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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Newspace parameters

Level: \( N \) \(=\) \( 261 = 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 261.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(15.3994985115\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
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

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

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

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.

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 \( T_{2}^{2} - 2 T_{2} - 1 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(261))\).

Hecke characteristic polynomials

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