Properties

Label 192.11.e.h
Level $192$
Weight $11$
Character orbit 192.e
Analytic conductor $121.989$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 192.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(121.988592513\)
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 \( q + (\beta_1 + 21) q^{3} + (\beta_{3} + 2 \beta_{2} - 4 \beta_1) q^{5} + ( - \beta_{3} + 5 \beta_{2} - 43 \beta_1 + 11278) q^{7} + (7 \beta_{3} - 62 \beta_{2} - 20 \beta_1 + 39753) q^{9} + (35 \beta_{3} + 189 \beta_{2} - 21 \beta_1) q^{11} + ( - 20 \beta_{3} + 100 \beta_{2} - 860 \beta_1 - 68810) q^{13} + (35 \beta_{3} + 1877 \beta_{2} + 743 \beta_1 + 295200) q^{15} + ( - 236 \beta_{3} + 2860 \beta_{2} + 4276 \beta_1) q^{17} + ( - 97 \beta_{3} + 485 \beta_{2} - 4171 \beta_1 - 392182) q^{19} + ( - 189 \beta_{3} + 1674 \beta_{2} + 12952 \beta_1 - 2407002) q^{21} + ( - 826 \beta_{3} - 4590 \beta_{2} + 366 \beta_1) q^{23} + ( - 420 \beta_{3} + 2100 \beta_{2} - 18060 \beta_1 - 8433095) q^{25} + ( - 1599 \beta_{3} + 5727 \beta_{2} + 39306 \beta_1 + 8654877) q^{27} + (2947 \beta_{3} + 34494 \beta_{2} + 16812 \beta_1) q^{29} + ( - 3045 \beta_{3} + 15225 \beta_{2} - 130935 \beta_1 + 5446462) q^{31} + (5033 \beta_{3} + 69146 \beta_{2} + 29456 \beta_1 + 6493536) q^{33} + (9598 \beta_{3} - 71914 \beta_{2} - 129502 \beta_1) q^{35} + (5684 \beta_{3} - 28420 \beta_{2} + 244412 \beta_1 + 17753542) q^{37} + ( - 3780 \beta_{3} + 33480 \beta_{2} - 35330 \beta_1 - 54321810) q^{39} + ( - 13414 \beta_{3} + 158308 \beta_{2} + 238792 \beta_1) q^{41} + (6783 \beta_{3} - 33915 \beta_{2} + 291669 \beta_1 - 117672166) q^{43} + (52581 \beta_{3} + 175386 \beta_{2} + 411492 \beta_1 - 78079680) q^{45} + (14460 \beta_{3} - 159940 \beta_{2} - 246700 \beta_1) q^{47} + ( - 22556 \beta_{3} + 112780 \beta_{2} - 969908 \beta_1 - 12514605) q^{49} + (98364 \beta_{3} - 346344 \beta_{2} - 78720 \beta_1 - 177144192) q^{51} + ( - 59339 \beta_{3} - 437750 \beta_{2} - 81716 \beta_1) q^{53} + ( - 37548 \beta_{3} + 187740 \beta_{2} + \cdots - 675339840) q^{55}+ \cdots + (2000271 \beta_{3} + 11748765 \beta_{2} + \cdots - 656728128) q^{99}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( ( -60\nu^{3} + 276\nu^{2} + 2172\nu - 4728 ) / 31 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -196\nu^{3} + 108\nu^{2} + 1540\nu + 2808 ) / 31 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -128\nu^{3} - 8736\nu^{2} + 19712\nu + 164208 ) / 31 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 4\beta_{3} - 47\beta_{2} + 145\beta _1 + 5184 ) / 10368 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -7\beta_{3} - 19\beta_{2} + 77\beta _1 + 50544 ) / 2592 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 16\beta_{3} - 2051\beta_{2} + 1309\beta _1 + 300672 ) / 10368 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/192\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(133\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
−4.10977 + 1.41421i
−4.10977 1.41421i
5.10977 1.41421i
5.10977 + 1.41421i
0 −200.269 137.627i 0 3630.47i 0 23226.5 0 21166.4 + 55125.0i 0
65.2 0 −200.269 + 137.627i 0 3630.47i 0 23226.5 0 21166.4 55125.0i 0
65.3 0 242.269 18.8335i 0 4818.41i 0 −670.530 0 58339.6 9125.53i 0
65.4 0 242.269 + 18.8335i 0 4818.41i 0 −670.530 0 58339.6 + 9125.53i 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 192.11.e.h 4
3.b odd 2 1 inner 192.11.e.h 4
4.b odd 2 1 192.11.e.g 4
8.b even 2 1 48.11.e.d 4
8.d odd 2 1 6.11.b.a 4
12.b even 2 1 192.11.e.g 4
24.f even 2 1 6.11.b.a 4
24.h odd 2 1 48.11.e.d 4
40.e odd 2 1 150.11.d.a 4
40.k even 4 2 150.11.b.a 8
72.l even 6 2 162.11.d.d 8
72.p odd 6 2 162.11.d.d 8
120.m even 2 1 150.11.d.a 4
120.q 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 8.d odd 2 1
6.11.b.a 4 24.f even 2 1
48.11.e.d 4 8.b even 2 1
48.11.e.d 4 24.h odd 2 1
150.11.b.a 8 40.k even 4 2
150.11.b.a 8 120.q odd 4 2
150.11.d.a 4 40.e odd 2 1
150.11.d.a 4 120.m even 2 1
162.11.d.d 8 72.l even 6 2
162.11.d.d 8 72.p odd 6 2
192.11.e.g 4 4.b odd 2 1
192.11.e.g 4 12.b even 2 1
192.11.e.h 4 1.a even 1 1 trivial
192.11.e.h 4 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{11}^{\mathrm{new}}(192, [\chi])\):

\( T_{5}^{4} + 36397440T_{5}^{2} + 306009247334400 \) Copy content Toggle raw display
\( T_{7}^{2} - 22556T_{7} - 15574076 \) 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
show more
show less