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Properties

Label 40.4.d.a
Level $40$
Weight $4$
Character orbit 40.d
Analytic conductor $2.360$
Analytic rank $0$
Dimension $12$
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] = [40,4,Mod(21,40)]
 
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([0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("40.21");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 40 = 2^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 40.d (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: \(2.36007640023\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4x^{11} + 7x^{10} - 12x^{9} + 21x^{8} - 68x^{6} + 336x^{4} - 768x^{3} + 1792x^{2} - 4096x + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{14}\cdot 5^{4} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{5} q^{3} + ( - \beta_{4} + 1) q^{4} - \beta_{8} q^{5} + (\beta_{11} + \beta_{7} + \beta_{5} + \cdots - 3) q^{6}+ \cdots + ( - \beta_{11} + \beta_{10} - 2 \beta_{4} + \cdots - 8) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{5} q^{3} + ( - \beta_{4} + 1) q^{4} - \beta_{8} q^{5} + (\beta_{11} + \beta_{7} + \beta_{5} + \cdots - 3) q^{6}+ \cdots + ( - \beta_{11} + 11 \beta_{10} + \cdots + 83) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{2} + 16 q^{4} - 36 q^{6} + 28 q^{7} - 40 q^{8} - 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2 q^{2} + 16 q^{4} - 36 q^{6} + 28 q^{7} - 40 q^{8} - 108 q^{9} + 30 q^{10} + 188 q^{12} + 68 q^{14} - 60 q^{15} - 56 q^{16} - 206 q^{18} + 20 q^{20} - 164 q^{22} + 604 q^{23} + 360 q^{24} - 300 q^{25} - 308 q^{26} - 436 q^{28} + 40 q^{30} - 264 q^{31} + 72 q^{32} - 232 q^{33} - 180 q^{34} + 440 q^{36} + 820 q^{38} + 600 q^{39} + 120 q^{40} + 40 q^{41} + 884 q^{42} - 472 q^{44} - 1268 q^{46} - 940 q^{47} + 424 q^{48} + 1308 q^{49} - 50 q^{50} + 1024 q^{52} - 1512 q^{54} + 440 q^{55} - 728 q^{56} - 680 q^{57} - 360 q^{58} - 820 q^{60} + 592 q^{62} - 1300 q^{63} - 2048 q^{64} + 2928 q^{66} - 2344 q^{68} + 1160 q^{70} - 1592 q^{71} - 152 q^{72} + 432 q^{73} - 420 q^{74} + 2256 q^{76} + 3320 q^{78} + 2016 q^{79} + 1600 q^{80} + 2508 q^{81} + 88 q^{82} + 1048 q^{84} - 244 q^{86} - 1968 q^{87} + 4080 q^{88} - 424 q^{89} - 2250 q^{90} - 900 q^{92} + 292 q^{94} - 1520 q^{95} - 5920 q^{96} - 1584 q^{97} - 7266 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4x^{11} + 7x^{10} - 12x^{9} + 21x^{8} - 68x^{6} + 336x^{4} - 768x^{3} + 1792x^{2} - 4096x + 4096 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 15 \nu^{11} + 32 \nu^{10} - 135 \nu^{9} + 144 \nu^{8} - 149 \nu^{7} + 204 \nu^{6} + 900 \nu^{5} + \cdots + 28672 ) / 15360 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5 \nu^{11} + 44 \nu^{10} + 131 \nu^{9} + 260 \nu^{8} - 1527 \nu^{7} - 832 \nu^{6} + 652 \nu^{5} + \cdots + 144896 ) / 7680 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 14 \nu^{11} - \nu^{10} - 90 \nu^{9} + 377 \nu^{8} - 158 \nu^{7} - 717 \nu^{6} + 588 \nu^{5} + \cdots + 48640 ) / 3840 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 4 \nu^{11} + \nu^{10} - 14 \nu^{9} + 63 \nu^{8} - 66 \nu^{7} - 83 \nu^{6} + 170 \nu^{5} + \cdots + 13504 ) / 960 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 93 \nu^{11} + 204 \nu^{10} - 379 \nu^{9} + 388 \nu^{8} - 817 \nu^{7} - 1672 \nu^{6} + \cdots + 191488 ) / 15360 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 103 \nu^{11} - 280 \nu^{10} + 865 \nu^{9} - 2168 \nu^{8} + 1955 \nu^{7} + 5140 \nu^{6} + \cdots - 494592 ) / 15360 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 121 \nu^{11} + 192 \nu^{10} - 479 \nu^{9} + 1392 \nu^{8} - 2333 \nu^{7} - 2996 \nu^{6} + \cdots + 370688 ) / 15360 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 23 \nu^{11} - 60 \nu^{10} + 81 \nu^{9} - 116 \nu^{8} + 51 \nu^{7} + 480 \nu^{6} - 684 \nu^{5} + \cdots - 30720 ) / 3072 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 63 \nu^{11} + 284 \nu^{10} - 569 \nu^{9} + 788 \nu^{8} - 427 \nu^{7} - 1952 \nu^{6} + \cdots + 188928 ) / 7680 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 21 \nu^{11} - 61 \nu^{10} + 23 \nu^{9} - 107 \nu^{8} + 165 \nu^{7} + 563 \nu^{6} - 1556 \nu^{5} + \cdots - 45184 ) / 1920 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 217 \nu^{11} + 420 \nu^{10} - 511 \nu^{9} + 1580 \nu^{8} - 1981 \nu^{7} - 2560 \nu^{6} + \cdots + 418816 ) / 15360 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{11} - \beta_{10} - \beta_{9} - 4\beta_{8} + 5\beta_{7} + 2\beta_{6} - 4\beta_{5} - \beta_{2} + 7\beta _1 + 15 ) / 40 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2 \beta_{11} - \beta_{10} - 3 \beta_{9} + 2 \beta_{8} + 5 \beta_{7} - 4 \beta_{6} - 4 \beta_{5} + \cdots + 5 ) / 40 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{11} - 4 \beta_{9} - 4 \beta_{8} + 3 \beta_{6} + 5 \beta_{5} - 8 \beta_{4} + 8 \beta_{3} + \cdots + 26 ) / 20 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2 \beta_{11} - \beta_{10} + 3 \beta_{9} + 10 \beta_{8} + 3 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + \cdots + 9 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 12 \beta_{11} - 53 \beta_{10} - 9 \beta_{9} + 84 \beta_{8} - 35 \beta_{7} + 8 \beta_{6} + 78 \beta_{5} + \cdots - 401 ) / 40 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 84 \beta_{11} + 4 \beta_{10} + 15 \beta_{9} + 104 \beta_{8} - 65 \beta_{7} + 15 \beta_{6} + 11 \beta_{5} + \cdots - 94 ) / 20 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 22 \beta_{11} - 53 \beta_{10} + 87 \beta_{9} - 124 \beta_{8} - 275 \beta_{7} + 66 \beta_{6} + \cdots + 2963 ) / 40 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 2 \beta_{11} - 39 \beta_{10} + 11 \beta_{9} - 90 \beta_{8} + 43 \beta_{7} + 36 \beta_{6} - 276 \beta_{5} + \cdots - 389 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 187 \beta_{11} - 264 \beta_{10} - 216 \beta_{9} - 212 \beta_{8} - 180 \beta_{7} - 163 \beta_{6} + \cdots + 422 ) / 20 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 342 \beta_{11} - 275 \beta_{10} - 439 \beta_{9} - 3722 \beta_{8} - 335 \beta_{7} + 1158 \beta_{6} + \cdots + 4883 ) / 40 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 1372 \beta_{11} - 1211 \beta_{10} + 1665 \beta_{9} + 2388 \beta_{8} + 2035 \beta_{7} - 440 \beta_{6} + \cdots + 15929 ) / 40 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
21.1
−1.86176 − 0.730647i
−1.86176 + 0.730647i
−0.650488 + 1.89126i
−0.650488 − 1.89126i
1.23537 + 1.57285i
1.23537 − 1.57285i
−0.428316 + 1.95360i
−0.428316 − 1.95360i
1.98839 − 0.215211i
1.98839 + 0.215211i
1.71681 − 1.02595i
1.71681 + 1.02595i
−2.59241 − 1.13111i − 6.25785i 5.44116 + 5.86462i − 5.00000i −7.07834 + 16.2229i −34.6280 −7.47214 − 21.3581i −12.1606 −5.65557 + 12.9620i
21.2 −2.59241 + 1.13111i 6.25785i 5.44116 − 5.86462i 5.00000i −7.07834 − 16.2229i −34.6280 −7.47214 + 21.3581i −12.1606 −5.65557 − 12.9620i
21.3 −2.54175 − 1.24077i 0.888401i 4.92097 + 6.30746i 5.00000i 1.10230 − 2.25809i 26.6173 −4.68175 − 22.1378i 26.2107 6.20386 − 12.7087i
21.4 −2.54175 + 1.24077i − 0.888401i 4.92097 − 6.30746i − 5.00000i 1.10230 + 2.25809i 26.6173 −4.68175 + 22.1378i 26.2107 6.20386 + 12.7087i
21.5 −0.337480 − 2.80822i − 7.99849i −7.77221 + 1.89544i 5.00000i −22.4615 + 2.69933i 9.93501 7.94578 + 21.1864i −36.9759 14.0411 − 1.68740i
21.6 −0.337480 + 2.80822i 7.99849i −7.77221 − 1.89544i − 5.00000i −22.4615 − 2.69933i 9.93501 7.94578 − 21.1864i −36.9759 14.0411 + 1.68740i
21.7 1.52528 − 2.38191i − 1.51777i −3.34703 − 7.26618i − 5.00000i −3.61520 − 2.31503i 5.13620 −22.4126 − 3.11063i 24.6964 −11.9096 − 7.62641i
21.8 1.52528 + 2.38191i 1.51777i −3.34703 + 7.26618i 5.00000i −3.61520 + 2.31503i 5.13620 −22.4126 + 3.11063i 24.6964 −11.9096 + 7.62641i
21.9 2.20360 − 1.77318i 9.57890i 1.71169 − 7.81474i 5.00000i 16.9851 + 21.1081i 21.5703 −10.0850 − 20.2557i −64.7554 8.86588 + 11.0180i
21.10 2.20360 + 1.77318i − 9.57890i 1.71169 + 7.81474i − 5.00000i 16.9851 − 21.1081i 21.5703 −10.0850 + 20.2557i −64.7554 8.86588 − 11.0180i
21.11 2.74276 − 0.690860i − 4.24443i 7.04543 − 3.78972i 5.00000i −2.93231 − 11.6414i −14.6308 16.7057 − 15.2617i 8.98481 3.45430 + 13.7138i
21.12 2.74276 + 0.690860i 4.24443i 7.04543 + 3.78972i − 5.00000i −2.93231 + 11.6414i −14.6308 16.7057 + 15.2617i 8.98481 3.45430 − 13.7138i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 21.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 40.4.d.a ✓ 12
3.b odd 2 1 360.4.k.c 12
4.b odd 2 1 160.4.d.a 12
5.b even 2 1 200.4.d.b 12
5.c odd 4 1 200.4.f.b 12
5.c odd 4 1 200.4.f.c 12
8.b even 2 1 inner 40.4.d.a ✓ 12
8.d odd 2 1 160.4.d.a 12
12.b even 2 1 1440.4.k.c 12
16.e even 4 1 1280.4.a.bb 6
16.e even 4 1 1280.4.a.bc 6
16.f odd 4 1 1280.4.a.ba 6
16.f odd 4 1 1280.4.a.bd 6
20.d odd 2 1 800.4.d.d 12
20.e even 4 1 800.4.f.b 12
20.e even 4 1 800.4.f.c 12
24.f even 2 1 1440.4.k.c 12
24.h odd 2 1 360.4.k.c 12
40.e odd 2 1 800.4.d.d 12
40.f even 2 1 200.4.d.b 12
40.i odd 4 1 200.4.f.b 12
40.i odd 4 1 200.4.f.c 12
40.k even 4 1 800.4.f.b 12
40.k even 4 1 800.4.f.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.4.d.a ✓ 12 1.a even 1 1 trivial
40.4.d.a ✓ 12 8.b even 2 1 inner
160.4.d.a 12 4.b odd 2 1
160.4.d.a 12 8.d odd 2 1
200.4.d.b 12 5.b even 2 1
200.4.d.b 12 40.f even 2 1
200.4.f.b 12 5.c odd 4 1
200.4.f.b 12 40.i odd 4 1
200.4.f.c 12 5.c odd 4 1
200.4.f.c 12 40.i odd 4 1
360.4.k.c 12 3.b odd 2 1
360.4.k.c 12 24.h odd 2 1
800.4.d.d 12 20.d odd 2 1
800.4.d.d 12 40.e odd 2 1
800.4.f.b 12 20.e even 4 1
800.4.f.b 12 40.k even 4 1
800.4.f.c 12 20.e even 4 1
800.4.f.c 12 40.k even 4 1
1280.4.a.ba 6 16.f odd 4 1
1280.4.a.bb 6 16.e even 4 1
1280.4.a.bc 6 16.e even 4 1
1280.4.a.bd 6 16.f odd 4 1
1440.4.k.c 12 12.b even 2 1
1440.4.k.c 12 24.f even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(40, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 2 T^{11} + \cdots + 262144 \) Copy content Toggle raw display
$3$ \( T^{12} + 216 T^{10} + \cdots + 7529536 \) Copy content Toggle raw display
$5$ \( (T^{2} + 25)^{6} \) Copy content Toggle raw display
$7$ \( (T^{6} - 14 T^{5} + \cdots + 14843128)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{6} - 14500 T^{4} + \cdots - 7473839808)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 17\!\cdots\!24 \) Copy content Toggle raw display
$23$ \( (T^{6} - 302 T^{5} + \cdots + 881168216)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots - 1437816300032)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 21\!\cdots\!04 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 71667547865600)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 25\!\cdots\!04 \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots - 72048375466472)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 86\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 61\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 87\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 36\!\cdots\!48)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots - 378730163491776)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots - 44\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 78\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots - 62\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots - 33\!\cdots\!28)^{2} \) Copy content Toggle raw display
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