LMFDB - Embedded newform 1104.6.a.r.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1104,6,Mod(1,1104)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1104, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1104.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1104 = 2^{4} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1104.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$177.063737074$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 113x^{3} - 257x^{2} + 1404x + 2197 $ x^5 - 113x^3 - 257x^2 + 1404*x + 2197
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	no (minimal twist has level 69)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$3.40352$ of defining polynomial
Character	$\chi$	$=$	1104.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.00000 q^{3} +70.5203 q^{5} +89.7132 q^{7} +81.0000 q^{9} +O(q^{10})$	$q-9.00000 q^{3} +70.5203 q^{5} +89.7132 q^{7} +81.0000 q^{9} +436.763 q^{11} -258.865 q^{13} -634.683 q^{15} +591.819 q^{17} -1663.69 q^{19} -807.419 q^{21} +529.000 q^{23} +1848.12 q^{25} -729.000 q^{27} -5121.48 q^{29} -1871.29 q^{31} -3930.87 q^{33} +6326.60 q^{35} +884.534 q^{37} +2329.78 q^{39} -18410.5 q^{41} -12334.9 q^{43} +5712.15 q^{45} -23871.6 q^{47} -8758.55 q^{49} -5326.37 q^{51} -14594.8 q^{53} +30800.7 q^{55} +14973.2 q^{57} -47739.1 q^{59} -15351.5 q^{61} +7266.77 q^{63} -18255.2 q^{65} +41174.4 q^{67} -4761.00 q^{69} +45608.3 q^{71} -7898.24 q^{73} -16633.1 q^{75} +39183.4 q^{77} -71435.2 q^{79} +6561.00 q^{81} -112580. q^{83} +41735.3 q^{85} +46093.3 q^{87} +137845. q^{89} -23223.6 q^{91} +16841.6 q^{93} -117324. q^{95} +41679.8 q^{97} +35377.8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 45 q^{3} + 94 q^{5} - 272 q^{7} + 405 q^{9}+O(q^{10}) $ 5 * q - 45 * q^3 + 94 * q^5 - 272 * q^7 + 405 * q^9	$ 5 q - 45 q^{3} + 94 q^{5} - 272 q^{7} + 405 q^{9} - 1100 q^{11} - 978 q^{13} - 846 q^{15} + 2522 q^{17} - 2060 q^{19} + 2448 q^{21} + 2645 q^{23} + 12035 q^{25} - 3645 q^{27} + 1526 q^{29} + 7392 q^{31} + 9900 q^{33} - 6056 q^{35} - 8210 q^{37} + 8802 q^{39} + 21250 q^{41} + 4548 q^{43} + 7614 q^{45} - 536 q^{47} - 27979 q^{49} - 22698 q^{51} - 11482 q^{53} + 77064 q^{55} + 18540 q^{57} - 74676 q^{59} - 44618 q^{61} - 22032 q^{63} - 24388 q^{65} + 1412 q^{67} - 23805 q^{69} - 37912 q^{71} + 46546 q^{73} - 108315 q^{75} + 157008 q^{77} - 50544 q^{79} + 32805 q^{81} - 89588 q^{83} + 147892 q^{85} - 13734 q^{87} + 280410 q^{89} + 27416 q^{91} - 66528 q^{93} - 203120 q^{95} + 90074 q^{97} - 89100 q^{99}+O(q^{100}) $ 5 * q - 45 * q^3 + 94 * q^5 - 272 * q^7 + 405 * q^9 - 1100 * q^11 - 978 * q^13 - 846 * q^15 + 2522 * q^17 - 2060 * q^19 + 2448 * q^21 + 2645 * q^23 + 12035 * q^25 - 3645 * q^27 + 1526 * q^29 + 7392 * q^31 + 9900 * q^33 - 6056 * q^35 - 8210 * q^37 + 8802 * q^39 + 21250 * q^41 + 4548 * q^43 + 7614 * q^45 - 536 * q^47 - 27979 * q^49 - 22698 * q^51 - 11482 * q^53 + 77064 * q^55 + 18540 * q^57 - 74676 * q^59 - 44618 * q^61 - 22032 * q^63 - 24388 * q^65 + 1412 * q^67 - 23805 * q^69 - 37912 * q^71 + 46546 * q^73 - 108315 * q^75 + 157008 * q^77 - 50544 * q^79 + 32805 * q^81 - 89588 * q^83 + 147892 * q^85 - 13734 * q^87 + 280410 * q^89 + 27416 * q^91 - 66528 * q^93 - 203120 * q^95 + 90074 * q^97 - 89100 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	0	0
$5$	70.5203	1.26151	0.630753	−	0.775984i	$-0.282746\pi$
$5$	70.5203	1.26151	0.630753	+	0.775984i	$0.282746\pi$
$6$	0	0
$7$	89.7132	0.692008	0.346004	−	0.938233i	$-0.387538\pi$
$7$	89.7132	0.692008	0.346004	+	0.938233i	$0.387538\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	436.763	1.08834	0.544170	−	0.838975i	$-0.316845\pi$
$11$	436.763	1.08834	0.544170	+	0.838975i	$0.316845\pi$
$12$	0	0
$13$	−258.865	−0.424830	−0.212415	−	0.977180i	$-0.568133\pi$
$13$	−258.865	−0.424830	−0.212415	+	0.977180i	$0.568133\pi$
$14$	0	0
$15$	−634.683	−0.728331
$16$	0	0
$17$	591.819	0.496669	0.248334	−	0.968674i	$-0.420117\pi$
$17$	591.819	0.496669	0.248334	+	0.968674i	$0.420117\pi$
$18$	0	0
$19$	−1663.69	−1.05727	−0.528637	−	0.848848i	$-0.677297\pi$
$19$	−1663.69	−1.05727	−0.528637	+	0.848848i	$0.677297\pi$
$20$	0	0
$21$	−807.419	−0.399531
$22$	0	0
$23$	529.000	0.208514
$23$	529.000	0.208514
$24$	0	0
$25$	1848.12	0.591398
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	−5121.48	−1.13084	−0.565419	−	0.824804i	$-0.691286\pi$
$29$	−5121.48	−1.13084	−0.565419	+	0.824804i	$0.691286\pi$
$30$	0	0
$31$	−1871.29	−0.349733	−0.174867	−	0.984592i	$-0.555949\pi$
$31$	−1871.29	−0.349733	−0.174867	+	0.984592i	$0.555949\pi$
$32$	0	0
$33$	−3930.87	−0.628353
$34$	0	0
$35$	6326.60	0.872972
$36$	0	0
$37$	884.534	0.106221	0.0531105	−	0.998589i	$-0.483086\pi$
$37$	884.534	0.106221	0.0531105	+	0.998589i	$0.483086\pi$
$38$	0	0
$39$	2329.78	0.245275
$40$	0	0
$41$	−18410.5	−1.71043	−0.855216	−	0.518271i	$-0.826576\pi$
$41$	−18410.5	−1.71043	−0.855216	+	0.518271i	$0.826576\pi$
$42$	0	0
$43$	−12334.9	−1.01734	−0.508669	−	0.860962i	$-0.669862\pi$
$43$	−12334.9	−1.01734	−0.508669	+	0.860962i	$0.669862\pi$
$44$	0	0
$45$	5712.15	0.420502
$46$	0	0
$47$	−23871.6	−1.57629	−0.788146	−	0.615489i	$-0.788959\pi$
$47$	−23871.6	−1.57629	−0.788146	+	0.615489i	$0.788959\pi$
$48$	0	0
$49$	−8758.55	−0.521125
$50$	0	0
$51$	−5326.37	−0.286752
$52$	0	0
$53$	−14594.8	−0.713690	−0.356845	−	0.934164i	$-0.616147\pi$
$53$	−14594.8	−0.713690	−0.356845	+	0.934164i	$0.616147\pi$
$54$	0	0
$55$	30800.7	1.37295
$56$	0	0
$57$	14973.2	0.610418
$58$	0	0
$59$	−47739.1	−1.78544	−0.892718	−	0.450616i	$-0.851205\pi$
$59$	−47739.1	−1.78544	−0.892718	+	0.450616i	$0.851205\pi$
$60$	0	0
$61$	−15351.5	−0.528233	−0.264116	−	0.964491i	$-0.585080\pi$
$61$	−15351.5	−0.528233	−0.264116	+	0.964491i	$0.585080\pi$
$62$	0	0
$63$	7266.77	0.230669
$64$	0	0
$65$	−18255.2	−0.535925
$66$	0	0
$67$	41174.4	1.12057	0.560287	−	0.828299i	$-0.310691\pi$
$67$	41174.4	1.12057	0.560287	+	0.828299i	$0.310691\pi$
$68$	0	0
$69$	−4761.00	−0.120386
$70$	0	0
$71$	45608.3	1.07374	0.536869	−	0.843666i	$-0.319607\pi$
$71$	45608.3	1.07374	0.536869	+	0.843666i	$0.319607\pi$
$72$	0	0
$73$	−7898.24	−0.173470	−0.0867348	−	0.996231i	$-0.527643\pi$
$73$	−7898.24	−0.173470	−0.0867348	+	0.996231i	$0.527643\pi$
$74$	0	0
$75$	−16633.1	−0.341444
$76$	0	0
$77$	39183.4	0.753140
$78$	0	0
$79$	−71435.2	−1.28779	−0.643894	−	0.765115i	$-0.722682\pi$
$79$	−71435.2	−1.28779	−0.643894	+	0.765115i	$0.722682\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−112580.	−1.79377	−0.896885	−	0.442264i	$-0.854175\pi$
$83$	−112580.	−1.79377	−0.896885	+	0.442264i	$0.854175\pi$
$84$	0	0
$85$	41735.3	0.626551
$86$	0	0
$87$	46093.3	0.652890
$88$	0	0
$89$	137845.	1.84466	0.922330	−	0.386404i	$-0.126283\pi$
$89$	137845.	1.84466	0.922330	+	0.386404i	$0.126283\pi$
$90$	0	0
$91$	−23223.6	−0.293985
$92$	0	0
$93$	16841.6	0.201918
$94$	0	0
$95$	−117324.	−1.33376
$96$	0	0
$97$	41679.8	0.449776	0.224888	−	0.974385i	$-0.427798\pi$
$97$	41679.8	0.449776	0.224888	+	0.974385i	$0.427798\pi$
$98$	0	0
$99$	35377.8	0.362780
$100$	0	0
$101$	37100.3	0.361888	0.180944	−	0.983493i	$-0.442085\pi$
$101$	37100.3	0.361888	0.180944	+	0.983493i	$0.442085\pi$
$102$	0	0
$103$	123323.	1.14538	0.572692	−	0.819771i	$-0.305899\pi$
$103$	123323.	1.14538	0.572692	+	0.819771i	$0.305899\pi$
$104$	0	0
$105$	−56939.4	−0.504011
$106$	0	0
$107$	−107455.	−0.907338	−0.453669	−	0.891170i	$-0.649885\pi$
$107$	−107455.	−0.907338	−0.453669	+	0.891170i	$0.649885\pi$
$108$	0	0
$109$	96652.7	0.779198	0.389599	−	0.920985i	$-0.372614\pi$
$109$	96652.7	0.779198	0.389599	+	0.920985i	$0.372614\pi$
$110$	0	0
$111$	−7960.80	−0.0613267
$112$	0	0
$113$	55788.7	0.411008	0.205504	−	0.978656i	$-0.434117\pi$
$113$	55788.7	0.411008	0.205504	+	0.978656i	$0.434117\pi$
$114$	0	0
$115$	37305.3	0.263042
$116$	0	0
$117$	−20968.1	−0.141610
$118$	0	0
$119$	53094.0	0.343699
$120$	0	0
$121$	29711.1	0.184483
$122$	0	0
$123$	165695.	0.987519
$124$	0	0
$125$	−90046.2	−0.515454
$126$	0	0
$127$	−167839.	−0.923388	−0.461694	−	0.887039i	$-0.652758\pi$
$127$	−167839.	−0.923388	−0.461694	+	0.887039i	$0.652758\pi$
$128$	0	0
$129$	111014.	0.587361
$130$	0	0
$131$	−129729.	−0.660476	−0.330238	−	0.943898i	$-0.607129\pi$
$131$	−129729.	−0.660476	−0.330238	+	0.943898i	$0.607129\pi$
$132$	0	0
$133$	−149255.	−0.731643
$134$	0	0
$135$	−51409.3	−0.242777
$136$	0	0
$137$	−226702.	−1.03194	−0.515970	−	0.856607i	$-0.672568\pi$
$137$	−226702.	−1.03194	−0.515970	+	0.856607i	$0.672568\pi$
$138$	0	0
$139$	419175.	1.84017	0.920086	−	0.391716i	$-0.128118\pi$
$139$	419175.	1.84017	0.920086	+	0.391716i	$0.128118\pi$
$140$	0	0
$141$	214844.	0.910072
$142$	0	0
$143$	−113063.	−0.462359
$144$	0	0
$145$	−361169.	−1.42656
$146$	0	0
$147$	78826.9	0.300872
$148$	0	0
$149$	503270.	1.85710	0.928551	−	0.371206i	$-0.121055\pi$
$149$	503270.	1.85710	0.928551	+	0.371206i	$0.121055\pi$
$150$	0	0
$151$	−517943.	−1.84858	−0.924292	−	0.381685i	$-0.875344\pi$
$151$	−517943.	−1.84858	−0.924292	+	0.381685i	$0.875344\pi$
$152$	0	0
$153$	47937.4	0.165556
$154$	0	0
$155$	−131964.	−0.441190
$156$	0	0
$157$	−39747.9	−0.128696	−0.0643480	−	0.997928i	$-0.520497\pi$
$157$	−39747.9	−0.128696	−0.0643480	+	0.997928i	$0.520497\pi$
$158$	0	0
$159$	131353.	0.412049
$160$	0	0
$161$	47458.3	0.144294
$162$	0	0
$163$	157828.	0.465279	0.232640	−	0.972563i	$-0.425264\pi$
$163$	157828.	0.465279	0.232640	+	0.972563i	$0.425264\pi$
$164$	0	0
$165$	−277206.	−0.792671
$166$	0	0
$167$	101000.	0.280241	0.140120	−	0.990134i	$-0.455251\pi$
$167$	101000.	0.280241	0.140120	+	0.990134i	$0.455251\pi$
$168$	0	0
$169$	−304282.	−0.819520
$170$	0	0
$171$	−134759.	−0.352425
$172$	0	0
$173$	61568.5	0.156402	0.0782012	−	0.996938i	$-0.475082\pi$
$173$	61568.5	0.156402	0.0782012	+	0.996938i	$0.475082\pi$
$174$	0	0
$175$	165801.	0.409252
$176$	0	0
$177$	429652.	1.03082
$178$	0	0
$179$	610536.	1.42423	0.712113	−	0.702065i	$-0.247738\pi$
$179$	610536.	1.42423	0.712113	+	0.702065i	$0.247738\pi$
$180$	0	0
$181$	121378.	0.275387	0.137694	−	0.990475i	$-0.456031\pi$
$181$	121378.	0.275387	0.137694	+	0.990475i	$0.456031\pi$
$182$	0	0
$183$	138163.	0.304975
$184$	0	0
$185$	62377.6	0.133998
$186$	0	0
$187$	258485.	0.540544
$188$	0	0
$189$	−65400.9	−0.133177
$190$	0	0
$191$	294581.	0.584280	0.292140	−	0.956376i	$-0.405633\pi$
$191$	294581.	0.584280	0.292140	+	0.956376i	$0.405633\pi$
$192$	0	0
$193$	−8895.53	−0.0171901	−0.00859505	−	0.999963i	$-0.502736\pi$
$193$	−8895.53	−0.0171901	−0.00859505	+	0.999963i	$0.502736\pi$
$194$	0	0
$195$	164297.	0.309417
$196$	0	0
$197$	−698113.	−1.28162	−0.640811	−	0.767699i	$-0.721402\pi$
$197$	−698113.	−1.28162	−0.640811	+	0.767699i	$0.721402\pi$
$198$	0	0
$199$	533271.	0.954586	0.477293	−	0.878744i	$-0.341618\pi$
$199$	533271.	0.954586	0.477293	+	0.878744i	$0.341618\pi$
$200$	0	0
$201$	−370570.	−0.646963
$202$	0	0
$203$	−459464.	−0.782550
$204$	0	0
$205$	−1.29831e6	−2.15772
$206$	0	0
$207$	42849.0	0.0695048
$208$	0	0
$209$	−726638.	−1.15067
$210$	0	0
$211$	658407.	1.01809	0.509047	−	0.860739i	$-0.329998\pi$
$211$	658407.	1.01809	0.509047	+	0.860739i	$0.329998\pi$
$212$	0	0
$213$	−410475.	−0.619923
$214$	0	0
$215$	−869864.	−1.28338
$216$	0	0
$217$	−167879.	−0.242018
$218$	0	0
$219$	71084.2	0.100153
$220$	0	0
$221$	−153201.	−0.211000
$222$	0	0
$223$	−79895.6	−0.107587	−0.0537936	−	0.998552i	$-0.517131\pi$
$223$	−79895.6	−0.107587	−0.0537936	+	0.998552i	$0.517131\pi$
$224$	0	0
$225$	149698.	0.197133
$226$	0	0
$227$	−716111.	−0.922393	−0.461196	−	0.887298i	$-0.652580\pi$
$227$	−716111.	−0.922393	−0.461196	+	0.887298i	$0.652580\pi$
$228$	0	0
$229$	−581299.	−0.732506	−0.366253	−	0.930515i	$-0.619360\pi$
$229$	−581299.	−0.732506	−0.366253	+	0.930515i	$0.619360\pi$
$230$	0	0
$231$	−352651.	−0.434825
$232$	0	0
$233$	122734.	0.148107	0.0740534	−	0.997254i	$-0.476406\pi$
$233$	122734.	0.148107	0.0740534	+	0.997254i	$0.476406\pi$
$234$	0	0
$235$	−1.68343e6	−1.98850
$236$	0	0
$237$	642917.	0.743505
$238$	0	0
$239$	−210076.	−0.237893	−0.118946	−	0.992901i	$-0.537952\pi$
$239$	−210076.	−0.237893	−0.118946	+	0.992901i	$0.537952\pi$
$240$	0	0
$241$	1.23224e6	1.36664	0.683319	−	0.730120i	$-0.260536\pi$
$241$	1.23224e6	1.36664	0.683319	+	0.730120i	$0.260536\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	−617656.	−0.657402
$246$	0	0
$247$	430670.	0.449162
$248$	0	0
$249$	1.01322e6	1.03563
$250$	0	0
$251$	545986.	0.547012	0.273506	−	0.961870i	$-0.411817\pi$
$251$	545986.	0.547012	0.273506	+	0.961870i	$0.411817\pi$
$252$	0	0
$253$	231048.	0.226934
$254$	0	0
$255$	−375618.	−0.361739
$256$	0	0
$257$	556705.	0.525765	0.262883	−	0.964828i	$-0.415327\pi$
$257$	556705.	0.525765	0.262883	+	0.964828i	$0.415327\pi$
$258$	0	0
$259$	79354.3	0.0735057
$260$	0	0
$261$	−414840.	−0.376946
$262$	0	0
$263$	−1.65392e6	−1.47443	−0.737217	−	0.675656i	$-0.763860\pi$
$263$	−1.65392e6	−1.47443	−0.737217	+	0.675656i	$0.763860\pi$
$264$	0	0
$265$	−1.02923e6	−0.900324
$266$	0	0
$267$	−1.24061e6	−1.06501
$268$	0	0
$269$	−2.06545e6	−1.74034	−0.870168	−	0.492755i	$-0.835990\pi$
$269$	−2.06545e6	−1.74034	−0.870168	+	0.492755i	$0.835990\pi$
$270$	0	0
$271$	−708326.	−0.585881	−0.292941	−	0.956131i	$-0.594634\pi$
$271$	−708326.	−0.585881	−0.292941	+	0.956131i	$0.594634\pi$
$272$	0	0
$273$	209012.	0.169733
$274$	0	0
$275$	807190.	0.643641
$276$	0	0
$277$	−17729.4	−0.0138833	−0.00694167	−	0.999976i	$-0.502210\pi$
$277$	−17729.4	−0.0138833	−0.00694167	+	0.999976i	$0.502210\pi$
$278$	0	0
$279$	−151574.	−0.116578
$280$	0	0
$281$	754660.	0.570145	0.285073	−	0.958506i	$-0.407982\pi$
$281$	754660.	0.570145	0.285073	+	0.958506i	$0.407982\pi$
$282$	0	0
$283$	417633.	0.309976	0.154988	−	0.987916i	$-0.450466\pi$
$283$	417633.	0.309976	0.154988	+	0.987916i	$0.450466\pi$
$284$	0	0
$285$	1.05591e6	0.770046
$286$	0	0
$287$	−1.65166e6	−1.18363
$288$	0	0
$289$	−1.06961e6	−0.753320
$290$	0	0
$291$	−375118.	−0.259678
$292$	0	0
$293$	−2.68010e6	−1.82382	−0.911908	−	0.410394i	$-0.865391\pi$
$293$	−2.68010e6	−1.82382	−0.911908	+	0.410394i	$0.865391\pi$
$294$	0	0
$295$	−3.36658e6	−2.25234
$296$	0	0
$297$	−318400.	−0.209451
$298$	0	0
$299$	−136940.	−0.0885831
$300$	0	0
$301$	−1.10661e6	−0.704007
$302$	0	0
$303$	−333903.	−0.208936
$304$	0	0
$305$	−1.08259e6	−0.666369
$306$	0	0
$307$	2.42927e6	1.47106	0.735528	−	0.677494i	$-0.236934\pi$
$307$	2.42927e6	1.47106	0.735528	+	0.677494i	$0.236934\pi$
$308$	0	0
$309$	−1.10991e6	−0.661288
$310$	0	0
$311$	−1.71783e6	−1.00712	−0.503559	−	0.863961i	$-0.667976\pi$
$311$	−1.71783e6	−1.00712	−0.503559	+	0.863961i	$0.667976\pi$
$312$	0	0
$313$	1.56583e6	0.903406	0.451703	−	0.892168i	$-0.350817\pi$
$313$	1.56583e6	0.903406	0.451703	+	0.892168i	$0.350817\pi$
$314$	0	0
$315$	512455.	0.290991
$316$	0	0
$317$	673750.	0.376574	0.188287	−	0.982114i	$-0.439706\pi$
$317$	673750.	0.376574	0.188287	+	0.982114i	$0.439706\pi$
$318$	0	0
$319$	−2.23688e6	−1.23074
$320$	0	0
$321$	967099.	0.523852
$322$	0	0
$323$	−984603.	−0.525115
$324$	0	0
$325$	−478413.	−0.251243
$326$	0	0
$327$	−869874.	−0.449870
$328$	0	0
$329$	−2.14159e6	−1.09081
$330$	0	0
$331$	−1.69768e6	−0.851699	−0.425850	−	0.904794i	$-0.640025\pi$
$331$	−1.69768e6	−0.851699	−0.425850	+	0.904794i	$0.640025\pi$
$332$	0	0
$333$	71647.2	0.0354070
$334$	0	0
$335$	2.90363e6	1.41361
$336$	0	0
$337$	−2.37281e6	−1.13812	−0.569061	−	0.822295i	$-0.692693\pi$
$337$	−2.37281e6	−1.13812	−0.569061	+	0.822295i	$0.692693\pi$
$338$	0	0
$339$	−502099.	−0.237296
$340$	0	0
$341$	−817310.	−0.380628
$342$	0	0
$343$	−2.29357e6	−1.05263
$344$	0	0
$345$	−335747.	−0.151867
$346$	0	0
$347$	−3.98316e6	−1.77584	−0.887921	−	0.459996i	$-0.847851\pi$
$347$	−3.98316e6	−1.77584	−0.887921	+	0.459996i	$0.847851\pi$
$348$	0	0
$349$	−2.10463e6	−0.924935	−0.462468	−	0.886636i	$-0.653036\pi$
$349$	−2.10463e6	−0.924935	−0.462468	+	0.886636i	$0.653036\pi$
$350$	0	0
$351$	188712.	0.0817585
$352$	0	0
$353$	3.41107e6	1.45698	0.728490	−	0.685056i	$-0.240222\pi$
$353$	3.41107e6	1.45698	0.728490	+	0.685056i	$0.240222\pi$
$354$	0	0
$355$	3.21631e6	1.35453
$356$	0	0
$357$	−477846.	−0.198435
$358$	0	0
$359$	1.86760e6	0.764800	0.382400	−	0.923997i	$-0.375098\pi$
$359$	1.86760e6	0.764800	0.382400	+	0.923997i	$0.375098\pi$
$360$	0	0
$361$	291759.	0.117830
$362$	0	0
$363$	−267400.	−0.106511
$364$	0	0
$365$	−556987.	−0.218833
$366$	0	0
$367$	437210.	0.169444	0.0847218	−	0.996405i	$-0.473000\pi$
$367$	437210.	0.169444	0.0847218	+	0.996405i	$0.473000\pi$
$368$	0	0
$369$	−1.49125e6	−0.570144
$370$	0	0
$371$	−1.30935e6	−0.493879
$372$	0	0
$373$	−921734.	−0.343031	−0.171516	−	0.985181i	$-0.554866\pi$
$373$	−921734.	−0.343031	−0.171516	+	0.985181i	$0.554866\pi$
$374$	0	0
$375$	810415.	0.297598
$376$	0	0
$377$	1.32577e6	0.480414
$378$	0	0
$379$	3.80141e6	1.35940	0.679699	−	0.733491i	$-0.262111\pi$
$379$	3.80141e6	1.35940	0.679699	+	0.733491i	$0.262111\pi$
$380$	0	0
$381$	1.51055e6	0.533118
$382$	0	0
$383$	4.52311e6	1.57558	0.787790	−	0.615944i	$-0.211225\pi$
$383$	4.52311e6	1.57558	0.787790	+	0.615944i	$0.211225\pi$
$384$	0	0
$385$	2.76323e6	0.950090
$386$	0	0
$387$	−999130.	−0.339113
$388$	0	0
$389$	4.48168e6	1.50164	0.750822	−	0.660505i	$-0.229658\pi$
$389$	4.48168e6	1.50164	0.750822	+	0.660505i	$0.229658\pi$
$390$	0	0
$391$	313072.	0.103563
$392$	0	0
$393$	1.16756e6	0.381326
$394$	0	0
$395$	−5.03763e6	−1.62455
$396$	0	0
$397$	1.76656e6	0.562539	0.281269	−	0.959629i	$-0.409245\pi$
$397$	1.76656e6	0.562539	0.281269	+	0.959629i	$0.409245\pi$
$398$	0	0
$399$	1.34329e6	0.422414
$400$	0	0
$401$	−522373.	−0.162226	−0.0811128	−	0.996705i	$-0.525847\pi$
$401$	−522373.	−0.162226	−0.0811128	+	0.996705i	$0.525847\pi$
$402$	0	0
$403$	484411.	0.148577
$404$	0	0
$405$	462684.	0.140167
$406$	0	0
$407$	386332.	0.115604
$408$	0	0
$409$	−1.60642e6	−0.474845	−0.237422	−	0.971407i	$-0.576303\pi$
$409$	−1.60642e6	−0.474845	−0.237422	+	0.971407i	$0.576303\pi$
$410$	0	0
$411$	2.04032e6	0.595790
$412$	0	0
$413$	−4.28283e6	−1.23554
$414$	0	0
$415$	−7.93919e6	−2.26285
$416$	0	0
$417$	−3.77258e6	−1.06242
$418$	0	0
$419$	5.30658e6	1.47666	0.738329	−	0.674441i	$-0.235615\pi$
$419$	5.30658e6	1.47666	0.738329	+	0.674441i	$0.235615\pi$
$420$	0	0
$421$	−2.30575e6	−0.634025	−0.317012	−	0.948421i	$-0.602680\pi$
$421$	−2.30575e6	−0.634025	−0.317012	+	0.948421i	$0.602680\pi$
$422$	0	0
$423$	−1.93360e6	−0.525430
$424$	0	0
$425$	1.09375e6	0.293729
$426$	0	0
$427$	−1.37723e6	−0.365541
$428$	0	0
$429$	1.01756e6	0.266943
$430$	0	0
$431$	1.27091e6	0.329549	0.164775	−	0.986331i	$-0.447310\pi$
$431$	1.27091e6	0.329549	0.164775	+	0.986331i	$0.447310\pi$
$432$	0	0
$433$	−3.71880e6	−0.953199	−0.476599	−	0.879121i	$-0.658131\pi$
$433$	−3.71880e6	−0.953199	−0.476599	+	0.879121i	$0.658131\pi$
$434$	0	0
$435$	3.25052e6	0.823625
$436$	0	0
$437$	−880091.	−0.220457
$438$	0	0
$439$	3.51069e6	0.869423	0.434711	−	0.900570i	$-0.356850\pi$
$439$	3.51069e6	0.869423	0.434711	+	0.900570i	$0.356850\pi$
$440$	0	0
$441$	−709442.	−0.173708
$442$	0	0
$443$	−2.67040e6	−0.646497	−0.323248	−	0.946314i	$-0.604775\pi$
$443$	−2.67040e6	−0.646497	−0.323248	+	0.946314i	$0.604775\pi$
$444$	0	0
$445$	9.72088e6	2.32705
$446$	0	0
$447$	−4.52943e6	−1.07220
$448$	0	0
$449$	808771.	0.189326	0.0946628	−	0.995509i	$-0.469823\pi$
$449$	808771.	0.189326	0.0946628	+	0.995509i	$0.469823\pi$
$450$	0	0
$451$	−8.04103e6	−1.86153
$452$	0	0
$453$	4.66148e6	1.06728
$454$	0	0
$455$	−1.63774e6	−0.370864
$456$	0	0
$457$	−6.03865e6	−1.35254	−0.676269	−	0.736655i	$-0.736404\pi$
$457$	−6.03865e6	−1.35254	−0.676269	+	0.736655i	$0.736404\pi$
$458$	0	0
$459$	−431436.	−0.0955839
$460$	0	0
$461$	−6.06334e6	−1.32880	−0.664399	−	0.747378i	$-0.731313\pi$
$461$	−6.06334e6	−1.32880	−0.664399	+	0.747378i	$0.731313\pi$
$462$	0	0
$463$	987185.	0.214016	0.107008	−	0.994258i	$-0.465873\pi$
$463$	987185.	0.214016	0.107008	+	0.994258i	$0.465873\pi$
$464$	0	0
$465$	1.18768e6	0.254721
$466$	0	0
$467$	2.75869e6	0.585344	0.292672	−	0.956213i	$-0.405456\pi$
$467$	2.75869e6	0.585344	0.292672	+	0.956213i	$0.405456\pi$
$468$	0	0
$469$	3.69389e6	0.775446
$470$	0	0
$471$	357731.	0.0743027
$472$	0	0
$473$	−5.38745e6	−1.10721
$474$	0	0
$475$	−3.07469e6	−0.625270
$476$	0	0
$477$	−1.18218e6	−0.237897
$478$	0	0
$479$	1.43871e6	0.286507	0.143254	−	0.989686i	$-0.454244\pi$
$479$	1.43871e6	0.286507	0.143254	+	0.989686i	$0.454244\pi$
$480$	0	0
$481$	−228975.	−0.0451258
$482$	0	0
$483$	−427124.	−0.0833080
$484$	0	0
$485$	2.93927e6	0.567395
$486$	0	0
$487$	9.54793e6	1.82426	0.912130	−	0.409901i	$-0.134437\pi$
$487$	9.54793e6	1.82426	0.912130	+	0.409901i	$0.134437\pi$
$488$	0	0
$489$	−1.42045e6	−0.268629
$490$	0	0
$491$	−8.94131e6	−1.67378	−0.836889	−	0.547373i	$-0.815628\pi$
$491$	−8.94131e6	−1.67378	−0.836889	+	0.547373i	$0.815628\pi$
$492$	0	0
$493$	−3.03099e6	−0.561652
$494$	0	0
$495$	2.49486e6	0.457649
$496$	0	0
$497$	4.09167e6	0.743035
$498$	0	0
$499$	−2.34135e6	−0.420934	−0.210467	−	0.977601i	$-0.567499\pi$
$499$	−2.34135e6	−0.420934	−0.210467	+	0.977601i	$0.567499\pi$
$500$	0	0
$501$	−909002.	−0.161797
$502$	0	0
$503$	−41307.7	−0.00727966	−0.00363983	−	0.999993i	$-0.501159\pi$
$503$	−41307.7	−0.00727966	−0.00363983	+	0.999993i	$0.501159\pi$
$504$	0	0
$505$	2.61633e6	0.456524
$506$	0	0
$507$	2.73854e6	0.473150
$508$	0	0
$509$	3.09166e6	0.528929	0.264464	−	0.964395i	$-0.414805\pi$
$509$	3.09166e6	0.528929	0.264464	+	0.964395i	$0.414805\pi$
$510$	0	0
$511$	−708576.	−0.120042
$512$	0	0
$513$	1.21283e6	0.203473
$514$	0	0
$515$	8.69678e6	1.44491
$516$	0	0
$517$	−1.04262e7	−1.71554
$518$	0	0
$519$	−554117.	−0.0902990
$520$	0	0
$521$	6.02002e6	0.971635	0.485818	−	0.874060i	$-0.338522\pi$
$521$	6.02002e6	0.971635	0.485818	+	0.874060i	$0.338522\pi$
$522$	0	0
$523$	6.39458e6	1.02225	0.511125	−	0.859506i	$-0.329229\pi$
$523$	6.39458e6	1.02225	0.511125	+	0.859506i	$0.329229\pi$
$524$	0	0
$525$	−1.49220e6	−0.236282
$526$	0	0
$527$	−1.10746e6	−0.173701
$528$	0	0
$529$	279841.	0.0434783
$530$	0	0
$531$	−3.86687e6	−0.595145
$532$	0	0
$533$	4.76583e6	0.726642
$534$	0	0
$535$	−7.57779e6	−1.14461
$536$	0	0
$537$	−5.49483e6	−0.822278
$538$	0	0
$539$	−3.82541e6	−0.567161
$540$	0	0
$541$	−3.24248e6	−0.476304	−0.238152	−	0.971228i	$-0.576542\pi$
$541$	−3.24248e6	−0.476304	−0.238152	+	0.971228i	$0.576542\pi$
$542$	0	0
$543$	−1.09240e6	−0.158995
$544$	0	0
$545$	6.81598e6	0.982963
$546$	0	0
$547$	−9.39508e6	−1.34255	−0.671277	−	0.741206i	$-0.734254\pi$
$547$	−9.39508e6	−1.34255	−0.671277	+	0.741206i	$0.734254\pi$
$548$	0	0
$549$	−1.24347e6	−0.176078
$550$	0	0
$551$	8.52055e6	1.19561
$552$	0	0
$553$	−6.40868e6	−0.891160
$554$	0	0
$555$	−561398.	−0.0773640
$556$	0	0
$557$	2.95696e6	0.403839	0.201919	−	0.979402i	$-0.435282\pi$
$557$	2.95696e6	0.403839	0.201919	+	0.979402i	$0.435282\pi$
$558$	0	0
$559$	3.19308e6	0.432196
$560$	0	0
$561$	−2.32636e6	−0.312083
$562$	0	0
$563$	1.35314e7	1.79917	0.899584	−	0.436747i	$-0.143870\pi$
$563$	1.35314e7	1.79917	0.899584	+	0.436747i	$0.143870\pi$
$564$	0	0
$565$	3.93424e6	0.518489
$566$	0	0
$567$	588608.	0.0768898
$568$	0	0
$569$	1.32776e7	1.71924	0.859622	−	0.510931i	$-0.170699\pi$
$569$	1.32776e7	1.71924	0.859622	+	0.510931i	$0.170699\pi$
$570$	0	0
$571$	−7.74960e6	−0.994693	−0.497346	−	0.867552i	$-0.665692\pi$
$571$	−7.74960e6	−0.994693	−0.497346	+	0.867552i	$0.665692\pi$
$572$	0	0
$573$	−2.65123e6	−0.337334
$574$	0	0
$575$	977654.	0.123315
$576$	0	0
$577$	−4.83479e6	−0.604558	−0.302279	−	0.953220i	$-0.597747\pi$
$577$	−4.83479e6	−0.604558	−0.302279	+	0.953220i	$0.597747\pi$
$578$	0	0
$579$	80059.7	0.00992471
$580$	0	0
$581$	−1.00999e7	−1.24130
$582$	0	0
$583$	−6.37448e6	−0.776736
$584$	0	0
$585$	−1.47867e6	−0.178642
$586$	0	0
$587$	−5.13380e6	−0.614955	−0.307478	−	0.951555i	$-0.599485\pi$
$587$	−5.13380e6	−0.614955	−0.307478	+	0.951555i	$0.599485\pi$
$588$	0	0
$589$	3.11324e6	0.369764
$590$	0	0
$591$	6.28301e6	0.739945
$592$	0	0
$593$	1.29134e7	1.50801	0.754003	−	0.656871i	$-0.228121\pi$
$593$	1.29134e7	1.50801	0.754003	+	0.656871i	$0.228121\pi$
$594$	0	0
$595$	3.74421e6	0.433578
$596$	0	0
$597$	−4.79944e6	−0.551130
$598$	0	0
$599$	5.79980e6	0.660460	0.330230	−	0.943901i	$-0.392874\pi$
$599$	5.79980e6	0.660460	0.330230	+	0.943901i	$0.392874\pi$
$600$	0	0
$601$	1.06993e7	1.20829	0.604144	−	0.796875i	$-0.293515\pi$
$601$	1.06993e7	1.20829	0.604144	+	0.796875i	$0.293515\pi$
$602$	0	0
$603$	3.33513e6	0.373525
$604$	0	0
$605$	2.09524e6	0.232726
$606$	0	0
$607$	1.48509e7	1.63600	0.817998	−	0.575221i	$-0.195084\pi$
$607$	1.48509e7	1.63600	0.817998	+	0.575221i	$0.195084\pi$
$608$	0	0
$609$	4.13518e6	0.451805
$610$	0	0
$611$	6.17951e6	0.669655
$612$	0	0
$613$	−7.64415e6	−0.821633	−0.410816	−	0.911718i	$-0.634756\pi$
$613$	−7.64415e6	−0.821633	−0.410816	+	0.911718i	$0.634756\pi$
$614$	0	0
$615$	1.16848e7	1.24576
$616$	0	0
$617$	−8.02025e6	−0.848155	−0.424077	−	0.905626i	$-0.639402\pi$
$617$	−8.02025e6	−0.848155	−0.424077	+	0.905626i	$0.639402\pi$
$618$	0	0
$619$	−9.46459e6	−0.992831	−0.496416	−	0.868085i	$-0.665351\pi$
$619$	−9.46459e6	−0.992831	−0.496416	+	0.868085i	$0.665351\pi$
$620$	0	0
$621$	−385641.	−0.0401286
$622$	0	0
$623$	1.23665e7	1.27652
$624$	0	0
$625$	−1.21255e7	−1.24165
$626$	0	0
$627$	6.53974e6	0.664342
$628$	0	0
$629$	523484.	0.0527566
$630$	0	0
$631$	−8.76525e6	−0.876377	−0.438189	−	0.898883i	$-0.644380\pi$
$631$	−8.76525e6	−0.876377	−0.438189	+	0.898883i	$0.644380\pi$
$632$	0	0
$633$	−5.92566e6	−0.587797
$634$	0	0
$635$	−1.18361e7	−1.16486
$636$	0	0
$637$	2.26728e6	0.221389
$638$	0	0
$639$	3.69427e6	0.357913
$640$	0	0
$641$	−8.04677e6	−0.773529	−0.386764	−	0.922179i	$-0.626407\pi$
$641$	−8.04677e6	−0.773529	−0.386764	+	0.922179i	$0.626407\pi$
$642$	0	0
$643$	−1.81157e7	−1.72794	−0.863970	−	0.503543i	$-0.832030\pi$
$643$	−1.81157e7	−1.72794	−0.863970	+	0.503543i	$0.832030\pi$
$644$	0	0
$645$	7.82877e6	0.740959
$646$	0	0
$647$	−6.09047e6	−0.571992	−0.285996	−	0.958231i	$-0.592324\pi$
$647$	−6.09047e6	−0.571992	−0.285996	+	0.958231i	$0.592324\pi$
$648$	0	0
$649$	−2.08507e7	−1.94316
$650$	0	0
$651$	1.51091e6	0.139729
$652$	0	0
$653$	1.73408e7	1.59142	0.795712	−	0.605675i	$-0.207097\pi$
$653$	1.73408e7	1.59142	0.795712	+	0.605675i	$0.207097\pi$
$654$	0	0
$655$	−9.14850e6	−0.833195
$656$	0	0
$657$	−639757.	−0.0578232
$658$	0	0
$659$	−1.27618e7	−1.14471	−0.572357	−	0.820004i	$-0.693971\pi$
$659$	−1.27618e7	−1.14471	−0.572357	+	0.820004i	$0.693971\pi$
$660$	0	0
$661$	−6.44195e6	−0.573474	−0.286737	−	0.958009i	$-0.592571\pi$
$661$	−6.44195e6	−0.573474	−0.286737	+	0.958009i	$0.592571\pi$
$662$	0	0
$663$	1.37881e6	0.121821
$664$	0	0
$665$	−1.05255e7	−0.922972
$666$	0	0
$667$	−2.70926e6	−0.235796
$668$	0	0
$669$	719060.	0.0621155
$670$	0	0
$671$	−6.70496e6	−0.574897
$672$	0	0
$673$	8.64864e6	0.736055	0.368028	−	0.929815i	$-0.380033\pi$
$673$	8.64864e6	0.736055	0.368028	+	0.929815i	$0.380033\pi$
$674$	0	0
$675$	−1.34728e6	−0.113815
$676$	0	0
$677$	−1.22020e7	−1.02320	−0.511600	−	0.859223i	$-0.670947\pi$
$677$	−1.22020e7	−1.02320	−0.511600	+	0.859223i	$0.670947\pi$
$678$	0	0
$679$	3.73923e6	0.311249
$680$	0	0
$681$	6.44500e6	0.532544
$682$	0	0
$683$	1.10784e7	0.908708	0.454354	−	0.890821i	$-0.349870\pi$
$683$	1.10784e7	0.908708	0.454354	+	0.890821i	$0.349870\pi$
$684$	0	0
$685$	−1.59871e7	−1.30180
$686$	0	0
$687$	5.23170e6	0.422913
$688$	0	0
$689$	3.77809e6	0.303196
$690$	0	0
$691$	1.24814e7	0.994415	0.497207	−	0.867632i	$-0.334359\pi$
$691$	1.24814e7	0.994415	0.497207	+	0.867632i	$0.334359\pi$
$692$	0	0
$693$	3.17386e6	0.251047
$694$	0	0
$695$	2.95604e7	2.32139
$696$	0	0
$697$	−1.08957e7	−0.849518
$698$	0	0
$699$	−1.10461e6	−0.0855094
$700$	0	0
$701$	−4.71570e6	−0.362452	−0.181226	−	0.983441i	$-0.558007\pi$
$701$	−4.71570e6	−0.362452	−0.181226	+	0.983441i	$0.558007\pi$
$702$	0	0
$703$	−1.47159e6	−0.112305
$704$	0	0
$705$	1.51509e7	1.14806
$706$	0	0
$707$	3.32839e6	0.250430
$708$	0	0
$709$	1.05958e7	0.791620	0.395810	−	0.918332i	$-0.370464\pi$
$709$	1.05958e7	0.791620	0.395810	+	0.918332i	$0.370464\pi$
$710$	0	0
$711$	−5.78625e6	−0.429263
$712$	0	0
$713$	−989912.	−0.0729244
$714$	0	0
$715$	−7.97322e6	−0.583268
$716$	0	0
$717$	1.89068e6	0.137347
$718$	0	0
$719$	−2.95783e6	−0.213379	−0.106689	−	0.994292i	$-0.534025\pi$
$719$	−2.95783e6	−0.213379	−0.106689	+	0.994292i	$0.534025\pi$
$720$	0	0
$721$	1.10637e7	0.792615
$722$	0	0
$723$	−1.10902e7	−0.789029
$724$	0	0
$725$	−9.46510e6	−0.668776
$726$	0	0
$727$	1.30768e7	0.917625	0.458813	−	0.888533i	$-0.348275\pi$
$727$	1.30768e7	0.917625	0.458813	+	0.888533i	$0.348275\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	−7.30005e6	−0.505280
$732$	0	0
$733$	5.06819e6	0.348412	0.174206	−	0.984709i	$-0.444264\pi$
$733$	5.06819e6	0.348412	0.174206	+	0.984709i	$0.444264\pi$
$734$	0	0
$735$	5.55890e6	0.379551
$736$	0	0
$737$	1.79835e7	1.21956
$738$	0	0
$739$	−5.39832e6	−0.363620	−0.181810	−	0.983334i	$-0.558196\pi$
$739$	−5.39832e6	−0.363620	−0.181810	+	0.983334i	$0.558196\pi$
$740$	0	0
$741$	−3.87603e6	−0.259324
$742$	0	0
$743$	−3.71690e6	−0.247007	−0.123503	−	0.992344i	$-0.539413\pi$
$743$	−3.71690e6	−0.247007	−0.123503	+	0.992344i	$0.539413\pi$
$744$	0	0
$745$	3.54908e7	2.34274
$746$	0	0
$747$	−9.11900e6	−0.597923
$748$	0	0
$749$	−9.64017e6	−0.627885
$750$	0	0
$751$	−1.00711e7	−0.651594	−0.325797	−	0.945440i	$-0.605633\pi$
$751$	−1.00711e7	−0.651594	−0.325797	+	0.945440i	$0.605633\pi$
$752$	0	0
$753$	−4.91387e6	−0.315818
$754$	0	0
$755$	−3.65255e7	−2.33200
$756$	0	0
$757$	−8.49996e6	−0.539110	−0.269555	−	0.962985i	$-0.586877\pi$
$757$	−8.49996e6	−0.539110	−0.269555	+	0.962985i	$0.586877\pi$
$758$	0	0
$759$	−2.07943e6	−0.131021
$760$	0	0
$761$	2.62780e7	1.64487	0.822434	−	0.568861i	$-0.192616\pi$
$761$	2.62780e7	1.64487	0.822434	+	0.568861i	$0.192616\pi$
$762$	0	0
$763$	8.67102e6	0.539211
$764$	0	0
$765$	3.38056e6	0.208850
$766$	0	0
$767$	1.23580e7	0.758506
$768$	0	0
$769$	899889.	0.0548748	0.0274374	−	0.999624i	$-0.491265\pi$
$769$	899889.	0.0548748	0.0274374	+	0.999624i	$0.491265\pi$
$770$	0	0
$771$	−5.01034e6	−0.303551
$772$	0	0
$773$	−1.33758e7	−0.805140	−0.402570	−	0.915389i	$-0.631883\pi$
$773$	−1.33758e7	−0.805140	−0.402570	+	0.915389i	$0.631883\pi$
$774$	0	0
$775$	−3.45836e6	−0.206831
$776$	0	0
$777$	−714189.	−0.0424385
$778$	0	0
$779$	3.06293e7	1.80840
$780$	0	0
$781$	1.99200e7	1.16859
$782$	0	0
$783$	3.73356e6	0.217630
$784$	0	0
$785$	−2.80304e6	−0.162351
$786$	0	0
$787$	−1.67690e7	−0.965098	−0.482549	−	0.875869i	$-0.660289\pi$
$787$	−1.67690e7	−0.965098	−0.482549	+	0.875869i	$0.660289\pi$
$788$	0	0
$789$	1.48853e7	0.851264
$790$	0	0
$791$	5.00498e6	0.284421
$792$	0	0
$793$	3.97396e6	0.224409
$794$	0	0
$795$	9.26309e6	0.519802
$796$	0	0
$797$	2.78280e7	1.55180	0.775902	−	0.630854i	$-0.217295\pi$
$797$	2.78280e7	1.55180	0.775902	+	0.630854i	$0.217295\pi$
$798$	0	0
$799$	−1.41277e7	−0.782894
$800$	0	0
$801$	1.11654e7	0.614886
$802$	0	0
$803$	−3.44966e6	−0.188794
$804$	0	0
$805$	3.34677e6	0.182027
$806$	0	0
$807$	1.85890e7	1.00478
$808$	0	0
$809$	−1.98606e7	−1.06689	−0.533447	−	0.845834i	$-0.679103\pi$
$809$	−1.98606e7	−1.06689	−0.533447	+	0.845834i	$0.679103\pi$
$810$	0	0
$811$	−1.95245e7	−1.04238	−0.521192	−	0.853439i	$-0.674512\pi$
$811$	−1.95245e7	−1.04238	−0.521192	+	0.853439i	$0.674512\pi$
$812$	0	0
$813$	6.37493e6	0.338259
$814$	0	0
$815$	1.11301e7	0.586953
$816$	0	0
$817$	2.05215e7	1.07561
$818$	0	0
$819$	−1.88111e6	−0.0979952
$820$	0	0
$821$	−2.39444e7	−1.23978	−0.619891	−	0.784688i	$-0.712823\pi$
$821$	−2.39444e7	−1.23978	−0.619891	+	0.784688i	$0.712823\pi$
$822$	0	0
$823$	2.78310e7	1.43229	0.716143	−	0.697953i	$-0.245906\pi$
$823$	2.78310e7	1.43229	0.716143	+	0.697953i	$0.245906\pi$
$824$	0	0
$825$	−7.26471e6	−0.371607
$826$	0	0
$827$	1.94204e7	0.987405	0.493703	−	0.869631i	$-0.335643\pi$
$827$	1.94204e7	0.987405	0.493703	+	0.869631i	$0.335643\pi$
$828$	0	0
$829$	2.66039e7	1.34449	0.672247	−	0.740327i	$-0.265329\pi$
$829$	2.66039e7	1.34449	0.672247	+	0.740327i	$0.265329\pi$
$830$	0	0
$831$	159564.	0.00801555
$832$	0	0
$833$	−5.18348e6	−0.258826
$834$	0	0
$835$	7.12257e6	0.353525
$836$	0	0
$837$	1.36417e6	0.0673062
$838$	0	0
$839$	1.41085e7	0.691954	0.345977	−	0.938243i	$-0.387547\pi$
$839$	1.41085e7	0.691954	0.345977	+	0.938243i	$0.387547\pi$
$840$	0	0
$841$	5.71843e6	0.278796
$842$	0	0
$843$	−6.79194e6	−0.329174
$844$	0	0
$845$	−2.14581e7	−1.03383
$846$	0	0
$847$	2.66548e6	0.127664
$848$	0	0
$849$	−3.75869e6	−0.178965
$850$	0	0
$851$	467918.	0.0221486
$852$	0	0
$853$	−1.87882e7	−0.884122	−0.442061	−	0.896985i	$-0.645752\pi$
$853$	−1.87882e7	−0.884122	−0.442061	+	0.896985i	$0.645752\pi$
$854$	0	0
$855$	−9.50323e6	−0.444586
$856$	0	0
$857$	3.67369e7	1.70864	0.854320	−	0.519748i	$-0.173974\pi$
$857$	3.67369e7	1.70864	0.854320	+	0.519748i	$0.173974\pi$
$858$	0	0
$859$	−2.03228e6	−0.0939723	−0.0469861	−	0.998896i	$-0.514962\pi$
$859$	−2.03228e6	−0.0939723	−0.0469861	+	0.998896i	$0.514962\pi$
$860$	0	0
$861$	1.48650e7	0.683371
$862$	0	0
$863$	1.21735e7	0.556400	0.278200	−	0.960523i	$-0.410262\pi$
$863$	1.21735e7	0.556400	0.278200	+	0.960523i	$0.410262\pi$
$864$	0	0
$865$	4.34183e6	0.197303
$866$	0	0
$867$	9.62646e6	0.434930
$868$	0	0
$869$	−3.12003e7	−1.40155
$870$	0	0
$871$	−1.06586e7	−0.476053
$872$	0	0
$873$	3.37606e6	0.149925
$874$	0	0
$875$	−8.07833e6	−0.356698
$876$	0	0
$877$	4.13453e7	1.81521	0.907605	−	0.419825i	$-0.137909\pi$
$877$	4.13453e7	1.81521	0.907605	+	0.419825i	$0.137909\pi$
$878$	0	0
$879$	2.41209e7	1.05298
$880$	0	0
$881$	1.92417e7	0.835227	0.417613	−	0.908625i	$-0.362867\pi$
$881$	1.92417e7	0.835227	0.417613	+	0.908625i	$0.362867\pi$
$882$	0	0
$883$	−4.59010e6	−0.198117	−0.0990583	−	0.995082i	$-0.531583\pi$
$883$	−4.59010e6	−0.198117	−0.0990583	+	0.995082i	$0.531583\pi$
$884$	0	0
$885$	3.02992e7	1.30039
$886$	0	0
$887$	1.34822e7	0.575374	0.287687	−	0.957724i	$-0.407114\pi$
$887$	1.34822e7	0.575374	0.287687	+	0.957724i	$0.407114\pi$
$888$	0	0
$889$	−1.50574e7	−0.638992
$890$	0	0
$891$	2.86560e6	0.120927
$892$	0	0
$893$	3.97149e7	1.66657
$894$	0	0
$895$	4.30552e7	1.79667
$896$	0	0
$897$	1.23246e6	0.0511435
$898$	0	0
$899$	9.58377e6	0.395492
$900$	0	0
$901$	−8.63750e6	−0.354467
$902$	0	0
$903$	9.95945e6	0.406458
$904$	0	0
$905$	8.55962e6	0.347403
$906$	0	0
$907$	−5.13039e6	−0.207077	−0.103539	−	0.994625i	$-0.533017\pi$
$907$	−5.13039e6	−0.207077	−0.103539	+	0.994625i	$0.533017\pi$
$908$	0	0
$909$	3.00513e6	0.120629
$910$	0	0
$911$	−3.88471e7	−1.55082	−0.775411	−	0.631456i	$-0.782457\pi$
$911$	−3.88471e7	−1.55082	−0.775411	+	0.631456i	$0.782457\pi$
$912$	0	0
$913$	−4.91709e7	−1.95223
$914$	0	0
$915$	9.74332e6	0.384728
$916$	0	0
$917$	−1.16384e7	−0.457055
$918$	0	0
$919$	−2.43482e7	−0.950993	−0.475496	−	0.879718i	$-0.657732\pi$
$919$	−2.43482e7	−0.950993	−0.475496	+	0.879718i	$0.657732\pi$
$920$	0	0
$921$	−2.18634e7	−0.849315
$922$	0	0
$923$	−1.18064e7	−0.456156
$924$	0	0
$925$	1.63472e6	0.0628188
$926$	0	0
$927$	9.98917e6	0.381795
$928$	0	0
$929$	2.17027e7	0.825040	0.412520	−	0.910948i	$-0.364649\pi$
$929$	2.17027e7	0.825040	0.412520	+	0.910948i	$0.364649\pi$
$930$	0	0
$931$	1.45715e7	0.550972
$932$	0	0
$933$	1.54605e7	0.581460
$934$	0	0
$935$	1.82284e7	0.681900
$936$	0	0
$937$	−2.01218e6	−0.0748718	−0.0374359	−	0.999299i	$-0.511919\pi$
$937$	−2.01218e6	−0.0748718	−0.0374359	+	0.999299i	$0.511919\pi$
$938$	0	0
$939$	−1.40925e7	−0.521582
$940$	0	0
$941$	−1.17136e7	−0.431237	−0.215618	−	0.976478i	$-0.569177\pi$
$941$	−1.17136e7	−0.431237	−0.215618	+	0.976478i	$0.569177\pi$
$942$	0	0
$943$	−9.73916e6	−0.356650
$944$	0	0
$945$	−4.61209e6	−0.168004
$946$	0	0
$947$	−571455.	−0.0207065	−0.0103533	−	0.999946i	$-0.503296\pi$
$947$	−571455.	−0.0207065	−0.0103533	+	0.999946i	$0.503296\pi$
$948$	0	0
$949$	2.04458e6	0.0736950
$950$	0	0
$951$	−6.06375e6	−0.217415
$952$	0	0
$953$	2.03678e7	0.726461	0.363231	−	0.931699i	$-0.381674\pi$
$953$	2.03678e7	0.726461	0.363231	+	0.931699i	$0.381674\pi$
$954$	0	0
$955$	2.07739e7	0.737073
$956$	0	0
$957$	2.01319e7	0.710566
$958$	0	0
$959$	−2.03382e7	−0.714110
$960$	0	0
$961$	−2.51274e7	−0.877687
$962$	0	0
$963$	−8.70389e6	−0.302446
$964$	0	0
$965$	−627316.	−0.0216854
$966$	0	0
$967$	3.93183e7	1.35216	0.676081	−	0.736827i	$-0.263677\pi$
$967$	3.93183e7	1.35216	0.676081	+	0.736827i	$0.263677\pi$
$968$	0	0
$969$	8.86142e6	0.303175
$970$	0	0
$971$	−4.86930e7	−1.65736	−0.828682	−	0.559719i	$-0.810909\pi$
$971$	−4.86930e7	−1.65736	−0.828682	+	0.559719i	$0.810909\pi$
$972$	0	0
$973$	3.76055e7	1.27341
$974$	0	0
$975$	4.30572e6	0.145055
$976$	0	0
$977$	−1.11758e7	−0.374577	−0.187288	−	0.982305i	$-0.559970\pi$
$977$	−1.11758e7	−0.374577	−0.187288	+	0.982305i	$0.559970\pi$
$978$	0	0
$979$	6.02056e7	2.00762
$980$	0	0
$981$	7.82887e6	0.259733
$982$	0	0
$983$	−2.09439e7	−0.691310	−0.345655	−	0.938362i	$-0.612343\pi$
$983$	−2.09439e7	−0.691310	−0.345655	+	0.938362i	$0.612343\pi$
$984$	0	0
$985$	−4.92311e7	−1.61677
$986$	0	0
$987$	1.92744e7	0.629777
$988$	0	0
$989$	−6.52518e6	−0.212130
$990$	0	0
$991$	1.95125e7	0.631144	0.315572	−	0.948902i	$-0.397804\pi$
$991$	1.95125e7	0.631144	0.315572	+	0.948902i	$0.397804\pi$
$992$	0	0
$993$	1.52791e7	0.491729
$994$	0	0
$995$	3.76064e7	1.20422
$996$	0	0
$997$	1.32696e7	0.422786	0.211393	−	0.977401i	$-0.432200\pi$
$997$	1.32696e7	0.422786	0.211393	+	0.977401i	$0.432200\pi$
$998$	0	0
$999$	−644825.	−0.0204422

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1104.6.a.r.1.4		5
4.3	odd	2		69.6.a.e.1.5	✓	5
12.11	even	2		207.6.a.f.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.6.a.e.1.5	✓	5	4.3	odd	2
207.6.a.f.1.1		5	12.11	even	2
1104.6.a.r.1.4		5	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1104 = 2^{4} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1104.a (trivial)

\(f(q)\)	\(=\)	\(q-9.00000 q^{3} +70.5203 q^{5} +89.7132 q^{7} +81.0000 q^{9} +O(q^{10})\)	\(q-9.00000 q^{3} +70.5203 q^{5} +89.7132 q^{7} +81.0000 q^{9} +436.763 q^{11} -258.865 q^{13} -634.683 q^{15} +591.819 q^{17} -1663.69 q^{19} -807.419 q^{21} +529.000 q^{23} +1848.12 q^{25} -729.000 q^{27} -5121.48 q^{29} -1871.29 q^{31} -3930.87 q^{33} +6326.60 q^{35} +884.534 q^{37} +2329.78 q^{39} -18410.5 q^{41} -12334.9 q^{43} +5712.15 q^{45} -23871.6 q^{47} -8758.55 q^{49} -5326.37 q^{51} -14594.8 q^{53} +30800.7 q^{55} +14973.2 q^{57} -47739.1 q^{59} -15351.5 q^{61} +7266.77 q^{63} -18255.2 q^{65} +41174.4 q^{67} -4761.00 q^{69} +45608.3 q^{71} -7898.24 q^{73} -16633.1 q^{75} +39183.4 q^{77} -71435.2 q^{79} +6561.00 q^{81} -112580. q^{83} +41735.3 q^{85} +46093.3 q^{87} +137845. q^{89} -23223.6 q^{91} +16841.6 q^{93} -117324. q^{95} +41679.8 q^{97} +35377.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 45 q^{3} + 94 q^{5} - 272 q^{7} + 405 q^{9}+O(q^{10}) \) 5 * q - 45 * q^3 + 94 * q^5 - 272 * q^7 + 405 * q^9	\( 5 q - 45 q^{3} + 94 q^{5} - 272 q^{7} + 405 q^{9} - 1100 q^{11} - 978 q^{13} - 846 q^{15} + 2522 q^{17} - 2060 q^{19} + 2448 q^{21} + 2645 q^{23} + 12035 q^{25} - 3645 q^{27} + 1526 q^{29} + 7392 q^{31} + 9900 q^{33} - 6056 q^{35} - 8210 q^{37} + 8802 q^{39} + 21250 q^{41} + 4548 q^{43} + 7614 q^{45} - 536 q^{47} - 27979 q^{49} - 22698 q^{51} - 11482 q^{53} + 77064 q^{55} + 18540 q^{57} - 74676 q^{59} - 44618 q^{61} - 22032 q^{63} - 24388 q^{65} + 1412 q^{67} - 23805 q^{69} - 37912 q^{71} + 46546 q^{73} - 108315 q^{75} + 157008 q^{77} - 50544 q^{79} + 32805 q^{81} - 89588 q^{83} + 147892 q^{85} - 13734 q^{87} + 280410 q^{89} + 27416 q^{91} - 66528 q^{93} - 203120 q^{95} + 90074 q^{97} - 89100 q^{99}+O(q^{100}) \) 5 * q - 45 * q^3 + 94 * q^5 - 272 * q^7 + 405 * q^9 - 1100 * q^11 - 978 * q^13 - 846 * q^15 + 2522 * q^17 - 2060 * q^19 + 2448 * q^21 + 2645 * q^23 + 12035 * q^25 - 3645 * q^27 + 1526 * q^29 + 7392 * q^31 + 9900 * q^33 - 6056 * q^35 - 8210 * q^37 + 8802 * q^39 + 21250 * q^41 + 4548 * q^43 + 7614 * q^45 - 536 * q^47 - 27979 * q^49 - 22698 * q^51 - 11482 * q^53 + 77064 * q^55 + 18540 * q^57 - 74676 * q^59 - 44618 * q^61 - 22032 * q^63 - 24388 * q^65 + 1412 * q^67 - 23805 * q^69 - 37912 * q^71 + 46546 * q^73 - 108315 * q^75 + 157008 * q^77 - 50544 * q^79 + 32805 * q^81 - 89588 * q^83 + 147892 * q^85 - 13734 * q^87 + 280410 * q^89 + 27416 * q^91 - 66528 * q^93 - 203120 * q^95 + 90074 * q^97 - 89100 * q^99

Introduction

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