Properties

Label 1911.2.a.h
Level $1911$
Weight $2$
Character orbit 1911.a
Self dual yes
Analytic conductor $15.259$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1911 = 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1911.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(15.2594118263\)
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 39)
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.

\(f(q)\) \(=\) \( q + (\beta - 1) q^{2} - q^{3} + ( - 2 \beta + 1) q^{4} + 2 \beta q^{5} + ( - \beta + 1) q^{6} + (\beta - 3) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta - 1) q^{2} - q^{3} + ( - 2 \beta + 1) q^{4} + 2 \beta q^{5} + ( - \beta + 1) q^{6} + (\beta - 3) q^{8} + q^{9} + ( - 2 \beta + 4) q^{10} - 2 q^{11} + (2 \beta - 1) q^{12} + q^{13} - 2 \beta q^{15} + 3 q^{16} + ( - 4 \beta - 2) q^{17} + (\beta - 1) q^{18} + 2 \beta q^{19} + (2 \beta - 8) q^{20} + ( - 2 \beta + 2) q^{22} - 4 q^{23} + ( - \beta + 3) q^{24} + 3 q^{25} + (\beta - 1) q^{26} - q^{27} + 2 q^{29} + (2 \beta - 4) q^{30} + ( - 2 \beta + 4) q^{31} + (\beta + 3) q^{32} + 2 q^{33} + (2 \beta - 6) q^{34} + ( - 2 \beta + 1) q^{36} + ( - 4 \beta - 2) q^{37} + ( - 2 \beta + 4) q^{38} - q^{39} + ( - 6 \beta + 4) q^{40} + (2 \beta - 8) q^{41} + ( - 4 \beta + 4) q^{43} + (4 \beta - 2) q^{44} + 2 \beta q^{45} + ( - 4 \beta + 4) q^{46} + (4 \beta + 6) q^{47} - 3 q^{48} + (3 \beta - 3) q^{50} + (4 \beta + 2) q^{51} + ( - 2 \beta + 1) q^{52} - 2 q^{53} + ( - \beta + 1) q^{54} - 4 \beta q^{55} - 2 \beta q^{57} + (2 \beta - 2) q^{58} + ( - 4 \beta - 2) q^{59} + ( - 2 \beta + 8) q^{60} + ( - 8 \beta - 2) q^{61} + (6 \beta - 8) q^{62} + (2 \beta - 7) q^{64} + 2 \beta q^{65} + (2 \beta - 2) q^{66} + (2 \beta + 4) q^{67} + 14 q^{68} + 4 q^{69} + 2 q^{71} + (\beta - 3) q^{72} + (4 \beta - 6) q^{73} + (2 \beta - 6) q^{74} - 3 q^{75} + (2 \beta - 8) q^{76} + ( - \beta + 1) q^{78} - 8 \beta q^{79} + 6 \beta q^{80} + q^{81} + ( - 10 \beta + 12) q^{82} + ( - 4 \beta + 2) q^{83} + ( - 4 \beta - 16) q^{85} + (8 \beta - 12) q^{86} - 2 q^{87} + ( - 2 \beta + 6) q^{88} + ( - 2 \beta - 12) q^{89} + ( - 2 \beta + 4) q^{90} + (8 \beta - 4) q^{92} + (2 \beta - 4) q^{93} + (2 \beta + 2) q^{94} + 8 q^{95} + ( - \beta - 3) q^{96} + ( - 4 \beta + 2) q^{97} - 2 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 6 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 6 q^{8} + 2 q^{9} + 8 q^{10} - 4 q^{11} - 2 q^{12} + 2 q^{13} + 6 q^{16} - 4 q^{17} - 2 q^{18} - 16 q^{20} + 4 q^{22} - 8 q^{23} + 6 q^{24} + 6 q^{25} - 2 q^{26} - 2 q^{27} + 4 q^{29} - 8 q^{30} + 8 q^{31} + 6 q^{32} + 4 q^{33} - 12 q^{34} + 2 q^{36} - 4 q^{37} + 8 q^{38} - 2 q^{39} + 8 q^{40} - 16 q^{41} + 8 q^{43} - 4 q^{44} + 8 q^{46} + 12 q^{47} - 6 q^{48} - 6 q^{50} + 4 q^{51} + 2 q^{52} - 4 q^{53} + 2 q^{54} - 4 q^{58} - 4 q^{59} + 16 q^{60} - 4 q^{61} - 16 q^{62} - 14 q^{64} - 4 q^{66} + 8 q^{67} + 28 q^{68} + 8 q^{69} + 4 q^{71} - 6 q^{72} - 12 q^{73} - 12 q^{74} - 6 q^{75} - 16 q^{76} + 2 q^{78} + 2 q^{81} + 24 q^{82} + 4 q^{83} - 32 q^{85} - 24 q^{86} - 4 q^{87} + 12 q^{88} - 24 q^{89} + 8 q^{90} - 8 q^{92} - 8 q^{93} + 4 q^{94} + 16 q^{95} - 6 q^{96} + 4 q^{97} - 4 q^{99}+O(q^{100}) \) 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 −1.00000 3.82843 −2.82843 2.41421 0 −4.41421 1.00000 6.82843
1.2 0.414214 −1.00000 −1.82843 2.82843 −0.414214 0 −1.58579 1.00000 1.17157
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(-1\)
\(13\) \(-1\)

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 1911.2.a.h 2
3.b odd 2 1 5733.2.a.u 2
7.b odd 2 1 39.2.a.b 2
21.c even 2 1 117.2.a.c 2
28.d even 2 1 624.2.a.k 2
35.c odd 2 1 975.2.a.l 2
35.f even 4 2 975.2.c.h 4
56.e even 2 1 2496.2.a.bi 2
56.h odd 2 1 2496.2.a.bf 2
63.l odd 6 2 1053.2.e.m 4
63.o even 6 2 1053.2.e.e 4
77.b even 2 1 4719.2.a.p 2
84.h odd 2 1 1872.2.a.w 2
91.b odd 2 1 507.2.a.h 2
91.i even 4 2 507.2.b.e 4
91.n odd 6 2 507.2.e.h 4
91.t odd 6 2 507.2.e.d 4
91.bc even 12 4 507.2.j.f 8
105.g even 2 1 2925.2.a.v 2
105.k odd 4 2 2925.2.c.u 4
168.e odd 2 1 7488.2.a.co 2
168.i even 2 1 7488.2.a.cl 2
273.g even 2 1 1521.2.a.f 2
273.o odd 4 2 1521.2.b.j 4
364.h even 2 1 8112.2.a.bm 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.a.b 2 7.b odd 2 1
117.2.a.c 2 21.c even 2 1
507.2.a.h 2 91.b odd 2 1
507.2.b.e 4 91.i even 4 2
507.2.e.d 4 91.t odd 6 2
507.2.e.h 4 91.n odd 6 2
507.2.j.f 8 91.bc even 12 4
624.2.a.k 2 28.d even 2 1
975.2.a.l 2 35.c odd 2 1
975.2.c.h 4 35.f even 4 2
1053.2.e.e 4 63.o even 6 2
1053.2.e.m 4 63.l odd 6 2
1521.2.a.f 2 273.g even 2 1
1521.2.b.j 4 273.o odd 4 2
1872.2.a.w 2 84.h odd 2 1
1911.2.a.h 2 1.a even 1 1 trivial
2496.2.a.bf 2 56.h odd 2 1
2496.2.a.bi 2 56.e even 2 1
2925.2.a.v 2 105.g even 2 1
2925.2.c.u 4 105.k odd 4 2
4719.2.a.p 2 77.b even 2 1
5733.2.a.u 2 3.b odd 2 1
7488.2.a.cl 2 168.i even 2 1
7488.2.a.co 2 168.e odd 2 1
8112.2.a.bm 2 364.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1911))\):

\( T_{2}^{2} + 2T_{2} - 1 \) Copy content Toggle raw display
\( T_{5}^{2} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T - 1 \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 8 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T + 2)^{2} \) Copy content Toggle raw display
$13$ \( (T - 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 4T - 28 \) Copy content Toggle raw display
$19$ \( T^{2} - 8 \) Copy content Toggle raw display
$23$ \( (T + 4)^{2} \) Copy content Toggle raw display
$29$ \( (T - 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 8T + 8 \) Copy content Toggle raw display
$37$ \( T^{2} + 4T - 28 \) Copy content Toggle raw display
$41$ \( T^{2} + 16T + 56 \) Copy content Toggle raw display
$43$ \( T^{2} - 8T - 16 \) Copy content Toggle raw display
$47$ \( T^{2} - 12T + 4 \) Copy content Toggle raw display
$53$ \( (T + 2)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 4T - 28 \) Copy content Toggle raw display
$61$ \( T^{2} + 4T - 124 \) Copy content Toggle raw display
$67$ \( T^{2} - 8T + 8 \) Copy content Toggle raw display
$71$ \( (T - 2)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 12T + 4 \) Copy content Toggle raw display
$79$ \( T^{2} - 128 \) Copy content Toggle raw display
$83$ \( T^{2} - 4T - 28 \) Copy content Toggle raw display
$89$ \( T^{2} + 24T + 136 \) Copy content Toggle raw display
$97$ \( T^{2} - 4T - 28 \) Copy content Toggle raw display
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