# Properties

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

# Learn more

## 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.

 $$\operatorname{Tr}(f)(q) =$$ $$240 q + 3 q^{3} - 3 q^{6} - 3 q^{7} + 30 q^{8} + 3 q^{9}+O(q^{10})$$ 240 * q + 3 * q^3 - 3 * q^6 - 3 * q^7 + 30 * q^8 + 3 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$240 q + 3 q^{3} - 3 q^{6} - 3 q^{7} + 30 q^{8} + 3 q^{9} - 3 q^{11} + 6 q^{13} - 18 q^{14} - 69 q^{15} + 9 q^{17} + 60 q^{18} - 45 q^{19} + 12 q^{20} - 48 q^{21} - 12 q^{22} + 3 q^{24} - 18 q^{25} + 84 q^{26} - 9 q^{27} + 6 q^{28} + 12 q^{29} + 9 q^{31} - 69 q^{33} - 24 q^{34} + 36 q^{35} + 3 q^{36} + 24 q^{38} - 30 q^{41} - 6 q^{42} + 12 q^{43} - 48 q^{44} + 48 q^{45} - 12 q^{46} - 54 q^{47} + 6 q^{48} - 81 q^{49} + 21 q^{50} + 45 q^{51} - 3 q^{52} + 81 q^{53} - 27 q^{54} - 72 q^{55} - 30 q^{57} - 24 q^{58} + 114 q^{59} + 78 q^{60} - 66 q^{61} - 45 q^{62} + 9 q^{63} + 30 q^{64} - 135 q^{66} - 9 q^{67} - 42 q^{68} - 54 q^{69} + 39 q^{70} - 102 q^{71} - 6 q^{72} + 12 q^{74} + 72 q^{77} + 36 q^{79} - 117 q^{81} + 30 q^{82} - 36 q^{83} - 36 q^{84} - 90 q^{85} + 3 q^{86} - 216 q^{87} - 3 q^{88} + 18 q^{89} - 24 q^{90} + 102 q^{91} + 30 q^{92} + 147 q^{93} + 18 q^{94} - 66 q^{95} - 24 q^{97} + 12 q^{98} - 195 q^{99}+O(q^{100})$$ 240 * q + 3 * q^3 - 3 * q^6 - 3 * q^7 + 30 * q^8 + 3 * q^9 - 3 * q^11 + 6 * q^13 - 18 * q^14 - 69 * q^15 + 9 * q^17 + 60 * q^18 - 45 * q^19 + 12 * q^20 - 48 * q^21 - 12 * q^22 + 3 * q^24 - 18 * q^25 + 84 * q^26 - 9 * q^27 + 6 * q^28 + 12 * q^29 + 9 * q^31 - 69 * q^33 - 24 * q^34 + 36 * q^35 + 3 * q^36 + 24 * q^38 - 30 * q^41 - 6 * q^42 + 12 * q^43 - 48 * q^44 + 48 * q^45 - 12 * q^46 - 54 * q^47 + 6 * q^48 - 81 * q^49 + 21 * q^50 + 45 * q^51 - 3 * q^52 + 81 * q^53 - 27 * q^54 - 72 * q^55 - 30 * q^57 - 24 * q^58 + 114 * q^59 + 78 * q^60 - 66 * q^61 - 45 * q^62 + 9 * q^63 + 30 * q^64 - 135 * q^66 - 9 * q^67 - 42 * q^68 - 54 * q^69 + 39 * q^70 - 102 * q^71 - 6 * q^72 + 12 * q^74 + 72 * q^77 + 36 * q^79 - 117 * q^81 + 30 * q^82 - 36 * q^83 - 36 * q^84 - 90 * q^85 + 3 * q^86 - 216 * q^87 - 3 * q^88 + 18 * q^89 - 24 * q^90 + 102 * q^91 + 30 * q^92 + 147 * q^93 + 18 * q^94 - 66 * q^95 - 24 * q^97 + 12 * q^98 - 195 * 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 −0.873028 3.04461i −0.719340 + 0.694658i 2.51211 1.33571i −2.49587 + 1.94999i −1.97806 4.44279i 0.913545 + 0.406737i −5.96333 + 3.72630i −2.17951 1.82882i
13.2 −0.374607 0.927184i −0.582769 2.03236i −0.719340 + 0.694658i −2.69676 + 1.43389i −1.66606 + 1.30167i 0.507726 + 1.14037i 0.913545 + 0.406737i −1.24671 + 0.779029i 2.33971 + 1.96325i
13.3 −0.374607 0.927184i −0.419952 1.46455i −0.719340 + 0.694658i −1.24254 + 0.660669i −1.20059 + 0.938002i −1.61772 3.63346i 0.913545 + 0.406737i 0.575603 0.359677i 1.07802 + 0.904570i
13.4 −0.374607 0.927184i −0.414539 1.44567i −0.719340 + 0.694658i 1.07843 0.573409i −1.18511 + 0.925911i 1.15655 + 2.59766i 0.913545 + 0.406737i 0.626029 0.391186i −0.935641 0.785096i
13.5 −0.374607 0.927184i −0.270648 0.943861i −0.719340 + 0.694658i 1.29541 0.688784i −0.773746 + 0.604517i 1.11507 + 2.50449i 0.913545 + 0.406737i 1.72652 1.07885i −1.12390 0.943064i
13.6 −0.374607 0.927184i 0.116237 + 0.405365i −0.719340 + 0.694658i 3.30844 1.75913i 0.332305 0.259625i −0.632649 1.42095i 0.913545 + 0.406737i 2.39333 1.49552i −2.87040 2.40855i
13.7 −0.374607 0.927184i 0.222302 + 0.775258i −0.719340 + 0.694658i −2.70784 + 1.43978i 0.635531 0.496531i −1.35747 3.04893i 0.913545 + 0.406737i 1.99254 1.24508i 2.34932 + 1.97131i
13.8 −0.374607 0.927184i 0.411485 + 1.43502i −0.719340 + 0.694658i −0.764836 + 0.406670i 1.17638 0.919089i 0.622864 + 1.39897i 0.913545 + 0.406737i 0.654189 0.408783i 0.663571 + 0.556802i
13.9 −0.374607 0.927184i 0.581094 + 2.02651i −0.719340 + 0.694658i 0.405909 0.215825i 1.66127 1.29793i 0.326823 + 0.734057i 0.913545 + 0.406737i −1.22494 + 0.765430i −0.352166 0.295502i
13.10 −0.374607 0.927184i 0.937688 + 3.27011i −0.719340 + 0.694658i −2.61697 + 1.39147i 2.68073 2.09441i −0.352914 0.792659i 0.913545 + 0.406737i −7.27020 + 4.54292i 2.27048 + 1.90516i
29.1 0.438371 0.898794i −2.63027 + 1.06270i −0.615661 0.788011i 1.03456 0.296655i −0.197888 + 2.82992i 0.0323920 0.152392i −0.978148 + 0.207912i 3.63096 3.50638i 0.186889 1.05990i
29.2 0.438371 0.898794i −2.41147 + 0.974297i −0.615661 0.788011i −2.76916 + 0.794045i −0.181426 + 2.59452i 0.709936 3.33998i −0.978148 + 0.207912i 2.70791 2.61500i −0.500239 + 2.83699i
29.3 0.438371 0.898794i −1.31507 + 0.531324i −0.615661 0.788011i 2.95966 0.848670i −0.0989393 + 1.41490i 0.318233 1.49717i −0.978148 + 0.207912i −0.710907 + 0.686515i 0.534652 3.03216i
29.4 0.438371 0.898794i −1.03102 + 0.416557i −0.615661 0.788011i 1.04375 0.299291i −0.0775682 + 1.10928i −0.760853 + 3.57953i −0.978148 + 0.207912i −1.26855 + 1.22502i 0.188549 1.06932i
29.5 0.438371 0.898794i 0.264611 0.106910i −0.615661 0.788011i −2.79611 + 0.801772i 0.0199080 0.284697i 0.419245 1.97239i −0.978148 + 0.207912i −2.09943 + 2.02740i −0.505107 + 2.86460i
29.6 0.438371 0.898794i 0.332827 0.134471i −0.615661 0.788011i −2.40319 + 0.689104i 0.0250402 0.358091i −0.957149 + 4.50303i −0.978148 + 0.207912i −2.06533 + 1.99446i −0.434127 + 2.46206i
29.7 0.438371 0.898794i 0.856586 0.346083i −0.615661 0.788011i 2.52606 0.724335i 0.0644450 0.921607i 1.07271 5.04669i −0.978148 + 0.207912i −1.54405 + 1.49108i 0.456322 2.58793i
29.8 0.438371 0.898794i 1.88473 0.761481i −0.615661 0.788011i 0.993014 0.284742i 0.141797 2.02780i 0.0468569 0.220445i −0.978148 + 0.207912i 0.814339 0.786398i 0.179384 1.01734i
29.9 0.438371 0.898794i 2.45510 0.991923i −0.615661 0.788011i 2.52718 0.724656i 0.184709 2.64146i −0.616865 + 2.90212i −0.978148 + 0.207912i 2.88556 2.78656i 0.456524 2.58908i
29.10 0.438371 0.898794i 2.88213 1.16445i −0.615661 0.788011i −3.70984 + 1.06378i 0.216836 3.10090i 0.625748 2.94391i −0.978148 + 0.207912i 4.79268 4.62824i −0.670169 + 3.80072i
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.b yes 240
11.d odd 10 1 418.2.v.a 240
19.f odd 18 1 418.2.v.a 240
209.w even 90 1 inner 418.2.v.b yes 240

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{240} - 3 T_{3}^{239} + 3 T_{3}^{238} + 3 T_{3}^{237} + 72 T_{3}^{236} - 267 T_{3}^{235} + 1519 T_{3}^{234} - 2982 T_{3}^{233} + 5622 T_{3}^{232} - 32062 T_{3}^{231} + 147558 T_{3}^{230} - 197475 T_{3}^{229} + \cdots + 14\!\cdots\!61$$ acting on $$S_{2}^{\mathrm{new}}(418, [\chi])$$.