Properties

Label 261.4.a.b
Level $261$
Weight $4$
Character orbit 261.a
Self dual yes
Analytic conductor $15.399$
Analytic rank $1$
Dimension $2$
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] = [261,4,Mod(1,261)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(261, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("261.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 261 = 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 261.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: \(15.3994985115\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 29)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{2} + (2 \beta - 5) q^{4} + (4 \beta + 5) q^{5} + ( - 10 \beta - 8) q^{7} + ( - 11 \beta - 9) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 1) q^{2} + (2 \beta - 5) q^{4} + (4 \beta + 5) q^{5} + ( - 10 \beta - 8) q^{7} + ( - 11 \beta - 9) q^{8} + (9 \beta + 13) q^{10} + ( - 37 \beta + 13) q^{11} + (26 \beta - 13) q^{13} + ( - 18 \beta - 28) q^{14} + ( - 36 \beta + 9) q^{16} + (18 \beta - 30) q^{17} + ( - 32 \beta - 110) q^{19} + ( - 10 \beta - 9) q^{20} + ( - 24 \beta - 61) q^{22} + (48 \beta - 26) q^{23} + (40 \beta - 68) q^{25} + (13 \beta + 39) q^{26} + 34 \beta q^{28} - 29 q^{29} + (63 \beta - 147) q^{31} + (61 \beta + 9) q^{32} + ( - 12 \beta + 6) q^{34} + ( - 82 \beta - 120) q^{35} + (56 \beta + 156) q^{37} + ( - 142 \beta - 174) q^{38} + ( - 91 \beta - 133) q^{40} + (138 \beta - 20) q^{41} + ( - 171 \beta - 161) q^{43} + (211 \beta - 213) q^{44} + (22 \beta + 70) q^{46} + ( - 207 \beta + 65) q^{47} + (160 \beta - 79) q^{49} + ( - 28 \beta + 12) q^{50} + ( - 156 \beta + 169) q^{52} + ( - 122 \beta - 501) q^{53} + ( - 133 \beta - 231) q^{55} + (178 \beta + 292) q^{56} + ( - 29 \beta - 29) q^{58} + (248 \beta + 450) q^{59} + (178 \beta - 474) q^{61} + ( - 84 \beta - 21) q^{62} + (358 \beta + 59) q^{64} + (78 \beta + 143) q^{65} + ( - 484 \beta + 160) q^{67} + ( - 150 \beta + 222) q^{68} + ( - 202 \beta - 284) q^{70} + ( - 34 \beta + 330) q^{71} + (640 \beta + 324) q^{73} + (212 \beta + 268) q^{74} + ( - 60 \beta + 422) q^{76} + (166 \beta + 636) q^{77} + (341 \beta + 129) q^{79} + ( - 144 \beta - 243) q^{80} + (118 \beta + 256) q^{82} + (64 \beta - 606) q^{83} + ( - 30 \beta - 6) q^{85} + ( - 332 \beta - 503) q^{86} + (190 \beta + 697) q^{88} + (522 \beta - 380) q^{89} + ( - 78 \beta - 416) q^{91} + ( - 292 \beta + 322) q^{92} + ( - 142 \beta - 349) q^{94} + ( - 600 \beta - 806) q^{95} + (578 \beta + 12) q^{97} + (81 \beta + 241) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 10 q^{4} + 10 q^{5} - 16 q^{7} - 18 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 10 q^{4} + 10 q^{5} - 16 q^{7} - 18 q^{8} + 26 q^{10} + 26 q^{11} - 26 q^{13} - 56 q^{14} + 18 q^{16} - 60 q^{17} - 220 q^{19} - 18 q^{20} - 122 q^{22} - 52 q^{23} - 136 q^{25} + 78 q^{26} - 58 q^{29} - 294 q^{31} + 18 q^{32} + 12 q^{34} - 240 q^{35} + 312 q^{37} - 348 q^{38} - 266 q^{40} - 40 q^{41} - 322 q^{43} - 426 q^{44} + 140 q^{46} + 130 q^{47} - 158 q^{49} + 24 q^{50} + 338 q^{52} - 1002 q^{53} - 462 q^{55} + 584 q^{56} - 58 q^{58} + 900 q^{59} - 948 q^{61} - 42 q^{62} + 118 q^{64} + 286 q^{65} + 320 q^{67} + 444 q^{68} - 568 q^{70} + 660 q^{71} + 648 q^{73} + 536 q^{74} + 844 q^{76} + 1272 q^{77} + 258 q^{79} - 486 q^{80} + 512 q^{82} - 1212 q^{83} - 12 q^{85} - 1006 q^{86} + 1394 q^{88} - 760 q^{89} - 832 q^{91} + 644 q^{92} - 698 q^{94} - 1612 q^{95} + 24 q^{97} + 482 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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
−1.41421
1.41421
−0.414214 0 −7.82843 −0.656854 0 6.14214 6.55635 0 0.272078
1.2 2.41421 0 −2.17157 10.6569 0 −22.1421 −24.5563 0 25.7279
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(29\) \(1\)

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 261.4.a.b 2
3.b odd 2 1 29.4.a.a 2
12.b even 2 1 464.4.a.f 2
15.d odd 2 1 725.4.a.b 2
21.c even 2 1 1421.4.a.c 2
24.f even 2 1 1856.4.a.h 2
24.h odd 2 1 1856.4.a.n 2
87.d odd 2 1 841.4.a.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.4.a.a 2 3.b odd 2 1
261.4.a.b 2 1.a even 1 1 trivial
464.4.a.f 2 12.b even 2 1
725.4.a.b 2 15.d odd 2 1
841.4.a.a 2 87.d odd 2 1
1421.4.a.c 2 21.c even 2 1
1856.4.a.h 2 24.f even 2 1
1856.4.a.n 2 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 2T_{2} - 1 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(261))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T - 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 10T - 7 \) Copy content Toggle raw display
$7$ \( T^{2} + 16T - 136 \) Copy content Toggle raw display
$11$ \( T^{2} - 26T - 2569 \) Copy content Toggle raw display
$13$ \( T^{2} + 26T - 1183 \) Copy content Toggle raw display
$17$ \( T^{2} + 60T + 252 \) Copy content Toggle raw display
$19$ \( T^{2} + 220T + 10052 \) Copy content Toggle raw display
$23$ \( T^{2} + 52T - 3932 \) Copy content Toggle raw display
$29$ \( (T + 29)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 294T + 13671 \) Copy content Toggle raw display
$37$ \( T^{2} - 312T + 18064 \) Copy content Toggle raw display
$41$ \( T^{2} + 40T - 37688 \) Copy content Toggle raw display
$43$ \( T^{2} + 322T - 32561 \) Copy content Toggle raw display
$47$ \( T^{2} - 130T - 81473 \) Copy content Toggle raw display
$53$ \( T^{2} + 1002 T + 221233 \) Copy content Toggle raw display
$59$ \( T^{2} - 900T + 79492 \) Copy content Toggle raw display
$61$ \( T^{2} + 948T + 161308 \) Copy content Toggle raw display
$67$ \( T^{2} - 320T - 442912 \) Copy content Toggle raw display
$71$ \( T^{2} - 660T + 106588 \) Copy content Toggle raw display
$73$ \( T^{2} - 648T - 714224 \) Copy content Toggle raw display
$79$ \( T^{2} - 258T - 215921 \) Copy content Toggle raw display
$83$ \( T^{2} + 1212 T + 359044 \) Copy content Toggle raw display
$89$ \( T^{2} + 760T - 400568 \) Copy content Toggle raw display
$97$ \( T^{2} - 24T - 668024 \) Copy content Toggle raw display
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