↑

Properties

Label 276.2.i.b
Level $276$
Weight $2$
Character orbit 276.i
Analytic conductor $2.204$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [276,2,Mod(13,276)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(276, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 0, 14]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("276.13");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 276 = 2^{2} \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 276.i (of order \(11\), degree \(10\), 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: \(2.20387109579\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(2\) over \(\Q(\zeta_{11})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 4 x^{19} + 18 x^{18} - 58 x^{17} + 169 x^{16} - 363 x^{15} + 800 x^{14} - 1682 x^{13} + \cdots + 529 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{3} + (\beta_{18} - \beta_{17} - 2 \beta_{16} + \cdots + 1) q^{5}+ \cdots - \beta_{13} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{3} + (\beta_{18} - \beta_{17} - 2 \beta_{16} + \cdots + 1) q^{5}+ \cdots + ( - \beta_{19} - \beta_{18} + \beta_{16} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{3} - 4 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 2 q^{3} - 4 q^{7} - 2 q^{9} + 14 q^{13} - 11 q^{15} + 3 q^{17} + 7 q^{19} + 4 q^{21} + 24 q^{23} + 12 q^{25} + 2 q^{27} - 26 q^{29} + 33 q^{31} - 11 q^{33} - 2 q^{35} - 18 q^{37} - 14 q^{39} + 4 q^{41} - 40 q^{43} - 54 q^{47} + 30 q^{49} - 14 q^{51} - 14 q^{53} + 11 q^{55} - 29 q^{57} + 4 q^{59} + 12 q^{61} - 4 q^{63} - 33 q^{65} + 15 q^{67} - 2 q^{69} - 33 q^{71} + 15 q^{73} + 10 q^{75} + 66 q^{77} - 42 q^{79} - 2 q^{81} - 14 q^{83} - 13 q^{85} + 4 q^{87} - 66 q^{89} - 16 q^{91} - 22 q^{93} - 31 q^{95} - 24 q^{97} + 11 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 4 x^{19} + 18 x^{18} - 58 x^{17} + 169 x^{16} - 363 x^{15} + 800 x^{14} - 1682 x^{13} + \cdots + 529 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 14\!\cdots\!02 \nu^{19} + \cdots + 72\!\cdots\!66 ) / 79\!\cdots\!93 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 13\!\cdots\!54 \nu^{19} + \cdots + 87\!\cdots\!80 ) / 79\!\cdots\!93 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 14\!\cdots\!52 \nu^{19} + \cdots - 80\!\cdots\!73 ) / 79\!\cdots\!93 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 15\!\cdots\!37 \nu^{19} + \cdots + 75\!\cdots\!19 ) / 79\!\cdots\!93 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 21\!\cdots\!09 \nu^{19} + \cdots - 53\!\cdots\!68 ) / 79\!\cdots\!93 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 56\!\cdots\!04 \nu^{19} + \cdots - 27\!\cdots\!81 ) / 79\!\cdots\!93 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 61\!\cdots\!85 \nu^{19} + \cdots - 56\!\cdots\!99 ) / 79\!\cdots\!93 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 78\!\cdots\!10 \nu^{19} + \cdots + 81\!\cdots\!22 ) / 79\!\cdots\!93 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 10\!\cdots\!41 \nu^{19} + \cdots - 16\!\cdots\!06 ) / 79\!\cdots\!93 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 10\!\cdots\!92 \nu^{19} + \cdots + 19\!\cdots\!74 ) / 79\!\cdots\!93 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 10\!\cdots\!31 \nu^{19} + \cdots - 12\!\cdots\!41 ) / 79\!\cdots\!93 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 11\!\cdots\!59 \nu^{19} + \cdots + 34\!\cdots\!32 ) / 79\!\cdots\!93 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 11\!\cdots\!85 \nu^{19} + \cdots + 22\!\cdots\!45 ) / 79\!\cdots\!93 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 11\!\cdots\!36 \nu^{19} + \cdots + 35\!\cdots\!90 ) / 79\!\cdots\!93 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 13\!\cdots\!32 \nu^{19} + \cdots - 16\!\cdots\!32 ) / 79\!\cdots\!93 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 16\!\cdots\!60 \nu^{19} + \cdots - 16\!\cdots\!99 ) / 79\!\cdots\!93 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 28\!\cdots\!86 \nu^{19} + \cdots + 26\!\cdots\!97 ) / 79\!\cdots\!93 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 34\!\cdots\!21 \nu^{19} + \cdots - 15\!\cdots\!70 ) / 79\!\cdots\!93 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( - \beta_{18} + \beta_{16} + 2 \beta_{13} + \beta_{12} + \beta_{11} - 2 \beta_{10} - \beta_{8} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{17} - \beta_{16} - 2 \beta_{15} + \beta_{14} - \beta_{13} - 2 \beta_{12} - 3 \beta_{11} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7 \beta_{18} + 2 \beta_{17} - 3 \beta_{16} + 6 \beta_{15} - \beta_{14} - 20 \beta_{13} - 7 \beta_{12} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{19} - 8 \beta_{18} - 2 \beta_{17} - 3 \beta_{16} + 32 \beta_{15} - 10 \beta_{14} + 28 \beta_{12} + \cdots - 38 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 9 \beta_{19} - 36 \beta_{18} - 15 \beta_{17} - 64 \beta_{16} - 36 \beta_{15} + 41 \beta_{14} + \cdots + 23 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 85 \beta_{19} + 117 \beta_{18} + 179 \beta_{17} + 62 \beta_{16} - 174 \beta_{15} + 117 \beta_{14} + \cdots + 259 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 311 \beta_{18} + 387 \beta_{17} + 767 \beta_{16} + 441 \beta_{15} - 333 \beta_{14} + \beta_{13} + \cdots - 206 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 691 \beta_{19} - 763 \beta_{18} - 1198 \beta_{17} - 179 \beta_{16} + 896 \beta_{15} - 436 \beta_{14} + \cdots - 1447 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 256 \beta_{19} - 1399 \beta_{18} - 2084 \beta_{17} - 3849 \beta_{16} - 4539 \beta_{15} + 3140 \beta_{14} + \cdots + 2194 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 3164 \beta_{19} + 7685 \beta_{18} + 7909 \beta_{17} + 5601 \beta_{16} - 3164 \beta_{15} - 8847 \beta_{13} + \cdots + 8194 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 5377 \beta_{19} + 5377 \beta_{18} + 15222 \beta_{16} + 44148 \beta_{15} - 25611 \beta_{14} + \cdots - 28754 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 20599 \beta_{19} - 73401 \beta_{18} - 68326 \beta_{17} - 99139 \beta_{16} + 25334 \beta_{14} + \cdots - 57891 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 104214 \beta_{19} + 105031 \beta_{17} - 94267 \beta_{16} - 365033 \beta_{15} + 198481 \beta_{14} + \cdots + 294137 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 199298 \beta_{19} + 613836 \beta_{18} + 642307 \beta_{17} + 963973 \beta_{16} + 255176 \beta_{15} + \cdots + 358660 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 935502 \beta_{19} - 440370 \beta_{18} - 1118995 \beta_{17} + 501913 \beta_{16} + 2676823 \beta_{15} + \cdots - 2534956 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 1180538 \beta_{19} - 4594806 \beta_{18} - 4583484 \beta_{17} - 6986005 \beta_{16} - 4594806 \beta_{15} + \cdots - 774424 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 6997327 \beta_{19} + 7813292 \beta_{18} + 10594230 \beta_{17} + 2780938 \beta_{16} - 18386211 \beta_{15} + \cdots + 21215296 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 31797283 \beta_{18} + 23043869 \beta_{17} + 50800630 \beta_{16} + 55446365 \beta_{15} - 38425439 \beta_{14} + \cdots - 14895668 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/276\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(139\) \(185\)
\(\chi(n)\) \(\beta_{7}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
1.79494 + 0.527041i
−0.453683 − 0.133213i
1.61721 − 1.86636i
−1.25953 + 1.45357i
−0.634872 + 0.408008i
1.55029 − 0.996309i
−0.0489501 − 0.340455i
−0.410543 − 2.85539i
1.79494 − 0.527041i
−0.453683 + 0.133213i
−0.0489501 + 0.340455i
−0.410543 + 2.85539i
0.753595 + 1.65014i
−0.908456 − 1.98924i
−0.634872 − 0.408008i
1.55029 + 0.996309i
0.753595 − 1.65014i
−0.908456 + 1.98924i
1.61721 + 1.86636i
−1.25953 − 1.45357i
0 −0.841254 − 0.540641i 0 −0.143141 + 0.313435i 0 1.55877 + 0.457697i 0 0.415415 + 0.909632i 0
13.2 0 −0.841254 − 0.540641i 0 1.63926 − 3.58947i 0 −4.27493 − 1.25523i 0 0.415415 + 0.909632i 0
25.1 0 0.142315 + 0.989821i 0 −2.42853 − 0.713080i 0 −2.20327 + 2.54270i 0 −0.959493 + 0.281733i 0
25.2 0 0.142315 + 0.989821i 0 1.44496 + 0.424279i 0 2.15021 − 2.48148i 0 −0.959493 + 0.281733i 0
49.1 0 −0.415415 + 0.909632i 0 0.0154463 − 0.0178260i 0 −3.20986 + 2.06285i 0 −0.654861 − 0.755750i 0
49.2 0 −0.415415 + 0.909632i 0 0.542284 − 0.625829i 0 3.79055 − 2.43604i 0 −0.654861 − 0.755750i 0
73.1 0 0.959493 − 0.281733i 0 −3.55432 − 2.28422i 0 0.00534053 + 0.0371442i 0 0.841254 − 0.540641i 0
73.2 0 0.959493 − 0.281733i 0 2.17941 + 1.40062i 0 −0.529416 − 3.68217i 0 0.841254 − 0.540641i 0
85.1 0 −0.841254 + 0.540641i 0 −0.143141 − 0.313435i 0 1.55877 − 0.457697i 0 0.415415 − 0.909632i 0
85.2 0 −0.841254 + 0.540641i 0 1.63926 + 3.58947i 0 −4.27493 + 1.25523i 0 0.415415 − 0.909632i 0
121.1 0 0.959493 + 0.281733i 0 −3.55432 + 2.28422i 0 0.00534053 − 0.0371442i 0 0.841254 + 0.540641i 0
121.2 0 0.959493 + 0.281733i 0 2.17941 − 1.40062i 0 −0.529416 + 3.68217i 0 0.841254 + 0.540641i 0
133.1 0 0.654861 − 0.755750i 0 −0.0513473 + 0.357128i 0 1.38384 + 3.03019i 0 −0.142315 − 0.989821i 0
133.2 0 0.654861 − 0.755750i 0 0.355980 − 2.47589i 0 −0.671250 − 1.46983i 0 −0.142315 − 0.989821i 0
169.1 0 −0.415415 − 0.909632i 0 0.0154463 + 0.0178260i 0 −3.20986 − 2.06285i 0 −0.654861 + 0.755750i 0
169.2 0 −0.415415 − 0.909632i 0 0.542284 + 0.625829i 0 3.79055 + 2.43604i 0 −0.654861 + 0.755750i 0
193.1 0 0.654861 + 0.755750i 0 −0.0513473 − 0.357128i 0 1.38384 − 3.03019i 0 −0.142315 + 0.989821i 0
193.2 0 0.654861 + 0.755750i 0 0.355980 + 2.47589i 0 −0.671250 + 1.46983i 0 −0.142315 + 0.989821i 0
265.1 0 0.142315 − 0.989821i 0 −2.42853 + 0.713080i 0 −2.20327 − 2.54270i 0 −0.959493 − 0.281733i 0
265.2 0 0.142315 − 0.989821i 0 1.44496 − 0.424279i 0 2.15021 + 2.48148i 0 −0.959493 − 0.281733i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 276.2.i.b ✓ 20
3.b odd 2 1 828.2.q.b 20
23.c even 11 1 inner 276.2.i.b ✓ 20
23.c even 11 1 6348.2.a.r 10
23.d odd 22 1 6348.2.a.q 10
69.h odd 22 1 828.2.q.b 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
276.2.i.b ✓ 20 1.a even 1 1 trivial
276.2.i.b ✓ 20 23.c even 11 1 inner
828.2.q.b 20 3.b odd 2 1
828.2.q.b 20 69.h odd 22 1
6348.2.a.q 10 23.d odd 22 1
6348.2.a.r 10 23.c even 11 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{20} - T_{5}^{18} + 44 T_{5}^{17} + 111 T_{5}^{16} - 858 T_{5}^{15} + 1781 T_{5}^{14} + 2024 T_{5}^{13} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(276, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( (T^{10} - T^{9} + T^{8} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{20} - T^{18} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{20} + 4 T^{19} + \cdots + 1067089 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 4097152081 \) Copy content Toggle raw display
$13$ \( T^{20} - 14 T^{19} + \cdots + 279841 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 132871729 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 542843401 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 41426511213649 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 1195983889 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 5844755401 \) Copy content Toggle raw display
$37$ \( T^{20} + 18 T^{19} + \cdots + 2582449 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 12\!\cdots\!61 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 197657083729249 \) Copy content Toggle raw display
$47$ \( (T^{10} + 27 T^{9} + \cdots - 12485353)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 1080239689 \) Copy content Toggle raw display
$59$ \( T^{20} - 4 T^{19} + \cdots + 10883401 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 4209025129 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 3426093642841 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 694529152074241 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 41412657001 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 30821664721 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 26\!\cdots\!89 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 67730052449929 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 48\!\cdots\!61 \) Copy content Toggle raw display
show more
show less