LMFDB - Embedded newform 8464.2.a.ch.1.15

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8464,2,Mod(1,8464)] 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("8464.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8464, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8464 = 2^{4} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8464.a (trivial)

Newform invariants

comment:select newform

sage:traces = [15,0,-1,0,0,0,10,0,16,0,23,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$67.5853802708$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - x^{14} - 30 x^{13} + 28 x^{12} + 354 x^{11} - 302 x^{10} - 2111 x^{9} + 1596 x^{8} + 6777 x^{7} + \cdots - 419 $ x^15 - x^14 - 30x^13 + 28x^12 + 354x^11 - 302x^10 - 2111x^9 + 1596x^8 + 6777x^7 - 4292x^6 - 11506x^5 + 5297x^4 + 9480x^3 - 1825x^2 - 3136*x - 419
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 184)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Root			$-3.13479$ of defining polynomial
Character	$\chi$	$=$	8464.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.13479 q^{3} +3.00137 q^{5} -0.714288 q^{7} +6.82690 q^{9} +1.17733 q^{11} +4.39302 q^{13} +9.40866 q^{15} +2.25057 q^{17} +8.02507 q^{19} -2.23914 q^{21} +4.00822 q^{25} +11.9965 q^{27} -8.34446 q^{29} -3.19860 q^{31} +3.69067 q^{33} -2.14384 q^{35} -4.74615 q^{37} +13.7712 q^{39} -10.8973 q^{41} +8.08417 q^{43} +20.4900 q^{45} -1.20581 q^{47} -6.48979 q^{49} +7.05508 q^{51} -0.460183 q^{53} +3.53359 q^{55} +25.1569 q^{57} -4.53574 q^{59} -2.56511 q^{61} -4.87637 q^{63} +13.1851 q^{65} -3.89321 q^{67} +8.18786 q^{71} -0.0978311 q^{73} +12.5649 q^{75} -0.840950 q^{77} -10.9981 q^{79} +17.1258 q^{81} +1.88740 q^{83} +6.75481 q^{85} -26.1581 q^{87} +5.90204 q^{89} -3.13788 q^{91} -10.0269 q^{93} +24.0862 q^{95} -17.4863 q^{97} +8.03748 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - q^{3} + 10 q^{7} + 16 q^{9} + 23 q^{11} + 10 q^{15} + 29 q^{19} - q^{21} + 23 q^{25} - q^{27} - 2 q^{29} - 20 q^{31} - 18 q^{33} + 18 q^{35} - 24 q^{37} + 19 q^{39} + 9 q^{41} + 48 q^{43} - 4 q^{45}+ \cdots + 63 q^{99}+O(q^{100}) $ 15 * q - q^3 + 10 * q^7 + 16 * q^9 + 23 * q^11 + 10 * q^15 + 29 * q^19 - q^21 + 23 * q^25 - q^27 - 2 * q^29 - 20 * q^31 - 18 * q^33 + 18 * q^35 - 24 * q^37 + 19 * q^39 + 9 * q^41 + 48 * q^43 - 4 * q^45 + 36 * q^47 + 25 * q^49 + 35 * q^51 + 5 * q^53 + 10 * q^55 - 23 * q^57 + 22 * q^59 - 12 * q^61 + 35 * q^63 + 26 * q^65 + 58 * q^67 - 2 * q^71 + 5 * q^73 + 17 * q^75 + 26 * q^77 + 26 * q^79 - 21 * q^81 + 68 * q^83 - 72 * q^85 - 19 * q^87 + 6 * q^89 + 71 * q^91 - 55 * q^93 + 12 * q^95 - 40 * q^97 + 63 * 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
$3$	3.13479	1.80987	0.904935	−	0.425549i	$-0.139919\pi$
$3$	3.13479	1.80987	0.904935	+	0.425549i	$0.139919\pi$
$4$	0	0
$5$	3.00137	1.34225	0.671127	−	0.741343i	$-0.265811\pi$
$5$	3.00137	1.34225	0.671127	+	0.741343i	$0.265811\pi$
$6$	0	0
$7$	−0.714288	−0.269975	−0.134988	−	0.990847i	$-0.543100\pi$
$7$	−0.714288	−0.269975	−0.134988	+	0.990847i	$0.543100\pi$
$8$	0	0
$9$	6.82690	2.27563
$10$	0	0
$11$	1.17733	0.354977	0.177489	−	0.984123i	$-0.443203\pi$
$11$	1.17733	0.354977	0.177489	+	0.984123i	$0.443203\pi$
$12$	0	0
$13$	4.39302	1.21840	0.609202	−	0.793015i	$-0.291490\pi$
$13$	4.39302	1.21840	0.609202	+	0.793015i	$0.291490\pi$
$14$	0	0
$15$	9.40866	2.42931
$16$	0	0
$17$	2.25057	0.545845	0.272922	−	0.962036i	$-0.412010\pi$
$17$	2.25057	0.545845	0.272922	+	0.962036i	$0.412010\pi$
$18$	0	0
$19$	8.02507	1.84108	0.920538	−	0.390652i	$-0.127750\pi$
$19$	8.02507	1.84108	0.920538	+	0.390652i	$0.127750\pi$
$20$	0	0
$21$	−2.23914	−0.488621
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	4.00822	0.801644
$26$	0	0
$27$	11.9965	2.30873
$28$	0	0
$29$	−8.34446	−1.54953	−0.774764	−	0.632251i	$-0.782131\pi$
$29$	−8.34446	−1.54953	−0.774764	+	0.632251i	$0.782131\pi$
$30$	0	0
$31$	−3.19860	−0.574485	−0.287243	−	0.957858i	$-0.592739\pi$
$31$	−3.19860	−0.574485	−0.287243	+	0.957858i	$0.592739\pi$
$32$	0	0
$33$	3.69067	0.642463
$34$	0	0
$35$	−2.14384	−0.362376
$36$	0	0
$37$	−4.74615	−0.780262	−0.390131	−	0.920759i	$-0.627570\pi$
$37$	−4.74615	−0.780262	−0.390131	+	0.920759i	$0.627570\pi$
$38$	0	0
$39$	13.7712	2.20515
$40$	0	0
$41$	−10.8973	−1.70187	−0.850937	−	0.525268i	$-0.823965\pi$
$41$	−10.8973	−1.70187	−0.850937	+	0.525268i	$0.823965\pi$
$42$	0	0
$43$	8.08417	1.23282	0.616412	−	0.787423i	$-0.288585\pi$
$43$	8.08417	1.23282	0.616412	+	0.787423i	$0.288585\pi$
$44$	0	0
$45$	20.4900	3.05448
$46$	0	0
$47$	−1.20581	−0.175886	−0.0879428	−	0.996126i	$-0.528029\pi$
$47$	−1.20581	−0.175886	−0.0879428	+	0.996126i	$0.528029\pi$
$48$	0	0
$49$	−6.48979	−0.927113
$50$	0	0
$51$	7.05508	0.987908
$52$	0	0
$53$	−0.460183	−0.0632110	−0.0316055	−	0.999500i	$-0.510062\pi$
$53$	−0.460183	−0.0632110	−0.0316055	+	0.999500i	$0.510062\pi$
$54$	0	0
$55$	3.53359	0.476469
$56$	0	0
$57$	25.1569	3.33211
$58$	0	0
$59$	−4.53574	−0.590503	−0.295252	−	0.955420i	$-0.595403\pi$
$59$	−4.53574	−0.590503	−0.295252	+	0.955420i	$0.595403\pi$
$60$	0	0
$61$	−2.56511	−0.328429	−0.164214	−	0.986425i	$-0.552509\pi$
$61$	−2.56511	−0.328429	−0.164214	+	0.986425i	$0.552509\pi$
$62$	0	0
$63$	−4.87637	−0.614365
$64$	0	0
$65$	13.1851	1.63541
$66$	0	0
$67$	−3.89321	−0.475632	−0.237816	−	0.971310i	$-0.576431\pi$
$67$	−3.89321	−0.475632	−0.237816	+	0.971310i	$0.576431\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	8.18786	0.971720	0.485860	−	0.874037i	$-0.338507\pi$
$71$	8.18786	0.971720	0.485860	+	0.874037i	$0.338507\pi$
$72$	0	0
$73$	−0.0978311	−0.0114503	−0.00572513	−	0.999984i	$-0.501822\pi$
$73$	−0.0978311	−0.0114503	−0.00572513	+	0.999984i	$0.501822\pi$
$74$	0	0
$75$	12.5649	1.45087
$76$	0	0
$77$	−0.840950	−0.0958351
$78$	0	0
$79$	−10.9981	−1.23738	−0.618691	−	0.785635i	$-0.712337\pi$
$79$	−10.9981	−1.23738	−0.618691	+	0.785635i	$0.712337\pi$
$80$	0	0
$81$	17.1258	1.90287
$82$	0	0
$83$	1.88740	0.207169	0.103584	−	0.994621i	$-0.466969\pi$
$83$	1.88740	0.207169	0.103584	+	0.994621i	$0.466969\pi$
$84$	0	0
$85$	6.75481	0.732662
$86$	0	0
$87$	−26.1581	−2.80444
$88$	0	0
$89$	5.90204	0.625615	0.312808	−	0.949816i	$-0.398730\pi$
$89$	5.90204	0.625615	0.312808	+	0.949816i	$0.398730\pi$
$90$	0	0
$91$	−3.13788	−0.328939
$92$	0	0
$93$	−10.0269	−1.03974
$94$	0	0
$95$	24.0862	2.47119
$96$	0	0
$97$	−17.4863	−1.77547	−0.887734	−	0.460356i	$-0.847722\pi$
$97$	−17.4863	−1.77547	−0.887734	+	0.460356i	$0.847722\pi$
$98$	0	0
$99$	8.03748	0.807798
$100$	0	0
$101$	−3.98854	−0.396875	−0.198437	−	0.980114i	$-0.563587\pi$
$101$	−3.98854	−0.396875	−0.198437	+	0.980114i	$0.563587\pi$
$102$	0	0
$103$	1.69287	0.166803	0.0834017	−	0.996516i	$-0.473422\pi$
$103$	1.69287	0.166803	0.0834017	+	0.996516i	$0.473422\pi$
$104$	0	0
$105$	−6.72049	−0.655853
$106$	0	0
$107$	8.15395	0.788272	0.394136	−	0.919052i	$-0.371044\pi$
$107$	8.15395	0.788272	0.394136	+	0.919052i	$0.371044\pi$
$108$	0	0
$109$	1.89599	0.181603	0.0908014	−	0.995869i	$-0.471057\pi$
$109$	1.89599	0.181603	0.0908014	+	0.995869i	$0.471057\pi$
$110$	0	0
$111$	−14.8782	−1.41217
$112$	0	0
$113$	11.2490	1.05822	0.529108	−	0.848555i	$-0.322527\pi$
$113$	11.2490	1.05822	0.529108	+	0.848555i	$0.322527\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	29.9907	2.77264
$118$	0	0
$119$	−1.60756	−0.147365
$120$	0	0
$121$	−9.61390	−0.873991
$122$	0	0
$123$	−34.1608	−3.08017
$124$	0	0
$125$	−2.97669	−0.266244
$126$	0	0
$127$	−4.42777	−0.392901	−0.196451	−	0.980514i	$-0.562942\pi$
$127$	−4.42777	−0.392901	−0.196451	+	0.980514i	$0.562942\pi$
$128$	0	0
$129$	25.3422	2.23125
$130$	0	0
$131$	−8.42261	−0.735887	−0.367943	−	0.929848i	$-0.619938\pi$
$131$	−8.42261	−0.735887	−0.367943	+	0.929848i	$0.619938\pi$
$132$	0	0
$133$	−5.73221	−0.497046
$134$	0	0
$135$	36.0060	3.09890
$136$	0	0
$137$	−2.06372	−0.176315	−0.0881576	−	0.996107i	$-0.528098\pi$
$137$	−2.06372	−0.176315	−0.0881576	+	0.996107i	$0.528098\pi$
$138$	0	0
$139$	−10.3441	−0.877379	−0.438689	−	0.898639i	$-0.644557\pi$
$139$	−10.3441	−0.877379	−0.438689	+	0.898639i	$0.644557\pi$
$140$	0	0
$141$	−3.77996	−0.318330
$142$	0	0
$143$	5.17202	0.432506
$144$	0	0
$145$	−25.0448	−2.07986
$146$	0	0
$147$	−20.3441	−1.67796
$148$	0	0
$149$	−10.9073	−0.893557	−0.446778	−	0.894645i	$-0.647429\pi$
$149$	−10.9073	−0.893557	−0.446778	+	0.894645i	$0.647429\pi$
$150$	0	0
$151$	−14.2619	−1.16062	−0.580309	−	0.814397i	$-0.697068\pi$
$151$	−14.2619	−1.16062	−0.580309	+	0.814397i	$0.697068\pi$
$152$	0	0
$153$	15.3644	1.24214
$154$	0	0
$155$	−9.60018	−0.771105
$156$	0	0
$157$	−15.8086	−1.26166	−0.630832	−	0.775919i	$-0.717286\pi$
$157$	−15.8086	−1.26166	−0.630832	+	0.775919i	$0.717286\pi$
$158$	0	0
$159$	−1.44258	−0.114404
$160$	0	0
$161$	0	0
$162$	0	0
$163$	11.5696	0.906203	0.453101	−	0.891459i	$-0.350318\pi$
$163$	11.5696	0.906203	0.453101	+	0.891459i	$0.350318\pi$
$164$	0	0
$165$	11.0771	0.862348
$166$	0	0
$167$	−14.4651	−1.11934	−0.559670	−	0.828716i	$-0.689072\pi$
$167$	−14.4651	−1.11934	−0.559670	+	0.828716i	$0.689072\pi$
$168$	0	0
$169$	6.29863	0.484510
$170$	0	0
$171$	54.7863	4.18961
$172$	0	0
$173$	6.89447	0.524177	0.262089	−	0.965044i	$-0.415589\pi$
$173$	6.89447	0.524177	0.262089	+	0.965044i	$0.415589\pi$
$174$	0	0
$175$	−2.86302	−0.216424
$176$	0	0
$177$	−14.2186	−1.06873
$178$	0	0
$179$	7.15659	0.534909	0.267454	−	0.963571i	$-0.413818\pi$
$179$	7.15659	0.534909	0.267454	+	0.963571i	$0.413818\pi$
$180$	0	0
$181$	20.8783	1.55187	0.775937	−	0.630811i	$-0.217278\pi$
$181$	20.8783	1.55187	0.775937	+	0.630811i	$0.217278\pi$
$182$	0	0
$183$	−8.04108	−0.594414
$184$	0	0
$185$	−14.2449	−1.04731
$186$	0	0
$187$	2.64966	0.193762
$188$	0	0
$189$	−8.56897	−0.623301
$190$	0	0
$191$	−6.42056	−0.464576	−0.232288	−	0.972647i	$-0.574621\pi$
$191$	−6.42056	−0.464576	−0.232288	+	0.972647i	$0.574621\pi$
$192$	0	0
$193$	−5.54810	−0.399361	−0.199680	−	0.979861i	$-0.563990\pi$
$193$	−5.54810	−0.399361	−0.199680	+	0.979861i	$0.563990\pi$
$194$	0	0
$195$	41.3324	2.95988
$196$	0	0
$197$	−10.5787	−0.753698	−0.376849	−	0.926275i	$-0.622992\pi$
$197$	−10.5787	−0.753698	−0.376849	+	0.926275i	$0.622992\pi$
$198$	0	0
$199$	−9.28579	−0.658252	−0.329126	−	0.944286i	$-0.606754\pi$
$199$	−9.28579	−0.658252	−0.329126	+	0.944286i	$0.606754\pi$
$200$	0	0
$201$	−12.2044	−0.860832
$202$	0	0
$203$	5.96035	0.418334
$204$	0	0
$205$	−32.7069	−2.28435
$206$	0	0
$207$	0	0
$208$	0	0
$209$	9.44812	0.653540
$210$	0	0
$211$	−7.91171	−0.544665	−0.272332	−	0.962203i	$-0.587795\pi$
$211$	−7.91171	−0.544665	−0.272332	+	0.962203i	$0.587795\pi$
$212$	0	0
$213$	25.6672	1.75869
$214$	0	0
$215$	24.2636	1.65476
$216$	0	0
$217$	2.28472	0.155097
$218$	0	0
$219$	−0.306680	−0.0207235
$220$	0	0
$221$	9.88682	0.665060
$222$	0	0
$223$	20.9608	1.40364	0.701820	−	0.712355i	$-0.252371\pi$
$223$	20.9608	1.40364	0.701820	+	0.712355i	$0.252371\pi$
$224$	0	0
$225$	27.3637	1.82425
$226$	0	0
$227$	5.20395	0.345399	0.172699	−	0.984975i	$-0.444751\pi$
$227$	5.20395	0.345399	0.172699	+	0.984975i	$0.444751\pi$
$228$	0	0
$229$	−1.86791	−0.123435	−0.0617174	−	0.998094i	$-0.519658\pi$
$229$	−1.86791	−0.123435	−0.0617174	+	0.998094i	$0.519658\pi$
$230$	0	0
$231$	−2.63620	−0.173449
$232$	0	0
$233$	8.66375	0.567581	0.283791	−	0.958886i	$-0.408408\pi$
$233$	8.66375	0.567581	0.283791	+	0.958886i	$0.408408\pi$
$234$	0	0
$235$	−3.61909	−0.236083
$236$	0	0
$237$	−34.4767	−2.23950
$238$	0	0
$239$	12.0242	0.777781	0.388891	−	0.921284i	$-0.372858\pi$
$239$	12.0242	0.777781	0.388891	+	0.921284i	$0.372858\pi$
$240$	0	0
$241$	−5.15016	−0.331751	−0.165876	−	0.986147i	$-0.553045\pi$
$241$	−5.15016	−0.331751	−0.165876	+	0.986147i	$0.553045\pi$
$242$	0	0
$243$	17.6963	1.13522
$244$	0	0
$245$	−19.4783	−1.24442
$246$	0	0
$247$	35.2543	2.24318
$248$	0	0
$249$	5.91659	0.374949
$250$	0	0
$251$	16.5531	1.04482	0.522410	−	0.852694i	$-0.325033\pi$
$251$	16.5531	1.04482	0.522410	+	0.852694i	$0.325033\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	21.1749	1.32602
$256$	0	0
$257$	30.7927	1.92080	0.960399	−	0.278629i	$-0.0898801\pi$
$257$	30.7927	1.92080	0.960399	+	0.278629i	$0.0898801\pi$
$258$	0	0
$259$	3.39012	0.210652
$260$	0	0
$261$	−56.9668	−3.52615
$262$	0	0
$263$	23.3987	1.44283	0.721414	−	0.692504i	$-0.243493\pi$
$263$	23.3987	1.44283	0.721414	+	0.692504i	$0.243493\pi$
$264$	0	0
$265$	−1.38118	−0.0848452
$266$	0	0
$267$	18.5017	1.13228
$268$	0	0
$269$	−24.7314	−1.50790	−0.753950	−	0.656932i	$-0.771854\pi$
$269$	−24.7314	−1.50790	−0.753950	+	0.656932i	$0.771854\pi$
$270$	0	0
$271$	13.6623	0.829926	0.414963	−	0.909838i	$-0.363795\pi$
$271$	13.6623	0.829926	0.414963	+	0.909838i	$0.363795\pi$
$272$	0	0
$273$	−9.83659	−0.595338
$274$	0	0
$275$	4.71898	0.284565
$276$	0	0
$277$	−5.17833	−0.311136	−0.155568	−	0.987825i	$-0.549721\pi$
$277$	−5.17833	−0.311136	−0.155568	+	0.987825i	$0.549721\pi$
$278$	0	0
$279$	−21.8365	−1.30732
$280$	0	0
$281$	−14.7949	−0.882592	−0.441296	−	0.897362i	$-0.645481\pi$
$281$	−14.7949	−0.882592	−0.441296	+	0.897362i	$0.645481\pi$
$282$	0	0
$283$	21.7302	1.29173	0.645863	−	0.763454i	$-0.276498\pi$
$283$	21.7302	1.29173	0.645863	+	0.763454i	$0.276498\pi$
$284$	0	0
$285$	75.5051	4.47254
$286$	0	0
$287$	7.78382	0.459464
$288$	0	0
$289$	−11.9349	−0.702054
$290$	0	0
$291$	−54.8160	−3.21337
$292$	0	0
$293$	10.2925	0.601294	0.300647	−	0.953736i	$-0.402797\pi$
$293$	10.2925	0.601294	0.300647	+	0.953736i	$0.402797\pi$
$294$	0	0
$295$	−13.6134	−0.792605
$296$	0	0
$297$	14.1238	0.819546
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−5.77443	−0.332832
$302$	0	0
$303$	−12.5032	−0.718292
$304$	0	0
$305$	−7.69885	−0.440835
$306$	0	0
$307$	8.16049	0.465744	0.232872	−	0.972507i	$-0.425188\pi$
$307$	8.16049	0.465744	0.232872	+	0.972507i	$0.425188\pi$
$308$	0	0
$309$	5.30679	0.301893
$310$	0	0
$311$	−7.91033	−0.448554	−0.224277	−	0.974525i	$-0.572002\pi$
$311$	−7.91033	−0.448554	−0.224277	+	0.974525i	$0.572002\pi$
$312$	0	0
$313$	−5.12138	−0.289477	−0.144739	−	0.989470i	$-0.546234\pi$
$313$	−5.12138	−0.289477	−0.144739	+	0.989470i	$0.546234\pi$
$314$	0	0
$315$	−14.6358	−0.824634
$316$	0	0
$317$	−22.0604	−1.23903	−0.619517	−	0.784983i	$-0.712672\pi$
$317$	−22.0604	−1.23903	−0.619517	+	0.784983i	$0.712672\pi$
$318$	0	0
$319$	−9.82415	−0.550047
$320$	0	0
$321$	25.5609	1.42667
$322$	0	0
$323$	18.0610	1.00494
$324$	0	0
$325$	17.6082	0.976727
$326$	0	0
$327$	5.94352	0.328677
$328$	0	0
$329$	0.861296	0.0474848
$330$	0	0
$331$	4.29964	0.236330	0.118165	−	0.992994i	$-0.462299\pi$
$331$	4.29964	0.236330	0.118165	+	0.992994i	$0.462299\pi$
$332$	0	0
$333$	−32.4015	−1.77559
$334$	0	0
$335$	−11.6850	−0.638418
$336$	0	0
$337$	−6.16205	−0.335668	−0.167834	−	0.985815i	$-0.553677\pi$
$337$	−6.16205	−0.335668	−0.167834	+	0.985815i	$0.553677\pi$
$338$	0	0
$339$	35.2632	1.91523
$340$	0	0
$341$	−3.76579	−0.203929
$342$	0	0
$343$	9.63560	0.520273
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−10.6240	−0.570324	−0.285162	−	0.958479i	$-0.592047\pi$
$347$	−10.6240	−0.570324	−0.285162	+	0.958479i	$0.592047\pi$
$348$	0	0
$349$	−4.46741	−0.239135	−0.119567	−	0.992826i	$-0.538151\pi$
$349$	−4.46741	−0.239135	−0.119567	+	0.992826i	$0.538151\pi$
$350$	0	0
$351$	52.7009	2.81297
$352$	0	0
$353$	23.5298	1.25237	0.626183	−	0.779676i	$-0.284616\pi$
$353$	23.5298	1.25237	0.626183	+	0.779676i	$0.284616\pi$
$354$	0	0
$355$	24.5748	1.30429
$356$	0	0
$357$	−5.03936	−0.266711
$358$	0	0
$359$	23.5290	1.24181	0.620906	−	0.783885i	$-0.286765\pi$
$359$	23.5290	1.24181	0.620906	+	0.783885i	$0.286765\pi$
$360$	0	0
$361$	45.4017	2.38956
$362$	0	0
$363$	−30.1376	−1.58181
$364$	0	0
$365$	−0.293627	−0.0153692
$366$	0	0
$367$	25.7412	1.34368	0.671839	−	0.740697i	$-0.265505\pi$
$367$	25.7412	1.34368	0.671839	+	0.740697i	$0.265505\pi$
$368$	0	0
$369$	−74.3948	−3.87284
$370$	0	0
$371$	0.328703	0.0170654
$372$	0	0
$373$	15.6759	0.811669	0.405834	−	0.913947i	$-0.366981\pi$
$373$	15.6759	0.811669	0.405834	+	0.913947i	$0.366981\pi$
$374$	0	0
$375$	−9.33131	−0.481867
$376$	0	0
$377$	−36.6574	−1.88795
$378$	0	0
$379$	−28.9204	−1.48554	−0.742770	−	0.669547i	$-0.766488\pi$
$379$	−28.9204	−1.48554	−0.742770	+	0.669547i	$0.766488\pi$
$380$	0	0
$381$	−13.8801	−0.711101
$382$	0	0
$383$	15.7980	0.807241	0.403620	−	0.914927i	$-0.367752\pi$
$383$	15.7980	0.807241	0.403620	+	0.914927i	$0.367752\pi$
$384$	0	0
$385$	−2.52400	−0.128635
$386$	0	0
$387$	55.1898	2.80546
$388$	0	0
$389$	−11.7455	−0.595521	−0.297761	−	0.954641i	$-0.596240\pi$
$389$	−11.7455	−0.595521	−0.297761	+	0.954641i	$0.596240\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−26.4031	−1.33186
$394$	0	0
$395$	−33.0093	−1.66088
$396$	0	0
$397$	2.54792	0.127876	0.0639381	−	0.997954i	$-0.479634\pi$
$397$	2.54792	0.127876	0.0639381	+	0.997954i	$0.479634\pi$
$398$	0	0
$399$	−17.9693	−0.899588
$400$	0	0
$401$	−30.5301	−1.52460	−0.762299	−	0.647225i	$-0.775930\pi$
$401$	−30.5301	−1.52460	−0.762299	+	0.647225i	$0.775930\pi$
$402$	0	0
$403$	−14.0515	−0.699956
$404$	0	0
$405$	51.4010	2.55414
$406$	0	0
$407$	−5.58776	−0.276975
$408$	0	0
$409$	26.4561	1.30817	0.654085	−	0.756421i	$-0.273054\pi$
$409$	26.4561	1.30817	0.654085	+	0.756421i	$0.273054\pi$
$410$	0	0
$411$	−6.46932	−0.319108
$412$	0	0
$413$	3.23982	0.159421
$414$	0	0
$415$	5.66478	0.278073
$416$	0	0
$417$	−32.4267	−1.58794
$418$	0	0
$419$	−4.55765	−0.222656	−0.111328	−	0.993784i	$-0.535510\pi$
$419$	−4.55765	−0.222656	−0.111328	+	0.993784i	$0.535510\pi$
$420$	0	0
$421$	23.8139	1.16062	0.580310	−	0.814396i	$-0.302931\pi$
$421$	23.8139	1.16062	0.580310	+	0.814396i	$0.302931\pi$
$422$	0	0
$423$	−8.23195	−0.400251
$424$	0	0
$425$	9.02080	0.437573
$426$	0	0
$427$	1.83223	0.0886677
$428$	0	0
$429$	16.2132	0.782780
$430$	0	0
$431$	0.774749	0.0373184	0.0186592	−	0.999826i	$-0.494060\pi$
$431$	0.774749	0.0373184	0.0186592	+	0.999826i	$0.494060\pi$
$432$	0	0
$433$	18.1654	0.872971	0.436486	−	0.899711i	$-0.356223\pi$
$433$	18.1654	0.872971	0.436486	+	0.899711i	$0.356223\pi$
$434$	0	0
$435$	−78.5102	−3.76428
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−2.05771	−0.0982091	−0.0491046	−	0.998794i	$-0.515637\pi$
$439$	−2.05771	−0.0982091	−0.0491046	+	0.998794i	$0.515637\pi$
$440$	0	0
$441$	−44.3052	−2.10977
$442$	0	0
$443$	−16.5506	−0.786345	−0.393173	−	0.919465i	$-0.628622\pi$
$443$	−16.5506	−0.786345	−0.393173	+	0.919465i	$0.628622\pi$
$444$	0	0
$445$	17.7142	0.839734
$446$	0	0
$447$	−34.1919	−1.61722
$448$	0	0
$449$	40.8954	1.92997	0.964986	−	0.262300i	$-0.0844810\pi$
$449$	40.8954	1.92997	0.964986	+	0.262300i	$0.0844810\pi$
$450$	0	0
$451$	−12.8297	−0.604126
$452$	0	0
$453$	−44.7081	−2.10057
$454$	0	0
$455$	−9.41794	−0.441520
$456$	0	0
$457$	36.1888	1.69284	0.846420	−	0.532516i	$-0.178753\pi$
$457$	36.1888	1.69284	0.846420	+	0.532516i	$0.178753\pi$
$458$	0	0
$459$	26.9991	1.26021
$460$	0	0
$461$	27.0757	1.26104	0.630521	−	0.776172i	$-0.282841\pi$
$461$	27.0757	1.26104	0.630521	+	0.776172i	$0.282841\pi$
$462$	0	0
$463$	27.3838	1.27264	0.636318	−	0.771427i	$-0.280457\pi$
$463$	27.3838	1.27264	0.636318	+	0.771427i	$0.280457\pi$
$464$	0	0
$465$	−30.0945	−1.39560
$466$	0	0
$467$	−6.16892	−0.285463	−0.142732	−	0.989761i	$-0.545589\pi$
$467$	−6.16892	−0.285463	−0.142732	+	0.989761i	$0.545589\pi$
$468$	0	0
$469$	2.78088	0.128409
$470$	0	0
$471$	−49.5567	−2.28345
$472$	0	0
$473$	9.51771	0.437625
$474$	0	0
$475$	32.1663	1.47589
$476$	0	0
$477$	−3.14162	−0.143845
$478$	0	0
$479$	−9.75228	−0.445593	−0.222796	−	0.974865i	$-0.571519\pi$
$479$	−9.75228	−0.445593	−0.222796	+	0.974865i	$0.571519\pi$
$480$	0	0
$481$	−20.8499	−0.950675
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−52.4830	−2.38313
$486$	0	0
$487$	5.80554	0.263074	0.131537	−	0.991311i	$-0.458009\pi$
$487$	5.80554	0.263074	0.131537	+	0.991311i	$0.458009\pi$
$488$	0	0
$489$	36.2683	1.64011
$490$	0	0
$491$	10.2489	0.462527	0.231263	−	0.972891i	$-0.425714\pi$
$491$	10.2489	0.462527	0.231263	+	0.972891i	$0.425714\pi$
$492$	0	0
$493$	−18.7798	−0.845801
$494$	0	0
$495$	24.1235	1.08427
$496$	0	0
$497$	−5.84849	−0.262341
$498$	0	0
$499$	−12.5733	−0.562856	−0.281428	−	0.959582i	$-0.590808\pi$
$499$	−12.5733	−0.562856	−0.281428	+	0.959582i	$0.590808\pi$
$500$	0	0
$501$	−45.3449	−2.02586
$502$	0	0
$503$	21.6637	0.965938	0.482969	−	0.875638i	$-0.339558\pi$
$503$	21.6637	0.965938	0.482969	+	0.875638i	$0.339558\pi$
$504$	0	0
$505$	−11.9711	−0.532706
$506$	0	0
$507$	19.7449	0.876900
$508$	0	0
$509$	−15.8729	−0.703552	−0.351776	−	0.936084i	$-0.614422\pi$
$509$	−15.8729	−0.703552	−0.351776	+	0.936084i	$0.614422\pi$
$510$	0	0
$511$	0.0698796	0.00309129
$512$	0	0
$513$	96.2729	4.25055
$514$	0	0
$515$	5.08093	0.223893
$516$	0	0
$517$	−1.41963	−0.0624354
$518$	0	0
$519$	21.6127	0.948693
$520$	0	0
$521$	27.2918	1.19567	0.597837	−	0.801617i	$-0.296027\pi$
$521$	27.2918	1.19567	0.597837	+	0.801617i	$0.296027\pi$
$522$	0	0
$523$	39.2610	1.71676	0.858381	−	0.513013i	$-0.171471\pi$
$523$	39.2610	1.71676	0.858381	+	0.513013i	$0.171471\pi$
$524$	0	0
$525$	−8.97498	−0.391700
$526$	0	0
$527$	−7.19869	−0.313580
$528$	0	0
$529$	0	0
$530$	0	0
$531$	−30.9650	−1.34377
$532$	0	0
$533$	−47.8721	−2.07357
$534$	0	0
$535$	24.4730	1.05806
$536$	0	0
$537$	22.4344	0.968116
$538$	0	0
$539$	−7.64060	−0.329104
$540$	0	0
$541$	−5.47677	−0.235465	−0.117732	−	0.993045i	$-0.537563\pi$
$541$	−5.47677	−0.235465	−0.117732	+	0.993045i	$0.537563\pi$
$542$	0	0
$543$	65.4491	2.80869
$544$	0	0
$545$	5.69056	0.243757
$546$	0	0
$547$	−33.1757	−1.41849	−0.709244	−	0.704963i	$-0.750964\pi$
$547$	−33.1757	−1.41849	−0.709244	+	0.704963i	$0.750964\pi$
$548$	0	0
$549$	−17.5117	−0.747383
$550$	0	0
$551$	−66.9649	−2.85280
$552$	0	0
$553$	7.85580	0.334063
$554$	0	0
$555$	−44.6549	−1.89549
$556$	0	0
$557$	−1.98685	−0.0841856	−0.0420928	−	0.999114i	$-0.513403\pi$
$557$	−1.98685	−0.0841856	−0.0420928	+	0.999114i	$0.513403\pi$
$558$	0	0
$559$	35.5139	1.50208
$560$	0	0
$561$	8.30612	0.350685
$562$	0	0
$563$	23.8401	1.00474	0.502369	−	0.864653i	$-0.332462\pi$
$563$	23.8401	1.00474	0.502369	+	0.864653i	$0.332462\pi$
$564$	0	0
$565$	33.7624	1.42039
$566$	0	0
$567$	−12.2328	−0.513729
$568$	0	0
$569$	42.5310	1.78299	0.891497	−	0.453027i	$-0.149656\pi$
$569$	42.5310	1.78299	0.891497	+	0.453027i	$0.149656\pi$
$570$	0	0
$571$	−38.1421	−1.59620	−0.798099	−	0.602526i	$-0.794161\pi$
$571$	−38.1421	−1.59620	−0.798099	+	0.602526i	$0.794161\pi$
$572$	0	0
$573$	−20.1271	−0.840822
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−11.4950	−0.478541	−0.239271	−	0.970953i	$-0.576908\pi$
$577$	−11.4950	−0.478541	−0.239271	+	0.970953i	$0.576908\pi$
$578$	0	0
$579$	−17.3921	−0.722792
$580$	0	0
$581$	−1.34815	−0.0559305
$582$	0	0
$583$	−0.541785	−0.0224385
$584$	0	0
$585$	90.0132	3.72159
$586$	0	0
$587$	9.57620	0.395252	0.197626	−	0.980277i	$-0.436677\pi$
$587$	9.57620	0.395252	0.197626	+	0.980277i	$0.436677\pi$
$588$	0	0
$589$	−25.6690	−1.05767
$590$	0	0
$591$	−33.1619	−1.36410
$592$	0	0
$593$	−15.0152	−0.616601	−0.308300	−	0.951289i	$-0.599760\pi$
$593$	−15.0152	−0.616601	−0.308300	+	0.951289i	$0.599760\pi$
$594$	0	0
$595$	−4.82488	−0.197801
$596$	0	0
$597$	−29.1090	−1.19135
$598$	0	0
$599$	−45.0600	−1.84110	−0.920551	−	0.390623i	$-0.872260\pi$
$599$	−45.0600	−1.84110	−0.920551	+	0.390623i	$0.872260\pi$
$600$	0	0
$601$	23.8747	0.973867	0.486934	−	0.873439i	$-0.338115\pi$
$601$	23.8747	0.973867	0.486934	+	0.873439i	$0.338115\pi$
$602$	0	0
$603$	−26.5786	−1.08236
$604$	0	0
$605$	−28.8549	−1.17312
$606$	0	0
$607$	33.6302	1.36501	0.682504	−	0.730882i	$-0.260891\pi$
$607$	33.6302	1.36501	0.682504	+	0.730882i	$0.260891\pi$
$608$	0	0
$609$	18.6844	0.757131
$610$	0	0
$611$	−5.29715	−0.214300
$612$	0	0
$613$	−44.7903	−1.80906	−0.904532	−	0.426405i	$-0.859780\pi$
$613$	−44.7903	−1.80906	−0.904532	+	0.426405i	$0.859780\pi$
$614$	0	0
$615$	−102.529	−4.13437
$616$	0	0
$617$	10.5276	0.423823	0.211912	−	0.977289i	$-0.432031\pi$
$617$	10.5276	0.423823	0.211912	+	0.977289i	$0.432031\pi$
$618$	0	0
$619$	42.4920	1.70790	0.853949	−	0.520357i	$-0.174201\pi$
$619$	42.4920	1.70790	0.853949	+	0.520357i	$0.174201\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−4.21576	−0.168901
$624$	0	0
$625$	−28.9753	−1.15901
$626$	0	0
$627$	29.6179	1.18282
$628$	0	0
$629$	−10.6816	−0.425902
$630$	0	0
$631$	−37.1922	−1.48060	−0.740299	−	0.672278i	$-0.765316\pi$
$631$	−37.1922	−1.48060	−0.740299	+	0.672278i	$0.765316\pi$
$632$	0	0
$633$	−24.8015	−0.985773
$634$	0	0
$635$	−13.2894	−0.527373
$636$	0	0
$637$	−28.5098	−1.12960
$638$	0	0
$639$	55.8977	2.21128
$640$	0	0
$641$	−14.2895	−0.564402	−0.282201	−	0.959355i	$-0.591065\pi$
$641$	−14.2895	−0.564402	−0.282201	+	0.959355i	$0.591065\pi$
$642$	0	0
$643$	27.0549	1.06694	0.533470	−	0.845819i	$-0.320888\pi$
$643$	27.0549	1.06694	0.533470	+	0.845819i	$0.320888\pi$
$644$	0	0
$645$	76.0612	2.99491
$646$	0	0
$647$	−39.1191	−1.53793	−0.768965	−	0.639290i	$-0.779228\pi$
$647$	−39.1191	−1.53793	−0.768965	+	0.639290i	$0.779228\pi$
$648$	0	0
$649$	−5.34004	−0.209615
$650$	0	0
$651$	7.16212	0.280705
$652$	0	0
$653$	15.9867	0.625606	0.312803	−	0.949818i	$-0.398732\pi$
$653$	15.9867	0.625606	0.312803	+	0.949818i	$0.398732\pi$
$654$	0	0
$655$	−25.2794	−0.987746
$656$	0	0
$657$	−0.667883	−0.0260566
$658$	0	0
$659$	38.3067	1.49222	0.746109	−	0.665824i	$-0.231920\pi$
$659$	38.3067	1.49222	0.746109	+	0.665824i	$0.231920\pi$
$660$	0	0
$661$	−17.6495	−0.686486	−0.343243	−	0.939247i	$-0.611525\pi$
$661$	−17.6495	−0.686486	−0.343243	+	0.939247i	$0.611525\pi$
$662$	0	0
$663$	30.9931	1.20367
$664$	0	0
$665$	−17.2045	−0.667161
$666$	0	0
$667$	0	0
$668$	0	0
$669$	65.7077	2.54041
$670$	0	0
$671$	−3.01997	−0.116585
$672$	0	0
$673$	14.9681	0.576978	0.288489	−	0.957483i	$-0.406847\pi$
$673$	14.9681	0.576978	0.288489	+	0.957483i	$0.406847\pi$
$674$	0	0
$675$	48.0847	1.85078
$676$	0	0
$677$	26.8985	1.03379	0.516897	−	0.856048i	$-0.327087\pi$
$677$	26.8985	1.03379	0.516897	+	0.856048i	$0.327087\pi$
$678$	0	0
$679$	12.4903	0.479333
$680$	0	0
$681$	16.3133	0.625127
$682$	0	0
$683$	24.3824	0.932967	0.466484	−	0.884530i	$-0.345521\pi$
$683$	24.3824	0.932967	0.466484	+	0.884530i	$0.345521\pi$
$684$	0	0
$685$	−6.19398	−0.236660
$686$	0	0
$687$	−5.85550	−0.223401
$688$	0	0
$689$	−2.02159	−0.0770165
$690$	0	0
$691$	−35.9497	−1.36759	−0.683795	−	0.729674i	$-0.739672\pi$
$691$	−35.9497	−1.36759	−0.683795	+	0.729674i	$0.739672\pi$
$692$	0	0
$693$	−5.74108	−0.218086
$694$	0	0
$695$	−31.0466	−1.17766
$696$	0	0
$697$	−24.5252	−0.928959
$698$	0	0
$699$	27.1590	1.02725
$700$	0	0
$701$	8.95591	0.338260	0.169130	−	0.985594i	$-0.445904\pi$
$701$	8.95591	0.338260	0.169130	+	0.985594i	$0.445904\pi$
$702$	0	0
$703$	−38.0882	−1.43652
$704$	0	0
$705$	−11.3451	−0.427280
$706$	0	0
$707$	2.84897	0.107146
$708$	0	0
$709$	−47.7568	−1.79354	−0.896772	−	0.442493i	$-0.854094\pi$
$709$	−47.7568	−1.79354	−0.896772	+	0.442493i	$0.854094\pi$
$710$	0	0
$711$	−75.0828	−2.81583
$712$	0	0
$713$	0	0
$714$	0	0
$715$	15.5231	0.580532
$716$	0	0
$717$	37.6933	1.40768
$718$	0	0
$719$	−0.623599	−0.0232563	−0.0116282	−	0.999932i	$-0.503701\pi$
$719$	−0.623599	−0.0232563	−0.0116282	+	0.999932i	$0.503701\pi$
$720$	0	0
$721$	−1.20920	−0.0450328
$722$	0	0
$723$	−16.1447	−0.600427
$724$	0	0
$725$	−33.4464	−1.24217
$726$	0	0
$727$	−49.9418	−1.85224	−0.926120	−	0.377230i	$-0.876877\pi$
$727$	−49.9418	−1.85224	−0.926120	+	0.377230i	$0.876877\pi$
$728$	0	0
$729$	4.09677	0.151732
$730$	0	0
$731$	18.1940	0.672931
$732$	0	0
$733$	−39.5908	−1.46232	−0.731160	−	0.682207i	$-0.761021\pi$
$733$	−39.5908	−1.46232	−0.731160	+	0.682207i	$0.761021\pi$
$734$	0	0
$735$	−61.0602	−2.25224
$736$	0	0
$737$	−4.58358	−0.168838
$738$	0	0
$739$	35.1757	1.29396	0.646979	−	0.762508i	$-0.276032\pi$
$739$	35.1757	1.29396	0.646979	+	0.762508i	$0.276032\pi$
$740$	0	0
$741$	110.515	4.05986
$742$	0	0
$743$	−12.6253	−0.463178	−0.231589	−	0.972814i	$-0.574392\pi$
$743$	−12.6253	−0.463178	−0.231589	+	0.972814i	$0.574392\pi$
$744$	0	0
$745$	−32.7367	−1.19938
$746$	0	0
$747$	12.8851	0.471440
$748$	0	0
$749$	−5.82427	−0.212814
$750$	0	0
$751$	10.4413	0.381008	0.190504	−	0.981686i	$-0.438988\pi$
$751$	10.4413	0.381008	0.190504	+	0.981686i	$0.438988\pi$
$752$	0	0
$753$	51.8904	1.89099
$754$	0	0
$755$	−42.8053	−1.55784
$756$	0	0
$757$	−34.3072	−1.24692	−0.623459	−	0.781856i	$-0.714273\pi$
$757$	−34.3072	−1.24692	−0.623459	+	0.781856i	$0.714273\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	32.0445	1.16161	0.580805	−	0.814042i	$-0.302738\pi$
$761$	32.0445	1.16161	0.580805	+	0.814042i	$0.302738\pi$
$762$	0	0
$763$	−1.35428	−0.0490283
$764$	0	0
$765$	46.1144	1.66727
$766$	0	0
$767$	−19.9256	−0.719472
$768$	0	0
$769$	18.0785	0.651926	0.325963	−	0.945382i	$-0.394311\pi$
$769$	18.0785	0.651926	0.325963	+	0.945382i	$0.394311\pi$
$770$	0	0
$771$	96.5287	3.47640
$772$	0	0
$773$	−40.6017	−1.46034	−0.730170	−	0.683265i	$-0.760559\pi$
$773$	−40.6017	−1.46034	−0.730170	+	0.683265i	$0.760559\pi$
$774$	0	0
$775$	−12.8207	−0.460533
$776$	0	0
$777$	10.6273	0.381252
$778$	0	0
$779$	−87.4516	−3.13328
$780$	0	0
$781$	9.63978	0.344938
$782$	0	0
$783$	−100.104	−3.57744
$784$	0	0
$785$	−47.4475	−1.69347
$786$	0	0
$787$	−14.3031	−0.509851	−0.254926	−	0.966961i	$-0.582051\pi$
$787$	−14.3031	−0.509851	−0.254926	+	0.966961i	$0.582051\pi$
$788$	0	0
$789$	73.3501	2.61133
$790$	0	0
$791$	−8.03502	−0.285692
$792$	0	0
$793$	−11.2686	−0.400159
$794$	0	0
$795$	−4.32970	−0.153559
$796$	0	0
$797$	−27.4560	−0.972540	−0.486270	−	0.873809i	$-0.661643\pi$
$797$	−27.4560	−0.972540	−0.486270	+	0.873809i	$0.661643\pi$
$798$	0	0
$799$	−2.71377	−0.0960062
$800$	0	0
$801$	40.2926	1.42367
$802$	0	0
$803$	−0.115179	−0.00406458
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−77.5277	−2.72910
$808$	0	0
$809$	14.8454	0.521936	0.260968	−	0.965347i	$-0.415958\pi$
$809$	14.8454	0.521936	0.260968	+	0.965347i	$0.415958\pi$
$810$	0	0
$811$	−37.5648	−1.31908	−0.659539	−	0.751670i	$-0.729249\pi$
$811$	−37.5648	−1.31908	−0.659539	+	0.751670i	$0.729249\pi$
$812$	0	0
$813$	42.8284	1.50206
$814$	0	0
$815$	34.7247	1.21635
$816$	0	0
$817$	64.8760	2.26973
$818$	0	0
$819$	−21.4220	−0.748545
$820$	0	0
$821$	48.0724	1.67774	0.838869	−	0.544333i	$-0.183217\pi$
$821$	48.0724	1.67774	0.838869	+	0.544333i	$0.183217\pi$
$822$	0	0
$823$	13.6897	0.477194	0.238597	−	0.971119i	$-0.423313\pi$
$823$	13.6897	0.477194	0.238597	+	0.971119i	$0.423313\pi$
$824$	0	0
$825$	14.7930	0.515027
$826$	0	0
$827$	−11.5023	−0.399974	−0.199987	−	0.979799i	$-0.564090\pi$
$827$	−11.5023	−0.399974	−0.199987	+	0.979799i	$0.564090\pi$
$828$	0	0
$829$	−12.2812	−0.426542	−0.213271	−	0.976993i	$-0.568412\pi$
$829$	−12.2812	−0.426542	−0.213271	+	0.976993i	$0.568412\pi$
$830$	0	0
$831$	−16.2330	−0.563115
$832$	0	0
$833$	−14.6058	−0.506060
$834$	0	0
$835$	−43.4150	−1.50244
$836$	0	0
$837$	−38.3720	−1.32633
$838$	0	0
$839$	−12.3134	−0.425107	−0.212554	−	0.977149i	$-0.568178\pi$
$839$	−12.3134	−0.425107	−0.212554	+	0.977149i	$0.568178\pi$
$840$	0	0
$841$	40.6300	1.40103
$842$	0	0
$843$	−46.3790	−1.59738
$844$	0	0
$845$	18.9045	0.650335
$846$	0	0
$847$	6.86710	0.235956
$848$	0	0
$849$	68.1195	2.33786
$850$	0	0
$851$	0	0
$852$	0	0
$853$	35.6446	1.22045	0.610225	−	0.792229i	$-0.291079\pi$
$853$	35.6446	1.22045	0.610225	+	0.792229i	$0.291079\pi$
$854$	0	0
$855$	164.434	5.62353
$856$	0	0
$857$	13.0385	0.445388	0.222694	−	0.974888i	$-0.428515\pi$
$857$	13.0385	0.445388	0.222694	+	0.974888i	$0.428515\pi$
$858$	0	0
$859$	34.4240	1.17453	0.587266	−	0.809394i	$-0.300204\pi$
$859$	34.4240	1.17453	0.587266	+	0.809394i	$0.300204\pi$
$860$	0	0
$861$	24.4006	0.831571
$862$	0	0
$863$	24.3365	0.828423	0.414212	−	0.910181i	$-0.364057\pi$
$863$	24.3365	0.828423	0.414212	+	0.910181i	$0.364057\pi$
$864$	0	0
$865$	20.6929	0.703579
$866$	0	0
$867$	−37.4134	−1.27063
$868$	0	0
$869$	−12.9483	−0.439242
$870$	0	0
$871$	−17.1030	−0.579512
$872$	0	0
$873$	−119.377	−4.04031
$874$	0	0
$875$	2.12622	0.0718792
$876$	0	0
$877$	18.8497	0.636510	0.318255	−	0.948005i	$-0.396903\pi$
$877$	18.8497	0.636510	0.318255	+	0.948005i	$0.396903\pi$
$878$	0	0
$879$	32.2648	1.08826
$880$	0	0
$881$	−40.4299	−1.36212	−0.681058	−	0.732229i	$-0.738480\pi$
$881$	−40.4299	−1.36212	−0.681058	+	0.732229i	$0.738480\pi$
$882$	0	0
$883$	24.6286	0.828817	0.414408	−	0.910091i	$-0.363989\pi$
$883$	24.6286	0.828817	0.414408	+	0.910091i	$0.363989\pi$
$884$	0	0
$885$	−42.6752	−1.43451
$886$	0	0
$887$	8.78585	0.295000	0.147500	−	0.989062i	$-0.452877\pi$
$887$	8.78585	0.295000	0.147500	+	0.989062i	$0.452877\pi$
$888$	0	0
$889$	3.16270	0.106074
$890$	0	0
$891$	20.1627	0.675476
$892$	0	0
$893$	−9.67672	−0.323819
$894$	0	0
$895$	21.4796	0.717983
$896$	0	0
$897$	0	0
$898$	0	0
$899$	26.6906	0.890181
$900$	0	0
$901$	−1.03568	−0.0345034
$902$	0	0
$903$	−18.1016	−0.602384
$904$	0	0
$905$	62.6636	2.08301
$906$	0	0
$907$	6.65030	0.220820	0.110410	−	0.993886i	$-0.464784\pi$
$907$	6.65030	0.220820	0.110410	+	0.993886i	$0.464784\pi$
$908$	0	0
$909$	−27.2294	−0.903141
$910$	0	0
$911$	26.0295	0.862396	0.431198	−	0.902257i	$-0.358091\pi$
$911$	26.0295	0.862396	0.431198	+	0.902257i	$0.358091\pi$
$912$	0	0
$913$	2.22208	0.0735402
$914$	0	0
$915$	−24.1343	−0.797854
$916$	0	0
$917$	6.01617	0.198671
$918$	0	0
$919$	40.9766	1.35169	0.675847	−	0.737042i	$-0.263778\pi$
$919$	40.9766	1.35169	0.675847	+	0.737042i	$0.263778\pi$
$920$	0	0
$921$	25.5814	0.842936
$922$	0	0
$923$	35.9694	1.18395
$924$	0	0
$925$	−19.0236	−0.625492
$926$	0	0
$927$	11.5571	0.379583
$928$	0	0
$929$	−8.78214	−0.288133	−0.144066	−	0.989568i	$-0.546018\pi$
$929$	−8.78214	−0.288133	−0.144066	+	0.989568i	$0.546018\pi$
$930$	0	0
$931$	−52.0810	−1.70689
$932$	0	0
$933$	−24.7972	−0.811824
$934$	0	0
$935$	7.95261	0.260078
$936$	0	0
$937$	12.1587	0.397206	0.198603	−	0.980080i	$-0.436360\pi$
$937$	12.1587	0.397206	0.198603	+	0.980080i	$0.436360\pi$
$938$	0	0
$939$	−16.0544	−0.523917
$940$	0	0
$941$	22.7011	0.740035	0.370018	−	0.929025i	$-0.379352\pi$
$941$	22.7011	0.740035	0.370018	+	0.929025i	$0.379352\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−25.7186	−0.836627
$946$	0	0
$947$	−4.33698	−0.140933	−0.0704664	−	0.997514i	$-0.522449\pi$
$947$	−4.33698	−0.140933	−0.0704664	+	0.997514i	$0.522449\pi$
$948$	0	0
$949$	−0.429774	−0.0139511
$950$	0	0
$951$	−69.1546	−2.24249
$952$	0	0
$953$	13.2316	0.428613	0.214306	−	0.976766i	$-0.431251\pi$
$953$	13.2316	0.428613	0.214306	+	0.976766i	$0.431251\pi$
$954$	0	0
$955$	−19.2705	−0.623578
$956$	0	0
$957$	−30.7966	−0.995514
$958$	0	0
$959$	1.47409	0.0476008
$960$	0	0
$961$	−20.7690	−0.669967
$962$	0	0
$963$	55.6662	1.79382
$964$	0	0
$965$	−16.6519	−0.536044
$966$	0	0
$967$	−20.8725	−0.671215	−0.335607	−	0.942002i	$-0.608942\pi$
$967$	−20.8725	−0.671215	−0.335607	+	0.942002i	$0.608942\pi$
$968$	0	0
$969$	56.6175	1.81882
$970$	0	0
$971$	15.4136	0.494645	0.247323	−	0.968933i	$-0.420449\pi$
$971$	15.4136	0.494645	0.247323	+	0.968933i	$0.420449\pi$
$972$	0	0
$973$	7.38870	0.236871
$974$	0	0
$975$	55.1980	1.76775
$976$	0	0
$977$	40.6854	1.30164	0.650821	−	0.759231i	$-0.274425\pi$
$977$	40.6854	1.30164	0.650821	+	0.759231i	$0.274425\pi$
$978$	0	0
$979$	6.94863	0.222079
$980$	0	0
$981$	12.9437	0.413261
$982$	0	0
$983$	−19.6080	−0.625398	−0.312699	−	0.949852i	$-0.601233\pi$
$983$	−19.6080	−0.625398	−0.312699	+	0.949852i	$0.601233\pi$
$984$	0	0
$985$	−31.7505	−1.01165
$986$	0	0
$987$	2.69998	0.0859414
$988$	0	0
$989$	0	0
$990$	0	0
$991$	26.5462	0.843267	0.421634	−	0.906766i	$-0.361457\pi$
$991$	26.5462	0.843267	0.421634	+	0.906766i	$0.361457\pi$
$992$	0	0
$993$	13.4785	0.427726
$994$	0	0
$995$	−27.8701	−0.883541
$996$	0	0
$997$	−34.6295	−1.09673	−0.548363	−	0.836240i	$-0.684749\pi$
$997$	−34.6295	−1.09673	−0.548363	+	0.836240i	$0.684749\pi$
$998$	0	0
$999$	−56.9372	−1.80141

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8464.2.a.ch.1.15		15
4.3	odd	2		4232.2.a.ba.1.1		15
23.11	odd	22		368.2.m.e.305.1		30
23.21	odd	22		368.2.m.e.257.1		30
23.22	odd	2		8464.2.a.cg.1.15		15
92.11	even	22		184.2.i.b.121.3	yes	30
92.67	even	22		184.2.i.b.73.3	✓	30
92.91	even	2		4232.2.a.bb.1.1		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
184.2.i.b.73.3	✓	30	92.67	even	22
184.2.i.b.121.3	yes	30	92.11	even	22
368.2.m.e.257.1		30	23.21	odd	22
368.2.m.e.305.1		30	23.11	odd	22
4232.2.a.ba.1.1		15	4.3	odd	2
4232.2.a.bb.1.1		15	92.91	even	2
8464.2.a.cg.1.15		15	23.22	odd	2
8464.2.a.ch.1.15		15	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8464 = 2^{4} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8464.a (trivial)

\(f(q)\)	\(=\)	\(q+3.13479 q^{3} +3.00137 q^{5} -0.714288 q^{7} +6.82690 q^{9} +1.17733 q^{11} +4.39302 q^{13} +9.40866 q^{15} +2.25057 q^{17} +8.02507 q^{19} -2.23914 q^{21} +4.00822 q^{25} +11.9965 q^{27} -8.34446 q^{29} -3.19860 q^{31} +3.69067 q^{33} -2.14384 q^{35} -4.74615 q^{37} +13.7712 q^{39} -10.8973 q^{41} +8.08417 q^{43} +20.4900 q^{45} -1.20581 q^{47} -6.48979 q^{49} +7.05508 q^{51} -0.460183 q^{53} +3.53359 q^{55} +25.1569 q^{57} -4.53574 q^{59} -2.56511 q^{61} -4.87637 q^{63} +13.1851 q^{65} -3.89321 q^{67} +8.18786 q^{71} -0.0978311 q^{73} +12.5649 q^{75} -0.840950 q^{77} -10.9981 q^{79} +17.1258 q^{81} +1.88740 q^{83} +6.75481 q^{85} -26.1581 q^{87} +5.90204 q^{89} -3.13788 q^{91} -10.0269 q^{93} +24.0862 q^{95} -17.4863 q^{97} +8.03748 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - q^{3} + 10 q^{7} + 16 q^{9} + 23 q^{11} + 10 q^{15} + 29 q^{19} - q^{21} + 23 q^{25} - q^{27} - 2 q^{29} - 20 q^{31} - 18 q^{33} + 18 q^{35} - 24 q^{37} + 19 q^{39} + 9 q^{41} + 48 q^{43} - 4 q^{45}+ \cdots + 63 q^{99}+O(q^{100}) \) 15 * q - q^3 + 10 * q^7 + 16 * q^9 + 23 * q^11 + 10 * q^15 + 29 * q^19 - q^21 + 23 * q^25 - q^27 - 2 * q^29 - 20 * q^31 - 18 * q^33 + 18 * q^35 - 24 * q^37 + 19 * q^39 + 9 * q^41 + 48 * q^43 - 4 * q^45 + 36 * q^47 + 25 * q^49 + 35 * q^51 + 5 * q^53 + 10 * q^55 - 23 * q^57 + 22 * q^59 - 12 * q^61 + 35 * q^63 + 26 * q^65 + 58 * q^67 - 2 * q^71 + 5 * q^73 + 17 * q^75 + 26 * q^77 + 26 * q^79 - 21 * q^81 + 68 * q^83 - 72 * q^85 - 19 * q^87 + 6 * q^89 + 71 * q^91 - 55 * q^93 + 12 * q^95 - 40 * q^97 + 63 * q^99

Introduction

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