LMFDB - Embedded newform 189.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [189,2,Mod(1,189)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(189, 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("189.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 189 = 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	189.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:	$1.50917259820$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	189.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205 q^{2} +1.00000 q^{4} -1.73205 q^{5} +1.00000 q^{7} +1.73205 q^{8} +O(q^{10})$	\(q-1.73205 q^{2} +1.00000 q^{4} -1.73205 q^{5} +1.00000 q^{7} +1.73205 q^{8} +3.00000 q^{10} +1.73205 q^{11} +2.00000 q^{13} -1.73205 q^{14} -5.00000 q^{16} +6.92820 q^{17} +5.00000 q^{19} -1.73205 q^{20} -3.00000 q^{22} -1.73205 q^{23} -2.00000 q^{25} -3.46410 q^{26} +1.00000 q^{28} +10.3923 q^{29} +5.00000 q^{31} +5.19615 q^{32} -12.0000 q^{34} -1.73205 q^{35} -7.00000 q^{37} -8.66025 q^{38} -3.00000 q^{40} -5.19615 q^{41} -4.00000 q^{43} +1.73205 q^{44} +3.00000 q^{46} -6.92820 q^{47} +1.00000 q^{49} +3.46410 q^{50} +2.00000 q^{52} -13.8564 q^{53} -3.00000 q^{55} +1.73205 q^{56} -18.0000 q^{58} +6.92820 q^{59} +8.00000 q^{61} -8.66025 q^{62} +1.00000 q^{64} -3.46410 q^{65} +14.0000 q^{67} +6.92820 q^{68} +3.00000 q^{70} -5.19615 q^{71} -4.00000 q^{73} +12.1244 q^{74} +5.00000 q^{76} +1.73205 q^{77} +8.00000 q^{79} +8.66025 q^{80} +9.00000 q^{82} +10.3923 q^{83} -12.0000 q^{85} +6.92820 q^{86} +3.00000 q^{88} +8.66025 q^{89} +2.00000 q^{91} -1.73205 q^{92} +12.0000 q^{94} -8.66025 q^{95} -4.00000 q^{97} -1.73205 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} + 2 q^{7}+O(q^{10}) $ 2 * q + 2 * q^4 + 2 * q^7	$ 2 q + 2 q^{4} + 2 q^{7} + 6 q^{10} + 4 q^{13} - 10 q^{16} + 10 q^{19} - 6 q^{22} - 4 q^{25} + 2 q^{28} + 10 q^{31} - 24 q^{34} - 14 q^{37} - 6 q^{40} - 8 q^{43} + 6 q^{46} + 2 q^{49} + 4 q^{52} - 6 q^{55} - 36 q^{58} + 16 q^{61} + 2 q^{64} + 28 q^{67} + 6 q^{70} - 8 q^{73} + 10 q^{76} + 16 q^{79} + 18 q^{82} - 24 q^{85} + 6 q^{88} + 4 q^{91} + 24 q^{94} - 8 q^{97}+O(q^{100}) $ 2 * q + 2 * q^4 + 2 * q^7 + 6 * q^10 + 4 * q^13 - 10 * q^16 + 10 * q^19 - 6 * q^22 - 4 * q^25 + 2 * q^28 + 10 * q^31 - 24 * q^34 - 14 * q^37 - 6 * q^40 - 8 * q^43 + 6 * q^46 + 2 * q^49 + 4 * q^52 - 6 * q^55 - 36 * q^58 + 16 * q^61 + 2 * q^64 + 28 * q^67 + 6 * q^70 - 8 * q^73 + 10 * q^76 + 16 * q^79 + 18 * q^82 - 24 * q^85 + 6 * q^88 + 4 * q^91 + 24 * q^94 - 8 * q^97

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 $

$2$	−1.73205	−1.22474	−0.612372	−	0.790569i	$-0.709785\pi$
$2$	−1.73205	−1.22474	−0.612372	+	0.790569i	$0.709785\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.73205	−0.774597	−0.387298	−	0.921954i	$-0.626592\pi$
$5$	−1.73205	−0.774597	−0.387298	+	0.921954i	$0.626592\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.73205	0.612372
$9$	0	0
$10$	3.00000	0.948683
$11$	1.73205	0.522233	0.261116	−	0.965307i	$-0.415909\pi$
$11$	1.73205	0.522233	0.261116	+	0.965307i	$0.415909\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	−1.73205	−0.462910
$15$	0	0
$16$	−5.00000	−1.25000
$17$	6.92820	1.68034	0.840168	−	0.542326i	$-0.182456\pi$
$17$	6.92820	1.68034	0.840168	+	0.542326i	$0.182456\pi$
$18$	0	0
$19$	5.00000	1.14708	0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000	1.14708	0.573539	+	0.819178i	$0.305570\pi$
$20$	−1.73205	−0.387298
$21$	0	0
$22$	−3.00000	−0.639602
$23$	−1.73205	−0.361158	−0.180579	−	0.983561i	$-0.557797\pi$
$23$	−1.73205	−0.361158	−0.180579	+	0.983561i	$0.557797\pi$
$24$	0	0
$25$	−2.00000	−0.400000
$26$	−3.46410	−0.679366
$27$	0	0
$28$	1.00000	0.188982
$29$	10.3923	1.92980	0.964901	−	0.262613i	$-0.0845842\pi$
$29$	10.3923	1.92980	0.964901	+	0.262613i	$0.0845842\pi$
$30$	0	0
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	5.19615	0.918559
$33$	0	0
$34$	−12.0000	−2.05798
$35$	−1.73205	−0.292770
$36$	0	0
$37$	−7.00000	−1.15079	−0.575396	−	0.817875i	$-0.695152\pi$
$37$	−7.00000	−1.15079	−0.575396	+	0.817875i	$0.695152\pi$
$38$	−8.66025	−1.40488
$39$	0	0
$40$	−3.00000	−0.474342
$41$	−5.19615	−0.811503	−0.405751	−	0.913984i	$-0.632990\pi$
$41$	−5.19615	−0.811503	−0.405751	+	0.913984i	$0.632990\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	1.73205	0.261116
$45$	0	0
$46$	3.00000	0.442326
$47$	−6.92820	−1.01058	−0.505291	−	0.862949i	$-0.668615\pi$
$47$	−6.92820	−1.01058	−0.505291	+	0.862949i	$0.668615\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	3.46410	0.489898
$51$	0	0
$52$	2.00000	0.277350
$53$	−13.8564	−1.90332	−0.951662	−	0.307148i	$-0.900625\pi$
$53$	−13.8564	−1.90332	−0.951662	+	0.307148i	$0.900625\pi$
$54$	0	0
$55$	−3.00000	−0.404520
$56$	1.73205	0.231455
$57$	0	0
$58$	−18.0000	−2.36352
$59$	6.92820	0.901975	0.450988	−	0.892530i	$-0.351072\pi$
$59$	6.92820	0.901975	0.450988	+	0.892530i	$0.351072\pi$
$60$	0	0
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	−8.66025	−1.09985
$63$	0	0
$64$	1.00000	0.125000
$65$	−3.46410	−0.429669
$66$	0	0
$67$	14.0000	1.71037	0.855186	−	0.518321i	$-0.173443\pi$
$67$	14.0000	1.71037	0.855186	+	0.518321i	$0.173443\pi$
$68$	6.92820	0.840168
$69$	0	0
$70$	3.00000	0.358569
$71$	−5.19615	−0.616670	−0.308335	−	0.951278i	$-0.599772\pi$
$71$	−5.19615	−0.616670	−0.308335	+	0.951278i	$0.599772\pi$
$72$	0	0
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	12.1244	1.40943
$75$	0	0
$76$	5.00000	0.573539
$77$	1.73205	0.197386
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	8.66025	0.968246
$81$	0	0
$82$	9.00000	0.993884
$83$	10.3923	1.14070	0.570352	−	0.821401i	$-0.306807\pi$
$83$	10.3923	1.14070	0.570352	+	0.821401i	$0.306807\pi$
$84$	0	0
$85$	−12.0000	−1.30158
$86$	6.92820	0.747087
$87$	0	0
$88$	3.00000	0.319801
$89$	8.66025	0.917985	0.458993	−	0.888440i	$-0.348210\pi$
$89$	8.66025	0.917985	0.458993	+	0.888440i	$0.348210\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	−1.73205	−0.180579
$93$	0	0
$94$	12.0000	1.23771
$95$	−8.66025	−0.888523
$96$	0	0
$97$	−4.00000	−0.406138	−0.203069	−	0.979164i	$-0.565092\pi$
$97$	−4.00000	−0.406138	−0.203069	+	0.979164i	$0.565092\pi$
$98$	−1.73205	−0.174964
$99$	0	0
$100$	−2.00000	−0.200000
$101$	−13.8564	−1.37876	−0.689382	−	0.724398i	$-0.742118\pi$
$101$	−13.8564	−1.37876	−0.689382	+	0.724398i	$0.742118\pi$
$102$	0	0
$103$	5.00000	0.492665	0.246332	−	0.969185i	$-0.420775\pi$
$103$	5.00000	0.492665	0.246332	+	0.969185i	$0.420775\pi$
$104$	3.46410	0.339683
$105$	0	0
$106$	24.0000	2.33109
$107$	3.46410	0.334887	0.167444	−	0.985882i	$-0.446449\pi$
$107$	3.46410	0.334887	0.167444	+	0.985882i	$0.446449\pi$
$108$	0	0
$109$	−7.00000	−0.670478	−0.335239	−	0.942133i	$-0.608817\pi$
$109$	−7.00000	−0.670478	−0.335239	+	0.942133i	$0.608817\pi$
$110$	5.19615	0.495434
$111$	0	0
$112$	−5.00000	−0.472456
$113$	−10.3923	−0.977626	−0.488813	−	0.872389i	$-0.662570\pi$
$113$	−10.3923	−0.977626	−0.488813	+	0.872389i	$0.662570\pi$
$114$	0	0
$115$	3.00000	0.279751
$116$	10.3923	0.964901
$117$	0	0
$118$	−12.0000	−1.10469
$119$	6.92820	0.635107
$120$	0	0
$121$	−8.00000	−0.727273
$122$	−13.8564	−1.25450
$123$	0	0
$124$	5.00000	0.449013
$125$	12.1244	1.08444
$126$	0	0
$127$	−10.0000	−0.887357	−0.443678	−	0.896186i	$-0.646327\pi$
$127$	−10.0000	−0.887357	−0.443678	+	0.896186i	$0.646327\pi$
$128$	−12.1244	−1.07165
$129$	0	0
$130$	6.00000	0.526235
$131$	−6.92820	−0.605320	−0.302660	−	0.953099i	$-0.597875\pi$
$131$	−6.92820	−0.605320	−0.302660	+	0.953099i	$0.597875\pi$
$132$	0	0
$133$	5.00000	0.433555
$134$	−24.2487	−2.09477
$135$	0	0
$136$	12.0000	1.02899
$137$	−3.46410	−0.295958	−0.147979	−	0.988990i	$-0.547277\pi$
$137$	−3.46410	−0.295958	−0.147979	+	0.988990i	$0.547277\pi$
$138$	0	0
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	−1.73205	−0.146385
$141$	0	0
$142$	9.00000	0.755263
$143$	3.46410	0.289683
$144$	0	0
$145$	−18.0000	−1.49482
$146$	6.92820	0.573382
$147$	0	0
$148$	−7.00000	−0.575396
$149$	3.46410	0.283790	0.141895	−	0.989882i	$-0.454680\pi$
$149$	3.46410	0.283790	0.141895	+	0.989882i	$0.454680\pi$
$150$	0	0
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	8.66025	0.702439
$153$	0	0
$154$	−3.00000	−0.241747
$155$	−8.66025	−0.695608
$156$	0	0
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	−13.8564	−1.10236
$159$	0	0
$160$	−9.00000	−0.711512
$161$	−1.73205	−0.136505
$162$	0	0
$163$	2.00000	0.156652	0.0783260	−	0.996928i	$-0.475042\pi$
$163$	2.00000	0.156652	0.0783260	+	0.996928i	$0.475042\pi$
$164$	−5.19615	−0.405751
$165$	0	0
$166$	−18.0000	−1.39707
$167$	10.3923	0.804181	0.402090	−	0.915600i	$-0.368284\pi$
$167$	10.3923	0.804181	0.402090	+	0.915600i	$0.368284\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	20.7846	1.59411
$171$	0	0
$172$	−4.00000	−0.304997
$173$	−1.73205	−0.131685	−0.0658427	−	0.997830i	$-0.520974\pi$
$173$	−1.73205	−0.131685	−0.0658427	+	0.997830i	$0.520974\pi$
$174$	0	0
$175$	−2.00000	−0.151186
$176$	−8.66025	−0.652791
$177$	0	0
$178$	−15.0000	−1.12430
$179$	−3.46410	−0.258919	−0.129460	−	0.991585i	$-0.541324\pi$
$179$	−3.46410	−0.258919	−0.129460	+	0.991585i	$0.541324\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	−3.46410	−0.256776
$183$	0	0
$184$	−3.00000	−0.221163
$185$	12.1244	0.891400
$186$	0	0
$187$	12.0000	0.877527
$188$	−6.92820	−0.505291
$189$	0	0
$190$	15.0000	1.08821
$191$	8.66025	0.626634	0.313317	−	0.949649i	$-0.398560\pi$
$191$	8.66025	0.626634	0.313317	+	0.949649i	$0.398560\pi$
$192$	0	0
$193$	−22.0000	−1.58359	−0.791797	−	0.610784i	$-0.790854\pi$
$193$	−22.0000	−1.58359	−0.791797	+	0.610784i	$0.790854\pi$
$194$	6.92820	0.497416
$195$	0	0
$196$	1.00000	0.0714286
$197$	10.3923	0.740421	0.370211	−	0.928948i	$-0.379286\pi$
$197$	10.3923	0.740421	0.370211	+	0.928948i	$0.379286\pi$
$198$	0	0
$199$	−25.0000	−1.77220	−0.886102	−	0.463491i	$-0.846597\pi$
$199$	−25.0000	−1.77220	−0.886102	+	0.463491i	$0.846597\pi$
$200$	−3.46410	−0.244949
$201$	0	0
$202$	24.0000	1.68863
$203$	10.3923	0.729397
$204$	0	0
$205$	9.00000	0.628587
$206$	−8.66025	−0.603388
$207$	0	0
$208$	−10.0000	−0.693375
$209$	8.66025	0.599042
$210$	0	0
$211$	2.00000	0.137686	0.0688428	−	0.997628i	$-0.478069\pi$
$211$	2.00000	0.137686	0.0688428	+	0.997628i	$0.478069\pi$
$212$	−13.8564	−0.951662
$213$	0	0
$214$	−6.00000	−0.410152
$215$	6.92820	0.472500
$216$	0	0
$217$	5.00000	0.339422
$218$	12.1244	0.821165
$219$	0	0
$220$	−3.00000	−0.202260
$221$	13.8564	0.932083
$222$	0	0
$223$	−19.0000	−1.27233	−0.636167	−	0.771551i	$-0.719481\pi$
$223$	−19.0000	−1.27233	−0.636167	+	0.771551i	$0.719481\pi$
$224$	5.19615	0.347183
$225$	0	0
$226$	18.0000	1.19734
$227$	−3.46410	−0.229920	−0.114960	−	0.993370i	$-0.536674\pi$
$227$	−3.46410	−0.229920	−0.114960	+	0.993370i	$0.536674\pi$
$228$	0	0
$229$	−4.00000	−0.264327	−0.132164	−	0.991228i	$-0.542192\pi$
$229$	−4.00000	−0.264327	−0.132164	+	0.991228i	$0.542192\pi$
$230$	−5.19615	−0.342624
$231$	0	0
$232$	18.0000	1.18176
$233$	−17.3205	−1.13470	−0.567352	−	0.823475i	$-0.692032\pi$
$233$	−17.3205	−1.13470	−0.567352	+	0.823475i	$0.692032\pi$
$234$	0	0
$235$	12.0000	0.782794
$236$	6.92820	0.450988
$237$	0	0
$238$	−12.0000	−0.777844
$239$	10.3923	0.672222	0.336111	−	0.941822i	$-0.390888\pi$
$239$	10.3923	0.672222	0.336111	+	0.941822i	$0.390888\pi$
$240$	0	0
$241$	8.00000	0.515325	0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000	0.515325	0.257663	+	0.966235i	$0.417048\pi$
$242$	13.8564	0.890724
$243$	0	0
$244$	8.00000	0.512148
$245$	−1.73205	−0.110657
$246$	0	0
$247$	10.0000	0.636285
$248$	8.66025	0.549927
$249$	0	0
$250$	−21.0000	−1.32816
$251$	−20.7846	−1.31191	−0.655956	−	0.754799i	$-0.727735\pi$
$251$	−20.7846	−1.31191	−0.655956	+	0.754799i	$0.727735\pi$
$252$	0	0
$253$	−3.00000	−0.188608
$254$	17.3205	1.08679
$255$	0	0
$256$	19.0000	1.18750
$257$	−1.73205	−0.108042	−0.0540212	−	0.998540i	$-0.517204\pi$
$257$	−1.73205	−0.108042	−0.0540212	+	0.998540i	$0.517204\pi$
$258$	0	0
$259$	−7.00000	−0.434959
$260$	−3.46410	−0.214834
$261$	0	0
$262$	12.0000	0.741362
$263$	−8.66025	−0.534014	−0.267007	−	0.963695i	$-0.586035\pi$
$263$	−8.66025	−0.534014	−0.267007	+	0.963695i	$0.586035\pi$
$264$	0	0
$265$	24.0000	1.47431
$266$	−8.66025	−0.530994
$267$	0	0
$268$	14.0000	0.855186
$269$	−19.0526	−1.16166	−0.580828	−	0.814027i	$-0.697271\pi$
$269$	−19.0526	−1.16166	−0.580828	+	0.814027i	$0.697271\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	−34.6410	−2.10042
$273$	0	0
$274$	6.00000	0.362473
$275$	−3.46410	−0.208893
$276$	0	0
$277$	−19.0000	−1.14160	−0.570800	−	0.821089i	$-0.693367\pi$
$277$	−19.0000	−1.14160	−0.570800	+	0.821089i	$0.693367\pi$
$278$	−34.6410	−2.07763
$279$	0	0
$280$	−3.00000	−0.179284
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	−5.19615	−0.308335
$285$	0	0
$286$	−6.00000	−0.354787
$287$	−5.19615	−0.306719
$288$	0	0
$289$	31.0000	1.82353
$290$	31.1769	1.83077
$291$	0	0
$292$	−4.00000	−0.234082
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	−12.0000	−0.698667
$296$	−12.1244	−0.704714
$297$	0	0
$298$	−6.00000	−0.347571
$299$	−3.46410	−0.200334
$300$	0	0
$301$	−4.00000	−0.230556
$302$	17.3205	0.996683
$303$	0	0
$304$	−25.0000	−1.43385
$305$	−13.8564	−0.793416
$306$	0	0
$307$	17.0000	0.970241	0.485121	−	0.874447i	$-0.338776\pi$
$307$	17.0000	0.970241	0.485121	+	0.874447i	$0.338776\pi$
$308$	1.73205	0.0986928
$309$	0	0
$310$	15.0000	0.851943
$311$	−3.46410	−0.196431	−0.0982156	−	0.995165i	$-0.531313\pi$
$311$	−3.46410	−0.196431	−0.0982156	+	0.995165i	$0.531313\pi$
$312$	0	0
$313$	2.00000	0.113047	0.0565233	−	0.998401i	$-0.481998\pi$
$313$	2.00000	0.113047	0.0565233	+	0.998401i	$0.481998\pi$
$314$	17.3205	0.977453
$315$	0	0
$316$	8.00000	0.450035
$317$	−6.92820	−0.389127	−0.194563	−	0.980890i	$-0.562329\pi$
$317$	−6.92820	−0.389127	−0.194563	+	0.980890i	$0.562329\pi$
$318$	0	0
$319$	18.0000	1.00781
$320$	−1.73205	−0.0968246
$321$	0	0
$322$	3.00000	0.167183
$323$	34.6410	1.92748
$324$	0	0
$325$	−4.00000	−0.221880
$326$	−3.46410	−0.191859
$327$	0	0
$328$	−9.00000	−0.496942
$329$	−6.92820	−0.381964
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	10.3923	0.570352
$333$	0	0
$334$	−18.0000	−0.984916
$335$	−24.2487	−1.32485
$336$	0	0
$337$	−13.0000	−0.708155	−0.354078	−	0.935216i	$-0.615205\pi$
$337$	−13.0000	−0.708155	−0.354078	+	0.935216i	$0.615205\pi$
$338$	15.5885	0.847900
$339$	0	0
$340$	−12.0000	−0.650791
$341$	8.66025	0.468979
$342$	0	0
$343$	1.00000	0.0539949
$344$	−6.92820	−0.373544
$345$	0	0
$346$	3.00000	0.161281
$347$	32.9090	1.76665	0.883323	−	0.468765i	$-0.155301\pi$
$347$	32.9090	1.76665	0.883323	+	0.468765i	$0.155301\pi$
$348$	0	0
$349$	14.0000	0.749403	0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000	0.749403	0.374701	+	0.927146i	$0.377745\pi$
$350$	3.46410	0.185164
$351$	0	0
$352$	9.00000	0.479702
$353$	32.9090	1.75157	0.875784	−	0.482704i	$-0.160345\pi$
$353$	32.9090	1.75157	0.875784	+	0.482704i	$0.160345\pi$
$354$	0	0
$355$	9.00000	0.477670
$356$	8.66025	0.458993
$357$	0	0
$358$	6.00000	0.317110
$359$	−17.3205	−0.914141	−0.457071	−	0.889430i	$-0.651101\pi$
$359$	−17.3205	−0.914141	−0.457071	+	0.889430i	$0.651101\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	−3.46410	−0.182069
$363$	0	0
$364$	2.00000	0.104828
$365$	6.92820	0.362639
$366$	0	0
$367$	−25.0000	−1.30499	−0.652495	−	0.757793i	$-0.726278\pi$
$367$	−25.0000	−1.30499	−0.652495	+	0.757793i	$0.726278\pi$
$368$	8.66025	0.451447
$369$	0	0
$370$	−21.0000	−1.09174
$371$	−13.8564	−0.719389
$372$	0	0
$373$	−19.0000	−0.983783	−0.491891	−	0.870657i	$-0.663694\pi$
$373$	−19.0000	−0.983783	−0.491891	+	0.870657i	$0.663694\pi$
$374$	−20.7846	−1.07475
$375$	0	0
$376$	−12.0000	−0.618853
$377$	20.7846	1.07046
$378$	0	0
$379$	8.00000	0.410932	0.205466	−	0.978664i	$-0.434129\pi$
$379$	8.00000	0.410932	0.205466	+	0.978664i	$0.434129\pi$
$380$	−8.66025	−0.444262
$381$	0	0
$382$	−15.0000	−0.767467
$383$	−38.1051	−1.94708	−0.973540	−	0.228515i	$-0.926613\pi$
$383$	−38.1051	−1.94708	−0.973540	+	0.228515i	$0.926613\pi$
$384$	0	0
$385$	−3.00000	−0.152894
$386$	38.1051	1.93950
$387$	0	0
$388$	−4.00000	−0.203069
$389$	6.92820	0.351274	0.175637	−	0.984455i	$-0.443802\pi$
$389$	6.92820	0.351274	0.175637	+	0.984455i	$0.443802\pi$
$390$	0	0
$391$	−12.0000	−0.606866
$392$	1.73205	0.0874818
$393$	0	0
$394$	−18.0000	−0.906827
$395$	−13.8564	−0.697191
$396$	0	0
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	43.3013	2.17050
$399$	0	0
$400$	10.0000	0.500000
$401$	3.46410	0.172989	0.0864945	−	0.996252i	$-0.472434\pi$
$401$	3.46410	0.172989	0.0864945	+	0.996252i	$0.472434\pi$
$402$	0	0
$403$	10.0000	0.498135
$404$	−13.8564	−0.689382
$405$	0	0
$406$	−18.0000	−0.893325
$407$	−12.1244	−0.600982
$408$	0	0
$409$	−4.00000	−0.197787	−0.0988936	−	0.995098i	$-0.531530\pi$
$409$	−4.00000	−0.197787	−0.0988936	+	0.995098i	$0.531530\pi$
$410$	−15.5885	−0.769859
$411$	0	0
$412$	5.00000	0.246332
$413$	6.92820	0.340915
$414$	0	0
$415$	−18.0000	−0.883585
$416$	10.3923	0.509525
$417$	0	0
$418$	−15.0000	−0.733674
$419$	−31.1769	−1.52309	−0.761546	−	0.648111i	$-0.775559\pi$
$419$	−31.1769	−1.52309	−0.761546	+	0.648111i	$0.775559\pi$
$420$	0	0
$421$	17.0000	0.828529	0.414265	−	0.910156i	$-0.364039\pi$
$421$	17.0000	0.828529	0.414265	+	0.910156i	$0.364039\pi$
$422$	−3.46410	−0.168630
$423$	0	0
$424$	−24.0000	−1.16554
$425$	−13.8564	−0.672134
$426$	0	0
$427$	8.00000	0.387147
$428$	3.46410	0.167444
$429$	0	0
$430$	−12.0000	−0.578691
$431$	22.5167	1.08459	0.542295	−	0.840188i	$-0.317556\pi$
$431$	22.5167	1.08459	0.542295	+	0.840188i	$0.317556\pi$
$432$	0	0
$433$	26.0000	1.24948	0.624740	−	0.780833i	$-0.285205\pi$
$433$	26.0000	1.24948	0.624740	+	0.780833i	$0.285205\pi$
$434$	−8.66025	−0.415705
$435$	0	0
$436$	−7.00000	−0.335239
$437$	−8.66025	−0.414276
$438$	0	0
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	−5.19615	−0.247717
$441$	0	0
$442$	−24.0000	−1.14156
$443$	19.0526	0.905214	0.452607	−	0.891710i	$-0.350494\pi$
$443$	19.0526	0.905214	0.452607	+	0.891710i	$0.350494\pi$
$444$	0	0
$445$	−15.0000	−0.711068
$446$	32.9090	1.55828
$447$	0	0
$448$	1.00000	0.0472456
$449$	−10.3923	−0.490443	−0.245222	−	0.969467i	$-0.578861\pi$
$449$	−10.3923	−0.490443	−0.245222	+	0.969467i	$0.578861\pi$
$450$	0	0
$451$	−9.00000	−0.423793
$452$	−10.3923	−0.488813
$453$	0	0
$454$	6.00000	0.281594
$455$	−3.46410	−0.162400
$456$	0	0
$457$	17.0000	0.795226	0.397613	−	0.917553i	$-0.369839\pi$
$457$	17.0000	0.795226	0.397613	+	0.917553i	$0.369839\pi$
$458$	6.92820	0.323734
$459$	0	0
$460$	3.00000	0.139876
$461$	−36.3731	−1.69406	−0.847031	−	0.531543i	$-0.821612\pi$
$461$	−36.3731	−1.69406	−0.847031	+	0.531543i	$0.821612\pi$
$462$	0	0
$463$	−22.0000	−1.02243	−0.511213	−	0.859454i	$-0.670804\pi$
$463$	−22.0000	−1.02243	−0.511213	+	0.859454i	$0.670804\pi$
$464$	−51.9615	−2.41225
$465$	0	0
$466$	30.0000	1.38972
$467$	13.8564	0.641198	0.320599	−	0.947215i	$-0.396116\pi$
$467$	13.8564	0.641198	0.320599	+	0.947215i	$0.396116\pi$
$468$	0	0
$469$	14.0000	0.646460
$470$	−20.7846	−0.958723
$471$	0	0
$472$	12.0000	0.552345
$473$	−6.92820	−0.318559
$474$	0	0
$475$	−10.0000	−0.458831
$476$	6.92820	0.317554
$477$	0	0
$478$	−18.0000	−0.823301
$479$	27.7128	1.26623	0.633115	−	0.774057i	$-0.281776\pi$
$479$	27.7128	1.26623	0.633115	+	0.774057i	$0.281776\pi$
$480$	0	0
$481$	−14.0000	−0.638345
$482$	−13.8564	−0.631142
$483$	0	0
$484$	−8.00000	−0.363636
$485$	6.92820	0.314594
$486$	0	0
$487$	−4.00000	−0.181257	−0.0906287	−	0.995885i	$-0.528888\pi$
$487$	−4.00000	−0.181257	−0.0906287	+	0.995885i	$0.528888\pi$
$488$	13.8564	0.627250
$489$	0	0
$490$	3.00000	0.135526
$491$	−36.3731	−1.64149	−0.820747	−	0.571292i	$-0.806442\pi$
$491$	−36.3731	−1.64149	−0.820747	+	0.571292i	$0.806442\pi$
$492$	0	0
$493$	72.0000	3.24272
$494$	−17.3205	−0.779287
$495$	0	0
$496$	−25.0000	−1.12253
$497$	−5.19615	−0.233079
$498$	0	0
$499$	32.0000	1.43252	0.716258	−	0.697835i	$-0.245853\pi$
$499$	32.0000	1.43252	0.716258	+	0.697835i	$0.245853\pi$
$500$	12.1244	0.542218
$501$	0	0
$502$	36.0000	1.60676
$503$	31.1769	1.39011	0.695055	−	0.718957i	$-0.255380\pi$
$503$	31.1769	1.39011	0.695055	+	0.718957i	$0.255380\pi$
$504$	0	0
$505$	24.0000	1.06799
$506$	5.19615	0.230997
$507$	0	0
$508$	−10.0000	−0.443678
$509$	13.8564	0.614174	0.307087	−	0.951681i	$-0.400646\pi$
$509$	13.8564	0.614174	0.307087	+	0.951681i	$0.400646\pi$
$510$	0	0
$511$	−4.00000	−0.176950
$512$	−8.66025	−0.382733
$513$	0	0
$514$	3.00000	0.132324
$515$	−8.66025	−0.381616
$516$	0	0
$517$	−12.0000	−0.527759
$518$	12.1244	0.532714
$519$	0	0
$520$	−6.00000	−0.263117
$521$	−39.8372	−1.74530	−0.872649	−	0.488348i	$-0.837600\pi$
$521$	−39.8372	−1.74530	−0.872649	+	0.488348i	$0.837600\pi$
$522$	0	0
$523$	11.0000	0.480996	0.240498	−	0.970650i	$-0.422689\pi$
$523$	11.0000	0.480996	0.240498	+	0.970650i	$0.422689\pi$
$524$	−6.92820	−0.302660
$525$	0	0
$526$	15.0000	0.654031
$527$	34.6410	1.50899
$528$	0	0
$529$	−20.0000	−0.869565
$530$	−41.5692	−1.80565
$531$	0	0
$532$	5.00000	0.216777
$533$	−10.3923	−0.450141
$534$	0	0
$535$	−6.00000	−0.259403
$536$	24.2487	1.04738
$537$	0	0
$538$	33.0000	1.42273
$539$	1.73205	0.0746047
$540$	0	0
$541$	−31.0000	−1.33279	−0.666397	−	0.745597i	$-0.732164\pi$
$541$	−31.0000	−1.33279	−0.666397	+	0.745597i	$0.732164\pi$
$542$	−34.6410	−1.48796
$543$	0	0
$544$	36.0000	1.54349
$545$	12.1244	0.519350
$546$	0	0
$547$	−40.0000	−1.71028	−0.855138	−	0.518400i	$-0.826528\pi$
$547$	−40.0000	−1.71028	−0.855138	+	0.518400i	$0.826528\pi$
$548$	−3.46410	−0.147979
$549$	0	0
$550$	6.00000	0.255841
$551$	51.9615	2.21364
$552$	0	0
$553$	8.00000	0.340195
$554$	32.9090	1.39817
$555$	0	0
$556$	20.0000	0.848189
$557$	17.3205	0.733893	0.366947	−	0.930242i	$-0.380403\pi$
$557$	17.3205	0.733893	0.366947	+	0.930242i	$0.380403\pi$
$558$	0	0
$559$	−8.00000	−0.338364
$560$	8.66025	0.365963
$561$	0	0
$562$	0	0
$563$	−3.46410	−0.145994	−0.0729972	−	0.997332i	$-0.523256\pi$
$563$	−3.46410	−0.145994	−0.0729972	+	0.997332i	$0.523256\pi$
$564$	0	0
$565$	18.0000	0.757266
$566$	6.92820	0.291214
$567$	0	0
$568$	−9.00000	−0.377632
$569$	24.2487	1.01656	0.508279	−	0.861192i	$-0.330282\pi$
$569$	24.2487	1.01656	0.508279	+	0.861192i	$0.330282\pi$
$570$	0	0
$571$	26.0000	1.08807	0.544033	−	0.839064i	$-0.316897\pi$
$571$	26.0000	1.08807	0.544033	+	0.839064i	$0.316897\pi$
$572$	3.46410	0.144841
$573$	0	0
$574$	9.00000	0.375653
$575$	3.46410	0.144463
$576$	0	0
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	−53.6936	−2.23336
$579$	0	0
$580$	−18.0000	−0.747409
$581$	10.3923	0.431145
$582$	0	0
$583$	−24.0000	−0.993978
$584$	−6.92820	−0.286691
$585$	0	0
$586$	0	0
$587$	31.1769	1.28681	0.643404	−	0.765526i	$-0.277521\pi$
$587$	31.1769	1.28681	0.643404	+	0.765526i	$0.277521\pi$
$588$	0	0
$589$	25.0000	1.03011
$590$	20.7846	0.855689
$591$	0	0
$592$	35.0000	1.43849
$593$	29.4449	1.20916	0.604578	−	0.796546i	$-0.293342\pi$
$593$	29.4449	1.20916	0.604578	+	0.796546i	$0.293342\pi$
$594$	0	0
$595$	−12.0000	−0.491952
$596$	3.46410	0.141895
$597$	0	0
$598$	6.00000	0.245358
$599$	1.73205	0.0707697	0.0353848	−	0.999374i	$-0.488734\pi$
$599$	1.73205	0.0707697	0.0353848	+	0.999374i	$0.488734\pi$
$600$	0	0
$601$	14.0000	0.571072	0.285536	−	0.958368i	$-0.407828\pi$
$601$	14.0000	0.571072	0.285536	+	0.958368i	$0.407828\pi$
$602$	6.92820	0.282372
$603$	0	0
$604$	−10.0000	−0.406894
$605$	13.8564	0.563343
$606$	0	0
$607$	−28.0000	−1.13648	−0.568242	−	0.822861i	$-0.692376\pi$
$607$	−28.0000	−1.13648	−0.568242	+	0.822861i	$0.692376\pi$
$608$	25.9808	1.05366
$609$	0	0
$610$	24.0000	0.971732
$611$	−13.8564	−0.560570
$612$	0	0
$613$	11.0000	0.444286	0.222143	−	0.975014i	$-0.428695\pi$
$613$	11.0000	0.444286	0.222143	+	0.975014i	$0.428695\pi$
$614$	−29.4449	−1.18830
$615$	0	0
$616$	3.00000	0.120873
$617$	−20.7846	−0.836757	−0.418378	−	0.908273i	$-0.637401\pi$
$617$	−20.7846	−0.836757	−0.418378	+	0.908273i	$0.637401\pi$
$618$	0	0
$619$	17.0000	0.683288	0.341644	−	0.939829i	$-0.389016\pi$
$619$	17.0000	0.683288	0.341644	+	0.939829i	$0.389016\pi$
$620$	−8.66025	−0.347804
$621$	0	0
$622$	6.00000	0.240578
$623$	8.66025	0.346966
$624$	0	0
$625$	−11.0000	−0.440000
$626$	−3.46410	−0.138453
$627$	0	0
$628$	−10.0000	−0.399043
$629$	−48.4974	−1.93372
$630$	0	0
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	13.8564	0.551178
$633$	0	0
$634$	12.0000	0.476581
$635$	17.3205	0.687343
$636$	0	0
$637$	2.00000	0.0792429
$638$	−31.1769	−1.23431
$639$	0	0
$640$	21.0000	0.830098
$641$	38.1051	1.50506	0.752531	−	0.658557i	$-0.228833\pi$
$641$	38.1051	1.50506	0.752531	+	0.658557i	$0.228833\pi$
$642$	0	0
$643$	−43.0000	−1.69575	−0.847877	−	0.530193i	$-0.822120\pi$
$643$	−43.0000	−1.69575	−0.847877	+	0.530193i	$0.822120\pi$
$644$	−1.73205	−0.0682524
$645$	0	0
$646$	−60.0000	−2.36067
$647$	−13.8564	−0.544752	−0.272376	−	0.962191i	$-0.587809\pi$
$647$	−13.8564	−0.544752	−0.272376	+	0.962191i	$0.587809\pi$
$648$	0	0
$649$	12.0000	0.471041
$650$	6.92820	0.271746
$651$	0	0
$652$	2.00000	0.0783260
$653$	13.8564	0.542243	0.271122	−	0.962545i	$-0.412605\pi$
$653$	13.8564	0.542243	0.271122	+	0.962545i	$0.412605\pi$
$654$	0	0
$655$	12.0000	0.468879
$656$	25.9808	1.01438
$657$	0	0
$658$	12.0000	0.467809
$659$	−25.9808	−1.01207	−0.506033	−	0.862514i	$-0.668889\pi$
$659$	−25.9808	−1.01207	−0.506033	+	0.862514i	$0.668889\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	−34.6410	−1.34636
$663$	0	0
$664$	18.0000	0.698535
$665$	−8.66025	−0.335830
$666$	0	0
$667$	−18.0000	−0.696963
$668$	10.3923	0.402090
$669$	0	0
$670$	42.0000	1.62260
$671$	13.8564	0.534921
$672$	0	0
$673$	−10.0000	−0.385472	−0.192736	−	0.981251i	$-0.561736\pi$
$673$	−10.0000	−0.385472	−0.192736	+	0.981251i	$0.561736\pi$
$674$	22.5167	0.867309
$675$	0	0
$676$	−9.00000	−0.346154
$677$	8.66025	0.332841	0.166420	−	0.986055i	$-0.446779\pi$
$677$	8.66025	0.332841	0.166420	+	0.986055i	$0.446779\pi$
$678$	0	0
$679$	−4.00000	−0.153506
$680$	−20.7846	−0.797053
$681$	0	0
$682$	−15.0000	−0.574380
$683$	−29.4449	−1.12668	−0.563338	−	0.826226i	$-0.690483\pi$
$683$	−29.4449	−1.12668	−0.563338	+	0.826226i	$0.690483\pi$
$684$	0	0
$685$	6.00000	0.229248
$686$	−1.73205	−0.0661300
$687$	0	0
$688$	20.0000	0.762493
$689$	−27.7128	−1.05577
$690$	0	0
$691$	8.00000	0.304334	0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000	0.304334	0.152167	+	0.988355i	$0.451375\pi$
$692$	−1.73205	−0.0658427
$693$	0	0
$694$	−57.0000	−2.16369
$695$	−34.6410	−1.31401
$696$	0	0
$697$	−36.0000	−1.36360
$698$	−24.2487	−0.917827
$699$	0	0
$700$	−2.00000	−0.0755929
$701$	−31.1769	−1.17754	−0.588768	−	0.808302i	$-0.700387\pi$
$701$	−31.1769	−1.17754	−0.588768	+	0.808302i	$0.700387\pi$
$702$	0	0
$703$	−35.0000	−1.32005
$704$	1.73205	0.0652791
$705$	0	0
$706$	−57.0000	−2.14522
$707$	−13.8564	−0.521124
$708$	0	0
$709$	−19.0000	−0.713560	−0.356780	−	0.934188i	$-0.616125\pi$
$709$	−19.0000	−0.713560	−0.356780	+	0.934188i	$0.616125\pi$
$710$	−15.5885	−0.585024
$711$	0	0
$712$	15.0000	0.562149
$713$	−8.66025	−0.324329
$714$	0	0
$715$	−6.00000	−0.224387
$716$	−3.46410	−0.129460
$717$	0	0
$718$	30.0000	1.11959
$719$	−6.92820	−0.258378	−0.129189	−	0.991620i	$-0.541237\pi$
$719$	−6.92820	−0.258378	−0.129189	+	0.991620i	$0.541237\pi$
$720$	0	0
$721$	5.00000	0.186210
$722$	−10.3923	−0.386762
$723$	0	0
$724$	2.00000	0.0743294
$725$	−20.7846	−0.771921
$726$	0	0
$727$	−40.0000	−1.48352	−0.741759	−	0.670667i	$-0.766008\pi$
$727$	−40.0000	−1.48352	−0.741759	+	0.670667i	$0.766008\pi$
$728$	3.46410	0.128388
$729$	0	0
$730$	−12.0000	−0.444140
$731$	−27.7128	−1.02500
$732$	0	0
$733$	32.0000	1.18195	0.590973	−	0.806691i	$-0.298744\pi$
$733$	32.0000	1.18195	0.590973	+	0.806691i	$0.298744\pi$
$734$	43.3013	1.59828
$735$	0	0
$736$	−9.00000	−0.331744
$737$	24.2487	0.893213
$738$	0	0
$739$	38.0000	1.39785	0.698926	−	0.715194i	$-0.253662\pi$
$739$	38.0000	1.39785	0.698926	+	0.715194i	$0.253662\pi$
$740$	12.1244	0.445700
$741$	0	0
$742$	24.0000	0.881068
$743$	15.5885	0.571885	0.285943	−	0.958247i	$-0.407693\pi$
$743$	15.5885	0.571885	0.285943	+	0.958247i	$0.407693\pi$
$744$	0	0
$745$	−6.00000	−0.219823
$746$	32.9090	1.20488
$747$	0	0
$748$	12.0000	0.438763
$749$	3.46410	0.126576
$750$	0	0
$751$	−28.0000	−1.02173	−0.510867	−	0.859660i	$-0.670676\pi$
$751$	−28.0000	−1.02173	−0.510867	+	0.859660i	$0.670676\pi$
$752$	34.6410	1.26323
$753$	0	0
$754$	−36.0000	−1.31104
$755$	17.3205	0.630358
$756$	0	0
$757$	−10.0000	−0.363456	−0.181728	−	0.983349i	$-0.558169\pi$
$757$	−10.0000	−0.363456	−0.181728	+	0.983349i	$0.558169\pi$
$758$	−13.8564	−0.503287
$759$	0	0
$760$	−15.0000	−0.544107
$761$	−27.7128	−1.00459	−0.502294	−	0.864697i	$-0.667511\pi$
$761$	−27.7128	−1.00459	−0.502294	+	0.864697i	$0.667511\pi$
$762$	0	0
$763$	−7.00000	−0.253417
$764$	8.66025	0.313317
$765$	0	0
$766$	66.0000	2.38468
$767$	13.8564	0.500326
$768$	0	0
$769$	−22.0000	−0.793340	−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000	−0.793340	−0.396670	+	0.917961i	$0.629834\pi$
$770$	5.19615	0.187256
$771$	0	0
$772$	−22.0000	−0.791797
$773$	43.3013	1.55744	0.778719	−	0.627373i	$-0.215870\pi$
$773$	43.3013	1.55744	0.778719	+	0.627373i	$0.215870\pi$
$774$	0	0
$775$	−10.0000	−0.359211
$776$	−6.92820	−0.248708
$777$	0	0
$778$	−12.0000	−0.430221
$779$	−25.9808	−0.930857
$780$	0	0
$781$	−9.00000	−0.322045
$782$	20.7846	0.743256
$783$	0	0
$784$	−5.00000	−0.178571
$785$	17.3205	0.618195
$786$	0	0
$787$	−4.00000	−0.142585	−0.0712923	−	0.997455i	$-0.522712\pi$
$787$	−4.00000	−0.142585	−0.0712923	+	0.997455i	$0.522712\pi$
$788$	10.3923	0.370211
$789$	0	0
$790$	24.0000	0.853882
$791$	−10.3923	−0.369508
$792$	0	0
$793$	16.0000	0.568177
$794$	−3.46410	−0.122936
$795$	0	0
$796$	−25.0000	−0.886102
$797$	−25.9808	−0.920286	−0.460143	−	0.887845i	$-0.652202\pi$
$797$	−25.9808	−0.920286	−0.460143	+	0.887845i	$0.652202\pi$
$798$	0	0
$799$	−48.0000	−1.69812
$800$	−10.3923	−0.367423
$801$	0	0
$802$	−6.00000	−0.211867
$803$	−6.92820	−0.244491
$804$	0	0
$805$	3.00000	0.105736
$806$	−17.3205	−0.610089
$807$	0	0
$808$	−24.0000	−0.844317
$809$	38.1051	1.33970	0.669852	−	0.742494i	$-0.266357\pi$
$809$	38.1051	1.33970	0.669852	+	0.742494i	$0.266357\pi$
$810$	0	0
$811$	−43.0000	−1.50993	−0.754967	−	0.655763i	$-0.772347\pi$
$811$	−43.0000	−1.50993	−0.754967	+	0.655763i	$0.772347\pi$
$812$	10.3923	0.364698
$813$	0	0
$814$	21.0000	0.736050
$815$	−3.46410	−0.121342
$816$	0	0
$817$	−20.0000	−0.699711
$818$	6.92820	0.242239
$819$	0	0
$820$	9.00000	0.314294
$821$	−6.92820	−0.241796	−0.120898	−	0.992665i	$-0.538577\pi$
$821$	−6.92820	−0.241796	−0.120898	+	0.992665i	$0.538577\pi$
$822$	0	0
$823$	14.0000	0.488009	0.244005	−	0.969774i	$-0.421539\pi$
$823$	14.0000	0.488009	0.244005	+	0.969774i	$0.421539\pi$
$824$	8.66025	0.301694
$825$	0	0
$826$	−12.0000	−0.417533
$827$	−5.19615	−0.180688	−0.0903440	−	0.995911i	$-0.528797\pi$
$827$	−5.19615	−0.180688	−0.0903440	+	0.995911i	$0.528797\pi$
$828$	0	0
$829$	32.0000	1.11141	0.555703	−	0.831381i	$-0.312449\pi$
$829$	32.0000	1.11141	0.555703	+	0.831381i	$0.312449\pi$
$830$	31.1769	1.08217
$831$	0	0
$832$	2.00000	0.0693375
$833$	6.92820	0.240048
$834$	0	0
$835$	−18.0000	−0.622916
$836$	8.66025	0.299521
$837$	0	0
$838$	54.0000	1.86540
$839$	−10.3923	−0.358782	−0.179391	−	0.983778i	$-0.557413\pi$
$839$	−10.3923	−0.358782	−0.179391	+	0.983778i	$0.557413\pi$
$840$	0	0
$841$	79.0000	2.72414
$842$	−29.4449	−1.01474
$843$	0	0
$844$	2.00000	0.0688428
$845$	15.5885	0.536259
$846$	0	0
$847$	−8.00000	−0.274883
$848$	69.2820	2.37915
$849$	0	0
$850$	24.0000	0.823193
$851$	12.1244	0.415618
$852$	0	0
$853$	44.0000	1.50653	0.753266	−	0.657716i	$-0.228477\pi$
$853$	44.0000	1.50653	0.753266	+	0.657716i	$0.228477\pi$
$854$	−13.8564	−0.474156
$855$	0	0
$856$	6.00000	0.205076
$857$	−39.8372	−1.36081	−0.680406	−	0.732835i	$-0.738196\pi$
$857$	−39.8372	−1.36081	−0.680406	+	0.732835i	$0.738196\pi$
$858$	0	0
$859$	41.0000	1.39890	0.699451	−	0.714681i	$-0.253428\pi$
$859$	41.0000	1.39890	0.699451	+	0.714681i	$0.253428\pi$
$860$	6.92820	0.236250
$861$	0	0
$862$	−39.0000	−1.32835
$863$	−17.3205	−0.589597	−0.294798	−	0.955559i	$-0.595253\pi$
$863$	−17.3205	−0.589597	−0.294798	+	0.955559i	$0.595253\pi$
$864$	0	0
$865$	3.00000	0.102003
$866$	−45.0333	−1.53029
$867$	0	0
$868$	5.00000	0.169711
$869$	13.8564	0.470046
$870$	0	0
$871$	28.0000	0.948744
$872$	−12.1244	−0.410582
$873$	0	0
$874$	15.0000	0.507383
$875$	12.1244	0.409878
$876$	0	0
$877$	−22.0000	−0.742887	−0.371444	−	0.928456i	$-0.621137\pi$
$877$	−22.0000	−0.742887	−0.371444	+	0.928456i	$0.621137\pi$
$878$	27.7128	0.935262
$879$	0	0
$880$	15.0000	0.505650
$881$	5.19615	0.175063	0.0875314	−	0.996162i	$-0.472102\pi$
$881$	5.19615	0.175063	0.0875314	+	0.996162i	$0.472102\pi$
$882$	0	0
$883$	56.0000	1.88455	0.942275	−	0.334840i	$-0.108682\pi$
$883$	56.0000	1.88455	0.942275	+	0.334840i	$0.108682\pi$
$884$	13.8564	0.466041
$885$	0	0
$886$	−33.0000	−1.10866
$887$	3.46410	0.116313	0.0581566	−	0.998307i	$-0.481478\pi$
$887$	3.46410	0.116313	0.0581566	+	0.998307i	$0.481478\pi$
$888$	0	0
$889$	−10.0000	−0.335389
$890$	25.9808	0.870877
$891$	0	0
$892$	−19.0000	−0.636167
$893$	−34.6410	−1.15922
$894$	0	0
$895$	6.00000	0.200558
$896$	−12.1244	−0.405046
$897$	0	0
$898$	18.0000	0.600668
$899$	51.9615	1.73301
$900$	0	0
$901$	−96.0000	−3.19822
$902$	15.5885	0.519039
$903$	0	0
$904$	−18.0000	−0.598671
$905$	−3.46410	−0.115151
$906$	0	0
$907$	−10.0000	−0.332045	−0.166022	−	0.986122i	$-0.553092\pi$
$907$	−10.0000	−0.332045	−0.166022	+	0.986122i	$0.553092\pi$
$908$	−3.46410	−0.114960
$909$	0	0
$910$	6.00000	0.198898
$911$	10.3923	0.344312	0.172156	−	0.985070i	$-0.444927\pi$
$911$	10.3923	0.344312	0.172156	+	0.985070i	$0.444927\pi$
$912$	0	0
$913$	18.0000	0.595713
$914$	−29.4449	−0.973950
$915$	0	0
$916$	−4.00000	−0.132164
$917$	−6.92820	−0.228789
$918$	0	0
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	5.19615	0.171312
$921$	0	0
$922$	63.0000	2.07479
$923$	−10.3923	−0.342067
$924$	0	0
$925$	14.0000	0.460317
$926$	38.1051	1.25221
$927$	0	0
$928$	54.0000	1.77264
$929$	−48.4974	−1.59115	−0.795574	−	0.605856i	$-0.792831\pi$
$929$	−48.4974	−1.59115	−0.795574	+	0.605856i	$0.792831\pi$
$930$	0	0
$931$	5.00000	0.163868
$932$	−17.3205	−0.567352
$933$	0	0
$934$	−24.0000	−0.785304
$935$	−20.7846	−0.679729
$936$	0	0
$937$	44.0000	1.43742	0.718709	−	0.695311i	$-0.244734\pi$
$937$	44.0000	1.43742	0.718709	+	0.695311i	$0.244734\pi$
$938$	−24.2487	−0.791748
$939$	0	0
$940$	12.0000	0.391397
$941$	32.9090	1.07280	0.536401	−	0.843963i	$-0.319784\pi$
$941$	32.9090	1.07280	0.536401	+	0.843963i	$0.319784\pi$
$942$	0	0
$943$	9.00000	0.293080
$944$	−34.6410	−1.12747
$945$	0	0
$946$	12.0000	0.390154
$947$	8.66025	0.281420	0.140710	−	0.990051i	$-0.455061\pi$
$947$	8.66025	0.281420	0.140710	+	0.990051i	$0.455061\pi$
$948$	0	0
$949$	−8.00000	−0.259691
$950$	17.3205	0.561951
$951$	0	0
$952$	12.0000	0.388922
$953$	10.3923	0.336640	0.168320	−	0.985732i	$-0.446166\pi$
$953$	10.3923	0.336640	0.168320	+	0.985732i	$0.446166\pi$
$954$	0	0
$955$	−15.0000	−0.485389
$956$	10.3923	0.336111
$957$	0	0
$958$	−48.0000	−1.55081
$959$	−3.46410	−0.111862
$960$	0	0
$961$	−6.00000	−0.193548
$962$	24.2487	0.781810
$963$	0	0
$964$	8.00000	0.257663
$965$	38.1051	1.22665
$966$	0	0
$967$	2.00000	0.0643157	0.0321578	−	0.999483i	$-0.489762\pi$
$967$	2.00000	0.0643157	0.0321578	+	0.999483i	$0.489762\pi$
$968$	−13.8564	−0.445362
$969$	0	0
$970$	−12.0000	−0.385297
$971$	−38.1051	−1.22285	−0.611426	−	0.791302i	$-0.709404\pi$
$971$	−38.1051	−1.22285	−0.611426	+	0.791302i	$0.709404\pi$
$972$	0	0
$973$	20.0000	0.641171
$974$	6.92820	0.221994
$975$	0	0
$976$	−40.0000	−1.28037
$977$	27.7128	0.886611	0.443306	−	0.896370i	$-0.353806\pi$
$977$	27.7128	0.886611	0.443306	+	0.896370i	$0.353806\pi$
$978$	0	0
$979$	15.0000	0.479402
$980$	−1.73205	−0.0553283
$981$	0	0
$982$	63.0000	2.01041
$983$	−55.4256	−1.76780	−0.883901	−	0.467673i	$-0.845092\pi$
$983$	−55.4256	−1.76780	−0.883901	+	0.467673i	$0.845092\pi$
$984$	0	0
$985$	−18.0000	−0.573528
$986$	−124.708	−3.97150
$987$	0	0
$988$	10.0000	0.318142
$989$	6.92820	0.220304
$990$	0	0
$991$	2.00000	0.0635321	0.0317660	−	0.999495i	$-0.489887\pi$
$991$	2.00000	0.0635321	0.0317660	+	0.999495i	$0.489887\pi$
$992$	25.9808	0.824890
$993$	0	0
$994$	9.00000	0.285463
$995$	43.3013	1.37274
$996$	0	0
$997$	26.0000	0.823428	0.411714	−	0.911313i	$-0.364930\pi$
$997$	26.0000	0.823428	0.411714	+	0.911313i	$0.364930\pi$
$998$	−55.4256	−1.75447
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.2.a.e.1.1	✓	2
3.2	odd	2	inner	189.2.a.e.1.2	yes	2
4.3	odd	2		3024.2.a.bg.1.1		2
5.4	even	2		4725.2.a.ba.1.2		2
7.6	odd	2		1323.2.a.t.1.1		2
9.2	odd	6		567.2.f.k.190.1		4
9.4	even	3		567.2.f.k.379.2		4
9.5	odd	6		567.2.f.k.379.1		4
9.7	even	3		567.2.f.k.190.2		4
12.11	even	2		3024.2.a.bg.1.2		2
15.14	odd	2		4725.2.a.ba.1.1		2
21.20	even	2		1323.2.a.t.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.a.e.1.1	✓	2	1.1	even	1	trivial
189.2.a.e.1.2	yes	2	3.2	odd	2	inner
567.2.f.k.190.1		4	9.2	odd	6
567.2.f.k.190.2		4	9.7	even	3
567.2.f.k.379.1		4	9.5	odd	6
567.2.f.k.379.2		4	9.4	even	3
1323.2.a.t.1.1		2	7.6	odd	2
1323.2.a.t.1.2		2	21.20	even	2
3024.2.a.bg.1.1		2	4.3	odd	2
3024.2.a.bg.1.2		2	12.11	even	2
4725.2.a.ba.1.1		2	15.14	odd	2
4725.2.a.ba.1.2		2	5.4	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:	\( N \)	\(=\)	\( 189 = 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	189.a (trivial)

\(f(q)\)	\(=\)	\(q-1.73205 q^{2} +1.00000 q^{4} -1.73205 q^{5} +1.00000 q^{7} +1.73205 q^{8} +O(q^{10})\)	\(q-1.73205 q^{2} +1.00000 q^{4} -1.73205 q^{5} +1.00000 q^{7} +1.73205 q^{8} +3.00000 q^{10} +1.73205 q^{11} +2.00000 q^{13} -1.73205 q^{14} -5.00000 q^{16} +6.92820 q^{17} +5.00000 q^{19} -1.73205 q^{20} -3.00000 q^{22} -1.73205 q^{23} -2.00000 q^{25} -3.46410 q^{26} +1.00000 q^{28} +10.3923 q^{29} +5.00000 q^{31} +5.19615 q^{32} -12.0000 q^{34} -1.73205 q^{35} -7.00000 q^{37} -8.66025 q^{38} -3.00000 q^{40} -5.19615 q^{41} -4.00000 q^{43} +1.73205 q^{44} +3.00000 q^{46} -6.92820 q^{47} +1.00000 q^{49} +3.46410 q^{50} +2.00000 q^{52} -13.8564 q^{53} -3.00000 q^{55} +1.73205 q^{56} -18.0000 q^{58} +6.92820 q^{59} +8.00000 q^{61} -8.66025 q^{62} +1.00000 q^{64} -3.46410 q^{65} +14.0000 q^{67} +6.92820 q^{68} +3.00000 q^{70} -5.19615 q^{71} -4.00000 q^{73} +12.1244 q^{74} +5.00000 q^{76} +1.73205 q^{77} +8.00000 q^{79} +8.66025 q^{80} +9.00000 q^{82} +10.3923 q^{83} -12.0000 q^{85} +6.92820 q^{86} +3.00000 q^{88} +8.66025 q^{89} +2.00000 q^{91} -1.73205 q^{92} +12.0000 q^{94} -8.66025 q^{95} -4.00000 q^{97} -1.73205 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} + 2 q^{7}+O(q^{10}) \) 2 * q + 2 * q^4 + 2 * q^7	\( 2 q + 2 q^{4} + 2 q^{7} + 6 q^{10} + 4 q^{13} - 10 q^{16} + 10 q^{19} - 6 q^{22} - 4 q^{25} + 2 q^{28} + 10 q^{31} - 24 q^{34} - 14 q^{37} - 6 q^{40} - 8 q^{43} + 6 q^{46} + 2 q^{49} + 4 q^{52} - 6 q^{55} - 36 q^{58} + 16 q^{61} + 2 q^{64} + 28 q^{67} + 6 q^{70} - 8 q^{73} + 10 q^{76} + 16 q^{79} + 18 q^{82} - 24 q^{85} + 6 q^{88} + 4 q^{91} + 24 q^{94} - 8 q^{97}+O(q^{100}) \) 2 * q + 2 * q^4 + 2 * q^7 + 6 * q^10 + 4 * q^13 - 10 * q^16 + 10 * q^19 - 6 * q^22 - 4 * q^25 + 2 * q^28 + 10 * q^31 - 24 * q^34 - 14 * q^37 - 6 * q^40 - 8 * q^43 + 6 * q^46 + 2 * q^49 + 4 * q^52 - 6 * q^55 - 36 * q^58 + 16 * q^61 + 2 * q^64 + 28 * q^67 + 6 * q^70 - 8 * q^73 + 10 * q^76 + 16 * q^79 + 18 * q^82 - 24 * q^85 + 6 * q^88 + 4 * q^91 + 24 * q^94 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more