LMFDB - Embedded newform 2151.2.a.g.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:	$8$
Coefficient field:	8.8.2585660609.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 4x^{6} + 15x^{5} + x^{4} - 19x^{3} + 6x^{2} + 3x - 1 $ x^8 - 3x^7 - 4x^6 + 15x^5 + x^4 - 19x^3 + 6x^2 + 3x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 717)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.417244$ of defining polynomial
Character	$\chi$	$=$	2151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.711649 q^{2} -1.49356 q^{4} -0.309852 q^{5} -4.19834 q^{7} -2.48618 q^{8} +O(q^{10})$	\(q+0.711649 q^{2} -1.49356 q^{4} -0.309852 q^{5} -4.19834 q^{7} -2.48618 q^{8} -0.220506 q^{10} -1.06352 q^{11} +3.09563 q^{13} -2.98775 q^{14} +1.21782 q^{16} -1.63596 q^{17} -5.86500 q^{19} +0.462781 q^{20} -0.756853 q^{22} +6.91896 q^{23} -4.90399 q^{25} +2.20300 q^{26} +6.27046 q^{28} +7.66121 q^{29} -5.79991 q^{31} +5.83903 q^{32} -1.16423 q^{34} +1.30086 q^{35} +4.34450 q^{37} -4.17382 q^{38} +0.770348 q^{40} +11.0321 q^{41} -1.61761 q^{43} +1.58843 q^{44} +4.92387 q^{46} +4.79638 q^{47} +10.6261 q^{49} -3.48992 q^{50} -4.62350 q^{52} -8.64828 q^{53} +0.329534 q^{55} +10.4379 q^{56} +5.45209 q^{58} -4.12511 q^{59} +11.3840 q^{61} -4.12750 q^{62} +1.71970 q^{64} -0.959187 q^{65} -11.9102 q^{67} +2.44341 q^{68} +0.925758 q^{70} +7.05266 q^{71} +1.39390 q^{73} +3.09176 q^{74} +8.75970 q^{76} +4.46502 q^{77} +12.7555 q^{79} -0.377344 q^{80} +7.85095 q^{82} +7.68673 q^{83} +0.506906 q^{85} -1.15117 q^{86} +2.64411 q^{88} +9.42458 q^{89} -12.9965 q^{91} -10.3339 q^{92} +3.41334 q^{94} +1.81728 q^{95} -2.84754 q^{97} +7.56204 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 5 q^{2} + 7 q^{4} + 13 q^{5} - 7 q^{7} + 15 q^{8}+O(q^{10}) $ 8 * q + 5 * q^2 + 7 * q^4 + 13 * q^5 - 7 * q^7 + 15 * q^8	$ 8 q + 5 q^{2} + 7 q^{4} + 13 q^{5} - 7 q^{7} + 15 q^{8} + 4 q^{10} + 19 q^{11} - 3 q^{13} + 8 q^{14} + 9 q^{16} + 13 q^{17} - 6 q^{19} + 18 q^{20} + 3 q^{22} + 18 q^{23} + 7 q^{25} - 2 q^{28} + 10 q^{29} - 2 q^{31} + 20 q^{32} - 4 q^{34} + 7 q^{35} - 8 q^{37} - 9 q^{38} + 29 q^{40} + 22 q^{41} - 24 q^{43} + 7 q^{44} + 30 q^{46} + 17 q^{47} + 15 q^{49} - 24 q^{50} + 22 q^{52} + 32 q^{55} - 19 q^{56} + 18 q^{58} + 24 q^{59} + 10 q^{61} - 30 q^{62} + 33 q^{64} + 17 q^{65} - 48 q^{67} + 21 q^{68} + 31 q^{70} + 17 q^{71} + 2 q^{73} - 9 q^{74} + 10 q^{76} - 10 q^{77} + 17 q^{79} - 8 q^{80} - 17 q^{82} + 37 q^{83} + 28 q^{85} + q^{86} + 15 q^{88} + 41 q^{89} - 39 q^{91} + 38 q^{92} + 2 q^{94} - 16 q^{95} - 20 q^{97} - 46 q^{98}+O(q^{100}) $ 8 * q + 5 * q^2 + 7 * q^4 + 13 * q^5 - 7 * q^7 + 15 * q^8 + 4 * q^10 + 19 * q^11 - 3 * q^13 + 8 * q^14 + 9 * q^16 + 13 * q^17 - 6 * q^19 + 18 * q^20 + 3 * q^22 + 18 * q^23 + 7 * q^25 - 2 * q^28 + 10 * q^29 - 2 * q^31 + 20 * q^32 - 4 * q^34 + 7 * q^35 - 8 * q^37 - 9 * q^38 + 29 * q^40 + 22 * q^41 - 24 * q^43 + 7 * q^44 + 30 * q^46 + 17 * q^47 + 15 * q^49 - 24 * q^50 + 22 * q^52 + 32 * q^55 - 19 * q^56 + 18 * q^58 + 24 * q^59 + 10 * q^61 - 30 * q^62 + 33 * q^64 + 17 * q^65 - 48 * q^67 + 21 * q^68 + 31 * q^70 + 17 * q^71 + 2 * q^73 - 9 * q^74 + 10 * q^76 - 10 * q^77 + 17 * q^79 - 8 * q^80 - 17 * q^82 + 37 * q^83 + 28 * q^85 + q^86 + 15 * q^88 + 41 * q^89 - 39 * q^91 + 38 * q^92 + 2 * q^94 - 16 * q^95 - 20 * q^97 - 46 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.711649	0.503212	0.251606	−	0.967830i	$-0.419041\pi$
$2$	0.711649	0.503212	0.251606	+	0.967830i	$0.419041\pi$
$3$	0	0
$3$	0	0
$4$	−1.49356	−0.746778
$5$	−0.309852	−0.138570	−0.0692849	−	0.997597i	$-0.522072\pi$
$5$	−0.309852	−0.138570	−0.0692849	+	0.997597i	$0.522072\pi$
$6$	0	0
$7$	−4.19834	−1.58682	−0.793412	−	0.608685i	$-0.791697\pi$
$7$	−4.19834	−1.58682	−0.793412	+	0.608685i	$0.791697\pi$
$8$	−2.48618	−0.878999
$9$	0	0
$10$	−0.220506	−0.0697300
$11$	−1.06352	−0.320664	−0.160332	−	0.987063i	$-0.551256\pi$
$11$	−1.06352	−0.320664	−0.160332	+	0.987063i	$0.551256\pi$
$12$	0	0
$13$	3.09563	0.858574	0.429287	−	0.903168i	$-0.358765\pi$
$13$	3.09563	0.858574	0.429287	+	0.903168i	$0.358765\pi$
$14$	−2.98775	−0.798509
$15$	0	0
$16$	1.21782	0.304455
$17$	−1.63596	−0.396780	−0.198390	−	0.980123i	$-0.563571\pi$
$17$	−1.63596	−0.396780	−0.198390	+	0.980123i	$0.563571\pi$
$18$	0	0
$19$	−5.86500	−1.34552	−0.672761	−	0.739859i	$-0.734892\pi$
$19$	−5.86500	−1.34552	−0.672761	+	0.739859i	$0.734892\pi$
$20$	0.462781	0.103481
$21$	0	0
$22$	−0.756853	−0.161362
$23$	6.91896	1.44270	0.721351	−	0.692569i	$-0.243521\pi$
$23$	6.91896	1.44270	0.721351	+	0.692569i	$0.243521\pi$
$24$	0	0
$25$	−4.90399	−0.980798
$26$	2.20300	0.432045
$27$	0	0
$28$	6.27046	1.18501
$29$	7.66121	1.42265	0.711326	−	0.702862i	$-0.248095\pi$
$29$	7.66121	1.42265	0.711326	+	0.702862i	$0.248095\pi$
$30$	0	0
$31$	−5.79991	−1.04169	−0.520847	−	0.853650i	$-0.674384\pi$
$31$	−5.79991	−1.04169	−0.520847	+	0.853650i	$0.674384\pi$
$32$	5.83903	1.03220
$33$	0	0
$34$	−1.16423	−0.199664
$35$	1.30086	0.219886
$36$	0	0
$37$	4.34450	0.714232	0.357116	−	0.934060i	$-0.383760\pi$
$37$	4.34450	0.714232	0.357116	+	0.934060i	$0.383760\pi$
$38$	−4.17382	−0.677083
$39$	0	0
$40$	0.770348	0.121803
$41$	11.0321	1.72292	0.861459	−	0.507827i	$-0.169551\pi$
$41$	11.0321	1.72292	0.861459	+	0.507827i	$0.169551\pi$
$42$	0	0
$43$	−1.61761	−0.246684	−0.123342	−	0.992364i	$-0.539361\pi$
$43$	−1.61761	−0.246684	−0.123342	+	0.992364i	$0.539361\pi$
$44$	1.58843	0.239465
$45$	0	0
$46$	4.92387	0.725985
$47$	4.79638	0.699624	0.349812	−	0.936820i	$-0.386246\pi$
$47$	4.79638	0.699624	0.349812	+	0.936820i	$0.386246\pi$
$48$	0	0
$49$	10.6261	1.51801
$50$	−3.48992	−0.493549
$51$	0	0
$52$	−4.62350	−0.641164
$53$	−8.64828	−1.18793	−0.593967	−	0.804490i	$-0.702439\pi$
$53$	−8.64828	−1.18793	−0.593967	+	0.804490i	$0.702439\pi$
$54$	0	0
$55$	0.329534	0.0444343
$56$	10.4379	1.39482
$57$	0	0
$58$	5.45209	0.715895
$59$	−4.12511	−0.537044	−0.268522	−	0.963274i	$-0.586535\pi$
$59$	−4.12511	−0.537044	−0.268522	+	0.963274i	$0.586535\pi$
$60$	0	0
$61$	11.3840	1.45757	0.728783	−	0.684745i	$-0.240086\pi$
$61$	11.3840	1.45757	0.728783	+	0.684745i	$0.240086\pi$
$62$	−4.12750	−0.524193
$63$	0	0
$64$	1.71970	0.214962
$65$	−0.959187	−0.118972
$66$	0	0
$67$	−11.9102	−1.45506	−0.727529	−	0.686077i	$-0.759331\pi$
$67$	−11.9102	−1.45506	−0.727529	+	0.686077i	$0.759331\pi$
$68$	2.44341	0.296306
$69$	0	0
$70$	0.925758	0.110649
$71$	7.05266	0.836997	0.418499	−	0.908217i	$-0.362556\pi$
$71$	7.05266	0.836997	0.418499	+	0.908217i	$0.362556\pi$
$72$	0	0
$73$	1.39390	0.163144	0.0815720	−	0.996667i	$-0.474006\pi$
$73$	1.39390	0.163144	0.0815720	+	0.996667i	$0.474006\pi$
$74$	3.09176	0.359410
$75$	0	0
$76$	8.75970	1.00481
$77$	4.46502	0.508837
$78$	0	0
$79$	12.7555	1.43510	0.717551	−	0.696506i	$-0.245263\pi$
$79$	12.7555	1.43510	0.717551	+	0.696506i	$0.245263\pi$
$80$	−0.377344	−0.0421883
$81$	0	0
$82$	7.85095	0.866993
$83$	7.68673	0.843728	0.421864	−	0.906659i	$-0.361376\pi$
$83$	7.68673	0.843728	0.421864	+	0.906659i	$0.361376\pi$
$84$	0	0
$85$	0.506906	0.0549817
$86$	−1.15117	−0.124134
$87$	0	0
$88$	2.64411	0.281863
$89$	9.42458	0.999004	0.499502	−	0.866313i	$-0.333516\pi$
$89$	9.42458	0.999004	0.499502	+	0.866313i	$0.333516\pi$
$90$	0	0
$91$	−12.9965	−1.36241
$92$	−10.3339	−1.07738
$93$	0	0
$94$	3.41334	0.352059
$95$	1.81728	0.186449
$96$	0	0
$97$	−2.84754	−0.289124	−0.144562	−	0.989496i	$-0.546177\pi$
$97$	−2.84754	−0.289124	−0.144562	+	0.989496i	$0.546177\pi$
$98$	7.56204	0.763881
$99$	0	0
$100$	7.32439	0.732439
$101$	−8.99104	−0.894642	−0.447321	−	0.894373i	$-0.647622\pi$
$101$	−8.99104	−0.894642	−0.447321	+	0.894373i	$0.647622\pi$
$102$	0	0
$103$	17.4683	1.72120	0.860601	−	0.509279i	$-0.170088\pi$
$103$	17.4683	1.72120	0.860601	+	0.509279i	$0.170088\pi$
$104$	−7.69632	−0.754686
$105$	0	0
$106$	−6.15454	−0.597782
$107$	−3.92688	−0.379626	−0.189813	−	0.981820i	$-0.560788\pi$
$107$	−3.92688	−0.379626	−0.189813	+	0.981820i	$0.560788\pi$
$108$	0	0
$109$	18.9932	1.81922	0.909608	−	0.415467i	$-0.136382\pi$
$109$	18.9932	1.81922	0.909608	+	0.415467i	$0.136382\pi$
$110$	0.234512	0.0223599
$111$	0	0
$112$	−5.11283	−0.483117
$113$	10.7573	1.01196	0.505980	−	0.862545i	$-0.331131\pi$
$113$	10.7573	1.01196	0.505980	+	0.862545i	$0.331131\pi$
$114$	0	0
$115$	−2.14385	−0.199915
$116$	−11.4425	−1.06241
$117$	0	0
$118$	−2.93563	−0.270247
$119$	6.86834	0.629620
$120$	0	0
$121$	−9.86892	−0.897175
$122$	8.10138	0.733464
$123$	0	0
$124$	8.66249	0.777915
$125$	3.06877	0.274479
$126$	0	0
$127$	−8.71405	−0.773247	−0.386624	−	0.922238i	$-0.626359\pi$
$127$	−8.71405	−0.773247	−0.386624	+	0.922238i	$0.626359\pi$
$128$	−10.4542	−0.924033
$129$	0	0
$130$	−0.682604	−0.0598683
$131$	8.70686	0.760722	0.380361	−	0.924838i	$-0.375800\pi$
$131$	8.70686	0.760722	0.380361	+	0.924838i	$0.375800\pi$
$132$	0	0
$133$	24.6233	2.13511
$134$	−8.47585	−0.732202
$135$	0	0
$136$	4.06731	0.348769
$137$	16.3421	1.39620	0.698101	−	0.715999i	$-0.254029\pi$
$137$	16.3421	1.39620	0.698101	+	0.715999i	$0.254029\pi$
$138$	0	0
$139$	−10.8378	−0.919247	−0.459624	−	0.888114i	$-0.652016\pi$
$139$	−10.8378	−0.919247	−0.459624	+	0.888114i	$0.652016\pi$
$140$	−1.94291	−0.164206
$141$	0	0
$142$	5.01902	0.421187
$143$	−3.29227	−0.275313
$144$	0	0
$145$	−2.37384	−0.197137
$146$	0.991970	0.0820960
$147$	0	0
$148$	−6.48876	−0.533372
$149$	−20.8899	−1.71137	−0.855684	−	0.517498i	$-0.826863\pi$
$149$	−20.8899	−1.71137	−0.855684	+	0.517498i	$0.826863\pi$
$150$	0	0
$151$	−12.1235	−0.986597	−0.493299	−	0.869860i	$-0.664209\pi$
$151$	−12.1235	−0.986597	−0.493299	+	0.869860i	$0.664209\pi$
$152$	14.5815	1.18271
$153$	0	0
$154$	3.17753	0.256053
$155$	1.79711	0.144347
$156$	0	0
$157$	−21.1431	−1.68740	−0.843700	−	0.536815i	$-0.819627\pi$
$157$	−21.1431	−1.68740	−0.843700	+	0.536815i	$0.819627\pi$
$158$	9.07741	0.722160
$159$	0	0
$160$	−1.80923	−0.143032
$161$	−29.0482	−2.28932
$162$	0	0
$163$	19.6345	1.53790	0.768948	−	0.639311i	$-0.220780\pi$
$163$	19.6345	1.53790	0.768948	+	0.639311i	$0.220780\pi$
$164$	−16.4770	−1.28664
$165$	0	0
$166$	5.47025	0.424574
$167$	−18.2792	−1.41449	−0.707243	−	0.706971i	$-0.750061\pi$
$167$	−18.2792	−1.41449	−0.707243	+	0.706971i	$0.750061\pi$
$168$	0	0
$169$	−3.41706	−0.262851
$170$	0.360739	0.0276674
$171$	0	0
$172$	2.41600	0.184218
$173$	2.25763	0.171645	0.0858223	−	0.996310i	$-0.472648\pi$
$173$	2.25763	0.171645	0.0858223	+	0.996310i	$0.472648\pi$
$174$	0	0
$175$	20.5886	1.55635
$176$	−1.29518	−0.0976278
$177$	0	0
$178$	6.70699	0.502710
$179$	−24.3298	−1.81849	−0.909246	−	0.416258i	$-0.863341\pi$
$179$	−24.3298	−1.81849	−0.909246	+	0.416258i	$0.863341\pi$
$180$	0	0
$181$	1.55124	0.115303	0.0576515	−	0.998337i	$-0.481639\pi$
$181$	1.55124	0.115303	0.0576515	+	0.998337i	$0.481639\pi$
$182$	−9.24896	−0.685579
$183$	0	0
$184$	−17.2018	−1.26813
$185$	−1.34615	−0.0989710
$186$	0	0
$187$	1.73988	0.127233
$188$	−7.16366	−0.522463
$189$	0	0
$190$	1.29326	0.0938233
$191$	1.96793	0.142394	0.0711972	−	0.997462i	$-0.477318\pi$
$191$	1.96793	0.142394	0.0711972	+	0.997462i	$0.477318\pi$
$192$	0	0
$193$	1.01740	0.0732342	0.0366171	−	0.999329i	$-0.488342\pi$
$193$	1.01740	0.0732342	0.0366171	+	0.999329i	$0.488342\pi$
$194$	−2.02645	−0.145491
$195$	0	0
$196$	−15.8706	−1.13362
$197$	15.1745	1.08114	0.540569	−	0.841299i	$-0.318209\pi$
$197$	15.1745	1.08114	0.540569	+	0.841299i	$0.318209\pi$
$198$	0	0
$199$	−3.06179	−0.217045	−0.108522	−	0.994094i	$-0.534612\pi$
$199$	−3.06179	−0.217045	−0.108522	+	0.994094i	$0.534612\pi$
$200$	12.1922	0.862121
$201$	0	0
$202$	−6.39846	−0.450194
$203$	−32.1644	−2.25750
$204$	0	0
$205$	−3.41830	−0.238745
$206$	12.4313	0.866129
$207$	0	0
$208$	3.76993	0.261398
$209$	6.23755	0.431460
$210$	0	0
$211$	10.3580	0.713074	0.356537	−	0.934281i	$-0.383957\pi$
$211$	10.3580	0.713074	0.356537	+	0.934281i	$0.383957\pi$
$212$	12.9167	0.887122
$213$	0	0
$214$	−2.79456	−0.191032
$215$	0.501220	0.0341829
$216$	0	0
$217$	24.3500	1.65299
$218$	13.5165	0.915451
$219$	0	0
$220$	−0.492177	−0.0331826
$221$	−5.06435	−0.340665
$222$	0	0
$223$	−0.771196	−0.0516431	−0.0258216	−	0.999667i	$-0.508220\pi$
$223$	−0.771196	−0.0516431	−0.0258216	+	0.999667i	$0.508220\pi$
$224$	−24.5143	−1.63793
$225$	0	0
$226$	7.65541	0.509230
$227$	21.0118	1.39460	0.697302	−	0.716777i	$-0.254384\pi$
$227$	21.0118	1.39460	0.697302	+	0.716777i	$0.254384\pi$
$228$	0	0
$229$	25.8503	1.70824	0.854119	−	0.520077i	$-0.174097\pi$
$229$	25.8503	1.70824	0.854119	+	0.520077i	$0.174097\pi$
$230$	−1.52567	−0.100600
$231$	0	0
$232$	−19.0472	−1.25051
$233$	−17.9134	−1.17354	−0.586771	−	0.809753i	$-0.699601\pi$
$233$	−17.9134	−1.17354	−0.586771	+	0.809753i	$0.699601\pi$
$234$	0	0
$235$	−1.48617	−0.0969467
$236$	6.16109	0.401053
$237$	0	0
$238$	4.88785	0.316832
$239$	−1.00000	−0.0646846
$239$	−1.00000	−0.0646846
$240$	0	0
$241$	9.53136	0.613969	0.306984	−	0.951715i	$-0.400680\pi$
$241$	9.53136	0.613969	0.306984	+	0.951715i	$0.400680\pi$
$242$	−7.02321	−0.451469
$243$	0	0
$244$	−17.0026	−1.08848
$245$	−3.29251	−0.210351
$246$	0	0
$247$	−18.1559	−1.15523
$248$	14.4197	0.915649
$249$	0	0
$250$	2.18389	0.138121
$251$	1.93621	0.122212	0.0611062	−	0.998131i	$-0.480537\pi$
$251$	1.93621	0.122212	0.0611062	+	0.998131i	$0.480537\pi$
$252$	0	0
$253$	−7.35846	−0.462622
$254$	−6.20135	−0.389107
$255$	0	0
$256$	−10.8791	−0.679946
$257$	0.0592320	0.00369479	0.00184740	−	0.999998i	$-0.499412\pi$
$257$	0.0592320	0.00369479	0.00184740	+	0.999998i	$0.499412\pi$
$258$	0	0
$259$	−18.2397	−1.13336
$260$	1.43260	0.0888460
$261$	0	0
$262$	6.19622	0.382804
$263$	24.0362	1.48213	0.741067	−	0.671431i	$-0.234320\pi$
$263$	24.0362	1.48213	0.741067	+	0.671431i	$0.234320\pi$
$264$	0	0
$265$	2.67968	0.164612
$266$	17.5231	1.07441
$267$	0	0
$268$	17.7885	1.08661
$269$	15.8995	0.969407	0.484703	−	0.874679i	$-0.338928\pi$
$269$	15.8995	0.969407	0.484703	+	0.874679i	$0.338928\pi$
$270$	0	0
$271$	−0.308408	−0.0187344	−0.00936722	−	0.999956i	$-0.502982\pi$
$271$	−0.308408	−0.0187344	−0.00936722	+	0.999956i	$0.502982\pi$
$272$	−1.99231	−0.120802
$273$	0	0
$274$	11.6299	0.702585
$275$	5.21550	0.314506
$276$	0	0
$277$	16.9466	1.01822	0.509112	−	0.860700i	$-0.329974\pi$
$277$	16.9466	1.01822	0.509112	+	0.860700i	$0.329974\pi$
$278$	−7.71268	−0.462576
$279$	0	0
$280$	−3.23419	−0.193280
$281$	16.5825	0.989231	0.494615	−	0.869112i	$-0.335309\pi$
$281$	16.5825	0.989231	0.494615	+	0.869112i	$0.335309\pi$
$282$	0	0
$283$	11.3402	0.674105	0.337052	−	0.941486i	$-0.390570\pi$
$283$	11.3402	0.674105	0.337052	+	0.941486i	$0.390570\pi$
$284$	−10.5335	−0.625051
$285$	0	0
$286$	−2.34294	−0.138541
$287$	−46.3164	−2.73397
$288$	0	0
$289$	−14.3236	−0.842566
$290$	−1.68934	−0.0992015
$291$	0	0
$292$	−2.08187	−0.121832
$293$	−1.88336	−0.110027	−0.0550134	−	0.998486i	$-0.517520\pi$
$293$	−1.88336	−0.110027	−0.0550134	+	0.998486i	$0.517520\pi$
$294$	0	0
$295$	1.27817	0.0744181
$296$	−10.8012	−0.627809
$297$	0	0
$298$	−14.8663	−0.861181
$299$	21.4186	1.23867
$300$	0	0
$301$	6.79130	0.391444
$302$	−8.62768	−0.496467
$303$	0	0
$304$	−7.14252	−0.409652
$305$	−3.52734	−0.201975
$306$	0	0
$307$	21.8785	1.24867	0.624335	−	0.781157i	$-0.285370\pi$
$307$	21.8785	1.24867	0.624335	+	0.781157i	$0.285370\pi$
$308$	−6.66876	−0.379988
$309$	0	0
$310$	1.27891	0.0726373
$311$	0.911901	0.0517092	0.0258546	−	0.999666i	$-0.491769\pi$
$311$	0.911901	0.0517092	0.0258546	+	0.999666i	$0.491769\pi$
$312$	0	0
$313$	−8.29642	−0.468941	−0.234471	−	0.972123i	$-0.575336\pi$
$313$	−8.29642	−0.468941	−0.234471	+	0.972123i	$0.575336\pi$
$314$	−15.0464	−0.849119
$315$	0	0
$316$	−19.0510	−1.07170
$317$	7.20688	0.404779	0.202389	−	0.979305i	$-0.435129\pi$
$317$	7.20688	0.404779	0.202389	+	0.979305i	$0.435129\pi$
$318$	0	0
$319$	−8.14786	−0.456193
$320$	−0.532851	−0.0297873
$321$	0	0
$322$	−20.6721	−1.15201
$323$	9.59493	0.533876
$324$	0	0
$325$	−15.1810	−0.842088
$326$	13.9729	0.773887
$327$	0	0
$328$	−27.4277	−1.51444
$329$	−20.1368	−1.11018
$330$	0	0
$331$	0.528456	0.0290466	0.0145233	−	0.999895i	$-0.495377\pi$
$331$	0.528456	0.0290466	0.0145233	+	0.999895i	$0.495377\pi$
$332$	−11.4806	−0.630078
$333$	0	0
$334$	−13.0084	−0.711786
$335$	3.69038	0.201627
$336$	0	0
$337$	−9.78750	−0.533159	−0.266579	−	0.963813i	$-0.585893\pi$
$337$	−9.78750	−0.533159	−0.266579	+	0.963813i	$0.585893\pi$
$338$	−2.43174	−0.132269
$339$	0	0
$340$	−0.757093	−0.0410591
$341$	6.16833	0.334034
$342$	0	0
$343$	−15.2235	−0.821994
$344$	4.02169	0.216835
$345$	0	0
$346$	1.60664	0.0863736
$347$	0.314769	0.0168977	0.00844885	−	0.999964i	$-0.497311\pi$
$347$	0.314769	0.0168977	0.00844885	+	0.999964i	$0.497311\pi$
$348$	0	0
$349$	31.7429	1.69916	0.849579	−	0.527461i	$-0.176856\pi$
$349$	31.7429	1.69916	0.849579	+	0.527461i	$0.176856\pi$
$350$	14.6519	0.783176
$351$	0	0
$352$	−6.20993	−0.330990
$353$	6.32810	0.336811	0.168405	−	0.985718i	$-0.446138\pi$
$353$	6.32810	0.336811	0.168405	+	0.985718i	$0.446138\pi$
$354$	0	0
$355$	−2.18528	−0.115983
$356$	−14.0761	−0.746034
$357$	0	0
$358$	−17.3143	−0.915087
$359$	13.0530	0.688910	0.344455	−	0.938803i	$-0.388064\pi$
$359$	13.0530	0.688910	0.344455	+	0.938803i	$0.388064\pi$
$360$	0	0
$361$	15.3982	0.810432
$362$	1.10394	0.0580219
$363$	0	0
$364$	19.4110	1.01742
$365$	−0.431903	−0.0226069
$366$	0	0
$367$	−26.7594	−1.39683	−0.698415	−	0.715693i	$-0.746111\pi$
$367$	−26.7594	−1.39683	−0.698415	+	0.715693i	$0.746111\pi$
$368$	8.42606	0.439239
$369$	0	0
$370$	−0.957987	−0.0498033
$371$	36.3085	1.88504
$372$	0	0
$373$	−18.1064	−0.937512	−0.468756	−	0.883328i	$-0.655298\pi$
$373$	−18.1064	−0.937512	−0.468756	+	0.883328i	$0.655298\pi$
$374$	1.23819	0.0640250
$375$	0	0
$376$	−11.9247	−0.614968
$377$	23.7163	1.22145
$378$	0	0
$379$	−19.6799	−1.01089	−0.505445	−	0.862859i	$-0.668672\pi$
$379$	−19.6799	−1.01089	−0.505445	+	0.862859i	$0.668672\pi$
$380$	−2.71421	−0.139236
$381$	0	0
$382$	1.40048	0.0716546
$383$	−1.56845	−0.0801441	−0.0400720	−	0.999197i	$-0.512759\pi$
$383$	−1.56845	−0.0801441	−0.0400720	+	0.999197i	$0.512759\pi$
$384$	0	0
$385$	−1.38350	−0.0705094
$386$	0.724033	0.0368523
$387$	0	0
$388$	4.25297	0.215912
$389$	8.80566	0.446465	0.223232	−	0.974765i	$-0.428339\pi$
$389$	8.80566	0.446465	0.223232	+	0.974765i	$0.428339\pi$
$390$	0	0
$391$	−11.3192	−0.572435
$392$	−26.4184	−1.33433
$393$	0	0
$394$	10.7989	0.544042
$395$	−3.95230	−0.198862
$396$	0	0
$397$	−6.24404	−0.313379	−0.156690	−	0.987648i	$-0.550082\pi$
$397$	−6.24404	−0.313379	−0.156690	+	0.987648i	$0.550082\pi$
$398$	−2.17892	−0.109219
$399$	0	0
$400$	−5.97219	−0.298609
$401$	2.89416	0.144527	0.0722637	−	0.997386i	$-0.476978\pi$
$401$	2.89416	0.144527	0.0722637	+	0.997386i	$0.476978\pi$
$402$	0	0
$403$	−17.9544	−0.894372
$404$	13.4286	0.668099
$405$	0	0
$406$	−22.8898	−1.13600
$407$	−4.62047	−0.229028
$408$	0	0
$409$	−11.8662	−0.586744	−0.293372	−	0.955998i	$-0.594777\pi$
$409$	−11.8662	−0.586744	−0.293372	+	0.955998i	$0.594777\pi$
$410$	−2.43263	−0.120139
$411$	0	0
$412$	−26.0899	−1.28536
$413$	17.3186	0.852195
$414$	0	0
$415$	−2.38175	−0.116915
$416$	18.0755	0.886224
$417$	0	0
$418$	4.43894	0.217116
$419$	−23.8835	−1.16678	−0.583392	−	0.812191i	$-0.698275\pi$
$419$	−23.8835	−1.16678	−0.583392	+	0.812191i	$0.698275\pi$
$420$	0	0
$421$	25.4207	1.23893	0.619464	−	0.785025i	$-0.287350\pi$
$421$	25.4207	1.23893	0.619464	+	0.785025i	$0.287350\pi$
$422$	7.37126	0.358827
$423$	0	0
$424$	21.5012	1.04419
$425$	8.02276	0.389161
$426$	0	0
$427$	−47.7938	−2.31290
$428$	5.86501	0.283496
$429$	0	0
$430$	0.356693	0.0172013
$431$	−33.8543	−1.63071	−0.815353	−	0.578965i	$-0.803457\pi$
$431$	−33.8543	−1.63071	−0.815353	+	0.578965i	$0.803457\pi$
$432$	0	0
$433$	9.14549	0.439504	0.219752	−	0.975556i	$-0.429475\pi$
$433$	9.14549	0.439504	0.219752	+	0.975556i	$0.429475\pi$
$434$	17.3287	0.831802
$435$	0	0
$436$	−28.3674	−1.35855
$437$	−40.5797	−1.94119
$438$	0	0
$439$	−2.56405	−0.122376	−0.0611878	−	0.998126i	$-0.519489\pi$
$439$	−2.56405	−0.122376	−0.0611878	+	0.998126i	$0.519489\pi$
$440$	−0.819282	−0.0390577
$441$	0	0
$442$	−3.60404	−0.171427
$443$	−8.44657	−0.401308	−0.200654	−	0.979662i	$-0.564307\pi$
$443$	−8.44657	−0.401308	−0.200654	+	0.979662i	$0.564307\pi$
$444$	0	0
$445$	−2.92022	−0.138432
$446$	−0.548821	−0.0259874
$447$	0	0
$448$	−7.21987	−0.341107
$449$	27.6378	1.30431	0.652153	−	0.758087i	$-0.273866\pi$
$449$	27.6378	1.30431	0.652153	+	0.758087i	$0.273866\pi$
$450$	0	0
$451$	−11.7328	−0.552477
$452$	−16.0666	−0.755709
$453$	0	0
$454$	14.9530	0.701781
$455$	4.02700	0.188788
$456$	0	0
$457$	−31.4026	−1.46895	−0.734476	−	0.678634i	$-0.762572\pi$
$457$	−31.4026	−1.46895	−0.734476	+	0.678634i	$0.762572\pi$
$458$	18.3964	0.859606
$459$	0	0
$460$	3.20196	0.149292
$461$	−4.62584	−0.215447	−0.107723	−	0.994181i	$-0.534356\pi$
$461$	−4.62584	−0.215447	−0.107723	+	0.994181i	$0.534356\pi$
$462$	0	0
$463$	−21.3014	−0.989962	−0.494981	−	0.868904i	$-0.664825\pi$
$463$	−21.3014	−0.989962	−0.494981	+	0.868904i	$0.664825\pi$
$464$	9.32999	0.433134
$465$	0	0
$466$	−12.7480	−0.590540
$467$	−18.5084	−0.856466	−0.428233	−	0.903668i	$-0.640864\pi$
$467$	−18.5084	−0.856466	−0.428233	+	0.903668i	$0.640864\pi$
$468$	0	0
$469$	50.0029	2.30892
$470$	−1.05763	−0.0487847
$471$	0	0
$472$	10.2558	0.472061
$473$	1.72037	0.0791025
$474$	0	0
$475$	28.7619	1.31969
$476$	−10.2583	−0.470186
$477$	0	0
$478$	−0.711649	−0.0325501
$479$	28.3367	1.29473	0.647367	−	0.762178i	$-0.275870\pi$
$479$	28.3367	1.29473	0.647367	+	0.762178i	$0.275870\pi$
$480$	0	0
$481$	13.4490	0.613221
$482$	6.78298	0.308956
$483$	0	0
$484$	14.7398	0.669990
$485$	0.882316	0.0400639
$486$	0	0
$487$	34.2775	1.55326	0.776631	−	0.629955i	$-0.216927\pi$
$487$	34.2775	1.55326	0.776631	+	0.629955i	$0.216927\pi$
$488$	−28.3026	−1.28120
$489$	0	0
$490$	−2.34311	−0.105851
$491$	−29.8401	−1.34666	−0.673332	−	0.739340i	$-0.735138\pi$
$491$	−29.8401	−1.34666	−0.673332	+	0.739340i	$0.735138\pi$
$492$	0	0
$493$	−12.5335	−0.564479
$494$	−12.9206	−0.581326
$495$	0	0
$496$	−7.06326	−0.317150
$497$	−29.6095	−1.32817
$498$	0	0
$499$	15.0147	0.672150	0.336075	−	0.941835i	$-0.390900\pi$
$499$	15.0147	0.672150	0.336075	+	0.941835i	$0.390900\pi$
$500$	−4.58338	−0.204975
$501$	0	0
$502$	1.37790	0.0614988
$503$	42.0242	1.87377	0.936883	−	0.349644i	$-0.113697\pi$
$503$	42.0242	1.87377	0.936883	+	0.349644i	$0.113697\pi$
$504$	0	0
$505$	2.78589	0.123970
$506$	−5.23664	−0.232797
$507$	0	0
$508$	13.0149	0.577444
$509$	3.07761	0.136413	0.0682064	−	0.997671i	$-0.478272\pi$
$509$	3.07761	0.136413	0.0682064	+	0.997671i	$0.478272\pi$
$510$	0	0
$511$	−5.85208	−0.258881
$512$	13.1664	0.581876
$513$	0	0
$514$	0.0421524	0.00185926
$515$	−5.41258	−0.238507
$516$	0	0
$517$	−5.10105	−0.224344
$518$	−12.9803	−0.570320
$519$	0	0
$520$	2.38472	0.104577
$521$	−24.5303	−1.07469	−0.537345	−	0.843363i	$-0.680573\pi$
$521$	−24.5303	−1.07469	−0.537345	+	0.843363i	$0.680573\pi$
$522$	0	0
$523$	22.3722	0.978267	0.489133	−	0.872209i	$-0.337313\pi$
$523$	22.3722	0.978267	0.489133	+	0.872209i	$0.337313\pi$
$524$	−13.0042	−0.568090
$525$	0	0
$526$	17.1053	0.745827
$527$	9.48845	0.413323
$528$	0	0
$529$	24.8720	1.08139
$530$	1.90699	0.0828345
$531$	0	0
$532$	−36.7762	−1.59445
$533$	34.1512	1.47925
$534$	0	0
$535$	1.21675	0.0526047
$536$	29.6109	1.27899
$537$	0	0
$538$	11.3148	0.487817
$539$	−11.3011	−0.486771
$540$	0	0
$541$	21.7122	0.933480	0.466740	−	0.884395i	$-0.345428\pi$
$541$	21.7122	0.933480	0.466740	+	0.884395i	$0.345428\pi$
$542$	−0.219478	−0.00942739
$543$	0	0
$544$	−9.55245	−0.409558
$545$	−5.88507	−0.252089
$546$	0	0
$547$	20.9706	0.896638	0.448319	−	0.893874i	$-0.352023\pi$
$547$	20.9706	0.896638	0.448319	+	0.893874i	$0.352023\pi$
$548$	−24.4079	−1.04265
$549$	0	0
$550$	3.71160	0.158263
$551$	−44.9330	−1.91421
$552$	0	0
$553$	−53.5518	−2.27725
$554$	12.0601	0.512383
$555$	0	0
$556$	16.1868	0.686474
$557$	−21.7092	−0.919848	−0.459924	−	0.887958i	$-0.652123\pi$
$557$	−21.7092	−0.919848	−0.459924	+	0.887958i	$0.652123\pi$
$558$	0	0
$559$	−5.00754	−0.211796
$560$	1.58422	0.0669455
$561$	0	0
$562$	11.8009	0.497792
$563$	37.4659	1.57900	0.789499	−	0.613752i	$-0.210340\pi$
$563$	37.4659	1.57900	0.789499	+	0.613752i	$0.210340\pi$
$564$	0	0
$565$	−3.33316	−0.140227
$566$	8.07024	0.339217
$567$	0	0
$568$	−17.5342	−0.735720
$569$	−20.9148	−0.876792	−0.438396	−	0.898782i	$-0.644453\pi$
$569$	−20.9148	−0.876792	−0.438396	+	0.898782i	$0.644453\pi$
$570$	0	0
$571$	30.8439	1.29078	0.645388	−	0.763855i	$-0.276696\pi$
$571$	30.8439	1.29078	0.645388	+	0.763855i	$0.276696\pi$
$572$	4.91719	0.205598
$573$	0	0
$574$	−32.9610	−1.37576
$575$	−33.9305	−1.41500
$576$	0	0
$577$	2.04326	0.0850622	0.0425311	−	0.999095i	$-0.486458\pi$
$577$	2.04326	0.0850622	0.0425311	+	0.999095i	$0.486458\pi$
$578$	−10.1934	−0.423989
$579$	0	0
$580$	3.54546	0.147217
$581$	−32.2715	−1.33885
$582$	0	0
$583$	9.19763	0.380927
$584$	−3.46550	−0.143403
$585$	0	0
$586$	−1.34029	−0.0553668
$587$	−7.07648	−0.292078	−0.146039	−	0.989279i	$-0.546652\pi$
$587$	−7.07648	−0.292078	−0.146039	+	0.989279i	$0.546652\pi$
$588$	0	0
$589$	34.0165	1.40162
$590$	0.909610	0.0374481
$591$	0	0
$592$	5.29083	0.217452
$593$	27.0220	1.10966	0.554831	−	0.831963i	$-0.312783\pi$
$593$	27.0220	1.10966	0.554831	+	0.831963i	$0.312783\pi$
$594$	0	0
$595$	−2.12817	−0.0872463
$596$	31.2003	1.27801
$597$	0	0
$598$	15.2425	0.623312
$599$	30.4062	1.24236	0.621182	−	0.783666i	$-0.286653\pi$
$599$	30.4062	1.24236	0.621182	+	0.783666i	$0.286653\pi$
$600$	0	0
$601$	−3.96818	−0.161865	−0.0809327	−	0.996720i	$-0.525790\pi$
$601$	−3.96818	−0.161865	−0.0809327	+	0.996720i	$0.525790\pi$
$602$	4.83302	0.196979
$603$	0	0
$604$	18.1071	0.736769
$605$	3.05790	0.124321
$606$	0	0
$607$	−43.1226	−1.75029	−0.875147	−	0.483858i	$-0.839235\pi$
$607$	−43.1226	−1.75029	−0.875147	+	0.483858i	$0.839235\pi$
$608$	−34.2459	−1.38885
$609$	0	0
$610$	−2.51023	−0.101636
$611$	14.8478	0.600679
$612$	0	0
$613$	−26.0558	−1.05238	−0.526192	−	0.850366i	$-0.676381\pi$
$613$	−26.0558	−1.05238	−0.526192	+	0.850366i	$0.676381\pi$
$614$	15.5698	0.628345
$615$	0	0
$616$	−11.1009	−0.447267
$617$	39.3030	1.58228	0.791140	−	0.611635i	$-0.209488\pi$
$617$	39.3030	1.58228	0.791140	+	0.611635i	$0.209488\pi$
$618$	0	0
$619$	1.28304	0.0515696	0.0257848	−	0.999668i	$-0.491792\pi$
$619$	1.28304	0.0515696	0.0257848	+	0.999668i	$0.491792\pi$
$620$	−2.68409	−0.107796
$621$	0	0
$622$	0.648953	0.0260207
$623$	−39.5676	−1.58524
$624$	0	0
$625$	23.5691	0.942764
$626$	−5.90414	−0.235977
$627$	0	0
$628$	31.5783	1.26011
$629$	−7.10745	−0.283393
$630$	0	0
$631$	12.5132	0.498142	0.249071	−	0.968485i	$-0.419875\pi$
$631$	12.5132	0.498142	0.249071	+	0.968485i	$0.419875\pi$
$632$	−31.7124	−1.26145
$633$	0	0
$634$	5.12877	0.203689
$635$	2.70006	0.107149
$636$	0	0
$637$	32.8944	1.30333
$638$	−5.79842	−0.229561
$639$	0	0
$640$	3.23926	0.128043
$641$	7.80075	0.308111	0.154056	−	0.988062i	$-0.450767\pi$
$641$	7.80075	0.308111	0.154056	+	0.988062i	$0.450767\pi$
$642$	0	0
$643$	−18.1134	−0.714323	−0.357162	−	0.934043i	$-0.616255\pi$
$643$	−18.1134	−0.714323	−0.357162	+	0.934043i	$0.616255\pi$
$644$	43.3851	1.70961
$645$	0	0
$646$	6.82822	0.268653
$647$	8.38465	0.329635	0.164817	−	0.986324i	$-0.447297\pi$
$647$	8.38465	0.329635	0.164817	+	0.986324i	$0.447297\pi$
$648$	0	0
$649$	4.38714	0.172210
$650$	−10.8035	−0.423749
$651$	0	0
$652$	−29.3253	−1.14847
$653$	−36.2780	−1.41967	−0.709834	−	0.704369i	$-0.751230\pi$
$653$	−36.2780	−1.41967	−0.709834	+	0.704369i	$0.751230\pi$
$654$	0	0
$655$	−2.69783	−0.105413
$656$	13.4351	0.524552
$657$	0	0
$658$	−14.3304	−0.558655
$659$	41.1303	1.60221	0.801104	−	0.598525i	$-0.204246\pi$
$659$	41.1303	1.60221	0.801104	+	0.598525i	$0.204246\pi$
$660$	0	0
$661$	−15.1622	−0.589740	−0.294870	−	0.955537i	$-0.595276\pi$
$661$	−15.1622	−0.589740	−0.294870	+	0.955537i	$0.595276\pi$
$662$	0.376075	0.0146166
$663$	0	0
$664$	−19.1106	−0.741636
$665$	−7.62956	−0.295862
$666$	0	0
$667$	53.0076	2.05246
$668$	27.3010	1.05631
$669$	0	0
$670$	2.62626	0.101461
$671$	−12.1071	−0.467388
$672$	0	0
$673$	12.6062	0.485932	0.242966	−	0.970035i	$-0.421880\pi$
$673$	12.6062	0.485932	0.242966	+	0.970035i	$0.421880\pi$
$674$	−6.96526	−0.268292
$675$	0	0
$676$	5.10357	0.196291
$677$	−5.74831	−0.220925	−0.110463	−	0.993880i	$-0.535233\pi$
$677$	−5.74831	−0.220925	−0.110463	+	0.993880i	$0.535233\pi$
$678$	0	0
$679$	11.9550	0.458790
$680$	−1.26026	−0.0483289
$681$	0	0
$682$	4.38968	0.168090
$683$	36.0725	1.38028	0.690139	−	0.723677i	$-0.257550\pi$
$683$	36.0725	1.38028	0.690139	+	0.723677i	$0.257550\pi$
$684$	0	0
$685$	−5.06364	−0.193472
$686$	−10.8338	−0.413637
$687$	0	0
$688$	−1.96996	−0.0751042
$689$	−26.7719	−1.01993
$690$	0	0
$691$	−21.3335	−0.811566	−0.405783	−	0.913969i	$-0.633001\pi$
$691$	−21.3335	−0.811566	−0.405783	+	0.913969i	$0.633001\pi$
$692$	−3.37190	−0.128180
$693$	0	0
$694$	0.224005	0.00850312
$695$	3.35810	0.127380
$696$	0	0
$697$	−18.0481	−0.683619
$698$	22.5898	0.855036
$699$	0	0
$700$	−30.7503	−1.16225
$701$	−20.2170	−0.763585	−0.381792	−	0.924248i	$-0.624693\pi$
$701$	−20.2170	−0.763585	−0.381792	+	0.924248i	$0.624693\pi$
$702$	0	0
$703$	−25.4805	−0.961015
$704$	−1.82893	−0.0689305
$705$	0	0
$706$	4.50339	0.169487
$707$	37.7475	1.41964
$708$	0	0
$709$	47.1596	1.77111	0.885557	−	0.464530i	$-0.153777\pi$
$709$	47.1596	1.77111	0.885557	+	0.464530i	$0.153777\pi$
$710$	−1.55515	−0.0583638
$711$	0	0
$712$	−23.4313	−0.878123
$713$	−40.1293	−1.50286
$714$	0	0
$715$	1.02012	0.0381501
$716$	36.3379	1.35801
$717$	0	0
$718$	9.28913	0.346667
$719$	37.6263	1.40322	0.701612	−	0.712560i	$-0.252464\pi$
$719$	37.6263	1.40322	0.701612	+	0.712560i	$0.252464\pi$
$720$	0	0
$721$	−73.3379	−2.73125
$722$	10.9581	0.407819
$723$	0	0
$724$	−2.31687	−0.0861058
$725$	−37.5705	−1.39533
$726$	0	0
$727$	5.18124	0.192161	0.0960807	−	0.995374i	$-0.469369\pi$
$727$	5.18124	0.192161	0.0960807	+	0.995374i	$0.469369\pi$
$728$	32.3118	1.19755
$729$	0	0
$730$	−0.307363	−0.0113760
$731$	2.64636	0.0978791
$732$	0	0
$733$	−12.8952	−0.476295	−0.238148	−	0.971229i	$-0.576540\pi$
$733$	−12.8952	−0.476295	−0.238148	+	0.971229i	$0.576540\pi$
$734$	−19.0433	−0.702901
$735$	0	0
$736$	40.4000	1.48916
$737$	12.6667	0.466584
$738$	0	0
$739$	−9.45746	−0.347898	−0.173949	−	0.984755i	$-0.555653\pi$
$739$	−9.45746	−0.347898	−0.173949	+	0.984755i	$0.555653\pi$
$740$	2.01055	0.0739093
$741$	0	0
$742$	25.8389	0.948575
$743$	−22.0948	−0.810581	−0.405290	−	0.914188i	$-0.632830\pi$
$743$	−22.0948	−0.810581	−0.405290	+	0.914188i	$0.632830\pi$
$744$	0	0
$745$	6.47277	0.237144
$746$	−12.8854	−0.471767
$747$	0	0
$748$	−2.59861	−0.0950147
$749$	16.4864	0.602399
$750$	0	0
$751$	16.6555	0.607769	0.303885	−	0.952709i	$-0.401716\pi$
$751$	16.6555	0.607769	0.303885	+	0.952709i	$0.401716\pi$
$752$	5.84113	0.213004
$753$	0	0
$754$	16.8777	0.614649
$755$	3.75649	0.136713
$756$	0	0
$757$	25.3571	0.921619	0.460809	−	0.887499i	$-0.347559\pi$
$757$	25.3571	0.921619	0.460809	+	0.887499i	$0.347559\pi$
$758$	−14.0052	−0.508691
$759$	0	0
$760$	−4.51809	−0.163888
$761$	−26.2708	−0.952317	−0.476158	−	0.879360i	$-0.657971\pi$
$761$	−26.2708	−0.952317	−0.476158	+	0.879360i	$0.657971\pi$
$762$	0	0
$763$	−79.7399	−2.88678
$764$	−2.93921	−0.106337
$765$	0	0
$766$	−1.11619	−0.0403294
$767$	−12.7698	−0.461092
$768$	0	0
$769$	51.8160	1.86853	0.934267	−	0.356574i	$-0.116055\pi$
$769$	51.8160	1.86853	0.934267	+	0.356574i	$0.116055\pi$
$770$	−0.984563	−0.0354812
$771$	0	0
$772$	−1.51955	−0.0546897
$773$	34.0422	1.22441	0.612206	−	0.790698i	$-0.290282\pi$
$773$	34.0422	1.22441	0.612206	+	0.790698i	$0.290282\pi$
$774$	0	0
$775$	28.4427	1.02169
$776$	7.07952	0.254140
$777$	0	0
$778$	6.26654	0.224666
$779$	−64.7030	−2.31823
$780$	0	0
$781$	−7.50066	−0.268395
$782$	−8.05528	−0.288056
$783$	0	0
$784$	12.9407	0.462167
$785$	6.55121	0.233823
$786$	0	0
$787$	−8.27449	−0.294954	−0.147477	−	0.989065i	$-0.547115\pi$
$787$	−8.27449	−0.294954	−0.147477	+	0.989065i	$0.547115\pi$
$788$	−22.6640	−0.807371
$789$	0	0
$790$	−2.81265	−0.100070
$791$	−45.1628	−1.60580
$792$	0	0
$793$	35.2406	1.25143
$794$	−4.44356	−0.157696
$795$	0	0
$796$	4.57296	0.162084
$797$	20.7719	0.735779	0.367890	−	0.929869i	$-0.380080\pi$
$797$	20.7719	0.735779	0.367890	+	0.929869i	$0.380080\pi$
$798$	0	0
$799$	−7.84670	−0.277596
$800$	−28.6346	−1.01238
$801$	0	0
$802$	2.05963	0.0727279
$803$	−1.48245	−0.0523144
$804$	0	0
$805$	9.00062	0.317230
$806$	−12.7772	−0.450058
$807$	0	0
$808$	22.3534	0.786390
$809$	−10.6449	−0.374255	−0.187127	−	0.982336i	$-0.559918\pi$
$809$	−10.6449	−0.374255	−0.187127	+	0.982336i	$0.559918\pi$
$810$	0	0
$811$	−46.7208	−1.64059	−0.820294	−	0.571942i	$-0.806190\pi$
$811$	−46.7208	−1.64059	−0.820294	+	0.571942i	$0.806190\pi$
$812$	48.0393	1.68585
$813$	0	0
$814$	−3.28815	−0.115250
$815$	−6.08379	−0.213106
$816$	0	0
$817$	9.48730	0.331919
$818$	−8.44454	−0.295256
$819$	0	0
$820$	5.10542	0.178289
$821$	14.2115	0.495985	0.247992	−	0.968762i	$-0.420229\pi$
$821$	14.2115	0.495985	0.247992	+	0.968762i	$0.420229\pi$
$822$	0	0
$823$	8.98657	0.313252	0.156626	−	0.987658i	$-0.449938\pi$
$823$	8.98657	0.313252	0.156626	+	0.987658i	$0.449938\pi$
$824$	−43.4294	−1.51294
$825$	0	0
$826$	12.3248	0.428834
$827$	48.3946	1.68285	0.841423	−	0.540378i	$-0.181719\pi$
$827$	48.3946	1.68285	0.841423	+	0.540378i	$0.181719\pi$
$828$	0	0
$829$	−19.5661	−0.679560	−0.339780	−	0.940505i	$-0.610353\pi$
$829$	−19.5661	−0.679560	−0.339780	+	0.940505i	$0.610353\pi$
$830$	−1.69497	−0.0588331
$831$	0	0
$832$	5.32355	0.184561
$833$	−17.3839	−0.602316
$834$	0	0
$835$	5.66383	0.196005
$836$	−9.31613	−0.322205
$837$	0	0
$838$	−16.9966	−0.587139
$839$	57.3202	1.97891	0.989456	−	0.144833i	$-0.0462646\pi$
$839$	57.3202	1.97891	0.989456	+	0.144833i	$0.0462646\pi$
$840$	0	0
$841$	29.6942	1.02394
$842$	18.0906	0.623443
$843$	0	0
$844$	−15.4702	−0.532508
$845$	1.05878	0.0364232
$846$	0	0
$847$	41.4331	1.42366
$848$	−10.5321	−0.361673
$849$	0	0
$850$	5.70939	0.195830
$851$	30.0594	1.03042
$852$	0	0
$853$	−25.5676	−0.875417	−0.437709	−	0.899117i	$-0.644210\pi$
$853$	−25.5676	−0.875417	−0.437709	+	0.899117i	$0.644210\pi$
$854$	−34.0124	−1.16388
$855$	0	0
$856$	9.76295	0.333691
$857$	−20.0681	−0.685514	−0.342757	−	0.939424i	$-0.611361\pi$
$857$	−20.0681	−0.685514	−0.342757	+	0.939424i	$0.611361\pi$
$858$	0	0
$859$	8.88891	0.303286	0.151643	−	0.988435i	$-0.451544\pi$
$859$	8.88891	0.303286	0.151643	+	0.988435i	$0.451544\pi$
$860$	−0.748600	−0.0255271
$861$	0	0
$862$	−24.0924	−0.820590
$863$	−10.8293	−0.368633	−0.184316	−	0.982867i	$-0.559007\pi$
$863$	−10.8293	−0.368633	−0.184316	+	0.982867i	$0.559007\pi$
$864$	0	0
$865$	−0.699531	−0.0237848
$866$	6.50838	0.221164
$867$	0	0
$868$	−36.3681	−1.23441
$869$	−13.5657	−0.460185
$870$	0	0
$871$	−36.8695	−1.24927
$872$	−47.2206	−1.59909
$873$	0	0
$874$	−28.8785	−0.976829
$875$	−12.8837	−0.435550
$876$	0	0
$877$	−28.2847	−0.955106	−0.477553	−	0.878603i	$-0.658476\pi$
$877$	−28.2847	−0.955106	−0.477553	+	0.878603i	$0.658476\pi$
$878$	−1.82471	−0.0615808
$879$	0	0
$880$	0.401313	0.0135283
$881$	1.05729	0.0356210	0.0178105	−	0.999841i	$-0.494330\pi$
$881$	1.05729	0.0356210	0.0178105	+	0.999841i	$0.494330\pi$
$882$	0	0
$883$	−46.6966	−1.57147	−0.785733	−	0.618566i	$-0.787714\pi$
$883$	−46.6966	−1.57147	−0.785733	+	0.618566i	$0.787714\pi$
$884$	7.56389	0.254401
$885$	0	0
$886$	−6.01099	−0.201943
$887$	25.9247	0.870465	0.435232	−	0.900318i	$-0.356666\pi$
$887$	25.9247	0.870465	0.435232	+	0.900318i	$0.356666\pi$
$888$	0	0
$889$	36.5846	1.22701
$890$	−2.07817	−0.0696605
$891$	0	0
$892$	1.15182	0.0385659
$893$	−28.1307	−0.941359
$894$	0	0
$895$	7.53862	0.251988
$896$	43.8905	1.46628
$897$	0	0
$898$	19.6684	0.656342
$899$	−44.4344	−1.48197
$900$	0	0
$901$	14.1483	0.471348
$902$	−8.34965	−0.278013
$903$	0	0
$904$	−26.7446	−0.889512
$905$	−0.480656	−0.0159775
$906$	0	0
$907$	−28.2503	−0.938036	−0.469018	−	0.883189i	$-0.655392\pi$
$907$	−28.2503	−0.938036	−0.469018	+	0.883189i	$0.655392\pi$
$908$	−31.3823	−1.04146
$909$	0	0
$910$	2.86581	0.0950006
$911$	14.4287	0.478043	0.239021	−	0.971014i	$-0.423173\pi$
$911$	14.4287	0.478043	0.239021	+	0.971014i	$0.423173\pi$
$912$	0	0
$913$	−8.17500	−0.270553
$914$	−22.3476	−0.739194
$915$	0	0
$916$	−38.6089	−1.27567
$917$	−36.5544	−1.20713
$918$	0	0
$919$	8.78663	0.289844	0.144922	−	0.989443i	$-0.453707\pi$
$919$	8.78663	0.289844	0.144922	+	0.989443i	$0.453707\pi$
$920$	5.33001	0.175725
$921$	0	0
$922$	−3.29197	−0.108415
$923$	21.8325	0.718624
$924$	0	0
$925$	−21.3054	−0.700517
$926$	−15.1591	−0.498160
$927$	0	0
$928$	44.7341	1.46847
$929$	40.5112	1.32913	0.664565	−	0.747230i	$-0.268617\pi$
$929$	40.5112	1.32913	0.664565	+	0.747230i	$0.268617\pi$
$930$	0	0
$931$	−62.3220	−2.04252
$932$	26.7546	0.876376
$933$	0	0
$934$	−13.1715	−0.430984
$935$	−0.539106	−0.0176306
$936$	0	0
$937$	−9.15460	−0.299068	−0.149534	−	0.988757i	$-0.547777\pi$
$937$	−9.15460	−0.299068	−0.149534	+	0.988757i	$0.547777\pi$
$938$	35.5845	1.16188
$939$	0	0
$940$	2.21967	0.0723977
$941$	12.8924	0.420280	0.210140	−	0.977671i	$-0.432608\pi$
$941$	12.8924	0.420280	0.210140	+	0.977671i	$0.432608\pi$
$942$	0	0
$943$	76.3304	2.48566
$944$	−5.02365	−0.163506
$945$	0	0
$946$	1.22430	0.0398053
$947$	56.1939	1.82606	0.913029	−	0.407895i	$-0.133737\pi$
$947$	56.1939	1.82606	0.913029	+	0.407895i	$0.133737\pi$
$948$	0	0
$949$	4.31501	0.140071
$950$	20.4684	0.664082
$951$	0	0
$952$	−17.0760	−0.553435
$953$	−27.2856	−0.883868	−0.441934	−	0.897048i	$-0.645707\pi$
$953$	−27.2856	−0.883868	−0.441934	+	0.897048i	$0.645707\pi$
$954$	0	0
$955$	−0.609767	−0.0197316
$956$	1.49356	0.0483051
$957$	0	0
$958$	20.1657	0.651526
$959$	−68.6099	−2.21553
$960$	0	0
$961$	2.63896	0.0851278
$962$	9.57095	0.308580
$963$	0	0
$964$	−14.2356	−0.458498
$965$	−0.315244	−0.0101480
$966$	0	0
$967$	−30.9244	−0.994460	−0.497230	−	0.867619i	$-0.665650\pi$
$967$	−30.9244	−0.994460	−0.497230	+	0.867619i	$0.665650\pi$
$968$	24.5360	0.788616
$969$	0	0
$970$	0.627899	0.0201606
$971$	15.5085	0.497692	0.248846	−	0.968543i	$-0.419949\pi$
$971$	15.5085	0.497692	0.248846	+	0.968543i	$0.419949\pi$
$972$	0	0
$973$	45.5007	1.45868
$974$	24.3936	0.781620
$975$	0	0
$976$	13.8636	0.443764
$977$	57.5528	1.84128	0.920639	−	0.390416i	$-0.127669\pi$
$977$	57.5528	1.84128	0.920639	+	0.390416i	$0.127669\pi$
$978$	0	0
$979$	−10.0232	−0.320344
$980$	4.91755	0.157085
$981$	0	0
$982$	−21.2357	−0.677657
$983$	41.3072	1.31750	0.658748	−	0.752363i	$-0.271086\pi$
$983$	41.3072	1.31750	0.658748	+	0.752363i	$0.271086\pi$
$984$	0	0
$985$	−4.70185	−0.149813
$986$	−8.91944	−0.284053
$987$	0	0
$988$	27.1168	0.862701
$989$	−11.1922	−0.355891
$990$	0	0
$991$	−1.77458	−0.0563713	−0.0281857	−	0.999603i	$-0.508973\pi$
$991$	−1.77458	−0.0563713	−0.0281857	+	0.999603i	$0.508973\pi$
$992$	−33.8659	−1.07524
$993$	0	0
$994$	−21.0716	−0.668350
$995$	0.948701	0.0300758
$996$	0	0
$997$	27.2945	0.864425	0.432212	−	0.901772i	$-0.357733\pi$
$997$	27.2945	0.864425	0.432212	+	0.901772i	$0.357733\pi$
$998$	10.6852	0.338234
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.2.a.g.1.4		8
3.2	odd	2		717.2.a.f.1.5	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.2.a.f.1.5	✓	8	3.2	odd	2
2151.2.a.g.1.4		8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+0.711649 q^{2} -1.49356 q^{4} -0.309852 q^{5} -4.19834 q^{7} -2.48618 q^{8} +O(q^{10})\)	\(q+0.711649 q^{2} -1.49356 q^{4} -0.309852 q^{5} -4.19834 q^{7} -2.48618 q^{8} -0.220506 q^{10} -1.06352 q^{11} +3.09563 q^{13} -2.98775 q^{14} +1.21782 q^{16} -1.63596 q^{17} -5.86500 q^{19} +0.462781 q^{20} -0.756853 q^{22} +6.91896 q^{23} -4.90399 q^{25} +2.20300 q^{26} +6.27046 q^{28} +7.66121 q^{29} -5.79991 q^{31} +5.83903 q^{32} -1.16423 q^{34} +1.30086 q^{35} +4.34450 q^{37} -4.17382 q^{38} +0.770348 q^{40} +11.0321 q^{41} -1.61761 q^{43} +1.58843 q^{44} +4.92387 q^{46} +4.79638 q^{47} +10.6261 q^{49} -3.48992 q^{50} -4.62350 q^{52} -8.64828 q^{53} +0.329534 q^{55} +10.4379 q^{56} +5.45209 q^{58} -4.12511 q^{59} +11.3840 q^{61} -4.12750 q^{62} +1.71970 q^{64} -0.959187 q^{65} -11.9102 q^{67} +2.44341 q^{68} +0.925758 q^{70} +7.05266 q^{71} +1.39390 q^{73} +3.09176 q^{74} +8.75970 q^{76} +4.46502 q^{77} +12.7555 q^{79} -0.377344 q^{80} +7.85095 q^{82} +7.68673 q^{83} +0.506906 q^{85} -1.15117 q^{86} +2.64411 q^{88} +9.42458 q^{89} -12.9965 q^{91} -10.3339 q^{92} +3.41334 q^{94} +1.81728 q^{95} -2.84754 q^{97} +7.56204 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 5 q^{2} + 7 q^{4} + 13 q^{5} - 7 q^{7} + 15 q^{8}+O(q^{10}) \) 8 * q + 5 * q^2 + 7 * q^4 + 13 * q^5 - 7 * q^7 + 15 * q^8	\( 8 q + 5 q^{2} + 7 q^{4} + 13 q^{5} - 7 q^{7} + 15 q^{8} + 4 q^{10} + 19 q^{11} - 3 q^{13} + 8 q^{14} + 9 q^{16} + 13 q^{17} - 6 q^{19} + 18 q^{20} + 3 q^{22} + 18 q^{23} + 7 q^{25} - 2 q^{28} + 10 q^{29} - 2 q^{31} + 20 q^{32} - 4 q^{34} + 7 q^{35} - 8 q^{37} - 9 q^{38} + 29 q^{40} + 22 q^{41} - 24 q^{43} + 7 q^{44} + 30 q^{46} + 17 q^{47} + 15 q^{49} - 24 q^{50} + 22 q^{52} + 32 q^{55} - 19 q^{56} + 18 q^{58} + 24 q^{59} + 10 q^{61} - 30 q^{62} + 33 q^{64} + 17 q^{65} - 48 q^{67} + 21 q^{68} + 31 q^{70} + 17 q^{71} + 2 q^{73} - 9 q^{74} + 10 q^{76} - 10 q^{77} + 17 q^{79} - 8 q^{80} - 17 q^{82} + 37 q^{83} + 28 q^{85} + q^{86} + 15 q^{88} + 41 q^{89} - 39 q^{91} + 38 q^{92} + 2 q^{94} - 16 q^{95} - 20 q^{97} - 46 q^{98}+O(q^{100}) \) 8 * q + 5 * q^2 + 7 * q^4 + 13 * q^5 - 7 * q^7 + 15 * q^8 + 4 * q^10 + 19 * q^11 - 3 * q^13 + 8 * q^14 + 9 * q^16 + 13 * q^17 - 6 * q^19 + 18 * q^20 + 3 * q^22 + 18 * q^23 + 7 * q^25 - 2 * q^28 + 10 * q^29 - 2 * q^31 + 20 * q^32 - 4 * q^34 + 7 * q^35 - 8 * q^37 - 9 * q^38 + 29 * q^40 + 22 * q^41 - 24 * q^43 + 7 * q^44 + 30 * q^46 + 17 * q^47 + 15 * q^49 - 24 * q^50 + 22 * q^52 + 32 * q^55 - 19 * q^56 + 18 * q^58 + 24 * q^59 + 10 * q^61 - 30 * q^62 + 33 * q^64 + 17 * q^65 - 48 * q^67 + 21 * q^68 + 31 * q^70 + 17 * q^71 + 2 * q^73 - 9 * q^74 + 10 * q^76 - 10 * q^77 + 17 * q^79 - 8 * q^80 - 17 * q^82 + 37 * q^83 + 28 * q^85 + q^86 + 15 * q^88 + 41 * q^89 - 39 * q^91 + 38 * q^92 + 2 * q^94 - 16 * q^95 - 20 * q^97 - 46 * q^98

Introduction

L-functions

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Database

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