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Properties

Label 98.7.h.a.61.2
Level $98$
Weight $7$
Character 98.61
Analytic conductor $22.545$
Analytic rank $0$
Dimension $336$
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] = [98,7,Mod(3,98)] 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("98.3"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(98, base_ring=CyclotomicField(42)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 7, names="a")
 
Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 98.h (of order \(42\), degree \(12\), 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: \(22.5453001947\)
Analytic rank: \(0\)
Dimension: \(336\)
Relative dimension: \(28\) over \(\Q(\zeta_{42})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label 61.2
Character \(\chi\) \(=\) 98.61
Dual form 98.7.h.a.45.2

$q$-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+(-4.67391 + 3.18662i) q^{2} +(-27.5931 - 29.7383i) q^{3} +(11.6909 - 29.7880i) q^{4} +(-36.0641 + 116.917i) q^{5} +(223.732 + 51.0654i) q^{6} +(-37.4745 - 340.947i) q^{7} +(40.2806 + 176.481i) q^{8} +(-68.5083 + 914.179i) q^{9} +(-204.009 - 661.382i) q^{10} +(-184.997 - 2468.61i) q^{11} +(-1208.43 + 474.274i) q^{12} +(1144.73 - 2377.07i) q^{13} +(1261.62 + 1474.14i) q^{14} +(4472.03 - 2153.61i) q^{15} +(-750.645 - 696.497i) q^{16} +(414.265 - 2748.47i) q^{17} +(-2592.94 - 4491.10i) q^{18} +(-103.100 - 59.5251i) q^{19} +(3061.09 + 2441.14i) q^{20} +(-9105.13 + 10522.2i) q^{21} +(8731.19 + 10948.6i) q^{22} +(-11747.2 + 1770.60i) q^{23} +(4136.77 - 6067.52i) q^{24} +(541.036 + 368.872i) q^{25} +(2224.42 + 14758.0i) q^{26} +(5954.67 - 4748.69i) q^{27} +(-10594.2 - 2869.69i) q^{28} +(-8252.27 + 10348.0i) q^{29} +(-14039.1 + 24316.4i) q^{30} +(22333.4 - 12894.2i) q^{31} +(5727.92 + 863.345i) q^{32} +(-68307.6 + 73618.1i) q^{33} +(6822.08 + 14166.2i) q^{34} +(41213.9 + 7914.53i) q^{35} +(26430.6 + 12728.3i) q^{36} +(18264.3 + 46536.6i) q^{37} +(671.566 - 50.3269i) q^{38} +(-102277. + 31548.2i) q^{39} +(-22086.3 - 1655.14i) q^{40} +(-98683.3 + 22523.8i) q^{41} +(9026.34 - 78194.4i) q^{42} +(29143.4 - 127686. i) q^{43} +(-75697.7 - 23349.7i) q^{44} +(-104412. - 40978.8i) q^{45} +(49263.1 - 45709.4i) q^{46} +(-53484.7 - 78447.7i) q^{47} +41541.4i q^{48} +(-114840. + 25553.6i) q^{49} -3704.21 q^{50} +(-93165.5 + 63519.2i) q^{51} +(-57425.0 - 61889.4i) q^{52} +(-79415.4 + 202347. i) q^{53} +(-12699.3 + 41170.2i) q^{54} +(295294. + 67399.0i) q^{55} +(58661.1 - 20347.1i) q^{56} +(1074.69 + 4708.51i) q^{57} +(5595.19 - 74662.6i) q^{58} +(86755.5 + 281254. i) q^{59} +(-11869.7 - 158390. i) q^{60} +(-17759.8 + 6970.20i) q^{61} +(-63295.6 + 131435. i) q^{62} +(314254. - 10900.7i) q^{63} +(-29522.9 + 14217.5i) q^{64} +(236635. + 219566. i) q^{65} +(84671.0 - 561755. i) q^{66} +(-9471.58 - 16405.3i) q^{67} +(-77028.1 - 44472.2i) q^{68} +(376796. + 300484. i) q^{69} +(-217851. + 94341.2i) q^{70} +(-78130.4 - 97972.5i) q^{71} +(-164095. + 24733.3i) q^{72} +(183827. - 269625. i) q^{73} +(-233660. - 159307. i) q^{74} +(-3959.23 - 26267.8i) q^{75} +(-2978.47 + 2375.25i) q^{76} +(-834733. + 155584. i) q^{77} +(377500. - 473370. i) q^{78} +(-209447. + 362773. i) q^{79} +(108504. - 62644.6i) q^{80} +(355316. + 53555.2i) q^{81} +(389462. - 419741. i) q^{82} +(1776.87 + 3689.70i) q^{83} +(206988. + 394237. i) q^{84} +(306402. + 147556. i) q^{85} +(270672. + 689661. i) q^{86} +(535438. - 40125.5i) q^{87} +(428211. - 132086. i) q^{88} +(26224.8 + 1965.28i) q^{89} +(618598. - 141191. i) q^{90} +(-853351. - 301214. i) q^{91} +(-84592.6 + 370625. i) q^{92} +(-999700. - 308367. i) q^{93} +(499966. + 196222. i) q^{94} +(10677.7 - 9907.47i) q^{95} +(-132377. - 194161. i) q^{96} -751107. i q^{97} +(455324. - 485388. i) q^{98} +2.26943e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 336 q + 896 q^{4} + 336 q^{5} - 784 q^{6} - 652 q^{7} - 11732 q^{9} + 2016 q^{10} + 9772 q^{11} - 2064 q^{14} + 3948 q^{15} + 28672 q^{16} + 11032 q^{17} + 5600 q^{18} + 32004 q^{19} + 25088 q^{20} - 9756 q^{21}+ \cdots + 7151816 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/98\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(e\left(\frac{11}{42}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.67391 + 3.18662i −0.584239 + 0.398327i
\(3\) −27.5931 − 29.7383i −1.02197 − 1.10142i −0.994806 − 0.101786i \(-0.967544\pi\)
−0.0271597 − 0.999631i \(-0.508646\pi\)
\(4\) 11.6909 − 29.7880i 0.182671 − 0.465437i
\(5\) −36.0641 + 116.917i −0.288513 + 0.935335i 0.689054 + 0.724710i \(0.258026\pi\)
−0.977567 + 0.210625i \(0.932450\pi\)
\(6\) 223.732 + 51.0654i 1.03580 + 0.236414i
\(7\) −37.4745 − 340.947i −0.109255 − 0.994014i
\(8\) 40.2806 + 176.481i 0.0786730 + 0.344689i
\(9\) −68.5083 + 914.179i −0.0939757 + 1.25402i
\(10\) −204.009 − 661.382i −0.204009 − 0.661382i
\(11\) −184.997 − 2468.61i −0.138991 − 1.85471i −0.436146 − 0.899876i \(-0.643657\pi\)
0.297155 − 0.954829i \(-0.403962\pi\)
\(12\) −1208.43 + 474.274i −0.699323 + 0.274464i
\(13\) 1144.73 − 2377.07i 0.521044 − 1.08196i −0.459954 − 0.887943i \(-0.652134\pi\)
0.980998 − 0.194017i \(-0.0621517\pi\)
\(14\) 1261.62 + 1474.14i 0.459774 + 0.537222i
\(15\) 4472.03 − 2153.61i 1.32504 − 0.638108i
\(16\) −750.645 − 696.497i −0.183263 − 0.170043i
\(17\) 414.265 − 2748.47i 0.0843201 − 0.559427i −0.906284 − 0.422669i \(-0.861093\pi\)
0.990604 − 0.136759i \(-0.0436685\pi\)
\(18\) −2592.94 − 4491.10i −0.444606 − 0.770080i
\(19\) −103.100 − 59.5251i −0.0150314 − 0.00867839i 0.492465 − 0.870332i \(-0.336096\pi\)
−0.507497 + 0.861654i \(0.669429\pi\)
\(20\) 3061.09 + 2441.14i 0.382637 + 0.305143i
\(21\) −9105.13 + 10522.2i −0.983169 + 1.13618i
\(22\) 8731.19 + 10948.6i 0.819984 + 1.02823i
\(23\) −11747.2 + 1770.60i −0.965495 + 0.145525i −0.612822 − 0.790221i \(-0.709966\pi\)
−0.352674 + 0.935746i \(0.614727\pi\)
\(24\) 4136.77 − 6067.52i 0.299245 − 0.438912i
\(25\) 541.036 + 368.872i 0.0346263 + 0.0236078i
\(26\) 2224.42 + 14758.0i 0.126560 + 0.839670i
\(27\) 5954.67 − 4748.69i 0.302529 − 0.241258i
\(28\) −10594.2 − 2869.69i −0.482608 − 0.130726i
\(29\) −8252.27 + 10348.0i −0.338360 + 0.424290i −0.921679 − 0.387953i \(-0.873182\pi\)
0.583319 + 0.812243i \(0.301754\pi\)
\(30\) −14039.1 + 24316.4i −0.519967 + 0.900609i
\(31\) 22333.4 − 12894.2i 0.749671 − 0.432823i −0.0759041 − 0.997115i \(-0.524184\pi\)
0.825575 + 0.564292i \(0.190851\pi\)
\(32\) 5727.92 + 863.345i 0.174802 + 0.0263472i
\(33\) −68307.6 + 73618.1i −1.90076 + 2.04853i
\(34\) 6822.08 + 14166.2i 0.173572 + 0.360426i
\(35\) 41213.9 + 7914.53i 0.961258 + 0.184596i
\(36\) 26430.6 + 12728.3i 0.566500 + 0.272812i
\(37\) 18264.3 + 46536.6i 0.360576 + 0.918733i 0.989923 + 0.141608i \(0.0452273\pi\)
−0.629347 + 0.777125i \(0.716678\pi\)
\(38\) 671.566 − 50.3269i 0.0122388 − 0.000917169i
\(39\) −102277. + 31548.2i −1.72418 + 0.531839i
\(40\) −22086.3 − 1655.14i −0.345098 − 0.0258615i
\(41\) −98683.3 + 22523.8i −1.43183 + 0.326806i −0.866962 − 0.498374i \(-0.833931\pi\)
−0.564870 + 0.825180i \(0.691074\pi\)
\(42\) 9026.34 − 78194.4i 0.121833 − 1.05543i
\(43\) 29143.4 − 127686.i 0.366552 − 1.60597i −0.369626 − 0.929181i \(-0.620514\pi\)
0.736178 − 0.676788i \(-0.236629\pi\)
\(44\) −75697.7 − 23349.7i −0.888638 − 0.274108i
\(45\) −104412. − 40978.8i −1.14581 − 0.449699i
\(46\) 49263.1 − 45709.4i 0.506113 − 0.469605i
\(47\) −53484.7 − 78447.7i −0.515153 − 0.755591i 0.477230 − 0.878778i \(-0.341641\pi\)
−0.992383 + 0.123188i \(0.960688\pi\)
\(48\) 41541.4i 0.375627i
\(49\) −114840. + 25553.6i −0.976127 + 0.217202i
\(50\) −3704.21 −0.0296337
\(51\) −93165.5 + 63519.2i −0.702335 + 0.478844i
\(52\) −57425.0 − 61889.4i −0.408405 − 0.440156i
\(53\) −79415.4 + 202347.i −0.533430 + 1.35916i 0.369445 + 0.929253i \(0.379548\pi\)
−0.902875 + 0.429904i \(0.858547\pi\)
\(54\) −12699.3 + 41170.2i −0.0806491 + 0.261458i
\(55\) 295294. + 67399.0i 1.77487 + 0.405103i
\(56\) 58661.1 − 20347.1i 0.334030 − 0.115861i
\(57\) 1074.69 + 4708.51i 0.00580306 + 0.0254249i
\(58\) 5595.19 − 74662.6i 0.0286768 − 0.382665i
\(59\) 86755.5 + 281254.i 0.422416 + 1.36944i 0.877750 + 0.479119i \(0.159044\pi\)
−0.455333 + 0.890321i \(0.650480\pi\)
\(60\) −11869.7 − 158390.i −0.0549524 − 0.733288i
\(61\) −17759.8 + 6970.20i −0.0782435 + 0.0307083i −0.404140 − 0.914697i \(-0.632429\pi\)
0.325896 + 0.945405i \(0.394334\pi\)
\(62\) −63295.6 + 131435.i −0.265582 + 0.551486i
\(63\) 314254. − 10900.7i 1.25678 − 0.0435947i
\(64\) −29522.9 + 14217.5i −0.112621 + 0.0542355i
\(65\) 236635. + 219566.i 0.861667 + 0.799511i
\(66\) 84671.0 − 561755.i 0.294512 − 1.95396i
\(67\) −9471.58 − 16405.3i −0.0314918 − 0.0545455i 0.849850 − 0.527025i \(-0.176692\pi\)
−0.881342 + 0.472479i \(0.843359\pi\)
\(68\) −77028.1 − 44472.2i −0.244975 − 0.141437i
\(69\) 376796. + 300484.i 1.14699 + 0.914692i
\(70\) −217851. + 94341.2i −0.635134 + 0.275047i
\(71\) −78130.4 − 97972.5i −0.218296 − 0.273734i 0.660610 − 0.750729i \(-0.270298\pi\)
−0.878906 + 0.476995i \(0.841726\pi\)
\(72\) −164095. + 24733.3i −0.439640 + 0.0662650i
\(73\) 183827. − 269625.i 0.472543 − 0.693093i −0.513779 − 0.857923i \(-0.671755\pi\)
0.986322 + 0.164829i \(0.0527073\pi\)
\(74\) −233660. − 159307.i −0.576619 − 0.393132i
\(75\) −3959.23 − 26267.8i −0.00938484 − 0.0622644i
\(76\) −2978.47 + 2375.25i −0.00678504 + 0.00541089i
\(77\) −834733. + 155584.i −1.82842 + 0.340795i
\(78\) 377500. − 473370.i 0.795487 − 0.997509i
\(79\) −209447. + 362773.i −0.424809 + 0.735790i −0.996403 − 0.0847464i \(-0.972992\pi\)
0.571594 + 0.820537i \(0.306325\pi\)
\(80\) 108504. − 62644.6i 0.211921 − 0.122353i
\(81\) 355316. + 53555.2i 0.668589 + 0.100774i
\(82\) 389462. − 419741.i 0.706356 − 0.761271i
\(83\) 1776.87 + 3689.70i 0.00310757 + 0.00645293i 0.902517 − 0.430654i \(-0.141717\pi\)
−0.899409 + 0.437107i \(0.856003\pi\)
\(84\) 206988. + 394237.i 0.349226 + 0.665150i
\(85\) 306402. + 147556.i 0.498925 + 0.240270i
\(86\) 270672. + 689661.i 0.425548 + 1.08428i
\(87\) 535438. − 40125.5i 0.813114 − 0.0609344i
\(88\) 428211. − 132086.i 0.628362 − 0.193824i
\(89\) 26224.8 + 1965.28i 0.0372000 + 0.00278775i 0.0933171 − 0.995636i \(-0.470253\pi\)
−0.0561171 + 0.998424i \(0.517872\pi\)
\(90\) 618598. − 141191.i 0.848557 − 0.193678i
\(91\) −853351. − 301214.i −1.13241 − 0.399716i
\(92\) −84592.6 + 370625.i −0.108635 + 0.475960i
\(93\) −999700. − 308367.i −1.24286 − 0.383370i
\(94\) 499966. + 196222.i 0.601945 + 0.236246i
\(95\) 10677.7 − 9907.47i 0.0124540 − 0.0115556i
\(96\) −132377. − 194161.i −0.149623 − 0.219456i
\(97\) − 751107.i − 0.822975i −0.911415 − 0.411488i \(-0.865009\pi\)
0.911415 − 0.411488i \(-0.134991\pi\)
\(98\) 455324. − 485388.i 0.483774 − 0.515716i
\(99\) 2.26943e6 2.33890
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 98.7.h.a.61.2 yes 336
49.45 odd 42 inner 98.7.h.a.45.2 ✓ 336
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
98.7.h.a.45.2 ✓ 336 49.45 odd 42 inner
98.7.h.a.61.2 yes 336 1.1 even 1 trivial