Properties

Label 363.8.a.g
Level $363$
Weight $8$
Character orbit 363.a
Self dual yes
Analytic conductor $113.396$
Analytic rank $1$
Dimension $4$
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] = [363,8,Mod(1,363)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(363, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("363.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 363.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: \(113.395764251\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{89}, \sqrt{449})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 269x^{2} + 8100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2}\cdot 5 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 27 q^{3} + (\beta_{2} + 7) q^{4} + ( - 2 \beta_{2} + 48) q^{5} + 27 \beta_1 q^{6} + (\beta_{3} + 4 \beta_1) q^{7} + (\beta_{3} - 47 \beta_1) q^{8} + 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + 27 q^{3} + (\beta_{2} + 7) q^{4} + ( - 2 \beta_{2} + 48) q^{5} + 27 \beta_1 q^{6} + (\beta_{3} + 4 \beta_1) q^{7} + (\beta_{3} - 47 \beta_1) q^{8} + 729 q^{9} + ( - 2 \beta_{3} - 100 \beta_1) q^{10} + (27 \beta_{2} + 189) q^{12} + (4 \beta_{3} + 18 \beta_1) q^{13} + (64 \beta_{2} + 540) q^{14} + ( - 54 \beta_{2} + 1296) q^{15} + ( - 115 \beta_{2} - 7241) q^{16} + ( - 19 \beta_{3} + 722 \beta_1) q^{17} + 729 \beta_1 q^{18} + ( - 51 \beta_{3} - 1034 \beta_1) q^{19} + (36 \beta_{2} - 19644) q^{20} + (27 \beta_{3} + 108 \beta_1) q^{21} + ( - 350 \beta_{2} - 14058) q^{23} + (27 \beta_{3} - 1269 \beta_1) q^{24} + ( - 196 \beta_{2} - 35861) q^{25} + (258 \beta_{2} + 2430) q^{26} + 19683 q^{27} + ( - 64 \beta_{3} + 4764 \beta_1) q^{28} + ( - 135 \beta_{3} - 442 \beta_1) q^{29} + ( - 54 \beta_{3} - 2700 \beta_1) q^{30} + (2188 \beta_{2} - 860) q^{31} + ( - 243 \beta_{3} - 9735 \beta_1) q^{32} + ( - 418 \beta_{2} + 97470) q^{34} + (190 \beta_{3} - 9280 \beta_1) q^{35} + (729 \beta_{2} + 5103) q^{36} + (1340 \beta_{2} - 190154) q^{37} + ( - 4094 \beta_{2} - 139590) q^{38} + (108 \beta_{3} + 486 \beta_1) q^{39} + (292 \beta_{3} - 4180 \beta_1) q^{40} + (367 \beta_{3} - 19514 \beta_1) q^{41} + (1728 \beta_{2} + 14580) q^{42} + (243 \beta_{3} - 6802 \beta_1) q^{43} + ( - 1458 \beta_{2} + 34992) q^{45} + ( - 350 \beta_{3} - 39958 \beta_1) q^{46} + (382 \beta_{2} - 285306) q^{47} + ( - 3105 \beta_{2} - 195507) q^{48} + ( - 4004 \beta_{2} - 221983) q^{49} + ( - 196 \beta_{3} - 50365 \beta_1) q^{50} + ( - 513 \beta_{3} + 19494 \beta_1) q^{51} + ( - 254 \beta_{3} + 19218 \beta_1) q^{52} + (10014 \beta_{2} - 253788) q^{53} + 19683 \beta_1 q^{54} + ( - 7268 \beta_{2} + 574020) q^{56} + ( - 1377 \beta_{3} - 27918 \beta_1) q^{57} + ( - 8542 \beta_{2} - 59670) q^{58} + ( - 2988 \beta_{2} - 974460) q^{59} + (972 \beta_{2} - 530388) q^{60} + (1910 \beta_{3} - 138950 \beta_1) q^{61} + (2188 \beta_{3} + 161052 \beta_1) q^{62} + (729 \beta_{3} + 2916 \beta_1) q^{63} + ( - 9595 \beta_{2} - 387377) q^{64} + (756 \beta_{3} - 37320 \beta_1) q^{65} + ( - 12968 \beta_{2} + 340924) q^{67} + (2014 \beta_{3} - 25878 \beta_1) q^{68} + ( - 9450 \beta_{2} - 379566) q^{69} + (2120 \beta_{2} - 1252800) q^{70} + (15074 \beta_{2} - 2069718) q^{71} + (729 \beta_{3} - 34263 \beta_1) q^{72} + ( - 1394 \beta_{3} + 296468 \beta_1) q^{73} + (1340 \beta_{3} - 90994 \beta_1) q^{74} + ( - 5292 \beta_{2} - 968247) q^{75} + (2434 \beta_{3} - 310194 \beta_1) q^{76} + (6966 \beta_{2} + 65610) q^{78} + (629 \beta_{3} - 460104 \beta_1) q^{79} + (8732 \beta_{2} + 1950132) q^{80} + 531441 q^{81} + (2506 \beta_{2} - 2634390) q^{82} + ( - 8666 \beta_{3} - 382624 \beta_1) q^{83} + ( - 1728 \beta_{3} + 128628 \beta_1) q^{84} + ( - 5206 \beta_{3} + 96520 \beta_1) q^{85} + (7778 \beta_{2} - 918270) q^{86} + ( - 3645 \beta_{3} - 11934 \beta_1) q^{87} + (79880 \beta_{2} - 1284246) q^{89} + ( - 1458 \beta_{3} - 72900 \beta_1) q^{90} + ( - 15888 \beta_{2} + 2407320) q^{91} + ( - 16158 \beta_{2} - 3594906) q^{92} + (59076 \beta_{2} - 23220) q^{93} + (382 \beta_{3} - 257038 \beta_1) q^{94} + ( - 8030 \beta_{3} + 556280 \beta_1) q^{95} + ( - 6561 \beta_{3} - 262845 \beta_1) q^{96} + ( - 148668 \beta_{2} - 1279234) q^{97} + ( - 4004 \beta_{3} - 518279 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 108 q^{3} + 26 q^{4} + 196 q^{5} + 2916 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 108 q^{3} + 26 q^{4} + 196 q^{5} + 2916 q^{9} + 702 q^{12} + 2032 q^{14} + 5292 q^{15} - 28734 q^{16} - 78648 q^{20} - 55532 q^{23} - 143052 q^{25} + 9204 q^{26} + 78732 q^{27} - 7816 q^{31} + 390716 q^{34} + 18954 q^{36} - 763296 q^{37} - 550172 q^{38} + 54864 q^{42} + 142884 q^{45} - 1141988 q^{47} - 775818 q^{48} - 879924 q^{49} - 1035180 q^{53} + 2310616 q^{56} - 221596 q^{58} - 3891864 q^{59} - 2123496 q^{60} - 1530318 q^{64} + 1389632 q^{67} - 1499364 q^{69} - 5015440 q^{70} - 8309020 q^{71} - 3862404 q^{75} + 248508 q^{78} + 7783064 q^{80} + 2125764 q^{81} - 10542572 q^{82} - 3688636 q^{86} - 5296744 q^{89} + 9661056 q^{91} - 14347308 q^{92} - 211032 q^{93} - 4819600 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 269x^{2} + 8100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 135 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 209\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 135 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 209\beta_1 \) 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
−15.3118
−5.87782
5.87782
15.3118
−15.3118 27.0000 106.451 −150.902 −413.419 −450.951 329.950 729.000 2310.59
1.2 −5.87782 27.0000 −93.4512 248.902 −158.701 1001.88 1301.65 729.000 −1463.00
1.3 5.87782 27.0000 −93.4512 248.902 158.701 −1001.88 −1301.65 729.000 1463.00
1.4 15.3118 27.0000 106.451 −150.902 413.419 450.951 −329.950 729.000 −2310.59
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(11\) \(1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.8.a.g 4
11.b odd 2 1 inner 363.8.a.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.8.a.g 4 1.a even 1 1 trivial
363.8.a.g 4 11.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 269T_{2}^{2} + 8100 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(363))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 269T^{2} + 8100 \) Copy content Toggle raw display
$3$ \( (T - 27)^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - 98 T - 37560)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 1207124 T^{2} + \cdots + 204123240000 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 19331316 T^{2} + \cdots + 53728313601600 \) Copy content Toggle raw display
$17$ \( T^{4} - 576262856 T^{2} + \cdots + 75\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{4} - 3411058184 T^{2} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T^{2} + 27766 T - 1031067936)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} - 21975535016 T^{2} + \cdots + 65\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{2} + 3908 T - 47822945280)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 381648 T + 18475306076)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 265364846984 T^{2} + \cdots + 58\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{4} - 83697885896 T^{2} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( (T^{2} + 570994 T + 80050719768)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 517590 T - 934849376064)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 1945932 T + 857468446560)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} - 9615216642500 T^{2} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{2} - 694816 T - 1559358273552)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 4154510 T + 2044949023416)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 26031180027296 T^{2} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{4} - 57456944294724 T^{2} + \cdots + 51\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{4} - 129351300110864 T^{2} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{2} + 2648372 T - 61992462497004)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 2409800 T - 219354462031316)^{2} \) Copy content Toggle raw display
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