Properties

Label 4.9.b.b
Level 4
Weight 9
Character orbit 4.b
Analytic conductor 1.630
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 4 = 2^{2} \)
Weight: \( k \) = \( 9 \)
Character orbit: \([\chi]\) = 4.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(1.62951444024\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-39}) \)
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 \(\beta = 2\sqrt{-39}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -10 - \beta ) q^{2} \) \( -8 \beta q^{3} \) \( + ( -56 + 20 \beta ) q^{4} \) \( + 610 q^{5} \) \( + ( -1248 + 80 \beta ) q^{6} \) \( -112 \beta q^{7} \) \( + ( 3680 - 144 \beta ) q^{8} \) \( -3423 q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -10 - \beta ) q^{2} \) \( -8 \beta q^{3} \) \( + ( -56 + 20 \beta ) q^{4} \) \( + 610 q^{5} \) \( + ( -1248 + 80 \beta ) q^{6} \) \( -112 \beta q^{7} \) \( + ( 3680 - 144 \beta ) q^{8} \) \( -3423 q^{9} \) \( + ( -6100 - 610 \beta ) q^{10} \) \( + 1480 \beta q^{11} \) \( + ( 24960 + 448 \beta ) q^{12} \) \( -5470 q^{13} \) \( + ( -17472 + 1120 \beta ) q^{14} \) \( -4880 \beta q^{15} \) \( + ( -59264 - 2240 \beta ) q^{16} \) \( + 73090 q^{17} \) \( + ( 34230 + 3423 \beta ) q^{18} \) \( + 1560 \beta q^{19} \) \( + ( -34160 + 12200 \beta ) q^{20} \) \( -139776 q^{21} \) \( + ( 230880 - 14800 \beta ) q^{22} \) \( + 18992 \beta q^{23} \) \( + ( -179712 - 29440 \beta ) q^{24} \) \( -18525 q^{25} \) \( + ( 54700 + 5470 \beta ) q^{26} \) \( -25104 \beta q^{27} \) \( + ( 349440 + 6272 \beta ) q^{28} \) \( -128222 q^{29} \) \( + ( -761280 + 48800 \beta ) q^{30} \) \( + 5440 \beta q^{31} \) \( + ( 243200 + 81664 \beta ) q^{32} \) \( + 1847040 q^{33} \) \( + ( -730900 - 73090 \beta ) q^{34} \) \( -68320 \beta q^{35} \) \( + ( 191688 - 68460 \beta ) q^{36} \) \( -3472030 q^{37} \) \( + ( 243360 - 15600 \beta ) q^{38} \) \( + 43760 \beta q^{39} \) \( + ( 2244800 - 87840 \beta ) q^{40} \) \( + 2146882 q^{41} \) \( + ( 1397760 + 139776 \beta ) q^{42} \) \( + 474632 \beta q^{43} \) \( + ( -4617600 - 82880 \beta ) q^{44} \) \( -2088030 q^{45} \) \( + ( 2962752 - 189920 \beta ) q^{46} \) \( -610592 \beta q^{47} \) \( + ( -2795520 + 474112 \beta ) q^{48} \) \( + 3807937 q^{49} \) \( + ( 185250 + 18525 \beta ) q^{50} \) \( -584720 \beta q^{51} \) \( + ( 306320 - 109400 \beta ) q^{52} \) \( + 824290 q^{53} \) \( + ( -3916224 + 251040 \beta ) q^{54} \) \( + 902800 \beta q^{55} \) \( + ( -2515968 - 412160 \beta ) q^{56} \) \( + 1946880 q^{57} \) \( + ( 1282220 + 128222 \beta ) q^{58} \) \( + 298280 \beta q^{59} \) \( + ( 15225600 + 273280 \beta ) q^{60} \) \( -14746078 q^{61} \) \( + ( 848640 - 54400 \beta ) q^{62} \) \( + 383376 \beta q^{63} \) \( + ( 10307584 - 1059840 \beta ) q^{64} \) \( -3336700 q^{65} \) \( + ( -18470400 - 1847040 \beta ) q^{66} \) \( -1221512 \beta q^{67} \) \( + ( -4093040 + 1461800 \beta ) q^{68} \) \( + 23702016 q^{69} \) \( + ( -10657920 + 683200 \beta ) q^{70} \) \( + 95760 \beta q^{71} \) \( + ( -12596640 + 492912 \beta ) q^{72} \) \( -5725630 q^{73} \) \( + ( 34720300 + 3472030 \beta ) q^{74} \) \( + 148200 \beta q^{75} \) \( + ( -4867200 - 87360 \beta ) q^{76} \) \( + 25858560 q^{77} \) \( + ( 6826560 - 437600 \beta ) q^{78} \) \( -2875360 \beta q^{79} \) \( + ( -36151040 - 1366400 \beta ) q^{80} \) \( -53788095 q^{81} \) \( + ( -21468820 - 2146882 \beta ) q^{82} \) \( + 4160152 \beta q^{83} \) \( + ( 7827456 - 2795520 \beta ) q^{84} \) \( + 44584900 q^{85} \) \( + ( 74042592 - 4746320 \beta ) q^{86} \) \( + 1025776 \beta q^{87} \) \( + ( 33246720 + 5446400 \beta ) q^{88} \) \( -83324222 q^{89} \) \( + ( 20880300 + 2088030 \beta ) q^{90} \) \( + 612640 \beta q^{91} \) \( + ( -59255040 - 1063552 \beta ) q^{92} \) \( + 6789120 q^{93} \) \( + ( -95252352 + 6105920 \beta ) q^{94} \) \( + 951600 \beta q^{95} \) \( + ( 101916672 - 1945600 \beta ) q^{96} \) \( + 120619010 q^{97} \) \( + ( -38079370 - 3807937 \beta ) q^{98} \) \( -5066040 \beta q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 20q^{2} \) \(\mathstrut -\mathstrut 112q^{4} \) \(\mathstrut +\mathstrut 1220q^{5} \) \(\mathstrut -\mathstrut 2496q^{6} \) \(\mathstrut +\mathstrut 7360q^{8} \) \(\mathstrut -\mathstrut 6846q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 20q^{2} \) \(\mathstrut -\mathstrut 112q^{4} \) \(\mathstrut +\mathstrut 1220q^{5} \) \(\mathstrut -\mathstrut 2496q^{6} \) \(\mathstrut +\mathstrut 7360q^{8} \) \(\mathstrut -\mathstrut 6846q^{9} \) \(\mathstrut -\mathstrut 12200q^{10} \) \(\mathstrut +\mathstrut 49920q^{12} \) \(\mathstrut -\mathstrut 10940q^{13} \) \(\mathstrut -\mathstrut 34944q^{14} \) \(\mathstrut -\mathstrut 118528q^{16} \) \(\mathstrut +\mathstrut 146180q^{17} \) \(\mathstrut +\mathstrut 68460q^{18} \) \(\mathstrut -\mathstrut 68320q^{20} \) \(\mathstrut -\mathstrut 279552q^{21} \) \(\mathstrut +\mathstrut 461760q^{22} \) \(\mathstrut -\mathstrut 359424q^{24} \) \(\mathstrut -\mathstrut 37050q^{25} \) \(\mathstrut +\mathstrut 109400q^{26} \) \(\mathstrut +\mathstrut 698880q^{28} \) \(\mathstrut -\mathstrut 256444q^{29} \) \(\mathstrut -\mathstrut 1522560q^{30} \) \(\mathstrut +\mathstrut 486400q^{32} \) \(\mathstrut +\mathstrut 3694080q^{33} \) \(\mathstrut -\mathstrut 1461800q^{34} \) \(\mathstrut +\mathstrut 383376q^{36} \) \(\mathstrut -\mathstrut 6944060q^{37} \) \(\mathstrut +\mathstrut 486720q^{38} \) \(\mathstrut +\mathstrut 4489600q^{40} \) \(\mathstrut +\mathstrut 4293764q^{41} \) \(\mathstrut +\mathstrut 2795520q^{42} \) \(\mathstrut -\mathstrut 9235200q^{44} \) \(\mathstrut -\mathstrut 4176060q^{45} \) \(\mathstrut +\mathstrut 5925504q^{46} \) \(\mathstrut -\mathstrut 5591040q^{48} \) \(\mathstrut +\mathstrut 7615874q^{49} \) \(\mathstrut +\mathstrut 370500q^{50} \) \(\mathstrut +\mathstrut 612640q^{52} \) \(\mathstrut +\mathstrut 1648580q^{53} \) \(\mathstrut -\mathstrut 7832448q^{54} \) \(\mathstrut -\mathstrut 5031936q^{56} \) \(\mathstrut +\mathstrut 3893760q^{57} \) \(\mathstrut +\mathstrut 2564440q^{58} \) \(\mathstrut +\mathstrut 30451200q^{60} \) \(\mathstrut -\mathstrut 29492156q^{61} \) \(\mathstrut +\mathstrut 1697280q^{62} \) \(\mathstrut +\mathstrut 20615168q^{64} \) \(\mathstrut -\mathstrut 6673400q^{65} \) \(\mathstrut -\mathstrut 36940800q^{66} \) \(\mathstrut -\mathstrut 8186080q^{68} \) \(\mathstrut +\mathstrut 47404032q^{69} \) \(\mathstrut -\mathstrut 21315840q^{70} \) \(\mathstrut -\mathstrut 25193280q^{72} \) \(\mathstrut -\mathstrut 11451260q^{73} \) \(\mathstrut +\mathstrut 69440600q^{74} \) \(\mathstrut -\mathstrut 9734400q^{76} \) \(\mathstrut +\mathstrut 51717120q^{77} \) \(\mathstrut +\mathstrut 13653120q^{78} \) \(\mathstrut -\mathstrut 72302080q^{80} \) \(\mathstrut -\mathstrut 107576190q^{81} \) \(\mathstrut -\mathstrut 42937640q^{82} \) \(\mathstrut +\mathstrut 15654912q^{84} \) \(\mathstrut +\mathstrut 89169800q^{85} \) \(\mathstrut +\mathstrut 148085184q^{86} \) \(\mathstrut +\mathstrut 66493440q^{88} \) \(\mathstrut -\mathstrut 166648444q^{89} \) \(\mathstrut +\mathstrut 41760600q^{90} \) \(\mathstrut -\mathstrut 118510080q^{92} \) \(\mathstrut +\mathstrut 13578240q^{93} \) \(\mathstrut -\mathstrut 190504704q^{94} \) \(\mathstrut +\mathstrut 203833344q^{96} \) \(\mathstrut +\mathstrut 241238020q^{97} \) \(\mathstrut -\mathstrut 76158740q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
0.500000 + 3.12250i
0.500000 3.12250i
−10.0000 12.4900i 99.9200i −56.0000 + 249.800i 610.000 −1248.00 + 999.200i 1398.88i 3680.00 1798.56i −3423.00 −6100.00 7618.90i
3.2 −10.0000 + 12.4900i 99.9200i −56.0000 249.800i 610.000 −1248.00 999.200i 1398.88i 3680.00 + 1798.56i −3423.00 −6100.00 + 7618.90i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{2} \) \(\mathstrut +\mathstrut 9984 \) acting on \(S_{9}^{\mathrm{new}}(4, [\chi])\).