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Properties

Label 41.4.c.b
Level $41$
Weight $4$
Character orbit 41.c
Analytic conductor $2.419$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [41,4,Mod(9,41)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(41, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("41.9");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 41 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 41.c (of order \(4\), degree \(2\), minimal)

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: no
Analytic conductor: \(2.41907831024\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 101 x^{14} + 4134 x^{12} + 88066 x^{10} + 1046785 x^{8} + 6974625 x^{6} + 25242848 x^{4} + \cdots + 27878400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{8} q^{3} + (\beta_{2} - 5) q^{4} + ( - \beta_{12} + \beta_{4} + \cdots + \beta_1) q^{5}+ \cdots + (\beta_{14} + 2 \beta_{12} + \cdots - 2 \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{8} q^{3} + (\beta_{2} - 5) q^{4} + ( - \beta_{12} + \beta_{4} + \cdots + \beta_1) q^{5}+ \cdots + ( - 14 \beta_{14} + 16 \beta_{13} + \cdots + 150) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 74 q^{4} + 52 q^{6} - 52 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 74 q^{4} + 52 q^{6} - 52 q^{7} - 192 q^{10} + 28 q^{11} - 42 q^{12} + 132 q^{13} + 78 q^{14} + 184 q^{15} + 42 q^{16} + 188 q^{17} - 118 q^{18} + 48 q^{19} + 634 q^{22} + 140 q^{23} - 324 q^{24} + 8 q^{25} - 266 q^{26} + 288 q^{27} - 372 q^{28} - 312 q^{29} - 1034 q^{30} - 12 q^{31} + 446 q^{34} - 888 q^{35} - 412 q^{37} + 388 q^{38} + 1364 q^{40} - 92 q^{41} - 1404 q^{42} + 188 q^{44} + 2836 q^{45} - 644 q^{47} + 1342 q^{48} - 3600 q^{51} - 2346 q^{52} + 376 q^{53} + 3158 q^{54} + 1308 q^{55} + 494 q^{56} - 56 q^{57} + 1934 q^{58} - 2124 q^{59} - 2478 q^{60} + 2672 q^{63} + 1470 q^{64} + 2204 q^{65} + 972 q^{66} - 3464 q^{67} - 1022 q^{68} - 312 q^{69} - 1158 q^{70} + 716 q^{71} - 3546 q^{72} - 2664 q^{75} + 2362 q^{76} + 1376 q^{78} + 1000 q^{79} + 3920 q^{81} - 1300 q^{82} + 328 q^{83} + 1980 q^{85} + 7148 q^{86} - 4578 q^{88} + 644 q^{89} - 11136 q^{92} - 6648 q^{93} + 3060 q^{94} + 3076 q^{95} - 2104 q^{96} - 3032 q^{97} - 3954 q^{98} + 2776 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 101 x^{14} + 4134 x^{12} + 88066 x^{10} + 1046785 x^{8} + 6974625 x^{6} + 25242848 x^{4} + \cdots + 27878400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 101741 \nu^{15} - 10493467 \nu^{13} - 436374294 \nu^{11} - 9302545550 \nu^{9} + \cdots - 996374642432 \nu ) / 14732762880 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2225 \nu^{15} + 214396 \nu^{13} + 8201550 \nu^{11} + 157730144 \nu^{9} + 1591924937 \nu^{7} + \cdots + 13415441216 \nu ) / 156731520 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 313 \nu^{14} - 30200 \nu^{12} - 1158082 \nu^{10} - 22338536 \nu^{8} - 226003449 \nu^{6} + \cdots - 1879680000 ) / 4749440 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 162619 \nu^{14} + 15952620 \nu^{12} + 616384486 \nu^{10} + 11816636288 \nu^{8} + \cdots + 729047517440 ) / 2455460480 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 529651 \nu^{15} - 504723 \nu^{14} - 50265110 \nu^{13} - 48969510 \nu^{12} + \cdots - 3343503609600 ) / 14732762880 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 529651 \nu^{15} - 504723 \nu^{14} + 50265110 \nu^{13} - 48969510 \nu^{12} + \cdots - 3343503609600 ) / 14732762880 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 404179 \nu^{15} - 772971 \nu^{14} - 38984873 \nu^{13} - 73139895 \nu^{12} + \cdots - 3457287863040 ) / 7366381440 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 263678 \nu^{14} + 24920125 \nu^{12} + 934029672 \nu^{10} + 17584566986 \nu^{8} + \cdots + 1309696921280 ) / 1227730240 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 404179 \nu^{15} - 772971 \nu^{14} + 38984873 \nu^{13} - 73139895 \nu^{12} + \cdots - 3457287863040 ) / 7366381440 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 4649 \nu^{15} + 447637 \nu^{13} + 17100861 \nu^{11} + 328330046 \nu^{9} + 3309227591 \nu^{7} + \cdots + 28092823952 \nu ) / 39182880 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 816791 \nu^{15} + 8611908 \nu^{14} + 78525571 \nu^{13} + 827748030 \nu^{12} + \cdots + 50363750680320 ) / 14732762880 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 3065047 \nu^{15} - 295549673 \nu^{13} - 11303015658 \nu^{11} - 217023463546 \nu^{9} + \cdots - 17204770490368 \nu ) / 14732762880 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 816791 \nu^{15} + 8611908 \nu^{14} - 78525571 \nu^{13} + 827748030 \nu^{12} + \cdots + 50363750680320 ) / 14732762880 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} - 2\beta_{4} - \beta_{3} - 20\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{11} - \beta_{10} - \beta_{9} - 2\beta_{8} - 2\beta_{7} - \beta_{6} + \beta_{5} - 26\beta_{2} + 251 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 28 \beta_{15} - 36 \beta_{14} - 28 \beta_{13} - 34 \beta_{12} - 4 \beta_{11} + 4 \beta_{9} + \cdots + 439 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2 \beta_{15} + 2 \beta_{13} + 42 \beta_{11} + 34 \beta_{10} + 42 \beta_{9} + 118 \beta_{8} + 118 \beta_{7} + \cdots - 5447 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 685 \beta_{15} + 1081 \beta_{14} + 685 \beta_{13} + 979 \beta_{12} + 190 \beta_{11} + \cdots - 9962 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 88 \beta_{15} - 88 \beta_{13} - 1333 \beta_{11} - 955 \beta_{10} - 1333 \beta_{9} - 4340 \beta_{8} + \cdots + 123181 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 16486 \beta_{15} - 30120 \beta_{14} - 16486 \beta_{13} - 26376 \beta_{12} - 6436 \beta_{11} + \cdots + 230577 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2692 \beta_{15} + 2692 \beta_{13} + 38112 \beta_{11} + 25528 \beta_{10} + 38112 \beta_{9} + \cdots - 2846889 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 397725 \beta_{15} + 805069 \beta_{14} + 397725 \beta_{13} + 685113 \beta_{12} + 190644 \beta_{11} + \cdots - 5415932 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 70688 \beta_{15} - 70688 \beta_{13} - 1032853 \beta_{11} - 669161 \beta_{10} - 1032853 \beta_{9} + \cdots + 66800335 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 9640204 \beta_{15} - 20974656 \beta_{14} - 9640204 \beta_{13} - 17399474 \beta_{12} + \cdots + 128692183 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 1696502 \beta_{15} + 1696502 \beta_{13} + 27108074 \beta_{11} + 17376018 \beta_{10} + 27108074 \beta_{9} + \cdots - 1585762339 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 234616137 \beta_{15} + 537439917 \beta_{14} + 234616137 \beta_{13} + 435432835 \beta_{12} + \cdots - 3085912698 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/41\mathbb{Z}\right)^\times\).

\(n\) \(6\)
\(\chi(n)\) \(\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
− 4.94073i
− 4.29161i
− 2.19938i
− 1.13005i
2.07540i
2.31096i
4.19083i
4.98458i
− 4.98458i
− 4.19083i
− 2.31096i
− 2.07540i
1.13005i
2.19938i
4.29161i
4.94073i
− 4.94073i 5.15011 + 5.15011i −16.4108 − 20.7027i 25.4453 − 25.4453i −1.20106 − 1.20106i 41.5557i 26.0472i −102.287
9.2 − 4.29161i −3.60976 − 3.60976i −10.4179 1.26262i −15.4917 + 15.4917i 5.36238 + 5.36238i 10.3767i − 0.939259i 5.41866
9.3 − 2.19938i 3.66008 + 3.66008i 3.16271 8.01186i 8.04991 − 8.04991i 3.70861 + 3.70861i − 24.5511i − 0.207652i 17.6212
9.4 − 1.13005i −3.13023 − 3.13023i 6.72299 − 4.47007i −3.53731 + 3.53731i −20.1932 − 20.1932i − 16.6377i − 7.40334i −5.05140
9.5 2.07540i −2.27891 − 2.27891i 3.69271 19.9761i 4.72965 − 4.72965i 4.92912 + 4.92912i 24.2671i − 16.6132i −41.4585
9.6 2.31096i 4.90735 + 4.90735i 2.65945 − 5.66312i −11.3407 + 11.3407i −12.7262 − 12.7262i 24.6336i 21.1641i 13.0873
9.7 4.19083i −6.64684 − 6.64684i −9.56307 − 6.72226i 27.8558 − 27.8558i −14.9969 − 14.9969i − 6.55056i 61.3611i 28.1718
9.8 4.98458i 1.94821 + 1.94821i −16.8460 2.30757i −9.71099 + 9.71099i 9.11723 + 9.11723i − 44.0938i − 19.4090i −11.5023
32.1 − 4.98458i 1.94821 − 1.94821i −16.8460 − 2.30757i −9.71099 − 9.71099i 9.11723 − 9.11723i 44.0938i 19.4090i −11.5023
32.2 − 4.19083i −6.64684 + 6.64684i −9.56307 6.72226i 27.8558 + 27.8558i −14.9969 + 14.9969i 6.55056i − 61.3611i 28.1718
32.3 − 2.31096i 4.90735 − 4.90735i 2.65945 5.66312i −11.3407 − 11.3407i −12.7262 + 12.7262i − 24.6336i − 21.1641i 13.0873
32.4 − 2.07540i −2.27891 + 2.27891i 3.69271 − 19.9761i 4.72965 + 4.72965i 4.92912 − 4.92912i − 24.2671i 16.6132i −41.4585
32.5 1.13005i −3.13023 + 3.13023i 6.72299 4.47007i −3.53731 − 3.53731i −20.1932 + 20.1932i 16.6377i 7.40334i −5.05140
32.6 2.19938i 3.66008 − 3.66008i 3.16271 − 8.01186i 8.04991 + 8.04991i 3.70861 − 3.70861i 24.5511i 0.207652i 17.6212
32.7 4.29161i −3.60976 + 3.60976i −10.4179 − 1.26262i −15.4917 − 15.4917i 5.36238 − 5.36238i − 10.3767i 0.939259i 5.41866
32.8 4.94073i 5.15011 − 5.15011i −16.4108 20.7027i 25.4453 + 25.4453i −1.20106 + 1.20106i − 41.5557i − 26.0472i −102.287
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 41.4.c.b ✓ 16
41.c even 4 1 inner 41.4.c.b ✓ 16
41.e odd 8 2 1681.4.a.g 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
41.4.c.b ✓ 16 1.a even 1 1 trivial
41.4.c.b ✓ 16 41.c even 4 1 inner
1681.4.a.g 16 41.e odd 8 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 101 T_{2}^{14} + 4134 T_{2}^{12} + 88066 T_{2}^{10} + 1046785 T_{2}^{8} + 6974625 T_{2}^{6} + \cdots + 27878400 \) acting on \(S_{4}^{\mathrm{new}}(41, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 101 T^{14} + \cdots + 27878400 \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 243561990400 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 2698791840000 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 16\!\cdots\!04 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 81\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 14\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots - 31271701696000)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 49\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 33\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 33\!\cdots\!80)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 50\!\cdots\!61 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots - 12\!\cdots\!60)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 96\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 27\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 10\!\cdots\!04 \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 22\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 69\!\cdots\!00 \) Copy content Toggle raw display
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