↑

Properties

Label 121.4.g.a.4.20
Level $121$
Weight $4$
Character 121.4
Analytic conductor $7.139$
Analytic rank $0$
Dimension $1280$
Inner twists $2$

Related objects

Downloads

Learn more

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 4.20
Character \(\chi\) \(=\) 121.4
Dual form 121.4.g.a.91.20

$q$-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+(1.28494 + 0.0734758i) q^{2} +(-0.222015 + 0.683293i) q^{3} +(-6.30217 - 0.723107i) q^{4} +(2.87383 + 5.44652i) q^{5} +(-0.335483 + 0.861681i) q^{6} +(-1.37445 - 0.580847i) q^{7} +(-18.1904 - 3.14797i) q^{8} +(21.4259 + 15.5668i) q^{9} +(3.29253 + 7.20963i) q^{10} +(-22.9762 + 28.3389i) q^{11} +(1.89327 - 4.14569i) q^{12} +(-78.6305 + 44.4047i) q^{13} +(-1.72341 - 0.847345i) q^{14} +(-4.35960 + 0.754459i) q^{15} +(26.2870 + 6.11278i) q^{16} +(-55.6348 + 19.8505i) q^{17} +(26.3873 + 21.5768i) q^{18} +(-0.145153 + 5.08101i) q^{19} +(-14.1730 - 36.4030i) q^{20} +(0.702037 - 0.810194i) q^{21} +(-31.6054 + 34.7257i) q^{22} +(81.4962 + 94.0516i) q^{23} +(6.18952 - 11.7304i) q^{24} +(49.1497 - 71.8792i) q^{25} +(-104.299 + 51.2801i) q^{26} +(-31.0871 + 22.5861i) q^{27} +(8.24199 + 4.65447i) q^{28} +(-44.3211 - 64.8175i) q^{29} +(-5.65728 + 0.649113i) q^{30} +(18.1321 - 89.4855i) q^{31} +(175.032 + 51.3940i) q^{32} +(-14.2627 - 21.9911i) q^{33} +(-72.9462 + 21.4189i) q^{34} +(-0.786342 - 9.15522i) q^{35} +(-123.773 - 113.598i) q^{36} +(237.132 - 217.637i) q^{37} +(-0.559845 + 6.51816i) q^{38} +(-12.8842 - 63.5862i) q^{39} +(-35.1306 - 108.121i) q^{40} +(21.2788 + 80.9529i) q^{41} +(0.961608 - 0.989472i) q^{42} +(53.2364 + 370.268i) q^{43} +(165.292 - 161.982i) q^{44} +(-23.2105 + 161.433i) q^{45} +(97.8076 + 126.839i) q^{46} +(-444.954 + 363.837i) q^{47} +(-10.0129 + 16.6046i) q^{48} +(-237.498 - 244.380i) q^{49} +(68.4361 - 88.7495i) q^{50} +(-1.21190 - 42.4220i) q^{51} +(527.652 - 222.988i) q^{52} +(299.694 - 69.6909i) q^{53} +(-41.6048 + 26.7378i) q^{54} +(-220.378 - 43.6990i) q^{55} +(23.1732 + 14.8925i) q^{56} +(-3.43959 - 1.22724i) q^{57} +(-52.1876 - 86.5434i) q^{58} +(140.272 - 533.651i) q^{59} +(28.0205 - 1.60227i) q^{60} +(488.999 - 27.9620i) q^{61} +(29.8738 - 113.652i) q^{62} +(-20.4068 - 33.8409i) q^{63} +(17.7792 + 6.34359i) q^{64} +(-467.822 - 300.651i) q^{65} +(-16.7110 - 29.3054i) q^{66} +(350.624 - 225.332i) q^{67} +(364.974 - 84.8711i) q^{68} +(-82.3582 + 34.8049i) q^{69} +(-0.337719 - 11.8217i) q^{70} +(-393.908 + 510.828i) q^{71} +(-340.740 - 350.613i) q^{72} +(-19.7937 + 32.8241i) q^{73} +(320.692 - 262.229i) q^{74} +(38.2026 + 49.5420i) q^{75} +(4.58889 - 31.9164i) q^{76} +(48.0402 - 25.6047i) q^{77} +(-11.8835 - 82.6514i) q^{78} +(-421.129 + 433.332i) q^{79} +(42.2511 + 160.740i) q^{80} +(212.436 + 653.809i) q^{81} +(21.3940 + 105.584i) q^{82} +(-8.64538 + 100.656i) q^{83} +(-5.01021 + 4.59833i) q^{84} +(-268.001 - 245.969i) q^{85} +(41.2002 + 479.685i) q^{86} +(54.1293 - 15.8938i) q^{87} +(507.155 - 443.167i) q^{88} +(545.978 + 160.314i) q^{89} +(-41.6856 + 205.727i) q^{90} +(133.866 - 15.3597i) q^{91} +(-445.594 - 651.660i) q^{92} +(57.1192 + 32.2567i) q^{93} +(-598.475 + 434.818i) q^{94} +(-28.0910 + 13.8114i) q^{95} +(-73.9769 + 108.188i) q^{96} +(-596.515 + 1130.52i) q^{97} +(-287.216 - 331.465i) q^{98} +(-933.431 + 249.520i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.28494 + 0.0734758i 0.454297 + 0.0259776i 0.282740 − 0.959197i \(-0.408757\pi\)
0.171557 + 0.985174i \(0.445120\pi\)
\(3\) −0.222015 + 0.683293i −0.0427269 + 0.131500i −0.970144 − 0.242528i \(-0.922023\pi\)
0.927418 + 0.374028i \(0.122023\pi\)
\(4\) −6.30217 − 0.723107i −0.787771 − 0.0903883i
\(5\) 2.87383 + 5.44652i 0.257043 + 0.487152i 0.978753 − 0.205045i \(-0.0657341\pi\)
−0.721709 + 0.692196i \(0.756643\pi\)
\(6\) −0.335483 + 0.861681i −0.0228267 + 0.0586300i
\(7\) −1.37445 − 0.580847i −0.0742132 − 0.0313628i 0.351848 − 0.936057i \(-0.385554\pi\)
−0.426061 + 0.904695i \(0.640099\pi\)
\(8\) −18.1904 − 3.14797i −0.803908 − 0.139122i
\(9\) 21.4259 + 15.5668i 0.793550 + 0.576548i
\(10\) 3.29253 + 7.20963i 0.104119 + 0.227989i
\(11\) −22.9762 + 28.3389i −0.629780 + 0.776773i
\(12\) 1.89327 − 4.14569i 0.0455450 − 0.0997297i
\(13\) −78.6305 + 44.4047i −1.67755 + 0.947357i −0.706754 + 0.707460i \(0.749841\pi\)
−0.970798 + 0.239897i \(0.922886\pi\)
\(14\) −1.72341 − 0.847345i −0.0329001 − 0.0161759i
\(15\) −4.35960 + 0.754459i −0.0750430 + 0.0129867i
\(16\) 26.2870 + 6.11278i 0.410734 + 0.0955122i
\(17\) −55.6348 + 19.8505i −0.793731 + 0.283203i −0.701625 − 0.712546i \(-0.747542\pi\)
−0.0921063 + 0.995749i \(0.529360\pi\)
\(18\) 26.3873 + 21.5768i 0.345530 + 0.282538i
\(19\) −0.145153 + 5.08101i −0.00175265 + 0.0613508i 0.998245 + 0.0592180i \(0.0188607\pi\)
−0.999998 + 0.00213278i \(0.999321\pi\)
\(20\) −14.1730 − 36.4030i −0.158459 − 0.406998i
\(21\) 0.702037 − 0.810194i 0.00729510 − 0.00841899i
\(22\) −31.6054 + 34.7257i −0.306286 + 0.336525i
\(23\) 81.4962 + 94.0516i 0.738832 + 0.852657i 0.993436 − 0.114388i \(-0.0364907\pi\)
−0.254604 + 0.967045i \(0.581945\pi\)
\(24\) 6.18952 − 11.7304i 0.0526429 − 0.0997695i
\(25\) 49.1497 − 71.8792i 0.393198 − 0.575034i
\(26\) −104.299 + 51.2801i −0.786716 + 0.386802i
\(27\) −31.0871 + 22.5861i −0.221582 + 0.160989i
\(28\) 8.24199 + 4.65447i 0.0556282 + 0.0314147i
\(29\) −44.3211 − 64.8175i −0.283801 − 0.415045i 0.657060 − 0.753838i \(-0.271800\pi\)
−0.940861 + 0.338793i \(0.889981\pi\)
\(30\) −5.65728 + 0.649113i −0.0344291 + 0.00395038i
\(31\) 18.1321 − 89.4855i 0.105052 − 0.518454i −0.892653 − 0.450745i \(-0.851158\pi\)
0.997705 − 0.0677089i \(-0.0215689\pi\)
\(32\) 175.032 + 51.3940i 0.966923 + 0.283914i
\(33\) −14.2627 − 21.9911i −0.0752370 − 0.116005i
\(34\) −72.9462 + 21.4189i −0.367946 + 0.108039i
\(35\) −0.786342 − 9.15522i −0.00379760 − 0.0442147i
\(36\) −123.773 − 113.598i −0.573023 − 0.525916i
\(37\) 237.132 − 217.637i 1.05363 − 0.967010i 0.0541235 − 0.998534i \(-0.482764\pi\)
0.999503 + 0.0315246i \(0.0100363\pi\)
\(38\) −0.559845 + 6.51816i −0.00238997 + 0.0278259i
\(39\) −12.8842 − 63.5862i −0.0529007 − 0.261075i
\(40\) −35.1306 − 108.121i −0.138866 − 0.427385i
\(41\) 21.2788 + 80.9529i 0.0810534 + 0.308359i 0.995305 − 0.0967877i \(-0.0308568\pi\)
−0.914252 + 0.405147i \(0.867220\pi\)
\(42\) 0.961608 − 0.989472i 0.00353284 − 0.00363521i
\(43\) 53.2364 + 370.268i 0.188802 + 1.31315i 0.835116 + 0.550074i \(0.185401\pi\)
−0.646314 + 0.763072i \(0.723690\pi\)
\(44\) 165.292 − 161.982i 0.566334 − 0.554995i
\(45\) −23.2105 + 161.433i −0.0768894 + 0.534777i
\(46\) 97.8076 + 126.839i 0.313499 + 0.406552i
\(47\) −444.954 + 363.837i −1.38092 + 1.12917i −0.404197 + 0.914672i \(0.632449\pi\)
−0.976724 + 0.214501i \(0.931187\pi\)
\(48\) −10.0129 + 16.6046i −0.0301092 + 0.0499305i
\(49\) −237.498 − 244.380i −0.692414 − 0.712477i
\(50\) 68.4361 − 88.7495i 0.193566 − 0.251021i
\(51\) −1.21190 − 42.4220i −0.00332745 − 0.116476i
\(52\) 527.652 − 222.988i 1.40716 − 0.594670i
\(53\) 299.694 − 69.6909i 0.776720 − 0.180618i 0.180646 − 0.983548i \(-0.442181\pi\)
0.596074 + 0.802930i \(0.296727\pi\)
\(54\) −41.6048 + 26.7378i −0.104846 + 0.0673805i
\(55\) −220.378 − 43.6990i −0.540287 − 0.107134i
\(56\) 23.1732 + 14.8925i 0.0552974 + 0.0355374i
\(57\) −3.43959 − 1.22724i −0.00799273 − 0.00285180i
\(58\) −52.1876 − 86.5434i −0.118148 − 0.195926i
\(59\) 140.272 − 533.651i 0.309523 − 1.17755i −0.612517 − 0.790457i \(-0.709843\pi\)
0.922041 − 0.387092i \(-0.126521\pi\)
\(60\) 28.0205 − 1.60227i 0.0602905 − 0.00344754i
\(61\) 488.999 − 27.9620i 1.02639 − 0.0586912i 0.464227 − 0.885716i \(-0.346332\pi\)
0.562165 + 0.827025i \(0.309969\pi\)
\(62\) 29.8738 − 113.652i 0.0611931 − 0.232803i
\(63\) −20.4068 − 33.8409i −0.0408098 − 0.0676754i
\(64\) 17.7792 + 6.34359i 0.0347249 + 0.0123898i
\(65\) −467.822 − 300.651i −0.892710 − 0.573710i
\(66\) −16.7110 − 29.3054i −0.0311664 − 0.0546552i
\(67\) 350.624 − 225.332i 0.639337 − 0.410877i −0.180419 − 0.983590i \(-0.557745\pi\)
0.819756 + 0.572713i \(0.194109\pi\)
\(68\) 364.974 − 84.8711i 0.650877 − 0.151355i
\(69\) −82.3582 + 34.8049i −0.143692 + 0.0607249i
\(70\) −0.337719 − 11.8217i −0.000576646 − 0.0201852i
\(71\) −393.908 + 510.828i −0.658426 + 0.853862i −0.996176 − 0.0873742i \(-0.972152\pi\)
0.337750 + 0.941236i \(0.390334\pi\)
\(72\) −340.740 − 350.613i −0.557731 − 0.573892i
\(73\) −19.7937 + 32.8241i −0.0317353 + 0.0526270i −0.871812 − 0.489840i \(-0.837055\pi\)
0.840077 + 0.542467i \(0.182510\pi\)
\(74\) 320.692 − 262.229i 0.503779 − 0.411938i
\(75\) 38.2026 + 49.5420i 0.0588167 + 0.0762748i
\(76\) 4.58889 − 31.9164i 0.00692608 − 0.0481719i
\(77\) 48.0402 − 25.6047i 0.0710998 − 0.0378952i
\(78\) −11.8835 − 82.6514i −0.0172505 − 0.119980i
\(79\) −421.129 + 433.332i −0.599756 + 0.617135i −0.947230 − 0.320555i \(-0.896131\pi\)
0.347474 + 0.937690i \(0.387040\pi\)
\(80\) 42.2511 + 160.740i 0.0590477 + 0.224641i
\(81\) 212.436 + 653.809i 0.291407 + 0.896858i
\(82\) 21.3940 + 105.584i 0.0288118 + 0.142192i
\(83\) −8.64538 + 100.656i −0.0114332 + 0.133114i −0.999823 − 0.0188269i \(-0.994007\pi\)
0.988390 + 0.151941i \(0.0485523\pi\)
\(84\) −5.01021 + 4.59833i −0.00650785 + 0.00597285i
\(85\) −268.001 − 245.969i −0.341986 − 0.313872i
\(86\) 41.2002 + 479.685i 0.0516596 + 0.601462i
\(87\) 54.1293 − 15.8938i 0.0667042 − 0.0195861i
\(88\) 507.155 − 443.167i 0.614351 − 0.536838i
\(89\) 545.978 + 160.314i 0.650264 + 0.190935i 0.590197 − 0.807259i \(-0.299050\pi\)
0.0600677 + 0.998194i \(0.480868\pi\)
\(90\) −41.6856 + 205.727i −0.0488228 + 0.240950i
\(91\) 133.866 − 15.3597i 0.154208 − 0.0176938i
\(92\) −445.594 − 651.660i −0.504960 − 0.738481i
\(93\) 57.1192 + 32.2567i 0.0636880 + 0.0359663i
\(94\) −598.475 + 434.818i −0.656681 + 0.477106i
\(95\) −28.0910 + 13.8114i −0.0303376 + 0.0149160i
\(96\) −73.9769 + 108.188i −0.0786483 + 0.115019i
\(97\) −596.515 + 1130.52i −0.624401 + 1.18337i 0.345165 + 0.938542i \(0.387823\pi\)
−0.969566 + 0.244829i \(0.921268\pi\)
\(98\) −287.216 − 331.465i −0.296053 − 0.341663i
\(99\) −933.431 + 249.520i −0.947610 + 0.253310i
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.4.20 ✓ 1280
121.91 even 55 inner 121.4.g.a.91.20 yes 1280
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
121.4.g.a.4.20 ✓ 1280 1.1 even 1 trivial
121.4.g.a.91.20 yes 1280 121.91 even 55 inner