LMFDB - Embedded newform 8400.2.a.cx.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8400,2,Mod(1,8400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8400, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8400.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8400 = 2^{4} \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8400.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:	$67.0743376979$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	8400.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} -6.47214 q^{11} -4.47214 q^{13} +2.00000 q^{17} +2.47214 q^{19} -1.00000 q^{21} +4.00000 q^{23} -1.00000 q^{27} -2.00000 q^{29} -1.52786 q^{31} +6.47214 q^{33} +6.94427 q^{37} +4.47214 q^{39} -2.00000 q^{41} +8.94427 q^{43} +12.9443 q^{47} +1.00000 q^{49} -2.00000 q^{51} +3.52786 q^{53} -2.47214 q^{57} +8.94427 q^{59} -2.00000 q^{61} +1.00000 q^{63} -4.00000 q^{67} -4.00000 q^{69} -5.52786 q^{71} +12.4721 q^{73} -6.47214 q^{77} -12.9443 q^{79} +1.00000 q^{81} -16.9443 q^{83} +2.00000 q^{87} -2.00000 q^{89} -4.47214 q^{91} +1.52786 q^{93} -8.47214 q^{97} -6.47214 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 2 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} + 2 q^{7} + 2 q^{9} - 4 q^{11} + 4 q^{17} - 4 q^{19} - 2 q^{21} + 8 q^{23} - 2 q^{27} - 4 q^{29} - 12 q^{31} + 4 q^{33} - 4 q^{37} - 4 q^{41} + 8 q^{47} + 2 q^{49} - 4 q^{51} + 16 q^{53} + 4 q^{57} - 4 q^{61} + 2 q^{63} - 8 q^{67} - 8 q^{69} - 20 q^{71} + 16 q^{73} - 4 q^{77} - 8 q^{79} + 2 q^{81} - 16 q^{83} + 4 q^{87} - 4 q^{89} + 12 q^{93} - 8 q^{97} - 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^7 + 2 * q^9 - 4 * q^11 + 4 * q^17 - 4 * q^19 - 2 * q^21 + 8 * q^23 - 2 * q^27 - 4 * q^29 - 12 * q^31 + 4 * q^33 - 4 * q^37 - 4 * q^41 + 8 * q^47 + 2 * q^49 - 4 * q^51 + 16 * q^53 + 4 * q^57 - 4 * q^61 + 2 * q^63 - 8 * q^67 - 8 * q^69 - 20 * q^71 + 16 * q^73 - 4 * q^77 - 8 * q^79 + 2 * q^81 - 16 * q^83 + 4 * q^87 - 4 * q^89 + 12 * q^93 - 8 * q^97 - 4 * q^99

Display 100 coefficients 10 coefficients

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
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−6.47214	−1.95142	−0.975711	−	0.219061i	$-0.929701\pi$
$11$	−6.47214	−1.95142	−0.975711	+	0.219061i	$0.929701\pi$
$12$	0	0
$13$	−4.47214	−1.24035	−0.620174	−	0.784465i	$-0.712938\pi$
$13$	−4.47214	−1.24035	−0.620174	+	0.784465i	$0.712938\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	2.47214	0.567147	0.283573	−	0.958951i	$-0.408480\pi$
$19$	2.47214	0.567147	0.283573	+	0.958951i	$0.408480\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	−1.52786	−0.274412	−0.137206	−	0.990543i	$-0.543812\pi$
$31$	−1.52786	−0.274412	−0.137206	+	0.990543i	$0.543812\pi$
$32$	0	0
$33$	6.47214	1.12665
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.94427	1.14163	0.570816	−	0.821078i	$-0.306627\pi$
$37$	6.94427	1.14163	0.570816	+	0.821078i	$0.306627\pi$
$38$	0	0
$39$	4.47214	0.716115
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	8.94427	1.36399	0.681994	−	0.731357i	$-0.261113\pi$
$43$	8.94427	1.36399	0.681994	+	0.731357i	$0.261113\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12.9443	1.88812	0.944058	−	0.329779i	$-0.106974\pi$
$47$	12.9443	1.88812	0.944058	+	0.329779i	$0.106974\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.00000	−0.280056
$52$	0	0
$53$	3.52786	0.484589	0.242295	−	0.970203i	$-0.422100\pi$
$53$	3.52786	0.484589	0.242295	+	0.970203i	$0.422100\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−2.47214	−0.327442
$58$	0	0
$59$	8.94427	1.16445	0.582223	−	0.813029i	$-0.302183\pi$
$59$	8.94427	1.16445	0.582223	+	0.813029i	$0.302183\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	0	0
$69$	−4.00000	−0.481543
$70$	0	0
$71$	−5.52786	−0.656037	−0.328018	−	0.944671i	$-0.606381\pi$
$71$	−5.52786	−0.656037	−0.328018	+	0.944671i	$0.606381\pi$
$72$	0	0
$73$	12.4721	1.45975	0.729877	−	0.683579i	$-0.239578\pi$
$73$	12.4721	1.45975	0.729877	+	0.683579i	$0.239578\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−6.47214	−0.737568
$78$	0	0
$79$	−12.9443	−1.45634	−0.728172	−	0.685394i	$-0.759630\pi$
$79$	−12.9443	−1.45634	−0.728172	+	0.685394i	$0.759630\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−16.9443	−1.85988	−0.929938	−	0.367717i	$-0.880140\pi$
$83$	−16.9443	−1.85988	−0.929938	+	0.367717i	$0.880140\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	2.00000	0.214423
$88$	0	0
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	−4.47214	−0.468807
$92$	0	0
$93$	1.52786	0.158432
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−8.47214	−0.860215	−0.430108	−	0.902778i	$-0.641524\pi$
$97$	−8.47214	−0.860215	−0.430108	+	0.902778i	$0.641524\pi$
$98$	0	0
$99$	−6.47214	−0.650474
$100$	0	0
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−12.9443	−1.25137	−0.625685	−	0.780076i	$-0.715180\pi$
$107$	−12.9443	−1.25137	−0.625685	+	0.780076i	$0.715180\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	−6.94427	−0.659121
$112$	0	0
$113$	−0.472136	−0.0444148	−0.0222074	−	0.999753i	$-0.507069\pi$
$113$	−0.472136	−0.0444148	−0.0222074	+	0.999753i	$0.507069\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−4.47214	−0.413449
$118$	0	0
$119$	2.00000	0.183340
$120$	0	0
$121$	30.8885	2.80805
$122$	0	0
$123$	2.00000	0.180334
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−4.94427	−0.438733	−0.219367	−	0.975643i	$-0.570399\pi$
$127$	−4.94427	−0.438733	−0.219367	+	0.975643i	$0.570399\pi$
$128$	0	0
$129$	−8.94427	−0.787499
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	2.47214	0.214361
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−3.52786	−0.301406	−0.150703	−	0.988579i	$-0.548154\pi$
$137$	−3.52786	−0.301406	−0.150703	+	0.988579i	$0.548154\pi$
$138$	0	0
$139$	−7.41641	−0.629052	−0.314526	−	0.949249i	$-0.601845\pi$
$139$	−7.41641	−0.629052	−0.314526	+	0.949249i	$0.601845\pi$
$140$	0	0
$141$	−12.9443	−1.09010
$142$	0	0
$143$	28.9443	2.42044
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−14.9443	−1.22428	−0.612141	−	0.790748i	$-0.709692\pi$
$149$	−14.9443	−1.22428	−0.612141	+	0.790748i	$0.709692\pi$
$150$	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
$153$	2.00000	0.161690
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0.472136	0.0376806	0.0188403	−	0.999823i	$-0.494003\pi$
$157$	0.472136	0.0376806	0.0188403	+	0.999823i	$0.494003\pi$
$158$	0	0
$159$	−3.52786	−0.279778
$160$	0	0
$161$	4.00000	0.315244
$162$	0	0
$163$	−16.9443	−1.32718	−0.663589	−	0.748097i	$-0.730968\pi$
$163$	−16.9443	−1.32718	−0.663589	+	0.748097i	$0.730968\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	7.00000	0.538462
$170$	0	0
$171$	2.47214	0.189049
$172$	0	0
$173$	2.94427	0.223849	0.111924	−	0.993717i	$-0.464299\pi$
$173$	2.94427	0.223849	0.111924	+	0.993717i	$0.464299\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−8.94427	−0.672293
$178$	0	0
$179$	−6.47214	−0.483750	−0.241875	−	0.970307i	$-0.577762\pi$
$179$	−6.47214	−0.483750	−0.241875	+	0.970307i	$0.577762\pi$
$180$	0	0
$181$	1.05573	0.0784717	0.0392358	−	0.999230i	$-0.487508\pi$
$181$	1.05573	0.0784717	0.0392358	+	0.999230i	$0.487508\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−12.9443	−0.946579
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−0.583592	−0.0422272	−0.0211136	−	0.999777i	$-0.506721\pi$
$191$	−0.583592	−0.0422272	−0.0211136	+	0.999777i	$0.506721\pi$
$192$	0	0
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−15.5279	−1.10631	−0.553157	−	0.833077i	$-0.686577\pi$
$197$	−15.5279	−1.10631	−0.553157	+	0.833077i	$0.686577\pi$
$198$	0	0
$199$	−27.4164	−1.94350	−0.971749	−	0.236017i	$-0.924158\pi$
$199$	−27.4164	−1.94350	−0.971749	+	0.236017i	$0.924158\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	0	0
$203$	−2.00000	−0.140372
$204$	0	0
$205$	0	0
$206$	0	0
$207$	4.00000	0.278019
$208$	0	0
$209$	−16.0000	−1.10674
$210$	0	0
$211$	16.9443	1.16649	0.583246	−	0.812296i	$-0.301782\pi$
$211$	16.9443	1.16649	0.583246	+	0.812296i	$0.301782\pi$
$212$	0	0
$213$	5.52786	0.378763
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−1.52786	−0.103718
$218$	0	0
$219$	−12.4721	−0.842789
$220$	0	0
$221$	−8.94427	−0.601657
$222$	0	0
$223$	−12.9443	−0.866813	−0.433406	−	0.901199i	$-0.642688\pi$
$223$	−12.9443	−0.866813	−0.433406	+	0.901199i	$0.642688\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0.944272	0.0626735	0.0313368	−	0.999509i	$-0.490024\pi$
$227$	0.944272	0.0626735	0.0313368	+	0.999509i	$0.490024\pi$
$228$	0	0
$229$	23.8885	1.57860	0.789300	−	0.614008i	$-0.210444\pi$
$229$	23.8885	1.57860	0.789300	+	0.614008i	$0.210444\pi$
$230$	0	0
$231$	6.47214	0.425835
$232$	0	0
$233$	9.41641	0.616889	0.308445	−	0.951242i	$-0.400192\pi$
$233$	9.41641	0.616889	0.308445	+	0.951242i	$0.400192\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	12.9443	0.840821
$238$	0	0
$239$	10.4721	0.677386	0.338693	−	0.940897i	$-0.390015\pi$
$239$	10.4721	0.677386	0.338693	+	0.940897i	$0.390015\pi$
$240$	0	0
$241$	−18.9443	−1.22031	−0.610154	−	0.792283i	$-0.708892\pi$
$241$	−18.9443	−1.22031	−0.610154	+	0.792283i	$0.708892\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−11.0557	−0.703459
$248$	0	0
$249$	16.9443	1.07380
$250$	0	0
$251$	−16.9443	−1.06951	−0.534756	−	0.845006i	$-0.679597\pi$
$251$	−16.9443	−1.06951	−0.534756	+	0.845006i	$0.679597\pi$
$252$	0	0
$253$	−25.8885	−1.62760
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−18.9443	−1.18171	−0.590856	−	0.806777i	$-0.701210\pi$
$257$	−18.9443	−1.18171	−0.590856	+	0.806777i	$0.701210\pi$
$258$	0	0
$259$	6.94427	0.431496
$260$	0	0
$261$	−2.00000	−0.123797
$262$	0	0
$263$	7.05573	0.435075	0.217537	−	0.976052i	$-0.430198\pi$
$263$	7.05573	0.435075	0.217537	+	0.976052i	$0.430198\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2.00000	0.122398
$268$	0	0
$269$	11.8885	0.724857	0.362429	−	0.932012i	$-0.381948\pi$
$269$	11.8885	0.724857	0.362429	+	0.932012i	$0.381948\pi$
$270$	0	0
$271$	1.52786	0.0928111	0.0464056	−	0.998923i	$-0.485223\pi$
$271$	1.52786	0.0928111	0.0464056	+	0.998923i	$0.485223\pi$
$272$	0	0
$273$	4.47214	0.270666
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−18.9443	−1.13825	−0.569125	−	0.822251i	$-0.692718\pi$
$277$	−18.9443	−1.13825	−0.569125	+	0.822251i	$0.692718\pi$
$278$	0	0
$279$	−1.52786	−0.0914708
$280$	0	0
$281$	−10.9443	−0.652881	−0.326440	−	0.945218i	$-0.605849\pi$
$281$	−10.9443	−0.652881	−0.326440	+	0.945218i	$0.605849\pi$
$282$	0	0
$283$	12.0000	0.713326	0.356663	−	0.934233i	$-0.383914\pi$
$283$	12.0000	0.713326	0.356663	+	0.934233i	$0.383914\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.00000	−0.118056
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	8.47214	0.496645
$292$	0	0
$293$	−5.05573	−0.295359	−0.147679	−	0.989035i	$-0.547180\pi$
$293$	−5.05573	−0.295359	−0.147679	+	0.989035i	$0.547180\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	6.47214	0.375551
$298$	0	0
$299$	−17.8885	−1.03452
$300$	0	0
$301$	8.94427	0.515539
$302$	0	0
$303$	14.0000	0.804279
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−15.0557	−0.859276	−0.429638	−	0.903001i	$-0.641359\pi$
$307$	−15.0557	−0.859276	−0.429638	+	0.903001i	$0.641359\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−25.8885	−1.46800	−0.734002	−	0.679147i	$-0.762350\pi$
$311$	−25.8885	−1.46800	−0.734002	+	0.679147i	$0.762350\pi$
$312$	0	0
$313$	17.4164	0.984434	0.492217	−	0.870473i	$-0.336187\pi$
$313$	17.4164	0.984434	0.492217	+	0.870473i	$0.336187\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−14.3607	−0.806576	−0.403288	−	0.915073i	$-0.632133\pi$
$317$	−14.3607	−0.806576	−0.403288	+	0.915073i	$0.632133\pi$
$318$	0	0
$319$	12.9443	0.724740
$320$	0	0
$321$	12.9443	0.722479
$322$	0	0
$323$	4.94427	0.275107
$324$	0	0
$325$	0	0
$326$	0	0
$327$	2.00000	0.110600
$328$	0	0
$329$	12.9443	0.713641
$330$	0	0
$331$	−0.944272	−0.0519019	−0.0259509	−	0.999663i	$-0.508261\pi$
$331$	−0.944272	−0.0519019	−0.0259509	+	0.999663i	$0.508261\pi$
$332$	0	0
$333$	6.94427	0.380544
$334$	0	0
$335$	0	0
$336$	0	0
$337$	23.8885	1.30129	0.650646	−	0.759381i	$-0.274498\pi$
$337$	23.8885	1.30129	0.650646	+	0.759381i	$0.274498\pi$
$338$	0	0
$339$	0.472136	0.0256429
$340$	0	0
$341$	9.88854	0.535495
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−8.00000	−0.429463	−0.214731	−	0.976673i	$-0.568888\pi$
$347$	−8.00000	−0.429463	−0.214731	+	0.976673i	$0.568888\pi$
$348$	0	0
$349$	−11.8885	−0.636379	−0.318190	−	0.948027i	$-0.603075\pi$
$349$	−11.8885	−0.636379	−0.318190	+	0.948027i	$0.603075\pi$
$350$	0	0
$351$	4.47214	0.238705
$352$	0	0
$353$	−7.88854	−0.419865	−0.209932	−	0.977716i	$-0.567324\pi$
$353$	−7.88854	−0.419865	−0.209932	+	0.977716i	$0.567324\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−2.00000	−0.105851
$358$	0	0
$359$	−18.4721	−0.974922	−0.487461	−	0.873145i	$-0.662077\pi$
$359$	−18.4721	−0.974922	−0.487461	+	0.873145i	$0.662077\pi$
$360$	0	0
$361$	−12.8885	−0.678344
$362$	0	0
$363$	−30.8885	−1.62123
$364$	0	0
$365$	0	0
$366$	0	0
$367$	3.05573	0.159508	0.0797539	−	0.996815i	$-0.474587\pi$
$367$	3.05573	0.159508	0.0797539	+	0.996815i	$0.474587\pi$
$368$	0	0
$369$	−2.00000	−0.104116
$370$	0	0
$371$	3.52786	0.183158
$372$	0	0
$373$	−6.00000	−0.310668	−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000	−0.310668	−0.155334	+	0.987862i	$0.549645\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	8.94427	0.460653
$378$	0	0
$379$	37.8885	1.94620	0.973102	−	0.230375i	$-0.0739953\pi$
$379$	37.8885	1.94620	0.973102	+	0.230375i	$0.0739953\pi$
$380$	0	0
$381$	4.94427	0.253303
$382$	0	0
$383$	−8.00000	−0.408781	−0.204390	−	0.978889i	$-0.565521\pi$
$383$	−8.00000	−0.408781	−0.204390	+	0.978889i	$0.565521\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	8.94427	0.454663
$388$	0	0
$389$	−6.94427	−0.352089	−0.176044	−	0.984382i	$-0.556330\pi$
$389$	−6.94427	−0.352089	−0.176044	+	0.984382i	$0.556330\pi$
$390$	0	0
$391$	8.00000	0.404577
$392$	0	0
$393$	4.00000	0.201773
$394$	0	0
$395$	0	0
$396$	0	0
$397$	13.4164	0.673350	0.336675	−	0.941621i	$-0.390698\pi$
$397$	13.4164	0.673350	0.336675	+	0.941621i	$0.390698\pi$
$398$	0	0
$399$	−2.47214	−0.123762
$400$	0	0
$401$	10.0000	0.499376	0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000	0.499376	0.249688	+	0.968326i	$0.419672\pi$
$402$	0	0
$403$	6.83282	0.340367
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−44.9443	−2.22780
$408$	0	0
$409$	11.8885	0.587851	0.293925	−	0.955828i	$-0.405038\pi$
$409$	11.8885	0.587851	0.293925	+	0.955828i	$0.405038\pi$
$410$	0	0
$411$	3.52786	0.174017
$412$	0	0
$413$	8.94427	0.440119
$414$	0	0
$415$	0	0
$416$	0	0
$417$	7.41641	0.363183
$418$	0	0
$419$	29.8885	1.46015	0.730075	−	0.683367i	$-0.239485\pi$
$419$	29.8885	1.46015	0.730075	+	0.683367i	$0.239485\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	0	0
$423$	12.9443	0.629372
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−2.00000	−0.0967868
$428$	0	0
$429$	−28.9443	−1.39744
$430$	0	0
$431$	18.4721	0.889771	0.444886	−	0.895587i	$-0.353244\pi$
$431$	18.4721	0.889771	0.444886	+	0.895587i	$0.353244\pi$
$432$	0	0
$433$	−16.4721	−0.791600	−0.395800	−	0.918337i	$-0.629533\pi$
$433$	−16.4721	−0.791600	−0.395800	+	0.918337i	$0.629533\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	9.88854	0.473033
$438$	0	0
$439$	−1.52786	−0.0729210	−0.0364605	−	0.999335i	$-0.511608\pi$
$439$	−1.52786	−0.0729210	−0.0364605	+	0.999335i	$0.511608\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−8.00000	−0.380091	−0.190046	−	0.981775i	$-0.560864\pi$
$443$	−8.00000	−0.380091	−0.190046	+	0.981775i	$0.560864\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	14.9443	0.706840
$448$	0	0
$449$	−14.0000	−0.660701	−0.330350	−	0.943858i	$-0.607167\pi$
$449$	−14.0000	−0.660701	−0.330350	+	0.943858i	$0.607167\pi$
$450$	0	0
$451$	12.9443	0.609522
$452$	0	0
$453$	−16.0000	−0.751746
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−6.94427	−0.324839	−0.162420	−	0.986722i	$-0.551930\pi$
$457$	−6.94427	−0.324839	−0.162420	+	0.986722i	$0.551930\pi$
$458$	0	0
$459$	−2.00000	−0.0933520
$460$	0	0
$461$	3.88854	0.181108	0.0905538	−	0.995892i	$-0.471136\pi$
$461$	3.88854	0.181108	0.0905538	+	0.995892i	$0.471136\pi$
$462$	0	0
$463$	20.9443	0.973363	0.486681	−	0.873580i	$-0.338207\pi$
$463$	20.9443	0.973363	0.486681	+	0.873580i	$0.338207\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−8.94427	−0.413892	−0.206946	−	0.978352i	$-0.566352\pi$
$467$	−8.94427	−0.413892	−0.206946	+	0.978352i	$0.566352\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	0	0
$471$	−0.472136	−0.0217549
$472$	0	0
$473$	−57.8885	−2.66172
$474$	0	0
$475$	0	0
$476$	0	0
$477$	3.52786	0.161530
$478$	0	0
$479$	17.8885	0.817348	0.408674	−	0.912680i	$-0.365991\pi$
$479$	17.8885	0.817348	0.408674	+	0.912680i	$0.365991\pi$
$480$	0	0
$481$	−31.0557	−1.41602
$482$	0	0
$483$	−4.00000	−0.182006
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−20.9443	−0.949076	−0.474538	−	0.880235i	$-0.657385\pi$
$487$	−20.9443	−0.949076	−0.474538	+	0.880235i	$0.657385\pi$
$488$	0	0
$489$	16.9443	0.766246
$490$	0	0
$491$	21.3050	0.961479	0.480740	−	0.876863i	$-0.340368\pi$
$491$	21.3050	0.961479	0.480740	+	0.876863i	$0.340368\pi$
$492$	0	0
$493$	−4.00000	−0.180151
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−5.52786	−0.247959
$498$	0	0
$499$	13.8885	0.621737	0.310868	−	0.950453i	$-0.399380\pi$
$499$	13.8885	0.621737	0.310868	+	0.950453i	$0.399380\pi$
$500$	0	0
$501$	8.00000	0.357414
$502$	0	0
$503$	32.0000	1.42681	0.713405	−	0.700752i	$-0.247152\pi$
$503$	32.0000	1.42681	0.713405	+	0.700752i	$0.247152\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−7.00000	−0.310881
$508$	0	0
$509$	−23.8885	−1.05884	−0.529421	−	0.848360i	$-0.677591\pi$
$509$	−23.8885	−1.05884	−0.529421	+	0.848360i	$0.677591\pi$
$510$	0	0
$511$	12.4721	0.551735
$512$	0	0
$513$	−2.47214	−0.109147
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−83.7771	−3.68451
$518$	0	0
$519$	−2.94427	−0.129239
$520$	0	0
$521$	−19.8885	−0.871333	−0.435666	−	0.900108i	$-0.643487\pi$
$521$	−19.8885	−0.871333	−0.435666	+	0.900108i	$0.643487\pi$
$522$	0	0
$523$	−8.94427	−0.391106	−0.195553	−	0.980693i	$-0.562650\pi$
$523$	−8.94427	−0.391106	−0.195553	+	0.980693i	$0.562650\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3.05573	−0.133110
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	8.94427	0.388148
$532$	0	0
$533$	8.94427	0.387419
$534$	0	0
$535$	0	0
$536$	0	0
$537$	6.47214	0.279293
$538$	0	0
$539$	−6.47214	−0.278775
$540$	0	0
$541$	−11.8885	−0.511128	−0.255564	−	0.966792i	$-0.582261\pi$
$541$	−11.8885	−0.511128	−0.255564	+	0.966792i	$0.582261\pi$
$542$	0	0
$543$	−1.05573	−0.0453056
$544$	0	0
$545$	0	0
$546$	0	0
$547$	5.88854	0.251776	0.125888	−	0.992044i	$-0.459822\pi$
$547$	5.88854	0.251776	0.125888	+	0.992044i	$0.459822\pi$
$548$	0	0
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	−4.94427	−0.210633
$552$	0	0
$553$	−12.9443	−0.550446
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−20.4721	−0.867432	−0.433716	−	0.901050i	$-0.642798\pi$
$557$	−20.4721	−0.867432	−0.433716	+	0.901050i	$0.642798\pi$
$558$	0	0
$559$	−40.0000	−1.69182
$560$	0	0
$561$	12.9443	0.546508
$562$	0	0
$563$	13.8885	0.585332	0.292666	−	0.956215i	$-0.405458\pi$
$563$	13.8885	0.585332	0.292666	+	0.956215i	$0.405458\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−39.8885	−1.67221	−0.836107	−	0.548566i	$-0.815174\pi$
$569$	−39.8885	−1.67221	−0.836107	+	0.548566i	$0.815174\pi$
$570$	0	0
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	0	0
$573$	0.583592	0.0243799
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−10.3607	−0.431321	−0.215660	−	0.976468i	$-0.569190\pi$
$577$	−10.3607	−0.431321	−0.215660	+	0.976468i	$0.569190\pi$
$578$	0	0
$579$	−14.0000	−0.581820
$580$	0	0
$581$	−16.9443	−0.702967
$582$	0	0
$583$	−22.8328	−0.945639
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−4.00000	−0.165098	−0.0825488	−	0.996587i	$-0.526306\pi$
$587$	−4.00000	−0.165098	−0.0825488	+	0.996587i	$0.526306\pi$
$588$	0	0
$589$	−3.77709	−0.155632
$590$	0	0
$591$	15.5279	0.638731
$592$	0	0
$593$	−23.8885	−0.980985	−0.490492	−	0.871445i	$-0.663183\pi$
$593$	−23.8885	−0.980985	−0.490492	+	0.871445i	$0.663183\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	27.4164	1.12208
$598$	0	0
$599$	−12.3607	−0.505044	−0.252522	−	0.967591i	$-0.581260\pi$
$599$	−12.3607	−0.505044	−0.252522	+	0.967591i	$0.581260\pi$
$600$	0	0
$601$	38.9443	1.58857	0.794285	−	0.607545i	$-0.207846\pi$
$601$	38.9443	1.58857	0.794285	+	0.607545i	$0.207846\pi$
$602$	0	0
$603$	−4.00000	−0.162893
$604$	0	0
$605$	0	0
$606$	0	0
$607$	38.8328	1.57618	0.788088	−	0.615563i	$-0.211071\pi$
$607$	38.8328	1.57618	0.788088	+	0.615563i	$0.211071\pi$
$608$	0	0
$609$	2.00000	0.0810441
$610$	0	0
$611$	−57.8885	−2.34192
$612$	0	0
$613$	6.94427	0.280477	0.140238	−	0.990118i	$-0.455213\pi$
$613$	6.94427	0.280477	0.140238	+	0.990118i	$0.455213\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−16.4721	−0.663143	−0.331572	−	0.943430i	$-0.607579\pi$
$617$	−16.4721	−0.663143	−0.331572	+	0.943430i	$0.607579\pi$
$618$	0	0
$619$	−39.4164	−1.58428	−0.792140	−	0.610340i	$-0.791033\pi$
$619$	−39.4164	−1.58428	−0.792140	+	0.610340i	$0.791033\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	0	0
$623$	−2.00000	−0.0801283
$624$	0	0
$625$	0	0
$626$	0	0
$627$	16.0000	0.638978
$628$	0	0
$629$	13.8885	0.553773
$630$	0	0
$631$	−30.8328	−1.22744	−0.613718	−	0.789526i	$-0.710327\pi$
$631$	−30.8328	−1.22744	−0.613718	+	0.789526i	$0.710327\pi$
$632$	0	0
$633$	−16.9443	−0.673474
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−4.47214	−0.177192
$638$	0	0
$639$	−5.52786	−0.218679
$640$	0	0
$641$	16.8328	0.664856	0.332428	−	0.943129i	$-0.392132\pi$
$641$	16.8328	0.664856	0.332428	+	0.943129i	$0.392132\pi$
$642$	0	0
$643$	−15.0557	−0.593740	−0.296870	−	0.954918i	$-0.595943\pi$
$643$	−15.0557	−0.593740	−0.296870	+	0.954918i	$0.595943\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.88854	−0.0742463	−0.0371232	−	0.999311i	$-0.511819\pi$
$647$	−1.88854	−0.0742463	−0.0371232	+	0.999311i	$0.511819\pi$
$648$	0	0
$649$	−57.8885	−2.27232
$650$	0	0
$651$	1.52786	0.0598817
$652$	0	0
$653$	22.5836	0.883764	0.441882	−	0.897073i	$-0.354311\pi$
$653$	22.5836	0.883764	0.441882	+	0.897073i	$0.354311\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	12.4721	0.486584
$658$	0	0
$659$	−21.3050	−0.829923	−0.414962	−	0.909839i	$-0.636205\pi$
$659$	−21.3050	−0.829923	−0.414962	+	0.909839i	$0.636205\pi$
$660$	0	0
$661$	−35.8885	−1.39590	−0.697951	−	0.716145i	$-0.745905\pi$
$661$	−35.8885	−1.39590	−0.697951	+	0.716145i	$0.745905\pi$
$662$	0	0
$663$	8.94427	0.347367
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−8.00000	−0.309761
$668$	0	0
$669$	12.9443	0.500454
$670$	0	0
$671$	12.9443	0.499708
$672$	0	0
$673$	−8.83282	−0.340480	−0.170240	−	0.985403i	$-0.554454\pi$
$673$	−8.83282	−0.340480	−0.170240	+	0.985403i	$0.554454\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−21.0557	−0.809237	−0.404619	−	0.914485i	$-0.632596\pi$
$677$	−21.0557	−0.809237	−0.404619	+	0.914485i	$0.632596\pi$
$678$	0	0
$679$	−8.47214	−0.325131
$680$	0	0
$681$	−0.944272	−0.0361846
$682$	0	0
$683$	−1.88854	−0.0722631	−0.0361316	−	0.999347i	$-0.511504\pi$
$683$	−1.88854	−0.0722631	−0.0361316	+	0.999347i	$0.511504\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−23.8885	−0.911405
$688$	0	0
$689$	−15.7771	−0.601059
$690$	0	0
$691$	−44.3607	−1.68756	−0.843780	−	0.536689i	$-0.819675\pi$
$691$	−44.3607	−1.68756	−0.843780	+	0.536689i	$0.819675\pi$
$692$	0	0
$693$	−6.47214	−0.245856
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−4.00000	−0.151511
$698$	0	0
$699$	−9.41641	−0.356161
$700$	0	0
$701$	−34.0000	−1.28416	−0.642081	−	0.766637i	$-0.721929\pi$
$701$	−34.0000	−1.28416	−0.642081	+	0.766637i	$0.721929\pi$
$702$	0	0
$703$	17.1672	0.647473
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−14.0000	−0.526524
$708$	0	0
$709$	25.7771	0.968079	0.484039	−	0.875046i	$-0.339169\pi$
$709$	25.7771	0.968079	0.484039	+	0.875046i	$0.339169\pi$
$710$	0	0
$711$	−12.9443	−0.485448
$712$	0	0
$713$	−6.11146	−0.228876
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−10.4721	−0.391089
$718$	0	0
$719$	6.83282	0.254821	0.127411	−	0.991850i	$-0.459333\pi$
$719$	6.83282	0.254821	0.127411	+	0.991850i	$0.459333\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	18.9443	0.704545
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−38.8328	−1.44023	−0.720115	−	0.693855i	$-0.755911\pi$
$727$	−38.8328	−1.44023	−0.720115	+	0.693855i	$0.755911\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	17.8885	0.661632
$732$	0	0
$733$	−10.5836	−0.390914	−0.195457	−	0.980712i	$-0.562619\pi$
$733$	−10.5836	−0.390914	−0.195457	+	0.980712i	$0.562619\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	25.8885	0.953617
$738$	0	0
$739$	5.88854	0.216614	0.108307	−	0.994118i	$-0.465457\pi$
$739$	5.88854	0.216614	0.108307	+	0.994118i	$0.465457\pi$
$740$	0	0
$741$	11.0557	0.406142
$742$	0	0
$743$	34.8328	1.27789	0.638946	−	0.769252i	$-0.279371\pi$
$743$	34.8328	1.27789	0.638946	+	0.769252i	$0.279371\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−16.9443	−0.619958
$748$	0	0
$749$	−12.9443	−0.472973
$750$	0	0
$751$	20.9443	0.764267	0.382134	−	0.924107i	$-0.375189\pi$
$751$	20.9443	0.764267	0.382134	+	0.924107i	$0.375189\pi$
$752$	0	0
$753$	16.9443	0.617484
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−31.8885	−1.15901	−0.579504	−	0.814969i	$-0.696754\pi$
$757$	−31.8885	−1.15901	−0.579504	+	0.814969i	$0.696754\pi$
$758$	0	0
$759$	25.8885	0.939695
$760$	0	0
$761$	−27.8885	−1.01096	−0.505479	−	0.862839i	$-0.668684\pi$
$761$	−27.8885	−1.01096	−0.505479	+	0.862839i	$0.668684\pi$
$762$	0	0
$763$	−2.00000	−0.0724049
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−40.0000	−1.44432
$768$	0	0
$769$	−52.8328	−1.90520	−0.952600	−	0.304226i	$-0.901602\pi$
$769$	−52.8328	−1.90520	−0.952600	+	0.304226i	$0.901602\pi$
$770$	0	0
$771$	18.9443	0.682261
$772$	0	0
$773$	42.9443	1.54460	0.772299	−	0.635259i	$-0.219107\pi$
$773$	42.9443	1.54460	0.772299	+	0.635259i	$0.219107\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−6.94427	−0.249124
$778$	0	0
$779$	−4.94427	−0.177147
$780$	0	0
$781$	35.7771	1.28020
$782$	0	0
$783$	2.00000	0.0714742
$784$	0	0
$785$	0	0
$786$	0	0
$787$	31.0557	1.10702	0.553509	−	0.832843i	$-0.313289\pi$
$787$	31.0557	1.10702	0.553509	+	0.832843i	$0.313289\pi$
$788$	0	0
$789$	−7.05573	−0.251191
$790$	0	0
$791$	−0.472136	−0.0167872
$792$	0	0
$793$	8.94427	0.317620
$794$	0	0
$795$	0	0
$796$	0	0
$797$	18.9443	0.671041	0.335520	−	0.942033i	$-0.391088\pi$
$797$	18.9443	0.671041	0.335520	+	0.942033i	$0.391088\pi$
$798$	0	0
$799$	25.8885	0.915871
$800$	0	0
$801$	−2.00000	−0.0706665
$802$	0	0
$803$	−80.7214	−2.84859
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−11.8885	−0.418497
$808$	0	0
$809$	38.9443	1.36921	0.684604	−	0.728915i	$-0.259975\pi$
$809$	38.9443	1.36921	0.684604	+	0.728915i	$0.259975\pi$
$810$	0	0
$811$	−55.4164	−1.94593	−0.972967	−	0.230946i	$-0.925818\pi$
$811$	−55.4164	−1.94593	−0.972967	+	0.230946i	$0.925818\pi$
$812$	0	0
$813$	−1.52786	−0.0535845
$814$	0	0
$815$	0	0
$816$	0	0
$817$	22.1115	0.773582
$818$	0	0
$819$	−4.47214	−0.156269
$820$	0	0
$821$	33.7771	1.17883	0.589414	−	0.807831i	$-0.299359\pi$
$821$	33.7771	1.17883	0.589414	+	0.807831i	$0.299359\pi$
$822$	0	0
$823$	44.9443	1.56666	0.783329	−	0.621607i	$-0.213520\pi$
$823$	44.9443	1.56666	0.783329	+	0.621607i	$0.213520\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	12.9443	0.450116	0.225058	−	0.974345i	$-0.427743\pi$
$827$	12.9443	0.450116	0.225058	+	0.974345i	$0.427743\pi$
$828$	0	0
$829$	−13.0557	−0.453444	−0.226722	−	0.973959i	$-0.572801\pi$
$829$	−13.0557	−0.453444	−0.226722	+	0.973959i	$0.572801\pi$
$830$	0	0
$831$	18.9443	0.657170
$832$	0	0
$833$	2.00000	0.0692959
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.52786	0.0528107
$838$	0	0
$839$	−54.8328	−1.89304	−0.946520	−	0.322647i	$-0.895427\pi$
$839$	−54.8328	−1.89304	−0.946520	+	0.322647i	$0.895427\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	10.9443	0.376941
$844$	0	0
$845$	0	0
$846$	0	0
$847$	30.8885	1.06134
$848$	0	0
$849$	−12.0000	−0.411839
$850$	0	0
$851$	27.7771	0.952186
$852$	0	0
$853$	31.3050	1.07186	0.535931	−	0.844262i	$-0.319961\pi$
$853$	31.3050	1.07186	0.535931	+	0.844262i	$0.319961\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−36.8328	−1.25819	−0.629093	−	0.777330i	$-0.716573\pi$
$857$	−36.8328	−1.25819	−0.629093	+	0.777330i	$0.716573\pi$
$858$	0	0
$859$	−50.4721	−1.72209	−0.861044	−	0.508531i	$-0.830189\pi$
$859$	−50.4721	−1.72209	−0.861044	+	0.508531i	$0.830189\pi$
$860$	0	0
$861$	2.00000	0.0681598
$862$	0	0
$863$	21.8885	0.745095	0.372547	−	0.928013i	$-0.378484\pi$
$863$	21.8885	0.745095	0.372547	+	0.928013i	$0.378484\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	13.0000	0.441503
$868$	0	0
$869$	83.7771	2.84194
$870$	0	0
$871$	17.8885	0.606130
$872$	0	0
$873$	−8.47214	−0.286738
$874$	0	0
$875$	0	0
$876$	0	0
$877$	56.8328	1.91911	0.959554	−	0.281525i	$-0.0908402\pi$
$877$	56.8328	1.91911	0.959554	+	0.281525i	$0.0908402\pi$
$878$	0	0
$879$	5.05573	0.170525
$880$	0	0
$881$	−27.8885	−0.939589	−0.469794	−	0.882776i	$-0.655672\pi$
$881$	−27.8885	−0.939589	−0.469794	+	0.882776i	$0.655672\pi$
$882$	0	0
$883$	−37.8885	−1.27505	−0.637526	−	0.770429i	$-0.720042\pi$
$883$	−37.8885	−1.27505	−0.637526	+	0.770429i	$0.720042\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−30.8328	−1.03526	−0.517632	−	0.855603i	$-0.673186\pi$
$887$	−30.8328	−1.03526	−0.517632	+	0.855603i	$0.673186\pi$
$888$	0	0
$889$	−4.94427	−0.165826
$890$	0	0
$891$	−6.47214	−0.216825
$892$	0	0
$893$	32.0000	1.07084
$894$	0	0
$895$	0	0
$896$	0	0
$897$	17.8885	0.597281
$898$	0	0
$899$	3.05573	0.101914
$900$	0	0
$901$	7.05573	0.235060
$902$	0	0
$903$	−8.94427	−0.297647
$904$	0	0
$905$	0	0
$906$	0	0
$907$	53.8885	1.78934	0.894670	−	0.446728i	$-0.147411\pi$
$907$	53.8885	1.78934	0.894670	+	0.446728i	$0.147411\pi$
$908$	0	0
$909$	−14.0000	−0.464351
$910$	0	0
$911$	−46.2492	−1.53231	−0.766153	−	0.642659i	$-0.777831\pi$
$911$	−46.2492	−1.53231	−0.766153	+	0.642659i	$0.777831\pi$
$912$	0	0
$913$	109.666	3.62940
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−4.00000	−0.132092
$918$	0	0
$919$	35.0557	1.15638	0.578191	−	0.815902i	$-0.303759\pi$
$919$	35.0557	1.15638	0.578191	+	0.815902i	$0.303759\pi$
$920$	0	0
$921$	15.0557	0.496103
$922$	0	0
$923$	24.7214	0.813713
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−16.1115	−0.528600	−0.264300	−	0.964441i	$-0.585141\pi$
$929$	−16.1115	−0.528600	−0.264300	+	0.964441i	$0.585141\pi$
$930$	0	0
$931$	2.47214	0.0810210
$932$	0	0
$933$	25.8885	0.847553
$934$	0	0
$935$	0	0
$936$	0	0
$937$	52.4721	1.71419	0.857095	−	0.515158i	$-0.172267\pi$
$937$	52.4721	1.71419	0.857095	+	0.515158i	$0.172267\pi$
$938$	0	0
$939$	−17.4164	−0.568363
$940$	0	0
$941$	−30.0000	−0.977972	−0.488986	−	0.872292i	$-0.662633\pi$
$941$	−30.0000	−0.977972	−0.488986	+	0.872292i	$0.662633\pi$
$942$	0	0
$943$	−8.00000	−0.260516
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−17.8885	−0.581300	−0.290650	−	0.956830i	$-0.593871\pi$
$947$	−17.8885	−0.581300	−0.290650	+	0.956830i	$0.593871\pi$
$948$	0	0
$949$	−55.7771	−1.81060
$950$	0	0
$951$	14.3607	0.465677
$952$	0	0
$953$	33.4164	1.08246	0.541232	−	0.840873i	$-0.317958\pi$
$953$	33.4164	1.08246	0.541232	+	0.840873i	$0.317958\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−12.9443	−0.418429
$958$	0	0
$959$	−3.52786	−0.113921
$960$	0	0
$961$	−28.6656	−0.924698
$962$	0	0
$963$	−12.9443	−0.417123
$964$	0	0
$965$	0	0
$966$	0	0
$967$	25.8885	0.832519	0.416260	−	0.909246i	$-0.363341\pi$
$967$	25.8885	0.832519	0.416260	+	0.909246i	$0.363341\pi$
$968$	0	0
$969$	−4.94427	−0.158833
$970$	0	0
$971$	−40.9443	−1.31396	−0.656982	−	0.753906i	$-0.728167\pi$
$971$	−40.9443	−1.31396	−0.656982	+	0.753906i	$0.728167\pi$
$972$	0	0
$973$	−7.41641	−0.237759
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−30.5836	−0.978456	−0.489228	−	0.872156i	$-0.662721\pi$
$977$	−30.5836	−0.978456	−0.489228	+	0.872156i	$0.662721\pi$
$978$	0	0
$979$	12.9443	0.413701
$980$	0	0
$981$	−2.00000	−0.0638551
$982$	0	0
$983$	−22.8328	−0.728254	−0.364127	−	0.931349i	$-0.618633\pi$
$983$	−22.8328	−0.728254	−0.364127	+	0.931349i	$0.618633\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−12.9443	−0.412021
$988$	0	0
$989$	35.7771	1.13765
$990$	0	0
$991$	−4.94427	−0.157060	−0.0785300	−	0.996912i	$-0.525023\pi$
$991$	−4.94427	−0.157060	−0.0785300	+	0.996912i	$0.525023\pi$
$992$	0	0
$993$	0.944272	0.0299656
$994$	0	0
$995$	0	0
$996$	0	0
$997$	5.41641	0.171539	0.0857697	−	0.996315i	$-0.472665\pi$
$997$	5.41641	0.171539	0.0857697	+	0.996315i	$0.472665\pi$
$998$	0	0
$999$	−6.94427	−0.219707

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8400.2.a.cx.1.1		2
4.3	odd	2		525.2.a.g.1.2		2
5.4	even	2		1680.2.a.v.1.1		2
12.11	even	2		1575.2.a.r.1.1		2
15.14	odd	2		5040.2.a.bw.1.2		2
20.3	even	4		525.2.d.c.274.2		4
20.7	even	4		525.2.d.c.274.3		4
20.19	odd	2		105.2.a.b.1.1	✓	2
28.27	even	2		3675.2.a.y.1.2		2
40.19	odd	2		6720.2.a.cx.1.1		2
40.29	even	2		6720.2.a.cs.1.2		2
60.23	odd	4		1575.2.d.d.1324.4		4
60.47	odd	4		1575.2.d.d.1324.1		4
60.59	even	2		315.2.a.d.1.2		2
140.19	even	6		735.2.i.i.361.2		4
140.39	odd	6		735.2.i.k.226.2		4
140.59	even	6		735.2.i.i.226.2		4
140.79	odd	6		735.2.i.k.361.2		4
140.139	even	2		735.2.a.k.1.1		2
420.419	odd	2		2205.2.a.w.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.a.b.1.1	✓	2	20.19	odd	2
315.2.a.d.1.2		2	60.59	even	2
525.2.a.g.1.2		2	4.3	odd	2
525.2.d.c.274.2		4	20.3	even	4
525.2.d.c.274.3		4	20.7	even	4
735.2.a.k.1.1		2	140.139	even	2
735.2.i.i.226.2		4	140.59	even	6
735.2.i.i.361.2		4	140.19	even	6
735.2.i.k.226.2		4	140.39	odd	6
735.2.i.k.361.2		4	140.79	odd	6
1575.2.a.r.1.1		2	12.11	even	2
1575.2.d.d.1324.1		4	60.47	odd	4
1575.2.d.d.1324.4		4	60.23	odd	4
1680.2.a.v.1.1		2	5.4	even	2
2205.2.a.w.1.2		2	420.419	odd	2
3675.2.a.y.1.2		2	28.27	even	2
5040.2.a.bw.1.2		2	15.14	odd	2
6720.2.a.cs.1.2		2	40.29	even	2
6720.2.a.cx.1.1		2	40.19	odd	2
8400.2.a.cx.1.1		2	1.1	even	1	trivial

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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8400 = 2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8400.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} -6.47214 q^{11} -4.47214 q^{13} +2.00000 q^{17} +2.47214 q^{19} -1.00000 q^{21} +4.00000 q^{23} -1.00000 q^{27} -2.00000 q^{29} -1.52786 q^{31} +6.47214 q^{33} +6.94427 q^{37} +4.47214 q^{39} -2.00000 q^{41} +8.94427 q^{43} +12.9443 q^{47} +1.00000 q^{49} -2.00000 q^{51} +3.52786 q^{53} -2.47214 q^{57} +8.94427 q^{59} -2.00000 q^{61} +1.00000 q^{63} -4.00000 q^{67} -4.00000 q^{69} -5.52786 q^{71} +12.4721 q^{73} -6.47214 q^{77} -12.9443 q^{79} +1.00000 q^{81} -16.9443 q^{83} +2.00000 q^{87} -2.00000 q^{89} -4.47214 q^{91} +1.52786 q^{93} -8.47214 q^{97} -6.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} + 2 q^{7} + 2 q^{9} - 4 q^{11} + 4 q^{17} - 4 q^{19} - 2 q^{21} + 8 q^{23} - 2 q^{27} - 4 q^{29} - 12 q^{31} + 4 q^{33} - 4 q^{37} - 4 q^{41} + 8 q^{47} + 2 q^{49} - 4 q^{51} + 16 q^{53} + 4 q^{57} - 4 q^{61} + 2 q^{63} - 8 q^{67} - 8 q^{69} - 20 q^{71} + 16 q^{73} - 4 q^{77} - 8 q^{79} + 2 q^{81} - 16 q^{83} + 4 q^{87} - 4 q^{89} + 12 q^{93} - 8 q^{97} - 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^7 + 2 * q^9 - 4 * q^11 + 4 * q^17 - 4 * q^19 - 2 * q^21 + 8 * q^23 - 2 * q^27 - 4 * q^29 - 12 * q^31 + 4 * q^33 - 4 * q^37 - 4 * q^41 + 8 * q^47 + 2 * q^49 - 4 * q^51 + 16 * q^53 + 4 * q^57 - 4 * q^61 + 2 * q^63 - 8 * q^67 - 8 * q^69 - 20 * q^71 + 16 * q^73 - 4 * q^77 - 8 * q^79 + 2 * q^81 - 16 * q^83 + 4 * q^87 - 4 * q^89 + 12 * q^93 - 8 * q^97 - 4 * q^99

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