Properties

Label 48.11.e.d
Level $48$
Weight $11$
Character orbit 48.e
Analytic conductor $30.497$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 48.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(30.4971481283\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{85})\)
Defining polynomial: \( x^{4} - 2x^{3} - 37x^{2} + 38x + 531 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{6} \)
Twist minimal: no (minimal twist has level 6)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 21) q^{3} + ( - \beta_{3} - 2 \beta_{2} + 4 \beta_1) q^{5} + ( - \beta_{3} + 5 \beta_{2} - 43 \beta_1 + 11278) q^{7} + (7 \beta_{3} - 62 \beta_{2} - 20 \beta_1 + 39753) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 21) q^{3} + ( - \beta_{3} - 2 \beta_{2} + 4 \beta_1) q^{5} + ( - \beta_{3} + 5 \beta_{2} - 43 \beta_1 + 11278) q^{7} + (7 \beta_{3} - 62 \beta_{2} - 20 \beta_1 + 39753) q^{9} + ( - 35 \beta_{3} - 189 \beta_{2} + 21 \beta_1) q^{11} + (20 \beta_{3} - 100 \beta_{2} + 860 \beta_1 + 68810) q^{13} + (35 \beta_{3} + 1877 \beta_{2} + 743 \beta_1 + 295200) q^{15} + ( - 236 \beta_{3} + 2860 \beta_{2} + 4276 \beta_1) q^{17} + (97 \beta_{3} - 485 \beta_{2} + 4171 \beta_1 + 392182) q^{19} + (189 \beta_{3} - 1674 \beta_{2} - 12952 \beta_1 + 2407002) q^{21} + ( - 826 \beta_{3} - 4590 \beta_{2} + 366 \beta_1) q^{23} + ( - 420 \beta_{3} + 2100 \beta_{2} - 18060 \beta_1 - 8433095) q^{25} + (1599 \beta_{3} - 5727 \beta_{2} - 39306 \beta_1 - 8654877) q^{27} + ( - 2947 \beta_{3} - 34494 \beta_{2} - 16812 \beta_1) q^{29} + ( - 3045 \beta_{3} + 15225 \beta_{2} - 130935 \beta_1 + 5446462) q^{31} + (5033 \beta_{3} + 69146 \beta_{2} + 29456 \beta_1 + 6493536) q^{33} + ( - 9598 \beta_{3} + 71914 \beta_{2} + 129502 \beta_1) q^{35} + ( - 5684 \beta_{3} + 28420 \beta_{2} - 244412 \beta_1 - 17753542) q^{37} + ( - 3780 \beta_{3} + 33480 \beta_{2} - 35330 \beta_1 - 54321810) q^{39} + ( - 13414 \beta_{3} + 158308 \beta_{2} + 238792 \beta_1) q^{41} + ( - 6783 \beta_{3} + 33915 \beta_{2} - 291669 \beta_1 + 117672166) q^{43} + ( - 52581 \beta_{3} - 175386 \beta_{2} - 411492 \beta_1 + 78079680) q^{45} + (14460 \beta_{3} - 159940 \beta_{2} - 246700 \beta_1) q^{47} + ( - 22556 \beta_{3} + 112780 \beta_{2} - 969908 \beta_1 - 12514605) q^{49} + ( - 98364 \beta_{3} + 346344 \beta_{2} + \cdots + 177144192) q^{51}+ \cdots + ( - 2000271 \beta_{3} - 11748765 \beta_{2} + \cdots + 656728128) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 84 q^{3} + 45112 q^{7} + 159012 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 84 q^{3} + 45112 q^{7} + 159012 q^{9} + 275240 q^{13} + 1180800 q^{15} + 1568728 q^{19} + 9628008 q^{21} - 33732380 q^{25} - 34619508 q^{27} + 21785848 q^{31} + 25974144 q^{33} - 71014168 q^{37} - 217287240 q^{39} + 470688664 q^{43} + 312318720 q^{45} - 50058420 q^{49} + 708576768 q^{51} - 2701359360 q^{55} - 1058753208 q^{57} - 1184038744 q^{61} + 905007096 q^{63} + 297365848 q^{67} + 596268288 q^{69} + 6534269000 q^{73} + 5150031180 q^{75} - 199282568 q^{79} + 1458964548 q^{81} - 12880512000 q^{85} - 210268800 q^{87} - 8317232080 q^{91} + 31744468392 q^{93} - 39176355064 q^{97} + 2626912512 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 37x^{2} + 38x + 531 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -4\nu^{3} - 180\nu^{2} + 1732\nu + 2760 ) / 31 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 260\nu^{3} - 204\nu^{2} - 5444\nu - 840 ) / 31 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -256\nu^{3} + 9312\nu^{2} + 3712\nu - 176016 ) / 31 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 4\beta_{3} + 7\beta_{2} + 199\beta _1 + 5184 ) / 10368 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} + \beta _1 + 5616 ) / 288 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 112\beta_{3} + 1411\beta_{2} + 4195\beta _1 + 300672 ) / 10368 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/48\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(31\) \(37\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
5.10977 1.41421i
5.10977 + 1.41421i
−4.10977 + 1.41421i
−4.10977 1.41421i
0 −242.269 18.8335i 0 4818.41i 0 −670.530 0 58339.6 + 9125.53i 0
17.2 0 −242.269 + 18.8335i 0 4818.41i 0 −670.530 0 58339.6 9125.53i 0
17.3 0 200.269 137.627i 0 3630.47i 0 23226.5 0 21166.4 55125.0i 0
17.4 0 200.269 + 137.627i 0 3630.47i 0 23226.5 0 21166.4 + 55125.0i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.11.e.d 4
3.b odd 2 1 inner 48.11.e.d 4
4.b odd 2 1 6.11.b.a 4
8.b even 2 1 192.11.e.h 4
8.d odd 2 1 192.11.e.g 4
12.b even 2 1 6.11.b.a 4
20.d odd 2 1 150.11.d.a 4
20.e even 4 2 150.11.b.a 8
24.f even 2 1 192.11.e.g 4
24.h odd 2 1 192.11.e.h 4
36.f odd 6 2 162.11.d.d 8
36.h even 6 2 162.11.d.d 8
60.h even 2 1 150.11.d.a 4
60.l odd 4 2 150.11.b.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.11.b.a 4 4.b odd 2 1
6.11.b.a 4 12.b even 2 1
48.11.e.d 4 1.a even 1 1 trivial
48.11.e.d 4 3.b odd 2 1 inner
150.11.b.a 8 20.e even 4 2
150.11.b.a 8 60.l odd 4 2
150.11.d.a 4 20.d odd 2 1
150.11.d.a 4 60.h even 2 1
162.11.d.d 8 36.f odd 6 2
162.11.d.d 8 36.h even 6 2
192.11.e.g 4 8.d odd 2 1
192.11.e.g 4 24.f even 2 1
192.11.e.h 4 8.b even 2 1
192.11.e.h 4 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 36397440T_{5}^{2} + 306009247334400 \) acting on \(S_{11}^{\mathrm{new}}(48, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 84 T^{3} + \cdots + 3486784401 \) Copy content Toggle raw display
$5$ \( T^{4} + \cdots + 306009247334400 \) Copy content Toggle raw display
$7$ \( (T^{2} - 22556 T - 15574076)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 58313211264 T^{2} + \cdots + 21\!\cdots\!24 \) Copy content Toggle raw display
$13$ \( (T^{2} - 137620 T - 52372127900)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 7560967182336 T^{2} + \cdots + 32\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( (T^{2} - 784364 T - 1189491369116)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 33055507478016 T^{2} + \cdots + 61\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( T^{4} + 907304099736960 T^{2} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{2} - 10892924 T - 12\!\cdots\!56)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 35507084 T - 42\!\cdots\!96)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 28\!\cdots\!04 \) Copy content Toggle raw display
$43$ \( (T^{2} - 235344332 T + 72\!\cdots\!16)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 37\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 20\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( (T^{2} + 592019372 T + 32\!\cdots\!36)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 148682924 T - 35\!\cdots\!96)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 49\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{2} - 3267134500 T + 23\!\cdots\!40)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 99641284 T - 36\!\cdots\!76)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 57\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 15\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( (T^{2} + 19588177532 T + 95\!\cdots\!96)^{2} \) Copy content Toggle raw display
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