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Properties

Label 340.2.bi.a.333.7
Level $340$
Weight $2$
Character 340.333
Analytic conductor $2.715$
Analytic rank $0$
Dimension $72$
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] = [340,2,Mod(37,340)] 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("340.37"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(340, base_ring=CyclotomicField(16)) chi = DirichletCharacter(H, H._module([0, 4, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 340 = 2^{2} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 340.bi (of order \(16\), degree \(8\), 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: \(2.71491366872\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(9\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label 333.7
Character \(\chi\) \(=\) 340.333
Dual form 340.2.bi.a.97.7

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.296491 + 1.49056i) q^{3} +(2.22866 - 0.181908i) q^{5} +(3.53511 - 2.36209i) q^{7} +(0.637769 - 0.264172i) q^{9} +(-1.78929 - 2.67786i) q^{11} -4.88344 q^{13} +(0.931922 + 3.26802i) q^{15} +(-3.99677 + 1.01284i) q^{17} +(4.80104 + 1.98866i) q^{19} +(4.56897 + 4.56897i) q^{21} +(3.80720 + 0.757298i) q^{23} +(4.93382 - 0.810820i) q^{25} +(3.11587 + 4.66322i) q^{27} +(-6.58513 + 1.30986i) q^{29} +(-2.09957 + 3.14223i) q^{31} +(3.46101 - 3.46101i) q^{33} +(7.44887 - 5.90735i) q^{35} +(-10.4471 + 2.07806i) q^{37} +(-1.44790 - 7.27907i) q^{39} +(-4.07539 - 0.810645i) q^{41} +(-2.50045 + 6.03662i) q^{43} +(1.37331 - 0.704765i) q^{45} +7.93989i q^{47} +(4.23879 - 10.2333i) q^{49} +(-2.69471 - 5.65714i) q^{51} +(-2.02084 + 0.837059i) q^{53} +(-4.47483 - 5.64254i) q^{55} +(-1.54075 + 7.74588i) q^{57} +(-3.66951 - 8.85898i) q^{59} +(1.78113 - 8.95434i) q^{61} +(1.63059 - 2.44035i) q^{63} +(-10.8835 + 0.888336i) q^{65} +(0.847172 + 0.847172i) q^{67} +5.89940i q^{69} +(-4.58999 - 3.06693i) q^{71} +(10.8023 + 7.21787i) q^{73} +(2.67141 + 7.11377i) q^{75} +(-12.6507 - 5.24008i) q^{77} +(11.7115 - 7.82536i) q^{79} +(-4.56262 + 4.56262i) q^{81} +(-2.14367 - 5.17528i) q^{83} +(-8.72318 + 2.98431i) q^{85} +(-3.90487 - 9.42719i) q^{87} +(-1.27573 + 1.27573i) q^{89} +(-17.2635 + 11.5351i) q^{91} +(-5.30620 - 2.19790i) q^{93} +(11.0616 + 3.55869i) q^{95} +(7.04095 + 4.70461i) q^{97} +(-1.84857 - 1.23517i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q + 24 q^{15} + 8 q^{25} + 48 q^{27} - 32 q^{31} - 16 q^{33} - 32 q^{37} + 32 q^{39} - 40 q^{41} + 32 q^{53} - 16 q^{55} + 72 q^{57} - 112 q^{59} - 48 q^{63} + 32 q^{67} - 16 q^{71} - 96 q^{73} + 24 q^{75}+ \cdots - 96 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(137\) \(171\) \(241\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(1\) \(e\left(\frac{3}{16}\right)\)

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 0
\(3\) 0.296491 + 1.49056i 0.171179 + 0.860577i 0.966948 + 0.254975i \(0.0820674\pi\)
−0.795768 + 0.605601i \(0.792933\pi\)
\(4\) 0 0
\(5\) 2.22866 − 0.181908i 0.996685 − 0.0813516i
\(6\) 0 0
\(7\) 3.53511 − 2.36209i 1.33615 − 0.892785i 0.337329 − 0.941387i \(-0.390476\pi\)
0.998818 + 0.0486015i \(0.0154764\pi\)
\(8\) 0 0
\(9\) 0.637769 − 0.264172i 0.212590 − 0.0880575i
\(10\) 0 0
\(11\) −1.78929 − 2.67786i −0.539490 − 0.807404i 0.457143 − 0.889393i \(-0.348873\pi\)
−0.996633 + 0.0819890i \(0.973873\pi\)
\(12\) 0 0
\(13\) −4.88344 −1.35442 −0.677211 − 0.735789i \(-0.736812\pi\)
−0.677211 + 0.735789i \(0.736812\pi\)
\(14\) 0 0
\(15\) 0.931922 + 3.26802i 0.240621 + 0.843799i
\(16\) 0 0
\(17\) −3.99677 + 1.01284i −0.969359 + 0.245649i
\(18\) 0 0
\(19\) 4.80104 + 1.98866i 1.10143 + 0.456229i 0.857980 − 0.513683i \(-0.171719\pi\)
0.243455 + 0.969912i \(0.421719\pi\)
\(20\) 0 0
\(21\) 4.56897 + 4.56897i 0.997031 + 0.997031i
\(22\) 0 0
\(23\) 3.80720 + 0.757298i 0.793855 + 0.157908i 0.575327 − 0.817923i \(-0.304875\pi\)
0.218528 + 0.975831i \(0.429875\pi\)
\(24\) 0 0
\(25\) 4.93382 − 0.810820i 0.986764 − 0.162164i
\(26\) 0 0
\(27\) 3.11587 + 4.66322i 0.599649 + 0.897438i
\(28\) 0 0
\(29\) −6.58513 + 1.30986i −1.22283 + 0.243236i −0.763957 − 0.645267i \(-0.776746\pi\)
−0.458871 + 0.888503i \(0.651746\pi\)
\(30\) 0 0
\(31\) −2.09957 + 3.14223i −0.377094 + 0.564361i −0.970670 − 0.240417i \(-0.922716\pi\)
0.593575 + 0.804778i \(0.297716\pi\)
\(32\) 0 0
\(33\) 3.46101 − 3.46101i 0.602484 − 0.602484i
\(34\) 0 0
\(35\) 7.44887 − 5.90735i 1.25909 − 0.998524i
\(36\) 0 0
\(37\) −10.4471 + 2.07806i −1.71749 + 0.341630i −0.952992 − 0.302994i \(-0.902014\pi\)
−0.764499 + 0.644625i \(0.777014\pi\)
\(38\) 0 0
\(39\) −1.44790 − 7.27907i −0.231849 − 1.16558i
\(40\) 0 0
\(41\) −4.07539 − 0.810645i −0.636469 − 0.126602i −0.133699 − 0.991022i \(-0.542686\pi\)
−0.502770 + 0.864420i \(0.667686\pi\)
\(42\) 0 0
\(43\) −2.50045 + 6.03662i −0.381315 + 0.920575i 0.610397 + 0.792095i \(0.291010\pi\)
−0.991712 + 0.128480i \(0.958990\pi\)
\(44\) 0 0
\(45\) 1.37331 − 0.704765i 0.204721 − 0.105060i
\(46\) 0 0
\(47\) 7.93989i 1.15815i 0.815274 + 0.579076i \(0.196586\pi\)
−0.815274 + 0.579076i \(0.803414\pi\)
\(48\) 0 0
\(49\) 4.23879 − 10.2333i 0.605541 − 1.46191i
\(50\) 0 0
\(51\) −2.69471 − 5.65714i −0.377334 − 0.792157i
\(52\) 0 0
\(53\) −2.02084 + 0.837059i −0.277584 + 0.114979i −0.517132 − 0.855905i \(-0.673001\pi\)
0.239549 + 0.970884i \(0.423001\pi\)
\(54\) 0 0
\(55\) −4.47483 − 5.64254i −0.603386 − 0.760840i
\(56\) 0 0
\(57\) −1.54075 + 7.74588i −0.204077 + 1.02597i
\(58\) 0 0
\(59\) −3.66951 − 8.85898i −0.477730 − 1.15334i −0.960671 − 0.277689i \(-0.910432\pi\)
0.482942 − 0.875653i \(-0.339568\pi\)
\(60\) 0 0
\(61\) 1.78113 − 8.95434i 0.228050 − 1.14649i −0.681797 − 0.731542i \(-0.738801\pi\)
0.909847 − 0.414944i \(-0.136199\pi\)
\(62\) 0 0
\(63\) 1.63059 − 2.44035i 0.205435 − 0.307455i
\(64\) 0 0
\(65\) −10.8835 + 0.888336i −1.34993 + 0.110184i
\(66\) 0 0
\(67\) 0.847172 + 0.847172i 0.103499 + 0.103499i 0.756960 − 0.653461i \(-0.226684\pi\)
−0.653461 + 0.756960i \(0.726684\pi\)
\(68\) 0 0
\(69\) 5.89940i 0.710204i
\(70\) 0 0
\(71\) −4.58999 − 3.06693i −0.544731 − 0.363978i 0.252552 − 0.967583i \(-0.418730\pi\)
−0.797283 + 0.603606i \(0.793730\pi\)
\(72\) 0 0
\(73\) 10.8023 + 7.21787i 1.26431 + 0.844788i 0.993047 − 0.117717i \(-0.0375574\pi\)
0.271267 + 0.962504i \(0.412557\pi\)
\(74\) 0 0
\(75\) 2.67141 + 7.11377i 0.308468 + 0.821427i
\(76\) 0 0
\(77\) −12.6507 − 5.24008i −1.44168 − 0.597162i
\(78\) 0 0
\(79\) 11.7115 − 7.82536i 1.31764 − 0.880421i 0.319893 − 0.947454i \(-0.396353\pi\)
0.997751 + 0.0670325i \(0.0213531\pi\)
\(80\) 0 0
\(81\) −4.56262 + 4.56262i −0.506958 + 0.506958i
\(82\) 0 0
\(83\) −2.14367 − 5.17528i −0.235299 − 0.568061i 0.761487 − 0.648181i \(-0.224470\pi\)
−0.996785 + 0.0801194i \(0.974470\pi\)
\(84\) 0 0
\(85\) −8.72318 + 2.98431i −0.946162 + 0.323694i
\(86\) 0 0
\(87\) −3.90487 − 9.42719i −0.418646 − 1.01070i
\(88\) 0 0
\(89\) −1.27573 + 1.27573i −0.135227 + 0.135227i −0.771480 − 0.636253i \(-0.780483\pi\)
0.636253 + 0.771480i \(0.280483\pi\)
\(90\) 0 0
\(91\) −17.2635 + 11.5351i −1.80971 + 1.20921i
\(92\) 0 0
\(93\) −5.30620 − 2.19790i −0.550227 − 0.227912i
\(94\) 0 0
\(95\) 11.0616 + 3.55869i 1.13490 + 0.365114i
\(96\) 0 0
\(97\) 7.04095 + 4.70461i 0.714900 + 0.477681i 0.859061 − 0.511873i \(-0.171048\pi\)
−0.144161 + 0.989554i \(0.546048\pi\)
\(98\) 0 0
\(99\) −1.84857 − 1.23517i −0.185788 − 0.124140i
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.bi.a.333.7 yes 72
5.2 odd 4 340.2.bd.a.197.7 ✓ 72
17.12 odd 16 340.2.bd.a.233.7 yes 72
85.12 even 16 inner 340.2.bi.a.97.7 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
340.2.bd.a.197.7 ✓ 72 5.2 odd 4
340.2.bd.a.233.7 yes 72 17.12 odd 16
340.2.bi.a.97.7 yes 72 85.12 even 16 inner
340.2.bi.a.333.7 yes 72 1.1 even 1 trivial