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Properties

Label 350.3.w.a
Level $350$
Weight $3$
Character orbit 350.w
Analytic conductor $9.537$
Analytic rank $0$
Dimension $320$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 350.w (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.53680925261\)
Analytic rank: \(0\)
Dimension: \(320\)
Relative dimension: \(20\) over \(\Q(\zeta_{60})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 320 q - 40 q^{2} - 6 q^{5} + 2 q^{7} - 160 q^{8} - 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 320 q - 40 q^{2} - 6 q^{5} + 2 q^{7} - 160 q^{8} - 40 q^{9} - 16 q^{11} - 30 q^{14} + 52 q^{15} - 160 q^{16} + 94 q^{17} + 496 q^{18} - 40 q^{19} + 16 q^{20} - 68 q^{21} - 32 q^{22} - 16 q^{23} - 62 q^{25} + 144 q^{27} - 8 q^{28} + 200 q^{29} - 46 q^{30} - 84 q^{31} - 640 q^{32} + 222 q^{33} - 252 q^{35} - 576 q^{36} + 214 q^{37} - 16 q^{38} + 320 q^{39} - 4 q^{40} - 128 q^{41} - 136 q^{42} + 100 q^{43} + 40 q^{44} - 214 q^{45} - 48 q^{46} - 110 q^{47} + 172 q^{50} - 56 q^{51} - 262 q^{53} - 184 q^{55} + 48 q^{56} - 244 q^{57} - 180 q^{58} + 520 q^{59} - 96 q^{60} - 216 q^{61} + 552 q^{62} + 968 q^{63} - 150 q^{65} + 16 q^{66} - 190 q^{67} - 88 q^{68} + 1060 q^{69} + 114 q^{70} + 340 q^{71} - 208 q^{72} + 134 q^{73} - 84 q^{75} - 64 q^{76} - 98 q^{77} + 532 q^{78} - 80 q^{79} - 56 q^{80} - 112 q^{81} + 256 q^{82} - 1216 q^{83} - 380 q^{84} - 48 q^{85} + 40 q^{86} - 334 q^{87} - 52 q^{88} + 990 q^{89} + 672 q^{90} - 42 q^{91} - 256 q^{92} + 306 q^{93} + 432 q^{95} - 576 q^{97} + 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1 1.09905 + 0.889993i −4.16727 − 2.70626i 0.415823 + 1.95630i 3.21899 − 3.82598i −2.17149 − 6.68315i −0.176682 + 6.99777i −1.28408 + 2.52015i 6.38168 + 14.3335i 6.94293 − 1.34006i
23.2 1.09905 + 0.889993i −4.13484 − 2.68520i 0.415823 + 1.95630i 1.14363 − 4.86745i −2.15459 − 6.63114i 0.841404 − 6.94925i −1.28408 + 2.52015i 6.22599 + 13.9838i 5.58891 − 4.33175i
23.3 1.09905 + 0.889993i −3.60662 − 2.34217i 0.415823 + 1.95630i −4.84107 + 1.25062i −1.87935 − 5.78403i 6.89609 + 1.20166i −1.28408 + 2.52015i 3.86135 + 8.67274i −6.43362 − 2.93403i
23.4 1.09905 + 0.889993i −3.42731 − 2.22572i 0.415823 + 1.95630i −4.95458 + 0.672400i −1.78591 − 5.49647i −2.79533 − 6.41764i −1.28408 + 2.52015i 3.13201 + 7.03460i −6.04377 − 3.67054i
23.5 1.09905 + 0.889993i −3.34989 − 2.17545i 0.415823 + 1.95630i 1.84942 + 4.64539i −1.74557 − 5.37231i 5.67887 + 4.09272i −1.28408 + 2.52015i 2.82859 + 6.35313i −2.10175 + 6.75149i
23.6 1.09905 + 0.889993i −2.87744 − 1.86863i 0.415823 + 1.95630i 4.84955 + 1.21730i −1.49938 − 4.61462i −6.83709 + 1.50142i −1.28408 + 2.52015i 1.12724 + 2.53183i 4.24651 + 5.65395i
23.7 1.09905 + 0.889993i −2.04025 − 1.32495i 0.415823 + 1.95630i 0.343406 + 4.98819i −1.06314 − 3.27200i −4.29280 − 5.52918i −1.28408 + 2.52015i −1.25351 − 2.81543i −4.06204 + 5.78790i
23.8 1.09905 + 0.889993i −1.28221 − 0.832674i 0.415823 + 1.95630i −2.89680 − 4.07536i −0.668134 − 2.05631i 0.580781 + 6.97587i −1.28408 + 2.52015i −2.70992 − 6.08659i 0.443312 − 7.05716i
23.9 1.09905 + 0.889993i −0.728724 − 0.473239i 0.415823 + 1.95630i 4.99621 − 0.194586i −0.379725 − 1.16867i 4.41728 − 5.43025i −1.28408 + 2.52015i −3.35355 − 7.53219i 5.66427 + 4.23274i
23.10 1.09905 + 0.889993i −0.372709 − 0.242040i 0.415823 + 1.95630i −4.59473 + 1.97191i −0.194212 − 0.597722i −3.14459 + 6.25392i −1.28408 + 2.52015i −3.58030 − 8.04149i −6.80483 − 1.92206i
23.11 1.09905 + 0.889993i 0.191153 + 0.124136i 0.415823 + 1.95630i 0.676110 − 4.95408i 0.0996063 + 0.306557i −6.82490 − 1.55589i −1.28408 + 2.52015i −3.63950 − 8.17445i 5.15217 − 4.84305i
23.12 1.09905 + 0.889993i 0.604979 + 0.392878i 0.415823 + 1.95630i −4.15914 − 2.77517i 0.315244 + 0.970220i 5.11960 − 4.77386i −1.28408 + 2.52015i −3.44898 − 7.74654i −2.10123 − 6.75166i
23.13 1.09905 + 0.889993i 0.894977 + 0.581205i 0.415823 + 1.95630i −1.45717 + 4.78295i 0.466356 + 1.43530i 6.77471 − 1.76162i −1.28408 + 2.52015i −3.19745 − 7.18158i −5.85830 + 3.95984i
23.14 1.09905 + 0.889993i 2.31506 + 1.50342i 0.415823 + 1.95630i −3.99632 + 3.00490i 1.20634 + 3.71272i −6.62384 − 2.26378i −1.28408 + 2.52015i −0.561394 − 1.26091i −7.06650 − 0.254165i
23.15 1.09905 + 0.889993i 2.65474 + 1.72401i 0.415823 + 1.95630i 2.92674 + 4.05391i 1.38334 + 4.25748i 1.01529 + 6.92598i −1.28408 + 2.52015i 0.414820 + 0.931702i −0.391318 + 7.06023i
23.16 1.09905 + 0.889993i 2.88965 + 1.87656i 0.415823 + 1.95630i 1.71976 − 4.69494i 1.50575 + 4.63421i 6.26337 + 3.12573i −1.28408 + 2.52015i 1.16798 + 2.62333i 6.06856 − 3.62940i
23.17 1.09905 + 0.889993i 2.91951 + 1.89595i 0.415823 + 1.95630i 4.88005 + 1.08861i 1.52130 + 4.68209i −6.93272 − 0.968186i −1.28408 + 2.52015i 1.26828 + 2.84859i 4.39457 + 5.53965i
23.18 1.09905 + 0.889993i 4.38001 + 2.84441i 0.415823 + 1.95630i −4.56381 − 2.04246i 2.28234 + 7.02433i −4.20782 + 5.59412i −1.28408 + 2.52015i 7.43318 + 16.6952i −3.19808 − 6.30653i
23.19 1.09905 + 0.889993i 4.39549 + 2.85447i 0.415823 + 1.95630i 4.57174 − 2.02464i 2.29041 + 7.04916i 1.17918 − 6.89997i −1.28408 + 2.52015i 7.51175 + 16.8717i 6.82649 + 1.84364i
23.20 1.09905 + 0.889993i 4.74169 + 3.07929i 0.415823 + 1.95630i −2.40553 + 4.38331i 2.47081 + 7.60437i 4.97800 − 4.92134i −1.28408 + 2.52015i 9.34097 + 20.9802i −6.54492 + 2.67658i
See next 80 embeddings (of 320 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 347.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
25.f odd 20 1 inner
175.w odd 60 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.3.w.a ✓ 320
7.c even 3 1 inner 350.3.w.a ✓ 320
25.f odd 20 1 inner 350.3.w.a ✓ 320
175.w odd 60 1 inner 350.3.w.a ✓ 320
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.3.w.a ✓ 320 1.a even 1 1 trivial
350.3.w.a ✓ 320 7.c even 3 1 inner
350.3.w.a ✓ 320 25.f odd 20 1 inner
350.3.w.a ✓ 320 175.w odd 60 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{320} + 20 T_{3}^{318} - 48 T_{3}^{317} + 2550 T_{3}^{316} + 16 T_{3}^{315} + 27952 T_{3}^{314} - 5574 T_{3}^{313} + 1794867 T_{3}^{312} + 3167608 T_{3}^{311} - 18490260 T_{3}^{310} + 184626824 T_{3}^{309} + \cdots + 26\!\cdots\!25 \) acting on \(S_{3}^{\mathrm{new}}(350, [\chi])\). Copy content Toggle raw display