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Properties

Label 41.6.a.b
Level $41$
Weight $6$
Character orbit 41.a
Self dual yes
Analytic conductor $6.576$
Analytic rank $0$
Dimension $10$
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] = [41,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("41.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 41 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(6.57573661233\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 3 x^{9} - 259 x^{8} + 639 x^{7} + 22422 x^{6} - 38356 x^{5} - 735592 x^{4} + 422608 x^{3} + \cdots - 24923264 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{7} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{4} + 3) q^{3} + (\beta_{5} - \beta_{4} + 21) q^{4} + (\beta_{9} - \beta_{8} - \beta_{5} + \cdots + 3) q^{5}+ \cdots + (2 \beta_{9} - 6 \beta_{8} + \cdots + 122) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{4} + 3) q^{3} + (\beta_{5} - \beta_{4} + 21) q^{4} + (\beta_{9} - \beta_{8} - \beta_{5} + \cdots + 3) q^{5}+ \cdots + (924 \beta_{9} - 1106 \beta_{8} + \cdots - 489) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 3 q^{2} + 28 q^{3} + 207 q^{4} + 32 q^{5} + 54 q^{6} + 342 q^{7} + 249 q^{8} + 1194 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 3 q^{2} + 28 q^{3} + 207 q^{4} + 32 q^{5} + 54 q^{6} + 342 q^{7} + 249 q^{8} + 1194 q^{9} + 102 q^{10} + 846 q^{11} + 4204 q^{12} + 1504 q^{13} + 3468 q^{14} + 1966 q^{15} + 5859 q^{16} + 560 q^{17} - 3713 q^{18} + 4240 q^{19} - 6182 q^{20} - 3096 q^{21} - 2628 q^{22} - 1508 q^{23} - 10166 q^{24} + 11734 q^{25} - 22014 q^{26} + 1882 q^{27} - 8662 q^{28} - 124 q^{29} - 45234 q^{30} + 10384 q^{31} - 6619 q^{32} - 22772 q^{33} + 802 q^{34} - 17890 q^{35} - 9657 q^{36} + 5524 q^{37} - 46098 q^{38} + 16844 q^{39} - 61738 q^{40} + 16810 q^{41} + 15476 q^{42} + 24160 q^{43} - 21594 q^{44} + 94688 q^{45} + 42404 q^{46} + 58984 q^{47} + 49296 q^{48} + 70326 q^{49} + 6817 q^{50} + 7336 q^{51} + 64374 q^{52} + 23456 q^{53} - 120694 q^{54} + 96426 q^{55} + 80184 q^{56} + 78004 q^{57} - 13378 q^{58} + 52428 q^{59} + 29422 q^{60} + 113540 q^{61} - 113008 q^{62} - 11036 q^{63} + 37363 q^{64} - 22340 q^{65} - 46224 q^{66} + 85506 q^{67} - 71406 q^{68} - 80004 q^{69} - 71946 q^{70} + 75236 q^{71} - 248911 q^{72} - 85148 q^{73} - 23462 q^{74} + 79652 q^{75} + 113376 q^{76} - 172896 q^{77} - 292760 q^{78} + 178200 q^{79} - 401850 q^{80} - 54126 q^{81} + 5043 q^{82} - 125412 q^{83} - 123276 q^{84} - 245912 q^{85} - 18848 q^{86} - 292760 q^{87} - 135952 q^{88} - 62696 q^{89} - 531830 q^{90} + 30056 q^{91} - 236372 q^{92} - 41700 q^{93} + 419014 q^{94} + 28002 q^{95} + 172086 q^{96} + 154548 q^{97} + 288367 q^{98} + 21952 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 3 x^{9} - 259 x^{8} + 639 x^{7} + 22422 x^{6} - 38356 x^{5} - 735592 x^{4} + 422608 x^{3} + \cdots - 24923264 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 821329 \nu^{9} + 14348689 \nu^{8} - 183370315 \nu^{7} - 3097677769 \nu^{6} + \cdots + 15206885360704 ) / 352204901760 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2881033 \nu^{9} + 23722737 \nu^{8} + 591371205 \nu^{7} - 5263588677 \nu^{6} + \cdots - 17307130815808 ) / 704409803520 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 4143687 \nu^{9} + 33647123 \nu^{8} + 908305915 \nu^{7} - 7674998483 \nu^{6} + \cdots + 64872509422848 ) / 704409803520 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 4143687 \nu^{9} + 33647123 \nu^{8} + 908305915 \nu^{7} - 7674998483 \nu^{6} + \cdots + 27538789836288 ) / 704409803520 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2620843 \nu^{9} - 37365137 \nu^{8} - 571171605 \nu^{7} + 7707530737 \nu^{6} + \cdots - 10943765787072 ) / 352204901760 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 3190523 \nu^{9} + 13685523 \nu^{8} + 805119975 \nu^{7} - 3191507079 \nu^{6} + \cdots + 19092607083712 ) / 140881960704 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 35596703 \nu^{9} - 244847827 \nu^{8} - 8529091815 \nu^{7} + 56442184127 \nu^{6} + \cdots - 315886449014592 ) / 1408819607040 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 62543397 \nu^{9} - 300516313 \nu^{8} - 15215639805 \nu^{7} + 66099051413 \nu^{6} + \cdots - 285269596432448 ) / 1408819607040 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - \beta_{4} + 53 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{8} + 3\beta_{7} + \beta_{6} + 2\beta_{5} + 4\beta_{4} + 3\beta_{3} + 3\beta_{2} + 87\beta _1 + 19 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 8 \beta_{9} + 8 \beta_{8} - 6 \beta_{7} + 12 \beta_{6} + 111 \beta_{5} - 91 \beta_{4} - 2 \beta_{3} + \cdots + 4661 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 56 \beta_{9} + 524 \beta_{8} + 317 \beta_{7} + 213 \beta_{6} + 228 \beta_{5} + 534 \beta_{4} + \cdots + 1699 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 1736 \beta_{9} + 1944 \beta_{8} - 1112 \beta_{7} + 2188 \beta_{6} + 11637 \beta_{5} - 8125 \beta_{4} + \cdots + 456741 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 11032 \beta_{9} + 60812 \beta_{8} + 31863 \beta_{7} + 31321 \beta_{6} + 25010 \beta_{5} + \cdots + 182511 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 260616 \beta_{9} + 310264 \beta_{8} - 151094 \beta_{7} + 299680 \beta_{6} + 1226879 \beta_{5} + \cdots + 46342725 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1569464 \beta_{9} + 6846492 \beta_{8} + 3276301 \beta_{7} + 4038625 \beta_{6} + 2840028 \beta_{5} + \cdots + 26557931 \) 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
−10.2151
−9.84411
−4.78711
−3.43528
−2.52657
2.00833
3.67603
8.15557
9.20306
10.7652
−10.2151 3.12029 72.3478 −81.8458 −31.8740 −15.0493 −412.156 −233.264 836.061
1.2 −9.84411 29.0260 64.9065 85.4409 −285.735 −100.276 −323.935 599.510 −841.090
1.3 −4.78711 −20.9873 −9.08361 −34.1675 100.468 −209.673 196.672 197.467 163.563
1.4 −3.43528 12.2840 −20.1989 41.7210 −42.1989 27.9809 179.318 −92.1041 −143.323
1.5 −2.52657 −18.1625 −25.6165 −62.7567 45.8887 238.318 145.572 86.8759 158.559
1.6 2.00833 −26.2398 −27.9666 74.6642 −52.6982 126.297 −120.433 445.529 149.950
1.7 3.67603 23.5698 −18.4868 17.9916 86.6435 214.269 −185.591 312.537 66.1377
1.8 8.15557 19.5942 34.5133 37.0474 159.802 −182.955 20.4971 140.932 302.143
1.9 9.20306 −7.24444 52.6962 60.0346 −66.6710 92.4544 190.469 −190.518 552.502
1.10 10.7652 13.0398 83.8885 −106.130 140.375 150.634 558.588 −72.9648 −1142.50
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
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 41.6.a.b ✓ 10
3.b odd 2 1 369.6.a.e 10
4.b odd 2 1 656.6.a.g 10
5.b even 2 1 1025.6.a.b 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
41.6.a.b ✓ 10 1.a even 1 1 trivial
369.6.a.e 10 3.b odd 2 1
656.6.a.g 10 4.b odd 2 1
1025.6.a.b 10 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} - 3 T_{2}^{9} - 259 T_{2}^{8} + 639 T_{2}^{7} + 22422 T_{2}^{6} - 38356 T_{2}^{5} + \cdots - 24923264 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(41))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - 3 T^{9} + \cdots - 24923264 \) Copy content Toggle raw display
$3$ \( T^{10} + \cdots + 485481280320 \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 19\!\cdots\!04 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 14\!\cdots\!56 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots - 11\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 84\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots - 16\!\cdots\!28 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots - 18\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots - 59\!\cdots\!08 \) Copy content Toggle raw display
$41$ \( (T - 1681)^{10} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots - 75\!\cdots\!64 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots - 35\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots - 47\!\cdots\!48 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 16\!\cdots\!20 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots - 71\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 99\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots - 94\!\cdots\!72 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 22\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 45\!\cdots\!20 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 51\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 75\!\cdots\!60 \) Copy content Toggle raw display
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