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Properties

Label 536.2.i.b
Level $536$
Weight $2$
Character orbit 536.i
Analytic conductor $4.280$
Analytic rank $0$
Dimension $14$
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] = [536,2,Mod(297,536)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(536, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("536.297");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 536 = 2^{3} \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 536.i (of order \(3\), 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: \(4.27998154834\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 3 x^{13} + 18 x^{12} - 11 x^{11} + 111 x^{10} - 15 x^{9} + 568 x^{8} + 431 x^{7} + 768 x^{6} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + ( - \beta_{8} + \beta_{6} - 1) q^{5} + (\beta_{11} - \beta_{10} - \beta_{4}) q^{7} + ( - \beta_{6} - \beta_{5} - \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + ( - \beta_{8} + \beta_{6} - 1) q^{5} + (\beta_{11} - \beta_{10} - \beta_{4}) q^{7} + ( - \beta_{6} - \beta_{5} - \beta_{2} + 1) q^{9} + ( - \beta_{13} - \beta_{12} - \beta_{9} + \cdots - 1) q^{11}+ \cdots + (\beta_{13} + 6 \beta_{11} - 2 \beta_{10} + \cdots + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 6 q^{3} - 10 q^{5} - 5 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 6 q^{3} - 10 q^{5} - 5 q^{7} + 12 q^{9} - 4 q^{11} + 9 q^{13} + 4 q^{15} - 2 q^{17} - 8 q^{19} - 8 q^{21} - 6 q^{23} + 28 q^{25} + 66 q^{27} + 2 q^{29} + 25 q^{31} + 9 q^{33} - 22 q^{35} - 18 q^{37} + 9 q^{39} - 22 q^{43} - 48 q^{45} - 5 q^{47} - 20 q^{49} + 27 q^{51} - 26 q^{53} - 34 q^{55} - 8 q^{57} - 22 q^{59} + 19 q^{61} + 4 q^{63} - 3 q^{65} + 28 q^{67} - 27 q^{69} - 14 q^{71} - 2 q^{73} - 14 q^{75} - 2 q^{77} + 23 q^{79} + 102 q^{81} + 10 q^{83} - 12 q^{85} - 6 q^{87} - 22 q^{89} + 14 q^{91} + 4 q^{93} - 11 q^{95} + 17 q^{97} + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 3 x^{13} + 18 x^{12} - 11 x^{11} + 111 x^{10} - 15 x^{9} + 568 x^{8} + 431 x^{7} + 768 x^{6} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 5867455815 \nu^{13} - 33912227754 \nu^{12} + 97338305874 \nu^{11} - 988840441113 \nu^{10} + \cdots + 1540257310437 ) / 13124682983942 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 173856754087 \nu^{13} - 929074064682 \nu^{12} + 6655485130990 \nu^{11} + \cdots + 35636953102811 ) / 91872780887594 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1741309387263 \nu^{13} - 11374165466116 \nu^{12} + 53357468546430 \nu^{11} + \cdots - 9253175085947 ) / 91872780887594 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1395940492272 \nu^{13} - 6696873438301 \nu^{12} + 33280706773736 \nu^{11} + \cdots + 16809372014257 ) / 45936390443797 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1555705480116 \nu^{13} + 7525900022344 \nu^{12} - 37308229437068 \nu^{11} + \cdots + 161565825269754 ) / 45936390443797 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5176521670557 \nu^{13} - 20593799020738 \nu^{12} + 106870659808538 \nu^{11} + \cdots - 344817753900019 ) / 91872780887594 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 3134861778246 \nu^{13} + 16104498036496 \nu^{12} - 78812014078624 \nu^{11} + \cdots + 185215014735721 ) / 45936390443797 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1540257310437 \nu^{13} - 4614904475496 \nu^{12} + 27758543815620 \nu^{11} + \cdots + 1183990375827 ) / 13124682983942 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 11168341227297 \nu^{13} + 38321455401376 \nu^{12} - 215195939758350 \nu^{11} + \cdots - 4533190371807 ) / 91872780887594 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 21497210064427 \nu^{13} + 64025432827464 \nu^{12} - 384951186782626 \nu^{11} + \cdots - 177112049898831 ) / 45936390443797 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 1272737684007 \nu^{13} + 3995792491180 \nu^{12} - 23422603985607 \nu^{11} + \cdots - 813307669773 ) / 2702140614341 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 29127018429540 \nu^{13} + 85985114796348 \nu^{12} - 517589458293419 \nu^{11} + \cdots - 205292041635316 ) / 45936390443797 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{11} - 4\beta_{9} + \beta_{3} + \beta_{2} + \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + 8\beta_{2} - 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{12} + 10\beta_{11} - \beta_{10} + 32\beta_{9} + 10\beta_{6} + 7\beta_{5} - 7\beta_{3} - 16\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 6 \beta_{13} - 12 \beta_{12} + 22 \beta_{11} - 11 \beta_{10} + 68 \beta_{9} - 12 \beta_{7} - 11 \beta_{4} + \cdots + 68 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{8} - 34\beta_{7} - 111\beta_{6} - 47\beta_{5} - 23\beta_{4} - 199\beta_{2} + 297 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 24 \beta_{13} + 145 \beta_{12} - 320 \beta_{11} + 122 \beta_{10} - 785 \beta_{9} - 24 \beta_{8} + \cdots + 776 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 45 \beta_{13} + 465 \beta_{12} - 1286 \beta_{11} + 343 \beta_{10} - 2938 \beta_{9} + 465 \beta_{7} + \cdots - 2938 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -53\beta_{8} + 1751\beta_{7} + 4105\beta_{6} + 452\beta_{5} + 1408\beta_{4} + 8223\beta_{2} - 8641 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 956 \beta_{13} - 5856 \beta_{12} + 15035 \beta_{11} - 4448 \beta_{10} + 30132 \beta_{9} + \cdots - 25869 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 3090 \beta_{13} - 20891 \beta_{12} + 49850 \beta_{11} - 16443 \beta_{10} + 93574 \beta_{9} + \cdots + 93574 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 15511\beta_{8} - 70741\beta_{7} - 175089\beta_{6} + 505\beta_{5} - 54298\beta_{4} - 287491\beta_{2} + 315856 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 54803 \beta_{13} + 245830 \beta_{12} - 588124 \beta_{11} + 191532 \beta_{10} - 1008909 \beta_{9} + \cdots + 969463 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/536\mathbb{Z}\right)^\times\).

\(n\) \(135\) \(269\) \(337\)
\(\chi(n)\) \(1\) \(1\) \(-1 - \beta_{9}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
297.1
−1.13321 + 1.96278i
−0.532311 + 0.921990i
−0.253439 + 0.438970i
−0.0638045 + 0.110513i
0.321460 − 0.556784i
1.49725 − 2.59331i
1.66406 − 2.88224i
−1.13321 − 1.96278i
−0.532311 − 0.921990i
−0.253439 − 0.438970i
−0.0638045 − 0.110513i
0.321460 + 0.556784i
1.49725 + 2.59331i
1.66406 + 2.88224i
0 −2.26642 0 −4.09775 0 −0.162026 − 0.280637i 0 2.13666 0
297.2 0 −1.06462 0 −0.160134 0 2.37979 + 4.12191i 0 −1.86658 0
297.3 0 −0.506879 0 3.46318 0 −1.66398 − 2.88210i 0 −2.74307 0
297.4 0 −0.127609 0 −1.25851 0 −1.69567 − 2.93698i 0 −2.98372 0
297.5 0 0.642919 0 −1.73223 0 −0.340592 − 0.589922i 0 −2.58665 0
297.6 0 2.99449 0 2.11918 0 −2.13397 − 3.69614i 0 5.96699 0
297.7 0 3.32812 0 −3.33374 0 1.11644 + 1.93373i 0 8.07637 0
305.1 0 −2.26642 0 −4.09775 0 −0.162026 + 0.280637i 0 2.13666 0
305.2 0 −1.06462 0 −0.160134 0 2.37979 − 4.12191i 0 −1.86658 0
305.3 0 −0.506879 0 3.46318 0 −1.66398 + 2.88210i 0 −2.74307 0
305.4 0 −0.127609 0 −1.25851 0 −1.69567 + 2.93698i 0 −2.98372 0
305.5 0 0.642919 0 −1.73223 0 −0.340592 + 0.589922i 0 −2.58665 0
305.6 0 2.99449 0 2.11918 0 −2.13397 + 3.69614i 0 5.96699 0
305.7 0 3.32812 0 −3.33374 0 1.11644 − 1.93373i 0 8.07637 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 297.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
67.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 536.2.i.b ✓ 14
4.b odd 2 1 1072.2.i.g 14
67.c even 3 1 inner 536.2.i.b ✓ 14
268.g odd 6 1 1072.2.i.g 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
536.2.i.b ✓ 14 1.a even 1 1 trivial
536.2.i.b ✓ 14 67.c even 3 1 inner
1072.2.i.g 14 4.b odd 2 1
1072.2.i.g 14 268.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{7} - 3T_{3}^{6} - 9T_{3}^{5} + 19T_{3}^{4} + 27T_{3}^{3} - 6T_{3}^{2} - 9T_{3} - 1 \) acting on \(S_{2}^{\mathrm{new}}(536, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( (T^{7} - 3 T^{6} - 9 T^{5} + \cdots - 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{7} + 5 T^{6} - 12 T^{5} + \cdots + 35)^{2} \) Copy content Toggle raw display
$7$ \( T^{14} + 5 T^{13} + \cdots + 12769 \) Copy content Toggle raw display
$11$ \( T^{14} + 4 T^{13} + \cdots + 1343281 \) Copy content Toggle raw display
$13$ \( T^{14} - 9 T^{13} + \cdots + 3094081 \) Copy content Toggle raw display
$17$ \( T^{14} + 2 T^{13} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 296252944 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 8978699536 \) Copy content Toggle raw display
$29$ \( T^{14} - 2 T^{13} + \cdots + 75394489 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 1406625025 \) Copy content Toggle raw display
$37$ \( T^{14} + 18 T^{13} + \cdots + 29235649 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 19145426689 \) Copy content Toggle raw display
$43$ \( (T^{7} + 11 T^{6} + \cdots - 20347)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} + 5 T^{13} + \cdots + 478864 \) Copy content Toggle raw display
$53$ \( (T^{7} + 13 T^{6} + \cdots - 64469)^{2} \) Copy content Toggle raw display
$59$ \( (T^{7} + 11 T^{6} + \cdots + 87409)^{2} \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 59483795449 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 6060711605323 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 26184094225 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 10922758144 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 363537849481 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 71987036416 \) Copy content Toggle raw display
$89$ \( (T^{7} + 11 T^{6} + \cdots + 384041)^{2} \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 329918128225 \) Copy content Toggle raw display
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