LMFDB - Embedded newform 3920.2.a.bn.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3920,2,Mod(1,3920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3920, 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("3920.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3920 = 2^{4} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3920.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:	$31.3013575923$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1960)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	3920.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.414214 q^{3} -1.00000 q^{5} -2.82843 q^{9} +O(q^{10})$	$q+0.414214 q^{3} -1.00000 q^{5} -2.82843 q^{9} +1.00000 q^{11} +2.41421 q^{13} -0.414214 q^{15} -0.414214 q^{17} -2.00000 q^{19} +2.24264 q^{23} +1.00000 q^{25} -2.41421 q^{27} +1.00000 q^{29} -1.75736 q^{31} +0.414214 q^{33} -7.89949 q^{37} +1.00000 q^{39} +7.41421 q^{41} +0.343146 q^{43} +2.82843 q^{45} -10.4142 q^{47} -0.171573 q^{51} +9.41421 q^{53} -1.00000 q^{55} -0.828427 q^{57} -10.2426 q^{59} -1.17157 q^{61} -2.41421 q^{65} -1.41421 q^{67} +0.928932 q^{69} -14.4853 q^{71} -5.17157 q^{73} +0.414214 q^{75} +14.6569 q^{79} +7.48528 q^{81} -11.3137 q^{83} +0.414214 q^{85} +0.414214 q^{87} -1.65685 q^{89} -0.727922 q^{93} +2.00000 q^{95} -0.0710678 q^{97} -2.82843 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{5}+O(q^{10}) $ 2 * q - 2 * q^3 - 2 * q^5	$ 2 q - 2 q^{3} - 2 q^{5} + 2 q^{11} + 2 q^{13} + 2 q^{15} + 2 q^{17} - 4 q^{19} - 4 q^{23} + 2 q^{25} - 2 q^{27} + 2 q^{29} - 12 q^{31} - 2 q^{33} + 4 q^{37} + 2 q^{39} + 12 q^{41} + 12 q^{43} - 18 q^{47} - 6 q^{51} + 16 q^{53} - 2 q^{55} + 4 q^{57} - 12 q^{59} - 8 q^{61} - 2 q^{65} + 16 q^{69} - 12 q^{71} - 16 q^{73} - 2 q^{75} + 18 q^{79} - 2 q^{81} - 2 q^{85} - 2 q^{87} + 8 q^{89} + 24 q^{93} + 4 q^{95} + 14 q^{97}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^11 + 2 * q^13 + 2 * q^15 + 2 * q^17 - 4 * q^19 - 4 * q^23 + 2 * q^25 - 2 * q^27 + 2 * q^29 - 12 * q^31 - 2 * q^33 + 4 * q^37 + 2 * q^39 + 12 * q^41 + 12 * q^43 - 18 * q^47 - 6 * q^51 + 16 * q^53 - 2 * q^55 + 4 * q^57 - 12 * q^59 - 8 * q^61 - 2 * q^65 + 16 * q^69 - 12 * q^71 - 16 * q^73 - 2 * q^75 + 18 * q^79 - 2 * q^81 - 2 * q^85 - 2 * q^87 + 8 * q^89 + 24 * q^93 + 4 * q^95 + 14 * 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 $

$2$	0	0
$2$	0	0
$3$	0.414214	0.239146	0.119573	−	0.992825i	$-0.461847\pi$
$3$	0.414214	0.239146	0.119573	+	0.992825i	$0.461847\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.82843	−0.942809
$10$	0	0
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	0	0
$13$	2.41421	0.669582	0.334791	−	0.942292i	$-0.391334\pi$
$13$	2.41421	0.669582	0.334791	+	0.942292i	$0.391334\pi$
$14$	0	0
$15$	−0.414214	−0.106949
$16$	0	0
$17$	−0.414214	−0.100462	−0.0502308	−	0.998738i	$-0.515996\pi$
$17$	−0.414214	−0.100462	−0.0502308	+	0.998738i	$0.515996\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.24264	0.467623	0.233811	−	0.972282i	$-0.424880\pi$
$23$	2.24264	0.467623	0.233811	+	0.972282i	$0.424880\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−2.41421	−0.464616
$28$	0	0
$29$	1.00000	0.185695	0.0928477	−	0.995680i	$-0.470403\pi$
$29$	1.00000	0.185695	0.0928477	+	0.995680i	$0.470403\pi$
$30$	0	0
$31$	−1.75736	−0.315631	−0.157816	−	0.987469i	$-0.550445\pi$
$31$	−1.75736	−0.315631	−0.157816	+	0.987469i	$0.550445\pi$
$32$	0	0
$33$	0.414214	0.0721053
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−7.89949	−1.29867	−0.649334	−	0.760503i	$-0.724952\pi$
$37$	−7.89949	−1.29867	−0.649334	+	0.760503i	$0.724952\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	7.41421	1.15791	0.578953	−	0.815361i	$-0.303462\pi$
$41$	7.41421	1.15791	0.578953	+	0.815361i	$0.303462\pi$
$42$	0	0
$43$	0.343146	0.0523292	0.0261646	−	0.999658i	$-0.491671\pi$
$43$	0.343146	0.0523292	0.0261646	+	0.999658i	$0.491671\pi$
$44$	0	0
$45$	2.82843	0.421637
$46$	0	0
$47$	−10.4142	−1.51907	−0.759535	−	0.650467i	$-0.774573\pi$
$47$	−10.4142	−1.51907	−0.759535	+	0.650467i	$0.774573\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−0.171573	−0.0240250
$52$	0	0
$53$	9.41421	1.29314	0.646571	−	0.762854i	$-0.276202\pi$
$53$	9.41421	1.29314	0.646571	+	0.762854i	$0.276202\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	−0.828427	−0.109728
$58$	0	0
$59$	−10.2426	−1.33348	−0.666739	−	0.745291i	$-0.732310\pi$
$59$	−10.2426	−1.33348	−0.666739	+	0.745291i	$0.732310\pi$
$60$	0	0
$61$	−1.17157	−0.150005	−0.0750023	−	0.997183i	$-0.523896\pi$
$61$	−1.17157	−0.150005	−0.0750023	+	0.997183i	$0.523896\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.41421	−0.299446
$66$	0	0
$67$	−1.41421	−0.172774	−0.0863868	−	0.996262i	$-0.527532\pi$
$67$	−1.41421	−0.172774	−0.0863868	+	0.996262i	$0.527532\pi$
$68$	0	0
$69$	0.928932	0.111830
$70$	0	0
$71$	−14.4853	−1.71909	−0.859543	−	0.511063i	$-0.829252\pi$
$71$	−14.4853	−1.71909	−0.859543	+	0.511063i	$0.829252\pi$
$72$	0	0
$73$	−5.17157	−0.605287	−0.302643	−	0.953104i	$-0.597869\pi$
$73$	−5.17157	−0.605287	−0.302643	+	0.953104i	$0.597869\pi$
$74$	0	0
$75$	0.414214	0.0478293
$76$	0	0
$77$	0	0
$78$	0	0
$79$	14.6569	1.64902	0.824512	−	0.565844i	$-0.191449\pi$
$79$	14.6569	1.64902	0.824512	+	0.565844i	$0.191449\pi$
$80$	0	0
$81$	7.48528	0.831698
$82$	0	0
$83$	−11.3137	−1.24184	−0.620920	−	0.783874i	$-0.713241\pi$
$83$	−11.3137	−1.24184	−0.620920	+	0.783874i	$0.713241\pi$
$84$	0	0
$85$	0.414214	0.0449278
$86$	0	0
$87$	0.414214	0.0444084
$88$	0	0
$89$	−1.65685	−0.175626	−0.0878131	−	0.996137i	$-0.527988\pi$
$89$	−1.65685	−0.175626	−0.0878131	+	0.996137i	$0.527988\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−0.727922	−0.0754820
$94$	0	0
$95$	2.00000	0.205196
$96$	0	0
$97$	−0.0710678	−0.00721584	−0.00360792	−	0.999993i	$-0.501148\pi$
$97$	−0.0710678	−0.00721584	−0.00360792	+	0.999993i	$0.501148\pi$
$98$	0	0
$99$	−2.82843	−0.284268
$100$	0	0
$101$	−6.48528	−0.645310	−0.322655	−	0.946517i	$-0.604575\pi$
$101$	−6.48528	−0.645310	−0.322655	+	0.946517i	$0.604575\pi$
$102$	0	0
$103$	−15.2426	−1.50190	−0.750951	−	0.660358i	$-0.770405\pi$
$103$	−15.2426	−1.50190	−0.750951	+	0.660358i	$0.770405\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.48528	−0.240261	−0.120131	−	0.992758i	$-0.538331\pi$
$107$	−2.48528	−0.240261	−0.120131	+	0.992758i	$0.538331\pi$
$108$	0	0
$109$	4.31371	0.413178	0.206589	−	0.978428i	$-0.433764\pi$
$109$	4.31371	0.413178	0.206589	+	0.978428i	$0.433764\pi$
$110$	0	0
$111$	−3.27208	−0.310572
$112$	0	0
$113$	5.07107	0.477046	0.238523	−	0.971137i	$-0.423337\pi$
$113$	5.07107	0.477046	0.238523	+	0.971137i	$0.423337\pi$
$114$	0	0
$115$	−2.24264	−0.209127
$116$	0	0
$117$	−6.82843	−0.631288
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	3.07107	0.276909
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−12.2426	−1.08636	−0.543179	−	0.839617i	$-0.682780\pi$
$127$	−12.2426	−1.08636	−0.543179	+	0.839617i	$0.682780\pi$
$128$	0	0
$129$	0.142136	0.0125143
$130$	0	0
$131$	−11.7574	−1.02725	−0.513623	−	0.858016i	$-0.671697\pi$
$131$	−11.7574	−1.02725	−0.513623	+	0.858016i	$0.671697\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	2.41421	0.207782
$136$	0	0
$137$	3.31371	0.283109	0.141555	−	0.989930i	$-0.454790\pi$
$137$	3.31371	0.283109	0.141555	+	0.989930i	$0.454790\pi$
$138$	0	0
$139$	1.41421	0.119952	0.0599760	−	0.998200i	$-0.480898\pi$
$139$	1.41421	0.119952	0.0599760	+	0.998200i	$0.480898\pi$
$140$	0	0
$141$	−4.31371	−0.363280
$142$	0	0
$143$	2.41421	0.201887
$144$	0	0
$145$	−1.00000	−0.0830455
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−14.8284	−1.21479	−0.607396	−	0.794399i	$-0.707786\pi$
$149$	−14.8284	−1.21479	−0.607396	+	0.794399i	$0.707786\pi$
$150$	0	0
$151$	−4.65685	−0.378969	−0.189485	−	0.981884i	$-0.560682\pi$
$151$	−4.65685	−0.378969	−0.189485	+	0.981884i	$0.560682\pi$
$152$	0	0
$153$	1.17157	0.0947161
$154$	0	0
$155$	1.75736	0.141154
$156$	0	0
$157$	−6.48528	−0.517582	−0.258791	−	0.965933i	$-0.583324\pi$
$157$	−6.48528	−0.517582	−0.258791	+	0.965933i	$0.583324\pi$
$158$	0	0
$159$	3.89949	0.309250
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−17.0711	−1.33711	−0.668555	−	0.743663i	$-0.733087\pi$
$163$	−17.0711	−1.33711	−0.668555	+	0.743663i	$0.733087\pi$
$164$	0	0
$165$	−0.414214	−0.0322465
$166$	0	0
$167$	−10.4142	−0.805876	−0.402938	−	0.915227i	$-0.632011\pi$
$167$	−10.4142	−0.805876	−0.402938	+	0.915227i	$0.632011\pi$
$168$	0	0
$169$	−7.17157	−0.551659
$170$	0	0
$171$	5.65685	0.432590
$172$	0	0
$173$	11.7279	0.891657	0.445829	−	0.895118i	$-0.352909\pi$
$173$	11.7279	0.891657	0.445829	+	0.895118i	$0.352909\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−4.24264	−0.318896
$178$	0	0
$179$	24.1421	1.80447	0.902234	−	0.431247i	$-0.141926\pi$
$179$	24.1421	1.80447	0.902234	+	0.431247i	$0.141926\pi$
$180$	0	0
$181$	5.55635	0.413000	0.206500	−	0.978447i	$-0.433793\pi$
$181$	5.55635	0.413000	0.206500	+	0.978447i	$0.433793\pi$
$182$	0	0
$183$	−0.485281	−0.0358730
$184$	0	0
$185$	7.89949	0.580782
$186$	0	0
$187$	−0.414214	−0.0302903
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−8.17157	−0.591274	−0.295637	−	0.955300i	$-0.595532\pi$
$191$	−8.17157	−0.591274	−0.295637	+	0.955300i	$0.595532\pi$
$192$	0	0
$193$	23.3137	1.67816	0.839079	−	0.544010i	$-0.183095\pi$
$193$	23.3137	1.67816	0.839079	+	0.544010i	$0.183095\pi$
$194$	0	0
$195$	−1.00000	−0.0716115
$196$	0	0
$197$	−3.75736	−0.267701	−0.133850	−	0.991002i	$-0.542734\pi$
$197$	−3.75736	−0.267701	−0.133850	+	0.991002i	$0.542734\pi$
$198$	0	0
$199$	−24.3848	−1.72859	−0.864295	−	0.502984i	$-0.832235\pi$
$199$	−24.3848	−1.72859	−0.864295	+	0.502984i	$0.832235\pi$
$200$	0	0
$201$	−0.585786	−0.0413182
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−7.41421	−0.517831
$206$	0	0
$207$	−6.34315	−0.440879
$208$	0	0
$209$	−2.00000	−0.138343
$210$	0	0
$211$	11.1421	0.767056	0.383528	−	0.923529i	$-0.374709\pi$
$211$	11.1421	0.767056	0.383528	+	0.923529i	$0.374709\pi$
$212$	0	0
$213$	−6.00000	−0.411113
$214$	0	0
$215$	−0.343146	−0.0234023
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−2.14214	−0.144752
$220$	0	0
$221$	−1.00000	−0.0672673
$222$	0	0
$223$	−19.3848	−1.29810	−0.649050	−	0.760745i	$-0.724834\pi$
$223$	−19.3848	−1.29810	−0.649050	+	0.760745i	$0.724834\pi$
$224$	0	0
$225$	−2.82843	−0.188562
$226$	0	0
$227$	−4.41421	−0.292982	−0.146491	−	0.989212i	$-0.546798\pi$
$227$	−4.41421	−0.292982	−0.146491	+	0.989212i	$0.546798\pi$
$228$	0	0
$229$	−23.8995	−1.57932	−0.789662	−	0.613543i	$-0.789744\pi$
$229$	−23.8995	−1.57932	−0.789662	+	0.613543i	$0.789744\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	20.4853	1.34204	0.671018	−	0.741441i	$-0.265857\pi$
$233$	20.4853	1.34204	0.671018	+	0.741441i	$0.265857\pi$
$234$	0	0
$235$	10.4142	0.679348
$236$	0	0
$237$	6.07107	0.394358
$238$	0	0
$239$	−6.65685	−0.430596	−0.215298	−	0.976548i	$-0.569072\pi$
$239$	−6.65685	−0.430596	−0.215298	+	0.976548i	$0.569072\pi$
$240$	0	0
$241$	−26.3848	−1.69959	−0.849796	−	0.527111i	$-0.823275\pi$
$241$	−26.3848	−1.69959	−0.849796	+	0.527111i	$0.823275\pi$
$242$	0	0
$243$	10.3431	0.663513
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−4.82843	−0.307225
$248$	0	0
$249$	−4.68629	−0.296982
$250$	0	0
$251$	23.5563	1.48686	0.743432	−	0.668812i	$-0.233197\pi$
$251$	23.5563	1.48686	0.743432	+	0.668812i	$0.233197\pi$
$252$	0	0
$253$	2.24264	0.140994
$254$	0	0
$255$	0.171573	0.0107443
$256$	0	0
$257$	−6.48528	−0.404541	−0.202270	−	0.979330i	$-0.564832\pi$
$257$	−6.48528	−0.404541	−0.202270	+	0.979330i	$0.564832\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−2.82843	−0.175075
$262$	0	0
$263$	14.9706	0.923124	0.461562	−	0.887108i	$-0.347289\pi$
$263$	14.9706	0.923124	0.461562	+	0.887108i	$0.347289\pi$
$264$	0	0
$265$	−9.41421	−0.578311
$266$	0	0
$267$	−0.686292	−0.0420004
$268$	0	0
$269$	21.4142	1.30565	0.652824	−	0.757510i	$-0.273584\pi$
$269$	21.4142	1.30565	0.652824	+	0.757510i	$0.273584\pi$
$270$	0	0
$271$	−13.6569	−0.829595	−0.414797	−	0.909914i	$-0.636148\pi$
$271$	−13.6569	−0.829595	−0.414797	+	0.909914i	$0.636148\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.00000	0.0603023
$276$	0	0
$277$	1.07107	0.0643542	0.0321771	−	0.999482i	$-0.489756\pi$
$277$	1.07107	0.0643542	0.0321771	+	0.999482i	$0.489756\pi$
$278$	0	0
$279$	4.97056	0.297580
$280$	0	0
$281$	−4.31371	−0.257334	−0.128667	−	0.991688i	$-0.541070\pi$
$281$	−4.31371	−0.257334	−0.128667	+	0.991688i	$0.541070\pi$
$282$	0	0
$283$	23.0416	1.36968	0.684841	−	0.728692i	$-0.259871\pi$
$283$	23.0416	1.36968	0.684841	+	0.728692i	$0.259871\pi$
$284$	0	0
$285$	0.828427	0.0490718
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−16.8284	−0.989907
$290$	0	0
$291$	−0.0294373	−0.00172564
$292$	0	0
$293$	−15.5858	−0.910531	−0.455266	−	0.890356i	$-0.650456\pi$
$293$	−15.5858	−0.910531	−0.455266	+	0.890356i	$0.650456\pi$
$294$	0	0
$295$	10.2426	0.596350
$296$	0	0
$297$	−2.41421	−0.140087
$298$	0	0
$299$	5.41421	0.313112
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−2.68629	−0.154323
$304$	0	0
$305$	1.17157	0.0670841
$306$	0	0
$307$	26.0711	1.48795	0.743977	−	0.668205i	$-0.232937\pi$
$307$	26.0711	1.48795	0.743977	+	0.668205i	$0.232937\pi$
$308$	0	0
$309$	−6.31371	−0.359174
$310$	0	0
$311$	−10.0000	−0.567048	−0.283524	−	0.958965i	$-0.591504\pi$
$311$	−10.0000	−0.567048	−0.283524	+	0.958965i	$0.591504\pi$
$312$	0	0
$313$	1.72792	0.0976679	0.0488340	−	0.998807i	$-0.484449\pi$
$313$	1.72792	0.0976679	0.0488340	+	0.998807i	$0.484449\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	23.6569	1.32870	0.664351	−	0.747421i	$-0.268708\pi$
$317$	23.6569	1.32870	0.664351	+	0.747421i	$0.268708\pi$
$318$	0	0
$319$	1.00000	0.0559893
$320$	0	0
$321$	−1.02944	−0.0574576
$322$	0	0
$323$	0.828427	0.0460949
$324$	0	0
$325$	2.41421	0.133916
$326$	0	0
$327$	1.78680	0.0988101
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−26.4853	−1.45576	−0.727881	−	0.685703i	$-0.759495\pi$
$331$	−26.4853	−1.45576	−0.727881	+	0.685703i	$0.759495\pi$
$332$	0	0
$333$	22.3431	1.22440
$334$	0	0
$335$	1.41421	0.0772667
$336$	0	0
$337$	1.07107	0.0583448	0.0291724	−	0.999574i	$-0.490713\pi$
$337$	1.07107	0.0583448	0.0291724	+	0.999574i	$0.490713\pi$
$338$	0	0
$339$	2.10051	0.114084
$340$	0	0
$341$	−1.75736	−0.0951663
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−0.928932	−0.0500120
$346$	0	0
$347$	24.8701	1.33509	0.667547	−	0.744567i	$-0.267344\pi$
$347$	24.8701	1.33509	0.667547	+	0.744567i	$0.267344\pi$
$348$	0	0
$349$	−29.3137	−1.56913	−0.784563	−	0.620049i	$-0.787113\pi$
$349$	−29.3137	−1.56913	−0.784563	+	0.620049i	$0.787113\pi$
$350$	0	0
$351$	−5.82843	−0.311098
$352$	0	0
$353$	−12.2132	−0.650043	−0.325022	−	0.945707i	$-0.605372\pi$
$353$	−12.2132	−0.650043	−0.325022	+	0.945707i	$0.605372\pi$
$354$	0	0
$355$	14.4853	0.768799
$356$	0	0
$357$	0	0
$358$	0	0
$359$	18.3431	0.968114	0.484057	−	0.875036i	$-0.339163\pi$
$359$	18.3431	0.968114	0.484057	+	0.875036i	$0.339163\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	−4.14214	−0.217406
$364$	0	0
$365$	5.17157	0.270692
$366$	0	0
$367$	−2.75736	−0.143933	−0.0719665	−	0.997407i	$-0.522927\pi$
$367$	−2.75736	−0.143933	−0.0719665	+	0.997407i	$0.522927\pi$
$368$	0	0
$369$	−20.9706	−1.09168
$370$	0	0
$371$	0	0
$372$	0	0
$373$	17.1716	0.889110	0.444555	−	0.895751i	$-0.353362\pi$
$373$	17.1716	0.889110	0.444555	+	0.895751i	$0.353362\pi$
$374$	0	0
$375$	−0.414214	−0.0213899
$376$	0	0
$377$	2.41421	0.124338
$378$	0	0
$379$	16.6274	0.854093	0.427047	−	0.904230i	$-0.359554\pi$
$379$	16.6274	0.854093	0.427047	+	0.904230i	$0.359554\pi$
$380$	0	0
$381$	−5.07107	−0.259799
$382$	0	0
$383$	−17.1716	−0.877426	−0.438713	−	0.898627i	$-0.644566\pi$
$383$	−17.1716	−0.877426	−0.438713	+	0.898627i	$0.644566\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−0.970563	−0.0493365
$388$	0	0
$389$	−28.7990	−1.46017	−0.730083	−	0.683358i	$-0.760519\pi$
$389$	−28.7990	−1.46017	−0.730083	+	0.683358i	$0.760519\pi$
$390$	0	0
$391$	−0.928932	−0.0469781
$392$	0	0
$393$	−4.87006	−0.245662
$394$	0	0
$395$	−14.6569	−0.737466
$396$	0	0
$397$	−8.41421	−0.422297	−0.211149	−	0.977454i	$-0.567720\pi$
$397$	−8.41421	−0.422297	−0.211149	+	0.977454i	$0.567720\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	26.4558	1.32114	0.660571	−	0.750764i	$-0.270314\pi$
$401$	26.4558	1.32114	0.660571	+	0.750764i	$0.270314\pi$
$402$	0	0
$403$	−4.24264	−0.211341
$404$	0	0
$405$	−7.48528	−0.371947
$406$	0	0
$407$	−7.89949	−0.391563
$408$	0	0
$409$	29.1127	1.43953	0.719765	−	0.694218i	$-0.244249\pi$
$409$	29.1127	1.43953	0.719765	+	0.694218i	$0.244249\pi$
$410$	0	0
$411$	1.37258	0.0677045
$412$	0	0
$413$	0	0
$414$	0	0
$415$	11.3137	0.555368
$416$	0	0
$417$	0.585786	0.0286861
$418$	0	0
$419$	31.4142	1.53468	0.767342	−	0.641238i	$-0.221579\pi$
$419$	31.4142	1.53468	0.767342	+	0.641238i	$0.221579\pi$
$420$	0	0
$421$	−3.68629	−0.179659	−0.0898294	−	0.995957i	$-0.528632\pi$
$421$	−3.68629	−0.179659	−0.0898294	+	0.995957i	$0.528632\pi$
$422$	0	0
$423$	29.4558	1.43219
$424$	0	0
$425$	−0.414214	−0.0200923
$426$	0	0
$427$	0	0
$428$	0	0
$429$	1.00000	0.0482805
$430$	0	0
$431$	−11.3431	−0.546380	−0.273190	−	0.961960i	$-0.588079\pi$
$431$	−11.3431	−0.546380	−0.273190	+	0.961960i	$0.588079\pi$
$432$	0	0
$433$	−26.9706	−1.29612	−0.648061	−	0.761588i	$-0.724420\pi$
$433$	−26.9706	−1.29612	−0.648061	+	0.761588i	$0.724420\pi$
$434$	0	0
$435$	−0.414214	−0.0198600
$436$	0	0
$437$	−4.48528	−0.214560
$438$	0	0
$439$	12.2426	0.584309	0.292155	−	0.956371i	$-0.405628\pi$
$439$	12.2426	0.584309	0.292155	+	0.956371i	$0.405628\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−38.7696	−1.84200	−0.920999	−	0.389566i	$-0.872625\pi$
$443$	−38.7696	−1.84200	−0.920999	+	0.389566i	$0.872625\pi$
$444$	0	0
$445$	1.65685	0.0785424
$446$	0	0
$447$	−6.14214	−0.290513
$448$	0	0
$449$	−10.5147	−0.496220	−0.248110	−	0.968732i	$-0.579809\pi$
$449$	−10.5147	−0.496220	−0.248110	+	0.968732i	$0.579809\pi$
$450$	0	0
$451$	7.41421	0.349122
$452$	0	0
$453$	−1.92893	−0.0906291
$454$	0	0
$455$	0	0
$456$	0	0
$457$	19.0711	0.892107	0.446053	−	0.895006i	$-0.352829\pi$
$457$	19.0711	0.892107	0.446053	+	0.895006i	$0.352829\pi$
$458$	0	0
$459$	1.00000	0.0466760
$460$	0	0
$461$	24.2843	1.13103	0.565516	−	0.824738i	$-0.308677\pi$
$461$	24.2843	1.13103	0.565516	+	0.824738i	$0.308677\pi$
$462$	0	0
$463$	25.4558	1.18303	0.591517	−	0.806293i	$-0.298529\pi$
$463$	25.4558	1.18303	0.591517	+	0.806293i	$0.298529\pi$
$464$	0	0
$465$	0.727922	0.0337566
$466$	0	0
$467$	−5.58579	−0.258479	−0.129240	−	0.991613i	$-0.541254\pi$
$467$	−5.58579	−0.258479	−0.129240	+	0.991613i	$0.541254\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−2.68629	−0.123778
$472$	0	0
$473$	0.343146	0.0157779
$474$	0	0
$475$	−2.00000	−0.0917663
$476$	0	0
$477$	−26.6274	−1.21919
$478$	0	0
$479$	−24.5269	−1.12066	−0.560332	−	0.828268i	$-0.689326\pi$
$479$	−24.5269	−1.12066	−0.560332	+	0.828268i	$0.689326\pi$
$480$	0	0
$481$	−19.0711	−0.869566
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0.0710678	0.00322702
$486$	0	0
$487$	−26.7279	−1.21116	−0.605579	−	0.795785i	$-0.707058\pi$
$487$	−26.7279	−1.21116	−0.605579	+	0.795785i	$0.707058\pi$
$488$	0	0
$489$	−7.07107	−0.319765
$490$	0	0
$491$	−5.48528	−0.247547	−0.123774	−	0.992310i	$-0.539500\pi$
$491$	−5.48528	−0.247547	−0.123774	+	0.992310i	$0.539500\pi$
$492$	0	0
$493$	−0.414214	−0.0186552
$494$	0	0
$495$	2.82843	0.127128
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−12.7990	−0.572961	−0.286481	−	0.958086i	$-0.592485\pi$
$499$	−12.7990	−0.572961	−0.286481	+	0.958086i	$0.592485\pi$
$500$	0	0
$501$	−4.31371	−0.192722
$502$	0	0
$503$	−31.0416	−1.38408	−0.692039	−	0.721860i	$-0.743287\pi$
$503$	−31.0416	−1.38408	−0.692039	+	0.721860i	$0.743287\pi$
$504$	0	0
$505$	6.48528	0.288591
$506$	0	0
$507$	−2.97056	−0.131927
$508$	0	0
$509$	7.75736	0.343839	0.171919	−	0.985111i	$-0.445003\pi$
$509$	7.75736	0.343839	0.171919	+	0.985111i	$0.445003\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	4.82843	0.213180
$514$	0	0
$515$	15.2426	0.671671
$516$	0	0
$517$	−10.4142	−0.458017
$518$	0	0
$519$	4.85786	0.213237
$520$	0	0
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	0	0
$523$	32.4853	1.42048	0.710241	−	0.703959i	$-0.248586\pi$
$523$	32.4853	1.42048	0.710241	+	0.703959i	$0.248586\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0.727922	0.0317088
$528$	0	0
$529$	−17.9706	−0.781329
$530$	0	0
$531$	28.9706	1.25722
$532$	0	0
$533$	17.8995	0.775313
$534$	0	0
$535$	2.48528	0.107448
$536$	0	0
$537$	10.0000	0.431532
$538$	0	0
$539$	0	0
$540$	0	0
$541$	10.3137	0.443421	0.221710	−	0.975113i	$-0.428836\pi$
$541$	10.3137	0.443421	0.221710	+	0.975113i	$0.428836\pi$
$542$	0	0
$543$	2.30152	0.0987675
$544$	0	0
$545$	−4.31371	−0.184779
$546$	0	0
$547$	27.1127	1.15926	0.579628	−	0.814881i	$-0.303198\pi$
$547$	27.1127	1.15926	0.579628	+	0.814881i	$0.303198\pi$
$548$	0	0
$549$	3.31371	0.141426
$550$	0	0
$551$	−2.00000	−0.0852029
$552$	0	0
$553$	0	0
$554$	0	0
$555$	3.27208	0.138892
$556$	0	0
$557$	−26.1421	−1.10768	−0.553839	−	0.832624i	$-0.686838\pi$
$557$	−26.1421	−1.10768	−0.553839	+	0.832624i	$0.686838\pi$
$558$	0	0
$559$	0.828427	0.0350387
$560$	0	0
$561$	−0.171573	−0.00724381
$562$	0	0
$563$	39.9411	1.68332	0.841659	−	0.540010i	$-0.181579\pi$
$563$	39.9411	1.68332	0.841659	+	0.540010i	$0.181579\pi$
$564$	0	0
$565$	−5.07107	−0.213341
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−5.85786	−0.245574	−0.122787	−	0.992433i	$-0.539183\pi$
$569$	−5.85786	−0.245574	−0.122787	+	0.992433i	$0.539183\pi$
$570$	0	0
$571$	−27.4558	−1.14899	−0.574496	−	0.818508i	$-0.694802\pi$
$571$	−27.4558	−1.14899	−0.574496	+	0.818508i	$0.694802\pi$
$572$	0	0
$573$	−3.38478	−0.141401
$574$	0	0
$575$	2.24264	0.0935246
$576$	0	0
$577$	−33.5858	−1.39819	−0.699097	−	0.715027i	$-0.746415\pi$
$577$	−33.5858	−1.39819	−0.699097	+	0.715027i	$0.746415\pi$
$578$	0	0
$579$	9.65685	0.401325
$580$	0	0
$581$	0	0
$582$	0	0
$583$	9.41421	0.389897
$584$	0	0
$585$	6.82843	0.282321
$586$	0	0
$587$	42.1421	1.73939	0.869696	−	0.493588i	$-0.164315\pi$
$587$	42.1421	1.73939	0.869696	+	0.493588i	$0.164315\pi$
$588$	0	0
$589$	3.51472	0.144821
$590$	0	0
$591$	−1.55635	−0.0640197
$592$	0	0
$593$	40.5563	1.66545	0.832725	−	0.553687i	$-0.186780\pi$
$593$	40.5563	1.66545	0.832725	+	0.553687i	$0.186780\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−10.1005	−0.413386
$598$	0	0
$599$	13.6863	0.559207	0.279603	−	0.960116i	$-0.409797\pi$
$599$	13.6863	0.559207	0.279603	+	0.960116i	$0.409797\pi$
$600$	0	0
$601$	38.2843	1.56165	0.780824	−	0.624751i	$-0.214800\pi$
$601$	38.2843	1.56165	0.780824	+	0.624751i	$0.214800\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	0	0
$605$	10.0000	0.406558
$606$	0	0
$607$	−5.92893	−0.240648	−0.120324	−	0.992735i	$-0.538393\pi$
$607$	−5.92893	−0.240648	−0.120324	+	0.992735i	$0.538393\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−25.1421	−1.01714
$612$	0	0
$613$	6.68629	0.270057	0.135028	−	0.990842i	$-0.456887\pi$
$613$	6.68629	0.270057	0.135028	+	0.990842i	$0.456887\pi$
$614$	0	0
$615$	−3.07107	−0.123837
$616$	0	0
$617$	−8.58579	−0.345651	−0.172825	−	0.984952i	$-0.555290\pi$
$617$	−8.58579	−0.345651	−0.172825	+	0.984952i	$0.555290\pi$
$618$	0	0
$619$	0.928932	0.0373369	0.0186685	−	0.999826i	$-0.494057\pi$
$619$	0.928932	0.0373369	0.0186685	+	0.999826i	$0.494057\pi$
$620$	0	0
$621$	−5.41421	−0.217265
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−0.828427	−0.0330842
$628$	0	0
$629$	3.27208	0.130466
$630$	0	0
$631$	−7.62742	−0.303643	−0.151821	−	0.988408i	$-0.548514\pi$
$631$	−7.62742	−0.303643	−0.151821	+	0.988408i	$0.548514\pi$
$632$	0	0
$633$	4.61522	0.183439
$634$	0	0
$635$	12.2426	0.485834
$636$	0	0
$637$	0	0
$638$	0	0
$639$	40.9706	1.62077
$640$	0	0
$641$	−24.2843	−0.959171	−0.479586	−	0.877495i	$-0.659213\pi$
$641$	−24.2843	−0.959171	−0.479586	+	0.877495i	$0.659213\pi$
$642$	0	0
$643$	−12.2132	−0.481642	−0.240821	−	0.970570i	$-0.577417\pi$
$643$	−12.2132	−0.481642	−0.240821	+	0.970570i	$0.577417\pi$
$644$	0	0
$645$	−0.142136	−0.00559658
$646$	0	0
$647$	10.1421	0.398728	0.199364	−	0.979925i	$-0.436112\pi$
$647$	10.1421	0.398728	0.199364	+	0.979925i	$0.436112\pi$
$648$	0	0
$649$	−10.2426	−0.402059
$650$	0	0
$651$	0	0
$652$	0	0
$653$	34.2843	1.34165	0.670824	−	0.741617i	$-0.265941\pi$
$653$	34.2843	1.34165	0.670824	+	0.741617i	$0.265941\pi$
$654$	0	0
$655$	11.7574	0.459398
$656$	0	0
$657$	14.6274	0.570670
$658$	0	0
$659$	−0.514719	−0.0200506	−0.0100253	−	0.999950i	$-0.503191\pi$
$659$	−0.514719	−0.0200506	−0.0100253	+	0.999950i	$0.503191\pi$
$660$	0	0
$661$	−25.1716	−0.979061	−0.489530	−	0.871986i	$-0.662832\pi$
$661$	−25.1716	−0.979061	−0.489530	+	0.871986i	$0.662832\pi$
$662$	0	0
$663$	−0.414214	−0.0160867
$664$	0	0
$665$	0	0
$666$	0	0
$667$	2.24264	0.0868354
$668$	0	0
$669$	−8.02944	−0.310436
$670$	0	0
$671$	−1.17157	−0.0452281
$672$	0	0
$673$	45.1716	1.74124	0.870618	−	0.491959i	$-0.163719\pi$
$673$	45.1716	1.74124	0.870618	+	0.491959i	$0.163719\pi$
$674$	0	0
$675$	−2.41421	−0.0929231
$676$	0	0
$677$	21.9289	0.842797	0.421399	−	0.906875i	$-0.361539\pi$
$677$	21.9289	0.842797	0.421399	+	0.906875i	$0.361539\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−1.82843	−0.0700655
$682$	0	0
$683$	18.8284	0.720450	0.360225	−	0.932866i	$-0.382700\pi$
$683$	18.8284	0.720450	0.360225	+	0.932866i	$0.382700\pi$
$684$	0	0
$685$	−3.31371	−0.126610
$686$	0	0
$687$	−9.89949	−0.377689
$688$	0	0
$689$	22.7279	0.865865
$690$	0	0
$691$	18.1421	0.690159	0.345080	−	0.938573i	$-0.387852\pi$
$691$	18.1421	0.690159	0.345080	+	0.938573i	$0.387852\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.41421	−0.0536442
$696$	0	0
$697$	−3.07107	−0.116325
$698$	0	0
$699$	8.48528	0.320943
$700$	0	0
$701$	−8.85786	−0.334557	−0.167278	−	0.985910i	$-0.553498\pi$
$701$	−8.85786	−0.334557	−0.167278	+	0.985910i	$0.553498\pi$
$702$	0	0
$703$	15.7990	0.595870
$704$	0	0
$705$	4.31371	0.162464
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−32.6569	−1.22645	−0.613227	−	0.789907i	$-0.710129\pi$
$709$	−32.6569	−1.22645	−0.613227	+	0.789907i	$0.710129\pi$
$710$	0	0
$711$	−41.4558	−1.55472
$712$	0	0
$713$	−3.94113	−0.147596
$714$	0	0
$715$	−2.41421	−0.0902865
$716$	0	0
$717$	−2.75736	−0.102975
$718$	0	0
$719$	9.69848	0.361692	0.180846	−	0.983511i	$-0.442116\pi$
$719$	9.69848	0.361692	0.180846	+	0.983511i	$0.442116\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−10.9289	−0.406451
$724$	0	0
$725$	1.00000	0.0371391
$726$	0	0
$727$	−18.6863	−0.693036	−0.346518	−	0.938043i	$-0.612636\pi$
$727$	−18.6863	−0.693036	−0.346518	+	0.938043i	$0.612636\pi$
$728$	0	0
$729$	−18.1716	−0.673021
$730$	0	0
$731$	−0.142136	−0.00525708
$732$	0	0
$733$	32.6985	1.20775	0.603873	−	0.797081i	$-0.293623\pi$
$733$	32.6985	1.20775	0.603873	+	0.797081i	$0.293623\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.41421	−0.0520932
$738$	0	0
$739$	−34.7990	−1.28010	−0.640051	−	0.768333i	$-0.721087\pi$
$739$	−34.7990	−1.28010	−0.640051	+	0.768333i	$0.721087\pi$
$740$	0	0
$741$	−2.00000	−0.0734718
$742$	0	0
$743$	−13.8995	−0.509923	−0.254962	−	0.966951i	$-0.582063\pi$
$743$	−13.8995	−0.509923	−0.254962	+	0.966951i	$0.582063\pi$
$744$	0	0
$745$	14.8284	0.543272
$746$	0	0
$747$	32.0000	1.17082
$748$	0	0
$749$	0	0
$750$	0	0
$751$	19.6274	0.716215	0.358107	−	0.933680i	$-0.383422\pi$
$751$	19.6274	0.716215	0.358107	+	0.933680i	$0.383422\pi$
$752$	0	0
$753$	9.75736	0.355578
$754$	0	0
$755$	4.65685	0.169480
$756$	0	0
$757$	36.7696	1.33641	0.668206	−	0.743976i	$-0.267062\pi$
$757$	36.7696	1.33641	0.668206	+	0.743976i	$0.267062\pi$
$758$	0	0
$759$	0.928932	0.0337181
$760$	0	0
$761$	35.5563	1.28892	0.644458	−	0.764639i	$-0.277083\pi$
$761$	35.5563	1.28892	0.644458	+	0.764639i	$0.277083\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−1.17157	−0.0423583
$766$	0	0
$767$	−24.7279	−0.892874
$768$	0	0
$769$	−1.51472	−0.0546222	−0.0273111	−	0.999627i	$-0.508694\pi$
$769$	−1.51472	−0.0546222	−0.0273111	+	0.999627i	$0.508694\pi$
$770$	0	0
$771$	−2.68629	−0.0967444
$772$	0	0
$773$	15.3848	0.553352	0.276676	−	0.960963i	$-0.410767\pi$
$773$	15.3848	0.553352	0.276676	+	0.960963i	$0.410767\pi$
$774$	0	0
$775$	−1.75736	−0.0631262
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−14.8284	−0.531284
$780$	0	0
$781$	−14.4853	−0.518324
$782$	0	0
$783$	−2.41421	−0.0862770
$784$	0	0
$785$	6.48528	0.231470
$786$	0	0
$787$	−44.8406	−1.59840	−0.799198	−	0.601068i	$-0.794742\pi$
$787$	−44.8406	−1.59840	−0.799198	+	0.601068i	$0.794742\pi$
$788$	0	0
$789$	6.20101	0.220762
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−2.82843	−0.100440
$794$	0	0
$795$	−3.89949	−0.138301
$796$	0	0
$797$	50.0711	1.77361	0.886804	−	0.462146i	$-0.152920\pi$
$797$	50.0711	1.77361	0.886804	+	0.462146i	$0.152920\pi$
$798$	0	0
$799$	4.31371	0.152608
$800$	0	0
$801$	4.68629	0.165582
$802$	0	0
$803$	−5.17157	−0.182501
$804$	0	0
$805$	0	0
$806$	0	0
$807$	8.87006	0.312241
$808$	0	0
$809$	−18.9411	−0.665935	−0.332967	−	0.942938i	$-0.608050\pi$
$809$	−18.9411	−0.665935	−0.332967	+	0.942938i	$0.608050\pi$
$810$	0	0
$811$	8.44365	0.296497	0.148248	−	0.988950i	$-0.452637\pi$
$811$	8.44365	0.296497	0.148248	+	0.988950i	$0.452637\pi$
$812$	0	0
$813$	−5.65685	−0.198395
$814$	0	0
$815$	17.0711	0.597973
$816$	0	0
$817$	−0.686292	−0.0240103
$818$	0	0
$819$	0	0
$820$	0	0
$821$	4.51472	0.157565	0.0787824	−	0.996892i	$-0.474897\pi$
$821$	4.51472	0.157565	0.0787824	+	0.996892i	$0.474897\pi$
$822$	0	0
$823$	−12.0416	−0.419745	−0.209872	−	0.977729i	$-0.567305\pi$
$823$	−12.0416	−0.419745	−0.209872	+	0.977729i	$0.567305\pi$
$824$	0	0
$825$	0.414214	0.0144211
$826$	0	0
$827$	5.95837	0.207193	0.103596	−	0.994619i	$-0.466965\pi$
$827$	5.95837	0.207193	0.103596	+	0.994619i	$0.466965\pi$
$828$	0	0
$829$	4.58579	0.159271	0.0796355	−	0.996824i	$-0.474624\pi$
$829$	4.58579	0.159271	0.0796355	+	0.996824i	$0.474624\pi$
$830$	0	0
$831$	0.443651	0.0153901
$832$	0	0
$833$	0	0
$834$	0	0
$835$	10.4142	0.360399
$836$	0	0
$837$	4.24264	0.146647
$838$	0	0
$839$	−16.2426	−0.560758	−0.280379	−	0.959889i	$-0.590460\pi$
$839$	−16.2426	−0.560758	−0.280379	+	0.959889i	$0.590460\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	0	0
$843$	−1.78680	−0.0615405
$844$	0	0
$845$	7.17157	0.246710
$846$	0	0
$847$	0	0
$848$	0	0
$849$	9.54416	0.327555
$850$	0	0
$851$	−17.7157	−0.607287
$852$	0	0
$853$	−13.3137	−0.455853	−0.227926	−	0.973678i	$-0.573195\pi$
$853$	−13.3137	−0.455853	−0.227926	+	0.973678i	$0.573195\pi$
$854$	0	0
$855$	−5.65685	−0.193460
$856$	0	0
$857$	37.5147	1.28148	0.640739	−	0.767759i	$-0.278628\pi$
$857$	37.5147	1.28148	0.640739	+	0.767759i	$0.278628\pi$
$858$	0	0
$859$	2.54416	0.0868055	0.0434027	−	0.999058i	$-0.486180\pi$
$859$	2.54416	0.0868055	0.0434027	+	0.999058i	$0.486180\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	54.1838	1.84444	0.922218	−	0.386669i	$-0.126375\pi$
$863$	54.1838	1.84444	0.922218	+	0.386669i	$0.126375\pi$
$864$	0	0
$865$	−11.7279	−0.398761
$866$	0	0
$867$	−6.97056	−0.236733
$868$	0	0
$869$	14.6569	0.497200
$870$	0	0
$871$	−3.41421	−0.115686
$872$	0	0
$873$	0.201010	0.00680316
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−6.97056	−0.235379	−0.117690	−	0.993050i	$-0.537549\pi$
$877$	−6.97056	−0.235379	−0.117690	+	0.993050i	$0.537549\pi$
$878$	0	0
$879$	−6.45584	−0.217750
$880$	0	0
$881$	6.34315	0.213706	0.106853	−	0.994275i	$-0.465923\pi$
$881$	6.34315	0.213706	0.106853	+	0.994275i	$0.465923\pi$
$882$	0	0
$883$	27.1127	0.912415	0.456207	−	0.889873i	$-0.349207\pi$
$883$	27.1127	0.912415	0.456207	+	0.889873i	$0.349207\pi$
$884$	0	0
$885$	4.24264	0.142615
$886$	0	0
$887$	−38.2843	−1.28546	−0.642730	−	0.766093i	$-0.722198\pi$
$887$	−38.2843	−1.28546	−0.642730	+	0.766093i	$0.722198\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	7.48528	0.250766
$892$	0	0
$893$	20.8284	0.696997
$894$	0	0
$895$	−24.1421	−0.806983
$896$	0	0
$897$	2.24264	0.0748796
$898$	0	0
$899$	−1.75736	−0.0586112
$900$	0	0
$901$	−3.89949	−0.129911
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−5.55635	−0.184699
$906$	0	0
$907$	−51.6985	−1.71662	−0.858310	−	0.513131i	$-0.828485\pi$
$907$	−51.6985	−1.71662	−0.858310	+	0.513131i	$0.828485\pi$
$908$	0	0
$909$	18.3431	0.608404
$910$	0	0
$911$	−43.3137	−1.43505	−0.717524	−	0.696534i	$-0.754724\pi$
$911$	−43.3137	−1.43505	−0.717524	+	0.696534i	$0.754724\pi$
$912$	0	0
$913$	−11.3137	−0.374429
$914$	0	0
$915$	0.485281	0.0160429
$916$	0	0
$917$	0	0
$918$	0	0
$919$	50.5980	1.66907	0.834537	−	0.550952i	$-0.185735\pi$
$919$	50.5980	1.66907	0.834537	+	0.550952i	$0.185735\pi$
$920$	0	0
$921$	10.7990	0.355839
$922$	0	0
$923$	−34.9706	−1.15107
$924$	0	0
$925$	−7.89949	−0.259734
$926$	0	0
$927$	43.1127	1.41601
$928$	0	0
$929$	30.5269	1.00156	0.500778	−	0.865576i	$-0.333047\pi$
$929$	30.5269	1.00156	0.500778	+	0.865576i	$0.333047\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−4.14214	−0.135607
$934$	0	0
$935$	0.414214	0.0135462
$936$	0	0
$937$	−28.2132	−0.921685	−0.460843	−	0.887482i	$-0.652453\pi$
$937$	−28.2132	−0.921685	−0.460843	+	0.887482i	$0.652453\pi$
$938$	0	0
$939$	0.715729	0.0233569
$940$	0	0
$941$	24.0000	0.782378	0.391189	−	0.920310i	$-0.372064\pi$
$941$	24.0000	0.782378	0.391189	+	0.920310i	$0.372064\pi$
$942$	0	0
$943$	16.6274	0.541463
$944$	0	0
$945$	0	0
$946$	0	0
$947$	9.89949	0.321690	0.160845	−	0.986980i	$-0.448578\pi$
$947$	9.89949	0.321690	0.160845	+	0.986980i	$0.448578\pi$
$948$	0	0
$949$	−12.4853	−0.405289
$950$	0	0
$951$	9.79899	0.317754
$952$	0	0
$953$	21.6152	0.700186	0.350093	−	0.936715i	$-0.386150\pi$
$953$	21.6152	0.700186	0.350093	+	0.936715i	$0.386150\pi$
$954$	0	0
$955$	8.17157	0.264426
$956$	0	0
$957$	0.414214	0.0133896
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−27.9117	−0.900377
$962$	0	0
$963$	7.02944	0.226520
$964$	0	0
$965$	−23.3137	−0.750495
$966$	0	0
$967$	−23.1716	−0.745148	−0.372574	−	0.928003i	$-0.621525\pi$
$967$	−23.1716	−0.745148	−0.372574	+	0.928003i	$0.621525\pi$
$968$	0	0
$969$	0.343146	0.0110234
$970$	0	0
$971$	−48.1838	−1.54629	−0.773145	−	0.634229i	$-0.781318\pi$
$971$	−48.1838	−1.54629	−0.773145	+	0.634229i	$0.781318\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	1.00000	0.0320256
$976$	0	0
$977$	−54.0833	−1.73028	−0.865138	−	0.501533i	$-0.832770\pi$
$977$	−54.0833	−1.73028	−0.865138	+	0.501533i	$0.832770\pi$
$978$	0	0
$979$	−1.65685	−0.0529533
$980$	0	0
$981$	−12.2010	−0.389548
$982$	0	0
$983$	3.38478	0.107958	0.0539788	−	0.998542i	$-0.482810\pi$
$983$	3.38478	0.107958	0.0539788	+	0.998542i	$0.482810\pi$
$984$	0	0
$985$	3.75736	0.119719
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0.769553	0.0244703
$990$	0	0
$991$	−17.3137	−0.549988	−0.274994	−	0.961446i	$-0.588676\pi$
$991$	−17.3137	−0.549988	−0.274994	+	0.961446i	$0.588676\pi$
$992$	0	0
$993$	−10.9706	−0.348140
$994$	0	0
$995$	24.3848	0.773049
$996$	0	0
$997$	6.89949	0.218509	0.109255	−	0.994014i	$-0.465154\pi$
$997$	6.89949	0.218509	0.109255	+	0.994014i	$0.465154\pi$
$998$	0	0
$999$	19.0711	0.603382

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3920.2.a.bn.1.2		2
4.3	odd	2		1960.2.a.u.1.1	yes	2
7.6	odd	2		3920.2.a.by.1.1		2
20.19	odd	2		9800.2.a.bs.1.2		2
28.3	even	6		1960.2.q.v.961.1		4
28.11	odd	6		1960.2.q.p.961.2		4
28.19	even	6		1960.2.q.v.361.1		4
28.23	odd	6		1960.2.q.p.361.2		4
28.27	even	2		1960.2.a.q.1.2	✓	2
140.139	even	2		9800.2.a.ca.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1960.2.a.q.1.2	✓	2	28.27	even	2
1960.2.a.u.1.1	yes	2	4.3	odd	2
1960.2.q.p.361.2		4	28.23	odd	6
1960.2.q.p.961.2		4	28.11	odd	6
1960.2.q.v.361.1		4	28.19	even	6
1960.2.q.v.961.1		4	28.3	even	6
3920.2.a.bn.1.2		2	1.1	even	1	trivial
3920.2.a.by.1.1		2	7.6	odd	2
9800.2.a.bs.1.2		2	20.19	odd	2
9800.2.a.ca.1.1		2	140.139	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:	\( N \)	\(=\)	\( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3920.a (trivial)

\(f(q)\)	\(=\)	\(q+0.414214 q^{3} -1.00000 q^{5} -2.82843 q^{9} +O(q^{10})\)	\(q+0.414214 q^{3} -1.00000 q^{5} -2.82843 q^{9} +1.00000 q^{11} +2.41421 q^{13} -0.414214 q^{15} -0.414214 q^{17} -2.00000 q^{19} +2.24264 q^{23} +1.00000 q^{25} -2.41421 q^{27} +1.00000 q^{29} -1.75736 q^{31} +0.414214 q^{33} -7.89949 q^{37} +1.00000 q^{39} +7.41421 q^{41} +0.343146 q^{43} +2.82843 q^{45} -10.4142 q^{47} -0.171573 q^{51} +9.41421 q^{53} -1.00000 q^{55} -0.828427 q^{57} -10.2426 q^{59} -1.17157 q^{61} -2.41421 q^{65} -1.41421 q^{67} +0.928932 q^{69} -14.4853 q^{71} -5.17157 q^{73} +0.414214 q^{75} +14.6569 q^{79} +7.48528 q^{81} -11.3137 q^{83} +0.414214 q^{85} +0.414214 q^{87} -1.65685 q^{89} -0.727922 q^{93} +2.00000 q^{95} -0.0710678 q^{97} -2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5}+O(q^{10}) \) 2 * q - 2 * q^3 - 2 * q^5	\( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{11} + 2 q^{13} + 2 q^{15} + 2 q^{17} - 4 q^{19} - 4 q^{23} + 2 q^{25} - 2 q^{27} + 2 q^{29} - 12 q^{31} - 2 q^{33} + 4 q^{37} + 2 q^{39} + 12 q^{41} + 12 q^{43} - 18 q^{47} - 6 q^{51} + 16 q^{53} - 2 q^{55} + 4 q^{57} - 12 q^{59} - 8 q^{61} - 2 q^{65} + 16 q^{69} - 12 q^{71} - 16 q^{73} - 2 q^{75} + 18 q^{79} - 2 q^{81} - 2 q^{85} - 2 q^{87} + 8 q^{89} + 24 q^{93} + 4 q^{95} + 14 q^{97}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^11 + 2 * q^13 + 2 * q^15 + 2 * q^17 - 4 * q^19 - 4 * q^23 + 2 * q^25 - 2 * q^27 + 2 * q^29 - 12 * q^31 - 2 * q^33 + 4 * q^37 + 2 * q^39 + 12 * q^41 + 12 * q^43 - 18 * q^47 - 6 * q^51 + 16 * q^53 - 2 * q^55 + 4 * q^57 - 12 * q^59 - 8 * q^61 - 2 * q^65 + 16 * q^69 - 12 * q^71 - 16 * q^73 - 2 * q^75 + 18 * q^79 - 2 * q^81 - 2 * q^85 - 2 * q^87 + 8 * q^89 + 24 * q^93 + 4 * q^95 + 14 * q^97

Introduction

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