# 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 $4$

# Related objects

## 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.

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

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{3}^{\mathrm{new}}(350, [\chi])$$.