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Properties

Label 8001.2.a.v
Level $8001$
Weight $2$
Character orbit 8001.a
Self dual yes
Analytic conductor $63.888$
Analytic rank $0$
Dimension $19$
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] = [8001,2,Mod(1,8001)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8001, 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("8001.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8001.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: \(63.8883066572\)
Analytic rank: \(0\)
Dimension: \(19\)
Coefficient field: \(\mathbb{Q}[x]/(x^{19} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{19} - 4 x^{18} - 22 x^{17} + 101 x^{16} + 178 x^{15} - 1035 x^{14} - 583 x^{13} + 5572 x^{12} + \cdots + 210 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 2667)
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 \(1,\beta_1,\ldots,\beta_{18}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + \beta_{10} q^{5} + q^{7} + ( - \beta_{3} - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + \beta_{10} q^{5} + q^{7} + ( - \beta_{3} - \beta_1) q^{8} + (\beta_{17} - \beta_{12} + \cdots + \beta_1) q^{10}+ \cdots - \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8} + 9 q^{11} + 24 q^{13} - 4 q^{14} + 20 q^{16} - 17 q^{17} + 23 q^{19} - 5 q^{20} - 3 q^{22} + 17 q^{23} + 38 q^{25} - 28 q^{26} + 22 q^{28} - 2 q^{29} + 16 q^{31} - 17 q^{32} + 29 q^{34} - 5 q^{35} + 56 q^{37} - 2 q^{38} - 13 q^{40} + 7 q^{41} + 19 q^{43} + 29 q^{44} + 10 q^{46} - 25 q^{47} + 19 q^{49} + 9 q^{50} + 16 q^{52} - 18 q^{53} + 10 q^{55} - 9 q^{56} + 31 q^{58} - 11 q^{59} + 26 q^{61} - 26 q^{62} + 45 q^{64} - 27 q^{65} + 24 q^{67} - 14 q^{68} + 32 q^{71} + 51 q^{73} + 12 q^{76} + 9 q^{77} + 30 q^{79} + 30 q^{80} - 52 q^{82} - q^{83} + 44 q^{85} + 24 q^{86} - 30 q^{88} - 5 q^{89} + 24 q^{91} + 88 q^{92} + 7 q^{94} + 24 q^{95} + 5 q^{97} - 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{19} - 4 x^{18} - 22 x^{17} + 101 x^{16} + 178 x^{15} - 1035 x^{14} - 583 x^{13} + 5572 x^{12} + \cdots + 210 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11967908 \nu^{18} + 10330919 \nu^{17} - 422890858 \nu^{16} - 185012986 \nu^{15} + \cdots - 9272820218 ) / 579103703 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 65114179 \nu^{18} - 132926940 \nu^{17} - 1671831523 \nu^{16} + 3285269092 \nu^{15} + \cdots - 6128233325 ) / 2895518515 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 138645363 \nu^{18} - 343698145 \nu^{17} - 3866824621 \nu^{16} + 9112645479 \nu^{15} + \cdots - 133199717500 ) / 5791037030 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 318716503 \nu^{18} - 1140866135 \nu^{17} - 7501449651 \nu^{16} + 29542139284 \nu^{15} + \cdots + 54702171430 ) / 11582074060 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 93609108 \nu^{18} + 195011795 \nu^{17} + 2566480776 \nu^{16} - 4911478344 \nu^{15} + \cdots + 40044597930 ) / 2895518515 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 93609108 \nu^{18} - 195011795 \nu^{17} - 2566480776 \nu^{16} + 4911478344 \nu^{15} + \cdots - 19775968325 ) / 2895518515 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 37953488 \nu^{18} + 92196111 \nu^{17} + 991696752 \nu^{16} - 2324375148 \nu^{15} + \cdots + 2068831656 ) / 1158207406 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 240105167 \nu^{18} + 881338715 \nu^{17} + 5495659804 \nu^{16} - 21651992771 \nu^{15} + \cdots + 53652831630 ) / 5791037030 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 290773938 \nu^{18} - 662122355 \nu^{17} - 7697023186 \nu^{16} + 16815829469 \nu^{15} + \cdots + 19309791560 ) / 5791037030 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 485961931 \nu^{18} + 1113420900 \nu^{17} + 13007283232 \nu^{16} - 28691667118 \nu^{15} + \cdots + 74147034150 ) / 5791037030 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 244538816 \nu^{18} + 640007340 \nu^{17} + 6214873087 \nu^{16} - 15917066553 \nu^{15} + \cdots + 28681664520 ) / 2895518515 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 1036415051 \nu^{18} - 3169187735 \nu^{17} - 25673615907 \nu^{16} + 80069329128 \nu^{15} + \cdots - 23929479290 ) / 11582074060 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 1177420721 \nu^{18} + 3022806135 \nu^{17} + 30214864317 \nu^{16} - 76041669698 \nu^{15} + \cdots + 101431035930 ) / 11582074060 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 727508506 \nu^{18} - 1789420580 \nu^{17} - 19108483507 \nu^{16} + 45513393513 \nu^{15} + \cdots - 153741088340 ) / 5791037030 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 3366745827 \nu^{18} - 9263333215 \nu^{17} - 85049828069 \nu^{16} + 231373969696 \nu^{15} + \cdots - 509625256950 ) / 11582074060 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} + \beta_{8} + 7\beta_{2} + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{15} - \beta_{14} + \beta_{7} - \beta_{5} + 9\beta_{3} + 30\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{17} - \beta_{16} - 2 \beta_{15} - \beta_{14} + \beta_{13} + 2 \beta_{11} + 11 \beta_{9} + \cdots + 80 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{18} + \beta_{17} - \beta_{16} - 14 \beta_{15} - 12 \beta_{14} + 2 \beta_{13} + 2 \beta_{11} + \cdots + 14 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2 \beta_{18} + 13 \beta_{17} - 14 \beta_{16} - 31 \beta_{15} - 13 \beta_{14} + 16 \beta_{13} + \cdots + 508 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 17 \beta_{18} + 17 \beta_{17} - 16 \beta_{16} - 145 \beta_{15} - 109 \beta_{14} + 34 \beta_{13} + \cdots + 141 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 38 \beta_{18} + 123 \beta_{17} - 137 \beta_{16} - 338 \beta_{15} - 125 \beta_{14} + 181 \beta_{13} + \cdots + 3420 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 196 \beta_{18} + 201 \beta_{17} - 175 \beta_{16} - 1338 \beta_{15} - 902 \beta_{14} + 401 \beta_{13} + \cdots + 1270 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 472 \beta_{18} + 1052 \beta_{17} - 1168 \beta_{16} - 3209 \beta_{15} - 1087 \beta_{14} + 1783 \beta_{13} + \cdots + 23845 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 1930 \beta_{18} + 2048 \beta_{17} - 1628 \beta_{16} - 11637 \beta_{15} - 7176 \beta_{14} + 4070 \beta_{13} + \cdots + 10927 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 4884 \beta_{18} + 8685 \beta_{17} - 9323 \beta_{16} - 28414 \beta_{15} - 9063 \beta_{14} + 16327 \beta_{13} + \cdots + 170076 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 17519 \beta_{18} + 19249 \beta_{17} - 13913 \beta_{16} - 97749 \beta_{15} - 56041 \beta_{14} + \cdots + 92083 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 45824 \beta_{18} + 70814 \beta_{17} - 71931 \beta_{16} - 241742 \beta_{15} - 74068 \beta_{14} + \cdots + 1232416 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 151648 \beta_{18} + 171997 \beta_{17} - 113356 \beta_{16} - 803134 \beta_{15} - 433642 \beta_{14} + \cdots + 767873 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 405545 \beta_{18} + 574450 \beta_{17} - 545365 \beta_{16} - 2006325 \beta_{15} - 599014 \beta_{14} + \cdots + 9034683 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
2.79341
2.49323
2.42782
2.26573
1.88777
1.49024
1.35771
1.20823
0.823573
0.395540
−0.249163
−0.407951
−0.782325
−1.30029
−1.48888
−1.91759
−1.94177
−2.35941
−2.69590
−2.79341 0 5.80316 1.63704 0 1.00000 −10.6238 0 −4.57293
1.2 −2.49323 0 4.21621 0.262973 0 1.00000 −5.52554 0 −0.655653
1.3 −2.42782 0 3.89430 −4.13136 0 1.00000 −4.59902 0 10.0302
1.4 −2.26573 0 3.13354 0.263451 0 1.00000 −2.56830 0 −0.596909
1.5 −1.88777 0 1.56369 3.95777 0 1.00000 0.823654 0 −7.47138
1.6 −1.49024 0 0.220813 −2.04744 0 1.00000 2.65141 0 3.05118
1.7 −1.35771 0 −0.156613 −4.06757 0 1.00000 2.92806 0 5.52260
1.8 −1.20823 0 −0.540168 −0.486834 0 1.00000 3.06912 0 0.588209
1.9 −0.823573 0 −1.32173 2.87784 0 1.00000 2.73569 0 −2.37011
1.10 −0.395540 0 −1.84355 −1.95228 0 1.00000 1.52028 0 0.772204
1.11 0.249163 0 −1.93792 −0.989651 0 1.00000 −0.981182 0 −0.246584
1.12 0.407951 0 −1.83358 1.08992 0 1.00000 −1.56391 0 0.444635
1.13 0.782325 0 −1.38797 2.85652 0 1.00000 −2.65049 0 2.23472
1.14 1.30029 0 −0.309249 −3.89453 0 1.00000 −3.00269 0 −5.06401
1.15 1.48888 0 0.216778 2.91159 0 1.00000 −2.65501 0 4.33502
1.16 1.91759 0 1.67713 −1.63006 0 1.00000 −0.619124 0 −3.12578
1.17 1.94177 0 1.77048 −3.66642 0 1.00000 −0.445683 0 −7.11935
1.18 2.35941 0 3.56679 3.48367 0 1.00000 3.69670 0 8.21939
1.19 2.69590 0 5.26787 −1.47463 0 1.00000 8.80983 0 −3.97544
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.19
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(-1\)
\(127\) \(-1\)

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 8001.2.a.v 19
3.b odd 2 1 2667.2.a.q ✓ 19
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2667.2.a.q ✓ 19 3.b odd 2 1
8001.2.a.v 19 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8001))\):

\( T_{2}^{19} + 4 T_{2}^{18} - 22 T_{2}^{17} - 101 T_{2}^{16} + 178 T_{2}^{15} + 1035 T_{2}^{14} + \cdots - 210 \) Copy content Toggle raw display
\( T_{5}^{19} + 5 T_{5}^{18} - 54 T_{5}^{17} - 282 T_{5}^{16} + 1147 T_{5}^{15} + 6431 T_{5}^{14} + \cdots - 45312 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{19} + 4 T^{18} + \cdots - 210 \) Copy content Toggle raw display
$3$ \( T^{19} \) Copy content Toggle raw display
$5$ \( T^{19} + 5 T^{18} + \cdots - 45312 \) Copy content Toggle raw display
$7$ \( (T - 1)^{19} \) Copy content Toggle raw display
$11$ \( T^{19} + \cdots + 6681231360 \) Copy content Toggle raw display
$13$ \( T^{19} - 24 T^{18} + \cdots + 61687808 \) Copy content Toggle raw display
$17$ \( T^{19} + 17 T^{18} + \cdots - 15267840 \) Copy content Toggle raw display
$19$ \( T^{19} + \cdots + 863515648 \) Copy content Toggle raw display
$23$ \( T^{19} - 17 T^{18} + \cdots - 31104000 \) Copy content Toggle raw display
$29$ \( T^{19} + \cdots - 342307676160 \) Copy content Toggle raw display
$31$ \( T^{19} + \cdots + 754255360 \) Copy content Toggle raw display
$37$ \( T^{19} + \cdots + 3880343600 \) Copy content Toggle raw display
$41$ \( T^{19} + \cdots - 840044212224 \) Copy content Toggle raw display
$43$ \( T^{19} + \cdots + 2627365310464 \) Copy content Toggle raw display
$47$ \( T^{19} + \cdots - 4860699264 \) Copy content Toggle raw display
$53$ \( T^{19} + \cdots - 81132625920 \) Copy content Toggle raw display
$59$ \( T^{19} + \cdots + 6371078307840 \) Copy content Toggle raw display
$61$ \( T^{19} + \cdots + 23219840000 \) Copy content Toggle raw display
$67$ \( T^{19} + \cdots + 745488293888 \) Copy content Toggle raw display
$71$ \( T^{19} + \cdots - 182187011747328 \) Copy content Toggle raw display
$73$ \( T^{19} + \cdots - 49767477174016 \) Copy content Toggle raw display
$79$ \( T^{19} + \cdots + 166522093490176 \) Copy content Toggle raw display
$83$ \( T^{19} + \cdots + 10295566761984 \) Copy content Toggle raw display
$89$ \( T^{19} + \cdots - 652944665997312 \) Copy content Toggle raw display
$97$ \( T^{19} + \cdots - 50408474451328 \) Copy content Toggle raw display
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