LMFDB - Embedded newform 2169.2.a.h.1.10

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			1.10
Root			$-1.32986$ 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)

$f(q)$	$=$	$q+1.32986 q^{2} -0.231473 q^{4} +3.40432 q^{5} -3.83334 q^{7} -2.96755 q^{8} +O(q^{10})$	\(q+1.32986 q^{2} -0.231473 q^{4} +3.40432 q^{5} -3.83334 q^{7} -2.96755 q^{8} +4.52727 q^{10} -4.78120 q^{11} +1.75719 q^{13} -5.09781 q^{14} -3.48347 q^{16} +6.01977 q^{17} -4.20086 q^{19} -0.788008 q^{20} -6.35832 q^{22} -4.28435 q^{23} +6.58942 q^{25} +2.33681 q^{26} +0.887315 q^{28} -4.96202 q^{29} +4.59803 q^{31} +1.30256 q^{32} +8.00545 q^{34} -13.0499 q^{35} -10.0806 q^{37} -5.58655 q^{38} -10.1025 q^{40} -1.12501 q^{41} -9.09669 q^{43} +1.10672 q^{44} -5.69758 q^{46} -7.19156 q^{47} +7.69453 q^{49} +8.76300 q^{50} -0.406741 q^{52} -6.07855 q^{53} -16.2767 q^{55} +11.3756 q^{56} -6.59879 q^{58} -1.36242 q^{59} -10.9181 q^{61} +6.11473 q^{62} +8.69917 q^{64} +5.98204 q^{65} +9.85684 q^{67} -1.39341 q^{68} -17.3546 q^{70} -11.9533 q^{71} -0.162452 q^{73} -13.4057 q^{74} +0.972384 q^{76} +18.3280 q^{77} +14.5772 q^{79} -11.8589 q^{80} -1.49611 q^{82} +5.77030 q^{83} +20.4932 q^{85} -12.0973 q^{86} +14.1884 q^{88} -4.73628 q^{89} -6.73591 q^{91} +0.991709 q^{92} -9.56377 q^{94} -14.3011 q^{95} +0.439068 q^{97} +10.2326 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.32986	0.940353	0.470176	−	0.882572i	$-0.344190\pi$
$2$	1.32986	0.940353	0.470176	+	0.882572i	$0.344190\pi$
$3$	0	0
$3$	0	0
$4$	−0.231473	−0.115736
$5$	3.40432	1.52246	0.761230	−	0.648482i	$-0.224596\pi$
$5$	3.40432	1.52246	0.761230	+	0.648482i	$0.224596\pi$
$6$	0	0
$7$	−3.83334	−1.44887	−0.724434	−	0.689344i	$-0.757899\pi$
$7$	−3.83334	−1.44887	−0.724434	+	0.689344i	$0.757899\pi$
$8$	−2.96755	−1.04919
$9$	0	0
$10$	4.52727	1.43165
$11$	−4.78120	−1.44159	−0.720793	−	0.693151i	$-0.756222\pi$
$11$	−4.78120	−1.44159	−0.720793	+	0.693151i	$0.756222\pi$
$12$	0	0
$13$	1.75719	0.487356	0.243678	−	0.969856i	$-0.421646\pi$
$13$	1.75719	0.487356	0.243678	+	0.969856i	$0.421646\pi$
$14$	−5.09781	−1.36245
$15$	0	0
$16$	−3.48347	−0.870869
$17$	6.01977	1.46001	0.730004	−	0.683443i	$-0.239518\pi$
$17$	6.01977	1.46001	0.730004	+	0.683443i	$0.239518\pi$
$18$	0	0
$19$	−4.20086	−0.963743	−0.481871	−	0.876242i	$-0.660043\pi$
$19$	−4.20086	−0.963743	−0.481871	+	0.876242i	$0.660043\pi$
$20$	−0.788008	−0.176204
$21$	0	0
$22$	−6.35832	−1.35560
$23$	−4.28435	−0.893348	−0.446674	−	0.894697i	$-0.647392\pi$
$23$	−4.28435	−0.893348	−0.446674	+	0.894697i	$0.647392\pi$
$24$	0	0
$25$	6.58942	1.31788
$26$	2.33681	0.458287
$27$	0	0
$28$	0.887315	0.167687
$29$	−4.96202	−0.921423	−0.460712	−	0.887550i	$-0.652406\pi$
$29$	−4.96202	−0.921423	−0.460712	+	0.887550i	$0.652406\pi$
$30$	0	0
$31$	4.59803	0.825830	0.412915	−	0.910770i	$-0.364511\pi$
$31$	4.59803	0.825830	0.412915	+	0.910770i	$0.364511\pi$
$32$	1.30256	0.230262
$33$	0	0
$34$	8.00545	1.37292
$35$	−13.0499	−2.20584
$36$	0	0
$37$	−10.0806	−1.65723	−0.828617	−	0.559816i	$-0.810872\pi$
$37$	−10.0806	−1.65723	−0.828617	+	0.559816i	$0.810872\pi$
$38$	−5.58655	−0.906258
$39$	0	0
$40$	−10.1025	−1.59734
$41$	−1.12501	−0.175698	−0.0878488	−	0.996134i	$-0.527999\pi$
$41$	−1.12501	−0.175698	−0.0878488	+	0.996134i	$0.527999\pi$
$42$	0	0
$43$	−9.09669	−1.38723	−0.693616	−	0.720345i	$-0.743984\pi$
$43$	−9.09669	−1.38723	−0.693616	+	0.720345i	$0.743984\pi$
$44$	1.10672	0.166844
$45$	0	0
$46$	−5.69758	−0.840062
$47$	−7.19156	−1.04900	−0.524499	−	0.851411i	$-0.675747\pi$
$47$	−7.19156	−1.04900	−0.524499	+	0.851411i	$0.675747\pi$
$48$	0	0
$49$	7.69453	1.09922
$50$	8.76300	1.23928
$51$	0	0
$52$	−0.406741	−0.0564049
$53$	−6.07855	−0.834953	−0.417477	−	0.908688i	$-0.637085\pi$
$53$	−6.07855	−0.834953	−0.417477	+	0.908688i	$0.637085\pi$
$54$	0	0
$55$	−16.2767	−2.19475
$56$	11.3756	1.52013
$57$	0	0
$58$	−6.59879	−0.866463
$59$	−1.36242	−0.177372	−0.0886862	−	0.996060i	$-0.528267\pi$
$59$	−1.36242	−0.177372	−0.0886862	+	0.996060i	$0.528267\pi$
$60$	0	0
$61$	−10.9181	−1.39791	−0.698957	−	0.715164i	$-0.746352\pi$
$61$	−10.9181	−1.39791	−0.698957	+	0.715164i	$0.746352\pi$
$62$	6.11473	0.776571
$63$	0	0
$64$	8.69917	1.08740
$65$	5.98204	0.741980
$66$	0	0
$67$	9.85684	1.20421	0.602103	−	0.798419i	$-0.294330\pi$
$67$	9.85684	1.20421	0.602103	+	0.798419i	$0.294330\pi$
$68$	−1.39341	−0.168976
$69$	0	0
$70$	−17.3546	−2.07427
$71$	−11.9533	−1.41859	−0.709296	−	0.704911i	$-0.750987\pi$
$71$	−11.9533	−1.41859	−0.709296	+	0.704911i	$0.750987\pi$
$72$	0	0
$73$	−0.162452	−0.0190135	−0.00950676	−	0.999955i	$-0.503026\pi$
$73$	−0.162452	−0.0190135	−0.00950676	+	0.999955i	$0.503026\pi$
$74$	−13.4057	−1.55838
$75$	0	0
$76$	0.972384	0.111540
$77$	18.3280	2.08867
$78$	0	0
$79$	14.5772	1.64006	0.820031	−	0.572319i	$-0.193956\pi$
$79$	14.5772	1.64006	0.820031	+	0.572319i	$0.193956\pi$
$80$	−11.8589	−1.32586
$81$	0	0
$82$	−1.49611	−0.165218
$83$	5.77030	0.633372	0.316686	−	0.948530i	$-0.397430\pi$
$83$	5.77030	0.633372	0.316686	+	0.948530i	$0.397430\pi$
$84$	0	0
$85$	20.4932	2.22280
$86$	−12.0973	−1.30449
$87$	0	0
$88$	14.1884	1.51249
$89$	−4.73628	−0.502044	−0.251022	−	0.967981i	$-0.580767\pi$
$89$	−4.73628	−0.502044	−0.251022	+	0.967981i	$0.580767\pi$
$90$	0	0
$91$	−6.73591	−0.706115
$92$	0.991709	0.103393
$93$	0	0
$94$	−9.56377	−0.986428
$95$	−14.3011	−1.46726
$96$	0	0
$97$	0.439068	0.0445806	0.0222903	−	0.999752i	$-0.492904\pi$
$97$	0.439068	0.0445806	0.0222903	+	0.999752i	$0.492904\pi$
$98$	10.2326	1.03365
$99$	0	0
$100$	−1.52527	−0.152527
$101$	4.54961	0.452703	0.226352	−	0.974046i	$-0.427320\pi$
$101$	4.54961	0.452703	0.226352	+	0.974046i	$0.427320\pi$
$102$	0	0
$103$	10.9977	1.08364	0.541819	−	0.840495i	$-0.317736\pi$
$103$	10.9977	1.08364	0.541819	+	0.840495i	$0.317736\pi$
$104$	−5.21454	−0.511327
$105$	0	0
$106$	−8.08362	−0.785151
$107$	−0.634979	−0.0613858	−0.0306929	−	0.999529i	$-0.509771\pi$
$107$	−0.634979	−0.0613858	−0.0306929	+	0.999529i	$0.509771\pi$
$108$	0	0
$109$	5.56974	0.533484	0.266742	−	0.963768i	$-0.414053\pi$
$109$	5.56974	0.533484	0.266742	+	0.963768i	$0.414053\pi$
$110$	−21.6458	−2.06384
$111$	0	0
$112$	13.3534	1.26177
$113$	−4.78347	−0.449991	−0.224995	−	0.974360i	$-0.572237\pi$
$113$	−4.78347	−0.449991	−0.224995	+	0.974360i	$0.572237\pi$
$114$	0	0
$115$	−14.5853	−1.36009
$116$	1.14857	0.106642
$117$	0	0
$118$	−1.81183	−0.166793
$119$	−23.0758	−2.11536
$120$	0	0
$121$	11.8598	1.07817
$122$	−14.5195	−1.31453
$123$	0	0
$124$	−1.06432	−0.0955785
$125$	5.41088	0.483964
$126$	0	0
$127$	−6.09612	−0.540944	−0.270472	−	0.962728i	$-0.587180\pi$
$127$	−6.09612	−0.540944	−0.270472	+	0.962728i	$0.587180\pi$
$128$	8.96356	0.792274
$129$	0	0
$130$	7.95527	0.697723
$131$	6.72698	0.587739	0.293869	−	0.955846i	$-0.405057\pi$
$131$	6.72698	0.587739	0.293869	+	0.955846i	$0.405057\pi$
$132$	0	0
$133$	16.1033	1.39634
$134$	13.1082	1.13238
$135$	0	0
$136$	−17.8639	−1.53182
$137$	9.94109	0.849325	0.424663	−	0.905352i	$-0.360393\pi$
$137$	9.94109	0.849325	0.424663	+	0.905352i	$0.360393\pi$
$138$	0	0
$139$	3.28633	0.278743	0.139372	−	0.990240i	$-0.455492\pi$
$139$	3.28633	0.278743	0.139372	+	0.990240i	$0.455492\pi$
$140$	3.02071	0.255296
$141$	0	0
$142$	−15.8962	−1.33398
$143$	−8.40146	−0.702566
$144$	0	0
$145$	−16.8923	−1.40283
$146$	−0.216038	−0.0178794
$147$	0	0
$148$	2.33337	0.191802
$149$	−9.42710	−0.772298	−0.386149	−	0.922436i	$-0.626195\pi$
$149$	−9.42710	−0.772298	−0.386149	+	0.922436i	$0.626195\pi$
$150$	0	0
$151$	−2.34993	−0.191234	−0.0956171	−	0.995418i	$-0.530482\pi$
$151$	−2.34993	−0.191234	−0.0956171	+	0.995418i	$0.530482\pi$
$152$	12.4662	1.01115
$153$	0	0
$154$	24.3736	1.96408
$155$	15.6532	1.25729
$156$	0	0
$157$	19.8666	1.58552	0.792762	−	0.609531i	$-0.208642\pi$
$157$	19.8666	1.58552	0.792762	+	0.609531i	$0.208642\pi$
$158$	19.3856	1.54224
$159$	0	0
$160$	4.43433	0.350565
$161$	16.4234	1.29434
$162$	0	0
$163$	−2.10105	−0.164567	−0.0822836	−	0.996609i	$-0.526221\pi$
$163$	−2.10105	−0.164567	−0.0822836	+	0.996609i	$0.526221\pi$
$164$	0.260410	0.0203346
$165$	0	0
$166$	7.67368	0.595593
$167$	−21.9087	−1.69535	−0.847673	−	0.530518i	$-0.821997\pi$
$167$	−21.9087	−1.69535	−0.847673	+	0.530518i	$0.821997\pi$
$168$	0	0
$169$	−9.91229	−0.762484
$170$	27.2531	2.09022
$171$	0	0
$172$	2.10564	0.160553
$173$	5.60734	0.426318	0.213159	−	0.977018i	$-0.431625\pi$
$173$	5.60734	0.426318	0.213159	+	0.977018i	$0.431625\pi$
$174$	0	0
$175$	−25.2595	−1.90944
$176$	16.6552	1.25543
$177$	0	0
$178$	−6.29859	−0.472099
$179$	23.8279	1.78098	0.890492	−	0.454999i	$-0.150360\pi$
$179$	23.8279	1.78098	0.890492	+	0.454999i	$0.150360\pi$
$180$	0	0
$181$	6.57832	0.488963	0.244481	−	0.969654i	$-0.421382\pi$
$181$	6.57832	0.488963	0.244481	+	0.969654i	$0.421382\pi$
$182$	−8.95781	−0.663997
$183$	0	0
$184$	12.7140	0.937288
$185$	−34.3175	−2.52307
$186$	0	0
$187$	−28.7817	−2.10473
$188$	1.66465	0.121407
$189$	0	0
$190$	−19.0184	−1.37974
$191$	−19.1620	−1.38652	−0.693258	−	0.720689i	$-0.743826\pi$
$191$	−19.1620	−1.38652	−0.693258	+	0.720689i	$0.743826\pi$
$192$	0	0
$193$	6.37070	0.458573	0.229286	−	0.973359i	$-0.426361\pi$
$193$	6.37070	0.458573	0.229286	+	0.973359i	$0.426361\pi$
$194$	0.583899	0.0419215
$195$	0	0
$196$	−1.78107	−0.127220
$197$	12.1752	0.867450	0.433725	−	0.901045i	$-0.357199\pi$
$197$	12.1752	0.867450	0.433725	+	0.901045i	$0.357199\pi$
$198$	0	0
$199$	−1.12984	−0.0800919	−0.0400459	−	0.999198i	$-0.512750\pi$
$199$	−1.12984	−0.0800919	−0.0400459	+	0.999198i	$0.512750\pi$
$200$	−19.5544	−1.38270
$201$	0	0
$202$	6.05035	0.425701
$203$	19.0211	1.33502
$204$	0	0
$205$	−3.82991	−0.267492
$206$	14.6254	1.01900
$207$	0	0
$208$	−6.12112	−0.424423
$209$	20.0851	1.38932
$210$	0	0
$211$	27.7361	1.90943	0.954716	−	0.297520i	$-0.0961596\pi$
$211$	27.7361	1.90943	0.954716	+	0.297520i	$0.0961596\pi$
$212$	1.40702	0.0966345
$213$	0	0
$214$	−0.844433	−0.0577243
$215$	−30.9681	−2.11201
$216$	0	0
$217$	−17.6258	−1.19652
$218$	7.40697	0.501664
$219$	0	0
$220$	3.76762	0.254013
$221$	10.5779	0.711544
$222$	0	0
$223$	−9.10518	−0.609728	−0.304864	−	0.952396i	$-0.598611\pi$
$223$	−9.10518	−0.609728	−0.304864	+	0.952396i	$0.598611\pi$
$224$	−4.99316	−0.333619
$225$	0	0
$226$	−6.36134	−0.423150
$227$	−24.1258	−1.60129	−0.800644	−	0.599141i	$-0.795509\pi$
$227$	−24.1258	−1.60129	−0.800644	+	0.599141i	$0.795509\pi$
$228$	0	0
$229$	−29.6552	−1.95967	−0.979835	−	0.199809i	$-0.935968\pi$
$229$	−29.6552	−1.95967	−0.979835	+	0.199809i	$0.935968\pi$
$230$	−19.3964	−1.27896
$231$	0	0
$232$	14.7250	0.966744
$233$	5.36189	0.351269	0.175635	−	0.984455i	$-0.443802\pi$
$233$	5.36189	0.351269	0.175635	+	0.984455i	$0.443802\pi$
$234$	0	0
$235$	−24.4824	−1.59706
$236$	0.315364	0.0205284
$237$	0	0
$238$	−30.6876	−1.98918
$239$	−2.14131	−0.138510	−0.0692549	−	0.997599i	$-0.522062\pi$
$239$	−2.14131	−0.138510	−0.0692549	+	0.997599i	$0.522062\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	15.7719	1.01386
$243$	0	0
$244$	2.52723	0.161789
$245$	26.1947	1.67352
$246$	0	0
$247$	−7.38170	−0.469686
$248$	−13.6449	−0.866449
$249$	0	0
$250$	7.19572	0.455097
$251$	28.1462	1.77657	0.888285	−	0.459293i	$-0.151897\pi$
$251$	28.1462	1.77657	0.888285	+	0.459293i	$0.151897\pi$
$252$	0	0
$253$	20.4843	1.28784
$254$	−8.10699	−0.508678
$255$	0	0
$256$	−5.47806	−0.342379
$257$	−1.23067	−0.0767674	−0.0383837	−	0.999263i	$-0.512221\pi$
$257$	−1.23067	−0.0767674	−0.0383837	+	0.999263i	$0.512221\pi$
$258$	0	0
$259$	38.6423	2.40111
$260$	−1.38468	−0.0858741
$261$	0	0
$262$	8.94594	0.552682
$263$	15.0159	0.925919	0.462960	−	0.886379i	$-0.346787\pi$
$263$	15.0159	0.925919	0.462960	+	0.886379i	$0.346787\pi$
$264$	0	0
$265$	−20.6934	−1.27118
$266$	21.4152	1.31305
$267$	0	0
$268$	−2.28159	−0.139370
$269$	0.940463	0.0573410	0.0286705	−	0.999589i	$-0.490873\pi$
$269$	0.940463	0.0573410	0.0286705	+	0.999589i	$0.490873\pi$
$270$	0	0
$271$	5.11230	0.310550	0.155275	−	0.987871i	$-0.450374\pi$
$271$	5.11230	0.310550	0.155275	+	0.987871i	$0.450374\pi$
$272$	−20.9697	−1.27148
$273$	0	0
$274$	13.2203	0.798665
$275$	−31.5053	−1.89984
$276$	0	0
$277$	2.39186	0.143713	0.0718565	−	0.997415i	$-0.477108\pi$
$277$	2.39186	0.143713	0.0718565	+	0.997415i	$0.477108\pi$
$278$	4.37036	0.262117
$279$	0	0
$280$	38.7263	2.31434
$281$	−7.49067	−0.446856	−0.223428	−	0.974720i	$-0.571725\pi$
$281$	−7.49067	−0.446856	−0.223428	+	0.974720i	$0.571725\pi$
$282$	0	0
$283$	−17.9408	−1.06647	−0.533235	−	0.845967i	$-0.679024\pi$
$283$	−17.9408	−1.06647	−0.533235	+	0.845967i	$0.679024\pi$
$284$	2.76686	0.164183
$285$	0	0
$286$	−11.1728	−0.660660
$287$	4.31256	0.254563
$288$	0	0
$289$	19.2376	1.13162
$290$	−22.4644	−1.31916
$291$	0	0
$292$	0.0376031	0.00220056
$293$	−12.6436	−0.738644	−0.369322	−	0.929301i	$-0.620410\pi$
$293$	−12.6436	−0.738644	−0.369322	+	0.929301i	$0.620410\pi$
$294$	0	0
$295$	−4.63813	−0.270042
$296$	29.9145	1.73875
$297$	0	0
$298$	−12.5367	−0.726233
$299$	−7.52840	−0.435379
$300$	0	0
$301$	34.8708	2.00992
$302$	−3.12507	−0.179828
$303$	0	0
$304$	14.6336	0.839293
$305$	−37.1686	−2.12827
$306$	0	0
$307$	4.93084	0.281418	0.140709	−	0.990051i	$-0.455062\pi$
$307$	4.93084	0.281418	0.140709	+	0.990051i	$0.455062\pi$
$308$	−4.24243	−0.241735
$309$	0	0
$310$	20.8165	1.18230
$311$	−33.2247	−1.88400	−0.942001	−	0.335609i	$-0.891058\pi$
$311$	−33.2247	−1.88400	−0.942001	+	0.335609i	$0.891058\pi$
$312$	0	0
$313$	18.2269	1.03024	0.515122	−	0.857117i	$-0.327747\pi$
$313$	18.2269	1.03024	0.515122	+	0.857117i	$0.327747\pi$
$314$	26.4197	1.49095
$315$	0	0
$316$	−3.37422	−0.189815
$317$	−15.7434	−0.884235	−0.442117	−	0.896957i	$-0.645773\pi$
$317$	−15.7434	−0.884235	−0.442117	+	0.896957i	$0.645773\pi$
$318$	0	0
$319$	23.7244	1.32831
$320$	29.6148	1.65552
$321$	0	0
$322$	21.8408	1.21714
$323$	−25.2882	−1.40707
$324$	0	0
$325$	11.5788	0.642279
$326$	−2.79411	−0.154751
$327$	0	0
$328$	3.33853	0.184339
$329$	27.5677	1.51986
$330$	0	0
$331$	−7.35834	−0.404451	−0.202225	−	0.979339i	$-0.564817\pi$
$331$	−7.35834	−0.404451	−0.202225	+	0.979339i	$0.564817\pi$
$332$	−1.33567	−0.0733042
$333$	0	0
$334$	−29.1355	−1.59422
$335$	33.5559	1.83335
$336$	0	0
$337$	−21.0122	−1.14461	−0.572303	−	0.820042i	$-0.693950\pi$
$337$	−21.0122	−1.14461	−0.572303	+	0.820042i	$0.693950\pi$
$338$	−13.1820	−0.717004
$339$	0	0
$340$	−4.74363	−0.257259
$341$	−21.9841	−1.19050
$342$	0	0
$343$	−2.66238	−0.143755
$344$	26.9949	1.45547
$345$	0	0
$346$	7.45697	0.400889
$347$	0.799138	0.0429000	0.0214500	−	0.999770i	$-0.493172\pi$
$347$	0.799138	0.0429000	0.0214500	+	0.999770i	$0.493172\pi$
$348$	0	0
$349$	−1.25641	−0.0672542	−0.0336271	−	0.999434i	$-0.510706\pi$
$349$	−1.25641	−0.0672542	−0.0336271	+	0.999434i	$0.510706\pi$
$350$	−33.5916	−1.79555
$351$	0	0
$352$	−6.22779	−0.331942
$353$	−30.7996	−1.63930	−0.819649	−	0.572865i	$-0.805832\pi$
$353$	−30.7996	−1.63930	−0.819649	+	0.572865i	$0.805832\pi$
$354$	0	0
$355$	−40.6928	−2.15975
$356$	1.09632	0.0581048
$357$	0	0
$358$	31.6878	1.67475
$359$	14.3153	0.755535	0.377768	−	0.925900i	$-0.376692\pi$
$359$	14.3153	0.755535	0.377768	+	0.925900i	$0.376692\pi$
$360$	0	0
$361$	−1.35280	−0.0712002
$362$	8.74825	0.459798
$363$	0	0
$364$	1.55918	0.0817232
$365$	−0.553038	−0.0289473
$366$	0	0
$367$	3.37048	0.175938	0.0879688	−	0.996123i	$-0.471962\pi$
$367$	3.37048	0.175938	0.0879688	+	0.996123i	$0.471962\pi$
$368$	14.9244	0.777989
$369$	0	0
$370$	−45.6374	−2.37258
$371$	23.3012	1.20974
$372$	0	0
$373$	5.72387	0.296371	0.148185	−	0.988960i	$-0.452657\pi$
$373$	5.72387	0.296371	0.148185	+	0.988960i	$0.452657\pi$
$374$	−38.2756	−1.97919
$375$	0	0
$376$	21.3413	1.10059
$377$	−8.71920	−0.449061
$378$	0	0
$379$	0.963739	0.0495040	0.0247520	−	0.999694i	$-0.492120\pi$
$379$	0.963739	0.0495040	0.0247520	+	0.999694i	$0.492120\pi$
$380$	3.31031	0.169815
$381$	0	0
$382$	−25.4828	−1.30382
$383$	−17.5403	−0.896267	−0.448134	−	0.893967i	$-0.647911\pi$
$383$	−17.5403	−0.896267	−0.448134	+	0.893967i	$0.647911\pi$
$384$	0	0
$385$	62.3943	3.17991
$386$	8.47213	0.431220
$387$	0	0
$388$	−0.101632	−0.00515960
$389$	16.8395	0.853796	0.426898	−	0.904300i	$-0.359606\pi$
$389$	16.8395	0.853796	0.426898	+	0.904300i	$0.359606\pi$
$390$	0	0
$391$	−25.7908	−1.30429
$392$	−22.8339	−1.15328
$393$	0	0
$394$	16.1914	0.815709
$395$	49.6254	2.49693
$396$	0	0
$397$	−2.82968	−0.142017	−0.0710087	−	0.997476i	$-0.522622\pi$
$397$	−2.82968	−0.142017	−0.0710087	+	0.997476i	$0.522622\pi$
$398$	−1.50252	−0.0753146
$399$	0	0
$400$	−22.9541	−1.14770
$401$	13.6028	0.679290	0.339645	−	0.940554i	$-0.389693\pi$
$401$	13.6028	0.679290	0.339645	+	0.940554i	$0.389693\pi$
$402$	0	0
$403$	8.07960	0.402473
$404$	−1.05311	−0.0523943
$405$	0	0
$406$	25.2954	1.25539
$407$	48.1971	2.38904
$408$	0	0
$409$	24.6420	1.21847	0.609234	−	0.792990i	$-0.291477\pi$
$409$	24.6420	1.21847	0.609234	+	0.792990i	$0.291477\pi$
$410$	−5.09324	−0.251537
$411$	0	0
$412$	−2.54567	−0.125416
$413$	5.22264	0.256989
$414$	0	0
$415$	19.6439	0.964284
$416$	2.28884	0.112220
$417$	0	0
$418$	26.7104	1.30645
$419$	7.82622	0.382336	0.191168	−	0.981557i	$-0.438772\pi$
$419$	7.82622	0.382336	0.191168	+	0.981557i	$0.438772\pi$
$420$	0	0
$421$	21.3683	1.04143	0.520715	−	0.853731i	$-0.325666\pi$
$421$	21.3683	1.04143	0.520715	+	0.853731i	$0.325666\pi$
$422$	36.8851	1.79554
$423$	0	0
$424$	18.0384	0.876021
$425$	39.6667	1.92412
$426$	0	0
$427$	41.8527	2.02539
$428$	0.146980	0.00710457
$429$	0	0
$430$	−41.1832	−1.98603
$431$	−33.9239	−1.63406	−0.817028	−	0.576598i	$-0.804380\pi$
$431$	−33.9239	−1.63406	−0.817028	+	0.576598i	$0.804380\pi$
$432$	0	0
$433$	34.5154	1.65871	0.829353	−	0.558725i	$-0.188709\pi$
$433$	34.5154	1.65871	0.829353	+	0.558725i	$0.188709\pi$
$434$	−23.4399	−1.12515
$435$	0	0
$436$	−1.28924	−0.0617435
$437$	17.9979	0.860957
$438$	0	0
$439$	2.28531	0.109072	0.0545360	−	0.998512i	$-0.482632\pi$
$439$	2.28531	0.109072	0.0545360	+	0.998512i	$0.482632\pi$
$440$	48.3020	2.30271
$441$	0	0
$442$	14.0671	0.669103
$443$	−18.3736	−0.872957	−0.436478	−	0.899715i	$-0.643774\pi$
$443$	−18.3736	−0.872957	−0.436478	+	0.899715i	$0.643774\pi$
$444$	0	0
$445$	−16.1238	−0.764342
$446$	−12.1086	−0.573360
$447$	0	0
$448$	−33.3469	−1.57549
$449$	−26.0138	−1.22767	−0.613833	−	0.789436i	$-0.710373\pi$
$449$	−26.0138	−1.22767	−0.613833	+	0.789436i	$0.710373\pi$
$450$	0	0
$451$	5.37891	0.253283
$452$	1.10724	0.0520803
$453$	0	0
$454$	−32.0840	−1.50578
$455$	−22.9312	−1.07503
$456$	0	0
$457$	9.10237	0.425791	0.212895	−	0.977075i	$-0.431711\pi$
$457$	9.10237	0.425791	0.212895	+	0.977075i	$0.431711\pi$
$458$	−39.4372	−1.84278
$459$	0	0
$460$	3.37610	0.157411
$461$	−12.3824	−0.576708	−0.288354	−	0.957524i	$-0.593108\pi$
$461$	−12.3824	−0.576708	−0.288354	+	0.957524i	$0.593108\pi$
$462$	0	0
$463$	25.3698	1.17904	0.589518	−	0.807755i	$-0.299318\pi$
$463$	25.3698	1.17904	0.589518	+	0.807755i	$0.299318\pi$
$464$	17.2851	0.802439
$465$	0	0
$466$	7.13057	0.330317
$467$	21.2025	0.981137	0.490568	−	0.871403i	$-0.336789\pi$
$467$	21.2025	0.981137	0.490568	+	0.871403i	$0.336789\pi$
$468$	0	0
$469$	−37.7847	−1.74473
$470$	−32.5582	−1.50180
$471$	0	0
$472$	4.04306	0.186097
$473$	43.4931	1.99981
$474$	0	0
$475$	−27.6812	−1.27010
$476$	5.34143	0.244824
$477$	0	0
$478$	−2.84764	−0.130248
$479$	26.5866	1.21477	0.607385	−	0.794407i	$-0.292218\pi$
$479$	26.5866	1.21477	0.607385	+	0.794407i	$0.292218\pi$
$480$	0	0
$481$	−17.7134	−0.807663
$482$	1.32986	0.0605735
$483$	0	0
$484$	−2.74523	−0.124783
$485$	1.49473	0.0678721
$486$	0	0
$487$	−17.1209	−0.775822	−0.387911	−	0.921697i	$-0.626803\pi$
$487$	−17.1209	−0.775822	−0.387911	+	0.921697i	$0.626803\pi$
$488$	32.3998	1.46667
$489$	0	0
$490$	34.8352	1.57370
$491$	5.11790	0.230968	0.115484	−	0.993309i	$-0.463158\pi$
$491$	5.11790	0.230968	0.115484	+	0.993309i	$0.463158\pi$
$492$	0	0
$493$	−29.8702	−1.34529
$494$	−9.81662	−0.441671
$495$	0	0
$496$	−16.0171	−0.719189
$497$	45.8210	2.05535
$498$	0	0
$499$	29.2980	1.31156	0.655780	−	0.754952i	$-0.272340\pi$
$499$	29.2980	1.31156	0.655780	+	0.754952i	$0.272340\pi$
$500$	−1.25247	−0.0560122
$501$	0	0
$502$	37.4305	1.67060
$503$	−12.8905	−0.574761	−0.287380	−	0.957817i	$-0.592784\pi$
$503$	−12.8905	−0.574761	−0.287380	+	0.957817i	$0.592784\pi$
$504$	0	0
$505$	15.4884	0.689223
$506$	27.2412	1.21102
$507$	0	0
$508$	1.41109	0.0626068
$509$	11.2706	0.499559	0.249780	−	0.968303i	$-0.419642\pi$
$509$	11.2706	0.499559	0.249780	+	0.968303i	$0.419642\pi$
$510$	0	0
$511$	0.622733	0.0275481
$512$	−25.2122	−1.11423
$513$	0	0
$514$	−1.63662	−0.0721884
$515$	37.4398	1.64980
$516$	0	0
$517$	34.3843	1.51222
$518$	51.3888	2.25789
$519$	0	0
$520$	−17.7520	−0.778475
$521$	−41.5318	−1.81954	−0.909771	−	0.415110i	$-0.863743\pi$
$521$	−41.5318	−1.81954	−0.909771	+	0.415110i	$0.863743\pi$
$522$	0	0
$523$	−8.02470	−0.350896	−0.175448	−	0.984489i	$-0.556137\pi$
$523$	−8.02470	−0.350896	−0.175448	+	0.984489i	$0.556137\pi$
$524$	−1.55711	−0.0680228
$525$	0	0
$526$	19.9690	0.870691
$527$	27.6790	1.20572
$528$	0	0
$529$	−4.64439	−0.201930
$530$	−27.5193	−1.19536
$531$	0	0
$532$	−3.72748	−0.161607
$533$	−1.97686	−0.0856273
$534$	0	0
$535$	−2.16167	−0.0934573
$536$	−29.2506	−1.26344
$537$	0	0
$538$	1.25068	0.0539208
$539$	−36.7891	−1.58462
$540$	0	0
$541$	9.87706	0.424648	0.212324	−	0.977199i	$-0.431897\pi$
$541$	9.87706	0.424648	0.212324	+	0.977199i	$0.431897\pi$
$542$	6.79864	0.292027
$543$	0	0
$544$	7.84110	0.336184
$545$	18.9612	0.812208
$546$	0	0
$547$	19.3803	0.828641	0.414320	−	0.910131i	$-0.364019\pi$
$547$	19.3803	0.828641	0.414320	+	0.910131i	$0.364019\pi$
$548$	−2.30109	−0.0982978
$549$	0	0
$550$	−41.8976	−1.78652
$551$	20.8447	0.888015
$552$	0	0
$553$	−55.8794	−2.37623
$554$	3.18084	0.135141
$555$	0	0
$556$	−0.760696	−0.0322607
$557$	18.9414	0.802571	0.401285	−	0.915953i	$-0.368564\pi$
$557$	18.9414	0.802571	0.401285	+	0.915953i	$0.368564\pi$
$558$	0	0
$559$	−15.9846	−0.676077
$560$	45.4592	1.92100
$561$	0	0
$562$	−9.96154	−0.420202
$563$	−12.0839	−0.509276	−0.254638	−	0.967036i	$-0.581956\pi$
$563$	−12.0839	−0.509276	−0.254638	+	0.967036i	$0.581956\pi$
$564$	0	0
$565$	−16.2845	−0.685093
$566$	−23.8588	−1.00286
$567$	0	0
$568$	35.4719	1.48837
$569$	−8.80054	−0.368938	−0.184469	−	0.982838i	$-0.559057\pi$
$569$	−8.80054	−0.368938	−0.184469	+	0.982838i	$0.559057\pi$
$570$	0	0
$571$	−35.4541	−1.48371	−0.741854	−	0.670562i	$-0.766053\pi$
$571$	−35.4541	−1.48371	−0.741854	+	0.670562i	$0.766053\pi$
$572$	1.94471	0.0813124
$573$	0	0
$574$	5.73511	0.239379
$575$	−28.2313	−1.17733
$576$	0	0
$577$	−22.5404	−0.938368	−0.469184	−	0.883100i	$-0.655452\pi$
$577$	−22.5404	−0.938368	−0.469184	+	0.883100i	$0.655452\pi$
$578$	25.5833	1.06413
$579$	0	0
$580$	3.91011	0.162358
$581$	−22.1195	−0.917673
$582$	0	0
$583$	29.0628	1.20366
$584$	0.482082	0.0199487
$585$	0	0
$586$	−16.8142	−0.694586
$587$	−29.8831	−1.23341	−0.616704	−	0.787195i	$-0.711533\pi$
$587$	−29.8831	−1.23341	−0.616704	+	0.787195i	$0.711533\pi$
$588$	0	0
$589$	−19.3156	−0.795887
$590$	−6.16806	−0.253935
$591$	0	0
$592$	35.1154	1.44323
$593$	−30.9659	−1.27162	−0.635808	−	0.771847i	$-0.719333\pi$
$593$	−30.9659	−1.27162	−0.635808	+	0.771847i	$0.719333\pi$
$594$	0	0
$595$	−78.5576	−3.22055
$596$	2.18212	0.0893830
$597$	0	0
$598$	−10.0117	−0.409410
$599$	22.4899	0.918912	0.459456	−	0.888201i	$-0.348044\pi$
$599$	22.4899	0.918912	0.459456	+	0.888201i	$0.348044\pi$
$600$	0	0
$601$	−22.2188	−0.906322	−0.453161	−	0.891429i	$-0.649704\pi$
$601$	−22.2188	−0.906322	−0.453161	+	0.891429i	$0.649704\pi$
$602$	46.3732	1.89003
$603$	0	0
$604$	0.543944	0.0221328
$605$	40.3747	1.64147
$606$	0	0
$607$	17.3047	0.702378	0.351189	−	0.936305i	$-0.385778\pi$
$607$	17.3047	0.702378	0.351189	+	0.936305i	$0.385778\pi$
$608$	−5.47186	−0.221913
$609$	0	0
$610$	−49.4290	−2.00132
$611$	−12.6369	−0.511235
$612$	0	0
$613$	13.8909	0.561050	0.280525	−	0.959847i	$-0.409491\pi$
$613$	13.8909	0.561050	0.280525	+	0.959847i	$0.409491\pi$
$614$	6.55732	0.264632
$615$	0	0
$616$	−54.3891	−2.19140
$617$	−47.6668	−1.91899	−0.959496	−	0.281722i	$-0.909094\pi$
$617$	−47.6668	−1.91899	−0.959496	+	0.281722i	$0.909094\pi$
$618$	0	0
$619$	−36.3518	−1.46110	−0.730551	−	0.682858i	$-0.760737\pi$
$619$	−36.3518	−1.46110	−0.730551	+	0.682858i	$0.760737\pi$
$620$	−3.62328	−0.145514
$621$	0	0
$622$	−44.1843	−1.77163
$623$	18.1558	0.727396
$624$	0	0
$625$	−14.5267	−0.581067
$626$	24.2392	0.968792
$627$	0	0
$628$	−4.59857	−0.183503
$629$	−60.6826	−2.41957
$630$	0	0
$631$	−42.0906	−1.67560	−0.837801	−	0.545976i	$-0.816159\pi$
$631$	−42.0906	−1.67560	−0.837801	+	0.545976i	$0.816159\pi$
$632$	−43.2585	−1.72073
$633$	0	0
$634$	−20.9365	−0.831493
$635$	−20.7532	−0.823565
$636$	0	0
$637$	13.5207	0.535711
$638$	31.5501	1.24908
$639$	0	0
$640$	30.5149	1.20621
$641$	−13.6204	−0.537972	−0.268986	−	0.963144i	$-0.586689\pi$
$641$	−13.6204	−0.537972	−0.268986	+	0.963144i	$0.586689\pi$
$642$	0	0
$643$	−15.5929	−0.614925	−0.307462	−	0.951560i	$-0.599480\pi$
$643$	−15.5929	−0.614925	−0.307462	+	0.951560i	$0.599480\pi$
$644$	−3.80156	−0.149803
$645$	0	0
$646$	−33.6297	−1.32314
$647$	−30.8612	−1.21328	−0.606639	−	0.794977i	$-0.707483\pi$
$647$	−30.8612	−1.21328	−0.606639	+	0.794977i	$0.707483\pi$
$648$	0	0
$649$	6.51402	0.255697
$650$	15.3982	0.603969
$651$	0	0
$652$	0.486336	0.0190464
$653$	−41.4375	−1.62157	−0.810787	−	0.585342i	$-0.800960\pi$
$653$	−41.4375	−1.62157	−0.810787	+	0.585342i	$0.800960\pi$
$654$	0	0
$655$	22.9008	0.894809
$656$	3.91896	0.153009
$657$	0	0
$658$	36.6612	1.42920
$659$	5.89065	0.229467	0.114733	−	0.993396i	$-0.463399\pi$
$659$	5.89065	0.229467	0.114733	+	0.993396i	$0.463399\pi$
$660$	0	0
$661$	−12.1895	−0.474118	−0.237059	−	0.971495i	$-0.576184\pi$
$661$	−12.1895	−0.474118	−0.237059	+	0.971495i	$0.576184\pi$
$662$	−9.78556	−0.380327
$663$	0	0
$664$	−17.1236	−0.664525
$665$	54.8209	2.12586
$666$	0	0
$667$	21.2590	0.823151
$668$	5.07127	0.196213
$669$	0	0
$670$	44.6246	1.72400
$671$	52.2014	2.01521
$672$	0	0
$673$	−13.4033	−0.516658	−0.258329	−	0.966057i	$-0.583172\pi$
$673$	−13.4033	−0.516658	−0.258329	+	0.966057i	$0.583172\pi$
$674$	−27.9432	−1.07633
$675$	0	0
$676$	2.29443	0.0882471
$677$	9.52226	0.365970	0.182985	−	0.983116i	$-0.441424\pi$
$677$	9.52226	0.365970	0.182985	+	0.983116i	$0.441424\pi$
$678$	0	0
$679$	−1.68310	−0.0645914
$680$	−60.8146	−2.33213
$681$	0	0
$682$	−29.2357	−1.11949
$683$	4.11578	0.157486	0.0787429	−	0.996895i	$-0.474909\pi$
$683$	4.11578	0.157486	0.0787429	+	0.996895i	$0.474909\pi$
$684$	0	0
$685$	33.8427	1.29306
$686$	−3.54059	−0.135180
$687$	0	0
$688$	31.6881	1.20810
$689$	−10.6812	−0.406920
$690$	0	0
$691$	39.0627	1.48601	0.743007	−	0.669283i	$-0.233399\pi$
$691$	39.0627	1.48601	0.743007	+	0.669283i	$0.233399\pi$
$692$	−1.29795	−0.0493405
$693$	0	0
$694$	1.06274	0.0403411
$695$	11.1877	0.424375
$696$	0	0
$697$	−6.77232	−0.256520
$698$	−1.67085	−0.0632427
$699$	0	0
$700$	5.84689	0.220992
$701$	−4.81641	−0.181913	−0.0909566	−	0.995855i	$-0.528992\pi$
$701$	−4.81641	−0.181913	−0.0909566	+	0.995855i	$0.528992\pi$
$702$	0	0
$703$	42.3470	1.59715
$704$	−41.5924	−1.56757
$705$	0	0
$706$	−40.9592	−1.54152
$707$	−17.4402	−0.655908
$708$	0	0
$709$	39.1628	1.47079	0.735394	−	0.677640i	$-0.236997\pi$
$709$	39.1628	1.47079	0.735394	+	0.677640i	$0.236997\pi$
$710$	−54.1157	−2.03093
$711$	0	0
$712$	14.0551	0.526738
$713$	−19.6995	−0.737753
$714$	0	0
$715$	−28.6013	−1.06963
$716$	−5.51552	−0.206125
$717$	0	0
$718$	19.0374	0.710470
$719$	32.1852	1.20030	0.600152	−	0.799886i	$-0.295107\pi$
$719$	32.1852	1.20030	0.600152	+	0.799886i	$0.295107\pi$
$720$	0	0
$721$	−42.1581	−1.57005
$722$	−1.79904	−0.0669533
$723$	0	0
$724$	−1.52270	−0.0565908
$725$	−32.6968	−1.21433
$726$	0	0
$727$	−1.69429	−0.0628377	−0.0314188	−	0.999506i	$-0.510003\pi$
$727$	−1.69429	−0.0628377	−0.0314188	+	0.999506i	$0.510003\pi$
$728$	19.9891	0.740846
$729$	0	0
$730$	−0.735462	−0.0272207
$731$	−54.7600	−2.02537
$732$	0	0
$733$	18.2300	0.673340	0.336670	−	0.941623i	$-0.390699\pi$
$733$	18.2300	0.673340	0.336670	+	0.941623i	$0.390699\pi$
$734$	4.48226	0.165443
$735$	0	0
$736$	−5.58061	−0.205704
$737$	−47.1275	−1.73596
$738$	0	0
$739$	−12.4675	−0.458623	−0.229312	−	0.973353i	$-0.573648\pi$
$739$	−12.4675	−0.458623	−0.229312	+	0.973353i	$0.573648\pi$
$740$	7.94356	0.292011
$741$	0	0
$742$	30.9873	1.13758
$743$	15.6839	0.575385	0.287693	−	0.957723i	$-0.407112\pi$
$743$	15.6839	0.575385	0.287693	+	0.957723i	$0.407112\pi$
$744$	0	0
$745$	−32.0929	−1.17579
$746$	7.61194	0.278693
$747$	0	0
$748$	6.66218	0.243593
$749$	2.43409	0.0889399
$750$	0	0
$751$	−3.58156	−0.130693	−0.0653465	−	0.997863i	$-0.520815\pi$
$751$	−3.58156	−0.130693	−0.0653465	+	0.997863i	$0.520815\pi$
$752$	25.0516	0.913539
$753$	0	0
$754$	−11.5953	−0.422276
$755$	−7.99990	−0.291146
$756$	0	0
$757$	−15.7026	−0.570720	−0.285360	−	0.958420i	$-0.592113\pi$
$757$	−15.7026	−0.570720	−0.285360	+	0.958420i	$0.592113\pi$
$758$	1.28164	0.0465512
$759$	0	0
$760$	42.4391	1.53943
$761$	36.1137	1.30912	0.654559	−	0.756011i	$-0.272854\pi$
$761$	36.1137	1.30912	0.654559	+	0.756011i	$0.272854\pi$
$762$	0	0
$763$	−21.3507	−0.772948
$764$	4.43549	0.160470
$765$	0	0
$766$	−23.3261	−0.842807
$767$	−2.39403	−0.0864436
$768$	0	0
$769$	−50.2736	−1.81291	−0.906456	−	0.422300i	$-0.861223\pi$
$769$	−50.2736	−1.81291	−0.906456	+	0.422300i	$0.861223\pi$
$770$	82.9757	2.99024
$771$	0	0
$772$	−1.47464	−0.0530736
$773$	27.4904	0.988762	0.494381	−	0.869245i	$-0.335395\pi$
$773$	27.4904	0.988762	0.494381	+	0.869245i	$0.335395\pi$
$774$	0	0
$775$	30.2983	1.08835
$776$	−1.30295	−0.0467733
$777$	0	0
$778$	22.3942	0.802870
$779$	4.72602	0.169327
$780$	0	0
$781$	57.1509	2.04502
$782$	−34.2981	−1.22650
$783$	0	0
$784$	−26.8037	−0.957275
$785$	67.6322	2.41390
$786$	0	0
$787$	43.4981	1.55054	0.775271	−	0.631629i	$-0.217614\pi$
$787$	43.4981	1.55054	0.775271	+	0.631629i	$0.217614\pi$
$788$	−2.81824	−0.100396
$789$	0	0
$790$	65.9949	2.34799
$791$	18.3367	0.651977
$792$	0	0
$793$	−19.1851	−0.681282
$794$	−3.76307	−0.133547
$795$	0	0
$796$	0.261526	0.00926955
$797$	−38.2429	−1.35463	−0.677316	−	0.735692i	$-0.736857\pi$
$797$	−38.2429	−1.35463	−0.677316	+	0.735692i	$0.736857\pi$
$798$	0	0
$799$	−43.2915	−1.53154
$800$	8.58310	0.303458
$801$	0	0
$802$	18.0898	0.638772
$803$	0.776713	0.0274096
$804$	0	0
$805$	55.9105	1.97058
$806$	10.7447	0.378467
$807$	0	0
$808$	−13.5012	−0.474970
$809$	28.3274	0.995939	0.497969	−	0.867195i	$-0.334079\pi$
$809$	28.3274	0.995939	0.497969	+	0.867195i	$0.334079\pi$
$810$	0	0
$811$	−39.3902	−1.38318	−0.691589	−	0.722292i	$-0.743089\pi$
$811$	−39.3902	−1.38318	−0.691589	+	0.722292i	$0.743089\pi$
$812$	−4.40287	−0.154510
$813$	0	0
$814$	64.0954	2.24654
$815$	−7.15266	−0.250547
$816$	0	0
$817$	38.2139	1.33694
$818$	32.7704	1.14579
$819$	0	0
$820$	0.886519	0.0309586
$821$	−39.6673	−1.38440	−0.692199	−	0.721707i	$-0.743358\pi$
$821$	−39.6673	−1.38440	−0.692199	+	0.721707i	$0.743358\pi$
$822$	0	0
$823$	−9.49313	−0.330910	−0.165455	−	0.986217i	$-0.552909\pi$
$823$	−9.49313	−0.330910	−0.165455	+	0.986217i	$0.552909\pi$
$824$	−32.6363	−1.13694
$825$	0	0
$826$	6.94538	0.241661
$827$	8.26358	0.287353	0.143676	−	0.989625i	$-0.454108\pi$
$827$	8.26358	0.287353	0.143676	+	0.989625i	$0.454108\pi$
$828$	0	0
$829$	−32.7576	−1.13772	−0.568860	−	0.822435i	$-0.692615\pi$
$829$	−32.7576	−1.13772	−0.568860	+	0.822435i	$0.692615\pi$
$830$	26.1237	0.906767
$831$	0	0
$832$	15.2861	0.529949
$833$	46.3193	1.60487
$834$	0	0
$835$	−74.5843	−2.58110
$836$	−4.64916	−0.160795
$837$	0	0
$838$	10.4078	0.359531
$839$	−28.8263	−0.995195	−0.497597	−	0.867408i	$-0.665784\pi$
$839$	−28.8263	−0.995195	−0.497597	+	0.867408i	$0.665784\pi$
$840$	0	0
$841$	−4.37839	−0.150979
$842$	28.4169	0.979311
$843$	0	0
$844$	−6.42015	−0.220991
$845$	−33.7446	−1.16085
$846$	0	0
$847$	−45.4629	−1.56212
$848$	21.1745	0.727135
$849$	0	0
$850$	52.7512	1.80935
$851$	43.1886	1.48049
$852$	0	0
$853$	1.74141	0.0596248	0.0298124	−	0.999556i	$-0.490509\pi$
$853$	1.74141	0.0596248	0.0298124	+	0.999556i	$0.490509\pi$
$854$	55.6582	1.90458
$855$	0	0
$856$	1.88433	0.0644051
$857$	23.6164	0.806720	0.403360	−	0.915041i	$-0.367842\pi$
$857$	23.6164	0.806720	0.403360	+	0.915041i	$0.367842\pi$
$858$	0	0
$859$	18.0458	0.615714	0.307857	−	0.951433i	$-0.400388\pi$
$859$	18.0458	0.615714	0.307857	+	0.951433i	$0.400388\pi$
$860$	7.16827	0.244436
$861$	0	0
$862$	−45.1140	−1.53659
$863$	−8.06392	−0.274499	−0.137249	−	0.990537i	$-0.543826\pi$
$863$	−8.06392	−0.274499	−0.137249	+	0.990537i	$0.543826\pi$
$864$	0	0
$865$	19.0892	0.649052
$866$	45.9007	1.55977
$867$	0	0
$868$	4.07990	0.138481
$869$	−69.6964	−2.36429
$870$	0	0
$871$	17.3203	0.586877
$872$	−16.5285	−0.559724
$873$	0	0
$874$	23.9347	0.809604
$875$	−20.7418	−0.701200
$876$	0	0
$877$	−14.4702	−0.488625	−0.244312	−	0.969697i	$-0.578562\pi$
$877$	−14.4702	−0.488625	−0.244312	+	0.969697i	$0.578562\pi$
$878$	3.03914	0.102566
$879$	0	0
$880$	56.6996	1.91134
$881$	−10.0284	−0.337867	−0.168933	−	0.985627i	$-0.554032\pi$
$881$	−10.0284	−0.337867	−0.168933	+	0.985627i	$0.554032\pi$
$882$	0	0
$883$	0.730440	0.0245813	0.0122906	−	0.999924i	$-0.496088\pi$
$883$	0.730440	0.0245813	0.0122906	+	0.999924i	$0.496088\pi$
$884$	−2.44849	−0.0823515
$885$	0	0
$886$	−24.4343	−0.820887
$887$	29.6147	0.994366	0.497183	−	0.867646i	$-0.334368\pi$
$887$	29.6147	0.994366	0.497183	+	0.867646i	$0.334368\pi$
$888$	0	0
$889$	23.3685	0.783756
$890$	−21.4424	−0.718752
$891$	0	0
$892$	2.10760	0.0705677
$893$	30.2107	1.01096
$894$	0	0
$895$	81.1180	2.71148
$896$	−34.3604	−1.14790
$897$	0	0
$898$	−34.5947	−1.15444
$899$	−22.8155	−0.760939
$900$	0	0
$901$	−36.5915	−1.21904
$902$	7.15320	0.238175
$903$	0	0
$904$	14.1952	0.472124
$905$	22.3947	0.744426
$906$	0	0
$907$	−3.96710	−0.131725	−0.0658627	−	0.997829i	$-0.520980\pi$
$907$	−3.96710	−0.131725	−0.0658627	+	0.997829i	$0.520980\pi$
$908$	5.58447	0.185327
$909$	0	0
$910$	−30.4953	−1.01091
$911$	−25.8759	−0.857307	−0.428653	−	0.903469i	$-0.641012\pi$
$911$	−25.8759	−0.857307	−0.428653	+	0.903469i	$0.641012\pi$
$912$	0	0
$913$	−27.5889	−0.913060
$914$	12.1049	0.400394
$915$	0	0
$916$	6.86437	0.226805
$917$	−25.7868	−0.851556
$918$	0	0
$919$	16.5327	0.545364	0.272682	−	0.962104i	$-0.412089\pi$
$919$	16.5327	0.545364	0.272682	+	0.962104i	$0.412089\pi$
$920$	43.2825	1.42698
$921$	0	0
$922$	−16.4669	−0.542309
$923$	−21.0041	−0.691360
$924$	0	0
$925$	−66.4250	−2.18404
$926$	33.7383	1.10871
$927$	0	0
$928$	−6.46332	−0.212169
$929$	1.50189	0.0492755	0.0246378	−	0.999696i	$-0.492157\pi$
$929$	1.50189	0.0492755	0.0246378	+	0.999696i	$0.492157\pi$
$930$	0	0
$931$	−32.3236	−1.05936
$932$	−1.24113	−0.0406546
$933$	0	0
$934$	28.1964	0.922615
$935$	−97.9822	−3.20436
$936$	0	0
$937$	−8.04782	−0.262911	−0.131455	−	0.991322i	$-0.541965\pi$
$937$	−8.04782	−0.262911	−0.131455	+	0.991322i	$0.541965\pi$
$938$	−50.2483	−1.64067
$939$	0	0
$940$	5.66701	0.184837
$941$	2.66537	0.0868887	0.0434443	−	0.999056i	$-0.486167\pi$
$941$	2.66537	0.0868887	0.0434443	+	0.999056i	$0.486167\pi$
$942$	0	0
$943$	4.81995	0.156959
$944$	4.74597	0.154468
$945$	0	0
$946$	57.8397	1.88053
$947$	−39.0257	−1.26816	−0.634082	−	0.773266i	$-0.718622\pi$
$947$	−39.0257	−1.26816	−0.634082	+	0.773266i	$0.718622\pi$
$948$	0	0
$949$	−0.285458	−0.00926636
$950$	−36.8121	−1.19434
$951$	0	0
$952$	68.4786	2.21941
$953$	40.4838	1.31140	0.655700	−	0.755022i	$-0.272374\pi$
$953$	40.4838	1.31140	0.655700	+	0.755022i	$0.272374\pi$
$954$	0	0
$955$	−65.2338	−2.11092
$956$	0.495655	0.0160306
$957$	0	0
$958$	35.3564	1.14231
$959$	−38.1076	−1.23056
$960$	0	0
$961$	−9.85816	−0.318005
$962$	−23.5564	−0.759489
$963$	0	0
$964$	−0.231473	−0.00745524
$965$	21.6879	0.698159
$966$	0	0
$967$	26.3391	0.847009	0.423504	−	0.905894i	$-0.360800\pi$
$967$	26.3391	0.847009	0.423504	+	0.905894i	$0.360800\pi$
$968$	−35.1946	−1.13120
$969$	0	0
$970$	1.98778	0.0638238
$971$	9.17879	0.294561	0.147281	−	0.989095i	$-0.452948\pi$
$971$	9.17879	0.294561	0.147281	+	0.989095i	$0.452948\pi$
$972$	0	0
$973$	−12.5976	−0.403862
$974$	−22.7684	−0.729546
$975$	0	0
$976$	38.0328	1.21740
$977$	27.6516	0.884653	0.442326	−	0.896854i	$-0.354153\pi$
$977$	27.6516	0.884653	0.442326	+	0.896854i	$0.354153\pi$
$978$	0	0
$979$	22.6451	0.723740
$980$	−6.06335	−0.193687
$981$	0	0
$982$	6.80610	0.217191
$983$	27.0845	0.863861	0.431930	−	0.901907i	$-0.357833\pi$
$983$	27.0845	0.863861	0.431930	+	0.901907i	$0.357833\pi$
$984$	0	0
$985$	41.4485	1.32066
$986$	−39.7232	−1.26504
$987$	0	0
$988$	1.70866	0.0543598
$989$	38.9734	1.23928
$990$	0	0
$991$	−41.0027	−1.30249	−0.651246	−	0.758867i	$-0.725753\pi$
$991$	−41.0027	−1.30249	−0.651246	+	0.758867i	$0.725753\pi$
$992$	5.98920	0.190157
$993$	0	0
$994$	60.9355	1.93276
$995$	−3.84632	−0.121937
$996$	0	0
$997$	−37.9223	−1.20101	−0.600505	−	0.799621i	$-0.705034\pi$
$997$	−37.9223	−1.20101	−0.600505	+	0.799621i	$0.705034\pi$
$998$	38.9623	1.23333
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2169.2.a.h.1.10		12
3.2	odd	2		241.2.a.b.1.3	✓	12
12.11	even	2		3856.2.a.n.1.10		12
15.14	odd	2		6025.2.a.h.1.10		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
241.2.a.b.1.3	✓	12	3.2	odd	2
2169.2.a.h.1.10		12	1.1	even	1	trivial
3856.2.a.n.1.10		12	12.11	even	2
6025.2.a.h.1.10		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)

\(f(q)\)	\(=\)	\(q+1.32986 q^{2} -0.231473 q^{4} +3.40432 q^{5} -3.83334 q^{7} -2.96755 q^{8} +O(q^{10})\)	\(q+1.32986 q^{2} -0.231473 q^{4} +3.40432 q^{5} -3.83334 q^{7} -2.96755 q^{8} +4.52727 q^{10} -4.78120 q^{11} +1.75719 q^{13} -5.09781 q^{14} -3.48347 q^{16} +6.01977 q^{17} -4.20086 q^{19} -0.788008 q^{20} -6.35832 q^{22} -4.28435 q^{23} +6.58942 q^{25} +2.33681 q^{26} +0.887315 q^{28} -4.96202 q^{29} +4.59803 q^{31} +1.30256 q^{32} +8.00545 q^{34} -13.0499 q^{35} -10.0806 q^{37} -5.58655 q^{38} -10.1025 q^{40} -1.12501 q^{41} -9.09669 q^{43} +1.10672 q^{44} -5.69758 q^{46} -7.19156 q^{47} +7.69453 q^{49} +8.76300 q^{50} -0.406741 q^{52} -6.07855 q^{53} -16.2767 q^{55} +11.3756 q^{56} -6.59879 q^{58} -1.36242 q^{59} -10.9181 q^{61} +6.11473 q^{62} +8.69917 q^{64} +5.98204 q^{65} +9.85684 q^{67} -1.39341 q^{68} -17.3546 q^{70} -11.9533 q^{71} -0.162452 q^{73} -13.4057 q^{74} +0.972384 q^{76} +18.3280 q^{77} +14.5772 q^{79} -11.8589 q^{80} -1.49611 q^{82} +5.77030 q^{83} +20.4932 q^{85} -12.0973 q^{86} +14.1884 q^{88} -4.73628 q^{89} -6.73591 q^{91} +0.991709 q^{92} -9.56377 q^{94} -14.3011 q^{95} +0.439068 q^{97} +10.2326 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{2} + 13 q^{4} - 6 q^{5} + 3 q^{7} - 9 q^{8}+O(q^{10}) \) 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

Modular forms

Varieties

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Representations

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Database

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