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Properties

Label 98.7.f.a.13.14
Level $98$
Weight $7$
Character 98.13
Analytic conductor $22.545$
Analytic rank $0$
Dimension $168$
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(13,98)] 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("98.13"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(98, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([11])) N = Newforms(chi, 7, names="a")
 
Level: \( N \) \(=\) \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 98.f (of order \(14\), degree \(6\), 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: \(168\)
Relative dimension: \(28\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label 13.14
Character \(\chi\) \(=\) 98.13
Dual form 98.7.f.a.83.14

$q$-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+(-1.25877 - 5.51503i) q^{2} +(37.0915 - 29.5795i) q^{3} +(-28.8310 + 13.8843i) q^{4} +(-127.979 + 102.060i) q^{5} +(-209.821 - 167.327i) q^{6} +(-315.608 - 134.316i) q^{7} +(112.864 + 141.527i) q^{8} +(338.615 - 1483.57i) q^{9} +(723.958 + 577.337i) q^{10} +(115.416 + 505.670i) q^{11} +(-658.695 + 1367.79i) q^{12} +(-1068.47 + 243.871i) q^{13} +(-343.478 + 1909.66i) q^{14} +(-1728.05 + 7571.10i) q^{15} +(638.454 - 800.595i) q^{16} +(-1729.53 + 3591.40i) q^{17} -8608.17 q^{18} +8590.74i q^{19} +(2272.73 - 4719.38i) q^{20} +(-15679.4 + 4353.54i) q^{21} +(2643.50 - 1273.04i) q^{22} +(7783.38 - 3748.28i) q^{23} +(8372.57 + 1910.98i) q^{24} +(2485.51 - 10889.7i) q^{25} +(2689.91 + 5585.67i) q^{26} +(-16317.6 - 33883.9i) q^{27} +(10964.2 - 509.527i) q^{28} +(-26189.0 - 12612.0i) q^{29} +43930.0 q^{30} +56799.5i q^{31} +(-5218.97 - 2513.32i) q^{32} +(19238.4 + 15342.1i) q^{33} +(21983.8 + 5017.65i) q^{34} +(54099.4 - 15021.3i) q^{35} +(10835.7 + 47474.3i) q^{36} +(-70363.1 - 33885.1i) q^{37} +(47378.2 - 10813.8i) q^{38} +(-32417.6 + 40650.4i) q^{39} +(-28888.3 - 6593.58i) q^{40} +(9597.84 - 7654.02i) q^{41} +(43746.6 + 80991.9i) q^{42} +(24973.4 - 31315.7i) q^{43} +(-10348.4 - 12976.5i) q^{44} +(108077. + 224425. i) q^{45} +(-30469.3 - 38207.3i) q^{46} +(-122524. + 27965.4i) q^{47} -48580.4i q^{48} +(81567.5 + 84782.2i) q^{49} -63185.9 q^{50} +(42081.0 + 184369. i) q^{51} +(27419.1 - 21866.0i) q^{52} +(-63422.1 + 30542.5i) q^{53} +(-166330. + 132644. i) q^{54} +(-66379.4 - 52935.8i) q^{55} +(-16611.4 - 59826.3i) q^{56} +(254110. + 318643. i) q^{57} +(-36589.4 + 160308. i) q^{58} +(-199224. - 158876. i) q^{59} +(-55297.7 - 242275. i) q^{60} +(83916.1 - 174254. i) q^{61} +(313251. - 71497.4i) q^{62} +(-306137. + 422745. i) q^{63} +(-7291.57 + 31946.4i) q^{64} +(111852. - 140258. i) q^{65} +(60395.4 - 125412. i) q^{66} -103606. q^{67} -127557. i q^{68} +(177825. - 369258. i) q^{69} +(-150941. - 279451. i) q^{70} +(507036. - 244175. i) q^{71} +(248182. - 119518. i) q^{72} +(438345. + 100049. i) q^{73} +(-98306.2 + 430708. i) q^{74} +(-229922. - 477437. i) q^{75} +(-119276. - 247680. i) q^{76} +(31493.3 - 175096. i) q^{77} +(264994. + 127614. i) q^{78} -698903. q^{79} +167620. i q^{80} +(-608033. - 292813. i) q^{81} +(-54293.6 - 43297.7i) q^{82} +(-867491. - 197999. i) q^{83} +(391606. - 343213. i) q^{84} +(-145195. - 636139. i) q^{85} +(-204143. - 98309.9i) q^{86} +(-1.34444e6 + 306861. i) q^{87} +(-58539.5 + 73406.2i) q^{88} +(-676283. - 154357. i) q^{89} +(1.10166e6 - 878548. i) q^{90} +(369973. + 66544.8i) q^{91} +(-172361. + 216133. i) q^{92} +(1.68010e6 + 2.10678e6i) q^{93} +(308460. + 640523. i) q^{94} +(-876769. - 1.09943e6i) q^{95} +(-267922. + 61151.5i) q^{96} -571990. i q^{97} +(364901. - 556568. i) q^{98} +789279. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 168 q - 896 q^{4} + 784 q^{6} - 308 q^{7} + 9884 q^{9} - 3976 q^{11} - 2688 q^{14} - 3948 q^{15} - 28672 q^{16} + 6272 q^{17} + 11200 q^{18} - 25088 q^{20} - 19488 q^{21} + 16800 q^{22} + 81536 q^{23} + 111524 q^{25}+ \cdots - 7151816 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(e\left(\frac{11}{14}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.25877 − 5.51503i −0.157346 − 0.689378i
\(3\) 37.0915 − 29.5795i 1.37376 − 1.09554i 0.389071 − 0.921208i \(-0.372796\pi\)
0.984688 − 0.174328i \(-0.0557755\pi\)
\(4\) −28.8310 + 13.8843i −0.450484 + 0.216942i
\(5\) −127.979 + 102.060i −1.02383 + 0.816478i −0.983170 − 0.182695i \(-0.941518\pi\)
−0.0406615 + 0.999173i \(0.512947\pi\)
\(6\) −209.821 − 167.327i −0.971394 − 0.774661i
\(7\) −315.608 − 134.316i −0.920139 − 0.391591i
\(8\) 112.864 + 141.527i 0.220437 + 0.276419i
\(9\) 338.615 − 1483.57i 0.464493 − 2.03508i
\(10\) 723.958 + 577.337i 0.723958 + 0.577337i
\(11\) 115.416 + 505.670i 0.0867137 + 0.379917i 0.999600 − 0.0282975i \(-0.00900857\pi\)
−0.912886 + 0.408215i \(0.866151\pi\)
\(12\) −658.695 + 1367.79i −0.381189 + 0.791548i
\(13\) −1068.47 + 243.871i −0.486332 + 0.111002i −0.458654 − 0.888615i \(-0.651668\pi\)
−0.0276776 + 0.999617i \(0.508811\pi\)
\(14\) −343.478 + 1909.66i −0.125174 + 0.695939i
\(15\) −1728.05 + 7571.10i −0.512016 + 2.24329i
\(16\) 638.454 − 800.595i 0.155872 − 0.195458i
\(17\) −1729.53 + 3591.40i −0.352031 + 0.731000i −0.999517 − 0.0310775i \(-0.990106\pi\)
0.647486 + 0.762077i \(0.275820\pi\)
\(18\) −8608.17 −1.47602
\(19\) 8590.74i 1.25248i 0.779631 + 0.626239i \(0.215406\pi\)
−0.779631 + 0.626239i \(0.784594\pi\)
\(20\) 2272.73 − 4719.38i 0.284092 − 0.589923i
\(21\) −15679.4 + 4353.54i −1.69305 + 0.470094i
\(22\) 2643.50 − 1273.04i 0.248263 − 0.119557i
\(23\) 7783.38 − 3748.28i 0.639712 − 0.308069i −0.0857649 − 0.996315i \(-0.527333\pi\)
0.725477 + 0.688246i \(0.241619\pi\)
\(24\) 8372.57 + 1910.98i 0.605654 + 0.138237i
\(25\) 2485.51 − 10889.7i 0.159073 − 0.696944i
\(26\) 2689.91 + 5585.67i 0.153045 + 0.317801i
\(27\) −16317.6 − 33883.9i −0.829020 − 1.72148i
\(28\) 10964.2 − 509.527i 0.499461 − 0.0232110i
\(29\) −26189.0 − 12612.0i −1.07380 − 0.517117i −0.188473 − 0.982078i \(-0.560354\pi\)
−0.885331 + 0.464962i \(0.846068\pi\)
\(30\) 43930.0 1.62704
\(31\) 56799.5i 1.90660i 0.302025 + 0.953300i \(0.402337\pi\)
−0.302025 + 0.953300i \(0.597663\pi\)
\(32\) −5218.97 − 2513.32i −0.159270 − 0.0767005i
\(33\) 19238.4 + 15342.1i 0.535337 + 0.426917i
\(34\) 21983.8 + 5017.65i 0.559326 + 0.127663i
\(35\) 54099.4 − 15021.3i 1.26179 − 0.350350i
\(36\) 10835.7 + 47474.3i 0.232247 + 1.01754i
\(37\) −70363.1 − 33885.1i −1.38912 − 0.668965i −0.418197 − 0.908357i \(-0.637338\pi\)
−0.970923 + 0.239392i \(0.923052\pi\)
\(38\) 47378.2 − 10813.8i 0.863431 − 0.197072i
\(39\) −32417.6 + 40650.4i −0.546496 + 0.685284i
\(40\) −28888.3 − 6593.58i −0.451380 − 0.103025i
\(41\) 9597.84 − 7654.02i 0.139259 − 0.111055i −0.551383 − 0.834252i \(-0.685900\pi\)
0.690642 + 0.723197i \(0.257328\pi\)
\(42\) 43746.6 + 80991.9i 0.590467 + 1.09319i
\(43\) 24973.4 − 31315.7i 0.314104 − 0.393873i −0.599570 − 0.800322i \(-0.704662\pi\)
0.913673 + 0.406449i \(0.133233\pi\)
\(44\) −10348.4 − 12976.5i −0.121483 − 0.152335i
\(45\) 108077. + 224425.i 1.18603 + 2.46282i
\(46\) −30469.3 − 38207.3i −0.313032 − 0.392530i
\(47\) −122524. + 27965.4i −1.18013 + 0.269356i −0.767200 − 0.641408i \(-0.778351\pi\)
−0.412927 + 0.910764i \(0.635494\pi\)
\(48\) − 48580.4i − 0.439276i
\(49\) 81567.5 + 84782.2i 0.693313 + 0.720637i
\(50\) −63185.9 −0.505487
\(51\) 42081.0 + 184369.i 0.317231 + 1.38988i
\(52\) 27419.1 − 21866.0i 0.195004 − 0.155510i
\(53\) −63422.1 + 30542.5i −0.426003 + 0.205152i −0.634586 − 0.772852i \(-0.718829\pi\)
0.208583 + 0.978005i \(0.433115\pi\)
\(54\) −166330. + 132644.i −1.05631 + 0.842376i
\(55\) −66379.4 − 52935.8i −0.398974 − 0.318171i
\(56\) −16611.4 − 59826.3i −0.0945893 − 0.340665i
\(57\) 254110. + 318643.i 1.37213 + 1.72060i
\(58\) −36589.4 + 160308.i −0.187530 + 0.821623i
\(59\) −199224. − 158876.i −0.970030 − 0.773573i 0.00399980 − 0.999992i \(-0.498727\pi\)
−0.974030 + 0.226419i \(0.927298\pi\)
\(60\) −55297.7 − 242275.i −0.256008 − 1.12164i
\(61\) 83916.1 − 174254.i 0.369705 − 0.767701i −0.630257 − 0.776387i \(-0.717050\pi\)
0.999962 + 0.00868549i \(0.00276471\pi\)
\(62\) 313251. − 71497.4i 1.31437 − 0.299996i
\(63\) −306137. + 422745.i −1.22432 + 1.69066i
\(64\) −7291.57 + 31946.4i −0.0278151 + 0.121866i
\(65\) 111852. − 140258.i 0.407291 − 0.510727i
\(66\) 60395.4 − 125412.i 0.210074 − 0.436223i
\(67\) −103606. −0.344478 −0.172239 − 0.985055i \(-0.555100\pi\)
−0.172239 + 0.985055i \(0.555100\pi\)
\(68\) − 127557.i − 0.405674i
\(69\) 177825. − 369258.i 0.541309 − 1.12404i
\(70\) −150941. − 279451.i −0.440062 − 0.814726i
\(71\) 507036. − 244175.i 1.41665 − 0.682224i 0.440189 − 0.897905i \(-0.354911\pi\)
0.976464 + 0.215681i \(0.0691972\pi\)
\(72\) 248182. − 119518.i 0.664926 − 0.320211i
\(73\) 438345. + 100049.i 1.12680 + 0.257185i 0.745004 − 0.667060i \(-0.232448\pi\)
0.381798 + 0.924246i \(0.375305\pi\)
\(74\) −98306.2 + 430708.i −0.242597 + 1.06289i
\(75\) −229922. − 477437.i −0.544999 − 1.13170i
\(76\) −119276. − 247680.i −0.271715 − 0.564222i
\(77\) 31493.3 − 175096.i 0.0689837 − 0.383533i
\(78\) 264994. + 127614.i 0.558409 + 0.268915i
\(79\) −698903. −1.41754 −0.708771 − 0.705439i \(-0.750750\pi\)
−0.708771 + 0.705439i \(0.750750\pi\)
\(80\) 167620.i 0.327382i
\(81\) −608033. − 292813.i −1.14412 − 0.550980i
\(82\) −54293.6 − 43297.7i −0.0984707 − 0.0785278i
\(83\) −867491. − 197999.i −1.51716 − 0.346281i −0.618801 − 0.785548i \(-0.712381\pi\)
−0.898357 + 0.439267i \(0.855238\pi\)
\(84\) 391606. − 343213.i 0.660710 − 0.579064i
\(85\) −145195. − 636139.i −0.236425 − 1.03585i
\(86\) −204143. − 98309.9i −0.320951 − 0.154562i
\(87\) −1.34444e6 + 306861.i −2.04167 + 0.465997i
\(88\) −58539.5 + 73406.2i −0.0859016 + 0.107717i
\(89\) −676283. − 154357.i −0.959309 − 0.218956i −0.285928 − 0.958251i \(-0.592302\pi\)
−0.673382 + 0.739295i \(0.735159\pi\)
\(90\) 1.10166e6 − 878548.i 1.51120 − 1.20514i
\(91\) 369973. + 66544.8i 0.490960 + 0.0883059i
\(92\) −172361. + 216133.i −0.221347 + 0.277561i
\(93\) 1.68010e6 + 2.10678e6i 2.08875 + 2.61921i
\(94\) 308460. + 640523.i 0.371377 + 0.771172i
\(95\) −876769. − 1.09943e6i −1.02262 − 1.28233i
\(96\) −267922. + 61151.5i −0.302827 + 0.0691183i
\(97\) − 571990.i − 0.626720i −0.949634 − 0.313360i \(-0.898545\pi\)
0.949634 − 0.313360i \(-0.101455\pi\)
\(98\) 364901. − 556568.i 0.387701 − 0.591344i
\(99\) 789279. 0.813439
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.f.a.13.14 ✓ 168
49.34 odd 14 inner 98.7.f.a.83.14 yes 168
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
98.7.f.a.13.14 ✓ 168 1.1 even 1 trivial
98.7.f.a.83.14 yes 168 49.34 odd 14 inner