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Properties

Label 40.5.g.a
Level $40$
Weight $5$
Character orbit 40.g
Analytic conductor $4.135$
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] = [40,5,Mod(11,40)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(40, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("40.11");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 40 = 2^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 40.g (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: \(4.13479852335\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 6 x^{15} + 14 x^{14} - 84 x^{13} + 628 x^{12} - 1392 x^{11} + 2016 x^{10} - 18048 x^{9} + \cdots + 4294967296 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{25}\cdot 5^{5} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{3} q^{3} + \beta_{2} q^{4} - \beta_{5} q^{5} + ( - \beta_{5} - \beta_{4} - \beta_{3} + 5) q^{6} + ( - \beta_{12} + \beta_{6} + \beta_{2} + \cdots - 2) q^{7}+ \cdots + (\beta_{15} - \beta_{13} - \beta_{7} + \cdots + 27) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_{3} q^{3} + \beta_{2} q^{4} - \beta_{5} q^{5} + ( - \beta_{5} - \beta_{4} - \beta_{3} + 5) q^{6} + ( - \beta_{12} + \beta_{6} + \beta_{2} + \cdots - 2) q^{7}+ \cdots + (2 \beta_{15} + 13 \beta_{14} + \cdots - 473) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 6 q^{2} + 8 q^{4} + 84 q^{6} - 216 q^{8} + 432 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 6 q^{2} + 8 q^{4} + 84 q^{6} - 216 q^{8} + 432 q^{9} - 50 q^{10} + 192 q^{11} + 260 q^{12} + 972 q^{14} - 824 q^{16} - 806 q^{18} - 704 q^{19} - 300 q^{20} - 1100 q^{22} - 1256 q^{24} - 2000 q^{25} + 108 q^{26} + 3648 q^{27} - 940 q^{28} - 1400 q^{30} + 4104 q^{32} - 992 q^{33} + 924 q^{34} - 2400 q^{36} - 420 q^{38} + 2200 q^{40} - 2208 q^{41} + 7820 q^{42} + 5568 q^{43} + 3816 q^{44} - 652 q^{46} + 4520 q^{48} - 2480 q^{49} + 750 q^{50} - 17792 q^{51} - 4640 q^{52} + 6488 q^{54} - 12552 q^{56} + 8608 q^{57} + 1040 q^{58} + 14016 q^{59} - 3500 q^{60} + 12480 q^{62} - 3712 q^{64} - 13312 q^{66} - 18880 q^{67} + 12360 q^{68} - 1800 q^{70} - 16456 q^{72} - 7360 q^{73} - 11508 q^{74} + 9312 q^{76} - 26760 q^{78} - 2400 q^{80} + 10384 q^{81} - 16640 q^{82} + 10560 q^{83} - 12520 q^{84} + 25764 q^{86} + 41200 q^{88} + 6816 q^{89} + 8550 q^{90} + 24576 q^{91} + 30420 q^{92} - 28388 q^{94} + 6784 q^{96} + 448 q^{97} + 23934 q^{98} + 2624 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 6 x^{15} + 14 x^{14} - 84 x^{13} + 628 x^{12} - 1392 x^{11} + 2016 x^{10} - 18048 x^{9} + \cdots + 4294967296 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2289 \nu^{15} + 7142 \nu^{14} + 74706 \nu^{13} + 421780 \nu^{12} - 1941812 \nu^{11} + \cdots + 8046621229056 ) / 240249733120 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3681 \nu^{15} + 118210 \nu^{14} + 173214 \nu^{13} - 1771620 \nu^{12} - 3818412 \nu^{11} + \cdots - 2641673322496 ) / 360374599680 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 20271 \nu^{15} + 62410 \nu^{14} + 117966 \nu^{13} + 764940 \nu^{12} + \cdots + 14240769376256 ) / 720749199360 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{15} - 6 \nu^{14} + 14 \nu^{13} - 84 \nu^{12} + 628 \nu^{11} - 1392 \nu^{10} + \cdots - 1509949440 ) / 33554432 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 11295 \nu^{15} + 24614 \nu^{14} + 7410 \nu^{13} - 680172 \nu^{12} + 756108 \nu^{11} + \cdots + 1713960386560 ) / 144149839872 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 19191 \nu^{15} + 64458 \nu^{14} - 31746 \nu^{13} + 580620 \nu^{12} - 6736108 \nu^{11} + \cdots + 12855105552384 ) / 240249733120 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 66083 \nu^{15} + 423186 \nu^{14} - 713706 \nu^{13} + 7074300 \nu^{12} + \cdots + 113365393342464 ) / 720749199360 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 75559 \nu^{15} - 192730 \nu^{14} - 234174 \nu^{13} - 4977900 \nu^{12} + 14228332 \nu^{11} + \cdots - 11708886155264 ) / 720749199360 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 82047 \nu^{15} - 376186 \nu^{14} - 68238 \nu^{13} - 5159340 \nu^{12} + 28811916 \nu^{11} + \cdots - 78527571427328 ) / 720749199360 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 34271 \nu^{15} + 60250 \nu^{14} - 128114 \nu^{13} + 2734060 \nu^{12} + \cdots + 21273778323456 ) / 240249733120 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 135479 \nu^{15} + 12554 \nu^{14} + 250878 \nu^{13} + 10420620 \nu^{12} + \cdots + 102587441348608 ) / 720749199360 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 11223 \nu^{15} + 42690 \nu^{14} - 66962 \nu^{13} + 1259420 \nu^{12} - 4311244 \nu^{11} + \cdots + 16960557416448 ) / 60062433280 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 140761 \nu^{15} - 349990 \nu^{14} + 1436094 \nu^{13} - 9959700 \nu^{12} + \cdots - 81238232662016 ) / 720749199360 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{12} + \beta_{11} - \beta_{5} + \beta_{4} + \beta_{3} + 14 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{15} - \beta_{14} + \beta_{13} - 2 \beta_{11} + 2 \beta_{10} + \beta_{9} - 2 \beta_{8} + \cdots - 60 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 2 \beta_{15} - 4 \beta_{14} + 14 \beta_{13} + 4 \beta_{12} + 2 \beta_{11} + 2 \beta_{10} + 4 \beta_{9} + \cdots - 256 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 10 \beta_{15} + 2 \beta_{14} - 2 \beta_{13} + 20 \beta_{12} + 20 \beta_{11} + 8 \beta_{10} + \cdots - 120 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 100 \beta_{15} - 112 \beta_{14} + 116 \beta_{13} - 212 \beta_{12} - 48 \beta_{11} + 12 \beta_{10} + \cdots - 2840 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 640 \beta_{15} + 552 \beta_{14} + 88 \beta_{13} + 176 \beta_{12} + 520 \beta_{11} + 152 \beta_{10} + \cdots - 35248 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1192 \beta_{15} - 488 \beta_{14} - 2008 \beta_{13} + 1688 \beta_{12} - 328 \beta_{11} + 128 \beta_{10} + \cdots + 5840 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 4520 \beta_{15} - 3368 \beta_{14} + 2904 \beta_{13} - 3184 \beta_{12} - 5072 \beta_{11} - 480 \beta_{10} + \cdots + 104064 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 41712 \beta_{15} + 30592 \beta_{14} - 24688 \beta_{13} + 7968 \beta_{12} - 56720 \beta_{11} + \cdots + 847936 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 43664 \beta_{15} - 9936 \beta_{14} - 144304 \beta_{13} + 83648 \beta_{12} - 40640 \beta_{11} + \cdots - 1224448 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 166336 \beta_{15} + 220832 \beta_{14} - 511808 \beta_{13} - 363616 \beta_{12} - 340160 \beta_{11} + \cdots + 13644352 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 724672 \beta_{15} + 1462464 \beta_{14} - 2066432 \beta_{13} + 2001664 \beta_{12} - 450944 \beta_{11} + \cdots - 17607808 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 926464 \beta_{15} + 1938176 \beta_{14} - 7844352 \beta_{13} + 1821120 \beta_{12} - 4952128 \beta_{11} + \cdots - 413712768 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/40\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(21\) \(31\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
3.79794 + 1.25525i
3.79794 − 1.25525i
3.55671 + 1.83026i
3.55671 − 1.83026i
3.30316 + 2.25591i
3.30316 − 2.25591i
1.64589 + 3.64569i
1.64589 − 3.64569i
−1.30255 + 3.78198i
−1.30255 − 3.78198i
−1.60069 + 3.66576i
−1.60069 − 3.66576i
−2.66926 + 2.97910i
−2.66926 − 2.97910i
−3.73120 + 1.44159i
−3.73120 − 1.44159i
−3.79794 − 1.25525i −0.715641 12.8487 + 9.53475i − 11.1803i 2.71796 + 0.898310i − 25.0983i −36.8300 − 52.3407i −80.4879 −14.0342 + 42.4622i
11.2 −3.79794 + 1.25525i −0.715641 12.8487 − 9.53475i 11.1803i 2.71796 − 0.898310i 25.0983i −36.8300 + 52.3407i −80.4879 −14.0342 − 42.4622i
11.3 −3.55671 − 1.83026i 13.1346 9.30033 + 13.0194i 11.1803i −46.7159 − 24.0397i 61.6422i −9.24975 − 63.3281i 91.5179 20.4629 − 39.7652i
11.4 −3.55671 + 1.83026i 13.1346 9.30033 − 13.0194i − 11.1803i −46.7159 + 24.0397i − 61.6422i −9.24975 + 63.3281i 91.5179 20.4629 + 39.7652i
11.5 −3.30316 − 2.25591i −15.2196 5.82176 + 14.9033i 11.1803i 50.2726 + 34.3339i − 47.5956i 14.3902 − 62.3612i 150.635 25.2218 − 36.9305i
11.6 −3.30316 + 2.25591i −15.2196 5.82176 − 14.9033i − 11.1803i 50.2726 − 34.3339i 47.5956i 14.3902 + 62.3612i 150.635 25.2218 + 36.9305i
11.7 −1.64589 − 3.64569i −5.51460 −10.5821 + 12.0008i − 11.1803i 9.07645 + 20.1045i 78.3513i 61.1682 + 18.8268i −50.5892 −40.7600 + 18.4017i
11.8 −1.64589 + 3.64569i −5.51460 −10.5821 − 12.0008i 11.1803i 9.07645 − 20.1045i − 78.3513i 61.1682 − 18.8268i −50.5892 −40.7600 − 18.4017i
11.9 1.30255 − 3.78198i −10.2034 −12.6067 − 9.85244i 11.1803i −13.2905 + 38.5890i 43.3025i −53.6826 + 34.8450i 23.1094 42.2838 + 14.5630i
11.10 1.30255 + 3.78198i −10.2034 −12.6067 + 9.85244i − 11.1803i −13.2905 − 38.5890i − 43.3025i −53.6826 − 34.8450i 23.1094 42.2838 − 14.5630i
11.11 1.60069 − 3.66576i 15.9375 −10.8756 − 11.7355i − 11.1803i 25.5111 − 58.4232i 56.7751i −60.4279 + 21.0824i 173.005 −40.9844 − 17.8963i
11.12 1.60069 + 3.66576i 15.9375 −10.8756 + 11.7355i 11.1803i 25.5111 + 58.4232i − 56.7751i −60.4279 − 21.0824i 173.005 −40.9844 + 17.8963i
11.13 2.66926 − 2.97910i −4.51805 −1.75006 − 15.9040i − 11.1803i −12.0599 + 13.4597i − 50.0881i −52.0510 − 37.2384i −60.5872 −33.3073 − 29.8433i
11.14 2.66926 + 2.97910i −4.51805 −1.75006 + 15.9040i 11.1803i −12.0599 − 13.4597i 50.0881i −52.0510 + 37.2384i −60.5872 −33.3073 + 29.8433i
11.15 3.73120 − 1.44159i 7.09911 11.8437 − 10.7577i 11.1803i 26.4882 − 10.2340i 2.59084i 28.6828 − 57.2127i −30.6027 16.1174 + 41.7160i
11.16 3.73120 + 1.44159i 7.09911 11.8437 + 10.7577i − 11.1803i 26.4882 + 10.2340i − 2.59084i 28.6828 + 57.2127i −30.6027 16.1174 − 41.7160i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 40.5.g.a ✓ 16
3.b odd 2 1 360.5.g.a 16
4.b odd 2 1 160.5.g.a 16
5.b even 2 1 200.5.g.h 16
5.c odd 4 2 200.5.e.e 32
8.b even 2 1 160.5.g.a 16
8.d odd 2 1 inner 40.5.g.a ✓ 16
12.b even 2 1 1440.5.g.a 16
20.d odd 2 1 800.5.g.h 16
20.e even 4 2 800.5.e.e 32
24.f even 2 1 360.5.g.a 16
24.h odd 2 1 1440.5.g.a 16
40.e odd 2 1 200.5.g.h 16
40.f even 2 1 800.5.g.h 16
40.i odd 4 2 800.5.e.e 32
40.k even 4 2 200.5.e.e 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.5.g.a ✓ 16 1.a even 1 1 trivial
40.5.g.a ✓ 16 8.d odd 2 1 inner
160.5.g.a 16 4.b odd 2 1
160.5.g.a 16 8.b even 2 1
200.5.e.e 32 5.c odd 4 2
200.5.e.e 32 40.k even 4 2
200.5.g.h 16 5.b even 2 1
200.5.g.h 16 40.e odd 2 1
360.5.g.a 16 3.b odd 2 1
360.5.g.a 16 24.f even 2 1
800.5.e.e 32 20.e even 4 2
800.5.e.e 32 40.i odd 4 2
800.5.g.h 16 20.d odd 2 1
800.5.g.h 16 40.f even 2 1
1440.5.g.a 16 12.b even 2 1
1440.5.g.a 16 24.h odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(40, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + \cdots + 4294967296 \) Copy content Toggle raw display
$3$ \( (T^{8} - 432 T^{6} + \cdots - 4114800)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 125)^{8} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 75\!\cdots\!72)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 14\!\cdots\!52)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 73\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots - 39\!\cdots\!48)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 96\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 62\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 95\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots - 11\!\cdots\!44)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 82\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 59\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 35\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots - 37\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots - 47\!\cdots\!48)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots - 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
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