LMFDB - Embedded newform 162.7.b.a.161.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [162,7,Mod(161,162)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$37.2687615464$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 1 $ x^4 + 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.4
Root			$-0.517638i$ of defining polynomial
Character	$\chi$	$=$	162.161
Dual form			162.7.b.a.161.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.65685i q^{2} -32.0000 q^{4} -24.8168i q^{5} -6.19615 q^{7} -181.019i q^{8} +O(q^{10})$	\(q+5.65685i q^{2} -32.0000 q^{4} -24.8168i q^{5} -6.19615 q^{7} -181.019i q^{8} +140.385 q^{10} +130.897i q^{11} -922.007 q^{13} -35.0507i q^{14} +1024.00 q^{16} -3384.11i q^{17} +5403.30 q^{19} +794.136i q^{20} -740.463 q^{22} +2361.73i q^{23} +15009.1 q^{25} -5215.66i q^{26} +198.277 q^{28} +34937.6i q^{29} +4660.33 q^{31} +5792.62i q^{32} +19143.4 q^{34} +153.768i q^{35} +6916.74 q^{37} +30565.7i q^{38} -4492.31 q^{40} +53851.9i q^{41} +123133. q^{43} -4188.69i q^{44} -13360.0 q^{46} +96052.5i q^{47} -117611. q^{49} +84904.5i q^{50} +29504.2 q^{52} +132310. i q^{53} +3248.43 q^{55} +1121.62i q^{56} -197637. q^{58} +29077.1i q^{59} +320666. q^{61} +26362.8i q^{62} -32768.0 q^{64} +22881.2i q^{65} -529819. q^{67} +108292. i q^{68} -869.845 q^{70} +628518. i q^{71} -68777.7 q^{73} +39127.0i q^{74} -172905. q^{76} -811.056i q^{77} +471754. q^{79} -25412.4i q^{80} -304633. q^{82} +33550.0i q^{83} -83982.7 q^{85} +696545. i q^{86} +23694.8 q^{88} +1.01543e6i q^{89} +5712.90 q^{91} -75575.5i q^{92} -543355. q^{94} -134092. i q^{95} +418189. q^{97} -665306. i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 128 q^{4} - 4 q^{7}+O(q^{10}) $ 4 * q - 128 * q^4 - 4 * q^7	$ 4 q - 128 q^{4} - 4 q^{7} + 2640 q^{10} + 3836 q^{13} + 4096 q^{16} + 10244 q^{19} - 27072 q^{22} - 25700 q^{25} + 128 q^{28} - 40096 q^{31} + 60528 q^{34} - 20200 q^{37} - 84480 q^{40} + 184940 q^{43} + 162720 q^{46} - 470484 q^{49} - 122752 q^{52} + 949860 q^{55} + 24624 q^{58} + 609056 q^{61} - 131072 q^{64} - 2008972 q^{67} + 8160 q^{70} + 525824 q^{73} - 327808 q^{76} - 848716 q^{79} + 705792 q^{82} - 987840 q^{85} + 866304 q^{88} + 35260 q^{91} - 1965408 q^{94} + 3621728 q^{97}+O(q^{100}) $ 4 * q - 128 * q^4 - 4 * q^7 + 2640 * q^10 + 3836 * q^13 + 4096 * q^16 + 10244 * q^19 - 27072 * q^22 - 25700 * q^25 + 128 * q^28 - 40096 * q^31 + 60528 * q^34 - 20200 * q^37 - 84480 * q^40 + 184940 * q^43 + 162720 * q^46 - 470484 * q^49 - 122752 * q^52 + 949860 * q^55 + 24624 * q^58 + 609056 * q^61 - 131072 * q^64 - 2008972 * q^67 + 8160 * q^70 + 525824 * q^73 - 327808 * q^76 - 848716 * q^79 + 705792 * q^82 - 987840 * q^85 + 866304 * q^88 + 35260 * q^91 - 1965408 * q^94 + 3621728 * q^97

Display 100 coefficients 10 coefficients

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
$5$	− 24.8168i	− 0.198534i	−0.995061	−	0.0992670i	$-0.968350\pi$
$5$	− 24.8168i	− 0.198534i	0.995061	−	0.0992670i	$-0.0316498\pi$
$6$	0	0
$7$	−6.19615	−0.0180646	−0.00903229	−	0.999959i	$-0.502875\pi$
$7$	−6.19615	−0.0180646	−0.00903229	+	0.999959i	$0.502875\pi$
$8$	− 181.019i	− 0.353553i
$9$	0	0
$10$	140.385	0.140385
$11$	130.897i	0.0983446i	0.998790	+	0.0491723i	$0.0156583\pi$
$11$	130.897i	0.0983446i	−0.998790	+	0.0491723i	$0.984342\pi$
$12$	0	0
$13$	−922.007	−0.419666	−0.209833	−	0.977737i	$-0.567292\pi$
$13$	−922.007	−0.419666	−0.209833	+	0.977737i	$0.567292\pi$
$14$	− 35.0507i	− 0.0127736i
$15$	0	0
$16$	1024.00	0.250000
$17$	− 3384.11i	− 0.688808i	−0.938822	−	0.344404i	$-0.888081\pi$
$17$	− 3384.11i	− 0.688808i	0.938822	−	0.344404i	$-0.111919\pi$
$18$	0	0
$19$	5403.30	0.787767	0.393884	−	0.919160i	$-0.371131\pi$
$19$	5403.30	0.787767	0.393884	+	0.919160i	$0.371131\pi$
$20$	794.136i	0.0992670i
$21$	0	0
$22$	−740.463	−0.0695401
$23$	2361.73i	0.194110i	0.995279	+	0.0970549i	$0.0309422\pi$
$23$	2361.73i	0.194110i	−0.995279	+	0.0970549i	$0.969058\pi$
$24$	0	0
$25$	15009.1	0.960584
$26$	− 5215.66i	− 0.296749i
$27$	0	0
$28$	198.277	0.00903229
$29$	34937.6i	1.43252i	0.697836	+	0.716258i	$0.254147\pi$
$29$	34937.6i	1.43252i	−0.697836	+	0.716258i	$0.745853\pi$
$30$	0	0
$31$	4660.33	0.156434	0.0782170	−	0.996936i	$-0.475077\pi$
$31$	4660.33	0.156434	0.0782170	+	0.996936i	$0.475077\pi$
$32$	5792.62i	0.176777i
$33$	0	0
$34$	19143.4	0.487061
$35$	153.768i	0.00358643i
$36$	0	0
$37$	6916.74	0.136551	0.0682757	−	0.997666i	$-0.478250\pi$
$37$	6916.74	0.136551	0.0682757	+	0.997666i	$0.478250\pi$
$38$	30565.7i	0.557036i
$39$	0	0
$40$	−4492.31	−0.0701924
$41$	53851.9i	0.781357i	0.920527	+	0.390679i	$0.127760\pi$
$41$	53851.9i	0.781357i	−0.920527	+	0.390679i	$0.872240\pi$
$42$	0	0
$43$	123133.	1.54870	0.774352	−	0.632755i	$-0.218076\pi$
$43$	123133.	1.54870	0.774352	+	0.632755i	$0.218076\pi$
$44$	− 4188.69i	− 0.0491723i
$45$	0	0
$46$	−13360.0	−0.137256
$47$	96052.5i	0.925156i	0.886578	+	0.462578i	$0.153076\pi$
$47$	96052.5i	0.925156i	−0.886578	+	0.462578i	$0.846924\pi$
$48$	0	0
$49$	−117611.	−0.999674
$50$	84904.5i	0.679236i
$51$	0	0
$52$	29504.2	0.209833
$53$	132310.i	0.888722i	0.895848	+	0.444361i	$0.146569\pi$
$53$	132310.i	0.888722i	−0.895848	+	0.444361i	$0.853431\pi$
$54$	0	0
$55$	3248.43	0.0195247
$56$	1121.62i	0.00638680i
$57$	0	0
$58$	−197637.	−1.01294
$59$	29077.1i	0.141578i	0.997491	+	0.0707889i	$0.0225517\pi$
$59$	29077.1i	0.141578i	−0.997491	+	0.0707889i	$0.977448\pi$
$60$	0	0
$61$	320666.	1.41274	0.706372	−	0.707841i	$-0.250331\pi$
$61$	320666.	1.41274	0.706372	+	0.707841i	$0.250331\pi$
$62$	26362.8i	0.110616i
$63$	0	0
$64$	−32768.0	−0.125000
$65$	22881.2i	0.0833181i
$66$	0	0
$67$	−529819.	−1.76158	−0.880791	−	0.473504i	$-0.842989\pi$
$67$	−529819.	−1.76158	−0.880791	+	0.473504i	$0.842989\pi$
$68$	108292.i	0.344404i
$69$	0	0
$70$	−869.845	−0.00253599
$71$	628518.i	1.75607i	0.478594	+	0.878036i	$0.341146\pi$
$71$	628518.i	1.75607i	−0.478594	+	0.878036i	$0.658854\pi$
$72$	0	0
$73$	−68777.7	−0.176799	−0.0883994	−	0.996085i	$-0.528175\pi$
$73$	−68777.7	−0.176799	−0.0883994	+	0.996085i	$0.528175\pi$
$74$	39127.0i	0.0965564i
$75$	0	0
$76$	−172905.	−0.393884
$77$	− 811.056i	− 0.00177655i
$78$	0	0
$79$	471754.	0.956829	0.478415	−	0.878134i	$-0.341212\pi$
$79$	471754.	0.956829	0.478415	+	0.878134i	$0.341212\pi$
$80$	− 25412.4i	− 0.0496335i
$81$	0	0
$82$	−304633.	−0.552503
$83$	33550.0i	0.0586756i	0.999570	+	0.0293378i	$0.00933985\pi$
$83$	33550.0i	0.0586756i	−0.999570	+	0.0293378i	$0.990660\pi$
$84$	0	0
$85$	−83982.7	−0.136752
$86$	696545.i	1.09510i
$87$	0	0
$88$	23694.8	0.0347701
$89$	1.01543e6i	1.44039i	0.693772	+	0.720195i	$0.255947\pi$
$89$	1.01543e6i	1.44039i	−0.693772	+	0.720195i	$0.744053\pi$
$90$	0	0
$91$	5712.90	0.00758110
$92$	− 75575.5i	− 0.0970549i
$93$	0	0
$94$	−543355.	−0.654184
$95$	− 134092.i	− 0.156399i
$96$	0	0
$97$	418189.	0.458202	0.229101	−	0.973403i	$-0.426421\pi$
$97$	418189.	0.458202	0.229101	+	0.973403i	$0.426421\pi$
$98$	− 665306.i	− 0.706876i
$99$	0	0
$100$	−480292.	−0.480292
$101$	671665.i	0.651912i	0.945385	+	0.325956i	$0.105686\pi$
$101$	671665.i	0.651912i	−0.945385	+	0.325956i	$0.894314\pi$
$102$	0	0
$103$	701999.	0.642428	0.321214	−	0.947007i	$-0.395909\pi$
$103$	701999.	0.642428	0.321214	+	0.947007i	$0.395909\pi$
$104$	166901.i	0.148374i
$105$	0	0
$106$	−748460.	−0.628421
$107$	− 350565.i	− 0.286166i	−0.989711	−	0.143083i	$-0.954298\pi$
$107$	− 350565.i	− 0.286166i	0.989711	−	0.143083i	$-0.0457016\pi$
$108$	0	0
$109$	724369.	0.559345	0.279673	−	0.960095i	$-0.409774\pi$
$109$	724369.	0.559345	0.279673	+	0.960095i	$0.409774\pi$
$110$	18375.9i	0.0138061i
$111$	0	0
$112$	−6344.86	−0.00451615
$113$	− 1.26346e6i	− 0.875644i	−0.899062	−	0.437822i	$-0.855750\pi$
$113$	− 1.26346e6i	− 0.875644i	0.899062	−	0.437822i	$-0.144250\pi$
$114$	0	0
$115$	58610.6	0.0385374
$116$	− 1.11800e6i	− 0.716258i
$117$	0	0
$118$	−164485.	−0.100111
$119$	20968.5i	0.0124430i
$120$	0	0
$121$	1.75443e6	0.990328
$122$	1.81396e6i	0.998961i
$123$	0	0
$124$	−149130.	−0.0782170
$125$	− 760240.i	− 0.389243i
$126$	0	0
$127$	2.67398e6	1.30541	0.652706	−	0.757611i	$-0.273634\pi$
$127$	2.67398e6	1.30541	0.652706	+	0.757611i	$0.273634\pi$
$128$	− 185364.i	− 0.0883883i
$129$	0	0
$130$	−129436.	−0.0589148
$131$	− 3.58815e6i	− 1.59609i	−0.602601	−	0.798043i	$-0.705869\pi$
$131$	− 3.58815e6i	− 1.59609i	0.602601	−	0.798043i	$-0.294131\pi$
$132$	0	0
$133$	−33479.6	−0.0142307
$134$	− 2.99711e6i	− 1.24563i
$135$	0	0
$136$	−612590.	−0.243530
$137$	− 1.53054e6i	− 0.595226i	−0.954687	−	0.297613i	$-0.903810\pi$
$137$	− 1.53054e6i	− 0.595226i	0.954687	−	0.297613i	$-0.0961905\pi$
$138$	0	0
$139$	3.52766e6	1.31354	0.656769	−	0.754092i	$-0.271923\pi$
$139$	3.52766e6	1.31354	0.656769	+	0.754092i	$0.271923\pi$
$140$	− 4920.59i	− 0.00179322i
$141$	0	0
$142$	−3.55543e6	−1.24173
$143$	− 120688.i	− 0.0412719i
$144$	0	0
$145$	867039.	0.284403
$146$	− 389066.i	− 0.125016i
$147$	0	0
$148$	−221336.	−0.0682757
$149$	− 2.39531e6i	− 0.724106i	−0.932158	−	0.362053i	$-0.882076\pi$
$149$	− 2.39531e6i	− 0.724106i	0.932158	−	0.362053i	$-0.117924\pi$
$150$	0	0
$151$	−3.01675e6	−0.876210	−0.438105	−	0.898924i	$-0.644350\pi$
$151$	−3.01675e6	−0.876210	−0.438105	+	0.898924i	$0.644350\pi$
$152$	− 978101.i	− 0.278518i
$153$	0	0
$154$	4588.02	0.00125621
$155$	− 115654.i	− 0.0310575i
$156$	0	0
$157$	5.58451e6	1.44307	0.721533	−	0.692380i	$-0.243438\pi$
$157$	5.58451e6	1.44307	0.721533	+	0.692380i	$0.243438\pi$
$158$	2.66864e6i	0.676580i
$159$	0	0
$160$	143754.	0.0350962
$161$	− 14633.7i	− 0.00350651i
$162$	0	0
$163$	−2.46039e6	−0.568121	−0.284061	−	0.958806i	$-0.591682\pi$
$163$	−2.46039e6	−0.568121	−0.284061	+	0.958806i	$0.591682\pi$
$164$	− 1.72326e6i	− 0.390679i
$165$	0	0
$166$	−189787.	−0.0414899
$167$	− 4.02275e6i	− 0.863722i	−0.901940	−	0.431861i	$-0.857857\pi$
$167$	− 4.02275e6i	− 0.863722i	0.901940	−	0.431861i	$-0.142143\pi$
$168$	0	0
$169$	−3.97671e6	−0.823880
$170$	− 475078.i	− 0.0966981i
$171$	0	0
$172$	−3.94025e6	−0.774352
$173$	− 4.13858e6i	− 0.799307i	−0.916666	−	0.399653i	$-0.869131\pi$
$173$	− 4.13858e6i	− 0.799307i	0.916666	−	0.399653i	$-0.130869\pi$
$174$	0	0
$175$	−92998.8	−0.0173526
$176$	134038.i	0.0245861i
$177$	0	0
$178$	−5.74414e6	−1.01851
$179$	4.64610e6i	0.810083i	0.914298	+	0.405042i	$0.132743\pi$
$179$	4.64610e6i	0.810083i	−0.914298	+	0.405042i	$0.867257\pi$
$180$	0	0
$181$	2.01455e6	0.339737	0.169868	−	0.985467i	$-0.445666\pi$
$181$	2.01455e6	0.339737	0.169868	+	0.985467i	$0.445666\pi$
$182$	32317.0i	0.00536065i
$183$	0	0
$184$	427520.	0.0686282
$185$	− 171651.i	− 0.0271101i
$186$	0	0
$187$	442969.	0.0677405
$188$	− 3.07368e6i	− 0.462578i
$189$	0	0
$190$	758540.	0.110591
$191$	− 3.80606e6i	− 0.546230i	−0.961981	−	0.273115i	$-0.911946\pi$
$191$	− 3.80606e6i	− 0.546230i	0.961981	−	0.273115i	$-0.0880540\pi$
$192$	0	0
$193$	−1.29548e6	−0.180201	−0.0901005	−	0.995933i	$-0.528719\pi$
$193$	−1.29548e6	−0.180201	−0.0901005	+	0.995933i	$0.528719\pi$
$194$	2.36563e6i	0.323998i
$195$	0	0
$196$	3.76354e6	0.499837
$197$	5.99463e6i	0.784086i	0.919947	+	0.392043i	$0.128231\pi$
$197$	5.99463e6i	0.784086i	−0.919947	+	0.392043i	$0.871769\pi$
$198$	0	0
$199$	−1.04921e6	−0.133139	−0.0665694	−	0.997782i	$-0.521205\pi$
$199$	−1.04921e6	−0.133139	−0.0665694	+	0.997782i	$0.521205\pi$
$200$	− 2.71694e6i	− 0.339618i
$201$	0	0
$202$	−3.79951e6	−0.460971
$203$	− 216479.i	− 0.0258778i
$204$	0	0
$205$	1.33643e6	0.155126
$206$	3.97110e6i	0.454265i
$207$	0	0
$208$	−944135.	−0.104917
$209$	707273.i	0.0774726i
$210$	0	0
$211$	−5.54207e6	−0.589963	−0.294982	−	0.955503i	$-0.595314\pi$
$211$	−5.54207e6	−0.589963	−0.294982	+	0.955503i	$0.595314\pi$
$212$	− 4.23393e6i	− 0.444361i
$213$	0	0
$214$	1.98310e6	0.202350
$215$	− 3.05576e6i	− 0.307471i
$216$	0	0
$217$	−28876.1	−0.00282592
$218$	4.09765e6i	0.395517i
$219$	0	0
$220$	−103950.	−0.00976237
$221$	3.12018e6i	0.289069i
$222$	0	0
$223$	4.13766e6	0.373113	0.186557	−	0.982444i	$-0.440267\pi$
$223$	4.13766e6	0.373113	0.186557	+	0.982444i	$0.440267\pi$
$224$	− 35891.9i	− 0.00319340i
$225$	0	0
$226$	7.14723e6	0.619174
$227$	1.47605e7i	1.26190i	0.775825	+	0.630948i	$0.217334\pi$
$227$	1.47605e7i	1.26190i	−0.775825	+	0.630948i	$0.782666\pi$
$228$	0	0
$229$	−1.46347e7	−1.21865	−0.609323	−	0.792922i	$-0.708559\pi$
$229$	−1.46347e7	−1.21865	−0.609323	+	0.792922i	$0.708559\pi$
$230$	331551.i	0.0272501i
$231$	0	0
$232$	6.32439e6	0.506471
$233$	− 1.51778e6i	− 0.119989i	−0.998199	−	0.0599946i	$-0.980892\pi$
$233$	− 1.51778e6i	− 0.119989i	0.998199	−	0.0599946i	$-0.0191083\pi$
$234$	0	0
$235$	2.38371e6	0.183675
$236$	− 930468.i	− 0.0707889i
$237$	0	0
$238$	−118616.	−0.00879855
$239$	1.39608e7i	1.02262i	0.859395	+	0.511312i	$0.170840\pi$
$239$	1.39608e7i	1.02262i	−0.859395	+	0.511312i	$0.829160\pi$
$240$	0	0
$241$	2.53164e7	1.80863	0.904317	−	0.426861i	$-0.140381\pi$
$241$	2.53164e7	1.80863	0.904317	+	0.426861i	$0.140381\pi$
$242$	9.92454e6i	0.700268i
$243$	0	0
$244$	−1.02613e7	−0.706372
$245$	2.91871e6i	0.198469i
$246$	0	0
$247$	−4.98188e6	−0.330599
$248$	− 843609.i	− 0.0553078i
$249$	0	0
$250$	4.30056e6	0.275236
$251$	− 3.58400e6i	− 0.226645i	−0.993558	−	0.113323i	$-0.963851\pi$
$251$	− 3.58400e6i	− 0.226645i	0.993558	−	0.113323i	$-0.0361494\pi$
$252$	0	0
$253$	−309143.	−0.0190896
$254$	1.51263e7i	0.923066i
$255$	0	0
$256$	1.04858e6	0.0625000
$257$	2.31906e7i	1.36620i	0.730327	+	0.683098i	$0.239368\pi$
$257$	2.31906e7i	1.36620i	−0.730327	+	0.683098i	$0.760632\pi$
$258$	0	0
$259$	−42857.2	−0.00246674
$260$	− 732199.i	− 0.0416590i
$261$	0	0
$262$	2.02976e7	1.12860
$263$	1.99760e7i	1.09810i	0.835790	+	0.549049i	$0.185010\pi$
$263$	1.99760e7i	1.09810i	−0.835790	+	0.549049i	$0.814990\pi$
$264$	0	0
$265$	3.28351e6	0.176442
$266$	− 189389.i	− 0.0100626i
$267$	0	0
$268$	1.69542e7	0.880791
$269$	− 1.05394e7i	− 0.541453i	−0.962656	−	0.270726i	$-0.912736\pi$
$269$	− 1.05394e7i	− 0.541453i	0.962656	−	0.270726i	$-0.0872639\pi$
$270$	0	0
$271$	−2.76838e7	−1.39097	−0.695485	−	0.718540i	$-0.744811\pi$
$271$	−2.76838e7	−1.39097	−0.695485	+	0.718540i	$0.744811\pi$
$272$	− 3.46533e6i	− 0.172202i
$273$	0	0
$274$	8.65802e6	0.420888
$275$	1.96464e6i	0.0944683i
$276$	0	0
$277$	8.16595e6	0.384209	0.192104	−	0.981374i	$-0.438469\pi$
$277$	8.16595e6	0.384209	0.192104	+	0.981374i	$0.438469\pi$
$278$	1.99555e7i	0.928811i
$279$	0	0
$280$	27835.1	0.00126800
$281$	4.45538e6i	0.200801i	0.994947	+	0.100400i	$0.0320124\pi$
$281$	4.45538e6i	0.200801i	−0.994947	+	0.100400i	$0.967988\pi$
$282$	0	0
$283$	−3.18827e7	−1.40668	−0.703341	−	0.710853i	$-0.748309\pi$
$283$	−3.18827e7	−1.40668	−0.703341	+	0.710853i	$0.748309\pi$
$284$	− 2.01126e7i	− 0.878036i
$285$	0	0
$286$	682712.	0.0291837
$287$	− 333675.i	− 0.0141149i
$288$	0	0
$289$	1.26854e7	0.525544
$290$	4.90471e6i	0.201103i
$291$	0	0
$292$	2.20089e6	0.0883994
$293$	3.57172e7i	1.41995i	0.704225	+	0.709977i	$0.251295\pi$
$293$	3.57172e7i	1.41995i	−0.704225	+	0.709977i	$0.748705\pi$
$294$	0	0
$295$	721600.	0.0281080
$296$	− 1.25206e6i	− 0.0482782i
$297$	0	0
$298$	1.35499e7	0.512020
$299$	− 2.17754e6i	− 0.0814614i
$300$	0	0
$301$	−762950.	−0.0279767
$302$	− 1.70653e7i	− 0.619574i
$303$	0	0
$304$	5.53297e6	0.196942
$305$	− 7.95789e6i	− 0.280478i
$306$	0	0
$307$	−5.01121e6	−0.173192	−0.0865959	−	0.996244i	$-0.527599\pi$
$307$	−5.01121e6	−0.173192	−0.0865959	+	0.996244i	$0.527599\pi$
$308$	25953.8i	0 0.000888277i
$309$	0	0
$310$	654239.	0.0219610
$311$	− 2.91619e7i	− 0.969471i	−0.874661	−	0.484736i	$-0.838916\pi$
$311$	− 2.91619e7i	− 0.969471i	0.874661	−	0.484736i	$-0.161084\pi$
$312$	0	0
$313$	−3.06826e7	−1.00060	−0.500298	−	0.865853i	$-0.666776\pi$
$313$	−3.06826e7	−1.00060	−0.500298	+	0.865853i	$0.666776\pi$
$314$	3.15908e7i	1.02040i
$315$	0	0
$316$	−1.50961e7	−0.478415
$317$	− 5.07411e7i	− 1.59288i	−0.604719	−	0.796439i	$-0.706715\pi$
$317$	− 5.07411e7i	− 1.59288i	0.604719	−	0.796439i	$-0.293285\pi$
$318$	0	0
$319$	−4.57322e6	−0.140880
$320$	813195.i	0.0248168i
$321$	0	0
$322$	82780.5	0.00247948
$323$	− 1.82854e7i	− 0.542620i
$324$	0	0
$325$	−1.38385e7	−0.403125
$326$	− 1.39181e7i	− 0.401723i
$327$	0	0
$328$	9.74824e6	0.276252
$329$	− 595156.i	− 0.0167126i
$330$	0	0
$331$	−4.21426e7	−1.16208	−0.581042	−	0.813873i	$-0.697355\pi$
$331$	−4.21426e7	−1.16208	−0.581042	+	0.813873i	$0.697355\pi$
$332$	− 1.07360e6i	− 0.0293378i
$333$	0	0
$334$	2.27561e7	0.610744
$335$	1.31484e7i	0.349734i
$336$	0	0
$337$	61432.7	0.00160513	0.000802565	−	1.00000i	$-0.499745\pi$
$337$	61432.7	0.00160513	0.000802565		1.00000i	$0.499745\pi$
$338$	− 2.24957e7i	− 0.582571i
$339$	0	0
$340$	2.68745e6	0.0683759
$341$	610021.i	0.0153844i
$342$	0	0
$343$	1.45770e6	0.0361233
$344$	− 2.22894e7i	− 0.547550i
$345$	0	0
$346$	2.34114e7	0.565195
$347$	2.43289e7i	0.582282i	0.956680	+	0.291141i	$0.0940349\pi$
$347$	2.43289e7i	0.582282i	−0.956680	+	0.291141i	$0.905965\pi$
$348$	0	0
$349$	−2.52028e7	−0.592888	−0.296444	−	0.955050i	$-0.595801\pi$
$349$	−2.52028e7	−0.592888	−0.296444	+	0.955050i	$0.595801\pi$
$350$	− 526081.i	− 0.0122701i
$351$	0	0
$352$	−758234.	−0.0173850
$353$	− 6.91107e7i	− 1.57116i	−0.618758	−	0.785581i	$-0.712364\pi$
$353$	− 6.91107e7i	− 1.57116i	0.618758	−	0.785581i	$-0.287636\pi$
$354$	0	0
$355$	1.55978e7	0.348640
$356$	− 3.24937e7i	− 0.720195i
$357$	0	0
$358$	−2.62823e7	−0.572815
$359$	− 1.41061e7i	− 0.304876i	−0.988313	−	0.152438i	$-0.951288\pi$
$359$	− 1.41061e7i	− 0.304876i	0.988313	−	0.152438i	$-0.0487124\pi$
$360$	0	0
$361$	−1.78503e7	−0.379423
$362$	1.13960e7i	0.240230i
$363$	0	0
$364$	−182813.	−0.00379055
$365$	1.70684e6i	0.0351006i
$366$	0	0
$367$	−8.14446e7	−1.64765	−0.823823	−	0.566847i	$-0.808163\pi$
$367$	−8.14446e7	−1.64765	−0.823823	+	0.566847i	$0.808163\pi$
$368$	2.41842e6i	0.0485275i
$369$	0	0
$370$	971005.	0.0191697
$371$	− 819814.i	− 0.0160544i
$372$	0	0
$373$	7.59721e7	1.46395	0.731977	−	0.681329i	$-0.238598\pi$
$373$	7.59721e7	1.46395	0.731977	+	0.681329i	$0.238598\pi$
$374$	2.50581e6i	0.0478998i
$375$	0	0
$376$	1.73874e7	0.327092
$377$	− 3.22127e7i	− 0.601179i
$378$	0	0
$379$	−4.48086e7	−0.823083	−0.411541	−	0.911391i	$-0.635009\pi$
$379$	−4.48086e7	−0.823083	−0.411541	+	0.911391i	$0.635009\pi$
$380$	4.29095e6i	0.0781993i
$381$	0	0
$382$	2.15303e7	0.386243
$383$	− 8.96262e7i	− 1.59529i	−0.603130	−	0.797643i	$-0.706080\pi$
$383$	− 8.96262e7i	− 1.59529i	0.603130	−	0.797643i	$-0.293920\pi$
$384$	0	0
$385$	−20127.8	−0.000352706 0
$386$	− 7.32832e6i	− 0.127421i
$387$	0	0
$388$	−1.33820e7	−0.229101
$389$	3.50684e7i	0.595754i	0.954604	+	0.297877i	$0.0962786\pi$
$389$	3.50684e7i	0.595754i	−0.954604	+	0.297877i	$0.903721\pi$
$390$	0	0
$391$	7.99237e6	0.133704
$392$	2.12898e7i	0.353438i
$393$	0	0
$394$	−3.39107e7	−0.554432
$395$	− 1.17074e7i	− 0.189963i
$396$	0	0
$397$	8.17666e7	1.30679	0.653393	−	0.757019i	$-0.273345\pi$
$397$	8.17666e7	1.30679	0.653393	+	0.757019i	$0.273345\pi$
$398$	− 5.93525e6i	− 0.0941433i
$399$	0	0
$400$	1.53693e7	0.240146
$401$	6.10769e7i	0.947204i	0.880739	+	0.473602i	$0.157047\pi$
$401$	6.10769e7i	0.947204i	−0.880739	+	0.473602i	$0.842953\pi$
$402$	0	0
$403$	−4.29685e6	−0.0656501
$404$	− 2.14933e7i	− 0.325956i
$405$	0	0
$406$	1.22459e6	0.0182984
$407$	905378.i	0.0134291i
$408$	0	0
$409$	6.69282e7	0.978226	0.489113	−	0.872221i	$-0.337321\pi$
$409$	6.69282e7	0.978226	0.489113	+	0.872221i	$0.337321\pi$
$410$	7.55999e6i	0.109691i
$411$	0	0
$412$	−2.24640e7	−0.321214
$413$	− 180166.i	− 0.00255755i
$414$	0	0
$415$	832601.	0.0116491
$416$	− 5.34084e6i	− 0.0741872i
$417$	0	0
$418$	−4.00094e6	−0.0547814
$419$	− 9.73994e7i	− 1.32408i	−0.749469	−	0.662040i	$-0.769691\pi$
$419$	− 9.73994e7i	− 1.32408i	0.749469	−	0.662040i	$-0.230309\pi$
$420$	0	0
$421$	7.59346e7	1.01764	0.508819	−	0.860874i	$-0.330082\pi$
$421$	7.59346e7	1.01764	0.508819	+	0.860874i	$0.330082\pi$
$422$	− 3.13507e7i	− 0.417167i
$423$	0	0
$424$	2.39507e7	0.314211
$425$	− 5.07926e7i	− 0.661658i
$426$	0	0
$427$	−1.98690e6	−0.0255206
$428$	1.12181e7i	0.143083i
$429$	0	0
$430$	1.72860e7	0.217415
$431$	− 1.10356e8i	− 1.37836i	−0.724590	−	0.689180i	$-0.757971\pi$
$431$	− 1.10356e8i	− 1.37836i	0.724590	−	0.689180i	$-0.242029\pi$
$432$	0	0
$433$	7.04803e7	0.868168	0.434084	−	0.900872i	$-0.357072\pi$
$433$	7.04803e7	0.868168	0.434084	+	0.900872i	$0.357072\pi$
$434$	− 163348.i	− 0.00199822i
$435$	0	0
$436$	−2.31798e7	−0.279673
$437$	1.27611e7i	0.152913i
$438$	0	0
$439$	−1.11974e7	−0.132349	−0.0661747	−	0.997808i	$-0.521079\pi$
$439$	−1.11974e7	−0.132349	−0.0661747	+	0.997808i	$0.521079\pi$
$440$	− 588029.i	− 0.00690304i
$441$	0	0
$442$	−1.76504e7	−0.204403
$443$	1.54810e8i	1.78069i	0.455290	+	0.890343i	$0.349536\pi$
$443$	1.54810e8i	1.78069i	−0.455290	+	0.890343i	$0.650464\pi$
$444$	0	0
$445$	2.51997e7	0.285966
$446$	2.34062e7i	0.263831i
$447$	0	0
$448$	203036.	0.00225807
$449$	9.20846e7i	1.01730i	0.860974	+	0.508648i	$0.169855\pi$
$449$	9.20846e7i	1.01730i	−0.860974	+	0.508648i	$0.830145\pi$
$450$	0	0
$451$	−7.04904e6	−0.0768423
$452$	4.04309e7i	0.437822i
$453$	0	0
$454$	−8.34980e7	−0.892295
$455$	− 141776.i	− 0.00150511i
$456$	0	0
$457$	8.36663e7	0.876601	0.438300	−	0.898829i	$-0.355581\pi$
$457$	8.36663e7	0.876601	0.438300	+	0.898829i	$0.355581\pi$
$458$	− 8.27864e7i	− 0.861712i
$459$	0	0
$460$	−1.87554e6	−0.0192687
$461$	− 1.19014e8i	− 1.21477i	−0.794406	−	0.607387i	$-0.792218\pi$
$461$	− 1.19014e8i	− 1.21477i	0.794406	−	0.607387i	$-0.207782\pi$
$462$	0	0
$463$	−1.13573e8	−1.14428	−0.572141	−	0.820155i	$-0.693887\pi$
$463$	−1.13573e8	−1.14428	−0.572141	+	0.820155i	$0.693887\pi$
$464$	3.57761e7i	0.358129i
$465$	0	0
$466$	8.58588e6	0.0848452
$467$	1.34722e8i	1.32278i	0.750044	+	0.661388i	$0.230032\pi$
$467$	1.34722e8i	1.32278i	−0.750044	+	0.661388i	$0.769968\pi$
$468$	0	0
$469$	3.28284e6	0.0318223
$470$	1.34843e7i	0.129878i
$471$	0	0
$472$	5.26352e6	0.0500553
$473$	1.61177e7i	0.152307i
$474$	0	0
$475$	8.10988e7	0.756717
$476$	− 670991.i	− 0.00622151i
$477$	0	0
$478$	−7.89742e7	−0.723105
$479$	1.60975e8i	1.46471i	0.680921	+	0.732357i	$0.261580\pi$
$479$	1.60975e8i	1.46471i	−0.680921	+	0.732357i	$0.738420\pi$
$480$	0	0
$481$	−6.37728e6	−0.0573060
$482$	1.43211e8i	1.27890i
$483$	0	0
$484$	−5.61417e7	−0.495164
$485$	− 1.03781e7i	− 0.0909687i
$486$	0	0
$487$	1.14964e7	0.0995351	0.0497676	−	0.998761i	$-0.484152\pi$
$487$	1.14964e7	0.0995351	0.0497676	+	0.998761i	$0.484152\pi$
$488$	− 5.80468e7i	− 0.499481i
$489$	0	0
$490$	−1.65107e7	−0.140339
$491$	− 1.39301e8i	− 1.17682i	−0.808564	−	0.588408i	$-0.799755\pi$
$491$	− 1.39301e8i	− 1.17682i	0.808564	−	0.588408i	$-0.200245\pi$
$492$	0	0
$493$	1.18233e8	0.986728
$494$	− 2.81818e7i	− 0.233769i
$495$	0	0
$496$	4.77217e6	0.0391085
$497$	− 3.89439e6i	− 0.0317227i
$498$	0	0
$499$	−9.34696e7	−0.752261	−0.376130	−	0.926567i	$-0.622746\pi$
$499$	−9.34696e7	−0.752261	−0.376130	+	0.926567i	$0.622746\pi$
$500$	2.43277e7i	0.194621i
$501$	0	0
$502$	2.02742e7	0.160263
$503$	2.32808e8i	1.82934i	0.404200	+	0.914671i	$0.367550\pi$
$503$	2.32808e8i	1.82934i	−0.404200	+	0.914671i	$0.632450\pi$
$504$	0	0
$505$	1.66686e7	0.129427
$506$	− 1.74878e6i	− 0.0134984i
$507$	0	0
$508$	−8.55675e7	−0.652706
$509$	1.71929e8i	1.30375i	0.758324	+	0.651877i	$0.226018\pi$
$509$	1.71929e8i	1.30375i	−0.758324	+	0.651877i	$0.773982\pi$
$510$	0	0
$511$	426157.	0.00319380
$512$	5.93164e6i	0.0441942i
$513$	0	0
$514$	−1.31186e8	−0.966046
$515$	− 1.74213e7i	− 0.127544i
$516$	0	0
$517$	−1.25730e7	−0.0909841
$518$	− 242437.i	− 0.00174425i
$519$	0	0
$520$	4.14194e6	0.0294574
$521$	2.02142e8i	1.42937i	0.699447	+	0.714684i	$0.253430\pi$
$521$	2.02142e8i	1.42937i	−0.699447	+	0.714684i	$0.746570\pi$
$522$	0	0
$523$	−8.40337e7	−0.587419	−0.293710	−	0.955895i	$-0.594890\pi$
$523$	−8.40337e7	−0.587419	−0.293710	+	0.955895i	$0.594890\pi$
$524$	1.14821e8i	0.798043i
$525$	0	0
$526$	−1.13001e8	−0.776473
$527$	− 1.57711e7i	− 0.107753i
$528$	0	0
$529$	1.42458e8	0.962321
$530$	1.85743e7i	0.124763i
$531$	0	0
$532$	1.07135e6	0.00711534
$533$	− 4.96519e7i	− 0.327910i
$534$	0	0
$535$	−8.69990e6	−0.0568137
$536$	9.59075e7i	0.622814i
$537$	0	0
$538$	5.96201e7	0.382865
$539$	− 1.53948e7i	− 0.0983125i
$540$	0	0
$541$	5.27746e7	0.333299	0.166649	−	0.986016i	$-0.446705\pi$
$541$	5.27746e7	0.333299	0.166649	+	0.986016i	$0.446705\pi$
$542$	− 1.56603e8i	− 0.983565i
$543$	0	0
$544$	1.96029e7	0.121765
$545$	− 1.79765e7i	− 0.111049i
$546$	0	0
$547$	6.24645e7	0.381655	0.190828	−	0.981624i	$-0.438883\pi$
$547$	6.24645e7	0.381655	0.190828	+	0.981624i	$0.438883\pi$
$548$	4.89771e7i	0.297613i
$549$	0	0
$550$	−1.11137e7	−0.0667991
$551$	1.88778e8i	1.12849i
$552$	0	0
$553$	−2.92306e6	−0.0172847
$554$	4.61936e7i	0.271677i
$555$	0	0
$556$	−1.12885e8	−0.656769
$557$	1.98446e8i	1.14836i	0.818730	+	0.574178i	$0.194678\pi$
$557$	1.98446e8i	1.14836i	−0.818730	+	0.574178i	$0.805322\pi$
$558$	0	0
$559$	−1.13529e8	−0.649939
$560$	157459.i	0 0.000896609i
$561$	0	0
$562$	−2.52034e7	−0.141988
$563$	− 9.87875e7i	− 0.553575i	−0.960931	−	0.276788i	$-0.910730\pi$
$563$	− 9.87875e7i	− 0.553575i	0.960931	−	0.276788i	$-0.0892698\pi$
$564$	0	0
$565$	−3.13551e7	−0.173845
$566$	− 1.80356e8i	− 0.994674i
$567$	0	0
$568$	1.13774e8	0.620865
$569$	− 1.64119e8i	− 0.890888i	−0.895310	−	0.445444i	$-0.853046\pi$
$569$	− 1.64119e8i	− 0.890888i	0.895310	−	0.445444i	$-0.146954\pi$
$570$	0	0
$571$	−1.95215e8	−1.04859	−0.524295	−	0.851537i	$-0.675671\pi$
$571$	−1.95215e8	−1.04859	−0.524295	+	0.851537i	$0.675671\pi$
$572$	3.86200e6i	0.0206360i
$573$	0	0
$574$	1.88755e6	0.00998074
$575$	3.54476e7i	0.186459i
$576$	0	0
$577$	1.93319e8	1.00634	0.503172	−	0.864186i	$-0.332166\pi$
$577$	1.93319e8	1.00634	0.503172	+	0.864186i	$0.332166\pi$
$578$	7.17592e7i	0.371616i
$579$	0	0
$580$	−2.77452e7	−0.142202
$581$	− 207881.i	− 0.00105995i
$582$	0	0
$583$	−1.73190e7	−0.0874010
$584$	1.24501e7i	0.0625078i
$585$	0	0
$586$	−2.02047e8	−1.00406
$587$	3.34879e7i	0.165567i	0.996568	+	0.0827835i	$0.0263810\pi$
$587$	3.34879e7i	0.165567i	−0.996568	+	0.0827835i	$0.973619\pi$
$588$	0	0
$589$	2.51811e7	0.123234
$590$	4.08198e6i	0.0198754i
$591$	0	0
$592$	7.08274e6	0.0341379
$593$	− 3.75231e8i	− 1.79943i	−0.436479	−	0.899715i	$-0.643775\pi$
$593$	− 3.75231e8i	− 1.79943i	0.436479	−	0.899715i	$-0.356225\pi$
$594$	0	0
$595$	520369.	0.00247036
$596$	7.66498e7i	0.362053i
$597$	0	0
$598$	1.23180e7	0.0576019
$599$	− 2.96984e8i	− 1.38183i	−0.722938	−	0.690913i	$-0.757209\pi$
$599$	− 2.96984e8i	− 1.38183i	0.722938	−	0.690913i	$-0.242791\pi$
$600$	0	0
$601$	1.55589e8	0.716729	0.358365	−	0.933582i	$-0.383334\pi$
$601$	1.55589e8	0.716729	0.358365	+	0.933582i	$0.383334\pi$
$602$	− 4.31590e6i	− 0.0197825i
$603$	0	0
$604$	9.65360e7	0.438105
$605$	− 4.35392e7i	− 0.196614i
$606$	0	0
$607$	2.22809e8	0.996245	0.498122	−	0.867107i	$-0.334023\pi$
$607$	2.22809e8	0.996245	0.498122	+	0.867107i	$0.334023\pi$
$608$	3.12992e7i	0.139259i
$609$	0	0
$610$	4.50166e7	0.198328
$611$	− 8.85611e7i	− 0.388257i
$612$	0	0
$613$	1.22144e8	0.530262	0.265131	−	0.964212i	$-0.414585\pi$
$613$	1.22144e8	0.530262	0.265131	+	0.964212i	$0.414585\pi$
$614$	− 2.83477e7i	− 0.122465i
$615$	0	0
$616$	−146817.	−0.000628107 0
$617$	3.77153e8i	1.60569i	0.596187	+	0.802846i	$0.296682\pi$
$617$	3.77153e8i	1.60569i	−0.596187	+	0.802846i	$0.703318\pi$
$618$	0	0
$619$	−4.36144e8	−1.83890	−0.919450	−	0.393207i	$-0.871366\pi$
$619$	−4.36144e8	−1.83890	−0.919450	+	0.393207i	$0.871366\pi$
$620$	3.70093e6i	0.0155287i
$621$	0	0
$622$	1.64965e8	0.685520
$623$	− 6.29176e6i	− 0.0260200i
$624$	0	0
$625$	2.15651e8	0.883306
$626$	− 1.73567e8i	− 0.707528i
$627$	0	0
$628$	−1.78704e8	−0.721533
$629$	− 2.34070e7i	− 0.0940577i
$630$	0	0
$631$	−8.46175e7	−0.336800	−0.168400	−	0.985719i	$-0.553860\pi$
$631$	−8.46175e7	−0.336800	−0.168400	+	0.985719i	$0.553860\pi$
$632$	− 8.53966e7i	− 0.338290i
$633$	0	0
$634$	2.87035e8	1.12633
$635$	− 6.63596e7i	− 0.259169i
$636$	0	0
$637$	1.08438e8	0.419529
$638$	− 2.58700e7i	− 0.0996173i
$639$	0	0
$640$	−4.60013e6	−0.0175481
$641$	− 1.14758e8i	− 0.435722i	−0.975980	−	0.217861i	$-0.930092\pi$
$641$	− 1.14758e8i	− 0.435722i	0.975980	−	0.217861i	$-0.0699080\pi$
$642$	0	0
$643$	2.07687e7	0.0781225	0.0390612	−	0.999237i	$-0.487563\pi$
$643$	2.07687e7	0.0781225	0.0390612	+	0.999237i	$0.487563\pi$
$644$	468277.i	0.00175326i
$645$	0	0
$646$	1.03438e8	0.383690
$647$	− 5.18773e8i	− 1.91542i	−0.287734	−	0.957710i	$-0.592902\pi$
$647$	− 5.18773e8i	− 1.91542i	0.287734	−	0.957710i	$-0.407098\pi$
$648$	0	0
$649$	−3.80610e6	−0.0139234
$650$	− 7.82825e7i	− 0.285052i
$651$	0	0
$652$	7.87325e7	0.284061
$653$	− 1.24966e8i	− 0.448800i	−0.974497	−	0.224400i	$-0.927958\pi$
$653$	− 1.24966e8i	− 0.448800i	0.974497	−	0.224400i	$-0.0720423\pi$
$654$	0	0
$655$	−8.90461e7	−0.316877
$656$	5.51444e7i	0.195339i
$657$	0	0
$658$	3.36671e6	0.0118176
$659$	5.05254e8i	1.76544i	0.469897	+	0.882721i	$0.344291\pi$
$659$	5.05254e8i	1.76544i	−0.469897	+	0.882721i	$0.655709\pi$
$660$	0	0
$661$	4.18934e8	1.45058	0.725289	−	0.688445i	$-0.241706\pi$
$661$	4.18934e8	1.45058	0.725289	+	0.688445i	$0.241706\pi$
$662$	− 2.38395e8i	− 0.821718i
$663$	0	0
$664$	6.07319e6	0.0207450
$665$	830856.i	0.00282528i
$666$	0	0
$667$	−8.25134e7	−0.278065
$668$	1.28728e8i	0.431861i
$669$	0	0
$670$	−7.43785e7	−0.247299
$671$	4.19741e7i	0.138936i
$672$	0	0
$673$	−3.98249e8	−1.30650	−0.653250	−	0.757142i	$-0.726595\pi$
$673$	−3.98249e8	−1.30650	−0.653250	+	0.757142i	$0.726595\pi$
$674$	347516.i	0.00113500i
$675$	0	0
$676$	1.27255e8	0.411940
$677$	− 5.46289e8i	− 1.76058i	−0.474433	−	0.880292i	$-0.657347\pi$
$677$	− 5.46289e8i	− 1.76058i	0.474433	−	0.880292i	$-0.342653\pi$
$678$	0	0
$679$	−2.59116e6	−0.00827723
$680$	1.52025e7i	0.0483491i
$681$	0	0
$682$	−3.45080e6	−0.0108784
$683$	− 3.24992e7i	− 0.102002i	−0.998699	−	0.0510012i	$-0.983759\pi$
$683$	− 3.24992e7i	− 0.102002i	0.998699	−	0.0510012i	$-0.0162412\pi$
$684$	0	0
$685$	−3.79829e7	−0.118173
$686$	8.24602e6i	0.0255430i
$687$	0	0
$688$	1.26088e8	0.387176
$689$	− 1.21991e8i	− 0.372967i
$690$	0	0
$691$	−2.87700e8	−0.871979	−0.435989	−	0.899952i	$-0.643601\pi$
$691$	−2.87700e8	−0.871979	−0.435989	+	0.899952i	$0.643601\pi$
$692$	1.32435e8i	0.399653i
$693$	0	0
$694$	−1.37625e8	−0.411736
$695$	− 8.75451e7i	− 0.260782i
$696$	0	0
$697$	1.82241e8	0.538205
$698$	− 1.42569e8i	− 0.419235i
$699$	0	0
$700$	2.97596e6	0.00867628
$701$	− 2.82907e8i	− 0.821278i	−0.911798	−	0.410639i	$-0.865306\pi$
$701$	− 2.82907e8i	− 0.821278i	0.911798	−	0.410639i	$-0.134694\pi$
$702$	0	0
$703$	3.73732e7	0.107571
$704$	− 4.28922e6i	− 0.0122931i
$705$	0	0
$706$	3.90949e8	1.11098
$707$	− 4.16174e6i	− 0.0117765i
$708$	0	0
$709$	−2.39950e8	−0.673258	−0.336629	−	0.941637i	$-0.609287\pi$
$709$	−2.39950e8	−0.673258	−0.336629	+	0.941637i	$0.609287\pi$
$710$	8.82343e7i	0.246526i
$711$	0	0
$712$	1.83812e8	0.509254
$713$	1.10065e7i	0.0303654i
$714$	0	0
$715$	−2.99508e6	−0.00819388
$716$	− 1.48675e8i	− 0.405042i
$717$	0	0
$718$	7.97960e7	0.215580
$719$	2.12836e7i	0.0572611i	0.999590	+	0.0286305i	$0.00911463\pi$
$719$	2.12836e7i	0.0572611i	−0.999590	+	0.0286305i	$0.990885\pi$
$720$	0	0
$721$	−4.34969e6	−0.0116052
$722$	− 1.00976e8i	− 0.268292i
$723$	0	0
$724$	−6.44657e7	−0.169868
$725$	5.24383e8i	1.37605i
$726$	0	0
$727$	−1.77737e8	−0.462567	−0.231283	−	0.972886i	$-0.574292\pi$
$727$	−1.77737e8	−0.462567	−0.231283	+	0.972886i	$0.574292\pi$
$728$	− 1.03414e6i	− 0.00268032i
$729$	0	0
$730$	−9.65535e6	−0.0248199
$731$	− 4.16695e8i	− 1.06676i
$732$	0	0
$733$	−5.22617e8	−1.32700	−0.663501	−	0.748176i	$-0.730930\pi$
$733$	−5.22617e8	−1.32700	−0.663501	+	0.748176i	$0.730930\pi$
$734$	− 4.60720e8i	− 1.16506i
$735$	0	0
$736$	−1.36806e7	−0.0343141
$737$	− 6.93515e7i	− 0.173242i
$738$	0	0
$739$	−5.12661e8	−1.27027	−0.635137	−	0.772400i	$-0.719056\pi$
$739$	−5.12661e8	−1.27027	−0.635137	+	0.772400i	$0.719056\pi$
$740$	5.49283e6i	0.0135551i
$741$	0	0
$742$	4.63757e6	0.0113522
$743$	− 1.09391e8i	− 0.266696i	−0.991069	−	0.133348i	$-0.957427\pi$
$743$	− 1.09391e8i	− 0.266696i	0.991069	−	0.133348i	$-0.0425728\pi$
$744$	0	0
$745$	−5.94437e7	−0.143760
$746$	4.29763e8i	1.03517i
$747$	0	0
$748$	−1.41750e7	−0.0338703
$749$	2.17216e6i	0.00516947i
$750$	0	0
$751$	−3.61516e8	−0.853509	−0.426755	−	0.904367i	$-0.640343\pi$
$751$	−3.61516e8	−0.853509	−0.426755	+	0.904367i	$0.640343\pi$
$752$	9.83578e7i	0.231289i
$753$	0	0
$754$	1.82223e8	0.425098
$755$	7.48659e7i	0.173958i
$756$	0	0
$757$	8.46631e7	0.195167	0.0975835	−	0.995227i	$-0.468889\pi$
$757$	8.46631e7	0.195167	0.0975835	+	0.995227i	$0.468889\pi$
$758$	− 2.53476e8i	− 0.582007i
$759$	0	0
$760$	−2.42733e7	−0.0552953
$761$	9.89661e7i	0.224560i	0.993677	+	0.112280i	$0.0358154\pi$
$761$	9.89661e7i	0.224560i	−0.993677	+	0.112280i	$0.964185\pi$
$762$	0	0
$763$	−4.48830e6	−0.0101043
$764$	1.21794e8i	0.273115i
$765$	0	0
$766$	5.07002e8	1.12804
$767$	− 2.68093e7i	− 0.0594155i
$768$	0	0
$769$	−5.41515e7	−0.119078	−0.0595390	−	0.998226i	$-0.518963\pi$
$769$	−5.41515e7	−0.119078	−0.0595390	+	0.998226i	$0.518963\pi$
$770$	− 113860.i	0 0.000249401i
$771$	0	0
$772$	4.14552e7	0.0901005
$773$	− 9.36821e7i	− 0.202823i	−0.994845	−	0.101412i	$-0.967664\pi$
$773$	− 9.36821e7i	− 0.202823i	0.994845	−	0.101412i	$-0.0323359\pi$
$774$	0	0
$775$	6.99474e7	0.150268
$776$	− 7.57003e7i	− 0.161999i
$777$	0	0
$778$	−1.98377e8	−0.421262
$779$	2.90978e8i	0.615528i
$780$	0	0
$781$	−8.22708e7	−0.172700
$782$	4.52117e7i	0.0945432i
$783$	0	0
$784$	−1.20433e8	−0.249918
$785$	− 1.38589e8i	− 0.286498i
$786$	0	0
$787$	−3.19347e8	−0.655147	−0.327574	−	0.944826i	$-0.606231\pi$
$787$	−3.19347e8	−0.655147	−0.327574	+	0.944826i	$0.606231\pi$
$788$	− 1.91828e8i	− 0.392043i
$789$	0	0
$790$	6.62271e7	0.134324
$791$	7.82862e6i	0.0158181i
$792$	0	0
$793$	−2.95656e8	−0.592881
$794$	4.62542e8i	0.924037i
$795$	0	0
$796$	3.35748e7	0.0665694
$797$	2.49663e8i	0.493149i	0.969124	+	0.246575i	$0.0793051\pi$
$797$	2.49663e8i	0.493149i	−0.969124	+	0.246575i	$0.920695\pi$
$798$	0	0
$799$	3.25053e8	0.637255
$800$	8.69422e7i	0.169809i
$801$	0	0
$802$	−3.45503e8	−0.669775
$803$	− 9.00277e6i	− 0.0173872i
$804$	0	0
$805$	−363160.	−0.000696162 0
$806$	− 2.43067e7i	− 0.0464216i
$807$	0	0
$808$	1.21584e8	0.230486
$809$	3.16268e8i	0.597323i	0.954359	+	0.298661i	$0.0965401\pi$
$809$	3.16268e8i	0.597323i	−0.954359	+	0.298661i	$0.903460\pi$
$810$	0	0
$811$	1.23239e8	0.231038	0.115519	−	0.993305i	$-0.463147\pi$
$811$	1.23239e8	0.231038	0.115519	+	0.993305i	$0.463147\pi$
$812$	6.92732e6i	0.0129389i
$813$	0	0
$814$	−5.12159e6	−0.00949580
$815$	6.10589e7i	0.112791i
$816$	0	0
$817$	6.65323e8	1.22002
$818$	3.78603e8i	0.691710i
$819$	0	0
$820$	−4.27658e7	−0.0775630
$821$	5.58447e8i	1.00914i	0.863370	+	0.504571i	$0.168349\pi$
$821$	5.58447e8i	1.00914i	−0.863370	+	0.504571i	$0.831651\pi$
$822$	0	0
$823$	−4.67148e8	−0.838021	−0.419010	−	0.907981i	$-0.637623\pi$
$823$	−4.67148e8	−0.838021	−0.419010	+	0.907981i	$0.637623\pi$
$824$	− 1.27075e8i	− 0.227133i
$825$	0	0
$826$	1.01917e6	0.00180846
$827$	2.65712e8i	0.469780i	0.972022	+	0.234890i	$0.0754730\pi$
$827$	2.65712e8i	0.469780i	−0.972022	+	0.234890i	$0.924527\pi$
$828$	0	0
$829$	−4.35459e8	−0.764335	−0.382167	−	0.924093i	$-0.624822\pi$
$829$	−4.35459e8	−0.764335	−0.382167	+	0.924093i	$0.624822\pi$
$830$	4.70990e6i	0.00823716i
$831$	0	0
$832$	3.02123e7	0.0524583
$833$	3.98007e8i	0.688583i
$834$	0	0
$835$	−9.98317e7	−0.171478
$836$	− 2.26327e7i	− 0.0387363i
$837$	0	0
$838$	5.50974e8	0.936266
$839$	9.18868e8i	1.55585i	0.628358	+	0.777925i	$0.283727\pi$
$839$	9.18868e8i	1.55585i	−0.628358	+	0.777925i	$0.716273\pi$
$840$	0	0
$841$	−6.25815e8	−1.05210
$842$	4.29551e8i	0.719579i
$843$	0	0
$844$	1.77346e8	0.294982
$845$	9.86891e7i	0.163568i
$846$	0	0
$847$	−1.08707e7	−0.0178899
$848$	1.35486e8i	0.222180i
$849$	0	0
$850$	2.87326e8	0.467863
$851$	1.63355e7i	0.0265060i
$852$	0	0
$853$	6.96242e8	1.12179	0.560897	−	0.827885i	$-0.310456\pi$
$853$	6.96242e8	1.12179	0.560897	+	0.827885i	$0.310456\pi$
$854$	− 1.12396e7i	− 0.0180458i
$855$	0	0
$856$	−6.34591e7	−0.101175
$857$	− 2.58458e8i	− 0.410627i	−0.978696	−	0.205314i	$-0.934179\pi$
$857$	− 2.58458e8i	− 0.410627i	0.978696	−	0.205314i	$-0.0658214\pi$
$858$	0	0
$859$	3.93823e8	0.621329	0.310664	−	0.950520i	$-0.399448\pi$
$859$	3.93823e8	0.621329	0.310664	+	0.950520i	$0.399448\pi$
$860$	9.77843e7i	0.153735i
$861$	0	0
$862$	6.24265e8	0.974647
$863$	− 4.97185e8i	− 0.773545i	−0.922175	−	0.386773i	$-0.873590\pi$
$863$	− 4.97185e8i	− 0.773545i	0.922175	−	0.386773i	$-0.126410\pi$
$864$	0	0
$865$	−1.02706e8	−0.158690
$866$	3.98697e8i	0.613888i
$867$	0	0
$868$	924035.	0.00141296
$869$	6.17510e7i	0.0940990i
$870$	0	0
$871$	4.88497e8	0.739277
$872$	− 1.31125e8i	− 0.197758i
$873$	0	0
$874$	−7.21879e7	−0.108126
$875$	4.71056e6i	0.00703151i
$876$	0	0
$877$	5.74356e7	0.0851495	0.0425748	−	0.999093i	$-0.486444\pi$
$877$	5.74356e7	0.0851495	0.0425748	+	0.999093i	$0.486444\pi$
$878$	− 6.33418e7i	− 0.0935852i
$879$	0	0
$880$	3.32639e6	0.00488119
$881$	2.82313e8i	0.412861i	0.978461	+	0.206430i	$0.0661847\pi$
$881$	2.82313e8i	0.412861i	−0.978461	+	0.206430i	$0.933815\pi$
$882$	0	0
$883$	5.96066e8	0.865789	0.432895	−	0.901445i	$-0.357492\pi$
$883$	5.96066e8	0.865789	0.432895	+	0.901445i	$0.357492\pi$
$884$	− 9.98456e7i	− 0.144535i
$885$	0	0
$886$	−8.75737e8	−1.25914
$887$	8.12349e8i	1.16405i	0.813171	+	0.582025i	$0.197739\pi$
$887$	8.12349e8i	1.16405i	−0.813171	+	0.582025i	$0.802261\pi$
$888$	0	0
$889$	−1.65684e7	−0.0235817
$890$	1.42551e8i	0.202209i
$891$	0	0
$892$	−1.32405e8	−0.186557
$893$	5.19000e8i	0.728808i
$894$	0	0
$895$	1.15301e8	0.160829
$896$	1.14854e6i	0.00159670i
$897$	0	0
$898$	−5.20909e8	−0.719338
$899$	1.62821e8i	0.224094i
$900$	0	0
$901$	4.47753e8	0.612158
$902$	− 3.98754e7i	− 0.0543357i
$903$	0	0
$904$	−2.28711e8	−0.309587
$905$	− 4.99946e7i	− 0.0674493i
$906$	0	0
$907$	9.88678e8	1.32505	0.662526	−	0.749039i	$-0.269484\pi$
$907$	9.88678e8	1.32505	0.662526	+	0.749039i	$0.269484\pi$
$908$	− 4.72336e8i	− 0.630948i
$909$	0	0
$910$	802004.	0.00106427
$911$	5.13556e8i	0.679255i	0.940560	+	0.339628i	$0.110301\pi$
$911$	5.13556e8i	0.679255i	−0.940560	+	0.339628i	$0.889699\pi$
$912$	0	0
$913$	−4.39158e6	−0.00577043
$914$	4.73288e8i	0.619850i
$915$	0	0
$916$	4.68310e8	0.609323
$917$	2.22327e7i	0.0288326i
$918$	0	0
$919$	−5.30833e8	−0.683929	−0.341965	−	0.939713i	$-0.611092\pi$
$919$	−5.30833e8	−0.683929	−0.341965	+	0.939713i	$0.611092\pi$
$920$	− 1.06096e7i	− 0.0136250i
$921$	0	0
$922$	6.73245e8	0.858975
$923$	− 5.79498e8i	− 0.736965i
$924$	0	0
$925$	1.03814e8	0.131169
$926$	− 6.42467e8i	− 0.809129i
$927$	0	0
$928$	−2.02380e8	−0.253235
$929$	9.13708e8i	1.13962i	0.821776	+	0.569810i	$0.192983\pi$
$929$	9.13708e8i	1.13962i	−0.821776	+	0.569810i	$0.807017\pi$
$930$	0	0
$931$	−6.35485e8	−0.787510
$932$	4.85691e7i	0.0599946i
$933$	0	0
$934$	−7.62100e8	−0.935344
$935$	− 1.09931e7i	− 0.0134488i
$936$	0	0
$937$	9.47917e8	1.15226	0.576131	−	0.817357i	$-0.304562\pi$
$937$	9.47917e8	1.15226	0.576131	+	0.817357i	$0.304562\pi$
$938$	1.85705e7i	0.0225017i
$939$	0	0
$940$	−7.62788e7	−0.0918375
$941$	− 9.96546e8i	− 1.19599i	−0.801499	−	0.597996i	$-0.795964\pi$
$941$	− 9.96546e8i	− 1.19599i	0.801499	−	0.597996i	$-0.204036\pi$
$942$	0	0
$943$	−1.27184e8	−0.151669
$944$	2.97750e7i	0.0353945i
$945$	0	0
$946$	−9.11754e7	−0.107697
$947$	− 1.48745e9i	− 1.75143i	−0.482830	−	0.875714i	$-0.660391\pi$
$947$	− 1.48745e9i	− 1.75143i	0.482830	−	0.875714i	$-0.339609\pi$
$948$	0	0
$949$	6.34136e7	0.0741965
$950$	4.58764e8i	0.535080i
$951$	0	0
$952$	3.79570e6	0.00439927
$953$	− 4.53743e8i	− 0.524241i	−0.965035	−	0.262121i	$-0.915578\pi$
$953$	− 4.53743e8i	− 0.524241i	0.965035	−	0.262121i	$-0.0844218\pi$
$954$	0	0
$955$	−9.44541e7	−0.108445
$956$	− 4.46745e8i	− 0.511312i
$957$	0	0
$958$	−9.10614e8	−1.03571
$959$	9.48343e6i	0.0107525i
$960$	0	0
$961$	−8.65785e8	−0.975528
$962$	− 3.60754e7i	− 0.0405215i
$963$	0	0
$964$	−8.10125e8	−0.904317
$965$	3.21495e7i	0.0357760i
$966$	0	0
$967$	1.00928e9	1.11617	0.558086	−	0.829783i	$-0.311536\pi$
$967$	1.00928e9	1.11617	0.558086	+	0.829783i	$0.311536\pi$
$968$	− 3.17585e8i	− 0.350134i
$969$	0	0
$970$	5.87073e7	0.0643246
$971$	6.88293e8i	0.751823i	0.926656	+	0.375912i	$0.122670\pi$
$971$	6.88293e8i	0.751823i	−0.926656	+	0.375912i	$0.877330\pi$
$972$	0	0
$973$	−2.18579e7	−0.0237285
$974$	6.50337e7i	0.0703820i
$975$	0	0
$976$	3.28362e8	0.353186
$977$	− 7.69746e8i	− 0.825399i	−0.910867	−	0.412699i	$-0.864586\pi$
$977$	− 7.69746e8i	− 0.825399i	0.910867	−	0.412699i	$-0.135414\pi$
$978$	0	0
$979$	−1.32916e8	−0.141654
$980$	− 9.33988e7i	− 0.0992346i
$981$	0	0
$982$	7.88003e8	0.832135
$983$	− 8.65228e8i	− 0.910898i	−0.890262	−	0.455449i	$-0.849479\pi$
$983$	− 8.65228e8i	− 0.910898i	0.890262	−	0.455449i	$-0.150521\pi$
$984$	0	0
$985$	1.48767e8	0.155668
$986$	6.68826e8i	0.697722i
$987$	0	0
$988$	1.59420e8	0.165300
$989$	2.90807e8i	0.300619i
$990$	0	0
$991$	1.48536e9	1.52620	0.763101	−	0.646280i	$-0.223676\pi$
$991$	1.48536e9	1.52620	0.763101	+	0.646280i	$0.223676\pi$
$992$	2.69955e7i	0.0276539i
$993$	0	0
$994$	2.20300e7	0.0224313
$995$	2.60381e7i	0.0264326i
$996$	0	0
$997$	−5.88180e8	−0.593505	−0.296753	−	0.954954i	$-0.595904\pi$
$997$	−5.88180e8	−0.593505	−0.296753	+	0.954954i	$0.595904\pi$
$998$	− 5.28744e8i	− 0.531929i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.7.b.a.161.4	yes	4
3.2	odd	2	inner	162.7.b.a.161.1	✓	4
9.2	odd	6		162.7.d.f.53.2		8
9.4	even	3		162.7.d.f.107.2		8
9.5	odd	6		162.7.d.f.107.3		8
9.7	even	3		162.7.d.f.53.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.7.b.a.161.1	✓	4	3.2	odd	2	inner
162.7.b.a.161.4	yes	4	1.1	even	1	trivial
162.7.d.f.53.2		8	9.2	odd	6
162.7.d.f.53.3		8	9.7	even	3
162.7.d.f.107.2		8	9.4	even	3
162.7.d.f.107.3		8	9.5	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+5.65685i q^{2} -32.0000 q^{4} -24.8168i q^{5} -6.19615 q^{7} -181.019i q^{8} +O(q^{10})\)	\(q+5.65685i q^{2} -32.0000 q^{4} -24.8168i q^{5} -6.19615 q^{7} -181.019i q^{8} +140.385 q^{10} +130.897i q^{11} -922.007 q^{13} -35.0507i q^{14} +1024.00 q^{16} -3384.11i q^{17} +5403.30 q^{19} +794.136i q^{20} -740.463 q^{22} +2361.73i q^{23} +15009.1 q^{25} -5215.66i q^{26} +198.277 q^{28} +34937.6i q^{29} +4660.33 q^{31} +5792.62i q^{32} +19143.4 q^{34} +153.768i q^{35} +6916.74 q^{37} +30565.7i q^{38} -4492.31 q^{40} +53851.9i q^{41} +123133. q^{43} -4188.69i q^{44} -13360.0 q^{46} +96052.5i q^{47} -117611. q^{49} +84904.5i q^{50} +29504.2 q^{52} +132310. i q^{53} +3248.43 q^{55} +1121.62i q^{56} -197637. q^{58} +29077.1i q^{59} +320666. q^{61} +26362.8i q^{62} -32768.0 q^{64} +22881.2i q^{65} -529819. q^{67} +108292. i q^{68} -869.845 q^{70} +628518. i q^{71} -68777.7 q^{73} +39127.0i q^{74} -172905. q^{76} -811.056i q^{77} +471754. q^{79} -25412.4i q^{80} -304633. q^{82} +33550.0i q^{83} -83982.7 q^{85} +696545. i q^{86} +23694.8 q^{88} +1.01543e6i q^{89} +5712.90 q^{91} -75575.5i q^{92} -543355. q^{94} -134092. i q^{95} +418189. q^{97} -665306. i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 128 q^{4} - 4 q^{7}+O(q^{10}) \) 4 * q - 128 * q^4 - 4 * q^7	\( 4 q - 128 q^{4} - 4 q^{7} + 2640 q^{10} + 3836 q^{13} + 4096 q^{16} + 10244 q^{19} - 27072 q^{22} - 25700 q^{25} + 128 q^{28} - 40096 q^{31} + 60528 q^{34} - 20200 q^{37} - 84480 q^{40} + 184940 q^{43} + 162720 q^{46} - 470484 q^{49} - 122752 q^{52} + 949860 q^{55} + 24624 q^{58} + 609056 q^{61} - 131072 q^{64} - 2008972 q^{67} + 8160 q^{70} + 525824 q^{73} - 327808 q^{76} - 848716 q^{79} + 705792 q^{82} - 987840 q^{85} + 866304 q^{88} + 35260 q^{91} - 1965408 q^{94} + 3621728 q^{97}+O(q^{100}) \) 4 * q - 128 * q^4 - 4 * q^7 + 2640 * q^10 + 3836 * q^13 + 4096 * q^16 + 10244 * q^19 - 27072 * q^22 - 25700 * q^25 + 128 * q^28 - 40096 * q^31 + 60528 * q^34 - 20200 * q^37 - 84480 * q^40 + 184940 * q^43 + 162720 * q^46 - 470484 * q^49 - 122752 * q^52 + 949860 * q^55 + 24624 * q^58 + 609056 * q^61 - 131072 * q^64 - 2008972 * q^67 + 8160 * q^70 + 525824 * q^73 - 327808 * q^76 - 848716 * q^79 + 705792 * q^82 - 987840 * q^85 + 866304 * q^88 + 35260 * q^91 - 1965408 * q^94 + 3621728 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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