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Properties

Label 175.2.x.a.73.9
Level $175$
Weight $2$
Character 175.73
Analytic conductor $1.397$
Analytic rank $0$
Dimension $288$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [175,2,Mod(3,175)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("175.3"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(175, base_ring=CyclotomicField(60)) chi = DirichletCharacter(H, H._module([21, 10])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 175.x (of order \(60\), degree \(16\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.39738203537\)
Analytic rank: \(0\)
Dimension: \(288\)
Relative dimension: \(18\) over \(\Q(\zeta_{60})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label 73.9
Character \(\chi\) \(=\) 175.73
Dual form 175.2.x.a.12.9

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.195749 + 0.0751409i) q^{2} +(0.218665 + 0.0114597i) q^{3} +(-1.45362 + 1.30884i) q^{4} +(-2.12373 + 0.699832i) q^{5} +(-0.0436645 + 0.0141875i) q^{6} +(2.12696 + 1.57354i) q^{7} +(0.376577 - 0.739075i) q^{8} +(-2.93588 - 0.308574i) q^{9} +(0.363132 - 0.296570i) q^{10} +(0.570376 + 5.42676i) q^{11} +(-0.332854 + 0.269540i) q^{12} +(-4.88233 + 0.773285i) q^{13} +(-0.534587 - 0.148197i) q^{14} +(-0.472405 + 0.128691i) q^{15} +(0.390743 - 3.71767i) q^{16} +(2.32593 + 3.58162i) q^{17} +(0.597882 - 0.160202i) q^{18} +(4.50759 - 5.00619i) q^{19} +(2.17112 - 3.79692i) q^{20} +(0.447059 + 0.368453i) q^{21} +(-0.519422 - 1.01942i) q^{22} +(1.52232 + 3.96579i) q^{23} +(0.0908139 - 0.157294i) q^{24} +(4.02047 - 2.97251i) q^{25} +(0.897604 - 0.518232i) q^{26} +(-1.28725 - 0.203880i) q^{27} +(-5.15131 + 0.496528i) q^{28} +(-2.62602 - 0.853244i) q^{29} +(0.0828028 - 0.0606882i) q^{30} +(0.817196 - 3.84460i) q^{31} +(0.632234 + 2.35953i) q^{32} +(0.0625319 + 1.19318i) q^{33} +(-0.724425 - 0.526326i) q^{34} +(-5.61831 - 1.85327i) q^{35} +(4.67153 - 3.39406i) q^{36} +(1.71732 + 2.12071i) q^{37} +(-0.506186 + 1.31866i) q^{38} +(-1.07646 + 0.113140i) q^{39} +(-0.282521 + 1.83314i) q^{40} +(2.57626 - 3.54591i) q^{41} +(-0.115197 - 0.0385318i) q^{42} +(2.80833 + 2.80833i) q^{43} +(-7.93190 - 7.14191i) q^{44} +(6.45098 - 1.39930i) q^{45} +(-0.595986 - 0.661909i) q^{46} +(1.33852 + 0.869244i) q^{47} +(0.128045 - 0.808447i) q^{48} +(2.04792 + 6.69373i) q^{49} +(-0.563645 + 0.883967i) q^{50} +(0.467556 + 0.809830i) q^{51} +(6.08493 - 7.51427i) q^{52} +(-0.156527 + 2.98672i) q^{53} +(0.267297 - 0.0568156i) q^{54} +(-5.00915 - 11.1258i) q^{55} +(1.96393 - 0.979422i) q^{56} +(1.04302 - 1.04302i) q^{57} +(0.578153 - 0.0302997i) q^{58} +(-0.516841 + 0.230112i) q^{59} +(0.518260 - 0.805373i) q^{60} +(2.18819 - 4.91476i) q^{61} +(0.128922 + 0.813982i) q^{62} +(-5.75895 - 5.27606i) q^{63} +(4.09340 + 5.63409i) q^{64} +(9.82758 - 5.05906i) q^{65} +(-0.101897 - 0.228865i) q^{66} +(-10.8242 + 7.02934i) q^{67} +(-8.06880 - 2.16203i) q^{68} +(0.287432 + 0.884624i) q^{69} +(1.23903 - 0.0593900i) q^{70} +(-2.85288 + 8.78027i) q^{71} +(-1.33365 + 2.05364i) q^{72} +(8.01419 + 6.48976i) q^{73} +(-0.495515 - 0.286086i) q^{74} +(0.913200 - 0.603910i) q^{75} +13.1768i q^{76} +(-7.32608 + 12.4400i) q^{77} +(0.202213 - 0.103033i) q^{78} +(0.632240 + 2.97446i) q^{79} +(1.77191 + 8.16879i) q^{80} +(8.38350 + 1.78197i) q^{81} +(-0.237856 + 0.887690i) q^{82} +(-3.18975 - 1.62526i) q^{83} +(-1.13210 + 0.0495406i) q^{84} +(-7.44619 - 5.97864i) q^{85} +(-0.760747 - 0.338706i) q^{86} +(-0.564439 - 0.216668i) q^{87} +(4.22558 + 1.62205i) q^{88} +(-6.81322 - 3.03344i) q^{89} +(-1.15763 + 0.758643i) q^{90} +(-11.6013 - 6.03781i) q^{91} +(-7.40347 - 3.77226i) q^{92} +(0.222750 - 0.831315i) q^{93} +(-0.327329 - 0.0695760i) q^{94} +(-6.06942 + 13.7864i) q^{95} +(0.111208 + 0.523192i) q^{96} +(8.84458 - 4.50654i) q^{97} +(-0.903852 - 1.15641i) q^{98} -16.1083i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 288 q - 8 q^{2} - 24 q^{3} - 10 q^{4} - 30 q^{5} - 10 q^{7} - 36 q^{8} - 10 q^{9} - 36 q^{10} - 6 q^{11} - 36 q^{12} - 20 q^{14} - 28 q^{15} - 30 q^{16} - 42 q^{17} - 14 q^{18} - 30 q^{19} - 12 q^{21} + 32 q^{22}+ \cdots + 222 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/175\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{11}{20}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.195749 + 0.0751409i −0.138415 + 0.0531327i −0.426590 − 0.904445i \(-0.640285\pi\)
0.288174 + 0.957578i \(0.406952\pi\)
\(3\) 0.218665 + 0.0114597i 0.126246 + 0.00661629i 0.115354 − 0.993324i \(-0.463200\pi\)
0.0108918 + 0.999941i \(0.496533\pi\)
\(4\) −1.45362 + 1.30884i −0.726809 + 0.654422i
\(5\) −2.12373 + 0.699832i −0.949762 + 0.312974i
\(6\) −0.0436645 + 0.0141875i −0.0178260 + 0.00579200i
\(7\) 2.12696 + 1.57354i 0.803916 + 0.594743i
\(8\) 0.376577 − 0.739075i 0.133140 − 0.261302i
\(9\) −2.93588 − 0.308574i −0.978628 − 0.102858i
\(10\) 0.363132 − 0.296570i 0.114832 − 0.0937838i
\(11\) 0.570376 + 5.42676i 0.171975 + 1.63623i 0.651461 + 0.758682i \(0.274156\pi\)
−0.479487 + 0.877549i \(0.659177\pi\)
\(12\) −0.332854 + 0.269540i −0.0960868 + 0.0778095i
\(13\) −4.88233 + 0.773285i −1.35411 + 0.214471i −0.790953 − 0.611877i \(-0.790415\pi\)
−0.563161 + 0.826347i \(0.690415\pi\)
\(14\) −0.534587 − 0.148197i −0.142875 − 0.0396074i
\(15\) −0.472405 + 0.128691i −0.121975 + 0.0332280i
\(16\) 0.390743 − 3.71767i 0.0976858 − 0.929418i
\(17\) 2.32593 + 3.58162i 0.564122 + 0.868671i 0.999501 − 0.0315758i \(-0.0100526\pi\)
−0.435380 + 0.900247i \(0.643386\pi\)
\(18\) 0.597882 − 0.160202i 0.140922 − 0.0377600i
\(19\) 4.50759 − 5.00619i 1.03411 − 1.14850i 0.0453550 − 0.998971i \(-0.485558\pi\)
0.988758 − 0.149527i \(-0.0477752\pi\)
\(20\) 2.17112 − 3.79692i 0.485478 − 0.849017i
\(21\) 0.447059 + 0.368453i 0.0975563 + 0.0804031i
\(22\) −0.519422 − 1.01942i −0.110741 − 0.217342i
\(23\) 1.52232 + 3.96579i 0.317426 + 0.826923i 0.995745 + 0.0921554i \(0.0293757\pi\)
−0.678318 + 0.734768i \(0.737291\pi\)
\(24\) 0.0908139 − 0.157294i 0.0185373 − 0.0321076i
\(25\) 4.02047 − 2.97251i 0.804094 − 0.594502i
\(26\) 0.897604 − 0.518232i 0.176035 − 0.101634i
\(27\) −1.28725 − 0.203880i −0.247731 − 0.0392367i
\(28\) −5.15131 + 0.496528i −0.973506 + 0.0938350i
\(29\) −2.62602 − 0.853244i −0.487639 − 0.158443i 0.0548701 − 0.998494i \(-0.482526\pi\)
−0.542509 + 0.840050i \(0.682526\pi\)
\(30\) 0.0828028 − 0.0606882i 0.0151177 − 0.0110801i
\(31\) 0.817196 − 3.84460i 0.146773 − 0.690511i −0.841803 − 0.539785i \(-0.818505\pi\)
0.988576 − 0.150726i \(-0.0481612\pi\)
\(32\) 0.632234 + 2.35953i 0.111764 + 0.417110i
\(33\) 0.0625319 + 1.19318i 0.0108854 + 0.207706i
\(34\) −0.724425 − 0.526326i −0.124238 − 0.0902641i
\(35\) −5.61831 − 1.85327i −0.949668 − 0.313259i
\(36\) 4.67153 − 3.39406i 0.778588 − 0.565677i
\(37\) 1.71732 + 2.12071i 0.282326 + 0.348643i 0.898505 − 0.438962i \(-0.144654\pi\)
−0.616180 + 0.787605i \(0.711321\pi\)
\(38\) −0.506186 + 1.31866i −0.0821142 + 0.213915i
\(39\) −1.07646 + 0.113140i −0.172371 + 0.0181169i
\(40\) −0.282521 + 1.83314i −0.0446705 + 0.289845i
\(41\) 2.57626 − 3.54591i 0.402344 − 0.553778i −0.558987 − 0.829177i \(-0.688810\pi\)
0.961330 + 0.275398i \(0.0888097\pi\)
\(42\) −0.115197 − 0.0385318i −0.0177753 − 0.00594559i
\(43\) 2.80833 + 2.80833i 0.428266 + 0.428266i 0.888037 − 0.459771i \(-0.152069\pi\)
−0.459771 + 0.888037i \(0.652069\pi\)
\(44\) −7.93190 − 7.14191i −1.19578 − 1.07668i
\(45\) 6.45098 − 1.39930i 0.961655 − 0.208595i
\(46\) −0.595986 − 0.661909i −0.0878733 − 0.0975932i
\(47\) 1.33852 + 0.869244i 0.195243 + 0.126792i 0.638558 − 0.769573i \(-0.279531\pi\)
−0.443315 + 0.896366i \(0.646198\pi\)
\(48\) 0.128045 − 0.808447i 0.0184818 − 0.116689i
\(49\) 2.04792 + 6.69373i 0.292561 + 0.956247i
\(50\) −0.563645 + 0.883967i −0.0797114 + 0.125012i
\(51\) 0.467556 + 0.809830i 0.0654709 + 0.113399i
\(52\) 6.08493 − 7.51427i 0.843828 − 1.04204i
\(53\) −0.156527 + 2.98672i −0.0215007 + 0.410257i 0.966584 + 0.256348i \(0.0825195\pi\)
−0.988085 + 0.153909i \(0.950814\pi\)
\(54\) 0.267297 − 0.0568156i 0.0363745 − 0.00773163i
\(55\) −5.00915 − 11.1258i −0.675434 − 1.50021i
\(56\) 1.96393 − 0.979422i 0.262441 − 0.130881i
\(57\) 1.04302 − 1.04302i 0.138152 − 0.138152i
\(58\) 0.578153 − 0.0302997i 0.0759152 − 0.00397855i
\(59\) −0.516841 + 0.230112i −0.0672869 + 0.0299581i −0.440104 − 0.897947i \(-0.645058\pi\)
0.372817 + 0.927905i \(0.378392\pi\)
\(60\) 0.518260 − 0.805373i 0.0669071 − 0.103973i
\(61\) 2.18819 − 4.91476i 0.280170 − 0.629271i −0.717569 − 0.696488i \(-0.754745\pi\)
0.997738 + 0.0672165i \(0.0214118\pi\)
\(62\) 0.128922 + 0.813982i 0.0163731 + 0.103376i
\(63\) −5.75895 − 5.27606i −0.725560 − 0.664721i
\(64\) 4.09340 + 5.63409i 0.511675 + 0.704261i
\(65\) 9.82758 − 5.05906i 1.21896 − 0.627499i
\(66\) −0.101897 − 0.228865i −0.0125427 − 0.0281713i
\(67\) −10.8242 + 7.02934i −1.32239 + 0.858771i −0.996248 − 0.0865493i \(-0.972416\pi\)
−0.326144 + 0.945320i \(0.605749\pi\)
\(68\) −8.06880 − 2.16203i −0.978486 − 0.262185i
\(69\) 0.287432 + 0.884624i 0.0346027 + 0.106496i
\(70\) 1.23903 − 0.0593900i 0.148093 − 0.00709846i
\(71\) −2.85288 + 8.78027i −0.338575 + 1.04203i 0.626359 + 0.779535i \(0.284544\pi\)
−0.964934 + 0.262492i \(0.915456\pi\)
\(72\) −1.33365 + 2.05364i −0.157172 + 0.242023i
\(73\) 8.01419 + 6.48976i 0.937989 + 0.759569i 0.970799 − 0.239893i \(-0.0771123\pi\)
−0.0328099 + 0.999462i \(0.510446\pi\)
\(74\) −0.495515 − 0.286086i −0.0576025 − 0.0332568i
\(75\) 0.913200 − 0.603910i 0.105447 − 0.0697336i
\(76\) 13.1768i 1.51148i
\(77\) −7.32608 + 12.4400i −0.834884 + 1.41767i
\(78\) 0.202213 − 0.103033i 0.0228962 − 0.0116662i
\(79\) 0.632240 + 2.97446i 0.0711326 + 0.334653i 0.999296 − 0.0375208i \(-0.0119460\pi\)
−0.928163 + 0.372173i \(0.878613\pi\)
\(80\) 1.77191 + 8.16879i 0.198106 + 0.913299i
\(81\) 8.38350 + 1.78197i 0.931499 + 0.197996i
\(82\) −0.237856 + 0.887690i −0.0262668 + 0.0980290i
\(83\) −3.18975 − 1.62526i −0.350121 − 0.178395i 0.270081 − 0.962838i \(-0.412949\pi\)
−0.620202 + 0.784442i \(0.712949\pi\)
\(84\) −1.13210 + 0.0495406i −0.123522 + 0.00540532i
\(85\) −7.44619 − 5.97864i −0.807653 − 0.648475i
\(86\) −0.760747 − 0.338706i −0.0820335 − 0.0365236i
\(87\) −0.564439 − 0.216668i −0.0605143 − 0.0232292i
\(88\) 4.22558 + 1.62205i 0.450448 + 0.172911i
\(89\) −6.81322 − 3.03344i −0.722200 − 0.321544i 0.0125135 − 0.999922i \(-0.496017\pi\)
−0.734713 + 0.678378i \(0.762683\pi\)
\(90\) −1.15763 + 0.758643i −0.122025 + 0.0799680i
\(91\) −11.6013 − 6.03781i −1.21615 − 0.632934i
\(92\) −7.40347 − 3.77226i −0.771865 − 0.393285i
\(93\) 0.222750 − 0.831315i 0.0230981 − 0.0862034i
\(94\) −0.327329 − 0.0695760i −0.0337614 − 0.00717621i
\(95\) −6.06942 + 13.7864i −0.622710 + 1.41445i
\(96\) 0.111208 + 0.523192i 0.0113501 + 0.0533980i
\(97\) 8.84458 − 4.50654i 0.898031 − 0.457570i 0.0568871 − 0.998381i \(-0.481882\pi\)
0.841144 + 0.540811i \(0.181882\pi\)
\(98\) −0.903852 − 1.15641i −0.0913028 − 0.116815i
\(99\) − 16.1083i − 1.61895i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 175.2.x.a.73.9 yes 288
5.2 odd 4 875.2.bb.b.157.9 288
5.3 odd 4 875.2.bb.a.157.10 288
5.4 even 2 875.2.bb.c.843.10 288
7.5 odd 6 inner 175.2.x.a.173.10 yes 288
25.9 even 10 875.2.bb.a.668.10 288
25.12 odd 20 inner 175.2.x.a.87.10 yes 288
25.13 odd 20 875.2.bb.c.332.9 288
25.16 even 5 875.2.bb.b.668.9 288
35.12 even 12 875.2.bb.b.782.9 288
35.19 odd 6 875.2.bb.c.593.9 288
35.33 even 12 875.2.bb.a.782.10 288
175.12 even 60 inner 175.2.x.a.12.9 ✓ 288
175.138 even 60 875.2.bb.c.82.10 288
175.159 odd 30 875.2.bb.a.418.10 288
175.166 odd 30 875.2.bb.b.418.9 288
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
175.2.x.a.12.9 ✓ 288 175.12 even 60 inner
175.2.x.a.73.9 yes 288 1.1 even 1 trivial
175.2.x.a.87.10 yes 288 25.12 odd 20 inner
175.2.x.a.173.10 yes 288 7.5 odd 6 inner
875.2.bb.a.157.10 288 5.3 odd 4
875.2.bb.a.418.10 288 175.159 odd 30
875.2.bb.a.668.10 288 25.9 even 10
875.2.bb.a.782.10 288 35.33 even 12
875.2.bb.b.157.9 288 5.2 odd 4
875.2.bb.b.418.9 288 175.166 odd 30
875.2.bb.b.668.9 288 25.16 even 5
875.2.bb.b.782.9 288 35.12 even 12
875.2.bb.c.82.10 288 175.138 even 60
875.2.bb.c.332.9 288 25.13 odd 20
875.2.bb.c.593.9 288 35.19 odd 6
875.2.bb.c.843.10 288 5.4 even 2