LMFDB - Embedded newform 4641.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4641,2,Mod(1,4641)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4641, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4641.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4641 = 3 \cdot 7 \cdot 13 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4641.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:	$37.0585715781$
Analytic rank:	$1$
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	$\chi$	$=$	4641.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} +1.00000 q^{5} -2.00000 q^{6} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

\(q+2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} +1.00000 q^{5} -2.00000 q^{6} -1.00000 q^{7} +1.00000 q^{9} +2.00000 q^{10} +2.00000 q^{11} -2.00000 q^{12} +1.00000 q^{13} -2.00000 q^{14} -1.00000 q^{15} -4.00000 q^{16} +1.00000 q^{17} +2.00000 q^{18} -7.00000 q^{19} +2.00000 q^{20} +1.00000 q^{21} +4.00000 q^{22} -9.00000 q^{23} -4.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} -2.00000 q^{28} +3.00000 q^{29} -2.00000 q^{30} +3.00000 q^{31} -8.00000 q^{32} -2.00000 q^{33} +2.00000 q^{34} -1.00000 q^{35} +2.00000 q^{36} -2.00000 q^{37} -14.0000 q^{38} -1.00000 q^{39} -10.0000 q^{41} +2.00000 q^{42} +3.00000 q^{43} +4.00000 q^{44} +1.00000 q^{45} -18.0000 q^{46} +9.00000 q^{47} +4.00000 q^{48} +1.00000 q^{49} -8.00000 q^{50} -1.00000 q^{51} +2.00000 q^{52} +5.00000 q^{53} -2.00000 q^{54} +2.00000 q^{55} +7.00000 q^{57} +6.00000 q^{58} -12.0000 q^{59} -2.00000 q^{60} -10.0000 q^{61} +6.00000 q^{62} -1.00000 q^{63} -8.00000 q^{64} +1.00000 q^{65} -4.00000 q^{66} -8.00000 q^{67} +2.00000 q^{68} +9.00000 q^{69} -2.00000 q^{70} +2.00000 q^{71} +13.0000 q^{73} -4.00000 q^{74} +4.00000 q^{75} -14.0000 q^{76} -2.00000 q^{77} -2.00000 q^{78} +1.00000 q^{79} -4.00000 q^{80} +1.00000 q^{81} -20.0000 q^{82} +1.00000 q^{83} +2.00000 q^{84} +1.00000 q^{85} +6.00000 q^{86} -3.00000 q^{87} -11.0000 q^{89} +2.00000 q^{90} -1.00000 q^{91} -18.0000 q^{92} -3.00000 q^{93} +18.0000 q^{94} -7.00000 q^{95} +8.00000 q^{96} +17.0000 q^{97} +2.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

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 $

$2$	2.00000	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$2$	2.00000	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	2.00000	1.00000
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	−2.00000	−0.816497
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	2.00000	0.632456
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	−2.00000	−0.577350
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−2.00000	−0.534522
$15$	−1.00000	−0.258199
$16$	−4.00000	−1.00000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	2.00000	0.471405
$19$	−7.00000	−1.60591	−0.802955	−	0.596040i	$-0.796740\pi$
$19$	−7.00000	−1.60591	−0.802955	+	0.596040i	$0.796740\pi$
$20$	2.00000	0.447214
$21$	1.00000	0.218218
$22$	4.00000	0.852803
$23$	−9.00000	−1.87663	−0.938315	−	0.345782i	$-0.887614\pi$
$23$	−9.00000	−1.87663	−0.938315	+	0.345782i	$0.887614\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	2.00000	0.392232
$27$	−1.00000	−0.192450
$28$	−2.00000	−0.377964
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	−2.00000	−0.365148
$31$	3.00000	0.538816	0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000	0.538816	0.269408	+	0.963026i	$0.413172\pi$
$32$	−8.00000	−1.41421
$33$	−2.00000	−0.348155
$34$	2.00000	0.342997
$35$	−1.00000	−0.169031
$36$	2.00000	0.333333
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	−14.0000	−2.27110
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	2.00000	0.308607
$43$	3.00000	0.457496	0.228748	−	0.973486i	$-0.426537\pi$
$43$	3.00000	0.457496	0.228748	+	0.973486i	$0.426537\pi$
$44$	4.00000	0.603023
$45$	1.00000	0.149071
$46$	−18.0000	−2.65396
$47$	9.00000	1.31278	0.656392	−	0.754420i	$-0.272082\pi$
$47$	9.00000	1.31278	0.656392	+	0.754420i	$0.272082\pi$
$48$	4.00000	0.577350
$49$	1.00000	0.142857
$50$	−8.00000	−1.13137
$51$	−1.00000	−0.140028
$52$	2.00000	0.277350
$53$	5.00000	0.686803	0.343401	−	0.939189i	$-0.388421\pi$
$53$	5.00000	0.686803	0.343401	+	0.939189i	$0.388421\pi$
$54$	−2.00000	−0.272166
$55$	2.00000	0.269680
$56$	0	0
$57$	7.00000	0.927173
$58$	6.00000	0.787839
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	−2.00000	−0.258199
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	6.00000	0.762001
$63$	−1.00000	−0.125988
$64$	−8.00000	−1.00000
$65$	1.00000	0.124035
$66$	−4.00000	−0.492366
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	2.00000	0.242536
$69$	9.00000	1.08347
$70$	−2.00000	−0.239046
$71$	2.00000	0.237356	0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000	0.237356	0.118678	+	0.992933i	$0.462134\pi$
$72$	0	0
$73$	13.0000	1.52153	0.760767	−	0.649025i	$-0.224823\pi$
$73$	13.0000	1.52153	0.760767	+	0.649025i	$0.224823\pi$
$74$	−4.00000	−0.464991
$75$	4.00000	0.461880
$76$	−14.0000	−1.60591
$77$	−2.00000	−0.227921
$78$	−2.00000	−0.226455
$79$	1.00000	0.112509	0.0562544	−	0.998416i	$-0.482084\pi$
$79$	1.00000	0.112509	0.0562544	+	0.998416i	$0.482084\pi$
$80$	−4.00000	−0.447214
$81$	1.00000	0.111111
$82$	−20.0000	−2.20863
$83$	1.00000	0.109764	0.0548821	−	0.998493i	$-0.482522\pi$
$83$	1.00000	0.109764	0.0548821	+	0.998493i	$0.482522\pi$
$84$	2.00000	0.218218
$85$	1.00000	0.108465
$86$	6.00000	0.646997
$87$	−3.00000	−0.321634
$88$	0	0
$89$	−11.0000	−1.16600	−0.582999	−	0.812473i	$-0.698121\pi$
$89$	−11.0000	−1.16600	−0.582999	+	0.812473i	$0.698121\pi$
$90$	2.00000	0.210819
$91$	−1.00000	−0.104828
$92$	−18.0000	−1.87663
$93$	−3.00000	−0.311086
$94$	18.0000	1.85656
$95$	−7.00000	−0.718185
$96$	8.00000	0.816497
$97$	17.0000	1.72609	0.863044	−	0.505128i	$-0.168555\pi$
$97$	17.0000	1.72609	0.863044	+	0.505128i	$0.168555\pi$
$98$	2.00000	0.202031
$99$	2.00000	0.201008
$100$	−8.00000	−0.800000
$101$	12.0000	1.19404	0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000	1.19404	0.597022	+	0.802225i	$0.296350\pi$
$102$	−2.00000	−0.198030
$103$	6.00000	0.591198	0.295599	−	0.955312i	$-0.404481\pi$
$103$	6.00000	0.591198	0.295599	+	0.955312i	$0.404481\pi$
$104$	0	0
$105$	1.00000	0.0975900
$106$	10.0000	0.971286
$107$	−20.0000	−1.93347	−0.966736	−	0.255774i	$-0.917670\pi$
$107$	−20.0000	−1.93347	−0.966736	+	0.255774i	$0.917670\pi$
$108$	−2.00000	−0.192450
$109$	−6.00000	−0.574696	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000	−0.574696	−0.287348	+	0.957826i	$0.592774\pi$
$110$	4.00000	0.381385
$111$	2.00000	0.189832
$112$	4.00000	0.377964
$113$	−15.0000	−1.41108	−0.705541	−	0.708669i	$-0.749296\pi$
$113$	−15.0000	−1.41108	−0.705541	+	0.708669i	$0.749296\pi$
$114$	14.0000	1.31122
$115$	−9.00000	−0.839254
$116$	6.00000	0.557086
$117$	1.00000	0.0924500
$118$	−24.0000	−2.20938
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−20.0000	−1.81071
$123$	10.0000	0.901670
$124$	6.00000	0.538816
$125$	−9.00000	−0.804984
$126$	−2.00000	−0.178174
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	0	0
$129$	−3.00000	−0.264135
$130$	2.00000	0.175412
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	−4.00000	−0.348155
$133$	7.00000	0.606977
$134$	−16.0000	−1.38219
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	18.0000	1.53226
$139$	6.00000	0.508913	0.254457	−	0.967084i	$-0.418103\pi$
$139$	6.00000	0.508913	0.254457	+	0.967084i	$0.418103\pi$
$140$	−2.00000	−0.169031
$141$	−9.00000	−0.757937
$142$	4.00000	0.335673
$143$	2.00000	0.167248
$144$	−4.00000	−0.333333
$145$	3.00000	0.249136
$146$	26.0000	2.15178
$147$	−1.00000	−0.0824786
$148$	−4.00000	−0.328798
$149$	22.0000	1.80231	0.901155	−	0.433497i	$-0.142720\pi$
$149$	22.0000	1.80231	0.901155	+	0.433497i	$0.142720\pi$
$150$	8.00000	0.653197
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	0	0
$153$	1.00000	0.0808452
$154$	−4.00000	−0.322329
$155$	3.00000	0.240966
$156$	−2.00000	−0.160128
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	2.00000	0.159111
$159$	−5.00000	−0.396526
$160$	−8.00000	−0.632456
$161$	9.00000	0.709299
$162$	2.00000	0.157135
$163$	−6.00000	−0.469956	−0.234978	−	0.972001i	$-0.575502\pi$
$163$	−6.00000	−0.469956	−0.234978	+	0.972001i	$0.575502\pi$
$164$	−20.0000	−1.56174
$165$	−2.00000	−0.155700
$166$	2.00000	0.155230
$167$	−23.0000	−1.77979	−0.889897	−	0.456162i	$-0.849224\pi$
$167$	−23.0000	−1.77979	−0.889897	+	0.456162i	$0.849224\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	2.00000	0.153393
$171$	−7.00000	−0.535303
$172$	6.00000	0.457496
$173$	4.00000	0.304114	0.152057	−	0.988372i	$-0.451410\pi$
$173$	4.00000	0.304114	0.152057	+	0.988372i	$0.451410\pi$
$174$	−6.00000	−0.454859
$175$	4.00000	0.302372
$176$	−8.00000	−0.603023
$177$	12.0000	0.901975
$178$	−22.0000	−1.64897
$179$	5.00000	0.373718	0.186859	−	0.982387i	$-0.440169\pi$
$179$	5.00000	0.373718	0.186859	+	0.982387i	$0.440169\pi$
$180$	2.00000	0.149071
$181$	−12.0000	−0.891953	−0.445976	−	0.895045i	$-0.647144\pi$
$181$	−12.0000	−0.891953	−0.445976	+	0.895045i	$0.647144\pi$
$182$	−2.00000	−0.148250
$183$	10.0000	0.739221
$184$	0	0
$185$	−2.00000	−0.147043
$186$	−6.00000	−0.439941
$187$	2.00000	0.146254
$188$	18.0000	1.31278
$189$	1.00000	0.0727393
$190$	−14.0000	−1.01567
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	8.00000	0.577350
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	34.0000	2.44106
$195$	−1.00000	−0.0716115
$196$	2.00000	0.142857
$197$	10.0000	0.712470	0.356235	−	0.934396i	$-0.384060\pi$
$197$	10.0000	0.712470	0.356235	+	0.934396i	$0.384060\pi$
$198$	4.00000	0.284268
$199$	2.00000	0.141776	0.0708881	−	0.997484i	$-0.477417\pi$
$199$	2.00000	0.141776	0.0708881	+	0.997484i	$0.477417\pi$
$200$	0	0
$201$	8.00000	0.564276
$202$	24.0000	1.68863
$203$	−3.00000	−0.210559
$204$	−2.00000	−0.140028
$205$	−10.0000	−0.698430
$206$	12.0000	0.836080
$207$	−9.00000	−0.625543
$208$	−4.00000	−0.277350
$209$	−14.0000	−0.968400
$210$	2.00000	0.138013
$211$	5.00000	0.344214	0.172107	−	0.985078i	$-0.444942\pi$
$211$	5.00000	0.344214	0.172107	+	0.985078i	$0.444942\pi$
$212$	10.0000	0.686803
$213$	−2.00000	−0.137038
$214$	−40.0000	−2.73434
$215$	3.00000	0.204598
$216$	0	0
$217$	−3.00000	−0.203653
$218$	−12.0000	−0.812743
$219$	−13.0000	−0.878459
$220$	4.00000	0.269680
$221$	1.00000	0.0672673
$222$	4.00000	0.268462
$223$	−13.0000	−0.870544	−0.435272	−	0.900299i	$-0.643348\pi$
$223$	−13.0000	−0.870544	−0.435272	+	0.900299i	$0.643348\pi$
$224$	8.00000	0.534522
$225$	−4.00000	−0.266667
$226$	−30.0000	−1.99557
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	14.0000	0.927173
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	−18.0000	−1.18688
$231$	2.00000	0.131590
$232$	0	0
$233$	−17.0000	−1.11371	−0.556854	−	0.830611i	$-0.687992\pi$
$233$	−17.0000	−1.11371	−0.556854	+	0.830611i	$0.687992\pi$
$234$	2.00000	0.130744
$235$	9.00000	0.587095
$236$	−24.0000	−1.56227
$237$	−1.00000	−0.0649570
$238$	−2.00000	−0.129641
$239$	−26.0000	−1.68180	−0.840900	−	0.541190i	$-0.817974\pi$
$239$	−26.0000	−1.68180	−0.840900	+	0.541190i	$0.817974\pi$
$240$	4.00000	0.258199
$241$	25.0000	1.61039	0.805196	−	0.593009i	$-0.202060\pi$
$241$	25.0000	1.61039	0.805196	+	0.593009i	$0.202060\pi$
$242$	−14.0000	−0.899954
$243$	−1.00000	−0.0641500
$244$	−20.0000	−1.28037
$245$	1.00000	0.0638877
$246$	20.0000	1.27515
$247$	−7.00000	−0.445399
$248$	0	0
$249$	−1.00000	−0.0633724
$250$	−18.0000	−1.13842
$251$	14.0000	0.883672	0.441836	−	0.897096i	$-0.354327\pi$
$251$	14.0000	0.883672	0.441836	+	0.897096i	$0.354327\pi$
$252$	−2.00000	−0.125988
$253$	−18.0000	−1.13165
$254$	−32.0000	−2.00786
$255$	−1.00000	−0.0626224
$256$	16.0000	1.00000
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	−6.00000	−0.373544
$259$	2.00000	0.124274
$260$	2.00000	0.124035
$261$	3.00000	0.185695
$262$	0	0
$263$	11.0000	0.678289	0.339145	−	0.940734i	$-0.389862\pi$
$263$	11.0000	0.678289	0.339145	+	0.940734i	$0.389862\pi$
$264$	0	0
$265$	5.00000	0.307148
$266$	14.0000	0.858395
$267$	11.0000	0.673189
$268$	−16.0000	−0.977356
$269$	26.0000	1.58525	0.792624	−	0.609711i	$-0.208714\pi$
$269$	26.0000	1.58525	0.792624	+	0.609711i	$0.208714\pi$
$270$	−2.00000	−0.121716
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	−4.00000	−0.242536
$273$	1.00000	0.0605228
$274$	−36.0000	−2.17484
$275$	−8.00000	−0.482418
$276$	18.0000	1.08347
$277$	19.0000	1.14160	0.570800	−	0.821089i	$-0.306633\pi$
$277$	19.0000	1.14160	0.570800	+	0.821089i	$0.306633\pi$
$278$	12.0000	0.719712
$279$	3.00000	0.179605
$280$	0	0
$281$	4.00000	0.238620	0.119310	−	0.992857i	$-0.461932\pi$
$281$	4.00000	0.238620	0.119310	+	0.992857i	$0.461932\pi$
$282$	−18.0000	−1.07188
$283$	−14.0000	−0.832214	−0.416107	−	0.909316i	$-0.636606\pi$
$283$	−14.0000	−0.832214	−0.416107	+	0.909316i	$0.636606\pi$
$284$	4.00000	0.237356
$285$	7.00000	0.414644
$286$	4.00000	0.236525
$287$	10.0000	0.590281
$288$	−8.00000	−0.471405
$289$	1.00000	0.0588235
$290$	6.00000	0.352332
$291$	−17.0000	−0.996558
$292$	26.0000	1.52153
$293$	19.0000	1.10999	0.554996	−	0.831853i	$-0.312720\pi$
$293$	19.0000	1.10999	0.554996	+	0.831853i	$0.312720\pi$
$294$	−2.00000	−0.116642
$295$	−12.0000	−0.698667
$296$	0	0
$297$	−2.00000	−0.116052
$298$	44.0000	2.54885
$299$	−9.00000	−0.520483
$300$	8.00000	0.461880
$301$	−3.00000	−0.172917
$302$	−20.0000	−1.15087
$303$	−12.0000	−0.689382
$304$	28.0000	1.60591
$305$	−10.0000	−0.572598
$306$	2.00000	0.114332
$307$	7.00000	0.399511	0.199756	−	0.979846i	$-0.435985\pi$
$307$	7.00000	0.399511	0.199756	+	0.979846i	$0.435985\pi$
$308$	−4.00000	−0.227921
$309$	−6.00000	−0.341328
$310$	6.00000	0.340777
$311$	18.0000	1.02069	0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000	1.02069	0.510343	+	0.859971i	$0.329518\pi$
$312$	0	0
$313$	−2.00000	−0.113047	−0.0565233	−	0.998401i	$-0.518002\pi$
$313$	−2.00000	−0.113047	−0.0565233	+	0.998401i	$0.518002\pi$
$314$	−20.0000	−1.12867
$315$	−1.00000	−0.0563436
$316$	2.00000	0.112509
$317$	−24.0000	−1.34797	−0.673987	−	0.738743i	$-0.735420\pi$
$317$	−24.0000	−1.34797	−0.673987	+	0.738743i	$0.735420\pi$
$318$	−10.0000	−0.560772
$319$	6.00000	0.335936
$320$	−8.00000	−0.447214
$321$	20.0000	1.11629
$322$	18.0000	1.00310
$323$	−7.00000	−0.389490
$324$	2.00000	0.111111
$325$	−4.00000	−0.221880
$326$	−12.0000	−0.664619
$327$	6.00000	0.331801
$328$	0	0
$329$	−9.00000	−0.496186
$330$	−4.00000	−0.220193
$331$	−26.0000	−1.42909	−0.714545	−	0.699590i	$-0.753366\pi$
$331$	−26.0000	−1.42909	−0.714545	+	0.699590i	$0.753366\pi$
$332$	2.00000	0.109764
$333$	−2.00000	−0.109599
$334$	−46.0000	−2.51701
$335$	−8.00000	−0.437087
$336$	−4.00000	−0.218218
$337$	−21.0000	−1.14394	−0.571971	−	0.820274i	$-0.693821\pi$
$337$	−21.0000	−1.14394	−0.571971	+	0.820274i	$0.693821\pi$
$338$	2.00000	0.108786
$339$	15.0000	0.814688
$340$	2.00000	0.108465
$341$	6.00000	0.324918
$342$	−14.0000	−0.757033
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	9.00000	0.484544
$346$	8.00000	0.430083
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	−6.00000	−0.321634
$349$	31.0000	1.65939	0.829696	−	0.558216i	$-0.188514\pi$
$349$	31.0000	1.65939	0.829696	+	0.558216i	$0.188514\pi$
$350$	8.00000	0.427618
$351$	−1.00000	−0.0533761
$352$	−16.0000	−0.852803
$353$	26.0000	1.38384	0.691920	−	0.721974i	$-0.256765\pi$
$353$	26.0000	1.38384	0.691920	+	0.721974i	$0.256765\pi$
$354$	24.0000	1.27559
$355$	2.00000	0.106149
$356$	−22.0000	−1.16600
$357$	1.00000	0.0529256
$358$	10.0000	0.528516
$359$	2.00000	0.105556	0.0527780	−	0.998606i	$-0.483192\pi$
$359$	2.00000	0.105556	0.0527780	+	0.998606i	$0.483192\pi$
$360$	0	0
$361$	30.0000	1.57895
$362$	−24.0000	−1.26141
$363$	7.00000	0.367405
$364$	−2.00000	−0.104828
$365$	13.0000	0.680451
$366$	20.0000	1.04542
$367$	6.00000	0.313197	0.156599	−	0.987662i	$-0.449947\pi$
$367$	6.00000	0.313197	0.156599	+	0.987662i	$0.449947\pi$
$368$	36.0000	1.87663
$369$	−10.0000	−0.520579
$370$	−4.00000	−0.207950
$371$	−5.00000	−0.259587
$372$	−6.00000	−0.311086
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	4.00000	0.206835
$375$	9.00000	0.464758
$376$	0	0
$377$	3.00000	0.154508
$378$	2.00000	0.102869
$379$	−10.0000	−0.513665	−0.256833	−	0.966456i	$-0.582679\pi$
$379$	−10.0000	−0.513665	−0.256833	+	0.966456i	$0.582679\pi$
$380$	−14.0000	−0.718185
$381$	16.0000	0.819705
$382$	24.0000	1.22795
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	0	0
$385$	−2.00000	−0.101929
$386$	−4.00000	−0.203595
$387$	3.00000	0.152499
$388$	34.0000	1.72609
$389$	10.0000	0.507020	0.253510	−	0.967333i	$-0.418415\pi$
$389$	10.0000	0.507020	0.253510	+	0.967333i	$0.418415\pi$
$390$	−2.00000	−0.101274
$391$	−9.00000	−0.455150
$392$	0	0
$393$	0	0
$394$	20.0000	1.00759
$395$	1.00000	0.0503155
$396$	4.00000	0.201008
$397$	−3.00000	−0.150566	−0.0752828	−	0.997162i	$-0.523986\pi$
$397$	−3.00000	−0.150566	−0.0752828	+	0.997162i	$0.523986\pi$
$398$	4.00000	0.200502
$399$	−7.00000	−0.350438
$400$	16.0000	0.800000
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	16.0000	0.798007
$403$	3.00000	0.149441
$404$	24.0000	1.19404
$405$	1.00000	0.0496904
$406$	−6.00000	−0.297775
$407$	−4.00000	−0.198273
$408$	0	0
$409$	−13.0000	−0.642809	−0.321404	−	0.946942i	$-0.604155\pi$
$409$	−13.0000	−0.642809	−0.321404	+	0.946942i	$0.604155\pi$
$410$	−20.0000	−0.987730
$411$	18.0000	0.887875
$412$	12.0000	0.591198
$413$	12.0000	0.590481
$414$	−18.0000	−0.884652
$415$	1.00000	0.0490881
$416$	−8.00000	−0.392232
$417$	−6.00000	−0.293821
$418$	−28.0000	−1.36952
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	2.00000	0.0975900
$421$	−4.00000	−0.194948	−0.0974740	−	0.995238i	$-0.531076\pi$
$421$	−4.00000	−0.194948	−0.0974740	+	0.995238i	$0.531076\pi$
$422$	10.0000	0.486792
$423$	9.00000	0.437595
$424$	0	0
$425$	−4.00000	−0.194029
$426$	−4.00000	−0.193801
$427$	10.0000	0.483934
$428$	−40.0000	−1.93347
$429$	−2.00000	−0.0965609
$430$	6.00000	0.289346
$431$	−8.00000	−0.385346	−0.192673	−	0.981263i	$-0.561716\pi$
$431$	−8.00000	−0.385346	−0.192673	+	0.981263i	$0.561716\pi$
$432$	4.00000	0.192450
$433$	6.00000	0.288342	0.144171	−	0.989553i	$-0.453949\pi$
$433$	6.00000	0.288342	0.144171	+	0.989553i	$0.453949\pi$
$434$	−6.00000	−0.288009
$435$	−3.00000	−0.143839
$436$	−12.0000	−0.574696
$437$	63.0000	3.01370
$438$	−26.0000	−1.24233
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	2.00000	0.0951303
$443$	25.0000	1.18779	0.593893	−	0.804544i	$-0.297590\pi$
$443$	25.0000	1.18779	0.593893	+	0.804544i	$0.297590\pi$
$444$	4.00000	0.189832
$445$	−11.0000	−0.521450
$446$	−26.0000	−1.23114
$447$	−22.0000	−1.04056
$448$	8.00000	0.377964
$449$	10.0000	0.471929	0.235965	−	0.971762i	$-0.424175\pi$
$449$	10.0000	0.471929	0.235965	+	0.971762i	$0.424175\pi$
$450$	−8.00000	−0.377124
$451$	−20.0000	−0.941763
$452$	−30.0000	−1.41108
$453$	10.0000	0.469841
$454$	−24.0000	−1.12638
$455$	−1.00000	−0.0468807
$456$	0	0
$457$	22.0000	1.02912	0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000	1.02912	0.514558	+	0.857455i	$0.327956\pi$
$458$	12.0000	0.560723
$459$	−1.00000	−0.0466760
$460$	−18.0000	−0.839254
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	4.00000	0.186097
$463$	22.0000	1.02243	0.511213	−	0.859454i	$-0.329196\pi$
$463$	22.0000	1.02243	0.511213	+	0.859454i	$0.329196\pi$
$464$	−12.0000	−0.557086
$465$	−3.00000	−0.139122
$466$	−34.0000	−1.57502
$467$	−10.0000	−0.462745	−0.231372	−	0.972865i	$-0.574322\pi$
$467$	−10.0000	−0.462745	−0.231372	+	0.972865i	$0.574322\pi$
$468$	2.00000	0.0924500
$469$	8.00000	0.369406
$470$	18.0000	0.830278
$471$	10.0000	0.460776
$472$	0	0
$473$	6.00000	0.275880
$474$	−2.00000	−0.0918630
$475$	28.0000	1.28473
$476$	−2.00000	−0.0916698
$477$	5.00000	0.228934
$478$	−52.0000	−2.37842
$479$	33.0000	1.50781	0.753904	−	0.656984i	$-0.228168\pi$
$479$	33.0000	1.50781	0.753904	+	0.656984i	$0.228168\pi$
$480$	8.00000	0.365148
$481$	−2.00000	−0.0911922
$482$	50.0000	2.27744
$483$	−9.00000	−0.409514
$484$	−14.0000	−0.636364
$485$	17.0000	0.771930
$486$	−2.00000	−0.0907218
$487$	38.0000	1.72194	0.860972	−	0.508652i	$-0.169856\pi$
$487$	38.0000	1.72194	0.860972	+	0.508652i	$0.169856\pi$
$488$	0	0
$489$	6.00000	0.271329
$490$	2.00000	0.0903508
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	20.0000	0.901670
$493$	3.00000	0.135113
$494$	−14.0000	−0.629890
$495$	2.00000	0.0898933
$496$	−12.0000	−0.538816
$497$	−2.00000	−0.0897123
$498$	−2.00000	−0.0896221
$499$	−24.0000	−1.07439	−0.537194	−	0.843459i	$-0.680516\pi$
$499$	−24.0000	−1.07439	−0.537194	+	0.843459i	$0.680516\pi$
$500$	−18.0000	−0.804984
$501$	23.0000	1.02756
$502$	28.0000	1.24970
$503$	−44.0000	−1.96186	−0.980932	−	0.194354i	$-0.937739\pi$
$503$	−44.0000	−1.96186	−0.980932	+	0.194354i	$0.937739\pi$
$504$	0	0
$505$	12.0000	0.533993
$506$	−36.0000	−1.60040
$507$	−1.00000	−0.0444116
$508$	−32.0000	−1.41977
$509$	−29.0000	−1.28540	−0.642701	−	0.766117i	$-0.722186\pi$
$509$	−29.0000	−1.28540	−0.642701	+	0.766117i	$0.722186\pi$
$510$	−2.00000	−0.0885615
$511$	−13.0000	−0.575086
$512$	32.0000	1.41421
$513$	7.00000	0.309058
$514$	4.00000	0.176432
$515$	6.00000	0.264392
$516$	−6.00000	−0.264135
$517$	18.0000	0.791639
$518$	4.00000	0.175750
$519$	−4.00000	−0.175581
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	6.00000	0.262613
$523$	−14.0000	−0.612177	−0.306089	−	0.952003i	$-0.599020\pi$
$523$	−14.0000	−0.612177	−0.306089	+	0.952003i	$0.599020\pi$
$524$	0	0
$525$	−4.00000	−0.174574
$526$	22.0000	0.959246
$527$	3.00000	0.130682
$528$	8.00000	0.348155
$529$	58.0000	2.52174
$530$	10.0000	0.434372
$531$	−12.0000	−0.520756
$532$	14.0000	0.606977
$533$	−10.0000	−0.433148
$534$	22.0000	0.952033
$535$	−20.0000	−0.864675
$536$	0	0
$537$	−5.00000	−0.215766
$538$	52.0000	2.24188
$539$	2.00000	0.0861461
$540$	−2.00000	−0.0860663
$541$	0	0		−	1.00000i	$-0.5\pi$
$541$	0	0			1.00000i	$0.5\pi$
$542$	40.0000	1.71815
$543$	12.0000	0.514969
$544$	−8.00000	−0.342997
$545$	−6.00000	−0.257012
$546$	2.00000	0.0855921
$547$	−5.00000	−0.213785	−0.106892	−	0.994271i	$-0.534090\pi$
$547$	−5.00000	−0.213785	−0.106892	+	0.994271i	$0.534090\pi$
$548$	−36.0000	−1.53784
$549$	−10.0000	−0.426790
$550$	−16.0000	−0.682242
$551$	−21.0000	−0.894630
$552$	0	0
$553$	−1.00000	−0.0425243
$554$	38.0000	1.61447
$555$	2.00000	0.0848953
$556$	12.0000	0.508913
$557$	−26.0000	−1.10166	−0.550828	−	0.834619i	$-0.685688\pi$
$557$	−26.0000	−1.10166	−0.550828	+	0.834619i	$0.685688\pi$
$558$	6.00000	0.254000
$559$	3.00000	0.126886
$560$	4.00000	0.169031
$561$	−2.00000	−0.0844401
$562$	8.00000	0.337460
$563$	26.0000	1.09577	0.547885	−	0.836554i	$-0.315433\pi$
$563$	26.0000	1.09577	0.547885	+	0.836554i	$0.315433\pi$
$564$	−18.0000	−0.757937
$565$	−15.0000	−0.631055
$566$	−28.0000	−1.17693
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	13.0000	0.544988	0.272494	−	0.962157i	$-0.412151\pi$
$569$	13.0000	0.544988	0.272494	+	0.962157i	$0.412151\pi$
$570$	14.0000	0.586395
$571$	−27.0000	−1.12991	−0.564957	−	0.825120i	$-0.691107\pi$
$571$	−27.0000	−1.12991	−0.564957	+	0.825120i	$0.691107\pi$
$572$	4.00000	0.167248
$573$	−12.0000	−0.501307
$574$	20.0000	0.834784
$575$	36.0000	1.50130
$576$	−8.00000	−0.333333
$577$	30.0000	1.24892	0.624458	−	0.781058i	$-0.285320\pi$
$577$	30.0000	1.24892	0.624458	+	0.781058i	$0.285320\pi$
$578$	2.00000	0.0831890
$579$	2.00000	0.0831172
$580$	6.00000	0.249136
$581$	−1.00000	−0.0414870
$582$	−34.0000	−1.40935
$583$	10.0000	0.414158
$584$	0	0
$585$	1.00000	0.0413449
$586$	38.0000	1.56977
$587$	5.00000	0.206372	0.103186	−	0.994662i	$-0.467096\pi$
$587$	5.00000	0.206372	0.103186	+	0.994662i	$0.467096\pi$
$588$	−2.00000	−0.0824786
$589$	−21.0000	−0.865290
$590$	−24.0000	−0.988064
$591$	−10.0000	−0.411345
$592$	8.00000	0.328798
$593$	39.0000	1.60154	0.800769	−	0.598973i	$-0.204424\pi$
$593$	39.0000	1.60154	0.800769	+	0.598973i	$0.204424\pi$
$594$	−4.00000	−0.164122
$595$	−1.00000	−0.0409960
$596$	44.0000	1.80231
$597$	−2.00000	−0.0818546
$598$	−18.0000	−0.736075
$599$	33.0000	1.34834	0.674172	−	0.738575i	$-0.264501\pi$
$599$	33.0000	1.34834	0.674172	+	0.738575i	$0.264501\pi$
$600$	0	0
$601$	−36.0000	−1.46847	−0.734235	−	0.678895i	$-0.762459\pi$
$601$	−36.0000	−1.46847	−0.734235	+	0.678895i	$0.762459\pi$
$602$	−6.00000	−0.244542
$603$	−8.00000	−0.325785
$604$	−20.0000	−0.813788
$605$	−7.00000	−0.284590
$606$	−24.0000	−0.974933
$607$	−28.0000	−1.13648	−0.568242	−	0.822861i	$-0.692376\pi$
$607$	−28.0000	−1.13648	−0.568242	+	0.822861i	$0.692376\pi$
$608$	56.0000	2.27110
$609$	3.00000	0.121566
$610$	−20.0000	−0.809776
$611$	9.00000	0.364101
$612$	2.00000	0.0808452
$613$	−14.0000	−0.565455	−0.282727	−	0.959200i	$-0.591239\pi$
$613$	−14.0000	−0.565455	−0.282727	+	0.959200i	$0.591239\pi$
$614$	14.0000	0.564994
$615$	10.0000	0.403239
$616$	0	0
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	−12.0000	−0.482711
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	6.00000	0.240966
$621$	9.00000	0.361158
$622$	36.0000	1.44347
$623$	11.0000	0.440706
$624$	4.00000	0.160128
$625$	11.0000	0.440000
$626$	−4.00000	−0.159872
$627$	14.0000	0.559106
$628$	−20.0000	−0.798087
$629$	−2.00000	−0.0797452
$630$	−2.00000	−0.0796819
$631$	32.0000	1.27390	0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000	1.27390	0.636950	+	0.770905i	$0.280196\pi$
$632$	0	0
$633$	−5.00000	−0.198732
$634$	−48.0000	−1.90632
$635$	−16.0000	−0.634941
$636$	−10.0000	−0.396526
$637$	1.00000	0.0396214
$638$	12.0000	0.475085
$639$	2.00000	0.0791188
$640$	0	0
$641$	−35.0000	−1.38242	−0.691208	−	0.722655i	$-0.742921\pi$
$641$	−35.0000	−1.38242	−0.691208	+	0.722655i	$0.742921\pi$
$642$	40.0000	1.57867
$643$	4.00000	0.157745	0.0788723	−	0.996885i	$-0.474868\pi$
$643$	4.00000	0.157745	0.0788723	+	0.996885i	$0.474868\pi$
$644$	18.0000	0.709299
$645$	−3.00000	−0.118125
$646$	−14.0000	−0.550823
$647$	−16.0000	−0.629025	−0.314512	−	0.949253i	$-0.601841\pi$
$647$	−16.0000	−0.629025	−0.314512	+	0.949253i	$0.601841\pi$
$648$	0	0
$649$	−24.0000	−0.942082
$650$	−8.00000	−0.313786
$651$	3.00000	0.117579
$652$	−12.0000	−0.469956
$653$	18.0000	0.704394	0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000	0.704394	0.352197	+	0.935926i	$0.385435\pi$
$654$	12.0000	0.469237
$655$	0	0
$656$	40.0000	1.56174
$657$	13.0000	0.507178
$658$	−18.0000	−0.701713
$659$	−45.0000	−1.75295	−0.876476	−	0.481446i	$-0.840112\pi$
$659$	−45.0000	−1.75295	−0.876476	+	0.481446i	$0.840112\pi$
$660$	−4.00000	−0.155700
$661$	−25.0000	−0.972387	−0.486194	−	0.873851i	$-0.661615\pi$
$661$	−25.0000	−0.972387	−0.486194	+	0.873851i	$0.661615\pi$
$662$	−52.0000	−2.02104
$663$	−1.00000	−0.0388368
$664$	0	0
$665$	7.00000	0.271448
$666$	−4.00000	−0.154997
$667$	−27.0000	−1.04544
$668$	−46.0000	−1.77979
$669$	13.0000	0.502609
$670$	−16.0000	−0.618134
$671$	−20.0000	−0.772091
$672$	−8.00000	−0.308607
$673$	19.0000	0.732396	0.366198	−	0.930537i	$-0.380659\pi$
$673$	19.0000	0.732396	0.366198	+	0.930537i	$0.380659\pi$
$674$	−42.0000	−1.61778
$675$	4.00000	0.153960
$676$	2.00000	0.0769231
$677$	28.0000	1.07613	0.538064	−	0.842904i	$-0.319156\pi$
$677$	28.0000	1.07613	0.538064	+	0.842904i	$0.319156\pi$
$678$	30.0000	1.15214
$679$	−17.0000	−0.652400
$680$	0	0
$681$	12.0000	0.459841
$682$	12.0000	0.459504
$683$	−10.0000	−0.382639	−0.191320	−	0.981528i	$-0.561277\pi$
$683$	−10.0000	−0.382639	−0.191320	+	0.981528i	$0.561277\pi$
$684$	−14.0000	−0.535303
$685$	−18.0000	−0.687745
$686$	−2.00000	−0.0763604
$687$	−6.00000	−0.228914
$688$	−12.0000	−0.457496
$689$	5.00000	0.190485
$690$	18.0000	0.685248
$691$	−15.0000	−0.570627	−0.285313	−	0.958434i	$-0.592098\pi$
$691$	−15.0000	−0.570627	−0.285313	+	0.958434i	$0.592098\pi$
$692$	8.00000	0.304114
$693$	−2.00000	−0.0759737
$694$	−24.0000	−0.911028
$695$	6.00000	0.227593
$696$	0	0
$697$	−10.0000	−0.378777
$698$	62.0000	2.34673
$699$	17.0000	0.642999
$700$	8.00000	0.302372
$701$	−21.0000	−0.793159	−0.396580	−	0.918000i	$-0.629803\pi$
$701$	−21.0000	−0.793159	−0.396580	+	0.918000i	$0.629803\pi$
$702$	−2.00000	−0.0754851
$703$	14.0000	0.528020
$704$	−16.0000	−0.603023
$705$	−9.00000	−0.338960
$706$	52.0000	1.95705
$707$	−12.0000	−0.451306
$708$	24.0000	0.901975
$709$	−26.0000	−0.976450	−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000	−0.976450	−0.488225	+	0.872718i	$0.662356\pi$
$710$	4.00000	0.150117
$711$	1.00000	0.0375029
$712$	0	0
$713$	−27.0000	−1.01116
$714$	2.00000	0.0748481
$715$	2.00000	0.0747958
$716$	10.0000	0.373718
$717$	26.0000	0.970988
$718$	4.00000	0.149279
$719$	−48.0000	−1.79010	−0.895049	−	0.445968i	$-0.852860\pi$
$719$	−48.0000	−1.79010	−0.895049	+	0.445968i	$0.852860\pi$
$720$	−4.00000	−0.149071
$721$	−6.00000	−0.223452
$722$	60.0000	2.23297
$723$	−25.0000	−0.929760
$724$	−24.0000	−0.891953
$725$	−12.0000	−0.445669
$726$	14.0000	0.519589
$727$	−40.0000	−1.48352	−0.741759	−	0.670667i	$-0.766008\pi$
$727$	−40.0000	−1.48352	−0.741759	+	0.670667i	$0.766008\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	26.0000	0.962303
$731$	3.00000	0.110959
$732$	20.0000	0.739221
$733$	−21.0000	−0.775653	−0.387826	−	0.921732i	$-0.626774\pi$
$733$	−21.0000	−0.775653	−0.387826	+	0.921732i	$0.626774\pi$
$734$	12.0000	0.442928
$735$	−1.00000	−0.0368856
$736$	72.0000	2.65396
$737$	−16.0000	−0.589368
$738$	−20.0000	−0.736210
$739$	−2.00000	−0.0735712	−0.0367856	−	0.999323i	$-0.511712\pi$
$739$	−2.00000	−0.0735712	−0.0367856	+	0.999323i	$0.511712\pi$
$740$	−4.00000	−0.147043
$741$	7.00000	0.257151
$742$	−10.0000	−0.367112
$743$	−28.0000	−1.02722	−0.513610	−	0.858024i	$-0.671692\pi$
$743$	−28.0000	−1.02722	−0.513610	+	0.858024i	$0.671692\pi$
$744$	0	0
$745$	22.0000	0.806018
$746$	52.0000	1.90386
$747$	1.00000	0.0365881
$748$	4.00000	0.146254
$749$	20.0000	0.730784
$750$	18.0000	0.657267
$751$	−23.0000	−0.839282	−0.419641	−	0.907690i	$-0.637844\pi$
$751$	−23.0000	−0.839282	−0.419641	+	0.907690i	$0.637844\pi$
$752$	−36.0000	−1.31278
$753$	−14.0000	−0.510188
$754$	6.00000	0.218507
$755$	−10.0000	−0.363937
$756$	2.00000	0.0727393
$757$	−31.0000	−1.12671	−0.563357	−	0.826214i	$-0.690490\pi$
$757$	−31.0000	−1.12671	−0.563357	+	0.826214i	$0.690490\pi$
$758$	−20.0000	−0.726433
$759$	18.0000	0.653359
$760$	0	0
$761$	7.00000	0.253750	0.126875	−	0.991919i	$-0.459505\pi$
$761$	7.00000	0.253750	0.126875	+	0.991919i	$0.459505\pi$
$762$	32.0000	1.15924
$763$	6.00000	0.217215
$764$	24.0000	0.868290
$765$	1.00000	0.0361551
$766$	48.0000	1.73431
$767$	−12.0000	−0.433295
$768$	−16.0000	−0.577350
$769$	−39.0000	−1.40638	−0.703188	−	0.711004i	$-0.748241\pi$
$769$	−39.0000	−1.40638	−0.703188	+	0.711004i	$0.748241\pi$
$770$	−4.00000	−0.144150
$771$	−2.00000	−0.0720282
$772$	−4.00000	−0.143963
$773$	−6.00000	−0.215805	−0.107903	−	0.994161i	$-0.534413\pi$
$773$	−6.00000	−0.215805	−0.107903	+	0.994161i	$0.534413\pi$
$774$	6.00000	0.215666
$775$	−12.0000	−0.431053
$776$	0	0
$777$	−2.00000	−0.0717496
$778$	20.0000	0.717035
$779$	70.0000	2.50801
$780$	−2.00000	−0.0716115
$781$	4.00000	0.143131
$782$	−18.0000	−0.643679
$783$	−3.00000	−0.107211
$784$	−4.00000	−0.142857
$785$	−10.0000	−0.356915
$786$	0	0
$787$	−13.0000	−0.463400	−0.231700	−	0.972787i	$-0.574429\pi$
$787$	−13.0000	−0.463400	−0.231700	+	0.972787i	$0.574429\pi$
$788$	20.0000	0.712470
$789$	−11.0000	−0.391610
$790$	2.00000	0.0711568
$791$	15.0000	0.533339
$792$	0	0
$793$	−10.0000	−0.355110
$794$	−6.00000	−0.212932
$795$	−5.00000	−0.177332
$796$	4.00000	0.141776
$797$	42.0000	1.48772	0.743858	−	0.668338i	$-0.232994\pi$
$797$	42.0000	1.48772	0.743858	+	0.668338i	$0.232994\pi$
$798$	−14.0000	−0.495595
$799$	9.00000	0.318397
$800$	32.0000	1.13137
$801$	−11.0000	−0.388666
$802$	−36.0000	−1.27120
$803$	26.0000	0.917520
$804$	16.0000	0.564276
$805$	9.00000	0.317208
$806$	6.00000	0.211341
$807$	−26.0000	−0.915243
$808$	0	0
$809$	33.0000	1.16022	0.580109	−	0.814539i	$-0.303010\pi$
$809$	33.0000	1.16022	0.580109	+	0.814539i	$0.303010\pi$
$810$	2.00000	0.0702728
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	−6.00000	−0.210559
$813$	−20.0000	−0.701431
$814$	−8.00000	−0.280400
$815$	−6.00000	−0.210171
$816$	4.00000	0.140028
$817$	−21.0000	−0.734697
$818$	−26.0000	−0.909069
$819$	−1.00000	−0.0349428
$820$	−20.0000	−0.698430
$821$	8.00000	0.279202	0.139601	−	0.990208i	$-0.455418\pi$
$821$	8.00000	0.279202	0.139601	+	0.990208i	$0.455418\pi$
$822$	36.0000	1.25564
$823$	32.0000	1.11545	0.557725	−	0.830026i	$-0.311674\pi$
$823$	32.0000	1.11545	0.557725	+	0.830026i	$0.311674\pi$
$824$	0	0
$825$	8.00000	0.278524
$826$	24.0000	0.835067
$827$	32.0000	1.11275	0.556375	−	0.830932i	$-0.312192\pi$
$827$	32.0000	1.11275	0.556375	+	0.830932i	$0.312192\pi$
$828$	−18.0000	−0.625543
$829$	6.00000	0.208389	0.104194	−	0.994557i	$-0.466774\pi$
$829$	6.00000	0.208389	0.104194	+	0.994557i	$0.466774\pi$
$830$	2.00000	0.0694210
$831$	−19.0000	−0.659103
$832$	−8.00000	−0.277350
$833$	1.00000	0.0346479
$834$	−12.0000	−0.415526
$835$	−23.0000	−0.795948
$836$	−28.0000	−0.968400
$837$	−3.00000	−0.103695
$838$	24.0000	0.829066
$839$	16.0000	0.552381	0.276191	−	0.961103i	$-0.410928\pi$
$839$	16.0000	0.552381	0.276191	+	0.961103i	$0.410928\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	−8.00000	−0.275698
$843$	−4.00000	−0.137767
$844$	10.0000	0.344214
$845$	1.00000	0.0344010
$846$	18.0000	0.618853
$847$	7.00000	0.240523
$848$	−20.0000	−0.686803
$849$	14.0000	0.480479
$850$	−8.00000	−0.274398
$851$	18.0000	0.617032
$852$	−4.00000	−0.137038
$853$	−25.0000	−0.855984	−0.427992	−	0.903783i	$-0.640779\pi$
$853$	−25.0000	−0.855984	−0.427992	+	0.903783i	$0.640779\pi$
$854$	20.0000	0.684386
$855$	−7.00000	−0.239395
$856$	0	0
$857$	28.0000	0.956462	0.478231	−	0.878234i	$-0.341278\pi$
$857$	28.0000	0.956462	0.478231	+	0.878234i	$0.341278\pi$
$858$	−4.00000	−0.136558
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	6.00000	0.204598
$861$	−10.0000	−0.340799
$862$	−16.0000	−0.544962
$863$	30.0000	1.02121	0.510606	−	0.859815i	$-0.329421\pi$
$863$	30.0000	1.02121	0.510606	+	0.859815i	$0.329421\pi$
$864$	8.00000	0.272166
$865$	4.00000	0.136004
$866$	12.0000	0.407777
$867$	−1.00000	−0.0339618
$868$	−6.00000	−0.203653
$869$	2.00000	0.0678454
$870$	−6.00000	−0.203419
$871$	−8.00000	−0.271070
$872$	0	0
$873$	17.0000	0.575363
$874$	126.000	4.26201
$875$	9.00000	0.304256
$876$	−26.0000	−0.878459
$877$	−2.00000	−0.0675352	−0.0337676	−	0.999430i	$-0.510751\pi$
$877$	−2.00000	−0.0675352	−0.0337676	+	0.999430i	$0.510751\pi$
$878$	−16.0000	−0.539974
$879$	−19.0000	−0.640854
$880$	−8.00000	−0.269680
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	2.00000	0.0673435
$883$	28.0000	0.942275	0.471138	−	0.882060i	$-0.343844\pi$
$883$	28.0000	0.942275	0.471138	+	0.882060i	$0.343844\pi$
$884$	2.00000	0.0672673
$885$	12.0000	0.403376
$886$	50.0000	1.67978
$887$	−30.0000	−1.00730	−0.503651	−	0.863907i	$-0.668010\pi$
$887$	−30.0000	−1.00730	−0.503651	+	0.863907i	$0.668010\pi$
$888$	0	0
$889$	16.0000	0.536623
$890$	−22.0000	−0.737442
$891$	2.00000	0.0670025
$892$	−26.0000	−0.870544
$893$	−63.0000	−2.10821
$894$	−44.0000	−1.47158
$895$	5.00000	0.167132
$896$	0	0
$897$	9.00000	0.300501
$898$	20.0000	0.667409
$899$	9.00000	0.300167
$900$	−8.00000	−0.266667
$901$	5.00000	0.166574
$902$	−40.0000	−1.33185
$903$	3.00000	0.0998337
$904$	0	0
$905$	−12.0000	−0.398893
$906$	20.0000	0.664455
$907$	−27.0000	−0.896520	−0.448260	−	0.893903i	$-0.647956\pi$
$907$	−27.0000	−0.896520	−0.448260	+	0.893903i	$0.647956\pi$
$908$	−24.0000	−0.796468
$909$	12.0000	0.398015
$910$	−2.00000	−0.0662994
$911$	−3.00000	−0.0993944	−0.0496972	−	0.998764i	$-0.515826\pi$
$911$	−3.00000	−0.0993944	−0.0496972	+	0.998764i	$0.515826\pi$
$912$	−28.0000	−0.927173
$913$	2.00000	0.0661903
$914$	44.0000	1.45539
$915$	10.0000	0.330590
$916$	12.0000	0.396491
$917$	0	0
$918$	−2.00000	−0.0660098
$919$	40.0000	1.31948	0.659739	−	0.751495i	$-0.270667\pi$
$919$	40.0000	1.31948	0.659739	+	0.751495i	$0.270667\pi$
$920$	0	0
$921$	−7.00000	−0.230658
$922$	12.0000	0.395199
$923$	2.00000	0.0658308
$924$	4.00000	0.131590
$925$	8.00000	0.263038
$926$	44.0000	1.44593
$927$	6.00000	0.197066
$928$	−24.0000	−0.787839
$929$	−39.0000	−1.27955	−0.639774	−	0.768563i	$-0.720972\pi$
$929$	−39.0000	−1.27955	−0.639774	+	0.768563i	$0.720972\pi$
$930$	−6.00000	−0.196748
$931$	−7.00000	−0.229416
$932$	−34.0000	−1.11371
$933$	−18.0000	−0.589294
$934$	−20.0000	−0.654420
$935$	2.00000	0.0654070
$936$	0	0
$937$	38.0000	1.24141	0.620703	−	0.784046i	$-0.286847\pi$
$937$	38.0000	1.24141	0.620703	+	0.784046i	$0.286847\pi$
$938$	16.0000	0.522419
$939$	2.00000	0.0652675
$940$	18.0000	0.587095
$941$	−57.0000	−1.85815	−0.929073	−	0.369895i	$-0.879394\pi$
$941$	−57.0000	−1.85815	−0.929073	+	0.369895i	$0.879394\pi$
$942$	20.0000	0.651635
$943$	90.0000	2.93080
$944$	48.0000	1.56227
$945$	1.00000	0.0325300
$946$	12.0000	0.390154
$947$	−32.0000	−1.03986	−0.519930	−	0.854209i	$-0.674042\pi$
$947$	−32.0000	−1.03986	−0.519930	+	0.854209i	$0.674042\pi$
$948$	−2.00000	−0.0649570
$949$	13.0000	0.421998
$950$	56.0000	1.81688
$951$	24.0000	0.778253
$952$	0	0
$953$	47.0000	1.52248	0.761240	−	0.648471i	$-0.224591\pi$
$953$	47.0000	1.52248	0.761240	+	0.648471i	$0.224591\pi$
$954$	10.0000	0.323762
$955$	12.0000	0.388311
$956$	−52.0000	−1.68180
$957$	−6.00000	−0.193952
$958$	66.0000	2.13236
$959$	18.0000	0.581250
$960$	8.00000	0.258199
$961$	−22.0000	−0.709677
$962$	−4.00000	−0.128965
$963$	−20.0000	−0.644491
$964$	50.0000	1.61039
$965$	−2.00000	−0.0643823
$966$	−18.0000	−0.579141
$967$	2.00000	0.0643157	0.0321578	−	0.999483i	$-0.489762\pi$
$967$	2.00000	0.0643157	0.0321578	+	0.999483i	$0.489762\pi$
$968$	0	0
$969$	7.00000	0.224872
$970$	34.0000	1.09167
$971$	62.0000	1.98967	0.994837	−	0.101482i	$-0.0323585\pi$
$971$	62.0000	1.98967	0.994837	+	0.101482i	$0.0323585\pi$
$972$	−2.00000	−0.0641500
$973$	−6.00000	−0.192351
$974$	76.0000	2.43520
$975$	4.00000	0.128103
$976$	40.0000	1.28037
$977$	10.0000	0.319928	0.159964	−	0.987123i	$-0.448862\pi$
$977$	10.0000	0.319928	0.159964	+	0.987123i	$0.448862\pi$
$978$	12.0000	0.383718
$979$	−22.0000	−0.703123
$980$	2.00000	0.0638877
$981$	−6.00000	−0.191565
$982$	−40.0000	−1.27645
$983$	33.0000	1.05254	0.526268	−	0.850319i	$-0.323591\pi$
$983$	33.0000	1.05254	0.526268	+	0.850319i	$0.323591\pi$
$984$	0	0
$985$	10.0000	0.318626
$986$	6.00000	0.191079
$987$	9.00000	0.286473
$988$	−14.0000	−0.445399
$989$	−27.0000	−0.858550
$990$	4.00000	0.127128
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	−24.0000	−0.762001
$993$	26.0000	0.825085
$994$	−4.00000	−0.126872
$995$	2.00000	0.0634043
$996$	−2.00000	−0.0633724
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	−48.0000	−1.51941
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4641.2.a.f.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4641.2.a.f.1.1	✓	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:	\( N \)	\(=\)	\( 4641 = 3 \cdot 7 \cdot 13 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4641.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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