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Properties

Label 4010.2.a.l
Level $4010$
Weight $2$
Character orbit 4010.a
Self dual yes
Analytic conductor $32.020$
Analytic rank $0$
Dimension $17$
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] = [4010,2,Mod(1,4010)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4010, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("4010.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4010 = 2 \cdot 5 \cdot 401 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4010.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: \(32.0200112105\)
Analytic rank: \(0\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{17} - 3 x^{16} - 24 x^{15} + 70 x^{14} + 228 x^{13} - 638 x^{12} - 1075 x^{11} + 2854 x^{10} + \cdots - 2 \) 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 \(1,\beta_1,\ldots,\beta_{16}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{17} - 3 x^{16} - 24 x^{15} + 70 x^{14} + 228 x^{13} - 638 x^{12} - 1075 x^{11} + 2854 x^{10} + \cdots - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 517292 \nu^{16} - 36703983 \nu^{15} + 120153563 \nu^{14} + 741135014 \nu^{13} - 2742036456 \nu^{12} + \cdots - 113062883 ) / 309515047 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4837565 \nu^{16} - 18840353 \nu^{15} - 132711821 \nu^{14} + 583174545 \nu^{13} + \cdots - 1191999696 ) / 309515047 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 13533000 \nu^{16} + 13049122 \nu^{15} + 426684545 \nu^{14} - 396189248 \nu^{13} + \cdots + 30057660 ) / 309515047 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 29164450 \nu^{16} - 75243059 \nu^{15} - 712942292 \nu^{14} + 1640652448 \nu^{13} + \cdots - 133252285 ) / 309515047 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 31959553 \nu^{16} - 24013167 \nu^{15} - 1004560849 \nu^{14} + 653567956 \nu^{13} + \cdots + 2506119991 ) / 309515047 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 42578708 \nu^{16} + 160512035 \nu^{15} + 898763550 \nu^{14} - 3638281524 \nu^{13} + \cdots + 734658957 ) / 309515047 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 42697450 \nu^{16} + 88292181 \nu^{15} + 1139626837 \nu^{14} - 2036841696 \nu^{13} + \cdots - 455720149 ) / 309515047 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 80578876 \nu^{16} + 238642459 \nu^{15} + 1945071424 \nu^{14} - 5575538085 \nu^{13} + \cdots - 171215480 ) / 309515047 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 113055721 \nu^{16} - 299359609 \nu^{15} - 2829478710 \nu^{14} + 6970241055 \nu^{13} + \cdots + 1635727447 ) / 309515047 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 127487096 \nu^{16} - 350251889 \nu^{15} - 3118269472 \nu^{14} + 7992434926 \nu^{13} + \cdots + 2741181689 ) / 309515047 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 131491739 \nu^{16} - 366833857 \nu^{15} - 3252935367 \nu^{14} + 8598825523 \nu^{13} + \cdots + 1300063361 ) / 309515047 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 143660013 \nu^{16} - 441098466 \nu^{15} - 3444899317 \nu^{14} + 10415391985 \nu^{13} + \cdots - 820821813 ) / 309515047 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 171291908 \nu^{16} + 481721894 \nu^{15} + 4204915171 \nu^{14} - 11215905374 \nu^{13} + \cdots - 456913120 ) / 309515047 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 377743308 \nu^{16} - 1097265728 \nu^{15} - 9106434527 \nu^{14} + 25302804816 \nu^{13} + \cdots + 2910598480 ) / 309515047 \) 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
\(\nu^{3}\)\(=\) \( -\beta_{11} - \beta_{10} + \beta_{7} + \beta_{3} + \beta_{2} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{16} + \beta_{15} + \beta_{14} - \beta_{12} - 2 \beta_{11} + \beta_{8} + 2 \beta_{7} + \beta_{5} + \cdots + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{16} + \beta_{15} + \beta_{14} - \beta_{12} - 11 \beta_{11} - 9 \beta_{10} + \beta_{9} + \beta_{8} + \cdots + 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 11 \beta_{16} + 9 \beta_{15} + 12 \beta_{14} - 2 \beta_{13} - 11 \beta_{12} - 24 \beta_{11} + 2 \beta_{9} + \cdots + 100 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 16 \beta_{16} + 13 \beta_{15} + 16 \beta_{14} - 2 \beta_{13} - 18 \beta_{12} - 99 \beta_{11} - 68 \beta_{10} + \cdots + 47 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 99 \beta_{16} + 64 \beta_{15} + 117 \beta_{14} - 34 \beta_{13} - 102 \beta_{12} - 233 \beta_{11} + \cdots + 682 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 184 \beta_{16} + 125 \beta_{15} + 195 \beta_{14} - 43 \beta_{13} - 224 \beta_{12} - 849 \beta_{11} + \cdots + 532 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 849 \beta_{16} + 424 \beta_{15} + 1080 \beta_{14} - 407 \beta_{13} - 922 \beta_{12} - 2116 \beta_{11} + \cdots + 4905 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1848 \beta_{16} + 1058 \beta_{15} + 2117 \beta_{14} - 622 \beta_{13} - 2383 \beta_{12} - 7176 \beta_{11} + \cdots + 5326 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 7182 \beta_{16} + 2736 \beta_{15} + 9779 \beta_{14} - 4253 \beta_{13} - 8293 \beta_{12} - 18641 \beta_{11} + \cdots + 36564 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 17293 \beta_{16} + 8332 \beta_{15} + 21517 \beta_{14} - 7563 \beta_{13} - 23325 \beta_{12} + \cdots + 50206 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 60478 \beta_{16} + 17465 \beta_{15} + 87776 \beta_{14} - 41619 \beta_{13} - 74174 \beta_{12} + \cdots + 279909 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 155277 \beta_{16} + 62631 \beta_{15} + 209671 \beta_{14} - 83436 \beta_{13} - 217305 \beta_{12} + \cdots + 457476 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 508006 \beta_{16} + 110854 \beta_{15} + 783879 \beta_{14} - 393189 \beta_{13} - 658089 \beta_{12} + \cdots + 2188243 \) 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
−2.56168
−2.39414
−2.24738
−1.71266
−1.70497
−0.548857
−0.291319
−0.103424
0.204745
0.278065
0.507814
1.40160
1.51896
2.15213
2.65670
2.89857
2.94584
−1.00000 −2.56168 1.00000 −1.00000 2.56168 0.101571 −1.00000 3.56218 1.00000
1.2 −1.00000 −2.39414 1.00000 −1.00000 2.39414 4.25395 −1.00000 2.73189 1.00000
1.3 −1.00000 −2.24738 1.00000 −1.00000 2.24738 2.64455 −1.00000 2.05073 1.00000
1.4 −1.00000 −1.71266 1.00000 −1.00000 1.71266 −2.30195 −1.00000 −0.0668018 1.00000
1.5 −1.00000 −1.70497 1.00000 −1.00000 1.70497 −1.19610 −1.00000 −0.0930693 1.00000
1.6 −1.00000 −0.548857 1.00000 −1.00000 0.548857 −1.86676 −1.00000 −2.69876 1.00000
1.7 −1.00000 −0.291319 1.00000 −1.00000 0.291319 −2.64183 −1.00000 −2.91513 1.00000
1.8 −1.00000 −0.103424 1.00000 −1.00000 0.103424 1.84281 −1.00000 −2.98930 1.00000
1.9 −1.00000 0.204745 1.00000 −1.00000 −0.204745 0.0646599 −1.00000 −2.95808 1.00000
1.10 −1.00000 0.278065 1.00000 −1.00000 −0.278065 4.59715 −1.00000 −2.92268 1.00000
1.11 −1.00000 0.507814 1.00000 −1.00000 −0.507814 1.00840 −1.00000 −2.74213 1.00000
1.12 −1.00000 1.40160 1.00000 −1.00000 −1.40160 −1.83860 −1.00000 −1.03552 1.00000
1.13 −1.00000 1.51896 1.00000 −1.00000 −1.51896 1.25913 −1.00000 −0.692760 1.00000
1.14 −1.00000 2.15213 1.00000 −1.00000 −2.15213 −4.40972 −1.00000 1.63168 1.00000
1.15 −1.00000 2.65670 1.00000 −1.00000 −2.65670 4.81257 −1.00000 4.05806 1.00000
1.16 −1.00000 2.89857 1.00000 −1.00000 −2.89857 −4.89781 −1.00000 5.40172 1.00000
1.17 −1.00000 2.94584 1.00000 −1.00000 −2.94584 2.56799 −1.00000 5.67796 1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.17
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(1\)
\(401\) \(-1\)

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 4010.2.a.l ✓ 17
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4010.2.a.l ✓ 17 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4010))\):

\( T_{3}^{17} - 3 T_{3}^{16} - 24 T_{3}^{15} + 70 T_{3}^{14} + 228 T_{3}^{13} - 638 T_{3}^{12} - 1075 T_{3}^{11} + \cdots - 2 \) Copy content Toggle raw display
\( T_{7}^{17} - 4 T_{7}^{16} - 65 T_{7}^{15} + 254 T_{7}^{14} + 1582 T_{7}^{13} - 5951 T_{7}^{12} + \cdots + 5296 \) Copy content Toggle raw display
\( T_{11}^{17} + 8 T_{11}^{16} - 68 T_{11}^{15} - 652 T_{11}^{14} + 1185 T_{11}^{13} + 18356 T_{11}^{12} + \cdots + 241764 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{17} \) Copy content Toggle raw display
$3$ \( T^{17} - 3 T^{16} + \cdots - 2 \) Copy content Toggle raw display
$5$ \( (T + 1)^{17} \) Copy content Toggle raw display
$7$ \( T^{17} - 4 T^{16} + \cdots + 5296 \) Copy content Toggle raw display
$11$ \( T^{17} + 8 T^{16} + \cdots + 241764 \) Copy content Toggle raw display
$13$ \( T^{17} - 14 T^{16} + \cdots + 5251072 \) Copy content Toggle raw display
$17$ \( T^{17} + 8 T^{16} + \cdots - 35766 \) Copy content Toggle raw display
$19$ \( T^{17} + \cdots + 1149648896 \) Copy content Toggle raw display
$23$ \( T^{17} - T^{16} + \cdots + 1859328 \) Copy content Toggle raw display
$29$ \( T^{17} + 18 T^{16} + \cdots - 69858816 \) Copy content Toggle raw display
$31$ \( T^{17} - 8 T^{16} + \cdots + 96966032 \) Copy content Toggle raw display
$37$ \( T^{17} + \cdots + 535927808 \) Copy content Toggle raw display
$41$ \( T^{17} + 23 T^{16} + \cdots - 17784 \) Copy content Toggle raw display
$43$ \( T^{17} + \cdots - 132078690304 \) Copy content Toggle raw display
$47$ \( T^{17} + \cdots + 18355699968 \) Copy content Toggle raw display
$53$ \( T^{17} + \cdots - 1094032879872 \) Copy content Toggle raw display
$59$ \( T^{17} + \cdots - 314716048896 \) Copy content Toggle raw display
$61$ \( T^{17} + \cdots + 18293613032 \) Copy content Toggle raw display
$67$ \( T^{17} + \cdots + 591287684914 \) Copy content Toggle raw display
$71$ \( T^{17} + \cdots + 17757907723296 \) Copy content Toggle raw display
$73$ \( T^{17} + \cdots + 13670954012672 \) Copy content Toggle raw display
$79$ \( T^{17} + \cdots - 765288518073728 \) Copy content Toggle raw display
$83$ \( T^{17} + \cdots - 136284853248 \) Copy content Toggle raw display
$89$ \( T^{17} + \cdots - 6985897104 \) Copy content Toggle raw display
$97$ \( T^{17} + \cdots + 229360294738 \) Copy content Toggle raw display
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