Properties

Label 847.2.u.a
Level $847$
Weight $2$
Character orbit 847.u
Analytic conductor $6.763$
Analytic rank $0$
Dimension $1720$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 847.u (of order \(33\), degree \(20\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.76332905120\)
Analytic rank: \(0\)
Dimension: \(1720\)
Relative dimension: \(86\) over \(\Q(\zeta_{33})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{33}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1720q - 11q^{2} - 20q^{3} + 73q^{4} - 7q^{5} + 8q^{6} - 24q^{7} - 32q^{8} - 844q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 1720q - 11q^{2} - 20q^{3} + 73q^{4} - 7q^{5} + 8q^{6} - 24q^{7} - 32q^{8} - 844q^{9} - 39q^{10} - 22q^{11} + 49q^{12} - 28q^{13} - 8q^{14} - 26q^{15} + 61q^{16} - 17q^{17} - 2q^{18} - 13q^{19} - 76q^{20} + 13q^{21} - 132q^{22} + 23q^{23} - 36q^{24} + 75q^{25} + 7q^{26} + 28q^{27} - 38q^{28} - 20q^{29} + 51q^{30} - 5q^{31} - 23q^{32} + 59q^{33} + 4q^{34} - 16q^{35} - 18q^{36} - 52q^{37} - 13q^{38} - 46q^{39} + 110q^{40} - 52q^{41} + 33q^{42} - 48q^{43} - 4q^{44} + 60q^{45} - 45q^{46} - 19q^{47} - 2q^{48} - 92q^{49} - 68q^{50} + 127q^{51} - 75q^{52} - 97q^{53} - 213q^{54} - 60q^{55} - 18q^{56} - 18q^{57} + 54q^{58} - 7q^{59} - 20q^{60} - 23q^{61} - 64q^{62} + 125q^{63} - 412q^{64} + 19q^{65} - 46q^{66} + 7q^{67} - 27q^{68} - 6q^{69} - 40q^{70} - 24q^{71} - 11q^{72} - 25q^{73} + 29q^{74} + 81q^{75} + 184q^{76} - 284q^{77} + 2q^{78} - 55q^{79} + 7q^{80} - 772q^{81} + 27q^{82} + 8q^{83} - q^{84} - 232q^{85} - 37q^{86} - 38q^{87} + 23q^{88} - 126q^{89} + 448q^{90} + 17q^{91} + 416q^{92} + 2q^{93} + 144q^{94} + 69q^{95} - 47q^{96} - 62q^{97} + 96q^{98} + 330q^{99} + O(q^{100}) \)

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 −1.58299 2.22300i −0.529306 0.916785i −1.78173 + 5.14797i −2.73161 + 2.60458i −1.20013 + 2.62791i 0.944173 2.47155i 9.02743 2.65069i 0.939671 1.62756i 10.1141 + 1.94933i
23.2 −1.54654 2.17181i 0.114913 + 0.199036i −1.67084 + 4.82758i 0.314136 0.299528i 0.254550 0.557387i 2.59071 + 0.536841i 7.95224 2.33499i 1.47359 2.55233i −1.13634 0.219012i
23.3 −1.54389 2.16809i −1.53393 2.65685i −1.66287 + 4.80455i 1.34609 1.28349i −3.39205 + 7.42756i −1.07048 2.41952i 7.87637 2.31271i −3.20589 + 5.55276i −4.86092 0.936866i
23.4 −1.54361 2.16770i 1.30997 + 2.26894i −1.66206 + 4.80220i 0.519449 0.495293i 2.89630 6.34201i 2.54979 0.706084i 7.86862 2.31044i −1.93206 + 3.34643i −1.87548 0.361469i
23.5 −1.53915 2.16144i −0.737739 1.27780i −1.64868 + 4.76356i 1.21935 1.16265i −1.62639 + 3.56131i −2.34511 + 1.22493i 7.74177 2.27319i 0.411483 0.712709i −4.38976 0.846058i
23.6 −1.41974 1.99374i −1.04317 1.80682i −1.30522 + 3.77118i −1.72874 + 1.64835i −2.12130 + 4.64501i −0.295823 + 2.62916i 4.67495 1.37269i −0.676390 + 1.17154i 5.74074 + 1.10644i
23.7 −1.41639 1.98904i 0.754247 + 1.30639i −1.29599 + 3.74452i 2.88425 2.75013i 1.53017 3.35060i −0.521292 2.59389i 4.59784 1.35005i 0.362222 0.627387i −9.55536 1.84164i
23.8 −1.39839 1.96377i 0.752732 + 1.30377i −1.24675 + 3.60224i −1.42488 + 1.35862i 1.50769 3.30138i −2.64184 0.143811i 4.19115 1.23063i 0.366789 0.635297i 4.66055 + 0.898247i
23.9 −1.30730 1.83585i 1.04438 + 1.80891i −1.00717 + 2.91001i −0.634596 + 0.605086i 1.95557 4.28211i −0.553085 + 2.58730i 2.33411 0.685355i −0.681441 + 1.18029i 1.94046 + 0.373992i
23.10 −1.28860 1.80958i 0.333684 + 0.577958i −0.959972 + 2.77366i −1.44180 + 1.37475i 0.615878 1.34859i −0.464073 2.60473i 1.99315 0.585241i 1.27731 2.21237i 4.34563 + 0.837552i
23.11 −1.28779 1.80845i −1.05900 1.83424i −0.957945 + 2.76780i 2.85294 2.72028i −1.95336 + 4.27725i 2.60448 0.465469i 1.97870 0.580997i −0.742952 + 1.28683i −8.59347 1.65626i
23.12 −1.26662 1.77872i −0.0320353 0.0554867i −0.905377 + 2.61591i 0.962361 0.917609i −0.0581187 + 0.127262i 1.16758 + 2.37418i 1.60942 0.472568i 1.49795 2.59452i −2.85111 0.549507i
23.13 −1.26589 1.77769i 1.32996 + 2.30355i −0.903582 + 2.61073i −3.13876 + 2.99281i 2.41143 5.28030i 1.96223 + 1.77473i 1.59701 0.468923i −2.03757 + 3.52918i 9.29362 + 1.79120i
23.14 −1.24481 1.74809i −0.628335 1.08831i −0.852131 + 2.46207i −0.293339 + 0.279699i −1.12030 + 2.45312i −2.19146 1.48239i 1.24649 0.366003i 0.710390 1.23043i 0.854089 + 0.164612i
23.15 −1.23063 1.72817i −1.49574 2.59069i −0.818005 + 2.36347i −1.74478 + 1.66364i −2.63647 + 5.77306i 2.63832 0.198173i 1.01990 0.299469i −2.97445 + 5.15190i 5.02223 + 0.967955i
23.16 −1.16442 1.63519i 1.21406 + 2.10281i −0.663855 + 1.91808i 2.54551 2.42714i 2.02483 4.43376i −2.62336 + 0.343509i 0.0572360 0.0168060i −1.44787 + 2.50779i −6.93286 1.33620i
23.17 −1.08714 1.52668i 1.52438 + 2.64031i −0.494736 + 1.42945i 0.554663 0.528870i 2.37369 5.19765i 0.743974 2.53900i −0.876412 + 0.257338i −3.14749 + 5.45162i −1.41042 0.271835i
23.18 −1.06888 1.50103i −1.66808 2.88919i −0.456452 + 1.31883i 1.17953 1.12468i −2.55379 + 5.59203i −0.367949 + 2.62004i −1.06864 + 0.313782i −4.06496 + 7.04072i −2.94896 0.568365i
23.19 −1.05144 1.47654i −0.136956 0.237214i −0.420511 + 1.21499i 1.03151 0.983547i −0.206256 + 0.451637i 1.54090 2.15073i −1.24233 + 0.364781i 1.46249 2.53310i −2.53682 0.488933i
23.20 −1.02678 1.44191i 0.607575 + 1.05235i −0.370686 + 1.07103i 2.00114 1.90808i 0.893547 1.95660i 0.986575 + 2.45493i −1.47193 + 0.432196i 0.761704 1.31931i −4.80599 0.926279i
See next 80 embeddings (of 1720 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 837.86
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
121.e even 11 1 inner
847.u even 33 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 847.2.u.a 1720
7.c even 3 1 inner 847.2.u.a 1720
121.e even 11 1 inner 847.2.u.a 1720
847.u even 33 1 inner 847.2.u.a 1720
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
847.2.u.a 1720 1.a even 1 1 trivial
847.2.u.a 1720 7.c even 3 1 inner
847.2.u.a 1720 121.e even 11 1 inner
847.2.u.a 1720 847.u even 33 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(847, [\chi])\).