2 * q^47 - 14 * q^49 + 4 * q^50 - 2 * q^52 + 4 * q^53 - 4 * q^55 - 7 * q^58 - 2 * q^59 + 10 * q^61 - 13 * q^62 + 4 * q^64 - 6 * q^65 - 8 * q^67 - 5 * q^68 - 8 * q^71 + 12 * q^73 + 6 * q^74 - 8 * q^76 - 12 * q^79 - 3 * q^80 + 20 * q^82 + 4 * q^83 + 10 * q^85 + 10 * q^86 + 10 * q^88 - 8 * q^89 - 12 * q^92 - 14 * q^94 + 6 * q^95 + 6 * q^97 + 7 * q^98

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