LMFDB - Embedded newform 1350.2.a.w.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1350,2,Mod(1,1350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1350, 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("1350.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1350 = 2 \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1350.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:	$10.7798042729$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{19}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 19 $ x^2 - 19
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 270)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-4.35890$ of defining polynomial
Character	$\chi$	$=$	1350.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -4.35890 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} -4.35890 q^{7} -1.00000 q^{8} -4.35890 q^{11} +4.35890 q^{14} +1.00000 q^{16} -4.00000 q^{17} +6.00000 q^{19} +4.35890 q^{22} +2.00000 q^{23} -4.35890 q^{28} +7.00000 q^{31} -1.00000 q^{32} +4.00000 q^{34} +8.71780 q^{37} -6.00000 q^{38} +8.71780 q^{41} +8.71780 q^{43} -4.35890 q^{44} -2.00000 q^{46} -2.00000 q^{47} +12.0000 q^{49} +3.00000 q^{53} +4.35890 q^{56} -8.71780 q^{59} -4.00000 q^{61} -7.00000 q^{62} +1.00000 q^{64} -8.71780 q^{67} -4.00000 q^{68} -4.35890 q^{73} -8.71780 q^{74} +6.00000 q^{76} +19.0000 q^{77} -8.71780 q^{82} +5.00000 q^{83} -8.71780 q^{86} +4.35890 q^{88} +8.71780 q^{89} +2.00000 q^{92} +2.00000 q^{94} +4.35890 q^{97} -12.0000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 2 q^{16} - 8 q^{17} + 12 q^{19} + 4 q^{23} + 14 q^{31} - 2 q^{32} + 8 q^{34} - 12 q^{38} - 4 q^{46} - 4 q^{47} + 24 q^{49} + 6 q^{53} - 8 q^{61} - 14 q^{62} + 2 q^{64} - 8 q^{68} + 12 q^{76} + 38 q^{77} + 10 q^{83} + 4 q^{92} + 4 q^{94} - 24 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 + 2 * q^16 - 8 * q^17 + 12 * q^19 + 4 * q^23 + 14 * q^31 - 2 * q^32 + 8 * q^34 - 12 * q^38 - 4 * q^46 - 4 * q^47 + 24 * q^49 + 6 * q^53 - 8 * q^61 - 14 * q^62 + 2 * q^64 - 8 * q^68 + 12 * q^76 + 38 * q^77 + 10 * q^83 + 4 * q^92 + 4 * q^94 - 24 * 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
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.35890	−1.64751	−0.823754	−	0.566947i	$-0.808125\pi$
$7$	−4.35890	−1.64751	−0.823754	+	0.566947i	$0.808125\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	0	0
$11$	−4.35890	−1.31426	−0.657129	−	0.753778i	$-0.728229\pi$
$11$	−4.35890	−1.31426	−0.657129	+	0.753778i	$0.728229\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	4.35890	1.16496
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	0	0
$21$	0	0
$22$	4.35890	0.929320
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	2.00000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	−4.35890	−0.823754
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	7.00000	1.25724	0.628619	−	0.777714i	$-0.283621\pi$
$31$	7.00000	1.25724	0.628619	+	0.777714i	$0.283621\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	4.00000	0.685994
$35$	0	0
$36$	0	0
$37$	8.71780	1.43320	0.716599	−	0.697486i	$-0.245698\pi$
$37$	8.71780	1.43320	0.716599	+	0.697486i	$0.245698\pi$
$38$	−6.00000	−0.973329
$39$	0	0
$40$	0	0
$41$	8.71780	1.36149	0.680746	−	0.732520i	$-0.261656\pi$
$41$	8.71780	1.36149	0.680746	+	0.732520i	$0.261656\pi$
$42$	0	0
$43$	8.71780	1.32945	0.664726	−	0.747087i	$-0.268548\pi$
$43$	8.71780	1.32945	0.664726	+	0.747087i	$0.268548\pi$
$44$	−4.35890	−0.657129
$45$	0	0
$46$	−2.00000	−0.294884
$47$	−2.00000	−0.291730	−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−2.00000	−0.291730	−0.145865	+	0.989305i	$0.546597\pi$
$48$	0	0
$49$	12.0000	1.71429
$50$	0	0
$51$	0	0
$52$	0	0
$53$	3.00000	0.412082	0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000	0.412082	0.206041	+	0.978543i	$0.433942\pi$
$54$	0	0
$55$	0	0
$56$	4.35890	0.582482
$57$	0	0
$58$	0	0
$59$	−8.71780	−1.13496	−0.567480	−	0.823387i	$-0.692082\pi$
$59$	−8.71780	−1.13496	−0.567480	+	0.823387i	$0.692082\pi$
$60$	0	0
$61$	−4.00000	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$62$	−7.00000	−0.889001
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−8.71780	−1.06505	−0.532524	−	0.846415i	$-0.678756\pi$
$67$	−8.71780	−1.06505	−0.532524	+	0.846415i	$0.678756\pi$
$68$	−4.00000	−0.485071
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−4.35890	−0.510171	−0.255085	−	0.966919i	$-0.582104\pi$
$73$	−4.35890	−0.510171	−0.255085	+	0.966919i	$0.582104\pi$
$74$	−8.71780	−1.01342
$75$	0	0
$76$	6.00000	0.688247
$77$	19.0000	2.16525
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	0	0
$82$	−8.71780	−0.962720
$83$	5.00000	0.548821	0.274411	−	0.961613i	$-0.411517\pi$
$83$	5.00000	0.548821	0.274411	+	0.961613i	$0.411517\pi$
$84$	0	0
$85$	0	0
$86$	−8.71780	−0.940064
$87$	0	0
$88$	4.35890	0.464660
$89$	8.71780	0.924085	0.462042	−	0.886858i	$-0.347117\pi$
$89$	8.71780	0.924085	0.462042	+	0.886858i	$0.347117\pi$
$90$	0	0
$91$	0	0
$92$	2.00000	0.208514
$93$	0	0
$94$	2.00000	0.206284
$95$	0	0
$96$	0	0
$97$	4.35890	0.442579	0.221290	−	0.975208i	$-0.428973\pi$
$97$	4.35890	0.442579	0.221290	+	0.975208i	$0.428973\pi$
$98$	−12.0000	−1.21218
$99$	0	0
$100$	0	0
$101$	4.35890	0.433727	0.216863	−	0.976202i	$-0.430417\pi$
$101$	4.35890	0.433727	0.216863	+	0.976202i	$0.430417\pi$
$102$	0	0
$103$	−8.71780	−0.858990	−0.429495	−	0.903069i	$-0.641308\pi$
$103$	−8.71780	−0.858990	−0.429495	+	0.903069i	$0.641308\pi$
$104$	0	0
$105$	0	0
$106$	−3.00000	−0.291386
$107$	−15.0000	−1.45010	−0.725052	−	0.688694i	$-0.758184\pi$
$107$	−15.0000	−1.45010	−0.725052	+	0.688694i	$0.758184\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	0	0
$112$	−4.35890	−0.411877
$113$	18.0000	1.69330	0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000	1.69330	0.846649	+	0.532152i	$0.178617\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	8.71780	0.802538
$119$	17.4356	1.59832
$120$	0	0
$121$	8.00000	0.727273
$122$	4.00000	0.362143
$123$	0	0
$124$	7.00000	0.628619
$125$	0	0
$126$	0	0
$127$	−4.35890	−0.386790	−0.193395	−	0.981121i	$-0.561950\pi$
$127$	−4.35890	−0.386790	−0.193395	+	0.981121i	$0.561950\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	−13.0767	−1.14252	−0.571258	−	0.820770i	$-0.693544\pi$
$131$	−13.0767	−1.14252	−0.571258	+	0.820770i	$0.693544\pi$
$132$	0	0
$133$	−26.1534	−2.26779
$134$	8.71780	0.753103
$135$	0	0
$136$	4.00000	0.342997
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	4.35890	0.360745
$147$	0	0
$148$	8.71780	0.716599
$149$	4.35890	0.357095	0.178547	−	0.983931i	$-0.442860\pi$
$149$	4.35890	0.357095	0.178547	+	0.983931i	$0.442860\pi$
$150$	0	0
$151$	−11.0000	−0.895167	−0.447584	−	0.894242i	$-0.647715\pi$
$151$	−11.0000	−0.895167	−0.447584	+	0.894242i	$0.647715\pi$
$152$	−6.00000	−0.486664
$153$	0	0
$154$	−19.0000	−1.53106
$155$	0	0
$156$	0	0
$157$	17.4356	1.39151	0.695756	−	0.718278i	$-0.255069\pi$
$157$	17.4356	1.39151	0.695756	+	0.718278i	$0.255069\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−8.71780	−0.687059
$162$	0	0
$163$	17.4356	1.36566	0.682831	−	0.730577i	$-0.260749\pi$
$163$	17.4356	1.36566	0.682831	+	0.730577i	$0.260749\pi$
$164$	8.71780	0.680746
$165$	0	0
$166$	−5.00000	−0.388075
$167$	18.0000	1.39288	0.696441	−	0.717614i	$-0.254766\pi$
$167$	18.0000	1.39288	0.696441	+	0.717614i	$0.254766\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	0	0
$172$	8.71780	0.664726
$173$	19.0000	1.44454	0.722272	−	0.691609i	$-0.243098\pi$
$173$	19.0000	1.44454	0.722272	+	0.691609i	$0.243098\pi$
$174$	0	0
$175$	0	0
$176$	−4.35890	−0.328564
$177$	0	0
$178$	−8.71780	−0.653427
$179$	−21.7945	−1.62900	−0.814499	−	0.580166i	$-0.802988\pi$
$179$	−21.7945	−1.62900	−0.814499	+	0.580166i	$0.802988\pi$
$180$	0	0
$181$	8.00000	0.594635	0.297318	−	0.954779i	$-0.403908\pi$
$181$	8.00000	0.594635	0.297318	+	0.954779i	$0.403908\pi$
$182$	0	0
$183$	0	0
$184$	−2.00000	−0.147442
$185$	0	0
$186$	0	0
$187$	17.4356	1.27502
$188$	−2.00000	−0.145865
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	21.7945	1.56880	0.784401	−	0.620254i	$-0.212970\pi$
$193$	21.7945	1.56880	0.784401	+	0.620254i	$0.212970\pi$
$194$	−4.35890	−0.312951
$195$	0	0
$196$	12.0000	0.857143
$197$	−5.00000	−0.356235	−0.178118	−	0.984009i	$-0.557001\pi$
$197$	−5.00000	−0.356235	−0.178118	+	0.984009i	$0.557001\pi$
$198$	0	0
$199$	−3.00000	−0.212664	−0.106332	−	0.994331i	$-0.533911\pi$
$199$	−3.00000	−0.212664	−0.106332	+	0.994331i	$0.533911\pi$
$200$	0	0
$201$	0	0
$202$	−4.35890	−0.306691
$203$	0	0
$204$	0	0
$205$	0	0
$206$	8.71780	0.607398
$207$	0	0
$208$	0	0
$209$	−26.1534	−1.80907
$210$	0	0
$211$	6.00000	0.413057	0.206529	−	0.978441i	$-0.433783\pi$
$211$	6.00000	0.413057	0.206529	+	0.978441i	$0.433783\pi$
$212$	3.00000	0.206041
$213$	0	0
$214$	15.0000	1.02538
$215$	0	0
$216$	0	0
$217$	−30.5123	−2.07131
$218$	−10.0000	−0.677285
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	8.71780	0.583787	0.291893	−	0.956451i	$-0.405715\pi$
$223$	8.71780	0.583787	0.291893	+	0.956451i	$0.405715\pi$
$224$	4.35890	0.291241
$225$	0	0
$226$	−18.0000	−1.19734
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	0	0
$229$	−22.0000	−1.45380	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000	−1.45380	−0.726900	+	0.686743i	$0.759040\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−14.0000	−0.917170	−0.458585	−	0.888650i	$-0.651644\pi$
$233$	−14.0000	−0.917170	−0.458585	+	0.888650i	$0.651644\pi$
$234$	0	0
$235$	0	0
$236$	−8.71780	−0.567480
$237$	0	0
$238$	−17.4356	−1.13018
$239$	8.71780	0.563907	0.281954	−	0.959428i	$-0.409018\pi$
$239$	8.71780	0.563907	0.281954	+	0.959428i	$0.409018\pi$
$240$	0	0
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	−8.00000	−0.514259
$243$	0	0
$244$	−4.00000	−0.256074
$245$	0	0
$246$	0	0
$247$	0	0
$248$	−7.00000	−0.444500
$249$	0	0
$250$	0	0
$251$	26.1534	1.65079	0.825394	−	0.564557i	$-0.190953\pi$
$251$	26.1534	1.65079	0.825394	+	0.564557i	$0.190953\pi$
$252$	0	0
$253$	−8.71780	−0.548083
$254$	4.35890	0.273502
$255$	0	0
$256$	1.00000	0.0625000
$257$	12.0000	0.748539	0.374270	−	0.927320i	$-0.377893\pi$
$257$	12.0000	0.748539	0.374270	+	0.927320i	$0.377893\pi$
$258$	0	0
$259$	−38.0000	−2.36121
$260$	0	0
$261$	0	0
$262$	13.0767	0.807881
$263$	2.00000	0.123325	0.0616626	−	0.998097i	$-0.480360\pi$
$263$	2.00000	0.123325	0.0616626	+	0.998097i	$0.480360\pi$
$264$	0	0
$265$	0	0
$266$	26.1534	1.60357
$267$	0	0
$268$	−8.71780	−0.532524
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	−5.00000	−0.303728	−0.151864	−	0.988401i	$-0.548528\pi$
$271$	−5.00000	−0.303728	−0.151864	+	0.988401i	$0.548528\pi$
$272$	−4.00000	−0.242536
$273$	0	0
$274$	−18.0000	−1.08742
$275$	0	0
$276$	0	0
$277$	−17.4356	−1.04760	−0.523802	−	0.851840i	$-0.675487\pi$
$277$	−17.4356	−1.04760	−0.523802	+	0.851840i	$0.675487\pi$
$278$	−4.00000	−0.239904
$279$	0	0
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	−8.71780	−0.518219	−0.259110	−	0.965848i	$-0.583429\pi$
$283$	−8.71780	−0.518219	−0.259110	+	0.965848i	$0.583429\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−38.0000	−2.24307
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	0	0
$292$	−4.35890	−0.255085
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	0	0
$296$	−8.71780	−0.506712
$297$	0	0
$298$	−4.35890	−0.252504
$299$	0	0
$300$	0	0
$301$	−38.0000	−2.19028
$302$	11.0000	0.632979
$303$	0	0
$304$	6.00000	0.344124
$305$	0	0
$306$	0	0
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	19.0000	1.08263
$309$	0	0
$310$	0	0
$311$	26.1534	1.48302	0.741511	−	0.670940i	$-0.234109\pi$
$311$	26.1534	1.48302	0.741511	+	0.670940i	$0.234109\pi$
$312$	0	0
$313$	21.7945	1.23190	0.615949	−	0.787786i	$-0.288773\pi$
$313$	21.7945	1.23190	0.615949	+	0.787786i	$0.288773\pi$
$314$	−17.4356	−0.983948
$315$	0	0
$316$	0	0
$317$	17.0000	0.954815	0.477408	−	0.878682i	$-0.341577\pi$
$317$	17.0000	0.954815	0.477408	+	0.878682i	$0.341577\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	8.71780	0.485824
$323$	−24.0000	−1.33540
$324$	0	0
$325$	0	0
$326$	−17.4356	−0.965668
$327$	0	0
$328$	−8.71780	−0.481360
$329$	8.71780	0.480628
$330$	0	0
$331$	10.0000	0.549650	0.274825	−	0.961494i	$-0.411380\pi$
$331$	10.0000	0.549650	0.274825	+	0.961494i	$0.411380\pi$
$332$	5.00000	0.274411
$333$	0	0
$334$	−18.0000	−0.984916
$335$	0	0
$336$	0	0
$337$	−17.4356	−0.949777	−0.474889	−	0.880046i	$-0.657512\pi$
$337$	−17.4356	−0.949777	−0.474889	+	0.880046i	$0.657512\pi$
$338$	13.0000	0.707107
$339$	0	0
$340$	0	0
$341$	−30.5123	−1.65233
$342$	0	0
$343$	−21.7945	−1.17679
$344$	−8.71780	−0.470032
$345$	0	0
$346$	−19.0000	−1.02145
$347$	−27.0000	−1.44944	−0.724718	−	0.689046i	$-0.758030\pi$
$347$	−27.0000	−1.44944	−0.724718	+	0.689046i	$0.758030\pi$
$348$	0	0
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	0	0
$351$	0	0
$352$	4.35890	0.232330
$353$	−26.0000	−1.38384	−0.691920	−	0.721974i	$-0.743235\pi$
$353$	−26.0000	−1.38384	−0.691920	+	0.721974i	$0.743235\pi$
$354$	0	0
$355$	0	0
$356$	8.71780	0.462042
$357$	0	0
$358$	21.7945	1.15187
$359$	26.1534	1.38032	0.690162	−	0.723655i	$-0.257539\pi$
$359$	26.1534	1.38032	0.690162	+	0.723655i	$0.257539\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	−8.00000	−0.420471
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−4.35890	−0.227533	−0.113766	−	0.993508i	$-0.536292\pi$
$367$	−4.35890	−0.227533	−0.113766	+	0.993508i	$0.536292\pi$
$368$	2.00000	0.104257
$369$	0	0
$370$	0	0
$371$	−13.0767	−0.678908
$372$	0	0
$373$	0	0		−	1.00000i	$-0.5\pi$
$373$	0	0			1.00000i	$0.5\pi$
$374$	−17.4356	−0.901573
$375$	0	0
$376$	2.00000	0.103142
$377$	0	0
$378$	0	0
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	4.00000	0.204390	0.102195	−	0.994764i	$-0.467413\pi$
$383$	4.00000	0.204390	0.102195	+	0.994764i	$0.467413\pi$
$384$	0	0
$385$	0	0
$386$	−21.7945	−1.10931
$387$	0	0
$388$	4.35890	0.221290
$389$	−4.35890	−0.221005	−0.110502	−	0.993876i	$-0.535246\pi$
$389$	−4.35890	−0.221005	−0.110502	+	0.993876i	$0.535246\pi$
$390$	0	0
$391$	−8.00000	−0.404577
$392$	−12.0000	−0.606092
$393$	0	0
$394$	5.00000	0.251896
$395$	0	0
$396$	0	0
$397$	34.8712	1.75013	0.875067	−	0.484001i	$-0.160817\pi$
$397$	34.8712	1.75013	0.875067	+	0.484001i	$0.160817\pi$
$398$	3.00000	0.150376
$399$	0	0
$400$	0	0
$401$	−34.8712	−1.74138	−0.870692	−	0.491828i	$-0.836329\pi$
$401$	−34.8712	−1.74138	−0.870692	+	0.491828i	$0.836329\pi$
$402$	0	0
$403$	0	0
$404$	4.35890	0.216863
$405$	0	0
$406$	0	0
$407$	−38.0000	−1.88359
$408$	0	0
$409$	−31.0000	−1.53285	−0.766426	−	0.642333i	$-0.777967\pi$
$409$	−31.0000	−1.53285	−0.766426	+	0.642333i	$0.777967\pi$
$410$	0	0
$411$	0	0
$412$	−8.71780	−0.429495
$413$	38.0000	1.86986
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	26.1534	1.27920
$419$	8.71780	0.425892	0.212946	−	0.977064i	$-0.431694\pi$
$419$	8.71780	0.425892	0.212946	+	0.977064i	$0.431694\pi$
$420$	0	0
$421$	16.0000	0.779792	0.389896	−	0.920859i	$-0.372511\pi$
$421$	16.0000	0.779792	0.389896	+	0.920859i	$0.372511\pi$
$422$	−6.00000	−0.292075
$423$	0	0
$424$	−3.00000	−0.145693
$425$	0	0
$426$	0	0
$427$	17.4356	0.843768
$428$	−15.0000	−0.725052
$429$	0	0
$430$	0	0
$431$	8.71780	0.419922	0.209961	−	0.977710i	$-0.432666\pi$
$431$	8.71780	0.419922	0.209961	+	0.977710i	$0.432666\pi$
$432$	0	0
$433$	21.7945	1.04738	0.523688	−	0.851910i	$-0.324556\pi$
$433$	21.7945	1.04738	0.523688	+	0.851910i	$0.324556\pi$
$434$	30.5123	1.46464
$435$	0	0
$436$	10.0000	0.478913
$437$	12.0000	0.574038
$438$	0	0
$439$	9.00000	0.429547	0.214773	−	0.976664i	$-0.431099\pi$
$439$	9.00000	0.429547	0.214773	+	0.976664i	$0.431099\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	0	0
$445$	0	0
$446$	−8.71780	−0.412800
$447$	0	0
$448$	−4.35890	−0.205939
$449$	26.1534	1.23425	0.617127	−	0.786863i	$-0.288296\pi$
$449$	26.1534	1.23425	0.617127	+	0.786863i	$0.288296\pi$
$450$	0	0
$451$	−38.0000	−1.78935
$452$	18.0000	0.846649
$453$	0	0
$454$	4.00000	0.187729
$455$	0	0
$456$	0	0
$457$	−4.35890	−0.203901	−0.101950	−	0.994789i	$-0.532508\pi$
$457$	−4.35890	−0.203901	−0.101950	+	0.994789i	$0.532508\pi$
$458$	22.0000	1.02799
$459$	0	0
$460$	0	0
$461$	39.2301	1.82713	0.913564	−	0.406696i	$-0.133319\pi$
$461$	39.2301	1.82713	0.913564	+	0.406696i	$0.133319\pi$
$462$	0	0
$463$	−13.0767	−0.607726	−0.303863	−	0.952716i	$-0.598276\pi$
$463$	−13.0767	−0.607726	−0.303863	+	0.952716i	$0.598276\pi$
$464$	0	0
$465$	0	0
$466$	14.0000	0.648537
$467$	21.0000	0.971764	0.485882	−	0.874024i	$-0.338498\pi$
$467$	21.0000	0.971764	0.485882	+	0.874024i	$0.338498\pi$
$468$	0	0
$469$	38.0000	1.75468
$470$	0	0
$471$	0	0
$472$	8.71780	0.401269
$473$	−38.0000	−1.74724
$474$	0	0
$475$	0	0
$476$	17.4356	0.799159
$477$	0	0
$478$	−8.71780	−0.398743
$479$	8.71780	0.398326	0.199163	−	0.979966i	$-0.436178\pi$
$479$	8.71780	0.398326	0.199163	+	0.979966i	$0.436178\pi$
$480$	0	0
$481$	0	0
$482$	14.0000	0.637683
$483$	0	0
$484$	8.00000	0.363636
$485$	0	0
$486$	0	0
$487$	26.1534	1.18512	0.592562	−	0.805525i	$-0.298117\pi$
$487$	26.1534	1.18512	0.592562	+	0.805525i	$0.298117\pi$
$488$	4.00000	0.181071
$489$	0	0
$490$	0	0
$491$	−13.0767	−0.590143	−0.295072	−	0.955475i	$-0.595343\pi$
$491$	−13.0767	−0.590143	−0.295072	+	0.955475i	$0.595343\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	7.00000	0.314309
$497$	0	0
$498$	0	0
$499$	30.0000	1.34298	0.671492	−	0.741012i	$-0.265654\pi$
$499$	30.0000	1.34298	0.671492	+	0.741012i	$0.265654\pi$
$500$	0	0
$501$	0	0
$502$	−26.1534	−1.16728
$503$	−30.0000	−1.33763	−0.668817	−	0.743427i	$-0.733199\pi$
$503$	−30.0000	−1.33763	−0.668817	+	0.743427i	$0.733199\pi$
$504$	0	0
$505$	0	0
$506$	8.71780	0.387553
$507$	0	0
$508$	−4.35890	−0.193395
$509$	−13.0767	−0.579614	−0.289807	−	0.957085i	$-0.593591\pi$
$509$	−13.0767	−0.579614	−0.289807	+	0.957085i	$0.593591\pi$
$510$	0	0
$511$	19.0000	0.840511
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−12.0000	−0.529297
$515$	0	0
$516$	0	0
$517$	8.71780	0.383408
$518$	38.0000	1.66962
$519$	0	0
$520$	0	0
$521$	17.4356	0.763867	0.381934	−	0.924190i	$-0.375258\pi$
$521$	17.4356	0.763867	0.381934	+	0.924190i	$0.375258\pi$
$522$	0	0
$523$	−34.8712	−1.52481	−0.762405	−	0.647100i	$-0.775982\pi$
$523$	−34.8712	−1.52481	−0.762405	+	0.647100i	$0.775982\pi$
$524$	−13.0767	−0.571258
$525$	0	0
$526$	−2.00000	−0.0872041
$527$	−28.0000	−1.21970
$528$	0	0
$529$	−19.0000	−0.826087
$530$	0	0
$531$	0	0
$532$	−26.1534	−1.13389
$533$	0	0
$534$	0	0
$535$	0	0
$536$	8.71780	0.376552
$537$	0	0
$538$	0	0
$539$	−52.3068	−2.25301
$540$	0	0
$541$	40.0000	1.71973	0.859867	−	0.510518i	$-0.170546\pi$
$541$	40.0000	1.71973	0.859867	+	0.510518i	$0.170546\pi$
$542$	5.00000	0.214768
$543$	0	0
$544$	4.00000	0.171499
$545$	0	0
$546$	0	0
$547$	−17.4356	−0.745492	−0.372746	−	0.927933i	$-0.621584\pi$
$547$	−17.4356	−0.745492	−0.372746	+	0.927933i	$0.621584\pi$
$548$	18.0000	0.768922
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	17.4356	0.740767
$555$	0	0
$556$	4.00000	0.169638
$557$	33.0000	1.39825	0.699127	−	0.714997i	$-0.253572\pi$
$557$	33.0000	1.39825	0.699127	+	0.714997i	$0.253572\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−11.0000	−0.463595	−0.231797	−	0.972764i	$-0.574461\pi$
$563$	−11.0000	−0.463595	−0.231797	+	0.972764i	$0.574461\pi$
$564$	0	0
$565$	0	0
$566$	8.71780	0.366436
$567$	0	0
$568$	0	0
$569$	−17.4356	−0.730938	−0.365469	−	0.930823i	$-0.619091\pi$
$569$	−17.4356	−0.730938	−0.365469	+	0.930823i	$0.619091\pi$
$570$	0	0
$571$	16.0000	0.669579	0.334790	−	0.942293i	$-0.391335\pi$
$571$	16.0000	0.669579	0.334790	+	0.942293i	$0.391335\pi$
$572$	0	0
$573$	0	0
$574$	38.0000	1.58609
$575$	0	0
$576$	0	0
$577$	−17.4356	−0.725853	−0.362927	−	0.931818i	$-0.618222\pi$
$577$	−17.4356	−0.725853	−0.362927	+	0.931818i	$0.618222\pi$
$578$	1.00000	0.0415945
$579$	0	0
$580$	0	0
$581$	−21.7945	−0.904188
$582$	0	0
$583$	−13.0767	−0.541581
$584$	4.35890	0.180373
$585$	0	0
$586$	−6.00000	−0.247858
$587$	−3.00000	−0.123823	−0.0619116	−	0.998082i	$-0.519720\pi$
$587$	−3.00000	−0.123823	−0.0619116	+	0.998082i	$0.519720\pi$
$588$	0	0
$589$	42.0000	1.73058
$590$	0	0
$591$	0	0
$592$	8.71780	0.358299
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	0	0
$595$	0	0
$596$	4.35890	0.178547
$597$	0	0
$598$	0	0
$599$	−26.1534	−1.06860	−0.534299	−	0.845295i	$-0.679424\pi$
$599$	−26.1534	−1.06860	−0.534299	+	0.845295i	$0.679424\pi$
$600$	0	0
$601$	−13.0000	−0.530281	−0.265141	−	0.964210i	$-0.585418\pi$
$601$	−13.0000	−0.530281	−0.265141	+	0.964210i	$0.585418\pi$
$602$	38.0000	1.54876
$603$	0	0
$604$	−11.0000	−0.447584
$605$	0	0
$606$	0	0
$607$	−8.71780	−0.353845	−0.176922	−	0.984225i	$-0.556614\pi$
$607$	−8.71780	−0.353845	−0.176922	+	0.984225i	$0.556614\pi$
$608$	−6.00000	−0.243332
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	8.71780	0.352109	0.176054	−	0.984380i	$-0.443667\pi$
$613$	8.71780	0.352109	0.176054	+	0.984380i	$0.443667\pi$
$614$	0	0
$615$	0	0
$616$	−19.0000	−0.765532
$617$	2.00000	0.0805170	0.0402585	−	0.999189i	$-0.487182\pi$
$617$	2.00000	0.0805170	0.0402585	+	0.999189i	$0.487182\pi$
$618$	0	0
$619$	32.0000	1.28619	0.643094	−	0.765787i	$-0.277650\pi$
$619$	32.0000	1.28619	0.643094	+	0.765787i	$0.277650\pi$
$620$	0	0
$621$	0	0
$622$	−26.1534	−1.04866
$623$	−38.0000	−1.52244
$624$	0	0
$625$	0	0
$626$	−21.7945	−0.871083
$627$	0	0
$628$	17.4356	0.695756
$629$	−34.8712	−1.39041
$630$	0	0
$631$	5.00000	0.199047	0.0995234	−	0.995035i	$-0.468268\pi$
$631$	5.00000	0.199047	0.0995234	+	0.995035i	$0.468268\pi$
$632$	0	0
$633$	0	0
$634$	−17.0000	−0.675156
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	8.71780	0.344332	0.172166	−	0.985068i	$-0.444923\pi$
$641$	8.71780	0.344332	0.172166	+	0.985068i	$0.444923\pi$
$642$	0	0
$643$	0	0		−	1.00000i	$-0.5\pi$
$643$	0	0			1.00000i	$0.5\pi$
$644$	−8.71780	−0.343529
$645$	0	0
$646$	24.0000	0.944267
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	0	0
$649$	38.0000	1.49163
$650$	0	0
$651$	0	0
$652$	17.4356	0.682831
$653$	35.0000	1.36966	0.684828	−	0.728705i	$-0.259877\pi$
$653$	35.0000	1.36966	0.684828	+	0.728705i	$0.259877\pi$
$654$	0	0
$655$	0	0
$656$	8.71780	0.340373
$657$	0	0
$658$	−8.71780	−0.339855
$659$	4.35890	0.169799	0.0848993	−	0.996390i	$-0.472943\pi$
$659$	4.35890	0.169799	0.0848993	+	0.996390i	$0.472943\pi$
$660$	0	0
$661$	−22.0000	−0.855701	−0.427850	−	0.903850i	$-0.640729\pi$
$661$	−22.0000	−0.855701	−0.427850	+	0.903850i	$0.640729\pi$
$662$	−10.0000	−0.388661
$663$	0	0
$664$	−5.00000	−0.194038
$665$	0	0
$666$	0	0
$667$	0	0
$668$	18.0000	0.696441
$669$	0	0
$670$	0	0
$671$	17.4356	0.673094
$672$	0	0
$673$	13.0767	0.504070	0.252035	−	0.967718i	$-0.418900\pi$
$673$	13.0767	0.504070	0.252035	+	0.967718i	$0.418900\pi$
$674$	17.4356	0.671594
$675$	0	0
$676$	−13.0000	−0.500000
$677$	26.0000	0.999261	0.499631	−	0.866239i	$-0.333469\pi$
$677$	26.0000	0.999261	0.499631	+	0.866239i	$0.333469\pi$
$678$	0	0
$679$	−19.0000	−0.729153
$680$	0	0
$681$	0	0
$682$	30.5123	1.16838
$683$	20.0000	0.765279	0.382639	−	0.923898i	$-0.375015\pi$
$683$	20.0000	0.765279	0.382639	+	0.923898i	$0.375015\pi$
$684$	0	0
$685$	0	0
$686$	21.7945	0.832118
$687$	0	0
$688$	8.71780	0.332363
$689$	0	0
$690$	0	0
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	19.0000	0.722272
$693$	0	0
$694$	27.0000	1.02491
$695$	0	0
$696$	0	0
$697$	−34.8712	−1.32084
$698$	14.0000	0.529908
$699$	0	0
$700$	0	0
$701$	21.7945	0.823167	0.411583	−	0.911372i	$-0.364976\pi$
$701$	21.7945	0.823167	0.411583	+	0.911372i	$0.364976\pi$
$702$	0	0
$703$	52.3068	1.97279
$704$	−4.35890	−0.164282
$705$	0	0
$706$	26.0000	0.978523
$707$	−19.0000	−0.714569
$708$	0	0
$709$	12.0000	0.450669	0.225335	−	0.974281i	$-0.427652\pi$
$709$	12.0000	0.450669	0.225335	+	0.974281i	$0.427652\pi$
$710$	0	0
$711$	0	0
$712$	−8.71780	−0.326713
$713$	14.0000	0.524304
$714$	0	0
$715$	0	0
$716$	−21.7945	−0.814499
$717$	0	0
$718$	−26.1534	−0.976036
$719$	34.8712	1.30048	0.650238	−	0.759731i	$-0.274669\pi$
$719$	34.8712	1.30048	0.650238	+	0.759731i	$0.274669\pi$
$720$	0	0
$721$	38.0000	1.41519
$722$	−17.0000	−0.632674
$723$	0	0
$724$	8.00000	0.297318
$725$	0	0
$726$	0	0
$727$	39.2301	1.45496	0.727482	−	0.686127i	$-0.240691\pi$
$727$	39.2301	1.45496	0.727482	+	0.686127i	$0.240691\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−34.8712	−1.28976
$732$	0	0
$733$	−8.71780	−0.321999	−0.161000	−	0.986954i	$-0.551472\pi$
$733$	−8.71780	−0.321999	−0.161000	+	0.986954i	$0.551472\pi$
$734$	4.35890	0.160890
$735$	0	0
$736$	−2.00000	−0.0737210
$737$	38.0000	1.39975
$738$	0	0
$739$	16.0000	0.588570	0.294285	−	0.955718i	$-0.404919\pi$
$739$	16.0000	0.588570	0.294285	+	0.955718i	$0.404919\pi$
$740$	0	0
$741$	0	0
$742$	13.0767	0.480061
$743$	−48.0000	−1.76095	−0.880475	−	0.474093i	$-0.842776\pi$
$743$	−48.0000	−1.76095	−0.880475	+	0.474093i	$0.842776\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	17.4356	0.637509
$749$	65.3835	2.38906
$750$	0	0
$751$	35.0000	1.27717	0.638584	−	0.769552i	$-0.279520\pi$
$751$	35.0000	1.27717	0.638584	+	0.769552i	$0.279520\pi$
$752$	−2.00000	−0.0729325
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	8.71780	0.316854	0.158427	−	0.987371i	$-0.449358\pi$
$757$	8.71780	0.316854	0.158427	+	0.987371i	$0.449358\pi$
$758$	4.00000	0.145287
$759$	0	0
$760$	0	0
$761$	52.3068	1.89612	0.948060	−	0.318092i	$-0.103042\pi$
$761$	52.3068	1.89612	0.948060	+	0.318092i	$0.103042\pi$
$762$	0	0
$763$	−43.5890	−1.57803
$764$	0	0
$765$	0	0
$766$	−4.00000	−0.144526
$767$	0	0
$768$	0	0
$769$	9.00000	0.324548	0.162274	−	0.986746i	$-0.448117\pi$
$769$	9.00000	0.324548	0.162274	+	0.986746i	$0.448117\pi$
$770$	0	0
$771$	0	0
$772$	21.7945	0.784401
$773$	−18.0000	−0.647415	−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000	−0.647415	−0.323708	+	0.946157i	$0.604929\pi$
$774$	0	0
$775$	0	0
$776$	−4.35890	−0.156475
$777$	0	0
$778$	4.35890	0.156274
$779$	52.3068	1.87409
$780$	0	0
$781$	0	0
$782$	8.00000	0.286079
$783$	0	0
$784$	12.0000	0.428571
$785$	0	0
$786$	0	0
$787$	−17.4356	−0.621512	−0.310756	−	0.950490i	$-0.600582\pi$
$787$	−17.4356	−0.621512	−0.310756	+	0.950490i	$0.600582\pi$
$788$	−5.00000	−0.178118
$789$	0	0
$790$	0	0
$791$	−78.4602	−2.78972
$792$	0	0
$793$	0	0
$794$	−34.8712	−1.23753
$795$	0	0
$796$	−3.00000	−0.106332
$797$	−37.0000	−1.31061	−0.655304	−	0.755366i	$-0.727459\pi$
$797$	−37.0000	−1.31061	−0.655304	+	0.755366i	$0.727459\pi$
$798$	0	0
$799$	8.00000	0.283020
$800$	0	0
$801$	0	0
$802$	34.8712	1.23134
$803$	19.0000	0.670495
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	−4.35890	−0.153346
$809$	26.1534	0.919504	0.459752	−	0.888047i	$-0.347938\pi$
$809$	26.1534	0.919504	0.459752	+	0.888047i	$0.347938\pi$
$810$	0	0
$811$	−14.0000	−0.491606	−0.245803	−	0.969320i	$-0.579052\pi$
$811$	−14.0000	−0.491606	−0.245803	+	0.969320i	$0.579052\pi$
$812$	0	0
$813$	0	0
$814$	38.0000	1.33190
$815$	0	0
$816$	0	0
$817$	52.3068	1.82998
$818$	31.0000	1.08389
$819$	0	0
$820$	0	0
$821$	−34.8712	−1.21701	−0.608506	−	0.793549i	$-0.708231\pi$
$821$	−34.8712	−1.21701	−0.608506	+	0.793549i	$0.708231\pi$
$822$	0	0
$823$	39.2301	1.36747	0.683737	−	0.729728i	$-0.260353\pi$
$823$	39.2301	1.36747	0.683737	+	0.729728i	$0.260353\pi$
$824$	8.71780	0.303699
$825$	0	0
$826$	−38.0000	−1.32219
$827$	−16.0000	−0.556375	−0.278187	−	0.960527i	$-0.589734\pi$
$827$	−16.0000	−0.556375	−0.278187	+	0.960527i	$0.589734\pi$
$828$	0	0
$829$	−42.0000	−1.45872	−0.729360	−	0.684130i	$-0.760182\pi$
$829$	−42.0000	−1.45872	−0.729360	+	0.684130i	$0.760182\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−48.0000	−1.66310
$834$	0	0
$835$	0	0
$836$	−26.1534	−0.904534
$837$	0	0
$838$	−8.71780	−0.301151
$839$	−34.8712	−1.20389	−0.601944	−	0.798539i	$-0.705607\pi$
$839$	−34.8712	−1.20389	−0.601944	+	0.798539i	$0.705607\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	−16.0000	−0.551396
$843$	0	0
$844$	6.00000	0.206529
$845$	0	0
$846$	0	0
$847$	−34.8712	−1.19819
$848$	3.00000	0.103020
$849$	0	0
$850$	0	0
$851$	17.4356	0.597685
$852$	0	0
$853$	−8.71780	−0.298492	−0.149246	−	0.988800i	$-0.547685\pi$
$853$	−8.71780	−0.298492	−0.149246	+	0.988800i	$0.547685\pi$
$854$	−17.4356	−0.596634
$855$	0	0
$856$	15.0000	0.512689
$857$	−48.0000	−1.63965	−0.819824	−	0.572615i	$-0.805929\pi$
$857$	−48.0000	−1.63965	−0.819824	+	0.572615i	$0.805929\pi$
$858$	0	0
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	0	0
$861$	0	0
$862$	−8.71780	−0.296929
$863$	48.0000	1.63394	0.816970	−	0.576681i	$-0.195652\pi$
$863$	48.0000	1.63394	0.816970	+	0.576681i	$0.195652\pi$
$864$	0	0
$865$	0	0
$866$	−21.7945	−0.740607
$867$	0	0
$868$	−30.5123	−1.03565
$869$	0	0
$870$	0	0
$871$	0	0
$872$	−10.0000	−0.338643
$873$	0	0
$874$	−12.0000	−0.405906
$875$	0	0
$876$	0	0
$877$	−8.71780	−0.294379	−0.147190	−	0.989108i	$-0.547023\pi$
$877$	−8.71780	−0.294379	−0.147190	+	0.989108i	$0.547023\pi$
$878$	−9.00000	−0.303735
$879$	0	0
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	26.1534	0.880132	0.440066	−	0.897965i	$-0.354955\pi$
$883$	26.1534	0.880132	0.440066	+	0.897965i	$0.354955\pi$
$884$	0	0
$885$	0	0
$886$	12.0000	0.403148
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	0	0
$889$	19.0000	0.637240
$890$	0	0
$891$	0	0
$892$	8.71780	0.291893
$893$	−12.0000	−0.401565
$894$	0	0
$895$	0	0
$896$	4.35890	0.145621
$897$	0	0
$898$	−26.1534	−0.872750
$899$	0	0
$900$	0	0
$901$	−12.0000	−0.399778
$902$	38.0000	1.26526
$903$	0	0
$904$	−18.0000	−0.598671
$905$	0	0
$906$	0	0
$907$	−26.1534	−0.868409	−0.434205	−	0.900814i	$-0.642971\pi$
$907$	−26.1534	−0.868409	−0.434205	+	0.900814i	$0.642971\pi$
$908$	−4.00000	−0.132745
$909$	0	0
$910$	0	0
$911$	−26.1534	−0.866501	−0.433250	−	0.901274i	$-0.642633\pi$
$911$	−26.1534	−0.866501	−0.433250	+	0.901274i	$0.642633\pi$
$912$	0	0
$913$	−21.7945	−0.721292
$914$	4.35890	0.144180
$915$	0	0
$916$	−22.0000	−0.726900
$917$	57.0000	1.88231
$918$	0	0
$919$	39.0000	1.28649	0.643246	−	0.765660i	$-0.277587\pi$
$919$	39.0000	1.28649	0.643246	+	0.765660i	$0.277587\pi$
$920$	0	0
$921$	0	0
$922$	−39.2301	−1.29197
$923$	0	0
$924$	0	0
$925$	0	0
$926$	13.0767	0.429727
$927$	0	0
$928$	0	0
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	0	0
$931$	72.0000	2.35970
$932$	−14.0000	−0.458585
$933$	0	0
$934$	−21.0000	−0.687141
$935$	0	0
$936$	0	0
$937$	−47.9479	−1.56639	−0.783195	−	0.621777i	$-0.786411\pi$
$937$	−47.9479	−1.56639	−0.783195	+	0.621777i	$0.786411\pi$
$938$	−38.0000	−1.24074
$939$	0	0
$940$	0	0
$941$	47.9479	1.56306	0.781528	−	0.623870i	$-0.214440\pi$
$941$	47.9479	1.56306	0.781528	+	0.623870i	$0.214440\pi$
$942$	0	0
$943$	17.4356	0.567781
$944$	−8.71780	−0.283740
$945$	0	0
$946$	38.0000	1.23549
$947$	−27.0000	−0.877382	−0.438691	−	0.898638i	$-0.644558\pi$
$947$	−27.0000	−0.877382	−0.438691	+	0.898638i	$0.644558\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	−17.4356	−0.565091
$953$	−32.0000	−1.03658	−0.518291	−	0.855204i	$-0.673432\pi$
$953$	−32.0000	−1.03658	−0.518291	+	0.855204i	$0.673432\pi$
$954$	0	0
$955$	0	0
$956$	8.71780	0.281954
$957$	0	0
$958$	−8.71780	−0.281659
$959$	−78.4602	−2.53361
$960$	0	0
$961$	18.0000	0.580645
$962$	0	0
$963$	0	0
$964$	−14.0000	−0.450910
$965$	0	0
$966$	0	0
$967$	30.5123	0.981209	0.490605	−	0.871382i	$-0.336776\pi$
$967$	30.5123	0.981209	0.490605	+	0.871382i	$0.336776\pi$
$968$	−8.00000	−0.257130
$969$	0	0
$970$	0	0
$971$	39.2301	1.25895	0.629477	−	0.777019i	$-0.283269\pi$
$971$	39.2301	1.25895	0.629477	+	0.777019i	$0.283269\pi$
$972$	0	0
$973$	−17.4356	−0.558960
$974$	−26.1534	−0.838009
$975$	0	0
$976$	−4.00000	−0.128037
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	0	0
$979$	−38.0000	−1.21449
$980$	0	0
$981$	0	0
$982$	13.0767	0.417294
$983$	34.0000	1.08443	0.542216	−	0.840239i	$-0.317586\pi$
$983$	34.0000	1.08443	0.542216	+	0.840239i	$0.317586\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	17.4356	0.554420
$990$	0	0
$991$	−53.0000	−1.68360	−0.841800	−	0.539789i	$-0.818504\pi$
$991$	−53.0000	−1.68360	−0.841800	+	0.539789i	$0.818504\pi$
$992$	−7.00000	−0.222250
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−52.3068	−1.65657	−0.828286	−	0.560305i	$-0.810684\pi$
$997$	−52.3068	−1.65657	−0.828286	+	0.560305i	$0.810684\pi$
$998$	−30.0000	−0.949633
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1350.2.a.w.1.1		2
3.2	odd	2		1350.2.a.x.1.1		2
5.2	odd	4		270.2.c.c.109.1	✓	4
5.3	odd	4		270.2.c.c.109.3	yes	4
5.4	even	2		1350.2.a.x.1.2		2
15.2	even	4		270.2.c.c.109.4	yes	4
15.8	even	4		270.2.c.c.109.2	yes	4
15.14	odd	2	inner	1350.2.a.w.1.2		2
20.3	even	4		2160.2.f.m.1729.1		4
20.7	even	4		2160.2.f.m.1729.2		4
45.2	even	12		810.2.i.h.109.3		8
45.7	odd	12		810.2.i.h.109.2		8
45.13	odd	12		810.2.i.h.379.2		8
45.22	odd	12		810.2.i.h.379.4		8
45.23	even	12		810.2.i.h.379.3		8
45.32	even	12		810.2.i.h.379.1		8
45.38	even	12		810.2.i.h.109.1		8
45.43	odd	12		810.2.i.h.109.4		8
60.23	odd	4		2160.2.f.m.1729.4		4
60.47	odd	4		2160.2.f.m.1729.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.2.c.c.109.1	✓	4	5.2	odd	4
270.2.c.c.109.2	yes	4	15.8	even	4
270.2.c.c.109.3	yes	4	5.3	odd	4
270.2.c.c.109.4	yes	4	15.2	even	4
810.2.i.h.109.1		8	45.38	even	12
810.2.i.h.109.2		8	45.7	odd	12
810.2.i.h.109.3		8	45.2	even	12
810.2.i.h.109.4		8	45.43	odd	12
810.2.i.h.379.1		8	45.32	even	12
810.2.i.h.379.2		8	45.13	odd	12
810.2.i.h.379.3		8	45.23	even	12
810.2.i.h.379.4		8	45.22	odd	12
1350.2.a.w.1.1		2	1.1	even	1	trivial
1350.2.a.w.1.2		2	15.14	odd	2	inner
1350.2.a.x.1.1		2	3.2	odd	2
1350.2.a.x.1.2		2	5.4	even	2
2160.2.f.m.1729.1		4	20.3	even	4
2160.2.f.m.1729.2		4	20.7	even	4
2160.2.f.m.1729.3		4	60.47	odd	4
2160.2.f.m.1729.4		4	60.23	odd	4

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:	\( N \)	\(=\)	\( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1350.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -4.35890 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -4.35890 q^{7} -1.00000 q^{8} -4.35890 q^{11} +4.35890 q^{14} +1.00000 q^{16} -4.00000 q^{17} +6.00000 q^{19} +4.35890 q^{22} +2.00000 q^{23} -4.35890 q^{28} +7.00000 q^{31} -1.00000 q^{32} +4.00000 q^{34} +8.71780 q^{37} -6.00000 q^{38} +8.71780 q^{41} +8.71780 q^{43} -4.35890 q^{44} -2.00000 q^{46} -2.00000 q^{47} +12.0000 q^{49} +3.00000 q^{53} +4.35890 q^{56} -8.71780 q^{59} -4.00000 q^{61} -7.00000 q^{62} +1.00000 q^{64} -8.71780 q^{67} -4.00000 q^{68} -4.35890 q^{73} -8.71780 q^{74} +6.00000 q^{76} +19.0000 q^{77} -8.71780 q^{82} +5.00000 q^{83} -8.71780 q^{86} +4.35890 q^{88} +8.71780 q^{89} +2.00000 q^{92} +2.00000 q^{94} +4.35890 q^{97} -12.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 2 q^{16} - 8 q^{17} + 12 q^{19} + 4 q^{23} + 14 q^{31} - 2 q^{32} + 8 q^{34} - 12 q^{38} - 4 q^{46} - 4 q^{47} + 24 q^{49} + 6 q^{53} - 8 q^{61} - 14 q^{62} + 2 q^{64} - 8 q^{68} + 12 q^{76} + 38 q^{77} + 10 q^{83} + 4 q^{92} + 4 q^{94} - 24 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 + 2 * q^16 - 8 * q^17 + 12 * q^19 + 4 * q^23 + 14 * q^31 - 2 * q^32 + 8 * q^34 - 12 * q^38 - 4 * q^46 - 4 * q^47 + 24 * q^49 + 6 * q^53 - 8 * q^61 - 14 * q^62 + 2 * q^64 - 8 * q^68 + 12 * q^76 + 38 * q^77 + 10 * q^83 + 4 * q^92 + 4 * q^94 - 24 * q^98

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