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Properties

Label 196.9.h.b
Level $196$
Weight $9$
Character orbit 196.h
Analytic conductor $79.846$
Analytic rank $0$
Dimension $12$
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] = [196,9,Mod(117,196)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(196, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("196.117");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 196 = 2^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 196.h (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: \(79.8462075720\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 720x^{10} + 409912x^{8} + 71803008x^{6} + 9498639424x^{4} + 342190245888x^{2} + 9948826238976 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{26}\cdot 3^{7}\cdot 7^{8} \)
Twist minimal: no (minimal twist has level 28)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + (\beta_{6} + \beta_{5} + \cdots - 3 \beta_{2}) q^{5}+ \cdots + (\beta_{7} - 3039 \beta_1 + 3039) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + (\beta_{6} + \beta_{5} + \cdots - 3 \beta_{2}) q^{5}+ \cdots + (3849 \beta_{8} + 5217 \beta_{4} - 99316302) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 18234 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 18234 q^{9} - 24492 q^{11} - 306816 q^{15} + 11604 q^{23} + 678714 q^{25} + 2528664 q^{29} - 3184332 q^{37} - 5634240 q^{39} - 15566760 q^{43} - 6877824 q^{51} + 8340660 q^{53} - 35901696 q^{57} - 84095232 q^{65} - 16579500 q^{67} - 124177704 q^{71} - 186114540 q^{79} - 131284998 q^{81} - 526421760 q^{85} - 638128512 q^{93} - 85912896 q^{95} - 1191795624 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 720x^{10} + 409912x^{8} + 71803008x^{6} + 9498639424x^{4} + 342190245888x^{2} + 9948826238976 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 694852079 \nu^{10} - 474744671280 \nu^{8} - 270282691241288 \nu^{6} + \cdots - 45\!\cdots\!60 ) / 17\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4849745401537 \nu^{11} + \cdots - 10\!\cdots\!84 \nu ) / 45\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 6050433751297 \nu^{11} + \cdots + 32\!\cdots\!08 \nu ) / 45\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 24300 \nu^{10} - 13834530 \nu^{8} - 7876305363 \nu^{6} - 320579080560 \nu^{4} + \cdots + 34\!\cdots\!28 ) / 4070568422468 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 8633723452673 \nu^{11} + \cdots - 17\!\cdots\!08 \nu ) / 57\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 9234067627553 \nu^{11} + \cdots + 52\!\cdots\!76 \nu ) / 57\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2941993195 \nu^{10} - 2070691811280 \nu^{8} + \cdots - 98\!\cdots\!12 ) / 89\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 10869930 \nu^{10} + 6188492703 \nu^{8} + 3285796593234 \nu^{6} + 143402146714056 \nu^{4} + \cdots - 36\!\cdots\!96 ) / 187246147433528 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 2743489059209 \nu^{10} + \cdots - 10\!\cdots\!16 ) / 36\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 210393334975939 \nu^{11} + \cdots + 22\!\cdots\!04 \nu ) / 28\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 665995508664019 \nu^{11} + \cdots + 60\!\cdots\!12 \nu ) / 28\!\cdots\!16 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} - \beta_{5} - 4\beta_{3} + 4\beta_{2} ) / 84 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{4} - 10080\beta_1 ) / 42 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{11} - \beta_{10} + 325\beta_{6} + 650\beta_{5} - 2808\beta_{3} - 5616\beta_{2} ) / 84 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 23\beta_{9} - 299\beta_{7} + 2109968\beta _1 - 2109968 ) / 21 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -69\beta_{11} + 138\beta_{10} - 72302\beta_{6} - 36151\beta_{5} + 783248\beta_{3} + 391624\beta_{2} ) / 21 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -16560\beta_{8} - 161036\beta_{4} + 1038635136 ) / 21 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 22558 \beta_{11} - 22558 \beta_{10} + 18002614 \beta_{6} - 18002614 \beta_{5} + \cdots + 208964880 \beta_{2} ) / 21 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -9427976\beta_{9} + 9427976\beta_{8} + 85085096\beta_{7} + 85085096\beta_{4} - 534808136576\beta_1 ) / 21 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 19089264 \beta_{11} - 9544632 \beta_{10} + 9296209192 \beta_{6} + 18592418384 \beta_{5} + \cdots - 220364881280 \beta_{2} ) / 21 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 5064127488\beta_{9} - 44733894176\beta_{7} + 279037620126720\beta _1 - 279037620126720 ) / 21 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 4642500880 \beta_{11} + 9285001760 \beta_{10} - 9708459294368 \beta_{6} + \cdots + 57896426381184 \beta_{2} ) / 21 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/196\mathbb{Z}\right)^\times\).

\(n\) \(99\) \(101\)
\(\chi(n)\) \(1\) \(1 - \beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
117.1
−11.4531 + 19.8373i
6.26546 − 10.8521i
3.09370 − 5.35844i
−3.09370 + 5.35844i
−6.26546 + 10.8521i
11.4531 − 19.8373i
−11.4531 − 19.8373i
6.26546 + 10.8521i
3.09370 + 5.35844i
−3.09370 − 5.35844i
−6.26546 − 10.8521i
11.4531 + 19.8373i
0 −127.153 + 73.4116i 0 834.906 + 482.033i 0 0 0 7498.03 − 12987.0i 0
117.2 0 −67.7219 + 39.0993i 0 −594.021 − 342.958i 0 0 0 −222.993 + 386.236i 0
117.3 0 −29.0850 + 16.7923i 0 −288.955 − 166.829i 0 0 0 −2716.54 + 4705.19i 0
117.4 0 29.0850 − 16.7923i 0 288.955 + 166.829i 0 0 0 −2716.54 + 4705.19i 0
117.5 0 67.7219 − 39.0993i 0 594.021 + 342.958i 0 0 0 −222.993 + 386.236i 0
117.6 0 127.153 − 73.4116i 0 −834.906 − 482.033i 0 0 0 7498.03 − 12987.0i 0
129.1 0 −127.153 − 73.4116i 0 834.906 − 482.033i 0 0 0 7498.03 + 12987.0i 0
129.2 0 −67.7219 − 39.0993i 0 −594.021 + 342.958i 0 0 0 −222.993 − 386.236i 0
129.3 0 −29.0850 − 16.7923i 0 −288.955 + 166.829i 0 0 0 −2716.54 − 4705.19i 0
129.4 0 29.0850 + 16.7923i 0 288.955 − 166.829i 0 0 0 −2716.54 − 4705.19i 0
129.5 0 67.7219 + 39.0993i 0 594.021 − 342.958i 0 0 0 −222.993 − 386.236i 0
129.6 0 127.153 + 73.4116i 0 −834.906 + 482.033i 0 0 0 7498.03 + 12987.0i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 117.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 196.9.h.b 12
7.b odd 2 1 inner 196.9.h.b 12
7.c even 3 1 28.9.b.a ✓ 6
7.c even 3 1 inner 196.9.h.b 12
7.d odd 6 1 28.9.b.a ✓ 6
7.d odd 6 1 inner 196.9.h.b 12
21.g even 6 1 252.9.d.b 6
21.h odd 6 1 252.9.d.b 6
28.f even 6 1 112.9.c.d 6
28.g odd 6 1 112.9.c.d 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.9.b.a ✓ 6 7.c even 3 1
28.9.b.a ✓ 6 7.d odd 6 1
112.9.c.d 6 28.f even 6 1
112.9.c.d 6 28.g odd 6 1
196.9.h.b 12 1.a even 1 1 trivial
196.9.h.b 12 7.b odd 2 1 inner
196.9.h.b 12 7.c even 3 1 inner
196.9.h.b 12 7.d odd 6 1 inner
252.9.d.b 6 21.g even 6 1
252.9.d.b 6 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} - 28800 T_{3}^{10} + 666406368 T_{3}^{8} - 4397999929344 T_{3}^{6} + \cdots + 22\!\cdots\!84 \) acting on \(S_{9}^{\mathrm{new}}(196, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 22\!\cdots\!84 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots + 49\!\cdots\!64)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots + 11\!\cdots\!68)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots + 36\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} + \cdots - 10\!\cdots\!44)^{4} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 97\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 61\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} + \cdots - 76\!\cdots\!00)^{4} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 52\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 49\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{3} + \cdots - 22\!\cdots\!40)^{4} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 70\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 29\!\cdots\!24)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots + 50\!\cdots\!32)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 12\!\cdots\!28)^{2} \) Copy content Toggle raw display
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