LMFDB - Embedded newform 2151.2.a.k.1.19

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			1.19
Root			$2.61567$ 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)

$f(q)$	$=$	$q+2.61567 q^{2} +4.84173 q^{4} +2.80971 q^{5} +3.73688 q^{7} +7.43303 q^{8} +O(q^{10})$	\(q+2.61567 q^{2} +4.84173 q^{4} +2.80971 q^{5} +3.73688 q^{7} +7.43303 q^{8} +7.34928 q^{10} -6.02355 q^{11} -0.651730 q^{13} +9.77444 q^{14} +9.75890 q^{16} +1.63490 q^{17} -6.09296 q^{19} +13.6039 q^{20} -15.7556 q^{22} +1.60504 q^{23} +2.89448 q^{25} -1.70471 q^{26} +18.0930 q^{28} +2.35372 q^{29} -6.90597 q^{31} +10.6600 q^{32} +4.27637 q^{34} +10.4995 q^{35} -10.0889 q^{37} -15.9372 q^{38} +20.8847 q^{40} +8.60185 q^{41} +8.78231 q^{43} -29.1644 q^{44} +4.19825 q^{46} -6.67446 q^{47} +6.96425 q^{49} +7.57100 q^{50} -3.15550 q^{52} -9.57641 q^{53} -16.9244 q^{55} +27.7763 q^{56} +6.15656 q^{58} -9.10915 q^{59} -9.50698 q^{61} -18.0637 q^{62} +8.36524 q^{64} -1.83117 q^{65} -8.35366 q^{67} +7.91576 q^{68} +27.4634 q^{70} +9.36039 q^{71} +8.06381 q^{73} -26.3893 q^{74} -29.5005 q^{76} -22.5092 q^{77} -4.58730 q^{79} +27.4197 q^{80} +22.4996 q^{82} +9.85175 q^{83} +4.59361 q^{85} +22.9716 q^{86} -44.7732 q^{88} +18.5981 q^{89} -2.43544 q^{91} +7.77117 q^{92} -17.4582 q^{94} -17.1195 q^{95} +6.39154 q^{97} +18.2162 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.61567	1.84956	0.924779	−	0.380504i	$-0.124250\pi$
$2$	2.61567	1.84956	0.924779	+	0.380504i	$0.124250\pi$
$3$	0	0
$3$	0	0
$4$	4.84173	2.42087
$5$	2.80971	1.25654	0.628271	−	0.777995i	$-0.283763\pi$
$5$	2.80971	1.25654	0.628271	+	0.777995i	$0.283763\pi$
$6$	0	0
$7$	3.73688	1.41241	0.706203	−	0.708009i	$-0.250406\pi$
$7$	3.73688	1.41241	0.706203	+	0.708009i	$0.250406\pi$
$8$	7.43303	2.62797
$9$	0	0
$10$	7.34928	2.32405
$11$	−6.02355	−1.81617	−0.908084	−	0.418789i	$-0.862455\pi$
$11$	−6.02355	−1.81617	−0.908084	+	0.418789i	$0.862455\pi$
$12$	0	0
$13$	−0.651730	−0.180757	−0.0903787	−	0.995907i	$-0.528808\pi$
$13$	−0.651730	−0.180757	−0.0903787	+	0.995907i	$0.528808\pi$
$14$	9.77444	2.61233
$15$	0	0
$16$	9.75890	2.43972
$17$	1.63490	0.396522	0.198261	−	0.980149i	$-0.436471\pi$
$17$	1.63490	0.396522	0.198261	+	0.980149i	$0.436471\pi$
$18$	0	0
$19$	−6.09296	−1.39782	−0.698910	−	0.715209i	$-0.746331\pi$
$19$	−6.09296	−1.39782	−0.698910	+	0.715209i	$0.746331\pi$
$20$	13.6039	3.04192
$21$	0	0
$22$	−15.7556	−3.35911
$23$	1.60504	0.334674	0.167337	−	0.985900i	$-0.446483\pi$
$23$	1.60504	0.334674	0.167337	+	0.985900i	$0.446483\pi$
$24$	0	0
$25$	2.89448	0.578896
$26$	−1.70471	−0.334321
$27$	0	0
$28$	18.0930	3.41925
$29$	2.35372	0.437075	0.218537	−	0.975829i	$-0.429871\pi$
$29$	2.35372	0.437075	0.218537	+	0.975829i	$0.429871\pi$
$30$	0	0
$31$	−6.90597	−1.24035	−0.620174	−	0.784464i	$-0.712938\pi$
$31$	−6.90597	−1.24035	−0.620174	+	0.784464i	$0.712938\pi$
$32$	10.6600	1.88444
$33$	0	0
$34$	4.27637	0.733391
$35$	10.4995	1.77475
$36$	0	0
$37$	−10.0889	−1.65861	−0.829305	−	0.558796i	$-0.811264\pi$
$37$	−10.0889	−1.65861	−0.829305	+	0.558796i	$0.811264\pi$
$38$	−15.9372	−2.58535
$39$	0	0
$40$	20.8847	3.30216
$41$	8.60185	1.34338	0.671691	−	0.740831i	$-0.265568\pi$
$41$	8.60185	1.34338	0.671691	+	0.740831i	$0.265568\pi$
$42$	0	0
$43$	8.78231	1.33929	0.669645	−	0.742681i	$-0.266446\pi$
$43$	8.78231	1.33929	0.669645	+	0.742681i	$0.266446\pi$
$44$	−29.1644	−4.39670
$45$	0	0
$46$	4.19825	0.618999
$47$	−6.67446	−0.973570	−0.486785	−	0.873522i	$-0.661830\pi$
$47$	−6.67446	−0.973570	−0.486785	+	0.873522i	$0.661830\pi$
$48$	0	0
$49$	6.96425	0.994893
$50$	7.57100	1.07070
$51$	0	0
$52$	−3.15550	−0.437589
$53$	−9.57641	−1.31542	−0.657711	−	0.753271i	$-0.728475\pi$
$53$	−9.57641	−1.31542	−0.657711	+	0.753271i	$0.728475\pi$
$54$	0	0
$55$	−16.9244	−2.28209
$56$	27.7763	3.71177
$57$	0	0
$58$	6.15656	0.808396
$59$	−9.10915	−1.18591	−0.592955	−	0.805235i	$-0.702039\pi$
$59$	−9.10915	−1.18591	−0.592955	+	0.805235i	$0.702039\pi$
$60$	0	0
$61$	−9.50698	−1.21724	−0.608622	−	0.793460i	$-0.708277\pi$
$61$	−9.50698	−1.21724	−0.608622	+	0.793460i	$0.708277\pi$
$62$	−18.0637	−2.29410
$63$	0	0
$64$	8.36524	1.04565
$65$	−1.83117	−0.227129
$66$	0	0
$67$	−8.35366	−1.02056	−0.510281	−	0.860008i	$-0.670459\pi$
$67$	−8.35366	−1.02056	−0.510281	+	0.860008i	$0.670459\pi$
$68$	7.91576	0.959927
$69$	0	0
$70$	27.4634	3.28250
$71$	9.36039	1.11087	0.555437	−	0.831558i	$-0.312551\pi$
$71$	9.36039	1.11087	0.555437	+	0.831558i	$0.312551\pi$
$72$	0	0
$73$	8.06381	0.943798	0.471899	−	0.881653i	$-0.343569\pi$
$73$	8.06381	0.943798	0.471899	+	0.881653i	$0.343569\pi$
$74$	−26.3893	−3.06770
$75$	0	0
$76$	−29.5005	−3.38394
$77$	−22.5092	−2.56517
$78$	0	0
$79$	−4.58730	−0.516111	−0.258056	−	0.966130i	$-0.583082\pi$
$79$	−4.58730	−0.516111	−0.258056	+	0.966130i	$0.583082\pi$
$80$	27.4197	3.06561
$81$	0	0
$82$	22.4996	2.48466
$83$	9.85175	1.08137	0.540685	−	0.841225i	$-0.318165\pi$
$83$	9.85175	1.08137	0.540685	+	0.841225i	$0.318165\pi$
$84$	0	0
$85$	4.59361	0.498247
$86$	22.9716	2.47709
$87$	0	0
$88$	−44.7732	−4.77284
$89$	18.5981	1.97139	0.985697	−	0.168526i	$-0.0539009\pi$
$89$	18.5981	1.97139	0.985697	+	0.168526i	$0.0539009\pi$
$90$	0	0
$91$	−2.43544	−0.255303
$92$	7.77117	0.810200
$93$	0	0
$94$	−17.4582	−1.80067
$95$	−17.1195	−1.75642
$96$	0	0
$97$	6.39154	0.648962	0.324481	−	0.945892i	$-0.394810\pi$
$97$	6.39154	0.648962	0.324481	+	0.945892i	$0.394810\pi$
$98$	18.2162	1.84011
$99$	0	0
$100$	14.0143	1.40143
$101$	−1.17857	−0.117272	−0.0586359	−	0.998279i	$-0.518675\pi$
$101$	−1.17857	−0.117272	−0.0586359	+	0.998279i	$0.518675\pi$
$102$	0	0
$103$	2.70231	0.266267	0.133133	−	0.991098i	$-0.457496\pi$
$103$	2.70231	0.266267	0.133133	+	0.991098i	$0.457496\pi$
$104$	−4.84433	−0.475026
$105$	0	0
$106$	−25.0487	−2.43295
$107$	10.4572	1.01093	0.505466	−	0.862847i	$-0.331321\pi$
$107$	10.4572	1.01093	0.505466	+	0.862847i	$0.331321\pi$
$108$	0	0
$109$	20.7374	1.98628	0.993141	−	0.116920i	$-0.0373021\pi$
$109$	20.7374	1.98628	0.993141	+	0.116920i	$0.0373021\pi$
$110$	−44.2687	−4.22086
$111$	0	0
$112$	36.4678	3.44588
$113$	5.76532	0.542356	0.271178	−	0.962529i	$-0.412587\pi$
$113$	5.76532	0.542356	0.271178	+	0.962529i	$0.412587\pi$
$114$	0	0
$115$	4.50970	0.420531
$116$	11.3961	1.05810
$117$	0	0
$118$	−23.8265	−2.19341
$119$	6.10943	0.560051
$120$	0	0
$121$	25.2831	2.29846
$122$	−24.8671	−2.25136
$123$	0	0
$124$	−33.4368	−3.00272
$125$	−5.91590	−0.529135
$126$	0	0
$127$	6.92357	0.614368	0.307184	−	0.951650i	$-0.400613\pi$
$127$	6.92357	0.614368	0.307184	+	0.951650i	$0.400613\pi$
$128$	0.560710	0.0495603
$129$	0	0
$130$	−4.78975	−0.420089
$131$	8.85105	0.773319	0.386660	−	0.922222i	$-0.373629\pi$
$131$	8.85105	0.773319	0.386660	+	0.922222i	$0.373629\pi$
$132$	0	0
$133$	−22.7686	−1.97429
$134$	−21.8504	−1.88759
$135$	0	0
$136$	12.1523	1.04205
$137$	8.08553	0.690794	0.345397	−	0.938457i	$-0.387744\pi$
$137$	8.08553	0.690794	0.345397	+	0.938457i	$0.387744\pi$
$138$	0	0
$139$	−6.34824	−0.538450	−0.269225	−	0.963077i	$-0.586768\pi$
$139$	−6.34824	−0.538450	−0.269225	+	0.963077i	$0.586768\pi$
$140$	50.8360	4.29642
$141$	0	0
$142$	24.4837	2.05463
$143$	3.92573	0.328286
$144$	0	0
$145$	6.61328	0.549203
$146$	21.0923	1.74561
$147$	0	0
$148$	−48.8479	−4.01527
$149$	−12.2210	−1.00118	−0.500592	−	0.865683i	$-0.666884\pi$
$149$	−12.2210	−1.00118	−0.500592	+	0.865683i	$0.666884\pi$
$150$	0	0
$151$	−13.7944	−1.12257	−0.561286	−	0.827622i	$-0.689693\pi$
$151$	−13.7944	−1.12257	−0.561286	+	0.827622i	$0.689693\pi$
$152$	−45.2892	−3.67344
$153$	0	0
$154$	−58.8768	−4.74443
$155$	−19.4038	−1.55855
$156$	0	0
$157$	18.1273	1.44671	0.723356	−	0.690475i	$-0.242598\pi$
$157$	18.1273	1.44671	0.723356	+	0.690475i	$0.242598\pi$
$158$	−11.9989	−0.954578
$159$	0	0
$160$	29.9515	2.36787
$161$	5.99783	0.472696
$162$	0	0
$163$	−4.78122	−0.374494	−0.187247	−	0.982313i	$-0.559956\pi$
$163$	−4.78122	−0.374494	−0.187247	+	0.982313i	$0.559956\pi$
$164$	41.6478	3.25215
$165$	0	0
$166$	25.7689	2.00006
$167$	−5.39807	−0.417715	−0.208857	−	0.977946i	$-0.566974\pi$
$167$	−5.39807	−0.417715	−0.208857	+	0.977946i	$0.566974\pi$
$168$	0	0
$169$	−12.5752	−0.967327
$170$	12.0154	0.921536
$171$	0	0
$172$	42.5216	3.24224
$173$	16.5400	1.25751	0.628757	−	0.777602i	$-0.283564\pi$
$173$	16.5400	1.25751	0.628757	+	0.777602i	$0.283564\pi$
$174$	0	0
$175$	10.8163	0.817637
$176$	−58.7832	−4.43095
$177$	0	0
$178$	48.6465	3.64621
$179$	10.8315	0.809584	0.404792	−	0.914409i	$-0.367344\pi$
$179$	10.8315	0.809584	0.404792	+	0.914409i	$0.367344\pi$
$180$	0	0
$181$	−4.87344	−0.362240	−0.181120	−	0.983461i	$-0.557972\pi$
$181$	−4.87344	−0.362240	−0.181120	+	0.983461i	$0.557972\pi$
$182$	−6.37030	−0.472198
$183$	0	0
$184$	11.9303	0.879514
$185$	−28.3470	−2.08411
$186$	0	0
$187$	−9.84791	−0.720151
$188$	−32.3159	−2.35688
$189$	0	0
$190$	−44.7789	−3.24860
$191$	−6.92857	−0.501333	−0.250667	−	0.968073i	$-0.580650\pi$
$191$	−6.92857	−0.501333	−0.250667	+	0.968073i	$0.580650\pi$
$192$	0	0
$193$	18.4673	1.32931	0.664654	−	0.747151i	$-0.268579\pi$
$193$	18.4673	1.32931	0.664654	+	0.747151i	$0.268579\pi$
$194$	16.7182	1.20029
$195$	0	0
$196$	33.7190	2.40850
$197$	10.4671	0.745753	0.372877	−	0.927881i	$-0.378371\pi$
$197$	10.4671	0.745753	0.372877	+	0.927881i	$0.378371\pi$
$198$	0	0
$199$	−1.98162	−0.140473	−0.0702367	−	0.997530i	$-0.522375\pi$
$199$	−1.98162	−0.140473	−0.0702367	+	0.997530i	$0.522375\pi$
$200$	21.5148	1.52132
$201$	0	0
$202$	−3.08274	−0.216901
$203$	8.79557	0.617328
$204$	0	0
$205$	24.1687	1.68802
$206$	7.06835	0.492476
$207$	0	0
$208$	−6.36017	−0.440998
$209$	36.7012	2.53868
$210$	0	0
$211$	−12.8748	−0.886335	−0.443167	−	0.896439i	$-0.646145\pi$
$211$	−12.8748	−0.886335	−0.443167	+	0.896439i	$0.646145\pi$
$212$	−46.3664	−3.18446
$213$	0	0
$214$	27.3525	1.86978
$215$	24.6758	1.68287
$216$	0	0
$217$	−25.8068	−1.75188
$218$	54.2422	3.67375
$219$	0	0
$220$	−81.9435	−5.52463
$221$	−1.06552	−0.0716743
$222$	0	0
$223$	−7.79435	−0.521949	−0.260974	−	0.965346i	$-0.584044\pi$
$223$	−7.79435	−0.521949	−0.260974	+	0.965346i	$0.584044\pi$
$224$	39.8351	2.66159
$225$	0	0
$226$	15.0802	1.00312
$227$	6.83248	0.453488	0.226744	−	0.973954i	$-0.427192\pi$
$227$	6.83248	0.453488	0.226744	+	0.973954i	$0.427192\pi$
$228$	0	0
$229$	−16.7361	−1.10595	−0.552976	−	0.833197i	$-0.686508\pi$
$229$	−16.7361	−1.10595	−0.552976	+	0.833197i	$0.686508\pi$
$230$	11.7959	0.777797
$231$	0	0
$232$	17.4953	1.14862
$233$	−0.0111558	−0.000730842 0	−0.000365421	−	1.00000i	$-0.500116\pi$
$233$	−0.0111558	−0.000730842 0	−0.000365421		1.00000i	$0.500116\pi$
$234$	0	0
$235$	−18.7533	−1.22333
$236$	−44.1041	−2.87093
$237$	0	0
$238$	15.9803	1.03585
$239$	1.00000	0.0646846
$239$	1.00000	0.0646846
$240$	0	0
$241$	−30.3328	−1.95391	−0.976955	−	0.213447i	$-0.931531\pi$
$241$	−30.3328	−1.95391	−0.976955	+	0.213447i	$0.931531\pi$
$242$	66.1322	4.25114
$243$	0	0
$244$	−46.0303	−2.94678
$245$	19.5675	1.25012
$246$	0	0
$247$	3.97097	0.252666
$248$	−51.3323	−3.25960
$249$	0	0
$250$	−15.4741	−0.978665
$251$	12.7475	0.804616	0.402308	−	0.915504i	$-0.368208\pi$
$251$	12.7475	0.804616	0.402308	+	0.915504i	$0.368208\pi$
$252$	0	0
$253$	−9.66803	−0.607824
$254$	18.1098	1.13631
$255$	0	0
$256$	−15.2638	−0.953990
$257$	13.4247	0.837407	0.418704	−	0.908123i	$-0.362485\pi$
$257$	13.4247	0.837407	0.418704	+	0.908123i	$0.362485\pi$
$258$	0	0
$259$	−37.7011	−2.34263
$260$	−8.86605	−0.549849
$261$	0	0
$262$	23.1514	1.43030
$263$	6.07882	0.374836	0.187418	−	0.982280i	$-0.439988\pi$
$263$	6.07882	0.374836	0.187418	+	0.982280i	$0.439988\pi$
$264$	0	0
$265$	−26.9070	−1.65288
$266$	−59.5553	−3.65157
$267$	0	0
$268$	−40.4462	−2.47064
$269$	−28.1289	−1.71505	−0.857526	−	0.514441i	$-0.828000\pi$
$269$	−28.1289	−1.71505	−0.857526	+	0.514441i	$0.828000\pi$
$270$	0	0
$271$	−8.81586	−0.535526	−0.267763	−	0.963485i	$-0.586284\pi$
$271$	−8.81586	−0.535526	−0.267763	+	0.963485i	$0.586284\pi$
$272$	15.9549	0.967405
$273$	0	0
$274$	21.1491	1.27766
$275$	−17.4350	−1.05137
$276$	0	0
$277$	−4.90099	−0.294472	−0.147236	−	0.989101i	$-0.547038\pi$
$277$	−4.90099	−0.294472	−0.147236	+	0.989101i	$0.547038\pi$
$278$	−16.6049	−0.995895
$279$	0	0
$280$	78.0435	4.66399
$281$	13.1553	0.784780	0.392390	−	0.919799i	$-0.371648\pi$
$281$	13.1553	0.784780	0.392390	+	0.919799i	$0.371648\pi$
$282$	0	0
$283$	−25.9060	−1.53995	−0.769976	−	0.638073i	$-0.779732\pi$
$283$	−25.9060	−1.53995	−0.769976	+	0.638073i	$0.779732\pi$
$284$	45.3205	2.68928
$285$	0	0
$286$	10.2684	0.607183
$287$	32.1440	1.89740
$288$	0	0
$289$	−14.3271	−0.842770
$290$	17.2982	1.01578
$291$	0	0
$292$	39.0428	2.28481
$293$	−6.11106	−0.357012	−0.178506	−	0.983939i	$-0.557126\pi$
$293$	−6.11106	−0.357012	−0.178506	+	0.983939i	$0.557126\pi$
$294$	0	0
$295$	−25.5941	−1.49015
$296$	−74.9914	−4.35879
$297$	0	0
$298$	−31.9661	−1.85175
$299$	−1.04605	−0.0604948
$300$	0	0
$301$	32.8184	1.89162
$302$	−36.0816	−2.07626
$303$	0	0
$304$	−59.4606	−3.41030
$305$	−26.7119	−1.52952
$306$	0	0
$307$	22.7608	1.29903	0.649514	−	0.760350i	$-0.274972\pi$
$307$	22.7608	1.29903	0.649514	+	0.760350i	$0.274972\pi$
$308$	−108.984	−6.20992
$309$	0	0
$310$	−50.7539	−2.88263
$311$	−0.0355362	−0.00201507	−0.00100754	−	0.999999i	$-0.500321\pi$
$311$	−0.0355362	−0.00201507	−0.00100754	+	0.999999i	$0.500321\pi$
$312$	0	0
$313$	19.1138	1.08038	0.540188	−	0.841544i	$-0.318353\pi$
$313$	19.1138	1.08038	0.540188	+	0.841544i	$0.318353\pi$
$314$	47.4149	2.67578
$315$	0	0
$316$	−22.2105	−1.24944
$317$	−18.9213	−1.06273	−0.531364	−	0.847143i	$-0.678320\pi$
$317$	−18.9213	−1.06273	−0.531364	+	0.847143i	$0.678320\pi$
$318$	0	0
$319$	−14.1777	−0.793801
$320$	23.5039	1.31391
$321$	0	0
$322$	15.6884	0.874278
$323$	−9.96140	−0.554267
$324$	0	0
$325$	−1.88642	−0.104640
$326$	−12.5061	−0.692649
$327$	0	0
$328$	63.9378	3.53037
$329$	−24.9416	−1.37508
$330$	0	0
$331$	−8.15827	−0.448419	−0.224209	−	0.974541i	$-0.571980\pi$
$331$	−8.15827	−0.448419	−0.224209	+	0.974541i	$0.571980\pi$
$332$	47.6995	2.61785
$333$	0	0
$334$	−14.1196	−0.772588
$335$	−23.4714	−1.28238
$336$	0	0
$337$	12.7368	0.693816	0.346908	−	0.937899i	$-0.387232\pi$
$337$	12.7368	0.693816	0.346908	+	0.937899i	$0.387232\pi$
$338$	−32.8927	−1.78913
$339$	0	0
$340$	22.2410	1.20619
$341$	41.5984	2.25268
$342$	0	0
$343$	−0.133596	−0.00721352
$344$	65.2792	3.51962
$345$	0	0
$346$	43.2632	2.32584
$347$	17.4737	0.938037	0.469019	−	0.883188i	$-0.344608\pi$
$347$	17.4737	0.938037	0.469019	+	0.883188i	$0.344608\pi$
$348$	0	0
$349$	17.3324	0.927781	0.463890	−	0.885893i	$-0.346453\pi$
$349$	17.3324	0.927781	0.463890	+	0.885893i	$0.346453\pi$
$350$	28.2919	1.51227
$351$	0	0
$352$	−64.2110	−3.42246
$353$	9.60581	0.511266	0.255633	−	0.966774i	$-0.417716\pi$
$353$	9.60581	0.511266	0.255633	+	0.966774i	$0.417716\pi$
$354$	0	0
$355$	26.3000	1.39586
$356$	90.0470	4.77248
$357$	0	0
$358$	28.3316	1.49737
$359$	−16.4121	−0.866199	−0.433099	−	0.901346i	$-0.642580\pi$
$359$	−16.4121	−0.866199	−0.433099	+	0.901346i	$0.642580\pi$
$360$	0	0
$361$	18.1242	0.953903
$362$	−12.7473	−0.669984
$363$	0	0
$364$	−11.7917	−0.618054
$365$	22.6570	1.18592
$366$	0	0
$367$	18.9634	0.989883	0.494942	−	0.868926i	$-0.335189\pi$
$367$	18.9634	0.989883	0.494942	+	0.868926i	$0.335189\pi$
$368$	15.6634	0.816512
$369$	0	0
$370$	−74.1464	−3.85469
$371$	−35.7859	−1.85791
$372$	0	0
$373$	−16.6031	−0.859677	−0.429838	−	0.902906i	$-0.641429\pi$
$373$	−16.6031	−0.859677	−0.429838	+	0.902906i	$0.641429\pi$
$374$	−25.7589	−1.33196
$375$	0	0
$376$	−49.6115	−2.55852
$377$	−1.53399	−0.0790045
$378$	0	0
$379$	−1.05770	−0.0543306	−0.0271653	−	0.999631i	$-0.508648\pi$
$379$	−1.05770	−0.0543306	−0.0271653	+	0.999631i	$0.508648\pi$
$380$	−82.8878	−4.25206
$381$	0	0
$382$	−18.1228	−0.927245
$383$	24.6868	1.26144	0.630719	−	0.776011i	$-0.282760\pi$
$383$	24.6868	1.26144	0.630719	+	0.776011i	$0.282760\pi$
$384$	0	0
$385$	−63.2445	−3.22324
$386$	48.3045	2.45863
$387$	0	0
$388$	30.9461	1.57105
$389$	−15.9721	−0.809818	−0.404909	−	0.914357i	$-0.632697\pi$
$389$	−15.9721	−0.809818	−0.404909	+	0.914357i	$0.632697\pi$
$390$	0	0
$391$	2.62408	0.132706
$392$	51.7655	2.61455
$393$	0	0
$394$	27.3786	1.37931
$395$	−12.8890	−0.648515
$396$	0	0
$397$	11.2477	0.564505	0.282252	−	0.959340i	$-0.408918\pi$
$397$	11.2477	0.564505	0.282252	+	0.959340i	$0.408918\pi$
$398$	−5.18327	−0.259814
$399$	0	0
$400$	28.2469	1.41235
$401$	−15.0646	−0.752292	−0.376146	−	0.926560i	$-0.622751\pi$
$401$	−15.0646	−0.752292	−0.376146	+	0.926560i	$0.622751\pi$
$402$	0	0
$403$	4.50083	0.224202
$404$	−5.70630	−0.283899
$405$	0	0
$406$	23.0063	1.14178
$407$	60.7712	3.01231
$408$	0	0
$409$	−4.66906	−0.230870	−0.115435	−	0.993315i	$-0.536826\pi$
$409$	−4.66906	−0.230870	−0.115435	+	0.993315i	$0.536826\pi$
$410$	63.2174	3.12208
$411$	0	0
$412$	13.0839	0.644596
$413$	−34.0398	−1.67499
$414$	0	0
$415$	27.6806	1.35879
$416$	−6.94744	−0.340626
$417$	0	0
$418$	95.9983	4.69543
$419$	−11.9766	−0.585095	−0.292547	−	0.956251i	$-0.594503\pi$
$419$	−11.9766	−0.585095	−0.292547	+	0.956251i	$0.594503\pi$
$420$	0	0
$421$	5.13637	0.250331	0.125166	−	0.992136i	$-0.460054\pi$
$421$	5.13637	0.250331	0.125166	+	0.992136i	$0.460054\pi$
$422$	−33.6761	−1.63933
$423$	0	0
$424$	−71.1818	−3.45689
$425$	4.73219	0.229545
$426$	0	0
$427$	−35.5264	−1.71924
$428$	50.6307	2.44733
$429$	0	0
$430$	64.5437	3.11257
$431$	32.7602	1.57801	0.789003	−	0.614390i	$-0.210598\pi$
$431$	32.7602	1.57801	0.789003	+	0.614390i	$0.210598\pi$
$432$	0	0
$433$	−32.9522	−1.58358	−0.791792	−	0.610791i	$-0.790852\pi$
$433$	−32.9522	−1.58358	−0.791792	+	0.610791i	$0.790852\pi$
$434$	−67.5020	−3.24020
$435$	0	0
$436$	100.405	4.80852
$437$	−9.77944	−0.467814
$438$	0	0
$439$	14.1383	0.674786	0.337393	−	0.941364i	$-0.390455\pi$
$439$	14.1383	0.674786	0.337393	+	0.941364i	$0.390455\pi$
$440$	−125.800	−5.99727
$441$	0	0
$442$	−2.78704	−0.132566
$443$	−4.55179	−0.216262	−0.108131	−	0.994137i	$-0.534487\pi$
$443$	−4.55179	−0.216262	−0.108131	+	0.994137i	$0.534487\pi$
$444$	0	0
$445$	52.2553	2.47714
$446$	−20.3875	−0.965374
$447$	0	0
$448$	31.2599	1.47689
$449$	−19.4088	−0.915956	−0.457978	−	0.888963i	$-0.651426\pi$
$449$	−19.4088	−0.915956	−0.457978	+	0.888963i	$0.651426\pi$
$450$	0	0
$451$	−51.8136	−2.43981
$452$	27.9141	1.31297
$453$	0	0
$454$	17.8715	0.838752
$455$	−6.84287	−0.320799
$456$	0	0
$457$	23.2730	1.08866	0.544332	−	0.838870i	$-0.316783\pi$
$457$	23.2730	1.08866	0.544332	+	0.838870i	$0.316783\pi$
$458$	−43.7761	−2.04552
$459$	0	0
$460$	21.8347	1.01805
$461$	−20.7529	−0.966558	−0.483279	−	0.875466i	$-0.660554\pi$
$461$	−20.7529	−0.966558	−0.483279	+	0.875466i	$0.660554\pi$
$462$	0	0
$463$	−21.0156	−0.976678	−0.488339	−	0.872654i	$-0.662397\pi$
$463$	−21.0156	−0.976678	−0.488339	+	0.872654i	$0.662397\pi$
$464$	22.9697	1.06634
$465$	0	0
$466$	−0.0291799	−0.00135173
$467$	−14.2561	−0.659693	−0.329846	−	0.944035i	$-0.606997\pi$
$467$	−14.2561	−0.659693	−0.329846	+	0.944035i	$0.606997\pi$
$468$	0	0
$469$	−31.2166	−1.44145
$470$	−49.0525	−2.26262
$471$	0	0
$472$	−67.7086	−3.11654
$473$	−52.9007	−2.43237
$474$	0	0
$475$	−17.6359	−0.809193
$476$	29.5802	1.35581
$477$	0	0
$478$	2.61567	0.119638
$479$	−0.790719	−0.0361289	−0.0180644	−	0.999837i	$-0.505750\pi$
$479$	−0.790719	−0.0361289	−0.0180644	+	0.999837i	$0.505750\pi$
$480$	0	0
$481$	6.57526	0.299806
$482$	−79.3407	−3.61387
$483$	0	0
$484$	122.414	5.56427
$485$	17.9584	0.815448
$486$	0	0
$487$	−28.5642	−1.29437	−0.647184	−	0.762333i	$-0.724054\pi$
$487$	−28.5642	−1.29437	−0.647184	+	0.762333i	$0.724054\pi$
$488$	−70.6657	−3.19889
$489$	0	0
$490$	51.1822	2.31218
$491$	−41.2183	−1.86016	−0.930079	−	0.367361i	$-0.880261\pi$
$491$	−41.2183	−1.86016	−0.930079	+	0.367361i	$0.880261\pi$
$492$	0	0
$493$	3.84811	0.173310
$494$	10.3867	0.467321
$495$	0	0
$496$	−67.3946	−3.02611
$497$	34.9786	1.56901
$498$	0	0
$499$	−4.13535	−0.185124	−0.0925619	−	0.995707i	$-0.529506\pi$
$499$	−4.13535	−0.185124	−0.0925619	+	0.995707i	$0.529506\pi$
$500$	−28.6432	−1.28096
$501$	0	0
$502$	33.3433	1.48818
$503$	26.8822	1.19862	0.599309	−	0.800518i	$-0.295442\pi$
$503$	26.8822	1.19862	0.599309	+	0.800518i	$0.295442\pi$
$504$	0	0
$505$	−3.31143	−0.147357
$506$	−25.2884	−1.12421
$507$	0	0
$508$	33.5221	1.48730
$509$	20.0950	0.890696	0.445348	−	0.895358i	$-0.353080\pi$
$509$	20.0950	0.890696	0.445348	+	0.895358i	$0.353080\pi$
$510$	0	0
$511$	30.1335	1.33303
$512$	−41.0466	−1.81402
$513$	0	0
$514$	35.1145	1.54883
$515$	7.59271	0.334575
$516$	0	0
$517$	40.2039	1.76817
$518$	−98.6137	−4.33284
$519$	0	0
$520$	−13.6112	−0.596889
$521$	28.1803	1.23460	0.617302	−	0.786727i	$-0.288226\pi$
$521$	28.1803	1.23460	0.617302	+	0.786727i	$0.288226\pi$
$522$	0	0
$523$	−4.66666	−0.204059	−0.102029	−	0.994781i	$-0.532534\pi$
$523$	−4.66666	−0.204059	−0.102029	+	0.994781i	$0.532534\pi$
$524$	42.8544	1.87210
$525$	0	0
$526$	15.9002	0.693281
$527$	−11.2906	−0.491826
$528$	0	0
$529$	−20.4238	−0.887993
$530$	−70.3797	−3.05710
$531$	0	0
$532$	−110.240	−4.77949
$533$	−5.60608	−0.242826
$534$	0	0
$535$	29.3816	1.27028
$536$	−62.0930	−2.68201
$537$	0	0
$538$	−73.5760	−3.17209
$539$	−41.9495	−1.80689
$540$	0	0
$541$	−25.0475	−1.07687	−0.538437	−	0.842666i	$-0.680985\pi$
$541$	−25.0475	−1.07687	−0.538437	+	0.842666i	$0.680985\pi$
$542$	−23.0594	−0.990486
$543$	0	0
$544$	17.4281	0.747222
$545$	58.2661	2.49585
$546$	0	0
$547$	41.9321	1.79289	0.896445	−	0.443156i	$-0.146141\pi$
$547$	41.9321	1.79289	0.896445	+	0.443156i	$0.146141\pi$
$548$	39.1480	1.67232
$549$	0	0
$550$	−45.6043	−1.94457
$551$	−14.3411	−0.610952
$552$	0	0
$553$	−17.1422	−0.728959
$554$	−12.8194	−0.544643
$555$	0	0
$556$	−30.7364	−1.30352
$557$	1.75216	0.0742414	0.0371207	−	0.999311i	$-0.488181\pi$
$557$	1.75216	0.0742414	0.0371207	+	0.999311i	$0.488181\pi$
$558$	0	0
$559$	−5.72370	−0.242087
$560$	102.464	4.32989
$561$	0	0
$562$	34.4100	1.45150
$563$	42.8126	1.80434	0.902168	−	0.431385i	$-0.141975\pi$
$563$	42.8126	1.80434	0.902168	+	0.431385i	$0.141975\pi$
$564$	0	0
$565$	16.1989	0.681492
$566$	−67.7615	−2.84823
$567$	0	0
$568$	69.5761	2.91935
$569$	4.33539	0.181749	0.0908745	−	0.995862i	$-0.471034\pi$
$569$	4.33539	0.181749	0.0908745	+	0.995862i	$0.471034\pi$
$570$	0	0
$571$	19.8015	0.828669	0.414334	−	0.910125i	$-0.364014\pi$
$571$	19.8015	0.828669	0.414334	+	0.910125i	$0.364014\pi$
$572$	19.0073	0.794735
$573$	0	0
$574$	84.0782	3.50936
$575$	4.64575	0.193741
$576$	0	0
$577$	6.90635	0.287515	0.143757	−	0.989613i	$-0.454081\pi$
$577$	6.90635	0.287515	0.143757	+	0.989613i	$0.454081\pi$
$578$	−37.4749	−1.55875
$579$	0	0
$580$	32.0197	1.32955
$581$	36.8148	1.52733
$582$	0	0
$583$	57.6839	2.38902
$584$	59.9386	2.48028
$585$	0	0
$586$	−15.9845	−0.660315
$587$	22.6076	0.933117	0.466558	−	0.884490i	$-0.345494\pi$
$587$	22.6076	0.933117	0.466558	+	0.884490i	$0.345494\pi$
$588$	0	0
$589$	42.0778	1.73378
$590$	−66.9457	−2.75611
$591$	0	0
$592$	−98.4569	−4.04655
$593$	32.5567	1.33694	0.668472	−	0.743737i	$-0.266948\pi$
$593$	32.5567	1.33694	0.668472	+	0.743737i	$0.266948\pi$
$594$	0	0
$595$	17.1657	0.703727
$596$	−59.1709	−2.42373
$597$	0	0
$598$	−2.73613	−0.111889
$599$	−11.6681	−0.476746	−0.238373	−	0.971174i	$-0.576614\pi$
$599$	−11.6681	−0.476746	−0.238373	+	0.971174i	$0.576614\pi$
$600$	0	0
$601$	−37.3270	−1.52260	−0.761301	−	0.648399i	$-0.775439\pi$
$601$	−37.3270	−1.52260	−0.761301	+	0.648399i	$0.775439\pi$
$602$	85.8422	3.49867
$603$	0	0
$604$	−66.7888	−2.71760
$605$	71.0382	2.88811
$606$	0	0
$607$	5.15815	0.209363	0.104682	−	0.994506i	$-0.466618\pi$
$607$	5.15815	0.209363	0.104682	+	0.994506i	$0.466618\pi$
$608$	−64.9509	−2.63411
$609$	0	0
$610$	−69.8695	−2.82893
$611$	4.34995	0.175980
$612$	0	0
$613$	−27.0102	−1.09093	−0.545466	−	0.838133i	$-0.683647\pi$
$613$	−27.0102	−1.09093	−0.545466	+	0.838133i	$0.683647\pi$
$614$	59.5348	2.40263
$615$	0	0
$616$	−167.312	−6.74119
$617$	43.4026	1.74732	0.873662	−	0.486533i	$-0.161739\pi$
$617$	43.4026	1.74732	0.873662	+	0.486533i	$0.161739\pi$
$618$	0	0
$619$	−10.2805	−0.413209	−0.206605	−	0.978425i	$-0.566241\pi$
$619$	−10.2805	−0.413209	−0.206605	+	0.978425i	$0.566241\pi$
$620$	−93.9479	−3.77304
$621$	0	0
$622$	−0.0929509	−0.00372699
$623$	69.4988	2.78441
$624$	0	0
$625$	−31.0944	−1.24378
$626$	49.9954	1.99822
$627$	0	0
$628$	87.7673	3.50230
$629$	−16.4944	−0.657676
$630$	0	0
$631$	−28.2438	−1.12437	−0.562185	−	0.827012i	$-0.690039\pi$
$631$	−28.2438	−1.12437	−0.562185	+	0.827012i	$0.690039\pi$
$632$	−34.0975	−1.35633
$633$	0	0
$634$	−49.4920	−1.96558
$635$	19.4532	0.771978
$636$	0	0
$637$	−4.53881	−0.179834
$638$	−37.0843	−1.46818
$639$	0	0
$640$	1.57543	0.0622745
$641$	−19.7227	−0.779000	−0.389500	−	0.921026i	$-0.627352\pi$
$641$	−19.7227	−0.779000	−0.389500	+	0.921026i	$0.627352\pi$
$642$	0	0
$643$	33.4104	1.31758	0.658788	−	0.752328i	$-0.271069\pi$
$643$	33.4104	1.31758	0.658788	+	0.752328i	$0.271069\pi$
$644$	29.0399	1.14433
$645$	0	0
$646$	−26.0557	−1.02515
$647$	−43.6081	−1.71441	−0.857205	−	0.514976i	$-0.827801\pi$
$647$	−43.6081	−1.71441	−0.857205	+	0.514976i	$0.827801\pi$
$648$	0	0
$649$	54.8694	2.15381
$650$	−4.93425	−0.193537
$651$	0	0
$652$	−23.1494	−0.906600
$653$	−34.6094	−1.35437	−0.677186	−	0.735812i	$-0.736801\pi$
$653$	−34.6094	−1.35437	−0.677186	+	0.735812i	$0.736801\pi$
$654$	0	0
$655$	24.8689	0.971708
$656$	83.9445	3.27748
$657$	0	0
$658$	−65.2391	−2.54328
$659$	−40.3435	−1.57156	−0.785779	−	0.618507i	$-0.787738\pi$
$659$	−40.3435	−1.57156	−0.785779	+	0.618507i	$0.787738\pi$
$660$	0	0
$661$	−5.19152	−0.201927	−0.100963	−	0.994890i	$-0.532192\pi$
$661$	−5.19152	−0.201927	−0.100963	+	0.994890i	$0.532192\pi$
$662$	−21.3393	−0.829377
$663$	0	0
$664$	73.2284	2.84181
$665$	−63.9733	−2.48078
$666$	0	0
$667$	3.77781	0.146278
$668$	−26.1360	−1.01123
$669$	0	0
$670$	−61.3934	−2.37183
$671$	57.2657	2.21072
$672$	0	0
$673$	50.4690	1.94544	0.972719	−	0.231987i	$-0.0745226\pi$
$673$	50.4690	1.94544	0.972719	+	0.231987i	$0.0745226\pi$
$674$	33.3152	1.28325
$675$	0	0
$676$	−60.8860	−2.34177
$677$	−20.6464	−0.793506	−0.396753	−	0.917925i	$-0.629863\pi$
$677$	−20.6464	−0.793506	−0.396753	+	0.917925i	$0.629863\pi$
$678$	0	0
$679$	23.8844	0.916599
$680$	34.1444	1.30938
$681$	0	0
$682$	108.808	4.16646
$683$	−40.6049	−1.55370	−0.776852	−	0.629684i	$-0.783184\pi$
$683$	−40.6049	−1.55370	−0.776852	+	0.629684i	$0.783184\pi$
$684$	0	0
$685$	22.7180	0.868011
$686$	−0.349444	−0.0133418
$687$	0	0
$688$	85.7057	3.26750
$689$	6.24124	0.237772
$690$	0	0
$691$	−16.2838	−0.619463	−0.309732	−	0.950824i	$-0.600239\pi$
$691$	−16.2838	−0.619463	−0.309732	+	0.950824i	$0.600239\pi$
$692$	80.0823	3.04427
$693$	0	0
$694$	45.7054	1.73495
$695$	−17.8367	−0.676585
$696$	0	0
$697$	14.0632	0.532681
$698$	45.3358	1.71598
$699$	0	0
$700$	52.3697	1.97939
$701$	46.6948	1.76364	0.881819	−	0.471589i	$-0.156319\pi$
$701$	46.6948	1.76364	0.881819	+	0.471589i	$0.156319\pi$
$702$	0	0
$703$	61.4715	2.31844
$704$	−50.3884	−1.89908
$705$	0	0
$706$	25.1256	0.945616
$707$	−4.40416	−0.165635
$708$	0	0
$709$	−34.6969	−1.30307	−0.651535	−	0.758619i	$-0.725875\pi$
$709$	−34.6969	−1.30307	−0.651535	+	0.758619i	$0.725875\pi$
$710$	68.7921	2.58172
$711$	0	0
$712$	138.240	5.18077
$713$	−11.0843	−0.415112
$714$	0	0
$715$	11.0302	0.412505
$716$	52.4432	1.95989
$717$	0	0
$718$	−42.9287	−1.60208
$719$	−25.9025	−0.966002	−0.483001	−	0.875620i	$-0.660453\pi$
$719$	−25.9025	−0.966002	−0.483001	+	0.875620i	$0.660453\pi$
$720$	0	0
$721$	10.0982	0.376077
$722$	47.4068	1.76430
$723$	0	0
$724$	−23.5959	−0.876934
$725$	6.81280	0.253021
$726$	0	0
$727$	−3.79662	−0.140809	−0.0704044	−	0.997519i	$-0.522429\pi$
$727$	−3.79662	−0.140809	−0.0704044	+	0.997519i	$0.522429\pi$
$728$	−18.1027	−0.670929
$729$	0	0
$730$	59.2632	2.19343
$731$	14.3582	0.531058
$732$	0	0
$733$	43.6187	1.61109	0.805546	−	0.592533i	$-0.201872\pi$
$733$	43.6187	1.61109	0.805546	+	0.592533i	$0.201872\pi$
$734$	49.6021	1.83085
$735$	0	0
$736$	17.1097	0.630672
$737$	50.3187	1.85351
$738$	0	0
$739$	16.0631	0.590889	0.295445	−	0.955360i	$-0.404532\pi$
$739$	16.0631	0.590889	0.295445	+	0.955360i	$0.404532\pi$
$740$	−137.249	−5.04536
$741$	0	0
$742$	−93.6040	−3.43631
$743$	−3.03523	−0.111352	−0.0556759	−	0.998449i	$-0.517731\pi$
$743$	−3.03523	−0.111352	−0.0556759	+	0.998449i	$0.517731\pi$
$744$	0	0
$745$	−34.3375	−1.25803
$746$	−43.4283	−1.59002
$747$	0	0
$748$	−47.6809	−1.74339
$749$	39.0771	1.42785
$750$	0	0
$751$	36.3757	1.32737	0.663684	−	0.748013i	$-0.268992\pi$
$751$	36.3757	1.32737	0.663684	+	0.748013i	$0.268992\pi$
$752$	−65.1353	−2.37524
$753$	0	0
$754$	−4.01241	−0.146124
$755$	−38.7583	−1.41056
$756$	0	0
$757$	5.22971	0.190077	0.0950384	−	0.995474i	$-0.469703\pi$
$757$	5.22971	0.190077	0.0950384	+	0.995474i	$0.469703\pi$
$758$	−2.76660	−0.100488
$759$	0	0
$760$	−127.249	−4.61582
$761$	48.1497	1.74543	0.872713	−	0.488234i	$-0.162359\pi$
$761$	48.1497	1.74543	0.872713	+	0.488234i	$0.162359\pi$
$762$	0	0
$763$	77.4931	2.80544
$764$	−33.5463	−1.21366
$765$	0	0
$766$	64.5726	2.33310
$767$	5.93671	0.214362
$768$	0	0
$769$	19.4642	0.701895	0.350948	−	0.936395i	$-0.385859\pi$
$769$	19.4642	0.701895	0.350948	+	0.936395i	$0.385859\pi$
$770$	−165.427	−5.96157
$771$	0	0
$772$	89.4139	3.21808
$773$	45.5889	1.63972	0.819859	−	0.572566i	$-0.194052\pi$
$773$	45.5889	1.63972	0.819859	+	0.572566i	$0.194052\pi$
$774$	0	0
$775$	−19.9892	−0.718033
$776$	47.5085	1.70546
$777$	0	0
$778$	−41.7778	−1.49781
$779$	−52.4107	−1.87781
$780$	0	0
$781$	−56.3828	−2.01753
$782$	6.86374	0.245447
$783$	0	0
$784$	67.9634	2.42726
$785$	50.9324	1.81785
$786$	0	0
$787$	20.4385	0.728553	0.364277	−	0.931291i	$-0.381316\pi$
$787$	20.4385	0.728553	0.364277	+	0.931291i	$0.381316\pi$
$788$	50.6791	1.80537
$789$	0	0
$790$	−33.7133	−1.19947
$791$	21.5443	0.766027
$792$	0	0
$793$	6.19599	0.220026
$794$	29.4202	1.04408
$795$	0	0
$796$	−9.59447	−0.340067
$797$	2.96900	0.105167	0.0525837	−	0.998617i	$-0.483254\pi$
$797$	2.96900	0.105167	0.0525837	+	0.998617i	$0.483254\pi$
$798$	0	0
$799$	−10.9121	−0.386042
$800$	30.8551	1.09089
$801$	0	0
$802$	−39.4041	−1.39141
$803$	−48.5728	−1.71410
$804$	0	0
$805$	16.8522	0.593961
$806$	11.7727	0.414675
$807$	0	0
$808$	−8.76032	−0.308187
$809$	−41.1587	−1.44706	−0.723531	−	0.690292i	$-0.757482\pi$
$809$	−41.1587	−1.44706	−0.723531	+	0.690292i	$0.757482\pi$
$810$	0	0
$811$	1.67476	0.0588087	0.0294044	−	0.999568i	$-0.490639\pi$
$811$	1.67476	0.0588087	0.0294044	+	0.999568i	$0.490639\pi$
$812$	42.5858	1.49447
$813$	0	0
$814$	158.957	5.57145
$815$	−13.4339	−0.470567
$816$	0	0
$817$	−53.5103	−1.87209
$818$	−12.2127	−0.427008
$819$	0	0
$820$	117.018	4.08646
$821$	3.51494	0.122672	0.0613362	−	0.998117i	$-0.480464\pi$
$821$	3.51494	0.122672	0.0613362	+	0.998117i	$0.480464\pi$
$822$	0	0
$823$	−28.6281	−0.997914	−0.498957	−	0.866627i	$-0.666284\pi$
$823$	−28.6281	−0.997914	−0.498957	+	0.866627i	$0.666284\pi$
$824$	20.0864	0.699741
$825$	0	0
$826$	−89.0369	−3.09799
$827$	−29.3296	−1.01989	−0.509945	−	0.860207i	$-0.670334\pi$
$827$	−29.3296	−1.01989	−0.509945	+	0.860207i	$0.670334\pi$
$828$	0	0
$829$	18.4506	0.640817	0.320408	−	0.947280i	$-0.396180\pi$
$829$	18.4506	0.640817	0.320408	+	0.947280i	$0.396180\pi$
$830$	72.4033	2.51315
$831$	0	0
$832$	−5.45188	−0.189010
$833$	11.3859	0.394497
$834$	0	0
$835$	−15.1670	−0.524876
$836$	177.697	6.14579
$837$	0	0
$838$	−31.3268	−1.08217
$839$	13.8785	0.479140	0.239570	−	0.970879i	$-0.422994\pi$
$839$	13.8785	0.479140	0.239570	+	0.970879i	$0.422994\pi$
$840$	0	0
$841$	−23.4600	−0.808965
$842$	13.4351	0.463002
$843$	0	0
$844$	−62.3361	−2.14570
$845$	−35.3328	−1.21549
$846$	0	0
$847$	94.4798	3.24636
$848$	−93.4552	−3.20927
$849$	0	0
$850$	12.3779	0.424557
$851$	−16.1931	−0.555094
$852$	0	0
$853$	−39.3799	−1.34834	−0.674171	−	0.738575i	$-0.735499\pi$
$853$	−39.3799	−1.34834	−0.674171	+	0.738575i	$0.735499\pi$
$854$	−92.9254	−3.17984
$855$	0	0
$856$	77.7284	2.65670
$857$	10.7445	0.367026	0.183513	−	0.983017i	$-0.441253\pi$
$857$	10.7445	0.367026	0.183513	+	0.983017i	$0.441253\pi$
$858$	0	0
$859$	−40.7933	−1.39185	−0.695925	−	0.718114i	$-0.745005\pi$
$859$	−40.7933	−1.39185	−0.695925	+	0.718114i	$0.745005\pi$
$860$	119.473	4.07401
$861$	0	0
$862$	85.6900	2.91861
$863$	−14.6436	−0.498475	−0.249238	−	0.968442i	$-0.580180\pi$
$863$	−14.6436	−0.498475	−0.249238	+	0.968442i	$0.580180\pi$
$864$	0	0
$865$	46.4726	1.58012
$866$	−86.1922	−2.92893
$867$	0	0
$868$	−124.949	−4.24106
$869$	27.6318	0.937345
$870$	0	0
$871$	5.44433	0.184474
$872$	154.142	5.21990
$873$	0	0
$874$	−25.5798	−0.865249
$875$	−22.1070	−0.747353
$876$	0	0
$877$	2.46280	0.0831628	0.0415814	−	0.999135i	$-0.486760\pi$
$877$	2.46280	0.0831628	0.0415814	+	0.999135i	$0.486760\pi$
$878$	36.9812	1.24806
$879$	0	0
$880$	−165.164	−5.56767
$881$	−18.9071	−0.636996	−0.318498	−	0.947924i	$-0.603178\pi$
$881$	−18.9071	−0.636996	−0.318498	+	0.947924i	$0.603178\pi$
$882$	0	0
$883$	25.7316	0.865938	0.432969	−	0.901409i	$-0.357466\pi$
$883$	25.7316	0.865938	0.432969	+	0.901409i	$0.357466\pi$
$884$	−5.15894	−0.173514
$885$	0	0
$886$	−11.9060	−0.399990
$887$	−9.26612	−0.311126	−0.155563	−	0.987826i	$-0.549719\pi$
$887$	−9.26612	−0.311126	−0.155563	+	0.987826i	$0.549719\pi$
$888$	0	0
$889$	25.8725	0.867737
$890$	136.683	4.58161
$891$	0	0
$892$	−37.7382	−1.26357
$893$	40.6672	1.36088
$894$	0	0
$895$	30.4334	1.01728
$896$	2.09531	0.0699993
$897$	0	0
$898$	−50.7669	−1.69411
$899$	−16.2547	−0.542125
$900$	0	0
$901$	−15.6565	−0.521594
$902$	−135.527	−4.51257
$903$	0	0
$904$	42.8538	1.42530
$905$	−13.6930	−0.455170
$906$	0	0
$907$	−1.45986	−0.0484740	−0.0242370	−	0.999706i	$-0.507716\pi$
$907$	−1.45986	−0.0484740	−0.0242370	+	0.999706i	$0.507716\pi$
$908$	33.0811	1.09783
$909$	0	0
$910$	−17.8987	−0.593336
$911$	−40.8089	−1.35206	−0.676030	−	0.736874i	$-0.736301\pi$
$911$	−40.8089	−1.35206	−0.676030	+	0.736874i	$0.736301\pi$
$912$	0	0
$913$	−59.3425	−1.96395
$914$	60.8745	2.01355
$915$	0	0
$916$	−81.0317	−2.67736
$917$	33.0753	1.09224
$918$	0	0
$919$	−10.1810	−0.335839	−0.167919	−	0.985801i	$-0.553705\pi$
$919$	−10.1810	−0.335839	−0.167919	+	0.985801i	$0.553705\pi$
$920$	33.5207	1.10515
$921$	0	0
$922$	−54.2827	−1.78770
$923$	−6.10045	−0.200799
$924$	0	0
$925$	−29.2022	−0.960163
$926$	−54.9699	−1.80642
$927$	0	0
$928$	25.0907	0.823641
$929$	6.89974	0.226373	0.113187	−	0.993574i	$-0.463894\pi$
$929$	6.89974	0.226373	0.113187	+	0.993574i	$0.463894\pi$
$930$	0	0
$931$	−42.4329	−1.39068
$932$	−0.0540135	−0.00176927
$933$	0	0
$934$	−37.2892	−1.22014
$935$	−27.6698	−0.904899
$936$	0	0
$937$	36.1115	1.17971	0.589856	−	0.807509i	$-0.299185\pi$
$937$	36.1115	1.17971	0.589856	+	0.807509i	$0.299185\pi$
$938$	−81.6524	−2.66604
$939$	0	0
$940$	−90.7984	−2.96152
$941$	37.6486	1.22731	0.613655	−	0.789574i	$-0.289698\pi$
$941$	37.6486	1.22731	0.613655	+	0.789574i	$0.289698\pi$
$942$	0	0
$943$	13.8063	0.449595
$944$	−88.8953	−2.89330
$945$	0	0
$946$	−138.371	−4.49882
$947$	−7.65789	−0.248848	−0.124424	−	0.992229i	$-0.539708\pi$
$947$	−7.65789	−0.248848	−0.124424	+	0.992229i	$0.539708\pi$
$948$	0	0
$949$	−5.25543	−0.170599
$950$	−46.1298	−1.49665
$951$	0	0
$952$	45.4116	1.47180
$953$	−3.61700	−0.117166	−0.0585830	−	0.998283i	$-0.518658\pi$
$953$	−3.61700	−0.117166	−0.0585830	+	0.998283i	$0.518658\pi$
$954$	0	0
$955$	−19.4673	−0.629946
$956$	4.84173	0.156593
$957$	0	0
$958$	−2.06826	−0.0668225
$959$	30.2146	0.975682
$960$	0	0
$961$	16.6924	0.538464
$962$	17.1987	0.554509
$963$	0	0
$964$	−146.863	−4.73015
$965$	51.8879	1.67033
$966$	0	0
$967$	40.7102	1.30915	0.654576	−	0.755996i	$-0.272847\pi$
$967$	40.7102	1.30915	0.654576	+	0.755996i	$0.272847\pi$
$968$	187.930	6.04030
$969$	0	0
$970$	46.9732	1.50822
$971$	26.5443	0.851848	0.425924	−	0.904759i	$-0.359949\pi$
$971$	26.5443	0.851848	0.425924	+	0.904759i	$0.359949\pi$
$972$	0	0
$973$	−23.7226	−0.760511
$974$	−74.7146	−2.39401
$975$	0	0
$976$	−92.7777	−2.96974
$977$	17.4506	0.558295	0.279147	−	0.960248i	$-0.409948\pi$
$977$	17.4506	0.558295	0.279147	+	0.960248i	$0.409948\pi$
$978$	0	0
$979$	−112.026	−3.58038
$980$	94.7407	3.02638
$981$	0	0
$982$	−107.814	−3.44047
$983$	46.7209	1.49016	0.745082	−	0.666973i	$-0.232410\pi$
$983$	46.7209	1.49016	0.745082	+	0.666973i	$0.232410\pi$
$984$	0	0
$985$	29.4097	0.937070
$986$	10.0654	0.320547
$987$	0	0
$988$	19.2263	0.611672
$989$	14.0960	0.448225
$990$	0	0
$991$	−55.4287	−1.76075	−0.880375	−	0.474279i	$-0.842709\pi$
$991$	−55.4287	−1.76075	−0.880375	+	0.474279i	$0.842709\pi$
$992$	−73.6176	−2.33736
$993$	0	0
$994$	91.4926	2.90197
$995$	−5.56778	−0.176511
$996$	0	0
$997$	37.5805	1.19019	0.595093	−	0.803657i	$-0.297115\pi$
$997$	37.5805	1.19019	0.595093	+	0.803657i	$0.297115\pi$
$998$	−10.8167	−0.342397
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.2.a.k.1.19	yes	20
3.2	odd	2		2151.2.a.j.1.2	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.2.a.j.1.2	✓	20	3.2	odd	2
2151.2.a.k.1.19	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)

\(f(q)\)	\(=\)	\(q+2.61567 q^{2} +4.84173 q^{4} +2.80971 q^{5} +3.73688 q^{7} +7.43303 q^{8} +O(q^{10})\)	\(q+2.61567 q^{2} +4.84173 q^{4} +2.80971 q^{5} +3.73688 q^{7} +7.43303 q^{8} +7.34928 q^{10} -6.02355 q^{11} -0.651730 q^{13} +9.77444 q^{14} +9.75890 q^{16} +1.63490 q^{17} -6.09296 q^{19} +13.6039 q^{20} -15.7556 q^{22} +1.60504 q^{23} +2.89448 q^{25} -1.70471 q^{26} +18.0930 q^{28} +2.35372 q^{29} -6.90597 q^{31} +10.6600 q^{32} +4.27637 q^{34} +10.4995 q^{35} -10.0889 q^{37} -15.9372 q^{38} +20.8847 q^{40} +8.60185 q^{41} +8.78231 q^{43} -29.1644 q^{44} +4.19825 q^{46} -6.67446 q^{47} +6.96425 q^{49} +7.57100 q^{50} -3.15550 q^{52} -9.57641 q^{53} -16.9244 q^{55} +27.7763 q^{56} +6.15656 q^{58} -9.10915 q^{59} -9.50698 q^{61} -18.0637 q^{62} +8.36524 q^{64} -1.83117 q^{65} -8.35366 q^{67} +7.91576 q^{68} +27.4634 q^{70} +9.36039 q^{71} +8.06381 q^{73} -26.3893 q^{74} -29.5005 q^{76} -22.5092 q^{77} -4.58730 q^{79} +27.4197 q^{80} +22.4996 q^{82} +9.85175 q^{83} +4.59361 q^{85} +22.9716 q^{86} -44.7732 q^{88} +18.5981 q^{89} -2.43544 q^{91} +7.77117 q^{92} -17.4582 q^{94} -17.1195 q^{95} +6.39154 q^{97} +18.2162 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 4 q^{2} + 18 q^{4} + 16 q^{5} - 4 q^{7} + 12 q^{8}+O(q^{10}) \) 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

Fields

Representations

Groups

Database

Properties

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