Properties

Label 1006.2.a.j
Level $1006$
Weight $2$
Character orbit 1006.a
Self dual yes
Analytic conductor $8.033$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1006,2,Mod(1,1006)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1006, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1006.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1006 = 2 \cdot 503 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1006.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: \(8.03295044334\)
Analytic rank: \(0\)
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: \( x^{12} - 5 x^{11} - 13 x^{10} + 94 x^{9} + 15 x^{8} - 616 x^{7} + 368 x^{6} + 1643 x^{5} - 1463 x^{4} - 1556 x^{3} + 1284 x^{2} + 576 x - 208 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + q^{2} + \beta_1 q^{3} + q^{4} - \beta_{9} q^{5} + \beta_1 q^{6} + (\beta_{10} + 1) q^{7} + q^{8} + (\beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + \beta_1 q^{3} + q^{4} - \beta_{9} q^{5} + \beta_1 q^{6} + (\beta_{10} + 1) q^{7} + q^{8} + (\beta_{2} + 1) q^{9} - \beta_{9} q^{10} + ( - \beta_{8} + 2) q^{11} + \beta_1 q^{12} + \beta_{11} q^{13} + (\beta_{10} + 1) q^{14} + ( - \beta_{10} - \beta_{4} - \beta_{3} - 1) q^{15} + q^{16} + ( - \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} - 2 \beta_{2} + \cdots + 1) q^{17}+ \cdots + (\beta_{10} - \beta_{9} - 4 \beta_{8} + \beta_{7} - \beta_{6} - \beta_{5} - 3 \beta_{4} - \beta_{3} + \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{2} + 5 q^{3} + 12 q^{4} + 5 q^{5} + 5 q^{6} + 8 q^{7} + 12 q^{8} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{2} + 5 q^{3} + 12 q^{4} + 5 q^{5} + 5 q^{6} + 8 q^{7} + 12 q^{8} + 15 q^{9} + 5 q^{10} + 18 q^{11} + 5 q^{12} + 4 q^{13} + 8 q^{14} + 2 q^{15} + 12 q^{16} - 2 q^{17} + 15 q^{18} + 6 q^{19} + 5 q^{20} + q^{21} + 18 q^{22} + 13 q^{23} + 5 q^{24} + q^{25} + 4 q^{26} + 8 q^{27} + 8 q^{28} + 20 q^{29} + 2 q^{30} + 7 q^{31} + 12 q^{32} - 8 q^{33} - 2 q^{34} + q^{35} + 15 q^{36} + 10 q^{37} + 6 q^{38} + 7 q^{39} + 5 q^{40} + 2 q^{41} + q^{42} + 8 q^{43} + 18 q^{44} - 7 q^{45} + 13 q^{46} + 12 q^{47} + 5 q^{48} + 4 q^{49} + q^{50} + 2 q^{51} + 4 q^{52} + 12 q^{53} + 8 q^{54} + 8 q^{55} + 8 q^{56} - 10 q^{57} + 20 q^{58} + 6 q^{59} + 2 q^{60} - 10 q^{61} + 7 q^{62} + 7 q^{63} + 12 q^{64} + 4 q^{65} - 8 q^{66} + 7 q^{67} - 2 q^{68} - 12 q^{69} + q^{70} + 22 q^{71} + 15 q^{72} - 23 q^{73} + 10 q^{74} - 34 q^{75} + 6 q^{76} - 19 q^{77} + 7 q^{78} + 13 q^{79} + 5 q^{80} - 28 q^{81} + 2 q^{82} + q^{83} + q^{84} - 28 q^{85} + 8 q^{86} - 22 q^{87} + 18 q^{88} - 3 q^{89} - 7 q^{90} - 21 q^{91} + 13 q^{92} - 33 q^{93} + 12 q^{94} - 2 q^{95} + 5 q^{96} - 70 q^{97} + 4 q^{98} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 5 x^{11} - 13 x^{10} + 94 x^{9} + 15 x^{8} - 616 x^{7} + 368 x^{6} + 1643 x^{5} - 1463 x^{4} - 1556 x^{3} + 1284 x^{2} + 576 x - 208 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{11} + 543 \nu^{10} - 917 \nu^{9} - 9770 \nu^{8} + 14883 \nu^{7} + 57076 \nu^{6} - 82700 \nu^{5} - 116237 \nu^{4} + 172813 \nu^{3} + 42460 \nu^{2} - 79464 \nu + 1864 ) / 7192 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 115 \nu^{11} - 485 \nu^{10} - 2969 \nu^{9} + 10988 \nu^{8} + 28617 \nu^{7} - 87062 \nu^{6} - 124940 \nu^{5} + 283161 \nu^{4} + 244729 \nu^{3} - 325906 \nu^{2} - 184320 \nu + 56136 ) / 7192 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 577 \nu^{11} - 1339 \nu^{10} - 11285 \nu^{9} + 24612 \nu^{8} + 77557 \nu^{7} - 157540 \nu^{6} - 221932 \nu^{5} + 406595 \nu^{4} + 245943 \nu^{3} - 348940 \nu^{2} + \cdots + 65052 ) / 3596 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1269 \nu^{11} + 3163 \nu^{10} + 25539 \nu^{9} - 60212 \nu^{8} - 183731 \nu^{7} + 402142 \nu^{6} + 568804 \nu^{5} - 1096351 \nu^{4} - 729423 \nu^{3} + 1023786 \nu^{2} + \cdots - 193432 ) / 7192 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1533 \nu^{11} + 3651 \nu^{10} + 30291 \nu^{9} - 66456 \nu^{8} - 214779 \nu^{7} + 419298 \nu^{6} + 663784 \nu^{5} - 1062087 \nu^{4} - 869847 \nu^{3} + 882834 \nu^{2} + \cdots - 131744 ) / 7192 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 969 \nu^{11} - 2445 \nu^{10} - 18341 \nu^{9} + 44290 \nu^{8} + 120335 \nu^{7} - 276728 \nu^{6} - 319484 \nu^{5} + 683499 \nu^{4} + 302929 \nu^{3} - 523112 \nu^{2} + \cdots + 72944 ) / 3596 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1163 \nu^{11} - 3185 \nu^{10} - 21833 \nu^{9} + 58386 \nu^{8} + 141625 \nu^{7} - 369564 \nu^{6} - 370074 \nu^{5} + 924969 \nu^{4} + 351689 \nu^{3} - 713098 \nu^{2} + \cdots + 85748 ) / 3596 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2572 \nu^{11} - 6743 \nu^{10} - 48993 \nu^{9} + 123571 \nu^{8} + 325094 \nu^{7} - 783065 \nu^{6} - 882020 \nu^{5} + 1969696 \nu^{4} + 887759 \nu^{3} - 1550579 \nu^{2} + \cdots + 209308 ) / 3596 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 646 \nu^{11} + 1630 \nu^{10} + 12527 \nu^{9} - 30126 \nu^{8} - 85018 \nu^{7} + 192876 \nu^{6} + 237562 \nu^{5} - 490727 \nu^{4} - 249300 \nu^{3} + 391294 \nu^{2} + \cdots - 53424 ) / 899 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{5} + \beta_{4} + 6\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{10} + 2\beta_{8} + \beta_{5} + \beta_{4} - \beta_{3} + 8\beta_{2} + 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -3\beta_{9} + 6\beta_{8} + \beta_{7} + 12\beta_{6} + 9\beta_{5} + 15\beta_{4} + 38\beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2 \beta_{11} - 18 \beta_{10} + 39 \beta_{8} + 6 \beta_{6} + 10 \beta_{5} + 24 \beta_{4} - 12 \beta_{3} + 57 \beta_{2} - \beta _1 + 162 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - \beta_{11} - 9 \beta_{10} - 42 \beta_{9} + 111 \beta_{8} + 14 \beta_{7} + 124 \beta_{6} + 71 \beta_{5} + 179 \beta_{4} + \beta_{3} - 2 \beta_{2} + 252 \beta _1 + 116 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 32 \beta_{11} - 220 \beta_{10} - 7 \beta_{9} + 517 \beta_{8} + 7 \beta_{7} + 116 \beta_{6} + 85 \beta_{5} + 353 \beta_{4} - 110 \beta_{3} + 407 \beta_{2} - 12 \beta _1 + 1195 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 21 \beta_{11} - 197 \beta_{10} - 440 \beta_{9} + 1468 \beta_{8} + 156 \beta_{7} + 1222 \beta_{6} + 563 \beta_{5} + 1943 \beta_{4} + 15 \beta_{3} - 20 \beta_{2} + 1751 \beta _1 + 1222 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 373 \beta_{11} - 2361 \beta_{10} - 167 \beta_{9} + 5950 \beta_{8} + 155 \beta_{7} + 1609 \beta_{6} + 741 \beta_{5} + 4347 \beta_{4} - 921 \beta_{3} + 2987 \beta_{2} - 60 \beta _1 + 9512 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 323 \beta_{11} - 2948 \beta_{10} - 4203 \beta_{9} + 17093 \beta_{8} + 1615 \beta_{7} + 11846 \beta_{6} + 4629 \beta_{5} + 20163 \beta_{4} + 142 \beta_{3} - 21 \beta_{2} + 12753 \beta _1 + 12839 \) 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
−2.76581
−2.25121
−2.17256
−0.867509
−0.665271
0.264006
1.29494
1.74625
2.31191
2.44300
2.49390
3.16835
1.00000 −2.76581 1.00000 −2.37847 −2.76581 −2.36325 1.00000 4.64970 −2.37847
1.2 1.00000 −2.25121 1.00000 1.33593 −2.25121 4.39363 1.00000 2.06796 1.33593
1.3 1.00000 −2.17256 1.00000 2.95208 −2.17256 2.27467 1.00000 1.72004 2.95208
1.4 1.00000 −0.867509 1.00000 −3.26212 −0.867509 1.33041 1.00000 −2.24743 −3.26212
1.5 1.00000 −0.665271 1.00000 2.52521 −0.665271 −3.86663 1.00000 −2.55741 2.52521
1.6 1.00000 0.264006 1.00000 1.70058 0.264006 2.77338 1.00000 −2.93030 1.70058
1.7 1.00000 1.29494 1.00000 3.51641 1.29494 1.29880 1.00000 −1.32313 3.51641
1.8 1.00000 1.74625 1.00000 −2.58578 1.74625 0.635213 1.00000 0.0494057 −2.58578
1.9 1.00000 2.31191 1.00000 0.313584 2.31191 −2.05708 1.00000 2.34493 0.313584
1.10 1.00000 2.44300 1.00000 −1.37594 2.44300 4.82526 1.00000 2.96827 −1.37594
1.11 1.00000 2.49390 1.00000 1.94778 2.49390 −1.87622 1.00000 3.21954 1.94778
1.12 1.00000 3.16835 1.00000 0.310718 3.16835 0.631809 1.00000 7.03843 0.310718
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(503\) \(1\)

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 1006.2.a.j 12
3.b odd 2 1 9054.2.a.bi 12
4.b odd 2 1 8048.2.a.q 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1006.2.a.j 12 1.a even 1 1 trivial
8048.2.a.q 12 4.b odd 2 1
9054.2.a.bi 12 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1006))\):

\( T_{3}^{12} - 5 T_{3}^{11} - 13 T_{3}^{10} + 94 T_{3}^{9} + 15 T_{3}^{8} - 616 T_{3}^{7} + 368 T_{3}^{6} + 1643 T_{3}^{5} - 1463 T_{3}^{4} - 1556 T_{3}^{3} + 1284 T_{3}^{2} + 576 T_{3} - 208 \) Copy content Toggle raw display
\( T_{5}^{12} - 5 T_{5}^{11} - 18 T_{5}^{10} + 121 T_{5}^{9} + 41 T_{5}^{8} - 982 T_{5}^{7} + 746 T_{5}^{6} + 2939 T_{5}^{5} - 4199 T_{5}^{4} - 1577 T_{5}^{3} + 5038 T_{5}^{2} - 2320 T_{5} + 312 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 5 T^{11} - 13 T^{10} + 94 T^{9} + \cdots - 208 \) Copy content Toggle raw display
$5$ \( T^{12} - 5 T^{11} - 18 T^{10} + 121 T^{9} + \cdots + 312 \) Copy content Toggle raw display
$7$ \( T^{12} - 8 T^{11} - 12 T^{10} + \cdots + 3271 \) Copy content Toggle raw display
$11$ \( T^{12} - 18 T^{11} + 104 T^{10} + \cdots - 2736 \) Copy content Toggle raw display
$13$ \( T^{12} - 4 T^{11} - 61 T^{10} + 294 T^{9} + \cdots + 784 \) Copy content Toggle raw display
$17$ \( T^{12} + 2 T^{11} - 118 T^{10} + \cdots + 4992 \) Copy content Toggle raw display
$19$ \( T^{12} - 6 T^{11} - 49 T^{10} + 235 T^{9} + \cdots - 424 \) Copy content Toggle raw display
$23$ \( T^{12} - 13 T^{11} - 39 T^{10} + \cdots + 2974977 \) Copy content Toggle raw display
$29$ \( T^{12} - 20 T^{11} + 51 T^{10} + \cdots - 1632384 \) Copy content Toggle raw display
$31$ \( T^{12} - 7 T^{11} - 90 T^{10} + \cdots + 211328 \) Copy content Toggle raw display
$37$ \( T^{12} - 10 T^{11} + \cdots + 388308680 \) Copy content Toggle raw display
$41$ \( T^{12} - 2 T^{11} - 182 T^{10} + \cdots + 11636352 \) Copy content Toggle raw display
$43$ \( T^{12} - 8 T^{11} - 183 T^{10} + \cdots + 22568624 \) Copy content Toggle raw display
$47$ \( T^{12} - 12 T^{11} + \cdots - 421292919 \) Copy content Toggle raw display
$53$ \( T^{12} - 12 T^{11} - 91 T^{10} + \cdots + 1253640 \) Copy content Toggle raw display
$59$ \( T^{12} - 6 T^{11} - 431 T^{10} + \cdots + 96691200 \) Copy content Toggle raw display
$61$ \( T^{12} + 10 T^{11} - 367 T^{10} + \cdots - 19632496 \) Copy content Toggle raw display
$67$ \( T^{12} - 7 T^{11} - 250 T^{10} + \cdots + 112948624 \) Copy content Toggle raw display
$71$ \( T^{12} - 22 T^{11} - 85 T^{10} + \cdots - 1152 \) Copy content Toggle raw display
$73$ \( T^{12} + 23 T^{11} + 38 T^{10} + \cdots - 248 \) Copy content Toggle raw display
$79$ \( T^{12} - 13 T^{11} - 208 T^{10} + \cdots - 11347520 \) Copy content Toggle raw display
$83$ \( T^{12} - T^{11} - 405 T^{10} + \cdots - 184141104 \) Copy content Toggle raw display
$89$ \( T^{12} + 3 T^{11} + \cdots + 1840063488 \) Copy content Toggle raw display
$97$ \( T^{12} + 70 T^{11} + \cdots + 2828620760 \) Copy content Toggle raw display
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