# Properties

 Label 2400.2.d.g Level 2400 Weight 2 Character orbit 2400.d Analytic conductor 19.164 Analytic rank 0 Dimension 8 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2400 = 2^{5} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2400.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$19.1640964851$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.214798336.3 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 600) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} + ( \beta_{2} + \beta_{7} ) q^{7} + q^{9} +O(q^{10})$$ $$q - q^{3} + ( \beta_{2} + \beta_{7} ) q^{7} + q^{9} + \beta_{5} q^{11} + ( -\beta_{1} - \beta_{3} ) q^{13} + ( -\beta_{5} - 2 \beta_{7} ) q^{17} + ( \beta_{2} - \beta_{5} + 2 \beta_{6} ) q^{19} + ( -\beta_{2} - \beta_{7} ) q^{21} + ( -\beta_{2} - 2 \beta_{5} + \beta_{6} ) q^{23} - q^{27} + ( -3 \beta_{2} + 2 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{29} + ( -1 - \beta_{1} + 2 \beta_{3} ) q^{31} -\beta_{5} q^{33} + 2 \beta_{4} q^{37} + ( \beta_{1} + \beta_{3} ) q^{39} + ( \beta_{1} + 2 \beta_{3} - \beta_{4} ) q^{41} + ( -1 - \beta_{1} - 3 \beta_{4} ) q^{43} + ( 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{47} + ( -4 \beta_{1} - 2 \beta_{4} ) q^{49} + ( \beta_{5} + 2 \beta_{7} ) q^{51} + ( -1 - 3 \beta_{3} ) q^{53} + ( -\beta_{2} + \beta_{5} - 2 \beta_{6} ) q^{57} + ( 2 \beta_{6} - 4 \beta_{7} ) q^{59} + ( 4 \beta_{2} + 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{61} + ( \beta_{2} + \beta_{7} ) q^{63} + ( 3 + 3 \beta_{1} + 2 \beta_{3} - \beta_{4} ) q^{67} + ( \beta_{2} + 2 \beta_{5} - \beta_{6} ) q^{69} + ( 5 - 2 \beta_{1} - \beta_{3} ) q^{71} + ( -2 \beta_{2} - 2 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} ) q^{73} + ( 3 + 4 \beta_{1} - \beta_{3} + 2 \beta_{4} ) q^{77} + ( -2 - 2 \beta_{1} - 4 \beta_{4} ) q^{79} + q^{81} + ( -4 - 2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} ) q^{83} + ( 3 \beta_{2} - 2 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{87} + ( -2 \beta_{1} + 4 \beta_{3} - 2 \beta_{4} ) q^{89} + ( -5 \beta_{2} + \beta_{5} - 2 \beta_{7} ) q^{91} + ( 1 + \beta_{1} - 2 \beta_{3} ) q^{93} + ( \beta_{2} + 4 \beta_{5} + 2 \beta_{7} ) q^{97} + \beta_{5} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 8q^{3} + 8q^{9} + O(q^{10})$$ $$8q - 8q^{3} + 8q^{9} - 8q^{27} - 8q^{31} - 8q^{43} - 8q^{53} + 24q^{67} + 40q^{71} + 24q^{77} - 16q^{79} + 8q^{81} - 32q^{83} + 8q^{93} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2 x^{7} - 2 x^{5} + 9 x^{4} - 4 x^{3} - 16 x + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{7} - \nu^{6} - 2 \nu^{5} - 2 \nu^{4} + 7 \nu^{3} + \nu^{2} - 4 \nu - 12$$$$)/4$$ $$\beta_{2}$$ $$=$$ $$($$$$-5 \nu^{7} + 2 \nu^{6} + 4 \nu^{5} + 18 \nu^{4} - 21 \nu^{3} - 12 \nu^{2} - 20 \nu + 56$$$$)/8$$ $$\beta_{3}$$ $$=$$ $$($$$$-5 \nu^{7} + 4 \nu^{6} + 8 \nu^{5} + 18 \nu^{4} - 25 \nu^{3} - 18 \nu^{2} - 20 \nu + 64$$$$)/8$$ $$\beta_{4}$$ $$=$$ $$($$$$-5 \nu^{7} + 6 \nu^{6} + 4 \nu^{5} + 18 \nu^{4} - 29 \nu^{3} - 8 \nu^{2} - 12 \nu + 64$$$$)/8$$ $$\beta_{5}$$ $$=$$ $$($$$$7 \nu^{7} - 4 \nu^{6} - 8 \nu^{5} - 22 \nu^{4} + 35 \nu^{3} + 30 \nu^{2} + 12 \nu - 88$$$$)/8$$ $$\beta_{6}$$ $$=$$ $$($$$$7 \nu^{7} - 5 \nu^{6} - 6 \nu^{5} - 22 \nu^{4} + 37 \nu^{3} + 17 \nu^{2} + 24 \nu - 88$$$$)/4$$ $$\beta_{7}$$ $$=$$ $$($$$$8 \nu^{7} - 5 \nu^{6} - 6 \nu^{5} - 24 \nu^{4} + 34 \nu^{3} + 15 \nu^{2} + 28 \nu - 84$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - \beta_{1} + 1$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{5} + \beta_{2} - \beta_{1} + 1$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{7} + 3 \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_{1} + 7$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$\beta_{7} + \beta_{6} + 2 \beta_{4} + 4 \beta_{2} - 1$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$\beta_{6} + 3 \beta_{5} - 3 \beta_{4} + 7 \beta_{3} + \beta_{2} - 5 \beta_{1} + 1$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$-2 \beta_{7} + 2 \beta_{6} - \beta_{5} + 4 \beta_{4} + 2 \beta_{3} - 7 \beta_{2} + 3 \beta_{1} + 1$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$14 \beta_{7} - 7 \beta_{6} + 5 \beta_{5} + 7 \beta_{4} + 7 \beta_{3} + 13 \beta_{2} + 3 \beta_{1} + 1$$$$)/4$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2400\mathbb{Z}\right)^\times$$.

 $$n$$ $$577$$ $$901$$ $$1601$$ $$1951$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 −0.565036 − 1.29643i 1.41216 − 0.0762223i −1.08003 + 0.912978i 1.23291 + 0.692769i 1.23291 − 0.692769i −1.08003 − 0.912978i 1.41216 + 0.0762223i −0.565036 + 1.29643i
0 −1.00000 0 0 0 4.72294i 0 1.00000 0
49.2 0 −1.00000 0 0 0 1.97676i 0 1.00000 0
49.3 0 −1.00000 0 0 0 1.33411i 0 1.00000 0
49.4 0 −1.00000 0 0 0 0.0802864i 0 1.00000 0
49.5 0 −1.00000 0 0 0 0.0802864i 0 1.00000 0
49.6 0 −1.00000 0 0 0 1.33411i 0 1.00000 0
49.7 0 −1.00000 0 0 0 1.97676i 0 1.00000 0
49.8 0 −1.00000 0 0 0 4.72294i 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 49.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2400.2.d.g 8
3.b odd 2 1 7200.2.d.s 8
4.b odd 2 1 600.2.d.h 8
5.b even 2 1 2400.2.d.h 8
5.c odd 4 1 2400.2.k.d 8
5.c odd 4 1 2400.2.k.e 8
8.b even 2 1 2400.2.d.h 8
8.d odd 2 1 600.2.d.g 8
12.b even 2 1 1800.2.d.s 8
15.d odd 2 1 7200.2.d.t 8
15.e even 4 1 7200.2.k.r 8
15.e even 4 1 7200.2.k.s 8
20.d odd 2 1 600.2.d.g 8
20.e even 4 1 600.2.k.d 8
20.e even 4 1 600.2.k.e yes 8
24.f even 2 1 1800.2.d.t 8
24.h odd 2 1 7200.2.d.t 8
40.e odd 2 1 600.2.d.h 8
40.f even 2 1 inner 2400.2.d.g 8
40.i odd 4 1 2400.2.k.d 8
40.i odd 4 1 2400.2.k.e 8
40.k even 4 1 600.2.k.d 8
40.k even 4 1 600.2.k.e yes 8
60.h even 2 1 1800.2.d.t 8
60.l odd 4 1 1800.2.k.q 8
60.l odd 4 1 1800.2.k.t 8
120.i odd 2 1 7200.2.d.s 8
120.m even 2 1 1800.2.d.s 8
120.q odd 4 1 1800.2.k.q 8
120.q odd 4 1 1800.2.k.t 8
120.w even 4 1 7200.2.k.r 8
120.w even 4 1 7200.2.k.s 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.2.d.g 8 8.d odd 2 1
600.2.d.g 8 20.d odd 2 1
600.2.d.h 8 4.b odd 2 1
600.2.d.h 8 40.e odd 2 1
600.2.k.d 8 20.e even 4 1
600.2.k.d 8 40.k even 4 1
600.2.k.e yes 8 20.e even 4 1
600.2.k.e yes 8 40.k even 4 1
1800.2.d.s 8 12.b even 2 1
1800.2.d.s 8 120.m even 2 1
1800.2.d.t 8 24.f even 2 1
1800.2.d.t 8 60.h even 2 1
1800.2.k.q 8 60.l odd 4 1
1800.2.k.q 8 120.q odd 4 1
1800.2.k.t 8 60.l odd 4 1
1800.2.k.t 8 120.q odd 4 1
2400.2.d.g 8 1.a even 1 1 trivial
2400.2.d.g 8 40.f even 2 1 inner
2400.2.d.h 8 5.b even 2 1
2400.2.d.h 8 8.b even 2 1
2400.2.k.d 8 5.c odd 4 1
2400.2.k.d 8 40.i odd 4 1
2400.2.k.e 8 5.c odd 4 1
2400.2.k.e 8 40.i odd 4 1
7200.2.d.s 8 3.b odd 2 1
7200.2.d.s 8 120.i odd 2 1
7200.2.d.t 8 15.d odd 2 1
7200.2.d.t 8 24.h odd 2 1
7200.2.k.r 8 15.e even 4 1
7200.2.k.r 8 120.w even 4 1
7200.2.k.s 8 15.e even 4 1
7200.2.k.s 8 120.w even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2400, [\chi])$$:

 $$T_{7}^{8} + 28 T_{7}^{6} + 134 T_{7}^{4} + 156 T_{7}^{2} + 1$$ $$T_{11}^{8} + 32 T_{11}^{6} + 336 T_{11}^{4} + 1344 T_{11}^{2} + 1600$$ $$T_{13}^{4} - 22 T_{13}^{2} - 32 T_{13} + 9$$ $$T_{37}^{4} - 64 T_{37}^{2} - 128 T_{37} + 64$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 + T )^{8}$$
$5$ 1
$7$ $$1 - 28 T^{2} + 330 T^{4} - 2224 T^{6} + 13203 T^{8} - 108976 T^{10} + 792330 T^{12} - 3294172 T^{14} + 5764801 T^{16}$$
$11$ $$1 - 56 T^{2} + 1612 T^{4} - 29896 T^{6} + 388998 T^{8} - 3617416 T^{10} + 23601292 T^{12} - 99207416 T^{14} + 214358881 T^{16}$$
$13$ $$( 1 + 30 T^{2} - 32 T^{3} + 451 T^{4} - 416 T^{5} + 5070 T^{6} + 28561 T^{8} )^{2}$$
$17$ $$1 - 56 T^{2} + 1484 T^{4} - 24968 T^{6} + 374950 T^{8} - 7215752 T^{10} + 123945164 T^{12} - 1351703864 T^{14} + 6975757441 T^{16}$$
$19$ $$1 - 36 T^{2} + 1546 T^{4} - 35120 T^{6} + 832243 T^{8} - 12678320 T^{10} + 201476266 T^{12} - 1693651716 T^{14} + 16983563041 T^{16}$$
$23$ $$1 - 56 T^{2} + 1324 T^{4} - 36616 T^{6} + 1094310 T^{8} - 19369864 T^{10} + 370509484 T^{12} - 8290009784 T^{14} + 78310985281 T^{16}$$
$29$ $$1 - 88 T^{2} + 4780 T^{4} - 171048 T^{6} + 5385990 T^{8} - 143851368 T^{10} + 3380803180 T^{12} - 52344452248 T^{14} + 500246412961 T^{16}$$
$31$ $$( 1 + 4 T + 54 T^{2} + 168 T^{3} + 2099 T^{4} + 5208 T^{5} + 51894 T^{6} + 119164 T^{7} + 923521 T^{8} )^{2}$$
$37$ $$( 1 + 84 T^{2} - 128 T^{3} + 3542 T^{4} - 4736 T^{5} + 114996 T^{6} + 1874161 T^{8} )^{2}$$
$41$ $$( 1 + 100 T^{2} + 56 T^{3} + 5166 T^{4} + 2296 T^{5} + 168100 T^{6} + 2825761 T^{8} )^{2}$$
$43$ $$( 1 + 4 T + 58 T^{2} + 336 T^{3} + 3379 T^{4} + 14448 T^{5} + 107242 T^{6} + 318028 T^{7} + 3418801 T^{8} )^{2}$$
$47$ $$1 - 232 T^{2} + 26076 T^{4} - 1910872 T^{6} + 102863686 T^{8} - 4221116248 T^{10} + 127242561756 T^{12} - 2500777956328 T^{14} + 23811286661761 T^{16}$$
$53$ $$( 1 + 4 T + 92 T^{2} - 44 T^{3} + 3982 T^{4} - 2332 T^{5} + 258428 T^{6} + 595508 T^{7} + 7890481 T^{8} )^{2}$$
$59$ $$1 - 40 T^{2} + 2364 T^{4} - 112984 T^{6} + 20250598 T^{8} - 393297304 T^{10} + 28645441404 T^{12} - 1687221345640 T^{14} + 146830437604321 T^{16}$$
$61$ $$1 - 252 T^{2} + 32650 T^{4} - 2942672 T^{6} + 202734451 T^{8} - 10949682512 T^{10} + 452066708650 T^{12} - 12983134338972 T^{14} + 191707312997281 T^{16}$$
$67$ $$( 1 - 12 T + 154 T^{2} - 1520 T^{3} + 16755 T^{4} - 101840 T^{5} + 691306 T^{6} - 3609156 T^{7} + 20151121 T^{8} )^{2}$$
$71$ $$( 1 - 20 T + 380 T^{2} - 4188 T^{3} + 43342 T^{4} - 297348 T^{5} + 1915580 T^{6} - 7158220 T^{7} + 25411681 T^{8} )^{2}$$
$73$ $$1 - 184 T^{2} + 15196 T^{4} - 1235336 T^{6} + 104948486 T^{8} - 6583105544 T^{10} + 431539670236 T^{12} - 27845497637176 T^{14} + 806460091894081 T^{16}$$
$79$ $$( 1 + 8 T + 132 T^{2} + 1032 T^{3} + 16454 T^{4} + 81528 T^{5} + 823812 T^{6} + 3944312 T^{7} + 38950081 T^{8} )^{2}$$
$83$ $$( 1 + 16 T + 276 T^{2} + 2576 T^{3} + 30070 T^{4} + 213808 T^{5} + 1901364 T^{6} + 9148592 T^{7} + 47458321 T^{8} )^{2}$$
$89$ $$( 1 + 132 T^{2} - 64 T^{3} + 18534 T^{4} - 5696 T^{5} + 1045572 T^{6} + 62742241 T^{8} )^{2}$$
$97$ $$1 - 356 T^{2} + 80362 T^{4} - 11927760 T^{6} + 1353370099 T^{8} - 112228293840 T^{10} + 7114390079722 T^{12} - 296538033754724 T^{14} + 7837433594376961 T^{16}$$