Properties

Label 1421.2.a.j
Level $1421$
Weight $2$
Character orbit 1421.a
Self dual yes
Analytic conductor $11.347$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1421 = 7^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1421.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.3467421272\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
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

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

\(f(q)\) \(=\) \( q + (\beta - 1) q^{2} + (\beta - 1) q^{3} + ( - 2 \beta + 1) q^{4} + q^{5} + ( - 2 \beta + 3) q^{6} + (\beta - 3) q^{8} - 2 \beta q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\beta - 1) q^{2} + (\beta - 1) q^{3} + ( - 2 \beta + 1) q^{4} + q^{5} + ( - 2 \beta + 3) q^{6} + (\beta - 3) q^{8} - 2 \beta q^{9} + (\beta - 1) q^{10} + (\beta + 1) q^{11} + (3 \beta - 5) q^{12} + ( - 2 \beta + 1) q^{13} + (\beta - 1) q^{15} + 3 q^{16} + (2 \beta + 2) q^{17} + (2 \beta - 4) q^{18} - 6 q^{19} + ( - 2 \beta + 1) q^{20} + q^{22} + ( - 4 \beta - 2) q^{23} + ( - 4 \beta + 5) q^{24} - 4 q^{25} + (3 \beta - 5) q^{26} + ( - \beta - 1) q^{27} + q^{29} + ( - 2 \beta + 3) q^{30} + (5 \beta - 3) q^{31} + (\beta + 3) q^{32} + q^{33} + 2 q^{34} + ( - 2 \beta + 8) q^{36} - 4 q^{37} + ( - 6 \beta + 6) q^{38} + (3 \beta - 5) q^{39} + (\beta - 3) q^{40} + ( - 6 \beta - 4) q^{41} + (\beta + 5) q^{43} + ( - \beta - 3) q^{44} - 2 \beta q^{45} + (2 \beta - 6) q^{46} + ( - 3 \beta - 1) q^{47} + (3 \beta - 3) q^{48} + ( - 4 \beta + 4) q^{50} + 2 q^{51} + ( - 4 \beta + 9) q^{52} + ( - 6 \beta + 1) q^{53} - q^{54} + (\beta + 1) q^{55} + ( - 6 \beta + 6) q^{57} + (\beta - 1) q^{58} + ( - 4 \beta - 2) q^{59} + (3 \beta - 5) q^{60} + ( - 2 \beta + 2) q^{61} + ( - 8 \beta + 13) q^{62} + (2 \beta - 7) q^{64} + ( - 2 \beta + 1) q^{65} + (\beta - 1) q^{66} - 4 \beta q^{67} + ( - 2 \beta - 6) q^{68} + (2 \beta - 6) q^{69} + (2 \beta - 6) q^{71} + (6 \beta - 4) q^{72} - 4 q^{73} + ( - 4 \beta + 4) q^{74} + ( - 4 \beta + 4) q^{75} + (12 \beta - 6) q^{76} + ( - 8 \beta + 11) q^{78} + (\beta - 1) q^{79} + 3 q^{80} + (6 \beta - 1) q^{81} + (2 \beta - 8) q^{82} + (4 \beta - 2) q^{83} + (2 \beta + 2) q^{85} + (4 \beta - 3) q^{86} + (\beta - 1) q^{87} + ( - 2 \beta - 1) q^{88} + ( - 6 \beta + 4) q^{89} + (2 \beta - 4) q^{90} + 14 q^{92} + ( - 8 \beta + 13) q^{93} + (2 \beta - 5) q^{94} - 6 q^{95} + (2 \beta - 1) q^{96} + (6 \beta + 4) q^{97} + ( - 2 \beta - 4) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 6 q^{6} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 6 q^{6} - 6 q^{8} - 2 q^{10} + 2 q^{11} - 10 q^{12} + 2 q^{13} - 2 q^{15} + 6 q^{16} + 4 q^{17} - 8 q^{18} - 12 q^{19} + 2 q^{20} + 2 q^{22} - 4 q^{23} + 10 q^{24} - 8 q^{25} - 10 q^{26} - 2 q^{27} + 2 q^{29} + 6 q^{30} - 6 q^{31} + 6 q^{32} + 2 q^{33} + 4 q^{34} + 16 q^{36} - 8 q^{37} + 12 q^{38} - 10 q^{39} - 6 q^{40} - 8 q^{41} + 10 q^{43} - 6 q^{44} - 12 q^{46} - 2 q^{47} - 6 q^{48} + 8 q^{50} + 4 q^{51} + 18 q^{52} + 2 q^{53} - 2 q^{54} + 2 q^{55} + 12 q^{57} - 2 q^{58} - 4 q^{59} - 10 q^{60} + 4 q^{61} + 26 q^{62} - 14 q^{64} + 2 q^{65} - 2 q^{66} - 12 q^{68} - 12 q^{69} - 12 q^{71} - 8 q^{72} - 8 q^{73} + 8 q^{74} + 8 q^{75} - 12 q^{76} + 22 q^{78} - 2 q^{79} + 6 q^{80} - 2 q^{81} - 16 q^{82} - 4 q^{83} + 4 q^{85} - 6 q^{86} - 2 q^{87} - 2 q^{88} + 8 q^{89} - 8 q^{90} + 28 q^{92} + 26 q^{93} - 10 q^{94} - 12 q^{95} - 2 q^{96} + 8 q^{97} - 8 q^{99}+O(q^{100}) \) 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.

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
−2.41421 −2.41421 3.82843 1.00000 5.82843 0 −4.41421 2.82843 −2.41421
1.2 0.414214 0.414214 −1.82843 1.00000 0.171573 0 −1.58579 −2.82843 0.414214
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)
\(29\) \(-1\)

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 1421.2.a.j 2
7.b odd 2 1 29.2.a.a 2
21.c even 2 1 261.2.a.d 2
28.d even 2 1 464.2.a.h 2
35.c odd 2 1 725.2.a.b 2
35.f even 4 2 725.2.b.b 4
56.e even 2 1 1856.2.a.w 2
56.h odd 2 1 1856.2.a.r 2
77.b even 2 1 3509.2.a.j 2
84.h odd 2 1 4176.2.a.bq 2
91.b odd 2 1 4901.2.a.g 2
105.g even 2 1 6525.2.a.o 2
119.d odd 2 1 8381.2.a.e 2
203.c odd 2 1 841.2.a.d 2
203.g even 4 2 841.2.b.a 4
203.n odd 14 6 841.2.d.j 12
203.o odd 14 6 841.2.d.f 12
203.r even 28 12 841.2.e.k 24
609.h even 2 1 7569.2.a.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.2.a.a 2 7.b odd 2 1
261.2.a.d 2 21.c even 2 1
464.2.a.h 2 28.d even 2 1
725.2.a.b 2 35.c odd 2 1
725.2.b.b 4 35.f even 4 2
841.2.a.d 2 203.c odd 2 1
841.2.b.a 4 203.g even 4 2
841.2.d.f 12 203.o odd 14 6
841.2.d.j 12 203.n odd 14 6
841.2.e.k 24 203.r even 28 12
1421.2.a.j 2 1.a even 1 1 trivial
1856.2.a.r 2 56.h odd 2 1
1856.2.a.w 2 56.e even 2 1
3509.2.a.j 2 77.b even 2 1
4176.2.a.bq 2 84.h odd 2 1
4901.2.a.g 2 91.b odd 2 1
6525.2.a.o 2 105.g even 2 1
7569.2.a.c 2 609.h even 2 1
8381.2.a.e 2 119.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1421))\):

\( T_{2}^{2} + 2T_{2} - 1 \) Copy content Toggle raw display
\( T_{3}^{2} + 2T_{3} - 1 \) Copy content Toggle raw display
\( T_{5} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T - 1 \) Copy content Toggle raw display
$3$ \( T^{2} + 2T - 1 \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 2T - 1 \) Copy content Toggle raw display
$13$ \( T^{2} - 2T - 7 \) Copy content Toggle raw display
$17$ \( T^{2} - 4T - 4 \) Copy content Toggle raw display
$19$ \( (T + 6)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 4T - 28 \) Copy content Toggle raw display
$29$ \( (T - 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 6T - 41 \) Copy content Toggle raw display
$37$ \( (T + 4)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 8T - 56 \) Copy content Toggle raw display
$43$ \( T^{2} - 10T + 23 \) Copy content Toggle raw display
$47$ \( T^{2} + 2T - 17 \) Copy content Toggle raw display
$53$ \( T^{2} - 2T - 71 \) Copy content Toggle raw display
$59$ \( T^{2} + 4T - 28 \) Copy content Toggle raw display
$61$ \( T^{2} - 4T - 4 \) Copy content Toggle raw display
$67$ \( T^{2} - 32 \) Copy content Toggle raw display
$71$ \( T^{2} + 12T + 28 \) Copy content Toggle raw display
$73$ \( (T + 4)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 2T - 1 \) Copy content Toggle raw display
$83$ \( T^{2} + 4T - 28 \) Copy content Toggle raw display
$89$ \( T^{2} - 8T - 56 \) Copy content Toggle raw display
$97$ \( T^{2} - 8T - 56 \) Copy content Toggle raw display
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