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

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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:

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 \(S_{9}^{\mathrm{new}}(3, \chi)\).