Properties

Label 32.11.d.b
Level $32$
Weight $11$
Character orbit 32.d
Analytic conductor $20.331$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [32,11,Mod(15,32)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(32, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("32.15");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 32 = 2^{5} \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 32.d (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(20.3314320856\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 5x^{6} - 51x^{5} + 30855x^{4} - 121569x^{3} + 12144527x^{2} + 279415575x + 3348211684 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{57}\cdot 3^{3} \)
Twist minimal: no (minimal twist has level 8)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 60) q^{3} + \beta_{3} q^{5} + ( - \beta_{4} + \beta_{3}) q^{7} + (9 \beta_{2} - 150 \beta_1 - 4437) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 60) q^{3} + \beta_{3} q^{5} + ( - \beta_{4} + \beta_{3}) q^{7} + (9 \beta_{2} - 150 \beta_1 - 4437) q^{9} + ( - \beta_{5} + 2 \beta_{2} - 204 \beta_1 - 17916) q^{11} + (\beta_{6} - 3 \beta_{4} + 10 \beta_{3}) q^{13} + (\beta_{7} - 2 \beta_{6} + 4 \beta_{4} - 4 \beta_{3}) q^{15} + (8 \beta_{5} - 245 \beta_{2} + 742 \beta_1 - 46350) q^{17} + ( - \beta_{5} - 798 \beta_{2} + 3564 \beta_1 - 219164) q^{19} + (8 \beta_{7} - \beta_{6} + 155 \beta_{4} - 167 \beta_{3}) q^{21} + (13 \beta_{7} + 6 \beta_{6} + 42 \beta_{4} - 554 \beta_{3}) q^{23} + ( - 40 \beta_{5} - 2970 \beta_{2} - 12300 \beta_1 - 2349095) q^{25} + (27 \beta_{5} - 2646 \beta_{2} - 37287 \beta_1 - 3618648) q^{27} + (56 \beta_{7} - 24 \beta_{6} - 912 \beta_{4} + 723 \beta_{3}) q^{29} + (63 \beta_{7} + 34 \beta_{6} + 227 \beta_{4} + 5405 \beta_{3}) q^{31} + ( - 48 \beta_{5} - 11403 \beta_{2} + 7346 \beta_1 - 9288588) q^{33} + (26 \beta_{5} - 15444 \beta_{2} + 22658 \beta_1 - 6717120) q^{35} + (96 \beta_{7} + 23 \beta_{6} + 3803 \beta_{4} - 3906 \beta_{3}) q^{37} + (104 \beta_{7} - 112 \beta_{6} + 485 \beta_{4} - 26597 \beta_{3}) q^{39} + (520 \beta_{5} - 6734 \beta_{2} - 111780 \beta_1 + 11583666) q^{41} + ( - 330 \beta_{5} - 8364 \beta_{2} + 139815 \beta_1 + 1273828) q^{43} + ( - 168 \beta_{7} + 255 \beta_{6} - 3957 \beta_{4} + 6288 \beta_{3}) q^{45} + ( - 203 \beta_{7} - 234 \beta_{6} - 1713 \beta_{4} + 69297 \beta_{3}) q^{47} + ( - 256 \beta_{5} + 63192 \beta_{2} + 507120 \beta_1 - 10054031) q^{49} + ( - 303 \beta_{5} + 116190 \beta_{2} - 491465 \beta_1 + 34589352) q^{51} + ( - 728 \beta_{7} - 231 \beta_{6} - 11091 \beta_{4} + 14504 \beta_{3}) q^{53} + ( - 1092 \beta_{7} + 872 \beta_{6} - 5633 \beta_{4} - 104447 \beta_{3}) q^{55} + ( - 2448 \beta_{5} + 137709 \beta_{2} - 1661822 \beta_1 + 175089108) q^{57} + (2384 \beta_{5} + 125888 \beta_{2} + 868761 \beta_1 + 120705252) q^{59} + ( - 840 \beta_{7} - 1543 \beta_{6} + 31677 \beta_{4} - 48994 \beta_{3}) q^{61} + ( - 1245 \beta_{7} + 762 \beta_{6} + 8160 \beta_{4} + 93216 \beta_{3}) q^{63} + (2888 \beta_{5} + 10698 \beta_{2} + 6133004 \beta_1 - 104878560) q^{65} + (2055 \beta_{5} - 48078 \beta_{2} + 1721316 \beta_1 + 153197860) q^{67} + ( - 280 \beta_{7} + 1289 \beta_{6} - 52195 \beta_{4} + 15583 \beta_{3}) q^{69} + (1339 \beta_{7} - 3414 \beta_{6} + 19434 \beta_{4} + 57878 \beta_{3}) q^{71} + (3328 \beta_{5} - 143871 \beta_{2} - 9329766 \beta_1 + 350009090) q^{73} + ( - 11070 \beta_{5} - 54180 \beta_{2} - 5269565 \beta_1 - 560694300) q^{75} + (2360 \beta_{7} + 5571 \beta_{6} + 26079 \beta_{4} + 60149 \beta_{3}) q^{77} + (4884 \beta_{7} - 712 \beta_{6} - 58670 \beta_{4} - 487634 \beta_{3}) q^{79} + ( - 6480 \beta_{5} - 235467 \beta_{2} + \cdots - 1488633147) q^{81}+ \cdots + (22248 \beta_{5} + 1144656 \beta_{2} - 13797525 \beta_1 + 1705843404) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 480 q^{3} - 35496 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 480 q^{3} - 35496 q^{9} - 143328 q^{11} - 370800 q^{17} - 1753312 q^{19} - 18792760 q^{25} - 28949184 q^{27} - 74308704 q^{33} - 53736960 q^{35} + 92669328 q^{41} + 10190624 q^{43} - 80432248 q^{49} + 276714816 q^{51} + 1400712864 q^{57} + 965642016 q^{59} - 839028480 q^{65} + 1225582880 q^{67} + 2800072720 q^{73} - 4485554400 q^{75} - 11909065176 q^{81} - 1853422560 q^{83} + 6162596112 q^{89} - 7645985280 q^{91} - 9697863536 q^{97} + 13646747232 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 5x^{6} - 51x^{5} + 30855x^{4} - 121569x^{3} + 12144527x^{2} + 279415575x + 3348211684 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 39 \nu^{7} - 640 \nu^{6} + 6467 \nu^{5} + 38788 \nu^{4} - 639715 \nu^{3} - 3220592 \nu^{2} + 460044665 \nu + 5414621036 ) / 29360128 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 73 \nu^{7} + 640 \nu^{6} - 5907 \nu^{5} - 42820 \nu^{4} + 4083379 \nu^{3} - 64144 \nu^{2} - 988760041 \nu + 25703063380 ) / 14680064 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 51 \nu^{7} - 2688 \nu^{6} + 81793 \nu^{5} - 1067988 \nu^{4} + 10212255 \nu^{3} - 64144848 \nu^{2} + 270376211 \nu - 56308982748 ) / 14680064 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 33 \nu^{7} - 128 \nu^{6} - 8411 \nu^{5} + 472156 \nu^{4} + 3373179 \nu^{3} - 8362768 \nu^{2} + 510138431 \nu + 15161255732 ) / 2097152 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 21393 \nu^{7} - 315776 \nu^{6} + 265557 \nu^{5} + 1444508 \nu^{4} + 408149835 \nu^{3} + 7067850992 \nu^{2} + 197013680079 \nu + 3049502010868 ) / 29360128 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 8361 \nu^{7} + 131712 \nu^{6} + 1075 \nu^{5} - 492860 \nu^{4} - 362453715 \nu^{3} + 11780215184 \nu^{2} - 122965473463 \nu - 1096096173780 ) / 7340032 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 6745 \nu^{7} - 2688 \nu^{6} + 246845 \nu^{5} - 6817668 \nu^{4} + 200294499 \nu^{3} - 218631312 \nu^{2} + 201729495303 \nu + 1657680830100 ) / 7340032 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} + 3\beta_{4} - 3\beta_{3} - 128\beta_{2} - 256\beta _1 + 12288 ) / 32768 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{7} + 16\beta_{6} + 32\beta_{5} + 89\beta_{4} + 167\beta_{3} - 384\beta_{2} - 5088\beta _1 - 4096 ) / 32768 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 37\beta_{7} + 256\beta_{5} + 6255\beta_{4} + 14225\beta_{3} + 10624\beta_{2} - 242688\beta _1 + 552960 ) / 32768 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 981 \beta_{7} + 1104 \beta_{6} - 4192 \beta_{5} + 127505 \beta_{4} + 751 \beta_{3} - 37248 \beta_{2} + 2555808 \beta _1 - 503222272 ) / 32768 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 16333 \beta_{7} + 40896 \beta_{6} - 150144 \beta_{5} + 880167 \beta_{4} + 3698137 \beta_{3} + 5472640 \beta_{2} + 84888704 \beta _1 + 597946368 ) / 32768 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 988899 \beta_{7} + 418192 \beta_{6} - 2574048 \beta_{5} + 1391737 \beta_{4} - 8143737 \beta_{3} + 154908288 \beta_{2} + 468765984 \beta _1 - 292505522176 ) / 32768 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 3554037 \beta_{7} + 304512 \beta_{6} - 6332928 \beta_{5} - 175360929 \beta_{4} - 465103967 \beta_{3} + 3324098944 \beta_{2} + 14361714944 \beta _1 - 9084365475840 ) / 32768 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/32\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(31\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
15.1
−8.43092 + 11.1953i
−8.43092 11.1953i
16.5295 8.54130i
16.5295 + 8.54130i
4.83019 + 15.8950i
4.83019 15.8950i
−11.4287 6.91461i
−11.4287 + 6.91461i
0 −352.099 0 3773.58i 0 15618.2i 0 64924.4 0
15.2 0 −352.099 0 3773.58i 0 15618.2i 0 64924.4 0
15.3 0 −203.867 0 2088.87i 0 25367.2i 0 −17487.2 0
15.4 0 −203.867 0 2088.87i 0 25367.2i 0 −17487.2 0
15.5 0 119.281 0 948.910i 0 15458.1i 0 −44821.0 0
15.6 0 119.281 0 948.910i 0 15458.1i 0 −44821.0 0
15.7 0 196.684 0 5381.00i 0 6613.83i 0 −20364.2 0
15.8 0 196.684 0 5381.00i 0 6613.83i 0 −20364.2 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 15.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 32.11.d.b 8
3.b odd 2 1 288.11.b.b 8
4.b odd 2 1 8.11.d.b 8
8.b even 2 1 8.11.d.b 8
8.d odd 2 1 inner 32.11.d.b 8
12.b even 2 1 72.11.b.b 8
16.e even 4 2 256.11.c.m 16
16.f odd 4 2 256.11.c.m 16
24.f even 2 1 288.11.b.b 8
24.h odd 2 1 72.11.b.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.11.d.b 8 4.b odd 2 1
8.11.d.b 8 8.b even 2 1
32.11.d.b 8 1.a even 1 1 trivial
32.11.d.b 8 8.d odd 2 1 inner
72.11.b.b 8 12.b even 2 1
72.11.b.b 8 24.h odd 2 1
256.11.c.m 16 16.e even 4 2
256.11.c.m 16 16.f odd 4 2
288.11.b.b 8 3.b odd 2 1
288.11.b.b 8 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 240T_{3}^{3} - 80424T_{3}^{2} - 9637056T_{3} + 1684044432 \) acting on \(S_{11}^{\mathrm{new}}(32, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 240 T^{3} - 80424 T^{2} + \cdots + 1684044432)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} + 48458880 T^{6} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{8} + 1170117120 T^{6} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( (T^{4} + 71664 T^{3} + \cdots + 56\!\cdots\!48)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 643617006720 T^{6} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{4} + 185400 T^{3} + \cdots + 29\!\cdots\!32)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 876656 T^{3} + \cdots + 20\!\cdots\!48)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 126385472340480 T^{6} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{4} - 46334664 T^{3} + \cdots + 68\!\cdots\!88)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 5095312 T^{3} + \cdots - 51\!\cdots\!16)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{4} - 482821008 T^{3} + \cdots + 47\!\cdots\!56)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{4} - 612791440 T^{3} + \cdots - 13\!\cdots\!88)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 81\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{4} - 1400036360 T^{3} + \cdots + 23\!\cdots\!12)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 57\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{4} + 926711280 T^{3} + \cdots + 77\!\cdots\!72)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 3081298056 T^{3} + \cdots + 20\!\cdots\!28)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 4848931768 T^{3} + \cdots + 57\!\cdots\!84)^{2} \) Copy content Toggle raw display
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