# Properties

 Label 6.7.b.a Level 6 Weight 7 Character orbit 6.b Analytic conductor 1.380 Analytic rank 0 Dimension 2 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$6 = 2 \cdot 3$$ Weight: $$k$$ = $$7$$ Character orbit: $$[\chi]$$ = 6.b (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic conductor: $$1.38032450172$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ 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 $$\beta = 4\sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$+ \beta q^{2}$$ $$+ ( 21 + 3 \beta ) q^{3}$$ $$-32 q^{4}$$ $$-30 \beta q^{5}$$ $$+ ( -96 + 21 \beta ) q^{6}$$ $$+ 2 q^{7}$$ $$-32 \beta q^{8}$$ $$+ ( 153 + 126 \beta ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ \beta q^{2}$$ $$+ ( 21 + 3 \beta ) q^{3}$$ $$-32 q^{4}$$ $$-30 \beta q^{5}$$ $$+ ( -96 + 21 \beta ) q^{6}$$ $$+ 2 q^{7}$$ $$-32 \beta q^{8}$$ $$+ ( 153 + 126 \beta ) q^{9}$$ $$+ 960 q^{10}$$ $$-6 \beta q^{11}$$ $$+ ( -672 - 96 \beta ) q^{12}$$ $$-2950 q^{13}$$ $$+ 2 \beta q^{14}$$ $$+ ( 2880 - 630 \beta ) q^{15}$$ $$+ 1024 q^{16}$$ $$+ 792 \beta q^{17}$$ $$+ ( -4032 + 153 \beta ) q^{18}$$ $$+ 5258 q^{19}$$ $$+ 960 \beta q^{20}$$ $$+ ( 42 + 6 \beta ) q^{21}$$ $$+ 192 q^{22}$$ $$-1812 \beta q^{23}$$ $$+ ( 3072 - 672 \beta ) q^{24}$$ $$-13175 q^{25}$$ $$-2950 \beta q^{26}$$ $$+ ( -8883 + 3105 \beta ) q^{27}$$ $$-64 q^{28}$$ $$+ 390 \beta q^{29}$$ $$+ ( 20160 + 2880 \beta ) q^{30}$$ $$+ 22898 q^{31}$$ $$+ 1024 \beta q^{32}$$ $$+ ( 576 - 126 \beta ) q^{33}$$ $$-25344 q^{34}$$ $$-60 \beta q^{35}$$ $$+ ( -4896 - 4032 \beta ) q^{36}$$ $$+ 34058 q^{37}$$ $$+ 5258 \beta q^{38}$$ $$+ ( -61950 - 8850 \beta ) q^{39}$$ $$-30720 q^{40}$$ $$-2964 \beta q^{41}$$ $$+ ( -192 + 42 \beta ) q^{42}$$ $$-6406 q^{43}$$ $$+ 192 \beta q^{44}$$ $$+ ( 120960 - 4590 \beta ) q^{45}$$ $$+ 57984 q^{46}$$ $$+ 31800 \beta q^{47}$$ $$+ ( 21504 + 3072 \beta ) q^{48}$$ $$-117645 q^{49}$$ $$-13175 \beta q^{50}$$ $$+ ( -76032 + 16632 \beta ) q^{51}$$ $$+ 94400 q^{52}$$ $$-34038 \beta q^{53}$$ $$+ ( -99360 - 8883 \beta ) q^{54}$$ $$-5760 q^{55}$$ $$-64 \beta q^{56}$$ $$+ ( 110418 + 15774 \beta ) q^{57}$$ $$-12480 q^{58}$$ $$-57774 \beta q^{59}$$ $$+ ( -92160 + 20160 \beta ) q^{60}$$ $$-62566 q^{61}$$ $$+ 22898 \beta q^{62}$$ $$+ ( 306 + 252 \beta ) q^{63}$$ $$-32768 q^{64}$$ $$+ 88500 \beta q^{65}$$ $$+ ( 4032 + 576 \beta ) q^{66}$$ $$+ 438698 q^{67}$$ $$-25344 \beta q^{68}$$ $$+ ( 173952 - 38052 \beta ) q^{69}$$ $$+ 1920 q^{70}$$ $$-12060 \beta q^{71}$$ $$+ ( 129024 - 4896 \beta ) q^{72}$$ $$-730510 q^{73}$$ $$+ 34058 \beta q^{74}$$ $$+ ( -276675 - 39525 \beta ) q^{75}$$ $$-168256 q^{76}$$ $$-12 \beta q^{77}$$ $$+ ( 283200 - 61950 \beta ) q^{78}$$ $$+ 340562 q^{79}$$ $$-30720 \beta q^{80}$$ $$+ ( -484623 + 38556 \beta ) q^{81}$$ $$+ 94848 q^{82}$$ $$+ 87726 \beta q^{83}$$ $$+ ( -1344 - 192 \beta ) q^{84}$$ $$+ 760320 q^{85}$$ $$-6406 \beta q^{86}$$ $$+ ( -37440 + 8190 \beta ) q^{87}$$ $$-6144 q^{88}$$ $$-68364 \beta q^{89}$$ $$+ ( 146880 + 120960 \beta ) q^{90}$$ $$-5900 q^{91}$$ $$+ 57984 \beta q^{92}$$ $$+ ( 480858 + 68694 \beta ) q^{93}$$ $$-1017600 q^{94}$$ $$-157740 \beta q^{95}$$ $$+ ( -98304 + 21504 \beta ) q^{96}$$ $$-281086 q^{97}$$ $$-117645 \beta q^{98}$$ $$+ ( 24192 - 918 \beta ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q$$ $$\mathstrut +\mathstrut 42q^{3}$$ $$\mathstrut -\mathstrut 64q^{4}$$ $$\mathstrut -\mathstrut 192q^{6}$$ $$\mathstrut +\mathstrut 4q^{7}$$ $$\mathstrut +\mathstrut 306q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$2q$$ $$\mathstrut +\mathstrut 42q^{3}$$ $$\mathstrut -\mathstrut 64q^{4}$$ $$\mathstrut -\mathstrut 192q^{6}$$ $$\mathstrut +\mathstrut 4q^{7}$$ $$\mathstrut +\mathstrut 306q^{9}$$ $$\mathstrut +\mathstrut 1920q^{10}$$ $$\mathstrut -\mathstrut 1344q^{12}$$ $$\mathstrut -\mathstrut 5900q^{13}$$ $$\mathstrut +\mathstrut 5760q^{15}$$ $$\mathstrut +\mathstrut 2048q^{16}$$ $$\mathstrut -\mathstrut 8064q^{18}$$ $$\mathstrut +\mathstrut 10516q^{19}$$ $$\mathstrut +\mathstrut 84q^{21}$$ $$\mathstrut +\mathstrut 384q^{22}$$ $$\mathstrut +\mathstrut 6144q^{24}$$ $$\mathstrut -\mathstrut 26350q^{25}$$ $$\mathstrut -\mathstrut 17766q^{27}$$ $$\mathstrut -\mathstrut 128q^{28}$$ $$\mathstrut +\mathstrut 40320q^{30}$$ $$\mathstrut +\mathstrut 45796q^{31}$$ $$\mathstrut +\mathstrut 1152q^{33}$$ $$\mathstrut -\mathstrut 50688q^{34}$$ $$\mathstrut -\mathstrut 9792q^{36}$$ $$\mathstrut +\mathstrut 68116q^{37}$$ $$\mathstrut -\mathstrut 123900q^{39}$$ $$\mathstrut -\mathstrut 61440q^{40}$$ $$\mathstrut -\mathstrut 384q^{42}$$ $$\mathstrut -\mathstrut 12812q^{43}$$ $$\mathstrut +\mathstrut 241920q^{45}$$ $$\mathstrut +\mathstrut 115968q^{46}$$ $$\mathstrut +\mathstrut 43008q^{48}$$ $$\mathstrut -\mathstrut 235290q^{49}$$ $$\mathstrut -\mathstrut 152064q^{51}$$ $$\mathstrut +\mathstrut 188800q^{52}$$ $$\mathstrut -\mathstrut 198720q^{54}$$ $$\mathstrut -\mathstrut 11520q^{55}$$ $$\mathstrut +\mathstrut 220836q^{57}$$ $$\mathstrut -\mathstrut 24960q^{58}$$ $$\mathstrut -\mathstrut 184320q^{60}$$ $$\mathstrut -\mathstrut 125132q^{61}$$ $$\mathstrut +\mathstrut 612q^{63}$$ $$\mathstrut -\mathstrut 65536q^{64}$$ $$\mathstrut +\mathstrut 8064q^{66}$$ $$\mathstrut +\mathstrut 877396q^{67}$$ $$\mathstrut +\mathstrut 347904q^{69}$$ $$\mathstrut +\mathstrut 3840q^{70}$$ $$\mathstrut +\mathstrut 258048q^{72}$$ $$\mathstrut -\mathstrut 1461020q^{73}$$ $$\mathstrut -\mathstrut 553350q^{75}$$ $$\mathstrut -\mathstrut 336512q^{76}$$ $$\mathstrut +\mathstrut 566400q^{78}$$ $$\mathstrut +\mathstrut 681124q^{79}$$ $$\mathstrut -\mathstrut 969246q^{81}$$ $$\mathstrut +\mathstrut 189696q^{82}$$ $$\mathstrut -\mathstrut 2688q^{84}$$ $$\mathstrut +\mathstrut 1520640q^{85}$$ $$\mathstrut -\mathstrut 74880q^{87}$$ $$\mathstrut -\mathstrut 12288q^{88}$$ $$\mathstrut +\mathstrut 293760q^{90}$$ $$\mathstrut -\mathstrut 11800q^{91}$$ $$\mathstrut +\mathstrut 961716q^{93}$$ $$\mathstrut -\mathstrut 2035200q^{94}$$ $$\mathstrut -\mathstrut 196608q^{96}$$ $$\mathstrut -\mathstrut 562172q^{97}$$ $$\mathstrut +\mathstrut 48384q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/6\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 − 1.41421i 1.41421i
5.65685i 21.0000 16.9706i −32.0000 169.706i −96.0000 118.794i 2.00000 181.019i 153.000 712.764i 960.000
5.2 5.65685i 21.0000 + 16.9706i −32.0000 169.706i −96.0000 + 118.794i 2.00000 181.019i 153.000 + 712.764i 960.000
 $$n$$: 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
3.b Odd 1 yes

## Hecke kernels

There are no other newforms in $$S_{7}^{\mathrm{new}}(6, [\chi])$$.