LMFDB - Embedded newform 1386.2.a.q.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1386,2,Mod(1,1386)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1386, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1386.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1386.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:	$11.0672657201$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1304.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 11x - 2 $ x^3 - 11*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.182370$ of defining polynomial
Character	$\chi$	$=$	1386.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +2.89219 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} +2.89219 q^{5} +1.00000 q^{7} -1.00000 q^{8} -2.89219 q^{10} -1.00000 q^{11} -0.364739 q^{13} -1.00000 q^{14} +1.00000 q^{16} +2.89219 q^{17} +7.25693 q^{19} +2.89219 q^{20} +1.00000 q^{22} -8.14911 q^{23} +3.36474 q^{25} +0.364739 q^{26} +1.00000 q^{28} +2.36474 q^{29} +10.6766 q^{31} -1.00000 q^{32} -2.89219 q^{34} +2.89219 q^{35} -3.78437 q^{37} -7.25693 q^{38} -2.89219 q^{40} -2.89219 q^{41} +11.4196 q^{43} -1.00000 q^{44} +8.14911 q^{46} +6.52745 q^{47} +1.00000 q^{49} -3.36474 q^{50} -0.364739 q^{52} -11.7844 q^{53} -2.89219 q^{55} -1.00000 q^{56} -2.36474 q^{58} +10.5139 q^{59} -11.9335 q^{61} -10.6766 q^{62} +1.00000 q^{64} -1.05489 q^{65} -6.14911 q^{67} +2.89219 q^{68} -2.89219 q^{70} -13.9335 q^{71} -1.10781 q^{73} +3.78437 q^{74} +7.25693 q^{76} -1.00000 q^{77} +10.1491 q^{79} +2.89219 q^{80} +2.89219 q^{82} +3.10781 q^{83} +8.36474 q^{85} -11.4196 q^{86} +1.00000 q^{88} -2.14911 q^{89} -0.364739 q^{91} -8.14911 q^{92} -6.52745 q^{94} +20.9884 q^{95} +6.72948 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{4} - 2 q^{5} + 3 q^{7} - 3 q^{8}+O(q^{10}) $ 3 * q - 3 * q^2 + 3 * q^4 - 2 * q^5 + 3 * q^7 - 3 * q^8	$ 3 q - 3 q^{2} + 3 q^{4} - 2 q^{5} + 3 q^{7} - 3 q^{8} + 2 q^{10} - 3 q^{11} - 3 q^{14} + 3 q^{16} - 2 q^{17} + 10 q^{19} - 2 q^{20} + 3 q^{22} - 2 q^{23} + 9 q^{25} + 3 q^{28} + 6 q^{29} - 3 q^{32} + 2 q^{34} - 2 q^{35} + 10 q^{37} - 10 q^{38} + 2 q^{40} + 2 q^{41} + 14 q^{43} - 3 q^{44} + 2 q^{46} + 10 q^{47} + 3 q^{49} - 9 q^{50} - 14 q^{53} + 2 q^{55} - 3 q^{56} - 6 q^{58} + 8 q^{59} + 8 q^{61} + 3 q^{64} + 16 q^{65} + 4 q^{67} - 2 q^{68} + 2 q^{70} + 2 q^{71} - 14 q^{73} - 10 q^{74} + 10 q^{76} - 3 q^{77} + 8 q^{79} - 2 q^{80} - 2 q^{82} + 20 q^{83} + 24 q^{85} - 14 q^{86} + 3 q^{88} + 16 q^{89} - 2 q^{92} - 10 q^{94} + 18 q^{97} - 3 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^4 - 2 * q^5 + 3 * q^7 - 3 * q^8 + 2 * q^10 - 3 * q^11 - 3 * q^14 + 3 * q^16 - 2 * q^17 + 10 * q^19 - 2 * q^20 + 3 * q^22 - 2 * q^23 + 9 * q^25 + 3 * q^28 + 6 * q^29 - 3 * q^32 + 2 * q^34 - 2 * q^35 + 10 * q^37 - 10 * q^38 + 2 * q^40 + 2 * q^41 + 14 * q^43 - 3 * q^44 + 2 * q^46 + 10 * q^47 + 3 * q^49 - 9 * q^50 - 14 * q^53 + 2 * q^55 - 3 * q^56 - 6 * q^58 + 8 * q^59 + 8 * q^61 + 3 * q^64 + 16 * q^65 + 4 * q^67 - 2 * q^68 + 2 * q^70 + 2 * q^71 - 14 * q^73 - 10 * q^74 + 10 * q^76 - 3 * q^77 + 8 * q^79 - 2 * q^80 - 2 * q^82 + 20 * q^83 + 24 * q^85 - 14 * q^86 + 3 * q^88 + 16 * q^89 - 2 * q^92 - 10 * q^94 + 18 * q^97 - 3 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	2.89219	1.29342	0.646712	−	0.762734i	$-0.276143\pi$
$5$	2.89219	1.29342	0.646712	+	0.762734i	$0.276143\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−2.89219	−0.914589
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−0.364739	−0.101160	−0.0505802	−	0.998720i	$-0.516107\pi$
$13$	−0.364739	−0.101160	−0.0505802	+	0.998720i	$0.516107\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	2.89219	0.701458	0.350729	−	0.936477i	$-0.385934\pi$
$17$	2.89219	0.701458	0.350729	+	0.936477i	$0.385934\pi$
$18$	0	0
$19$	7.25693	1.66485	0.832426	−	0.554136i	$-0.186951\pi$
$19$	7.25693	1.66485	0.832426	+	0.554136i	$0.186951\pi$
$20$	2.89219	0.646712
$21$	0	0
$22$	1.00000	0.213201
$23$	−8.14911	−1.69921	−0.849604	−	0.527422i	$-0.823159\pi$
$23$	−8.14911	−1.69921	−0.849604	+	0.527422i	$0.823159\pi$
$24$	0	0
$25$	3.36474	0.672948
$26$	0.364739	0.0715312
$27$	0	0
$28$	1.00000	0.188982
$29$	2.36474	0.439121	0.219561	−	0.975599i	$-0.429538\pi$
$29$	2.36474	0.439121	0.219561	+	0.975599i	$0.429538\pi$
$30$	0	0
$31$	10.6766	1.91757	0.958783	−	0.284139i	$-0.0917076\pi$
$31$	10.6766	1.91757	0.958783	+	0.284139i	$0.0917076\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−2.89219	−0.496006
$35$	2.89219	0.488869
$36$	0	0
$37$	−3.78437	−0.622147	−0.311073	−	0.950386i	$-0.600689\pi$
$37$	−3.78437	−0.622147	−0.311073	+	0.950386i	$0.600689\pi$
$38$	−7.25693	−1.17723
$39$	0	0
$40$	−2.89219	−0.457295
$41$	−2.89219	−0.451684	−0.225842	−	0.974164i	$-0.572513\pi$
$41$	−2.89219	−0.451684	−0.225842	+	0.974164i	$0.572513\pi$
$42$	0	0
$43$	11.4196	1.74148	0.870739	−	0.491746i	$-0.163641\pi$
$43$	11.4196	1.74148	0.870739	+	0.491746i	$0.163641\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	8.14911	1.20152
$47$	6.52745	0.952126	0.476063	−	0.879411i	$-0.342063\pi$
$47$	6.52745	0.952126	0.476063	+	0.879411i	$0.342063\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−3.36474	−0.475846
$51$	0	0
$52$	−0.364739	−0.0505802
$53$	−11.7844	−1.61871	−0.809354	−	0.587321i	$-0.800183\pi$
$53$	−11.7844	−1.61871	−0.809354	+	0.587321i	$0.800183\pi$
$54$	0	0
$55$	−2.89219	−0.389982
$56$	−1.00000	−0.133631
$57$	0	0
$58$	−2.36474	−0.310505
$59$	10.5139	1.36879	0.684393	−	0.729113i	$-0.260067\pi$
$59$	10.5139	1.36879	0.684393	+	0.729113i	$0.260067\pi$
$60$	0	0
$61$	−11.9335	−1.52793	−0.763963	−	0.645260i	$-0.776749\pi$
$61$	−11.9335	−1.52793	−0.763963	+	0.645260i	$0.776749\pi$
$62$	−10.6766	−1.35592
$63$	0	0
$64$	1.00000	0.125000
$65$	−1.05489	−0.130843
$66$	0	0
$67$	−6.14911	−0.751233	−0.375617	−	0.926775i	$-0.622569\pi$
$67$	−6.14911	−0.751233	−0.375617	+	0.926775i	$0.622569\pi$
$68$	2.89219	0.350729
$69$	0	0
$70$	−2.89219	−0.345682
$71$	−13.9335	−1.65360	−0.826800	−	0.562496i	$-0.809841\pi$
$71$	−13.9335	−1.65360	−0.826800	+	0.562496i	$0.809841\pi$
$72$	0	0
$73$	−1.10781	−0.129660	−0.0648299	−	0.997896i	$-0.520650\pi$
$73$	−1.10781	−0.129660	−0.0648299	+	0.997896i	$0.520650\pi$
$74$	3.78437	0.439924
$75$	0	0
$76$	7.25693	0.832426
$77$	−1.00000	−0.113961
$78$	0	0
$79$	10.1491	1.14186	0.570932	−	0.820997i	$-0.306582\pi$
$79$	10.1491	1.14186	0.570932	+	0.820997i	$0.306582\pi$
$80$	2.89219	0.323356
$81$	0	0
$82$	2.89219	0.319389
$83$	3.10781	0.341127	0.170563	−	0.985347i	$-0.445441\pi$
$83$	3.10781	0.341127	0.170563	+	0.985347i	$0.445441\pi$
$84$	0	0
$85$	8.36474	0.907283
$86$	−11.4196	−1.23141
$87$	0	0
$88$	1.00000	0.106600
$89$	−2.14911	−0.227805	−0.113903	−	0.993492i	$-0.536335\pi$
$89$	−2.14911	−0.227805	−0.113903	+	0.993492i	$0.536335\pi$
$90$	0	0
$91$	−0.364739	−0.0382351
$92$	−8.14911	−0.849604
$93$	0	0
$94$	−6.52745	−0.673255
$95$	20.9884	2.15336
$96$	0	0
$97$	6.72948	0.683275	0.341638	−	0.939832i	$-0.389018\pi$
$97$	6.72948	0.683275	0.341638	+	0.939832i	$0.389018\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	3.36474	0.336474
$101$	1.27052	0.126422	0.0632108	−	0.998000i	$-0.479866\pi$
$101$	1.27052	0.126422	0.0632108	+	0.998000i	$0.479866\pi$
$102$	0	0
$103$	−0.892186	−0.0879097	−0.0439548	−	0.999034i	$-0.513996\pi$
$103$	−0.892186	−0.0879097	−0.0439548	+	0.999034i	$0.513996\pi$
$104$	0.364739	0.0357656
$105$	0	0
$106$	11.7844	1.14460
$107$	−5.78437	−0.559196	−0.279598	−	0.960117i	$-0.590201\pi$
$107$	−5.78437	−0.559196	−0.279598	+	0.960117i	$0.590201\pi$
$108$	0	0
$109$	8.00000	0.766261	0.383131	−	0.923694i	$-0.374846\pi$
$109$	8.00000	0.766261	0.383131	+	0.923694i	$0.374846\pi$
$110$	2.89219	0.275759
$111$	0	0
$112$	1.00000	0.0944911
$113$	19.9335	1.87518	0.937592	−	0.347737i	$-0.113050\pi$
$113$	19.9335	1.87518	0.937592	+	0.347737i	$0.113050\pi$
$114$	0	0
$115$	−23.5687	−2.19780
$116$	2.36474	0.219561
$117$	0	0
$118$	−10.5139	−0.967878
$119$	2.89219	0.265126
$120$	0	0
$121$	1.00000	0.0909091
$122$	11.9335	1.08041
$123$	0	0
$124$	10.6766	0.958783
$125$	−4.72948	−0.423017
$126$	0	0
$127$	10.1491	0.900588	0.450294	−	0.892880i	$-0.351319\pi$
$127$	10.1491	0.900588	0.450294	+	0.892880i	$0.351319\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	1.05489	0.0925203
$131$	20.8922	1.82536	0.912679	−	0.408676i	$-0.134010\pi$
$131$	20.8922	1.82536	0.912679	+	0.408676i	$0.134010\pi$
$132$	0	0
$133$	7.25693	0.629255
$134$	6.14911	0.531202
$135$	0	0
$136$	−2.89219	−0.248003
$137$	7.93348	0.677803	0.338902	−	0.940822i	$-0.389945\pi$
$137$	7.93348	0.677803	0.338902	+	0.940822i	$0.389945\pi$
$138$	0	0
$139$	18.8257	1.59677	0.798386	−	0.602146i	$-0.205687\pi$
$139$	18.8257	1.59677	0.798386	+	0.602146i	$0.205687\pi$
$140$	2.89219	0.244434
$141$	0	0
$142$	13.9335	1.16927
$143$	0.364739	0.0305010
$144$	0	0
$145$	6.83927	0.567970
$146$	1.10781	0.0916833
$147$	0	0
$148$	−3.78437	−0.311073
$149$	1.27052	0.104085	0.0520426	−	0.998645i	$-0.483427\pi$
$149$	1.27052	0.104085	0.0520426	+	0.998645i	$0.483427\pi$
$150$	0	0
$151$	−8.29822	−0.675300	−0.337650	−	0.941272i	$-0.609632\pi$
$151$	−8.29822	−0.675300	−0.337650	+	0.941272i	$0.609632\pi$
$152$	−7.25693	−0.588614
$153$	0	0
$154$	1.00000	0.0805823
$155$	30.8786	2.48023
$156$	0	0
$157$	−18.8922	−1.50776	−0.753880	−	0.657012i	$-0.771820\pi$
$157$	−18.8922	−1.50776	−0.753880	+	0.657012i	$0.771820\pi$
$158$	−10.1491	−0.807420
$159$	0	0
$160$	−2.89219	−0.228647
$161$	−8.14911	−0.642240
$162$	0	0
$163$	−20.2982	−1.58988	−0.794940	−	0.606688i	$-0.792498\pi$
$163$	−20.2982	−1.58988	−0.794940	+	0.606688i	$0.792498\pi$
$164$	−2.89219	−0.225842
$165$	0	0
$166$	−3.10781	−0.241213
$167$	−1.05489	−0.0816301	−0.0408151	−	0.999167i	$-0.512995\pi$
$167$	−1.05489	−0.0816301	−0.0408151	+	0.999167i	$0.512995\pi$
$168$	0	0
$169$	−12.8670	−0.989767
$170$	−8.36474	−0.641546
$171$	0	0
$172$	11.4196	0.870739
$173$	16.5139	1.25552	0.627762	−	0.778405i	$-0.283971\pi$
$173$	16.5139	1.25552	0.627762	+	0.778405i	$0.283971\pi$
$174$	0	0
$175$	3.36474	0.254350
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	2.14911	0.161083
$179$	−20.9884	−1.56874	−0.784372	−	0.620290i	$-0.787015\pi$
$179$	−20.9884	−1.56874	−0.784372	+	0.620290i	$0.787015\pi$
$180$	0	0
$181$	−1.10781	−0.0823432	−0.0411716	−	0.999152i	$-0.513109\pi$
$181$	−1.10781	−0.0823432	−0.0411716	+	0.999152i	$0.513109\pi$
$182$	0.364739	0.0270363
$183$	0	0
$184$	8.14911	0.600760
$185$	−10.9451	−0.804700
$186$	0	0
$187$	−2.89219	−0.211498
$188$	6.52745	0.476063
$189$	0	0
$190$	−20.9884	−1.52266
$191$	10.6902	0.773512	0.386756	−	0.922182i	$-0.373595\pi$
$191$	10.6902	0.773512	0.386756	+	0.922182i	$0.373595\pi$
$192$	0	0
$193$	−21.5687	−1.55255	−0.776276	−	0.630393i	$-0.782894\pi$
$193$	−21.5687	−1.55255	−0.776276	+	0.630393i	$0.782894\pi$
$194$	−6.72948	−0.483148
$195$	0	0
$196$	1.00000	0.0714286
$197$	4.90578	0.349523	0.174761	−	0.984611i	$-0.444085\pi$
$197$	4.90578	0.349523	0.174761	+	0.984611i	$0.444085\pi$
$198$	0	0
$199$	−7.10781	−0.503860	−0.251930	−	0.967746i	$-0.581065\pi$
$199$	−7.10781	−0.503860	−0.251930	+	0.967746i	$0.581065\pi$
$200$	−3.36474	−0.237923
$201$	0	0
$202$	−1.27052	−0.0893936
$203$	2.36474	0.165972
$204$	0	0
$205$	−8.36474	−0.584219
$206$	0.892186	0.0621615
$207$	0	0
$208$	−0.364739	−0.0252901
$209$	−7.25693	−0.501972
$210$	0	0
$211$	8.87859	0.611227	0.305614	−	0.952156i	$-0.401138\pi$
$211$	8.87859	0.611227	0.305614	+	0.952156i	$0.401138\pi$
$212$	−11.7844	−0.809354
$213$	0	0
$214$	5.78437	0.395412
$215$	33.0277	2.25247
$216$	0	0
$217$	10.6766	0.724772
$218$	−8.00000	−0.541828
$219$	0	0
$220$	−2.89219	−0.194991
$221$	−1.05489	−0.0709598
$222$	0	0
$223$	22.2453	1.48966	0.744828	−	0.667257i	$-0.232532\pi$
$223$	22.2453	1.48966	0.744828	+	0.667257i	$0.232532\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	−19.9335	−1.32596
$227$	−8.46093	−0.561572	−0.280786	−	0.959770i	$-0.590595\pi$
$227$	−8.46093	−0.561572	−0.280786	+	0.959770i	$0.590595\pi$
$228$	0	0
$229$	−2.16271	−0.142916	−0.0714579	−	0.997444i	$-0.522765\pi$
$229$	−2.16271	−0.142916	−0.0714579	+	0.997444i	$0.522765\pi$
$230$	23.5687	1.55408
$231$	0	0
$232$	−2.36474	−0.155253
$233$	3.45896	0.226604	0.113302	−	0.993561i	$-0.463857\pi$
$233$	3.45896	0.226604	0.113302	+	0.993561i	$0.463857\pi$
$234$	0	0
$235$	18.8786	1.23150
$236$	10.5139	0.684393
$237$	0	0
$238$	−2.89219	−0.187473
$239$	−16.2982	−1.05424	−0.527122	−	0.849790i	$-0.676729\pi$
$239$	−16.2982	−1.05424	−0.527122	+	0.849790i	$0.676729\pi$
$240$	0	0
$241$	4.67656	0.301244	0.150622	−	0.988591i	$-0.451872\pi$
$241$	4.67656	0.301244	0.150622	+	0.988591i	$0.451872\pi$
$242$	−1.00000	−0.0642824
$243$	0	0
$244$	−11.9335	−0.763963
$245$	2.89219	0.184775
$246$	0	0
$247$	−2.64688	−0.168417
$248$	−10.6766	−0.677962
$249$	0	0
$250$	4.72948	0.299118
$251$	10.0826	0.636408	0.318204	−	0.948022i	$-0.396920\pi$
$251$	10.0826	0.636408	0.318204	+	0.948022i	$0.396920\pi$
$252$	0	0
$253$	8.14911	0.512330
$254$	−10.1491	−0.636812
$255$	0	0
$256$	1.00000	0.0625000
$257$	−9.41963	−0.587581	−0.293790	−	0.955870i	$-0.594917\pi$
$257$	−9.41963	−0.587581	−0.293790	+	0.955870i	$0.594917\pi$
$258$	0	0
$259$	−3.78437	−0.235149
$260$	−1.05489	−0.0654217
$261$	0	0
$262$	−20.8922	−1.29072
$263$	−6.87859	−0.424152	−0.212076	−	0.977253i	$-0.568022\pi$
$263$	−6.87859	−0.424152	−0.212076	+	0.977253i	$0.568022\pi$
$264$	0	0
$265$	−34.0826	−2.09368
$266$	−7.25693	−0.444951
$267$	0	0
$268$	−6.14911	−0.375617
$269$	−26.8922	−1.63965	−0.819823	−	0.572617i	$-0.805928\pi$
$269$	−26.8922	−1.63965	−0.819823	+	0.572617i	$0.805928\pi$
$270$	0	0
$271$	−21.7844	−1.32331	−0.661653	−	0.749810i	$-0.730145\pi$
$271$	−21.7844	−1.32331	−0.661653	+	0.749810i	$0.730145\pi$
$272$	2.89219	0.175365
$273$	0	0
$274$	−7.93348	−0.479279
$275$	−3.36474	−0.202901
$276$	0	0
$277$	3.27052	0.196507	0.0982533	−	0.995161i	$-0.468674\pi$
$277$	3.27052	0.196507	0.0982533	+	0.995161i	$0.468674\pi$
$278$	−18.8257	−1.12909
$279$	0	0
$280$	−2.89219	−0.172841
$281$	−10.2982	−0.614340	−0.307170	−	0.951655i	$-0.599382\pi$
$281$	−10.2982	−0.614340	−0.307170	+	0.951655i	$0.599382\pi$
$282$	0	0
$283$	−28.3118	−1.68296	−0.841481	−	0.540286i	$-0.818316\pi$
$283$	−28.3118	−1.68296	−0.841481	+	0.540286i	$0.818316\pi$
$284$	−13.9335	−0.826800
$285$	0	0
$286$	−0.364739	−0.0215675
$287$	−2.89219	−0.170720
$288$	0	0
$289$	−8.63526	−0.507957
$290$	−6.83927	−0.401615
$291$	0	0
$292$	−1.10781	−0.0648299
$293$	−11.7844	−0.688450	−0.344225	−	0.938887i	$-0.611858\pi$
$293$	−11.7844	−0.688450	−0.344225	+	0.938887i	$0.611858\pi$
$294$	0	0
$295$	30.4080	1.77042
$296$	3.78437	0.219962
$297$	0	0
$298$	−1.27052	−0.0735993
$299$	2.97230	0.171893
$300$	0	0
$301$	11.4196	0.658217
$302$	8.29822	0.477509
$303$	0	0
$304$	7.25693	0.416213
$305$	−34.5139	−1.97626
$306$	0	0
$307$	4.01360	0.229068	0.114534	−	0.993419i	$-0.463463\pi$
$307$	4.01360	0.229068	0.114534	+	0.993419i	$0.463463\pi$
$308$	−1.00000	−0.0569803
$309$	0	0
$310$	−30.8786	−1.75379
$311$	6.52745	0.370138	0.185069	−	0.982726i	$-0.440749\pi$
$311$	6.52745	0.370138	0.185069	+	0.982726i	$0.440749\pi$
$312$	0	0
$313$	0.513850	0.0290445	0.0145223	−	0.999895i	$-0.495377\pi$
$313$	0.513850	0.0290445	0.0145223	+	0.999895i	$0.495377\pi$
$314$	18.8922	1.06615
$315$	0	0
$316$	10.1491	0.570932
$317$	−7.48615	−0.420464	−0.210232	−	0.977652i	$-0.567422\pi$
$317$	−7.48615	−0.420464	−0.210232	+	0.977652i	$0.567422\pi$
$318$	0	0
$319$	−2.36474	−0.132400
$320$	2.89219	0.161678
$321$	0	0
$322$	8.14911	0.454132
$323$	20.9884	1.16782
$324$	0	0
$325$	−1.22725	−0.0680757
$326$	20.2982	1.12421
$327$	0	0
$328$	2.89219	0.159694
$329$	6.52745	0.359870
$330$	0	0
$331$	−13.4196	−0.737610	−0.368805	−	0.929507i	$-0.620233\pi$
$331$	−13.4196	−0.737610	−0.368805	+	0.929507i	$0.620233\pi$
$332$	3.10781	0.170563
$333$	0	0
$334$	1.05489	0.0577212
$335$	−17.7844	−0.971664
$336$	0	0
$337$	−8.94511	−0.487271	−0.243636	−	0.969867i	$-0.578340\pi$
$337$	−8.94511	−0.487271	−0.243636	+	0.969867i	$0.578340\pi$
$338$	12.8670	0.699871
$339$	0	0
$340$	8.36474	0.453642
$341$	−10.6766	−0.578168
$342$	0	0
$343$	1.00000	0.0539949
$344$	−11.4196	−0.615705
$345$	0	0
$346$	−16.5139	−0.887790
$347$	−5.78437	−0.310521	−0.155261	−	0.987874i	$-0.549622\pi$
$347$	−5.78437	−0.310521	−0.155261	+	0.987874i	$0.549622\pi$
$348$	0	0
$349$	22.1491	1.18561	0.592807	−	0.805344i	$-0.298020\pi$
$349$	22.1491	1.18561	0.592807	+	0.805344i	$0.298020\pi$
$350$	−3.36474	−0.179853
$351$	0	0
$352$	1.00000	0.0533002
$353$	−9.41963	−0.501356	−0.250678	−	0.968070i	$-0.580654\pi$
$353$	−9.41963	−0.501356	−0.250678	+	0.968070i	$0.580654\pi$
$354$	0	0
$355$	−40.2982	−2.13881
$356$	−2.14911	−0.113903
$357$	0	0
$358$	20.9884	1.10927
$359$	18.4473	0.973613	0.486806	−	0.873510i	$-0.338162\pi$
$359$	18.4473	0.973613	0.486806	+	0.873510i	$0.338162\pi$
$360$	0	0
$361$	33.6630	1.77173
$362$	1.10781	0.0582254
$363$	0	0
$364$	−0.364739	−0.0191175
$365$	−3.20400	−0.167705
$366$	0	0
$367$	−18.6766	−0.974908	−0.487454	−	0.873149i	$-0.662074\pi$
$367$	−18.6766	−0.974908	−0.487454	+	0.873149i	$0.662074\pi$
$368$	−8.14911	−0.424802
$369$	0	0
$370$	10.9451	0.569009
$371$	−11.7844	−0.611814
$372$	0	0
$373$	−19.2040	−0.994346	−0.497173	−	0.867652i	$-0.665629\pi$
$373$	−19.2040	−0.994346	−0.497173	+	0.867652i	$0.665629\pi$
$374$	2.89219	0.149551
$375$	0	0
$376$	−6.52745	−0.336627
$377$	−0.862513	−0.0444217
$378$	0	0
$379$	−31.8670	−1.63690	−0.818448	−	0.574581i	$-0.805165\pi$
$379$	−31.8670	−1.63690	−0.818448	+	0.574581i	$0.805165\pi$
$380$	20.9884	1.07668
$381$	0	0
$382$	−10.6902	−0.546956
$383$	−29.0413	−1.48394	−0.741970	−	0.670433i	$-0.766109\pi$
$383$	−29.0413	−1.48394	−0.741970	+	0.670433i	$0.766109\pi$
$384$	0	0
$385$	−2.89219	−0.147399
$386$	21.5687	1.09782
$387$	0	0
$388$	6.72948	0.341638
$389$	26.5964	1.34849	0.674247	−	0.738506i	$-0.264468\pi$
$389$	26.5964	1.34849	0.674247	+	0.738506i	$0.264468\pi$
$390$	0	0
$391$	−23.5687	−1.19192
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	−4.90578	−0.247150
$395$	29.3531	1.47692
$396$	0	0
$397$	−8.37834	−0.420497	−0.210248	−	0.977648i	$-0.567427\pi$
$397$	−8.37834	−0.420497	−0.210248	+	0.977648i	$0.567427\pi$
$398$	7.10781	0.356283
$399$	0	0
$400$	3.36474	0.168237
$401$	7.93348	0.396179	0.198090	−	0.980184i	$-0.436526\pi$
$401$	7.93348	0.396179	0.198090	+	0.980184i	$0.436526\pi$
$402$	0	0
$403$	−3.89416	−0.193982
$404$	1.27052	0.0632108
$405$	0	0
$406$	−2.36474	−0.117360
$407$	3.78437	0.187584
$408$	0	0
$409$	−24.6766	−1.22018	−0.610089	−	0.792333i	$-0.708866\pi$
$409$	−24.6766	−1.22018	−0.610089	+	0.792333i	$0.708866\pi$
$410$	8.36474	0.413105
$411$	0	0
$412$	−0.892186	−0.0439548
$413$	10.5139	0.517353
$414$	0	0
$415$	8.98838	0.441222
$416$	0.364739	0.0178828
$417$	0	0
$418$	7.25693	0.354948
$419$	2.54104	0.124138	0.0620690	−	0.998072i	$-0.480230\pi$
$419$	2.54104	0.124138	0.0620690	+	0.998072i	$0.480230\pi$
$420$	0	0
$421$	2.43126	0.118492	0.0592461	−	0.998243i	$-0.481130\pi$
$421$	2.43126	0.118492	0.0592461	+	0.998243i	$0.481130\pi$
$422$	−8.87859	−0.432203
$423$	0	0
$424$	11.7844	0.572300
$425$	9.73145	0.472045
$426$	0	0
$427$	−11.9335	−0.577502
$428$	−5.78437	−0.279598
$429$	0	0
$430$	−33.0277	−1.59274
$431$	−2.58037	−0.124292	−0.0621460	−	0.998067i	$-0.519794\pi$
$431$	−2.58037	−0.124292	−0.0621460	+	0.998067i	$0.519794\pi$
$432$	0	0
$433$	37.1375	1.78471	0.892357	−	0.451331i	$-0.149050\pi$
$433$	37.1375	1.78471	0.892357	+	0.451331i	$0.149050\pi$
$434$	−10.6766	−0.512491
$435$	0	0
$436$	8.00000	0.383131
$437$	−59.1375	−2.82893
$438$	0	0
$439$	18.5139	0.883618	0.441809	−	0.897109i	$-0.354337\pi$
$439$	18.5139	0.883618	0.441809	+	0.897109i	$0.354337\pi$
$440$	2.89219	0.137880
$441$	0	0
$442$	1.05489	0.0501762
$443$	21.4196	1.01768	0.508839	−	0.860862i	$-0.330075\pi$
$443$	21.4196	1.01768	0.508839	+	0.860862i	$0.330075\pi$
$444$	0	0
$445$	−6.21563	−0.294649
$446$	−22.2453	−1.05335
$447$	0	0
$448$	1.00000	0.0472456
$449$	−4.06652	−0.191911	−0.0959554	−	0.995386i	$-0.530591\pi$
$449$	−4.06652	−0.191911	−0.0959554	+	0.995386i	$0.530591\pi$
$450$	0	0
$451$	2.89219	0.136188
$452$	19.9335	0.937592
$453$	0	0
$454$	8.46093	0.397091
$455$	−1.05489	−0.0494542
$456$	0	0
$457$	−16.8393	−0.787708	−0.393854	−	0.919173i	$-0.628858\pi$
$457$	−16.8393	−0.787708	−0.393854	+	0.919173i	$0.628858\pi$
$458$	2.16271	0.101057
$459$	0	0
$460$	−23.5687	−1.09890
$461$	15.4590	0.719995	0.359998	−	0.932953i	$-0.382777\pi$
$461$	15.4590	0.719995	0.359998	+	0.932953i	$0.382777\pi$
$462$	0	0
$463$	19.5687	0.909437	0.454718	−	0.890635i	$-0.349740\pi$
$463$	19.5687	0.909437	0.454718	+	0.890635i	$0.349740\pi$
$464$	2.36474	0.109780
$465$	0	0
$466$	−3.45896	−0.160233
$467$	−16.2982	−0.754192	−0.377096	−	0.926174i	$-0.623077\pi$
$467$	−16.2982	−0.754192	−0.377096	+	0.926174i	$0.623077\pi$
$468$	0	0
$469$	−6.14911	−0.283940
$470$	−18.8786	−0.870804
$471$	0	0
$472$	−10.5139	−0.483939
$473$	−11.4196	−0.525075
$474$	0	0
$475$	24.4177	1.12036
$476$	2.89219	0.132563
$477$	0	0
$478$	16.2982	0.745463
$479$	0.431256	0.0197046	0.00985231	−	0.999951i	$-0.496864\pi$
$479$	0.431256	0.0197046	0.00985231	+	0.999951i	$0.496864\pi$
$480$	0	0
$481$	1.38031	0.0629367
$482$	−4.67656	−0.213011
$483$	0	0
$484$	1.00000	0.0454545
$485$	19.4629	0.883765
$486$	0	0
$487$	−14.0826	−0.638143	−0.319072	−	0.947731i	$-0.603371\pi$
$487$	−14.0826	−0.638143	−0.319072	+	0.947731i	$0.603371\pi$
$488$	11.9335	0.540203
$489$	0	0
$490$	−2.89219	−0.130656
$491$	23.5687	1.06364	0.531821	−	0.846857i	$-0.321508\pi$
$491$	23.5687	1.06364	0.531821	+	0.846857i	$0.321508\pi$
$492$	0	0
$493$	6.83927	0.308025
$494$	2.64688	0.119089
$495$	0	0
$496$	10.6766	0.479392
$497$	−13.9335	−0.625002
$498$	0	0
$499$	28.9884	1.29770	0.648849	−	0.760917i	$-0.275251\pi$
$499$	28.9884	1.29770	0.648849	+	0.760917i	$0.275251\pi$
$500$	−4.72948	−0.211509
$501$	0	0
$502$	−10.0826	−0.450008
$503$	−25.0549	−1.11714	−0.558571	−	0.829457i	$-0.688650\pi$
$503$	−25.0549	−1.11714	−0.558571	+	0.829457i	$0.688650\pi$
$504$	0	0
$505$	3.67458	0.163517
$506$	−8.14911	−0.362272
$507$	0	0
$508$	10.1491	0.450294
$509$	−29.0020	−1.28549	−0.642745	−	0.766080i	$-0.722204\pi$
$509$	−29.0020	−1.28549	−0.642745	+	0.766080i	$0.722204\pi$
$510$	0	0
$511$	−1.10781	−0.0490068
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	9.41963	0.415482
$515$	−2.58037	−0.113705
$516$	0	0
$517$	−6.52745	−0.287077
$518$	3.78437	0.166276
$519$	0	0
$520$	1.05489	0.0462601
$521$	−20.9884	−0.919517	−0.459759	−	0.888044i	$-0.652064\pi$
$521$	−20.9884	−0.919517	−0.459759	+	0.888044i	$0.652064\pi$
$522$	0	0
$523$	−15.8806	−0.694408	−0.347204	−	0.937790i	$-0.612869\pi$
$523$	−15.8806	−0.694408	−0.347204	+	0.937790i	$0.612869\pi$
$524$	20.8922	0.912679
$525$	0	0
$526$	6.87859	0.299921
$527$	30.8786	1.34509
$528$	0	0
$529$	43.4080	1.88730
$530$	34.0826	1.48045
$531$	0	0
$532$	7.25693	0.314628
$533$	1.05489	0.0456925
$534$	0	0
$535$	−16.7295	−0.723278
$536$	6.14911	0.265601
$537$	0	0
$538$	26.8922	1.15940
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−14.4745	−0.622308	−0.311154	−	0.950359i	$-0.600716\pi$
$541$	−14.4745	−0.622308	−0.311154	+	0.950359i	$0.600716\pi$
$542$	21.7844	0.935719
$543$	0	0
$544$	−2.89219	−0.124001
$545$	23.1375	0.991101
$546$	0	0
$547$	−29.0710	−1.24298	−0.621492	−	0.783420i	$-0.713473\pi$
$547$	−29.0710	−1.24298	−0.621492	+	0.783420i	$0.713473\pi$
$548$	7.93348	0.338902
$549$	0	0
$550$	3.36474	0.143473
$551$	17.1607	0.731072
$552$	0	0
$553$	10.1491	0.431584
$554$	−3.27052	−0.138951
$555$	0	0
$556$	18.8257	0.798386
$557$	−37.5022	−1.58902	−0.794510	−	0.607251i	$-0.792272\pi$
$557$	−37.5022	−1.58902	−0.794510	+	0.607251i	$0.792272\pi$
$558$	0	0
$559$	−4.16519	−0.176169
$560$	2.89219	0.122217
$561$	0	0
$562$	10.2982	0.434404
$563$	27.1078	1.14246	0.571229	−	0.820791i	$-0.306467\pi$
$563$	27.1078	1.14246	0.571229	+	0.820791i	$0.306467\pi$
$564$	0	0
$565$	57.6513	2.42541
$566$	28.3118	1.19003
$567$	0	0
$568$	13.9335	0.584636
$569$	−46.2982	−1.94092	−0.970461	−	0.241257i	$-0.922440\pi$
$569$	−46.2982	−1.94092	−0.970461	+	0.241257i	$0.922440\pi$
$570$	0	0
$571$	5.63526	0.235828	0.117914	−	0.993024i	$-0.462379\pi$
$571$	5.63526	0.235828	0.117914	+	0.993024i	$0.462379\pi$
$572$	0.364739	0.0152505
$573$	0	0
$574$	2.89219	0.120718
$575$	−27.4196	−1.14348
$576$	0	0
$577$	0.513850	0.0213919	0.0106959	−	0.999943i	$-0.496595\pi$
$577$	0.513850	0.0213919	0.0106959	+	0.999943i	$0.496595\pi$
$578$	8.63526	0.359179
$579$	0	0
$580$	6.83927	0.283985
$581$	3.10781	0.128934
$582$	0	0
$583$	11.7844	0.488059
$584$	1.10781	0.0458417
$585$	0	0
$586$	11.7844	0.486808
$587$	−17.7844	−0.734040	−0.367020	−	0.930213i	$-0.619622\pi$
$587$	−17.7844	−0.734040	−0.367020	+	0.930213i	$0.619622\pi$
$588$	0	0
$589$	77.4790	3.19247
$590$	−30.4080	−1.25188
$591$	0	0
$592$	−3.78437	−0.155537
$593$	−26.0297	−1.06891	−0.534455	−	0.845197i	$-0.679483\pi$
$593$	−26.0297	−1.06891	−0.534455	+	0.845197i	$0.679483\pi$
$594$	0	0
$595$	8.36474	0.342921
$596$	1.27052	0.0520426
$597$	0	0
$598$	−2.97230	−0.121546
$599$	−25.9335	−1.05961	−0.529807	−	0.848118i	$-0.677736\pi$
$599$	−25.9335	−1.05961	−0.529807	+	0.848118i	$0.677736\pi$
$600$	0	0
$601$	−33.2730	−1.35723	−0.678617	−	0.734492i	$-0.737420\pi$
$601$	−33.2730	−1.35723	−0.678617	+	0.734492i	$0.737420\pi$
$602$	−11.4196	−0.465429
$603$	0	0
$604$	−8.29822	−0.337650
$605$	2.89219	0.117584
$606$	0	0
$607$	−34.4080	−1.39658	−0.698289	−	0.715816i	$-0.746055\pi$
$607$	−34.4080	−1.39658	−0.698289	+	0.715816i	$0.746055\pi$
$608$	−7.25693	−0.294307
$609$	0	0
$610$	34.5139	1.39742
$611$	−2.38082	−0.0963175
$612$	0	0
$613$	7.33704	0.296340	0.148170	−	0.988962i	$-0.452662\pi$
$613$	7.33704	0.296340	0.148170	+	0.988962i	$0.452662\pi$
$614$	−4.01360	−0.161976
$615$	0	0
$616$	1.00000	0.0402911
$617$	7.70178	0.310062	0.155031	−	0.987910i	$-0.450452\pi$
$617$	7.70178	0.310062	0.155031	+	0.987910i	$0.450452\pi$
$618$	0	0
$619$	−20.2982	−0.815854	−0.407927	−	0.913014i	$-0.633748\pi$
$619$	−20.2982	−0.815854	−0.407927	+	0.913014i	$0.633748\pi$
$620$	30.8786	1.24011
$621$	0	0
$622$	−6.52745	−0.261727
$623$	−2.14911	−0.0861023
$624$	0	0
$625$	−30.5022	−1.22009
$626$	−0.513850	−0.0205376
$627$	0	0
$628$	−18.8922	−0.753880
$629$	−10.9451	−0.436410
$630$	0	0
$631$	35.2433	1.40301	0.701507	−	0.712662i	$-0.252511\pi$
$631$	35.2433	1.40301	0.701507	+	0.712662i	$0.252511\pi$
$632$	−10.1491	−0.403710
$633$	0	0
$634$	7.48615	0.297313
$635$	29.3531	1.16484
$636$	0	0
$637$	−0.364739	−0.0144515
$638$	2.36474	0.0936209
$639$	0	0
$640$	−2.89219	−0.114324
$641$	−25.7572	−1.01735	−0.508674	−	0.860959i	$-0.669864\pi$
$641$	−25.7572	−1.01735	−0.508674	+	0.860959i	$0.669864\pi$
$642$	0	0
$643$	−5.48615	−0.216353	−0.108176	−	0.994132i	$-0.534501\pi$
$643$	−5.48615	−0.216353	−0.108176	+	0.994132i	$0.534501\pi$
$644$	−8.14911	−0.321120
$645$	0	0
$646$	−20.9884	−0.825777
$647$	−17.4726	−0.686917	−0.343458	−	0.939168i	$-0.611598\pi$
$647$	−17.4726	−0.686917	−0.343458	+	0.939168i	$0.611598\pi$
$648$	0	0
$649$	−10.5139	−0.412705
$650$	1.22725	0.0481368
$651$	0	0
$652$	−20.2982	−0.794940
$653$	8.54104	0.334237	0.167118	−	0.985937i	$-0.446554\pi$
$653$	8.54104	0.334237	0.167118	+	0.985937i	$0.446554\pi$
$654$	0	0
$655$	60.4241	2.36096
$656$	−2.89219	−0.112921
$657$	0	0
$658$	−6.52745	−0.254466
$659$	43.5416	1.69614	0.848069	−	0.529886i	$-0.177765\pi$
$659$	43.5416	1.69614	0.848069	+	0.529886i	$0.177765\pi$
$660$	0	0
$661$	18.8650	0.733763	0.366882	−	0.930268i	$-0.380425\pi$
$661$	18.8650	0.733763	0.366882	+	0.930268i	$0.380425\pi$
$662$	13.4196	0.521569
$663$	0	0
$664$	−3.10781	−0.120607
$665$	20.9884	0.813894
$666$	0	0
$667$	−19.2705	−0.746158
$668$	−1.05489	−0.0408151
$669$	0	0
$670$	17.7844	0.687070
$671$	11.9335	0.460687
$672$	0	0
$673$	−8.08259	−0.311561	−0.155781	−	0.987792i	$-0.549789\pi$
$673$	−8.08259	−0.311561	−0.155781	+	0.987792i	$0.549789\pi$
$674$	8.94511	0.344553
$675$	0	0
$676$	−12.8670	−0.494883
$677$	6.62364	0.254567	0.127284	−	0.991866i	$-0.459374\pi$
$677$	6.62364	0.254567	0.127284	+	0.991866i	$0.459374\pi$
$678$	0	0
$679$	6.72948	0.258254
$680$	−8.36474	−0.320773
$681$	0	0
$682$	10.6766	0.408827
$683$	−13.7179	−0.524899	−0.262450	−	0.964946i	$-0.584530\pi$
$683$	−13.7179	−0.524899	−0.262450	+	0.964946i	$0.584530\pi$
$684$	0	0
$685$	22.9451	0.876687
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	11.4196	0.435369
$689$	4.29822	0.163749
$690$	0	0
$691$	−18.3808	−0.699239	−0.349620	−	0.936892i	$-0.613689\pi$
$691$	−18.3808	−0.699239	−0.349620	+	0.936892i	$0.613689\pi$
$692$	16.5139	0.627762
$693$	0	0
$694$	5.78437	0.219572
$695$	54.4473	2.06531
$696$	0	0
$697$	−8.36474	−0.316837
$698$	−22.1491	−0.838356
$699$	0	0
$700$	3.36474	0.127175
$701$	31.3259	1.18316	0.591582	−	0.806245i	$-0.298504\pi$
$701$	31.3259	1.18316	0.591582	+	0.806245i	$0.298504\pi$
$702$	0	0
$703$	−27.4629	−1.03578
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	9.41963	0.354513
$707$	1.27052	0.0477829
$708$	0	0
$709$	−41.1103	−1.54393	−0.771965	−	0.635665i	$-0.780726\pi$
$709$	−41.1103	−1.54393	−0.771965	+	0.635665i	$0.780726\pi$
$710$	40.2982	1.51237
$711$	0	0
$712$	2.14911	0.0805413
$713$	−87.0045	−3.25834
$714$	0	0
$715$	1.05489	0.0394508
$716$	−20.9884	−0.784372
$717$	0	0
$718$	−18.4473	−0.688448
$719$	−1.17433	−0.0437952	−0.0218976	−	0.999760i	$-0.506971\pi$
$719$	−1.17433	−0.0437952	−0.0218976	+	0.999760i	$0.506971\pi$
$720$	0	0
$721$	−0.892186	−0.0332267
$722$	−33.6630	−1.25281
$723$	0	0
$724$	−1.10781	−0.0411716
$725$	7.95673	0.295506
$726$	0	0
$727$	43.2730	1.60491	0.802453	−	0.596715i	$-0.203528\pi$
$727$	43.2730	1.60491	0.802453	+	0.596715i	$0.203528\pi$
$728$	0.364739	0.0135181
$729$	0	0
$730$	3.20400	0.118586
$731$	33.0277	1.22157
$732$	0	0
$733$	−14.9058	−0.550558	−0.275279	−	0.961364i	$-0.588770\pi$
$733$	−14.9058	−0.550558	−0.275279	+	0.961364i	$0.588770\pi$
$734$	18.6766	0.689364
$735$	0	0
$736$	8.14911	0.300380
$737$	6.14911	0.226505
$738$	0	0
$739$	24.9058	0.916174	0.458087	−	0.888907i	$-0.348535\pi$
$739$	24.9058	0.916174	0.458087	+	0.888907i	$0.348535\pi$
$740$	−10.9451	−0.402350
$741$	0	0
$742$	11.7844	0.432618
$743$	11.5687	0.424416	0.212208	−	0.977225i	$-0.431935\pi$
$743$	11.5687	0.424416	0.212208	+	0.977225i	$0.431935\pi$
$744$	0	0
$745$	3.67458	0.134626
$746$	19.2040	0.703109
$747$	0	0
$748$	−2.89219	−0.105749
$749$	−5.78437	−0.211356
$750$	0	0
$751$	34.3808	1.25457	0.627287	−	0.778788i	$-0.284165\pi$
$751$	34.3808	1.25457	0.627287	+	0.778788i	$0.284165\pi$
$752$	6.52745	0.238031
$753$	0	0
$754$	0.862513	0.0314109
$755$	−24.0000	−0.873449
$756$	0	0
$757$	2.62364	0.0953577	0.0476789	−	0.998863i	$-0.484818\pi$
$757$	2.62364	0.0953577	0.0476789	+	0.998863i	$0.484818\pi$
$758$	31.8670	1.15746
$759$	0	0
$760$	−20.9884	−0.761328
$761$	45.7315	1.65776	0.828882	−	0.559424i	$-0.188977\pi$
$761$	45.7315	1.65776	0.828882	+	0.559424i	$0.188977\pi$
$762$	0	0
$763$	8.00000	0.289619
$764$	10.6902	0.386756
$765$	0	0
$766$	29.0413	1.04930
$767$	−3.83481	−0.138467
$768$	0	0
$769$	14.5668	0.525291	0.262646	−	0.964892i	$-0.415405\pi$
$769$	14.5668	0.525291	0.262646	+	0.964892i	$0.415405\pi$
$770$	2.89219	0.104227
$771$	0	0
$772$	−21.5687	−0.776276
$773$	38.0297	1.36783	0.683916	−	0.729561i	$-0.260275\pi$
$773$	38.0297	1.36783	0.683916	+	0.729561i	$0.260275\pi$
$774$	0	0
$775$	35.9238	1.29042
$776$	−6.72948	−0.241574
$777$	0	0
$778$	−26.5964	−0.953529
$779$	−20.9884	−0.751987
$780$	0	0
$781$	13.9335	0.498579
$782$	23.5687	0.842817
$783$	0	0
$784$	1.00000	0.0357143
$785$	−54.6397	−1.95017
$786$	0	0
$787$	−26.2020	−0.934002	−0.467001	−	0.884257i	$-0.654665\pi$
$787$	−26.2020	−0.934002	−0.467001	+	0.884257i	$0.654665\pi$
$788$	4.90578	0.174761
$789$	0	0
$790$	−29.3531	−1.04434
$791$	19.9335	0.708753
$792$	0	0
$793$	4.35261	0.154566
$794$	8.37834	0.297336
$795$	0	0
$796$	−7.10781	−0.251930
$797$	−13.8373	−0.490142	−0.245071	−	0.969505i	$-0.578811\pi$
$797$	−13.8373	−0.490142	−0.245071	+	0.969505i	$0.578811\pi$
$798$	0	0
$799$	18.8786	0.667876
$800$	−3.36474	−0.118961
$801$	0	0
$802$	−7.93348	−0.280141
$803$	1.10781	0.0390939
$804$	0	0
$805$	−23.5687	−0.830689
$806$	3.89416	0.137166
$807$	0	0
$808$	−1.27052	−0.0446968
$809$	28.0826	0.987331	0.493666	−	0.869652i	$-0.335657\pi$
$809$	28.0826	0.987331	0.493666	+	0.869652i	$0.335657\pi$
$810$	0	0
$811$	56.1516	1.97175	0.985875	−	0.167486i	$-0.0535648\pi$
$811$	56.1516	1.97175	0.985875	+	0.167486i	$0.0535648\pi$
$812$	2.36474	0.0829861
$813$	0	0
$814$	−3.78437	−0.132642
$815$	−58.7062	−2.05639
$816$	0	0
$817$	82.8714	2.89930
$818$	24.6766	0.862796
$819$	0	0
$820$	−8.36474	−0.292109
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	0	0
$823$	35.2433	1.22851	0.614253	−	0.789109i	$-0.289458\pi$
$823$	35.2433	1.22851	0.614253	+	0.789109i	$0.289458\pi$
$824$	0.892186	0.0310808
$825$	0	0
$826$	−10.5139	−0.365824
$827$	0.431256	0.0149963	0.00749813	−	0.999972i	$-0.497613\pi$
$827$	0.431256	0.0149963	0.00749813	+	0.999972i	$0.497613\pi$
$828$	0	0
$829$	−2.86499	−0.0995053	−0.0497527	−	0.998762i	$-0.515843\pi$
$829$	−2.86499	−0.0995053	−0.0497527	+	0.998762i	$0.515843\pi$
$830$	−8.98838	−0.311991
$831$	0	0
$832$	−0.364739	−0.0126451
$833$	2.89219	0.100208
$834$	0	0
$835$	−3.05095	−0.105582
$836$	−7.25693	−0.250986
$837$	0	0
$838$	−2.54104	−0.0877789
$839$	−1.17433	−0.0405424	−0.0202712	−	0.999795i	$-0.506453\pi$
$839$	−1.17433	−0.0405424	−0.0202712	+	0.999795i	$0.506453\pi$
$840$	0	0
$841$	−23.4080	−0.807173
$842$	−2.43126	−0.0837866
$843$	0	0
$844$	8.87859	0.305614
$845$	−37.2137	−1.28019
$846$	0	0
$847$	1.00000	0.0343604
$848$	−11.7844	−0.404677
$849$	0	0
$850$	−9.73145	−0.333786
$851$	30.8393	1.05716
$852$	0	0
$853$	−51.8004	−1.77361	−0.886807	−	0.462140i	$-0.847082\pi$
$853$	−51.8004	−1.77361	−0.886807	+	0.462140i	$0.847082\pi$
$854$	11.9335	0.408355
$855$	0	0
$856$	5.78437	0.197706
$857$	−12.7824	−0.436638	−0.218319	−	0.975877i	$-0.570057\pi$
$857$	−12.7824	−0.436638	−0.218319	+	0.975877i	$0.570057\pi$
$858$	0	0
$859$	16.7567	0.571730	0.285865	−	0.958270i	$-0.407719\pi$
$859$	16.7567	0.571730	0.285865	+	0.958270i	$0.407719\pi$
$860$	33.0277	1.12624
$861$	0	0
$862$	2.58037	0.0878877
$863$	−26.5571	−0.904015	−0.452007	−	0.892014i	$-0.649292\pi$
$863$	−26.5571	−0.904015	−0.452007	+	0.892014i	$0.649292\pi$
$864$	0	0
$865$	47.7611	1.62393
$866$	−37.1375	−1.26198
$867$	0	0
$868$	10.6766	0.362386
$869$	−10.1491	−0.344285
$870$	0	0
$871$	2.24282	0.0759951
$872$	−8.00000	−0.270914
$873$	0	0
$874$	59.1375	2.00036
$875$	−4.72948	−0.159886
$876$	0	0
$877$	−15.5687	−0.525719	−0.262860	−	0.964834i	$-0.584666\pi$
$877$	−15.5687	−0.525719	−0.262860	+	0.964834i	$0.584666\pi$
$878$	−18.5139	−0.624812
$879$	0	0
$880$	−2.89219	−0.0974956
$881$	45.6907	1.53936	0.769679	−	0.638431i	$-0.220416\pi$
$881$	45.6907	1.53936	0.769679	+	0.638431i	$0.220416\pi$
$882$	0	0
$883$	−55.0045	−1.85105	−0.925524	−	0.378690i	$-0.876375\pi$
$883$	−55.0045	−1.85105	−0.925524	+	0.378690i	$0.876375\pi$
$884$	−1.05489	−0.0354799
$885$	0	0
$886$	−21.4196	−0.719607
$887$	−30.8393	−1.03548	−0.517741	−	0.855538i	$-0.673227\pi$
$887$	−30.8393	−1.03548	−0.517741	+	0.855538i	$0.673227\pi$
$888$	0	0
$889$	10.1491	0.340390
$890$	6.21563	0.208348
$891$	0	0
$892$	22.2453	0.744828
$893$	47.3692	1.58515
$894$	0	0
$895$	−60.7023	−2.02905
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	4.06652	0.135701
$899$	25.2473	0.842044
$900$	0	0
$901$	−34.0826	−1.13546
$902$	−2.89219	−0.0962993
$903$	0	0
$904$	−19.9335	−0.662978
$905$	−3.20400	−0.106505
$906$	0	0
$907$	−25.4196	−0.844045	−0.422023	−	0.906585i	$-0.638680\pi$
$907$	−25.4196	−0.844045	−0.422023	+	0.906585i	$0.638680\pi$
$908$	−8.46093	−0.280786
$909$	0	0
$910$	1.05489	0.0349694
$911$	−50.5571	−1.67503	−0.837516	−	0.546413i	$-0.815993\pi$
$911$	−50.5571	−1.67503	−0.837516	+	0.546413i	$0.815993\pi$
$912$	0	0
$913$	−3.10781	−0.102854
$914$	16.8393	0.556993
$915$	0	0
$916$	−2.16271	−0.0714579
$917$	20.8922	0.689921
$918$	0	0
$919$	12.7295	0.419907	0.209953	−	0.977711i	$-0.432669\pi$
$919$	12.7295	0.419907	0.209953	+	0.977711i	$0.432669\pi$
$920$	23.5687	0.777038
$921$	0	0
$922$	−15.4590	−0.509114
$923$	5.08209	0.167279
$924$	0	0
$925$	−12.7334	−0.418672
$926$	−19.5687	−0.643069
$927$	0	0
$928$	−2.36474	−0.0776264
$929$	6.44733	0.211530	0.105765	−	0.994391i	$-0.466271\pi$
$929$	6.44733	0.211530	0.105765	+	0.994391i	$0.466271\pi$
$930$	0	0
$931$	7.25693	0.237836
$932$	3.45896	0.113302
$933$	0	0
$934$	16.2982	0.533294
$935$	−8.36474	−0.273556
$936$	0	0
$937$	39.1904	1.28029	0.640147	−	0.768252i	$-0.278873\pi$
$937$	39.1904	1.28029	0.640147	+	0.768252i	$0.278873\pi$
$938$	6.14911	0.200776
$939$	0	0
$940$	18.8786	0.615752
$941$	37.1103	1.20976	0.604881	−	0.796316i	$-0.293221\pi$
$941$	37.1103	1.20976	0.604881	+	0.796316i	$0.293221\pi$
$942$	0	0
$943$	23.5687	0.767504
$944$	10.5139	0.342197
$945$	0	0
$946$	11.4196	0.371284
$947$	49.3259	1.60288	0.801439	−	0.598077i	$-0.204068\pi$
$947$	49.3259	1.60288	0.801439	+	0.598077i	$0.204068\pi$
$948$	0	0
$949$	0.404063	0.0131164
$950$	−24.4177	−0.792213
$951$	0	0
$952$	−2.89219	−0.0937363
$953$	−24.2156	−0.784421	−0.392211	−	0.919875i	$-0.628290\pi$
$953$	−24.2156	−0.784421	−0.392211	+	0.919875i	$0.628290\pi$
$954$	0	0
$955$	30.9179	1.00048
$956$	−16.2982	−0.527122
$957$	0	0
$958$	−0.431256	−0.0139333
$959$	7.93348	0.256186
$960$	0	0
$961$	82.9889	2.67706
$962$	−1.38031	−0.0445029
$963$	0	0
$964$	4.67656	0.150622
$965$	−62.3808	−2.00811
$966$	0	0
$967$	−43.8670	−1.41067	−0.705333	−	0.708876i	$-0.749203\pi$
$967$	−43.8670	−1.41067	−0.705333	+	0.708876i	$0.749203\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	−19.4629	−0.624916
$971$	−28.2982	−0.908133	−0.454067	−	0.890968i	$-0.650027\pi$
$971$	−28.2982	−0.908133	−0.454067	+	0.890968i	$0.650027\pi$
$972$	0	0
$973$	18.8257	0.603523
$974$	14.0826	0.451235
$975$	0	0
$976$	−11.9335	−0.381981
$977$	28.2982	0.905340	0.452670	−	0.891678i	$-0.350472\pi$
$977$	28.2982	0.905340	0.452670	+	0.891678i	$0.350472\pi$
$978$	0	0
$979$	2.14911	0.0686859
$980$	2.89219	0.0923875
$981$	0	0
$982$	−23.5687	−0.752109
$983$	32.6372	1.04097	0.520483	−	0.853872i	$-0.325752\pi$
$983$	32.6372	1.04097	0.520483	+	0.853872i	$0.325752\pi$
$984$	0	0
$985$	14.1884	0.452081
$986$	−6.83927	−0.217807
$987$	0	0
$988$	−2.64688	−0.0842086
$989$	−93.0599	−2.95913
$990$	0	0
$991$	−31.8670	−1.01229	−0.506144	−	0.862449i	$-0.668929\pi$
$991$	−31.8670	−1.01229	−0.506144	+	0.862449i	$0.668929\pi$
$992$	−10.6766	−0.338981
$993$	0	0
$994$	13.9335	0.441943
$995$	−20.5571	−0.651705
$996$	0	0
$997$	−37.4196	−1.18509	−0.592546	−	0.805537i	$-0.701877\pi$
$997$	−37.4196	−1.18509	−0.592546	+	0.805537i	$0.701877\pi$
$998$	−28.9884	−0.917611
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1386.2.a.q.1.3	✓	3
3.2	odd	2		1386.2.a.r.1.1	yes	3
7.6	odd	2		9702.2.a.dx.1.1		3
21.20	even	2		9702.2.a.dy.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1386.2.a.q.1.3	✓	3	1.1	even	1	trivial
1386.2.a.r.1.1	yes	3	3.2	odd	2
9702.2.a.dx.1.1		3	7.6	odd	2
9702.2.a.dy.1.3		3	21.20	even	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:	\( N \)	\(=\)	\( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1386.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.89219 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.89219 q^{5} +1.00000 q^{7} -1.00000 q^{8} -2.89219 q^{10} -1.00000 q^{11} -0.364739 q^{13} -1.00000 q^{14} +1.00000 q^{16} +2.89219 q^{17} +7.25693 q^{19} +2.89219 q^{20} +1.00000 q^{22} -8.14911 q^{23} +3.36474 q^{25} +0.364739 q^{26} +1.00000 q^{28} +2.36474 q^{29} +10.6766 q^{31} -1.00000 q^{32} -2.89219 q^{34} +2.89219 q^{35} -3.78437 q^{37} -7.25693 q^{38} -2.89219 q^{40} -2.89219 q^{41} +11.4196 q^{43} -1.00000 q^{44} +8.14911 q^{46} +6.52745 q^{47} +1.00000 q^{49} -3.36474 q^{50} -0.364739 q^{52} -11.7844 q^{53} -2.89219 q^{55} -1.00000 q^{56} -2.36474 q^{58} +10.5139 q^{59} -11.9335 q^{61} -10.6766 q^{62} +1.00000 q^{64} -1.05489 q^{65} -6.14911 q^{67} +2.89219 q^{68} -2.89219 q^{70} -13.9335 q^{71} -1.10781 q^{73} +3.78437 q^{74} +7.25693 q^{76} -1.00000 q^{77} +10.1491 q^{79} +2.89219 q^{80} +2.89219 q^{82} +3.10781 q^{83} +8.36474 q^{85} -11.4196 q^{86} +1.00000 q^{88} -2.14911 q^{89} -0.364739 q^{91} -8.14911 q^{92} -6.52745 q^{94} +20.9884 q^{95} +6.72948 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} - 2 q^{5} + 3 q^{7} - 3 q^{8}+O(q^{10}) \) 3 * q - 3 * q^2 + 3 * q^4 - 2 * q^5 + 3 * q^7 - 3 * q^8	\( 3 q - 3 q^{2} + 3 q^{4} - 2 q^{5} + 3 q^{7} - 3 q^{8} + 2 q^{10} - 3 q^{11} - 3 q^{14} + 3 q^{16} - 2 q^{17} + 10 q^{19} - 2 q^{20} + 3 q^{22} - 2 q^{23} + 9 q^{25} + 3 q^{28} + 6 q^{29} - 3 q^{32} + 2 q^{34} - 2 q^{35} + 10 q^{37} - 10 q^{38} + 2 q^{40} + 2 q^{41} + 14 q^{43} - 3 q^{44} + 2 q^{46} + 10 q^{47} + 3 q^{49} - 9 q^{50} - 14 q^{53} + 2 q^{55} - 3 q^{56} - 6 q^{58} + 8 q^{59} + 8 q^{61} + 3 q^{64} + 16 q^{65} + 4 q^{67} - 2 q^{68} + 2 q^{70} + 2 q^{71} - 14 q^{73} - 10 q^{74} + 10 q^{76} - 3 q^{77} + 8 q^{79} - 2 q^{80} - 2 q^{82} + 20 q^{83} + 24 q^{85} - 14 q^{86} + 3 q^{88} + 16 q^{89} - 2 q^{92} - 10 q^{94} + 18 q^{97} - 3 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^4 - 2 * q^5 + 3 * q^7 - 3 * q^8 + 2 * q^10 - 3 * q^11 - 3 * q^14 + 3 * q^16 - 2 * q^17 + 10 * q^19 - 2 * q^20 + 3 * q^22 - 2 * q^23 + 9 * q^25 + 3 * q^28 + 6 * q^29 - 3 * q^32 + 2 * q^34 - 2 * q^35 + 10 * q^37 - 10 * q^38 + 2 * q^40 + 2 * q^41 + 14 * q^43 - 3 * q^44 + 2 * q^46 + 10 * q^47 + 3 * q^49 - 9 * q^50 - 14 * q^53 + 2 * q^55 - 3 * q^56 - 6 * q^58 + 8 * q^59 + 8 * q^61 + 3 * q^64 + 16 * q^65 + 4 * q^67 - 2 * q^68 + 2 * q^70 + 2 * q^71 - 14 * q^73 - 10 * q^74 + 10 * q^76 - 3 * q^77 + 8 * q^79 - 2 * q^80 - 2 * q^82 + 20 * q^83 + 24 * q^85 - 14 * q^86 + 3 * q^88 + 16 * q^89 - 2 * q^92 - 10 * q^94 + 18 * q^97 - 3 * q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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