LMFDB - Embedded newform 162.7.b.c.161.1

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			161.1
Root			$4.28281i$ of defining polynomial
Character	$\chi$	$=$	162.161
Dual form			162.7.b.c.161.12

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-5.65685i q^{2} -32.0000 q^{4} -181.232i q^{5} -208.612 q^{7} +181.019i q^{8} -1025.20 q^{10} +2658.11i q^{11} +877.198 q^{13} +1180.09i q^{14} +1024.00 q^{16} +4428.53i q^{17} -4194.21 q^{19} +5799.42i q^{20} +15036.5 q^{22} -12008.8i q^{23} -17220.0 q^{25} -4962.18i q^{26} +6675.58 q^{28} +2642.86i q^{29} +5805.01 q^{31} -5792.62i q^{32} +25051.5 q^{34} +37807.1i q^{35} +41579.5 q^{37} +23726.1i q^{38} +32806.5 q^{40} +74733.6i q^{41} +146106. q^{43} -85059.5i q^{44} -67932.1 q^{46} -25772.5i q^{47} -74130.1 q^{49} +97411.2i q^{50} -28070.4 q^{52} +197505. i q^{53} +481734. q^{55} -37762.8i q^{56} +14950.3 q^{58} +36166.9i q^{59} +23934.6 q^{61} -32838.1i q^{62} -32768.0 q^{64} -158976. i q^{65} +353079. q^{67} -141713. i q^{68} +213869. q^{70} -496781. i q^{71} -382139. q^{73} -235209. i q^{74} +134215. q^{76} -554513. i q^{77} +386657. q^{79} -185582. i q^{80} +422757. q^{82} +415335. i q^{83} +802590. q^{85} -826500. i q^{86} -481169. q^{88} +405654. i q^{89} -182994. q^{91} +384282. i q^{92} -145791. q^{94} +760126. i q^{95} +356487. q^{97} +419343. i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	− 181.232i	− 1.44986i	−0.688825	−	0.724928i	$-0.741873\pi$
$5$	− 181.232i	− 1.44986i	0.688825	−	0.724928i	$-0.258127\pi$
$6$	0	0
$7$	−208.612	−0.608198	−0.304099	−	0.952640i	$-0.598355\pi$
$7$	−208.612	−0.608198	−0.304099	+	0.952640i	$0.598355\pi$
$8$	181.019i	0.353553i
$9$	0	0
$10$	−1025.20	−1.02520
$11$	2658.11i	1.99708i	0.0540453	+	0.998538i	$0.482788\pi$
$11$	2658.11i	1.99708i	−0.0540453	+	0.998538i	$0.517212\pi$
$12$	0	0
$13$	877.198	0.399271	0.199636	−	0.979870i	$-0.436024\pi$
$13$	877.198	0.399271	0.199636	+	0.979870i	$0.436024\pi$
$14$	1180.09i	0.430061i
$15$	0	0
$16$	1024.00	0.250000
$17$	4428.53i	0.901389i	0.892678	+	0.450695i	$0.148824\pi$
$17$	4428.53i	0.901389i	−0.892678	+	0.450695i	$0.851176\pi$
$18$	0	0
$19$	−4194.21	−0.611491	−0.305745	−	0.952113i	$-0.598906\pi$
$19$	−4194.21	−0.611491	−0.305745	+	0.952113i	$0.598906\pi$
$20$	5799.42i	0.724928i
$21$	0	0
$22$	15036.5	1.41215
$23$	− 12008.8i	− 0.986998i	−0.869746	−	0.493499i	$-0.835718\pi$
$23$	− 12008.8i	− 0.986998i	0.869746	−	0.493499i	$-0.164282\pi$
$24$	0	0
$25$	−17220.0	−1.10208
$26$	− 4962.18i	− 0.282327i
$27$	0	0
$28$	6675.58	0.304099
$29$	2642.86i	0.108363i	0.998531	+	0.0541815i	$0.0172549\pi$
$29$	2642.86i	0.108363i	−0.998531	+	0.0541815i	$0.982745\pi$
$30$	0	0
$31$	5805.01	0.194858	0.0974289	−	0.995242i	$-0.468938\pi$
$31$	5805.01	0.194858	0.0974289	+	0.995242i	$0.468938\pi$
$32$	− 5792.62i	− 0.176777i
$33$	0	0
$34$	25051.5	0.637379
$35$	37807.1i	0.881799i
$36$	0	0
$37$	41579.5	0.820869	0.410435	−	0.911890i	$-0.365377\pi$
$37$	41579.5	0.820869	0.410435	+	0.911890i	$0.365377\pi$
$38$	23726.1i	0.432389i
$39$	0	0
$40$	32806.5	0.512601
$41$	74733.6i	1.08434i	0.840270	+	0.542168i	$0.182396\pi$
$41$	74733.6i	1.08434i	−0.840270	+	0.542168i	$0.817604\pi$
$42$	0	0
$43$	146106.	1.83765	0.918824	−	0.394667i	$-0.129140\pi$
$43$	146106.	1.83765	0.918824	+	0.394667i	$0.129140\pi$
$44$	− 85059.5i	− 0.998538i
$45$	0	0
$46$	−67932.1	−0.697913
$47$	− 25772.5i	− 0.248235i	−0.992268	−	0.124118i	$-0.960390\pi$
$47$	− 25772.5i	− 0.248235i	0.992268	−	0.124118i	$-0.0396100\pi$
$48$	0	0
$49$	−74130.1	−0.630096
$50$	97411.2i	0.779290i
$51$	0	0
$52$	−28070.4	−0.199636
$53$	197505.i	1.32663i	0.748340	+	0.663316i	$0.230851\pi$
$53$	197505.i	1.32663i	−0.748340	+	0.663316i	$0.769149\pi$
$54$	0	0
$55$	481734.	2.89547
$56$	− 37762.8i	− 0.215030i
$57$	0	0
$58$	14950.3	0.0766242
$59$	36166.9i	0.176099i	0.996116	+	0.0880493i	$0.0280633\pi$
$59$	36166.9i	0.176099i	−0.996116	+	0.0880493i	$0.971937\pi$
$60$	0	0
$61$	23934.6	0.105448	0.0527239	−	0.998609i	$-0.483210\pi$
$61$	23934.6	0.105448	0.0527239	+	0.998609i	$0.483210\pi$
$62$	− 32838.1i	− 0.137785i
$63$	0	0
$64$	−32768.0	−0.125000
$65$	− 158976.i	− 0.578885i
$66$	0	0
$67$	353079.	1.17395	0.586973	−	0.809607i	$-0.300319\pi$
$67$	353079.	1.17395	0.586973	+	0.809607i	$0.300319\pi$
$68$	− 141713.i	− 0.450695i
$69$	0	0
$70$	213869.	0.623526
$71$	− 496781.i	− 1.38800i	−0.719974	−	0.694001i	$-0.755846\pi$
$71$	− 496781.i	− 1.38800i	0.719974	−	0.694001i	$-0.244154\pi$
$72$	0	0
$73$	−382139.	−0.982319	−0.491160	−	0.871070i	$-0.663427\pi$
$73$	−382139.	−0.982319	−0.491160	+	0.871070i	$0.663427\pi$
$74$	− 235209.i	− 0.580442i
$75$	0	0
$76$	134215.	0.305745
$77$	− 554513.i	− 1.21462i
$78$	0	0
$79$	386657.	0.784232	0.392116	−	0.919916i	$-0.371743\pi$
$79$	386657.	0.784232	0.392116	+	0.919916i	$0.371743\pi$
$80$	− 185582.i	− 0.362464i
$81$	0	0
$82$	422757.	0.766742
$83$	415335.i	0.726380i	0.931715	+	0.363190i	$0.118312\pi$
$83$	415335.i	0.726380i	−0.931715	+	0.363190i	$0.881688\pi$
$84$	0	0
$85$	802590.	1.30688
$86$	− 826500.i	− 1.29941i
$87$	0	0
$88$	−481169.	−0.706073
$89$	405654.i	0.575421i	0.957717	+	0.287710i	$0.0928941\pi$
$89$	405654.i	0.575421i	−0.957717	+	0.287710i	$0.907106\pi$
$90$	0	0
$91$	−182994.	−0.242836
$92$	384282.i	0.493499i
$93$	0	0
$94$	−145791.	−0.175529
$95$	760126.i	0.886573i
$96$	0	0
$97$	356487.	0.390597	0.195298	−	0.980744i	$-0.437432\pi$
$97$	356487.	0.390597	0.195298	+	0.980744i	$0.437432\pi$
$98$	419343.i	0.445545i
$99$	0	0
$100$	551041.	0.551041
$101$	1.63924e6i	1.59103i	0.605932	+	0.795516i	$0.292800\pi$
$101$	1.63924e6i	1.59103i	−0.605932	+	0.795516i	$0.707200\pi$
$102$	0	0
$103$	925474.	0.846940	0.423470	−	0.905910i	$-0.360812\pi$
$103$	925474.	0.846940	0.423470	+	0.905910i	$0.360812\pi$
$104$	158790.i	0.141164i
$105$	0	0
$106$	1.11726e6	0.938070
$107$	266819.i	0.217804i	0.994053	+	0.108902i	$0.0347334\pi$
$107$	266819.i	0.217804i	−0.994053	+	0.108902i	$0.965267\pi$
$108$	0	0
$109$	169960.	0.131240	0.0656200	−	0.997845i	$-0.479097\pi$
$109$	169960.	0.131240	0.0656200	+	0.997845i	$0.479097\pi$
$110$	− 2.72510e6i	− 2.04741i
$111$	0	0
$112$	−213619.	−0.152049
$113$	1.91032e6i	1.32395i	0.749527	+	0.661974i	$0.230281\pi$
$113$	1.91032e6i	1.32395i	−0.749527	+	0.661974i	$0.769719\pi$
$114$	0	0
$115$	−2.17638e6	−1.43101
$116$	− 84571.6i	− 0.0541815i
$117$	0	0
$118$	204591.	0.124520
$119$	− 923843.i	− 0.548223i
$120$	0	0
$121$	−5.29398e6	−2.98832
$122$	− 135395.i	− 0.0745628i
$123$	0	0
$124$	−185760.	−0.0974289
$125$	289070.i	0.148004i
$126$	0	0
$127$	−1.75663e6	−0.857571	−0.428786	−	0.903406i	$-0.641058\pi$
$127$	−1.75663e6	−0.857571	−0.428786	+	0.903406i	$0.641058\pi$
$128$	185364.i	0.0883883i
$129$	0	0
$130$	−899306.	−0.409334
$131$	2.39656e6i	1.06604i	0.846101	+	0.533022i	$0.178944\pi$
$131$	2.39656e6i	1.06604i	−0.846101	+	0.533022i	$0.821056\pi$
$132$	0	0
$133$	874963.	0.371907
$134$	− 1.99732e6i	− 0.830105i
$135$	0	0
$136$	−801649.	−0.318689
$137$	1.33362e6i	0.518644i	0.965791	+	0.259322i	$0.0834990\pi$
$137$	1.33362e6i	0.518644i	−0.965791	+	0.259322i	$0.916501\pi$
$138$	0	0
$139$	2.17756e6	0.810824	0.405412	−	0.914134i	$-0.367128\pi$
$139$	2.17756e6	0.810824	0.405412	+	0.914134i	$0.367128\pi$
$140$	− 1.20983e6i	− 0.440900i
$141$	0	0
$142$	−2.81022e6	−0.981465
$143$	2.33169e6i	0.797375i
$144$	0	0
$145$	478971.	0.157111
$146$	2.16170e6i	0.694604i
$147$	0	0
$148$	−1.33054e6	−0.410435
$149$	− 3.86850e6i	− 1.16945i	−0.811230	−	0.584727i	$-0.801201\pi$
$149$	− 3.86850e6i	− 1.16945i	0.811230	−	0.584727i	$-0.198799\pi$
$150$	0	0
$151$	961759.	0.279341	0.139671	−	0.990198i	$-0.455396\pi$
$151$	961759.	0.279341	0.139671	+	0.990198i	$0.455396\pi$
$152$	− 759234.i	− 0.216195i
$153$	0	0
$154$	−3.13680e6	−0.858864
$155$	− 1.05205e6i	− 0.282516i
$156$	0	0
$157$	−1.59235e6	−0.411470	−0.205735	−	0.978608i	$-0.565959\pi$
$157$	−1.59235e6	−0.411470	−0.205735	+	0.978608i	$0.565959\pi$
$158$	− 2.18726e6i	− 0.554536i
$159$	0	0
$160$	−1.04981e6	−0.256301
$161$	2.50518e6i	0.600290i
$162$	0	0
$163$	702280.	0.162161	0.0810807	−	0.996708i	$-0.474163\pi$
$163$	702280.	0.162161	0.0810807	+	0.996708i	$0.474163\pi$
$164$	− 2.39147e6i	− 0.542168i
$165$	0	0
$166$	2.34949e6	0.513629
$167$	3.78498e6i	0.812671i	0.913724	+	0.406335i	$0.133194\pi$
$167$	3.78498e6i	0.812671i	−0.913724	+	0.406335i	$0.866806\pi$
$168$	0	0
$169$	−4.05733e6	−0.840583
$170$	− 4.54014e6i	− 0.924107i
$171$	0	0
$172$	−4.67539e6	−0.918824
$173$	4.37292e6i	0.844566i	0.906464	+	0.422283i	$0.138771\pi$
$173$	4.37292e6i	0.844566i	−0.906464	+	0.422283i	$0.861229\pi$
$174$	0	0
$175$	3.59230e6	0.670284
$176$	2.72190e6i	0.499269i
$177$	0	0
$178$	2.29473e6	0.406884
$179$	− 49288.2i	− 0.00859377i	−0.999991	−	0.00429688i	$-0.998632\pi$
$179$	− 49288.2i	− 0.00859377i	0.999991	−	0.00429688i	$-0.00136775\pi$
$180$	0	0
$181$	−4.52205e6	−0.762605	−0.381303	−	0.924450i	$-0.624524\pi$
$181$	−4.52205e6	−0.762605	−0.381303	+	0.924450i	$0.624524\pi$
$182$	1.03517e6i	0.171711i
$183$	0	0
$184$	2.17383e6	0.348957
$185$	− 7.53553e6i	− 1.19014i
$186$	0	0
$187$	−1.17715e7	−1.80014
$188$	824721.i	0.124118i
$189$	0	0
$190$	4.29992e6	0.626902
$191$	1.24512e6i	0.178695i	0.996001	+	0.0893473i	$0.0284781\pi$
$191$	1.24512e6i	0.178695i	−0.996001	+	0.0893473i	$0.971522\pi$
$192$	0	0
$193$	−1.09534e7	−1.52362	−0.761812	−	0.647798i	$-0.775690\pi$
$193$	−1.09534e7	−1.52362	−0.761812	+	0.647798i	$0.775690\pi$
$194$	− 2.01660e6i	− 0.276194i
$195$	0	0
$196$	2.37216e6	0.315048
$197$	1.07910e7i	1.41144i	0.708492	+	0.705719i	$0.249376\pi$
$197$	1.07910e7i	1.41144i	−0.708492	+	0.705719i	$0.750624\pi$
$198$	0	0
$199$	5.05948e6	0.642017	0.321008	−	0.947076i	$-0.395978\pi$
$199$	5.05948e6	0.642017	0.321008	+	0.947076i	$0.395978\pi$
$200$	− 3.11716e6i	− 0.389645i
$201$	0	0
$202$	9.27295e6	1.12503
$203$	− 551333.i	− 0.0659061i
$204$	0	0
$205$	1.35441e7	1.57213
$206$	− 5.23527e6i	− 0.598877i
$207$	0	0
$208$	898251.	0.0998178
$209$	− 1.11487e7i	− 1.22119i
$210$	0	0
$211$	−3.45831e6	−0.368143	−0.184071	−	0.982913i	$-0.558928\pi$
$211$	−3.45831e6	−0.368143	−0.184071	+	0.982913i	$0.558928\pi$
$212$	− 6.32016e6i	− 0.663316i
$213$	0	0
$214$	1.50935e6	0.154010
$215$	− 2.64791e7i	− 2.66432i
$216$	0	0
$217$	−1.21099e6	−0.118512
$218$	− 961437.i	− 0.0928007i
$219$	0	0
$220$	−1.54155e7	−1.44774
$221$	3.88470e6i	0.359899i
$222$	0	0
$223$	7.56010e6	0.681730	0.340865	−	0.940112i	$-0.389280\pi$
$223$	7.56010e6	0.681730	0.340865	+	0.940112i	$0.389280\pi$
$224$	1.20841e6i	0.107515i
$225$	0	0
$226$	1.08064e7	0.936173
$227$	− 5.13440e6i	− 0.438947i	−0.975618	−	0.219473i	$-0.929566\pi$
$227$	− 5.13440e6i	− 0.438947i	0.975618	−	0.219473i	$-0.0704339\pi$
$228$	0	0
$229$	1.86558e7	1.55349	0.776744	−	0.629816i	$-0.216870\pi$
$229$	1.86558e7	1.55349	0.776744	+	0.629816i	$0.216870\pi$
$230$	1.23115e7i	1.01187i
$231$	0	0
$232$	−478409.	−0.0383121
$233$	421669.i	0.0333352i	0.999861	+	0.0166676i	$0.00530571\pi$
$233$	421669.i	0.0333352i	−0.999861	+	0.0166676i	$0.994694\pi$
$234$	0	0
$235$	−4.67081e6	−0.359905
$236$	− 1.15734e6i	− 0.0880493i
$237$	0	0
$238$	−5.22604e6	−0.387652
$239$	2.22929e7i	1.63295i	0.577380	+	0.816475i	$0.304075\pi$
$239$	2.22929e7i	1.63295i	−0.577380	+	0.816475i	$0.695925\pi$
$240$	0	0
$241$	9.83193e6	0.702405	0.351202	−	0.936300i	$-0.385773\pi$
$241$	9.83193e6	0.702405	0.351202	+	0.936300i	$0.385773\pi$
$242$	2.99473e7i	2.11306i
$243$	0	0
$244$	−765908.	−0.0527239
$245$	1.34347e7i	0.913548i
$246$	0	0
$247$	−3.67916e6	−0.244150
$248$	1.05082e6i	0.0688926i
$249$	0	0
$250$	1.63523e6	0.104655
$251$	1.55511e7i	0.983422i	0.870759	+	0.491711i	$0.163628\pi$
$251$	1.55511e7i	0.983422i	−0.870759	+	0.491711i	$0.836372\pi$
$252$	0	0
$253$	3.19207e7	1.97111
$254$	9.93703e6i	0.606395i
$255$	0	0
$256$	1.04858e6	0.0625000
$257$	2.13244e6i	0.125626i	0.998025	+	0.0628128i	$0.0200071\pi$
$257$	2.13244e6i	0.125626i	−0.998025	+	0.0628128i	$0.979993\pi$
$258$	0	0
$259$	−8.67398e6	−0.499251
$260$	5.08725e6i	0.289443i
$261$	0	0
$262$	1.35570e7	0.753807
$263$	− 1.05223e6i	− 0.0578419i	−0.999582	−	0.0289209i	$-0.990793\pi$
$263$	− 1.05223e6i	− 0.0578419i	0.999582	−	0.0289209i	$-0.00920711\pi$
$264$	0	0
$265$	3.57942e7	1.92342
$266$	− 4.94954e6i	− 0.262978i
$267$	0	0
$268$	−1.12985e7	−0.586973
$269$	− 9.68762e6i	− 0.497692i	−0.968543	−	0.248846i	$-0.919949\pi$
$269$	− 9.68762e6i	− 0.497692i	0.968543	−	0.248846i	$-0.0800512\pi$
$270$	0	0
$271$	−2.05637e7	−1.03322	−0.516610	−	0.856221i	$-0.672806\pi$
$271$	−2.05637e7	−1.03322	−0.516610	+	0.856221i	$0.672806\pi$
$272$	4.53481e6i	0.225347i
$273$	0	0
$274$	7.54407e6	0.366736
$275$	− 4.57727e7i	− 2.20094i
$276$	0	0
$277$	2.01392e7	0.947552	0.473776	−	0.880645i	$-0.342891\pi$
$277$	2.01392e7	0.947552	0.473776	+	0.880645i	$0.342891\pi$
$278$	− 1.23182e7i	− 0.573339i
$279$	0	0
$280$	−6.84382e6	−0.311763
$281$	2.03138e7i	0.915531i	0.889073	+	0.457766i	$0.151350\pi$
$281$	2.03138e7i	0.915531i	−0.889073	+	0.457766i	$0.848650\pi$
$282$	0	0
$283$	2.26151e7	0.997789	0.498894	−	0.866663i	$-0.333740\pi$
$283$	2.26151e7	0.997789	0.498894	+	0.866663i	$0.333740\pi$
$284$	1.58970e7i	0.694001i
$285$	0	0
$286$	1.31900e7	0.563829
$287$	− 1.55903e7i	− 0.659491i
$288$	0	0
$289$	4.52573e6	0.187497
$290$	− 2.70947e6i	− 0.111094i
$291$	0	0
$292$	1.22284e7	0.491160
$293$	3.86418e6i	0.153622i	0.997046	+	0.0768111i	$0.0244738\pi$
$293$	3.86418e6i	0.153622i	−0.997046	+	0.0768111i	$0.975526\pi$
$294$	0	0
$295$	6.55461e6	0.255318
$296$	7.52669e6i	0.290221i
$297$	0	0
$298$	−2.18835e7	−0.826930
$299$	− 1.05341e7i	− 0.394080i
$300$	0	0
$301$	−3.04794e7	−1.11765
$302$	− 5.44053e6i	− 0.197524i
$303$	0	0
$304$	−4.29488e6	−0.152873
$305$	− 4.33772e6i	− 0.152884i
$306$	0	0
$307$	−3.02526e6	−0.104556	−0.0522779	−	0.998633i	$-0.516648\pi$
$307$	−3.02526e6	−0.104556	−0.0522779	+	0.998633i	$0.516648\pi$
$308$	1.77444e7i	0.607309i
$309$	0	0
$310$	−5.95131e6	−0.199769
$311$	− 3.81270e7i	− 1.26751i	−0.773534	−	0.633755i	$-0.781513\pi$
$311$	− 3.81270e7i	− 1.26751i	0.773534	−	0.633755i	$-0.218487\pi$
$312$	0	0
$313$	−2.26297e7	−0.737981	−0.368990	−	0.929433i	$-0.620297\pi$
$313$	−2.26297e7	−0.737981	−0.368990	+	0.929433i	$0.620297\pi$
$314$	9.00767e6i	0.290954i
$315$	0	0
$316$	−1.23730e7	−0.392116
$317$	− 2.39028e7i	− 0.750361i	−0.926952	−	0.375180i	$-0.877581\pi$
$317$	− 2.39028e7i	− 0.750361i	0.926952	−	0.375180i	$-0.122419\pi$
$318$	0	0
$319$	−7.02502e6	−0.216409
$320$	5.93861e6i	0.181232i
$321$	0	0
$322$	1.41714e7	0.424469
$323$	− 1.85742e7i	− 0.551191i
$324$	0	0
$325$	−1.51054e7	−0.440029
$326$	− 3.97269e6i	− 0.114665i
$327$	0	0
$328$	−1.35282e7	−0.383371
$329$	5.37646e6i	0.150976i
$330$	0	0
$331$	−2.83825e7	−0.782647	−0.391324	−	0.920253i	$-0.627983\pi$
$331$	−2.83825e7	−0.782647	−0.391324	+	0.920253i	$0.627983\pi$
$332$	− 1.32907e7i	− 0.363190i
$333$	0	0
$334$	2.14111e7	0.574645
$335$	− 6.39893e7i	− 1.70205i
$336$	0	0
$337$	−3.12301e7	−0.815988	−0.407994	−	0.912985i	$-0.633772\pi$
$337$	−3.12301e7	−0.815988	−0.407994	+	0.912985i	$0.633772\pi$
$338$	2.29517e7i	0.594382i
$339$	0	0
$340$	−2.56829e7	−0.653442
$341$	1.54303e7i	0.389146i
$342$	0	0
$343$	4.00074e7	0.991420
$344$	2.64480e7i	0.649707i
$345$	0	0
$346$	2.47370e7	0.597198
$347$	2.00866e7i	0.480748i	0.970680	+	0.240374i	$0.0772702\pi$
$347$	2.00866e7i	0.480748i	−0.970680	+	0.240374i	$0.922730\pi$
$348$	0	0
$349$	−3.68004e7	−0.865717	−0.432859	−	0.901462i	$-0.642495\pi$
$349$	−3.68004e7	−0.865717	−0.432859	+	0.901462i	$0.642495\pi$
$350$	− 2.03211e7i	− 0.473962i
$351$	0	0
$352$	1.53974e7	0.353037
$353$	− 2.36735e7i	− 0.538193i	−0.963113	−	0.269097i	$-0.913275\pi$
$353$	− 2.36735e7i	− 0.538193i	0.963113	−	0.269097i	$-0.0867251\pi$
$354$	0	0
$355$	−9.00326e7	−2.01240
$356$	− 1.29809e7i	− 0.287710i
$357$	0	0
$358$	−278816.	−0.00607671
$359$	3.67159e7i	0.793544i	0.917917	+	0.396772i	$0.129870\pi$
$359$	3.67159e7i	0.793544i	−0.917917	+	0.396772i	$0.870130\pi$
$360$	0	0
$361$	−2.94545e7	−0.626079
$362$	2.55806e7i	0.539243i
$363$	0	0
$364$	5.85581e6	0.121418
$365$	6.92558e7i	1.42422i
$366$	0	0
$367$	8.30040e7	1.67919	0.839597	−	0.543210i	$-0.182791\pi$
$367$	8.30040e7	1.67919	0.839597	+	0.543210i	$0.182791\pi$
$368$	− 1.22970e7i	− 0.246750i
$369$	0	0
$370$	−4.26274e7	−0.841558
$371$	− 4.12019e7i	− 0.806854i
$372$	0	0
$373$	−6.13650e7	−1.18248	−0.591241	−	0.806495i	$-0.701362\pi$
$373$	−6.13650e7	−1.18248	−0.591241	+	0.806495i	$0.701362\pi$
$374$	6.65897e7i	1.27289i
$375$	0	0
$376$	4.66533e6	0.0877644
$377$	2.31832e6i	0.0432662i
$378$	0	0
$379$	8.94442e6	0.164299	0.0821495	−	0.996620i	$-0.473822\pi$
$379$	8.94442e6	0.164299	0.0821495	+	0.996620i	$0.473822\pi$
$380$	− 2.43240e7i	− 0.443287i
$381$	0	0
$382$	7.04347e6	0.126356
$383$	− 8.77768e7i	− 1.56237i	−0.624300	−	0.781184i	$-0.714616\pi$
$383$	− 8.77768e7i	− 1.56237i	0.624300	−	0.781184i	$-0.285384\pi$
$384$	0	0
$385$	−1.00495e8	−1.76102
$386$	6.19619e7i	1.07737i
$387$	0	0
$388$	−1.14076e7	−0.195298
$389$	9.34599e7i	1.58773i	0.608094	+	0.793865i	$0.291934\pi$
$389$	9.34599e7i	1.58773i	−0.608094	+	0.793865i	$0.708066\pi$
$390$	0	0
$391$	5.31813e7	0.889670
$392$	− 1.34190e7i	− 0.222772i
$393$	0	0
$394$	6.10430e7	0.998038
$395$	− 7.00746e7i	− 1.13702i
$396$	0	0
$397$	−1.21583e8	−1.94313	−0.971566	−	0.236767i	$-0.923912\pi$
$397$	−1.21583e8	−1.94313	−0.971566	+	0.236767i	$0.923912\pi$
$398$	− 2.86207e7i	− 0.453974i
$399$	0	0
$400$	−1.76333e7	−0.275520
$401$	− 2.14773e7i	− 0.333079i	−0.986035	−	0.166540i	$-0.946741\pi$
$401$	− 2.14773e7i	− 0.333079i	0.986035	−	0.166540i	$-0.0532594\pi$
$402$	0	0
$403$	5.09214e6	0.0778011
$404$	− 5.24557e7i	− 0.795516i
$405$	0	0
$406$	−3.11881e6	−0.0466026
$407$	1.10523e8i	1.63934i
$408$	0	0
$409$	1.16186e8	1.69818	0.849090	−	0.528249i	$-0.177151\pi$
$409$	1.16186e8	1.69818	0.849090	+	0.528249i	$0.177151\pi$
$410$	− 7.66171e7i	− 1.11166i
$411$	0	0
$412$	−2.96152e7	−0.423470
$413$	− 7.54485e6i	− 0.107103i
$414$	0	0
$415$	7.52720e7	1.05315
$416$	− 5.08128e6i	− 0.0705818i
$417$	0	0
$418$	−6.30665e7	−0.863514
$419$	− 3.15544e7i	− 0.428962i	−0.976728	−	0.214481i	$-0.931194\pi$
$419$	− 3.15544e7i	− 0.428962i	0.976728	−	0.214481i	$-0.0688059\pi$
$420$	0	0
$421$	−1.50063e7	−0.201107	−0.100553	−	0.994932i	$-0.532061\pi$
$421$	−1.50063e7	−0.201107	−0.100553	+	0.994932i	$0.532061\pi$
$422$	1.95631e7i	0.260316i
$423$	0	0
$424$	−3.57522e7	−0.469035
$425$	− 7.62593e7i	− 0.993405i
$426$	0	0
$427$	−4.99305e6	−0.0641331
$428$	− 8.53820e6i	− 0.108902i
$429$	0	0
$430$	−1.49788e8	−1.88396
$431$	6.49322e7i	0.811014i	0.914092	+	0.405507i	$0.132905\pi$
$431$	6.49322e7i	0.811014i	−0.914092	+	0.405507i	$0.867095\pi$
$432$	0	0
$433$	8.31669e7	1.02444	0.512220	−	0.858854i	$-0.328823\pi$
$433$	8.31669e7	1.02444	0.512220	+	0.858854i	$0.328823\pi$
$434$	6.85041e6i	0.0838007i
$435$	0	0
$436$	−5.43871e6	−0.0656200
$437$	5.03675e7i	0.603540i
$438$	0	0
$439$	3.27151e7	0.386683	0.193341	−	0.981132i	$-0.438067\pi$
$439$	3.27151e7	0.386683	0.193341	+	0.981132i	$0.438067\pi$
$440$	8.72032e7i	1.02370i
$441$	0	0
$442$	2.19752e7	0.254487
$443$	1.07382e8i	1.23516i	0.786509	+	0.617578i	$0.211886\pi$
$443$	1.07382e8i	1.23516i	−0.786509	+	0.617578i	$0.788114\pi$
$444$	0	0
$445$	7.35175e7	0.834277
$446$	− 4.27664e7i	− 0.482056i
$447$	0	0
$448$	6.83579e6	0.0760247
$449$	3.79594e7i	0.419354i	0.977771	+	0.209677i	$0.0672413\pi$
$449$	3.79594e7i	0.419354i	−0.977771	+	0.209677i	$0.932759\pi$
$450$	0	0
$451$	−1.98650e8	−2.16550
$452$	− 6.11303e7i	− 0.661974i
$453$	0	0
$454$	−2.90445e7	−0.310382
$455$	3.31644e7i	0.352077i
$456$	0	0
$457$	2.54471e7	0.266618	0.133309	−	0.991074i	$-0.457440\pi$
$457$	2.54471e7	0.266618	0.133309	+	0.991074i	$0.457440\pi$
$458$	− 1.05533e8i	− 1.09848i
$459$	0	0
$460$	6.96442e7	0.715503
$461$	6.51436e7i	0.664919i	0.943117	+	0.332460i	$0.107879\pi$
$461$	6.51436e7i	0.664919i	−0.943117	+	0.332460i	$0.892121\pi$
$462$	0	0
$463$	2.89828e7	0.292010	0.146005	−	0.989284i	$-0.453358\pi$
$463$	2.89828e7	0.292010	0.146005	+	0.989284i	$0.453358\pi$
$464$	2.70629e6i	0.0270907i
$465$	0	0
$466$	2.38532e6	0.0235716
$467$	6.09985e7i	0.598919i	0.954109	+	0.299460i	$0.0968064\pi$
$467$	6.09985e7i	0.598919i	−0.954109	+	0.299460i	$0.903194\pi$
$468$	0	0
$469$	−7.36566e7	−0.713991
$470$	2.64221e7i	0.254492i
$471$	0	0
$472$	−6.54692e6	−0.0622602
$473$	3.88365e8i	3.66992i
$474$	0	0
$475$	7.22245e7	0.673913
$476$	2.95630e7i	0.274111i
$477$	0	0
$478$	1.26108e8	1.15467
$479$	− 9.92875e7i	− 0.903416i	−0.892166	−	0.451708i	$-0.850815\pi$
$479$	− 9.92875e7i	− 0.903416i	0.892166	−	0.451708i	$-0.149185\pi$
$480$	0	0
$481$	3.64735e7	0.327749
$482$	− 5.56178e7i	− 0.496675i
$483$	0	0
$484$	1.69408e8	1.49416
$485$	− 6.46069e7i	− 0.566309i
$486$	0	0
$487$	1.05049e8	0.909504	0.454752	−	0.890618i	$-0.349728\pi$
$487$	1.05049e8	0.909504	0.454752	+	0.890618i	$0.349728\pi$
$488$	4.33263e6i	0.0372814i
$489$	0	0
$490$	7.59984e7	0.645976
$491$	− 1.89765e8i	− 1.60314i	−0.597902	−	0.801569i	$-0.703999\pi$
$491$	− 1.89765e8i	− 1.60314i	0.597902	−	0.801569i	$-0.296001\pi$
$492$	0	0
$493$	−1.17040e7	−0.0976772
$494$	2.08125e7i	0.172640i
$495$	0	0
$496$	5.94433e6	0.0487144
$497$	1.03634e8i	0.844180i
$498$	0	0
$499$	−2.02005e8	−1.62577	−0.812887	−	0.582421i	$-0.802106\pi$
$499$	−2.02005e8	−1.62577	−0.812887	+	0.582421i	$0.802106\pi$
$500$	− 9.25025e6i	− 0.0740020i
$501$	0	0
$502$	8.79703e7	0.695384
$503$	− 1.88347e8i	− 1.47998i	−0.672619	−	0.739989i	$-0.734831\pi$
$503$	− 1.88347e8i	− 1.47998i	0.672619	−	0.739989i	$-0.265169\pi$
$504$	0	0
$505$	2.97083e8	2.30677
$506$	− 1.80571e8i	− 1.39379i
$507$	0	0
$508$	5.62123e7	0.428786
$509$	− 1.65782e8i	− 1.25714i	−0.777751	−	0.628572i	$-0.783640\pi$
$509$	− 1.65782e8i	− 1.25714i	0.777751	−	0.628572i	$-0.216360\pi$
$510$	0	0
$511$	7.97187e7	0.597444
$512$	− 5.93164e6i	− 0.0441942i
$513$	0	0
$514$	1.20629e7	0.0888307
$515$	− 1.67726e8i	− 1.22794i
$516$	0	0
$517$	6.85062e7	0.495745
$518$	4.90674e7i	0.353024i
$519$	0	0
$520$	2.87778e7	0.204667
$521$	2.76446e8i	1.95478i	0.211451	+	0.977389i	$0.432181\pi$
$521$	2.76446e8i	1.95478i	−0.211451	+	0.977389i	$0.567819\pi$
$522$	0	0
$523$	8.78008e7	0.613753	0.306876	−	0.951749i	$-0.400716\pi$
$523$	8.78008e7	0.613753	0.306876	+	0.951749i	$0.400716\pi$
$524$	− 7.66900e7i	− 0.533022i
$525$	0	0
$526$	−5.95230e6	−0.0409004
$527$	2.57076e7i	0.175643i
$528$	0	0
$529$	3.82442e6	0.0258344
$530$	− 2.02483e8i	− 1.36007i
$531$	0	0
$532$	−2.79988e7	−0.185954
$533$	6.55562e7i	0.432944i
$534$	0	0
$535$	4.83561e7	0.315784
$536$	6.39142e7i	0.415053i
$537$	0	0
$538$	−5.48015e7	−0.351921
$539$	− 1.97046e8i	− 1.25835i
$540$	0	0
$541$	−1.60020e8	−1.01061	−0.505305	−	0.862941i	$-0.668620\pi$
$541$	−1.60020e8	−1.01061	−0.505305	+	0.862941i	$0.668620\pi$
$542$	1.16326e8i	0.730597i
$543$	0	0
$544$	2.56528e7	0.159345
$545$	− 3.08021e7i	− 0.190279i
$546$	0	0
$547$	−2.20616e7	−0.134795	−0.0673977	−	0.997726i	$-0.521470\pi$
$547$	−2.20616e7	−0.134795	−0.0673977	+	0.997726i	$0.521470\pi$
$548$	− 4.26757e7i	− 0.259322i
$549$	0	0
$550$	−2.58930e8	−1.55630
$551$	− 1.10847e7i	− 0.0662629i
$552$	0	0
$553$	−8.06612e7	−0.476968
$554$	− 1.13925e8i	− 0.670020i
$555$	0	0
$556$	−6.96820e7	−0.405412
$557$	1.71224e8i	0.990830i	0.868657	+	0.495415i	$0.164984\pi$
$557$	1.71224e8i	0.990830i	−0.868657	+	0.495415i	$0.835016\pi$
$558$	0	0
$559$	1.28164e8	0.733720
$560$	3.87145e7i	0.220450i
$561$	0	0
$562$	1.14912e8	0.647378
$563$	2.22205e8i	1.24517i	0.782551	+	0.622586i	$0.213918\pi$
$563$	2.22205e8i	1.24517i	−0.782551	+	0.622586i	$0.786082\pi$
$564$	0	0
$565$	3.46211e8	1.91953
$566$	− 1.27930e8i	− 0.705543i
$567$	0	0
$568$	8.99270e7	0.490733
$569$	− 2.60657e8i	− 1.41492i	−0.706752	−	0.707461i	$-0.749840\pi$
$569$	− 2.60657e8i	− 1.41492i	0.706752	−	0.707461i	$-0.250160\pi$
$570$	0	0
$571$	1.21893e8	0.654741	0.327370	−	0.944896i	$-0.393838\pi$
$571$	1.21893e8	0.654741	0.327370	+	0.944896i	$0.393838\pi$
$572$	− 7.46141e7i	− 0.398688i
$573$	0	0
$574$	−8.81921e7	−0.466331
$575$	2.06792e8i	1.08775i
$576$	0	0
$577$	1.22005e8	0.635111	0.317555	−	0.948240i	$-0.397138\pi$
$577$	1.22005e8	0.635111	0.317555	+	0.948240i	$0.397138\pi$
$578$	− 2.56014e7i	− 0.132581i
$579$	0	0
$580$	−1.53271e7	−0.0785553
$581$	− 8.66438e7i	− 0.441783i
$582$	0	0
$583$	−5.24990e8	−2.64939
$584$	− 6.91745e7i	− 0.347302i
$585$	0	0
$586$	2.18591e7	0.108627
$587$	− 1.33655e8i	− 0.660799i	−0.943841	−	0.330400i	$-0.892816\pi$
$587$	− 1.33655e8i	− 0.660799i	0.943841	−	0.330400i	$-0.107184\pi$
$588$	0	0
$589$	−2.43474e7	−0.119154
$590$	− 3.70785e7i	− 0.180537i
$591$	0	0
$592$	4.25774e7	0.205217
$593$	1.98748e7i	0.0953100i	0.998864	+	0.0476550i	$0.0151748\pi$
$593$	1.98748e7i	0.0953100i	−0.998864	+	0.0476550i	$0.984825\pi$
$594$	0	0
$595$	−1.67430e8	−0.794844
$596$	1.23792e8i	0.584727i
$597$	0	0
$598$	−5.95899e7	−0.278657
$599$	− 3.90247e8i	− 1.81576i	−0.419228	−	0.907881i	$-0.637699\pi$
$599$	− 3.90247e8i	− 1.81576i	0.419228	−	0.907881i	$-0.362301\pi$
$600$	0	0
$601$	1.57814e8	0.726980	0.363490	−	0.931598i	$-0.381585\pi$
$601$	1.57814e8	0.726980	0.363490	+	0.931598i	$0.381585\pi$
$602$	1.72418e8i	0.790300i
$603$	0	0
$604$	−3.07763e7	−0.139671
$605$	9.59439e8i	4.33263i
$606$	0	0
$607$	3.48590e8	1.55865	0.779325	−	0.626621i	$-0.215562\pi$
$607$	3.48590e8	1.55865	0.779325	+	0.626621i	$0.215562\pi$
$608$	2.42955e7i	0.108097i
$609$	0	0
$610$	−2.45379e7	−0.108105
$611$	− 2.26076e7i	− 0.0991132i
$612$	0	0
$613$	1.41726e7	0.0615273	0.0307636	−	0.999527i	$-0.490206\pi$
$613$	1.41726e7	0.0615273	0.0307636	+	0.999527i	$0.490206\pi$
$614$	1.71135e7i	0.0739321i
$615$	0	0
$616$	1.00378e8	0.429432
$617$	2.09693e8i	0.892745i	0.894847	+	0.446373i	$0.147284\pi$
$617$	2.09693e8i	0.892745i	−0.894847	+	0.446373i	$0.852716\pi$
$618$	0	0
$619$	−1.67204e8	−0.704977	−0.352488	−	0.935816i	$-0.614664\pi$
$619$	−1.67204e8	−0.704977	−0.352488	+	0.935816i	$0.614664\pi$
$620$	3.36657e7i	0.141258i
$621$	0	0
$622$	−2.15679e8	−0.896265
$623$	− 8.46242e7i	− 0.349970i
$624$	0	0
$625$	−2.16674e8	−0.887497
$626$	1.28013e8i	0.521831i
$627$	0	0
$628$	5.09551e7	0.205735
$629$	1.84136e8i	0.739923i
$630$	0	0
$631$	−3.36933e8	−1.34108	−0.670541	−	0.741872i	$-0.733938\pi$
$631$	−3.36933e8	−1.34108	−0.670541	+	0.741872i	$0.733938\pi$
$632$	6.99924e7i	0.277268i
$633$	0	0
$634$	−1.35214e8	−0.530585
$635$	3.18358e8i	1.24335i
$636$	0	0
$637$	−6.50268e7	−0.251579
$638$	3.97395e7i	0.153024i
$639$	0	0
$640$	3.35938e7	0.128150
$641$	− 9.97125e7i	− 0.378596i	−0.981920	−	0.189298i	$-0.939379\pi$
$641$	− 9.97125e7i	− 0.378596i	0.981920	−	0.189298i	$-0.0606212\pi$
$642$	0	0
$643$	−2.69385e8	−1.01330	−0.506652	−	0.862151i	$-0.669117\pi$
$643$	−2.69385e8	−1.01330	−0.506652	+	0.862151i	$0.669117\pi$
$644$	− 8.01657e7i	− 0.300145i
$645$	0	0
$646$	−1.05071e8	−0.389751
$647$	− 7.78039e7i	− 0.287269i	−0.989631	−	0.143634i	$-0.954121\pi$
$647$	− 7.78039e7i	− 0.287269i	0.989631	−	0.143634i	$-0.0458790\pi$
$648$	0	0
$649$	−9.61357e7	−0.351682
$650$	8.54490e7i	0.311148i
$651$	0	0
$652$	−2.24729e7	−0.0810807
$653$	− 2.25239e8i	− 0.808917i	−0.914556	−	0.404459i	$-0.867460\pi$
$653$	− 2.25239e8i	− 0.808917i	0.914556	−	0.404459i	$-0.132540\pi$
$654$	0	0
$655$	4.34334e8	1.54561
$656$	7.65272e7i	0.271084i
$657$	0	0
$658$	3.04138e7	0.106756
$659$	− 2.29251e8i	− 0.801041i	−0.916288	−	0.400521i	$-0.868829\pi$
$659$	− 2.29251e8i	− 0.801041i	0.916288	−	0.400521i	$-0.131171\pi$
$660$	0	0
$661$	−2.10989e8	−0.730561	−0.365280	−	0.930898i	$-0.619027\pi$
$661$	−2.10989e8	−0.730561	−0.365280	+	0.930898i	$0.619027\pi$
$662$	1.60555e8i	0.553415i
$663$	0	0
$664$	−7.51836e7	−0.256814
$665$	− 1.58571e8i	− 0.539212i
$666$	0	0
$667$	3.17376e7	0.106954
$668$	− 1.21119e8i	− 0.406335i
$669$	0	0
$670$	−3.61978e8	−1.20353
$671$	6.36209e7i	0.210587i
$672$	0	0
$673$	−5.75429e8	−1.88776	−0.943880	−	0.330289i	$-0.892854\pi$
$673$	−5.75429e8	−1.88776	−0.943880	+	0.330289i	$0.892854\pi$
$674$	1.76664e8i	0.576991i
$675$	0	0
$676$	1.29835e8	0.420291
$677$	− 3.53147e8i	− 1.13812i	−0.822295	−	0.569062i	$-0.807306\pi$
$677$	− 3.53147e8i	− 1.13812i	0.822295	−	0.569062i	$-0.192694\pi$
$678$	0	0
$679$	−7.43674e7	−0.237560
$680$	1.45284e8i	0.462053i
$681$	0	0
$682$	8.72872e7	0.275168
$683$	− 4.65139e8i	− 1.45989i	−0.683506	−	0.729945i	$-0.739546\pi$
$683$	− 4.65139e8i	− 1.45989i	0.683506	−	0.729945i	$-0.260454\pi$
$684$	0	0
$685$	2.41694e8	0.751959
$686$	− 2.26316e8i	− 0.701040i
$687$	0	0
$688$	1.49612e8	0.459412
$689$	1.73251e8i	0.529686i
$690$	0	0
$691$	−1.19264e8	−0.361471	−0.180736	−	0.983532i	$-0.557848\pi$
$691$	−1.19264e8	−0.361471	−0.180736	+	0.983532i	$0.557848\pi$
$692$	− 1.39934e8i	− 0.422283i
$693$	0	0
$694$	1.13627e8	0.339941
$695$	− 3.94644e8i	− 1.17558i
$696$	0	0
$697$	−3.30959e8	−0.977409
$698$	2.08174e8i	0.612155i
$699$	0	0
$700$	−1.14954e8	−0.335142
$701$	− 3.55114e8i	− 1.03089i	−0.856921	−	0.515447i	$-0.827626\pi$
$701$	− 3.55114e8i	− 1.03089i	0.856921	−	0.515447i	$-0.172374\pi$
$702$	0	0
$703$	−1.74393e8	−0.501954
$704$	− 8.71009e7i	− 0.249635i
$705$	0	0
$706$	−1.33918e8	−0.380560
$707$	− 3.41965e8i	− 0.967662i
$708$	0	0
$709$	6.02652e8	1.69094	0.845470	−	0.534023i	$-0.179320\pi$
$709$	6.02652e8	1.69094	0.845470	+	0.534023i	$0.179320\pi$
$710$	5.09301e8i	1.42298i
$711$	0	0
$712$	−7.34312e7	−0.203442
$713$	− 6.97112e7i	− 0.192324i
$714$	0	0
$715$	4.22577e8	1.15608
$716$	1.57722e6i	0.00429688i
$717$	0	0
$718$	2.07697e8	0.561120
$719$	− 3.36046e8i	− 0.904092i	−0.891995	−	0.452046i	$-0.850694\pi$
$719$	− 3.36046e8i	− 0.904092i	0.891995	−	0.452046i	$-0.149306\pi$
$720$	0	0
$721$	−1.93065e8	−0.515107
$722$	1.66620e8i	0.442705i
$723$	0	0
$724$	1.44706e8	0.381303
$725$	− 4.55102e7i	− 0.119425i
$726$	0	0
$727$	−5.12483e8	−1.33375	−0.666877	−	0.745168i	$-0.732369\pi$
$727$	−5.12483e8	−1.33375	−0.666877	+	0.745168i	$0.732369\pi$
$728$	− 3.31254e7i	− 0.0858554i
$729$	0	0
$730$	3.91770e8	1.00708
$731$	6.47034e8i	1.65644i
$732$	0	0
$733$	−3.58786e7	−0.0911010	−0.0455505	−	0.998962i	$-0.514504\pi$
$733$	−3.58786e7	−0.0911010	−0.0455505	+	0.998962i	$0.514504\pi$
$734$	− 4.69542e8i	− 1.18737i
$735$	0	0
$736$	−6.95624e7	−0.174478
$737$	9.38524e8i	2.34446i
$738$	0	0
$739$	−3.58948e7	−0.0889403	−0.0444701	−	0.999011i	$-0.514160\pi$
$739$	−3.58948e7	−0.0889403	−0.0444701	+	0.999011i	$0.514160\pi$
$740$	2.41137e8i	0.595071i
$741$	0	0
$742$	−2.33073e8	−0.570532
$743$	4.35860e8i	1.06263i	0.847176	+	0.531313i	$0.178301\pi$
$743$	4.35860e8i	1.06263i	−0.847176	+	0.531313i	$0.821699\pi$
$744$	0	0
$745$	−7.01095e8	−1.69554
$746$	3.47133e8i	0.836140i
$747$	0	0
$748$	3.76688e8	0.900072
$749$	− 5.56616e7i	− 0.132468i
$750$	0	0
$751$	−4.29118e8	−1.01311	−0.506556	−	0.862207i	$-0.669082\pi$
$751$	−4.29118e8	−1.01311	−0.506556	+	0.862207i	$0.669082\pi$
$752$	− 2.63911e7i	− 0.0620588i
$753$	0	0
$754$	1.31144e7	0.0305938
$755$	− 1.74301e8i	− 0.405005i
$756$	0	0
$757$	−8.26773e8	−1.90589	−0.952947	−	0.303136i	$-0.901966\pi$
$757$	−8.26773e8	−1.90589	−0.952947	+	0.303136i	$0.901966\pi$
$758$	− 5.05973e7i	− 0.116177i
$759$	0	0
$760$	−1.37597e8	−0.313451
$761$	− 3.72557e8i	− 0.845353i	−0.906281	−	0.422677i	$-0.861091\pi$
$761$	− 3.72557e8i	− 0.845353i	0.906281	−	0.422677i	$-0.138909\pi$
$762$	0	0
$763$	−3.54556e7	−0.0798199
$764$	− 3.98439e7i	− 0.0893473i
$765$	0	0
$766$	−4.96541e8	−1.10476
$767$	3.17256e7i	0.0703111i
$768$	0	0
$769$	4.10019e8	0.901624	0.450812	−	0.892619i	$-0.351134\pi$
$769$	4.10019e8	0.901624	0.450812	+	0.892619i	$0.351134\pi$
$770$	5.68488e8i	1.24523i
$771$	0	0
$772$	3.50510e8	0.761812
$773$	− 6.77485e8i	− 1.46677i	−0.679815	−	0.733384i	$-0.737940\pi$
$773$	− 6.77485e8i	− 1.46677i	0.679815	−	0.733384i	$-0.262060\pi$
$774$	0	0
$775$	−9.99624e7	−0.214749
$776$	6.45311e7i	0.138097i
$777$	0	0
$778$	5.28689e8	1.12269
$779$	− 3.13449e8i	− 0.663062i
$780$	0	0
$781$	1.32050e9	2.77195
$782$	− 3.00839e8i	− 0.629091i
$783$	0	0
$784$	−7.59092e7	−0.157524
$785$	2.88584e8i	0.596573i
$786$	0	0
$787$	5.41300e8	1.11049	0.555244	−	0.831688i	$-0.312625\pi$
$787$	5.41300e8	1.11049	0.555244	+	0.831688i	$0.312625\pi$
$788$	− 3.45311e8i	− 0.705719i
$789$	0	0
$790$	−3.96402e8	−0.803997
$791$	− 3.98516e8i	− 0.805222i
$792$	0	0
$793$	2.09954e7	0.0421022
$794$	6.87779e8i	1.37400i
$795$	0	0
$796$	−1.61903e8	−0.321008
$797$	− 6.14153e8i	− 1.21311i	−0.795040	−	0.606557i	$-0.792550\pi$
$797$	− 6.14153e8i	− 1.21311i	0.795040	−	0.606557i	$-0.207450\pi$
$798$	0	0
$799$	1.14134e8	0.223757
$800$	9.97491e7i	0.194822i
$801$	0	0
$802$	−1.21494e8	−0.235523
$803$	− 1.01577e9i	− 1.96177i
$804$	0	0
$805$	4.54019e8	0.870334
$806$	− 2.88055e7i	− 0.0550137i
$807$	0	0
$808$	−2.96735e8	−0.562515
$809$	1.54965e8i	0.292677i	0.989235	+	0.146338i	$0.0467489\pi$
$809$	1.54965e8i	0.292677i	−0.989235	+	0.146338i	$0.953251\pi$
$810$	0	0
$811$	8.47410e8	1.58866	0.794330	−	0.607487i	$-0.207822\pi$
$811$	8.47410e8	1.58866	0.794330	+	0.607487i	$0.207822\pi$
$812$	1.76426e7i	0.0329530i
$813$	0	0
$814$	6.25212e8	1.15919
$815$	− 1.27276e8i	− 0.235111i
$816$	0	0
$817$	−6.12799e8	−1.12370
$818$	− 6.57247e8i	− 1.20079i
$819$	0	0
$820$	−4.33412e8	−0.786066
$821$	− 2.59494e8i	− 0.468919i	−0.972126	−	0.234460i	$-0.924668\pi$
$821$	− 2.59494e8i	− 0.468919i	0.972126	−	0.234460i	$-0.0753321\pi$
$822$	0	0
$823$	−3.88119e8	−0.696251	−0.348126	−	0.937448i	$-0.613182\pi$
$823$	−3.88119e8	−0.696251	−0.348126	+	0.937448i	$0.613182\pi$
$824$	1.67529e8i	0.299439i
$825$	0	0
$826$	−4.26801e7	−0.0757331
$827$	7.11216e8i	1.25743i	0.777635	+	0.628716i	$0.216419\pi$
$827$	7.11216e8i	1.25743i	−0.777635	+	0.628716i	$0.783581\pi$
$828$	0	0
$829$	9.26204e8	1.62571	0.812855	−	0.582466i	$-0.197912\pi$
$829$	9.26204e8	1.62571	0.812855	+	0.582466i	$0.197912\pi$
$830$	− 4.25803e8i	− 0.744687i
$831$	0	0
$832$	−2.87440e7	−0.0499089
$833$	− 3.28287e8i	− 0.567961i
$834$	0	0
$835$	6.85960e8	1.17826
$836$	3.56758e8i	0.610597i
$837$	0	0
$838$	−1.78499e8	−0.303322
$839$	6.69627e8i	1.13383i	0.823777	+	0.566914i	$0.191863\pi$
$839$	6.69627e8i	1.13383i	−0.823777	+	0.566914i	$0.808137\pi$
$840$	0	0
$841$	5.87839e8	0.988257
$842$	8.48882e7i	0.142204i
$843$	0	0
$844$	1.10666e8	0.184071
$845$	7.35318e8i	1.21872i
$846$	0	0
$847$	1.10439e9	1.81749
$848$	2.02245e8i	0.331658i
$849$	0	0
$850$	−4.31388e8	−0.702443
$851$	− 4.99320e8i	− 0.810197i
$852$	0	0
$853$	−2.03848e8	−0.328443	−0.164221	−	0.986424i	$-0.552511\pi$
$853$	−2.03848e8	−0.328443	−0.164221	+	0.986424i	$0.552511\pi$
$854$	2.82449e7i	0.0453489i
$855$	0	0
$856$	−4.82994e7	−0.0770052
$857$	− 3.51763e8i	− 0.558866i	−0.960165	−	0.279433i	$-0.909854\pi$
$857$	− 3.51763e8i	− 0.558866i	0.960165	−	0.279433i	$-0.0901464\pi$
$858$	0	0
$859$	3.79275e8	0.598377	0.299189	−	0.954194i	$-0.403284\pi$
$859$	3.79275e8	0.598377	0.299189	+	0.954194i	$0.403284\pi$
$860$	8.47330e8i	1.33216i
$861$	0	0
$862$	3.67312e8	0.573473
$863$	− 5.09276e7i	− 0.0792358i	−0.999215	−	0.0396179i	$-0.987386\pi$
$863$	− 5.09276e7i	− 0.0792358i	0.999215	−	0.0396179i	$-0.0126141\pi$
$864$	0	0
$865$	7.92514e8	1.22450
$866$	− 4.70463e8i	− 0.724389i
$867$	0	0
$868$	3.87518e7	0.0592560
$869$	1.02778e9i	1.56617i
$870$	0	0
$871$	3.09721e8	0.468723
$872$	3.07660e7i	0.0464003i
$873$	0	0
$874$	2.84922e8	0.426767
$875$	− 6.03035e7i	− 0.0900157i
$876$	0	0
$877$	−8.95187e8	−1.32713	−0.663567	−	0.748117i	$-0.730958\pi$
$877$	−8.95187e8	−1.32713	−0.663567	+	0.748117i	$0.730958\pi$
$878$	− 1.85065e8i	− 0.273426i
$879$	0	0
$880$	4.93296e8	0.723868
$881$	− 5.82577e8i	− 0.851973i	−0.904730	−	0.425986i	$-0.859927\pi$
$881$	− 5.82577e8i	− 0.851973i	0.904730	−	0.425986i	$-0.140073\pi$
$882$	0	0
$883$	−2.10539e7	−0.0305810	−0.0152905	−	0.999883i	$-0.504867\pi$
$883$	−2.10539e7	−0.0305810	−0.0152905	+	0.999883i	$0.504867\pi$
$884$	− 1.24310e8i	− 0.179949i
$885$	0	0
$886$	6.07447e8	0.873387
$887$	1.12209e9i	1.60788i	0.594707	+	0.803942i	$0.297268\pi$
$887$	1.12209e9i	1.60788i	−0.594707	+	0.803942i	$0.702732\pi$
$888$	0	0
$889$	3.66455e8	0.521573
$890$	− 4.15878e8i	− 0.589923i
$891$	0	0
$892$	−2.41923e8	−0.340865
$893$	1.08096e8i	0.151794i
$894$	0	0
$895$	−8.93259e6	−0.0124597
$896$	− 3.86691e7i	− 0.0537576i
$897$	0	0
$898$	2.14731e8	0.296528
$899$	1.53418e7i	0.0211154i
$900$	0	0
$901$	−8.74656e8	−1.19581
$902$	1.12373e9i	1.53124i
$903$	0	0
$904$	−3.45805e8	−0.468086
$905$	8.19540e8i	1.10567i
$906$	0	0
$907$	6.31370e8	0.846179	0.423089	−	0.906088i	$-0.360946\pi$
$907$	6.31370e8	0.846179	0.423089	+	0.906088i	$0.360946\pi$
$908$	1.64301e8i	0.219473i
$909$	0	0
$910$	1.87606e8	0.248956
$911$	1.31089e8i	0.173385i	0.996235	+	0.0866924i	$0.0276297\pi$
$911$	1.31089e8i	0.173385i	−0.996235	+	0.0866924i	$0.972370\pi$
$912$	0	0
$913$	−1.10401e9	−1.45064
$914$	− 1.43951e8i	− 0.188528i
$915$	0	0
$916$	−5.96987e8	−0.776744
$917$	− 4.99951e8i	− 0.648365i
$918$	0	0
$919$	−7.88351e8	−1.01572	−0.507859	−	0.861440i	$-0.669563\pi$
$919$	−7.88351e8	−1.01572	−0.507859	+	0.861440i	$0.669563\pi$
$920$	− 3.93967e8i	− 0.505937i
$921$	0	0
$922$	3.68508e8	0.470169
$923$	− 4.35776e8i	− 0.554189i
$924$	0	0
$925$	−7.16000e8	−0.904665
$926$	− 1.63952e8i	− 0.206482i
$927$	0	0
$928$	1.53091e7	0.0191560
$929$	− 9.54397e8i	− 1.19037i	−0.803589	−	0.595185i	$-0.797079\pi$
$929$	− 9.54397e8i	− 1.19037i	0.803589	−	0.595185i	$-0.202921\pi$
$930$	0	0
$931$	3.10918e8	0.385297
$932$	− 1.34934e7i	− 0.0166676i
$933$	0	0
$934$	3.45060e8	0.423500
$935$	2.13337e9i	2.60995i
$936$	0	0
$937$	−4.59379e8	−0.558409	−0.279205	−	0.960232i	$-0.590071\pi$
$937$	−4.59379e8	−0.558409	−0.279205	+	0.960232i	$0.590071\pi$
$938$	4.16664e8i	0.504868i
$939$	0	0
$940$	1.49466e8	0.179953
$941$	1.46910e8i	0.176313i	0.996107	+	0.0881563i	$0.0280975\pi$
$941$	1.46910e8i	0.176313i	−0.996107	+	0.0881563i	$0.971902\pi$
$942$	0	0
$943$	8.97461e8	1.07024
$944$	3.70350e7i	0.0440246i
$945$	0	0
$946$	2.19693e9	2.59503
$947$	4.47375e8i	0.526771i	0.964691	+	0.263385i	$0.0848391\pi$
$947$	4.47375e8i	0.526771i	−0.964691	+	0.263385i	$0.915161\pi$
$948$	0	0
$949$	−3.35212e8	−0.392212
$950$	− 4.08563e8i	− 0.476528i
$951$	0	0
$952$	1.67233e8	0.193826
$953$	1.69556e8i	0.195900i	0.995191	+	0.0979500i	$0.0312285\pi$
$953$	1.69556e8i	0.195900i	−0.995191	+	0.0979500i	$0.968771\pi$
$954$	0	0
$955$	2.25656e8	0.259081
$956$	− 7.13373e8i	− 0.816475i
$957$	0	0
$958$	−5.61655e8	−0.638812
$959$	− 2.78208e8i	− 0.315438i
$960$	0	0
$961$	−8.53806e8	−0.962030
$962$	− 2.06325e8i	− 0.231754i
$963$	0	0
$964$	−3.14622e8	−0.351202
$965$	1.98511e9i	2.20904i
$966$	0	0
$967$	−3.84263e8	−0.424962	−0.212481	−	0.977165i	$-0.568154\pi$
$967$	−3.84263e8	−0.424962	−0.212481	+	0.977165i	$0.568154\pi$
$968$	− 9.58314e8i	− 1.05653i
$969$	0	0
$970$	−3.65472e8	−0.400441
$971$	3.23298e8i	0.353139i	0.984288	+	0.176569i	$0.0565000\pi$
$971$	3.23298e8i	0.353139i	−0.984288	+	0.176569i	$0.943500\pi$
$972$	0	0
$973$	−4.54265e8	−0.493141
$974$	− 5.94246e8i	− 0.643117i
$975$	0	0
$976$	2.45091e7	0.0263619
$977$	1.24151e9i	1.33127i	0.746278	+	0.665634i	$0.231839\pi$
$977$	1.24151e9i	1.33127i	−0.746278	+	0.665634i	$0.768161\pi$
$978$	0	0
$979$	−1.07827e9	−1.14916
$980$	− 4.29912e8i	− 0.456774i
$981$	0	0
$982$	−1.07347e9	−1.13359
$983$	6.90841e8i	0.727306i	0.931534	+	0.363653i	$0.118471\pi$
$983$	6.90841e8i	0.727306i	−0.931534	+	0.363653i	$0.881529\pi$
$984$	0	0
$985$	1.95567e9	2.04638
$986$	6.62078e7i	0.0690682i
$987$	0	0
$988$	1.17733e8	0.122075
$989$	− 1.75456e9i	− 1.81376i
$990$	0	0
$991$	−1.68628e9	−1.73264	−0.866321	−	0.499487i	$-0.833522\pi$
$991$	−1.68628e9	−1.73264	−0.866321	+	0.499487i	$0.833522\pi$
$992$	− 3.36262e7i	− 0.0344463i
$993$	0	0
$994$	5.86245e8	0.596925
$995$	− 9.16939e8i	− 0.930832i
$996$	0	0
$997$	−1.56802e9	−1.58222	−0.791110	−	0.611674i	$-0.790497\pi$
$997$	−1.56802e9	−1.58222	−0.791110	+	0.611674i	$0.790497\pi$
$998$	1.14271e9i	1.14960i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.7.b.c.161.1		12
3.2	odd	2	inner	162.7.b.c.161.12		12
9.2	odd	6		54.7.d.a.17.4		12
9.4	even	3		54.7.d.a.35.4		12
9.5	odd	6		18.7.d.a.11.1	yes	12
9.7	even	3		18.7.d.a.5.1	✓	12
36.7	odd	6		144.7.q.c.113.5		12
36.11	even	6		432.7.q.b.17.2		12
36.23	even	6		144.7.q.c.65.5		12
36.31	odd	6		432.7.q.b.305.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.7.d.a.5.1	✓	12	9.7	even	3
18.7.d.a.11.1	yes	12	9.5	odd	6
54.7.d.a.17.4		12	9.2	odd	6
54.7.d.a.35.4		12	9.4	even	3
144.7.q.c.65.5		12	36.23	even	6
144.7.q.c.113.5		12	36.7	odd	6
162.7.b.c.161.1		12	1.1	even	1	trivial
162.7.b.c.161.12		12	3.2	odd	2	inner
432.7.q.b.17.2		12	36.11	even	6
432.7.q.b.305.2		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)

\(f(q)\)	\(=\)	\(q-5.65685i q^{2} -32.0000 q^{4} -181.232i q^{5} -208.612 q^{7} +181.019i q^{8} -1025.20 q^{10} +2658.11i q^{11} +877.198 q^{13} +1180.09i q^{14} +1024.00 q^{16} +4428.53i q^{17} -4194.21 q^{19} +5799.42i q^{20} +15036.5 q^{22} -12008.8i q^{23} -17220.0 q^{25} -4962.18i q^{26} +6675.58 q^{28} +2642.86i q^{29} +5805.01 q^{31} -5792.62i q^{32} +25051.5 q^{34} +37807.1i q^{35} +41579.5 q^{37} +23726.1i q^{38} +32806.5 q^{40} +74733.6i q^{41} +146106. q^{43} -85059.5i q^{44} -67932.1 q^{46} -25772.5i q^{47} -74130.1 q^{49} +97411.2i q^{50} -28070.4 q^{52} +197505. i q^{53} +481734. q^{55} -37762.8i q^{56} +14950.3 q^{58} +36166.9i q^{59} +23934.6 q^{61} -32838.1i q^{62} -32768.0 q^{64} -158976. i q^{65} +353079. q^{67} -141713. i q^{68} +213869. q^{70} -496781. i q^{71} -382139. q^{73} -235209. i q^{74} +134215. q^{76} -554513. i q^{77} +386657. q^{79} -185582. i q^{80} +422757. q^{82} +415335. i q^{83} +802590. q^{85} -826500. i q^{86} -481169. q^{88} +405654. i q^{89} -182994. q^{91} +384282. i q^{92} -145791. q^{94} +760126. i q^{95} +356487. q^{97} +419343. i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 384 q^{4} - 480 q^{7} - 3360 q^{13} + 12288 q^{16} - 2820 q^{19} + 7200 q^{22} - 16188 q^{25} + 15360 q^{28} - 42960 q^{31} + 54720 q^{34} - 25536 q^{37} - 142860 q^{43} - 135072 q^{46} + 271908 q^{49}+ \cdots + 77748 q^{97}+O(q^{100}) \) 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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