Properties

Label 418.2.n.b
Level $418$
Weight $2$
Character orbit 418.n
Analytic conductor $3.338$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 418 = 2 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 418.n (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.33774680449\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{15})\)
Defining polynomial: \( x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/418\mathbb{Z}\right)^\times\).

\(n\) \(287\) \(343\)
\(\chi(n)\) \(-1 - \zeta_{15}^{5}\) \(-\zeta_{15}^{2} - \zeta_{15}^{7}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
0.669131 0.743145i
−0.104528 + 0.994522i
−0.978148 0.207912i
−0.978148 + 0.207912i
0.913545 + 0.406737i
0.913545 0.406737i
0.669131 + 0.743145i
−0.104528 0.994522i
−0.978148 0.207912i −1.82709 + 0.813473i 0.913545 + 0.406737i −0.924716 1.02700i 1.95630 0.415823i 2.42705 1.76336i −0.809017 0.587785i 0.669131 0.743145i 0.690983 + 1.19682i
125.1 0.913545 0.406737i −1.33826 1.48629i 0.669131 0.743145i 0.378188 + 3.59821i −1.82709 0.813473i −0.927051 2.85317i 0.309017 0.951057i −0.104528 + 0.994522i 1.80902 + 3.13331i
159.1 0.669131 0.743145i 0.209057 1.98904i −0.104528 0.994522i 1.35177 0.287327i −1.33826 1.48629i 2.42705 1.76336i −0.809017 0.587785i −0.978148 0.207912i 0.690983 1.19682i
163.1 0.669131 + 0.743145i 0.209057 + 1.98904i −0.104528 + 0.994522i 1.35177 + 0.287327i −1.33826 + 1.48629i 2.42705 + 1.76336i −0.809017 + 0.587785i −0.978148 + 0.207912i 0.690983 + 1.19682i
201.1 −0.104528 0.994522i 1.95630 + 0.415823i −0.978148 + 0.207912i −3.30524 + 1.47159i 0.209057 1.98904i −0.927051 + 2.85317i 0.309017 + 0.951057i 0.913545 + 0.406737i 1.80902 + 3.13331i
235.1 −0.104528 + 0.994522i 1.95630 0.415823i −0.978148 0.207912i −3.30524 1.47159i 0.209057 + 1.98904i −0.927051 2.85317i 0.309017 0.951057i 0.913545 0.406737i 1.80902 3.13331i
273.1 −0.978148 + 0.207912i −1.82709 0.813473i 0.913545 0.406737i −0.924716 + 1.02700i 1.95630 + 0.415823i 2.42705 + 1.76336i −0.809017 + 0.587785i 0.669131 + 0.743145i 0.690983 1.19682i
311.1 0.913545 + 0.406737i −1.33826 + 1.48629i 0.669131 + 0.743145i 0.378188 3.59821i −1.82709 + 0.813473i −0.927051 + 2.85317i 0.309017 + 0.951057i −0.104528 0.994522i 1.80902 3.13331i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 311.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner
19.c even 3 1 inner
209.n even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 418.2.n.b 8
11.c even 5 1 inner 418.2.n.b 8
19.c even 3 1 inner 418.2.n.b 8
209.n even 15 1 inner 418.2.n.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.n.b 8 1.a even 1 1 trivial
418.2.n.b 8 11.c even 5 1 inner
418.2.n.b 8 19.c even 3 1 inner
418.2.n.b 8 209.n even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 2T_{3}^{7} - 8T_{3}^{5} - 16T_{3}^{4} - 32T_{3}^{3} + 128T_{3} + 256 \) acting on \(S_{2}^{\mathrm{new}}(418, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{7} + T^{5} - T^{4} + T^{3} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{8} + 2 T^{7} - 8 T^{5} - 16 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$5$ \( T^{8} + 5 T^{7} + 15 T^{6} + 50 T^{5} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( (T^{4} - 3 T^{3} + 9 T^{2} - 27 T + 81)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + T^{3} + 21 T^{2} + 11 T + 121)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 6 T^{7} + 20 T^{6} + \cdots + 14641 \) Copy content Toggle raw display
$17$ \( T^{8} - 4 T^{7} + 10 T^{6} - 26 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{8} + 8 T^{7} + 45 T^{6} + \cdots + 130321 \) Copy content Toggle raw display
$23$ \( (T^{4} + 45 T^{2} + 2025)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 11 T^{7} + 70 T^{6} + \cdots + 923521 \) Copy content Toggle raw display
$31$ \( (T^{4} + 8 T^{3} + 34 T^{2} + 77 T + 121)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 5 T^{3} + 40 T^{2} - 50 T + 25)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 11 T^{7} + 60 T^{6} + \cdots + 130321 \) Copy content Toggle raw display
$43$ \( (T^{4} - 10 T^{3} + 95 T^{2} - 50 T + 25)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 3 T^{7} - 70 T^{6} + \cdots + 12117361 \) Copy content Toggle raw display
$53$ \( T^{8} - 24 T^{7} + \cdots + 236421376 \) Copy content Toggle raw display
$59$ \( T^{8} + 19 T^{7} + 120 T^{6} + \cdots + 38950081 \) Copy content Toggle raw display
$61$ \( T^{8} + 13 T^{7} + 105 T^{6} + \cdots + 130321 \) Copy content Toggle raw display
$67$ \( (T^{4} - 5 T^{3} + 80 T^{2} + 275 T + 3025)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 6 T^{7} - 40 T^{6} + \cdots + 25411681 \) Copy content Toggle raw display
$73$ \( T^{8} - 16 T^{7} + 120 T^{6} + \cdots + 3748096 \) Copy content Toggle raw display
$79$ \( T^{8} + 12 T^{7} + 432 T^{5} + \cdots + 1679616 \) Copy content Toggle raw display
$83$ \( (T^{4} - 19 T^{3} + 411 T^{2} - 2759 T + 7921)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 5 T^{2} + 25)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 34 T^{7} + \cdots + 3544535296 \) Copy content Toggle raw display
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