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Properties

Label 75.22.b.i
Level $75$
Weight $22$
Character orbit 75.b
Analytic conductor $209.608$
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] = [75,22,Mod(49,75)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(75, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.49");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 75.b (of order \(2\), degree \(1\), 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: \(209.608008215\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 1060747 x^{10} + 401129033703 x^{8} + \cdots + 46\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{39}\cdot 3^{12}\cdot 5^{14}\cdot 7^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{6} - 15 \beta_{5}) q^{2} - 59049 \beta_{5} q^{3} + (\beta_{2} + 243 \beta_1 - 731822) q^{4} + ( - 59049 \beta_1 - 885735) q^{6} + ( - \beta_{11} + \cdots - 221197120 \beta_{5}) q^{7}+ \cdots - 3486784401 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} - 15 \beta_{5}) q^{2} - 59049 \beta_{5} q^{3} + (\beta_{2} + 243 \beta_1 - 731822) q^{4} + ( - 59049 \beta_1 - 885735) q^{6} + ( - \beta_{11} + \cdots - 221197120 \beta_{5}) q^{7}+ \cdots + ( - 76709256822 \beta_{7} + \cdots + 45\!\cdots\!69) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 8780896 q^{4} - 10865016 q^{6} - 41841412812 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 8780896 q^{4} - 10865016 q^{6} - 41841412812 q^{9} - 156620225032 q^{11} + 627651460296 q^{14} + 3993403225600 q^{16} - 13939277995244 q^{19} - 156732987630492 q^{21} - 502184750631552 q^{24} - 548628652527256 q^{26} + 77\!\cdots\!92 q^{29}+ \cdots + 54\!\cdots\!32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 1060747 x^{10} + 401129033703 x^{8} + \cdots + 46\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 97404947564203 \nu^{10} + \cdots - 70\!\cdots\!00 ) / 34\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 10\!\cdots\!33 \nu^{10} + \cdots + 46\!\cdots\!00 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 18\!\cdots\!23 \nu^{10} + \cdots - 46\!\cdots\!00 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 10\!\cdots\!37 \nu^{10} + \cdots - 15\!\cdots\!00 ) / 68\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 11\!\cdots\!23 \nu^{11} + \cdots + 30\!\cdots\!00 \nu ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 11\!\cdots\!23 \nu^{11} + \cdots + 41\!\cdots\!00 \nu ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 20\!\cdots\!77 \nu^{10} + \cdots + 24\!\cdots\!00 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 12\!\cdots\!37 \nu^{11} + \cdots + 48\!\cdots\!00 \nu ) / 64\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 29\!\cdots\!87 \nu^{11} + \cdots + 11\!\cdots\!00 \nu ) / 80\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 56\!\cdots\!31 \nu^{11} + \cdots - 22\!\cdots\!00 \nu ) / 80\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 62\!\cdots\!83 \nu^{11} + \cdots + 48\!\cdots\!00 \nu ) / 80\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} + \beta_{5} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + 275\beta _1 - 2828750 ) / 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{11} + \beta_{9} + 454\beta_{8} - 4870675\beta_{6} - 781801849\beta_{5} ) / 64 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -121\beta_{7} - 88\beta_{4} + 231\beta_{3} - 1628146\beta_{2} - 751841447\beta _1 + 3444795464265 ) / 64 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 338183 \beta_{11} - 6908 \beta_{10} - 237687 \beta_{9} - 156769854 \beta_{8} + \cdots + 266494147996751 \beta_{5} ) / 32 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 44250355 \beta_{7} + 23343320 \beta_{4} - 48994797 \beta_{3} + 317451890184 \beta_{2} + \cdots - 60\!\cdots\!47 ) / 32 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 293178270541 \beta_{11} + 9473137872 \beta_{10} + 195159302285 \beta_{9} + \cdots - 28\!\cdots\!25 \beta_{5} ) / 64 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 48330441340947 \beta_{7} - 19825547400120 \beta_{4} + 24588167856333 \beta_{3} + \cdots + 45\!\cdots\!43 ) / 64 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 56\!\cdots\!69 \beta_{11} + \cdots + 69\!\cdots\!85 \beta_{5} ) / 32 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 59\!\cdots\!27 \beta_{7} + \cdots - 44\!\cdots\!81 ) / 16 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 43\!\cdots\!97 \beta_{11} + \cdots - 64\!\cdots\!13 \beta_{5} ) / 64 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\) \(26\) \(52\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
− 646.238i
− 571.187i
− 388.716i
− 370.009i
− 144.571i
− 89.2274i
89.2274i
144.571i
370.009i
388.716i
571.187i
646.238i
− 2568.95i 59049.0i −4.50235e6 0 1.51694e8 5.58323e8i 6.17885e9i −3.48678e9 0
49.2 − 2300.75i − 59049.0i −3.19629e6 0 −1.35857e8 2.10374e8i 2.52885e9i −3.48678e9 0
49.3 − 1570.86i − 59049.0i −370456. 0 −9.27578e7 − 1.39580e9i − 2.71240e9i −3.48678e9 0
49.4 − 1464.04i 59049.0i −46252.4 0 8.64499e7 − 1.40408e8i − 3.00259e9i −3.48678e9 0
49.5 − 594.286i − 59049.0i 1.74398e6 0 −3.50920e7 9.49356e8i − 2.28273e9i −3.48678e9 0
49.6 − 340.910i 59049.0i 1.98093e6 0 2.01304e7 6.73159e8i − 1.39026e9i −3.48678e9 0
49.7 340.910i − 59049.0i 1.98093e6 0 2.01304e7 − 6.73159e8i 1.39026e9i −3.48678e9 0
49.8 594.286i 59049.0i 1.74398e6 0 −3.50920e7 − 9.49356e8i 2.28273e9i −3.48678e9 0
49.9 1464.04i − 59049.0i −46252.4 0 8.64499e7 1.40408e8i 3.00259e9i −3.48678e9 0
49.10 1570.86i 59049.0i −370456. 0 −9.27578e7 1.39580e9i 2.71240e9i −3.48678e9 0
49.11 2300.75i 59049.0i −3.19629e6 0 −1.35857e8 − 2.10374e8i − 2.52885e9i −3.48678e9 0
49.12 2568.95i − 59049.0i −4.50235e6 0 1.51694e8 − 5.58323e8i − 6.17885e9i −3.48678e9 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.22.b.i 12
5.b even 2 1 inner 75.22.b.i 12
5.c odd 4 1 75.22.a.i ✓ 6
5.c odd 4 1 75.22.a.j yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.22.a.i ✓ 6 5.c odd 4 1
75.22.a.j yes 6 5.c odd 4 1
75.22.b.i 12 1.a even 1 1 trivial
75.22.b.i 12 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 16973360 T_{2}^{10} + 102849689765632 T_{2}^{8} + \cdots + 75\!\cdots\!84 \) acting on \(S_{22}^{\mathrm{new}}(75, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + \cdots + 75\!\cdots\!84 \) Copy content Toggle raw display
$3$ \( (T^{2} + 3486784401)^{6} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 21\!\cdots\!25 \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots - 99\!\cdots\!52)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 20\!\cdots\!09 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 72\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots - 69\!\cdots\!75)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 53\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots - 38\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 43\!\cdots\!25)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 69\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 85\!\cdots\!61 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 20\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 36\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 14\!\cdots\!41)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 92\!\cdots\!01 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 54\!\cdots\!84)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots - 37\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 26\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots - 85\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 64\!\cdots\!81 \) Copy content Toggle raw display
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