Properties

Label 8034.2.a.x
Level $8034$
Weight $2$
Character orbit 8034.a
Self dual yes
Analytic conductor $64.152$
Analytic rank $0$
Dimension $13$
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] = [8034,2,Mod(1,8034)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8034, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("8034.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8034.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: \(64.1518129839\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - x^{12} - 40 x^{11} + 34 x^{10} + 604 x^{9} - 381 x^{8} - 4352 x^{7} + 1474 x^{6} + 15809 x^{5} - 368 x^{4} - 27369 x^{3} - 7621 x^{2} + 17300 x + 8832 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} + \beta_1 q^{5} - q^{6} + \beta_{3} q^{7} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{3} + q^{4} + \beta_1 q^{5} - q^{6} + \beta_{3} q^{7} - q^{8} + q^{9} - \beta_1 q^{10} + \beta_{4} q^{11} + q^{12} + q^{13} - \beta_{3} q^{14} + \beta_1 q^{15} + q^{16} + ( - \beta_{11} + \beta_{3} - \beta_1 + 2) q^{17} - q^{18} + (\beta_{12} - \beta_{5} + \beta_1) q^{19} + \beta_1 q^{20} + \beta_{3} q^{21} - \beta_{4} q^{22} + (\beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_1 + 3) q^{23} - q^{24} + (\beta_{7} + \beta_{6} + \beta_{2} + 2) q^{25} - q^{26} + q^{27} + \beta_{3} q^{28} + (\beta_{11} - \beta_{8} + \beta_{6} + \beta_{3} + 1) q^{29} - \beta_1 q^{30} + ( - \beta_{12} - \beta_{5} + \beta_1) q^{31} - q^{32} + \beta_{4} q^{33} + (\beta_{11} - \beta_{3} + \beta_1 - 2) q^{34} + ( - \beta_{12} + \beta_{8} + \beta_{2} + 3) q^{35} + q^{36} + (\beta_{8} + \beta_{7} - \beta_{6} - \beta_{3} - \beta_{2}) q^{37} + ( - \beta_{12} + \beta_{5} - \beta_1) q^{38} + q^{39} - \beta_1 q^{40} + (2 \beta_{11} + \beta_{10} - \beta_{9} - \beta_{5} - \beta_{4} - \beta_{2} + 2 \beta_1) q^{41} - \beta_{3} q^{42} + ( - \beta_{11} - \beta_{10} - \beta_{8} + \beta_{5} + \beta_{3} + \beta_{2} - \beta_1) q^{43} + \beta_{4} q^{44} + \beta_1 q^{45} + ( - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_1 - 3) q^{46} + ( - 2 \beta_{11} - 2 \beta_{10} + \beta_{8} - \beta_{6} + \beta_{4} - \beta_1 + 1) q^{47} + q^{48} + (\beta_{10} - \beta_{7} + \beta_{5} + \beta_{4} - 1) q^{49} + ( - \beta_{7} - \beta_{6} - \beta_{2} - 2) q^{50} + ( - \beta_{11} + \beta_{3} - \beta_1 + 2) q^{51} + q^{52} + (2 \beta_{12} + \beta_{10} - \beta_{8} + \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} + 2) q^{53} - q^{54} + ( - \beta_{12} + \beta_{11} + \beta_{10} - \beta_{7} - \beta_{2} + 2 \beta_1 - 1) q^{55} - \beta_{3} q^{56} + (\beta_{12} - \beta_{5} + \beta_1) q^{57} + ( - \beta_{11} + \beta_{8} - \beta_{6} - \beta_{3} - 1) q^{58} + (2 \beta_{12} + \beta_{9} + 2 \beta_{6} - \beta_{4} + \beta_{3} + \beta_1 + 2) q^{59} + \beta_1 q^{60} + (\beta_{12} - 2 \beta_{11} - \beta_{10} + 2 \beta_{9} - \beta_{8} - \beta_{6} + \beta_{4} - \beta_1 + 1) q^{61} + (\beta_{12} + \beta_{5} - \beta_1) q^{62} + \beta_{3} q^{63} + q^{64} + \beta_1 q^{65} - \beta_{4} q^{66} + ( - 2 \beta_{12} - \beta_{10} - \beta_{9} - \beta_{7} - 2 \beta_{6} + \beta_{4} - 2 \beta_{2} + \cdots - 2) q^{67}+ \cdots + \beta_{4} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q - 13 q^{2} + 13 q^{3} + 13 q^{4} + q^{5} - 13 q^{6} - q^{7} - 13 q^{8} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q - 13 q^{2} + 13 q^{3} + 13 q^{4} + q^{5} - 13 q^{6} - q^{7} - 13 q^{8} + 13 q^{9} - q^{10} + 6 q^{11} + 13 q^{12} + 13 q^{13} + q^{14} + q^{15} + 13 q^{16} + 18 q^{17} - 13 q^{18} - q^{19} + q^{20} - q^{21} - 6 q^{22} + 20 q^{23} - 13 q^{24} + 16 q^{25} - 13 q^{26} + 13 q^{27} - q^{28} + 25 q^{29} - q^{30} - 5 q^{31} - 13 q^{32} + 6 q^{33} - 18 q^{34} + 26 q^{35} + 13 q^{36} - 6 q^{37} + q^{38} + 13 q^{39} - q^{40} + 13 q^{41} + q^{42} - 2 q^{43} + 6 q^{44} + q^{45} - 20 q^{46} + 5 q^{47} + 13 q^{48} - 16 q^{50} + 18 q^{51} + 13 q^{52} + 21 q^{53} - 13 q^{54} + 2 q^{55} + q^{56} - q^{57} - 25 q^{58} + 22 q^{59} + q^{60} + 2 q^{61} + 5 q^{62} - q^{63} + 13 q^{64} + q^{65} - 6 q^{66} - q^{67} + 18 q^{68} + 20 q^{69} - 26 q^{70} + 32 q^{71} - 13 q^{72} - 13 q^{73} + 6 q^{74} + 16 q^{75} - q^{76} + 33 q^{77} - 13 q^{78} - 7 q^{79} + q^{80} + 13 q^{81} - 13 q^{82} + 17 q^{83} - q^{84} - 25 q^{85} + 2 q^{86} + 25 q^{87} - 6 q^{88} - 12 q^{89} - q^{90} - q^{91} + 20 q^{92} - 5 q^{93} - 5 q^{94} + 36 q^{95} - 13 q^{96} - 42 q^{97} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{13} - x^{12} - 40 x^{11} + 34 x^{10} + 604 x^{9} - 381 x^{8} - 4352 x^{7} + 1474 x^{6} + 15809 x^{5} - 368 x^{4} - 27369 x^{3} - 7621 x^{2} + 17300 x + 8832 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 82571179 \nu^{12} - 4674469543 \nu^{11} + 4045229380 \nu^{10} + 153227692402 \nu^{9} - 190916406966 \nu^{8} - 1693362162947 \nu^{7} + \cdots - 94191942330 ) / 316533270094 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 821635533 \nu^{12} - 14688528363 \nu^{11} + 23628366040 \nu^{10} + 567415004562 \nu^{9} - 132389839580 \nu^{8} + \cdots + 74380301896924 ) / 633066540188 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 464768053 \nu^{12} - 3020129323 \nu^{11} - 17668924332 \nu^{10} + 110747092838 \nu^{9} + 253508520438 \nu^{8} + \cdots + 10932434824650 ) / 316533270094 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1800405167 \nu^{12} - 9821990403 \nu^{11} - 66844233676 \nu^{10} + 372531445822 \nu^{9} + 923116143424 \nu^{8} + \cdots + 37558928545592 ) / 633066540188 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 459562957 \nu^{12} + 518271920 \nu^{11} - 16642314467 \nu^{10} - 17632719959 \nu^{9} + 219058129306 \nu^{8} + 220995431293 \nu^{7} + \cdots - 261207897518 ) / 158266635047 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1001697093 \nu^{12} + 3637925703 \nu^{11} + 29239399554 \nu^{10} - 117962252484 \nu^{9} - 247199851646 \nu^{8} + \cdots - 1599125153292 ) / 316533270094 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1301185969 \nu^{12} - 3095768311 \nu^{11} + 48687375005 \nu^{10} + 117305988087 \nu^{9} - 652269947584 \nu^{8} + \cdots + 11198108957142 ) / 158266635047 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 5709491399 \nu^{12} + 1884380107 \nu^{11} + 220983218852 \nu^{10} - 60804918910 \nu^{9} - 3140305882384 \nu^{8} + \cdots - 22176589418488 ) / 633066540188 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1572124490 \nu^{12} + 1168195578 \nu^{11} + 59893192506 \nu^{10} - 46050449196 \nu^{9} - 844559885012 \nu^{8} + \cdots - 13075070948772 ) / 158266635047 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 4905042985 \nu^{12} - 8698615733 \nu^{11} - 187812103994 \nu^{10} + 306055612136 \nu^{9} + 2649761260912 \nu^{8} + \cdots + 40291809896428 ) / 316533270094 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 5235281189 \nu^{12} - 6247478525 \nu^{11} - 196255326952 \nu^{10} + 205900657400 \nu^{9} + 2669456439700 \nu^{8} + \cdots + 19623283268508 ) / 316533270094 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{6} + \beta_{2} + 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} - 2\beta_{8} + \beta_{6} - \beta_{5} - 2\beta_{4} + 3\beta_{3} + \beta_{2} + 9\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{11} - 2\beta_{10} - \beta_{8} + 15\beta_{7} + 12\beta_{6} - \beta_{5} + \beta_{3} + 16\beta_{2} - 4\beta _1 + 75 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{12} + 2 \beta_{11} + 16 \beta_{10} + 7 \beta_{9} - 33 \beta_{8} + 16 \beta_{6} - 16 \beta_{5} - 37 \beta_{4} + 49 \beta_{3} + 14 \beta_{2} + 100 \beta _1 + 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 4 \beta_{12} - 12 \beta_{11} - 36 \beta_{10} + 7 \beta_{9} - 22 \beta_{8} + 198 \beta_{7} + 139 \beta_{6} - 14 \beta_{5} + 2 \beta_{4} + 15 \beta_{3} + 215 \beta_{2} - 86 \beta _1 + 896 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 52 \beta_{12} + 42 \beta_{11} + 216 \beta_{10} + 178 \beta_{9} - 477 \beta_{8} - 9 \beta_{7} + 201 \beta_{6} - 229 \beta_{5} - 535 \beta_{4} + 693 \beta_{3} + 163 \beta_{2} + 1187 \beta _1 + 237 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 136 \beta_{12} - 70 \beta_{11} - 520 \beta_{10} + 190 \beta_{9} - 367 \beta_{8} + 2549 \beta_{7} + 1644 \beta_{6} - 154 \beta_{5} + 57 \beta_{4} + 166 \beta_{3} + 2789 \beta_{2} - 1416 \beta _1 + 11090 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1011 \beta_{12} + 650 \beta_{11} + 2812 \beta_{10} + 3318 \beta_{9} - 6682 \beta_{8} - 241 \beta_{7} + 2403 \beta_{6} - 3190 \beta_{5} - 7268 \beta_{4} + 9510 \beta_{3} + 1815 \beta_{2} + 14464 \beta _1 + 2129 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 3017 \beta_{12} + 419 \beta_{11} - 7116 \beta_{10} + 3536 \beta_{9} - 5573 \beta_{8} + 32694 \beta_{7} + 19796 \beta_{6} - 1499 \beta_{5} + 1131 \beta_{4} + 1557 \beta_{3} + 36022 \beta_{2} + \cdots + 139500 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 17512 \beta_{12} + 8931 \beta_{11} + 36263 \beta_{10} + 54796 \beta_{9} - 92393 \beta_{8} - 4410 \beta_{7} + 28728 \beta_{6} - 44052 \beta_{5} - 97071 \beta_{4} + 129454 \beta_{3} + 20001 \beta_{2} + \cdots + 17574 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 56146 \beta_{12} + 22414 \beta_{11} - 95843 \beta_{10} + 56188 \beta_{9} - 80875 \beta_{8} + 420209 \beta_{7} + 241672 \beta_{6} - 12416 \beta_{5} + 19472 \beta_{4} + 11516 \beta_{3} + \cdots + 1771965 \) 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
−3.67523
−3.54523
−2.08649
−2.01149
−1.22001
−0.975286
−0.757993
1.20657
1.95504
1.98373
3.21537
3.25869
3.65233
−1.00000 1.00000 1.00000 −3.67523 −1.00000 −3.11602 −1.00000 1.00000 3.67523
1.2 −1.00000 1.00000 1.00000 −3.54523 −1.00000 0.166701 −1.00000 1.00000 3.54523
1.3 −1.00000 1.00000 1.00000 −2.08649 −1.00000 −3.62308 −1.00000 1.00000 2.08649
1.4 −1.00000 1.00000 1.00000 −2.01149 −1.00000 0.821182 −1.00000 1.00000 2.01149
1.5 −1.00000 1.00000 1.00000 −1.22001 −1.00000 3.96620 −1.00000 1.00000 1.22001
1.6 −1.00000 1.00000 1.00000 −0.975286 −1.00000 −3.46524 −1.00000 1.00000 0.975286
1.7 −1.00000 1.00000 1.00000 −0.757993 −1.00000 1.64575 −1.00000 1.00000 0.757993
1.8 −1.00000 1.00000 1.00000 1.20657 −1.00000 −1.77104 −1.00000 1.00000 −1.20657
1.9 −1.00000 1.00000 1.00000 1.95504 −1.00000 −2.89499 −1.00000 1.00000 −1.95504
1.10 −1.00000 1.00000 1.00000 1.98373 −1.00000 3.61896 −1.00000 1.00000 −1.98373
1.11 −1.00000 1.00000 1.00000 3.21537 −1.00000 −0.928130 −1.00000 1.00000 −3.21537
1.12 −1.00000 1.00000 1.00000 3.25869 −1.00000 3.01501 −1.00000 1.00000 −3.25869
1.13 −1.00000 1.00000 1.00000 3.65233 −1.00000 1.56471 −1.00000 1.00000 −3.65233
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(13\) \(-1\)
\(103\) \(-1\)

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 8034.2.a.x 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8034.2.a.x 13 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8034))\):

\( T_{5}^{13} - T_{5}^{12} - 40 T_{5}^{11} + 34 T_{5}^{10} + 604 T_{5}^{9} - 381 T_{5}^{8} - 4352 T_{5}^{7} + 1474 T_{5}^{6} + 15809 T_{5}^{5} - 368 T_{5}^{4} - 27369 T_{5}^{3} - 7621 T_{5}^{2} + 17300 T_{5} + 8832 \) Copy content Toggle raw display
\( T_{7}^{13} + T_{7}^{12} - 45 T_{7}^{11} - 44 T_{7}^{10} + 762 T_{7}^{9} + 669 T_{7}^{8} - 6037 T_{7}^{7} - 4081 T_{7}^{6} + 22787 T_{7}^{5} + 8272 T_{7}^{4} - 37493 T_{7}^{3} - 1775 T_{7}^{2} + 18319 T_{7} - 2840 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{13} \) Copy content Toggle raw display
$3$ \( (T - 1)^{13} \) Copy content Toggle raw display
$5$ \( T^{13} - T^{12} - 40 T^{11} + 34 T^{10} + \cdots + 8832 \) Copy content Toggle raw display
$7$ \( T^{13} + T^{12} - 45 T^{11} - 44 T^{10} + \cdots - 2840 \) Copy content Toggle raw display
$11$ \( T^{13} - 6 T^{12} - 48 T^{11} + \cdots + 1296 \) Copy content Toggle raw display
$13$ \( (T - 1)^{13} \) Copy content Toggle raw display
$17$ \( T^{13} - 18 T^{12} + 50 T^{11} + \cdots + 4990080 \) Copy content Toggle raw display
$19$ \( T^{13} + T^{12} - 158 T^{11} + \cdots + 2495488 \) Copy content Toggle raw display
$23$ \( T^{13} - 20 T^{12} + 53 T^{11} + \cdots + 4424592 \) Copy content Toggle raw display
$29$ \( T^{13} - 25 T^{12} + 137 T^{11} + \cdots + 4049256 \) Copy content Toggle raw display
$31$ \( T^{13} + 5 T^{12} - 174 T^{11} + \cdots - 9725696 \) Copy content Toggle raw display
$37$ \( T^{13} + 6 T^{12} + \cdots - 1312715840 \) Copy content Toggle raw display
$41$ \( T^{13} - 13 T^{12} - 204 T^{11} + \cdots - 12873600 \) Copy content Toggle raw display
$43$ \( T^{13} + 2 T^{12} + \cdots + 3969021440 \) Copy content Toggle raw display
$47$ \( T^{13} - 5 T^{12} - 325 T^{11} + \cdots + 3985992 \) Copy content Toggle raw display
$53$ \( T^{13} - 21 T^{12} - 100 T^{11} + \cdots - 48406272 \) Copy content Toggle raw display
$59$ \( T^{13} - 22 T^{12} - 167 T^{11} + \cdots - 28914768 \) Copy content Toggle raw display
$61$ \( T^{13} - 2 T^{12} + \cdots + 1289907712 \) Copy content Toggle raw display
$67$ \( T^{13} + T^{12} - 546 T^{11} + \cdots - 19748422250 \) Copy content Toggle raw display
$71$ \( T^{13} - 32 T^{12} + \cdots - 361413120 \) Copy content Toggle raw display
$73$ \( T^{13} + 13 T^{12} + \cdots - 6590780212 \) Copy content Toggle raw display
$79$ \( T^{13} + 7 T^{12} - 361 T^{11} + \cdots + 818214400 \) Copy content Toggle raw display
$83$ \( T^{13} - 17 T^{12} + \cdots - 257420472 \) Copy content Toggle raw display
$89$ \( T^{13} + 12 T^{12} + \cdots + 169980344064 \) Copy content Toggle raw display
$97$ \( T^{13} + 42 T^{12} + 647 T^{11} + \cdots - 58931008 \) Copy content Toggle raw display
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