Properties

Label 495.4.c.d
Level $495$
Weight $4$
Character orbit 495.c
Analytic conductor $29.206$
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] = [495,4,Mod(199,495)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(495, 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("495.199");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 495.c (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: \(29.2059454528\)
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: \( x^{14} + 66x^{12} + 1705x^{10} + 22060x^{8} + 151880x^{6} + 537860x^{4} + 825344x^{2} + 262144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{4}\cdot 5 \)
Twist minimal: no (minimal twist has level 165)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_1 - 2) q^{4} + ( - \beta_{3} + 1) q^{5} + (\beta_{6} - \beta_{2}) q^{7} + (\beta_{13} - \beta_{6} - \beta_{4} + \beta_{3}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + (\beta_1 - 2) q^{4} + ( - \beta_{3} + 1) q^{5} + (\beta_{6} - \beta_{2}) q^{7} + (\beta_{13} - \beta_{6} - \beta_{4} + \beta_{3}) q^{8} + (\beta_{12} + \beta_{11} + \beta_{10} - \beta_{9} - \beta_{7} + \beta_{4} - \beta_{3} + 3 \beta_{2} + 2 \beta_1 - 4) q^{10} - 11 q^{11} + ( - \beta_{12} + \beta_{11} - \beta_{10} + \beta_{9} - 2 \beta_{8} - 2 \beta_{7} - \beta_{6} - \beta_{4} + \cdots - \beta_1) q^{13}+ \cdots + (36 \beta_{13} - 3 \beta_{12} - 21 \beta_{11} - 3 \beta_{10} + 3 \beta_{9} - 29 \beta_{8} + \cdots + 21 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 26 q^{4} + 14 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 26 q^{4} + 14 q^{5} - 48 q^{10} - 154 q^{11} + 84 q^{14} - 86 q^{16} - 116 q^{19} - 442 q^{20} + 162 q^{25} + 400 q^{26} + 128 q^{29} - 696 q^{31} - 412 q^{34} - 672 q^{35} + 1612 q^{40} + 664 q^{41} + 286 q^{44} - 656 q^{46} + 834 q^{49} - 1908 q^{50} - 154 q^{55} + 3236 q^{56} - 664 q^{59} + 44 q^{61} - 1122 q^{64} + 2328 q^{65} + 1220 q^{70} + 1032 q^{71} + 3256 q^{74} + 5588 q^{76} - 3492 q^{79} + 510 q^{80} - 1068 q^{85} - 2540 q^{86} - 4452 q^{89} + 2144 q^{91} - 9472 q^{94} + 932 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 66x^{12} + 1705x^{10} + 22060x^{8} + 151880x^{6} + 537860x^{4} + 825344x^{2} + 262144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 1221 \nu^{12} + 72330 \nu^{10} + 1598477 \nu^{8} + 16399708 \nu^{6} + 79067688 \nu^{4} + 154270420 \nu^{2} + 63510464 ) / 1168832 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 51749 \nu^{13} - 3102858 \nu^{11} - 69715565 \nu^{9} - 732372828 \nu^{7} - 3661312872 \nu^{5} - 7592389012 \nu^{3} - 2918278144 \nu ) / 598441984 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 737029 \nu^{13} - 1877968 \nu^{12} - 46201930 \nu^{11} - 124343200 \nu^{10} - 1093595725 \nu^{9} - 3133573200 \nu^{8} - 12139022300 \nu^{7} + \cdots - 77504233472 ) / 5984419840 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1198531 \nu^{13} - 1877968 \nu^{12} + 79829830 \nu^{11} - 124343200 \nu^{10} + 2035085755 \nu^{9} - 3133573200 \nu^{8} + 24747089540 \nu^{7} + \cdots - 77504233472 ) / 5984419840 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1850421 \nu^{13} + 7538672 \nu^{12} + 108940970 \nu^{11} + 476675040 \nu^{10} + 2392182525 \nu^{9} + 11432902000 \nu^{8} + \cdots + 469051588608 ) / 5984419840 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 107095 \nu^{13} + 6260846 \nu^{11} + 136053103 \nu^{9} + 1379053300 \nu^{7} + 6867744632 \nu^{5} + 16450198620 \nu^{3} + 14796939264 \nu ) / 299220992 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 682901 \nu^{13} + 4062552 \nu^{12} + 39549290 \nu^{11} + 240543920 \nu^{10} + 837221725 \nu^{9} + 5299023640 \nu^{8} + 7885287580 \nu^{7} + \cdots + 167469785088 ) / 1496104960 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 294303 \nu^{13} - 17109502 \nu^{11} - 365782519 \nu^{9} - 3539609684 \nu^{7} - 15350839224 \nu^{5} - 23344585084 \nu^{3} + \cdots + 2440846336 \nu ) / 598441984 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 155247 \nu^{13} + 363768 \nu^{12} + 9308574 \nu^{11} + 22077424 \nu^{10} + 209146695 \nu^{9} + 506003384 \nu^{8} + 2197118484 \nu^{7} + \cdots + 24391294976 ) / 299220992 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 155247 \nu^{13} - 261384 \nu^{12} + 9308574 \nu^{11} - 14955536 \nu^{10} + 209146695 \nu^{9} - 312416840 \nu^{8} + 2197118484 \nu^{7} + \cdots + 551346176 ) / 299220992 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 4086959 \nu^{13} + 18718992 \nu^{12} - 239390750 \nu^{11} + 1104516640 \nu^{10} - 5192523655 \nu^{9} + 24214541200 \nu^{8} + \cdots + 681009594368 ) / 5984419840 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 2659527 \nu^{13} - 4703936 \nu^{12} + 156670030 \nu^{11} - 286924160 \nu^{10} + 3417332575 \nu^{9} - 6563509440 \nu^{8} + 34079895860 \nu^{7} + \cdots - 292313759744 ) / 2992209920 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 569239 \nu^{13} + 34131438 \nu^{11} + 766871215 \nu^{9} + 8056101108 \nu^{7} + 40274441592 \nu^{5} + 83516279132 \nu^{3} + \cdots + 39282363392 \nu ) / 598441984 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{13} + 11\beta_{2} ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{10} - \beta_{9} + 2\beta _1 - 29 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 19 \beta_{13} + 6 \beta_{12} + 6 \beta_{11} + 6 \beta_{10} - 6 \beta_{9} + 12 \beta_{5} + 6 \beta_{4} + 12 \beta_{3} - 155 \beta_{2} - 6 \beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - \beta_{13} - 5 \beta_{12} - 8 \beta_{11} - 26 \beta_{10} + 25 \beta_{9} + 4 \beta_{8} + 5 \beta_{7} + 4 \beta_{4} + 4 \beta_{3} + 2 \beta_{2} - 45 \beta _1 + 421 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 347 \beta_{13} - 138 \beta_{12} - 162 \beta_{11} - 138 \beta_{10} + 138 \beta_{9} + 60 \beta_{8} + 24 \beta_{7} + 24 \beta_{6} - 300 \beta_{5} - 150 \beta_{4} - 312 \beta_{3} + 2587 \beta_{2} + 162 \beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 29 \beta_{13} + 139 \beta_{12} + 244 \beta_{11} + 591 \beta_{10} - 532 \beta_{9} - 122 \beta_{8} - 139 \beta_{7} - 12 \beta_{5} - 122 \beta_{4} - 110 \beta_{3} - 34 \beta_{2} + 898 \beta _1 - 7209 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 6819 \beta_{13} + 2866 \beta_{12} + 3706 \beta_{11} + 2866 \beta_{10} - 2866 \beta_{9} - 2148 \beta_{8} - 840 \beta_{7} - 448 \beta_{6} + 6572 \beta_{5} + 3438 \beta_{4} + 6840 \beta_{3} - 47171 \beta_{2} - 3706 \beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 805 \beta_{13} - 3023 \beta_{12} - 5888 \beta_{11} - 12887 \beta_{10} + 11140 \beta_{9} + 2944 \beta_{8} + 3023 \beta_{7} + 492 \beta_{5} + 2944 \beta_{4} + 2452 \beta_{3} + 392 \beta_{2} - 17490 \beta _1 + 133657 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 140779 \beta_{13} - 59274 \beta_{12} - 80322 \beta_{11} - 59274 \beta_{10} + 59274 \beta_{9} + 57396 \beta_{8} + 21048 \beta_{7} + 4128 \beta_{6} - 139596 \beta_{5} - 77814 \beta_{4} + \cdots + 80322 \beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 21263 \beta_{13} + 60991 \beta_{12} + 132016 \beta_{11} + 276527 \beta_{10} - 234084 \beta_{9} - 66008 \beta_{8} - 60991 \beta_{7} - 13944 \beta_{5} - 66008 \beta_{4} - 52064 \beta_{3} + \cdots - 2578069 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 2975435 \beta_{13} + 1235098 \beta_{12} + 1704610 \beta_{11} + 1235098 \beta_{10} - 1235098 \beta_{9} - 1377588 \beta_{8} - 469512 \beta_{7} + 41312 \beta_{6} + 2939708 \beta_{5} + \cdots - 1704610 \beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 176823 \beta_{13} - 399537 \beta_{12} - 957104 \beta_{11} - 1963517 \beta_{10} + 1645232 \beta_{9} + 478552 \beta_{8} + 399537 \beta_{7} + 114364 \beta_{5} + 478552 \beta_{4} + \cdots + 16938411 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 63436523 \beta_{13} - 25880442 \beta_{12} - 35866674 \beta_{11} - 25880442 \beta_{10} + 25880442 \beta_{9} + 31431012 \beta_{8} + 9986232 \beta_{7} - 3396000 \beta_{6} + \cdots + 35866674 \beta_1 ) / 12 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/495\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(56\) \(397\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
4.15324i
2.70507i
4.57884i
1.98683i
3.20690i
2.41169i
0.647712i
0.647712i
2.41169i
3.20690i
1.98683i
4.57884i
2.70507i
4.15324i
5.15324i 0 −18.5558 7.01850 8.70291i 0 17.2148i 54.3968i 0 −44.8482 36.1680i
199.2 3.70507i 0 −5.72754 10.6368 + 3.44353i 0 30.9104i 8.41961i 0 12.7585 39.4102i
199.3 3.57884i 0 −4.80807 0.925309 11.1420i 0 7.85216i 11.4234i 0 −39.8753 3.31153i
199.4 2.98683i 0 −0.921158 −7.35339 + 8.42185i 0 6.37827i 21.1433i 0 25.1547 + 21.9633i
199.5 2.20690i 0 3.12958 −4.62093 + 10.1807i 0 1.50972i 24.5619i 0 22.4678 + 10.1979i
199.6 1.41169i 0 6.00714 11.1339 + 1.01772i 0 15.2844i 19.7737i 0 1.43670 15.7176i
199.7 0.352288i 0 7.87589 −10.7402 3.10602i 0 19.8486i 5.59288i 0 −1.09421 + 3.78365i
199.8 0.352288i 0 7.87589 −10.7402 + 3.10602i 0 19.8486i 5.59288i 0 −1.09421 3.78365i
199.9 1.41169i 0 6.00714 11.1339 1.01772i 0 15.2844i 19.7737i 0 1.43670 + 15.7176i
199.10 2.20690i 0 3.12958 −4.62093 10.1807i 0 1.50972i 24.5619i 0 22.4678 10.1979i
199.11 2.98683i 0 −0.921158 −7.35339 8.42185i 0 6.37827i 21.1433i 0 25.1547 21.9633i
199.12 3.57884i 0 −4.80807 0.925309 + 11.1420i 0 7.85216i 11.4234i 0 −39.8753 + 3.31153i
199.13 3.70507i 0 −5.72754 10.6368 3.44353i 0 30.9104i 8.41961i 0 12.7585 + 39.4102i
199.14 5.15324i 0 −18.5558 7.01850 + 8.70291i 0 17.2148i 54.3968i 0 −44.8482 + 36.1680i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.14
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 495.4.c.d 14
3.b odd 2 1 165.4.c.b 14
5.b even 2 1 inner 495.4.c.d 14
5.c odd 4 1 2475.4.a.bo 7
5.c odd 4 1 2475.4.a.bs 7
15.d odd 2 1 165.4.c.b 14
15.e even 4 1 825.4.a.ba 7
15.e even 4 1 825.4.a.bd 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.c.b 14 3.b odd 2 1
165.4.c.b 14 15.d odd 2 1
495.4.c.d 14 1.a even 1 1 trivial
495.4.c.d 14 5.b even 2 1 inner
825.4.a.ba 7 15.e even 4 1
825.4.a.bd 7 15.e even 4 1
2475.4.a.bo 7 5.c odd 4 1
2475.4.a.bs 7 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(495, [\chi])\):

\( T_{2}^{14} + 69T_{2}^{12} + 1798T_{2}^{10} + 22642T_{2}^{8} + 143537T_{2}^{6} + 424913T_{2}^{4} + 454864T_{2}^{2} + 50176 \) Copy content Toggle raw display
\( T_{29}^{7} - 64 T_{29}^{6} - 70264 T_{29}^{5} + 1539392 T_{29}^{4} + 1292528592 T_{29}^{3} + 32015548800 T_{29}^{2} - 1446089536512 T_{29} - 38672535237120 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + 69 T^{12} + 1798 T^{10} + \cdots + 50176 \) Copy content Toggle raw display
$3$ \( T^{14} \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 476837158203125 \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 148986826592256 \) Copy content Toggle raw display
$11$ \( (T + 11)^{14} \) Copy content Toggle raw display
$13$ \( T^{14} + 17988 T^{12} + \cdots + 24\!\cdots\!76 \) Copy content Toggle raw display
$17$ \( T^{14} + 54892 T^{12} + \cdots + 46\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( (T^{7} + 58 T^{6} + \cdots - 46691778560000)^{2} \) Copy content Toggle raw display
$23$ \( T^{14} + 103772 T^{12} + \cdots + 40\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( (T^{7} - 64 T^{6} + \cdots - 38672535237120)^{2} \) Copy content Toggle raw display
$31$ \( (T^{7} + 348 T^{6} + \cdots + 197687760429056)^{2} \) Copy content Toggle raw display
$37$ \( T^{14} + 562632 T^{12} + \cdots + 94\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( (T^{7} - 332 T^{6} + \cdots - 26\!\cdots\!92)^{2} \) Copy content Toggle raw display
$43$ \( T^{14} + 1022676 T^{12} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{14} + 887148 T^{12} + \cdots + 74\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{14} + 1430852 T^{12} + \cdots + 61\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( (T^{7} + 332 T^{6} + \cdots + 30\!\cdots\!20)^{2} \) Copy content Toggle raw display
$61$ \( (T^{7} - 22 T^{6} + \cdots + 39\!\cdots\!12)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + 3775064 T^{12} + \cdots + 17\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( (T^{7} - 516 T^{6} + \cdots - 765697934131200)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + 4480568 T^{12} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( (T^{7} + 1746 T^{6} + \cdots + 55\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + 4133328 T^{12} + \cdots + 58\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( (T^{7} + 2226 T^{6} + \cdots + 70\!\cdots\!40)^{2} \) Copy content Toggle raw display
$97$ \( T^{14} + 3938712 T^{12} + \cdots + 40\!\cdots\!76 \) Copy content Toggle raw display
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