LMFDB - Embedded newform 800.6.d.d.401.16

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			401.16
Root			$4.12326 - 3.87282i$ of defining polynomial
Character	$\chi$	$=$	800.401
Dual form			800.6.d.d.401.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+18.2848i q^{3} +2.25573 q^{7} -91.3327 q^{9} -419.261i q^{11} -106.889i q^{13} +849.098 q^{17} +335.414i q^{19} +41.2454i q^{21} -3541.78 q^{23} +2773.20i q^{27} -5208.57i q^{29} -5637.83 q^{31} +7666.09 q^{33} -61.9860i q^{37} +1954.44 q^{39} +16286.1 q^{41} +2417.19i q^{43} +22781.9 q^{47} -16801.9 q^{49} +15525.6i q^{51} +13667.5i q^{53} -6132.98 q^{57} +23407.1i q^{59} +33444.7i q^{61} -206.022 q^{63} +66162.9i q^{67} -64760.7i q^{69} +51421.4 q^{71} +21271.6 q^{73} -945.738i q^{77} -38418.7 q^{79} -72901.2 q^{81} +93166.7i q^{83} +95237.5 q^{87} -60678.0 q^{89} -241.112i q^{91} -103086. i q^{93} +157428. q^{97} +38292.2i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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}+ \cdots - 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

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
$3$	18.2848i	1.17297i	0.809961	+	0.586484i	$0.199488\pi$
$3$	18.2848i	1.17297i	−0.809961	+	0.586484i	$0.800512\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.25573	0.0173997	0.00869984	−	0.999962i	$-0.497231\pi$
$7$	2.25573	0.0173997	0.00869984	+	0.999962i	$0.497231\pi$
$8$	0	0
$9$	−91.3327	−0.375855
$10$	0	0
$11$	− 419.261i	− 1.04473i	−0.852723	−	0.522363i	$-0.825050\pi$
$11$	− 419.261i	− 1.04473i	0.852723	−	0.522363i	$-0.174950\pi$
$12$	0	0
$13$	− 106.889i	− 0.175418i	−0.996146	−	0.0877090i	$-0.972045\pi$
$13$	− 106.889i	− 0.175418i	0.996146	−	0.0877090i	$-0.0279546\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	849.098	0.712584	0.356292	−	0.934375i	$-0.384041\pi$
$17$	849.098	0.712584	0.356292	+	0.934375i	$0.384041\pi$
$18$	0	0
$19$	335.414i	0.213156i	0.994304	+	0.106578i	$0.0339894\pi$
$19$	335.414i	0.213156i	−0.994304	+	0.106578i	$0.966011\pi$
$20$	0	0
$21$	41.2454i	0.0204093i
$22$	0	0
$23$	−3541.78	−1.39605	−0.698027	−	0.716071i	$-0.745938\pi$
$23$	−3541.78	−1.39605	−0.698027	+	0.716071i	$0.745938\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	2773.20i	0.732103i
$28$	0	0
$29$	− 5208.57i	− 1.15007i	−0.818129	−	0.575034i	$-0.804989\pi$
$29$	− 5208.57i	− 1.15007i	0.818129	−	0.575034i	$-0.195011\pi$
$30$	0	0
$31$	−5637.83	−1.05368	−0.526839	−	0.849965i	$-0.676623\pi$
$31$	−5637.83	−1.05368	−0.526839	+	0.849965i	$0.676623\pi$
$32$	0	0
$33$	7666.09	1.22543
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 61.9860i	− 0.00744370i	−0.999993	−	0.00372185i	$-0.998815\pi$
$37$	− 61.9860i	− 0.00744370i	0.999993	−	0.00372185i	$-0.00118470\pi$
$38$	0	0
$39$	1954.44	0.205760
$40$	0	0
$41$	16286.1	1.51306	0.756531	−	0.653958i	$-0.226893\pi$
$41$	16286.1	1.51306	0.756531	+	0.653958i	$0.226893\pi$
$42$	0	0
$43$	2417.19i	0.199361i	0.995020	+	0.0996803i	$0.0317820\pi$
$43$	2417.19i	0.199361i	−0.995020	+	0.0996803i	$0.968218\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	22781.9	1.50434	0.752170	−	0.658970i	$-0.229007\pi$
$47$	22781.9	1.50434	0.752170	+	0.658970i	$0.229007\pi$
$48$	0	0
$49$	−16801.9	−0.999697
$50$	0	0
$51$	15525.6i	0.835838i
$52$	0	0
$53$	13667.5i	0.668344i	0.942512	+	0.334172i	$0.108457\pi$
$53$	13667.5i	0.668344i	−0.942512	+	0.334172i	$0.891543\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−6132.98	−0.250025
$58$	0	0
$59$	23407.1i	0.875423i	0.899116	+	0.437711i	$0.144211\pi$
$59$	23407.1i	0.875423i	−0.899116	+	0.437711i	$0.855789\pi$
$60$	0	0
$61$	33444.7i	1.15081i	0.817869	+	0.575405i	$0.195155\pi$
$61$	33444.7i	1.15081i	−0.817869	+	0.575405i	$0.804845\pi$
$62$	0	0
$63$	−206.022	−0.00653975
$64$	0	0
$65$	0	0
$66$	0	0
$67$	66162.9i	1.80064i	0.435226	+	0.900321i	$0.356668\pi$
$67$	66162.9i	1.80064i	−0.435226	+	0.900321i	$0.643332\pi$
$68$	0	0
$69$	− 64760.7i	− 1.63753i
$70$	0	0
$71$	51421.4	1.21059	0.605296	−	0.796001i	$-0.293055\pi$
$71$	51421.4	1.21059	0.605296	+	0.796001i	$0.293055\pi$
$72$	0	0
$73$	21271.6	0.467189	0.233595	−	0.972334i	$-0.424951\pi$
$73$	21271.6	0.467189	0.233595	+	0.972334i	$0.424951\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 945.738i	− 0.0181779i
$78$	0	0
$79$	−38418.7	−0.692589	−0.346294	−	0.938126i	$-0.612560\pi$
$79$	−38418.7	−0.692589	−0.346294	+	0.938126i	$0.612560\pi$
$80$	0	0
$81$	−72901.2	−1.23459
$82$	0	0
$83$	93166.7i	1.48445i	0.670151	+	0.742225i	$0.266229\pi$
$83$	93166.7i	1.48445i	−0.670151	+	0.742225i	$0.733771\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	95237.5	1.34899
$88$	0	0
$89$	−60678.0	−0.812000	−0.406000	−	0.913873i	$-0.633077\pi$
$89$	−60678.0	−0.812000	−0.406000	+	0.913873i	$0.633077\pi$
$90$	0	0
$91$	− 241.112i	− 0.00305222i
$92$	0	0
$93$	− 103086.i	− 1.23593i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	157428.	1.69884	0.849418	−	0.527721i	$-0.176953\pi$
$97$	157428.	1.69884	0.849418	+	0.527721i	$0.176953\pi$
$98$	0	0
$99$	38292.2i	0.392665i
$100$	0	0
$101$	124508.i	1.21449i	0.794516	+	0.607243i	$0.207725\pi$
$101$	124508.i	1.21449i	−0.794516	+	0.607243i	$0.792275\pi$
$102$	0	0
$103$	10346.4	0.0960938	0.0480469	−	0.998845i	$-0.484700\pi$
$103$	10346.4	0.0960938	0.0480469	+	0.998845i	$0.484700\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	70765.9i	0.597537i	0.954326	+	0.298769i	$0.0965758\pi$
$107$	70765.9i	0.597537i	−0.954326	+	0.298769i	$0.903424\pi$
$108$	0	0
$109$	− 128453.i	− 1.03557i	−0.855511	−	0.517784i	$-0.826757\pi$
$109$	− 128453.i	− 1.03557i	0.855511	−	0.517784i	$-0.173243\pi$
$110$	0	0
$111$	1133.40	0.00873123
$112$	0	0
$113$	−115196.	−0.848672	−0.424336	−	0.905505i	$-0.639492\pi$
$113$	−115196.	−0.848672	−0.424336	+	0.905505i	$0.639492\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	9762.45i	0.0659317i
$118$	0	0
$119$	1915.33	0.0123987
$120$	0	0
$121$	−14728.7	−0.0914537
$122$	0	0
$123$	297787.i	1.77477i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−146026.	−0.803378	−0.401689	−	0.915776i	$-0.631577\pi$
$127$	−146026.	−0.803378	−0.401689	+	0.915776i	$0.631577\pi$
$128$	0	0
$129$	−44197.7	−0.233844
$130$	0	0
$131$	− 12998.0i	− 0.0661757i	−0.999452	−	0.0330878i	$-0.989466\pi$
$131$	− 12998.0i	− 0.0661757i	0.999452	−	0.0330878i	$-0.0105341\pi$
$132$	0	0
$133$	756.603i	0.00370885i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	189204.	0.861249	0.430624	−	0.902531i	$-0.358293\pi$
$137$	189204.	0.861249	0.430624	+	0.902531i	$0.358293\pi$
$138$	0	0
$139$	44334.4i	0.194627i	0.995254	+	0.0973136i	$0.0310250\pi$
$139$	44334.4i	0.194627i	−0.995254	+	0.0973136i	$0.968975\pi$
$140$	0	0
$141$	416562.i	1.76454i
$142$	0	0
$143$	−44814.3	−0.183264
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 307219.i	− 1.17261i
$148$	0	0
$149$	352240.i	1.29979i	0.760024	+	0.649895i	$0.225187\pi$
$149$	352240.i	1.29979i	−0.760024	+	0.649895i	$0.774813\pi$
$150$	0	0
$151$	−446273.	−1.59279	−0.796395	−	0.604776i	$-0.793262\pi$
$151$	−446273.	−1.59279	−0.796395	+	0.604776i	$0.793262\pi$
$152$	0	0
$153$	−77550.5	−0.267828
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 383316.i	− 1.24110i	−0.784165	−	0.620552i	$-0.786909\pi$
$157$	− 383316.i	− 1.24110i	0.784165	−	0.620552i	$-0.213091\pi$
$158$	0	0
$159$	−249907.	−0.783946
$160$	0	0
$161$	−7989.29	−0.0242909
$162$	0	0
$163$	405200.i	1.19454i	0.802040	+	0.597270i	$0.203748\pi$
$163$	405200.i	1.19454i	−0.802040	+	0.597270i	$0.796252\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	193042.	0.535625	0.267812	−	0.963471i	$-0.413699\pi$
$167$	193042.	0.535625	0.267812	+	0.963471i	$0.413699\pi$
$168$	0	0
$169$	359868.	0.969229
$170$	0	0
$171$	− 30634.3i	− 0.0801157i
$172$	0	0
$173$	− 599397.i	− 1.52265i	−0.648372	−	0.761323i	$-0.724550\pi$
$173$	− 599397.i	− 1.52265i	0.648372	−	0.761323i	$-0.275450\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−427994.	−1.02684
$178$	0	0
$179$	421278.i	0.982736i	0.870952	+	0.491368i	$0.163503\pi$
$179$	421278.i	0.982736i	−0.870952	+	0.491368i	$0.836497\pi$
$180$	0	0
$181$	183362.i	0.416018i	0.978127	+	0.208009i	$0.0666983\pi$
$181$	183362.i	0.416018i	−0.978127	+	0.208009i	$0.933302\pi$
$182$	0	0
$183$	−611529.	−1.34986
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 355994.i	− 0.744455i
$188$	0	0
$189$	6255.58i	0.0127384i
$190$	0	0
$191$	−982533.	−1.94878	−0.974392	−	0.224856i	$-0.927809\pi$
$191$	−982533.	−1.94878	−0.974392	+	0.224856i	$0.927809\pi$
$192$	0	0
$193$	−164978.	−0.318810	−0.159405	−	0.987213i	$-0.550958\pi$
$193$	−164978.	−0.318810	−0.159405	+	0.987213i	$0.550958\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 506605.i	− 0.930044i	−0.885299	−	0.465022i	$-0.846046\pi$
$197$	− 506605.i	− 0.930044i	0.885299	−	0.465022i	$-0.153954\pi$
$198$	0	0
$199$	−475645.	−0.851433	−0.425716	−	0.904857i	$-0.639978\pi$
$199$	−475645.	−0.851433	−0.425716	+	0.904857i	$0.639978\pi$
$200$	0	0
$201$	−1.20977e6	−2.11210
$202$	0	0
$203$	− 11749.1i	− 0.0200108i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	323481.	0.524714
$208$	0	0
$209$	140626.	0.222690
$210$	0	0
$211$	260466.i	0.402759i	0.979513	+	0.201379i	$0.0645424\pi$
$211$	260466.i	0.402759i	−0.979513	+	0.201379i	$0.935458\pi$
$212$	0	0
$213$	940228.i	1.41999i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−12717.4	−0.0183337
$218$	0	0
$219$	388946.i	0.547998i
$220$	0	0
$221$	− 90759.2i	− 0.125000i
$222$	0	0
$223$	−624180.	−0.840519	−0.420260	−	0.907404i	$-0.638061\pi$
$223$	−624180.	−0.840519	−0.420260	+	0.907404i	$0.638061\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 656014.i	− 0.844984i	−0.906367	−	0.422492i	$-0.861155\pi$
$227$	− 656014.i	− 0.844984i	0.906367	−	0.422492i	$-0.138845\pi$
$228$	0	0
$229$	807777.i	1.01790i	0.860798	+	0.508948i	$0.169965\pi$
$229$	807777.i	1.01790i	−0.860798	+	0.508948i	$0.830035\pi$
$230$	0	0
$231$	17292.6	0.0213221
$232$	0	0
$233$	−1.30452e6	−1.57421	−0.787103	−	0.616822i	$-0.788420\pi$
$233$	−1.30452e6	−1.57421	−0.787103	+	0.616822i	$0.788420\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 702478.i	− 0.812385i
$238$	0	0
$239$	1.21861e6	1.37997	0.689985	−	0.723824i	$-0.257617\pi$
$239$	1.21861e6	1.37997	0.689985	+	0.723824i	$0.257617\pi$
$240$	0	0
$241$	−983578.	−1.09085	−0.545426	−	0.838159i	$-0.683632\pi$
$241$	−983578.	−1.09085	−0.545426	+	0.838159i	$0.683632\pi$
$242$	0	0
$243$	− 659093.i	− 0.716030i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	35852.1	0.0373914
$248$	0	0
$249$	−1.70353e6	−1.74121
$250$	0	0
$251$	1.60324e6i	1.60625i	0.595809	+	0.803126i	$0.296832\pi$
$251$	1.60324e6i	1.60625i	−0.595809	+	0.803126i	$0.703168\pi$
$252$	0	0
$253$	1.48493e6i	1.45849i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.07709e6	1.01723	0.508617	−	0.860993i	$-0.330157\pi$
$257$	1.07709e6	1.01723	0.508617	+	0.860993i	$0.330157\pi$
$258$	0	0
$259$	− 139.823i	0 0.000129518i
$260$	0	0
$261$	475713.i	0.432259i
$262$	0	0
$263$	812889.	0.724673	0.362336	−	0.932047i	$-0.381979\pi$
$263$	812889.	0.724673	0.362336	+	0.932047i	$0.381979\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 1.10948e6i	− 0.952451i
$268$	0	0
$269$	− 113957.i	− 0.0960201i	−0.998847	−	0.0480101i	$-0.984712\pi$
$269$	− 113957.i	− 0.0960201i	0.998847	−	0.0480101i	$-0.0152880\pi$
$270$	0	0
$271$	−372728.	−0.308297	−0.154148	−	0.988048i	$-0.549263\pi$
$271$	−372728.	−0.308297	−0.154148	+	0.988048i	$0.549263\pi$
$272$	0	0
$273$	4408.68	0.00358015
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 914263.i	− 0.715932i	−0.933735	−	0.357966i	$-0.883470\pi$
$277$	− 914263.i	− 0.715932i	0.933735	−	0.357966i	$-0.116530\pi$
$278$	0	0
$279$	514918.	0.396030
$280$	0	0
$281$	−511184.	−0.386199	−0.193100	−	0.981179i	$-0.561854\pi$
$281$	−511184.	−0.386199	−0.193100	+	0.981179i	$0.561854\pi$
$282$	0	0
$283$	− 533078.i	− 0.395662i	−0.980236	−	0.197831i	$-0.936610\pi$
$283$	− 533078.i	− 0.395662i	0.980236	−	0.197831i	$-0.0633897\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	36736.9	0.0263268
$288$	0	0
$289$	−698889.	−0.492225
$290$	0	0
$291$	2.87853e6i	1.99268i
$292$	0	0
$293$	2.54648e6i	1.73289i	0.499273	+	0.866445i	$0.333600\pi$
$293$	2.54648e6i	1.73289i	−0.499273	+	0.866445i	$0.666400\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.16269e6	0.764847
$298$	0	0
$299$	378577.i	0.244893i
$300$	0	0
$301$	5452.51i	0.00346881i
$302$	0	0
$303$	−2.27659e6	−1.42455
$304$	0	0
$305$	0	0
$306$	0	0
$307$	740673.i	0.448518i	0.974530	+	0.224259i	$0.0719962\pi$
$307$	740673.i	0.448518i	−0.974530	+	0.224259i	$0.928004\pi$
$308$	0	0
$309$	189181.i	0.112715i
$310$	0	0
$311$	−1.08876e6	−0.638310	−0.319155	−	0.947702i	$-0.603399\pi$
$311$	−1.08876e6	−0.638310	−0.319155	+	0.947702i	$0.603399\pi$
$312$	0	0
$313$	1.08277e6	0.624705	0.312353	−	0.949966i	$-0.398883\pi$
$313$	1.08277e6	0.624705	0.312353	+	0.949966i	$0.398883\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.93243e6i	1.63900i	0.573078	+	0.819501i	$0.305749\pi$
$317$	2.93243e6i	1.63900i	−0.573078	+	0.819501i	$0.694251\pi$
$318$	0	0
$319$	−2.18375e6	−1.20151
$320$	0	0
$321$	−1.29394e6	−0.700892
$322$	0	0
$323$	284800.i	0.151892i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	2.34874e6	1.21469
$328$	0	0
$329$	51389.8	0.0261750
$330$	0	0
$331$	357214.i	0.179209i	0.995977	+	0.0896043i	$0.0285602\pi$
$331$	357214.i	0.179209i	−0.995977	+	0.0896043i	$0.971440\pi$
$332$	0	0
$333$	5661.34i	0.00279775i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	1.45195e6	0.696429	0.348215	−	0.937415i	$-0.386788\pi$
$337$	1.45195e6	0.696429	0.348215	+	0.937415i	$0.386788\pi$
$338$	0	0
$339$	− 2.10633e6i	− 0.995465i
$340$	0	0
$341$	2.36372e6i	1.10081i
$342$	0	0
$343$	−75812.5	−0.0347941
$344$	0	0
$345$	0	0
$346$	0	0
$347$	393735.i	0.175541i	0.996141	+	0.0877707i	$0.0279743\pi$
$347$	393735.i	0.175541i	−0.996141	+	0.0877707i	$0.972026\pi$
$348$	0	0
$349$	1.73875e6i	0.764140i	0.924133	+	0.382070i	$0.124789\pi$
$349$	1.73875e6i	0.764140i	−0.924133	+	0.382070i	$0.875211\pi$
$350$	0	0
$351$	296424.	0.128424
$352$	0	0
$353$	−2.05124e6	−0.876154	−0.438077	−	0.898937i	$-0.644340\pi$
$353$	−2.05124e6	−0.876154	−0.438077	+	0.898937i	$0.644340\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	35021.4i	0.0145433i
$358$	0	0
$359$	−3.56728e6	−1.46084	−0.730418	−	0.683000i	$-0.760675\pi$
$359$	−3.56728e6	−1.46084	−0.730418	+	0.683000i	$0.760675\pi$
$360$	0	0
$361$	2.36360e6	0.954564
$362$	0	0
$363$	− 269311.i	− 0.107272i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	3.72721e6	1.44450	0.722251	−	0.691631i	$-0.243107\pi$
$367$	3.72721e6	1.44450	0.722251	+	0.691631i	$0.243107\pi$
$368$	0	0
$369$	−1.48745e6	−0.568691
$370$	0	0
$371$	30830.2i	0.0116290i
$372$	0	0
$373$	1.71134e6i	0.636891i	0.947941	+	0.318446i	$0.103161\pi$
$373$	1.71134e6i	0.636891i	−0.947941	+	0.318446i	$0.896839\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−556738.	−0.201743
$378$	0	0
$379$	− 3.60174e6i	− 1.28799i	−0.765028	−	0.643997i	$-0.777275\pi$
$379$	− 3.60174e6i	− 1.28799i	0.765028	−	0.643997i	$-0.222725\pi$
$380$	0	0
$381$	− 2.67005e6i	− 0.942337i
$382$	0	0
$383$	3.80875e6	1.32674	0.663369	−	0.748293i	$-0.269126\pi$
$383$	3.80875e6	1.32674	0.663369	+	0.748293i	$0.269126\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 220768.i	− 0.0749306i
$388$	0	0
$389$	431800.i	0.144680i	0.997380	+	0.0723401i	$0.0230467\pi$
$389$	431800.i	0.144680i	−0.997380	+	0.0723401i	$0.976953\pi$
$390$	0	0
$391$	−3.00732e6	−0.994805
$392$	0	0
$393$	237665.	0.0776220
$394$	0	0
$395$	0	0
$396$	0	0
$397$	2.28465e6i	0.727518i	0.931493	+	0.363759i	$0.118507\pi$
$397$	2.28465e6i	0.727518i	−0.931493	+	0.363759i	$0.881493\pi$
$398$	0	0
$399$	−13834.3	−0.00435036
$400$	0	0
$401$	1.63473e6	0.507674	0.253837	−	0.967247i	$-0.418307\pi$
$401$	1.63473e6	0.507674	0.253837	+	0.967247i	$0.418307\pi$
$402$	0	0
$403$	602621.i	0.184834i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−25988.3	−0.00777663
$408$	0	0
$409$	5.21834e6	1.54250	0.771248	−	0.636534i	$-0.219633\pi$
$409$	5.21834e6	1.54250	0.771248	+	0.636534i	$0.219633\pi$
$410$	0	0
$411$	3.45955e6i	1.01022i
$412$	0	0
$413$	52800.0i	0.0152321i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−810644.	−0.228292
$418$	0	0
$419$	666477.i	0.185460i	0.995691	+	0.0927299i	$0.0295593\pi$
$419$	666477.i	0.185460i	−0.995691	+	0.0927299i	$0.970441\pi$
$420$	0	0
$421$	6.82007e6i	1.87536i	0.347504	+	0.937678i	$0.387029\pi$
$421$	6.82007e6i	1.87536i	−0.347504	+	0.937678i	$0.612971\pi$
$422$	0	0
$423$	−2.08073e6	−0.565413
$424$	0	0
$425$	0	0
$426$	0	0
$427$	75442.2i	0.0200237i
$428$	0	0
$429$	− 819419.i	− 0.214963i
$430$	0	0
$431$	1.38707e6	0.359671	0.179836	−	0.983697i	$-0.442443\pi$
$431$	1.38707e6	0.359671	0.179836	+	0.983697i	$0.442443\pi$
$432$	0	0
$433$	1.33169e6	0.341336	0.170668	−	0.985329i	$-0.445407\pi$
$433$	1.33169e6	0.341336	0.170668	+	0.985329i	$0.445407\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 1.18797e6i	− 0.297577i
$438$	0	0
$439$	2.78241e6	0.689064	0.344532	−	0.938775i	$-0.388038\pi$
$439$	2.78241e6	0.689064	0.344532	+	0.938775i	$0.388038\pi$
$440$	0	0
$441$	1.53456e6	0.375741
$442$	0	0
$443$	4.63595e6i	1.12235i	0.827696	+	0.561177i	$0.189651\pi$
$443$	4.63595e6i	1.12235i	−0.827696	+	0.561177i	$0.810349\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−6.44063e6	−1.52461
$448$	0	0
$449$	1.18197e6	0.276688	0.138344	−	0.990384i	$-0.455822\pi$
$449$	1.18197e6	0.276688	0.138344	+	0.990384i	$0.455822\pi$
$450$	0	0
$451$	− 6.82811e6i	− 1.58074i
$452$	0	0
$453$	− 8.16001e6i	− 1.86829i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1.55240e6	0.347708	0.173854	−	0.984771i	$-0.444378\pi$
$457$	1.55240e6	0.347708	0.173854	+	0.984771i	$0.444378\pi$
$458$	0	0
$459$	2.35472e6i	0.521684i
$460$	0	0
$461$	− 7.43437e6i	− 1.62926i	−0.579978	−	0.814632i	$-0.696939\pi$
$461$	− 7.43437e6i	− 1.62926i	0.579978	−	0.814632i	$-0.303061\pi$
$462$	0	0
$463$	−20179.5	−0.00437480	−0.00218740	−	0.999998i	$-0.500696\pi$
$463$	−20179.5	−0.00437480	−0.00218740	+	0.999998i	$0.500696\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 3.58698e6i	− 0.761092i	−0.924762	−	0.380546i	$-0.875736\pi$
$467$	− 3.58698e6i	− 0.761092i	0.924762	−	0.380546i	$-0.124264\pi$
$468$	0	0
$469$	149245.i	0.0313306i
$470$	0	0
$471$	7.00885e6	1.45578
$472$	0	0
$473$	1.01343e6	0.208277
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 1.24829e6i	− 0.251200i
$478$	0	0
$479$	−183482.	−0.0365388	−0.0182694	−	0.999833i	$-0.505816\pi$
$479$	−183482.	−0.0365388	−0.0182694	+	0.999833i	$0.505816\pi$
$480$	0	0
$481$	−6625.61	−0.00130576
$482$	0	0
$483$	− 146082.i	− 0.0284925i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−6.68968e6	−1.27815	−0.639077	−	0.769143i	$-0.720683\pi$
$487$	−6.68968e6	−1.27815	−0.639077	+	0.769143i	$0.720683\pi$
$488$	0	0
$489$	−7.40899e6	−1.40116
$490$	0	0
$491$	− 9.90038e6i	− 1.85331i	−0.375913	−	0.926655i	$-0.622671\pi$
$491$	− 9.90038e6i	− 1.85331i	0.375913	−	0.926655i	$-0.377329\pi$
$492$	0	0
$493$	− 4.42259e6i	− 0.819520i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	115993.	0.0210639
$498$	0	0
$499$	5.95907e6i	1.07134i	0.844428	+	0.535670i	$0.179941\pi$
$499$	5.95907e6i	1.07134i	−0.844428	+	0.535670i	$0.820059\pi$
$500$	0	0
$501$	3.52973e6i	0.628271i
$502$	0	0
$503$	3.69353e6	0.650910	0.325455	−	0.945557i	$-0.394482\pi$
$503$	3.69353e6	0.650910	0.325455	+	0.945557i	$0.394482\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	6.58010e6i	1.13687i
$508$	0	0
$509$	− 7.94222e6i	− 1.35877i	−0.733780	−	0.679387i	$-0.762246\pi$
$509$	− 7.94222e6i	− 1.35877i	0.733780	−	0.679387i	$-0.237754\pi$
$510$	0	0
$511$	47982.9	0.00812894
$512$	0	0
$513$	−930172.	−0.156052
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 9.55157e6i	− 1.57162i
$518$	0	0
$519$	1.09598e7	1.78602
$520$	0	0
$521$	−1.06312e7	−1.71588	−0.857940	−	0.513749i	$-0.828256\pi$
$521$	−1.06312e7	−1.71588	−0.857940	+	0.513749i	$0.828256\pi$
$522$	0	0
$523$	− 3.13913e6i	− 0.501828i	−0.968009	−	0.250914i	$-0.919269\pi$
$523$	− 3.13913e6i	− 0.501828i	0.968009	−	0.250914i	$-0.0807312\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4.78708e6	−0.750834
$528$	0	0
$529$	6.10788e6	0.948967
$530$	0	0
$531$	− 2.13783e6i	− 0.329032i
$532$	0	0
$533$	− 1.74080e6i	− 0.265418i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−7.70298e6	−1.15272
$538$	0	0
$539$	7.04438e6i	1.04441i
$540$	0	0
$541$	− 1.20044e7i	− 1.76339i	−0.471821	−	0.881694i	$-0.656403\pi$
$541$	− 1.20044e7i	− 1.76339i	0.471821	−	0.881694i	$-0.343597\pi$
$542$	0	0
$543$	−3.35273e6	−0.487976
$544$	0	0
$545$	0	0
$546$	0	0
$547$	9.93035e6i	1.41905i	0.704682	+	0.709523i	$0.251090\pi$
$547$	9.93035e6i	1.41905i	−0.704682	+	0.709523i	$0.748910\pi$
$548$	0	0
$549$	− 3.05460e6i	− 0.432537i
$550$	0	0
$551$	1.74703e6	0.245144
$552$	0	0
$553$	−86662.2	−0.0120508
$554$	0	0
$555$	0	0
$556$	0	0
$557$	8.72764e6i	1.19195i	0.803002	+	0.595976i	$0.203235\pi$
$557$	8.72764e6i	1.19195i	−0.803002	+	0.595976i	$0.796765\pi$
$558$	0	0
$559$	258370.	0.0349714
$560$	0	0
$561$	6.50926e6	0.873222
$562$	0	0
$563$	− 2.13884e6i	− 0.284386i	−0.989839	−	0.142193i	$-0.954585\pi$
$563$	− 2.13884e6i	− 0.284386i	0.989839	−	0.142193i	$-0.0454154\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−164445.	−0.0214814
$568$	0	0
$569$	2.98366e6	0.386339	0.193169	−	0.981165i	$-0.438123\pi$
$569$	2.98366e6	0.386339	0.193169	+	0.981165i	$0.438123\pi$
$570$	0	0
$571$	− 8.50264e6i	− 1.09135i	−0.837997	−	0.545674i	$-0.816274\pi$
$571$	− 8.50264e6i	− 1.09135i	0.837997	−	0.545674i	$-0.183726\pi$
$572$	0	0
$573$	− 1.79654e7i	− 2.28586i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−1.55926e7	−1.94975	−0.974874	−	0.222756i	$-0.928495\pi$
$577$	−1.55926e7	−1.94975	−0.974874	+	0.222756i	$0.928495\pi$
$578$	0	0
$579$	− 3.01658e6i	− 0.373954i
$580$	0	0
$581$	210159.i	0.0258289i
$582$	0	0
$583$	5.73025e6	0.698236
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 1.01432e7i	− 1.21501i	−0.794314	−	0.607507i	$-0.792170\pi$
$587$	− 1.01432e7i	− 1.21501i	0.794314	−	0.607507i	$-0.207830\pi$
$588$	0	0
$589$	− 1.89101e6i	− 0.224598i
$590$	0	0
$591$	9.26315e6	1.09091
$592$	0	0
$593$	1.44098e7	1.68276	0.841381	−	0.540442i	$-0.181743\pi$
$593$	1.44098e7	1.68276	0.841381	+	0.540442i	$0.181743\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 8.69706e6i	− 0.998704i
$598$	0	0
$599$	3.13415e6	0.356905	0.178452	−	0.983949i	$-0.442891\pi$
$599$	3.13415e6	0.356905	0.178452	+	0.983949i	$0.442891\pi$
$600$	0	0
$601$	1.57244e7	1.77578	0.887889	−	0.460058i	$-0.152171\pi$
$601$	1.57244e7	1.77578	0.887889	+	0.460058i	$0.152171\pi$
$602$	0	0
$603$	− 6.04283e6i	− 0.676780i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	1.03887e7	1.14443	0.572217	−	0.820102i	$-0.306083\pi$
$607$	1.03887e7	1.14443	0.572217	+	0.820102i	$0.306083\pi$
$608$	0	0
$609$	214830.	0.0234721
$610$	0	0
$611$	− 2.43513e6i	− 0.263888i
$612$	0	0
$613$	1.46355e7i	1.57310i	0.617525	+	0.786551i	$0.288135\pi$
$613$	1.46355e7i	1.57310i	−0.617525	+	0.786551i	$0.711865\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−6.06227e6	−0.641095	−0.320547	−	0.947233i	$-0.603867\pi$
$617$	−6.06227e6	−0.641095	−0.320547	+	0.947233i	$0.603867\pi$
$618$	0	0
$619$	888836.i	0.0932384i	0.998913	+	0.0466192i	$0.0148447\pi$
$619$	888836.i	0.0932384i	−0.998913	+	0.0466192i	$0.985155\pi$
$620$	0	0
$621$	− 9.82207e6i	− 1.02205i
$622$	0	0
$623$	−136873.	−0.0141285
$624$	0	0
$625$	0	0
$626$	0	0
$627$	2.57132e6i	0.261208i
$628$	0	0
$629$	− 52632.2i	− 0.00530426i
$630$	0	0
$631$	−5.86047e6	−0.585948	−0.292974	−	0.956120i	$-0.594645\pi$
$631$	−5.86047e6	−0.585948	−0.292974	+	0.956120i	$0.594645\pi$
$632$	0	0
$633$	−4.76256e6	−0.472423
$634$	0	0
$635$	0	0
$636$	0	0
$637$	1.79594e6i	0.175365i
$638$	0	0
$639$	−4.69645e6	−0.455007
$640$	0	0
$641$	6.88679e6	0.662021	0.331011	−	0.943627i	$-0.392610\pi$
$641$	6.88679e6	0.662021	0.331011	+	0.943627i	$0.392610\pi$
$642$	0	0
$643$	7.66403e6i	0.731021i	0.930807	+	0.365510i	$0.119106\pi$
$643$	7.66403e6i	0.731021i	−0.930807	+	0.365510i	$0.880894\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−3.93185e6	−0.369263	−0.184631	−	0.982808i	$-0.559109\pi$
$647$	−3.93185e6	−0.369263	−0.184631	+	0.982808i	$0.559109\pi$
$648$	0	0
$649$	9.81369e6	0.914577
$650$	0	0
$651$	− 232535.i	− 0.0215048i
$652$	0	0
$653$	− 5.55582e6i	− 0.509876i	−0.966957	−	0.254938i	$-0.917945\pi$
$653$	− 5.55582e6i	− 0.509876i	0.966957	−	0.254938i	$-0.0820551\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−1.94279e6	−0.175595
$658$	0	0
$659$	2.17299e6i	0.194914i	0.995240	+	0.0974572i	$0.0310709\pi$
$659$	2.17299e6i	0.194914i	−0.995240	+	0.0974572i	$0.968929\pi$
$660$	0	0
$661$	6.93458e6i	0.617329i	0.951171	+	0.308665i	$0.0998821\pi$
$661$	6.93458e6i	0.617329i	−0.951171	+	0.308665i	$0.900118\pi$
$662$	0	0
$663$	1.65951e6	0.146621
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.84476e7i	1.60556i
$668$	0	0
$669$	− 1.14130e7i	− 0.985902i
$670$	0	0
$671$	1.40221e7	1.20228
$672$	0	0
$673$	7.72402e6	0.657364	0.328682	−	0.944441i	$-0.393396\pi$
$673$	7.72402e6	0.657364	0.328682	+	0.944441i	$0.393396\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.38586e7i	1.16211i	0.813865	+	0.581054i	$0.197360\pi$
$677$	1.38586e7i	1.16211i	−0.813865	+	0.581054i	$0.802640\pi$
$678$	0	0
$679$	355113.	0.0295592
$680$	0	0
$681$	1.19951e7	0.991139
$682$	0	0
$683$	− 778871.i	− 0.0638872i	−0.999490	−	0.0319436i	$-0.989830\pi$
$683$	− 778871.i	− 0.0638872i	0.999490	−	0.0319436i	$-0.0101697\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−1.47700e7	−1.19396
$688$	0	0
$689$	1.46090e6	0.117239
$690$	0	0
$691$	− 6.98899e6i	− 0.556826i	−0.960461	−	0.278413i	$-0.910192\pi$
$691$	− 6.98899e6i	− 0.556826i	0.960461	−	0.278413i	$-0.0898084\pi$
$692$	0	0
$693$	86376.8i	0.00683225i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1.38285e7	1.07818
$698$	0	0
$699$	− 2.38529e7i	− 1.84649i
$700$	0	0
$701$	− 4.14587e6i	− 0.318655i	−0.987226	−	0.159328i	$-0.949067\pi$
$701$	− 4.14587e6i	− 0.318655i	0.987226	−	0.159328i	$-0.0509326\pi$
$702$	0	0
$703$	20791.0	0.00158667
$704$	0	0
$705$	0	0
$706$	0	0
$707$	280855.i	0.0211317i
$708$	0	0
$709$	1.52511e7i	1.13943i	0.821843	+	0.569714i	$0.192946\pi$
$709$	1.52511e7i	1.13943i	−0.821843	+	0.569714i	$0.807054\pi$
$710$	0	0
$711$	3.50889e6	0.260313
$712$	0	0
$713$	1.99680e7	1.47099
$714$	0	0
$715$	0	0
$716$	0	0
$717$	2.22820e7i	1.61866i
$718$	0	0
$719$	1.60151e6	0.115534	0.0577668	−	0.998330i	$-0.481602\pi$
$719$	1.60151e6	0.115534	0.0577668	+	0.998330i	$0.481602\pi$
$720$	0	0
$721$	23338.6	0.00167200
$722$	0	0
$723$	− 1.79845e7i	− 1.27954i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	6.15453e6	0.431876	0.215938	−	0.976407i	$-0.430719\pi$
$727$	6.15453e6	0.431876	0.215938	+	0.976407i	$0.430719\pi$
$728$	0	0
$729$	−5.66362e6	−0.394708
$730$	0	0
$731$	2.05243e6i	0.142061i
$732$	0	0
$733$	− 7.15476e6i	− 0.491853i	−0.969289	−	0.245926i	$-0.920908\pi$
$733$	− 7.15476e6i	− 0.491853i	0.969289	−	0.245926i	$-0.0790921\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2.77395e7	1.88118
$738$	0	0
$739$	− 2.74026e7i	− 1.84578i	−0.385063	−	0.922890i	$-0.625820\pi$
$739$	− 2.74026e7i	− 1.84578i	0.385063	−	0.922890i	$-0.374180\pi$
$740$	0	0
$741$	655547.i	0.0438589i
$742$	0	0
$743$	1.20016e7	0.797569	0.398785	−	0.917045i	$-0.369432\pi$
$743$	1.20016e7	0.797569	0.398785	+	0.917045i	$0.369432\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 8.50916e6i	− 0.557937i
$748$	0	0
$749$	159629.i	0.0103970i
$750$	0	0
$751$	−3.38961e6	−0.219305	−0.109653	−	0.993970i	$-0.534974\pi$
$751$	−3.38961e6	−0.219305	−0.109653	+	0.993970i	$0.534974\pi$
$752$	0	0
$753$	−2.93148e7	−1.88408
$754$	0	0
$755$	0	0
$756$	0	0
$757$	3.47336e6i	0.220298i	0.993915	+	0.110149i	$0.0351328\pi$
$757$	3.47336e6i	0.220298i	−0.993915	+	0.110149i	$0.964867\pi$
$758$	0	0
$759$	−2.71516e7	−1.71077
$760$	0	0
$761$	−1.16257e6	−0.0727710	−0.0363855	−	0.999338i	$-0.511584\pi$
$761$	−1.16257e6	−0.0727710	−0.0363855	+	0.999338i	$0.511584\pi$
$762$	0	0
$763$	− 289755.i	− 0.0180186i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.50196e6	0.153565
$768$	0	0
$769$	−1.71387e7	−1.04511	−0.522556	−	0.852605i	$-0.675021\pi$
$769$	−1.71387e7	−1.04511	−0.522556	+	0.852605i	$0.675021\pi$
$770$	0	0
$771$	1.96944e7i	1.19318i
$772$	0	0
$773$	3.17392e7i	1.91050i	0.295800	+	0.955250i	$0.404414\pi$
$773$	3.17392e7i	1.91050i	−0.295800	+	0.955250i	$0.595586\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	2556.64	0.000151921 0
$778$	0	0
$779$	5.46258e6i	0.322518i
$780$	0	0
$781$	− 2.15590e7i	− 1.26474i
$782$	0	0
$783$	1.44444e7	0.841968
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 2.67032e6i	− 0.153683i	−0.997043	−	0.0768416i	$-0.975516\pi$
$787$	− 2.67032e6i	− 0.153683i	0.997043	−	0.0768416i	$-0.0244836\pi$
$788$	0	0
$789$	1.48635e7i	0.850018i
$790$	0	0
$791$	−259850.	−0.0147666
$792$	0	0
$793$	3.57487e6	0.201873
$794$	0	0
$795$	0	0
$796$	0	0
$797$	3.37393e7i	1.88144i	0.339188	+	0.940719i	$0.389848\pi$
$797$	3.37393e7i	1.88144i	−0.339188	+	0.940719i	$0.610152\pi$
$798$	0	0
$799$	1.93441e7	1.07197
$800$	0	0
$801$	5.54189e6	0.305194
$802$	0	0
$803$	− 8.91835e6i	− 0.488085i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	2.08369e6	0.112629
$808$	0	0
$809$	1.25443e6	0.0673870	0.0336935	−	0.999432i	$-0.489273\pi$
$809$	1.25443e6	0.0673870	0.0336935	+	0.999432i	$0.489273\pi$
$810$	0	0
$811$	1.12827e7i	0.602365i	0.953567	+	0.301182i	$0.0973813\pi$
$811$	1.12827e7i	0.602365i	−0.953567	+	0.301182i	$0.902619\pi$
$812$	0	0
$813$	− 6.81525e6i	− 0.361622i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−810760.	−0.0424949
$818$	0	0
$819$	22021.4i	0.00114719i
$820$	0	0
$821$	1.21845e7i	0.630886i	0.948945	+	0.315443i	$0.102153\pi$
$821$	1.21845e7i	0.630886i	−0.948945	+	0.315443i	$0.897847\pi$
$822$	0	0
$823$	2.12650e7	1.09437	0.547187	−	0.837010i	$-0.315698\pi$
$823$	2.12650e7	1.09437	0.547187	+	0.837010i	$0.315698\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	6.06316e6i	0.308273i	0.988050	+	0.154136i	$0.0492596\pi$
$827$	6.06316e6i	0.308273i	−0.988050	+	0.154136i	$0.950740\pi$
$828$	0	0
$829$	− 727353.i	− 0.0367586i	−0.999831	−	0.0183793i	$-0.994149\pi$
$829$	− 727353.i	− 0.0367586i	0.999831	−	0.0183793i	$-0.00585064\pi$
$830$	0	0
$831$	1.67171e7	0.839765
$832$	0	0
$833$	−1.42665e7	−0.712368
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 1.56348e7i	− 0.771401i
$838$	0	0
$839$	1.36723e7	0.670560	0.335280	−	0.942118i	$-0.391169\pi$
$839$	1.36723e7	0.670560	0.335280	+	0.942118i	$0.391169\pi$
$840$	0	0
$841$	−6.61809e6	−0.322658
$842$	0	0
$843$	− 9.34688e6i	− 0.452999i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−33223.9	−0.00159126
$848$	0	0
$849$	9.74720e6	0.464099
$850$	0	0
$851$	219541.i	0.0103918i
$852$	0	0
$853$	3.66971e7i	1.72687i	0.504462	+	0.863434i	$0.331691\pi$
$853$	3.66971e7i	1.72687i	−0.504462	+	0.863434i	$0.668309\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1.73498e7	−0.806941	−0.403470	−	0.914993i	$-0.632196\pi$
$857$	−1.73498e7	−0.806941	−0.403470	+	0.914993i	$0.632196\pi$
$858$	0	0
$859$	− 1.90085e7i	− 0.878954i	−0.898254	−	0.439477i	$-0.855164\pi$
$859$	− 1.90085e7i	− 0.878954i	0.898254	−	0.439477i	$-0.144836\pi$
$860$	0	0
$861$	671726.i	0.0308805i
$862$	0	0
$863$	−3.46920e7	−1.58563	−0.792816	−	0.609460i	$-0.791386\pi$
$863$	−3.46920e7	−1.58563	−0.792816	+	0.609460i	$0.791386\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 1.27790e7i	− 0.577364i
$868$	0	0
$869$	1.61075e7i	0.723566i
$870$	0	0
$871$	7.07207e6	0.315865
$872$	0	0
$873$	−1.43783e7	−0.638515
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 3.99003e7i	− 1.75177i	−0.482520	−	0.875885i	$-0.660279\pi$
$877$	− 3.99003e7i	− 1.75177i	0.482520	−	0.875885i	$-0.339721\pi$
$878$	0	0
$879$	−4.65618e7	−2.03262
$880$	0	0
$881$	−3.50875e7	−1.52304	−0.761522	−	0.648139i	$-0.775548\pi$
$881$	−3.50875e7	−1.52304	−0.761522	+	0.648139i	$0.775548\pi$
$882$	0	0
$883$	1.47881e7i	0.638279i	0.947708	+	0.319139i	$0.103394\pi$
$883$	1.47881e7i	0.638279i	−0.947708	+	0.319139i	$0.896606\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.17291e7	0.500560	0.250280	−	0.968173i	$-0.419477\pi$
$887$	1.17291e7	0.500560	0.250280	+	0.968173i	$0.419477\pi$
$888$	0	0
$889$	−329394.	−0.0139785
$890$	0	0
$891$	3.05646e7i	1.28981i
$892$	0	0
$893$	7.64139e6i	0.320659i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−6.92219e6	−0.287252
$898$	0	0
$899$	2.93651e7i	1.21180i
$900$	0	0
$901$	1.16051e7i	0.476251i
$902$	0	0
$903$	−99697.9	−0.00406880
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 1.36136e7i	− 0.549483i	−0.961518	−	0.274742i	$-0.911408\pi$
$907$	− 1.36136e7i	− 0.549483i	0.961518	−	0.274742i	$-0.0885923\pi$
$908$	0	0
$909$	− 1.13716e7i	− 0.456470i
$910$	0	0
$911$	−1.26888e7	−0.506551	−0.253276	−	0.967394i	$-0.581508\pi$
$911$	−1.26888e7	−0.506551	−0.253276	+	0.967394i	$0.581508\pi$
$912$	0	0
$913$	3.90611e7	1.55084
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 29319.9i	− 0.00115144i
$918$	0	0
$919$	4.25253e7	1.66096	0.830479	−	0.557050i	$-0.188067\pi$
$919$	4.25253e7	1.66096	0.830479	+	0.557050i	$0.188067\pi$
$920$	0	0
$921$	−1.35430e7	−0.526098
$922$	0	0
$923$	− 5.49637e6i	− 0.212360i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−944963.	−0.0361173
$928$	0	0
$929$	4.25450e6	0.161737	0.0808686	−	0.996725i	$-0.474231\pi$
$929$	4.25450e6	0.161737	0.0808686	+	0.996725i	$0.474231\pi$
$930$	0	0
$931$	− 5.63560e6i	− 0.213092i
$932$	0	0
$933$	− 1.99078e7i	− 0.748718i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1.07844e7	0.401279	0.200640	−	0.979665i	$-0.435698\pi$
$937$	1.07844e7	0.401279	0.200640	+	0.979665i	$0.435698\pi$
$938$	0	0
$939$	1.97982e7i	0.732760i
$940$	0	0
$941$	− 112960.i	− 0.00415862i	−0.999998	−	0.00207931i	$-0.999338\pi$
$941$	− 112960.i	− 0.00415862i	0.999998	−	0.00207931i	$-0.000661865\pi$
$942$	0	0
$943$	−5.76817e7	−2.11232
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1.62952e7i	− 0.590454i	−0.955427	−	0.295227i	$-0.904605\pi$
$947$	− 1.62952e7i	− 0.590454i	0.955427	−	0.295227i	$-0.0953953\pi$
$948$	0	0
$949$	− 2.27370e6i	− 0.0819534i
$950$	0	0
$951$	−5.36188e7	−1.92250
$952$	0	0
$953$	−1.07203e7	−0.382363	−0.191181	−	0.981555i	$-0.561232\pi$
$953$	−1.07203e7	−0.382363	−0.191181	+	0.981555i	$0.561232\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 3.99294e7i	− 1.40933i
$958$	0	0
$959$	426792.	0.0149855
$960$	0	0
$961$	3.15601e6	0.110238
$962$	0	0
$963$	− 6.46324e6i	− 0.224587i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	3.92793e7	1.35082	0.675411	−	0.737442i	$-0.263966\pi$
$967$	3.92793e7	1.35082	0.675411	+	0.737442i	$0.263966\pi$
$968$	0	0
$969$	−5.20750e6	−0.178164
$970$	0	0
$971$	− 2.76288e7i	− 0.940405i	−0.882559	−	0.470202i	$-0.844181\pi$
$971$	− 2.76288e7i	− 0.940405i	0.882559	−	0.470202i	$-0.155819\pi$
$972$	0	0
$973$	100006.i	0.00338645i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−2.56548e7	−0.859868	−0.429934	−	0.902860i	$-0.641463\pi$
$977$	−2.56548e7	−0.859868	−0.429934	+	0.902860i	$0.641463\pi$
$978$	0	0
$979$	2.54399e7i	0.848318i
$980$	0	0
$981$	1.17320e7i	0.389223i
$982$	0	0
$983$	5.21860e6	0.172254	0.0861271	−	0.996284i	$-0.472551\pi$
$983$	5.21860e6	0.172254	0.0861271	+	0.996284i	$0.472551\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	939650.i	0.0307025i
$988$	0	0
$989$	− 8.56115e6i	− 0.278318i
$990$	0	0
$991$	−4.76772e7	−1.54215	−0.771075	−	0.636744i	$-0.780281\pi$
$991$	−4.76772e7	−1.54215	−0.771075	+	0.636744i	$0.780281\pi$
$992$	0	0
$993$	−6.53158e6	−0.210206
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 2.96368e7i	− 0.944263i	−0.881528	−	0.472132i	$-0.843485\pi$
$997$	− 2.96368e7i	− 0.944263i	0.881528	−	0.472132i	$-0.156515\pi$
$998$	0	0
$999$	171900.	0.00544955

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.6.d.d.401.16		20
4.3	odd	2		200.6.d.c.101.5	✓	20
5.2	odd	4		800.6.f.d.49.32		40
5.3	odd	4		800.6.f.d.49.9		40
5.4	even	2		800.6.d.b.401.5		20
8.3	odd	2		200.6.d.c.101.6	yes	20
8.5	even	2	inner	800.6.d.d.401.5		20
20.3	even	4		200.6.f.d.149.12		40
20.7	even	4		200.6.f.d.149.29		40
20.19	odd	2		200.6.d.d.101.16	yes	20
40.3	even	4		200.6.f.d.149.30		40
40.13	odd	4		800.6.f.d.49.31		40
40.19	odd	2		200.6.d.d.101.15	yes	20
40.27	even	4		200.6.f.d.149.11		40
40.29	even	2		800.6.d.b.401.16		20
40.37	odd	4		800.6.f.d.49.10		40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
200.6.d.c.101.5	✓	20	4.3	odd	2
200.6.d.c.101.6	yes	20	8.3	odd	2
200.6.d.d.101.15	yes	20	40.19	odd	2
200.6.d.d.101.16	yes	20	20.19	odd	2
200.6.f.d.149.11		40	40.27	even	4
200.6.f.d.149.12		40	20.3	even	4
200.6.f.d.149.29		40	20.7	even	4
200.6.f.d.149.30		40	40.3	even	4
800.6.d.b.401.5		20	5.4	even	2
800.6.d.b.401.16		20	40.29	even	2
800.6.d.d.401.5		20	8.5	even	2	inner
800.6.d.d.401.16		20	1.1	even	1	trivial
800.6.f.d.49.9		40	5.3	odd	4
800.6.f.d.49.10		40	40.37	odd	4
800.6.f.d.49.31		40	40.13	odd	4
800.6.f.d.49.32		40	5.2	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)

\(f(q)\)	\(=\)	\(q+18.2848i q^{3} +2.25573 q^{7} -91.3327 q^{9} -419.261i q^{11} -106.889i q^{13} +849.098 q^{17} +335.414i q^{19} +41.2454i q^{21} -3541.78 q^{23} +2773.20i q^{27} -5208.57i q^{29} -5637.83 q^{31} +7666.09 q^{33} -61.9860i q^{37} +1954.44 q^{39} +16286.1 q^{41} +2417.19i q^{43} +22781.9 q^{47} -16801.9 q^{49} +15525.6i q^{51} +13667.5i q^{53} -6132.98 q^{57} +23407.1i q^{59} +33444.7i q^{61} -206.022 q^{63} +66162.9i q^{67} -64760.7i q^{69} +51421.4 q^{71} +21271.6 q^{73} -945.738i q^{77} -38418.7 q^{79} -72901.2 q^{81} +93166.7i q^{83} +95237.5 q^{87} -60678.0 q^{89} -241.112i q^{91} -103086. i q^{93} +157428. q^{97} +38292.2i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 196 q^{7} - 1620 q^{9} + 7184 q^{23} - 7160 q^{31} - 2836 q^{33} - 22452 q^{39} - 5804 q^{41} - 44180 q^{47} + 62652 q^{49} + 43696 q^{57} + 1240 q^{63} + 7724 q^{71} - 105136 q^{73} + 7780 q^{79}+ \cdots - 73688 q^{97}+O(q^{100}) \) 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

Related objects

Downloads

Learn more