Properties

Label 192.9.e.f
Level $192$
Weight $9$
Character orbit 192.e
Analytic conductor $78.217$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 192.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(78.2166931317\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-14}) \)
Defining polynomial: \( x^{2} + 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: no (minimal twist has level 3)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 6\sqrt{-14}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 3 \beta + 45) q^{3} + 10 \beta q^{5} + 1750 q^{7} + ( - 270 \beta - 2511) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 3 \beta + 45) q^{3} + 10 \beta q^{5} + 1750 q^{7} + ( - 270 \beta - 2511) q^{9} + 310 \beta q^{11} - 25730 q^{13} + (450 \beta + 15120) q^{15} + 3336 \beta q^{17} + 18938 q^{19} + ( - 5250 \beta + 78750) q^{21} + 20956 \beta q^{23} + 340225 q^{25} + ( - 4617 \beta - 521235) q^{27} - 20530 \beta q^{29} + 351478 q^{31} + (13950 \beta + 468720) q^{33} + 17500 \beta q^{35} - 1335170 q^{37} + (77190 \beta - 1157850) q^{39} + 83540 \beta q^{41} - 3526150 q^{43} + ( - 25110 \beta + 1360800) q^{45} + 181784 \beta q^{47} - 2702301 q^{49} + (150120 \beta + 5044032) q^{51} + 294066 \beta q^{53} - 1562400 q^{55} + ( - 56814 \beta + 852210) q^{57} + 610910 \beta q^{59} - 753602 q^{61} + ( - 472500 \beta - 4394250) q^{63} - 257300 \beta q^{65} + 2268890 q^{67} + (943020 \beta + 31685472) q^{69} - 758220 \beta q^{71} + 27672770 q^{73} + ( - 1020675 \beta + 15310125) q^{75} + 542500 \beta q^{77} + 22980982 q^{79} + (1355940 \beta - 30436479) q^{81} - 2066606 \beta q^{83} - 16813440 q^{85} + ( - 923850 \beta - 31041360) q^{87} + 3234540 \beta q^{89} - 45027500 q^{91} + ( - 1054434 \beta + 15816510) q^{93} + 189380 \beta q^{95} + 147271010 q^{97} + ( - 778410 \beta + 42184800) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 90 q^{3} + 3500 q^{7} - 5022 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 90 q^{3} + 3500 q^{7} - 5022 q^{9} - 51460 q^{13} + 30240 q^{15} + 37876 q^{19} + 157500 q^{21} + 680450 q^{25} - 1042470 q^{27} + 702956 q^{31} + 937440 q^{33} - 2670340 q^{37} - 2315700 q^{39} - 7052300 q^{43} + 2721600 q^{45} - 5404602 q^{49} + 10088064 q^{51} - 3124800 q^{55} + 1704420 q^{57} - 1507204 q^{61} - 8788500 q^{63} + 4537780 q^{67} + 63370944 q^{69} + 55345540 q^{73} + 30620250 q^{75} + 45961964 q^{79} - 60872958 q^{81} - 33626880 q^{85} - 62082720 q^{87} - 90055000 q^{91} + 31633020 q^{93} + 294542020 q^{97} + 84369600 q^{99}+O(q^{100}) \) 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
3.74166i
3.74166i
0 45.0000 67.3498i 0 224.499i 0 1750.00 0 −2511.00 6061.48i 0
65.2 0 45.0000 + 67.3498i 0 224.499i 0 1750.00 0 −2511.00 + 6061.48i 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 192.9.e.f 2
3.b odd 2 1 inner 192.9.e.f 2
4.b odd 2 1 192.9.e.e 2
8.b even 2 1 48.9.e.b 2
8.d odd 2 1 3.9.b.a 2
12.b even 2 1 192.9.e.e 2
24.f even 2 1 3.9.b.a 2
24.h odd 2 1 48.9.e.b 2
40.e odd 2 1 75.9.c.c 2
40.k even 4 2 75.9.d.b 4
72.l even 6 2 81.9.d.d 4
72.p odd 6 2 81.9.d.d 4
120.m even 2 1 75.9.c.c 2
120.q odd 4 2 75.9.d.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.9.b.a 2 8.d odd 2 1
3.9.b.a 2 24.f even 2 1
48.9.e.b 2 8.b even 2 1
48.9.e.b 2 24.h odd 2 1
75.9.c.c 2 40.e odd 2 1
75.9.c.c 2 120.m even 2 1
75.9.d.b 4 40.k even 4 2
75.9.d.b 4 120.q odd 4 2
81.9.d.d 4 72.l even 6 2
81.9.d.d 4 72.p odd 6 2
192.9.e.e 2 4.b odd 2 1
192.9.e.e 2 12.b even 2 1
192.9.e.f 2 1.a even 1 1 trivial
192.9.e.f 2 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 \(S_{9}^{\mathrm{new}}(192, [\chi])\):

\( T_{5}^{2} + 50400 \) Copy content Toggle raw display
\( T_{7} - 1750 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 90T + 6561 \) Copy content Toggle raw display
$5$ \( T^{2} + 50400 \) Copy content Toggle raw display
$7$ \( (T - 1750)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 48434400 \) Copy content Toggle raw display
$13$ \( (T + 25730)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 5608963584 \) Copy content Toggle raw display
$19$ \( (T - 18938)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 221333583744 \) Copy content Toggle raw display
$29$ \( T^{2} + 212426373600 \) Copy content Toggle raw display
$31$ \( (T - 351478)^{2} \) Copy content Toggle raw display
$37$ \( (T + 1335170)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 3517381526400 \) Copy content Toggle raw display
$43$ \( (T + 3526150)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 16654893018624 \) Copy content Toggle raw display
$53$ \( T^{2} + 43583305427424 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 188098358162400 \) Copy content Toggle raw display
$61$ \( (T + 753602)^{2} \) Copy content Toggle raw display
$67$ \( (T - 2268890)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots + 289748374473600 \) Copy content Toggle raw display
$73$ \( (T - 27672770)^{2} \) Copy content Toggle raw display
$79$ \( (T - 22980982)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 21\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{2} + 52\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T - 147271010)^{2} \) Copy content Toggle raw display
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