Properties

Label 10.5.c.b
Level 10
Weight 5
Character orbit 10.c
Analytic conductor 1.034
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 10 = 2 \cdot 5 \)
Weight: \( k \) = \( 5 \)
Character orbit: \([\chi]\) = 10.c (of order \(4\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(1.03369963084\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( 2 + 2 i ) q^{2} \) \( + ( 1 - i ) q^{3} \) \( + 8 i q^{4} \) \( + ( -15 - 20 i ) q^{5} \) \( + 4 q^{6} \) \( + ( -19 - 19 i ) q^{7} \) \( + ( -16 + 16 i ) q^{8} \) \( + 79 i q^{9} \) \(+O(q^{10})\) \( q\) \( + ( 2 + 2 i ) q^{2} \) \( + ( 1 - i ) q^{3} \) \( + 8 i q^{4} \) \( + ( -15 - 20 i ) q^{5} \) \( + 4 q^{6} \) \( + ( -19 - 19 i ) q^{7} \) \( + ( -16 + 16 i ) q^{8} \) \( + 79 i q^{9} \) \( + ( 10 - 70 i ) q^{10} \) \( + 202 q^{11} \) \( + ( 8 + 8 i ) q^{12} \) \( + ( -99 + 99 i ) q^{13} \) \( -76 i q^{14} \) \( + ( -35 - 5 i ) q^{15} \) \( -64 q^{16} \) \( + ( -239 - 239 i ) q^{17} \) \( + ( -158 + 158 i ) q^{18} \) \( + 40 i q^{19} \) \( + ( 160 - 120 i ) q^{20} \) \( -38 q^{21} \) \( + ( 404 + 404 i ) q^{22} \) \( + ( 541 - 541 i ) q^{23} \) \( + 32 i q^{24} \) \( + ( -175 + 600 i ) q^{25} \) \( -396 q^{26} \) \( + ( 160 + 160 i ) q^{27} \) \( + ( 152 - 152 i ) q^{28} \) \( -200 i q^{29} \) \( + ( -60 - 80 i ) q^{30} \) \( -758 q^{31} \) \( + ( -128 - 128 i ) q^{32} \) \( + ( 202 - 202 i ) q^{33} \) \( -956 i q^{34} \) \( + ( -95 + 665 i ) q^{35} \) \( -632 q^{36} \) \( + ( 141 + 141 i ) q^{37} \) \( + ( -80 + 80 i ) q^{38} \) \( + 198 i q^{39} \) \( + ( 560 + 80 i ) q^{40} \) \( + 1042 q^{41} \) \( + ( -76 - 76 i ) q^{42} \) \( + ( -759 + 759 i ) q^{43} \) \( + 1616 i q^{44} \) \( + ( 1580 - 1185 i ) q^{45} \) \( + 2164 q^{46} \) \( + ( -459 - 459 i ) q^{47} \) \( + ( -64 + 64 i ) q^{48} \) \( -1679 i q^{49} \) \( + ( -1550 + 850 i ) q^{50} \) \( -478 q^{51} \) \( + ( -792 - 792 i ) q^{52} \) \( + ( -1819 + 1819 i ) q^{53} \) \( + 640 i q^{54} \) \( + ( -3030 - 4040 i ) q^{55} \) \( + 608 q^{56} \) \( + ( 40 + 40 i ) q^{57} \) \( + ( 400 - 400 i ) q^{58} \) \( + 4600 i q^{59} \) \( + ( 40 - 280 i ) q^{60} \) \( + 2082 q^{61} \) \( + ( -1516 - 1516 i ) q^{62} \) \( + ( 1501 - 1501 i ) q^{63} \) \( -512 i q^{64} \) \( + ( 3465 + 495 i ) q^{65} \) \( + 808 q^{66} \) \( + ( 5081 + 5081 i ) q^{67} \) \( + ( 1912 - 1912 i ) q^{68} \) \( -1082 i q^{69} \) \( + ( -1520 + 1140 i ) q^{70} \) \( -3478 q^{71} \) \( + ( -1264 - 1264 i ) q^{72} \) \( + ( -3479 + 3479 i ) q^{73} \) \( + 564 i q^{74} \) \( + ( 425 + 775 i ) q^{75} \) \( -320 q^{76} \) \( + ( -3838 - 3838 i ) q^{77} \) \( + ( -396 + 396 i ) q^{78} \) \( -7680 i q^{79} \) \( + ( 960 + 1280 i ) q^{80} \) \( -6079 q^{81} \) \( + ( 2084 + 2084 i ) q^{82} \) \( + ( 6081 - 6081 i ) q^{83} \) \( -304 i q^{84} \) \( + ( -1195 + 8365 i ) q^{85} \) \( -3036 q^{86} \) \( + ( -200 - 200 i ) q^{87} \) \( + ( -3232 + 3232 i ) q^{88} \) \( -5680 i q^{89} \) \( + ( 5530 + 790 i ) q^{90} \) \( + 3762 q^{91} \) \( + ( 4328 + 4328 i ) q^{92} \) \( + ( -758 + 758 i ) q^{93} \) \( -1836 i q^{94} \) \( + ( 800 - 600 i ) q^{95} \) \( -256 q^{96} \) \( + ( 561 + 561 i ) q^{97} \) \( + ( 3358 - 3358 i ) q^{98} \) \( + 15958 i q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 4q^{2} \) \(\mathstrut +\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 30q^{5} \) \(\mathstrut +\mathstrut 8q^{6} \) \(\mathstrut -\mathstrut 38q^{7} \) \(\mathstrut -\mathstrut 32q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 4q^{2} \) \(\mathstrut +\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 30q^{5} \) \(\mathstrut +\mathstrut 8q^{6} \) \(\mathstrut -\mathstrut 38q^{7} \) \(\mathstrut -\mathstrut 32q^{8} \) \(\mathstrut +\mathstrut 20q^{10} \) \(\mathstrut +\mathstrut 404q^{11} \) \(\mathstrut +\mathstrut 16q^{12} \) \(\mathstrut -\mathstrut 198q^{13} \) \(\mathstrut -\mathstrut 70q^{15} \) \(\mathstrut -\mathstrut 128q^{16} \) \(\mathstrut -\mathstrut 478q^{17} \) \(\mathstrut -\mathstrut 316q^{18} \) \(\mathstrut +\mathstrut 320q^{20} \) \(\mathstrut -\mathstrut 76q^{21} \) \(\mathstrut +\mathstrut 808q^{22} \) \(\mathstrut +\mathstrut 1082q^{23} \) \(\mathstrut -\mathstrut 350q^{25} \) \(\mathstrut -\mathstrut 792q^{26} \) \(\mathstrut +\mathstrut 320q^{27} \) \(\mathstrut +\mathstrut 304q^{28} \) \(\mathstrut -\mathstrut 120q^{30} \) \(\mathstrut -\mathstrut 1516q^{31} \) \(\mathstrut -\mathstrut 256q^{32} \) \(\mathstrut +\mathstrut 404q^{33} \) \(\mathstrut -\mathstrut 190q^{35} \) \(\mathstrut -\mathstrut 1264q^{36} \) \(\mathstrut +\mathstrut 282q^{37} \) \(\mathstrut -\mathstrut 160q^{38} \) \(\mathstrut +\mathstrut 1120q^{40} \) \(\mathstrut +\mathstrut 2084q^{41} \) \(\mathstrut -\mathstrut 152q^{42} \) \(\mathstrut -\mathstrut 1518q^{43} \) \(\mathstrut +\mathstrut 3160q^{45} \) \(\mathstrut +\mathstrut 4328q^{46} \) \(\mathstrut -\mathstrut 918q^{47} \) \(\mathstrut -\mathstrut 128q^{48} \) \(\mathstrut -\mathstrut 3100q^{50} \) \(\mathstrut -\mathstrut 956q^{51} \) \(\mathstrut -\mathstrut 1584q^{52} \) \(\mathstrut -\mathstrut 3638q^{53} \) \(\mathstrut -\mathstrut 6060q^{55} \) \(\mathstrut +\mathstrut 1216q^{56} \) \(\mathstrut +\mathstrut 80q^{57} \) \(\mathstrut +\mathstrut 800q^{58} \) \(\mathstrut +\mathstrut 80q^{60} \) \(\mathstrut +\mathstrut 4164q^{61} \) \(\mathstrut -\mathstrut 3032q^{62} \) \(\mathstrut +\mathstrut 3002q^{63} \) \(\mathstrut +\mathstrut 6930q^{65} \) \(\mathstrut +\mathstrut 1616q^{66} \) \(\mathstrut +\mathstrut 10162q^{67} \) \(\mathstrut +\mathstrut 3824q^{68} \) \(\mathstrut -\mathstrut 3040q^{70} \) \(\mathstrut -\mathstrut 6956q^{71} \) \(\mathstrut -\mathstrut 2528q^{72} \) \(\mathstrut -\mathstrut 6958q^{73} \) \(\mathstrut +\mathstrut 850q^{75} \) \(\mathstrut -\mathstrut 640q^{76} \) \(\mathstrut -\mathstrut 7676q^{77} \) \(\mathstrut -\mathstrut 792q^{78} \) \(\mathstrut +\mathstrut 1920q^{80} \) \(\mathstrut -\mathstrut 12158q^{81} \) \(\mathstrut +\mathstrut 4168q^{82} \) \(\mathstrut +\mathstrut 12162q^{83} \) \(\mathstrut -\mathstrut 2390q^{85} \) \(\mathstrut -\mathstrut 6072q^{86} \) \(\mathstrut -\mathstrut 400q^{87} \) \(\mathstrut -\mathstrut 6464q^{88} \) \(\mathstrut +\mathstrut 11060q^{90} \) \(\mathstrut +\mathstrut 7524q^{91} \) \(\mathstrut +\mathstrut 8656q^{92} \) \(\mathstrut -\mathstrut 1516q^{93} \) \(\mathstrut +\mathstrut 1600q^{95} \) \(\mathstrut -\mathstrut 512q^{96} \) \(\mathstrut +\mathstrut 1122q^{97} \) \(\mathstrut +\mathstrut 6716q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/10\mathbb{Z}\right)^\times\).

\(n\) \(7\)
\(\chi(n)\) \(i\)

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
1.00000i
1.00000i
2.00000 2.00000i 1.00000 + 1.00000i 8.00000i −15.0000 + 20.0000i 4.00000 −19.0000 + 19.0000i −16.0000 16.0000i 79.0000i 10.0000 + 70.0000i
7.1 2.00000 + 2.00000i 1.00000 1.00000i 8.00000i −15.0000 20.0000i 4.00000 −19.0000 19.0000i −16.0000 + 16.0000i 79.0000i 10.0000 70.0000i
\(n\): 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
5.c Odd 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{2} \) \(\mathstrut -\mathstrut 2 T_{3} \) \(\mathstrut +\mathstrut 2 \) acting on \(S_{5}^{\mathrm{new}}(10, [\chi])\).