Properties

Label 1936.4.a.w
Level $1936$
Weight $4$
Character orbit 1936.a
Self dual yes
Analytic conductor $114.228$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1936 = 2^{4} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1936.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(114.227697771\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 11)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 4\sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta ) q^{3} + ( 1 + 2 \beta ) q^{5} + ( 10 - \beta ) q^{7} + ( 22 + 2 \beta ) q^{9} +O(q^{10})\) \( q + ( 1 + \beta ) q^{3} + ( 1 + 2 \beta ) q^{5} + ( 10 - \beta ) q^{7} + ( 22 + 2 \beta ) q^{9} + ( -40 + 5 \beta ) q^{13} + ( 97 + 3 \beta ) q^{15} + ( 62 - 3 \beta ) q^{17} + ( 36 + 15 \beta ) q^{19} + ( -38 + 9 \beta ) q^{21} + ( 49 + 9 \beta ) q^{23} + ( 68 + 4 \beta ) q^{25} + ( 91 - 3 \beta ) q^{27} + ( -72 + 14 \beta ) q^{29} + ( 17 - 7 \beta ) q^{31} + ( -86 + 19 \beta ) q^{35} + ( 27 - 2 \beta ) q^{37} + ( 200 - 35 \beta ) q^{39} + ( -268 + \beta ) q^{41} + ( -30 - 4 \beta ) q^{43} + ( 214 + 46 \beta ) q^{45} + ( 136 + 30 \beta ) q^{47} + ( -195 - 20 \beta ) q^{49} + ( -82 + 59 \beta ) q^{51} + ( -246 - 14 \beta ) q^{53} + ( 756 + 51 \beta ) q^{57} + ( -317 + 33 \beta ) q^{59} + ( -420 - 46 \beta ) q^{61} + ( 124 - 2 \beta ) q^{63} + ( 440 - 75 \beta ) q^{65} + ( -377 + 5 \beta ) q^{67} + ( 481 + 58 \beta ) q^{69} + ( 339 - 19 \beta ) q^{71} + ( 200 + 117 \beta ) q^{73} + ( 260 + 72 \beta ) q^{75} + ( 158 + 164 \beta ) q^{79} + ( -647 + 34 \beta ) q^{81} + ( 234 + 30 \beta ) q^{83} + ( -226 + 121 \beta ) q^{85} + ( 600 - 58 \beta ) q^{87} + ( -921 - 82 \beta ) q^{89} + ( -640 + 90 \beta ) q^{91} + ( -319 + 10 \beta ) q^{93} + ( 1476 + 87 \beta ) q^{95} + ( 1097 + 36 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{5} + 20 q^{7} + 44 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{3} + 2 q^{5} + 20 q^{7} + 44 q^{9} - 80 q^{13} + 194 q^{15} + 124 q^{17} + 72 q^{19} - 76 q^{21} + 98 q^{23} + 136 q^{25} + 182 q^{27} - 144 q^{29} + 34 q^{31} - 172 q^{35} + 54 q^{37} + 400 q^{39} - 536 q^{41} - 60 q^{43} + 428 q^{45} + 272 q^{47} - 390 q^{49} - 164 q^{51} - 492 q^{53} + 1512 q^{57} - 634 q^{59} - 840 q^{61} + 248 q^{63} + 880 q^{65} - 754 q^{67} + 962 q^{69} + 678 q^{71} + 400 q^{73} + 520 q^{75} + 316 q^{79} - 1294 q^{81} + 468 q^{83} - 452 q^{85} + 1200 q^{87} - 1842 q^{89} - 1280 q^{91} - 638 q^{93} + 2952 q^{95} + 2194 q^{97} + O(q^{100}) \)

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
−1.73205
1.73205
0 −5.92820 0 −12.8564 0 16.9282 0 8.14359 0
1.2 0 7.92820 0 14.8564 0 3.07180 0 35.8564 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(11\) \(-1\)

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 1936.4.a.w 2
4.b odd 2 1 121.4.a.c 2
11.b odd 2 1 176.4.a.i 2
12.b even 2 1 1089.4.a.v 2
33.d even 2 1 1584.4.a.bc 2
44.c even 2 1 11.4.a.a 2
44.g even 10 4 121.4.c.c 8
44.h odd 10 4 121.4.c.f 8
88.b odd 2 1 704.4.a.n 2
88.g even 2 1 704.4.a.p 2
132.d odd 2 1 99.4.a.c 2
220.g even 2 1 275.4.a.b 2
220.i odd 4 2 275.4.b.c 4
308.g odd 2 1 539.4.a.e 2
572.b even 2 1 1859.4.a.a 2
660.g odd 2 1 2475.4.a.q 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.4.a.a 2 44.c even 2 1
99.4.a.c 2 132.d odd 2 1
121.4.a.c 2 4.b odd 2 1
121.4.c.c 8 44.g even 10 4
121.4.c.f 8 44.h odd 10 4
176.4.a.i 2 11.b odd 2 1
275.4.a.b 2 220.g even 2 1
275.4.b.c 4 220.i odd 4 2
539.4.a.e 2 308.g odd 2 1
704.4.a.n 2 88.b odd 2 1
704.4.a.p 2 88.g even 2 1
1089.4.a.v 2 12.b even 2 1
1584.4.a.bc 2 33.d even 2 1
1859.4.a.a 2 572.b even 2 1
1936.4.a.w 2 1.a even 1 1 trivial
2475.4.a.q 2 660.g odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1936))\):

\( T_{3}^{2} - 2 T_{3} - 47 \)
\( T_{5}^{2} - 2 T_{5} - 191 \)
\( T_{7}^{2} - 20 T_{7} + 52 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -47 - 2 T + T^{2} \)
$5$ \( -191 - 2 T + T^{2} \)
$7$ \( 52 - 20 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 400 + 80 T + T^{2} \)
$17$ \( 3412 - 124 T + T^{2} \)
$19$ \( -9504 - 72 T + T^{2} \)
$23$ \( -1487 - 98 T + T^{2} \)
$29$ \( -4224 + 144 T + T^{2} \)
$31$ \( -2063 - 34 T + T^{2} \)
$37$ \( 537 - 54 T + T^{2} \)
$41$ \( 71776 + 536 T + T^{2} \)
$43$ \( 132 + 60 T + T^{2} \)
$47$ \( -24704 - 272 T + T^{2} \)
$53$ \( 51108 + 492 T + T^{2} \)
$59$ \( 48217 + 634 T + T^{2} \)
$61$ \( 74832 + 840 T + T^{2} \)
$67$ \( 140929 + 754 T + T^{2} \)
$71$ \( 97593 - 678 T + T^{2} \)
$73$ \( -617072 - 400 T + T^{2} \)
$79$ \( -1266044 - 316 T + T^{2} \)
$83$ \( 11556 - 468 T + T^{2} \)
$89$ \( 525489 + 1842 T + T^{2} \)
$97$ \( 1141201 - 2194 T + T^{2} \)
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