Properties

Label 4205.2.a.q
Level $4205$
Weight $2$
Character orbit 4205.a
Self dual yes
Analytic conductor $33.577$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4205,2,Mod(1,4205)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4205, 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("4205.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4205 = 5 \cdot 29^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4205.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: \(33.5770940499\)
Analytic rank: \(1\)
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: \( x^{12} - x^{11} - 14 x^{10} + 11 x^{9} + 72 x^{8} - 41 x^{7} - 164 x^{6} + 62 x^{5} + 156 x^{4} + \cdots - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 145)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{11} q^{3} + \beta_{2} q^{4} - q^{5} + ( - \beta_{10} - \beta_{8} - \beta_{6} + \cdots - 2) q^{6}+ \cdots + (\beta_{11} + \beta_{10} + \beta_{8} + \cdots + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_{11} q^{3} + \beta_{2} q^{4} - q^{5} + ( - \beta_{10} - \beta_{8} - \beta_{6} + \cdots - 2) q^{6}+ \cdots + ( - \beta_{11} + 2 \beta_{10} + \cdots + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - q^{2} + q^{3} + 5 q^{4} - 12 q^{5} - 9 q^{6} - 5 q^{7} - 6 q^{8} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - q^{2} + q^{3} + 5 q^{4} - 12 q^{5} - 9 q^{6} - 5 q^{7} - 6 q^{8} + 7 q^{9} + q^{10} - q^{11} + q^{12} - 5 q^{13} + 5 q^{14} - q^{15} - 9 q^{16} - 20 q^{17} + 5 q^{18} + 8 q^{19} - 5 q^{20} + 3 q^{22} - 8 q^{23} + 10 q^{24} + 12 q^{25} + 17 q^{26} + 25 q^{27} + 10 q^{28} + 9 q^{30} + 7 q^{31} - 3 q^{32} + q^{33} - 8 q^{34} + 5 q^{35} - 8 q^{36} + 14 q^{38} - 29 q^{39} + 6 q^{40} + 4 q^{41} + 13 q^{42} + 15 q^{43} - 2 q^{44} - 7 q^{45} - 24 q^{46} + 39 q^{47} - 2 q^{48} - 19 q^{49} - q^{50} - 32 q^{51} - 32 q^{52} + 12 q^{53} - 34 q^{54} + q^{55} + 19 q^{56} + 10 q^{57} - 19 q^{59} - q^{60} - 28 q^{61} - 13 q^{62} - 40 q^{63} - 34 q^{64} + 5 q^{65} - 48 q^{66} - 38 q^{67} - 18 q^{68} - 18 q^{69} - 5 q^{70} - 20 q^{71} + 6 q^{72} + 5 q^{73} - 12 q^{74} + q^{75} + 19 q^{76} - 32 q^{77} - 14 q^{78} - 13 q^{79} + 9 q^{80} - 32 q^{81} - 34 q^{82} + 15 q^{83} + 32 q^{84} + 20 q^{85} - 9 q^{86} - 10 q^{88} + 22 q^{89} - 5 q^{90} - 46 q^{91} - 31 q^{92} - 6 q^{93} - 13 q^{94} - 8 q^{95} - 3 q^{96} - 53 q^{97} + 8 q^{98} + 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} - 14 x^{10} + 11 x^{9} + 72 x^{8} - 41 x^{7} - 164 x^{6} + 62 x^{5} + 156 x^{4} + \cdots - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 30 \nu^{11} + 64 \nu^{10} + 366 \nu^{9} - 717 \nu^{8} - 1542 \nu^{7} + 2644 \nu^{6} + 2498 \nu^{5} + \cdots + 51 ) / 139 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 36 \nu^{11} - 49 \nu^{10} - 467 \nu^{9} + 499 \nu^{8} + 2184 \nu^{7} - 1616 \nu^{6} - 4332 \nu^{5} + \cdots + 50 ) / 139 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 20 \nu^{11} + 50 \nu^{10} - 383 \nu^{9} - 634 \nu^{8} + 2557 \nu^{7} + 2778 \nu^{6} - 7318 \nu^{5} + \cdots + 661 ) / 139 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 51 \nu^{11} + 81 \nu^{10} + 650 \nu^{9} - 927 \nu^{8} - 2955 \nu^{7} + 3633 \nu^{6} + 5720 \nu^{5} + \cdots - 372 ) / 139 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 50 \nu^{11} + 14 \nu^{10} + 749 \nu^{9} - 83 \nu^{8} - 4099 \nu^{7} - 134 \nu^{6} + 9816 \nu^{5} + \cdots - 54 ) / 139 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 50 \nu^{11} + 14 \nu^{10} + 749 \nu^{9} - 83 \nu^{8} - 4099 \nu^{7} - 134 \nu^{6} + 9816 \nu^{5} + \cdots - 332 ) / 139 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 69 \nu^{11} - 36 \nu^{10} - 953 \nu^{9} + 273 \nu^{8} + 4742 \nu^{7} - 271 \nu^{6} - 9971 \nu^{5} + \cdots - 298 ) / 139 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 74 \nu^{11} - 93 \nu^{10} - 1014 \nu^{9} + 1018 \nu^{8} + 5138 \nu^{7} - 3677 \nu^{6} - 11731 \nu^{5} + \cdots + 458 ) / 139 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 82 \nu^{11} - 73 \nu^{10} - 1195 \nu^{9} + 820 \nu^{8} + 6411 \nu^{7} - 3094 \nu^{6} - 15242 \nu^{5} + \cdots + 778 ) / 139 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{8} + \beta_{7} + \beta_{2} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{8} + \beta_{5} - \beta_{3} + 5\beta_{2} + \beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{11} + \beta_{9} - 4\beta_{8} + 7\beta_{7} + \beta_{6} + \beta_{5} - \beta_{3} + 7\beta_{2} + 18\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{10} + \beta_{9} + 8 \beta_{8} + \beta_{7} + \beta_{6} + 8 \beta_{5} - 2 \beta_{4} - 9 \beta_{3} + \cdots + 37 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 9 \beta_{11} + 2 \beta_{10} + 10 \beta_{9} - 9 \beta_{8} + 40 \beta_{7} + 10 \beta_{6} + 11 \beta_{5} + \cdots + 21 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2 \beta_{11} + 10 \beta_{10} + 12 \beta_{9} + 53 \beta_{8} + 13 \beta_{7} + 10 \beta_{6} + 52 \beta_{5} + \cdots + 181 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 59 \beta_{11} + 24 \beta_{10} + 74 \beta_{9} + 15 \beta_{8} + 217 \beta_{7} + 70 \beta_{6} + 87 \beta_{5} + \cdots + 160 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 25 \beta_{11} + 74 \beta_{10} + 102 \beta_{9} + 333 \beta_{8} + 113 \beta_{7} + 72 \beta_{6} + \cdots + 915 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 346 \beta_{11} + 197 \beta_{10} + 491 \beta_{9} + 375 \beta_{8} + 1161 \beta_{7} + 426 \beta_{6} + \cdots + 1074 \) Copy content Toggle raw display

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
2.40450
2.19433
1.91653
1.18316
0.623326
0.218677
0.0975264
−0.886465
−1.10366
−1.48615
−2.00441
−2.15738
−2.40450 −0.433927 3.78164 −1.00000 1.04338 −1.22316 −4.28397 −2.81171 2.40450
1.2 −2.19433 2.38477 2.81508 −1.00000 −5.23297 1.90240 −1.78854 2.68713 2.19433
1.3 −1.91653 −1.12311 1.67310 −1.00000 2.15247 −1.69505 0.626513 −1.73863 1.91653
1.4 −1.18316 2.97173 −0.600128 −1.00000 −3.51603 −3.71603 3.07637 5.83115 1.18316
1.5 −0.623326 −0.691067 −1.61147 −1.00000 0.430760 2.77627 2.25112 −2.52243 0.623326
1.6 −0.218677 −1.19342 −1.95218 −1.00000 0.260974 1.69290 0.864252 −1.57575 0.218677
1.7 −0.0975264 2.38706 −1.99049 −1.00000 −0.232802 −1.61211 0.389178 2.69808 0.0975264
1.8 0.886465 −2.17820 −1.21418 −1.00000 −1.93090 −4.18862 −2.84926 1.74457 −0.886465
1.9 1.10366 −1.50704 −0.781928 −1.00000 −1.66326 −1.41268 −3.07031 −0.728845 −1.10366
1.10 1.48615 2.28644 0.208632 −1.00000 3.39798 −0.688115 −2.66224 2.22780 −1.48615
1.11 2.00441 −2.59136 2.01764 −1.00000 −5.19413 −0.000698765 0 0.0353668 3.71514 −2.00441
1.12 2.15738 0.688118 2.65427 −1.00000 1.48453 3.16489 1.41152 −2.52649 −2.15738
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( +1 \)
\(29\) \( +1 \)

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 4205.2.a.q 12
29.b even 2 1 4205.2.a.r 12
29.d even 7 2 145.2.k.a 24
145.n even 14 2 725.2.l.d 24
145.p odd 28 4 725.2.r.c 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.2.k.a 24 29.d even 7 2
725.2.l.d 24 145.n even 14 2
725.2.r.c 48 145.p odd 28 4
4205.2.a.q 12 1.a even 1 1 trivial
4205.2.a.r 12 29.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4205))\):

\( T_{2}^{12} + T_{2}^{11} - 14 T_{2}^{10} - 11 T_{2}^{9} + 72 T_{2}^{8} + 41 T_{2}^{7} - 164 T_{2}^{6} + \cdots - 1 \) Copy content Toggle raw display
\( T_{3}^{12} - T_{3}^{11} - 21 T_{3}^{10} + 11 T_{3}^{9} + 169 T_{3}^{8} - 3 T_{3}^{7} - 629 T_{3}^{6} + \cdots - 91 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + T^{11} + \cdots - 1 \) Copy content Toggle raw display
$3$ \( T^{12} - T^{11} + \cdots - 91 \) Copy content Toggle raw display
$5$ \( (T + 1)^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + 5 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{12} + T^{11} + \cdots + 113 \) Copy content Toggle raw display
$13$ \( T^{12} + 5 T^{11} + \cdots - 83 \) Copy content Toggle raw display
$17$ \( T^{12} + 20 T^{11} + \cdots + 147139 \) Copy content Toggle raw display
$19$ \( T^{12} - 8 T^{11} + \cdots - 6833 \) Copy content Toggle raw display
$23$ \( T^{12} + 8 T^{11} + \cdots - 7439083 \) Copy content Toggle raw display
$29$ \( T^{12} \) Copy content Toggle raw display
$31$ \( T^{12} - 7 T^{11} + \cdots + 54674341 \) Copy content Toggle raw display
$37$ \( T^{12} - 266 T^{10} + \cdots + 478631 \) Copy content Toggle raw display
$41$ \( T^{12} - 4 T^{11} + \cdots - 15490663 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots - 252553097 \) Copy content Toggle raw display
$47$ \( T^{12} - 39 T^{11} + \cdots + 502207 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots - 600156257 \) Copy content Toggle raw display
$59$ \( T^{12} + 19 T^{11} + \cdots - 13085407 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 5237693287 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 1486528457 \) Copy content Toggle raw display
$71$ \( T^{12} + 20 T^{11} + \cdots - 44250121 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots - 5489455061 \) Copy content Toggle raw display
$79$ \( T^{12} + 13 T^{11} + \cdots + 75905257 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 27453317717 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 506469104821 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots - 102994949681 \) Copy content Toggle raw display
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