LMFDB - Embedded newform 1458.2.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1458,2,Mod(1,1458)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1458, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1458.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1458 = 2 \cdot 3^{6} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1458.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:	$11.6421886147$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\Q(\zeta_{36})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 6x^{4} + 9x^{2} - 3 $ x^6 - 6x^4 + 9x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.28558$ of defining polynomial
Character	$\chi$	$=$	1458.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -3.07935 q^{5} -0.254239 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -3.07935 q^{5} -0.254239 q^{7} -1.00000 q^{8} +3.07935 q^{10} -3.45369 q^{11} +5.63995 q^{13} +0.254239 q^{14} +1.00000 q^{16} +5.01396 q^{17} -0.904786 q^{19} -3.07935 q^{20} +3.45369 q^{22} +5.64985 q^{23} +4.48238 q^{25} -5.63995 q^{26} -0.254239 q^{28} -1.76677 q^{29} -10.6587 q^{31} -1.00000 q^{32} -5.01396 q^{34} +0.782889 q^{35} +8.83464 q^{37} +0.904786 q^{38} +3.07935 q^{40} -5.71415 q^{41} -8.56574 q^{43} -3.45369 q^{44} -5.64985 q^{46} -12.5309 q^{47} -6.93536 q^{49} -4.48238 q^{50} +5.63995 q^{52} +2.65282 q^{53} +10.6351 q^{55} +0.254239 q^{56} +1.76677 q^{58} +3.70138 q^{59} -3.77884 q^{61} +10.6587 q^{62} +1.00000 q^{64} -17.3673 q^{65} -7.58603 q^{67} +5.01396 q^{68} -0.782889 q^{70} -8.42685 q^{71} -6.99035 q^{73} -8.83464 q^{74} -0.904786 q^{76} +0.878062 q^{77} +13.1091 q^{79} -3.07935 q^{80} +5.71415 q^{82} -6.92438 q^{83} -15.4397 q^{85} +8.56574 q^{86} +3.45369 q^{88} -6.80373 q^{89} -1.43389 q^{91} +5.64985 q^{92} +12.5309 q^{94} +2.78615 q^{95} +8.62609 q^{97} +6.93536 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{2} + 6 q^{4} - 6 q^{5} - 6 q^{8}+O(q^{10}) $ 6 * q - 6 * q^2 + 6 * q^4 - 6 * q^5 - 6 * q^8	$ 6 q - 6 q^{2} + 6 q^{4} - 6 q^{5} - 6 q^{8} + 6 q^{10} - 6 q^{11} + 6 q^{13} + 6 q^{16} - 12 q^{17} + 6 q^{19} - 6 q^{20} + 6 q^{22} - 12 q^{23} + 6 q^{25} - 6 q^{26} - 6 q^{29} - 6 q^{31} - 6 q^{32} + 12 q^{34} - 12 q^{35} - 6 q^{38} + 6 q^{40} - 24 q^{41} - 6 q^{43} - 6 q^{44} + 12 q^{46} - 18 q^{47} + 6 q^{49} - 6 q^{50} + 6 q^{52} - 24 q^{53} - 18 q^{55} + 6 q^{58} - 12 q^{59} - 6 q^{61} + 6 q^{62} + 6 q^{64} - 12 q^{65} - 24 q^{67} - 12 q^{68} + 12 q^{70} + 6 q^{71} - 24 q^{73} + 6 q^{76} - 12 q^{77} - 12 q^{79} - 6 q^{80} + 24 q^{82} - 18 q^{83} + 6 q^{86} + 6 q^{88} - 12 q^{89} - 30 q^{91} - 12 q^{92} + 18 q^{94} + 6 q^{95} + 6 q^{97} - 6 q^{98}+O(q^{100}) $ 6 * q - 6 * q^2 + 6 * q^4 - 6 * q^5 - 6 * q^8 + 6 * q^10 - 6 * q^11 + 6 * q^13 + 6 * q^16 - 12 * q^17 + 6 * q^19 - 6 * q^20 + 6 * q^22 - 12 * q^23 + 6 * q^25 - 6 * q^26 - 6 * q^29 - 6 * q^31 - 6 * q^32 + 12 * q^34 - 12 * q^35 - 6 * q^38 + 6 * q^40 - 24 * q^41 - 6 * q^43 - 6 * q^44 + 12 * q^46 - 18 * q^47 + 6 * q^49 - 6 * q^50 + 6 * q^52 - 24 * q^53 - 18 * q^55 + 6 * q^58 - 12 * q^59 - 6 * q^61 + 6 * q^62 + 6 * q^64 - 12 * q^65 - 24 * q^67 - 12 * q^68 + 12 * q^70 + 6 * q^71 - 24 * q^73 + 6 * q^76 - 12 * q^77 - 12 * q^79 - 6 * q^80 + 24 * q^82 - 18 * q^83 + 6 * q^86 + 6 * q^88 - 12 * q^89 - 30 * q^91 - 12 * q^92 + 18 * q^94 + 6 * q^95 + 6 * q^97 - 6 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−3.07935	−1.37713	−0.688563	−	0.725177i	$-0.741758\pi$
$5$	−3.07935	−1.37713	−0.688563	+	0.725177i	$0.741758\pi$
$6$	0	0
$7$	−0.254239	−0.0960931	−0.0480466	−	0.998845i	$-0.515300\pi$
$7$	−0.254239	−0.0960931	−0.0480466	+	0.998845i	$0.515300\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	3.07935	0.973775
$11$	−3.45369	−1.04133	−0.520664	−	0.853762i	$-0.674316\pi$
$11$	−3.45369	−1.04133	−0.520664	+	0.853762i	$0.674316\pi$
$12$	0	0
$13$	5.63995	1.56424	0.782120	−	0.623128i	$-0.214139\pi$
$13$	5.63995	1.56424	0.782120	+	0.623128i	$0.214139\pi$
$14$	0.254239	0.0679481
$15$	0	0
$16$	1.00000	0.250000
$17$	5.01396	1.21606	0.608032	−	0.793912i	$-0.291959\pi$
$17$	5.01396	1.21606	0.608032	+	0.793912i	$0.291959\pi$
$18$	0	0
$19$	−0.904786	−0.207572	−0.103786	−	0.994600i	$-0.533096\pi$
$19$	−0.904786	−0.207572	−0.103786	+	0.994600i	$0.533096\pi$
$20$	−3.07935	−0.688563
$21$	0	0
$22$	3.45369	0.736330
$23$	5.64985	1.17807	0.589037	−	0.808106i	$-0.299507\pi$
$23$	5.64985	1.17807	0.589037	+	0.808106i	$0.299507\pi$
$24$	0	0
$25$	4.48238	0.896476
$26$	−5.63995	−1.10608
$27$	0	0
$28$	−0.254239	−0.0480466
$29$	−1.76677	−0.328080	−0.164040	−	0.986454i	$-0.552453\pi$
$29$	−1.76677	−0.328080	−0.164040	+	0.986454i	$0.552453\pi$
$30$	0	0
$31$	−10.6587	−1.91435	−0.957177	−	0.289505i	$-0.906509\pi$
$31$	−10.6587	−1.91435	−0.957177	+	0.289505i	$0.906509\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−5.01396	−0.859887
$35$	0.782889	0.132332
$36$	0	0
$37$	8.83464	1.45241	0.726203	−	0.687481i	$-0.241283\pi$
$37$	8.83464	1.45241	0.726203	+	0.687481i	$0.241283\pi$
$38$	0.904786	0.146776
$39$	0	0
$40$	3.07935	0.486888
$41$	−5.71415	−0.892401	−0.446200	−	0.894933i	$-0.647223\pi$
$41$	−5.71415	−0.892401	−0.446200	+	0.894933i	$0.647223\pi$
$42$	0	0
$43$	−8.56574	−1.30626	−0.653131	−	0.757245i	$-0.726545\pi$
$43$	−8.56574	−1.30626	−0.653131	+	0.757245i	$0.726545\pi$
$44$	−3.45369	−0.520664
$45$	0	0
$46$	−5.64985	−0.833024
$47$	−12.5309	−1.82782	−0.913908	−	0.405920i	$-0.866951\pi$
$47$	−12.5309	−1.82782	−0.913908	+	0.405920i	$0.866951\pi$
$48$	0	0
$49$	−6.93536	−0.990766
$50$	−4.48238	−0.633904
$51$	0	0
$52$	5.63995	0.782120
$53$	2.65282	0.364393	0.182196	−	0.983262i	$-0.441679\pi$
$53$	2.65282	0.364393	0.182196	+	0.983262i	$0.441679\pi$
$54$	0	0
$55$	10.6351	1.43404
$56$	0.254239	0.0339741
$57$	0	0
$58$	1.76677	0.231988
$59$	3.70138	0.481879	0.240939	−	0.970540i	$-0.422544\pi$
$59$	3.70138	0.481879	0.240939	+	0.970540i	$0.422544\pi$
$60$	0	0
$61$	−3.77884	−0.483831	−0.241915	−	0.970297i	$-0.577776\pi$
$61$	−3.77884	−0.483831	−0.241915	+	0.970297i	$0.577776\pi$
$62$	10.6587	1.35365
$63$	0	0
$64$	1.00000	0.125000
$65$	−17.3673	−2.15415
$66$	0	0
$67$	−7.58603	−0.926782	−0.463391	−	0.886154i	$-0.653367\pi$
$67$	−7.58603	−0.926782	−0.463391	+	0.886154i	$0.653367\pi$
$68$	5.01396	0.608032
$69$	0	0
$70$	−0.782889	−0.0935731
$71$	−8.42685	−1.00008	−0.500041	−	0.866002i	$-0.666682\pi$
$71$	−8.42685	−1.00008	−0.500041	+	0.866002i	$0.666682\pi$
$72$	0	0
$73$	−6.99035	−0.818159	−0.409080	−	0.912499i	$-0.634150\pi$
$73$	−6.99035	−0.818159	−0.409080	+	0.912499i	$0.634150\pi$
$74$	−8.83464	−1.02701
$75$	0	0
$76$	−0.904786	−0.103786
$77$	0.878062	0.100064
$78$	0	0
$79$	13.1091	1.47488	0.737442	−	0.675410i	$-0.236033\pi$
$79$	13.1091	1.47488	0.737442	+	0.675410i	$0.236033\pi$
$80$	−3.07935	−0.344281
$81$	0	0
$82$	5.71415	0.631023
$83$	−6.92438	−0.760050	−0.380025	−	0.924976i	$-0.624085\pi$
$83$	−6.92438	−0.760050	−0.380025	+	0.924976i	$0.624085\pi$
$84$	0	0
$85$	−15.4397	−1.67467
$86$	8.56574	0.923667
$87$	0	0
$88$	3.45369	0.368165
$89$	−6.80373	−0.721194	−0.360597	−	0.932722i	$-0.617427\pi$
$89$	−6.80373	−0.721194	−0.360597	+	0.932722i	$0.617427\pi$
$90$	0	0
$91$	−1.43389	−0.150313
$92$	5.64985	0.589037
$93$	0	0
$94$	12.5309	1.29246
$95$	2.78615	0.285853
$96$	0	0
$97$	8.62609	0.875847	0.437924	−	0.899012i	$-0.355714\pi$
$97$	8.62609	0.875847	0.437924	+	0.899012i	$0.355714\pi$
$98$	6.93536	0.700577
$99$	0	0
$100$	4.48238	0.448238
$101$	2.48818	0.247583	0.123792	−	0.992308i	$-0.460495\pi$
$101$	2.48818	0.247583	0.123792	+	0.992308i	$0.460495\pi$
$102$	0	0
$103$	1.02862	0.101353	0.0506767	−	0.998715i	$-0.483862\pi$
$103$	1.02862	0.101353	0.0506767	+	0.998715i	$0.483862\pi$
$104$	−5.63995	−0.553042
$105$	0	0
$106$	−2.65282	−0.257665
$107$	7.61531	0.736199	0.368100	−	0.929786i	$-0.380009\pi$
$107$	7.61531	0.736199	0.368100	+	0.929786i	$0.380009\pi$
$108$	0	0
$109$	−14.6099	−1.39937	−0.699687	−	0.714449i	$-0.746677\pi$
$109$	−14.6099	−1.39937	−0.699687	+	0.714449i	$0.746677\pi$
$110$	−10.6351	−1.01402
$111$	0	0
$112$	−0.254239	−0.0240233
$113$	−4.58711	−0.431519	−0.215760	−	0.976446i	$-0.569223\pi$
$113$	−4.58711	−0.431519	−0.215760	+	0.976446i	$0.569223\pi$
$114$	0	0
$115$	−17.3978	−1.62236
$116$	−1.76677	−0.164040
$117$	0	0
$118$	−3.70138	−0.340740
$119$	−1.27474	−0.116855
$120$	0	0
$121$	0.928001	0.0843638
$122$	3.77884	0.342120
$123$	0	0
$124$	−10.6587	−0.957177
$125$	1.59393	0.142566
$126$	0	0
$127$	−21.0106	−1.86439	−0.932195	−	0.361958i	$-0.882108\pi$
$127$	−21.0106	−1.86439	−0.932195	+	0.361958i	$0.882108\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	17.3673	1.52322
$131$	17.4582	1.52533	0.762664	−	0.646795i	$-0.223891\pi$
$131$	17.4582	1.52533	0.762664	+	0.646795i	$0.223891\pi$
$132$	0	0
$133$	0.230032	0.0199463
$134$	7.58603	0.655334
$135$	0	0
$136$	−5.01396	−0.429944
$137$	0.262350	0.0224141	0.0112070	−	0.999937i	$-0.496433\pi$
$137$	0.262350	0.0224141	0.0112070	+	0.999937i	$0.496433\pi$
$138$	0	0
$139$	11.8111	1.00181	0.500904	−	0.865503i	$-0.333001\pi$
$139$	11.8111	1.00181	0.500904	+	0.865503i	$0.333001\pi$
$140$	0.782889	0.0661662
$141$	0	0
$142$	8.42685	0.707165
$143$	−19.4786	−1.62889
$144$	0	0
$145$	5.44049	0.451808
$146$	6.99035	0.578526
$147$	0	0
$148$	8.83464	0.726203
$149$	−4.63410	−0.379641	−0.189820	−	0.981819i	$-0.560791\pi$
$149$	−4.63410	−0.379641	−0.189820	+	0.981819i	$0.560791\pi$
$150$	0	0
$151$	−7.31199	−0.595042	−0.297521	−	0.954715i	$-0.596160\pi$
$151$	−7.31199	−0.595042	−0.297521	+	0.954715i	$0.596160\pi$
$152$	0.904786	0.0733879
$153$	0	0
$154$	−0.878062	−0.0707563
$155$	32.8217	2.63631
$156$	0	0
$157$	−3.33359	−0.266049	−0.133025	−	0.991113i	$-0.542469\pi$
$157$	−3.33359	−0.266049	−0.133025	+	0.991113i	$0.542469\pi$
$158$	−13.1091	−1.04290
$159$	0	0
$160$	3.07935	0.243444
$161$	−1.43641	−0.113205
$162$	0	0
$163$	−2.00066	−0.156704	−0.0783519	−	0.996926i	$-0.524966\pi$
$163$	−2.00066	−0.156704	−0.0783519	+	0.996926i	$0.524966\pi$
$164$	−5.71415	−0.446200
$165$	0	0
$166$	6.92438	0.537436
$167$	−13.8751	−1.07369	−0.536844	−	0.843681i	$-0.680384\pi$
$167$	−13.8751	−1.07369	−0.536844	+	0.843681i	$0.680384\pi$
$168$	0	0
$169$	18.8090	1.44684
$170$	15.4397	1.18417
$171$	0	0
$172$	−8.56574	−0.653131
$173$	−9.62111	−0.731479	−0.365740	−	0.930717i	$-0.619184\pi$
$173$	−9.62111	−0.731479	−0.365740	+	0.930717i	$0.619184\pi$
$174$	0	0
$175$	−1.13959	−0.0861452
$176$	−3.45369	−0.260332
$177$	0	0
$178$	6.80373	0.509961
$179$	0.0919653	0.00687381	0.00343691	−	0.999994i	$-0.498906\pi$
$179$	0.0919653	0.00687381	0.00343691	+	0.999994i	$0.498906\pi$
$180$	0	0
$181$	4.70338	0.349599	0.174800	−	0.984604i	$-0.444072\pi$
$181$	4.70338	0.349599	0.174800	+	0.984604i	$0.444072\pi$
$182$	1.43389	0.106287
$183$	0	0
$184$	−5.64985	−0.416512
$185$	−27.2049	−2.00015
$186$	0	0
$187$	−17.3167	−1.26632
$188$	−12.5309	−0.913908
$189$	0	0
$190$	−2.78615	−0.202129
$191$	−11.2965	−0.817387	−0.408694	−	0.912672i	$-0.634016\pi$
$191$	−11.2965	−0.817387	−0.408694	+	0.912672i	$0.634016\pi$
$192$	0	0
$193$	0.231072	0.0166330	0.00831648	−	0.999965i	$-0.497353\pi$
$193$	0.231072	0.0166330	0.00831648	+	0.999965i	$0.497353\pi$
$194$	−8.62609	−0.619317
$195$	0	0
$196$	−6.93536	−0.495383
$197$	−12.2292	−0.871295	−0.435647	−	0.900117i	$-0.643481\pi$
$197$	−12.2292	−0.871295	−0.435647	+	0.900117i	$0.643481\pi$
$198$	0	0
$199$	−8.31187	−0.589213	−0.294606	−	0.955619i	$-0.595189\pi$
$199$	−8.31187	−0.589213	−0.294606	+	0.955619i	$0.595189\pi$
$200$	−4.48238	−0.316952
$201$	0	0
$202$	−2.48818	−0.175068
$203$	0.449180	0.0315263
$204$	0	0
$205$	17.5959	1.22895
$206$	−1.02862	−0.0716676
$207$	0	0
$208$	5.63995	0.391060
$209$	3.12486	0.216151
$210$	0	0
$211$	−8.73953	−0.601654	−0.300827	−	0.953679i	$-0.597263\pi$
$211$	−8.73953	−0.601654	−0.300827	+	0.953679i	$0.597263\pi$
$212$	2.65282	0.182196
$213$	0	0
$214$	−7.61531	−0.520572
$215$	26.3769	1.79889
$216$	0	0
$217$	2.70984	0.183956
$218$	14.6099	0.989507
$219$	0	0
$220$	10.6351	0.717020
$221$	28.2785	1.90222
$222$	0	0
$223$	−5.48534	−0.367326	−0.183663	−	0.982989i	$-0.558795\pi$
$223$	−5.48534	−0.367326	−0.183663	+	0.982989i	$0.558795\pi$
$224$	0.254239	0.0169870
$225$	0	0
$226$	4.58711	0.305130
$227$	13.9875	0.928382	0.464191	−	0.885735i	$-0.346345\pi$
$227$	13.9875	0.928382	0.464191	+	0.885735i	$0.346345\pi$
$228$	0	0
$229$	−16.4092	−1.08435	−0.542175	−	0.840265i	$-0.682399\pi$
$229$	−16.4092	−1.08435	−0.542175	+	0.840265i	$0.682399\pi$
$230$	17.3978	1.14718
$231$	0	0
$232$	1.76677	0.115994
$233$	−7.02588	−0.460281	−0.230140	−	0.973157i	$-0.573919\pi$
$233$	−7.02588	−0.460281	−0.230140	+	0.973157i	$0.573919\pi$
$234$	0	0
$235$	38.5869	2.51713
$236$	3.70138	0.240939
$237$	0	0
$238$	1.27474	0.0826293
$239$	17.9301	1.15980	0.579900	−	0.814688i	$-0.303092\pi$
$239$	17.9301	1.15980	0.579900	+	0.814688i	$0.303092\pi$
$240$	0	0
$241$	−10.5554	−0.679932	−0.339966	−	0.940438i	$-0.610416\pi$
$241$	−10.5554	−0.679932	−0.339966	+	0.940438i	$0.610416\pi$
$242$	−0.928001	−0.0596542
$243$	0	0
$244$	−3.77884	−0.241915
$245$	21.3564	1.36441
$246$	0	0
$247$	−5.10295	−0.324693
$248$	10.6587	0.676826
$249$	0	0
$250$	−1.59393	−0.100809
$251$	25.9705	1.63924	0.819622	−	0.572905i	$-0.194184\pi$
$251$	25.9705	1.63924	0.819622	+	0.572905i	$0.194184\pi$
$252$	0	0
$253$	−19.5128	−1.22676
$254$	21.0106	1.31832
$255$	0	0
$256$	1.00000	0.0625000
$257$	−15.8817	−0.990675	−0.495338	−	0.868701i	$-0.664956\pi$
$257$	−15.8817	−0.990675	−0.495338	+	0.868701i	$0.664956\pi$
$258$	0	0
$259$	−2.24611	−0.139566
$260$	−17.3673	−1.07708
$261$	0	0
$262$	−17.4582	−1.07857
$263$	−20.8492	−1.28562	−0.642809	−	0.766026i	$-0.722231\pi$
$263$	−20.8492	−1.28562	−0.642809	+	0.766026i	$0.722231\pi$
$264$	0	0
$265$	−8.16895	−0.501815
$266$	−0.230032	−0.0141041
$267$	0	0
$268$	−7.58603	−0.463391
$269$	−12.0416	−0.734187	−0.367094	−	0.930184i	$-0.619647\pi$
$269$	−12.0416	−0.734187	−0.367094	+	0.930184i	$0.619647\pi$
$270$	0	0
$271$	−21.5767	−1.31069	−0.655346	−	0.755329i	$-0.727477\pi$
$271$	−21.5767	−1.31069	−0.655346	+	0.755329i	$0.727477\pi$
$272$	5.01396	0.304016
$273$	0	0
$274$	−0.262350	−0.0158492
$275$	−15.4808	−0.933525
$276$	0	0
$277$	27.1547	1.63157	0.815783	−	0.578358i	$-0.196306\pi$
$277$	27.1547	1.63157	0.815783	+	0.578358i	$0.196306\pi$
$278$	−11.8111	−0.708385
$279$	0	0
$280$	−0.782889	−0.0467866
$281$	15.5144	0.925510	0.462755	−	0.886486i	$-0.346861\pi$
$281$	15.5144	0.925510	0.462755	+	0.886486i	$0.346861\pi$
$282$	0	0
$283$	−20.2163	−1.20173	−0.600866	−	0.799350i	$-0.705177\pi$
$283$	−20.2163	−1.20173	−0.600866	+	0.799350i	$0.705177\pi$
$284$	−8.42685	−0.500041
$285$	0	0
$286$	19.4786	1.15180
$287$	1.45276	0.0857536
$288$	0	0
$289$	8.13981	0.478812
$290$	−5.44049	−0.319477
$291$	0	0
$292$	−6.99035	−0.409080
$293$	2.81379	0.164383	0.0821917	−	0.996617i	$-0.473808\pi$
$293$	2.81379	0.164383	0.0821917	+	0.996617i	$0.473808\pi$
$294$	0	0
$295$	−11.3978	−0.663608
$296$	−8.83464	−0.513503
$297$	0	0
$298$	4.63410	0.268446
$299$	31.8648	1.84279
$300$	0	0
$301$	2.17774	0.125523
$302$	7.31199	0.420758
$303$	0	0
$304$	−0.904786	−0.0518931
$305$	11.6364	0.666296
$306$	0	0
$307$	−7.23426	−0.412881	−0.206440	−	0.978459i	$-0.566188\pi$
$307$	−7.23426	−0.412881	−0.206440	+	0.978459i	$0.566188\pi$
$308$	0.878062	0.0500322
$309$	0	0
$310$	−32.8217	−1.86415
$311$	5.85695	0.332117	0.166059	−	0.986116i	$-0.446896\pi$
$311$	5.85695	0.332117	0.166059	+	0.986116i	$0.446896\pi$
$312$	0	0
$313$	−12.8465	−0.726126	−0.363063	−	0.931765i	$-0.618269\pi$
$313$	−12.8465	−0.726126	−0.363063	+	0.931765i	$0.618269\pi$
$314$	3.33359	0.188125
$315$	0	0
$316$	13.1091	0.737442
$317$	23.5707	1.32386	0.661931	−	0.749565i	$-0.269737\pi$
$317$	23.5707	1.32386	0.661931	+	0.749565i	$0.269737\pi$
$318$	0	0
$319$	6.10187	0.341639
$320$	−3.07935	−0.172141
$321$	0	0
$322$	1.43641	0.0800479
$323$	−4.53656	−0.252421
$324$	0	0
$325$	25.2804	1.40230
$326$	2.00066	0.110806
$327$	0	0
$328$	5.71415	0.315511
$329$	3.18583	0.175641
$330$	0	0
$331$	15.1665	0.833625	0.416813	−	0.908992i	$-0.363147\pi$
$331$	15.1665	0.833625	0.416813	+	0.908992i	$0.363147\pi$
$332$	−6.92438	−0.380025
$333$	0	0
$334$	13.8751	0.759213
$335$	23.3600	1.27630
$336$	0	0
$337$	−14.3623	−0.782362	−0.391181	−	0.920314i	$-0.627933\pi$
$337$	−14.3623	−0.782362	−0.391181	+	0.920314i	$0.627933\pi$
$338$	−18.8090	−1.02307
$339$	0	0
$340$	−15.4397	−0.837337
$341$	36.8118	1.99347
$342$	0	0
$343$	3.54291	0.191299
$344$	8.56574	0.461834
$345$	0	0
$346$	9.62111	0.517234
$347$	−11.1041	−0.596102	−0.298051	−	0.954550i	$-0.596336\pi$
$347$	−11.1041	−0.596102	−0.298051	+	0.954550i	$0.596336\pi$
$348$	0	0
$349$	14.3382	0.767508	0.383754	−	0.923435i	$-0.374631\pi$
$349$	14.3382	0.767508	0.383754	+	0.923435i	$0.374631\pi$
$350$	1.13959	0.0609138
$351$	0	0
$352$	3.45369	0.184083
$353$	32.0581	1.70628	0.853139	−	0.521683i	$-0.174696\pi$
$353$	32.0581	1.70628	0.853139	+	0.521683i	$0.174696\pi$
$354$	0	0
$355$	25.9492	1.37724
$356$	−6.80373	−0.360597
$357$	0	0
$358$	−0.0919653	−0.00486052
$359$	7.89430	0.416645	0.208323	−	0.978060i	$-0.433200\pi$
$359$	7.89430	0.416645	0.208323	+	0.978060i	$0.433200\pi$
$360$	0	0
$361$	−18.1814	−0.956914
$362$	−4.70338	−0.247204
$363$	0	0
$364$	−1.43389	−0.0751563
$365$	21.5257	1.12671
$366$	0	0
$367$	−9.60650	−0.501455	−0.250728	−	0.968058i	$-0.580670\pi$
$367$	−9.60650	−0.501455	−0.250728	+	0.968058i	$0.580670\pi$
$368$	5.64985	0.294519
$369$	0	0
$370$	27.2049	1.41432
$371$	−0.674449	−0.0350156
$372$	0	0
$373$	2.37060	0.122745	0.0613726	−	0.998115i	$-0.480452\pi$
$373$	2.37060	0.122745	0.0613726	+	0.998115i	$0.480452\pi$
$374$	17.3167	0.895425
$375$	0	0
$376$	12.5309	0.646231
$377$	−9.96447	−0.513196
$378$	0	0
$379$	12.1691	0.625087	0.312543	−	0.949904i	$-0.398819\pi$
$379$	12.1691	0.625087	0.312543	+	0.949904i	$0.398819\pi$
$380$	2.78615	0.142927
$381$	0	0
$382$	11.2965	0.577980
$383$	−6.68477	−0.341576	−0.170788	−	0.985308i	$-0.554631\pi$
$383$	−6.68477	−0.341576	−0.170788	+	0.985308i	$0.554631\pi$
$384$	0	0
$385$	−2.70386	−0.137801
$386$	−0.231072	−0.0117613
$387$	0	0
$388$	8.62609	0.437924
$389$	−21.4226	−1.08617	−0.543084	−	0.839678i	$-0.682743\pi$
$389$	−21.4226	−1.08617	−0.543084	+	0.839678i	$0.682743\pi$
$390$	0	0
$391$	28.3281	1.43261
$392$	6.93536	0.350289
$393$	0	0
$394$	12.2292	0.616099
$395$	−40.3673	−2.03110
$396$	0	0
$397$	21.1756	1.06277	0.531387	−	0.847129i	$-0.321671\pi$
$397$	21.1756	1.06277	0.531387	+	0.847129i	$0.321671\pi$
$398$	8.31187	0.416636
$399$	0	0
$400$	4.48238	0.224119
$401$	34.9397	1.74480	0.872402	−	0.488789i	$-0.162561\pi$
$401$	34.9397	1.74480	0.872402	+	0.488789i	$0.162561\pi$
$402$	0	0
$403$	−60.1143	−2.99451
$404$	2.48818	0.123792
$405$	0	0
$406$	−0.449180	−0.0222924
$407$	−30.5121	−1.51243
$408$	0	0
$409$	28.2265	1.39571	0.697854	−	0.716240i	$-0.254138\pi$
$409$	28.2265	1.39571	0.697854	+	0.716240i	$0.254138\pi$
$410$	−17.5959	−0.868998
$411$	0	0
$412$	1.02862	0.0506767
$413$	−0.941034	−0.0463053
$414$	0	0
$415$	21.3226	1.04668
$416$	−5.63995	−0.276521
$417$	0	0
$418$	−3.12486	−0.152842
$419$	−8.56513	−0.418434	−0.209217	−	0.977869i	$-0.567092\pi$
$419$	−8.56513	−0.418434	−0.209217	+	0.977869i	$0.567092\pi$
$420$	0	0
$421$	36.9844	1.80251	0.901256	−	0.433288i	$-0.142647\pi$
$421$	36.9844	1.80251	0.901256	+	0.433288i	$0.142647\pi$
$422$	8.73953	0.425434
$423$	0	0
$424$	−2.65282	−0.128832
$425$	22.4745	1.09017
$426$	0	0
$427$	0.960726	0.0464928
$428$	7.61531	0.368100
$429$	0	0
$430$	−26.3769	−1.27201
$431$	−17.1539	−0.826273	−0.413137	−	0.910669i	$-0.635567\pi$
$431$	−17.1539	−0.826273	−0.413137	+	0.910669i	$0.635567\pi$
$432$	0	0
$433$	5.42645	0.260779	0.130389	−	0.991463i	$-0.458377\pi$
$433$	5.42645	0.260779	0.130389	+	0.991463i	$0.458377\pi$
$434$	−2.70984	−0.130077
$435$	0	0
$436$	−14.6099	−0.699687
$437$	−5.11190	−0.244536
$438$	0	0
$439$	−19.2822	−0.920289	−0.460145	−	0.887844i	$-0.652202\pi$
$439$	−19.2822	−0.920289	−0.460145	+	0.887844i	$0.652202\pi$
$440$	−10.6351	−0.507010
$441$	0	0
$442$	−28.2785	−1.34507
$443$	36.2168	1.72071	0.860356	−	0.509694i	$-0.170241\pi$
$443$	36.2168	1.72071	0.860356	+	0.509694i	$0.170241\pi$
$444$	0	0
$445$	20.9511	0.993175
$446$	5.48534	0.259739
$447$	0	0
$448$	−0.254239	−0.0120116
$449$	37.4745	1.76853	0.884266	−	0.466983i	$-0.154659\pi$
$449$	37.4745	1.76853	0.884266	+	0.466983i	$0.154659\pi$
$450$	0	0
$451$	19.7349	0.929282
$452$	−4.58711	−0.215760
$453$	0	0
$454$	−13.9875	−0.656466
$455$	4.41545	0.206999
$456$	0	0
$457$	40.0694	1.87437	0.937183	−	0.348837i	$-0.113423\pi$
$457$	40.0694	1.87437	0.937183	+	0.348837i	$0.113423\pi$
$458$	16.4092	0.766751
$459$	0	0
$460$	−17.3978	−0.811178
$461$	−2.78651	−0.129781	−0.0648903	−	0.997892i	$-0.520670\pi$
$461$	−2.78651	−0.129781	−0.0648903	+	0.997892i	$0.520670\pi$
$462$	0	0
$463$	−5.26722	−0.244788	−0.122394	−	0.992482i	$-0.539057\pi$
$463$	−5.26722	−0.244788	−0.122394	+	0.992482i	$0.539057\pi$
$464$	−1.76677	−0.0820201
$465$	0	0
$466$	7.02588	0.325468
$467$	−5.17863	−0.239639	−0.119819	−	0.992796i	$-0.538232\pi$
$467$	−5.17863	−0.239639	−0.119819	+	0.992796i	$0.538232\pi$
$468$	0	0
$469$	1.92866	0.0890574
$470$	−38.5869	−1.77988
$471$	0	0
$472$	−3.70138	−0.170370
$473$	29.5834	1.36025
$474$	0	0
$475$	−4.05560	−0.186083
$476$	−1.27474	−0.0584277
$477$	0	0
$478$	−17.9301	−0.820102
$479$	−20.2892	−0.927039	−0.463519	−	0.886087i	$-0.653414\pi$
$479$	−20.2892	−0.927039	−0.463519	+	0.886087i	$0.653414\pi$
$480$	0	0
$481$	49.8269	2.27191
$482$	10.5554	0.480785
$483$	0	0
$484$	0.928001	0.0421819
$485$	−26.5627	−1.20615
$486$	0	0
$487$	−11.0515	−0.500791	−0.250396	−	0.968144i	$-0.580561\pi$
$487$	−11.0515	−0.500791	−0.250396	+	0.968144i	$0.580561\pi$
$488$	3.77884	0.171060
$489$	0	0
$490$	−21.3564	−0.964783
$491$	−25.3811	−1.14543	−0.572717	−	0.819753i	$-0.694111\pi$
$491$	−25.3811	−1.14543	−0.572717	+	0.819753i	$0.694111\pi$
$492$	0	0
$493$	−8.85850	−0.398967
$494$	5.10295	0.229592
$495$	0	0
$496$	−10.6587	−0.478588
$497$	2.14243	0.0961011
$498$	0	0
$499$	−12.7222	−0.569522	−0.284761	−	0.958599i	$-0.591914\pi$
$499$	−12.7222	−0.569522	−0.284761	+	0.958599i	$0.591914\pi$
$500$	1.59393	0.0712829
$501$	0	0
$502$	−25.9705	−1.15912
$503$	10.5666	0.471143	0.235572	−	0.971857i	$-0.424304\pi$
$503$	10.5666	0.471143	0.235572	+	0.971857i	$0.424304\pi$
$504$	0	0
$505$	−7.66197	−0.340953
$506$	19.5128	0.867452
$507$	0	0
$508$	−21.0106	−0.932195
$509$	−40.0110	−1.77346	−0.886728	−	0.462291i	$-0.847028\pi$
$509$	−40.0110	−1.77346	−0.886728	+	0.462291i	$0.847028\pi$
$510$	0	0
$511$	1.77722	0.0786195
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	15.8817	0.700513
$515$	−3.16749	−0.139576
$516$	0	0
$517$	43.2778	1.90336
$518$	2.24611	0.0986882
$519$	0	0
$520$	17.3673	0.761609
$521$	−29.9472	−1.31201	−0.656006	−	0.754756i	$-0.727755\pi$
$521$	−29.9472	−1.31201	−0.656006	+	0.754756i	$0.727755\pi$
$522$	0	0
$523$	17.9761	0.786040	0.393020	−	0.919530i	$-0.371430\pi$
$523$	17.9761	0.786040	0.393020	+	0.919530i	$0.371430\pi$
$524$	17.4582	0.762664
$525$	0	0
$526$	20.8492	0.909070
$527$	−53.4421	−2.32798
$528$	0	0
$529$	8.92076	0.387859
$530$	8.16895	0.354836
$531$	0	0
$532$	0.230032	0.00997314
$533$	−32.2275	−1.39593
$534$	0	0
$535$	−23.4502	−1.01384
$536$	7.58603	0.327667
$537$	0	0
$538$	12.0416	0.519149
$539$	23.9526	1.03171
$540$	0	0
$541$	−25.6164	−1.10134	−0.550668	−	0.834724i	$-0.685627\pi$
$541$	−25.6164	−1.10134	−0.550668	+	0.834724i	$0.685627\pi$
$542$	21.5767	0.926799
$543$	0	0
$544$	−5.01396	−0.214972
$545$	44.9889	1.92711
$546$	0	0
$547$	10.4807	0.448122	0.224061	−	0.974575i	$-0.428068\pi$
$547$	10.4807	0.448122	0.224061	+	0.974575i	$0.428068\pi$
$548$	0.262350	0.0112070
$549$	0	0
$550$	15.4808	0.660102
$551$	1.59855	0.0681004
$552$	0	0
$553$	−3.33283	−0.141726
$554$	−27.1547	−1.15369
$555$	0	0
$556$	11.8111	0.500904
$557$	−20.6161	−0.873534	−0.436767	−	0.899575i	$-0.643877\pi$
$557$	−20.6161	−0.873534	−0.436767	+	0.899575i	$0.643877\pi$
$558$	0	0
$559$	−48.3103	−2.04331
$560$	0.782889	0.0330831
$561$	0	0
$562$	−15.5144	−0.654435
$563$	−37.2709	−1.57078	−0.785390	−	0.619001i	$-0.787537\pi$
$563$	−37.2709	−1.57078	−0.785390	+	0.619001i	$0.787537\pi$
$564$	0	0
$565$	14.1253	0.594257
$566$	20.2163	0.849753
$567$	0	0
$568$	8.42685	0.353583
$569$	−24.4629	−1.02554	−0.512768	−	0.858527i	$-0.671380\pi$
$569$	−24.4629	−1.02554	−0.512768	+	0.858527i	$0.671380\pi$
$570$	0	0
$571$	−41.8879	−1.75295	−0.876477	−	0.481445i	$-0.840112\pi$
$571$	−41.8879	−1.75295	−0.876477	+	0.481445i	$0.840112\pi$
$572$	−19.4786	−0.814443
$573$	0	0
$574$	−1.45276	−0.0606370
$575$	25.3248	1.05612
$576$	0	0
$577$	22.2497	0.926267	0.463134	−	0.886288i	$-0.346725\pi$
$577$	22.2497	0.926267	0.463134	+	0.886288i	$0.346725\pi$
$578$	−8.13981	−0.338571
$579$	0	0
$580$	5.44049	0.225904
$581$	1.76045	0.0730356
$582$	0	0
$583$	−9.16202	−0.379452
$584$	6.99035	0.289263
$585$	0	0
$586$	−2.81379	−0.116237
$587$	−7.57240	−0.312546	−0.156273	−	0.987714i	$-0.549948\pi$
$587$	−7.57240	−0.312546	−0.156273	+	0.987714i	$0.549948\pi$
$588$	0	0
$589$	9.64382	0.397367
$590$	11.3978	0.469242
$591$	0	0
$592$	8.83464	0.363101
$593$	−11.2028	−0.460045	−0.230022	−	0.973185i	$-0.573880\pi$
$593$	−11.2028	−0.460045	−0.230022	+	0.973185i	$0.573880\pi$
$594$	0	0
$595$	3.92537	0.160925
$596$	−4.63410	−0.189820
$597$	0	0
$598$	−31.8648	−1.30305
$599$	−8.21125	−0.335503	−0.167751	−	0.985829i	$-0.553651\pi$
$599$	−8.21125	−0.335503	−0.167751	+	0.985829i	$0.553651\pi$
$600$	0	0
$601$	12.2246	0.498651	0.249325	−	0.968420i	$-0.419791\pi$
$601$	12.2246	0.498651	0.249325	+	0.968420i	$0.419791\pi$
$602$	−2.17774	−0.0887581
$603$	0	0
$604$	−7.31199	−0.297521
$605$	−2.85764	−0.116180
$606$	0	0
$607$	27.2686	1.10680	0.553398	−	0.832917i	$-0.313331\pi$
$607$	27.2686	1.10680	0.553398	+	0.832917i	$0.313331\pi$
$608$	0.904786	0.0366939
$609$	0	0
$610$	−11.6364	−0.471142
$611$	−70.6735	−2.85914
$612$	0	0
$613$	−32.6777	−1.31984	−0.659920	−	0.751336i	$-0.729410\pi$
$613$	−32.6777	−1.31984	−0.659920	+	0.751336i	$0.729410\pi$
$614$	7.23426	0.291951
$615$	0	0
$616$	−0.878062	−0.0353781
$617$	−32.0461	−1.29013	−0.645063	−	0.764129i	$-0.723169\pi$
$617$	−32.0461	−1.29013	−0.645063	+	0.764129i	$0.723169\pi$
$618$	0	0
$619$	−13.1247	−0.527526	−0.263763	−	0.964587i	$-0.584964\pi$
$619$	−13.1247	−0.527526	−0.263763	+	0.964587i	$0.584964\pi$
$620$	32.8217	1.31815
$621$	0	0
$622$	−5.85695	−0.234842
$623$	1.72977	0.0693018
$624$	0	0
$625$	−27.3202	−1.09281
$626$	12.8465	0.513449
$627$	0	0
$628$	−3.33359	−0.133025
$629$	44.2965	1.76622
$630$	0	0
$631$	−1.88376	−0.0749913	−0.0374957	−	0.999297i	$-0.511938\pi$
$631$	−1.88376	−0.0749913	−0.0374957	+	0.999297i	$0.511938\pi$
$632$	−13.1091	−0.521450
$633$	0	0
$634$	−23.5707	−0.936112
$635$	64.6989	2.56750
$636$	0	0
$637$	−39.1151	−1.54980
$638$	−6.10187	−0.241575
$639$	0	0
$640$	3.07935	0.121722
$641$	−16.6098	−0.656047	−0.328024	−	0.944670i	$-0.606383\pi$
$641$	−16.6098	−0.656047	−0.328024	+	0.944670i	$0.606383\pi$
$642$	0	0
$643$	16.1442	0.636664	0.318332	−	0.947979i	$-0.396877\pi$
$643$	16.1442	0.636664	0.318332	+	0.947979i	$0.396877\pi$
$644$	−1.43641	−0.0566024
$645$	0	0
$646$	4.53656	0.178489
$647$	−25.3119	−0.995114	−0.497557	−	0.867431i	$-0.665769\pi$
$647$	−25.3119	−0.995114	−0.497557	+	0.867431i	$0.665769\pi$
$648$	0	0
$649$	−12.7834	−0.501794
$650$	−25.2804	−0.991578
$651$	0	0
$652$	−2.00066	−0.0783519
$653$	38.9253	1.52327	0.761633	−	0.648009i	$-0.224398\pi$
$653$	38.9253	1.52327	0.761633	+	0.648009i	$0.224398\pi$
$654$	0	0
$655$	−53.7598	−2.10057
$656$	−5.71415	−0.223100
$657$	0	0
$658$	−3.18583	−0.124197
$659$	−0.534527	−0.0208222	−0.0104111	−	0.999946i	$-0.503314\pi$
$659$	−0.534527	−0.0208222	−0.0104111	+	0.999946i	$0.503314\pi$
$660$	0	0
$661$	−18.4187	−0.716405	−0.358202	−	0.933644i	$-0.616610\pi$
$661$	−18.4187	−0.716405	−0.358202	+	0.933644i	$0.616610\pi$
$662$	−15.1665	−0.589462
$663$	0	0
$664$	6.92438	0.268718
$665$	−0.708347	−0.0274685
$666$	0	0
$667$	−9.98196	−0.386503
$668$	−13.8751	−0.536844
$669$	0	0
$670$	−23.3600	−0.902477
$671$	13.0509	0.503826
$672$	0	0
$673$	31.7472	1.22376	0.611882	−	0.790949i	$-0.290413\pi$
$673$	31.7472	1.22376	0.611882	+	0.790949i	$0.290413\pi$
$674$	14.3623	0.553214
$675$	0	0
$676$	18.8090	0.723422
$677$	−27.1056	−1.04175	−0.520877	−	0.853632i	$-0.674395\pi$
$677$	−27.1056	−1.04175	−0.520877	+	0.853632i	$0.674395\pi$
$678$	0	0
$679$	−2.19309	−0.0841629
$680$	15.4397	0.592087
$681$	0	0
$682$	−36.8118	−1.40960
$683$	−9.36887	−0.358490	−0.179245	−	0.983804i	$-0.557365\pi$
$683$	−9.36887	−0.358490	−0.179245	+	0.983804i	$0.557365\pi$
$684$	0	0
$685$	−0.807867	−0.0308670
$686$	−3.54291	−0.135269
$687$	0	0
$688$	−8.56574	−0.326566
$689$	14.9618	0.569997
$690$	0	0
$691$	40.1096	1.52584	0.762920	−	0.646493i	$-0.223765\pi$
$691$	40.1096	1.52584	0.762920	+	0.646493i	$0.223765\pi$
$692$	−9.62111	−0.365740
$693$	0	0
$694$	11.1041	0.421508
$695$	−36.3706	−1.37962
$696$	0	0
$697$	−28.6505	−1.08522
$698$	−14.3382	−0.542710
$699$	0	0
$700$	−1.13959	−0.0430726
$701$	10.7919	0.407605	0.203803	−	0.979012i	$-0.434670\pi$
$701$	10.7919	0.407605	0.203803	+	0.979012i	$0.434670\pi$
$702$	0	0
$703$	−7.99346	−0.301479
$704$	−3.45369	−0.130166
$705$	0	0
$706$	−32.0581	−1.20652
$707$	−0.632591	−0.0237910
$708$	0	0
$709$	−6.31373	−0.237117	−0.118559	−	0.992947i	$-0.537827\pi$
$709$	−6.31373	−0.237117	−0.118559	+	0.992947i	$0.537827\pi$
$710$	−25.9492	−0.973855
$711$	0	0
$712$	6.80373	0.254981
$713$	−60.2198	−2.25525
$714$	0	0
$715$	59.9815	2.24318
$716$	0.0919653	0.00343691
$717$	0	0
$718$	−7.89430	−0.294613
$719$	11.0750	0.413027	0.206513	−	0.978444i	$-0.433788\pi$
$719$	11.0750	0.413027	0.206513	+	0.978444i	$0.433788\pi$
$720$	0	0
$721$	−0.261516	−0.00973936
$722$	18.1814	0.676640
$723$	0	0
$724$	4.70338	0.174800
$725$	−7.91932	−0.294116
$726$	0	0
$727$	13.7856	0.511280	0.255640	−	0.966772i	$-0.417714\pi$
$727$	13.7856	0.511280	0.255640	+	0.966772i	$0.417714\pi$
$728$	1.43389	0.0531436
$729$	0	0
$730$	−21.5257	−0.796703
$731$	−42.9483	−1.58850
$732$	0	0
$733$	−8.56268	−0.316270	−0.158135	−	0.987418i	$-0.550548\pi$
$733$	−8.56268	−0.316270	−0.158135	+	0.987418i	$0.550548\pi$
$734$	9.60650	0.354582
$735$	0	0
$736$	−5.64985	−0.208256
$737$	26.1998	0.965084
$738$	0	0
$739$	−11.8089	−0.434398	−0.217199	−	0.976127i	$-0.569692\pi$
$739$	−11.8089	−0.434398	−0.217199	+	0.976127i	$0.569692\pi$
$740$	−27.2049	−1.00007
$741$	0	0
$742$	0.674449	0.0247598
$743$	−20.5103	−0.752451	−0.376226	−	0.926528i	$-0.622778\pi$
$743$	−20.5103	−0.752451	−0.376226	+	0.926528i	$0.622778\pi$
$744$	0	0
$745$	14.2700	0.522813
$746$	−2.37060	−0.0867939
$747$	0	0
$748$	−17.3167	−0.633161
$749$	−1.93610	−0.0707437
$750$	0	0
$751$	21.9422	0.800682	0.400341	−	0.916366i	$-0.368892\pi$
$751$	21.9422	0.800682	0.400341	+	0.916366i	$0.368892\pi$
$752$	−12.5309	−0.456954
$753$	0	0
$754$	9.96447	0.362885
$755$	22.5162	0.819447
$756$	0	0
$757$	38.7972	1.41011	0.705054	−	0.709153i	$-0.250923\pi$
$757$	38.7972	1.41011	0.705054	+	0.709153i	$0.250923\pi$
$758$	−12.1691	−0.442003
$759$	0	0
$760$	−2.78615	−0.101064
$761$	23.7838	0.862164	0.431082	−	0.902313i	$-0.358132\pi$
$761$	23.7838	0.862164	0.431082	+	0.902313i	$0.358132\pi$
$762$	0	0
$763$	3.71440	0.134470
$764$	−11.2965	−0.408694
$765$	0	0
$766$	6.68477	0.241531
$767$	20.8756	0.753774
$768$	0	0
$769$	27.4212	0.988834	0.494417	−	0.869225i	$-0.335382\pi$
$769$	27.4212	0.988834	0.494417	+	0.869225i	$0.335382\pi$
$770$	2.70386	0.0974403
$771$	0	0
$772$	0.231072	0.00831648
$773$	32.0681	1.15341	0.576705	−	0.816952i	$-0.304338\pi$
$773$	32.0681	1.15341	0.576705	+	0.816952i	$0.304338\pi$
$774$	0	0
$775$	−47.7762	−1.71617
$776$	−8.62609	−0.309659
$777$	0	0
$778$	21.4226	0.768037
$779$	5.17009	0.185238
$780$	0	0
$781$	29.1037	1.04141
$782$	−28.3281	−1.01301
$783$	0	0
$784$	−6.93536	−0.247692
$785$	10.2653	0.366383
$786$	0	0
$787$	30.0863	1.07246	0.536230	−	0.844072i	$-0.319848\pi$
$787$	30.0863	1.07246	0.536230	+	0.844072i	$0.319848\pi$
$788$	−12.2292	−0.435647
$789$	0	0
$790$	40.3673	1.43621
$791$	1.16622	0.0414661
$792$	0	0
$793$	−21.3124	−0.756827
$794$	−21.1756	−0.751495
$795$	0	0
$796$	−8.31187	−0.294606
$797$	−27.7482	−0.982893	−0.491446	−	0.870908i	$-0.663532\pi$
$797$	−27.7482	−0.982893	−0.491446	+	0.870908i	$0.663532\pi$
$798$	0	0
$799$	−62.8293	−2.22274
$800$	−4.48238	−0.158476
$801$	0	0
$802$	−34.9397	−1.23376
$803$	24.1425	0.851972
$804$	0	0
$805$	4.42320	0.155897
$806$	60.1143	2.11744
$807$	0	0
$808$	−2.48818	−0.0875339
$809$	−5.28719	−0.185888	−0.0929439	−	0.995671i	$-0.529628\pi$
$809$	−5.28719	−0.185888	−0.0929439	+	0.995671i	$0.529628\pi$
$810$	0	0
$811$	1.82768	0.0641785	0.0320893	−	0.999485i	$-0.489784\pi$
$811$	1.82768	0.0641785	0.0320893	+	0.999485i	$0.489784\pi$
$812$	0.449180	0.0157631
$813$	0	0
$814$	30.5121	1.06945
$815$	6.16073	0.215801
$816$	0	0
$817$	7.75016	0.271144
$818$	−28.2265	−0.986915
$819$	0	0
$820$	17.5959	0.614474
$821$	22.7975	0.795638	0.397819	−	0.917464i	$-0.369767\pi$
$821$	22.7975	0.795638	0.397819	+	0.917464i	$0.369767\pi$
$822$	0	0
$823$	22.1731	0.772906	0.386453	−	0.922309i	$-0.373700\pi$
$823$	22.1731	0.772906	0.386453	+	0.922309i	$0.373700\pi$
$824$	−1.02862	−0.0358338
$825$	0	0
$826$	0.941034	0.0327428
$827$	16.3304	0.567863	0.283931	−	0.958845i	$-0.408361\pi$
$827$	16.3304	0.567863	0.283931	+	0.958845i	$0.408361\pi$
$828$	0	0
$829$	−26.3084	−0.913729	−0.456865	−	0.889536i	$-0.651028\pi$
$829$	−26.3084	−0.913729	−0.456865	+	0.889536i	$0.651028\pi$
$830$	−21.3226	−0.740117
$831$	0	0
$832$	5.63995	0.195530
$833$	−34.7736	−1.20484
$834$	0	0
$835$	42.7263	1.47860
$836$	3.12486	0.108075
$837$	0	0
$838$	8.56513	0.295878
$839$	−57.0239	−1.96868	−0.984341	−	0.176274i	$-0.943596\pi$
$839$	−57.0239	−1.96868	−0.984341	+	0.176274i	$0.943596\pi$
$840$	0	0
$841$	−25.8785	−0.892363
$842$	−36.9844	−1.27457
$843$	0	0
$844$	−8.73953	−0.300827
$845$	−57.9194	−1.99249
$846$	0	0
$847$	−0.235934	−0.00810678
$848$	2.65282	0.0910982
$849$	0	0
$850$	−22.4745	−0.770868
$851$	49.9144	1.71104
$852$	0	0
$853$	−31.8313	−1.08988	−0.544941	−	0.838474i	$-0.683448\pi$
$853$	−31.8313	−1.08988	−0.544941	+	0.838474i	$0.683448\pi$
$854$	−0.960726	−0.0328754
$855$	0	0
$856$	−7.61531	−0.260286
$857$	−12.9559	−0.442564	−0.221282	−	0.975210i	$-0.571024\pi$
$857$	−12.9559	−0.442564	−0.221282	+	0.975210i	$0.571024\pi$
$858$	0	0
$859$	25.4856	0.869556	0.434778	−	0.900538i	$-0.356827\pi$
$859$	25.4856	0.869556	0.434778	+	0.900538i	$0.356827\pi$
$860$	26.3769	0.899444
$861$	0	0
$862$	17.1539	0.584263
$863$	16.1395	0.549396	0.274698	−	0.961531i	$-0.411422\pi$
$863$	16.1395	0.549396	0.274698	+	0.961531i	$0.411422\pi$
$864$	0	0
$865$	29.6267	1.00734
$866$	−5.42645	−0.184398
$867$	0	0
$868$	2.70984	0.0919781
$869$	−45.2747	−1.53584
$870$	0	0
$871$	−42.7848	−1.44971
$872$	14.6099	0.494754
$873$	0	0
$874$	5.11190	0.172913
$875$	−0.405240	−0.0136996
$876$	0	0
$877$	21.1661	0.714729	0.357364	−	0.933965i	$-0.383675\pi$
$877$	21.1661	0.714729	0.357364	+	0.933965i	$0.383675\pi$
$878$	19.2822	0.650743
$879$	0	0
$880$	10.6351	0.358510
$881$	4.52441	0.152431	0.0762157	−	0.997091i	$-0.475716\pi$
$881$	4.52441	0.152431	0.0762157	+	0.997091i	$0.475716\pi$
$882$	0	0
$883$	30.5817	1.02916	0.514578	−	0.857444i	$-0.327949\pi$
$883$	30.5817	1.02916	0.514578	+	0.857444i	$0.327949\pi$
$884$	28.2785	0.951108
$885$	0	0
$886$	−36.2168	−1.21673
$887$	22.3827	0.751538	0.375769	−	0.926713i	$-0.377379\pi$
$887$	22.3827	0.751538	0.375769	+	0.926713i	$0.377379\pi$
$888$	0	0
$889$	5.34170	0.179155
$890$	−20.9511	−0.702281
$891$	0	0
$892$	−5.48534	−0.183663
$893$	11.3378	0.379404
$894$	0	0
$895$	−0.283193	−0.00946611
$896$	0.254239	0.00849351
$897$	0	0
$898$	−37.4745	−1.25054
$899$	18.8314	0.628062
$900$	0	0
$901$	13.3011	0.443125
$902$	−19.7349	−0.657101
$903$	0	0
$904$	4.58711	0.152565
$905$	−14.4833	−0.481442
$906$	0	0
$907$	−41.0516	−1.36309	−0.681547	−	0.731774i	$-0.738693\pi$
$907$	−41.0516	−1.36309	−0.681547	+	0.731774i	$0.738693\pi$
$908$	13.9875	0.464191
$909$	0	0
$910$	−4.41545	−0.146371
$911$	−9.65713	−0.319955	−0.159977	−	0.987121i	$-0.551142\pi$
$911$	−9.65713	−0.319955	−0.159977	+	0.987121i	$0.551142\pi$
$912$	0	0
$913$	23.9147	0.791461
$914$	−40.0694	−1.32538
$915$	0	0
$916$	−16.4092	−0.542175
$917$	−4.43854	−0.146574
$918$	0	0
$919$	−17.8488	−0.588776	−0.294388	−	0.955686i	$-0.595116\pi$
$919$	−17.8488	−0.588776	−0.294388	+	0.955686i	$0.595116\pi$
$920$	17.3978	0.573590
$921$	0	0
$922$	2.78651	0.0917688
$923$	−47.5269	−1.56437
$924$	0	0
$925$	39.6002	1.30205
$926$	5.26722	0.173091
$927$	0	0
$928$	1.76677	0.0579970
$929$	−9.92774	−0.325718	−0.162859	−	0.986649i	$-0.552072\pi$
$929$	−9.92774	−0.325718	−0.162859	+	0.986649i	$0.552072\pi$
$930$	0	0
$931$	6.27502	0.205656
$932$	−7.02588	−0.230140
$933$	0	0
$934$	5.17863	0.169450
$935$	53.3241	1.74388
$936$	0	0
$937$	14.9391	0.488041	0.244020	−	0.969770i	$-0.421534\pi$
$937$	14.9391	0.488041	0.244020	+	0.969770i	$0.421534\pi$
$938$	−1.92866	−0.0629731
$939$	0	0
$940$	38.5869	1.25857
$941$	25.7255	0.838627	0.419313	−	0.907842i	$-0.362271\pi$
$941$	25.7255	0.838627	0.419313	+	0.907842i	$0.362271\pi$
$942$	0	0
$943$	−32.2841	−1.05131
$944$	3.70138	0.120470
$945$	0	0
$946$	−29.5834	−0.961840
$947$	4.72784	0.153634	0.0768171	−	0.997045i	$-0.475524\pi$
$947$	4.72784	0.153634	0.0768171	+	0.997045i	$0.475524\pi$
$948$	0	0
$949$	−39.4252	−1.27980
$950$	4.05560	0.131581
$951$	0	0
$952$	1.27474	0.0413146
$953$	10.9627	0.355118	0.177559	−	0.984110i	$-0.443180\pi$
$953$	10.9627	0.355118	0.177559	+	0.984110i	$0.443180\pi$
$954$	0	0
$955$	34.7859	1.12564
$956$	17.9301	0.579900
$957$	0	0
$958$	20.2892	0.655515
$959$	−0.0666996	−0.00215384
$960$	0	0
$961$	82.6072	2.66475
$962$	−49.8269	−1.60648
$963$	0	0
$964$	−10.5554	−0.339966
$965$	−0.711552	−0.0229057
$966$	0	0
$967$	7.15478	0.230082	0.115041	−	0.993361i	$-0.463300\pi$
$967$	7.15478	0.230082	0.115041	+	0.993361i	$0.463300\pi$
$968$	−0.928001	−0.0298271
$969$	0	0
$970$	26.5627	0.852878
$971$	9.66743	0.310243	0.155121	−	0.987895i	$-0.450423\pi$
$971$	9.66743	0.310243	0.155121	+	0.987895i	$0.450423\pi$
$972$	0	0
$973$	−3.00285	−0.0962669
$974$	11.0515	0.354113
$975$	0	0
$976$	−3.77884	−0.120958
$977$	10.5927	0.338890	0.169445	−	0.985540i	$-0.445802\pi$
$977$	10.5927	0.338890	0.169445	+	0.985540i	$0.445802\pi$
$978$	0	0
$979$	23.4980	0.751000
$980$	21.3564	0.682205
$981$	0	0
$982$	25.3811	0.809945
$983$	−10.2035	−0.325442	−0.162721	−	0.986672i	$-0.552027\pi$
$983$	−10.2035	−0.325442	−0.162721	+	0.986672i	$0.552027\pi$
$984$	0	0
$985$	37.6580	1.19988
$986$	8.85850	0.282112
$987$	0	0
$988$	−5.10295	−0.162346
$989$	−48.3951	−1.53887
$990$	0	0
$991$	55.3945	1.75966	0.879832	−	0.475285i	$-0.157655\pi$
$991$	55.3945	1.75966	0.879832	+	0.475285i	$0.157655\pi$
$992$	10.6587	0.338413
$993$	0	0
$994$	−2.14243	−0.0679537
$995$	25.5951	0.811420
$996$	0	0
$997$	−17.7707	−0.562805	−0.281402	−	0.959590i	$-0.590800\pi$
$997$	−17.7707	−0.562805	−0.281402	+	0.959590i	$0.590800\pi$
$998$	12.7222	0.402713
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1458.2.a.e.1.2	✓	6
3.2	odd	2		1458.2.a.h.1.5	yes	6
9.2	odd	6		1458.2.c.e.973.2		12
9.4	even	3		1458.2.c.h.487.5		12
9.5	odd	6		1458.2.c.e.487.2		12
9.7	even	3		1458.2.c.h.973.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1458.2.a.e.1.2	✓	6	1.1	even	1	trivial
1458.2.a.h.1.5	yes	6	3.2	odd	2
1458.2.c.e.487.2		12	9.5	odd	6
1458.2.c.e.973.2		12	9.2	odd	6
1458.2.c.h.487.5		12	9.4	even	3
1458.2.c.h.973.5		12	9.7	even	3

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:	\( N \)	\(=\)	\( 1458 = 2 \cdot 3^{6} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1458.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -3.07935 q^{5} -0.254239 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -3.07935 q^{5} -0.254239 q^{7} -1.00000 q^{8} +3.07935 q^{10} -3.45369 q^{11} +5.63995 q^{13} +0.254239 q^{14} +1.00000 q^{16} +5.01396 q^{17} -0.904786 q^{19} -3.07935 q^{20} +3.45369 q^{22} +5.64985 q^{23} +4.48238 q^{25} -5.63995 q^{26} -0.254239 q^{28} -1.76677 q^{29} -10.6587 q^{31} -1.00000 q^{32} -5.01396 q^{34} +0.782889 q^{35} +8.83464 q^{37} +0.904786 q^{38} +3.07935 q^{40} -5.71415 q^{41} -8.56574 q^{43} -3.45369 q^{44} -5.64985 q^{46} -12.5309 q^{47} -6.93536 q^{49} -4.48238 q^{50} +5.63995 q^{52} +2.65282 q^{53} +10.6351 q^{55} +0.254239 q^{56} +1.76677 q^{58} +3.70138 q^{59} -3.77884 q^{61} +10.6587 q^{62} +1.00000 q^{64} -17.3673 q^{65} -7.58603 q^{67} +5.01396 q^{68} -0.782889 q^{70} -8.42685 q^{71} -6.99035 q^{73} -8.83464 q^{74} -0.904786 q^{76} +0.878062 q^{77} +13.1091 q^{79} -3.07935 q^{80} +5.71415 q^{82} -6.92438 q^{83} -15.4397 q^{85} +8.56574 q^{86} +3.45369 q^{88} -6.80373 q^{89} -1.43389 q^{91} +5.64985 q^{92} +12.5309 q^{94} +2.78615 q^{95} +8.62609 q^{97} +6.93536 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{2} + 6 q^{4} - 6 q^{5} - 6 q^{8}+O(q^{10}) \) 6 * q - 6 * q^2 + 6 * q^4 - 6 * q^5 - 6 * q^8	\( 6 q - 6 q^{2} + 6 q^{4} - 6 q^{5} - 6 q^{8} + 6 q^{10} - 6 q^{11} + 6 q^{13} + 6 q^{16} - 12 q^{17} + 6 q^{19} - 6 q^{20} + 6 q^{22} - 12 q^{23} + 6 q^{25} - 6 q^{26} - 6 q^{29} - 6 q^{31} - 6 q^{32} + 12 q^{34} - 12 q^{35} - 6 q^{38} + 6 q^{40} - 24 q^{41} - 6 q^{43} - 6 q^{44} + 12 q^{46} - 18 q^{47} + 6 q^{49} - 6 q^{50} + 6 q^{52} - 24 q^{53} - 18 q^{55} + 6 q^{58} - 12 q^{59} - 6 q^{61} + 6 q^{62} + 6 q^{64} - 12 q^{65} - 24 q^{67} - 12 q^{68} + 12 q^{70} + 6 q^{71} - 24 q^{73} + 6 q^{76} - 12 q^{77} - 12 q^{79} - 6 q^{80} + 24 q^{82} - 18 q^{83} + 6 q^{86} + 6 q^{88} - 12 q^{89} - 30 q^{91} - 12 q^{92} + 18 q^{94} + 6 q^{95} + 6 q^{97} - 6 q^{98}+O(q^{100}) \) 6 * q - 6 * q^2 + 6 * q^4 - 6 * q^5 - 6 * q^8 + 6 * q^10 - 6 * q^11 + 6 * q^13 + 6 * q^16 - 12 * q^17 + 6 * q^19 - 6 * q^20 + 6 * q^22 - 12 * q^23 + 6 * q^25 - 6 * q^26 - 6 * q^29 - 6 * q^31 - 6 * q^32 + 12 * q^34 - 12 * q^35 - 6 * q^38 + 6 * q^40 - 24 * q^41 - 6 * q^43 - 6 * q^44 + 12 * q^46 - 18 * q^47 + 6 * q^49 - 6 * q^50 + 6 * q^52 - 24 * q^53 - 18 * q^55 + 6 * q^58 - 12 * q^59 - 6 * q^61 + 6 * q^62 + 6 * q^64 - 12 * q^65 - 24 * q^67 - 12 * q^68 + 12 * q^70 + 6 * q^71 - 24 * q^73 + 6 * q^76 - 12 * q^77 - 12 * q^79 - 6 * q^80 + 24 * q^82 - 18 * q^83 + 6 * q^86 + 6 * q^88 - 12 * q^89 - 30 * q^91 - 12 * q^92 + 18 * q^94 + 6 * q^95 + 6 * q^97 - 6 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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