Properties

Label 294.2.j.a
Level $294$
Weight $2$
Character orbit 294.j
Analytic conductor $2.348$
Analytic rank $0$
Dimension $120$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 294.j (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.34760181943\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(20\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 120 q + 20 q^{4} - 14 q^{6} + 4 q^{7} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 120 q + 20 q^{4} - 14 q^{6} + 4 q^{7} - 10 q^{9} + 12 q^{15} - 20 q^{16} - 8 q^{18} - 12 q^{21} - 10 q^{22} - 24 q^{25} + 42 q^{27} - 4 q^{28} + 8 q^{30} + 10 q^{36} - 58 q^{37} + 8 q^{39} - 14 q^{40} - 2 q^{42} - 12 q^{43} + 14 q^{45} - 60 q^{46} - 36 q^{49} - 24 q^{51} - 42 q^{52} - 14 q^{55} + 34 q^{57} + 50 q^{58} - 12 q^{60} - 154 q^{61} + 20 q^{64} - 28 q^{67} + 28 q^{69} - 38 q^{70} + 8 q^{72} - 84 q^{75} - 16 q^{78} + 16 q^{79} + 22 q^{81} - 30 q^{84} - 32 q^{85} - 56 q^{87} - 4 q^{88} - 42 q^{90} + 88 q^{91} + 14 q^{93} + 112 q^{94} - 224 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1 −0.781831 + 0.623490i −1.59440 + 0.676684i 0.222521 0.974928i −2.59923 + 1.25172i 0.824644 1.52314i −2.01156 + 1.71861i 0.433884 + 0.900969i 2.08420 2.15780i 1.25172 2.59923i
41.2 −0.781831 + 0.623490i −1.54527 0.782396i 0.222521 0.974928i 0.415766 0.200223i 1.69596 0.351758i −2.57018 0.627843i 0.433884 + 0.900969i 1.77571 + 2.41802i −0.200223 + 0.415766i
41.3 −0.781831 + 0.623490i −0.995627 + 1.41730i 0.222521 0.974928i 1.32694 0.639022i −0.105257 1.72885i 1.06200 2.42325i 0.433884 + 0.900969i −1.01745 2.82220i −0.639022 + 1.32694i
41.4 −0.781831 + 0.623490i −0.580532 1.63186i 0.222521 0.974928i −3.14996 + 1.51694i 1.47133 + 0.913887i 2.47808 + 0.926885i 0.433884 + 0.900969i −2.32596 + 1.89470i 1.51694 3.14996i
41.5 −0.781831 + 0.623490i −0.0435112 1.73150i 0.222521 0.974928i 3.37524 1.62543i 1.11359 + 1.32662i −0.0659865 2.64493i 0.433884 + 0.900969i −2.99621 + 0.150680i −1.62543 + 3.37524i
41.6 −0.781831 + 0.623490i 0.169937 + 1.72369i 0.222521 0.974928i −0.210425 + 0.101335i −1.20757 1.24168i 0.981077 + 2.45713i 0.433884 + 0.900969i −2.94224 + 0.585838i 0.101335 0.210425i
41.7 −0.781831 + 0.623490i 1.06385 + 1.36683i 0.222521 0.974928i −2.67532 + 1.28836i −1.68396 0.405333i −1.08169 2.41453i 0.433884 + 0.900969i −0.736454 + 2.90820i 1.28836 2.67532i
41.8 −0.781831 + 0.623490i 1.34258 1.09430i 0.222521 0.974928i −2.13620 + 1.02874i −0.367384 + 1.69264i −1.53826 2.15262i 0.433884 + 0.900969i 0.605021 2.93836i 1.02874 2.13620i
41.9 −0.781831 + 0.623490i 1.38755 1.03668i 0.222521 0.974928i 1.24567 0.599884i −0.438472 + 1.67563i 2.35011 + 1.21532i 0.433884 + 0.900969i 0.850594 2.87689i −0.599884 + 1.24567i
41.10 −0.781831 + 0.623490i 1.57726 + 0.715728i 0.222521 0.974928i 2.99870 1.44410i −1.67940 + 0.423824i −2.37504 + 1.16585i 0.433884 + 0.900969i 1.97547 + 2.25777i −1.44410 + 2.99870i
41.11 0.781831 0.623490i −1.69264 + 0.367384i 0.222521 0.974928i 2.13620 1.02874i −1.09430 + 1.34258i −1.53826 2.15262i −0.433884 0.900969i 2.73006 1.24370i 1.02874 2.13620i
41.12 0.781831 0.623490i −1.67563 + 0.438472i 0.222521 0.974928i −1.24567 + 0.599884i −1.03668 + 1.38755i 2.35011 + 1.21532i −0.433884 0.900969i 2.61548 1.46944i −0.599884 + 1.24567i
41.13 0.781831 0.623490i −1.32662 1.11359i 0.222521 0.974928i −3.37524 + 1.62543i −1.73150 0.0435112i −0.0659865 2.64493i −0.433884 0.900969i 0.519818 + 2.95462i −1.62543 + 3.37524i
41.14 0.781831 0.623490i −0.913887 1.47133i 0.222521 0.974928i 3.14996 1.51694i −1.63186 0.580532i 2.47808 + 0.926885i −0.433884 0.900969i −1.32962 + 2.68926i 1.51694 3.14996i
41.15 0.781831 0.623490i −0.423824 + 1.67940i 0.222521 0.974928i −2.99870 + 1.44410i 0.715728 + 1.57726i −2.37504 + 1.16585i −0.433884 0.900969i −2.64075 1.42354i −1.44410 + 2.99870i
41.16 0.781831 0.623490i 0.351758 1.69596i 0.222521 0.974928i −0.415766 + 0.200223i −0.782396 1.54527i −2.57018 0.627843i −0.433884 0.900969i −2.75253 1.19313i −0.200223 + 0.415766i
41.17 0.781831 0.623490i 0.405333 + 1.68396i 0.222521 0.974928i 2.67532 1.28836i 1.36683 + 1.06385i −1.08169 2.41453i −0.433884 0.900969i −2.67141 + 1.36512i 1.28836 2.67532i
41.18 0.781831 0.623490i 1.24168 + 1.20757i 0.222521 0.974928i 0.210425 0.101335i 1.72369 + 0.169937i 0.981077 + 2.45713i −0.433884 0.900969i 0.0835604 + 2.99884i 0.101335 0.210425i
41.19 0.781831 0.623490i 1.52314 0.824644i 0.222521 0.974928i 2.59923 1.25172i 0.676684 1.59440i −2.01156 + 1.71861i −0.433884 0.900969i 1.63993 2.51210i 1.25172 2.59923i
41.20 0.781831 0.623490i 1.72885 + 0.105257i 0.222521 0.974928i −1.32694 + 0.639022i 1.41730 0.995627i 1.06200 2.42325i −0.433884 0.900969i 2.97784 + 0.363947i −0.639022 + 1.32694i
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 251.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
49.f odd 14 1 inner
147.k even 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 294.2.j.a 120
3.b odd 2 1 inner 294.2.j.a 120
49.f odd 14 1 inner 294.2.j.a 120
147.k even 14 1 inner 294.2.j.a 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
294.2.j.a 120 1.a even 1 1 trivial
294.2.j.a 120 3.b odd 2 1 inner
294.2.j.a 120 49.f odd 14 1 inner
294.2.j.a 120 147.k even 14 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(294, [\chi])\).