Properties

Label 230.4.b.a
Level $230$
Weight $4$
Character orbit 230.b
Analytic conductor $13.570$
Analytic rank $0$
Dimension $14$
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] = [230,4,Mod(139,230)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(230, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("230.139");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 230.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: \(13.5704393013\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 212 x^{12} + 17560 x^{10} + 728073 x^{8} + 16036416 x^{6} + 183184060 x^{4} + 961600400 x^{2} + 1560250000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 \beta_{8} q^{2} + ( - \beta_{8} + \beta_1) q^{3} - 4 q^{4} + ( - \beta_{8} - \beta_{7}) q^{5} + (2 \beta_{4} - 2) q^{6} + ( - \beta_{13} - \beta_{12} + 6 \beta_{8} + \beta_1) q^{7} + 8 \beta_{8} q^{8} + ( - \beta_{11} - \beta_{10} - \beta_{9} - \beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 \beta_{8} q^{2} + ( - \beta_{8} + \beta_1) q^{3} - 4 q^{4} + ( - \beta_{8} - \beta_{7}) q^{5} + (2 \beta_{4} - 2) q^{6} + ( - \beta_{13} - \beta_{12} + 6 \beta_{8} + \beta_1) q^{7} + 8 \beta_{8} q^{8} + ( - \beta_{11} - \beta_{10} - \beta_{9} - \beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} - 3) q^{9} + (2 \beta_{11} - 2 \beta_{9} + 2 \beta_{4} + 2 \beta_{2} + 2 \beta_1 - 2) q^{10} + (2 \beta_{11} + 2 \beta_{10} - 2 \beta_{9} - 2 \beta_{7} + 2 \beta_{6} + \beta_{5} + \beta_{4} + \cdots - 9) q^{11}+ \cdots + (19 \beta_{11} + 19 \beta_{10} + 3 \beta_{9} + 3 \beta_{7} - 3 \beta_{6} - 10 \beta_{5} + \cdots + 208) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 56 q^{4} - 6 q^{5} - 36 q^{6} - 68 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 56 q^{4} - 6 q^{5} - 36 q^{6} - 68 q^{9} - 40 q^{10} - 146 q^{11} + 176 q^{14} - 206 q^{15} + 224 q^{16} + 154 q^{19} + 24 q^{20} - 220 q^{21} + 144 q^{24} - 286 q^{25} - 180 q^{26} + 790 q^{29} - 232 q^{30} - 320 q^{31} - 200 q^{34} - 426 q^{35} + 272 q^{36} + 1616 q^{39} + 160 q^{40} - 1904 q^{41} + 584 q^{44} + 622 q^{45} + 644 q^{46} + 610 q^{49} + 200 q^{50} - 1834 q^{51} + 192 q^{54} + 854 q^{55} - 704 q^{56} + 2814 q^{59} + 824 q^{60} - 3742 q^{61} - 896 q^{64} + 1730 q^{65} - 612 q^{66} + 414 q^{69} + 348 q^{70} - 3808 q^{71} + 268 q^{74} + 2904 q^{75} - 616 q^{76} - 1528 q^{79} - 96 q^{80} - 4618 q^{81} + 880 q^{84} + 2574 q^{85} - 2024 q^{86} + 2336 q^{89} + 2092 q^{90} - 3866 q^{91} + 456 q^{94} + 838 q^{95} - 576 q^{96} + 3342 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 212 x^{12} + 17560 x^{10} + 728073 x^{8} + 16036416 x^{6} + 183184060 x^{4} + 961600400 x^{2} + 1560250000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3186868379 \nu^{12} + 534359144753 \nu^{10} + 31796999102640 \nu^{8} + 844061002553517 \nu^{6} + \cdots + 86\!\cdots\!50 ) / 23\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 3499703585 \nu^{12} - 585068510510 \nu^{10} - 35830636596552 \nu^{8} + \cdots - 34\!\cdots\!60 ) / 94\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 289581 \nu^{12} - 56463152 \nu^{10} - 4132854900 \nu^{8} - 142227122813 \nu^{6} - 2338196775636 \nu^{4} - 16286584250280 \nu^{2} + \cdots - 31018745255000 ) / 613694015800 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 130865103661 \nu^{12} - 25582406911102 \nu^{10} + \cdots - 11\!\cdots\!00 ) / 94\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 5109264790403 \nu^{13} - 553302086253686 \nu^{11} + \cdots + 10\!\cdots\!00 \nu ) / 37\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 9297613576608 \nu^{13} - 185139331460725 \nu^{12} + \cdots - 22\!\cdots\!00 ) / 37\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 78528469 \nu^{13} - 15504190478 \nu^{11} - 1155930465240 \nu^{9} - 40849681155237 \nu^{7} - 697518061615754 \nu^{5} + \cdots - 11\!\cdots\!00 \nu ) / 24\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 14406878367011 \nu^{13} - 185139331460725 \nu^{12} + \cdots - 22\!\cdots\!00 ) / 37\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 17013028715369 \nu^{13} + 229642540842400 \nu^{12} + \cdots + 26\!\cdots\!00 ) / 37\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 17013028715369 \nu^{13} + 229642540842400 \nu^{12} + \cdots + 26\!\cdots\!00 ) / 37\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 3805919420404 \nu^{13} + 748011503552023 \nu^{11} + \cdots + 21\!\cdots\!00 \nu ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 41766387715051 \nu^{13} + \cdots - 51\!\cdots\!00 \nu ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{11} - \beta_{10} - \beta_{9} - \beta_{7} + \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} - 29 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{13} + 4\beta_{12} - 7\beta_{11} + 7\beta_{10} - 25\beta_{8} + 7\beta_{6} - 52\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 84 \beta_{11} + 84 \beta_{10} + 73 \beta_{9} + 73 \beta_{7} - 73 \beta_{6} - 4 \beta_{5} + 123 \beta_{4} - 72 \beta_{3} - 36 \beta_{2} + 1316 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 393 \beta_{13} - 384 \beta_{12} + 729 \beta_{11} - 729 \beta_{10} - 171 \beta_{9} + 2915 \beta_{8} + 171 \beta_{7} - 576 \beta_{6} + 3297 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 5919 \beta_{11} - 5919 \beta_{10} - 4836 \beta_{9} - 4836 \beta_{7} + 4836 \beta_{6} + 954 \beta_{5} - 11426 \beta_{4} + 4734 \beta_{3} + 1077 \beta_{2} - 72393 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 31526 \beta_{13} + 27528 \beta_{12} - 60661 \beta_{11} + 60661 \beta_{10} + 19979 \beta_{9} - 270718 \beta_{8} - 19979 \beta_{7} + 41481 \beta_{6} - 220815 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 409088 \beta_{11} + 409088 \beta_{10} + 317283 \beta_{9} + 317283 \beta_{7} - 317283 \beta_{6} - 103040 \beta_{5} + 941783 \beta_{4} - 316493 \beta_{3} - 16393 \beta_{2} + 4397417 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 2366262 \beta_{13} - 1851602 \beta_{12} + 4677821 \beta_{11} - 4677821 \beta_{10} - 1734830 \beta_{9} + 22387525 \beta_{8} + 1734830 \beta_{7} - 2909981 \beta_{6} + \cdots + 15130036 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 28340982 \beta_{11} - 28340982 \beta_{10} - 21095829 \beta_{9} - 21095829 \beta_{7} + 21095829 \beta_{6} + 8946672 \beta_{5} - 72874659 \beta_{4} + 21654816 \beta_{3} + \cdots - 282821788 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 172412469 \beta_{13} + 123616692 \beta_{12} - 348171957 \beta_{11} + 348171957 \beta_{10} + 135862743 \beta_{9} - 1737485295 \beta_{8} - 135862743 \beta_{7} + \cdots - 1049728261 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1972644007 \beta_{11} + 1972644007 \beta_{10} + 1427099788 \beta_{9} + 1427099788 \beta_{7} - 1427099788 \beta_{6} - 710128842 \beta_{5} + 5441110618 \beta_{4} + \cdots + 18839621909 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 12379981858 \beta_{13} - 8321998624 \beta_{12} + 25422950533 \beta_{11} - 25422950533 \beta_{10} - 10161400047 \beta_{9} + 130037778814 \beta_{8} + \cdots + 73387455835 \beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(47\) \(51\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
139.1
6.80605i
6.41782i
3.85861i
1.66599i
3.36381i
4.96965i
8.41501i
8.41501i
4.96965i
3.36381i
1.66599i
3.85861i
6.41782i
6.80605i
2.00000i 7.80605i −4.00000 −4.43387 10.2636i −15.6121 21.1583i 8.00000i −33.9344 −20.5271 + 8.86773i
139.2 2.00000i 7.41782i −4.00000 2.14945 10.9718i −14.8356 26.7462i 8.00000i −28.0240 −21.9436 4.29890i
139.3 2.00000i 4.85861i −4.00000 8.38650 + 7.39369i −9.71722 11.5462i 8.00000i 3.39393 14.7874 16.7730i
139.4 2.00000i 2.66599i −4.00000 −3.67145 + 10.5603i −5.33198 1.62258i 8.00000i 19.8925 21.1206 + 7.34291i
139.5 2.00000i 2.36381i −4.00000 −6.34330 9.20666i 4.72762 7.27979i 8.00000i 21.4124 −18.4133 + 12.6866i
139.6 2.00000i 3.96965i −4.00000 10.8788 2.57927i 7.93930 5.88989i 8.00000i 11.2419 −5.15854 21.7575i
139.7 2.00000i 7.41501i −4.00000 −9.96609 + 5.06726i 14.8300 26.6333i 8.00000i −27.9823 10.1345 + 19.9322i
139.8 2.00000i 7.41501i −4.00000 −9.96609 5.06726i 14.8300 26.6333i 8.00000i −27.9823 10.1345 19.9322i
139.9 2.00000i 3.96965i −4.00000 10.8788 + 2.57927i 7.93930 5.88989i 8.00000i 11.2419 −5.15854 + 21.7575i
139.10 2.00000i 2.36381i −4.00000 −6.34330 + 9.20666i 4.72762 7.27979i 8.00000i 21.4124 −18.4133 12.6866i
139.11 2.00000i 2.66599i −4.00000 −3.67145 10.5603i −5.33198 1.62258i 8.00000i 19.8925 21.1206 7.34291i
139.12 2.00000i 4.85861i −4.00000 8.38650 7.39369i −9.71722 11.5462i 8.00000i 3.39393 14.7874 + 16.7730i
139.13 2.00000i 7.41782i −4.00000 2.14945 + 10.9718i −14.8356 26.7462i 8.00000i −28.0240 −21.9436 + 4.29890i
139.14 2.00000i 7.80605i −4.00000 −4.43387 + 10.2636i −15.6121 21.1583i 8.00000i −33.9344 −20.5271 8.86773i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 139.14
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 230.4.b.a 14
5.b even 2 1 inner 230.4.b.a 14
5.c odd 4 1 1150.4.a.y 7
5.c odd 4 1 1150.4.a.z 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.4.b.a 14 1.a even 1 1 trivial
230.4.b.a 14 5.b even 2 1 inner
1150.4.a.y 7 5.c odd 4 1
1150.4.a.z 7 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{14} + 223 T_{3}^{12} + 19539 T_{3}^{10} + 852894 T_{3}^{8} + 19553159 T_{3}^{6} + 231696639 T_{3}^{4} + 1302477653 T_{3}^{2} + 2723378596 \) acting on \(S_{4}^{\mathrm{new}}(230, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 4)^{7} \) Copy content Toggle raw display
$3$ \( T^{14} + 223 T^{12} + \cdots + 2723378596 \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 476837158203125 \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 146581095556624 \) Copy content Toggle raw display
$11$ \( (T^{7} + 73 T^{6} - 631 T^{5} + \cdots - 1520797760)^{2} \) Copy content Toggle raw display
$13$ \( T^{14} + 17875 T^{12} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( T^{14} + 35676 T^{12} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{7} - 77 T^{6} + \cdots + 3897418182928)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 529)^{7} \) Copy content Toggle raw display
$29$ \( (T^{7} - 395 T^{6} + \cdots + 43908716160740)^{2} \) Copy content Toggle raw display
$31$ \( (T^{7} + 160 T^{6} + \cdots - 108456344870975)^{2} \) Copy content Toggle raw display
$37$ \( T^{14} + 539049 T^{12} + \cdots + 32\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( (T^{7} + 952 T^{6} + \cdots - 44\!\cdots\!03)^{2} \) Copy content Toggle raw display
$43$ \( T^{14} + 694656 T^{12} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{14} + 798808 T^{12} + \cdots + 34\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{14} + 966577 T^{12} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( (T^{7} - 1407 T^{6} + \cdots - 30\!\cdots\!40)^{2} \) Copy content Toggle raw display
$61$ \( (T^{7} + 1871 T^{6} + \cdots - 35\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + 1206121 T^{12} + \cdots + 73\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{7} + 1904 T^{6} + \cdots + 63\!\cdots\!93)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + 1885088 T^{12} + \cdots + 16\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( (T^{7} + 764 T^{6} + \cdots + 25\!\cdots\!32)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + 2532285 T^{12} + \cdots + 42\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( (T^{7} - 1168 T^{6} + \cdots + 40\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{14} + 6164419 T^{12} + \cdots + 57\!\cdots\!44 \) Copy content Toggle raw display
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