Properties

Label 272.3.d.b
Level $272$
Weight $3$
Character orbit 272.d
Analytic conductor $7.411$
Analytic rank $0$
Dimension $12$
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] = [272,3,Mod(239,272)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(272, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("272.239");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 272 = 2^{4} \cdot 17 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 272.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: \(7.41146319060\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} + 25 x^{10} - 16 x^{9} + 294 x^{8} - 156 x^{7} + 1960 x^{6} + 2136 x^{5} + 2980 x^{4} + 976 x^{3} + 528 x^{2} - 64 x + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{22} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + (\beta_1 - 1) q^{5} + \beta_{4} q^{7} + (\beta_{10} - \beta_{8} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + (\beta_1 - 1) q^{5} + \beta_{4} q^{7} + (\beta_{10} - \beta_{8} - 2) q^{9} + (\beta_{9} + \beta_{2}) q^{11} + ( - \beta_{11} - \beta_{10} + 2) q^{13} + (\beta_{9} + \beta_{7} - \beta_{6} + \beta_{4} - \beta_{3} + 2 \beta_{2}) q^{15} - \beta_{8} q^{17} + (\beta_{6} - 2 \beta_{3}) q^{19} + (\beta_{11} - 6 \beta_{8} + \beta_{5} - 2 \beta_1 - 1) q^{21} + (\beta_{9} - \beta_{7} + 2 \beta_{6} + \beta_{4}) q^{23} + ( - \beta_{10} + 2 \beta_{8} - \beta_{5} + 2) q^{25} + (\beta_{9} - 2 \beta_{3}) q^{27} + (2 \beta_{11} - 4 \beta_{8} + 3 \beta_1 - 7) q^{29} + (\beta_{9} + \beta_{7} - 2 \beta_{6} + \beta_{4} - 2 \beta_{2}) q^{31} + (2 \beta_{11} - 2 \beta_{10} - \beta_{8} - \beta_{5} - 4 \beta_1 + 14) q^{33} + (\beta_{9} + 2 \beta_{7} - 2 \beta_{6} - \beta_{4} - 3 \beta_{3} - 2 \beta_{2}) q^{35} + ( - 2 \beta_{11} + 4 \beta_{10} - 6 \beta_{8} - 5 \beta_1 - 9) q^{37} + (\beta_{9} - \beta_{7} + 2 \beta_{3} - 8 \beta_{2}) q^{39} + ( - 2 \beta_{11} + 2 \beta_{8} + 2 \beta_{5} + 2) q^{41} + ( - \beta_{9} - \beta_{6} + 3 \beta_{4} - \beta_{3} + 8 \beta_{2}) q^{43} + ( - 2 \beta_{10} - 2 \beta_{8} - 2 \beta_{5} - 5 \beta_1 + 17) q^{45} + ( - \beta_{9} - \beta_{7} + 3 \beta_{6} + 3 \beta_{4} - \beta_{3} + 6 \beta_{2}) q^{47} + ( - 2 \beta_{11} + 4 \beta_{10} + 7 \beta_{8} - \beta_{5} + 4 \beta_1 - 21) q^{49} + ( - \beta_{4} - \beta_{3} + \beta_{2}) q^{51} + ( - 3 \beta_{10} - 4 \beta_{8} + \beta_{5} + 6 \beta_1 - 1) q^{53} + ( - 3 \beta_{9} - \beta_{7} + \beta_{6} - \beta_{4} + 5 \beta_{3} - 14 \beta_{2}) q^{55} + ( - 2 \beta_{11} - 2 \beta_{10} + 12 \beta_{8} + 4 \beta_1 - 8) q^{57} + (\beta_{9} + \beta_{6} - \beta_{4} + 3 \beta_{3} + 8 \beta_{2}) q^{59} + (2 \beta_{10} + 4 \beta_{8} + 9 \beta_1 + 7) q^{61} + ( - \beta_{9} + 3 \beta_{7} - 3 \beta_{4} - 6 \beta_{3} + 6 \beta_{2}) q^{63} + (2 \beta_{11} + 2 \beta_{10} + 4 \beta_{8} + 2 \beta_{5} + 4 \beta_1 - 2) q^{65} + ( - 4 \beta_{7} + \beta_{6} - 6 \beta_{4} + 8 \beta_{3} + 2 \beta_{2}) q^{67} + (5 \beta_{11} - 3 \beta_{10} - 6 \beta_{8} + 2 \beta_{5} + 6 \beta_1 - 6) q^{69} + (2 \beta_{9} + 2 \beta_{7} - 4 \beta_{6} + \beta_{4} + 8 \beta_{3} + 2 \beta_{2}) q^{71} + (4 \beta_{11} - 2 \beta_{10} + 8 \beta_{8} + 4 \beta_{5} - 2 \beta_1 + 16) q^{73} + ( - 4 \beta_{9} - 4 \beta_{7} + 2 \beta_{6} + 4 \beta_{4} + 4 \beta_{3} - 11 \beta_{2}) q^{75} + ( - 9 \beta_{11} + 5 \beta_{10} + 2 \beta_{8} - 4 \beta_{5} - 6 \beta_1 + 12) q^{77} + ( - 5 \beta_{9} - 3 \beta_{7} - 9 \beta_{4} + 6 \beta_{3} - 4 \beta_{2}) q^{79} + (6 \beta_{10} + \beta_{8} - \beta_{5} - 4 \beta_1 - 19) q^{81} + ( - 3 \beta_{9} + 4 \beta_{7} + \beta_{6} - \beta_{4} - 5 \beta_{3} + 4 \beta_{2}) q^{83} + (\beta_{10} - \beta_{5} - \beta_1 + 6) q^{85} + ( - \beta_{9} + 5 \beta_{7} - 3 \beta_{6} - 3 \beta_{4} - 9 \beta_{3} + 14 \beta_{2}) q^{87} + (2 \beta_{11} - 8 \beta_{10} + 21 \beta_{8} + \beta_{5} + 4 \beta_1 + 4) q^{89} + ( - 5 \beta_{9} + 2 \beta_{7} - 4 \beta_{6} + 2 \beta_{4} + 6 \beta_{3} - 6 \beta_{2}) q^{91} + (\beta_{11} + 3 \beta_{10} - 12 \beta_{8} - 2 \beta_{5} - 18 \beta_1 - 12) q^{93} + ( - 2 \beta_{9} - 4 \beta_{7} - 2 \beta_{6} + 2 \beta_{4} - 2 \beta_{3} + 4 \beta_{2}) q^{95} + ( - 2 \beta_{11} - 10 \beta_{10} + 20 \beta_{8} + 2 \beta_1 + 8) q^{97} + ( - 4 \beta_{9} - 6 \beta_{7} + 6 \beta_{6} - 6 \beta_{4} + 4 \beta_{3} - 21 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{5} - 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{5} - 28 q^{9} + 32 q^{13} - 16 q^{21} + 28 q^{25} - 92 q^{29} + 168 q^{33} - 116 q^{37} + 32 q^{41} + 212 q^{45} - 260 q^{49} - 80 q^{57} + 76 q^{61} - 40 q^{65} - 80 q^{69} + 184 q^{73} + 160 q^{77} - 252 q^{81} + 68 q^{85} + 72 q^{89} - 160 q^{93} + 144 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} + 25 x^{10} - 16 x^{9} + 294 x^{8} - 156 x^{7} + 1960 x^{6} + 2136 x^{5} + 2980 x^{4} + 976 x^{3} + 528 x^{2} - 64 x + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 154524933 \nu^{11} + 478470931 \nu^{10} - 3898157409 \nu^{9} + 2843759544 \nu^{8} - 45535904778 \nu^{7} + 29009461680 \nu^{6} + \cdots - 30127498968 ) / 39666231960 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 854388159 \nu^{11} - 1561421493 \nu^{10} + 17825980707 \nu^{9} + 12893673988 \nu^{8} + 222884377654 \nu^{7} + 165035138660 \nu^{6} + \cdots + 89227289184 ) / 158664927840 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1472487891 \nu^{11} - 3475305217 \nu^{10} + 33418610343 \nu^{9} + 1518635812 \nu^{8} + 405027996766 \nu^{7} + 48997291940 \nu^{6} + \cdots + 51072357216 ) / 158664927840 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2357301861 \nu^{11} - 1165878247 \nu^{10} + 38217782753 \nu^{9} + 118201104972 \nu^{8} + 528185910946 \nu^{7} + \cdots + 613533279136 ) / 158664927840 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 116341408 \nu^{11} + 373742662 \nu^{10} - 2949227696 \nu^{9} + 2346295329 \nu^{8} - 33658113765 \nu^{7} + 23952776007 \nu^{6} + \cdots + 78430328582 ) / 5949934794 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1075350649 \nu^{11} + 5442385203 \nu^{10} - 34670140997 \nu^{9} + 75858352572 \nu^{8} - 378966455394 \nu^{7} + \cdots + 276973569376 ) / 39666231960 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 149042703 \nu^{11} - 292603176 \nu^{10} + 3247596644 \nu^{9} + 1513474161 \nu^{8} + 40974795138 \nu^{7} + 22285243110 \nu^{6} + \cdots + 27351173848 ) / 4958278995 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1117537 \nu^{11} - 3584989 \nu^{10} + 28327961 \nu^{9} - 22460841 \nu^{8} + 323543127 \nu^{7} - 229980045 \nu^{6} + 2129151100 \nu^{5} + \cdots - 63777578 ) / 33691590 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 5343614297 \nu^{11} + 16067870099 \nu^{10} - 135092448541 \nu^{9} + 90458903436 \nu^{8} - 1604931624562 \nu^{7} + \cdots - 148594773792 ) / 79332463920 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 413871677 \nu^{11} - 1342367993 \nu^{10} + 10504360891 \nu^{9} - 8541536301 \nu^{8} + 119140931715 \nu^{7} - 87675314073 \nu^{6} + \cdots + 38648292212 ) / 5949934794 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1723724581 \nu^{11} - 5364724411 \nu^{10} + 43520193917 \nu^{9} - 32138337348 \nu^{8} + 506683696134 \nu^{7} - 327933428520 \nu^{6} + \cdots - 382439746496 ) / 23799739176 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} - \beta_{2} + \beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{10} + 2 \beta_{9} - 2 \beta_{8} + 3 \beta_{7} - \beta_{6} + \beta_{5} - 2 \beta_{4} + 4 \beta_{3} + 2 \beta_{2} - 4 \beta _1 - 27 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{11} - \beta_{10} + 4\beta_{8} + \beta_{5} - 26\beta _1 - 51 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 2 \beta_{11} + 7 \beta_{10} - 22 \beta_{9} - 8 \beta_{8} - 33 \beta_{7} + 5 \beta_{6} + 17 \beta_{5} + 40 \beta_{4} - 78 \beta_{3} - 42 \beta_{2} - 66 \beta _1 - 367 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 34 \beta_{11} + 27 \beta_{10} + 2 \beta_{9} - 108 \beta_{8} - 17 \beta_{7} - 61 \beta_{6} - 35 \beta_{5} + 148 \beta_{4} - 442 \beta_{3} + 166 \beta_{2} + 370 \beta _1 + 909 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 70\beta_{11} - 5\beta_{10} - 184\beta_{8} - 287\beta_{5} + 1086\beta _1 + 5417 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 574 \beta_{11} + 591 \beta_{10} - 62 \beta_{9} - 2320 \beta_{8} + 265 \beta_{7} + 1165 \beta_{6} - 811 \beta_{5} - 3052 \beta_{4} + 7226 \beta_{3} - 422 \beta_{2} + 5542 \beta _1 + 15893 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 1622 \beta_{11} - 1219 \beta_{10} + 2298 \beta_{9} + 6864 \beta_{8} + 4385 \beta_{7} + 2841 \beta_{6} + 4919 \beta_{5} - 14484 \beta_{4} + 25698 \beta_{3} + 14250 \beta_{2} - 18158 \beta _1 - 83785 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -9838\beta_{11} - 11771\beta_{10} + 45656\beta_{8} + 16391\beta_{5} - 86614\beta _1 - 277049 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 32782 \beta_{11} - 35923 \beta_{10} - 21050 \beta_{9} + 160952 \beta_{8} - 52209 \beta_{7} - 68705 \beta_{6} + 85135 \beta_{5} + 263996 \beta_{4} - 456826 \beta_{3} - 258698 \beta_{2} + \cdots - 1344609 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 170270 \beta_{11} + 223075 \beta_{10} + 32934 \beta_{9} - 858536 \beta_{8} - 63481 \beta_{7} - 393345 \beta_{6} - 311199 \beta_{5} + 1083996 \beta_{4} - 2055754 \beta_{3} - 660506 \beta_{2} + \cdots + 4827633 ) / 8 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/272\mathbb{Z}\right)^\times\).

\(n\) \(69\) \(239\) \(241\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
239.1
−1.64327 2.84622i
1.68421 2.91714i
−0.458115 + 0.793479i
0.146455 + 0.253667i
2.09244 + 3.62421i
−0.321718 + 0.557232i
−0.321718 0.557232i
2.09244 3.62421i
0.146455 0.253667i
−0.458115 0.793479i
1.68421 + 2.91714i
−1.64327 + 2.84622i
0 4.83337i 0 −8.57307 0 8.54360i 0 −14.3615 0
239.2 0 4.31458i 0 4.73683 0 6.12094i 0 −9.61562 0
239.3 0 4.14258i 0 −3.83246 0 13.9064i 0 −8.16093 0
239.4 0 2.14999i 0 −1.41418 0 7.84997i 0 4.37755 0
239.5 0 1.99259i 0 6.36975 0 6.28119i 0 5.02960 0
239.6 0 0.518786i 0 −3.28687 0 4.36794i 0 8.73086 0
239.7 0 0.518786i 0 −3.28687 0 4.36794i 0 8.73086 0
239.8 0 1.99259i 0 6.36975 0 6.28119i 0 5.02960 0
239.9 0 2.14999i 0 −1.41418 0 7.84997i 0 4.37755 0
239.10 0 4.14258i 0 −3.83246 0 13.9064i 0 −8.16093 0
239.11 0 4.31458i 0 4.73683 0 6.12094i 0 −9.61562 0
239.12 0 4.83337i 0 −8.57307 0 8.54360i 0 −14.3615 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 239.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 272.3.d.b 12
3.b odd 2 1 2448.3.k.f 12
4.b odd 2 1 inner 272.3.d.b 12
8.b even 2 1 1088.3.d.b 12
8.d odd 2 1 1088.3.d.b 12
12.b even 2 1 2448.3.k.f 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
272.3.d.b 12 1.a even 1 1 trivial
272.3.d.b 12 4.b odd 2 1 inner
1088.3.d.b 12 8.b even 2 1
1088.3.d.b 12 8.d odd 2 1
2448.3.k.f 12 3.b odd 2 1
2448.3.k.f 12 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 68T_{3}^{10} + 1700T_{3}^{8} + 18928T_{3}^{6} + 90304T_{3}^{4} + 159936T_{3}^{2} + 36864 \) acting on \(S_{3}^{\mathrm{new}}(272, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 68 T^{10} + 1700 T^{8} + \cdots + 36864 \) Copy content Toggle raw display
$5$ \( (T^{6} + 6 T^{5} - 64 T^{4} - 336 T^{3} + \cdots + 4608)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} + 424 T^{10} + \cdots + 24531077376 \) Copy content Toggle raw display
$11$ \( T^{12} + 1076 T^{10} + \cdots + 895794103296 \) Copy content Toggle raw display
$13$ \( (T^{6} - 16 T^{5} - 488 T^{4} + \cdots - 377600)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 17)^{6} \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 137783273324544 \) Copy content Toggle raw display
$23$ \( T^{12} + 4840 T^{10} + \cdots + 28\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( (T^{6} + 46 T^{5} - 2080 T^{4} + \cdots + 380420352)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + 4120 T^{10} + \cdots + 1647793264896 \) Copy content Toggle raw display
$37$ \( (T^{6} + 58 T^{5} - 4424 T^{4} + \cdots + 1464176512)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} - 16 T^{5} - 6748 T^{4} + \cdots - 617041728)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + 10384 T^{10} + \cdots + 11\!\cdots\!64 \) Copy content Toggle raw display
$47$ \( T^{12} + 13184 T^{10} + \cdots + 96\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( (T^{6} - 6988 T^{4} + \cdots + 4699279296)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + 11792 T^{10} + \cdots + 14\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( (T^{6} - 38 T^{5} - 7544 T^{4} + \cdots - 4119920768)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + 35440 T^{10} + \cdots + 47\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( T^{12} + 46520 T^{10} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( (T^{6} - 92 T^{5} + \cdots - 285567998400)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + 48904 T^{10} + \cdots + 78\!\cdots\!84 \) Copy content Toggle raw display
$83$ \( T^{12} + 40208 T^{10} + \cdots + 24\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( (T^{6} - 36 T^{5} - 28336 T^{4} + \cdots - 48635889408)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} - 72 T^{5} - 30268 T^{4} + \cdots + 48604036288)^{2} \) Copy content Toggle raw display
show more
show less