Properties

Label 66.6.b.a
Level $66$
Weight $6$
Character orbit 66.b
Analytic conductor $10.585$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [66,6,Mod(65,66)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(66, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("66.65");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 66 = 2 \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 66.b (of order \(2\), degree \(1\), 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: \(10.5853321077\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 105 x^{8} - 594 x^{7} - 24399 x^{6} - 88320 x^{5} - 5353086 x^{4} + 44410248 x^{3} + 2052482496 x^{2} + 30651651920 x + 814545531072 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{7} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 4 q^{2} + ( - \beta_{3} + 1) q^{3} + 16 q^{4} + (\beta_{3} - \beta_1) q^{5} + (4 \beta_{3} - 4) q^{6} + (\beta_{5} - \beta_{3} + \beta_1) q^{7} - 64 q^{8} + (\beta_{8} - \beta_{3} - \beta_1 - 20) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 q^{2} + ( - \beta_{3} + 1) q^{3} + 16 q^{4} + (\beta_{3} - \beta_1) q^{5} + (4 \beta_{3} - 4) q^{6} + (\beta_{5} - \beta_{3} + \beta_1) q^{7} - 64 q^{8} + (\beta_{8} - \beta_{3} - \beta_1 - 20) q^{9} + ( - 4 \beta_{3} + 4 \beta_1) q^{10} + ( - \beta_{8} - \beta_{6} + 7 \beta_{3} - \beta_{2} - \beta_1 - 3) q^{11} + ( - 16 \beta_{3} + 16) q^{12} + ( - \beta_{5} - \beta_{4} + 8 \beta_{3} + \beta_{2} + \beta_1) q^{13} + ( - 4 \beta_{5} + 4 \beta_{3} - 4 \beta_1) q^{14} + (\beta_{9} + 2 \beta_{6} + \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} - 4 \beta_1 + 176) q^{15} + 256 q^{16} + ( - \beta_{8} + \beta_{7} + \beta_{6} - 8 \beta_{3} - 208) q^{17} + ( - 4 \beta_{8} + 4 \beta_{3} + 4 \beta_1 + 80) q^{18} + (2 \beta_{9} - 2 \beta_{8} - 2 \beta_{6} + 2 \beta_{5} - 3 \beta_{4} + 25 \beta_{3} + \cdots - 10 \beta_1) q^{19}+ \cdots + ( - 81 \beta_{9} - 91 \beta_{8} - 2 \beta_{7} + 159 \beta_{6} + \cdots + 10569) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 40 q^{2} + 10 q^{3} + 160 q^{4} - 40 q^{6} - 640 q^{8} - 200 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 40 q^{2} + 10 q^{3} + 160 q^{4} - 40 q^{6} - 640 q^{8} - 200 q^{9} - 30 q^{11} + 160 q^{12} + 1760 q^{15} + 2560 q^{16} - 2076 q^{17} + 800 q^{18} - 1716 q^{21} + 120 q^{22} - 640 q^{24} - 3582 q^{25} - 2402 q^{27} - 1776 q^{29} - 7040 q^{30} + 2168 q^{31} - 10240 q^{32} + 868 q^{33} + 8304 q^{34} + 24252 q^{35} - 3200 q^{36} + 32 q^{37} + 19200 q^{39} - 2664 q^{41} + 6864 q^{42} - 480 q^{44} - 31702 q^{45} + 2560 q^{48} - 42086 q^{49} + 14328 q^{50} + 17412 q^{51} + 9608 q^{54} - 43012 q^{55} + 47904 q^{57} + 7104 q^{58} + 28160 q^{60} - 8672 q^{62} - 58980 q^{63} + 40960 q^{64} + 18372 q^{65} - 3472 q^{66} + 82160 q^{67} - 33216 q^{68} - 11026 q^{69} - 97008 q^{70} + 12800 q^{72} - 128 q^{74} - 83298 q^{75} + 75624 q^{77} - 76800 q^{78} + 111268 q^{81} + 10656 q^{82} - 528 q^{83} - 27456 q^{84} - 160020 q^{87} + 1920 q^{88} + 126808 q^{90} + 225108 q^{91} + 159206 q^{93} - 401400 q^{95} - 10240 q^{96} + 129668 q^{97} + 168344 q^{98} + 105682 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 105 x^{8} - 594 x^{7} - 24399 x^{6} - 88320 x^{5} - 5353086 x^{4} + 44410248 x^{3} + 2052482496 x^{2} + 30651651920 x + 814545531072 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 3127 \nu^{9} + 1365206 \nu^{8} - 16943185 \nu^{7} - 60199912 \nu^{6} + 5169753971 \nu^{5} - 41410816306 \nu^{4} + \cdots + 23\!\cdots\!84 ) / 123836634785916 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 26543 \nu^{9} - 1689037 \nu^{8} + 53963357 \nu^{7} + 839196257 \nu^{6} + 16274224607 \nu^{5} + 372089278835 \nu^{4} + \cdots - 74\!\cdots\!28 ) / 123836634785916 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - \nu^{9} - \nu^{8} - 106 \nu^{7} + 488 \nu^{6} + 24887 \nu^{5} + 113207 \nu^{4} + 5466293 \nu^{3} - 38943955 \nu^{2} - 2091426451 \nu - 29256293970 ) / 3486784401 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - \nu^{9} - \nu^{8} - 106 \nu^{7} + 488 \nu^{6} + 24887 \nu^{5} + 113207 \nu^{4} + 5466293 \nu^{3} - 38943955 \nu^{2} + 29289633158 \nu - 29256293970 ) / 3486784401 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 63443 \nu^{9} - 539683 \nu^{8} - 44456431 \nu^{7} + 414026519 \nu^{6} - 9867182017 \nu^{5} + 229745992085 \nu^{4} + \cdots - 37\!\cdots\!32 ) / 123836634785916 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 51820 \nu^{9} + 795035 \nu^{8} + 7840004 \nu^{7} - 287835565 \nu^{6} + 1152269150 \nu^{5} + 3654223361 \nu^{4} + \cdots + 595936000291086 ) / 20639439130986 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 312371 \nu^{9} - 26430983 \nu^{8} + 309433381 \nu^{7} + 2090444911 \nu^{6} + 80280242719 \nu^{5} + 303429635653 \nu^{4} + \cdots - 43\!\cdots\!64 ) / 123836634785916 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 316517 \nu^{9} - 6945538 \nu^{8} - 321697 \nu^{7} - 1148268088 \nu^{6} + 494321183 \nu^{5} + 140467258982 \nu^{4} + \cdots - 28\!\cdots\!88 ) / 123836634785916 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 401737 \nu^{9} + 5098568 \nu^{8} + 79251461 \nu^{7} - 444633994 \nu^{6} + 1673401301 \nu^{5} + 312715240688 \nu^{4} + \cdots + 246521024182368 ) / 123836634785916 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} - \beta_{3} ) / 9 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 6\beta_{9} + 3\beta_{8} - 6\beta_{6} - 3\beta_{5} - 5\beta_{4} + 8\beta_{3} - 3\beta_{2} + 9\beta _1 - 189 ) / 9 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 15 \beta_{9} + 15 \beta_{8} - 18 \beta_{7} - 12 \beta_{6} - 87 \beta_{5} - 20 \beta_{4} + 485 \beta_{3} + 57 \beta_{2} - 126 \beta _1 + 1611 ) / 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 387 \beta_{9} + 504 \beta_{8} - 54 \beta_{7} + 585 \beta_{6} + 2196 \beta_{5} - 17 \beta_{4} + 251 \beta_{3} + 1170 \beta_{2} + 351 \beta _1 + 107703 ) / 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 7950 \beta_{9} + 3801 \beta_{8} + 2430 \beta_{7} + 6330 \beta_{6} - 7932 \beta_{5} + 10543 \beta_{4} + 27359 \beta_{3} + 17826 \beta_{2} + 93861 \beta _1 + 115803 ) / 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 14466 \beta_{9} - 361200 \beta_{8} + 33084 \beta_{7} - 320754 \beta_{6} + 75819 \beta_{5} - 34535 \beta_{4} - 638281 \beta_{3} + 73587 \beta_{2} - 26856 \beta _1 + 13928130 ) / 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1544175 \beta_{9} - 2047752 \beta_{8} + 219726 \beta_{7} + 720828 \beta_{6} - 6784353 \beta_{5} + 1772443 \beta_{4} - 54773380 \beta_{3} + 2267451 \beta_{2} - 15705117 \beta _1 - 205732440 ) / 9 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 33658143 \beta_{9} - 52964700 \beta_{8} - 6946074 \beta_{7} + 15096081 \beta_{6} + 28990887 \beta_{5} - 60370808 \beta_{4} - 821704708 \beta_{3} + 14932689 \beta_{2} + \cdots - 14412341529 ) / 9 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 659979951 \beta_{9} + 210575157 \beta_{8} - 44230932 \beta_{7} + 143797218 \beta_{6} + 419613465 \beta_{5} - 1889934644 \beta_{4} - 19924532443 \beta_{3} + \cdots - 189048409821 ) / 9 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/66\mathbb{Z}\right)^\times\).

\(n\) \(13\) \(23\)
\(\chi(n)\) \(-1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
15.7713 4.98091i
15.7713 + 4.98091i
6.93803 14.4132i
6.93803 + 14.4132i
−1.11022 15.4450i
−1.11022 + 15.4450i
−7.54857 13.0354i
−7.54857 + 13.0354i
−14.0505 4.05977i
−14.0505 + 4.05977i
−4.00000 −14.7713 4.98091i 16.0000 33.4485i 59.0851 + 19.9237i 26.3904i −64.0000 193.381 + 147.149i 133.794i
65.2 −4.00000 −14.7713 + 4.98091i 16.0000 33.4485i 59.0851 19.9237i 26.3904i −64.0000 193.381 147.149i 133.794i
65.3 −4.00000 −5.93803 14.4132i 16.0000 8.81389i 23.7521 + 57.6527i 143.107i −64.0000 −172.480 + 171.172i 35.2556i
65.4 −4.00000 −5.93803 + 14.4132i 16.0000 8.81389i 23.7521 57.6527i 143.107i −64.0000 −172.480 171.172i 35.2556i
65.5 −4.00000 2.11022 15.4450i 16.0000 100.562i −8.44086 + 61.7799i 204.166i −64.0000 −234.094 65.1844i 402.247i
65.6 −4.00000 2.11022 + 15.4450i 16.0000 100.562i −8.44086 61.7799i 204.166i −64.0000 −234.094 + 65.1844i 402.247i
65.7 −4.00000 8.54857 13.0354i 16.0000 77.4379i −34.1943 + 52.1417i 53.9762i −64.0000 −96.8440 222.868i 309.752i
65.8 −4.00000 8.54857 + 13.0354i 16.0000 77.4379i −34.1943 52.1417i 53.9762i −64.0000 −96.8440 + 222.868i 309.752i
65.9 −4.00000 15.0505 4.05977i 16.0000 10.4989i −60.2021 + 16.2391i 198.254i −64.0000 210.037 122.203i 41.9956i
65.10 −4.00000 15.0505 + 4.05977i 16.0000 10.4989i −60.2021 16.2391i 198.254i −64.0000 210.037 + 122.203i 41.9956i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
33.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 66.6.b.a 10
3.b odd 2 1 66.6.b.b yes 10
11.b odd 2 1 66.6.b.b yes 10
33.d even 2 1 inner 66.6.b.a 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.6.b.a 10 1.a even 1 1 trivial
66.6.b.a 10 33.d even 2 1 inner
66.6.b.b yes 10 3.b odd 2 1
66.6.b.b yes 10 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{5} + 1038T_{17}^{4} - 4533096T_{17}^{3} - 5102614656T_{17}^{2} + 2497107681792T_{17} + 2183764681359360 \) acting on \(S_{6}^{\mathrm{new}}(66, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 4)^{10} \) Copy content Toggle raw display
$3$ \( T^{10} - 10 T^{9} + \cdots + 847288609443 \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 580964238766592 \) Copy content Toggle raw display
$7$ \( T^{10} + 105078 T^{8} + \cdots + 68\!\cdots\!08 \) Copy content Toggle raw display
$11$ \( T^{10} + 30 T^{9} + \cdots + 10\!\cdots\!51 \) Copy content Toggle raw display
$13$ \( T^{10} + 858042 T^{8} + \cdots + 20\!\cdots\!52 \) Copy content Toggle raw display
$17$ \( (T^{5} + 1038 T^{4} + \cdots + 21\!\cdots\!60)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} + 16308588 T^{8} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{10} + 42452896 T^{8} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{5} + 888 T^{4} + \cdots + 12\!\cdots\!92)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} - 1084 T^{4} + \cdots + 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( (T^{5} - 16 T^{4} + \cdots - 44\!\cdots\!44)^{2} \) Copy content Toggle raw display
$41$ \( (T^{5} + 1332 T^{4} + \cdots + 36\!\cdots\!36)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + 433628292 T^{8} + \cdots + 15\!\cdots\!48 \) Copy content Toggle raw display
$47$ \( T^{10} + 1150695994 T^{8} + \cdots + 14\!\cdots\!92 \) Copy content Toggle raw display
$53$ \( T^{10} + 1572253978 T^{8} + \cdots + 80\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{10} + 1197786766 T^{8} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{10} + 8294004138 T^{8} + \cdots + 43\!\cdots\!68 \) Copy content Toggle raw display
$67$ \( (T^{5} - 41080 T^{4} + \cdots - 21\!\cdots\!24)^{2} \) Copy content Toggle raw display
$71$ \( T^{10} + 12332773360 T^{8} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{10} + 9701788080 T^{8} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{10} + 6232034694 T^{8} + \cdots + 91\!\cdots\!88 \) Copy content Toggle raw display
$83$ \( (T^{5} + 264 T^{4} + \cdots - 92\!\cdots\!32)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + 37766069974 T^{8} + \cdots + 20\!\cdots\!48 \) Copy content Toggle raw display
$97$ \( (T^{5} - 64834 T^{4} + \cdots + 32\!\cdots\!16)^{2} \) Copy content Toggle raw display
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