LMFDB - Embedded newform 2250.2.a.e.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2250,2,Mod(1,2250)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2250, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2250.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2250 = 2 \cdot 3^{2} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2250.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-2,0,2,0,0,1,-2,0,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$17.9663404548$
Analytic rank:	$0$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 750)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	2250.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -0.618034 q^{7} -1.00000 q^{8} +1.61803 q^{11} +4.85410 q^{13} +0.618034 q^{14} +1.00000 q^{16} -0.763932 q^{17} +5.85410 q^{19} -1.61803 q^{22} -4.85410 q^{23} -4.85410 q^{26} -0.618034 q^{28} +2.76393 q^{29} -2.47214 q^{31} -1.00000 q^{32} +0.763932 q^{34} -9.56231 q^{37} -5.85410 q^{38} +9.38197 q^{41} +5.70820 q^{43} +1.61803 q^{44} +4.85410 q^{46} -1.61803 q^{47} -6.61803 q^{49} +4.85410 q^{52} -5.38197 q^{53} +0.618034 q^{56} -2.76393 q^{58} +13.0902 q^{59} -9.70820 q^{61} +2.47214 q^{62} +1.00000 q^{64} -3.70820 q^{67} -0.763932 q^{68} +3.52786 q^{71} +12.9443 q^{73} +9.56231 q^{74} +5.85410 q^{76} -1.00000 q^{77} +13.4164 q^{79} -9.38197 q^{82} +14.9443 q^{83} -5.70820 q^{86} -1.61803 q^{88} +18.0902 q^{89} -3.00000 q^{91} -4.85410 q^{92} +1.61803 q^{94} +9.70820 q^{97} +6.61803 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} + q^{7} - 2 q^{8} + q^{11} + 3 q^{13} - q^{14} + 2 q^{16} - 6 q^{17} + 5 q^{19} - q^{22} - 3 q^{23} - 3 q^{26} + q^{28} + 10 q^{29} + 4 q^{31} - 2 q^{32} + 6 q^{34} + q^{37}+ \cdots + 11 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 + q^7 - 2 * q^8 + q^11 + 3 * q^13 - q^14 + 2 * q^16 - 6 * q^17 + 5 * q^19 - q^22 - 3 * q^23 - 3 * q^26 + q^28 + 10 * q^29 + 4 * q^31 - 2 * q^32 + 6 * q^34 + q^37 - 5 * q^38 + 21 * q^41 - 2 * q^43 + q^44 + 3 * q^46 - q^47 - 11 * q^49 + 3 * q^52 - 13 * q^53 - q^56 - 10 * q^58 + 15 * q^59 - 6 * q^61 - 4 * q^62 + 2 * q^64 + 6 * q^67 - 6 * q^68 + 16 * q^71 + 8 * q^73 - q^74 + 5 * q^76 - 2 * q^77 - 21 * q^82 + 12 * q^83 + 2 * q^86 - q^88 + 25 * q^89 - 6 * q^91 - 3 * q^92 + q^94 + 6 * q^97 + 11 * q^98

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.618034	−0.233595	−0.116797	−	0.993156i	$-0.537263\pi$
$7$	−0.618034	−0.233595	−0.116797	+	0.993156i	$0.537263\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	0	0
$11$	1.61803	0.487856	0.243928	−	0.969793i	$-0.421564\pi$
$11$	1.61803	0.487856	0.243928	+	0.969793i	$0.421564\pi$
$12$	0	0
$13$	4.85410	1.34629	0.673143	−	0.739512i	$-0.264944\pi$
$13$	4.85410	1.34629	0.673143	+	0.739512i	$0.264944\pi$
$14$	0.618034	0.165177
$15$	0	0
$16$	1.00000	0.250000
$17$	−0.763932	−0.185281	−0.0926404	−	0.995700i	$-0.529531\pi$
$17$	−0.763932	−0.185281	−0.0926404	+	0.995700i	$0.529531\pi$
$18$	0	0
$19$	5.85410	1.34302	0.671512	−	0.740994i	$-0.265645\pi$
$19$	5.85410	1.34302	0.671512	+	0.740994i	$0.265645\pi$
$20$	0	0
$21$	0	0
$22$	−1.61803	−0.344966
$23$	−4.85410	−1.01215	−0.506075	−	0.862489i	$-0.668904\pi$
$23$	−4.85410	−1.01215	−0.506075	+	0.862489i	$0.668904\pi$
$24$	0	0
$25$	0	0
$26$	−4.85410	−0.951968
$27$	0	0
$28$	−0.618034	−0.116797
$29$	2.76393	0.513249	0.256625	−	0.966511i	$-0.417390\pi$
$29$	2.76393	0.513249	0.256625	+	0.966511i	$0.417390\pi$
$30$	0	0
$31$	−2.47214	−0.444009	−0.222004	−	0.975046i	$-0.571260\pi$
$31$	−2.47214	−0.444009	−0.222004	+	0.975046i	$0.571260\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	0.763932	0.131013
$35$	0	0
$36$	0	0
$37$	−9.56231	−1.57203	−0.786017	−	0.618205i	$-0.787860\pi$
$37$	−9.56231	−1.57203	−0.786017	+	0.618205i	$0.787860\pi$
$38$	−5.85410	−0.949661
$39$	0	0
$40$	0	0
$41$	9.38197	1.46522	0.732608	−	0.680650i	$-0.238303\pi$
$41$	9.38197	1.46522	0.732608	+	0.680650i	$0.238303\pi$
$42$	0	0
$43$	5.70820	0.870493	0.435246	−	0.900311i	$-0.356661\pi$
$43$	5.70820	0.870493	0.435246	+	0.900311i	$0.356661\pi$
$44$	1.61803	0.243928
$45$	0	0
$46$	4.85410	0.715698
$47$	−1.61803	−0.236015	−0.118007	−	0.993013i	$-0.537651\pi$
$47$	−1.61803	−0.236015	−0.118007	+	0.993013i	$0.537651\pi$
$48$	0	0
$49$	−6.61803	−0.945433
$50$	0	0
$51$	0	0
$52$	4.85410	0.673143
$53$	−5.38197	−0.739270	−0.369635	−	0.929177i	$-0.620517\pi$
$53$	−5.38197	−0.739270	−0.369635	+	0.929177i	$0.620517\pi$
$54$	0	0
$55$	0	0
$56$	0.618034	0.0825883
$57$	0	0
$58$	−2.76393	−0.362922
$59$	13.0902	1.70419	0.852097	−	0.523383i	$-0.175330\pi$
$59$	13.0902	1.70419	0.852097	+	0.523383i	$0.175330\pi$
$60$	0	0
$61$	−9.70820	−1.24301	−0.621504	−	0.783411i	$-0.713478\pi$
$61$	−9.70820	−1.24301	−0.621504	+	0.783411i	$0.713478\pi$
$62$	2.47214	0.313962
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−3.70820	−0.453029	−0.226515	−	0.974008i	$-0.572733\pi$
$67$	−3.70820	−0.453029	−0.226515	+	0.974008i	$0.572733\pi$
$68$	−0.763932	−0.0926404
$69$	0	0
$70$	0	0
$71$	3.52786	0.418680	0.209340	−	0.977843i	$-0.432868\pi$
$71$	3.52786	0.418680	0.209340	+	0.977843i	$0.432868\pi$
$72$	0	0
$73$	12.9443	1.51501	0.757506	−	0.652828i	$-0.226418\pi$
$73$	12.9443	1.51501	0.757506	+	0.652828i	$0.226418\pi$
$74$	9.56231	1.11160
$75$	0	0
$76$	5.85410	0.671512
$77$	−1.00000	−0.113961
$78$	0	0
$79$	13.4164	1.50946	0.754732	−	0.656033i	$-0.227767\pi$
$79$	13.4164	1.50946	0.754732	+	0.656033i	$0.227767\pi$
$80$	0	0
$81$	0	0
$82$	−9.38197	−1.03606
$83$	14.9443	1.64035	0.820173	−	0.572115i	$-0.193877\pi$
$83$	14.9443	1.64035	0.820173	+	0.572115i	$0.193877\pi$
$84$	0	0
$85$	0	0
$86$	−5.70820	−0.615531
$87$	0	0
$88$	−1.61803	−0.172483
$89$	18.0902	1.91755	0.958777	−	0.284159i	$-0.0917144\pi$
$89$	18.0902	1.91755	0.958777	+	0.284159i	$0.0917144\pi$
$90$	0	0
$91$	−3.00000	−0.314485
$92$	−4.85410	−0.506075
$93$	0	0
$94$	1.61803	0.166887
$95$	0	0
$96$	0	0
$97$	9.70820	0.985719	0.492859	−	0.870109i	$-0.335952\pi$
$97$	9.70820	0.985719	0.492859	+	0.870109i	$0.335952\pi$
$98$	6.61803	0.668522
$99$	0	0
$100$	0	0
$101$	−12.6525	−1.25897	−0.629484	−	0.777013i	$-0.716734\pi$
$101$	−12.6525	−1.25897	−0.629484	+	0.777013i	$0.716734\pi$
$102$	0	0
$103$	−8.56231	−0.843669	−0.421835	−	0.906673i	$-0.638614\pi$
$103$	−8.56231	−0.843669	−0.421835	+	0.906673i	$0.638614\pi$
$104$	−4.85410	−0.475984
$105$	0	0
$106$	5.38197	0.522743
$107$	−6.94427	−0.671328	−0.335664	−	0.941982i	$-0.608961\pi$
$107$	−6.94427	−0.671328	−0.335664	+	0.941982i	$0.608961\pi$
$108$	0	0
$109$	−6.18034	−0.591969	−0.295985	−	0.955193i	$-0.595648\pi$
$109$	−6.18034	−0.591969	−0.295985	+	0.955193i	$0.595648\pi$
$110$	0	0
$111$	0	0
$112$	−0.618034	−0.0583987
$113$	3.23607	0.304424	0.152212	−	0.988348i	$-0.451360\pi$
$113$	3.23607	0.304424	0.152212	+	0.988348i	$0.451360\pi$
$114$	0	0
$115$	0	0
$116$	2.76393	0.256625
$117$	0	0
$118$	−13.0902	−1.20505
$119$	0.472136	0.0432806
$120$	0	0
$121$	−8.38197	−0.761997
$122$	9.70820	0.878939
$123$	0	0
$124$	−2.47214	−0.222004
$125$	0	0
$126$	0	0
$127$	11.4164	1.01304	0.506521	−	0.862228i	$-0.330931\pi$
$127$	11.4164	1.01304	0.506521	+	0.862228i	$0.330931\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	−3.90983	−0.341603	−0.170802	−	0.985305i	$-0.554636\pi$
$131$	−3.90983	−0.341603	−0.170802	+	0.985305i	$0.554636\pi$
$132$	0	0
$133$	−3.61803	−0.313723
$134$	3.70820	0.320340
$135$	0	0
$136$	0.763932	0.0655066
$137$	3.05573	0.261068	0.130534	−	0.991444i	$-0.458331\pi$
$137$	3.05573	0.261068	0.130534	+	0.991444i	$0.458331\pi$
$138$	0	0
$139$	12.5623	1.06552	0.532760	−	0.846266i	$-0.321155\pi$
$139$	12.5623	1.06552	0.532760	+	0.846266i	$0.321155\pi$
$140$	0	0
$141$	0	0
$142$	−3.52786	−0.296052
$143$	7.85410	0.656793
$144$	0	0
$145$	0	0
$146$	−12.9443	−1.07128
$147$	0	0
$148$	−9.56231	−0.786017
$149$	13.4164	1.09911	0.549557	−	0.835456i	$-0.314796\pi$
$149$	13.4164	1.09911	0.549557	+	0.835456i	$0.314796\pi$
$150$	0	0
$151$	10.2918	0.837534	0.418767	−	0.908094i	$-0.362462\pi$
$151$	10.2918	0.837534	0.418767	+	0.908094i	$0.362462\pi$
$152$	−5.85410	−0.474830
$153$	0	0
$154$	1.00000	0.0805823
$155$	0	0
$156$	0	0
$157$	−10.9443	−0.873448	−0.436724	−	0.899596i	$-0.643861\pi$
$157$	−10.9443	−0.873448	−0.436724	+	0.899596i	$0.643861\pi$
$158$	−13.4164	−1.06735
$159$	0	0
$160$	0	0
$161$	3.00000	0.236433
$162$	0	0
$163$	−4.94427	−0.387265	−0.193633	−	0.981074i	$-0.562027\pi$
$163$	−4.94427	−0.387265	−0.193633	+	0.981074i	$0.562027\pi$
$164$	9.38197	0.732608
$165$	0	0
$166$	−14.9443	−1.15990
$167$	7.85410	0.607769	0.303884	−	0.952709i	$-0.401716\pi$
$167$	7.85410	0.607769	0.303884	+	0.952709i	$0.401716\pi$
$168$	0	0
$169$	10.5623	0.812485
$170$	0	0
$171$	0	0
$172$	5.70820	0.435246
$173$	14.6180	1.11139	0.555694	−	0.831387i	$-0.312453\pi$
$173$	14.6180	1.11139	0.555694	+	0.831387i	$0.312453\pi$
$174$	0	0
$175$	0	0
$176$	1.61803	0.121964
$177$	0	0
$178$	−18.0902	−1.35592
$179$	−22.0344	−1.64693	−0.823466	−	0.567366i	$-0.807962\pi$
$179$	−22.0344	−1.64693	−0.823466	+	0.567366i	$0.807962\pi$
$180$	0	0
$181$	25.4164	1.88919	0.944593	−	0.328243i	$-0.106456\pi$
$181$	25.4164	1.88919	0.944593	+	0.328243i	$0.106456\pi$
$182$	3.00000	0.222375
$183$	0	0
$184$	4.85410	0.357849
$185$	0	0
$186$	0	0
$187$	−1.23607	−0.0903902
$188$	−1.61803	−0.118007
$189$	0	0
$190$	0	0
$191$	16.9443	1.22604	0.613022	−	0.790066i	$-0.289954\pi$
$191$	16.9443	1.22604	0.613022	+	0.790066i	$0.289954\pi$
$192$	0	0
$193$	−4.29180	−0.308930	−0.154465	−	0.987998i	$-0.549365\pi$
$193$	−4.29180	−0.308930	−0.154465	+	0.987998i	$0.549365\pi$
$194$	−9.70820	−0.697008
$195$	0	0
$196$	−6.61803	−0.472717
$197$	−6.94427	−0.494759	−0.247379	−	0.968919i	$-0.579569\pi$
$197$	−6.94427	−0.494759	−0.247379	+	0.968919i	$0.579569\pi$
$198$	0	0
$199$	−6.18034	−0.438113	−0.219056	−	0.975712i	$-0.570298\pi$
$199$	−6.18034	−0.438113	−0.219056	+	0.975712i	$0.570298\pi$
$200$	0	0
$201$	0	0
$202$	12.6525	0.890225
$203$	−1.70820	−0.119892
$204$	0	0
$205$	0	0
$206$	8.56231	0.596564
$207$	0	0
$208$	4.85410	0.336571
$209$	9.47214	0.655201
$210$	0	0
$211$	−17.2705	−1.18895	−0.594475	−	0.804114i	$-0.702640\pi$
$211$	−17.2705	−1.18895	−0.594475	+	0.804114i	$0.702640\pi$
$212$	−5.38197	−0.369635
$213$	0	0
$214$	6.94427	0.474701
$215$	0	0
$216$	0	0
$217$	1.52786	0.103718
$218$	6.18034	0.418585
$219$	0	0
$220$	0	0
$221$	−3.70820	−0.249441
$222$	0	0
$223$	24.0000	1.60716	0.803579	−	0.595198i	$-0.202926\pi$
$223$	24.0000	1.60716	0.803579	+	0.595198i	$0.202926\pi$
$224$	0.618034	0.0412941
$225$	0	0
$226$	−3.23607	−0.215260
$227$	28.1803	1.87039	0.935197	−	0.354127i	$-0.115222\pi$
$227$	28.1803	1.87039	0.935197	+	0.354127i	$0.115222\pi$
$228$	0	0
$229$	1.70820	0.112881	0.0564406	−	0.998406i	$-0.482025\pi$
$229$	1.70820	0.112881	0.0564406	+	0.998406i	$0.482025\pi$
$230$	0	0
$231$	0	0
$232$	−2.76393	−0.181461
$233$	19.4164	1.27201	0.636006	−	0.771684i	$-0.280586\pi$
$233$	19.4164	1.27201	0.636006	+	0.771684i	$0.280586\pi$
$234$	0	0
$235$	0	0
$236$	13.0902	0.852097
$237$	0	0
$238$	−0.472136	−0.0306040
$239$	−8.94427	−0.578557	−0.289278	−	0.957245i	$-0.593415\pi$
$239$	−8.94427	−0.578557	−0.289278	+	0.957245i	$0.593415\pi$
$240$	0	0
$241$	−7.14590	−0.460308	−0.230154	−	0.973154i	$-0.573923\pi$
$241$	−7.14590	−0.460308	−0.230154	+	0.973154i	$0.573923\pi$
$242$	8.38197	0.538813
$243$	0	0
$244$	−9.70820	−0.621504
$245$	0	0
$246$	0	0
$247$	28.4164	1.80809
$248$	2.47214	0.156981
$249$	0	0
$250$	0	0
$251$	−20.9443	−1.32199	−0.660995	−	0.750390i	$-0.729866\pi$
$251$	−20.9443	−1.32199	−0.660995	+	0.750390i	$0.729866\pi$
$252$	0	0
$253$	−7.85410	−0.493783
$254$	−11.4164	−0.716329
$255$	0	0
$256$	1.00000	0.0625000
$257$	−13.5279	−0.843845	−0.421922	−	0.906632i	$-0.638645\pi$
$257$	−13.5279	−0.843845	−0.421922	+	0.906632i	$0.638645\pi$
$258$	0	0
$259$	5.90983	0.367219
$260$	0	0
$261$	0	0
$262$	3.90983	0.241550
$263$	14.6180	0.901387	0.450693	−	0.892679i	$-0.351177\pi$
$263$	14.6180	0.901387	0.450693	+	0.892679i	$0.351177\pi$
$264$	0	0
$265$	0	0
$266$	3.61803	0.221836
$267$	0	0
$268$	−3.70820	−0.226515
$269$	13.4164	0.818013	0.409006	−	0.912532i	$-0.365875\pi$
$269$	13.4164	0.818013	0.409006	+	0.912532i	$0.365875\pi$
$270$	0	0
$271$	25.4164	1.54394	0.771968	−	0.635661i	$-0.219272\pi$
$271$	25.4164	1.54394	0.771968	+	0.635661i	$0.219272\pi$
$272$	−0.763932	−0.0463202
$273$	0	0
$274$	−3.05573	−0.184603
$275$	0	0
$276$	0	0
$277$	−12.8541	−0.772328	−0.386164	−	0.922430i	$-0.626200\pi$
$277$	−12.8541	−0.772328	−0.386164	+	0.922430i	$0.626200\pi$
$278$	−12.5623	−0.753437
$279$	0	0
$280$	0	0
$281$	6.09017	0.363309	0.181655	−	0.983362i	$-0.441855\pi$
$281$	6.09017	0.363309	0.181655	+	0.983362i	$0.441855\pi$
$282$	0	0
$283$	30.1803	1.79403	0.897017	−	0.441995i	$-0.145729\pi$
$283$	30.1803	1.79403	0.897017	+	0.441995i	$0.145729\pi$
$284$	3.52786	0.209340
$285$	0	0
$286$	−7.85410	−0.464423
$287$	−5.79837	−0.342267
$288$	0	0
$289$	−16.4164	−0.965671
$290$	0	0
$291$	0	0
$292$	12.9443	0.757506
$293$	−22.0902	−1.29052	−0.645261	−	0.763962i	$-0.723251\pi$
$293$	−22.0902	−1.29052	−0.645261	+	0.763962i	$0.723251\pi$
$294$	0	0
$295$	0	0
$296$	9.56231	0.555798
$297$	0	0
$298$	−13.4164	−0.777192
$299$	−23.5623	−1.36264
$300$	0	0
$301$	−3.52786	−0.203343
$302$	−10.2918	−0.592226
$303$	0	0
$304$	5.85410	0.335756
$305$	0	0
$306$	0	0
$307$	20.7639	1.18506	0.592530	−	0.805548i	$-0.298129\pi$
$307$	20.7639	1.18506	0.592530	+	0.805548i	$0.298129\pi$
$308$	−1.00000	−0.0569803
$309$	0	0
$310$	0	0
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	0	0
$313$	2.29180	0.129540	0.0647700	−	0.997900i	$-0.479369\pi$
$313$	2.29180	0.129540	0.0647700	+	0.997900i	$0.479369\pi$
$314$	10.9443	0.617621
$315$	0	0
$316$	13.4164	0.754732
$317$	1.14590	0.0643600	0.0321800	−	0.999482i	$-0.489755\pi$
$317$	1.14590	0.0643600	0.0321800	+	0.999482i	$0.489755\pi$
$318$	0	0
$319$	4.47214	0.250392
$320$	0	0
$321$	0	0
$322$	−3.00000	−0.167183
$323$	−4.47214	−0.248836
$324$	0	0
$325$	0	0
$326$	4.94427	0.273838
$327$	0	0
$328$	−9.38197	−0.518032
$329$	1.00000	0.0551318
$330$	0	0
$331$	−2.47214	−0.135881	−0.0679404	−	0.997689i	$-0.521643\pi$
$331$	−2.47214	−0.135881	−0.0679404	+	0.997689i	$0.521643\pi$
$332$	14.9443	0.820173
$333$	0	0
$334$	−7.85410	−0.429757
$335$	0	0
$336$	0	0
$337$	−19.8885	−1.08340	−0.541699	−	0.840573i	$-0.682219\pi$
$337$	−19.8885	−1.08340	−0.541699	+	0.840573i	$0.682219\pi$
$338$	−10.5623	−0.574514
$339$	0	0
$340$	0	0
$341$	−4.00000	−0.216612
$342$	0	0
$343$	8.41641	0.454443
$344$	−5.70820	−0.307766
$345$	0	0
$346$	−14.6180	−0.785870
$347$	26.0689	1.39945	0.699726	−	0.714412i	$-0.253306\pi$
$347$	26.0689	1.39945	0.699726	+	0.714412i	$0.253306\pi$
$348$	0	0
$349$	−26.8328	−1.43633	−0.718164	−	0.695874i	$-0.755017\pi$
$349$	−26.8328	−1.43633	−0.718164	+	0.695874i	$0.755017\pi$
$350$	0	0
$351$	0	0
$352$	−1.61803	−0.0862415
$353$	−15.7082	−0.836063	−0.418032	−	0.908432i	$-0.637280\pi$
$353$	−15.7082	−0.836063	−0.418032	+	0.908432i	$0.637280\pi$
$354$	0	0
$355$	0	0
$356$	18.0902	0.958777
$357$	0	0
$358$	22.0344	1.16456
$359$	−4.47214	−0.236030	−0.118015	−	0.993012i	$-0.537653\pi$
$359$	−4.47214	−0.236030	−0.118015	+	0.993012i	$0.537653\pi$
$360$	0	0
$361$	15.2705	0.803711
$362$	−25.4164	−1.33586
$363$	0	0
$364$	−3.00000	−0.157243
$365$	0	0
$366$	0	0
$367$	4.58359	0.239262	0.119631	−	0.992818i	$-0.461829\pi$
$367$	4.58359	0.239262	0.119631	+	0.992818i	$0.461829\pi$
$368$	−4.85410	−0.253038
$369$	0	0
$370$	0	0
$371$	3.32624	0.172690
$372$	0	0
$373$	30.5066	1.57957	0.789785	−	0.613383i	$-0.210192\pi$
$373$	30.5066	1.57957	0.789785	+	0.613383i	$0.210192\pi$
$374$	1.23607	0.0639156
$375$	0	0
$376$	1.61803	0.0834437
$377$	13.4164	0.690980
$378$	0	0
$379$	−15.9787	−0.820771	−0.410386	−	0.911912i	$-0.634606\pi$
$379$	−15.9787	−0.820771	−0.410386	+	0.911912i	$0.634606\pi$
$380$	0	0
$381$	0	0
$382$	−16.9443	−0.866944
$383$	−37.0902	−1.89522	−0.947610	−	0.319431i	$-0.896508\pi$
$383$	−37.0902	−1.89522	−0.947610	+	0.319431i	$0.896508\pi$
$384$	0	0
$385$	0	0
$386$	4.29180	0.218447
$387$	0	0
$388$	9.70820	0.492859
$389$	−12.7639	−0.647157	−0.323579	−	0.946201i	$-0.604886\pi$
$389$	−12.7639	−0.647157	−0.323579	+	0.946201i	$0.604886\pi$
$390$	0	0
$391$	3.70820	0.187532
$392$	6.61803	0.334261
$393$	0	0
$394$	6.94427	0.349847
$395$	0	0
$396$	0	0
$397$	17.2705	0.866782	0.433391	−	0.901206i	$-0.357317\pi$
$397$	17.2705	0.866782	0.433391	+	0.901206i	$0.357317\pi$
$398$	6.18034	0.309792
$399$	0	0
$400$	0	0
$401$	−2.32624	−0.116167	−0.0580834	−	0.998312i	$-0.518499\pi$
$401$	−2.32624	−0.116167	−0.0580834	+	0.998312i	$0.518499\pi$
$402$	0	0
$403$	−12.0000	−0.597763
$404$	−12.6525	−0.629484
$405$	0	0
$406$	1.70820	0.0847767
$407$	−15.4721	−0.766925
$408$	0	0
$409$	−14.1459	−0.699470	−0.349735	−	0.936849i	$-0.613728\pi$
$409$	−14.1459	−0.699470	−0.349735	+	0.936849i	$0.613728\pi$
$410$	0	0
$411$	0	0
$412$	−8.56231	−0.421835
$413$	−8.09017	−0.398091
$414$	0	0
$415$	0	0
$416$	−4.85410	−0.237992
$417$	0	0
$418$	−9.47214	−0.463297
$419$	−11.0557	−0.540108	−0.270054	−	0.962845i	$-0.587042\pi$
$419$	−11.0557	−0.540108	−0.270054	+	0.962845i	$0.587042\pi$
$420$	0	0
$421$	18.1803	0.886056	0.443028	−	0.896508i	$-0.353904\pi$
$421$	18.1803	0.886056	0.443028	+	0.896508i	$0.353904\pi$
$422$	17.2705	0.840715
$423$	0	0
$424$	5.38197	0.261371
$425$	0	0
$426$	0	0
$427$	6.00000	0.290360
$428$	−6.94427	−0.335664
$429$	0	0
$430$	0	0
$431$	−15.8197	−0.762006	−0.381003	−	0.924574i	$-0.624421\pi$
$431$	−15.8197	−0.762006	−0.381003	+	0.924574i	$0.624421\pi$
$432$	0	0
$433$	17.4164	0.836979	0.418490	−	0.908222i	$-0.362560\pi$
$433$	17.4164	0.836979	0.418490	+	0.908222i	$0.362560\pi$
$434$	−1.52786	−0.0733398
$435$	0	0
$436$	−6.18034	−0.295985
$437$	−28.4164	−1.35934
$438$	0	0
$439$	1.70820	0.0815281	0.0407641	−	0.999169i	$-0.487021\pi$
$439$	1.70820	0.0815281	0.0407641	+	0.999169i	$0.487021\pi$
$440$	0	0
$441$	0	0
$442$	3.70820	0.176381
$443$	−13.5967	−0.646001	−0.323000	−	0.946399i	$-0.604692\pi$
$443$	−13.5967	−0.646001	−0.323000	+	0.946399i	$0.604692\pi$
$444$	0	0
$445$	0	0
$446$	−24.0000	−1.13643
$447$	0	0
$448$	−0.618034	−0.0291994
$449$	−28.2148	−1.33154	−0.665769	−	0.746158i	$-0.731896\pi$
$449$	−28.2148	−1.33154	−0.665769	+	0.746158i	$0.731896\pi$
$450$	0	0
$451$	15.1803	0.714814
$452$	3.23607	0.152212
$453$	0	0
$454$	−28.1803	−1.32257
$455$	0	0
$456$	0	0
$457$	−38.8328	−1.81652	−0.908261	−	0.418404i	$-0.862590\pi$
$457$	−38.8328	−1.81652	−0.908261	+	0.418404i	$0.862590\pi$
$458$	−1.70820	−0.0798191
$459$	0	0
$460$	0	0
$461$	−38.1803	−1.77824	−0.889118	−	0.457678i	$-0.848681\pi$
$461$	−38.1803	−1.77824	−0.889118	+	0.457678i	$0.848681\pi$
$462$	0	0
$463$	−31.7771	−1.47681	−0.738403	−	0.674359i	$-0.764420\pi$
$463$	−31.7771	−1.47681	−0.738403	+	0.674359i	$0.764420\pi$
$464$	2.76393	0.128312
$465$	0	0
$466$	−19.4164	−0.899448
$467$	−15.2361	−0.705041	−0.352521	−	0.935804i	$-0.614675\pi$
$467$	−15.2361	−0.705041	−0.352521	+	0.935804i	$0.614675\pi$
$468$	0	0
$469$	2.29180	0.105825
$470$	0	0
$471$	0	0
$472$	−13.0902	−0.602524
$473$	9.23607	0.424675
$474$	0	0
$475$	0	0
$476$	0.472136	0.0216403
$477$	0	0
$478$	8.94427	0.409101
$479$	−8.29180	−0.378862	−0.189431	−	0.981894i	$-0.560664\pi$
$479$	−8.29180	−0.378862	−0.189431	+	0.981894i	$0.560664\pi$
$480$	0	0
$481$	−46.4164	−2.11641
$482$	7.14590	0.325487
$483$	0	0
$484$	−8.38197	−0.380998
$485$	0	0
$486$	0	0
$487$	−17.3262	−0.785127	−0.392563	−	0.919725i	$-0.628412\pi$
$487$	−17.3262	−0.785127	−0.392563	+	0.919725i	$0.628412\pi$
$488$	9.70820	0.439470
$489$	0	0
$490$	0	0
$491$	18.9787	0.856497	0.428249	−	0.903661i	$-0.359131\pi$
$491$	18.9787	0.856497	0.428249	+	0.903661i	$0.359131\pi$
$492$	0	0
$493$	−2.11146	−0.0950952
$494$	−28.4164	−1.27851
$495$	0	0
$496$	−2.47214	−0.111002
$497$	−2.18034	−0.0978016
$498$	0	0
$499$	−5.72949	−0.256487	−0.128244	−	0.991743i	$-0.540934\pi$
$499$	−5.72949	−0.256487	−0.128244	+	0.991743i	$0.540934\pi$
$500$	0	0
$501$	0	0
$502$	20.9443	0.934789
$503$	13.0344	0.581177	0.290589	−	0.956848i	$-0.406149\pi$
$503$	13.0344	0.581177	0.290589	+	0.956848i	$0.406149\pi$
$504$	0	0
$505$	0	0
$506$	7.85410	0.349157
$507$	0	0
$508$	11.4164	0.506521
$509$	−30.6525	−1.35865	−0.679324	−	0.733839i	$-0.737727\pi$
$509$	−30.6525	−1.35865	−0.679324	+	0.733839i	$0.737727\pi$
$510$	0	0
$511$	−8.00000	−0.353899
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	13.5279	0.596689
$515$	0	0
$516$	0	0
$517$	−2.61803	−0.115141
$518$	−5.90983	−0.259663
$519$	0	0
$520$	0	0
$521$	38.4508	1.68456	0.842281	−	0.539038i	$-0.181212\pi$
$521$	38.4508	1.68456	0.842281	+	0.539038i	$0.181212\pi$
$522$	0	0
$523$	29.5279	1.29116	0.645582	−	0.763691i	$-0.276615\pi$
$523$	29.5279	1.29116	0.645582	+	0.763691i	$0.276615\pi$
$524$	−3.90983	−0.170802
$525$	0	0
$526$	−14.6180	−0.637377
$527$	1.88854	0.0822663
$528$	0	0
$529$	0.562306	0.0244481
$530$	0	0
$531$	0	0
$532$	−3.61803	−0.156862
$533$	45.5410	1.97260
$534$	0	0
$535$	0	0
$536$	3.70820	0.160170
$537$	0	0
$538$	−13.4164	−0.578422
$539$	−10.7082	−0.461235
$540$	0	0
$541$	45.4164	1.95260	0.976302	−	0.216413i	$-0.0694357\pi$
$541$	45.4164	1.95260	0.976302	+	0.216413i	$0.0694357\pi$
$542$	−25.4164	−1.09173
$543$	0	0
$544$	0.763932	0.0327533
$545$	0	0
$546$	0	0
$547$	23.1246	0.988737	0.494369	−	0.869252i	$-0.335399\pi$
$547$	23.1246	0.988737	0.494369	+	0.869252i	$0.335399\pi$
$548$	3.05573	0.130534
$549$	0	0
$550$	0	0
$551$	16.1803	0.689306
$552$	0	0
$553$	−8.29180	−0.352603
$554$	12.8541	0.546118
$555$	0	0
$556$	12.5623	0.532760
$557$	−11.6180	−0.492272	−0.246136	−	0.969235i	$-0.579161\pi$
$557$	−11.6180	−0.492272	−0.246136	+	0.969235i	$0.579161\pi$
$558$	0	0
$559$	27.7082	1.17193
$560$	0	0
$561$	0	0
$562$	−6.09017	−0.256898
$563$	−2.94427	−0.124086	−0.0620431	−	0.998073i	$-0.519762\pi$
$563$	−2.94427	−0.124086	−0.0620431	+	0.998073i	$0.519762\pi$
$564$	0	0
$565$	0	0
$566$	−30.1803	−1.26857
$567$	0	0
$568$	−3.52786	−0.148026
$569$	14.6738	0.615156	0.307578	−	0.951523i	$-0.400481\pi$
$569$	14.6738	0.615156	0.307578	+	0.951523i	$0.400481\pi$
$570$	0	0
$571$	−26.2148	−1.09705	−0.548527	−	0.836133i	$-0.684811\pi$
$571$	−26.2148	−1.09705	−0.548527	+	0.836133i	$0.684811\pi$
$572$	7.85410	0.328397
$573$	0	0
$574$	5.79837	0.242019
$575$	0	0
$576$	0	0
$577$	−33.7082	−1.40329	−0.701645	−	0.712526i	$-0.747551\pi$
$577$	−33.7082	−1.40329	−0.701645	+	0.712526i	$0.747551\pi$
$578$	16.4164	0.682833
$579$	0	0
$580$	0	0
$581$	−9.23607	−0.383177
$582$	0	0
$583$	−8.70820	−0.360657
$584$	−12.9443	−0.535638
$585$	0	0
$586$	22.0902	0.912537
$587$	15.8197	0.652947	0.326474	−	0.945206i	$-0.394140\pi$
$587$	15.8197	0.652947	0.326474	+	0.945206i	$0.394140\pi$
$588$	0	0
$589$	−14.4721	−0.596314
$590$	0	0
$591$	0	0
$592$	−9.56231	−0.393008
$593$	−1.63932	−0.0673188	−0.0336594	−	0.999433i	$-0.510716\pi$
$593$	−1.63932	−0.0673188	−0.0336594	+	0.999433i	$0.510716\pi$
$594$	0	0
$595$	0	0
$596$	13.4164	0.549557
$597$	0	0
$598$	23.5623	0.963534
$599$	−48.5410	−1.98333	−0.991666	−	0.128834i	$-0.958876\pi$
$599$	−48.5410	−1.98333	−0.991666	+	0.128834i	$0.958876\pi$
$600$	0	0
$601$	−32.2705	−1.31634	−0.658171	−	0.752869i	$-0.728670\pi$
$601$	−32.2705	−1.31634	−0.658171	+	0.752869i	$0.728670\pi$
$602$	3.52786	0.143785
$603$	0	0
$604$	10.2918	0.418767
$605$	0	0
$606$	0	0
$607$	20.6869	0.839656	0.419828	−	0.907604i	$-0.362090\pi$
$607$	20.6869	0.839656	0.419828	+	0.907604i	$0.362090\pi$
$608$	−5.85410	−0.237415
$609$	0	0
$610$	0	0
$611$	−7.85410	−0.317743
$612$	0	0
$613$	−22.3820	−0.903999	−0.452000	−	0.892018i	$-0.649289\pi$
$613$	−22.3820	−0.903999	−0.452000	+	0.892018i	$0.649289\pi$
$614$	−20.7639	−0.837964
$615$	0	0
$616$	1.00000	0.0402911
$617$	−29.3050	−1.17977	−0.589886	−	0.807486i	$-0.700828\pi$
$617$	−29.3050	−1.17977	−0.589886	+	0.807486i	$0.700828\pi$
$618$	0	0
$619$	−18.0902	−0.727105	−0.363553	−	0.931574i	$-0.618436\pi$
$619$	−18.0902	−0.727105	−0.363553	+	0.931574i	$0.618436\pi$
$620$	0	0
$621$	0	0
$622$	12.0000	0.481156
$623$	−11.1803	−0.447931
$624$	0	0
$625$	0	0
$626$	−2.29180	−0.0915986
$627$	0	0
$628$	−10.9443	−0.436724
$629$	7.30495	0.291267
$630$	0	0
$631$	−1.81966	−0.0724395	−0.0362198	−	0.999344i	$-0.511532\pi$
$631$	−1.81966	−0.0724395	−0.0362198	+	0.999344i	$0.511532\pi$
$632$	−13.4164	−0.533676
$633$	0	0
$634$	−1.14590	−0.0455094
$635$	0	0
$636$	0	0
$637$	−32.1246	−1.27282
$638$	−4.47214	−0.177054
$639$	0	0
$640$	0	0
$641$	1.74265	0.0688304	0.0344152	−	0.999408i	$-0.489043\pi$
$641$	1.74265	0.0688304	0.0344152	+	0.999408i	$0.489043\pi$
$642$	0	0
$643$	−7.05573	−0.278251	−0.139125	−	0.990275i	$-0.544429\pi$
$643$	−7.05573	−0.278251	−0.139125	+	0.990275i	$0.544429\pi$
$644$	3.00000	0.118217
$645$	0	0
$646$	4.47214	0.175954
$647$	5.09017	0.200115	0.100058	−	0.994982i	$-0.468097\pi$
$647$	5.09017	0.200115	0.100058	+	0.994982i	$0.468097\pi$
$648$	0	0
$649$	21.1803	0.831401
$650$	0	0
$651$	0	0
$652$	−4.94427	−0.193633
$653$	−13.1459	−0.514439	−0.257219	−	0.966353i	$-0.582806\pi$
$653$	−13.1459	−0.514439	−0.257219	+	0.966353i	$0.582806\pi$
$654$	0	0
$655$	0	0
$656$	9.38197	0.366304
$657$	0	0
$658$	−1.00000	−0.0389841
$659$	1.25735	0.0489796	0.0244898	−	0.999700i	$-0.492204\pi$
$659$	1.25735	0.0489796	0.0244898	+	0.999700i	$0.492204\pi$
$660$	0	0
$661$	2.65248	0.103169	0.0515847	−	0.998669i	$-0.483573\pi$
$661$	2.65248	0.103169	0.0515847	+	0.998669i	$0.483573\pi$
$662$	2.47214	0.0960823
$663$	0	0
$664$	−14.9443	−0.579950
$665$	0	0
$666$	0	0
$667$	−13.4164	−0.519485
$668$	7.85410	0.303884
$669$	0	0
$670$	0	0
$671$	−15.7082	−0.606408
$672$	0	0
$673$	−21.5279	−0.829838	−0.414919	−	0.909858i	$-0.636190\pi$
$673$	−21.5279	−0.829838	−0.414919	+	0.909858i	$0.636190\pi$
$674$	19.8885	0.766078
$675$	0	0
$676$	10.5623	0.406243
$677$	7.85410	0.301858	0.150929	−	0.988545i	$-0.451774\pi$
$677$	7.85410	0.301858	0.150929	+	0.988545i	$0.451774\pi$
$678$	0	0
$679$	−6.00000	−0.230259
$680$	0	0
$681$	0	0
$682$	4.00000	0.153168
$683$	24.9443	0.954466	0.477233	−	0.878777i	$-0.341640\pi$
$683$	24.9443	0.954466	0.477233	+	0.878777i	$0.341640\pi$
$684$	0	0
$685$	0	0
$686$	−8.41641	−0.321340
$687$	0	0
$688$	5.70820	0.217623
$689$	−26.1246	−0.995268
$690$	0	0
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	14.6180	0.555694
$693$	0	0
$694$	−26.0689	−0.989561
$695$	0	0
$696$	0	0
$697$	−7.16718	−0.271476
$698$	26.8328	1.01564
$699$	0	0
$700$	0	0
$701$	25.2361	0.953153	0.476577	−	0.879133i	$-0.341878\pi$
$701$	25.2361	0.953153	0.476577	+	0.879133i	$0.341878\pi$
$702$	0	0
$703$	−55.9787	−2.11128
$704$	1.61803	0.0609820
$705$	0	0
$706$	15.7082	0.591186
$707$	7.81966	0.294089
$708$	0	0
$709$	−2.76393	−0.103802	−0.0519008	−	0.998652i	$-0.516528\pi$
$709$	−2.76393	−0.103802	−0.0519008	+	0.998652i	$0.516528\pi$
$710$	0	0
$711$	0	0
$712$	−18.0902	−0.677958
$713$	12.0000	0.449404
$714$	0	0
$715$	0	0
$716$	−22.0344	−0.823466
$717$	0	0
$718$	4.47214	0.166899
$719$	27.8885	1.04007	0.520034	−	0.854146i	$-0.325919\pi$
$719$	27.8885	1.04007	0.520034	+	0.854146i	$0.325919\pi$
$720$	0	0
$721$	5.29180	0.197077
$722$	−15.2705	−0.568310
$723$	0	0
$724$	25.4164	0.944593
$725$	0	0
$726$	0	0
$727$	44.1033	1.63570	0.817851	−	0.575430i	$-0.195165\pi$
$727$	44.1033	1.63570	0.817851	+	0.575430i	$0.195165\pi$
$728$	3.00000	0.111187
$729$	0	0
$730$	0	0
$731$	−4.36068	−0.161286
$732$	0	0
$733$	−31.3262	−1.15706	−0.578530	−	0.815661i	$-0.696374\pi$
$733$	−31.3262	−1.15706	−0.578530	+	0.815661i	$0.696374\pi$
$734$	−4.58359	−0.169183
$735$	0	0
$736$	4.85410	0.178925
$737$	−6.00000	−0.221013
$738$	0	0
$739$	10.7295	0.394691	0.197345	−	0.980334i	$-0.436768\pi$
$739$	10.7295	0.394691	0.197345	+	0.980334i	$0.436768\pi$
$740$	0	0
$741$	0	0
$742$	−3.32624	−0.122110
$743$	−4.85410	−0.178080	−0.0890399	−	0.996028i	$-0.528380\pi$
$743$	−4.85410	−0.178080	−0.0890399	+	0.996028i	$0.528380\pi$
$744$	0	0
$745$	0	0
$746$	−30.5066	−1.11693
$747$	0	0
$748$	−1.23607	−0.0451951
$749$	4.29180	0.156819
$750$	0	0
$751$	−47.5967	−1.73683	−0.868415	−	0.495838i	$-0.834861\pi$
$751$	−47.5967	−1.73683	−0.868415	+	0.495838i	$0.834861\pi$
$752$	−1.61803	−0.0590036
$753$	0	0
$754$	−13.4164	−0.488597
$755$	0	0
$756$	0	0
$757$	−21.7984	−0.792275	−0.396138	−	0.918191i	$-0.629650\pi$
$757$	−21.7984	−0.792275	−0.396138	+	0.918191i	$0.629650\pi$
$758$	15.9787	0.580373
$759$	0	0
$760$	0	0
$761$	10.0344	0.363748	0.181874	−	0.983322i	$-0.441784\pi$
$761$	10.0344	0.363748	0.181874	+	0.983322i	$0.441784\pi$
$762$	0	0
$763$	3.81966	0.138281
$764$	16.9443	0.613022
$765$	0	0
$766$	37.0902	1.34012
$767$	63.5410	2.29433
$768$	0	0
$769$	−15.7295	−0.567220	−0.283610	−	0.958940i	$-0.591532\pi$
$769$	−15.7295	−0.567220	−0.283610	+	0.958940i	$0.591532\pi$
$770$	0	0
$771$	0	0
$772$	−4.29180	−0.154465
$773$	−2.94427	−0.105898	−0.0529491	−	0.998597i	$-0.516862\pi$
$773$	−2.94427	−0.105898	−0.0529491	+	0.998597i	$0.516862\pi$
$774$	0	0
$775$	0	0
$776$	−9.70820	−0.348504
$777$	0	0
$778$	12.7639	0.457609
$779$	54.9230	1.96782
$780$	0	0
$781$	5.70820	0.204256
$782$	−3.70820	−0.132605
$783$	0	0
$784$	−6.61803	−0.236358
$785$	0	0
$786$	0	0
$787$	−25.4164	−0.905997	−0.452999	−	0.891511i	$-0.649646\pi$
$787$	−25.4164	−0.905997	−0.452999	+	0.891511i	$0.649646\pi$
$788$	−6.94427	−0.247379
$789$	0	0
$790$	0	0
$791$	−2.00000	−0.0711118
$792$	0	0
$793$	−47.1246	−1.67344
$794$	−17.2705	−0.612907
$795$	0	0
$796$	−6.18034	−0.219056
$797$	5.09017	0.180303	0.0901515	−	0.995928i	$-0.471265\pi$
$797$	5.09017	0.180303	0.0901515	+	0.995928i	$0.471265\pi$
$798$	0	0
$799$	1.23607	0.0437289
$800$	0	0
$801$	0	0
$802$	2.32624	0.0821423
$803$	20.9443	0.739107
$804$	0	0
$805$	0	0
$806$	12.0000	0.422682
$807$	0	0
$808$	12.6525	0.445113
$809$	−15.9787	−0.561782	−0.280891	−	0.959740i	$-0.590630\pi$
$809$	−15.9787	−0.561782	−0.280891	+	0.959740i	$0.590630\pi$
$810$	0	0
$811$	53.1033	1.86471	0.932355	−	0.361544i	$-0.117750\pi$
$811$	53.1033	1.86471	0.932355	+	0.361544i	$0.117750\pi$
$812$	−1.70820	−0.0599462
$813$	0	0
$814$	15.4721	0.542298
$815$	0	0
$816$	0	0
$817$	33.4164	1.16909
$818$	14.1459	0.494600
$819$	0	0
$820$	0	0
$821$	16.9443	0.591359	0.295680	−	0.955287i	$-0.404454\pi$
$821$	16.9443	0.591359	0.295680	+	0.955287i	$0.404454\pi$
$822$	0	0
$823$	−8.03444	−0.280063	−0.140032	−	0.990147i	$-0.544720\pi$
$823$	−8.03444	−0.280063	−0.140032	+	0.990147i	$0.544720\pi$
$824$	8.56231	0.298282
$825$	0	0
$826$	8.09017	0.281493
$827$	−36.5410	−1.27066	−0.635328	−	0.772243i	$-0.719135\pi$
$827$	−36.5410	−1.27066	−0.635328	+	0.772243i	$0.719135\pi$
$828$	0	0
$829$	−40.2492	−1.39791	−0.698957	−	0.715164i	$-0.746352\pi$
$829$	−40.2492	−1.39791	−0.698957	+	0.715164i	$0.746352\pi$
$830$	0	0
$831$	0	0
$832$	4.85410	0.168286
$833$	5.05573	0.175171
$834$	0	0
$835$	0	0
$836$	9.47214	0.327601
$837$	0	0
$838$	11.0557	0.381914
$839$	16.1803	0.558607	0.279304	−	0.960203i	$-0.409896\pi$
$839$	16.1803	0.558607	0.279304	+	0.960203i	$0.409896\pi$
$840$	0	0
$841$	−21.3607	−0.736575
$842$	−18.1803	−0.626536
$843$	0	0
$844$	−17.2705	−0.594475
$845$	0	0
$846$	0	0
$847$	5.18034	0.177999
$848$	−5.38197	−0.184817
$849$	0	0
$850$	0	0
$851$	46.4164	1.59113
$852$	0	0
$853$	6.43769	0.220422	0.110211	−	0.993908i	$-0.464847\pi$
$853$	6.43769	0.220422	0.110211	+	0.993908i	$0.464847\pi$
$854$	−6.00000	−0.205316
$855$	0	0
$856$	6.94427	0.237350
$857$	8.18034	0.279435	0.139718	−	0.990191i	$-0.455381\pi$
$857$	8.18034	0.279435	0.139718	+	0.990191i	$0.455381\pi$
$858$	0	0
$859$	0.978714	0.0333933	0.0166966	−	0.999861i	$-0.494685\pi$
$859$	0.978714	0.0333933	0.0166966	+	0.999861i	$0.494685\pi$
$860$	0	0
$861$	0	0
$862$	15.8197	0.538820
$863$	2.90983	0.0990518	0.0495259	−	0.998773i	$-0.484229\pi$
$863$	2.90983	0.0990518	0.0495259	+	0.998773i	$0.484229\pi$
$864$	0	0
$865$	0	0
$866$	−17.4164	−0.591834
$867$	0	0
$868$	1.52786	0.0518591
$869$	21.7082	0.736400
$870$	0	0
$871$	−18.0000	−0.609907
$872$	6.18034	0.209293
$873$	0	0
$874$	28.4164	0.961199
$875$	0	0
$876$	0	0
$877$	−7.72949	−0.261006	−0.130503	−	0.991448i	$-0.541659\pi$
$877$	−7.72949	−0.261006	−0.130503	+	0.991448i	$0.541659\pi$
$878$	−1.70820	−0.0576491
$879$	0	0
$880$	0	0
$881$	8.72949	0.294104	0.147052	−	0.989129i	$-0.453022\pi$
$881$	8.72949	0.294104	0.147052	+	0.989129i	$0.453022\pi$
$882$	0	0
$883$	38.4721	1.29469	0.647345	−	0.762197i	$-0.275879\pi$
$883$	38.4721	1.29469	0.647345	+	0.762197i	$0.275879\pi$
$884$	−3.70820	−0.124720
$885$	0	0
$886$	13.5967	0.456792
$887$	−34.9098	−1.17216	−0.586079	−	0.810254i	$-0.699329\pi$
$887$	−34.9098	−1.17216	−0.586079	+	0.810254i	$0.699329\pi$
$888$	0	0
$889$	−7.05573	−0.236642
$890$	0	0
$891$	0	0
$892$	24.0000	0.803579
$893$	−9.47214	−0.316973
$894$	0	0
$895$	0	0
$896$	0.618034	0.0206471
$897$	0	0
$898$	28.2148	0.941539
$899$	−6.83282	−0.227887
$900$	0	0
$901$	4.11146	0.136972
$902$	−15.1803	−0.505450
$903$	0	0
$904$	−3.23607	−0.107630
$905$	0	0
$906$	0	0
$907$	13.1246	0.435796	0.217898	−	0.975972i	$-0.430080\pi$
$907$	13.1246	0.435796	0.217898	+	0.975972i	$0.430080\pi$
$908$	28.1803	0.935197
$909$	0	0
$910$	0	0
$911$	35.8885	1.18904	0.594520	−	0.804081i	$-0.297342\pi$
$911$	35.8885	1.18904	0.594520	+	0.804081i	$0.297342\pi$
$912$	0	0
$913$	24.1803	0.800252
$914$	38.8328	1.28448
$915$	0	0
$916$	1.70820	0.0564406
$917$	2.41641	0.0797968
$918$	0	0
$919$	−29.5967	−0.976307	−0.488153	−	0.872758i	$-0.662329\pi$
$919$	−29.5967	−0.976307	−0.488153	+	0.872758i	$0.662329\pi$
$920$	0	0
$921$	0	0
$922$	38.1803	1.25740
$923$	17.1246	0.563663
$924$	0	0
$925$	0	0
$926$	31.7771	1.04426
$927$	0	0
$928$	−2.76393	−0.0907305
$929$	30.4508	0.999060	0.499530	−	0.866297i	$-0.333506\pi$
$929$	30.4508	0.999060	0.499530	+	0.866297i	$0.333506\pi$
$930$	0	0
$931$	−38.7426	−1.26974
$932$	19.4164	0.636006
$933$	0	0
$934$	15.2361	0.498539
$935$	0	0
$936$	0	0
$937$	−27.1246	−0.886122	−0.443061	−	0.896491i	$-0.646108\pi$
$937$	−27.1246	−0.886122	−0.443061	+	0.896491i	$0.646108\pi$
$938$	−2.29180	−0.0748298
$939$	0	0
$940$	0	0
$941$	−17.1246	−0.558246	−0.279123	−	0.960255i	$-0.590044\pi$
$941$	−17.1246	−0.558246	−0.279123	+	0.960255i	$0.590044\pi$
$942$	0	0
$943$	−45.5410	−1.48302
$944$	13.0902	0.426049
$945$	0	0
$946$	−9.23607	−0.300290
$947$	29.8885	0.971247	0.485624	−	0.874168i	$-0.338593\pi$
$947$	29.8885	0.971247	0.485624	+	0.874168i	$0.338593\pi$
$948$	0	0
$949$	62.8328	2.03964
$950$	0	0
$951$	0	0
$952$	−0.472136	−0.0153020
$953$	−60.1803	−1.94943	−0.974716	−	0.223446i	$-0.928269\pi$
$953$	−60.1803	−1.94943	−0.974716	+	0.223446i	$0.928269\pi$
$954$	0	0
$955$	0	0
$956$	−8.94427	−0.289278
$957$	0	0
$958$	8.29180	0.267896
$959$	−1.88854	−0.0609843
$960$	0	0
$961$	−24.8885	−0.802856
$962$	46.4164	1.49653
$963$	0	0
$964$	−7.14590	−0.230154
$965$	0	0
$966$	0	0
$967$	13.3262	0.428543	0.214271	−	0.976774i	$-0.431262\pi$
$967$	13.3262	0.428543	0.214271	+	0.976774i	$0.431262\pi$
$968$	8.38197	0.269407
$969$	0	0
$970$	0	0
$971$	2.02129	0.0648662	0.0324331	−	0.999474i	$-0.489674\pi$
$971$	2.02129	0.0648662	0.0324331	+	0.999474i	$0.489674\pi$
$972$	0	0
$973$	−7.76393	−0.248900
$974$	17.3262	0.555168
$975$	0	0
$976$	−9.70820	−0.310752
$977$	−4.58359	−0.146642	−0.0733211	−	0.997308i	$-0.523360\pi$
$977$	−4.58359	−0.146642	−0.0733211	+	0.997308i	$0.523360\pi$
$978$	0	0
$979$	29.2705	0.935490
$980$	0	0
$981$	0	0
$982$	−18.9787	−0.605635
$983$	−24.4508	−0.779861	−0.389930	−	0.920844i	$-0.627501\pi$
$983$	−24.4508	−0.779861	−0.389930	+	0.920844i	$0.627501\pi$
$984$	0	0
$985$	0	0
$986$	2.11146	0.0672425
$987$	0	0
$988$	28.4164	0.904046
$989$	−27.7082	−0.881070
$990$	0	0
$991$	42.0000	1.33417	0.667087	−	0.744980i	$-0.267541\pi$
$991$	42.0000	1.33417	0.667087	+	0.744980i	$0.267541\pi$
$992$	2.47214	0.0784904
$993$	0	0
$994$	2.18034	0.0691562
$995$	0	0
$996$	0	0
$997$	22.1459	0.701368	0.350684	−	0.936494i	$-0.385949\pi$
$997$	22.1459	0.701368	0.350684	+	0.936494i	$0.385949\pi$
$998$	5.72949	0.181364
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2250.2.a.e.1.1		2
3.2	odd	2		750.2.a.g.1.1	yes	2
5.2	odd	4		2250.2.c.d.1999.1		4
5.3	odd	4		2250.2.c.d.1999.4		4
5.4	even	2		2250.2.a.l.1.2		2
12.11	even	2		6000.2.a.f.1.2		2
15.2	even	4		750.2.c.d.499.3		4
15.8	even	4		750.2.c.d.499.2		4
15.14	odd	2		750.2.a.b.1.2	✓	2
60.23	odd	4		6000.2.f.h.1249.1		4
60.47	odd	4		6000.2.f.h.1249.4		4
60.59	even	2		6000.2.a.w.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
750.2.a.b.1.2	✓	2	15.14	odd	2
750.2.a.g.1.1	yes	2	3.2	odd	2
750.2.c.d.499.2		4	15.8	even	4
750.2.c.d.499.3		4	15.2	even	4
2250.2.a.e.1.1		2	1.1	even	1	trivial
2250.2.a.l.1.2		2	5.4	even	2
2250.2.c.d.1999.1		4	5.2	odd	4
2250.2.c.d.1999.4		4	5.3	odd	4
6000.2.a.f.1.2		2	12.11	even	2
6000.2.a.w.1.1		2	60.59	even	2
6000.2.f.h.1249.1		4	60.23	odd	4
6000.2.f.h.1249.4		4	60.47	odd	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2250 = 2 \cdot 3^{2} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2250.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -0.618034 q^{7} -1.00000 q^{8} +1.61803 q^{11} +4.85410 q^{13} +0.618034 q^{14} +1.00000 q^{16} -0.763932 q^{17} +5.85410 q^{19} -1.61803 q^{22} -4.85410 q^{23} -4.85410 q^{26} -0.618034 q^{28} +2.76393 q^{29} -2.47214 q^{31} -1.00000 q^{32} +0.763932 q^{34} -9.56231 q^{37} -5.85410 q^{38} +9.38197 q^{41} +5.70820 q^{43} +1.61803 q^{44} +4.85410 q^{46} -1.61803 q^{47} -6.61803 q^{49} +4.85410 q^{52} -5.38197 q^{53} +0.618034 q^{56} -2.76393 q^{58} +13.0902 q^{59} -9.70820 q^{61} +2.47214 q^{62} +1.00000 q^{64} -3.70820 q^{67} -0.763932 q^{68} +3.52786 q^{71} +12.9443 q^{73} +9.56231 q^{74} +5.85410 q^{76} -1.00000 q^{77} +13.4164 q^{79} -9.38197 q^{82} +14.9443 q^{83} -5.70820 q^{86} -1.61803 q^{88} +18.0902 q^{89} -3.00000 q^{91} -4.85410 q^{92} +1.61803 q^{94} +9.70820 q^{97} +6.61803 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + q^{7} - 2 q^{8} + q^{11} + 3 q^{13} - q^{14} + 2 q^{16} - 6 q^{17} + 5 q^{19} - q^{22} - 3 q^{23} - 3 q^{26} + q^{28} + 10 q^{29} + 4 q^{31} - 2 q^{32} + 6 q^{34} + q^{37}+ \cdots + 11 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 + q^7 - 2 * q^8 + q^11 + 3 * q^13 - q^14 + 2 * q^16 - 6 * q^17 + 5 * q^19 - q^22 - 3 * q^23 - 3 * q^26 + q^28 + 10 * q^29 + 4 * q^31 - 2 * q^32 + 6 * q^34 + q^37 - 5 * q^38 + 21 * q^41 - 2 * q^43 + q^44 + 3 * q^46 - q^47 - 11 * q^49 + 3 * q^52 - 13 * q^53 - q^56 - 10 * q^58 + 15 * q^59 - 6 * q^61 - 4 * q^62 + 2 * q^64 + 6 * q^67 - 6 * q^68 + 16 * q^71 + 8 * q^73 - q^74 + 5 * q^76 - 2 * q^77 - 21 * q^82 + 12 * q^83 + 2 * q^86 - q^88 + 25 * q^89 - 6 * q^91 - 3 * q^92 + q^94 + 6 * q^97 + 11 * q^98

Introduction

L-functions

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