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Properties

Label 340.2.l.b.103.11
Level $340$
Weight $2$
Character 340.103
Analytic conductor $2.715$
Analytic rank $0$
Dimension $88$
Inner twists $4$

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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] = [340,2,Mod(103,340)] 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("340.103"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(340, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 3, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 340 = 2^{2} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 340.l (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [88] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.71491366872\)
Analytic rank: \(0\)
Dimension: \(88\)
Relative dimension: \(44\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 103.11
Character \(\chi\) \(=\) 340.103
Dual form 340.2.l.b.307.11

$q$-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+(-1.15649 - 0.813954i) q^{2} +(-0.897363 + 0.897363i) q^{3} +(0.674957 + 1.88267i) q^{4} +(0.344172 + 2.20942i) q^{5} +(1.76821 - 0.307383i) q^{6} +(-0.526200 - 0.526200i) q^{7} +(0.751820 - 2.72668i) q^{8} +1.38948i q^{9} +(1.40034 - 2.83532i) q^{10} +1.18506i q^{11} +(-2.29512 - 1.08375i) q^{12} +(-1.71494 - 1.71494i) q^{13} +(0.180245 + 1.03685i) q^{14} +(-2.29150 - 1.67381i) q^{15} +(-3.08887 + 2.54144i) q^{16} +(0.707107 - 0.707107i) q^{17} +(1.13097 - 1.60692i) q^{18} -6.00407 q^{19} +(-3.92730 + 2.13923i) q^{20} +0.944386 q^{21} +(0.964581 - 1.37051i) q^{22} +(-1.31333 + 1.31333i) q^{23} +(1.77216 + 3.12148i) q^{24} +(-4.76309 + 1.52084i) q^{25} +(0.587435 + 3.37920i) q^{26} +(-3.93896 - 3.93896i) q^{27} +(0.635497 - 1.34582i) q^{28} +2.97836i q^{29} +(1.28771 + 3.80092i) q^{30} +6.14015i q^{31} +(5.64087 - 0.424963i) q^{32} +(-1.06343 - 1.06343i) q^{33} +(-1.39332 + 0.242212i) q^{34} +(0.981495 - 1.34370i) q^{35} +(-2.61592 + 0.937839i) q^{36} +(-1.87541 + 1.87541i) q^{37} +(6.94367 + 4.88704i) q^{38} +3.07785 q^{39} +(6.28314 + 0.722642i) q^{40} +6.31197 q^{41} +(-1.09218 - 0.768687i) q^{42} +(-6.33722 + 6.33722i) q^{43} +(-2.23106 + 0.799862i) q^{44} +(-3.06994 + 0.478220i) q^{45} +(2.58785 - 0.449868i) q^{46} +(3.78064 + 3.78064i) q^{47} +(0.491241 - 5.05243i) q^{48} -6.44623i q^{49} +(6.74638 + 2.11809i) q^{50} +1.26906i q^{51} +(2.07115 - 4.38617i) q^{52} +(-3.34328 - 3.34328i) q^{53} +(1.34925 + 7.76151i) q^{54} +(-2.61829 + 0.407863i) q^{55} +(-1.83039 + 1.03917i) q^{56} +(5.38783 - 5.38783i) q^{57} +(2.42425 - 3.44446i) q^{58} +2.97588 q^{59} +(1.60455 - 5.44388i) q^{60} -6.50923 q^{61} +(4.99780 - 7.10104i) q^{62} +(0.731144 - 0.731144i) q^{63} +(-6.86953 - 4.09994i) q^{64} +(3.19879 - 4.37926i) q^{65} +(0.364266 + 2.09542i) q^{66} +(0.952830 + 0.952830i) q^{67} +(1.80851 + 0.853979i) q^{68} -2.35707i q^{69} +(-2.22881 + 0.755091i) q^{70} -6.13959i q^{71} +(3.78866 + 1.04464i) q^{72} +(-4.17212 - 4.17212i) q^{73} +(3.69540 - 0.642404i) q^{74} +(2.90947 - 5.63897i) q^{75} +(-4.05249 - 11.3037i) q^{76} +(0.623577 - 0.623577i) q^{77} +(-3.55951 - 2.50523i) q^{78} +10.1743 q^{79} +(-6.67821 - 5.94992i) q^{80} +2.90091 q^{81} +(-7.29976 - 5.13766i) q^{82} +(-1.15689 + 1.15689i) q^{83} +(0.637420 + 1.77796i) q^{84} +(1.80566 + 1.31893i) q^{85} +(12.4872 - 2.17075i) q^{86} +(-2.67267 - 2.67267i) q^{87} +(3.23126 + 0.890949i) q^{88} +15.6449i q^{89} +(3.93962 + 1.94574i) q^{90} +1.80480i q^{91} +(-3.35901 - 1.58612i) q^{92} +(-5.50994 - 5.50994i) q^{93} +(-1.29502 - 7.44955i) q^{94} +(-2.06643 - 13.2655i) q^{95} +(-4.68056 + 5.44325i) q^{96} +(7.83790 - 7.83790i) q^{97} +(-5.24693 + 7.45502i) q^{98} -1.64661 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 88 q - 4 q^{2} + 16 q^{5} - 20 q^{6} - 16 q^{8} - 4 q^{10} - 28 q^{12} - 16 q^{13} + 4 q^{16} + 28 q^{18} + 16 q^{20} - 48 q^{21} - 16 q^{22} - 24 q^{25} - 52 q^{26} + 16 q^{28} + 36 q^{30} - 4 q^{32} + 56 q^{33}+ \cdots - 32 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/340\mathbb{Z}\right)^\times\).

\(n\) \(137\) \(171\) \(241\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.15649 − 0.813954i −0.817765 − 0.575553i
\(3\) −0.897363 + 0.897363i −0.518093 + 0.518093i −0.916994 − 0.398901i \(-0.869392\pi\)
0.398901 + 0.916994i \(0.369392\pi\)
\(4\) 0.674957 + 1.88267i 0.337479 + 0.941333i
\(5\) 0.344172 + 2.20942i 0.153918 + 0.988084i
\(6\) 1.76821 − 0.307383i 0.721868 − 0.125488i
\(7\) −0.526200 − 0.526200i −0.198885 − 0.198885i 0.600637 − 0.799522i \(-0.294914\pi\)
−0.799522 + 0.600637i \(0.794914\pi\)
\(8\) 0.751820 − 2.72668i 0.265809 − 0.964026i
\(9\) 1.38948i 0.463160i
\(10\) 1.40034 − 2.83532i 0.442825 − 0.896608i
\(11\) 1.18506i 0.357308i 0.983912 + 0.178654i \(0.0571742\pi\)
−0.983912 + 0.178654i \(0.942826\pi\)
\(12\) −2.29512 − 1.08375i −0.662543 − 0.312853i
\(13\) −1.71494 − 1.71494i −0.475638 − 0.475638i 0.428095 − 0.903734i \(-0.359185\pi\)
−0.903734 + 0.428095i \(0.859185\pi\)
\(14\) 0.180245 + 1.03685i 0.0481724 + 0.277110i
\(15\) −2.29150 − 1.67381i −0.591663 − 0.432175i
\(16\) −3.08887 + 2.54144i −0.772216 + 0.635360i
\(17\) 0.707107 − 0.707107i 0.171499 − 0.171499i
\(18\) 1.13097 − 1.60692i 0.266573 − 0.378756i
\(19\) −6.00407 −1.37743 −0.688714 − 0.725033i \(-0.741824\pi\)
−0.688714 + 0.725033i \(0.741824\pi\)
\(20\) −3.92730 + 2.13923i −0.878172 + 0.478346i
\(21\) 0.944386 0.206082
\(22\) 0.964581 − 1.37051i 0.205649 − 0.292194i
\(23\) −1.31333 + 1.31333i −0.273849 + 0.273849i −0.830647 − 0.556799i \(-0.812029\pi\)
0.556799 + 0.830647i \(0.312029\pi\)
\(24\) 1.77216 + 3.12148i 0.361741 + 0.637168i
\(25\) −4.76309 + 1.52084i −0.952618 + 0.304168i
\(26\) 0.587435 + 3.37920i 0.115205 + 0.662715i
\(27\) −3.93896 − 3.93896i −0.758053 − 0.758053i
\(28\) 0.635497 − 1.34582i 0.120098 − 0.254337i
\(29\) 2.97836i 0.553068i 0.961004 + 0.276534i \(0.0891859\pi\)
−0.961004 + 0.276534i \(0.910814\pi\)
\(30\) 1.28771 + 3.80092i 0.235102 + 0.693951i
\(31\) 6.14015i 1.10280i 0.834240 + 0.551402i \(0.185907\pi\)
−0.834240 + 0.551402i \(0.814093\pi\)
\(32\) 5.64087 − 0.424963i 0.997174 − 0.0751236i
\(33\) −1.06343 − 1.06343i −0.185119 − 0.185119i
\(34\) −1.39332 + 0.242212i −0.238952 + 0.0415391i
\(35\) 0.981495 − 1.34370i 0.165903 − 0.227127i
\(36\) −2.61592 + 0.937839i −0.435987 + 0.156306i
\(37\) −1.87541 + 1.87541i −0.308316 + 0.308316i −0.844256 − 0.535940i \(-0.819957\pi\)
0.535940 + 0.844256i \(0.319957\pi\)
\(38\) 6.94367 + 4.88704i 1.12641 + 0.792782i
\(39\) 3.07785 0.492850
\(40\) 6.28314 + 0.722642i 0.993451 + 0.114260i
\(41\) 6.31197 0.985765 0.492882 − 0.870096i \(-0.335943\pi\)
0.492882 + 0.870096i \(0.335943\pi\)
\(42\) −1.09218 − 0.768687i −0.168526 − 0.118611i
\(43\) −6.33722 + 6.33722i −0.966417 + 0.966417i −0.999454 − 0.0330370i \(-0.989482\pi\)
0.0330370 + 0.999454i \(0.489482\pi\)
\(44\) −2.23106 + 0.799862i −0.336346 + 0.120584i
\(45\) −3.06994 + 0.478220i −0.457640 + 0.0712888i
\(46\) 2.58785 − 0.449868i 0.381558 − 0.0663295i
\(47\) 3.78064 + 3.78064i 0.551462 + 0.551462i 0.926863 − 0.375400i \(-0.122495\pi\)
−0.375400 + 0.926863i \(0.622495\pi\)
\(48\) 0.491241 − 5.05243i 0.0709045 − 0.729255i
\(49\) − 6.44623i − 0.920889i
\(50\) 6.74638 + 2.11809i 0.954083 + 0.299544i
\(51\) 1.26906i 0.177704i
\(52\) 2.07115 − 4.38617i 0.287216 − 0.608252i
\(53\) −3.34328 − 3.34328i −0.459235 − 0.459235i 0.439169 − 0.898404i \(-0.355273\pi\)
−0.898404 + 0.439169i \(0.855273\pi\)
\(54\) 1.34925 + 7.76151i 0.183610 + 1.05621i
\(55\) −2.61829 + 0.407863i −0.353050 + 0.0549962i
\(56\) −1.83039 + 1.03917i −0.244596 + 0.138865i
\(57\) 5.38783 − 5.38783i 0.713635 − 0.713635i
\(58\) 2.42425 − 3.44446i 0.318320 − 0.452280i
\(59\) 2.97588 0.387426 0.193713 − 0.981058i \(-0.437947\pi\)
0.193713 + 0.981058i \(0.437947\pi\)
\(60\) 1.60455 − 5.44388i 0.207147 − 0.702802i
\(61\) −6.50923 −0.833422 −0.416711 − 0.909039i \(-0.636817\pi\)
−0.416711 + 0.909039i \(0.636817\pi\)
\(62\) 4.99780 − 7.10104i 0.634721 − 0.901834i
\(63\) 0.731144 − 0.731144i 0.0921155 − 0.0921155i
\(64\) −6.86953 − 4.09994i −0.858692 − 0.512493i
\(65\) 3.19879 − 4.37926i 0.396761 − 0.543180i
\(66\) 0.364266 + 2.09542i 0.0448380 + 0.257929i
\(67\) 0.952830 + 0.952830i 0.116407 + 0.116407i 0.762911 − 0.646504i \(-0.223770\pi\)
−0.646504 + 0.762911i \(0.723770\pi\)
\(68\) 1.80851 + 0.853979i 0.219314 + 0.103560i
\(69\) − 2.35707i − 0.283758i
\(70\) −2.22881 + 0.755091i −0.266393 + 0.0902507i
\(71\) − 6.13959i − 0.728635i −0.931275 − 0.364318i \(-0.881302\pi\)
0.931275 − 0.364318i \(-0.118698\pi\)
\(72\) 3.78866 + 1.04464i 0.446498 + 0.123112i
\(73\) −4.17212 − 4.17212i −0.488310 − 0.488310i 0.419463 − 0.907772i \(-0.362218\pi\)
−0.907772 + 0.419463i \(0.862218\pi\)
\(74\) 3.69540 − 0.642404i 0.429582 − 0.0746779i
\(75\) 2.90947 − 5.63897i 0.335957 − 0.651132i
\(76\) −4.05249 − 11.3037i −0.464852 − 1.29662i
\(77\) 0.623577 − 0.623577i 0.0710632 − 0.0710632i
\(78\) −3.55951 − 2.50523i −0.403035 − 0.283661i
\(79\) 10.1743 1.14469 0.572347 − 0.820012i \(-0.306033\pi\)
0.572347 + 0.820012i \(0.306033\pi\)
\(80\) −6.67821 − 5.94992i −0.746647 − 0.665221i
\(81\) 2.90091 0.322324
\(82\) −7.29976 − 5.13766i −0.806124 − 0.567359i
\(83\) −1.15689 + 1.15689i −0.126985 + 0.126985i −0.767743 − 0.640758i \(-0.778620\pi\)
0.640758 + 0.767743i \(0.278620\pi\)
\(84\) 0.637420 + 1.77796i 0.0695482 + 0.193992i
\(85\) 1.80566 + 1.31893i 0.195852 + 0.143058i
\(86\) 12.4872 − 2.17075i 1.34653 − 0.234078i
\(87\) −2.67267 − 2.67267i −0.286541 − 0.286541i
\(88\) 3.23126 + 0.890949i 0.344454 + 0.0949755i
\(89\) 15.6449i 1.65836i 0.558983 + 0.829179i \(0.311192\pi\)
−0.558983 + 0.829179i \(0.688808\pi\)
\(90\) 3.93962 + 1.94574i 0.415273 + 0.205099i
\(91\) 1.80480i 0.189195i
\(92\) −3.35901 − 1.58612i −0.350201 − 0.165365i
\(93\) −5.50994 − 5.50994i −0.571354 − 0.571354i
\(94\) −1.29502 − 7.44955i −0.133571 − 0.768362i
\(95\) −2.06643 − 13.2655i −0.212011 − 1.36101i
\(96\) −4.68056 + 5.44325i −0.477708 + 0.555550i
\(97\) 7.83790 − 7.83790i 0.795819 − 0.795819i −0.186615 − 0.982433i \(-0.559752\pi\)
0.982433 + 0.186615i \(0.0597516\pi\)
\(98\) −5.24693 + 7.45502i −0.530020 + 0.753071i
\(99\) −1.64661 −0.165491
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 340.2.l.b.103.11 ✓ 88
4.3 odd 2 inner 340.2.l.b.103.32 yes 88
5.2 odd 4 inner 340.2.l.b.307.32 yes 88
20.7 even 4 inner 340.2.l.b.307.11 yes 88
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
340.2.l.b.103.11 ✓ 88 1.1 even 1 trivial
340.2.l.b.103.32 yes 88 4.3 odd 2 inner
340.2.l.b.307.11 yes 88 20.7 even 4 inner
340.2.l.b.307.32 yes 88 5.2 odd 4 inner