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Properties

Label 322.2.i.d.29.1
Level $322$
Weight $2$
Character 322.29
Analytic conductor $2.571$
Analytic rank $0$
Dimension $40$
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] = [322,2,Mod(29,322)] 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("322.29"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(322, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([0, 18])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 322 = 2 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 322.i (of order \(11\), degree \(10\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [40,-4,0] 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: \(2.57118294509\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(4\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label 29.1
Character \(\chi\) \(=\) 322.29
Dual form 322.2.i.d.211.1

$q$-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+(0.415415 - 0.909632i) q^{2} +(-2.49681 - 0.733130i) q^{3} +(-0.654861 - 0.755750i) q^{4} +(-1.90645 + 1.22520i) q^{5} +(-1.70409 + 1.96663i) q^{6} +(0.142315 - 0.989821i) q^{7} +(-0.959493 + 0.281733i) q^{8} +(3.17283 + 2.03906i) q^{9} +(0.322515 + 2.24314i) q^{10} +(2.57290 + 5.63387i) q^{11} +(1.08100 + 2.36706i) q^{12} +(-0.405547 - 2.82064i) q^{13} +(-0.841254 - 0.540641i) q^{14} +(5.65829 - 1.66142i) q^{15} +(-0.142315 + 0.989821i) q^{16} +(-3.20248 + 3.69586i) q^{17} +(3.17283 - 2.03906i) q^{18} +(3.80365 + 4.38965i) q^{19} +(2.17441 + 0.638464i) q^{20} +(-1.08100 + 2.36706i) q^{21} +6.19357 q^{22} +(3.07454 - 3.68066i) q^{23} +2.60222 q^{24} +(0.0563685 - 0.123430i) q^{25} +(-2.73422 - 0.802838i) q^{26} +(-1.31480 - 1.51736i) q^{27} +(-0.841254 + 0.540641i) q^{28} +(-4.57668 + 5.28177i) q^{29} +(0.839255 - 5.83715i) q^{30} +(-6.66582 + 1.95726i) q^{31} +(0.841254 + 0.540641i) q^{32} +(-2.29369 - 15.9530i) q^{33} +(2.03151 + 4.44840i) q^{34} +(0.941416 + 2.06141i) q^{35} +(-0.536748 - 3.73317i) q^{36} +(-7.09832 - 4.56181i) q^{37} +(5.57306 - 1.63640i) q^{38} +(-1.05532 + 7.33993i) q^{39} +(1.48405 - 1.71268i) q^{40} +(-1.48345 + 0.953356i) q^{41} +(1.70409 + 1.96663i) q^{42} +(7.43289 + 2.18249i) q^{43} +(2.57290 - 5.63387i) q^{44} -8.54712 q^{45} +(-2.07083 - 4.32570i) q^{46} +0.254073 q^{47} +(1.08100 - 2.36706i) q^{48} +(-0.959493 - 0.281733i) q^{49} +(-0.0888594 - 0.102549i) q^{50} +(10.7055 - 6.88004i) q^{51} +(-1.86612 + 2.15362i) q^{52} +(-1.47999 + 10.2935i) q^{53} +(-1.92643 + 0.565650i) q^{54} +(-11.8078 - 7.58838i) q^{55} +(0.142315 + 0.989821i) q^{56} +(-6.27882 - 13.7487i) q^{57} +(2.90325 + 6.35723i) q^{58} +(-0.441520 - 3.07084i) q^{59} +(-4.96102 - 3.18825i) q^{60} +(-6.24521 + 1.83376i) q^{61} +(-0.988695 + 6.87652i) q^{62} +(2.46984 - 2.85035i) q^{63} +(0.841254 - 0.540641i) q^{64} +(4.22902 + 4.88054i) q^{65} +(-15.4642 - 4.54069i) q^{66} +(-4.96443 + 10.8706i) q^{67} +4.89033 q^{68} +(-10.3749 + 6.93587i) q^{69} +2.26621 q^{70} +(0.394376 - 0.863562i) q^{71} +(-3.61878 - 1.06257i) q^{72} +(-2.31619 - 2.67303i) q^{73} +(-7.09832 + 4.56181i) q^{74} +(-0.231232 + 0.266856i) q^{75} +(0.826613 - 5.74922i) q^{76} +(5.94268 - 1.74493i) q^{77} +(6.23824 + 4.00907i) q^{78} +(-1.28937 - 8.96779i) q^{79} +(-0.941416 - 2.06141i) q^{80} +(-2.52990 - 5.53970i) q^{81} +(0.250955 + 1.74543i) q^{82} +(5.47724 + 3.52001i) q^{83} +(2.49681 - 0.733130i) q^{84} +(1.57720 - 10.9697i) q^{85} +(5.07300 - 5.85456i) q^{86} +(15.2994 - 9.83230i) q^{87} +(-4.05592 - 4.68079i) q^{88} +(12.6959 + 3.72785i) q^{89} +(-3.55060 + 7.77473i) q^{90} -2.84965 q^{91} +(-4.79505 + 0.0867377i) q^{92} +18.0782 q^{93} +(0.105546 - 0.231113i) q^{94} +(-12.6297 - 3.70842i) q^{95} +(-1.70409 - 1.96663i) q^{96} +(-2.77410 + 1.78281i) q^{97} +(-0.654861 + 0.755750i) q^{98} +(-3.32439 + 23.1216i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40 q - 4 q^{2} - 4 q^{4} + 9 q^{5} + 4 q^{7} - 4 q^{8} - 8 q^{9} - 2 q^{10} + 6 q^{11} + 2 q^{13} + 4 q^{14} - 3 q^{15} - 4 q^{16} - 13 q^{17} - 8 q^{18} - 22 q^{19} - 2 q^{20} - 16 q^{22} - 9 q^{23} + 22 q^{24}+ \cdots + 112 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/322\mathbb{Z}\right)^\times\).

\(n\) \(185\) \(281\)
\(\chi(n)\) \(1\) \(e\left(\frac{9}{11}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.415415 − 0.909632i 0.293743 − 0.643207i
\(3\) −2.49681 − 0.733130i −1.44154 − 0.423273i −0.534802 − 0.844977i \(-0.679614\pi\)
−0.906733 + 0.421704i \(0.861432\pi\)
\(4\) −0.654861 − 0.755750i −0.327430 − 0.377875i
\(5\) −1.90645 + 1.22520i −0.852592 + 0.547928i −0.892382 − 0.451280i \(-0.850967\pi\)
0.0397900 + 0.999208i \(0.487331\pi\)
\(6\) −1.70409 + 1.96663i −0.695693 + 0.802872i
\(7\) 0.142315 − 0.989821i 0.0537900 − 0.374117i
\(8\) −0.959493 + 0.281733i −0.339232 + 0.0996075i
\(9\) 3.17283 + 2.03906i 1.05761 + 0.679685i
\(10\) 0.322515 + 2.24314i 0.101988 + 0.709343i
\(11\) 2.57290 + 5.63387i 0.775759 + 1.69867i 0.713542 + 0.700612i \(0.247090\pi\)
0.0622164 + 0.998063i \(0.480183\pi\)
\(12\) 1.08100 + 2.36706i 0.312058 + 0.683312i
\(13\) −0.405547 − 2.82064i −0.112478 − 0.782305i −0.965495 − 0.260421i \(-0.916139\pi\)
0.853017 − 0.521884i \(-0.174771\pi\)
\(14\) −0.841254 − 0.540641i −0.224834 − 0.144492i
\(15\) 5.65829 − 1.66142i 1.46097 − 0.428978i
\(16\) −0.142315 + 0.989821i −0.0355787 + 0.247455i
\(17\) −3.20248 + 3.69586i −0.776716 + 0.896378i −0.996868 − 0.0790875i \(-0.974799\pi\)
0.220152 + 0.975466i \(0.429345\pi\)
\(18\) 3.17283 − 2.03906i 0.747844 − 0.480610i
\(19\) 3.80365 + 4.38965i 0.872618 + 1.00705i 0.999885 + 0.0151861i \(0.00483407\pi\)
−0.127267 + 0.991869i \(0.540620\pi\)
\(20\) 2.17441 + 0.638464i 0.486213 + 0.142765i
\(21\) −1.08100 + 2.36706i −0.235894 + 0.516536i
\(22\) 6.19357 1.32047
\(23\) 3.07454 − 3.68066i 0.641085 − 0.767470i
\(24\) 2.60222 0.531176
\(25\) 0.0563685 − 0.123430i 0.0112737 − 0.0246860i
\(26\) −2.73422 − 0.802838i −0.536224 − 0.157449i
\(27\) −1.31480 − 1.51736i −0.253033 − 0.292016i
\(28\) −0.841254 + 0.540641i −0.158982 + 0.102172i
\(29\) −4.57668 + 5.28177i −0.849869 + 0.980801i −0.999969 − 0.00786993i \(-0.997495\pi\)
0.150100 + 0.988671i \(0.452040\pi\)
\(30\) 0.839255 − 5.83715i 0.153226 − 1.06571i
\(31\) −6.66582 + 1.95726i −1.19722 + 0.351535i −0.818788 − 0.574096i \(-0.805354\pi\)
−0.378429 + 0.925630i \(0.623536\pi\)
\(32\) 0.841254 + 0.540641i 0.148714 + 0.0955727i
\(33\) −2.29369 − 15.9530i −0.399281 − 2.77706i
\(34\) 2.03151 + 4.44840i 0.348402 + 0.762894i
\(35\) 0.941416 + 2.06141i 0.159128 + 0.348443i
\(36\) −0.536748 − 3.73317i −0.0894580 − 0.622194i
\(37\) −7.09832 − 4.56181i −1.16696 − 0.749958i −0.194013 − 0.980999i \(-0.562150\pi\)
−0.972944 + 0.231041i \(0.925787\pi\)
\(38\) 5.57306 − 1.63640i 0.904070 − 0.265459i
\(39\) −1.05532 + 7.33993i −0.168987 + 1.17533i
\(40\) 1.48405 − 1.71268i 0.234649 − 0.270799i
\(41\) −1.48345 + 0.953356i −0.231676 + 0.148889i −0.651329 − 0.758796i \(-0.725788\pi\)
0.419653 + 0.907685i \(0.362152\pi\)
\(42\) 1.70409 + 1.96663i 0.262947 + 0.303457i
\(43\) 7.43289 + 2.18249i 1.13351 + 0.332827i 0.794085 − 0.607807i \(-0.207951\pi\)
0.339421 + 0.940635i \(0.389769\pi\)
\(44\) 2.57290 − 5.63387i 0.387879 − 0.849337i
\(45\) −8.54712 −1.27413
\(46\) −2.07083 − 4.32570i −0.305328 − 0.637789i
\(47\) 0.254073 0.0370604 0.0185302 − 0.999828i \(-0.494101\pi\)
0.0185302 + 0.999828i \(0.494101\pi\)
\(48\) 1.08100 − 2.36706i 0.156029 − 0.341656i
\(49\) −0.959493 − 0.281733i −0.137070 − 0.0402475i
\(50\) −0.0888594 − 0.102549i −0.0125666 − 0.0145026i
\(51\) 10.7055 − 6.88004i 1.49908 − 0.963398i
\(52\) −1.86612 + 2.15362i −0.258784 + 0.298653i
\(53\) −1.47999 + 10.2935i −0.203292 + 1.41392i 0.591139 + 0.806570i \(0.298678\pi\)
−0.794431 + 0.607355i \(0.792231\pi\)
\(54\) −1.92643 + 0.565650i −0.262153 + 0.0769752i
\(55\) −11.8078 − 7.58838i −1.59216 − 1.02322i
\(56\) 0.142315 + 0.989821i 0.0190176 + 0.132270i
\(57\) −6.27882 − 13.7487i −0.831650 − 1.82106i
\(58\) 2.90325 + 6.35723i 0.381215 + 0.834745i
\(59\) −0.441520 − 3.07084i −0.0574810 − 0.399789i −0.998167 − 0.0605121i \(-0.980727\pi\)
0.940686 − 0.339277i \(-0.110182\pi\)
\(60\) −4.96102 − 3.18825i −0.640464 − 0.411601i
\(61\) −6.24521 + 1.83376i −0.799618 + 0.234789i −0.655918 − 0.754832i \(-0.727718\pi\)
−0.143700 + 0.989621i \(0.545900\pi\)
\(62\) −0.988695 + 6.87652i −0.125564 + 0.873319i
\(63\) 2.46984 − 2.85035i 0.311171 − 0.359110i
\(64\) 0.841254 − 0.540641i 0.105157 − 0.0675801i
\(65\) 4.22902 + 4.88054i 0.524545 + 0.605357i
\(66\) −15.4642 − 4.54069i −1.90351 − 0.558921i
\(67\) −4.96443 + 10.8706i −0.606502 + 1.32805i 0.318439 + 0.947943i \(0.396841\pi\)
−0.924941 + 0.380110i \(0.875886\pi\)
\(68\) 4.89033 0.593039
\(69\) −10.3749 + 6.93587i −1.24900 + 0.834981i
\(70\) 2.26621 0.270863
\(71\) 0.394376 − 0.863562i 0.0468038 − 0.102486i −0.884786 − 0.465998i \(-0.845695\pi\)
0.931589 + 0.363512i \(0.118423\pi\)
\(72\) −3.61878 − 1.06257i −0.426477 − 0.125225i
\(73\) −2.31619 − 2.67303i −0.271090 − 0.312855i 0.603838 − 0.797107i \(-0.293637\pi\)
−0.874929 + 0.484252i \(0.839092\pi\)
\(74\) −7.09832 + 4.56181i −0.825163 + 0.530300i
\(75\) −0.231232 + 0.266856i −0.0267003 + 0.0308138i
\(76\) 0.826613 − 5.74922i 0.0948190 − 0.659481i
\(77\) 5.94268 − 1.74493i 0.677232 − 0.198853i
\(78\) 6.23824 + 4.00907i 0.706341 + 0.453938i
\(79\) −1.28937 − 8.96779i −0.145066 − 1.00896i −0.924148 − 0.382034i \(-0.875224\pi\)
0.779083 − 0.626921i \(-0.215685\pi\)
\(80\) −0.941416 − 2.06141i −0.105254 − 0.230473i
\(81\) −2.52990 − 5.53970i −0.281099 − 0.615522i
\(82\) 0.250955 + 1.74543i 0.0277134 + 0.192751i
\(83\) 5.47724 + 3.52001i 0.601206 + 0.386371i 0.805550 − 0.592528i \(-0.201870\pi\)
−0.204345 + 0.978899i \(0.565506\pi\)
\(84\) 2.49681 − 0.733130i 0.272425 − 0.0799911i
\(85\) 1.57720 − 10.9697i 0.171072 − 1.18983i
\(86\) 5.07300 − 5.85456i 0.547036 − 0.631313i
\(87\) 15.2994 − 9.83230i 1.64026 − 1.05413i
\(88\) −4.05592 − 4.68079i −0.432363 − 0.498973i
\(89\) 12.6959 + 3.72785i 1.34576 + 0.395151i 0.873721 − 0.486427i \(-0.161700\pi\)
0.472039 + 0.881578i \(0.343518\pi\)
\(90\) −3.55060 + 7.77473i −0.374266 + 0.819529i
\(91\) −2.84965 −0.298724
\(92\) −4.79505 + 0.0867377i −0.499918 + 0.00904303i
\(93\) 18.0782 1.87463
\(94\) 0.105546 − 0.231113i 0.0108862 − 0.0238375i
\(95\) −12.6297 − 3.70842i −1.29578 − 0.380475i
\(96\) −1.70409 − 1.96663i −0.173923 − 0.200718i
\(97\) −2.77410 + 1.78281i −0.281667 + 0.181017i −0.673845 − 0.738873i \(-0.735358\pi\)
0.392177 + 0.919890i \(0.371722\pi\)
\(98\) −0.654861 + 0.755750i −0.0661509 + 0.0763422i
\(99\) −3.32439 + 23.1216i −0.334113 + 2.32381i
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 322.2.i.d.29.1 ✓ 40
23.2 even 11 7406.2.a.bu.1.16 20
23.4 even 11 inner 322.2.i.d.211.1 yes 40
23.21 odd 22 7406.2.a.bv.1.16 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
322.2.i.d.29.1 ✓ 40 1.1 even 1 trivial
322.2.i.d.211.1 yes 40 23.4 even 11 inner
7406.2.a.bu.1.16 20 23.2 even 11
7406.2.a.bv.1.16 20 23.21 odd 22