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Properties

Label 162.11.d.d
Level $162$
Weight $11$
Character orbit 162.d
Analytic conductor $102.928$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [162,11,Mod(53,162)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(162, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("162.53");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 162.d (of order \(6\), degree \(2\), 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: \(102.927874933\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.17318914560000.97
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 82x^{6} + 260x^{5} + 2477x^{4} - 5392x^{3} - 31616x^{2} + 34356x + 161859 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{24}\cdot 3^{16} \)
Twist minimal: no (minimal twist has level 6)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} - 512 \beta_1 q^{4} + (26 \beta_{5} + \beta_{4} + 26 \beta_{3}) q^{5} + ( - \beta_{7} + 11278 \beta_1 + 11278) q^{7} + 512 \beta_{5} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} - 512 \beta_1 q^{4} + (26 \beta_{5} + \beta_{4} + 26 \beta_{3}) q^{5} + ( - \beta_{7} + 11278 \beta_1 + 11278) q^{7} + 512 \beta_{5} q^{8} + ( - 8 \beta_{6} - 13440) q^{10} + (3766 \beta_{3} + 35 \beta_{2}) q^{11} + (20 \beta_{7} - 20 \beta_{6} + 68810 \beta_1) q^{13} + ( - 11294 \beta_{5} + 64 \beta_{4} - 11294 \beta_{3}) q^{14} + ( - 262144 \beta_1 - 262144) q^{16} + (73832 \beta_{5} - 236 \beta_{4} + 236 \beta_{2}) q^{17} + ( - 97 \beta_{6} - 392182) q^{19} + (13312 \beta_{3} + 512 \beta_{2}) q^{20} + (280 \beta_{7} - 280 \beta_{6} + 1932672 \beta_1) q^{22} + ( - 91988 \beta_{5} - 826 \beta_{4} - 91988 \beta_{3}) q^{23} + (420 \beta_{7} + 8433095 \beta_1 + 8433095) q^{25} + ( - 68490 \beta_{5} - 1280 \beta_{4} + 1280 \beta_{2}) q^{26} + ( - 512 \beta_{6} + 5774336) q^{28} + ( - 763022 \beta_{3} - 2947 \beta_{2}) q^{29} + (3045 \beta_{7} - 3045 \beta_{6} - 5446462 \beta_1) q^{31} + (262144 \beta_{5} + 262144 \beta_{3}) q^{32} + (1888 \beta_{7} - 37771776 \beta_1 - 37771776) q^{34} + ( - 1937092 \beta_{5} + 9598 \beta_{4} - 9598 \beta_{2}) q^{35} + ( - 5684 \beta_{6} - 17753542) q^{37} + (390630 \beta_{3} + 6208 \beta_{2}) q^{38} + (4096 \beta_{7} - 4096 \beta_{6} + 6881280 \beta_1) q^{40} + ( - 4094500 \beta_{5} + 13414 \beta_{4} - 4094500 \beta_{3}) q^{41} + ( - 6783 \beta_{7} + 117672166 \beta_1 + 117672166) q^{43} + ( - 1928192 \beta_{5} - 17920 \beta_{4} + 17920 \beta_{2}) q^{44} + (6608 \beta_{6} + 47203584) q^{46} + (4156680 \beta_{3} - 14460 \beta_{2}) q^{47} + ( - 22556 \beta_{7} + 22556 \beta_{6} - 12514605 \beta_1) q^{49} + ( - 8426375 \beta_{5} - 26880 \beta_{4} - 8426375 \beta_{3}) q^{50} + (10240 \beta_{7} + 35230720 \beta_1 + 35230720) q^{52} + (9200542 \beta_{5} + 59339 \beta_{4} - 59339 \beta_{2}) q^{53} + (37548 \beta_{6} + 675339840) q^{55} + ( - 5782528 \beta_{3} + 32768 \beta_{2}) q^{56} + ( - 23576 \beta_{7} + 23576 \beta_{6} + \cdots - 391044480 \beta_1) q^{58}+ \cdots + (12153709 \beta_{5} + 1443584 \beta_{4} + \cdots - 1443584 \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2048 q^{4} + 45112 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2048 q^{4} + 45112 q^{7} - 107520 q^{10} - 275240 q^{13} - 1048576 q^{16} - 3137456 q^{19} - 7730688 q^{22} + 33732380 q^{25} + 46194688 q^{28} + 21785848 q^{31} - 151087104 q^{34} - 142028336 q^{37} - 27525120 q^{40} + 470688664 q^{43} + 377628672 q^{46} + 50058420 q^{49} + 140922880 q^{52} + 5402718720 q^{55} + 1564177920 q^{58} + 1184038744 q^{61} - 1073741824 q^{64} + 297365848 q^{67} + 3962250240 q^{70} + 13068538000 q^{73} - 803188736 q^{76} - 199282568 q^{79} + 16757336064 q^{82} + 12880512000 q^{85} + 3958112256 q^{88} + 16634464160 q^{91} - 8505477120 q^{94} + 39176355064 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} - 82x^{6} + 260x^{5} + 2477x^{4} - 5392x^{3} - 31616x^{2} + 34356x + 161859 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{6} - 6\nu^{5} - 211\nu^{4} + 432\nu^{3} + 6275\nu^{2} - 6492\nu - 57465 ) / 13746 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 6736 \nu^{7} - 4259056 \nu^{6} + 13438472 \nu^{5} + 450282296 \nu^{4} - 945290272 \nu^{3} - 28122112960 \nu^{2} + 28839704256 \nu + 431997408168 ) / 45423657 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 26944 \nu^{7} + 94304 \nu^{6} + 2362304 \nu^{5} - 6141520 \nu^{4} - 80967040 \nu^{3} + 127639232 \nu^{2} + 884063664 \nu - 463512000 ) / 45423657 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 25180 \nu^{7} + 88776484 \nu^{6} - 268154120 \nu^{5} - 5691779540 \nu^{4} + 11865395680 \nu^{3} + 113195424340 \nu^{2} + \cdots - 606875544888 ) / 45423657 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1616 \nu^{7} - 5656 \nu^{6} - 108904 \nu^{5} + 286400 \nu^{4} + 2507000 \nu^{3} - 4049728 \nu^{2} - 17321976 \nu + 9345624 ) / 574983 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 484992 \nu^{7} + 1697472 \nu^{6} + 42521472 \nu^{5} - 110547360 \nu^{4} - 1457406720 \nu^{3} + 2297506176 \nu^{2} + 28995159168 \nu - 14884222608 ) / 5047073 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3716352 \nu^{7} - 13007232 \nu^{6} - 310292784 \nu^{5} + 808250040 \nu^{4} + 7873873344 \nu^{3} - 12625563672 \nu^{2} - 47236696128 \nu + 25749860040 ) / 5047073 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{6} - 162\beta_{3} + 1296 ) / 2592 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} - 160\beta_{3} - 8\beta_{2} - 5184\beta _1 + 55728 ) / 2592 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{7} + 14\beta_{6} + 162\beta_{5} - 5223\beta_{3} - 6\beta_{2} - 3888\beta _1 + 41472 ) / 1296 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 12 \beta_{7} + 55 \beta_{6} + 640 \beta_{5} + 32 \beta_{4} - 20648 \beta_{3} - 384 \beta_{2} - 679104 \beta _1 + 1242864 ) / 2592 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 460 \beta_{7} + 945 \beta_{6} + 70288 \beta_{5} + 80 \beta_{4} - 390772 \beta_{3} - 940 \beta_{2} - 1684800 \beta _1 + 2969136 ) / 2592 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 675 \beta_{7} + 1349 \beta_{6} + 104200 \beta_{5} + 1808 \beta_{4} - 559098 \beta_{3} - 7820 \beta_{2} - 20470320 \beta _1 + 12974256 ) / 1296 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 24290 \beta_{7} + 33167 \beta_{6} + 5772368 \beta_{5} + 12376 \beta_{4} - 12520766 \beta_{3} - 51464 \beta_{2} - 137404512 \beta _1 + 80524368 ) / 2592 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).

\(n\) \(83\)
\(\chi(n)\) \(-\beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
−5.33452 + 0.707107i
3.88503 + 0.707107i
6.33452 − 0.707107i
−2.88503 − 0.707107i
−5.33452 − 0.707107i
3.88503 − 0.707107i
6.33452 + 0.707107i
−2.88503 + 0.707107i
−19.5959 + 11.3137i 0 256.000 − 443.405i −3144.08 − 1815.24i 0 11613.3 + 20114.8i 11585.2i 0 82148.2
53.2 −19.5959 + 11.3137i 0 256.000 − 443.405i 4172.87 + 2409.21i 0 −335.265 − 580.696i 11585.2i 0 −109028.
53.3 19.5959 − 11.3137i 0 256.000 − 443.405i −4172.87 − 2409.21i 0 −335.265 − 580.696i − 11585.2i 0 −109028.
53.4 19.5959 − 11.3137i 0 256.000 − 443.405i 3144.08 + 1815.24i 0 11613.3 + 20114.8i − 11585.2i 0 82148.2
107.1 −19.5959 − 11.3137i 0 256.000 + 443.405i −3144.08 + 1815.24i 0 11613.3 − 20114.8i − 11585.2i 0 82148.2
107.2 −19.5959 − 11.3137i 0 256.000 + 443.405i 4172.87 − 2409.21i 0 −335.265 + 580.696i − 11585.2i 0 −109028.
107.3 19.5959 + 11.3137i 0 256.000 + 443.405i −4172.87 + 2409.21i 0 −335.265 + 580.696i 11585.2i 0 −109028.
107.4 19.5959 + 11.3137i 0 256.000 + 443.405i 3144.08 − 1815.24i 0 11613.3 − 20114.8i 11585.2i 0 82148.2
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 162.11.d.d 8
3.b odd 2 1 inner 162.11.d.d 8
9.c even 3 1 6.11.b.a ✓ 4
9.c even 3 1 inner 162.11.d.d 8
9.d odd 6 1 6.11.b.a ✓ 4
9.d odd 6 1 inner 162.11.d.d 8
36.f odd 6 1 48.11.e.d 4
36.h even 6 1 48.11.e.d 4
45.h odd 6 1 150.11.d.a 4
45.j even 6 1 150.11.d.a 4
45.k odd 12 2 150.11.b.a 8
45.l even 12 2 150.11.b.a 8
72.j odd 6 1 192.11.e.g 4
72.l even 6 1 192.11.e.h 4
72.n even 6 1 192.11.e.g 4
72.p odd 6 1 192.11.e.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.11.b.a ✓ 4 9.c even 3 1
6.11.b.a ✓ 4 9.d odd 6 1
48.11.e.d 4 36.f odd 6 1
48.11.e.d 4 36.h even 6 1
150.11.b.a 8 45.k odd 12 2
150.11.b.a 8 45.l even 12 2
150.11.d.a 4 45.h odd 6 1
150.11.d.a 4 45.j even 6 1
162.11.d.d 8 1.a even 1 1 trivial
162.11.d.d 8 3.b odd 2 1 inner
162.11.d.d 8 9.c even 3 1 inner
162.11.d.d 8 9.d odd 6 1 inner
192.11.e.g 4 72.j odd 6 1
192.11.e.g 4 72.n even 6 1
192.11.e.h 4 72.l even 6 1
192.11.e.h 4 72.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 36397440 T_{5}^{6} + \cdots + 93\!\cdots\!00 \) acting on \(S_{11}^{\mathrm{new}}(162, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 512 T^{2} + 262144)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 36397440 T^{6} + \cdots + 93\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{4} - 22556 T^{3} + \cdots + 242551843253776)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 58313211264 T^{6} + \cdots + 45\!\cdots\!76 \) Copy content Toggle raw display
$13$ \( (T^{4} + 137620 T^{3} + \cdots + 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 7560967182336 T^{2} + \cdots + 32\!\cdots\!84)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 784364 T - 1189491369116)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} - 33055507478016 T^{6} + \cdots + 37\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( T^{8} - 907304099736960 T^{6} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{4} - 10892924 T^{3} + \cdots + 16\!\cdots\!36)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 35507084 T - 42\!\cdots\!96)^{4} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 82\!\cdots\!16 \) Copy content Toggle raw display
$43$ \( (T^{4} - 235344332 T^{3} + \cdots + 52\!\cdots\!56)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 67\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 37\!\cdots\!04)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 41\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( (T^{4} - 592019372 T^{3} + \cdots + 10\!\cdots\!96)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 148682924 T^{3} + \cdots + 12\!\cdots\!16)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 49\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 3267134500 T + 23\!\cdots\!40)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 99641284 T^{3} + \cdots + 13\!\cdots\!76)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 32\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 15\!\cdots\!84)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 19588177532 T^{3} + \cdots + 90\!\cdots\!16)^{2} \) Copy content Toggle raw display
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