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Properties

Label 261.3.f.c
Level $261$
Weight $3$
Character orbit 261.f
Analytic conductor $7.112$
Analytic rank $0$
Dimension $12$
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] = [261,3,Mod(46,261)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(261, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("261.46");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 261 = 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 261.f (of order \(4\), 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: \(7.11173489980\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 36x^{9} + 263x^{8} - 576x^{7} + 648x^{6} - 144x^{5} + 373x^{4} - 684x^{3} + 648x^{2} - 216x + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 87)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{9} + \beta_1) q^{2} + ( - \beta_{9} - \beta_{8} + \cdots - \beta_{2}) q^{4}+ \cdots + ( - 2 \beta_{10} - 2 \beta_{8} + \cdots - 3) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{9} + \beta_1) q^{2} + ( - \beta_{9} - \beta_{8} + \cdots - \beta_{2}) q^{4}+ \cdots + (10 \beta_{11} - 26 \beta_{9} + \cdots + 72) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{7} - 36 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{7} - 36 q^{8} - 8 q^{10} - 12 q^{11} - 12 q^{14} - 32 q^{16} + 40 q^{17} + 36 q^{19} + 52 q^{20} - 80 q^{23} - 56 q^{25} + 84 q^{26} + 76 q^{29} + 52 q^{31} - 264 q^{32} + 112 q^{37} - 196 q^{40} + 112 q^{41} + 260 q^{43} + 192 q^{44} - 176 q^{46} + 212 q^{47} + 24 q^{49} + 60 q^{50} + 612 q^{52} + 328 q^{53} - 292 q^{55} - 528 q^{56} - 156 q^{58} + 260 q^{59} + 112 q^{61} - 432 q^{65} - 356 q^{68} - 508 q^{70} - 92 q^{73} + 204 q^{74} - 348 q^{76} + 84 q^{77} - 272 q^{79} + 492 q^{82} - 584 q^{83} + 336 q^{85} + 232 q^{88} - 312 q^{89} + 520 q^{94} - 228 q^{95} - 216 q^{97} + 864 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 36x^{9} + 263x^{8} - 576x^{7} + 648x^{6} - 144x^{5} + 373x^{4} - 684x^{3} + 648x^{2} - 216x + 36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 9000633077 \nu^{11} + 3796372744 \nu^{10} + 3929084070 \nu^{9} - 323758941696 \nu^{8} + \cdots - 1369496846124 ) / 4488249714288 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1800816895 \nu^{11} + 1102774950 \nu^{10} + 1104810516 \nu^{9} - 64415517876 \nu^{8} + \cdots - 271590945300 ) / 561031214286 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 21978234733 \nu^{11} + 25718982966 \nu^{10} + 9018497706 \nu^{9} - 751045215924 \nu^{8} + \cdots - 372243616656 ) / 4488249714288 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 23289714499 \nu^{11} - 19565272458 \nu^{10} - 6957651078 \nu^{9} + 854822765124 \nu^{8} + \cdots + 6060868313208 ) / 2244124857144 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 53979959312 \nu^{11} - 36384769893 \nu^{10} - 34541182566 \nu^{9} - 1961135517066 \nu^{8} + \cdots - 15702455535642 ) / 4488249714288 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 7544192925 \nu^{11} + 1800816895 \nu^{10} + 1102774950 \nu^{9} - 270486134784 \nu^{8} + \cdots - 821181857868 ) / 561031214286 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 64077985441 \nu^{11} + 27014156540 \nu^{10} + 32172893838 \nu^{9} - 2276499130560 \nu^{8} + \cdots - 7303643739900 ) / 2244124857144 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 53390964321 \nu^{11} + 18315880439 \nu^{10} + 3126902768 \nu^{9} - 1920770449042 \nu^{8} + \cdots - 2043675961182 ) / 1496083238096 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 105946515512 \nu^{11} + 34904081879 \nu^{10} + 28395614322 \nu^{9} + \cdots + 27096775332222 ) / 2244124857144 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 223156378785 \nu^{11} - 79602605479 \nu^{10} - 4591081560 \nu^{9} + 8048166273162 \nu^{8} + \cdots + 8687495038398 ) / 2244124857144 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{9} - \beta_{8} + 6\beta_{7} - \beta_{6} - 2\beta_{3} + \beta_{2} - 2\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} + \beta_{8} - 9\beta_{7} + 6\beta_{6} - \beta_{5} + 2\beta_{4} + 16\beta_{3} - 2\beta_{2} + 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{11} - 2\beta_{10} + 21\beta_{9} - 21\beta_{6} + 14\beta_{5} - 19\beta_{4} - 48\beta_{3} + 48\beta _1 - 91 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 19 \beta_{11} - 138 \beta_{9} - 34 \beta_{8} + 245 \beta_{7} - 34 \beta_{5} + 52 \beta_{4} + \cdots + 245 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 52 \beta_{11} + 52 \beta_{10} + 452 \beta_{9} + 261 \beta_{8} - 1787 \beta_{7} + 452 \beta_{6} + \cdots + 1060 \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 382 \beta_{10} - 799 \beta_{8} + 5578 \beta_{7} - 2988 \beta_{6} + 799 \beta_{5} - 1182 \beta_{4} + \cdots - 5578 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1182 \beta_{11} + 1182 \beta_{10} - 9781 \beta_{9} + 9781 \beta_{6} - 5437 \beta_{5} + 8041 \beta_{4} + \cdots + 37656 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 8041 \beta_{11} + 64482 \beta_{9} + 17581 \beta_{8} - 122115 \beta_{7} + 17581 \beta_{5} - 25940 \beta_{4} + \cdots - 122115 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 25940 \beta_{11} - 25940 \beta_{10} - 211575 \beta_{9} - 116492 \beta_{8} + 808363 \beta_{7} + \cdots - 497946 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 172297 \beta_{10} + 381454 \beta_{8} - 2647613 \beta_{7} + 1392366 \beta_{6} - 381454 \beta_{5} + \cdots + 2647613 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/261\mathbb{Z}\right)^\times\).

\(n\) \(118\) \(146\)
\(\chi(n)\) \(\beta_{7}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
46.1
−3.28696 + 3.28696i
−0.730190 + 0.730190i
0.216889 − 0.216889i
1.73805 − 1.73805i
0.547134 − 0.547134i
1.51509 − 1.51509i
−3.28696 − 3.28696i
−0.730190 − 0.730190i
0.216889 + 0.216889i
1.73805 + 1.73805i
0.547134 + 0.547134i
1.51509 + 1.51509i
−2.06222 + 2.06222i 0 − 4.50550i − 0.728172i 0 −2.14276 1.04246 + 1.04246i 0 1.50165 + 1.50165i
46.2 −1.95493 + 1.95493i 0 − 3.64354i 8.96002i 0 1.83147 −0.696854 − 0.696854i 0 −17.5163 − 17.5163i
46.3 −1.00786 + 1.00786i 0 1.96845i − 4.29032i 0 5.25580 −6.01534 − 6.01534i 0 4.32402 + 4.32402i
46.4 0.513301 − 0.513301i 0 3.47305i 0.330294i 0 −7.63777 3.83592 + 3.83592i 0 0.169540 + 0.169540i
46.5 1.77188 − 1.77188i 0 − 2.27911i 8.44377i 0 −11.8177 3.04921 + 3.04921i 0 14.9613 + 14.9613i
46.6 2.73983 − 2.73983i 0 − 11.0133i − 2.71560i 0 8.51099 −19.2154 − 19.2154i 0 −7.44029 − 7.44029i
244.1 −2.06222 − 2.06222i 0 4.50550i 0.728172i 0 −2.14276 1.04246 − 1.04246i 0 1.50165 − 1.50165i
244.2 −1.95493 − 1.95493i 0 3.64354i − 8.96002i 0 1.83147 −0.696854 + 0.696854i 0 −17.5163 + 17.5163i
244.3 −1.00786 − 1.00786i 0 − 1.96845i 4.29032i 0 5.25580 −6.01534 + 6.01534i 0 4.32402 − 4.32402i
244.4 0.513301 + 0.513301i 0 − 3.47305i − 0.330294i 0 −7.63777 3.83592 − 3.83592i 0 0.169540 − 0.169540i
244.5 1.77188 + 1.77188i 0 2.27911i − 8.44377i 0 −11.8177 3.04921 − 3.04921i 0 14.9613 − 14.9613i
244.6 2.73983 + 2.73983i 0 11.0133i 2.71560i 0 8.51099 −19.2154 + 19.2154i 0 −7.44029 + 7.44029i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 46.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 261.3.f.c 12
3.b odd 2 1 87.3.e.b ✓ 12
29.c odd 4 1 inner 261.3.f.c 12
87.f even 4 1 87.3.e.b ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
87.3.e.b ✓ 12 3.b odd 2 1
87.3.e.b ✓ 12 87.f even 4 1
261.3.f.c 12 1.a even 1 1 trivial
261.3.f.c 12 29.c odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 12 T_{2}^{9} + 200 T_{2}^{8} + 168 T_{2}^{7} + 72 T_{2}^{6} + 240 T_{2}^{5} + 6760 T_{2}^{4} + \cdots + 6561 \) acting on \(S_{3}^{\mathrm{new}}(261, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 12 T^{9} + \cdots + 6561 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 178 T^{10} + \cdots + 44944 \) Copy content Toggle raw display
$7$ \( (T^{6} + 6 T^{5} + \cdots - 15845)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 30257210916 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 7629726742416 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 259850160025 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 17912397148416 \) Copy content Toggle raw display
$23$ \( (T^{6} + 40 T^{5} + \cdots + 2421352)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 35\!\cdots\!41 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 162975249662976 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 86\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 582330257510400 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 18\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 29\!\cdots\!25 \) Copy content Toggle raw display
$53$ \( (T^{6} - 164 T^{5} + \cdots - 523816704)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - 130 T^{5} + \cdots + 128264549260)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 12\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 367573013284 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 34\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{6} + 292 T^{5} + \cdots - 14476799808)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 21\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 45\!\cdots\!16 \) Copy content Toggle raw display
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