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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 4.17
Character \(\chi\) \(=\) 121.4
Dual form 121.4.g.a.91.17

$q$-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+(-0.0563200 - 0.00322050i) q^{2} +(-1.78138 + 5.48253i) q^{3} +(-7.94469 - 0.911569i) q^{4} +(7.94342 + 15.0544i) q^{5} +(0.117984 - 0.303039i) q^{6} +(-23.2081 - 9.80784i) q^{7} +(0.889196 + 0.153881i) q^{8} +(-5.04131 - 3.66272i) q^{9} +(-0.398891 - 0.873449i) q^{10} +(-5.60883 - 36.0491i) q^{11} +(19.1502 - 41.9331i) q^{12} +(33.9972 - 19.1991i) q^{13} +(1.27550 + 0.627120i) q^{14} +(-96.6866 + 16.7323i) q^{15} +(62.2624 + 14.4785i) q^{16} +(-116.538 + 41.5805i) q^{17} +(0.272131 + 0.222520i) q^{18} +(1.21280 - 42.4536i) q^{19} +(-49.3848 - 126.844i) q^{20} +(95.1143 - 109.768i) q^{21} +(0.199793 + 2.04835i) q^{22} +(-100.442 - 115.916i) q^{23} +(-2.42766 + 4.60092i) q^{24} +(-92.9832 + 135.984i) q^{25} +(-1.97655 + 0.971806i) q^{26} +(-96.8588 + 70.3721i) q^{27} +(175.441 + 99.0761i) q^{28} +(11.1838 + 16.3558i) q^{29} +(5.49928 - 0.630984i) q^{30} +(-46.8927 + 231.425i) q^{31} +(-10.3869 - 3.04986i) q^{32} +(207.632 + 33.4667i) q^{33} +(6.69731 - 1.96651i) q^{34} +(-36.7003 - 427.294i) q^{35} +(36.7128 + 33.6947i) q^{36} +(68.3986 - 62.7757i) q^{37} +(-0.205027 + 2.38708i) q^{38} +(44.6976 + 220.591i) q^{39} +(4.74665 + 14.6087i) q^{40} +(16.8990 + 64.2905i) q^{41} +(-5.71034 + 5.87581i) q^{42} +(-19.8391 - 137.984i) q^{43} +(11.6991 + 291.512i) q^{44} +(15.0951 - 104.989i) q^{45} +(5.28359 + 6.85188i) q^{46} +(-258.828 + 211.642i) q^{47} +(-190.292 + 315.563i) q^{48} +(203.374 + 209.267i) q^{49} +(5.67475 - 7.35915i) q^{50} +(-20.3685 - 712.991i) q^{51} +(-287.599 + 121.540i) q^{52} +(-125.413 + 29.1635i) q^{53} +(5.68172 - 3.65142i) q^{54} +(498.147 - 370.791i) q^{55} +(-19.1273 - 12.2924i) q^{56} +(230.592 + 82.2752i) q^{57} +(-0.577198 - 0.957176i) q^{58} +(-172.259 + 655.340i) q^{59} +(783.398 - 44.7963i) q^{60} +(26.5418 - 1.51771i) q^{61} +(3.38630 - 12.8828i) q^{62} +(81.0760 + 134.449i) q^{63} +(-481.074 - 171.647i) q^{64} +(559.086 + 359.303i) q^{65} +(-11.5860 - 2.55352i) q^{66} +(445.070 - 286.029i) q^{67} +(963.758 - 224.112i) q^{68} +(814.440 - 344.185i) q^{69} +(0.690863 + 24.1834i) q^{70} +(-344.197 + 446.363i) q^{71} +(-3.91909 - 4.03264i) q^{72} +(-235.981 + 391.331i) q^{73} +(-4.05438 + 3.31525i) q^{74} +(-579.895 - 752.021i) q^{75} +(-48.3347 + 336.175i) q^{76} +(-223.394 + 891.644i) q^{77} +(-1.80696 - 12.5677i) q^{78} +(-165.940 + 170.748i) q^{79} +(276.610 + 1052.33i) q^{80} +(-265.266 - 816.404i) q^{81} +(-0.744706 - 3.67527i) q^{82} +(95.0521 - 1106.67i) q^{83} +(-855.714 + 785.368i) q^{84} +(-1551.68 - 1424.12i) q^{85} +(0.672963 + 7.83516i) q^{86} +(-109.594 + 32.1796i) q^{87} +(0.559947 - 32.9179i) q^{88} +(-610.682 - 179.313i) q^{89} +(-1.18827 + 5.86435i) q^{90} +(-977.314 + 112.136i) q^{91} +(692.316 + 1012.48i) q^{92} +(-1185.26 - 669.346i) q^{93} +(15.2588 - 11.0862i) q^{94} +(648.749 - 318.969i) q^{95} +(35.2239 - 51.5133i) q^{96} +(-444.326 + 842.092i) q^{97} +(-10.7801 - 12.4409i) q^{98} +(-103.762 + 202.278i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.0563200 − 0.00322050i −0.0199121 − 0.00113862i 0.0471299 − 0.998889i \(-0.484993\pi\)
−0.0670420 + 0.997750i \(0.521356\pi\)
\(3\) −1.78138 + 5.48253i −0.342827 + 1.05511i 0.619910 + 0.784673i \(0.287169\pi\)
−0.962737 + 0.270440i \(0.912831\pi\)
\(4\) −7.94469 − 0.911569i −0.993087 − 0.113946i
\(5\) 7.94342 + 15.0544i 0.710481 + 1.34651i 0.928781 + 0.370629i \(0.120858\pi\)
−0.218300 + 0.975882i \(0.570051\pi\)
\(6\) 0.117984 − 0.303039i 0.00802778 − 0.0206192i
\(7\) −23.2081 − 9.80784i −1.25312 − 0.529574i −0.341221 − 0.939983i \(-0.610840\pi\)
−0.911901 + 0.410410i \(0.865386\pi\)
\(8\) 0.889196 + 0.153881i 0.0392973 + 0.00680066i
\(9\) −5.04131 − 3.66272i −0.186715 − 0.135656i
\(10\) −0.398891 − 0.873449i −0.0126140 − 0.0276209i
\(11\) −5.60883 − 36.0491i −0.153739 − 0.988112i
\(12\) 19.1502 − 41.9331i 0.460683 − 1.00875i
\(13\) 33.9972 − 19.1991i 0.725318 − 0.409606i −0.0843950 − 0.996432i \(-0.526896\pi\)
0.809713 + 0.586827i \(0.199623\pi\)
\(14\) 1.27550 + 0.627120i 0.0243493 + 0.0119718i
\(15\) −96.6866 + 16.7323i −1.66429 + 0.288017i
\(16\) 62.2624 + 14.4785i 0.972850 + 0.226227i
\(17\) −116.538 + 41.5805i −1.66262 + 0.593221i −0.989269 − 0.146104i \(-0.953327\pi\)
−0.673349 + 0.739325i \(0.735145\pi\)
\(18\) 0.272131 + 0.222520i 0.00356344 + 0.00291381i
\(19\) 1.21280 − 42.4536i 0.0146440 − 0.512607i −0.958853 − 0.283905i \(-0.908370\pi\)
0.973497 − 0.228702i \(-0.0734481\pi\)
\(20\) −49.3848 − 126.844i −0.552139 − 1.41816i
\(21\) 95.1143 − 109.768i 0.988364 − 1.14063i
\(22\) 0.199793 + 2.04835i 0.00193618 + 0.0198505i
\(23\) −100.442 − 115.916i −0.910592 − 1.05088i −0.998500 − 0.0547508i \(-0.982564\pi\)
0.0879078 − 0.996129i \(-0.471982\pi\)
\(24\) −2.42766 + 4.60092i −0.0206476 + 0.0391316i
\(25\) −92.9832 + 135.984i −0.743865 + 1.08787i
\(26\) −1.97655 + 0.971806i −0.0149090 + 0.00733026i
\(27\) −96.8588 + 70.3721i −0.690389 + 0.501597i
\(28\) 175.441 + 99.0761i 1.18412 + 0.668701i
\(29\) 11.1838 + 16.3558i 0.0716131 + 0.104731i 0.859616 − 0.510941i \(-0.170703\pi\)
−0.788003 + 0.615672i \(0.788885\pi\)
\(30\) 5.49928 − 0.630984i 0.0334676 − 0.00384004i
\(31\) −46.8927 + 231.425i −0.271683 + 1.34081i 0.579084 + 0.815268i \(0.303410\pi\)
−0.850767 + 0.525542i \(0.823862\pi\)
\(32\) −10.3869 − 3.04986i −0.0573798 − 0.0168482i
\(33\) 207.632 + 33.4667i 1.09527 + 0.176540i
\(34\) 6.69731 − 1.96651i 0.0337817 − 0.00991921i
\(35\) −36.7003 − 427.294i −0.177242 − 2.06359i
\(36\) 36.7128 + 33.6947i 0.169967 + 0.155994i
\(37\) 68.3986 − 62.7757i 0.303910 − 0.278926i −0.509681 − 0.860363i \(-0.670237\pi\)
0.813591 + 0.581437i \(0.197509\pi\)
\(38\) −0.205027 + 2.38708i −0.000875256 + 0.0101904i
\(39\) 44.6976 + 220.591i 0.183522 + 0.905715i
\(40\) 4.74665 + 14.6087i 0.0187628 + 0.0577460i
\(41\) 16.8990 + 64.2905i 0.0643703 + 0.244890i 0.991550 − 0.129728i \(-0.0414103\pi\)
−0.927179 + 0.374618i \(0.877774\pi\)
\(42\) −5.71034 + 5.87581i −0.0209792 + 0.0215871i
\(43\) −19.8391 − 137.984i −0.0703590 − 0.489358i −0.994283 − 0.106781i \(-0.965946\pi\)
0.923924 − 0.382577i \(-0.124963\pi\)
\(44\) 11.6991 + 291.512i 0.0400844 + 0.998798i
\(45\) 15.0951 − 104.989i 0.0500054 − 0.347795i
\(46\) 5.28359 + 6.85188i 0.0169353 + 0.0219621i
\(47\) −258.828 + 211.642i −0.803275 + 0.656834i −0.942791 − 0.333385i \(-0.891809\pi\)
0.139516 + 0.990220i \(0.455445\pi\)
\(48\) −190.292 + 315.563i −0.572213 + 0.948909i
\(49\) 203.374 + 209.267i 0.592928 + 0.610109i
\(50\) 5.67475 − 7.35915i 0.0160506 − 0.0208148i
\(51\) −20.3685 − 712.991i −0.0559247 − 1.95762i
\(52\) −287.599 + 121.540i −0.766976 + 0.324127i
\(53\) −125.413 + 29.1635i −0.325034 + 0.0755833i −0.385838 − 0.922566i \(-0.626088\pi\)
0.0608049 + 0.998150i \(0.480633\pi\)
\(54\) 5.68172 − 3.65142i 0.0143182 − 0.00920177i
\(55\) 498.147 − 370.791i 1.22127 − 0.909045i
\(56\) −19.1273 − 12.2924i −0.0456428 − 0.0293329i
\(57\) 230.592 + 82.2752i 0.535837 + 0.191186i
\(58\) −0.577198 − 0.957176i −0.00130672 − 0.00216695i
\(59\) −172.259 + 655.340i −0.380105 + 1.44607i 0.452283 + 0.891874i \(0.350610\pi\)
−0.832388 + 0.554194i \(0.813027\pi\)
\(60\) 783.398 − 44.7963i 1.68560 − 0.0963864i
\(61\) 26.5418 − 1.51771i 0.0557103 − 0.00318563i −0.0292115 − 0.999573i \(-0.509300\pi\)
0.0849217 + 0.996388i \(0.472936\pi\)
\(62\) 3.38630 − 12.8828i 0.00693646 − 0.0263890i
\(63\) 81.0760 + 134.449i 0.162137 + 0.268873i
\(64\) −481.074 − 171.647i −0.939597 − 0.335248i
\(65\) 559.086 + 359.303i 1.06686 + 0.685631i
\(66\) −11.5860 − 2.55352i −0.0216082 − 0.00476238i
\(67\) 445.070 − 286.029i 0.811551 − 0.521552i −0.0678151 − 0.997698i \(-0.521603\pi\)
0.879367 + 0.476145i \(0.157966\pi\)
\(68\) 963.758 − 224.112i 1.71872 − 0.399671i
\(69\) 814.440 − 344.185i 1.42097 − 0.600508i
\(70\) 0.690863 + 24.1834i 0.00117963 + 0.0412924i
\(71\) −344.197 + 446.363i −0.575333 + 0.746106i −0.986194 − 0.165593i \(-0.947046\pi\)
0.410861 + 0.911698i \(0.365228\pi\)
\(72\) −3.91909 − 4.03264i −0.00641484 − 0.00660072i
\(73\) −235.981 + 391.331i −0.378349 + 0.627422i −0.985993 − 0.166788i \(-0.946660\pi\)
0.607644 + 0.794210i \(0.292115\pi\)
\(74\) −4.05438 + 3.31525i −0.00636908 + 0.00520797i
\(75\) −579.895 − 752.021i −0.892807 − 1.15781i
\(76\) −48.3347 + 336.175i −0.0729522 + 0.507394i
\(77\) −223.394 + 891.644i −0.330625 + 1.31964i
\(78\) −1.80696 − 12.5677i −0.00262305 − 0.0182437i
\(79\) −165.940 + 170.748i −0.236326 + 0.243174i −0.824912 − 0.565261i \(-0.808776\pi\)
0.588586 + 0.808434i \(0.299685\pi\)
\(80\) 276.610 + 1052.33i 0.386575 + 1.47068i
\(81\) −265.266 − 816.404i −0.363876 − 1.11990i
\(82\) −0.744706 − 3.67527i −0.00100291 − 0.00494958i
\(83\) 95.0521 − 1106.67i 0.125703 − 1.46353i −0.613312 − 0.789841i \(-0.710163\pi\)
0.739015 − 0.673689i \(-0.235291\pi\)
\(84\) −855.714 + 785.368i −1.11150 + 1.02013i
\(85\) −1551.68 − 1424.12i −1.98004 − 1.81726i
\(86\) 0.672963 + 7.83516i 0.000843807 + 0.00982427i
\(87\) −109.594 + 32.1796i −0.135054 + 0.0396554i
\(88\) 0.559947 − 32.9179i 0.000678302 − 0.0398756i
\(89\) −610.682 − 179.313i −0.727328 − 0.213563i −0.102952 − 0.994686i \(-0.532829\pi\)
−0.624376 + 0.781124i \(0.714647\pi\)
\(90\) −1.18827 + 5.86435i −0.00139172 + 0.00686841i
\(91\) −977.314 + 112.136i −1.12583 + 0.129177i
\(92\) 692.316 + 1012.48i 0.784553 + 1.14737i
\(93\) −1185.26 − 669.346i −1.32157 − 0.746322i
\(94\) 15.2588 − 11.0862i 0.0167428 − 0.0121644i
\(95\) 648.749 − 318.969i 0.700635 − 0.344479i
\(96\) 35.2239 − 51.5133i 0.0374481 − 0.0547662i
\(97\) −444.326 + 842.092i −0.465098 + 0.881458i 0.534249 + 0.845327i \(0.320594\pi\)
−0.999347 + 0.0361312i \(0.988497\pi\)
\(98\) −10.7801 − 12.4409i −0.0111118 − 0.0128237i
\(99\) −103.762 + 202.278i −0.105338 + 0.205351i
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.4.17 ✓ 1280
121.91 even 55 inner 121.4.g.a.91.17 yes 1280
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
121.4.g.a.4.17 ✓ 1280 1.1 even 1 trivial
121.4.g.a.91.17 yes 1280 121.91 even 55 inner