Properties

Label 384.9.b.c
Level $384$
Weight $9$
Character orbit 384.b
Analytic conductor $156.433$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [384,9,Mod(319,384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(384, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.319");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 384.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: \(156.433386263\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} + 2 x^{14} + 4280 x^{13} - 146457 x^{12} - 603388 x^{11} + 10658890 x^{10} - 278740622 x^{9} + 17482533993 x^{8} - 147565541732 x^{7} + \cdots + 10\!\cdots\!36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{120}\cdot 3^{24} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{2} q^{5} + \beta_{5} q^{7} + 2187 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{2} q^{5} + \beta_{5} q^{7} + 2187 q^{9} + (\beta_{10} + 53 \beta_1) q^{11} + ( - \beta_{11} - 26 \beta_{3} + 16 \beta_{2}) q^{13} + ( - 3 \beta_{9} + 2 \beta_{8} + 5 \beta_{5}) q^{15} + (\beta_{4} - 13458) q^{17} + ( - \beta_{12} + 3 \beta_{10} + 967 \beta_1) q^{19} + (\beta_{15} - \beta_{11} + 35 \beta_{3} - 78 \beta_{2}) q^{21} + ( - 2 \beta_{14} + 25 \beta_{9} + \beta_{8} - 54 \beta_{5}) q^{23} + (\beta_{7} + 3 \beta_{6} + 4 \beta_{4} - 127311) q^{25} + 2187 \beta_1 q^{27} + (3 \beta_{15} - 182 \beta_{3} + 438 \beta_{2}) q^{29} + (13 \beta_{14} + 5 \beta_{9} - 10 \beta_{8} + 76 \beta_{5}) q^{31} + (4 \beta_{7} + \beta_{6} + 2 \beta_{4} + 115020) q^{33} + (13 \beta_{13} + 2 \beta_{12} - 35 \beta_{10} - 19044 \beta_1) q^{35} + ( - 10 \beta_{15} - \beta_{11} - 999 \beta_{3} - 496 \beta_{2}) q^{37} + ( - 27 \beta_{14} + 144 \beta_{9} - 63 \beta_{5}) q^{39} + (19 \beta_{7} + 10 \beta_{6} + 20 \beta_{4} + 25506) q^{41} + (21 \beta_{13} - 3 \beta_{12} - 165 \beta_{10} - 4986 \beta_1) q^{43} - 2187 \beta_{2} q^{45} + (16 \beta_{14} - 575 \beta_{9} + 187 \beta_{8} - 240 \beta_{5}) q^{47} + ( - 37 \beta_{7} + 27 \beta_{6} + 56 \beta_{4} - 2993807) q^{49} + (18 \beta_{13} - 9 \beta_{12} + 63 \beta_{10} - 13437 \beta_1) q^{51} + ( - 45 \beta_{15} + 132 \beta_{11} - 7704 \beta_{3} + 5812 \beta_{2}) q^{53} + ( - 9 \beta_{14} + 309 \beta_{9} - 324 \beta_{8} + 919 \beta_{5}) q^{55} + (18 \beta_{7} - 36 \beta_{6} + 9 \beta_{4} + 2112156) q^{57} + (49 \beta_{13} - 58 \beta_{12} + 356 \beta_{10} + 84083 \beta_1) q^{59} + (104 \beta_{15} + 5 \beta_{11} - 9171 \beta_{3} - 6502 \beta_{2}) q^{61} + 2187 \beta_{5} q^{63} + (11 \beta_{7} - 106 \beta_{6} - 288 \beta_{4} + 8941728) q^{65} + ( - 3 \beta_{13} - 94 \beta_{12} - 480 \beta_{10} + 34147 \beta_1) q^{67} + ( - 36 \beta_{15} - 126 \beta_{11} - 13167 \beta_{3} + 8154 \beta_{2}) q^{69} + (254 \beta_{14} + 506 \beta_{9} - 748 \beta_{8} - 8136 \beta_{5}) q^{71} + ( - 163 \beta_{7} - 33 \beta_{6} - 238 \beta_{4} - 5142946) q^{73} + (63 \beta_{13} + 117 \beta_{12} + 990 \beta_{10} - 126924 \beta_1) q^{75} + (165 \beta_{15} - 858 \beta_{11} - 50596 \beta_{3} - 18211 \beta_{2}) q^{77} + ( - 548 \beta_{14} + 1130 \beta_{9} + 2342 \beta_{8} + \cdots + 1071 \beta_{5}) q^{79}+ \cdots + (2187 \beta_{10} + 115911 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 34992 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 34992 q^{9} - 215328 q^{17} - 2036976 q^{25} + 1840320 q^{33} + 408096 q^{41} - 47900912 q^{49} + 33794496 q^{57} + 143067648 q^{65} - 82287136 q^{73} + 76527504 q^{81} - 267298080 q^{89} + 492934880 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + 2 x^{14} + 4280 x^{13} - 146457 x^{12} - 603388 x^{11} + 10658890 x^{10} - 278740622 x^{9} + 17482533993 x^{8} - 147565541732 x^{7} + \cdots + 10\!\cdots\!36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 21\!\cdots\!66 \nu^{15} + \cdots + 21\!\cdots\!44 ) / 23\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 25\!\cdots\!79 \nu^{15} + \cdots - 73\!\cdots\!08 ) / 92\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 93\!\cdots\!14 \nu^{15} + \cdots - 28\!\cdots\!04 ) / 11\!\cdots\!65 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 47\!\cdots\!27 \nu^{15} + \cdots + 40\!\cdots\!12 ) / 64\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 11\!\cdots\!26 \nu^{15} + \cdots + 32\!\cdots\!44 ) / 92\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 36\!\cdots\!37 \nu^{15} + \cdots + 22\!\cdots\!88 ) / 21\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 30\!\cdots\!67 \nu^{15} + \cdots + 26\!\cdots\!16 ) / 12\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 55\!\cdots\!86 \nu^{15} + \cdots + 16\!\cdots\!44 ) / 23\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 17\!\cdots\!14 \nu^{15} + \cdots + 52\!\cdots\!36 ) / 66\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 16\!\cdots\!54 \nu^{15} + \cdots + 14\!\cdots\!96 ) / 37\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 10\!\cdots\!51 \nu^{15} + \cdots - 30\!\cdots\!16 ) / 17\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 40\!\cdots\!46 \nu^{15} + \cdots - 13\!\cdots\!44 ) / 45\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 64\!\cdots\!62 \nu^{15} + \cdots + 59\!\cdots\!28 ) / 37\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 55\!\cdots\!94 \nu^{15} + \cdots - 14\!\cdots\!76 ) / 30\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 96\!\cdots\!73 \nu^{15} + \cdots + 29\!\cdots\!28 ) / 46\!\cdots\!80 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 8 \beta_{15} - 216 \beta_{14} + 117 \beta_{13} + 36 \beta_{12} + 208 \beta_{11} - 684 \beta_{10} + 1548 \beta_{9} - 1188 \beta_{8} - 61 \beta_{7} + 32 \beta_{6} - 2736 \beta_{5} + 145 \beta_{4} - 2420 \beta_{3} + \cdots + 331776 ) / 2654208 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 264 \beta_{15} - 237 \beta_{13} - 228 \beta_{12} - 48 \beta_{11} - 2868 \beta_{10} + 84696 \beta_{3} - 24408 \beta_{2} + 1653677 \beta_1 ) / 442368 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 8408 \beta_{15} - 38896 \beta_{11} - 14647 \beta_{7} + 11168 \beta_{6} + 62323 \beta_{4} - 1401820 \beta_{3} + 6184776 \beta_{2} - 1065332736 ) / 1327104 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 44352 \beta_{14} - 906591 \beta_{9} - 101365 \beta_{8} - 1284 \beta_{7} - 22344 \beta_{6} + 2849450 \beta_{5} - 52212 \beta_{4} + 3812520960 ) / 110592 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 4997192 \beta_{15} - 17724312 \beta_{14} + 18799029 \beta_{13} + 3604644 \beta_{12} - 12727120 \beta_{11} - 8924076 \beta_{10} + 351630108 \beta_{9} + \cdots + 736327397376 ) / 2654208 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 598963\beta_{15} + 604430\beta_{11} + 149566413\beta_{3} - 114827825\beta_{2} ) / 3072 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 2620967912 \beta_{15} - 7713501048 \beta_{14} - 7905120633 \beta_{13} - 2079841716 \beta_{12} - 5092533136 \beta_{11} - 3189941604 \beta_{10} + \cdots - 385003974610944 ) / 2654208 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 9815092560 \beta_{14} - 185372224617 \beta_{9} + 11733550541 \beta_{8} - 883275090 \beta_{7} + 4019616120 \beta_{6} + 529598247158 \beta_{5} + \cdots - 621453306571776 ) / 110592 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 1279858131128 \beta_{15} - 2180103583408 \beta_{11} + 637283578531 \beta_{7} - 1232406546272 \beta_{6} - 4798437117775 \beta_{4} + \cdots + 18\!\cdots\!24 ) / 1327104 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 7627901680104 \beta_{15} + 15441707690697 \beta_{13} + 7200043504788 \beta_{12} + 9954113438352 \beta_{11} + 36311405891748 \beta_{10} + \cdots - 39\!\cdots\!17 \beta_1 ) / 442368 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 601574666263448 \beta_{15} + \cdots + 86\!\cdots\!96 ) / 2654208 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( -3434913029488\beta_{7} + 10623601165857\beta_{6} + 35074591332841\beta_{4} - 1624639113985587840 ) / 768 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 27\!\cdots\!72 \beta_{15} + \cdots + 39\!\cdots\!36 ) / 2654208 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 14\!\cdots\!96 \beta_{15} + \cdots - 76\!\cdots\!51 \beta_1 ) / 442368 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 12\!\cdots\!72 \beta_{15} + \cdots + 17\!\cdots\!24 ) / 1327104 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
319.1
5.48360 20.4651i
−4.13855 15.4453i
−2.50847 + 9.36173i
0.797390 + 2.97590i
0.797390 2.97590i
−2.50847 9.36173i
−4.13855 + 15.4453i
5.48360 + 20.4651i
−20.4651 + 5.48360i
15.4453 + 4.13855i
9.36173 2.50847i
−2.97590 0.797390i
−2.97590 + 0.797390i
9.36173 + 2.50847i
15.4453 4.13855i
−20.4651 5.48360i
0 −46.7654 0 1083.75i 0 4332.97i 0 2187.00 0
319.2 0 −46.7654 0 907.072i 0 1761.88i 0 2187.00 0
319.3 0 −46.7654 0 215.156i 0 640.287i 0 2187.00 0
319.4 0 −46.7654 0 167.830i 0 3570.09i 0 2187.00 0
319.5 0 −46.7654 0 167.830i 0 3570.09i 0 2187.00 0
319.6 0 −46.7654 0 215.156i 0 640.287i 0 2187.00 0
319.7 0 −46.7654 0 907.072i 0 1761.88i 0 2187.00 0
319.8 0 −46.7654 0 1083.75i 0 4332.97i 0 2187.00 0
319.9 0 46.7654 0 1083.75i 0 4332.97i 0 2187.00 0
319.10 0 46.7654 0 907.072i 0 1761.88i 0 2187.00 0
319.11 0 46.7654 0 215.156i 0 640.287i 0 2187.00 0
319.12 0 46.7654 0 167.830i 0 3570.09i 0 2187.00 0
319.13 0 46.7654 0 167.830i 0 3570.09i 0 2187.00 0
319.14 0 46.7654 0 215.156i 0 640.287i 0 2187.00 0
319.15 0 46.7654 0 907.072i 0 1761.88i 0 2187.00 0
319.16 0 46.7654 0 1083.75i 0 4332.97i 0 2187.00 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 319.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.9.b.c 16
4.b odd 2 1 inner 384.9.b.c 16
8.b even 2 1 inner 384.9.b.c 16
8.d odd 2 1 inner 384.9.b.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.9.b.c 16 1.a even 1 1 trivial
384.9.b.c 16 4.b odd 2 1 inner
384.9.b.c 16 8.b even 2 1 inner
384.9.b.c 16 8.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 2071744T_{5}^{6} + 1116378502656T_{5}^{4} + 74558195297075200T_{5}^{2} + 1260034623602728960000 \) acting on \(S_{9}^{\mathrm{new}}(384, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{2} - 2187)^{8} \) Copy content Toggle raw display
$5$ \( (T^{8} + 2071744 T^{6} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + 35034432 T^{6} + \cdots + 30\!\cdots\!56)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} - 672057024 T^{6} + \cdots + 13\!\cdots\!04)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 5452084224 T^{6} + \cdots + 43\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 53832 T^{3} + \cdots + 15\!\cdots\!36)^{4} \) Copy content Toggle raw display
$19$ \( (T^{8} - 81692003520 T^{6} + \cdots + 12\!\cdots\!64)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 232142554368 T^{6} + \cdots + 32\!\cdots\!96)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 1006791742144 T^{6} + \cdots + 12\!\cdots\!64)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 2311568773440 T^{6} + \cdots + 13\!\cdots\!04)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 7665737345280 T^{6} + \cdots + 20\!\cdots\!16)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 102024 T^{3} + \cdots - 98\!\cdots\!64)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} - 69312972148416 T^{6} + \cdots + 16\!\cdots\!76)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 76848245427456 T^{6} + \cdots + 51\!\cdots\!76)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 316051586123968 T^{6} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 587728140311232 T^{6} + \cdots + 14\!\cdots\!96)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 835172191745280 T^{6} + \cdots + 20\!\cdots\!24)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} - 741259303895232 T^{6} + \cdots + 43\!\cdots\!04)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 87\!\cdots\!76)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 20571784 T^{3} + \cdots - 79\!\cdots\!92)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 85\!\cdots\!76)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 67\!\cdots\!76)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 66824520 T^{3} + \cdots - 44\!\cdots\!24)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} - 123233720 T^{3} + \cdots + 33\!\cdots\!00)^{4} \) Copy content Toggle raw display
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