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Properties

Label 275.4.b.e
Level $275$
Weight $4$
Character orbit 275.b
Analytic conductor $16.226$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 51x^{6} + 835x^{4} + 4881x^{2} + 9216 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 55)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{4} - \beta_{3}) q^{3} + (\beta_{2} - 5) q^{4} + (\beta_{6} - 2 \beta_{5} + 2 \beta_{2} - 6) q^{6} + (\beta_{7} + 2 \beta_{4} + \cdots - 2 \beta_1) q^{7}+ \cdots + (3 \beta_{6} - 3 \beta_{5} - 13) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{4} - \beta_{3}) q^{3} + (\beta_{2} - 5) q^{4} + (\beta_{6} - 2 \beta_{5} + 2 \beta_{2} - 6) q^{6} + (\beta_{7} + 2 \beta_{4} + \cdots - 2 \beta_1) q^{7}+ \cdots + (33 \beta_{6} - 33 \beta_{5} - 143) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 38 q^{4} - 38 q^{6} - 98 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 38 q^{4} - 38 q^{6} - 98 q^{9} + 88 q^{11} + 98 q^{14} - 74 q^{16} + 410 q^{19} - 362 q^{21} + 774 q^{24} - 172 q^{26} + 158 q^{29} + 98 q^{31} + 1878 q^{34} + 224 q^{36} + 52 q^{39} + 1472 q^{41} - 418 q^{44} - 668 q^{46} - 74 q^{49} + 826 q^{51} + 2978 q^{54} - 514 q^{56} + 1684 q^{59} - 2194 q^{61} + 330 q^{64} - 418 q^{66} + 116 q^{69} - 1042 q^{71} - 6514 q^{74} - 5650 q^{76} + 2236 q^{79} + 368 q^{81} + 1210 q^{84} + 6904 q^{86} + 362 q^{89} - 4380 q^{91} - 2068 q^{94} + 7962 q^{96} - 1078 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 51x^{6} + 835x^{4} + 4881x^{2} + 9216 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 54\nu^{5} + 889\nu^{3} + 3876\nu ) / 288 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} - 45\nu^{5} - 583\nu^{3} - 1851\nu ) / 144 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} + 42\nu^{4} + 505\nu^{2} + 1536 ) / 24 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} + 50\nu^{4} + 721\nu^{2} + 2432 ) / 32 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -11\nu^{7} - 522\nu^{5} - 7043\nu^{3} - 21540\nu ) / 288 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - 4\beta_{4} + 3\beta_{3} - 17\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{6} - 3\beta_{5} - 27\beta_{2} + 239 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -34\beta_{7} + 152\beta_{4} - 70\beta_{3} + 353\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -168\beta_{6} + 150\beta_{5} + 629\beta_{2} - 5009 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 947\beta_{7} - 4652\beta_{4} + 1401\beta_{3} - 7825\beta_1 \) 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
− 4.90807i
− 4.18087i
− 2.30365i
− 2.03085i
2.03085i
2.30365i
4.18087i
4.90807i
− 4.90807i − 1.10798i −16.0892 0 −5.43806 17.0703i 39.7022i 25.7724 0
199.2 − 4.18087i − 9.33168i −9.47970 0 −39.0146 − 14.2911i 6.18645i −60.0803 0
199.3 − 2.30365i 6.23583i 2.69320 0 14.3652 − 13.1506i − 24.6334i −11.8856 0
199.4 − 2.03085i 5.45953i 3.87566 0 11.0875 27.2109i − 24.1176i −2.80648 0
199.5 2.03085i − 5.45953i 3.87566 0 11.0875 − 27.2109i 24.1176i −2.80648 0
199.6 2.30365i − 6.23583i 2.69320 0 14.3652 13.1506i 24.6334i −11.8856 0
199.7 4.18087i 9.33168i −9.47970 0 −39.0146 14.2911i − 6.18645i −60.0803 0
199.8 4.90807i 1.10798i −16.0892 0 −5.43806 − 17.0703i − 39.7022i 25.7724 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.8
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.e 8
5.b even 2 1 inner 275.4.b.e 8
5.c odd 4 1 55.4.a.d ✓ 4
5.c odd 4 1 275.4.a.e 4
15.e even 4 1 495.4.a.n 4
15.e even 4 1 2475.4.a.bc 4
20.e even 4 1 880.4.a.z 4
55.e even 4 1 605.4.a.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.4.a.d ✓ 4 5.c odd 4 1
275.4.a.e 4 5.c odd 4 1
275.4.b.e 8 1.a even 1 1 trivial
275.4.b.e 8 5.b even 2 1 inner
495.4.a.n 4 15.e even 4 1
605.4.a.j 4 55.e even 4 1
880.4.a.z 4 20.e even 4 1
2475.4.a.bc 4 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 51T_{2}^{6} + 835T_{2}^{4} + 4881T_{2}^{2} + 9216 \) acting on \(S_{4}^{\mathrm{new}}(275, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 51 T^{6} + \cdots + 9216 \) Copy content Toggle raw display
$3$ \( T^{8} + 157 T^{6} + \cdots + 123904 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 7620591616 \) Copy content Toggle raw display
$11$ \( (T - 11)^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 8336108621824 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 39661519104 \) Copy content Toggle raw display
$19$ \( (T^{4} - 205 T^{3} + \cdots - 5224000)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 49092378624 \) Copy content Toggle raw display
$29$ \( (T^{4} - 79 T^{3} + \cdots + 1106175480)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 49 T^{3} + \cdots + 126259200)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 16\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( (T^{4} - 736 T^{3} + \cdots + 329735184)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 67\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 39\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 42\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( (T^{4} - 842 T^{3} + \cdots - 123367943040)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 1097 T^{3} + \cdots + 4578287464)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( (T^{4} + 521 T^{3} + \cdots + 1139751168)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 19\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( (T^{4} - 1118 T^{3} + \cdots - 289419632640)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 11\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T^{4} - 181 T^{3} + \cdots + 27515045400)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 20\!\cdots\!24 \) Copy content Toggle raw display
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