Properties

Label 10.5.c.a
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.

\(f(q)\) \(=\) \( q\) \( + ( -2 - 2 i ) q^{2} \) \( + ( 9 - 9 i ) q^{3} \) \( + 8 i q^{4} \) \( + ( -15 + 20 i ) q^{5} \) \( -36 q^{6} \) \( + ( 29 + 29 i ) q^{7} \) \( + ( 16 - 16 i ) q^{8} \) \( -81 i q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -2 - 2 i ) q^{2} \) \( + ( 9 - 9 i ) q^{3} \) \( + 8 i q^{4} \) \( + ( -15 + 20 i ) q^{5} \) \( -36 q^{6} \) \( + ( 29 + 29 i ) q^{7} \) \( + ( 16 - 16 i ) q^{8} \) \( -81 i q^{9} \) \( + ( 70 - 10 i ) q^{10} \) \( -118 q^{11} \) \( + ( 72 + 72 i ) q^{12} \) \( + ( 69 - 69 i ) q^{13} \) \( -116 i q^{14} \) \( + ( 45 + 315 i ) q^{15} \) \( -64 q^{16} \) \( + ( -271 - 271 i ) q^{17} \) \( + ( -162 + 162 i ) q^{18} \) \( + 280 i q^{19} \) \( + ( -160 - 120 i ) q^{20} \) \( + 522 q^{21} \) \( + ( 236 + 236 i ) q^{22} \) \( + ( 269 - 269 i ) q^{23} \) \( -288 i q^{24} \) \( + ( -175 - 600 i ) q^{25} \) \( -276 q^{26} \) \( + ( -232 + 232 i ) q^{28} \) \( + 680 i q^{29} \) \( + ( 540 - 720 i ) q^{30} \) \( + 202 q^{31} \) \( + ( 128 + 128 i ) q^{32} \) \( + ( -1062 + 1062 i ) q^{33} \) \( + 1084 i q^{34} \) \( + ( -1015 + 145 i ) q^{35} \) \( + 648 q^{36} \) \( + ( -651 - 651 i ) q^{37} \) \( + ( 560 - 560 i ) q^{38} \) \( -1242 i q^{39} \) \( + ( 80 + 560 i ) q^{40} \) \( + 1682 q^{41} \) \( + ( -1044 - 1044 i ) q^{42} \) \( + ( 1089 - 1089 i ) q^{43} \) \( -944 i q^{44} \) \( + ( 1620 + 1215 i ) q^{45} \) \( -1076 q^{46} \) \( + ( 1269 + 1269 i ) q^{47} \) \( + ( -576 + 576 i ) q^{48} \) \( -719 i q^{49} \) \( + ( -850 + 1550 i ) q^{50} \) \( -4878 q^{51} \) \( + ( 552 + 552 i ) q^{52} \) \( + ( -611 + 611 i ) q^{53} \) \( + ( 1770 - 2360 i ) q^{55} \) \( + 928 q^{56} \) \( + ( 2520 + 2520 i ) q^{57} \) \( + ( 1360 - 1360 i ) q^{58} \) \( + 1160 i q^{59} \) \( + ( -2520 + 360 i ) q^{60} \) \( -5598 q^{61} \) \( + ( -404 - 404 i ) q^{62} \) \( + ( 2349 - 2349 i ) q^{63} \) \( -512 i q^{64} \) \( + ( 345 + 2415 i ) q^{65} \) \( + 4248 q^{66} \) \( + ( -751 - 751 i ) q^{67} \) \( + ( 2168 - 2168 i ) q^{68} \) \( -4842 i q^{69} \) \( + ( 2320 + 1740 i ) q^{70} \) \( + 6442 q^{71} \) \( + ( -1296 - 1296 i ) q^{72} \) \( + ( -2951 + 2951 i ) q^{73} \) \( + 2604 i q^{74} \) \( + ( -6975 - 3825 i ) q^{75} \) \( -2240 q^{76} \) \( + ( -3422 - 3422 i ) q^{77} \) \( + ( -2484 + 2484 i ) q^{78} \) \( + 10560 i q^{79} \) \( + ( 960 - 1280 i ) q^{80} \) \( + 6561 q^{81} \) \( + ( -3364 - 3364 i ) q^{82} \) \( + ( -6231 + 6231 i ) q^{83} \) \( + 4176 i q^{84} \) \( + ( 9485 - 1355 i ) q^{85} \) \( -4356 q^{86} \) \( + ( 6120 + 6120 i ) q^{87} \) \( + ( -1888 + 1888 i ) q^{88} \) \( -14480 i q^{89} \) \( + ( -810 - 5670 i ) q^{90} \) \( + 4002 q^{91} \) \( + ( 2152 + 2152 i ) q^{92} \) \( + ( 1818 - 1818 i ) q^{93} \) \( -5076 i q^{94} \) \( + ( -5600 - 4200 i ) q^{95} \) \( + 2304 q^{96} \) \( + ( -7311 - 7311 i ) q^{97} \) \( + ( -1438 + 1438 i ) q^{98} \) \( + 9558 i q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 4q^{2} \) \(\mathstrut +\mathstrut 18q^{3} \) \(\mathstrut -\mathstrut 30q^{5} \) \(\mathstrut -\mathstrut 72q^{6} \) \(\mathstrut +\mathstrut 58q^{7} \) \(\mathstrut +\mathstrut 32q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 4q^{2} \) \(\mathstrut +\mathstrut 18q^{3} \) \(\mathstrut -\mathstrut 30q^{5} \) \(\mathstrut -\mathstrut 72q^{6} \) \(\mathstrut +\mathstrut 58q^{7} \) \(\mathstrut +\mathstrut 32q^{8} \) \(\mathstrut +\mathstrut 140q^{10} \) \(\mathstrut -\mathstrut 236q^{11} \) \(\mathstrut +\mathstrut 144q^{12} \) \(\mathstrut +\mathstrut 138q^{13} \) \(\mathstrut +\mathstrut 90q^{15} \) \(\mathstrut -\mathstrut 128q^{16} \) \(\mathstrut -\mathstrut 542q^{17} \) \(\mathstrut -\mathstrut 324q^{18} \) \(\mathstrut -\mathstrut 320q^{20} \) \(\mathstrut +\mathstrut 1044q^{21} \) \(\mathstrut +\mathstrut 472q^{22} \) \(\mathstrut +\mathstrut 538q^{23} \) \(\mathstrut -\mathstrut 350q^{25} \) \(\mathstrut -\mathstrut 552q^{26} \) \(\mathstrut -\mathstrut 464q^{28} \) \(\mathstrut +\mathstrut 1080q^{30} \) \(\mathstrut +\mathstrut 404q^{31} \) \(\mathstrut +\mathstrut 256q^{32} \) \(\mathstrut -\mathstrut 2124q^{33} \) \(\mathstrut -\mathstrut 2030q^{35} \) \(\mathstrut +\mathstrut 1296q^{36} \) \(\mathstrut -\mathstrut 1302q^{37} \) \(\mathstrut +\mathstrut 1120q^{38} \) \(\mathstrut +\mathstrut 160q^{40} \) \(\mathstrut +\mathstrut 3364q^{41} \) \(\mathstrut -\mathstrut 2088q^{42} \) \(\mathstrut +\mathstrut 2178q^{43} \) \(\mathstrut +\mathstrut 3240q^{45} \) \(\mathstrut -\mathstrut 2152q^{46} \) \(\mathstrut +\mathstrut 2538q^{47} \) \(\mathstrut -\mathstrut 1152q^{48} \) \(\mathstrut -\mathstrut 1700q^{50} \) \(\mathstrut -\mathstrut 9756q^{51} \) \(\mathstrut +\mathstrut 1104q^{52} \) \(\mathstrut -\mathstrut 1222q^{53} \) \(\mathstrut +\mathstrut 3540q^{55} \) \(\mathstrut +\mathstrut 1856q^{56} \) \(\mathstrut +\mathstrut 5040q^{57} \) \(\mathstrut +\mathstrut 2720q^{58} \) \(\mathstrut -\mathstrut 5040q^{60} \) \(\mathstrut -\mathstrut 11196q^{61} \) \(\mathstrut -\mathstrut 808q^{62} \) \(\mathstrut +\mathstrut 4698q^{63} \) \(\mathstrut +\mathstrut 690q^{65} \) \(\mathstrut +\mathstrut 8496q^{66} \) \(\mathstrut -\mathstrut 1502q^{67} \) \(\mathstrut +\mathstrut 4336q^{68} \) \(\mathstrut +\mathstrut 4640q^{70} \) \(\mathstrut +\mathstrut 12884q^{71} \) \(\mathstrut -\mathstrut 2592q^{72} \) \(\mathstrut -\mathstrut 5902q^{73} \) \(\mathstrut -\mathstrut 13950q^{75} \) \(\mathstrut -\mathstrut 4480q^{76} \) \(\mathstrut -\mathstrut 6844q^{77} \) \(\mathstrut -\mathstrut 4968q^{78} \) \(\mathstrut +\mathstrut 1920q^{80} \) \(\mathstrut +\mathstrut 13122q^{81} \) \(\mathstrut -\mathstrut 6728q^{82} \) \(\mathstrut -\mathstrut 12462q^{83} \) \(\mathstrut +\mathstrut 18970q^{85} \) \(\mathstrut -\mathstrut 8712q^{86} \) \(\mathstrut +\mathstrut 12240q^{87} \) \(\mathstrut -\mathstrut 3776q^{88} \) \(\mathstrut -\mathstrut 1620q^{90} \) \(\mathstrut +\mathstrut 8004q^{91} \) \(\mathstrut +\mathstrut 4304q^{92} \) \(\mathstrut +\mathstrut 3636q^{93} \) \(\mathstrut -\mathstrut 11200q^{95} \) \(\mathstrut +\mathstrut 4608q^{96} \) \(\mathstrut -\mathstrut 14622q^{97} \) \(\mathstrut -\mathstrut 2876q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 9.00000 + 9.00000i 8.00000i −15.0000 20.0000i −36.0000 29.0000 29.0000i 16.0000 + 16.0000i 81.0000i 70.0000 + 10.0000i
7.1 −2.00000 2.00000i 9.00000 9.00000i 8.00000i −15.0000 + 20.0000i −36.0000 29.0000 + 29.0000i 16.0000 16.0000i 81.0000i 70.0000 10.0000i
\(n\): 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 \(T_{3}^{2} \) \(\mathstrut -\mathstrut 18 T_{3} \) \(\mathstrut +\mathstrut 162 \) acting on \(S_{5}^{\mathrm{new}}(10, [\chi])\).