Properties

Label 8034.2.a.w
Level $8034$
Weight $2$
Character orbit 8034.a
Self dual yes
Analytic conductor $64.152$
Analytic rank $1$
Dimension $12$
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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} - 8 x^{10} + 94 x^{9} - 7 x^{8} - 580 x^{7} + 180 x^{6} + 1787 x^{5} - 308 x^{4} - 2790 x^{3} - 352 x^{2} + 1768 x + 768 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} + \beta_{11} q^{5} - q^{6} - \beta_{9} q^{7} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{3} + q^{4} + \beta_{11} q^{5} - q^{6} - \beta_{9} q^{7} - q^{8} + q^{9} - \beta_{11} q^{10} + ( - \beta_{11} + \beta_{9} + \beta_{7} - \beta_1 - 1) q^{11} + q^{12} + q^{13} + \beta_{9} q^{14} + \beta_{11} q^{15} + q^{16} + ( - 2 \beta_{11} + \beta_{9} - \beta_{8} + \beta_{5} + \beta_{4} + \beta_{3} - \beta_1 - 3) q^{17} - q^{18} + ( - \beta_{11} + \beta_{9} - \beta_{8} - \beta_{6} + \beta_{4} - \beta_{3} - \beta_1) q^{19} + \beta_{11} q^{20} - \beta_{9} q^{21} + (\beta_{11} - \beta_{9} - \beta_{7} + \beta_1 + 1) q^{22} + ( - \beta_{8} + \beta_{7} - \beta_{4} - 2) q^{23} - q^{24} + (\beta_{11} + \beta_{10} - \beta_{7} - \beta_{5} - \beta_{2} + \beta_1 + 1) q^{25} - q^{26} + q^{27} - \beta_{9} q^{28} + ( - \beta_{10} + \beta_{6} - \beta_{2} + \beta_1 - 2) q^{29} - \beta_{11} q^{30} + ( - \beta_{11} - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_1) q^{31} - q^{32} + ( - \beta_{11} + \beta_{9} + \beta_{7} - \beta_1 - 1) q^{33} + (2 \beta_{11} - \beta_{9} + \beta_{8} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_1 + 3) q^{34} + (2 \beta_{11} + \beta_{10} - \beta_{9} + 3 \beta_{8} - \beta_{6} - 3 \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{35} + q^{36} + ( - \beta_{10} - \beta_{9} + 2 \beta_{8} - \beta_{7} + \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 + 1) q^{37} + (\beta_{11} - \beta_{9} + \beta_{8} + \beta_{6} - \beta_{4} + \beta_{3} + \beta_1) q^{38} + q^{39} - \beta_{11} q^{40} + (3 \beta_{11} + \beta_{10} - 2 \beta_{9} + 2 \beta_{8} - \beta_{7} - 2 \beta_{5} - \beta_{4} + \cdots + 3 \beta_1) q^{41}+ \cdots + ( - \beta_{11} + \beta_{9} + \beta_{7} - \beta_1 - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{2} + 12 q^{3} + 12 q^{4} - 4 q^{5} - 12 q^{6} - 12 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{2} + 12 q^{3} + 12 q^{4} - 4 q^{5} - 12 q^{6} - 12 q^{8} + 12 q^{9} + 4 q^{10} - 9 q^{11} + 12 q^{12} + 12 q^{13} - 4 q^{15} + 12 q^{16} - 20 q^{17} - 12 q^{18} + 4 q^{19} - 4 q^{20} + 9 q^{22} - 30 q^{23} - 12 q^{24} + 14 q^{25} - 12 q^{26} + 12 q^{27} - 29 q^{29} + 4 q^{30} + 6 q^{31} - 12 q^{32} - 9 q^{33} + 20 q^{34} - 22 q^{35} + 12 q^{36} + 7 q^{37} - 4 q^{38} + 12 q^{39} + 4 q^{40} - 8 q^{41} - 8 q^{43} - 9 q^{44} - 4 q^{45} + 30 q^{46} - 16 q^{47} + 12 q^{48} + 10 q^{49} - 14 q^{50} - 20 q^{51} + 12 q^{52} - 9 q^{53} - 12 q^{54} - 20 q^{55} + 4 q^{57} + 29 q^{58} - 29 q^{59} - 4 q^{60} - 26 q^{61} - 6 q^{62} + 12 q^{64} - 4 q^{65} + 9 q^{66} + 12 q^{67} - 20 q^{68} - 30 q^{69} + 22 q^{70} - 35 q^{71} - 12 q^{72} + 18 q^{73} - 7 q^{74} + 14 q^{75} + 4 q^{76} - 25 q^{77} - 12 q^{78} - 37 q^{79} - 4 q^{80} + 12 q^{81} + 8 q^{82} - 24 q^{83} - 17 q^{85} + 8 q^{86} - 29 q^{87} + 9 q^{88} + 15 q^{89} + 4 q^{90} - 30 q^{92} + 6 q^{93} + 16 q^{94} - 54 q^{95} - 12 q^{96} - 11 q^{97} - 10 q^{98} - 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} - 8 x^{10} + 94 x^{9} - 7 x^{8} - 580 x^{7} + 180 x^{6} + 1787 x^{5} - 308 x^{4} - 2790 x^{3} - 352 x^{2} + 1768 x + 768 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 81 \nu^{11} - 874 \nu^{10} + 1400 \nu^{9} + 10622 \nu^{8} - 27271 \nu^{7} - 48976 \nu^{6} + 141564 \nu^{5} + 120931 \nu^{4} - 292064 \nu^{3} - 190830 \nu^{2} + 207480 \nu + 143896 ) / 2936 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 103 \nu^{11} - 835 \nu^{10} + 276 \nu^{9} + 11894 \nu^{8} - 17071 \nu^{7} - 64829 \nu^{6} + 108598 \nu^{5} + 175289 \nu^{4} - 254607 \nu^{3} - 253616 \nu^{2} + 199594 \nu + 165236 ) / 1468 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 75 \nu^{11} + 551 \nu^{10} - 5 \nu^{9} - 7606 \nu^{8} + 8967 \nu^{7} + 39381 \nu^{6} - 56903 \nu^{5} - 97565 \nu^{4} + 125519 \nu^{3} + 123561 \nu^{2} - 89922 \nu - 70548 ) / 734 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 268 \nu^{11} + 1827 \nu^{10} + 814 \nu^{9} - 26572 \nu^{8} + 21678 \nu^{7} + 146843 \nu^{6} - 158256 \nu^{5} - 389932 \nu^{4} + 369489 \nu^{3} + 514220 \nu^{2} + \cdots - 281980 ) / 1468 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 176 \nu^{11} - 1156 \nu^{10} - 551 \nu^{9} + 16048 \nu^{8} - 13086 \nu^{7} - 82784 \nu^{6} + 93363 \nu^{5} + 199984 \nu^{4} - 211208 \nu^{3} - 240617 \nu^{2} + 152708 \nu + 131084 ) / 734 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 464 \nu^{11} - 3081 \nu^{10} - 1486 \nu^{9} + 43376 \nu^{8} - 34366 \nu^{7} - 228825 \nu^{6} + 247440 \nu^{5} + 569936 \nu^{4} - 561859 \nu^{3} - 702716 \nu^{2} + \cdots + 376680 ) / 1468 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 927 \nu^{11} + 6414 \nu^{10} + 1920 \nu^{9} - 90898 \nu^{8} + 85377 \nu^{7} + 488408 \nu^{6} - 593500 \nu^{5} - 1267853 \nu^{4} + 1378000 \nu^{3} + 1679930 \nu^{2} + \cdots - 977728 ) / 2936 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1151 \nu^{11} + 7952 \nu^{10} + 2688 \nu^{9} - 113458 \nu^{8} + 101765 \nu^{7} + 614922 \nu^{6} - 714928 \nu^{5} - 1610725 \nu^{4} + 1662758 \nu^{3} + \cdots - 1227304 ) / 2936 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1227 \nu^{11} - 8618 \nu^{10} - 1900 \nu^{9} + 121322 \nu^{8} - 121245 \nu^{7} - 645932 \nu^{6} + 821112 \nu^{5} + 1658113 \nu^{4} - 1877140 \nu^{3} - 2177110 \nu^{2} + \cdots + 1262856 ) / 2936 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1415 \nu^{11} + 9686 \nu^{10} + 2964 \nu^{9} - 134594 \nu^{8} + 124697 \nu^{7} + 704600 \nu^{6} - 850752 \nu^{5} - 1763901 \nu^{4} + 1927456 \nu^{3} + \cdots - 1251440 ) / 2936 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1555 \nu^{11} - 10372 \nu^{10} - 4912 \nu^{9} + 147226 \nu^{8} - 118241 \nu^{7} - 788534 \nu^{6} + 860768 \nu^{5} + 2017465 \nu^{4} - 1999642 \nu^{3} - 2588182 \nu^{2} + \cdots + 1437152 ) / 2936 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} - \beta_{9} + \beta_{8} - \beta_{4} + \beta_{2} + \beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} - 3\beta_{9} + 4\beta_{8} - 2\beta_{7} + 3\beta_{6} - 4\beta_{4} + 4\beta_{3} + 4\beta_{2} + 4\beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8 \beta_{11} - 9 \beta_{9} + 10 \beta_{8} + 2 \beta_{6} + \beta_{5} - 11 \beta_{4} + 5 \beta_{3} + 11 \beta_{2} + 11 \beta _1 + 34 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 11 \beta_{11} - 2 \beta_{10} - 24 \beta_{9} + 33 \beta_{8} - 8 \beta_{7} + 20 \beta_{6} + 3 \beta_{5} - 37 \beta_{4} + 36 \beta_{3} + 37 \beta_{2} + 34 \beta _1 + 71 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 55 \beta_{11} - 5 \beta_{10} - 75 \beta_{9} + 90 \beta_{8} + 3 \beta_{7} + 29 \beta_{6} + 18 \beta_{5} - 111 \beta_{4} + 80 \beta_{3} + 109 \beta_{2} + 102 \beta _1 + 271 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 95 \beta_{11} - 34 \beta_{10} - 207 \beta_{9} + 284 \beta_{8} - 23 \beta_{7} + 151 \beta_{6} + 56 \beta_{5} - 360 \beta_{4} + 352 \beta_{3} + 350 \beta_{2} + 305 \beta _1 + 692 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 380 \beta_{11} - 98 \beta_{10} - 641 \beta_{9} + 820 \beta_{8} + 64 \beta_{7} + 313 \beta_{6} + 235 \beta_{5} - 1117 \beta_{4} + 964 \beta_{3} + 1062 \beta_{2} + 937 \beta _1 + 2375 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 786 \beta_{11} - 434 \beta_{10} - 1845 \beta_{9} + 2568 \beta_{8} + 72 \beta_{7} + 1232 \beta_{6} + 749 \beta_{5} - 3584 \beta_{4} + 3542 \beta_{3} + 3356 \beta_{2} + 2835 \beta _1 + 6734 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2788 \beta_{11} - 1331 \beta_{10} - 5677 \beta_{9} + 7715 \beta_{8} + 920 \beta_{7} + 3041 \beta_{6} + 2744 \beta_{5} - 11286 \beta_{4} + 10557 \beta_{3} + 10348 \beta_{2} + 8791 \beta _1 + 22049 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 6699 \beta_{11} - 4984 \beta_{10} - 16860 \beta_{9} + 24258 \beta_{8} + 2411 \beta_{7} + 10443 \beta_{6} + 8847 \beta_{5} - 36083 \beta_{4} + 35919 \beta_{3} + 32539 \beta_{2} + 27032 \beta _1 + 66028 \) 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
2.37328
−0.829811
1.78941
−2.08335
1.38143
3.10048
−1.02386
−1.61063
3.23048
−2.20360
2.65870
−0.782524
−1.00000 1.00000 1.00000 −3.83399 −1.00000 −3.29790 −1.00000 1.00000 3.83399
1.2 −1.00000 1.00000 1.00000 −3.54947 −1.00000 1.31288 −1.00000 1.00000 3.54947
1.3 −1.00000 1.00000 1.00000 −2.74714 −1.00000 3.39290 −1.00000 1.00000 2.74714
1.4 −1.00000 1.00000 1.00000 −1.71945 −1.00000 2.07930 −1.00000 1.00000 1.71945
1.5 −1.00000 1.00000 1.00000 −1.60911 −1.00000 2.84255 −1.00000 1.00000 1.60911
1.6 −1.00000 1.00000 1.00000 −1.18783 −1.00000 0.107097 −1.00000 1.00000 1.18783
1.7 −1.00000 1.00000 1.00000 −0.812175 −1.00000 −2.80411 −1.00000 1.00000 0.812175
1.8 −1.00000 1.00000 1.00000 1.40749 −1.00000 −0.998879 −1.00000 1.00000 −1.40749
1.9 −1.00000 1.00000 1.00000 1.42840 −1.00000 −2.28162 −1.00000 1.00000 −1.42840
1.10 −1.00000 1.00000 1.00000 2.09722 −1.00000 −0.0373018 −1.00000 1.00000 −2.09722
1.11 −1.00000 1.00000 1.00000 2.30797 −1.00000 4.49849 −1.00000 1.00000 −2.30797
1.12 −1.00000 1.00000 1.00000 4.21809 −1.00000 −4.81342 −1.00000 1.00000 −4.21809
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
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.w 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8034.2.a.w 12 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8034))\):

\( T_{5}^{12} + 4 T_{5}^{11} - 29 T_{5}^{10} - 127 T_{5}^{9} + 235 T_{5}^{8} + 1255 T_{5}^{7} - 585 T_{5}^{6} - 5293 T_{5}^{5} - 638 T_{5}^{4} + 9852 T_{5}^{3} + 4208 T_{5}^{2} - 6656 T_{5} - 4096 \) Copy content Toggle raw display
\( T_{7}^{12} - 47 T_{7}^{10} + 6 T_{7}^{9} + 761 T_{7}^{8} - 144 T_{7}^{7} - 5271 T_{7}^{6} + 1175 T_{7}^{5} + 14923 T_{7}^{4} - 3397 T_{7}^{3} - 11911 T_{7}^{2} + 848 T_{7} + 48 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{12} \) Copy content Toggle raw display
$3$ \( (T - 1)^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 4 T^{11} - 29 T^{10} + \cdots - 4096 \) Copy content Toggle raw display
$7$ \( T^{12} - 47 T^{10} + 6 T^{9} + 761 T^{8} + \cdots + 48 \) Copy content Toggle raw display
$11$ \( T^{12} + 9 T^{11} - 24 T^{10} + \cdots + 8268 \) Copy content Toggle raw display
$13$ \( (T - 1)^{12} \) Copy content Toggle raw display
$17$ \( T^{12} + 20 T^{11} + 104 T^{10} + \cdots - 1291856 \) Copy content Toggle raw display
$19$ \( T^{12} - 4 T^{11} - 102 T^{10} + \cdots - 28672 \) Copy content Toggle raw display
$23$ \( T^{12} + 30 T^{11} + 292 T^{10} + \cdots + 5057504 \) Copy content Toggle raw display
$29$ \( T^{12} + 29 T^{11} + 233 T^{10} + \cdots - 10293248 \) Copy content Toggle raw display
$31$ \( T^{12} - 6 T^{11} - 170 T^{10} + \cdots + 155392 \) Copy content Toggle raw display
$37$ \( T^{12} - 7 T^{11} - 178 T^{10} + \cdots + 10528208 \) Copy content Toggle raw display
$41$ \( T^{12} + 8 T^{11} - 282 T^{10} + \cdots - 123048656 \) Copy content Toggle raw display
$43$ \( T^{12} + 8 T^{11} - 273 T^{10} + \cdots + 5504384 \) Copy content Toggle raw display
$47$ \( T^{12} + 16 T^{11} + \cdots - 1167890192 \) Copy content Toggle raw display
$53$ \( T^{12} + 9 T^{11} - 272 T^{10} + \cdots + 4951728 \) Copy content Toggle raw display
$59$ \( T^{12} + 29 T^{11} + 43 T^{10} + \cdots + 48294912 \) Copy content Toggle raw display
$61$ \( T^{12} + 26 T^{11} + \cdots - 108370312 \) Copy content Toggle raw display
$67$ \( T^{12} - 12 T^{11} + \cdots - 418980608 \) Copy content Toggle raw display
$71$ \( T^{12} + 35 T^{11} + 283 T^{10} + \cdots + 12376064 \) Copy content Toggle raw display
$73$ \( T^{12} - 18 T^{11} + \cdots + 27188417904 \) Copy content Toggle raw display
$79$ \( T^{12} + 37 T^{11} + \cdots - 519477248 \) Copy content Toggle raw display
$83$ \( T^{12} + 24 T^{11} + \cdots + 776561664 \) Copy content Toggle raw display
$89$ \( T^{12} - 15 T^{11} + \cdots + 180564352 \) Copy content Toggle raw display
$97$ \( T^{12} + 11 T^{11} - 138 T^{10} + \cdots - 83823616 \) Copy content Toggle raw display
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