# Properties

 Label 418.2.v.a Level $418$ Weight $2$ Character orbit 418.v Analytic conductor $3.338$ Analytic rank $0$ Dimension $240$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$418 = 2 \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 418.v (of order $$90$$, degree $$24$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.33774680449$$ Analytic rank: $$0$$ Dimension: $$240$$ Relative dimension: $$10$$ over $$\Q(\zeta_{90})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{90}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$240 q + 3 q^{3} - 12 q^{6} + 33 q^{7} - 30 q^{8} + 3 q^{9}+O(q^{10})$$ 240 * q + 3 * q^3 - 12 * q^6 + 33 * q^7 - 30 * q^8 + 3 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$240 q + 3 q^{3} - 12 q^{6} + 33 q^{7} - 30 q^{8} + 3 q^{9} - 3 q^{11} - 6 q^{13} - 18 q^{14} + 21 q^{15} + 21 q^{17} - 60 q^{18} + 45 q^{19} + 12 q^{20} + 48 q^{21} - 12 q^{22} + 12 q^{24} - 18 q^{25} - 96 q^{26} - 9 q^{27} - 6 q^{28} + 18 q^{29} + 9 q^{31} + 87 q^{33} - 24 q^{34} - 36 q^{35} - 12 q^{36} - 36 q^{38} + 60 q^{41} - 6 q^{42} - 12 q^{43} + 18 q^{44} + 48 q^{45} + 12 q^{46} - 54 q^{47} + 6 q^{48} - 81 q^{49} - 21 q^{50} - 75 q^{51} + 3 q^{52} - 39 q^{53} + 27 q^{54} - 126 q^{55} - 90 q^{57} - 24 q^{58} + 69 q^{59} - 42 q^{60} + 66 q^{61} - 45 q^{62} - 9 q^{63} + 30 q^{64} - 12 q^{66} - 9 q^{67} + 12 q^{68} - 54 q^{69} + 9 q^{70} + 48 q^{71} + 6 q^{72} - 12 q^{74} + 72 q^{77} - 36 q^{79} - 27 q^{81} + 45 q^{82} + 36 q^{83} + 36 q^{84} - 210 q^{85} + 3 q^{86} + 216 q^{87} + 3 q^{88} + 18 q^{89} - 96 q^{90} - 108 q^{91} - 30 q^{92} + 147 q^{93} - 18 q^{94} + 66 q^{95} - 9 q^{97} - 12 q^{98} - 42 q^{99}+O(q^{100})$$ 240 * q + 3 * q^3 - 12 * q^6 + 33 * q^7 - 30 * q^8 + 3 * q^9 - 3 * q^11 - 6 * q^13 - 18 * q^14 + 21 * q^15 + 21 * q^17 - 60 * q^18 + 45 * q^19 + 12 * q^20 + 48 * q^21 - 12 * q^22 + 12 * q^24 - 18 * q^25 - 96 * q^26 - 9 * q^27 - 6 * q^28 + 18 * q^29 + 9 * q^31 + 87 * q^33 - 24 * q^34 - 36 * q^35 - 12 * q^36 - 36 * q^38 + 60 * q^41 - 6 * q^42 - 12 * q^43 + 18 * q^44 + 48 * q^45 + 12 * q^46 - 54 * q^47 + 6 * q^48 - 81 * q^49 - 21 * q^50 - 75 * q^51 + 3 * q^52 - 39 * q^53 + 27 * q^54 - 126 * q^55 - 90 * q^57 - 24 * q^58 + 69 * q^59 - 42 * q^60 + 66 * q^61 - 45 * q^62 - 9 * q^63 + 30 * q^64 - 12 * q^66 - 9 * q^67 + 12 * q^68 - 54 * q^69 + 9 * q^70 + 48 * q^71 + 6 * q^72 - 12 * q^74 + 72 * q^77 - 36 * q^79 - 27 * q^81 + 45 * q^82 + 36 * q^83 + 36 * q^84 - 210 * q^85 + 3 * q^86 + 216 * q^87 + 3 * q^88 + 18 * q^89 - 96 * q^90 - 108 * q^91 - 30 * q^92 + 147 * q^93 - 18 * q^94 + 66 * q^95 - 9 * q^97 - 12 * q^98 - 42 * q^99

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1 0.374607 + 0.927184i −0.866547 3.02201i −0.719340 + 0.694658i −2.04892 + 1.08943i 2.47734 1.93551i 1.73061 + 3.88702i −0.913545 0.406737i −5.83749 + 3.64767i −1.77764 1.49162i
13.2 0.374607 + 0.927184i −0.630231 2.19788i −0.719340 + 0.694658i 1.48095 0.787437i 1.80175 1.40768i −1.06207 2.38544i −0.913545 0.406737i −1.88932 + 1.18058i 1.28487 + 1.07814i
13.3 0.374607 + 0.927184i −0.301053 1.04990i −0.719340 + 0.694658i −0.793784 + 0.422062i 0.860670 0.672429i −0.868426 1.95052i −0.913545 0.406737i 1.53250 0.957610i −0.688686 0.577876i
13.4 0.374607 + 0.927184i −0.185288 0.646175i −0.719340 + 0.694658i −3.75384 + 1.99595i 0.529713 0.413857i −0.225011 0.505382i −0.913545 0.406737i 2.16093 1.35030i −3.25683 2.73280i
13.5 0.374607 + 0.927184i −0.0175505 0.0612060i −0.719340 + 0.694658i 2.65042 1.40925i 0.0501746 0.0392007i 1.63684 + 3.67640i −0.913545 0.406737i 2.54071 1.58761i 2.29950 + 1.92951i
13.6 0.374607 + 0.927184i −0.0174768 0.0609489i −0.719340 + 0.694658i −0.867424 + 0.461218i 0.0499639 0.0390361i 0.714147 + 1.60400i −0.913545 0.406737i 2.54073 1.58763i −0.752576 0.631487i
13.7 0.374607 + 0.927184i 0.195526 + 0.681879i −0.719340 + 0.694658i 2.12313 1.12889i −0.558982 + 0.436725i −0.266302 0.598123i −0.913545 0.406737i 2.11742 1.32311i 1.84203 + 1.54564i
13.8 0.374607 + 0.927184i 0.682065 + 2.37864i −0.719340 + 0.694658i −1.17948 + 0.627143i −1.94993 + 1.52345i 1.45204 + 3.26133i −0.913545 0.406737i −2.64858 + 1.65502i −1.02332 0.858667i
13.9 0.374607 + 0.927184i 0.765882 + 2.67095i −0.719340 + 0.694658i −2.13408 + 1.13471i −2.18956 + 1.71067i −1.72666 3.87814i −0.913545 0.406737i −4.00325 + 2.50151i −1.85152 1.55361i
13.10 0.374607 + 0.927184i 0.885806 + 3.08917i −0.719340 + 0.694658i 3.09438 1.64531i −2.53240 + 1.97853i −0.131685 0.295769i −0.913545 0.406737i −6.21419 + 3.88306i 2.68468 + 2.25272i
29.1 −0.438371 + 0.898794i −2.91493 + 1.17771i −0.615661 0.788011i −0.450149 + 0.129078i 0.219304 3.13619i 0.395058 1.85860i 0.978148 0.207912i 4.95179 4.78189i 0.0813176 0.461175i
29.2 −0.438371 + 0.898794i −2.15354 + 0.870087i −0.615661 0.788011i −2.44299 + 0.700517i 0.162021 2.31701i −0.805999 + 3.79193i 0.978148 0.207912i 1.72267 1.66356i 0.441317 2.50284i
29.3 −0.438371 + 0.898794i −0.914225 + 0.369371i −0.615661 0.788011i 0.610535 0.175068i 0.0687815 0.983622i 0.321044 1.51039i 0.978148 0.207912i −1.45865 + 1.40860i −0.110291 + 0.625490i
29.4 −0.438371 + 0.898794i −0.913816 + 0.369206i −0.615661 0.788011i 0.341675 0.0979736i 0.0687508 0.983182i −0.672800 + 3.16528i 0.978148 0.207912i −1.45927 + 1.40920i −0.0617222 + 0.350044i
29.5 −0.438371 + 0.898794i −0.559221 + 0.225940i −0.615661 0.788011i −0.807264 + 0.231479i 0.0420729 0.601670i 0.675252 3.17681i 0.978148 0.207912i −1.89634 + 1.83127i 0.145829 0.827038i
29.6 −0.438371 + 0.898794i 0.467532 0.188895i −0.615661 0.788011i 3.20340 0.918561i −0.0351747 + 0.503021i −0.476423 + 2.24139i 0.978148 0.207912i −1.97511 + 1.90735i −0.578682 + 3.28187i
29.7 −0.438371 + 0.898794i 1.43643 0.580356i −0.615661 0.788011i −3.49602 + 1.00247i −0.108070 + 1.54547i 0.232262 1.09270i 0.978148 0.207912i −0.431498 + 0.416692i 0.631543 3.58166i
29.8 −0.438371 + 0.898794i 2.01881 0.815653i −0.615661 0.788011i 3.65522 1.04812i −0.151885 + 2.17205i 0.300599 1.41421i 0.978148 0.207912i 1.25229 1.20932i −0.660301 + 3.74475i
29.9 −0.438371 + 0.898794i 2.12470 0.858435i −0.615661 0.788011i −0.690733 + 0.198064i −0.159852 + 2.28598i 0.729217 3.43070i 0.978148 0.207912i 1.61943 1.56386i 0.124778 0.707652i
29.10 −0.438371 + 0.898794i 3.07495 1.24236i −0.615661 0.788011i −0.517758 + 0.148465i −0.231343 + 3.30836i −0.797517 + 3.75202i 0.978148 0.207912i 5.75383 5.55641i 0.0935310 0.530441i
See next 80 embeddings (of 240 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 413.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
209.w even 90 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 418.2.v.a 240
11.d odd 10 1 418.2.v.b yes 240
19.f odd 18 1 418.2.v.b yes 240
209.w even 90 1 inner 418.2.v.a 240

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.v.a 240 1.a even 1 1 trivial
418.2.v.a 240 209.w even 90 1 inner
418.2.v.b yes 240 11.d odd 10 1
418.2.v.b yes 240 19.f odd 18 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{240} - 3 T_{3}^{239} + 3 T_{3}^{238} + 3 T_{3}^{237} - 18 T_{3}^{236} + 3 T_{3}^{235} + 1249 T_{3}^{234} - 912 T_{3}^{233} + 2517 T_{3}^{232} + 13303 T_{3}^{231} - 87447 T_{3}^{230} + 308985 T_{3}^{229} - 424811 T_{3}^{228} + \cdots + 14\!\cdots\!61$$ acting on $$S_{2}^{\mathrm{new}}(418, [\chi])$$.