Properties

Label 800.6.c.g
Level 800
Weight 6
Character orbit 800.c
Analytic conductor 128.307
Analytic rank 0
Dimension 4
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 800.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(128.307055850\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{70})\)
Defining polynomial: \(x^{4} + 1225\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 160)
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.

\(f(q)\) \(=\) \( q + ( 2 \beta_{1} - \beta_{2} ) q^{3} + ( -26 \beta_{1} - \beta_{2} ) q^{7} + ( -53 + 4 \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( 2 \beta_{1} - \beta_{2} ) q^{3} + ( -26 \beta_{1} - \beta_{2} ) q^{7} + ( -53 + 4 \beta_{3} ) q^{9} + ( 160 + 5 \beta_{3} ) q^{11} + ( -25 \beta_{1} - 40 \beta_{2} ) q^{13} + ( -145 \beta_{1} + 40 \beta_{2} ) q^{17} + ( -360 - 20 \beta_{3} ) q^{19} + ( -72 - 24 \beta_{3} ) q^{21} + ( 422 \beta_{1} - 97 \beta_{2} ) q^{23} + ( -740 \beta_{1} - 158 \beta_{2} ) q^{27} + ( -54 - 40 \beta_{3} ) q^{29} + ( 4920 - 65 \beta_{3} ) q^{31} + ( -1080 \beta_{1} - 120 \beta_{2} ) q^{33} + ( -1635 \beta_{1} - 560 \beta_{2} ) q^{37} + ( -11000 + 55 \beta_{3} ) q^{39} + ( -5310 + 404 \beta_{3} ) q^{41} + ( 6418 \beta_{1} + 579 \beta_{2} ) q^{43} + ( -7074 \beta_{1} + 383 \beta_{2} ) q^{47} + ( 13823 - 52 \beta_{3} ) q^{49} + ( 12360 - 225 \beta_{3} ) q^{51} + ( 7835 \beta_{1} - 1080 \beta_{2} ) q^{53} + ( 4880 \beta_{1} + 200 \beta_{2} ) q^{57} + ( -15400 - 10 \beta_{3} ) q^{59} + ( 12270 + 592 \beta_{3} ) q^{61} + ( 258 \beta_{1} - 363 \beta_{2} ) q^{63} + ( -8646 \beta_{1} + 495 \beta_{2} ) q^{67} + ( -30536 + 616 \beta_{3} ) q^{69} + ( -6200 + 1495 \beta_{3} ) q^{71} + ( -1795 \beta_{1} - 3720 \beta_{2} ) q^{73} + ( -5560 \beta_{1} - 680 \beta_{2} ) q^{77} + ( -35920 - 110 \beta_{3} ) q^{79} + ( -51199 + 548 \beta_{3} ) q^{81} + ( -7982 \beta_{1} - 2817 \beta_{2} ) q^{83} + ( 11092 \beta_{1} - 266 \beta_{2} ) q^{87} + ( 20374 - 1640 \beta_{3} ) q^{89} + ( -13800 - 1065 \beta_{3} ) q^{91} + ( 28040 \beta_{1} - 5440 \beta_{2} ) q^{93} + ( 47535 \beta_{1} - 4840 \beta_{2} ) q^{97} + ( 13920 + 375 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 212q^{9} + O(q^{10}) \) \( 4q - 212q^{9} + 640q^{11} - 1440q^{19} - 288q^{21} - 216q^{29} + 19680q^{31} - 44000q^{39} - 21240q^{41} + 55292q^{49} + 49440q^{51} - 61600q^{59} + 49080q^{61} - 122144q^{69} - 24800q^{71} - 143680q^{79} - 204796q^{81} + 81496q^{89} - 55200q^{91} + 55680q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 1225\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu^{2} \)\(/35\)
\(\beta_{2}\)\(=\)\((\)\( 2 \nu^{3} + 70 \nu \)\()/35\)
\(\beta_{3}\)\(=\)\((\)\( -4 \nu^{3} + 140 \nu \)\()/35\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + 2 \beta_{2}\)\()/8\)
\(\nu^{2}\)\(=\)\(35 \beta_{1}\)\(/2\)
\(\nu^{3}\)\(=\)\((\)\(-35 \beta_{3} + 70 \beta_{2}\)\()/8\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/800\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(577\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
−4.18330 + 4.18330i
4.18330 + 4.18330i
4.18330 4.18330i
−4.18330 4.18330i
0 20.7332i 0 0 0 35.2668i 0 −186.866 0
449.2 0 12.7332i 0 0 0 68.7332i 0 80.8656 0
449.3 0 12.7332i 0 0 0 68.7332i 0 80.8656 0
449.4 0 20.7332i 0 0 0 35.2668i 0 −186.866 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.6.c.g 4
4.b odd 2 1 800.6.c.f 4
5.b even 2 1 inner 800.6.c.g 4
5.c odd 4 1 160.6.a.e yes 2
5.c odd 4 1 800.6.a.g 2
20.d odd 2 1 800.6.c.f 4
20.e even 4 1 160.6.a.a 2
20.e even 4 1 800.6.a.l 2
40.i odd 4 1 320.6.a.r 2
40.k even 4 1 320.6.a.v 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.6.a.a 2 20.e even 4 1
160.6.a.e yes 2 5.c odd 4 1
320.6.a.r 2 40.i odd 4 1
320.6.a.v 2 40.k even 4 1
800.6.a.g 2 5.c odd 4 1
800.6.a.l 2 20.e even 4 1
800.6.c.f 4 4.b odd 2 1
800.6.c.f 4 20.d odd 2 1
800.6.c.g 4 1.a even 1 1 trivial
800.6.c.g 4 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(800, [\chi])\):

\( T_{3}^{4} + 592 T_{3}^{2} + 69696 \)
\( T_{11}^{2} - 320 T_{11} - 2400 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 380 T^{2} + 136278 T^{4} - 22438620 T^{6} + 3486784401 T^{8} \)
$5$ 1
$7$ \( 1 - 61260 T^{2} + 1500118918 T^{4} - 17304433753740 T^{6} + 79792266297612001 T^{8} \)
$11$ \( ( 1 - 320 T + 319702 T^{2} - 51536320 T^{3} + 25937424601 T^{4} )^{2} \)
$13$ \( 1 - 584172 T^{2} + 356551215094 T^{4} - 80533070900414028 T^{6} + \)\(19\!\cdots\!01\)\( T^{8} \)
$17$ \( 1 - 4615228 T^{2} + 9206362973894 T^{4} - 9304271497181437372 T^{6} + \)\(40\!\cdots\!01\)\( T^{8} \)
$19$ \( ( 1 + 720 T + 4633798 T^{2} + 1782791280 T^{3} + 6131066257801 T^{4} )^{2} \)
$23$ \( 1 - 19051660 T^{2} + 166087805861318 T^{4} - \)\(78\!\cdots\!40\)\( T^{6} + \)\(17\!\cdots\!01\)\( T^{8} \)
$29$ \( ( 1 + 108 T + 39233214 T^{2} + 2215204092 T^{3} + 420707233300201 T^{4} )^{2} \)
$31$ \( ( 1 - 9840 T + 76732702 T^{2} - 281710845840 T^{3} + 819628286980801 T^{4} )^{2} \)
$37$ \( 1 - 80374028 T^{2} + 7476476186271894 T^{4} - \)\(38\!\cdots\!72\)\( T^{6} + \)\(23\!\cdots\!01\)\( T^{8} \)
$41$ \( ( 1 + 10620 T + 77106582 T^{2} + 1230392854620 T^{3} + 13422659310152401 T^{4} )^{2} \)
$43$ \( 1 - 70773020 T^{2} - 17388341663539722 T^{4} - \)\(15\!\cdots\!80\)\( T^{6} + \)\(46\!\cdots\!01\)\( T^{8} \)
$47$ \( 1 - 434902380 T^{2} + 119597691857197478 T^{4} - \)\(22\!\cdots\!20\)\( T^{6} + \)\(27\!\cdots\!01\)\( T^{8} \)
$53$ \( 1 - 528500172 T^{2} + 98825823286833494 T^{4} - \)\(92\!\cdots\!28\)\( T^{6} + \)\(30\!\cdots\!01\)\( T^{8} \)
$59$ \( ( 1 + 30800 T + 1666896598 T^{2} + 22019668409200 T^{3} + 511116753300641401 T^{4} )^{2} \)
$61$ \( ( 1 - 24540 T + 1447225822 T^{2} - 20726393226540 T^{3} + 713342911662882601 T^{4} )^{2} \)
$67$ \( 1 - 4665259900 T^{2} + 9004780480727533398 T^{4} - \)\(85\!\cdots\!00\)\( T^{6} + \)\(33\!\cdots\!01\)\( T^{8} \)
$71$ \( ( 1 + 12400 T + 1143670702 T^{2} + 22372443952400 T^{3} + 3255243551009881201 T^{4} )^{2} \)
$73$ \( 1 - 517006172 T^{2} + 8462322739873838694 T^{4} - \)\(22\!\cdots\!28\)\( T^{6} + \)\(18\!\cdots\!01\)\( T^{8} \)
$79$ \( ( 1 + 71840 T + 7430807198 T^{2} + 221055731704160 T^{3} + 9468276082626847201 T^{4} )^{2} \)
$83$ \( 1 - 10802590140 T^{2} + 57941034568344378518 T^{4} - \)\(16\!\cdots\!60\)\( T^{6} + \)\(24\!\cdots\!01\)\( T^{8} \)
$89$ \( ( 1 - 40748 T + 8570866774 T^{2} - 227539254427852 T^{3} + 31181719929966183601 T^{4} )^{2} \)
$97$ \( 1 - 3154415228 T^{2} - 87162631860647574906 T^{4} - \)\(23\!\cdots\!72\)\( T^{6} + \)\(54\!\cdots\!01\)\( T^{8} \)
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