↑

Properties

Label 340.2.bd.a.197.7
Level $340$
Weight $2$
Character 340.197
Analytic conductor $2.715$
Analytic rank $0$
Dimension $72$
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] = [340,2,Mod(57,340)] 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("340.57"); 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, 15])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 340 = 2^{2} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 340.bd (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 197.7
Character \(\chi\) \(=\) 340.197
Dual form 340.2.bd.a.233.7

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.49056 - 0.296491i) q^{3} +(2.12862 - 0.684809i) q^{5} +(2.36209 + 3.53511i) q^{7} +(-0.637769 + 0.264172i) q^{9} +(-1.78929 - 2.67786i) q^{11} +4.88344i q^{13} +(2.96981 - 1.65187i) q^{15} +(-1.01284 - 3.99677i) q^{17} +(-4.80104 - 1.98866i) q^{19} +(4.56897 + 4.56897i) q^{21} +(0.757298 - 3.80720i) q^{23} +(4.06207 - 2.91540i) q^{25} +(-4.66322 + 3.11587i) q^{27} +(6.58513 - 1.30986i) q^{29} +(-2.09957 + 3.14223i) q^{31} +(-3.46101 - 3.46101i) q^{33} +(7.44887 + 5.90735i) q^{35} +(-2.07806 - 10.4471i) q^{37} +(1.44790 + 7.27907i) q^{39} +(-4.07539 - 0.810645i) q^{41} +(6.03662 + 2.50045i) q^{43} +(-1.17666 + 0.999073i) q^{45} -7.93989 q^{47} +(-4.23879 + 10.2333i) q^{49} +(-2.69471 - 5.65714i) q^{51} +(0.837059 + 2.02084i) q^{53} +(-5.64254 - 4.47483i) q^{55} +(-7.74588 - 1.54075i) q^{57} +(3.66951 + 8.85898i) q^{59} +(1.78113 - 8.95434i) q^{61} +(-2.44035 - 1.63059i) q^{63} +(3.34422 + 10.3950i) q^{65} +(-0.847172 + 0.847172i) q^{67} -5.89940i q^{69} +(-4.58999 - 3.06693i) q^{71} +(7.21787 - 10.8023i) q^{73} +(5.19038 - 5.54996i) q^{75} +(5.24008 - 12.6507i) q^{77} +(-11.7115 + 7.82536i) q^{79} +(-4.56262 + 4.56262i) q^{81} +(-5.17528 + 2.14367i) q^{83} +(-4.89298 - 7.81401i) q^{85} +(9.42719 - 3.90487i) q^{87} +(1.27573 - 1.27573i) q^{89} +(-17.2635 + 11.5351i) q^{91} +(-2.19790 + 5.30620i) q^{93} +(-11.5815 - 0.945304i) q^{95} +(-4.70461 + 7.04095i) q^{97} +(1.84857 + 1.23517i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q + 24 q^{15} + 8 q^{25} - 48 q^{27} - 32 q^{31} + 16 q^{33} + 32 q^{37} - 32 q^{39} - 40 q^{41} + 80 q^{47} - 40 q^{53} + 16 q^{55} + 8 q^{57} + 112 q^{59} - 48 q^{63} - 32 q^{67} - 16 q^{71} + 8 q^{73}+ \cdots + 96 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(3\) 1.49056 − 0.296491i 0.860577 − 0.171179i 0.254975 − 0.966948i \(-0.417933\pi\)
0.605601 + 0.795768i \(0.292933\pi\)
\(4\) 0 0
\(5\) 2.12862 − 0.684809i 0.951949 − 0.306256i
\(6\) 0 0
\(7\) 2.36209 + 3.53511i 0.892785 + 1.33615i 0.941387 + 0.337329i \(0.109524\pi\)
−0.0486015 + 0.998818i \(0.515476\pi\)
\(8\) 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
\(13\) 4.88344i 1.35442i 0.735789 + 0.677211i \(0.236812\pi\)
−0.735789 + 0.677211i \(0.763188\pi\)
\(14\) 0 0
\(15\) 2.96981 − 1.65187i 0.766801 − 0.426511i
\(16\) 0 0
\(17\) −1.01284 − 3.99677i −0.245649 − 0.969359i
\(18\) 0 0
\(19\) −4.80104 − 1.98866i −1.10143 − 0.456229i −0.243455 − 0.969912i \(-0.578281\pi\)
−0.857980 + 0.513683i \(0.828281\pi\)
\(20\) 0 0
\(21\) 4.56897 + 4.56897i 0.997031 + 0.997031i
\(22\) 0 0
\(23\) 0.757298 − 3.80720i 0.157908 − 0.793855i −0.817923 − 0.575327i \(-0.804875\pi\)
0.975831 − 0.218528i \(-0.0701254\pi\)
\(24\) 0 0
\(25\) 4.06207 − 2.91540i 0.812415 − 0.583080i
\(26\) 0 0
\(27\) −4.66322 + 3.11587i −0.897438 + 0.599649i
\(28\) 0 0
\(29\) 6.58513 − 1.30986i 1.22283 − 0.243236i 0.458871 − 0.888503i \(-0.348254\pi\)
0.763957 + 0.645267i \(0.223254\pi\)
\(30\) 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
\(37\) −2.07806 − 10.4471i −0.341630 − 1.71749i −0.644625 − 0.764499i \(-0.722986\pi\)
0.302994 − 0.952992i \(-0.402014\pi\)
\(38\) 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
\(43\) 6.03662 + 2.50045i 0.920575 + 0.381315i 0.792095 − 0.610397i \(-0.208990\pi\)
0.128480 + 0.991712i \(0.458990\pi\)
\(44\) 0 0
\(45\) −1.17666 + 0.999073i −0.175406 + 0.148933i
\(46\) 0 0
\(47\) −7.93989 −1.15815 −0.579076 − 0.815274i \(-0.696586\pi\)
−0.579076 + 0.815274i \(0.696586\pi\)
\(48\) 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
\(53\) 0.837059 + 2.02084i 0.114979 + 0.277584i 0.970884 − 0.239549i \(-0.0769995\pi\)
−0.855905 + 0.517132i \(0.826999\pi\)
\(54\) 0 0
\(55\) −5.64254 − 4.47483i −0.760840 − 0.603386i
\(56\) 0 0
\(57\) −7.74588 − 1.54075i −1.02597 − 0.204077i
\(58\) 0 0
\(59\) 3.66951 + 8.85898i 0.477730 + 1.15334i 0.960671 + 0.277689i \(0.0895683\pi\)
−0.482942 + 0.875653i \(0.660432\pi\)
\(60\) 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
\(63\) −2.44035 − 1.63059i −0.307455 − 0.205435i
\(64\) 0 0
\(65\) 3.34422 + 10.3950i 0.414800 + 1.28934i
\(66\) 0 0
\(67\) −0.847172 + 0.847172i −0.103499 + 0.103499i −0.756960 − 0.653461i \(-0.773316\pi\)
0.653461 + 0.756960i \(0.273316\pi\)
\(68\) 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
\(73\) 7.21787 − 10.8023i 0.844788 − 1.26431i −0.117717 − 0.993047i \(-0.537557\pi\)
0.962504 − 0.271267i \(-0.0874426\pi\)
\(74\) 0 0
\(75\) 5.19038 − 5.54996i 0.599334 − 0.640854i
\(76\) 0 0
\(77\) 5.24008 − 12.6507i 0.597162 − 1.44168i
\(78\) 0 0
\(79\) −11.7115 + 7.82536i −1.31764 + 0.880421i −0.997751 − 0.0670325i \(-0.978647\pi\)
−0.319893 + 0.947454i \(0.603647\pi\)
\(80\) 0 0
\(81\) −4.56262 + 4.56262i −0.506958 + 0.506958i
\(82\) 0 0
\(83\) −5.17528 + 2.14367i −0.568061 + 0.235299i −0.648181 − 0.761487i \(-0.724470\pi\)
0.0801194 + 0.996785i \(0.474470\pi\)
\(84\) 0 0
\(85\) −4.89298 − 7.81401i −0.530718 − 0.847549i
\(86\) 0 0
\(87\) 9.42719 − 3.90487i 1.01070 − 0.418646i
\(88\) 0 0
\(89\) 1.27573 − 1.27573i 0.135227 − 0.135227i −0.636253 − 0.771480i \(-0.719517\pi\)
0.771480 + 0.636253i \(0.219517\pi\)
\(90\) 0 0
\(91\) −17.2635 + 11.5351i −1.80971 + 1.20921i
\(92\) 0 0
\(93\) −2.19790 + 5.30620i −0.227912 + 0.550227i
\(94\) 0 0
\(95\) −11.5815 − 0.945304i −1.18823 − 0.0969862i
\(96\) 0 0
\(97\) −4.70461 + 7.04095i −0.477681 + 0.714900i −0.989554 − 0.144161i \(-0.953952\pi\)
0.511873 + 0.859061i \(0.328952\pi\)
\(98\) 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.bd.a.197.7 ✓ 72
5.3 odd 4 340.2.bi.a.333.7 yes 72
17.12 odd 16 340.2.bi.a.97.7 yes 72
85.63 even 16 inner 340.2.bd.a.233.7 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
340.2.bd.a.197.7 ✓ 72 1.1 even 1 trivial
340.2.bd.a.233.7 yes 72 85.63 even 16 inner
340.2.bi.a.97.7 yes 72 17.12 odd 16
340.2.bi.a.333.7 yes 72 5.3 odd 4