# Properties

 Label 784.2.x.o Level $784$ Weight $2$ Character orbit 784.x Analytic conductor $6.260$ Analytic rank $0$ Dimension $48$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$784 = 2^{4} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 784.x (of order $$12$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.26027151847$$ Analytic rank: $$0$$ Dimension: $$48$$ Relative dimension: $$12$$ over $$\Q(\zeta_{12})$$ Twist minimal: no (minimal twist has level 112) Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$48 q - 4 q^{2} - 4 q^{4} - 4 q^{5} + 4 q^{6} - 4 q^{8}+O(q^{10})$$ 48 * q - 4 * q^2 - 4 * q^4 - 4 * q^5 + 4 * q^6 - 4 * q^8 $$\operatorname{Tr}(f)(q) =$$ $$48 q - 4 q^{2} - 4 q^{4} - 4 q^{5} + 4 q^{6} - 4 q^{8} + 2 q^{10} - 4 q^{11} - 2 q^{12} + 24 q^{13} - 40 q^{15} + 16 q^{16} - 8 q^{17} + 18 q^{18} + 4 q^{19} + 16 q^{20} - 18 q^{24} + 10 q^{26} + 24 q^{27} + 24 q^{29} - 4 q^{30} - 28 q^{31} + 16 q^{32} - 16 q^{33} + 44 q^{34} - 72 q^{36} - 24 q^{37} - 20 q^{38} - 26 q^{40} - 40 q^{43} + 6 q^{44} + 28 q^{45} - 14 q^{46} + 20 q^{47} - 56 q^{48} + 56 q^{50} + 24 q^{51} + 16 q^{52} - 16 q^{53} - 64 q^{54} - 6 q^{58} + 20 q^{59} + 46 q^{60} - 8 q^{61} - 24 q^{62} + 80 q^{64} + 8 q^{65} + 20 q^{66} + 48 q^{67} + 40 q^{69} - 32 q^{72} - 8 q^{74} + 4 q^{75} + 36 q^{76} + 116 q^{78} - 36 q^{79} + 28 q^{80} - 2 q^{82} + 8 q^{83} - 20 q^{86} - 42 q^{88} + 20 q^{90} + 76 q^{92} + 8 q^{93} + 72 q^{94} - 4 q^{95} + 120 q^{96} + 48 q^{97} - 24 q^{99}+O(q^{100})$$ 48 * q - 4 * q^2 - 4 * q^4 - 4 * q^5 + 4 * q^6 - 4 * q^8 + 2 * q^10 - 4 * q^11 - 2 * q^12 + 24 * q^13 - 40 * q^15 + 16 * q^16 - 8 * q^17 + 18 * q^18 + 4 * q^19 + 16 * q^20 - 18 * q^24 + 10 * q^26 + 24 * q^27 + 24 * q^29 - 4 * q^30 - 28 * q^31 + 16 * q^32 - 16 * q^33 + 44 * q^34 - 72 * q^36 - 24 * q^37 - 20 * q^38 - 26 * q^40 - 40 * q^43 + 6 * q^44 + 28 * q^45 - 14 * q^46 + 20 * q^47 - 56 * q^48 + 56 * q^50 + 24 * q^51 + 16 * q^52 - 16 * q^53 - 64 * q^54 - 6 * q^58 + 20 * q^59 + 46 * q^60 - 8 * q^61 - 24 * q^62 + 80 * q^64 + 8 * q^65 + 20 * q^66 + 48 * q^67 + 40 * q^69 - 32 * q^72 - 8 * q^74 + 4 * q^75 + 36 * q^76 + 116 * q^78 - 36 * q^79 + 28 * q^80 - 2 * q^82 + 8 * q^83 - 20 * q^86 - 42 * q^88 + 20 * q^90 + 76 * q^92 + 8 * q^93 + 72 * q^94 - 4 * q^95 + 120 * q^96 + 48 * q^97 - 24 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
165.1 −1.41419 + 0.00789795i −0.814813 3.04092i 1.99988 0.0223384i −0.501787 + 1.87270i 1.17632 + 4.29401i 0 −2.82803 + 0.0473857i −5.98523 + 3.45557i 0.694833 2.65231i
165.2 −1.32577 0.492271i 0.221819 + 0.827840i 1.51534 + 1.30528i 1.10095 4.10878i 0.113440 1.20672i 0 −1.36644 2.47646i 1.96196 1.13274i −3.48224 + 4.90534i
165.3 −1.08254 0.909999i 0.672759 + 2.51077i 0.343804 + 1.97023i −0.780276 + 2.91203i 1.55651 3.33023i 0 1.42072 2.44572i −3.25328 + 1.87828i 3.49463 2.44235i
165.4 −0.743894 + 1.20276i 0.222846 + 0.831674i −0.893243 1.78945i −0.543265 + 2.02749i −1.16607 0.350647i 0 2.81674 + 0.256804i 1.95606 1.12933i −2.03445 2.16165i
165.5 −0.504093 1.32132i −0.589510 2.20008i −1.49178 + 1.33214i 0.622192 2.32205i −2.60985 + 1.88798i 0 2.51218 + 1.29960i −1.89476 + 1.09394i −3.38182 + 0.348414i
165.6 −0.478757 + 1.33071i −0.659252 2.46036i −1.54158 1.27417i 0.502490 1.87532i 3.58965 + 0.300642i 0 2.43360 1.44138i −3.02070 + 1.74400i 2.25494 + 1.56649i
165.7 0.0582062 1.41302i 0.523249 + 1.95279i −1.99322 0.164492i 0.256983 0.959072i 2.78978 0.625694i 0 −0.348448 + 2.80688i −0.941530 + 0.543593i −1.34023 0.418944i
165.8 0.438190 + 1.34461i −0.0145698 0.0543752i −1.61598 + 1.17839i 0.337028 1.25781i 0.0667293 0.0434174i 0 −2.29259 1.65651i 2.59533 1.49842i 1.83895 0.0979855i
165.9 0.559929 1.29865i −0.224854 0.839165i −1.37296 1.45430i −0.847576 + 3.16320i −1.21568 0.177868i 0 −2.65738 + 0.968682i 1.94444 1.12262i 3.63329 + 2.87187i
165.10 1.01114 0.988731i −0.430984 1.60845i 0.0448227 1.99950i 0.522232 1.94900i −2.02612 1.20025i 0 −1.93164 2.06610i 0.196696 0.113563i −1.39898 2.48706i
165.11 1.07234 + 0.922005i 0.781062 + 2.91496i 0.299815 + 1.97740i −0.199758 + 0.745506i −1.85005 + 3.84597i 0 −1.50167 + 2.39687i −5.28887 + 3.05353i −0.901568 + 0.615256i
165.12 1.40944 0.116052i 0.312249 + 1.16533i 1.97306 0.327139i 0.262843 0.980942i 0.575337 + 1.60623i 0 2.74296 0.690063i 1.33758 0.772254i 0.256621 1.41309i
373.1 −1.40462 + 0.164410i 0.839165 + 0.224854i 1.94594 0.461868i 3.16320 0.847576i −1.21568 0.177868i 0 −2.65738 + 0.968682i −1.94444 1.12262i −4.30375 + 1.71059i
373.2 −1.36184 0.381311i 1.60845 + 0.430984i 1.70920 + 1.03857i −1.94900 + 0.522232i −2.02612 1.20025i 0 −1.93164 2.06610i −0.196696 0.113563i 2.85335 + 0.0319777i
373.3 −1.25281 + 0.656100i −1.95279 0.523249i 1.13907 1.64394i −0.959072 + 0.256983i 2.78978 0.625694i 0 −0.348448 + 2.80688i 0.941530 + 0.543593i 1.03293 0.951197i
373.4 −0.892252 + 1.09722i 2.20008 + 0.589510i −0.407774 1.95799i −2.32205 + 0.622192i −2.60985 + 1.88798i 0 2.51218 + 1.29960i 1.89476 + 1.09394i 1.38918 3.10295i
373.5 −0.805226 1.16259i −1.16533 0.312249i −0.703221 + 1.87229i −0.980942 + 0.262843i 0.575337 + 1.60623i 0 2.74296 0.690063i −1.33758 0.772254i 1.09546 + 0.928784i
373.6 −0.246810 + 1.39251i −2.51077 0.672759i −1.87817 0.687371i 2.91203 0.780276i 1.55651 3.33023i 0 1.42072 2.44572i 3.25328 + 1.87828i 0.367824 + 4.24761i
373.7 0.236566 + 1.39429i −0.827840 0.221819i −1.88807 + 0.659683i −4.10878 + 1.10095i 0.113440 1.20672i 0 −1.36644 2.47646i −1.96196 1.13274i −2.50703 5.46838i
373.8 0.262311 1.38967i −2.91496 0.781062i −1.86239 0.729053i 0.745506 0.199758i −1.85005 + 3.84597i 0 −1.50167 + 2.39687i 5.28887 + 3.05353i −0.0820438 1.08841i
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 765.12 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
16.e even 4 1 inner
112.w even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.2.x.o 48
7.b odd 2 1 112.2.w.c 48
7.c even 3 1 784.2.m.k 24
7.c even 3 1 inner 784.2.x.o 48
7.d odd 6 1 112.2.w.c 48
7.d odd 6 1 784.2.m.j 24
16.e even 4 1 inner 784.2.x.o 48
28.d even 2 1 448.2.ba.c 48
28.f even 6 1 448.2.ba.c 48
56.e even 2 1 896.2.ba.e 48
56.h odd 2 1 896.2.ba.f 48
56.j odd 6 1 896.2.ba.f 48
56.m even 6 1 896.2.ba.e 48
112.j even 4 1 448.2.ba.c 48
112.j even 4 1 896.2.ba.e 48
112.l odd 4 1 112.2.w.c 48
112.l odd 4 1 896.2.ba.f 48
112.v even 12 1 448.2.ba.c 48
112.v even 12 1 896.2.ba.e 48
112.w even 12 1 784.2.m.k 24
112.w even 12 1 inner 784.2.x.o 48
112.x odd 12 1 112.2.w.c 48
112.x odd 12 1 784.2.m.j 24
112.x odd 12 1 896.2.ba.f 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.w.c 48 7.b odd 2 1
112.2.w.c 48 7.d odd 6 1
112.2.w.c 48 112.l odd 4 1
112.2.w.c 48 112.x odd 12 1
448.2.ba.c 48 28.d even 2 1
448.2.ba.c 48 28.f even 6 1
448.2.ba.c 48 112.j even 4 1
448.2.ba.c 48 112.v even 12 1
784.2.m.j 24 7.d odd 6 1
784.2.m.j 24 112.x odd 12 1
784.2.m.k 24 7.c even 3 1
784.2.m.k 24 112.w even 12 1
784.2.x.o 48 1.a even 1 1 trivial
784.2.x.o 48 7.c even 3 1 inner
784.2.x.o 48 16.e even 4 1 inner
784.2.x.o 48 112.w even 12 1 inner
896.2.ba.e 48 56.e even 2 1
896.2.ba.e 48 56.m even 6 1
896.2.ba.e 48 112.j even 4 1
896.2.ba.e 48 112.v even 12 1
896.2.ba.f 48 56.h odd 2 1
896.2.ba.f 48 56.j odd 6 1
896.2.ba.f 48 112.l odd 4 1
896.2.ba.f 48 112.x odd 12 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(784, [\chi])$$:

 $$T_{3}^{48} - 8 T_{3}^{45} - 162 T_{3}^{44} + 24 T_{3}^{43} + 32 T_{3}^{42} + 1116 T_{3}^{41} + 18293 T_{3}^{40} - 864 T_{3}^{39} - 3456 T_{3}^{38} - 103652 T_{3}^{37} - 1004406 T_{3}^{36} + 80664 T_{3}^{35} + 285960 T_{3}^{34} + \cdots + 194481$$ T3^48 - 8*T3^45 - 162*T3^44 + 24*T3^43 + 32*T3^42 + 1116*T3^41 + 18293*T3^40 - 864*T3^39 - 3456*T3^38 - 103652*T3^37 - 1004406*T3^36 + 80664*T3^35 + 285960*T3^34 + 4993772*T3^33 + 39410418*T3^32 - 1452176*T3^31 - 11629504*T3^30 - 170262984*T3^29 - 810174874*T3^28 + 64119872*T3^27 + 355069616*T3^26 + 2600062796*T3^25 + 11537897125*T3^24 + 1984752360*T3^23 - 2773815616*T3^22 - 24249331460*T3^21 - 64702303250*T3^20 - 7236391992*T3^19 + 23368692216*T3^18 + 65972795356*T3^17 + 234349610834*T3^16 + 190766251144*T3^15 + 99662977088*T3^14 + 76994795064*T3^13 - 81276505406*T3^12 - 112640447672*T3^11 - 57072227344*T3^10 - 48101100980*T3^9 + 35520287189*T3^8 + 63048361336*T3^7 + 36558066720*T3^6 + 37059622236*T3^5 + 16283664742*T3^4 - 1424310720*T3^3 + 51653448*T3^2 - 4482324*T3 + 194481 $$T_{5}^{48} + 4 T_{5}^{47} + 8 T_{5}^{46} + 40 T_{5}^{45} - 186 T_{5}^{44} - 1252 T_{5}^{43} - 2720 T_{5}^{42} - 8532 T_{5}^{41} + 51205 T_{5}^{40} + 210948 T_{5}^{39} + 370704 T_{5}^{38} + 503116 T_{5}^{37} + \cdots + 7076456545921$$ T5^48 + 4*T5^47 + 8*T5^46 + 40*T5^45 - 186*T5^44 - 1252*T5^43 - 2720*T5^42 - 8532*T5^41 + 51205*T5^40 + 210948*T5^39 + 370704*T5^38 + 503116*T5^37 - 4555342*T5^36 - 14985936*T5^35 - 22857208*T5^34 - 29465460*T5^33 + 223100506*T5^32 + 784694564*T5^31 + 1270822072*T5^30 + 1829502560*T5^29 - 4195098370*T5^28 - 18803433516*T5^27 - 31908388592*T5^26 - 50054715956*T5^25 + 48736044477*T5^24 + 330373629196*T5^23 + 611622231568*T5^22 + 1034125569620*T5^21 + 368291005726*T5^20 - 2437232688992*T5^19 - 5445326025288*T5^18 - 10186694208860*T5^17 - 8475915109990*T5^16 + 10974994868812*T5^15 + 34771359820296*T5^14 + 72447386780784*T5^13 + 120525275954450*T5^12 + 123137216065548*T5^11 + 102636522262288*T5^10 + 76961123401348*T5^9 + 11185067515717*T5^8 - 32643885656724*T5^7 - 32979000086688*T5^6 - 31132378205732*T5^5 - 9282580951194*T5^4 + 13097937657896*T5^3 + 15849643582472*T5^2 + 14977270384132*T5 + 7076456545921