LMFDB - Embedded newform 8048.2.a.x.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8048,2,Mod(1,8048)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8048, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8048.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8048 = 2^{4} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8048.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:	$64.2636035467$
Analytic rank:	$1$
Dimension:	$33$
Twist minimal:	no (minimal twist has level 4024)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Character	$\chi$	$=$	8048.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.649284 q^{3} -2.92377 q^{5} +1.90965 q^{7} -2.57843 q^{9} +O(q^{10})$	$q-0.649284 q^{3} -2.92377 q^{5} +1.90965 q^{7} -2.57843 q^{9} -3.79600 q^{11} +1.26781 q^{13} +1.89835 q^{15} -3.64607 q^{17} -4.46389 q^{19} -1.23991 q^{21} +9.12945 q^{23} +3.54841 q^{25} +3.62199 q^{27} +2.77231 q^{29} +4.74950 q^{31} +2.46468 q^{33} -5.58338 q^{35} +1.24878 q^{37} -0.823169 q^{39} +8.26383 q^{41} +6.09603 q^{43} +7.53873 q^{45} +1.58222 q^{47} -3.35322 q^{49} +2.36733 q^{51} +9.70315 q^{53} +11.0986 q^{55} +2.89833 q^{57} -7.05146 q^{59} +1.40971 q^{61} -4.92391 q^{63} -3.70678 q^{65} -6.44588 q^{67} -5.92761 q^{69} +10.2738 q^{71} -4.81030 q^{73} -2.30393 q^{75} -7.24905 q^{77} -16.6520 q^{79} +5.38359 q^{81} -4.24583 q^{83} +10.6603 q^{85} -1.80001 q^{87} +14.4821 q^{89} +2.42108 q^{91} -3.08378 q^{93} +13.0514 q^{95} -7.34573 q^{97} +9.78772 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 33 q - 10 q^{3} - 12 q^{7} + 47 q^{9}+O(q^{10}) $ 33 * q - 10 * q^3 - 12 * q^7 + 47 * q^9	$ 33 q - 10 q^{3} - 12 q^{7} + 47 q^{9} - 22 q^{11} - 17 q^{13} - 22 q^{15} + 9 q^{17} - 16 q^{19} + 6 q^{21} - 36 q^{23} + 47 q^{25} - 34 q^{27} + 13 q^{29} - 21 q^{31} + 14 q^{33} - 33 q^{35} - 55 q^{37} - 37 q^{39} + 42 q^{41} - 23 q^{43} + 5 q^{45} - 20 q^{47} + 55 q^{49} - 53 q^{51} - 32 q^{53} - 35 q^{55} + 21 q^{57} - 20 q^{59} - 15 q^{61} - 48 q^{63} + 34 q^{65} - 66 q^{67} - 4 q^{69} - 61 q^{71} + 19 q^{73} - 59 q^{75} + 2 q^{77} - 62 q^{79} + 77 q^{81} - 36 q^{83} - 14 q^{85} - 43 q^{87} + 34 q^{89} - 41 q^{91} - 11 q^{93} - 61 q^{95} - 8 q^{97} - 98 q^{99}+O(q^{100}) $ 33 * q - 10 * q^3 - 12 * q^7 + 47 * q^9 - 22 * q^11 - 17 * q^13 - 22 * q^15 + 9 * q^17 - 16 * q^19 + 6 * q^21 - 36 * q^23 + 47 * q^25 - 34 * q^27 + 13 * q^29 - 21 * q^31 + 14 * q^33 - 33 * q^35 - 55 * q^37 - 37 * q^39 + 42 * q^41 - 23 * q^43 + 5 * q^45 - 20 * q^47 + 55 * q^49 - 53 * q^51 - 32 * q^53 - 35 * q^55 + 21 * q^57 - 20 * q^59 - 15 * q^61 - 48 * q^63 + 34 * q^65 - 66 * q^67 - 4 * q^69 - 61 * q^71 + 19 * q^73 - 59 * q^75 + 2 * q^77 - 62 * q^79 + 77 * q^81 - 36 * q^83 - 14 * q^85 - 43 * q^87 + 34 * q^89 - 41 * q^91 - 11 * q^93 - 61 * q^95 - 8 * q^97 - 98 * 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
$3$	−0.649284	−0.374864	−0.187432	−	0.982278i	$-0.560016\pi$
$3$	−0.649284	−0.374864	−0.187432	+	0.982278i	$0.560016\pi$
$4$	0	0
$5$	−2.92377	−1.30755	−0.653774	−	0.756690i	$-0.726815\pi$
$5$	−2.92377	−1.30755	−0.653774	+	0.756690i	$0.726815\pi$
$6$	0	0
$7$	1.90965	0.721782	0.360891	−	0.932608i	$-0.382473\pi$
$7$	1.90965	0.721782	0.360891	+	0.932608i	$0.382473\pi$
$8$	0	0
$9$	−2.57843	−0.859477
$10$	0	0
$11$	−3.79600	−1.14454	−0.572269	−	0.820066i	$-0.693937\pi$
$11$	−3.79600	−1.14454	−0.572269	+	0.820066i	$0.693937\pi$
$12$	0	0
$13$	1.26781	0.351627	0.175814	−	0.984423i	$-0.443744\pi$
$13$	1.26781	0.351627	0.175814	+	0.984423i	$0.443744\pi$
$14$	0	0
$15$	1.89835	0.490153
$16$	0	0
$17$	−3.64607	−0.884301	−0.442151	−	0.896941i	$-0.645784\pi$
$17$	−3.64607	−0.884301	−0.442151	+	0.896941i	$0.645784\pi$
$18$	0	0
$19$	−4.46389	−1.02409	−0.512044	−	0.858959i	$-0.671111\pi$
$19$	−4.46389	−1.02409	−0.512044	+	0.858959i	$0.671111\pi$
$20$	0	0
$21$	−1.23991	−0.270570
$22$	0	0
$23$	9.12945	1.90362	0.951811	−	0.306684i	$-0.0992195\pi$
$23$	9.12945	1.90362	0.951811	+	0.306684i	$0.0992195\pi$
$24$	0	0
$25$	3.54841	0.709682
$26$	0	0
$27$	3.62199	0.697052
$28$	0	0
$29$	2.77231	0.514805	0.257402	−	0.966304i	$-0.417133\pi$
$29$	2.77231	0.514805	0.257402	+	0.966304i	$0.417133\pi$
$30$	0	0
$31$	4.74950	0.853036	0.426518	−	0.904479i	$-0.359740\pi$
$31$	4.74950	0.853036	0.426518	+	0.904479i	$0.359740\pi$
$32$	0	0
$33$	2.46468	0.429046
$34$	0	0
$35$	−5.58338	−0.943764
$36$	0	0
$37$	1.24878	0.205298	0.102649	−	0.994718i	$-0.467268\pi$
$37$	1.24878	0.205298	0.102649	+	0.994718i	$0.467268\pi$
$38$	0	0
$39$	−0.823169	−0.131812
$40$	0	0
$41$	8.26383	1.29059	0.645296	−	0.763932i	$-0.276734\pi$
$41$	8.26383	1.29059	0.645296	+	0.763932i	$0.276734\pi$
$42$	0	0
$43$	6.09603	0.929637	0.464818	−	0.885406i	$-0.346120\pi$
$43$	6.09603	0.929637	0.464818	+	0.885406i	$0.346120\pi$
$44$	0	0
$45$	7.53873	1.12381
$46$	0	0
$47$	1.58222	0.230790	0.115395	−	0.993320i	$-0.463187\pi$
$47$	1.58222	0.230790	0.115395	+	0.993320i	$0.463187\pi$
$48$	0	0
$49$	−3.35322	−0.479031
$50$	0	0
$51$	2.36733	0.331493
$52$	0	0
$53$	9.70315	1.33283	0.666415	−	0.745581i	$-0.267828\pi$
$53$	9.70315	1.33283	0.666415	+	0.745581i	$0.267828\pi$
$54$	0	0
$55$	11.0986	1.49654
$56$	0	0
$57$	2.89833	0.383894
$58$	0	0
$59$	−7.05146	−0.918022	−0.459011	−	0.888431i	$-0.651796\pi$
$59$	−7.05146	−0.918022	−0.459011	+	0.888431i	$0.651796\pi$
$60$	0	0
$61$	1.40971	0.180495	0.0902474	−	0.995919i	$-0.471234\pi$
$61$	1.40971	0.180495	0.0902474	+	0.995919i	$0.471234\pi$
$62$	0	0
$63$	−4.92391	−0.620355
$64$	0	0
$65$	−3.70678	−0.459769
$66$	0	0
$67$	−6.44588	−0.787489	−0.393745	−	0.919220i	$-0.628821\pi$
$67$	−6.44588	−0.787489	−0.393745	+	0.919220i	$0.628821\pi$
$68$	0	0
$69$	−5.92761	−0.713600
$70$	0	0
$71$	10.2738	1.21928	0.609639	−	0.792679i	$-0.291315\pi$
$71$	10.2738	1.21928	0.609639	+	0.792679i	$0.291315\pi$
$72$	0	0
$73$	−4.81030	−0.563003	−0.281502	−	0.959561i	$-0.590832\pi$
$73$	−4.81030	−0.563003	−0.281502	+	0.959561i	$0.590832\pi$
$74$	0	0
$75$	−2.30393	−0.266034
$76$	0	0
$77$	−7.24905	−0.826106
$78$	0	0
$79$	−16.6520	−1.87350	−0.936748	−	0.350005i	$-0.886180\pi$
$79$	−16.6520	−1.87350	−0.936748	+	0.350005i	$0.886180\pi$
$80$	0	0
$81$	5.38359	0.598177
$82$	0	0
$83$	−4.24583	−0.466040	−0.233020	−	0.972472i	$-0.574861\pi$
$83$	−4.24583	−0.466040	−0.233020	+	0.972472i	$0.574861\pi$
$84$	0	0
$85$	10.6603	1.15627
$86$	0	0
$87$	−1.80001	−0.192982
$88$	0	0
$89$	14.4821	1.53510	0.767551	−	0.640988i	$-0.221475\pi$
$89$	14.4821	1.53510	0.767551	+	0.640988i	$0.221475\pi$
$90$	0	0
$91$	2.42108	0.253798
$92$	0	0
$93$	−3.08378	−0.319773
$94$	0	0
$95$	13.0514	1.33904
$96$	0	0
$97$	−7.34573	−0.745846	−0.372923	−	0.927862i	$-0.621644\pi$
$97$	−7.34573	−0.745846	−0.372923	+	0.927862i	$0.621644\pi$
$98$	0	0
$99$	9.78772	0.983703
$100$	0	0
$101$	4.70072	0.467739	0.233869	−	0.972268i	$-0.424861\pi$
$101$	4.70072	0.467739	0.233869	+	0.972268i	$0.424861\pi$
$102$	0	0
$103$	2.26044	0.222728	0.111364	−	0.993780i	$-0.464478\pi$
$103$	2.26044	0.222728	0.111364	+	0.993780i	$0.464478\pi$
$104$	0	0
$105$	3.62520	0.353784
$106$	0	0
$107$	−13.4572	−1.30096	−0.650478	−	0.759526i	$-0.725431\pi$
$107$	−13.4572	−1.30096	−0.650478	+	0.759526i	$0.725431\pi$
$108$	0	0
$109$	−15.4456	−1.47942	−0.739709	−	0.672927i	$-0.765037\pi$
$109$	−15.4456	−1.47942	−0.739709	+	0.672927i	$0.765037\pi$
$110$	0	0
$111$	−0.810814	−0.0769591
$112$	0	0
$113$	−6.64048	−0.624683	−0.312342	−	0.949970i	$-0.601113\pi$
$113$	−6.64048	−0.624683	−0.312342	+	0.949970i	$0.601113\pi$
$114$	0	0
$115$	−26.6924	−2.48908
$116$	0	0
$117$	−3.26896	−0.302215
$118$	0	0
$119$	−6.96273	−0.638273
$120$	0	0
$121$	3.40963	0.309966
$122$	0	0
$123$	−5.36557	−0.483797
$124$	0	0
$125$	4.24411	0.379605
$126$	0	0
$127$	3.11813	0.276690	0.138345	−	0.990384i	$-0.455822\pi$
$127$	3.11813	0.276690	0.138345	+	0.990384i	$0.455822\pi$
$128$	0	0
$129$	−3.95806	−0.348488
$130$	0	0
$131$	−17.6238	−1.53980	−0.769901	−	0.638163i	$-0.779694\pi$
$131$	−17.6238	−1.53980	−0.769901	+	0.638163i	$0.779694\pi$
$132$	0	0
$133$	−8.52449	−0.739167
$134$	0	0
$135$	−10.5898	−0.911428
$136$	0	0
$137$	19.5644	1.67150	0.835750	−	0.549110i	$-0.185033\pi$
$137$	19.5644	1.67150	0.835750	+	0.549110i	$0.185033\pi$
$138$	0	0
$139$	−7.60405	−0.644967	−0.322483	−	0.946575i	$-0.604518\pi$
$139$	−7.60405	−0.644967	−0.322483	+	0.946575i	$0.604518\pi$
$140$	0	0
$141$	−1.02731	−0.0865151
$142$	0	0
$143$	−4.81261	−0.402450
$144$	0	0
$145$	−8.10558	−0.673132
$146$	0	0
$147$	2.17719	0.179572
$148$	0	0
$149$	11.5856	0.949127	0.474564	−	0.880221i	$-0.342606\pi$
$149$	11.5856	0.949127	0.474564	+	0.880221i	$0.342606\pi$
$150$	0	0
$151$	−5.87029	−0.477718	−0.238859	−	0.971054i	$-0.576773\pi$
$151$	−5.87029	−0.477718	−0.238859	+	0.971054i	$0.576773\pi$
$152$	0	0
$153$	9.40113	0.760037
$154$	0	0
$155$	−13.8864	−1.11539
$156$	0	0
$157$	11.0186	0.879382	0.439691	−	0.898149i	$-0.355088\pi$
$157$	11.0186	0.879382	0.439691	+	0.898149i	$0.355088\pi$
$158$	0	0
$159$	−6.30010	−0.499630
$160$	0	0
$161$	17.4341	1.37400
$162$	0	0
$163$	−10.2705	−0.804449	−0.402225	−	0.915541i	$-0.631763\pi$
$163$	−10.2705	−0.804449	−0.402225	+	0.915541i	$0.631763\pi$
$164$	0	0
$165$	−7.20616	−0.560999
$166$	0	0
$167$	0.169266	0.0130982	0.00654909	−	0.999979i	$-0.497915\pi$
$167$	0.169266	0.0130982	0.00654909	+	0.999979i	$0.497915\pi$
$168$	0	0
$169$	−11.3927	−0.876358
$170$	0	0
$171$	11.5098	0.880179
$172$	0	0
$173$	2.30188	0.175009	0.0875043	−	0.996164i	$-0.472111\pi$
$173$	2.30188	0.175009	0.0875043	+	0.996164i	$0.472111\pi$
$174$	0	0
$175$	6.77624	0.512235
$176$	0	0
$177$	4.57840	0.344134
$178$	0	0
$179$	−23.0819	−1.72522	−0.862612	−	0.505866i	$-0.831173\pi$
$179$	−23.0819	−1.72522	−0.862612	+	0.505866i	$0.831173\pi$
$180$	0	0
$181$	19.4781	1.44780	0.723898	−	0.689907i	$-0.242349\pi$
$181$	19.4781	1.44780	0.723898	+	0.689907i	$0.242349\pi$
$182$	0	0
$183$	−0.915302	−0.0676611
$184$	0	0
$185$	−3.65115	−0.268438
$186$	0	0
$187$	13.8405	1.01212
$188$	0	0
$189$	6.91674	0.503119
$190$	0	0
$191$	23.3292	1.68804	0.844021	−	0.536310i	$-0.180182\pi$
$191$	23.3292	1.68804	0.844021	+	0.536310i	$0.180182\pi$
$192$	0	0
$193$	2.48731	0.179040	0.0895202	−	0.995985i	$-0.471467\pi$
$193$	2.48731	0.179040	0.0895202	+	0.995985i	$0.471467\pi$
$194$	0	0
$195$	2.40675	0.172351
$196$	0	0
$197$	−11.8559	−0.844699	−0.422350	−	0.906433i	$-0.638795\pi$
$197$	−11.8559	−0.844699	−0.422350	+	0.906433i	$0.638795\pi$
$198$	0	0
$199$	−25.3537	−1.79728	−0.898638	−	0.438691i	$-0.855442\pi$
$199$	−25.3537	−1.79728	−0.898638	+	0.438691i	$0.855442\pi$
$200$	0	0
$201$	4.18521	0.295202
$202$	0	0
$203$	5.29415	0.371576
$204$	0	0
$205$	−24.1615	−1.68751
$206$	0	0
$207$	−23.5397	−1.63612
$208$	0	0
$209$	16.9449	1.17211
$210$	0	0
$211$	−15.8161	−1.08882	−0.544412	−	0.838818i	$-0.683247\pi$
$211$	−15.8161	−1.08882	−0.544412	+	0.838818i	$0.683247\pi$
$212$	0	0
$213$	−6.67062	−0.457064
$214$	0	0
$215$	−17.8234	−1.21554
$216$	0	0
$217$	9.06991	0.615706
$218$	0	0
$219$	3.12325	0.211050
$220$	0	0
$221$	−4.62252	−0.310944
$222$	0	0
$223$	20.9037	1.39981	0.699907	−	0.714234i	$-0.253225\pi$
$223$	20.9037	1.39981	0.699907	+	0.714234i	$0.253225\pi$
$224$	0	0
$225$	−9.14933	−0.609955
$226$	0	0
$227$	24.4607	1.62352	0.811758	−	0.583995i	$-0.198511\pi$
$227$	24.4607	1.62352	0.811758	+	0.583995i	$0.198511\pi$
$228$	0	0
$229$	−5.07631	−0.335452	−0.167726	−	0.985834i	$-0.553642\pi$
$229$	−5.07631	−0.335452	−0.167726	+	0.985834i	$0.553642\pi$
$230$	0	0
$231$	4.70669	0.309678
$232$	0	0
$233$	11.0189	0.721873	0.360936	−	0.932590i	$-0.382457\pi$
$233$	11.0189	0.721873	0.360936	+	0.932590i	$0.382457\pi$
$234$	0	0
$235$	−4.62604	−0.301769
$236$	0	0
$237$	10.8119	0.702307
$238$	0	0
$239$	−2.50016	−0.161722	−0.0808611	−	0.996725i	$-0.525767\pi$
$239$	−2.50016	−0.161722	−0.0808611	+	0.996725i	$0.525767\pi$
$240$	0	0
$241$	−16.3262	−1.05166	−0.525831	−	0.850589i	$-0.676246\pi$
$241$	−16.3262	−1.05166	−0.525831	+	0.850589i	$0.676246\pi$
$242$	0	0
$243$	−14.3614	−0.921287
$244$	0	0
$245$	9.80403	0.626356
$246$	0	0
$247$	−5.65937	−0.360097
$248$	0	0
$249$	2.75675	0.174702
$250$	0	0
$251$	−19.9630	−1.26005	−0.630026	−	0.776574i	$-0.716956\pi$
$251$	−19.9630	−1.26005	−0.630026	+	0.776574i	$0.716956\pi$
$252$	0	0
$253$	−34.6554	−2.17877
$254$	0	0
$255$	−6.92153	−0.433443
$256$	0	0
$257$	−8.64384	−0.539188	−0.269594	−	0.962974i	$-0.586889\pi$
$257$	−8.64384	−0.539188	−0.269594	+	0.962974i	$0.586889\pi$
$258$	0	0
$259$	2.38474	0.148181
$260$	0	0
$261$	−7.14820	−0.442463
$262$	0	0
$263$	0.118392	0.00730038	0.00365019	−	0.999993i	$-0.498838\pi$
$263$	0.118392	0.00730038	0.00365019	+	0.999993i	$0.498838\pi$
$264$	0	0
$265$	−28.3697	−1.74274
$266$	0	0
$267$	−9.40301	−0.575455
$268$	0	0
$269$	−4.67830	−0.285241	−0.142621	−	0.989777i	$-0.545553\pi$
$269$	−4.67830	−0.285241	−0.142621	+	0.989777i	$0.545553\pi$
$270$	0	0
$271$	−20.1728	−1.22541	−0.612705	−	0.790312i	$-0.709919\pi$
$271$	−20.1728	−1.22541	−0.612705	+	0.790312i	$0.709919\pi$
$272$	0	0
$273$	−1.57197	−0.0951398
$274$	0	0
$275$	−13.4698	−0.812258
$276$	0	0
$277$	−3.39604	−0.204049	−0.102024	−	0.994782i	$-0.532532\pi$
$277$	−3.39604	−0.204049	−0.102024	+	0.994782i	$0.532532\pi$
$278$	0	0
$279$	−12.2463	−0.733164
$280$	0	0
$281$	2.89693	0.172816	0.0864081	−	0.996260i	$-0.472461\pi$
$281$	2.89693	0.172816	0.0864081	+	0.996260i	$0.472461\pi$
$282$	0	0
$283$	21.8412	1.29832	0.649162	−	0.760650i	$-0.275120\pi$
$283$	21.8412	1.29832	0.649162	+	0.760650i	$0.275120\pi$
$284$	0	0
$285$	−8.47405	−0.501960
$286$	0	0
$287$	15.7811	0.931526
$288$	0	0
$289$	−3.70619	−0.218011
$290$	0	0
$291$	4.76947	0.279591
$292$	0	0
$293$	3.00629	0.175629	0.0878147	−	0.996137i	$-0.472012\pi$
$293$	3.00629	0.175629	0.0878147	+	0.996137i	$0.472012\pi$
$294$	0	0
$295$	20.6168	1.20036
$296$	0	0
$297$	−13.7491	−0.797802
$298$	0	0
$299$	11.5744	0.669365
$300$	0	0
$301$	11.6413	0.670995
$302$	0	0
$303$	−3.05210	−0.175339
$304$	0	0
$305$	−4.12166	−0.236006
$306$	0	0
$307$	30.7196	1.75326	0.876631	−	0.481163i	$-0.159786\pi$
$307$	30.7196	1.75326	0.876631	+	0.481163i	$0.159786\pi$
$308$	0	0
$309$	−1.46767	−0.0834928
$310$	0	0
$311$	8.78128	0.497941	0.248970	−	0.968511i	$-0.419908\pi$
$311$	8.78128	0.497941	0.248970	+	0.968511i	$0.419908\pi$
$312$	0	0
$313$	13.1310	0.742209	0.371104	−	0.928591i	$-0.378979\pi$
$313$	13.1310	0.742209	0.371104	+	0.928591i	$0.378979\pi$
$314$	0	0
$315$	14.3964	0.811143
$316$	0	0
$317$	18.2280	1.02378	0.511892	−	0.859050i	$-0.328945\pi$
$317$	18.2280	1.02378	0.511892	+	0.859050i	$0.328945\pi$
$318$	0	0
$319$	−10.5237	−0.589213
$320$	0	0
$321$	8.73754	0.487682
$322$	0	0
$323$	16.2757	0.905602
$324$	0	0
$325$	4.49871	0.249543
$326$	0	0
$327$	10.0286	0.554581
$328$	0	0
$329$	3.02149	0.166580
$330$	0	0
$331$	−18.4672	−1.01505	−0.507526	−	0.861637i	$-0.669440\pi$
$331$	−18.4672	−1.01505	−0.507526	+	0.861637i	$0.669440\pi$
$332$	0	0
$333$	−3.21990	−0.176449
$334$	0	0
$335$	18.8462	1.02968
$336$	0	0
$337$	29.6384	1.61451	0.807254	−	0.590204i	$-0.200953\pi$
$337$	29.6384	1.61451	0.807254	+	0.590204i	$0.200953\pi$
$338$	0	0
$339$	4.31156	0.234172
$340$	0	0
$341$	−18.0291	−0.976331
$342$	0	0
$343$	−19.7711	−1.06754
$344$	0	0
$345$	17.3309	0.933067
$346$	0	0
$347$	−22.3202	−1.19821	−0.599105	−	0.800670i	$-0.704477\pi$
$347$	−22.3202	−1.19821	−0.599105	+	0.800670i	$0.704477\pi$
$348$	0	0
$349$	−3.58420	−0.191858	−0.0959289	−	0.995388i	$-0.530582\pi$
$349$	−3.58420	−0.191858	−0.0959289	+	0.995388i	$0.530582\pi$
$350$	0	0
$351$	4.59199	0.245102
$352$	0	0
$353$	−20.3245	−1.08176	−0.540881	−	0.841099i	$-0.681909\pi$
$353$	−20.3245	−1.08176	−0.540881	+	0.841099i	$0.681909\pi$
$354$	0	0
$355$	−30.0382	−1.59426
$356$	0	0
$357$	4.52079	0.239266
$358$	0	0
$359$	−20.8194	−1.09880	−0.549402	−	0.835558i	$-0.685144\pi$
$359$	−20.8194	−1.09880	−0.549402	+	0.835558i	$0.685144\pi$
$360$	0	0
$361$	0.926334	0.0487544
$362$	0	0
$363$	−2.21382	−0.116195
$364$	0	0
$365$	14.0642	0.736154
$366$	0	0
$367$	16.3899	0.855548	0.427774	−	0.903886i	$-0.359298\pi$
$367$	16.3899	0.855548	0.427774	+	0.903886i	$0.359298\pi$
$368$	0	0
$369$	−21.3077	−1.10923
$370$	0	0
$371$	18.5297	0.962012
$372$	0	0
$373$	−32.2454	−1.66961	−0.834803	−	0.550549i	$-0.814418\pi$
$373$	−32.2454	−1.66961	−0.834803	+	0.550549i	$0.814418\pi$
$374$	0	0
$375$	−2.75563	−0.142300
$376$	0	0
$377$	3.51476	0.181019
$378$	0	0
$379$	−13.6498	−0.701143	−0.350571	−	0.936536i	$-0.614013\pi$
$379$	−13.6498	−0.701143	−0.350571	+	0.936536i	$0.614013\pi$
$380$	0	0
$381$	−2.02455	−0.103721
$382$	0	0
$383$	28.3953	1.45093	0.725465	−	0.688259i	$-0.241625\pi$
$383$	28.3953	1.45093	0.725465	+	0.688259i	$0.241625\pi$
$384$	0	0
$385$	21.1945	1.08017
$386$	0	0
$387$	−15.7182	−0.799001
$388$	0	0
$389$	−25.9741	−1.31694	−0.658469	−	0.752607i	$-0.728796\pi$
$389$	−25.9741	−1.31694	−0.658469	+	0.752607i	$0.728796\pi$
$390$	0	0
$391$	−33.2866	−1.68338
$392$	0	0
$393$	11.4429	0.577217
$394$	0	0
$395$	48.6865	2.44969
$396$	0	0
$397$	21.7097	1.08958	0.544790	−	0.838572i	$-0.316609\pi$
$397$	21.7097	1.08958	0.544790	+	0.838572i	$0.316609\pi$
$398$	0	0
$399$	5.53482	0.277087
$400$	0	0
$401$	−30.2178	−1.50901	−0.754503	−	0.656296i	$-0.772122\pi$
$401$	−30.2178	−1.50901	−0.754503	+	0.656296i	$0.772122\pi$
$402$	0	0
$403$	6.02146	0.299951
$404$	0	0
$405$	−15.7404	−0.782145
$406$	0	0
$407$	−4.74038	−0.234972
$408$	0	0
$409$	−29.7393	−1.47051	−0.735257	−	0.677788i	$-0.762939\pi$
$409$	−29.7393	−1.47051	−0.735257	+	0.677788i	$0.762939\pi$
$410$	0	0
$411$	−12.7029	−0.626586
$412$	0	0
$413$	−13.4659	−0.662612
$414$	0	0
$415$	12.4138	0.609370
$416$	0	0
$417$	4.93719	0.241775
$418$	0	0
$419$	8.56794	0.418571	0.209286	−	0.977855i	$-0.432886\pi$
$419$	8.56794	0.418571	0.209286	+	0.977855i	$0.432886\pi$
$420$	0	0
$421$	−38.3028	−1.86677	−0.933383	−	0.358883i	$-0.883158\pi$
$421$	−38.3028	−1.86677	−0.933383	+	0.358883i	$0.883158\pi$
$422$	0	0
$423$	−4.07964	−0.198359
$424$	0	0
$425$	−12.9377	−0.627573
$426$	0	0
$427$	2.69206	0.130278
$428$	0	0
$429$	3.12475	0.150864
$430$	0	0
$431$	−17.6208	−0.848763	−0.424381	−	0.905484i	$-0.639508\pi$
$431$	−17.6208	−0.848763	−0.424381	+	0.905484i	$0.639508\pi$
$432$	0	0
$433$	23.8451	1.14592	0.572960	−	0.819583i	$-0.305795\pi$
$433$	23.8451	1.14592	0.572960	+	0.819583i	$0.305795\pi$
$434$	0	0
$435$	5.26282	0.252333
$436$	0	0
$437$	−40.7529	−1.94948
$438$	0	0
$439$	−22.3019	−1.06441	−0.532207	−	0.846614i	$-0.678637\pi$
$439$	−22.3019	−1.06441	−0.532207	+	0.846614i	$0.678637\pi$
$440$	0	0
$441$	8.64604	0.411716
$442$	0	0
$443$	14.8788	0.706911	0.353456	−	0.935451i	$-0.385007\pi$
$443$	14.8788	0.706911	0.353456	+	0.935451i	$0.385007\pi$
$444$	0	0
$445$	−42.3423	−2.00722
$446$	0	0
$447$	−7.52233	−0.355794
$448$	0	0
$449$	−12.4180	−0.586043	−0.293021	−	0.956106i	$-0.594661\pi$
$449$	−12.4180	−0.586043	−0.293021	+	0.956106i	$0.594661\pi$
$450$	0	0
$451$	−31.3695	−1.47713
$452$	0	0
$453$	3.81149	0.179079
$454$	0	0
$455$	−7.07867	−0.331853
$456$	0	0
$457$	18.9508	0.886480	0.443240	−	0.896403i	$-0.353829\pi$
$457$	18.9508	0.886480	0.443240	+	0.896403i	$0.353829\pi$
$458$	0	0
$459$	−13.2060	−0.616404
$460$	0	0
$461$	−2.48209	−0.115602	−0.0578012	−	0.998328i	$-0.518409\pi$
$461$	−2.48209	−0.115602	−0.0578012	+	0.998328i	$0.518409\pi$
$462$	0	0
$463$	9.11042	0.423397	0.211699	−	0.977335i	$-0.432101\pi$
$463$	9.11042	0.423397	0.211699	+	0.977335i	$0.432101\pi$
$464$	0	0
$465$	9.01624	0.418118
$466$	0	0
$467$	−37.2288	−1.72274	−0.861372	−	0.507974i	$-0.830395\pi$
$467$	−37.2288	−1.72274	−0.861372	+	0.507974i	$0.830395\pi$
$468$	0	0
$469$	−12.3094	−0.568395
$470$	0	0
$471$	−7.15422	−0.329649
$472$	0	0
$473$	−23.1406	−1.06400
$474$	0	0
$475$	−15.8397	−0.726776
$476$	0	0
$477$	−25.0189	−1.14554
$478$	0	0
$479$	−2.96384	−0.135421	−0.0677106	−	0.997705i	$-0.521569\pi$
$479$	−2.96384	−0.135421	−0.0677106	+	0.997705i	$0.521569\pi$
$480$	0	0
$481$	1.58322	0.0721885
$482$	0	0
$483$	−11.3197	−0.515064
$484$	0	0
$485$	21.4772	0.975229
$486$	0	0
$487$	−24.4702	−1.10885	−0.554424	−	0.832234i	$-0.687062\pi$
$487$	−24.4702	−1.10885	−0.554424	+	0.832234i	$0.687062\pi$
$488$	0	0
$489$	6.66848	0.301559
$490$	0	0
$491$	17.0876	0.771151	0.385576	−	0.922676i	$-0.374003\pi$
$491$	17.0876	0.771151	0.385576	+	0.922676i	$0.374003\pi$
$492$	0	0
$493$	−10.1080	−0.455242
$494$	0	0
$495$	−28.6170	−1.28624
$496$	0	0
$497$	19.6194	0.880052
$498$	0	0
$499$	−9.05669	−0.405433	−0.202716	−	0.979237i	$-0.564977\pi$
$499$	−9.05669	−0.405433	−0.202716	+	0.979237i	$0.564977\pi$
$500$	0	0
$501$	−0.109902	−0.00491004
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−13.7438	−0.611591
$506$	0	0
$507$	7.39707	0.328515
$508$	0	0
$509$	10.4770	0.464385	0.232193	−	0.972670i	$-0.425410\pi$
$509$	10.4770	0.464385	0.232193	+	0.972670i	$0.425410\pi$
$510$	0	0
$511$	−9.18601	−0.406365
$512$	0	0
$513$	−16.1682	−0.713841
$514$	0	0
$515$	−6.60901	−0.291228
$516$	0	0
$517$	−6.00610	−0.264148
$518$	0	0
$519$	−1.49457	−0.0656045
$520$	0	0
$521$	42.0358	1.84162	0.920812	−	0.390007i	$-0.127527\pi$
$521$	42.0358	1.84162	0.920812	+	0.390007i	$0.127527\pi$
$522$	0	0
$523$	21.3746	0.934647	0.467323	−	0.884086i	$-0.345218\pi$
$523$	21.3746	0.934647	0.467323	+	0.884086i	$0.345218\pi$
$524$	0	0
$525$	−4.39970	−0.192019
$526$	0	0
$527$	−17.3170	−0.754341
$528$	0	0
$529$	60.3469	2.62378
$530$	0	0
$531$	18.1817	0.789019
$532$	0	0
$533$	10.4770	0.453808
$534$	0	0
$535$	39.3457	1.70106
$536$	0	0
$537$	14.9867	0.646725
$538$	0	0
$539$	12.7288	0.548269
$540$	0	0
$541$	−9.32108	−0.400744	−0.200372	−	0.979720i	$-0.564215\pi$
$541$	−9.32108	−0.400744	−0.200372	+	0.979720i	$0.564215\pi$
$542$	0	0
$543$	−12.6468	−0.542727
$544$	0	0
$545$	45.1592	1.93441
$546$	0	0
$547$	24.7564	1.05851	0.529253	−	0.848464i	$-0.322472\pi$
$547$	24.7564	1.05851	0.529253	+	0.848464i	$0.322472\pi$
$548$	0	0
$549$	−3.63484	−0.155131
$550$	0	0
$551$	−12.3753	−0.527205
$552$	0	0
$553$	−31.7996	−1.35225
$554$	0	0
$555$	2.37063	0.100628
$556$	0	0
$557$	−7.60566	−0.322262	−0.161131	−	0.986933i	$-0.551514\pi$
$557$	−7.60566	−0.322262	−0.161131	+	0.986933i	$0.551514\pi$
$558$	0	0
$559$	7.72861	0.326885
$560$	0	0
$561$	−8.98640	−0.379406
$562$	0	0
$563$	−1.31552	−0.0554427	−0.0277213	−	0.999616i	$-0.508825\pi$
$563$	−1.31552	−0.0554427	−0.0277213	+	0.999616i	$0.508825\pi$
$564$	0	0
$565$	19.4152	0.816804
$566$	0	0
$567$	10.2808	0.431753
$568$	0	0
$569$	−34.3068	−1.43822	−0.719108	−	0.694899i	$-0.755449\pi$
$569$	−34.3068	−1.43822	−0.719108	+	0.694899i	$0.755449\pi$
$570$	0	0
$571$	−13.8306	−0.578794	−0.289397	−	0.957209i	$-0.593455\pi$
$571$	−13.8306	−0.578794	−0.289397	+	0.957209i	$0.593455\pi$
$572$	0	0
$573$	−15.1473	−0.632787
$574$	0	0
$575$	32.3950	1.35097
$576$	0	0
$577$	−36.8822	−1.53542	−0.767712	−	0.640795i	$-0.778605\pi$
$577$	−36.8822	−1.53542	−0.767712	+	0.640795i	$0.778605\pi$
$578$	0	0
$579$	−1.61497	−0.0671159
$580$	0	0
$581$	−8.10806	−0.336379
$582$	0	0
$583$	−36.8332	−1.52547
$584$	0	0
$585$	9.55767	0.395161
$586$	0	0
$587$	−15.7123	−0.648516	−0.324258	−	0.945969i	$-0.605115\pi$
$587$	−15.7123	−0.648516	−0.324258	+	0.945969i	$0.605115\pi$
$588$	0	0
$589$	−21.2013	−0.873583
$590$	0	0
$591$	7.69786	0.316648
$592$	0	0
$593$	23.6896	0.972814	0.486407	−	0.873732i	$-0.338307\pi$
$593$	23.6896	0.972814	0.486407	+	0.873732i	$0.338307\pi$
$594$	0	0
$595$	20.3574	0.834572
$596$	0	0
$597$	16.4618	0.673735
$598$	0	0
$599$	−12.9584	−0.529468	−0.264734	−	0.964322i	$-0.585284\pi$
$599$	−12.9584	−0.529468	−0.264734	+	0.964322i	$0.585284\pi$
$600$	0	0
$601$	−34.6005	−1.41138	−0.705692	−	0.708519i	$-0.749364\pi$
$601$	−34.6005	−1.41138	−0.705692	+	0.708519i	$0.749364\pi$
$602$	0	0
$603$	16.6202	0.676829
$604$	0	0
$605$	−9.96895	−0.405295
$606$	0	0
$607$	−20.2430	−0.821640	−0.410820	−	0.911717i	$-0.634757\pi$
$607$	−20.2430	−0.821640	−0.410820	+	0.911717i	$0.634757\pi$
$608$	0	0
$609$	−3.43741	−0.139291
$610$	0	0
$611$	2.00595	0.0811521
$612$	0	0
$613$	−39.9202	−1.61236	−0.806181	−	0.591669i	$-0.798469\pi$
$613$	−39.9202	−1.61236	−0.806181	+	0.591669i	$0.798469\pi$
$614$	0	0
$615$	15.6877	0.632588
$616$	0	0
$617$	49.2647	1.98332	0.991662	−	0.128870i	$-0.0411349\pi$
$617$	49.2647	1.98332	0.991662	+	0.128870i	$0.0411349\pi$
$618$	0	0
$619$	−15.1002	−0.606926	−0.303463	−	0.952843i	$-0.598143\pi$
$619$	−15.1002	−0.606926	−0.303463	+	0.952843i	$0.598143\pi$
$620$	0	0
$621$	33.0668	1.32692
$622$	0	0
$623$	27.6558	1.10801
$624$	0	0
$625$	−30.1508	−1.20603
$626$	0	0
$627$	−11.0021	−0.439381
$628$	0	0
$629$	−4.55314	−0.181546
$630$	0	0
$631$	−15.5662	−0.619680	−0.309840	−	0.950789i	$-0.600276\pi$
$631$	−15.5662	−0.619680	−0.309840	+	0.950789i	$0.600276\pi$
$632$	0	0
$633$	10.2691	0.408161
$634$	0	0
$635$	−9.11669	−0.361785
$636$	0	0
$637$	−4.25124	−0.168440
$638$	0	0
$639$	−26.4903	−1.04794
$640$	0	0
$641$	5.14200	0.203097	0.101548	−	0.994831i	$-0.467620\pi$
$641$	5.14200	0.203097	0.101548	+	0.994831i	$0.467620\pi$
$642$	0	0
$643$	−6.79831	−0.268099	−0.134050	−	0.990975i	$-0.542798\pi$
$643$	−6.79831	−0.268099	−0.134050	+	0.990975i	$0.542798\pi$
$644$	0	0
$645$	11.5724	0.455664
$646$	0	0
$647$	−12.4590	−0.489812	−0.244906	−	0.969547i	$-0.578757\pi$
$647$	−12.4590	−0.489812	−0.244906	+	0.969547i	$0.578757\pi$
$648$	0	0
$649$	26.7674	1.05071
$650$	0	0
$651$	−5.88895	−0.230806
$652$	0	0
$653$	−1.09746	−0.0429471	−0.0214735	−	0.999769i	$-0.506836\pi$
$653$	−1.09746	−0.0429471	−0.0214735	+	0.999769i	$0.506836\pi$
$654$	0	0
$655$	51.5280	2.01337
$656$	0	0
$657$	12.4030	0.483888
$658$	0	0
$659$	−27.0878	−1.05519	−0.527596	−	0.849496i	$-0.676906\pi$
$659$	−27.0878	−1.05519	−0.527596	+	0.849496i	$0.676906\pi$
$660$	0	0
$661$	−16.6931	−0.649287	−0.324644	−	0.945836i	$-0.605244\pi$
$661$	−16.6931	−0.649287	−0.324644	+	0.945836i	$0.605244\pi$
$662$	0	0
$663$	3.00133	0.116562
$664$	0	0
$665$	24.9236	0.966497
$666$	0	0
$667$	25.3097	0.979994
$668$	0	0
$669$	−13.5724	−0.524740
$670$	0	0
$671$	−5.35126	−0.206583
$672$	0	0
$673$	42.7644	1.64845	0.824224	−	0.566264i	$-0.191612\pi$
$673$	42.7644	1.64845	0.824224	+	0.566264i	$0.191612\pi$
$674$	0	0
$675$	12.8523	0.494685
$676$	0	0
$677$	6.94186	0.266797	0.133399	−	0.991062i	$-0.457411\pi$
$677$	6.94186	0.266797	0.133399	+	0.991062i	$0.457411\pi$
$678$	0	0
$679$	−14.0278	−0.538338
$680$	0	0
$681$	−15.8820	−0.608598
$682$	0	0
$683$	−16.6251	−0.636143	−0.318072	−	0.948067i	$-0.603035\pi$
$683$	−16.6251	−0.636143	−0.318072	+	0.948067i	$0.603035\pi$
$684$	0	0
$685$	−57.2018	−2.18557
$686$	0	0
$687$	3.29597	0.125749
$688$	0	0
$689$	12.3017	0.468659
$690$	0	0
$691$	−2.92763	−0.111372	−0.0556861	−	0.998448i	$-0.517735\pi$
$691$	−2.92763	−0.111372	−0.0556861	+	0.998448i	$0.517735\pi$
$692$	0	0
$693$	18.6912	0.710019
$694$	0	0
$695$	22.2325	0.843325
$696$	0	0
$697$	−30.1305	−1.14127
$698$	0	0
$699$	−7.15440	−0.270604
$700$	0	0
$701$	0.193799	0.00731970	0.00365985	−	0.999993i	$-0.498835\pi$
$701$	0.193799	0.00731970	0.00365985	+	0.999993i	$0.498835\pi$
$702$	0	0
$703$	−5.57443	−0.210243
$704$	0	0
$705$	3.00361	0.113123
$706$	0	0
$707$	8.97674	0.337605
$708$	0	0
$709$	38.5430	1.44751	0.723756	−	0.690056i	$-0.242414\pi$
$709$	38.5430	1.44751	0.723756	+	0.690056i	$0.242414\pi$
$710$	0	0
$711$	42.9360	1.61023
$712$	0	0
$713$	43.3604	1.62386
$714$	0	0
$715$	14.0709	0.526223
$716$	0	0
$717$	1.62332	0.0606239
$718$	0	0
$719$	29.6071	1.10416	0.552078	−	0.833792i	$-0.313835\pi$
$719$	29.6071	1.10416	0.552078	+	0.833792i	$0.313835\pi$
$720$	0	0
$721$	4.31667	0.160761
$722$	0	0
$723$	10.6003	0.394231
$724$	0	0
$725$	9.83728	0.365347
$726$	0	0
$727$	−13.7172	−0.508743	−0.254371	−	0.967107i	$-0.581869\pi$
$727$	−13.7172	−0.508743	−0.254371	+	0.967107i	$0.581869\pi$
$728$	0	0
$729$	−6.82612	−0.252819
$730$	0	0
$731$	−22.2266	−0.822079
$732$	0	0
$733$	−51.6865	−1.90908	−0.954541	−	0.298079i	$-0.903654\pi$
$733$	−51.6865	−1.90908	−0.954541	+	0.298079i	$0.903654\pi$
$734$	0	0
$735$	−6.36560	−0.234799
$736$	0	0
$737$	24.4686	0.901311
$738$	0	0
$739$	38.4545	1.41457	0.707285	−	0.706929i	$-0.249920\pi$
$739$	38.4545	1.41457	0.707285	+	0.706929i	$0.249920\pi$
$740$	0	0
$741$	3.67454	0.134987
$742$	0	0
$743$	25.5027	0.935604	0.467802	−	0.883833i	$-0.345046\pi$
$743$	25.5027	0.935604	0.467802	+	0.883833i	$0.345046\pi$
$744$	0	0
$745$	−33.8735	−1.24103
$746$	0	0
$747$	10.9476	0.400550
$748$	0	0
$749$	−25.6986	−0.939005
$750$	0	0
$751$	−20.1189	−0.734150	−0.367075	−	0.930191i	$-0.619641\pi$
$751$	−20.1189	−0.734150	−0.367075	+	0.930191i	$0.619641\pi$
$752$	0	0
$753$	12.9617	0.472349
$754$	0	0
$755$	17.1634	0.624639
$756$	0	0
$757$	32.3402	1.17543	0.587713	−	0.809070i	$-0.300029\pi$
$757$	32.3402	1.17543	0.587713	+	0.809070i	$0.300029\pi$
$758$	0	0
$759$	22.5012	0.816742
$760$	0	0
$761$	16.1196	0.584334	0.292167	−	0.956367i	$-0.405624\pi$
$761$	16.1196	0.584334	0.292167	+	0.956367i	$0.405624\pi$
$762$	0	0
$763$	−29.4957	−1.06782
$764$	0	0
$765$	−27.4867	−0.993784
$766$	0	0
$767$	−8.93991	−0.322802
$768$	0	0
$769$	−27.3213	−0.985231	−0.492616	−	0.870247i	$-0.663959\pi$
$769$	−27.3213	−0.985231	−0.492616	+	0.870247i	$0.663959\pi$
$770$	0	0
$771$	5.61231	0.202122
$772$	0	0
$773$	−9.13547	−0.328580	−0.164290	−	0.986412i	$-0.552533\pi$
$773$	−9.13547	−0.328580	−0.164290	+	0.986412i	$0.552533\pi$
$774$	0	0
$775$	16.8532	0.605384
$776$	0	0
$777$	−1.54837	−0.0555476
$778$	0	0
$779$	−36.8888	−1.32168
$780$	0	0
$781$	−38.9994	−1.39551
$782$	0	0
$783$	10.0413	0.358845
$784$	0	0
$785$	−32.2159	−1.14983
$786$	0	0
$787$	9.33974	0.332926	0.166463	−	0.986048i	$-0.446765\pi$
$787$	9.33974	0.332926	0.166463	+	0.986048i	$0.446765\pi$
$788$	0	0
$789$	−0.0768702	−0.00273665
$790$	0	0
$791$	−12.6810	−0.450885
$792$	0	0
$793$	1.78724	0.0634669
$794$	0	0
$795$	18.4200	0.653291
$796$	0	0
$797$	7.62797	0.270197	0.135098	−	0.990832i	$-0.456865\pi$
$797$	7.62797	0.270197	0.135098	+	0.990832i	$0.456865\pi$
$798$	0	0
$799$	−5.76888	−0.204088
$800$	0	0
$801$	−37.3411	−1.31938
$802$	0	0
$803$	18.2599	0.644378
$804$	0	0
$805$	−50.9733	−1.79657
$806$	0	0
$807$	3.03755	0.106927
$808$	0	0
$809$	24.0887	0.846913	0.423457	−	0.905916i	$-0.360817\pi$
$809$	24.0887	0.846913	0.423457	+	0.905916i	$0.360817\pi$
$810$	0	0
$811$	11.1940	0.393076	0.196538	−	0.980496i	$-0.437030\pi$
$811$	11.1940	0.393076	0.196538	+	0.980496i	$0.437030\pi$
$812$	0	0
$813$	13.0979	0.459363
$814$	0	0
$815$	30.0286	1.05186
$816$	0	0
$817$	−27.2120	−0.952029
$818$	0	0
$819$	−6.24258	−0.218133
$820$	0	0
$821$	−7.64165	−0.266695	−0.133348	−	0.991069i	$-0.542573\pi$
$821$	−7.64165	−0.266695	−0.133348	+	0.991069i	$0.542573\pi$
$822$	0	0
$823$	54.9967	1.91707	0.958533	−	0.284983i	$-0.0919880\pi$
$823$	54.9967	1.91707	0.958533	+	0.284983i	$0.0919880\pi$
$824$	0	0
$825$	8.74571	0.304486
$826$	0	0
$827$	−31.9183	−1.10991	−0.554954	−	0.831881i	$-0.687264\pi$
$827$	−31.9183	−1.10991	−0.554954	+	0.831881i	$0.687264\pi$
$828$	0	0
$829$	−12.0931	−0.420012	−0.210006	−	0.977700i	$-0.567348\pi$
$829$	−12.0931	−0.420012	−0.210006	+	0.977700i	$0.567348\pi$
$830$	0	0
$831$	2.20500	0.0764905
$832$	0	0
$833$	12.2261	0.423608
$834$	0	0
$835$	−0.494893	−0.0171265
$836$	0	0
$837$	17.2026	0.594610
$838$	0	0
$839$	−41.0022	−1.41555	−0.707776	−	0.706437i	$-0.750301\pi$
$839$	−41.0022	−1.41555	−0.707776	+	0.706437i	$0.750301\pi$
$840$	0	0
$841$	−21.3143	−0.734976
$842$	0	0
$843$	−1.88093	−0.0647826
$844$	0	0
$845$	33.3095	1.14588
$846$	0	0
$847$	6.51121	0.223728
$848$	0	0
$849$	−14.1811	−0.486695
$850$	0	0
$851$	11.4007	0.390811
$852$	0	0
$853$	19.1215	0.654709	0.327355	−	0.944902i	$-0.393843\pi$
$853$	19.1215	0.654709	0.327355	+	0.944902i	$0.393843\pi$
$854$	0	0
$855$	−33.6521	−1.15088
$856$	0	0
$857$	26.7002	0.912062	0.456031	−	0.889964i	$-0.349271\pi$
$857$	26.7002	0.912062	0.456031	+	0.889964i	$0.349271\pi$
$858$	0	0
$859$	18.3963	0.627672	0.313836	−	0.949477i	$-0.398386\pi$
$859$	18.3963	0.627672	0.313836	+	0.949477i	$0.398386\pi$
$860$	0	0
$861$	−10.2464	−0.349196
$862$	0	0
$863$	−54.9965	−1.87210	−0.936052	−	0.351862i	$-0.885548\pi$
$863$	−54.9965	−1.87210	−0.936052	+	0.351862i	$0.885548\pi$
$864$	0	0
$865$	−6.73016	−0.228832
$866$	0	0
$867$	2.40637	0.0817245
$868$	0	0
$869$	63.2110	2.14429
$870$	0	0
$871$	−8.17215	−0.276903
$872$	0	0
$873$	18.9405	0.641037
$874$	0	0
$875$	8.10479	0.273992
$876$	0	0
$877$	9.83069	0.331959	0.165979	−	0.986129i	$-0.446921\pi$
$877$	9.83069	0.331959	0.165979	+	0.986129i	$0.446921\pi$
$878$	0	0
$879$	−1.95194	−0.0658372
$880$	0	0
$881$	2.95845	0.0996726	0.0498363	−	0.998757i	$-0.484130\pi$
$881$	2.95845	0.0996726	0.0498363	+	0.998757i	$0.484130\pi$
$882$	0	0
$883$	−29.7455	−1.00102	−0.500508	−	0.865732i	$-0.666853\pi$
$883$	−29.7455	−1.00102	−0.500508	+	0.865732i	$0.666853\pi$
$884$	0	0
$885$	−13.3862	−0.449971
$886$	0	0
$887$	4.04360	0.135771	0.0678854	−	0.997693i	$-0.478375\pi$
$887$	4.04360	0.135771	0.0678854	+	0.997693i	$0.478375\pi$
$888$	0	0
$889$	5.95456	0.199710
$890$	0	0
$891$	−20.4361	−0.684636
$892$	0	0
$893$	−7.06285	−0.236349
$894$	0	0
$895$	67.4862	2.25581
$896$	0	0
$897$	−7.51508	−0.250921
$898$	0	0
$899$	13.1671	0.439147
$900$	0	0
$901$	−35.3783	−1.17862
$902$	0	0
$903$	−7.55852	−0.251532
$904$	0	0
$905$	−56.9494	−1.89306
$906$	0	0
$907$	−41.1144	−1.36518	−0.682591	−	0.730801i	$-0.739147\pi$
$907$	−41.1144	−1.36518	−0.682591	+	0.730801i	$0.739147\pi$
$908$	0	0
$909$	−12.1205	−0.402011
$910$	0	0
$911$	9.69984	0.321370	0.160685	−	0.987006i	$-0.448630\pi$
$911$	9.69984	0.321370	0.160685	+	0.987006i	$0.448630\pi$
$912$	0	0
$913$	16.1172	0.533400
$914$	0	0
$915$	2.67613	0.0884701
$916$	0	0
$917$	−33.6555	−1.11140
$918$	0	0
$919$	53.1616	1.75364	0.876820	−	0.480819i	$-0.159661\pi$
$919$	53.1616	1.75364	0.876820	+	0.480819i	$0.159661\pi$
$920$	0	0
$921$	−19.9458	−0.657235
$922$	0	0
$923$	13.0252	0.428731
$924$	0	0
$925$	4.43119	0.145697
$926$	0	0
$927$	−5.82839	−0.191430
$928$	0	0
$929$	−26.3532	−0.864621	−0.432311	−	0.901725i	$-0.642302\pi$
$929$	−26.3532	−0.864621	−0.432311	+	0.901725i	$0.642302\pi$
$930$	0	0
$931$	14.9684	0.490570
$932$	0	0
$933$	−5.70154	−0.186660
$934$	0	0
$935$	−40.4663	−1.32339
$936$	0	0
$937$	−53.1013	−1.73474	−0.867372	−	0.497660i	$-0.834193\pi$
$937$	−53.1013	−1.73474	−0.867372	+	0.497660i	$0.834193\pi$
$938$	0	0
$939$	−8.52576	−0.278228
$940$	0	0
$941$	29.2627	0.953936	0.476968	−	0.878921i	$-0.341736\pi$
$941$	29.2627	0.953936	0.476968	+	0.878921i	$0.341736\pi$
$942$	0	0
$943$	75.4442	2.45680
$944$	0	0
$945$	−20.2229	−0.657852
$946$	0	0
$947$	−49.5823	−1.61121	−0.805604	−	0.592454i	$-0.798159\pi$
$947$	−49.5823	−1.61121	−0.805604	+	0.592454i	$0.798159\pi$
$948$	0	0
$949$	−6.09855	−0.197967
$950$	0	0
$951$	−11.8351	−0.383780
$952$	0	0
$953$	−26.7995	−0.868121	−0.434060	−	0.900884i	$-0.642920\pi$
$953$	−26.7995	−0.868121	−0.434060	+	0.900884i	$0.642920\pi$
$954$	0	0
$955$	−68.2092	−2.20720
$956$	0	0
$957$	6.83286	0.220875
$958$	0	0
$959$	37.3613	1.20646
$960$	0	0
$961$	−8.44223	−0.272330
$962$	0	0
$963$	34.6984	1.11814
$964$	0	0
$965$	−7.27231	−0.234104
$966$	0	0
$967$	−10.4565	−0.336258	−0.168129	−	0.985765i	$-0.553773\pi$
$967$	−10.4565	−0.336258	−0.168129	+	0.985765i	$0.553773\pi$
$968$	0	0
$969$	−10.5675	−0.339478
$970$	0	0
$971$	−4.10178	−0.131632	−0.0658162	−	0.997832i	$-0.520965\pi$
$971$	−4.10178	−0.131632	−0.0658162	+	0.997832i	$0.520965\pi$
$972$	0	0
$973$	−14.5211	−0.465525
$974$	0	0
$975$	−2.92094	−0.0935449
$976$	0	0
$977$	40.0544	1.28145	0.640726	−	0.767769i	$-0.278633\pi$
$977$	40.0544	1.28145	0.640726	+	0.767769i	$0.278633\pi$
$978$	0	0
$979$	−54.9741	−1.75698
$980$	0	0
$981$	39.8253	1.27152
$982$	0	0
$983$	−29.2160	−0.931845	−0.465923	−	0.884826i	$-0.654277\pi$
$983$	−29.2160	−0.931845	−0.465923	+	0.884826i	$0.654277\pi$
$984$	0	0
$985$	34.6639	1.10448
$986$	0	0
$987$	−1.96181	−0.0624450
$988$	0	0
$989$	55.6535	1.76968
$990$	0	0
$991$	16.2779	0.517084	0.258542	−	0.966000i	$-0.416758\pi$
$991$	16.2779	0.517084	0.258542	+	0.966000i	$0.416758\pi$
$992$	0	0
$993$	11.9905	0.380506
$994$	0	0
$995$	74.1283	2.35003
$996$	0	0
$997$	42.5090	1.34627	0.673137	−	0.739518i	$-0.264947\pi$
$997$	42.5090	1.34627	0.673137	+	0.739518i	$0.264947\pi$
$998$	0	0
$999$	4.52307	0.143104

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.x.1.16		33
4.3	odd	2		4024.2.a.g.1.18	✓	33

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.g.1.18	✓	33	4.3	odd	2
8048.2.a.x.1.16		33	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8048 = 2^{4} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8048.a (trivial)

\(f(q)\)	\(=\)	\(q-0.649284 q^{3} -2.92377 q^{5} +1.90965 q^{7} -2.57843 q^{9} +O(q^{10})\)	\(q-0.649284 q^{3} -2.92377 q^{5} +1.90965 q^{7} -2.57843 q^{9} -3.79600 q^{11} +1.26781 q^{13} +1.89835 q^{15} -3.64607 q^{17} -4.46389 q^{19} -1.23991 q^{21} +9.12945 q^{23} +3.54841 q^{25} +3.62199 q^{27} +2.77231 q^{29} +4.74950 q^{31} +2.46468 q^{33} -5.58338 q^{35} +1.24878 q^{37} -0.823169 q^{39} +8.26383 q^{41} +6.09603 q^{43} +7.53873 q^{45} +1.58222 q^{47} -3.35322 q^{49} +2.36733 q^{51} +9.70315 q^{53} +11.0986 q^{55} +2.89833 q^{57} -7.05146 q^{59} +1.40971 q^{61} -4.92391 q^{63} -3.70678 q^{65} -6.44588 q^{67} -5.92761 q^{69} +10.2738 q^{71} -4.81030 q^{73} -2.30393 q^{75} -7.24905 q^{77} -16.6520 q^{79} +5.38359 q^{81} -4.24583 q^{83} +10.6603 q^{85} -1.80001 q^{87} +14.4821 q^{89} +2.42108 q^{91} -3.08378 q^{93} +13.0514 q^{95} -7.34573 q^{97} +9.78772 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 33 q - 10 q^{3} - 12 q^{7} + 47 q^{9}+O(q^{10}) \) 33 * q - 10 * q^3 - 12 * q^7 + 47 * q^9	\( 33 q - 10 q^{3} - 12 q^{7} + 47 q^{9} - 22 q^{11} - 17 q^{13} - 22 q^{15} + 9 q^{17} - 16 q^{19} + 6 q^{21} - 36 q^{23} + 47 q^{25} - 34 q^{27} + 13 q^{29} - 21 q^{31} + 14 q^{33} - 33 q^{35} - 55 q^{37} - 37 q^{39} + 42 q^{41} - 23 q^{43} + 5 q^{45} - 20 q^{47} + 55 q^{49} - 53 q^{51} - 32 q^{53} - 35 q^{55} + 21 q^{57} - 20 q^{59} - 15 q^{61} - 48 q^{63} + 34 q^{65} - 66 q^{67} - 4 q^{69} - 61 q^{71} + 19 q^{73} - 59 q^{75} + 2 q^{77} - 62 q^{79} + 77 q^{81} - 36 q^{83} - 14 q^{85} - 43 q^{87} + 34 q^{89} - 41 q^{91} - 11 q^{93} - 61 q^{95} - 8 q^{97} - 98 q^{99}+O(q^{100}) \) 33 * q - 10 * q^3 - 12 * q^7 + 47 * q^9 - 22 * q^11 - 17 * q^13 - 22 * q^15 + 9 * q^17 - 16 * q^19 + 6 * q^21 - 36 * q^23 + 47 * q^25 - 34 * q^27 + 13 * q^29 - 21 * q^31 + 14 * q^33 - 33 * q^35 - 55 * q^37 - 37 * q^39 + 42 * q^41 - 23 * q^43 + 5 * q^45 - 20 * q^47 + 55 * q^49 - 53 * q^51 - 32 * q^53 - 35 * q^55 + 21 * q^57 - 20 * q^59 - 15 * q^61 - 48 * q^63 + 34 * q^65 - 66 * q^67 - 4 * q^69 - 61 * q^71 + 19 * q^73 - 59 * q^75 + 2 * q^77 - 62 * q^79 + 77 * q^81 - 36 * q^83 - 14 * q^85 - 43 * q^87 + 34 * q^89 - 41 * q^91 - 11 * q^93 - 61 * q^95 - 8 * q^97 - 98 * q^99

Introduction

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