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Properties

Label 230.3.f.a
Level $230$
Weight $3$
Character orbit 230.f
Analytic conductor $6.267$
Analytic rank $0$
Dimension $20$
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] = [230,3,Mod(47,230)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(230, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("230.47");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 230.f (of order \(4\), degree \(2\), 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: \(6.26704608029\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 52 x^{17} + 1020 x^{16} - 1316 x^{15} + 1352 x^{14} - 18724 x^{13} + 250686 x^{12} + \cdots + 88804 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{8} - 1) q^{2} + \beta_1 q^{3} + 2 \beta_{8} q^{4} - \beta_{7} q^{5} + (\beta_{2} - \beta_1) q^{6} - \beta_{12} q^{7} + ( - 2 \beta_{8} + 2) q^{8} + (\beta_{14} + \beta_{13} + \cdots - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{8} - 1) q^{2} + \beta_1 q^{3} + 2 \beta_{8} q^{4} - \beta_{7} q^{5} + (\beta_{2} - \beta_1) q^{6} - \beta_{12} q^{7} + ( - 2 \beta_{8} + 2) q^{8} + (\beta_{14} + \beta_{13} + \cdots - \beta_1) q^{9}+ \cdots + ( - 4 \beta_{19} + 8 \beta_{18} + \cdots - 12 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 20 q^{2} + 4 q^{5} + 8 q^{7} + 40 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 20 q^{2} + 4 q^{5} + 8 q^{7} + 40 q^{8} + 4 q^{10} + 56 q^{11} - 4 q^{13} - 48 q^{15} - 80 q^{16} - 72 q^{17} - 28 q^{18} - 16 q^{20} + 8 q^{21} - 56 q^{22} + 36 q^{25} + 8 q^{26} + 156 q^{27} - 16 q^{28} + 84 q^{30} - 212 q^{31} + 80 q^{32} - 100 q^{33} + 56 q^{36} + 72 q^{37} + 88 q^{38} + 24 q^{40} - 12 q^{41} - 8 q^{42} + 120 q^{43} - 32 q^{45} + 8 q^{47} - 28 q^{50} + 64 q^{51} - 8 q^{52} - 244 q^{53} + 68 q^{55} + 32 q^{56} - 384 q^{57} - 188 q^{58} - 72 q^{60} + 328 q^{61} + 212 q^{62} + 172 q^{63} + 20 q^{65} + 200 q^{66} + 56 q^{67} + 144 q^{68} - 28 q^{70} - 92 q^{71} - 56 q^{72} + 144 q^{73} - 124 q^{75} - 176 q^{76} + 292 q^{77} - 208 q^{78} - 16 q^{80} - 84 q^{81} + 12 q^{82} - 72 q^{83} - 20 q^{85} - 240 q^{86} - 208 q^{87} + 112 q^{88} - 56 q^{90} - 192 q^{91} + 256 q^{93} - 96 q^{95} - 276 q^{97} + 104 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 52 x^{17} + 1020 x^{16} - 1316 x^{15} + 1352 x^{14} - 18724 x^{13} + 250686 x^{12} + \cdots + 88804 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 88\!\cdots\!61 \nu^{19} + \cdots + 10\!\cdots\!08 ) / 87\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 18\!\cdots\!24 \nu^{19} + \cdots - 22\!\cdots\!44 ) / 51\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 65\!\cdots\!41 \nu^{19} + \cdots - 27\!\cdots\!80 ) / 80\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 30\!\cdots\!81 \nu^{19} + \cdots + 97\!\cdots\!60 ) / 35\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 28\!\cdots\!89 \nu^{19} + \cdots + 15\!\cdots\!68 ) / 24\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 62\!\cdots\!91 \nu^{19} + \cdots + 28\!\cdots\!52 ) / 51\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 17\!\cdots\!23 \nu^{19} + \cdots + 12\!\cdots\!24 ) / 13\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 12\!\cdots\!71 \nu^{19} + \cdots - 65\!\cdots\!24 ) / 35\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 97\!\cdots\!51 \nu^{19} + \cdots + 39\!\cdots\!28 ) / 23\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 15\!\cdots\!99 \nu^{19} + \cdots - 47\!\cdots\!84 ) / 35\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 16\!\cdots\!24 \nu^{19} + \cdots + 22\!\cdots\!00 ) / 35\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 57\!\cdots\!61 \nu^{19} + \cdots + 36\!\cdots\!00 ) / 11\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 12\!\cdots\!73 \nu^{19} + \cdots + 15\!\cdots\!16 ) / 23\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 25\!\cdots\!11 \nu^{19} + \cdots - 14\!\cdots\!92 ) / 35\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 26\!\cdots\!94 \nu^{19} + \cdots - 19\!\cdots\!16 ) / 35\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 36\!\cdots\!85 \nu^{19} + \cdots - 21\!\cdots\!68 ) / 35\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 16\!\cdots\!78 \nu^{19} + \cdots + 63\!\cdots\!92 ) / 11\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 23\!\cdots\!81 \nu^{19} + \cdots + 31\!\cdots\!60 ) / 17\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{14} + \beta_{13} - \beta_{10} - 10\beta_{8} - \beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2 \beta_{19} - \beta_{18} - \beta_{17} + 2 \beta_{15} - 4 \beta_{14} - 3 \beta_{13} + 2 \beta_{11} + \cdots + 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 7 \beta_{19} - 6 \beta_{18} + 7 \beta_{16} - 7 \beta_{15} + 33 \beta_{14} - 3 \beta_{12} - \beta_{11} + \cdots - 202 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 30 \beta_{19} + 84 \beta_{18} + 28 \beta_{17} - 88 \beta_{16} + 10 \beta_{15} - 30 \beta_{14} + \cdots + 321 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 255 \beta_{19} - 349 \beta_{18} - 110 \beta_{17} + 244 \beta_{16} + 244 \beta_{15} - 976 \beta_{14} + \cdots + 1116 \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 2937 \beta_{19} + 866 \beta_{18} + 787 \beta_{17} + 470 \beta_{16} - 3066 \beta_{15} + 6737 \beta_{14} + \cdots - 12085 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 13270 \beta_{19} + 8973 \beta_{18} - 12905 \beta_{16} + 12905 \beta_{15} - 34504 \beta_{14} + \cdots + 137518 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 25919 \beta_{19} - 97647 \beta_{18} - 23234 \beta_{17} + 101044 \beta_{16} - 17220 \beta_{15} + \cdots - 424345 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 299718 \beta_{19} + 461512 \beta_{18} + 150960 \beta_{17} - 313546 \beta_{16} - 313546 \beta_{15} + \cdots - 1217859 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 3196192 \beta_{19} - 801439 \beta_{18} - 712957 \beta_{17} - 585768 \beta_{16} + 3285114 \beta_{15} + \cdots + 14393681 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 15480507 \beta_{19} - 9838398 \beta_{18} + 15092379 \beta_{16} - 15092379 \beta_{15} + \cdots - 127905718 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 25322218 \beta_{19} + 104126788 \beta_{18} + 22434272 \beta_{17} - 106565104 \beta_{16} + \cdots + 479184449 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 321015807 \beta_{19} - 511029377 \beta_{18} - 169583230 \beta_{17} + 344231680 \beta_{16} + \cdots + 1312419952 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 3388147513 \beta_{19} + 810535394 \beta_{18} + 716587555 \beta_{17} + 635027174 \beta_{16} + \cdots - 15789324785 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 16745667574 \beta_{19} + 10453501581 \beta_{18} - 16341965065 \beta_{16} + 16341965065 \beta_{15} + \cdots + 130865658022 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 26138406939 \beta_{19} - 110226681007 \beta_{18} - 23084709966 \beta_{17} + 112357677276 \beta_{16} + \cdots - 517330362901 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 340205444706 \beta_{19} + 546816069288 \beta_{18} + 182691867016 \beta_{17} + \cdots - 1399124240059 \beta_1 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 3586311908268 \beta_{19} - 846442183307 \beta_{18} - 747173987357 \beta_{17} - 674413682784 \beta_{16} + \cdots + 16897845076885 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(\beta_{8}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
−4.03398 + 4.03398i
−2.63948 + 2.63948i
−2.41818 + 2.41818i
−0.454110 + 0.454110i
−0.0649756 + 0.0649756i
0.608645 − 0.608645i
1.21198 − 1.21198i
1.94658 − 1.94658i
2.91022 − 2.91022i
2.93330 − 2.93330i
−4.03398 − 4.03398i
−2.63948 − 2.63948i
−2.41818 − 2.41818i
−0.454110 − 0.454110i
−0.0649756 − 0.0649756i
0.608645 + 0.608645i
1.21198 + 1.21198i
1.94658 + 1.94658i
2.91022 + 2.91022i
2.93330 + 2.93330i
−1.00000 − 1.00000i −4.03398 + 4.03398i 2.00000i 4.54274 − 2.08891i 8.06796 −1.88621 − 1.88621i 2.00000 − 2.00000i − 23.5460i −6.63165 − 2.45382i
47.2 −1.00000 − 1.00000i −2.63948 + 2.63948i 2.00000i 1.86266 + 4.64010i 5.27895 5.81530 + 5.81530i 2.00000 − 2.00000i − 4.93366i 2.77744 − 6.50275i
47.3 −1.00000 − 1.00000i −2.41818 + 2.41818i 2.00000i −4.87518 + 1.11024i 4.83636 3.71756 + 3.71756i 2.00000 − 2.00000i − 2.69521i 5.98542 + 3.76494i
47.4 −1.00000 − 1.00000i −0.454110 + 0.454110i 2.00000i 4.61282 − 1.92922i 0.908220 −8.21299 − 8.21299i 2.00000 − 2.00000i 8.58757i −6.54204 − 2.68360i
47.5 −1.00000 − 1.00000i −0.0649756 + 0.0649756i 2.00000i −4.95703 − 0.654141i 0.129951 −3.31044 − 3.31044i 2.00000 − 2.00000i 8.99156i 4.30288 + 5.61117i
47.6 −1.00000 − 1.00000i 0.608645 − 0.608645i 2.00000i 4.88434 + 1.06922i −1.21729 3.48181 + 3.48181i 2.00000 − 2.00000i 8.25910i −3.81512 − 5.95356i
47.7 −1.00000 − 1.00000i 1.21198 − 1.21198i 2.00000i −0.802406 + 4.93519i −2.42397 −5.44701 − 5.44701i 2.00000 − 2.00000i 6.06220i 5.73760 − 4.13279i
47.8 −1.00000 − 1.00000i 1.94658 − 1.94658i 2.00000i 0.283956 − 4.99193i −3.89316 9.76397 + 9.76397i 2.00000 − 2.00000i 1.42166i −5.27589 + 4.70797i
47.9 −1.00000 − 1.00000i 2.91022 − 2.91022i 2.00000i −3.94243 − 3.07527i −5.82043 −1.42062 − 1.42062i 2.00000 − 2.00000i − 7.93871i 0.867158 + 7.01769i
47.10 −1.00000 − 1.00000i 2.93330 − 2.93330i 2.00000i 0.390531 + 4.98473i −5.86660 1.49862 + 1.49862i 2.00000 − 2.00000i − 8.20851i 4.59419 − 5.37526i
93.1 −1.00000 + 1.00000i −4.03398 − 4.03398i − 2.00000i 4.54274 + 2.08891i 8.06796 −1.88621 + 1.88621i 2.00000 + 2.00000i 23.5460i −6.63165 + 2.45382i
93.2 −1.00000 + 1.00000i −2.63948 − 2.63948i − 2.00000i 1.86266 − 4.64010i 5.27895 5.81530 − 5.81530i 2.00000 + 2.00000i 4.93366i 2.77744 + 6.50275i
93.3 −1.00000 + 1.00000i −2.41818 − 2.41818i − 2.00000i −4.87518 − 1.11024i 4.83636 3.71756 − 3.71756i 2.00000 + 2.00000i 2.69521i 5.98542 − 3.76494i
93.4 −1.00000 + 1.00000i −0.454110 − 0.454110i − 2.00000i 4.61282 + 1.92922i 0.908220 −8.21299 + 8.21299i 2.00000 + 2.00000i − 8.58757i −6.54204 + 2.68360i
93.5 −1.00000 + 1.00000i −0.0649756 − 0.0649756i − 2.00000i −4.95703 + 0.654141i 0.129951 −3.31044 + 3.31044i 2.00000 + 2.00000i − 8.99156i 4.30288 − 5.61117i
93.6 −1.00000 + 1.00000i 0.608645 + 0.608645i − 2.00000i 4.88434 − 1.06922i −1.21729 3.48181 − 3.48181i 2.00000 + 2.00000i − 8.25910i −3.81512 + 5.95356i
93.7 −1.00000 + 1.00000i 1.21198 + 1.21198i − 2.00000i −0.802406 − 4.93519i −2.42397 −5.44701 + 5.44701i 2.00000 + 2.00000i − 6.06220i 5.73760 + 4.13279i
93.8 −1.00000 + 1.00000i 1.94658 + 1.94658i − 2.00000i 0.283956 + 4.99193i −3.89316 9.76397 − 9.76397i 2.00000 + 2.00000i − 1.42166i −5.27589 − 4.70797i
93.9 −1.00000 + 1.00000i 2.91022 + 2.91022i − 2.00000i −3.94243 + 3.07527i −5.82043 −1.42062 + 1.42062i 2.00000 + 2.00000i 7.93871i 0.867158 − 7.01769i
93.10 −1.00000 + 1.00000i 2.93330 + 2.93330i − 2.00000i 0.390531 − 4.98473i −5.86660 1.49862 − 1.49862i 2.00000 + 2.00000i 8.20851i 4.59419 + 5.37526i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 230.3.f.a ✓ 20
5.c odd 4 1 inner 230.3.f.a ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.3.f.a ✓ 20 1.a even 1 1 trivial
230.3.f.a ✓ 20 5.c odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{20} - 52 T_{3}^{17} + 1020 T_{3}^{16} - 1316 T_{3}^{15} + 1352 T_{3}^{14} - 18724 T_{3}^{13} + \cdots + 88804 \) acting on \(S_{3}^{\mathrm{new}}(230, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T + 2)^{10} \) Copy content Toggle raw display
$3$ \( T^{20} - 52 T^{17} + \cdots + 88804 \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 95367431640625 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 195628000677264 \) Copy content Toggle raw display
$11$ \( (T^{10} - 28 T^{9} + \cdots - 1481201960)^{2} \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 46\!\cdots\!36 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
$23$ \( (T^{4} + 529)^{5} \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 13\!\cdots\!25 \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots - 1910123254695)^{2} \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 65\!\cdots\!04 \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots + 621850944556593)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 94\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 87\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots - 38679925234168)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots + 24\!\cdots\!29)^{2} \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 70\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 19\!\cdots\!24 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 74\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 13\!\cdots\!64 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 17\!\cdots\!76 \) Copy content Toggle raw display
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