# Properties

 Label 10.4.b.a Level 10 Weight 4 Character orbit 10.b Analytic conductor 0.590 Analytic rank 0 Dimension 2 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$10 = 2 \cdot 5$$ Weight: $$k$$ = $$4$$ Character orbit: $$[\chi]$$ = 10.b (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic conductor: $$0.590019100057$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ 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.

 $$f(q)$$ $$=$$ $$q$$ $$+ \beta q^{2}$$ $$-\beta q^{3}$$ $$-4 q^{4}$$ $$+ ( -5 - 5 \beta ) q^{5}$$ $$+ 4 q^{6}$$ $$+ 13 \beta q^{7}$$ $$-4 \beta q^{8}$$ $$+ 23 q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ \beta q^{2}$$ $$-\beta q^{3}$$ $$-4 q^{4}$$ $$+ ( -5 - 5 \beta ) q^{5}$$ $$+ 4 q^{6}$$ $$+ 13 \beta q^{7}$$ $$-4 \beta q^{8}$$ $$+ 23 q^{9}$$ $$+ ( 20 - 5 \beta ) q^{10}$$ $$-28 q^{11}$$ $$+ 4 \beta q^{12}$$ $$-6 \beta q^{13}$$ $$-52 q^{14}$$ $$+ ( -20 + 5 \beta ) q^{15}$$ $$+ 16 q^{16}$$ $$-32 \beta q^{17}$$ $$+ 23 \beta q^{18}$$ $$+ 60 q^{19}$$ $$+ ( 20 + 20 \beta ) q^{20}$$ $$+ 52 q^{21}$$ $$-28 \beta q^{22}$$ $$+ 29 \beta q^{23}$$ $$-16 q^{24}$$ $$+ ( -75 + 50 \beta ) q^{25}$$ $$+ 24 q^{26}$$ $$-50 \beta q^{27}$$ $$-52 \beta q^{28}$$ $$-90 q^{29}$$ $$+ ( -20 - 20 \beta ) q^{30}$$ $$-128 q^{31}$$ $$+ 16 \beta q^{32}$$ $$+ 28 \beta q^{33}$$ $$+ 128 q^{34}$$ $$+ ( 260 - 65 \beta ) q^{35}$$ $$-92 q^{36}$$ $$+ 118 \beta q^{37}$$ $$+ 60 \beta q^{38}$$ $$-24 q^{39}$$ $$+ ( -80 + 20 \beta ) q^{40}$$ $$+ 242 q^{41}$$ $$+ 52 \beta q^{42}$$ $$-181 \beta q^{43}$$ $$+ 112 q^{44}$$ $$+ ( -115 - 115 \beta ) q^{45}$$ $$-116 q^{46}$$ $$+ 113 \beta q^{47}$$ $$-16 \beta q^{48}$$ $$-333 q^{49}$$ $$+ ( -200 - 75 \beta ) q^{50}$$ $$-128 q^{51}$$ $$+ 24 \beta q^{52}$$ $$+ 54 \beta q^{53}$$ $$+ 200 q^{54}$$ $$+ ( 140 + 140 \beta ) q^{55}$$ $$+ 208 q^{56}$$ $$-60 \beta q^{57}$$ $$-90 \beta q^{58}$$ $$+ 20 q^{59}$$ $$+ ( 80 - 20 \beta ) q^{60}$$ $$+ 542 q^{61}$$ $$-128 \beta q^{62}$$ $$+ 299 \beta q^{63}$$ $$-64 q^{64}$$ $$+ ( -120 + 30 \beta ) q^{65}$$ $$-112 q^{66}$$ $$-217 \beta q^{67}$$ $$+ 128 \beta q^{68}$$ $$+ 116 q^{69}$$ $$+ ( 260 + 260 \beta ) q^{70}$$ $$-1128 q^{71}$$ $$-92 \beta q^{72}$$ $$-316 \beta q^{73}$$ $$-472 q^{74}$$ $$+ ( 200 + 75 \beta ) q^{75}$$ $$-240 q^{76}$$ $$-364 \beta q^{77}$$ $$-24 \beta q^{78}$$ $$+ 720 q^{79}$$ $$+ ( -80 - 80 \beta ) q^{80}$$ $$+ 421 q^{81}$$ $$+ 242 \beta q^{82}$$ $$+ 239 \beta q^{83}$$ $$-208 q^{84}$$ $$+ ( -640 + 160 \beta ) q^{85}$$ $$+ 724 q^{86}$$ $$+ 90 \beta q^{87}$$ $$+ 112 \beta q^{88}$$ $$+ 490 q^{89}$$ $$+ ( 460 - 115 \beta ) q^{90}$$ $$+ 312 q^{91}$$ $$-116 \beta q^{92}$$ $$+ 128 \beta q^{93}$$ $$-452 q^{94}$$ $$+ ( -300 - 300 \beta ) q^{95}$$ $$+ 64 q^{96}$$ $$+ 728 \beta q^{97}$$ $$-333 \beta q^{98}$$ $$-644 q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q$$ $$\mathstrut -\mathstrut 8q^{4}$$ $$\mathstrut -\mathstrut 10q^{5}$$ $$\mathstrut +\mathstrut 8q^{6}$$ $$\mathstrut +\mathstrut 46q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$2q$$ $$\mathstrut -\mathstrut 8q^{4}$$ $$\mathstrut -\mathstrut 10q^{5}$$ $$\mathstrut +\mathstrut 8q^{6}$$ $$\mathstrut +\mathstrut 46q^{9}$$ $$\mathstrut +\mathstrut 40q^{10}$$ $$\mathstrut -\mathstrut 56q^{11}$$ $$\mathstrut -\mathstrut 104q^{14}$$ $$\mathstrut -\mathstrut 40q^{15}$$ $$\mathstrut +\mathstrut 32q^{16}$$ $$\mathstrut +\mathstrut 120q^{19}$$ $$\mathstrut +\mathstrut 40q^{20}$$ $$\mathstrut +\mathstrut 104q^{21}$$ $$\mathstrut -\mathstrut 32q^{24}$$ $$\mathstrut -\mathstrut 150q^{25}$$ $$\mathstrut +\mathstrut 48q^{26}$$ $$\mathstrut -\mathstrut 180q^{29}$$ $$\mathstrut -\mathstrut 40q^{30}$$ $$\mathstrut -\mathstrut 256q^{31}$$ $$\mathstrut +\mathstrut 256q^{34}$$ $$\mathstrut +\mathstrut 520q^{35}$$ $$\mathstrut -\mathstrut 184q^{36}$$ $$\mathstrut -\mathstrut 48q^{39}$$ $$\mathstrut -\mathstrut 160q^{40}$$ $$\mathstrut +\mathstrut 484q^{41}$$ $$\mathstrut +\mathstrut 224q^{44}$$ $$\mathstrut -\mathstrut 230q^{45}$$ $$\mathstrut -\mathstrut 232q^{46}$$ $$\mathstrut -\mathstrut 666q^{49}$$ $$\mathstrut -\mathstrut 400q^{50}$$ $$\mathstrut -\mathstrut 256q^{51}$$ $$\mathstrut +\mathstrut 400q^{54}$$ $$\mathstrut +\mathstrut 280q^{55}$$ $$\mathstrut +\mathstrut 416q^{56}$$ $$\mathstrut +\mathstrut 40q^{59}$$ $$\mathstrut +\mathstrut 160q^{60}$$ $$\mathstrut +\mathstrut 1084q^{61}$$ $$\mathstrut -\mathstrut 128q^{64}$$ $$\mathstrut -\mathstrut 240q^{65}$$ $$\mathstrut -\mathstrut 224q^{66}$$ $$\mathstrut +\mathstrut 232q^{69}$$ $$\mathstrut +\mathstrut 520q^{70}$$ $$\mathstrut -\mathstrut 2256q^{71}$$ $$\mathstrut -\mathstrut 944q^{74}$$ $$\mathstrut +\mathstrut 400q^{75}$$ $$\mathstrut -\mathstrut 480q^{76}$$ $$\mathstrut +\mathstrut 1440q^{79}$$ $$\mathstrut -\mathstrut 160q^{80}$$ $$\mathstrut +\mathstrut 842q^{81}$$ $$\mathstrut -\mathstrut 416q^{84}$$ $$\mathstrut -\mathstrut 1280q^{85}$$ $$\mathstrut +\mathstrut 1448q^{86}$$ $$\mathstrut +\mathstrut 980q^{89}$$ $$\mathstrut +\mathstrut 920q^{90}$$ $$\mathstrut +\mathstrut 624q^{91}$$ $$\mathstrut -\mathstrut 904q^{94}$$ $$\mathstrut -\mathstrut 600q^{95}$$ $$\mathstrut +\mathstrut 128q^{96}$$ $$\mathstrut -\mathstrut 1288q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/10\mathbb{Z}\right)^\times$$.

 $$n$$ $$7$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
9.1
 − 1.00000i 1.00000i
2.00000i 2.00000i −4.00000 −5.00000 + 10.0000i 4.00000 26.0000i 8.00000i 23.0000 20.0000 + 10.0000i
9.2 2.00000i 2.00000i −4.00000 −5.00000 10.0000i 4.00000 26.0000i 8.00000i 23.0000 20.0000 10.0000i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
5.b Even 1 yes

## Hecke kernels

There are no other newforms in $$S_{4}^{\mathrm{new}}(10, [\chi])$$.