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Properties

Label 121.4.g.a.4.9
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 4.9
Character \(\chi\) \(=\) 121.4
Dual form 121.4.g.a.91.9

$q$-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+(-3.07035 - 0.175569i) q^{2} +(0.824583 - 2.53780i) q^{3} +(1.44837 + 0.166184i) q^{4} +(1.29762 + 2.45926i) q^{5} +(-2.97732 + 7.64718i) q^{6} +(-0.429079 - 0.181330i) q^{7} +(19.8248 + 3.43081i) q^{8} +(16.0829 + 11.6849i) q^{9} +(-3.55238 - 7.77862i) q^{10} +(-19.8722 - 30.5957i) q^{11} +(1.61604 - 3.53864i) q^{12} +(3.28781 - 1.85671i) q^{13} +(1.28559 + 0.632081i) q^{14} +(7.31113 - 1.26524i) q^{15} +(-71.6265 - 16.6560i) q^{16} +(37.1797 - 13.2657i) q^{17} +(-47.3287 - 38.7005i) q^{18} +(1.66696 - 58.3514i) q^{19} +(1.47074 + 3.77756i) q^{20} +(-0.813993 + 0.939397i) q^{21} +(55.6429 + 97.4284i) q^{22} +(-108.164 - 124.828i) q^{23} +(25.0539 - 47.4824i) q^{24} +(66.1912 - 96.8016i) q^{25} +(-10.4207 + 5.12351i) q^{26} +(101.203 - 73.5283i) q^{27} +(-0.591329 - 0.333939i) q^{28} +(-59.9311 - 87.6465i) q^{29} +(-22.6699 + 2.60112i) q^{30} +(42.6509 - 210.490i) q^{31} +(62.5583 + 18.3688i) q^{32} +(-94.0322 + 25.2030i) q^{33} +(-116.484 + 34.2027i) q^{34} +(-0.110843 - 1.29052i) q^{35} +(21.3521 + 19.5968i) q^{36} +(198.001 - 181.724i) q^{37} +(-15.3628 + 178.866i) q^{38} +(-2.00090 - 9.87483i) q^{39} +(17.2878 + 53.2062i) q^{40} +(124.329 + 472.995i) q^{41} +(2.66417 - 2.74137i) q^{42} +(18.0065 + 125.238i) q^{43} +(-23.6976 - 47.6162i) q^{44} +(-7.86680 + 54.7148i) q^{45} +(310.186 + 402.257i) q^{46} +(-253.579 + 207.351i) q^{47} +(-101.332 + 168.040i) q^{48} +(-238.898 - 245.821i) q^{49} +(-220.226 + 285.594i) q^{50} +(-3.00799 - 105.293i) q^{51} +(5.07051 - 2.14281i) q^{52} +(224.285 - 52.1551i) q^{53} +(-323.638 + 207.989i) q^{54} +(49.4564 - 88.5725i) q^{55} +(-7.88429 - 5.06692i) q^{56} +(-146.710 - 52.3460i) q^{57} +(168.622 + 279.627i) q^{58} +(-6.66138 + 25.3425i) q^{59} +(10.7994 - 0.617535i) q^{60} +(365.250 - 20.8858i) q^{61} +(-167.909 + 638.791i) q^{62} +(-4.78202 - 7.93009i) q^{63} +(365.237 + 130.316i) q^{64} +(8.83247 + 5.67629i) q^{65} +(293.136 - 60.8730i) q^{66} +(-568.560 + 365.391i) q^{67} +(56.0543 - 13.0349i) q^{68} +(-405.980 + 171.569i) q^{69} +(0.113751 + 3.98180i) q^{70} +(300.020 - 389.072i) q^{71} +(278.752 + 286.829i) q^{72} +(223.925 - 371.337i) q^{73} +(-639.837 + 523.192i) q^{74} +(-191.083 - 247.801i) q^{75} +(12.1115 - 84.2371i) q^{76} +(2.97881 + 16.7314i) q^{77} +(4.40975 + 30.6705i) q^{78} +(898.926 - 924.973i) q^{79} +(-51.9824 - 197.762i) q^{80} +(62.7145 + 193.015i) q^{81} +(-298.689 - 1474.09i) q^{82} +(-55.1033 + 641.556i) q^{83} +(-1.33507 + 1.22532i) q^{84} +(80.8690 + 74.2209i) q^{85} +(-33.2984 - 387.686i) q^{86} +(-271.848 + 79.8217i) q^{87} +(-288.993 - 674.730i) q^{88} +(-395.357 - 116.087i) q^{89} +(33.7601 - 166.612i) q^{90} +(-1.74741 + 0.200497i) q^{91} +(-135.917 - 198.772i) q^{92} +(-499.014 - 281.806i) q^{93} +(814.981 - 592.119i) q^{94} +(145.665 - 71.6184i) q^{95} +(98.2008 - 143.614i) q^{96} +(816.159 - 1546.79i) q^{97} +(690.343 + 796.698i) q^{98} +(37.9058 - 724.274i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.07035 − 0.175569i −1.08553 − 0.0620730i −0.494851 − 0.868978i \(-0.664777\pi\)
−0.590681 + 0.806905i \(0.701141\pi\)
\(3\) 0.824583 − 2.53780i 0.158691 − 0.488401i −0.839825 − 0.542857i \(-0.817343\pi\)
0.998516 + 0.0544563i \(0.0173425\pi\)
\(4\) 1.44837 + 0.166184i 0.181046 + 0.0207731i
\(5\) 1.29762 + 2.45926i 0.116063 + 0.219963i 0.935599 − 0.353063i \(-0.114860\pi\)
−0.819537 + 0.573027i \(0.805769\pi\)
\(6\) −2.97732 + 7.64718i −0.202581 + 0.520324i
\(7\) −0.429079 − 0.181330i −0.0231681 − 0.00979092i 0.377655 − 0.925946i \(-0.376730\pi\)
−0.400823 + 0.916155i \(0.631276\pi\)
\(8\) 19.8248 + 3.43081i 0.876139 + 0.151622i
\(9\) 16.0829 + 11.6849i 0.595665 + 0.432776i
\(10\) −3.55238 − 7.77862i −0.112336 − 0.245982i
\(11\) −19.8722 − 30.5957i −0.544699 − 0.838632i
\(12\) 1.61604 − 3.53864i 0.0388759 − 0.0851263i
\(13\) 3.28781 − 1.85671i 0.0701442 − 0.0396122i −0.456262 − 0.889846i \(-0.650812\pi\)
0.526406 + 0.850233i \(0.323539\pi\)
\(14\) 1.28559 + 0.632081i 0.0245420 + 0.0120665i
\(15\) 7.31113 − 1.26524i 0.125848 − 0.0217789i
\(16\) −71.6265 − 16.6560i −1.11916 − 0.260250i
\(17\) 37.1797 − 13.2657i 0.530435 − 0.189259i −0.0572330 − 0.998361i \(-0.518228\pi\)
0.587668 + 0.809102i \(0.300046\pi\)
\(18\) −47.3287 − 38.7005i −0.619749 − 0.506767i
\(19\) 1.66696 − 58.3514i 0.0201278 − 0.704564i −0.925394 − 0.379006i \(-0.876266\pi\)
0.945522 − 0.325558i \(-0.105552\pi\)
\(20\) 1.47074 + 3.77756i 0.0164433 + 0.0422344i
\(21\) −0.813993 + 0.939397i −0.00845846 + 0.00976159i
\(22\) 55.6429 + 97.4284i 0.539232 + 0.944173i
\(23\) −108.164 − 124.828i −0.980601 − 1.13167i −0.991285 − 0.131731i \(-0.957946\pi\)
0.0106840 − 0.999943i \(-0.496599\pi\)
\(24\) 25.0539 − 47.4824i 0.213088 − 0.403846i
\(25\) 66.1912 − 96.8016i 0.529530 − 0.774413i
\(26\) −10.4207 + 5.12351i −0.0786026 + 0.0386463i
\(27\) 101.203 − 73.5283i 0.721353 − 0.524094i
\(28\) −0.591329 − 0.333939i −0.00399110 − 0.00225388i
\(29\) −59.9311 − 87.6465i −0.383756 − 0.561226i 0.584434 − 0.811441i \(-0.301317\pi\)
−0.968190 + 0.250216i \(0.919498\pi\)
\(30\) −22.6699 + 2.60112i −0.137964 + 0.0158299i
\(31\) 42.6509 − 210.490i 0.247107 − 1.21952i −0.643749 − 0.765237i \(-0.722622\pi\)
0.890856 − 0.454286i \(-0.150105\pi\)
\(32\) 62.5583 + 18.3688i 0.345589 + 0.101474i
\(33\) −94.0322 + 25.2030i −0.496027 + 0.132948i
\(34\) −116.484 + 34.2027i −0.587553 + 0.172521i
\(35\) −0.110843 − 1.29052i −0.000535309 − 0.00623249i
\(36\) 21.3521 + 19.5968i 0.0988524 + 0.0907259i
\(37\) 198.001 − 181.724i 0.879761 − 0.807437i −0.102543 − 0.994729i \(-0.532698\pi\)
0.982304 + 0.187291i \(0.0599708\pi\)
\(38\) −15.3628 + 178.866i −0.0655838 + 0.763578i
\(39\) −2.00090 − 9.87483i −0.00821540 − 0.0405446i
\(40\) 17.2878 + 53.2062i 0.0683359 + 0.210316i
\(41\) 124.329 + 472.995i 0.473582 + 1.80169i 0.588711 + 0.808344i \(0.299636\pi\)
−0.115128 + 0.993351i \(0.536728\pi\)
\(42\) 2.66417 − 2.74137i 0.00978787 − 0.0100715i
\(43\) 18.0065 + 125.238i 0.0638597 + 0.444154i 0.996517 + 0.0833902i \(0.0265748\pi\)
−0.932657 + 0.360764i \(0.882516\pi\)
\(44\) −23.6976 − 47.6162i −0.0811944 − 0.163146i
\(45\) −7.86680 + 54.7148i −0.0260603 + 0.181253i
\(46\) 310.186 + 402.257i 0.994228 + 1.28934i
\(47\) −253.579 + 207.351i −0.786986 + 0.643515i −0.938632 − 0.344920i \(-0.887906\pi\)
0.151646 + 0.988435i \(0.451543\pi\)
\(48\) −101.332 + 168.040i −0.304708 + 0.505301i
\(49\) −238.898 − 245.821i −0.696497 − 0.716678i
\(50\) −220.226 + 285.594i −0.622892 + 0.807781i
\(51\) −3.00799 − 105.293i −0.00825889 − 0.289099i
\(52\) 5.07051 − 2.14281i 0.0135222 − 0.00571452i
\(53\) 224.285 − 52.1551i 0.581280 − 0.135171i 0.0744619 − 0.997224i \(-0.476276\pi\)
0.506818 + 0.862053i \(0.330822\pi\)
\(54\) −323.638 + 207.989i −0.815584 + 0.524144i
\(55\) 49.4564 − 88.5725i 0.121249 − 0.217148i
\(56\) −7.88429 − 5.06692i −0.0188140 − 0.0120910i
\(57\) −146.710 − 52.3460i −0.340916 − 0.121638i
\(58\) 168.622 + 279.627i 0.381743 + 0.633050i
\(59\) −6.66138 + 25.3425i −0.0146989 + 0.0559206i −0.974092 − 0.226152i \(-0.927385\pi\)
0.959393 + 0.282072i \(0.0910218\pi\)
\(60\) 10.7994 − 0.617535i 0.0232367 − 0.00132872i
\(61\) 365.250 − 20.8858i 0.766647 − 0.0438385i 0.330611 − 0.943767i \(-0.392745\pi\)
0.436037 + 0.899929i \(0.356382\pi\)
\(62\) −167.909 + 638.791i −0.343942 + 1.30849i
\(63\) −4.78202 − 7.93009i −0.00956314 − 0.0158587i
\(64\) 365.237 + 130.316i 0.713353 + 0.254524i
\(65\) 8.83247 + 5.67629i 0.0168544 + 0.0108316i
\(66\) 293.136 − 60.8730i 0.546706 − 0.113529i
\(67\) −568.560 + 365.391i −1.03673 + 0.666264i −0.944175 − 0.329444i \(-0.893139\pi\)
−0.0925516 + 0.995708i \(0.529502\pi\)
\(68\) 56.0543 − 13.0349i 0.0999645 − 0.0232458i
\(69\) −405.980 + 171.569i −0.708323 + 0.299340i
\(70\) 0.113751 + 3.98180i 0.000194226 + 0.00679880i
\(71\) 300.020 − 389.072i 0.501490 − 0.650343i −0.471316 − 0.881964i \(-0.656221\pi\)
0.972806 + 0.231621i \(0.0744028\pi\)
\(72\) 278.752 + 286.829i 0.456267 + 0.469487i
\(73\) 223.925 − 371.337i 0.359019 − 0.595366i −0.623434 − 0.781876i \(-0.714263\pi\)
0.982453 + 0.186510i \(0.0597175\pi\)
\(74\) −639.837 + 523.192i −1.00513 + 0.821890i
\(75\) −191.083 − 247.801i −0.294192 − 0.381515i
\(76\) 12.1115 − 84.2371i 0.0182800 − 0.127140i
\(77\) 2.97881 + 16.7314i 0.00440866 + 0.0247626i
\(78\) 4.40975 + 30.6705i 0.00640135 + 0.0445224i
\(79\) 898.926 − 924.973i 1.28022 − 1.31731i 0.356470 − 0.934307i \(-0.383980\pi\)
0.923746 − 0.383005i \(-0.125111\pi\)
\(80\) −51.9824 − 197.762i −0.0726476 − 0.276380i
\(81\) 62.7145 + 193.015i 0.0860281 + 0.264767i
\(82\) −298.689 − 1474.09i −0.402252 − 1.98519i
\(83\) −55.1033 + 641.556i −0.0728720 + 0.848433i 0.865072 + 0.501647i \(0.167272\pi\)
−0.937944 + 0.346786i \(0.887273\pi\)
\(84\) −1.33507 + 1.22532i −0.00173415 + 0.00159159i
\(85\) 80.8690 + 74.2209i 0.103194 + 0.0947104i
\(86\) −33.2984 − 387.686i −0.0417518 − 0.486107i
\(87\) −271.848 + 79.8217i −0.335002 + 0.0983654i
\(88\) −288.993 − 674.730i −0.350077 − 0.817346i
\(89\) −395.357 − 116.087i −0.470873 − 0.138261i 0.0376827 − 0.999290i \(-0.488002\pi\)
−0.508556 + 0.861029i \(0.669821\pi\)
\(90\) 33.7601 − 166.612i 0.0395403 − 0.195139i
\(91\) −1.74741 + 0.200497i −0.00201295 + 0.000230964i
\(92\) −135.917 − 198.772i −0.154025 − 0.225255i
\(93\) −499.014 − 281.806i −0.556402 − 0.314215i
\(94\) 814.981 − 592.119i 0.894244 − 0.649706i
\(95\) 145.665 − 71.6184i 0.157314 − 0.0773463i
\(96\) 98.2008 − 143.614i 0.104402 − 0.152683i
\(97\) 816.159 − 1546.79i 0.854313 − 1.61910i 0.0709997 − 0.997476i \(-0.477381\pi\)
0.783313 − 0.621627i \(-0.213528\pi\)
\(98\) 690.343 + 796.698i 0.711583 + 0.821211i
\(99\) 37.9058 − 724.274i 0.0384816 − 0.735276i
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.4.9 ✓ 1280
121.91 even 55 inner 121.4.g.a.91.9 yes 1280
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
121.4.g.a.4.9 ✓ 1280 1.1 even 1 trivial
121.4.g.a.91.9 yes 1280 121.91 even 55 inner