Properties

Label 4.7.b.a
Level 4
Weight 7
Character orbit 4.b
Analytic conductor 0.920
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 4 = 2^{2} \)
Weight: \( k \) = \( 7 \)
Character orbit: \([\chi]\) = 4.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(0.920216334479\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-15}) \)
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 \(\beta = 2\sqrt{-15}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( 2 + \beta ) q^{2} \) \( -4 \beta q^{3} \) \( + ( -56 + 4 \beta ) q^{4} \) \( + 10 q^{5} \) \( + ( 240 - 8 \beta ) q^{6} \) \( + 40 \beta q^{7} \) \( + ( -352 - 48 \beta ) q^{8} \) \( -231 q^{9} \) \(+O(q^{10})\) \( q\) \( + ( 2 + \beta ) q^{2} \) \( -4 \beta q^{3} \) \( + ( -56 + 4 \beta ) q^{4} \) \( + 10 q^{5} \) \( + ( 240 - 8 \beta ) q^{6} \) \( + 40 \beta q^{7} \) \( + ( -352 - 48 \beta ) q^{8} \) \( -231 q^{9} \) \( + ( 20 + 10 \beta ) q^{10} \) \( -124 \beta q^{11} \) \( + ( 960 + 224 \beta ) q^{12} \) \( + 1466 q^{13} \) \( + ( -2400 + 80 \beta ) q^{14} \) \( -40 \beta q^{15} \) \( + ( 2176 - 448 \beta ) q^{16} \) \( -4766 q^{17} \) \( + ( -462 - 231 \beta ) q^{18} \) \( + 972 \beta q^{19} \) \( + ( -560 + 40 \beta ) q^{20} \) \( + 9600 q^{21} \) \( + ( 7440 - 248 \beta ) q^{22} \) \( -1352 \beta q^{23} \) \( + ( -11520 + 1408 \beta ) q^{24} \) \( -15525 q^{25} \) \( + ( 2932 + 1466 \beta ) q^{26} \) \( -1992 \beta q^{27} \) \( + ( -9600 - 2240 \beta ) q^{28} \) \( + 25498 q^{29} \) \( + ( 2400 - 80 \beta ) q^{30} \) \( + 5408 \beta q^{31} \) \( + ( 31232 + 1280 \beta ) q^{32} \) \( -29760 q^{33} \) \( + ( -9532 - 4766 \beta ) q^{34} \) \( + 400 \beta q^{35} \) \( + ( 12936 - 924 \beta ) q^{36} \) \( + 1994 q^{37} \) \( + ( -58320 + 1944 \beta ) q^{38} \) \( -5864 \beta q^{39} \) \( + ( -3520 - 480 \beta ) q^{40} \) \( + 29362 q^{41} \) \( + ( 19200 + 9600 \beta ) q^{42} \) \( -2780 \beta q^{43} \) \( + ( 29760 + 6944 \beta ) q^{44} \) \( -2310 q^{45} \) \( + ( 81120 - 2704 \beta ) q^{46} \) \( -976 \beta q^{47} \) \( + ( -107520 - 8704 \beta ) q^{48} \) \( + 21649 q^{49} \) \( + ( -31050 - 15525 \beta ) q^{50} \) \( + 19064 \beta q^{51} \) \( + ( -82096 + 5864 \beta ) q^{52} \) \( -192854 q^{53} \) \( + ( 119520 - 3984 \beta ) q^{54} \) \( -1240 \beta q^{55} \) \( + ( 115200 - 14080 \beta ) q^{56} \) \( + 233280 q^{57} \) \( + ( 50996 + 25498 \beta ) q^{58} \) \( -10124 \beta q^{59} \) \( + ( 9600 + 2240 \beta ) q^{60} \) \( -10918 q^{61} \) \( + ( -324480 + 10816 \beta ) q^{62} \) \( -9240 \beta q^{63} \) \( + ( -14336 + 33792 \beta ) q^{64} \) \( + 14660 q^{65} \) \( + ( -59520 - 29760 \beta ) q^{66} \) \( -50884 \beta q^{67} \) \( + ( 266896 - 19064 \beta ) q^{68} \) \( -324480 q^{69} \) \( + ( -24000 + 800 \beta ) q^{70} \) \( + 68712 \beta q^{71} \) \( + ( 81312 + 11088 \beta ) q^{72} \) \( + 288626 q^{73} \) \( + ( 3988 + 1994 \beta ) q^{74} \) \( + 62100 \beta q^{75} \) \( + ( -233280 - 54432 \beta ) q^{76} \) \( + 297600 q^{77} \) \( + ( 351840 - 11728 \beta ) q^{78} \) \( -40112 \beta q^{79} \) \( + ( 21760 - 4480 \beta ) q^{80} \) \( -646479 q^{81} \) \( + ( 58724 + 29362 \beta ) q^{82} \) \( -26356 \beta q^{83} \) \( + ( -537600 + 38400 \beta ) q^{84} \) \( -47660 q^{85} \) \( + ( 166800 - 5560 \beta ) q^{86} \) \( -101992 \beta q^{87} \) \( + ( -357120 + 43648 \beta ) q^{88} \) \( + 310738 q^{89} \) \( + ( -4620 - 2310 \beta ) q^{90} \) \( + 58640 \beta q^{91} \) \( + ( 324480 + 75712 \beta ) q^{92} \) \( + 1297920 q^{93} \) \( + ( 58560 - 1952 \beta ) q^{94} \) \( + 9720 \beta q^{95} \) \( + ( 307200 - 124928 \beta ) q^{96} \) \( -1457086 q^{97} \) \( + ( 43298 + 21649 \beta ) q^{98} \) \( + 28644 \beta q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 4q^{2} \) \(\mathstrut -\mathstrut 112q^{4} \) \(\mathstrut +\mathstrut 20q^{5} \) \(\mathstrut +\mathstrut 480q^{6} \) \(\mathstrut -\mathstrut 704q^{8} \) \(\mathstrut -\mathstrut 462q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 4q^{2} \) \(\mathstrut -\mathstrut 112q^{4} \) \(\mathstrut +\mathstrut 20q^{5} \) \(\mathstrut +\mathstrut 480q^{6} \) \(\mathstrut -\mathstrut 704q^{8} \) \(\mathstrut -\mathstrut 462q^{9} \) \(\mathstrut +\mathstrut 40q^{10} \) \(\mathstrut +\mathstrut 1920q^{12} \) \(\mathstrut +\mathstrut 2932q^{13} \) \(\mathstrut -\mathstrut 4800q^{14} \) \(\mathstrut +\mathstrut 4352q^{16} \) \(\mathstrut -\mathstrut 9532q^{17} \) \(\mathstrut -\mathstrut 924q^{18} \) \(\mathstrut -\mathstrut 1120q^{20} \) \(\mathstrut +\mathstrut 19200q^{21} \) \(\mathstrut +\mathstrut 14880q^{22} \) \(\mathstrut -\mathstrut 23040q^{24} \) \(\mathstrut -\mathstrut 31050q^{25} \) \(\mathstrut +\mathstrut 5864q^{26} \) \(\mathstrut -\mathstrut 19200q^{28} \) \(\mathstrut +\mathstrut 50996q^{29} \) \(\mathstrut +\mathstrut 4800q^{30} \) \(\mathstrut +\mathstrut 62464q^{32} \) \(\mathstrut -\mathstrut 59520q^{33} \) \(\mathstrut -\mathstrut 19064q^{34} \) \(\mathstrut +\mathstrut 25872q^{36} \) \(\mathstrut +\mathstrut 3988q^{37} \) \(\mathstrut -\mathstrut 116640q^{38} \) \(\mathstrut -\mathstrut 7040q^{40} \) \(\mathstrut +\mathstrut 58724q^{41} \) \(\mathstrut +\mathstrut 38400q^{42} \) \(\mathstrut +\mathstrut 59520q^{44} \) \(\mathstrut -\mathstrut 4620q^{45} \) \(\mathstrut +\mathstrut 162240q^{46} \) \(\mathstrut -\mathstrut 215040q^{48} \) \(\mathstrut +\mathstrut 43298q^{49} \) \(\mathstrut -\mathstrut 62100q^{50} \) \(\mathstrut -\mathstrut 164192q^{52} \) \(\mathstrut -\mathstrut 385708q^{53} \) \(\mathstrut +\mathstrut 239040q^{54} \) \(\mathstrut +\mathstrut 230400q^{56} \) \(\mathstrut +\mathstrut 466560q^{57} \) \(\mathstrut +\mathstrut 101992q^{58} \) \(\mathstrut +\mathstrut 19200q^{60} \) \(\mathstrut -\mathstrut 21836q^{61} \) \(\mathstrut -\mathstrut 648960q^{62} \) \(\mathstrut -\mathstrut 28672q^{64} \) \(\mathstrut +\mathstrut 29320q^{65} \) \(\mathstrut -\mathstrut 119040q^{66} \) \(\mathstrut +\mathstrut 533792q^{68} \) \(\mathstrut -\mathstrut 648960q^{69} \) \(\mathstrut -\mathstrut 48000q^{70} \) \(\mathstrut +\mathstrut 162624q^{72} \) \(\mathstrut +\mathstrut 577252q^{73} \) \(\mathstrut +\mathstrut 7976q^{74} \) \(\mathstrut -\mathstrut 466560q^{76} \) \(\mathstrut +\mathstrut 595200q^{77} \) \(\mathstrut +\mathstrut 703680q^{78} \) \(\mathstrut +\mathstrut 43520q^{80} \) \(\mathstrut -\mathstrut 1292958q^{81} \) \(\mathstrut +\mathstrut 117448q^{82} \) \(\mathstrut -\mathstrut 1075200q^{84} \) \(\mathstrut -\mathstrut 95320q^{85} \) \(\mathstrut +\mathstrut 333600q^{86} \) \(\mathstrut -\mathstrut 714240q^{88} \) \(\mathstrut +\mathstrut 621476q^{89} \) \(\mathstrut -\mathstrut 9240q^{90} \) \(\mathstrut +\mathstrut 648960q^{92} \) \(\mathstrut +\mathstrut 2595840q^{93} \) \(\mathstrut +\mathstrut 117120q^{94} \) \(\mathstrut +\mathstrut 614400q^{96} \) \(\mathstrut -\mathstrut 2914172q^{97} \) \(\mathstrut +\mathstrut 86596q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 1.93649i
0.500000 + 1.93649i
2.00000 7.74597i 30.9839i −56.0000 30.9839i 10.0000 240.000 + 61.9677i 309.839i −352.000 + 371.806i −231.000 20.0000 77.4597i
3.2 2.00000 + 7.74597i 30.9839i −56.0000 + 30.9839i 10.0000 240.000 61.9677i 309.839i −352.000 371.806i −231.000 20.0000 + 77.4597i
\(n\): 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

There are no other newforms in \(S_{7}^{\mathrm{new}}(4, [\chi])\).