LMFDB - Embedded newform 162.7.b.c.161.11

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [162,7,Mod(161,162)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("162.161"); S:= CuspForms(chi, 7); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(162, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 7, names="a")

Level:	$ N $	$=$	$ 162 = 2 \cdot 3^{4} $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	162.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$37.2687615464$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 370x^{10} + 51793x^{8} + 3491832x^{6} + 117603792x^{4} + 1832032512x^{2} + 10453017600 $ x^12 + 370x^10 + 51793x^8 + 3491832x^6 + 117603792x^4 + 1832032512*x^2 + 10453017600
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{16}\cdot 3^{42} $
Twist minimal:	no (minimal twist has level 18)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.11
Root			$-8.15670i$ of defining polynomial
Character	$\chi$	$=$	162.161
Dual form			162.7.b.c.161.2

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+5.65685i q^{2} -32.0000 q^{4} +110.679i q^{5} +326.338 q^{7} -181.019i q^{8} -626.092 q^{10} +625.637i q^{11} -2028.01 q^{13} +1846.05i q^{14} +1024.00 q^{16} +4126.55i q^{17} +12194.5 q^{19} -3541.71i q^{20} -3539.14 q^{22} +21450.5i q^{23} +3375.26 q^{25} -11472.2i q^{26} -10442.8 q^{28} -24751.4i q^{29} -40531.8 q^{31} +5792.62i q^{32} -23343.3 q^{34} +36118.6i q^{35} -30093.7 q^{37} +68982.5i q^{38} +20035.0 q^{40} -59229.9i q^{41} -87935.7 q^{43} -20020.4i q^{44} -121342. q^{46} +129770. i q^{47} -11152.5 q^{49} +19093.4i q^{50} +64896.3 q^{52} +165599. i q^{53} -69244.6 q^{55} -59073.5i q^{56} +140015. q^{58} -53250.1i q^{59} -266982. q^{61} -229282. i q^{62} -32768.0 q^{64} -224457. i q^{65} -407280. q^{67} -132049. i q^{68} -204318. q^{70} -186563. i q^{71} -242102. q^{73} -170236. i q^{74} -390224. q^{76} +204169. i q^{77} -126373. q^{79} +113335. i q^{80} +335055. q^{82} -68971.7i q^{83} -456720. q^{85} -497439. i q^{86} +113252. q^{88} -413126. i q^{89} -661817. q^{91} -686414. i q^{92} -734091. q^{94} +1.34967e6i q^{95} +978443. q^{97} -63088.0i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 384 q^{4} - 480 q^{7} - 3360 q^{13} + 12288 q^{16} - 2820 q^{19} + 7200 q^{22} - 16188 q^{25} + 15360 q^{28} - 42960 q^{31} + 54720 q^{34} - 25536 q^{37} - 142860 q^{43} - 135072 q^{46} + 271908 q^{49}+ \cdots + 77748 q^{97}+O(q^{100}) $ 12 * q - 384 * q^4 - 480 * q^7 - 3360 * q^13 + 12288 * q^16 - 2820 * q^19 + 7200 * q^22 - 16188 * q^25 + 15360 * q^28 - 42960 * q^31 + 54720 * q^34 - 25536 * q^37 - 142860 * q^43 - 135072 * q^46 + 271908 * q^49 + 107520 * q^52 + 580392 * q^55 - 318528 * q^58 - 271488 * q^61 - 393216 * q^64 + 579876 * q^67 - 311904 * q^70 - 977700 * q^73 + 90240 * q^76 + 1529592 * q^79 + 1073088 * q^82 - 3239136 * q^85 - 230400 * q^88 + 355584 * q^91 + 1473696 * q^94 + 77748 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/162\mathbb{Z}\right)^\times$.

$n$	$83$
$\chi(n)$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	5.65685i	0.707107i
$2$	5.65685i	0.707107i
$3$	0	0
$3$	0	0
$4$	−32.0000	−0.500000
$5$	110.679i	0.885428i	0.896663	+	0.442714i	$0.145984\pi$
$5$	110.679i	0.885428i	−0.896663	+	0.442714i	$0.854016\pi$
$6$	0	0
$7$	326.338	0.951423	0.475711	−	0.879601i	$-0.342191\pi$
$7$	326.338	0.951423	0.475711	+	0.879601i	$0.342191\pi$
$8$	− 181.019i	− 0.353553i
$9$	0	0
$10$	−626.092	−0.626092
$11$	625.637i	0.470050i	0.971989	+	0.235025i	$0.0755172\pi$
$11$	625.637i	0.470050i	−0.971989	+	0.235025i	$0.924483\pi$
$12$	0	0
$13$	−2028.01	−0.923082	−0.461541	−	0.887119i	$-0.652703\pi$
$13$	−2028.01	−0.923082	−0.461541	+	0.887119i	$0.652703\pi$
$14$	1846.05i	0.672758i
$15$	0	0
$16$	1024.00	0.250000
$17$	4126.55i	0.839924i	0.907542	+	0.419962i	$0.137957\pi$
$17$	4126.55i	0.839924i	−0.907542	+	0.419962i	$0.862043\pi$
$18$	0	0
$19$	12194.5	1.77788	0.888942	−	0.458020i	$-0.151441\pi$
$19$	12194.5	1.77788	0.888942	+	0.458020i	$0.151441\pi$
$20$	− 3541.71i	− 0.442714i
$21$	0	0
$22$	−3539.14	−0.332376
$23$	21450.5i	1.76300i	0.472182	+	0.881501i	$0.343467\pi$
$23$	21450.5i	1.76300i	−0.472182	+	0.881501i	$0.656533\pi$
$24$	0	0
$25$	3375.26	0.216017
$26$	− 11472.2i	− 0.652717i
$27$	0	0
$28$	−10442.8	−0.475711
$29$	− 24751.4i	− 1.01486i	−0.861693	−	0.507429i	$-0.830596\pi$
$29$	− 24751.4i	− 1.01486i	0.861693	−	0.507429i	$-0.169404\pi$
$30$	0	0
$31$	−40531.8	−1.36054	−0.680269	−	0.732963i	$-0.738137\pi$
$31$	−40531.8	−1.36054	−0.680269	+	0.732963i	$0.738137\pi$
$32$	5792.62i	0.176777i
$33$	0	0
$34$	−23343.3	−0.593916
$35$	36118.6i	0.842417i
$36$	0	0
$37$	−30093.7	−0.594115	−0.297058	−	0.954860i	$-0.596005\pi$
$37$	−30093.7	−0.594115	−0.297058	+	0.954860i	$0.596005\pi$
$38$	68982.5i	1.25715i
$39$	0	0
$40$	20035.0	0.313046
$41$	− 59229.9i	− 0.859388i	−0.902975	−	0.429694i	$-0.858621\pi$
$41$	− 59229.9i	− 0.859388i	0.902975	−	0.429694i	$-0.141379\pi$
$42$	0	0
$43$	−87935.7	−1.10601	−0.553006	−	0.833177i	$-0.686519\pi$
$43$	−87935.7	−1.10601	−0.553006	+	0.833177i	$0.686519\pi$
$44$	− 20020.4i	− 0.235025i
$45$	0	0
$46$	−121342.	−1.24663
$47$	129770.i	1.24992i	0.780658	+	0.624958i	$0.214884\pi$
$47$	129770.i	1.24992i	−0.780658	+	0.624958i	$0.785116\pi$
$48$	0	0
$49$	−11152.5	−0.0947946
$50$	19093.4i	0.152747i
$51$	0	0
$52$	64896.3	0.461541
$53$	165599.i	1.11232i	0.831074	+	0.556161i	$0.187726\pi$
$53$	165599.i	1.11232i	−0.831074	+	0.556161i	$0.812274\pi$
$54$	0	0
$55$	−69244.6	−0.416196
$56$	− 59073.5i	− 0.336379i
$57$	0	0
$58$	140015.	0.717613
$59$	− 53250.1i	− 0.259277i	−0.991561	−	0.129639i	$-0.958618\pi$
$59$	− 53250.1i	− 0.259277i	0.991561	−	0.129639i	$-0.0413818\pi$
$60$	0	0
$61$	−266982.	−1.17623	−0.588115	−	0.808778i	$-0.700130\pi$
$61$	−266982.	−1.17623	−0.588115	+	0.808778i	$0.700130\pi$
$62$	− 229282.i	− 0.962046i
$63$	0	0
$64$	−32768.0	−0.125000
$65$	− 224457.i	− 0.817323i
$66$	0	0
$67$	−407280.	−1.35416	−0.677079	−	0.735911i	$-0.736754\pi$
$67$	−407280.	−1.35416	−0.677079	+	0.735911i	$0.736754\pi$
$68$	− 132049.i	− 0.419962i
$69$	0	0
$70$	−204318.	−0.595679
$71$	− 186563.i	− 0.521254i	−0.965440	−	0.260627i	$-0.916071\pi$
$71$	− 186563.i	− 0.521254i	0.965440	−	0.260627i	$-0.0839293\pi$
$72$	0	0
$73$	−242102.	−0.622344	−0.311172	−	0.950354i	$-0.600721\pi$
$73$	−242102.	−0.622344	−0.311172	+	0.950354i	$0.600721\pi$
$74$	− 170236.i	− 0.420103i
$75$	0	0
$76$	−390224.	−0.888942
$77$	204169.i	0.447217i
$78$	0	0
$79$	−126373.	−0.256314	−0.128157	−	0.991754i	$-0.540906\pi$
$79$	−126373.	−0.256314	−0.128157	+	0.991754i	$0.540906\pi$
$80$	113335.i	0.221357i
$81$	0	0
$82$	335055.	0.607679
$83$	− 68971.7i	− 0.120625i	−0.998180	−	0.0603124i	$-0.980790\pi$
$83$	− 68971.7i	− 0.120625i	0.998180	−	0.0603124i	$-0.0192097\pi$
$84$	0	0
$85$	−456720.	−0.743693
$86$	− 497439.i	− 0.782069i
$87$	0	0
$88$	113252.	0.166188
$89$	− 413126.i	− 0.586020i	−0.956109	−	0.293010i	$-0.905343\pi$
$89$	− 413126.i	− 0.586020i	0.956109	−	0.293010i	$-0.0946569\pi$
$90$	0	0
$91$	−661817.	−0.878241
$92$	− 686414.i	− 0.881501i
$93$	0	0
$94$	−734091.	−0.883825
$95$	1.34967e6i	1.57419i
$96$	0	0
$97$	978443.	1.07206	0.536031	−	0.844198i	$-0.319923\pi$
$97$	978443.	1.07206	0.536031	+	0.844198i	$0.319923\pi$
$98$	− 63088.0i	− 0.0670299i
$99$	0	0
$100$	−108008.	−0.108008
$101$	1.12624e6i	1.09312i	0.837420	+	0.546560i	$0.184063\pi$
$101$	1.12624e6i	1.09312i	−0.837420	+	0.546560i	$0.815937\pi$
$102$	0	0
$103$	1.05758e6	0.967839	0.483919	−	0.875113i	$-0.339213\pi$
$103$	1.05758e6	0.967839	0.483919	+	0.875113i	$0.339213\pi$
$104$	367109.i	0.326359i
$105$	0	0
$106$	−936771.	−0.786531
$107$	607244.i	0.495692i	0.968799	+	0.247846i	$0.0797227\pi$
$107$	607244.i	0.495692i	−0.968799	+	0.247846i	$0.920277\pi$
$108$	0	0
$109$	961305.	0.742303	0.371152	−	0.928572i	$-0.378963\pi$
$109$	961305.	0.742303	0.371152	+	0.928572i	$0.378963\pi$
$110$	− 391707.i	− 0.294295i
$111$	0	0
$112$	334170.	0.237856
$113$	− 1.33878e6i	− 0.927839i	−0.885877	−	0.463919i	$-0.846443\pi$
$113$	− 1.33878e6i	− 0.927839i	0.885877	−	0.463919i	$-0.153557\pi$
$114$	0	0
$115$	−2.37410e6	−1.56101
$116$	792044.i	0.507429i
$117$	0	0
$118$	301228.	0.183337
$119$	1.34665e6i	0.799123i
$120$	0	0
$121$	1.38014e6	0.779053
$122$	− 1.51028e6i	− 0.831720i
$123$	0	0
$124$	1.29702e6	0.680269
$125$	2.10292e6i	1.07670i
$126$	0	0
$127$	−1.62823e6	−0.794885	−0.397443	−	0.917627i	$-0.630102\pi$
$127$	−1.62823e6	−0.794885	−0.397443	+	0.917627i	$0.630102\pi$
$128$	− 185364.i	− 0.0883883i
$129$	0	0
$130$	1.26972e6	0.577934
$131$	1.38103e6i	0.614313i	0.951659	+	0.307156i	$0.0993774\pi$
$131$	1.38103e6i	0.614313i	−0.951659	+	0.307156i	$0.900623\pi$
$132$	0	0
$133$	3.97953e6	1.69152
$134$	− 2.30393e6i	− 0.957534i
$135$	0	0
$136$	746985.	0.296958
$137$	3.54766e6i	1.37968i	0.723960	+	0.689842i	$0.242320\pi$
$137$	3.54766e6i	1.37968i	−0.723960	+	0.689842i	$0.757680\pi$
$138$	0	0
$139$	3.22032e6	1.19910	0.599548	−	0.800339i	$-0.295347\pi$
$139$	3.22032e6	1.19910	0.599548	+	0.800339i	$0.295347\pi$
$140$	− 1.15580e6i	− 0.421208i
$141$	0	0
$142$	1.05536e6	0.368582
$143$	− 1.26880e6i	− 0.433895i
$144$	0	0
$145$	2.73945e6	0.898585
$146$	− 1.36954e6i	− 0.440063i
$147$	0	0
$148$	962999.	0.297058
$149$	913857.i	0.276261i	0.990414	+	0.138130i	$0.0441093\pi$
$149$	913857.i	0.276261i	−0.990414	+	0.138130i	$0.955891\pi$
$150$	0	0
$151$	1.59890e6	0.464398	0.232199	−	0.972668i	$-0.425408\pi$
$151$	1.59890e6	0.464398	0.232199	+	0.972668i	$0.425408\pi$
$152$	− 2.20744e6i	− 0.628577i
$153$	0	0
$154$	−1.15495e6	−0.316230
$155$	− 4.48600e6i	− 1.20466i
$156$	0	0
$157$	−5.67074e6	−1.46535	−0.732673	−	0.680580i	$-0.761728\pi$
$157$	−5.67074e6	−1.46535	−0.732673	+	0.680580i	$0.761728\pi$
$158$	− 714872.i	− 0.181241i
$159$	0	0
$160$	−641119.	−0.156523
$161$	7.00010e6i	1.67736i
$162$	0	0
$163$	5.58510e6	1.28964	0.644820	−	0.764335i	$-0.276932\pi$
$163$	5.58510e6	1.28964	0.644820	+	0.764335i	$0.276932\pi$
$164$	1.89536e6i	0.429694i
$165$	0	0
$166$	390163.	0.0852946
$167$	5.92882e6i	1.27297i	0.771288	+	0.636486i	$0.219613\pi$
$167$	5.92882e6i	1.27297i	−0.771288	+	0.636486i	$0.780387\pi$
$168$	0	0
$169$	−713983.	−0.147920
$170$	− 2.58360e6i	− 0.525870i
$171$	0	0
$172$	2.81394e6	0.553006
$173$	− 2.24430e6i	− 0.433453i	−0.976232	−	0.216727i	$-0.930462\pi$
$173$	− 2.24430e6i	− 0.433453i	0.976232	−	0.216727i	$-0.0695380\pi$
$174$	0	0
$175$	1.10148e6	0.205523
$176$	640652.i	0.117513i
$177$	0	0
$178$	2.33699e6	0.414379
$179$	− 777450.i	− 0.135554i	−0.997700	−	0.0677772i	$-0.978409\pi$
$179$	− 777450.i	− 0.135554i	0.997700	−	0.0677772i	$-0.0215907\pi$
$180$	0	0
$181$	4.10696e6	0.692604	0.346302	−	0.938123i	$-0.387437\pi$
$181$	4.10696e6	0.692604	0.346302	+	0.938123i	$0.387437\pi$
$182$	− 3.74380e6i	− 0.621010i
$183$	0	0
$184$	3.88295e6	0.623316
$185$	− 3.33073e6i	− 0.526047i
$186$	0	0
$187$	−2.58172e6	−0.394807
$188$	− 4.15264e6i	− 0.624958i
$189$	0	0
$190$	−7.63489e6	−1.11312
$191$	− 4.26795e6i	− 0.612518i	−0.951948	−	0.306259i	$-0.900923\pi$
$191$	− 4.26795e6i	− 0.612518i	0.951948	−	0.306259i	$-0.0990773\pi$
$192$	0	0
$193$	−8.10012e6	−1.12673	−0.563365	−	0.826208i	$-0.690493\pi$
$193$	−8.10012e6	−1.12673	−0.563365	+	0.826208i	$0.690493\pi$
$194$	5.53491e6i	0.758063i
$195$	0	0
$196$	356880.	0.0473973
$197$	− 4.08228e6i	− 0.533955i	−0.963703	−	0.266977i	$-0.913975\pi$
$197$	− 4.08228e6i	− 0.533955i	0.963703	−	0.266977i	$-0.0860249\pi$
$198$	0	0
$199$	1.02814e6	0.130464	0.0652322	−	0.997870i	$-0.479221\pi$
$199$	1.02814e6	0.130464	0.0652322	+	0.997870i	$0.479221\pi$
$200$	− 610987.i	− 0.0763734i
$201$	0	0
$202$	−6.37098e6	−0.772952
$203$	− 8.07732e6i	− 0.965560i
$204$	0	0
$205$	6.55548e6	0.760927
$206$	5.98260e6i	0.684365i
$207$	0	0
$208$	−2.07668e6	−0.230770
$209$	7.62933e6i	0.835695i
$210$	0	0
$211$	1.54504e7	1.64472	0.822359	−	0.568969i	$-0.192657\pi$
$211$	1.54504e7	1.64472	0.822359	+	0.568969i	$0.192657\pi$
$212$	− 5.29918e6i	− 0.556161i
$213$	0	0
$214$	−3.43509e6	−0.350507
$215$	− 9.73259e6i	− 0.979294i
$216$	0	0
$217$	−1.32271e7	−1.29445
$218$	5.43796e6i	0.524888i
$219$	0	0
$220$	2.21583e6	0.208098
$221$	− 8.36868e6i	− 0.775318i
$222$	0	0
$223$	−2.81129e6	−0.253508	−0.126754	−	0.991934i	$-0.540456\pi$
$223$	−2.81129e6	−0.253508	−0.126754	+	0.991934i	$0.540456\pi$
$224$	1.89035e6i	0.168189i
$225$	0	0
$226$	7.57326e6	0.656081
$227$	7.45718e6i	0.637525i	0.947835	+	0.318762i	$0.103267\pi$
$227$	7.45718e6i	0.637525i	−0.947835	+	0.318762i	$0.896733\pi$
$228$	0	0
$229$	1.26246e7	1.05127	0.525633	−	0.850712i	$-0.323829\pi$
$229$	1.26246e7	1.05127	0.525633	+	0.850712i	$0.323829\pi$
$230$	− 1.34300e7i	− 1.10380i
$231$	0	0
$232$	−4.48048e6	−0.358807
$233$	1.64283e7i	1.29875i	0.760469	+	0.649374i	$0.224969\pi$
$233$	1.64283e7i	1.29875i	−0.760469	+	0.649374i	$0.775031\pi$
$234$	0	0
$235$	−1.43628e7	−1.10671
$236$	1.70400e6i	0.129639i
$237$	0	0
$238$	−7.61780e6	−0.565065
$239$	9.75791e6i	0.714765i	0.933958	+	0.357382i	$0.116331\pi$
$239$	9.75791e6i	0.714765i	−0.933958	+	0.357382i	$0.883669\pi$
$240$	0	0
$241$	−1.47403e7	−1.05307	−0.526534	−	0.850154i	$-0.676509\pi$
$241$	−1.47403e7	−1.05307	−0.526534	+	0.850154i	$0.676509\pi$
$242$	7.80725e6i	0.550873i
$243$	0	0
$244$	8.54341e6	0.588115
$245$	− 1.23434e6i	− 0.0839338i
$246$	0	0
$247$	−2.47306e7	−1.64113
$248$	7.33704e6i	0.481023i
$249$	0	0
$250$	−1.18959e7	−0.761339
$251$	− 1.31485e7i	− 0.831486i	−0.909482	−	0.415743i	$-0.863522\pi$
$251$	− 1.31485e7i	− 0.831486i	0.909482	−	0.415743i	$-0.136478\pi$
$252$	0	0
$253$	−1.34202e7	−0.828700
$254$	− 9.21065e6i	− 0.562069i
$255$	0	0
$256$	1.04858e6	0.0625000
$257$	2.15267e7i	1.26817i	0.773262	+	0.634087i	$0.218624\pi$
$257$	2.15267e7i	1.26817i	−0.773262	+	0.634087i	$0.781376\pi$
$258$	0	0
$259$	−9.82073e6	−0.565255
$260$	7.18263e6i	0.408661i
$261$	0	0
$262$	−7.81229e6	−0.434385
$263$	8.49122e6i	0.466770i	0.972384	+	0.233385i	$0.0749803\pi$
$263$	8.49122e6i	0.466770i	−0.972384	+	0.233385i	$0.925020\pi$
$264$	0	0
$265$	−1.83283e7	−0.984882
$266$	2.25116e7i	1.19608i
$267$	0	0
$268$	1.30330e7	0.677079
$269$	− 883900.i	− 0.0454094i	−0.999742	−	0.0227047i	$-0.992772\pi$
$269$	− 883900.i	− 0.0454094i	0.999742	−	0.0227047i	$-0.00722776\pi$
$270$	0	0
$271$	−6.17663e6	−0.310345	−0.155172	−	0.987887i	$-0.549593\pi$
$271$	−6.17663e6	−0.310345	−0.155172	+	0.987887i	$0.549593\pi$
$272$	4.22558e6i	0.209981i
$273$	0	0
$274$	−2.00686e7	−0.975584
$275$	2.11169e6i	0.101539i
$276$	0	0
$277$	−5.80573e6	−0.273160	−0.136580	−	0.990629i	$-0.543611\pi$
$277$	−5.80573e6	−0.273160	−0.136580	+	0.990629i	$0.543611\pi$
$278$	1.82169e7i	0.847889i
$279$	0	0
$280$	6.53817e6	0.297839
$281$	− 1.35800e7i	− 0.612039i	−0.952025	−	0.306020i	$-0.901003\pi$
$281$	− 1.35800e7i	− 0.612039i	0.952025	−	0.306020i	$-0.0989973\pi$
$282$	0	0
$283$	5.06204e6	0.223340	0.111670	−	0.993745i	$-0.464380\pi$
$283$	5.06204e6	0.223340	0.111670	+	0.993745i	$0.464380\pi$
$284$	5.97000e6i	0.260627i
$285$	0	0
$286$	7.17741e6	0.306810
$287$	− 1.93290e7i	− 0.817642i
$288$	0	0
$289$	7.10918e6	0.294528
$290$	1.54967e7i	0.635395i
$291$	0	0
$292$	7.74727e6	0.311172
$293$	− 3.76667e7i	− 1.49746i	−0.662875	−	0.748730i	$-0.730664\pi$
$293$	− 3.76667e7i	− 1.49746i	0.662875	−	0.748730i	$-0.269336\pi$
$294$	0	0
$295$	5.89365e6	0.229572
$296$	5.44755e6i	0.210051i
$297$	0	0
$298$	−5.16956e6	−0.195346
$299$	− 4.35017e7i	− 1.62740i
$300$	0	0
$301$	−2.86968e7	−1.05229
$302$	9.04475e6i	0.328379i
$303$	0	0
$304$	1.24872e7	0.444471
$305$	− 2.95491e7i	− 1.04147i
$306$	0	0
$307$	4.60441e7	1.59132	0.795662	−	0.605740i	$-0.207123\pi$
$307$	4.60441e7	1.59132	0.795662	+	0.605740i	$0.207123\pi$
$308$	− 6.53341e6i	− 0.223608i
$309$	0	0
$310$	2.53766e7	0.851823
$311$	− 2.85310e7i	− 0.948498i	−0.880391	−	0.474249i	$-0.842720\pi$
$311$	− 2.85310e7i	− 0.948498i	0.880391	−	0.474249i	$-0.157280\pi$
$312$	0	0
$313$	2.20513e7	0.719120	0.359560	−	0.933122i	$-0.382927\pi$
$313$	2.20513e7	0.719120	0.359560	+	0.933122i	$0.382927\pi$
$314$	− 3.20785e7i	− 1.03616i
$315$	0	0
$316$	4.04392e6	0.128157
$317$	1.73192e7i	0.543689i	0.962341	+	0.271844i	$0.0876336\pi$
$317$	1.73192e7i	0.543689i	−0.962341	+	0.271844i	$0.912366\pi$
$318$	0	0
$319$	1.54854e7	0.477035
$320$	− 3.62671e6i	− 0.110679i
$321$	0	0
$322$	−3.95985e7	−1.18607
$323$	5.03212e7i	1.49329i
$324$	0	0
$325$	−6.84506e6	−0.199401
$326$	3.15941e7i	0.911913i
$327$	0	0
$328$	−1.07218e7	−0.303840
$329$	4.23489e7i	1.18920i
$330$	0	0
$331$	1.41600e7	0.390461	0.195231	−	0.980757i	$-0.437454\pi$
$331$	1.41600e7	0.390461	0.195231	+	0.980757i	$0.437454\pi$
$332$	2.20709e6i	0.0603124i
$333$	0	0
$334$	−3.35385e7	−0.900127
$335$	− 4.50772e7i	− 1.19901i
$336$	0	0
$337$	4.81393e7	1.25780	0.628898	−	0.777488i	$-0.283506\pi$
$337$	4.81393e7	1.25780	0.628898	+	0.777488i	$0.283506\pi$
$338$	− 4.03890e6i	− 0.104595i
$339$	0	0
$340$	1.46150e7	0.371846
$341$	− 2.53582e7i	− 0.639521i
$342$	0	0
$343$	−4.20328e7	−1.04161
$344$	1.59181e7i	0.391034i
$345$	0	0
$346$	1.26957e7	0.306498
$347$	− 5.48516e7i	− 1.31281i	−0.754410	−	0.656404i	$-0.772077\pi$
$347$	− 5.48516e7i	− 1.31281i	0.754410	−	0.656404i	$-0.227923\pi$
$348$	0	0
$349$	−1.71985e7	−0.404589	−0.202295	−	0.979325i	$-0.564840\pi$
$349$	−1.71985e7	−0.404589	−0.202295	+	0.979325i	$0.564840\pi$
$350$	6.23089e6i	0.145327i
$351$	0	0
$352$	−3.62408e6	−0.0830939
$353$	− 7.22010e6i	− 0.164142i	−0.996626	−	0.0820709i	$-0.973847\pi$
$353$	− 7.22010e6i	− 0.164142i	0.996626	−	0.0820709i	$-0.0261534\pi$
$354$	0	0
$355$	2.06485e7	0.461533
$356$	1.32200e7i	0.293010i
$357$	0	0
$358$	4.39792e6	0.0958514
$359$	− 1.62223e6i	− 0.0350614i	−0.999846	−	0.0175307i	$-0.994420\pi$
$359$	− 1.62223e6i	− 0.0350614i	0.999846	−	0.0175307i	$-0.00558048\pi$
$360$	0	0
$361$	1.01660e8	2.16087
$362$	2.32325e7i	0.489745i
$363$	0	0
$364$	2.11781e7	0.439120
$365$	− 2.67955e7i	− 0.551041i
$366$	0	0
$367$	2.17291e7	0.439586	0.219793	−	0.975547i	$-0.429462\pi$
$367$	2.17291e7	0.439586	0.219793	+	0.975547i	$0.429462\pi$
$368$	2.19653e7i	0.440751i
$369$	0	0
$370$	1.88415e7	0.371971
$371$	5.40413e7i	1.05829i
$372$	0	0
$373$	6.07342e7	1.17033	0.585163	−	0.810915i	$-0.301030\pi$
$373$	6.07342e7	1.17033	0.585163	+	0.810915i	$0.301030\pi$
$374$	− 1.46044e7i	− 0.279170i
$375$	0	0
$376$	2.34909e7	0.441912
$377$	5.01961e7i	0.936797i
$378$	0	0
$379$	2.01121e7	0.369437	0.184719	−	0.982791i	$-0.440863\pi$
$379$	2.01121e7	0.369437	0.184719	+	0.982791i	$0.440863\pi$
$380$	− 4.31894e7i	− 0.787094i
$381$	0	0
$382$	2.41431e7	0.433116
$383$	− 4.72489e7i	− 0.840998i	−0.907293	−	0.420499i	$-0.861855\pi$
$383$	− 4.72489e7i	− 0.840998i	0.907293	−	0.420499i	$-0.138145\pi$
$384$	0	0
$385$	−2.25971e7	−0.395978
$386$	− 4.58212e7i	− 0.796718i
$387$	0	0
$388$	−3.13102e7	−0.536031
$389$	1.93628e7i	0.328942i	0.986382	+	0.164471i	$0.0525917\pi$
$389$	1.93628e7i	0.328942i	−0.986382	+	0.164471i	$0.947408\pi$
$390$	0	0
$391$	−8.85163e7	−1.48079
$392$	2.01882e6i	0.0335149i
$393$	0	0
$394$	2.30929e7	0.377563
$395$	− 1.39867e7i	− 0.226947i
$396$	0	0
$397$	−3.66656e7	−0.585986	−0.292993	−	0.956115i	$-0.594651\pi$
$397$	−3.66656e7	−0.585986	−0.292993	+	0.956115i	$0.594651\pi$
$398$	5.81602e6i	0.0922522i
$399$	0	0
$400$	3.45627e6	0.0540042
$401$	1.21073e8i	1.87765i	0.344391	+	0.938826i	$0.388085\pi$
$401$	1.21073e8i	1.87765i	−0.344391	+	0.938826i	$0.611915\pi$
$402$	0	0
$403$	8.21989e7	1.25589
$404$	− 3.60397e7i	− 0.546560i
$405$	0	0
$406$	4.56922e7	0.682754
$407$	− 1.88277e7i	− 0.279264i
$408$	0	0
$409$	3.98285e7	0.582135	0.291067	−	0.956703i	$-0.405990\pi$
$409$	3.98285e7	0.582135	0.291067	+	0.956703i	$0.405990\pi$
$410$	3.70834e7i	0.538056i
$411$	0	0
$412$	−3.38427e7	−0.483919
$413$	− 1.73775e7i	− 0.246683i
$414$	0	0
$415$	7.63369e6	0.106805
$416$	− 1.17475e7i	− 0.163179i
$417$	0	0
$418$	−4.31580e7	−0.590925
$419$	− 5.41631e7i	− 0.736311i	−0.929764	−	0.368155i	$-0.879989\pi$
$419$	− 5.41631e7i	− 0.736311i	0.929764	−	0.368155i	$-0.120011\pi$
$420$	0	0
$421$	5.41272e7	0.725386	0.362693	−	0.931909i	$-0.381857\pi$
$421$	5.41272e7	0.725386	0.362693	+	0.931909i	$0.381857\pi$
$422$	8.74005e7i	1.16299i
$423$	0	0
$424$	2.99767e7	0.393265
$425$	1.39282e7i	0.181438i
$426$	0	0
$427$	−8.71263e7	−1.11909
$428$	− 1.94318e7i	− 0.247846i
$429$	0	0
$430$	5.50559e7	0.692466
$431$	− 8.08186e7i	− 1.00944i	−0.863284	−	0.504719i	$-0.831596\pi$
$431$	− 8.08186e7i	− 1.00944i	0.863284	−	0.504719i	$-0.168404\pi$
$432$	0	0
$433$	−1.33718e7	−0.164713	−0.0823563	−	0.996603i	$-0.526245\pi$
$433$	−1.33718e7	−0.164713	−0.0823563	+	0.996603i	$0.526245\pi$
$434$	− 7.48236e7i	− 0.915312i
$435$	0	0
$436$	−3.07617e7	−0.371152
$437$	2.61578e8i	3.13441i
$438$	0	0
$439$	−7.63606e7	−0.902560	−0.451280	−	0.892382i	$-0.649032\pi$
$439$	−7.63606e7	−0.902560	−0.451280	+	0.892382i	$0.649032\pi$
$440$	1.25346e7i	0.147147i
$441$	0	0
$442$	4.73404e7	0.548233
$443$	− 1.11204e8i	− 1.27911i	−0.768743	−	0.639557i	$-0.779118\pi$
$443$	− 1.11204e8i	− 1.27911i	0.768743	−	0.639557i	$-0.220882\pi$
$444$	0	0
$445$	4.57242e7	0.518879
$446$	− 1.59031e7i	− 0.179257i
$447$	0	0
$448$	−1.06934e7	−0.118928
$449$	1.36515e8i	1.50814i	0.656793	+	0.754071i	$0.271913\pi$
$449$	1.36515e8i	1.50814i	−0.656793	+	0.754071i	$0.728087\pi$
$450$	0	0
$451$	3.70564e7	0.403956
$452$	4.28408e7i	0.463919i
$453$	0	0
$454$	−4.21842e7	−0.450798
$455$	− 7.32489e7i	− 0.777619i
$456$	0	0
$457$	−1.08409e8	−1.13584	−0.567920	−	0.823084i	$-0.692252\pi$
$457$	−1.08409e8	−1.13584	−0.567920	+	0.823084i	$0.692252\pi$
$458$	7.14157e7i	0.743357i
$459$	0	0
$460$	7.59713e7	0.780506
$461$	1.64142e8i	1.67540i	0.546132	+	0.837699i	$0.316100\pi$
$461$	1.64142e8i	1.67540i	−0.546132	+	0.837699i	$0.683900\pi$
$462$	0	0
$463$	3.73969e7	0.376784	0.188392	−	0.982094i	$-0.439672\pi$
$463$	3.73969e7	0.376784	0.188392	+	0.982094i	$0.439672\pi$
$464$	− 2.53454e7i	− 0.253715i
$465$	0	0
$466$	−9.29326e7	−0.918354
$467$	− 1.45951e8i	− 1.43304i	−0.697569	−	0.716518i	$-0.745735\pi$
$467$	− 1.45951e8i	− 1.43304i	0.697569	−	0.716518i	$-0.254265\pi$
$468$	0	0
$469$	−1.32911e8	−1.28838
$470$	− 8.12481e7i	− 0.782564i
$471$	0	0
$472$	−9.63931e6	−0.0916684
$473$	− 5.50158e7i	− 0.519881i
$474$	0	0
$475$	4.11596e7	0.384053
$476$	− 4.30928e7i	− 0.399561i
$477$	0	0
$478$	−5.51991e7	−0.505415
$479$	1.32972e8i	1.20991i	0.796258	+	0.604957i	$0.206810\pi$
$479$	1.32972e8i	1.20991i	−0.796258	+	0.604957i	$0.793190\pi$
$480$	0	0
$481$	6.10304e7	0.548417
$482$	− 8.33840e7i	− 0.744632i
$483$	0	0
$484$	−4.41645e7	−0.389526
$485$	1.08293e8i	0.949235i
$486$	0	0
$487$	−4.62190e7	−0.400160	−0.200080	−	0.979780i	$-0.564120\pi$
$487$	−4.62190e7	−0.400160	−0.200080	+	0.979780i	$0.564120\pi$
$488$	4.83288e7i	0.415860i
$489$	0	0
$490$	6.98249e6	0.0593502
$491$	9.15791e7i	0.773663i	0.922150	+	0.386832i	$0.126431\pi$
$491$	9.15791e7i	0.773663i	−0.922150	+	0.386832i	$0.873569\pi$
$492$	0	0
$493$	1.02138e8	0.852404
$494$	− 1.39897e8i	− 1.16046i
$495$	0	0
$496$	−4.15046e7	−0.340135
$497$	− 6.08824e7i	− 0.495933i
$498$	0	0
$499$	−5.71899e7	−0.460275	−0.230138	−	0.973158i	$-0.573918\pi$
$499$	−5.71899e7	−0.460275	−0.230138	+	0.973158i	$0.573918\pi$
$500$	− 6.72935e7i	− 0.538348i
$501$	0	0
$502$	7.43791e7	0.587949
$503$	1.85206e8i	1.45530i	0.685950	+	0.727649i	$0.259387\pi$
$503$	1.85206e8i	1.45530i	−0.685950	+	0.727649i	$0.740613\pi$
$504$	0	0
$505$	−1.24651e8	−0.967879
$506$	− 7.59161e7i	− 0.585979i
$507$	0	0
$508$	5.21033e7	0.397443
$509$	− 1.25335e8i	− 0.950430i	−0.879870	−	0.475215i	$-0.842370\pi$
$509$	− 1.25335e8i	− 0.950430i	0.879870	−	0.475215i	$-0.157630\pi$
$510$	0	0
$511$	−7.90072e7	−0.592112
$512$	5.93164e6i	0.0441942i
$513$	0	0
$514$	−1.21774e8	−0.896734
$515$	1.17052e8i	0.856952i
$516$	0	0
$517$	−8.11890e7	−0.587524
$518$	− 5.55544e7i	− 0.399696i
$519$	0	0
$520$	−4.06311e7	−0.288967
$521$	− 1.64203e8i	− 1.16109i	−0.814227	−	0.580546i	$-0.802839\pi$
$521$	− 1.64203e8i	− 1.16109i	0.814227	−	0.580546i	$-0.197161\pi$
$522$	0	0
$523$	−1.39702e8	−0.976559	−0.488280	−	0.872687i	$-0.662375\pi$
$523$	−1.39702e8	−0.976559	−0.488280	+	0.872687i	$0.662375\pi$
$524$	− 4.41930e7i	− 0.307156i
$525$	0	0
$526$	−4.80336e7	−0.330056
$527$	− 1.67256e8i	− 1.14275i
$528$	0	0
$529$	−3.12086e8	−2.10818
$530$	− 1.03680e8i	− 0.696417i
$531$	0	0
$532$	−1.27345e8	−0.845760
$533$	1.20119e8i	0.793285i
$534$	0	0
$535$	−6.72089e7	−0.438900
$536$	7.37256e7i	0.478767i
$537$	0	0
$538$	5.00009e6	0.0321093
$539$	− 6.97741e6i	− 0.0445582i
$540$	0	0
$541$	1.45535e8	0.919130	0.459565	−	0.888144i	$-0.348005\pi$
$541$	1.45535e8	0.919130	0.459565	+	0.888144i	$0.348005\pi$
$542$	− 3.49403e7i	− 0.219447i
$543$	0	0
$544$	−2.39035e7	−0.148479
$545$	1.06396e8i	0.657257i
$546$	0	0
$547$	−8.57155e6	−0.0523718	−0.0261859	−	0.999657i	$-0.508336\pi$
$547$	−8.57155e6	−0.0523718	−0.0261859	+	0.999657i	$0.508336\pi$
$548$	− 1.13525e8i	− 0.689842i
$549$	0	0
$550$	−1.19455e7	−0.0717987
$551$	− 3.01831e8i	− 1.80430i
$552$	0	0
$553$	−4.12402e7	−0.243863
$554$	− 3.28422e7i	− 0.193153i
$555$	0	0
$556$	−1.03050e8	−0.599548
$557$	8.68427e7i	0.502536i	0.967918	+	0.251268i	$0.0808476\pi$
$557$	8.68427e7i	0.502536i	−0.967918	+	0.251268i	$0.919152\pi$
$558$	0	0
$559$	1.78335e8	1.02094
$560$	3.69855e7i	0.210604i
$561$	0	0
$562$	7.68198e7	0.432777
$563$	6.07412e6i	0.0340375i	0.999855	+	0.0170188i	$0.00541750\pi$
$563$	6.07412e6i	0.0340375i	−0.999855	+	0.0170188i	$0.994582\pi$
$564$	0	0
$565$	1.48174e8	0.821535
$566$	2.86352e7i	0.157925i
$567$	0	0
$568$	−3.37714e7	−0.184291
$569$	− 1.56056e8i	− 0.847116i	−0.905869	−	0.423558i	$-0.860781\pi$
$569$	− 1.56056e8i	− 0.847116i	0.905869	−	0.423558i	$-0.139219\pi$
$570$	0	0
$571$	1.02008e8	0.547933	0.273967	−	0.961739i	$-0.411664\pi$
$571$	1.02008e8	0.547933	0.273967	+	0.961739i	$0.411664\pi$
$572$	4.06015e7i	0.216947i
$573$	0	0
$574$	1.09341e8	0.578160
$575$	7.24009e7i	0.380838i
$576$	0	0
$577$	2.90791e8	1.51375	0.756874	−	0.653561i	$-0.226726\pi$
$577$	2.90791e8	1.51375	0.756874	+	0.653561i	$0.226726\pi$
$578$	4.02156e7i	0.208262i
$579$	0	0
$580$	−8.76623e7	−0.449292
$581$	− 2.25081e7i	− 0.114765i
$582$	0	0
$583$	−1.03605e8	−0.522848
$584$	4.38252e7i	0.220032i
$585$	0	0
$586$	2.13075e8	1.05886
$587$	− 1.08186e8i	− 0.534880i	−0.963574	−	0.267440i	$-0.913822\pi$
$587$	− 1.08186e8i	− 0.534880i	0.963574	−	0.267440i	$-0.0861778\pi$
$588$	0	0
$589$	−4.94265e8	−2.41888
$590$	3.33395e7i	0.162332i
$591$	0	0
$592$	−3.08160e7	−0.148529
$593$	2.75079e8i	1.31915i	0.751641	+	0.659573i	$0.229263\pi$
$593$	2.75079e8i	1.31915i	−0.751641	+	0.659573i	$0.770737\pi$
$594$	0	0
$595$	−1.49045e8	−0.707566
$596$	− 2.92434e7i	− 0.138130i
$597$	0	0
$598$	2.46083e8	1.15074
$599$	3.39727e8i	1.58070i	0.612654	+	0.790351i	$0.290102\pi$
$599$	3.39727e8i	1.58070i	−0.612654	+	0.790351i	$0.709898\pi$
$600$	0	0
$601$	1.61656e8	0.744679	0.372340	−	0.928097i	$-0.378556\pi$
$601$	1.61656e8	0.744679	0.372340	+	0.928097i	$0.378556\pi$
$602$	− 1.62333e8i	− 0.744078i
$603$	0	0
$604$	−5.11648e7	−0.232199
$605$	1.52752e8i	0.689795i
$606$	0	0
$607$	2.27189e8	1.01583	0.507916	−	0.861406i	$-0.330416\pi$
$607$	2.27189e8	1.01583	0.507916	+	0.861406i	$0.330416\pi$
$608$	7.06381e7i	0.314288i
$609$	0	0
$610$	1.67155e8	0.736428
$611$	− 2.63175e8i	− 1.15378i
$612$	0	0
$613$	−1.32276e8	−0.574247	−0.287124	−	0.957894i	$-0.592699\pi$
$613$	−1.32276e8	−0.574247	−0.287124	+	0.957894i	$0.592699\pi$
$614$	2.60465e8i	1.12524i
$615$	0	0
$616$	3.69586e7	0.158115
$617$	3.56815e8i	1.51910i	0.650447	+	0.759551i	$0.274581\pi$
$617$	3.56815e8i	1.51910i	−0.650447	+	0.759551i	$0.725419\pi$
$618$	0	0
$619$	3.12067e8	1.31576	0.657880	−	0.753123i	$-0.271453\pi$
$619$	3.12067e8	1.31576	0.657880	+	0.753123i	$0.271453\pi$
$620$	1.43552e8i	0.602329i
$621$	0	0
$622$	1.61396e8	0.670689
$623$	− 1.34819e8i	− 0.557553i
$624$	0	0
$625$	−1.80010e8	−0.737320
$626$	1.24741e8i	0.508495i
$627$	0	0
$628$	1.81464e8	0.732673
$629$	− 1.24183e8i	− 0.499012i
$630$	0	0
$631$	1.95005e7	0.0776172	0.0388086	−	0.999247i	$-0.487644\pi$
$631$	1.95005e7	0.0776172	0.0388086	+	0.999247i	$0.487644\pi$
$632$	2.28759e7i	0.0906206i
$633$	0	0
$634$	−9.79723e7	−0.384446
$635$	− 1.80210e8i	− 0.703814i
$636$	0	0
$637$	2.26174e7	0.0875031
$638$	8.75985e7i	0.337314i
$639$	0	0
$640$	2.05158e7	0.0782615
$641$	4.28268e8i	1.62608i	0.582208	+	0.813040i	$0.302189\pi$
$641$	4.28268e8i	1.62608i	−0.582208	+	0.813040i	$0.697811\pi$
$642$	0	0
$643$	3.22005e8	1.21124	0.605619	−	0.795755i	$-0.292926\pi$
$643$	3.22005e8	1.21124	0.605619	+	0.795755i	$0.292926\pi$
$644$	− 2.24003e8i	− 0.838680i
$645$	0	0
$646$	−2.84660e8	−1.05591
$647$	8.47366e7i	0.312866i	0.987689	+	0.156433i	$0.0499995\pi$
$647$	8.47366e7i	0.312866i	−0.987689	+	0.156433i	$0.950001\pi$
$648$	0	0
$649$	3.33153e7	0.121873
$650$	− 3.87215e7i	− 0.140998i
$651$	0	0
$652$	−1.78723e8	−0.644820
$653$	− 4.37807e8i	− 1.57233i	−0.618019	−	0.786163i	$-0.712065\pi$
$653$	− 4.37807e8i	− 1.57233i	0.618019	−	0.786163i	$-0.287935\pi$
$654$	0	0
$655$	−1.52850e8	−0.543930
$656$	− 6.06514e7i	− 0.214847i
$657$	0	0
$658$	−2.39562e8	−0.840891
$659$	− 2.72708e8i	− 0.952888i	−0.879205	−	0.476444i	$-0.841926\pi$
$659$	− 2.72708e8i	− 0.952888i	0.879205	−	0.476444i	$-0.158074\pi$
$660$	0	0
$661$	2.94289e8	1.01899	0.509495	−	0.860474i	$-0.329832\pi$
$661$	2.94289e8	1.01899	0.509495	+	0.860474i	$0.329832\pi$
$662$	8.01008e7i	0.276098i
$663$	0	0
$664$	−1.24852e7	−0.0426473
$665$	4.40449e8i	1.49772i
$666$	0	0
$667$	5.30928e8	1.78920
$668$	− 1.89722e8i	− 0.636486i
$669$	0	0
$670$	2.54995e8	0.847828
$671$	− 1.67034e8i	− 0.552887i
$672$	0	0
$673$	2.97051e7	0.0974508	0.0487254	−	0.998812i	$-0.484484\pi$
$673$	2.97051e7	0.0974508	0.0487254	+	0.998812i	$0.484484\pi$
$674$	2.72317e8i	0.889396i
$675$	0	0
$676$	2.28475e7	0.0739601
$677$	5.39379e8i	1.73831i	0.494536	+	0.869157i	$0.335338\pi$
$677$	5.39379e8i	1.73831i	−0.494536	+	0.869157i	$0.664662\pi$
$678$	0	0
$679$	3.19303e8	1.01998
$680$	8.26752e7i	0.262935i
$681$	0	0
$682$	1.43448e8	0.452210
$683$	− 5.86183e8i	− 1.83980i	−0.392152	−	0.919901i	$-0.628269\pi$
$683$	− 5.86183e8i	− 1.83980i	0.392152	−	0.919901i	$-0.371731\pi$
$684$	0	0
$685$	−3.92649e8	−1.22161
$686$	− 2.37774e8i	− 0.736531i
$687$	0	0
$688$	−9.00462e7	−0.276503
$689$	− 3.35837e8i	− 1.02676i
$690$	0	0
$691$	−4.69056e8	−1.42164	−0.710822	−	0.703372i	$-0.751677\pi$
$691$	−4.69056e8	−1.42164	−0.710822	+	0.703372i	$0.751677\pi$
$692$	7.18176e7i	0.216727i
$693$	0	0
$694$	3.10288e8	0.928295
$695$	3.56420e8i	1.06171i
$696$	0	0
$697$	2.44415e8	0.721821
$698$	− 9.72895e7i	− 0.286088i
$699$	0	0
$700$	−3.52472e7	−0.102762
$701$	5.94604e7i	0.172613i	0.996269	+	0.0863065i	$0.0275064\pi$
$701$	5.94604e7i	0.172613i	−0.996269	+	0.0863065i	$0.972494\pi$
$702$	0	0
$703$	−3.66978e8	−1.05627
$704$	− 2.05009e7i	− 0.0587563i
$705$	0	0
$706$	4.08431e7	0.116066
$707$	3.67535e8i	1.04002i
$708$	0	0
$709$	6.68042e8	1.87441	0.937206	−	0.348777i	$-0.113403\pi$
$709$	6.68042e8	1.87441	0.937206	+	0.348777i	$0.113403\pi$
$710$	1.16805e8i	0.326353i
$711$	0	0
$712$	−7.47838e7	−0.207189
$713$	− 8.69425e8i	− 2.39863i
$714$	0	0
$715$	1.40429e8	0.384183
$716$	2.48784e7i	0.0677772i
$717$	0	0
$718$	9.17673e6	0.0247922
$719$	− 6.40752e8i	− 1.72387i	−0.507022	−	0.861933i	$-0.669254\pi$
$719$	− 6.40752e8i	− 1.72387i	0.507022	−	0.861933i	$-0.330746\pi$
$720$	0	0
$721$	3.45130e8	0.920824
$722$	5.75076e8i	1.52797i
$723$	0	0
$724$	−1.31423e8	−0.346302
$725$	− 8.35424e7i	− 0.219226i
$726$	0	0
$727$	3.29279e8	0.856960	0.428480	−	0.903551i	$-0.359049\pi$
$727$	3.29279e8	0.856960	0.428480	+	0.903551i	$0.359049\pi$
$728$	1.19802e8i	0.310505i
$729$	0	0
$730$	1.51578e8	0.389645
$731$	− 3.62871e8i	− 0.928966i
$732$	0	0
$733$	3.62595e8	0.920683	0.460342	−	0.887742i	$-0.347727\pi$
$733$	3.62595e8	0.920683	0.460342	+	0.887742i	$0.347727\pi$
$734$	1.22918e8i	0.310834i
$735$	0	0
$736$	−1.24254e8	−0.311658
$737$	− 2.54810e8i	− 0.636522i
$738$	0	0
$739$	1.76953e7	0.0438454	0.0219227	−	0.999760i	$-0.493021\pi$
$739$	1.76953e7	0.0438454	0.0219227	+	0.999760i	$0.493021\pi$
$740$	1.06583e8i	0.263023i
$741$	0	0
$742$	−3.05704e8	−0.748324
$743$	− 1.28993e8i	− 0.314486i	−0.987560	−	0.157243i	$-0.949739\pi$
$743$	− 1.28993e8i	− 0.314486i	0.987560	−	0.157243i	$-0.0502605\pi$
$744$	0	0
$745$	−1.01144e8	−0.244609
$746$	3.43565e8i	0.827546i
$747$	0	0
$748$	8.26150e7	0.197403
$749$	1.98167e8i	0.471613i
$750$	0	0
$751$	−3.37629e8	−0.797114	−0.398557	−	0.917144i	$-0.630489\pi$
$751$	−3.37629e8	−0.797114	−0.398557	+	0.917144i	$0.630489\pi$
$752$	1.32885e8i	0.312479i
$753$	0	0
$754$	−2.83952e8	−0.662416
$755$	1.76964e8i	0.411192i
$756$	0	0
$757$	−9.58119e7	−0.220868	−0.110434	−	0.993883i	$-0.535224\pi$
$757$	−9.58119e7	−0.220868	−0.110434	+	0.993883i	$0.535224\pi$
$758$	1.13771e8i	0.261232i
$759$	0	0
$760$	2.44316e8	0.556560
$761$	3.34633e8i	0.759303i	0.925130	+	0.379651i	$0.123956\pi$
$761$	3.34633e8i	0.759303i	−0.925130	+	0.379651i	$0.876044\pi$
$762$	0	0
$763$	3.13710e8	0.706244
$764$	1.36574e8i	0.306259i
$765$	0	0
$766$	2.67280e8	0.594676
$767$	1.07992e8i	0.239334i
$768$	0	0
$769$	−2.42574e8	−0.533415	−0.266708	−	0.963778i	$-0.585936\pi$
$769$	−2.42574e8	−0.533415	−0.266708	+	0.963778i	$0.585936\pi$
$770$	− 1.27829e8i	− 0.279999i
$771$	0	0
$772$	2.59204e8	0.563365
$773$	2.36476e8i	0.511974i	0.966680	+	0.255987i	$0.0824005\pi$
$773$	2.36476e8i	0.511974i	−0.966680	+	0.255987i	$0.917599\pi$
$774$	0	0
$775$	−1.36805e8	−0.293899
$776$	− 1.77117e8i	− 0.379031i
$777$	0	0
$778$	−1.09533e8	−0.232597
$779$	− 7.22279e8i	− 1.52789i
$780$	0	0
$781$	1.16720e8	0.245016
$782$	− 5.00724e8i	− 1.04708i
$783$	0	0
$784$	−1.14201e7	−0.0236986
$785$	− 6.27629e8i	− 1.29746i
$786$	0	0
$787$	−4.96838e8	−1.01927	−0.509636	−	0.860390i	$-0.670220\pi$
$787$	−4.96838e8	−1.01927	−0.509636	+	0.860390i	$0.670220\pi$
$788$	1.30633e8i	0.266977i
$789$	0	0
$790$	7.91210e7	0.160476
$791$	− 4.36893e8i	− 0.882767i
$792$	0	0
$793$	5.41442e8	1.08576
$794$	− 2.07412e8i	− 0.414355i
$795$	0	0
$796$	−3.29004e7	−0.0652322
$797$	1.06050e8i	0.209476i	0.994500	+	0.104738i	$0.0334004\pi$
$797$	1.06050e8i	0.209476i	−0.994500	+	0.104738i	$0.966600\pi$
$798$	0	0
$799$	−5.35503e8	−1.04984
$800$	1.95516e7i	0.0381867i
$801$	0	0
$802$	−6.84894e8	−1.32770
$803$	− 1.51468e8i	− 0.292533i
$804$	0	0
$805$	−7.74761e8	−1.48518
$806$	4.64987e8i	0.888047i
$807$	0	0
$808$	2.03872e8	0.386476
$809$	− 2.99505e8i	− 0.565664i	−0.959169	−	0.282832i	$-0.908726\pi$
$809$	− 2.99505e8i	− 0.565664i	0.959169	−	0.282832i	$-0.0912739\pi$
$810$	0	0
$811$	−4.31983e8	−0.809848	−0.404924	−	0.914350i	$-0.632702\pi$
$811$	−4.31983e8	−0.809848	−0.404924	+	0.914350i	$0.632702\pi$
$812$	2.58474e8i	0.482780i
$813$	0	0
$814$	1.06506e8	0.197470
$815$	6.18151e8i	1.14188i
$816$	0	0
$817$	−1.07233e9	−1.96636
$818$	2.25304e8i	0.411632i
$819$	0	0
$820$	−2.09775e8	−0.380463
$821$	2.64725e8i	0.478371i	0.970974	+	0.239186i	$0.0768804\pi$
$821$	2.64725e8i	0.478371i	−0.970974	+	0.239186i	$0.923120\pi$
$822$	0	0
$823$	−3.66538e8	−0.657537	−0.328768	−	0.944411i	$-0.606634\pi$
$823$	−3.66538e8	−0.657537	−0.328768	+	0.944411i	$0.606634\pi$
$824$	− 1.91443e8i	− 0.342183i
$825$	0	0
$826$	9.83023e7	0.174431
$827$	5.77816e8i	1.02158i	0.859705	+	0.510791i	$0.170647\pi$
$827$	5.77816e8i	1.02158i	−0.859705	+	0.510791i	$0.829353\pi$
$828$	0	0
$829$	−3.22169e8	−0.565484	−0.282742	−	0.959196i	$-0.591244\pi$
$829$	−3.22169e8	−0.565484	−0.282742	+	0.959196i	$0.591244\pi$
$830$	4.31827e7i	0.0755223i
$831$	0	0
$832$	6.64538e7	0.115385
$833$	− 4.60213e7i	− 0.0796203i
$834$	0	0
$835$	−6.56193e8	−1.12713
$836$	− 2.44139e8i	− 0.417847i
$837$	0	0
$838$	3.06393e8	0.520650
$839$	− 8.83088e8i	− 1.49526i	−0.664113	−	0.747632i	$-0.731191\pi$
$839$	− 8.83088e8i	− 1.49526i	0.664113	−	0.747632i	$-0.268809\pi$
$840$	0	0
$841$	−1.78078e7	−0.0299380
$842$	3.06190e8i	0.512925i
$843$	0	0
$844$	−4.94412e8	−0.822359
$845$	− 7.90226e7i	− 0.130973i
$846$	0	0
$847$	4.50392e8	0.741209
$848$	1.69574e8i	0.278081i
$849$	0	0
$850$	−7.87896e7	−0.128296
$851$	− 6.45524e8i	− 1.04743i
$852$	0	0
$853$	4.61854e8	0.744145	0.372073	−	0.928204i	$-0.378647\pi$
$853$	4.61854e8	0.744145	0.372073	+	0.928204i	$0.378647\pi$
$854$	− 4.92861e8i	− 0.791317i
$855$	0	0
$856$	1.09923e8	0.175254
$857$	− 4.26122e8i	− 0.677004i	−0.940966	−	0.338502i	$-0.890080\pi$
$857$	− 4.26122e8i	− 0.677004i	0.940966	−	0.338502i	$-0.109920\pi$
$858$	0	0
$859$	−3.09587e8	−0.488431	−0.244216	−	0.969721i	$-0.578531\pi$
$859$	−3.09587e8	−0.488431	−0.244216	+	0.969721i	$0.578531\pi$
$860$	3.11443e8i	0.489647i
$861$	0	0
$862$	4.57179e8	0.713781
$863$	3.39614e8i	0.528388i	0.964470	+	0.264194i	$0.0851059\pi$
$863$	3.39614e8i	0.528388i	−0.964470	+	0.264194i	$0.914894\pi$
$864$	0	0
$865$	2.48396e8	0.383792
$866$	− 7.56424e7i	− 0.116469i
$867$	0	0
$868$	4.23266e8	0.647223
$869$	− 7.90634e7i	− 0.120480i
$870$	0	0
$871$	8.25969e8	1.25000
$872$	− 1.74015e8i	− 0.262444i
$873$	0	0
$874$	−1.47971e9	−2.21636
$875$	6.86263e8i	1.02439i
$876$	0	0
$877$	5.82927e8	0.864203	0.432101	−	0.901825i	$-0.357772\pi$
$877$	5.82927e8	0.864203	0.432101	+	0.901825i	$0.357772\pi$
$878$	− 4.31961e8i	− 0.638206i
$879$	0	0
$880$	−7.09065e7	−0.104049
$881$	5.04493e8i	0.737781i	0.929473	+	0.368890i	$0.120262\pi$
$881$	5.04493e8i	0.737781i	−0.929473	+	0.368890i	$0.879738\pi$
$882$	0	0
$883$	−5.92257e8	−0.860256	−0.430128	−	0.902768i	$-0.641532\pi$
$883$	−5.92257e8	−0.860256	−0.430128	+	0.902768i	$0.641532\pi$
$884$	2.67798e8i	0.387659i
$885$	0	0
$886$	6.29065e8	0.904471
$887$	− 3.81132e8i	− 0.546141i	−0.961994	−	0.273071i	$-0.911961\pi$
$887$	− 3.81132e8i	− 0.546141i	0.961994	−	0.273071i	$-0.0880393\pi$
$888$	0	0
$889$	−5.31353e8	−0.756272
$890$	2.58655e8i	0.366903i
$891$	0	0
$892$	8.99614e7	0.126754
$893$	1.58248e9i	2.22221i
$894$	0	0
$895$	8.60470e7	0.120024
$896$	− 6.04913e7i	− 0.0840947i
$897$	0	0
$898$	−7.72247e8	−1.06642
$899$	1.00322e9i	1.38075i
$900$	0	0
$901$	−6.83353e8	−0.934267
$902$	2.09623e8i	0.285640i
$903$	0	0
$904$	−2.42344e8	−0.328041
$905$	4.54552e8i	0.613251i
$906$	0	0
$907$	−1.32768e9	−1.77939	−0.889695	−	0.456556i	$-0.849083\pi$
$907$	−1.32768e9	−1.77939	−0.889695	+	0.456556i	$0.849083\pi$
$908$	− 2.38630e8i	− 0.318762i
$909$	0	0
$910$	4.14359e8	0.549860
$911$	1.12864e9i	1.49279i	0.665503	+	0.746396i	$0.268217\pi$
$911$	1.12864e9i	1.49279i	−0.665503	+	0.746396i	$0.731783\pi$
$912$	0	0
$913$	4.31512e7	0.0566997
$914$	− 6.13255e8i	− 0.803161i
$915$	0	0
$916$	−4.03988e8	−0.525633
$917$	4.50683e8i	0.584471i
$918$	0	0
$919$	−1.60932e8	−0.207347	−0.103673	−	0.994611i	$-0.533060\pi$
$919$	−1.60932e8	−0.207347	−0.103673	+	0.994611i	$0.533060\pi$
$920$	4.29759e8i	0.551901i
$921$	0	0
$922$	−9.28530e8	−1.18469
$923$	3.78351e8i	0.481160i
$924$	0	0
$925$	−1.01574e8	−0.128339
$926$	2.11549e8i	0.266427i
$927$	0	0
$928$	1.43375e8	0.179403
$929$	− 2.29903e8i	− 0.286747i	−0.989669	−	0.143373i	$-0.954205\pi$
$929$	− 2.29903e8i	− 0.286747i	0.989669	−	0.143373i	$-0.0457949\pi$
$930$	0	0
$931$	−1.35999e8	−0.168534
$932$	− 5.25706e8i	− 0.649374i
$933$	0	0
$934$	8.25624e8	1.01331
$935$	− 2.85741e8i	− 0.349573i
$936$	0	0
$937$	−7.89016e8	−0.959107	−0.479553	−	0.877513i	$-0.659201\pi$
$937$	−7.89016e8	−0.959107	−0.479553	+	0.877513i	$0.659201\pi$
$938$	− 7.51859e8i	− 0.911019i
$939$	0	0
$940$	4.59609e8	0.553356
$941$	− 2.44392e8i	− 0.293304i	−0.989188	−	0.146652i	$-0.953150\pi$
$941$	− 2.44392e8i	− 0.293304i	0.989188	−	0.146652i	$-0.0468498\pi$
$942$	0	0
$943$	1.27051e9	1.51510
$944$	− 5.45282e7i	− 0.0648194i
$945$	0	0
$946$	3.11216e8	0.367612
$947$	− 4.77586e8i	− 0.562343i	−0.959657	−	0.281172i	$-0.909277\pi$
$947$	− 4.77586e8i	− 0.562343i	0.959657	−	0.281172i	$-0.0907230\pi$
$948$	0	0
$949$	4.90986e8	0.574474
$950$	2.32834e8i	0.271566i
$951$	0	0
$952$	2.43770e8	0.282533
$953$	− 1.14002e9i	− 1.31715i	−0.752517	−	0.658573i	$-0.771160\pi$
$953$	− 1.14002e9i	− 1.31715i	0.752517	−	0.658573i	$-0.228840\pi$
$954$	0	0
$955$	4.72370e8	0.542341
$956$	− 3.12253e8i	− 0.357382i
$957$	0	0
$958$	−7.52204e8	−0.855538
$959$	1.15773e9i	1.31266i
$960$	0	0
$961$	7.55322e8	0.851064
$962$	3.45240e8i	0.387789i
$963$	0	0
$964$	4.71691e8	0.526534
$965$	− 8.96510e8i	− 0.997638i
$966$	0	0
$967$	3.02793e8	0.334863	0.167431	−	0.985884i	$-0.446453\pi$
$967$	3.02793e8	0.334863	0.167431	+	0.985884i	$0.446453\pi$
$968$	− 2.49832e8i	− 0.275437i
$969$	0	0
$970$	−6.12595e8	−0.671210
$971$	− 2.05629e8i	− 0.224608i	−0.993674	−	0.112304i	$-0.964177\pi$
$971$	− 2.05629e8i	− 0.224608i	0.993674	−	0.112304i	$-0.0358231\pi$
$972$	0	0
$973$	1.05091e9	1.14085
$974$	− 2.61454e8i	− 0.282956i
$975$	0	0
$976$	−2.73389e8	−0.294057
$977$	5.87684e7i	0.0630173i	0.999503	+	0.0315087i	$0.0100312\pi$
$977$	5.87684e7i	0.0630173i	−0.999503	+	0.0315087i	$0.989969\pi$
$978$	0	0
$979$	2.58467e8	0.275459
$980$	3.94989e7i	0.0419669i
$981$	0	0
$982$	−5.18050e8	−0.547063
$983$	8.12264e7i	0.0855138i	0.999086	+	0.0427569i	$0.0136141\pi$
$983$	8.12264e7i	0.0855138i	−0.999086	+	0.0427569i	$0.986386\pi$
$984$	0	0
$985$	4.51821e8	0.472779
$986$	5.77778e8i	0.602741i
$987$	0	0
$988$	7.91379e8	0.820566
$989$	− 1.88626e9i	− 1.94990i
$990$	0	0
$991$	2.17703e8	0.223688	0.111844	−	0.993726i	$-0.464324\pi$
$991$	2.17703e8	0.223688	0.111844	+	0.993726i	$0.464324\pi$
$992$	− 2.34785e8i	− 0.240511i
$993$	0	0
$994$	3.44403e8	0.350678
$995$	1.13793e8i	0.115517i
$996$	0	0
$997$	2.19903e8	0.221894	0.110947	−	0.993826i	$-0.464612\pi$
$997$	2.19903e8	0.221894	0.110947	+	0.993826i	$0.464612\pi$
$998$	− 3.23515e8i	− 0.325464i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.7.b.c.161.11		12
3.2	odd	2	inner	162.7.b.c.161.2		12
9.2	odd	6		54.7.d.a.17.3		12
9.4	even	3		54.7.d.a.35.3		12
9.5	odd	6		18.7.d.a.11.4	yes	12
9.7	even	3		18.7.d.a.5.4	✓	12
36.7	odd	6		144.7.q.c.113.6		12
36.11	even	6		432.7.q.b.17.6		12
36.23	even	6		144.7.q.c.65.6		12
36.31	odd	6		432.7.q.b.305.6		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.7.d.a.5.4	✓	12	9.7	even	3
18.7.d.a.11.4	yes	12	9.5	odd	6
54.7.d.a.17.3		12	9.2	odd	6
54.7.d.a.35.3		12	9.4	even	3
144.7.q.c.65.6		12	36.23	even	6
144.7.q.c.113.6		12	36.7	odd	6
162.7.b.c.161.2		12	3.2	odd	2	inner
162.7.b.c.161.11		12	1.1	even	1	trivial
432.7.q.b.17.6		12	36.11	even	6
432.7.q.b.305.6		12	36.31	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	162.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+5.65685i q^{2} -32.0000 q^{4} +110.679i q^{5} +326.338 q^{7} -181.019i q^{8} -626.092 q^{10} +625.637i q^{11} -2028.01 q^{13} +1846.05i q^{14} +1024.00 q^{16} +4126.55i q^{17} +12194.5 q^{19} -3541.71i q^{20} -3539.14 q^{22} +21450.5i q^{23} +3375.26 q^{25} -11472.2i q^{26} -10442.8 q^{28} -24751.4i q^{29} -40531.8 q^{31} +5792.62i q^{32} -23343.3 q^{34} +36118.6i q^{35} -30093.7 q^{37} +68982.5i q^{38} +20035.0 q^{40} -59229.9i q^{41} -87935.7 q^{43} -20020.4i q^{44} -121342. q^{46} +129770. i q^{47} -11152.5 q^{49} +19093.4i q^{50} +64896.3 q^{52} +165599. i q^{53} -69244.6 q^{55} -59073.5i q^{56} +140015. q^{58} -53250.1i q^{59} -266982. q^{61} -229282. i q^{62} -32768.0 q^{64} -224457. i q^{65} -407280. q^{67} -132049. i q^{68} -204318. q^{70} -186563. i q^{71} -242102. q^{73} -170236. i q^{74} -390224. q^{76} +204169. i q^{77} -126373. q^{79} +113335. i q^{80} +335055. q^{82} -68971.7i q^{83} -456720. q^{85} -497439. i q^{86} +113252. q^{88} -413126. i q^{89} -661817. q^{91} -686414. i q^{92} -734091. q^{94} +1.34967e6i q^{95} +978443. q^{97} -63088.0i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 384 q^{4} - 480 q^{7} - 3360 q^{13} + 12288 q^{16} - 2820 q^{19} + 7200 q^{22} - 16188 q^{25} + 15360 q^{28} - 42960 q^{31} + 54720 q^{34} - 25536 q^{37} - 142860 q^{43} - 135072 q^{46} + 271908 q^{49}+ \cdots + 77748 q^{97}+O(q^{100}) \) 12 * q - 384 * q^4 - 480 * q^7 - 3360 * q^13 + 12288 * q^16 - 2820 * q^19 + 7200 * q^22 - 16188 * q^25 + 15360 * q^28 - 42960 * q^31 + 54720 * q^34 - 25536 * q^37 - 142860 * q^43 - 135072 * q^46 + 271908 * q^49 + 107520 * q^52 + 580392 * q^55 - 318528 * q^58 - 271488 * q^61 - 393216 * q^64 + 579876 * q^67 - 311904 * q^70 - 977700 * q^73 + 90240 * q^76 + 1529592 * q^79 + 1073088 * q^82 - 3239136 * q^85 - 230400 * q^88 + 355584 * q^91 + 1473696 * q^94 + 77748 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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