LMFDB - Embedded newform 8624.2.a.bh.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8624,2,Mod(1,8624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8624, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8624.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8624 = 2^{4} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8624.a (trivial)

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:	yes
Analytic conductor:	$68.8629867032$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 154)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	8624.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.41421 q^{3} +0.585786 q^{5} +2.82843 q^{9} -1.00000 q^{11} +3.82843 q^{13} -1.41421 q^{15} -3.65685 q^{17} -0.585786 q^{19} +6.24264 q^{23} -4.65685 q^{25} +0.414214 q^{27} +2.65685 q^{29} -4.00000 q^{31} +2.41421 q^{33} -9.41421 q^{37} -9.24264 q^{39} +5.41421 q^{41} +5.65685 q^{43} +1.65685 q^{45} -10.4853 q^{47} +8.82843 q^{51} +7.89949 q^{53} -0.585786 q^{55} +1.41421 q^{57} -5.58579 q^{59} -11.8284 q^{61} +2.24264 q^{65} -2.75736 q^{67} -15.0711 q^{69} +11.0711 q^{71} +9.41421 q^{73} +11.2426 q^{75} +13.2426 q^{79} -9.48528 q^{81} -12.1421 q^{83} -2.14214 q^{85} -6.41421 q^{87} -12.4853 q^{89} +9.65685 q^{93} -0.343146 q^{95} +3.82843 q^{97} -2.82843 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 4 q^{5} - 2 q^{11} + 2 q^{13} + 4 q^{17} - 4 q^{19} + 4 q^{23} + 2 q^{25} - 2 q^{27} - 6 q^{29} - 8 q^{31} + 2 q^{33} - 16 q^{37} - 10 q^{39} + 8 q^{41} - 8 q^{45} - 4 q^{47} + 12 q^{51}+ \cdots + 2 q^{97}+O(q^{100}) $ 2 * q - 2 * q^3 + 4 * q^5 - 2 * q^11 + 2 * q^13 + 4 * q^17 - 4 * q^19 + 4 * q^23 + 2 * q^25 - 2 * q^27 - 6 * q^29 - 8 * q^31 + 2 * q^33 - 16 * q^37 - 10 * q^39 + 8 * q^41 - 8 * q^45 - 4 * q^47 + 12 * q^51 - 4 * q^53 - 4 * q^55 - 14 * q^59 - 18 * q^61 - 4 * q^65 - 14 * q^67 - 16 * q^69 + 8 * q^71 + 16 * q^73 + 14 * q^75 + 18 * q^79 - 2 * q^81 + 4 * q^83 + 24 * q^85 - 10 * q^87 - 8 * q^89 + 8 * q^93 - 12 * q^95 + 2 * q^97

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
$3$	−2.41421	−1.39385	−0.696923	−	0.717146i	$-0.745448\pi$
$3$	−2.41421	−1.39385	−0.696923	+	0.717146i	$0.745448\pi$
$4$	0	0
$5$	0.585786	0.261972	0.130986	−	0.991384i	$-0.458186\pi$
$5$	0.585786	0.261972	0.130986	+	0.991384i	$0.458186\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	2.82843	0.942809
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	3.82843	1.06181	0.530907	−	0.847430i	$-0.321851\pi$
$13$	3.82843	1.06181	0.530907	+	0.847430i	$0.321851\pi$
$14$	0	0
$15$	−1.41421	−0.365148
$16$	0	0
$17$	−3.65685	−0.886917	−0.443459	−	0.896295i	$-0.646249\pi$
$17$	−3.65685	−0.886917	−0.443459	+	0.896295i	$0.646249\pi$
$18$	0	0
$19$	−0.585786	−0.134389	−0.0671943	−	0.997740i	$-0.521405\pi$
$19$	−0.585786	−0.134389	−0.0671943	+	0.997740i	$0.521405\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.24264	1.30168	0.650840	−	0.759215i	$-0.274417\pi$
$23$	6.24264	1.30168	0.650840	+	0.759215i	$0.274417\pi$
$24$	0	0
$25$	−4.65685	−0.931371
$26$	0	0
$27$	0.414214	0.0797154
$28$	0	0
$29$	2.65685	0.493365	0.246683	−	0.969096i	$-0.420659\pi$
$29$	2.65685	0.493365	0.246683	+	0.969096i	$0.420659\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	2.41421	0.420261
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−9.41421	−1.54769	−0.773844	−	0.633377i	$-0.781668\pi$
$37$	−9.41421	−1.54769	−0.773844	+	0.633377i	$0.781668\pi$
$38$	0	0
$39$	−9.24264	−1.48001
$40$	0	0
$41$	5.41421	0.845558	0.422779	−	0.906233i	$-0.361055\pi$
$41$	5.41421	0.845558	0.422779	+	0.906233i	$0.361055\pi$
$42$	0	0
$43$	5.65685	0.862662	0.431331	−	0.902194i	$-0.358044\pi$
$43$	5.65685	0.862662	0.431331	+	0.902194i	$0.358044\pi$
$44$	0	0
$45$	1.65685	0.246989
$46$	0	0
$47$	−10.4853	−1.52944	−0.764718	−	0.644365i	$-0.777122\pi$
$47$	−10.4853	−1.52944	−0.764718	+	0.644365i	$0.777122\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	8.82843	1.23623
$52$	0	0
$53$	7.89949	1.08508	0.542540	−	0.840030i	$-0.317463\pi$
$53$	7.89949	1.08508	0.542540	+	0.840030i	$0.317463\pi$
$54$	0	0
$55$	−0.585786	−0.0789874
$56$	0	0
$57$	1.41421	0.187317
$58$	0	0
$59$	−5.58579	−0.727207	−0.363604	−	0.931554i	$-0.618454\pi$
$59$	−5.58579	−0.727207	−0.363604	+	0.931554i	$0.618454\pi$
$60$	0	0
$61$	−11.8284	−1.51447	−0.757237	−	0.653140i	$-0.773451\pi$
$61$	−11.8284	−1.51447	−0.757237	+	0.653140i	$0.773451\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.24264	0.278165
$66$	0	0
$67$	−2.75736	−0.336865	−0.168433	−	0.985713i	$-0.553871\pi$
$67$	−2.75736	−0.336865	−0.168433	+	0.985713i	$0.553871\pi$
$68$	0	0
$69$	−15.0711	−1.81434
$70$	0	0
$71$	11.0711	1.31389	0.656947	−	0.753937i	$-0.271848\pi$
$71$	11.0711	1.31389	0.656947	+	0.753937i	$0.271848\pi$
$72$	0	0
$73$	9.41421	1.10185	0.550925	−	0.834555i	$-0.314275\pi$
$73$	9.41421	1.10185	0.550925	+	0.834555i	$0.314275\pi$
$74$	0	0
$75$	11.2426	1.29819
$76$	0	0
$77$	0	0
$78$	0	0
$79$	13.2426	1.48991	0.744957	−	0.667113i	$-0.232470\pi$
$79$	13.2426	1.48991	0.744957	+	0.667113i	$0.232470\pi$
$80$	0	0
$81$	−9.48528	−1.05392
$82$	0	0
$83$	−12.1421	−1.33277	−0.666386	−	0.745607i	$-0.732160\pi$
$83$	−12.1421	−1.33277	−0.666386	+	0.745607i	$0.732160\pi$
$84$	0	0
$85$	−2.14214	−0.232347
$86$	0	0
$87$	−6.41421	−0.687676
$88$	0	0
$89$	−12.4853	−1.32344	−0.661719	−	0.749752i	$-0.730173\pi$
$89$	−12.4853	−1.32344	−0.661719	+	0.749752i	$0.730173\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	9.65685	1.00137
$94$	0	0
$95$	−0.343146	−0.0352060
$96$	0	0
$97$	3.82843	0.388718	0.194359	−	0.980930i	$-0.437737\pi$
$97$	3.82843	0.388718	0.194359	+	0.980930i	$0.437737\pi$
$98$	0	0
$99$	−2.82843	−0.284268
$100$	0	0
$101$	−6.17157	−0.614094	−0.307047	−	0.951694i	$-0.599341\pi$
$101$	−6.17157	−0.614094	−0.307047	+	0.951694i	$0.599341\pi$
$102$	0	0
$103$	13.4142	1.32174	0.660871	−	0.750500i	$-0.270187\pi$
$103$	13.4142	1.32174	0.660871	+	0.750500i	$0.270187\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.07107	0.296891	0.148446	−	0.988921i	$-0.452573\pi$
$107$	3.07107	0.296891	0.148446	+	0.988921i	$0.452573\pi$
$108$	0	0
$109$	16.4853	1.57900	0.789502	−	0.613748i	$-0.210339\pi$
$109$	16.4853	1.57900	0.789502	+	0.613748i	$0.210339\pi$
$110$	0	0
$111$	22.7279	2.15724
$112$	0	0
$113$	−8.17157	−0.768717	−0.384358	−	0.923184i	$-0.625577\pi$
$113$	−8.17157	−0.768717	−0.384358	+	0.923184i	$0.625577\pi$
$114$	0	0
$115$	3.65685	0.341003
$116$	0	0
$117$	10.8284	1.00109
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−13.0711	−1.17858
$124$	0	0
$125$	−5.65685	−0.505964
$126$	0	0
$127$	−15.7279	−1.39563	−0.697814	−	0.716279i	$-0.745844\pi$
$127$	−15.7279	−1.39563	−0.697814	+	0.716279i	$0.745844\pi$
$128$	0	0
$129$	−13.6569	−1.20242
$130$	0	0
$131$	−0.585786	−0.0511804	−0.0255902	−	0.999673i	$-0.508147\pi$
$131$	−0.585786	−0.0511804	−0.0255902	+	0.999673i	$0.508147\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0.242641	0.0208832
$136$	0	0
$137$	−16.6569	−1.42309	−0.711546	−	0.702640i	$-0.752004\pi$
$137$	−16.6569	−1.42309	−0.711546	+	0.702640i	$0.752004\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	25.3137	2.13180
$142$	0	0
$143$	−3.82843	−0.320149
$144$	0	0
$145$	1.55635	0.129248
$146$	0	0
$147$	0	0
$148$	0	0
$149$	17.6569	1.44651	0.723253	−	0.690583i	$-0.242646\pi$
$149$	17.6569	1.44651	0.723253	+	0.690583i	$0.242646\pi$
$150$	0	0
$151$	15.7279	1.27992	0.639960	−	0.768408i	$-0.278951\pi$
$151$	15.7279	1.27992	0.639960	+	0.768408i	$0.278951\pi$
$152$	0	0
$153$	−10.3431	−0.836194
$154$	0	0
$155$	−2.34315	−0.188206
$156$	0	0
$157$	−17.6569	−1.40917	−0.704585	−	0.709619i	$-0.748867\pi$
$157$	−17.6569	−1.40917	−0.704585	+	0.709619i	$0.748867\pi$
$158$	0	0
$159$	−19.0711	−1.51243
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−9.72792	−0.761950	−0.380975	−	0.924585i	$-0.624412\pi$
$163$	−9.72792	−0.761950	−0.380975	+	0.924585i	$0.624412\pi$
$164$	0	0
$165$	1.41421	0.110096
$166$	0	0
$167$	13.7279	1.06230	0.531149	−	0.847278i	$-0.321760\pi$
$167$	13.7279	1.06230	0.531149	+	0.847278i	$0.321760\pi$
$168$	0	0
$169$	1.65685	0.127450
$170$	0	0
$171$	−1.65685	−0.126703
$172$	0	0
$173$	9.82843	0.747241	0.373621	−	0.927582i	$-0.378116\pi$
$173$	9.82843	0.747241	0.373621	+	0.927582i	$0.378116\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	13.4853	1.01362
$178$	0	0
$179$	−0.899495	−0.0672314	−0.0336157	−	0.999435i	$-0.510702\pi$
$179$	−0.899495	−0.0672314	−0.0336157	+	0.999435i	$0.510702\pi$
$180$	0	0
$181$	7.65685	0.569129	0.284565	−	0.958657i	$-0.408151\pi$
$181$	7.65685	0.569129	0.284565	+	0.958657i	$0.408151\pi$
$182$	0	0
$183$	28.5563	2.11095
$184$	0	0
$185$	−5.51472	−0.405450
$186$	0	0
$187$	3.65685	0.267416
$188$	0	0
$189$	0	0
$190$	0	0
$191$	7.17157	0.518917	0.259458	−	0.965754i	$-0.416456\pi$
$191$	7.17157	0.518917	0.259458	+	0.965754i	$0.416456\pi$
$192$	0	0
$193$	21.8995	1.57636	0.788180	−	0.615445i	$-0.211024\pi$
$193$	21.8995	1.57636	0.788180	+	0.615445i	$0.211024\pi$
$194$	0	0
$195$	−5.41421	−0.387720
$196$	0	0
$197$	−0.514719	−0.0366722	−0.0183361	−	0.999832i	$-0.505837\pi$
$197$	−0.514719	−0.0366722	−0.0183361	+	0.999832i	$0.505837\pi$
$198$	0	0
$199$	0.100505	0.00712462	0.00356231	−	0.999994i	$-0.498866\pi$
$199$	0.100505	0.00712462	0.00356231	+	0.999994i	$0.498866\pi$
$200$	0	0
$201$	6.65685	0.469538
$202$	0	0
$203$	0	0
$204$	0	0
$205$	3.17157	0.221512
$206$	0	0
$207$	17.6569	1.22724
$208$	0	0
$209$	0.585786	0.0405197
$210$	0	0
$211$	−7.41421	−0.510416	−0.255208	−	0.966886i	$-0.582144\pi$
$211$	−7.41421	−0.510416	−0.255208	+	0.966886i	$0.582144\pi$
$212$	0	0
$213$	−26.7279	−1.83137
$214$	0	0
$215$	3.31371	0.225993
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−22.7279	−1.53581
$220$	0	0
$221$	−14.0000	−0.941742
$222$	0	0
$223$	8.58579	0.574947	0.287473	−	0.957789i	$-0.407185\pi$
$223$	8.58579	0.574947	0.287473	+	0.957789i	$0.407185\pi$
$224$	0	0
$225$	−13.1716	−0.878105
$226$	0	0
$227$	−28.8284	−1.91341	−0.956705	−	0.291059i	$-0.905992\pi$
$227$	−28.8284	−1.91341	−0.956705	+	0.291059i	$0.905992\pi$
$228$	0	0
$229$	−23.3137	−1.54061	−0.770307	−	0.637674i	$-0.779897\pi$
$229$	−23.3137	−1.54061	−0.770307	+	0.637674i	$0.779897\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.41421	0.0926482	0.0463241	−	0.998926i	$-0.485249\pi$
$233$	1.41421	0.0926482	0.0463241	+	0.998926i	$0.485249\pi$
$234$	0	0
$235$	−6.14214	−0.400669
$236$	0	0
$237$	−31.9706	−2.07671
$238$	0	0
$239$	−20.2132	−1.30748	−0.653742	−	0.756718i	$-0.726802\pi$
$239$	−20.2132	−1.30748	−0.653742	+	0.756718i	$0.726802\pi$
$240$	0	0
$241$	−12.2426	−0.788618	−0.394309	−	0.918978i	$-0.629016\pi$
$241$	−12.2426	−0.788618	−0.394309	+	0.918978i	$0.629016\pi$
$242$	0	0
$243$	21.6569	1.38929
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−2.24264	−0.142696
$248$	0	0
$249$	29.3137	1.85768
$250$	0	0
$251$	−26.1421	−1.65008	−0.825038	−	0.565077i	$-0.808847\pi$
$251$	−26.1421	−1.65008	−0.825038	+	0.565077i	$0.808847\pi$
$252$	0	0
$253$	−6.24264	−0.392471
$254$	0	0
$255$	5.17157	0.323856
$256$	0	0
$257$	25.1421	1.56832	0.784162	−	0.620557i	$-0.213093\pi$
$257$	25.1421	1.56832	0.784162	+	0.620557i	$0.213093\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	7.51472	0.465149
$262$	0	0
$263$	17.0416	1.05083	0.525416	−	0.850845i	$-0.323910\pi$
$263$	17.0416	1.05083	0.525416	+	0.850845i	$0.323910\pi$
$264$	0	0
$265$	4.62742	0.284260
$266$	0	0
$267$	30.1421	1.84467
$268$	0	0
$269$	−2.34315	−0.142864	−0.0714321	−	0.997445i	$-0.522757\pi$
$269$	−2.34315	−0.142864	−0.0714321	+	0.997445i	$0.522757\pi$
$270$	0	0
$271$	4.55635	0.276779	0.138389	−	0.990378i	$-0.455808\pi$
$271$	4.55635	0.276779	0.138389	+	0.990378i	$0.455808\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.65685	0.280819
$276$	0	0
$277$	1.82843	0.109860	0.0549298	−	0.998490i	$-0.482507\pi$
$277$	1.82843	0.109860	0.0549298	+	0.998490i	$0.482507\pi$
$278$	0	0
$279$	−11.3137	−0.677334
$280$	0	0
$281$	8.72792	0.520664	0.260332	−	0.965519i	$-0.416168\pi$
$281$	8.72792	0.520664	0.260332	+	0.965519i	$0.416168\pi$
$282$	0	0
$283$	−23.4142	−1.39183	−0.695915	−	0.718124i	$-0.745001\pi$
$283$	−23.4142	−1.39183	−0.695915	+	0.718124i	$0.745001\pi$
$284$	0	0
$285$	0.828427	0.0490718
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−3.62742	−0.213377
$290$	0	0
$291$	−9.24264	−0.541813
$292$	0	0
$293$	10.8284	0.632603	0.316302	−	0.948659i	$-0.397559\pi$
$293$	10.8284	0.632603	0.316302	+	0.948659i	$0.397559\pi$
$294$	0	0
$295$	−3.27208	−0.190508
$296$	0	0
$297$	−0.414214	−0.0240351
$298$	0	0
$299$	23.8995	1.38214
$300$	0	0
$301$	0	0
$302$	0	0
$303$	14.8995	0.855954
$304$	0	0
$305$	−6.92893	−0.396750
$306$	0	0
$307$	9.89949	0.564994	0.282497	−	0.959268i	$-0.408837\pi$
$307$	9.89949	0.564994	0.282497	+	0.959268i	$0.408837\pi$
$308$	0	0
$309$	−32.3848	−1.84231
$310$	0	0
$311$	16.7279	0.948553	0.474277	−	0.880376i	$-0.342710\pi$
$311$	16.7279	0.948553	0.474277	+	0.880376i	$0.342710\pi$
$312$	0	0
$313$	20.6569	1.16759	0.583797	−	0.811900i	$-0.301566\pi$
$313$	20.6569	1.16759	0.583797	+	0.811900i	$0.301566\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8.68629	0.487871	0.243935	−	0.969791i	$-0.421562\pi$
$317$	8.68629	0.487871	0.243935	+	0.969791i	$0.421562\pi$
$318$	0	0
$319$	−2.65685	−0.148755
$320$	0	0
$321$	−7.41421	−0.413821
$322$	0	0
$323$	2.14214	0.119192
$324$	0	0
$325$	−17.8284	−0.988943
$326$	0	0
$327$	−39.7990	−2.20089
$328$	0	0
$329$	0	0
$330$	0	0
$331$	24.0711	1.32307	0.661533	−	0.749916i	$-0.269906\pi$
$331$	24.0711	1.32307	0.661533	+	0.749916i	$0.269906\pi$
$332$	0	0
$333$	−26.6274	−1.45917
$334$	0	0
$335$	−1.61522	−0.0882491
$336$	0	0
$337$	−28.2426	−1.53847	−0.769237	−	0.638963i	$-0.779364\pi$
$337$	−28.2426	−1.53847	−0.769237	+	0.638963i	$0.779364\pi$
$338$	0	0
$339$	19.7279	1.07147
$340$	0	0
$341$	4.00000	0.216612
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−8.82843	−0.475307
$346$	0	0
$347$	−17.4142	−0.934844	−0.467422	−	0.884034i	$-0.654817\pi$
$347$	−17.4142	−0.934844	−0.467422	+	0.884034i	$0.654817\pi$
$348$	0	0
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	0	0
$351$	1.58579	0.0846430
$352$	0	0
$353$	2.68629	0.142977	0.0714884	−	0.997441i	$-0.477225\pi$
$353$	2.68629	0.142977	0.0714884	+	0.997441i	$0.477225\pi$
$354$	0	0
$355$	6.48528	0.344203
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−19.2426	−1.01559	−0.507794	−	0.861479i	$-0.669539\pi$
$359$	−19.2426	−1.01559	−0.507794	+	0.861479i	$0.669539\pi$
$360$	0	0
$361$	−18.6569	−0.981940
$362$	0	0
$363$	−2.41421	−0.126713
$364$	0	0
$365$	5.51472	0.288654
$366$	0	0
$367$	−10.7279	−0.559993	−0.279996	−	0.960001i	$-0.590333\pi$
$367$	−10.7279	−0.559993	−0.279996	+	0.960001i	$0.590333\pi$
$368$	0	0
$369$	15.3137	0.797200
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−19.9706	−1.03404	−0.517018	−	0.855974i	$-0.672958\pi$
$373$	−19.9706	−1.03404	−0.517018	+	0.855974i	$0.672958\pi$
$374$	0	0
$375$	13.6569	0.705237
$376$	0	0
$377$	10.1716	0.523863
$378$	0	0
$379$	−27.8701	−1.43159	−0.715794	−	0.698311i	$-0.753935\pi$
$379$	−27.8701	−1.43159	−0.715794	+	0.698311i	$0.753935\pi$
$380$	0	0
$381$	37.9706	1.94529
$382$	0	0
$383$	−6.38478	−0.326247	−0.163123	−	0.986606i	$-0.552157\pi$
$383$	−6.38478	−0.326247	−0.163123	+	0.986606i	$0.552157\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	16.0000	0.813326
$388$	0	0
$389$	−22.7279	−1.15235	−0.576176	−	0.817326i	$-0.695456\pi$
$389$	−22.7279	−1.15235	−0.576176	+	0.817326i	$0.695456\pi$
$390$	0	0
$391$	−22.8284	−1.15448
$392$	0	0
$393$	1.41421	0.0713376
$394$	0	0
$395$	7.75736	0.390315
$396$	0	0
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	18.3137	0.914543	0.457271	−	0.889327i	$-0.348827\pi$
$401$	18.3137	0.914543	0.457271	+	0.889327i	$0.348827\pi$
$402$	0	0
$403$	−15.3137	−0.762830
$404$	0	0
$405$	−5.55635	−0.276097
$406$	0	0
$407$	9.41421	0.466645
$408$	0	0
$409$	−2.72792	−0.134887	−0.0674435	−	0.997723i	$-0.521484\pi$
$409$	−2.72792	−0.134887	−0.0674435	+	0.997723i	$0.521484\pi$
$410$	0	0
$411$	40.2132	1.98357
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−7.11270	−0.349149
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−26.1421	−1.27713	−0.638563	−	0.769569i	$-0.720471\pi$
$419$	−26.1421	−1.27713	−0.638563	+	0.769569i	$0.720471\pi$
$420$	0	0
$421$	0.686292	0.0334478	0.0167239	−	0.999860i	$-0.494676\pi$
$421$	0.686292	0.0334478	0.0167239	+	0.999860i	$0.494676\pi$
$422$	0	0
$423$	−29.6569	−1.44197
$424$	0	0
$425$	17.0294	0.826049
$426$	0	0
$427$	0	0
$428$	0	0
$429$	9.24264	0.446239
$430$	0	0
$431$	17.5858	0.847078	0.423539	−	0.905878i	$-0.360788\pi$
$431$	17.5858	0.847078	0.423539	+	0.905878i	$0.360788\pi$
$432$	0	0
$433$	−26.1421	−1.25631	−0.628155	−	0.778088i	$-0.716190\pi$
$433$	−26.1421	−1.25631	−0.628155	+	0.778088i	$0.716190\pi$
$434$	0	0
$435$	−3.75736	−0.180152
$436$	0	0
$437$	−3.65685	−0.174931
$438$	0	0
$439$	−27.3848	−1.30700	−0.653502	−	0.756925i	$-0.726701\pi$
$439$	−27.3848	−1.30700	−0.653502	+	0.756925i	$0.726701\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−12.6274	−0.599947	−0.299973	−	0.953948i	$-0.596978\pi$
$443$	−12.6274	−0.599947	−0.299973	+	0.953948i	$0.596978\pi$
$444$	0	0
$445$	−7.31371	−0.346703
$446$	0	0
$447$	−42.6274	−2.01621
$448$	0	0
$449$	22.3431	1.05444	0.527219	−	0.849729i	$-0.323235\pi$
$449$	22.3431	1.05444	0.527219	+	0.849729i	$0.323235\pi$
$450$	0	0
$451$	−5.41421	−0.254945
$452$	0	0
$453$	−37.9706	−1.78401
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−11.6569	−0.545285	−0.272642	−	0.962115i	$-0.587898\pi$
$457$	−11.6569	−0.545285	−0.272642	+	0.962115i	$0.587898\pi$
$458$	0	0
$459$	−1.51472	−0.0707010
$460$	0	0
$461$	8.31371	0.387208	0.193604	−	0.981080i	$-0.437982\pi$
$461$	8.31371	0.387208	0.193604	+	0.981080i	$0.437982\pi$
$462$	0	0
$463$	−12.8284	−0.596188	−0.298094	−	0.954537i	$-0.596351\pi$
$463$	−12.8284	−0.596188	−0.298094	+	0.954537i	$0.596351\pi$
$464$	0	0
$465$	5.65685	0.262330
$466$	0	0
$467$	−34.0000	−1.57333	−0.786666	−	0.617379i	$-0.788195\pi$
$467$	−34.0000	−1.57333	−0.786666	+	0.617379i	$0.788195\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	42.6274	1.96417
$472$	0	0
$473$	−5.65685	−0.260102
$474$	0	0
$475$	2.72792	0.125166
$476$	0	0
$477$	22.3431	1.02302
$478$	0	0
$479$	−11.9289	−0.545047	−0.272523	−	0.962149i	$-0.587858\pi$
$479$	−11.9289	−0.545047	−0.272523	+	0.962149i	$0.587858\pi$
$480$	0	0
$481$	−36.0416	−1.64336
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.24264	0.101833
$486$	0	0
$487$	9.65685	0.437594	0.218797	−	0.975770i	$-0.429787\pi$
$487$	9.65685	0.437594	0.218797	+	0.975770i	$0.429787\pi$
$488$	0	0
$489$	23.4853	1.06204
$490$	0	0
$491$	−19.1716	−0.865201	−0.432600	−	0.901586i	$-0.642404\pi$
$491$	−19.1716	−0.865201	−0.432600	+	0.901586i	$0.642404\pi$
$492$	0	0
$493$	−9.71573	−0.437574
$494$	0	0
$495$	−1.65685	−0.0744701
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−22.1421	−0.991218	−0.495609	−	0.868546i	$-0.665055\pi$
$499$	−22.1421	−0.991218	−0.495609	+	0.868546i	$0.665055\pi$
$500$	0	0
$501$	−33.1421	−1.48068
$502$	0	0
$503$	4.21320	0.187857	0.0939287	−	0.995579i	$-0.470057\pi$
$503$	4.21320	0.187857	0.0939287	+	0.995579i	$0.470057\pi$
$504$	0	0
$505$	−3.61522	−0.160875
$506$	0	0
$507$	−4.00000	−0.177646
$508$	0	0
$509$	−9.31371	−0.412823	−0.206411	−	0.978465i	$-0.566179\pi$
$509$	−9.31371	−0.412823	−0.206411	+	0.978465i	$0.566179\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−0.242641	−0.0107128
$514$	0	0
$515$	7.85786	0.346259
$516$	0	0
$517$	10.4853	0.461142
$518$	0	0
$519$	−23.7279	−1.04154
$520$	0	0
$521$	28.2843	1.23916	0.619578	−	0.784935i	$-0.287304\pi$
$521$	28.2843	1.23916	0.619578	+	0.784935i	$0.287304\pi$
$522$	0	0
$523$	−12.7279	−0.556553	−0.278277	−	0.960501i	$-0.589763\pi$
$523$	−12.7279	−0.556553	−0.278277	+	0.960501i	$0.589763\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	14.6274	0.637180
$528$	0	0
$529$	15.9706	0.694372
$530$	0	0
$531$	−15.7990	−0.685618
$532$	0	0
$533$	20.7279	0.897826
$534$	0	0
$535$	1.79899	0.0777771
$536$	0	0
$537$	2.17157	0.0937103
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−12.8579	−0.552803	−0.276401	−	0.961042i	$-0.589142\pi$
$541$	−12.8579	−0.552803	−0.276401	+	0.961042i	$0.589142\pi$
$542$	0	0
$543$	−18.4853	−0.793279
$544$	0	0
$545$	9.65685	0.413654
$546$	0	0
$547$	−34.8701	−1.49094	−0.745468	−	0.666541i	$-0.767774\pi$
$547$	−34.8701	−1.49094	−0.745468	+	0.666541i	$0.767774\pi$
$548$	0	0
$549$	−33.4558	−1.42786
$550$	0	0
$551$	−1.55635	−0.0663027
$552$	0	0
$553$	0	0
$554$	0	0
$555$	13.3137	0.565135
$556$	0	0
$557$	7.51472	0.318409	0.159204	−	0.987246i	$-0.449107\pi$
$557$	7.51472	0.318409	0.159204	+	0.987246i	$0.449107\pi$
$558$	0	0
$559$	21.6569	0.915987
$560$	0	0
$561$	−8.82843	−0.372736
$562$	0	0
$563$	9.07107	0.382300	0.191150	−	0.981561i	$-0.438778\pi$
$563$	9.07107	0.382300	0.191150	+	0.981561i	$0.438778\pi$
$564$	0	0
$565$	−4.78680	−0.201382
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−4.00000	−0.167689	−0.0838444	−	0.996479i	$-0.526720\pi$
$569$	−4.00000	−0.167689	−0.0838444	+	0.996479i	$0.526720\pi$
$570$	0	0
$571$	−26.3848	−1.10417	−0.552084	−	0.833788i	$-0.686167\pi$
$571$	−26.3848	−1.10417	−0.552084	+	0.833788i	$0.686167\pi$
$572$	0	0
$573$	−17.3137	−0.723291
$574$	0	0
$575$	−29.0711	−1.21235
$576$	0	0
$577$	9.68629	0.403246	0.201623	−	0.979463i	$-0.435378\pi$
$577$	9.68629	0.403246	0.201623	+	0.979463i	$0.435378\pi$
$578$	0	0
$579$	−52.8701	−2.19720
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−7.89949	−0.327164
$584$	0	0
$585$	6.34315	0.262257
$586$	0	0
$587$	−25.1005	−1.03601	−0.518004	−	0.855378i	$-0.673325\pi$
$587$	−25.1005	−1.03601	−0.518004	+	0.855378i	$0.673325\pi$
$588$	0	0
$589$	2.34315	0.0965476
$590$	0	0
$591$	1.24264	0.0511154
$592$	0	0
$593$	−23.6985	−0.973180	−0.486590	−	0.873630i	$-0.661759\pi$
$593$	−23.6985	−0.973180	−0.486590	+	0.873630i	$0.661759\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−0.242641	−0.00993062
$598$	0	0
$599$	−42.6274	−1.74171	−0.870855	−	0.491541i	$-0.836434\pi$
$599$	−42.6274	−1.74171	−0.870855	+	0.491541i	$0.836434\pi$
$600$	0	0
$601$	−31.9411	−1.30291	−0.651453	−	0.758689i	$-0.725840\pi$
$601$	−31.9411	−1.30291	−0.651453	+	0.758689i	$0.725840\pi$
$602$	0	0
$603$	−7.79899	−0.317599
$604$	0	0
$605$	0.585786	0.0238156
$606$	0	0
$607$	−42.9706	−1.74412	−0.872061	−	0.489398i	$-0.837217\pi$
$607$	−42.9706	−1.74412	−0.872061	+	0.489398i	$0.837217\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−40.1421	−1.62398
$612$	0	0
$613$	28.6274	1.15625	0.578125	−	0.815948i	$-0.303785\pi$
$613$	28.6274	1.15625	0.578125	+	0.815948i	$0.303785\pi$
$614$	0	0
$615$	−7.65685	−0.308754
$616$	0	0
$617$	−41.9706	−1.68967	−0.844836	−	0.535026i	$-0.820302\pi$
$617$	−41.9706	−1.68967	−0.844836	+	0.535026i	$0.820302\pi$
$618$	0	0
$619$	−21.9411	−0.881888	−0.440944	−	0.897535i	$-0.645356\pi$
$619$	−21.9411	−0.881888	−0.440944	+	0.897535i	$0.645356\pi$
$620$	0	0
$621$	2.58579	0.103764
$622$	0	0
$623$	0	0
$624$	0	0
$625$	19.9706	0.798823
$626$	0	0
$627$	−1.41421	−0.0564782
$628$	0	0
$629$	34.4264	1.37267
$630$	0	0
$631$	−23.2721	−0.926447	−0.463223	−	0.886242i	$-0.653307\pi$
$631$	−23.2721	−0.926447	−0.463223	+	0.886242i	$0.653307\pi$
$632$	0	0
$633$	17.8995	0.711441
$634$	0	0
$635$	−9.21320	−0.365615
$636$	0	0
$637$	0	0
$638$	0	0
$639$	31.3137	1.23875
$640$	0	0
$641$	15.2843	0.603692	0.301846	−	0.953357i	$-0.402397\pi$
$641$	15.2843	0.603692	0.301846	+	0.953357i	$0.402397\pi$
$642$	0	0
$643$	1.58579	0.0625373	0.0312687	−	0.999511i	$-0.490045\pi$
$643$	1.58579	0.0625373	0.0312687	+	0.999511i	$0.490045\pi$
$644$	0	0
$645$	−8.00000	−0.315000
$646$	0	0
$647$	30.1838	1.18665	0.593323	−	0.804964i	$-0.297816\pi$
$647$	30.1838	1.18665	0.593323	+	0.804964i	$0.297816\pi$
$648$	0	0
$649$	5.58579	0.219261
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−18.3848	−0.719452	−0.359726	−	0.933058i	$-0.617130\pi$
$653$	−18.3848	−0.719452	−0.359726	+	0.933058i	$0.617130\pi$
$654$	0	0
$655$	−0.343146	−0.0134078
$656$	0	0
$657$	26.6274	1.03883
$658$	0	0
$659$	12.0000	0.467454	0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000	0.467454	0.233727	+	0.972302i	$0.424908\pi$
$660$	0	0
$661$	−18.9706	−0.737869	−0.368935	−	0.929455i	$-0.620277\pi$
$661$	−18.9706	−0.737869	−0.368935	+	0.929455i	$0.620277\pi$
$662$	0	0
$663$	33.7990	1.31264
$664$	0	0
$665$	0	0
$666$	0	0
$667$	16.5858	0.642204
$668$	0	0
$669$	−20.7279	−0.801388
$670$	0	0
$671$	11.8284	0.456631
$672$	0	0
$673$	5.55635	0.214182	0.107091	−	0.994249i	$-0.465846\pi$
$673$	5.55635	0.214182	0.107091	+	0.994249i	$0.465846\pi$
$674$	0	0
$675$	−1.92893	−0.0742446
$676$	0	0
$677$	35.3137	1.35722	0.678608	−	0.734501i	$-0.262584\pi$
$677$	35.3137	1.35722	0.678608	+	0.734501i	$0.262584\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	69.5980	2.66700
$682$	0	0
$683$	−13.5858	−0.519846	−0.259923	−	0.965629i	$-0.583697\pi$
$683$	−13.5858	−0.519846	−0.259923	+	0.965629i	$0.583697\pi$
$684$	0	0
$685$	−9.75736	−0.372810
$686$	0	0
$687$	56.2843	2.14738
$688$	0	0
$689$	30.2426	1.15215
$690$	0	0
$691$	28.0711	1.06787	0.533937	−	0.845524i	$-0.320712\pi$
$691$	28.0711	1.06787	0.533937	+	0.845524i	$0.320712\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−19.7990	−0.749940
$698$	0	0
$699$	−3.41421	−0.129137
$700$	0	0
$701$	−26.1127	−0.986263	−0.493132	−	0.869955i	$-0.664148\pi$
$701$	−26.1127	−0.986263	−0.493132	+	0.869955i	$0.664148\pi$
$702$	0	0
$703$	5.51472	0.207992
$704$	0	0
$705$	14.8284	0.558471
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−12.0416	−0.452233	−0.226116	−	0.974100i	$-0.572603\pi$
$709$	−12.0416	−0.452233	−0.226116	+	0.974100i	$0.572603\pi$
$710$	0	0
$711$	37.4558	1.40470
$712$	0	0
$713$	−24.9706	−0.935155
$714$	0	0
$715$	−2.24264	−0.0838700
$716$	0	0
$717$	48.7990	1.82243
$718$	0	0
$719$	−1.51472	−0.0564895	−0.0282447	−	0.999601i	$-0.508992\pi$
$719$	−1.51472	−0.0564895	−0.0282447	+	0.999601i	$0.508992\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	29.5563	1.09921
$724$	0	0
$725$	−12.3726	−0.459506
$726$	0	0
$727$	36.4264	1.35098	0.675490	−	0.737369i	$-0.263932\pi$
$727$	36.4264	1.35098	0.675490	+	0.737369i	$0.263932\pi$
$728$	0	0
$729$	−23.8284	−0.882534
$730$	0	0
$731$	−20.6863	−0.765110
$732$	0	0
$733$	13.0000	0.480166	0.240083	−	0.970752i	$-0.422825\pi$
$733$	13.0000	0.480166	0.240083	+	0.970752i	$0.422825\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2.75736	0.101569
$738$	0	0
$739$	52.4264	1.92854	0.964268	−	0.264928i	$-0.0853481\pi$
$739$	52.4264	1.92854	0.964268	+	0.264928i	$0.0853481\pi$
$740$	0	0
$741$	5.41421	0.198896
$742$	0	0
$743$	13.3137	0.488433	0.244216	−	0.969721i	$-0.421469\pi$
$743$	13.3137	0.488433	0.244216	+	0.969721i	$0.421469\pi$
$744$	0	0
$745$	10.3431	0.378944
$746$	0	0
$747$	−34.3431	−1.25655
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−45.6569	−1.66604	−0.833021	−	0.553241i	$-0.813391\pi$
$751$	−45.6569	−1.66604	−0.833021	+	0.553241i	$0.813391\pi$
$752$	0	0
$753$	63.1127	2.29995
$754$	0	0
$755$	9.21320	0.335303
$756$	0	0
$757$	8.34315	0.303237	0.151618	−	0.988439i	$-0.451552\pi$
$757$	8.34315	0.303237	0.151618	+	0.988439i	$0.451552\pi$
$758$	0	0
$759$	15.0711	0.547045
$760$	0	0
$761$	−30.9706	−1.12268	−0.561341	−	0.827585i	$-0.689714\pi$
$761$	−30.9706	−1.12268	−0.561341	+	0.827585i	$0.689714\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−6.05887	−0.219059
$766$	0	0
$767$	−21.3848	−0.772160
$768$	0	0
$769$	−22.9706	−0.828340	−0.414170	−	0.910200i	$-0.635928\pi$
$769$	−22.9706	−0.828340	−0.414170	+	0.910200i	$0.635928\pi$
$770$	0	0
$771$	−60.6985	−2.18600
$772$	0	0
$773$	21.9411	0.789167	0.394584	−	0.918860i	$-0.370889\pi$
$773$	21.9411	0.789167	0.394584	+	0.918860i	$0.370889\pi$
$774$	0	0
$775$	18.6274	0.669117
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−3.17157	−0.113633
$780$	0	0
$781$	−11.0711	−0.396154
$782$	0	0
$783$	1.10051	0.0393288
$784$	0	0
$785$	−10.3431	−0.369163
$786$	0	0
$787$	13.5563	0.483232	0.241616	−	0.970372i	$-0.422323\pi$
$787$	13.5563	0.483232	0.241616	+	0.970372i	$0.422323\pi$
$788$	0	0
$789$	−41.1421	−1.46470
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−45.2843	−1.60809
$794$	0	0
$795$	−11.1716	−0.396215
$796$	0	0
$797$	7.65685	0.271220	0.135610	−	0.990762i	$-0.456701\pi$
$797$	7.65685	0.271220	0.135610	+	0.990762i	$0.456701\pi$
$798$	0	0
$799$	38.3431	1.35648
$800$	0	0
$801$	−35.3137	−1.24775
$802$	0	0
$803$	−9.41421	−0.332220
$804$	0	0
$805$	0	0
$806$	0	0
$807$	5.65685	0.199131
$808$	0	0
$809$	−46.6274	−1.63933	−0.819666	−	0.572841i	$-0.805841\pi$
$809$	−46.6274	−1.63933	−0.819666	+	0.572841i	$0.805841\pi$
$810$	0	0
$811$	−24.2843	−0.852736	−0.426368	−	0.904550i	$-0.640207\pi$
$811$	−24.2843	−0.852736	−0.426368	+	0.904550i	$0.640207\pi$
$812$	0	0
$813$	−11.0000	−0.385787
$814$	0	0
$815$	−5.69848	−0.199609
$816$	0	0
$817$	−3.31371	−0.115932
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−29.4853	−1.02904	−0.514522	−	0.857477i	$-0.672031\pi$
$821$	−29.4853	−1.02904	−0.514522	+	0.857477i	$0.672031\pi$
$822$	0	0
$823$	43.4558	1.51478	0.757388	−	0.652965i	$-0.226475\pi$
$823$	43.4558	1.51478	0.757388	+	0.652965i	$0.226475\pi$
$824$	0	0
$825$	−11.2426	−0.391419
$826$	0	0
$827$	21.3553	0.742598	0.371299	−	0.928513i	$-0.378913\pi$
$827$	21.3553	0.742598	0.371299	+	0.928513i	$0.378913\pi$
$828$	0	0
$829$	−21.7574	−0.755664	−0.377832	−	0.925874i	$-0.623330\pi$
$829$	−21.7574	−0.755664	−0.377832	+	0.925874i	$0.623330\pi$
$830$	0	0
$831$	−4.41421	−0.153127
$832$	0	0
$833$	0	0
$834$	0	0
$835$	8.04163	0.278292
$836$	0	0
$837$	−1.65685	−0.0572693
$838$	0	0
$839$	−18.4853	−0.638183	−0.319091	−	0.947724i	$-0.603378\pi$
$839$	−18.4853	−0.638183	−0.319091	+	0.947724i	$0.603378\pi$
$840$	0	0
$841$	−21.9411	−0.756591
$842$	0	0
$843$	−21.0711	−0.725726
$844$	0	0
$845$	0.970563	0.0333884
$846$	0	0
$847$	0	0
$848$	0	0
$849$	56.5269	1.94000
$850$	0	0
$851$	−58.7696	−2.01459
$852$	0	0
$853$	−14.0000	−0.479351	−0.239675	−	0.970853i	$-0.577041\pi$
$853$	−14.0000	−0.479351	−0.239675	+	0.970853i	$0.577041\pi$
$854$	0	0
$855$	−0.970563	−0.0331925
$856$	0	0
$857$	34.6274	1.18285	0.591425	−	0.806360i	$-0.298566\pi$
$857$	34.6274	1.18285	0.591425	+	0.806360i	$0.298566\pi$
$858$	0	0
$859$	−5.72792	−0.195434	−0.0977171	−	0.995214i	$-0.531154\pi$
$859$	−5.72792	−0.195434	−0.0977171	+	0.995214i	$0.531154\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−35.2132	−1.19867	−0.599336	−	0.800498i	$-0.704569\pi$
$863$	−35.2132	−1.19867	−0.599336	+	0.800498i	$0.704569\pi$
$864$	0	0
$865$	5.75736	0.195756
$866$	0	0
$867$	8.75736	0.297416
$868$	0	0
$869$	−13.2426	−0.449226
$870$	0	0
$871$	−10.5563	−0.357688
$872$	0	0
$873$	10.8284	0.366487
$874$	0	0
$875$	0	0
$876$	0	0
$877$	19.6274	0.662771	0.331385	−	0.943495i	$-0.392484\pi$
$877$	19.6274	0.662771	0.331385	+	0.943495i	$0.392484\pi$
$878$	0	0
$879$	−26.1421	−0.881752
$880$	0	0
$881$	−6.45584	−0.217503	−0.108751	−	0.994069i	$-0.534685\pi$
$881$	−6.45584	−0.217503	−0.108751	+	0.994069i	$0.534685\pi$
$882$	0	0
$883$	−15.7279	−0.529287	−0.264643	−	0.964346i	$-0.585254\pi$
$883$	−15.7279	−0.529287	−0.264643	+	0.964346i	$0.585254\pi$
$884$	0	0
$885$	7.89949	0.265539
$886$	0	0
$887$	−3.10051	−0.104105	−0.0520524	−	0.998644i	$-0.516576\pi$
$887$	−3.10051	−0.104105	−0.0520524	+	0.998644i	$0.516576\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	9.48528	0.317769
$892$	0	0
$893$	6.14214	0.205539
$894$	0	0
$895$	−0.526912	−0.0176127
$896$	0	0
$897$	−57.6985	−1.92650
$898$	0	0
$899$	−10.6274	−0.354444
$900$	0	0
$901$	−28.8873	−0.962376
$902$	0	0
$903$	0	0
$904$	0	0
$905$	4.48528	0.149096
$906$	0	0
$907$	−10.6863	−0.354832	−0.177416	−	0.984136i	$-0.556774\pi$
$907$	−10.6863	−0.354832	−0.177416	+	0.984136i	$0.556774\pi$
$908$	0	0
$909$	−17.4558	−0.578974
$910$	0	0
$911$	30.4853	1.01002	0.505011	−	0.863113i	$-0.331488\pi$
$911$	30.4853	1.01002	0.505011	+	0.863113i	$0.331488\pi$
$912$	0	0
$913$	12.1421	0.401846
$914$	0	0
$915$	16.7279	0.553008
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−22.1421	−0.730402	−0.365201	−	0.930929i	$-0.619000\pi$
$919$	−22.1421	−0.730402	−0.365201	+	0.930929i	$0.619000\pi$
$920$	0	0
$921$	−23.8995	−0.787515
$922$	0	0
$923$	42.3848	1.39511
$924$	0	0
$925$	43.8406	1.44147
$926$	0	0
$927$	37.9411	1.24615
$928$	0	0
$929$	40.4558	1.32731	0.663657	−	0.748037i	$-0.269004\pi$
$929$	40.4558	1.32731	0.663657	+	0.748037i	$0.269004\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−40.3848	−1.32214
$934$	0	0
$935$	2.14214	0.0700553
$936$	0	0
$937$	19.4142	0.634235	0.317117	−	0.948386i	$-0.397285\pi$
$937$	19.4142	0.634235	0.317117	+	0.948386i	$0.397285\pi$
$938$	0	0
$939$	−49.8701	−1.62745
$940$	0	0
$941$	−24.6569	−0.803790	−0.401895	−	0.915686i	$-0.631648\pi$
$941$	−24.6569	−0.803790	−0.401895	+	0.915686i	$0.631648\pi$
$942$	0	0
$943$	33.7990	1.10065
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−34.8284	−1.13177	−0.565886	−	0.824484i	$-0.691466\pi$
$947$	−34.8284	−1.13177	−0.565886	+	0.824484i	$0.691466\pi$
$948$	0	0
$949$	36.0416	1.16996
$950$	0	0
$951$	−20.9706	−0.680017
$952$	0	0
$953$	5.55635	0.179988	0.0899939	−	0.995942i	$-0.471315\pi$
$953$	5.55635	0.179988	0.0899939	+	0.995942i	$0.471315\pi$
$954$	0	0
$955$	4.20101	0.135941
$956$	0	0
$957$	6.41421	0.207342
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	8.68629	0.279912
$964$	0	0
$965$	12.8284	0.412962
$966$	0	0
$967$	36.2843	1.16682	0.583412	−	0.812177i	$-0.301717\pi$
$967$	36.2843	1.16682	0.583412	+	0.812177i	$0.301717\pi$
$968$	0	0
$969$	−5.17157	−0.166135
$970$	0	0
$971$	22.2721	0.714745	0.357372	−	0.933962i	$-0.383673\pi$
$971$	22.2721	0.714745	0.357372	+	0.933962i	$0.383673\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	43.0416	1.37844
$976$	0	0
$977$	40.0000	1.27971	0.639857	−	0.768494i	$-0.278994\pi$
$977$	40.0000	1.27971	0.639857	+	0.768494i	$0.278994\pi$
$978$	0	0
$979$	12.4853	0.399031
$980$	0	0
$981$	46.6274	1.48870
$982$	0	0
$983$	32.4264	1.03424	0.517121	−	0.855912i	$-0.327004\pi$
$983$	32.4264	1.03424	0.517121	+	0.855912i	$0.327004\pi$
$984$	0	0
$985$	−0.301515	−0.00960707
$986$	0	0
$987$	0	0
$988$	0	0
$989$	35.3137	1.12291
$990$	0	0
$991$	1.61522	0.0513093	0.0256546	−	0.999671i	$-0.491833\pi$
$991$	1.61522	0.0513093	0.0256546	+	0.999671i	$0.491833\pi$
$992$	0	0
$993$	−58.1127	−1.84415
$994$	0	0
$995$	0.0588745	0.00186645
$996$	0	0
$997$	−13.1716	−0.417148	−0.208574	−	0.978007i	$-0.566882\pi$
$997$	−13.1716	−0.417148	−0.208574	+	0.978007i	$0.566882\pi$
$998$	0	0
$999$	−3.89949	−0.123375

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8624.2.a.bh.1.1		2
4.3	odd	2		1078.2.a.x.1.2		2
7.3	odd	6		1232.2.q.f.177.1		4
7.5	odd	6		1232.2.q.f.529.1		4
7.6	odd	2		8624.2.a.cc.1.2		2
12.11	even	2		9702.2.a.ch.1.2		2
28.3	even	6		154.2.e.e.23.2	✓	4
28.11	odd	6		1078.2.e.m.177.1		4
28.19	even	6		154.2.e.e.67.2	yes	4
28.23	odd	6		1078.2.e.m.67.1		4
28.27	even	2		1078.2.a.t.1.1		2
84.47	odd	6		1386.2.k.t.991.2		4
84.59	odd	6		1386.2.k.t.793.2		4
84.83	odd	2		9702.2.a.cx.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
154.2.e.e.23.2	✓	4	28.3	even	6
154.2.e.e.67.2	yes	4	28.19	even	6
1078.2.a.t.1.1		2	28.27	even	2
1078.2.a.x.1.2		2	4.3	odd	2
1078.2.e.m.67.1		4	28.23	odd	6
1078.2.e.m.177.1		4	28.11	odd	6
1232.2.q.f.177.1		4	7.3	odd	6
1232.2.q.f.529.1		4	7.5	odd	6
1386.2.k.t.793.2		4	84.59	odd	6
1386.2.k.t.991.2		4	84.47	odd	6
8624.2.a.bh.1.1		2	1.1	even	1	trivial
8624.2.a.cc.1.2		2	7.6	odd	2
9702.2.a.ch.1.2		2	12.11	even	2
9702.2.a.cx.1.1		2	84.83	odd	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:	\( N \)	\(=\)	\( 8624 = 2^{4} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8624.a (trivial)

\(f(q)\)	\(=\)	\(q-2.41421 q^{3} +0.585786 q^{5} +2.82843 q^{9} -1.00000 q^{11} +3.82843 q^{13} -1.41421 q^{15} -3.65685 q^{17} -0.585786 q^{19} +6.24264 q^{23} -4.65685 q^{25} +0.414214 q^{27} +2.65685 q^{29} -4.00000 q^{31} +2.41421 q^{33} -9.41421 q^{37} -9.24264 q^{39} +5.41421 q^{41} +5.65685 q^{43} +1.65685 q^{45} -10.4853 q^{47} +8.82843 q^{51} +7.89949 q^{53} -0.585786 q^{55} +1.41421 q^{57} -5.58579 q^{59} -11.8284 q^{61} +2.24264 q^{65} -2.75736 q^{67} -15.0711 q^{69} +11.0711 q^{71} +9.41421 q^{73} +11.2426 q^{75} +13.2426 q^{79} -9.48528 q^{81} -12.1421 q^{83} -2.14214 q^{85} -6.41421 q^{87} -12.4853 q^{89} +9.65685 q^{93} -0.343146 q^{95} +3.82843 q^{97} -2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 4 q^{5} - 2 q^{11} + 2 q^{13} + 4 q^{17} - 4 q^{19} + 4 q^{23} + 2 q^{25} - 2 q^{27} - 6 q^{29} - 8 q^{31} + 2 q^{33} - 16 q^{37} - 10 q^{39} + 8 q^{41} - 8 q^{45} - 4 q^{47} + 12 q^{51}+ \cdots + 2 q^{97}+O(q^{100}) \) 2 * q - 2 * q^3 + 4 * q^5 - 2 * q^11 + 2 * q^13 + 4 * q^17 - 4 * q^19 + 4 * q^23 + 2 * q^25 - 2 * q^27 - 6 * q^29 - 8 * q^31 + 2 * q^33 - 16 * q^37 - 10 * q^39 + 8 * q^41 - 8 * q^45 - 4 * q^47 + 12 * q^51 - 4 * q^53 - 4 * q^55 - 14 * q^59 - 18 * q^61 - 4 * q^65 - 14 * q^67 - 16 * q^69 + 8 * q^71 + 16 * q^73 + 14 * q^75 + 18 * q^79 - 2 * q^81 + 4 * q^83 + 24 * q^85 - 10 * q^87 - 8 * q^89 + 8 * q^93 - 12 * q^95 + 2 * q^97

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