LMFDB - Embedded newform 800.6.d.d.401.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,6,Mod(401,800)]

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, 1, 0]))

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

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

chi := DirichletCharacter("800.401");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	800.d (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:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{19} - 130 x^{17} + 144 x^{16} + 1560 x^{15} - 12320 x^{14} - 56128 x^{13} + \cdots + 11\!\cdots\!24 $ x^20 - x^19 - 130x^17 + 144x^16 + 1560x^15 - 12320x^14 - 56128x^13 + 672256x^12 - 4183040x^11 + 25763840x^10 - 133857280x^9 + 688390144x^8 - 1839202304x^7 - 12918456320x^6 + 52344913920x^5 + 154618822656x^4 - 4466765987840x^3 - 35184372088832x + 1125899906842624
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{99}\cdot 5^{4}\cdot 31 $
Twist minimal:	no (minimal twist has level 200)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			401.12
Root			$0.858917 - 5.59127i$ of defining polynomial
Character	$\chi$	$=$	800.401
Dual form			800.6.d.d.401.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.58812i q^{3} +27.4015 q^{7} +236.302 q^{9} +O(q^{10})$	$q+2.58812i q^{3} +27.4015 q^{7} +236.302 q^{9} +313.336i q^{11} -627.354i q^{13} -1598.17 q^{17} +219.049i q^{19} +70.9184i q^{21} +1007.11 q^{23} +1240.49i q^{27} +6140.83i q^{29} +2244.61 q^{31} -810.951 q^{33} +3591.40i q^{37} +1623.67 q^{39} -608.697 q^{41} +11410.6i q^{43} +16909.4 q^{47} -16056.2 q^{49} -4136.26i q^{51} -12783.9i q^{53} -566.923 q^{57} -29917.1i q^{59} +17985.1i q^{61} +6475.03 q^{63} -48881.2i q^{67} +2606.52i q^{69} -37296.0 q^{71} -26092.7 q^{73} +8585.89i q^{77} +77636.5 q^{79} +54210.8 q^{81} +58419.8i q^{83} -15893.2 q^{87} -81513.8 q^{89} -17190.5i q^{91} +5809.30i q^{93} -28396.6 q^{97} +74041.9i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 196 q^{7} - 1620 q^{9}+O(q^{10}) $ 20 * q + 196 * q^7 - 1620 * q^9	$ 20 q + 196 q^{7} - 1620 q^{9} + 7184 q^{23} - 7160 q^{31} - 2836 q^{33} - 22452 q^{39} - 5804 q^{41} - 44180 q^{47} + 62652 q^{49} + 43696 q^{57} + 1240 q^{63} + 7724 q^{71} - 105136 q^{73} + 7780 q^{79} + 96984 q^{81} + 106188 q^{87} - 3160 q^{89} - 73688 q^{97}+O(q^{100}) $ 20 * q + 196 * q^7 - 1620 * q^9 + 7184 * q^23 - 7160 * q^31 - 2836 * q^33 - 22452 * q^39 - 5804 * q^41 - 44180 * q^47 + 62652 * q^49 + 43696 * q^57 + 1240 * q^63 + 7724 * q^71 - 105136 * q^73 + 7780 * q^79 + 96984 * q^81 + 106188 * q^87 - 3160 * q^89 - 73688 * q^97

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
$3$	2.58812i	0.166028i	0.996548	+	0.0830139i	$0.0264546\pi$
$3$	2.58812i	0.166028i	−0.996548	+	0.0830139i	$0.973545\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	27.4015	0.211363	0.105682	−	0.994400i	$-0.466298\pi$
$7$	27.4015	0.211363	0.105682	+	0.994400i	$0.466298\pi$
$8$	0	0
$9$	236.302	0.972435
$10$	0	0
$11$	313.336i	0.780780i	0.920650	+	0.390390i	$0.127660\pi$
$11$	313.336i	0.780780i	−0.920650	+	0.390390i	$0.872340\pi$
$12$	0	0
$13$	− 627.354i	− 1.02957i	−0.857320	−	0.514783i	$-0.827872\pi$
$13$	− 627.354i	− 1.02957i	0.857320	−	0.514783i	$-0.172128\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1598.17	−1.34122	−0.670611	−	0.741809i	$-0.733968\pi$
$17$	−1598.17	−1.34122	−0.670611	+	0.741809i	$0.733968\pi$
$18$	0	0
$19$	219.049i	0.139205i	0.997575	+	0.0696027i	$0.0221732\pi$
$19$	219.049i	0.139205i	−0.997575	+	0.0696027i	$0.977827\pi$
$20$	0	0
$21$	70.9184i	0.0350922i
$22$	0	0
$23$	1007.11	0.396970	0.198485	−	0.980104i	$-0.436398\pi$
$23$	1007.11	0.396970	0.198485	+	0.980104i	$0.436398\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1240.49i	0.327479i
$28$	0	0
$29$	6140.83i	1.35591i	0.735102	+	0.677957i	$0.237134\pi$
$29$	6140.83i	1.35591i	−0.735102	+	0.677957i	$0.762866\pi$
$30$	0	0
$31$	2244.61	0.419504	0.209752	−	0.977755i	$-0.432734\pi$
$31$	2244.61	0.419504	0.209752	+	0.977755i	$0.432734\pi$
$32$	0	0
$33$	−810.951	−0.129631
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3591.40i	0.431280i	0.976473	+	0.215640i	$0.0691837\pi$
$37$	3591.40i	0.431280i	−0.976473	+	0.215640i	$0.930816\pi$
$38$	0	0
$39$	1623.67	0.170937
$40$	0	0
$41$	−608.697	−0.0565511	−0.0282756	−	0.999600i	$-0.509002\pi$
$41$	−608.697	−0.0565511	−0.0282756	+	0.999600i	$0.509002\pi$
$42$	0	0
$43$	11410.6i	0.941105i	0.882372	+	0.470553i	$0.155945\pi$
$43$	11410.6i	0.941105i	−0.882372	+	0.470553i	$0.844055\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	16909.4	1.11656	0.558280	−	0.829652i	$-0.311461\pi$
$47$	16909.4	1.11656	0.558280	+	0.829652i	$0.311461\pi$
$48$	0	0
$49$	−16056.2	−0.955326
$50$	0	0
$51$	− 4136.26i	− 0.222680i
$52$	0	0
$53$	− 12783.9i	− 0.625134i	−0.949896	−	0.312567i	$-0.898811\pi$
$53$	− 12783.9i	− 0.625134i	0.949896	−	0.312567i	$-0.101189\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−566.923	−0.0231120
$58$	0	0
$59$	− 29917.1i	− 1.11889i	−0.828866	−	0.559447i	$-0.811014\pi$
$59$	− 29917.1i	− 1.11889i	0.828866	−	0.559447i	$-0.188986\pi$
$60$	0	0
$61$	17985.1i	0.618852i	0.950923	+	0.309426i	$0.100137\pi$
$61$	17985.1i	0.618852i	−0.950923	+	0.309426i	$0.899863\pi$
$62$	0	0
$63$	6475.03	0.205537
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 48881.2i	− 1.33032i	−0.746703	−	0.665158i	$-0.768364\pi$
$67$	− 48881.2i	− 1.33032i	0.746703	−	0.665158i	$-0.231636\pi$
$68$	0	0
$69$	2606.52i	0.0659081i
$70$	0	0
$71$	−37296.0	−0.878044	−0.439022	−	0.898476i	$-0.644675\pi$
$71$	−37296.0	−0.878044	−0.439022	+	0.898476i	$0.644675\pi$
$72$	0	0
$73$	−26092.7	−0.573075	−0.286538	−	0.958069i	$-0.592504\pi$
$73$	−26092.7	−0.573075	−0.286538	+	0.958069i	$0.592504\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	8585.89i	0.165028i
$78$	0	0
$79$	77636.5	1.39958	0.699791	−	0.714348i	$-0.253277\pi$
$79$	77636.5	1.39958	0.699791	+	0.714348i	$0.253277\pi$
$80$	0	0
$81$	54210.8	0.918064
$82$	0	0
$83$	58419.8i	0.930818i	0.885096	+	0.465409i	$0.154093\pi$
$83$	58419.8i	0.930818i	−0.885096	+	0.465409i	$0.845907\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−15893.2	−0.225120
$88$	0	0
$89$	−81513.8	−1.09083	−0.545414	−	0.838167i	$-0.683627\pi$
$89$	−81513.8	−1.09083	−0.545414	+	0.838167i	$0.683627\pi$
$90$	0	0
$91$	− 17190.5i	− 0.217613i
$92$	0	0
$93$	5809.30i	0.0696493i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−28396.6	−0.306434	−0.153217	−	0.988193i	$-0.548963\pi$
$97$	−28396.6	−0.306434	−0.153217	+	0.988193i	$0.548963\pi$
$98$	0	0
$99$	74041.9i	0.759258i
$100$	0	0
$101$	136025.i	1.32683i	0.748253	+	0.663413i	$0.230893\pi$
$101$	136025.i	1.32683i	−0.748253	+	0.663413i	$0.769107\pi$
$102$	0	0
$103$	166600.	1.54733	0.773665	−	0.633595i	$-0.218421\pi$
$103$	166600.	1.54733	0.773665	+	0.633595i	$0.218421\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	67774.4i	0.572277i	0.958188	+	0.286138i	$0.0923717\pi$
$107$	67774.4i	0.572277i	−0.958188	+	0.286138i	$0.907628\pi$
$108$	0	0
$109$	55567.6i	0.447977i	0.974592	+	0.223988i	$0.0719077\pi$
$109$	55567.6i	0.447977i	−0.974592	+	0.223988i	$0.928092\pi$
$110$	0	0
$111$	−9294.96	−0.0716044
$112$	0	0
$113$	−170497.	−1.25609	−0.628045	−	0.778177i	$-0.716145\pi$
$113$	−170497.	−1.25609	−0.628045	+	0.778177i	$0.716145\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 148245.i	− 1.00119i
$118$	0	0
$119$	−43792.3	−0.283485
$120$	0	0
$121$	62871.4	0.390382
$122$	0	0
$123$	− 1575.38i	− 0.00938906i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	313764.	1.72621	0.863106	−	0.505022i	$-0.168516\pi$
$127$	313764.	1.72621	0.863106	+	0.505022i	$0.168516\pi$
$128$	0	0
$129$	−29532.0	−0.156250
$130$	0	0
$131$	353658.i	1.80055i	0.435322	+	0.900275i	$0.356634\pi$
$131$	353658.i	1.80055i	−0.435322	+	0.900275i	$0.643366\pi$
$132$	0	0
$133$	6002.26i	0.0294229i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−72750.1	−0.331156	−0.165578	−	0.986197i	$-0.552949\pi$
$137$	−72750.1	−0.331156	−0.165578	+	0.986197i	$0.552949\pi$
$138$	0	0
$139$	259480.i	1.13911i	0.821952	+	0.569557i	$0.192885\pi$
$139$	259480.i	1.13911i	−0.821952	+	0.569557i	$0.807115\pi$
$140$	0	0
$141$	43763.4i	0.185380i
$142$	0	0
$143$	196573.	0.803865
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 41555.2i	− 0.158611i
$148$	0	0
$149$	− 206318.i	− 0.761328i	−0.924713	−	0.380664i	$-0.875695\pi$
$149$	− 206318.i	− 0.761328i	0.924713	−	0.380664i	$-0.124305\pi$
$150$	0	0
$151$	−435318.	−1.55369	−0.776845	−	0.629692i	$-0.783181\pi$
$151$	−435318.	−1.55369	−0.776845	+	0.629692i	$0.783181\pi$
$152$	0	0
$153$	−377650.	−1.30425
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 341575.i	− 1.10595i	−0.833196	−	0.552977i	$-0.813492\pi$
$157$	− 341575.i	− 1.10595i	0.833196	−	0.552977i	$-0.186508\pi$
$158$	0	0
$159$	33086.2	0.103790
$160$	0	0
$161$	27596.4	0.0839049
$162$	0	0
$163$	458266.i	1.35098i	0.737370	+	0.675489i	$0.236068\pi$
$163$	458266.i	1.35098i	−0.737370	+	0.675489i	$0.763932\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−443800.	−1.23139	−0.615696	−	0.787984i	$-0.711125\pi$
$167$	−443800.	−1.23139	−0.615696	+	0.787984i	$0.711125\pi$
$168$	0	0
$169$	−22280.4	−0.0600075
$170$	0	0
$171$	51761.5i	0.135368i
$172$	0	0
$173$	731330.i	1.85780i	0.370336	+	0.928898i	$0.379243\pi$
$173$	731330.i	1.85780i	−0.370336	+	0.928898i	$0.620757\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	77428.9	0.185768
$178$	0	0
$179$	318951.i	0.744032i	0.928226	+	0.372016i	$0.121333\pi$
$179$	318951.i	0.744032i	−0.928226	+	0.372016i	$0.878667\pi$
$180$	0	0
$181$	586939.i	1.33167i	0.746099	+	0.665835i	$0.231924\pi$
$181$	586939.i	1.33167i	−0.746099	+	0.665835i	$0.768076\pi$
$182$	0	0
$183$	−46547.4	−0.102747
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 500765.i	− 1.04720i
$188$	0	0
$189$	33991.3i	0.0692171i
$190$	0	0
$191$	97320.1	0.193027	0.0965137	−	0.995332i	$-0.469231\pi$
$191$	97320.1	0.193027	0.0965137	+	0.995332i	$0.469231\pi$
$192$	0	0
$193$	−52528.5	−0.101508	−0.0507541	−	0.998711i	$-0.516162\pi$
$193$	−52528.5	−0.101508	−0.0507541	+	0.998711i	$0.516162\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 446496.i	− 0.819695i	−0.912154	−	0.409847i	$-0.865582\pi$
$197$	− 446496.i	− 0.819695i	0.912154	−	0.409847i	$-0.134418\pi$
$198$	0	0
$199$	518787.	0.928659	0.464329	−	0.885663i	$-0.346295\pi$
$199$	518787.	0.928659	0.464329	+	0.885663i	$0.346295\pi$
$200$	0	0
$201$	126510.	0.220870
$202$	0	0
$203$	168268.i	0.286591i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	237982.	0.386028
$208$	0	0
$209$	−68635.8	−0.108689
$210$	0	0
$211$	1.12592e6i	1.74101i	0.492163	+	0.870503i	$0.336206\pi$
$211$	1.12592e6i	1.74101i	−0.492163	+	0.870503i	$0.663794\pi$
$212$	0	0
$213$	− 96526.5i	− 0.145780i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	61505.6	0.0886677
$218$	0	0
$219$	− 67530.9i	− 0.0951464i
$220$	0	0
$221$	1.00262e6i	1.38088i
$222$	0	0
$223$	−138775.	−0.186874	−0.0934368	−	0.995625i	$-0.529785\pi$
$223$	−138775.	−0.186874	−0.0934368	+	0.995625i	$0.529785\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 966829.i	− 1.24533i	−0.782488	−	0.622666i	$-0.786050\pi$
$227$	− 966829.i	− 1.24533i	0.782488	−	0.622666i	$-0.213950\pi$
$228$	0	0
$229$	840679.i	1.05935i	0.848199	+	0.529677i	$0.177687\pi$
$229$	840679.i	1.05935i	−0.848199	+	0.529677i	$0.822313\pi$
$230$	0	0
$231$	−22221.3	−0.0273993
$232$	0	0
$233$	977446.	1.17951	0.589757	−	0.807581i	$-0.299224\pi$
$233$	977446.	1.17951	0.589757	+	0.807581i	$0.299224\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	200932.i	0.232370i
$238$	0	0
$239$	−1.17543e6	−1.33108	−0.665538	−	0.746364i	$-0.731798\pi$
$239$	−1.17543e6	−1.33108	−0.665538	+	0.746364i	$0.731798\pi$
$240$	0	0
$241$	−615094.	−0.682180	−0.341090	−	0.940031i	$-0.610796\pi$
$241$	−615094.	−0.682180	−0.341090	+	0.940031i	$0.610796\pi$
$242$	0	0
$243$	441743.i	0.479903i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	137421.	0.143321
$248$	0	0
$249$	−151197.	−0.154542
$250$	0	0
$251$	1.26252e6i	1.26490i	0.774602	+	0.632448i	$0.217950\pi$
$251$	1.26252e6i	1.26490i	−0.774602	+	0.632448i	$0.782050\pi$
$252$	0	0
$253$	315564.i	0.309946i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	346343.	0.327095	0.163547	−	0.986535i	$-0.447706\pi$
$257$	346343.	0.327095	0.163547	+	0.986535i	$0.447706\pi$
$258$	0	0
$259$	98409.8i	0.0911567i
$260$	0	0
$261$	1.45109e6i	1.31854i
$262$	0	0
$263$	−1.93221e6	−1.72252	−0.861262	−	0.508162i	$-0.830325\pi$
$263$	−1.93221e6	−1.72252	−0.861262	+	0.508162i	$0.830325\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 210967.i	− 0.181108i
$268$	0	0
$269$	2.22703e6i	1.87649i	0.345976	+	0.938243i	$0.387548\pi$
$269$	2.22703e6i	1.87649i	−0.345976	+	0.938243i	$0.612452\pi$
$270$	0	0
$271$	512488.	0.423897	0.211948	−	0.977281i	$-0.432019\pi$
$271$	512488.	0.423897	0.211948	+	0.977281i	$0.432019\pi$
$272$	0	0
$273$	44491.0	0.0361298
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 825684.i	− 0.646568i	−0.946302	−	0.323284i	$-0.895213\pi$
$277$	− 825684.i	− 0.646568i	0.946302	−	0.323284i	$-0.104787\pi$
$278$	0	0
$279$	530404.	0.407940
$280$	0	0
$281$	−2.18865e6	−1.65352	−0.826762	−	0.562552i	$-0.809820\pi$
$281$	−2.18865e6	−1.65352	−0.826762	+	0.562552i	$0.809820\pi$
$282$	0	0
$283$	1.64409e6i	1.22028i	0.792293	+	0.610141i	$0.208887\pi$
$283$	1.64409e6i	1.22028i	−0.792293	+	0.610141i	$0.791113\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−16679.2	−0.0119528
$288$	0	0
$289$	1.13429e6	0.798879
$290$	0	0
$291$	− 73493.9i	− 0.0508767i
$292$	0	0
$293$	1.79254e6i	1.21983i	0.792465	+	0.609917i	$0.208797\pi$
$293$	1.79254e6i	1.21983i	−0.792465	+	0.609917i	$0.791203\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−388690.	−0.255689
$298$	0	0
$299$	− 631815.i	− 0.408707i
$300$	0	0
$301$	312669.i	0.198915i
$302$	0	0
$303$	−352048.	−0.220290
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 1.30565e6i	− 0.790646i	−0.918542	−	0.395323i	$-0.870633\pi$
$307$	− 1.30565e6i	− 0.790646i	0.918542	−	0.395323i	$-0.129367\pi$
$308$	0	0
$309$	431182.i	0.256900i
$310$	0	0
$311$	996126.	0.584001	0.292000	−	0.956418i	$-0.405679\pi$
$311$	996126.	0.584001	0.292000	+	0.956418i	$0.405679\pi$
$312$	0	0
$313$	887076.	0.511800	0.255900	−	0.966703i	$-0.417628\pi$
$313$	887076.	0.511800	0.255900	+	0.966703i	$0.417628\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 1.23286e6i	− 0.689075i	−0.938773	−	0.344538i	$-0.888036\pi$
$317$	− 1.23286e6i	− 0.689075i	0.938773	−	0.344538i	$-0.111964\pi$
$318$	0	0
$319$	−1.92414e6	−1.05867
$320$	0	0
$321$	−175408.	−0.0950139
$322$	0	0
$323$	− 350077.i	− 0.186706i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−143815.	−0.0743766
$328$	0	0
$329$	463342.	0.236000
$330$	0	0
$331$	3.32521e6i	1.66820i	0.551612	+	0.834101i	$0.314013\pi$
$331$	3.32521e6i	1.66820i	−0.551612	+	0.834101i	$0.685987\pi$
$332$	0	0
$333$	848653.i	0.419391i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	902443.	0.432858	0.216429	−	0.976298i	$-0.430559\pi$
$337$	902443.	0.432858	0.216429	+	0.976298i	$0.430559\pi$
$338$	0	0
$339$	− 441267.i	− 0.208546i
$340$	0	0
$341$	703316.i	0.327540i
$342$	0	0
$343$	−900501.	−0.413284
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 1.99415e6i	− 0.889066i	−0.895763	−	0.444533i	$-0.853370\pi$
$347$	− 1.99415e6i	− 0.889066i	0.895763	−	0.444533i	$-0.146630\pi$
$348$	0	0
$349$	− 3.06586e6i	− 1.34737i	−0.739017	−	0.673687i	$-0.764710\pi$
$349$	− 3.06586e6i	− 1.34737i	0.739017	−	0.673687i	$-0.235290\pi$
$350$	0	0
$351$	778226.	0.337162
$352$	0	0
$353$	2.66290e6	1.13741	0.568705	−	0.822541i	$-0.307444\pi$
$353$	2.66290e6	1.13741	0.568705	+	0.822541i	$0.307444\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 113340.i	− 0.0470665i
$358$	0	0
$359$	−39323.6	−0.0161034	−0.00805169	−	0.999968i	$-0.502563\pi$
$359$	−39323.6	−0.0161034	−0.00805169	+	0.999968i	$0.502563\pi$
$360$	0	0
$361$	2.42812e6	0.980622
$362$	0	0
$363$	162719.i	0.0648143i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−851976.	−0.330189	−0.165094	−	0.986278i	$-0.552793\pi$
$367$	−851976.	−0.330189	−0.165094	+	0.986278i	$0.552793\pi$
$368$	0	0
$369$	−143836.	−0.0549923
$370$	0	0
$371$	− 350298.i	− 0.132130i
$372$	0	0
$373$	1.78163e6i	0.663047i	0.943447	+	0.331524i	$0.107563\pi$
$373$	1.78163e6i	0.663047i	−0.943447	+	0.331524i	$0.892437\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.85248e6	1.39600
$378$	0	0
$379$	4.29296e6i	1.53518i	0.640941	+	0.767590i	$0.278544\pi$
$379$	4.29296e6i	1.53518i	−0.640941	+	0.767590i	$0.721456\pi$
$380$	0	0
$381$	812059.i	0.286599i
$382$	0	0
$383$	−1.06598e6	−0.371322	−0.185661	−	0.982614i	$-0.559443\pi$
$383$	−1.06598e6	−0.371322	−0.185661	+	0.982614i	$0.559443\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2.69635e6i	0.915164i
$388$	0	0
$389$	251531.i	0.0842786i	0.999112	+	0.0421393i	$0.0134173\pi$
$389$	251531.i	0.0842786i	−0.999112	+	0.0421393i	$0.986583\pi$
$390$	0	0
$391$	−1.60954e6	−0.532425
$392$	0	0
$393$	−915308.	−0.298941
$394$	0	0
$395$	0	0
$396$	0	0
$397$	560036.i	0.178336i	0.996017	+	0.0891682i	$0.0284209\pi$
$397$	560036.i	0.178336i	−0.996017	+	0.0891682i	$0.971579\pi$
$398$	0	0
$399$	−15534.6	−0.00488503
$400$	0	0
$401$	319882.	0.0993410	0.0496705	−	0.998766i	$-0.484183\pi$
$401$	319882.	0.0993410	0.0496705	+	0.998766i	$0.484183\pi$
$402$	0	0
$403$	− 1.40816e6i	− 0.431907i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.12531e6	−0.336735
$408$	0	0
$409$	−221487.	−0.0654695	−0.0327348	−	0.999464i	$-0.510422\pi$
$409$	−221487.	−0.0654695	−0.0327348	+	0.999464i	$0.510422\pi$
$410$	0	0
$411$	− 188286.i	− 0.0549811i
$412$	0	0
$413$	− 819773.i	− 0.236493i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−671565.	−0.189125
$418$	0	0
$419$	1.61245e6i	0.448695i	0.974509	+	0.224347i	$0.0720250\pi$
$419$	1.61245e6i	0.448695i	−0.974509	+	0.224347i	$0.927975\pi$
$420$	0	0
$421$	− 3.39591e6i	− 0.933795i	−0.884312	−	0.466897i	$-0.845372\pi$
$421$	− 3.39591e6i	− 0.933795i	0.884312	−	0.466897i	$-0.154628\pi$
$422$	0	0
$423$	3.99571e6	1.08578
$424$	0	0
$425$	0	0
$426$	0	0
$427$	492818.i	0.130803i
$428$	0	0
$429$	508754.i	0.133464i
$430$	0	0
$431$	−2.08841e6	−0.541530	−0.270765	−	0.962645i	$-0.587277\pi$
$431$	−2.08841e6	−0.541530	−0.270765	+	0.962645i	$0.587277\pi$
$432$	0	0
$433$	−4.35432e6	−1.11609	−0.558047	−	0.829810i	$-0.688449\pi$
$433$	−4.35432e6	−1.11609	−0.558047	+	0.829810i	$0.688449\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	220606.i	0.0552604i
$438$	0	0
$439$	−3.21572e6	−0.796375	−0.398187	−	0.917304i	$-0.630361\pi$
$439$	−3.21572e6	−0.796375	−0.398187	+	0.917304i	$0.630361\pi$
$440$	0	0
$441$	−3.79410e6	−0.928992
$442$	0	0
$443$	− 3.36028e6i	− 0.813516i	−0.913536	−	0.406758i	$-0.866659\pi$
$443$	− 3.36028e6i	− 0.813516i	0.913536	−	0.406758i	$-0.133341\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	533976.	0.126402
$448$	0	0
$449$	963097.	0.225452	0.112726	−	0.993626i	$-0.464042\pi$
$449$	963097.	0.225452	0.112726	+	0.993626i	$0.464042\pi$
$450$	0	0
$451$	− 190727.i	− 0.0441540i
$452$	0	0
$453$	− 1.12665e6i	− 0.257956i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−6.88516e6	−1.54214	−0.771069	−	0.636752i	$-0.780278\pi$
$457$	−6.88516e6	−1.54214	−0.771069	+	0.636752i	$0.780278\pi$
$458$	0	0
$459$	− 1.98251e6i	− 0.439222i
$460$	0	0
$461$	686048.i	0.150349i	0.997170	+	0.0751747i	$0.0239515\pi$
$461$	686048.i	0.150349i	−0.997170	+	0.0751747i	$0.976049\pi$
$462$	0	0
$463$	−104740.	−0.0227070	−0.0113535	−	0.999936i	$-0.503614\pi$
$463$	−104740.	−0.0227070	−0.0113535	+	0.999936i	$0.503614\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	5.79633e6i	1.22987i	0.788576	+	0.614937i	$0.210819\pi$
$467$	5.79633e6i	1.22987i	−0.788576	+	0.614937i	$0.789181\pi$
$468$	0	0
$469$	− 1.33942e6i	− 0.281180i
$470$	0	0
$471$	884037.	0.183619
$472$	0	0
$473$	−3.57536e6	−0.734797
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 3.02085e6i	− 0.607902i
$478$	0	0
$479$	9.29136e6	1.85029	0.925146	−	0.379611i	$-0.123942\pi$
$479$	9.29136e6	1.85029	0.925146	+	0.379611i	$0.123942\pi$
$480$	0	0
$481$	2.25308e6	0.444031
$482$	0	0
$483$	71422.7i	0.0139306i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	6.53708e6	1.24900	0.624498	−	0.781026i	$-0.285304\pi$
$487$	6.53708e6	1.24900	0.624498	+	0.781026i	$0.285304\pi$
$488$	0	0
$489$	−1.18605e6	−0.224300
$490$	0	0
$491$	− 6.62569e6i	− 1.24030i	−0.784483	−	0.620151i	$-0.787071\pi$
$491$	− 6.62569e6i	− 1.24030i	0.784483	−	0.620151i	$-0.212929\pi$
$492$	0	0
$493$	− 9.81410e6i	− 1.81858i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.02197e6	−0.185586
$498$	0	0
$499$	− 5.25473e6i	− 0.944710i	−0.881408	−	0.472355i	$-0.843404\pi$
$499$	− 5.25473e6i	− 0.944710i	0.881408	−	0.472355i	$-0.156596\pi$
$500$	0	0
$501$	− 1.14861e6i	− 0.204445i
$502$	0	0
$503$	8.78835e6	1.54877	0.774386	−	0.632713i	$-0.218059\pi$
$503$	8.78835e6	1.54877	0.774386	+	0.632713i	$0.218059\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 57664.2i	− 0.00996292i
$508$	0	0
$509$	− 5.24405e6i	− 0.897164i	−0.893742	−	0.448582i	$-0.851929\pi$
$509$	− 5.24405e6i	− 0.897164i	0.893742	−	0.448582i	$-0.148071\pi$
$510$	0	0
$511$	−714979.	−0.121127
$512$	0	0
$513$	−271727.	−0.0455869
$514$	0	0
$515$	0	0
$516$	0	0
$517$	5.29831e6i	0.871788i
$518$	0	0
$519$	−1.89277e6	−0.308446
$520$	0	0
$521$	1.02485e7	1.65412	0.827059	−	0.562115i	$-0.190012\pi$
$521$	1.02485e7	1.65412	0.827059	+	0.562115i	$0.190012\pi$
$522$	0	0
$523$	5.94878e6i	0.950985i	0.879720	+	0.475493i	$0.157730\pi$
$523$	5.94878e6i	0.950985i	−0.879720	+	0.475493i	$0.842270\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3.58726e6	−0.562648
$528$	0	0
$529$	−5.42207e6	−0.842415
$530$	0	0
$531$	− 7.06945e6i	− 1.08805i
$532$	0	0
$533$	381868.i	0.0582231i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−825483.	−0.123530
$538$	0	0
$539$	− 5.03097e6i	− 0.745899i
$540$	0	0
$541$	− 1.29778e7i	− 1.90637i	−0.302385	−	0.953186i	$-0.597783\pi$
$541$	− 1.29778e7i	− 1.90637i	0.302385	−	0.953186i	$-0.402217\pi$
$542$	0	0
$543$	−1.51907e6	−0.221094
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 6.10894e6i	− 0.872966i	−0.899712	−	0.436483i	$-0.856224\pi$
$547$	− 6.10894e6i	− 0.872966i	0.899712	−	0.436483i	$-0.143776\pi$
$548$	0	0
$549$	4.24990e6i	0.601794i
$550$	0	0
$551$	−1.34514e6	−0.188751
$552$	0	0
$553$	2.12736e6	0.295820
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.01541e7i	1.38676i	0.720570	+	0.693382i	$0.243880\pi$
$557$	1.01541e7i	1.38676i	−0.720570	+	0.693382i	$0.756120\pi$
$558$	0	0
$559$	7.15850e6	0.968931
$560$	0	0
$561$	1.29604e6	0.173864
$562$	0	0
$563$	229520.i	0.0305175i	0.999884	+	0.0152588i	$0.00485721\pi$
$563$	229520.i	0.0305175i	−0.999884	+	0.0152588i	$0.995143\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.48546e6	0.194045
$568$	0	0
$569$	−5.11894e6	−0.662826	−0.331413	−	0.943486i	$-0.607525\pi$
$569$	−5.11894e6	−0.662826	−0.331413	+	0.943486i	$0.607525\pi$
$570$	0	0
$571$	− 1.09781e7i	− 1.40909i	−0.709660	−	0.704544i	$-0.751152\pi$
$571$	− 1.09781e7i	− 1.40909i	0.709660	−	0.704544i	$-0.248848\pi$
$572$	0	0
$573$	251876.i	0.0320479i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−3.09452e6	−0.386949	−0.193474	−	0.981105i	$-0.561976\pi$
$577$	−3.09452e6	−0.386949	−0.193474	+	0.981105i	$0.561976\pi$
$578$	0	0
$579$	− 135950.i	− 0.0168532i
$580$	0	0
$581$	1.60079e6i	0.196741i
$582$	0	0
$583$	4.00565e6	0.488093
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.24448e7i	1.49071i	0.666666	+	0.745357i	$0.267721\pi$
$587$	1.24448e7i	1.49071i	−0.666666	+	0.745357i	$0.732279\pi$
$588$	0	0
$589$	491677.i	0.0583972i
$590$	0	0
$591$	1.15558e6	0.136092
$592$	0	0
$593$	−7.81508e6	−0.912634	−0.456317	−	0.889817i	$-0.650832\pi$
$593$	−7.81508e6	−0.912634	−0.456317	+	0.889817i	$0.650832\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.34268e6i	0.154183i
$598$	0	0
$599$	8.68576e6	0.989101	0.494550	−	0.869149i	$-0.335333\pi$
$599$	8.68576e6	0.989101	0.494550	+	0.869149i	$0.335333\pi$
$600$	0	0
$601$	−3.99770e6	−0.451465	−0.225733	−	0.974189i	$-0.572478\pi$
$601$	−3.99770e6	−0.451465	−0.225733	+	0.974189i	$0.572478\pi$
$602$	0	0
$603$	− 1.15507e7i	− 1.29365i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−2.96166e6	−0.326259	−0.163130	−	0.986605i	$-0.552159\pi$
$607$	−2.96166e6	−0.326259	−0.163130	+	0.986605i	$0.552159\pi$
$608$	0	0
$609$	−435498.	−0.0475820
$610$	0	0
$611$	− 1.06082e7i	− 1.14957i
$612$	0	0
$613$	− 1.42575e7i	− 1.53247i	−0.642561	−	0.766235i	$-0.722128\pi$
$613$	− 1.42575e7i	− 1.53247i	0.642561	−	0.766235i	$-0.277872\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	4.11383e6	0.435044	0.217522	−	0.976055i	$-0.430203\pi$
$617$	4.11383e6	0.435044	0.217522	+	0.976055i	$0.430203\pi$
$618$	0	0
$619$	− 6.74663e6i	− 0.707718i	−0.935299	−	0.353859i	$-0.884869\pi$
$619$	− 6.74663e6i	− 0.707718i	0.935299	−	0.353859i	$-0.115131\pi$
$620$	0	0
$621$	1.24931e6i	0.129999i
$622$	0	0
$623$	−2.23360e6	−0.230561
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 177638.i	− 0.0180454i
$628$	0	0
$629$	− 5.73967e6i	− 0.578442i
$630$	0	0
$631$	−5.01676e6	−0.501591	−0.250795	−	0.968040i	$-0.580692\pi$
$631$	−5.01676e6	−0.501591	−0.250795	+	0.968040i	$0.580692\pi$
$632$	0	0
$633$	−2.91401e6	−0.289055
$634$	0	0
$635$	0	0
$636$	0	0
$637$	1.00729e7i	0.983571i
$638$	0	0
$639$	−8.81311e6	−0.853841
$640$	0	0
$641$	−1.79114e6	−0.172180	−0.0860902	−	0.996287i	$-0.527437\pi$
$641$	−1.79114e6	−0.172180	−0.0860902	+	0.996287i	$0.527437\pi$
$642$	0	0
$643$	− 4.49516e6i	− 0.428763i	−0.976750	−	0.214382i	$-0.931226\pi$
$643$	− 4.49516e6i	− 0.428763i	0.976750	−	0.214382i	$-0.0687736\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−2.47054e6	−0.232023	−0.116012	−	0.993248i	$-0.537011\pi$
$647$	−2.47054e6	−0.232023	−0.116012	+	0.993248i	$0.537011\pi$
$648$	0	0
$649$	9.37410e6	0.873610
$650$	0	0
$651$	159184.i	0.0147213i
$652$	0	0
$653$	− 1.26068e7i	− 1.15697i	−0.815693	−	0.578484i	$-0.803644\pi$
$653$	− 1.26068e7i	− 1.15697i	0.815693	−	0.578484i	$-0.196356\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−6.16574e6	−0.557278
$658$	0	0
$659$	7.51240e6i	0.673853i	0.941531	+	0.336927i	$0.109387\pi$
$659$	7.51240e6i	0.673853i	−0.941531	+	0.336927i	$0.890613\pi$
$660$	0	0
$661$	− 1.66707e7i	− 1.48405i	−0.670371	−	0.742026i	$-0.733865\pi$
$661$	− 1.66707e7i	− 1.48405i	0.670371	−	0.742026i	$-0.266135\pi$
$662$	0	0
$663$	−2.59490e6	−0.229264
$664$	0	0
$665$	0	0
$666$	0	0
$667$	6.18450e6i	0.538257i
$668$	0	0
$669$	− 359165.i	− 0.0310262i
$670$	0	0
$671$	−5.63537e6	−0.483188
$672$	0	0
$673$	1.75455e7	1.49323	0.746617	−	0.665254i	$-0.231677\pi$
$673$	1.75455e7	1.49323	0.746617	+	0.665254i	$0.231677\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	681435.i	0.0571416i	0.999592	+	0.0285708i	$0.00909561\pi$
$677$	681435.i	0.0571416i	−0.999592	+	0.0285708i	$0.990904\pi$
$678$	0	0
$679$	−778111.	−0.0647690
$680$	0	0
$681$	2.50227e6	0.206760
$682$	0	0
$683$	− 1.08945e7i	− 0.893629i	−0.894627	−	0.446815i	$-0.852558\pi$
$683$	− 1.08945e7i	− 0.893629i	0.894627	−	0.446815i	$-0.147442\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−2.17578e6	−0.175882
$688$	0	0
$689$	−8.02003e6	−0.643617
$690$	0	0
$691$	1.31179e7i	1.04513i	0.852599	+	0.522565i	$0.175025\pi$
$691$	1.31179e7i	1.04513i	−0.852599	+	0.522565i	$0.824975\pi$
$692$	0	0
$693$	2.02886e6i	0.160479i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	972801.	0.0758477
$698$	0	0
$699$	2.52975e6i	0.195832i
$700$	0	0
$701$	− 1.34915e7i	− 1.03697i	−0.855087	−	0.518485i	$-0.826496\pi$
$701$	− 1.34915e7i	− 1.03697i	0.855087	−	0.518485i	$-0.173504\pi$
$702$	0	0
$703$	−786690.	−0.0600365
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.72728e6i	0.280443i
$708$	0	0
$709$	− 1.23849e7i	− 0.925290i	−0.886544	−	0.462645i	$-0.846900\pi$
$709$	− 1.23849e7i	− 0.925290i	0.886544	−	0.462645i	$-0.153100\pi$
$710$	0	0
$711$	1.83456e7	1.36100
$712$	0	0
$713$	2.26057e6	0.166530
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 3.04216e6i	− 0.220996i
$718$	0	0
$719$	−1.59608e7	−1.15142	−0.575710	−	0.817654i	$-0.695274\pi$
$719$	−1.59608e7	−1.15142	−0.575710	+	0.817654i	$0.695274\pi$
$720$	0	0
$721$	4.56511e6	0.327049
$722$	0	0
$723$	− 1.59194e6i	− 0.113261i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1.58056e7	1.10911	0.554554	−	0.832148i	$-0.312889\pi$
$727$	1.58056e7	1.10911	0.554554	+	0.832148i	$0.312889\pi$
$728$	0	0
$729$	1.20299e7	0.838387
$730$	0	0
$731$	− 1.82361e7i	− 1.26223i
$732$	0	0
$733$	2.47037e7i	1.69825i	0.528190	+	0.849126i	$0.322871\pi$
$733$	2.47037e7i	1.69825i	−0.528190	+	0.849126i	$0.677129\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.53162e7	1.03868
$738$	0	0
$739$	− 8.37792e6i	− 0.564319i	−0.959367	−	0.282160i	$-0.908949\pi$
$739$	− 8.37792e6i	− 0.564319i	0.959367	−	0.282160i	$-0.0910508\pi$
$740$	0	0
$741$	355662.i	0.0237953i
$742$	0	0
$743$	1.37134e6	0.0911323	0.0455662	−	0.998961i	$-0.485491\pi$
$743$	1.37134e6	0.0911323	0.0455662	+	0.998961i	$0.485491\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	1.38047e7i	0.905160i
$748$	0	0
$749$	1.85712e6i	0.120958i
$750$	0	0
$751$	2.32593e7	1.50486	0.752431	−	0.658671i	$-0.228881\pi$
$751$	2.32593e7	1.50486	0.752431	+	0.658671i	$0.228881\pi$
$752$	0	0
$753$	−3.26756e6	−0.210008
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 9.50255e6i	− 0.602699i	−0.953514	−	0.301350i	$-0.902563\pi$
$757$	− 9.50255e6i	− 0.602699i	0.953514	−	0.301350i	$-0.0974371\pi$
$758$	0	0
$759$	−816718.	−0.0514597
$760$	0	0
$761$	−1.85691e6	−0.116233	−0.0581164	−	0.998310i	$-0.518509\pi$
$761$	−1.85691e6	−0.116233	−0.0581164	+	0.998310i	$0.518509\pi$
$762$	0	0
$763$	1.52264e6i	0.0946858i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.87686e7	−1.15198
$768$	0	0
$769$	3.11565e7	1.89991	0.949955	−	0.312386i	$-0.101128\pi$
$769$	3.11565e7	1.89991	0.949955	+	0.312386i	$0.101128\pi$
$770$	0	0
$771$	896377.i	0.0543069i
$772$	0	0
$773$	3.47037e6i	0.208894i	0.994530	+	0.104447i	$0.0333073\pi$
$773$	3.47037e6i	0.208894i	−0.994530	+	0.104447i	$0.966693\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−254696.	−0.0151346
$778$	0	0
$779$	− 133334.i	− 0.00787223i
$780$	0	0
$781$	− 1.16862e7i	− 0.685560i
$782$	0	0
$783$	−7.61764e6	−0.444034
$784$	0	0
$785$	0	0
$786$	0	0
$787$	8.99343e6i	0.517593i	0.965932	+	0.258797i	$0.0833260\pi$
$787$	8.99343e6i	0.517593i	−0.965932	+	0.258797i	$0.916674\pi$
$788$	0	0
$789$	− 5.00079e6i	− 0.285987i
$790$	0	0
$791$	−4.67188e6	−0.265491
$792$	0	0
$793$	1.12830e7	0.637150
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 6.61412e6i	− 0.368830i	−0.982848	−	0.184415i	$-0.940961\pi$
$797$	− 6.61412e6i	− 0.368830i	0.982848	−	0.184415i	$-0.0590390\pi$
$798$	0	0
$799$	−2.70240e7	−1.49756
$800$	0	0
$801$	−1.92618e7	−1.06076
$802$	0	0
$803$	− 8.17578e6i	− 0.447446i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−5.76382e6	−0.311549
$808$	0	0
$809$	−1.45192e7	−0.779957	−0.389978	−	0.920824i	$-0.627518\pi$
$809$	−1.45192e7	−0.779957	−0.389978	+	0.920824i	$0.627518\pi$
$810$	0	0
$811$	1.23447e7i	0.659068i	0.944144	+	0.329534i	$0.106892\pi$
$811$	1.23447e7i	0.659068i	−0.944144	+	0.329534i	$0.893108\pi$
$812$	0	0
$813$	1.32638e6i	0.0703787i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−2.49948e6	−0.131007
$818$	0	0
$819$	− 4.06214e6i	− 0.211614i
$820$	0	0
$821$	1.98036e7i	1.02538i	0.858573	+	0.512692i	$0.171352\pi$
$821$	1.98036e7i	1.02538i	−0.858573	+	0.512692i	$0.828648\pi$
$822$	0	0
$823$	−1.28299e7	−0.660275	−0.330137	−	0.943933i	$-0.607095\pi$
$823$	−1.28299e7	−0.660275	−0.330137	+	0.943933i	$0.607095\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 3.83991e7i	− 1.95235i	−0.216989	−	0.976174i	$-0.569624\pi$
$827$	− 3.83991e7i	− 1.95235i	0.216989	−	0.976174i	$-0.430376\pi$
$828$	0	0
$829$	− 1.17104e7i	− 0.591813i	−0.955217	−	0.295907i	$-0.904378\pi$
$829$	− 1.17104e7i	− 0.591813i	0.955217	−	0.295907i	$-0.0956217\pi$
$830$	0	0
$831$	2.13697e6	0.107348
$832$	0	0
$833$	2.56605e7	1.28130
$834$	0	0
$835$	0	0
$836$	0	0
$837$	2.78441e6i	0.137379i
$838$	0	0
$839$	2.39114e7	1.17273	0.586367	−	0.810046i	$-0.300558\pi$
$839$	2.39114e7	1.17273	0.586367	+	0.810046i	$0.300558\pi$
$840$	0	0
$841$	−1.71987e7	−0.838503
$842$	0	0
$843$	− 5.66448e6i	− 0.274531i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1.72277e6	0.0825125
$848$	0	0
$849$	−4.25511e6	−0.202601
$850$	0	0
$851$	3.61694e6i	0.171205i
$852$	0	0
$853$	1.27294e6i	0.0599010i	0.999551	+	0.0299505i	$0.00953496\pi$
$853$	1.27294e6i	0.0599010i	−0.999551	+	0.0299505i	$0.990465\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1.88053e7	0.874636	0.437318	−	0.899307i	$-0.355928\pi$
$857$	1.88053e7	0.874636	0.437318	+	0.899307i	$0.355928\pi$
$858$	0	0
$859$	1.00554e7i	0.464960i	0.972601	+	0.232480i	$0.0746840\pi$
$859$	1.00554e7i	0.464960i	−0.972601	+	0.232480i	$0.925316\pi$
$860$	0	0
$861$	− 43167.8i	− 0.00198450i
$862$	0	0
$863$	−2.96146e7	−1.35357	−0.676783	−	0.736183i	$-0.736626\pi$
$863$	−2.96146e7	−1.35357	−0.676783	+	0.736183i	$0.736626\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	2.93569e6i	0.132636i
$868$	0	0
$869$	2.43263e7i	1.09277i
$870$	0	0
$871$	−3.06658e7	−1.36965
$872$	0	0
$873$	−6.71017e6	−0.297988
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1.42067e7i	0.623727i	0.950127	+	0.311863i	$0.100953\pi$
$877$	1.42067e7i	0.623727i	−0.950127	+	0.311863i	$0.899047\pi$
$878$	0	0
$879$	−4.63932e6	−0.202526
$880$	0	0
$881$	1.88875e7	0.819851	0.409925	−	0.912119i	$-0.365555\pi$
$881$	1.88875e7	0.819851	0.409925	+	0.912119i	$0.365555\pi$
$882$	0	0
$883$	9.53131e6i	0.411387i	0.978616	+	0.205694i	$0.0659450\pi$
$883$	9.53131e6i	0.411387i	−0.978616	+	0.205694i	$0.934055\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−3.28115e7	−1.40029	−0.700143	−	0.714002i	$-0.746881\pi$
$887$	−3.28115e7	−1.40029	−0.700143	+	0.714002i	$0.746881\pi$
$888$	0	0
$889$	8.59762e6	0.364858
$890$	0	0
$891$	1.69862e7i	0.716806i
$892$	0	0
$893$	3.70397e6i	0.155431i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	1.63521e6	0.0678568
$898$	0	0
$899$	1.37837e7i	0.568811i
$900$	0	0
$901$	2.04308e7i	0.838444i
$902$	0	0
$903$	−809223.	−0.0330255
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 1.53129e7i	− 0.618073i	−0.951050	−	0.309037i	$-0.899993\pi$
$907$	− 1.53129e7i	− 0.618073i	0.951050	−	0.309037i	$-0.100007\pi$
$908$	0	0
$909$	3.21429e7i	1.29025i
$910$	0	0
$911$	5.70920e6	0.227918	0.113959	−	0.993485i	$-0.463647\pi$
$911$	5.70920e6	0.227918	0.113959	+	0.993485i	$0.463647\pi$
$912$	0	0
$913$	−1.83050e7	−0.726764
$914$	0	0
$915$	0	0
$916$	0	0
$917$	9.69077e6i	0.380570i
$918$	0	0
$919$	9.51702e6	0.371717	0.185858	−	0.982577i	$-0.440493\pi$
$919$	9.51702e6	0.371717	0.185858	+	0.982577i	$0.440493\pi$
$920$	0	0
$921$	3.37919e6	0.131269
$922$	0	0
$923$	2.33978e7i	0.904005i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	3.93680e7	1.50468
$928$	0	0
$929$	−2.79927e7	−1.06416	−0.532078	−	0.846695i	$-0.678589\pi$
$929$	−2.79927e7	−1.06416	−0.532078	+	0.846695i	$0.678589\pi$
$930$	0	0
$931$	− 3.51708e6i	− 0.132987i
$932$	0	0
$933$	2.57809e6i	0.0969604i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−6.87551e6	−0.255832	−0.127916	−	0.991785i	$-0.540829\pi$
$937$	−6.87551e6	−0.255832	−0.127916	+	0.991785i	$0.540829\pi$
$938$	0	0
$939$	2.29586e6i	0.0849730i
$940$	0	0
$941$	2.68027e7i	0.986743i	0.869819	+	0.493371i	$0.164236\pi$
$941$	2.68027e7i	0.986743i	−0.869819	+	0.493371i	$0.835764\pi$
$942$	0	0
$943$	−613025.	−0.0224491
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 8.91973e6i	− 0.323204i	−0.986856	−	0.161602i	$-0.948334\pi$
$947$	− 8.91973e6i	− 0.323204i	0.986856	−	0.161602i	$-0.0516661\pi$
$948$	0	0
$949$	1.63694e7i	0.590019i
$950$	0	0
$951$	3.19080e6	0.114406
$952$	0	0
$953$	3.25154e7	1.15973	0.579865	−	0.814713i	$-0.303105\pi$
$953$	3.25154e7	1.15973	0.579865	+	0.814713i	$0.303105\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 4.97991e6i	− 0.175769i
$958$	0	0
$959$	−1.99347e6	−0.0699942
$960$	0	0
$961$	−2.35909e7	−0.824017
$962$	0	0
$963$	1.60152e7i	0.556502i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	1.43390e7	0.493121	0.246561	−	0.969127i	$-0.420700\pi$
$967$	1.43390e7	0.493121	0.246561	+	0.969127i	$0.420700\pi$
$968$	0	0
$969$	906041.	0.0309983
$970$	0	0
$971$	4.37060e7i	1.48762i	0.668390	+	0.743811i	$0.266984\pi$
$971$	4.37060e7i	1.48762i	−0.668390	+	0.743811i	$0.733016\pi$
$972$	0	0
$973$	7.11015e6i	0.240767i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−7.03004e6	−0.235625	−0.117813	−	0.993036i	$-0.537588\pi$
$977$	−7.03004e6	−0.235625	−0.117813	+	0.993036i	$0.537588\pi$
$978$	0	0
$979$	− 2.55412e7i	− 0.851696i
$980$	0	0
$981$	1.31307e7i	0.435628i
$982$	0	0
$983$	3.83715e7	1.26656	0.633279	−	0.773924i	$-0.281709\pi$
$983$	3.83715e7	1.26656	0.633279	+	0.773924i	$0.281709\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1.19918e6i	0.0391826i
$988$	0	0
$989$	1.14918e7i	0.373591i
$990$	0	0
$991$	5.24440e7	1.69634	0.848168	−	0.529727i	$-0.177706\pi$
$991$	5.24440e7	1.69634	0.848168	+	0.529727i	$0.177706\pi$
$992$	0	0
$993$	−8.60602e6	−0.276968
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 4.88081e7i	− 1.55508i	−0.628832	−	0.777542i	$-0.716467\pi$
$997$	− 4.88081e7i	− 1.55508i	0.628832	−	0.777542i	$-0.283533\pi$
$998$	0	0
$999$	−4.45509e6	−0.141235

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.6.d.d.401.12		20
4.3	odd	2		200.6.d.c.101.9	✓	20
5.2	odd	4		800.6.f.d.49.23		40
5.3	odd	4		800.6.f.d.49.18		40
5.4	even	2		800.6.d.b.401.9		20
8.3	odd	2		200.6.d.c.101.10	yes	20
8.5	even	2	inner	800.6.d.d.401.9		20
20.3	even	4		200.6.f.d.149.2		40
20.7	even	4		200.6.f.d.149.39		40
20.19	odd	2		200.6.d.d.101.12	yes	20
40.3	even	4		200.6.f.d.149.40		40
40.13	odd	4		800.6.f.d.49.24		40
40.19	odd	2		200.6.d.d.101.11	yes	20
40.27	even	4		200.6.f.d.149.1		40
40.29	even	2		800.6.d.b.401.12		20
40.37	odd	4		800.6.f.d.49.17		40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
200.6.d.c.101.9	✓	20	4.3	odd	2
200.6.d.c.101.10	yes	20	8.3	odd	2
200.6.d.d.101.11	yes	20	40.19	odd	2
200.6.d.d.101.12	yes	20	20.19	odd	2
200.6.f.d.149.1		40	40.27	even	4
200.6.f.d.149.2		40	20.3	even	4
200.6.f.d.149.39		40	20.7	even	4
200.6.f.d.149.40		40	40.3	even	4
800.6.d.b.401.9		20	5.4	even	2
800.6.d.b.401.12		20	40.29	even	2
800.6.d.d.401.9		20	8.5	even	2	inner
800.6.d.d.401.12		20	1.1	even	1	trivial
800.6.f.d.49.17		40	40.37	odd	4
800.6.f.d.49.18		40	5.3	odd	4
800.6.f.d.49.23		40	5.2	odd	4
800.6.f.d.49.24		40	40.13	odd	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:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	800.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.58812i q^{3} +27.4015 q^{7} +236.302 q^{9} +O(q^{10})\)	\(q+2.58812i q^{3} +27.4015 q^{7} +236.302 q^{9} +313.336i q^{11} -627.354i q^{13} -1598.17 q^{17} +219.049i q^{19} +70.9184i q^{21} +1007.11 q^{23} +1240.49i q^{27} +6140.83i q^{29} +2244.61 q^{31} -810.951 q^{33} +3591.40i q^{37} +1623.67 q^{39} -608.697 q^{41} +11410.6i q^{43} +16909.4 q^{47} -16056.2 q^{49} -4136.26i q^{51} -12783.9i q^{53} -566.923 q^{57} -29917.1i q^{59} +17985.1i q^{61} +6475.03 q^{63} -48881.2i q^{67} +2606.52i q^{69} -37296.0 q^{71} -26092.7 q^{73} +8585.89i q^{77} +77636.5 q^{79} +54210.8 q^{81} +58419.8i q^{83} -15893.2 q^{87} -81513.8 q^{89} -17190.5i q^{91} +5809.30i q^{93} -28396.6 q^{97} +74041.9i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 196 q^{7} - 1620 q^{9}+O(q^{10}) \) 20 * q + 196 * q^7 - 1620 * q^9	\( 20 q + 196 q^{7} - 1620 q^{9} + 7184 q^{23} - 7160 q^{31} - 2836 q^{33} - 22452 q^{39} - 5804 q^{41} - 44180 q^{47} + 62652 q^{49} + 43696 q^{57} + 1240 q^{63} + 7724 q^{71} - 105136 q^{73} + 7780 q^{79} + 96984 q^{81} + 106188 q^{87} - 3160 q^{89} - 73688 q^{97}+O(q^{100}) \) 20 * q + 196 * q^7 - 1620 * q^9 + 7184 * q^23 - 7160 * q^31 - 2836 * q^33 - 22452 * q^39 - 5804 * q^41 - 44180 * q^47 + 62652 * q^49 + 43696 * q^57 + 1240 * q^63 + 7724 * q^71 - 105136 * q^73 + 7780 * q^79 + 96984 * q^81 + 106188 * q^87 - 3160 * q^89 - 73688 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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