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Properties

Label 275.4.b.g
Level $275$
Weight $4$
Character orbit 275.b
Analytic conductor $16.226$
Analytic rank $0$
Dimension $10$
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] = [275,4,Mod(199,275)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(275, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("275.199");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 275.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: \(16.2255252516\)
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: \( x^{10} + 52x^{8} + 916x^{6} + 6481x^{4} + 16872x^{2} + 7056 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 5^{2} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{6} q^{3} + (\beta_{2} - 2) q^{4} + (\beta_{3} - \beta_{2} + 1) q^{6} + ( - \beta_{9} - \beta_{8} + \cdots + \beta_1) q^{7}+ \cdots + (\beta_{5} - 2 \beta_{4} - 2 \beta_{3} + \cdots - 6) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{6} q^{3} + (\beta_{2} - 2) q^{4} + (\beta_{3} - \beta_{2} + 1) q^{6} + ( - \beta_{9} - \beta_{8} + \cdots + \beta_1) q^{7}+ \cdots + (11 \beta_{5} - 22 \beta_{4} + \cdots - 66) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 24 q^{4} + 16 q^{6} - 62 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 24 q^{4} + 16 q^{6} - 62 q^{9} + 110 q^{11} - 94 q^{14} - 112 q^{16} - 410 q^{19} - 188 q^{21} - 546 q^{24} - 708 q^{26} - 502 q^{29} - 578 q^{31} - 1044 q^{34} - 1444 q^{36} - 1076 q^{39} - 924 q^{41} - 264 q^{44} - 1944 q^{46} - 150 q^{49} - 1640 q^{51} - 562 q^{54} - 1054 q^{56} + 44 q^{59} + 1440 q^{61} + 634 q^{64} + 176 q^{66} + 1028 q^{69} + 1902 q^{71} - 1746 q^{74} + 580 q^{76} + 1176 q^{79} + 3770 q^{81} - 2642 q^{84} - 822 q^{86} + 102 q^{89} + 4344 q^{91} + 1730 q^{94} + 1206 q^{96} - 682 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 52x^{8} + 916x^{6} + 6481x^{4} + 16872x^{2} + 7056 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -17\nu^{8} - 752\nu^{6} - 8996\nu^{4} - 15761\nu^{2} + 59340 ) / 4176 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 85\nu^{8} + 3760\nu^{6} + 49156\nu^{4} + 179029\nu^{2} + 24852 ) / 8352 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -35\nu^{8} - 1712\nu^{6} - 26300\nu^{4} - 131363\nu^{2} - 114060 ) / 2784 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -37\nu^{9} - 2128\nu^{7} - 42916\nu^{5} - 347749\nu^{3} - 863508\nu ) / 50112 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -263\nu^{9} - 12248\nu^{7} - 177740\nu^{5} - 861143\nu^{3} - 1008708\nu ) / 87696 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 293\nu^{9} + 14312\nu^{7} + 222356\nu^{5} + 1208789\nu^{3} + 1935708\nu ) / 87696 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 913\nu^{9} + 42352\nu^{7} + 589012\nu^{5} + 2225521\nu^{3} - 1283388\nu ) / 175392 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{9} + 2\beta_{8} + 2\beta_{6} - 17\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{4} + 5\beta_{3} - 24\beta_{2} + 163 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 24\beta_{9} - 65\beta_{8} - 15\beta_{7} - 64\beta_{6} + 335\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -17\beta_{5} - 95\beta_{4} - 185\beta_{3} + 536\beta_{2} - 3145 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -502\beta_{9} + 1756\beta_{8} + 694\beta_{7} + 1588\beta_{6} - 7087\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 752\beta_{5} + 3144\beta_{4} + 5292\beta_{3} - 11937\beta_{2} + 65626 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 10433\beta_{9} - 44398\beta_{8} - 22516\beta_{7} - 37250\beta_{6} + 155473\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/275\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
− 4.83592i
− 3.95823i
− 2.68861i
− 2.28165i
− 0.715355i
0.715355i
2.28165i
2.68861i
3.95823i
4.83592i
− 4.83592i 5.01496i −15.3861 0 24.2519 − 11.6188i 35.7185i 1.85020 0
199.2 − 3.95823i − 0.384445i −7.66762 0 −1.52172 27.0299i − 1.31563i 26.8522 0
199.3 − 2.68861i − 5.92895i 0.771363 0 −15.9407 − 13.6511i − 23.5828i −8.15246 0
199.4 − 2.28165i − 2.58679i 2.79408 0 −5.90215 − 27.1421i − 24.6283i 20.3085 0
199.5 − 0.715355i 9.94276i 7.48827 0 7.11261 1.15778i − 11.0796i −71.8585 0
199.6 0.715355i − 9.94276i 7.48827 0 7.11261 − 1.15778i 11.0796i −71.8585 0
199.7 2.28165i 2.58679i 2.79408 0 −5.90215 27.1421i 24.6283i 20.3085 0
199.8 2.68861i 5.92895i 0.771363 0 −15.9407 13.6511i 23.5828i −8.15246 0
199.9 3.95823i 0.384445i −7.66762 0 −1.52172 − 27.0299i 1.31563i 26.8522 0
199.10 4.83592i − 5.01496i −15.3861 0 24.2519 11.6188i − 35.7185i 1.85020 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.10
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 275.4.b.g 10
5.b even 2 1 inner 275.4.b.g 10
5.c odd 4 1 275.4.a.f ✓ 5
5.c odd 4 1 275.4.a.i yes 5
15.e even 4 1 2475.4.a.bg 5
15.e even 4 1 2475.4.a.bk 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
275.4.a.f ✓ 5 5.c odd 4 1
275.4.a.i yes 5 5.c odd 4 1
275.4.b.g 10 1.a even 1 1 trivial
275.4.b.g 10 5.b even 2 1 inner
2475.4.a.bg 5 15.e even 4 1
2475.4.a.bk 5 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + 52T_{2}^{8} + 916T_{2}^{6} + 6481T_{2}^{4} + 16872T_{2}^{2} + 7056 \) acting on \(S_{4}^{\mathrm{new}}(275, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 52 T^{8} + \cdots + 7056 \) Copy content Toggle raw display
$3$ \( T^{10} + 166 T^{8} + \cdots + 86436 \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 18150556176 \) Copy content Toggle raw display
$11$ \( (T - 11)^{10} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 12606483414721 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 26\!\cdots\!04 \) Copy content Toggle raw display
$19$ \( (T^{5} + 205 T^{4} + \cdots - 1257816875)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 11\!\cdots\!21 \) Copy content Toggle raw display
$29$ \( (T^{5} + 251 T^{4} + \cdots + 5895839025)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} + 289 T^{4} + \cdots + 1327232025)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 75\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( (T^{5} + 462 T^{4} + \cdots + 932789172)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 59\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 81\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 52\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( (T^{5} - 22 T^{4} + \cdots - 178871706300)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots - 1242903344018)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 44\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots - 32394319745136)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 16\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots - 426963412494900)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 56\!\cdots\!49 \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots + 26515175239875)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 11\!\cdots\!89 \) Copy content Toggle raw display
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