LMFDB - Embedded newform 1480.2.a.g.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1480,2,Mod(1,1480)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1480, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1480.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1480 = 2^{3} \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1480.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:	$11.8178594991$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.583504.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 8x^{3} - 2x^{2} + 15x + 8 $ x^5 - 8x^3 - 2x^2 + 15*x + 8
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.44679$ of defining polynomial
Character	$\chi$	$=$	1480.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.44679 q^{3} +1.00000 q^{5} -4.02963 q^{7} +2.98676 q^{9} +O(q^{10})$	$q-2.44679 q^{3} +1.00000 q^{5} -4.02963 q^{7} +2.98676 q^{9} +1.17769 q^{11} +5.60200 q^{13} -2.44679 q^{15} -3.00455 q^{17} +1.62448 q^{19} +9.85964 q^{21} -0.296128 q^{23} +1.00000 q^{25} +0.0323921 q^{27} +0.296128 q^{29} -1.77199 q^{31} -2.88157 q^{33} -4.02963 q^{35} -1.00000 q^{37} -13.7069 q^{39} -1.17769 q^{41} -5.07995 q^{43} +2.98676 q^{45} -4.94968 q^{47} +9.23792 q^{49} +7.35150 q^{51} -8.75086 q^{53} +1.17769 q^{55} -3.97476 q^{57} +6.72482 q^{59} -8.22469 q^{61} -12.0355 q^{63} +5.60200 q^{65} -13.5238 q^{67} +0.724561 q^{69} +12.2719 q^{71} -2.74015 q^{73} -2.44679 q^{75} -4.74568 q^{77} +6.18305 q^{79} -9.03954 q^{81} -14.4851 q^{83} -3.00455 q^{85} -0.724561 q^{87} -5.13921 q^{89} -22.5740 q^{91} +4.33567 q^{93} +1.62448 q^{95} -3.32201 q^{97} +3.51749 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{3} + 5 q^{5} - 5 q^{7} + 6 q^{9}+O(q^{10}) $ 5 * q - 5 * q^3 + 5 * q^5 - 5 * q^7 + 6 * q^9	$ 5 q - 5 q^{3} + 5 q^{5} - 5 q^{7} + 6 q^{9} - 7 q^{11} - 6 q^{13} - 5 q^{15} - 12 q^{19} - q^{21} - 6 q^{23} + 5 q^{25} - 17 q^{27} + 6 q^{29} - 10 q^{31} + 3 q^{33} - 5 q^{35} - 5 q^{37} + 4 q^{39} + 7 q^{41} - 22 q^{43} + 6 q^{45} - 13 q^{47} + 14 q^{49} - 10 q^{51} - 11 q^{53} - 7 q^{55} - 8 q^{57} - 10 q^{59} - 10 q^{63} - 6 q^{65} - 24 q^{67} + 26 q^{69} - 13 q^{71} - 13 q^{73} - 5 q^{75} - 19 q^{77} + 2 q^{79} + q^{81} - 13 q^{83} - 26 q^{87} + 8 q^{89} - 8 q^{91} - 20 q^{93} - 12 q^{95} - 20 q^{97} + 2 q^{99}+O(q^{100}) $ 5 * q - 5 * q^3 + 5 * q^5 - 5 * q^7 + 6 * q^9 - 7 * q^11 - 6 * q^13 - 5 * q^15 - 12 * q^19 - q^21 - 6 * q^23 + 5 * q^25 - 17 * q^27 + 6 * q^29 - 10 * q^31 + 3 * q^33 - 5 * q^35 - 5 * q^37 + 4 * q^39 + 7 * q^41 - 22 * q^43 + 6 * q^45 - 13 * q^47 + 14 * q^49 - 10 * q^51 - 11 * q^53 - 7 * q^55 - 8 * q^57 - 10 * q^59 - 10 * q^63 - 6 * q^65 - 24 * q^67 + 26 * q^69 - 13 * q^71 - 13 * q^73 - 5 * q^75 - 19 * q^77 + 2 * q^79 + q^81 - 13 * q^83 - 26 * q^87 + 8 * q^89 - 8 * q^91 - 20 * q^93 - 12 * q^95 - 20 * q^97 + 2 * 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
$3$	−2.44679	−1.41265	−0.706326	−	0.707886i	$-0.749649\pi$
$3$	−2.44679	−1.41265	−0.706326	+	0.707886i	$0.749649\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−4.02963	−1.52306	−0.761529	−	0.648131i	$-0.775551\pi$
$7$	−4.02963	−1.52306	−0.761529	+	0.648131i	$0.775551\pi$
$8$	0	0
$9$	2.98676	0.995587
$10$	0	0
$11$	1.17769	0.355088	0.177544	−	0.984113i	$-0.443185\pi$
$11$	1.17769	0.355088	0.177544	+	0.984113i	$0.443185\pi$
$12$	0	0
$13$	5.60200	1.55371	0.776857	−	0.629677i	$-0.216813\pi$
$13$	5.60200	1.55371	0.776857	+	0.629677i	$0.216813\pi$
$14$	0	0
$15$	−2.44679	−0.631757
$16$	0	0
$17$	−3.00455	−0.728711	−0.364356	−	0.931260i	$-0.618711\pi$
$17$	−3.00455	−0.728711	−0.364356	+	0.931260i	$0.618711\pi$
$18$	0	0
$19$	1.62448	0.372681	0.186341	−	0.982485i	$-0.440337\pi$
$19$	1.62448	0.372681	0.186341	+	0.982485i	$0.440337\pi$
$20$	0	0
$21$	9.85964	2.15155
$22$	0	0
$23$	−0.296128	−0.0617469	−0.0308735	−	0.999523i	$-0.509829\pi$
$23$	−0.296128	−0.0617469	−0.0308735	+	0.999523i	$0.509829\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0.0323921	0.00623386
$28$	0	0
$29$	0.296128	0.0549896	0.0274948	−	0.999622i	$-0.491247\pi$
$29$	0.296128	0.0549896	0.0274948	+	0.999622i	$0.491247\pi$
$30$	0	0
$31$	−1.77199	−0.318258	−0.159129	−	0.987258i	$-0.550869\pi$
$31$	−1.77199	−0.318258	−0.159129	+	0.987258i	$0.550869\pi$
$32$	0	0
$33$	−2.88157	−0.501616
$34$	0	0
$35$	−4.02963	−0.681132
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	0	0
$39$	−13.7069	−2.19486
$40$	0	0
$41$	−1.17769	−0.183925	−0.0919625	−	0.995762i	$-0.529314\pi$
$41$	−1.17769	−0.183925	−0.0919625	+	0.995762i	$0.529314\pi$
$42$	0	0
$43$	−5.07995	−0.774685	−0.387343	−	0.921936i	$-0.626607\pi$
$43$	−5.07995	−0.774685	−0.387343	+	0.921936i	$0.626607\pi$
$44$	0	0
$45$	2.98676	0.445240
$46$	0	0
$47$	−4.94968	−0.721985	−0.360993	−	0.932569i	$-0.617562\pi$
$47$	−4.94968	−0.721985	−0.360993	+	0.932569i	$0.617562\pi$
$48$	0	0
$49$	9.23792	1.31970
$50$	0	0
$51$	7.35150	1.02942
$52$	0	0
$53$	−8.75086	−1.20202	−0.601012	−	0.799240i	$-0.705236\pi$
$53$	−8.75086	−1.20202	−0.601012	+	0.799240i	$0.705236\pi$
$54$	0	0
$55$	1.17769	0.158800
$56$	0	0
$57$	−3.97476	−0.526469
$58$	0	0
$59$	6.72482	0.875497	0.437748	−	0.899098i	$-0.355776\pi$
$59$	6.72482	0.875497	0.437748	+	0.899098i	$0.355776\pi$
$60$	0	0
$61$	−8.22469	−1.05306	−0.526532	−	0.850156i	$-0.676508\pi$
$61$	−8.22469	−1.05306	−0.526532	+	0.850156i	$0.676508\pi$
$62$	0	0
$63$	−12.0355	−1.51634
$64$	0	0
$65$	5.60200	0.694842
$66$	0	0
$67$	−13.5238	−1.65220	−0.826100	−	0.563523i	$-0.809445\pi$
$67$	−13.5238	−1.65220	−0.826100	+	0.563523i	$0.809445\pi$
$68$	0	0
$69$	0.724561	0.0872269
$70$	0	0
$71$	12.2719	1.45641	0.728206	−	0.685359i	$-0.240354\pi$
$71$	12.2719	1.45641	0.728206	+	0.685359i	$0.240354\pi$
$72$	0	0
$73$	−2.74015	−0.320711	−0.160355	−	0.987059i	$-0.551264\pi$
$73$	−2.74015	−0.320711	−0.160355	+	0.987059i	$0.551264\pi$
$74$	0	0
$75$	−2.44679	−0.282531
$76$	0	0
$77$	−4.74568	−0.540820
$78$	0	0
$79$	6.18305	0.695647	0.347824	−	0.937560i	$-0.386921\pi$
$79$	6.18305	0.695647	0.347824	+	0.937560i	$0.386921\pi$
$80$	0	0
$81$	−9.03954	−1.00439
$82$	0	0
$83$	−14.4851	−1.58995	−0.794973	−	0.606645i	$-0.792515\pi$
$83$	−14.4851	−1.58995	−0.794973	+	0.606645i	$0.792515\pi$
$84$	0	0
$85$	−3.00455	−0.325890
$86$	0	0
$87$	−0.724561	−0.0776811
$88$	0	0
$89$	−5.13921	−0.544755	−0.272378	−	0.962190i	$-0.587810\pi$
$89$	−5.13921	−0.544755	−0.272378	+	0.962190i	$0.587810\pi$
$90$	0	0
$91$	−22.5740	−2.36640
$92$	0	0
$93$	4.33567	0.449588
$94$	0	0
$95$	1.62448	0.166668
$96$	0	0
$97$	−3.32201	−0.337299	−0.168649	−	0.985676i	$-0.553941\pi$
$97$	−3.32201	−0.337299	−0.168649	+	0.985676i	$0.553941\pi$
$98$	0	0
$99$	3.51749	0.353521
$100$	0	0
$101$	−13.9694	−1.39001	−0.695003	−	0.719007i	$-0.744597\pi$
$101$	−13.9694	−1.39001	−0.695003	+	0.719007i	$0.744597\pi$
$102$	0	0
$103$	−7.19489	−0.708933	−0.354467	−	0.935069i	$-0.615338\pi$
$103$	−7.19489	−0.708933	−0.354467	+	0.935069i	$0.615338\pi$
$104$	0	0
$105$	9.85964	0.962203
$106$	0	0
$107$	−2.11468	−0.204434	−0.102217	−	0.994762i	$-0.532594\pi$
$107$	−2.11468	−0.204434	−0.102217	+	0.994762i	$0.532594\pi$
$108$	0	0
$109$	−1.97352	−0.189029	−0.0945146	−	0.995523i	$-0.530130\pi$
$109$	−1.97352	−0.189029	−0.0945146	+	0.995523i	$0.530130\pi$
$110$	0	0
$111$	2.44679	0.232239
$112$	0	0
$113$	14.7969	1.39197	0.695987	−	0.718054i	$-0.254967\pi$
$113$	14.7969	1.39197	0.695987	+	0.718054i	$0.254967\pi$
$114$	0	0
$115$	−0.296128	−0.0276141
$116$	0	0
$117$	16.7318	1.54686
$118$	0	0
$119$	12.1072	1.10987
$120$	0	0
$121$	−9.61304	−0.873912
$122$	0	0
$123$	2.88157	0.259822
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	2.20633	0.195780	0.0978902	−	0.995197i	$-0.468791\pi$
$127$	2.20633	0.195780	0.0978902	+	0.995197i	$0.468791\pi$
$128$	0	0
$129$	12.4296	1.09436
$130$	0	0
$131$	−13.4433	−1.17455	−0.587273	−	0.809389i	$-0.699798\pi$
$131$	−13.4433	−1.17455	−0.587273	+	0.809389i	$0.699798\pi$
$132$	0	0
$133$	−6.54606	−0.567615
$134$	0	0
$135$	0.0323921	0.00278787
$136$	0	0
$137$	−18.0328	−1.54064	−0.770322	−	0.637655i	$-0.779905\pi$
$137$	−18.0328	−1.54064	−0.770322	+	0.637655i	$0.779905\pi$
$138$	0	0
$139$	15.5238	1.31671	0.658356	−	0.752707i	$-0.271252\pi$
$139$	15.5238	1.31671	0.658356	+	0.752707i	$0.271252\pi$
$140$	0	0
$141$	12.1108	1.01991
$142$	0	0
$143$	6.59744	0.551706
$144$	0	0
$145$	0.296128	0.0245921
$146$	0	0
$147$	−22.6032	−1.86428
$148$	0	0
$149$	−6.75322	−0.553245	−0.276622	−	0.960979i	$-0.589215\pi$
$149$	−6.75322	−0.553245	−0.276622	+	0.960979i	$0.589215\pi$
$150$	0	0
$151$	−0.547291	−0.0445379	−0.0222690	−	0.999752i	$-0.507089\pi$
$151$	−0.547291	−0.0445379	−0.0222690	+	0.999752i	$0.507089\pi$
$152$	0	0
$153$	−8.97389	−0.725496
$154$	0	0
$155$	−1.77199	−0.142329
$156$	0	0
$157$	2.34338	0.187022	0.0935112	−	0.995618i	$-0.470191\pi$
$157$	2.34338	0.187022	0.0935112	+	0.995618i	$0.470191\pi$
$158$	0	0
$159$	21.4115	1.69804
$160$	0	0
$161$	1.19329	0.0940441
$162$	0	0
$163$	−23.1218	−1.81104	−0.905521	−	0.424301i	$-0.860520\pi$
$163$	−23.1218	−1.81104	−0.905521	+	0.424301i	$0.860520\pi$
$164$	0	0
$165$	−2.88157	−0.224330
$166$	0	0
$167$	1.38127	0.106886	0.0534428	−	0.998571i	$-0.482981\pi$
$167$	1.38127	0.106886	0.0534428	+	0.998571i	$0.482981\pi$
$168$	0	0
$169$	18.3824	1.41403
$170$	0	0
$171$	4.85194	0.371037
$172$	0	0
$173$	−1.55956	−0.118571	−0.0592856	−	0.998241i	$-0.518882\pi$
$173$	−1.55956	−0.118571	−0.0592856	+	0.998241i	$0.518882\pi$
$174$	0	0
$175$	−4.02963	−0.304611
$176$	0	0
$177$	−16.4542	−1.23677
$178$	0	0
$179$	16.8075	1.25625	0.628127	−	0.778111i	$-0.283822\pi$
$179$	16.8075	1.25625	0.628127	+	0.778111i	$0.283822\pi$
$180$	0	0
$181$	−14.7267	−1.09463	−0.547315	−	0.836927i	$-0.684350\pi$
$181$	−14.7267	−1.09463	−0.547315	+	0.836927i	$0.684350\pi$
$182$	0	0
$183$	20.1240	1.48761
$184$	0	0
$185$	−1.00000	−0.0735215
$186$	0	0
$187$	−3.53845	−0.258757
$188$	0	0
$189$	−0.130528	−0.00949453
$190$	0	0
$191$	−20.1417	−1.45740	−0.728700	−	0.684833i	$-0.759875\pi$
$191$	−20.1417	−1.45740	−0.728700	+	0.684833i	$0.759875\pi$
$192$	0	0
$193$	23.9057	1.72077	0.860384	−	0.509647i	$-0.170224\pi$
$193$	23.9057	1.72077	0.860384	+	0.509647i	$0.170224\pi$
$194$	0	0
$195$	−13.7069	−0.981571
$196$	0	0
$197$	−3.44623	−0.245533	−0.122767	−	0.992436i	$-0.539177\pi$
$197$	−3.44623	−0.245533	−0.122767	+	0.992436i	$0.539177\pi$
$198$	0	0
$199$	−24.3788	−1.72816	−0.864082	−	0.503351i	$-0.832100\pi$
$199$	−24.3788	−1.72816	−0.864082	+	0.503351i	$0.832100\pi$
$200$	0	0
$201$	33.0899	2.33398
$202$	0	0
$203$	−1.19329	−0.0837522
$204$	0	0
$205$	−1.17769	−0.0822538
$206$	0	0
$207$	−0.884463	−0.0614744
$208$	0	0
$209$	1.91314	0.132335
$210$	0	0
$211$	−6.98579	−0.480922	−0.240461	−	0.970659i	$-0.577299\pi$
$211$	−6.98579	−0.480922	−0.240461	+	0.970659i	$0.577299\pi$
$212$	0	0
$213$	−30.0268	−2.05740
$214$	0	0
$215$	−5.07995	−0.346450
$216$	0	0
$217$	7.14045	0.484725
$218$	0	0
$219$	6.70457	0.453053
$220$	0	0
$221$	−16.8315	−1.13221
$222$	0	0
$223$	0.545824	0.0365511	0.0182755	−	0.999833i	$-0.494182\pi$
$223$	0.545824	0.0365511	0.0182755	+	0.999833i	$0.494182\pi$
$224$	0	0
$225$	2.98676	0.199117
$226$	0	0
$227$	−0.650399	−0.0431685	−0.0215842	−	0.999767i	$-0.506871\pi$
$227$	−0.650399	−0.0431685	−0.0215842	+	0.999767i	$0.506871\pi$
$228$	0	0
$229$	28.8995	1.90973	0.954865	−	0.297041i	$-0.0959998\pi$
$229$	28.8995	1.90973	0.954865	+	0.297041i	$0.0959998\pi$
$230$	0	0
$231$	11.6117	0.763991
$232$	0	0
$233$	12.8787	0.843710	0.421855	−	0.906663i	$-0.361379\pi$
$233$	12.8787	0.843710	0.421855	+	0.906663i	$0.361379\pi$
$234$	0	0
$235$	−4.94968	−0.322882
$236$	0	0
$237$	−15.1286	−0.982708
$238$	0	0
$239$	−7.05982	−0.456662	−0.228331	−	0.973584i	$-0.573327\pi$
$239$	−7.05982	−0.456662	−0.228331	+	0.973584i	$0.573327\pi$
$240$	0	0
$241$	−10.0328	−0.646268	−0.323134	−	0.946353i	$-0.604737\pi$
$241$	−10.0328	−0.646268	−0.323134	+	0.946353i	$0.604737\pi$
$242$	0	0
$243$	22.0206	1.41263
$244$	0	0
$245$	9.23792	0.590189
$246$	0	0
$247$	9.10034	0.579041
$248$	0	0
$249$	35.4419	2.24604
$250$	0	0
$251$	24.3965	1.53989	0.769945	−	0.638110i	$-0.220283\pi$
$251$	24.3965	1.53989	0.769945	+	0.638110i	$0.220283\pi$
$252$	0	0
$253$	−0.348748	−0.0219256
$254$	0	0
$255$	7.35150	0.460369
$256$	0	0
$257$	14.4001	0.898255	0.449127	−	0.893468i	$-0.351735\pi$
$257$	14.4001	0.898255	0.449127	+	0.893468i	$0.351735\pi$
$258$	0	0
$259$	4.02963	0.250389
$260$	0	0
$261$	0.884463	0.0547469
$262$	0	0
$263$	16.6475	1.02653	0.513263	−	0.858231i	$-0.328436\pi$
$263$	16.6475	1.02653	0.513263	+	0.858231i	$0.328436\pi$
$264$	0	0
$265$	−8.75086	−0.537561
$266$	0	0
$267$	12.5746	0.769550
$268$	0	0
$269$	−6.76410	−0.412415	−0.206207	−	0.978508i	$-0.566112\pi$
$269$	−6.76410	−0.412415	−0.206207	+	0.978508i	$0.566112\pi$
$270$	0	0
$271$	29.2728	1.77820	0.889098	−	0.457717i	$-0.151333\pi$
$271$	29.2728	1.77820	0.889098	+	0.457717i	$0.151333\pi$
$272$	0	0
$273$	55.2337	3.34290
$274$	0	0
$275$	1.17769	0.0710177
$276$	0	0
$277$	1.96046	0.117793	0.0588963	−	0.998264i	$-0.481242\pi$
$277$	1.96046	0.117793	0.0588963	+	0.998264i	$0.481242\pi$
$278$	0	0
$279$	−5.29250	−0.316854
$280$	0	0
$281$	−32.1268	−1.91652	−0.958261	−	0.285894i	$-0.907709\pi$
$281$	−32.1268	−1.91652	−0.958261	+	0.285894i	$0.907709\pi$
$282$	0	0
$283$	25.0540	1.48931	0.744653	−	0.667452i	$-0.232615\pi$
$283$	25.0540	1.48931	0.744653	+	0.667452i	$0.232615\pi$
$284$	0	0
$285$	−3.97476	−0.235444
$286$	0	0
$287$	4.74568	0.280128
$288$	0	0
$289$	−7.97265	−0.468980
$290$	0	0
$291$	8.12824	0.476486
$292$	0	0
$293$	0.674408	0.0393994	0.0196997	−	0.999806i	$-0.493729\pi$
$293$	0.674408	0.0393994	0.0196997	+	0.999806i	$0.493729\pi$
$294$	0	0
$295$	6.72482	0.391534
$296$	0	0
$297$	0.0381480	0.00221357
$298$	0	0
$299$	−1.65891	−0.0959371
$300$	0	0
$301$	20.4703	1.17989
$302$	0	0
$303$	34.1801	1.96360
$304$	0	0
$305$	−8.22469	−0.470944
$306$	0	0
$307$	8.87824	0.506708	0.253354	−	0.967374i	$-0.418466\pi$
$307$	8.87824	0.506708	0.253354	+	0.967374i	$0.418466\pi$
$308$	0	0
$309$	17.6044	1.00148
$310$	0	0
$311$	10.3016	0.584149	0.292074	−	0.956396i	$-0.405655\pi$
$311$	10.3016	0.584149	0.292074	+	0.956396i	$0.405655\pi$
$312$	0	0
$313$	−15.2633	−0.862730	−0.431365	−	0.902177i	$-0.641968\pi$
$313$	−15.2633	−0.862730	−0.431365	+	0.902177i	$0.641968\pi$
$314$	0	0
$315$	−12.0355	−0.678126
$316$	0	0
$317$	17.2751	0.970266	0.485133	−	0.874440i	$-0.338771\pi$
$317$	17.2751	0.970266	0.485133	+	0.874440i	$0.338771\pi$
$318$	0	0
$319$	0.348748	0.0195261
$320$	0	0
$321$	5.17417	0.288794
$322$	0	0
$323$	−4.88084	−0.271577
$324$	0	0
$325$	5.60200	0.310743
$326$	0	0
$327$	4.82879	0.267033
$328$	0	0
$329$	19.9454	1.09962
$330$	0	0
$331$	−8.17696	−0.449446	−0.224723	−	0.974423i	$-0.572148\pi$
$331$	−8.17696	−0.449446	−0.224723	+	0.974423i	$0.572148\pi$
$332$	0	0
$333$	−2.98676	−0.163674
$334$	0	0
$335$	−13.5238	−0.738886
$336$	0	0
$337$	16.2723	0.886407	0.443204	−	0.896421i	$-0.353842\pi$
$337$	16.2723	0.886407	0.443204	+	0.896421i	$0.353842\pi$
$338$	0	0
$339$	−36.2048	−1.96638
$340$	0	0
$341$	−2.08686	−0.113010
$342$	0	0
$343$	−9.01801	−0.486927
$344$	0	0
$345$	0.724561	0.0390091
$346$	0	0
$347$	−32.2492	−1.73123	−0.865614	−	0.500712i	$-0.833072\pi$
$347$	−32.2492	−1.73123	−0.865614	+	0.500712i	$0.833072\pi$
$348$	0	0
$349$	25.6373	1.37233	0.686165	−	0.727446i	$-0.259293\pi$
$349$	25.6373	1.37233	0.686165	+	0.727446i	$0.259293\pi$
$350$	0	0
$351$	0.181461	0.00968565
$352$	0	0
$353$	−0.408344	−0.0217340	−0.0108670	−	0.999941i	$-0.503459\pi$
$353$	−0.408344	−0.0217340	−0.0108670	+	0.999941i	$0.503459\pi$
$354$	0	0
$355$	12.2719	0.651327
$356$	0	0
$357$	−29.6238	−1.56786
$358$	0	0
$359$	−28.3792	−1.49780	−0.748899	−	0.662684i	$-0.769417\pi$
$359$	−28.3792	−1.49780	−0.748899	+	0.662684i	$0.769417\pi$
$360$	0	0
$361$	−16.3611	−0.861109
$362$	0	0
$363$	23.5210	1.23453
$364$	0	0
$365$	−2.74015	−0.143426
$366$	0	0
$367$	27.3615	1.42826	0.714129	−	0.700015i	$-0.246823\pi$
$367$	27.3615	1.42826	0.714129	+	0.700015i	$0.246823\pi$
$368$	0	0
$369$	−3.51749	−0.183113
$370$	0	0
$371$	35.2627	1.83075
$372$	0	0
$373$	−8.74534	−0.452817	−0.226408	−	0.974032i	$-0.572698\pi$
$373$	−8.74534	−0.452817	−0.226408	+	0.974032i	$0.572698\pi$
$374$	0	0
$375$	−2.44679	−0.126351
$376$	0	0
$377$	1.65891	0.0854381
$378$	0	0
$379$	−2.13879	−0.109862	−0.0549311	−	0.998490i	$-0.517494\pi$
$379$	−2.13879	−0.109862	−0.0549311	+	0.998490i	$0.517494\pi$
$380$	0	0
$381$	−5.39843	−0.276570
$382$	0	0
$383$	−12.8905	−0.658674	−0.329337	−	0.944212i	$-0.606825\pi$
$383$	−12.8905	−0.658674	−0.329337	+	0.944212i	$0.606825\pi$
$384$	0	0
$385$	−4.74568	−0.241862
$386$	0	0
$387$	−15.1726	−0.771267
$388$	0	0
$389$	−22.6843	−1.15014	−0.575070	−	0.818104i	$-0.695025\pi$
$389$	−22.6843	−1.15014	−0.575070	+	0.818104i	$0.695025\pi$
$390$	0	0
$391$	0.889732	0.0449957
$392$	0	0
$393$	32.8929	1.65922
$394$	0	0
$395$	6.18305	0.311103
$396$	0	0
$397$	18.2430	0.915590	0.457795	−	0.889058i	$-0.348639\pi$
$397$	18.2430	0.915590	0.457795	+	0.889058i	$0.348639\pi$
$398$	0	0
$399$	16.0168	0.801843
$400$	0	0
$401$	−15.9586	−0.796936	−0.398468	−	0.917182i	$-0.630458\pi$
$401$	−15.9586	−0.796936	−0.398468	+	0.917182i	$0.630458\pi$
$402$	0	0
$403$	−9.92666	−0.494482
$404$	0	0
$405$	−9.03954	−0.449178
$406$	0	0
$407$	−1.17769	−0.0583762
$408$	0	0
$409$	24.8296	1.22775	0.613873	−	0.789405i	$-0.289611\pi$
$409$	24.8296	1.22775	0.613873	+	0.789405i	$0.289611\pi$
$410$	0	0
$411$	44.1224	2.17640
$412$	0	0
$413$	−27.0985	−1.33343
$414$	0	0
$415$	−14.4851	−0.711045
$416$	0	0
$417$	−37.9834	−1.86006
$418$	0	0
$419$	−37.6102	−1.83738	−0.918690	−	0.394979i	$-0.870752\pi$
$419$	−37.6102	−1.83738	−0.918690	+	0.394979i	$0.870752\pi$
$420$	0	0
$421$	15.1100	0.736417	0.368209	−	0.929743i	$-0.379971\pi$
$421$	15.1100	0.736417	0.368209	+	0.929743i	$0.379971\pi$
$422$	0	0
$423$	−14.7835	−0.718799
$424$	0	0
$425$	−3.00455	−0.145742
$426$	0	0
$427$	33.1424	1.60388
$428$	0	0
$429$	−16.1425	−0.779369
$430$	0	0
$431$	−0.865719	−0.0417002	−0.0208501	−	0.999783i	$-0.506637\pi$
$431$	−0.865719	−0.0417002	−0.0208501	+	0.999783i	$0.506637\pi$
$432$	0	0
$433$	−5.55956	−0.267175	−0.133588	−	0.991037i	$-0.542650\pi$
$433$	−5.55956	−0.267175	−0.133588	+	0.991037i	$0.542650\pi$
$434$	0	0
$435$	−0.724561	−0.0347401
$436$	0	0
$437$	−0.481054	−0.0230119
$438$	0	0
$439$	21.2925	1.01624	0.508118	−	0.861288i	$-0.330341\pi$
$439$	21.2925	1.01624	0.508118	+	0.861288i	$0.330341\pi$
$440$	0	0
$441$	27.5915	1.31388
$442$	0	0
$443$	−26.1283	−1.24139	−0.620697	−	0.784051i	$-0.713150\pi$
$443$	−26.1283	−1.24139	−0.620697	+	0.784051i	$0.713150\pi$
$444$	0	0
$445$	−5.13921	−0.243622
$446$	0	0
$447$	16.5237	0.781543
$448$	0	0
$449$	−16.2599	−0.767354	−0.383677	−	0.923467i	$-0.625342\pi$
$449$	−16.2599	−0.767354	−0.383677	+	0.923467i	$0.625342\pi$
$450$	0	0
$451$	−1.38696	−0.0653096
$452$	0	0
$453$	1.33910	0.0629166
$454$	0	0
$455$	−22.5740	−1.05828
$456$	0	0
$457$	−23.1519	−1.08300	−0.541500	−	0.840701i	$-0.682143\pi$
$457$	−23.1519	−1.08300	−0.541500	+	0.840701i	$0.682143\pi$
$458$	0	0
$459$	−0.0973239	−0.00454269
$460$	0	0
$461$	−20.8787	−0.972417	−0.486208	−	0.873843i	$-0.661620\pi$
$461$	−20.8787	−0.972417	−0.486208	+	0.873843i	$0.661620\pi$
$462$	0	0
$463$	17.7785	0.826239	0.413120	−	0.910677i	$-0.364439\pi$
$463$	17.7785	0.826239	0.413120	+	0.910677i	$0.364439\pi$
$464$	0	0
$465$	4.33567	0.201062
$466$	0	0
$467$	−31.6443	−1.46432	−0.732161	−	0.681132i	$-0.761488\pi$
$467$	−31.6443	−1.46432	−0.732161	+	0.681132i	$0.761488\pi$
$468$	0	0
$469$	54.4961	2.51640
$470$	0	0
$471$	−5.73376	−0.264198
$472$	0	0
$473$	−5.98263	−0.275082
$474$	0	0
$475$	1.62448	0.0745363
$476$	0	0
$477$	−26.1367	−1.19672
$478$	0	0
$479$	−28.2547	−1.29099	−0.645495	−	0.763764i	$-0.723349\pi$
$479$	−28.2547	−1.29099	−0.645495	+	0.763764i	$0.723349\pi$
$480$	0	0
$481$	−5.60200	−0.255429
$482$	0	0
$483$	−2.91971	−0.132852
$484$	0	0
$485$	−3.32201	−0.150844
$486$	0	0
$487$	−5.75156	−0.260628	−0.130314	−	0.991473i	$-0.541599\pi$
$487$	−5.75156	−0.260628	−0.130314	+	0.991473i	$0.541599\pi$
$488$	0	0
$489$	56.5742	2.55837
$490$	0	0
$491$	−3.55420	−0.160399	−0.0801994	−	0.996779i	$-0.525556\pi$
$491$	−3.55420	−0.160399	−0.0801994	+	0.996779i	$0.525556\pi$
$492$	0	0
$493$	−0.889732	−0.0400715
$494$	0	0
$495$	3.51749	0.158100
$496$	0	0
$497$	−49.4514	−2.21820
$498$	0	0
$499$	−16.1842	−0.724503	−0.362251	−	0.932080i	$-0.617992\pi$
$499$	−16.1842	−0.724503	−0.362251	+	0.932080i	$0.617992\pi$
$500$	0	0
$501$	−3.37966	−0.150992
$502$	0	0
$503$	13.1447	0.586095	0.293047	−	0.956098i	$-0.405331\pi$
$503$	13.1447	0.586095	0.293047	+	0.956098i	$0.405331\pi$
$504$	0	0
$505$	−13.9694	−0.621630
$506$	0	0
$507$	−44.9778	−1.99753
$508$	0	0
$509$	−26.1677	−1.15986	−0.579932	−	0.814665i	$-0.696921\pi$
$509$	−26.1677	−1.15986	−0.579932	+	0.814665i	$0.696921\pi$
$510$	0	0
$511$	11.0418	0.488461
$512$	0	0
$513$	0.0526204	0.00232325
$514$	0	0
$515$	−7.19489	−0.317045
$516$	0	0
$517$	−5.82921	−0.256368
$518$	0	0
$519$	3.81591	0.167500
$520$	0	0
$521$	9.98369	0.437393	0.218697	−	0.975793i	$-0.429819\pi$
$521$	9.98369	0.437393	0.218697	+	0.975793i	$0.429819\pi$
$522$	0	0
$523$	32.1916	1.40764	0.703819	−	0.710379i	$-0.251477\pi$
$523$	32.1916	1.40764	0.703819	+	0.710379i	$0.251477\pi$
$524$	0	0
$525$	9.85964	0.430310
$526$	0	0
$527$	5.32403	0.231918
$528$	0	0
$529$	−22.9123	−0.996187
$530$	0	0
$531$	20.0854	0.871633
$532$	0	0
$533$	−6.59744	−0.285767
$534$	0	0
$535$	−2.11468	−0.0914256
$536$	0	0
$537$	−41.1244	−1.77465
$538$	0	0
$539$	10.8795	0.468611
$540$	0	0
$541$	−29.3904	−1.26359	−0.631795	−	0.775135i	$-0.717682\pi$
$541$	−29.3904	−1.26359	−0.631795	+	0.775135i	$0.717682\pi$
$542$	0	0
$543$	36.0332	1.54633
$544$	0	0
$545$	−1.97352	−0.0845364
$546$	0	0
$547$	−31.6104	−1.35156	−0.675781	−	0.737102i	$-0.736194\pi$
$547$	−31.6104	−1.35156	−0.675781	+	0.737102i	$0.736194\pi$
$548$	0	0
$549$	−24.5652	−1.04842
$550$	0	0
$551$	0.481054	0.0204936
$552$	0	0
$553$	−24.9154	−1.05951
$554$	0	0
$555$	2.44679	0.103860
$556$	0	0
$557$	−39.6014	−1.67796	−0.838982	−	0.544158i	$-0.816849\pi$
$557$	−39.6014	−1.67796	−0.838982	+	0.544158i	$0.816849\pi$
$558$	0	0
$559$	−28.4579	−1.20364
$560$	0	0
$561$	8.65782	0.365534
$562$	0	0
$563$	−37.9329	−1.59868	−0.799341	−	0.600878i	$-0.794818\pi$
$563$	−37.9329	−1.59868	−0.799341	+	0.600878i	$0.794818\pi$
$564$	0	0
$565$	14.7969	0.622510
$566$	0	0
$567$	36.4260	1.52975
$568$	0	0
$569$	3.36977	0.141268	0.0706341	−	0.997502i	$-0.477498\pi$
$569$	3.36977	0.141268	0.0706341	+	0.997502i	$0.477498\pi$
$570$	0	0
$571$	−7.53861	−0.315481	−0.157740	−	0.987481i	$-0.550421\pi$
$571$	−7.53861	−0.315481	−0.157740	+	0.987481i	$0.550421\pi$
$572$	0	0
$573$	49.2824	2.05880
$574$	0	0
$575$	−0.296128	−0.0123494
$576$	0	0
$577$	−33.5226	−1.39556	−0.697781	−	0.716311i	$-0.745829\pi$
$577$	−33.5226	−1.39556	−0.697781	+	0.716311i	$0.745829\pi$
$578$	0	0
$579$	−58.4920	−2.43085
$580$	0	0
$581$	58.3696	2.42158
$582$	0	0
$583$	−10.3058	−0.426825
$584$	0	0
$585$	16.7318	0.691776
$586$	0	0
$587$	12.3785	0.510917	0.255458	−	0.966820i	$-0.417774\pi$
$587$	12.3785	0.510917	0.255458	+	0.966820i	$0.417774\pi$
$588$	0	0
$589$	−2.87856	−0.118609
$590$	0	0
$591$	8.43218	0.346853
$592$	0	0
$593$	32.6703	1.34161	0.670805	−	0.741634i	$-0.265949\pi$
$593$	32.6703	1.34161	0.670805	+	0.741634i	$0.265949\pi$
$594$	0	0
$595$	12.1072	0.496349
$596$	0	0
$597$	59.6496	2.44130
$598$	0	0
$599$	20.1132	0.821801	0.410901	−	0.911680i	$-0.365214\pi$
$599$	20.1132	0.821801	0.410901	+	0.911680i	$0.365214\pi$
$600$	0	0
$601$	−2.58291	−0.105359	−0.0526796	−	0.998611i	$-0.516776\pi$
$601$	−2.58291	−0.105359	−0.0526796	+	0.998611i	$0.516776\pi$
$602$	0	0
$603$	−40.3925	−1.64491
$604$	0	0
$605$	−9.61304	−0.390825
$606$	0	0
$607$	15.1629	0.615443	0.307721	−	0.951477i	$-0.400434\pi$
$607$	15.1629	0.615443	0.307721	+	0.951477i	$0.400434\pi$
$608$	0	0
$609$	2.91971	0.118313
$610$	0	0
$611$	−27.7281	−1.12176
$612$	0	0
$613$	−38.2949	−1.54672	−0.773358	−	0.633970i	$-0.781424\pi$
$613$	−38.2949	−1.54672	−0.773358	+	0.633970i	$0.781424\pi$
$614$	0	0
$615$	2.88157	0.116196
$616$	0	0
$617$	9.89672	0.398427	0.199213	−	0.979956i	$-0.436161\pi$
$617$	9.89672	0.398427	0.199213	+	0.979956i	$0.436161\pi$
$618$	0	0
$619$	35.3052	1.41904	0.709519	−	0.704686i	$-0.248912\pi$
$619$	35.3052	1.41904	0.709519	+	0.704686i	$0.248912\pi$
$620$	0	0
$621$	−0.00959221	−0.000384922 0
$622$	0	0
$623$	20.7091	0.829694
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−4.68105	−0.186943
$628$	0	0
$629$	3.00455	0.119799
$630$	0	0
$631$	4.03196	0.160510	0.0802549	−	0.996774i	$-0.474427\pi$
$631$	4.03196	0.160510	0.0802549	+	0.996774i	$0.474427\pi$
$632$	0	0
$633$	17.0927	0.679375
$634$	0	0
$635$	2.20633	0.0875557
$636$	0	0
$637$	51.7508	2.05044
$638$	0	0
$639$	36.6534	1.44998
$640$	0	0
$641$	−12.6080	−0.497985	−0.248992	−	0.968505i	$-0.580099\pi$
$641$	−12.6080	−0.497985	−0.248992	+	0.968505i	$0.580099\pi$
$642$	0	0
$643$	−10.9553	−0.432035	−0.216017	−	0.976390i	$-0.569307\pi$
$643$	−10.9553	−0.432035	−0.216017	+	0.976390i	$0.569307\pi$
$644$	0	0
$645$	12.4296	0.489413
$646$	0	0
$647$	17.3667	0.682755	0.341378	−	0.939926i	$-0.389106\pi$
$647$	17.3667	0.682755	0.341378	+	0.939926i	$0.389106\pi$
$648$	0	0
$649$	7.91978	0.310879
$650$	0	0
$651$	−17.4711	−0.684748
$652$	0	0
$653$	43.5326	1.70356	0.851780	−	0.523899i	$-0.175523\pi$
$653$	43.5326	1.70356	0.851780	+	0.523899i	$0.175523\pi$
$654$	0	0
$655$	−13.4433	−0.525273
$656$	0	0
$657$	−8.18418	−0.319295
$658$	0	0
$659$	4.21899	0.164348	0.0821742	−	0.996618i	$-0.473814\pi$
$659$	4.21899	0.164348	0.0821742	+	0.996618i	$0.473814\pi$
$660$	0	0
$661$	44.6587	1.73702	0.868511	−	0.495671i	$-0.165078\pi$
$661$	44.6587	1.73702	0.868511	+	0.495671i	$0.165078\pi$
$662$	0	0
$663$	41.1831	1.59942
$664$	0	0
$665$	−6.54606	−0.253845
$666$	0	0
$667$	−0.0876917	−0.00339544
$668$	0	0
$669$	−1.33551	−0.0516340
$670$	0	0
$671$	−9.68617	−0.373930
$672$	0	0
$673$	26.7726	1.03201	0.516004	−	0.856586i	$-0.327419\pi$
$673$	26.7726	1.03201	0.516004	+	0.856586i	$0.327419\pi$
$674$	0	0
$675$	0.0323921	0.00124677
$676$	0	0
$677$	−11.5201	−0.442752	−0.221376	−	0.975189i	$-0.571055\pi$
$677$	−11.5201	−0.442752	−0.221376	+	0.975189i	$0.571055\pi$
$678$	0	0
$679$	13.3865	0.513725
$680$	0	0
$681$	1.59139	0.0609820
$682$	0	0
$683$	−16.1654	−0.618553	−0.309276	−	0.950972i	$-0.600087\pi$
$683$	−16.1654	−0.618553	−0.309276	+	0.950972i	$0.600087\pi$
$684$	0	0
$685$	−18.0328	−0.688997
$686$	0	0
$687$	−70.7108	−2.69778
$688$	0	0
$689$	−49.0223	−1.86760
$690$	0	0
$691$	36.9079	1.40404	0.702021	−	0.712156i	$-0.252281\pi$
$691$	36.9079	1.40404	0.702021	+	0.712156i	$0.252281\pi$
$692$	0	0
$693$	−14.1742	−0.538433
$694$	0	0
$695$	15.5238	0.588851
$696$	0	0
$697$	3.53845	0.134028
$698$	0	0
$699$	−31.5114	−1.19187
$700$	0	0
$701$	−5.93275	−0.224077	−0.112038	−	0.993704i	$-0.535738\pi$
$701$	−5.93275	−0.224077	−0.112038	+	0.993704i	$0.535738\pi$
$702$	0	0
$703$	−1.62448	−0.0612684
$704$	0	0
$705$	12.1108	0.456119
$706$	0	0
$707$	56.2915	2.11706
$708$	0	0
$709$	−11.9746	−0.449717	−0.224859	−	0.974391i	$-0.572192\pi$
$709$	−11.9746	−0.449717	−0.224859	+	0.974391i	$0.572192\pi$
$710$	0	0
$711$	18.4673	0.692578
$712$	0	0
$713$	0.524734	0.0196514
$714$	0	0
$715$	6.59744	0.246730
$716$	0	0
$717$	17.2739	0.645105
$718$	0	0
$719$	−13.2521	−0.494221	−0.247110	−	0.968987i	$-0.579481\pi$
$719$	−13.2521	−0.494221	−0.247110	+	0.968987i	$0.579481\pi$
$720$	0	0
$721$	28.9927	1.07975
$722$	0	0
$723$	24.5481	0.912953
$724$	0	0
$725$	0.296128	0.0109979
$726$	0	0
$727$	6.62112	0.245564	0.122782	−	0.992434i	$-0.460818\pi$
$727$	6.62112	0.245564	0.122782	+	0.992434i	$0.460818\pi$
$728$	0	0
$729$	−26.7612	−0.991155
$730$	0	0
$731$	15.2630	0.564522
$732$	0	0
$733$	−28.5783	−1.05556	−0.527782	−	0.849380i	$-0.676976\pi$
$733$	−28.5783	−1.05556	−0.527782	+	0.849380i	$0.676976\pi$
$734$	0	0
$735$	−22.6032	−0.833732
$736$	0	0
$737$	−15.9270	−0.586677
$738$	0	0
$739$	−32.4227	−1.19269	−0.596345	−	0.802728i	$-0.703381\pi$
$739$	−32.4227	−1.19269	−0.596345	+	0.802728i	$0.703381\pi$
$740$	0	0
$741$	−22.2666	−0.817983
$742$	0	0
$743$	37.1096	1.36142	0.680709	−	0.732554i	$-0.261672\pi$
$743$	37.1096	1.36142	0.680709	+	0.732554i	$0.261672\pi$
$744$	0	0
$745$	−6.75322	−0.247419
$746$	0	0
$747$	−43.2635	−1.58293
$748$	0	0
$749$	8.52138	0.311364
$750$	0	0
$751$	46.4524	1.69507	0.847537	−	0.530736i	$-0.178085\pi$
$751$	46.4524	1.69507	0.847537	+	0.530736i	$0.178085\pi$
$752$	0	0
$753$	−59.6929	−2.17533
$754$	0	0
$755$	−0.547291	−0.0199180
$756$	0	0
$757$	15.2429	0.554013	0.277006	−	0.960868i	$-0.410658\pi$
$757$	15.2429	0.554013	0.277006	+	0.960868i	$0.410658\pi$
$758$	0	0
$759$	0.853312	0.0309733
$760$	0	0
$761$	−38.4577	−1.39409	−0.697046	−	0.717027i	$-0.745502\pi$
$761$	−38.4577	−1.39409	−0.697046	+	0.717027i	$0.745502\pi$
$762$	0	0
$763$	7.95257	0.287902
$764$	0	0
$765$	−8.97389	−0.324452
$766$	0	0
$767$	37.6724	1.36027
$768$	0	0
$769$	52.5072	1.89346	0.946728	−	0.322033i	$-0.104366\pi$
$769$	52.5072	1.89346	0.946728	+	0.322033i	$0.104366\pi$
$770$	0	0
$771$	−35.2340	−1.26892
$772$	0	0
$773$	13.4944	0.485361	0.242680	−	0.970106i	$-0.421973\pi$
$773$	13.4944	0.485361	0.242680	+	0.970106i	$0.421973\pi$
$774$	0	0
$775$	−1.77199	−0.0636516
$776$	0	0
$777$	−9.85964	−0.353713
$778$	0	0
$779$	−1.91314	−0.0685454
$780$	0	0
$781$	14.4526	0.517155
$782$	0	0
$783$	0.00959221	0.000342797 0
$784$	0	0
$785$	2.34338	0.0836390
$786$	0	0
$787$	−45.5231	−1.62272	−0.811362	−	0.584545i	$-0.801273\pi$
$787$	−45.5231	−1.62272	−0.811362	+	0.584545i	$0.801273\pi$
$788$	0	0
$789$	−40.7328	−1.45013
$790$	0	0
$791$	−59.6260	−2.12006
$792$	0	0
$793$	−46.0747	−1.63616
$794$	0	0
$795$	21.4115	0.759387
$796$	0	0
$797$	−19.3244	−0.684504	−0.342252	−	0.939608i	$-0.611190\pi$
$797$	−19.3244	−0.684504	−0.342252	+	0.939608i	$0.611190\pi$
$798$	0	0
$799$	14.8716	0.526119
$800$	0	0
$801$	−15.3496	−0.542351
$802$	0	0
$803$	−3.22706	−0.113881
$804$	0	0
$805$	1.19329	0.0420578
$806$	0	0
$807$	16.5503	0.582598
$808$	0	0
$809$	1.79985	0.0632795	0.0316398	−	0.999499i	$-0.489927\pi$
$809$	1.79985	0.0632795	0.0316398	+	0.999499i	$0.489927\pi$
$810$	0	0
$811$	14.0856	0.494611	0.247306	−	0.968938i	$-0.420455\pi$
$811$	14.0856	0.494611	0.247306	+	0.968938i	$0.420455\pi$
$812$	0	0
$813$	−71.6243	−2.51197
$814$	0	0
$815$	−23.1218	−0.809923
$816$	0	0
$817$	−8.25228	−0.288711
$818$	0	0
$819$	−67.4231	−2.35595
$820$	0	0
$821$	−10.1677	−0.354856	−0.177428	−	0.984134i	$-0.556778\pi$
$821$	−10.1677	−0.354856	−0.177428	+	0.984134i	$0.556778\pi$
$822$	0	0
$823$	26.4606	0.922359	0.461179	−	0.887307i	$-0.347426\pi$
$823$	26.4606	0.922359	0.461179	+	0.887307i	$0.347426\pi$
$824$	0	0
$825$	−2.88157	−0.100323
$826$	0	0
$827$	34.6037	1.20329	0.601645	−	0.798764i	$-0.294512\pi$
$827$	34.6037	1.20329	0.601645	+	0.798764i	$0.294512\pi$
$828$	0	0
$829$	−18.0945	−0.628448	−0.314224	−	0.949349i	$-0.601744\pi$
$829$	−18.0945	−0.628448	−0.314224	+	0.949349i	$0.601744\pi$
$830$	0	0
$831$	−4.79682	−0.166400
$832$	0	0
$833$	−27.7558	−0.961683
$834$	0	0
$835$	1.38127	0.0478007
$836$	0	0
$837$	−0.0573983	−0.00198398
$838$	0	0
$839$	16.7290	0.577549	0.288775	−	0.957397i	$-0.406752\pi$
$839$	16.7290	0.577549	0.288775	+	0.957397i	$0.406752\pi$
$840$	0	0
$841$	−28.9123	−0.996976
$842$	0	0
$843$	78.6073	2.70738
$844$	0	0
$845$	18.3824	0.632373
$846$	0	0
$847$	38.7370	1.33102
$848$	0	0
$849$	−61.3017	−2.10387
$850$	0	0
$851$	0.296128	0.0101511
$852$	0	0
$853$	44.7962	1.53379	0.766896	−	0.641772i	$-0.221800\pi$
$853$	44.7962	1.53379	0.766896	+	0.641772i	$0.221800\pi$
$854$	0	0
$855$	4.85194	0.165933
$856$	0	0
$857$	−34.0394	−1.16276	−0.581382	−	0.813631i	$-0.697488\pi$
$857$	−34.0394	−1.16276	−0.581382	+	0.813631i	$0.697488\pi$
$858$	0	0
$859$	−33.7341	−1.15099	−0.575497	−	0.817804i	$-0.695191\pi$
$859$	−33.7341	−1.15099	−0.575497	+	0.817804i	$0.695191\pi$
$860$	0	0
$861$	−11.6117	−0.395724
$862$	0	0
$863$	48.1190	1.63799	0.818995	−	0.573801i	$-0.194532\pi$
$863$	48.1190	1.63799	0.818995	+	0.573801i	$0.194532\pi$
$864$	0	0
$865$	−1.55956	−0.0530267
$866$	0	0
$867$	19.5074	0.662505
$868$	0	0
$869$	7.28175	0.247016
$870$	0	0
$871$	−75.7605	−2.56705
$872$	0	0
$873$	−9.92204	−0.335810
$874$	0	0
$875$	−4.02963	−0.136226
$876$	0	0
$877$	44.5082	1.50293	0.751467	−	0.659771i	$-0.229347\pi$
$877$	44.5082	1.50293	0.751467	+	0.659771i	$0.229347\pi$
$878$	0	0
$879$	−1.65013	−0.0556576
$880$	0	0
$881$	−42.8253	−1.44282	−0.721411	−	0.692507i	$-0.756506\pi$
$881$	−42.8253	−1.44282	−0.721411	+	0.692507i	$0.756506\pi$
$882$	0	0
$883$	15.1555	0.510023	0.255011	−	0.966938i	$-0.417921\pi$
$883$	15.1555	0.510023	0.255011	+	0.966938i	$0.417921\pi$
$884$	0	0
$885$	−16.4542	−0.553102
$886$	0	0
$887$	16.2381	0.545221	0.272611	−	0.962124i	$-0.412113\pi$
$887$	16.2381	0.545221	0.272611	+	0.962124i	$0.412113\pi$
$888$	0	0
$889$	−8.89071	−0.298185
$890$	0	0
$891$	−10.6458	−0.356648
$892$	0	0
$893$	−8.04066	−0.269070
$894$	0	0
$895$	16.8075	0.561814
$896$	0	0
$897$	4.05899	0.135526
$898$	0	0
$899$	−0.524734	−0.0175009
$900$	0	0
$901$	26.2924	0.875928
$902$	0	0
$903$	−50.0865	−1.66677
$904$	0	0
$905$	−14.7267	−0.489533
$906$	0	0
$907$	−39.6804	−1.31757	−0.658783	−	0.752333i	$-0.728928\pi$
$907$	−39.6804	−1.31757	−0.658783	+	0.752333i	$0.728928\pi$
$908$	0	0
$909$	−41.7232	−1.38387
$910$	0	0
$911$	−8.68438	−0.287726	−0.143863	−	0.989598i	$-0.545953\pi$
$911$	−8.68438	−0.287726	−0.143863	+	0.989598i	$0.545953\pi$
$912$	0	0
$913$	−17.0590	−0.564571
$914$	0	0
$915$	20.1240	0.665280
$916$	0	0
$917$	54.1715	1.78890
$918$	0	0
$919$	13.0577	0.430732	0.215366	−	0.976533i	$-0.430906\pi$
$919$	13.0577	0.430732	0.215366	+	0.976533i	$0.430906\pi$
$920$	0	0
$921$	−21.7231	−0.715802
$922$	0	0
$923$	68.7474	2.26285
$924$	0	0
$925$	−1.00000	−0.0328798
$926$	0	0
$927$	−21.4894	−0.705805
$928$	0	0
$929$	28.0027	0.918739	0.459369	−	0.888245i	$-0.348075\pi$
$929$	28.0027	0.918739	0.459369	+	0.888245i	$0.348075\pi$
$930$	0	0
$931$	15.0068	0.491829
$932$	0	0
$933$	−25.2057	−0.825199
$934$	0	0
$935$	−3.53845	−0.115720
$936$	0	0
$937$	−39.3559	−1.28570	−0.642851	−	0.765991i	$-0.722249\pi$
$937$	−39.3559	−1.28570	−0.642851	+	0.765991i	$0.722249\pi$
$938$	0	0
$939$	37.3459	1.21874
$940$	0	0
$941$	−1.58139	−0.0515520	−0.0257760	−	0.999668i	$-0.508206\pi$
$941$	−1.58139	−0.0515520	−0.0257760	+	0.999668i	$0.508206\pi$
$942$	0	0
$943$	0.348748	0.0113568
$944$	0	0
$945$	−0.130528	−0.00424608
$946$	0	0
$947$	14.6564	0.476268	0.238134	−	0.971232i	$-0.423464\pi$
$947$	14.6564	0.476268	0.238134	+	0.971232i	$0.423464\pi$
$948$	0	0
$949$	−15.3503	−0.498293
$950$	0	0
$951$	−42.2684	−1.37065
$952$	0	0
$953$	−5.57726	−0.180665	−0.0903327	−	0.995912i	$-0.528793\pi$
$953$	−5.57726	−0.180665	−0.0903327	+	0.995912i	$0.528793\pi$
$954$	0	0
$955$	−20.1417	−0.651769
$956$	0	0
$957$	−0.853312	−0.0275837
$958$	0	0
$959$	72.6655	2.34649
$960$	0	0
$961$	−27.8601	−0.898712
$962$	0	0
$963$	−6.31604	−0.203532
$964$	0	0
$965$	23.9057	0.769551
$966$	0	0
$967$	13.2964	0.427583	0.213792	−	0.976879i	$-0.431419\pi$
$967$	13.2964	0.427583	0.213792	+	0.976879i	$0.431419\pi$
$968$	0	0
$969$	11.9424	0.383644
$970$	0	0
$971$	9.69118	0.311005	0.155502	−	0.987836i	$-0.450300\pi$
$971$	9.69118	0.311005	0.155502	+	0.987836i	$0.450300\pi$
$972$	0	0
$973$	−62.5552	−2.00543
$974$	0	0
$975$	−13.7069	−0.438972
$976$	0	0
$977$	−36.9029	−1.18063	−0.590315	−	0.807173i	$-0.700996\pi$
$977$	−36.9029	−1.18063	−0.590315	+	0.807173i	$0.700996\pi$
$978$	0	0
$979$	−6.05242	−0.193436
$980$	0	0
$981$	−5.89444	−0.188195
$982$	0	0
$983$	10.0346	0.320055	0.160028	−	0.987113i	$-0.448842\pi$
$983$	10.0346	0.320055	0.160028	+	0.987113i	$0.448842\pi$
$984$	0	0
$985$	−3.44623	−0.109806
$986$	0	0
$987$	−48.8021	−1.55339
$988$	0	0
$989$	1.50431	0.0478344
$990$	0	0
$991$	13.7094	0.435492	0.217746	−	0.976005i	$-0.430130\pi$
$991$	13.7094	0.435492	0.217746	+	0.976005i	$0.430130\pi$
$992$	0	0
$993$	20.0073	0.634912
$994$	0	0
$995$	−24.3788	−0.772858
$996$	0	0
$997$	45.3620	1.43663	0.718314	−	0.695719i	$-0.244914\pi$
$997$	45.3620	1.43663	0.718314	+	0.695719i	$0.244914\pi$
$998$	0	0
$999$	−0.0323921	−0.00102484

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1480.2.a.g.1.2	✓	5
4.3	odd	2		2960.2.a.bb.1.4		5
5.4	even	2		7400.2.a.r.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1480.2.a.g.1.2	✓	5	1.1	even	1	trivial
2960.2.a.bb.1.4		5	4.3	odd	2
7400.2.a.r.1.4		5	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1480 = 2^{3} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1480.a (trivial)

\(f(q)\)	\(=\)	\(q-2.44679 q^{3} +1.00000 q^{5} -4.02963 q^{7} +2.98676 q^{9} +O(q^{10})\)	\(q-2.44679 q^{3} +1.00000 q^{5} -4.02963 q^{7} +2.98676 q^{9} +1.17769 q^{11} +5.60200 q^{13} -2.44679 q^{15} -3.00455 q^{17} +1.62448 q^{19} +9.85964 q^{21} -0.296128 q^{23} +1.00000 q^{25} +0.0323921 q^{27} +0.296128 q^{29} -1.77199 q^{31} -2.88157 q^{33} -4.02963 q^{35} -1.00000 q^{37} -13.7069 q^{39} -1.17769 q^{41} -5.07995 q^{43} +2.98676 q^{45} -4.94968 q^{47} +9.23792 q^{49} +7.35150 q^{51} -8.75086 q^{53} +1.17769 q^{55} -3.97476 q^{57} +6.72482 q^{59} -8.22469 q^{61} -12.0355 q^{63} +5.60200 q^{65} -13.5238 q^{67} +0.724561 q^{69} +12.2719 q^{71} -2.74015 q^{73} -2.44679 q^{75} -4.74568 q^{77} +6.18305 q^{79} -9.03954 q^{81} -14.4851 q^{83} -3.00455 q^{85} -0.724561 q^{87} -5.13921 q^{89} -22.5740 q^{91} +4.33567 q^{93} +1.62448 q^{95} -3.32201 q^{97} +3.51749 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{3} + 5 q^{5} - 5 q^{7} + 6 q^{9}+O(q^{10}) \) 5 * q - 5 * q^3 + 5 * q^5 - 5 * q^7 + 6 * q^9	\( 5 q - 5 q^{3} + 5 q^{5} - 5 q^{7} + 6 q^{9} - 7 q^{11} - 6 q^{13} - 5 q^{15} - 12 q^{19} - q^{21} - 6 q^{23} + 5 q^{25} - 17 q^{27} + 6 q^{29} - 10 q^{31} + 3 q^{33} - 5 q^{35} - 5 q^{37} + 4 q^{39} + 7 q^{41} - 22 q^{43} + 6 q^{45} - 13 q^{47} + 14 q^{49} - 10 q^{51} - 11 q^{53} - 7 q^{55} - 8 q^{57} - 10 q^{59} - 10 q^{63} - 6 q^{65} - 24 q^{67} + 26 q^{69} - 13 q^{71} - 13 q^{73} - 5 q^{75} - 19 q^{77} + 2 q^{79} + q^{81} - 13 q^{83} - 26 q^{87} + 8 q^{89} - 8 q^{91} - 20 q^{93} - 12 q^{95} - 20 q^{97} + 2 q^{99}+O(q^{100}) \) 5 * q - 5 * q^3 + 5 * q^5 - 5 * q^7 + 6 * q^9 - 7 * q^11 - 6 * q^13 - 5 * q^15 - 12 * q^19 - q^21 - 6 * q^23 + 5 * q^25 - 17 * q^27 + 6 * q^29 - 10 * q^31 + 3 * q^33 - 5 * q^35 - 5 * q^37 + 4 * q^39 + 7 * q^41 - 22 * q^43 + 6 * q^45 - 13 * q^47 + 14 * q^49 - 10 * q^51 - 11 * q^53 - 7 * q^55 - 8 * q^57 - 10 * q^59 - 10 * q^63 - 6 * q^65 - 24 * q^67 + 26 * q^69 - 13 * q^71 - 13 * q^73 - 5 * q^75 - 19 * q^77 + 2 * q^79 + q^81 - 13 * q^83 - 26 * q^87 + 8 * q^89 - 8 * q^91 - 20 * q^93 - 12 * q^95 - 20 * q^97 + 2 * q^99

Introduction

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