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Properties

Label 112.10.i.c
Level $112$
Weight $10$
Character orbit 112.i
Analytic conductor $57.684$
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] = [112,10,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, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("112.65");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
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: \(57.6840136504\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + 430 x^{8} + 61 x^{7} + 146753 x^{6} + 23608 x^{5} + 16136944 x^{4} + 30575648 x^{3} + \cdots + 761760000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{18}\cdot 3^{3}\cdot 7^{4} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{7} - 32 \beta_{2}) q^{3} + (\beta_{9} - \beta_{6} - 307 \beta_{2} + 307) q^{5} + ( - \beta_{9} - 5 \beta_{8} + \cdots - 604) q^{7}+ \cdots + ( - 7 \beta_{9} - 8 \beta_{8} + \cdots - 7120) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{7} - 32 \beta_{2}) q^{3} + (\beta_{9} - \beta_{6} - 307 \beta_{2} + 307) q^{5} + ( - \beta_{9} - 5 \beta_{8} + \cdots - 604) q^{7}+ \cdots + (309295 \beta_{6} + 1021071 \beta_{5} + \cdots + 187313765) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 161 q^{3} + 1533 q^{5} + 1036 q^{7} - 35734 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 161 q^{3} + 1533 q^{5} + 1036 q^{7} - 35734 q^{9} - 42213 q^{11} - 319676 q^{13} - 151394 q^{15} + 324681 q^{17} + 16121 q^{19} - 1557857 q^{21} - 2638863 q^{23} - 1304092 q^{25} + 18331558 q^{27} + 15292500 q^{29} - 19179237 q^{31} + 1689359 q^{33} + 43746759 q^{35} + 39566985 q^{37} + 44299486 q^{39} - 53436852 q^{41} - 101835992 q^{43} + 85098230 q^{45} - 32509659 q^{47} - 49024598 q^{49} - 44168403 q^{51} - 25714707 q^{53} + 144695222 q^{55} - 121710346 q^{57} - 46776513 q^{59} - 113075039 q^{61} - 318071530 q^{63} - 338113566 q^{65} + 126707879 q^{67} + 1323616182 q^{69} + 1188736032 q^{71} - 859257651 q^{73} + 169061732 q^{75} + 1911891891 q^{77} + 527065417 q^{79} + 551662715 q^{81} + 144863208 q^{83} - 1197360222 q^{85} + 340781350 q^{87} + 1661554797 q^{89} - 726641384 q^{91} - 423057489 q^{93} + 1197123495 q^{95} + 869770188 q^{97} + 1900777180 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} + 430 x^{8} + 61 x^{7} + 146753 x^{6} + 23608 x^{5} + 16136944 x^{4} + 30575648 x^{3} + \cdots + 761760000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 39319426631371 \nu^{9} + 453020550895311 \nu^{8} + \cdots - 59\!\cdots\!00 ) / 75\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 439915122044593 \nu^{9} - 359425911506973 \nu^{8} + \cdots + 45\!\cdots\!00 ) / 45\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 807596969249 \nu^{9} - 294192051695673 \nu^{8} + \cdots - 61\!\cdots\!60 ) / 46\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 439915122044593 \nu^{9} - 359425911506973 \nu^{8} + \cdots + 23\!\cdots\!00 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4726405667711 \nu^{9} + 5835972479847 \nu^{8} - 22347050770504 \nu^{7} + \cdots - 27\!\cdots\!40 ) / 77\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 10228081391315 \nu^{9} + \cdots + 13\!\cdots\!84 ) / 69\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 47\!\cdots\!43 \nu^{9} + \cdots + 11\!\cdots\!00 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 20\!\cdots\!55 \nu^{9} + \cdots + 94\!\cdots\!00 ) / 15\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 10\!\cdots\!87 \nu^{9} + \cdots + 10\!\cdots\!00 ) / 53\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} - 2\beta_{2} + 2 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{8} + \beta_{7} - \beta_{3} - 1375\beta_{2} - \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 55\beta_{5} - 240\beta_{4} - \beta_{3} - 240\beta _1 - 1285 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 56 \beta_{9} - 338 \beta_{8} - 282 \beta_{7} - 56 \beta_{6} - 282 \beta_{5} - 289 \beta_{4} + \cdots - 350852 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 56\beta_{9} - 583\beta_{8} + 23497\beta_{7} + 583\beta_{3} + 353373\beta_{2} + 65593\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 24080\beta_{6} + 70129\beta_{5} + 80342\beta_{4} + 107929\beta_{3} + 80342\beta _1 + 99053241 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 37800 \beta_{9} + 230772 \beta_{8} - 8028612 \beta_{7} + 37800 \beta_{6} - 8028612 \beta_{5} + \cdots + 99991970 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 8259384 \beta_{9} + 33731905 \beta_{8} + 17331073 \beta_{7} - 33731905 \beta_{3} + \cdots - 24089881 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 16400832 \beta_{6} + 2567150827 \beta_{5} - 5756927664 \beta_{4} - 81699181 \beta_{3} + \cdots - 31091870113 ) / 8 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(15\) \(17\) \(85\)
\(\chi(n)\) \(1\) \(-1 + \beta_{2}\) \(1\)

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
−8.71912 + 15.1020i
5.89912 − 10.2176i
−0.371984 + 0.644295i
8.80924 − 15.2580i
−5.11725 + 8.86334i
−8.71912 − 15.1020i
5.89912 + 10.2176i
−0.371984 − 0.644295i
8.80924 + 15.2580i
−5.11725 − 8.86334i
0 −113.728 − 196.982i 0 −162.760 + 281.909i 0 −5234.95 + 3598.46i 0 −16026.5 + 27758.7i 0
65.2 0 −104.977 − 181.826i 0 983.791 − 1703.98i 0 5768.52 − 2660.41i 0 −12198.9 + 21129.2i 0
65.3 0 1.70307 + 2.94981i 0 −828.924 + 1435.74i 0 −2822.68 − 5690.88i 0 9835.70 − 17035.9i 0
65.4 0 56.7670 + 98.3234i 0 −239.755 + 415.269i 0 −1428.40 + 6189.77i 0 3396.51 − 5882.92i 0
65.5 0 79.7348 + 138.105i 0 1014.15 − 1756.56i 0 4235.51 + 4734.35i 0 −2873.78 + 4977.54i 0
81.1 0 −113.728 + 196.982i 0 −162.760 − 281.909i 0 −5234.95 − 3598.46i 0 −16026.5 − 27758.7i 0
81.2 0 −104.977 + 181.826i 0 983.791 + 1703.98i 0 5768.52 + 2660.41i 0 −12198.9 − 21129.2i 0
81.3 0 1.70307 − 2.94981i 0 −828.924 − 1435.74i 0 −2822.68 + 5690.88i 0 9835.70 + 17035.9i 0
81.4 0 56.7670 − 98.3234i 0 −239.755 − 415.269i 0 −1428.40 − 6189.77i 0 3396.51 + 5882.92i 0
81.5 0 79.7348 − 138.105i 0 1014.15 + 1756.56i 0 4235.51 − 4734.35i 0 −2873.78 − 4977.54i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.5
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 112.10.i.c 10
4.b odd 2 1 7.10.c.a ✓ 10
7.c even 3 1 inner 112.10.i.c 10
12.b even 2 1 63.10.e.b 10
28.d even 2 1 49.10.c.g 10
28.f even 6 1 49.10.a.f 5
28.f even 6 1 49.10.c.g 10
28.g odd 6 1 7.10.c.a ✓ 10
28.g odd 6 1 49.10.a.e 5
84.n even 6 1 63.10.e.b 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.10.c.a ✓ 10 4.b odd 2 1
7.10.c.a ✓ 10 28.g odd 6 1
49.10.a.e 5 28.g odd 6 1
49.10.a.f 5 28.f even 6 1
49.10.c.g 10 28.d even 2 1
49.10.c.g 10 28.f even 6 1
63.10.e.b 10 12.b even 2 1
63.10.e.b 10 84.n even 6 1
112.10.i.c 10 1.a even 1 1 trivial
112.10.i.c 10 7.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} + 161 T_{3}^{9} + 80035 T_{3}^{8} + 1158738 T_{3}^{7} + 2840894541 T_{3}^{6} + \cdots + 86\!\cdots\!89 \) acting on \(S_{10}^{\mathrm{new}}(112, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + \cdots + 86\!\cdots\!89 \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 10\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 10\!\cdots\!07 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 91\!\cdots\!25 \) Copy content Toggle raw display
$13$ \( (T^{5} + \cdots + 79\!\cdots\!92)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 29\!\cdots\!81 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 39\!\cdots\!01 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 42\!\cdots\!01 \) Copy content Toggle raw display
$29$ \( (T^{5} + \cdots + 73\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 14\!\cdots\!69 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 47\!\cdots\!25 \) Copy content Toggle raw display
$41$ \( (T^{5} + \cdots - 20\!\cdots\!40)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} + \cdots - 72\!\cdots\!36)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 34\!\cdots\!25 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 47\!\cdots\!21 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 79\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 27\!\cdots\!41 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 20\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots - 25\!\cdots\!92)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 87\!\cdots\!29 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 26\!\cdots\!21 \) Copy content Toggle raw display
$83$ \( (T^{5} + \cdots - 39\!\cdots\!36)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 21\!\cdots\!41 \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots - 84\!\cdots\!56)^{2} \) Copy content Toggle raw display
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