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Properties

Label 121.4.g.a.4.15
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 4.15
Character \(\chi\) \(=\) 121.4
Dual form 121.4.g.a.91.15

$q$-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+(-0.575357 - 0.0329001i) q^{2} +(2.68812 - 8.27318i) q^{3} +(-7.61790 - 0.874073i) q^{4} +(0.0851125 + 0.161306i) q^{5} +(-1.81882 + 4.67159i) q^{6} +(-4.04277 - 1.70849i) q^{7} +(8.89711 + 1.53971i) q^{8} +(-39.3761 - 28.6084i) q^{9} +(-0.0436631 - 0.0956088i) q^{10} +(12.2740 - 34.3562i) q^{11} +(-27.7092 + 60.6747i) q^{12} +(-29.4298 + 16.6198i) q^{13} +(2.26983 + 1.11600i) q^{14} +(1.56331 - 0.270541i) q^{15} +(54.6805 + 12.7154i) q^{16} +(-126.153 + 45.0113i) q^{17} +(21.7141 + 17.7555i) q^{18} +(-3.41829 + 119.656i) q^{19} +(-0.507385 - 1.30321i) q^{20} +(-25.0021 + 28.8539i) q^{21} +(-8.19223 + 19.3633i) q^{22} +(-0.608269 - 0.701980i) q^{23} +(36.6548 - 69.4685i) q^{24} +(70.5366 - 103.157i) q^{25} +(17.4795 - 8.59407i) q^{26} +(-152.515 + 110.809i) q^{27} +(29.3041 + 16.5488i) q^{28} +(-43.2281 - 63.2190i) q^{29} +(-0.908361 + 0.104225i) q^{30} +(58.9980 - 291.167i) q^{31} +(-100.351 - 29.4658i) q^{32} +(-251.241 - 193.898i) q^{33} +(74.0639 - 21.7471i) q^{34} +(-0.0685005 - 0.797537i) q^{35} +(274.957 + 252.354i) q^{36} +(-174.350 + 160.017i) q^{37} +(5.90343 - 68.7324i) q^{38} +(58.3876 + 288.154i) q^{39} +(0.508891 + 1.56621i) q^{40} +(-45.1271 - 171.681i) q^{41} +(15.3344 - 15.7788i) q^{42} +(-51.8654 - 360.732i) q^{43} +(-123.532 + 250.994i) q^{44} +(1.26331 - 8.78653i) q^{45} +(0.326877 + 0.423901i) q^{46} +(327.562 - 267.846i) q^{47} +(252.185 - 418.201i) q^{48} +(-225.625 - 232.162i) q^{49} +(-43.9776 + 57.0312i) q^{50} +(33.2723 + 1164.68i) q^{51} +(238.720 - 100.884i) q^{52} +(387.142 - 90.0261i) q^{53} +(91.3964 - 58.7369i) q^{54} +(6.58653 - 0.944280i) q^{55} +(-33.3384 - 21.4253i) q^{56} +(980.746 + 349.929i) q^{57} +(22.7917 + 37.7957i) q^{58} +(72.8011 - 276.964i) q^{59} +(-12.1456 + 0.694510i) q^{60} +(308.868 - 17.6617i) q^{61} +(-43.5243 + 165.584i) q^{62} +(110.311 + 182.931i) q^{63} +(-366.229 - 130.670i) q^{64} +(-5.18572 - 3.33266i) q^{65} +(138.174 + 119.827i) q^{66} +(-504.408 + 324.163i) q^{67} +(1000.36 - 232.625i) q^{68} +(-7.44271 + 3.14531i) q^{69} +(0.0131732 + 0.461122i) q^{70} +(-196.797 + 255.211i) q^{71} +(-306.285 - 315.160i) q^{72} +(-53.8230 + 89.2554i) q^{73} +(105.578 - 86.3306i) q^{74} +(-663.822 - 860.859i) q^{75} +(130.628 - 908.538i) q^{76} +(-108.318 + 117.924i) q^{77} +(-24.1134 - 167.713i) q^{78} +(-265.361 + 273.050i) q^{79} +(2.60292 + 9.90254i) q^{80} +(100.674 + 309.841i) q^{81} +(20.3159 + 100.263i) q^{82} +(-27.0879 + 315.379i) q^{83} +(215.684 - 197.953i) q^{84} +(-17.9978 - 16.5182i) q^{85} +(17.9730 + 209.256i) q^{86} +(-639.225 + 187.693i) q^{87} +(162.101 - 286.773i) q^{88} +(-360.717 - 105.916i) q^{89} +(-1.01593 + 5.01383i) q^{90} +(147.373 - 16.9094i) q^{91} +(4.02015 + 5.87928i) q^{92} +(-2250.28 - 1270.79i) q^{93} +(-197.277 + 143.330i) q^{94} +(-19.5921 + 9.63281i) q^{95} +(-513.532 + 751.017i) q^{96} +(-158.224 + 299.869i) q^{97} +(122.177 + 140.999i) q^{98} +(-1466.18 + 1001.68i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.575357 − 0.0329001i −0.203419 − 0.0116319i −0.0449179 − 0.998991i \(-0.514303\pi\)
−0.158502 + 0.987359i \(0.550666\pi\)
\(3\) 2.68812 − 8.27318i 0.517329 − 1.59217i −0.261675 − 0.965156i \(-0.584275\pi\)
0.779004 − 0.627019i \(-0.215725\pi\)
\(4\) −7.61790 − 0.874073i −0.952238 − 0.109259i
\(5\) 0.0851125 + 0.161306i 0.00761269 + 0.0144277i 0.888210 − 0.459438i \(-0.151949\pi\)
−0.880597 + 0.473866i \(0.842858\pi\)
\(6\) −1.81882 + 4.67159i −0.123755 + 0.317862i
\(7\) −4.04277 − 1.70849i −0.218289 − 0.0922497i 0.277383 − 0.960759i \(-0.410533\pi\)
−0.495672 + 0.868510i \(0.665078\pi\)
\(8\) 8.89711 + 1.53971i 0.393200 + 0.0680460i
\(9\) −39.3761 − 28.6084i −1.45837 − 1.05957i
\(10\) −0.0436631 − 0.0956088i −0.00138075 − 0.00302342i
\(11\) 12.2740 − 34.3562i 0.336431 − 0.941708i
\(12\) −27.7092 + 60.6747i −0.666580 + 1.45961i
\(13\) −29.4298 + 16.6198i −0.627874 + 0.354577i −0.772598 − 0.634895i \(-0.781043\pi\)
0.144724 + 0.989472i \(0.453771\pi\)
\(14\) 2.26983 + 1.11600i 0.0433312 + 0.0213045i
\(15\) 1.56331 − 0.270541i 0.0269096 − 0.00465689i
\(16\) 54.6805 + 12.7154i 0.854383 + 0.198678i
\(17\) −126.153 + 45.0113i −1.79980 + 0.642167i −0.800731 + 0.599024i \(0.795555\pi\)
−0.999069 + 0.0431436i \(0.986263\pi\)
\(18\) 21.7141 + 17.7555i 0.284337 + 0.232501i
\(19\) −3.41829 + 119.656i −0.0412742 + 1.44479i 0.670256 + 0.742130i \(0.266184\pi\)
−0.711530 + 0.702656i \(0.751997\pi\)
\(20\) −0.507385 − 1.30321i −0.00567274 − 0.0145703i
\(21\) −25.0021 + 28.8539i −0.259805 + 0.299831i
\(22\) −8.19223 + 19.3633i −0.0793904 + 0.187648i
\(23\) −0.608269 − 0.701980i −0.00551447 − 0.00636404i 0.752986 − 0.658037i \(-0.228613\pi\)
−0.758500 + 0.651673i \(0.774067\pi\)
\(24\) 36.6548 − 69.4685i 0.311755 − 0.590841i
\(25\) 70.5366 − 103.157i 0.564293 − 0.825252i
\(26\) 17.4795 − 8.59407i 0.131846 − 0.0648244i
\(27\) −152.515 + 110.809i −1.08710 + 0.789821i
\(28\) 29.3041 + 16.5488i 0.197784 + 0.111694i
\(29\) −43.2281 − 63.2190i −0.276802 − 0.404810i 0.661872 − 0.749617i \(-0.269762\pi\)
−0.938673 + 0.344808i \(0.887944\pi\)
\(30\) −0.908361 + 0.104225i −0.00552811 + 0.000634291i
\(31\) 58.9980 − 291.167i 0.341818 − 1.68694i −0.328213 − 0.944604i \(-0.606446\pi\)
0.670031 − 0.742333i \(-0.266281\pi\)
\(32\) −100.351 − 29.4658i −0.554368 − 0.162777i
\(33\) −251.241 − 193.898i −1.32532 − 1.02283i
\(34\) 74.0639 − 21.7471i 0.373584 − 0.109694i
\(35\) −0.0685005 − 0.797537i −0.000330820 − 0.00385167i
\(36\) 274.957 + 252.354i 1.27295 + 1.16830i
\(37\) −174.350 + 160.017i −0.774673 + 0.710988i −0.962520 − 0.271210i \(-0.912576\pi\)
0.187847 + 0.982198i \(0.439849\pi\)
\(38\) 5.90343 − 68.7324i 0.0252016 − 0.293417i
\(39\) 58.3876 + 288.154i 0.239731 + 1.18312i
\(40\) 0.508891 + 1.56621i 0.00201157 + 0.00619097i
\(41\) −45.1271 − 171.681i −0.171894 − 0.653954i −0.996144 − 0.0877283i \(-0.972039\pi\)
0.824250 − 0.566226i \(-0.191597\pi\)
\(42\) 15.3344 − 15.7788i 0.0563370 − 0.0579694i
\(43\) −51.8654 − 360.732i −0.183940 − 1.27933i −0.847335 − 0.531059i \(-0.821794\pi\)
0.663395 − 0.748269i \(-0.269115\pi\)
\(44\) −123.532 + 250.994i −0.423252 + 0.859972i
\(45\) 1.26331 − 8.78653i 0.00418497 − 0.0291071i
\(46\) 0.326877 + 0.423901i 0.00104772 + 0.00135871i
\(47\) 327.562 − 267.846i 1.01659 − 0.831264i 0.0308503 − 0.999524i \(-0.490178\pi\)
0.985743 + 0.168260i \(0.0538148\pi\)
\(48\) 252.185 − 418.201i 0.758327 − 1.25754i
\(49\) −225.625 − 232.162i −0.657797 − 0.676858i
\(50\) −43.9776 + 57.0312i −0.124387 + 0.161309i
\(51\) 33.2723 + 1164.68i 0.0913540 + 3.19781i
\(52\) 238.720 − 100.884i 0.636626 − 0.269040i
\(53\) 387.142 − 90.0261i 1.00336 − 0.233321i 0.307538 − 0.951536i \(-0.400495\pi\)
0.695822 + 0.718214i \(0.255040\pi\)
\(54\) 91.3964 − 58.7369i 0.230324 − 0.148020i
\(55\) 6.58653 − 0.944280i 0.0161478 − 0.00231503i
\(56\) −33.3384 − 21.4253i −0.0795541 − 0.0511263i
\(57\) 980.746 + 349.929i 2.27900 + 0.813145i
\(58\) 22.7917 + 37.7957i 0.0515981 + 0.0855659i
\(59\) 72.8011 − 276.964i 0.160642 − 0.611147i −0.837283 − 0.546769i \(-0.815857\pi\)
0.997926 − 0.0643775i \(-0.0205062\pi\)
\(60\) −12.1456 + 0.694510i −0.0261331 + 0.00149435i
\(61\) 308.868 − 17.6617i 0.648304 − 0.0370714i 0.270156 − 0.962817i \(-0.412925\pi\)
0.378148 + 0.925745i \(0.376561\pi\)
\(62\) −43.5243 + 165.584i −0.0891548 + 0.339180i
\(63\) 110.311 + 182.931i 0.220602 + 0.365827i
\(64\) −366.229 − 130.670i −0.715290 − 0.255215i
\(65\) −5.18572 − 3.33266i −0.00989552 − 0.00635947i
\(66\) 138.174 + 119.827i 0.257698 + 0.223479i
\(67\) −504.408 + 324.163i −0.919749 + 0.591087i −0.912585 − 0.408887i \(-0.865917\pi\)
−0.00716441 + 0.999974i \(0.502281\pi\)
\(68\) 1000.36 − 232.625i 1.78400 − 0.414851i
\(69\) −7.44271 + 3.14531i −0.0129855 + 0.00548770i
\(70\) 0.0131732 + 0.461122i 2.24928e−5 + 0.000787352i
\(71\) −196.797 + 255.211i −0.328951 + 0.426592i −0.927112 − 0.374785i \(-0.877717\pi\)
0.598161 + 0.801376i \(0.295898\pi\)
\(72\) −306.285 − 315.160i −0.501334 − 0.515860i
\(73\) −53.8230 + 89.2554i −0.0862945 + 0.143103i −0.896502 − 0.443040i \(-0.853900\pi\)
0.810207 + 0.586143i \(0.199354\pi\)
\(74\) 105.578 − 86.3306i 0.165854 − 0.135618i
\(75\) −663.822 − 860.859i −1.02202 − 1.32538i
\(76\) 130.628 − 908.538i 0.197159 − 1.37127i
\(77\) −108.318 + 117.924i −0.160311 + 0.174529i
\(78\) −24.1134 − 167.713i −0.0350039 − 0.243458i
\(79\) −265.361 + 273.050i −0.377917 + 0.388868i −0.879229 − 0.476399i \(-0.841942\pi\)
0.501312 + 0.865267i \(0.332851\pi\)
\(80\) 2.60292 + 9.90254i 0.00363769 + 0.0138392i
\(81\) 100.674 + 309.841i 0.138098 + 0.425022i
\(82\) 20.3159 + 100.263i 0.0273599 + 0.135027i
\(83\) −27.0879 + 315.379i −0.0358227 + 0.417076i 0.956352 + 0.292218i \(0.0943932\pi\)
−0.992175 + 0.124859i \(0.960152\pi\)
\(84\) 215.684 − 197.953i 0.280155 − 0.257124i
\(85\) −17.9978 − 16.5182i −0.0229663 − 0.0210783i
\(86\) 17.9730 + 209.256i 0.0225358 + 0.262380i
\(87\) −639.225 + 187.693i −0.787725 + 0.231297i
\(88\) 162.101 − 286.773i 0.196364 − 0.347387i
\(89\) −360.717 − 105.916i −0.429618 − 0.126147i 0.0597726 − 0.998212i \(-0.480962\pi\)
−0.489390 + 0.872065i \(0.662781\pi\)
\(90\) −1.01593 + 5.01383i −0.00118988 + 0.00587227i
\(91\) 147.373 − 16.9094i 0.169768 − 0.0194790i
\(92\) 4.02015 + 5.87928i 0.00455576 + 0.00666258i
\(93\) −2250.28 − 1270.79i −2.50907 − 1.41694i
\(94\) −197.277 + 143.330i −0.216464 + 0.157270i
\(95\) −19.5921 + 9.63281i −0.0211591 + 0.0104032i
\(96\) −513.532 + 751.017i −0.545960 + 0.798441i
\(97\) −158.224 + 299.869i −0.165621 + 0.313887i −0.953207 − 0.302318i \(-0.902240\pi\)
0.787586 + 0.616205i \(0.211331\pi\)
\(98\) 122.177 + 140.999i 0.125936 + 0.145337i
\(99\) −1466.18 + 1001.68i −1.48845 + 1.01689i
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.4.15 ✓ 1280
121.91 even 55 inner 121.4.g.a.91.15 yes 1280
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
121.4.g.a.4.15 ✓ 1280 1.1 even 1 trivial
121.4.g.a.91.15 yes 1280 121.91 even 55 inner