# Properties

 Label 2400.2.k.e Level 2400 Weight 2 Character orbit 2400.k 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.k (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: $$\Z[a_1, \ldots, a_{11}]$$ 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.

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

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1201.1
 1.41216 − 0.0762223i 1.23291 + 0.692769i −1.08003 − 0.912978i −0.565036 + 1.29643i 1.41216 + 0.0762223i 1.23291 − 0.692769i −1.08003 + 0.912978i −0.565036 − 1.29643i
0 1.00000i 0 0 0 −1.97676 0 −1.00000 0
1201.2 0 1.00000i 0 0 0 −0.0802864 0 −1.00000 0
1201.3 0 1.00000i 0 0 0 1.33411 0 −1.00000 0
1201.4 0 1.00000i 0 0 0 4.72294 0 −1.00000 0
1201.5 0 1.00000i 0 0 0 −1.97676 0 −1.00000 0
1201.6 0 1.00000i 0 0 0 −0.0802864 0 −1.00000 0
1201.7 0 1.00000i 0 0 0 1.33411 0 −1.00000 0
1201.8 0 1.00000i 0 0 0 4.72294 0 −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1201.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} - 4 T_{7}^{3} - 6 T_{7}^{2} + 12 T_{7} + 1$$ acting on $$S_{2}^{\mathrm{new}}(2400, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 + T^{2} )^{4}$$
$5$ 1
$7$ $$( 1 - 4 T + 22 T^{2} - 72 T^{3} + 211 T^{4} - 504 T^{5} + 1078 T^{6} - 1372 T^{7} + 2401 T^{8} )^{2}$$
$11$ $$1 - 56 T^{2} + 1612 T^{4} - 29896 T^{6} + 388998 T^{8} - 3617416 T^{10} + 23601292 T^{12} - 99207416 T^{14} + 214358881 T^{16}$$
$13$ $$1 - 60 T^{2} + 1802 T^{4} - 36176 T^{6} + 538099 T^{8} - 6113744 T^{10} + 51466922 T^{12} - 289608540 T^{14} + 815730721 T^{16}$$
$17$ $$( 1 + 28 T^{2} - 104 T^{3} + 350 T^{4} - 1768 T^{5} + 8092 T^{6} + 83521 T^{8} )^{2}$$
$19$ $$1 - 36 T^{2} + 1546 T^{4} - 35120 T^{6} + 832243 T^{8} - 12678320 T^{10} + 201476266 T^{12} - 1693651716 T^{14} + 16983563041 T^{16}$$
$23$ $$( 1 - 4 T + 36 T^{2} - 124 T^{3} + 510 T^{4} - 2852 T^{5} + 19044 T^{6} - 48668 T^{7} + 279841 T^{8} )^{2}$$
$29$ $$1 - 88 T^{2} + 4780 T^{4} - 171048 T^{6} + 5385990 T^{8} - 143851368 T^{10} + 3380803180 T^{12} - 52344452248 T^{14} + 500246412961 T^{16}$$
$31$ $$( 1 + 4 T + 54 T^{2} + 168 T^{3} + 2099 T^{4} + 5208 T^{5} + 51894 T^{6} + 119164 T^{7} + 923521 T^{8} )^{2}$$
$37$ $$1 - 168 T^{2} + 14140 T^{4} - 808664 T^{6} + 34400998 T^{8} - 1107061016 T^{10} + 26500636540 T^{12} - 431042036712 T^{14} + 3512479453921 T^{16}$$
$41$ $$( 1 + 100 T^{2} + 56 T^{3} + 5166 T^{4} + 2296 T^{5} + 168100 T^{6} + 2825761 T^{8} )^{2}$$
$43$ $$1 - 100 T^{2} + 7434 T^{4} - 377968 T^{6} + 18442035 T^{8} - 698862832 T^{10} + 25415366634 T^{12} - 632136304900 T^{14} + 11688200277601 T^{16}$$
$47$ $$( 1 + 116 T^{2} - 256 T^{3} + 6310 T^{4} - 12032 T^{5} + 256244 T^{6} + 4879681 T^{8} )^{2}$$
$53$ $$1 - 168 T^{2} + 16780 T^{4} - 1266264 T^{6} + 74218758 T^{8} - 3556935576 T^{10} + 132402271180 T^{12} - 3723612669672 T^{14} + 62259690411361 T^{16}$$
$59$ $$1 - 40 T^{2} + 2364 T^{4} - 112984 T^{6} + 20250598 T^{8} - 393297304 T^{10} + 28645441404 T^{12} - 1687221345640 T^{14} + 146830437604321 T^{16}$$
$61$ $$1 - 252 T^{2} + 32650 T^{4} - 2942672 T^{6} + 202734451 T^{8} - 10949682512 T^{10} + 452066708650 T^{12} - 12983134338972 T^{14} + 191707312997281 T^{16}$$
$67$ $$1 - 164 T^{2} + 20746 T^{4} - 1788592 T^{6} + 137741171 T^{8} - 8028989488 T^{10} + 418055156266 T^{12} - 14835174675716 T^{14} + 406067677556641 T^{16}$$
$71$ $$( 1 - 20 T + 380 T^{2} - 4188 T^{3} + 43342 T^{4} - 297348 T^{5} + 1915580 T^{6} - 7158220 T^{7} + 25411681 T^{8} )^{2}$$
$73$ $$( 1 - 8 T + 124 T^{2} - 888 T^{3} + 7014 T^{4} - 64824 T^{5} + 660796 T^{6} - 3112136 T^{7} + 28398241 T^{8} )^{2}$$
$79$ $$( 1 - 8 T + 132 T^{2} - 1032 T^{3} + 16454 T^{4} - 81528 T^{5} + 823812 T^{6} - 3944312 T^{7} + 38950081 T^{8} )^{2}$$
$83$ $$1 - 296 T^{2} + 53884 T^{4} - 6923736 T^{6} + 654380710 T^{8} - 47697617304 T^{10} + 2557244168764 T^{12} - 96774350517224 T^{14} + 2252292232139041 T^{16}$$
$89$ $$( 1 + 132 T^{2} + 64 T^{3} + 18534 T^{4} + 5696 T^{5} + 1045572 T^{6} + 62742241 T^{8} )^{2}$$
$97$ $$( 1 - 4 T + 186 T^{2} - 816 T^{3} + 26147 T^{4} - 79152 T^{5} + 1750074 T^{6} - 3650692 T^{7} + 88529281 T^{8} )^{2}$$