LMFDB - Embedded newform 2169.2.a.h.1.8

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			1.8
Root			$-0.342147$ 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)

$f(q)$	$=$	$q+0.342147 q^{2} -1.88294 q^{4} +0.548903 q^{5} +1.82459 q^{7} -1.32853 q^{8} +O(q^{10})$	\(q+0.342147 q^{2} -1.88294 q^{4} +0.548903 q^{5} +1.82459 q^{7} -1.32853 q^{8} +0.187805 q^{10} -5.99218 q^{11} +3.70515 q^{13} +0.624278 q^{14} +3.31132 q^{16} +1.64451 q^{17} -3.15821 q^{19} -1.03355 q^{20} -2.05021 q^{22} +5.46015 q^{23} -4.69871 q^{25} +1.26771 q^{26} -3.43559 q^{28} -7.24801 q^{29} -9.41700 q^{31} +3.79003 q^{32} +0.562662 q^{34} +1.00152 q^{35} +1.27680 q^{37} -1.08057 q^{38} -0.729236 q^{40} +5.81239 q^{41} +7.82887 q^{43} +11.2829 q^{44} +1.86817 q^{46} -2.61568 q^{47} -3.67086 q^{49} -1.60765 q^{50} -6.97656 q^{52} -8.81076 q^{53} -3.28912 q^{55} -2.42403 q^{56} -2.47989 q^{58} +7.78270 q^{59} +1.03194 q^{61} -3.22200 q^{62} -5.32589 q^{64} +2.03377 q^{65} -8.39801 q^{67} -3.09650 q^{68} +0.342668 q^{70} -13.1691 q^{71} -13.3963 q^{73} +0.436852 q^{74} +5.94670 q^{76} -10.9333 q^{77} -10.6362 q^{79} +1.81759 q^{80} +1.98869 q^{82} -6.25636 q^{83} +0.902673 q^{85} +2.67862 q^{86} +7.96082 q^{88} +3.94129 q^{89} +6.76039 q^{91} -10.2811 q^{92} -0.894948 q^{94} -1.73355 q^{95} -2.22451 q^{97} -1.25598 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.342147	0.241934	0.120967	−	0.992657i	$-0.461400\pi$
$2$	0.342147	0.241934	0.120967	+	0.992657i	$0.461400\pi$
$3$	0	0
$3$	0	0
$4$	−1.88294	−0.941468
$5$	0.548903	0.245477	0.122738	−	0.992439i	$-0.460832\pi$
$5$	0.548903	0.245477	0.122738	+	0.992439i	$0.460832\pi$
$6$	0	0
$7$	1.82459	0.689631	0.344815	−	0.938671i	$-0.387942\pi$
$7$	1.82459	0.689631	0.344815	+	0.938671i	$0.387942\pi$
$8$	−1.32853	−0.469708
$9$	0	0
$10$	0.187805	0.0593893
$11$	−5.99218	−1.80671	−0.903355	−	0.428894i	$-0.858904\pi$
$11$	−5.99218	−1.80671	−0.903355	+	0.428894i	$0.858904\pi$
$12$	0	0
$13$	3.70515	1.02762	0.513812	−	0.857903i	$-0.328233\pi$
$13$	3.70515	1.02762	0.513812	+	0.857903i	$0.328233\pi$
$14$	0.624278	0.166845
$15$	0	0
$16$	3.31132	0.827829
$17$	1.64451	0.398851	0.199426	−	0.979913i	$-0.436092\pi$
$17$	1.64451	0.398851	0.199426	+	0.979913i	$0.436092\pi$
$18$	0	0
$19$	−3.15821	−0.724543	−0.362271	−	0.932073i	$-0.617999\pi$
$19$	−3.15821	−0.724543	−0.362271	+	0.932073i	$0.617999\pi$
$20$	−1.03355	−0.231108
$21$	0	0
$22$	−2.05021	−0.437105
$23$	5.46015	1.13852	0.569260	−	0.822158i	$-0.307230\pi$
$23$	5.46015	1.13852	0.569260	+	0.822158i	$0.307230\pi$
$24$	0	0
$25$	−4.69871	−0.939741
$26$	1.26771	0.248618
$27$	0	0
$28$	−3.43559	−0.649265
$29$	−7.24801	−1.34592	−0.672961	−	0.739678i	$-0.734978\pi$
$29$	−7.24801	−1.34592	−0.672961	+	0.739678i	$0.734978\pi$
$30$	0	0
$31$	−9.41700	−1.69134	−0.845672	−	0.533703i	$-0.820800\pi$
$31$	−9.41700	−1.69134	−0.845672	+	0.533703i	$0.820800\pi$
$32$	3.79003	0.669988
$33$	0	0
$34$	0.562662	0.0964958
$35$	1.00152	0.169288
$36$	0	0
$37$	1.27680	0.209904	0.104952	−	0.994477i	$-0.466531\pi$
$37$	1.27680	0.209904	0.104952	+	0.994477i	$0.466531\pi$
$38$	−1.08057	−0.175292
$39$	0	0
$40$	−0.729236	−0.115302
$41$	5.81239	0.907743	0.453872	−	0.891067i	$-0.350042\pi$
$41$	5.81239	0.907743	0.453872	+	0.891067i	$0.350042\pi$
$42$	0	0
$43$	7.82887	1.19389	0.596946	−	0.802281i	$-0.296381\pi$
$43$	7.82887	1.19389	0.596946	+	0.802281i	$0.296381\pi$
$44$	11.2829	1.70096
$45$	0	0
$46$	1.86817	0.275447
$47$	−2.61568	−0.381537	−0.190768	−	0.981635i	$-0.561098\pi$
$47$	−2.61568	−0.381537	−0.190768	+	0.981635i	$0.561098\pi$
$48$	0	0
$49$	−3.67086	−0.524409
$50$	−1.60765	−0.227356
$51$	0	0
$52$	−6.97656	−0.967475
$53$	−8.81076	−1.21025	−0.605126	−	0.796130i	$-0.706877\pi$
$53$	−8.81076	−1.21025	−0.605126	+	0.796130i	$0.706877\pi$
$54$	0	0
$55$	−3.28912	−0.443505
$56$	−2.42403	−0.323925
$57$	0	0
$58$	−2.47989	−0.325625
$59$	7.78270	1.01322	0.506610	−	0.862175i	$-0.330898\pi$
$59$	7.78270	1.01322	0.506610	+	0.862175i	$0.330898\pi$
$60$	0	0
$61$	1.03194	0.132127	0.0660635	−	0.997815i	$-0.478956\pi$
$61$	1.03194	0.132127	0.0660635	+	0.997815i	$0.478956\pi$
$62$	−3.22200	−0.409194
$63$	0	0
$64$	−5.32589	−0.665736
$65$	2.03377	0.252258
$66$	0	0
$67$	−8.39801	−1.02598	−0.512990	−	0.858395i	$-0.671462\pi$
$67$	−8.39801	−1.02598	−0.512990	+	0.858395i	$0.671462\pi$
$68$	−3.09650	−0.375505
$69$	0	0
$70$	0.342668	0.0409567
$71$	−13.1691	−1.56288	−0.781440	−	0.623981i	$-0.785514\pi$
$71$	−13.1691	−1.56288	−0.781440	+	0.623981i	$0.785514\pi$
$72$	0	0
$73$	−13.3963	−1.56792	−0.783959	−	0.620813i	$-0.786803\pi$
$73$	−13.3963	−1.56792	−0.783959	+	0.620813i	$0.786803\pi$
$74$	0.436852	0.0507830
$75$	0	0
$76$	5.94670	0.682133
$77$	−10.9333	−1.24596
$78$	0	0
$79$	−10.6362	−1.19666	−0.598332	−	0.801248i	$-0.704170\pi$
$79$	−10.6362	−1.19666	−0.598332	+	0.801248i	$0.704170\pi$
$80$	1.81759	0.203213
$81$	0	0
$82$	1.98869	0.219614
$83$	−6.25636	−0.686725	−0.343362	−	0.939203i	$-0.611566\pi$
$83$	−6.25636	−0.686725	−0.343362	+	0.939203i	$0.611566\pi$
$84$	0	0
$85$	0.902673	0.0979087
$86$	2.67862	0.288844
$87$	0	0
$88$	7.96082	0.848626
$89$	3.94129	0.417776	0.208888	−	0.977940i	$-0.433016\pi$
$89$	3.94129	0.417776	0.208888	+	0.977940i	$0.433016\pi$
$90$	0	0
$91$	6.76039	0.708681
$92$	−10.2811	−1.07188
$93$	0	0
$94$	−0.894948	−0.0923069
$95$	−1.73355	−0.177858
$96$	0	0
$97$	−2.22451	−0.225865	−0.112933	−	0.993603i	$-0.536024\pi$
$97$	−2.22451	−0.225865	−0.112933	+	0.993603i	$0.536024\pi$
$98$	−1.25598	−0.126873
$99$	0	0
$100$	8.84736	0.884736
$101$	4.24027	0.421923	0.210962	−	0.977494i	$-0.432341\pi$
$101$	4.24027	0.421923	0.210962	+	0.977494i	$0.432341\pi$
$102$	0	0
$103$	0.164858	0.0162439	0.00812196	−	0.999967i	$-0.497415\pi$
$103$	0.164858	0.0162439	0.00812196	+	0.999967i	$0.497415\pi$
$104$	−4.92242	−0.482683
$105$	0	0
$106$	−3.01458	−0.292801
$107$	−9.28536	−0.897650	−0.448825	−	0.893620i	$-0.648157\pi$
$107$	−9.28536	−0.897650	−0.448825	+	0.893620i	$0.648157\pi$
$108$	0	0
$109$	12.1952	1.16809	0.584044	−	0.811722i	$-0.301470\pi$
$109$	12.1952	1.16809	0.584044	+	0.811722i	$0.301470\pi$
$110$	−1.12536	−0.107299
$111$	0	0
$112$	6.04180	0.570897
$113$	−13.1834	−1.24019	−0.620093	−	0.784528i	$-0.712905\pi$
$113$	−13.1834	−1.24019	−0.620093	+	0.784528i	$0.712905\pi$
$114$	0	0
$115$	2.99709	0.279480
$116$	13.6475	1.26714
$117$	0	0
$118$	2.66283	0.245133
$119$	3.00055	0.275060
$120$	0	0
$121$	24.9062	2.26420
$122$	0.353077	0.0319661
$123$	0	0
$124$	17.7316	1.59235
$125$	−5.32365	−0.476161
$126$	0	0
$127$	−11.7563	−1.04321	−0.521603	−	0.853188i	$-0.674666\pi$
$127$	−11.7563	−1.04321	−0.521603	+	0.853188i	$0.674666\pi$
$128$	−9.40229	−0.831053
$129$	0	0
$130$	0.695847	0.0610298
$131$	−14.0839	−1.23051	−0.615257	−	0.788326i	$-0.710948\pi$
$131$	−14.0839	−1.23051	−0.615257	+	0.788326i	$0.710948\pi$
$132$	0	0
$133$	−5.76244	−0.499667
$134$	−2.87335	−0.248220
$135$	0	0
$136$	−2.18478	−0.187343
$137$	12.3993	1.05935	0.529673	−	0.848202i	$-0.322315\pi$
$137$	12.3993	1.05935	0.529673	+	0.848202i	$0.322315\pi$
$138$	0	0
$139$	−10.5456	−0.894464	−0.447232	−	0.894418i	$-0.647590\pi$
$139$	−10.5456	−0.894464	−0.447232	+	0.894418i	$0.647590\pi$
$140$	−1.88580	−0.159380
$141$	0	0
$142$	−4.50575	−0.378114
$143$	−22.2019	−1.85662
$144$	0	0
$145$	−3.97845	−0.330393
$146$	−4.58350	−0.379333
$147$	0	0
$148$	−2.40412	−0.197618
$149$	9.30604	0.762380	0.381190	−	0.924497i	$-0.375514\pi$
$149$	9.30604	0.762380	0.381190	+	0.924497i	$0.375514\pi$
$150$	0	0
$151$	10.2870	0.837143	0.418572	−	0.908184i	$-0.362531\pi$
$151$	10.2870	0.837143	0.418572	+	0.908184i	$0.362531\pi$
$152$	4.19579	0.340323
$153$	0	0
$154$	−3.74079	−0.301441
$155$	−5.16902	−0.415186
$156$	0	0
$157$	−4.23848	−0.338268	−0.169134	−	0.985593i	$-0.554097\pi$
$157$	−4.23848	−0.338268	−0.169134	+	0.985593i	$0.554097\pi$
$158$	−3.63914	−0.289514
$159$	0	0
$160$	2.08036	0.164467
$161$	9.96254	0.785159
$162$	0	0
$163$	−6.42582	−0.503309	−0.251655	−	0.967817i	$-0.580975\pi$
$163$	−6.42582	−0.503309	−0.251655	+	0.967817i	$0.580975\pi$
$164$	−10.9444	−0.854611
$165$	0	0
$166$	−2.14060	−0.166142
$167$	−3.96260	−0.306636	−0.153318	−	0.988177i	$-0.548996\pi$
$167$	−3.96260	−0.306636	−0.153318	+	0.988177i	$0.548996\pi$
$168$	0	0
$169$	0.728137	0.0560105
$170$	0.308847	0.0236875
$171$	0	0
$172$	−14.7413	−1.12401
$173$	8.45221	0.642610	0.321305	−	0.946976i	$-0.395879\pi$
$173$	8.45221	0.642610	0.321305	+	0.946976i	$0.395879\pi$
$174$	0	0
$175$	−8.57322	−0.648075
$176$	−19.8420	−1.49565
$177$	0	0
$178$	1.34850	0.101074
$179$	10.5847	0.791138	0.395569	−	0.918436i	$-0.370547\pi$
$179$	10.5847	0.791138	0.395569	+	0.918436i	$0.370547\pi$
$180$	0	0
$181$	−19.4914	−1.44879	−0.724394	−	0.689387i	$-0.757880\pi$
$181$	−19.4914	−1.44879	−0.724394	+	0.689387i	$0.757880\pi$
$182$	2.31305	0.171454
$183$	0	0
$184$	−7.25400	−0.534772
$185$	0.700837	0.0515265
$186$	0	0
$187$	−9.85417	−0.720608
$188$	4.92517	0.359205
$189$	0	0
$190$	−0.593128	−0.0430301
$191$	2.21537	0.160298	0.0801491	−	0.996783i	$-0.474460\pi$
$191$	2.21537	0.160298	0.0801491	+	0.996783i	$0.474460\pi$
$192$	0	0
$193$	−8.84755	−0.636861	−0.318430	−	0.947946i	$-0.603156\pi$
$193$	−8.84755	−0.636861	−0.318430	+	0.947946i	$0.603156\pi$
$194$	−0.761111	−0.0546446
$195$	0	0
$196$	6.91200	0.493714
$197$	−6.19744	−0.441549	−0.220775	−	0.975325i	$-0.570859\pi$
$197$	−6.19744	−0.441549	−0.220775	+	0.975325i	$0.570859\pi$
$198$	0	0
$199$	20.2303	1.43409	0.717043	−	0.697028i	$-0.245495\pi$
$199$	20.2303	1.43409	0.717043	+	0.697028i	$0.245495\pi$
$200$	6.24239	0.441404
$201$	0	0
$202$	1.45080	0.102078
$203$	−13.2247	−0.928190
$204$	0	0
$205$	3.19044	0.222830
$206$	0.0564056	0.00392996
$207$	0	0
$208$	12.2689	0.850697
$209$	18.9245	1.30904
$210$	0	0
$211$	1.96502	0.135278	0.0676388	−	0.997710i	$-0.478453\pi$
$211$	1.96502	0.135278	0.0676388	+	0.997710i	$0.478453\pi$
$212$	16.5901	1.13941
$213$	0	0
$214$	−3.17696	−0.217172
$215$	4.29729	0.293073
$216$	0	0
$217$	−17.1822	−1.16640
$218$	4.17255	0.282600
$219$	0	0
$220$	6.19321	0.417546
$221$	6.09314	0.409869
$222$	0	0
$223$	21.4906	1.43911	0.719557	−	0.694433i	$-0.244345\pi$
$223$	21.4906	1.43911	0.719557	+	0.694433i	$0.244345\pi$
$224$	6.91525	0.462045
$225$	0	0
$226$	−4.51065	−0.300044
$227$	−20.7806	−1.37925	−0.689627	−	0.724165i	$-0.742226\pi$
$227$	−20.7806	−1.37925	−0.689627	+	0.724165i	$0.742226\pi$
$228$	0	0
$229$	28.4761	1.88176	0.940878	−	0.338746i	$-0.110003\pi$
$229$	28.4761	1.88176	0.940878	+	0.338746i	$0.110003\pi$
$230$	1.02545	0.0676159
$231$	0	0
$232$	9.62924	0.632190
$233$	2.19220	0.143616	0.0718079	−	0.997418i	$-0.477123\pi$
$233$	2.19220	0.143616	0.0718079	+	0.997418i	$0.477123\pi$
$234$	0	0
$235$	−1.43576	−0.0936584
$236$	−14.6543	−0.953915
$237$	0	0
$238$	1.02663	0.0665465
$239$	−25.2449	−1.63296	−0.816478	−	0.577377i	$-0.804076\pi$
$239$	−25.2449	−1.63296	−0.816478	+	0.577377i	$0.804076\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	8.52158	0.547788
$243$	0	0
$244$	−1.94309	−0.124393
$245$	−2.01495	−0.128730
$246$	0	0
$247$	−11.7016	−0.744557
$248$	12.5108	0.794437
$249$	0	0
$250$	−1.82147	−0.115200
$251$	−21.2167	−1.33919	−0.669593	−	0.742728i	$-0.733532\pi$
$251$	−21.2167	−1.33919	−0.669593	+	0.742728i	$0.733532\pi$
$252$	0	0
$253$	−32.7182	−2.05698
$254$	−4.02240	−0.252388
$255$	0	0
$256$	7.43481	0.464676
$257$	−19.1130	−1.19224	−0.596119	−	0.802896i	$-0.703292\pi$
$257$	−19.1130	−1.19224	−0.596119	+	0.802896i	$0.703292\pi$
$258$	0	0
$259$	2.32963	0.144756
$260$	−3.82945	−0.237493
$261$	0	0
$262$	−4.81876	−0.297704
$263$	12.7792	0.787999	0.394000	−	0.919111i	$-0.371091\pi$
$263$	12.7792	0.787999	0.394000	+	0.919111i	$0.371091\pi$
$264$	0	0
$265$	−4.83625	−0.297089
$266$	−1.97160	−0.120887
$267$	0	0
$268$	15.8129	0.965927
$269$	−1.65545	−0.100934	−0.0504672	−	0.998726i	$-0.516071\pi$
$269$	−1.65545	−0.100934	−0.0504672	+	0.998726i	$0.516071\pi$
$270$	0	0
$271$	−17.0226	−1.03405	−0.517023	−	0.855971i	$-0.672960\pi$
$271$	−17.0226	−1.03405	−0.517023	+	0.855971i	$0.672960\pi$
$272$	5.44548	0.330181
$273$	0	0
$274$	4.24239	0.256292
$275$	28.1555	1.69784
$276$	0	0
$277$	16.8245	1.01089	0.505443	−	0.862860i	$-0.331329\pi$
$277$	16.8245	1.01089	0.505443	+	0.862860i	$0.331329\pi$
$278$	−3.60814	−0.216402
$279$	0	0
$280$	−1.33056	−0.0795161
$281$	16.1664	0.964405	0.482203	−	0.876060i	$-0.339837\pi$
$281$	16.1664	0.964405	0.482203	+	0.876060i	$0.339837\pi$
$282$	0	0
$283$	2.53878	0.150915	0.0754573	−	0.997149i	$-0.475958\pi$
$283$	2.53878	0.150915	0.0754573	+	0.997149i	$0.475958\pi$
$284$	24.7965	1.47140
$285$	0	0
$286$	−7.59632	−0.449180
$287$	10.6052	0.626008
$288$	0	0
$289$	−14.2956	−0.840918
$290$	−1.36122	−0.0799334
$291$	0	0
$292$	25.2243	1.47614
$293$	22.8971	1.33767	0.668833	−	0.743413i	$-0.266794\pi$
$293$	22.8971	1.33767	0.668833	+	0.743413i	$0.266794\pi$
$294$	0	0
$295$	4.27194	0.248722
$296$	−1.69627	−0.0985935
$297$	0	0
$298$	3.18403	0.184446
$299$	20.2307	1.16997
$300$	0	0
$301$	14.2845	0.823345
$302$	3.51966	0.202534
$303$	0	0
$304$	−10.4578	−0.599798
$305$	0.566437	0.0324341
$306$	0	0
$307$	−4.71512	−0.269106	−0.134553	−	0.990906i	$-0.542960\pi$
$307$	−4.71512	−0.269106	−0.134553	+	0.990906i	$0.542960\pi$
$308$	20.5867	1.17303
$309$	0	0
$310$	−1.76856	−0.100448
$311$	−4.63908	−0.263058	−0.131529	−	0.991312i	$-0.541989\pi$
$311$	−4.63908	−0.263058	−0.131529	+	0.991312i	$0.541989\pi$
$312$	0	0
$313$	−5.04057	−0.284910	−0.142455	−	0.989801i	$-0.545500\pi$
$313$	−5.04057	−0.284910	−0.142455	+	0.989801i	$0.545500\pi$
$314$	−1.45018	−0.0818386
$315$	0	0
$316$	20.0272	1.12662
$317$	28.0992	1.57821	0.789105	−	0.614259i	$-0.210545\pi$
$317$	28.0992	1.57821	0.789105	+	0.614259i	$0.210545\pi$
$318$	0	0
$319$	43.4314	2.43169
$320$	−2.92339	−0.163423
$321$	0	0
$322$	3.40865	0.189957
$323$	−5.19369	−0.288985
$324$	0	0
$325$	−17.4094	−0.965700
$326$	−2.19858	−0.121768
$327$	0	0
$328$	−7.72197	−0.426374
$329$	−4.77256	−0.263120
$330$	0	0
$331$	29.6777	1.63124	0.815618	−	0.578590i	$-0.196397\pi$
$331$	29.6777	1.63124	0.815618	+	0.578590i	$0.196397\pi$
$332$	11.7803	0.646529
$333$	0	0
$334$	−1.35579	−0.0741857
$335$	−4.60969	−0.251854
$336$	0	0
$337$	−12.2225	−0.665799	−0.332900	−	0.942962i	$-0.608027\pi$
$337$	−12.2225	−0.665799	−0.332900	+	0.942962i	$0.608027\pi$
$338$	0.249130	0.0135509
$339$	0	0
$340$	−1.69968	−0.0921779
$341$	56.4284	3.05577
$342$	0	0
$343$	−19.4700	−1.05128
$344$	−10.4009	−0.560780
$345$	0	0
$346$	2.89190	0.155469
$347$	17.9815	0.965296	0.482648	−	0.875814i	$-0.339675\pi$
$347$	17.9815	0.965296	0.482648	+	0.875814i	$0.339675\pi$
$348$	0	0
$349$	26.2848	1.40699	0.703496	−	0.710699i	$-0.251621\pi$
$349$	26.2848	1.40699	0.703496	+	0.710699i	$0.251621\pi$
$350$	−2.93330	−0.156792
$351$	0	0
$352$	−22.7105	−1.21047
$353$	13.7610	0.732424	0.366212	−	0.930531i	$-0.380654\pi$
$353$	13.7610	0.732424	0.366212	+	0.930531i	$0.380654\pi$
$354$	0	0
$355$	−7.22853	−0.383651
$356$	−7.42120	−0.393323
$357$	0	0
$358$	3.62152	0.191403
$359$	1.32555	0.0699598	0.0349799	−	0.999388i	$-0.488863\pi$
$359$	1.32555	0.0699598	0.0349799	+	0.999388i	$0.488863\pi$
$360$	0	0
$361$	−9.02572	−0.475038
$362$	−6.66894	−0.350511
$363$	0	0
$364$	−12.7294	−0.667200
$365$	−7.35326	−0.384887
$366$	0	0
$367$	6.05979	0.316319	0.158159	−	0.987414i	$-0.449444\pi$
$367$	6.05979	0.316319	0.158159	+	0.987414i	$0.449444\pi$
$368$	18.0803	0.942500
$369$	0	0
$370$	0.239789	0.0124660
$371$	−16.0760	−0.834627
$372$	0	0
$373$	20.4445	1.05858	0.529288	−	0.848442i	$-0.322459\pi$
$373$	20.4445	1.05858	0.529288	+	0.848442i	$0.322459\pi$
$374$	−3.37157	−0.174340
$375$	0	0
$376$	3.47503	0.179211
$377$	−26.8550	−1.38310
$378$	0	0
$379$	35.4042	1.81859	0.909296	−	0.416150i	$-0.136621\pi$
$379$	35.4042	1.81859	0.909296	+	0.416150i	$0.136621\pi$
$380$	3.26416	0.167448
$381$	0	0
$382$	0.757981	0.0387817
$383$	−6.79327	−0.347120	−0.173560	−	0.984823i	$-0.555527\pi$
$383$	−6.79327	−0.347120	−0.173560	+	0.984823i	$0.555527\pi$
$384$	0	0
$385$	−6.00131	−0.305855
$386$	−3.02716	−0.154079
$387$	0	0
$388$	4.18862	0.212645
$389$	−24.6869	−1.25168	−0.625838	−	0.779953i	$-0.715243\pi$
$389$	−24.6869	−1.25168	−0.625838	+	0.779953i	$0.715243\pi$
$390$	0	0
$391$	8.97924	0.454100
$392$	4.87687	0.246319
$393$	0	0
$394$	−2.12043	−0.106826
$395$	−5.83823	−0.293753
$396$	0	0
$397$	14.3804	0.721730	0.360865	−	0.932618i	$-0.382482\pi$
$397$	14.3804	0.721730	0.360865	+	0.932618i	$0.382482\pi$
$398$	6.92173	0.346955
$399$	0	0
$400$	−15.5589	−0.777945
$401$	−7.62276	−0.380663	−0.190331	−	0.981720i	$-0.560956\pi$
$401$	−7.62276	−0.380663	−0.190331	+	0.981720i	$0.560956\pi$
$402$	0	0
$403$	−34.8914	−1.73806
$404$	−7.98416	−0.397227
$405$	0	0
$406$	−4.52478	−0.224561
$407$	−7.65079	−0.379236
$408$	0	0
$409$	5.12423	0.253377	0.126688	−	0.991943i	$-0.459565\pi$
$409$	5.12423	0.253377	0.126688	+	0.991943i	$0.459565\pi$
$410$	1.09160	0.0539102
$411$	0	0
$412$	−0.310416	−0.0152931
$413$	14.2002	0.698748
$414$	0	0
$415$	−3.43413	−0.168575
$416$	14.0426	0.688496
$417$	0	0
$418$	6.47498	0.316701
$419$	−13.3513	−0.652254	−0.326127	−	0.945326i	$-0.605744\pi$
$419$	−13.3513	−0.652254	−0.326127	+	0.945326i	$0.605744\pi$
$420$	0	0
$421$	27.8059	1.35518	0.677588	−	0.735442i	$-0.263025\pi$
$421$	27.8059	1.35518	0.677588	+	0.735442i	$0.263025\pi$
$422$	0.672326	0.0327283
$423$	0	0
$424$	11.7054	0.568465
$425$	−7.72705	−0.374817
$426$	0	0
$427$	1.88288	0.0911188
$428$	17.4837	0.845109
$429$	0	0
$430$	1.47030	0.0709044
$431$	−18.5717	−0.894567	−0.447283	−	0.894392i	$-0.647608\pi$
$431$	−18.5717	−0.894567	−0.447283	+	0.894392i	$0.647608\pi$
$432$	0	0
$433$	−7.19376	−0.345710	−0.172855	−	0.984947i	$-0.555299\pi$
$433$	−7.19376	−0.345710	−0.172855	+	0.984947i	$0.555299\pi$
$434$	−5.87883	−0.282193
$435$	0	0
$436$	−22.9628	−1.09972
$437$	−17.2443	−0.824906
$438$	0	0
$439$	−16.2866	−0.777315	−0.388657	−	0.921382i	$-0.627061\pi$
$439$	−16.2866	−0.777315	−0.388657	+	0.921382i	$0.627061\pi$
$440$	4.36971	0.208318
$441$	0	0
$442$	2.08475	0.0991614
$443$	37.0915	1.76227	0.881135	−	0.472864i	$-0.156780\pi$
$443$	37.0915	1.76227	0.881135	+	0.472864i	$0.156780\pi$
$444$	0	0
$445$	2.16338	0.102554
$446$	7.35293	0.348171
$447$	0	0
$448$	−9.71757	−0.459112
$449$	17.1059	0.807275	0.403638	−	0.914919i	$-0.367746\pi$
$449$	17.1059	0.807275	0.403638	+	0.914919i	$0.367746\pi$
$450$	0	0
$451$	−34.8289	−1.64003
$452$	24.8234	1.16760
$453$	0	0
$454$	−7.11001	−0.333689
$455$	3.71079	0.173965
$456$	0	0
$457$	−9.35857	−0.437776	−0.218888	−	0.975750i	$-0.570243\pi$
$457$	−9.35857	−0.437776	−0.218888	+	0.975750i	$0.570243\pi$
$458$	9.74302	0.455262
$459$	0	0
$460$	−5.64333	−0.263122
$461$	15.8670	0.738999	0.369500	−	0.929231i	$-0.379529\pi$
$461$	15.8670	0.738999	0.369500	+	0.929231i	$0.379529\pi$
$462$	0	0
$463$	−6.99749	−0.325201	−0.162601	−	0.986692i	$-0.551988\pi$
$463$	−6.99749	−0.325201	−0.162601	+	0.986692i	$0.551988\pi$
$464$	−24.0005	−1.11419
$465$	0	0
$466$	0.750055	0.0347456
$467$	24.2782	1.12346	0.561729	−	0.827321i	$-0.310136\pi$
$467$	24.2782	1.12346	0.561729	+	0.827321i	$0.310136\pi$
$468$	0	0
$469$	−15.3229	−0.707547
$470$	−0.491240	−0.0226592
$471$	0	0
$472$	−10.3396	−0.475918
$473$	−46.9120	−2.15702
$474$	0	0
$475$	14.8395	0.680882
$476$	−5.64984	−0.258960
$477$	0	0
$478$	−8.63746	−0.395068
$479$	−8.93171	−0.408100	−0.204050	−	0.978960i	$-0.565411\pi$
$479$	−8.93171	−0.408100	−0.204050	+	0.978960i	$0.565411\pi$
$480$	0	0
$481$	4.73072	0.215702
$482$	0.342147	0.0155844
$483$	0	0
$484$	−46.8968	−2.13167
$485$	−1.22104	−0.0554447
$486$	0	0
$487$	34.5963	1.56771	0.783855	−	0.620944i	$-0.213250\pi$
$487$	34.5963	1.56771	0.783855	+	0.620944i	$0.213250\pi$
$488$	−1.37097	−0.0620611
$489$	0	0
$490$	−0.689408	−0.0311443
$491$	−34.3798	−1.55154	−0.775770	−	0.631016i	$-0.782638\pi$
$491$	−34.3798	−1.55154	−0.775770	+	0.631016i	$0.782638\pi$
$492$	0	0
$493$	−11.9194	−0.536823
$494$	−4.00368	−0.180134
$495$	0	0
$496$	−31.1827	−1.40014
$497$	−24.0281	−1.07781
$498$	0	0
$499$	38.3126	1.71511	0.857554	−	0.514394i	$-0.171983\pi$
$499$	38.3126	1.71511	0.857554	+	0.514394i	$0.171983\pi$
$500$	10.0241	0.448291
$501$	0	0
$502$	−7.25923	−0.323995
$503$	12.6073	0.562132	0.281066	−	0.959688i	$-0.409312\pi$
$503$	12.6073	0.562132	0.281066	+	0.959688i	$0.409312\pi$
$504$	0	0
$505$	2.32750	0.103572
$506$	−11.1944	−0.497653
$507$	0	0
$508$	22.1364	0.982146
$509$	19.0748	0.845476	0.422738	−	0.906252i	$-0.361069\pi$
$509$	19.0748	0.845476	0.422738	+	0.906252i	$0.361069\pi$
$510$	0	0
$511$	−24.4428	−1.08128
$512$	21.3484	0.943474
$513$	0	0
$514$	−6.53947	−0.288444
$515$	0.0904909	0.00398750
$516$	0	0
$517$	15.6737	0.689326
$518$	0.797076	0.0350215
$519$	0	0
$520$	−2.70193	−0.118487
$521$	−11.5960	−0.508028	−0.254014	−	0.967201i	$-0.581751\pi$
$521$	−11.5960	−0.508028	−0.254014	+	0.967201i	$0.581751\pi$
$522$	0	0
$523$	−26.2727	−1.14883	−0.574413	−	0.818566i	$-0.694770\pi$
$523$	−26.2727	−1.14883	−0.574413	+	0.818566i	$0.694770\pi$
$524$	26.5190	1.15849
$525$	0	0
$526$	4.37236	0.190644
$527$	−15.4863	−0.674594
$528$	0	0
$529$	6.81324	0.296228
$530$	−1.65471	−0.0718760
$531$	0	0
$532$	10.8503	0.470420
$533$	21.5358	0.932819
$534$	0	0
$535$	−5.09676	−0.220352
$536$	11.1570	0.481911
$537$	0	0
$538$	−0.566406	−0.0244195
$539$	21.9965	0.947455
$540$	0	0
$541$	−15.1441	−0.651097	−0.325548	−	0.945525i	$-0.605549\pi$
$541$	−15.1441	−0.651097	−0.325548	+	0.945525i	$0.605549\pi$
$542$	−5.82421	−0.250171
$543$	0	0
$544$	6.23272	0.267226
$545$	6.69397	0.286738
$546$	0	0
$547$	34.7331	1.48508	0.742540	−	0.669802i	$-0.233621\pi$
$547$	34.7331	1.48508	0.742540	+	0.669802i	$0.233621\pi$
$548$	−23.3471	−0.997339
$549$	0	0
$550$	9.63331	0.410766
$551$	22.8907	0.975178
$552$	0	0
$553$	−19.4067	−0.825256
$554$	5.75645	0.244568
$555$	0	0
$556$	19.8566	0.842109
$557$	−34.1470	−1.44685	−0.723427	−	0.690401i	$-0.757434\pi$
$557$	−34.1470	−1.44685	−0.723427	+	0.690401i	$0.757434\pi$
$558$	0	0
$559$	29.0071	1.22687
$560$	3.31636	0.140142
$561$	0	0
$562$	5.53128	0.233323
$563$	19.2900	0.812975	0.406488	−	0.913656i	$-0.366753\pi$
$563$	19.2900	0.812975	0.406488	+	0.913656i	$0.366753\pi$
$564$	0	0
$565$	−7.23638	−0.304437
$566$	0.868635	0.0365114
$567$	0	0
$568$	17.4955	0.734097
$569$	33.4263	1.40130	0.700652	−	0.713503i	$-0.252893\pi$
$569$	33.4263	1.40130	0.700652	+	0.713503i	$0.252893\pi$
$570$	0	0
$571$	3.07936	0.128867	0.0644336	−	0.997922i	$-0.479476\pi$
$571$	3.07936	0.128867	0.0644336	+	0.997922i	$0.479476\pi$
$572$	41.8048	1.74795
$573$	0	0
$574$	3.62855	0.151453
$575$	−25.6556	−1.06991
$576$	0	0
$577$	−5.95435	−0.247883	−0.123941	−	0.992290i	$-0.539553\pi$
$577$	−5.95435	−0.247883	−0.123941	+	0.992290i	$0.539553\pi$
$578$	−4.89120	−0.203447
$579$	0	0
$580$	7.49117	0.311054
$581$	−11.4153	−0.473587
$582$	0	0
$583$	52.7957	2.18657
$584$	17.7974	0.736463
$585$	0	0
$586$	7.83419	0.323627
$587$	−42.0432	−1.73531	−0.867655	−	0.497167i	$-0.834374\pi$
$587$	−42.0432	−1.73531	−0.867655	+	0.497167i	$0.834374\pi$
$588$	0	0
$589$	29.7409	1.22545
$590$	1.46163	0.0601745
$591$	0	0
$592$	4.22788	0.173765
$593$	−40.9908	−1.68329	−0.841645	−	0.540031i	$-0.818413\pi$
$593$	−40.9908	−1.68329	−0.841645	+	0.540031i	$0.818413\pi$
$594$	0	0
$595$	1.64701	0.0675208
$596$	−17.5227	−0.717757
$597$	0	0
$598$	6.92186	0.283056
$599$	27.0862	1.10671	0.553356	−	0.832945i	$-0.313347\pi$
$599$	27.0862	1.10671	0.553356	+	0.832945i	$0.313347\pi$
$600$	0	0
$601$	−3.03033	−0.123610	−0.0618049	−	0.998088i	$-0.519686\pi$
$601$	−3.03033	−0.123610	−0.0618049	+	0.998088i	$0.519686\pi$
$602$	4.88740	0.199195
$603$	0	0
$604$	−19.3697	−0.788144
$605$	13.6711	0.555809
$606$	0	0
$607$	0.560838	0.0227637	0.0113818	−	0.999935i	$-0.496377\pi$
$607$	0.560838	0.0227637	0.0113818	+	0.999935i	$0.496377\pi$
$608$	−11.9697	−0.485435
$609$	0	0
$610$	0.193805	0.00784693
$611$	−9.69150	−0.392076
$612$	0	0
$613$	23.7388	0.958801	0.479401	−	0.877596i	$-0.340854\pi$
$613$	23.7388	0.958801	0.479401	+	0.877596i	$0.340854\pi$
$614$	−1.61326	−0.0651061
$615$	0	0
$616$	14.5252	0.585239
$617$	10.2411	0.412290	0.206145	−	0.978521i	$-0.433908\pi$
$617$	10.2411	0.412290	0.206145	+	0.978521i	$0.433908\pi$
$618$	0	0
$619$	−27.8348	−1.11878	−0.559388	−	0.828906i	$-0.688964\pi$
$619$	−27.8348	−1.11878	−0.559388	+	0.828906i	$0.688964\pi$
$620$	9.73293	0.390884
$621$	0	0
$622$	−1.58725	−0.0636428
$623$	7.19125	0.288111
$624$	0	0
$625$	20.5714	0.822855
$626$	−1.72462	−0.0689295
$627$	0	0
$628$	7.98079	0.318468
$629$	2.09970	0.0837204
$630$	0	0
$631$	23.0564	0.917861	0.458931	−	0.888472i	$-0.348233\pi$
$631$	23.0564	0.917861	0.458931	+	0.888472i	$0.348233\pi$
$632$	14.1305	0.562082
$633$	0	0
$634$	9.61406	0.381823
$635$	−6.45309	−0.256083
$636$	0	0
$637$	−13.6011	−0.538895
$638$	14.8599	0.588310
$639$	0	0
$640$	−5.16094	−0.204004
$641$	−35.6143	−1.40668	−0.703340	−	0.710853i	$-0.748309\pi$
$641$	−35.6143	−1.40668	−0.703340	+	0.710853i	$0.748309\pi$
$642$	0	0
$643$	−28.2806	−1.11528	−0.557639	−	0.830084i	$-0.688293\pi$
$643$	−28.2806	−1.11528	−0.557639	+	0.830084i	$0.688293\pi$
$644$	−18.7588	−0.739201
$645$	0	0
$646$	−1.77700	−0.0699153
$647$	−14.5953	−0.573802	−0.286901	−	0.957960i	$-0.592625\pi$
$647$	−14.5953	−0.573802	−0.286901	+	0.957960i	$0.592625\pi$
$648$	0	0
$649$	−46.6353	−1.83060
$650$	−5.95658	−0.233636
$651$	0	0
$652$	12.0994	0.473850
$653$	28.0718	1.09853	0.549267	−	0.835647i	$-0.314907\pi$
$653$	28.0718	1.09853	0.549267	+	0.835647i	$0.314907\pi$
$654$	0	0
$655$	−7.73068	−0.302063
$656$	19.2467	0.751457
$657$	0	0
$658$	−1.63292	−0.0636577
$659$	2.33181	0.0908343	0.0454172	−	0.998968i	$-0.485538\pi$
$659$	2.33181	0.0908343	0.0454172	+	0.998968i	$0.485538\pi$
$660$	0	0
$661$	−21.3409	−0.830063	−0.415031	−	0.909807i	$-0.636229\pi$
$661$	−21.3409	−0.830063	−0.415031	+	0.909807i	$0.636229\pi$
$662$	10.1542	0.394652
$663$	0	0
$664$	8.31179	0.322560
$665$	−3.16302	−0.122657
$666$	0	0
$667$	−39.5752	−1.53236
$668$	7.46133	0.288687
$669$	0	0
$670$	−1.57719	−0.0609322
$671$	−6.18360	−0.238715
$672$	0	0
$673$	−25.0948	−0.967333	−0.483667	−	0.875252i	$-0.660695\pi$
$673$	−25.0948	−0.967333	−0.483667	+	0.875252i	$0.660695\pi$
$674$	−4.18187	−0.161080
$675$	0	0
$676$	−1.37103	−0.0527321
$677$	7.78669	0.299267	0.149633	−	0.988742i	$-0.452191\pi$
$677$	7.78669	0.299267	0.149633	+	0.988742i	$0.452191\pi$
$678$	0	0
$679$	−4.05883	−0.155764
$680$	−1.19923	−0.0459885
$681$	0	0
$682$	19.3068	0.739295
$683$	−2.68470	−0.102727	−0.0513636	−	0.998680i	$-0.516357\pi$
$683$	−2.68470	−0.102727	−0.0513636	+	0.998680i	$0.516357\pi$
$684$	0	0
$685$	6.80602	0.260045
$686$	−6.66159	−0.254341
$687$	0	0
$688$	25.9239	0.988339
$689$	−32.6452	−1.24368
$690$	0	0
$691$	−46.3260	−1.76232	−0.881162	−	0.472814i	$-0.843238\pi$
$691$	−46.3260	−1.76232	−0.881162	+	0.472814i	$0.843238\pi$
$692$	−15.9150	−0.604996
$693$	0	0
$694$	6.15230	0.233538
$695$	−5.78849	−0.219570
$696$	0	0
$697$	9.55851	0.362054
$698$	8.99326	0.340400
$699$	0	0
$700$	16.1428	0.610141
$701$	−4.33233	−0.163630	−0.0818150	−	0.996648i	$-0.526072\pi$
$701$	−4.33233	−0.163630	−0.0818150	+	0.996648i	$0.526072\pi$
$702$	0	0
$703$	−4.03239	−0.152084
$704$	31.9137	1.20279
$705$	0	0
$706$	4.70829	0.177199
$707$	7.73677	0.290971
$708$	0	0
$709$	−45.2491	−1.69936	−0.849682	−	0.527295i	$-0.823206\pi$
$709$	−45.2491	−1.69936	−0.849682	+	0.527295i	$0.823206\pi$
$710$	−2.47322	−0.0928183
$711$	0	0
$712$	−5.23614	−0.196233
$713$	−51.4182	−1.92563
$714$	0	0
$715$	−12.1867	−0.455757
$716$	−19.9303	−0.744831
$717$	0	0
$718$	0.453533	0.0169257
$719$	−34.3560	−1.28126	−0.640632	−	0.767848i	$-0.721328\pi$
$719$	−34.3560	−1.28126	−0.640632	+	0.767848i	$0.721328\pi$
$720$	0	0
$721$	0.300798	0.0112023
$722$	−3.08812	−0.114928
$723$	0	0
$724$	36.7011	1.36399
$725$	34.0563	1.26482
$726$	0	0
$727$	−29.1579	−1.08141	−0.540703	−	0.841213i	$-0.681842\pi$
$727$	−29.1579	−1.08141	−0.540703	+	0.841213i	$0.681842\pi$
$728$	−8.98141	−0.332873
$729$	0	0
$730$	−2.51590	−0.0931175
$731$	12.8746	0.476185
$732$	0	0
$733$	36.8326	1.36044	0.680222	−	0.733007i	$-0.261884\pi$
$733$	36.8326	1.36044	0.680222	+	0.733007i	$0.261884\pi$
$734$	2.07334	0.0765283
$735$	0	0
$736$	20.6941	0.762795
$737$	50.3224	1.85365
$738$	0	0
$739$	24.4258	0.898517	0.449258	−	0.893402i	$-0.351688\pi$
$739$	24.4258	0.898517	0.449258	+	0.893402i	$0.351688\pi$
$740$	−1.31963	−0.0485106
$741$	0	0
$742$	−5.50037	−0.201925
$743$	24.4245	0.896049	0.448024	−	0.894021i	$-0.352128\pi$
$743$	24.4245	0.896049	0.448024	+	0.894021i	$0.352128\pi$
$744$	0	0
$745$	5.10811	0.187147
$746$	6.99503	0.256106
$747$	0	0
$748$	18.5548	0.678429
$749$	−16.9420	−0.619047
$750$	0	0
$751$	−30.0598	−1.09690	−0.548449	−	0.836184i	$-0.684781\pi$
$751$	−30.0598	−1.09690	−0.548449	+	0.836184i	$0.684781\pi$
$752$	−8.66136	−0.315847
$753$	0	0
$754$	−9.18835	−0.334620
$755$	5.64656	0.205499
$756$	0	0
$757$	5.79087	0.210473	0.105236	−	0.994447i	$-0.466440\pi$
$757$	5.79087	0.210473	0.105236	+	0.994447i	$0.466440\pi$
$758$	12.1134	0.439980
$759$	0	0
$760$	2.30308	0.0835415
$761$	−2.42691	−0.0879754	−0.0439877	−	0.999032i	$-0.514006\pi$
$761$	−2.42691	−0.0879754	−0.0439877	+	0.999032i	$0.514006\pi$
$762$	0	0
$763$	22.2512	0.805549
$764$	−4.17139	−0.150916
$765$	0	0
$766$	−2.32430	−0.0839803
$767$	28.8361	1.04121
$768$	0	0
$769$	22.0750	0.796045	0.398022	−	0.917376i	$-0.369697\pi$
$769$	22.0750	0.796045	0.398022	+	0.917376i	$0.369697\pi$
$770$	−2.05333	−0.0739968
$771$	0	0
$772$	16.6594	0.599584
$773$	−28.7802	−1.03515	−0.517576	−	0.855637i	$-0.673166\pi$
$773$	−28.7802	−1.03515	−0.517576	+	0.855637i	$0.673166\pi$
$774$	0	0
$775$	44.2477	1.58943
$776$	2.95534	0.106091
$777$	0	0
$778$	−8.44655	−0.302824
$779$	−18.3567	−0.657699
$780$	0	0
$781$	78.9113	2.82367
$782$	3.07222	0.109862
$783$	0	0
$784$	−12.1554	−0.434121
$785$	−2.32651	−0.0830369
$786$	0	0
$787$	−53.4388	−1.90489	−0.952443	−	0.304715i	$-0.901439\pi$
$787$	−53.4388	−1.90489	−0.952443	+	0.304715i	$0.901439\pi$
$788$	11.6694	0.415704
$789$	0	0
$790$	−1.99753	−0.0710690
$791$	−24.0543	−0.855271
$792$	0	0
$793$	3.82351	0.135777
$794$	4.92020	0.174611
$795$	0	0
$796$	−38.0923	−1.35015
$797$	−10.4149	−0.368914	−0.184457	−	0.982841i	$-0.559053\pi$
$797$	−10.4149	−0.368914	−0.184457	+	0.982841i	$0.559053\pi$
$798$	0	0
$799$	−4.30151	−0.152176
$800$	−17.8082	−0.629616
$801$	0	0
$802$	−2.60810	−0.0920954
$803$	80.2730	2.83277
$804$	0	0
$805$	5.46847	0.192738
$806$	−11.9380	−0.420498
$807$	0	0
$808$	−5.63335	−0.198181
$809$	25.9467	0.912237	0.456118	−	0.889919i	$-0.349239\pi$
$809$	25.9467	0.912237	0.456118	+	0.889919i	$0.349239\pi$
$810$	0	0
$811$	−11.1712	−0.392275	−0.196137	−	0.980576i	$-0.562840\pi$
$811$	−11.1712	−0.392275	−0.196137	+	0.980576i	$0.562840\pi$
$812$	24.9012	0.873861
$813$	0	0
$814$	−2.61769	−0.0917501
$815$	−3.52715	−0.123551
$816$	0	0
$817$	−24.7252	−0.865026
$818$	1.75324	0.0613005
$819$	0	0
$820$	−6.00739	−0.209787
$821$	−18.3232	−0.639484	−0.319742	−	0.947505i	$-0.603596\pi$
$821$	−18.3232	−0.639484	−0.319742	+	0.947505i	$0.603596\pi$
$822$	0	0
$823$	21.1230	0.736301	0.368150	−	0.929766i	$-0.379991\pi$
$823$	21.1230	0.736301	0.368150	+	0.929766i	$0.379991\pi$
$824$	−0.219019	−0.00762989
$825$	0	0
$826$	4.85857	0.169051
$827$	−47.5383	−1.65307	−0.826534	−	0.562887i	$-0.809690\pi$
$827$	−47.5383	−1.65307	−0.826534	+	0.562887i	$0.809690\pi$
$828$	0	0
$829$	15.8701	0.551190	0.275595	−	0.961274i	$-0.411125\pi$
$829$	15.8701	0.551190	0.275595	+	0.961274i	$0.411125\pi$
$830$	−1.17498	−0.0407841
$831$	0	0
$832$	−19.7332	−0.684126
$833$	−6.03676	−0.209161
$834$	0	0
$835$	−2.17508	−0.0752719
$836$	−35.6337	−1.23242
$837$	0	0
$838$	−4.56811	−0.157803
$839$	−34.3114	−1.18456	−0.592281	−	0.805732i	$-0.701772\pi$
$839$	−34.3114	−1.18456	−0.592281	+	0.805732i	$0.701772\pi$
$840$	0	0
$841$	23.5337	0.811507
$842$	9.51370	0.327864
$843$	0	0
$844$	−3.70001	−0.127360
$845$	0.399676	0.0137493
$846$	0	0
$847$	45.4437	1.56146
$848$	−29.1752	−1.00188
$849$	0	0
$850$	−2.64379	−0.0906811
$851$	6.97150	0.238980
$852$	0	0
$853$	−45.8466	−1.56976	−0.784879	−	0.619649i	$-0.787275\pi$
$853$	−45.8466	−1.56976	−0.784879	+	0.619649i	$0.787275\pi$
$854$	0.644221	0.0220448
$855$	0	0
$856$	12.3359	0.421633
$857$	−21.7254	−0.742125	−0.371063	−	0.928608i	$-0.621007\pi$
$857$	−21.7254	−0.742125	−0.371063	+	0.928608i	$0.621007\pi$
$858$	0	0
$859$	52.5363	1.79252	0.896258	−	0.443534i	$-0.146275\pi$
$859$	52.5363	1.79252	0.896258	+	0.443534i	$0.146275\pi$
$860$	−8.09152	−0.275919
$861$	0	0
$862$	−6.35425	−0.216427
$863$	−32.9971	−1.12323	−0.561617	−	0.827397i	$-0.689821\pi$
$863$	−32.9971	−1.12323	−0.561617	+	0.827397i	$0.689821\pi$
$864$	0	0
$865$	4.63944	0.157746
$866$	−2.46132	−0.0836392
$867$	0	0
$868$	32.3529	1.09813
$869$	63.7339	2.16202
$870$	0	0
$871$	−31.1159	−1.05432
$872$	−16.2017	−0.548660
$873$	0	0
$874$	−5.90008	−0.199573
$875$	−9.71348	−0.328376
$876$	0	0
$877$	−21.4221	−0.723372	−0.361686	−	0.932300i	$-0.617799\pi$
$877$	−21.4221	−0.723372	−0.361686	+	0.932300i	$0.617799\pi$
$878$	−5.57239	−0.188059
$879$	0	0
$880$	−10.8913	−0.367147
$881$	39.1162	1.31786	0.658929	−	0.752205i	$-0.271010\pi$
$881$	39.1162	1.31786	0.658929	+	0.752205i	$0.271010\pi$
$882$	0	0
$883$	−2.84717	−0.0958150	−0.0479075	−	0.998852i	$-0.515255\pi$
$883$	−2.84717	−0.0958150	−0.0479075	+	0.998852i	$0.515255\pi$
$884$	−11.4730	−0.385878
$885$	0	0
$886$	12.6907	0.426354
$887$	46.9262	1.57563	0.787814	−	0.615913i	$-0.211213\pi$
$887$	46.9262	1.57563	0.787814	+	0.615913i	$0.211213\pi$
$888$	0	0
$889$	−21.4505	−0.719428
$890$	0.740196	0.0248114
$891$	0	0
$892$	−40.4653	−1.35488
$893$	8.26088	0.276440
$894$	0	0
$895$	5.80997	0.194206
$896$	−17.1553	−0.573120
$897$	0	0
$898$	5.85272	0.195308
$899$	68.2546	2.27642
$900$	0	0
$901$	−14.4893	−0.482710
$902$	−11.9166	−0.396779
$903$	0	0
$904$	17.5145	0.582525
$905$	−10.6989	−0.355644
$906$	0	0
$907$	15.7164	0.521856	0.260928	−	0.965358i	$-0.415972\pi$
$907$	15.7164	0.521856	0.260928	+	0.965358i	$0.415972\pi$
$908$	39.1285	1.29852
$909$	0	0
$910$	1.26964	0.0420881
$911$	−0.0451186	−0.00149485	−0.000747423	−	1.00000i	$-0.500238\pi$
$911$	−0.0451186	−0.00149485	−0.000747423		1.00000i	$0.500238\pi$
$912$	0	0
$913$	37.4892	1.24071
$914$	−3.20201	−0.105913
$915$	0	0
$916$	−53.6187	−1.77161
$917$	−25.6973	−0.848601
$918$	0	0
$919$	−7.88987	−0.260263	−0.130131	−	0.991497i	$-0.541540\pi$
$919$	−7.88987	−0.260263	−0.130131	+	0.991497i	$0.541540\pi$
$920$	−3.98174	−0.131274
$921$	0	0
$922$	5.42884	0.178789
$923$	−48.7933	−1.60605
$924$	0	0
$925$	−5.99929	−0.197255
$926$	−2.39417	−0.0786773
$927$	0	0
$928$	−27.4702	−0.901752
$929$	−25.6276	−0.840815	−0.420408	−	0.907335i	$-0.638113\pi$
$929$	−25.6276	−0.840815	−0.420408	+	0.907335i	$0.638113\pi$
$930$	0	0
$931$	11.5934	0.379957
$932$	−4.12777	−0.135210
$933$	0	0
$934$	8.30670	0.271803
$935$	−5.40898	−0.176893
$936$	0	0
$937$	−57.8627	−1.89029	−0.945146	−	0.326649i	$-0.894081\pi$
$937$	−57.8627	−1.89029	−0.945146	+	0.326649i	$0.894081\pi$
$938$	−5.24269	−0.171180
$939$	0	0
$940$	2.70344	0.0881764
$941$	14.1272	0.460533	0.230266	−	0.973128i	$-0.426040\pi$
$941$	14.1272	0.460533	0.230266	+	0.973128i	$0.426040\pi$
$942$	0	0
$943$	31.7365	1.03348
$944$	25.7710	0.838774
$945$	0	0
$946$	−16.0508	−0.521857
$947$	−22.5318	−0.732185	−0.366092	−	0.930578i	$-0.619305\pi$
$947$	−22.5318	−0.732185	−0.366092	+	0.930578i	$0.619305\pi$
$948$	0	0
$949$	−49.6353	−1.61123
$950$	5.07729	0.164729
$951$	0	0
$952$	−3.98633	−0.129198
$953$	34.0142	1.10183	0.550914	−	0.834562i	$-0.314279\pi$
$953$	34.0142	1.10183	0.550914	+	0.834562i	$0.314279\pi$
$954$	0	0
$955$	1.21602	0.0393495
$956$	47.5345	1.53737
$957$	0	0
$958$	−3.05596	−0.0987335
$959$	22.6237	0.730557
$960$	0	0
$961$	57.6799	1.86064
$962$	1.61860	0.0521858
$963$	0	0
$964$	−1.88294	−0.0606453
$965$	−4.85644	−0.156334
$966$	0	0
$967$	−62.1874	−1.99981	−0.999906	−	0.0137093i	$-0.995636\pi$
$967$	−62.1874	−1.99981	−0.999906	+	0.0137093i	$0.995636\pi$
$968$	−33.0888	−1.06351
$969$	0	0
$970$	−0.417776	−0.0134140
$971$	−18.4346	−0.591594	−0.295797	−	0.955251i	$-0.595585\pi$
$971$	−18.4346	−0.591594	−0.295797	+	0.955251i	$0.595585\pi$
$972$	0	0
$973$	−19.2414	−0.616850
$974$	11.8370	0.379283
$975$	0	0
$976$	3.41710	0.109379
$977$	−25.2228	−0.806949	−0.403474	−	0.914991i	$-0.632198\pi$
$977$	−25.2228	−0.806949	−0.403474	+	0.914991i	$0.632198\pi$
$978$	0	0
$979$	−23.6169	−0.754800
$980$	3.79402	0.121195
$981$	0	0
$982$	−11.7630	−0.375371
$983$	−12.7405	−0.406358	−0.203179	−	0.979142i	$-0.565127\pi$
$983$	−12.7405	−0.406358	−0.203179	+	0.979142i	$0.565127\pi$
$984$	0	0
$985$	−3.40179	−0.108390
$986$	−4.07819	−0.129876
$987$	0	0
$988$	22.0334	0.700977
$989$	42.7468	1.35927
$990$	0	0
$991$	15.6775	0.498011	0.249005	−	0.968502i	$-0.419896\pi$
$991$	15.6775	0.498011	0.249005	+	0.968502i	$0.419896\pi$
$992$	−35.6907	−1.13318
$993$	0	0
$994$	−8.22116	−0.260759
$995$	11.1045	0.352035
$996$	0	0
$997$	28.1048	0.890088	0.445044	−	0.895509i	$-0.353188\pi$
$997$	28.1048	0.890088	0.445044	+	0.895509i	$0.353188\pi$
$998$	13.1085	0.414944
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2169.2.a.h.1.8		12
3.2	odd	2		241.2.a.b.1.5	✓	12
12.11	even	2		3856.2.a.n.1.3		12
15.14	odd	2		6025.2.a.h.1.8		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
241.2.a.b.1.5	✓	12	3.2	odd	2
2169.2.a.h.1.8		12	1.1	even	1	trivial
3856.2.a.n.1.3		12	12.11	even	2
6025.2.a.h.1.8		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)

\(f(q)\)	\(=\)	\(q+0.342147 q^{2} -1.88294 q^{4} +0.548903 q^{5} +1.82459 q^{7} -1.32853 q^{8} +O(q^{10})\)	\(q+0.342147 q^{2} -1.88294 q^{4} +0.548903 q^{5} +1.82459 q^{7} -1.32853 q^{8} +0.187805 q^{10} -5.99218 q^{11} +3.70515 q^{13} +0.624278 q^{14} +3.31132 q^{16} +1.64451 q^{17} -3.15821 q^{19} -1.03355 q^{20} -2.05021 q^{22} +5.46015 q^{23} -4.69871 q^{25} +1.26771 q^{26} -3.43559 q^{28} -7.24801 q^{29} -9.41700 q^{31} +3.79003 q^{32} +0.562662 q^{34} +1.00152 q^{35} +1.27680 q^{37} -1.08057 q^{38} -0.729236 q^{40} +5.81239 q^{41} +7.82887 q^{43} +11.2829 q^{44} +1.86817 q^{46} -2.61568 q^{47} -3.67086 q^{49} -1.60765 q^{50} -6.97656 q^{52} -8.81076 q^{53} -3.28912 q^{55} -2.42403 q^{56} -2.47989 q^{58} +7.78270 q^{59} +1.03194 q^{61} -3.22200 q^{62} -5.32589 q^{64} +2.03377 q^{65} -8.39801 q^{67} -3.09650 q^{68} +0.342668 q^{70} -13.1691 q^{71} -13.3963 q^{73} +0.436852 q^{74} +5.94670 q^{76} -10.9333 q^{77} -10.6362 q^{79} +1.81759 q^{80} +1.98869 q^{82} -6.25636 q^{83} +0.902673 q^{85} +2.67862 q^{86} +7.96082 q^{88} +3.94129 q^{89} +6.76039 q^{91} -10.2811 q^{92} -0.894948 q^{94} -1.73355 q^{95} -2.22451 q^{97} -1.25598 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{2} + 13 q^{4} - 6 q^{5} + 3 q^{7} - 9 q^{8}+O(q^{10}) \) 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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Varieties

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Database

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