Properties

Label 175.2.q.a
Level 175
Weight 2
Character orbit 175.q
Analytic conductor 1.397
Analytic rank 0
Dimension 144
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 175.q (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.39738203537\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(18\) over \(\Q(\zeta_{15})\)
Coefficient ring index: multiple of None
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 144q - 3q^{2} - 3q^{3} + 13q^{4} - 3q^{5} - 12q^{6} - 22q^{7} - 2q^{8} + 11q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 144q - 3q^{2} - 3q^{3} + 13q^{4} - 3q^{5} - 12q^{6} - 22q^{7} - 2q^{8} + 11q^{9} - 3q^{10} - 6q^{11} - 11q^{12} - 12q^{13} - 6q^{14} - 64q^{15} + 13q^{16} + 9q^{17} - 18q^{18} - 11q^{19} - 24q^{20} - 3q^{21} - 52q^{22} - 17q^{23} + 46q^{24} - 3q^{25} + 44q^{26} - 84q^{27} + 62q^{28} - 24q^{29} - 27q^{30} - 21q^{31} - 16q^{32} - 18q^{33} - 36q^{34} + 24q^{35} - 104q^{36} - 5q^{37} - 12q^{38} + 25q^{39} + q^{40} + 38q^{41} - 58q^{42} + 20q^{43} - 7q^{44} - 45q^{45} + 21q^{46} - q^{47} - 12q^{48} - 38q^{49} + 66q^{50} - 8q^{51} + 50q^{52} + 37q^{53} + 15q^{54} - 28q^{55} - 60q^{56} + 136q^{57} + 53q^{58} - 39q^{59} + 9q^{60} - 13q^{61} + 124q^{62} + 75q^{63} + 42q^{64} - 9q^{65} + 7q^{66} - 13q^{67} - 110q^{68} + 50q^{69} - 5q^{70} + 22q^{71} - 18q^{72} - 41q^{73} - 10q^{74} + 27q^{75} - 276q^{76} + 37q^{77} + 2q^{78} + 9q^{79} - 94q^{80} + 57q^{81} - 108q^{82} + 86q^{83} - 29q^{84} - 58q^{85} - 17q^{86} - 7q^{87} - 26q^{88} - 42q^{89} + 376q^{90} - 34q^{91} - 62q^{92} + 98q^{93} - 11q^{94} + 45q^{95} + 13q^{96} + 96q^{97} - 86q^{98} - 68q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −2.26854 1.01002i −1.03919 1.15413i 2.78788 + 3.09625i 0.598087 2.15460i 1.19174 + 3.66780i 1.63141 2.08291i −1.66242 5.11641i 0.0614698 0.584846i −3.53297 + 4.28371i
11.2 −2.25839 1.00550i 0.712831 + 0.791679i 2.75102 + 3.05531i −2.23601 0.0162832i −0.813815 2.50467i −2.33148 + 1.25068i −1.61290 4.96398i 0.194958 1.85490i 5.03340 + 2.28508i
11.3 −2.16145 0.962341i 2.09781 + 2.32985i 2.40752 + 2.67382i 2.08640 + 0.804329i −2.29220 7.05468i −0.0714197 2.64479i −1.16834 3.59577i −0.713827 + 6.79161i −3.73561 3.74634i
11.4 −1.94331 0.865215i −1.55527 1.72730i 1.68958 + 1.87647i 0.566195 + 2.16320i 1.52788 + 4.70232i −2.64421 + 0.0903750i −0.345128 1.06219i −0.251123 + 2.38928i 0.771342 4.69363i
11.5 −1.40506 0.625574i 0.893605 + 0.992448i 0.244597 + 0.271652i −1.97083 + 1.05633i −0.634720 1.95347i 2.64452 + 0.0807455i 0.776821 + 2.39081i 0.127161 1.20985i 3.42995 0.251317i
11.6 −1.26136 0.561596i −0.564524 0.626968i −0.0626100 0.0695354i 2.11758 + 0.718244i 0.359969 + 1.10787i 2.36555 + 1.18498i 0.893265 + 2.74919i 0.239184 2.27569i −2.26767 2.09519i
11.7 −0.853279 0.379904i −1.30628 1.45077i −0.754503 0.837961i −1.24736 1.85583i 0.563466 + 1.73417i −0.508729 + 2.59638i 0.902719 + 2.77828i −0.0847812 + 0.806639i 0.359309 + 2.05742i
11.8 −0.677204 0.301511i 0.992974 + 1.10281i −0.970565 1.07792i −0.0581969 2.23531i −0.339937 1.04622i −1.08929 2.41111i 0.790409 + 2.43263i 0.0833944 0.793444i −0.634559 + 1.53131i
11.9 −0.428333 0.190706i 1.73082 + 1.92227i −1.19116 1.32292i 0.335942 + 2.21069i −0.374778 1.15345i −1.07014 + 2.41967i 0.547701 + 1.68565i −0.385798 + 3.67063i 0.277696 1.01098i
11.10 0.189312 + 0.0842871i −0.609162 0.676543i −1.30953 1.45438i −1.65641 + 1.50211i −0.0582978 0.179422i −2.20881 1.45642i −0.253398 0.779878i 0.226953 2.15932i −0.440185 + 0.144753i
11.11 0.529445 + 0.235724i −0.229691 0.255097i −1.11352 1.23668i 1.94658 + 1.10038i −0.0614760 0.189204i 1.74489 1.98881i −0.656210 2.01961i 0.301269 2.86638i 0.771220 + 1.04145i
11.12 0.669019 + 0.297867i 1.50997 + 1.67699i −0.979399 1.08773i 1.33545 1.79348i 0.510679 + 1.57171i 1.21370 + 2.35094i −0.783844 2.41242i −0.218706 + 2.08085i 1.42766 0.802086i
11.13 0.956389 + 0.425812i −1.22328 1.35859i −0.604896 0.671805i 1.72038 1.42838i −0.591430 1.82023i −2.07580 + 1.64044i −0.939473 2.89140i −0.0357700 + 0.340329i 2.25358 0.633531i
11.14 1.37473 + 0.612069i 2.04628 + 2.27262i 0.176990 + 0.196568i −2.23021 + 0.161703i 1.42208 + 4.37671i 0.781664 2.52765i −0.807034 2.48380i −0.663974 + 6.31729i −3.16491 1.14275i
11.15 1.53521 + 0.683522i −1.65355 1.83645i 0.551421 + 0.612415i −1.53440 1.62653i −1.28330 3.94959i 2.14197 1.55304i −0.610656 1.87941i −0.324749 + 3.08978i −1.24387 3.54587i
11.16 1.86868 + 0.831989i −0.100275 0.111367i 1.46149 + 1.62315i −0.993637 + 2.00317i −0.0947260 0.291537i 1.42653 + 2.22823i 0.116408 + 0.358267i 0.311238 2.96123i −3.52340 + 2.91658i
11.17 2.21264 + 0.985130i 0.483361 + 0.536826i 2.58702 + 2.87318i −1.04758 1.97550i 0.540658 + 1.66398i −2.59815 + 0.499637i 1.39679 + 4.29888i 0.259040 2.46460i −0.371785 5.40306i
11.18 2.44336 + 1.08785i −1.77288 1.96899i 3.44830 + 3.82973i 1.66349 + 1.49426i −2.18982 6.73956i −1.54320 2.14908i 2.60627 + 8.02126i −0.420206 + 3.99799i 2.43896 + 5.46063i
16.1 −2.26854 + 1.01002i −1.03919 + 1.15413i 2.78788 3.09625i 0.598087 + 2.15460i 1.19174 3.66780i 1.63141 + 2.08291i −1.66242 + 5.11641i 0.0614698 + 0.584846i −3.53297 4.28371i
16.2 −2.25839 + 1.00550i 0.712831 0.791679i 2.75102 3.05531i −2.23601 + 0.0162832i −0.813815 + 2.50467i −2.33148 1.25068i −1.61290 + 4.96398i 0.194958 + 1.85490i 5.03340 2.28508i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 156.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
25.d even 5 1 inner
175.q even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 175.2.q.a 144
5.b even 2 1 875.2.q.a 144
5.c odd 4 2 875.2.u.b 288
7.c even 3 1 inner 175.2.q.a 144
25.d even 5 1 inner 175.2.q.a 144
25.e even 10 1 875.2.q.a 144
25.f odd 20 2 875.2.u.b 288
35.j even 6 1 875.2.q.a 144
35.l odd 12 2 875.2.u.b 288
175.q even 15 1 inner 175.2.q.a 144
175.t even 30 1 875.2.q.a 144
175.w odd 60 2 875.2.u.b 288
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.2.q.a 144 1.a even 1 1 trivial
175.2.q.a 144 7.c even 3 1 inner
175.2.q.a 144 25.d even 5 1 inner
175.2.q.a 144 175.q even 15 1 inner
875.2.q.a 144 5.b even 2 1
875.2.q.a 144 25.e even 10 1
875.2.q.a 144 35.j even 6 1
875.2.q.a 144 175.t even 30 1
875.2.u.b 288 5.c odd 4 2
875.2.u.b 288 25.f odd 20 2
875.2.u.b 288 35.l odd 12 2
875.2.u.b 288 175.w odd 60 2

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(175, [\chi])\).

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database