LMFDB - Embedded newform 6000.2.f.p.1249.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6000,2,Mod(1249,6000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6000, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

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

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

chi := DirichletCharacter("6000.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$6000 = 2^{4} \cdot 3 \cdot 5^{3}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	6000.f (of order $2$ , degree $1$ , not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$47.9102412128$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.566105760000.21
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} + 45x^{6} + 728x^{4} + 4965x^{2} + 11881$ x^8 + 45x^6 + 728x^4 + 4965*x^2 + 11881
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 3000)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.3
Root			$2.35902i$ of defining polynomial
Character	$\chi$	$=$	6000.1249
Dual form			6000.2.f.p.1249.6

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +3.35902i q^{7} -1.00000 q^{9} +0.740989 q^{11} +3.81698i q^{13} -1.61803i q^{17} +0.839922 q^{19} +3.35902 q^{21} -2.74099i q^{23} +1.00000i q^{27} +3.05305 q^{29} -1.77811 q^{31} -0.740989i q^{33} +4.89297i q^{37} +3.81698 q^{39} +5.11311 q^{41} -0.145898i q^{43} +2.65516i q^{47} -4.28303 q^{49} -1.61803 q^{51} -10.6711i q^{53} -0.839922i q^{57} -14.9602 q^{59} +14.9930 q^{61} -3.35902i q^{63} +8.41099i q^{67} -2.74099 q^{69} -3.43501 q^{71} +5.75517i q^{73} +2.48900i q^{77} +0.0327846 q^{79} +1.00000 q^{81} +10.5492i q^{83} -3.05305i q^{87} -1.60385 q^{89} -12.8213 q^{91} +1.77811i q^{93} +3.11311i q^{97} -0.740989 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$8 q - 8 q^{9} - 6 q^{11} + 6 q^{21} - 30 q^{29} - 12 q^{31} - 6 q^{39} + 26 q^{41} - 38 q^{49} - 4 q^{51} + 16 q^{59} + 26 q^{61} - 10 q^{69} + 18 q^{71} + 42 q^{79} + 8 q^{81} - 24 q^{89} - 34 q^{91} + 6 q^{99}+O(q^{100})$ 8 * q - 8 * q^9 - 6 * q^11 + 6 * q^21 - 30 * q^29 - 12 * q^31 - 6 * q^39 + 26 * q^41 - 38 * q^49 - 4 * q^51 + 16 * q^59 + 26 * q^61 - 10 * q^69 + 18 * q^71 + 42 * q^79 + 8 * q^81 - 24 * q^89 - 34 * q^91 + 6 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6000\mathbb{Z}\right)^\times$ .

$n$	$751$	$4001$	$4501$	$5377$
$\chi(n)$	$1$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$ -expansion $a_n$ , the Satake parameters $\alpha_p$ , and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$ .

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	0	0
$2$	0	0
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.35902i	1.26959i	0.772680	+	0.634796i	$0.218916\pi$
$7$	3.35902i	1.26959i	−0.772680	+	0.634796i	$0.781084\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	0.740989	0.223416	0.111708	−	0.993741i	$-0.464368\pi$
$11$	0.740989	0.223416	0.111708	+	0.993741i	$0.464368\pi$
$12$	0	0
$13$	3.81698i	1.05864i	0.848422	+	0.529320i	$0.177553\pi$
$13$	3.81698i	1.05864i	−0.848422	+	0.529320i	$0.822447\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 1.61803i	− 0.392431i	−0.980561	−	0.196215i	$-0.937135\pi$
$17$	− 1.61803i	− 0.392431i	0.980561	−	0.196215i	$-0.0628652\pi$
$18$	0	0
$19$	0.839922	0.192691	0.0963457	−	0.995348i	$-0.469285\pi$
$19$	0.839922	0.192691	0.0963457	+	0.995348i	$0.469285\pi$
$20$	0	0
$21$	3.35902	0.732999
$22$	0	0
$23$	− 2.74099i	− 0.571536i	−0.958299	−	0.285768i	$-0.907751\pi$
$23$	− 2.74099i	− 0.571536i	0.958299	−	0.285768i	$-0.0922486\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	3.05305	0.566937	0.283468	−	0.958982i	$-0.408515\pi$
$29$	3.05305	0.566937	0.283468	+	0.958982i	$0.408515\pi$
$30$	0	0
$31$	−1.77811	−0.319358	−0.159679	−	0.987169i	$-0.551046\pi$
$31$	−1.77811	−0.319358	−0.159679	+	0.987169i	$0.551046\pi$
$32$	0	0
$33$	− 0.740989i	− 0.128990i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.89297i	0.804399i	0.915552	+	0.402200i	$0.131754\pi$
$37$	4.89297i	0.804399i	−0.915552	+	0.402200i	$0.868246\pi$
$38$	0	0
$39$	3.81698	0.611206
$40$	0	0
$41$	5.11311	0.798534	0.399267	−	0.916835i	$-0.369265\pi$
$41$	5.11311	0.798534	0.399267	+	0.916835i	$0.369265\pi$
$42$	0	0
$43$	− 0.145898i	− 0.0222492i	−0.999938	−	0.0111246i	$-0.996459\pi$
$43$	− 0.145898i	− 0.0222492i	0.999938	−	0.0111246i	$-0.00354115\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.65516i	0.387294i	0.981071	+	0.193647i	$0.0620317\pi$
$47$	2.65516i	0.387294i	−0.981071	+	0.193647i	$0.937968\pi$
$48$	0	0
$49$	−4.28303	−0.611862
$50$	0	0
$51$	−1.61803	−0.226570
$52$	0	0
$53$	− 10.6711i	− 1.46579i	−0.680344	−	0.732893i	$-0.738170\pi$
$53$	− 10.6711i	− 1.46579i	0.680344	−	0.732893i	$-0.261830\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 0.839922i	− 0.111250i
$58$	0	0
$59$	−14.9602	−1.94765	−0.973826	−	0.227296i	$-0.927012\pi$
$59$	−14.9602	−1.94765	−0.973826	+	0.227296i	$0.927012\pi$
$60$	0	0
$61$	14.9930	1.91965	0.959827	−	0.280592i	$-0.0905307\pi$
$61$	14.9930	1.91965	0.959827	+	0.280592i	$0.0905307\pi$
$62$	0	0
$63$	− 3.35902i	− 0.423197i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	8.41099i	1.02757i	0.857920	+	0.513783i	$0.171756\pi$
$67$	8.41099i	1.02757i	−0.857920	+	0.513783i	$0.828244\pi$
$68$	0	0
$69$	−2.74099	−0.329976
$70$	0	0
$71$	−3.43501	−0.407661	−0.203831	−	0.979006i	$-0.565339\pi$
$71$	−3.43501	−0.407661	−0.203831	+	0.979006i	$0.565339\pi$
$72$	0	0
$73$	5.75517i	0.673592i	0.941578	+	0.336796i	$0.109343\pi$
$73$	5.75517i	0.673592i	−0.941578	+	0.336796i	$0.890657\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.48900i	0.283648i
$78$	0	0
$79$	0.0327846	0.00368856	0.00184428	−	0.999998i	$-0.499413\pi$
$79$	0.0327846	0.00368856	0.00184428	+	0.999998i	$0.499413\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	10.5492i	1.15793i	0.815354	+	0.578963i	$0.196542\pi$
$83$	10.5492i	1.15793i	−0.815354	+	0.578963i	$0.803458\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 3.05305i	− 0.327321i
$88$	0	0
$89$	−1.60385	−0.170008	−0.0850041	−	0.996381i	$-0.527090\pi$
$89$	−1.60385	−0.170008	−0.0850041	+	0.996381i	$0.527090\pi$
$90$	0	0
$91$	−12.8213	−1.34404
$92$	0	0
$93$	1.77811i	0.184382i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	3.11311i	0.316089i	0.987432	+	0.158044i	$0.0505189\pi$
$97$	3.11311i	0.316089i	−0.987432	+	0.158044i	$0.949481\pi$
$98$	0	0
$99$	−0.740989	−0.0744722
$100$	0	0
$101$	13.8071	1.37386	0.686931	−	0.726723i	$-0.258958\pi$
$101$	13.8071	1.37386	0.686931	+	0.726723i	$0.258958\pi$
$102$	0	0
$103$	12.6252i	1.24400i	0.783018	+	0.621999i	$0.213679\pi$
$103$	12.6252i	1.24400i	−0.783018	+	0.621999i	$0.786321\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	9.14322i	0.883908i	0.897038	+	0.441954i	$0.145715\pi$
$107$	9.14322i	0.883908i	−0.897038	+	0.441954i	$0.854285\pi$
$108$	0	0
$109$	−9.23781	−0.884822	−0.442411	−	0.896812i	$-0.645877\pi$
$109$	−9.23781	−0.884822	−0.442411	+	0.896812i	$0.645877\pi$
$110$	0	0
$111$	4.89297	0.464420
$112$	0	0
$113$	− 7.90107i	− 0.743270i	−0.928379	−	0.371635i	$-0.878797\pi$
$113$	− 7.90107i	− 0.743270i	0.928379	−	0.371635i	$-0.121203\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 3.81698i	− 0.352880i
$118$	0	0
$119$	5.43501	0.498227
$120$	0	0
$121$	−10.4509	−0.950085
$122$	0	0
$123$	− 5.11311i	− 0.461034i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	7.31206i	0.648840i	0.945913	+	0.324420i	$0.105169\pi$
$127$	7.31206i	0.648840i	−0.945913	+	0.324420i	$0.894831\pi$
$128$	0	0
$129$	−0.145898	−0.0128456
$130$	0	0
$131$	−15.2192	−1.32971	−0.664854	−	0.746973i	$-0.731506\pi$
$131$	−15.2192	−1.32971	−0.664854	+	0.746973i	$0.731506\pi$
$132$	0	0
$133$	2.82132i	0.244639i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	10.4492i	0.892735i	0.894850	+	0.446367i	$0.147283\pi$
$137$	10.4492i	0.892735i	−0.894850	+	0.446367i	$0.852717\pi$
$138$	0	0
$139$	−16.1362	−1.36865	−0.684327	−	0.729175i	$-0.739904\pi$
$139$	−16.1362	−1.36865	−0.684327	+	0.729175i	$0.739904\pi$
$140$	0	0
$141$	2.65516	0.223605
$142$	0	0
$143$	2.82834i	0.236517i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	4.28303i	0.353259i
$148$	0	0
$149$	−2.12188	−0.173831	−0.0869155	−	0.996216i	$-0.527701\pi$
$149$	−2.12188	−0.173831	−0.0869155	+	0.996216i	$0.527701\pi$
$150$	0	0
$151$	−18.3732	−1.49519	−0.747595	−	0.664155i	$-0.768792\pi$
$151$	−18.3732	−1.49519	−0.747595	+	0.664155i	$0.768792\pi$
$152$	0	0
$153$	1.61803i	0.130810i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	17.9585i	1.43324i	0.697464	+	0.716620i	$0.254312\pi$
$157$	17.9585i	1.43324i	−0.697464	+	0.716620i	$0.745688\pi$
$158$	0	0
$159$	−10.6711	−0.846272
$160$	0	0
$161$	9.20704	0.725617
$162$	0	0
$163$	− 7.89123i	− 0.618088i	−0.951048	−	0.309044i	$-0.899991\pi$
$163$	− 7.89123i	− 0.618088i	0.951048	−	0.309044i	$-0.100009\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	7.60927i	0.588823i	0.955679	+	0.294412i	$0.0951236\pi$
$167$	7.60927i	0.588823i	−0.955679	+	0.294412i	$0.904876\pi$
$168$	0	0
$169$	−1.56933	−0.120717
$170$	0	0
$171$	−0.839922	−0.0642305
$172$	0	0
$173$	− 12.6410i	− 0.961076i	−0.876974	−	0.480538i	$-0.840441\pi$
$173$	− 12.6410i	− 0.961076i	0.876974	−	0.480538i	$-0.159559\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	14.9602i	1.12448i
$178$	0	0
$179$	4.61370	0.344844	0.172422	−	0.985023i	$-0.444841\pi$
$179$	4.61370	0.344844	0.172422	+	0.985023i	$0.444841\pi$
$180$	0	0
$181$	−21.5881	−1.60463	−0.802314	−	0.596902i	$-0.796398\pi$
$181$	−21.5881	−1.60463	−0.802314	+	0.596902i	$0.796398\pi$
$182$	0	0
$183$	− 14.9930i	− 1.10831i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 1.19894i	− 0.0876755i
$188$	0	0
$189$	−3.35902	−0.244333
$190$	0	0
$191$	−8.59617	−0.621997	−0.310998	−	0.950410i	$-0.600663\pi$
$191$	−8.59617	−0.621997	−0.310998	+	0.950410i	$0.600663\pi$
$192$	0	0
$193$	25.7383i	1.85268i	0.376684	+	0.926342i	$0.377064\pi$
$193$	25.7383i	1.85268i	−0.376684	+	0.926342i	$0.622936\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	12.3678i	0.881168i	0.897711	+	0.440584i	$0.145229\pi$
$197$	12.3678i	0.881168i	−0.897711	+	0.440584i	$0.854771\pi$
$198$	0	0
$199$	25.4050	1.80092	0.900458	−	0.434943i	$-0.143231\pi$
$199$	25.4050	1.80092	0.900458	+	0.434943i	$0.143231\pi$
$200$	0	0
$201$	8.41099	0.593266
$202$	0	0
$203$	10.2553i	0.719778i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	2.74099i	0.190512i
$208$	0	0
$209$	0.622373	0.0430504
$210$	0	0
$211$	−22.7761	−1.56797	−0.783986	−	0.620779i	$-0.786816\pi$
$211$	−22.7761	−1.56797	−0.783986	+	0.620779i	$0.786816\pi$
$212$	0	0
$213$	3.43501i	0.235363i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 5.97272i	− 0.405455i
$218$	0	0
$219$	5.75517	0.388898
$220$	0	0
$221$	6.17600	0.415443
$222$	0	0
$223$	− 8.15024i	− 0.545780i	−0.962045	−	0.272890i	$-0.912020\pi$
$223$	− 8.15024i	− 0.545780i	0.962045	−	0.272890i	$-0.0879795\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 8.07773i	− 0.536138i	−0.963400	−	0.268069i	$-0.913614\pi$
$227$	− 8.07773i	− 0.536138i	0.963400	−	0.268069i	$-0.0863855\pi$
$228$	0	0
$229$	23.1443	1.52942	0.764709	−	0.644376i	$-0.222883\pi$
$229$	23.1443	1.52942	0.764709	+	0.644376i	$0.222883\pi$
$230$	0	0
$231$	2.48900	0.163764
$232$	0	0
$233$	7.20436i	0.471973i	0.971756	+	0.235987i	$0.0758322\pi$
$233$	7.20436i	0.471973i	−0.971756	+	0.235987i	$0.924168\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 0.0327846i	− 0.00212959i
$238$	0	0
$239$	13.0213	0.842280	0.421140	−	0.906996i	$-0.361630\pi$
$239$	13.0213	0.842280	0.421140	+	0.906996i	$0.361630\pi$
$240$	0	0
$241$	17.2351	1.11021	0.555106	−	0.831780i	$-0.312678\pi$
$241$	17.2351	1.11021	0.555106	+	0.831780i	$0.312678\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3.20596i	0.203991i
$248$	0	0
$249$	10.5492	0.668529
$250$	0	0
$251$	−11.5420	−0.728527	−0.364264	−	0.931296i	$-0.618679\pi$
$251$	−11.5420	−0.728527	−0.364264	+	0.931296i	$0.618679\pi$
$252$	0	0
$253$	− 2.03104i	− 0.127690i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	28.8390i	1.79893i	0.436997	+	0.899463i	$0.356042\pi$
$257$	28.8390i	1.79893i	−0.436997	+	0.899463i	$0.643958\pi$
$258$	0	0
$259$	−16.4356	−1.02126
$260$	0	0
$261$	−3.05305	−0.188979
$262$	0	0
$263$	12.6562i	0.780417i	0.920727	+	0.390208i	$0.127597\pi$
$263$	12.6562i	0.780417i	−0.920727	+	0.390208i	$0.872403\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	1.60385i	0.0981543i
$268$	0	0
$269$	−15.2884	−0.932153	−0.466077	−	0.884744i	$-0.654333\pi$
$269$	−15.2884	−0.932153	−0.466077	+	0.884744i	$0.654333\pi$
$270$	0	0
$271$	16.4404	0.998685	0.499342	−	0.866405i	$-0.333575\pi$
$271$	16.4404	0.998685	0.499342	+	0.866405i	$0.333575\pi$
$272$	0	0
$273$	12.8213i	0.775981i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	25.8329i	1.55215i	0.630641	+	0.776074i	$0.282792\pi$
$277$	25.8329i	1.55215i	−0.630641	+	0.776074i	$0.717208\pi$
$278$	0	0
$279$	1.77811	0.106453
$280$	0	0
$281$	−12.7170	−0.758631	−0.379315	−	0.925267i	$-0.623840\pi$
$281$	−12.7170	−0.758631	−0.379315	+	0.925267i	$0.623840\pi$
$282$	0	0
$283$	− 18.6209i	− 1.10689i	−0.832884	−	0.553447i	$-0.813312\pi$
$283$	− 18.6209i	− 1.10689i	0.832884	−	0.553447i	$-0.186688\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	17.1751i	1.01381i
$288$	0	0
$289$	14.3820	0.845998
$290$	0	0
$291$	3.11311	0.182494
$292$	0	0
$293$	− 25.9541i	− 1.51626i	−0.652106	−	0.758128i	$-0.726114\pi$
$293$	− 25.9541i	− 1.51626i	0.652106	−	0.758128i	$-0.273886\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0.740989i	0.0429965i
$298$	0	0
$299$	10.4623	0.605050
$300$	0	0
$301$	0.490075	0.0282474
$302$	0	0
$303$	− 13.8071i	− 0.793199i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 5.68526i	− 0.324475i	−0.986752	−	0.162237i	$-0.948129\pi$
$307$	− 5.68526i	− 0.324475i	0.986752	−	0.162237i	$-0.0518711\pi$
$308$	0	0
$309$	12.6252	0.718222
$310$	0	0
$311$	25.6472	1.45432	0.727160	−	0.686468i	$-0.240840\pi$
$311$	25.6472	1.45432	0.727160	+	0.686468i	$0.240840\pi$
$312$	0	0
$313$	4.66934i	0.263927i	0.991255	+	0.131963i	$0.0421281\pi$
$313$	4.66934i	0.263927i	−0.991255	+	0.131963i	$0.957872\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 9.49240i	− 0.533146i	−0.963815	−	0.266573i	$-0.914109\pi$
$317$	− 9.49240i	− 0.533146i	0.963815	−	0.266573i	$-0.0858914\pi$
$318$	0	0
$319$	2.26227	0.126663
$320$	0	0
$321$	9.14322	0.510325
$322$	0	0
$323$	− 1.35902i	− 0.0756180i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	9.23781i	0.510852i
$328$	0	0
$329$	−8.91873	−0.491706
$330$	0	0
$331$	−22.2875	−1.22503	−0.612516	−	0.790458i	$-0.709843\pi$
$331$	−22.2875	−1.22503	−0.612516	+	0.790458i	$0.709843\pi$
$332$	0	0
$333$	− 4.89297i	− 0.268133i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 24.7958i	− 1.35071i	−0.737492	−	0.675356i	$-0.763990\pi$
$337$	− 24.7958i	− 1.35071i	0.737492	−	0.675356i	$-0.236010\pi$
$338$	0	0
$339$	−7.90107	−0.429127
$340$	0	0
$341$	−1.31756	−0.0713499
$342$	0	0
$343$	9.12636i	0.492777i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	31.4013i	1.68571i	0.538141	+	0.842855i	$0.319127\pi$
$347$	31.4013i	1.68571i	−0.538141	+	0.842855i	$0.680873\pi$
$348$	0	0
$349$	−6.39548	−0.342342	−0.171171	−	0.985241i	$-0.554755\pi$
$349$	−6.39548	−0.342342	−0.171171	+	0.985241i	$0.554755\pi$
$350$	0	0
$351$	−3.81698	−0.203735
$352$	0	0
$353$	1.13780i	0.0605590i	0.999541	+	0.0302795i	$0.00963974\pi$
$353$	1.13780i	0.0605590i	−0.999541	+	0.0302795i	$0.990360\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 5.43501i	− 0.287651i
$358$	0	0
$359$	−14.6403	−0.772686	−0.386343	−	0.922355i	$-0.626262\pi$
$359$	−14.6403	−0.772686	−0.386343	+	0.922355i	$0.626262\pi$
$360$	0	0
$361$	−18.2945	−0.962870
$362$	0	0
$363$	10.4509i	0.548532i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 6.57715i	− 0.343325i	−0.985156	−	0.171662i	$-0.945086\pi$
$367$	− 6.57715i	− 0.343325i	0.985156	−	0.171662i	$-0.0549138\pi$
$368$	0	0
$369$	−5.11311	−0.266178
$370$	0	0
$371$	35.8444	1.86095
$372$	0	0
$373$	− 23.9224i	− 1.23866i	−0.785133	−	0.619328i	$-0.787405\pi$
$373$	− 23.9224i	− 1.23866i	0.785133	−	0.619328i	$-0.212595\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	11.6534i	0.600181i
$378$	0	0
$379$	−9.34150	−0.479840	−0.239920	−	0.970793i	$-0.577121\pi$
$379$	−9.34150	−0.479840	−0.239920	+	0.970793i	$0.577121\pi$
$380$	0	0
$381$	7.31206	0.374608
$382$	0	0
$383$	31.2794i	1.59830i	0.601129	+	0.799152i	$0.294718\pi$
$383$	31.2794i	1.59830i	−0.601129	+	0.799152i	$0.705282\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0.145898i	0.00741641i
$388$	0	0
$389$	8.53220	0.432600	0.216300	−	0.976327i	$-0.430601\pi$
$389$	8.53220	0.432600	0.216300	+	0.976327i	$0.430601\pi$
$390$	0	0
$391$	−4.43501	−0.224288
$392$	0	0
$393$	15.2192i	0.767707i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	9.84802i	0.494258i	0.968983	+	0.247129i	$0.0794871\pi$
$397$	9.84802i	0.494258i	−0.968983	+	0.247129i	$0.920513\pi$
$398$	0	0
$399$	2.82132	0.141243
$400$	0	0
$401$	31.6924	1.58264	0.791322	−	0.611400i	$-0.209393\pi$
$401$	31.6924	1.58264	0.791322	+	0.611400i	$0.209393\pi$
$402$	0	0
$403$	− 6.78702i	− 0.338085i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.62563i	0.179716i
$408$	0	0
$409$	2.26711	0.112101	0.0560507	−	0.998428i	$-0.482149\pi$
$409$	2.26711	0.112101	0.0560507	+	0.998428i	$0.482149\pi$
$410$	0	0
$411$	10.4492	0.515421
$412$	0	0
$413$	− 50.2516i	− 2.47272i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	16.1362i	0.790193i
$418$	0	0
$419$	34.2678	1.67409	0.837046	−	0.547132i	$-0.184280\pi$
$419$	34.2678	1.67409	0.837046	+	0.547132i	$0.184280\pi$
$420$	0	0
$421$	−24.7923	−1.20830	−0.604151	−	0.796870i	$-0.706488\pi$
$421$	−24.7923	−1.20830	−0.604151	+	0.796870i	$0.706488\pi$
$422$	0	0
$423$	− 2.65516i	− 0.129098i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	50.3618i	2.43718i
$428$	0	0
$429$	2.82834	0.136553
$430$	0	0
$431$	37.9842	1.82964	0.914818	−	0.403867i	$-0.132334\pi$
$431$	37.9842	1.82964	0.914818	+	0.403867i	$0.132334\pi$
$432$	0	0
$433$	12.7267i	0.611607i	0.952095	+	0.305804i	$0.0989251\pi$
$433$	12.7267i	0.611607i	−0.952095	+	0.305804i	$0.901075\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 2.30222i	− 0.110130i
$438$	0	0
$439$	6.49441	0.309961	0.154981	−	0.987917i	$-0.450468\pi$
$439$	6.49441	0.309961	0.154981	+	0.987917i	$0.450468\pi$
$440$	0	0
$441$	4.28303	0.203954
$442$	0	0
$443$	41.4017i	1.96705i	0.180761	+	0.983527i	$0.442144\pi$
$443$	41.4017i	1.96705i	−0.180761	+	0.983527i	$0.557856\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	2.12188i	0.100361i
$448$	0	0
$449$	−2.07532	−0.0979406	−0.0489703	−	0.998800i	$-0.515594\pi$
$449$	−2.07532	−0.0979406	−0.0489703	+	0.998800i	$0.515594\pi$
$450$	0	0
$451$	3.78876	0.178406
$452$	0	0
$453$	18.3732i	0.863248i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	14.5579i	0.680989i	0.940247	+	0.340494i	$0.110594\pi$
$457$	14.5579i	0.680989i	−0.940247	+	0.340494i	$0.889406\pi$
$458$	0	0
$459$	1.61803	0.0755234
$460$	0	0
$461$	2.08042	0.0968946	0.0484473	−	0.998826i	$-0.484573\pi$
$461$	2.08042	0.0968946	0.0484473	+	0.998826i	$0.484573\pi$
$462$	0	0
$463$	− 35.4935i	− 1.64952i	−0.565482	−	0.824761i	$-0.691310\pi$
$463$	− 35.4935i	− 1.64952i	0.565482	−	0.824761i	$-0.308690\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 23.0690i	− 1.06750i	−0.845641	−	0.533752i	$-0.820781\pi$
$467$	− 23.0690i	− 1.06750i	0.845641	−	0.533752i	$-0.179219\pi$
$468$	0	0
$469$	−28.2527	−1.30459
$470$	0	0
$471$	17.9585	0.827482
$472$	0	0
$473$	− 0.108109i	− 0.00497085i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	10.6711i	0.488595i
$478$	0	0
$479$	−24.9761	−1.14119	−0.570594	−	0.821232i	$-0.693287\pi$
$479$	−24.9761	−1.14119	−0.570594	+	0.821232i	$0.693287\pi$
$480$	0	0
$481$	−18.6764	−0.851569
$482$	0	0
$483$	− 9.20704i	− 0.418935i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	5.93443i	0.268915i	0.990919	+	0.134457i	$0.0429291\pi$
$487$	5.93443i	0.268915i	−0.990919	+	0.134457i	$0.957071\pi$
$488$	0	0
$489$	−7.89123	−0.356854
$490$	0	0
$491$	9.04428	0.408163	0.204081	−	0.978954i	$-0.434579\pi$
$491$	9.04428	0.408163	0.204081	+	0.978954i	$0.434579\pi$
$492$	0	0
$493$	− 4.93993i	− 0.222483i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 11.5383i	− 0.517563i
$498$	0	0
$499$	13.0203	0.582867	0.291433	−	0.956591i	$-0.405868\pi$
$499$	13.0203	0.582867	0.291433	+	0.956591i	$0.405868\pi$
$500$	0	0
$501$	7.60927	0.339957
$502$	0	0
$503$	4.22730i	0.188486i	0.995549	+	0.0942431i	$0.0300431\pi$
$503$	4.22730i	0.188486i	−0.995549	+	0.0942431i	$0.969957\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.56933i	0.0696962i
$508$	0	0
$509$	35.5543	1.57592	0.787959	−	0.615727i	$-0.211138\pi$
$509$	35.5543	1.57592	0.787959	+	0.615727i	$0.211138\pi$
$510$	0	0
$511$	−19.3317	−0.855186
$512$	0	0
$513$	0.839922i	0.0370835i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1.96744i	0.0865280i
$518$	0	0
$519$	−12.6410	−0.554877
$520$	0	0
$521$	−12.4994	−0.547609	−0.273805	−	0.961785i	$-0.588282\pi$
$521$	−12.4994	−0.547609	−0.273805	+	0.961785i	$0.588282\pi$
$522$	0	0
$523$	− 18.7346i	− 0.819208i	−0.912263	−	0.409604i	$-0.865667\pi$
$523$	− 18.7346i	− 0.819208i	0.912263	−	0.409604i	$-0.134333\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.87705i	0.125326i
$528$	0	0
$529$	15.4870	0.673347
$530$	0	0
$531$	14.9602	0.649217
$532$	0	0
$533$	19.5166i	0.845360i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 4.61370i	− 0.199096i
$538$	0	0
$539$	−3.17368	−0.136700
$540$	0	0
$541$	−18.0771	−0.777194	−0.388597	−	0.921408i	$-0.627040\pi$
$541$	−18.0771	−0.777194	−0.388597	+	0.921408i	$0.627040\pi$
$542$	0	0
$543$	21.5881i	0.926433i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 16.0246i	− 0.685162i	−0.939488	−	0.342581i	$-0.888699\pi$
$547$	− 16.0246i	− 0.685162i	0.939488	−	0.342581i	$-0.111301\pi$
$548$	0	0
$549$	−14.9930	−0.639885
$550$	0	0
$551$	2.56432	0.109244
$552$	0	0
$553$	0.110124i	0.00468296i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 11.9093i	− 0.504613i	−0.967647	−	0.252307i	$-0.918811\pi$
$557$	− 11.9093i	− 0.504613i	0.967647	−	0.252307i	$-0.0811892\pi$
$558$	0	0
$559$	0.556890	0.0235539
$560$	0	0
$561$	−1.19894	−0.0506195
$562$	0	0
$563$	− 7.78702i	− 0.328184i	−0.986445	−	0.164092i	$-0.947531\pi$
$563$	− 7.78702i	− 0.328184i	0.986445	−	0.164092i	$-0.0524693\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	3.35902i	0.141066i
$568$	0	0
$569$	0.387468	0.0162435	0.00812176	−	0.999967i	$-0.497415\pi$
$569$	0.387468	0.0162435	0.00812176	+	0.999967i	$0.497415\pi$
$570$	0	0
$571$	29.9652	1.25400	0.627002	−	0.779017i	$-0.284282\pi$
$571$	29.9652	1.25400	0.627002	+	0.779017i	$0.284282\pi$
$572$	0	0
$573$	8.59617i	0.359110i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 5.08583i	− 0.211726i	−0.994381	−	0.105863i	$-0.966240\pi$
$577$	− 5.08583i	− 0.211726i	0.994381	−	0.105863i	$-0.0337605\pi$
$578$	0	0
$579$	25.7383	1.06965
$580$	0	0
$581$	−35.4350	−1.47009
$582$	0	0
$583$	− 7.90715i	− 0.327481i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 9.68159i	− 0.399602i	−0.979837	−	0.199801i	$-0.935970\pi$
$587$	− 9.68159i	− 0.399602i	0.979837	−	0.199801i	$-0.0640296\pi$
$588$	0	0
$589$	−1.49348	−0.0615376
$590$	0	0
$591$	12.3678	0.508743
$592$	0	0
$593$	− 0.120359i	− 0.00494257i	−0.999997	−	0.00247128i	$-0.999213\pi$
$593$	− 0.120359i	− 0.00494257i	0.999997	−	0.00247128i	$-0.000786635\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 25.4050i	− 1.03976i
$598$	0	0
$599$	3.53936	0.144614	0.0723072	−	0.997382i	$-0.476964\pi$
$599$	3.53936	0.144614	0.0723072	+	0.997382i	$0.476964\pi$
$600$	0	0
$601$	−20.6295	−0.841496	−0.420748	−	0.907178i	$-0.638232\pi$
$601$	−20.6295	−0.841496	−0.420748	+	0.907178i	$0.638232\pi$
$602$	0	0
$603$	− 8.41099i	− 0.342522i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 2.77610i	− 0.112678i	−0.998412	−	0.0563391i	$-0.982057\pi$
$607$	− 2.77610i	− 0.112678i	0.998412	−	0.0563391i	$-0.0179428\pi$
$608$	0	0
$609$	10.2553	0.415564
$610$	0	0
$611$	−10.1347	−0.410005
$612$	0	0
$613$	− 2.42451i	− 0.0979249i	−0.998801	−	0.0489624i	$-0.984409\pi$
$613$	− 2.42451i	− 0.0979249i	0.998801	−	0.0489624i	$-0.0155915\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0.991902i	0.0399325i	0.999801	+	0.0199662i	$0.00635587\pi$
$617$	0.991902i	0.0399325i	−0.999801	+	0.0199662i	$0.993644\pi$
$618$	0	0
$619$	30.9783	1.24512	0.622561	−	0.782571i	$-0.286092\pi$
$619$	30.9783	1.24512	0.622561	+	0.782571i	$0.286092\pi$
$620$	0	0
$621$	2.74099	0.109992
$622$	0	0
$623$	− 5.38738i	− 0.215841i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 0.622373i	− 0.0248552i
$628$	0	0
$629$	7.91699	0.315671
$630$	0	0
$631$	33.9231	1.35046	0.675228	−	0.737609i	$-0.264045\pi$
$631$	33.9231	1.35046	0.675228	+	0.737609i	$0.264045\pi$
$632$	0	0
$633$	22.7761i	0.905269i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 16.3482i	− 0.647741i
$638$	0	0
$639$	3.43501	0.135887
$640$	0	0
$641$	24.2405	0.957444	0.478722	−	0.877967i	$-0.341100\pi$
$641$	24.2405	0.957444	0.478722	+	0.877967i	$0.341100\pi$
$642$	0	0
$643$	− 2.43662i	− 0.0960908i	−0.998845	−	0.0480454i	$-0.984701\pi$
$643$	− 2.43662i	− 0.0960908i	0.998845	−	0.0480454i	$-0.0152992\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	16.6481i	0.654506i	0.944937	+	0.327253i	$0.106123\pi$
$647$	16.6481i	0.654506i	−0.944937	+	0.327253i	$0.893877\pi$
$648$	0	0
$649$	−11.0853	−0.435137
$650$	0	0
$651$	−5.97272	−0.234089
$652$	0	0
$653$	14.8700i	0.581909i	0.956737	+	0.290955i	$0.0939728\pi$
$653$	14.8700i	0.581909i	−0.956737	+	0.290955i	$0.906027\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 5.75517i	− 0.224531i
$658$	0	0
$659$	−11.3710	−0.442953	−0.221477	−	0.975166i	$-0.571088\pi$
$659$	−11.3710	−0.442953	−0.221477	+	0.975166i	$0.571088\pi$
$660$	0	0
$661$	−43.7035	−1.69987	−0.849935	−	0.526888i	$-0.823359\pi$
$661$	−43.7035	−1.69987	−0.849935	+	0.526888i	$0.823359\pi$
$662$	0	0
$663$	− 6.17600i	− 0.239856i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 8.36837i	− 0.324024i
$668$	0	0
$669$	−8.15024	−0.315106
$670$	0	0
$671$	11.1096	0.428882
$672$	0	0
$673$	14.6438i	0.564477i	0.959344	+	0.282238i	$0.0910769\pi$
$673$	14.6438i	0.564477i	−0.959344	+	0.282238i	$0.908923\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 15.0525i	− 0.578515i	−0.957251	−	0.289258i	$-0.906592\pi$
$677$	− 15.0525i	− 0.578515i	0.957251	−	0.289258i	$-0.0934084\pi$
$678$	0	0
$679$	−10.4570	−0.401304
$680$	0	0
$681$	−8.07773	−0.309539
$682$	0	0
$683$	21.7186i	0.831040i	0.909584	+	0.415520i	$0.136400\pi$
$683$	21.7186i	0.831040i	−0.909584	+	0.415520i	$0.863600\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 23.1443i	− 0.883010i
$688$	0	0
$689$	40.7313	1.55174
$690$	0	0
$691$	−20.4367	−0.777448	−0.388724	−	0.921354i	$-0.627084\pi$
$691$	−20.4367	−0.777448	−0.388724	+	0.921354i	$0.627084\pi$
$692$	0	0
$693$	− 2.48900i	− 0.0945492i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 8.27319i	− 0.313369i
$698$	0	0
$699$	7.20436	0.272494
$700$	0	0
$701$	19.9308	0.752774	0.376387	−	0.926462i	$-0.377166\pi$
$701$	19.9308	0.752774	0.376387	+	0.926462i	$0.377166\pi$
$702$	0	0
$703$	4.10971i	0.155001i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	46.3785i	1.74424i
$708$	0	0
$709$	37.0980	1.39324	0.696622	−	0.717438i	$-0.254685\pi$
$709$	37.0980	1.39324	0.696622	+	0.717438i	$0.254685\pi$
$710$	0	0
$711$	−0.0327846	−0.00122952
$712$	0	0
$713$	4.87378i	0.182525i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 13.0213i	− 0.486291i
$718$	0	0
$719$	−28.7975	−1.07397	−0.536983	−	0.843593i	$-0.680436\pi$
$719$	−28.7975	−1.07397	−0.536983	+	0.843593i	$0.680436\pi$
$720$	0	0
$721$	−42.4083	−1.57937
$722$	0	0
$723$	− 17.2351i	− 0.640981i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	37.7221i	1.39904i	0.714615	+	0.699518i	$0.246602\pi$
$727$	37.7221i	1.39904i	−0.714615	+	0.699518i	$0.753398\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−0.236068	−0.00873129
$732$	0	0
$733$	− 24.5732i	− 0.907633i	−0.891095	−	0.453816i	$-0.850062\pi$
$733$	− 24.5732i	− 0.907633i	0.891095	−	0.453816i	$-0.149938\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	6.23245i	0.229575i
$738$	0	0
$739$	1.83465	0.0674885	0.0337443	−	0.999431i	$-0.489257\pi$
$739$	1.83465	0.0674885	0.0337443	+	0.999431i	$0.489257\pi$
$740$	0	0
$741$	3.20596	0.117774
$742$	0	0
$743$	0.728992i	0.0267441i	0.999911	+	0.0133721i	$0.00425659\pi$
$743$	0.728992i	0.0267441i	−0.999911	+	0.0133721i	$0.995743\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 10.5492i	− 0.385975i
$748$	0	0
$749$	−30.7123	−1.12220
$750$	0	0
$751$	−31.3690	−1.14467	−0.572336	−	0.820019i	$-0.693963\pi$
$751$	−31.3690	−1.14467	−0.572336	+	0.820019i	$0.693963\pi$
$752$	0	0
$753$	11.5420i	0.420615i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	37.6006i	1.36662i	0.730129	+	0.683309i	$0.239460\pi$
$757$	37.6006i	1.36662i	−0.730129	+	0.683309i	$0.760540\pi$
$758$	0	0
$759$	−2.03104	−0.0737221
$760$	0	0
$761$	−26.6105	−0.964629	−0.482315	−	0.875998i	$-0.660204\pi$
$761$	−26.6105	−0.964629	−0.482315	+	0.875998i	$0.660204\pi$
$762$	0	0
$763$	− 31.0300i	− 1.12336i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 57.1027i	− 2.06186i
$768$	0	0
$769$	−31.1520	−1.12337	−0.561686	−	0.827351i	$-0.689847\pi$
$769$	−31.1520	−1.12337	−0.561686	+	0.827351i	$0.689847\pi$
$770$	0	0
$771$	28.8390	1.03861
$772$	0	0
$773$	− 55.2028i	− 1.98551i	−0.120176	−	0.992753i	$-0.538346\pi$
$773$	− 55.2028i	− 1.98551i	0.120176	−	0.992753i	$-0.461654\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	16.4356i	0.589624i
$778$	0	0
$779$	4.29462	0.153871
$780$	0	0
$781$	−2.54531	−0.0910782
$782$	0	0
$783$	3.05305i	0.109107i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1.46243i	0.0521302i	0.999660	+	0.0260651i	$0.00829771\pi$
$787$	1.46243i	0.0521302i	−0.999660	+	0.0260651i	$0.991702\pi$
$788$	0	0
$789$	12.6562	0.450574
$790$	0	0
$791$	26.5399	0.943649
$792$	0	0
$793$	57.2279i	2.03222i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	36.6287i	1.29745i	0.761021	+	0.648727i	$0.224698\pi$
$797$	36.6287i	1.29745i	−0.761021	+	0.648727i	$0.775302\pi$
$798$	0	0
$799$	4.29613	0.151986
$800$	0	0
$801$	1.60385	0.0566694
$802$	0	0
$803$	4.26451i	0.150491i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	15.2884i	0.538179i
$808$	0	0
$809$	−9.23673	−0.324746	−0.162373	−	0.986729i	$-0.551915\pi$
$809$	−9.23673	−0.324746	−0.162373	+	0.986729i	$0.551915\pi$
$810$	0	0
$811$	−12.0011	−0.421415	−0.210707	−	0.977549i	$-0.567577\pi$
$811$	−12.0011	−0.421415	−0.210707	+	0.977549i	$0.567577\pi$
$812$	0	0
$813$	− 16.4404i	− 0.576591i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 0.122543i	− 0.00428724i
$818$	0	0
$819$	12.8213	0.448013
$820$	0	0
$821$	−33.1991	−1.15866	−0.579328	−	0.815095i	$-0.696685\pi$
$821$	−33.1991	−1.15866	−0.579328	+	0.815095i	$0.696685\pi$
$822$	0	0
$823$	19.6745i	0.685809i	0.939370	+	0.342905i	$0.111411\pi$
$823$	19.6745i	0.685809i	−0.939370	+	0.342905i	$0.888589\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 17.7712i	− 0.617966i	−0.951068	−	0.308983i	$-0.900011\pi$
$827$	− 17.7712i	− 0.617966i	0.951068	−	0.308983i	$-0.0999887\pi$
$828$	0	0
$829$	5.87987	0.204216	0.102108	−	0.994773i	$-0.467441\pi$
$829$	5.87987	0.204216	0.102108	+	0.994773i	$0.467441\pi$
$830$	0	0
$831$	25.8329	0.896133
$832$	0	0
$833$	6.93009i	0.240113i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 1.77811i	− 0.0614605i
$838$	0	0
$839$	9.56664	0.330277	0.165139	−	0.986270i	$-0.447193\pi$
$839$	9.56664	0.330277	0.165139	+	0.986270i	$0.447193\pi$
$840$	0	0
$841$	−19.6789	−0.678583
$842$	0	0
$843$	12.7170i	0.437996i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 35.1049i	− 1.20622i
$848$	0	0
$849$	−18.6209	−0.639066
$850$	0	0
$851$	13.4116	0.459743
$852$	0	0
$853$	− 44.2548i	− 1.51526i	−0.652687	−	0.757628i	$-0.726358\pi$
$853$	− 44.2548i	− 1.51526i	0.652687	−	0.757628i	$-0.273642\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 19.5920i	− 0.669250i	−0.942351	−	0.334625i	$-0.891390\pi$
$857$	− 19.5920i	− 0.669250i	0.942351	−	0.334625i	$-0.108610\pi$
$858$	0	0
$859$	−35.7011	−1.21811	−0.609053	−	0.793130i	$-0.708450\pi$
$859$	−35.7011	−1.21811	−0.609053	+	0.793130i	$0.708450\pi$
$860$	0	0
$861$	17.1751	0.585325
$862$	0	0
$863$	0.815321i	0.0277539i	0.999904	+	0.0138769i	$0.00441731\pi$
$863$	0.815321i	0.0277539i	−0.999904	+	0.0138769i	$0.995583\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 14.3820i	− 0.488437i
$868$	0	0
$869$	0.0242930	0.000824085 0
$870$	0	0
$871$	−32.1046	−1.08782
$872$	0	0
$873$	− 3.11311i	− 0.105363i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 25.4021i	− 0.857769i	−0.903359	−	0.428885i	$-0.858907\pi$
$877$	− 25.4021i	− 0.857769i	0.903359	−	0.428885i	$-0.141093\pi$
$878$	0	0
$879$	−25.9541	−0.875411
$880$	0	0
$881$	−24.6263	−0.829680	−0.414840	−	0.909894i	$-0.636162\pi$
$881$	−24.6263	−0.829680	−0.414840	+	0.909894i	$0.636162\pi$
$882$	0	0
$883$	38.7763i	1.30493i	0.757820	+	0.652464i	$0.226264\pi$
$883$	38.7763i	1.30493i	−0.757820	+	0.652464i	$0.773736\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 22.5001i	− 0.755479i	−0.925912	−	0.377739i	$-0.876702\pi$
$887$	− 22.5001i	− 0.755479i	0.925912	−	0.377739i	$-0.123298\pi$
$888$	0	0
$889$	−24.5614	−0.823762
$890$	0	0
$891$	0.740989	0.0248241
$892$	0	0
$893$	2.23013i	0.0746283i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 10.4623i	− 0.349326i
$898$	0	0
$899$	−5.42866	−0.181056
$900$	0	0
$901$	−17.2662	−0.575220
$902$	0	0
$903$	− 0.490075i	− 0.0163087i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 42.6695i	− 1.41682i	−0.705803	−	0.708408i	$-0.749414\pi$
$907$	− 42.6695i	− 1.41682i	0.705803	−	0.708408i	$-0.250586\pi$
$908$	0	0
$909$	−13.8071	−0.457954
$910$	0	0
$911$	−3.31314	−0.109769	−0.0548845	−	0.998493i	$-0.517479\pi$
$911$	−3.31314	−0.109769	−0.0548845	+	0.998493i	$0.517479\pi$
$912$	0	0
$913$	7.81684i	0.258700i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 51.1217i	− 1.68819i
$918$	0	0
$919$	−10.3050	−0.339932	−0.169966	−	0.985450i	$-0.554366\pi$
$919$	−10.3050	−0.339932	−0.169966	+	0.985450i	$0.554366\pi$
$920$	0	0
$921$	−5.68526	−0.187336
$922$	0	0
$923$	− 13.1114i	− 0.431566i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 12.6252i	− 0.414666i
$928$	0	0
$929$	1.66298	0.0545607	0.0272804	−	0.999628i	$-0.491315\pi$
$929$	1.66298	0.0545607	0.0272804	+	0.999628i	$0.491315\pi$
$930$	0	0
$931$	−3.59741	−0.117900
$932$	0	0
$933$	− 25.6472i	− 0.839652i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1.19367i	0.0389954i	0.999810	+	0.0194977i	$0.00620671\pi$
$937$	1.19367i	0.0389954i	−0.999810	+	0.0194977i	$0.993793\pi$
$938$	0	0
$939$	4.66934	0.152378
$940$	0	0
$941$	28.8387	0.940116	0.470058	−	0.882636i	$-0.344233\pi$
$941$	28.8387	0.940116	0.470058	+	0.882636i	$0.344233\pi$
$942$	0	0
$943$	− 14.0150i	− 0.456391i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 28.4650i	− 0.924989i	−0.886622	−	0.462495i	$-0.846954\pi$
$947$	− 28.4650i	− 0.924989i	0.886622	−	0.462495i	$-0.153046\pi$
$948$	0	0
$949$	−21.9674	−0.713091
$950$	0	0
$951$	−9.49240	−0.307812
$952$	0	0
$953$	38.7750i	1.25605i	0.778195	+	0.628023i	$0.216136\pi$
$953$	38.7750i	1.25605i	−0.778195	+	0.628023i	$0.783864\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 2.26227i	− 0.0731289i
$958$	0	0
$959$	−35.0991	−1.13341
$960$	0	0
$961$	−27.8383	−0.898010
$962$	0	0
$963$	− 9.14322i	− 0.294636i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	8.30570i	0.267093i	0.991043	+	0.133547i	$0.0426366\pi$
$967$	8.30570i	0.267093i	−0.991043	+	0.133547i	$0.957363\pi$
$968$	0	0
$969$	−1.35902	−0.0436581
$970$	0	0
$971$	28.9609	0.929398	0.464699	−	0.885469i	$-0.346162\pi$
$971$	28.9609	0.929398	0.464699	+	0.885469i	$0.346162\pi$
$972$	0	0
$973$	− 54.2018i	− 1.73763i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	49.0411i	1.56896i	0.620152	+	0.784481i	$0.287071\pi$
$977$	49.0411i	1.56896i	−0.620152	+	0.784481i	$0.712929\pi$
$978$	0	0
$979$	−1.18844	−0.0379826
$980$	0	0
$981$	9.23781	0.294941
$982$	0	0
$983$	22.1108i	0.705226i	0.935769	+	0.352613i	$0.114707\pi$
$983$	22.1108i	0.705226i	−0.935769	+	0.352613i	$0.885293\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	8.91873i	0.283886i
$988$	0	0
$989$	−0.399905	−0.0127162
$990$	0	0
$991$	−6.69228	−0.212587	−0.106294	−	0.994335i	$-0.533898\pi$
$991$	−6.69228	−0.212587	−0.106294	+	0.994335i	$0.533898\pi$
$992$	0	0
$993$	22.2875i	0.707273i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	49.5535i	1.56938i	0.619891	+	0.784688i	$0.287177\pi$
$997$	49.5535i	1.56938i	−0.619891	+	0.784688i	$0.712823\pi$
$998$	0	0
$999$	−4.89297	−0.154807

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6000.2.f.p.1249.3		8
4.3	odd	2		3000.2.f.h.1249.6		8
5.2	odd	4		6000.2.a.bc.1.2		4
5.3	odd	4		6000.2.a.bl.1.3		4
5.4	even	2	inner	6000.2.f.p.1249.6		8
20.3	even	4		3000.2.a.j.1.2	✓	4
20.7	even	4		3000.2.a.o.1.3	yes	4
20.19	odd	2		3000.2.f.h.1249.3		8
60.23	odd	4		9000.2.a.t.1.2		4
60.47	odd	4		9000.2.a.y.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3000.2.a.j.1.2	✓	4	20.3	even	4
3000.2.a.o.1.3	yes	4	20.7	even	4
3000.2.f.h.1249.3		8	20.19	odd	2
3000.2.f.h.1249.6		8	4.3	odd	2
6000.2.a.bc.1.2		4	5.2	odd	4
6000.2.a.bl.1.3		4	5.3	odd	4
6000.2.f.p.1249.3		8	1.1	even	1	trivial
6000.2.f.p.1249.6		8	5.4	even	2	inner
9000.2.a.t.1.2		4	60.23	odd	4
9000.2.a.y.1.3		4	60.47	odd	4

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Hilbert	Bianchi

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

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	6000.2.f.p.1249.3

Level	$6000$
Weight	$2$
Character	6000.1249
Analytic conductor	$47.910$
Analytic rank	$0$
Dimension	$8$
Inner twists	$2$