LMFDB - Embedded newform 3200.2.c.w.2049.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3200,2,Mod(2049,3200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3200, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3200.2049");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3200 = 2^{7} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3200.c (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$25.5521286468$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 640)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2049.4
Root			$1.61803i$ of defining polynomial
Character	$\chi$	$=$	3200.2049
Dual form			3200.2.c.w.2049.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.23607i q^{3} +1.23607i q^{7} -7.47214 q^{9} +O(q^{10})$	$q+3.23607i q^{3} +1.23607i q^{7} -7.47214 q^{9} +2.00000 q^{11} +4.47214i q^{13} +4.47214i q^{17} +4.47214 q^{19} -4.00000 q^{21} +9.23607i q^{23} -14.4721i q^{27} +2.00000 q^{29} +2.47214 q^{31} +6.47214i q^{33} +10.9443i q^{37} -14.4721 q^{39} +3.52786 q^{41} -5.70820i q^{43} -2.76393i q^{47} +5.47214 q^{49} -14.4721 q^{51} -8.47214i q^{53} +14.4721i q^{57} +0.472136 q^{59} -6.00000 q^{61} -9.23607i q^{63} -5.70820i q^{67} -29.8885 q^{69} -6.47214 q^{71} -4.47214i q^{73} +2.47214i q^{77} -4.94427 q^{79} +24.4164 q^{81} -9.70820i q^{83} +6.47214i q^{87} -2.94427 q^{89} -5.52786 q^{91} +8.00000i q^{93} +7.52786i q^{97} -14.9443 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{9}+O(q^{10}) $ 4 * q - 12 * q^9	$ 4 q - 12 q^{9} + 8 q^{11} - 16 q^{21} + 8 q^{29} - 8 q^{31} - 40 q^{39} + 32 q^{41} + 4 q^{49} - 40 q^{51} - 16 q^{59} - 24 q^{61} - 48 q^{69} - 8 q^{71} + 16 q^{79} + 44 q^{81} + 24 q^{89} - 40 q^{91} - 24 q^{99}+O(q^{100}) $ 4 * q - 12 * q^9 + 8 * q^11 - 16 * q^21 + 8 * q^29 - 8 * q^31 - 40 * q^39 + 32 * q^41 + 4 * q^49 - 40 * q^51 - 16 * q^59 - 24 * q^61 - 48 * q^69 - 8 * q^71 + 16 * q^79 + 44 * q^81 + 24 * q^89 - 40 * q^91 - 24 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3200\mathbb{Z}\right)^\times$.

$n$	$901$	$1151$	$2177$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	3.23607i	1.86834i	0.356822	+	0.934172i	$0.383860\pi$
$3$	3.23607i	1.86834i	−0.356822	+	0.934172i	$0.616140\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.23607i	0.467190i	0.972334	+	0.233595i	$0.0750489\pi$
$7$	1.23607i	0.467190i	−0.972334	+	0.233595i	$0.924951\pi$
$8$	0	0
$9$	−7.47214	−2.49071
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	4.47214i	1.24035i	0.784465	+	0.620174i	$0.212938\pi$
$13$	4.47214i	1.24035i	−0.784465	+	0.620174i	$0.787062\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.47214i	1.08465i	0.840168	+	0.542326i	$0.182456\pi$
$17$	4.47214i	1.08465i	−0.840168	+	0.542326i	$0.817544\pi$
$18$	0	0
$19$	4.47214	1.02598	0.512989	−	0.858395i	$-0.328538\pi$
$19$	4.47214	1.02598	0.512989	+	0.858395i	$0.328538\pi$
$20$	0	0
$21$	−4.00000	−0.872872
$22$	0	0
$23$	9.23607i	1.92585i	0.269763	+	0.962927i	$0.413055\pi$
$23$	9.23607i	1.92585i	−0.269763	+	0.962927i	$0.586945\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 14.4721i	− 2.78516i
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	2.47214	0.444009	0.222004	−	0.975046i	$-0.428740\pi$
$31$	2.47214	0.444009	0.222004	+	0.975046i	$0.428740\pi$
$32$	0	0
$33$	6.47214i	1.12665i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	10.9443i	1.79923i	0.436687	+	0.899614i	$0.356152\pi$
$37$	10.9443i	1.79923i	−0.436687	+	0.899614i	$0.643848\pi$
$38$	0	0
$39$	−14.4721	−2.31740
$40$	0	0
$41$	3.52786	0.550960	0.275480	−	0.961307i	$-0.411163\pi$
$41$	3.52786	0.550960	0.275480	+	0.961307i	$0.411163\pi$
$42$	0	0
$43$	− 5.70820i	− 0.870493i	−0.900311	−	0.435246i	$-0.856661\pi$
$43$	− 5.70820i	− 0.870493i	0.900311	−	0.435246i	$-0.143339\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 2.76393i	− 0.403161i	−0.979472	−	0.201580i	$-0.935392\pi$
$47$	− 2.76393i	− 0.403161i	0.979472	−	0.201580i	$-0.0646078\pi$
$48$	0	0
$49$	5.47214	0.781734
$50$	0	0
$51$	−14.4721	−2.02650
$52$	0	0
$53$	− 8.47214i	− 1.16374i	−0.813283	−	0.581869i	$-0.802322\pi$
$53$	− 8.47214i	− 1.16374i	0.813283	−	0.581869i	$-0.197678\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	14.4721i	1.91688i
$58$	0	0
$59$	0.472136	0.0614669	0.0307334	−	0.999528i	$-0.490216\pi$
$59$	0.472136	0.0614669	0.0307334	+	0.999528i	$0.490216\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	0	0
$63$	− 9.23607i	− 1.16364i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 5.70820i	− 0.697368i	−0.937240	−	0.348684i	$-0.886629\pi$
$67$	− 5.70820i	− 0.697368i	0.937240	−	0.348684i	$-0.113371\pi$
$68$	0	0
$69$	−29.8885	−3.59816
$70$	0	0
$71$	−6.47214	−0.768101	−0.384051	−	0.923312i	$-0.625471\pi$
$71$	−6.47214	−0.768101	−0.384051	+	0.923312i	$0.625471\pi$
$72$	0	0
$73$	− 4.47214i	− 0.523424i	−0.965146	−	0.261712i	$-0.915713\pi$
$73$	− 4.47214i	− 0.523424i	0.965146	−	0.261712i	$-0.0842870\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.47214i	0.281726i
$78$	0	0
$79$	−4.94427	−0.556274	−0.278137	−	0.960541i	$-0.589717\pi$
$79$	−4.94427	−0.556274	−0.278137	+	0.960541i	$0.589717\pi$
$80$	0	0
$81$	24.4164	2.71293
$82$	0	0
$83$	− 9.70820i	− 1.06561i	−0.846237	−	0.532807i	$-0.821137\pi$
$83$	− 9.70820i	− 1.06561i	0.846237	−	0.532807i	$-0.178863\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	6.47214i	0.693886i
$88$	0	0
$89$	−2.94427	−0.312092	−0.156046	−	0.987750i	$-0.549875\pi$
$89$	−2.94427	−0.312092	−0.156046	+	0.987750i	$0.549875\pi$
$90$	0	0
$91$	−5.52786	−0.579478
$92$	0	0
$93$	8.00000i	0.829561i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	7.52786i	0.764339i	0.924092	+	0.382169i	$0.124823\pi$
$97$	7.52786i	0.764339i	−0.924092	+	0.382169i	$0.875177\pi$
$98$	0	0
$99$	−14.9443	−1.50196
$100$	0	0
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	0	0
$103$	1.23607i	0.121793i	0.998144	+	0.0608967i	$0.0193960\pi$
$103$	1.23607i	0.121793i	−0.998144	+	0.0608967i	$0.980604\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 12.7639i	− 1.23394i	−0.786988	−	0.616968i	$-0.788361\pi$
$107$	− 12.7639i	− 1.23394i	0.786988	−	0.616968i	$-0.211639\pi$
$108$	0	0
$109$	2.94427	0.282010	0.141005	−	0.990009i	$-0.454967\pi$
$109$	2.94427	0.282010	0.141005	+	0.990009i	$0.454967\pi$
$110$	0	0
$111$	−35.4164	−3.36158
$112$	0	0
$113$	14.9443i	1.40584i	0.711269	+	0.702919i	$0.248121\pi$
$113$	14.9443i	1.40584i	−0.711269	+	0.702919i	$0.751879\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 33.4164i	− 3.08935i
$118$	0	0
$119$	−5.52786	−0.506738
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	11.4164i	1.02938i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	2.18034i	0.193474i	0.995310	+	0.0967369i	$0.0308406\pi$
$127$	2.18034i	0.193474i	−0.995310	+	0.0967369i	$0.969159\pi$
$128$	0	0
$129$	18.4721	1.62638
$130$	0	0
$131$	2.94427	0.257242	0.128621	−	0.991694i	$-0.458945\pi$
$131$	2.94427	0.257242	0.128621	+	0.991694i	$0.458945\pi$
$132$	0	0
$133$	5.52786i	0.479327i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 2.00000i	− 0.170872i	−0.996344	−	0.0854358i	$-0.972772\pi$
$137$	− 2.00000i	− 0.170872i	0.996344	−	0.0854358i	$-0.0272282\pi$
$138$	0	0
$139$	11.5279	0.977781	0.488890	−	0.872345i	$-0.337402\pi$
$139$	11.5279	0.977781	0.488890	+	0.872345i	$0.337402\pi$
$140$	0	0
$141$	8.94427	0.753244
$142$	0	0
$143$	8.94427i	0.747958i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	17.7082i	1.46055i
$148$	0	0
$149$	10.9443	0.896590	0.448295	−	0.893886i	$-0.352031\pi$
$149$	10.9443	0.896590	0.448295	+	0.893886i	$0.352031\pi$
$150$	0	0
$151$	3.41641	0.278023	0.139012	−	0.990291i	$-0.455607\pi$
$151$	3.41641	0.278023	0.139012	+	0.990291i	$0.455607\pi$
$152$	0	0
$153$	− 33.4164i	− 2.70156i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 22.9443i	− 1.83115i	−0.402145	−	0.915576i	$-0.631735\pi$
$157$	− 22.9443i	− 1.83115i	0.402145	−	0.915576i	$-0.368265\pi$
$158$	0	0
$159$	27.4164	2.17426
$160$	0	0
$161$	−11.4164	−0.899739
$162$	0	0
$163$	− 1.70820i	− 0.133797i	−0.997760	−	0.0668984i	$-0.978690\pi$
$163$	− 1.70820i	− 0.133797i	0.997760	−	0.0668984i	$-0.0213103\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 16.6525i	− 1.28861i	−0.764770	−	0.644304i	$-0.777147\pi$
$167$	− 16.6525i	− 1.28861i	0.764770	−	0.644304i	$-0.222853\pi$
$168$	0	0
$169$	−7.00000	−0.538462
$170$	0	0
$171$	−33.4164	−2.55542
$172$	0	0
$173$	6.94427i	0.527963i	0.964528	+	0.263982i	$0.0850358\pi$
$173$	6.94427i	0.527963i	−0.964528	+	0.263982i	$0.914964\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1.52786i	0.114841i
$178$	0	0
$179$	−11.5279	−0.861633	−0.430817	−	0.902440i	$-0.641774\pi$
$179$	−11.5279	−0.861633	−0.430817	+	0.902440i	$0.641774\pi$
$180$	0	0
$181$	−6.94427	−0.516164	−0.258082	−	0.966123i	$-0.583090\pi$
$181$	−6.94427	−0.516164	−0.258082	+	0.966123i	$0.583090\pi$
$182$	0	0
$183$	− 19.4164i	− 1.43530i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	8.94427i	0.654070i
$188$	0	0
$189$	17.8885	1.30120
$190$	0	0
$191$	15.4164	1.11549	0.557746	−	0.830012i	$-0.311666\pi$
$191$	15.4164	1.11549	0.557746	+	0.830012i	$0.311666\pi$
$192$	0	0
$193$	− 20.4721i	− 1.47362i	−0.676102	−	0.736808i	$-0.736332\pi$
$193$	− 20.4721i	− 1.47362i	0.676102	−	0.736808i	$-0.263668\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	8.47214i	0.603615i	0.953369	+	0.301807i	$0.0975899\pi$
$197$	8.47214i	0.603615i	−0.953369	+	0.301807i	$0.902410\pi$
$198$	0	0
$199$	16.9443	1.20115	0.600574	−	0.799569i	$-0.294939\pi$
$199$	16.9443	1.20115	0.600574	+	0.799569i	$0.294939\pi$
$200$	0	0
$201$	18.4721	1.30292
$202$	0	0
$203$	2.47214i	0.173510i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 69.0132i	− 4.79675i
$208$	0	0
$209$	8.94427	0.618688
$210$	0	0
$211$	18.9443	1.30418	0.652089	−	0.758143i	$-0.273893\pi$
$211$	18.9443	1.30418	0.652089	+	0.758143i	$0.273893\pi$
$212$	0	0
$213$	− 20.9443i	− 1.43508i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	3.05573i	0.207436i
$218$	0	0
$219$	14.4721	0.977936
$220$	0	0
$221$	−20.0000	−1.34535
$222$	0	0
$223$	13.2361i	0.886353i	0.896434	+	0.443176i	$0.146148\pi$
$223$	13.2361i	0.886353i	−0.896434	+	0.443176i	$0.853852\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	12.1803i	0.808438i	0.914662	+	0.404219i	$0.132457\pi$
$227$	12.1803i	0.808438i	−0.914662	+	0.404219i	$0.867543\pi$
$228$	0	0
$229$	14.9443	0.987545	0.493773	−	0.869591i	$-0.335618\pi$
$229$	14.9443	0.987545	0.493773	+	0.869591i	$0.335618\pi$
$230$	0	0
$231$	−8.00000	−0.526361
$232$	0	0
$233$	24.4721i	1.60322i	0.597845	+	0.801611i	$0.296024\pi$
$233$	24.4721i	1.60322i	−0.597845	+	0.801611i	$0.703976\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 16.0000i	− 1.03931i
$238$	0	0
$239$	4.94427	0.319818	0.159909	−	0.987132i	$-0.448880\pi$
$239$	4.94427	0.319818	0.159909	+	0.987132i	$0.448880\pi$
$240$	0	0
$241$	−18.3607	−1.18272	−0.591358	−	0.806409i	$-0.701408\pi$
$241$	−18.3607	−1.18272	−0.591358	+	0.806409i	$0.701408\pi$
$242$	0	0
$243$	35.5967i	2.28353i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	20.0000i	1.27257i
$248$	0	0
$249$	31.4164	1.99093
$250$	0	0
$251$	2.00000	0.126239	0.0631194	−	0.998006i	$-0.479895\pi$
$251$	2.00000	0.126239	0.0631194	+	0.998006i	$0.479895\pi$
$252$	0	0
$253$	18.4721i	1.16133i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 14.9443i	− 0.932198i	−0.884733	−	0.466099i	$-0.845659\pi$
$257$	− 14.9443i	− 0.932198i	0.884733	−	0.466099i	$-0.154341\pi$
$258$	0	0
$259$	−13.5279	−0.840581
$260$	0	0
$261$	−14.9443	−0.925027
$262$	0	0
$263$	4.29180i	0.264643i	0.991207	+	0.132322i	$0.0422432\pi$
$263$	4.29180i	0.264643i	−0.991207	+	0.132322i	$0.957757\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 9.52786i	− 0.583096i
$268$	0	0
$269$	−11.8885	−0.724857	−0.362429	−	0.932012i	$-0.618052\pi$
$269$	−11.8885	−0.724857	−0.362429	+	0.932012i	$0.618052\pi$
$270$	0	0
$271$	−20.3607	−1.23682	−0.618412	−	0.785854i	$-0.712224\pi$
$271$	−20.3607	−1.23682	−0.618412	+	0.785854i	$0.712224\pi$
$272$	0	0
$273$	− 17.8885i	− 1.08266i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 6.94427i	− 0.417241i	−0.977997	−	0.208620i	$-0.933103\pi$
$277$	− 6.94427i	− 0.417241i	0.977997	−	0.208620i	$-0.0668973\pi$
$278$	0	0
$279$	−18.4721	−1.10590
$280$	0	0
$281$	12.4721	0.744025	0.372013	−	0.928228i	$-0.378668\pi$
$281$	12.4721	0.744025	0.372013	+	0.928228i	$0.378668\pi$
$282$	0	0
$283$	− 21.7082i	− 1.29042i	−0.764006	−	0.645209i	$-0.776770\pi$
$283$	− 21.7082i	− 1.29042i	0.764006	−	0.645209i	$-0.223230\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.36068i	0.257403i
$288$	0	0
$289$	−3.00000	−0.176471
$290$	0	0
$291$	−24.3607	−1.42805
$292$	0	0
$293$	11.8885i	0.694536i	0.937766	+	0.347268i	$0.112891\pi$
$293$	11.8885i	0.694536i	−0.937766	+	0.347268i	$0.887109\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 28.9443i	− 1.67952i
$298$	0	0
$299$	−41.3050	−2.38873
$300$	0	0
$301$	7.05573	0.406685
$302$	0	0
$303$	− 32.3607i	− 1.85907i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 13.7082i	− 0.782369i	−0.920312	−	0.391184i	$-0.872066\pi$
$307$	− 13.7082i	− 0.782369i	0.920312	−	0.391184i	$-0.127934\pi$
$308$	0	0
$309$	−4.00000	−0.227552
$310$	0	0
$311$	−11.4164	−0.647365	−0.323683	−	0.946166i	$-0.604921\pi$
$311$	−11.4164	−0.647365	−0.323683	+	0.946166i	$0.604921\pi$
$312$	0	0
$313$	24.8328i	1.40363i	0.712357	+	0.701817i	$0.247628\pi$
$313$	24.8328i	1.40363i	−0.712357	+	0.701817i	$0.752372\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	11.5279i	0.647469i	0.946148	+	0.323735i	$0.104939\pi$
$317$	11.5279i	0.647469i	−0.946148	+	0.323735i	$0.895061\pi$
$318$	0	0
$319$	4.00000	0.223957
$320$	0	0
$321$	41.3050	2.30542
$322$	0	0
$323$	20.0000i	1.11283i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	9.52786i	0.526892i
$328$	0	0
$329$	3.41641	0.188353
$330$	0	0
$331$	19.8885	1.09317	0.546587	−	0.837403i	$-0.315927\pi$
$331$	19.8885	1.09317	0.546587	+	0.837403i	$0.315927\pi$
$332$	0	0
$333$	− 81.7771i	− 4.48136i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 22.9443i	− 1.24985i	−0.780683	−	0.624927i	$-0.785129\pi$
$337$	− 22.9443i	− 1.24985i	0.780683	−	0.624927i	$-0.214871\pi$
$338$	0	0
$339$	−48.3607	−2.62659
$340$	0	0
$341$	4.94427	0.267747
$342$	0	0
$343$	15.4164i	0.832408i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 12.7639i	− 0.685204i	−0.939481	−	0.342602i	$-0.888692\pi$
$347$	− 12.7639i	− 0.685204i	0.939481	−	0.342602i	$-0.111308\pi$
$348$	0	0
$349$	−2.94427	−0.157603	−0.0788016	−	0.996890i	$-0.525109\pi$
$349$	−2.94427	−0.157603	−0.0788016	+	0.996890i	$0.525109\pi$
$350$	0	0
$351$	64.7214	3.45457
$352$	0	0
$353$	22.9443i	1.22120i	0.791939	+	0.610600i	$0.209072\pi$
$353$	22.9443i	1.22120i	−0.791939	+	0.610600i	$0.790928\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 17.8885i	− 0.946762i
$358$	0	0
$359$	−32.9443	−1.73873	−0.869366	−	0.494169i	$-0.835473\pi$
$359$	−32.9443	−1.73873	−0.869366	+	0.494169i	$0.835473\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	− 22.6525i	− 1.18895i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 31.7082i	− 1.65515i	−0.561352	−	0.827577i	$-0.689718\pi$
$367$	− 31.7082i	− 1.65515i	0.561352	−	0.827577i	$-0.310282\pi$
$368$	0	0
$369$	−26.3607	−1.37228
$370$	0	0
$371$	10.4721	0.543686
$372$	0	0
$373$	22.9443i	1.18801i	0.804462	+	0.594005i	$0.202454\pi$
$373$	22.9443i	1.18801i	−0.804462	+	0.594005i	$0.797546\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	8.94427i	0.460653i
$378$	0	0
$379$	24.4721	1.25705	0.628525	−	0.777790i	$-0.283659\pi$
$379$	24.4721	1.25705	0.628525	+	0.777790i	$0.283659\pi$
$380$	0	0
$381$	−7.05573	−0.361476
$382$	0	0
$383$	− 23.7082i	− 1.21143i	−0.795681	−	0.605716i	$-0.792887\pi$
$383$	− 23.7082i	− 1.21143i	0.795681	−	0.605716i	$-0.207113\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	42.6525i	2.16815i
$388$	0	0
$389$	23.8885	1.21120	0.605599	−	0.795770i	$-0.292934\pi$
$389$	23.8885	1.21120	0.605599	+	0.795770i	$0.292934\pi$
$390$	0	0
$391$	−41.3050	−2.08888
$392$	0	0
$393$	9.52786i	0.480617i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	3.52786i	0.177058i	0.996074	+	0.0885292i	$0.0282167\pi$
$397$	3.52786i	0.177058i	−0.996074	+	0.0885292i	$0.971783\pi$
$398$	0	0
$399$	−17.8885	−0.895547
$400$	0	0
$401$	−2.00000	−0.0998752	−0.0499376	−	0.998752i	$-0.515902\pi$
$401$	−2.00000	−0.0998752	−0.0499376	+	0.998752i	$0.515902\pi$
$402$	0	0
$403$	11.0557i	0.550725i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	21.8885i	1.08497i
$408$	0	0
$409$	9.41641	0.465611	0.232806	−	0.972523i	$-0.425209\pi$
$409$	9.41641	0.465611	0.232806	+	0.972523i	$0.425209\pi$
$410$	0	0
$411$	6.47214	0.319247
$412$	0	0
$413$	0.583592i	0.0287167i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	37.3050i	1.82683i
$418$	0	0
$419$	4.47214	0.218478	0.109239	−	0.994016i	$-0.465159\pi$
$419$	4.47214	0.218478	0.109239	+	0.994016i	$0.465159\pi$
$420$	0	0
$421$	24.8328	1.21028	0.605139	−	0.796120i	$-0.293118\pi$
$421$	24.8328	1.21028	0.605139	+	0.796120i	$0.293118\pi$
$422$	0	0
$423$	20.6525i	1.00416i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 7.41641i	− 0.358905i
$428$	0	0
$429$	−28.9443	−1.39744
$430$	0	0
$431$	−20.3607	−0.980739	−0.490370	−	0.871515i	$-0.663138\pi$
$431$	−20.3607	−0.980739	−0.490370	+	0.871515i	$0.663138\pi$
$432$	0	0
$433$	8.47214i	0.407145i	0.979060	+	0.203572i	$0.0652552\pi$
$433$	8.47214i	0.407145i	−0.979060	+	0.203572i	$0.934745\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	41.3050i	1.97588i
$438$	0	0
$439$	32.9443	1.57234	0.786172	−	0.618008i	$-0.212060\pi$
$439$	32.9443	1.57234	0.786172	+	0.618008i	$0.212060\pi$
$440$	0	0
$441$	−40.8885	−1.94707
$442$	0	0
$443$	− 26.6525i	− 1.26630i	−0.774030	−	0.633149i	$-0.781762\pi$
$443$	− 26.6525i	− 1.26630i	0.774030	−	0.633149i	$-0.218238\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	35.4164i	1.67514i
$448$	0	0
$449$	−25.4164	−1.19947	−0.599737	−	0.800197i	$-0.704728\pi$
$449$	−25.4164	−1.19947	−0.599737	+	0.800197i	$0.704728\pi$
$450$	0	0
$451$	7.05573	0.332241
$452$	0	0
$453$	11.0557i	0.519443i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	31.8885i	1.49168i	0.666123	+	0.745842i	$0.267952\pi$
$457$	31.8885i	1.49168i	−0.666123	+	0.745842i	$0.732048\pi$
$458$	0	0
$459$	64.7214	3.02093
$460$	0	0
$461$	−29.7771	−1.38686	−0.693429	−	0.720525i	$-0.743901\pi$
$461$	−29.7771	−1.38686	−0.693429	+	0.720525i	$0.743901\pi$
$462$	0	0
$463$	39.1246i	1.81827i	0.416496	+	0.909137i	$0.363258\pi$
$463$	39.1246i	1.81827i	−0.416496	+	0.909137i	$0.636742\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 7.59675i	− 0.351536i	−0.984432	−	0.175768i	$-0.943759\pi$
$467$	− 7.59675i	− 0.351536i	0.984432	−	0.175768i	$-0.0562408\pi$
$468$	0	0
$469$	7.05573	0.325803
$470$	0	0
$471$	74.2492	3.42122
$472$	0	0
$473$	− 11.4164i	− 0.524927i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	63.3050i	2.89853i
$478$	0	0
$479$	−4.94427	−0.225910	−0.112955	−	0.993600i	$-0.536032\pi$
$479$	−4.94427	−0.225910	−0.112955	+	0.993600i	$0.536032\pi$
$480$	0	0
$481$	−48.9443	−2.23167
$482$	0	0
$483$	− 36.9443i	− 1.68102i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	17.2361i	0.781041i	0.920594	+	0.390520i	$0.127705\pi$
$487$	17.2361i	0.781041i	−0.920594	+	0.390520i	$0.872295\pi$
$488$	0	0
$489$	5.52786	0.249979
$490$	0	0
$491$	38.9443	1.75753	0.878765	−	0.477254i	$-0.158368\pi$
$491$	38.9443	1.75753	0.878765	+	0.477254i	$0.158368\pi$
$492$	0	0
$493$	8.94427i	0.402830i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 8.00000i	− 0.358849i
$498$	0	0
$499$	41.4164	1.85405	0.927027	−	0.374996i	$-0.122356\pi$
$499$	41.4164	1.85405	0.927027	+	0.374996i	$0.122356\pi$
$500$	0	0
$501$	53.8885	2.40756
$502$	0	0
$503$	35.1246i	1.56613i	0.621940	+	0.783065i	$0.286345\pi$
$503$	35.1246i	1.56613i	−0.621940	+	0.783065i	$0.713655\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 22.6525i	− 1.00603i
$508$	0	0
$509$	−36.8328	−1.63259	−0.816293	−	0.577638i	$-0.803974\pi$
$509$	−36.8328	−1.63259	−0.816293	+	0.577638i	$0.803974\pi$
$510$	0	0
$511$	5.52786	0.244538
$512$	0	0
$513$	− 64.7214i	− 2.85752i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 5.52786i	− 0.243115i
$518$	0	0
$519$	−22.4721	−0.986417
$520$	0	0
$521$	19.8885	0.871333	0.435666	−	0.900108i	$-0.356513\pi$
$521$	19.8885	0.871333	0.435666	+	0.900108i	$0.356513\pi$
$522$	0	0
$523$	38.0689i	1.66464i	0.554298	+	0.832318i	$0.312987\pi$
$523$	38.0689i	1.66464i	−0.554298	+	0.832318i	$0.687013\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	11.0557i	0.481595i
$528$	0	0
$529$	−62.3050	−2.70891
$530$	0	0
$531$	−3.52786	−0.153096
$532$	0	0
$533$	15.7771i	0.683382i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 37.3050i	− 1.60983i
$538$	0	0
$539$	10.9443	0.471403
$540$	0	0
$541$	−11.8885	−0.511128	−0.255564	−	0.966792i	$-0.582261\pi$
$541$	−11.8885	−0.511128	−0.255564	+	0.966792i	$0.582261\pi$
$542$	0	0
$543$	− 22.4721i	− 0.964372i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 10.6525i	− 0.455467i	−0.973724	−	0.227733i	$-0.926869\pi$
$547$	− 10.6525i	− 0.455467i	0.973724	−	0.227733i	$-0.0731315\pi$
$548$	0	0
$549$	44.8328	1.91342
$550$	0	0
$551$	8.94427	0.381039
$552$	0	0
$553$	− 6.11146i	− 0.259886i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 30.9443i	− 1.31115i	−0.755130	−	0.655575i	$-0.772426\pi$
$557$	− 30.9443i	− 1.31115i	0.755130	−	0.655575i	$-0.227574\pi$
$558$	0	0
$559$	25.5279	1.07971
$560$	0	0
$561$	−28.9443	−1.22203
$562$	0	0
$563$	− 17.7082i	− 0.746312i	−0.927769	−	0.373156i	$-0.878276\pi$
$563$	− 17.7082i	− 0.746312i	0.927769	−	0.373156i	$-0.121724\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	30.1803i	1.26746i
$568$	0	0
$569$	−10.3607	−0.434342	−0.217171	−	0.976134i	$-0.569683\pi$
$569$	−10.3607	−0.434342	−0.217171	+	0.976134i	$0.569683\pi$
$570$	0	0
$571$	−36.8328	−1.54141	−0.770703	−	0.637195i	$-0.780095\pi$
$571$	−36.8328	−1.54141	−0.770703	+	0.637195i	$0.780095\pi$
$572$	0	0
$573$	49.8885i	2.08412i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	15.8885i	0.661449i	0.943727	+	0.330724i	$0.107293\pi$
$577$	15.8885i	0.661449i	−0.943727	+	0.330724i	$0.892707\pi$
$578$	0	0
$579$	66.2492	2.75322
$580$	0	0
$581$	12.0000	0.497844
$582$	0	0
$583$	− 16.9443i	− 0.701760i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 22.6525i	− 0.934968i	−0.884002	−	0.467484i	$-0.845161\pi$
$587$	− 22.6525i	− 0.934968i	0.884002	−	0.467484i	$-0.154839\pi$
$588$	0	0
$589$	11.0557	0.455543
$590$	0	0
$591$	−27.4164	−1.12776
$592$	0	0
$593$	− 14.0000i	− 0.574911i	−0.957794	−	0.287456i	$-0.907191\pi$
$593$	− 14.0000i	− 0.574911i	0.957794	−	0.287456i	$-0.0928094\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	54.8328i	2.24416i
$598$	0	0
$599$	13.8885	0.567471	0.283735	−	0.958903i	$-0.408426\pi$
$599$	13.8885	0.567471	0.283735	+	0.958903i	$0.408426\pi$
$600$	0	0
$601$	22.3607	0.912111	0.456056	−	0.889951i	$-0.349262\pi$
$601$	22.3607	0.912111	0.456056	+	0.889951i	$0.349262\pi$
$602$	0	0
$603$	42.6525i	1.73694i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	47.1246i	1.91273i	0.292176	+	0.956364i	$0.405621\pi$
$607$	47.1246i	1.91273i	−0.292176	+	0.956364i	$0.594379\pi$
$608$	0	0
$609$	−8.00000	−0.324176
$610$	0	0
$611$	12.3607	0.500060
$612$	0	0
$613$	28.4721i	1.14998i	0.818161	+	0.574989i	$0.194994\pi$
$613$	28.4721i	1.14998i	−0.818161	+	0.574989i	$0.805006\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.41641i	0.0570224i	0.999593	+	0.0285112i	$0.00907663\pi$
$617$	1.41641i	0.0570224i	−0.999593	+	0.0285112i	$0.990923\pi$
$618$	0	0
$619$	11.5279	0.463344	0.231672	−	0.972794i	$-0.425580\pi$
$619$	11.5279	0.463344	0.231672	+	0.972794i	$0.425580\pi$
$620$	0	0
$621$	133.666	5.36382
$622$	0	0
$623$	− 3.63932i	− 0.145806i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	28.9443i	1.15592i
$628$	0	0
$629$	−48.9443	−1.95154
$630$	0	0
$631$	25.5279	1.01625	0.508124	−	0.861284i	$-0.330339\pi$
$631$	25.5279	1.01625	0.508124	+	0.861284i	$0.330339\pi$
$632$	0	0
$633$	61.3050i	2.43665i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	24.4721i	0.969621i
$638$	0	0
$639$	48.3607	1.91312
$640$	0	0
$641$	−8.47214	−0.334629	−0.167315	−	0.985904i	$-0.553510\pi$
$641$	−8.47214	−0.334629	−0.167315	+	0.985904i	$0.553510\pi$
$642$	0	0
$643$	0.180340i	0.00711191i	0.999994	+	0.00355596i	$0.00113190\pi$
$643$	0.180340i	0.00711191i	−0.999994	+	0.00355596i	$0.998868\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	17.2361i	0.677620i	0.940855	+	0.338810i	$0.110024\pi$
$647$	17.2361i	0.677620i	−0.940855	+	0.338810i	$0.889976\pi$
$648$	0	0
$649$	0.944272	0.0370659
$650$	0	0
$651$	−9.88854	−0.387563
$652$	0	0
$653$	− 22.5836i	− 0.883764i	−0.897073	−	0.441882i	$-0.854311\pi$
$653$	− 22.5836i	− 0.883764i	0.897073	−	0.441882i	$-0.145689\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	33.4164i	1.30370i
$658$	0	0
$659$	−5.41641	−0.210993	−0.105497	−	0.994420i	$-0.533643\pi$
$659$	−5.41641	−0.210993	−0.105497	+	0.994420i	$0.533643\pi$
$660$	0	0
$661$	0.111456	0.00433514	0.00216757	−	0.999998i	$-0.499310\pi$
$661$	0.111456	0.00433514	0.00216757	+	0.999998i	$0.499310\pi$
$662$	0	0
$663$	− 64.7214i	− 2.51357i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	18.4721i	0.715244i
$668$	0	0
$669$	−42.8328	−1.65601
$670$	0	0
$671$	−12.0000	−0.463255
$672$	0	0
$673$	39.3050i	1.51509i	0.652780	+	0.757547i	$0.273602\pi$
$673$	39.3050i	1.51509i	−0.652780	+	0.757547i	$0.726398\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 14.3607i	− 0.551926i	−0.961168	−	0.275963i	$-0.911003\pi$
$677$	− 14.3607i	− 0.551926i	0.961168	−	0.275963i	$-0.0889967\pi$
$678$	0	0
$679$	−9.30495	−0.357091
$680$	0	0
$681$	−39.4164	−1.51044
$682$	0	0
$683$	− 37.7082i	− 1.44286i	−0.692485	−	0.721432i	$-0.743484\pi$
$683$	− 37.7082i	− 1.44286i	0.692485	−	0.721432i	$-0.256516\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	48.3607i	1.84508i
$688$	0	0
$689$	37.8885	1.44344
$690$	0	0
$691$	30.0000	1.14125	0.570627	−	0.821209i	$-0.306700\pi$
$691$	30.0000	1.14125	0.570627	+	0.821209i	$0.306700\pi$
$692$	0	0
$693$	− 18.4721i	− 0.701698i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	15.7771i	0.597600i
$698$	0	0
$699$	−79.1935	−2.99537
$700$	0	0
$701$	−26.9443	−1.01767	−0.508836	−	0.860864i	$-0.669924\pi$
$701$	−26.9443	−1.01767	−0.508836	+	0.860864i	$0.669924\pi$
$702$	0	0
$703$	48.9443i	1.84597i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 12.3607i	− 0.464871i
$708$	0	0
$709$	18.0000	0.676004	0.338002	−	0.941145i	$-0.390249\pi$
$709$	18.0000	0.676004	0.338002	+	0.941145i	$0.390249\pi$
$710$	0	0
$711$	36.9443	1.38552
$712$	0	0
$713$	22.8328i	0.855096i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	16.0000i	0.597531i
$718$	0	0
$719$	−30.8328	−1.14987	−0.574935	−	0.818199i	$-0.694973\pi$
$719$	−30.8328	−1.14987	−0.574935	+	0.818199i	$0.694973\pi$
$720$	0	0
$721$	−1.52786	−0.0569006
$722$	0	0
$723$	− 59.4164i	− 2.20972i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 11.7082i	− 0.434233i	−0.976146	−	0.217117i	$-0.930335\pi$
$727$	− 11.7082i	− 0.434233i	0.976146	−	0.217117i	$-0.0696652\pi$
$728$	0	0
$729$	−41.9443	−1.55349
$730$	0	0
$731$	25.5279	0.944182
$732$	0	0
$733$	16.1115i	0.595090i	0.954708	+	0.297545i	$0.0961679\pi$
$733$	16.1115i	0.595090i	−0.954708	+	0.297545i	$0.903832\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 11.4164i	− 0.420529i
$738$	0	0
$739$	−49.1935	−1.80961	−0.904806	−	0.425824i	$-0.859984\pi$
$739$	−49.1935	−1.80961	−0.904806	+	0.425824i	$0.859984\pi$
$740$	0	0
$741$	−64.7214	−2.37760
$742$	0	0
$743$	− 0.652476i	− 0.0239370i	−0.999928	−	0.0119685i	$-0.996190\pi$
$743$	− 0.652476i	− 0.0239370i	0.999928	−	0.0119685i	$-0.00380979\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	72.5410i	2.65414i
$748$	0	0
$749$	15.7771	0.576482
$750$	0	0
$751$	28.3607	1.03490	0.517448	−	0.855715i	$-0.326882\pi$
$751$	28.3607	1.03490	0.517448	+	0.855715i	$0.326882\pi$
$752$	0	0
$753$	6.47214i	0.235858i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	15.8885i	0.577479i	0.957408	+	0.288739i	$0.0932361\pi$
$757$	15.8885i	0.577479i	−0.957408	+	0.288739i	$0.906764\pi$
$758$	0	0
$759$	−59.7771	−2.16977
$760$	0	0
$761$	−31.8885	−1.15596	−0.577979	−	0.816051i	$-0.696159\pi$
$761$	−31.8885	−1.15596	−0.577979	+	0.816051i	$0.696159\pi$
$762$	0	0
$763$	3.63932i	0.131752i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.11146i	0.0762403i
$768$	0	0
$769$	−2.94427	−0.106173	−0.0530866	−	0.998590i	$-0.516906\pi$
$769$	−2.94427	−0.106173	−0.0530866	+	0.998590i	$0.516906\pi$
$770$	0	0
$771$	48.3607	1.74167
$772$	0	0
$773$	− 26.3607i	− 0.948128i	−0.880490	−	0.474064i	$-0.842787\pi$
$773$	− 26.3607i	− 0.948128i	0.880490	−	0.474064i	$-0.157213\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 43.7771i	− 1.57049i
$778$	0	0
$779$	15.7771	0.565273
$780$	0	0
$781$	−12.9443	−0.463182
$782$	0	0
$783$	− 28.9443i	− 1.03438i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	14.0689i	0.501502i	0.968052	+	0.250751i	$0.0806775\pi$
$787$	14.0689i	0.501502i	−0.968052	+	0.250751i	$0.919323\pi$
$788$	0	0
$789$	−13.8885	−0.494445
$790$	0	0
$791$	−18.4721	−0.656794
$792$	0	0
$793$	− 26.8328i	− 0.952861i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 1.41641i	− 0.0501717i	−0.999685	−	0.0250859i	$-0.992014\pi$
$797$	− 1.41641i	− 0.0501717i	0.999685	−	0.0250859i	$-0.00798592\pi$
$798$	0	0
$799$	12.3607	0.437289
$800$	0	0
$801$	22.0000	0.777332
$802$	0	0
$803$	− 8.94427i	− 0.315637i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 38.4721i	− 1.35428i
$808$	0	0
$809$	47.8885	1.68367	0.841836	−	0.539734i	$-0.181475\pi$
$809$	47.8885	1.68367	0.841836	+	0.539734i	$0.181475\pi$
$810$	0	0
$811$	2.00000	0.0702295	0.0351147	−	0.999383i	$-0.488820\pi$
$811$	2.00000	0.0702295	0.0351147	+	0.999383i	$0.488820\pi$
$812$	0	0
$813$	− 65.8885i	− 2.31081i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 25.5279i	− 0.893107i
$818$	0	0
$819$	41.3050	1.44331
$820$	0	0
$821$	38.9443	1.35916	0.679582	−	0.733599i	$-0.262161\pi$
$821$	38.9443	1.35916	0.679582	+	0.733599i	$0.262161\pi$
$822$	0	0
$823$	− 11.7082i	− 0.408122i	−0.978958	−	0.204061i	$-0.934586\pi$
$823$	− 11.7082i	− 0.408122i	0.978958	−	0.204061i	$-0.0654141\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	16.1803i	0.562646i	0.959613	+	0.281323i	$0.0907732\pi$
$827$	16.1803i	0.562646i	−0.959613	+	0.281323i	$0.909227\pi$
$828$	0	0
$829$	38.0000	1.31979	0.659897	−	0.751356i	$-0.270600\pi$
$829$	38.0000	1.31979	0.659897	+	0.751356i	$0.270600\pi$
$830$	0	0
$831$	22.4721	0.779550
$832$	0	0
$833$	24.4721i	0.847909i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 35.7771i	− 1.23664i
$838$	0	0
$839$	4.00000	0.138095	0.0690477	−	0.997613i	$-0.478004\pi$
$839$	4.00000	0.138095	0.0690477	+	0.997613i	$0.478004\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	40.3607i	1.39010i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 8.65248i	− 0.297303i
$848$	0	0
$849$	70.2492	2.41095
$850$	0	0
$851$	−101.082	−3.46505
$852$	0	0
$853$	− 16.4721i	− 0.563995i	−0.959415	−	0.281998i	$-0.909003\pi$
$853$	− 16.4721i	− 0.563995i	0.959415	−	0.281998i	$-0.0909970\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	28.8328i	0.984910i	0.870338	+	0.492455i	$0.163900\pi$
$857$	28.8328i	0.984910i	−0.870338	+	0.492455i	$0.836100\pi$
$858$	0	0
$859$	22.5836	0.770542	0.385271	−	0.922803i	$-0.374108\pi$
$859$	22.5836	0.770542	0.385271	+	0.922803i	$0.374108\pi$
$860$	0	0
$861$	−14.1115	−0.480917
$862$	0	0
$863$	− 31.7082i	− 1.07936i	−0.841870	−	0.539680i	$-0.818545\pi$
$863$	− 31.7082i	− 1.07936i	0.841870	−	0.539680i	$-0.181455\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 9.70820i	− 0.329708i
$868$	0	0
$869$	−9.88854	−0.335446
$870$	0	0
$871$	25.5279	0.864979
$872$	0	0
$873$	− 56.2492i	− 1.90375i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 16.1115i	− 0.544045i	−0.962291	−	0.272023i	$-0.912307\pi$
$877$	− 16.1115i	− 0.544045i	0.962291	−	0.272023i	$-0.0876926\pi$
$878$	0	0
$879$	−38.4721	−1.29763
$880$	0	0
$881$	−15.5279	−0.523147	−0.261574	−	0.965184i	$-0.584241\pi$
$881$	−15.5279	−0.523147	−0.261574	+	0.965184i	$0.584241\pi$
$882$	0	0
$883$	14.2918i	0.480957i	0.970654	+	0.240479i	$0.0773044\pi$
$883$	14.2918i	0.480957i	−0.970654	+	0.240479i	$0.922696\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 37.5967i	− 1.26238i	−0.775630	−	0.631188i	$-0.782568\pi$
$887$	− 37.5967i	− 1.26238i	0.775630	−	0.631188i	$-0.217432\pi$
$888$	0	0
$889$	−2.69505	−0.0903890
$890$	0	0
$891$	48.8328	1.63596
$892$	0	0
$893$	− 12.3607i	− 0.413634i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 133.666i	− 4.46297i
$898$	0	0
$899$	4.94427	0.164901
$900$	0	0
$901$	37.8885	1.26225
$902$	0	0
$903$	22.8328i	0.759829i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 4.76393i	− 0.158184i	−0.996867	−	0.0790919i	$-0.974798\pi$
$907$	− 4.76393i	− 0.158184i	0.996867	−	0.0790919i	$-0.0252021\pi$
$908$	0	0
$909$	74.7214	2.47835
$910$	0	0
$911$	−9.30495	−0.308287	−0.154143	−	0.988048i	$-0.549262\pi$
$911$	−9.30495	−0.308287	−0.154143	+	0.988048i	$0.549262\pi$
$912$	0	0
$913$	− 19.4164i	− 0.642589i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	3.63932i	0.120181i
$918$	0	0
$919$	29.8885	0.985932	0.492966	−	0.870049i	$-0.335913\pi$
$919$	29.8885	0.985932	0.492966	+	0.870049i	$0.335913\pi$
$920$	0	0
$921$	44.3607	1.46173
$922$	0	0
$923$	− 28.9443i	− 0.952712i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 9.23607i	− 0.303352i
$928$	0	0
$929$	−15.5279	−0.509453	−0.254726	−	0.967013i	$-0.581985\pi$
$929$	−15.5279	−0.509453	−0.254726	+	0.967013i	$0.581985\pi$
$930$	0	0
$931$	24.4721	0.802042
$932$	0	0
$933$	− 36.9443i	− 1.20950i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	12.4721i	0.407447i	0.979028	+	0.203723i	$0.0653043\pi$
$937$	12.4721i	0.407447i	−0.979028	+	0.203723i	$0.934696\pi$
$938$	0	0
$939$	−80.3607	−2.62247
$940$	0	0
$941$	−8.83282	−0.287942	−0.143971	−	0.989582i	$-0.545987\pi$
$941$	−8.83282	−0.287942	−0.143971	+	0.989582i	$0.545987\pi$
$942$	0	0
$943$	32.5836i	1.06107i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	44.1803i	1.43567i	0.696214	+	0.717834i	$0.254866\pi$
$947$	44.1803i	1.43567i	−0.696214	+	0.717834i	$0.745134\pi$
$948$	0	0
$949$	20.0000	0.649227
$950$	0	0
$951$	−37.3050	−1.20970
$952$	0	0
$953$	− 25.7771i	− 0.835002i	−0.908677	−	0.417501i	$-0.862906\pi$
$953$	− 25.7771i	− 0.835002i	0.908677	−	0.417501i	$-0.137094\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	12.9443i	0.418429i
$958$	0	0
$959$	2.47214	0.0798294
$960$	0	0
$961$	−24.8885	−0.802856
$962$	0	0
$963$	95.3738i	3.07338i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 32.6525i	− 1.05003i	−0.851092	−	0.525016i	$-0.824059\pi$
$967$	− 32.6525i	− 1.05003i	0.851092	−	0.525016i	$-0.175941\pi$
$968$	0	0
$969$	−64.7214	−2.07915
$970$	0	0
$971$	11.8885	0.381522	0.190761	−	0.981637i	$-0.438905\pi$
$971$	11.8885	0.381522	0.190761	+	0.981637i	$0.438905\pi$
$972$	0	0
$973$	14.2492i	0.456809i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 23.3050i	− 0.745591i	−0.927913	−	0.372796i	$-0.878399\pi$
$977$	− 23.3050i	− 0.745591i	0.927913	−	0.372796i	$-0.121601\pi$
$978$	0	0
$979$	−5.88854	−0.188199
$980$	0	0
$981$	−22.0000	−0.702406
$982$	0	0
$983$	37.0132i	1.18054i	0.807207	+	0.590268i	$0.200978\pi$
$983$	37.0132i	1.18054i	−0.807207	+	0.590268i	$0.799022\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	11.0557i	0.351908i
$988$	0	0
$989$	52.7214	1.67644
$990$	0	0
$991$	26.4721	0.840915	0.420458	−	0.907312i	$-0.361870\pi$
$991$	26.4721	0.840915	0.420458	+	0.907312i	$0.361870\pi$
$992$	0	0
$993$	64.3607i	2.04242i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	61.4164i	1.94508i	0.232742	+	0.972539i	$0.425230\pi$
$997$	61.4164i	1.94508i	−0.232742	+	0.972539i	$0.574770\pi$
$998$	0	0
$999$	158.387	5.01114

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3200.2.c.w.2049.4		4
4.3	odd	2		3200.2.c.u.2049.1		4
5.2	odd	4		640.2.a.k.1.2	yes	2
5.3	odd	4		3200.2.a.be.1.1		2
5.4	even	2	inner	3200.2.c.w.2049.1		4
8.3	odd	2		3200.2.c.x.2049.4		4
8.5	even	2		3200.2.c.v.2049.1		4
15.2	even	4		5760.2.a.ci.1.1		2
20.3	even	4		3200.2.a.bl.1.2		2
20.7	even	4		640.2.a.i.1.1	✓	2
20.19	odd	2		3200.2.c.u.2049.4		4
40.3	even	4		3200.2.a.bf.1.1		2
40.13	odd	4		3200.2.a.bk.1.2		2
40.19	odd	2		3200.2.c.x.2049.1		4
40.27	even	4		640.2.a.l.1.2	yes	2
40.29	even	2		3200.2.c.v.2049.4		4
40.37	odd	4		640.2.a.j.1.1	yes	2
60.47	odd	4		5760.2.a.ch.1.2		2
80.27	even	4		1280.2.d.m.641.4		4
80.37	odd	4		1280.2.d.k.641.1		4
80.67	even	4		1280.2.d.m.641.1		4
80.77	odd	4		1280.2.d.k.641.4		4
120.77	even	4		5760.2.a.cd.1.1		2
120.107	odd	4		5760.2.a.bw.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
640.2.a.i.1.1	✓	2	20.7	even	4
640.2.a.j.1.1	yes	2	40.37	odd	4
640.2.a.k.1.2	yes	2	5.2	odd	4
640.2.a.l.1.2	yes	2	40.27	even	4
1280.2.d.k.641.1		4	80.37	odd	4
1280.2.d.k.641.4		4	80.77	odd	4
1280.2.d.m.641.1		4	80.67	even	4
1280.2.d.m.641.4		4	80.27	even	4
3200.2.a.be.1.1		2	5.3	odd	4
3200.2.a.bf.1.1		2	40.3	even	4
3200.2.a.bk.1.2		2	40.13	odd	4
3200.2.a.bl.1.2		2	20.3	even	4
3200.2.c.u.2049.1		4	4.3	odd	2
3200.2.c.u.2049.4		4	20.19	odd	2
3200.2.c.v.2049.1		4	8.5	even	2
3200.2.c.v.2049.4		4	40.29	even	2
3200.2.c.w.2049.1		4	5.4	even	2	inner
3200.2.c.w.2049.4		4	1.1	even	1	trivial
3200.2.c.x.2049.1		4	40.19	odd	2
3200.2.c.x.2049.4		4	8.3	odd	2
5760.2.a.bw.1.2		2	120.107	odd	4
5760.2.a.cd.1.1		2	120.77	even	4
5760.2.a.ch.1.2		2	60.47	odd	4
5760.2.a.ci.1.1		2	15.2	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3200 = 2^{7} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3200.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.23607i q^{3} +1.23607i q^{7} -7.47214 q^{9} +O(q^{10})\)	\(q+3.23607i q^{3} +1.23607i q^{7} -7.47214 q^{9} +2.00000 q^{11} +4.47214i q^{13} +4.47214i q^{17} +4.47214 q^{19} -4.00000 q^{21} +9.23607i q^{23} -14.4721i q^{27} +2.00000 q^{29} +2.47214 q^{31} +6.47214i q^{33} +10.9443i q^{37} -14.4721 q^{39} +3.52786 q^{41} -5.70820i q^{43} -2.76393i q^{47} +5.47214 q^{49} -14.4721 q^{51} -8.47214i q^{53} +14.4721i q^{57} +0.472136 q^{59} -6.00000 q^{61} -9.23607i q^{63} -5.70820i q^{67} -29.8885 q^{69} -6.47214 q^{71} -4.47214i q^{73} +2.47214i q^{77} -4.94427 q^{79} +24.4164 q^{81} -9.70820i q^{83} +6.47214i q^{87} -2.94427 q^{89} -5.52786 q^{91} +8.00000i q^{93} +7.52786i q^{97} -14.9443 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{9}+O(q^{10}) \) 4 * q - 12 * q^9	\( 4 q - 12 q^{9} + 8 q^{11} - 16 q^{21} + 8 q^{29} - 8 q^{31} - 40 q^{39} + 32 q^{41} + 4 q^{49} - 40 q^{51} - 16 q^{59} - 24 q^{61} - 48 q^{69} - 8 q^{71} + 16 q^{79} + 44 q^{81} + 24 q^{89} - 40 q^{91} - 24 q^{99}+O(q^{100}) \) 4 * q - 12 * q^9 + 8 * q^11 - 16 * q^21 + 8 * q^29 - 8 * q^31 - 40 * q^39 + 32 * q^41 + 4 * q^49 - 40 * q^51 - 16 * q^59 - 24 * q^61 - 48 * q^69 - 8 * q^71 + 16 * q^79 + 44 * q^81 + 24 * q^89 - 40 * q^91 - 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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