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Properties

Label 1900.4.c.g
Level $1900$
Weight $4$
Character orbit 1900.c
Analytic conductor $112.104$
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] = [1900,4,Mod(1749,1900)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1900.1749"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1900, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1900.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,0,0,0,0,0,0,-76] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(112.103629011\)
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: \( x^{10} - 2 x^{9} + 2 x^{8} - 4 x^{7} + 2060 x^{6} - 4388 x^{5} + 4664 x^{4} + 4136 x^{3} + 14884 x^{2} + \cdots + 14112 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 380)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + (\beta_{9} + \beta_{2} + \beta_1) q^{7} + ( - \beta_{5} - 2 \beta_{4} + \beta_{3} - 8) q^{9} + (\beta_{6} + \beta_{4} - \beta_{3} + 1) q^{11} + ( - \beta_{9} - 3 \beta_{7} + \cdots - 3 \beta_1) q^{13}+ \cdots + (29 \beta_{6} - 18 \beta_{5} + \cdots - 571) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 76 q^{9} + 8 q^{11} - 190 q^{19} + 482 q^{21} - 134 q^{29} + 876 q^{31} - 1310 q^{39} + 1220 q^{41} - 1500 q^{49} - 658 q^{51} - 1082 q^{59} + 688 q^{61} - 1226 q^{69} + 440 q^{71} - 4572 q^{79}+ \cdots - 5768 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2 x^{9} + 2 x^{8} - 4 x^{7} + 2060 x^{6} - 4388 x^{5} + 4664 x^{4} + 4136 x^{3} + 14884 x^{2} + \cdots + 14112 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 199086419535 \nu^{9} - 756752447660 \nu^{8} + 2411490943770 \nu^{7} - 1555143088784 \nu^{6} + \cdots - 68\!\cdots\!12 ) / 52\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 18162497929 \nu^{9} + 22726829234 \nu^{8} - 6422373878 \nu^{7} + 12155069284 \nu^{6} + \cdots + 175550599120344 ) / 153456930563916 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 37570157960 \nu^{9} + 75025752595 \nu^{8} - 178518024408 \nu^{7} + \cdots + 733488890990808 ) / 186340558541898 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 75446709598 \nu^{9} - 367456096715 \nu^{8} + 332744342118 \nu^{7} - 353938361702 \nu^{6} + \cdots - 29\!\cdots\!52 ) / 372681117083796 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 47482795026 \nu^{9} + 85754636671 \nu^{8} - 223694338794 \nu^{7} + \cdots - 14\!\cdots\!48 ) / 124227039027932 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 60885902290 \nu^{9} - 281853916249 \nu^{8} + 264063479082 \nu^{7} - 285816128702 \nu^{6} + \cdots - 41\!\cdots\!80 ) / 124227039027932 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 730326412927 \nu^{9} + 977429439082 \nu^{8} - 550957302824 \nu^{7} + \cdots + 76\!\cdots\!00 ) / 13\!\cdots\!86 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1174303465497 \nu^{9} - 1106053925741 \nu^{8} - 1228640710890 \nu^{7} + \cdots - 78\!\cdots\!52 ) / 13\!\cdots\!86 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 6162505500061 \nu^{9} + 8403611776450 \nu^{8} - 5243279777318 \nu^{7} + \cdots + 66\!\cdots\!64 ) / 26\!\cdots\!72 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{9} - \beta_{8} + 2\beta_{7} - \beta_{6} + \beta_{5} + 2\beta_{4} - \beta_{3} - 2\beta _1 + 3 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{9} - 2\beta_{8} - 2\beta_{7} - 27\beta_{2} - 4\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 19 \beta_{9} - 16 \beta_{8} + 41 \beta_{7} + 19 \beta_{6} - 16 \beta_{5} - 41 \beta_{4} + 34 \beta_{3} + \cdots - 9 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -44\beta_{6} - 91\beta_{5} + 109\beta_{4} + 175\beta_{3} - 2376 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 863 \beta_{9} + 689 \beta_{8} - 1828 \beta_{7} + 863 \beta_{6} - 689 \beta_{5} - 1828 \beta_{4} + \cdots + 195 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -2035\beta_{9} + 4055\beta_{8} + 5126\beta_{7} + 49992\beta_{2} + 8626\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 19634 \beta_{9} + 15233 \beta_{8} - 41527 \beta_{7} - 19634 \beta_{6} + 15233 \beta_{5} + 41527 \beta_{4} + \cdots - 12666 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 94598\beta_{6} + 180898\beta_{5} - 237652\beta_{4} - 351604\beta_{3} + 4810902 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 893245 \beta_{9} - 674947 \beta_{8} + 1891280 \beta_{7} - 893245 \beta_{6} + 674947 \beta_{5} + \cdots - 914337 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\) \(951\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1749.1
4.71645 − 4.71645i
1.55272 + 1.55272i
−1.05189 + 1.05189i
0.569506 + 0.569506i
−4.78679 − 4.78679i
−4.78679 + 4.78679i
0.569506 − 0.569506i
−1.05189 − 1.05189i
1.55272 − 1.55272i
4.71645 + 4.71645i
0 − 9.49462i 0 0 0 19.4388i 0 −63.1479 0
1749.2 0 − 6.92101i 0 0 0 9.10979i 0 −20.9004 0
1749.3 0 − 5.49584i 0 0 0 2.16482i 0 −3.20422 0
1749.4 0 − 1.67026i 0 0 0 − 34.7038i 0 24.2102 0
1749.5 0 − 1.39919i 0 0 0 28.1976i 0 25.0423 0
1749.6 0 1.39919i 0 0 0 − 28.1976i 0 25.0423 0
1749.7 0 1.67026i 0 0 0 34.7038i 0 24.2102 0
1749.8 0 5.49584i 0 0 0 − 2.16482i 0 −3.20422 0
1749.9 0 6.92101i 0 0 0 − 9.10979i 0 −20.9004 0
1749.10 0 9.49462i 0 0 0 − 19.4388i 0 −63.1479 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1749.10
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 1900.4.c.g 10
5.b even 2 1 inner 1900.4.c.g 10
5.c odd 4 1 380.4.a.e ✓ 5
5.c odd 4 1 1900.4.a.g 5
20.e even 4 1 1520.4.a.v 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.4.a.e ✓ 5 5.c odd 4 1
1520.4.a.v 5 20.e even 4 1
1900.4.a.g 5 5.c odd 4 1
1900.4.c.g 10 1.a even 1 1 trivial
1900.4.c.g 10 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} + 173T_{3}^{8} + 9292T_{3}^{6} + 171640T_{3}^{4} + 665552T_{3}^{2} + 712336 \) acting on \(S_{4}^{\mathrm{new}}(1900, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + 173 T^{8} + \cdots + 712336 \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 140727018496 \) Copy content Toggle raw display
$11$ \( (T^{5} - 4 T^{4} + \cdots - 1247616)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 50333746458384 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 677999107789056 \) Copy content Toggle raw display
$19$ \( (T + 19)^{10} \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 65\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( (T^{5} + 67 T^{4} + \cdots - 539901936)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} - 438 T^{4} + \cdots - 5721278208)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 69\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( (T^{5} - 610 T^{4} + \cdots - 67394592)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 23\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 40\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 90\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( (T^{5} + \cdots + 44682518051136)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} - 344 T^{4} + \cdots - 5529593984)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 47\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( (T^{5} - 220 T^{4} + \cdots - 191671984128)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 26\!\cdots\!64 \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots - 3862283076992)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 10\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots + 264722826885408)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 64\!\cdots\!24 \) Copy content Toggle raw display
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