# Properties

 Label 400.10.c.c Level 400 Weight 10 Character orbit 400.c Analytic conductor 206.014 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ = $$10$$ Character orbit: $$[\chi]$$ = 400.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$206.014334466$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 10) 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.

 $$f(q)$$ $$=$$ $$q + 174 i q^{3} -4658 i q^{7} -10593 q^{9} +O(q^{10})$$ $$q + 174 i q^{3} -4658 i q^{7} -10593 q^{9} -28992 q^{11} + 164446 i q^{13} -594822 i q^{17} -295780 q^{19} + 810492 q^{21} + 2544534 i q^{23} + 1581660 i q^{27} + 3722970 q^{29} -2335772 q^{31} -5044608 i q^{33} + 10840418 i q^{37} -28613604 q^{39} + 21593862 q^{41} + 10832294 i q^{43} -5172138 i q^{47} + 18656643 q^{49} + 103499028 q^{51} -98179674 i q^{53} -51465720 i q^{57} + 16162860 q^{59} -43928158 q^{61} + 49342194 i q^{63} + 81557422 i q^{67} -442748916 q^{69} -161307732 q^{71} + 247147966 i q^{73} + 135044736 i q^{77} -583345720 q^{79} -483710859 q^{81} -14571786 i q^{83} + 647796780 i q^{87} -470133690 q^{89} + 765989468 q^{91} -406424328 i q^{93} -117838462 i q^{97} + 307112256 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 21186q^{9} + O(q^{10})$$ $$2q - 21186q^{9} - 57984q^{11} - 591560q^{19} + 1620984q^{21} + 7445940q^{29} - 4671544q^{31} - 57227208q^{39} + 43187724q^{41} + 37313286q^{49} + 206998056q^{51} + 32325720q^{59} - 87856316q^{61} - 885497832q^{69} - 322615464q^{71} - 1166691440q^{79} - 967421718q^{81} - 940267380q^{89} + 1531978936q^{91} + 614224512q^{99} + O(q^{100})$$

## 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
0 174.000i 0 0 0 4658.00i 0 −10593.0 0
49.2 0 174.000i 0 0 0 4658.00i 0 −10593.0 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 400.10.c.c 2
4.b odd 2 1 50.10.b.d 2
5.b even 2 1 inner 400.10.c.c 2
5.c odd 4 1 80.10.a.a 1
5.c odd 4 1 400.10.a.j 1
20.d odd 2 1 50.10.b.d 2
20.e even 4 1 10.10.a.c 1
20.e even 4 1 50.10.a.a 1
40.i odd 4 1 320.10.a.i 1
40.k even 4 1 320.10.a.b 1
60.l odd 4 1 90.10.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.10.a.c 1 20.e even 4 1
50.10.a.a 1 20.e even 4 1
50.10.b.d 2 4.b odd 2 1
50.10.b.d 2 20.d odd 2 1
80.10.a.a 1 5.c odd 4 1
90.10.a.e 1 60.l odd 4 1
320.10.a.b 1 40.k even 4 1
320.10.a.i 1 40.i odd 4 1
400.10.a.j 1 5.c odd 4 1
400.10.c.c 2 1.a even 1 1 trivial
400.10.c.c 2 5.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 30276$$ acting on $$S_{10}^{\mathrm{new}}(400, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ $$1 - 9090 T^{2} + 387420489 T^{4}$$
$5$ 
$7$ $$1 - 59010250 T^{2} + 1628413597910449 T^{4}$$
$11$ $$( 1 + 28992 T + 2357947691 T^{2} )^{2}$$
$13$ $$1 + 5833488170 T^{2} +$$$$11\!\cdots\!29$$$$T^{4}$$
$17$ $$1 + 116637458690 T^{2} +$$$$14\!\cdots\!09$$$$T^{4}$$
$19$ $$( 1 + 295780 T + 322687697779 T^{2} )^{2}$$
$23$ $$1 + 2872347954230 T^{2} +$$$$32\!\cdots\!69$$$$T^{4}$$
$29$ $$( 1 - 3722970 T + 14507145975869 T^{2} )^{2}$$
$31$ $$( 1 + 2335772 T + 26439622160671 T^{2} )^{2}$$
$37$ $$1 - 142408817175430 T^{2} +$$$$16\!\cdots\!29$$$$T^{4}$$
$41$ $$( 1 - 21593862 T + 327381934393961 T^{2} )^{2}$$
$43$ $$1 - 887846630571250 T^{2} +$$$$25\!\cdots\!49$$$$T^{4}$$
$47$ $$1 - 2211509934714490 T^{2} +$$$$12\!\cdots\!89$$$$T^{4}$$
$53$ $$1 + 3039721203142010 T^{2} +$$$$10\!\cdots\!89$$$$T^{4}$$
$59$ $$( 1 - 16162860 T + 8662995818654939 T^{2} )^{2}$$
$61$ $$( 1 + 43928158 T + 11694146092834141 T^{2} )^{2}$$
$67$ $$1 - 47761455709303810 T^{2} +$$$$74\!\cdots\!09$$$$T^{4}$$
$71$ $$( 1 + 161307732 T + 45848500718449031 T^{2} )^{2}$$
$73$ $$1 - 56661056318598670 T^{2} +$$$$34\!\cdots\!69$$$$T^{4}$$
$79$ $$( 1 + 583345720 T + 119851595982618319 T^{2} )^{2}$$
$83$ $$1 - 373668173587851010 T^{2} +$$$$34\!\cdots\!09$$$$T^{4}$$
$89$ $$( 1 + 470133690 T + 350356403707485209 T^{2} )^{2}$$
$97$ $$1 - 1506576214182604990 T^{2} +$$$$57\!\cdots\!89$$$$T^{4}$$