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Properties

Label 930.4.d.c
Level $930$
Weight $4$
Character orbit 930.d
Analytic conductor $54.872$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [930,4,Mod(559,930)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(930, 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("930.559");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 930.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: \(54.8717763053\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 2763 x^{18} + 2652899 x^{16} + 1161420105 x^{14} + 247831438280 x^{12} + 26461073949176 x^{10} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \beta_{8} q^{2} - 3 \beta_{8} q^{3} - 4 q^{4} - \beta_{16} q^{5} + 6 q^{6} + ( - \beta_{8} + \beta_{7}) q^{7} - 8 \beta_{8} q^{8} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta_{8} q^{2} - 3 \beta_{8} q^{3} - 4 q^{4} - \beta_{16} q^{5} + 6 q^{6} + ( - \beta_{8} + \beta_{7}) q^{7} - 8 \beta_{8} q^{8} - 9 q^{9} + 2 \beta_{10} q^{10} + (\beta_{16} + \beta_{15} + \beta_{11} + \cdots - 5) q^{11}+ \cdots + ( - 9 \beta_{16} - 9 \beta_{15} + \cdots + 45) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 80 q^{4} - 2 q^{5} + 120 q^{6} - 180 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 80 q^{4} - 2 q^{5} + 120 q^{6} - 180 q^{9} + 8 q^{10} - 114 q^{11} + 52 q^{14} - 12 q^{15} + 320 q^{16} + 370 q^{19} + 8 q^{20} - 78 q^{21} - 480 q^{24} - 90 q^{25} - 368 q^{26} + 368 q^{29} - 12 q^{30} - 620 q^{31} + 712 q^{34} + 374 q^{35} + 720 q^{36} + 552 q^{39} - 32 q^{40} - 872 q^{41} + 456 q^{44} + 18 q^{45} - 1236 q^{46} + 1334 q^{49} + 416 q^{50} - 1068 q^{51} - 1080 q^{54} - 1290 q^{55} - 208 q^{56} + 3228 q^{59} + 48 q^{60} - 2604 q^{61} - 1280 q^{64} + 44 q^{65} - 684 q^{66} + 1854 q^{69} - 852 q^{70} - 2290 q^{71} + 2008 q^{74} - 624 q^{75} - 1480 q^{76} + 4342 q^{79} - 32 q^{80} + 1620 q^{81} + 312 q^{84} + 500 q^{85} - 4 q^{86} + 1390 q^{89} - 72 q^{90} - 5744 q^{91} + 2608 q^{94} - 1136 q^{95} + 1920 q^{96} + 1026 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 2763 x^{18} + 2652899 x^{16} + 1161420105 x^{14} + 247831438280 x^{12} + 26461073949176 x^{10} + \cdots + 34\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 54\!\cdots\!17 \nu^{18} + \cdots - 28\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 36\!\cdots\!79 \nu^{18} + \cdots + 18\!\cdots\!00 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 38\!\cdots\!83 \nu^{18} + \cdots - 21\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 96\!\cdots\!73 \nu^{18} + \cdots - 60\!\cdots\!00 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 31\!\cdots\!71 \nu^{18} + \cdots - 17\!\cdots\!00 ) / 69\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 33\!\cdots\!72 \nu^{18} + \cdots - 18\!\cdots\!00 ) / 69\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 15\!\cdots\!89 \nu^{19} + \cdots - 76\!\cdots\!00 \nu ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 15\!\cdots\!89 \nu^{19} + \cdots - 78\!\cdots\!00 \nu ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 12\!\cdots\!67 \nu^{19} + \cdots - 57\!\cdots\!00 \nu ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 11\!\cdots\!87 \nu^{19} + \cdots - 21\!\cdots\!00 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 11\!\cdots\!87 \nu^{19} + \cdots + 21\!\cdots\!00 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 40\!\cdots\!49 \nu^{19} + \cdots + 16\!\cdots\!00 ) / 82\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 40\!\cdots\!49 \nu^{19} + \cdots + 16\!\cdots\!00 ) / 82\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 11\!\cdots\!77 \nu^{19} + \cdots - 54\!\cdots\!00 \nu ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 10\!\cdots\!59 \nu^{19} + \cdots + 15\!\cdots\!00 ) / 82\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 10\!\cdots\!59 \nu^{19} + \cdots + 15\!\cdots\!00 ) / 82\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 25\!\cdots\!19 \nu^{19} + \cdots + 14\!\cdots\!00 \nu ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 69\!\cdots\!31 \nu^{19} + \cdots + 42\!\cdots\!00 \nu ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 21\!\cdots\!83 \nu^{19} + \cdots - 11\!\cdots\!00 \nu ) / 82\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( -\beta_{8} + \beta_{7} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( - 18 \beta_{16} - 18 \beta_{15} + 4 \beta_{11} - 4 \beta_{10} + 12 \beta_{6} - 4 \beta_{5} - 4 \beta_{4} + \cdots - 282 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 235 \beta_{19} - 148 \beta_{18} - 241 \beta_{17} + 27 \beta_{16} - 27 \beta_{15} + \cdots - 824 \beta_{7} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 24476 \beta_{16} + 24476 \beta_{15} + 2043 \beta_{13} + 2043 \beta_{12} - 4354 \beta_{11} + \cdots + 235134 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 445876 \beta_{19} + 191736 \beta_{18} + 322326 \beta_{17} - 90191 \beta_{16} + 90191 \beta_{15} + \cdots + 809791 \beta_{7} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 31357260 \beta_{16} - 31357260 \beta_{15} - 3624288 \beta_{13} - 3624288 \beta_{12} + \cdots - 259162202 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 659861794 \beta_{19} - 249280818 \beta_{18} - 419918317 \beta_{17} + 144540755 \beta_{16} + \cdots - 932167435 \beta_{7} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 40729598390 \beta_{16} + 40729598390 \beta_{15} + 5268444486 \beta_{13} + 5268444486 \beta_{12} + \cdots + 318245219188 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 908554023504 \beta_{19} + 328215539042 \beta_{18} + 550284205115 \beta_{17} + \cdots + 1162332928479 \beta_{7} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 53416120942924 \beta_{16} - 53416120942924 \beta_{15} - 7215915884820 \beta_{13} + \cdots - 408486270446774 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 12\!\cdots\!22 \beta_{19} - 433523440562956 \beta_{18} - 724371154543925 \beta_{17} + \cdots - 15\!\cdots\!15 \beta_{7} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 70\!\cdots\!16 \beta_{16} + \cdots + 53\!\cdots\!38 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 16\!\cdots\!06 \beta_{19} + \cdots + 19\!\cdots\!67 \beta_{7} \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 92\!\cdots\!76 \beta_{16} + \cdots - 70\!\cdots\!46 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 21\!\cdots\!26 \beta_{19} + \cdots - 25\!\cdots\!19 \beta_{7} \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 12\!\cdots\!68 \beta_{16} + \cdots + 92\!\cdots\!54 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 28\!\cdots\!10 \beta_{19} + \cdots + 34\!\cdots\!07 \beta_{7} \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 16\!\cdots\!24 \beta_{16} + \cdots - 12\!\cdots\!86 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 37\!\cdots\!22 \beta_{19} + \cdots - 45\!\cdots\!23 \beta_{7} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/930\mathbb{Z}\right)^\times\).

\(n\) \(187\) \(311\) \(871\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
559.1
− 22.8518i
36.3525i
7.62904i
− 2.24994i
− 2.45273i
− 7.05659i
2.56665i
9.07744i
− 22.4343i
14.4198i
22.8518i
− 36.3525i
− 7.62904i
2.24994i
2.45273i
7.05659i
− 2.56665i
− 9.07744i
22.4343i
− 14.4198i
− 2.00000i 3.00000i −4.00000 −11.0046 + 1.97458i 6.00000 − 22.8518i 8.00000i −9.00000 3.94915 + 22.0092i
559.2 − 2.00000i 3.00000i −4.00000 −10.3430 − 4.24534i 6.00000 36.3525i 8.00000i −9.00000 −8.49067 + 20.6859i
559.3 − 2.00000i 3.00000i −4.00000 −7.42511 + 8.35869i 6.00000 7.62904i 8.00000i −9.00000 16.7174 + 14.8502i
559.4 − 2.00000i 3.00000i −4.00000 −6.23400 − 9.28102i 6.00000 − 2.24994i 8.00000i −9.00000 −18.5620 + 12.4680i
559.5 − 2.00000i 3.00000i −4.00000 1.65380 + 11.0573i 6.00000 − 2.45273i 8.00000i −9.00000 22.1147 − 3.30759i
559.6 − 2.00000i 3.00000i −4.00000 2.19955 − 10.9618i 6.00000 − 7.05659i 8.00000i −9.00000 −21.9237 − 4.39910i
559.7 − 2.00000i 3.00000i −4.00000 3.61266 + 10.5806i 6.00000 2.56665i 8.00000i −9.00000 21.1612 − 7.22532i
559.8 − 2.00000i 3.00000i −4.00000 4.77279 − 10.1104i 6.00000 9.07744i 8.00000i −9.00000 −20.2208 − 9.54558i
559.9 − 2.00000i 3.00000i −4.00000 10.6574 + 3.37922i 6.00000 − 22.4343i 8.00000i −9.00000 6.75844 − 21.3149i
559.10 − 2.00000i 3.00000i −4.00000 11.1104 + 1.24819i 6.00000 14.4198i 8.00000i −9.00000 2.49638 − 22.2209i
559.11 2.00000i − 3.00000i −4.00000 −11.0046 − 1.97458i 6.00000 22.8518i − 8.00000i −9.00000 3.94915 − 22.0092i
559.12 2.00000i − 3.00000i −4.00000 −10.3430 + 4.24534i 6.00000 − 36.3525i − 8.00000i −9.00000 −8.49067 − 20.6859i
559.13 2.00000i − 3.00000i −4.00000 −7.42511 − 8.35869i 6.00000 − 7.62904i − 8.00000i −9.00000 16.7174 − 14.8502i
559.14 2.00000i − 3.00000i −4.00000 −6.23400 + 9.28102i 6.00000 2.24994i − 8.00000i −9.00000 −18.5620 − 12.4680i
559.15 2.00000i − 3.00000i −4.00000 1.65380 − 11.0573i 6.00000 2.45273i − 8.00000i −9.00000 22.1147 + 3.30759i
559.16 2.00000i − 3.00000i −4.00000 2.19955 + 10.9618i 6.00000 7.05659i − 8.00000i −9.00000 −21.9237 + 4.39910i
559.17 2.00000i − 3.00000i −4.00000 3.61266 − 10.5806i 6.00000 − 2.56665i − 8.00000i −9.00000 21.1612 + 7.22532i
559.18 2.00000i − 3.00000i −4.00000 4.77279 + 10.1104i 6.00000 − 9.07744i − 8.00000i −9.00000 −20.2208 + 9.54558i
559.19 2.00000i − 3.00000i −4.00000 10.6574 − 3.37922i 6.00000 22.4343i − 8.00000i −9.00000 6.75844 + 21.3149i
559.20 2.00000i − 3.00000i −4.00000 11.1104 − 1.24819i 6.00000 − 14.4198i − 8.00000i −9.00000 2.49638 + 22.2209i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 559.20
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.4.d.c ✓ 20
5.b even 2 1 inner 930.4.d.c ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.4.d.c ✓ 20 1.a even 1 1 trivial
930.4.d.c ✓ 20 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{20} + 2763 T_{7}^{18} + 2652899 T_{7}^{16} + 1161420105 T_{7}^{14} + 247831438280 T_{7}^{12} + \cdots + 34\!\cdots\!00 \) acting on \(S_{4}^{\mathrm{new}}(930, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 4)^{10} \) Copy content Toggle raw display
$3$ \( (T^{2} + 9)^{10} \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 93\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( (T^{10} + \cdots - 156520544587776)^{2} \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 39\!\cdots\!36 \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots - 23\!\cdots\!68)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 75\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots + 14\!\cdots\!56)^{2} \) Copy content Toggle raw display
$31$ \( (T + 31)^{20} \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 98\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots - 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 12\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots - 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots - 54\!\cdots\!96)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 87\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots - 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 17\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots - 72\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots - 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
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