# Properties

 Label 400.6.c.n Level $400$ Weight $6$ Character orbit 400.c Analytic conductor $64.154$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 400.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$64.1535279252$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{241})$$ Defining polynomial: $$x^{4} + 121 x^{2} + 3600$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}\cdot 5^{2}$$ Twist minimal: no (minimal twist has level 25) 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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} - 2 \beta_{2} ) q^{3} + ( -2 \beta_{1} + 20 \beta_{2} ) q^{7} + ( -98 - 4 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{1} - 2 \beta_{2} ) q^{3} + ( -2 \beta_{1} + 20 \beta_{2} ) q^{7} + ( -98 - 4 \beta_{3} ) q^{9} + ( 98 - 5 \beta_{3} ) q^{11} + ( -16 \beta_{1} - 36 \beta_{2} ) q^{13} + ( -68 \beta_{1} - 149 \beta_{2} ) q^{17} + ( -1590 - 7 \beta_{3} ) q^{19} + ( 1482 + 24 \beta_{3} ) q^{21} + ( 6 \beta_{1} - 156 \beta_{2} ) q^{23} + ( -55 \beta_{1} + 674 \beta_{2} ) q^{27} + ( 1960 + 8 \beta_{3} ) q^{29} + ( 548 + 110 \beta_{3} ) q^{31} + ( -152 \beta_{1} + 1009 \beta_{2} ) q^{33} + ( 192 \beta_{1} - 202 \beta_{2} ) q^{37} + ( 2056 - 4 \beta_{3} ) q^{39} + ( 13877 + 40 \beta_{3} ) q^{41} + ( -1064 \beta_{1} + 300 \beta_{2} ) q^{43} + ( -772 \beta_{1} + 2576 \beta_{2} ) q^{47} + ( 5843 - 80 \beta_{3} ) q^{49} + ( 8938 - 13 \beta_{3} ) q^{51} + ( -376 \beta_{1} - 2698 \beta_{2} ) q^{53} + ( -1940 \beta_{1} + 4867 \beta_{2} ) q^{57} + ( 5980 - 196 \beta_{3} ) q^{59} + ( -12198 + 200 \beta_{3} ) q^{61} + ( 2196 \beta_{1} - 3888 \beta_{2} ) q^{63} + ( 793 \beta_{1} - 4006 \beta_{2} ) q^{67} + ( -9246 - 168 \beta_{3} ) q^{69} + ( 43648 - 100 \beta_{3} ) q^{71} + ( -556 \beta_{1} - 7029 \beta_{2} ) q^{73} + ( 2304 \beta_{1} - 450 \beta_{2} ) q^{77} + ( 32740 + 502 \beta_{3} ) q^{79} + ( 23141 - 188 \beta_{3} ) q^{81} + ( -429 \beta_{1} - 9258 \beta_{2} ) q^{83} + ( 2360 \beta_{1} - 5848 \beta_{2} ) q^{87} + ( 36405 + 1044 \beta_{3} ) q^{89} + ( 10288 - 248 \beta_{3} ) q^{91} + ( 6048 \beta_{1} - 27606 \beta_{2} ) q^{93} + ( 5472 \beta_{1} + 12614 \beta_{2} ) q^{97} + ( 110896 + 98 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 392q^{9} + O(q^{10})$$ $$4q - 392q^{9} + 392q^{11} - 6360q^{19} + 5928q^{21} + 7840q^{29} + 2192q^{31} + 8224q^{39} + 55508q^{41} + 23372q^{49} + 35752q^{51} + 23920q^{59} - 48792q^{61} - 36984q^{69} + 174592q^{71} + 130960q^{79} + 92564q^{81} + 145620q^{89} + 41152q^{91} + 443584q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 121 x^{2} + 3600$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 181 \nu$$$$)/60$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} - 61 \nu$$$$)/12$$ $$\beta_{3}$$ $$=$$ $$10 \nu^{2} + 605$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + 5 \beta_{1}$$$$)/10$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 605$$$$)/10$$ $$\nu^{3}$$ $$=$$ $$($$$$-181 \beta_{2} - 305 \beta_{1}$$$$)/10$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$177$$ $$351$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 − 7.26209i − 8.26209i 8.26209i 7.26209i
0 25.5242i 0 0 0 131.048i 0 −408.483 0
49.2 0 5.52417i 0 0 0 68.9517i 0 212.483 0
49.3 0 5.52417i 0 0 0 68.9517i 0 212.483 0
49.4 0 25.5242i 0 0 0 131.048i 0 −408.483 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 400.6.c.n 4
4.b odd 2 1 25.6.b.b 4
5.b even 2 1 inner 400.6.c.n 4
5.c odd 4 1 400.6.a.o 2
5.c odd 4 1 400.6.a.w 2
12.b even 2 1 225.6.b.i 4
20.d odd 2 1 25.6.b.b 4
20.e even 4 1 25.6.a.b 2
20.e even 4 1 25.6.a.d yes 2
60.h even 2 1 225.6.b.i 4
60.l odd 4 1 225.6.a.l 2
60.l odd 4 1 225.6.a.s 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.6.a.b 2 20.e even 4 1
25.6.a.d yes 2 20.e even 4 1
25.6.b.b 4 4.b odd 2 1
25.6.b.b 4 20.d odd 2 1
225.6.a.l 2 60.l odd 4 1
225.6.a.s 2 60.l odd 4 1
225.6.b.i 4 12.b even 2 1
225.6.b.i 4 60.h even 2 1
400.6.a.o 2 5.c odd 4 1
400.6.a.w 2 5.c odd 4 1
400.6.c.n 4 1.a even 1 1 trivial
400.6.c.n 4 5.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 682 T_{3}^{2} + 19881$$ acting on $$S_{6}^{\mathrm{new}}(400, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$19881 + 682 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$81649296 + 21928 T^{2} + T^{4}$$
$11$ $$( -141021 - 196 T + T^{2} )^{2}$$
$13$ $$858255616 + 188192 T^{2} + T^{4}$$
$17$ $$312882490881 + 3338818 T^{2} + T^{4}$$
$19$ $$( 2232875 + 3180 T + T^{2} )^{2}$$
$23$ $$359668876176 + 1234152 T^{2} + T^{4}$$
$29$ $$( 3456000 - 3920 T + T^{2} )^{2}$$
$31$ $$( -72602196 - 1096 T + T^{2} )^{2}$$
$37$ $$61844446287376 + 19808648 T^{2} + T^{4}$$
$41$ $$( 182931129 - 27754 T + T^{2} )^{2}$$
$43$ $$73216315824138496 + 550170272 T^{2} + T^{4}$$
$47$ $$495608042209536 + 619053088 T^{2} + T^{4}$$
$53$ $$21876919639178256 + 432103432 T^{2} + T^{4}$$
$59$ $$( -195696000 - 11960 T + T^{2} )^{2}$$
$61$ $$( -92208796 + 24396 T + T^{2} )^{2}$$
$67$ $$62324269199220681 + 1105507018 T^{2} + T^{4}$$
$71$ $$( 1844897904 - 87296 T + T^{2} )^{2}$$
$73$ $$1347153105574224001 + 2619345602 T^{2} + T^{4}$$
$79$ $$( -446416500 - 65480 T + T^{2} )^{2}$$
$83$ $$4403325447203627961 + 4374235962 T^{2} + T^{4}$$
$89$ $$( -5241540375 - 72810 T + T^{2} )^{2}$$
$97$ $$10487144169973969936 + 22388071688 T^{2} + T^{4}$$