# Properties

 Label 3.9.b.a Level 3 Weight 9 Character orbit 3.b Analytic conductor 1.222 Analytic rank 0 Dimension 2 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$3$$ Weight: $$k$$ = $$9$$ Character orbit: $$[\chi]$$ = 3.b (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic Conductor $$1.22213583018$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-14})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2\cdot 3$$ Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 6\sqrt{-14}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$+ \beta q^{2}$$ $$+ ( 45 - 3 \beta ) q^{3}$$ $$-248 q^{4}$$ $$-10 \beta q^{5}$$ $$+ ( 1512 + 45 \beta ) q^{6}$$ $$-1750 q^{7}$$ $$+ 8 \beta q^{8}$$ $$+ ( -2511 - 270 \beta ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ \beta q^{2}$$ $$+ ( 45 - 3 \beta ) q^{3}$$ $$-248 q^{4}$$ $$-10 \beta q^{5}$$ $$+ ( 1512 + 45 \beta ) q^{6}$$ $$-1750 q^{7}$$ $$+ 8 \beta q^{8}$$ $$+ ( -2511 - 270 \beta ) q^{9}$$ $$+ 5040 q^{10}$$ $$+ 310 \beta q^{11}$$ $$+ ( -11160 + 744 \beta ) q^{12}$$ $$+ 25730 q^{13}$$ $$-1750 \beta q^{14}$$ $$+ ( -15120 - 450 \beta ) q^{15}$$ $$-67520 q^{16}$$ $$+ 3336 \beta q^{17}$$ $$+ ( 136080 - 2511 \beta ) q^{18}$$ $$+ 18938 q^{19}$$ $$+ 2480 \beta q^{20}$$ $$+ ( -78750 + 5250 \beta ) q^{21}$$ $$-156240 q^{22}$$ $$-20956 \beta q^{23}$$ $$+ ( 12096 + 360 \beta ) q^{24}$$ $$+ 340225 q^{25}$$ $$+ 25730 \beta q^{26}$$ $$+ ( -521235 - 4617 \beta ) q^{27}$$ $$+ 434000 q^{28}$$ $$+ 20530 \beta q^{29}$$ $$+ ( 226800 - 15120 \beta ) q^{30}$$ $$-351478 q^{31}$$ $$-65472 \beta q^{32}$$ $$+ ( 468720 + 13950 \beta ) q^{33}$$ $$-1681344 q^{34}$$ $$+ 17500 \beta q^{35}$$ $$+ ( 622728 + 66960 \beta ) q^{36}$$ $$+ 1335170 q^{37}$$ $$+ 18938 \beta q^{38}$$ $$+ ( 1157850 - 77190 \beta ) q^{39}$$ $$+ 40320 q^{40}$$ $$+ 83540 \beta q^{41}$$ $$+ ( -2646000 - 78750 \beta ) q^{42}$$ $$-3526150 q^{43}$$ $$-76880 \beta q^{44}$$ $$+ ( -1360800 + 25110 \beta ) q^{45}$$ $$+ 10561824 q^{46}$$ $$-181784 \beta q^{47}$$ $$+ ( -3038400 + 202560 \beta ) q^{48}$$ $$-2702301 q^{49}$$ $$+ 340225 \beta q^{50}$$ $$+ ( 5044032 + 150120 \beta ) q^{51}$$ $$-6381040 q^{52}$$ $$-294066 \beta q^{53}$$ $$+ ( 2326968 - 521235 \beta ) q^{54}$$ $$+ 1562400 q^{55}$$ $$-14000 \beta q^{56}$$ $$+ ( 852210 - 56814 \beta ) q^{57}$$ $$-10347120 q^{58}$$ $$+ 610910 \beta q^{59}$$ $$+ ( 3749760 + 111600 \beta ) q^{60}$$ $$+ 753602 q^{61}$$ $$-351478 \beta q^{62}$$ $$+ ( 4394250 + 472500 \beta ) q^{63}$$ $$+ 15712768 q^{64}$$ $$-257300 \beta q^{65}$$ $$+ ( -7030800 + 468720 \beta ) q^{66}$$ $$+ 2268890 q^{67}$$ $$-827328 \beta q^{68}$$ $$+ ( -31685472 - 943020 \beta ) q^{69}$$ $$-8820000 q^{70}$$ $$+ 758220 \beta q^{71}$$ $$+ ( 1088640 - 20088 \beta ) q^{72}$$ $$+ 27672770 q^{73}$$ $$+ 1335170 \beta q^{74}$$ $$+ ( 15310125 - 1020675 \beta ) q^{75}$$ $$-4696624 q^{76}$$ $$-542500 \beta q^{77}$$ $$+ ( 38903760 + 1157850 \beta ) q^{78}$$ $$-22980982 q^{79}$$ $$+ 675200 \beta q^{80}$$ $$+ ( -30436479 + 1355940 \beta ) q^{81}$$ $$-42104160 q^{82}$$ $$-2066606 \beta q^{83}$$ $$+ ( 19530000 - 1302000 \beta ) q^{84}$$ $$+ 16813440 q^{85}$$ $$-3526150 \beta q^{86}$$ $$+ ( 31041360 + 923850 \beta ) q^{87}$$ $$-1249920 q^{88}$$ $$+ 3234540 \beta q^{89}$$ $$+ ( -12655440 - 1360800 \beta ) q^{90}$$ $$-45027500 q^{91}$$ $$+ 5197088 \beta q^{92}$$ $$+ ( -15816510 + 1054434 \beta ) q^{93}$$ $$+ 91619136 q^{94}$$ $$-189380 \beta q^{95}$$ $$+ ( -98993664 - 2946240 \beta ) q^{96}$$ $$+ 147271010 q^{97}$$ $$-2702301 \beta q^{98}$$ $$+ ( 42184800 - 778410 \beta ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q$$ $$\mathstrut +\mathstrut 90q^{3}$$ $$\mathstrut -\mathstrut 496q^{4}$$ $$\mathstrut +\mathstrut 3024q^{6}$$ $$\mathstrut -\mathstrut 3500q^{7}$$ $$\mathstrut -\mathstrut 5022q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$2q$$ $$\mathstrut +\mathstrut 90q^{3}$$ $$\mathstrut -\mathstrut 496q^{4}$$ $$\mathstrut +\mathstrut 3024q^{6}$$ $$\mathstrut -\mathstrut 3500q^{7}$$ $$\mathstrut -\mathstrut 5022q^{9}$$ $$\mathstrut +\mathstrut 10080q^{10}$$ $$\mathstrut -\mathstrut 22320q^{12}$$ $$\mathstrut +\mathstrut 51460q^{13}$$ $$\mathstrut -\mathstrut 30240q^{15}$$ $$\mathstrut -\mathstrut 135040q^{16}$$ $$\mathstrut +\mathstrut 272160q^{18}$$ $$\mathstrut +\mathstrut 37876q^{19}$$ $$\mathstrut -\mathstrut 157500q^{21}$$ $$\mathstrut -\mathstrut 312480q^{22}$$ $$\mathstrut +\mathstrut 24192q^{24}$$ $$\mathstrut +\mathstrut 680450q^{25}$$ $$\mathstrut -\mathstrut 1042470q^{27}$$ $$\mathstrut +\mathstrut 868000q^{28}$$ $$\mathstrut +\mathstrut 453600q^{30}$$ $$\mathstrut -\mathstrut 702956q^{31}$$ $$\mathstrut +\mathstrut 937440q^{33}$$ $$\mathstrut -\mathstrut 3362688q^{34}$$ $$\mathstrut +\mathstrut 1245456q^{36}$$ $$\mathstrut +\mathstrut 2670340q^{37}$$ $$\mathstrut +\mathstrut 2315700q^{39}$$ $$\mathstrut +\mathstrut 80640q^{40}$$ $$\mathstrut -\mathstrut 5292000q^{42}$$ $$\mathstrut -\mathstrut 7052300q^{43}$$ $$\mathstrut -\mathstrut 2721600q^{45}$$ $$\mathstrut +\mathstrut 21123648q^{46}$$ $$\mathstrut -\mathstrut 6076800q^{48}$$ $$\mathstrut -\mathstrut 5404602q^{49}$$ $$\mathstrut +\mathstrut 10088064q^{51}$$ $$\mathstrut -\mathstrut 12762080q^{52}$$ $$\mathstrut +\mathstrut 4653936q^{54}$$ $$\mathstrut +\mathstrut 3124800q^{55}$$ $$\mathstrut +\mathstrut 1704420q^{57}$$ $$\mathstrut -\mathstrut 20694240q^{58}$$ $$\mathstrut +\mathstrut 7499520q^{60}$$ $$\mathstrut +\mathstrut 1507204q^{61}$$ $$\mathstrut +\mathstrut 8788500q^{63}$$ $$\mathstrut +\mathstrut 31425536q^{64}$$ $$\mathstrut -\mathstrut 14061600q^{66}$$ $$\mathstrut +\mathstrut 4537780q^{67}$$ $$\mathstrut -\mathstrut 63370944q^{69}$$ $$\mathstrut -\mathstrut 17640000q^{70}$$ $$\mathstrut +\mathstrut 2177280q^{72}$$ $$\mathstrut +\mathstrut 55345540q^{73}$$ $$\mathstrut +\mathstrut 30620250q^{75}$$ $$\mathstrut -\mathstrut 9393248q^{76}$$ $$\mathstrut +\mathstrut 77807520q^{78}$$ $$\mathstrut -\mathstrut 45961964q^{79}$$ $$\mathstrut -\mathstrut 60872958q^{81}$$ $$\mathstrut -\mathstrut 84208320q^{82}$$ $$\mathstrut +\mathstrut 39060000q^{84}$$ $$\mathstrut +\mathstrut 33626880q^{85}$$ $$\mathstrut +\mathstrut 62082720q^{87}$$ $$\mathstrut -\mathstrut 2499840q^{88}$$ $$\mathstrut -\mathstrut 25310880q^{90}$$ $$\mathstrut -\mathstrut 90055000q^{91}$$ $$\mathstrut -\mathstrut 31633020q^{93}$$ $$\mathstrut +\mathstrut 183238272q^{94}$$ $$\mathstrut -\mathstrut 197987328q^{96}$$ $$\mathstrut +\mathstrut 294542020q^{97}$$ $$\mathstrut +\mathstrut 84369600q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3\mathbb{Z}\right)^\times$$.

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

## 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 $$\iota_m(\alpha)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1
 − 3.74166i 3.74166i
22.4499i 45.0000 + 67.3498i −248.000 224.499i 1512.00 1010.25i −1750.00 179.600i −2511.00 + 6061.48i 5040.00
2.2 22.4499i 45.0000 67.3498i −248.000 224.499i 1512.00 + 1010.25i −1750.00 179.600i −2511.00 6061.48i 5040.00
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char. orbit Parity Mult. Type Proved
1.a Even 1 trivial yes
3.b Odd 1 yes

## Hecke kernels

There are no other newforms in $$S_{9}^{\mathrm{new}}(3, \chi)$$.