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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [121,4,Mod(4,121)] 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("121.4"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(121, base_ring=CyclotomicField(110)) chi = DirichletCharacter(H, H._module([2])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 121 = 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 121.g (of order \(55\), degree \(40\), 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: \(7.13923111069\)
Analytic rank: \(0\)
Dimension: \(1280\)
Relative dimension: \(32\) over \(\Q(\zeta_{55})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{55}]$

Embedding invariants

Embedding label 91.2
Character \(\chi\) \(=\) 121.91
Dual form 121.4.g.a.4.2

$q$-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+(-4.96162 + 0.283716i) q^{2} +(-0.441007 - 1.35728i) q^{3} +(16.5893 - 1.90345i) q^{4} +(-1.29295 + 2.45042i) q^{5} +(2.57319 + 6.60918i) q^{6} +(-18.6947 + 7.90044i) q^{7} +(-42.5943 + 7.37123i) q^{8} +(20.1957 - 14.6731i) q^{9} +(5.71992 - 12.5249i) q^{10} +(15.4856 - 33.0333i) q^{11} +(-9.89950 - 21.6769i) q^{12} +(0.348004 + 0.196527i) q^{13} +(90.5144 - 44.5029i) q^{14} +(3.89610 + 0.674247i) q^{15} +(79.1316 - 18.4013i) q^{16} +(-52.3308 - 18.6716i) q^{17} +(-96.0406 + 78.5320i) q^{18} +(3.82237 + 133.800i) q^{19} +(-16.7850 + 43.1118i) q^{20} +(18.9676 + 21.8897i) q^{21} +(-67.4615 + 168.292i) q^{22} +(-56.4848 + 65.1870i) q^{23} +(28.7892 + 54.5615i) q^{24} +(66.2226 + 96.8474i) q^{25} +(-1.78242 - 0.876358i) q^{26} +(-59.9953 - 43.5891i) q^{27} +(-295.094 + 166.647i) q^{28} +(-52.6510 + 76.9996i) q^{29} +(-19.5223 - 2.23997i) q^{30} +(39.9511 + 197.167i) q^{31} +(-55.5892 + 16.3225i) q^{32} +(-51.6646 - 6.45036i) q^{33} +(264.943 + 77.7942i) q^{34} +(4.81196 - 56.0247i) q^{35} +(307.104 - 281.857i) q^{36} +(-38.3693 - 35.2151i) q^{37} +(-56.9264 - 662.782i) q^{38} +(0.113270 - 0.559009i) q^{39} +(37.0098 - 113.904i) q^{40} +(-30.7556 + 117.007i) q^{41} +(-100.320 - 103.227i) q^{42} +(-70.2463 + 488.574i) q^{43} +(194.018 - 577.475i) q^{44} +(9.84300 + 68.4596i) q^{45} +(261.762 - 339.459i) q^{46} +(132.319 + 108.197i) q^{47} +(-59.8732 - 99.2886i) q^{48} +(48.0246 - 49.4162i) q^{49} +(-356.048 - 461.732i) q^{50} +(-2.26433 + 79.2618i) q^{51} +(6.14723 + 2.59784i) q^{52} +(-465.314 - 108.204i) q^{53} +(310.041 + 199.251i) q^{54} +(60.9232 + 80.6566i) q^{55} +(738.051 - 474.316i) q^{56} +(179.919 - 64.1949i) q^{57} +(239.388 - 396.981i) q^{58} +(-94.5495 - 359.704i) q^{59} +(65.9170 + 3.76927i) q^{60} +(208.652 + 11.9311i) q^{61} +(-254.162 - 966.931i) q^{62} +(-261.629 + 433.863i) q^{63} +(-340.965 + 121.656i) q^{64} +(-0.931527 + 0.598656i) q^{65} +(258.170 + 17.3462i) q^{66} +(595.768 + 382.877i) q^{67} +(-903.672 - 210.140i) q^{68} +(113.387 + 47.9178i) q^{69} +(-7.98005 + 279.338i) q^{70} +(230.687 + 299.160i) q^{71} +(-752.064 + 773.856i) q^{72} +(-368.952 - 611.838i) q^{73} +(200.365 + 163.838i) q^{74} +(102.244 - 132.593i) q^{75} +(318.092 + 2212.38i) q^{76} +(-28.5207 + 739.889i) q^{77} +(-0.403402 + 2.80572i) q^{78} +(-781.325 - 803.965i) q^{79} +(-57.2227 + 217.698i) q^{80} +(175.576 - 540.368i) q^{81} +(119.401 - 589.268i) q^{82} +(71.6707 + 834.447i) q^{83} +(356.325 + 327.032i) q^{84} +(113.414 - 104.091i) q^{85} +(209.919 - 2444.05i) q^{86} +(127.729 + 37.5047i) q^{87} +(-416.101 + 1521.18i) q^{88} +(352.499 - 103.503i) q^{89} +(-68.2603 - 336.878i) q^{90} +(-8.05848 - 0.924625i) q^{91} +(-812.964 + 1188.92i) q^{92} +(249.991 - 141.177i) q^{93} +(-687.212 - 499.289i) q^{94} +(-332.809 - 163.631i) q^{95} +(46.6693 + 68.2517i) q^{96} +(188.879 + 357.966i) q^{97} +(-224.260 + 258.810i) q^{98} +(-171.956 - 894.352i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 1280 q - 37 q^{2} - 32 q^{3} + 73 q^{4} - 33 q^{5} + 73 q^{6} - 9 q^{7} - 91 q^{8} - 2748 q^{9} + 48 q^{10} + 43 q^{11} + 323 q^{12} - 287 q^{13} - 120 q^{14} + 632 q^{15} + 785 q^{16} - 13 q^{17} + 443 q^{18}+ \cdots + 13103 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/121\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{54}{55}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.96162 + 0.283716i −1.75420 + 0.100309i −0.904369 − 0.426752i \(-0.859658\pi\)
−0.849828 + 0.527060i \(0.823294\pi\)
\(3\) −0.441007 − 1.35728i −0.0848717 − 0.261208i 0.899610 − 0.436694i \(-0.143851\pi\)
−0.984482 + 0.175485i \(0.943851\pi\)
\(4\) 16.5893 − 1.90345i 2.07366 − 0.237931i
\(5\) −1.29295 + 2.45042i −0.115645 + 0.219172i −0.935440 − 0.353485i \(-0.884996\pi\)
0.819795 + 0.572657i \(0.194087\pi\)
\(6\) 2.57319 + 6.60918i 0.175083 + 0.449698i
\(7\) −18.6947 + 7.90044i −1.00942 + 0.426584i −0.830343 − 0.557252i \(-0.811856\pi\)
−0.179074 + 0.983836i \(0.557310\pi\)
\(8\) −42.5943 + 7.37123i −1.88242 + 0.325765i
\(9\) 20.1957 − 14.6731i 0.747990 − 0.543447i
\(10\) 5.71992 − 12.5249i 0.180880 − 0.396071i
\(11\) 15.4856 − 33.0333i 0.424462 − 0.905446i
\(12\) −9.89950 − 21.6769i −0.238145 − 0.521465i
\(13\) 0.348004 + 0.196527i 0.00742454 + 0.00419283i 0.495441 − 0.868642i \(-0.335007\pi\)
−0.488016 + 0.872835i \(0.662279\pi\)
\(14\) 90.5144 − 44.5029i 1.72793 − 0.849565i
\(15\) 3.89610 + 0.674247i 0.0670646 + 0.0116060i
\(16\) 79.1316 − 18.4013i 1.23643 − 0.287520i
\(17\) −52.3308 − 18.6716i −0.746593 − 0.266384i −0.0647693 − 0.997900i \(-0.520631\pi\)
−0.681824 + 0.731516i \(0.738813\pi\)
\(18\) −96.0406 + 78.5320i −1.25761 + 1.02834i
\(19\) 3.82237 + 133.800i 0.0461532 + 1.61557i 0.611859 + 0.790967i \(0.290422\pi\)
−0.565706 + 0.824607i \(0.691396\pi\)
\(20\) −16.7850 + 43.1118i −0.187661 + 0.482005i
\(21\) 18.9676 + 21.8897i 0.197098 + 0.227464i
\(22\) −67.4615 + 168.292i −0.653765 + 1.63091i
\(23\) −56.4848 + 65.1870i −0.512083 + 0.590975i −0.951631 − 0.307244i \(-0.900593\pi\)
0.439548 + 0.898219i \(0.355139\pi\)
\(24\) 28.7892 + 54.5615i 0.244857 + 0.464055i
\(25\) 66.2226 + 96.8474i 0.529781 + 0.774779i
\(26\) −1.78242 − 0.876358i −0.0134447 − 0.00661031i
\(27\) −59.9953 − 43.5891i −0.427633 − 0.310694i
\(28\) −295.094 + 166.647i −1.99170 + 1.12476i
\(29\) −52.6510 + 76.9996i −0.337139 + 0.493051i −0.956370 − 0.292159i \(-0.905627\pi\)
0.619230 + 0.785209i \(0.287445\pi\)
\(30\) −19.5223 − 2.23997i −0.118809 − 0.0136320i
\(31\) 39.9511 + 197.167i 0.231466 + 1.14233i 0.912183 + 0.409782i \(0.134395\pi\)
−0.680718 + 0.732546i \(0.738332\pi\)
\(32\) −55.5892 + 16.3225i −0.307090 + 0.0901697i
\(33\) −51.6646 − 6.45036i −0.272535 − 0.0340262i
\(34\) 264.943 + 77.7942i 1.33639 + 0.392400i
\(35\) 4.81196 − 56.0247i 0.0232392 − 0.270569i
\(36\) 307.104 − 281.857i 1.42178 − 1.30490i
\(37\) −38.3693 − 35.2151i −0.170483 − 0.156468i 0.586392 − 0.810027i \(-0.300548\pi\)
−0.756875 + 0.653559i \(0.773275\pi\)
\(38\) −56.9264 − 662.782i −0.243018 − 2.82941i
\(39\) 0.113270 − 0.559009i 0.000465069 − 0.00229521i
\(40\) 37.0098 − 113.904i 0.146294 − 0.450247i
\(41\) −30.7556 + 117.007i −0.117152 + 0.445692i −0.999699 − 0.0245321i \(-0.992190\pi\)
0.882547 + 0.470224i \(0.155827\pi\)
\(42\) −100.320 − 103.227i −0.368566 − 0.379245i
\(43\) −70.2463 + 488.574i −0.249127 + 1.73272i 0.354153 + 0.935187i \(0.384769\pi\)
−0.603280 + 0.797529i \(0.706140\pi\)
\(44\) 194.018 − 577.475i 0.664757 − 1.97858i
\(45\) 9.84300 + 68.4596i 0.0326069 + 0.226786i
\(46\) 261.762 − 339.459i 0.839014 − 1.08805i
\(47\) 132.319 + 108.197i 0.410653 + 0.335789i 0.815667 − 0.578521i \(-0.196370\pi\)
−0.405015 + 0.914310i \(0.632734\pi\)
\(48\) −59.8732 − 99.2886i −0.180041 − 0.298564i
\(49\) 48.0246 − 49.4162i 0.140014 − 0.144070i
\(50\) −356.048 − 461.732i −1.00706 − 1.30597i
\(51\) −2.26433 + 79.2618i −0.00621704 + 0.217625i
\(52\) 6.14723 + 2.59784i 0.0163936 + 0.00692800i
\(53\) −465.314 − 108.204i −1.20596 − 0.280434i −0.425103 − 0.905145i \(-0.639762\pi\)
−0.780856 + 0.624711i \(0.785217\pi\)
\(54\) 310.041 + 199.251i 0.781318 + 0.502123i
\(55\) 60.9232 + 80.6566i 0.149361 + 0.197741i
\(56\) 738.051 − 474.316i 1.76118 − 1.13184i
\(57\) 179.919 − 64.1949i 0.418084 − 0.149172i
\(58\) 239.388 − 396.981i 0.541952 − 0.898726i
\(59\) −94.5495 − 359.704i −0.208632 − 0.793719i −0.986289 − 0.165029i \(-0.947228\pi\)
0.777657 − 0.628689i \(-0.216408\pi\)
\(60\) 65.9170 + 3.76927i 0.141831 + 0.00811018i
\(61\) 208.652 + 11.9311i 0.437953 + 0.0250431i 0.274675 − 0.961537i \(-0.411430\pi\)
0.163278 + 0.986580i \(0.447793\pi\)
\(62\) −254.162 − 966.931i −0.520622 − 1.98065i
\(63\) −261.629 + 433.863i −0.523209 + 0.867645i
\(64\) −340.965 + 121.656i −0.665947 + 0.237609i
\(65\) −0.931527 + 0.598656i −0.00177756 + 0.00114237i
\(66\) 258.170 + 17.3462i 0.481493 + 0.0323510i
\(67\) 595.768 + 382.877i 1.08634 + 0.698147i 0.956013 − 0.293324i \(-0.0947616\pi\)
0.130325 + 0.991471i \(0.458398\pi\)
\(68\) −903.672 − 210.140i −1.61156 − 0.374753i
\(69\) 113.387 + 47.9178i 0.197829 + 0.0836032i
\(70\) −7.98005 + 279.338i −0.0136257 + 0.476962i
\(71\) 230.687 + 299.160i 0.385598 + 0.500052i 0.944195 − 0.329387i \(-0.106842\pi\)
−0.558597 + 0.829439i \(0.688660\pi\)
\(72\) −752.064 + 773.856i −1.23100 + 1.26666i
\(73\) −368.952 − 611.838i −0.591541 − 0.980961i −0.997741 − 0.0671770i \(-0.978601\pi\)
0.406200 − 0.913784i \(-0.366854\pi\)
\(74\) 200.365 + 163.838i 0.314756 + 0.257375i
\(75\) 102.244 − 132.593i 0.157415 − 0.204140i
\(76\) 318.092 + 2212.38i 0.480101 + 3.33918i
\(77\) −28.5207 + 739.889i −0.0422109 + 1.09504i
\(78\) −0.403402 + 2.80572i −0.000585594 + 0.00407289i
\(79\) −781.325 − 803.965i −1.11273 − 1.14498i −0.988265 − 0.152752i \(-0.951186\pi\)
−0.124469 − 0.992223i \(-0.539723\pi\)
\(80\) −57.2227 + 217.698i −0.0799711 + 0.304242i
\(81\) 175.576 − 540.368i 0.240845 − 0.741245i
\(82\) 119.401 − 589.268i 0.160801 − 0.793582i
\(83\) 71.6707 + 834.447i 0.0947817 + 1.10352i 0.876991 + 0.480507i \(0.159547\pi\)
−0.782209 + 0.623016i \(0.785907\pi\)
\(84\) 356.325 + 327.032i 0.462836 + 0.424787i
\(85\) 113.414 − 104.091i 0.144724 − 0.132826i
\(86\) 209.919 − 2444.05i 0.263211 − 3.06452i
\(87\) 127.729 + 37.5047i 0.157403 + 0.0462176i
\(88\) −416.101 + 1521.18i −0.504052 + 1.84270i
\(89\) 352.499 − 103.503i 0.419830 − 0.123273i −0.0649926 − 0.997886i \(-0.520702\pi\)
0.484822 + 0.874613i \(0.338884\pi\)
\(90\) −68.2603 − 336.878i −0.0799474 − 0.394556i
\(91\) −8.05848 − 0.924625i −0.00928306 − 0.00106513i
\(92\) −812.964 + 1188.92i −0.921276 + 1.34732i
\(93\) 249.991 − 141.177i 0.278741 − 0.157412i
\(94\) −687.212 − 499.289i −0.754048 − 0.547848i
\(95\) −332.809 − 163.631i −0.359426 − 0.176718i
\(96\) 46.6693 + 68.2517i 0.0496163 + 0.0725615i
\(97\) 188.879 + 357.966i 0.197709 + 0.374701i 0.963216 − 0.268727i \(-0.0866028\pi\)
−0.765507 + 0.643427i \(0.777512\pi\)
\(98\) −224.260 + 258.810i −0.231160 + 0.266773i
\(99\) −171.956 − 894.352i −0.174568 − 0.907937i
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 121.4.g.a.91.2 yes 1280
121.4 even 55 inner 121.4.g.a.4.2 ✓ 1280
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
121.4.g.a.4.2 ✓ 1280 121.4 even 55 inner
121.4.g.a.91.2 yes 1280 1.1 even 1 trivial