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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [115,2,Mod(4,115)] 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("115.4"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(115, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([11, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 115 = 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 115.j (of order \(22\), degree \(10\), 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: \(0.918279623245\)
Analytic rank: \(0\)
Dimension: \(100\)
Relative dimension: \(10\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label 29.7
Character \(\chi\) \(=\) 115.29
Dual form 115.2.j.a.4.7

$q$-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+(0.477984 + 0.218288i) q^{2} +(0.663307 - 2.25902i) q^{3} +(-1.12890 - 1.30282i) q^{4} +(-2.15745 + 0.587726i) q^{5} +(0.810167 - 0.934983i) q^{6} +(3.03882 + 0.436916i) q^{7} +(-0.551291 - 1.87752i) q^{8} +(-2.13942 - 1.37492i) q^{9} +(-1.15952 - 0.190022i) q^{10} +(0.639358 + 1.40000i) q^{11} +(-3.69191 + 1.68604i) q^{12} +(2.46613 - 0.354576i) q^{13} +(1.35713 + 0.872177i) q^{14} +(-0.103368 + 5.26355i) q^{15} +(-0.344335 + 2.39490i) q^{16} +(-0.765539 - 0.663343i) q^{17} +(-0.722480 - 1.12420i) q^{18} +(4.34173 + 5.01062i) q^{19} +(3.20125 + 2.14729i) q^{20} +(3.00267 - 6.57493i) q^{21} +0.808742i q^{22} +(-4.70182 + 0.944904i) q^{23} -4.60703 q^{24} +(4.30916 - 2.53597i) q^{25} +(1.25617 + 0.368845i) q^{26} +(0.812918 - 0.704397i) q^{27} +(-2.86130 - 4.45227i) q^{28} +(1.91799 - 2.21348i) q^{29} +(-1.19838 + 2.49333i) q^{30} +(-8.97455 + 2.63517i) q^{31} +(-2.80320 + 4.36187i) q^{32} +(3.58671 - 0.515691i) q^{33} +(-0.221116 - 0.484176i) q^{34} +(-6.81287 + 0.843367i) q^{35} +(0.623916 + 4.33943i) q^{36} +(-4.50401 + 7.00838i) q^{37} +(0.981520 + 3.34275i) q^{38} +(0.834809 - 5.80622i) q^{39} +(2.29285 + 3.72665i) q^{40} +(-2.93712 + 1.88758i) q^{41} +(2.87046 - 2.48727i) q^{42} +(3.05193 - 10.3939i) q^{43} +(1.10218 - 2.41343i) q^{44} +(5.42376 + 1.70893i) q^{45} +(-2.45366 - 0.574703i) q^{46} -6.01433i q^{47} +(5.18172 + 2.36641i) q^{48} +(2.32706 + 0.683286i) q^{49} +(2.61328 - 0.271518i) q^{50} +(-2.00629 + 1.28937i) q^{51} +(-3.24597 - 2.81265i) q^{52} +(-0.472909 - 0.0679941i) q^{53} +(0.542324 - 0.159241i) q^{54} +(-2.20220 - 2.64466i) q^{55} +(-0.854952 - 5.94632i) q^{56} +(14.1990 - 6.48446i) q^{57} +(1.39994 - 0.639333i) q^{58} +(1.15651 + 8.04367i) q^{59} +(6.97417 - 5.80736i) q^{60} +(2.28378 - 0.670578i) q^{61} +(-4.86492 - 0.699470i) q^{62} +(-5.90058 - 5.11288i) q^{63} +(1.77884 - 1.14319i) q^{64} +(-5.11216 + 2.21439i) q^{65} +(1.82696 + 0.536444i) q^{66} +(4.81628 + 2.19952i) q^{67} +1.74621i q^{68} +(-0.984201 + 11.2483i) q^{69} +(-3.44054 - 1.08405i) q^{70} +(-3.69107 + 8.08231i) q^{71} +(-1.40201 + 4.77479i) q^{72} +(3.01277 - 2.61058i) q^{73} +(-3.68269 + 2.36672i) q^{74} +(-2.87051 - 11.4166i) q^{75} +(1.62657 - 11.3130i) q^{76} +(1.33121 + 4.53369i) q^{77} +(1.66646 - 2.59306i) q^{78} +(-1.08101 - 7.51856i) q^{79} +(-0.664661 - 5.36925i) q^{80} +(-4.22140 - 9.24357i) q^{81} +(-1.81594 + 0.261092i) q^{82} +(-8.17334 + 12.7180i) q^{83} +(-11.9557 + 3.51050i) q^{84} +(2.04147 + 0.981202i) q^{85} +(3.72765 - 4.30193i) q^{86} +(-3.72806 - 5.80098i) q^{87} +(2.27606 - 1.97222i) q^{88} +(7.12317 + 2.09155i) q^{89} +(2.21943 + 2.00078i) q^{90} +7.64904 q^{91} +(6.53894 + 5.05894i) q^{92} +22.0216i q^{93} +(1.31286 - 2.87476i) q^{94} +(-12.3119 - 8.25841i) q^{95} +(7.99415 + 9.22575i) q^{96} +(2.20825 + 3.43610i) q^{97} +(0.963144 + 0.834569i) q^{98} +(0.557033 - 3.87425i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 100 q - 14 q^{4} - 9 q^{5} - 18 q^{6} - 12 q^{9} - 13 q^{10} - 26 q^{11} - 26 q^{14} - 10 q^{15} - 18 q^{16} - 14 q^{19} + 49 q^{20} - 22 q^{21} - 68 q^{24} + 21 q^{25} - 42 q^{26} - 24 q^{29} + 19 q^{30}+ \cdots - 60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/115\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(-1\) \(e\left(\frac{9}{11}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.477984 + 0.218288i 0.337986 + 0.154353i 0.577176 − 0.816620i \(-0.304155\pi\)
−0.239190 + 0.970973i \(0.576882\pi\)
\(3\) 0.663307 − 2.25902i 0.382961 − 1.30424i −0.512341 − 0.858782i \(-0.671221\pi\)
0.895301 − 0.445461i \(-0.146960\pi\)
\(4\) −1.12890 − 1.30282i −0.564451 − 0.651411i
\(5\) −2.15745 + 0.587726i −0.964840 + 0.262839i
\(6\) 0.810167 − 0.934983i 0.330749 − 0.381705i
\(7\) 3.03882 + 0.436916i 1.14856 + 0.165139i 0.690195 − 0.723623i \(-0.257525\pi\)
0.458369 + 0.888762i \(0.348434\pi\)
\(8\) −0.551291 − 1.87752i −0.194911 − 0.663805i
\(9\) −2.13942 − 1.37492i −0.713139 − 0.458307i
\(10\) −1.15952 − 0.190022i −0.366672 − 0.0600901i
\(11\) 0.639358 + 1.40000i 0.192774 + 0.422116i 0.981195 − 0.193020i \(-0.0618284\pi\)
−0.788421 + 0.615136i \(0.789101\pi\)
\(12\) −3.69191 + 1.68604i −1.06576 + 0.486717i
\(13\) 2.46613 − 0.354576i 0.683982 − 0.0983417i 0.208440 − 0.978035i \(-0.433161\pi\)
0.475541 + 0.879693i \(0.342252\pi\)
\(14\) 1.35713 + 0.872177i 0.362709 + 0.233099i
\(15\) −0.103368 + 5.26355i −0.0266896 + 1.35904i
\(16\) −0.344335 + 2.39490i −0.0860837 + 0.598725i
\(17\) −0.765539 − 0.663343i −0.185671 − 0.160884i 0.557066 − 0.830468i \(-0.311927\pi\)
−0.742737 + 0.669584i \(0.766472\pi\)
\(18\) −0.722480 − 1.12420i −0.170290 − 0.264977i
\(19\) 4.34173 + 5.01062i 0.996061 + 1.14952i 0.988755 + 0.149543i \(0.0477802\pi\)
0.00730613 + 0.999973i \(0.497674\pi\)
\(20\) 3.20125 + 2.14729i 0.715821 + 0.480148i
\(21\) 3.00267 − 6.57493i 0.655236 − 1.43477i
\(22\) 0.808742i 0.172424i
\(23\) −4.70182 + 0.944904i −0.980398 + 0.197026i
\(24\) −4.60703 −0.940407
\(25\) 4.30916 − 2.53597i 0.861831 − 0.507195i
\(26\) 1.25617 + 0.368845i 0.246356 + 0.0723365i
\(27\) 0.812918 − 0.704397i 0.156446 − 0.135561i
\(28\) −2.86130 − 4.45227i −0.540735 − 0.841401i
\(29\) 1.91799 − 2.21348i 0.356161 − 0.411032i −0.549188 − 0.835699i \(-0.685063\pi\)
0.905350 + 0.424666i \(0.139609\pi\)
\(30\) −1.19838 + 2.49333i −0.218793 + 0.455218i
\(31\) −8.97455 + 2.63517i −1.61188 + 0.473290i −0.958819 − 0.284018i \(-0.908332\pi\)
−0.653058 + 0.757308i \(0.726514\pi\)
\(32\) −2.80320 + 4.36187i −0.495541 + 0.771077i
\(33\) 3.58671 − 0.515691i 0.624366 − 0.0897703i
\(34\) −0.221116 − 0.484176i −0.0379210 − 0.0830355i
\(35\) −6.81287 + 0.843367i −1.15159 + 0.142555i
\(36\) 0.623916 + 4.33943i 0.103986 + 0.723239i
\(37\) −4.50401 + 7.00838i −0.740455 + 1.15217i 0.242825 + 0.970070i \(0.421926\pi\)
−0.983280 + 0.182100i \(0.941711\pi\)
\(38\) 0.981520 + 3.34275i 0.159224 + 0.542266i
\(39\) 0.834809 − 5.80622i 0.133676 − 0.929740i
\(40\) 2.29285 + 3.72665i 0.362531 + 0.589235i
\(41\) −2.93712 + 1.88758i −0.458702 + 0.294790i −0.749512 − 0.661991i \(-0.769712\pi\)
0.290810 + 0.956781i \(0.406075\pi\)
\(42\) 2.87046 − 2.48727i 0.442921 − 0.383793i
\(43\) 3.05193 − 10.3939i 0.465415 − 1.58506i −0.308147 − 0.951339i \(-0.599709\pi\)
0.773562 − 0.633720i \(-0.218473\pi\)
\(44\) 1.10218 − 2.41343i 0.166159 − 0.363839i
\(45\) 5.42376 + 1.70893i 0.808526 + 0.254752i
\(46\) −2.45366 − 0.574703i −0.361772 − 0.0847354i
\(47\) − 6.01433i − 0.877281i −0.898663 − 0.438640i \(-0.855460\pi\)
0.898663 − 0.438640i \(-0.144540\pi\)
\(48\) 5.18172 + 2.36641i 0.747917 + 0.341562i
\(49\) 2.32706 + 0.683286i 0.332437 + 0.0976123i
\(50\) 2.61328 − 0.271518i 0.369574 − 0.0383984i
\(51\) −2.00629 + 1.28937i −0.280937 + 0.180547i
\(52\) −3.24597 − 2.81265i −0.450135 − 0.390044i
\(53\) −0.472909 − 0.0679941i −0.0649591 − 0.00933970i 0.109759 − 0.993958i \(-0.464992\pi\)
−0.174718 + 0.984619i \(0.555901\pi\)
\(54\) 0.542324 − 0.159241i 0.0738009 − 0.0216699i
\(55\) −2.20220 − 2.64466i −0.296944 − 0.356605i
\(56\) −0.854952 − 5.94632i −0.114248 − 0.794610i
\(57\) 14.1990 − 6.48446i 1.88070 − 0.858888i
\(58\) 1.39994 − 0.639333i 0.183822 − 0.0839486i
\(59\) 1.15651 + 8.04367i 0.150564 + 1.04720i 0.915277 + 0.402825i \(0.131972\pi\)
−0.764713 + 0.644371i \(0.777119\pi\)
\(60\) 6.97417 − 5.80736i 0.900361 − 0.749727i
\(61\) 2.28378 − 0.670578i 0.292408 − 0.0858587i −0.132239 − 0.991218i \(-0.542217\pi\)
0.424647 + 0.905359i \(0.360398\pi\)
\(62\) −4.86492 − 0.699470i −0.617846 − 0.0888328i
\(63\) −5.90058 − 5.11288i −0.743403 − 0.644162i
\(64\) 1.77884 − 1.14319i 0.222355 − 0.142899i
\(65\) −5.11216 + 2.21439i −0.634085 + 0.274661i
\(66\) 1.82696 + 0.536444i 0.224883 + 0.0660317i
\(67\) 4.81628 + 2.19952i 0.588402 + 0.268714i 0.687288 − 0.726385i \(-0.258801\pi\)
−0.0988867 + 0.995099i \(0.531528\pi\)
\(68\) 1.74621i 0.211759i
\(69\) −0.984201 + 11.2483i −0.118484 + 1.35413i
\(70\) −3.44054 − 1.08405i −0.411224 − 0.129569i
\(71\) −3.69107 + 8.08231i −0.438049 + 0.959194i 0.553903 + 0.832581i \(0.313138\pi\)
−0.991952 + 0.126613i \(0.959589\pi\)
\(72\) −1.40201 + 4.77479i −0.165228 + 0.562715i
\(73\) 3.01277 − 2.61058i 0.352619 − 0.305546i −0.460482 − 0.887669i \(-0.652323\pi\)
0.813101 + 0.582123i \(0.197778\pi\)
\(74\) −3.68269 + 2.36672i −0.428104 + 0.275126i
\(75\) −2.87051 − 11.4166i −0.331458 − 1.31827i
\(76\) 1.62657 − 11.3130i 0.186580 − 1.29769i
\(77\) 1.33121 + 4.53369i 0.151705 + 0.516661i
\(78\) 1.66646 − 2.59306i 0.188689 − 0.293606i
\(79\) −1.08101 − 7.51856i −0.121623 − 0.845904i −0.955718 − 0.294284i \(-0.904919\pi\)
0.834096 − 0.551620i \(-0.185990\pi\)
\(80\) −0.664661 − 5.36925i −0.0743113 − 0.600300i
\(81\) −4.22140 − 9.24357i −0.469044 − 1.02706i
\(82\) −1.81594 + 0.261092i −0.200536 + 0.0288328i
\(83\) −8.17334 + 12.7180i −0.897140 + 1.39598i 0.0210242 + 0.999779i \(0.493307\pi\)
−0.918165 + 0.396199i \(0.870329\pi\)
\(84\) −11.9557 + 3.51050i −1.30447 + 0.383027i
\(85\) 2.04147 + 0.981202i 0.221429 + 0.106426i
\(86\) 3.72765 − 4.30193i 0.401963 − 0.463890i
\(87\) −3.72806 − 5.80098i −0.399690 − 0.621931i
\(88\) 2.27606 − 1.97222i 0.242629 − 0.210239i
\(89\) 7.12317 + 2.09155i 0.755055 + 0.221704i 0.636535 − 0.771248i \(-0.280367\pi\)
0.118519 + 0.992952i \(0.462185\pi\)
\(90\) 2.21943 + 2.00078i 0.233949 + 0.210901i
\(91\) 7.64904 0.801837
\(92\) 6.53894 + 5.05894i 0.681732 + 0.527431i
\(93\) 22.0216i 2.28353i
\(94\) 1.31286 − 2.87476i 0.135411 − 0.296509i
\(95\) −12.3119 − 8.25841i −1.26318 − 0.847295i
\(96\) 7.99415 + 9.22575i 0.815900 + 0.941599i
\(97\) 2.20825 + 3.43610i 0.224214 + 0.348883i 0.935076 − 0.354448i \(-0.115331\pi\)
−0.710862 + 0.703332i \(0.751695\pi\)
\(98\) 0.963144 + 0.834569i 0.0972923 + 0.0843042i
\(99\) 0.557033 − 3.87425i 0.0559839 − 0.389377i
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 115.2.j.a.29.7 yes 100
5.2 odd 4 575.2.k.g.351.4 100
5.3 odd 4 575.2.k.g.351.7 100
5.4 even 2 inner 115.2.j.a.29.4 yes 100
23.4 even 11 inner 115.2.j.a.4.4 ✓ 100
115.4 even 22 inner 115.2.j.a.4.7 yes 100
115.27 odd 44 575.2.k.g.326.4 100
115.73 odd 44 575.2.k.g.326.7 100
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.2.j.a.4.4 ✓ 100 23.4 even 11 inner
115.2.j.a.4.7 yes 100 115.4 even 22 inner
115.2.j.a.29.4 yes 100 5.4 even 2 inner
115.2.j.a.29.7 yes 100 1.1 even 1 trivial
575.2.k.g.326.4 100 115.27 odd 44
575.2.k.g.326.7 100 115.73 odd 44
575.2.k.g.351.4 100 5.2 odd 4
575.2.k.g.351.7 100 5.3 odd 4