# Properties

 Label 5.6.b.a Level 5 Weight 6 Character orbit 5.b Analytic conductor 0.802 Analytic rank 0 Dimension 2 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$5$$ Weight: $$k$$ = $$6$$ Character orbit: $$[\chi]$$ = 5.b (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic conductor: $$0.801919099065$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-11})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{2}$$ Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{-11}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$- \beta q^{2}$$ $$+ 3 \beta q^{3}$$ $$-12 q^{4}$$ $$+ ( -45 - 5 \beta ) q^{5}$$ $$+ 132 q^{6}$$ $$+ 9 \beta q^{7}$$ $$-20 \beta q^{8}$$ $$-153 q^{9}$$ $$+O(q^{10})$$ $$q$$ $$- \beta q^{2}$$ $$+ 3 \beta q^{3}$$ $$-12 q^{4}$$ $$+ ( -45 - 5 \beta ) q^{5}$$ $$+ 132 q^{6}$$ $$+ 9 \beta q^{7}$$ $$-20 \beta q^{8}$$ $$-153 q^{9}$$ $$+ ( -220 + 45 \beta ) q^{10}$$ $$+ 252 q^{11}$$ $$-36 \beta q^{12}$$ $$+ 18 \beta q^{13}$$ $$+ 396 q^{14}$$ $$+ ( 660 - 135 \beta ) q^{15}$$ $$-1264 q^{16}$$ $$+ 104 \beta q^{17}$$ $$+ 153 \beta q^{18}$$ $$-220 q^{19}$$ $$+ ( 540 + 60 \beta ) q^{20}$$ $$-1188 q^{21}$$ $$-252 \beta q^{22}$$ $$-367 \beta q^{23}$$ $$+ 2640 q^{24}$$ $$+ ( 925 + 450 \beta ) q^{25}$$ $$+ 792 q^{26}$$ $$+ 270 \beta q^{27}$$ $$-108 \beta q^{28}$$ $$-6930 q^{29}$$ $$+ ( -5940 - 660 \beta ) q^{30}$$ $$+ 6752 q^{31}$$ $$+ 624 \beta q^{32}$$ $$+ 756 \beta q^{33}$$ $$+ 4576 q^{34}$$ $$+ ( 1980 - 405 \beta ) q^{35}$$ $$+ 1836 q^{36}$$ $$-2106 \beta q^{37}$$ $$+ 220 \beta q^{38}$$ $$-2376 q^{39}$$ $$+ ( -4400 + 900 \beta ) q^{40}$$ $$-198 q^{41}$$ $$+ 1188 \beta q^{42}$$ $$+ 63 \beta q^{43}$$ $$-3024 q^{44}$$ $$+ ( 6885 + 765 \beta ) q^{45}$$ $$-16148 q^{46}$$ $$+ 1589 \beta q^{47}$$ $$-3792 \beta q^{48}$$ $$+ 13243 q^{49}$$ $$+ ( 19800 - 925 \beta ) q^{50}$$ $$-13728 q^{51}$$ $$-216 \beta q^{52}$$ $$+ 878 \beta q^{53}$$ $$+ 11880 q^{54}$$ $$+ ( -11340 - 1260 \beta ) q^{55}$$ $$+ 7920 q^{56}$$ $$-660 \beta q^{57}$$ $$+ 6930 \beta q^{58}$$ $$-24660 q^{59}$$ $$+ ( -7920 + 1620 \beta ) q^{60}$$ $$-5698 q^{61}$$ $$-6752 \beta q^{62}$$ $$-1377 \beta q^{63}$$ $$-12992 q^{64}$$ $$+ ( 3960 - 810 \beta ) q^{65}$$ $$+ 33264 q^{66}$$ $$+ 6579 \beta q^{67}$$ $$-1248 \beta q^{68}$$ $$+ 48444 q^{69}$$ $$+ ( -17820 - 1980 \beta ) q^{70}$$ $$+ 53352 q^{71}$$ $$+ 3060 \beta q^{72}$$ $$-10692 \beta q^{73}$$ $$-92664 q^{74}$$ $$+ ( -59400 + 2775 \beta ) q^{75}$$ $$+ 2640 q^{76}$$ $$+ 2268 \beta q^{77}$$ $$+ 2376 \beta q^{78}$$ $$+ 51920 q^{79}$$ $$+ ( 56880 + 6320 \beta ) q^{80}$$ $$-72819 q^{81}$$ $$+ 198 \beta q^{82}$$ $$+ 9323 \beta q^{83}$$ $$+ 14256 q^{84}$$ $$+ ( 22880 - 4680 \beta ) q^{85}$$ $$+ 2772 q^{86}$$ $$-20790 \beta q^{87}$$ $$-5040 \beta q^{88}$$ $$-9990 q^{89}$$ $$+ ( 33660 - 6885 \beta ) q^{90}$$ $$-7128 q^{91}$$ $$+ 4404 \beta q^{92}$$ $$+ 20256 \beta q^{93}$$ $$+ 69916 q^{94}$$ $$+ ( 9900 + 1100 \beta ) q^{95}$$ $$-82368 q^{96}$$ $$+ 15264 \beta q^{97}$$ $$-13243 \beta q^{98}$$ $$-38556 q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q$$ $$\mathstrut -\mathstrut 24q^{4}$$ $$\mathstrut -\mathstrut 90q^{5}$$ $$\mathstrut +\mathstrut 264q^{6}$$ $$\mathstrut -\mathstrut 306q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$2q$$ $$\mathstrut -\mathstrut 24q^{4}$$ $$\mathstrut -\mathstrut 90q^{5}$$ $$\mathstrut +\mathstrut 264q^{6}$$ $$\mathstrut -\mathstrut 306q^{9}$$ $$\mathstrut -\mathstrut 440q^{10}$$ $$\mathstrut +\mathstrut 504q^{11}$$ $$\mathstrut +\mathstrut 792q^{14}$$ $$\mathstrut +\mathstrut 1320q^{15}$$ $$\mathstrut -\mathstrut 2528q^{16}$$ $$\mathstrut -\mathstrut 440q^{19}$$ $$\mathstrut +\mathstrut 1080q^{20}$$ $$\mathstrut -\mathstrut 2376q^{21}$$ $$\mathstrut +\mathstrut 5280q^{24}$$ $$\mathstrut +\mathstrut 1850q^{25}$$ $$\mathstrut +\mathstrut 1584q^{26}$$ $$\mathstrut -\mathstrut 13860q^{29}$$ $$\mathstrut -\mathstrut 11880q^{30}$$ $$\mathstrut +\mathstrut 13504q^{31}$$ $$\mathstrut +\mathstrut 9152q^{34}$$ $$\mathstrut +\mathstrut 3960q^{35}$$ $$\mathstrut +\mathstrut 3672q^{36}$$ $$\mathstrut -\mathstrut 4752q^{39}$$ $$\mathstrut -\mathstrut 8800q^{40}$$ $$\mathstrut -\mathstrut 396q^{41}$$ $$\mathstrut -\mathstrut 6048q^{44}$$ $$\mathstrut +\mathstrut 13770q^{45}$$ $$\mathstrut -\mathstrut 32296q^{46}$$ $$\mathstrut +\mathstrut 26486q^{49}$$ $$\mathstrut +\mathstrut 39600q^{50}$$ $$\mathstrut -\mathstrut 27456q^{51}$$ $$\mathstrut +\mathstrut 23760q^{54}$$ $$\mathstrut -\mathstrut 22680q^{55}$$ $$\mathstrut +\mathstrut 15840q^{56}$$ $$\mathstrut -\mathstrut 49320q^{59}$$ $$\mathstrut -\mathstrut 15840q^{60}$$ $$\mathstrut -\mathstrut 11396q^{61}$$ $$\mathstrut -\mathstrut 25984q^{64}$$ $$\mathstrut +\mathstrut 7920q^{65}$$ $$\mathstrut +\mathstrut 66528q^{66}$$ $$\mathstrut +\mathstrut 96888q^{69}$$ $$\mathstrut -\mathstrut 35640q^{70}$$ $$\mathstrut +\mathstrut 106704q^{71}$$ $$\mathstrut -\mathstrut 185328q^{74}$$ $$\mathstrut -\mathstrut 118800q^{75}$$ $$\mathstrut +\mathstrut 5280q^{76}$$ $$\mathstrut +\mathstrut 103840q^{79}$$ $$\mathstrut +\mathstrut 113760q^{80}$$ $$\mathstrut -\mathstrut 145638q^{81}$$ $$\mathstrut +\mathstrut 28512q^{84}$$ $$\mathstrut +\mathstrut 45760q^{85}$$ $$\mathstrut +\mathstrut 5544q^{86}$$ $$\mathstrut -\mathstrut 19980q^{89}$$ $$\mathstrut +\mathstrut 67320q^{90}$$ $$\mathstrut -\mathstrut 14256q^{91}$$ $$\mathstrut +\mathstrut 139832q^{94}$$ $$\mathstrut +\mathstrut 19800q^{95}$$ $$\mathstrut -\mathstrut 164736q^{96}$$ $$\mathstrut -\mathstrut 77112q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/5\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1
 0.5 + 1.65831i 0.5 − 1.65831i
6.63325i 19.8997i −12.0000 −45.0000 33.1662i 132.000 59.6992i 132.665i −153.000 −220.000 + 298.496i
4.2 6.63325i 19.8997i −12.0000 −45.0000 + 33.1662i 132.000 59.6992i 132.665i −153.000 −220.000 298.496i
 $$n$$: 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 $$S_{6}^{\mathrm{new}}(5, \chi)$$.