Properties

Label 41.4.a.b
Level $41$
Weight $4$
Character orbit 41.a
Self dual yes
Analytic conductor $2.419$
Analytic rank $0$
Dimension $7$
CM no
Inner twists $1$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [41,4,Mod(1,41)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(41, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("41.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 41 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 41.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: \(2.41907831024\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 49x^{5} + 33x^{4} + 720x^{3} - 320x^{2} - 3200x + 512 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{6} + \beta_{4} + 1) q^{3} + (\beta_{6} - \beta_{4} + \beta_{3} + \beta_1 + 6) q^{4} + (\beta_{5} - \beta_{3} + \beta_{2} + 1) q^{5} + (\beta_{6} - 3 \beta_{5} - \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 2) q^{6} + (\beta_{5} + \beta_{4} - \beta_{3} - 2 \beta_{2} - 3 \beta_1 + 6) q^{7} + (4 \beta_{5} - 4 \beta_{4} + 4 \beta_{3} + 4 \beta_{2} + 5 \beta_1 + 4) q^{8} + ( - 3 \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} - 6 \beta_1 + 18) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{6} + \beta_{4} + 1) q^{3} + (\beta_{6} - \beta_{4} + \beta_{3} + \beta_1 + 6) q^{4} + (\beta_{5} - \beta_{3} + \beta_{2} + 1) q^{5} + (\beta_{6} - 3 \beta_{5} - \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 2) q^{6} + (\beta_{5} + \beta_{4} - \beta_{3} - 2 \beta_{2} - 3 \beta_1 + 6) q^{7} + (4 \beta_{5} - 4 \beta_{4} + 4 \beta_{3} + 4 \beta_{2} + 5 \beta_1 + 4) q^{8} + ( - 3 \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} - 6 \beta_1 + 18) q^{9} + (5 \beta_{6} - 2 \beta_{5} + 3 \beta_{4} - \beta_{3} - 4 \beta_1 + 14) q^{10} + ( - 2 \beta_{6} + \beta_{5} - 3 \beta_{4} + 3 \beta_{3} + 4 \beta_{2} + \beta_1 + 6) q^{11} + ( - 9 \beta_{6} + 5 \beta_{5} + 5 \beta_{4} - 6 \beta_{2} + 2 \beta_1 - 34) q^{12} + ( - 8 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 6 \beta_{2} - 12) q^{13} + (\beta_{6} - 5 \beta_{5} + 7 \beta_{4} - 12 \beta_{3} - 6 \beta_{2} - 46) q^{14} + (7 \beta_{6} - 9 \beta_{5} + 2 \beta_{4} - 5 \beta_{3} + 6 \beta_{2} - 9 \beta_1 + 13) q^{15} + (9 \beta_{6} - 9 \beta_{4} + 9 \beta_{3} + 16 \beta_{2} + 9 \beta_1 + 14) q^{16} + (4 \beta_{6} - 2 \beta_{5} + 10 \beta_{4} - 4 \beta_{3} - 8 \beta_{2} + 2 \beta_1 - 12) q^{17} + ( - 13 \beta_{6} + 8 \beta_{5} - 3 \beta_{4} - \beta_{3} + 4 \beta_{2} + 15 \beta_1 - 86) q^{18} + (9 \beta_{6} + 2 \beta_{5} - 19 \beta_{4} + 2 \beta_{3} - 6 \beta_{2} + 6 \beta_1 + 9) q^{19} + ( - \beta_{6} + 10 \beta_{5} - 3 \beta_{4} + \beta_{3} + 16 \beta_1 - 66) q^{20} + ( - 4 \beta_{6} + 11 \beta_{5} + 4 \beta_{4} + 15 \beta_{3} + \beta_{2} + 6 \beta_1 + 15) q^{21} + (\beta_{6} - \beta_{5} - 9 \beta_{4} + 16 \beta_{3} + 10 \beta_{2} + 10 \beta_1 + 18) q^{22} + (18 \beta_{6} - 6 \beta_{5} - 4 \beta_{4} - 6 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 28) q^{23} + (9 \beta_{6} - 19 \beta_{5} + 11 \beta_{4} - 14 \beta_{3} - 14 \beta_{2} - 38 \beta_1 - 6) q^{24} + ( - 2 \beta_{5} + 2 \beta_{4} - 10 \beta_{3} - 2 \beta_{2} - 16 \beta_1 + 23) q^{25} + ( - 10 \beta_{6} - 32 \beta_{5} + 10 \beta_{4} - 10 \beta_{3} - 24 \beta_{2} + \cdots - 36) q^{26}+ \cdots + ( - 9 \beta_{6} - 24 \beta_{5} + 117 \beta_{4} - 100 \beta_{3} + \cdots + 145) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + q^{2} + 4 q^{3} + 43 q^{4} + 10 q^{5} + 18 q^{6} + 48 q^{7} + 33 q^{8} + 99 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + q^{2} + 4 q^{3} + 43 q^{4} + 10 q^{5} + 18 q^{6} + 48 q^{7} + 33 q^{8} + 99 q^{9} + 82 q^{10} + 34 q^{11} - 218 q^{12} - 60 q^{13} - 304 q^{14} + 46 q^{15} + 59 q^{16} - 82 q^{17} - 563 q^{18} + 144 q^{19} - 410 q^{20} + 84 q^{21} + 82 q^{22} + 204 q^{23} - 86 q^{24} + 169 q^{25} - 292 q^{26} + 46 q^{27} + 68 q^{29} - 444 q^{30} + 696 q^{31} + 625 q^{32} + 124 q^{33} + 182 q^{34} - 6 q^{35} + 831 q^{36} + 730 q^{37} + 34 q^{38} + 848 q^{39} + 778 q^{40} - 287 q^{41} - 4 q^{42} + 368 q^{43} + 230 q^{44} - 1510 q^{45} + 432 q^{46} - 26 q^{47} - 1770 q^{48} + 679 q^{49} - 1137 q^{50} + 424 q^{51} - 1308 q^{52} - 892 q^{53} + 1340 q^{54} + 358 q^{55} - 1976 q^{56} - 3788 q^{57} + 1416 q^{58} - 916 q^{59} + 220 q^{60} - 450 q^{61} + 208 q^{62} + 874 q^{63} + 939 q^{64} - 1092 q^{65} - 576 q^{66} - 142 q^{67} + 226 q^{68} - 3396 q^{69} + 2208 q^{70} + 390 q^{71} - 2875 q^{72} + 882 q^{73} + 1822 q^{74} - 292 q^{75} + 4278 q^{76} - 2572 q^{77} + 232 q^{78} + 2890 q^{79} + 38 q^{80} + 951 q^{81} - 41 q^{82} + 1368 q^{83} + 3540 q^{84} - 1740 q^{85} - 132 q^{86} - 812 q^{87} + 4506 q^{88} - 2006 q^{89} + 5746 q^{90} + 2284 q^{91} + 4112 q^{92} + 2508 q^{93} + 104 q^{94} - 1002 q^{95} - 1782 q^{96} - 1950 q^{97} - 931 q^{98} + 892 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - x^{6} - 49x^{5} + 33x^{4} + 720x^{3} - 320x^{2} - 3200x + 512 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} - 33\nu^{2} + 176 ) / 16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} + \nu^{5} + 37\nu^{4} - 17\nu^{3} - 260\nu^{2} - 16\nu - 448 ) / 64 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} - \nu^{4} - 37\nu^{3} + 17\nu^{2} + 292\nu - 32 ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} + 3\nu^{5} - 45\nu^{4} - 115\nu^{3} + 460\nu^{2} + 848\nu - 448 ) / 64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} + 3\nu^{5} - 41\nu^{4} - 131\nu^{3} + 392\nu^{2} + 1120\nu - 576 ) / 64 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - \beta_{4} + \beta_{3} + \beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{5} - 4\beta_{4} + 4\beta_{3} + 4\beta_{2} + 21\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 33\beta_{6} - 33\beta_{4} + 33\beta_{3} + 16\beta_{2} + 33\beta _1 + 286 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 16\beta_{6} + 148\beta_{5} - 148\beta_{4} + 164\beta_{3} + 164\beta_{2} + 501\beta _1 + 228 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 977\beta_{6} + 80\beta_{5} - 1041\beta_{4} + 993\beta_{3} + 688\beta_{2} + 1089\beta _1 + 6654 \) 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
−4.84699
−3.89242
−2.69942
0.158390
3.23284
3.61196
5.43564
−4.84699 −9.64203 15.4934 −14.2831 46.7349 10.2695 −36.3203 65.9687 69.2303
1.2 −3.89242 8.44968 7.15096 −10.0474 −32.8897 33.5824 3.30483 44.3971 39.1088
1.3 −2.69942 −3.27713 −0.713139 15.5245 8.84635 20.4426 23.5204 −16.2604 −41.9070
1.4 0.158390 7.99110 −7.97491 14.3619 1.26571 −13.0501 −2.53027 36.8576 2.27479
1.5 3.23284 −0.631321 2.45129 13.5140 −2.04096 25.2162 −17.9381 −26.6014 43.6887
1.6 3.61196 5.45499 5.04626 −10.3662 19.7032 −3.79729 −10.6688 2.75692 −37.4424
1.7 5.43564 −4.34529 21.5462 1.29640 −23.6194 −24.6633 73.6322 −8.11846 7.04677
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(41\) \(1\)

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 41.4.a.b 7
3.b odd 2 1 369.4.a.h 7
4.b odd 2 1 656.4.a.j 7
5.b even 2 1 1025.4.a.d 7
7.b odd 2 1 2009.4.a.c 7
41.b even 2 1 1681.4.a.c 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
41.4.a.b 7 1.a even 1 1 trivial
369.4.a.h 7 3.b odd 2 1
656.4.a.j 7 4.b odd 2 1
1025.4.a.d 7 5.b even 2 1
1681.4.a.c 7 41.b even 2 1
2009.4.a.c 7 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{7} - T_{2}^{6} - 49T_{2}^{5} + 33T_{2}^{4} + 720T_{2}^{3} - 320T_{2}^{2} - 3200T_{2} + 512 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(41))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} - T^{6} - 49 T^{5} + 33 T^{4} + \cdots + 512 \) Copy content Toggle raw display
$3$ \( T^{7} - 4 T^{6} - 136 T^{5} + \cdots - 31928 \) Copy content Toggle raw display
$5$ \( T^{7} - 10 T^{6} - 472 T^{5} + \cdots + 5811008 \) Copy content Toggle raw display
$7$ \( T^{7} - 48 T^{6} + \cdots + 217277488 \) Copy content Toggle raw display
$11$ \( T^{7} - 34 T^{6} + \cdots + 1089373224 \) Copy content Toggle raw display
$13$ \( T^{7} + 60 T^{6} + \cdots - 1344939417600 \) Copy content Toggle raw display
$17$ \( T^{7} + 82 T^{6} + \cdots + 359921618304 \) Copy content Toggle raw display
$19$ \( T^{7} - 144 T^{6} + \cdots - 10579181757624 \) Copy content Toggle raw display
$23$ \( T^{7} - 204 T^{6} + \cdots + 5799910821888 \) Copy content Toggle raw display
$29$ \( T^{7} + \cdots - 503094100381696 \) Copy content Toggle raw display
$31$ \( T^{7} + \cdots - 298979937161216 \) Copy content Toggle raw display
$37$ \( T^{7} - 730 T^{6} + \cdots + 58\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( (T + 41)^{7} \) Copy content Toggle raw display
$43$ \( T^{7} - 368 T^{6} + \cdots - 43\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{7} + 26 T^{6} + \cdots + 10\!\cdots\!32 \) Copy content Toggle raw display
$53$ \( T^{7} + 892 T^{6} + \cdots + 29\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{7} + 916 T^{6} + \cdots + 28\!\cdots\!12 \) Copy content Toggle raw display
$61$ \( T^{7} + 450 T^{6} + \cdots + 94\!\cdots\!64 \) Copy content Toggle raw display
$67$ \( T^{7} + 142 T^{6} + \cdots + 58\!\cdots\!72 \) Copy content Toggle raw display
$71$ \( T^{7} - 390 T^{6} + \cdots - 27\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( T^{7} - 882 T^{6} + \cdots + 32\!\cdots\!68 \) Copy content Toggle raw display
$79$ \( T^{7} - 2890 T^{6} + \cdots + 70\!\cdots\!76 \) Copy content Toggle raw display
$83$ \( T^{7} - 1368 T^{6} + \cdots + 36\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{7} + 2006 T^{6} + \cdots - 12\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( T^{7} + 1950 T^{6} + \cdots + 14\!\cdots\!04 \) Copy content Toggle raw display
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