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Properties

Label 2175.4.a.p
Level $2175$
Weight $4$
Character orbit 2175.a
Self dual yes
Analytic conductor $128.329$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2175,4,Mod(1,2175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2175, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2175.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2175.a (trivial)

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: yes
Analytic conductor: \(128.329154262\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 4 x^{9} - 65 x^{8} + 257 x^{7} + 1374 x^{6} - 5303 x^{5} - 10562 x^{4} + 39282 x^{3} + \cdots + 34560 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5}\cdot 5 \)
Twist minimal: no (minimal twist has level 435)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - 3 q^{3} + (\beta_{2} + 7) q^{4} + 3 \beta_1 q^{6} + (\beta_{4} - \beta_1 - 7) q^{7} + ( - \beta_{3} - 6 \beta_1 + 1) q^{8} + 9 q^{9} + ( - \beta_{6} - \beta_{2} + \beta_1) q^{11} + ( - 3 \beta_{2} - 21) q^{12}+ \cdots + ( - 9 \beta_{6} - 9 \beta_{2} + 9 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 4 q^{2} - 30 q^{3} + 66 q^{4} + 12 q^{6} - 75 q^{7} - 9 q^{8} + 90 q^{9} + 3 q^{11} - 198 q^{12} - 75 q^{13} + 105 q^{14} + 394 q^{16} - 131 q^{17} - 36 q^{18} + 264 q^{19} + 225 q^{21} - 111 q^{22}+ \cdots + 27 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 4 x^{9} - 65 x^{8} + 257 x^{7} + 1374 x^{6} - 5303 x^{5} - 10562 x^{4} + 39282 x^{3} + \cdots + 34560 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 15 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 22\nu + 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 162 \nu^{9} - 1567 \nu^{8} - 18481 \nu^{7} + 96736 \nu^{6} + 636461 \nu^{5} - 2005063 \nu^{4} + \cdots - 31329560 ) / 507560 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 299 \nu^{9} + 934 \nu^{8} + 17113 \nu^{7} - 42411 \nu^{6} - 223576 \nu^{5} + 393495 \nu^{4} + \cdots - 9520128 ) / 203024 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2113 \nu^{9} - 8298 \nu^{8} - 125519 \nu^{7} + 529389 \nu^{6} + 2363904 \nu^{5} - 10022937 \nu^{4} + \cdots - 35239920 ) / 1015120 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 867 \nu^{9} + 162 \nu^{8} + 55436 \nu^{7} - 19556 \nu^{6} - 1172746 \nu^{5} + 543178 \nu^{4} + \cdots + 14942600 ) / 253780 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 434 \nu^{9} + 125 \nu^{8} + 30399 \nu^{7} - 364 \nu^{6} - 705319 \nu^{5} - 183687 \nu^{4} + \cdots + 1547144 ) / 101512 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2549 \nu^{9} - 8599 \nu^{8} - 166642 \nu^{7} + 492097 \nu^{6} + 3545007 \nu^{5} - 8525706 \nu^{4} + \cdots - 69429600 ) / 507560 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 22\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{9} - \beta_{8} + \beta_{7} + 2\beta_{6} + 30\beta_{2} - 4\beta _1 + 339 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{9} - \beta_{8} + 2\beta_{7} + 4\beta_{5} - 5\beta_{4} + 37\beta_{3} - 6\beta_{2} + 553\beta _1 - 77 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 50 \beta_{9} - 37 \beta_{8} + 32 \beta_{7} + 90 \beta_{6} - 6 \beta_{5} + 19 \beta_{4} - 2 \beta_{3} + \cdots + 8623 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 127 \beta_{9} - 28 \beta_{8} + 87 \beta_{7} - 28 \beta_{6} + 206 \beta_{5} - 309 \beta_{4} + 1149 \beta_{3} + \cdots - 3943 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1918 \beta_{9} - 1126 \beta_{8} + 752 \beta_{7} + 3122 \beta_{6} - 366 \beta_{5} + 1096 \beta_{4} + \cdots + 231493 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 5558 \beta_{9} - 440 \beta_{8} + 2610 \beta_{7} - 1984 \beta_{6} + 7828 \beta_{5} - 13218 \beta_{4} + \cdots - 163767 \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
5.27006
5.08233
3.91143
3.24917
1.17947
0.529542
−2.22416
−2.88319
−4.58010
−5.53456
−5.27006 −3.00000 19.7735 0 15.8102 −28.6477 −62.0473 9.00000 0
1.2 −5.08233 −3.00000 17.8301 0 15.2470 12.9634 −49.9600 9.00000 0
1.3 −3.91143 −3.00000 7.29931 0 11.7343 −12.6446 2.74068 9.00000 0
1.4 −3.24917 −3.00000 2.55711 0 9.74751 −15.3171 17.6849 9.00000 0
1.5 −1.17947 −3.00000 −6.60885 0 3.53841 16.2789 17.2307 9.00000 0
1.6 −0.529542 −3.00000 −7.71959 0 1.58863 −30.0869 8.32418 9.00000 0
1.7 2.22416 −3.00000 −3.05310 0 −6.67249 −13.0672 −24.5839 9.00000 0
1.8 2.88319 −3.00000 0.312808 0 −8.64958 18.0264 −22.1637 9.00000 0
1.9 4.58010 −3.00000 12.9773 0 −13.7403 −26.8184 22.7966 9.00000 0
1.10 5.53456 −3.00000 22.6313 0 −16.6037 4.31330 80.9777 9.00000 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
\(3\) \( +1 \)
\(5\) \( +1 \)
\(29\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2175.4.a.p 10
5.b even 2 1 435.4.a.l ✓ 10
15.d odd 2 1 1305.4.a.q 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.4.a.l ✓ 10 5.b even 2 1
1305.4.a.q 10 15.d odd 2 1
2175.4.a.p 10 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2175))\):

\( T_{2}^{10} + 4 T_{2}^{9} - 65 T_{2}^{8} - 257 T_{2}^{7} + 1374 T_{2}^{6} + 5303 T_{2}^{5} - 10562 T_{2}^{4} + \cdots + 34560 \) Copy content Toggle raw display
\( T_{7}^{10} + 75 T_{7}^{9} + 919 T_{7}^{8} - 49883 T_{7}^{7} - 1159268 T_{7}^{6} + 8747012 T_{7}^{5} + \cdots + 959902838272 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 4 T^{9} + \cdots + 34560 \) Copy content Toggle raw display
$3$ \( (T + 3)^{10} \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 959902838272 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 16473693800448 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots - 774860220918784 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots - 53\!\cdots\!80 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 61\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 43\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( (T + 29)^{10} \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots - 41\!\cdots\!08 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots - 14\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots - 48\!\cdots\!04 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots - 15\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 11\!\cdots\!92 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 12\!\cdots\!68 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots - 47\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots - 52\!\cdots\!52 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 44\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots - 59\!\cdots\!48 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 19\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots - 10\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 44\!\cdots\!08 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots - 16\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 72\!\cdots\!92 \) Copy content Toggle raw display
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