Properties

Label 8034.2.a.u
Level $8034$
Weight $2$
Character orbit 8034.a
Self dual yes
Analytic conductor $64.152$
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] = [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: \(1\)
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: \( x^{11} - 4x^{10} - 18x^{9} + 64x^{8} + 85x^{7} - 249x^{6} - 109x^{5} + 230x^{4} + 97x^{3} - 53x^{2} - 32x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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.

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} + \beta_{9} q^{5} - q^{6} - \beta_{10} q^{7} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - q^{3} + q^{4} + \beta_{9} q^{5} - q^{6} - \beta_{10} q^{7} + q^{8} + q^{9} + \beta_{9} q^{10} + ( - \beta_{7} - \beta_{3} - \beta_1 - 1) q^{11} - q^{12} - q^{13} - \beta_{10} q^{14} - \beta_{9} q^{15} + q^{16} + ( - \beta_{9} + \beta_{4} + \beta_{2}) q^{17} + q^{18} + (\beta_{10} + \beta_{8} + \beta_{7} + \beta_{3} + \beta_1) q^{19} + \beta_{9} q^{20} + \beta_{10} q^{21} + ( - \beta_{7} - \beta_{3} - \beta_1 - 1) q^{22} + (\beta_{10} - \beta_{9} - \beta_{8} + \beta_{6} + \beta_1 - 1) q^{23} - q^{24} + ( - \beta_{10} - \beta_{8} - \beta_{6} - \beta_{5} - \beta_{2}) q^{25} - q^{26} - q^{27} - \beta_{10} q^{28} + (\beta_{10} - \beta_{9} + \beta_{6} + \beta_{5} + \beta_{3} + \beta_1 - 3) q^{29} - \beta_{9} q^{30} + (\beta_{10} - \beta_{9} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_1) q^{31} + q^{32} + (\beta_{7} + \beta_{3} + \beta_1 + 1) q^{33} + ( - \beta_{9} + \beta_{4} + \beta_{2}) q^{34} + (\beta_{10} + \beta_{8} + \beta_{7} - \beta_{6} + 2 \beta_{3} - \beta_{2} - 1) q^{35} + q^{36} + (2 \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} - 2 \beta_{4} - \beta_{2} - \beta_1 - 1) q^{37} + (\beta_{10} + \beta_{8} + \beta_{7} + \beta_{3} + \beta_1) q^{38} + q^{39} + \beta_{9} q^{40} + ( - \beta_{10} - \beta_{8} + \beta_{7} - \beta_{5} - 3 \beta_{4} - 2 \beta_{2} + 1) q^{41} + \beta_{10} q^{42} + (\beta_{9} + \beta_{7} - \beta_{3} + 2 \beta_1 - 1) q^{43} + ( - \beta_{7} - \beta_{3} - \beta_1 - 1) q^{44} + \beta_{9} q^{45} + (\beta_{10} - \beta_{9} - \beta_{8} + \beta_{6} + \beta_1 - 1) q^{46} + ( - \beta_{10} + \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{2} + 2) q^{47} - q^{48} + ( - \beta_{9} - \beta_{8} - \beta_{3} + \beta_{2} - 1) q^{49} + ( - \beta_{10} - \beta_{8} - \beta_{6} - \beta_{5} - \beta_{2}) q^{50} + (\beta_{9} - \beta_{4} - \beta_{2}) q^{51} - q^{52} + (2 \beta_{10} - 2 \beta_{9} + 2 \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} + 2 \beta_{4} + \beta_{3} + \cdots - 3) q^{53}+ \cdots + ( - \beta_{7} - \beta_{3} - \beta_1 - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q + 11 q^{2} - 11 q^{3} + 11 q^{4} - 2 q^{5} - 11 q^{6} - 2 q^{7} + 11 q^{8} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 11 q + 11 q^{2} - 11 q^{3} + 11 q^{4} - 2 q^{5} - 11 q^{6} - 2 q^{7} + 11 q^{8} + 11 q^{9} - 2 q^{10} - 10 q^{11} - 11 q^{12} - 11 q^{13} - 2 q^{14} + 2 q^{15} + 11 q^{16} + 6 q^{17} + 11 q^{18} - 3 q^{19} - 2 q^{20} + 2 q^{21} - 10 q^{22} + 3 q^{23} - 11 q^{24} - q^{25} - 11 q^{26} - 11 q^{27} - 2 q^{28} - 22 q^{29} + 2 q^{30} - 5 q^{31} + 11 q^{32} + 10 q^{33} + 6 q^{34} - 20 q^{35} + 11 q^{36} - 26 q^{37} - 3 q^{38} + 11 q^{39} - 2 q^{40} - 6 q^{41} + 2 q^{42} - 8 q^{43} - 10 q^{44} - 2 q^{45} + 3 q^{46} + 6 q^{47} - 11 q^{48} - 5 q^{49} - q^{50} - 6 q^{51} - 11 q^{52} - 25 q^{53} - 11 q^{54} - 2 q^{56} + 3 q^{57} - 22 q^{58} + 7 q^{59} + 2 q^{60} - 36 q^{61} - 5 q^{62} - 2 q^{63} + 11 q^{64} + 2 q^{65} + 10 q^{66} - 12 q^{67} + 6 q^{68} - 3 q^{69} - 20 q^{70} - 15 q^{71} + 11 q^{72} - 12 q^{73} - 26 q^{74} + q^{75} - 3 q^{76} - q^{77} + 11 q^{78} - 15 q^{79} - 2 q^{80} + 11 q^{81} - 6 q^{82} - 16 q^{83} + 2 q^{84} - 25 q^{85} - 8 q^{86} + 22 q^{87} - 10 q^{88} - 2 q^{89} - 2 q^{90} + 2 q^{91} + 3 q^{92} + 5 q^{93} + 6 q^{94} + 16 q^{95} - 11 q^{96} - 10 q^{97} - 5 q^{98} - 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{11} - 4x^{10} - 18x^{9} + 64x^{8} + 85x^{7} - 249x^{6} - 109x^{5} + 230x^{4} + 97x^{3} - 53x^{2} - 32x - 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 25115 \nu^{10} + 123880 \nu^{9} + 347054 \nu^{8} - 1973096 \nu^{7} - 475719 \nu^{6} + 7344639 \nu^{5} - 3375623 \nu^{4} - 4857162 \nu^{3} + 1771243 \nu^{2} + \cdots - 57834 ) / 57098 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 43620 \nu^{10} + 194155 \nu^{9} + 703885 \nu^{8} - 3133089 \nu^{7} - 2408751 \nu^{6} + 12312604 \nu^{5} - 266083 \nu^{4} - 11137378 \nu^{3} + \cdots + 326464 ) / 57098 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 115421 \nu^{10} - 551264 \nu^{9} - 1667268 \nu^{8} + 8763870 \nu^{7} + 3266695 \nu^{6} - 32593715 \nu^{5} + 12149715 \nu^{4} + 22005140 \nu^{3} + \cdots + 355594 ) / 114196 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 189461 \nu^{10} + 831576 \nu^{9} + 3083592 \nu^{8} - 13311542 \nu^{7} - 10877367 \nu^{6} + 51186895 \nu^{5} + 637757 \nu^{4} - 43024636 \nu^{3} + \cdots + 2166734 ) / 114196 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 140379 \nu^{10} + 618522 \nu^{9} + 2271966 \nu^{8} - 9891326 \nu^{7} - 7853363 \nu^{6} + 37899765 \nu^{5} - 344375 \nu^{4} - 31274154 \nu^{3} + \cdots + 1342942 ) / 57098 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 366281 \nu^{10} + 1630196 \nu^{9} + 5868288 \nu^{8} - 26121810 \nu^{7} - 19548755 \nu^{6} + 100542707 \nu^{5} - 4436099 \nu^{4} - 84025608 \nu^{3} + \cdots + 3395542 ) / 114196 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 510619 \nu^{10} - 2246898 \nu^{9} - 8303822 \nu^{8} + 36042260 \nu^{7} + 29219587 \nu^{6} - 139385317 \nu^{5} - 1255417 \nu^{4} + 119432904 \nu^{3} + \cdots - 5761410 ) / 114196 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 671471 \nu^{10} + 2966642 \nu^{9} + 10849434 \nu^{8} - 47518076 \nu^{7} - 37292383 \nu^{6} + 182903005 \nu^{5} - 2609191 \nu^{4} - 153749580 \nu^{3} + \cdots + 6971166 ) / 114196 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 460175 \nu^{10} + 2024699 \nu^{9} + 7470473 \nu^{8} - 32427051 \nu^{7} - 26090460 \nu^{6} + 124845689 \nu^{5} - 53294 \nu^{4} - 105229792 \nu^{3} + \cdots + 4665936 ) / 57098 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} - \beta_{9} + \beta_{8} + \beta_{5} + \beta_{4} + \beta_{3} + 2\beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{10} - 2\beta_{9} + \beta_{8} - \beta_{7} - 2\beta_{6} + 2\beta_{4} + 5\beta_{2} + 11\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 21 \beta_{10} - 19 \beta_{9} + 16 \beta_{8} + \beta_{7} - 8 \beta_{6} + 15 \beta_{5} + 19 \beta_{4} + 10 \beta_{3} + 39 \beta_{2} + 23 \beta _1 + 48 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 81 \beta_{10} - 58 \beta_{9} + 42 \beta_{8} - 5 \beta_{7} - 49 \beta_{6} + 9 \beta_{5} + 66 \beta_{4} + 3 \beta_{3} + 141 \beta_{2} + 173 \beta _1 + 91 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 441 \beta_{10} - 383 \beta_{9} + 300 \beta_{8} + 49 \beta_{7} - 213 \beta_{6} + 222 \beta_{5} + 394 \beta_{4} + 112 \beta_{3} + 796 \beta_{2} + 525 \beta _1 + 758 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1863 \beta_{10} - 1428 \beta_{9} + 1067 \beta_{8} + 109 \beta_{7} - 1067 \beta_{6} + 330 \beta_{5} + 1612 \beta_{4} + 82 \beta_{3} + 3287 \beta_{2} + 3131 \beta _1 + 2199 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 9266 \beta_{10} - 7869 \beta_{9} + 5979 \beta_{8} + 1309 \beta_{7} - 4789 \beta_{6} + 3571 \beta_{5} + 8308 \beta_{4} + 1410 \beta_{3} + 16589 \beta_{2} + 11579 \beta _1 + 13651 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 40813 \beta_{10} - 32453 \beta_{9} + 24224 \beta_{8} + 4636 \beta_{7} - 22775 \beta_{6} + 8756 \beta_{5} + 36143 \beta_{4} + 1962 \beta_{3} + 72387 \beta_{2} + 60981 \beta _1 + 49344 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 194994 \beta_{10} - 163463 \beta_{9} + 122607 \beta_{8} + 30061 \beta_{7} - 103522 \beta_{6} + 63091 \beta_{5} + 175406 \beta_{4} + 20202 \beta_{3} + 348138 \beta_{2} + 250349 \beta _1 + 264771 \) 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
−0.740393
−0.403660
1.71784
1.09038
3.12575
−3.29880
4.59586
−2.14636
−0.444701
−0.218590
0.722678
1.00000 −1.00000 1.00000 −3.99294 −1.00000 −2.56934 1.00000 1.00000 −3.99294
1.2 1.00000 −1.00000 1.00000 −2.98995 −1.00000 3.36753 1.00000 1.00000 −2.98995
1.3 1.00000 −1.00000 1.00000 −2.58633 −1.00000 3.26383 1.00000 1.00000 −2.58633
1.4 1.00000 −1.00000 1.00000 −1.28102 −1.00000 −0.150820 1.00000 1.00000 −1.28102
1.5 1.00000 −1.00000 1.00000 −0.539721 −1.00000 0.708670 1.00000 1.00000 −0.539721
1.6 1.00000 −1.00000 1.00000 0.113923 −1.00000 2.65133 1.00000 1.00000 0.113923
1.7 1.00000 −1.00000 1.00000 0.819538 −1.00000 −3.89413 1.00000 1.00000 0.819538
1.8 1.00000 −1.00000 1.00000 1.41909 −1.00000 −0.663444 1.00000 1.00000 1.41909
1.9 1.00000 −1.00000 1.00000 1.48470 −1.00000 −3.17363 1.00000 1.00000 1.48470
1.10 1.00000 −1.00000 1.00000 2.47956 −1.00000 −2.89002 1.00000 1.00000 2.47956
1.11 1.00000 −1.00000 1.00000 3.07315 −1.00000 1.35002 1.00000 1.00000 3.07315
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
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.u 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8034.2.a.u 11 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8034))\):

\( T_{5}^{11} + 2 T_{5}^{10} - 25 T_{5}^{9} - 33 T_{5}^{8} + 221 T_{5}^{7} + 141 T_{5}^{6} - 805 T_{5}^{5} - 45 T_{5}^{4} + 998 T_{5}^{3} - 190 T_{5}^{2} - 272 T_{5} + 32 \) Copy content Toggle raw display
\( T_{7}^{11} + 2 T_{7}^{10} - 34 T_{7}^{9} - 58 T_{7}^{8} + 417 T_{7}^{7} + 566 T_{7}^{6} - 2192 T_{7}^{5} - 1977 T_{7}^{4} + 4397 T_{7}^{3} + 1240 T_{7}^{2} - 1616 T_{7} - 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{11} \) Copy content Toggle raw display
$3$ \( (T + 1)^{11} \) Copy content Toggle raw display
$5$ \( T^{11} + 2 T^{10} - 25 T^{9} - 33 T^{8} + \cdots + 32 \) Copy content Toggle raw display
$7$ \( T^{11} + 2 T^{10} - 34 T^{9} - 58 T^{8} + \cdots - 256 \) Copy content Toggle raw display
$11$ \( T^{11} + 10 T^{10} - 20 T^{9} + \cdots - 13030 \) Copy content Toggle raw display
$13$ \( (T + 1)^{11} \) Copy content Toggle raw display
$17$ \( T^{11} - 6 T^{10} - 34 T^{9} + \cdots - 5632 \) Copy content Toggle raw display
$19$ \( T^{11} + 3 T^{10} - 62 T^{9} - 76 T^{8} + \cdots + 704 \) Copy content Toggle raw display
$23$ \( T^{11} - 3 T^{10} - 125 T^{9} + \cdots + 13021480 \) Copy content Toggle raw display
$29$ \( T^{11} + 22 T^{10} + 90 T^{9} + \cdots + 200864 \) Copy content Toggle raw display
$31$ \( T^{11} + 5 T^{10} - 140 T^{9} + \cdots + 9523904 \) Copy content Toggle raw display
$37$ \( T^{11} + 26 T^{10} + 82 T^{9} + \cdots - 22305200 \) Copy content Toggle raw display
$41$ \( T^{11} + 6 T^{10} - 297 T^{9} + \cdots + 164675840 \) Copy content Toggle raw display
$43$ \( T^{11} + 8 T^{10} - 216 T^{9} + \cdots + 93337088 \) Copy content Toggle raw display
$47$ \( T^{11} - 6 T^{10} - 180 T^{9} + \cdots - 130160 \) Copy content Toggle raw display
$53$ \( T^{11} + 25 T^{10} + 36 T^{9} + \cdots + 48160 \) Copy content Toggle raw display
$59$ \( T^{11} - 7 T^{10} - 137 T^{9} + \cdots + 279808 \) Copy content Toggle raw display
$61$ \( T^{11} + 36 T^{10} + 256 T^{9} + \cdots - 7048960 \) Copy content Toggle raw display
$67$ \( T^{11} + 12 T^{10} + \cdots - 325432148 \) Copy content Toggle raw display
$71$ \( T^{11} + 15 T^{10} + \cdots - 137830160 \) Copy content Toggle raw display
$73$ \( T^{11} + 12 T^{10} - 534 T^{9} + \cdots + 71448320 \) Copy content Toggle raw display
$79$ \( T^{11} + 15 T^{10} - 213 T^{9} + \cdots + 17148880 \) Copy content Toggle raw display
$83$ \( T^{11} + 16 T^{10} + \cdots + 7389723520 \) Copy content Toggle raw display
$89$ \( T^{11} + 2 T^{10} - 606 T^{9} + \cdots + 887083844 \) Copy content Toggle raw display
$97$ \( T^{11} + 10 T^{10} - 399 T^{9} + \cdots - 18389296 \) Copy content Toggle raw display
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