LMFDB - Embedded newform 6936.2.a.y.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6936,2,Mod(1,6936)] 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("6936.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6936, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6936 = 2^{3} \cdot 3 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6936.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,2,0,1,0,2,0,2,0,9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$55.3842388420$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 408)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	6936.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +2.56155 q^{5} -3.12311 q^{7} +1.00000 q^{9} +6.56155 q^{11} +0.561553 q^{13} +2.56155 q^{15} -7.68466 q^{19} -3.12311 q^{21} +3.43845 q^{23} +1.56155 q^{25} +1.00000 q^{27} -1.12311 q^{29} +8.24621 q^{31} +6.56155 q^{33} -8.00000 q^{35} +4.00000 q^{37} +0.561553 q^{39} -9.68466 q^{41} +7.68466 q^{43} +2.56155 q^{45} -9.12311 q^{47} +2.75379 q^{49} +6.00000 q^{53} +16.8078 q^{55} -7.68466 q^{57} +11.3693 q^{59} +4.00000 q^{61} -3.12311 q^{63} +1.43845 q^{65} +12.0000 q^{67} +3.43845 q^{69} +13.3693 q^{71} +8.24621 q^{73} +1.56155 q^{75} -20.4924 q^{77} -2.00000 q^{79} +1.00000 q^{81} -1.12311 q^{83} -1.12311 q^{87} +0.876894 q^{89} -1.75379 q^{91} +8.24621 q^{93} -19.6847 q^{95} +7.12311 q^{97} +6.56155 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + q^{5} + 2 q^{7} + 2 q^{9} + 9 q^{11} - 3 q^{13} + q^{15} - 3 q^{19} + 2 q^{21} + 11 q^{23} - q^{25} + 2 q^{27} + 6 q^{29} + 9 q^{33} - 16 q^{35} + 8 q^{37} - 3 q^{39} - 7 q^{41} + 3 q^{43}+ \cdots + 9 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + q^5 + 2 * q^7 + 2 * q^9 + 9 * q^11 - 3 * q^13 + q^15 - 3 * q^19 + 2 * q^21 + 11 * q^23 - q^25 + 2 * q^27 + 6 * q^29 + 9 * q^33 - 16 * q^35 + 8 * q^37 - 3 * q^39 - 7 * q^41 + 3 * q^43 + q^45 - 10 * q^47 + 22 * q^49 + 12 * q^53 + 13 * q^55 - 3 * q^57 - 2 * q^59 + 8 * q^61 + 2 * q^63 + 7 * q^65 + 24 * q^67 + 11 * q^69 + 2 * q^71 - q^75 - 8 * q^77 - 4 * q^79 + 2 * q^81 + 6 * q^83 + 6 * q^87 + 10 * q^89 - 20 * q^91 - 27 * q^95 + 6 * q^97 + 9 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	2.56155	1.14556	0.572781	−	0.819709i	$-0.305865\pi$
$5$	2.56155	1.14556	0.572781	+	0.819709i	$0.305865\pi$
$6$	0	0
$7$	−3.12311	−1.18042	−0.590211	−	0.807249i	$-0.700956\pi$
$7$	−3.12311	−1.18042	−0.590211	+	0.807249i	$0.700956\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	6.56155	1.97838	0.989191	−	0.146631i	$-0.0468429\pi$
$11$	6.56155	1.97838	0.989191	+	0.146631i	$0.0468429\pi$
$12$	0	0
$13$	0.561553	0.155747	0.0778734	−	0.996963i	$-0.475187\pi$
$13$	0.561553	0.155747	0.0778734	+	0.996963i	$0.475187\pi$
$14$	0	0
$15$	2.56155	0.661390
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−7.68466	−1.76298	−0.881491	−	0.472201i	$-0.843460\pi$
$19$	−7.68466	−1.76298	−0.881491	+	0.472201i	$0.843460\pi$
$20$	0	0
$21$	−3.12311	−0.681518
$22$	0	0
$23$	3.43845	0.716966	0.358483	−	0.933536i	$-0.383294\pi$
$23$	3.43845	0.716966	0.358483	+	0.933536i	$0.383294\pi$
$24$	0	0
$25$	1.56155	0.312311
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−1.12311	−0.208555	−0.104278	−	0.994548i	$-0.533253\pi$
$29$	−1.12311	−0.208555	−0.104278	+	0.994548i	$0.533253\pi$
$30$	0	0
$31$	8.24621	1.48106	0.740532	−	0.672022i	$-0.234574\pi$
$31$	8.24621	1.48106	0.740532	+	0.672022i	$0.234574\pi$
$32$	0	0
$33$	6.56155	1.14222
$34$	0	0
$35$	−8.00000	−1.35225
$36$	0	0
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	0	0
$39$	0.561553	0.0899204
$40$	0	0
$41$	−9.68466	−1.51249	−0.756245	−	0.654289i	$-0.772968\pi$
$41$	−9.68466	−1.51249	−0.756245	+	0.654289i	$0.772968\pi$
$42$	0	0
$43$	7.68466	1.17190	0.585950	−	0.810347i	$-0.300722\pi$
$43$	7.68466	1.17190	0.585950	+	0.810347i	$0.300722\pi$
$44$	0	0
$45$	2.56155	0.381854
$46$	0	0
$47$	−9.12311	−1.33074	−0.665371	−	0.746513i	$-0.731727\pi$
$47$	−9.12311	−1.33074	−0.665371	+	0.746513i	$0.731727\pi$
$48$	0	0
$49$	2.75379	0.393398
$50$	0	0
$51$	0	0
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	16.8078	2.26636
$56$	0	0
$57$	−7.68466	−1.01786
$58$	0	0
$59$	11.3693	1.48016	0.740079	−	0.672519i	$-0.234788\pi$
$59$	11.3693	1.48016	0.740079	+	0.672519i	$0.234788\pi$
$60$	0	0
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	0	0
$63$	−3.12311	−0.393474
$64$	0	0
$65$	1.43845	0.178417
$66$	0	0
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	0	0
$69$	3.43845	0.413940
$70$	0	0
$71$	13.3693	1.58665	0.793323	−	0.608801i	$-0.208349\pi$
$71$	13.3693	1.58665	0.793323	+	0.608801i	$0.208349\pi$
$72$	0	0
$73$	8.24621	0.965146	0.482573	−	0.875856i	$-0.339702\pi$
$73$	8.24621	0.965146	0.482573	+	0.875856i	$0.339702\pi$
$74$	0	0
$75$	1.56155	0.180313
$76$	0	0
$77$	−20.4924	−2.33533
$78$	0	0
$79$	−2.00000	−0.225018	−0.112509	−	0.993651i	$-0.535889\pi$
$79$	−2.00000	−0.225018	−0.112509	+	0.993651i	$0.535889\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−1.12311	−0.123277	−0.0616384	−	0.998099i	$-0.519633\pi$
$83$	−1.12311	−0.123277	−0.0616384	+	0.998099i	$0.519633\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−1.12311	−0.120410
$88$	0	0
$89$	0.876894	0.0929506	0.0464753	−	0.998919i	$-0.485201\pi$
$89$	0.876894	0.0929506	0.0464753	+	0.998919i	$0.485201\pi$
$90$	0	0
$91$	−1.75379	−0.183847
$92$	0	0
$93$	8.24621	0.855092
$94$	0	0
$95$	−19.6847	−2.01960
$96$	0	0
$97$	7.12311	0.723242	0.361621	−	0.932325i	$-0.382223\pi$
$97$	7.12311	0.723242	0.361621	+	0.932325i	$0.382223\pi$
$98$	0	0
$99$	6.56155	0.659461
$100$	0	0
$101$	3.12311	0.310761	0.155380	−	0.987855i	$-0.450340\pi$
$101$	3.12311	0.310761	0.155380	+	0.987855i	$0.450340\pi$
$102$	0	0
$103$	−0.315342	−0.0310715	−0.0155358	−	0.999879i	$-0.504945\pi$
$103$	−0.315342	−0.0310715	−0.0155358	+	0.999879i	$0.504945\pi$
$104$	0	0
$105$	−8.00000	−0.780720
$106$	0	0
$107$	−3.68466	−0.356209	−0.178105	−	0.984012i	$-0.556997\pi$
$107$	−3.68466	−0.356209	−0.178105	+	0.984012i	$0.556997\pi$
$108$	0	0
$109$	−10.2462	−0.981409	−0.490705	−	0.871326i	$-0.663261\pi$
$109$	−10.2462	−0.981409	−0.490705	+	0.871326i	$0.663261\pi$
$110$	0	0
$111$	4.00000	0.379663
$112$	0	0
$113$	−8.56155	−0.805403	−0.402702	−	0.915331i	$-0.631929\pi$
$113$	−8.56155	−0.805403	−0.402702	+	0.915331i	$0.631929\pi$
$114$	0	0
$115$	8.80776	0.821328
$116$	0	0
$117$	0.561553	0.0519156
$118$	0	0
$119$	0	0
$120$	0	0
$121$	32.0540	2.91400
$122$	0	0
$123$	−9.68466	−0.873236
$124$	0	0
$125$	−8.80776	−0.787790
$126$	0	0
$127$	−13.9309	−1.23616	−0.618082	−	0.786113i	$-0.712090\pi$
$127$	−13.9309	−1.23616	−0.618082	+	0.786113i	$0.712090\pi$
$128$	0	0
$129$	7.68466	0.676596
$130$	0	0
$131$	9.43845	0.824641	0.412320	−	0.911039i	$-0.364718\pi$
$131$	9.43845	0.824641	0.412320	+	0.911039i	$0.364718\pi$
$132$	0	0
$133$	24.0000	2.08106
$134$	0	0
$135$	2.56155	0.220463
$136$	0	0
$137$	−16.2462	−1.38801	−0.694004	−	0.719971i	$-0.744155\pi$
$137$	−16.2462	−1.38801	−0.694004	+	0.719971i	$0.744155\pi$
$138$	0	0
$139$	−5.12311	−0.434536	−0.217268	−	0.976112i	$-0.569715\pi$
$139$	−5.12311	−0.434536	−0.217268	+	0.976112i	$0.569715\pi$
$140$	0	0
$141$	−9.12311	−0.768304
$142$	0	0
$143$	3.68466	0.308127
$144$	0	0
$145$	−2.87689	−0.238913
$146$	0	0
$147$	2.75379	0.227129
$148$	0	0
$149$	−14.0000	−1.14692	−0.573462	−	0.819232i	$-0.694400\pi$
$149$	−14.0000	−1.14692	−0.573462	+	0.819232i	$0.694400\pi$
$150$	0	0
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	21.1231	1.69665
$156$	0	0
$157$	−3.43845	−0.274418	−0.137209	−	0.990542i	$-0.543813\pi$
$157$	−3.43845	−0.274418	−0.137209	+	0.990542i	$0.543813\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	0	0
$161$	−10.7386	−0.846323
$162$	0	0
$163$	11.3693	0.890514	0.445257	−	0.895403i	$-0.353112\pi$
$163$	11.3693	0.890514	0.445257	+	0.895403i	$0.353112\pi$
$164$	0	0
$165$	16.8078	1.30848
$166$	0	0
$167$	1.19224	0.0922580	0.0461290	−	0.998935i	$-0.485311\pi$
$167$	1.19224	0.0922580	0.0461290	+	0.998935i	$0.485311\pi$
$168$	0	0
$169$	−12.6847	−0.975743
$170$	0	0
$171$	−7.68466	−0.587661
$172$	0	0
$173$	−7.68466	−0.584254	−0.292127	−	0.956380i	$-0.594363\pi$
$173$	−7.68466	−0.584254	−0.292127	+	0.956380i	$0.594363\pi$
$174$	0	0
$175$	−4.87689	−0.368659
$176$	0	0
$177$	11.3693	0.854570
$178$	0	0
$179$	−6.87689	−0.514003	−0.257002	−	0.966411i	$-0.582735\pi$
$179$	−6.87689	−0.514003	−0.257002	+	0.966411i	$0.582735\pi$
$180$	0	0
$181$	1.12311	0.0834798	0.0417399	−	0.999129i	$-0.486710\pi$
$181$	1.12311	0.0834798	0.0417399	+	0.999129i	$0.486710\pi$
$182$	0	0
$183$	4.00000	0.295689
$184$	0	0
$185$	10.2462	0.753316
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−3.12311	−0.227173
$190$	0	0
$191$	19.3693	1.40151	0.700757	−	0.713400i	$-0.252846\pi$
$191$	19.3693	1.40151	0.700757	+	0.713400i	$0.252846\pi$
$192$	0	0
$193$	4.24621	0.305649	0.152824	−	0.988253i	$-0.451163\pi$
$193$	4.24621	0.305649	0.152824	+	0.988253i	$0.451163\pi$
$194$	0	0
$195$	1.43845	0.103009
$196$	0	0
$197$	12.3153	0.877432	0.438716	−	0.898626i	$-0.355433\pi$
$197$	12.3153	0.877432	0.438716	+	0.898626i	$0.355433\pi$
$198$	0	0
$199$	23.1231	1.63915	0.819577	−	0.572969i	$-0.194209\pi$
$199$	23.1231	1.63915	0.819577	+	0.572969i	$0.194209\pi$
$200$	0	0
$201$	12.0000	0.846415
$202$	0	0
$203$	3.50758	0.246184
$204$	0	0
$205$	−24.8078	−1.73265
$206$	0	0
$207$	3.43845	0.238989
$208$	0	0
$209$	−50.4233	−3.48785
$210$	0	0
$211$	−10.8769	−0.748796	−0.374398	−	0.927268i	$-0.622151\pi$
$211$	−10.8769	−0.748796	−0.374398	+	0.927268i	$0.622151\pi$
$212$	0	0
$213$	13.3693	0.916050
$214$	0	0
$215$	19.6847	1.34248
$216$	0	0
$217$	−25.7538	−1.74828
$218$	0	0
$219$	8.24621	0.557227
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−21.4384	−1.43562	−0.717812	−	0.696237i	$-0.754856\pi$
$223$	−21.4384	−1.43562	−0.717812	+	0.696237i	$0.754856\pi$
$224$	0	0
$225$	1.56155	0.104104
$226$	0	0
$227$	15.6847	1.04103	0.520514	−	0.853853i	$-0.325740\pi$
$227$	15.6847	1.04103	0.520514	+	0.853853i	$0.325740\pi$
$228$	0	0
$229$	−18.0000	−1.18947	−0.594737	−	0.803921i	$-0.702744\pi$
$229$	−18.0000	−1.18947	−0.594737	+	0.803921i	$0.702744\pi$
$230$	0	0
$231$	−20.4924	−1.34830
$232$	0	0
$233$	−3.43845	−0.225260	−0.112630	−	0.993637i	$-0.535928\pi$
$233$	−3.43845	−0.225260	−0.112630	+	0.993637i	$0.535928\pi$
$234$	0	0
$235$	−23.3693	−1.52445
$236$	0	0
$237$	−2.00000	−0.129914
$238$	0	0
$239$	22.2462	1.43899	0.719494	−	0.694499i	$-0.244374\pi$
$239$	22.2462	1.43899	0.719494	+	0.694499i	$0.244374\pi$
$240$	0	0
$241$	−5.36932	−0.345868	−0.172934	−	0.984933i	$-0.555325\pi$
$241$	−5.36932	−0.345868	−0.172934	+	0.984933i	$0.555325\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	7.05398	0.450662
$246$	0	0
$247$	−4.31534	−0.274579
$248$	0	0
$249$	−1.12311	−0.0711739
$250$	0	0
$251$	−6.24621	−0.394257	−0.197129	−	0.980378i	$-0.563162\pi$
$251$	−6.24621	−0.394257	−0.197129	+	0.980378i	$0.563162\pi$
$252$	0	0
$253$	22.5616	1.41843
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4.87689	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$257$	4.87689	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$258$	0	0
$259$	−12.4924	−0.776241
$260$	0	0
$261$	−1.12311	−0.0695185
$262$	0	0
$263$	4.00000	0.246651	0.123325	−	0.992366i	$-0.460644\pi$
$263$	4.00000	0.246651	0.123325	+	0.992366i	$0.460644\pi$
$264$	0	0
$265$	15.3693	0.944130
$266$	0	0
$267$	0.876894	0.0536651
$268$	0	0
$269$	23.6847	1.44408	0.722040	−	0.691852i	$-0.243205\pi$
$269$	23.6847	1.44408	0.722040	+	0.691852i	$0.243205\pi$
$270$	0	0
$271$	19.0540	1.15745	0.578723	−	0.815524i	$-0.303551\pi$
$271$	19.0540	1.15745	0.578723	+	0.815524i	$0.303551\pi$
$272$	0	0
$273$	−1.75379	−0.106144
$274$	0	0
$275$	10.2462	0.617870
$276$	0	0
$277$	21.1231	1.26916	0.634582	−	0.772855i	$-0.281172\pi$
$277$	21.1231	1.26916	0.634582	+	0.772855i	$0.281172\pi$
$278$	0	0
$279$	8.24621	0.493688
$280$	0	0
$281$	11.1231	0.663549	0.331774	−	0.943359i	$-0.392353\pi$
$281$	11.1231	0.663549	0.331774	+	0.943359i	$0.392353\pi$
$282$	0	0
$283$	−13.1231	−0.780088	−0.390044	−	0.920796i	$-0.627540\pi$
$283$	−13.1231	−0.780088	−0.390044	+	0.920796i	$0.627540\pi$
$284$	0	0
$285$	−19.6847	−1.16602
$286$	0	0
$287$	30.2462	1.78538
$288$	0	0
$289$	0	0
$290$	0	0
$291$	7.12311	0.417564
$292$	0	0
$293$	−3.12311	−0.182454	−0.0912269	−	0.995830i	$-0.529079\pi$
$293$	−3.12311	−0.182454	−0.0912269	+	0.995830i	$0.529079\pi$
$294$	0	0
$295$	29.1231	1.69561
$296$	0	0
$297$	6.56155	0.380740
$298$	0	0
$299$	1.93087	0.111665
$300$	0	0
$301$	−24.0000	−1.38334
$302$	0	0
$303$	3.12311	0.179418
$304$	0	0
$305$	10.2462	0.586696
$306$	0	0
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	0	0
$309$	−0.315342	−0.0179392
$310$	0	0
$311$	−6.63068	−0.375992	−0.187996	−	0.982170i	$-0.560199\pi$
$311$	−6.63068	−0.375992	−0.187996	+	0.982170i	$0.560199\pi$
$312$	0	0
$313$	11.6155	0.656548	0.328274	−	0.944582i	$-0.393533\pi$
$313$	11.6155	0.656548	0.328274	+	0.944582i	$0.393533\pi$
$314$	0	0
$315$	−8.00000	−0.450749
$316$	0	0
$317$	10.8769	0.610907	0.305454	−	0.952207i	$-0.401192\pi$
$317$	10.8769	0.610907	0.305454	+	0.952207i	$0.401192\pi$
$318$	0	0
$319$	−7.36932	−0.412603
$320$	0	0
$321$	−3.68466	−0.205658
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0.876894	0.0486413
$326$	0	0
$327$	−10.2462	−0.566617
$328$	0	0
$329$	28.4924	1.57084
$330$	0	0
$331$	2.56155	0.140796	0.0703978	−	0.997519i	$-0.477573\pi$
$331$	2.56155	0.140796	0.0703978	+	0.997519i	$0.477573\pi$
$332$	0	0
$333$	4.00000	0.219199
$334$	0	0
$335$	30.7386	1.67943
$336$	0	0
$337$	−30.4924	−1.66103	−0.830514	−	0.556998i	$-0.811953\pi$
$337$	−30.4924	−1.66103	−0.830514	+	0.556998i	$0.811953\pi$
$338$	0	0
$339$	−8.56155	−0.465000
$340$	0	0
$341$	54.1080	2.93011
$342$	0	0
$343$	13.2614	0.716046
$344$	0	0
$345$	8.80776	0.474194
$346$	0	0
$347$	−16.4924	−0.885360	−0.442680	−	0.896680i	$-0.645972\pi$
$347$	−16.4924	−0.885360	−0.442680	+	0.896680i	$0.645972\pi$
$348$	0	0
$349$	11.4384	0.612286	0.306143	−	0.951986i	$-0.400961\pi$
$349$	11.4384	0.612286	0.306143	+	0.951986i	$0.400961\pi$
$350$	0	0
$351$	0.561553	0.0299735
$352$	0	0
$353$	−2.00000	−0.106449	−0.0532246	−	0.998583i	$-0.516950\pi$
$353$	−2.00000	−0.106449	−0.0532246	+	0.998583i	$0.516950\pi$
$354$	0	0
$355$	34.2462	1.81760
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4.49242	0.237101	0.118550	−	0.992948i	$-0.462175\pi$
$359$	4.49242	0.237101	0.118550	+	0.992948i	$0.462175\pi$
$360$	0	0
$361$	40.0540	2.10810
$362$	0	0
$363$	32.0540	1.68240
$364$	0	0
$365$	21.1231	1.10563
$366$	0	0
$367$	11.1231	0.580621	0.290311	−	0.956932i	$-0.406241\pi$
$367$	11.1231	0.580621	0.290311	+	0.956932i	$0.406241\pi$
$368$	0	0
$369$	−9.68466	−0.504163
$370$	0	0
$371$	−18.7386	−0.972861
$372$	0	0
$373$	−2.49242	−0.129053	−0.0645264	−	0.997916i	$-0.520554\pi$
$373$	−2.49242	−0.129053	−0.0645264	+	0.997916i	$0.520554\pi$
$374$	0	0
$375$	−8.80776	−0.454831
$376$	0	0
$377$	−0.630683	−0.0324818
$378$	0	0
$379$	−32.4924	−1.66902	−0.834512	−	0.550990i	$-0.814250\pi$
$379$	−32.4924	−1.66902	−0.834512	+	0.550990i	$0.814250\pi$
$380$	0	0
$381$	−13.9309	−0.713700
$382$	0	0
$383$	1.75379	0.0896144	0.0448072	−	0.998996i	$-0.485733\pi$
$383$	1.75379	0.0896144	0.0448072	+	0.998996i	$0.485733\pi$
$384$	0	0
$385$	−52.4924	−2.67526
$386$	0	0
$387$	7.68466	0.390633
$388$	0	0
$389$	2.63068	0.133381	0.0666905	−	0.997774i	$-0.478756\pi$
$389$	2.63068	0.133381	0.0666905	+	0.997774i	$0.478756\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	9.43845	0.476107
$394$	0	0
$395$	−5.12311	−0.257771
$396$	0	0
$397$	6.24621	0.313488	0.156744	−	0.987639i	$-0.449900\pi$
$397$	6.24621	0.313488	0.156744	+	0.987639i	$0.449900\pi$
$398$	0	0
$399$	24.0000	1.20150
$400$	0	0
$401$	4.06913	0.203203	0.101601	−	0.994825i	$-0.467603\pi$
$401$	4.06913	0.203203	0.101601	+	0.994825i	$0.467603\pi$
$402$	0	0
$403$	4.63068	0.230671
$404$	0	0
$405$	2.56155	0.127285
$406$	0	0
$407$	26.2462	1.30098
$408$	0	0
$409$	−16.5616	−0.818916	−0.409458	−	0.912329i	$-0.634282\pi$
$409$	−16.5616	−0.818916	−0.409458	+	0.912329i	$0.634282\pi$
$410$	0	0
$411$	−16.2462	−0.801367
$412$	0	0
$413$	−35.5076	−1.74721
$414$	0	0
$415$	−2.87689	−0.141221
$416$	0	0
$417$	−5.12311	−0.250880
$418$	0	0
$419$	28.0000	1.36789	0.683945	−	0.729534i	$-0.260263\pi$
$419$	28.0000	1.36789	0.683945	+	0.729534i	$0.260263\pi$
$420$	0	0
$421$	−32.4233	−1.58021	−0.790107	−	0.612969i	$-0.789975\pi$
$421$	−32.4233	−1.58021	−0.790107	+	0.612969i	$0.789975\pi$
$422$	0	0
$423$	−9.12311	−0.443581
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−12.4924	−0.604551
$428$	0	0
$429$	3.68466	0.177897
$430$	0	0
$431$	−15.6155	−0.752174	−0.376087	−	0.926584i	$-0.622731\pi$
$431$	−15.6155	−0.752174	−0.376087	+	0.926584i	$0.622731\pi$
$432$	0	0
$433$	25.0540	1.20402	0.602009	−	0.798490i	$-0.294367\pi$
$433$	25.0540	1.20402	0.602009	+	0.798490i	$0.294367\pi$
$434$	0	0
$435$	−2.87689	−0.137937
$436$	0	0
$437$	−26.4233	−1.26400
$438$	0	0
$439$	35.6155	1.69984	0.849918	−	0.526915i	$-0.176651\pi$
$439$	35.6155	1.69984	0.849918	+	0.526915i	$0.176651\pi$
$440$	0	0
$441$	2.75379	0.131133
$442$	0	0
$443$	−19.3693	−0.920264	−0.460132	−	0.887851i	$-0.652198\pi$
$443$	−19.3693	−0.920264	−0.460132	+	0.887851i	$0.652198\pi$
$444$	0	0
$445$	2.24621	0.106481
$446$	0	0
$447$	−14.0000	−0.662177
$448$	0	0
$449$	−16.2462	−0.766706	−0.383353	−	0.923602i	$-0.625231\pi$
$449$	−16.2462	−0.766706	−0.383353	+	0.923602i	$0.625231\pi$
$450$	0	0
$451$	−63.5464	−2.99228
$452$	0	0
$453$	16.0000	0.751746
$454$	0	0
$455$	−4.49242	−0.210608
$456$	0	0
$457$	19.9309	0.932327	0.466163	−	0.884699i	$-0.345636\pi$
$457$	19.9309	0.932327	0.466163	+	0.884699i	$0.345636\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−26.9848	−1.25681	−0.628405	−	0.777887i	$-0.716292\pi$
$461$	−26.9848	−1.25681	−0.628405	+	0.777887i	$0.716292\pi$
$462$	0	0
$463$	9.75379	0.453297	0.226649	−	0.973977i	$-0.427223\pi$
$463$	9.75379	0.453297	0.226649	+	0.973977i	$0.427223\pi$
$464$	0	0
$465$	21.1231	0.979561
$466$	0	0
$467$	−9.12311	−0.422167	−0.211083	−	0.977468i	$-0.567699\pi$
$467$	−9.12311	−0.422167	−0.211083	+	0.977468i	$0.567699\pi$
$468$	0	0
$469$	−37.4773	−1.73054
$470$	0	0
$471$	−3.43845	−0.158435
$472$	0	0
$473$	50.4233	2.31847
$474$	0	0
$475$	−12.0000	−0.550598
$476$	0	0
$477$	6.00000	0.274721
$478$	0	0
$479$	−21.0540	−0.961981	−0.480990	−	0.876726i	$-0.659723\pi$
$479$	−21.0540	−0.961981	−0.480990	+	0.876726i	$0.659723\pi$
$480$	0	0
$481$	2.24621	0.102418
$482$	0	0
$483$	−10.7386	−0.488625
$484$	0	0
$485$	18.2462	0.828518
$486$	0	0
$487$	28.7386	1.30227	0.651136	−	0.758961i	$-0.274293\pi$
$487$	28.7386	1.30227	0.651136	+	0.758961i	$0.274293\pi$
$488$	0	0
$489$	11.3693	0.514138
$490$	0	0
$491$	−13.6155	−0.614460	−0.307230	−	0.951635i	$-0.599402\pi$
$491$	−13.6155	−0.614460	−0.307230	+	0.951635i	$0.599402\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	16.8078	0.755453
$496$	0	0
$497$	−41.7538	−1.87291
$498$	0	0
$499$	−38.1080	−1.70595	−0.852973	−	0.521955i	$-0.825203\pi$
$499$	−38.1080	−1.70595	−0.852973	+	0.521955i	$0.825203\pi$
$500$	0	0
$501$	1.19224	0.0532652
$502$	0	0
$503$	−9.68466	−0.431818	−0.215909	−	0.976414i	$-0.569271\pi$
$503$	−9.68466	−0.431818	−0.215909	+	0.976414i	$0.569271\pi$
$504$	0	0
$505$	8.00000	0.355995
$506$	0	0
$507$	−12.6847	−0.563345
$508$	0	0
$509$	−37.3693	−1.65637	−0.828183	−	0.560458i	$-0.810625\pi$
$509$	−37.3693	−1.65637	−0.828183	+	0.560458i	$0.810625\pi$
$510$	0	0
$511$	−25.7538	−1.13928
$512$	0	0
$513$	−7.68466	−0.339286
$514$	0	0
$515$	−0.807764	−0.0355943
$516$	0	0
$517$	−59.8617	−2.63272
$518$	0	0
$519$	−7.68466	−0.337319
$520$	0	0
$521$	15.3002	0.670313	0.335157	−	0.942162i	$-0.391211\pi$
$521$	15.3002	0.670313	0.335157	+	0.942162i	$0.391211\pi$
$522$	0	0
$523$	24.4924	1.07098	0.535489	−	0.844542i	$-0.320127\pi$
$523$	24.4924	1.07098	0.535489	+	0.844542i	$0.320127\pi$
$524$	0	0
$525$	−4.87689	−0.212845
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−11.1771	−0.485960
$530$	0	0
$531$	11.3693	0.493386
$532$	0	0
$533$	−5.43845	−0.235565
$534$	0	0
$535$	−9.43845	−0.408060
$536$	0	0
$537$	−6.87689	−0.296760
$538$	0	0
$539$	18.0691	0.778293
$540$	0	0
$541$	26.7386	1.14958	0.574792	−	0.818300i	$-0.305083\pi$
$541$	26.7386	1.14958	0.574792	+	0.818300i	$0.305083\pi$
$542$	0	0
$543$	1.12311	0.0481971
$544$	0	0
$545$	−26.2462	−1.12426
$546$	0	0
$547$	22.7386	0.972234	0.486117	−	0.873894i	$-0.338413\pi$
$547$	22.7386	0.972234	0.486117	+	0.873894i	$0.338413\pi$
$548$	0	0
$549$	4.00000	0.170716
$550$	0	0
$551$	8.63068	0.367679
$552$	0	0
$553$	6.24621	0.265616
$554$	0	0
$555$	10.2462	0.434927
$556$	0	0
$557$	20.7386	0.878724	0.439362	−	0.898310i	$-0.355205\pi$
$557$	20.7386	0.878724	0.439362	+	0.898310i	$0.355205\pi$
$558$	0	0
$559$	4.31534	0.182520
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−19.3693	−0.816319	−0.408160	−	0.912911i	$-0.633829\pi$
$563$	−19.3693	−0.816319	−0.408160	+	0.912911i	$0.633829\pi$
$564$	0	0
$565$	−21.9309	−0.922639
$566$	0	0
$567$	−3.12311	−0.131158
$568$	0	0
$569$	−8.87689	−0.372139	−0.186069	−	0.982537i	$-0.559575\pi$
$569$	−8.87689	−0.372139	−0.186069	+	0.982537i	$0.559575\pi$
$570$	0	0
$571$	−32.4924	−1.35977	−0.679883	−	0.733321i	$-0.737969\pi$
$571$	−32.4924	−1.35977	−0.679883	+	0.733321i	$0.737969\pi$
$572$	0	0
$573$	19.3693	0.809165
$574$	0	0
$575$	5.36932	0.223916
$576$	0	0
$577$	−41.0540	−1.70910	−0.854550	−	0.519370i	$-0.826167\pi$
$577$	−41.0540	−1.70910	−0.854550	+	0.519370i	$0.826167\pi$
$578$	0	0
$579$	4.24621	0.176467
$580$	0	0
$581$	3.50758	0.145519
$582$	0	0
$583$	39.3693	1.63051
$584$	0	0
$585$	1.43845	0.0594725
$586$	0	0
$587$	−40.4924	−1.67130	−0.835651	−	0.549261i	$-0.814909\pi$
$587$	−40.4924	−1.67130	−0.835651	+	0.549261i	$0.814909\pi$
$588$	0	0
$589$	−63.3693	−2.61109
$590$	0	0
$591$	12.3153	0.506585
$592$	0	0
$593$	−9.50758	−0.390429	−0.195215	−	0.980761i	$-0.562540\pi$
$593$	−9.50758	−0.390429	−0.195215	+	0.980761i	$0.562540\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	23.1231	0.946366
$598$	0	0
$599$	8.63068	0.352640	0.176320	−	0.984333i	$-0.443581\pi$
$599$	8.63068	0.352640	0.176320	+	0.984333i	$0.443581\pi$
$600$	0	0
$601$	−24.7386	−1.00911	−0.504555	−	0.863380i	$-0.668343\pi$
$601$	−24.7386	−1.00911	−0.504555	+	0.863380i	$0.668343\pi$
$602$	0	0
$603$	12.0000	0.488678
$604$	0	0
$605$	82.1080	3.33816
$606$	0	0
$607$	−28.2462	−1.14648	−0.573239	−	0.819388i	$-0.694313\pi$
$607$	−28.2462	−1.14648	−0.573239	+	0.819388i	$0.694313\pi$
$608$	0	0
$609$	3.50758	0.142134
$610$	0	0
$611$	−5.12311	−0.207259
$612$	0	0
$613$	20.5616	0.830473	0.415237	−	0.909713i	$-0.363699\pi$
$613$	20.5616	0.830473	0.415237	+	0.909713i	$0.363699\pi$
$614$	0	0
$615$	−24.8078	−1.00035
$616$	0	0
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	0	0
$619$	33.1231	1.33133	0.665665	−	0.746251i	$-0.268148\pi$
$619$	33.1231	1.33133	0.665665	+	0.746251i	$0.268148\pi$
$620$	0	0
$621$	3.43845	0.137980
$622$	0	0
$623$	−2.73863	−0.109721
$624$	0	0
$625$	−30.3693	−1.21477
$626$	0	0
$627$	−50.4233	−2.01371
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−3.19224	−0.127081	−0.0635405	−	0.997979i	$-0.520239\pi$
$631$	−3.19224	−0.127081	−0.0635405	+	0.997979i	$0.520239\pi$
$632$	0	0
$633$	−10.8769	−0.432318
$634$	0	0
$635$	−35.6847	−1.41610
$636$	0	0
$637$	1.54640	0.0612705
$638$	0	0
$639$	13.3693	0.528882
$640$	0	0
$641$	30.6695	1.21137	0.605686	−	0.795704i	$-0.292899\pi$
$641$	30.6695	1.21137	0.605686	+	0.795704i	$0.292899\pi$
$642$	0	0
$643$	16.4924	0.650398	0.325199	−	0.945646i	$-0.394569\pi$
$643$	16.4924	0.650398	0.325199	+	0.945646i	$0.394569\pi$
$644$	0	0
$645$	19.6847	0.775083
$646$	0	0
$647$	−33.1231	−1.30220	−0.651102	−	0.758990i	$-0.725693\pi$
$647$	−33.1231	−1.30220	−0.651102	+	0.758990i	$0.725693\pi$
$648$	0	0
$649$	74.6004	2.92832
$650$	0	0
$651$	−25.7538	−1.00937
$652$	0	0
$653$	−30.4233	−1.19056	−0.595278	−	0.803520i	$-0.702958\pi$
$653$	−30.4233	−1.19056	−0.595278	+	0.803520i	$0.702958\pi$
$654$	0	0
$655$	24.1771	0.944677
$656$	0	0
$657$	8.24621	0.321715
$658$	0	0
$659$	37.6155	1.46529	0.732646	−	0.680609i	$-0.238285\pi$
$659$	37.6155	1.46529	0.732646	+	0.680609i	$0.238285\pi$
$660$	0	0
$661$	21.6847	0.843435	0.421718	−	0.906727i	$-0.361427\pi$
$661$	21.6847	0.843435	0.421718	+	0.906727i	$0.361427\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	61.4773	2.38399
$666$	0	0
$667$	−3.86174	−0.149527
$668$	0	0
$669$	−21.4384	−0.828858
$670$	0	0
$671$	26.2462	1.01322
$672$	0	0
$673$	26.0000	1.00223	0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000	1.00223	0.501113	+	0.865382i	$0.332924\pi$
$674$	0	0
$675$	1.56155	0.0601042
$676$	0	0
$677$	−0.807764	−0.0310449	−0.0155224	−	0.999880i	$-0.504941\pi$
$677$	−0.807764	−0.0310449	−0.0155224	+	0.999880i	$0.504941\pi$
$678$	0	0
$679$	−22.2462	−0.853731
$680$	0	0
$681$	15.6847	0.601037
$682$	0	0
$683$	−12.8078	−0.490075	−0.245038	−	0.969514i	$-0.578800\pi$
$683$	−12.8078	−0.490075	−0.245038	+	0.969514i	$0.578800\pi$
$684$	0	0
$685$	−41.6155	−1.59005
$686$	0	0
$687$	−18.0000	−0.686743
$688$	0	0
$689$	3.36932	0.128361
$690$	0	0
$691$	−26.2462	−0.998453	−0.499226	−	0.866472i	$-0.666382\pi$
$691$	−26.2462	−0.998453	−0.499226	+	0.866472i	$0.666382\pi$
$692$	0	0
$693$	−20.4924	−0.778443
$694$	0	0
$695$	−13.1231	−0.497788
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−3.43845	−0.130054
$700$	0	0
$701$	−28.8769	−1.09067	−0.545333	−	0.838220i	$-0.683597\pi$
$701$	−28.8769	−1.09067	−0.545333	+	0.838220i	$0.683597\pi$
$702$	0	0
$703$	−30.7386	−1.15933
$704$	0	0
$705$	−23.3693	−0.880139
$706$	0	0
$707$	−9.75379	−0.366829
$708$	0	0
$709$	2.87689	0.108044	0.0540220	−	0.998540i	$-0.482796\pi$
$709$	2.87689	0.108044	0.0540220	+	0.998540i	$0.482796\pi$
$710$	0	0
$711$	−2.00000	−0.0750059
$712$	0	0
$713$	28.3542	1.06187
$714$	0	0
$715$	9.43845	0.352978
$716$	0	0
$717$	22.2462	0.830800
$718$	0	0
$719$	47.7926	1.78236	0.891182	−	0.453646i	$-0.149877\pi$
$719$	47.7926	1.78236	0.891182	+	0.453646i	$0.149877\pi$
$720$	0	0
$721$	0.984845	0.0366775
$722$	0	0
$723$	−5.36932	−0.199687
$724$	0	0
$725$	−1.75379	−0.0651341
$726$	0	0
$727$	36.4924	1.35343	0.676715	−	0.736246i	$-0.263403\pi$
$727$	36.4924	1.35343	0.676715	+	0.736246i	$0.263403\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−30.0000	−1.10808	−0.554038	−	0.832492i	$-0.686914\pi$
$733$	−30.0000	−1.10808	−0.554038	+	0.832492i	$0.686914\pi$
$734$	0	0
$735$	7.05398	0.260190
$736$	0	0
$737$	78.7386	2.90037
$738$	0	0
$739$	−19.1922	−0.705998	−0.352999	−	0.935624i	$-0.614838\pi$
$739$	−19.1922	−0.705998	−0.352999	+	0.935624i	$0.614838\pi$
$740$	0	0
$741$	−4.31534	−0.158528
$742$	0	0
$743$	−3.12311	−0.114576	−0.0572878	−	0.998358i	$-0.518245\pi$
$743$	−3.12311	−0.114576	−0.0572878	+	0.998358i	$0.518245\pi$
$744$	0	0
$745$	−35.8617	−1.31387
$746$	0	0
$747$	−1.12311	−0.0410923
$748$	0	0
$749$	11.5076	0.420478
$750$	0	0
$751$	50.9848	1.86046	0.930232	−	0.366973i	$-0.119606\pi$
$751$	50.9848	1.86046	0.930232	+	0.366973i	$0.119606\pi$
$752$	0	0
$753$	−6.24621	−0.227625
$754$	0	0
$755$	40.9848	1.49159
$756$	0	0
$757$	−39.3002	−1.42839	−0.714195	−	0.699947i	$-0.753207\pi$
$757$	−39.3002	−1.42839	−0.714195	+	0.699947i	$0.753207\pi$
$758$	0	0
$759$	22.5616	0.818932
$760$	0	0
$761$	−11.7538	−0.426075	−0.213037	−	0.977044i	$-0.568336\pi$
$761$	−11.7538	−0.426075	−0.213037	+	0.977044i	$0.568336\pi$
$762$	0	0
$763$	32.0000	1.15848
$764$	0	0
$765$	0	0
$766$	0	0
$767$	6.38447	0.230530
$768$	0	0
$769$	14.9460	0.538967	0.269484	−	0.963005i	$-0.413147\pi$
$769$	14.9460	0.538967	0.269484	+	0.963005i	$0.413147\pi$
$770$	0	0
$771$	4.87689	0.175637
$772$	0	0
$773$	31.1231	1.11942	0.559710	−	0.828688i	$-0.310912\pi$
$773$	31.1231	1.11942	0.559710	+	0.828688i	$0.310912\pi$
$774$	0	0
$775$	12.8769	0.462552
$776$	0	0
$777$	−12.4924	−0.448163
$778$	0	0
$779$	74.4233	2.66649
$780$	0	0
$781$	87.7235	3.13899
$782$	0	0
$783$	−1.12311	−0.0401365
$784$	0	0
$785$	−8.80776	−0.314363
$786$	0	0
$787$	−35.2311	−1.25585	−0.627926	−	0.778273i	$-0.716096\pi$
$787$	−35.2311	−1.25585	−0.627926	+	0.778273i	$0.716096\pi$
$788$	0	0
$789$	4.00000	0.142404
$790$	0	0
$791$	26.7386	0.950716
$792$	0	0
$793$	2.24621	0.0797653
$794$	0	0
$795$	15.3693	0.545093
$796$	0	0
$797$	−53.8617	−1.90788	−0.953940	−	0.299996i	$-0.903015\pi$
$797$	−53.8617	−1.90788	−0.953940	+	0.299996i	$0.903015\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0.876894	0.0309835
$802$	0	0
$803$	54.1080	1.90943
$804$	0	0
$805$	−27.5076	−0.969515
$806$	0	0
$807$	23.6847	0.833739
$808$	0	0
$809$	−21.1922	−0.745079	−0.372540	−	0.928016i	$-0.621513\pi$
$809$	−21.1922	−0.745079	−0.372540	+	0.928016i	$0.621513\pi$
$810$	0	0
$811$	−28.6307	−1.00536	−0.502680	−	0.864473i	$-0.667652\pi$
$811$	−28.6307	−1.00536	−0.502680	+	0.864473i	$0.667652\pi$
$812$	0	0
$813$	19.0540	0.668252
$814$	0	0
$815$	29.1231	1.02014
$816$	0	0
$817$	−59.0540	−2.06604
$818$	0	0
$819$	−1.75379	−0.0612823
$820$	0	0
$821$	33.7926	1.17937	0.589685	−	0.807633i	$-0.299252\pi$
$821$	33.7926	1.17937	0.589685	+	0.807633i	$0.299252\pi$
$822$	0	0
$823$	−2.63068	−0.0916998	−0.0458499	−	0.998948i	$-0.514600\pi$
$823$	−2.63068	−0.0916998	−0.0458499	+	0.998948i	$0.514600\pi$
$824$	0	0
$825$	10.2462	0.356727
$826$	0	0
$827$	9.43845	0.328207	0.164103	−	0.986443i	$-0.447527\pi$
$827$	9.43845	0.328207	0.164103	+	0.986443i	$0.447527\pi$
$828$	0	0
$829$	10.4924	0.364417	0.182208	−	0.983260i	$-0.441675\pi$
$829$	10.4924	0.364417	0.182208	+	0.983260i	$0.441675\pi$
$830$	0	0
$831$	21.1231	0.732752
$832$	0	0
$833$	0	0
$834$	0	0
$835$	3.05398	0.105687
$836$	0	0
$837$	8.24621	0.285031
$838$	0	0
$839$	41.0540	1.41734	0.708670	−	0.705540i	$-0.249295\pi$
$839$	41.0540	1.41734	0.708670	+	0.705540i	$0.249295\pi$
$840$	0	0
$841$	−27.7386	−0.956505
$842$	0	0
$843$	11.1231	0.383100
$844$	0	0
$845$	−32.4924	−1.11777
$846$	0	0
$847$	−100.108	−3.43975
$848$	0	0
$849$	−13.1231	−0.450384
$850$	0	0
$851$	13.7538	0.471474
$852$	0	0
$853$	−33.6155	−1.15097	−0.575487	−	0.817811i	$-0.695187\pi$
$853$	−33.6155	−1.15097	−0.575487	+	0.817811i	$0.695187\pi$
$854$	0	0
$855$	−19.6847	−0.673201
$856$	0	0
$857$	−6.00000	−0.204956	−0.102478	−	0.994735i	$-0.532677\pi$
$857$	−6.00000	−0.204956	−0.102478	+	0.994735i	$0.532677\pi$
$858$	0	0
$859$	16.4924	0.562714	0.281357	−	0.959603i	$-0.409215\pi$
$859$	16.4924	0.562714	0.281357	+	0.959603i	$0.409215\pi$
$860$	0	0
$861$	30.2462	1.03079
$862$	0	0
$863$	−1.26137	−0.0429374	−0.0214687	−	0.999770i	$-0.506834\pi$
$863$	−1.26137	−0.0429374	−0.0214687	+	0.999770i	$0.506834\pi$
$864$	0	0
$865$	−19.6847	−0.669298
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−13.1231	−0.445171
$870$	0	0
$871$	6.73863	0.228330
$872$	0	0
$873$	7.12311	0.241081
$874$	0	0
$875$	27.5076	0.929926
$876$	0	0
$877$	56.3542	1.90294	0.951472	−	0.307734i	$-0.0995707\pi$
$877$	56.3542	1.90294	0.951472	+	0.307734i	$0.0995707\pi$
$878$	0	0
$879$	−3.12311	−0.105340
$880$	0	0
$881$	−34.0000	−1.14549	−0.572745	−	0.819734i	$-0.694121\pi$
$881$	−34.0000	−1.14549	−0.572745	+	0.819734i	$0.694121\pi$
$882$	0	0
$883$	−21.4384	−0.721461	−0.360731	−	0.932670i	$-0.617473\pi$
$883$	−21.4384	−0.721461	−0.360731	+	0.932670i	$0.617473\pi$
$884$	0	0
$885$	29.1231	0.978962
$886$	0	0
$887$	−43.4384	−1.45852	−0.729260	−	0.684237i	$-0.760136\pi$
$887$	−43.4384	−1.45852	−0.729260	+	0.684237i	$0.760136\pi$
$888$	0	0
$889$	43.5076	1.45920
$890$	0	0
$891$	6.56155	0.219820
$892$	0	0
$893$	70.1080	2.34607
$894$	0	0
$895$	−17.6155	−0.588822
$896$	0	0
$897$	1.93087	0.0644699
$898$	0	0
$899$	−9.26137	−0.308884
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−24.0000	−0.798670
$904$	0	0
$905$	2.87689	0.0956312
$906$	0	0
$907$	−16.6307	−0.552213	−0.276106	−	0.961127i	$-0.589044\pi$
$907$	−16.6307	−0.552213	−0.276106	+	0.961127i	$0.589044\pi$
$908$	0	0
$909$	3.12311	0.103587
$910$	0	0
$911$	−32.5616	−1.07881	−0.539406	−	0.842046i	$-0.681351\pi$
$911$	−32.5616	−1.07881	−0.539406	+	0.842046i	$0.681351\pi$
$912$	0	0
$913$	−7.36932	−0.243889
$914$	0	0
$915$	10.2462	0.338729
$916$	0	0
$917$	−29.4773	−0.973425
$918$	0	0
$919$	10.0691	0.332150	0.166075	−	0.986113i	$-0.446891\pi$
$919$	10.0691	0.332150	0.166075	+	0.986113i	$0.446891\pi$
$920$	0	0
$921$	−20.0000	−0.659022
$922$	0	0
$923$	7.50758	0.247115
$924$	0	0
$925$	6.24621	0.205374
$926$	0	0
$927$	−0.315342	−0.0103572
$928$	0	0
$929$	−55.7926	−1.83050	−0.915248	−	0.402891i	$-0.868005\pi$
$929$	−55.7926	−1.83050	−0.915248	+	0.402891i	$0.868005\pi$
$930$	0	0
$931$	−21.1619	−0.693554
$932$	0	0
$933$	−6.63068	−0.217079
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−4.24621	−0.138718	−0.0693588	−	0.997592i	$-0.522095\pi$
$937$	−4.24621	−0.138718	−0.0693588	+	0.997592i	$0.522095\pi$
$938$	0	0
$939$	11.6155	0.379058
$940$	0	0
$941$	6.38447	0.208128	0.104064	−	0.994571i	$-0.466815\pi$
$941$	6.38447	0.208128	0.104064	+	0.994571i	$0.466815\pi$
$942$	0	0
$943$	−33.3002	−1.08440
$944$	0	0
$945$	−8.00000	−0.260240
$946$	0	0
$947$	2.24621	0.0729921	0.0364960	−	0.999334i	$-0.488380\pi$
$947$	2.24621	0.0729921	0.0364960	+	0.999334i	$0.488380\pi$
$948$	0	0
$949$	4.63068	0.150318
$950$	0	0
$951$	10.8769	0.352707
$952$	0	0
$953$	−32.1080	−1.04008	−0.520039	−	0.854142i	$-0.674083\pi$
$953$	−32.1080	−1.04008	−0.520039	+	0.854142i	$0.674083\pi$
$954$	0	0
$955$	49.6155	1.60552
$956$	0	0
$957$	−7.36932	−0.238216
$958$	0	0
$959$	50.7386	1.63844
$960$	0	0
$961$	37.0000	1.19355
$962$	0	0
$963$	−3.68466	−0.118736
$964$	0	0
$965$	10.8769	0.350140
$966$	0	0
$967$	5.43845	0.174889	0.0874443	−	0.996169i	$-0.472130\pi$
$967$	5.43845	0.174889	0.0874443	+	0.996169i	$0.472130\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	44.3542	1.42339	0.711696	−	0.702487i	$-0.247927\pi$
$971$	44.3542	1.42339	0.711696	+	0.702487i	$0.247927\pi$
$972$	0	0
$973$	16.0000	0.512936
$974$	0	0
$975$	0.876894	0.0280831
$976$	0	0
$977$	38.4924	1.23148	0.615741	−	0.787949i	$-0.288857\pi$
$977$	38.4924	1.23148	0.615741	+	0.787949i	$0.288857\pi$
$978$	0	0
$979$	5.75379	0.183892
$980$	0	0
$981$	−10.2462	−0.327136
$982$	0	0
$983$	−51.4384	−1.64063	−0.820316	−	0.571911i	$-0.806202\pi$
$983$	−51.4384	−1.64063	−0.820316	+	0.571911i	$0.806202\pi$
$984$	0	0
$985$	31.5464	1.00515
$986$	0	0
$987$	28.4924	0.906924
$988$	0	0
$989$	26.4233	0.840212
$990$	0	0
$991$	49.3693	1.56827	0.784134	−	0.620592i	$-0.213107\pi$
$991$	49.3693	1.56827	0.784134	+	0.620592i	$0.213107\pi$
$992$	0	0
$993$	2.56155	0.0812884
$994$	0	0
$995$	59.2311	1.87775
$996$	0	0
$997$	11.3693	0.360070	0.180035	−	0.983660i	$-0.442379\pi$
$997$	11.3693	0.360070	0.180035	+	0.983660i	$0.442379\pi$
$998$	0	0
$999$	4.00000	0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6936.2.a.y.1.2		2
17.16	even	2		408.2.a.e.1.1	✓	2
51.50	odd	2		1224.2.a.j.1.2		2
68.67	odd	2		816.2.a.l.1.1		2
136.67	odd	2		3264.2.a.bi.1.2		2
136.101	even	2		3264.2.a.bo.1.2		2
204.203	even	2		2448.2.a.ba.1.2		2
408.101	odd	2		9792.2.a.cm.1.1		2
408.203	even	2		9792.2.a.cn.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
408.2.a.e.1.1	✓	2	17.16	even	2
816.2.a.l.1.1		2	68.67	odd	2
1224.2.a.j.1.2		2	51.50	odd	2
2448.2.a.ba.1.2		2	204.203	even	2
3264.2.a.bi.1.2		2	136.67	odd	2
3264.2.a.bo.1.2		2	136.101	even	2
6936.2.a.y.1.2		2	1.1	even	1	trivial
9792.2.a.cm.1.1		2	408.101	odd	2
9792.2.a.cn.1.1		2	408.203	even	2

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:	\( N \)	\(=\)	\( 6936 = 2^{3} \cdot 3 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6936.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.56155 q^{5} -3.12311 q^{7} +1.00000 q^{9} +6.56155 q^{11} +0.561553 q^{13} +2.56155 q^{15} -7.68466 q^{19} -3.12311 q^{21} +3.43845 q^{23} +1.56155 q^{25} +1.00000 q^{27} -1.12311 q^{29} +8.24621 q^{31} +6.56155 q^{33} -8.00000 q^{35} +4.00000 q^{37} +0.561553 q^{39} -9.68466 q^{41} +7.68466 q^{43} +2.56155 q^{45} -9.12311 q^{47} +2.75379 q^{49} +6.00000 q^{53} +16.8078 q^{55} -7.68466 q^{57} +11.3693 q^{59} +4.00000 q^{61} -3.12311 q^{63} +1.43845 q^{65} +12.0000 q^{67} +3.43845 q^{69} +13.3693 q^{71} +8.24621 q^{73} +1.56155 q^{75} -20.4924 q^{77} -2.00000 q^{79} +1.00000 q^{81} -1.12311 q^{83} -1.12311 q^{87} +0.876894 q^{89} -1.75379 q^{91} +8.24621 q^{93} -19.6847 q^{95} +7.12311 q^{97} +6.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + q^{5} + 2 q^{7} + 2 q^{9} + 9 q^{11} - 3 q^{13} + q^{15} - 3 q^{19} + 2 q^{21} + 11 q^{23} - q^{25} + 2 q^{27} + 6 q^{29} + 9 q^{33} - 16 q^{35} + 8 q^{37} - 3 q^{39} - 7 q^{41} + 3 q^{43}+ \cdots + 9 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + q^5 + 2 * q^7 + 2 * q^9 + 9 * q^11 - 3 * q^13 + q^15 - 3 * q^19 + 2 * q^21 + 11 * q^23 - q^25 + 2 * q^27 + 6 * q^29 + 9 * q^33 - 16 * q^35 + 8 * q^37 - 3 * q^39 - 7 * q^41 + 3 * q^43 + q^45 - 10 * q^47 + 22 * q^49 + 12 * q^53 + 13 * q^55 - 3 * q^57 - 2 * q^59 + 8 * q^61 + 2 * q^63 + 7 * q^65 + 24 * q^67 + 11 * q^69 + 2 * q^71 - q^75 - 8 * q^77 - 4 * q^79 + 2 * q^81 + 6 * q^83 + 6 * q^87 + 10 * q^89 - 20 * q^91 - 27 * q^95 + 6 * q^97 + 9 * q^99

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