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Properties

Label 121.4.c.i.27.2
Level $121$
Weight $4$
Character 121.27
Analytic conductor $7.139$
Analytic rank $0$
Dimension $8$
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(3,121)] 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("121.3"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(121, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([8])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 121 = 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 121.c (of order \(5\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,7,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.13923111069\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.29283765625.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 10x^{6} - 19x^{5} + 109x^{4} + 171x^{3} + 810x^{2} + 729x + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 11)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 27.2
Root \(-1.09435 - 3.36805i\) of defining polynomial
Character \(\chi\) \(=\) 121.27
Dual form 121.4.c.i.9.2

$q$-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+(0.976313 + 3.00478i) q^{2} +(1.24700 + 0.906001i) q^{3} +(-1.60339 + 1.16493i) q^{4} +(-5.33576 + 16.4218i) q^{5} +(-1.50487 + 4.63152i) q^{6} +(-7.73807 + 5.62204i) q^{7} +(15.3824 + 11.1760i) q^{8} +(-7.60928 - 23.4190i) q^{9} -54.5532 q^{10} -3.05487 q^{12} +(2.00292 + 6.16434i) q^{13} +(-24.4478 - 17.7624i) q^{14} +(-21.5319 + 15.6438i) q^{15} +(-23.4628 + 72.2111i) q^{16} +(-29.2135 + 89.9099i) q^{17} +(62.9398 - 45.7285i) q^{18} +(-42.5672 - 30.9269i) q^{19} +(-10.5750 - 32.5464i) q^{20} -14.7430 q^{21} +82.2579 q^{23} +(9.05645 + 27.8729i) q^{24} +(-140.077 - 101.772i) q^{25} +(-16.5670 + 12.0367i) q^{26} +(24.5893 - 75.6779i) q^{27} +(5.85788 - 18.0287i) q^{28} +(121.265 - 88.1041i) q^{29} +(-68.0281 - 49.4253i) q^{30} +(40.6007 + 124.956i) q^{31} -87.7765 q^{32} -298.681 q^{34} +(-51.0354 - 157.071i) q^{35} +(39.4822 + 28.6855i) q^{36} +(99.3085 - 72.1518i) q^{37} +(51.3696 - 158.099i) q^{38} +(-3.08726 + 9.50160i) q^{39} +(-265.606 + 192.974i) q^{40} +(153.793 + 111.737i) q^{41} +(-14.3938 - 44.2994i) q^{42} +320.348 q^{43} +425.182 q^{45} +(80.3094 + 247.167i) q^{46} +(356.916 + 259.314i) q^{47} +(-94.6816 + 68.7902i) q^{48} +(-77.7224 + 239.205i) q^{49} +(169.044 - 520.263i) q^{50} +(-117.888 + 85.6505i) q^{51} +(-10.3925 - 7.55060i) q^{52} +(-38.1858 - 117.524i) q^{53} +251.403 q^{54} -181.862 q^{56} +(-25.0617 - 77.1318i) q^{57} +(383.126 + 278.358i) q^{58} +(58.6475 - 42.6099i) q^{59} +(16.3001 - 50.1664i) q^{60} +(-76.8425 + 236.497i) q^{61} +(-335.827 + 243.993i) q^{62} +(190.543 + 138.438i) q^{63} +(102.005 + 313.940i) q^{64} -111.916 q^{65} -128.926 q^{67} +(-57.8984 - 178.193i) q^{68} +(102.576 + 74.5257i) q^{69} +(422.137 - 306.700i) q^{70} +(306.810 - 944.264i) q^{71} +(144.680 - 445.280i) q^{72} +(418.233 - 303.864i) q^{73} +(313.757 + 227.958i) q^{74} +(-82.4713 - 253.820i) q^{75} +104.280 q^{76} -31.5644 q^{78} +(-177.589 - 546.562i) q^{79} +(-1060.64 - 770.603i) q^{80} +(-438.649 + 318.697i) q^{81} +(-185.596 + 571.204i) q^{82} +(-58.1187 + 178.871i) q^{83} +(23.6388 - 17.1746i) q^{84} +(-1320.60 - 959.475i) q^{85} +(312.760 + 962.577i) q^{86} +231.040 q^{87} -42.1254 q^{89} +(415.111 + 1277.58i) q^{90} +(-50.1549 - 36.4396i) q^{91} +(-131.892 + 95.8250i) q^{92} +(-62.5812 + 192.605i) q^{93} +(-430.722 + 1325.63i) q^{94} +(735.003 - 534.011i) q^{95} +(-109.458 - 79.5257i) q^{96} +(-397.278 - 1222.70i) q^{97} -794.640 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 7 q^{2} - 3 q^{3} + 3 q^{4} - 7 q^{5} + 29 q^{6} + 35 q^{7} - 47 q^{8} + 31 q^{9} - 40 q^{10} + 190 q^{12} + 65 q^{13} - 196 q^{14} - 121 q^{15} - 377 q^{16} + 31 q^{17} + 102 q^{18} - 148 q^{19}+ \cdots - 2740 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/121\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{2}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.976313 + 3.00478i 0.345179 + 1.06235i 0.961488 + 0.274847i \(0.0886271\pi\)
−0.616309 + 0.787504i \(0.711373\pi\)
\(3\) 1.24700 + 0.906001i 0.239986 + 0.174360i 0.701277 − 0.712889i \(-0.252614\pi\)
−0.461291 + 0.887249i \(0.652614\pi\)
\(4\) −1.60339 + 1.16493i −0.200424 + 0.145617i
\(5\) −5.33576 + 16.4218i −0.477245 + 1.46881i 0.365661 + 0.930748i \(0.380843\pi\)
−0.842906 + 0.538060i \(0.819157\pi\)
\(6\) −1.50487 + 4.63152i −0.102393 + 0.315135i
\(7\) −7.73807 + 5.62204i −0.417817 + 0.303562i −0.776759 − 0.629798i \(-0.783138\pi\)
0.358942 + 0.933360i \(0.383138\pi\)
\(8\) 15.3824 + 11.1760i 0.679811 + 0.493912i
\(9\) −7.60928 − 23.4190i −0.281825 − 0.867369i
\(10\) −54.5532 −1.72512
\(11\) 0 0
\(12\) −3.05487 −0.0734888
\(13\) 2.00292 + 6.16434i 0.0427314 + 0.131514i 0.970146 − 0.242521i \(-0.0779742\pi\)
−0.927415 + 0.374034i \(0.877974\pi\)
\(14\) −24.4478 − 17.7624i −0.466710 − 0.339085i
\(15\) −21.5319 + 15.6438i −0.370633 + 0.269281i
\(16\) −23.4628 + 72.2111i −0.366607 + 1.12830i
\(17\) −29.2135 + 89.9099i −0.416783 + 1.28273i 0.493863 + 0.869540i \(0.335585\pi\)
−0.910646 + 0.413187i \(0.864415\pi\)
\(18\) 62.9398 − 45.7285i 0.824170 − 0.598794i
\(19\) −42.5672 − 30.9269i −0.513978 − 0.373427i 0.300353 − 0.953828i \(-0.402896\pi\)
−0.814331 + 0.580401i \(0.802896\pi\)
\(20\) −10.5750 − 32.5464i −0.118232 − 0.363880i
\(21\) −14.7430 −0.153199
\(22\) 0 0
\(23\) 82.2579 0.745737 0.372868 − 0.927884i \(-0.378374\pi\)
0.372868 + 0.927884i \(0.378374\pi\)
\(24\) 9.05645 + 27.8729i 0.0770267 + 0.237064i
\(25\) −140.077 − 101.772i −1.12062 − 0.814177i
\(26\) −16.5670 + 12.0367i −0.124964 + 0.0907916i
\(27\) 24.5893 − 75.6779i 0.175267 − 0.539416i
\(28\) 5.85788 − 18.0287i 0.0395370 − 0.121682i
\(29\) 121.265 − 88.1041i 0.776494 − 0.564156i −0.127430 − 0.991848i \(-0.540673\pi\)
0.903925 + 0.427691i \(0.140673\pi\)
\(30\) −68.0281 − 49.4253i −0.414006 − 0.300793i
\(31\) 40.6007 + 124.956i 0.235229 + 0.723961i 0.997091 + 0.0762213i \(0.0242855\pi\)
−0.761862 + 0.647740i \(0.775714\pi\)
\(32\) −87.7765 −0.484902
\(33\) 0 0
\(34\) −298.681 −1.50657
\(35\) −51.0354 − 157.071i −0.246473 − 0.758566i
\(36\) 39.4822 + 28.6855i 0.182788 + 0.132803i
\(37\) 99.3085 − 72.1518i 0.441249 − 0.320586i −0.344882 − 0.938646i \(-0.612081\pi\)
0.786131 + 0.618060i \(0.212081\pi\)
\(38\) 51.3696 − 158.099i 0.219296 − 0.674924i
\(39\) −3.08726 + 9.50160i −0.0126758 + 0.0390121i
\(40\) −265.606 + 192.974i −1.04990 + 0.762796i
\(41\) 153.793 + 111.737i 0.585815 + 0.425619i 0.840816 − 0.541321i \(-0.182076\pi\)
−0.255001 + 0.966941i \(0.582076\pi\)
\(42\) −14.3938 − 44.2994i −0.0528811 − 0.162751i
\(43\) 320.348 1.13611 0.568054 − 0.822991i \(-0.307696\pi\)
0.568054 + 0.822991i \(0.307696\pi\)
\(44\) 0 0
\(45\) 425.182 1.40850
\(46\) 80.3094 + 247.167i 0.257413 + 0.792234i
\(47\) 356.916 + 259.314i 1.10769 + 0.804785i 0.982299 − 0.187322i \(-0.0599808\pi\)
0.125393 + 0.992107i \(0.459981\pi\)
\(48\) −94.6816 + 68.7902i −0.284711 + 0.206854i
\(49\) −77.7224 + 239.205i −0.226596 + 0.697390i
\(50\) 169.044 − 520.263i 0.478128 − 1.47153i
\(51\) −117.888 + 85.6505i −0.323678 + 0.235166i
\(52\) −10.3925 − 7.55060i −0.0277151 − 0.0201362i
\(53\) −38.1858 − 117.524i −0.0989665 − 0.304588i 0.889301 − 0.457323i \(-0.151192\pi\)
−0.988267 + 0.152736i \(0.951192\pi\)
\(54\) 251.403 0.633547
\(55\) 0 0
\(56\) −181.862 −0.433969
\(57\) −25.0617 − 77.1318i −0.0582368 − 0.179234i
\(58\) 383.126 + 278.358i 0.867361 + 0.630175i
\(59\) 58.6475 − 42.6099i 0.129411 − 0.0940227i −0.521197 − 0.853437i \(-0.674514\pi\)
0.650608 + 0.759414i \(0.274514\pi\)
\(60\) 16.3001 − 50.1664i 0.0350721 − 0.107941i
\(61\) −76.8425 + 236.497i −0.161290 + 0.496399i −0.998744 − 0.0501090i \(-0.984043\pi\)
0.837454 + 0.546508i \(0.184043\pi\)
\(62\) −335.827 + 243.993i −0.687905 + 0.499792i
\(63\) 190.543 + 138.438i 0.381051 + 0.276850i
\(64\) 102.005 + 313.940i 0.199229 + 0.613164i
\(65\) −111.916 −0.213562
\(66\) 0 0
\(67\) −128.926 −0.235086 −0.117543 − 0.993068i \(-0.537502\pi\)
−0.117543 + 0.993068i \(0.537502\pi\)
\(68\) −57.8984 − 178.193i −0.103253 − 0.317780i
\(69\) 102.576 + 74.5257i 0.178966 + 0.130027i
\(70\) 422.137 − 306.700i 0.720786 − 0.523682i
\(71\) 306.810 − 944.264i 0.512840 − 1.57836i −0.274338 − 0.961633i \(-0.588459\pi\)
0.787178 − 0.616726i \(-0.211541\pi\)
\(72\) 144.680 − 445.280i 0.236816 − 0.728844i
\(73\) 418.233 − 303.864i 0.670554 − 0.487186i −0.199657 − 0.979866i \(-0.563983\pi\)
0.870211 + 0.492680i \(0.163983\pi\)
\(74\) 313.757 + 227.958i 0.492885 + 0.358102i
\(75\) −82.4713 − 253.820i −0.126973 − 0.390782i
\(76\) 104.280 0.157391
\(77\) 0 0
\(78\) −31.5644 −0.0458200
\(79\) −177.589 − 546.562i −0.252915 − 0.778392i −0.994233 − 0.107239i \(-0.965799\pi\)
0.741318 − 0.671154i \(-0.234201\pi\)
\(80\) −1060.64 − 770.603i −1.48229 − 1.07695i
\(81\) −438.649 + 318.697i −0.601714 + 0.437171i
\(82\) −185.596 + 571.204i −0.249946 + 0.769256i
\(83\) −58.1187 + 178.871i −0.0768597 + 0.236550i −0.982103 − 0.188343i \(-0.939688\pi\)
0.905244 + 0.424893i \(0.139688\pi\)
\(84\) 23.6388 − 17.1746i 0.0307048 − 0.0223084i
\(85\) −1320.60 − 959.475i −1.68517 − 1.22435i
\(86\) 312.760 + 962.577i 0.392160 + 1.20695i
\(87\) 231.040 0.284714
\(88\) 0 0
\(89\) −42.1254 −0.0501717 −0.0250859 − 0.999685i \(-0.507986\pi\)
−0.0250859 + 0.999685i \(0.507986\pi\)
\(90\) 415.111 + 1277.58i 0.486184 + 1.49632i
\(91\) −50.1549 − 36.4396i −0.0577765 − 0.0419771i
\(92\) −131.892 + 95.8250i −0.149464 + 0.108592i
\(93\) −62.5812 + 192.605i −0.0697781 + 0.214755i
\(94\) −430.722 + 1325.63i −0.472613 + 1.45455i
\(95\) 735.003 − 534.011i 0.793786 − 0.576719i
\(96\) −109.458 − 79.5257i −0.116370 − 0.0845474i
\(97\) −397.278 − 1222.70i −0.415850 − 1.27985i −0.911488 − 0.411326i \(-0.865066\pi\)
0.495638 − 0.868529i \(-0.334934\pi\)
\(98\) −794.640 −0.819089
\(99\) 0 0
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.c.i.27.2 8
11.2 odd 10 11.4.c.a.9.1 yes 8
11.3 even 5 121.4.a.f.1.4 4
11.4 even 5 121.4.c.b.3.1 8
11.5 even 5 121.4.c.b.81.1 8
11.6 odd 10 121.4.c.h.81.2 8
11.7 odd 10 121.4.c.h.3.2 8
11.8 odd 10 121.4.a.g.1.1 4
11.9 even 5 inner 121.4.c.i.9.2 8
11.10 odd 2 11.4.c.a.5.1 ✓ 8
33.2 even 10 99.4.f.c.64.2 8
33.8 even 10 1089.4.a.y.1.4 4
33.14 odd 10 1089.4.a.bh.1.1 4
33.32 even 2 99.4.f.c.82.2 8
44.3 odd 10 1936.4.a.bl.1.3 4
44.19 even 10 1936.4.a.bk.1.3 4
44.35 even 10 176.4.m.c.97.1 8
44.43 even 2 176.4.m.c.49.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.4.c.a.5.1 ✓ 8 11.10 odd 2
11.4.c.a.9.1 yes 8 11.2 odd 10
99.4.f.c.64.2 8 33.2 even 10
99.4.f.c.82.2 8 33.32 even 2
121.4.a.f.1.4 4 11.3 even 5
121.4.a.g.1.1 4 11.8 odd 10
121.4.c.b.3.1 8 11.4 even 5
121.4.c.b.81.1 8 11.5 even 5
121.4.c.h.3.2 8 11.7 odd 10
121.4.c.h.81.2 8 11.6 odd 10
121.4.c.i.9.2 8 11.9 even 5 inner
121.4.c.i.27.2 8 1.1 even 1 trivial
176.4.m.c.49.1 8 44.43 even 2
176.4.m.c.97.1 8 44.35 even 10
1089.4.a.y.1.4 4 33.8 even 10
1089.4.a.bh.1.1 4 33.14 odd 10
1936.4.a.bk.1.3 4 44.19 even 10
1936.4.a.bl.1.3 4 44.3 odd 10