Properties

 Label 400.6.c.d Level $400$ Weight $6$ Character orbit 400.c Analytic conductor $64.154$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 8) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 10 \beta q^{3} + 12 \beta q^{7} - 157 q^{9}+O(q^{10})$$ q + 10*b * q^3 + 12*b * q^7 - 157 * q^9 $$q + 10 \beta q^{3} + 12 \beta q^{7} - 157 q^{9} - 124 q^{11} - 239 \beta q^{13} - 599 \beta q^{17} + 3044 q^{19} - 480 q^{21} + 92 \beta q^{23} + 860 \beta q^{27} + 3282 q^{29} + 5728 q^{31} - 1240 \beta q^{33} + 5163 \beta q^{37} + 9560 q^{39} - 8886 q^{41} - 4594 \beta q^{43} - 11832 \beta q^{47} + 16231 q^{49} + 23960 q^{51} - 5843 \beta q^{53} + 30440 \beta q^{57} + 16876 q^{59} - 18482 q^{61} - 1884 \beta q^{63} + 7766 \beta q^{67} - 3680 q^{69} + 31960 q^{71} + 2443 \beta q^{73} - 1488 \beta q^{77} + 44560 q^{79} - 72551 q^{81} + 33682 \beta q^{83} + 32820 \beta q^{87} - 71994 q^{89} + 11472 q^{91} + 57280 \beta q^{93} + 24433 \beta q^{97} + 19468 q^{99} +O(q^{100})$$ q + 10*b * q^3 + 12*b * q^7 - 157 * q^9 - 124 * q^11 - 239*b * q^13 - 599*b * q^17 + 3044 * q^19 - 480 * q^21 + 92*b * q^23 + 860*b * q^27 + 3282 * q^29 + 5728 * q^31 - 1240*b * q^33 + 5163*b * q^37 + 9560 * q^39 - 8886 * q^41 - 4594*b * q^43 - 11832*b * q^47 + 16231 * q^49 + 23960 * q^51 - 5843*b * q^53 + 30440*b * q^57 + 16876 * q^59 - 18482 * q^61 - 1884*b * q^63 + 7766*b * q^67 - 3680 * q^69 + 31960 * q^71 + 2443*b * q^73 - 1488*b * q^77 + 44560 * q^79 - 72551 * q^81 + 33682*b * q^83 + 32820*b * q^87 - 71994 * q^89 + 11472 * q^91 + 57280*b * q^93 + 24433*b * q^97 + 19468 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 314 q^{9}+O(q^{10})$$ 2 * q - 314 * q^9 $$2 q - 314 q^{9} - 248 q^{11} + 6088 q^{19} - 960 q^{21} + 6564 q^{29} + 11456 q^{31} + 19120 q^{39} - 17772 q^{41} + 32462 q^{49} + 47920 q^{51} + 33752 q^{59} - 36964 q^{61} - 7360 q^{69} + 63920 q^{71} + 89120 q^{79} - 145102 q^{81} - 143988 q^{89} + 22944 q^{91} + 38936 q^{99}+O(q^{100})$$ 2 * q - 314 * q^9 - 248 * q^11 + 6088 * q^19 - 960 * q^21 + 6564 * q^29 + 11456 * q^31 + 19120 * q^39 - 17772 * q^41 + 32462 * q^49 + 47920 * q^51 + 33752 * q^59 - 36964 * q^61 - 7360 * q^69 + 63920 * q^71 + 89120 * q^79 - 145102 * q^81 - 143988 * q^89 + 22944 * q^91 + 38936 * q^99

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
 − 1.00000i 1.00000i
0 20.0000i 0 0 0 24.0000i 0 −157.000 0
49.2 0 20.0000i 0 0 0 24.0000i 0 −157.000 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.d 2
4.b odd 2 1 200.6.c.a 2
5.b even 2 1 inner 400.6.c.d 2
5.c odd 4 1 16.6.a.a 1
5.c odd 4 1 400.6.a.l 1
15.e even 4 1 144.6.a.k 1
20.d odd 2 1 200.6.c.a 2
20.e even 4 1 8.6.a.a 1
20.e even 4 1 200.6.a.a 1
35.f even 4 1 784.6.a.l 1
40.i odd 4 1 64.6.a.g 1
40.k even 4 1 64.6.a.a 1
60.l odd 4 1 72.6.a.f 1
80.i odd 4 1 256.6.b.d 2
80.j even 4 1 256.6.b.f 2
80.s even 4 1 256.6.b.f 2
80.t odd 4 1 256.6.b.d 2
120.q odd 4 1 576.6.a.g 1
120.w even 4 1 576.6.a.h 1
140.j odd 4 1 392.6.a.b 1
140.w even 12 2 392.6.i.b 2
140.x odd 12 2 392.6.i.e 2
220.i odd 4 1 968.6.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.6.a.a 1 20.e even 4 1
16.6.a.a 1 5.c odd 4 1
64.6.a.a 1 40.k even 4 1
64.6.a.g 1 40.i odd 4 1
72.6.a.f 1 60.l odd 4 1
144.6.a.k 1 15.e even 4 1
200.6.a.a 1 20.e even 4 1
200.6.c.a 2 4.b odd 2 1
200.6.c.a 2 20.d odd 2 1
256.6.b.d 2 80.i odd 4 1
256.6.b.d 2 80.t odd 4 1
256.6.b.f 2 80.j even 4 1
256.6.b.f 2 80.s even 4 1
392.6.a.b 1 140.j odd 4 1
392.6.i.b 2 140.w even 12 2
392.6.i.e 2 140.x odd 12 2
400.6.a.l 1 5.c odd 4 1
400.6.c.d 2 1.a even 1 1 trivial
400.6.c.d 2 5.b even 2 1 inner
576.6.a.g 1 120.q odd 4 1
576.6.a.h 1 120.w even 4 1
784.6.a.l 1 35.f even 4 1
968.6.a.a 1 220.i odd 4 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 400$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 576$$
$11$ $$(T + 124)^{2}$$
$13$ $$T^{2} + 228484$$
$17$ $$T^{2} + 1435204$$
$19$ $$(T - 3044)^{2}$$
$23$ $$T^{2} + 33856$$
$29$ $$(T - 3282)^{2}$$
$31$ $$(T - 5728)^{2}$$
$37$ $$T^{2} + 106626276$$
$41$ $$(T + 8886)^{2}$$
$43$ $$T^{2} + 84419344$$
$47$ $$T^{2} + 559984896$$
$53$ $$T^{2} + 136562596$$
$59$ $$(T - 16876)^{2}$$
$61$ $$(T + 18482)^{2}$$
$67$ $$T^{2} + 241243024$$
$71$ $$(T - 31960)^{2}$$
$73$ $$T^{2} + 23872996$$
$79$ $$(T - 44560)^{2}$$
$83$ $$T^{2} + 4537908496$$
$89$ $$(T + 71994)^{2}$$
$97$ $$T^{2} + 2387885956$$