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Properties

Label 91.2.r.a
Level $91$
Weight $2$
Character orbit 91.r
Analytic conductor $0.727$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [91,2,Mod(25,91)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(91, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("91.25");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 91.r (of order \(6\), degree \(2\), 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: \(0.726638658394\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 11x^{14} + 85x^{12} - 334x^{10} + 952x^{8} - 1050x^{6} + 853x^{4} - 93x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{11} q^{2} + (\beta_{6} + \beta_{3} - 1) q^{3} + ( - \beta_{8} - \beta_{6} + 1) q^{4} - \beta_{13} q^{5} + (\beta_{13} + \beta_{12} + \cdots - \beta_1) q^{6}+ \cdots + ( - \beta_{9} + \beta_{7} - 2 \beta_{6} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{11} q^{2} + (\beta_{6} + \beta_{3} - 1) q^{3} + ( - \beta_{8} - \beta_{6} + 1) q^{4} - \beta_{13} q^{5} + (\beta_{13} + \beta_{12} + \cdots - \beta_1) q^{6}+ \cdots + ( - 2 \beta_{15} + \beta_{14} + \cdots + 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{3} + 6 q^{4} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{3} + 6 q^{4} - 12 q^{9} - 6 q^{10} + 18 q^{12} - 12 q^{13} - 26 q^{14} + 2 q^{16} + 8 q^{17} - 36 q^{22} - 12 q^{23} - 6 q^{26} + 32 q^{27} - 16 q^{29} + 38 q^{30} - 56 q^{36} + 34 q^{38} + 18 q^{39} - 4 q^{40} + 16 q^{42} + 16 q^{43} + 36 q^{48} + 40 q^{49} + 16 q^{51} - 42 q^{52} - 20 q^{53} + 24 q^{55} - 36 q^{56} - 12 q^{61} + 44 q^{62} + 88 q^{64} - 30 q^{65} + 2 q^{66} - 2 q^{68} - 56 q^{69} + 42 q^{74} + 8 q^{75} - 76 q^{77} + 20 q^{78} + 20 q^{79} - 24 q^{81} - 16 q^{82} - 68 q^{87} + 4 q^{88} - 216 q^{90} + 56 q^{91} + 12 q^{92} - 26 q^{94} - 16 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 11x^{14} + 85x^{12} - 334x^{10} + 952x^{8} - 1050x^{6} + 853x^{4} - 93x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 24498 \nu^{14} + 246060 \nu^{12} - 1852321 \nu^{10} + 6411671 \nu^{8} - 17193085 \nu^{6} + \cdots - 40872159 ) / 14163622 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 172099 \nu^{14} - 2170865 \nu^{12} + 17340370 \nu^{10} - 78484018 \nu^{8} + 236538400 \nu^{6} + \cdots - 23829231 ) / 42490866 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 99072 \nu^{14} + 1000291 \nu^{12} - 7490944 \nu^{10} + 25929344 \nu^{8} - 66564370 \nu^{6} + \cdots - 20454191 ) / 14163622 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 24498 \nu^{15} - 246060 \nu^{13} + 1852321 \nu^{11} - 6411671 \nu^{9} + 17193085 \nu^{7} + \cdots + 55035781 \nu ) / 14163622 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 539569 \nu^{14} - 5861765 \nu^{12} + 45125185 \nu^{10} - 174659083 \nu^{8} + 494434675 \nu^{6} + \cdots - 5618970 ) / 42490866 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 102312 \nu^{14} - 1048444 \nu^{12} + 7735924 \nu^{10} - 26777324 \nu^{8} + 67022271 \nu^{6} + \cdots + 23341327 ) / 7081811 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 515071 \nu^{14} + 5615705 \nu^{12} - 43272864 \nu^{10} + 168247412 \nu^{8} - 477241590 \nu^{6} + \cdots + 46491129 ) / 14163622 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 3598 \nu^{14} + 40712 \nu^{12} - 317920 \nu^{10} + 1287475 \nu^{8} - 3722089 \nu^{6} + \cdots + 366555 ) / 95271 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 123570 \nu^{15} + 1246351 \nu^{13} - 9343265 \nu^{11} + 32341015 \nu^{9} - 83757455 \nu^{7} + \cdots - 61326350 \nu ) / 14163622 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 539569 \nu^{15} - 5861765 \nu^{13} + 45125185 \nu^{11} - 174659083 \nu^{9} + 494434675 \nu^{7} + \cdots - 48109836 \nu ) / 42490866 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1079138 \nu^{15} - 11723530 \nu^{13} + 90250370 \nu^{11} - 349318166 \nu^{9} + \cdots - 11237940 \nu ) / 21245433 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 205171 \nu^{15} + 2261795 \nu^{13} - 17480377 \nu^{11} + 68898667 \nu^{9} - 196608895 \nu^{7} + \cdots + 19219314 \nu ) / 3862806 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 1137374 \nu^{15} - 12639109 \nu^{13} + 98087264 \nu^{11} - 390741062 \nu^{9} + \cdots - 167644965 \nu ) / 11588418 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 17689120 \nu^{15} + 191553668 \nu^{13} - 1470474691 \nu^{11} + 5653350919 \nu^{9} + \cdots - 359333319 \nu ) / 127472598 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} + 3\beta_{6} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{12} + 4\beta_{11} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5\beta_{8} + 14\beta_{6} + \beta_{3} - 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{13} - 6\beta_{12} + 19\beta_{11} + 6\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{7} + 8\beta_{4} - 24\beta_{2} - 61 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -\beta_{15} - \beta_{14} + 11\beta_{10} + 32\beta_{5} - 94\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -11\beta_{9} - 115\beta_{8} + 11\beta_{7} - 345\beta_{6} + 52\beta_{4} - 52\beta_{3} - 115\beta_{2} + 52 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 22 \beta_{15} + 11 \beta_{14} + 85 \beta_{13} + 145 \beta_{12} - 493 \beta_{11} + 85 \beta_{10} + \cdots - 482 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -85\beta_{9} - 553\beta_{8} - 1736\beta_{6} - 315\beta_{3} + 1736 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -85\beta_{15} + 170\beta_{14} + 570\beta_{13} + 698\beta_{12} - 2544\beta_{11} - 783\beta_{5} - 85\beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -570\beta_{7} - 1838\beta_{4} + 2672\beta_{2} + 6935 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 570\beta_{15} + 570\beta_{14} - 3548\beta_{10} - 4510\beta_{5} + 12015\beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 3548 \beta_{9} + 12977 \beta_{8} - 3548 \beta_{7} + 44495 \beta_{6} - 10466 \beta_{4} + 10466 \beta_{3} + \cdots - 10466 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 7096 \beta_{15} - 3548 \beta_{14} - 21110 \beta_{13} - 16347 \beta_{12} + 68116 \beta_{11} + \cdots + 64568 \beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/91\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(66\)
\(\chi(n)\) \(-1\) \(-\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1
1.97871 + 1.14241i
1.84073 + 1.06275i
0.929293 + 0.536527i
0.287846 + 0.166188i
−0.287846 − 0.166188i
−0.929293 − 0.536527i
−1.84073 − 1.06275i
−1.97871 − 1.14241i
1.97871 − 1.14241i
1.84073 − 1.06275i
0.929293 − 0.536527i
0.287846 − 0.166188i
−0.287846 + 0.166188i
−0.929293 + 0.536527i
−1.84073 + 1.06275i
−1.97871 + 1.14241i
−1.97871 + 1.14241i −1.57521 + 2.72835i 1.61019 − 2.78892i −1.84030 + 1.06250i − 7.19813i 2.62488 + 0.331665i 2.78832i −3.46258 − 5.99736i 2.42760 − 4.20473i
25.2 −1.84073 + 1.06275i 0.0894272 − 0.154892i 1.25885 − 2.18040i 3.12291 − 1.80301i 0.380153i −1.20931 + 2.35320i 1.10038i 1.48401 + 2.57037i −3.83229 + 6.63772i
25.3 −0.929293 + 0.536527i 1.21570 − 2.10566i −0.424277 + 0.734868i 0.541640 − 0.312716i 2.60903i 2.34996 − 1.21561i − 3.05665i −1.45586 − 2.52163i −0.335561 + 0.581209i
25.4 −0.287846 + 0.166188i −0.729919 + 1.26426i −0.944763 + 1.63638i −1.25195 + 0.722811i − 0.485214i −2.26391 − 1.36920i − 1.29278i 0.434437 + 0.752468i 0.240245 − 0.416116i
25.5 0.287846 − 0.166188i −0.729919 + 1.26426i −0.944763 + 1.63638i 1.25195 − 0.722811i 0.485214i 2.26391 + 1.36920i 1.29278i 0.434437 + 0.752468i 0.240245 − 0.416116i
25.6 0.929293 − 0.536527i 1.21570 − 2.10566i −0.424277 + 0.734868i −0.541640 + 0.312716i − 2.60903i −2.34996 + 1.21561i 3.05665i −1.45586 − 2.52163i −0.335561 + 0.581209i
25.7 1.84073 − 1.06275i 0.0894272 − 0.154892i 1.25885 − 2.18040i −3.12291 + 1.80301i − 0.380153i 1.20931 − 2.35320i − 1.10038i 1.48401 + 2.57037i −3.83229 + 6.63772i
25.8 1.97871 − 1.14241i −1.57521 + 2.72835i 1.61019 − 2.78892i 1.84030 − 1.06250i 7.19813i −2.62488 − 0.331665i − 2.78832i −3.46258 − 5.99736i 2.42760 − 4.20473i
51.1 −1.97871 − 1.14241i −1.57521 − 2.72835i 1.61019 + 2.78892i −1.84030 − 1.06250i 7.19813i 2.62488 − 0.331665i − 2.78832i −3.46258 + 5.99736i 2.42760 + 4.20473i
51.2 −1.84073 − 1.06275i 0.0894272 + 0.154892i 1.25885 + 2.18040i 3.12291 + 1.80301i − 0.380153i −1.20931 − 2.35320i − 1.10038i 1.48401 − 2.57037i −3.83229 − 6.63772i
51.3 −0.929293 − 0.536527i 1.21570 + 2.10566i −0.424277 − 0.734868i 0.541640 + 0.312716i − 2.60903i 2.34996 + 1.21561i 3.05665i −1.45586 + 2.52163i −0.335561 − 0.581209i
51.4 −0.287846 − 0.166188i −0.729919 − 1.26426i −0.944763 − 1.63638i −1.25195 − 0.722811i 0.485214i −2.26391 + 1.36920i 1.29278i 0.434437 − 0.752468i 0.240245 + 0.416116i
51.5 0.287846 + 0.166188i −0.729919 − 1.26426i −0.944763 − 1.63638i 1.25195 + 0.722811i − 0.485214i 2.26391 − 1.36920i − 1.29278i 0.434437 − 0.752468i 0.240245 + 0.416116i
51.6 0.929293 + 0.536527i 1.21570 + 2.10566i −0.424277 − 0.734868i −0.541640 − 0.312716i 2.60903i −2.34996 − 1.21561i − 3.05665i −1.45586 + 2.52163i −0.335561 − 0.581209i
51.7 1.84073 + 1.06275i 0.0894272 + 0.154892i 1.25885 + 2.18040i −3.12291 − 1.80301i 0.380153i 1.20931 + 2.35320i 1.10038i 1.48401 − 2.57037i −3.83229 − 6.63772i
51.8 1.97871 + 1.14241i −1.57521 − 2.72835i 1.61019 + 2.78892i 1.84030 + 1.06250i − 7.19813i −2.62488 + 0.331665i 2.78832i −3.46258 + 5.99736i 2.42760 + 4.20473i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
13.b even 2 1 inner
91.r even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 91.2.r.a ✓ 16
3.b odd 2 1 819.2.dl.e 16
7.b odd 2 1 637.2.r.f 16
7.c even 3 1 inner 91.2.r.a ✓ 16
7.c even 3 1 637.2.c.f 8
7.d odd 6 1 637.2.c.e 8
7.d odd 6 1 637.2.r.f 16
13.b even 2 1 inner 91.2.r.a ✓ 16
13.d odd 4 2 1183.2.e.i 16
21.h odd 6 1 819.2.dl.e 16
39.d odd 2 1 819.2.dl.e 16
91.b odd 2 1 637.2.r.f 16
91.r even 6 1 inner 91.2.r.a ✓ 16
91.r even 6 1 637.2.c.f 8
91.s odd 6 1 637.2.c.e 8
91.s odd 6 1 637.2.r.f 16
91.z odd 12 2 1183.2.e.i 16
91.z odd 12 2 8281.2.a.ck 8
91.bb even 12 2 8281.2.a.cj 8
273.w odd 6 1 819.2.dl.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.r.a ✓ 16 1.a even 1 1 trivial
91.2.r.a ✓ 16 7.c even 3 1 inner
91.2.r.a ✓ 16 13.b even 2 1 inner
91.2.r.a ✓ 16 91.r even 6 1 inner
637.2.c.e 8 7.d odd 6 1
637.2.c.e 8 91.s odd 6 1
637.2.c.f 8 7.c even 3 1
637.2.c.f 8 91.r even 6 1
637.2.r.f 16 7.b odd 2 1
637.2.r.f 16 7.d odd 6 1
637.2.r.f 16 91.b odd 2 1
637.2.r.f 16 91.s odd 6 1
819.2.dl.e 16 3.b odd 2 1
819.2.dl.e 16 21.h odd 6 1
819.2.dl.e 16 39.d odd 2 1
819.2.dl.e 16 273.w odd 6 1
1183.2.e.i 16 13.d odd 4 2
1183.2.e.i 16 91.z odd 12 2
8281.2.a.cj 8 91.bb even 12 2
8281.2.a.ck 8 91.z odd 12 2

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(91, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 11 T^{14} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( (T^{8} + 2 T^{7} + 11 T^{6} + \cdots + 4)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} - 20 T^{14} + \cdots + 2304 \) Copy content Toggle raw display
$7$ \( T^{16} - 20 T^{14} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( T^{16} - 52 T^{14} + \cdots + 729 \) Copy content Toggle raw display
$13$ \( (T^{8} + 6 T^{7} + \cdots + 28561)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 4 T^{7} + \cdots + 15129)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} - 44 T^{14} + \cdots + 10673289 \) Copy content Toggle raw display
$23$ \( (T^{8} + 6 T^{7} + 31 T^{6} + \cdots + 36)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 4 T^{3} + \cdots + 624)^{4} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 1136229264 \) Copy content Toggle raw display
$37$ \( T^{16} - 120 T^{14} + \cdots + 76527504 \) Copy content Toggle raw display
$41$ \( (T^{8} + 132 T^{6} + \cdots + 292032)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 4 T^{3} + \cdots - 104)^{4} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 57728231289 \) Copy content Toggle raw display
$53$ \( (T^{8} + 10 T^{7} + \cdots + 7569)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 12487392009 \) Copy content Toggle raw display
$61$ \( (T^{8} + 6 T^{7} + \cdots + 49729)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 66330457209 \) Copy content Toggle raw display
$71$ \( (T^{8} + 292 T^{6} + \cdots + 397488)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 8437677133824 \) Copy content Toggle raw display
$79$ \( (T^{8} - 10 T^{7} + \cdots + 64)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 296 T^{6} + \cdots + 5483712)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 58102628210064 \) Copy content Toggle raw display
$97$ \( (T^{8} + 104 T^{6} + \cdots + 192)^{2} \) Copy content Toggle raw display
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