Properties

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

Related objects

Downloads

Learn more

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: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.978025.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 99x^{2} + 100x + 2420 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 11 \)
Twist minimal: no (minimal twist has level 22)
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.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{3} + ( - \beta_{2} + 6) q^{5} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 + 1) q^{7} + ( - 2 \beta_{2} - \beta_1 + 25) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{3} + ( - \beta_{2} + 6) q^{5} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 + 1) q^{7} + ( - 2 \beta_{2} - \beta_1 + 25) q^{9} + ( - 2 \beta_{3} + \beta_{2} - 6 \beta_1 + 10) q^{13} + ( - 3 \beta_{3} - 5 \beta_{2} + 9 \beta_1 - 19) q^{15} + ( - 4 \beta_{3} - 3 \beta_1 - 14) q^{17} + (5 \beta_{3} + 7 \beta_{2} + 6 \beta_1 + 7) q^{19} + (14 \beta_{3} + 11 \beta_{2} - 2 \beta_1) q^{21} + (8 \beta_{3} + 2 \beta_{2} - 6 \beta_1 - 76) q^{23} + (3 \beta_{3} - 6 \beta_{2} + 7 \beta_1 - 6) q^{25} + ( - 6 \beta_{3} - 8 \beta_{2} + 4 \beta_1 - 75) q^{27} + ( - 5 \beta_{3} - 6 \beta_{2} - 9 \beta_1 - 143) q^{29} + (14 \beta_{3} - 7 \beta_{2} - 4 \beta_1 - 48) q^{31} + ( - 13 \beta_{3} + 3 \beta_{2} - 19 \beta_1 - 181) q^{35} + (5 \beta_{3} + 10 \beta_{2} + 27 \beta_1 + 93) q^{37} + (19 \beta_{3} + 15 \beta_{2} + 13 \beta_1 - 251) q^{39} + ( - 2 \beta_{3} + 22 \beta_{2} - 7 \beta_1 - 3) q^{41} + ( - 19 \beta_{3} - 12 \beta_{2} - 14 \beta_1 - 188) q^{43} + (9 \beta_{3} - 19 \beta_{2} + 5 \beta_1 + 329) q^{45} + (11 \beta_{3} + 20 \beta_{2} - 21 \beta_1 - 93) q^{47} + (4 \beta_{3} + 9 \beta_{2} + 2 \beta_1 + 209) q^{49} + (32 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 35) q^{51} + (18 \beta_{3} - 21 \beta_{2} - 8 \beta_1 - 34) q^{53} + ( - 19 \beta_{3} + 28 \beta_{2} - 29 \beta_1 + 260) q^{57} + ( - 9 \beta_{3} + 5 \beta_{2} + 26 \beta_1 - 255) q^{59} + (14 \beta_{3} - 13 \beta_{2} + 28 \beta_1 + 230) q^{61} + ( - 52 \beta_{3} + 19 \beta_{2} - 48 \beta_1 - 350) q^{63} + ( - 5 \beta_{3} + 22 \beta_{2} - 53 \beta_1 - 13) q^{65} + (30 \beta_{3} - 29 \beta_{2} + 51 \beta_1 - 72) q^{67} + ( - 58 \beta_{3} + 30 \beta_{2} - 106 \beta_1 - 412) q^{69} + ( - 28 \beta_{3} + 23 \beta_{2} - 32 \beta_1 + 300) q^{71} + (8 \beta_{3} - 54 \beta_{2} + 45 \beta_1 + 297) q^{73} + ( - 42 \beta_{3} - 41 \beta_{2} + 3 \beta_1 + 207) q^{75} + ( - 14 \beta_{3} + 41 \beta_{2} - 36 \beta_1 - 144) q^{79} + (24 \beta_{3} - 6 \beta_1 - 344) q^{81} + (17 \beta_{3} + 5 \beta_{2} - 92 \beta_1 + 384) q^{83} + ( - 31 \beta_{3} + 24 \beta_{2} - 11 \beta_1 - 181) q^{85} + (22 \beta_{3} - 17 \beta_{2} - 110 \beta_1 - 264) q^{87} + ( - 15 \beta_{3} - 76 \beta_{2} + 26 \beta_1 - 298) q^{89} + ( - 73 \beta_{3} - 45 \beta_{2} - 31 \beta_1 + 547) q^{91} + ( - 133 \beta_{3} - 13 \beta_{2} - 69 \beta_1 - 611) q^{93} + (11 \beta_{3} - 33 \beta_{2} - 15 \beta_1 - 447) q^{95} + (79 \beta_{3} + 69 \beta_{2} + 20 \beta_1 + 566) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} + 25 q^{5} + 3 q^{7} + 102 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} + 25 q^{5} + 3 q^{7} + 102 q^{9} + 41 q^{13} - 68 q^{15} - 52 q^{17} + 16 q^{19} - 25 q^{21} - 314 q^{23} - 21 q^{25} - 286 q^{27} - 561 q^{29} - 199 q^{31} - 714 q^{35} + 357 q^{37} - 1038 q^{39} - 32 q^{41} - 721 q^{43} + 1326 q^{45} - 403 q^{47} + 823 q^{49} - 174 q^{51} - 133 q^{53} + 1031 q^{57} - 1016 q^{59} + 919 q^{61} - 1367 q^{63} - 69 q^{65} - 289 q^{67} - 1620 q^{69} + 1205 q^{71} + 1234 q^{73} + 911 q^{75} - 603 q^{79} - 1400 q^{81} + 1514 q^{83} - 717 q^{85} - 1061 q^{87} - 1101 q^{89} + 2306 q^{91} - 2298 q^{93} - 1766 q^{95} + 2116 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 99x^{2} + 100x + 2420 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{2} + 7\nu - 54 ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} - 2\nu^{2} + 57\nu + 146 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} - 9\nu^{2} - 46\nu + 400 ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} + 11\beta _1 + 6 ) / 11 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -7\beta_{3} - 7\beta_{2} + 11\beta _1 + 552 ) / 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 71\beta_{3} - 17\beta_{2} + 605\beta _1 + 844 ) / 11 \) 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
−5.92695
−7.19378
6.92695
8.19378
0 −8.54499 0 12.7359 0 −23.4611 0 46.0168 0
1.2 0 −7.57575 0 5.40810 0 22.1498 0 30.3919 0
1.3 0 4.30892 0 −8.06215 0 26.0792 0 −8.43321 0
1.4 0 7.81182 0 14.9181 0 −21.7679 0 34.0245 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.bn 4
4.b odd 2 1 242.4.a.n 4
11.b odd 2 1 1936.4.a.bm 4
11.c even 5 2 176.4.m.b 8
12.b even 2 1 2178.4.a.by 4
44.c even 2 1 242.4.a.o 4
44.g even 10 2 242.4.c.n 8
44.g even 10 2 242.4.c.q 8
44.h odd 10 2 22.4.c.b 8
44.h odd 10 2 242.4.c.r 8
132.d odd 2 1 2178.4.a.bt 4
132.o even 10 2 198.4.f.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.4.c.b 8 44.h odd 10 2
176.4.m.b 8 11.c even 5 2
198.4.f.d 8 132.o even 10 2
242.4.a.n 4 4.b odd 2 1
242.4.a.o 4 44.c even 2 1
242.4.c.n 8 44.g even 10 2
242.4.c.q 8 44.g even 10 2
242.4.c.r 8 44.h odd 10 2
1936.4.a.bm 4 11.b odd 2 1
1936.4.a.bn 4 1.a even 1 1 trivial
2178.4.a.bt 4 132.d odd 2 1
2178.4.a.by 4 12.b even 2 1

Hecke kernels

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

\( T_{3}^{4} + 4T_{3}^{3} - 97T_{3}^{2} - 242T_{3} + 2179 \) Copy content Toggle raw display
\( T_{5}^{4} - 25T_{5}^{3} + 73T_{5}^{2} + 1710T_{5} - 8284 \) Copy content Toggle raw display
\( T_{7}^{4} - 3T_{7}^{3} - 1093T_{7}^{2} + 1496T_{7} + 295004 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 4 T^{3} - 97 T^{2} + \cdots + 2179 \) Copy content Toggle raw display
$5$ \( T^{4} - 25 T^{3} + 73 T^{2} + \cdots - 8284 \) Copy content Toggle raw display
$7$ \( T^{4} - 3 T^{3} - 1093 T^{2} + \cdots + 295004 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 41 T^{3} - 3151 T^{2} + \cdots + 126580 \) Copy content Toggle raw display
$17$ \( T^{4} + 52 T^{3} - 3079 T^{2} + \cdots - 2019455 \) Copy content Toggle raw display
$19$ \( T^{4} - 16 T^{3} - 12611 T^{2} + \cdots + 1910345 \) Copy content Toggle raw display
$23$ \( T^{4} + 314 T^{3} + \cdots - 257882816 \) Copy content Toggle raw display
$29$ \( T^{4} + 561 T^{3} + \cdots + 102492820 \) Copy content Toggle raw display
$31$ \( T^{4} + 199 T^{3} + \cdots - 922892780 \) Copy content Toggle raw display
$37$ \( T^{4} - 357 T^{3} + \cdots + 1010724084 \) Copy content Toggle raw display
$41$ \( T^{4} + 32 T^{3} + \cdots + 777346421 \) Copy content Toggle raw display
$43$ \( T^{4} + 721 T^{3} + \cdots - 2713875120 \) Copy content Toggle raw display
$47$ \( T^{4} + 403 T^{3} + \cdots + 314788720 \) Copy content Toggle raw display
$53$ \( T^{4} + 133 T^{3} + \cdots + 628715536 \) Copy content Toggle raw display
$59$ \( T^{4} + 1016 T^{3} + \cdots + 184247305 \) Copy content Toggle raw display
$61$ \( T^{4} - 919 T^{3} + \cdots - 4825318480 \) Copy content Toggle raw display
$67$ \( T^{4} + 289 T^{3} + \cdots + 35027256944 \) Copy content Toggle raw display
$71$ \( T^{4} - 1205 T^{3} + \cdots + 4828358036 \) Copy content Toggle raw display
$73$ \( T^{4} - 1234 T^{3} + \cdots + 17904806629 \) Copy content Toggle raw display
$79$ \( T^{4} + 603 T^{3} + \cdots + 26212027500 \) Copy content Toggle raw display
$83$ \( T^{4} - 1514 T^{3} + \cdots + 40525024445 \) Copy content Toggle raw display
$89$ \( T^{4} + 1101 T^{3} + \cdots - 64943655580 \) Copy content Toggle raw display
$97$ \( T^{4} - 2116 T^{3} + \cdots + 216660545541 \) Copy content Toggle raw display
show more
show less