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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 91.6
Character \(\chi\) \(=\) 121.91
Dual form 121.4.g.a.4.6

$q$-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+(-4.38498 + 0.250742i) q^{2} +(2.61878 + 8.05977i) q^{3} +(11.2173 - 1.28707i) q^{4} +(9.37767 - 17.7727i) q^{5} +(-13.5042 - 34.6853i) q^{6} +(-15.3011 + 6.46629i) q^{7} +(-14.2425 + 2.46476i) q^{8} +(-36.2585 + 26.3433i) q^{9} +(-36.6645 + 80.2841i) q^{10} +(-28.1921 - 23.1561i) q^{11} +(39.7492 + 87.0385i) q^{12} +(-53.0441 - 29.9554i) q^{13} +(65.4735 - 32.1912i) q^{14} +(167.802 + 29.0392i) q^{15} +(-26.1449 + 6.07973i) q^{16} +(-53.6413 - 19.1392i) q^{17} +(152.387 - 124.606i) q^{18} +(-0.363328 - 12.7181i) q^{19} +(82.3177 - 211.431i) q^{20} +(-92.1869 - 106.389i) q^{21} +(129.428 + 94.4699i) q^{22} +(-26.3891 + 30.4546i) q^{23} +(-57.1634 - 108.337i) q^{24} +(-157.371 - 230.148i) q^{25} +(240.108 + 118.053i) q^{26} +(-122.161 - 88.7549i) q^{27} +(-163.315 + 92.2280i) q^{28} +(-50.7441 + 74.2108i) q^{29} +(-743.088 - 85.2614i) q^{30} +(-62.0845 - 306.399i) q^{31} +(224.070 - 65.7929i) q^{32} +(112.804 - 287.863i) q^{33} +(240.015 + 70.4748i) q^{34} +(-28.5653 + 332.579i) q^{35} +(-372.817 + 342.169i) q^{36} +(181.392 + 166.480i) q^{37} +(4.78216 + 55.6776i) q^{38} +(102.523 - 505.970i) q^{39} +(-89.7561 + 276.241i) q^{40} +(-11.1905 + 42.5732i) q^{41} +(430.914 + 443.400i) q^{42} +(17.1696 - 119.417i) q^{43} +(-346.044 - 223.464i) q^{44} +(128.171 + 891.448i) q^{45} +(108.079 - 140.160i) q^{46} +(323.609 + 264.614i) q^{47} +(-117.469 - 194.800i) q^{48} +(-46.7396 + 48.0939i) q^{49} +(747.778 + 969.735i) q^{50} +(13.7827 - 482.458i) q^{51} +(-633.567 - 267.748i) q^{52} +(208.283 + 48.4341i) q^{53} +(557.927 + 358.558i) q^{54} +(-675.921 + 283.899i) q^{55} +(201.988 - 129.810i) q^{56} +(101.554 - 36.2343i) q^{57} +(203.904 - 338.137i) q^{58} +(-183.667 - 698.741i) q^{59} +(1919.66 + 109.770i) q^{60} +(-886.569 - 50.6959i) q^{61} +(349.067 + 1327.99i) q^{62} +(384.450 - 637.539i) q^{63} +(-763.794 + 272.521i) q^{64} +(-1029.82 + 661.823i) q^{65} +(-422.462 + 1290.56i) q^{66} +(137.878 + 88.6088i) q^{67} +(-626.345 - 145.650i) q^{68} +(-314.565 - 132.936i) q^{69} +(41.8664 - 1465.52i) q^{70} +(224.337 + 290.926i) q^{71} +(451.481 - 464.563i) q^{72} +(97.4490 + 161.601i) q^{73} +(-837.145 - 684.530i) q^{74} +(1442.82 - 1871.08i) q^{75} +(-20.4447 - 142.196i) q^{76} +(581.104 + 172.014i) q^{77} +(-322.692 + 2244.37i) q^{78} +(401.575 + 413.211i) q^{79} +(-137.125 + 521.678i) q^{80} +(21.4961 - 66.1582i) q^{81} +(38.3953 - 189.488i) q^{82} +(-25.1007 - 292.243i) q^{83} +(-1171.02 - 1074.75i) q^{84} +(-843.185 + 773.868i) q^{85} +(-45.3455 + 527.948i) q^{86} +(-731.010 - 214.644i) q^{87} +(458.601 + 260.313i) q^{88} +(-727.821 + 213.707i) q^{89} +(-785.551 - 3876.84i) q^{90} +(1005.33 + 115.351i) q^{91} +(-256.818 + 375.584i) q^{92} +(2306.92 - 1302.78i) q^{93} +(-1485.37 - 1079.18i) q^{94} +(-229.442 - 112.809i) q^{95} +(1117.07 + 1633.66i) q^{96} +(-705.942 - 1337.91i) q^{97} +(192.893 - 222.610i) q^{98} +(1632.21 + 96.9289i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.38498 + 0.250742i −1.55032 + 0.0886508i −0.811140 − 0.584852i \(-0.801153\pi\)
−0.739185 + 0.673503i \(0.764789\pi\)
\(3\) 2.61878 + 8.05977i 0.503984 + 1.55110i 0.802472 + 0.596690i \(0.203518\pi\)
−0.298487 + 0.954414i \(0.596482\pi\)
\(4\) 11.2173 − 1.28707i 1.40217 − 0.160884i
\(5\) 9.37767 − 17.7727i 0.838764 − 1.58963i 0.0314556 − 0.999505i \(-0.489986\pi\)
0.807309 − 0.590130i \(-0.200923\pi\)
\(6\) −13.5042 − 34.6853i −0.918846 − 2.36004i
\(7\) −15.3011 + 6.46629i −0.826180 + 0.349147i −0.761020 − 0.648728i \(-0.775301\pi\)
−0.0651601 + 0.997875i \(0.520756\pi\)
\(8\) −14.2425 + 2.46476i −0.629435 + 0.108928i
\(9\) −36.2585 + 26.3433i −1.34291 + 0.975678i
\(10\) −36.6645 + 80.2841i −1.15943 + 2.53881i
\(11\) −28.1921 − 23.1561i −0.772750 − 0.634711i
\(12\) 39.7492 + 87.0385i 0.956216 + 2.09382i
\(13\) −53.0441 − 29.9554i −1.13168 − 0.639086i −0.192082 − 0.981379i \(-0.561524\pi\)
−0.939593 + 0.342293i \(0.888797\pi\)
\(14\) 65.4735 − 32.1912i 1.24990 − 0.614532i
\(15\) 167.802 + 29.0392i 2.88841 + 0.499859i
\(16\) −26.1449 + 6.07973i −0.408514 + 0.0949958i
\(17\) −53.6413 − 19.1392i −0.765290 − 0.273055i −0.0755791 − 0.997140i \(-0.524081\pi\)
−0.689711 + 0.724085i \(0.742262\pi\)
\(18\) 152.387 − 124.606i 1.99545 − 1.63167i
\(19\) −0.363328 − 12.7181i −0.00438701 − 0.153565i −0.998832 − 0.0483109i \(-0.984616\pi\)
0.994445 − 0.105254i \(-0.0335657\pi\)
\(20\) 82.3177 − 211.431i 0.920340 − 2.36387i
\(21\) −92.1869 − 106.389i −0.957945 − 1.10553i
\(22\) 129.428 + 94.4699i 1.25428 + 0.915502i
\(23\) −26.3891 + 30.4546i −0.239239 + 0.276097i −0.862654 − 0.505794i \(-0.831199\pi\)
0.623415 + 0.781891i \(0.285745\pi\)
\(24\) −57.1634 − 108.337i −0.486184 − 0.921422i
\(25\) −157.371 − 230.148i −1.25897 − 1.84119i
\(26\) 240.108 + 118.053i 1.81112 + 0.890467i
\(27\) −122.161 − 88.7549i −0.870734 − 0.632626i
\(28\) −163.315 + 92.2280i −1.10227 + 0.622480i
\(29\) −50.7441 + 74.2108i −0.324929 + 0.475193i −0.953003 − 0.302960i \(-0.902025\pi\)
0.628074 + 0.778153i \(0.283843\pi\)
\(30\) −743.088 − 85.2614i −4.52229 − 0.518884i
\(31\) −62.0845 − 306.399i −0.359700 − 1.77519i −0.595466 − 0.803380i \(-0.703033\pi\)
0.235766 − 0.971810i \(-0.424240\pi\)
\(32\) 224.070 − 65.7929i 1.23782 − 0.363458i
\(33\) 112.804 − 287.863i 0.595048 − 1.51850i
\(34\) 240.015 + 70.4748i 1.21065 + 0.355480i
\(35\) −28.5653 + 332.579i −0.137955 + 1.60618i
\(36\) −372.817 + 342.169i −1.72601 + 1.58411i
\(37\) 181.392 + 166.480i 0.805964 + 0.739708i 0.969039 − 0.246906i \(-0.0794140\pi\)
−0.163075 + 0.986614i \(0.552141\pi\)
\(38\) 4.78216 + 55.6776i 0.0204150 + 0.237687i
\(39\) 102.523 − 505.970i 0.420943 − 2.07744i
\(40\) −89.7561 + 276.241i −0.354792 + 1.09194i
\(41\) −11.1905 + 42.5732i −0.0426260 + 0.162166i −0.985055 − 0.172240i \(-0.944899\pi\)
0.942429 + 0.334406i \(0.108536\pi\)
\(42\) 430.914 + 443.400i 1.58313 + 1.62900i
\(43\) 17.1696 − 119.417i 0.0608918 − 0.423512i −0.936459 − 0.350776i \(-0.885918\pi\)
0.997351 − 0.0727358i \(-0.0231730\pi\)
\(44\) −346.044 − 223.464i −1.18564 − 0.765647i
\(45\) 128.171 + 891.448i 0.424591 + 2.95309i
\(46\) 108.079 − 140.160i 0.346422 − 0.449249i
\(47\) 323.609 + 264.614i 1.00433 + 0.821233i 0.983929 − 0.178559i \(-0.0571437\pi\)
0.0203959 + 0.999792i \(0.493507\pi\)
\(48\) −117.469 − 194.800i −0.353233 − 0.585771i
\(49\) −46.7396 + 48.0939i −0.136267 + 0.140215i
\(50\) 747.778 + 969.735i 2.11503 + 2.74283i
\(51\) 13.7827 − 482.458i 0.0378425 − 1.32466i
\(52\) −633.567 − 267.748i −1.68961 − 0.714037i
\(53\) 208.283 + 48.4341i 0.539808 + 0.125527i 0.487531 − 0.873106i \(-0.337898\pi\)
0.0522771 + 0.998633i \(0.483352\pi\)
\(54\) 557.927 + 358.558i 1.40600 + 0.903584i
\(55\) −675.921 + 283.899i −1.65711 + 0.696018i
\(56\) 201.988 − 129.810i 0.481995 − 0.309760i
\(57\) 101.554 − 36.2343i 0.235985 − 0.0841991i
\(58\) 203.904 − 338.137i 0.461619 − 0.765509i
\(59\) −183.667 − 698.741i −0.405278 − 1.54184i −0.786641 − 0.617411i \(-0.788182\pi\)
0.381363 − 0.924425i \(-0.375455\pi\)
\(60\) 1919.66 + 109.770i 4.13045 + 0.236188i
\(61\) −886.569 − 50.6959i −1.86088 − 0.106409i −0.909741 − 0.415177i \(-0.863720\pi\)
−0.951137 + 0.308768i \(0.900083\pi\)
\(62\) 349.067 + 1327.99i 0.715024 + 2.72023i
\(63\) 384.450 − 637.539i 0.768828 − 1.27496i
\(64\) −763.794 + 272.521i −1.49178 + 0.532268i
\(65\) −1029.82 + 661.823i −1.96512 + 1.26291i
\(66\) −422.462 + 1290.56i −0.787902 + 2.40692i
\(67\) 137.878 + 88.6088i 0.251410 + 0.161571i 0.660276 − 0.751023i \(-0.270439\pi\)
−0.408866 + 0.912594i \(0.634076\pi\)
\(68\) −626.345 − 145.650i −1.11699 − 0.259746i
\(69\) −314.565 − 132.936i −0.548828 − 0.231937i
\(70\) 41.8664 − 1465.52i 0.0714857 − 2.50233i
\(71\) 224.337 + 290.926i 0.374985 + 0.486289i 0.941143 − 0.338007i \(-0.109753\pi\)
−0.566159 + 0.824296i \(0.691571\pi\)
\(72\) 451.481 − 464.563i 0.738994 − 0.760407i
\(73\) 97.4490 + 161.601i 0.156240 + 0.259095i 0.924614 − 0.380907i \(-0.124388\pi\)
−0.768373 + 0.640002i \(0.778933\pi\)
\(74\) −837.145 − 684.530i −1.31508 − 1.07534i
\(75\) 1442.82 − 1871.08i 2.22137 − 2.88072i
\(76\) −20.4447 − 142.196i −0.0308574 − 0.214618i
\(77\) 581.104 + 172.014i 0.860038 + 0.254582i
\(78\) −322.692 + 2244.37i −0.468432 + 3.25802i
\(79\) 401.575 + 413.211i 0.571908 + 0.588480i 0.939943 − 0.341332i \(-0.110878\pi\)
−0.368034 + 0.929812i \(0.619969\pi\)
\(80\) −137.125 + 521.678i −0.191638 + 0.729067i
\(81\) 21.4961 − 66.1582i 0.0294871 − 0.0907520i
\(82\) 38.3953 − 189.488i 0.0517080 − 0.255189i
\(83\) −25.1007 − 292.243i −0.0331948 − 0.386480i −0.994010 − 0.109290i \(-0.965142\pi\)
0.960815 − 0.277190i \(-0.0894031\pi\)
\(84\) −1171.02 − 1074.75i −1.52106 − 1.39601i
\(85\) −843.185 + 773.868i −1.07596 + 0.987503i
\(86\) −45.3455 + 527.948i −0.0568574 + 0.661978i
\(87\) −731.010 − 214.644i −0.900833 − 0.264509i
\(88\) 458.601 + 260.313i 0.555534 + 0.315335i
\(89\) −727.821 + 213.707i −0.866841 + 0.254527i −0.684771 − 0.728758i \(-0.740098\pi\)
−0.182070 + 0.983286i \(0.558280\pi\)
\(90\) −785.551 − 3876.84i −0.920048 − 4.54061i
\(91\) 1005.33 + 115.351i 1.15810 + 0.132880i
\(92\) −256.818 + 375.584i −0.291034 + 0.425623i
\(93\) 2306.92 − 1302.78i 2.57222 − 1.45260i
\(94\) −1485.37 − 1079.18i −1.62983 − 1.18414i
\(95\) −229.442 − 112.809i −0.247792 − 0.121831i
\(96\) 1117.07 + 1633.66i 1.18760 + 1.73682i
\(97\) −705.942 − 1337.91i −0.738943 − 1.40045i −0.909753 − 0.415149i \(-0.863729\pi\)
0.170810 − 0.985304i \(-0.445362\pi\)
\(98\) 192.893 − 222.610i 0.198828 − 0.229460i
\(99\) 1632.21 + 96.9289i 1.65700 + 0.0984012i
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.91.6 yes 1280
121.4 even 55 inner 121.4.g.a.4.6 ✓ 1280
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
121.4.g.a.4.6 ✓ 1280 121.4 even 55 inner
121.4.g.a.91.6 yes 1280 1.1 even 1 trivial