Properties

Label 230.6.a.i
Level $230$
Weight $6$
Character orbit 230.a
Self dual yes
Analytic conductor $36.888$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [230,6,Mod(1,230)]
 
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([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("230.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 230.a (trivial)

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: yes
Analytic conductor: \(36.8882785570\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} - 1156x^{4} + 593x^{3} + 338133x^{2} + 408388x - 13033476 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 4 q^{2} + (\beta_1 + 2) q^{3} + 16 q^{4} - 25 q^{5} + (4 \beta_1 + 8) q^{6} + ( - \beta_{2} + 2 \beta_1 + 17) q^{7} + 64 q^{8} + (\beta_{4} + 3 \beta_{3} + 6 \beta_1 + 147) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{2} + (\beta_1 + 2) q^{3} + 16 q^{4} - 25 q^{5} + (4 \beta_1 + 8) q^{6} + ( - \beta_{2} + 2 \beta_1 + 17) q^{7} + 64 q^{8} + (\beta_{4} + 3 \beta_{3} + 6 \beta_1 + 147) q^{9} - 100 q^{10} + (\beta_{5} - 2 \beta_{4} + 7 \beta_1 + 50) q^{11} + (16 \beta_1 + 32) q^{12} + (5 \beta_{5} + 3 \beta_{4} - 2 \beta_{3} + 16 \beta_1 + 82) q^{13} + ( - 4 \beta_{2} + 8 \beta_1 + 68) q^{14} + ( - 25 \beta_1 - 50) q^{15} + 256 q^{16} + ( - 27 \beta_{5} - 5 \beta_{4} - 8 \beta_{3} + 5 \beta_{2} + 7 \beta_1 - 125) q^{17} + (4 \beta_{4} + 12 \beta_{3} + 24 \beta_1 + 588) q^{18} + ( - 3 \beta_{5} - 9 \beta_{4} - 21 \beta_{3} + 8 \beta_{2} + 14 \beta_1 + 446) q^{19} - 400 q^{20} + (25 \beta_{5} + 9 \beta_{4} + 4 \beta_{3} - 4 \beta_{2} - 5 \beta_1 + 1014) q^{21} + (4 \beta_{5} - 8 \beta_{4} + 28 \beta_1 + 200) q^{22} - 529 q^{23} + (64 \beta_1 + 128) q^{24} + 625 q^{25} + (20 \beta_{5} + 12 \beta_{4} - 8 \beta_{3} + 64 \beta_1 + 328) q^{26} + ( - 4 \beta_{5} + 31 \beta_{4} + 20 \beta_{3} - 2 \beta_{2} + 109 \beta_1 + 2512) q^{27} + ( - 16 \beta_{2} + 32 \beta_1 + 272) q^{28} + (23 \beta_{5} - 33 \beta_{4} + 10 \beta_{3} + \beta_{2} - 22 \beta_1 + 569) q^{29} + ( - 100 \beta_1 - 200) q^{30} + (12 \beta_{5} + 2 \beta_{4} + 35 \beta_{3} + 23 \beta_{2} + 24 \beta_1 + 5167) q^{31} + 1024 q^{32} + (3 \beta_{5} - 31 \beta_{4} - 35 \beta_{3} + 34 \beta_{2} + 54 \beta_1 + 2472) q^{33} + ( - 108 \beta_{5} - 20 \beta_{4} - 32 \beta_{3} + 20 \beta_{2} + \cdots - 500) q^{34}+ \cdots + ( - 984 \beta_{5} - 357 \beta_{4} - 348 \beta_{3} + 528 \beta_{2} + \cdots - 924) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 24 q^{2} + 15 q^{3} + 96 q^{4} - 150 q^{5} + 60 q^{6} + 106 q^{7} + 384 q^{8} + 899 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 24 q^{2} + 15 q^{3} + 96 q^{4} - 150 q^{5} + 60 q^{6} + 106 q^{7} + 384 q^{8} + 899 q^{9} - 600 q^{10} + 321 q^{11} + 240 q^{12} + 527 q^{13} + 424 q^{14} - 375 q^{15} + 1536 q^{16} - 660 q^{17} + 3596 q^{18} + 2749 q^{19} - 2400 q^{20} + 6002 q^{21} + 1284 q^{22} - 3174 q^{23} + 960 q^{24} + 3750 q^{25} + 2108 q^{26} + 15372 q^{27} + 1696 q^{28} + 3337 q^{29} - 1500 q^{30} + 31094 q^{31} + 6144 q^{32} + 15087 q^{33} - 2640 q^{34} - 2650 q^{35} + 14384 q^{36} + 27037 q^{37} + 10996 q^{38} + 38528 q^{39} - 9600 q^{40} + 33608 q^{41} + 24008 q^{42} + 17024 q^{43} + 5136 q^{44} - 22475 q^{45} - 12696 q^{46} + 16864 q^{47} + 3840 q^{48} + 6002 q^{49} + 15000 q^{50} + 5719 q^{51} + 8432 q^{52} - 8475 q^{53} + 61488 q^{54} - 8025 q^{55} + 6784 q^{56} + 9566 q^{57} + 13348 q^{58} + 7899 q^{59} - 6000 q^{60} + 25437 q^{61} + 124376 q^{62} - 13333 q^{63} + 24576 q^{64} - 13175 q^{65} + 60348 q^{66} - 25517 q^{67} - 10560 q^{68} - 7935 q^{69} - 10600 q^{70} + 17204 q^{71} + 57536 q^{72} + 760 q^{73} + 108148 q^{74} + 9375 q^{75} + 43984 q^{76} + 102330 q^{77} + 154112 q^{78} + 66972 q^{79} - 38400 q^{80} + 115874 q^{81} + 134432 q^{82} + 58523 q^{83} + 96032 q^{84} + 16500 q^{85} + 68096 q^{86} - 70854 q^{87} + 20544 q^{88} + 38406 q^{89} - 89900 q^{90} + 25111 q^{91} - 50784 q^{92} + 130338 q^{93} + 67456 q^{94} - 68725 q^{95} + 15360 q^{96} + 82861 q^{97} + 24008 q^{98} - 2973 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} - 1156x^{4} + 593x^{3} + 338133x^{2} + 408388x - 13033476 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 473\nu^{5} - 432595\nu^{4} + 3301386\nu^{3} + 342922345\nu^{2} - 1629413963\nu - 39620612346 ) / 192701508 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -457\nu^{5} + 10559\nu^{4} - 948996\nu^{3} + 25766089\nu^{2} + 628918249\nu - 10563544722 ) / 96350754 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 457\nu^{5} - 10559\nu^{4} + 948996\nu^{3} + 6350829\nu^{2} - 693152085\nu - 1833585626 ) / 32116918 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 8222\nu^{5} - 84553\nu^{4} - 7593858\nu^{3} + 46230502\nu^{2} + 1479281155\nu - 1814631102 ) / 96350754 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + 3\beta_{3} + 2\beta _1 + 386 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -4\beta_{5} + 25\beta_{4} + 2\beta_{3} - 2\beta_{2} + 571\beta _1 + 1160 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -22\beta_{5} + 1014\beta_{4} + 2404\beta_{3} - 468\beta_{2} + 2573\beta _1 + 224818 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 7798\beta_{5} + 27895\beta_{4} + 9701\beta_{3} - 6660\beta_{2} + 362674\beta _1 + 1433662 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−24.9052
−19.3306
−7.76257
5.93160
20.8018
28.2649
4.00000 −22.9052 16.0000 −25.0000 −91.6207 10.2657 64.0000 281.647 −100.000
1.2 4.00000 −17.3306 16.0000 −25.0000 −69.3223 −200.644 64.0000 57.3489 −100.000
1.3 4.00000 −5.76257 16.0000 −25.0000 −23.0503 50.4462 64.0000 −209.793 −100.000
1.4 4.00000 7.93160 16.0000 −25.0000 31.7264 221.199 64.0000 −180.090 −100.000
1.5 4.00000 22.8018 16.0000 −25.0000 91.2074 −73.3684 64.0000 276.924 −100.000
1.6 4.00000 30.2649 16.0000 −25.0000 121.060 98.1017 64.0000 672.963 −100.000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(23\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 230.6.a.i 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.6.a.i 6 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} - 15T_{3}^{5} - 1066T_{3}^{4} + 9561T_{3}^{3} + 307311T_{3}^{2} - 900468T_{3} - 12520800 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(230))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 4)^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 15 T^{5} - 1066 T^{4} + \cdots - 12520800 \) Copy content Toggle raw display
$5$ \( (T + 25)^{6} \) Copy content Toggle raw display
$7$ \( T^{6} - 106 T^{5} + \cdots + 165428983104 \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots - 221236397079552 \) Copy content Toggle raw display
$13$ \( T^{6} - 527 T^{5} + \cdots + 66\!\cdots\!16 \) Copy content Toggle raw display
$17$ \( T^{6} + 660 T^{5} + \cdots - 22\!\cdots\!80 \) Copy content Toggle raw display
$19$ \( T^{6} - 2749 T^{5} + \cdots + 29\!\cdots\!28 \) Copy content Toggle raw display
$23$ \( (T + 529)^{6} \) Copy content Toggle raw display
$29$ \( T^{6} - 3337 T^{5} + \cdots - 13\!\cdots\!32 \) Copy content Toggle raw display
$31$ \( T^{6} - 31094 T^{5} + \cdots + 68\!\cdots\!60 \) Copy content Toggle raw display
$37$ \( T^{6} - 27037 T^{5} + \cdots - 36\!\cdots\!08 \) Copy content Toggle raw display
$41$ \( T^{6} - 33608 T^{5} + \cdots + 18\!\cdots\!46 \) Copy content Toggle raw display
$43$ \( T^{6} - 17024 T^{5} + \cdots + 13\!\cdots\!20 \) Copy content Toggle raw display
$47$ \( T^{6} - 16864 T^{5} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{6} + 8475 T^{5} + \cdots - 80\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{6} - 7899 T^{5} + \cdots - 29\!\cdots\!88 \) Copy content Toggle raw display
$61$ \( T^{6} - 25437 T^{5} + \cdots + 57\!\cdots\!92 \) Copy content Toggle raw display
$67$ \( T^{6} + 25517 T^{5} + \cdots + 13\!\cdots\!20 \) Copy content Toggle raw display
$71$ \( T^{6} - 17204 T^{5} + \cdots - 31\!\cdots\!80 \) Copy content Toggle raw display
$73$ \( T^{6} - 760 T^{5} + \cdots + 72\!\cdots\!48 \) Copy content Toggle raw display
$79$ \( T^{6} - 66972 T^{5} + \cdots + 93\!\cdots\!56 \) Copy content Toggle raw display
$83$ \( T^{6} - 58523 T^{5} + \cdots + 62\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T^{6} - 38406 T^{5} + \cdots + 77\!\cdots\!48 \) Copy content Toggle raw display
$97$ \( T^{6} - 82861 T^{5} + \cdots + 27\!\cdots\!76 \) Copy content Toggle raw display
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