LMFDB - Embedded newform 800.6.c.g.449.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,6,Mod(449,800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(800, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("800.449");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	800.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:	$128.307055850$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{70})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1225 $ x^4 + 1225
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 160)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.1
Root			$-4.18330 + 4.18330i$ of defining polynomial
Character	$\chi$	$=$	800.449
Dual form			800.6.c.g.449.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-20.7332i q^{3} +35.2668i q^{7} -186.866 q^{9} +O(q^{10})$	$q-20.7332i q^{3} +35.2668i q^{7} -186.866 q^{9} -7.33201 q^{11} -619.328i q^{13} +959.328i q^{17} +309.328 q^{19} +731.194 q^{21} -2467.12i q^{23} -1163.85i q^{27} +1284.66 q^{29} +7095.32 q^{31} +152.016i q^{33} -6100.59i q^{37} -12840.7 q^{39} -18830.4 q^{41} -3147.48i q^{43} +20556.8i q^{47} +15563.3 q^{49} +19889.9 q^{51} -33741.9i q^{53} -6413.36i q^{57} -15065.3 q^{59} -7542.11 q^{61} -6590.15i q^{63} +25574.9i q^{67} -51151.3 q^{69} -56232.3 q^{71} -58657.5i q^{73} -258.576i q^{77} -32238.7 q^{79} -69538.6 q^{81} -31173.4i q^{83} -26635.0i q^{87} +75258.9 q^{89} +21841.7 q^{91} -147109. i q^{93} -176059. i q^{97} +1370.10 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 212 q^{9}+O(q^{10}) $ 4 * q - 212 * q^9	$ 4 q - 212 q^{9} + 640 q^{11} - 1440 q^{19} - 288 q^{21} - 216 q^{29} + 19680 q^{31} - 44000 q^{39} - 21240 q^{41} + 55292 q^{49} + 49440 q^{51} - 61600 q^{59} + 49080 q^{61} - 122144 q^{69} - 24800 q^{71} - 143680 q^{79} - 204796 q^{81} + 81496 q^{89} - 55200 q^{91} + 55680 q^{99}+O(q^{100}) $ 4 * q - 212 * q^9 + 640 * q^11 - 1440 * q^19 - 288 * q^21 - 216 * q^29 + 19680 * q^31 - 44000 * q^39 - 21240 * q^41 + 55292 * q^49 + 49440 * q^51 - 61600 * q^59 + 49080 * q^61 - 122144 * q^69 - 24800 * q^71 - 143680 * q^79 - 204796 * q^81 + 81496 * q^89 - 55200 * q^91 + 55680 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/800\mathbb{Z}\right)^\times$.

$n$	$101$	$351$	$577$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	− 20.7332i	− 1.33004i	−0.746828	−	0.665018i	$-0.768424\pi$
$3$	− 20.7332i	− 1.33004i	0.746828	−	0.665018i	$-0.231576\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	35.2668i	0.272033i	0.990707	+	0.136016i	$0.0434300\pi$
$7$	35.2668i	0.272033i	−0.990707	+	0.136016i	$0.956570\pi$
$8$	0	0
$9$	−186.866	−0.768994
$10$	0	0
$11$	−7.33201	−0.0182701	−0.00913505	−	0.999958i	$-0.502908\pi$
$11$	−7.33201	−0.0182701	−0.00913505	+	0.999958i	$0.502908\pi$
$12$	0	0
$13$	− 619.328i	− 1.01639i	−0.861241	−	0.508197i	$-0.830312\pi$
$13$	− 619.328i	− 1.01639i	0.861241	−	0.508197i	$-0.169688\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	959.328i	0.805091i	0.915400	+	0.402545i	$0.131874\pi$
$17$	959.328i	0.805091i	−0.915400	+	0.402545i	$0.868126\pi$
$18$	0	0
$19$	309.328	0.196578	0.0982891	−	0.995158i	$-0.468663\pi$
$19$	309.328	0.196578	0.0982891	+	0.995158i	$0.468663\pi$
$20$	0	0
$21$	731.194	0.361813
$22$	0	0
$23$	− 2467.12i	− 0.972458i	−0.873831	−	0.486229i	$-0.838372\pi$
$23$	− 2467.12i	− 0.972458i	0.873831	−	0.486229i	$-0.161628\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1163.85i	− 0.307246i
$28$	0	0
$29$	1284.66	0.283656	0.141828	−	0.989891i	$-0.454702\pi$
$29$	1284.66	0.283656	0.141828	+	0.989891i	$0.454702\pi$
$30$	0	0
$31$	7095.32	1.32607	0.663037	−	0.748587i	$-0.269267\pi$
$31$	7095.32	1.32607	0.663037	+	0.748587i	$0.269267\pi$
$32$	0	0
$33$	152.016i	0.0242999i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 6100.59i	− 0.732601i	−0.930497	−	0.366301i	$-0.880624\pi$
$37$	− 6100.59i	− 0.732601i	0.930497	−	0.366301i	$-0.119376\pi$
$38$	0	0
$39$	−12840.7	−1.35184
$40$	0	0
$41$	−18830.4	−1.74945	−0.874723	−	0.484623i	$-0.838957\pi$
$41$	−18830.4	−1.74945	−0.874723	+	0.484623i	$0.838957\pi$
$42$	0	0
$43$	− 3147.48i	− 0.259592i	−0.991541	−	0.129796i	$-0.958568\pi$
$43$	− 3147.48i	− 0.259592i	0.991541	−	0.129796i	$-0.0414323\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	20556.8i	1.35741i	0.734411	+	0.678705i	$0.237459\pi$
$47$	20556.8i	1.35741i	−0.734411	+	0.678705i	$0.762541\pi$
$48$	0	0
$49$	15563.3	0.925998
$50$	0	0
$51$	19889.9	1.07080
$52$	0	0
$53$	− 33741.9i	− 1.64998i	−0.565146	−	0.824991i	$-0.691180\pi$
$53$	− 33741.9i	− 1.64998i	0.565146	−	0.824991i	$-0.308820\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 6413.36i	− 0.261456i
$58$	0	0
$59$	−15065.3	−0.563441	−0.281721	−	0.959496i	$-0.590905\pi$
$59$	−15065.3	−0.563441	−0.281721	+	0.959496i	$0.590905\pi$
$60$	0	0
$61$	−7542.11	−0.259518	−0.129759	−	0.991546i	$-0.541420\pi$
$61$	−7542.11	−0.259518	−0.129759	+	0.991546i	$0.541420\pi$
$62$	0	0
$63$	− 6590.15i	− 0.209192i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	25574.9i	0.696029i	0.937489	+	0.348015i	$0.113144\pi$
$67$	25574.9i	0.696029i	−0.937489	+	0.348015i	$0.886856\pi$
$68$	0	0
$69$	−51151.3	−1.29340
$70$	0	0
$71$	−56232.3	−1.32385	−0.661926	−	0.749569i	$-0.730261\pi$
$71$	−56232.3	−1.32385	−0.661926	+	0.749569i	$0.730261\pi$
$72$	0	0
$73$	− 58657.5i	− 1.28830i	−0.764900	−	0.644149i	$-0.777212\pi$
$73$	− 58657.5i	− 1.28830i	0.764900	−	0.644149i	$-0.222788\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 258.576i	− 0.00497006i
$78$	0	0
$79$	−32238.7	−0.581179	−0.290589	−	0.956848i	$-0.593851\pi$
$79$	−32238.7	−0.581179	−0.290589	+	0.956848i	$0.593851\pi$
$80$	0	0
$81$	−69538.6	−1.17764
$82$	0	0
$83$	− 31173.4i	− 0.496694i	−0.968671	−	0.248347i	$-0.920113\pi$
$83$	− 31173.4i	− 0.496694i	0.968671	−	0.248347i	$-0.0798874\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 26635.0i	− 0.377272i
$88$	0	0
$89$	75258.9	1.00712	0.503562	−	0.863959i	$-0.332023\pi$
$89$	75258.9	1.00712	0.503562	+	0.863959i	$0.332023\pi$
$90$	0	0
$91$	21841.7	0.276492
$92$	0	0
$93$	− 147109.i	− 1.76372i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 176059.i	− 1.89989i	−0.312419	−	0.949944i	$-0.601139\pi$
$97$	− 176059.i	− 1.89989i	0.312419	−	0.949944i	$-0.398861\pi$
$98$	0	0
$99$	1370.10	0.0140496
$100$	0	0
$101$	−102454.	−0.999370	−0.499685	−	0.866207i	$-0.666551\pi$
$101$	−102454.	−0.999370	−0.499685	+	0.866207i	$0.666551\pi$
$102$	0	0
$103$	150320.i	1.39612i	0.716038	+	0.698062i	$0.245954\pi$
$103$	150320.i	1.39612i	−0.716038	+	0.698062i	$0.754046\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 220857.i	− 1.86488i	−0.361321	−	0.932441i	$-0.617674\pi$
$107$	− 220857.i	− 1.86488i	0.361321	−	0.932441i	$-0.382326\pi$
$108$	0	0
$109$	24621.0	0.198490	0.0992450	−	0.995063i	$-0.468357\pi$
$109$	24621.0	0.198490	0.0992450	+	0.995063i	$0.468357\pi$
$110$	0	0
$111$	−126485.	−0.974386
$112$	0	0
$113$	64568.7i	0.475692i	0.971303	+	0.237846i	$0.0764413\pi$
$113$	64568.7i	0.475692i	−0.971303	+	0.237846i	$0.923559\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	115731.i	0.781602i
$118$	0	0
$119$	−33832.4	−0.219011
$120$	0	0
$121$	−160997.	−0.999666
$122$	0	0
$123$	390415.i	2.32682i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	159599.i	0.878054i	0.898474	+	0.439027i	$0.144677\pi$
$127$	159599.i	0.878054i	−0.898474	+	0.439027i	$0.855323\pi$
$128$	0	0
$129$	−65257.3	−0.345267
$130$	0	0
$131$	116112.	0.591153	0.295577	−	0.955319i	$-0.404488\pi$
$131$	116112.	0.591153	0.295577	+	0.955319i	$0.404488\pi$
$132$	0	0
$133$	10909.0i	0.0534757i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 1446.13i	− 0.00658272i	−0.999995	−	0.00329136i	$-0.998952\pi$
$137$	− 1446.13i	− 0.00658272i	0.999995	−	0.00329136i	$-0.00104767\pi$
$138$	0	0
$139$	135055.	0.592891	0.296446	−	0.955050i	$-0.404199\pi$
$139$	135055.	0.592891	0.296446	+	0.955050i	$0.404199\pi$
$140$	0	0
$141$	426209.	1.80540
$142$	0	0
$143$	4540.92i	0.0185696i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 322676.i	− 1.23161i
$148$	0	0
$149$	−16097.8	−0.0594019	−0.0297010	−	0.999559i	$-0.509455\pi$
$149$	−16097.8	−0.0594019	−0.0297010	+	0.999559i	$0.509455\pi$
$150$	0	0
$151$	−467353.	−1.66802	−0.834012	−	0.551746i	$-0.813962\pi$
$151$	−467353.	−1.66802	−0.834012	+	0.551746i	$0.813962\pi$
$152$	0	0
$153$	− 179265.i	− 0.619110i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	210478.i	0.681487i	0.940156	+	0.340743i	$0.110679\pi$
$157$	210478.i	0.681487i	−0.940156	+	0.340743i	$0.889321\pi$
$158$	0	0
$159$	−699577.	−2.19454
$160$	0	0
$161$	87007.4	0.264540
$162$	0	0
$163$	136230.i	0.401610i	0.979631	+	0.200805i	$0.0643557\pi$
$163$	136230.i	0.401610i	−0.979631	+	0.200805i	$0.935644\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	246713.i	0.684542i	0.939601	+	0.342271i	$0.111196\pi$
$167$	246713.i	0.684542i	−0.939601	+	0.342271i	$0.888804\pi$
$168$	0	0
$169$	−12274.2	−0.0330580
$170$	0	0
$171$	−57802.8	−0.151167
$172$	0	0
$173$	652180.i	1.65673i	0.560187	+	0.828366i	$0.310729\pi$
$173$	652180.i	1.65673i	−0.560187	+	0.828366i	$0.689271\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	312353.i	0.749397i
$178$	0	0
$179$	−786542.	−1.83480	−0.917402	−	0.397962i	$-0.869718\pi$
$179$	−786542.	−1.83480	−0.917402	+	0.397962i	$0.869718\pi$
$180$	0	0
$181$	−974.027	−0.00220991	−0.00110495	−	0.999999i	$-0.500352\pi$
$181$	−974.027	−0.00220991	−0.00110495	+	0.999999i	$0.500352\pi$
$182$	0	0
$183$	156372.i	0.345169i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 7033.80i	− 0.0147091i
$188$	0	0
$189$	41045.1	0.0835809
$190$	0	0
$191$	−469675.	−0.931567	−0.465783	−	0.884899i	$-0.654227\pi$
$191$	−469675.	−0.931567	−0.465783	+	0.884899i	$0.654227\pi$
$192$	0	0
$193$	644811.i	1.24606i	0.782198	+	0.623030i	$0.214099\pi$
$193$	644811.i	1.24606i	−0.782198	+	0.623030i	$0.785901\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 521778.i	− 0.957899i	−0.877842	−	0.478950i	$-0.841018\pi$
$197$	− 521778.i	− 0.957899i	0.877842	−	0.478950i	$-0.158982\pi$
$198$	0	0
$199$	−394042.	−0.705358	−0.352679	−	0.935744i	$-0.614729\pi$
$199$	−394042.	−0.705358	−0.352679	+	0.935744i	$0.614729\pi$
$200$	0	0
$201$	530250.	0.925744
$202$	0	0
$203$	45305.7i	0.0771637i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	461020.i	0.747814i
$208$	0	0
$209$	−2267.99	−0.00359150
$210$	0	0
$211$	750041.	1.15979	0.579894	−	0.814692i	$-0.303094\pi$
$211$	750041.	1.15979	0.579894	+	0.814692i	$0.303094\pi$
$212$	0	0
$213$	1.16587e6i	1.76077i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	250229.i	0.360735i
$218$	0	0
$219$	−1.21616e6	−1.71348
$220$	0	0
$221$	594139.	0.818290
$222$	0	0
$223$	1.00328e6i	1.35102i	0.737351	+	0.675509i	$0.236076\pi$
$223$	1.00328e6i	1.35102i	−0.737351	+	0.675509i	$0.763924\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	385367.i	0.496375i	0.968712	+	0.248188i	$0.0798349\pi$
$227$	385367.i	0.496375i	−0.968712	+	0.248188i	$0.920165\pi$
$228$	0	0
$229$	16601.6	0.0209200	0.0104600	−	0.999945i	$-0.496670\pi$
$229$	16601.6	0.0209200	0.0104600	+	0.999945i	$0.496670\pi$
$230$	0	0
$231$	−5361.12	−0.00661036
$232$	0	0
$233$	− 933464.i	− 1.12644i	−0.826307	−	0.563219i	$-0.809563\pi$
$233$	− 933464.i	− 1.12644i	0.826307	−	0.563219i	$-0.190437\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	668411.i	0.772988i
$238$	0	0
$239$	−1.34120e6	−1.51880	−0.759398	−	0.650626i	$-0.774507\pi$
$239$	−1.34120e6	−1.51880	−0.759398	+	0.650626i	$0.774507\pi$
$240$	0	0
$241$	390139.	0.432690	0.216345	−	0.976317i	$-0.430586\pi$
$241$	390139.	0.432690	0.216345	+	0.976317i	$0.430586\pi$
$242$	0	0
$243$	1.15894e6i	1.25906i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 191576.i	− 0.199801i
$248$	0	0
$249$	−646325.	−0.660621
$250$	0	0
$251$	805856.	0.807371	0.403686	−	0.914898i	$-0.367729\pi$
$251$	805856.	0.807371	0.403686	+	0.914898i	$0.367729\pi$
$252$	0	0
$253$	18088.9i	0.0177669i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	153621.i	0.145084i	0.997365	+	0.0725419i	$0.0231111\pi$
$257$	153621.i	0.145084i	−0.997365	+	0.0725419i	$0.976889\pi$
$258$	0	0
$259$	215148.	0.199291
$260$	0	0
$261$	−240058.	−0.218130
$262$	0	0
$263$	28007.0i	0.0249676i	0.999922	+	0.0124838i	$0.00397382\pi$
$263$	28007.0i	0.0249676i	−0.999922	+	0.0124838i	$0.996026\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 1.56036e6i	− 1.33951i
$268$	0	0
$269$	−715607.	−0.602968	−0.301484	−	0.953471i	$-0.597482\pi$
$269$	−715607.	−0.602968	−0.301484	+	0.953471i	$0.597482\pi$
$270$	0	0
$271$	−1.35674e6	−1.12221	−0.561104	−	0.827745i	$-0.689623\pi$
$271$	−1.35674e6	−1.12221	−0.561104	+	0.827745i	$0.689623\pi$
$272$	0	0
$273$	− 452849.i	− 0.367745i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 791037.i	− 0.619437i	−0.950828	−	0.309719i	$-0.899765\pi$
$277$	− 791037.i	− 0.619437i	0.950828	−	0.309719i	$-0.100235\pi$
$278$	0	0
$279$	−1.32587e6	−1.01974
$280$	0	0
$281$	−191136.	−0.144403	−0.0722016	−	0.997390i	$-0.523002\pi$
$281$	−191136.	−0.144403	−0.0722016	+	0.997390i	$0.523002\pi$
$282$	0	0
$283$	1.16924e6i	0.867834i	0.900953	+	0.433917i	$0.142869\pi$
$283$	1.16924e6i	0.867834i	−0.900953	+	0.433917i	$0.857131\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 664089.i	− 0.475906i
$288$	0	0
$289$	499547.	0.351829
$290$	0	0
$291$	−3.65026e6	−2.52692
$292$	0	0
$293$	932811.i	0.634782i	0.948295	+	0.317391i	$0.102807\pi$
$293$	932811.i	0.634782i	−0.948295	+	0.317391i	$0.897193\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	8533.32i	0.00561341i
$298$	0	0
$299$	−1.52796e6	−0.988401
$300$	0	0
$301$	111001.	0.0706175
$302$	0	0
$303$	2.12420e6i	1.32920i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 678740.i	− 0.411015i	−0.978656	−	0.205507i	$-0.934116\pi$
$307$	− 678740.i	− 0.411015i	0.978656	−	0.205507i	$-0.0658844\pi$
$308$	0	0
$309$	3.11662e6	1.85689
$310$	0	0
$311$	1.25645e6	0.736621	0.368311	−	0.929703i	$-0.379936\pi$
$311$	1.25645e6	0.736621	0.368311	+	0.929703i	$0.379936\pi$
$312$	0	0
$313$	− 3.13670e6i	− 1.80972i	−0.425708	−	0.904861i	$-0.639975\pi$
$313$	− 3.13670e6i	− 1.80972i	0.425708	−	0.904861i	$-0.360025\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	353375.i	0.197510i	0.995112	+	0.0987548i	$0.0314859\pi$
$317$	353375.i	0.197510i	−0.995112	+	0.0987548i	$0.968514\pi$
$318$	0	0
$319$	−9419.10	−0.00518242
$320$	0	0
$321$	−4.57907e6	−2.48036
$322$	0	0
$323$	296747.i	0.158263i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 510471.i	− 0.263999i
$328$	0	0
$329$	−724973.	−0.369260
$330$	0	0
$331$	−2.91561e6	−1.46271	−0.731356	−	0.681996i	$-0.761112\pi$
$331$	−2.91561e6	−1.46271	−0.731356	+	0.681996i	$0.761112\pi$
$332$	0	0
$333$	1.13999e6i	0.563366i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 3.11496e6i	− 1.49409i	−0.664772	−	0.747046i	$-0.731471\pi$
$337$	− 3.11496e6i	− 1.49409i	0.664772	−	0.747046i	$-0.268529\pi$
$338$	0	0
$339$	1.33871e6	0.632687
$340$	0	0
$341$	−52022.9	−0.0242275
$342$	0	0
$343$	1.14160e6i	0.523934i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4.09755e6i	1.82684i	0.407020	+	0.913419i	$0.366568\pi$
$347$	4.09755e6i	1.82684i	−0.407020	+	0.913419i	$0.633432\pi$
$348$	0	0
$349$	−2.46309e6	−1.08247	−0.541237	−	0.840870i	$-0.682044\pi$
$349$	−2.46309e6	−1.08247	−0.541237	+	0.840870i	$0.682044\pi$
$350$	0	0
$351$	−720802.	−0.312283
$352$	0	0
$353$	− 3.56682e6i	− 1.52350i	−0.647868	−	0.761752i	$-0.724339\pi$
$353$	− 3.56682e6i	− 1.52350i	0.647868	−	0.761752i	$-0.275661\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	701455.i	0.291292i
$358$	0	0
$359$	648318.	0.265492	0.132746	−	0.991150i	$-0.457620\pi$
$359$	648318.	0.265492	0.132746	+	0.991150i	$0.457620\pi$
$360$	0	0
$361$	−2.38042e6	−0.961357
$362$	0	0
$363$	3.33799e6i	1.32959i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 3.84494e6i	− 1.49013i	−0.666992	−	0.745065i	$-0.732419\pi$
$367$	− 3.84494e6i	− 1.49013i	0.666992	−	0.745065i	$-0.267581\pi$
$368$	0	0
$369$	3.51876e6	1.34531
$370$	0	0
$371$	1.18997e6	0.448849
$372$	0	0
$373$	− 2.40550e6i	− 0.895228i	−0.894227	−	0.447614i	$-0.852274\pi$
$373$	− 2.40550e6i	− 0.895228i	0.894227	−	0.447614i	$-0.147726\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 795623.i	− 0.288306i
$378$	0	0
$379$	−5.03689e6	−1.80121	−0.900605	−	0.434638i	$-0.856876\pi$
$379$	−5.03689e6	−1.80121	−0.900605	+	0.434638i	$0.856876\pi$
$380$	0	0
$381$	3.30900e6	1.16784
$382$	0	0
$383$	1.61371e6i	0.562121i	0.959690	+	0.281060i	$0.0906861\pi$
$383$	1.61371e6i	0.562121i	−0.959690	+	0.281060i	$0.909314\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	588155.i	0.199625i
$388$	0	0
$389$	−1.99508e6	−0.668477	−0.334238	−	0.942489i	$-0.608479\pi$
$389$	−1.99508e6	−0.668477	−0.334238	+	0.942489i	$0.608479\pi$
$390$	0	0
$391$	2.36678e6	0.782917
$392$	0	0
$393$	− 2.40738e6i	− 0.786255i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	3.61674e6i	1.15170i	0.817554	+	0.575852i	$0.195329\pi$
$397$	3.61674e6i	1.15170i	−0.817554	+	0.575852i	$0.804671\pi$
$398$	0	0
$399$	226179.	0.0711245
$400$	0	0
$401$	−2.64534e6	−0.821525	−0.410763	−	0.911742i	$-0.634738\pi$
$401$	−2.64534e6	−0.821525	−0.410763	+	0.911742i	$0.634738\pi$
$402$	0	0
$403$	− 4.39433e6i	− 1.34781i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	44729.6i	0.0133847i
$408$	0	0
$409$	−2.86360e6	−0.846456	−0.423228	−	0.906023i	$-0.639103\pi$
$409$	−2.86360e6	−0.846456	−0.423228	+	0.906023i	$0.639103\pi$
$410$	0	0
$411$	−29982.9	−0.00875525
$412$	0	0
$413$	− 531306.i	− 0.153274i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 2.80013e6i	− 0.788566i
$418$	0	0
$419$	−1.09856e6	−0.305695	−0.152847	−	0.988250i	$-0.548844\pi$
$419$	−1.09856e6	−0.305695	−0.152847	+	0.988250i	$0.548844\pi$
$420$	0	0
$421$	2.31987e6	0.637909	0.318955	−	0.947770i	$-0.396668\pi$
$421$	2.31987e6	0.637909	0.318955	+	0.947770i	$0.396668\pi$
$422$	0	0
$423$	− 3.84136e6i	− 1.04384i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 265986.i	− 0.0705975i
$428$	0	0
$429$	94147.7	0.0246983
$430$	0	0
$431$	5.05072e6	1.30967	0.654833	−	0.755774i	$-0.272739\pi$
$431$	5.05072e6	1.30967	0.654833	+	0.755774i	$0.272739\pi$
$432$	0	0
$433$	1.01025e6i	0.258946i	0.991583	+	0.129473i	$0.0413286\pi$
$433$	1.01025e6i	0.258946i	−0.991583	+	0.129473i	$0.958671\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 763149.i	− 0.191164i
$438$	0	0
$439$	5.14635e6	1.27450	0.637248	−	0.770659i	$-0.280073\pi$
$439$	5.14635e6	1.27450	0.637248	+	0.770659i	$0.280073\pi$
$440$	0	0
$441$	−2.90824e6	−0.712087
$442$	0	0
$443$	− 5.25603e6i	− 1.27247i	−0.771494	−	0.636236i	$-0.780490\pi$
$443$	− 5.25603e6i	− 1.27247i	0.771494	−	0.636236i	$-0.219510\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	333759.i	0.0790067i
$448$	0	0
$449$	4.10261e6	0.960384	0.480192	−	0.877163i	$-0.340567\pi$
$449$	4.10261e6	0.960384	0.480192	+	0.877163i	$0.340567\pi$
$450$	0	0
$451$	138065.	0.0319626
$452$	0	0
$453$	9.68972e6i	2.21853i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 4.01053e6i	− 0.898278i	−0.893462	−	0.449139i	$-0.851731\pi$
$457$	− 4.01053e6i	− 0.898278i	0.893462	−	0.449139i	$-0.148269\pi$
$458$	0	0
$459$	1.11651e6	0.247361
$460$	0	0
$461$	3.83280e6	0.839971	0.419986	−	0.907531i	$-0.362035\pi$
$461$	3.83280e6	0.839971	0.419986	+	0.907531i	$0.362035\pi$
$462$	0	0
$463$	− 1.14398e6i	− 0.248009i	−0.992282	−	0.124004i	$-0.960426\pi$
$463$	− 1.14398e6i	− 0.248009i	0.992282	−	0.124004i	$-0.0395737\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	7.12577e6i	1.51196i	0.654596	+	0.755979i	$0.272839\pi$
$467$	7.12577e6i	1.51196i	−0.654596	+	0.755979i	$0.727161\pi$
$468$	0	0
$469$	−901946.	−0.189343
$470$	0	0
$471$	4.36388e6	0.906401
$472$	0	0
$473$	23077.3i	0.00474277i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	6.30519e6i	1.26883i
$478$	0	0
$479$	−6.45039e6	−1.28454	−0.642269	−	0.766479i	$-0.722007\pi$
$479$	−6.45039e6	−1.28454	−0.642269	+	0.766479i	$0.722007\pi$
$480$	0	0
$481$	−3.77827e6	−0.744612
$482$	0	0
$483$	− 1.80394e6i	− 0.351848i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 1.46947e6i	− 0.280762i	−0.990098	−	0.140381i	$-0.955167\pi$
$487$	− 1.46947e6i	− 0.280762i	0.990098	−	0.140381i	$-0.0448327\pi$
$488$	0	0
$489$	2.82449e6	0.534155
$490$	0	0
$491$	−5.19488e6	−0.972460	−0.486230	−	0.873831i	$-0.661628\pi$
$491$	−5.19488e6	−0.972460	−0.486230	+	0.873831i	$0.661628\pi$
$492$	0	0
$493$	1.23241e6i	0.228369i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 1.98313e6i	− 0.360131i
$498$	0	0
$499$	7.07689e6	1.27230	0.636152	−	0.771564i	$-0.280525\pi$
$499$	7.07689e6	1.27230	0.636152	+	0.771564i	$0.280525\pi$
$500$	0	0
$501$	5.11514e6	0.910465
$502$	0	0
$503$	− 8.92098e6i	− 1.57215i	−0.618134	−	0.786073i	$-0.712111\pi$
$503$	− 8.92098e6i	− 1.57215i	0.618134	−	0.786073i	$-0.287889\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	254483.i	0.0439683i
$508$	0	0
$509$	−5.79714e6	−0.991789	−0.495895	−	0.868383i	$-0.665160\pi$
$509$	−5.79714e6	−0.991789	−0.495895	+	0.868383i	$0.665160\pi$
$510$	0	0
$511$	2.06866e6	0.350459
$512$	0	0
$513$	− 360010.i	− 0.0603978i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 150723.i	− 0.0248000i
$518$	0	0
$519$	1.35218e7	2.20351
$520$	0	0
$521$	7.45721e6	1.20360	0.601800	−	0.798647i	$-0.294450\pi$
$521$	7.45721e6	1.20360	0.601800	+	0.798647i	$0.294450\pi$
$522$	0	0
$523$	1.01015e7i	1.61485i	0.589971	+	0.807424i	$0.299139\pi$
$523$	1.01015e7i	1.61485i	−0.589971	+	0.807424i	$0.700861\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6.80674e6i	1.06761i
$528$	0	0
$529$	349660.	0.0543258
$530$	0	0
$531$	2.81519e6	0.433283
$532$	0	0
$533$	1.16622e7i	1.77813i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	1.63075e7i	2.44035i
$538$	0	0
$539$	−114110.	−0.0169181
$540$	0	0
$541$	1.23616e7	1.81585	0.907925	−	0.419132i	$-0.137666\pi$
$541$	1.23616e7	1.81585	0.907925	+	0.419132i	$0.137666\pi$
$542$	0	0
$543$	20194.7i	0.00293926i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 3.72735e6i	− 0.532637i	−0.963885	−	0.266319i	$-0.914193\pi$
$547$	− 3.72735e6i	− 0.532637i	0.963885	−	0.266319i	$-0.0858073\pi$
$548$	0	0
$549$	1.40936e6	0.199568
$550$	0	0
$551$	397380.	0.0557606
$552$	0	0
$553$	− 1.13696e6i	− 0.158100i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 1.68294e6i	− 0.229842i	−0.993375	−	0.114921i	$-0.963338\pi$
$557$	− 1.68294e6i	− 0.229842i	0.993375	−	0.114921i	$-0.0366615\pi$
$558$	0	0
$559$	−1.94932e6	−0.263848
$560$	0	0
$561$	−145833.	−0.0195636
$562$	0	0
$563$	− 1.31501e7i	− 1.74848i	−0.485498	−	0.874238i	$-0.661362\pi$
$563$	− 1.31501e7i	− 1.74848i	0.485498	−	0.874238i	$-0.338638\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 2.45240e6i	− 0.320357i
$568$	0	0
$569$	−6.91087e6	−0.894854	−0.447427	−	0.894320i	$-0.647660\pi$
$569$	−6.91087e6	−0.894854	−0.447427	+	0.894320i	$0.647660\pi$
$570$	0	0
$571$	1.49642e6	0.192072	0.0960361	−	0.995378i	$-0.469384\pi$
$571$	1.49642e6	0.192072	0.0960361	+	0.995378i	$0.469384\pi$
$572$	0	0
$573$	9.73787e6i	1.23902i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 2.81857e6i	− 0.352444i	−0.984350	−	0.176222i	$-0.943612\pi$
$577$	− 2.81857e6i	− 0.352444i	0.984350	−	0.176222i	$-0.0563876\pi$
$578$	0	0
$579$	1.33690e7	1.65730
$580$	0	0
$581$	1.09939e6	0.135117
$582$	0	0
$583$	247395.i	0.0301454i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 9.61995e6i	− 1.15233i	−0.817333	−	0.576166i	$-0.804548\pi$
$587$	− 9.61995e6i	− 1.15233i	0.817333	−	0.576166i	$-0.195452\pi$
$588$	0	0
$589$	2.19478e6	0.260677
$590$	0	0
$591$	−1.08181e7	−1.27404
$592$	0	0
$593$	− 7.39724e6i	− 0.863839i	−0.901912	−	0.431920i	$-0.857836\pi$
$593$	− 7.39724e6i	− 0.863839i	0.901912	−	0.431920i	$-0.142164\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	8.16975e6i	0.938152i
$598$	0	0
$599$	1.14938e7	1.30887	0.654434	−	0.756119i	$-0.272907\pi$
$599$	1.14938e7	1.30887	0.654434	+	0.756119i	$0.272907\pi$
$600$	0	0
$601$	−7.58526e6	−0.856612	−0.428306	−	0.903634i	$-0.640889\pi$
$601$	−7.58526e6	−0.856612	−0.428306	+	0.903634i	$0.640889\pi$
$602$	0	0
$603$	− 4.77908e6i	− 0.535243i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 2.36915e6i	− 0.260988i	−0.991449	−	0.130494i	$-0.958344\pi$
$607$	− 2.36915e6i	− 0.260988i	0.991449	−	0.130494i	$-0.0416563\pi$
$608$	0	0
$609$	939332.	0.102630
$610$	0	0
$611$	1.27314e7	1.37966
$612$	0	0
$613$	1.05992e7i	1.13926i	0.821902	+	0.569629i	$0.192913\pi$
$613$	1.05992e7i	1.13926i	−0.821902	+	0.569629i	$0.807087\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 1.22009e7i	− 1.29026i	−0.764072	−	0.645131i	$-0.776803\pi$
$617$	− 1.22009e7i	− 1.29026i	0.764072	−	0.645131i	$-0.223197\pi$
$618$	0	0
$619$	1.46019e7	1.53173	0.765865	−	0.643001i	$-0.222311\pi$
$619$	1.46019e7	1.53173	0.765865	+	0.643001i	$0.222311\pi$
$620$	0	0
$621$	−2.87135e6	−0.298784
$622$	0	0
$623$	2.65414e6i	0.273970i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	47022.8i	0.00477683i
$628$	0	0
$629$	5.85247e6	0.589811
$630$	0	0
$631$	−9.40997e6	−0.940838	−0.470419	−	0.882443i	$-0.655897\pi$
$631$	−9.40997e6	−0.940838	−0.470419	+	0.882443i	$0.655897\pi$
$632$	0	0
$633$	− 1.55507e7i	− 1.54256i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 9.63876e6i	− 0.941180i
$638$	0	0
$639$	1.05079e7	1.01804
$640$	0	0
$641$	−8.44166e6	−0.811489	−0.405744	−	0.913987i	$-0.632988\pi$
$641$	−8.44166e6	−0.811489	−0.405744	+	0.913987i	$0.632988\pi$
$642$	0	0
$643$	− 1.03157e7i	− 0.983942i	−0.870612	−	0.491971i	$-0.836277\pi$
$643$	− 1.03157e7i	− 0.983942i	0.870612	−	0.491971i	$-0.163723\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	3.52118e6i	0.330695i	0.986235	+	0.165348i	$0.0528746\pi$
$647$	3.52118e6i	0.330695i	−0.986235	+	0.165348i	$0.947125\pi$
$648$	0	0
$649$	110459.	0.0102941
$650$	0	0
$651$	5.18805e6	0.479791
$652$	0	0
$653$	2.20492e6i	0.202354i	0.994868	+	0.101177i	$0.0322608\pi$
$653$	2.20492e6i	0.202354i	−0.994868	+	0.101177i	$0.967739\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	1.09611e7i	0.990694i
$658$	0	0
$659$	1.26903e7	1.13830	0.569152	−	0.822232i	$-0.307271\pi$
$659$	1.26903e7	1.13830	0.569152	+	0.822232i	$0.307271\pi$
$660$	0	0
$661$	2.20351e6	0.196160	0.0980801	−	0.995179i	$-0.468730\pi$
$661$	2.20351e6	0.196160	0.0980801	+	0.995179i	$0.468730\pi$
$662$	0	0
$663$	− 1.23184e7i	− 1.08835i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 3.16940e6i	− 0.275843i
$668$	0	0
$669$	2.08013e7	1.79690
$670$	0	0
$671$	55298.8	0.00474143
$672$	0	0
$673$	− 2.02905e6i	− 0.172685i	−0.996266	−	0.0863424i	$-0.972482\pi$
$673$	− 2.02905e6i	− 0.172685i	0.996266	−	0.0863424i	$-0.0275179\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 1.33080e7i	− 1.11594i	−0.829860	−	0.557971i	$-0.811580\pi$
$677$	− 1.33080e7i	− 1.11594i	0.829860	−	0.557971i	$-0.188420\pi$
$678$	0	0
$679$	6.20903e6	0.516832
$680$	0	0
$681$	7.98989e6	0.660196
$682$	0	0
$683$	5.95063e6i	0.488102i	0.969762	+	0.244051i	$0.0784765\pi$
$683$	5.95063e6i	0.488102i	−0.969762	+	0.244051i	$0.921524\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 344204.i	− 0.0278243i
$688$	0	0
$689$	−2.08973e7	−1.67703
$690$	0	0
$691$	1.95555e7	1.55802	0.779010	−	0.627011i	$-0.215722\pi$
$691$	1.95555e7	1.55802	0.779010	+	0.627011i	$0.215722\pi$
$692$	0	0
$693$	48319.0i	0.00382195i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 1.80646e7i	− 1.40846i
$698$	0	0
$699$	−1.93537e7	−1.49820
$700$	0	0
$701$	−1.70838e7	−1.31308	−0.656538	−	0.754293i	$-0.727980\pi$
$701$	−1.70838e7	−1.31308	−0.656538	+	0.754293i	$0.727980\pi$
$702$	0	0
$703$	− 1.88708e6i	− 0.144013i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 3.61323e6i	− 0.271861i
$708$	0	0
$709$	−2.47096e7	−1.84608	−0.923038	−	0.384710i	$-0.874301\pi$
$709$	−2.47096e7	−1.84608	−0.923038	+	0.384710i	$0.874301\pi$
$710$	0	0
$711$	6.02430e6	0.446923
$712$	0	0
$713$	− 1.75050e7i	− 1.28955i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	2.78074e7i	2.02005i
$718$	0	0
$719$	−2.02894e7	−1.46369	−0.731843	−	0.681474i	$-0.761339\pi$
$719$	−2.02894e7	−1.46369	−0.731843	+	0.681474i	$0.761339\pi$
$720$	0	0
$721$	−5.30131e6	−0.379791
$722$	0	0
$723$	− 8.08882e6i	− 0.575492i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	2.59786e7i	1.82297i	0.411331	+	0.911486i	$0.365064\pi$
$727$	2.59786e7i	1.82297i	−0.411331	+	0.911486i	$0.634936\pi$
$728$	0	0
$729$	7.13072e6	0.496952
$730$	0	0
$731$	3.01946e6	0.208995
$732$	0	0
$733$	− 1.75705e7i	− 1.20788i	−0.797030	−	0.603940i	$-0.793597\pi$
$733$	− 1.75705e7i	− 1.20788i	0.797030	−	0.603940i	$-0.206403\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 187516.i	− 0.0127165i
$738$	0	0
$739$	2.26188e7	1.52356	0.761779	−	0.647837i	$-0.224326\pi$
$739$	2.26188e7	1.52356	0.761779	+	0.647837i	$0.224326\pi$
$740$	0	0
$741$	−3.97197e6	−0.265742
$742$	0	0
$743$	− 1.75610e7i	− 1.16701i	−0.812108	−	0.583507i	$-0.801680\pi$
$743$	− 1.75610e7i	− 1.16701i	0.812108	−	0.583507i	$-0.198320\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	5.82524e6i	0.381955i
$748$	0	0
$749$	7.78892e6	0.507309
$750$	0	0
$751$	1.36472e6	0.0882967	0.0441483	−	0.999025i	$-0.485943\pi$
$751$	1.36472e6	0.0882967	0.0441483	+	0.999025i	$0.485943\pi$
$752$	0	0
$753$	− 1.67080e7i	− 1.07383i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 2.89968e6i	− 0.183912i	−0.995763	−	0.0919560i	$-0.970688\pi$
$757$	− 2.89968e6i	− 0.183912i	0.995763	−	0.0919560i	$-0.0293119\pi$
$758$	0	0
$759$	375042.	0.0236306
$760$	0	0
$761$	−9.33991e6	−0.584630	−0.292315	−	0.956322i	$-0.594426\pi$
$761$	−9.33991e6	−0.584630	−0.292315	+	0.956322i	$0.594426\pi$
$762$	0	0
$763$	868303.i	0.0539958i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	9.33038e6i	0.572679i
$768$	0	0
$769$	1.86030e7	1.13440	0.567200	−	0.823580i	$-0.308026\pi$
$769$	1.86030e7	1.13440	0.567200	+	0.823580i	$0.308026\pi$
$770$	0	0
$771$	3.18506e6	0.192967
$772$	0	0
$773$	− 2.71314e7i	− 1.63314i	−0.577247	−	0.816570i	$-0.695873\pi$
$773$	− 2.71314e7i	− 1.63314i	0.577247	−	0.816570i	$-0.304127\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 4.46071e6i	− 0.265065i
$778$	0	0
$779$	−5.82478e6	−0.343903
$780$	0	0
$781$	412295.	0.0241869
$782$	0	0
$783$	− 1.49514e6i	− 0.0871521i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 9.16693e6i	− 0.527579i	−0.964580	−	0.263789i	$-0.915028\pi$
$787$	− 9.16693e6i	− 0.527579i	0.964580	−	0.263789i	$-0.0849723\pi$
$788$	0	0
$789$	580674.	0.0332078
$790$	0	0
$791$	−2.27713e6	−0.129404
$792$	0	0
$793$	4.67104e6i	0.263773i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 1.90109e7i	− 1.06012i	−0.847959	−	0.530062i	$-0.822169\pi$
$797$	− 1.90109e7i	− 1.06012i	0.847959	−	0.530062i	$-0.177831\pi$
$798$	0	0
$799$	−1.97207e7	−1.09284
$800$	0	0
$801$	−1.40633e7	−0.774472
$802$	0	0
$803$	430077.i	0.0235373i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.48368e7i	0.801969i
$808$	0	0
$809$	4.04747e6	0.217426	0.108713	−	0.994073i	$-0.465327\pi$
$809$	4.04747e6	0.217426	0.108713	+	0.994073i	$0.465327\pi$
$810$	0	0
$811$	984765.	0.0525752	0.0262876	−	0.999654i	$-0.491631\pi$
$811$	984765.	0.0525752	0.0262876	+	0.999654i	$0.491631\pi$
$812$	0	0
$813$	2.81296e7i	1.49258i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 973603.i	− 0.0510301i
$818$	0	0
$819$	−4.08147e6	−0.212621
$820$	0	0
$821$	3.53595e6	0.183083	0.0915415	−	0.995801i	$-0.470821\pi$
$821$	3.53595e6	0.183083	0.0915415	+	0.995801i	$0.470821\pi$
$822$	0	0
$823$	− 9.27174e6i	− 0.477157i	−0.971123	−	0.238579i	$-0.923319\pi$
$823$	− 9.27174e6i	− 0.477157i	0.971123	−	0.238579i	$-0.0766815\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3.88822e6i	0.197691i	0.995103	+	0.0988456i	$0.0315150\pi$
$827$	3.88822e6i	0.197691i	−0.995103	+	0.0988456i	$0.968485\pi$
$828$	0	0
$829$	−2.20046e7	−1.11206	−0.556028	−	0.831163i	$-0.687675\pi$
$829$	−2.20046e7	−1.11206	−0.556028	+	0.831163i	$0.687675\pi$
$830$	0	0
$831$	−1.64007e7	−0.823874
$832$	0	0
$833$	1.49303e7i	0.745513i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 8.25785e6i	− 0.407430i
$838$	0	0
$839$	−888779.	−0.0435902	−0.0217951	−	0.999762i	$-0.506938\pi$
$839$	−888779.	−0.0435902	−0.0217951	+	0.999762i	$0.506938\pi$
$840$	0	0
$841$	−1.88608e7	−0.919539
$842$	0	0
$843$	3.96286e6i	0.192061i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 5.67786e6i	− 0.271942i
$848$	0	0
$849$	2.42420e7	1.15425
$850$	0	0
$851$	−1.50509e7	−0.712424
$852$	0	0
$853$	− 2.03265e7i	− 0.956511i	−0.878221	−	0.478255i	$-0.841269\pi$
$853$	− 2.03265e7i	− 0.956511i	0.878221	−	0.478255i	$-0.158731\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1.96142e6i	0.0912258i	0.998959	+	0.0456129i	$0.0145241\pi$
$857$	1.96142e6i	0.0912258i	−0.998959	+	0.0456129i	$0.985476\pi$
$858$	0	0
$859$	−1.54216e7	−0.713095	−0.356548	−	0.934277i	$-0.616046\pi$
$859$	−1.54216e7	−0.713095	−0.356548	+	0.934277i	$0.616046\pi$
$860$	0	0
$861$	−1.37687e7	−0.632972
$862$	0	0
$863$	2.79115e7i	1.27572i	0.770152	+	0.637860i	$0.220180\pi$
$863$	2.79115e7i	1.27572i	−0.770152	+	0.637860i	$0.779820\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 1.03572e7i	− 0.467945i
$868$	0	0
$869$	236374.	0.0106182
$870$	0	0
$871$	1.58393e7	0.707440
$872$	0	0
$873$	3.28993e7i	1.46100i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1.72191e7i	0.755980i	0.925810	+	0.377990i	$0.123385\pi$
$877$	1.72191e7i	0.755980i	−0.925810	+	0.377990i	$0.876615\pi$
$878$	0	0
$879$	1.93402e7	0.844282
$880$	0	0
$881$	3.57166e7	1.55035	0.775177	−	0.631745i	$-0.217661\pi$
$881$	3.57166e7	1.55035	0.775177	+	0.631745i	$0.217661\pi$
$882$	0	0
$883$	− 3.78278e6i	− 0.163271i	−0.996662	−	0.0816356i	$-0.973986\pi$
$883$	− 3.78278e6i	− 0.163271i	0.996662	−	0.0816356i	$-0.0260144\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 2.45768e7i	− 1.04886i	−0.851455	−	0.524428i	$-0.824279\pi$
$887$	− 2.45768e7i	− 1.04886i	0.851455	−	0.524428i	$-0.175721\pi$
$888$	0	0
$889$	−5.62855e6	−0.238859
$890$	0	0
$891$	509857.	0.0215156
$892$	0	0
$893$	6.35880e6i	0.266837i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	3.16794e7i	1.31461i
$898$	0	0
$899$	9.11504e6	0.376149
$900$	0	0
$901$	3.23695e7	1.32839
$902$	0	0
$903$	− 2.30142e6i	− 0.0939238i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 1.47776e6i	− 0.0596467i	−0.999555	−	0.0298234i	$-0.990506\pi$
$907$	− 1.47776e6i	− 0.0596467i	0.999555	−	0.0298234i	$-0.00949448\pi$
$908$	0	0
$909$	1.91452e7	0.768510
$910$	0	0
$911$	−2.62169e7	−1.04661	−0.523305	−	0.852146i	$-0.675301\pi$
$911$	−2.62169e7	−1.04661	−0.523305	+	0.852146i	$0.675301\pi$
$912$	0	0
$913$	228564.i	0.00907466i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.09491e6i	0.160813i
$918$	0	0
$919$	−1.10734e6	−0.0432507	−0.0216254	−	0.999766i	$-0.506884\pi$
$919$	−1.10734e6	−0.0432507	−0.0216254	+	0.999766i	$0.506884\pi$
$920$	0	0
$921$	−1.40724e7	−0.546664
$922$	0	0
$923$	3.48262e7i	1.34556i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 2.80896e7i	− 1.07361i
$928$	0	0
$929$	−2.36543e7	−0.899229	−0.449615	−	0.893223i	$-0.648439\pi$
$929$	−2.36543e7	−0.899229	−0.449615	+	0.893223i	$0.648439\pi$
$930$	0	0
$931$	4.81415e6	0.182031
$932$	0	0
$933$	− 2.60502e7i	− 0.979732i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 3.28168e7i	− 1.22109i	−0.791983	−	0.610544i	$-0.790951\pi$
$937$	− 3.28168e7i	− 1.22109i	0.791983	−	0.610544i	$-0.209049\pi$
$938$	0	0
$939$	−6.50338e7	−2.40699
$940$	0	0
$941$	3.66896e7	1.35073	0.675366	−	0.737483i	$-0.263986\pi$
$941$	3.66896e7	1.35073	0.675366	+	0.737483i	$0.263986\pi$
$942$	0	0
$943$	4.64569e7i	1.70126i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	8.63334e6i	0.312827i	0.987692	+	0.156413i	$0.0499932\pi$
$947$	8.63334e6i	0.312827i	−0.987692	+	0.156413i	$0.950007\pi$
$948$	0	0
$949$	−3.63282e7	−1.30942
$950$	0	0
$951$	7.32660e6	0.262695
$952$	0	0
$953$	1.76257e7i	0.628659i	0.949314	+	0.314329i	$0.101780\pi$
$953$	1.76257e7i	0.628659i	−0.949314	+	0.314329i	$0.898220\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	195288.i	0.00689281i
$958$	0	0
$959$	51000.3	0.00179071
$960$	0	0
$961$	2.17144e7	0.758470
$962$	0	0
$963$	4.12706e7i	1.43408i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 4.87856e7i	− 1.67774i	−0.544330	−	0.838871i	$-0.683216\pi$
$967$	− 4.87856e7i	− 1.67774i	0.544330	−	0.838871i	$-0.316784\pi$
$968$	0	0
$969$	6.15252e6	0.210496
$970$	0	0
$971$	−3.81830e7	−1.29964	−0.649818	−	0.760090i	$-0.725155\pi$
$971$	−3.81830e7	−1.29964	−0.649818	+	0.760090i	$0.725155\pi$
$972$	0	0
$973$	4.76297e6i	0.161286i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3.64067e7i	1.22024i	0.792309	+	0.610120i	$0.208879\pi$
$977$	3.64067e7i	1.22024i	−0.792309	+	0.610120i	$0.791121\pi$
$978$	0	0
$979$	−551799.	−0.0184003
$980$	0	0
$981$	−4.60081e6	−0.152638
$982$	0	0
$983$	4.61248e7i	1.52248i	0.648472	+	0.761239i	$0.275408\pi$
$983$	4.61248e7i	1.52248i	−0.648472	+	0.761239i	$0.724592\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1.50310e7i	0.491129i
$988$	0	0
$989$	−7.76520e6	−0.252442
$990$	0	0
$991$	2.77541e7	0.897725	0.448862	−	0.893601i	$-0.351829\pi$
$991$	2.77541e7	0.897725	0.448862	+	0.893601i	$0.351829\pi$
$992$	0	0
$993$	6.04499e7i	1.94546i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	2.50394e7i	0.797784i	0.916998	+	0.398892i	$0.130605\pi$
$997$	2.50394e7i	0.797784i	−0.916998	+	0.398892i	$0.869395\pi$
$998$	0	0
$999$	−7.10015e6	−0.225089

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.6.c.g.449.1		4
4.3	odd	2		800.6.c.f.449.4		4
5.2	odd	4		800.6.a.g.1.1		2
5.3	odd	4		160.6.a.e.1.2	yes	2
5.4	even	2	inner	800.6.c.g.449.4		4
20.3	even	4		160.6.a.a.1.1	✓	2
20.7	even	4		800.6.a.l.1.2		2
20.19	odd	2		800.6.c.f.449.1		4
40.3	even	4		320.6.a.v.1.2		2
40.13	odd	4		320.6.a.r.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.6.a.a.1.1	✓	2	20.3	even	4
160.6.a.e.1.2	yes	2	5.3	odd	4
320.6.a.r.1.1		2	40.13	odd	4
320.6.a.v.1.2		2	40.3	even	4
800.6.a.g.1.1		2	5.2	odd	4
800.6.a.l.1.2		2	20.7	even	4
800.6.c.f.449.1		4	20.19	odd	2
800.6.c.f.449.4		4	4.3	odd	2
800.6.c.g.449.1		4	1.1	even	1	trivial
800.6.c.g.449.4		4	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	800.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-20.7332i q^{3} +35.2668i q^{7} -186.866 q^{9} +O(q^{10})\)	\(q-20.7332i q^{3} +35.2668i q^{7} -186.866 q^{9} -7.33201 q^{11} -619.328i q^{13} +959.328i q^{17} +309.328 q^{19} +731.194 q^{21} -2467.12i q^{23} -1163.85i q^{27} +1284.66 q^{29} +7095.32 q^{31} +152.016i q^{33} -6100.59i q^{37} -12840.7 q^{39} -18830.4 q^{41} -3147.48i q^{43} +20556.8i q^{47} +15563.3 q^{49} +19889.9 q^{51} -33741.9i q^{53} -6413.36i q^{57} -15065.3 q^{59} -7542.11 q^{61} -6590.15i q^{63} +25574.9i q^{67} -51151.3 q^{69} -56232.3 q^{71} -58657.5i q^{73} -258.576i q^{77} -32238.7 q^{79} -69538.6 q^{81} -31173.4i q^{83} -26635.0i q^{87} +75258.9 q^{89} +21841.7 q^{91} -147109. i q^{93} -176059. i q^{97} +1370.10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 212 q^{9}+O(q^{10}) \) 4 * q - 212 * q^9	\( 4 q - 212 q^{9} + 640 q^{11} - 1440 q^{19} - 288 q^{21} - 216 q^{29} + 19680 q^{31} - 44000 q^{39} - 21240 q^{41} + 55292 q^{49} + 49440 q^{51} - 61600 q^{59} + 49080 q^{61} - 122144 q^{69} - 24800 q^{71} - 143680 q^{79} - 204796 q^{81} + 81496 q^{89} - 55200 q^{91} + 55680 q^{99}+O(q^{100}) \) 4 * q - 212 * q^9 + 640 * q^11 - 1440 * q^19 - 288 * q^21 - 216 * q^29 + 19680 * q^31 - 44000 * q^39 - 21240 * q^41 + 55292 * q^49 + 49440 * q^51 - 61600 * q^59 + 49080 * q^61 - 122144 * q^69 - 24800 * q^71 - 143680 * q^79 - 204796 * q^81 + 81496 * q^89 - 55200 * q^91 + 55680 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more