Properties

Label 275.4.b.a
Level $275$
Weight $4$
Character orbit 275.b
Analytic conductor $16.226$
Analytic rank $0$
Dimension $2$
Inner twists $2$

Related objects

Downloads

Learn more

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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + i q^{2} + 3 i q^{3} + 7 q^{4} - 3 q^{6} - 9 i q^{7} + 15 i q^{8} + 18 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + i q^{2} + 3 i q^{3} + 7 q^{4} - 3 q^{6} - 9 i q^{7} + 15 i q^{8} + 18 q^{9} + 11 q^{11} + 21 i q^{12} - 2 i q^{13} + 9 q^{14} + 41 q^{16} + 21 i q^{17} + 18 i q^{18} + 85 q^{19} + 27 q^{21} + 11 i q^{22} - 22 i q^{23} - 45 q^{24} + 2 q^{26} + 135 i q^{27} - 63 i q^{28} + 165 q^{29} - 83 q^{31} + 161 i q^{32} + 33 i q^{33} - 21 q^{34} + 126 q^{36} + i q^{37} + 85 i q^{38} + 6 q^{39} - 478 q^{41} + 27 i q^{42} + 8 i q^{43} + 77 q^{44} + 22 q^{46} + 126 i q^{47} + 123 i q^{48} + 262 q^{49} - 63 q^{51} - 14 i q^{52} + 683 i q^{53} - 135 q^{54} + 135 q^{56} + 255 i q^{57} + 165 i q^{58} + 290 q^{59} + 257 q^{61} - 83 i q^{62} - 162 i q^{63} + 167 q^{64} - 33 q^{66} + 776 i q^{67} + 147 i q^{68} + 66 q^{69} - 313 q^{71} + 270 i q^{72} - 902 i q^{73} - q^{74} + 595 q^{76} - 99 i q^{77} + 6 i q^{78} - 830 q^{79} + 81 q^{81} - 478 i q^{82} - 842 i q^{83} + 189 q^{84} - 8 q^{86} + 495 i q^{87} + 165 i q^{88} - 25 q^{89} - 18 q^{91} - 154 i q^{92} - 249 i q^{93} - 126 q^{94} - 483 q^{96} - 1784 i q^{97} + 262 i q^{98} + 198 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 14 q^{4} - 6 q^{6} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 14 q^{4} - 6 q^{6} + 36 q^{9} + 22 q^{11} + 18 q^{14} + 82 q^{16} + 170 q^{19} + 54 q^{21} - 90 q^{24} + 4 q^{26} + 330 q^{29} - 166 q^{31} - 42 q^{34} + 252 q^{36} + 12 q^{39} - 956 q^{41} + 154 q^{44} + 44 q^{46} + 524 q^{49} - 126 q^{51} - 270 q^{54} + 270 q^{56} + 580 q^{59} + 514 q^{61} + 334 q^{64} - 66 q^{66} + 132 q^{69} - 626 q^{71} - 2 q^{74} + 1190 q^{76} - 1660 q^{79} + 162 q^{81} + 378 q^{84} - 16 q^{86} - 50 q^{89} - 36 q^{91} - 252 q^{94} - 966 q^{96} + 396 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
1.00000i
1.00000i
1.00000i 3.00000i 7.00000 0 −3.00000 9.00000i 15.0000i 18.0000 0
199.2 1.00000i 3.00000i 7.00000 0 −3.00000 9.00000i 15.0000i 18.0000 0
\(n\): e.g. 2-40 or 990-1000
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.a 2
5.b even 2 1 inner 275.4.b.a 2
5.c odd 4 1 55.4.a.a 1
5.c odd 4 1 275.4.a.a 1
15.e even 4 1 495.4.a.a 1
15.e even 4 1 2475.4.a.h 1
20.e even 4 1 880.4.a.j 1
55.e even 4 1 605.4.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.4.a.a 1 5.c odd 4 1
275.4.a.a 1 5.c odd 4 1
275.4.b.a 2 1.a even 1 1 trivial
275.4.b.a 2 5.b even 2 1 inner
495.4.a.a 1 15.e even 4 1
605.4.a.b 1 55.e even 4 1
880.4.a.j 1 20.e even 4 1
2475.4.a.h 1 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 1 \) acting on \(S_{4}^{\mathrm{new}}(275, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 81 \) Copy content Toggle raw display
$11$ \( (T - 11)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 4 \) Copy content Toggle raw display
$17$ \( T^{2} + 441 \) Copy content Toggle raw display
$19$ \( (T - 85)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 484 \) Copy content Toggle raw display
$29$ \( (T - 165)^{2} \) Copy content Toggle raw display
$31$ \( (T + 83)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 1 \) Copy content Toggle raw display
$41$ \( (T + 478)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 64 \) Copy content Toggle raw display
$47$ \( T^{2} + 15876 \) Copy content Toggle raw display
$53$ \( T^{2} + 466489 \) Copy content Toggle raw display
$59$ \( (T - 290)^{2} \) Copy content Toggle raw display
$61$ \( (T - 257)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 602176 \) Copy content Toggle raw display
$71$ \( (T + 313)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 813604 \) Copy content Toggle raw display
$79$ \( (T + 830)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 708964 \) Copy content Toggle raw display
$89$ \( (T + 25)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 3182656 \) Copy content Toggle raw display
show more
show less