Properties

Label 5.5.c.a
Level 5
Weight 5
Character orbit 5.c
Analytic conductor 0.517
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(0.516849815419\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q\) \( + ( -1 - i ) q^{2} \) \( + ( -6 + 6 i ) q^{3} \) \( -14 i q^{4} \) \( + ( 20 + 15 i ) q^{5} \) \( + 12 q^{6} \) \( + ( -26 - 26 i ) q^{7} \) \( + ( -30 + 30 i ) q^{8} \) \( + 9 i q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -1 - i ) q^{2} \) \( + ( -6 + 6 i ) q^{3} \) \( -14 i q^{4} \) \( + ( 20 + 15 i ) q^{5} \) \( + 12 q^{6} \) \( + ( -26 - 26 i ) q^{7} \) \( + ( -30 + 30 i ) q^{8} \) \( + 9 i q^{9} \) \( + ( -5 - 35 i ) q^{10} \) \( -8 q^{11} \) \( + ( 84 + 84 i ) q^{12} \) \( + ( 139 - 139 i ) q^{13} \) \( + 52 i q^{14} \) \( + ( -210 + 30 i ) q^{15} \) \( -164 q^{16} \) \( + ( -1 - i ) q^{17} \) \( + ( 9 - 9 i ) q^{18} \) \( + 180 i q^{19} \) \( + ( 210 - 280 i ) q^{20} \) \( + 312 q^{21} \) \( + ( 8 + 8 i ) q^{22} \) \( + ( -166 + 166 i ) q^{23} \) \( -360 i q^{24} \) \( + ( 175 + 600 i ) q^{25} \) \( -278 q^{26} \) \( + ( -540 - 540 i ) q^{27} \) \( + ( -364 + 364 i ) q^{28} \) \( -480 i q^{29} \) \( + ( 240 + 180 i ) q^{30} \) \( + 572 q^{31} \) \( + ( 644 + 644 i ) q^{32} \) \( + ( 48 - 48 i ) q^{33} \) \( + 2 i q^{34} \) \( + ( -130 - 910 i ) q^{35} \) \( + 126 q^{36} \) \( + ( -251 - 251 i ) q^{37} \) \( + ( 180 - 180 i ) q^{38} \) \( + 1668 i q^{39} \) \( + ( -1050 + 150 i ) q^{40} \) \( -1688 q^{41} \) \( + ( -312 - 312 i ) q^{42} \) \( + ( 1474 - 1474 i ) q^{43} \) \( + 112 i q^{44} \) \( + ( -135 + 180 i ) q^{45} \) \( + 332 q^{46} \) \( + ( 2474 + 2474 i ) q^{47} \) \( + ( 984 - 984 i ) q^{48} \) \( -1049 i q^{49} \) \( + ( 425 - 775 i ) q^{50} \) \( + 12 q^{51} \) \( + ( -1946 - 1946 i ) q^{52} \) \( + ( -3331 + 3331 i ) q^{53} \) \( + 1080 i q^{54} \) \( + ( -160 - 120 i ) q^{55} \) \( + 1560 q^{56} \) \( + ( -1080 - 1080 i ) q^{57} \) \( + ( -480 + 480 i ) q^{58} \) \( -3660 i q^{59} \) \( + ( 420 + 2940 i ) q^{60} \) \( + 1592 q^{61} \) \( + ( -572 - 572 i ) q^{62} \) \( + ( 234 - 234 i ) q^{63} \) \( + 1336 i q^{64} \) \( + ( 4865 - 695 i ) q^{65} \) \( -96 q^{66} \) \( + ( 874 + 874 i ) q^{67} \) \( + ( -14 + 14 i ) q^{68} \) \( -1992 i q^{69} \) \( + ( -780 + 1040 i ) q^{70} \) \( -6068 q^{71} \) \( + ( -270 - 270 i ) q^{72} \) \( + ( -791 + 791 i ) q^{73} \) \( + 502 i q^{74} \) \( + ( -4650 - 2550 i ) q^{75} \) \( + 2520 q^{76} \) \( + ( 208 + 208 i ) q^{77} \) \( + ( 1668 - 1668 i ) q^{78} \) \( + 9120 i q^{79} \) \( + ( -3280 - 2460 i ) q^{80} \) \( + 5751 q^{81} \) \( + ( 1688 + 1688 i ) q^{82} \) \( + ( 5654 - 5654 i ) q^{83} \) \( -4368 i q^{84} \) \( + ( -5 - 35 i ) q^{85} \) \( -2948 q^{86} \) \( + ( 2880 + 2880 i ) q^{87} \) \( + ( 240 - 240 i ) q^{88} \) \( + 2160 i q^{89} \) \( + ( 315 - 45 i ) q^{90} \) \( -7228 q^{91} \) \( + ( 2324 + 2324 i ) q^{92} \) \( + ( -3432 + 3432 i ) q^{93} \) \( -4948 i q^{94} \) \( + ( -2700 + 3600 i ) q^{95} \) \( -7728 q^{96} \) \( + ( -6551 - 6551 i ) q^{97} \) \( + ( -1049 + 1049 i ) q^{98} \) \( -72 i q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 12q^{3} \) \(\mathstrut +\mathstrut 40q^{5} \) \(\mathstrut +\mathstrut 24q^{6} \) \(\mathstrut -\mathstrut 52q^{7} \) \(\mathstrut -\mathstrut 60q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 12q^{3} \) \(\mathstrut +\mathstrut 40q^{5} \) \(\mathstrut +\mathstrut 24q^{6} \) \(\mathstrut -\mathstrut 52q^{7} \) \(\mathstrut -\mathstrut 60q^{8} \) \(\mathstrut -\mathstrut 10q^{10} \) \(\mathstrut -\mathstrut 16q^{11} \) \(\mathstrut +\mathstrut 168q^{12} \) \(\mathstrut +\mathstrut 278q^{13} \) \(\mathstrut -\mathstrut 420q^{15} \) \(\mathstrut -\mathstrut 328q^{16} \) \(\mathstrut -\mathstrut 2q^{17} \) \(\mathstrut +\mathstrut 18q^{18} \) \(\mathstrut +\mathstrut 420q^{20} \) \(\mathstrut +\mathstrut 624q^{21} \) \(\mathstrut +\mathstrut 16q^{22} \) \(\mathstrut -\mathstrut 332q^{23} \) \(\mathstrut +\mathstrut 350q^{25} \) \(\mathstrut -\mathstrut 556q^{26} \) \(\mathstrut -\mathstrut 1080q^{27} \) \(\mathstrut -\mathstrut 728q^{28} \) \(\mathstrut +\mathstrut 480q^{30} \) \(\mathstrut +\mathstrut 1144q^{31} \) \(\mathstrut +\mathstrut 1288q^{32} \) \(\mathstrut +\mathstrut 96q^{33} \) \(\mathstrut -\mathstrut 260q^{35} \) \(\mathstrut +\mathstrut 252q^{36} \) \(\mathstrut -\mathstrut 502q^{37} \) \(\mathstrut +\mathstrut 360q^{38} \) \(\mathstrut -\mathstrut 2100q^{40} \) \(\mathstrut -\mathstrut 3376q^{41} \) \(\mathstrut -\mathstrut 624q^{42} \) \(\mathstrut +\mathstrut 2948q^{43} \) \(\mathstrut -\mathstrut 270q^{45} \) \(\mathstrut +\mathstrut 664q^{46} \) \(\mathstrut +\mathstrut 4948q^{47} \) \(\mathstrut +\mathstrut 1968q^{48} \) \(\mathstrut +\mathstrut 850q^{50} \) \(\mathstrut +\mathstrut 24q^{51} \) \(\mathstrut -\mathstrut 3892q^{52} \) \(\mathstrut -\mathstrut 6662q^{53} \) \(\mathstrut -\mathstrut 320q^{55} \) \(\mathstrut +\mathstrut 3120q^{56} \) \(\mathstrut -\mathstrut 2160q^{57} \) \(\mathstrut -\mathstrut 960q^{58} \) \(\mathstrut +\mathstrut 840q^{60} \) \(\mathstrut +\mathstrut 3184q^{61} \) \(\mathstrut -\mathstrut 1144q^{62} \) \(\mathstrut +\mathstrut 468q^{63} \) \(\mathstrut +\mathstrut 9730q^{65} \) \(\mathstrut -\mathstrut 192q^{66} \) \(\mathstrut +\mathstrut 1748q^{67} \) \(\mathstrut -\mathstrut 28q^{68} \) \(\mathstrut -\mathstrut 1560q^{70} \) \(\mathstrut -\mathstrut 12136q^{71} \) \(\mathstrut -\mathstrut 540q^{72} \) \(\mathstrut -\mathstrut 1582q^{73} \) \(\mathstrut -\mathstrut 9300q^{75} \) \(\mathstrut +\mathstrut 5040q^{76} \) \(\mathstrut +\mathstrut 416q^{77} \) \(\mathstrut +\mathstrut 3336q^{78} \) \(\mathstrut -\mathstrut 6560q^{80} \) \(\mathstrut +\mathstrut 11502q^{81} \) \(\mathstrut +\mathstrut 3376q^{82} \) \(\mathstrut +\mathstrut 11308q^{83} \) \(\mathstrut -\mathstrut 10q^{85} \) \(\mathstrut -\mathstrut 5896q^{86} \) \(\mathstrut +\mathstrut 5760q^{87} \) \(\mathstrut +\mathstrut 480q^{88} \) \(\mathstrut +\mathstrut 630q^{90} \) \(\mathstrut -\mathstrut 14456q^{91} \) \(\mathstrut +\mathstrut 4648q^{92} \) \(\mathstrut -\mathstrut 6864q^{93} \) \(\mathstrut -\mathstrut 5400q^{95} \) \(\mathstrut -\mathstrut 15456q^{96} \) \(\mathstrut -\mathstrut 13102q^{97} \) \(\mathstrut -\mathstrut 2098q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(i\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
1.00000i
1.00000i
−1.00000 1.00000i −6.00000 + 6.00000i 14.0000i 20.0000 + 15.0000i 12.0000 −26.0000 26.0000i −30.0000 + 30.0000i 9.00000i −5.00000 35.0000i
3.1 −1.00000 + 1.00000i −6.00000 6.00000i 14.0000i 20.0000 15.0000i 12.0000 −26.0000 + 26.0000i −30.0000 30.0000i 9.00000i −5.00000 + 35.0000i
\(n\): 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

There are no other newforms in \(S_{5}^{\mathrm{new}}(5, [\chi])\).