Properties

Label 418.2.n.a
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.

\(f(q)\) \(=\) \( q + (\zeta_{15}^{6} + \zeta_{15}) q^{2} + (2 \zeta_{15}^{6} - 2 \zeta_{15}^{5} + 2 \zeta_{15}) q^{3} + \zeta_{15}^{7} q^{4} + ( - \zeta_{15}^{7} - 2 \zeta_{15}^{6} - \zeta_{15}^{4} + \zeta_{15}^{3} + 2 \zeta_{15}^{2} - 3 \zeta_{15} + 1) q^{5} + (2 \zeta_{15}^{7} + 2 \zeta_{15}) q^{6} + ( - \zeta_{15}^{7} - \zeta_{15}^{2}) q^{7} - \zeta_{15}^{3} q^{8} + (4 \zeta_{15}^{7} - 4 \zeta_{15}^{5} + 5 \zeta_{15} - 4) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{15}^{6} + \zeta_{15}) q^{2} + (2 \zeta_{15}^{6} - 2 \zeta_{15}^{5} + 2 \zeta_{15}) q^{3} + \zeta_{15}^{7} q^{4} + ( - \zeta_{15}^{7} - 2 \zeta_{15}^{6} - \zeta_{15}^{4} + \zeta_{15}^{3} + 2 \zeta_{15}^{2} - 3 \zeta_{15} + 1) q^{5} + (2 \zeta_{15}^{7} + 2 \zeta_{15}) q^{6} + ( - \zeta_{15}^{7} - \zeta_{15}^{2}) q^{7} - \zeta_{15}^{3} q^{8} + (4 \zeta_{15}^{7} - 4 \zeta_{15}^{5} + 5 \zeta_{15} - 4) q^{9} + ( - 2 \zeta_{15}^{5} + 3 \zeta_{15}^{4} + 3 \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 \zeta_{15}^{7} - 2 \zeta_{15}^{3} + 2 \zeta_{15}^{2}) q^{12} + ( - 3 \zeta_{15}^{7} + 3 \zeta_{15}^{5} - \zeta_{15} + 3) q^{13} + ( - \zeta_{15}^{7} + \zeta_{15}^{5} - \zeta_{15}^{4} + \zeta_{15}^{3} - \zeta_{15} + 1) q^{14} + ( - 6 \zeta_{15}^{7} - 4 \zeta_{15}^{5} + 4 \zeta_{15}^{4} - 4) q^{15} + ( - \zeta_{15}^{7} + \zeta_{15}^{6} - \zeta_{15}^{4} + \zeta_{15}^{3} - \zeta_{15}^{2} + 1) q^{16} + ( - 2 \zeta_{15}^{7} + 3 \zeta_{15}^{6} - 2 \zeta_{15}^{4} + 2 \zeta_{15}^{3} - 3 \zeta_{15}^{2} + \zeta_{15} + 2) q^{17} + (5 \zeta_{15}^{7} - 4 \zeta_{15}^{6} - 4 \zeta_{15}^{3} + 5 \zeta_{15}^{2}) q^{18} + ( - 2 \zeta_{15}^{7} + 2 \zeta_{15}^{6} - 2 \zeta_{15}^{5} - 2 \zeta_{15}^{4} - \zeta_{15}^{3} + 2 \zeta_{15}) q^{19} + (3 \zeta_{15}^{7} - 2 \zeta_{15}^{6} + 3 \zeta_{15}^{2} - 3) q^{20} + ( - 2 \zeta_{15}^{7} + 2 \zeta_{15}^{5} - 2 \zeta_{15}^{4} + 2 \zeta_{15}^{3} - 2 \zeta_{15}^{2} - 2 \zeta_{15} + 2) q^{21} + (\zeta_{15}^{7} - \zeta_{15}^{6} + 2 \zeta_{15}^{5} + \zeta_{15}^{4} - \zeta_{15}^{3} - \zeta_{15}^{2} - 1) q^{22} + ( - \zeta_{15}^{5} - 1) q^{23} + (2 \zeta_{15}^{6} - 2 \zeta_{15}^{5} - 2 \zeta_{15}^{2} + 2 \zeta_{15}) q^{24} + ( - 5 \zeta_{15}^{7} + 8 \zeta_{15}^{5} - 5 \zeta_{15}^{4} - 8 \zeta_{15} + 8) q^{25} + ( - \zeta_{15}^{7} + 3 \zeta_{15}^{6} + 3 \zeta_{15}^{3} - \zeta_{15}^{2}) q^{26} + (12 \zeta_{15}^{7} - 12 \zeta_{15}^{6} - 8 \zeta_{15}^{3} + 12 \zeta_{15}^{2} - 8) q^{27} + \zeta_{15}^{4} q^{28} - \zeta_{15}^{7} q^{29} + ( - 4 \zeta_{15}^{6} + 6 \zeta_{15}^{3} - 4) q^{30} + (5 \zeta_{15}^{7} + 3 \zeta_{15}^{6} + 5 \zeta_{15}^{2} - 5) q^{31} + (\zeta_{15}^{5} + 1) q^{32} + ( - 4 \zeta_{15}^{7} + 6 \zeta_{15}^{6} + 6 \zeta_{15}^{5} - 4 \zeta_{15}^{4} + 4 \zeta_{15}^{3} - 6 \zeta_{15}^{2} + \cdots + 4) q^{33} + \cdots + ( - 2 \zeta_{15}^{7} + 18 \zeta_{15}^{5} - 15 \zeta_{15}^{4} - 12 \zeta_{15} + 18) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - q^{2} + 6 q^{3} + q^{4} + 7 q^{5} + 4 q^{6} - 2 q^{7} + 2 q^{8} - 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - q^{2} + 6 q^{3} + q^{4} + 7 q^{5} + 4 q^{6} - 2 q^{7} + 2 q^{8} - 7 q^{9} - 2 q^{10} - 2 q^{11} + 8 q^{12} + 8 q^{13} - q^{14} - 18 q^{15} + q^{16} + 26 q^{18} + 4 q^{19} - 14 q^{20} - 4 q^{21} - 11 q^{22} - 4 q^{23} + 4 q^{24} + 14 q^{25} - 14 q^{26} + q^{28} - q^{29} - 36 q^{30} - 36 q^{31} + 4 q^{32} - 24 q^{33} + 10 q^{34} + 7 q^{35} + 13 q^{36} + 10 q^{37} + 17 q^{38} + 4 q^{39} + 8 q^{40} - 7 q^{41} + 4 q^{42} - 28 q^{43} - 4 q^{44} - 132 q^{45} + 2 q^{46} - q^{47} + 6 q^{48} + 12 q^{49} - 2 q^{50} + 20 q^{51} - 7 q^{52} - 14 q^{53} + 40 q^{54} + 12 q^{55} - 8 q^{56} + 8 q^{57} - 2 q^{58} + 25 q^{59} + 2 q^{60} - 15 q^{61} + 7 q^{62} + 13 q^{63} - 2 q^{64} + 88 q^{65} + 14 q^{66} + 22 q^{67} - 12 q^{69} + 8 q^{70} - 12 q^{71} + 7 q^{72} + 18 q^{73} + 5 q^{74} + 52 q^{75} + 8 q^{76} + 8 q^{77} - 28 q^{78} - 8 q^{80} + 41 q^{81} - 3 q^{82} - 10 q^{83} - 12 q^{84} - 25 q^{85} + 3 q^{86} - 8 q^{87} + 2 q^{88} + 12 q^{89} - 11 q^{90} - 7 q^{91} + q^{92} - 22 q^{93} - 2 q^{94} - 14 q^{95} + 12 q^{96} + 16 q^{97} - 24 q^{98} + 43 q^{99}+O(q^{100}) \) 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 2.95630 1.31623i 0.913545 + 0.406737i −2.57890 2.86416i 3.16535 0.672816i −0.809017 + 0.587785i 0.809017 + 0.587785i 4.99983 5.55288i −1.92705 3.33775i
125.1 −0.913545 + 0.406737i −0.827091 0.918578i 0.669131 0.743145i −0.298335 2.83847i 1.12920 + 0.502754i 0.309017 + 0.951057i −0.309017 + 0.951057i 0.153880 1.46407i 1.42705 + 2.47172i
159.1 −0.669131 + 0.743145i −0.338261 + 3.21834i −0.104528 0.994522i 3.76988 0.801313i −2.16535 2.40487i −0.809017 + 0.587785i 0.809017 + 0.587785i −7.30885 1.55354i −1.92705 + 3.33775i
163.1 −0.669131 0.743145i −0.338261 3.21834i −0.104528 + 0.994522i 3.76988 + 0.801313i −2.16535 + 2.40487i −0.809017 0.587785i 0.809017 0.587785i −7.30885 + 1.55354i −1.92705 3.33775i
201.1 0.104528 + 0.994522i 1.20906 + 0.256993i −0.978148 + 0.207912i 2.60735 1.16087i −0.129204 + 1.22930i 0.309017 0.951057i −0.309017 0.951057i −1.34486 0.598772i 1.42705 + 2.47172i
235.1 0.104528 0.994522i 1.20906 0.256993i −0.978148 0.207912i 2.60735 + 1.16087i −0.129204 1.22930i 0.309017 + 0.951057i −0.309017 + 0.951057i −1.34486 + 0.598772i 1.42705 2.47172i
273.1 0.978148 0.207912i 2.95630 + 1.31623i 0.913545 0.406737i −2.57890 + 2.86416i 3.16535 + 0.672816i −0.809017 0.587785i 0.809017 0.587785i 4.99983 + 5.55288i −1.92705 + 3.33775i
311.1 −0.913545 0.406737i −0.827091 + 0.918578i 0.669131 + 0.743145i −0.298335 + 2.83847i 1.12920 0.502754i 0.309017 0.951057i −0.309017 0.951057i 0.153880 + 1.46407i 1.42705 2.47172i
\(n\): 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.a 8
11.c even 5 1 inner 418.2.n.a 8
19.c even 3 1 inner 418.2.n.a 8
209.n even 15 1 inner 418.2.n.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.n.a 8 1.a even 1 1 trivial
418.2.n.a 8 11.c even 5 1 inner
418.2.n.a 8 19.c even 3 1 inner
418.2.n.a 8 209.n even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 6T_{3}^{7} + 20T_{3}^{6} - 64T_{3}^{5} + 144T_{3}^{4} - 64T_{3}^{3} - 256T_{3} + 256 \) 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
$3$ \( T^{8} - 6 T^{7} + 20 T^{6} - 64 T^{5} + \cdots + 256 \) Copy content Toggle raw display
$5$ \( T^{8} - 7 T^{7} + 15 T^{6} + \cdots + 14641 \) Copy content Toggle raw display
$7$ \( (T^{4} + T^{3} + T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + T^{3} + 21 T^{2} + 11 T + 121)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} - 8 T^{7} + 30 T^{6} + \cdots + 14641 \) Copy content Toggle raw display
$17$ \( T^{8} - 10 T^{6} + 50 T^{5} + \cdots + 625 \) Copy content Toggle raw display
$19$ \( T^{8} - 4 T^{7} - 3 T^{6} + \cdots + 130321 \) Copy content Toggle raw display
$23$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$29$ \( T^{8} + T^{7} - T^{5} - T^{4} - T^{3} + T + 1 \) Copy content Toggle raw display
$31$ \( (T^{4} + 18 T^{3} + 124 T^{2} + 7 T + 1)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 5 T^{3} + 10 T^{2} + 25)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 7 T^{7} + 30 T^{6} + 127 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( (T^{4} + 14 T^{3} + 167 T^{2} + 406 T + 841)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + T^{7} - T^{5} - T^{4} - T^{3} + T + 1 \) Copy content Toggle raw display
$53$ \( T^{8} + 14 T^{7} + 100 T^{6} + \cdots + 33362176 \) Copy content Toggle raw display
$59$ \( T^{8} - 25 T^{7} + 360 T^{6} + \cdots + 9150625 \) Copy content Toggle raw display
$61$ \( T^{8} + 15 T^{7} + 135 T^{6} + \cdots + 4100625 \) Copy content Toggle raw display
$67$ \( (T^{4} - 11 T^{3} + 152 T^{2} + 341 T + 961)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 12 T^{7} + \cdots + 1908029761 \) Copy content Toggle raw display
$73$ \( T^{8} - 18 T^{7} + 80 T^{6} + \cdots + 33362176 \) Copy content Toggle raw display
$79$ \( T^{8} - 160 T^{6} + \cdots + 40960000 \) Copy content Toggle raw display
$83$ \( (T^{4} + 5 T^{3} + 25 T^{2} + 125 T + 625)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 6 T^{3} + 107 T^{2} + 426 T + 5041)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 16 T^{7} + \cdots + 181063936 \) Copy content Toggle raw display
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