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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 103.13
Character \(\chi\) \(=\) 175.103
Dual form 175.2.x.a.17.13

$q$-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+(0.857433 + 0.0449361i) q^{2} +(-1.95506 - 2.41430i) q^{3} +(-1.25587 - 0.131998i) q^{4} +(-0.521374 + 2.17444i) q^{5} +(-1.56784 - 2.15795i) q^{6} +(-2.09857 - 1.61121i) q^{7} +(-2.76697 - 0.438245i) q^{8} +(-1.38284 + 6.50575i) q^{9} +(-0.544753 + 1.84100i) q^{10} +(2.41483 - 0.513288i) q^{11} +(2.13662 + 3.29011i) q^{12} +(-3.07959 - 1.56913i) q^{13} +(-1.72698 - 1.47580i) q^{14} +(6.26905 - 2.99240i) q^{15} +(0.117592 + 0.0249949i) q^{16} +(-1.30681 - 3.40436i) q^{17} +(-1.47803 + 5.51610i) q^{18} +(-0.742133 - 7.06093i) q^{19} +(0.941799 - 2.66199i) q^{20} +(0.212895 + 8.21659i) q^{21} +(2.09362 - 0.331597i) q^{22} +(-0.107322 + 2.04783i) q^{23} +(4.35154 + 7.53709i) q^{24} +(-4.45634 - 2.26739i) q^{25} +(-2.57003 - 1.48381i) q^{26} +(10.1063 - 5.14941i) q^{27} +(2.42286 + 2.30048i) q^{28} +(0.594101 - 0.817710i) q^{29} +(5.50976 - 2.28407i) q^{30} +(-1.48256 + 3.32988i) q^{31} +(5.51171 + 1.47686i) q^{32} +(-5.96037 - 4.82661i) q^{33} +(-0.967525 - 2.97773i) q^{34} +(4.59761 - 3.72317i) q^{35} +(2.59541 - 7.98786i) q^{36} +(0.757805 - 0.492125i) q^{37} +(-0.319038 - 6.08762i) q^{38} +(2.23243 + 10.5028i) q^{39} +(2.39556 - 5.78811i) q^{40} +(2.23957 - 0.727680i) q^{41} +(-0.186678 + 7.05474i) q^{42} +(1.54846 + 1.54846i) q^{43} +(-3.10047 + 0.325873i) q^{44} +(-13.4254 - 6.39882i) q^{45} +(-0.184043 + 1.75105i) q^{46} +(-2.51648 - 0.965986i) q^{47} +(-0.169554 - 0.332768i) q^{48} +(1.80801 + 6.76248i) q^{49} +(-3.71912 - 2.14438i) q^{50} +(-5.66425 + 9.81076i) q^{51} +(3.66045 + 2.37712i) q^{52} +(-0.683987 + 0.553882i) q^{53} +(8.89685 - 3.96114i) q^{54} +(-0.142917 + 5.51851i) q^{55} +(5.10058 + 5.37786i) q^{56} +(-15.5963 + 15.5963i) q^{57} +(0.546147 - 0.674435i) q^{58} +(-0.825876 + 0.917228i) q^{59} +(-8.26812 + 2.93057i) q^{60} +(-8.07138 + 7.26750i) q^{61} +(-1.42082 + 2.78852i) q^{62} +(13.3841 - 11.4247i) q^{63} +(4.43089 + 1.43968i) q^{64} +(5.01758 - 5.87826i) q^{65} +(-4.89372 - 4.40633i) q^{66} +(4.72283 - 1.81293i) q^{67} +(1.19182 + 4.44794i) q^{68} +(5.15389 - 3.74452i) q^{69} +(4.10945 - 2.98577i) q^{70} +(-11.9130 - 8.65529i) q^{71} +(6.67739 - 17.3952i) q^{72} +(4.49475 - 6.92130i) q^{73} +(0.671881 - 0.387911i) q^{74} +(3.23826 + 15.1918i) q^{75} +8.96558i q^{76} +(-5.89471 - 2.81362i) q^{77} +(1.44221 + 9.10574i) q^{78} +(-6.17313 - 13.8651i) q^{79} +(-0.115659 + 0.242664i) q^{80} +(-13.9624 - 6.21645i) q^{81} +(1.95298 - 0.523299i) q^{82} +(0.481396 - 3.03941i) q^{83} +(0.817199 - 10.3471i) q^{84} +(8.08390 - 1.06663i) q^{85} +(1.25812 + 1.39728i) q^{86} +(-3.13570 + 0.164335i) q^{87} +(-6.90671 + 0.361965i) q^{88} +(-6.29169 - 6.98763i) q^{89} +(-11.2238 - 6.08984i) q^{90} +(3.93454 + 8.25478i) q^{91} +(0.405091 - 2.55764i) q^{92} +(10.9378 - 2.93077i) q^{93} +(-2.11430 - 0.941349i) q^{94} +(15.7405 + 2.06766i) q^{95} +(-7.21015 - 16.1943i) q^{96} +(2.78673 + 17.5947i) q^{97} +(1.24637 + 5.87961i) q^{98} +16.4201i 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 \(\left(\mathbb{Z}/175\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{7}{20}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.857433 + 0.0449361i 0.606296 + 0.0317746i 0.353014 − 0.935618i \(-0.385157\pi\)
0.253282 + 0.967393i \(0.418490\pi\)
\(3\) −1.95506 − 2.41430i −1.12875 − 1.39390i −0.908208 − 0.418519i \(-0.862549\pi\)
−0.220546 − 0.975377i \(-0.570784\pi\)
\(4\) −1.25587 − 0.131998i −0.627936 − 0.0659988i
\(5\) −0.521374 + 2.17444i −0.233165 + 0.972437i
\(6\) −1.56784 − 2.15795i −0.640069 − 0.880979i
\(7\) −2.09857 − 1.61121i −0.793186 − 0.608980i
\(8\) −2.76697 − 0.438245i −0.978272 − 0.154943i
\(9\) −1.38284 + 6.50575i −0.460946 + 2.16858i
\(10\) −0.544753 + 1.84100i −0.172266 + 0.582176i
\(11\) 2.41483 − 0.513288i 0.728099 − 0.154762i 0.171074 − 0.985258i \(-0.445276\pi\)
0.557025 + 0.830496i \(0.311943\pi\)
\(12\) 2.13662 + 3.29011i 0.616790 + 0.949774i
\(13\) −3.07959 − 1.56913i −0.854124 − 0.435198i −0.0286171 − 0.999590i \(-0.509110\pi\)
−0.825506 + 0.564393i \(0.809110\pi\)
\(14\) −1.72698 − 1.47580i −0.461556 − 0.394425i
\(15\) 6.26905 − 2.99240i 1.61866 − 0.772634i
\(16\) 0.117592 + 0.0249949i 0.0293980 + 0.00624873i
\(17\) −1.30681 − 3.40436i −0.316949 − 0.825679i −0.995813 − 0.0914154i \(-0.970861\pi\)
0.678864 − 0.734264i \(-0.262472\pi\)
\(18\) −1.47803 + 5.51610i −0.348376 + 1.30016i
\(19\) −0.742133 − 7.06093i −0.170257 − 1.61989i −0.662246 − 0.749286i \(-0.730397\pi\)
0.491989 − 0.870601i \(-0.336270\pi\)
\(20\) 0.941799 − 2.66199i 0.210593 − 0.595240i
\(21\) 0.212895 + 8.21659i 0.0464576 + 1.79301i
\(22\) 2.09362 − 0.331597i 0.446361 − 0.0706967i
\(23\) −0.107322 + 2.04783i −0.0223782 + 0.427001i 0.964351 + 0.264627i \(0.0852487\pi\)
−0.986729 + 0.162375i \(0.948085\pi\)
\(24\) 4.35154 + 7.53709i 0.888254 + 1.53850i
\(25\) −4.45634 − 2.26739i −0.891268 − 0.453477i
\(26\) −2.57003 − 1.48381i −0.504024 − 0.290998i
\(27\) 10.1063 − 5.14941i 1.94496 − 0.991005i
\(28\) 2.42286 + 2.30048i 0.457878 + 0.434750i
\(29\) 0.594101 − 0.817710i 0.110322 − 0.151845i −0.750286 − 0.661114i \(-0.770084\pi\)
0.860608 + 0.509269i \(0.170084\pi\)
\(30\) 5.50976 − 2.28407i 1.00594 − 0.417013i
\(31\) −1.48256 + 3.32988i −0.266275 + 0.598063i −0.996355 − 0.0852978i \(-0.972816\pi\)
0.730081 + 0.683361i \(0.239482\pi\)
\(32\) 5.51171 + 1.47686i 0.974342 + 0.261074i
\(33\) −5.96037 − 4.82661i −1.03757 − 0.840205i
\(34\) −0.967525 − 2.97773i −0.165929 − 0.510677i
\(35\) 4.59761 − 3.72317i 0.777138 − 0.629330i
\(36\) 2.59541 − 7.98786i 0.432569 − 1.33131i
\(37\) 0.757805 − 0.492125i 0.124582 − 0.0809048i −0.480843 − 0.876806i \(-0.659669\pi\)
0.605426 + 0.795902i \(0.293003\pi\)
\(38\) −0.319038 − 6.08762i −0.0517549 − 0.987542i
\(39\) 2.23243 + 10.5028i 0.357475 + 1.68179i
\(40\) 2.39556 − 5.78811i 0.378772 − 0.915181i
\(41\) 2.23957 − 0.727680i 0.349762 − 0.113644i −0.128867 − 0.991662i \(-0.541134\pi\)
0.478629 + 0.878017i \(0.341134\pi\)
\(42\) −0.186678 + 7.05474i −0.0288051 + 1.08857i
\(43\) 1.54846 + 1.54846i 0.236138 + 0.236138i 0.815249 − 0.579111i \(-0.196600\pi\)
−0.579111 + 0.815249i \(0.696600\pi\)
\(44\) −3.10047 + 0.325873i −0.467414 + 0.0491272i
\(45\) −13.4254 − 6.39882i −2.00133 − 0.953880i
\(46\) −0.184043 + 1.75105i −0.0271356 + 0.258178i
\(47\) −2.51648 − 0.965986i −0.367066 − 0.140904i 0.167842 − 0.985814i \(-0.446320\pi\)
−0.534908 + 0.844910i \(0.679654\pi\)
\(48\) −0.169554 − 0.332768i −0.0244730 − 0.0480310i
\(49\) 1.80801 + 6.76248i 0.258287 + 0.966068i
\(50\) −3.71912 − 2.14438i −0.525963 − 0.303261i
\(51\) −5.66425 + 9.81076i −0.793153 + 1.37378i
\(52\) 3.66045 + 2.37712i 0.507613 + 0.329647i
\(53\) −0.683987 + 0.553882i −0.0939528 + 0.0760815i −0.675146 − 0.737684i \(-0.735919\pi\)
0.581193 + 0.813766i \(0.302586\pi\)
\(54\) 8.89685 − 3.96114i 1.21071 − 0.539042i
\(55\) −0.142917 + 5.51851i −0.0192709 + 0.744115i
\(56\) 5.10058 + 5.37786i 0.681594 + 0.718647i
\(57\) −15.5963 + 15.5963i −2.06578 + 2.06578i
\(58\) 0.546147 − 0.674435i 0.0717126 − 0.0885576i
\(59\) −0.825876 + 0.917228i −0.107520 + 0.119413i −0.794502 − 0.607261i \(-0.792268\pi\)
0.686982 + 0.726674i \(0.258935\pi\)
\(60\) −8.26812 + 2.93057i −1.06741 + 0.378335i
\(61\) −8.07138 + 7.26750i −1.03343 + 0.930508i −0.997627 − 0.0688502i \(-0.978067\pi\)
−0.0358074 + 0.999359i \(0.511400\pi\)
\(62\) −1.42082 + 2.78852i −0.180445 + 0.354143i
\(63\) 13.3841 − 11.4247i 1.68624 − 1.43938i
\(64\) 4.43089 + 1.43968i 0.553861 + 0.179960i
\(65\) 5.01758 − 5.87826i 0.622354 − 0.729108i
\(66\) −4.89372 − 4.40633i −0.602376 − 0.542382i
\(67\) 4.72283 − 1.81293i 0.576986 − 0.221484i −0.0523352 − 0.998630i \(-0.516666\pi\)
0.629321 + 0.777145i \(0.283333\pi\)
\(68\) 1.19182 + 4.44794i 0.144530 + 0.539392i
\(69\) 5.15389 − 3.74452i 0.620455 − 0.450787i
\(70\) 4.10945 − 2.98577i 0.491173 − 0.356867i
\(71\) −11.9130 − 8.65529i −1.41381 − 1.02719i −0.992754 − 0.120161i \(-0.961659\pi\)
−0.421057 − 0.907034i \(-0.638341\pi\)
\(72\) 6.67739 − 17.3952i 0.786938 − 2.05004i
\(73\) 4.49475 − 6.92130i 0.526070 − 0.810077i −0.471343 − 0.881950i \(-0.656231\pi\)
0.997414 + 0.0718724i \(0.0228974\pi\)
\(74\) 0.671881 − 0.387911i 0.0781046 − 0.0450937i
\(75\) 3.23826 + 15.1918i 0.373922 + 1.75420i
\(76\) 8.96558i 1.02842i
\(77\) −5.89471 − 2.81362i −0.671765 − 0.320642i
\(78\) 1.44221 + 9.10574i 0.163298 + 1.03102i
\(79\) −6.17313 − 13.8651i −0.694531 − 1.55994i −0.822881 − 0.568214i \(-0.807635\pi\)
0.128350 − 0.991729i \(-0.459032\pi\)
\(80\) −0.115659 + 0.242664i −0.0129311 + 0.0271307i
\(81\) −13.9624 − 6.21645i −1.55138 − 0.690717i
\(82\) 1.95298 − 0.523299i 0.215670 − 0.0577887i
\(83\) 0.481396 − 3.03941i 0.0528401 − 0.333619i −0.947080 − 0.320997i \(-0.895982\pi\)
0.999920 − 0.0126222i \(-0.00401788\pi\)
\(84\) 0.817199 − 10.3471i 0.0891638 − 1.12896i
\(85\) 8.08390 − 1.06663i 0.876822 − 0.115693i
\(86\) 1.25812 + 1.39728i 0.135667 + 0.150673i
\(87\) −3.13570 + 0.164335i −0.336182 + 0.0176186i
\(88\) −6.90671 + 0.361965i −0.736258 + 0.0385856i
\(89\) −6.29169 − 6.98763i −0.666918 − 0.740687i 0.310831 − 0.950465i \(-0.399393\pi\)
−0.977749 + 0.209778i \(0.932726\pi\)
\(90\) −11.2238 − 6.08984i −1.18309 − 0.641926i
\(91\) 3.93454 + 8.25478i 0.412452 + 0.865337i
\(92\) 0.405091 − 2.55764i 0.0422336 − 0.266653i
\(93\) 10.9378 − 2.93077i 1.13420 − 0.303907i
\(94\) −2.11430 − 0.941349i −0.218074 − 0.0970927i
\(95\) 15.7405 + 2.06766i 1.61494 + 0.212137i
\(96\) −7.21015 − 16.1943i −0.735883 − 1.65282i
\(97\) 2.78673 + 17.5947i 0.282949 + 1.78647i 0.562970 + 0.826477i \(0.309659\pi\)
−0.280021 + 0.959994i \(0.590341\pi\)
\(98\) 1.24637 + 5.87961i 0.125902 + 0.593931i
\(99\) 16.4201i 1.65028i
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.103.13 yes 288
5.2 odd 4 875.2.bb.b.257.6 288
5.3 odd 4 875.2.bb.a.257.13 288
5.4 even 2 875.2.bb.c.243.6 288
7.3 odd 6 inner 175.2.x.a.3.6 ✓ 288
25.6 even 5 875.2.bb.b.768.6 288
25.8 odd 20 875.2.bb.c.607.13 288
25.17 odd 20 inner 175.2.x.a.117.6 yes 288
25.19 even 10 875.2.bb.a.768.13 288
35.3 even 12 875.2.bb.a.507.13 288
35.17 even 12 875.2.bb.b.507.6 288
35.24 odd 6 875.2.bb.c.493.13 288
175.17 even 60 inner 175.2.x.a.17.13 yes 288
175.31 odd 30 875.2.bb.b.143.6 288
175.94 odd 30 875.2.bb.a.143.13 288
175.108 even 60 875.2.bb.c.857.6 288
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
175.2.x.a.3.6 ✓ 288 7.3 odd 6 inner
175.2.x.a.17.13 yes 288 175.17 even 60 inner
175.2.x.a.103.13 yes 288 1.1 even 1 trivial
175.2.x.a.117.6 yes 288 25.17 odd 20 inner
875.2.bb.a.143.13 288 175.94 odd 30
875.2.bb.a.257.13 288 5.3 odd 4
875.2.bb.a.507.13 288 35.3 even 12
875.2.bb.a.768.13 288 25.19 even 10
875.2.bb.b.143.6 288 175.31 odd 30
875.2.bb.b.257.6 288 5.2 odd 4
875.2.bb.b.507.6 288 35.17 even 12
875.2.bb.b.768.6 288 25.6 even 5
875.2.bb.c.243.6 288 5.4 even 2
875.2.bb.c.493.13 288 35.24 odd 6
875.2.bb.c.607.13 288 25.8 odd 20
875.2.bb.c.857.6 288 175.108 even 60