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Properties

Label 112.12.i.c
Level $112$
Weight $12$
Character orbit 112.i
Analytic conductor $86.054$
Analytic rank $0$
Dimension $12$
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] = [112,12,Mod(65,112)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(112, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("112.65");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 112.i (of order \(3\), 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: \(86.0544362227\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} + 1846 x^{10} + 9475 x^{9} + 2735534 x^{8} + 11305015 x^{7} + 1247863105 x^{6} + \cdots + 4089842896896 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{32}\cdot 3^{3}\cdot 7^{6} \)
Twist minimal: no (minimal twist has level 7)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} - 41 \beta_1) q^{3} + ( - \beta_{11} - \beta_{9} + \cdots - 1463) q^{5}+ \cdots + ( - 26 \beta_{11} + 15 \beta_{10} + \cdots - 28728) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} - 41 \beta_1) q^{3} + ( - \beta_{11} - \beta_{9} + \cdots - 1463) q^{5}+ \cdots + ( - 615584 \beta_{9} + \cdots - 30582413412) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 244 q^{3} - 8782 q^{5} + 504 q^{7} - 172348 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 244 q^{3} - 8782 q^{5} + 504 q^{7} - 172348 q^{9} + 1001572 q^{11} + 3864504 q^{13} + 1286512 q^{15} - 6704802 q^{17} - 4192212 q^{19} + 44745358 q^{21} + 33871872 q^{23} + 13695456 q^{25} - 73859384 q^{27} - 255125224 q^{29} + 331783920 q^{31} - 80899438 q^{33} - 1407354844 q^{35} - 833082774 q^{37} - 737605904 q^{39} + 3104076808 q^{41} + 1722177552 q^{43} - 7406493484 q^{45} + 1327587552 q^{47} + 11976558636 q^{49} + 13921261140 q^{51} + 6725755626 q^{53} - 26323921200 q^{55} - 16884487756 q^{57} + 26237179548 q^{59} - 14411013726 q^{61} - 45955779184 q^{63} - 16224702172 q^{65} + 4241860068 q^{67} + 46750854252 q^{69} + 37335334656 q^{71} + 6005568990 q^{73} - 17116276792 q^{75} - 51928077698 q^{77} - 11712395640 q^{79} - 12455008366 q^{81} + 100821781200 q^{83} + 138884613396 q^{85} - 119455310144 q^{87} - 48633519778 q^{89} + 160908361488 q^{91} + 266530114134 q^{93} + 72161225128 q^{95} - 401308415928 q^{97} - 367357472240 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} + 1846 x^{10} + 9475 x^{9} + 2735534 x^{8} + 11305015 x^{7} + 1247863105 x^{6} + \cdots + 4089842896896 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 25\!\cdots\!01 \nu^{11} + \cdots - 15\!\cdots\!84 ) / 12\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 21\!\cdots\!84 \nu^{11} + \cdots + 99\!\cdots\!39 ) / 20\!\cdots\!05 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 98\!\cdots\!09 \nu^{11} + \cdots + 28\!\cdots\!04 ) / 37\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 73\!\cdots\!97 \nu^{11} + \cdots + 14\!\cdots\!32 ) / 21\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 15\!\cdots\!13 \nu^{11} + \cdots - 92\!\cdots\!60 ) / 10\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 16\!\cdots\!71 \nu^{11} + \cdots + 99\!\cdots\!44 ) / 10\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 12\!\cdots\!27 \nu^{11} + \cdots - 75\!\cdots\!48 ) / 17\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 13\!\cdots\!09 \nu^{11} + \cdots - 43\!\cdots\!68 ) / 15\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 17\!\cdots\!43 \nu^{11} + \cdots + 16\!\cdots\!88 ) / 18\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 24\!\cdots\!31 \nu^{11} + \cdots + 14\!\cdots\!04 ) / 25\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 52\!\cdots\!37 \nu^{11} + \cdots - 31\!\cdots\!48 ) / 28\!\cdots\!60 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + 11\beta _1 + 11 ) / 64 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 8\beta_{10} + 5\beta_{6} + 5\beta_{5} + 39375\beta_1 ) / 64 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -24\beta_{9} - 40\beta_{8} - 1185\beta_{4} + 1137\beta_{3} + 48\beta_{2} - 181547 ) / 64 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 488 \beta_{11} - 11528 \beta_{10} + 488 \beta_{9} - 456 \beta_{8} + 456 \beta_{7} - 8809 \beta_{6} + \cdots - 44516699 ) / 64 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 46576 \beta_{11} - 123312 \beta_{10} + 73968 \beta_{7} - 1453665 \beta_{6} - 1655009 \beta_{5} - 365759035 \beta_1 ) / 64 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 762224 \beta_{9} + 1067888 \beta_{8} + 62721061 \beta_{4} - 15783557 \beta_{3} - 16258184 \beta_{2} + 56858316623 ) / 64 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 67356552 \beta_{11} + 232347792 \beta_{10} + 67356552 \beta_{9} + 112691448 \beta_{8} + \cdots + 665178860915 ) / 64 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 907385112 \beta_{11} + 22879593512 \beta_{10} - 1944560952 \beta_{7} + 27420754649 \beta_{6} + \cdots + 76762131483915 \beta_1 ) / 64 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 90884333472 \beta_{9} - 163349360896 \beta_{8} - 3640214102937 \beta_{4} + 2707524842745 \beta_{3} + \cdots - 11\!\cdots\!71 ) / 64 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 955450205984 \beta_{11} - 32305021839080 \beta_{10} + 955450205984 \beta_{9} - 3253017353568 \beta_{8} + \cdots - 10\!\cdots\!79 ) / 64 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 120638637710200 \beta_{11} - 650854588525104 \beta_{10} + 233861824957128 \beta_{7} + \cdots - 19\!\cdots\!75 \beta_1 ) / 64 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/112\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\) \(85\)
\(\chi(n)\) \(1\) \(-1 - \beta_{1}\) \(1\)

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
−10.4934 + 18.1751i
11.5012 − 19.9207i
−1.81957 + 3.15159i
−17.5066 + 30.3223i
19.2454 − 33.3340i
−0.427084 + 0.739732i
−10.4934 − 18.1751i
11.5012 + 19.9207i
−1.81957 − 3.15159i
−17.5066 − 30.3223i
19.2454 + 33.3340i
−0.427084 − 0.739732i
0 −311.801 − 540.056i 0 −5294.67 + 9170.64i 0 −23661.4 + 37649.3i 0 −105867. + 183367.i 0
65.2 0 −205.214 − 355.441i 0 −41.2685 + 71.4792i 0 43319.9 − 10035.5i 0 4347.82 − 7530.65i 0
65.3 0 23.0866 + 39.9871i 0 4304.93 − 7456.37i 0 −44340.4 + 3354.68i 0 87507.5 − 151567.i 0
65.4 0 82.0189 + 142.061i 0 1670.28 − 2893.01i 0 26385.1 − 35793.2i 0 75119.3 − 130110.i 0
65.5 0 167.747 + 290.547i 0 −539.162 + 933.856i 0 −44465.4 + 388.777i 0 32295.3 − 55937.0i 0
65.6 0 366.163 + 634.213i 0 −4491.11 + 7778.83i 0 43014.2 + 11274.1i 0 −179577. + 311037.i 0
81.1 0 −311.801 + 540.056i 0 −5294.67 − 9170.64i 0 −23661.4 − 37649.3i 0 −105867. − 183367.i 0
81.2 0 −205.214 + 355.441i 0 −41.2685 − 71.4792i 0 43319.9 + 10035.5i 0 4347.82 + 7530.65i 0
81.3 0 23.0866 − 39.9871i 0 4304.93 + 7456.37i 0 −44340.4 − 3354.68i 0 87507.5 + 151567.i 0
81.4 0 82.0189 − 142.061i 0 1670.28 + 2893.01i 0 26385.1 + 35793.2i 0 75119.3 + 130110.i 0
81.5 0 167.747 − 290.547i 0 −539.162 − 933.856i 0 −44465.4 − 388.777i 0 32295.3 + 55937.0i 0
81.6 0 366.163 − 634.213i 0 −4491.11 − 7778.83i 0 43014.2 − 11274.1i 0 −179577. − 311037.i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 112.12.i.c 12
4.b odd 2 1 7.12.c.a ✓ 12
7.c even 3 1 inner 112.12.i.c 12
12.b even 2 1 63.12.e.b 12
28.d even 2 1 49.12.c.i 12
28.f even 6 1 49.12.a.f 6
28.f even 6 1 49.12.c.i 12
28.g odd 6 1 7.12.c.a ✓ 12
28.g odd 6 1 49.12.a.g 6
84.n even 6 1 63.12.e.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.12.c.a ✓ 12 4.b odd 2 1
7.12.c.a ✓ 12 28.g odd 6 1
49.12.a.f 6 28.f even 6 1
49.12.a.g 6 28.g odd 6 1
49.12.c.i 12 28.d even 2 1
49.12.c.i 12 28.f even 6 1
63.12.e.b 12 12.b even 2 1
63.12.e.b 12 84.n even 6 1
112.12.i.c 12 1.a even 1 1 trivial
112.12.i.c 12 7.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} - 244 T_{3}^{11} + 647383 T_{3}^{10} - 70867572 T_{3}^{9} + 309342730734 T_{3}^{8} + \cdots + 22\!\cdots\!09 \) acting on \(S_{12}^{\mathrm{new}}(112, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 22\!\cdots\!09 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 59\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 59\!\cdots\!49 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 14\!\cdots\!25 \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots + 32\!\cdots\!44)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 29\!\cdots\!69 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 41\!\cdots\!21 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 77\!\cdots\!09 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 28\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 18\!\cdots\!01 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 14\!\cdots\!25 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 95\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots - 24\!\cdots\!68)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 59\!\cdots\!25 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 22\!\cdots\!01 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 24\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 19\!\cdots\!69 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 93\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 10\!\cdots\!44)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 13\!\cdots\!89 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 20\!\cdots\!09 \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 24\!\cdots\!56)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 95\!\cdots\!41 \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots - 18\!\cdots\!64)^{2} \) Copy content Toggle raw display
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