Properties

Label 162.11.d.c
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: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 862x^{6} + 2600x^{5} + 278207x^{4} - 560752x^{3} - 39833846x^{2} + 40114656x + 2136938124 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{20}\cdot 3^{12} \)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} - 512 \beta_1 q^{4} + ( - 49 \beta_{6} - \beta_{4} + 49 \beta_{3}) q^{5} + (5 \beta_{7} + 5 \beta_{5} - 2879 \beta_1 - 2879) q^{7} - 512 \beta_{6} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} - 512 \beta_1 q^{4} + ( - 49 \beta_{6} - \beta_{4} + 49 \beta_{3}) q^{5} + (5 \beta_{7} + 5 \beta_{5} - 2879 \beta_1 - 2879) q^{7} - 512 \beta_{6} q^{8} + ( - 32 \beta_{5} - 24960) q^{10} + ( - 6049 \beta_{3} - \beta_{2}) q^{11} + ( - 37 \beta_{7} - 337765 \beta_1) q^{13} + ( - 2859 \beta_{6} + 80 \beta_{4} + 2859 \beta_{3}) q^{14} + ( - 262144 \beta_1 - 262144) q^{16} + ( - 139 \beta_{6} + 245 \beta_{4} + 245 \beta_{2}) q^{17} + (1178 \beta_{5} - 212317) q^{19} + (25088 \beta_{3} + 512 \beta_{2}) q^{20} + ( - 32 \beta_{7} - 3096960 \beta_1) q^{22} + (161593 \beta_{6} - 1271 \beta_{4} - 161593 \beta_{3}) q^{23} + (3120 \beta_{7} + 3120 \beta_{5} + 11629895 \beta_1 + 11629895) q^{25} + ( - 337913 \beta_{6} - 592 \beta_{4} - 592 \beta_{2}) q^{26} + (2560 \beta_{5} - 1474048) q^{28} + (93542 \beta_{3} + 2534 \beta_{2}) q^{29} + ( - 8304 \beta_{7} - 10782958 \beta_1) q^{31} + ( - 262144 \beta_{6} + 262144 \beta_{3}) q^{32} + (7840 \beta_{7} + 7840 \beta_{5} - 102528 \beta_1 - 102528) q^{34} + ( - 3012829 \beta_{6} - 1021 \beta_{4} - 1021 \beta_{2}) q^{35} + (31273 \beta_{5} - 7906525) q^{37} + (207605 \beta_{3} - 18848 \beta_{2}) q^{38} + (16384 \beta_{7} + 12779520 \beta_1) q^{40} + (3705914 \beta_{6} + 11066 \beta_{4} - 3705914 \beta_{3}) q^{41} + ( - 5424 \beta_{7} - 5424 \beta_{5} - 135962378 \beta_1 - 135962378) q^{43} + ( - 3097088 \beta_{6} - 512 \beta_{4} - 512 \beta_{2}) q^{44} + ( - 40672 \beta_{5} + 82898304) q^{46} + (13854447 \beta_{3} + 16383 \beta_{2}) q^{47} + ( - 28790 \beta_{7} - 21952608 \beta_1) q^{49} + (11642375 \beta_{6} + 49920 \beta_{4} - 11642375 \beta_{3}) q^{50} + ( - 18944 \beta_{7} - 18944 \beta_{5} - 172935680 \beta_1 - 172935680) q^{52} + ( - 26256578 \beta_{6} + 30814 \beta_{4} + 30814 \beta_{2}) q^{53} + ( - 195120 \beta_{5} - 171155520) q^{55} + (1463808 \beta_{3} - 40960 \beta_{2}) q^{56} + (81088 \beta_{7} + 47569152 \beta_1) q^{58} + (7313221 \beta_{6} - 255995 \beta_{4} - 7313221 \beta_{3}) q^{59} + ( - 232413 \beta_{7} - 232413 \beta_{5} + \cdots - 406320875) q^{61}+ \cdots + ( - 22067768 \beta_{6} - 460640 \beta_{4} + \cdots - 460640 \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2048 q^{4} - 11516 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2048 q^{4} - 11516 q^{7} - 199680 q^{10} + 1351060 q^{13} - 1048576 q^{16} - 1698536 q^{19} + 12387840 q^{22} + 46519580 q^{25} - 11792384 q^{28} + 43131832 q^{31} - 410112 q^{34} - 63252200 q^{37} - 51118080 q^{40} - 543849512 q^{43} + 663186432 q^{46} + 87810432 q^{49} - 691742720 q^{52} - 1369244160 q^{55} - 190276608 q^{58} - 1625283500 q^{61} - 1073741824 q^{64} + 6665359156 q^{67} - 6169751040 q^{70} - 21976117832 q^{73} - 434825216 q^{76} + 1634877796 q^{79} + 15168092160 q^{82} + 19755152640 q^{85} - 6342574080 q^{88} + 7152849320 q^{91} - 28365519360 q^{94} - 21000351740 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} - 862x^{6} + 2600x^{5} + 278207x^{4} - 560752x^{3} - 39833846x^{2} + 40114656x + 2136938124 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{6} - 6\nu^{5} - 2161\nu^{4} + 4332\nu^{3} + 654065\nu^{2} - 656232\nu - 60780660 ) / 1489506 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 69136 \nu^{7} + 159893728 \nu^{6} - 543088808 \nu^{5} - 172869318992 \nu^{4} + 369370905424 \nu^{3} + 111966419712928 \nu^{2} + \cdots - 17\!\cdots\!56 ) / 552136786857 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 276544 \nu^{7} + 967904 \nu^{6} + 250726784 \nu^{5} - 629236720 \nu^{4} - 90067882240 \nu^{3} + 135731544032 \nu^{2} + \cdots - 5295548274240 ) / 552136786857 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2957980 \nu^{7} + 34538925208 \nu^{6} - 105562988192 \nu^{5} - 22417988016200 \nu^{4} + 45428450511820 \nu^{3} + \cdots - 34\!\cdots\!80 ) / 552136786857 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 414816 \nu^{7} - 1451856 \nu^{6} - 376090176 \nu^{5} + 943855080 \nu^{4} + 135101823360 \nu^{3} - 203597316048 \nu^{2} + \cdots + 14568963853644 ) / 61348531873 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 14096 \nu^{7} + 49336 \nu^{6} + 9209944 \nu^{5} - 23148200 \nu^{4} - 2014517000 \nu^{3} + 3044948368 \nu^{2} + 147278425056 \nu - 74147476704 ) / 642766923 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 35955552 \nu^{7} + 125844432 \nu^{6} + 31065504468 \nu^{5} - 77978372250 \nu^{4} - 8395277671272 \nu^{3} + \cdots - 365904406474344 ) / 61348531873 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -2\beta_{5} - 27\beta_{3} + 216 ) / 432 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{5} - 13\beta_{3} + 2\beta_{2} - 432\beta _1 + 46764 ) / 216 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 12\beta_{7} - 54\beta_{6} - 434\beta_{5} - 17535\beta_{3} + 6\beta_{2} - 1296\beta _1 + 140184 ) / 432 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 12 \beta_{7} - 52 \beta_{6} - 433 \beta_{5} + 8 \beta_{4} - 17306 \beta_{3} + 876 \beta_{2} - 562032 \beta _1 + 10170252 ) / 216 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 8720 \beta_{7} - 117008 \beta_{6} - 94570 \beta_{5} + 40 \beta_{4} - 6355757 \beta_{3} + 4370 \beta_{2} - 2808000 \beta _1 + 50617656 ) / 432 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 13050 \beta_{7} - 173216 \beta_{6} - 140773 \beta_{5} + 8704 \beta_{4} - 9420507 \beta_{3} + 292510 \beta_{2} - 307939320 \beta _1 + 2219464476 ) / 216 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 4034380 \beta_{7} - 89473090 \beta_{6} - 20656554 \beta_{5} + 60788 \beta_{4} - 1943503927 \beta_{3} + 2032282 \beta_{2} - 2145748752 \beta _1 + 15359253048 ) / 432 \) Copy content Toggle raw display

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
13.9807 + 0.707107i
−15.4302 + 0.707107i
−12.9807 0.707107i
16.4302 0.707107i
13.9807 0.707107i
−15.4302 0.707107i
−12.9807 + 0.707107i
16.4302 + 0.707107i
−19.5959 + 11.3137i 0 256.000 443.405i −2934.95 1694.49i 0 −9380.44 16247.4i 11585.2i 0 76684.0
53.2 −19.5959 + 11.3137i 0 256.000 443.405i 4845.55 + 2797.58i 0 6501.44 + 11260.8i 11585.2i 0 −126604.
53.3 19.5959 11.3137i 0 256.000 443.405i −4845.55 2797.58i 0 6501.44 + 11260.8i 11585.2i 0 −126604.
53.4 19.5959 11.3137i 0 256.000 443.405i 2934.95 + 1694.49i 0 −9380.44 16247.4i 11585.2i 0 76684.0
107.1 −19.5959 11.3137i 0 256.000 + 443.405i −2934.95 + 1694.49i 0 −9380.44 + 16247.4i 11585.2i 0 76684.0
107.2 −19.5959 11.3137i 0 256.000 + 443.405i 4845.55 2797.58i 0 6501.44 11260.8i 11585.2i 0 −126604.
107.3 19.5959 + 11.3137i 0 256.000 + 443.405i −4845.55 + 2797.58i 0 6501.44 11260.8i 11585.2i 0 −126604.
107.4 19.5959 + 11.3137i 0 256.000 + 443.405i 2934.95 1694.49i 0 −9380.44 + 16247.4i 11585.2i 0 76684.0
\(n\): 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.c 8
3.b odd 2 1 inner 162.11.d.c 8
9.c even 3 1 54.11.b.b 4
9.c even 3 1 inner 162.11.d.c 8
9.d odd 6 1 54.11.b.b 4
9.d odd 6 1 inner 162.11.d.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.11.b.b 4 9.c even 3 1
54.11.b.b 4 9.d odd 6 1
162.11.d.c 8 1.a even 1 1 trivial
162.11.d.c 8 3.b odd 2 1 inner
162.11.d.c 8 9.c even 3 1 inner
162.11.d.c 8 9.d odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 42791040 T_{5}^{6} + \cdots + 12\!\cdots\!00 \) 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
$5$ \( T^{8} - 42791040 T^{6} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{4} + 5758 T^{3} + \cdots + 59\!\cdots\!81)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 37505831040 T^{6} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{4} - 675530 T^{3} + \cdots + 10\!\cdots\!25)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 2422496398464 T^{2} + \cdots + 14\!\cdots\!24)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 424634 T - 13955764933751)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} - 92039316379776 T^{6} + \cdots + 13\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( T^{8} - 267980575117824 T^{6} + \cdots + 24\!\cdots\!56 \) Copy content Toggle raw display
$31$ \( (T^{4} - 21565916 T^{3} + \cdots + 33\!\cdots\!16)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 15813050 T - 98\!\cdots\!15)^{4} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{4} + 271924756 T^{3} + \cdots + 33\!\cdots\!76)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 11\!\cdots\!44)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 28\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( (T^{4} + 812641750 T^{3} + \cdots + 14\!\cdots\!25)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 3332679578 T^{3} + \cdots + 46\!\cdots\!41)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 29\!\cdots\!44)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 5494029458 T + 73\!\cdots\!01)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} - 817438898 T^{3} + \cdots + 10\!\cdots\!01)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 72\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 30\!\cdots\!24)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 10500175870 T^{3} + \cdots + 71\!\cdots\!25)^{2} \) Copy content Toggle raw display
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