Properties

 Label 400.4.c.e Level $400$ Weight $4$ Character orbit 400.c Analytic conductor $23.601$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

Related objects

Newspace parameters

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

Newform invariants

 Self dual: no Analytic conductor: $$23.6007640023$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 7 i q^{3} -6 i q^{7} -22 q^{9} +O(q^{10})$$ $$q + 7 i q^{3} -6 i q^{7} -22 q^{9} + 43 q^{11} + 28 i q^{13} + 91 i q^{17} -35 q^{19} + 42 q^{21} + 162 i q^{23} + 35 i q^{27} -160 q^{29} -42 q^{31} + 301 i q^{33} -314 i q^{37} -196 q^{39} -203 q^{41} + 92 i q^{43} -196 i q^{47} + 307 q^{49} -637 q^{51} -82 i q^{53} -245 i q^{57} -280 q^{59} -518 q^{61} + 132 i q^{63} -141 i q^{67} -1134 q^{69} -412 q^{71} + 763 i q^{73} -258 i q^{77} + 510 q^{79} -839 q^{81} + 777 i q^{83} -1120 i q^{87} + 945 q^{89} + 168 q^{91} -294 i q^{93} + 1246 i q^{97} -946 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 44 q^{9} + O(q^{10})$$ $$2 q - 44 q^{9} + 86 q^{11} - 70 q^{19} + 84 q^{21} - 320 q^{29} - 84 q^{31} - 392 q^{39} - 406 q^{41} + 614 q^{49} - 1274 q^{51} - 560 q^{59} - 1036 q^{61} - 2268 q^{69} - 824 q^{71} + 1020 q^{79} - 1678 q^{81} + 1890 q^{89} + 336 q^{91} - 1892 q^{99} + O(q^{100})$$

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 7.00000i 0 0 0 6.00000i 0 −22.0000 0
49.2 0 7.00000i 0 0 0 6.00000i 0 −22.0000 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 400.4.c.e 2
4.b odd 2 1 25.4.b.b 2
5.b even 2 1 inner 400.4.c.e 2
5.c odd 4 1 400.4.a.c 1
5.c odd 4 1 400.4.a.s 1
12.b even 2 1 225.4.b.f 2
20.d odd 2 1 25.4.b.b 2
20.e even 4 1 25.4.a.a 1
20.e even 4 1 25.4.a.b yes 1
40.i odd 4 1 1600.4.a.h 1
40.i odd 4 1 1600.4.a.bs 1
40.k even 4 1 1600.4.a.i 1
40.k even 4 1 1600.4.a.bt 1
60.h even 2 1 225.4.b.f 2
60.l odd 4 1 225.4.a.c 1
60.l odd 4 1 225.4.a.e 1
140.j odd 4 1 1225.4.a.h 1
140.j odd 4 1 1225.4.a.i 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.4.a.a 1 20.e even 4 1
25.4.a.b yes 1 20.e even 4 1
25.4.b.b 2 4.b odd 2 1
25.4.b.b 2 20.d odd 2 1
225.4.a.c 1 60.l odd 4 1
225.4.a.e 1 60.l odd 4 1
225.4.b.f 2 12.b even 2 1
225.4.b.f 2 60.h even 2 1
400.4.a.c 1 5.c odd 4 1
400.4.a.s 1 5.c odd 4 1
400.4.c.e 2 1.a even 1 1 trivial
400.4.c.e 2 5.b even 2 1 inner
1225.4.a.h 1 140.j odd 4 1
1225.4.a.i 1 140.j odd 4 1
1600.4.a.h 1 40.i odd 4 1
1600.4.a.i 1 40.k even 4 1
1600.4.a.bs 1 40.i odd 4 1
1600.4.a.bt 1 40.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(400, [\chi])$$:

 $$T_{3}^{2} + 49$$ $$T_{7}^{2} + 36$$ $$T_{11} - 43$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$49 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$36 + T^{2}$$
$11$ $$( -43 + T )^{2}$$
$13$ $$784 + T^{2}$$
$17$ $$8281 + T^{2}$$
$19$ $$( 35 + T )^{2}$$
$23$ $$26244 + T^{2}$$
$29$ $$( 160 + T )^{2}$$
$31$ $$( 42 + T )^{2}$$
$37$ $$98596 + T^{2}$$
$41$ $$( 203 + T )^{2}$$
$43$ $$8464 + T^{2}$$
$47$ $$38416 + T^{2}$$
$53$ $$6724 + T^{2}$$
$59$ $$( 280 + T )^{2}$$
$61$ $$( 518 + T )^{2}$$
$67$ $$19881 + T^{2}$$
$71$ $$( 412 + T )^{2}$$
$73$ $$582169 + T^{2}$$
$79$ $$( -510 + T )^{2}$$
$83$ $$603729 + T^{2}$$
$89$ $$( -945 + T )^{2}$$
$97$ $$1552516 + T^{2}$$