LMFDB - Embedded newform 2673.2.a.p.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2673,2,Mod(1,2673)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2673, 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("2673.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2673 = 3^{5} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2673.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:	$21.3440124603$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.864654912.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 8x^{4} + 14x^{3} + 14x^{2} - 16x - 9 $ x^6 - 2x^5 - 8x^4 + 14x^3 + 14x^2 - 16*x - 9
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.35962$ of defining polynomial
Character	$\chi$	$=$	2673.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.35962 q^{2} -0.151441 q^{4} -1.15144 q^{5} -1.76201 q^{7} -2.92514 q^{8} +O(q^{10})$	\(q+1.35962 q^{2} -0.151441 q^{4} -1.15144 q^{5} -1.76201 q^{7} -2.92514 q^{8} -1.56552 q^{10} +1.00000 q^{11} -2.03604 q^{13} -2.39566 q^{14} -3.67418 q^{16} +4.55879 q^{17} +7.04003 q^{19} +0.174375 q^{20} +1.35962 q^{22} -2.67546 q^{23} -3.67418 q^{25} -2.76824 q^{26} +0.266840 q^{28} -6.08280 q^{29} +8.49015 q^{31} +0.854787 q^{32} +6.19820 q^{34} +2.02885 q^{35} +4.43847 q^{37} +9.57175 q^{38} +3.36812 q^{40} +10.6416 q^{41} -11.0689 q^{43} -0.151441 q^{44} -3.63760 q^{46} +11.0468 q^{47} -3.89532 q^{49} -4.99548 q^{50} +0.308339 q^{52} +8.32753 q^{53} -1.15144 q^{55} +5.15412 q^{56} -8.27029 q^{58} -4.79132 q^{59} +12.3676 q^{61} +11.5434 q^{62} +8.51055 q^{64} +2.34438 q^{65} +5.52274 q^{67} -0.690385 q^{68} +2.75846 q^{70} -3.46550 q^{71} +6.74261 q^{73} +6.03462 q^{74} -1.06615 q^{76} -1.76201 q^{77} +1.60156 q^{79} +4.23061 q^{80} +14.4685 q^{82} +4.98062 q^{83} -5.24917 q^{85} -15.0494 q^{86} -2.92514 q^{88} -1.56779 q^{89} +3.58752 q^{91} +0.405173 q^{92} +15.0194 q^{94} -8.10618 q^{95} -3.69459 q^{97} -5.29615 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{2} + 8 q^{4} + 2 q^{5} - 2 q^{7} + 6 q^{8}+O(q^{10}) $ 6 * q + 2 * q^2 + 8 * q^4 + 2 * q^5 - 2 * q^7 + 6 * q^8	$ 6 q + 2 q^{2} + 8 q^{4} + 2 q^{5} - 2 q^{7} + 6 q^{8} + 8 q^{10} + 6 q^{11} - 8 q^{13} - 4 q^{14} + 8 q^{16} + 2 q^{17} - 4 q^{19} + 40 q^{20} + 2 q^{22} + 10 q^{23} + 8 q^{25} + 2 q^{26} - 12 q^{28} + 6 q^{29} - 8 q^{31} + 4 q^{32} - 10 q^{34} - 10 q^{35} + 2 q^{37} + 30 q^{38} + 18 q^{40} - 4 q^{41} + 2 q^{43} + 8 q^{44} + 4 q^{46} + 28 q^{47} + 6 q^{49} + 16 q^{50} - 12 q^{52} + 24 q^{53} + 2 q^{55} - 6 q^{56} + 6 q^{58} - 8 q^{59} + 2 q^{61} - 10 q^{62} + 18 q^{64} - 4 q^{65} + 12 q^{67} - 14 q^{68} - 10 q^{70} + 30 q^{71} + 16 q^{73} + 78 q^{74} + 2 q^{76} - 2 q^{77} - 12 q^{79} + 58 q^{80} - 16 q^{82} - 16 q^{85} - 24 q^{86} + 6 q^{88} - 6 q^{89} + 6 q^{91} + 32 q^{92} + 10 q^{94} + 6 q^{95} - 18 q^{97} - 6 q^{98}+O(q^{100}) $ 6 * q + 2 * q^2 + 8 * q^4 + 2 * q^5 - 2 * q^7 + 6 * q^8 + 8 * q^10 + 6 * q^11 - 8 * q^13 - 4 * q^14 + 8 * q^16 + 2 * q^17 - 4 * q^19 + 40 * q^20 + 2 * q^22 + 10 * q^23 + 8 * q^25 + 2 * q^26 - 12 * q^28 + 6 * q^29 - 8 * q^31 + 4 * q^32 - 10 * q^34 - 10 * q^35 + 2 * q^37 + 30 * q^38 + 18 * q^40 - 4 * q^41 + 2 * q^43 + 8 * q^44 + 4 * q^46 + 28 * q^47 + 6 * q^49 + 16 * q^50 - 12 * q^52 + 24 * q^53 + 2 * q^55 - 6 * q^56 + 6 * q^58 - 8 * q^59 + 2 * q^61 - 10 * q^62 + 18 * q^64 - 4 * q^65 + 12 * q^67 - 14 * q^68 - 10 * q^70 + 30 * q^71 + 16 * q^73 + 78 * q^74 + 2 * q^76 - 2 * q^77 - 12 * q^79 + 58 * q^80 - 16 * q^82 - 16 * q^85 - 24 * q^86 + 6 * q^88 - 6 * q^89 + 6 * q^91 + 32 * q^92 + 10 * q^94 + 6 * q^95 - 18 * 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 $

$2$	1.35962	0.961395	0.480697	−	0.876887i	$-0.340384\pi$
$2$	1.35962	0.961395	0.480697	+	0.876887i	$0.340384\pi$
$3$	0	0
$3$	0	0
$4$	−0.151441	−0.0757203
$5$	−1.15144	−0.514940	−0.257470	−	0.966286i	$-0.582889\pi$
$5$	−1.15144	−0.514940	−0.257470	+	0.966286i	$0.582889\pi$
$6$	0	0
$7$	−1.76201	−0.665977	−0.332989	−	0.942931i	$-0.608057\pi$
$7$	−1.76201	−0.665977	−0.332989	+	0.942931i	$0.608057\pi$
$8$	−2.92514	−1.03419
$9$	0	0
$10$	−1.56552	−0.495060
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−2.03604	−0.564696	−0.282348	−	0.959312i	$-0.591113\pi$
$13$	−2.03604	−0.564696	−0.282348	+	0.959312i	$0.591113\pi$
$14$	−2.39566	−0.640267
$15$	0	0
$16$	−3.67418	−0.918546
$17$	4.55879	1.10567	0.552834	−	0.833291i	$-0.313546\pi$
$17$	4.55879	1.10567	0.552834	+	0.833291i	$0.313546\pi$
$18$	0	0
$19$	7.04003	1.61509	0.807547	−	0.589804i	$-0.200795\pi$
$19$	7.04003	1.61509	0.807547	+	0.589804i	$0.200795\pi$
$20$	0.174375	0.0389914
$21$	0	0
$22$	1.35962	0.289871
$23$	−2.67546	−0.557872	−0.278936	−	0.960310i	$-0.589982\pi$
$23$	−2.67546	−0.557872	−0.278936	+	0.960310i	$0.589982\pi$
$24$	0	0
$25$	−3.67418	−0.734837
$26$	−2.76824	−0.542896
$27$	0	0
$28$	0.266840	0.0504280
$29$	−6.08280	−1.12955	−0.564774	−	0.825245i	$-0.691037\pi$
$29$	−6.08280	−1.12955	−0.564774	+	0.825245i	$0.691037\pi$
$30$	0	0
$31$	8.49015	1.52488	0.762438	−	0.647061i	$-0.224002\pi$
$31$	8.49015	1.52488	0.762438	+	0.647061i	$0.224002\pi$
$32$	0.854787	0.151106
$33$	0	0
$34$	6.19820	1.06298
$35$	2.02885	0.342938
$36$	0	0
$37$	4.43847	0.729680	0.364840	−	0.931070i	$-0.381124\pi$
$37$	4.43847	0.729680	0.364840	+	0.931070i	$0.381124\pi$
$38$	9.57175	1.55274
$39$	0	0
$40$	3.36812	0.532547
$41$	10.6416	1.66194	0.830969	−	0.556319i	$-0.187787\pi$
$41$	10.6416	1.66194	0.830969	+	0.556319i	$0.187787\pi$
$42$	0	0
$43$	−11.0689	−1.68799	−0.843994	−	0.536352i	$-0.819802\pi$
$43$	−11.0689	−1.68799	−0.843994	+	0.536352i	$0.819802\pi$
$44$	−0.151441	−0.0228305
$45$	0	0
$46$	−3.63760	−0.536335
$47$	11.0468	1.61134	0.805668	−	0.592367i	$-0.201807\pi$
$47$	11.0468	1.61134	0.805668	+	0.592367i	$0.201807\pi$
$48$	0	0
$49$	−3.89532	−0.556475
$50$	−4.99548	−0.706468
$51$	0	0
$52$	0.308339	0.0427590
$53$	8.32753	1.14387	0.571937	−	0.820298i	$-0.306192\pi$
$53$	8.32753	1.14387	0.571937	+	0.820298i	$0.306192\pi$
$54$	0	0
$55$	−1.15144	−0.155260
$56$	5.15412	0.688748
$57$	0	0
$58$	−8.27029	−1.08594
$59$	−4.79132	−0.623776	−0.311888	−	0.950119i	$-0.600961\pi$
$59$	−4.79132	−0.623776	−0.311888	+	0.950119i	$0.600961\pi$
$60$	0	0
$61$	12.3676	1.58350	0.791752	−	0.610843i	$-0.209169\pi$
$61$	12.3676	1.58350	0.791752	+	0.610843i	$0.209169\pi$
$62$	11.5434	1.46601
$63$	0	0
$64$	8.51055	1.06382
$65$	2.34438	0.290785
$66$	0	0
$67$	5.52274	0.674711	0.337355	−	0.941377i	$-0.390468\pi$
$67$	5.52274	0.674711	0.337355	+	0.941377i	$0.390468\pi$
$68$	−0.690385	−0.0837215
$69$	0	0
$70$	2.75846	0.329699
$71$	−3.46550	−0.411279	−0.205640	−	0.978628i	$-0.565927\pi$
$71$	−3.46550	−0.411279	−0.205640	+	0.978628i	$0.565927\pi$
$72$	0	0
$73$	6.74261	0.789162	0.394581	−	0.918861i	$-0.370890\pi$
$73$	6.74261	0.789162	0.394581	+	0.918861i	$0.370890\pi$
$74$	6.03462	0.701510
$75$	0	0
$76$	−1.06615	−0.122295
$77$	−1.76201	−0.200800
$78$	0	0
$79$	1.60156	0.180190	0.0900948	−	0.995933i	$-0.471283\pi$
$79$	1.60156	0.180190	0.0900948	+	0.995933i	$0.471283\pi$
$80$	4.23061	0.472996
$81$	0	0
$82$	14.4685	1.59778
$83$	4.98062	0.546694	0.273347	−	0.961916i	$-0.411869\pi$
$83$	4.98062	0.546694	0.273347	+	0.961916i	$0.411869\pi$
$84$	0	0
$85$	−5.24917	−0.569352
$86$	−15.0494	−1.62282
$87$	0	0
$88$	−2.92514	−0.311821
$89$	−1.56779	−0.166186	−0.0830929	−	0.996542i	$-0.526480\pi$
$89$	−1.56779	−0.166186	−0.0830929	+	0.996542i	$0.526480\pi$
$90$	0	0
$91$	3.58752	0.376075
$92$	0.405173	0.0422422
$93$	0	0
$94$	15.0194	1.54913
$95$	−8.10618	−0.831676
$96$	0	0
$97$	−3.69459	−0.375129	−0.187564	−	0.982252i	$-0.560059\pi$
$97$	−3.69459	−0.375129	−0.187564	+	0.982252i	$0.560059\pi$
$98$	−5.29615	−0.534992
$99$	0	0
$100$	0.556421	0.0556421
$101$	15.0609	1.49861	0.749306	−	0.662224i	$-0.230387\pi$
$101$	15.0609	1.49861	0.749306	+	0.662224i	$0.230387\pi$
$102$	0	0
$103$	5.23699	0.516016	0.258008	−	0.966143i	$-0.416934\pi$
$103$	5.23699	0.516016	0.258008	+	0.966143i	$0.416934\pi$
$104$	5.95570	0.584004
$105$	0	0
$106$	11.3223	1.09971
$107$	16.1009	1.55654	0.778268	−	0.627932i	$-0.216099\pi$
$107$	16.1009	1.55654	0.778268	+	0.627932i	$0.216099\pi$
$108$	0	0
$109$	2.38042	0.228003	0.114002	−	0.993481i	$-0.463633\pi$
$109$	2.38042	0.228003	0.114002	+	0.993481i	$0.463633\pi$
$110$	−1.56552	−0.149266
$111$	0	0
$112$	6.47395	0.611731
$113$	4.66624	0.438963	0.219481	−	0.975617i	$-0.429564\pi$
$113$	4.66624	0.438963	0.219481	+	0.975617i	$0.429564\pi$
$114$	0	0
$115$	3.08063	0.287271
$116$	0.921184	0.0855298
$117$	0	0
$118$	−6.51436	−0.599695
$119$	−8.03262	−0.736349
$120$	0	0
$121$	1.00000	0.0909091
$122$	16.8151	1.52237
$123$	0	0
$124$	−1.28575	−0.115464
$125$	9.98781	0.893337
$126$	0	0
$127$	−5.77466	−0.512418	−0.256209	−	0.966621i	$-0.582474\pi$
$127$	−5.77466	−0.512418	−0.256209	+	0.966621i	$0.582474\pi$
$128$	9.86152	0.871644
$129$	0	0
$130$	3.18746	0.279559
$131$	−0.506564	−0.0442587	−0.0221294	−	0.999755i	$-0.507045\pi$
$131$	−0.506564	−0.0442587	−0.0221294	+	0.999755i	$0.507045\pi$
$132$	0	0
$133$	−12.4046	−1.07562
$134$	7.50882	0.648663
$135$	0	0
$136$	−13.3351	−1.14347
$137$	−11.2887	−0.964462	−0.482231	−	0.876044i	$-0.660173\pi$
$137$	−11.2887	−0.964462	−0.482231	+	0.876044i	$0.660173\pi$
$138$	0	0
$139$	−20.6991	−1.75567	−0.877836	−	0.478961i	$-0.841014\pi$
$139$	−20.6991	−1.75567	−0.877836	+	0.478961i	$0.841014\pi$
$140$	−0.307250	−0.0259674
$141$	0	0
$142$	−4.71176	−0.395402
$143$	−2.03604	−0.170262
$144$	0	0
$145$	7.00399	0.581650
$146$	9.16736	0.758697
$147$	0	0
$148$	−0.672165	−0.0552516
$149$	−14.3313	−1.17407	−0.587033	−	0.809563i	$-0.699704\pi$
$149$	−14.3313	−1.17407	−0.587033	+	0.809563i	$0.699704\pi$
$150$	0	0
$151$	−7.55287	−0.614644	−0.307322	−	0.951606i	$-0.599433\pi$
$151$	−7.55287	−0.614644	−0.307322	+	0.951606i	$0.599433\pi$
$152$	−20.5930	−1.67032
$153$	0	0
$154$	−2.39566	−0.193048
$155$	−9.77590	−0.785219
$156$	0	0
$157$	4.20312	0.335446	0.167723	−	0.985834i	$-0.446359\pi$
$157$	4.20312	0.335446	0.167723	+	0.985834i	$0.446359\pi$
$158$	2.17751	0.173233
$159$	0	0
$160$	−0.984237	−0.0778107
$161$	4.71419	0.371530
$162$	0	0
$163$	−19.2770	−1.50989	−0.754945	−	0.655788i	$-0.772337\pi$
$163$	−19.2770	−1.50989	−0.754945	+	0.655788i	$0.772337\pi$
$164$	−1.61157	−0.125842
$165$	0	0
$166$	6.77173	0.525588
$167$	−4.47405	−0.346213	−0.173106	−	0.984903i	$-0.555380\pi$
$167$	−4.47405	−0.346213	−0.173106	+	0.984903i	$0.555380\pi$
$168$	0	0
$169$	−8.85454	−0.681118
$170$	−7.13686	−0.547372
$171$	0	0
$172$	1.67628	0.127815
$173$	21.9773	1.67090	0.835452	−	0.549564i	$-0.185206\pi$
$173$	21.9773	1.67090	0.835452	+	0.549564i	$0.185206\pi$
$174$	0	0
$175$	6.47395	0.489384
$176$	−3.67418	−0.276952
$177$	0	0
$178$	−2.13160	−0.159770
$179$	1.85355	0.138541	0.0692703	−	0.997598i	$-0.477933\pi$
$179$	1.85355	0.138541	0.0692703	+	0.997598i	$0.477933\pi$
$180$	0	0
$181$	−4.19946	−0.312143	−0.156072	−	0.987746i	$-0.549883\pi$
$181$	−4.19946	−0.312143	−0.156072	+	0.987746i	$0.549883\pi$
$182$	4.87766	0.361556
$183$	0	0
$184$	7.82608	0.576947
$185$	−5.11063	−0.375741
$186$	0	0
$187$	4.55879	0.333371
$188$	−1.67293	−0.122011
$189$	0	0
$190$	−11.0213	−0.799569
$191$	16.2611	1.17661	0.588305	−	0.808639i	$-0.299796\pi$
$191$	16.2611	1.17661	0.588305	+	0.808639i	$0.299796\pi$
$192$	0	0
$193$	−0.398226	−0.0286649	−0.0143325	−	0.999897i	$-0.504562\pi$
$193$	−0.398226	−0.0286649	−0.0143325	+	0.999897i	$0.504562\pi$
$194$	−5.02323	−0.360647
$195$	0	0
$196$	0.589910	0.0421364
$197$	−4.01620	−0.286142	−0.143071	−	0.989712i	$-0.545698\pi$
$197$	−4.01620	−0.286142	−0.143071	+	0.989712i	$0.545698\pi$
$198$	0	0
$199$	12.2557	0.868783	0.434391	−	0.900724i	$-0.356963\pi$
$199$	12.2557	0.868783	0.434391	+	0.900724i	$0.356963\pi$
$200$	10.7475	0.759962
$201$	0	0
$202$	20.4770	1.44076
$203$	10.7180	0.752253
$204$	0	0
$205$	−12.2532	−0.855798
$206$	7.12030	0.496095
$207$	0	0
$208$	7.48079	0.518700
$209$	7.04003	0.486969
$210$	0	0
$211$	−27.3871	−1.88541	−0.942704	−	0.333629i	$-0.891727\pi$
$211$	−27.3871	−1.88541	−0.942704	+	0.333629i	$0.891727\pi$
$212$	−1.26113	−0.0866145
$213$	0	0
$214$	21.8911	1.49645
$215$	12.7452	0.869212
$216$	0	0
$217$	−14.9597	−1.01553
$218$	3.23646	0.219201
$219$	0	0
$220$	0.174375	0.0117564
$221$	−9.28188	−0.624367
$222$	0	0
$223$	9.01164	0.603464	0.301732	−	0.953393i	$-0.402435\pi$
$223$	9.01164	0.603464	0.301732	+	0.953393i	$0.402435\pi$
$224$	−1.50614	−0.100633
$225$	0	0
$226$	6.34430	0.422016
$227$	7.41752	0.492318	0.246159	−	0.969229i	$-0.420831\pi$
$227$	7.41752	0.492318	0.246159	+	0.969229i	$0.420831\pi$
$228$	0	0
$229$	−19.3096	−1.27601	−0.638007	−	0.770031i	$-0.720241\pi$
$229$	−19.3096	−1.27601	−0.638007	+	0.770031i	$0.720241\pi$
$230$	4.18848	0.276180
$231$	0	0
$232$	17.7930	1.16817
$233$	13.1537	0.861729	0.430864	−	0.902417i	$-0.358209\pi$
$233$	13.1537	0.861729	0.430864	+	0.902417i	$0.358209\pi$
$234$	0	0
$235$	−12.7197	−0.829741
$236$	0.725600	0.0472325
$237$	0	0
$238$	−10.9213	−0.707922
$239$	7.62770	0.493395	0.246697	−	0.969093i	$-0.420655\pi$
$239$	7.62770	0.493395	0.246697	+	0.969093i	$0.420655\pi$
$240$	0	0
$241$	−8.62915	−0.555853	−0.277926	−	0.960602i	$-0.589647\pi$
$241$	−8.62915	−0.555853	−0.277926	+	0.960602i	$0.589647\pi$
$242$	1.35962	0.0873995
$243$	0	0
$244$	−1.87295	−0.119903
$245$	4.48523	0.286551
$246$	0	0
$247$	−14.3338	−0.912037
$248$	−24.8348	−1.57701
$249$	0	0
$250$	13.5796	0.858849
$251$	5.90523	0.372735	0.186367	−	0.982480i	$-0.440329\pi$
$251$	5.90523	0.372735	0.186367	+	0.982480i	$0.440329\pi$
$252$	0	0
$253$	−2.67546	−0.168205
$254$	−7.85133	−0.492636
$255$	0	0
$256$	−3.61321	−0.225826
$257$	25.5590	1.59433	0.797164	−	0.603763i	$-0.206333\pi$
$257$	25.5590	1.59433	0.797164	+	0.603763i	$0.206333\pi$
$258$	0	0
$259$	−7.82063	−0.485950
$260$	−0.355035	−0.0220183
$261$	0	0
$262$	−0.688733	−0.0425501
$263$	3.38444	0.208694	0.104347	−	0.994541i	$-0.466725\pi$
$263$	3.38444	0.208694	0.104347	+	0.994541i	$0.466725\pi$
$264$	0	0
$265$	−9.58865	−0.589026
$266$	−16.8655	−1.03409
$267$	0	0
$268$	−0.836368	−0.0510893
$269$	2.38419	0.145366	0.0726832	−	0.997355i	$-0.476844\pi$
$269$	2.38419	0.145366	0.0726832	+	0.997355i	$0.476844\pi$
$270$	0	0
$271$	−6.72471	−0.408497	−0.204249	−	0.978919i	$-0.565475\pi$
$271$	−6.72471	−0.408497	−0.204249	+	0.978919i	$0.565475\pi$
$272$	−16.7498	−1.01561
$273$	0	0
$274$	−15.3484	−0.927229
$275$	−3.67418	−0.221562
$276$	0	0
$277$	13.0326	0.783054	0.391527	−	0.920167i	$-0.371947\pi$
$277$	13.0326	0.783054	0.391527	+	0.920167i	$0.371947\pi$
$278$	−28.1428	−1.68789
$279$	0	0
$280$	−5.93466	−0.354664
$281$	−17.4618	−1.04168	−0.520841	−	0.853654i	$-0.674381\pi$
$281$	−17.4618	−1.04168	−0.520841	+	0.853654i	$0.674381\pi$
$282$	0	0
$283$	−1.24938	−0.0742682	−0.0371341	−	0.999310i	$-0.511823\pi$
$283$	−1.24938	−0.0742682	−0.0371341	+	0.999310i	$0.511823\pi$
$284$	0.524818	0.0311422
$285$	0	0
$286$	−2.76824	−0.163689
$287$	−18.7506	−1.10681
$288$	0	0
$289$	3.78252	0.222501
$290$	9.52274	0.559195
$291$	0	0
$292$	−1.02110	−0.0597556
$293$	22.0465	1.28797	0.643987	−	0.765037i	$-0.277279\pi$
$293$	22.0465	1.28797	0.643987	+	0.765037i	$0.277279\pi$
$294$	0	0
$295$	5.51692	0.321207
$296$	−12.9831	−0.754629
$297$	0	0
$298$	−19.4851	−1.12874
$299$	5.44735	0.315028
$300$	0	0
$301$	19.5035	1.12416
$302$	−10.2690	−0.590915
$303$	0	0
$304$	−25.8664	−1.48354
$305$	−14.2405	−0.815409
$306$	0	0
$307$	−16.7197	−0.954246	−0.477123	−	0.878837i	$-0.658320\pi$
$307$	−16.7197	−0.954246	−0.477123	+	0.878837i	$0.658320\pi$
$308$	0.266840	0.0152046
$309$	0	0
$310$	−13.2915	−0.754906
$311$	8.00879	0.454137	0.227069	−	0.973879i	$-0.427086\pi$
$311$	8.00879	0.454137	0.227069	+	0.973879i	$0.427086\pi$
$312$	0	0
$313$	2.39790	0.135537	0.0677686	−	0.997701i	$-0.478412\pi$
$313$	2.39790	0.135537	0.0677686	+	0.997701i	$0.478412\pi$
$314$	5.71464	0.322496
$315$	0	0
$316$	−0.242541	−0.0136440
$317$	−22.5099	−1.26428	−0.632140	−	0.774854i	$-0.717823\pi$
$317$	−22.5099	−1.26428	−0.632140	+	0.774854i	$0.717823\pi$
$318$	0	0
$319$	−6.08280	−0.340572
$320$	−9.79940	−0.547803
$321$	0	0
$322$	6.40949	0.357187
$323$	32.0940	1.78576
$324$	0	0
$325$	7.48079	0.414960
$326$	−26.2093	−1.45160
$327$	0	0
$328$	−31.1281	−1.71876
$329$	−19.4645	−1.07311
$330$	0	0
$331$	15.2233	0.836749	0.418374	−	0.908275i	$-0.362600\pi$
$331$	15.2233	0.836749	0.418374	+	0.908275i	$0.362600\pi$
$332$	−0.754268	−0.0413958
$333$	0	0
$334$	−6.08300	−0.332847
$335$	−6.35911	−0.347435
$336$	0	0
$337$	5.60680	0.305422	0.152711	−	0.988271i	$-0.451200\pi$
$337$	5.60680	0.305422	0.152711	+	0.988271i	$0.451200\pi$
$338$	−12.0388	−0.654823
$339$	0	0
$340$	0.794938	0.0431116
$341$	8.49015	0.459767
$342$	0	0
$343$	19.1977	1.03658
$344$	32.3780	1.74570
$345$	0	0
$346$	29.8807	1.60640
$347$	27.1156	1.45564	0.727822	−	0.685766i	$-0.240533\pi$
$347$	27.1156	1.45564	0.727822	+	0.685766i	$0.240533\pi$
$348$	0	0
$349$	−11.3996	−0.610207	−0.305104	−	0.952319i	$-0.598691\pi$
$349$	−11.3996	−0.610207	−0.305104	+	0.952319i	$0.598691\pi$
$350$	8.80209	0.470492
$351$	0	0
$352$	0.854787	0.0455603
$353$	−37.0293	−1.97087	−0.985435	−	0.170054i	$-0.945606\pi$
$353$	−37.0293	−1.97087	−0.985435	+	0.170054i	$0.945606\pi$
$354$	0	0
$355$	3.99032	0.211784
$356$	0.237428	0.0125836
$357$	0	0
$358$	2.52012	0.133192
$359$	−2.66771	−0.140796	−0.0703981	−	0.997519i	$-0.522427\pi$
$359$	−2.66771	−0.140796	−0.0703981	+	0.997519i	$0.522427\pi$
$360$	0	0
$361$	30.5620	1.60853
$362$	−5.70966	−0.300093
$363$	0	0
$364$	−0.543297	−0.0284765
$365$	−7.76371	−0.406371
$366$	0	0
$367$	35.1318	1.83386	0.916931	−	0.399045i	$-0.130658\pi$
$367$	35.1318	1.83386	0.916931	+	0.399045i	$0.130658\pi$
$368$	9.83013	0.512431
$369$	0	0
$370$	−6.94851	−0.361236
$371$	−14.6732	−0.761794
$372$	0	0
$373$	−21.5790	−1.11732	−0.558658	−	0.829398i	$-0.688684\pi$
$373$	−21.5790	−1.11732	−0.558658	+	0.829398i	$0.688684\pi$
$374$	6.19820	0.320501
$375$	0	0
$376$	−32.3133	−1.66643
$377$	12.3848	0.637852
$378$	0	0
$379$	−21.7397	−1.11669	−0.558346	−	0.829608i	$-0.688564\pi$
$379$	−21.7397	−1.11669	−0.558346	+	0.829608i	$0.688564\pi$
$380$	1.22760	0.0629748
$381$	0	0
$382$	22.1088	1.13119
$383$	38.3165	1.95788	0.978940	−	0.204147i	$-0.0654421\pi$
$383$	38.3165	1.95788	0.978940	+	0.204147i	$0.0654421\pi$
$384$	0	0
$385$	2.02885	0.103400
$386$	−0.541434	−0.0275583
$387$	0	0
$388$	0.559511	0.0284049
$389$	−14.0410	−0.711905	−0.355952	−	0.934504i	$-0.615843\pi$
$389$	−14.0410	−0.711905	−0.355952	+	0.934504i	$0.615843\pi$
$390$	0	0
$391$	−12.1968	−0.616821
$392$	11.3943	0.575501
$393$	0	0
$394$	−5.46049	−0.275096
$395$	−1.84410	−0.0927868
$396$	0	0
$397$	−7.70021	−0.386462	−0.193231	−	0.981153i	$-0.561897\pi$
$397$	−7.70021	−0.386462	−0.193231	+	0.981153i	$0.561897\pi$
$398$	16.6630	0.835243
$399$	0	0
$400$	13.4996	0.674982
$401$	19.3428	0.965934	0.482967	−	0.875639i	$-0.339559\pi$
$401$	19.3428	0.965934	0.482967	+	0.875639i	$0.339559\pi$
$402$	0	0
$403$	−17.2863	−0.861092
$404$	−2.28083	−0.113475
$405$	0	0
$406$	14.5723	0.723212
$407$	4.43847	0.220007
$408$	0	0
$409$	−9.47178	−0.468349	−0.234175	−	0.972195i	$-0.575239\pi$
$409$	−9.47178	−0.468349	−0.234175	+	0.972195i	$0.575239\pi$
$410$	−16.6596	−0.822759
$411$	0	0
$412$	−0.793093	−0.0390729
$413$	8.44235	0.415421
$414$	0	0
$415$	−5.73488	−0.281514
$416$	−1.74038	−0.0853292
$417$	0	0
$418$	9.57175	0.468169
$419$	21.0251	1.02714	0.513571	−	0.858047i	$-0.328322\pi$
$419$	21.0251	1.02714	0.513571	+	0.858047i	$0.328322\pi$
$420$	0	0
$421$	25.2852	1.23233	0.616163	−	0.787619i	$-0.288686\pi$
$421$	25.2852	1.23233	0.616163	+	0.787619i	$0.288686\pi$
$422$	−37.2360	−1.81262
$423$	0	0
$424$	−24.3592	−1.18299
$425$	−16.7498	−0.812486
$426$	0	0
$427$	−21.7918	−1.05458
$428$	−2.43834	−0.117861
$429$	0	0
$430$	17.3285	0.835656
$431$	41.0191	1.97582	0.987909	−	0.155034i	$-0.0495488\pi$
$431$	41.0191	1.97582	0.987909	+	0.155034i	$0.0495488\pi$
$432$	0	0
$433$	−2.40881	−0.115760	−0.0578801	−	0.998324i	$-0.518434\pi$
$433$	−2.40881	−0.115760	−0.0578801	+	0.998324i	$0.518434\pi$
$434$	−20.3395	−0.976327
$435$	0	0
$436$	−0.360493	−0.0172645
$437$	−18.8353	−0.901015
$438$	0	0
$439$	4.19407	0.200172	0.100086	−	0.994979i	$-0.468088\pi$
$439$	4.19407	0.200172	0.100086	+	0.994979i	$0.468088\pi$
$440$	3.36812	0.160569
$441$	0	0
$442$	−12.6198	−0.600263
$443$	39.7698	1.88952	0.944759	−	0.327765i	$-0.106296\pi$
$443$	39.7698	1.88952	0.944759	+	0.327765i	$0.106296\pi$
$444$	0	0
$445$	1.80522	0.0855757
$446$	12.2524	0.580167
$447$	0	0
$448$	−14.9957	−0.708479
$449$	−35.0930	−1.65614	−0.828069	−	0.560626i	$-0.810561\pi$
$449$	−35.0930	−1.65614	−0.828069	+	0.560626i	$0.810561\pi$
$450$	0	0
$451$	10.6416	0.501093
$452$	−0.706658	−0.0332384
$453$	0	0
$454$	10.0850	0.473312
$455$	−4.13082	−0.193656
$456$	0	0
$457$	33.5472	1.56927	0.784636	−	0.619957i	$-0.212850\pi$
$457$	33.5472	1.56927	0.784636	+	0.619957i	$0.212850\pi$
$458$	−26.2536	−1.22675
$459$	0	0
$460$	−0.466533	−0.0217522
$461$	−42.6476	−1.98630	−0.993149	−	0.116853i	$-0.962719\pi$
$461$	−42.6476	−1.98630	−0.993149	+	0.116853i	$0.962719\pi$
$462$	0	0
$463$	37.5848	1.74671	0.873356	−	0.487082i	$-0.161939\pi$
$463$	37.5848	1.74671	0.873356	+	0.487082i	$0.161939\pi$
$464$	22.3493	1.03754
$465$	0	0
$466$	17.8840	0.828461
$467$	−20.2141	−0.935398	−0.467699	−	0.883888i	$-0.654917\pi$
$467$	−20.2141	−0.935398	−0.467699	+	0.883888i	$0.654917\pi$
$468$	0	0
$469$	−9.73113	−0.449342
$470$	−17.2939	−0.797709
$471$	0	0
$472$	14.0153	0.645104
$473$	−11.0689	−0.508948
$474$	0	0
$475$	−25.8664	−1.18683
$476$	1.21647	0.0557566
$477$	0	0
$478$	10.3707	0.474347
$479$	40.3735	1.84471	0.922357	−	0.386340i	$-0.126261\pi$
$479$	40.3735	1.84471	0.922357	+	0.386340i	$0.126261\pi$
$480$	0	0
$481$	−9.03691	−0.412048
$482$	−11.7323	−0.534394
$483$	0	0
$484$	−0.151441	−0.00688367
$485$	4.25410	0.193169
$486$	0	0
$487$	2.44792	0.110926	0.0554629	−	0.998461i	$-0.482337\pi$
$487$	2.44792	0.110926	0.0554629	+	0.998461i	$0.482337\pi$
$488$	−36.1768	−1.63765
$489$	0	0
$490$	6.09820	0.275489
$491$	−3.45205	−0.155789	−0.0778943	−	0.996962i	$-0.524820\pi$
$491$	−3.45205	−0.155789	−0.0778943	+	0.996962i	$0.524820\pi$
$492$	0	0
$493$	−27.7302	−1.24891
$494$	−19.4885	−0.876828
$495$	0	0
$496$	−31.1944	−1.40067
$497$	6.10625	0.273903
$498$	0	0
$499$	17.7851	0.796168	0.398084	−	0.917349i	$-0.369675\pi$
$499$	17.7851	0.796168	0.398084	+	0.917349i	$0.369675\pi$
$500$	−1.51256	−0.0676437
$501$	0	0
$502$	8.02885	0.358345
$503$	−18.6642	−0.832195	−0.416097	−	0.909320i	$-0.636602\pi$
$503$	−18.6642	−0.832195	−0.416097	+	0.909320i	$0.636602\pi$
$504$	0	0
$505$	−17.3417	−0.771695
$506$	−3.63760	−0.161711
$507$	0	0
$508$	0.874518	0.0388005
$509$	−24.2871	−1.07650	−0.538252	−	0.842784i	$-0.680915\pi$
$509$	−24.2871	−1.07650	−0.538252	+	0.842784i	$0.680915\pi$
$510$	0	0
$511$	−11.8805	−0.525564
$512$	−24.6356	−1.08875
$513$	0	0
$514$	34.7505	1.53278
$515$	−6.03008	−0.265717
$516$	0	0
$517$	11.0468	0.485836
$518$	−10.6331	−0.467190
$519$	0	0
$520$	−6.85763	−0.300727
$521$	−19.8194	−0.868304	−0.434152	−	0.900840i	$-0.642952\pi$
$521$	−19.8194	−0.868304	−0.434152	+	0.900840i	$0.642952\pi$
$522$	0	0
$523$	−0.447131	−0.0195517	−0.00977585	−	0.999952i	$-0.503112\pi$
$523$	−0.447131	−0.0195517	−0.00977585	+	0.999952i	$0.503112\pi$
$524$	0.0767144	0.00335128
$525$	0	0
$526$	4.60155	0.200637
$527$	38.7048	1.68601
$528$	0	0
$529$	−15.8419	−0.688779
$530$	−13.0369	−0.566287
$531$	0	0
$532$	1.87856	0.0814459
$533$	−21.6667	−0.938490
$534$	0	0
$535$	−18.5393	−0.801522
$536$	−16.1548	−0.697780
$537$	0	0
$538$	3.24158	0.139755
$539$	−3.89532	−0.167783
$540$	0	0
$541$	17.0636	0.733623	0.366811	−	0.930295i	$-0.380449\pi$
$541$	17.0636	0.733623	0.366811	+	0.930295i	$0.380449\pi$
$542$	−9.14304	−0.392727
$543$	0	0
$544$	3.89679	0.167074
$545$	−2.74091	−0.117408
$546$	0	0
$547$	−13.4094	−0.573345	−0.286673	−	0.958029i	$-0.592549\pi$
$547$	−13.4094	−0.573345	−0.286673	+	0.958029i	$0.592549\pi$
$548$	1.70957	0.0730294
$549$	0	0
$550$	−4.99548	−0.213008
$551$	−42.8231	−1.82433
$552$	0	0
$553$	−2.82196	−0.120002
$554$	17.7194	0.752824
$555$	0	0
$556$	3.13468	0.132940
$557$	36.1065	1.52988	0.764940	−	0.644101i	$-0.222768\pi$
$557$	36.1065	1.52988	0.764940	+	0.644101i	$0.222768\pi$
$558$	0	0
$559$	22.5367	0.953201
$560$	−7.45437	−0.315004
$561$	0	0
$562$	−23.7413	−1.00147
$563$	−6.33596	−0.267029	−0.133515	−	0.991047i	$-0.542626\pi$
$563$	−6.33596	−0.267029	−0.133515	+	0.991047i	$0.542626\pi$
$564$	0	0
$565$	−5.37289	−0.226039
$566$	−1.69869	−0.0714011
$567$	0	0
$568$	10.1371	0.425342
$569$	13.4173	0.562484	0.281242	−	0.959637i	$-0.409254\pi$
$569$	13.4173	0.562484	0.281242	+	0.959637i	$0.409254\pi$
$570$	0	0
$571$	−21.1177	−0.883748	−0.441874	−	0.897077i	$-0.645686\pi$
$571$	−21.1177	−0.883748	−0.441874	+	0.897077i	$0.645686\pi$
$572$	0.308339	0.0128923
$573$	0	0
$574$	−25.4936	−1.06408
$575$	9.83013	0.409945
$576$	0	0
$577$	19.0617	0.793548	0.396774	−	0.917916i	$-0.370130\pi$
$577$	19.0617	0.793548	0.396774	+	0.917916i	$0.370130\pi$
$578$	5.14278	0.213912
$579$	0	0
$580$	−1.06069	−0.0440427
$581$	−8.77589	−0.364085
$582$	0	0
$583$	8.32753	0.344891
$584$	−19.7230	−0.816145
$585$	0	0
$586$	29.9749	1.23825
$587$	−17.1264	−0.706881	−0.353440	−	0.935457i	$-0.614988\pi$
$587$	−17.1264	−0.706881	−0.353440	+	0.935457i	$0.614988\pi$
$588$	0	0
$589$	59.7709	2.46282
$590$	7.50090	0.308807
$591$	0	0
$592$	−16.3078	−0.670245
$593$	−9.40862	−0.386366	−0.193183	−	0.981163i	$-0.561881\pi$
$593$	−9.40862	−0.386366	−0.193183	+	0.981163i	$0.561881\pi$
$594$	0	0
$595$	9.24909	0.379176
$596$	2.17034	0.0889006
$597$	0	0
$598$	7.40631	0.302866
$599$	10.8718	0.444210	0.222105	−	0.975023i	$-0.428707\pi$
$599$	10.8718	0.444210	0.222105	+	0.975023i	$0.428707\pi$
$600$	0	0
$601$	14.4014	0.587447	0.293723	−	0.955890i	$-0.405106\pi$
$601$	14.4014	0.587447	0.293723	+	0.955890i	$0.405106\pi$
$602$	26.5173	1.08076
$603$	0	0
$604$	1.14381	0.0465410
$605$	−1.15144	−0.0468127
$606$	0	0
$607$	−16.8005	−0.681912	−0.340956	−	0.940079i	$-0.610751\pi$
$607$	−16.8005	−0.681912	−0.340956	+	0.940079i	$0.610751\pi$
$608$	6.01773	0.244051
$609$	0	0
$610$	−19.3616	−0.783930
$611$	−22.4917	−0.909915
$612$	0	0
$613$	−11.4539	−0.462617	−0.231308	−	0.972880i	$-0.574301\pi$
$613$	−11.4539	−0.462617	−0.231308	+	0.972880i	$0.574301\pi$
$614$	−22.7324	−0.917407
$615$	0	0
$616$	5.15412	0.207665
$617$	7.70804	0.310314	0.155157	−	0.987890i	$-0.450412\pi$
$617$	7.70804	0.310314	0.155157	+	0.987890i	$0.450412\pi$
$618$	0	0
$619$	−18.6993	−0.751588	−0.375794	−	0.926703i	$-0.622630\pi$
$619$	−18.6993	−0.751588	−0.375794	+	0.926703i	$0.622630\pi$
$620$	1.48047	0.0594571
$621$	0	0
$622$	10.8889	0.436605
$623$	2.76247	0.110676
$624$	0	0
$625$	6.87055	0.274822
$626$	3.26022	0.130305
$627$	0	0
$628$	−0.636523	−0.0254000
$629$	20.2340	0.806784
$630$	0	0
$631$	−12.4754	−0.496638	−0.248319	−	0.968678i	$-0.579878\pi$
$631$	−12.4754	−0.496638	−0.248319	+	0.968678i	$0.579878\pi$
$632$	−4.68478	−0.186351
$633$	0	0
$634$	−30.6048	−1.21547
$635$	6.64918	0.263865
$636$	0	0
$637$	7.93104	0.314239
$638$	−8.27029	−0.327424
$639$	0	0
$640$	−11.3550	−0.448844
$641$	−32.7666	−1.29420	−0.647102	−	0.762403i	$-0.724019\pi$
$641$	−32.7666	−1.29420	−0.647102	+	0.762403i	$0.724019\pi$
$642$	0	0
$643$	23.3012	0.918910	0.459455	−	0.888201i	$-0.348045\pi$
$643$	23.3012	0.918910	0.459455	+	0.888201i	$0.348045\pi$
$644$	−0.713919	−0.0281324
$645$	0	0
$646$	43.6355	1.71682
$647$	−18.8724	−0.741951	−0.370976	−	0.928643i	$-0.620977\pi$
$647$	−18.8724	−0.741951	−0.370976	+	0.928643i	$0.620977\pi$
$648$	0	0
$649$	−4.79132	−0.188076
$650$	10.1710	0.398940
$651$	0	0
$652$	2.91932	0.114329
$653$	−6.09647	−0.238573	−0.119287	−	0.992860i	$-0.538061\pi$
$653$	−6.09647	−0.238573	−0.119287	+	0.992860i	$0.538061\pi$
$654$	0	0
$655$	0.583279	0.0227906
$656$	−39.0992	−1.52657
$657$	0	0
$658$	−26.4643	−1.03168
$659$	−24.5048	−0.954573	−0.477286	−	0.878748i	$-0.658380\pi$
$659$	−24.5048	−0.954573	−0.477286	+	0.878748i	$0.658380\pi$
$660$	0	0
$661$	−7.54055	−0.293294	−0.146647	−	0.989189i	$-0.546848\pi$
$661$	−7.54055	−0.293294	−0.146647	+	0.989189i	$0.546848\pi$
$662$	20.6979	0.804446
$663$	0	0
$664$	−14.5690	−0.565386
$665$	14.2832	0.553877
$666$	0	0
$667$	16.2743	0.630143
$668$	0.677553	0.0262153
$669$	0	0
$670$	−8.64596	−0.334023
$671$	12.3676	0.477444
$672$	0	0
$673$	−0.530040	−0.0204315	−0.0102158	−	0.999948i	$-0.503252\pi$
$673$	−0.530040	−0.0204315	−0.0102158	+	0.999948i	$0.503252\pi$
$674$	7.62311	0.293631
$675$	0	0
$676$	1.34094	0.0515745
$677$	20.4298	0.785179	0.392590	−	0.919714i	$-0.371579\pi$
$677$	20.4298	0.785179	0.392590	+	0.919714i	$0.371579\pi$
$678$	0	0
$679$	6.50990	0.249827
$680$	15.3545	0.588820
$681$	0	0
$682$	11.5434	0.442018
$683$	−20.3293	−0.777880	−0.388940	−	0.921263i	$-0.627159\pi$
$683$	−20.3293	−0.777880	−0.388940	+	0.921263i	$0.627159\pi$
$684$	0	0
$685$	12.9983	0.496640
$686$	26.1015	0.996559
$687$	0	0
$688$	40.6691	1.55050
$689$	−16.9552	−0.645941
$690$	0	0
$691$	−0.341668	−0.0129977	−0.00649883	−	0.999979i	$-0.502069\pi$
$691$	−0.341668	−0.0129977	−0.00649883	+	0.999979i	$0.502069\pi$
$692$	−3.32826	−0.126521
$693$	0	0
$694$	36.8669	1.39945
$695$	23.8338	0.904066
$696$	0	0
$697$	48.5127	1.83755
$698$	−15.4991	−0.586650
$699$	0	0
$700$	−0.980419	−0.0370563
$701$	19.0012	0.717664	0.358832	−	0.933402i	$-0.383175\pi$
$701$	19.0012	0.717664	0.358832	+	0.933402i	$0.383175\pi$
$702$	0	0
$703$	31.2470	1.17850
$704$	8.51055	0.320754
$705$	0	0
$706$	−50.3456	−1.89478
$707$	−26.5374	−0.998042
$708$	0	0
$709$	39.4286	1.48077	0.740386	−	0.672182i	$-0.234643\pi$
$709$	39.4286	1.48077	0.740386	+	0.672182i	$0.234643\pi$
$710$	5.42531	0.203608
$711$	0	0
$712$	4.58601	0.171868
$713$	−22.7151	−0.850685
$714$	0	0
$715$	2.34438	0.0876749
$716$	−0.280702	−0.0104903
$717$	0	0
$718$	−3.62706	−0.135361
$719$	41.5523	1.54964	0.774819	−	0.632183i	$-0.217841\pi$
$719$	41.5523	1.54964	0.774819	+	0.632183i	$0.217841\pi$
$720$	0	0
$721$	−9.22763	−0.343655
$722$	41.5526	1.54643
$723$	0	0
$724$	0.635969	0.0236356
$725$	22.3493	0.830034
$726$	0	0
$727$	−17.9625	−0.666193	−0.333096	−	0.942893i	$-0.608093\pi$
$727$	−17.9625	−0.666193	−0.333096	+	0.942893i	$0.608093\pi$
$728$	−10.4940	−0.388933
$729$	0	0
$730$	−10.5557	−0.390683
$731$	−50.4606	−1.86635
$732$	0	0
$733$	41.5208	1.53361	0.766804	−	0.641882i	$-0.221846\pi$
$733$	41.5208	1.53361	0.766804	+	0.641882i	$0.221846\pi$
$734$	47.7657	1.76307
$735$	0	0
$736$	−2.28695	−0.0842980
$737$	5.52274	0.203433
$738$	0	0
$739$	35.0156	1.28807	0.644035	−	0.764996i	$-0.277259\pi$
$739$	35.0156	1.28807	0.644035	+	0.764996i	$0.277259\pi$
$740$	0.773958	0.0284512
$741$	0	0
$742$	−19.9499	−0.732384
$743$	−7.68179	−0.281818	−0.140909	−	0.990023i	$-0.545002\pi$
$743$	−7.68179	−0.281818	−0.140909	+	0.990023i	$0.545002\pi$
$744$	0	0
$745$	16.5016	0.604573
$746$	−29.3391	−1.07418
$747$	0	0
$748$	−0.690385	−0.0252430
$749$	−28.3700	−1.03662
$750$	0	0
$751$	−30.6530	−1.11854	−0.559272	−	0.828984i	$-0.688919\pi$
$751$	−30.6530	−1.11854	−0.559272	+	0.828984i	$0.688919\pi$
$752$	−40.5878	−1.48009
$753$	0	0
$754$	16.8386	0.613227
$755$	8.69668	0.316505
$756$	0	0
$757$	37.1316	1.34957	0.674785	−	0.738014i	$-0.264236\pi$
$757$	37.1316	1.34957	0.674785	+	0.738014i	$0.264236\pi$
$758$	−29.5576	−1.07358
$759$	0	0
$760$	23.7117	0.860113
$761$	−49.3454	−1.78877	−0.894385	−	0.447298i	$-0.852386\pi$
$761$	−49.3454	−1.78877	−0.894385	+	0.447298i	$0.852386\pi$
$762$	0	0
$763$	−4.19433	−0.151845
$764$	−2.46259	−0.0890932
$765$	0	0
$766$	52.0957	1.88230
$767$	9.75532	0.352244
$768$	0	0
$769$	−49.6717	−1.79121	−0.895604	−	0.444853i	$-0.853256\pi$
$769$	−49.6717	−1.79121	−0.895604	+	0.444853i	$0.853256\pi$
$770$	2.75846	0.0994080
$771$	0	0
$772$	0.0603075	0.00217052
$773$	47.8613	1.72145	0.860726	−	0.509068i	$-0.170010\pi$
$773$	47.8613	1.72145	0.860726	+	0.509068i	$0.170010\pi$
$774$	0	0
$775$	−31.1944	−1.12053
$776$	10.8072	0.387955
$777$	0	0
$778$	−19.0903	−0.684421
$779$	74.9171	2.68418
$780$	0	0
$781$	−3.46550	−0.124005
$782$	−16.5830	−0.593008
$783$	0	0
$784$	14.3121	0.511148
$785$	−4.83964	−0.172734
$786$	0	0
$787$	46.4026	1.65408	0.827038	−	0.562147i	$-0.190024\pi$
$787$	46.4026	1.65408	0.827038	+	0.562147i	$0.190024\pi$
$788$	0.608216	0.0216668
$789$	0	0
$790$	−2.50727	−0.0892047
$791$	−8.22195	−0.292339
$792$	0	0
$793$	−25.1809	−0.894199
$794$	−10.4693	−0.371543
$795$	0	0
$796$	−1.85601	−0.0657845
$797$	7.77120	0.275270	0.137635	−	0.990483i	$-0.456050\pi$
$797$	7.77120	0.275270	0.137635	+	0.990483i	$0.456050\pi$
$798$	0	0
$799$	50.3598	1.78160
$800$	−3.14065	−0.111039
$801$	0	0
$802$	26.2988	0.928643
$803$	6.74261	0.237941
$804$	0	0
$805$	−5.42810	−0.191316
$806$	−23.5027	−0.827849
$807$	0	0
$808$	−44.0551	−1.54985
$809$	−12.1971	−0.428827	−0.214413	−	0.976743i	$-0.568784\pi$
$809$	−12.1971	−0.428827	−0.214413	+	0.976743i	$0.568784\pi$
$810$	0	0
$811$	36.9480	1.29742	0.648710	−	0.761036i	$-0.275309\pi$
$811$	36.9480	1.29742	0.648710	+	0.761036i	$0.275309\pi$
$812$	−1.62313	−0.0569609
$813$	0	0
$814$	6.03462	0.211513
$815$	22.1963	0.777503
$816$	0	0
$817$	−77.9252	−2.72626
$818$	−12.8780	−0.450269
$819$	0	0
$820$	1.85563	0.0648013
$821$	−38.8230	−1.35493	−0.677465	−	0.735555i	$-0.736922\pi$
$821$	−38.8230	−1.35493	−0.677465	+	0.735555i	$0.736922\pi$
$822$	0	0
$823$	−15.5999	−0.543779	−0.271889	−	0.962329i	$-0.587648\pi$
$823$	−15.5999	−0.543779	−0.271889	+	0.962329i	$0.587648\pi$
$824$	−15.3189	−0.533659
$825$	0	0
$826$	11.4784	0.399383
$827$	−5.95414	−0.207046	−0.103523	−	0.994627i	$-0.533012\pi$
$827$	−5.95414	−0.207046	−0.103523	+	0.994627i	$0.533012\pi$
$828$	0	0
$829$	23.3117	0.809649	0.404824	−	0.914394i	$-0.367333\pi$
$829$	23.3117	0.809649	0.404824	+	0.914394i	$0.367333\pi$
$830$	−7.79725	−0.270646
$831$	0	0
$832$	−17.3278	−0.600735
$833$	−17.7579	−0.615276
$834$	0	0
$835$	5.15161	0.178279
$836$	−1.06615	−0.0368734
$837$	0	0
$838$	28.5861	0.987489
$839$	−21.9479	−0.757726	−0.378863	−	0.925453i	$-0.623685\pi$
$839$	−21.9479	−0.757726	−0.378863	+	0.925453i	$0.623685\pi$
$840$	0	0
$841$	8.00051	0.275880
$842$	34.3782	1.18475
$843$	0	0
$844$	4.14753	0.142764
$845$	10.1955	0.350735
$846$	0	0
$847$	−1.76201	−0.0605434
$848$	−30.5969	−1.05070
$849$	0	0
$850$	−22.7733	−0.781119
$851$	−11.8749	−0.407068
$852$	0	0
$853$	−8.30825	−0.284469	−0.142235	−	0.989833i	$-0.545429\pi$
$853$	−8.30825	−0.284469	−0.142235	+	0.989833i	$0.545429\pi$
$854$	−29.6284	−1.01386
$855$	0	0
$856$	−47.0974	−1.60976
$857$	−0.489337	−0.0167155	−0.00835773	−	0.999965i	$-0.502660\pi$
$857$	−0.489337	−0.0167155	−0.00835773	+	0.999965i	$0.502660\pi$
$858$	0	0
$859$	19.6663	0.671007	0.335503	−	0.942039i	$-0.391094\pi$
$859$	19.6663	0.671007	0.335503	+	0.942039i	$0.391094\pi$
$860$	−1.93013	−0.0658170
$861$	0	0
$862$	55.7702	1.89954
$863$	−12.5631	−0.427654	−0.213827	−	0.976872i	$-0.568593\pi$
$863$	−12.5631	−0.427654	−0.213827	+	0.976872i	$0.568593\pi$
$864$	0	0
$865$	−25.3056	−0.860415
$866$	−3.27507	−0.111291
$867$	0	0
$868$	2.26551	0.0768964
$869$	1.60156	0.0543292
$870$	0	0
$871$	−11.2445	−0.381007
$872$	−6.96306	−0.235799
$873$	0	0
$874$	−25.6088	−0.866231
$875$	−17.5986	−0.594942
$876$	0	0
$877$	−14.5235	−0.490425	−0.245213	−	0.969469i	$-0.578858\pi$
$877$	−14.5235	−0.490425	−0.245213	+	0.969469i	$0.578858\pi$
$878$	5.70233	0.192444
$879$	0	0
$880$	4.23061	0.142614
$881$	−18.6158	−0.627181	−0.313591	−	0.949558i	$-0.601532\pi$
$881$	−18.6158	−0.627181	−0.313591	+	0.949558i	$0.601532\pi$
$882$	0	0
$883$	−53.5166	−1.80098	−0.900489	−	0.434880i	$-0.856791\pi$
$883$	−53.5166	−1.80098	−0.900489	+	0.434880i	$0.856791\pi$
$884$	1.40565	0.0472772
$885$	0	0
$886$	54.0717	1.81657
$887$	34.0317	1.14267	0.571337	−	0.820716i	$-0.306425\pi$
$887$	34.0317	1.14267	0.571337	+	0.820716i	$0.306425\pi$
$888$	0	0
$889$	10.1750	0.341259
$890$	2.45441	0.0822720
$891$	0	0
$892$	−1.36473	−0.0456945
$893$	77.7695	2.60246
$894$	0	0
$895$	−2.13425	−0.0713401
$896$	−17.3761	−0.580495
$897$	0	0
$898$	−47.7130	−1.59220
$899$	−51.6439	−1.72242
$900$	0	0
$901$	37.9634	1.26474
$902$	14.4685	0.481748
$903$	0	0
$904$	−13.6494	−0.453972
$905$	4.83543	0.160735
$906$	0	0
$907$	4.10133	0.136183	0.0680913	−	0.997679i	$-0.478309\pi$
$907$	4.10133	0.136183	0.0680913	+	0.997679i	$0.478309\pi$
$908$	−1.12331	−0.0372785
$909$	0	0
$910$	−5.61634	−0.186180
$911$	10.6782	0.353786	0.176893	−	0.984230i	$-0.443395\pi$
$911$	10.6782	0.353786	0.176893	+	0.984230i	$0.443395\pi$
$912$	0	0
$913$	4.98062	0.164834
$914$	45.6114	1.50869
$915$	0	0
$916$	2.92426	0.0966201
$917$	0.892571	0.0294753
$918$	0	0
$919$	−36.5044	−1.20417	−0.602085	−	0.798432i	$-0.705663\pi$
$919$	−36.5044	−1.20417	−0.602085	+	0.798432i	$0.705663\pi$
$920$	−9.01127	−0.297093
$921$	0	0
$922$	−57.9845	−1.90962
$923$	7.05591	0.232248
$924$	0	0
$925$	−16.3078	−0.536196
$926$	51.1009	1.67928
$927$	0	0
$928$	−5.19950	−0.170682
$929$	36.7547	1.20588	0.602941	−	0.797786i	$-0.293995\pi$
$929$	36.7547	1.20588	0.602941	+	0.797786i	$0.293995\pi$
$930$	0	0
$931$	−27.4232	−0.898759
$932$	−1.99201	−0.0652504
$933$	0	0
$934$	−27.4835	−0.899287
$935$	−5.24917	−0.171666
$936$	0	0
$937$	17.3065	0.565378	0.282689	−	0.959212i	$-0.408774\pi$
$937$	17.3065	0.565378	0.282689	+	0.959212i	$0.408774\pi$
$938$	−13.2306	−0.431995
$939$	0	0
$940$	1.92628	0.0628283
$941$	25.7259	0.838640	0.419320	−	0.907838i	$-0.362268\pi$
$941$	25.7259	0.838640	0.419320	+	0.907838i	$0.362268\pi$
$942$	0	0
$943$	−28.4711	−0.927148
$944$	17.6042	0.572967
$945$	0	0
$946$	−15.0494	−0.489299
$947$	8.40317	0.273066	0.136533	−	0.990635i	$-0.456404\pi$
$947$	8.40317	0.273066	0.136533	+	0.990635i	$0.456404\pi$
$948$	0	0
$949$	−13.7282	−0.445637
$950$	−35.1684	−1.14101
$951$	0	0
$952$	23.4965	0.761526
$953$	40.5039	1.31205	0.656024	−	0.754740i	$-0.272237\pi$
$953$	40.5039	1.31205	0.656024	+	0.754740i	$0.272237\pi$
$954$	0	0
$955$	−18.7236	−0.605883
$956$	−1.15514	−0.0373600
$957$	0	0
$958$	54.8925	1.77350
$959$	19.8909	0.642310
$960$	0	0
$961$	41.0826	1.32525
$962$	−12.2867	−0.396140
$963$	0	0
$964$	1.30680	0.0420893
$965$	0.458533	0.0147607
$966$	0	0
$967$	46.7661	1.50390	0.751948	−	0.659223i	$-0.229115\pi$
$967$	46.7661	1.50390	0.751948	+	0.659223i	$0.229115\pi$
$968$	−2.92514	−0.0940174
$969$	0	0
$970$	5.78395	0.185711
$971$	48.7704	1.56512	0.782559	−	0.622577i	$-0.213914\pi$
$971$	48.7704	1.56512	0.782559	+	0.622577i	$0.213914\pi$
$972$	0	0
$973$	36.4720	1.16924
$974$	3.32823	0.106643
$975$	0	0
$976$	−45.4407	−1.45452
$977$	−31.3533	−1.00308	−0.501540	−	0.865135i	$-0.667233\pi$
$977$	−31.3533	−1.00308	−0.501540	+	0.865135i	$0.667233\pi$
$978$	0	0
$979$	−1.56779	−0.0501069
$980$	−0.679246	−0.0216977
$981$	0	0
$982$	−4.69346	−0.149774
$983$	7.44661	0.237510	0.118755	−	0.992924i	$-0.462110\pi$
$983$	7.44661	0.237510	0.118755	+	0.992924i	$0.462110\pi$
$984$	0	0
$985$	4.62441	0.147346
$986$	−37.7025	−1.20069
$987$	0	0
$988$	2.17072	0.0690598
$989$	29.6143	0.941681
$990$	0	0
$991$	19.6236	0.623364	0.311682	−	0.950186i	$-0.399108\pi$
$991$	19.6236	0.623364	0.311682	+	0.950186i	$0.399108\pi$
$992$	7.25727	0.230419
$993$	0	0
$994$	8.30216	0.263329
$995$	−14.1117	−0.447371
$996$	0	0
$997$	−40.5124	−1.28304	−0.641521	−	0.767106i	$-0.721696\pi$
$997$	−40.5124	−1.28304	−0.641521	+	0.767106i	$0.721696\pi$
$998$	24.1809	0.765432
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2673.2.a.p.1.4	yes	6
3.2	odd	2		2673.2.a.j.1.3	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2673.2.a.j.1.3	✓	6	3.2	odd	2
2673.2.a.p.1.4	yes	6	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:	\( N \)	\(=\)	\( 2673 = 3^{5} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2673.a (trivial)

\(f(q)\)	\(=\)	\(q+1.35962 q^{2} -0.151441 q^{4} -1.15144 q^{5} -1.76201 q^{7} -2.92514 q^{8} +O(q^{10})\)	\(q+1.35962 q^{2} -0.151441 q^{4} -1.15144 q^{5} -1.76201 q^{7} -2.92514 q^{8} -1.56552 q^{10} +1.00000 q^{11} -2.03604 q^{13} -2.39566 q^{14} -3.67418 q^{16} +4.55879 q^{17} +7.04003 q^{19} +0.174375 q^{20} +1.35962 q^{22} -2.67546 q^{23} -3.67418 q^{25} -2.76824 q^{26} +0.266840 q^{28} -6.08280 q^{29} +8.49015 q^{31} +0.854787 q^{32} +6.19820 q^{34} +2.02885 q^{35} +4.43847 q^{37} +9.57175 q^{38} +3.36812 q^{40} +10.6416 q^{41} -11.0689 q^{43} -0.151441 q^{44} -3.63760 q^{46} +11.0468 q^{47} -3.89532 q^{49} -4.99548 q^{50} +0.308339 q^{52} +8.32753 q^{53} -1.15144 q^{55} +5.15412 q^{56} -8.27029 q^{58} -4.79132 q^{59} +12.3676 q^{61} +11.5434 q^{62} +8.51055 q^{64} +2.34438 q^{65} +5.52274 q^{67} -0.690385 q^{68} +2.75846 q^{70} -3.46550 q^{71} +6.74261 q^{73} +6.03462 q^{74} -1.06615 q^{76} -1.76201 q^{77} +1.60156 q^{79} +4.23061 q^{80} +14.4685 q^{82} +4.98062 q^{83} -5.24917 q^{85} -15.0494 q^{86} -2.92514 q^{88} -1.56779 q^{89} +3.58752 q^{91} +0.405173 q^{92} +15.0194 q^{94} -8.10618 q^{95} -3.69459 q^{97} -5.29615 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{2} + 8 q^{4} + 2 q^{5} - 2 q^{7} + 6 q^{8}+O(q^{10}) \) 6 * q + 2 * q^2 + 8 * q^4 + 2 * q^5 - 2 * q^7 + 6 * q^8	\( 6 q + 2 q^{2} + 8 q^{4} + 2 q^{5} - 2 q^{7} + 6 q^{8} + 8 q^{10} + 6 q^{11} - 8 q^{13} - 4 q^{14} + 8 q^{16} + 2 q^{17} - 4 q^{19} + 40 q^{20} + 2 q^{22} + 10 q^{23} + 8 q^{25} + 2 q^{26} - 12 q^{28} + 6 q^{29} - 8 q^{31} + 4 q^{32} - 10 q^{34} - 10 q^{35} + 2 q^{37} + 30 q^{38} + 18 q^{40} - 4 q^{41} + 2 q^{43} + 8 q^{44} + 4 q^{46} + 28 q^{47} + 6 q^{49} + 16 q^{50} - 12 q^{52} + 24 q^{53} + 2 q^{55} - 6 q^{56} + 6 q^{58} - 8 q^{59} + 2 q^{61} - 10 q^{62} + 18 q^{64} - 4 q^{65} + 12 q^{67} - 14 q^{68} - 10 q^{70} + 30 q^{71} + 16 q^{73} + 78 q^{74} + 2 q^{76} - 2 q^{77} - 12 q^{79} + 58 q^{80} - 16 q^{82} - 16 q^{85} - 24 q^{86} + 6 q^{88} - 6 q^{89} + 6 q^{91} + 32 q^{92} + 10 q^{94} + 6 q^{95} - 18 q^{97} - 6 q^{98}+O(q^{100}) \) 6 * q + 2 * q^2 + 8 * q^4 + 2 * q^5 - 2 * q^7 + 6 * q^8 + 8 * q^10 + 6 * q^11 - 8 * q^13 - 4 * q^14 + 8 * q^16 + 2 * q^17 - 4 * q^19 + 40 * q^20 + 2 * q^22 + 10 * q^23 + 8 * q^25 + 2 * q^26 - 12 * q^28 + 6 * q^29 - 8 * q^31 + 4 * q^32 - 10 * q^34 - 10 * q^35 + 2 * q^37 + 30 * q^38 + 18 * q^40 - 4 * q^41 + 2 * q^43 + 8 * q^44 + 4 * q^46 + 28 * q^47 + 6 * q^49 + 16 * q^50 - 12 * q^52 + 24 * q^53 + 2 * q^55 - 6 * q^56 + 6 * q^58 - 8 * q^59 + 2 * q^61 - 10 * q^62 + 18 * q^64 - 4 * q^65 + 12 * q^67 - 14 * q^68 - 10 * q^70 + 30 * q^71 + 16 * q^73 + 78 * q^74 + 2 * q^76 - 2 * q^77 - 12 * q^79 + 58 * q^80 - 16 * q^82 - 16 * q^85 - 24 * q^86 + 6 * q^88 - 6 * q^89 + 6 * q^91 + 32 * q^92 + 10 * q^94 + 6 * q^95 - 18 * q^97 - 6 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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