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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [95,4,Mod(2,95)] 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("95.2"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(95, base_ring=CyclotomicField(36)) chi = DirichletCharacter(H, H._module([9, 2])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 95 = 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 95.r (of order \(36\), degree \(12\), 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: \(5.60518145055\)
Analytic rank: \(0\)
Dimension: \(336\)
Relative dimension: \(28\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label 2.19
Character \(\chi\) \(=\) 95.2
Dual form 95.4.r.a.48.19

$q$-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+(1.01359 + 1.44756i) q^{2} +(0.774605 + 8.85377i) q^{3} +(1.66810 - 4.58306i) q^{4} +(9.73411 + 5.49973i) q^{5} +(-12.0313 + 10.0954i) q^{6} +(-4.72680 + 17.6406i) q^{7} +(21.9805 - 5.88966i) q^{8} +(-51.1995 + 9.02785i) q^{9} +(1.90524 + 19.6652i) q^{10} +(27.5375 - 47.6964i) q^{11} +(41.8695 + 11.2189i) q^{12} +(-21.0648 - 1.84293i) q^{13} +(-30.3270 + 11.0381i) q^{14} +(-41.1533 + 90.4438i) q^{15} +(0.915802 + 0.768449i) q^{16} +(-51.4230 + 36.0067i) q^{17} +(-64.9639 - 64.9639i) q^{18} +(-81.2724 - 15.9308i) q^{19} +(41.4430 - 35.4379i) q^{20} +(-159.848 - 28.1855i) q^{21} +(96.9555 - 8.48250i) q^{22} +(131.354 - 61.2513i) q^{23} +(69.1719 + 190.048i) q^{24} +(64.5059 + 107.070i) q^{25} +(-18.6834 - 32.3606i) q^{26} +(-57.4824 - 214.527i) q^{27} +(72.9634 + 51.0895i) q^{28} +(10.2127 + 57.9189i) q^{29} +(-172.636 + 32.1013i) q^{30} +(155.645 - 89.8620i) q^{31} +(15.6823 - 179.250i) q^{32} +(443.624 + 206.865i) q^{33} +(-104.244 - 37.9417i) q^{34} +(-143.030 + 145.720i) q^{35} +(-44.0306 + 249.710i) q^{36} +(234.482 - 234.482i) q^{37} +(-59.3165 - 133.794i) q^{38} -187.930i q^{39} +(246.352 + 63.5562i) q^{40} +(150.870 - 179.800i) q^{41} +(-121.220 - 259.958i) q^{42} +(-41.8821 + 89.8164i) q^{43} +(-172.660 - 205.768i) q^{44} +(-548.033 - 193.705i) q^{45} +(221.805 + 128.059i) q^{46} +(-350.939 + 501.193i) q^{47} +(-6.09429 + 8.70355i) q^{48} +(8.19712 + 4.73261i) q^{49} +(-89.6076 + 201.902i) q^{50} +(-358.628 - 427.396i) q^{51} +(-43.5844 + 93.4669i) q^{52} +(13.6586 + 29.2911i) q^{53} +(252.278 - 300.653i) q^{54} +(530.371 - 312.833i) q^{55} +415.589i q^{56} +(78.0936 - 731.908i) q^{57} +(-73.4897 + 73.4897i) q^{58} +(21.5382 - 122.149i) q^{59} +(345.862 + 339.477i) q^{60} +(480.710 + 174.964i) q^{61} +(287.842 + 134.223i) q^{62} +(82.7525 - 945.865i) q^{63} +(283.653 - 163.767i) q^{64} +(-194.911 - 133.790i) q^{65} +(150.204 + 851.851i) q^{66} +(-549.303 - 384.626i) q^{67} +(79.2425 + 295.737i) q^{68} +(644.053 + 1115.53i) q^{69} +(-355.913 - 59.3439i) q^{70} +(-84.8058 - 233.002i) q^{71} +(-1072.22 + 499.984i) q^{72} +(-591.673 + 51.7647i) q^{73} +(577.097 + 101.758i) q^{74} +(-898.007 + 654.058i) q^{75} +(-208.582 + 345.902i) q^{76} +(711.231 + 711.231i) q^{77} +(272.041 - 190.485i) q^{78} +(-329.373 - 276.377i) q^{79} +(4.68826 + 12.5168i) q^{80} +(535.793 - 195.013i) q^{81} +(413.193 + 36.1497i) q^{82} +(-685.056 - 183.560i) q^{83} +(-395.817 + 685.575i) q^{84} +(-698.584 + 67.6814i) q^{85} +(-172.466 + 30.4105i) q^{86} +(-504.890 + 135.285i) q^{87} +(324.373 - 1210.58i) q^{88} +(442.353 - 371.178i) q^{89} +(-275.082 - 989.650i) q^{90} +(132.079 - 362.885i) q^{91} +(-61.6074 - 704.176i) q^{92} +(916.181 + 1308.44i) q^{93} -1081.22 q^{94} +(-703.500 - 602.049i) q^{95} +1599.19 q^{96} +(7.66803 + 10.9511i) q^{97} +(1.45780 + 16.6628i) q^{98} +(-979.312 + 2690.64i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 336 q - 12 q^{2} - 12 q^{3} - 12 q^{5} + 12 q^{6} + 6 q^{7} - 18 q^{8} - 12 q^{10} - 12 q^{11} - 18 q^{12} - 12 q^{13} - 264 q^{15} - 996 q^{16} + 132 q^{17} + 1116 q^{20} - 600 q^{21} + 468 q^{22} + 564 q^{23}+ \cdots + 1290 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/95\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\)
\(\chi(n)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.01359 + 1.44756i 0.358360 + 0.511791i 0.957336 − 0.288976i \(-0.0933147\pi\)
−0.598977 + 0.800766i \(0.704426\pi\)
\(3\) 0.774605 + 8.85377i 0.149073 + 1.70391i 0.590847 + 0.806784i \(0.298794\pi\)
−0.441774 + 0.897126i \(0.645651\pi\)
\(4\) 1.66810 − 4.58306i 0.208512 − 0.572882i
\(5\) 9.73411 + 5.49973i 0.870646 + 0.491911i
\(6\) −12.0313 + 10.0954i −0.818623 + 0.686907i
\(7\) −4.72680 + 17.6406i −0.255223 + 0.952505i 0.712743 + 0.701425i \(0.247452\pi\)
−0.967966 + 0.251080i \(0.919214\pi\)
\(8\) 21.9805 − 5.88966i 0.971410 − 0.260289i
\(9\) −51.1995 + 9.02785i −1.89628 + 0.334365i
\(10\) 1.90524 + 19.6652i 0.0602489 + 0.621869i
\(11\) 27.5375 − 47.6964i 0.754808 − 1.30736i −0.190662 − 0.981656i \(-0.561064\pi\)
0.945470 − 0.325709i \(-0.105603\pi\)
\(12\) 41.8695 + 11.2189i 1.00722 + 0.269885i
\(13\) −21.0648 − 1.84293i −0.449409 − 0.0393182i −0.139795 − 0.990180i \(-0.544644\pi\)
−0.309614 + 0.950862i \(0.600200\pi\)
\(14\) −30.3270 + 11.0381i −0.578945 + 0.210719i
\(15\) −41.1533 + 90.4438i −0.708382 + 1.55683i
\(16\) 0.915802 + 0.768449i 0.0143094 + 0.0120070i
\(17\) −51.4230 + 36.0067i −0.733641 + 0.513701i −0.879673 − 0.475579i \(-0.842239\pi\)
0.146032 + 0.989280i \(0.453350\pi\)
\(18\) −64.9639 − 64.9639i −0.850674 − 0.850674i
\(19\) −81.2724 − 15.9308i −0.981325 − 0.192357i
\(20\) 41.4430 − 35.4379i 0.463347 − 0.396208i
\(21\) −159.848 − 28.1855i −1.66103 − 0.292884i
\(22\) 96.9555 − 8.48250i 0.939590 − 0.0822034i
\(23\) 131.354 − 61.2513i 1.19083 − 0.555295i 0.276798 − 0.960928i \(-0.410727\pi\)
0.914036 + 0.405633i \(0.132949\pi\)
\(24\) 69.1719 + 190.048i 0.588319 + 1.61639i
\(25\) 64.5059 + 107.070i 0.516048 + 0.856560i
\(26\) −18.6834 − 32.3606i −0.140927 − 0.244093i
\(27\) −57.4824 − 214.527i −0.409722 − 1.52910i
\(28\) 72.9634 + 51.0895i 0.492456 + 0.344822i
\(29\) 10.2127 + 57.9189i 0.0653946 + 0.370871i 0.999889 + 0.0148937i \(0.00474098\pi\)
−0.934494 + 0.355978i \(0.884148\pi\)
\(30\) −172.636 + 32.1013i −1.05063 + 0.195363i
\(31\) 155.645 − 89.8620i 0.901766 − 0.520635i 0.0239935 − 0.999712i \(-0.492362\pi\)
0.877773 + 0.479077i \(0.159029\pi\)
\(32\) 15.6823 − 179.250i 0.0866334 − 0.990225i
\(33\) 443.624 + 206.865i 2.34015 + 1.09123i
\(34\) −104.244 − 37.9417i −0.525815 − 0.191381i
\(35\) −143.030 + 145.720i −0.690756 + 0.703747i
\(36\) −44.0306 + 249.710i −0.203845 + 1.15606i
\(37\) 234.482 − 234.482i 1.04185 − 1.04185i 0.0427690 − 0.999085i \(-0.486382\pi\)
0.999085 − 0.0427690i \(-0.0136180\pi\)
\(38\) −59.3165 − 133.794i −0.253221 − 0.571166i
\(39\) − 187.930i − 0.771614i
\(40\) 246.352 + 63.5562i 0.973793 + 0.251228i
\(41\) 150.870 − 179.800i 0.574682 − 0.684880i −0.397903 − 0.917428i \(-0.630262\pi\)
0.972585 + 0.232548i \(0.0747062\pi\)
\(42\) −121.220 − 259.958i −0.445350 − 0.955057i
\(43\) −41.8821 + 89.8164i −0.148534 + 0.318532i −0.966471 − 0.256776i \(-0.917340\pi\)
0.817937 + 0.575308i \(0.195118\pi\)
\(44\) −172.660 − 205.768i −0.591580 − 0.705017i
\(45\) −548.033 − 193.705i −1.81546 − 0.641686i
\(46\) 221.805 + 128.059i 0.710941 + 0.410462i
\(47\) −350.939 + 501.193i −1.08914 + 1.55546i −0.287378 + 0.957817i \(0.592784\pi\)
−0.801765 + 0.597640i \(0.796105\pi\)
\(48\) −6.09429 + 8.70355i −0.0183257 + 0.0261719i
\(49\) 8.19712 + 4.73261i 0.0238983 + 0.0137977i
\(50\) −89.6076 + 201.902i −0.253449 + 0.571065i
\(51\) −358.628 − 427.396i −0.984666 − 1.17348i
\(52\) −43.5844 + 93.4669i −0.116232 + 0.249260i
\(53\) 13.6586 + 29.2911i 0.0353992 + 0.0759139i 0.923212 − 0.384291i \(-0.125554\pi\)
−0.887813 + 0.460204i \(0.847776\pi\)
\(54\) 252.278 − 300.653i 0.635753 − 0.757661i
\(55\) 530.371 − 312.833i 1.30028 − 0.766954i
\(56\) 415.589i 0.991705i
\(57\) 78.0936 − 731.908i 0.181469 − 1.70076i
\(58\) −73.4897 + 73.4897i −0.166374 + 0.166374i
\(59\) 21.5382 − 122.149i 0.0475261 − 0.269534i −0.951780 − 0.306782i \(-0.900748\pi\)
0.999306 + 0.0372476i \(0.0118590\pi\)
\(60\) 345.862 + 339.477i 0.744175 + 0.730438i
\(61\) 480.710 + 174.964i 1.00899 + 0.367243i 0.793046 − 0.609162i \(-0.208494\pi\)
0.215947 + 0.976405i \(0.430716\pi\)
\(62\) 287.842 + 134.223i 0.589613 + 0.274941i
\(63\) 82.7525 − 945.865i 0.165489 − 1.89155i
\(64\) 283.653 − 163.767i 0.554011 − 0.319858i
\(65\) −194.911 − 133.790i −0.371935 − 0.255301i
\(66\) 150.204 + 851.851i 0.280134 + 1.58872i
\(67\) −549.303 − 384.626i −1.00161 − 0.701336i −0.0469098 − 0.998899i \(-0.514937\pi\)
−0.954702 + 0.297563i \(0.903826\pi\)
\(68\) 79.2425 + 295.737i 0.141317 + 0.527403i
\(69\) 644.053 + 1115.53i 1.12369 + 1.94629i
\(70\) −355.913 − 59.3439i −0.607710 − 0.101328i
\(71\) −84.8058 − 233.002i −0.141755 − 0.389469i 0.848416 − 0.529330i \(-0.177557\pi\)
−0.990171 + 0.139861i \(0.955334\pi\)
\(72\) −1072.22 + 499.984i −1.75503 + 0.818385i
\(73\) −591.673 + 51.7647i −0.948631 + 0.0829945i −0.550958 − 0.834533i \(-0.685738\pi\)
−0.397673 + 0.917527i \(0.630182\pi\)
\(74\) 577.097 + 101.758i 0.906570 + 0.159853i
\(75\) −898.007 + 654.058i −1.38257 + 1.00699i
\(76\) −208.582 + 345.902i −0.314816 + 0.522075i
\(77\) 711.231 + 711.231i 1.05263 + 1.05263i
\(78\) 272.041 − 190.485i 0.394905 − 0.276515i
\(79\) −329.373 − 276.377i −0.469081 − 0.393606i 0.377378 − 0.926059i \(-0.376826\pi\)
−0.846459 + 0.532453i \(0.821270\pi\)
\(80\) 4.68826 + 12.5168i 0.00655204 + 0.0174928i
\(81\) 535.793 − 195.013i 0.734970 − 0.267507i
\(82\) 413.193 + 36.1497i 0.556458 + 0.0486838i
\(83\) −685.056 − 183.560i −0.905959 − 0.242751i −0.224385 − 0.974500i \(-0.572037\pi\)
−0.681574 + 0.731749i \(0.738704\pi\)
\(84\) −395.817 + 685.575i −0.514133 + 0.890505i
\(85\) −698.584 + 67.6814i −0.891436 + 0.0863656i
\(86\) −172.466 + 30.4105i −0.216250 + 0.0381307i
\(87\) −504.890 + 135.285i −0.622183 + 0.166713i
\(88\) 324.373 − 1210.58i 0.392936 − 1.46646i
\(89\) 442.353 − 371.178i 0.526846 − 0.442076i −0.340165 − 0.940366i \(-0.610483\pi\)
0.867011 + 0.498290i \(0.166038\pi\)
\(90\) −275.082 − 989.650i −0.322180 − 1.15909i
\(91\) 132.079 − 362.885i 0.152150 − 0.418029i
\(92\) −61.6074 − 704.176i −0.0698154 − 0.797993i
\(93\) 916.181 + 1308.44i 1.02154 + 1.45892i
\(94\) −1081.22 −1.18637
\(95\) −703.500 − 602.049i −0.759764 − 0.650199i
\(96\) 1599.19 1.70017
\(97\) 7.66803 + 10.9511i 0.00802650 + 0.0114630i 0.823145 − 0.567831i \(-0.192217\pi\)
−0.815119 + 0.579294i \(0.803328\pi\)
\(98\) 1.45780 + 16.6628i 0.00150266 + 0.0171755i
\(99\) −979.312 + 2690.64i −0.994188 + 2.73151i
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 95.4.r.a.2.19 ✓ 336
5.3 odd 4 inner 95.4.r.a.78.19 yes 336
19.10 odd 18 inner 95.4.r.a.67.19 yes 336
95.48 even 36 inner 95.4.r.a.48.19 yes 336
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.4.r.a.2.19 ✓ 336 1.1 even 1 trivial
95.4.r.a.48.19 yes 336 95.48 even 36 inner
95.4.r.a.67.19 yes 336 19.10 odd 18 inner
95.4.r.a.78.19 yes 336 5.3 odd 4 inner