Properties

Label 418.2.n.c
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.

\(f(q)\) \(=\) \( q + ( - \zeta_{15}^{6} - \zeta_{15}) q^{2} + (2 \zeta_{15}^{7} - 2 \zeta_{15}^{5} + 2 \zeta_{15}^{4} - 2 \zeta_{15}^{3} + 2 \zeta_{15} - 2) q^{3} + \zeta_{15}^{7} q^{4} + (\zeta_{15}^{7} - 2 \zeta_{15}^{6} + \zeta_{15}^{4} - \zeta_{15}^{3} + 2 \zeta_{15}^{2} - \zeta_{15} - 1) q^{5} + 2 \zeta_{15}^{4} q^{6} + ( - \zeta_{15}^{7} - 2 \zeta_{15}^{6} - 2 \zeta_{15}^{3} - \zeta_{15}^{2}) q^{7} + \zeta_{15}^{3} q^{8} + \zeta_{15} q^{9}+O(q^{10}) \) 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} + (\zeta_{15}^{7} - 2 \zeta_{15}^{6} + \zeta_{15}^{4} - \zeta_{15}^{3} + 2 \zeta_{15}^{2} - \zeta_{15} - 1) q^{5} + 2 \zeta_{15}^{4} q^{6} + ( - \zeta_{15}^{7} - 2 \zeta_{15}^{6} - 2 \zeta_{15}^{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} + 2 \zeta_{15}^{6} - \zeta_{15}^{3} + 2 \zeta_{15}^{2}) q^{11} + 2 q^{12} + (3 \zeta_{15}^{7} - 3 \zeta_{15}^{5} + 3 \zeta_{15} - 3) q^{13} + (3 \zeta_{15}^{7} - 2 \zeta_{15}^{6} - \zeta_{15}^{5} + 3 \zeta_{15}^{4} - 3 \zeta_{15}^{3} + \zeta_{15} - 3) q^{14} + ( - 2 \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} + (6 \zeta_{15}^{7} - 7 \zeta_{15}^{6} + 6 \zeta_{15}^{4} - 6 \zeta_{15}^{3} + 7 \zeta_{15}^{2} - \zeta_{15} - 6) q^{17} + ( - \zeta_{15}^{7} - \zeta_{15}^{2}) q^{18} + ( - 2 \zeta_{15}^{7} + 2 \zeta_{15}^{6} - 2 \zeta_{15}^{5} - 2 \zeta_{15}^{4} + 3 \zeta_{15}^{3} + 2 \zeta_{15}) q^{19} + (\zeta_{15}^{7} - 2 \zeta_{15}^{6} + \zeta_{15}^{2} - 1) q^{20} + (4 \zeta_{15}^{7} + 2 \zeta_{15}^{5} + 4 \zeta_{15}^{4} - 4 \zeta_{15}^{3} + 4 \zeta_{15}^{2} + 4 \zeta_{15} - 4) q^{21} + ( - \zeta_{15}^{7} - \zeta_{15}^{6} + 2 \zeta_{15}^{5} - \zeta_{15}^{4} + \zeta_{15}^{3} + 3 \zeta_{15}^{2} - 2 \zeta_{15} + 1) q^{22} + (\zeta_{15}^{5} + 1) q^{23} + ( - 2 \zeta_{15}^{6} - 2 \zeta_{15}) q^{24} + (3 \zeta_{15}^{7} + 3 \zeta_{15}^{4}) q^{25} + ( - 3 \zeta_{15}^{7} + 3 \zeta_{15}^{6} + 3 \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} + (2 \zeta_{15}^{7} + 3 \zeta_{15}^{4} + 2 \zeta_{15}) q^{28} + (\zeta_{15}^{7} - 2 \zeta_{15}^{5} + 2 \zeta_{15}^{4} - 2) q^{29} + (2 \zeta_{15}^{6} - 2 \zeta_{15}^{3} + 2) q^{30} + (3 \zeta_{15}^{7} + 5 \zeta_{15}^{6} + 3 \zeta_{15}^{2} - 3) q^{31} + ( - \zeta_{15}^{5} - 1) q^{32} + ( - 4 \zeta_{15}^{7} + 6 \zeta_{15}^{6} - 4 \zeta_{15}^{5} - 4 \zeta_{15}^{4} + 4 \zeta_{15}^{3} - 4 \zeta_{15}^{2} + \cdots + 4) q^{33}+ \cdots + (4 \zeta_{15}^{7} - 2 \zeta_{15}^{5} + \zeta_{15}^{4} + 2 \zeta_{15} - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{2} + 2 q^{3} + q^{4} + q^{5} + 2 q^{6} + 6 q^{7} - 2 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + q^{2} + 2 q^{3} + q^{4} + q^{5} + 2 q^{6} + 6 q^{7} - 2 q^{8} + q^{9} + 6 q^{10} + 2 q^{11} + 16 q^{12} - 6 q^{13} - 3 q^{14} - 8 q^{15} + q^{16} - 4 q^{17} - 2 q^{18} - 4 q^{19} - 2 q^{20} - 16 q^{21} - q^{22} + 4 q^{23} + 2 q^{24} + 6 q^{25} - 18 q^{26} + 8 q^{27} + 7 q^{28} - 5 q^{29} + 16 q^{30} - 28 q^{31} - 4 q^{32} + 18 q^{33} + 26 q^{34} - 3 q^{35} + q^{36} + 14 q^{37} - 13 q^{38} + 24 q^{39} - 4 q^{40} - 19 q^{41} + 14 q^{42} + 16 q^{43} + 4 q^{44} - 12 q^{45} + 2 q^{46} + q^{47} + 2 q^{48} - 44 q^{49} + 18 q^{50} - 18 q^{51} + 9 q^{52} + 4 q^{53} + 16 q^{54} + 14 q^{55} + 16 q^{56} - 26 q^{57} + 10 q^{58} - 13 q^{59} + 2 q^{60} - 3 q^{61} - q^{62} - 3 q^{63} - 2 q^{64} - 48 q^{65} - 22 q^{66} - 18 q^{67} + 8 q^{68} + 4 q^{69} + 2 q^{70} - 16 q^{71} + q^{72} - 4 q^{73} - 7 q^{74} + 36 q^{75} - 8 q^{76} + 24 q^{77} - 12 q^{78} + 12 q^{79} - 4 q^{80} - 11 q^{81} + 21 q^{82} - 34 q^{83} + 12 q^{84} + 31 q^{85} - 19 q^{86} - 18 q^{88} - 32 q^{89} + q^{90} + 3 q^{91} - q^{92} + 28 q^{93} - 2 q^{94} - 38 q^{95} - 4 q^{96} - 30 q^{97} - 8 q^{98} - q^{99}+O(q^{100}) \) 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 −1.75181 1.94558i −1.95630 + 0.415823i 0.190983 0.138757i −0.809017 0.587785i 0.669131 0.743145i 1.30902 + 2.26728i
125.1 0.913545 0.406737i 1.33826 + 1.48629i 0.669131 0.743145i 0.0399263 + 0.379874i 1.82709 + 0.813473i 1.30902 + 4.02874i 0.309017 0.951057i −0.104528 + 0.994522i 0.190983 + 0.330792i
159.1 0.669131 0.743145i −0.209057 + 1.98904i −0.104528 0.994522i 2.56082 0.544320i 1.33826 + 1.48629i 0.190983 0.138757i −0.809017 0.587785i −0.978148 0.207912i 1.30902 2.26728i
163.1 0.669131 + 0.743145i −0.209057 1.98904i −0.104528 + 0.994522i 2.56082 + 0.544320i 1.33826 1.48629i 0.190983 + 0.138757i −0.809017 + 0.587785i −0.978148 + 0.207912i 1.30902 + 2.26728i
201.1 −0.104528 0.994522i −1.95630 0.415823i −0.978148 + 0.207912i −0.348943 + 0.155360i −0.209057 + 1.98904i 1.30902 4.02874i 0.309017 + 0.951057i 0.913545 + 0.406737i 0.190983 + 0.330792i
235.1 −0.104528 + 0.994522i −1.95630 + 0.415823i −0.978148 0.207912i −0.348943 0.155360i −0.209057 1.98904i 1.30902 + 4.02874i 0.309017 0.951057i 0.913545 0.406737i 0.190983 0.330792i
273.1 −0.978148 + 0.207912i 1.82709 + 0.813473i 0.913545 0.406737i −1.75181 + 1.94558i −1.95630 0.415823i 0.190983 + 0.138757i −0.809017 + 0.587785i 0.669131 + 0.743145i 1.30902 2.26728i
311.1 0.913545 + 0.406737i 1.33826 1.48629i 0.669131 + 0.743145i 0.0399263 0.379874i 1.82709 0.813473i 1.30902 4.02874i 0.309017 + 0.951057i −0.104528 0.994522i 0.190983 0.330792i
\(n\): 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.c 8
11.c even 5 1 inner 418.2.n.c 8
19.c even 3 1 inner 418.2.n.c 8
209.n even 15 1 inner 418.2.n.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.n.c 8 1.a even 1 1 trivial
418.2.n.c 8 11.c even 5 1 inner
418.2.n.c 8 19.c even 3 1 inner
418.2.n.c 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
$5$ \( T^{8} - T^{7} - 5 T^{6} - 14 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( (T^{4} - 3 T^{3} + 19 T^{2} - 7 T + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - T^{3} - 9 T^{2} - 11 T + 121)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 6 T^{7} + 54 T^{5} + \cdots + 6561 \) Copy content Toggle raw display
$17$ \( T^{8} + 4 T^{7} - 30 T^{6} + \cdots + 2825761 \) 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
$29$ \( T^{8} + 5 T^{7} + 10 T^{6} + 25 T^{5} + \cdots + 625 \) Copy content Toggle raw display
$31$ \( (T^{4} + 14 T^{3} + 76 T^{2} - 31 T + 961)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 7 T^{3} + 24 T^{2} - 38 T + 361)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 19 T^{7} + 120 T^{6} + \cdots + 38950081 \) Copy content Toggle raw display
$43$ \( (T^{4} - 8 T^{3} + 93 T^{2} + 232 T + 841)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - T^{7} + T^{5} - T^{4} + T^{3} - T + 1 \) Copy content Toggle raw display
$53$ \( T^{8} - 4 T^{7} - 80 T^{6} + \cdots + 33362176 \) Copy content Toggle raw display
$59$ \( T^{8} + 13 T^{7} + 30 T^{6} + \cdots + 2825761 \) Copy content Toggle raw display
$61$ \( T^{8} + 3 T^{7} - 135 T^{6} + \cdots + 96059601 \) Copy content Toggle raw display
$67$ \( (T^{4} + 9 T^{3} + 122 T^{2} - 369 T + 1681)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 16 T^{7} + 70 T^{6} + \cdots + 12117361 \) Copy content Toggle raw display
$73$ \( T^{8} + 4 T^{7} - 80 T^{6} + \cdots + 33362176 \) Copy content Toggle raw display
$79$ \( T^{8} - 12 T^{7} - 432 T^{5} + \cdots + 1679616 \) Copy content Toggle raw display
$83$ \( (T^{4} + 17 T^{3} + 109 T^{2} - 87 T + 841)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 16 T^{3} + 237 T^{2} + 304 T + 361)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 30 T^{7} + \cdots + 100000000 \) Copy content Toggle raw display
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