LMFDB - Embedded newform 2169.2.a.h.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2169,2,Mod(1,2169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2169, 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("2169.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2169 = 3^{2} \cdot 241 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2169.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:	$17.3195521984$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 14 x^{10} + 44 x^{9} + 65 x^{8} - 219 x^{7} - 123 x^{6} + 444 x^{5} + 105 x^{4} + \cdots - 1 $ x^12 - 3x^11 - 14x^10 + 44x^9 + 65x^8 - 219x^7 - 123x^6 + 444x^5 + 105x^4 - 328x^3 - 45x^2 + 18*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 241)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$-2.59703$ of defining polynomial
Character	$\chi$	$=$	2169.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.59703 q^{2} +4.74454 q^{4} -3.49051 q^{5} -0.744578 q^{7} +7.12764 q^{8} +O(q^{10})$	\(q+2.59703 q^{2} +4.74454 q^{4} -3.49051 q^{5} -0.744578 q^{7} +7.12764 q^{8} -9.06493 q^{10} -6.28793 q^{11} -4.06994 q^{13} -1.93369 q^{14} +9.02158 q^{16} +1.60034 q^{17} +2.14421 q^{19} -16.5608 q^{20} -16.3299 q^{22} -9.25572 q^{23} +7.18363 q^{25} -10.5697 q^{26} -3.53268 q^{28} -3.13090 q^{29} -3.15223 q^{31} +9.17399 q^{32} +4.15612 q^{34} +2.59895 q^{35} +4.19928 q^{37} +5.56857 q^{38} -24.8791 q^{40} +4.52694 q^{41} +6.99535 q^{43} -29.8333 q^{44} -24.0373 q^{46} +4.82023 q^{47} -6.44560 q^{49} +18.6561 q^{50} -19.3100 q^{52} -3.71179 q^{53} +21.9481 q^{55} -5.30708 q^{56} -8.13103 q^{58} -3.34128 q^{59} +10.8630 q^{61} -8.18643 q^{62} +5.78192 q^{64} +14.2062 q^{65} +5.80682 q^{67} +7.59287 q^{68} +6.74955 q^{70} -15.7805 q^{71} -1.64682 q^{73} +10.9056 q^{74} +10.1733 q^{76} +4.68186 q^{77} -11.4813 q^{79} -31.4899 q^{80} +11.7566 q^{82} +4.74454 q^{83} -5.58599 q^{85} +18.1671 q^{86} -44.8181 q^{88} +7.95332 q^{89} +3.03039 q^{91} -43.9141 q^{92} +12.5182 q^{94} -7.48438 q^{95} -5.49759 q^{97} -16.7394 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 3 q^{2} + 13 q^{4} - 6 q^{5} + 3 q^{7} - 9 q^{8}+O(q^{10}) $ 12 * q - 3 * q^2 + 13 * q^4 - 6 * q^5 + 3 * q^7 - 9 * q^8	$ 12 q - 3 q^{2} + 13 q^{4} - 6 q^{5} + 3 q^{7} - 9 q^{8} - 7 q^{10} - 22 q^{11} - 5 q^{13} - 6 q^{14} + 15 q^{16} + 4 q^{17} - 6 q^{19} - 10 q^{20} - 12 q^{22} - 32 q^{23} + 4 q^{25} - 8 q^{26} - 11 q^{28} - 6 q^{29} + 8 q^{31} - q^{32} - 19 q^{34} - 15 q^{35} - 8 q^{37} + 10 q^{38} - 52 q^{40} + q^{41} - 2 q^{43} - 42 q^{44} - 25 q^{46} - 34 q^{47} - 9 q^{49} + 27 q^{50} - 41 q^{52} - 5 q^{53} - 3 q^{55} - q^{56} - 33 q^{58} - 26 q^{59} - 26 q^{61} + 17 q^{62} + 13 q^{64} + 25 q^{65} + 6 q^{67} + 35 q^{68} - 4 q^{70} - 94 q^{71} - 22 q^{73} - 26 q^{74} - 20 q^{76} + 7 q^{77} + 9 q^{79} - 19 q^{80} + 15 q^{82} + 8 q^{83} + 4 q^{85} - 9 q^{86} + 6 q^{88} + 3 q^{89} - 20 q^{91} - 36 q^{92} + 48 q^{94} - 33 q^{95} - 29 q^{97} - 28 q^{98}+O(q^{100}) $ 12 * q - 3 * q^2 + 13 * q^4 - 6 * q^5 + 3 * q^7 - 9 * q^8 - 7 * q^10 - 22 * q^11 - 5 * q^13 - 6 * q^14 + 15 * q^16 + 4 * q^17 - 6 * q^19 - 10 * q^20 - 12 * q^22 - 32 * q^23 + 4 * q^25 - 8 * q^26 - 11 * q^28 - 6 * q^29 + 8 * q^31 - q^32 - 19 * q^34 - 15 * q^35 - 8 * q^37 + 10 * q^38 - 52 * q^40 + q^41 - 2 * q^43 - 42 * q^44 - 25 * q^46 - 34 * q^47 - 9 * q^49 + 27 * q^50 - 41 * q^52 - 5 * q^53 - 3 * q^55 - q^56 - 33 * q^58 - 26 * q^59 - 26 * q^61 + 17 * q^62 + 13 * q^64 + 25 * q^65 + 6 * q^67 + 35 * q^68 - 4 * q^70 - 94 * q^71 - 22 * q^73 - 26 * q^74 - 20 * q^76 + 7 * q^77 + 9 * q^79 - 19 * q^80 + 15 * q^82 + 8 * q^83 + 4 * q^85 - 9 * q^86 + 6 * q^88 + 3 * q^89 - 20 * q^91 - 36 * q^92 + 48 * q^94 - 33 * q^95 - 29 * q^97 - 28 * 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 $

$2$	2.59703	1.83637	0.918187	−	0.396147i	$-0.129653\pi$
$2$	2.59703	1.83637	0.918187	+	0.396147i	$0.129653\pi$
$3$	0	0
$3$	0	0
$4$	4.74454	2.37227
$5$	−3.49051	−1.56100	−0.780501	−	0.625155i	$-0.785036\pi$
$5$	−3.49051	−1.56100	−0.780501	+	0.625155i	$0.785036\pi$
$6$	0	0
$7$	−0.744578	−0.281424	−0.140712	−	0.990051i	$-0.544939\pi$
$7$	−0.744578	−0.281424	−0.140712	+	0.990051i	$0.544939\pi$
$8$	7.12764	2.52000
$9$	0	0
$10$	−9.06493	−2.86658
$11$	−6.28793	−1.89588	−0.947941	−	0.318445i	$-0.896839\pi$
$11$	−6.28793	−1.89588	−0.947941	+	0.318445i	$0.896839\pi$
$12$	0	0
$13$	−4.06994	−1.12880	−0.564399	−	0.825502i	$-0.690892\pi$
$13$	−4.06994	−1.12880	−0.564399	+	0.825502i	$0.690892\pi$
$14$	−1.93369	−0.516800
$15$	0	0
$16$	9.02158	2.25539
$17$	1.60034	0.388139	0.194070	−	0.980988i	$-0.437831\pi$
$17$	1.60034	0.388139	0.194070	+	0.980988i	$0.437831\pi$
$18$	0	0
$19$	2.14421	0.491916	0.245958	−	0.969280i	$-0.420897\pi$
$19$	2.14421	0.491916	0.245958	+	0.969280i	$0.420897\pi$
$20$	−16.5608	−3.70312
$21$	0	0
$22$	−16.3299	−3.48155
$23$	−9.25572	−1.92995	−0.964975	−	0.262340i	$-0.915506\pi$
$23$	−9.25572	−1.92995	−0.964975	+	0.262340i	$0.915506\pi$
$24$	0	0
$25$	7.18363	1.43673
$26$	−10.5697	−2.07290
$27$	0	0
$28$	−3.53268	−0.667614
$29$	−3.13090	−0.581394	−0.290697	−	0.956815i	$-0.593887\pi$
$29$	−3.13090	−0.581394	−0.290697	+	0.956815i	$0.593887\pi$
$30$	0	0
$31$	−3.15223	−0.566158	−0.283079	−	0.959097i	$-0.591356\pi$
$31$	−3.15223	−0.566158	−0.283079	+	0.959097i	$0.591356\pi$
$32$	9.17399	1.62175
$33$	0	0
$34$	4.15612	0.712769
$35$	2.59895	0.439303
$36$	0	0
$37$	4.19928	0.690358	0.345179	−	0.938537i	$-0.387818\pi$
$37$	4.19928	0.690358	0.345179	+	0.938537i	$0.387818\pi$
$38$	5.56857	0.903342
$39$	0	0
$40$	−24.8791	−3.93372
$41$	4.52694	0.706990	0.353495	−	0.935436i	$-0.384993\pi$
$41$	4.52694	0.706990	0.353495	+	0.935436i	$0.384993\pi$
$42$	0	0
$43$	6.99535	1.06678	0.533391	−	0.845869i	$-0.320918\pi$
$43$	6.99535	1.06678	0.533391	+	0.845869i	$0.320918\pi$
$44$	−29.8333	−4.49754
$45$	0	0
$46$	−24.0373	−3.54411
$47$	4.82023	0.703102	0.351551	−	0.936169i	$-0.385654\pi$
$47$	4.82023	0.703102	0.351551	+	0.936169i	$0.385654\pi$
$48$	0	0
$49$	−6.44560	−0.920800
$50$	18.6561	2.63837
$51$	0	0
$52$	−19.3100	−2.67782
$53$	−3.71179	−0.509853	−0.254927	−	0.966960i	$-0.582051\pi$
$53$	−3.71179	−0.509853	−0.254927	+	0.966960i	$0.582051\pi$
$54$	0	0
$55$	21.9481	2.95948
$56$	−5.30708	−0.709189
$57$	0	0
$58$	−8.13103	−1.06766
$59$	−3.34128	−0.434997	−0.217499	−	0.976061i	$-0.569790\pi$
$59$	−3.34128	−0.434997	−0.217499	+	0.976061i	$0.569790\pi$
$60$	0	0
$61$	10.8630	1.39087	0.695434	−	0.718590i	$-0.255212\pi$
$61$	10.8630	1.39087	0.695434	+	0.718590i	$0.255212\pi$
$62$	−8.18643	−1.03968
$63$	0	0
$64$	5.78192	0.722740
$65$	14.2062	1.76206
$66$	0	0
$67$	5.80682	0.709416	0.354708	−	0.934977i	$-0.384580\pi$
$67$	5.80682	0.709416	0.354708	+	0.934977i	$0.384580\pi$
$68$	7.59287	0.920771
$69$	0	0
$70$	6.74955	0.806725
$71$	−15.7805	−1.87281	−0.936403	−	0.350927i	$-0.885866\pi$
$71$	−15.7805	−1.87281	−0.936403	+	0.350927i	$0.885866\pi$
$72$	0	0
$73$	−1.64682	−0.192745	−0.0963727	−	0.995345i	$-0.530724\pi$
$73$	−1.64682	−0.192745	−0.0963727	+	0.995345i	$0.530724\pi$
$74$	10.9056	1.26775
$75$	0	0
$76$	10.1733	1.16696
$77$	4.68186	0.533547
$78$	0	0
$79$	−11.4813	−1.29174	−0.645872	−	0.763446i	$-0.723506\pi$
$79$	−11.4813	−1.29174	−0.645872	+	0.763446i	$0.723506\pi$
$80$	−31.4899	−3.52067
$81$	0	0
$82$	11.7566	1.29830
$83$	4.74454	0.520781	0.260390	−	0.965503i	$-0.416149\pi$
$83$	4.74454	0.520781	0.260390	+	0.965503i	$0.416149\pi$
$84$	0	0
$85$	−5.58599	−0.605886
$86$	18.1671	1.95901
$87$	0	0
$88$	−44.8181	−4.77762
$89$	7.95332	0.843050	0.421525	−	0.906817i	$-0.361495\pi$
$89$	7.95332	0.843050	0.421525	+	0.906817i	$0.361495\pi$
$90$	0	0
$91$	3.03039	0.317671
$92$	−43.9141	−4.57836
$93$	0	0
$94$	12.5182	1.29116
$95$	−7.48438	−0.767882
$96$	0	0
$97$	−5.49759	−0.558195	−0.279098	−	0.960263i	$-0.590035\pi$
$97$	−5.49759	−0.558195	−0.279098	+	0.960263i	$0.590035\pi$
$98$	−16.7394	−1.69093
$99$	0	0
$100$	34.0830	3.40830
$101$	10.4008	1.03492	0.517459	−	0.855708i	$-0.326878\pi$
$101$	10.4008	1.03492	0.517459	+	0.855708i	$0.326878\pi$
$102$	0	0
$103$	−16.9464	−1.66978	−0.834888	−	0.550420i	$-0.814468\pi$
$103$	−16.9464	−1.66978	−0.834888	+	0.550420i	$0.814468\pi$
$104$	−29.0091	−2.84457
$105$	0	0
$106$	−9.63961	−0.936281
$107$	0.873633	0.0844573	0.0422287	−	0.999108i	$-0.486554\pi$
$107$	0.873633	0.0844573	0.0422287	+	0.999108i	$0.486554\pi$
$108$	0	0
$109$	−9.90349	−0.948582	−0.474291	−	0.880368i	$-0.657296\pi$
$109$	−9.90349	−0.948582	−0.474291	+	0.880368i	$0.657296\pi$
$110$	56.9996	5.43470
$111$	0	0
$112$	−6.71727	−0.634722
$113$	−5.23192	−0.492177	−0.246089	−	0.969247i	$-0.579145\pi$
$113$	−5.23192	−0.492177	−0.246089	+	0.969247i	$0.579145\pi$
$114$	0	0
$115$	32.3071	3.01266
$116$	−14.8547	−1.37922
$117$	0	0
$118$	−8.67738	−0.798818
$119$	−1.19158	−0.109232
$120$	0	0
$121$	28.5381	2.59437
$122$	28.2115	2.55415
$123$	0	0
$124$	−14.9559	−1.34308
$125$	−7.62196	−0.681729
$126$	0	0
$127$	−3.50226	−0.310775	−0.155388	−	0.987854i	$-0.549663\pi$
$127$	−3.50226	−0.310775	−0.155388	+	0.987854i	$0.549663\pi$
$128$	−3.33219	−0.294527
$129$	0	0
$130$	36.8937	3.23579
$131$	13.2883	1.16100	0.580500	−	0.814260i	$-0.302857\pi$
$131$	13.2883	1.16100	0.580500	+	0.814260i	$0.302857\pi$
$132$	0	0
$133$	−1.59653	−0.138437
$134$	15.0805	1.30275
$135$	0	0
$136$	11.4066	0.978111
$137$	9.37803	0.801219	0.400610	−	0.916249i	$-0.368798\pi$
$137$	9.37803	0.801219	0.400610	+	0.916249i	$0.368798\pi$
$138$	0	0
$139$	3.48685	0.295751	0.147875	−	0.989006i	$-0.452757\pi$
$139$	3.48685	0.295751	0.147875	+	0.989006i	$0.452757\pi$
$140$	12.3308	1.04215
$141$	0	0
$142$	−40.9825	−3.43917
$143$	25.5915	2.14007
$144$	0	0
$145$	10.9284	0.907557
$146$	−4.27683	−0.353953
$147$	0	0
$148$	19.9237	1.63771
$149$	8.39873	0.688051	0.344025	−	0.938960i	$-0.388209\pi$
$149$	8.39873	0.688051	0.344025	+	0.938960i	$0.388209\pi$
$150$	0	0
$151$	−6.68837	−0.544292	−0.272146	−	0.962256i	$-0.587733\pi$
$151$	−6.68837	−0.544292	−0.272146	+	0.962256i	$0.587733\pi$
$152$	15.2832	1.23963
$153$	0	0
$154$	12.1589	0.979792
$155$	11.0029	0.883773
$156$	0	0
$157$	−11.9755	−0.955752	−0.477876	−	0.878427i	$-0.658593\pi$
$157$	−11.9755	−0.955752	−0.477876	+	0.878427i	$0.658593\pi$
$158$	−29.8171	−2.37213
$159$	0	0
$160$	−32.0218	−2.53155
$161$	6.89161	0.543135
$162$	0	0
$163$	−13.7742	−1.07888	−0.539438	−	0.842025i	$-0.681363\pi$
$163$	−13.7742	−1.07888	−0.539438	+	0.842025i	$0.681363\pi$
$164$	21.4783	1.67717
$165$	0	0
$166$	12.3217	0.956348
$167$	−7.34847	−0.568642	−0.284321	−	0.958729i	$-0.591768\pi$
$167$	−7.34847	−0.568642	−0.284321	+	0.958729i	$0.591768\pi$
$168$	0	0
$169$	3.56442	0.274187
$170$	−14.5070	−1.11263
$171$	0	0
$172$	33.1897	2.53069
$173$	−14.4542	−1.09893	−0.549467	−	0.835515i	$-0.685169\pi$
$173$	−14.4542	−1.09893	−0.549467	+	0.835515i	$0.685169\pi$
$174$	0	0
$175$	−5.34877	−0.404329
$176$	−56.7270	−4.27596
$177$	0	0
$178$	20.6550	1.54816
$179$	−10.4351	−0.779953	−0.389976	−	0.920825i	$-0.627517\pi$
$179$	−10.4351	−0.779953	−0.389976	+	0.920825i	$0.627517\pi$
$180$	0	0
$181$	25.3704	1.88577	0.942883	−	0.333125i	$-0.108103\pi$
$181$	25.3704	1.88577	0.942883	+	0.333125i	$0.108103\pi$
$182$	7.87000	0.583363
$183$	0	0
$184$	−65.9714	−4.86348
$185$	−14.6576	−1.07765
$186$	0	0
$187$	−10.0628	−0.735867
$188$	22.8698	1.66795
$189$	0	0
$190$	−19.4371	−1.41012
$191$	16.4111	1.18747	0.593733	−	0.804662i	$-0.297654\pi$
$191$	16.4111	1.18747	0.593733	+	0.804662i	$0.297654\pi$
$192$	0	0
$193$	−8.05553	−0.579850	−0.289925	−	0.957049i	$-0.593630\pi$
$193$	−8.05553	−0.579850	−0.289925	+	0.957049i	$0.593630\pi$
$194$	−14.2774	−1.02506
$195$	0	0
$196$	−30.5814	−2.18439
$197$	−1.16210	−0.0827963	−0.0413981	−	0.999143i	$-0.513181\pi$
$197$	−1.16210	−0.0827963	−0.0413981	+	0.999143i	$0.513181\pi$
$198$	0	0
$199$	−15.8123	−1.12090	−0.560451	−	0.828188i	$-0.689372\pi$
$199$	−15.8123	−1.12090	−0.560451	+	0.828188i	$0.689372\pi$
$200$	51.2023	3.62055
$201$	0	0
$202$	27.0112	1.90050
$203$	2.33120	0.163618
$204$	0	0
$205$	−15.8013	−1.10361
$206$	−44.0102	−3.06633
$207$	0	0
$208$	−36.7173	−2.54589
$209$	−13.4827	−0.932615
$210$	0	0
$211$	−6.41779	−0.441819	−0.220910	−	0.975294i	$-0.570903\pi$
$211$	−6.41779	−0.441819	−0.220910	+	0.975294i	$0.570903\pi$
$212$	−17.6107	−1.20951
$213$	0	0
$214$	2.26885	0.155095
$215$	−24.4173	−1.66525
$216$	0	0
$217$	2.34708	0.159330
$218$	−25.7196	−1.74195
$219$	0	0
$220$	104.133	7.02067
$221$	−6.51329	−0.438131
$222$	0	0
$223$	15.6352	1.04701	0.523506	−	0.852022i	$-0.324624\pi$
$223$	15.6352	1.04701	0.523506	+	0.852022i	$0.324624\pi$
$224$	−6.83075	−0.456399
$225$	0	0
$226$	−13.5874	−0.903821
$227$	6.01020	0.398911	0.199456	−	0.979907i	$-0.436083\pi$
$227$	6.01020	0.398911	0.199456	+	0.979907i	$0.436083\pi$
$228$	0	0
$229$	−10.7079	−0.707595	−0.353798	−	0.935322i	$-0.615110\pi$
$229$	−10.7079	−0.707595	−0.353798	+	0.935322i	$0.615110\pi$
$230$	83.9024	5.53236
$231$	0	0
$232$	−22.3159	−1.46511
$233$	−2.77571	−0.181843	−0.0909214	−	0.995858i	$-0.528981\pi$
$233$	−2.77571	−0.181843	−0.0909214	+	0.995858i	$0.528981\pi$
$234$	0	0
$235$	−16.8250	−1.09754
$236$	−15.8528	−1.03193
$237$	0	0
$238$	−3.09456	−0.200590
$239$	−9.74488	−0.630344	−0.315172	−	0.949035i	$-0.602062\pi$
$239$	−9.74488	−0.630344	−0.315172	+	0.949035i	$0.602062\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	74.1141	4.76423
$243$	0	0
$244$	51.5400	3.29951
$245$	22.4984	1.43737
$246$	0	0
$247$	−8.72682	−0.555274
$248$	−22.4680	−1.42672
$249$	0	0
$250$	−19.7944	−1.25191
$251$	−1.47018	−0.0927970	−0.0463985	−	0.998923i	$-0.514774\pi$
$251$	−1.47018	−0.0927970	−0.0463985	+	0.998923i	$0.514774\pi$
$252$	0	0
$253$	58.1993	3.65896
$254$	−9.09545	−0.570699
$255$	0	0
$256$	−20.2176	−1.26360
$257$	−19.0717	−1.18966	−0.594829	−	0.803852i	$-0.702780\pi$
$257$	−19.0717	−1.18966	−0.594829	+	0.803852i	$0.702780\pi$
$258$	0	0
$259$	−3.12669	−0.194283
$260$	67.4017	4.18007
$261$	0	0
$262$	34.5099	2.13203
$263$	11.0150	0.679217	0.339608	−	0.940567i	$-0.389705\pi$
$263$	11.0150	0.679217	0.339608	+	0.940567i	$0.389705\pi$
$264$	0	0
$265$	12.9560	0.795882
$266$	−4.14624	−0.254222
$267$	0	0
$268$	27.5507	1.68293
$269$	9.48636	0.578393	0.289197	−	0.957270i	$-0.406612\pi$
$269$	9.48636	0.578393	0.289197	+	0.957270i	$0.406612\pi$
$270$	0	0
$271$	21.3477	1.29678	0.648390	−	0.761308i	$-0.275443\pi$
$271$	21.3477	1.29678	0.648390	+	0.761308i	$0.275443\pi$
$272$	14.4376	0.875407
$273$	0	0
$274$	24.3550	1.47134
$275$	−45.1701	−2.72386
$276$	0	0
$277$	−18.2788	−1.09827	−0.549135	−	0.835734i	$-0.685043\pi$
$277$	−18.2788	−1.09827	−0.549135	+	0.835734i	$0.685043\pi$
$278$	9.05544	0.543109
$279$	0	0
$280$	18.5244	1.10704
$281$	−11.5147	−0.686909	−0.343454	−	0.939169i	$-0.611597\pi$
$281$	−11.5147	−0.686909	−0.343454	+	0.939169i	$0.611597\pi$
$282$	0	0
$283$	25.1735	1.49641	0.748205	−	0.663468i	$-0.230916\pi$
$283$	25.1735	1.49641	0.748205	+	0.663468i	$0.230916\pi$
$284$	−74.8714	−4.44280
$285$	0	0
$286$	66.4618	3.92997
$287$	−3.37066	−0.198964
$288$	0	0
$289$	−14.4389	−0.849348
$290$	28.3814	1.66661
$291$	0	0
$292$	−7.81339	−0.457244
$293$	0.319785	0.0186820	0.00934102	−	0.999956i	$-0.497027\pi$
$293$	0.319785	0.0186820	0.00934102	+	0.999956i	$0.497027\pi$
$294$	0	0
$295$	11.6627	0.679031
$296$	29.9310	1.73970
$297$	0	0
$298$	21.8117	1.26352
$299$	37.6702	2.17853
$300$	0	0
$301$	−5.20859	−0.300218
$302$	−17.3699	−0.999523
$303$	0	0
$304$	19.3442	1.10946
$305$	−37.9174	−2.17115
$306$	0	0
$307$	−8.24844	−0.470763	−0.235382	−	0.971903i	$-0.575634\pi$
$307$	−8.24844	−0.470763	−0.235382	+	0.971903i	$0.575634\pi$
$308$	22.2132	1.26572
$309$	0	0
$310$	28.5748	1.62294
$311$	11.0098	0.624306	0.312153	−	0.950032i	$-0.398950\pi$
$311$	11.0098	0.624306	0.312153	+	0.950032i	$0.398950\pi$
$312$	0	0
$313$	−10.3244	−0.583570	−0.291785	−	0.956484i	$-0.594249\pi$
$313$	−10.3244	−0.583570	−0.291785	+	0.956484i	$0.594249\pi$
$314$	−31.1008	−1.75512
$315$	0	0
$316$	−54.4733	−3.06437
$317$	29.1691	1.63830	0.819149	−	0.573581i	$-0.194446\pi$
$317$	29.1691	1.63830	0.819149	+	0.573581i	$0.194446\pi$
$318$	0	0
$319$	19.6869	1.10225
$320$	−20.1818	−1.12820
$321$	0	0
$322$	17.8977	0.997398
$323$	3.43147	0.190932
$324$	0	0
$325$	−29.2369	−1.62177
$326$	−35.7719	−1.98122
$327$	0	0
$328$	32.2664	1.78162
$329$	−3.58903	−0.197870
$330$	0	0
$331$	14.3362	0.787988	0.393994	−	0.919113i	$-0.371093\pi$
$331$	14.3362	0.787988	0.393994	+	0.919113i	$0.371093\pi$
$332$	22.5107	1.23543
$333$	0	0
$334$	−19.0842	−1.04424
$335$	−20.2687	−1.10740
$336$	0	0
$337$	5.47393	0.298184	0.149092	−	0.988823i	$-0.452365\pi$
$337$	5.47393	0.298184	0.149092	+	0.988823i	$0.452365\pi$
$338$	9.25690	0.503509
$339$	0	0
$340$	−26.5030	−1.43733
$341$	19.8210	1.07337
$342$	0	0
$343$	10.0113	0.540559
$344$	49.8604	2.68829
$345$	0	0
$346$	−37.5380	−2.01805
$347$	24.3460	1.30696	0.653480	−	0.756944i	$-0.273309\pi$
$347$	24.3460	1.30696	0.653480	+	0.756944i	$0.273309\pi$
$348$	0	0
$349$	−32.0868	−1.71757	−0.858783	−	0.512339i	$-0.828779\pi$
$349$	−32.0868	−1.71757	−0.858783	+	0.512339i	$0.828779\pi$
$350$	−13.8909	−0.742499
$351$	0	0
$352$	−57.6854	−3.07464
$353$	−13.4079	−0.713631	−0.356816	−	0.934175i	$-0.616138\pi$
$353$	−13.4079	−0.713631	−0.356816	+	0.934175i	$0.616138\pi$
$354$	0	0
$355$	55.0821	2.92345
$356$	37.7348	1.99994
$357$	0	0
$358$	−27.1001	−1.43228
$359$	−26.1416	−1.37970	−0.689851	−	0.723951i	$-0.742324\pi$
$359$	−26.1416	−1.37970	−0.689851	+	0.723951i	$0.742324\pi$
$360$	0	0
$361$	−14.4024	−0.758019
$362$	65.8875	3.46297
$363$	0	0
$364$	14.3778	0.753602
$365$	5.74822	0.300876
$366$	0	0
$367$	4.22872	0.220737	0.110369	−	0.993891i	$-0.464797\pi$
$367$	4.22872	0.220737	0.110369	+	0.993891i	$0.464797\pi$
$368$	−83.5012	−4.35280
$369$	0	0
$370$	−38.0662	−1.97897
$371$	2.76372	0.143485
$372$	0	0
$373$	−35.5776	−1.84214	−0.921070	−	0.389398i	$-0.872683\pi$
$373$	−35.5776	−1.84214	−0.921070	+	0.389398i	$0.872683\pi$
$374$	−26.1334	−1.35133
$375$	0	0
$376$	34.3568	1.77182
$377$	12.7426	0.656277
$378$	0	0
$379$	−10.5187	−0.540310	−0.270155	−	0.962817i	$-0.587075\pi$
$379$	−10.5187	−0.540310	−0.270155	+	0.962817i	$0.587075\pi$
$380$	−35.5100	−1.82162
$381$	0	0
$382$	42.6201	2.18063
$383$	−14.8054	−0.756519	−0.378260	−	0.925700i	$-0.623477\pi$
$383$	−14.8054	−0.756519	−0.378260	+	0.925700i	$0.623477\pi$
$384$	0	0
$385$	−16.3420	−0.832867
$386$	−20.9204	−1.06482
$387$	0	0
$388$	−26.0835	−1.32419
$389$	8.90749	0.451628	0.225814	−	0.974170i	$-0.427496\pi$
$389$	8.90749	0.451628	0.225814	+	0.974170i	$0.427496\pi$
$390$	0	0
$391$	−14.8123	−0.749090
$392$	−45.9419	−2.32042
$393$	0	0
$394$	−3.01801	−0.152045
$395$	40.0754	2.01641
$396$	0	0
$397$	32.7082	1.64158	0.820789	−	0.571232i	$-0.193534\pi$
$397$	32.7082	1.64158	0.820789	+	0.571232i	$0.193534\pi$
$398$	−41.0648	−2.05839
$399$	0	0
$400$	64.8076	3.24038
$401$	16.3591	0.816933	0.408467	−	0.912773i	$-0.366064\pi$
$401$	16.3591	0.816933	0.408467	+	0.912773i	$0.366064\pi$
$402$	0	0
$403$	12.8294	0.639078
$404$	49.3470	2.45511
$405$	0	0
$406$	6.05419	0.300464
$407$	−26.4048	−1.30884
$408$	0	0
$409$	−27.1198	−1.34099	−0.670494	−	0.741915i	$-0.733918\pi$
$409$	−27.1198	−1.34099	−0.670494	+	0.741915i	$0.733918\pi$
$410$	−41.0364	−2.02664
$411$	0	0
$412$	−80.4028	−3.96116
$413$	2.48784	0.122419
$414$	0	0
$415$	−16.5608	−0.812939
$416$	−37.3376	−1.83063
$417$	0	0
$418$	−35.0148	−1.71263
$419$	1.69471	0.0827921	0.0413960	−	0.999143i	$-0.486819\pi$
$419$	1.69471	0.0827921	0.0413960	+	0.999143i	$0.486819\pi$
$420$	0	0
$421$	6.27194	0.305676	0.152838	−	0.988251i	$-0.451159\pi$
$421$	6.27194	0.305676	0.152838	+	0.988251i	$0.451159\pi$
$422$	−16.6672	−0.811345
$423$	0	0
$424$	−26.4563	−1.28483
$425$	11.4962	0.557650
$426$	0	0
$427$	−8.08837	−0.391424
$428$	4.14499	0.200356
$429$	0	0
$430$	−63.4124	−3.05802
$431$	−26.2043	−1.26222	−0.631108	−	0.775695i	$-0.717400\pi$
$431$	−26.2043	−1.26222	−0.631108	+	0.775695i	$0.717400\pi$
$432$	0	0
$433$	−1.61920	−0.0778136	−0.0389068	−	0.999243i	$-0.512388\pi$
$433$	−1.61920	−0.0778136	−0.0389068	+	0.999243i	$0.512388\pi$
$434$	6.09543	0.292590
$435$	0	0
$436$	−46.9875	−2.25029
$437$	−19.8462	−0.949374
$438$	0	0
$439$	−35.8451	−1.71080	−0.855398	−	0.517972i	$-0.826687\pi$
$439$	−35.8451	−1.71080	−0.855398	+	0.517972i	$0.826687\pi$
$440$	156.438	7.45788
$441$	0	0
$442$	−16.9152	−0.804573
$443$	−19.6847	−0.935250	−0.467625	−	0.883927i	$-0.654890\pi$
$443$	−19.6847	−0.935250	−0.467625	+	0.883927i	$0.654890\pi$
$444$	0	0
$445$	−27.7611	−1.31600
$446$	40.6051	1.92271
$447$	0	0
$448$	−4.30509	−0.203396
$449$	−23.0445	−1.08754	−0.543769	−	0.839235i	$-0.683003\pi$
$449$	−23.0445	−1.08754	−0.543769	+	0.839235i	$0.683003\pi$
$450$	0	0
$451$	−28.4651	−1.34037
$452$	−24.8230	−1.16758
$453$	0	0
$454$	15.6086	0.732550
$455$	−10.5776	−0.495885
$456$	0	0
$457$	9.75545	0.456341	0.228170	−	0.973621i	$-0.426726\pi$
$457$	9.75545	0.456341	0.228170	+	0.973621i	$0.426726\pi$
$458$	−27.8086	−1.29941
$459$	0	0
$460$	153.282	7.14683
$461$	−29.0597	−1.35345	−0.676724	−	0.736237i	$-0.736601\pi$
$461$	−29.0597	−1.35345	−0.676724	+	0.736237i	$0.736601\pi$
$462$	0	0
$463$	34.1246	1.58591	0.792953	−	0.609282i	$-0.208542\pi$
$463$	34.1246	1.58591	0.792953	+	0.609282i	$0.208542\pi$
$464$	−28.2457	−1.31127
$465$	0	0
$466$	−7.20859	−0.333931
$467$	12.0061	0.555577	0.277789	−	0.960642i	$-0.410399\pi$
$467$	12.0061	0.555577	0.277789	+	0.960642i	$0.410399\pi$
$468$	0	0
$469$	−4.32363	−0.199647
$470$	−43.6950	−2.01550
$471$	0	0
$472$	−23.8154	−1.09619
$473$	−43.9863	−2.02249
$474$	0	0
$475$	15.4032	0.706748
$476$	−5.65349	−0.259127
$477$	0	0
$478$	−25.3077	−1.15755
$479$	−7.47517	−0.341549	−0.170775	−	0.985310i	$-0.554627\pi$
$479$	−7.47517	−0.341549	−0.170775	+	0.985310i	$0.554627\pi$
$480$	0	0
$481$	−17.0908	−0.779275
$482$	2.59703	0.118291
$483$	0	0
$484$	135.400	6.15455
$485$	19.1894	0.871344
$486$	0	0
$487$	21.7985	0.987783	0.493891	−	0.869524i	$-0.335574\pi$
$487$	21.7985	0.987783	0.493891	+	0.869524i	$0.335574\pi$
$488$	77.4277	3.50499
$489$	0	0
$490$	58.4289	2.63955
$491$	12.8909	0.581758	0.290879	−	0.956760i	$-0.406052\pi$
$491$	12.8909	0.581758	0.290879	+	0.956760i	$0.406052\pi$
$492$	0	0
$493$	−5.01051	−0.225662
$494$	−22.6638	−1.01969
$495$	0	0
$496$	−28.4381	−1.27691
$497$	11.7498	0.527053
$498$	0	0
$499$	−24.9744	−1.11801	−0.559004	−	0.829165i	$-0.688816\pi$
$499$	−24.9744	−1.11801	−0.559004	+	0.829165i	$0.688816\pi$
$500$	−36.1627	−1.61725
$501$	0	0
$502$	−3.81810	−0.170410
$503$	−11.1640	−0.497779	−0.248890	−	0.968532i	$-0.580066\pi$
$503$	−11.1640	−0.497779	−0.248890	+	0.968532i	$0.580066\pi$
$504$	0	0
$505$	−36.3041	−1.61551
$506$	151.145	6.71922
$507$	0	0
$508$	−16.6166	−0.737242
$509$	−18.6656	−0.827337	−0.413668	−	0.910428i	$-0.635753\pi$
$509$	−18.6656	−0.827337	−0.413668	+	0.910428i	$0.635753\pi$
$510$	0	0
$511$	1.22618	0.0542432
$512$	−45.8413	−2.02592
$513$	0	0
$514$	−49.5296	−2.18466
$515$	59.1514	2.60652
$516$	0	0
$517$	−30.3092	−1.33300
$518$	−8.12010	−0.356777
$519$	0	0
$520$	101.256	4.44038
$521$	37.9249	1.66152	0.830760	−	0.556631i	$-0.187906\pi$
$521$	37.9249	1.66152	0.830760	+	0.556631i	$0.187906\pi$
$522$	0	0
$523$	23.8400	1.04245	0.521225	−	0.853419i	$-0.325475\pi$
$523$	23.8400	1.04245	0.521225	+	0.853419i	$0.325475\pi$
$524$	63.0467	2.75421
$525$	0	0
$526$	28.6063	1.24730
$527$	−5.04464	−0.219748
$528$	0	0
$529$	62.6683	2.72471
$530$	33.6471	1.46154
$531$	0	0
$532$	−7.57482	−0.328410
$533$	−18.4244	−0.798049
$534$	0	0
$535$	−3.04942	−0.131838
$536$	41.3889	1.78773
$537$	0	0
$538$	24.6363	1.06215
$539$	40.5295	1.74573
$540$	0	0
$541$	37.7424	1.62267	0.811337	−	0.584579i	$-0.198740\pi$
$541$	37.7424	1.62267	0.811337	+	0.584579i	$0.198740\pi$
$542$	55.4405	2.38137
$543$	0	0
$544$	14.6815	0.629464
$545$	34.5682	1.48074
$546$	0	0
$547$	−22.1817	−0.948423	−0.474211	−	0.880411i	$-0.657267\pi$
$547$	−22.1817	−0.948423	−0.474211	+	0.880411i	$0.657267\pi$
$548$	44.4944	1.90071
$549$	0	0
$550$	−117.308	−5.00203
$551$	−6.71332	−0.285997
$552$	0	0
$553$	8.54870	0.363528
$554$	−47.4706	−2.01683
$555$	0	0
$556$	16.5435	0.701601
$557$	−4.36041	−0.184757	−0.0923783	−	0.995724i	$-0.529447\pi$
$557$	−4.36041	−0.184757	−0.0923783	+	0.995724i	$0.529447\pi$
$558$	0	0
$559$	−28.4707	−1.20418
$560$	23.4467	0.990802
$561$	0	0
$562$	−29.9039	−1.26142
$563$	−5.45151	−0.229754	−0.114877	−	0.993380i	$-0.536647\pi$
$563$	−5.45151	−0.229754	−0.114877	+	0.993380i	$0.536647\pi$
$564$	0	0
$565$	18.2620	0.768289
$566$	65.3762	2.74797
$567$	0	0
$568$	−112.478	−4.71947
$569$	17.0647	0.715388	0.357694	−	0.933839i	$-0.383563\pi$
$569$	17.0647	0.715388	0.357694	+	0.933839i	$0.383563\pi$
$570$	0	0
$571$	−12.7567	−0.533853	−0.266926	−	0.963717i	$-0.586008\pi$
$571$	−12.7567	−0.533853	−0.266926	+	0.963717i	$0.586008\pi$
$572$	121.420	5.07682
$573$	0	0
$574$	−8.75370	−0.365372
$575$	−66.4896	−2.77281
$576$	0	0
$577$	−28.2866	−1.17759	−0.588794	−	0.808283i	$-0.700397\pi$
$577$	−28.2866	−1.17759	−0.588794	+	0.808283i	$0.700397\pi$
$578$	−37.4982	−1.55972
$579$	0	0
$580$	51.8504	2.15297
$581$	−3.53268	−0.146560
$582$	0	0
$583$	23.3395	0.966622
$584$	−11.7379	−0.485718
$585$	0	0
$586$	0.830490	0.0343072
$587$	−15.2711	−0.630308	−0.315154	−	0.949041i	$-0.602056\pi$
$587$	−15.2711	−0.630308	−0.315154	+	0.949041i	$0.602056\pi$
$588$	0	0
$589$	−6.75906	−0.278502
$590$	30.2884	1.24696
$591$	0	0
$592$	37.8841	1.55703
$593$	−14.5819	−0.598808	−0.299404	−	0.954126i	$-0.596788\pi$
$593$	−14.5819	−0.598808	−0.299404	+	0.954126i	$0.596788\pi$
$594$	0	0
$595$	4.15921	0.170511
$596$	39.8481	1.63224
$597$	0	0
$598$	97.8305	4.00059
$599$	−24.4860	−1.00047	−0.500235	−	0.865890i	$-0.666753\pi$
$599$	−24.4860	−1.00047	−0.500235	+	0.865890i	$0.666753\pi$
$600$	0	0
$601$	10.0479	0.409863	0.204931	−	0.978776i	$-0.434303\pi$
$601$	10.0479	0.409863	0.204931	+	0.978776i	$0.434303\pi$
$602$	−13.5268	−0.551312
$603$	0	0
$604$	−31.7332	−1.29121
$605$	−99.6123	−4.04982
$606$	0	0
$607$	−5.94024	−0.241107	−0.120553	−	0.992707i	$-0.538467\pi$
$607$	−5.94024	−0.241107	−0.120553	+	0.992707i	$0.538467\pi$
$608$	19.6710	0.797763
$609$	0	0
$610$	−98.4725	−3.98704
$611$	−19.6180	−0.793661
$612$	0	0
$613$	15.6857	0.633539	0.316769	−	0.948503i	$-0.397402\pi$
$613$	15.6857	0.633539	0.316769	+	0.948503i	$0.397402\pi$
$614$	−21.4214	−0.864497
$615$	0	0
$616$	33.3706	1.34454
$617$	6.45067	0.259694	0.129847	−	0.991534i	$-0.458551\pi$
$617$	6.45067	0.259694	0.129847	+	0.991534i	$0.458551\pi$
$618$	0	0
$619$	−3.10687	−0.124876	−0.0624379	−	0.998049i	$-0.519888\pi$
$619$	−3.10687	−0.124876	−0.0624379	+	0.998049i	$0.519888\pi$
$620$	52.2036	2.09655
$621$	0	0
$622$	28.5926	1.14646
$623$	−5.92187	−0.237255
$624$	0	0
$625$	−9.31364	−0.372545
$626$	−26.8127	−1.07165
$627$	0	0
$628$	−56.8184	−2.26730
$629$	6.72028	0.267955
$630$	0	0
$631$	−29.4635	−1.17292	−0.586462	−	0.809976i	$-0.699480\pi$
$631$	−29.4635	−1.17292	−0.586462	+	0.809976i	$0.699480\pi$
$632$	−81.8343	−3.25520
$633$	0	0
$634$	75.7528	3.00853
$635$	12.2246	0.485120
$636$	0	0
$637$	26.2332	1.03940
$638$	51.1274	2.02415
$639$	0	0
$640$	11.6310	0.459756
$641$	48.1471	1.90169	0.950847	−	0.309660i	$-0.100215\pi$
$641$	48.1471	1.90169	0.950847	+	0.309660i	$0.100215\pi$
$642$	0	0
$643$	−44.2183	−1.74380	−0.871899	−	0.489686i	$-0.837112\pi$
$643$	−44.2183	−1.74380	−0.871899	+	0.489686i	$0.837112\pi$
$644$	32.6975	1.28846
$645$	0	0
$646$	8.91161	0.350622
$647$	−45.0000	−1.76913	−0.884566	−	0.466415i	$-0.845546\pi$
$647$	−45.0000	−1.76913	−0.884566	+	0.466415i	$0.845546\pi$
$648$	0	0
$649$	21.0097	0.824704
$650$	−75.9291	−2.97818
$651$	0	0
$652$	−65.3521	−2.55939
$653$	32.3117	1.26445	0.632227	−	0.774784i	$-0.282141\pi$
$653$	32.3117	1.26445	0.632227	+	0.774784i	$0.282141\pi$
$654$	0	0
$655$	−46.3827	−1.81232
$656$	40.8402	1.59454
$657$	0	0
$658$	−9.32081	−0.363363
$659$	−38.4217	−1.49670	−0.748349	−	0.663305i	$-0.769153\pi$
$659$	−38.4217	−1.49670	−0.748349	+	0.663305i	$0.769153\pi$
$660$	0	0
$661$	18.8765	0.734210	0.367105	−	0.930179i	$-0.380349\pi$
$661$	18.8765	0.734210	0.367105	+	0.930179i	$0.380349\pi$
$662$	37.2315	1.44704
$663$	0	0
$664$	33.8174	1.31237
$665$	5.57271	0.216100
$666$	0	0
$667$	28.9788	1.12206
$668$	−34.8651	−1.34897
$669$	0	0
$670$	−52.6384	−2.03360
$671$	−68.3059	−2.63692
$672$	0	0
$673$	−4.88197	−0.188186	−0.0940931	−	0.995563i	$-0.529995\pi$
$673$	−4.88197	−0.188186	−0.0940931	+	0.995563i	$0.529995\pi$
$674$	14.2159	0.547577
$675$	0	0
$676$	16.9116	0.650444
$677$	14.1390	0.543405	0.271703	−	0.962381i	$-0.412413\pi$
$677$	14.1390	0.543405	0.271703	+	0.962381i	$0.412413\pi$
$678$	0	0
$679$	4.09338	0.157090
$680$	−39.8149	−1.52683
$681$	0	0
$682$	51.4757	1.97111
$683$	18.3055	0.700440	0.350220	−	0.936668i	$-0.386107\pi$
$683$	18.3055	0.700440	0.350220	+	0.936668i	$0.386107\pi$
$684$	0	0
$685$	−32.7341	−1.25070
$686$	25.9996	0.992669
$687$	0	0
$688$	63.1091	2.40601
$689$	15.1068	0.575522
$690$	0	0
$691$	12.6938	0.482895	0.241448	−	0.970414i	$-0.422378\pi$
$691$	12.6938	0.482895	0.241448	+	0.970414i	$0.422378\pi$
$692$	−68.5786	−2.60697
$693$	0	0
$694$	63.2271	2.40007
$695$	−12.1709	−0.461668
$696$	0	0
$697$	7.24465	0.274411
$698$	−83.3302	−3.15409
$699$	0	0
$700$	−25.3775	−0.959178
$701$	−13.7325	−0.518670	−0.259335	−	0.965787i	$-0.583503\pi$
$701$	−13.7325	−0.518670	−0.259335	+	0.965787i	$0.583503\pi$
$702$	0	0
$703$	9.00415	0.339598
$704$	−36.3563	−1.37023
$705$	0	0
$706$	−34.8207	−1.31049
$707$	−7.74421	−0.291251
$708$	0	0
$709$	37.0563	1.39168	0.695840	−	0.718197i	$-0.255032\pi$
$709$	37.0563	1.39168	0.695840	+	0.718197i	$0.255032\pi$
$710$	143.049	5.36855
$711$	0	0
$712$	56.6884	2.12449
$713$	29.1762	1.09266
$714$	0	0
$715$	−89.3273	−3.34065
$716$	−49.5095	−1.85026
$717$	0	0
$718$	−67.8905	−2.53365
$719$	21.7402	0.810771	0.405386	−	0.914146i	$-0.367137\pi$
$719$	21.7402	0.810771	0.405386	+	0.914146i	$0.367137\pi$
$720$	0	0
$721$	12.6179	0.469915
$722$	−37.4033	−1.39201
$723$	0	0
$724$	120.371	4.47354
$725$	−22.4912	−0.835304
$726$	0	0
$727$	22.3166	0.827679	0.413839	−	0.910350i	$-0.364188\pi$
$727$	22.3166	0.827679	0.413839	+	0.910350i	$0.364188\pi$
$728$	21.5995	0.800531
$729$	0	0
$730$	14.9283	0.552520
$731$	11.1949	0.414060
$732$	0	0
$733$	−45.3568	−1.67529	−0.837645	−	0.546215i	$-0.816068\pi$
$733$	−45.3568	−1.67529	−0.837645	+	0.546215i	$0.816068\pi$
$734$	10.9821	0.405356
$735$	0	0
$736$	−84.9118	−3.12989
$737$	−36.5129	−1.34497
$738$	0	0
$739$	−12.2783	−0.451663	−0.225831	−	0.974166i	$-0.572510\pi$
$739$	−12.2783	−0.451663	−0.225831	+	0.974166i	$0.572510\pi$
$740$	−69.5436	−2.55647
$741$	0	0
$742$	7.17744	0.263492
$743$	−23.7224	−0.870291	−0.435146	−	0.900360i	$-0.643303\pi$
$743$	−23.7224	−0.870291	−0.435146	+	0.900360i	$0.643303\pi$
$744$	0	0
$745$	−29.3158	−1.07405
$746$	−92.3960	−3.38286
$747$	0	0
$748$	−47.7435	−1.74567
$749$	−0.650488	−0.0237683
$750$	0	0
$751$	18.8204	0.686765	0.343383	−	0.939196i	$-0.388427\pi$
$751$	18.8204	0.686765	0.343383	+	0.939196i	$0.388427\pi$
$752$	43.4860	1.58577
$753$	0	0
$754$	33.0928	1.20517
$755$	23.3458	0.849640
$756$	0	0
$757$	20.4755	0.744195	0.372098	−	0.928194i	$-0.378639\pi$
$757$	20.4755	0.744195	0.372098	+	0.928194i	$0.378639\pi$
$758$	−27.3173	−0.992210
$759$	0	0
$760$	−53.3460	−1.93506
$761$	−6.54100	−0.237111	−0.118555	−	0.992947i	$-0.537826\pi$
$761$	−6.54100	−0.237111	−0.118555	+	0.992947i	$0.537826\pi$
$762$	0	0
$763$	7.37392	0.266954
$764$	77.8632	2.81699
$765$	0	0
$766$	−38.4499	−1.38925
$767$	13.5988	0.491024
$768$	0	0
$769$	−10.7564	−0.387885	−0.193942	−	0.981013i	$-0.562127\pi$
$769$	−10.7564	−0.387885	−0.193942	+	0.981013i	$0.562127\pi$
$770$	−42.4407	−1.52946
$771$	0	0
$772$	−38.2198	−1.37556
$773$	24.6229	0.885624	0.442812	−	0.896614i	$-0.353981\pi$
$773$	24.6229	0.885624	0.442812	+	0.896614i	$0.353981\pi$
$774$	0	0
$775$	−22.6445	−0.813413
$776$	−39.1848	−1.40665
$777$	0	0
$778$	23.1330	0.829357
$779$	9.70673	0.347780
$780$	0	0
$781$	99.2269	3.55062
$782$	−38.4679	−1.37561
$783$	0	0
$784$	−58.1495	−2.07677
$785$	41.8007	1.49193
$786$	0	0
$787$	−10.1240	−0.360883	−0.180441	−	0.983586i	$-0.557753\pi$
$787$	−10.1240	−0.360883	−0.180441	+	0.983586i	$0.557753\pi$
$788$	−5.51363	−0.196415
$789$	0	0
$790$	104.077	3.70289
$791$	3.89557	0.138511
$792$	0	0
$793$	−44.2119	−1.57001
$794$	84.9440	3.01455
$795$	0	0
$796$	−75.0219	−2.65908
$797$	8.60700	0.304876	0.152438	−	0.988313i	$-0.451288\pi$
$797$	8.60700	0.304876	0.152438	+	0.988313i	$0.451288\pi$
$798$	0	0
$799$	7.71400	0.272902
$800$	65.9025	2.33000
$801$	0	0
$802$	42.4849	1.50020
$803$	10.3551	0.365423
$804$	0	0
$805$	−24.0552	−0.847834
$806$	33.3183	1.17359
$807$	0	0
$808$	74.1332	2.60800
$809$	−23.7331	−0.834411	−0.417205	−	0.908812i	$-0.636990\pi$
$809$	−23.7331	−0.834411	−0.417205	+	0.908812i	$0.636990\pi$
$810$	0	0
$811$	35.1374	1.23384	0.616920	−	0.787026i	$-0.288380\pi$
$811$	35.1374	1.23384	0.616920	+	0.787026i	$0.288380\pi$
$812$	11.0605	0.388147
$813$	0	0
$814$	−68.5739	−2.40351
$815$	48.0788	1.68413
$816$	0	0
$817$	14.9995	0.524767
$818$	−70.4308	−2.46256
$819$	0	0
$820$	−74.9700	−2.61807
$821$	−18.3525	−0.640508	−0.320254	−	0.947332i	$-0.603768\pi$
$821$	−18.3525	−0.640508	−0.320254	+	0.947332i	$0.603768\pi$
$822$	0	0
$823$	−0.411745	−0.0143525	−0.00717626	−	0.999974i	$-0.502284\pi$
$823$	−0.411745	−0.0143525	−0.00717626	+	0.999974i	$0.502284\pi$
$824$	−120.788	−4.20784
$825$	0	0
$826$	6.46099	0.224807
$827$	10.9496	0.380756	0.190378	−	0.981711i	$-0.439029\pi$
$827$	10.9496	0.380756	0.190378	+	0.981711i	$0.439029\pi$
$828$	0	0
$829$	6.76398	0.234923	0.117461	−	0.993077i	$-0.462524\pi$
$829$	6.76398	0.234923	0.117461	+	0.993077i	$0.462524\pi$
$830$	−43.0089	−1.49286
$831$	0	0
$832$	−23.5321	−0.815827
$833$	−10.3152	−0.357399
$834$	0	0
$835$	25.6499	0.887651
$836$	−63.9690	−2.21241
$837$	0	0
$838$	4.40121	0.152037
$839$	−3.28222	−0.113315	−0.0566574	−	0.998394i	$-0.518044\pi$
$839$	−3.28222	−0.113315	−0.0566574	+	0.998394i	$0.518044\pi$
$840$	0	0
$841$	−19.1974	−0.661981
$842$	16.2884	0.561335
$843$	0	0
$844$	−30.4495	−1.04811
$845$	−12.4416	−0.428006
$846$	0	0
$847$	−21.2488	−0.730118
$848$	−33.4862	−1.14992
$849$	0	0
$850$	29.8560	1.02405
$851$	−38.8674	−1.33236
$852$	0	0
$853$	−36.1823	−1.23886	−0.619429	−	0.785053i	$-0.712636\pi$
$853$	−36.1823	−1.23886	−0.619429	+	0.785053i	$0.712636\pi$
$854$	−21.0057	−0.718800
$855$	0	0
$856$	6.22694	0.212832
$857$	25.4700	0.870037	0.435019	−	0.900421i	$-0.356742\pi$
$857$	25.4700	0.870037	0.435019	+	0.900421i	$0.356742\pi$
$858$	0	0
$859$	3.88904	0.132692	0.0663461	−	0.997797i	$-0.478866\pi$
$859$	3.88904	0.132692	0.0663461	+	0.997797i	$0.478866\pi$
$860$	−115.849	−3.95042
$861$	0	0
$862$	−68.0532	−2.31790
$863$	4.43810	0.151075	0.0755373	−	0.997143i	$-0.475933\pi$
$863$	4.43810	0.151075	0.0755373	+	0.997143i	$0.475933\pi$
$864$	0	0
$865$	50.4526	1.71544
$866$	−4.20509	−0.142895
$867$	0	0
$868$	11.1358	0.377975
$869$	72.1934	2.44899
$870$	0	0
$871$	−23.6334	−0.800788
$872$	−70.5885	−2.39043
$873$	0	0
$874$	−51.5411	−1.74341
$875$	5.67515	0.191855
$876$	0	0
$877$	29.5579	0.998101	0.499050	−	0.866573i	$-0.333682\pi$
$877$	29.5579	0.998101	0.499050	+	0.866573i	$0.333682\pi$
$878$	−93.0907	−3.14166
$879$	0	0
$880$	198.006	6.67478
$881$	48.9852	1.65035	0.825177	−	0.564874i	$-0.191075\pi$
$881$	48.9852	1.65035	0.825177	+	0.564874i	$0.191075\pi$
$882$	0	0
$883$	10.8101	0.363790	0.181895	−	0.983318i	$-0.441777\pi$
$883$	10.8101	0.363790	0.181895	+	0.983318i	$0.441777\pi$
$884$	−30.9026	−1.03937
$885$	0	0
$886$	−51.1218	−1.71747
$887$	13.9186	0.467340	0.233670	−	0.972316i	$-0.424926\pi$
$887$	13.9186	0.467340	0.233670	+	0.972316i	$0.424926\pi$
$888$	0	0
$889$	2.60770	0.0874596
$890$	−72.0963	−2.41667
$891$	0	0
$892$	74.1819	2.48379
$893$	10.3356	0.345867
$894$	0	0
$895$	36.4236	1.21751
$896$	2.48107	0.0828869
$897$	0	0
$898$	−59.8472	−1.99713
$899$	9.86933	0.329161
$900$	0	0
$901$	−5.94012	−0.197894
$902$	−73.9246	−2.46142
$903$	0	0
$904$	−37.2912	−1.24029
$905$	−88.5554	−2.94368
$906$	0	0
$907$	−45.3560	−1.50602	−0.753011	−	0.658008i	$-0.771399\pi$
$907$	−45.3560	−1.50602	−0.753011	+	0.658008i	$0.771399\pi$
$908$	28.5156	0.946325
$909$	0	0
$910$	−27.4703	−0.910630
$911$	−3.51366	−0.116413	−0.0582064	−	0.998305i	$-0.518538\pi$
$911$	−3.51366	−0.116413	−0.0582064	+	0.998305i	$0.518538\pi$
$912$	0	0
$913$	−29.8333	−0.987339
$914$	25.3352	0.838013
$915$	0	0
$916$	−50.8039	−1.67861
$917$	−9.89414	−0.326733
$918$	0	0
$919$	22.3344	0.736743	0.368372	−	0.929679i	$-0.379915\pi$
$919$	22.3344	0.736743	0.368372	+	0.929679i	$0.379915\pi$
$920$	230.274	7.59190
$921$	0	0
$922$	−75.4689	−2.48544
$923$	64.2259	2.11402
$924$	0	0
$925$	30.1661	0.991854
$926$	88.6226	2.91232
$927$	0	0
$928$	−28.7229	−0.942874
$929$	−18.1496	−0.595470	−0.297735	−	0.954648i	$-0.596231\pi$
$929$	−18.1496	−0.595470	−0.297735	+	0.954648i	$0.596231\pi$
$930$	0	0
$931$	−13.8207	−0.452956
$932$	−13.1695	−0.431380
$933$	0	0
$934$	31.1802	1.02025
$935$	35.1243	1.14869
$936$	0	0
$937$	50.6507	1.65469	0.827343	−	0.561696i	$-0.189851\pi$
$937$	50.6507	1.65469	0.827343	+	0.561696i	$0.189851\pi$
$938$	−11.2286	−0.366626
$939$	0	0
$940$	−79.8270	−2.60367
$941$	−44.9519	−1.46539	−0.732695	−	0.680558i	$-0.761738\pi$
$941$	−44.9519	−1.46539	−0.732695	+	0.680558i	$0.761738\pi$
$942$	0	0
$943$	−41.9001	−1.36446
$944$	−30.1436	−0.981090
$945$	0	0
$946$	−114.234	−3.71405
$947$	−18.5204	−0.601832	−0.300916	−	0.953651i	$-0.597292\pi$
$947$	−18.5204	−0.601832	−0.300916	+	0.953651i	$0.597292\pi$
$948$	0	0
$949$	6.70245	0.217571
$950$	40.0025	1.29785
$951$	0	0
$952$	−8.49313	−0.275264
$953$	−4.77064	−0.154536	−0.0772681	−	0.997010i	$-0.524620\pi$
$953$	−4.77064	−0.154536	−0.0772681	+	0.997010i	$0.524620\pi$
$954$	0	0
$955$	−57.2831	−1.85364
$956$	−46.2350	−1.49535
$957$	0	0
$958$	−19.4132	−0.627212
$959$	−6.98268	−0.225482
$960$	0	0
$961$	−21.0634	−0.679465
$962$	−44.3853	−1.43104
$963$	0	0
$964$	4.74454	0.152811
$965$	28.1179	0.905147
$966$	0	0
$967$	−42.9853	−1.38231	−0.691157	−	0.722705i	$-0.742898\pi$
$967$	−42.9853	−1.38231	−0.691157	+	0.722705i	$0.742898\pi$
$968$	203.409	6.53781
$969$	0	0
$970$	49.8352	1.60011
$971$	−38.4332	−1.23338	−0.616690	−	0.787206i	$-0.711527\pi$
$971$	−38.4332	−1.23338	−0.616690	+	0.787206i	$0.711527\pi$
$972$	0	0
$973$	−2.59623	−0.0832314
$974$	56.6112	1.81394
$975$	0	0
$976$	98.0016	3.13695
$977$	−11.7197	−0.374948	−0.187474	−	0.982270i	$-0.560030\pi$
$977$	−11.7197	−0.374948	−0.187474	+	0.982270i	$0.560030\pi$
$978$	0	0
$979$	−50.0099	−1.59832
$980$	106.745	3.40983
$981$	0	0
$982$	33.4780	1.06832
$983$	24.4745	0.780615	0.390307	−	0.920685i	$-0.372369\pi$
$983$	24.4745	0.780615	0.390307	+	0.920685i	$0.372369\pi$
$984$	0	0
$985$	4.05632	0.129245
$986$	−13.0124	−0.414400
$987$	0	0
$988$	−41.4047	−1.31726
$989$	−64.7470	−2.05884
$990$	0	0
$991$	22.7385	0.722313	0.361156	−	0.932505i	$-0.382382\pi$
$991$	22.7385	0.722313	0.361156	+	0.932505i	$0.382382\pi$
$992$	−28.9185	−0.918164
$993$	0	0
$994$	30.5146	0.967866
$995$	55.1928	1.74973
$996$	0	0
$997$	51.7206	1.63801	0.819005	−	0.573787i	$-0.194526\pi$
$997$	51.7206	1.63801	0.819005	+	0.573787i	$0.194526\pi$
$998$	−64.8592	−2.05308
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2169.2.a.h.1.12		12
3.2	odd	2		241.2.a.b.1.1	✓	12
12.11	even	2		3856.2.a.n.1.9		12
15.14	odd	2		6025.2.a.h.1.12		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
241.2.a.b.1.1	✓	12	3.2	odd	2
2169.2.a.h.1.12		12	1.1	even	1	trivial
3856.2.a.n.1.9		12	12.11	even	2
6025.2.a.h.1.12		12	15.14	odd	2

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:	\( N \)	\(=\)	\( 2169 = 3^{2} \cdot 241 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2169.a (trivial)

\(f(q)\)	\(=\)	\(q+2.59703 q^{2} +4.74454 q^{4} -3.49051 q^{5} -0.744578 q^{7} +7.12764 q^{8} +O(q^{10})\)	\(q+2.59703 q^{2} +4.74454 q^{4} -3.49051 q^{5} -0.744578 q^{7} +7.12764 q^{8} -9.06493 q^{10} -6.28793 q^{11} -4.06994 q^{13} -1.93369 q^{14} +9.02158 q^{16} +1.60034 q^{17} +2.14421 q^{19} -16.5608 q^{20} -16.3299 q^{22} -9.25572 q^{23} +7.18363 q^{25} -10.5697 q^{26} -3.53268 q^{28} -3.13090 q^{29} -3.15223 q^{31} +9.17399 q^{32} +4.15612 q^{34} +2.59895 q^{35} +4.19928 q^{37} +5.56857 q^{38} -24.8791 q^{40} +4.52694 q^{41} +6.99535 q^{43} -29.8333 q^{44} -24.0373 q^{46} +4.82023 q^{47} -6.44560 q^{49} +18.6561 q^{50} -19.3100 q^{52} -3.71179 q^{53} +21.9481 q^{55} -5.30708 q^{56} -8.13103 q^{58} -3.34128 q^{59} +10.8630 q^{61} -8.18643 q^{62} +5.78192 q^{64} +14.2062 q^{65} +5.80682 q^{67} +7.59287 q^{68} +6.74955 q^{70} -15.7805 q^{71} -1.64682 q^{73} +10.9056 q^{74} +10.1733 q^{76} +4.68186 q^{77} -11.4813 q^{79} -31.4899 q^{80} +11.7566 q^{82} +4.74454 q^{83} -5.58599 q^{85} +18.1671 q^{86} -44.8181 q^{88} +7.95332 q^{89} +3.03039 q^{91} -43.9141 q^{92} +12.5182 q^{94} -7.48438 q^{95} -5.49759 q^{97} -16.7394 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{2} + 13 q^{4} - 6 q^{5} + 3 q^{7} - 9 q^{8}+O(q^{10}) \) 12 * q - 3 * q^2 + 13 * q^4 - 6 * q^5 + 3 * q^7 - 9 * q^8	\( 12 q - 3 q^{2} + 13 q^{4} - 6 q^{5} + 3 q^{7} - 9 q^{8} - 7 q^{10} - 22 q^{11} - 5 q^{13} - 6 q^{14} + 15 q^{16} + 4 q^{17} - 6 q^{19} - 10 q^{20} - 12 q^{22} - 32 q^{23} + 4 q^{25} - 8 q^{26} - 11 q^{28} - 6 q^{29} + 8 q^{31} - q^{32} - 19 q^{34} - 15 q^{35} - 8 q^{37} + 10 q^{38} - 52 q^{40} + q^{41} - 2 q^{43} - 42 q^{44} - 25 q^{46} - 34 q^{47} - 9 q^{49} + 27 q^{50} - 41 q^{52} - 5 q^{53} - 3 q^{55} - q^{56} - 33 q^{58} - 26 q^{59} - 26 q^{61} + 17 q^{62} + 13 q^{64} + 25 q^{65} + 6 q^{67} + 35 q^{68} - 4 q^{70} - 94 q^{71} - 22 q^{73} - 26 q^{74} - 20 q^{76} + 7 q^{77} + 9 q^{79} - 19 q^{80} + 15 q^{82} + 8 q^{83} + 4 q^{85} - 9 q^{86} + 6 q^{88} + 3 q^{89} - 20 q^{91} - 36 q^{92} + 48 q^{94} - 33 q^{95} - 29 q^{97} - 28 q^{98}+O(q^{100}) \) 12 * q - 3 * q^2 + 13 * q^4 - 6 * q^5 + 3 * q^7 - 9 * q^8 - 7 * q^10 - 22 * q^11 - 5 * q^13 - 6 * q^14 + 15 * q^16 + 4 * q^17 - 6 * q^19 - 10 * q^20 - 12 * q^22 - 32 * q^23 + 4 * q^25 - 8 * q^26 - 11 * q^28 - 6 * q^29 + 8 * q^31 - q^32 - 19 * q^34 - 15 * q^35 - 8 * q^37 + 10 * q^38 - 52 * q^40 + q^41 - 2 * q^43 - 42 * q^44 - 25 * q^46 - 34 * q^47 - 9 * q^49 + 27 * q^50 - 41 * q^52 - 5 * q^53 - 3 * q^55 - q^56 - 33 * q^58 - 26 * q^59 - 26 * q^61 + 17 * q^62 + 13 * q^64 + 25 * q^65 + 6 * q^67 + 35 * q^68 - 4 * q^70 - 94 * q^71 - 22 * q^73 - 26 * q^74 - 20 * q^76 + 7 * q^77 + 9 * q^79 - 19 * q^80 + 15 * q^82 + 8 * q^83 + 4 * q^85 - 9 * q^86 + 6 * q^88 + 3 * q^89 - 20 * q^91 - 36 * q^92 + 48 * q^94 - 33 * q^95 - 29 * q^97 - 28 * q^98

Introduction

L-functions

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Groups

Database

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