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Properties

Label 121.4.g.a.4.1
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 4.1
Character \(\chi\) \(=\) 121.4
Dual form 121.4.g.a.91.1

$q$-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+(-5.43344 - 0.310695i) q^{2} +(0.873694 - 2.68895i) q^{3} +(21.4778 + 2.46435i) q^{4} +(-1.64335 - 3.11450i) q^{5} +(-5.58261 + 14.3388i) q^{6} +(25.7497 + 10.8819i) q^{7} +(-73.0320 - 12.6387i) q^{8} +(15.3763 + 11.1716i) q^{9} +(7.96139 + 17.4330i) q^{10} +(-36.3135 - 3.51098i) q^{11} +(25.3916 - 55.5998i) q^{12} +(-48.7390 + 27.5242i) q^{13} +(-136.528 - 67.1264i) q^{14} +(-9.81054 + 1.69778i) q^{15} +(224.432 + 52.1895i) q^{16} +(68.0343 - 24.2746i) q^{17} +(-80.0753 - 65.4773i) q^{18} +(-4.03914 + 141.388i) q^{19} +(-27.6205 - 70.9426i) q^{20} +(51.7583 - 59.7322i) q^{21} +(196.216 + 30.3591i) q^{22} +(71.8949 + 82.9711i) q^{23} +(-97.7924 + 185.337i) q^{24} +(63.5559 - 92.9475i) q^{25} +(273.372 - 134.408i) q^{26} +(105.233 - 76.4561i) q^{27} +(526.231 + 297.176i) q^{28} +(97.3534 + 142.375i) q^{29} +(53.8324 - 6.17669i) q^{30} +(18.8349 - 92.9537i) q^{31} +(-634.302 - 186.248i) q^{32} +(-41.1678 + 94.5779i) q^{33} +(-377.202 + 110.757i) q^{34} +(-8.42412 - 98.0802i) q^{35} +(302.720 + 277.834i) q^{36} +(-83.4284 + 76.5699i) q^{37} +(65.8750 - 766.969i) q^{38} +(31.4282 + 155.104i) q^{39} +(80.6542 + 248.228i) q^{40} +(52.7568 + 200.708i) q^{41} +(-299.784 + 308.470i) q^{42} +(-40.5489 - 282.024i) q^{43} +(-771.284 - 164.898i) q^{44} +(9.52508 - 66.2484i) q^{45} +(-364.858 - 473.156i) q^{46} +(262.690 - 214.800i) q^{47} +(336.420 - 557.891i) q^{48} +(305.581 + 314.435i) q^{49} +(-374.205 + 485.278i) q^{50} +(-5.83209 - 204.150i) q^{51} +(-1114.64 + 471.049i) q^{52} +(142.209 - 33.0693i) q^{53} +(-595.530 + 382.724i) q^{54} +(48.7410 + 118.868i) q^{55} +(-1743.02 - 1120.17i) q^{56} +(376.657 + 134.391i) q^{57} +(-484.728 - 803.832i) q^{58} +(-114.330 + 434.955i) q^{59} +(-214.893 + 12.2880i) q^{60} +(379.117 - 21.6787i) q^{61} +(-131.218 + 499.206i) q^{62} +(274.368 + 454.988i) q^{63} +(1652.41 + 589.579i) q^{64} +(165.819 + 106.566i) q^{65} +(253.067 - 501.092i) q^{66} +(723.887 - 465.214i) q^{67} +(1521.05 - 353.705i) q^{68} +(285.920 - 120.831i) q^{69} +(15.2989 + 535.530i) q^{70} +(-208.377 + 270.228i) q^{71} +(-981.770 - 1010.22i) q^{72} +(282.050 - 467.728i) q^{73} +(477.093 - 390.117i) q^{74} +(-194.403 - 252.107i) q^{75} +(-435.182 + 3026.76i) q^{76} +(-896.856 - 485.567i) q^{77} +(-122.573 - 852.515i) q^{78} +(-457.269 + 470.519i) q^{79} +(-206.277 - 784.761i) q^{80} +(44.9318 + 138.286i) q^{81} +(-224.292 - 1106.92i) q^{82} +(-50.8214 + 591.703i) q^{83} +(1258.86 - 1155.37i) q^{84} +(-187.408 - 172.001i) q^{85} +(132.696 + 1544.96i) q^{86} +(467.896 - 137.387i) q^{87} +(2607.68 + 715.369i) q^{88} +(-403.583 - 118.503i) q^{89} +(-72.3370 + 356.997i) q^{90} +(-1554.53 + 178.365i) q^{91} +(1339.68 + 1959.22i) q^{92} +(-233.492 - 131.859i) q^{93} +(-1494.05 + 1085.49i) q^{94} +(446.991 - 219.771i) q^{95} +(-1055.00 + 1542.89i) q^{96} +(-259.293 + 491.414i) q^{97} +(-1562.66 - 1803.41i) q^{98} +(-519.146 - 459.665i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.43344 − 0.310695i −1.92101 − 0.109847i −0.944932 − 0.327268i \(-0.893872\pi\)
−0.976078 + 0.217420i \(0.930236\pi\)
\(3\) 0.873694 − 2.68895i 0.168142 − 0.517489i −0.831112 − 0.556106i \(-0.812295\pi\)
0.999254 + 0.0386163i \(0.0122950\pi\)
\(4\) 21.4778 + 2.46435i 2.68473 + 0.308044i
\(5\) −1.64335 − 3.11450i −0.146986 − 0.278569i 0.799903 − 0.600129i \(-0.204884\pi\)
−0.946889 + 0.321559i \(0.895793\pi\)
\(6\) −5.58261 + 14.3388i −0.379848 + 0.975632i
\(7\) 25.7497 + 10.8819i 1.39035 + 0.587568i 0.950563 − 0.310533i \(-0.100508\pi\)
0.439790 + 0.898101i \(0.355053\pi\)
\(8\) −73.0320 − 12.6387i −3.22759 − 0.558556i
\(9\) 15.3763 + 11.1716i 0.569494 + 0.413761i
\(10\) 7.96139 + 17.4330i 0.251761 + 0.551281i
\(11\) −36.3135 − 3.51098i −0.995359 − 0.0962363i
\(12\) 25.3916 − 55.5998i 0.610827 − 1.33752i
\(13\) −48.7390 + 27.5242i −1.03983 + 0.587217i −0.914369 − 0.404882i \(-0.867313\pi\)
−0.125458 + 0.992099i \(0.540040\pi\)
\(14\) −136.528 − 67.1264i −2.60634 − 1.28145i
\(15\) −9.81054 + 1.69778i −0.168871 + 0.0292243i
\(16\) 224.432 + 52.1895i 3.50676 + 0.815461i
\(17\) 68.0343 − 24.2746i 0.970632 − 0.346321i 0.197315 − 0.980340i \(-0.436778\pi\)
0.773317 + 0.634019i \(0.218596\pi\)
\(18\) −80.0753 − 65.4773i −1.04855 − 0.857397i
\(19\) −4.03914 + 141.388i −0.0487706 + 1.70719i 0.495766 + 0.868456i \(0.334887\pi\)
−0.544536 + 0.838737i \(0.683294\pi\)
\(20\) −27.6205 − 70.9426i −0.308806 − 0.793162i
\(21\) 51.7583 − 59.7322i 0.537837 − 0.620697i
\(22\) 196.216 + 30.3591i 1.90152 + 0.294208i
\(23\) 71.8949 + 82.9711i 0.651788 + 0.752203i 0.981413 − 0.191909i \(-0.0614677\pi\)
−0.329625 + 0.944112i \(0.606922\pi\)
\(24\) −97.7924 + 185.337i −0.831742 + 1.57633i
\(25\) 63.5559 − 92.9475i 0.508447 − 0.743580i
\(26\) 273.372 − 134.408i 2.06202 − 1.01383i
\(27\) 105.233 − 76.4561i 0.750076 − 0.544962i
\(28\) 526.231 + 297.176i 3.55172 + 2.00575i
\(29\) 97.3534 + 142.375i 0.623382 + 0.911667i 0.999891 − 0.0147459i \(-0.00469394\pi\)
−0.376509 + 0.926413i \(0.622876\pi\)
\(30\) 53.8324 − 6.17669i 0.327614 − 0.0375902i
\(31\) 18.8349 − 92.9537i 0.109124 − 0.538548i −0.887804 − 0.460222i \(-0.847770\pi\)
0.996928 − 0.0783258i \(-0.0249574\pi\)
\(32\) −634.302 − 186.248i −3.50406 − 1.02888i
\(33\) −41.1678 + 94.5779i −0.217163 + 0.498906i
\(34\) −377.202 + 110.757i −1.90264 + 0.558664i
\(35\) −8.42412 − 98.0802i −0.0406839 − 0.473674i
\(36\) 302.720 + 277.834i 1.40148 + 1.28627i
\(37\) −83.4284 + 76.5699i −0.370690 + 0.340216i −0.839773 − 0.542938i \(-0.817312\pi\)
0.469083 + 0.883154i \(0.344585\pi\)
\(38\) 65.8750 − 766.969i 0.281219 − 3.27418i
\(39\) 31.4282 + 155.104i 0.129040 + 0.636836i
\(40\) 80.6542 + 248.228i 0.318814 + 0.981207i
\(41\) 52.7568 + 200.708i 0.200957 + 0.764519i 0.988876 + 0.148741i \(0.0475219\pi\)
−0.787920 + 0.615778i \(0.788842\pi\)
\(42\) −299.784 + 308.470i −1.10137 + 1.13329i
\(43\) −40.5489 − 282.024i −0.143806 − 1.00019i −0.926099 − 0.377280i \(-0.876859\pi\)
0.782293 − 0.622910i \(-0.214050\pi\)
\(44\) −771.284 − 164.898i −2.64262 − 0.564983i
\(45\) 9.52508 − 66.2484i 0.0315537 − 0.219461i
\(46\) −364.858 − 473.156i −1.16946 − 1.51659i
\(47\) 262.690 − 214.800i 0.815261 − 0.666636i −0.130518 − 0.991446i \(-0.541664\pi\)
0.945779 + 0.324810i \(0.105300\pi\)
\(48\) 336.420 − 557.891i 1.01163 − 1.67760i
\(49\) 305.581 + 314.435i 0.890906 + 0.916720i
\(50\) −374.205 + 485.278i −1.05841 + 1.37257i
\(51\) −5.83209 − 204.150i −0.0160129 − 0.560523i
\(52\) −1114.64 + 471.049i −2.97254 + 1.25621i
\(53\) 142.209 − 33.0693i 0.368565 − 0.0857061i −0.0381235 − 0.999273i \(-0.512138\pi\)
0.406688 + 0.913567i \(0.366683\pi\)
\(54\) −595.530 + 382.724i −1.50077 + 0.964484i
\(55\) 48.7410 + 118.868i 0.119495 + 0.291422i
\(56\) −1743.02 − 1120.17i −4.15930 − 2.67302i
\(57\) 376.657 + 134.391i 0.875254 + 0.312290i
\(58\) −484.728 − 803.832i −1.09738 − 1.81980i
\(59\) −114.330 + 434.955i −0.252279 + 0.959768i 0.713554 + 0.700601i \(0.247085\pi\)
−0.965832 + 0.259167i \(0.916552\pi\)
\(60\) −214.893 + 12.2880i −0.462376 + 0.0264396i
\(61\) 379.117 − 21.6787i 0.795753 − 0.0455028i 0.345517 − 0.938412i \(-0.387703\pi\)
0.450236 + 0.892910i \(0.351340\pi\)
\(62\) −131.218 + 499.206i −0.268786 + 1.02257i
\(63\) 274.368 + 454.988i 0.548684 + 0.909890i
\(64\) 1652.41 + 589.579i 3.22737 + 1.15152i
\(65\) 165.819 + 106.566i 0.316421 + 0.203351i
\(66\) 253.067 − 501.092i 0.471976 − 0.934549i
\(67\) 723.887 − 465.214i 1.31995 − 0.848283i 0.324721 − 0.945810i \(-0.394730\pi\)
0.995233 + 0.0975274i \(0.0310934\pi\)
\(68\) 1521.05 − 353.705i 2.71257 − 0.630781i
\(69\) 285.920 − 120.831i 0.498850 − 0.210816i
\(70\) 15.2989 + 535.530i 0.0261223 + 0.914401i
\(71\) −208.377 + 270.228i −0.348307 + 0.451693i −0.933168 − 0.359441i \(-0.882967\pi\)
0.584861 + 0.811134i \(0.301149\pi\)
\(72\) −981.770 − 1010.22i −1.60698 − 1.65355i
\(73\) 282.050 − 467.728i 0.452212 − 0.749910i −0.543791 − 0.839220i \(-0.683012\pi\)
0.996004 + 0.0893103i \(0.0284663\pi\)
\(74\) 477.093 − 390.117i 0.749471 − 0.612840i
\(75\) −194.403 − 252.107i −0.299303 − 0.388143i
\(76\) −435.182 + 3026.76i −0.656827 + 4.56833i
\(77\) −896.856 − 485.567i −1.32735 − 0.718643i
\(78\) −122.573 − 852.515i −0.177932 − 1.23754i
\(79\) −457.269 + 470.519i −0.651225 + 0.670095i −0.959664 − 0.281149i \(-0.909284\pi\)
0.308439 + 0.951244i \(0.400193\pi\)
\(80\) −206.277 − 784.761i −0.288281 − 1.09674i
\(81\) 44.9318 + 138.286i 0.0616348 + 0.189692i
\(82\) −224.292 − 1106.92i −0.302060 − 1.49072i
\(83\) −50.8214 + 591.703i −0.0672094 + 0.782504i 0.882691 + 0.469953i \(0.155729\pi\)
−0.949901 + 0.312551i \(0.898816\pi\)
\(84\) 1258.86 − 1155.37i 1.63515 − 1.50073i
\(85\) −187.408 − 172.001i −0.239144 − 0.219484i
\(86\) 132.696 + 1544.96i 0.166384 + 1.93717i
\(87\) 467.896 − 137.387i 0.576595 − 0.169304i
\(88\) 2607.68 + 715.369i 3.15885 + 0.866575i
\(89\) −403.583 − 118.503i −0.480671 − 0.141138i 0.0324160 − 0.999474i \(-0.489680\pi\)
−0.513087 + 0.858337i \(0.671498\pi\)
\(90\) −72.3370 + 356.997i −0.0847221 + 0.418120i
\(91\) −1554.53 + 178.365i −1.79076 + 0.205470i
\(92\) 1339.68 + 1959.22i 1.51816 + 2.22024i
\(93\) −233.492 − 131.859i −0.260344 − 0.147023i
\(94\) −1494.05 + 1085.49i −1.63935 + 1.19106i
\(95\) 446.991 − 219.771i 0.482740 − 0.237347i
\(96\) −1055.00 + 1542.89i −1.12162 + 1.64031i
\(97\) −259.293 + 491.414i −0.271414 + 0.514387i −0.981918 − 0.189306i \(-0.939376\pi\)
0.710504 + 0.703693i \(0.248467\pi\)
\(98\) −1562.66 − 1803.41i −1.61074 − 1.85889i
\(99\) −519.146 − 459.665i −0.527031 − 0.466647i
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.4.1 ✓ 1280
121.91 even 55 inner 121.4.g.a.91.1 yes 1280
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
121.4.g.a.4.1 ✓ 1280 1.1 even 1 trivial
121.4.g.a.91.1 yes 1280 121.91 even 55 inner