LMFDB - Embedded newform 1638.2.a.t.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1638,2,Mod(1,1638)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1638, 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("1638.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1638.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:	$13.0794958511$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1638.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{4} +3.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$

$q+1.00000 q^{2} +1.00000 q^{4} +3.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} +3.00000 q^{10} +3.00000 q^{11} +1.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -3.00000 q^{17} +5.00000 q^{19} +3.00000 q^{20} +3.00000 q^{22} -3.00000 q^{23} +4.00000 q^{25} +1.00000 q^{26} +1.00000 q^{28} -3.00000 q^{29} -10.0000 q^{31} +1.00000 q^{32} -3.00000 q^{34} +3.00000 q^{35} -7.00000 q^{37} +5.00000 q^{38} +3.00000 q^{40} +6.00000 q^{41} +5.00000 q^{43} +3.00000 q^{44} -3.00000 q^{46} +1.00000 q^{49} +4.00000 q^{50} +1.00000 q^{52} +6.00000 q^{53} +9.00000 q^{55} +1.00000 q^{56} -3.00000 q^{58} +6.00000 q^{59} -7.00000 q^{61} -10.0000 q^{62} +1.00000 q^{64} +3.00000 q^{65} -4.00000 q^{67} -3.00000 q^{68} +3.00000 q^{70} -6.00000 q^{71} -1.00000 q^{73} -7.00000 q^{74} +5.00000 q^{76} +3.00000 q^{77} -10.0000 q^{79} +3.00000 q^{80} +6.00000 q^{82} +6.00000 q^{83} -9.00000 q^{85} +5.00000 q^{86} +3.00000 q^{88} -6.00000 q^{89} +1.00000 q^{91} -3.00000 q^{92} +15.0000 q^{95} +2.00000 q^{97} +1.00000 q^{98} +O(q^{100})$

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
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	0	0
$10$	3.00000	0.948683
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	5.00000	1.14708	0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000	1.14708	0.573539	+	0.819178i	$0.305570\pi$
$20$	3.00000	0.670820
$21$	0	0
$22$	3.00000	0.639602
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	1.00000	0.196116
$27$	0	0
$28$	1.00000	0.188982
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	0	0
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−3.00000	−0.514496
$35$	3.00000	0.507093
$36$	0	0
$37$	−7.00000	−1.15079	−0.575396	−	0.817875i	$-0.695152\pi$
$37$	−7.00000	−1.15079	−0.575396	+	0.817875i	$0.695152\pi$
$38$	5.00000	0.811107
$39$	0	0
$40$	3.00000	0.474342
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	5.00000	0.762493	0.381246	−	0.924473i	$-0.375495\pi$
$43$	5.00000	0.762493	0.381246	+	0.924473i	$0.375495\pi$
$44$	3.00000	0.452267
$45$	0	0
$46$	−3.00000	−0.442326
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	4.00000	0.565685
$51$	0	0
$52$	1.00000	0.138675
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	9.00000	1.21356
$56$	1.00000	0.133631
$57$	0	0
$58$	−3.00000	−0.393919
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	−7.00000	−0.896258	−0.448129	−	0.893969i	$-0.647910\pi$
$61$	−7.00000	−0.896258	−0.448129	+	0.893969i	$0.647910\pi$
$62$	−10.0000	−1.27000
$63$	0	0
$64$	1.00000	0.125000
$65$	3.00000	0.372104
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	−3.00000	−0.363803
$69$	0	0
$70$	3.00000	0.358569
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	0	0
$73$	−1.00000	−0.117041	−0.0585206	−	0.998286i	$-0.518638\pi$
$73$	−1.00000	−0.117041	−0.0585206	+	0.998286i	$0.518638\pi$
$74$	−7.00000	−0.813733
$75$	0	0
$76$	5.00000	0.573539
$77$	3.00000	0.341882
$78$	0	0
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	3.00000	0.335410
$81$	0	0
$82$	6.00000	0.662589
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	−9.00000	−0.976187
$86$	5.00000	0.539164
$87$	0	0
$88$	3.00000	0.319801
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	−3.00000	−0.312772
$93$	0	0
$94$	0	0
$95$	15.0000	1.53897
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	4.00000	0.400000
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	−13.0000	−1.28093	−0.640464	−	0.767988i	$-0.721258\pi$
$103$	−13.0000	−1.28093	−0.640464	+	0.767988i	$0.721258\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	6.00000	0.582772
$107$	18.0000	1.74013	0.870063	−	0.492941i	$-0.164078\pi$
$107$	18.0000	1.74013	0.870063	+	0.492941i	$0.164078\pi$
$108$	0	0
$109$	−1.00000	−0.0957826	−0.0478913	−	0.998853i	$-0.515250\pi$
$109$	−1.00000	−0.0957826	−0.0478913	+	0.998853i	$0.515250\pi$
$110$	9.00000	0.858116
$111$	0	0
$112$	1.00000	0.0944911
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	−9.00000	−0.839254
$116$	−3.00000	−0.278543
$117$	0	0
$118$	6.00000	0.552345
$119$	−3.00000	−0.275010
$120$	0	0
$121$	−2.00000	−0.181818
$122$	−7.00000	−0.633750
$123$	0	0
$124$	−10.0000	−0.898027
$125$	−3.00000	−0.268328
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	3.00000	0.263117
$131$	15.0000	1.31056	0.655278	−	0.755388i	$-0.272551\pi$
$131$	15.0000	1.31056	0.655278	+	0.755388i	$0.272551\pi$
$132$	0	0
$133$	5.00000	0.433555
$134$	−4.00000	−0.345547
$135$	0	0
$136$	−3.00000	−0.257248
$137$	−9.00000	−0.768922	−0.384461	−	0.923141i	$-0.625613\pi$
$137$	−9.00000	−0.768922	−0.384461	+	0.923141i	$0.625613\pi$
$138$	0	0
$139$	14.0000	1.18746	0.593732	−	0.804663i	$-0.297654\pi$
$139$	14.0000	1.18746	0.593732	+	0.804663i	$0.297654\pi$
$140$	3.00000	0.253546
$141$	0	0
$142$	−6.00000	−0.503509
$143$	3.00000	0.250873
$144$	0	0
$145$	−9.00000	−0.747409
$146$	−1.00000	−0.0827606
$147$	0	0
$148$	−7.00000	−0.575396
$149$	−12.0000	−0.983078	−0.491539	−	0.870855i	$-0.663566\pi$
$149$	−12.0000	−0.983078	−0.491539	+	0.870855i	$0.663566\pi$
$150$	0	0
$151$	−13.0000	−1.05792	−0.528962	−	0.848645i	$-0.677419\pi$
$151$	−13.0000	−1.05792	−0.528962	+	0.848645i	$0.677419\pi$
$152$	5.00000	0.405554
$153$	0	0
$154$	3.00000	0.241747
$155$	−30.0000	−2.40966
$156$	0	0
$157$	5.00000	0.399043	0.199522	−	0.979893i	$-0.436061\pi$
$157$	5.00000	0.399043	0.199522	+	0.979893i	$0.436061\pi$
$158$	−10.0000	−0.795557
$159$	0	0
$160$	3.00000	0.237171
$161$	−3.00000	−0.236433
$162$	0	0
$163$	2.00000	0.156652	0.0783260	−	0.996928i	$-0.475042\pi$
$163$	2.00000	0.156652	0.0783260	+	0.996928i	$0.475042\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	6.00000	0.465690
$167$	21.0000	1.62503	0.812514	−	0.582941i	$-0.198098\pi$
$167$	21.0000	1.62503	0.812514	+	0.582941i	$0.198098\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−9.00000	−0.690268
$171$	0	0
$172$	5.00000	0.381246
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	3.00000	0.226134
$177$	0	0
$178$	−6.00000	−0.449719
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	1.00000	0.0741249
$183$	0	0
$184$	−3.00000	−0.221163
$185$	−21.0000	−1.54395
$186$	0	0
$187$	−9.00000	−0.658145
$188$	0	0
$189$	0	0
$190$	15.0000	1.08821
$191$	−15.0000	−1.08536	−0.542681	−	0.839939i	$-0.682591\pi$
$191$	−15.0000	−1.08536	−0.542681	+	0.839939i	$0.682591\pi$
$192$	0	0
$193$	−22.0000	−1.58359	−0.791797	−	0.610784i	$-0.790854\pi$
$193$	−22.0000	−1.58359	−0.791797	+	0.610784i	$0.790854\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	1.00000	0.0714286
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	0	0
$199$	5.00000	0.354441	0.177220	−	0.984171i	$-0.443289\pi$
$199$	5.00000	0.354441	0.177220	+	0.984171i	$0.443289\pi$
$200$	4.00000	0.282843
$201$	0	0
$202$	6.00000	0.422159
$203$	−3.00000	−0.210559
$204$	0	0
$205$	18.0000	1.25717
$206$	−13.0000	−0.905753
$207$	0	0
$208$	1.00000	0.0693375
$209$	15.0000	1.03757
$210$	0	0
$211$	23.0000	1.58339	0.791693	−	0.610920i	$-0.209200\pi$
$211$	23.0000	1.58339	0.791693	+	0.610920i	$0.209200\pi$
$212$	6.00000	0.412082
$213$	0	0
$214$	18.0000	1.23045
$215$	15.0000	1.02299
$216$	0	0
$217$	−10.0000	−0.678844
$218$	−1.00000	−0.0677285
$219$	0	0
$220$	9.00000	0.606780
$221$	−3.00000	−0.201802
$222$	0	0
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	0	0
$227$	18.0000	1.19470	0.597351	−	0.801980i	$-0.296220\pi$
$227$	18.0000	1.19470	0.597351	+	0.801980i	$0.296220\pi$
$228$	0	0
$229$	−22.0000	−1.45380	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000	−1.45380	−0.726900	+	0.686743i	$0.759040\pi$
$230$	−9.00000	−0.593442
$231$	0	0
$232$	−3.00000	−0.196960
$233$	−24.0000	−1.57229	−0.786146	−	0.618041i	$-0.787927\pi$
$233$	−24.0000	−1.57229	−0.786146	+	0.618041i	$0.787927\pi$
$234$	0	0
$235$	0	0
$236$	6.00000	0.390567
$237$	0	0
$238$	−3.00000	−0.194461
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	−2.00000	−0.128565
$243$	0	0
$244$	−7.00000	−0.448129
$245$	3.00000	0.191663
$246$	0	0
$247$	5.00000	0.318142
$248$	−10.0000	−0.635001
$249$	0	0
$250$	−3.00000	−0.189737
$251$	−9.00000	−0.568075	−0.284037	−	0.958813i	$-0.591674\pi$
$251$	−9.00000	−0.568075	−0.284037	+	0.958813i	$0.591674\pi$
$252$	0	0
$253$	−9.00000	−0.565825
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	−7.00000	−0.434959
$260$	3.00000	0.186052
$261$	0	0
$262$	15.0000	0.926703
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	0	0
$265$	18.0000	1.10573
$266$	5.00000	0.306570
$267$	0	0
$268$	−4.00000	−0.244339
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	0	0
$271$	2.00000	0.121491	0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000	0.121491	0.0607457	+	0.998153i	$0.480652\pi$
$272$	−3.00000	−0.181902
$273$	0	0
$274$	−9.00000	−0.543710
$275$	12.0000	0.723627
$276$	0	0
$277$	−28.0000	−1.68236	−0.841178	−	0.540758i	$-0.818138\pi$
$277$	−28.0000	−1.68236	−0.841178	+	0.540758i	$0.818138\pi$
$278$	14.0000	0.839664
$279$	0	0
$280$	3.00000	0.179284
$281$	−30.0000	−1.78965	−0.894825	−	0.446417i	$-0.852700\pi$
$281$	−30.0000	−1.78965	−0.894825	+	0.446417i	$0.852700\pi$
$282$	0	0
$283$	−16.0000	−0.951101	−0.475551	−	0.879688i	$-0.657751\pi$
$283$	−16.0000	−0.951101	−0.475551	+	0.879688i	$0.657751\pi$
$284$	−6.00000	−0.356034
$285$	0	0
$286$	3.00000	0.177394
$287$	6.00000	0.354169
$288$	0	0
$289$	−8.00000	−0.470588
$290$	−9.00000	−0.528498
$291$	0	0
$292$	−1.00000	−0.0585206
$293$	30.0000	1.75262	0.876309	−	0.481749i	$-0.159998\pi$
$293$	30.0000	1.75262	0.876309	+	0.481749i	$0.159998\pi$
$294$	0	0
$295$	18.0000	1.04800
$296$	−7.00000	−0.406867
$297$	0	0
$298$	−12.0000	−0.695141
$299$	−3.00000	−0.173494
$300$	0	0
$301$	5.00000	0.288195
$302$	−13.0000	−0.748066
$303$	0	0
$304$	5.00000	0.286770
$305$	−21.0000	−1.20246
$306$	0	0
$307$	−4.00000	−0.228292	−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000	−0.228292	−0.114146	+	0.993464i	$0.536413\pi$
$308$	3.00000	0.170941
$309$	0	0
$310$	−30.0000	−1.70389
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	5.00000	0.282166
$315$	0	0
$316$	−10.0000	−0.562544
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	0	0
$319$	−9.00000	−0.503903
$320$	3.00000	0.167705
$321$	0	0
$322$	−3.00000	−0.167183
$323$	−15.0000	−0.834622
$324$	0	0
$325$	4.00000	0.221880
$326$	2.00000	0.110770
$327$	0	0
$328$	6.00000	0.331295
$329$	0	0
$330$	0	0
$331$	14.0000	0.769510	0.384755	−	0.923019i	$-0.374286\pi$
$331$	14.0000	0.769510	0.384755	+	0.923019i	$0.374286\pi$
$332$	6.00000	0.329293
$333$	0	0
$334$	21.0000	1.14907
$335$	−12.0000	−0.655630
$336$	0	0
$337$	−25.0000	−1.36184	−0.680918	−	0.732359i	$-0.738419\pi$
$337$	−25.0000	−1.36184	−0.680918	+	0.732359i	$0.738419\pi$
$338$	1.00000	0.0543928
$339$	0	0
$340$	−9.00000	−0.488094
$341$	−30.0000	−1.62459
$342$	0	0
$343$	1.00000	0.0539949
$344$	5.00000	0.269582
$345$	0	0
$346$	18.0000	0.967686
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	0	0
$349$	−16.0000	−0.856460	−0.428230	−	0.903670i	$-0.640863\pi$
$349$	−16.0000	−0.856460	−0.428230	+	0.903670i	$0.640863\pi$
$350$	4.00000	0.213809
$351$	0	0
$352$	3.00000	0.159901
$353$	−30.0000	−1.59674	−0.798369	−	0.602168i	$-0.794304\pi$
$353$	−30.0000	−1.59674	−0.798369	+	0.602168i	$0.794304\pi$
$354$	0	0
$355$	−18.0000	−0.955341
$356$	−6.00000	−0.317999
$357$	0	0
$358$	12.0000	0.634220
$359$	−30.0000	−1.58334	−0.791670	−	0.610949i	$-0.790788\pi$
$359$	−30.0000	−1.58334	−0.791670	+	0.610949i	$0.790788\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	−22.0000	−1.15629
$363$	0	0
$364$	1.00000	0.0524142
$365$	−3.00000	−0.157027
$366$	0	0
$367$	32.0000	1.67039	0.835193	−	0.549957i	$-0.185356\pi$
$367$	32.0000	1.67039	0.835193	+	0.549957i	$0.185356\pi$
$368$	−3.00000	−0.156386
$369$	0	0
$370$	−21.0000	−1.09174
$371$	6.00000	0.311504
$372$	0	0
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	−9.00000	−0.465379
$375$	0	0
$376$	0	0
$377$	−3.00000	−0.154508
$378$	0	0
$379$	2.00000	0.102733	0.0513665	−	0.998680i	$-0.483642\pi$
$379$	2.00000	0.102733	0.0513665	+	0.998680i	$0.483642\pi$
$380$	15.0000	0.769484
$381$	0	0
$382$	−15.0000	−0.767467
$383$	9.00000	0.459879	0.229939	−	0.973205i	$-0.426147\pi$
$383$	9.00000	0.459879	0.229939	+	0.973205i	$0.426147\pi$
$384$	0	0
$385$	9.00000	0.458682
$386$	−22.0000	−1.11977
$387$	0	0
$388$	2.00000	0.101535
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	9.00000	0.455150
$392$	1.00000	0.0505076
$393$	0	0
$394$	6.00000	0.302276
$395$	−30.0000	−1.50946
$396$	0	0
$397$	8.00000	0.401508	0.200754	−	0.979642i	$-0.435661\pi$
$397$	8.00000	0.401508	0.200754	+	0.979642i	$0.435661\pi$
$398$	5.00000	0.250627
$399$	0	0
$400$	4.00000	0.200000
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	0	0
$403$	−10.0000	−0.498135
$404$	6.00000	0.298511
$405$	0	0
$406$	−3.00000	−0.148888
$407$	−21.0000	−1.04093
$408$	0	0
$409$	5.00000	0.247234	0.123617	−	0.992330i	$-0.460551\pi$
$409$	5.00000	0.247234	0.123617	+	0.992330i	$0.460551\pi$
$410$	18.0000	0.888957
$411$	0	0
$412$	−13.0000	−0.640464
$413$	6.00000	0.295241
$414$	0	0
$415$	18.0000	0.883585
$416$	1.00000	0.0490290
$417$	0	0
$418$	15.0000	0.733674
$419$	−3.00000	−0.146560	−0.0732798	−	0.997311i	$-0.523347\pi$
$419$	−3.00000	−0.146560	−0.0732798	+	0.997311i	$0.523347\pi$
$420$	0	0
$421$	26.0000	1.26716	0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000	1.26716	0.633581	+	0.773676i	$0.281584\pi$
$422$	23.0000	1.11962
$423$	0	0
$424$	6.00000	0.291386
$425$	−12.0000	−0.582086
$426$	0	0
$427$	−7.00000	−0.338754
$428$	18.0000	0.870063
$429$	0	0
$430$	15.0000	0.723364
$431$	−18.0000	−0.867029	−0.433515	−	0.901146i	$-0.642727\pi$
$431$	−18.0000	−0.867029	−0.433515	+	0.901146i	$0.642727\pi$
$432$	0	0
$433$	14.0000	0.672797	0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000	0.672797	0.336399	+	0.941720i	$0.390791\pi$
$434$	−10.0000	−0.480015
$435$	0	0
$436$	−1.00000	−0.0478913
$437$	−15.0000	−0.717547
$438$	0	0
$439$	35.0000	1.67046	0.835229	−	0.549902i	$-0.185335\pi$
$439$	35.0000	1.67046	0.835229	+	0.549902i	$0.185335\pi$
$440$	9.00000	0.429058
$441$	0	0
$442$	−3.00000	−0.142695
$443$	−6.00000	−0.285069	−0.142534	−	0.989790i	$-0.545525\pi$
$443$	−6.00000	−0.285069	−0.142534	+	0.989790i	$0.545525\pi$
$444$	0	0
$445$	−18.0000	−0.853282
$446$	8.00000	0.378811
$447$	0	0
$448$	1.00000	0.0472456
$449$	−21.0000	−0.991051	−0.495526	−	0.868593i	$-0.665025\pi$
$449$	−21.0000	−0.991051	−0.495526	+	0.868593i	$0.665025\pi$
$450$	0	0
$451$	18.0000	0.847587
$452$	0	0
$453$	0	0
$454$	18.0000	0.844782
$455$	3.00000	0.140642
$456$	0	0
$457$	20.0000	0.935561	0.467780	−	0.883845i	$-0.345054\pi$
$457$	20.0000	0.935561	0.467780	+	0.883845i	$0.345054\pi$
$458$	−22.0000	−1.02799
$459$	0	0
$460$	−9.00000	−0.419627
$461$	33.0000	1.53696	0.768482	−	0.639872i	$-0.221013\pi$
$461$	33.0000	1.53696	0.768482	+	0.639872i	$0.221013\pi$
$462$	0	0
$463$	−25.0000	−1.16185	−0.580924	−	0.813958i	$-0.697309\pi$
$463$	−25.0000	−1.16185	−0.580924	+	0.813958i	$0.697309\pi$
$464$	−3.00000	−0.139272
$465$	0	0
$466$	−24.0000	−1.11178
$467$	3.00000	0.138823	0.0694117	−	0.997588i	$-0.477888\pi$
$467$	3.00000	0.138823	0.0694117	+	0.997588i	$0.477888\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	0	0
$471$	0	0
$472$	6.00000	0.276172
$473$	15.0000	0.689701
$474$	0	0
$475$	20.0000	0.917663
$476$	−3.00000	−0.137505
$477$	0	0
$478$	−12.0000	−0.548867
$479$	15.0000	0.685367	0.342684	−	0.939451i	$-0.388664\pi$
$479$	15.0000	0.685367	0.342684	+	0.939451i	$0.388664\pi$
$480$	0	0
$481$	−7.00000	−0.319173
$482$	−22.0000	−1.00207
$483$	0	0
$484$	−2.00000	−0.0909091
$485$	6.00000	0.272446
$486$	0	0
$487$	32.0000	1.45006	0.725029	−	0.688718i	$-0.241826\pi$
$487$	32.0000	1.45006	0.725029	+	0.688718i	$0.241826\pi$
$488$	−7.00000	−0.316875
$489$	0	0
$490$	3.00000	0.135526
$491$	−36.0000	−1.62466	−0.812329	−	0.583200i	$-0.801800\pi$
$491$	−36.0000	−1.62466	−0.812329	+	0.583200i	$0.801800\pi$
$492$	0	0
$493$	9.00000	0.405340
$494$	5.00000	0.224961
$495$	0	0
$496$	−10.0000	−0.449013
$497$	−6.00000	−0.269137
$498$	0	0
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	−3.00000	−0.134164
$501$	0	0
$502$	−9.00000	−0.401690
$503$	12.0000	0.535054	0.267527	−	0.963550i	$-0.413794\pi$
$503$	12.0000	0.535054	0.267527	+	0.963550i	$0.413794\pi$
$504$	0	0
$505$	18.0000	0.800989
$506$	−9.00000	−0.400099
$507$	0	0
$508$	8.00000	0.354943
$509$	33.0000	1.46270	0.731350	−	0.682003i	$-0.238891\pi$
$509$	33.0000	1.46270	0.731350	+	0.682003i	$0.238891\pi$
$510$	0	0
$511$	−1.00000	−0.0442374
$512$	1.00000	0.0441942
$513$	0	0
$514$	6.00000	0.264649
$515$	−39.0000	−1.71855
$516$	0	0
$517$	0	0
$518$	−7.00000	−0.307562
$519$	0	0
$520$	3.00000	0.131559
$521$	3.00000	0.131432	0.0657162	−	0.997838i	$-0.479067\pi$
$521$	3.00000	0.131432	0.0657162	+	0.997838i	$0.479067\pi$
$522$	0	0
$523$	−34.0000	−1.48672	−0.743358	−	0.668894i	$-0.766768\pi$
$523$	−34.0000	−1.48672	−0.743358	+	0.668894i	$0.766768\pi$
$524$	15.0000	0.655278
$525$	0	0
$526$	−24.0000	−1.04645
$527$	30.0000	1.30682
$528$	0	0
$529$	−14.0000	−0.608696
$530$	18.0000	0.781870
$531$	0	0
$532$	5.00000	0.216777
$533$	6.00000	0.259889
$534$	0	0
$535$	54.0000	2.33462
$536$	−4.00000	−0.172774
$537$	0	0
$538$	−18.0000	−0.776035
$539$	3.00000	0.129219
$540$	0	0
$541$	−1.00000	−0.0429934	−0.0214967	−	0.999769i	$-0.506843\pi$
$541$	−1.00000	−0.0429934	−0.0214967	+	0.999769i	$0.506843\pi$
$542$	2.00000	0.0859074
$543$	0	0
$544$	−3.00000	−0.128624
$545$	−3.00000	−0.128506
$546$	0	0
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	−9.00000	−0.384461
$549$	0	0
$550$	12.0000	0.511682
$551$	−15.0000	−0.639021
$552$	0	0
$553$	−10.0000	−0.425243
$554$	−28.0000	−1.18961
$555$	0	0
$556$	14.0000	0.593732
$557$	−6.00000	−0.254228	−0.127114	−	0.991888i	$-0.540571\pi$
$557$	−6.00000	−0.254228	−0.127114	+	0.991888i	$0.540571\pi$
$558$	0	0
$559$	5.00000	0.211477
$560$	3.00000	0.126773
$561$	0	0
$562$	−30.0000	−1.26547
$563$	39.0000	1.64365	0.821827	−	0.569737i	$-0.192955\pi$
$563$	39.0000	1.64365	0.821827	+	0.569737i	$0.192955\pi$
$564$	0	0
$565$	0	0
$566$	−16.0000	−0.672530
$567$	0	0
$568$	−6.00000	−0.251754
$569$	42.0000	1.76073	0.880366	−	0.474295i	$-0.157297\pi$
$569$	42.0000	1.76073	0.880366	+	0.474295i	$0.157297\pi$
$570$	0	0
$571$	44.0000	1.84134	0.920671	−	0.390339i	$-0.127642\pi$
$571$	44.0000	1.84134	0.920671	+	0.390339i	$0.127642\pi$
$572$	3.00000	0.125436
$573$	0	0
$574$	6.00000	0.250435
$575$	−12.0000	−0.500435
$576$	0	0
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	−8.00000	−0.332756
$579$	0	0
$580$	−9.00000	−0.373705
$581$	6.00000	0.248922
$582$	0	0
$583$	18.0000	0.745484
$584$	−1.00000	−0.0413803
$585$	0	0
$586$	30.0000	1.23929
$587$	−6.00000	−0.247647	−0.123823	−	0.992304i	$-0.539516\pi$
$587$	−6.00000	−0.247647	−0.123823	+	0.992304i	$0.539516\pi$
$588$	0	0
$589$	−50.0000	−2.06021
$590$	18.0000	0.741048
$591$	0	0
$592$	−7.00000	−0.287698
$593$	−12.0000	−0.492781	−0.246390	−	0.969171i	$-0.579245\pi$
$593$	−12.0000	−0.492781	−0.246390	+	0.969171i	$0.579245\pi$
$594$	0	0
$595$	−9.00000	−0.368964
$596$	−12.0000	−0.491539
$597$	0	0
$598$	−3.00000	−0.122679
$599$	−9.00000	−0.367730	−0.183865	−	0.982952i	$-0.558861\pi$
$599$	−9.00000	−0.367730	−0.183865	+	0.982952i	$0.558861\pi$
$600$	0	0
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	5.00000	0.203785
$603$	0	0
$604$	−13.0000	−0.528962
$605$	−6.00000	−0.243935
$606$	0	0
$607$	−1.00000	−0.0405887	−0.0202944	−	0.999794i	$-0.506460\pi$
$607$	−1.00000	−0.0405887	−0.0202944	+	0.999794i	$0.506460\pi$
$608$	5.00000	0.202777
$609$	0	0
$610$	−21.0000	−0.850265
$611$	0	0
$612$	0	0
$613$	−49.0000	−1.97909	−0.989546	−	0.144220i	$-0.953933\pi$
$613$	−49.0000	−1.97909	−0.989546	+	0.144220i	$0.953933\pi$
$614$	−4.00000	−0.161427
$615$	0	0
$616$	3.00000	0.120873
$617$	33.0000	1.32853	0.664265	−	0.747497i	$-0.268745\pi$
$617$	33.0000	1.32853	0.664265	+	0.747497i	$0.268745\pi$
$618$	0	0
$619$	17.0000	0.683288	0.341644	−	0.939829i	$-0.389016\pi$
$619$	17.0000	0.683288	0.341644	+	0.939829i	$0.389016\pi$
$620$	−30.0000	−1.20483
$621$	0	0
$622$	0	0
$623$	−6.00000	−0.240385
$624$	0	0
$625$	−29.0000	−1.16000
$626$	14.0000	0.559553
$627$	0	0
$628$	5.00000	0.199522
$629$	21.0000	0.837325
$630$	0	0
$631$	−13.0000	−0.517522	−0.258761	−	0.965941i	$-0.583314\pi$
$631$	−13.0000	−0.517522	−0.258761	+	0.965941i	$0.583314\pi$
$632$	−10.0000	−0.397779
$633$	0	0
$634$	18.0000	0.714871
$635$	24.0000	0.952411
$636$	0	0
$637$	1.00000	0.0396214
$638$	−9.00000	−0.356313
$639$	0	0
$640$	3.00000	0.118585
$641$	42.0000	1.65890	0.829450	−	0.558581i	$-0.188654\pi$
$641$	42.0000	1.65890	0.829450	+	0.558581i	$0.188654\pi$
$642$	0	0
$643$	−25.0000	−0.985904	−0.492952	−	0.870057i	$-0.664082\pi$
$643$	−25.0000	−0.985904	−0.492952	+	0.870057i	$0.664082\pi$
$644$	−3.00000	−0.118217
$645$	0	0
$646$	−15.0000	−0.590167
$647$	−12.0000	−0.471769	−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000	−0.471769	−0.235884	+	0.971781i	$0.575799\pi$
$648$	0	0
$649$	18.0000	0.706562
$650$	4.00000	0.156893
$651$	0	0
$652$	2.00000	0.0783260
$653$	15.0000	0.586995	0.293498	−	0.955960i	$-0.405181\pi$
$653$	15.0000	0.586995	0.293498	+	0.955960i	$0.405181\pi$
$654$	0	0
$655$	45.0000	1.75830
$656$	6.00000	0.234261
$657$	0	0
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	−4.00000	−0.155582	−0.0777910	−	0.996970i	$-0.524787\pi$
$661$	−4.00000	−0.155582	−0.0777910	+	0.996970i	$0.524787\pi$
$662$	14.0000	0.544125
$663$	0	0
$664$	6.00000	0.232845
$665$	15.0000	0.581675
$666$	0	0
$667$	9.00000	0.348481
$668$	21.0000	0.812514
$669$	0	0
$670$	−12.0000	−0.463600
$671$	−21.0000	−0.810696
$672$	0	0
$673$	−13.0000	−0.501113	−0.250557	−	0.968102i	$-0.580614\pi$
$673$	−13.0000	−0.501113	−0.250557	+	0.968102i	$0.580614\pi$
$674$	−25.0000	−0.962964
$675$	0	0
$676$	1.00000	0.0384615
$677$	42.0000	1.61419	0.807096	−	0.590421i	$-0.201038\pi$
$677$	42.0000	1.61419	0.807096	+	0.590421i	$0.201038\pi$
$678$	0	0
$679$	2.00000	0.0767530
$680$	−9.00000	−0.345134
$681$	0	0
$682$	−30.0000	−1.14876
$683$	21.0000	0.803543	0.401771	−	0.915740i	$-0.368395\pi$
$683$	21.0000	0.803543	0.401771	+	0.915740i	$0.368395\pi$
$684$	0	0
$685$	−27.0000	−1.03162
$686$	1.00000	0.0381802
$687$	0	0
$688$	5.00000	0.190623
$689$	6.00000	0.228582
$690$	0	0
$691$	32.0000	1.21734	0.608669	−	0.793424i	$-0.291704\pi$
$691$	32.0000	1.21734	0.608669	+	0.793424i	$0.291704\pi$
$692$	18.0000	0.684257
$693$	0	0
$694$	−12.0000	−0.455514
$695$	42.0000	1.59315
$696$	0	0
$697$	−18.0000	−0.681799
$698$	−16.0000	−0.605609
$699$	0	0
$700$	4.00000	0.151186
$701$	−30.0000	−1.13308	−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000	−1.13308	−0.566542	+	0.824033i	$0.691719\pi$
$702$	0	0
$703$	−35.0000	−1.32005
$704$	3.00000	0.113067
$705$	0	0
$706$	−30.0000	−1.12906
$707$	6.00000	0.225653
$708$	0	0
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	−18.0000	−0.675528
$711$	0	0
$712$	−6.00000	−0.224860
$713$	30.0000	1.12351
$714$	0	0
$715$	9.00000	0.336581
$716$	12.0000	0.448461
$717$	0	0
$718$	−30.0000	−1.11959
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	−13.0000	−0.484145
$722$	6.00000	0.223297
$723$	0	0
$724$	−22.0000	−0.817624
$725$	−12.0000	−0.445669
$726$	0	0
$727$	−25.0000	−0.927199	−0.463599	−	0.886045i	$-0.653442\pi$
$727$	−25.0000	−0.927199	−0.463599	+	0.886045i	$0.653442\pi$
$728$	1.00000	0.0370625
$729$	0	0
$730$	−3.00000	−0.111035
$731$	−15.0000	−0.554795
$732$	0	0
$733$	−28.0000	−1.03420	−0.517102	−	0.855924i	$-0.672989\pi$
$733$	−28.0000	−1.03420	−0.517102	+	0.855924i	$0.672989\pi$
$734$	32.0000	1.18114
$735$	0	0
$736$	−3.00000	−0.110581
$737$	−12.0000	−0.442026
$738$	0	0
$739$	−34.0000	−1.25071	−0.625355	−	0.780340i	$-0.715046\pi$
$739$	−34.0000	−1.25071	−0.625355	+	0.780340i	$0.715046\pi$
$740$	−21.0000	−0.771975
$741$	0	0
$742$	6.00000	0.220267
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	−36.0000	−1.31894
$746$	14.0000	0.512576
$747$	0	0
$748$	−9.00000	−0.329073
$749$	18.0000	0.657706
$750$	0	0
$751$	44.0000	1.60558	0.802791	−	0.596260i	$-0.203347\pi$
$751$	44.0000	1.60558	0.802791	+	0.596260i	$0.203347\pi$
$752$	0	0
$753$	0	0
$754$	−3.00000	−0.109254
$755$	−39.0000	−1.41936
$756$	0	0
$757$	8.00000	0.290765	0.145382	−	0.989376i	$-0.453559\pi$
$757$	8.00000	0.290765	0.145382	+	0.989376i	$0.453559\pi$
$758$	2.00000	0.0726433
$759$	0	0
$760$	15.0000	0.544107
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	−1.00000	−0.0362024
$764$	−15.0000	−0.542681
$765$	0	0
$766$	9.00000	0.325183
$767$	6.00000	0.216647
$768$	0	0
$769$	53.0000	1.91123	0.955614	−	0.294620i	$-0.0951931\pi$
$769$	53.0000	1.91123	0.955614	+	0.294620i	$0.0951931\pi$
$770$	9.00000	0.324337
$771$	0	0
$772$	−22.0000	−0.791797
$773$	−39.0000	−1.40273	−0.701366	−	0.712801i	$-0.747426\pi$
$773$	−39.0000	−1.40273	−0.701366	+	0.712801i	$0.747426\pi$
$774$	0	0
$775$	−40.0000	−1.43684
$776$	2.00000	0.0717958
$777$	0	0
$778$	6.00000	0.215110
$779$	30.0000	1.07486
$780$	0	0
$781$	−18.0000	−0.644091
$782$	9.00000	0.321839
$783$	0	0
$784$	1.00000	0.0357143
$785$	15.0000	0.535373
$786$	0	0
$787$	17.0000	0.605985	0.302992	−	0.952993i	$-0.402014\pi$
$787$	17.0000	0.605985	0.302992	+	0.952993i	$0.402014\pi$
$788$	6.00000	0.213741
$789$	0	0
$790$	−30.0000	−1.06735
$791$	0	0
$792$	0	0
$793$	−7.00000	−0.248577
$794$	8.00000	0.283909
$795$	0	0
$796$	5.00000	0.177220
$797$	48.0000	1.70025	0.850124	−	0.526583i	$-0.176527\pi$
$797$	48.0000	1.70025	0.850124	+	0.526583i	$0.176527\pi$
$798$	0	0
$799$	0	0
$800$	4.00000	0.141421
$801$	0	0
$802$	18.0000	0.635602
$803$	−3.00000	−0.105868
$804$	0	0
$805$	−9.00000	−0.317208
$806$	−10.0000	−0.352235
$807$	0	0
$808$	6.00000	0.211079
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	0	0
$811$	−7.00000	−0.245803	−0.122902	−	0.992419i	$-0.539220\pi$
$811$	−7.00000	−0.245803	−0.122902	+	0.992419i	$0.539220\pi$
$812$	−3.00000	−0.105279
$813$	0	0
$814$	−21.0000	−0.736050
$815$	6.00000	0.210171
$816$	0	0
$817$	25.0000	0.874639
$818$	5.00000	0.174821
$819$	0	0
$820$	18.0000	0.628587
$821$	36.0000	1.25641	0.628204	−	0.778048i	$-0.283790\pi$
$821$	36.0000	1.25641	0.628204	+	0.778048i	$0.283790\pi$
$822$	0	0
$823$	−34.0000	−1.18517	−0.592583	−	0.805510i	$-0.701892\pi$
$823$	−34.0000	−1.18517	−0.592583	+	0.805510i	$0.701892\pi$
$824$	−13.0000	−0.452876
$825$	0	0
$826$	6.00000	0.208767
$827$	−21.0000	−0.730242	−0.365121	−	0.930960i	$-0.618972\pi$
$827$	−21.0000	−0.730242	−0.365121	+	0.930960i	$0.618972\pi$
$828$	0	0
$829$	−55.0000	−1.91023	−0.955114	−	0.296237i	$-0.904268\pi$
$829$	−55.0000	−1.91023	−0.955114	+	0.296237i	$0.904268\pi$
$830$	18.0000	0.624789
$831$	0	0
$832$	1.00000	0.0346688
$833$	−3.00000	−0.103944
$834$	0	0
$835$	63.0000	2.18020
$836$	15.0000	0.518786
$837$	0	0
$838$	−3.00000	−0.103633
$839$	48.0000	1.65714	0.828572	−	0.559883i	$-0.189154\pi$
$839$	48.0000	1.65714	0.828572	+	0.559883i	$0.189154\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	26.0000	0.896019
$843$	0	0
$844$	23.0000	0.791693
$845$	3.00000	0.103203
$846$	0	0
$847$	−2.00000	−0.0687208
$848$	6.00000	0.206041
$849$	0	0
$850$	−12.0000	−0.411597
$851$	21.0000	0.719871
$852$	0	0
$853$	8.00000	0.273915	0.136957	−	0.990577i	$-0.456268\pi$
$853$	8.00000	0.273915	0.136957	+	0.990577i	$0.456268\pi$
$854$	−7.00000	−0.239535
$855$	0	0
$856$	18.0000	0.615227
$857$	−54.0000	−1.84460	−0.922302	−	0.386469i	$-0.873695\pi$
$857$	−54.0000	−1.84460	−0.922302	+	0.386469i	$0.873695\pi$
$858$	0	0
$859$	14.0000	0.477674	0.238837	−	0.971060i	$-0.423234\pi$
$859$	14.0000	0.477674	0.238837	+	0.971060i	$0.423234\pi$
$860$	15.0000	0.511496
$861$	0	0
$862$	−18.0000	−0.613082
$863$	12.0000	0.408485	0.204242	−	0.978920i	$-0.434527\pi$
$863$	12.0000	0.408485	0.204242	+	0.978920i	$0.434527\pi$
$864$	0	0
$865$	54.0000	1.83606
$866$	14.0000	0.475739
$867$	0	0
$868$	−10.0000	−0.339422
$869$	−30.0000	−1.01768
$870$	0	0
$871$	−4.00000	−0.135535
$872$	−1.00000	−0.0338643
$873$	0	0
$874$	−15.0000	−0.507383
$875$	−3.00000	−0.101419
$876$	0	0
$877$	2.00000	0.0675352	0.0337676	−	0.999430i	$-0.489249\pi$
$877$	2.00000	0.0675352	0.0337676	+	0.999430i	$0.489249\pi$
$878$	35.0000	1.18119
$879$	0	0
$880$	9.00000	0.303390
$881$	−3.00000	−0.101073	−0.0505363	−	0.998722i	$-0.516093\pi$
$881$	−3.00000	−0.101073	−0.0505363	+	0.998722i	$0.516093\pi$
$882$	0	0
$883$	17.0000	0.572096	0.286048	−	0.958215i	$-0.407658\pi$
$883$	17.0000	0.572096	0.286048	+	0.958215i	$0.407658\pi$
$884$	−3.00000	−0.100901
$885$	0	0
$886$	−6.00000	−0.201574
$887$	42.0000	1.41022	0.705111	−	0.709097i	$-0.250897\pi$
$887$	42.0000	1.41022	0.705111	+	0.709097i	$0.250897\pi$
$888$	0	0
$889$	8.00000	0.268311
$890$	−18.0000	−0.603361
$891$	0	0
$892$	8.00000	0.267860
$893$	0	0
$894$	0	0
$895$	36.0000	1.20335
$896$	1.00000	0.0334077
$897$	0	0
$898$	−21.0000	−0.700779
$899$	30.0000	1.00056
$900$	0	0
$901$	−18.0000	−0.599667
$902$	18.0000	0.599334
$903$	0	0
$904$	0	0
$905$	−66.0000	−2.19391
$906$	0	0
$907$	44.0000	1.46100	0.730498	−	0.682915i	$-0.239288\pi$
$907$	44.0000	1.46100	0.730498	+	0.682915i	$0.239288\pi$
$908$	18.0000	0.597351
$909$	0	0
$910$	3.00000	0.0994490
$911$	27.0000	0.894550	0.447275	−	0.894397i	$-0.352395\pi$
$911$	27.0000	0.894550	0.447275	+	0.894397i	$0.352395\pi$
$912$	0	0
$913$	18.0000	0.595713
$914$	20.0000	0.661541
$915$	0	0
$916$	−22.0000	−0.726900
$917$	15.0000	0.495344
$918$	0	0
$919$	−34.0000	−1.12156	−0.560778	−	0.827966i	$-0.689498\pi$
$919$	−34.0000	−1.12156	−0.560778	+	0.827966i	$0.689498\pi$
$920$	−9.00000	−0.296721
$921$	0	0
$922$	33.0000	1.08680
$923$	−6.00000	−0.197492
$924$	0	0
$925$	−28.0000	−0.920634
$926$	−25.0000	−0.821551
$927$	0	0
$928$	−3.00000	−0.0984798
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	0	0
$931$	5.00000	0.163868
$932$	−24.0000	−0.786146
$933$	0	0
$934$	3.00000	0.0981630
$935$	−27.0000	−0.882994
$936$	0	0
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	−4.00000	−0.130605
$939$	0	0
$940$	0	0
$941$	−42.0000	−1.36916	−0.684580	−	0.728937i	$-0.740015\pi$
$941$	−42.0000	−1.36916	−0.684580	+	0.728937i	$0.740015\pi$
$942$	0	0
$943$	−18.0000	−0.586161
$944$	6.00000	0.195283
$945$	0	0
$946$	15.0000	0.487692
$947$	27.0000	0.877382	0.438691	−	0.898638i	$-0.355442\pi$
$947$	27.0000	0.877382	0.438691	+	0.898638i	$0.355442\pi$
$948$	0	0
$949$	−1.00000	−0.0324614
$950$	20.0000	0.648886
$951$	0	0
$952$	−3.00000	−0.0972306
$953$	36.0000	1.16615	0.583077	−	0.812417i	$-0.301849\pi$
$953$	36.0000	1.16615	0.583077	+	0.812417i	$0.301849\pi$
$954$	0	0
$955$	−45.0000	−1.45617
$956$	−12.0000	−0.388108
$957$	0	0
$958$	15.0000	0.484628
$959$	−9.00000	−0.290625
$960$	0	0
$961$	69.0000	2.22581
$962$	−7.00000	−0.225689
$963$	0	0
$964$	−22.0000	−0.708572
$965$	−66.0000	−2.12462
$966$	0	0
$967$	41.0000	1.31847	0.659236	−	0.751936i	$-0.270880\pi$
$967$	41.0000	1.31847	0.659236	+	0.751936i	$0.270880\pi$
$968$	−2.00000	−0.0642824
$969$	0	0
$970$	6.00000	0.192648
$971$	−48.0000	−1.54039	−0.770197	−	0.637806i	$-0.779842\pi$
$971$	−48.0000	−1.54039	−0.770197	+	0.637806i	$0.779842\pi$
$972$	0	0
$973$	14.0000	0.448819
$974$	32.0000	1.02535
$975$	0	0
$976$	−7.00000	−0.224065
$977$	57.0000	1.82359	0.911796	−	0.410644i	$-0.134696\pi$
$977$	57.0000	1.82359	0.911796	+	0.410644i	$0.134696\pi$
$978$	0	0
$979$	−18.0000	−0.575282
$980$	3.00000	0.0958315
$981$	0	0
$982$	−36.0000	−1.14881
$983$	−21.0000	−0.669796	−0.334898	−	0.942254i	$-0.608702\pi$
$983$	−21.0000	−0.669796	−0.334898	+	0.942254i	$0.608702\pi$
$984$	0	0
$985$	18.0000	0.573528
$986$	9.00000	0.286618
$987$	0	0
$988$	5.00000	0.159071
$989$	−15.0000	−0.476972
$990$	0	0
$991$	−34.0000	−1.08005	−0.540023	−	0.841650i	$-0.681584\pi$
$991$	−34.0000	−1.08005	−0.540023	+	0.841650i	$0.681584\pi$
$992$	−10.0000	−0.317500
$993$	0	0
$994$	−6.00000	−0.190308
$995$	15.0000	0.475532
$996$	0	0
$997$	−58.0000	−1.83688	−0.918439	−	0.395562i	$-0.870550\pi$
$997$	−58.0000	−1.83688	−0.918439	+	0.395562i	$0.870550\pi$
$998$	−4.00000	−0.126618
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1638.2.a.t.1.1	yes	1
3.2	odd	2		1638.2.a.b.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1638.2.a.b.1.1	✓	1	3.2	odd	2
1638.2.a.t.1.1	yes	1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1638.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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