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Properties

Label 121.4.g.a.4.2
Level $121$
Weight $4$
Character 121.4
Analytic conductor $7.139$
Analytic rank $0$
Dimension $1280$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

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

$q$-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 \(\left(\mathbb{Z}/121\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{1}{55}\right)\)

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 −0.849828 − 0.527060i \(-0.823294\pi\)
−0.904369 + 0.426752i \(0.859658\pi\)
\(3\) −0.441007 + 1.35728i −0.0848717 + 0.261208i −0.984482 − 0.175485i \(-0.943851\pi\)
0.899610 + 0.436694i \(0.143851\pi\)
\(4\) 16.5893 + 1.90345i 2.07366 + 0.237931i
\(5\) −1.29295 − 2.45042i −0.115645 − 0.219172i 0.819795 − 0.572657i \(-0.194087\pi\)
−0.935440 + 0.353485i \(0.884996\pi\)
\(6\) 2.57319 − 6.60918i 0.175083 − 0.449698i
\(7\) −18.6947 − 7.90044i −1.00942 − 0.426584i −0.179074 − 0.983836i \(-0.557310\pi\)
−0.830343 + 0.557252i \(0.811856\pi\)
\(8\) −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 −0.488016 − 0.872835i \(-0.662279\pi\)
0.495441 + 0.868642i \(0.335007\pi\)
\(14\) 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 −0.681824 − 0.731516i \(-0.738813\pi\)
−0.0647693 + 0.997900i \(0.520631\pi\)
\(18\) −96.0406 − 78.5320i −1.25761 − 1.02834i
\(19\) 3.82237 − 133.800i 0.0461532 − 1.61557i −0.565706 − 0.824607i \(-0.691396\pi\)
0.611859 − 0.790967i \(-0.290422\pi\)
\(20\) −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 0.439548 − 0.898219i \(-0.355139\pi\)
−0.951631 + 0.307244i \(0.900593\pi\)
\(24\) 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 0.619230 − 0.785209i \(-0.287445\pi\)
−0.956370 + 0.292159i \(0.905627\pi\)
\(30\) −19.5223 + 2.23997i −0.118809 + 0.0136320i
\(31\) 39.9511 − 197.167i 0.231466 − 1.14233i −0.680718 − 0.732546i \(-0.738332\pi\)
0.912183 − 0.409782i \(-0.134395\pi\)
\(32\) −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 −0.756875 − 0.653559i \(-0.773275\pi\)
0.586392 + 0.810027i \(0.300548\pi\)
\(38\) −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 0.882547 − 0.470224i \(-0.155827\pi\)
−0.999699 + 0.0245321i \(0.992190\pi\)
\(42\) −100.320 + 103.227i −0.368566 + 0.379245i
\(43\) −70.2463 − 488.574i −0.249127 − 1.73272i −0.603280 − 0.797529i \(-0.706140\pi\)
0.354153 − 0.935187i \(-0.384769\pi\)
\(44\) 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 −0.405015 − 0.914310i \(-0.632734\pi\)
0.815667 + 0.578521i \(0.196370\pi\)
\(48\) −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 −0.780856 − 0.624711i \(-0.785217\pi\)
−0.425103 + 0.905145i \(0.639762\pi\)
\(54\) 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 0.777657 + 0.628689i \(0.216408\pi\)
−0.986289 + 0.165029i \(0.947228\pi\)
\(60\) 65.9170 − 3.76927i 0.141831 − 0.00811018i
\(61\) 208.652 − 11.9311i 0.437953 − 0.0250431i 0.163278 − 0.986580i \(-0.447793\pi\)
0.274675 + 0.961537i \(0.411430\pi\)
\(62\) −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 0.130325 − 0.991471i \(-0.458398\pi\)
0.956013 + 0.293324i \(0.0947616\pi\)
\(68\) −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 −0.558597 − 0.829439i \(-0.688660\pi\)
0.944195 + 0.329387i \(0.106842\pi\)
\(72\) −752.064 − 773.856i −1.23100 − 1.26666i
\(73\) −368.952 + 611.838i −0.591541 + 0.980961i 0.406200 + 0.913784i \(0.366854\pi\)
−0.997741 + 0.0671770i \(0.978601\pi\)
\(74\) 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 −0.124469 + 0.992223i \(0.539723\pi\)
−0.988265 + 0.152752i \(0.951186\pi\)
\(80\) −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 −0.782209 − 0.623016i \(-0.785907\pi\)
0.876991 − 0.480507i \(-0.159547\pi\)
\(84\) 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 0.484822 − 0.874613i \(-0.338884\pi\)
−0.0649926 + 0.997886i \(0.520702\pi\)
\(90\) −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 −0.765507 − 0.643427i \(-0.777512\pi\)
0.963216 + 0.268727i \(0.0866028\pi\)
\(98\) −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.4.2 ✓ 1280
121.91 even 55 inner 121.4.g.a.91.2 yes 1280
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
121.4.g.a.4.2 ✓ 1280 1.1 even 1 trivial
121.4.g.a.91.2 yes 1280 121.91 even 55 inner