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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 2.16
Character \(\chi\) \(=\) 95.2
Dual form 95.4.r.a.48.16

$q$-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+(0.679956 + 0.971077i) q^{2} +(0.0660396 + 0.754836i) q^{3} +(2.25551 - 6.19696i) q^{4} +(-4.41803 + 10.2704i) q^{5} +(-0.688100 + 0.577385i) q^{6} +(-6.89475 + 25.7315i) q^{7} +(16.7120 - 4.47795i) q^{8} +(26.0244 - 4.58880i) q^{9} +(-12.9774 + 2.69316i) q^{10} +(-23.4613 + 40.6361i) q^{11} +(4.82664 + 1.29330i) q^{12} +(88.4036 + 7.73431i) q^{13} +(-29.6754 + 10.8010i) q^{14} +(-8.04423 - 2.65664i) q^{15} +(-24.7027 - 20.7280i) q^{16} +(25.6938 - 17.9910i) q^{17} +(22.1515 + 22.1515i) q^{18} +(-72.3688 + 40.2711i) q^{19} +(53.6803 + 50.5434i) q^{20} +(-19.8784 - 3.50510i) q^{21} +(-55.4134 + 4.84804i) q^{22} +(-111.941 + 52.1988i) q^{23} +(4.48377 + 12.3191i) q^{24} +(-85.9620 - 90.7499i) q^{25} +(52.5999 + 91.1057i) q^{26} +(10.4775 + 39.1024i) q^{27} +(143.906 + 100.764i) q^{28} +(-17.6741 - 100.235i) q^{29} +(-2.88992 - 9.61796i) q^{30} +(188.519 - 108.841i) q^{31} +(15.3952 - 175.968i) q^{32} +(-32.2230 - 15.0258i) q^{33} +(34.9413 + 12.7176i) q^{34} +(-233.812 - 184.495i) q^{35} +(30.2616 - 171.622i) q^{36} +(148.400 - 148.400i) q^{37} +(-88.3139 - 42.8931i) q^{38} +67.2410i q^{39} +(-27.8436 + 191.422i) q^{40} +(-91.2202 + 108.712i) q^{41} +(-10.1127 - 21.6868i) q^{42} +(143.674 - 308.110i) q^{43} +(198.903 + 237.044i) q^{44} +(-67.8478 + 287.554i) q^{45} +(-126.804 - 73.2102i) q^{46} +(-35.5485 + 50.7685i) q^{47} +(14.0149 - 20.0153i) q^{48} +(-317.528 - 183.325i) q^{49} +(29.6748 - 145.182i) q^{50} +(15.2771 + 18.2065i) q^{51} +(247.324 - 530.389i) q^{52} +(15.1639 + 32.5192i) q^{53} +(-30.8473 + 36.7623i) q^{54} +(-313.696 - 420.488i) q^{55} +460.899i q^{56} +(-35.1773 - 51.9671i) q^{57} +(85.3182 - 85.3182i) q^{58} +(67.2735 - 381.527i) q^{59} +(-34.6069 + 43.8577i) q^{60} +(-177.990 - 64.7829i) q^{61} +(233.878 + 109.059i) q^{62} +(-61.3546 + 701.286i) q^{63} +(-42.0679 + 24.2879i) q^{64} +(-470.004 + 873.769i) q^{65} +(-7.31896 - 41.5079i) q^{66} +(205.008 + 143.548i) q^{67} +(-53.5369 - 199.803i) q^{68} +(-46.7940 - 81.0497i) q^{69} +(20.1768 - 352.497i) q^{70} +(153.526 + 421.809i) q^{71} +(414.370 - 193.224i) q^{72} +(497.191 - 43.4986i) q^{73} +(245.013 + 43.2024i) q^{74} +(62.8244 - 70.8803i) q^{75} +(86.3298 + 539.298i) q^{76} +(-883.870 - 883.870i) q^{77} +(-65.2962 + 45.7209i) q^{78} +(-163.977 - 137.593i) q^{79} +(322.022 - 162.129i) q^{80} +(641.645 - 233.540i) q^{81} +(-167.593 - 14.6625i) q^{82} +(695.531 + 186.367i) q^{83} +(-66.5570 + 115.280i) q^{84} +(71.2586 + 343.370i) q^{85} +(396.891 - 69.9826i) q^{86} +(74.4938 - 19.9605i) q^{87} +(-210.117 + 784.167i) q^{88} +(-908.655 + 762.452i) q^{89} +(-325.371 + 129.639i) q^{90} +(-808.536 + 2221.43i) q^{91} +(70.9907 + 811.427i) q^{92} +(94.6072 + 135.113i) q^{93} -73.4715 q^{94} +(-93.8721 - 921.175i) q^{95} +133.843 q^{96} +(-554.241 - 791.538i) q^{97} +(-37.8823 - 432.997i) q^{98} +(-424.094 + 1165.19i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.679956 + 0.971077i 0.240401 + 0.343328i 0.921163 − 0.389176i \(-0.127240\pi\)
−0.680763 + 0.732504i \(0.738351\pi\)
\(3\) 0.0660396 + 0.754836i 0.0127093 + 0.145268i 0.999898 − 0.0142896i \(-0.00454868\pi\)
−0.987189 + 0.159558i \(0.948993\pi\)
\(4\) 2.25551 − 6.19696i 0.281939 − 0.774620i
\(5\) −4.41803 + 10.2704i −0.395161 + 0.918612i
\(6\) −0.688100 + 0.577385i −0.0468193 + 0.0392860i
\(7\) −6.89475 + 25.7315i −0.372281 + 1.38937i 0.484995 + 0.874517i \(0.338821\pi\)
−0.857277 + 0.514856i \(0.827845\pi\)
\(8\) 16.7120 − 4.47795i 0.738571 − 0.197900i
\(9\) 26.0244 − 4.58880i 0.963866 − 0.169956i
\(10\) −12.9774 + 2.69316i −0.410382 + 0.0851652i
\(11\) −23.4613 + 40.6361i −0.643076 + 1.11384i 0.341666 + 0.939821i \(0.389009\pi\)
−0.984742 + 0.174019i \(0.944325\pi\)
\(12\) 4.82664 + 1.29330i 0.116111 + 0.0311119i
\(13\) 88.4036 + 7.73431i 1.88606 + 0.165009i 0.971120 − 0.238590i \(-0.0766851\pi\)
0.914936 + 0.403598i \(0.132241\pi\)
\(14\) −29.6754 + 10.8010i −0.566507 + 0.206192i
\(15\) −8.04423 − 2.65664i −0.138467 − 0.0457294i
\(16\) −24.7027 − 20.7280i −0.385979 − 0.323875i
\(17\) 25.6938 − 17.9910i 0.366569 − 0.256674i −0.375750 − 0.926721i \(-0.622615\pi\)
0.742319 + 0.670047i \(0.233726\pi\)
\(18\) 22.1515 + 22.1515i 0.290065 + 0.290065i
\(19\) −72.3688 + 40.2711i −0.873818 + 0.486254i
\(20\) 53.6803 + 50.5434i 0.600164 + 0.565092i
\(21\) −19.8784 − 3.50510i −0.206563 − 0.0364227i
\(22\) −55.4134 + 4.84804i −0.537008 + 0.0469821i
\(23\) −111.941 + 52.1988i −1.01484 + 0.473226i −0.857590 − 0.514333i \(-0.828039\pi\)
−0.157246 + 0.987559i \(0.550262\pi\)
\(24\) 4.48377 + 12.3191i 0.0381353 + 0.104776i
\(25\) −85.9620 − 90.7499i −0.687696 − 0.725999i
\(26\) 52.5999 + 91.1057i 0.396757 + 0.687204i
\(27\) 10.4775 + 39.1024i 0.0746811 + 0.278714i
\(28\) 143.906 + 100.764i 0.971276 + 0.680095i
\(29\) −17.6741 − 100.235i −0.113172 − 0.641833i −0.987639 − 0.156747i \(-0.949899\pi\)
0.874466 − 0.485086i \(-0.161212\pi\)
\(30\) −2.88992 − 9.61796i −0.0175875 − 0.0585331i
\(31\) 188.519 − 108.841i 1.09223 − 0.630597i 0.158058 − 0.987430i \(-0.449477\pi\)
0.934168 + 0.356833i \(0.116143\pi\)
\(32\) 15.3952 − 175.968i 0.0850471 − 0.972092i
\(33\) −32.2230 − 15.0258i −0.169979 − 0.0792624i
\(34\) 34.9413 + 12.7176i 0.176247 + 0.0641485i
\(35\) −233.812 − 184.495i −1.12918 − 0.891008i
\(36\) 30.2616 − 171.622i 0.140100 − 0.794548i
\(37\) 148.400 − 148.400i 0.659372 − 0.659372i −0.295859 − 0.955232i \(-0.595606\pi\)
0.955232 + 0.295859i \(0.0956059\pi\)
\(38\) −88.3139 − 42.8931i −0.377011 − 0.183110i
\(39\) 67.2410i 0.276081i
\(40\) −27.8436 + 191.422i −0.110062 + 0.756662i
\(41\) −91.2202 + 108.712i −0.347468 + 0.414097i −0.911267 − 0.411815i \(-0.864895\pi\)
0.563799 + 0.825912i \(0.309339\pi\)
\(42\) −10.1127 − 21.6868i −0.0371530 − 0.0796749i
\(43\) 143.674 − 308.110i 0.509537 − 1.09271i −0.468344 − 0.883546i \(-0.655149\pi\)
0.977881 − 0.209160i \(-0.0670730\pi\)
\(44\) 198.903 + 237.044i 0.681495 + 0.812174i
\(45\) −67.8478 + 287.554i −0.224759 + 0.952579i
\(46\) −126.804 − 73.2102i −0.406439 − 0.234658i
\(47\) −35.5485 + 50.7685i −0.110325 + 0.157561i −0.870518 − 0.492137i \(-0.836216\pi\)
0.760192 + 0.649698i \(0.225105\pi\)
\(48\) 14.0149 − 20.0153i 0.0421432 − 0.0601867i
\(49\) −317.528 − 183.325i −0.925738 − 0.534475i
\(50\) 29.6748 − 145.182i 0.0839330 − 0.410636i
\(51\) 15.2771 + 18.2065i 0.0419454 + 0.0499886i
\(52\) 247.324 − 530.389i 0.659571 − 1.41446i
\(53\) 15.1639 + 32.5192i 0.0393005 + 0.0842802i 0.924978 − 0.380021i \(-0.124083\pi\)
−0.885677 + 0.464301i \(0.846305\pi\)
\(54\) −30.8473 + 36.7623i −0.0777367 + 0.0926430i
\(55\) −313.696 − 420.488i −0.769068 − 1.03088i
\(56\) 460.899i 1.09982i
\(57\) −35.1773 − 51.9671i −0.0817429 − 0.120758i
\(58\) 85.3182 − 85.3182i 0.193152 − 0.193152i
\(59\) 67.2735 − 381.527i 0.148445 − 0.841875i −0.816091 − 0.577924i \(-0.803863\pi\)
0.964536 − 0.263951i \(-0.0850258\pi\)
\(60\) −34.6069 + 43.8577i −0.0744623 + 0.0943668i
\(61\) −177.990 − 64.7829i −0.373594 − 0.135977i 0.148398 − 0.988928i \(-0.452588\pi\)
−0.521991 + 0.852951i \(0.674811\pi\)
\(62\) 233.878 + 109.059i 0.479073 + 0.223395i
\(63\) −61.3546 + 701.286i −0.122698 + 1.40244i
\(64\) −42.0679 + 24.2879i −0.0821638 + 0.0474373i
\(65\) −470.004 + 873.769i −0.896875 + 1.66735i
\(66\) −7.31896 − 41.5079i −0.0136500 − 0.0774131i
\(67\) 205.008 + 143.548i 0.373818 + 0.261750i 0.745355 − 0.666668i \(-0.232280\pi\)
−0.371537 + 0.928418i \(0.621169\pi\)
\(68\) −53.5369 − 199.803i −0.0954751 − 0.356318i
\(69\) −46.7940 − 81.0497i −0.0816426 − 0.141409i
\(70\) 20.1768 − 352.497i 0.0344512 − 0.601879i
\(71\) 153.526 + 421.809i 0.256622 + 0.705064i 0.999370 + 0.0354927i \(0.0113001\pi\)
−0.742748 + 0.669572i \(0.766478\pi\)
\(72\) 414.370 − 193.224i 0.678250 − 0.316273i
\(73\) 497.191 − 43.4986i 0.797148 − 0.0697414i 0.318698 − 0.947856i \(-0.396754\pi\)
0.478450 + 0.878115i \(0.341199\pi\)
\(74\) 245.013 + 43.2024i 0.384894 + 0.0678673i
\(75\) 62.8244 − 70.8803i 0.0967245 − 0.109127i
\(76\) 86.3298 + 539.298i 0.130299 + 0.813971i
\(77\) −883.870 − 883.870i −1.30813 − 1.30813i
\(78\) −65.2962 + 45.7209i −0.0947864 + 0.0663701i
\(79\) −163.977 − 137.593i −0.233530 − 0.195955i 0.518512 − 0.855071i \(-0.326486\pi\)
−0.752041 + 0.659116i \(0.770931\pi\)
\(80\) 322.022 − 162.129i 0.450039 − 0.226582i
\(81\) 641.645 − 233.540i 0.880172 − 0.320356i
\(82\) −167.593 − 14.6625i −0.225702 − 0.0197464i
\(83\) 695.531 + 186.367i 0.919812 + 0.246463i 0.687505 − 0.726180i \(-0.258706\pi\)
0.232307 + 0.972642i \(0.425373\pi\)
\(84\) −66.5570 + 115.280i −0.0864519 + 0.149739i
\(85\) 71.2586 + 343.370i 0.0909303 + 0.438162i
\(86\) 396.891 − 69.9826i 0.497649 − 0.0877490i
\(87\) 74.4938 − 19.9605i 0.0917996 − 0.0245976i
\(88\) −210.117 + 784.167i −0.254529 + 0.949914i
\(89\) −908.655 + 762.452i −1.08222 + 0.908087i −0.996103 − 0.0881982i \(-0.971889\pi\)
−0.0861132 + 0.996285i \(0.527445\pi\)
\(90\) −325.371 + 129.639i −0.381079 + 0.151835i
\(91\) −808.536 + 2221.43i −0.931402 + 2.55901i
\(92\) 70.9907 + 811.427i 0.0804488 + 0.919534i
\(93\) 94.6072 + 135.113i 0.105487 + 0.150651i
\(94\) −73.4715 −0.0806171
\(95\) −93.8721 − 921.175i −0.101380 − 0.994848i
\(96\) 133.843 0.142295
\(97\) −554.241 − 791.538i −0.580151 − 0.828541i 0.416600 − 0.909090i \(-0.363222\pi\)
−0.996751 + 0.0805488i \(0.974333\pi\)
\(98\) −37.8823 − 432.997i −0.0390479 − 0.446320i
\(99\) −424.094 + 1165.19i −0.430536 + 1.18289i
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.2.16 ✓ 336
5.3 odd 4 inner 95.4.r.a.78.16 yes 336
19.10 odd 18 inner 95.4.r.a.67.16 yes 336
95.48 even 36 inner 95.4.r.a.48.16 yes 336
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.4.r.a.2.16 ✓ 336 1.1 even 1 trivial
95.4.r.a.48.16 yes 336 95.48 even 36 inner
95.4.r.a.67.16 yes 336 19.10 odd 18 inner
95.4.r.a.78.16 yes 336 5.3 odd 4 inner