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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 91.1
Character \(\chi\) \(=\) 121.91
Dual form 121.4.g.a.4.1

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.43344 + 0.310695i −1.92101 + 0.109847i −0.976078 − 0.217420i \(-0.930236\pi\)
−0.944932 + 0.327268i \(0.893872\pi\)
\(3\) 0.873694 + 2.68895i 0.168142 + 0.517489i 0.999254 − 0.0386163i \(-0.0122950\pi\)
−0.831112 + 0.556106i \(0.812295\pi\)
\(4\) 21.4778 − 2.46435i 2.68473 − 0.308044i
\(5\) −1.64335 + 3.11450i −0.146986 + 0.278569i −0.946889 − 0.321559i \(-0.895793\pi\)
0.799903 + 0.600129i \(0.204884\pi\)
\(6\) −5.58261 − 14.3388i −0.379848 − 0.975632i
\(7\) 25.7497 − 10.8819i 1.39035 − 0.587568i 0.439790 − 0.898101i \(-0.355053\pi\)
0.950563 + 0.310533i \(0.100508\pi\)
\(8\) −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 −0.125458 − 0.992099i \(-0.540040\pi\)
−0.914369 + 0.404882i \(0.867313\pi\)
\(14\) −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 0.773317 − 0.634019i \(-0.218596\pi\)
0.197315 + 0.980340i \(0.436778\pi\)
\(18\) −80.0753 + 65.4773i −1.04855 + 0.857397i
\(19\) −4.03914 − 141.388i −0.0487706 − 1.70719i −0.544536 − 0.838737i \(-0.683294\pi\)
0.495766 − 0.868456i \(-0.334887\pi\)
\(20\) −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 −0.329625 − 0.944112i \(-0.606922\pi\)
0.981413 + 0.191909i \(0.0614677\pi\)
\(24\) −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 −0.376509 − 0.926413i \(-0.622876\pi\)
0.999891 + 0.0147459i \(0.00469394\pi\)
\(30\) 53.8324 + 6.17669i 0.327614 + 0.0375902i
\(31\) 18.8349 + 92.9537i 0.109124 + 0.538548i 0.996928 + 0.0783258i \(0.0249574\pi\)
−0.887804 + 0.460222i \(0.847770\pi\)
\(32\) −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 0.469083 − 0.883154i \(-0.344585\pi\)
−0.839773 + 0.542938i \(0.817312\pi\)
\(38\) 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 −0.787920 − 0.615778i \(-0.788842\pi\)
0.988876 − 0.148741i \(-0.0475219\pi\)
\(42\) −299.784 − 308.470i −1.10137 − 1.13329i
\(43\) −40.5489 + 282.024i −0.143806 + 1.00019i 0.782293 + 0.622910i \(0.214050\pi\)
−0.926099 + 0.377280i \(0.876859\pi\)
\(44\) −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 0.945779 − 0.324810i \(-0.105300\pi\)
−0.130518 + 0.991446i \(0.541664\pi\)
\(48\) 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 0.406688 − 0.913567i \(-0.366683\pi\)
−0.0381235 + 0.999273i \(0.512138\pi\)
\(54\) −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 −0.965832 − 0.259167i \(-0.916552\pi\)
0.713554 − 0.700601i \(-0.247085\pi\)
\(60\) −214.893 − 12.2880i −0.462376 − 0.0264396i
\(61\) 379.117 + 21.6787i 0.795753 + 0.0455028i 0.450236 − 0.892910i \(-0.351340\pi\)
0.345517 + 0.938412i \(0.387703\pi\)
\(62\) −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 0.995233 − 0.0975274i \(-0.0310934\pi\)
0.324721 + 0.945810i \(0.394730\pi\)
\(68\) 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 0.584861 − 0.811134i \(-0.301149\pi\)
−0.933168 + 0.359441i \(0.882967\pi\)
\(72\) −981.770 + 1010.22i −1.60698 + 1.65355i
\(73\) 282.050 + 467.728i 0.452212 + 0.749910i 0.996004 − 0.0893103i \(-0.0284663\pi\)
−0.543791 + 0.839220i \(0.683012\pi\)
\(74\) 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 0.308439 − 0.951244i \(-0.400193\pi\)
−0.959664 + 0.281149i \(0.909284\pi\)
\(80\) −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 −0.949901 − 0.312551i \(-0.898816\pi\)
0.882691 − 0.469953i \(-0.155729\pi\)
\(84\) 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 −0.513087 − 0.858337i \(-0.671498\pi\)
0.0324160 + 0.999474i \(0.489680\pi\)
\(90\) −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 0.710504 − 0.703693i \(-0.248467\pi\)
−0.981918 + 0.189306i \(0.939376\pi\)
\(98\) −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.91.1 yes 1280
121.4 even 55 inner 121.4.g.a.4.1 ✓ 1280
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
121.4.g.a.4.1 ✓ 1280 121.4 even 55 inner
121.4.g.a.91.1 yes 1280 1.1 even 1 trivial