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Properties

Label 1205.2.a.b
Level $1205$
Weight $2$
Character orbit 1205.a
Self dual yes
Analytic conductor $9.622$
Analytic rank $1$
Dimension $11$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1205,2,Mod(1,1205)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1205, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1205.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1205 = 5 \cdot 241 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1205.a (trivial)

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: yes
Analytic conductor: \(9.62197344356\)
Analytic rank: \(1\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{11} - 2x^{10} - 11x^{9} + 15x^{8} + 43x^{7} - 28x^{6} - 62x^{5} + 14x^{4} + 31x^{3} + x^{2} - 5x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1 - 1) q^{2} + ( - \beta_{3} - \beta_{2} - 1) q^{3} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{4} + q^{5} + ( - \beta_{10} + \beta_{6} - \beta_{5} + \cdots + 2) q^{6}+ \cdots + (\beta_{9} - \beta_{8} + \beta_{6} + \cdots + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1 - 1) q^{2} + ( - \beta_{3} - \beta_{2} - 1) q^{3} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{4} + q^{5} + ( - \beta_{10} + \beta_{6} - \beta_{5} + \cdots + 2) q^{6}+ \cdots + (3 \beta_{10} + 4 \beta_{9} - 4 \beta_{8} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q - 4 q^{2} - 8 q^{3} + 6 q^{4} + 11 q^{5} + 7 q^{6} - 9 q^{7} - 12 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 11 q - 4 q^{2} - 8 q^{3} + 6 q^{4} + 11 q^{5} + 7 q^{6} - 9 q^{7} - 12 q^{8} + 9 q^{9} - 4 q^{10} - 3 q^{11} - 28 q^{12} - 9 q^{13} + 2 q^{14} - 8 q^{15} - 16 q^{16} - 4 q^{17} - 6 q^{18} - 33 q^{19} + 6 q^{20} + 2 q^{21} + 6 q^{22} - 31 q^{23} + 32 q^{24} + 11 q^{25} - 20 q^{26} - 32 q^{27} - q^{28} + q^{29} + 7 q^{30} + 6 q^{31} + 7 q^{32} - 35 q^{33} + 9 q^{34} - 9 q^{35} + 33 q^{36} - 23 q^{37} + 20 q^{38} + 14 q^{39} - 12 q^{40} + 8 q^{41} - 26 q^{42} - 19 q^{43} + 9 q^{45} + 6 q^{46} - 35 q^{47} + 16 q^{48} + 4 q^{49} - 4 q^{50} - 3 q^{51} - 3 q^{52} + 14 q^{53} + 9 q^{54} - 3 q^{55} + 33 q^{56} + q^{57} - 11 q^{58} - 6 q^{59} - 28 q^{60} + 9 q^{61} - 23 q^{62} - 31 q^{63} + 18 q^{64} - 9 q^{65} - 36 q^{66} - 54 q^{67} + q^{68} + 17 q^{69} + 2 q^{70} - 5 q^{71} - 64 q^{72} + 17 q^{73} + 8 q^{74} - 8 q^{75} - 31 q^{76} - 18 q^{77} + 15 q^{78} - 16 q^{79} - 16 q^{80} + 43 q^{81} - 61 q^{82} - 29 q^{83} + 69 q^{84} - 4 q^{85} + 5 q^{86} + 5 q^{87} - 14 q^{88} - 5 q^{89} - 6 q^{90} - 54 q^{91} - 6 q^{92} - 25 q^{93} - 19 q^{94} - 33 q^{95} + 9 q^{96} + 6 q^{97} - 29 q^{98} + 60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{11} - 2x^{10} - 11x^{9} + 15x^{8} + 43x^{7} - 28x^{6} - 62x^{5} + 14x^{4} + 31x^{3} + x^{2} - 5x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{10} - 2\nu^{9} - 11\nu^{8} + 15\nu^{7} + 43\nu^{6} - 28\nu^{5} - 62\nu^{4} + 14\nu^{3} + 31\nu^{2} + \nu - 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( - 4 \nu^{10} + 9 \nu^{9} + 42 \nu^{8} - 71 \nu^{7} - 157 \nu^{6} + 155 \nu^{5} + 220 \nu^{4} + \cdots + 17 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 4 \nu^{10} - 9 \nu^{9} - 42 \nu^{8} + 71 \nu^{7} + 157 \nu^{6} - 155 \nu^{5} - 220 \nu^{4} + 118 \nu^{3} + \cdots - 19 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 8 \nu^{10} - 19 \nu^{9} - 80 \nu^{8} + 148 \nu^{7} + 280 \nu^{6} - 315 \nu^{5} - 349 \nu^{4} + 221 \nu^{3} + \cdots - 17 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( - 9 \nu^{10} + 23 \nu^{9} + 88 \nu^{8} - 187 \nu^{7} - 303 \nu^{6} + 441 \nu^{5} + 390 \nu^{4} + \cdots + 30 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 13 \nu^{10} - 30 \nu^{9} - 133 \nu^{8} + 235 \nu^{7} + 477 \nu^{6} - 505 \nu^{5} - 611 \nu^{4} + \cdots - 32 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( 14 \nu^{10} - 34 \nu^{9} - 140 \nu^{8} + 271 \nu^{7} + 492 \nu^{6} - 609 \nu^{5} - 630 \nu^{4} + \cdots - 39 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( 18 \nu^{10} - 43 \nu^{9} - 182 \nu^{8} + 342 \nu^{7} + 649 \nu^{6} - 764 \nu^{5} - 849 \nu^{4} + \cdots - 53 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( - 24 \nu^{10} + 58 \nu^{9} + 240 \nu^{8} - 460 \nu^{7} - 843 \nu^{6} + 1023 \nu^{5} + 1074 \nu^{4} + \cdots + 61 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} + \beta_{9} + \beta_{5} + \beta_{4} + \beta_{3} - 2\beta_{2} + 6\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{10} + 2\beta_{9} - \beta_{8} + \beta_{5} + 7\beta_{4} + 8\beta_{3} - 2\beta_{2} + 10\beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 10 \beta_{10} + 11 \beta_{9} - 3 \beta_{8} + \beta_{7} - 2 \beta_{6} + 10 \beta_{5} + 11 \beta_{4} + \cdots + 14 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 17 \beta_{10} + 27 \beta_{9} - 17 \beta_{8} + 3 \beta_{7} - 6 \beta_{6} + 18 \beta_{5} + 50 \beta_{4} + \cdots + 67 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 87 \beta_{10} + 104 \beta_{9} - 52 \beta_{8} + 17 \beta_{7} - 32 \beta_{6} + 89 \beta_{5} + 103 \beta_{4} + \cdots + 138 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 196 \beta_{10} + 283 \beta_{9} - 203 \beta_{8} + 52 \beta_{7} - 99 \beta_{6} + 211 \beta_{5} + \cdots + 503 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 754 \beta_{10} + 950 \beta_{9} - 622 \beta_{8} + 203 \beta_{7} - 374 \beta_{6} + 791 \beta_{5} + \cdots + 1232 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1956 \beta_{10} + 2710 \beta_{9} - 2112 \beta_{8} + 622 \beta_{7} - 1155 \beta_{6} + 2122 \beta_{5} + \cdots + 3987 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.07008
0.582513
−1.54550
0.745266
−1.19664
1.27000
−0.514683
2.52552
−0.385940
2.91366
−0.324121
−2.58701 −2.62253 4.69261 1.00000 6.78451 −4.29833 −6.96581 3.87767 −2.58701
1.2 −2.13419 −3.13727 2.55475 1.00000 6.69551 1.41692 −1.18395 6.84244 −2.13419
1.3 −1.89846 −0.0586609 1.60416 1.00000 0.111366 2.85624 0.751482 −2.99656 −1.89846
1.4 −1.59654 −1.29420 0.548929 1.00000 2.06623 −2.14214 2.31669 −1.32506 −1.59654
1.5 −1.36096 1.34441 −0.147774 1.00000 −1.82970 1.74128 2.92305 −1.19255 −1.36096
1.6 −0.517395 0.462298 −1.73230 1.00000 −0.239191 −3.59765 1.93107 −2.78628 −0.517395
1.7 0.428259 2.33128 −1.81659 1.00000 0.998390 −3.85771 −1.63449 2.43486 0.428259
1.8 1.12957 −1.80145 −0.724078 1.00000 −2.03485 2.49744 −3.07703 0.245209 1.12957
1.9 1.20514 0.933583 −0.547644 1.00000 1.12510 −2.09289 −3.07026 −2.12842 1.20514
1.10 1.57045 −3.37997 0.466308 1.00000 −5.30806 −2.36397 −2.40858 8.42418 1.57045
1.11 1.76114 −0.777505 1.10163 1.00000 −1.36930 0.840819 −1.58216 −2.39549 1.76114
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(241\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1205.2.a.b ✓ 11
5.b even 2 1 6025.2.a.g 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1205.2.a.b ✓ 11 1.a even 1 1 trivial
6025.2.a.g 11 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{11} + 4 T_{2}^{10} - 6 T_{2}^{9} - 36 T_{2}^{8} + 7 T_{2}^{7} + 121 T_{2}^{6} + 15 T_{2}^{5} + \cdots - 19 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1205))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{11} + 4 T^{10} + \cdots - 19 \) Copy content Toggle raw display
$3$ \( T^{11} + 8 T^{10} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( (T - 1)^{11} \) Copy content Toggle raw display
$7$ \( T^{11} + 9 T^{10} + \cdots - 9356 \) Copy content Toggle raw display
$11$ \( T^{11} + 3 T^{10} + \cdots + 2368 \) Copy content Toggle raw display
$13$ \( T^{11} + 9 T^{10} + \cdots + 38 \) Copy content Toggle raw display
$17$ \( T^{11} + 4 T^{10} + \cdots - 11574 \) Copy content Toggle raw display
$19$ \( T^{11} + 33 T^{10} + \cdots - 42248 \) Copy content Toggle raw display
$23$ \( T^{11} + 31 T^{10} + \cdots + 8444 \) Copy content Toggle raw display
$29$ \( T^{11} - T^{10} + \cdots + 87814 \) Copy content Toggle raw display
$31$ \( T^{11} - 6 T^{10} + \cdots - 2664644 \) Copy content Toggle raw display
$37$ \( T^{11} + 23 T^{10} + \cdots - 16646506 \) Copy content Toggle raw display
$41$ \( T^{11} - 8 T^{10} + \cdots + 27218 \) Copy content Toggle raw display
$43$ \( T^{11} + 19 T^{10} + \cdots - 1342504 \) Copy content Toggle raw display
$47$ \( T^{11} + 35 T^{10} + \cdots - 22906712 \) Copy content Toggle raw display
$53$ \( T^{11} + \cdots - 200270398 \) Copy content Toggle raw display
$59$ \( T^{11} + \cdots - 228526492 \) Copy content Toggle raw display
$61$ \( T^{11} - 9 T^{10} + \cdots + 7806014 \) Copy content Toggle raw display
$67$ \( T^{11} + 54 T^{10} + \cdots + 78262892 \) Copy content Toggle raw display
$71$ \( T^{11} + \cdots + 171774332 \) Copy content Toggle raw display
$73$ \( T^{11} + \cdots - 1586331898 \) Copy content Toggle raw display
$79$ \( T^{11} + 16 T^{10} + \cdots + 7778248 \) Copy content Toggle raw display
$83$ \( T^{11} + \cdots - 169996516 \) Copy content Toggle raw display
$89$ \( T^{11} + \cdots - 13243411022 \) Copy content Toggle raw display
$97$ \( T^{11} + \cdots - 4514064014 \) Copy content Toggle raw display
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