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Properties

Label 784.2.x.n.373.3
Level $784$
Weight $2$
Character 784.373
Analytic conductor $6.260$
Analytic rank $0$
Dimension $40$
Inner twists $8$

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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] = [784,2,Mod(165,784)] 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("784.165"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(784, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([0, 3, 8])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 784.x (of order \(12\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [40,0,0,8,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.26027151847\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(10\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 373.3
Character \(\chi\) \(=\) 784.373
Dual form 784.2.x.n.557.3

$q$-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+(-1.08279 - 0.909711i) q^{2} +(-1.23755 - 0.331601i) q^{3} +(0.344852 + 1.97004i) q^{4} +(-3.17421 + 0.850527i) q^{5} +(1.03834 + 1.48487i) q^{6} +(1.41877 - 2.44685i) q^{8} +(-1.17650 - 0.679252i) q^{9} +(4.21072 + 1.96667i) q^{10} +(0.110070 - 0.410787i) q^{11} +(0.226497 - 2.55239i) q^{12} +(-3.70160 + 3.70160i) q^{13} +4.21029 q^{15} +(-3.76215 + 1.35875i) q^{16} +(1.35248 + 2.34256i) q^{17} +(0.655973 + 1.80576i) q^{18} +(0.914248 + 3.41202i) q^{19} +(-2.77021 - 5.96003i) q^{20} +(-0.492880 + 0.344662i) q^{22} +(-4.36053 - 2.51755i) q^{23} +(-2.56718 + 2.55764i) q^{24} +(5.02207 - 2.89950i) q^{25} +(7.37543 - 0.640655i) q^{26} +(3.94859 + 3.94859i) q^{27} +(0.464798 - 0.464798i) q^{29} +(-4.55884 - 3.83014i) q^{30} +(-3.87569 - 6.71289i) q^{31} +(5.30968 + 1.95124i) q^{32} +(-0.272435 + 0.471871i) q^{33} +(0.666608 - 3.76685i) q^{34} +(0.932439 - 2.55200i) q^{36} +(5.24218 - 1.40464i) q^{37} +(2.11402 - 4.52619i) q^{38} +(5.80838 - 3.35347i) q^{39} +(-2.42236 + 8.97352i) q^{40} -10.5868i q^{41} +(8.35006 + 8.35006i) q^{43} +(0.847226 + 0.0751823i) q^{44} +(4.31217 + 1.15544i) q^{45} +(2.43127 + 6.69279i) q^{46} +(5.11933 - 8.86694i) q^{47} +(5.10643 - 0.433986i) q^{48} +(-8.07554 - 1.42910i) q^{50} +(-0.896966 - 3.34752i) q^{51} +(-8.56882 - 6.01581i) q^{52} +(3.30284 - 12.3264i) q^{53} +(-0.683403 - 7.86756i) q^{54} +1.39754i q^{55} -4.52572i q^{57} +(-0.926108 + 0.0804449i) q^{58} +(-2.63392 + 9.82994i) q^{59} +(1.45192 + 8.29445i) q^{60} +(-1.84061 - 6.86924i) q^{61} +(-1.91025 + 10.7944i) q^{62} +(-3.97418 - 6.94305i) q^{64} +(8.60134 - 14.8980i) q^{65} +(0.724255 - 0.263098i) q^{66} +(-0.243592 - 0.0652703i) q^{67} +(-4.14854 + 3.47228i) q^{68} +(4.56156 + 4.56156i) q^{69} +6.38433i q^{71} +(-3.33121 + 1.91502i) q^{72} +(1.22356 - 0.706421i) q^{73} +(-6.95397 - 3.24795i) q^{74} +(-7.17656 + 1.92295i) q^{75} +(-6.40655 + 2.97775i) q^{76} +(-9.33992 - 1.65286i) q^{78} +(-4.14366 + 7.17702i) q^{79} +(10.7862 - 7.51276i) q^{80} +(-1.53948 - 2.66646i) q^{81} +(-9.63096 + 11.4633i) q^{82} +(2.84836 - 2.84836i) q^{83} +(-6.28545 - 6.28545i) q^{85} +(-1.44519 - 16.6375i) q^{86} +(-0.729339 + 0.421084i) q^{87} +(-0.848971 - 0.852137i) q^{88} +(0.471871 + 0.272435i) q^{89} +(-3.61804 - 5.17393i) q^{90} +(3.45595 - 9.45861i) q^{92} +(2.57037 + 9.59274i) q^{93} +(-13.6095 + 4.94389i) q^{94} +(-5.80403 - 10.0529i) q^{95} +(-5.92397 - 4.17546i) q^{96} +3.67469 q^{97} +(-0.408525 + 0.408525i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40 q + 8 q^{4} - 4 q^{11} - 32 q^{15} - 16 q^{18} - 8 q^{29} - 8 q^{30} + 40 q^{32} + 80 q^{36} + 20 q^{37} + 120 q^{43} - 56 q^{44} + 64 q^{46} - 112 q^{50} + 16 q^{51} - 28 q^{53} + 72 q^{58} + 24 q^{60}+ \cdots - 216 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/784\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(687\) \(689\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(1\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.08279 − 0.909711i −0.765645 − 0.643263i
\(3\) −1.23755 − 0.331601i −0.714502 − 0.191450i −0.116785 − 0.993157i \(-0.537259\pi\)
−0.597717 + 0.801707i \(0.703925\pi\)
\(4\) 0.344852 + 1.97004i 0.172426 + 0.985022i
\(5\) −3.17421 + 0.850527i −1.41955 + 0.380367i −0.885324 − 0.464975i \(-0.846063\pi\)
−0.534225 + 0.845342i \(0.679397\pi\)
\(6\) 1.03834 + 1.48487i 0.423902 + 0.606195i
\(7\) 0 0
\(8\) 1.41877 − 2.44685i 0.501611 − 0.865093i
\(9\) −1.17650 − 0.679252i −0.392166 − 0.226417i
\(10\) 4.21072 + 1.96667i 1.33155 + 0.621917i
\(11\) 0.110070 − 0.410787i 0.0331874 − 0.123857i −0.947345 − 0.320215i \(-0.896245\pi\)
0.980532 + 0.196358i \(0.0629115\pi\)
\(12\) 0.226497 − 2.55239i 0.0653841 − 0.736811i
\(13\) −3.70160 + 3.70160i −1.02664 + 1.02664i −0.0270035 + 0.999635i \(0.508597\pi\)
−0.999635 + 0.0270035i \(0.991403\pi\)
\(14\) 0 0
\(15\) 4.21029 1.08709
\(16\) −3.76215 + 1.35875i −0.940539 + 0.339687i
\(17\) 1.35248 + 2.34256i 0.328024 + 0.568154i 0.982120 − 0.188258i \(-0.0602841\pi\)
−0.654096 + 0.756412i \(0.726951\pi\)
\(18\) 0.655973 + 1.80576i 0.154614 + 0.425621i
\(19\) 0.914248 + 3.41202i 0.209743 + 0.782771i 0.987951 + 0.154765i \(0.0494620\pi\)
−0.778208 + 0.628006i \(0.783871\pi\)
\(20\) −2.77021 − 5.96003i −0.619437 − 1.33270i
\(21\) 0 0
\(22\) −0.492880 + 0.344662i −0.105082 + 0.0734823i
\(23\) −4.36053 − 2.51755i −0.909232 − 0.524946i −0.0290483 − 0.999578i \(-0.509248\pi\)
−0.880184 + 0.474632i \(0.842581\pi\)
\(24\) −2.56718 + 2.55764i −0.524024 + 0.522077i
\(25\) 5.02207 − 2.89950i 1.00441 − 0.579899i
\(26\) 7.37543 − 0.640655i 1.44644 − 0.125643i
\(27\) 3.94859 + 3.94859i 0.759907 + 0.759907i
\(28\) 0 0
\(29\) 0.464798 − 0.464798i 0.0863108 − 0.0863108i −0.662633 − 0.748944i \(-0.730561\pi\)
0.748944 + 0.662633i \(0.230561\pi\)
\(30\) −4.55884 − 3.83014i −0.832326 − 0.699285i
\(31\) −3.87569 − 6.71289i −0.696094 − 1.20567i −0.969810 − 0.243860i \(-0.921586\pi\)
0.273716 − 0.961811i \(-0.411747\pi\)
\(32\) 5.30968 + 1.95124i 0.938627 + 0.344934i
\(33\) −0.272435 + 0.471871i −0.0474248 + 0.0821422i
\(34\) 0.666608 − 3.76685i 0.114322 − 0.646010i
\(35\) 0 0
\(36\) 0.932439 − 2.55200i 0.155406 − 0.425333i
\(37\) 5.24218 − 1.40464i 0.861809 − 0.230921i 0.199267 − 0.979945i \(-0.436144\pi\)
0.662543 + 0.749024i \(0.269477\pi\)
\(38\) 2.11402 − 4.52619i 0.342939 − 0.734245i
\(39\) 5.80838 − 3.35347i 0.930085 − 0.536985i
\(40\) −2.42236 + 8.97352i −0.383009 + 1.41884i
\(41\) − 10.5868i − 1.65339i −0.562653 − 0.826693i \(-0.690219\pi\)
0.562653 − 0.826693i \(-0.309781\pi\)
\(42\) 0 0
\(43\) 8.35006 + 8.35006i 1.27337 + 1.27337i 0.944307 + 0.329065i \(0.106733\pi\)
0.329065 + 0.944307i \(0.393267\pi\)
\(44\) 0.847226 + 0.0751823i 0.127724 + 0.0113342i
\(45\) 4.31217 + 1.15544i 0.642821 + 0.172243i
\(46\) 2.43127 + 6.69279i 0.358472 + 0.986798i
\(47\) 5.11933 − 8.86694i 0.746731 − 1.29338i −0.202651 − 0.979251i \(-0.564956\pi\)
0.949382 − 0.314125i \(-0.101711\pi\)
\(48\) 5.10643 − 0.433986i 0.737049 − 0.0626404i
\(49\) 0 0
\(50\) −8.07554 − 1.42910i −1.14205 − 0.202106i
\(51\) −0.896966 − 3.34752i −0.125600 − 0.468747i
\(52\) −8.56882 − 6.01581i −1.18828 − 0.834243i
\(53\) 3.30284 − 12.3264i 0.453680 − 1.69316i −0.238261 − 0.971201i \(-0.576577\pi\)
0.691941 − 0.721954i \(-0.256756\pi\)
\(54\) −0.683403 − 7.86756i −0.0929994 − 1.07064i
\(55\) 1.39754i 0.188444i
\(56\) 0 0
\(57\) − 4.52572i − 0.599446i
\(58\) −0.926108 + 0.0804449i −0.121604 + 0.0105629i
\(59\) −2.63392 + 9.82994i −0.342908 + 1.27975i 0.552130 + 0.833758i \(0.313815\pi\)
−0.895037 + 0.445991i \(0.852851\pi\)
\(60\) 1.45192 + 8.29445i 0.187443 + 1.07081i
\(61\) −1.84061 − 6.86924i −0.235666 − 0.879516i −0.977848 − 0.209318i \(-0.932876\pi\)
0.742182 − 0.670198i \(-0.233791\pi\)
\(62\) −1.91025 + 10.7944i −0.242602 + 1.37089i
\(63\) 0 0
\(64\) −3.97418 − 6.94305i −0.496772 − 0.867881i
\(65\) 8.60134 − 14.8980i 1.06686 − 1.84786i
\(66\) 0.724255 − 0.263098i 0.0891496 − 0.0323852i
\(67\) −0.243592 − 0.0652703i −0.0297595 − 0.00797403i 0.243909 − 0.969798i \(-0.421570\pi\)
−0.273668 + 0.961824i \(0.588237\pi\)
\(68\) −4.14854 + 3.47228i −0.503085 + 0.421075i
\(69\) 4.56156 + 4.56156i 0.549147 + 0.549147i
\(70\) 0 0
\(71\) 6.38433i 0.757680i 0.925462 + 0.378840i \(0.123677\pi\)
−0.925462 + 0.378840i \(0.876323\pi\)
\(72\) −3.33121 + 1.91502i −0.392587 + 0.225687i
\(73\) 1.22356 − 0.706421i 0.143206 − 0.0826803i −0.426685 − 0.904400i \(-0.640319\pi\)
0.569891 + 0.821720i \(0.306985\pi\)
\(74\) −6.95397 − 3.24795i −0.808383 − 0.377566i
\(75\) −7.17656 + 1.92295i −0.828678 + 0.222044i
\(76\) −6.40655 + 2.97775i −0.734882 + 0.341571i
\(77\) 0 0
\(78\) −9.33992 − 1.65286i −1.05754 − 0.187149i
\(79\) −4.14366 + 7.17702i −0.466198 + 0.807478i −0.999255 − 0.0386013i \(-0.987710\pi\)
0.533057 + 0.846079i \(0.321043\pi\)
\(80\) 10.7862 − 7.51276i 1.20594 − 0.839952i
\(81\) −1.53948 − 2.66646i −0.171053 − 0.296273i
\(82\) −9.63096 + 11.4633i −1.06356 + 1.26591i
\(83\) 2.84836 − 2.84836i 0.312648 − 0.312648i −0.533286 − 0.845935i \(-0.679043\pi\)
0.845935 + 0.533286i \(0.179043\pi\)
\(84\) 0 0
\(85\) −6.28545 − 6.28545i −0.681753 − 0.681753i
\(86\) −1.44519 − 16.6375i −0.155839 − 1.79406i
\(87\) −0.729339 + 0.421084i −0.0781934 + 0.0451450i
\(88\) −0.848971 − 0.852137i −0.0905006 − 0.0908382i
\(89\) 0.471871 + 0.272435i 0.0500182 + 0.0288780i 0.524801 − 0.851225i \(-0.324140\pi\)
−0.474782 + 0.880103i \(0.657473\pi\)
\(90\) −3.61804 − 5.17393i −0.381375 − 0.545380i
\(91\) 0 0
\(92\) 3.45595 − 9.45861i 0.360308 − 0.986129i
\(93\) 2.57037 + 9.59274i 0.266535 + 0.994721i
\(94\) −13.6095 + 4.94389i −1.40371 + 0.509923i
\(95\) −5.80403 − 10.0529i −0.595481 − 1.03140i
\(96\) −5.92397 − 4.17546i −0.604613 − 0.426156i
\(97\) 3.67469 0.373108 0.186554 − 0.982445i \(-0.440268\pi\)
0.186554 + 0.982445i \(0.440268\pi\)
\(98\) 0 0
\(99\) −0.408525 + 0.408525i −0.0410583 + 0.0410583i
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 784.2.x.n.373.3 40
7.2 even 3 784.2.m.i.197.6 yes 20
7.3 odd 6 inner 784.2.x.n.165.9 40
7.4 even 3 inner 784.2.x.n.165.10 40
7.5 odd 6 784.2.m.i.197.5 ✓ 20
7.6 odd 2 inner 784.2.x.n.373.4 40
16.13 even 4 inner 784.2.x.n.765.10 40
112.13 odd 4 inner 784.2.x.n.765.9 40
112.45 odd 12 inner 784.2.x.n.557.4 40
112.61 odd 12 784.2.m.i.589.5 yes 20
112.93 even 12 784.2.m.i.589.6 yes 20
112.109 even 12 inner 784.2.x.n.557.3 40
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
784.2.m.i.197.5 ✓ 20 7.5 odd 6
784.2.m.i.197.6 yes 20 7.2 even 3
784.2.m.i.589.5 yes 20 112.61 odd 12
784.2.m.i.589.6 yes 20 112.93 even 12
784.2.x.n.165.9 40 7.3 odd 6 inner
784.2.x.n.165.10 40 7.4 even 3 inner
784.2.x.n.373.3 40 1.1 even 1 trivial
784.2.x.n.373.4 40 7.6 odd 2 inner
784.2.x.n.557.3 40 112.109 even 12 inner
784.2.x.n.557.4 40 112.45 odd 12 inner
784.2.x.n.765.9 40 112.13 odd 4 inner
784.2.x.n.765.10 40 16.13 even 4 inner