Properties

Label 41.6.b.b
Level $41$
Weight $6$
Character orbit 41.b
Analytic conductor $6.576$
Analytic rank $0$
Dimension $14$
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] = [41,6,Mod(40,41)]
 
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([1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("41.40");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 41 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 41.b (of order \(2\), degree \(1\), 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: \(6.57573661233\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 2190 x^{12} + 1866680 x^{10} + 789625888 x^{8} + 174600029184 x^{6} + 19518056974976 x^{4} + 998258913197568 x^{2} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{24} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} + 1) q^{2} - \beta_1 q^{3} + (\beta_{4} + 20) q^{4} + ( - \beta_{6} - \beta_{3} - 3) q^{5} - \beta_{10} q^{6} + (\beta_{8} - \beta_1) q^{7} + ( - \beta_{5} + 2 \beta_{4} + 9 \beta_{3} + \beta_{2} + 4) q^{8} + (\beta_{6} + \beta_{5} + 3 \beta_{4} + \beta_{3} - 70) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + 1) q^{2} - \beta_1 q^{3} + (\beta_{4} + 20) q^{4} + ( - \beta_{6} - \beta_{3} - 3) q^{5} - \beta_{10} q^{6} + (\beta_{8} - \beta_1) q^{7} + ( - \beta_{5} + 2 \beta_{4} + 9 \beta_{3} + \beta_{2} + 4) q^{8} + (\beta_{6} + \beta_{5} + 3 \beta_{4} + \beta_{3} - 70) q^{9} + (\beta_{7} - \beta_{4} + 4 \beta_{3} - 70) q^{10} + (\beta_{10} - \beta_{9} - \beta_{8}) q^{11} + ( - \beta_{13} + \beta_{12} - \beta_{11} - 2 \beta_{10} + \beta_{9} - \beta_{8} - 11 \beta_1) q^{12} + (\beta_{12} + \beta_{9} - 2 \beta_{8} - 10 \beta_1) q^{13} + ( - 2 \beta_{13} - \beta_{10}) q^{14} + (\beta_{13} + 2 \beta_{12} + 4 \beta_{10} - \beta_{9} - 13 \beta_1) q^{15} + (8 \beta_{6} - 2 \beta_{5} + 23 \beta_{4} + 30 \beta_{3} - 2 \beta_{2} - 128) q^{16} + ( - \beta_{13} + \beta_{12} + 2 \beta_{11} + 4 \beta_{10} + 3 \beta_{8} - \beta_1) q^{17} + ( - \beta_{7} - 8 \beta_{6} + 2 \beta_{5} - 13 \beta_{4} - 15 \beta_{3} + 4 \beta_{2} + \cdots + 43) q^{18}+ \cdots + ( - 52 \beta_{13} + 52 \beta_{12} - 80 \beta_{11} - 66 \beta_{10} + \cdots - 1377 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 8 q^{2} + 284 q^{4} - 32 q^{5} + 12 q^{8} - 978 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 8 q^{2} + 284 q^{4} - 32 q^{5} + 12 q^{8} - 978 q^{9} - 1008 q^{10} - 1916 q^{16} + 680 q^{18} + 2944 q^{20} - 3876 q^{21} - 2416 q^{23} + 11274 q^{25} - 1088 q^{31} + 24332 q^{32} + 1228 q^{33} + 18828 q^{36} - 9776 q^{37} - 44688 q^{39} + 336 q^{40} + 15654 q^{41} - 80 q^{42} - 19464 q^{43} - 58608 q^{45} + 51376 q^{46} + 21062 q^{49} - 52472 q^{50} + 2736 q^{51} + 64916 q^{57} - 173592 q^{59} + 114860 q^{61} - 88816 q^{62} + 86868 q^{64} + 156832 q^{66} - 217524 q^{72} + 344616 q^{73} - 111824 q^{74} + 282132 q^{77} - 16432 q^{78} - 311840 q^{80} - 240854 q^{81} - 191496 q^{82} - 345048 q^{83} + 76424 q^{84} - 115696 q^{86} + 147744 q^{87} + 328800 q^{90} + 221232 q^{91} + 429360 q^{92} + 227656 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 2190 x^{12} + 1866680 x^{10} + 789625888 x^{8} + 174600029184 x^{6} + 19518056974976 x^{4} + 998258913197568 x^{2} + \cdots + 16\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 951542199786175 \nu^{12} + \cdots + 10\!\cdots\!08 ) / 41\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 90\!\cdots\!56 \nu^{12} + \cdots + 94\!\cdots\!72 ) / 41\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 12\!\cdots\!67 \nu^{12} + \cdots - 72\!\cdots\!24 ) / 37\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 13\!\cdots\!33 \nu^{12} + \cdots + 59\!\cdots\!92 ) / 41\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 35\!\cdots\!34 \nu^{12} + \cdots - 32\!\cdots\!00 ) / 41\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 11\!\cdots\!70 \nu^{12} + \cdots - 30\!\cdots\!28 ) / 41\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 77\!\cdots\!44 \nu^{13} + \cdots - 19\!\cdots\!72 \nu ) / 68\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 44\!\cdots\!28 \nu^{13} + \cdots + 80\!\cdots\!84 \nu ) / 20\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 90\!\cdots\!56 \nu^{13} + \cdots + 94\!\cdots\!16 \nu ) / 41\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 17\!\cdots\!81 \nu^{13} + \cdots + 24\!\cdots\!08 \nu ) / 68\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 26\!\cdots\!58 \nu^{13} + \cdots - 15\!\cdots\!04 \nu ) / 61\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 51\!\cdots\!26 \nu^{13} + \cdots - 64\!\cdots\!52 \nu ) / 11\!\cdots\!12 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{5} + 3\beta_{4} + \beta_{3} - 313 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 6\beta_{13} - \beta_{12} + 6\beta_{11} + 2\beta_{10} - \beta_{9} - 5\beta_{8} - 484\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -15\beta_{7} - 700\beta_{6} - 577\beta_{5} - 2903\beta_{4} + 495\beta_{3} - 220\beta_{2} + 152693 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -4962\beta_{13} + 1293\beta_{12} - 6174\beta_{11} - 525\beta_{10} + 2412\beta_{9} + 4848\beta_{8} + 276301\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 1461 \beta_{7} + 460498 \beta_{6} + 330355 \beta_{5} + 2285976 \beta_{4} - 1069295 \beta_{3} + 262557 \beta_{2} - 87910066 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 3668280 \beta_{13} - 1344526 \beta_{12} + 5114940 \beta_{11} - 632206 \beta_{10} - 2531392 \beta_{9} - 4279148 \beta_{8} - 171678406 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 7759566 \beta_{7} - 291376612 \beta_{6} - 203846806 \beta_{5} - 1709916104 \beta_{4} + 1188207702 \beta_{3} - 233416258 \beta_{2} + 55112234756 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 2650162152 \beta_{13} + 1235796804 \beta_{12} - 3955370328 \beta_{11} + 1170194604 \beta_{10} + 2152077600 \beta_{9} + 3507457440 \beta_{8} + \cdots + 112522350292 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 11169476796 \beta_{7} + 181529165896 \beta_{6} + 134044577740 \beta_{5} + 1255976128560 \beta_{4} - 1094654046476 \beta_{3} + \cdots - 36419542111336 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1904178626448 \beta_{13} - 1049290471912 \beta_{12} + 2970012759792 \beta_{11} - 1283727378424 \beta_{10} - 1686971474752 \beta_{9} + \cdots - 76387752215560 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 11251160774808 \beta_{7} - 113436868602064 \beta_{6} - 91896018213304 \beta_{5} - 915849688495328 \beta_{4} + 921891959103672 \beta_{3} + \cdots + 24\!\cdots\!84 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 13\!\cdots\!68 \beta_{13} + 846492202138512 \beta_{12} + \cdots + 53\!\cdots\!80 \beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(6\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
40.1
10.0030i
10.0030i
22.5889i
22.5889i
18.6202i
18.6202i
26.9717i
26.9717i
5.63262i
5.63262i
10.7247i
10.7247i
18.9407i
18.9407i
−9.74377 10.0030i 62.9411 −6.94248 97.4670i 167.078i −301.484 142.940 67.6460
40.2 −9.74377 10.0030i 62.9411 −6.94248 97.4670i 167.078i −301.484 142.940 67.6460
40.3 −6.92159 22.5889i 15.9084 101.152 156.351i 113.018i 111.379 −267.257 −700.133
40.4 −6.92159 22.5889i 15.9084 101.152 156.351i 113.018i 111.379 −267.257 −700.133
40.5 −5.18850 18.6202i −5.07946 −66.3253 96.6111i 37.9864i 192.387 −103.713 344.129
40.6 −5.18850 18.6202i −5.07946 −66.3253 96.6111i 37.9864i 192.387 −103.713 344.129
40.7 3.07208 26.9717i −22.5623 4.41952 82.8593i 47.7428i −167.620 −484.473 13.5771
40.8 3.07208 26.9717i −22.5623 4.41952 82.8593i 47.7428i −167.620 −484.473 13.5771
40.9 5.00419 5.63262i −6.95804 −85.7879 28.1867i 250.090i −194.954 211.274 −429.299
40.10 5.00419 5.63262i −6.95804 −85.7879 28.1867i 250.090i −194.954 211.274 −429.299
40.11 7.52334 10.7247i 24.6006 67.4831 80.6854i 11.4746i −55.6682 127.981 507.698
40.12 7.52334 10.7247i 24.6006 67.4831 80.6854i 11.4746i −55.6682 127.981 507.698
40.13 10.2543 18.9407i 73.1497 −29.9991 194.223i 5.54343i 421.959 −115.751 −307.618
40.14 10.2543 18.9407i 73.1497 −29.9991 194.223i 5.54343i 421.959 −115.751 −307.618
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 40.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 41.6.b.b 14
3.b odd 2 1 369.6.d.c 14
41.b even 2 1 inner 41.6.b.b 14
123.b odd 2 1 369.6.d.c 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
41.6.b.b 14 1.a even 1 1 trivial
41.6.b.b 14 41.b even 2 1 inner
369.6.d.c 14 3.b odd 2 1
369.6.d.c 14 123.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{7} - 4T_{2}^{6} - 175T_{2}^{5} + 634T_{2}^{4} + 8888T_{2}^{3} - 29424T_{2}^{2} - 131120T_{2} + 415008 \) acting on \(S_{6}^{\mathrm{new}}(41, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{7} - 4 T^{6} - 175 T^{5} + \cdots + 415008)^{2} \) Copy content Toggle raw display
$3$ \( T^{14} + 2190 T^{12} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( (T^{7} + 16 T^{6} - 13628 T^{5} + \cdots - 35749794816)^{2} \) Copy content Toggle raw display
$7$ \( T^{14} + 107118 T^{12} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{14} + 1257998 T^{12} + \cdots + 41\!\cdots\!56 \) Copy content Toggle raw display
$13$ \( T^{14} + 2334480 T^{12} + \cdots + 16\!\cdots\!96 \) Copy content Toggle raw display
$17$ \( T^{14} + 14722592 T^{12} + \cdots + 47\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( T^{14} + 22961662 T^{12} + \cdots + 92\!\cdots\!64 \) Copy content Toggle raw display
$23$ \( (T^{7} + 1208 T^{6} + \cdots - 14\!\cdots\!40)^{2} \) Copy content Toggle raw display
$29$ \( T^{14} + 74998960 T^{12} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{7} + 544 T^{6} + \cdots - 91\!\cdots\!32)^{2} \) Copy content Toggle raw display
$37$ \( (T^{7} + 4888 T^{6} + \cdots + 78\!\cdots\!88)^{2} \) Copy content Toggle raw display
$41$ \( T^{14} - 15654 T^{13} + \cdots + 28\!\cdots\!01 \) Copy content Toggle raw display
$43$ \( (T^{7} + 9732 T^{6} + \cdots - 21\!\cdots\!56)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} + 1886390350 T^{12} + \cdots + 85\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{14} + 3153818160 T^{12} + \cdots + 15\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( (T^{7} + 86796 T^{6} + \cdots - 88\!\cdots\!72)^{2} \) Copy content Toggle raw display
$61$ \( (T^{7} - 57430 T^{6} + \cdots + 52\!\cdots\!84)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + 9477383598 T^{12} + \cdots + 26\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( T^{14} + 7464196046 T^{12} + \cdots + 84\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( (T^{7} - 172308 T^{6} + \cdots + 89\!\cdots\!80)^{2} \) Copy content Toggle raw display
$79$ \( T^{14} + 34470532334 T^{12} + \cdots + 39\!\cdots\!56 \) Copy content Toggle raw display
$83$ \( (T^{7} + 172524 T^{6} + \cdots + 21\!\cdots\!68)^{2} \) Copy content Toggle raw display
$89$ \( T^{14} + 61868218976 T^{12} + \cdots + 48\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( T^{14} + 55897289888 T^{12} + \cdots + 65\!\cdots\!96 \) Copy content Toggle raw display
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