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Properties

Label 510.2.u.d
Level $510$
Weight $2$
Character orbit 510.u
Analytic conductor $4.072$
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] = [510,2,Mod(121,510)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(510, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 0, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("510.121");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 510 = 2 \cdot 3 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 510.u (of order \(8\), degree \(4\), 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: \(4.07237050309\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{8})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 32x^{14} + 400x^{12} + 2576x^{10} + 9314x^{8} + 18976x^{6} + 20176x^{4} + 8720x^{2} + 289 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} - \beta_{9} q^{3} + \beta_{5} q^{4} - \beta_1 q^{5} - \beta_{10} q^{6} + (\beta_{11} + \beta_{9} - \beta_{3} + \cdots + 1) q^{7}+ \cdots + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} - \beta_{9} q^{3} + \beta_{5} q^{4} - \beta_1 q^{5} - \beta_{10} q^{6} + (\beta_{11} + \beta_{9} - \beta_{3} + \cdots + 1) q^{7}+ \cdots + (\beta_{15} - \beta_{11} + \cdots + 2 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{7} + 8 q^{14} - 16 q^{16} + 16 q^{18} + 16 q^{22} + 8 q^{28} + 16 q^{29} + 16 q^{33} - 16 q^{34} + 16 q^{37} + 8 q^{39} + 8 q^{41} - 16 q^{42} - 16 q^{43} - 16 q^{44} + 16 q^{46} - 16 q^{49} - 16 q^{50} + 8 q^{51} + 16 q^{52} + 8 q^{56} - 16 q^{57} - 16 q^{58} - 32 q^{59} + 16 q^{61} - 8 q^{63} + 8 q^{65} - 32 q^{67} - 16 q^{68} - 16 q^{70} + 8 q^{71} + 24 q^{73} + 16 q^{74} - 64 q^{77} - 8 q^{78} - 16 q^{79} + 24 q^{82} - 32 q^{83} - 8 q^{85} + 32 q^{86} + 16 q^{87} - 96 q^{91} - 16 q^{92} - 16 q^{93} - 16 q^{94} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 32x^{14} + 400x^{12} + 2576x^{10} + 9314x^{8} + 18976x^{6} + 20176x^{4} + 8720x^{2} + 289 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 78 \nu^{15} + 511 \nu^{14} - 2332 \nu^{13} + 14809 \nu^{12} - 26316 \nu^{11} + 159647 \nu^{10} + \cdots + 51731 ) / 9248 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 3 \nu^{15} + 262 \nu^{14} - 91 \nu^{13} + 7732 \nu^{12} - 1037 \nu^{11} + 85510 \nu^{10} + \cdots + 33524 ) / 4624 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3 \nu^{15} + 262 \nu^{14} + 91 \nu^{13} + 7732 \nu^{12} + 1037 \nu^{11} + 85510 \nu^{10} + \cdots + 33524 ) / 4624 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 78 \nu^{15} - 511 \nu^{14} - 2332 \nu^{13} - 14809 \nu^{12} - 26316 \nu^{11} - 159647 \nu^{10} + \cdots - 51731 ) / 9248 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 13\nu^{15} + 382\nu^{13} + 4197\nu^{11} + 22404\nu^{9} + 61429\nu^{7} + 81142\nu^{5} + 38925\nu^{3} - 132\nu ) / 272 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 78 \nu^{15} + 645 \nu^{14} - 2332 \nu^{13} + 19293 \nu^{12} - 26316 \nu^{11} + 217209 \nu^{10} + \cdots + 76007 ) / 9248 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 78 \nu^{15} - 645 \nu^{14} - 2332 \nu^{13} - 19293 \nu^{12} - 26316 \nu^{11} - 217209 \nu^{10} + \cdots - 76007 ) / 9248 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 295 \nu^{15} + 92 \nu^{14} - 8631 \nu^{13} + 2768 \nu^{12} - 94401 \nu^{11} + 31467 \nu^{10} + \cdots + 6069 ) / 4624 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 512 \nu^{15} - 675 \nu^{14} + 14930 \nu^{13} - 19693 \nu^{12} + 162486 \nu^{11} - 214421 \nu^{10} + \cdots - 60401 ) / 9248 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 512 \nu^{15} - 675 \nu^{14} - 14930 \nu^{13} - 19693 \nu^{12} - 162486 \nu^{11} - 214421 \nu^{10} + \cdots - 60401 ) / 9248 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 88 \nu^{15} + 645 \nu^{14} + 2624 \nu^{13} + 18715 \nu^{12} + 29461 \nu^{11} + 202181 \nu^{10} + \cdots + 58089 ) / 4624 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 303 \nu^{15} - 645 \nu^{14} + 8936 \nu^{13} - 18715 \nu^{12} + 98736 \nu^{11} - 202181 \nu^{10} + \cdots - 58089 ) / 4624 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 131 \nu^{15} + 3866 \nu^{13} + 42755 \nu^{11} + 230458 \nu^{9} + 641631 \nu^{7} + 871706 \nu^{5} + \cdots + 16762 \nu ) / 1156 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 814 \nu^{15} - 809 \nu^{14} + 23745 \nu^{13} - 23599 \nu^{12} + 258383 \nu^{11} - 256955 \nu^{10} + \cdots - 85255 ) / 4624 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 593 \nu^{15} + 809 \nu^{14} + 17251 \nu^{13} + 23599 \nu^{12} + 187034 \nu^{11} + 256955 \nu^{10} + \cdots + 85255 ) / 4624 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} - \beta_{6} + \beta_{4} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{15} - \beta_{14} + \beta_{12} - \beta_{11} + 2 \beta_{10} + 2 \beta_{9} - 2 \beta_{4} + \beta_{3} + \cdots - 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 2 \beta_{13} - 2 \beta_{12} - 2 \beta_{11} + 5 \beta_{7} + 5 \beta_{6} + 6 \beta_{5} + \cdots - 11 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 5 \beta_{15} + 5 \beta_{14} - 4 \beta_{12} + 4 \beta_{11} - 10 \beta_{10} - 14 \beta_{9} - 4 \beta_{8} + \cdots + 27 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 6 \beta_{15} + 6 \beta_{14} + 30 \beta_{13} + 30 \beta_{12} + 30 \beta_{11} + 14 \beta_{10} + \cdots + 115 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 113 \beta_{15} - 113 \beta_{14} + 69 \beta_{12} - 69 \beta_{11} + 242 \beta_{10} + 362 \beta_{9} + \cdots - 480 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 116 \beta_{15} - 116 \beta_{14} - 388 \beta_{13} - 382 \beta_{12} - 382 \beta_{11} + \cdots - 1275 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 668 \beta_{15} + 668 \beta_{14} - 340 \beta_{12} + 340 \beta_{11} - 1488 \beta_{10} - 2232 \beta_{9} + \cdots + 2503 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1664 \beta_{15} + 1664 \beta_{14} + 4832 \beta_{13} + 4656 \beta_{12} + 4656 \beta_{11} + \cdots + 14719 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 16023 \beta_{15} - 16023 \beta_{14} + 7367 \beta_{12} - 7367 \beta_{11} + 36302 \beta_{10} + \cdots - 56392 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 21600 \beta_{15} - 21600 \beta_{14} - 59182 \beta_{13} - 56062 \beta_{12} - 56062 \beta_{11} + \cdots - 173557 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 96415 \beta_{15} + 96415 \beta_{14} - 42064 \beta_{12} + 42064 \beta_{11} - 219894 \beta_{10} + \cdots + 328909 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 269018 \beta_{15} + 269018 \beta_{14} + 718530 \beta_{13} + 673170 \beta_{12} + 673170 \beta_{11} + \cdots + 2067341 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 2321159 \beta_{15} - 2321159 \beta_{14} + 986451 \beta_{12} - 986451 \beta_{11} + 5307102 \beta_{10} + \cdots - 7796016 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 3288540 \beta_{15} - 3288540 \beta_{14} - 8682284 \beta_{13} - 8081714 \beta_{12} + \cdots - 24742389 \beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/510\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(307\) \(341\)
\(\chi(n)\) \(\beta_{2}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
− 0.189865i
2.03762i
1.61909i
− 3.46685i
− 1.67373i
2.43910i
1.05400i
− 1.81937i
1.67373i
− 2.43910i
− 1.05400i
1.81937i
0.189865i
− 2.03762i
− 1.61909i
3.46685i
0.707107 + 0.707107i −0.923880 + 0.382683i 1.00000i −0.382683 − 0.923880i −0.923880 − 0.382683i 0.438442 − 1.05849i −0.707107 + 0.707107i 0.707107 − 0.707107i 0.382683 − 0.923880i
121.2 0.707107 + 0.707107i −0.923880 + 0.382683i 1.00000i −0.382683 − 0.923880i −0.923880 − 0.382683i 1.64395 − 3.96885i −0.707107 + 0.707107i 0.707107 − 0.707107i 0.382683 − 0.923880i
121.3 0.707107 + 0.707107i 0.923880 − 0.382683i 1.00000i 0.382683 + 0.923880i 0.923880 + 0.382683i −1.41744 + 3.42200i −0.707107 + 0.707107i 0.707107 − 0.707107i −0.382683 + 0.923880i
121.4 0.707107 + 0.707107i 0.923880 − 0.382683i 1.00000i 0.382683 + 0.923880i 0.923880 + 0.382683i 1.33505 − 3.22309i −0.707107 + 0.707107i 0.707107 − 0.707107i −0.382683 + 0.923880i
151.1 −0.707107 − 0.707107i −0.382683 − 0.923880i 1.00000i 0.923880 − 0.382683i −0.382683 + 0.923880i −0.880271 − 0.364620i 0.707107 − 0.707107i −0.707107 + 0.707107i −0.923880 − 0.382683i
151.2 −0.707107 − 0.707107i −0.382683 − 0.923880i 1.00000i 0.923880 − 0.382683i −0.382683 + 0.923880i 4.49340 + 1.86123i 0.707107 − 0.707107i −0.707107 + 0.707107i −0.923880 − 0.382683i
151.3 −0.707107 − 0.707107i 0.382683 + 0.923880i 1.00000i −0.923880 + 0.382683i 0.382683 − 0.923880i −2.68368 − 1.11162i 0.707107 − 0.707107i −0.707107 + 0.707107i 0.923880 + 0.382683i
151.4 −0.707107 − 0.707107i 0.382683 + 0.923880i 1.00000i −0.923880 + 0.382683i 0.382683 − 0.923880i 1.07055 + 0.443437i 0.707107 − 0.707107i −0.707107 + 0.707107i 0.923880 + 0.382683i
331.1 −0.707107 + 0.707107i −0.382683 + 0.923880i − 1.00000i 0.923880 + 0.382683i −0.382683 − 0.923880i −0.880271 + 0.364620i 0.707107 + 0.707107i −0.707107 − 0.707107i −0.923880 + 0.382683i
331.2 −0.707107 + 0.707107i −0.382683 + 0.923880i − 1.00000i 0.923880 + 0.382683i −0.382683 − 0.923880i 4.49340 − 1.86123i 0.707107 + 0.707107i −0.707107 − 0.707107i −0.923880 + 0.382683i
331.3 −0.707107 + 0.707107i 0.382683 − 0.923880i − 1.00000i −0.923880 − 0.382683i 0.382683 + 0.923880i −2.68368 + 1.11162i 0.707107 + 0.707107i −0.707107 − 0.707107i 0.923880 − 0.382683i
331.4 −0.707107 + 0.707107i 0.382683 − 0.923880i − 1.00000i −0.923880 − 0.382683i 0.382683 + 0.923880i 1.07055 − 0.443437i 0.707107 + 0.707107i −0.707107 − 0.707107i 0.923880 − 0.382683i
451.1 0.707107 − 0.707107i −0.923880 − 0.382683i − 1.00000i −0.382683 + 0.923880i −0.923880 + 0.382683i 0.438442 + 1.05849i −0.707107 − 0.707107i 0.707107 + 0.707107i 0.382683 + 0.923880i
451.2 0.707107 − 0.707107i −0.923880 − 0.382683i − 1.00000i −0.382683 + 0.923880i −0.923880 + 0.382683i 1.64395 + 3.96885i −0.707107 − 0.707107i 0.707107 + 0.707107i 0.382683 + 0.923880i
451.3 0.707107 − 0.707107i 0.923880 + 0.382683i − 1.00000i 0.382683 − 0.923880i 0.923880 − 0.382683i −1.41744 − 3.42200i −0.707107 − 0.707107i 0.707107 + 0.707107i −0.382683 − 0.923880i
451.4 0.707107 − 0.707107i 0.923880 + 0.382683i − 1.00000i 0.382683 − 0.923880i 0.923880 − 0.382683i 1.33505 + 3.22309i −0.707107 − 0.707107i 0.707107 + 0.707107i −0.382683 − 0.923880i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 121.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.d even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 510.2.u.d ✓ 16
17.d even 8 1 inner 510.2.u.d ✓ 16
17.e odd 16 1 8670.2.a.cl 8
17.e odd 16 1 8670.2.a.cm 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
510.2.u.d ✓ 16 1.a even 1 1 trivial
510.2.u.d ✓ 16 17.d even 8 1 inner
8670.2.a.cl 8 17.e odd 16 1
8670.2.a.cm 8 17.e odd 16 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{16} - 8 T_{7}^{15} + 40 T_{7}^{14} - 128 T_{7}^{13} + 288 T_{7}^{12} - 448 T_{7}^{11} + \cdots + 984064 \) acting on \(S_{2}^{\mathrm{new}}(510, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{4} \) Copy content Toggle raw display
$3$ \( (T^{8} + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{8} + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} - 8 T^{15} + \cdots + 984064 \) Copy content Toggle raw display
$11$ \( T^{16} - 128 T^{13} + \cdots + 2408704 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 339591184 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 6975757441 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 251920384 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 858958864 \) Copy content Toggle raw display
$29$ \( T^{16} - 16 T^{15} + \cdots + 54878464 \) Copy content Toggle raw display
$31$ \( T^{16} - 40 T^{14} + \cdots + 74580496 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 13389266944 \) Copy content Toggle raw display
$41$ \( T^{16} - 8 T^{15} + \cdots + 1024 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 19679600656 \) Copy content Toggle raw display
$47$ \( T^{16} + 384 T^{14} + \cdots + 18939904 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 297051160576 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 35664354017296 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 16680239104 \) Copy content Toggle raw display
$67$ \( (T^{8} + 16 T^{7} + \cdots - 69884)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} - 8 T^{15} + \cdots + 1024 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 3897754624 \) Copy content Toggle raw display
$79$ \( T^{16} + 16 T^{15} + \cdots + 53173264 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 21040642392064 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 1117299736576 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 80852035634176 \) Copy content Toggle raw display
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