Properties

Label 403.2.be.b
Level 403
Weight 2
Character orbit 403.be
Analytic conductor 3.218
Analytic rank 0
Dimension 4
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 403 = 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 403.be (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.21797120146\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(249\) \(313\)
\(\chi(n)\) \(\zeta_{12}^{3}\) \(\zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
57.1
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
0.866025 + 0.500000i
0.366025 0.366025i 2.36603 + 1.36603i 1.73205i −1.36603 0.366025i 1.36603 0.366025i 2.86603 0.767949i 1.36603 + 1.36603i 2.23205 + 3.86603i −0.633975 + 0.366025i
99.1 0.366025 + 0.366025i 2.36603 1.36603i 1.73205i −1.36603 + 0.366025i 1.36603 + 0.366025i 2.86603 + 0.767949i 1.36603 1.36603i 2.23205 3.86603i −0.633975 0.366025i
161.1 −1.36603 + 1.36603i 0.633975 0.366025i 1.73205i 0.366025 + 1.36603i −0.366025 + 1.36603i 1.13397 4.23205i −0.366025 0.366025i −1.23205 + 2.13397i −2.36603 1.36603i
398.1 −1.36603 1.36603i 0.633975 + 0.366025i 1.73205i 0.366025 1.36603i −0.366025 1.36603i 1.13397 + 4.23205i −0.366025 + 0.366025i −1.23205 2.13397i −2.36603 + 1.36603i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
403.be even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 403.2.be.b yes 4
13.d odd 4 1 403.2.be.a 4
31.e odd 6 1 403.2.be.a 4
403.be even 12 1 inner 403.2.be.b yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.be.a 4 13.d odd 4 1
403.2.be.a 4 31.e odd 6 1
403.2.be.b yes 4 1.a even 1 1 trivial
403.2.be.b yes 4 403.be even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(403, [\chi])\):

\( T_{2}^{4} + 2 T_{2}^{3} + 2 T_{2}^{2} - 2 T_{2} + 1 \)
\( T_{7}^{4} - 8 T_{7}^{3} + 41 T_{7}^{2} - 130 T_{7} + 169 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 2 T^{2} + 2 T^{3} + T^{4} + 4 T^{5} + 8 T^{6} + 16 T^{7} + 16 T^{8} \)
$3$ \( 1 - 6 T + 20 T^{2} - 48 T^{3} + 91 T^{4} - 144 T^{5} + 180 T^{6} - 162 T^{7} + 81 T^{8} \)
$5$ \( ( 1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4} )( 1 + 4 T + 11 T^{2} + 20 T^{3} + 25 T^{4} ) \)
$7$ \( ( 1 - 4 T + 7 T^{2} )^{2}( 1 + 11 T^{2} + 49 T^{4} ) \)
$11$ \( 1 + 9 T^{2} - 48 T^{3} + 20 T^{4} - 528 T^{5} + 1089 T^{6} + 14641 T^{8} \)
$13$ \( 1 + 4 T + 3 T^{2} + 52 T^{3} + 169 T^{4} \)
$17$ \( ( 1 - 17 T^{2} + 289 T^{4} )^{2} \)
$19$ \( 1 - 12 T + 45 T^{2} + 108 T^{3} - 1348 T^{4} + 2052 T^{5} + 16245 T^{6} - 82308 T^{7} + 130321 T^{8} \)
$23$ \( ( 1 - 2 T + 20 T^{2} - 46 T^{3} + 529 T^{4} )^{2} \)
$29$ \( ( 1 - 4 T - 13 T^{2} - 116 T^{3} + 841 T^{4} )( 1 + 4 T - 13 T^{2} + 116 T^{3} + 841 T^{4} ) \)
$31$ \( ( 1 - 7 T + 31 T^{2} )^{2} \)
$37$ \( 1 + 12 T + 72 T^{2} + 288 T^{3} + 983 T^{4} + 10656 T^{5} + 98568 T^{6} + 607836 T^{7} + 1874161 T^{8} \)
$41$ \( 1 - 16 T + 164 T^{2} - 1204 T^{3} + 8239 T^{4} - 49364 T^{5} + 275684 T^{6} - 1102736 T^{7} + 2825761 T^{8} \)
$43$ \( 1 + 2 T - 56 T^{2} - 52 T^{3} + 1579 T^{4} - 2236 T^{5} - 103544 T^{6} + 159014 T^{7} + 3418801 T^{8} \)
$47$ \( 1 - 20 T + 200 T^{2} - 1460 T^{3} + 9982 T^{4} - 68620 T^{5} + 441800 T^{6} - 2076460 T^{7} + 4879681 T^{8} \)
$53$ \( 1 + 30 T + 477 T^{2} + 5310 T^{3} + 44420 T^{4} + 281430 T^{5} + 1339893 T^{6} + 4466310 T^{7} + 7890481 T^{8} \)
$59$ \( 1 - 2 T + 122 T^{2} - 608 T^{3} + 8287 T^{4} - 35872 T^{5} + 424682 T^{6} - 410758 T^{7} + 12117361 T^{8} \)
$61$ \( 1 - 182 T^{2} + 15291 T^{4} - 677222 T^{6} + 13845841 T^{8} \)
$67$ \( 1 + 22 T + 137 T^{2} - 834 T^{3} - 16648 T^{4} - 55878 T^{5} + 614993 T^{6} + 6616786 T^{7} + 20151121 T^{8} \)
$71$ \( 1 + 2 T + 65 T^{2} + 818 T^{3} + 2464 T^{4} + 58078 T^{5} + 327665 T^{6} + 715822 T^{7} + 25411681 T^{8} \)
$73$ \( 1 - 18 T + 90 T^{2} + 1356 T^{3} - 21121 T^{4} + 98988 T^{5} + 479610 T^{6} - 7002306 T^{7} + 28398241 T^{8} \)
$79$ \( 1 + 36 T + 694 T^{2} + 9432 T^{3} + 96531 T^{4} + 745128 T^{5} + 4331254 T^{6} + 17749404 T^{7} + 38950081 T^{8} \)
$83$ \( 1 - 34 T + 650 T^{2} - 8464 T^{3} + 86383 T^{4} - 702512 T^{5} + 4477850 T^{6} - 19440758 T^{7} + 47458321 T^{8} \)
$89$ \( 1 - 16 T + 128 T^{2} - 1840 T^{3} + 25774 T^{4} - 163760 T^{5} + 1013888 T^{6} - 11279504 T^{7} + 62742241 T^{8} \)
$97$ \( 1 + 20 T + 200 T^{2} + 2460 T^{3} + 29582 T^{4} + 238620 T^{5} + 1881800 T^{6} + 18253460 T^{7} + 88529281 T^{8} \)
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