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Properties

Label 196.10.a.g
Level $196$
Weight $10$
Character orbit 196.a
Self dual yes
Analytic conductor $100.947$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [196,10,Mod(1,196)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(196, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 10, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("196.1"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Level: \( N \) \(=\) \( 196 = 2^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 196.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(100.947023888\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 23330x^{8} + 114080917x^{6} - 121201507892x^{4} + 31086921022884x^{2} - 2278042380749088 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{4}\cdot 7^{12} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q - \beta_{6} q^{3} + (\beta_{7} - 3 \beta_{6} + 5 \beta_{5}) q^{5} + (\beta_1 + 9355) q^{9} + (2 \beta_{4} - \beta_{3} + \beta_1 + 2775) q^{11} + ( - \beta_{9} + 23 \beta_{7} + \cdots + 451 \beta_{5}) q^{13}+ \cdots + (121878 \beta_{4} + 8397 \beta_{3} + \cdots + 531665905) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 93550 q^{9} + 27740 q^{11} + 912088 q^{15} - 2921096 q^{23} + 3484998 q^{25} - 835748 q^{29} - 31928756 q^{37} - 4930944 q^{39} + 13414340 q^{43} + 206013028 q^{51} + 137301788 q^{53} + 72246332 q^{57}+ \cdots + 5316067012 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 23330x^{8} + 114080917x^{6} - 121201507892x^{4} + 31086921022884x^{2} - 2278042380749088 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 1451498581 \nu^{8} - 34917954627264 \nu^{6} + \cdots + 10\!\cdots\!24 ) / 19\!\cdots\!74 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 15002786537 \nu^{8} + 356174471418256 \nu^{6} + \cdots - 31\!\cdots\!08 ) / 16\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 150283455853 \nu^{8} + \cdots - 19\!\cdots\!04 ) / 23\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4668001487 \nu^{8} - 108051734518480 \nu^{6} + \cdots + 64\!\cdots\!76 ) / 47\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 22306332023 \nu^{9} - 522239233491520 \nu^{7} + \cdots + 11\!\cdots\!16 \nu ) / 12\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 50\!\cdots\!85 \nu^{9} + \cdots - 71\!\cdots\!52 \nu ) / 45\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 43\!\cdots\!27 \nu^{9} + \cdots + 64\!\cdots\!28 \nu ) / 13\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 46\!\cdots\!87 \nu^{9} + \cdots - 19\!\cdots\!08 \nu ) / 45\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 41\!\cdots\!09 \nu^{9} + \cdots + 38\!\cdots\!46 \nu ) / 34\!\cdots\!80 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -6\beta_{9} + \beta_{8} + 71\beta_{7} - 548\beta_{6} - 76\beta_{5} ) / 9408 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -58\beta_{4} - 91\beta_{3} - 85\beta_{2} - 126\beta _1 + 5487228 ) / 1176 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -84522\beta_{9} + 7507\beta_{8} + 574997\beta_{7} - 9107300\beta_{6} - 50057860\beta_{5} ) / 9408 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -188214\beta_{4} - 305767\beta_{3} - 76565\beta_{2} - 258888\beta _1 + 12392182168 ) / 196 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 1385842926 \beta_{9} + 81044489 \beta_{8} + 2455304335 \beta_{7} - 168057290620 \beta_{6} - 983959272332 \beta_{5} ) / 9408 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 19646179460 \beta_{4} - 32489847299 \beta_{3} - 1211464295 \beta_{2} - 23792689284 \beta _1 + 11\!\cdots\!14 ) / 1176 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 23373535569354 \beta_{9} + 1165576997867 \beta_{8} + 1344366896413 \beta_{7} + \cdots - 17\!\cdots\!68 \beta_{5} ) / 9408 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 28037583773468 \beta_{4} - 46643720637885 \beta_{3} + 1149897163235 \beta_{2} + \cdots + 16\!\cdots\!90 ) / 98 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 39\!\cdots\!62 \beta_{9} + \cdots - 29\!\cdots\!28 \beta_{5} ) / 9408 \) Copy content Toggle raw display

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

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−11.8009
13.8427
−31.8528
−130.574
70.2492
−70.2492
130.574
31.8528
−13.8427
11.8009
0 −254.961 0 −1875.78 0 0 0 45322.1 0
1.2 0 −242.271 0 −1124.63 0 0 0 39012.3 0
1.3 0 −115.545 0 1382.29 0 0 0 −6332.31 0
1.4 0 −70.3781 0 162.776 0 0 0 −14729.9 0
1.5 0 −56.4429 0 2188.04 0 0 0 −16497.2 0
1.6 0 56.4429 0 −2188.04 0 0 0 −16497.2 0
1.7 0 70.3781 0 −162.776 0 0 0 −14729.9 0
1.8 0 115.545 0 −1382.29 0 0 0 −6332.31 0
1.9 0 242.271 0 1124.63 0 0 0 39012.3 0
1.10 0 254.961 0 1875.78 0 0 0 45322.1 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(7\) \( +1 \)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 196.10.a.g ✓ 10
7.b odd 2 1 inner 196.10.a.g ✓ 10
7.c even 3 2 196.10.e.i 20
7.d odd 6 2 196.10.e.i 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
196.10.a.g ✓ 10 1.a even 1 1 trivial
196.10.a.g ✓ 10 7.b odd 2 1 inner
196.10.e.i 20 7.c even 3 2
196.10.e.i 20 7.d odd 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} - 145190 T_{3}^{8} + 6598203372 T_{3}^{6} - 97597172782728 T_{3}^{4} + \cdots - 80\!\cdots\!32 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(196))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + \cdots - 80\!\cdots\!32 \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots - 10\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{10} \) Copy content Toggle raw display
$11$ \( (T^{5} + \cdots + 59\!\cdots\!48)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots - 14\!\cdots\!68 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots - 25\!\cdots\!12 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots - 25\!\cdots\!08 \) Copy content Toggle raw display
$23$ \( (T^{5} + \cdots + 87\!\cdots\!88)^{2} \) Copy content Toggle raw display
$29$ \( (T^{5} + \cdots + 12\!\cdots\!92)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots - 13\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{5} + \cdots - 30\!\cdots\!56)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots - 17\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{5} + \cdots - 41\!\cdots\!04)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots - 26\!\cdots\!32 \) Copy content Toggle raw display
$53$ \( (T^{5} + \cdots - 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots - 45\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots - 71\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{5} + \cdots + 58\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots + 22\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots - 47\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots - 56\!\cdots\!32)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots - 11\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots - 17\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots - 50\!\cdots\!72 \) Copy content Toggle raw display
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