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Properties

Label 290.2.b.b
Level $290$
Weight $2$
Character orbit 290.b
Analytic conductor $2.316$
Analytic rank $0$
Dimension $10$
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] = [290,2,Mod(59,290)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(290, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("290.59");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 290 = 2 \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 290.b (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: \(2.31566165862\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 24x^{8} + 152x^{6} + 377x^{4} + 352x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} + (\beta_{5} - \beta_1) q^{3} - q^{4} + \beta_{3} q^{5} + ( - \beta_{4} + 1) q^{6} + (\beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} - \beta_{3} + \beta_{2} + \beta_1) q^{7} + \beta_{5} q^{8} + ( - \beta_{9} + \beta_{7} + \beta_{2} + \beta_1 - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} + (\beta_{5} - \beta_1) q^{3} - q^{4} + \beta_{3} q^{5} + ( - \beta_{4} + 1) q^{6} + (\beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} - \beta_{3} + \beta_{2} + \beta_1) q^{7} + \beta_{5} q^{8} + ( - \beta_{9} + \beta_{7} + \beta_{2} + \beta_1 - 2) q^{9} + \beta_{8} q^{10} + ( - \beta_{8} + \beta_{7}) q^{11} + ( - \beta_{5} + \beta_1) q^{12} + (\beta_{8} + \beta_{7} + \beta_{5} - \beta_{3} + \beta_{2} + \beta_1) q^{13} + ( - \beta_{9} + \beta_{7} - \beta_{3} + \beta_1 - 1) q^{14} + (\beta_{9} - 2 \beta_{8} - \beta_{6} - \beta_{5} + 2 \beta_{4} - \beta_{2} - \beta_1 + 1) q^{15} + q^{16} + ( - \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{3} - \beta_{2} - 3 \beta_1) q^{17} + ( - \beta_{6} + 2 \beta_{5} + \beta_{3} - \beta_{2} - \beta_1) q^{18} + (2 \beta_{9} - 2 \beta_{8} + 2 \beta_{4} - 2 \beta_{2} - 2 \beta_1) q^{19} - \beta_{3} q^{20} + (\beta_{9} - \beta_{7} - 3 \beta_{4} + 2 \beta_{3} + \beta_{2} - \beta_1) q^{21} + (\beta_{3} - \beta_{2}) q^{22} + (\beta_{8} + \beta_{7} + 2 \beta_{6} + 3 \beta_{5} + \beta_1) q^{23} + (\beta_{4} - 1) q^{24} + ( - \beta_{9} + 2 \beta_{8} + \beta_{7} + \beta_{6} - 2 \beta_{5} - \beta_{4} - \beta_{3} + 3 \beta_{2} + 3 \beta_1) q^{25} + ( - \beta_{8} + \beta_{7} + \beta_{4} - \beta_{3} - \beta_{2} + 1) q^{26} + (\beta_{8} + \beta_{7} - \beta_{6} - \beta_{3} + \beta_{2} + 2 \beta_1) q^{27} + ( - \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{3} - \beta_{2} - \beta_1) q^{28} - q^{29} + (\beta_{9} - \beta_{8} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1 - 1) q^{30} + ( - \beta_{9} + \beta_{8} - \beta_{3} + \beta_1 + 1) q^{31} - \beta_{5} q^{32} + ( - 2 \beta_{6} + 2 \beta_{5} - \beta_{3} + \beta_{2} + 2 \beta_1) q^{33} + (\beta_{9} - \beta_{7} - 2 \beta_{4} + \beta_{3} - \beta_1 - 1) q^{34} + ( - \beta_{9} + \beta_{7} - 2 \beta_{5} + 2 \beta_{4} - \beta_{3} - 2 \beta_{2} + \beta_1 + 1) q^{35} + (\beta_{9} - \beta_{7} - \beta_{2} - \beta_1 + 2) q^{36} + ( - 2 \beta_{8} - 2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{3} - 2 \beta_{2} - 4 \beta_1) q^{37} + 2 \beta_{6} q^{38} + (\beta_{9} + \beta_{8} - 2 \beta_{7} + \beta_{3} - \beta_1 - 1) q^{39} - \beta_{8} q^{40} + (2 \beta_{8} - 2 \beta_{7} - 2 \beta_{4}) q^{41} + (2 \beta_{8} + 2 \beta_{7} + \beta_{6} - \beta_{3} + \beta_{2} + 4 \beta_1) q^{42} + ( - 2 \beta_{8} - 2 \beta_{7} + \beta_{6} + \beta_{5} + \beta_{3} - \beta_{2} - \beta_1) q^{43} + (\beta_{8} - \beta_{7}) q^{44} + ( - \beta_{9} + \beta_{8} - 2 \beta_{7} - 3 \beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_{2} + \cdots + 3) q^{45}+ \cdots + (2 \beta_{8} - 2 \beta_{7} + 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 10 q^{4} + 6 q^{6} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 10 q^{4} + 6 q^{6} - 20 q^{9} - 10 q^{14} + 18 q^{15} + 10 q^{16} + 8 q^{19} - 12 q^{21} - 6 q^{24} - 4 q^{25} + 14 q^{26} - 10 q^{29} - 6 q^{30} + 10 q^{31} - 18 q^{34} + 18 q^{35} + 20 q^{36} - 10 q^{39} - 8 q^{41} + 26 q^{45} + 26 q^{46} - 32 q^{49} - 16 q^{50} - 64 q^{51} + 12 q^{54} + 16 q^{55} + 10 q^{56} + 18 q^{59} - 18 q^{60} + 10 q^{61} - 10 q^{64} + 20 q^{65} + 36 q^{66} - 14 q^{69} - 20 q^{70} - 12 q^{71} + 12 q^{74} + 20 q^{75} - 8 q^{76} + 34 q^{79} - 6 q^{81} + 12 q^{84} + 18 q^{85} + 2 q^{86} - 20 q^{89} - 34 q^{90} - 44 q^{91} - 28 q^{94} - 8 q^{95} + 6 q^{96} + 68 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 24x^{8} + 152x^{6} + 377x^{4} + 352x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 11 \nu^{9} + 200 \nu^{8} - 224 \nu^{7} + 4128 \nu^{6} - 968 \nu^{5} + 16384 \nu^{4} - 3059 \nu^{3} + 17480 \nu^{2} - 5240 \nu + 1696 ) / 1216 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11 \nu^{9} + 200 \nu^{8} + 224 \nu^{7} + 4128 \nu^{6} + 968 \nu^{5} + 16384 \nu^{4} + 3059 \nu^{3} + 17480 \nu^{2} + 5240 \nu + 1696 ) / 1216 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 17\nu^{8} + 360\nu^{6} + 1572\nu^{4} + 2033\nu^{2} + 360 ) / 76 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 45\nu^{9} + 944\nu^{7} + 3960\nu^{5} + 4389\nu^{3} - 424\nu ) / 608 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 55\nu^{9} + 1120\nu^{7} + 4232\nu^{5} + 4351\nu^{3} + 1272\nu ) / 608 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 117 \nu^{9} + 20 \nu^{8} - 2576 \nu^{7} + 352 \nu^{6} - 12728 \nu^{5} + 544 \nu^{4} - 20045 \nu^{3} - 76 \nu^{2} - 7288 \nu + 2176 ) / 1216 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 117 \nu^{9} - 20 \nu^{8} - 2576 \nu^{7} - 352 \nu^{6} - 12728 \nu^{5} - 544 \nu^{4} - 20045 \nu^{3} + 76 \nu^{2} - 7288 \nu - 2176 ) / 1216 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 32 \nu^{9} - 81 \nu^{8} - 700 \nu^{7} - 1760 \nu^{6} - 3424 \nu^{5} - 8344 \nu^{4} - 5776 \nu^{3} - 12217 \nu^{2} - 2828 \nu - 3128 ) / 304 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{9} + \beta_{7} - 2\beta_{4} + \beta_{2} + \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{8} - \beta_{7} + 4\beta_{6} - 7\beta_{5} - 2\beta_{3} + 2\beta_{2} - 8\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 18\beta_{9} - 6\beta_{8} - 12\beta_{7} + 41\beta_{4} - 4\beta_{3} - 22\beta_{2} - 18\beta _1 + 38 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 18\beta_{8} + 18\beta_{7} - 73\beta_{6} + 126\beta_{5} + 41\beta_{3} - 41\beta_{2} + 103\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -294\beta_{9} + 118\beta_{8} + 176\beta_{7} - 678\beta_{4} + 73\beta_{3} + 367\beta_{2} + 294\beta _1 - 531 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -294\beta_{8} - 294\beta_{7} + 1176\beta_{6} - 2036\beta_{5} - 678\beta_{3} + 678\beta_{2} - 1561\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 4681 \beta_{9} - 1944 \beta_{8} - 2737 \beta_{7} + 10810 \beta_{4} - 1176 \beta_{3} - 5857 \beta_{2} - 4681 \beta _1 + 8188 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 4681\beta_{8} + 4681\beta_{7} - 18636\beta_{6} + 32319\beta_{5} + 10810\beta_{3} - 10810\beta_{2} + 24472\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/290\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(117\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
59.1
3.97530i
1.68193i
− 0.485804i
− 1.35864i
− 1.81278i
1.81278i
1.35864i
0.485804i
− 1.68193i
− 3.97530i
− 1.00000i − 2.97530i −1.00000 −0.639550 + 2.14266i −2.97530 − 3.57331i 1.00000i −5.85241 2.14266 + 0.639550i
59.2 − 1.00000i − 0.681929i −1.00000 2.23558 − 0.0464742i −0.681929 − 0.936197i 1.00000i 2.53497 −0.0464742 − 2.23558i
59.3 − 1.00000i 1.48580i −1.00000 −1.29150 − 1.82539i 1.48580 4.37538i 1.00000i 0.792385 −1.82539 + 1.29150i
59.4 − 1.00000i 2.35864i −1.00000 1.32739 − 1.79945i 2.35864 − 4.21797i 1.00000i −2.56319 −1.79945 − 1.32739i
59.5 − 1.00000i 2.81278i −1.00000 −1.63193 + 1.52866i 2.81278 − 0.647892i 1.00000i −4.91176 1.52866 + 1.63193i
59.6 1.00000i − 2.81278i −1.00000 −1.63193 − 1.52866i 2.81278 0.647892i − 1.00000i −4.91176 1.52866 − 1.63193i
59.7 1.00000i − 2.35864i −1.00000 1.32739 + 1.79945i 2.35864 4.21797i − 1.00000i −2.56319 −1.79945 + 1.32739i
59.8 1.00000i − 1.48580i −1.00000 −1.29150 + 1.82539i 1.48580 − 4.37538i − 1.00000i 0.792385 −1.82539 − 1.29150i
59.9 1.00000i 0.681929i −1.00000 2.23558 + 0.0464742i −0.681929 0.936197i − 1.00000i 2.53497 −0.0464742 + 2.23558i
59.10 1.00000i 2.97530i −1.00000 −0.639550 − 2.14266i −2.97530 3.57331i − 1.00000i −5.85241 2.14266 − 0.639550i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 59.10
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 290.2.b.b ✓ 10
3.b odd 2 1 2610.2.e.i 10
4.b odd 2 1 2320.2.d.h 10
5.b even 2 1 inner 290.2.b.b ✓ 10
5.c odd 4 1 1450.2.a.t 5
5.c odd 4 1 1450.2.a.u 5
15.d odd 2 1 2610.2.e.i 10
20.d odd 2 1 2320.2.d.h 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
290.2.b.b ✓ 10 1.a even 1 1 trivial
290.2.b.b ✓ 10 5.b even 2 1 inner
1450.2.a.t 5 5.c odd 4 1
1450.2.a.u 5 5.c odd 4 1
2320.2.d.h 10 4.b odd 2 1
2320.2.d.h 10 20.d odd 2 1
2610.2.e.i 10 3.b odd 2 1
2610.2.e.i 10 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} + 25T_{3}^{8} + 224T_{3}^{6} + 849T_{3}^{4} + 1209T_{3}^{2} + 400 \) acting on \(S_{2}^{\mathrm{new}}(290, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{5} \) Copy content Toggle raw display
$3$ \( T^{10} + 25 T^{8} + 224 T^{6} + \cdots + 400 \) Copy content Toggle raw display
$5$ \( T^{10} + 2 T^{8} - 18 T^{7} + \cdots + 3125 \) Copy content Toggle raw display
$7$ \( T^{10} + 51 T^{8} + 877 T^{6} + \cdots + 1600 \) Copy content Toggle raw display
$11$ \( (T^{5} - 27 T^{3} + 4 T^{2} + 172 T - 16)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + 53 T^{8} + 356 T^{6} + \cdots + 64 \) Copy content Toggle raw display
$17$ \( T^{10} + 119 T^{8} + 5325 T^{6} + \cdots + 2930944 \) Copy content Toggle raw display
$19$ \( (T^{5} - 4 T^{4} - 56 T^{3} + 144 T^{2} + \cdots - 640)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + 167 T^{8} + 9813 T^{6} + \cdots + 2611456 \) Copy content Toggle raw display
$29$ \( (T + 1)^{10} \) Copy content Toggle raw display
$31$ \( (T^{5} - 5 T^{4} - 22 T^{3} + 175 T^{2} + \cdots + 184)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + 212 T^{8} + 12608 T^{6} + \cdots + 262144 \) Copy content Toggle raw display
$41$ \( (T^{5} + 4 T^{4} - 76 T^{3} - 504 T^{2} + \cdots + 608)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + 205 T^{8} + 11856 T^{6} + \cdots + 440896 \) Copy content Toggle raw display
$47$ \( T^{10} + 194 T^{8} + 11057 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$53$ \( T^{10} + 109 T^{8} + 1084 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$59$ \( (T^{5} - 9 T^{4} - 79 T^{3} + 448 T^{2} + \cdots - 4000)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} - 5 T^{4} - 169 T^{3} + 998 T^{2} + \cdots - 9808)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + 564 T^{8} + 112832 T^{6} + \cdots + 1048576 \) Copy content Toggle raw display
$71$ \( (T^{5} + 6 T^{4} - 208 T^{3} - 1192 T^{2} + \cdots + 62464)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + 543 T^{8} + \cdots + 3008303104 \) Copy content Toggle raw display
$79$ \( (T^{5} - 17 T^{4} + 48 T^{3} + 471 T^{2} + \cdots + 3020)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + 684 T^{8} + \cdots + 3399356416 \) Copy content Toggle raw display
$89$ \( (T^{5} + 10 T^{4} - 96 T^{3} - 1088 T^{2} + \cdots - 160)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + 623 T^{8} + \cdots + 3351946816 \) Copy content Toggle raw display
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