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Properties

Label 162.13.d.c
Level $162$
Weight $13$
Character orbit 162.d
Analytic conductor $148.067$
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] = [162,13,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, 13, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("162.53");
 
S:= CuspForms(chi, 13);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 13 \)
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: \(148.066998399\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4 x^{7} - 18478 x^{6} + 55448 x^{5} + 128029439 x^{4} - 256151296 x^{3} - 394230846230 x^{2} + 394358931120 x + 455189180292012 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{24}\cdot 3^{12} \)
Twist minimal: no (minimal twist has level 54)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} - 2048 \beta_1 q^{4} + (139 \beta_{6} + \beta_{5} + 139 \beta_{3} + \beta_{2}) q^{5} + (\beta_{7} + \beta_{4} - 67871 \beta_1 - 67871) q^{7} - 2048 \beta_{6} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} - 2048 \beta_1 q^{4} + (139 \beta_{6} + \beta_{5} + 139 \beta_{3} + \beta_{2}) q^{5} + (\beta_{7} + \beta_{4} - 67871 \beta_1 - 67871) q^{7} - 2048 \beta_{6} q^{8} + (32 \beta_{4} + 284928) q^{10} + ( - 3727 \beta_{3} + 59 \beta_{2}) q^{11} + (211 \beta_{7} + 174071 \beta_1) q^{13} + ( - 67879 \beta_{6} + 64 \beta_{5} - 67879 \beta_{3} + 64 \beta_{2}) q^{14} + ( - 4194304 \beta_1 - 4194304) q^{16} + (355945 \beta_{6} - 653 \beta_{5}) q^{17} + (802 \beta_{4} - 21969241) q^{19} + (284672 \beta_{3} + 2048 \beta_{2}) q^{20} + ( - 1888 \beta_{7} + 7617792 \beta_1) q^{22} + (321665 \beta_{6} + 5411 \beta_{5} + 321665 \beta_{3} + 5411 \beta_{2}) q^{23} + (8904 \beta_{7} + 8904 \beta_{4} + 226624751 \beta_1 + 226624751) q^{25} + (172383 \beta_{6} + 13504 \beta_{5}) q^{26} + (2048 \beta_{4} - 138999808) q^{28} + ( - 14836630 \beta_{3} - 9106 \beta_{2}) q^{29} + ( - 4296 \beta_{7} - 595633534 \beta_1) q^{31} + ( - 4194304 \beta_{6} - 4194304 \beta_{3}) q^{32} + ( - 20896 \beta_{7} - 20896 \beta_{4} + 728808192 \beta_1 + 728808192) q^{34} + (4037467 \beta_{6} - 58967 \beta_{5}) q^{35} + (144785 \beta_{4} - 143321785) q^{37} + ( - 21975657 \beta_{3} + 51328 \beta_{2}) q^{38} + ( - 65536 \beta_{7} - 583532544 \beta_1) q^{40} + (44146834 \beta_{6} - 177818 \beta_{5} + 44146834 \beta_{3} - 177818 \beta_{2}) q^{41} + (275352 \beta_{7} + 275352 \beta_{4} + 3779183086 \beta_1 + 3779183086) q^{43} + (7632896 \beta_{6} - 120832 \beta_{5}) q^{44} + (173152 \beta_{4} + 660155136) q^{46} + ( - 140053719 \beta_{3} - 120597 \beta_{2}) q^{47} + ( - 135742 \beta_{7} - 8372565024 \beta_1) q^{49} + (226553519 \beta_{6} + 569856 \beta_{5} + 226553519 \beta_{3} + 569856 \beta_{2}) q^{50} + (432128 \beta_{7} + 432128 \beta_{4} + 356497408 \beta_1 + 356497408) q^{52} + ( - 114882106 \beta_{6} + 967730 \beta_{5}) q^{53} + (143640 \beta_{4} + 24376536000) q^{55} + ( - 139016192 \beta_{3} + 131072 \beta_{2}) q^{56} + (291392 \beta_{7} + 30387749376 \beta_1) q^{58} + (1396558109 \beta_{6} - 71857 \beta_{5} + 1396558109 \beta_{3} - 71857 \beta_{2}) q^{59} + ( - 1595109 \beta_{7} - 1595109 \beta_{4} + \cdots - 14590716599) q^{61}+ \cdots + ( - 8371479088 \beta_{6} - 8687488 \beta_{5}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8192 q^{4} - 271484 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8192 q^{4} - 271484 q^{7} + 2279424 q^{10} - 696284 q^{13} - 16777216 q^{16} - 175753928 q^{19} - 30471168 q^{22} + 906499004 q^{25} - 1111998464 q^{28} + 2382534136 q^{31} + 2915232768 q^{34} - 1146574280 q^{37} + 2334130176 q^{40} + 15116732344 q^{43} + 5281241088 q^{46} + 33490260096 q^{49} + 1425989632 q^{52} + 195012288000 q^{55} - 121550997504 q^{58} - 58362866396 q^{61} - 68719476736 q^{64} - 308975155100 q^{67} + 33014547456 q^{70} - 357741406856 q^{73} - 179972022272 q^{76} + 905099168836 q^{79} + 722937556992 q^{82} + 720516135168 q^{85} + 62404952064 q^{88} - 1360962234040 q^{91} - 1147443557376 q^{94} + 5671281236356 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4 x^{7} - 18478 x^{6} + 55448 x^{5} + 128029439 x^{4} - 256151296 x^{3} - 394230846230 x^{2} + 394358931120 x + 455189180292012 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{6} - 6\nu^{5} - 46201\nu^{4} + 92412\nu^{3} + 298840097\nu^{2} - 298886304\nu - 591815920068 ) / 682946850 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1478416 \nu^{7} + 73747483088 \nu^{6} - 192540569512 \nu^{5} + \cdots - 79\!\cdots\!60 ) / 11\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 11827328 \nu^{7} + 41395648 \nu^{6} + 229490851072 \nu^{5} - 573830616800 \nu^{4} + \cdots - 22\!\cdots\!76 ) / 11\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 17740992 \nu^{7} - 62093472 \nu^{6} - 344236276608 \nu^{5} + 860745925200 \nu^{4} + \cdots + 61\!\cdots\!64 ) / 12\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1366928300 \nu^{7} - 340749556860364 \nu^{6} + \cdots + 33\!\cdots\!76 ) / 11\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 591904 \nu^{7} - 2071664 \nu^{6} - 8207041328 \nu^{5} + 20522782480 \nu^{4} + 37939744551952 \nu^{3} - 56930140646240 \nu^{2} + \cdots + 29\!\cdots\!28 ) / 6310088381691 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 32790319296 \nu^{7} + 114766117536 \nu^{6} + 605965096600104 \nu^{5} + \cdots - 32\!\cdots\!32 ) / 12\!\cdots\!25 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{4} - 27\beta_{3} + 432 ) / 864 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{4} - 14\beta_{3} + 4\beta_{2} - 864\beta _1 + 1996056 ) / 432 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 6\beta_{7} + 27\beta_{6} - 4621\beta_{4} - 187131\beta_{3} + 6\beta_{2} - 1296\beta _1 + 2993976 ) / 432 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 12 \beta_{7} + 56 \beta_{6} - 16 \beta_{5} - 9241 \beta_{4} - 378868 \beta_{3} + 36968 \beta_{2} - 23954400 \beta _1 + 9224770968 ) / 432 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 92440 \beta_{7} + 1247594 \beta_{6} - 40 \beta_{5} - 21358241 \beta_{4} - 1441382231 \beta_{3} + 92410 \beta_{2} - 59883840 \beta _1 + 23056937496 ) / 432 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 277290 \beta_{7} + 3788848 \beta_{6} - 369728 \beta_{5} - 64051621 \beta_{4} - 4355213814 \beta_{3} + 256299084 \beta_{2} - 276862944240 \beta _1 + 42641387030520 ) / 432 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 897693902 \beta_{7} + 20184591658 \beta_{6} - 1293908 \beta_{5} - 98738570505 \beta_{4} - 9327936462583 \beta_{3} + 896723366 \beta_{2} + \cdots + 149164158818520 ) / 432 \) 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
−66.2477 − 0.707107i
69.6972 − 0.707107i
67.2477 + 0.707107i
−68.6972 + 0.707107i
−66.2477 + 0.707107i
69.6972 + 0.707107i
67.2477 − 0.707107i
−68.6972 − 0.707107i
−39.1918 + 22.6274i 0 1024.00 − 1773.62i −23434.3 − 13529.8i 0 −19253.5 − 33348.0i 92681.9i 0 1.22458e6
53.2 −39.1918 + 22.6274i 0 1024.00 − 1773.62i 12529.2 + 7233.73i 0 −48617.5 − 84208.1i 92681.9i 0 −654723.
53.3 39.1918 − 22.6274i 0 1024.00 − 1773.62i −12529.2 − 7233.73i 0 −48617.5 − 84208.1i − 92681.9i 0 −654723.
53.4 39.1918 − 22.6274i 0 1024.00 − 1773.62i 23434.3 + 13529.8i 0 −19253.5 − 33348.0i − 92681.9i 0 1.22458e6
107.1 −39.1918 − 22.6274i 0 1024.00 + 1773.62i −23434.3 + 13529.8i 0 −19253.5 + 33348.0i − 92681.9i 0 1.22458e6
107.2 −39.1918 − 22.6274i 0 1024.00 + 1773.62i 12529.2 − 7233.73i 0 −48617.5 + 84208.1i − 92681.9i 0 −654723.
107.3 39.1918 + 22.6274i 0 1024.00 + 1773.62i −12529.2 + 7233.73i 0 −48617.5 + 84208.1i 92681.9i 0 −654723.
107.4 39.1918 + 22.6274i 0 1024.00 + 1773.62i 23434.3 − 13529.8i 0 −19253.5 + 33348.0i 92681.9i 0 1.22458e6
\(n\): 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 162.13.d.c 8
3.b odd 2 1 inner 162.13.d.c 8
9.c even 3 1 54.13.b.b ✓ 4
9.c even 3 1 inner 162.13.d.c 8
9.d odd 6 1 54.13.b.b ✓ 4
9.d odd 6 1 inner 162.13.d.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.13.b.b ✓ 4 9.c even 3 1
54.13.b.b ✓ 4 9.d odd 6 1
162.13.d.c 8 1.a even 1 1 trivial
162.13.d.c 8 3.b odd 2 1 inner
162.13.d.c 8 9.c even 3 1 inner
162.13.d.c 8 9.d odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 941530752 T_{5}^{6} + \cdots + 23\!\cdots\!00 \) acting on \(S_{13}^{\mathrm{new}}(162, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2048 T^{2} + 4194304)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 941530752 T^{6} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{4} + 135742 T^{3} + \cdots + 14\!\cdots\!25)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 3058161293952 T^{6} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{4} + 348142 T^{3} + \cdots + 14\!\cdots\!25)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 886383248261760 T^{2} + \cdots + 57\!\cdots\!36)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 43938482 T - 71954800437263)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 23\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 29\!\cdots\!16 \) Copy content Toggle raw display
$31$ \( (T^{4} - 1191267068 T^{3} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 286643570 T - 18\!\cdots\!75)^{4} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 86\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{4} - 7558366172 T^{3} + \cdots + 26\!\cdots\!04)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 14\!\cdots\!24)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 25\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( (T^{4} + 29181433198 T^{3} + \cdots + 39\!\cdots\!25)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 154487577550 T^{3} + \cdots + 31\!\cdots\!21)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 58\!\cdots\!84)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 89435351714 T - 41\!\cdots\!27)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} - 452549584418 T^{3} + \cdots + 24\!\cdots\!89)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 40\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 60\!\cdots\!44)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 2835640618178 T^{3} + \cdots + 40\!\cdots\!25)^{2} \) Copy content Toggle raw display
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