Properties

Label 7.11.b.b
Level $7$
Weight $11$
Character orbit 7.b
Analytic conductor $4.448$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 7 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 7.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.44750076872\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.373770240.2
Defining polynomial: \( x^{4} + 368x^{2} + 2760 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{7}\cdot 3\cdot 5\cdot 7 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + (\beta_{2} - 12) q^{2} - \beta_1 q^{3} + ( - 24 \beta_{2} - 144) q^{4} + ( - \beta_{3} + 2 \beta_1) q^{5} + ( - 3 \beta_{3} + 11 \beta_1) q^{6} + ( - 7 \beta_{3} + 147 \beta_{2} + 1225) q^{7} + ( - 880 \beta_{2} - 3648) q^{8} + ( - 90 \beta_{2} - 7191) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - 12) q^{2} - \beta_1 q^{3} + ( - 24 \beta_{2} - 144) q^{4} + ( - \beta_{3} + 2 \beta_1) q^{5} + ( - 3 \beta_{3} + 11 \beta_1) q^{6} + ( - 7 \beta_{3} + 147 \beta_{2} + 1225) q^{7} + ( - 880 \beta_{2} - 3648) q^{8} + ( - 90 \beta_{2} - 7191) q^{9} + (19 \beta_{3} - 267 \beta_1) q^{10} + (4204 \beta_{2} + 21498) q^{11} + (72 \beta_{3} + 168 \beta_1) q^{12} + (23 \beta_{3} + 1848 \beta_1) q^{13} + (91 \beta_{3} - 539 \beta_{2} - 1715 \beta_1 + 93492) q^{14} + ( - 21870 \beta_{2} + 132480) q^{15} + (31488 \beta_{2} - 456448) q^{16} + ( - 342 \beta_{3} - 3606 \beta_1) q^{17} + ( - 6111 \beta_{2} + 20052) q^{18} + ( - 942 \beta_{3} + 3033 \beta_1) q^{19} + ( - 24 \beta_{3} + 5544 \beta_1) q^{20} + ( - 441 \beta_{3} - 154350 \beta_{2} - 1372 \beta_1) q^{21} + ( - 28950 \beta_{2} + 2836168) q^{22} + (263824 \beta_{2} - 4815438) q^{23} + (2640 \beta_{3} + 4528 \beta_1) q^{24} + (95190 \beta_{2} + 4091065) q^{25} + (5245 \beta_{3} - 14693 \beta_1) q^{26} + (270 \beta_{3} - 51768 \beta_1) q^{27} + (840 \beta_{3} - 50568 \beta_{2} + 41160 \beta_1 - 2773008) q^{28} + ( - 562604 \beta_{2} + 5761002) q^{29} + (394920 \beta_{2} - 17686080) q^{30} + ( - 10834 \beta_{3} + 102402 \beta_1) q^{31} + (66816 \beta_{2} + 32388096) q^{32} + ( - 12612 \beta_{3} - 25702 \beta_1) q^{33} + ( - 6372 \beta_{3} - 44124 \beta_1) q^{34} + ( - 196 \beta_{3} + 360150 \beta_{2} - 33271 \beta_1 - 37867200) q^{35} + (185544 \beta_{2} + 2625264) q^{36} + ( - 2341884 \beta_{2} - 43729222) q^{37} + (21345 \beta_{3} - 264153 \beta_1) q^{38} + (673470 \beta_{2} + 122411520) q^{39} + ( - 2512 \beta_{3} + 206544 \beta_1) q^{40} + (38870 \beta_{3} + 346976 \beta_1) q^{41} + (1617 \beta_{3} + 1852200 \beta_{2} - 92953 \beta_1 - 113601600) q^{42} + (5694024 \beta_{2} + 40789562) q^{43} + ( - 1121328 \beta_{2} - 77355168) q^{44} + (6561 \beta_{3} + 7488 \beta_1) q^{45} + ( - 7981326 \beta_{2} + 251959720) q^{46} + (83878 \beta_{3} - 361394 \beta_1) q^{47} + ( - 94464 \beta_{3} + 424960 \beta_1) q^{48} + ( - 15092 \beta_{3} + 720300 \beta_{2} + \cdots - 247665551) q^{49}+ \cdots + ( - 32165784 \beta_{2} - 433065078) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 48 q^{2} - 576 q^{4} + 4900 q^{7} - 14592 q^{8} - 28764 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 48 q^{2} - 576 q^{4} + 4900 q^{7} - 14592 q^{8} - 28764 q^{9} + 85992 q^{11} + 373968 q^{14} + 529920 q^{15} - 1825792 q^{16} + 80208 q^{18} + 11344672 q^{22} - 19261752 q^{23} + 16364260 q^{25} - 11092032 q^{28} + 23044008 q^{29} - 70744320 q^{30} + 129552384 q^{32} - 151468800 q^{35} + 10501056 q^{36} - 174916888 q^{37} + 489646080 q^{39} - 454406400 q^{42} + 163158248 q^{43} - 309420672 q^{44} + 1007838880 q^{46} - 990662204 q^{49} + 83868240 q^{50} - 955445760 q^{51} + 1693931880 q^{53} - 398711040 q^{56} + 803623680 q^{57} - 1932834272 q^{58} + 1468938240 q^{60} - 74185020 q^{63} + 511688704 q^{64} - 481608960 q^{65} - 2064766168 q^{67} + 2877907200 q^{70} + 1129059912 q^{71} + 338095872 q^{72} - 4795503840 q^{74} + 1924696872 q^{77} - 3893057280 q^{78} + 10836295688 q^{79} - 15414934716 q^{81} + 10905753600 q^{84} - 5489441280 q^{85} + 14805307680 q^{86} - 11205085696 q^{88} + 3483782400 q^{91} - 15867056256 q^{92} + 27132433920 q^{93} - 21990620160 q^{95} + 14008509648 q^{98} - 1732260312 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 368x^{2} + 2760 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -3\nu^{3} - 1254\nu ) / 13 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{2} + 368 ) / 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -35\nu^{3} - 10990\nu ) / 13 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 3\beta_{3} - 35\beta_1 ) / 840 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 13\beta_{2} - 368 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -627\beta_{3} + 5495\beta_1 ) / 420 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6.1
18.9826i
18.9826i
2.76757i
2.76757i
−39.1293 252.583i 507.104 2873.50i 9883.42i −2763.01 + 16578.3i 20225.8 −4749.36 112438.i
6.2 −39.1293 252.583i 507.104 2873.50i 9883.42i −2763.01 16578.3i 20225.8 −4749.36 112438.i
6.3 15.1293 262.072i −795.104 1758.44i 3964.97i 5213.01 15978.1i −27521.8 −9632.64 26604.0i
6.4 15.1293 262.072i −795.104 1758.44i 3964.97i 5213.01 + 15978.1i −27521.8 −9632.64 26604.0i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7.11.b.b 4
3.b odd 2 1 63.11.d.c 4
4.b odd 2 1 112.11.c.b 4
7.b odd 2 1 inner 7.11.b.b 4
7.c even 3 2 49.11.d.b 8
7.d odd 6 2 49.11.d.b 8
21.c even 2 1 63.11.d.c 4
28.d even 2 1 112.11.c.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.11.b.b 4 1.a even 1 1 trivial
7.11.b.b 4 7.b odd 2 1 inner
49.11.d.b 8 7.c even 3 2
49.11.d.b 8 7.d odd 6 2
63.11.d.c 4 3.b odd 2 1
63.11.d.c 4 21.c even 2 1
112.11.c.b 4 4.b odd 2 1
112.11.c.b 4 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 24T_{2} - 592 \) acting on \(S_{11}^{\mathrm{new}}(7, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 24 T - 592)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 132480 T^{2} + \cdots + 4381776000 \) Copy content Toggle raw display
$5$ \( T^{4} + 11349120 T^{2} + \cdots + 25531635024000 \) Copy content Toggle raw display
$7$ \( T^{4} - 4900 T^{3} + \cdots + 79\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( (T^{2} - 42996 T - 12545617372)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 458156334720 T^{2} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{4} + 2988125614080 T^{2} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{4} + 10819263899520 T^{2} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T^{2} + 9630876 T - 28039440658492)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 11522004 T - 199771975916572)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{2} + 87458444 T - 21\!\cdots\!32)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 62\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{2} - 81579124 T - 22\!\cdots\!92)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 78\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{2} - 846965940 T + 75\!\cdots\!64)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{2} + 1032383084 T + 23\!\cdots\!88)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 564529956 T - 12\!\cdots\!32)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{2} - 5418147844 T + 59\!\cdots\!68)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 83\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 53\!\cdots\!00 \) Copy content Toggle raw display
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