Properties

Label 96.9.e.b
Level $96$
Weight $9$
Character orbit 96.e
Analytic conductor $39.108$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [96,9,Mod(65,96)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(96, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("96.65");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 96 = 2^{5} \cdot 3 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 96.e (of order \(2\), degree \(1\), 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: \(39.1083465659\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 1632 x^{14} - 6200 x^{13} + 1101040 x^{12} - 214728 x^{11} + 414852536 x^{10} + 1040392136 x^{9} + 100771140894 x^{8} + \cdots + 46\!\cdots\!49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{110}\cdot 3^{24}\cdot 7^{2} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} - \beta_{3} q^{5} + (\beta_{2} - \beta_1) q^{7} + ( - \beta_{4} + \beta_{3} + 197) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} - \beta_{3} q^{5} + (\beta_{2} - \beta_1) q^{7} + ( - \beta_{4} + \beta_{3} + 197) q^{9} + (\beta_{12} + \beta_{5} + 14 \beta_{2}) q^{11} + ( - \beta_{11} + 3154) q^{13} + ( - \beta_{12} + \beta_{8} - 3 \beta_{6} + 2 \beta_{5} - \beta_{2} + 2 \beta_1) q^{15} + ( - \beta_{7} - 2 \beta_{4} + 2 \beta_{3}) q^{17} + (\beta_{13} - \beta_{12} - \beta_{10} + 2 \beta_{8} + \beta_{6} + \beta_{5} - 72 \beta_{2} + 12 \beta_1) q^{19} + (3 \beta_{11} - \beta_{9} + 3 \beta_{7} - \beta_{4} - 37 \beta_{3} - 3878) q^{21} + (9 \beta_{12} + \beta_{10} + 4 \beta_{8} - 18 \beta_{6} + 22 \beta_{5} + 434 \beta_{2}) q^{23} + (\beta_{15} - \beta_{14} + 6 \beta_{11} - \beta_{7} + 25 \beta_{4} + \beta_{3} - 5575) q^{25} + ( - 3 \beta_{13} + 22 \beta_{12} - \beta_{10} + 4 \beta_{6} + 29 \beta_{5} + \cdots - 58 \beta_1) q^{27}+ \cdots + ( - 201 \beta_{13} - 101 \beta_{12} - 31 \beta_{10} - 540 \beta_{8} + \cdots + 21638 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 3152 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 3152 q^{9} + 50464 q^{13} - 62048 q^{21} - 89200 q^{25} + 1439936 q^{33} + 5092384 q^{37} + 6818176 q^{45} + 16955568 q^{49} + 6929632 q^{57} + 36642336 q^{61} + 45119616 q^{69} + 91112736 q^{73} + 52195088 q^{81} + 13526016 q^{85} - 54214624 q^{93} - 77696096 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 1632 x^{14} - 6200 x^{13} + 1101040 x^{12} - 214728 x^{11} + 414852536 x^{10} + 1040392136 x^{9} + 100771140894 x^{8} + \cdots + 46\!\cdots\!49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 41\!\cdots\!20 \nu^{15} + \cdots + 39\!\cdots\!61 ) / 40\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 45\!\cdots\!00 \nu^{15} + \cdots + 65\!\cdots\!01 ) / 40\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 71\!\cdots\!26 \nu^{15} + \cdots + 34\!\cdots\!11 ) / 50\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 18\!\cdots\!77 \nu^{15} + \cdots - 11\!\cdots\!48 ) / 25\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 32\!\cdots\!84 \nu^{15} + \cdots - 23\!\cdots\!70 ) / 20\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 52\!\cdots\!44 \nu^{15} + \cdots - 53\!\cdots\!57 ) / 20\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 19\!\cdots\!80 \nu^{15} + \cdots + 36\!\cdots\!03 ) / 25\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 35\!\cdots\!68 \nu^{15} + \cdots - 37\!\cdots\!45 ) / 40\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 25\!\cdots\!49 \nu^{15} + \cdots - 39\!\cdots\!58 ) / 25\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 44\!\cdots\!00 \nu^{15} + \cdots + 65\!\cdots\!89 ) / 20\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 24\!\cdots\!35 \nu^{15} + \cdots + 96\!\cdots\!77 ) / 86\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 31\!\cdots\!96 \nu^{15} + \cdots + 83\!\cdots\!04 ) / 10\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 90\!\cdots\!76 \nu^{15} + \cdots - 13\!\cdots\!93 ) / 20\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 52\!\cdots\!88 \nu^{15} + \cdots - 12\!\cdots\!51 ) / 50\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 18\!\cdots\!10 \nu^{15} + \cdots + 48\!\cdots\!90 ) / 12\!\cdots\!97 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 71 \beta_{15} + 2 \beta_{14} + 216 \beta_{11} + 430 \beta_{9} - 71 \beta_{7} + 2268 \beta_{6} - 3582 \beta_{4} + 21816 \beta_{3} + 4536 \beta_{2} + 1161216 ) / 2322432 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 1834 \beta_{15} - 2303 \beta_{14} + 504 \beta_{13} - 1976 \beta_{12} + 3024 \beta_{11} - 536 \beta_{10} + 623 \beta_{9} + 1216 \beta_{8} + 1190 \beta_{7} + 6017 \beta_{6} - 20672 \beta_{5} + \cdots - 464486400 ) / 2322432 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 3777 \beta_{15} - 4894 \beta_{14} - 1008 \beta_{13} - 2744 \beta_{12} - 28584 \beta_{11} - 728 \beta_{10} - 61738 \beta_{9} + 6496 \beta_{8} - 8823 \beta_{7} - 495775 \beta_{6} + \cdots - 970389504 ) / 774144 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 827680 \beta_{15} + 830879 \beta_{14} - 207648 \beta_{13} + 703280 \beta_{12} - 1965600 \beta_{11} + 135920 \beta_{10} - 1241639 \beta_{9} - 667264 \beta_{8} + \cdots + 102679363584 ) / 2322432 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1533617 \beta_{15} + 10607837 \beta_{14} + 2478420 \beta_{13} + 10374560 \beta_{12} + 23584392 \beta_{11} + 2814560 \beta_{10} + 59304619 \beta_{9} + \cdots + 1569941968896 ) / 2322432 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 98867818 \beta_{15} - 41627964 \beta_{14} + 42685776 \beta_{13} + 29582904 \beta_{12} + 328282416 \beta_{11} + 5298456 \beta_{10} + 258285076 \beta_{9} + \cdots - 7565010259968 ) / 774144 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 821256031 \beta_{15} - 5045882210 \beta_{14} - 1532670804 \beta_{13} - 5835338264 \beta_{12} - 7182562248 \beta_{11} - 1205741432 \beta_{10} + \cdots - 540357140219904 ) / 2322432 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 3551872106 \beta_{15} - 1292772326 \beta_{14} - 2889880992 \beta_{13} - 4245318728 \beta_{12} - 15423821712 \beta_{11} - 1446268904 \beta_{10} + \cdots + 115644902999040 ) / 82944 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 271146222143 \beta_{15} + 517907350830 \beta_{14} + 223964863920 \beta_{13} + 463939977312 \beta_{12} + 688249537128 \beta_{11} + \cdots + 42\!\cdots\!56 ) / 774144 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 29965600728266 \beta_{15} + 40862657381777 \beta_{14} + 43640997828120 \beta_{13} + 50619896914952 \beta_{12} + 177402157965648 \beta_{11} + \cdots + 27\!\cdots\!80 ) / 2322432 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 429355911461273 \beta_{15} - 167488782530134 \beta_{14} - 184683968541072 \beta_{13} + 140125336006040 \beta_{12} - 434226957772824 \beta_{11} + \cdots - 63\!\cdots\!04 ) / 2322432 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 24\!\cdots\!84 \beta_{15} + \cdots - 10\!\cdots\!44 ) / 774144 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 16\!\cdots\!59 \beta_{15} + \cdots - 15\!\cdots\!24 ) / 2322432 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 11\!\cdots\!70 \beta_{15} + \cdots + 12\!\cdots\!16 ) / 2322432 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 16\!\cdots\!57 \beta_{15} + \cdots + 41\!\cdots\!36 ) / 774144 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/96\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(37\) \(65\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
−0.195942 + 17.2691i
−0.195942 17.2691i
3.86965 + 12.4191i
3.86965 12.4191i
5.31078 + 16.8759i
5.31078 16.8759i
−6.98449 + 10.3192i
−6.98449 10.3192i
−6.98449 7.49075i
−6.98449 + 7.49075i
5.31078 19.7044i
5.31078 + 19.7044i
3.86965 9.59065i
3.86965 + 9.59065i
−0.195942 20.0975i
−0.195942 + 20.0975i
0 −80.3709 10.0760i 0 969.842i 0 878.697 0 6357.95 + 1619.63i 0
65.2 0 −80.3709 + 10.0760i 0 969.842i 0 878.697 0 6357.95 1619.63i 0
65.3 0 −69.8984 40.9294i 0 427.907i 0 −2483.92 0 3210.56 + 5721.80i 0
65.4 0 −69.8984 + 40.9294i 0 427.907i 0 −2483.92 0 3210.56 5721.80i 0
65.5 0 −41.6091 69.4959i 0 590.447i 0 595.059 0 −3098.37 + 5783.32i 0
65.6 0 −41.6091 + 69.4959i 0 590.447i 0 595.059 0 −3098.37 5783.32i 0
65.7 0 −20.9626 78.2405i 0 335.371i 0 4472.36 0 −5682.14 + 3280.25i 0
65.8 0 −20.9626 + 78.2405i 0 335.371i 0 4472.36 0 −5682.14 3280.25i 0
65.9 0 20.9626 78.2405i 0 335.371i 0 −4472.36 0 −5682.14 3280.25i 0
65.10 0 20.9626 + 78.2405i 0 335.371i 0 −4472.36 0 −5682.14 + 3280.25i 0
65.11 0 41.6091 69.4959i 0 590.447i 0 −595.059 0 −3098.37 5783.32i 0
65.12 0 41.6091 + 69.4959i 0 590.447i 0 −595.059 0 −3098.37 + 5783.32i 0
65.13 0 69.8984 40.9294i 0 427.907i 0 2483.92 0 3210.56 5721.80i 0
65.14 0 69.8984 + 40.9294i 0 427.907i 0 2483.92 0 3210.56 + 5721.80i 0
65.15 0 80.3709 10.0760i 0 969.842i 0 −878.697 0 6357.95 1619.63i 0
65.16 0 80.3709 + 10.0760i 0 969.842i 0 −878.697 0 6357.95 + 1619.63i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 96.9.e.b 16
3.b odd 2 1 inner 96.9.e.b 16
4.b odd 2 1 inner 96.9.e.b 16
8.b even 2 1 192.9.e.l 16
8.d odd 2 1 192.9.e.l 16
12.b even 2 1 inner 96.9.e.b 16
24.f even 2 1 192.9.e.l 16
24.h odd 2 1 192.9.e.l 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.9.e.b 16 1.a even 1 1 trivial
96.9.e.b 16 3.b odd 2 1 inner
96.9.e.b 16 4.b odd 2 1 inner
96.9.e.b 16 12.b even 2 1 inner
192.9.e.l 16 8.b even 2 1
192.9.e.l 16 8.d odd 2 1
192.9.e.l 16 24.f even 2 1
192.9.e.l 16 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 1584800T_{5}^{6} + 729577765248T_{5}^{4} + 123476140339865600T_{5}^{2} + 6753289812415982080000 \) acting on \(S_{9}^{\mathrm{new}}(96, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} - 1576 T^{14} + \cdots + 34\!\cdots\!81 \) Copy content Toggle raw display
$5$ \( (T^{8} + 1584800 T^{6} + \cdots + 67\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} - 27298096 T^{6} + \cdots + 33\!\cdots\!92)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 869687264 T^{6} + \cdots + 75\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 12616 T^{3} + \cdots + 41\!\cdots\!00)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} + 16128645120 T^{6} + \cdots + 39\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - 113875911856 T^{6} + \cdots + 43\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 346718655360 T^{6} + \cdots + 10\!\cdots\!16)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 1892527938464 T^{6} + \cdots + 30\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 2694902057392 T^{6} + \cdots + 57\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 1273096 T^{3} + \cdots + 18\!\cdots\!00)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} + 48994196190848 T^{6} + \cdots + 42\!\cdots\!32)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - 50795677404592 T^{6} + \cdots + 11\!\cdots\!52)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 52662402321920 T^{6} + \cdots + 51\!\cdots\!36)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 509421185089952 T^{6} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 259443682637792 T^{6} + \cdots + 20\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 9160584 T^{3} + \cdots + 50\!\cdots\!20)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 33\!\cdots\!52)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 22778184 T^{3} + \cdots - 34\!\cdots\!00)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 10\!\cdots\!04)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 19424024 T^{3} + \cdots + 56\!\cdots\!00)^{4} \) Copy content Toggle raw display
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