Properties

Label 1936.4.a.bl
Level $1936$
Weight $4$
Character orbit 1936.a
Self dual yes
Analytic conductor $114.228$
Analytic rank $0$
Dimension $4$
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] = [1936,4,Mod(1,1936)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1936, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1936.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1936 = 2^{4} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1936.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: \(114.227697771\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{5}, \sqrt{37})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 21x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 11)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} - 1) q^{3} + ( - 3 \beta_{3} + \beta_{2} - \beta_1 - 1) q^{5} + ( - 3 \beta_{3} - 2 \beta_{2} + 6 \beta_1 + 3) q^{7} + ( - 3 \beta_{2} - 17) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - 1) q^{3} + ( - 3 \beta_{3} + \beta_{2} - \beta_1 - 1) q^{5} + ( - 3 \beta_{3} - 2 \beta_{2} + 6 \beta_1 + 3) q^{7} + ( - 3 \beta_{2} - 17) q^{9} + (7 \beta_{3} - 11 \beta_{2} + 7 \beta_1 - 17) q^{13} + (2 \beta_{3} - 3 \beta_{2} - 25 \beta_1 + 10) q^{15} + (2 \beta_{3} - 9 \beta_{2} - 18 \beta_1 - 54) q^{17} + ( - 12 \beta_{3} + 6 \beta_{2} - 39 \beta_1 + 64) q^{19} + (9 \beta_{3} + 7 \beta_{2} - 39 \beta_1 - 21) q^{21} + (10 \beta_{3} - 36 \beta_{2} - 26 \beta_1 - 6) q^{23} + (19 \beta_{3} + 12 \beta_{2} + 32 \beta_1 - 18) q^{25} + ( - 38 \beta_{2} + 17) q^{27} + ( - 31 \beta_{3} + 32 \beta_{2} + 12 \beta_1 - 73) q^{29} + ( - 41 \beta_{3} + \beta_{2} + 89 \beta_1 - 43) q^{31} + ( - 4 \beta_{3} + \beta_{2} + 85 \beta_1 + 48) q^{35} + ( - 35 \beta_{3} + 18 \beta_{2} - 8 \beta_1 + 45) q^{37} + (5 \beta_{2} + 49 \beta_1 - 82) q^{39} + ( - 62 \beta_{3} + 47 \beta_{2} + 40 \beta_1 - 19) q^{41} + ( - 31 \beta_{3} + 39 \beta_{2} - 168 \beta_1 + 30) q^{43} + (54 \beta_{3} - 11 \beta_{2} + 95 \beta_1 - 10) q^{45} + (9 \beta_{3} + 28 \beta_{2} + 80 \beta_1 + 189) q^{47} + ( - 69 \beta_{3} - 43 \beta_{2} + 285 \beta_1 - 208) q^{49} + ( - 20 \beta_{3} - 36 \beta_{2} + 54 \beta_1 - 27) q^{51} + ( - 41 \beta_{3} + 39 \beta_{2} - 197 \beta_1 + 331) q^{53} + ( - 27 \beta_{3} + 52 \beta_{2} - 30 \beta_1 - 10) q^{57} + ( - 32 \beta_{3} + 34 \beta_{2} + 169 \beta_1 - 104) q^{59} + ( - 147 \beta_{3} + 105 \beta_{2} + 189 \beta_1 + 21) q^{61} + (33 \beta_{3} + 19 \beta_{2} - 3 \beta_1 + 3) q^{63} + (13 \beta_{3} - 44 \beta_{2} + 156 \beta_1 - 327) q^{65} + (63 \beta_{3} + 46 \beta_{2} - 309 \beta_1 + 151) q^{67} + ( - 36 \beta_{3} + 66 \beta_{2} + 142 \beta_1 - 318) q^{69} + ( - 77 \beta_{3} - 43 \beta_{2} - 253 \beta_1 - 33) q^{71} + (44 \beta_{3} - 95 \beta_{2} - 480 \beta_1 + 249) q^{73} + (13 \beta_{3} - 42 \beta_{2} + 107 \beta_1 + 126) q^{75} + (13 \beta_{3} + 115 \beta_{2} - 81 \beta_1 + 339) q^{79} + (174 \beta_{2} + 100) q^{81} + ( - 208 \beta_{3} + 194 \beta_{2} + 283 \beta_1 + 429) q^{83} + (197 \beta_{3} + 10 \beta_{2} + 306 \beta_1 - 17) q^{85} + (43 \beta_{3} - 137 \beta_{2} - 303 \beta_1 + 361) q^{87} + (25 \beta_{3} - 57 \beta_{2} - 428 \beta_1 + 652) q^{89} + ( - 8 \beta_{3} - 21 \beta_{2} + 23 \beta_1) q^{91} + (130 \beta_{3} - 45 \beta_{2} - 547 \beta_1 + 52) q^{93} + ( - 60 \beta_{3} + 217 \beta_{2} + 113 \beta_1 + 518) q^{95} + (50 \beta_{3} - 194 \beta_{2} - 471 \beta_1 - 317) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{3} - 11 q^{5} + 25 q^{7} - 62 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{3} - 11 q^{5} + 25 q^{7} - 62 q^{9} - 25 q^{13} - 2 q^{15} - 232 q^{17} + 154 q^{19} - 167 q^{21} + 6 q^{23} - 13 q^{25} + 144 q^{27} - 363 q^{29} - 37 q^{31} + 356 q^{35} + 93 q^{37} - 240 q^{39} - 152 q^{41} - 325 q^{43} + 226 q^{45} + 869 q^{47} - 245 q^{49} + 52 q^{51} + 811 q^{53} - 231 q^{57} - 178 q^{59} + 105 q^{61} + q^{63} - 895 q^{65} - 43 q^{67} - 1156 q^{69} - 629 q^{71} + 270 q^{73} + 815 q^{75} + 977 q^{79} + 52 q^{81} + 1686 q^{83} + 721 q^{85} + 1155 q^{87} + 1891 q^{89} + 80 q^{91} - 666 q^{93} + 1804 q^{95} - 1772 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 21x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} - 13\nu + 8 ) / 16 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 29\nu - 8 ) / 16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{2} + \nu - 10 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{3} - \beta_{2} - \beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 13\beta_{2} + 29\beta _1 - 8 \) 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
−4.15942
−1.92335
4.15942
1.92335
0 −4.54138 0 −8.63533 0 1.66258 0 −6.37586 0
1.2 0 −4.54138 0 6.17671 0 32.1271 0 −6.37586 0
1.3 0 1.54138 0 −17.2669 0 −9.56478 0 −24.6241 0
1.4 0 1.54138 0 8.72550 0 0.775116 0 −24.6241 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.bl 4
4.b odd 2 1 121.4.a.f 4
11.b odd 2 1 1936.4.a.bk 4
11.d odd 10 2 176.4.m.c 8
12.b even 2 1 1089.4.a.bh 4
44.c even 2 1 121.4.a.g 4
44.g even 10 2 11.4.c.a 8
44.g even 10 2 121.4.c.h 8
44.h odd 10 2 121.4.c.b 8
44.h odd 10 2 121.4.c.i 8
132.d odd 2 1 1089.4.a.y 4
132.n odd 10 2 99.4.f.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.4.c.a 8 44.g even 10 2
99.4.f.c 8 132.n odd 10 2
121.4.a.f 4 4.b odd 2 1
121.4.a.g 4 44.c even 2 1
121.4.c.b 8 44.h odd 10 2
121.4.c.h 8 44.g even 10 2
121.4.c.i 8 44.h odd 10 2
176.4.m.c 8 11.d odd 10 2
1089.4.a.y 4 132.d odd 2 1
1089.4.a.bh 4 12.b even 2 1
1936.4.a.bk 4 11.b odd 2 1
1936.4.a.bl 4 1.a even 1 1 trivial

Hecke kernels

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

\( T_{3}^{2} + 3T_{3} - 7 \) Copy content Toggle raw display
\( T_{5}^{4} + 11T_{5}^{3} - 183T_{5}^{2} - 826T_{5} + 8036 \) Copy content Toggle raw display
\( T_{7}^{4} - 25T_{7}^{3} - 251T_{7}^{2} + 720T_{7} - 396 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3 T - 7)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 11 T^{3} - 183 T^{2} + \cdots + 8036 \) Copy content Toggle raw display
$7$ \( T^{4} - 25 T^{3} - 251 T^{2} + \cdots - 396 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 25 T^{3} - 2215 T^{2} + \cdots - 53900 \) Copy content Toggle raw display
$17$ \( T^{4} + 232 T^{3} + 18185 T^{2} + \cdots + 3099789 \) Copy content Toggle raw display
$19$ \( T^{4} - 154 T^{3} + 501 T^{2} + \cdots - 9492329 \) Copy content Toggle raw display
$23$ \( T^{4} - 6 T^{3} - 21180 T^{2} + \cdots + 49883584 \) Copy content Toggle raw display
$29$ \( T^{4} + 363 T^{3} + \cdots - 98528364 \) Copy content Toggle raw display
$31$ \( T^{4} + 37 T^{3} - 57125 T^{2} + \cdots - 33222196 \) Copy content Toggle raw display
$37$ \( T^{4} - 93 T^{3} + \cdots + 163244164 \) Copy content Toggle raw display
$41$ \( T^{4} + 152 T^{3} + \cdots + 1659084581 \) Copy content Toggle raw display
$43$ \( T^{4} + 325 T^{3} + \cdots - 1288748736 \) Copy content Toggle raw display
$47$ \( T^{4} - 869 T^{3} + \cdots + 837687536 \) Copy content Toggle raw display
$53$ \( T^{4} - 811 T^{3} + \cdots - 847714576 \) Copy content Toggle raw display
$59$ \( T^{4} + 178 T^{3} + \cdots + 258148219 \) Copy content Toggle raw display
$61$ \( T^{4} - 105 T^{3} + \cdots + 47885889744 \) Copy content Toggle raw display
$67$ \( T^{4} + 43 T^{3} + \cdots - 8869996224 \) Copy content Toggle raw display
$71$ \( T^{4} + 629 T^{3} + \cdots - 744521796 \) Copy content Toggle raw display
$73$ \( T^{4} - 270 T^{3} + \cdots + 38050128809 \) Copy content Toggle raw display
$79$ \( T^{4} - 977 T^{3} + \cdots - 210591964 \) Copy content Toggle raw display
$83$ \( T^{4} - 1686 T^{3} + \cdots - 181259356509 \) Copy content Toggle raw display
$89$ \( T^{4} - 1891 T^{3} + \cdots - 2046678844 \) Copy content Toggle raw display
$97$ \( T^{4} + 1772 T^{3} + \cdots - 357121332099 \) Copy content Toggle raw display
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