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Properties

Label 91.3.d.a
Level $91$
Weight $3$
Character orbit 91.d
Analytic conductor $2.480$
Analytic rank $0$
Dimension $16$
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] = [91,3,Mod(27,91)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(91, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("91.27");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 91.d (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: \(2.47957040568\)
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: \( x^{16} + 112 x^{14} + 4962 x^{12} + 110020 x^{10} + 1267625 x^{8} + 7059464 x^{6} + 15193060 x^{4} + \cdots + 132496 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + \beta_1 q^{3} + (\beta_{4} + 2) q^{4} + \beta_{3} q^{5} + (\beta_{13} + \beta_{9} + \beta_{8}) q^{6} + ( - \beta_{11} + \beta_{2}) q^{7} + ( - \beta_{6} + \beta_{5} - 2 \beta_{2} - 1) q^{8} + ( - \beta_{11} + \beta_{10} - \beta_{7} + \cdots - 6) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + \beta_1 q^{3} + (\beta_{4} + 2) q^{4} + \beta_{3} q^{5} + (\beta_{13} + \beta_{9} + \beta_{8}) q^{6} + ( - \beta_{11} + \beta_{2}) q^{7} + ( - \beta_{6} + \beta_{5} - 2 \beta_{2} - 1) q^{8} + ( - \beta_{11} + \beta_{10} - \beta_{7} + \cdots - 6) q^{9}+ \cdots + (2 \beta_{15} + 2 \beta_{14} + \cdots - 55) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 2 q^{2} + 26 q^{4} + 2 q^{7} - 14 q^{8} - 80 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 2 q^{2} + 26 q^{4} + 2 q^{7} - 14 q^{8} - 80 q^{9} + 36 q^{11} - 80 q^{14} + 36 q^{15} + 50 q^{16} + 70 q^{18} + 32 q^{21} - 56 q^{22} + 6 q^{23} - 138 q^{25} + 70 q^{28} + 110 q^{29} + 100 q^{30} - 274 q^{32} + 148 q^{35} - 318 q^{36} - 48 q^{37} + 66 q^{42} + 46 q^{43} + 344 q^{44} - 60 q^{46} - 54 q^{49} + 330 q^{50} - 140 q^{51} - 66 q^{53} - 340 q^{56} - 256 q^{57} - 336 q^{58} + 692 q^{60} + 122 q^{63} - 30 q^{64} - 26 q^{65} + 296 q^{67} - 388 q^{70} + 76 q^{71} - 134 q^{72} - 164 q^{74} + 24 q^{77} + 260 q^{78} + 410 q^{79} + 488 q^{81} + 348 q^{84} + 556 q^{85} - 240 q^{86} - 688 q^{88} - 52 q^{91} - 376 q^{92} - 220 q^{93} - 6 q^{95} + 94 q^{98} - 712 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 112 x^{14} + 4962 x^{12} + 110020 x^{10} + 1267625 x^{8} + 7059464 x^{6} + 15193060 x^{4} + \cdots + 132496 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 485771222 \nu^{14} + 53712019005 \nu^{12} + 2243231077668 \nu^{10} + 43790460295427 \nu^{8} + \cdots - 32\!\cdots\!40 ) / 10\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 16663605 \nu^{15} + 2072461021 \nu^{13} + 103518116491 \nu^{11} + \cdots + 978006227641504 \nu ) / 22984371015280 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1144971171 \nu^{14} + 118215536214 \nu^{12} + 4758602594592 \nu^{10} + \cdots + 20\!\cdots\!84 ) / 522894440597620 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4663973589 \nu^{14} + 540998649623 \nu^{12} + 24449652424267 \nu^{10} + \cdots + 16\!\cdots\!44 ) / 20\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 5011319253 \nu^{14} - 559389249017 \nu^{12} - 24527653510535 \nu^{10} + \cdots + 15\!\cdots\!00 ) / 20\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 521868047 \nu^{14} + 62326673933 \nu^{12} + 2916686757285 \nu^{10} + \cdots + 200921425477136 ) / 160890597106960 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1487 \nu^{15} + 166193 \nu^{13} + 7350999 \nu^{11} + 162783743 \nu^{9} + 1873813254 \nu^{7} + \cdots + 17500676024 \nu ) / 542847760 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 5778688493 \nu^{15} + 653821140553 \nu^{13} + 29343557805096 \nu^{11} + \cdots + 61\!\cdots\!08 \nu ) / 10\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 31288922727 \nu^{15} + 11711514959 \nu^{14} + 3491560924139 \nu^{13} + \cdots + 94\!\cdots\!60 ) / 41\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 31288922727 \nu^{15} - 11711514959 \nu^{14} + 3491560924139 \nu^{13} + \cdots - 94\!\cdots\!60 ) / 41\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1377434797 \nu^{15} + 153751301625 \nu^{13} + 6792220721333 \nu^{11} + \cdots + 16\!\cdots\!92 \nu ) / 160890597106960 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 18258290281 \nu^{15} - 2055403460905 \nu^{13} - 91496778435555 \nu^{11} + \cdots - 18\!\cdots\!88 \nu ) / 20\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 60906601837 \nu^{15} + 4614366519 \nu^{14} + 6795019747657 \nu^{13} + 446600206059 \nu^{12} + \cdots - 20\!\cdots\!88 ) / 41\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 60906601837 \nu^{15} + 4614366519 \nu^{14} - 6795019747657 \nu^{13} + \cdots - 20\!\cdots\!88 ) / 41\!\cdots\!60 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{11} + \beta_{10} - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - 15 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{15} + 2\beta_{14} - \beta_{13} - 2\beta_{12} - 2\beta_{11} - 2\beta_{10} - 4\beta_{9} + 3\beta_{3} - 24\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 3 \beta_{15} - 3 \beta_{14} + 30 \beta_{11} - 30 \beta_{10} + 22 \beta_{7} - 37 \beta_{6} + \cdots + 359 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 80 \beta_{15} - 80 \beta_{14} + 49 \beta_{13} + 74 \beta_{12} + 90 \beta_{11} + 90 \beta_{10} + \cdots + 628 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 157 \beta_{15} + 157 \beta_{14} - 824 \beta_{11} + 824 \beta_{10} - 446 \beta_{7} + 1239 \beta_{6} + \cdots - 9401 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 2618 \beta_{15} + 2618 \beta_{14} - 1849 \beta_{13} - 2356 \beta_{12} - 3190 \beta_{11} + \cdots - 17158 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 6283 \beta_{15} - 6283 \beta_{14} + 22346 \beta_{11} - 22346 \beta_{10} + 8662 \beta_{7} + \cdots + 257313 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 80532 \beta_{15} - 80532 \beta_{14} + 62689 \beta_{13} + 72268 \beta_{12} + 103864 \beta_{11} + \cdots + 481146 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 224693 \beta_{15} + 224693 \beta_{14} - 609048 \beta_{11} + 609048 \beta_{10} - 156502 \beta_{7} + \cdots - 7232933 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 2419906 \beta_{15} + 2419906 \beta_{14} - 2007349 \beta_{13} - 2185196 \beta_{12} + \cdots - 13715914 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 7569935 \beta_{15} - 7569935 \beta_{14} + 16769198 \beta_{11} - 16769198 \beta_{10} + \cdots + 206729777 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 72042916 \beta_{15} - 72042916 \beta_{14} + 62118201 \beta_{13} + 65546660 \beta_{12} + \cdots + 395245234 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 246237285 \beta_{15} + 246237285 \beta_{14} - 466776512 \beta_{11} + 466776512 \beta_{10} + \cdots - 5972269133 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 2137116434 \beta_{15} + 2137116434 \beta_{14} - 1880724813 \beta_{13} - 1954695532 \beta_{12} + \cdots - 11474022906 \beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(15\) \(66\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
27.1
− 3.97223i
3.97223i
− 5.46933i
5.46933i
− 1.49244i
1.49244i
− 4.69084i
4.69084i
− 1.14486i
1.14486i
− 3.40207i
3.40207i
− 0.113751i
0.113751i
− 5.40174i
5.40174i
−3.74502 − 3.97223i 10.0252 6.95233i 14.8761i 5.95019 − 3.68718i −22.5646 −6.77858 − 26.0367i
27.2 −3.74502 3.97223i 10.0252 − 6.95233i − 14.8761i 5.95019 + 3.68718i −22.5646 −6.77858 26.0367i
27.3 −2.32577 − 5.46933i 1.40922 − 7.98567i 12.7204i −0.677488 + 6.96714i 6.02557 −20.9136 18.5728i
27.4 −2.32577 5.46933i 1.40922 7.98567i − 12.7204i −0.677488 − 6.96714i 6.02557 −20.9136 − 18.5728i
27.5 −1.35190 − 1.49244i −2.17238 4.34544i 2.01762i 4.53023 + 5.33639i 8.34441 6.77263 − 5.87459i
27.6 −1.35190 1.49244i −2.17238 − 4.34544i − 2.01762i 4.53023 − 5.33639i 8.34441 6.77263 5.87459i
27.7 −0.401182 − 4.69084i −3.83905 3.66457i 1.88188i −2.10719 − 6.67531i 3.14489 −13.0039 − 1.47016i
27.8 −0.401182 4.69084i −3.83905 − 3.66457i − 1.88188i −2.10719 + 6.67531i 3.14489 −13.0039 1.47016i
27.9 0.743508 − 1.14486i −3.44720 − 8.27504i − 0.851216i −6.81434 − 1.60150i −5.53705 7.68929 − 6.15256i
27.10 0.743508 1.14486i −3.44720 8.27504i 0.851216i −6.81434 + 1.60150i −5.53705 7.68929 6.15256i
27.11 1.77769 − 3.40207i −0.839804 − 2.35726i − 6.04784i 6.68337 − 2.08149i −8.60369 −2.57408 − 4.19048i
27.12 1.77769 3.40207i −0.839804 2.35726i 6.04784i 6.68337 + 2.08149i −8.60369 −2.57408 4.19048i
27.13 3.11844 − 0.113751i 5.72465 − 4.78167i − 0.354726i −1.15842 + 6.90348i 5.37822 8.98706 − 14.9113i
27.14 3.11844 0.113751i 5.72465 4.78167i 0.354726i −1.15842 − 6.90348i 5.37822 8.98706 14.9113i
27.15 3.18424 − 5.40174i 6.13935 5.26165i − 17.2004i −5.40635 + 4.44651i 6.81220 −20.1788 16.7543i
27.16 3.18424 5.40174i 6.13935 − 5.26165i 17.2004i −5.40635 − 4.44651i 6.81220 −20.1788 − 16.7543i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 27.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 91.3.d.a ✓ 16
3.b odd 2 1 819.3.f.b 16
4.b odd 2 1 1456.3.d.b 16
7.b odd 2 1 inner 91.3.d.a ✓ 16
21.c even 2 1 819.3.f.b 16
28.d even 2 1 1456.3.d.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.3.d.a ✓ 16 1.a even 1 1 trivial
91.3.d.a ✓ 16 7.b odd 2 1 inner
819.3.f.b 16 3.b odd 2 1
819.3.f.b 16 21.c even 2 1
1456.3.d.b 16 4.b odd 2 1
1456.3.d.b 16 28.d even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(91, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - T^{7} - 22 T^{6} + \cdots + 62)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} + 112 T^{14} + \cdots + 132496 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 188258796544 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 33232930569601 \) Copy content Toggle raw display
$11$ \( (T^{8} - 18 T^{7} + \cdots + 53034976)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 13)^{8} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 22\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 31\!\cdots\!44 \) Copy content Toggle raw display
$23$ \( (T^{8} - 3 T^{7} + \cdots + 15238536)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} - 55 T^{7} + \cdots - 24931981376)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 30\!\cdots\!84 \) Copy content Toggle raw display
$37$ \( (T^{8} + 24 T^{7} + \cdots - 19414500236)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 13\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( (T^{8} - 23 T^{7} + \cdots + 185342100648)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 26\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 13299651398784)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 32\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 15\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 9686936660288)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots - 10655331867088)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 94\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 11968419586832)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 20\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 35\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 47\!\cdots\!96 \) Copy content Toggle raw display
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