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Properties

Label 20.9.b.a
Level $20$
Weight $9$
Character orbit 20.b
Analytic conductor $8.148$
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] = [20,9,Mod(11,20)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(20, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("20.11");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 20.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: \(8.14757220122\)
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: \( x^{16} - 5 x^{15} + 26 x^{14} - 834 x^{13} + 4390 x^{12} - 61783 x^{11} + 466168 x^{10} + \cdots + 206161212459445 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{58}\cdot 5^{16} \)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{3} + \beta_1) q^{3} + (\beta_{2} - 3) q^{4} + (\beta_{4} - 2 \beta_1 - 1) q^{5} + (\beta_{5} - \beta_{4} + 5 \beta_{3} + \cdots + 271) q^{6}+ \cdots + ( - \beta_{14} - \beta_{11} + \cdots - 2374) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{3} + \beta_1) q^{3} + (\beta_{2} - 3) q^{4} + (\beta_{4} - 2 \beta_1 - 1) q^{5} + (\beta_{5} - \beta_{4} + 5 \beta_{3} + \cdots + 271) q^{6}+ \cdots + ( - 1088 \beta_{15} + 1626 \beta_{14} + \cdots - 445413) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 6 q^{2} - 52 q^{4} + 4368 q^{6} - 14184 q^{8} - 38800 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 6 q^{2} - 52 q^{4} + 4368 q^{6} - 14184 q^{8} - 38800 q^{9} + 8750 q^{10} - 64040 q^{12} + 51392 q^{13} + 68472 q^{14} - 81424 q^{16} + 27552 q^{17} - 616994 q^{18} + 172500 q^{20} + 414496 q^{21} - 389120 q^{22} + 163792 q^{24} + 1250000 q^{25} + 1037124 q^{26} + 1288520 q^{28} + 2764896 q^{29} - 805000 q^{30} - 4379904 q^{32} - 5521600 q^{33} + 3793964 q^{34} - 5468916 q^{36} + 9009472 q^{37} - 3087360 q^{38} + 385000 q^{40} - 8576448 q^{41} - 4067400 q^{42} + 16921200 q^{44} + 1580000 q^{45} - 7974152 q^{46} - 2696640 q^{48} - 32803600 q^{49} + 468750 q^{50} + 6679352 q^{52} + 2452032 q^{53} + 8898704 q^{54} + 34134768 q^{56} + 11957760 q^{57} + 52156572 q^{58} - 24185000 q^{60} + 8371712 q^{61} + 1290000 q^{62} - 47543872 q^{64} + 9060000 q^{65} + 19358000 q^{66} + 16095192 q^{68} + 7527264 q^{69} + 10500000 q^{70} - 42242664 q^{72} + 61907232 q^{73} - 138210876 q^{74} + 2570400 q^{76} - 156997440 q^{77} - 104032400 q^{78} + 37590000 q^{80} + 140586672 q^{81} + 83921012 q^{82} + 69761824 q^{84} - 106960000 q^{85} - 101724672 q^{86} + 44728480 q^{88} + 106647456 q^{89} + 32613750 q^{90} - 13876200 q^{92} + 105563840 q^{93} + 55264632 q^{94} - 453389952 q^{96} + 171851232 q^{97} - 285387714 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 5 x^{15} + 26 x^{14} - 834 x^{13} + 4390 x^{12} - 61783 x^{11} + 466168 x^{10} + \cdots + 206161212459445 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 71\!\cdots\!16 \nu^{15} + \cdots - 32\!\cdots\!90 ) / 13\!\cdots\!65 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 19\!\cdots\!64 \nu^{15} + \cdots + 74\!\cdots\!15 ) / 13\!\cdots\!65 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 48\!\cdots\!09 \nu^{15} + \cdots + 18\!\cdots\!35 ) / 21\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 87\!\cdots\!68 \nu^{15} + \cdots + 22\!\cdots\!10 ) / 13\!\cdots\!65 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 43\!\cdots\!91 \nu^{15} + \cdots + 14\!\cdots\!85 ) / 21\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 25\!\cdots\!87 \nu^{15} + \cdots - 35\!\cdots\!85 ) / 10\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 53\!\cdots\!87 \nu^{15} + \cdots + 37\!\cdots\!35 ) / 21\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 11\!\cdots\!73 \nu^{15} + \cdots - 97\!\cdots\!15 ) / 21\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 13\!\cdots\!67 \nu^{15} + \cdots + 14\!\cdots\!15 ) / 21\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 14\!\cdots\!15 \nu^{15} + \cdots + 25\!\cdots\!75 ) / 21\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 17\!\cdots\!97 \nu^{15} + \cdots - 77\!\cdots\!95 ) / 21\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 30\!\cdots\!06 \nu^{15} + \cdots - 14\!\cdots\!85 ) / 36\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 31\!\cdots\!53 \nu^{15} + \cdots + 10\!\cdots\!65 ) / 21\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 34\!\cdots\!43 \nu^{15} + \cdots - 78\!\cdots\!75 ) / 21\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 47\!\cdots\!67 \nu^{15} + \cdots - 12\!\cdots\!95 ) / 21\!\cdots\!80 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} + 123\beta _1 + 124 ) / 250 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{14} + 2\beta_{10} + 2\beta_{4} - 8\beta_{3} + 121\beta_{2} + 244\beta _1 - 719 ) / 500 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 15 \beta_{15} + 12 \beta_{14} + 15 \beta_{13} + 45 \beta_{12} + 15 \beta_{11} - 3 \beta_{10} + \cdots + 139713 ) / 1000 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 125 \beta_{15} - 137 \beta_{14} + 1510 \beta_{13} + 410 \beta_{12} + 765 \beta_{11} + \cdots - 130022 ) / 2000 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 375 \beta_{15} + 623 \beta_{14} + 700 \beta_{13} + 940 \beta_{12} + 1065 \beta_{11} + 2033 \beta_{10} + \cdots + 2485451 ) / 200 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 4755 \beta_{15} + 20720 \beta_{14} + 14515 \beta_{13} + 49705 \beta_{12} + 19165 \beta_{11} + \cdots + 34662356 ) / 1000 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 13715 \beta_{15} + 197313 \beta_{14} + 797250 \beta_{13} + 195550 \beta_{12} + 510675 \beta_{11} + \cdots - 1215185396 ) / 2000 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 899210 \beta_{15} - 681918 \beta_{14} + 3012380 \beta_{13} + 3162500 \beta_{12} + 3531030 \beta_{11} + \cdots - 6968504837 ) / 1000 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 11919945 \beta_{15} - 11166951 \beta_{14} + 23725000 \beta_{13} + 20836320 \beta_{12} + \cdots - 69182012056 ) / 1000 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 22322485 \beta_{15} - 5897839 \beta_{14} + 52435630 \beta_{13} + 32305090 \beta_{12} + \cdots - 28755514013 ) / 200 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 1147601620 \beta_{15} - 1613309315 \beta_{14} + 1334290815 \beta_{13} + 2779943605 \beta_{12} + \cdots - 6866702025759 ) / 1000 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 22323988735 \beta_{15} - 29642977867 \beta_{14} + 6550653720 \beta_{13} - 3130372960 \beta_{12} + \cdots - 199006383104476 ) / 2000 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 136002850625 \beta_{15} - 360807053871 \beta_{14} + 26931144990 \beta_{13} - 66284344590 \beta_{12} + \cdots - 11\!\cdots\!14 ) / 2000 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 690002435965 \beta_{15} - 2595594352266 \beta_{14} - 195256153965 \beta_{13} - 89071665735 \beta_{12} + \cdots - 73\!\cdots\!56 ) / 1000 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 3179941561745 \beta_{15} - 5554539006727 \beta_{14} - 3785987666310 \beta_{13} - 4667770832730 \beta_{12} + \cdots - 22\!\cdots\!84 ) / 400 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/20\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(17\)
\(\chi(n)\) \(-1\) \(1\)

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
9.29581 + 2.24761i
9.29581 − 2.24761i
6.86496 + 2.82928i
6.86496 − 2.82928i
3.05707 + 7.10588i
3.05707 − 7.10588i
2.32463 + 7.96873i
2.32463 − 7.96873i
−2.93811 + 7.65619i
−2.93811 − 7.65619i
−6.29294 + 5.63875i
−6.29294 − 5.63875i
−4.16577 + 5.52698i
−4.16577 − 5.52698i
−5.64565 + 3.35245i
−5.64565 − 3.35245i
−15.3556 − 4.49522i 98.1237i 215.586 + 138.053i 279.508 441.088 − 1506.74i 820.952i −2689.86 − 3088.99i −3067.27 −4292.01 − 1256.45i
11.2 −15.3556 + 4.49522i − 98.1237i 215.586 − 138.053i 279.508 441.088 + 1506.74i − 820.952i −2689.86 + 3088.99i −3067.27 −4292.01 + 1256.45i
11.3 −14.9660 − 5.65855i − 25.1248i 191.962 + 169.372i −279.508 −142.170 + 376.017i 2973.76i −1914.50 − 3621.04i 5929.74 4183.12 + 1581.61i
11.4 −14.9660 + 5.65855i 25.1248i 191.962 − 169.372i −279.508 −142.170 − 376.017i − 2973.76i −1914.50 + 3621.04i 5929.74 4183.12 − 1581.61i
11.5 −7.35022 − 14.2118i 110.171i −147.949 + 208.919i −279.508 1565.72 − 809.778i − 3540.70i 4056.56 + 567.011i −5576.56 2054.45 + 3972.31i
11.6 −7.35022 + 14.2118i − 110.171i −147.949 − 208.919i −279.508 1565.72 + 809.778i 3540.70i 4056.56 − 567.011i −5576.56 2054.45 − 3972.31i
11.7 −1.41320 − 15.9375i 39.9624i −252.006 + 45.0455i 279.508 636.899 − 56.4746i 2633.20i 1074.04 + 3952.68i 4964.01 −395.000 − 4454.66i
11.8 −1.41320 + 15.9375i − 39.9624i −252.006 − 45.0455i 279.508 636.899 + 56.4746i − 2633.20i 1074.04 − 3952.68i 4964.01 −395.000 + 4454.66i
11.9 4.64016 − 15.3124i − 75.7492i −212.938 − 142.104i −279.508 −1159.90 − 351.488i − 210.345i −3164.01 + 2601.20i 823.060 −1296.96 + 4279.94i
11.10 4.64016 + 15.3124i 75.7492i −212.938 + 142.104i −279.508 −1159.90 + 351.488i 210.345i −3164.01 − 2601.20i 823.060 −1296.96 − 4279.94i
11.11 11.3498 − 11.2775i 137.297i 1.63618 − 255.995i −279.508 1548.37 + 1558.29i 3940.57i −2868.41 − 2923.94i −12289.4 −3172.37 + 3152.16i
11.12 11.3498 + 11.2775i − 137.297i 1.63618 + 255.995i −279.508 1548.37 − 1558.29i − 3940.57i −2868.41 + 2923.94i −12289.4 −3172.37 − 3152.16i
11.13 11.5676 − 11.0540i 27.2434i 11.6196 − 255.736i 279.508 301.148 + 315.141i − 3325.58i −2692.49 − 3086.70i 5818.80 3233.25 − 3089.68i
11.14 11.5676 + 11.0540i − 27.2434i 11.6196 + 255.736i 279.508 301.148 − 315.141i 3325.58i −2692.49 + 3086.70i 5818.80 3233.25 + 3089.68i
11.15 14.5274 − 6.70489i − 150.211i 166.089 − 194.809i 279.508 −1007.15 − 2182.17i 2626.96i 1106.66 − 3943.67i −16002.3 4060.52 − 1874.07i
11.16 14.5274 + 6.70489i 150.211i 166.089 + 194.809i 279.508 −1007.15 + 2182.17i − 2626.96i 1106.66 + 3943.67i −16002.3 4060.52 + 1874.07i
\(n\): 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
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 20.9.b.a ✓ 16
3.b odd 2 1 180.9.c.a 16
4.b odd 2 1 inner 20.9.b.a ✓ 16
5.b even 2 1 100.9.b.d 16
5.c odd 4 2 100.9.d.c 32
8.b even 2 1 320.9.b.d 16
8.d odd 2 1 320.9.b.d 16
12.b even 2 1 180.9.c.a 16
20.d odd 2 1 100.9.b.d 16
20.e even 4 2 100.9.d.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.9.b.a ✓ 16 1.a even 1 1 trivial
20.9.b.a ✓ 16 4.b odd 2 1 inner
100.9.b.d 16 5.b even 2 1
100.9.b.d 16 20.d odd 2 1
100.9.d.c 32 5.c odd 4 2
100.9.d.c 32 20.e even 4 2
180.9.c.a 16 3.b odd 2 1
180.9.c.a 16 12.b even 2 1
320.9.b.d 16 8.b even 2 1
320.9.b.d 16 8.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(20, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + \cdots + 18\!\cdots\!16 \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( (T^{2} - 78125)^{8} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 69\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 36\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 92\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots - 61\!\cdots\!04)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 81\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots - 37\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots - 56\!\cdots\!64)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots - 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 60\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 83\!\cdots\!76)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 75\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots - 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots - 36\!\cdots\!24)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 40\!\cdots\!00)^{2} \) Copy content Toggle raw display
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