Properties

Label 403.2.bl.a
Level 403
Weight 2
Character orbit 403.bl
Analytic conductor 3.218
Analytic rank 1
Dimension 8
CM No
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 403 = 13 \cdot 31 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 403.bl (of order \(15\) and degree \(8\))

Newform invariants

Self dual: No
Analytic conductor: \(3.21797120146\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{15})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{15}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -2 + \zeta_{15} + \zeta_{15}^{2} - 2 \zeta_{15}^{3} + 2 \zeta_{15}^{4} - \zeta_{15}^{6} + 2 \zeta_{15}^{7} ) q^{2} + ( -\zeta_{15} + 2 \zeta_{15}^{2} + 2 \zeta_{15}^{5} - \zeta_{15}^{6} ) q^{3} + ( -3 \zeta_{15}^{4} - 3 \zeta_{15}^{7} ) q^{4} -3 q^{5} + ( -1 - 3 \zeta_{15} - 3 \zeta_{15}^{4} - \zeta_{15}^{5} ) q^{6} + ( -3 + 3 \zeta_{15} + 3 \zeta_{15}^{4} - 3 \zeta_{15}^{5} + 3 \zeta_{15}^{7} ) q^{7} + ( \zeta_{15}^{2} + 4 \zeta_{15}^{3} + 4 \zeta_{15}^{6} + \zeta_{15}^{7} ) q^{8} + 2 \zeta_{15}^{7} q^{9} +O(q^{10})\) \( q + ( -2 + \zeta_{15} + \zeta_{15}^{2} - 2 \zeta_{15}^{3} + 2 \zeta_{15}^{4} - \zeta_{15}^{6} + 2 \zeta_{15}^{7} ) q^{2} + ( -\zeta_{15} + 2 \zeta_{15}^{2} + 2 \zeta_{15}^{5} - \zeta_{15}^{6} ) q^{3} + ( -3 \zeta_{15}^{4} - 3 \zeta_{15}^{7} ) q^{4} -3 q^{5} + ( -1 - 3 \zeta_{15} - 3 \zeta_{15}^{4} - \zeta_{15}^{5} ) q^{6} + ( -3 + 3 \zeta_{15} + 3 \zeta_{15}^{4} - 3 \zeta_{15}^{5} + 3 \zeta_{15}^{7} ) q^{7} + ( \zeta_{15}^{2} + 4 \zeta_{15}^{3} + 4 \zeta_{15}^{6} + \zeta_{15}^{7} ) q^{8} + 2 \zeta_{15}^{7} q^{9} + ( 6 - 3 \zeta_{15} - 3 \zeta_{15}^{2} + 6 \zeta_{15}^{3} - 6 \zeta_{15}^{4} + 3 \zeta_{15}^{6} - 6 \zeta_{15}^{7} ) q^{10} + ( -1 - \zeta_{15} - \zeta_{15}^{3} + \zeta_{15}^{4} + \zeta_{15}^{5} - 2 \zeta_{15}^{6} + \zeta_{15}^{7} ) q^{11} + ( 9 - 6 \zeta_{15}^{2} + 9 \zeta_{15}^{3} + 6 \zeta_{15}^{6} - 6 \zeta_{15}^{7} ) q^{12} + ( 3 \zeta_{15} + 4 \zeta_{15}^{6} ) q^{13} + ( -3 \zeta_{15}^{2} - 3 \zeta_{15}^{3} - 3 \zeta_{15}^{6} - 3 \zeta_{15}^{7} ) q^{14} + ( 3 \zeta_{15} - 6 \zeta_{15}^{2} - 6 \zeta_{15}^{5} + 3 \zeta_{15}^{6} ) q^{15} + ( -5 \zeta_{15} - 3 \zeta_{15}^{2} - 3 \zeta_{15}^{5} - 5 \zeta_{15}^{6} ) q^{16} + ( -1 + \zeta_{15}^{4} - \zeta_{15}^{5} - 2 \zeta_{15}^{7} ) q^{17} + ( 2 - 2 \zeta_{15}^{2} - 2 \zeta_{15}^{6} - 2 \zeta_{15}^{7} ) q^{18} -6 \zeta_{15}^{4} q^{19} + ( 9 \zeta_{15}^{4} + 9 \zeta_{15}^{7} ) q^{20} + ( -3 - 3 \zeta_{15}^{2} - 3 \zeta_{15}^{3} + 3 \zeta_{15}^{6} - 3 \zeta_{15}^{7} ) q^{21} + ( 3 - 3 \zeta_{15}^{4} + 3 \zeta_{15}^{5} - 4 \zeta_{15}^{7} ) q^{22} + ( 2 - 7 \zeta_{15}^{2} + 2 \zeta_{15}^{3} - 2 \zeta_{15}^{4} + 2 \zeta_{15}^{5} + 2 \zeta_{15}^{6} - 2 \zeta_{15}^{7} ) q^{23} + ( -13 + 7 \zeta_{15} - 13 \zeta_{15}^{3} + 13 \zeta_{15}^{4} - 7 \zeta_{15}^{5} - 6 \zeta_{15}^{6} + 13 \zeta_{15}^{7} ) q^{24} + 4 q^{25} + ( -2 - 4 \zeta_{15} - \zeta_{15}^{2} + \zeta_{15}^{3} - 4 \zeta_{15}^{4} - 4 \zeta_{15}^{5} - \zeta_{15}^{7} ) q^{26} + ( 2 + \zeta_{15}^{3} + 2 \zeta_{15}^{6} ) q^{27} + ( 9 \zeta_{15}^{2} + 9 \zeta_{15}^{5} ) q^{28} + ( -5 + \zeta_{15} + 4 \zeta_{15}^{2} - 5 \zeta_{15}^{3} + 5 \zeta_{15}^{4} - 4 \zeta_{15}^{6} + 5 \zeta_{15}^{7} ) q^{29} + ( 3 + 9 \zeta_{15} + 9 \zeta_{15}^{4} + 3 \zeta_{15}^{5} ) q^{30} + ( 4 - 6 \zeta_{15}^{2} + 6 \zeta_{15}^{3} + 3 \zeta_{15}^{6} - 6 \zeta_{15}^{7} ) q^{31} + ( 6 + 3 \zeta_{15} + 3 \zeta_{15}^{4} + 6 \zeta_{15}^{5} ) q^{32} -5 \zeta_{15}^{4} q^{33} + ( -1 + \zeta_{15}^{2} + 2 \zeta_{15}^{6} + \zeta_{15}^{7} ) q^{34} + ( 9 - 9 \zeta_{15} - 9 \zeta_{15}^{4} + 9 \zeta_{15}^{5} - 9 \zeta_{15}^{7} ) q^{35} + ( -6 + 6 \zeta_{15} + 6 \zeta_{15}^{2} - 6 \zeta_{15}^{3} + 6 \zeta_{15}^{4} + 6 \zeta_{15}^{7} ) q^{36} + ( 5 - 5 \zeta_{15} - 5 \zeta_{15}^{2} + 5 \zeta_{15}^{3} - 5 \zeta_{15}^{4} - 5 \zeta_{15}^{7} ) q^{37} + ( 6 + 12 \zeta_{15}^{3} + 6 \zeta_{15}^{6} ) q^{38} + ( -8 + \zeta_{15}^{2} - 2 \zeta_{15}^{3} + 8 \zeta_{15}^{4} - 8 \zeta_{15}^{5} - 2 \zeta_{15}^{6} + 5 \zeta_{15}^{7} ) q^{39} + ( -3 \zeta_{15}^{2} - 12 \zeta_{15}^{3} - 12 \zeta_{15}^{6} - 3 \zeta_{15}^{7} ) q^{40} + ( 3 - 7 \zeta_{15} + 4 \zeta_{15}^{2} + 3 \zeta_{15}^{3} - 3 \zeta_{15}^{4} - 4 \zeta_{15}^{6} - 3 \zeta_{15}^{7} ) q^{41} + ( 12 - 3 \zeta_{15} + 12 \zeta_{15}^{3} - 12 \zeta_{15}^{4} + 3 \zeta_{15}^{5} + 9 \zeta_{15}^{6} - 12 \zeta_{15}^{7} ) q^{42} + ( -4 \zeta_{15} - 4 \zeta_{15}^{4} - 4 \zeta_{15}^{7} ) q^{43} + ( -3 + 3 \zeta_{15}^{2} + 6 \zeta_{15}^{6} + 3 \zeta_{15}^{7} ) q^{44} -6 \zeta_{15}^{7} q^{45} + ( 3 + 5 \zeta_{15} + 3 \zeta_{15}^{5} - 3 \zeta_{15}^{7} ) q^{46} + ( -4 + 4 \zeta_{15}^{2} + 2 \zeta_{15}^{6} + 4 \zeta_{15}^{7} ) q^{47} + ( 13 - 13 \zeta_{15}^{4} + 13 \zeta_{15}^{5} - 14 \zeta_{15}^{7} ) q^{48} + ( -2 \zeta_{15} - 2 \zeta_{15}^{6} ) q^{49} + ( -8 + 4 \zeta_{15} + 4 \zeta_{15}^{2} - 8 \zeta_{15}^{3} + 8 \zeta_{15}^{4} - 4 \zeta_{15}^{6} + 8 \zeta_{15}^{7} ) q^{50} + ( 5 + 5 \zeta_{15}^{6} ) q^{51} + ( 9 + 3 \zeta_{15} + 9 \zeta_{15}^{3} + 3 \zeta_{15}^{4} + 3 \zeta_{15}^{7} ) q^{52} + ( 2 \zeta_{15}^{2} - \zeta_{15}^{3} - \zeta_{15}^{6} + 2 \zeta_{15}^{7} ) q^{53} + ( -3 - \zeta_{15}^{2} - 3 \zeta_{15}^{3} + 3 \zeta_{15}^{4} - 3 \zeta_{15}^{5} - 3 \zeta_{15}^{6} + 3 \zeta_{15}^{7} ) q^{54} + ( 3 + 3 \zeta_{15} + 3 \zeta_{15}^{3} - 3 \zeta_{15}^{4} - 3 \zeta_{15}^{5} + 6 \zeta_{15}^{6} - 3 \zeta_{15}^{7} ) q^{55} + ( -3 - 12 \zeta_{15} - 12 \zeta_{15}^{4} - 3 \zeta_{15}^{5} ) q^{56} + ( 6 - 12 \zeta_{15}^{2} + 12 \zeta_{15}^{3} - 12 \zeta_{15}^{7} ) q^{57} + ( 5 - 5 \zeta_{15} - 11 \zeta_{15}^{4} + 5 \zeta_{15}^{5} - 11 \zeta_{15}^{7} ) q^{58} + ( -2 + 10 \zeta_{15} - 2 \zeta_{15}^{5} + 2 \zeta_{15}^{7} ) q^{59} + ( -27 + 18 \zeta_{15}^{2} - 27 \zeta_{15}^{3} - 18 \zeta_{15}^{6} + 18 \zeta_{15}^{7} ) q^{60} + ( 3 - 9 \zeta_{15} - 9 \zeta_{15}^{4} + 3 \zeta_{15}^{5} ) q^{61} + ( -11 + 7 \zeta_{15} + \zeta_{15}^{2} - 11 \zeta_{15}^{3} + 11 \zeta_{15}^{4} - 3 \zeta_{15}^{5} - 4 \zeta_{15}^{6} + 11 \zeta_{15}^{7} ) q^{62} -6 \zeta_{15}^{5} q^{63} + ( -5 - \zeta_{15}^{2} - 5 \zeta_{15}^{3} + \zeta_{15}^{6} - \zeta_{15}^{7} ) q^{64} + ( -9 \zeta_{15} - 12 \zeta_{15}^{6} ) q^{65} + ( 5 + 10 \zeta_{15}^{3} + 5 \zeta_{15}^{6} ) q^{66} + ( -2 + 2 \zeta_{15} + 2 \zeta_{15}^{2} - 2 \zeta_{15}^{3} + 2 \zeta_{15}^{4} + 9 \zeta_{15}^{5} + 2 \zeta_{15}^{7} ) q^{67} + ( 6 - 6 \zeta_{15} - 6 \zeta_{15}^{2} + 6 \zeta_{15}^{3} - 6 \zeta_{15}^{4} + 3 \zeta_{15}^{5} - 6 \zeta_{15}^{7} ) q^{68} + ( -11 + 11 \zeta_{15} - \zeta_{15}^{4} - 11 \zeta_{15}^{5} - \zeta_{15}^{7} ) q^{69} + ( 9 \zeta_{15}^{2} + 9 \zeta_{15}^{3} + 9 \zeta_{15}^{6} + 9 \zeta_{15}^{7} ) q^{70} + ( -4 + 4 \zeta_{15}^{4} - 4 \zeta_{15}^{5} - 7 \zeta_{15}^{7} ) q^{71} + ( -8 \zeta_{15} - 10 \zeta_{15}^{4} - 8 \zeta_{15}^{7} ) q^{72} + ( -8 - 2 \zeta_{15}^{3} - 8 \zeta_{15}^{6} ) q^{73} + ( 10 \zeta_{15} + 15 \zeta_{15}^{4} + 10 \zeta_{15}^{7} ) q^{74} + ( -4 \zeta_{15} + 8 \zeta_{15}^{2} + 8 \zeta_{15}^{5} - 4 \zeta_{15}^{6} ) q^{75} + ( -18 - 18 \zeta_{15}^{3} + 18 \zeta_{15}^{4} - 18 \zeta_{15}^{5} - 18 \zeta_{15}^{6} + 18 \zeta_{15}^{7} ) q^{76} + ( 6 - 6 \zeta_{15}^{2} + 3 \zeta_{15}^{6} - 6 \zeta_{15}^{7} ) q^{77} + ( 12 + \zeta_{15} - 9 \zeta_{15}^{2} + 3 \zeta_{15}^{5} - 3 \zeta_{15}^{6} - 12 \zeta_{15}^{7} ) q^{78} + ( -3 \zeta_{15}^{2} + 12 \zeta_{15}^{3} + 12 \zeta_{15}^{6} - 3 \zeta_{15}^{7} ) q^{79} + ( 15 \zeta_{15} + 9 \zeta_{15}^{2} + 9 \zeta_{15}^{5} + 15 \zeta_{15}^{6} ) q^{80} + ( -11 + 11 \zeta_{15}^{2} - 11 \zeta_{15}^{3} + 11 \zeta_{15}^{4} - 11 \zeta_{15}^{6} + 11 \zeta_{15}^{7} ) q^{81} + ( -3 + 3 \zeta_{15} + 13 \zeta_{15}^{4} - 3 \zeta_{15}^{5} + 13 \zeta_{15}^{7} ) q^{82} + ( -7 - \zeta_{15}^{2} - 7 \zeta_{15}^{3} + \zeta_{15}^{6} - \zeta_{15}^{7} ) q^{83} + ( -9 + 9 \zeta_{15}^{4} - 9 \zeta_{15}^{5} + 27 \zeta_{15}^{7} ) q^{84} + ( 3 - 3 \zeta_{15}^{4} + 3 \zeta_{15}^{5} + 6 \zeta_{15}^{7} ) q^{85} + ( 8 + 12 \zeta_{15}^{3} + 8 \zeta_{15}^{6} ) q^{86} + ( 2 - 9 \zeta_{15} - 9 \zeta_{15}^{4} + 2 \zeta_{15}^{5} ) q^{87} + ( 6 - 6 \zeta_{15} - 6 \zeta_{15}^{2} + 6 \zeta_{15}^{3} - 6 \zeta_{15}^{4} - 7 \zeta_{15}^{5} - 6 \zeta_{15}^{7} ) q^{88} + ( 3 \zeta_{15} - 3 \zeta_{15}^{5} + 3 \zeta_{15}^{6} ) q^{89} + ( -6 + 6 \zeta_{15}^{2} + 6 \zeta_{15}^{6} + 6 \zeta_{15}^{7} ) q^{90} + ( -9 + 9 \zeta_{15}^{2} - 9 \zeta_{15}^{3} - 3 \zeta_{15}^{4} - 9 \zeta_{15}^{6} + 9 \zeta_{15}^{7} ) q^{91} + ( -9 + 15 \zeta_{15}^{2} - 15 \zeta_{15}^{3} + 15 \zeta_{15}^{7} ) q^{92} + ( -6 + 14 \zeta_{15} + 5 \zeta_{15}^{2} - 6 \zeta_{15}^{3} + 6 \zeta_{15}^{4} - 4 \zeta_{15}^{5} + 8 \zeta_{15}^{6} + 6 \zeta_{15}^{7} ) q^{93} + ( 10 - 10 \zeta_{15} - 10 \zeta_{15}^{2} + 10 \zeta_{15}^{3} - 10 \zeta_{15}^{4} - 6 \zeta_{15}^{5} - 10 \zeta_{15}^{7} ) q^{94} + 18 \zeta_{15}^{4} q^{95} + ( -15 + 15 \zeta_{15}^{2} + 15 \zeta_{15}^{7} ) q^{96} + ( -1 + \zeta_{15} - \zeta_{15}^{5} ) q^{97} + ( 2 + 2 \zeta_{15} + 2 \zeta_{15}^{4} + 2 \zeta_{15}^{5} ) q^{98} + ( 2 - 4 \zeta_{15}^{2} + 4 \zeta_{15}^{3} - 4 \zeta_{15}^{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 4q^{2} - 5q^{3} - 6q^{4} - 24q^{5} - 10q^{6} - 3q^{7} - 14q^{8} + 2q^{9} + O(q^{10}) \) \( 8q - 4q^{2} - 5q^{3} - 6q^{4} - 24q^{5} - 10q^{6} - 3q^{7} - 14q^{8} + 2q^{9} + 12q^{10} - 5q^{11} + 30q^{12} - 5q^{13} + 6q^{14} + 15q^{15} + 14q^{16} - 5q^{17} + 16q^{18} - 6q^{19} + 18q^{20} - 30q^{21} + 5q^{22} - 11q^{23} - 5q^{24} + 32q^{25} - 12q^{26} + 10q^{27} - 27q^{28} - 7q^{29} + 30q^{30} + 2q^{31} + 30q^{32} - 5q^{33} - 10q^{34} + 9q^{35} - 12q^{36} + 10q^{37} + 12q^{38} - 10q^{39} + 42q^{40} + 17q^{41} + 15q^{42} - 12q^{43} - 30q^{44} - 6q^{45} + 14q^{46} - 28q^{47} + 25q^{48} + 2q^{49} - 16q^{50} + 30q^{51} + 63q^{52} + 8q^{53} + 5q^{54} + 15q^{55} - 36q^{56} - 7q^{58} + 4q^{59} - 90q^{60} - 6q^{61} - 16q^{62} + 24q^{63} - 34q^{64} + 15q^{65} + 10q^{66} - 40q^{67} - 35q^{69} - 18q^{70} - 19q^{71} - 26q^{72} - 44q^{73} + 35q^{74} - 20q^{75} + 36q^{76} + 30q^{77} + 70q^{78} - 54q^{79} - 42q^{80} - 11q^{81} + 17q^{82} - 46q^{83} + 15q^{85} + 24q^{86} - 10q^{87} + 40q^{88} + 9q^{89} - 48q^{90} - 21q^{91} - 12q^{92} - 5q^{93} + 44q^{94} + 18q^{95} - 90q^{96} - 3q^{97} + 12q^{98} + O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/403\mathbb{Z}\right)^\times\).

\(n\) \(249\) \(313\)
\(\chi(n)\) \(-1 - \zeta_{15}^{5}\) \(-\zeta_{15}^{2} - \zeta_{15}^{7}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1
0.913545 0.406737i
−0.978148 + 0.207912i
0.913545 + 0.406737i
0.669131 + 0.743145i
−0.104528 0.994522i
0.669131 0.743145i
−0.104528 + 0.994522i
−0.978148 0.207912i
−2.39169 1.06485i 0.233733 2.22382i 3.24803 + 3.60730i −3.00000 −2.92705 + 5.06980i −2.00739 2.22943i −2.30902 7.10642i −1.95630 0.415823i 7.17508 + 3.19455i
35.1 0.373619 + 0.0794152i 1.49622 + 1.66172i −1.69381 0.754131i −3.00000 0.427051 + 0.739674i −2.74064 1.22021i −1.19098 0.865300i −0.209057 + 1.98904i −1.12086 0.238246i
126.1 −2.39169 + 1.06485i 0.233733 + 2.22382i 3.24803 3.60730i −3.00000 −2.92705 5.06980i −2.00739 + 2.22943i −2.30902 + 7.10642i −1.95630 + 0.415823i 7.17508 3.19455i
159.1 −0.255585 + 0.283856i −2.18720 + 0.464905i 0.193806 + 1.84395i −3.00000 0.427051 0.739674i 0.313585 + 2.98357i −1.19098 0.865300i 1.82709 0.813473i 0.766755 0.851568i
250.1 0.273659 2.60369i −2.04275 0.909491i −4.74803 1.00922i −3.00000 −2.92705 + 5.06980i 2.93444 + 0.623735i −2.30902 + 7.10642i 1.33826 + 1.48629i −0.820977 + 7.81108i
256.1 −0.255585 0.283856i −2.18720 0.464905i 0.193806 1.84395i −3.00000 0.427051 + 0.739674i 0.313585 2.98357i −1.19098 + 0.865300i 1.82709 + 0.813473i 0.766755 + 0.851568i
295.1 0.273659 + 2.60369i −2.04275 + 0.909491i −4.74803 + 1.00922i −3.00000 −2.92705 5.06980i 2.93444 0.623735i −2.30902 7.10642i 1.33826 1.48629i −0.820977 7.81108i
380.1 0.373619 0.0794152i 1.49622 1.66172i −1.69381 + 0.754131i −3.00000 0.427051 0.739674i −2.74064 + 1.22021i −1.19098 + 0.865300i −0.209057 1.98904i −1.12086 + 0.238246i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 380.1
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
13.c Even 1 yes
31.d Even 1 yes
403.bl Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{8} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(403, [\chi])\).