LMFDB - Embedded newform 2151.2.a.k.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("2151.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$17.1758214748$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 4 x^{19} - 21 x^{18} + 96 x^{17} + 164 x^{16} - 936 x^{15} - 540 x^{14} + 4804 x^{13} + \cdots + 1 $ x^20 - 4x^19 - 21x^18 + 96x^17 + 164x^16 - 936x^15 - 540x^14 + 4804x^13 + 229x^12 - 14020x^11 + 3356x^10 + 23404x^9 - 9429x^8 - 21252x^7 + 10479x^6 + 9108x^5 - 4844x^4 - 1184x^3 + 640x^2 - 56*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.88454$ of defining polynomial
Character	$\chi$	$=$	2151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.88454 q^{2} +1.55150 q^{4} -1.57704 q^{5} -2.47195 q^{7} +0.845212 q^{8} +O(q^{10})$	\(q-1.88454 q^{2} +1.55150 q^{4} -1.57704 q^{5} -2.47195 q^{7} +0.845212 q^{8} +2.97201 q^{10} +3.83501 q^{11} -3.18691 q^{13} +4.65849 q^{14} -4.69584 q^{16} +4.51018 q^{17} +4.54355 q^{19} -2.44679 q^{20} -7.22723 q^{22} -1.92415 q^{23} -2.51293 q^{25} +6.00587 q^{26} -3.83524 q^{28} +0.0587099 q^{29} -9.58500 q^{31} +7.15910 q^{32} -8.49962 q^{34} +3.89837 q^{35} +0.789192 q^{37} -8.56252 q^{38} -1.33294 q^{40} +2.07589 q^{41} +2.55270 q^{43} +5.95002 q^{44} +3.62615 q^{46} -11.9701 q^{47} -0.889470 q^{49} +4.73573 q^{50} -4.94450 q^{52} +0.844934 q^{53} -6.04797 q^{55} -2.08932 q^{56} -0.110641 q^{58} +1.09245 q^{59} +2.97957 q^{61} +18.0633 q^{62} -4.09994 q^{64} +5.02589 q^{65} +1.93061 q^{67} +6.99755 q^{68} -7.34665 q^{70} +8.02351 q^{71} -6.42371 q^{73} -1.48727 q^{74} +7.04934 q^{76} -9.47994 q^{77} +8.18519 q^{79} +7.40555 q^{80} -3.91210 q^{82} +9.82889 q^{83} -7.11274 q^{85} -4.81067 q^{86} +3.24139 q^{88} -8.85934 q^{89} +7.87787 q^{91} -2.98533 q^{92} +22.5581 q^{94} -7.16538 q^{95} +16.4977 q^{97} +1.67625 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 4 q^{2} + 18 q^{4} + 16 q^{5} - 4 q^{7} + 12 q^{8}+O(q^{10}) $ 20 * q + 4 * q^2 + 18 * q^4 + 16 * q^5 - 4 * q^7 + 12 * q^8	$ 20 q + 4 q^{2} + 18 q^{4} + 16 q^{5} - 4 q^{7} + 12 q^{8} + 4 q^{10} + 12 q^{11} - 4 q^{13} + 20 q^{14} + 22 q^{16} + 24 q^{17} - 4 q^{19} + 40 q^{20} - 6 q^{22} + 12 q^{23} + 22 q^{25} + 30 q^{26} - 12 q^{28} + 24 q^{29} - 4 q^{31} + 28 q^{32} + 8 q^{34} + 20 q^{35} - 10 q^{37} + 26 q^{38} + 6 q^{40} + 66 q^{41} + 8 q^{43} + 36 q^{44} - 12 q^{46} + 28 q^{47} + 18 q^{49} + 28 q^{50} - 18 q^{52} + 28 q^{53} - 4 q^{55} + 60 q^{56} + 54 q^{59} - 4 q^{61} + 20 q^{62} + 22 q^{64} + 42 q^{65} + 12 q^{67} + 12 q^{68} + 20 q^{70} + 36 q^{71} + 14 q^{73} - 50 q^{76} + 8 q^{77} - 12 q^{79} + 88 q^{80} - 8 q^{82} + 20 q^{83} + 4 q^{85} + 18 q^{86} - 10 q^{88} + 130 q^{89} - 6 q^{91} - 46 q^{92} - 26 q^{94} - 2 q^{97} + 12 q^{98}+O(q^{100}) $ 20 * q + 4 * q^2 + 18 * q^4 + 16 * q^5 - 4 * q^7 + 12 * q^8 + 4 * q^10 + 12 * q^11 - 4 * q^13 + 20 * q^14 + 22 * q^16 + 24 * q^17 - 4 * q^19 + 40 * q^20 - 6 * q^22 + 12 * q^23 + 22 * q^25 + 30 * q^26 - 12 * q^28 + 24 * q^29 - 4 * q^31 + 28 * q^32 + 8 * q^34 + 20 * q^35 - 10 * q^37 + 26 * q^38 + 6 * q^40 + 66 * q^41 + 8 * q^43 + 36 * q^44 - 12 * q^46 + 28 * q^47 + 18 * q^49 + 28 * q^50 - 18 * q^52 + 28 * q^53 - 4 * q^55 + 60 * q^56 + 54 * q^59 - 4 * q^61 + 20 * q^62 + 22 * q^64 + 42 * q^65 + 12 * q^67 + 12 * q^68 + 20 * q^70 + 36 * q^71 + 14 * q^73 - 50 * q^76 + 8 * q^77 - 12 * q^79 + 88 * q^80 - 8 * q^82 + 20 * q^83 + 4 * q^85 + 18 * q^86 - 10 * q^88 + 130 * q^89 - 6 * q^91 - 46 * q^92 - 26 * q^94 - 2 * q^97 + 12 * 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 $

$2$	−1.88454	−1.33257	−0.666287	−	0.745696i	$-0.732117\pi$
$2$	−1.88454	−1.33257	−0.666287	+	0.745696i	$0.732117\pi$
$3$	0	0
$3$	0	0
$4$	1.55150	0.775751
$5$	−1.57704	−0.705275	−0.352638	−	0.935760i	$-0.614715\pi$
$5$	−1.57704	−0.705275	−0.352638	+	0.935760i	$0.614715\pi$
$6$	0	0
$7$	−2.47195	−0.934309	−0.467154	−	0.884176i	$-0.654721\pi$
$7$	−2.47195	−0.934309	−0.467154	+	0.884176i	$0.654721\pi$
$8$	0.845212	0.298828
$9$	0	0
$10$	2.97201	0.939831
$11$	3.83501	1.15630	0.578149	−	0.815931i	$-0.303775\pi$
$11$	3.83501	1.15630	0.578149	+	0.815931i	$0.303775\pi$
$12$	0	0
$13$	−3.18691	−0.883889	−0.441945	−	0.897042i	$-0.645711\pi$
$13$	−3.18691	−0.883889	−0.441945	+	0.897042i	$0.645711\pi$
$14$	4.65849	1.24503
$15$	0	0
$16$	−4.69584	−1.17396
$17$	4.51018	1.09388	0.546939	−	0.837172i	$-0.315793\pi$
$17$	4.51018	1.09388	0.546939	+	0.837172i	$0.315793\pi$
$18$	0	0
$19$	4.54355	1.04236	0.521181	−	0.853446i	$-0.325492\pi$
$19$	4.54355	1.04236	0.521181	+	0.853446i	$0.325492\pi$
$20$	−2.44679	−0.547118
$21$	0	0
$22$	−7.22723	−1.54085
$23$	−1.92415	−0.401213	−0.200607	−	0.979672i	$-0.564291\pi$
$23$	−1.92415	−0.401213	−0.200607	+	0.979672i	$0.564291\pi$
$24$	0	0
$25$	−2.51293	−0.502587
$26$	6.00587	1.17785
$27$	0	0
$28$	−3.83524	−0.724791
$29$	0.0587099	0.0109022	0.00545108	−	0.999985i	$-0.498265\pi$
$29$	0.0587099	0.0109022	0.00545108	+	0.999985i	$0.498265\pi$
$30$	0	0
$31$	−9.58500	−1.72152	−0.860758	−	0.509014i	$-0.830010\pi$
$31$	−9.58500	−1.72152	−0.860758	+	0.509014i	$0.830010\pi$
$32$	7.15910	1.26556
$33$	0	0
$34$	−8.49962	−1.45767
$35$	3.89837	0.658945
$36$	0	0
$37$	0.789192	0.129742	0.0648712	−	0.997894i	$-0.479336\pi$
$37$	0.789192	0.129742	0.0648712	+	0.997894i	$0.479336\pi$
$38$	−8.56252	−1.38902
$39$	0	0
$40$	−1.33294	−0.210756
$41$	2.07589	0.324199	0.162100	−	0.986774i	$-0.448173\pi$
$41$	2.07589	0.324199	0.162100	+	0.986774i	$0.448173\pi$
$42$	0	0
$43$	2.55270	0.389283	0.194641	−	0.980874i	$-0.437646\pi$
$43$	2.55270	0.389283	0.194641	+	0.980874i	$0.437646\pi$
$44$	5.95002	0.897000
$45$	0	0
$46$	3.62615	0.534646
$47$	−11.9701	−1.74601	−0.873007	−	0.487708i	$-0.837833\pi$
$47$	−11.9701	−1.74601	−0.873007	+	0.487708i	$0.837833\pi$
$48$	0	0
$49$	−0.889470	−0.127067
$50$	4.73573	0.669734
$51$	0	0
$52$	−4.94450	−0.685678
$53$	0.844934	0.116061	0.0580303	−	0.998315i	$-0.481518\pi$
$53$	0.844934	0.116061	0.0580303	+	0.998315i	$0.481518\pi$
$54$	0	0
$55$	−6.04797	−0.815508
$56$	−2.08932	−0.279197
$57$	0	0
$58$	−0.110641	−0.0145279
$59$	1.09245	0.142225	0.0711123	−	0.997468i	$-0.477345\pi$
$59$	1.09245	0.142225	0.0711123	+	0.997468i	$0.477345\pi$
$60$	0	0
$61$	2.97957	0.381494	0.190747	−	0.981639i	$-0.438909\pi$
$61$	2.97957	0.381494	0.190747	+	0.981639i	$0.438909\pi$
$62$	18.0633	2.29405
$63$	0	0
$64$	−4.09994	−0.512492
$65$	5.02589	0.623385
$66$	0	0
$67$	1.93061	0.235862	0.117931	−	0.993022i	$-0.462374\pi$
$67$	1.93061	0.235862	0.117931	+	0.993022i	$0.462374\pi$
$68$	6.99755	0.848578
$69$	0	0
$70$	−7.34665	−0.878092
$71$	8.02351	0.952215	0.476107	−	0.879387i	$-0.342047\pi$
$71$	8.02351	0.952215	0.476107	+	0.879387i	$0.342047\pi$
$72$	0	0
$73$	−6.42371	−0.751839	−0.375919	−	0.926652i	$-0.622673\pi$
$73$	−6.42371	−0.751839	−0.375919	+	0.926652i	$0.622673\pi$
$74$	−1.48727	−0.172891
$75$	0	0
$76$	7.04934	0.808614
$77$	−9.47994	−1.08034
$78$	0	0
$79$	8.18519	0.920906	0.460453	−	0.887684i	$-0.347687\pi$
$79$	8.18519	0.920906	0.460453	+	0.887684i	$0.347687\pi$
$80$	7.40555	0.827966
$81$	0	0
$82$	−3.91210	−0.432019
$83$	9.82889	1.07886	0.539431	−	0.842030i	$-0.318640\pi$
$83$	9.82889	1.07886	0.539431	+	0.842030i	$0.318640\pi$
$84$	0	0
$85$	−7.11274	−0.771485
$86$	−4.81067	−0.518748
$87$	0	0
$88$	3.24139	0.345534
$89$	−8.85934	−0.939088	−0.469544	−	0.882909i	$-0.655582\pi$
$89$	−8.85934	−0.939088	−0.469544	+	0.882909i	$0.655582\pi$
$90$	0	0
$91$	7.87787	0.825825
$92$	−2.98533	−0.311242
$93$	0	0
$94$	22.5581	2.32669
$95$	−7.16538	−0.735153
$96$	0	0
$97$	16.4977	1.67508	0.837541	−	0.546374i	$-0.183992\pi$
$97$	16.4977	1.67508	0.837541	+	0.546374i	$0.183992\pi$
$98$	1.67625	0.169326
$99$	0	0
$100$	−3.89882	−0.389882
$101$	8.00022	0.796052	0.398026	−	0.917374i	$-0.369695\pi$
$101$	8.00022	0.796052	0.398026	+	0.917374i	$0.369695\pi$
$102$	0	0
$103$	−4.66612	−0.459766	−0.229883	−	0.973218i	$-0.573834\pi$
$103$	−4.66612	−0.459766	−0.229883	+	0.973218i	$0.573834\pi$
$104$	−2.69361	−0.264130
$105$	0	0
$106$	−1.59232	−0.154659
$107$	1.59511	0.154205	0.0771023	−	0.997023i	$-0.475433\pi$
$107$	1.59511	0.154205	0.0771023	+	0.997023i	$0.475433\pi$
$108$	0	0
$109$	6.67236	0.639096	0.319548	−	0.947570i	$-0.396469\pi$
$109$	6.67236	0.639096	0.319548	+	0.947570i	$0.396469\pi$
$110$	11.3977	1.08672
$111$	0	0
$112$	11.6079	1.09684
$113$	−1.46613	−0.137922	−0.0689612	−	0.997619i	$-0.521968\pi$
$113$	−1.46613	−0.137922	−0.0689612	+	0.997619i	$0.521968\pi$
$114$	0	0
$115$	3.03447	0.282966
$116$	0.0910886	0.00845736
$117$	0	0
$118$	−2.05876	−0.189525
$119$	−11.1489	−1.02202
$120$	0	0
$121$	3.70727	0.337024
$122$	−5.61512	−0.508369
$123$	0	0
$124$	−14.8712	−1.33547
$125$	11.8482	1.05974
$126$	0	0
$127$	−15.0707	−1.33731	−0.668655	−	0.743572i	$-0.733130\pi$
$127$	−15.0707	−1.33731	−0.668655	+	0.743572i	$0.733130\pi$
$128$	−6.59168	−0.582628
$129$	0	0
$130$	−9.47151	−0.830706
$131$	10.1307	0.885122	0.442561	−	0.896738i	$-0.354070\pi$
$131$	10.1307	0.885122	0.442561	+	0.896738i	$0.354070\pi$
$132$	0	0
$133$	−11.2314	−0.973888
$134$	−3.63832	−0.314303
$135$	0	0
$136$	3.81205	0.326881
$137$	1.49257	0.127519	0.0637596	−	0.997965i	$-0.479691\pi$
$137$	1.49257	0.127519	0.0637596	+	0.997965i	$0.479691\pi$
$138$	0	0
$139$	22.4652	1.90547	0.952736	−	0.303800i	$-0.0982555\pi$
$139$	22.4652	1.90547	0.952736	+	0.303800i	$0.0982555\pi$
$140$	6.04833	0.511177
$141$	0	0
$142$	−15.1206	−1.26890
$143$	−12.2218	−1.02204
$144$	0	0
$145$	−0.0925881	−0.00768902
$146$	12.1058	1.00188
$147$	0	0
$148$	1.22443	0.100648
$149$	11.3282	0.928045	0.464022	−	0.885823i	$-0.346406\pi$
$149$	11.3282	0.928045	0.464022	+	0.885823i	$0.346406\pi$
$150$	0	0
$151$	10.2512	0.834231	0.417116	−	0.908853i	$-0.363041\pi$
$151$	10.2512	0.834231	0.417116	+	0.908853i	$0.363041\pi$
$152$	3.84027	0.311487
$153$	0	0
$154$	17.8653	1.43963
$155$	15.1160	1.21414
$156$	0	0
$157$	−2.23497	−0.178370	−0.0891849	−	0.996015i	$-0.528426\pi$
$157$	−2.23497	−0.178370	−0.0891849	+	0.996015i	$0.528426\pi$
$158$	−15.4253	−1.22717
$159$	0	0
$160$	−11.2902	−0.892569
$161$	4.75640	0.374857
$162$	0	0
$163$	−11.1937	−0.876762	−0.438381	−	0.898789i	$-0.644448\pi$
$163$	−11.1937	−0.876762	−0.438381	+	0.898789i	$0.644448\pi$
$164$	3.22075	0.251498
$165$	0	0
$166$	−18.5230	−1.43766
$167$	9.54997	0.738999	0.369499	−	0.929231i	$-0.379529\pi$
$167$	9.54997	0.738999	0.369499	+	0.929231i	$0.379529\pi$
$168$	0	0
$169$	−2.84362	−0.218740
$170$	13.4043	1.02806
$171$	0	0
$172$	3.96052	0.301987
$173$	−7.16166	−0.544491	−0.272246	−	0.962228i	$-0.587766\pi$
$173$	−7.16166	−0.544491	−0.272246	+	0.962228i	$0.587766\pi$
$174$	0	0
$175$	6.21184	0.469571
$176$	−18.0086	−1.35745
$177$	0	0
$178$	16.6958	1.25140
$179$	−0.114586	−0.00856454	−0.00428227	−	0.999991i	$-0.501363\pi$
$179$	−0.114586	−0.00856454	−0.00428227	+	0.999991i	$0.501363\pi$
$180$	0	0
$181$	−2.24629	−0.166966	−0.0834828	−	0.996509i	$-0.526604\pi$
$181$	−2.24629	−0.166966	−0.0834828	+	0.996509i	$0.526604\pi$
$182$	−14.8462	−1.10047
$183$	0	0
$184$	−1.62632	−0.119894
$185$	−1.24459	−0.0915041
$186$	0	0
$187$	17.2965	1.26485
$188$	−18.5716	−1.35447
$189$	0	0
$190$	13.5035	0.979645
$191$	7.90682	0.572117	0.286059	−	0.958212i	$-0.407655\pi$
$191$	7.90682	0.572117	0.286059	+	0.958212i	$0.407655\pi$
$192$	0	0
$193$	−10.5321	−0.758120	−0.379060	−	0.925372i	$-0.623753\pi$
$193$	−10.5321	−0.758120	−0.379060	+	0.925372i	$0.623753\pi$
$194$	−31.0905	−2.23217
$195$	0	0
$196$	−1.38002	−0.0985726
$197$	16.5832	1.18150	0.590751	−	0.806854i	$-0.298832\pi$
$197$	16.5832	1.18150	0.590751	+	0.806854i	$0.298832\pi$
$198$	0	0
$199$	12.2741	0.870089	0.435044	−	0.900409i	$-0.356733\pi$
$199$	12.2741	0.870089	0.435044	+	0.900409i	$0.356733\pi$
$200$	−2.12396	−0.150187
$201$	0	0
$202$	−15.0768	−1.06080
$203$	−0.145128	−0.0101860
$204$	0	0
$205$	−3.27377	−0.228650
$206$	8.79350	0.612672
$207$	0	0
$208$	14.9652	1.03765
$209$	17.4246	1.20528
$210$	0	0
$211$	−1.24295	−0.0855685	−0.0427842	−	0.999084i	$-0.513623\pi$
$211$	−1.24295	−0.0855685	−0.0427842	+	0.999084i	$0.513623\pi$
$212$	1.31092	0.0900342
$213$	0	0
$214$	−3.00605	−0.205489
$215$	−4.02571	−0.274551
$216$	0	0
$217$	23.6936	1.60843
$218$	−12.5743	−0.851642
$219$	0	0
$220$	−9.38344	−0.632632
$221$	−14.3735	−0.966867
$222$	0	0
$223$	17.4338	1.16745	0.583727	−	0.811950i	$-0.301594\pi$
$223$	17.4338	1.16745	0.583727	+	0.811950i	$0.301594\pi$
$224$	−17.6969	−1.18243
$225$	0	0
$226$	2.76299	0.183792
$227$	1.91090	0.126831	0.0634154	−	0.997987i	$-0.479801\pi$
$227$	1.91090	0.126831	0.0634154	+	0.997987i	$0.479801\pi$
$228$	0	0
$229$	−12.7477	−0.842389	−0.421195	−	0.906970i	$-0.638389\pi$
$229$	−12.7477	−0.842389	−0.421195	+	0.906970i	$0.638389\pi$
$230$	−5.71859	−0.377073
$231$	0	0
$232$	0.0496223	0.00325787
$233$	9.04128	0.592314	0.296157	−	0.955139i	$-0.404295\pi$
$233$	9.04128	0.592314	0.296157	+	0.955139i	$0.404295\pi$
$234$	0	0
$235$	18.8773	1.23142
$236$	1.69494	0.110331
$237$	0	0
$238$	21.0106	1.36192
$239$	1.00000	0.0646846
$239$	1.00000	0.0646846
$240$	0	0
$241$	21.3266	1.37377	0.686884	−	0.726767i	$-0.258978\pi$
$241$	21.3266	1.37377	0.686884	+	0.726767i	$0.258978\pi$
$242$	−6.98650	−0.449110
$243$	0	0
$244$	4.62281	0.295945
$245$	1.40273	0.0896174
$246$	0	0
$247$	−14.4799	−0.921333
$248$	−8.10135	−0.514437
$249$	0	0
$250$	−22.3285	−1.41218
$251$	29.4696	1.86010	0.930052	−	0.367427i	$-0.119761\pi$
$251$	29.4696	1.86010	0.930052	+	0.367427i	$0.119761\pi$
$252$	0	0
$253$	−7.37913	−0.463922
$254$	28.4014	1.78206
$255$	0	0
$256$	20.6222	1.28889
$257$	6.15087	0.383681	0.191840	−	0.981426i	$-0.438554\pi$
$257$	6.15087	0.383681	0.191840	+	0.981426i	$0.438554\pi$
$258$	0	0
$259$	−1.95084	−0.121219
$260$	7.79769	0.483592
$261$	0	0
$262$	−19.0917	−1.17949
$263$	−10.2302	−0.630823	−0.315412	−	0.948955i	$-0.602143\pi$
$263$	−10.2302	−0.630823	−0.315412	+	0.948955i	$0.602143\pi$
$264$	0	0
$265$	−1.33250	−0.0818547
$266$	21.1661	1.29778
$267$	0	0
$268$	2.99535	0.182970
$269$	27.2980	1.66439	0.832193	−	0.554486i	$-0.187085\pi$
$269$	27.2980	1.66439	0.832193	+	0.554486i	$0.187085\pi$
$270$	0	0
$271$	21.5829	1.31107	0.655534	−	0.755166i	$-0.272444\pi$
$271$	21.5829	1.31107	0.655534	+	0.755166i	$0.272444\pi$
$272$	−21.1791	−1.28417
$273$	0	0
$274$	−2.81282	−0.169929
$275$	−9.63712	−0.581140
$276$	0	0
$277$	−16.8035	−1.00962	−0.504812	−	0.863230i	$-0.668438\pi$
$277$	−16.8035	−1.00962	−0.504812	+	0.863230i	$0.668438\pi$
$278$	−42.3366	−2.53918
$279$	0	0
$280$	3.29495	0.196911
$281$	2.74082	0.163504	0.0817519	−	0.996653i	$-0.473948\pi$
$281$	2.74082	0.163504	0.0817519	+	0.996653i	$0.473948\pi$
$282$	0	0
$283$	19.1177	1.13643	0.568214	−	0.822881i	$-0.307635\pi$
$283$	19.1177	1.13643	0.568214	+	0.822881i	$0.307635\pi$
$284$	12.4485	0.738682
$285$	0	0
$286$	23.0325	1.36194
$287$	−5.13149	−0.302902
$288$	0	0
$289$	3.34168	0.196569
$290$	0.174486	0.0102462
$291$	0	0
$292$	−9.96641	−0.583240
$293$	26.6427	1.55648	0.778241	−	0.627966i	$-0.216112\pi$
$293$	26.6427	1.55648	0.778241	+	0.627966i	$0.216112\pi$
$294$	0	0
$295$	−1.72284	−0.100307
$296$	0.667034	0.0387706
$297$	0	0
$298$	−21.3485	−1.23669
$299$	6.13210	0.354628
$300$	0	0
$301$	−6.31014	−0.363710
$302$	−19.3188	−1.11167
$303$	0	0
$304$	−21.3358	−1.22369
$305$	−4.69891	−0.269059
$306$	0	0
$307$	15.0399	0.858373	0.429186	−	0.903216i	$-0.358800\pi$
$307$	15.0399	0.858373	0.429186	+	0.903216i	$0.358800\pi$
$308$	−14.7081	−0.838075
$309$	0	0
$310$	−28.4867	−1.61793
$311$	13.7719	0.780933	0.390467	−	0.920617i	$-0.372314\pi$
$311$	13.7719	0.780933	0.390467	+	0.920617i	$0.372314\pi$
$312$	0	0
$313$	−1.15686	−0.0653898	−0.0326949	−	0.999465i	$-0.510409\pi$
$313$	−1.15686	−0.0653898	−0.0326949	+	0.999465i	$0.510409\pi$
$314$	4.21189	0.237691
$315$	0	0
$316$	12.6993	0.714394
$317$	−19.8540	−1.11511	−0.557557	−	0.830139i	$-0.688261\pi$
$317$	−19.8540	−1.11511	−0.557557	+	0.830139i	$0.688261\pi$
$318$	0	0
$319$	0.225153	0.0126061
$320$	6.46578	0.361448
$321$	0	0
$322$	−8.96365	−0.499525
$323$	20.4922	1.14022
$324$	0	0
$325$	8.00849	0.444231
$326$	21.0951	1.16835
$327$	0	0
$328$	1.75457	0.0968797
$329$	29.5894	1.63132
$330$	0	0
$331$	14.5712	0.800904	0.400452	−	0.916318i	$-0.368853\pi$
$331$	14.5712	0.800904	0.400452	+	0.916318i	$0.368853\pi$
$332$	15.2496	0.836928
$333$	0	0
$334$	−17.9973	−0.984770
$335$	−3.04466	−0.166347
$336$	0	0
$337$	15.1805	0.826933	0.413466	−	0.910519i	$-0.364318\pi$
$337$	15.1805	0.826933	0.413466	+	0.910519i	$0.364318\pi$
$338$	5.35892	0.291487
$339$	0	0
$340$	−11.0354	−0.598481
$341$	−36.7585	−1.99059
$342$	0	0
$343$	19.5024	1.05303
$344$	2.15757	0.116328
$345$	0	0
$346$	13.4965	0.725574
$347$	−10.1274	−0.543667	−0.271834	−	0.962344i	$-0.587630\pi$
$347$	−10.1274	−0.543667	−0.271834	+	0.962344i	$0.587630\pi$
$348$	0	0
$349$	14.4379	0.772842	0.386421	−	0.922322i	$-0.373711\pi$
$349$	14.4379	0.772842	0.386421	+	0.922322i	$0.373711\pi$
$350$	−11.7065	−0.625738
$351$	0	0
$352$	27.4552	1.46337
$353$	24.6191	1.31034	0.655171	−	0.755481i	$-0.272597\pi$
$353$	24.6191	1.31034	0.655171	+	0.755481i	$0.272597\pi$
$354$	0	0
$355$	−12.6534	−0.671574
$356$	−13.7453	−0.728499
$357$	0	0
$358$	0.215942	0.0114129
$359$	−15.5002	−0.818070	−0.409035	−	0.912519i	$-0.634135\pi$
$359$	−15.5002	−0.818070	−0.409035	+	0.912519i	$0.634135\pi$
$360$	0	0
$361$	1.64388	0.0865198
$362$	4.23324	0.222494
$363$	0	0
$364$	12.2225	0.640635
$365$	10.1305	0.530253
$366$	0	0
$367$	−11.1151	−0.580205	−0.290103	−	0.956996i	$-0.593689\pi$
$367$	−11.1151	−0.580205	−0.290103	+	0.956996i	$0.593689\pi$
$368$	9.03552	0.471009
$369$	0	0
$370$	2.34548	0.121936
$371$	−2.08863	−0.108436
$372$	0	0
$373$	28.7318	1.48767	0.743837	−	0.668361i	$-0.233004\pi$
$373$	28.7318	1.48767	0.743837	+	0.668361i	$0.233004\pi$
$374$	−32.5961	−1.68550
$375$	0	0
$376$	−10.1172	−0.521757
$377$	−0.187103	−0.00963630
$378$	0	0
$379$	−23.0688	−1.18496	−0.592481	−	0.805584i	$-0.701852\pi$
$379$	−23.0688	−1.18496	−0.592481	+	0.805584i	$0.701852\pi$
$380$	−11.1171	−0.570296
$381$	0	0
$382$	−14.9007	−0.762388
$383$	−25.1173	−1.28343	−0.641717	−	0.766941i	$-0.721778\pi$
$383$	−25.1173	−1.28343	−0.641717	+	0.766941i	$0.721778\pi$
$384$	0	0
$385$	14.9503	0.761936
$386$	19.8483	1.01025
$387$	0	0
$388$	25.5962	1.29945
$389$	20.2979	1.02915	0.514573	−	0.857447i	$-0.327951\pi$
$389$	20.2979	1.02915	0.514573	+	0.857447i	$0.327951\pi$
$390$	0	0
$391$	−8.67826	−0.438879
$392$	−0.751791	−0.0379712
$393$	0	0
$394$	−31.2517	−1.57444
$395$	−12.9084	−0.649492
$396$	0	0
$397$	−5.09694	−0.255808	−0.127904	−	0.991787i	$-0.540825\pi$
$397$	−5.09694	−0.255808	−0.127904	+	0.991787i	$0.540825\pi$
$398$	−23.1311	−1.15946
$399$	0	0
$400$	11.8003	0.590017
$401$	22.6376	1.13047	0.565235	−	0.824930i	$-0.308786\pi$
$401$	22.6376	1.13047	0.565235	+	0.824930i	$0.308786\pi$
$402$	0	0
$403$	30.5465	1.52163
$404$	12.4124	0.617539
$405$	0	0
$406$	0.273500	0.0135736
$407$	3.02656	0.150021
$408$	0	0
$409$	−12.2093	−0.603712	−0.301856	−	0.953353i	$-0.597606\pi$
$409$	−12.2093	−0.603712	−0.301856	+	0.953353i	$0.597606\pi$
$410$	6.16955	0.304693
$411$	0	0
$412$	−7.23950	−0.356665
$413$	−2.70047	−0.132882
$414$	0	0
$415$	−15.5006	−0.760894
$416$	−22.8154	−1.11862
$417$	0	0
$418$	−32.8373	−1.60613
$419$	−36.9777	−1.80648	−0.903239	−	0.429137i	$-0.858818\pi$
$419$	−36.9777	−1.80648	−0.903239	+	0.429137i	$0.858818\pi$
$420$	0	0
$421$	−23.2814	−1.13466	−0.567332	−	0.823489i	$-0.692024\pi$
$421$	−23.2814	−1.13466	−0.567332	+	0.823489i	$0.692024\pi$
$422$	2.34240	0.114026
$423$	0	0
$424$	0.714149	0.0346821
$425$	−11.3338	−0.549769
$426$	0	0
$427$	−7.36534	−0.356434
$428$	2.47481	0.119625
$429$	0	0
$430$	7.58663	0.365860
$431$	−34.2064	−1.64766	−0.823832	−	0.566834i	$-0.808168\pi$
$431$	−34.2064	−1.64766	−0.823832	+	0.566834i	$0.808168\pi$
$432$	0	0
$433$	0.581765	0.0279578	0.0139789	−	0.999902i	$-0.495550\pi$
$433$	0.581765	0.0279578	0.0139789	+	0.999902i	$0.495550\pi$
$434$	−44.6516	−2.14335
$435$	0	0
$436$	10.3522	0.495780
$437$	−8.74249	−0.418210
$438$	0	0
$439$	−15.5628	−0.742771	−0.371386	−	0.928479i	$-0.621117\pi$
$439$	−15.5628	−0.742771	−0.371386	+	0.928479i	$0.621117\pi$
$440$	−5.11182	−0.243696
$441$	0	0
$442$	27.0875	1.28842
$443$	−37.1224	−1.76374	−0.881870	−	0.471493i	$-0.843715\pi$
$443$	−37.1224	−1.76374	−0.881870	+	0.471493i	$0.843715\pi$
$444$	0	0
$445$	13.9716	0.662316
$446$	−32.8548	−1.55572
$447$	0	0
$448$	10.1348	0.478826
$449$	−16.6269	−0.784670	−0.392335	−	0.919822i	$-0.628333\pi$
$449$	−16.6269	−0.784670	−0.392335	+	0.919822i	$0.628333\pi$
$450$	0	0
$451$	7.96104	0.374871
$452$	−2.27471	−0.106993
$453$	0	0
$454$	−3.60117	−0.169011
$455$	−12.4237	−0.582434
$456$	0	0
$457$	10.3822	0.485659	0.242829	−	0.970069i	$-0.421924\pi$
$457$	10.3822	0.485659	0.242829	+	0.970069i	$0.421924\pi$
$458$	24.0235	1.12255
$459$	0	0
$460$	4.70799	0.219511
$461$	13.7020	0.638164	0.319082	−	0.947727i	$-0.396625\pi$
$461$	13.7020	0.638164	0.319082	+	0.947727i	$0.396625\pi$
$462$	0	0
$463$	−41.4574	−1.92669	−0.963345	−	0.268265i	$-0.913550\pi$
$463$	−41.4574	−1.92669	−0.963345	+	0.268265i	$0.913550\pi$
$464$	−0.275693	−0.0127987
$465$	0	0
$466$	−17.0387	−0.789302
$467$	−19.8888	−0.920345	−0.460173	−	0.887829i	$-0.652212\pi$
$467$	−19.8888	−0.920345	−0.460173	+	0.887829i	$0.652212\pi$
$468$	0	0
$469$	−4.77237	−0.220367
$470$	−35.5751	−1.64096
$471$	0	0
$472$	0.923350	0.0425006
$473$	9.78961	0.450127
$474$	0	0
$475$	−11.4176	−0.523878
$476$	−17.2976	−0.792833
$477$	0	0
$478$	−1.88454	−0.0861970
$479$	−13.2603	−0.605878	−0.302939	−	0.953010i	$-0.597968\pi$
$479$	−13.2603	−0.605878	−0.302939	+	0.953010i	$0.597968\pi$
$480$	0	0
$481$	−2.51508	−0.114678
$482$	−40.1909	−1.83065
$483$	0	0
$484$	5.75184	0.261447
$485$	−26.0175	−1.18139
$486$	0	0
$487$	35.1123	1.59109	0.795545	−	0.605895i	$-0.207185\pi$
$487$	35.1123	1.59109	0.795545	+	0.605895i	$0.207185\pi$
$488$	2.51837	0.114001
$489$	0	0
$490$	−2.64351	−0.119422
$491$	6.60916	0.298267	0.149134	−	0.988817i	$-0.452352\pi$
$491$	6.60916	0.298267	0.149134	+	0.988817i	$0.452352\pi$
$492$	0	0
$493$	0.264792	0.0119256
$494$	27.2880	1.22774
$495$	0	0
$496$	45.0097	2.02099
$497$	−19.8337	−0.889663
$498$	0	0
$499$	−22.6811	−1.01534	−0.507672	−	0.861550i	$-0.669494\pi$
$499$	−22.6811	−1.01534	−0.507672	+	0.861550i	$0.669494\pi$
$500$	18.3826	0.822093
$501$	0	0
$502$	−55.5367	−2.47873
$503$	−30.0329	−1.33910	−0.669549	−	0.742768i	$-0.733513\pi$
$503$	−30.0329	−1.33910	−0.669549	+	0.742768i	$0.733513\pi$
$504$	0	0
$505$	−12.6167	−0.561436
$506$	13.9063	0.618210
$507$	0	0
$508$	−23.3823	−1.03742
$509$	39.2106	1.73798	0.868990	−	0.494829i	$-0.164769\pi$
$509$	39.2106	1.73798	0.868990	+	0.494829i	$0.164769\pi$
$510$	0	0
$511$	15.8791	0.702450
$512$	−25.6800	−1.13491
$513$	0	0
$514$	−11.5916	−0.511283
$515$	7.35867	0.324262
$516$	0	0
$517$	−45.9053	−2.01891
$518$	3.67645	0.161534
$519$	0	0
$520$	4.24794	0.186285
$521$	34.0212	1.49049	0.745247	−	0.666789i	$-0.232332\pi$
$521$	34.0212	1.49049	0.745247	+	0.666789i	$0.232332\pi$
$522$	0	0
$523$	−26.3066	−1.15031	−0.575153	−	0.818046i	$-0.695058\pi$
$523$	−26.3066	−1.15031	−0.575153	+	0.818046i	$0.695058\pi$
$524$	15.7178	0.686635
$525$	0	0
$526$	19.2793	0.840618
$527$	−43.2300	−1.88313
$528$	0	0
$529$	−19.2976	−0.839028
$530$	2.51115	0.109077
$531$	0	0
$532$	−17.4256	−0.755495
$533$	−6.61567	−0.286556
$534$	0	0
$535$	−2.51555	−0.108757
$536$	1.63177	0.0704819
$537$	0	0
$538$	−51.4442	−2.21792
$539$	−3.41112	−0.146928
$540$	0	0
$541$	16.1226	0.693163	0.346581	−	0.938020i	$-0.387342\pi$
$541$	16.1226	0.693163	0.346581	+	0.938020i	$0.387342\pi$
$542$	−40.6739	−1.74709
$543$	0	0
$544$	32.2888	1.38437
$545$	−10.5226	−0.450739
$546$	0	0
$547$	−0.430130	−0.0183910	−0.00919552	−	0.999958i	$-0.502927\pi$
$547$	−0.430130	−0.0183910	−0.00919552	+	0.999958i	$0.502927\pi$
$548$	2.31573	0.0989232
$549$	0	0
$550$	18.1616	0.774412
$551$	0.266752	0.0113640
$552$	0	0
$553$	−20.2334	−0.860410
$554$	31.6669	1.34540
$555$	0	0
$556$	34.8548	1.47817
$557$	13.6791	0.579602	0.289801	−	0.957087i	$-0.406411\pi$
$557$	13.6791	0.579602	0.289801	+	0.957087i	$0.406411\pi$
$558$	0	0
$559$	−8.13521	−0.344083
$560$	−18.3061	−0.773576
$561$	0	0
$562$	−5.16520	−0.217881
$563$	13.3807	0.563931	0.281965	−	0.959425i	$-0.409014\pi$
$563$	13.3807	0.563931	0.281965	+	0.959425i	$0.409014\pi$
$564$	0	0
$565$	2.31216	0.0972732
$566$	−36.0281	−1.51437
$567$	0	0
$568$	6.78156	0.284548
$569$	−5.69249	−0.238641	−0.119321	−	0.992856i	$-0.538072\pi$
$569$	−5.69249	−0.238641	−0.119321	+	0.992856i	$0.538072\pi$
$570$	0	0
$571$	14.9082	0.623889	0.311944	−	0.950100i	$-0.399020\pi$
$571$	14.9082	0.623889	0.311944	+	0.950100i	$0.399020\pi$
$572$	−18.9622	−0.792848
$573$	0	0
$574$	9.67051	0.403639
$575$	4.83527	0.201645
$576$	0	0
$577$	−26.0799	−1.08572	−0.542860	−	0.839823i	$-0.682659\pi$
$577$	−26.0799	−1.08572	−0.542860	+	0.839823i	$0.682659\pi$
$578$	−6.29754	−0.261943
$579$	0	0
$580$	−0.143651	−0.00596477
$581$	−24.2965	−1.00799
$582$	0	0
$583$	3.24033	0.134201
$584$	−5.42940	−0.224670
$585$	0	0
$586$	−50.2093	−2.07413
$587$	20.0301	0.826729	0.413365	−	0.910566i	$-0.364354\pi$
$587$	20.0301	0.826729	0.413365	+	0.910566i	$0.364354\pi$
$588$	0	0
$589$	−43.5499	−1.79444
$590$	3.24676	0.133667
$591$	0	0
$592$	−3.70592	−0.152312
$593$	28.0351	1.15126	0.575632	−	0.817709i	$-0.304756\pi$
$593$	28.0351	1.15126	0.575632	+	0.817709i	$0.304756\pi$
$594$	0	0
$595$	17.5823	0.720805
$596$	17.5758	0.719932
$597$	0	0
$598$	−11.5562	−0.472568
$599$	31.5499	1.28909	0.644547	−	0.764564i	$-0.277046\pi$
$599$	31.5499	1.28909	0.644547	+	0.764564i	$0.277046\pi$
$600$	0	0
$601$	2.70265	0.110243	0.0551217	−	0.998480i	$-0.482445\pi$
$601$	2.70265	0.110243	0.0551217	+	0.998480i	$0.482445\pi$
$602$	11.8917	0.484670
$603$	0	0
$604$	15.9048	0.647156
$605$	−5.84652	−0.237695
$606$	0	0
$607$	−39.4724	−1.60213	−0.801067	−	0.598575i	$-0.795734\pi$
$607$	−39.4724	−1.60213	−0.801067	+	0.598575i	$0.795734\pi$
$608$	32.5277	1.31917
$609$	0	0
$610$	8.85529	0.358540
$611$	38.1475	1.54328
$612$	0	0
$613$	17.3060	0.698982	0.349491	−	0.936940i	$-0.386354\pi$
$613$	17.3060	0.698982	0.349491	+	0.936940i	$0.386354\pi$
$614$	−28.3434	−1.14384
$615$	0	0
$616$	−8.01256	−0.322835
$617$	42.7513	1.72110	0.860551	−	0.509365i	$-0.170120\pi$
$617$	42.7513	1.72110	0.860551	+	0.509365i	$0.170120\pi$
$618$	0	0
$619$	2.29106	0.0920853	0.0460427	−	0.998939i	$-0.485339\pi$
$619$	2.29106	0.0920853	0.0460427	+	0.998939i	$0.485339\pi$
$620$	23.4525	0.941873
$621$	0	0
$622$	−25.9538	−1.04065
$623$	21.8998	0.877398
$624$	0	0
$625$	−6.12049	−0.244820
$626$	2.18016	0.0871367
$627$	0	0
$628$	−3.46756	−0.138371
$629$	3.55939	0.141922
$630$	0	0
$631$	40.3021	1.60440	0.802201	−	0.597054i	$-0.203662\pi$
$631$	40.3021	1.60440	0.802201	+	0.597054i	$0.203662\pi$
$632$	6.91822	0.275192
$633$	0	0
$634$	37.4158	1.48597
$635$	23.7672	0.943172
$636$	0	0
$637$	2.83466	0.112313
$638$	−0.424310	−0.0167986
$639$	0	0
$640$	10.3954	0.410913
$641$	12.3428	0.487511	0.243756	−	0.969837i	$-0.421621\pi$
$641$	12.3428	0.487511	0.243756	+	0.969837i	$0.421621\pi$
$642$	0	0
$643$	6.36474	0.251001	0.125501	−	0.992094i	$-0.459946\pi$
$643$	6.36474	0.251001	0.125501	+	0.992094i	$0.459946\pi$
$644$	7.37958	0.290796
$645$	0	0
$646$	−38.6185	−1.51942
$647$	−13.9746	−0.549399	−0.274700	−	0.961530i	$-0.588578\pi$
$647$	−13.9746	−0.549399	−0.274700	+	0.961530i	$0.588578\pi$
$648$	0	0
$649$	4.18954	0.164454
$650$	−15.0923	−0.591970
$651$	0	0
$652$	−17.3671	−0.680149
$653$	−0.742205	−0.0290447	−0.0145224	−	0.999895i	$-0.504623\pi$
$653$	−0.742205	−0.0290447	−0.0145224	+	0.999895i	$0.504623\pi$
$654$	0	0
$655$	−15.9765	−0.624255
$656$	−9.74805	−0.380597
$657$	0	0
$658$	−55.7625	−2.17385
$659$	−41.9051	−1.63239	−0.816196	−	0.577776i	$-0.803921\pi$
$659$	−41.9051	−1.63239	−0.816196	+	0.577776i	$0.803921\pi$
$660$	0	0
$661$	−30.6618	−1.19261	−0.596304	−	0.802759i	$-0.703365\pi$
$661$	−30.6618	−1.19261	−0.596304	+	0.802759i	$0.703365\pi$
$662$	−27.4600	−1.06726
$663$	0	0
$664$	8.30750	0.322393
$665$	17.7125	0.686859
$666$	0	0
$667$	−0.112967	−0.00437409
$668$	14.8168	0.573279
$669$	0	0
$670$	5.73778	0.221670
$671$	11.4267	0.441121
$672$	0	0
$673$	41.2232	1.58904	0.794519	−	0.607240i	$-0.207723\pi$
$673$	41.2232	1.58904	0.794519	+	0.607240i	$0.207723\pi$
$674$	−28.6082	−1.10195
$675$	0	0
$676$	−4.41188	−0.169688
$677$	−42.5360	−1.63479	−0.817395	−	0.576078i	$-0.804583\pi$
$677$	−42.5360	−1.63479	−0.817395	+	0.576078i	$0.804583\pi$
$678$	0	0
$679$	−40.7814	−1.56504
$680$	−6.01178	−0.230541
$681$	0	0
$682$	69.2730	2.65260
$683$	32.3571	1.23811	0.619055	−	0.785348i	$-0.287516\pi$
$683$	32.3571	1.23811	0.619055	+	0.785348i	$0.287516\pi$
$684$	0	0
$685$	−2.35385	−0.0899361
$686$	−36.7530	−1.40324
$687$	0	0
$688$	−11.9871	−0.457003
$689$	−2.69273	−0.102585
$690$	0	0
$691$	29.4308	1.11960	0.559801	−	0.828627i	$-0.310878\pi$
$691$	29.4308	1.11960	0.559801	+	0.828627i	$0.310878\pi$
$692$	−11.1113	−0.422390
$693$	0	0
$694$	19.0855	0.724476
$695$	−35.4286	−1.34388
$696$	0	0
$697$	9.36262	0.354635
$698$	−27.2088	−1.02987
$699$	0	0
$700$	9.63769	0.364271
$701$	27.9008	1.05380	0.526899	−	0.849928i	$-0.323354\pi$
$701$	27.9008	1.05380	0.526899	+	0.849928i	$0.323354\pi$
$702$	0	0
$703$	3.58574	0.135239
$704$	−15.7233	−0.592594
$705$	0	0
$706$	−46.3957	−1.74613
$707$	−19.7761	−0.743758
$708$	0	0
$709$	−13.6158	−0.511351	−0.255676	−	0.966763i	$-0.582298\pi$
$709$	−13.6158	−0.511351	−0.255676	+	0.966763i	$0.582298\pi$
$710$	23.8459	0.894921
$711$	0	0
$712$	−7.48802	−0.280625
$713$	18.4430	0.690695
$714$	0	0
$715$	19.2743	0.720819
$716$	−0.177780	−0.00664395
$717$	0	0
$718$	29.2108	1.09014
$719$	42.0015	1.56639	0.783195	−	0.621776i	$-0.213589\pi$
$719$	42.0015	1.56639	0.783195	+	0.621776i	$0.213589\pi$
$720$	0	0
$721$	11.5344	0.429564
$722$	−3.09796	−0.115294
$723$	0	0
$724$	−3.48513	−0.129524
$725$	−0.147534	−0.00547928
$726$	0	0
$727$	−45.9245	−1.70324	−0.851622	−	0.524156i	$-0.824381\pi$
$727$	−45.9245	−1.70324	−0.851622	+	0.524156i	$0.824381\pi$
$728$	6.65847	0.246779
$729$	0	0
$730$	−19.0913	−0.706601
$731$	11.5131	0.425828
$732$	0	0
$733$	2.25611	0.0833315	0.0416657	−	0.999132i	$-0.486734\pi$
$733$	2.25611	0.0833315	0.0416657	+	0.999132i	$0.486734\pi$
$734$	20.9469	0.773166
$735$	0	0
$736$	−13.7752	−0.507760
$737$	7.40390	0.272726
$738$	0	0
$739$	−11.6417	−0.428248	−0.214124	−	0.976806i	$-0.568690\pi$
$739$	−11.6417	−0.428248	−0.214124	+	0.976806i	$0.568690\pi$
$740$	−1.93098	−0.0709844
$741$	0	0
$742$	3.93612	0.144500
$743$	−9.19350	−0.337277	−0.168638	−	0.985678i	$-0.553937\pi$
$743$	−9.19350	−0.337277	−0.168638	+	0.985678i	$0.553937\pi$
$744$	0	0
$745$	−17.8651	−0.654527
$746$	−54.1463	−1.98244
$747$	0	0
$748$	26.8356	0.981208
$749$	−3.94302	−0.144075
$750$	0	0
$751$	53.6899	1.95917	0.979586	−	0.201027i	$-0.0644280\pi$
$751$	53.6899	1.95917	0.979586	+	0.201027i	$0.0644280\pi$
$752$	56.2096	2.04975
$753$	0	0
$754$	0.352604	0.0128411
$755$	−16.1666	−0.588363
$756$	0	0
$757$	−16.7983	−0.610546	−0.305273	−	0.952265i	$-0.598748\pi$
$757$	−16.7983	−0.610546	−0.305273	+	0.952265i	$0.598748\pi$
$758$	43.4741	1.57905
$759$	0	0
$760$	−6.05627	−0.219684
$761$	9.97766	0.361690	0.180845	−	0.983512i	$-0.442117\pi$
$761$	9.97766	0.361690	0.180845	+	0.983512i	$0.442117\pi$
$762$	0	0
$763$	−16.4937	−0.597113
$764$	12.2675	0.443821
$765$	0	0
$766$	47.3347	1.71027
$767$	−3.48153	−0.125711
$768$	0	0
$769$	−49.8609	−1.79803	−0.899016	−	0.437917i	$-0.855716\pi$
$769$	−49.8609	−1.79803	−0.899016	+	0.437917i	$0.855716\pi$
$770$	−28.1744	−1.01534
$771$	0	0
$772$	−16.3406	−0.588112
$773$	2.95716	0.106362	0.0531809	−	0.998585i	$-0.483064\pi$
$773$	2.95716	0.106362	0.0531809	+	0.998585i	$0.483064\pi$
$774$	0	0
$775$	24.0865	0.865211
$776$	13.9440	0.500561
$777$	0	0
$778$	−38.2523	−1.37141
$779$	9.43191	0.337933
$780$	0	0
$781$	30.7702	1.10104
$782$	16.3546	0.584838
$783$	0	0
$784$	4.17681	0.149172
$785$	3.52464	0.125800
$786$	0	0
$787$	−25.9607	−0.925399	−0.462699	−	0.886515i	$-0.653119\pi$
$787$	−25.9607	−0.925399	−0.462699	+	0.886515i	$0.653119\pi$
$788$	25.7288	0.916552
$789$	0	0
$790$	24.3264	0.865496
$791$	3.62421	0.128862
$792$	0	0
$793$	−9.49560	−0.337199
$794$	9.60541	0.340883
$795$	0	0
$796$	19.0433	0.674973
$797$	5.84720	0.207119	0.103559	−	0.994623i	$-0.466977\pi$
$797$	5.84720	0.207119	0.103559	+	0.994623i	$0.466977\pi$
$798$	0	0
$799$	−53.9871	−1.90993
$800$	−17.9903	−0.636055
$801$	0	0
$802$	−42.6616	−1.50643
$803$	−24.6350	−0.869349
$804$	0	0
$805$	−7.50106	−0.264378
$806$	−57.5662	−2.02768
$807$	0	0
$808$	6.76189	0.237882
$809$	1.10730	0.0389306	0.0194653	−	0.999811i	$-0.493804\pi$
$809$	1.10730	0.0389306	0.0194653	+	0.999811i	$0.493804\pi$
$810$	0	0
$811$	−30.9703	−1.08751	−0.543757	−	0.839243i	$-0.682999\pi$
$811$	−30.9703	−1.08751	−0.543757	+	0.839243i	$0.682999\pi$
$812$	−0.225166	−0.00790179
$813$	0	0
$814$	−5.70367	−0.199914
$815$	17.6530	0.618358
$816$	0	0
$817$	11.5983	0.405774
$818$	23.0090	0.804491
$819$	0	0
$820$	−5.07926	−0.177375
$821$	−44.0188	−1.53627	−0.768133	−	0.640290i	$-0.778814\pi$
$821$	−44.0188	−1.53627	−0.768133	+	0.640290i	$0.778814\pi$
$822$	0	0
$823$	3.54554	0.123590	0.0617949	−	0.998089i	$-0.480318\pi$
$823$	3.54554	0.123590	0.0617949	+	0.998089i	$0.480318\pi$
$824$	−3.94386	−0.137391
$825$	0	0
$826$	5.08916	0.177075
$827$	32.6215	1.13436	0.567180	−	0.823594i	$-0.308034\pi$
$827$	32.6215	1.13436	0.567180	+	0.823594i	$0.308034\pi$
$828$	0	0
$829$	43.5811	1.51363	0.756817	−	0.653627i	$-0.226754\pi$
$829$	43.5811	1.51363	0.756817	+	0.653627i	$0.226754\pi$
$830$	29.2115	1.01395
$831$	0	0
$832$	13.0661	0.452987
$833$	−4.01167	−0.138996
$834$	0	0
$835$	−15.0607	−0.521197
$836$	27.0342	0.934999
$837$	0	0
$838$	69.6861	2.40726
$839$	27.9176	0.963822	0.481911	−	0.876220i	$-0.339943\pi$
$839$	27.9176	0.963822	0.481911	+	0.876220i	$0.339943\pi$
$840$	0	0
$841$	−28.9966	−0.999881
$842$	43.8747	1.51202
$843$	0	0
$844$	−1.92845	−0.0663799
$845$	4.48451	0.154272
$846$	0	0
$847$	−9.16417	−0.314885
$848$	−3.96768	−0.136251
$849$	0	0
$850$	21.3590	0.732607
$851$	−1.51853	−0.0520544
$852$	0	0
$853$	15.6342	0.535303	0.267652	−	0.963516i	$-0.413752\pi$
$853$	15.6342	0.535303	0.267652	+	0.963516i	$0.413752\pi$
$854$	13.8803	0.474974
$855$	0	0
$856$	1.34820	0.0460806
$857$	19.1584	0.654438	0.327219	−	0.944949i	$-0.393889\pi$
$857$	19.1584	0.654438	0.327219	+	0.944949i	$0.393889\pi$
$858$	0	0
$859$	4.26808	0.145625	0.0728125	−	0.997346i	$-0.476803\pi$
$859$	4.26808	0.145625	0.0728125	+	0.997346i	$0.476803\pi$
$860$	−6.24591	−0.212984
$861$	0	0
$862$	64.4635	2.19563
$863$	−23.1367	−0.787582	−0.393791	−	0.919200i	$-0.628837\pi$
$863$	−23.1367	−0.787582	−0.393791	+	0.919200i	$0.628837\pi$
$864$	0	0
$865$	11.2943	0.384016
$866$	−1.09636	−0.0372559
$867$	0	0
$868$	36.7607	1.24774
$869$	31.3902	1.06484
$870$	0	0
$871$	−6.15268	−0.208475
$872$	5.63956	0.190980
$873$	0	0
$874$	16.4756	0.557295
$875$	−29.2882	−0.990122
$876$	0	0
$877$	0.378032	0.0127652	0.00638262	−	0.999980i	$-0.497968\pi$
$877$	0.378032	0.0127652	0.00638262	+	0.999980i	$0.497968\pi$
$878$	29.3287	0.989797
$879$	0	0
$880$	28.4003	0.957375
$881$	3.64768	0.122893	0.0614467	−	0.998110i	$-0.480429\pi$
$881$	3.64768	0.122893	0.0614467	+	0.998110i	$0.480429\pi$
$882$	0	0
$883$	−34.3198	−1.15495	−0.577477	−	0.816407i	$-0.695963\pi$
$883$	−34.3198	−1.15495	−0.577477	+	0.816407i	$0.695963\pi$
$884$	−22.3005	−0.750049
$885$	0	0
$886$	69.9588	2.35031
$887$	48.0103	1.61203	0.806014	−	0.591896i	$-0.201620\pi$
$887$	48.0103	1.61203	0.806014	+	0.591896i	$0.201620\pi$
$888$	0	0
$889$	37.2541	1.24946
$890$	−26.3300	−0.882584
$891$	0	0
$892$	27.0486	0.905654
$893$	−54.3866	−1.81998
$894$	0	0
$895$	0.180707	0.00604036
$896$	16.2943	0.544354
$897$	0	0
$898$	31.3340	1.04563
$899$	−0.562734	−0.0187682
$900$	0	0
$901$	3.81080	0.126956
$902$	−15.0029	−0.499543
$903$	0	0
$904$	−1.23919	−0.0412150
$905$	3.54250	0.117757
$906$	0	0
$907$	−18.2463	−0.605857	−0.302929	−	0.953013i	$-0.597964\pi$
$907$	−18.2463	−0.605857	−0.302929	+	0.953013i	$0.597964\pi$
$908$	2.96476	0.0983891
$909$	0	0
$910$	23.4131	0.776136
$911$	−3.39432	−0.112459	−0.0562294	−	0.998418i	$-0.517908\pi$
$911$	−3.39432	−0.112459	−0.0562294	+	0.998418i	$0.517908\pi$
$912$	0	0
$913$	37.6939	1.24748
$914$	−19.5657	−0.647176
$915$	0	0
$916$	−19.7780	−0.653485
$917$	−25.0425	−0.826977
$918$	0	0
$919$	−58.2249	−1.92066	−0.960332	−	0.278860i	$-0.910044\pi$
$919$	−58.2249	−1.92066	−0.960332	+	0.278860i	$0.910044\pi$
$920$	2.56477	0.0845580
$921$	0	0
$922$	−25.8220	−0.850401
$923$	−25.5702	−0.841653
$924$	0	0
$925$	−1.98319	−0.0652068
$926$	78.1283	2.56746
$927$	0	0
$928$	0.420310	0.0137974
$929$	42.0740	1.38040	0.690202	−	0.723617i	$-0.257522\pi$
$929$	42.0740	1.38040	0.690202	+	0.723617i	$0.257522\pi$
$930$	0	0
$931$	−4.04136	−0.132450
$932$	14.0276	0.459489
$933$	0	0
$934$	37.4814	1.22643
$935$	−27.2774	−0.892067
$936$	0	0
$937$	23.9489	0.782375	0.391188	−	0.920311i	$-0.372064\pi$
$937$	23.9489	0.782375	0.391188	+	0.920311i	$0.372064\pi$
$938$	8.99373	0.293656
$939$	0	0
$940$	29.2882	0.955276
$941$	−3.92678	−0.128009	−0.0640047	−	0.997950i	$-0.520387\pi$
$941$	−3.92678	−0.128009	−0.0640047	+	0.997950i	$0.520387\pi$
$942$	0	0
$943$	−3.99433	−0.130073
$944$	−5.12996	−0.166966
$945$	0	0
$946$	−18.4489	−0.599827
$947$	−8.59682	−0.279359	−0.139680	−	0.990197i	$-0.544607\pi$
$947$	−8.59682	−0.279359	−0.139680	+	0.990197i	$0.544607\pi$
$948$	0	0
$949$	20.4718	0.664542
$950$	21.5171	0.698105
$951$	0	0
$952$	−9.42320	−0.305408
$953$	10.1361	0.328342	0.164171	−	0.986432i	$-0.447505\pi$
$953$	10.1361	0.328342	0.164171	+	0.986432i	$0.447505\pi$
$954$	0	0
$955$	−12.4694	−0.403500
$956$	1.55150	0.0501792
$957$	0	0
$958$	24.9896	0.807377
$959$	−3.68957	−0.119142
$960$	0	0
$961$	60.8722	1.96362
$962$	4.73978	0.152817
$963$	0	0
$964$	33.0883	1.06570
$965$	16.6096	0.534683
$966$	0	0
$967$	37.1978	1.19620	0.598100	−	0.801421i	$-0.295922\pi$
$967$	37.1978	1.19620	0.598100	+	0.801421i	$0.295922\pi$
$968$	3.13343	0.100712
$969$	0	0
$970$	49.0311	1.57429
$971$	−24.0240	−0.770967	−0.385483	−	0.922715i	$-0.625965\pi$
$971$	−24.0240	−0.770967	−0.385483	+	0.922715i	$0.625965\pi$
$972$	0	0
$973$	−55.5328	−1.78030
$974$	−66.1706	−2.12024
$975$	0	0
$976$	−13.9916	−0.447860
$977$	−54.0203	−1.72826	−0.864132	−	0.503265i	$-0.832132\pi$
$977$	−54.0203	−1.72826	−0.864132	+	0.503265i	$0.832132\pi$
$978$	0	0
$979$	−33.9756	−1.08587
$980$	2.17635	0.0695208
$981$	0	0
$982$	−12.4552	−0.397463
$983$	−27.5324	−0.878146	−0.439073	−	0.898451i	$-0.644693\pi$
$983$	−27.5324	−0.878146	−0.439073	+	0.898451i	$0.644693\pi$
$984$	0	0
$985$	−26.1524	−0.833284
$986$	−0.499012	−0.0158918
$987$	0	0
$988$	−22.4656	−0.714726
$989$	−4.91178	−0.156185
$990$	0	0
$991$	−0.915480	−0.0290812	−0.0145406	−	0.999894i	$-0.504629\pi$
$991$	−0.915480	−0.0290812	−0.0145406	+	0.999894i	$0.504629\pi$
$992$	−68.6199	−2.17868
$993$	0	0
$994$	37.3774	1.18554
$995$	−19.3568	−0.613652
$996$	0	0
$997$	59.5160	1.88489	0.942446	−	0.334359i	$-0.108520\pi$
$997$	59.5160	1.88489	0.942446	+	0.334359i	$0.108520\pi$
$998$	42.7435	1.35302
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.2.a.k.1.4	yes	20
3.2	odd	2		2151.2.a.j.1.17	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.2.a.j.1.17	✓	20	3.2	odd	2
2151.2.a.k.1.4	yes	20	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-1.88454 q^{2} +1.55150 q^{4} -1.57704 q^{5} -2.47195 q^{7} +0.845212 q^{8} +O(q^{10})\)	\(q-1.88454 q^{2} +1.55150 q^{4} -1.57704 q^{5} -2.47195 q^{7} +0.845212 q^{8} +2.97201 q^{10} +3.83501 q^{11} -3.18691 q^{13} +4.65849 q^{14} -4.69584 q^{16} +4.51018 q^{17} +4.54355 q^{19} -2.44679 q^{20} -7.22723 q^{22} -1.92415 q^{23} -2.51293 q^{25} +6.00587 q^{26} -3.83524 q^{28} +0.0587099 q^{29} -9.58500 q^{31} +7.15910 q^{32} -8.49962 q^{34} +3.89837 q^{35} +0.789192 q^{37} -8.56252 q^{38} -1.33294 q^{40} +2.07589 q^{41} +2.55270 q^{43} +5.95002 q^{44} +3.62615 q^{46} -11.9701 q^{47} -0.889470 q^{49} +4.73573 q^{50} -4.94450 q^{52} +0.844934 q^{53} -6.04797 q^{55} -2.08932 q^{56} -0.110641 q^{58} +1.09245 q^{59} +2.97957 q^{61} +18.0633 q^{62} -4.09994 q^{64} +5.02589 q^{65} +1.93061 q^{67} +6.99755 q^{68} -7.34665 q^{70} +8.02351 q^{71} -6.42371 q^{73} -1.48727 q^{74} +7.04934 q^{76} -9.47994 q^{77} +8.18519 q^{79} +7.40555 q^{80} -3.91210 q^{82} +9.82889 q^{83} -7.11274 q^{85} -4.81067 q^{86} +3.24139 q^{88} -8.85934 q^{89} +7.87787 q^{91} -2.98533 q^{92} +22.5581 q^{94} -7.16538 q^{95} +16.4977 q^{97} +1.67625 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 4 q^{2} + 18 q^{4} + 16 q^{5} - 4 q^{7} + 12 q^{8}+O(q^{10}) \) 20 * q + 4 * q^2 + 18 * q^4 + 16 * q^5 - 4 * q^7 + 12 * q^8	\( 20 q + 4 q^{2} + 18 q^{4} + 16 q^{5} - 4 q^{7} + 12 q^{8} + 4 q^{10} + 12 q^{11} - 4 q^{13} + 20 q^{14} + 22 q^{16} + 24 q^{17} - 4 q^{19} + 40 q^{20} - 6 q^{22} + 12 q^{23} + 22 q^{25} + 30 q^{26} - 12 q^{28} + 24 q^{29} - 4 q^{31} + 28 q^{32} + 8 q^{34} + 20 q^{35} - 10 q^{37} + 26 q^{38} + 6 q^{40} + 66 q^{41} + 8 q^{43} + 36 q^{44} - 12 q^{46} + 28 q^{47} + 18 q^{49} + 28 q^{50} - 18 q^{52} + 28 q^{53} - 4 q^{55} + 60 q^{56} + 54 q^{59} - 4 q^{61} + 20 q^{62} + 22 q^{64} + 42 q^{65} + 12 q^{67} + 12 q^{68} + 20 q^{70} + 36 q^{71} + 14 q^{73} - 50 q^{76} + 8 q^{77} - 12 q^{79} + 88 q^{80} - 8 q^{82} + 20 q^{83} + 4 q^{85} + 18 q^{86} - 10 q^{88} + 130 q^{89} - 6 q^{91} - 46 q^{92} - 26 q^{94} - 2 q^{97} + 12 q^{98}+O(q^{100}) \) 20 * q + 4 * q^2 + 18 * q^4 + 16 * q^5 - 4 * q^7 + 12 * q^8 + 4 * q^10 + 12 * q^11 - 4 * q^13 + 20 * q^14 + 22 * q^16 + 24 * q^17 - 4 * q^19 + 40 * q^20 - 6 * q^22 + 12 * q^23 + 22 * q^25 + 30 * q^26 - 12 * q^28 + 24 * q^29 - 4 * q^31 + 28 * q^32 + 8 * q^34 + 20 * q^35 - 10 * q^37 + 26 * q^38 + 6 * q^40 + 66 * q^41 + 8 * q^43 + 36 * q^44 - 12 * q^46 + 28 * q^47 + 18 * q^49 + 28 * q^50 - 18 * q^52 + 28 * q^53 - 4 * q^55 + 60 * q^56 + 54 * q^59 - 4 * q^61 + 20 * q^62 + 22 * q^64 + 42 * q^65 + 12 * q^67 + 12 * q^68 + 20 * q^70 + 36 * q^71 + 14 * q^73 - 50 * q^76 + 8 * q^77 - 12 * q^79 + 88 * q^80 - 8 * q^82 + 20 * q^83 + 4 * q^85 + 18 * q^86 - 10 * q^88 + 130 * q^89 - 6 * q^91 - 46 * q^92 - 26 * q^94 - 2 * q^97 + 12 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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