LMFDB - Embedded newform 2960.2.a.bc.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2960,2,Mod(1,2960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2960, 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("2960.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2960 = 2^{4} \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2960.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:	$23.6357189983$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.693982032.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 16x^{4} + 12x^{3} + 60x^{2} - 18x - 44 $ x^6 - x^5 - 16x^4 + 12x^3 + 60x^2 - 18x - 44
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1480)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.854190$ of defining polynomial
Character	$\chi$	$=$	2960.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.854190 q^{3} +1.00000 q^{5} -3.23895 q^{7} -2.27036 q^{9} +O(q^{10})$	$q+0.854190 q^{3} +1.00000 q^{5} -3.23895 q^{7} -2.27036 q^{9} -6.36999 q^{11} +5.87920 q^{13} +0.854190 q^{15} -1.09963 q^{17} +1.96671 q^{19} -2.76668 q^{21} +8.26208 q^{23} +1.00000 q^{25} -4.50189 q^{27} +1.07119 q^{29} +1.84779 q^{31} -5.44118 q^{33} -3.23895 q^{35} -1.00000 q^{37} +5.02195 q^{39} +8.14495 q^{41} +11.4567 q^{43} -2.27036 q^{45} -7.73987 q^{47} +3.49079 q^{49} -0.939290 q^{51} +10.0784 q^{53} -6.36999 q^{55} +1.67994 q^{57} -4.85239 q^{59} +2.77957 q^{61} +7.35358 q^{63} +5.87920 q^{65} +8.52041 q^{67} +7.05738 q^{69} +5.44118 q^{71} +14.7914 q^{73} +0.854190 q^{75} +20.6321 q^{77} +6.90600 q^{79} +2.96561 q^{81} -15.0905 q^{83} -1.09963 q^{85} +0.914999 q^{87} -10.3121 q^{89} -19.0424 q^{91} +1.57836 q^{93} +1.96671 q^{95} +13.6946 q^{97} +14.4622 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{3} + 6 q^{5} - 8 q^{7} + 15 q^{9}+O(q^{10}) $ 6 * q - q^3 + 6 * q^5 - 8 * q^7 + 15 * q^9	$ 6 q - q^{3} + 6 q^{5} - 8 q^{7} + 15 q^{9} + 10 q^{13} - q^{15} + 3 q^{17} + 6 q^{19} + 9 q^{21} - 4 q^{23} + 6 q^{25} - 7 q^{27} + 3 q^{29} + 3 q^{31} + 9 q^{33} - 8 q^{35} - 6 q^{37} + 16 q^{39} + 10 q^{41} + 11 q^{43} + 15 q^{45} - 23 q^{47} + 8 q^{49} + 16 q^{51} + 10 q^{53} + 4 q^{57} - 6 q^{59} + q^{61} - 23 q^{63} + 10 q^{65} + 4 q^{67} - 2 q^{69} - 9 q^{71} + 11 q^{73} - q^{75} + 32 q^{77} + 14 q^{79} + 46 q^{81} - 11 q^{83} + 3 q^{85} - 32 q^{87} + 30 q^{89} + 4 q^{91} + 2 q^{93} + 6 q^{95} + 29 q^{97} + 39 q^{99}+O(q^{100}) $ 6 * q - q^3 + 6 * q^5 - 8 * q^7 + 15 * q^9 + 10 * q^13 - q^15 + 3 * q^17 + 6 * q^19 + 9 * q^21 - 4 * q^23 + 6 * q^25 - 7 * q^27 + 3 * q^29 + 3 * q^31 + 9 * q^33 - 8 * q^35 - 6 * q^37 + 16 * q^39 + 10 * q^41 + 11 * q^43 + 15 * q^45 - 23 * q^47 + 8 * q^49 + 16 * q^51 + 10 * q^53 + 4 * q^57 - 6 * q^59 + q^61 - 23 * q^63 + 10 * q^65 + 4 * q^67 - 2 * q^69 - 9 * q^71 + 11 * q^73 - q^75 + 32 * q^77 + 14 * q^79 + 46 * q^81 - 11 * q^83 + 3 * q^85 - 32 * q^87 + 30 * q^89 + 4 * q^91 + 2 * q^93 + 6 * q^95 + 29 * q^97 + 39 * 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
$3$	0.854190	0.493167	0.246583	−	0.969122i	$-0.420692\pi$
$3$	0.854190	0.493167	0.246583	+	0.969122i	$0.420692\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−3.23895	−1.22421	−0.612104	−	0.790777i	$-0.709677\pi$
$7$	−3.23895	−1.22421	−0.612104	+	0.790777i	$0.709677\pi$
$8$	0	0
$9$	−2.27036	−0.756787
$10$	0	0
$11$	−6.36999	−1.92062	−0.960312	−	0.278929i	$-0.910020\pi$
$11$	−6.36999	−1.92062	−0.960312	+	0.278929i	$0.910020\pi$
$12$	0	0
$13$	5.87920	1.63060	0.815298	−	0.579042i	$-0.196573\pi$
$13$	5.87920	1.63060	0.815298	+	0.579042i	$0.196573\pi$
$14$	0	0
$15$	0.854190	0.220551
$16$	0	0
$17$	−1.09963	−0.266699	−0.133349	−	0.991069i	$-0.542573\pi$
$17$	−1.09963	−0.266699	−0.133349	+	0.991069i	$0.542573\pi$
$18$	0	0
$19$	1.96671	0.451194	0.225597	−	0.974221i	$-0.427567\pi$
$19$	1.96671	0.451194	0.225597	+	0.974221i	$0.427567\pi$
$20$	0	0
$21$	−2.76668	−0.603738
$22$	0	0
$23$	8.26208	1.72276	0.861381	−	0.507960i	$-0.169600\pi$
$23$	8.26208	1.72276	0.861381	+	0.507960i	$0.169600\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−4.50189	−0.866389
$28$	0	0
$29$	1.07119	0.198915	0.0994575	−	0.995042i	$-0.468289\pi$
$29$	1.07119	0.198915	0.0994575	+	0.995042i	$0.468289\pi$
$30$	0	0
$31$	1.84779	0.331872	0.165936	−	0.986137i	$-0.446935\pi$
$31$	1.84779	0.331872	0.165936	+	0.986137i	$0.446935\pi$
$32$	0	0
$33$	−5.44118	−0.947187
$34$	0	0
$35$	−3.23895	−0.547482
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	0	0
$39$	5.02195	0.804155
$40$	0	0
$41$	8.14495	1.27203	0.636014	−	0.771678i	$-0.280582\pi$
$41$	8.14495	1.27203	0.636014	+	0.771678i	$0.280582\pi$
$42$	0	0
$43$	11.4567	1.74713	0.873567	−	0.486704i	$-0.161801\pi$
$43$	11.4567	1.74713	0.873567	+	0.486704i	$0.161801\pi$
$44$	0	0
$45$	−2.27036	−0.338445
$46$	0	0
$47$	−7.73987	−1.12898	−0.564488	−	0.825441i	$-0.690926\pi$
$47$	−7.73987	−1.12898	−0.564488	+	0.825441i	$0.690926\pi$
$48$	0	0
$49$	3.49079	0.498684
$50$	0	0
$51$	−0.939290	−0.131527
$52$	0	0
$53$	10.0784	1.38437	0.692185	−	0.721720i	$-0.256648\pi$
$53$	10.0784	1.38437	0.692185	+	0.721720i	$0.256648\pi$
$54$	0	0
$55$	−6.36999	−0.858929
$56$	0	0
$57$	1.67994	0.222514
$58$	0	0
$59$	−4.85239	−0.631728	−0.315864	−	0.948805i	$-0.602294\pi$
$59$	−4.85239	−0.631728	−0.315864	+	0.948805i	$0.602294\pi$
$60$	0	0
$61$	2.77957	0.355887	0.177944	−	0.984041i	$-0.443056\pi$
$61$	2.77957	0.355887	0.177944	+	0.984041i	$0.443056\pi$
$62$	0	0
$63$	7.35358	0.926464
$64$	0	0
$65$	5.87920	0.729225
$66$	0	0
$67$	8.52041	1.04093	0.520467	−	0.853882i	$-0.325758\pi$
$67$	8.52041	1.04093	0.520467	+	0.853882i	$0.325758\pi$
$68$	0	0
$69$	7.05738	0.849609
$70$	0	0
$71$	5.44118	0.645749	0.322874	−	0.946442i	$-0.395351\pi$
$71$	5.44118	0.645749	0.322874	+	0.946442i	$0.395351\pi$
$72$	0	0
$73$	14.7914	1.73120	0.865599	−	0.500739i	$-0.166938\pi$
$73$	14.7914	1.73120	0.865599	+	0.500739i	$0.166938\pi$
$74$	0	0
$75$	0.854190	0.0986333
$76$	0	0
$77$	20.6321	2.35124
$78$	0	0
$79$	6.90600	0.776986	0.388493	−	0.921452i	$-0.372996\pi$
$79$	6.90600	0.776986	0.388493	+	0.921452i	$0.372996\pi$
$80$	0	0
$81$	2.96561	0.329513
$82$	0	0
$83$	−15.0905	−1.65640	−0.828198	−	0.560436i	$-0.810634\pi$
$83$	−15.0905	−1.65640	−0.828198	+	0.560436i	$0.810634\pi$
$84$	0	0
$85$	−1.09963	−0.119271
$86$	0	0
$87$	0.914999	0.0980982
$88$	0	0
$89$	−10.3121	−1.09308	−0.546540	−	0.837433i	$-0.684055\pi$
$89$	−10.3121	−1.09308	−0.546540	+	0.837433i	$0.684055\pi$
$90$	0	0
$91$	−19.0424	−1.99619
$92$	0	0
$93$	1.57836	0.163668
$94$	0	0
$95$	1.96671	0.201780
$96$	0	0
$97$	13.6946	1.39047	0.695236	−	0.718781i	$-0.255300\pi$
$97$	13.6946	1.39047	0.695236	+	0.718781i	$0.255300\pi$
$98$	0	0
$99$	14.4622	1.45350
$100$	0	0
$101$	−9.02875	−0.898394	−0.449197	−	0.893433i	$-0.648290\pi$
$101$	−9.02875	−0.898394	−0.449197	+	0.893433i	$0.648290\pi$
$102$	0	0
$103$	12.5149	1.23313	0.616567	−	0.787303i	$-0.288523\pi$
$103$	12.5149	1.23313	0.616567	+	0.787303i	$0.288523\pi$
$104$	0	0
$105$	−2.76668	−0.270000
$106$	0	0
$107$	7.47267	0.722410	0.361205	−	0.932486i	$-0.382365\pi$
$107$	7.47267	0.722410	0.361205	+	0.932486i	$0.382365\pi$
$108$	0	0
$109$	−15.7316	−1.50681	−0.753407	−	0.657554i	$-0.771591\pi$
$109$	−15.7316	−1.50681	−0.753407	+	0.657554i	$0.771591\pi$
$110$	0	0
$111$	−0.854190	−0.0810761
$112$	0	0
$113$	13.0701	1.22953	0.614765	−	0.788710i	$-0.289251\pi$
$113$	13.0701	1.22953	0.614765	+	0.788710i	$0.289251\pi$
$114$	0	0
$115$	8.26208	0.770443
$116$	0	0
$117$	−13.3479	−1.23401
$118$	0	0
$119$	3.56164	0.326495
$120$	0	0
$121$	29.5767	2.68879
$122$	0	0
$123$	6.95733	0.627321
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−13.3191	−1.18188	−0.590940	−	0.806716i	$-0.701243\pi$
$127$	−13.3191	−1.18188	−0.590940	+	0.806716i	$0.701243\pi$
$128$	0	0
$129$	9.78621	0.861628
$130$	0	0
$131$	10.5612	0.922736	0.461368	−	0.887209i	$-0.347359\pi$
$131$	10.5612	0.922736	0.461368	+	0.887209i	$0.347359\pi$
$132$	0	0
$133$	−6.37007	−0.552355
$134$	0	0
$135$	−4.50189	−0.387461
$136$	0	0
$137$	−3.01862	−0.257898	−0.128949	−	0.991651i	$-0.541160\pi$
$137$	−3.01862	−0.257898	−0.128949	+	0.991651i	$0.541160\pi$
$138$	0	0
$139$	−5.39014	−0.457186	−0.228593	−	0.973522i	$-0.573412\pi$
$139$	−5.39014	−0.457186	−0.228593	+	0.973522i	$0.573412\pi$
$140$	0	0
$141$	−6.61132	−0.556774
$142$	0	0
$143$	−37.4504	−3.13176
$144$	0	0
$145$	1.07119	0.0889575
$146$	0	0
$147$	2.98180	0.245934
$148$	0	0
$149$	6.06566	0.496919	0.248459	−	0.968642i	$-0.420076\pi$
$149$	6.06566	0.496919	0.248459	+	0.968642i	$0.420076\pi$
$150$	0	0
$151$	−11.6918	−0.951466	−0.475733	−	0.879590i	$-0.657817\pi$
$151$	−11.6918	−0.951466	−0.475733	+	0.879590i	$0.657817\pi$
$152$	0	0
$153$	2.49655	0.201834
$154$	0	0
$155$	1.84779	0.148418
$156$	0	0
$157$	−10.3312	−0.824522	−0.412261	−	0.911066i	$-0.635261\pi$
$157$	−10.3312	−0.824522	−0.412261	+	0.911066i	$0.635261\pi$
$158$	0	0
$159$	8.60884	0.682725
$160$	0	0
$161$	−26.7604	−2.10902
$162$	0	0
$163$	−19.2779	−1.50996	−0.754982	−	0.655745i	$-0.772355\pi$
$163$	−19.2779	−1.50996	−0.754982	+	0.655745i	$0.772355\pi$
$164$	0	0
$165$	−5.44118	−0.423595
$166$	0	0
$167$	−18.5499	−1.43544	−0.717719	−	0.696333i	$-0.754814\pi$
$167$	−18.5499	−1.43544	−0.717719	+	0.696333i	$0.754814\pi$
$168$	0	0
$169$	21.5649	1.65884
$170$	0	0
$171$	−4.46514	−0.341458
$172$	0	0
$173$	2.78506	0.211744	0.105872	−	0.994380i	$-0.466237\pi$
$173$	2.78506	0.211744	0.105872	+	0.994380i	$0.466237\pi$
$174$	0	0
$175$	−3.23895	−0.244842
$176$	0	0
$177$	−4.14486	−0.311547
$178$	0	0
$179$	19.1974	1.43488	0.717440	−	0.696620i	$-0.245314\pi$
$179$	19.1974	1.43488	0.717440	+	0.696620i	$0.245314\pi$
$180$	0	0
$181$	−14.4382	−1.07318	−0.536592	−	0.843842i	$-0.680289\pi$
$181$	−14.4382	−1.07318	−0.536592	+	0.843842i	$0.680289\pi$
$182$	0	0
$183$	2.37428	0.175512
$184$	0	0
$185$	−1.00000	−0.0735215
$186$	0	0
$187$	7.00461	0.512228
$188$	0	0
$189$	14.5814	1.06064
$190$	0	0
$191$	−1.43957	−0.104164	−0.0520818	−	0.998643i	$-0.516586\pi$
$191$	−1.43957	−0.104164	−0.0520818	+	0.998643i	$0.516586\pi$
$192$	0	0
$193$	−14.6573	−1.05506	−0.527528	−	0.849537i	$-0.676881\pi$
$193$	−14.6573	−1.05506	−0.527528	+	0.849537i	$0.676881\pi$
$194$	0	0
$195$	5.02195	0.359629
$196$	0	0
$197$	14.7144	1.04836	0.524178	−	0.851609i	$-0.324373\pi$
$197$	14.7144	1.04836	0.524178	+	0.851609i	$0.324373\pi$
$198$	0	0
$199$	21.5759	1.52947	0.764736	−	0.644344i	$-0.222869\pi$
$199$	21.5759	1.52947	0.764736	+	0.644344i	$0.222869\pi$
$200$	0	0
$201$	7.27804	0.513354
$202$	0	0
$203$	−3.46953	−0.243513
$204$	0	0
$205$	8.14495	0.568868
$206$	0	0
$207$	−18.7579	−1.30376
$208$	0	0
$209$	−12.5279	−0.866574
$210$	0	0
$211$	20.4200	1.40577	0.702885	−	0.711303i	$-0.251895\pi$
$211$	20.4200	1.40577	0.702885	+	0.711303i	$0.251895\pi$
$212$	0	0
$213$	4.64780	0.318462
$214$	0	0
$215$	11.4567	0.781342
$216$	0	0
$217$	−5.98488	−0.406280
$218$	0	0
$219$	12.6346	0.853769
$220$	0	0
$221$	−6.46492	−0.434878
$222$	0	0
$223$	4.58752	0.307203	0.153602	−	0.988133i	$-0.450913\pi$
$223$	4.58752	0.307203	0.153602	+	0.988133i	$0.450913\pi$
$224$	0	0
$225$	−2.27036	−0.151357
$226$	0	0
$227$	0.637189	0.0422917	0.0211459	−	0.999776i	$-0.493269\pi$
$227$	0.637189	0.0422917	0.0211459	+	0.999776i	$0.493269\pi$
$228$	0	0
$229$	−2.46270	−0.162740	−0.0813698	−	0.996684i	$-0.525929\pi$
$229$	−2.46270	−0.162740	−0.0813698	+	0.996684i	$0.525929\pi$
$230$	0	0
$231$	17.6237	1.15955
$232$	0	0
$233$	−21.3732	−1.40021	−0.700103	−	0.714042i	$-0.746863\pi$
$233$	−21.3732	−1.40021	−0.700103	+	0.714042i	$0.746863\pi$
$234$	0	0
$235$	−7.73987	−0.504894
$236$	0	0
$237$	5.89903	0.383183
$238$	0	0
$239$	−16.7667	−1.08455	−0.542275	−	0.840201i	$-0.682437\pi$
$239$	−16.7667	−1.08455	−0.542275	+	0.840201i	$0.682437\pi$
$240$	0	0
$241$	−11.0186	−0.709772	−0.354886	−	0.934910i	$-0.615480\pi$
$241$	−11.0186	−0.709772	−0.354886	+	0.934910i	$0.615480\pi$
$242$	0	0
$243$	16.0389	1.02889
$244$	0	0
$245$	3.49079	0.223018
$246$	0	0
$247$	11.5627	0.735715
$248$	0	0
$249$	−12.8901	−0.816879
$250$	0	0
$251$	11.2011	0.707010	0.353505	−	0.935433i	$-0.384990\pi$
$251$	11.2011	0.707010	0.353505	+	0.935433i	$0.384990\pi$
$252$	0	0
$253$	−52.6293	−3.30878
$254$	0	0
$255$	−0.939290	−0.0588206
$256$	0	0
$257$	−1.32174	−0.0824479	−0.0412239	−	0.999150i	$-0.513126\pi$
$257$	−1.32174	−0.0824479	−0.0412239	+	0.999150i	$0.513126\pi$
$258$	0	0
$259$	3.23895	0.201258
$260$	0	0
$261$	−2.43199	−0.150536
$262$	0	0
$263$	2.12447	0.131000	0.0655001	−	0.997853i	$-0.479136\pi$
$263$	2.12447	0.131000	0.0655001	+	0.997853i	$0.479136\pi$
$264$	0	0
$265$	10.0784	0.619109
$266$	0	0
$267$	−8.80848	−0.539070
$268$	0	0
$269$	−2.82374	−0.172167	−0.0860833	−	0.996288i	$-0.527435\pi$
$269$	−2.82374	−0.172167	−0.0860833	+	0.996288i	$0.527435\pi$
$270$	0	0
$271$	3.11252	0.189072	0.0945360	−	0.995521i	$-0.469863\pi$
$271$	3.11252	0.189072	0.0945360	+	0.995521i	$0.469863\pi$
$272$	0	0
$273$	−16.2658	−0.984453
$274$	0	0
$275$	−6.36999	−0.384125
$276$	0	0
$277$	22.6414	1.36039	0.680194	−	0.733032i	$-0.261896\pi$
$277$	22.6414	1.36039	0.680194	+	0.733032i	$0.261896\pi$
$278$	0	0
$279$	−4.19514	−0.251156
$280$	0	0
$281$	25.1921	1.50283	0.751417	−	0.659828i	$-0.229371\pi$
$281$	25.1921	1.50283	0.751417	+	0.659828i	$0.229371\pi$
$282$	0	0
$283$	4.46289	0.265291	0.132646	−	0.991164i	$-0.457653\pi$
$283$	4.46289	0.265291	0.132646	+	0.991164i	$0.457653\pi$
$284$	0	0
$285$	1.67994	0.0995112
$286$	0	0
$287$	−26.3811	−1.55723
$288$	0	0
$289$	−15.7908	−0.928872
$290$	0	0
$291$	11.6978	0.685735
$292$	0	0
$293$	0.235948	0.0137842	0.00689211	−	0.999976i	$-0.497806\pi$
$293$	0.235948	0.0137842	0.00689211	+	0.999976i	$0.497806\pi$
$294$	0	0
$295$	−4.85239	−0.282517
$296$	0	0
$297$	28.6770	1.66401
$298$	0	0
$299$	48.5744	2.80913
$300$	0	0
$301$	−37.1077	−2.13885
$302$	0	0
$303$	−7.71227	−0.443058
$304$	0	0
$305$	2.77957	0.159158
$306$	0	0
$307$	33.5022	1.91207	0.956037	−	0.293248i	$-0.0947361\pi$
$307$	33.5022	1.91207	0.956037	+	0.293248i	$0.0947361\pi$
$308$	0	0
$309$	10.6901	0.608140
$310$	0	0
$311$	24.6378	1.39708	0.698540	−	0.715571i	$-0.253833\pi$
$311$	24.6378	1.39708	0.698540	+	0.715571i	$0.253833\pi$
$312$	0	0
$313$	−15.8269	−0.894589	−0.447295	−	0.894387i	$-0.647612\pi$
$313$	−15.8269	−0.894589	−0.447295	+	0.894387i	$0.647612\pi$
$314$	0	0
$315$	7.35358	0.414327
$316$	0	0
$317$	2.90662	0.163252	0.0816259	−	0.996663i	$-0.473989\pi$
$317$	2.90662	0.163252	0.0816259	+	0.996663i	$0.473989\pi$
$318$	0	0
$319$	−6.82347	−0.382041
$320$	0	0
$321$	6.38308	0.356269
$322$	0	0
$323$	−2.16265	−0.120333
$324$	0	0
$325$	5.87920	0.326119
$326$	0	0
$327$	−13.4378	−0.743111
$328$	0	0
$329$	25.0691	1.38210
$330$	0	0
$331$	11.5525	0.634982	0.317491	−	0.948261i	$-0.397160\pi$
$331$	11.5525	0.634982	0.317491	+	0.948261i	$0.397160\pi$
$332$	0	0
$333$	2.27036	0.124415
$334$	0	0
$335$	8.52041	0.465519
$336$	0	0
$337$	6.26806	0.341443	0.170722	−	0.985319i	$-0.445390\pi$
$337$	6.26806	0.341443	0.170722	+	0.985319i	$0.445390\pi$
$338$	0	0
$339$	11.1643	0.606363
$340$	0	0
$341$	−11.7704	−0.637401
$342$	0	0
$343$	11.3662	0.613714
$344$	0	0
$345$	7.05738	0.379957
$346$	0	0
$347$	−28.2636	−1.51727	−0.758635	−	0.651516i	$-0.774134\pi$
$347$	−28.2636	−1.51727	−0.758635	+	0.651516i	$0.774134\pi$
$348$	0	0
$349$	31.4788	1.68502	0.842512	−	0.538678i	$-0.181076\pi$
$349$	31.4788	1.68502	0.842512	+	0.538678i	$0.181076\pi$
$350$	0	0
$351$	−26.4675	−1.41273
$352$	0	0
$353$	−17.9124	−0.953382	−0.476691	−	0.879071i	$-0.658164\pi$
$353$	−17.9124	−0.953382	−0.476691	+	0.879071i	$0.658164\pi$
$354$	0	0
$355$	5.44118	0.288788
$356$	0	0
$357$	3.04231	0.161016
$358$	0	0
$359$	13.5609	0.715715	0.357858	−	0.933776i	$-0.383507\pi$
$359$	13.5609	0.715715	0.357858	+	0.933776i	$0.383507\pi$
$360$	0	0
$361$	−15.1321	−0.796424
$362$	0	0
$363$	25.2641	1.32602
$364$	0	0
$365$	14.7914	0.774215
$366$	0	0
$367$	4.44561	0.232059	0.116029	−	0.993246i	$-0.462983\pi$
$367$	4.44561	0.232059	0.116029	+	0.993246i	$0.462983\pi$
$368$	0	0
$369$	−18.4920	−0.962653
$370$	0	0
$371$	−32.6433	−1.69476
$372$	0	0
$373$	5.76748	0.298629	0.149314	−	0.988790i	$-0.452293\pi$
$373$	5.76748	0.298629	0.149314	+	0.988790i	$0.452293\pi$
$374$	0	0
$375$	0.854190	0.0441102
$376$	0	0
$377$	6.29774	0.324350
$378$	0	0
$379$	10.5409	0.541450	0.270725	−	0.962657i	$-0.412737\pi$
$379$	10.5409	0.541450	0.270725	+	0.962657i	$0.412737\pi$
$380$	0	0
$381$	−11.3770	−0.582864
$382$	0	0
$383$	−20.8711	−1.06646	−0.533232	−	0.845969i	$-0.679023\pi$
$383$	−20.8711	−1.06646	−0.533232	+	0.845969i	$0.679023\pi$
$384$	0	0
$385$	20.6321	1.05151
$386$	0	0
$387$	−26.0109	−1.32221
$388$	0	0
$389$	4.21482	0.213700	0.106850	−	0.994275i	$-0.465924\pi$
$389$	4.21482	0.213700	0.106850	+	0.994275i	$0.465924\pi$
$390$	0	0
$391$	−9.08520	−0.459458
$392$	0	0
$393$	9.02127	0.455063
$394$	0	0
$395$	6.90600	0.347479
$396$	0	0
$397$	−11.5238	−0.578365	−0.289182	−	0.957274i	$-0.593383\pi$
$397$	−11.5238	−0.578365	−0.289182	+	0.957274i	$0.593383\pi$
$398$	0	0
$399$	−5.44125	−0.272403
$400$	0	0
$401$	26.6242	1.32955	0.664774	−	0.747045i	$-0.268528\pi$
$401$	26.6242	1.32955	0.664774	+	0.747045i	$0.268528\pi$
$402$	0	0
$403$	10.8635	0.541149
$404$	0	0
$405$	2.96561	0.147363
$406$	0	0
$407$	6.36999	0.315749
$408$	0	0
$409$	−12.1278	−0.599683	−0.299842	−	0.953989i	$-0.596934\pi$
$409$	−12.1278	−0.599683	−0.299842	+	0.953989i	$0.596934\pi$
$410$	0	0
$411$	−2.57847	−0.127187
$412$	0	0
$413$	15.7167	0.773366
$414$	0	0
$415$	−15.0905	−0.740763
$416$	0	0
$417$	−4.60420	−0.225469
$418$	0	0
$419$	−1.69404	−0.0827591	−0.0413795	−	0.999144i	$-0.513175\pi$
$419$	−1.69404	−0.0827591	−0.0413795	+	0.999144i	$0.513175\pi$
$420$	0	0
$421$	−9.25713	−0.451165	−0.225582	−	0.974224i	$-0.572428\pi$
$421$	−9.25713	−0.451165	−0.225582	+	0.974224i	$0.572428\pi$
$422$	0	0
$423$	17.5723	0.854394
$424$	0	0
$425$	−1.09963	−0.0533397
$426$	0	0
$427$	−9.00288	−0.435680
$428$	0	0
$429$	−31.9897	−1.54448
$430$	0	0
$431$	13.1342	0.632650	0.316325	−	0.948651i	$-0.397551\pi$
$431$	13.1342	0.632650	0.316325	+	0.948651i	$0.397551\pi$
$432$	0	0
$433$	19.1311	0.919380	0.459690	−	0.888079i	$-0.347960\pi$
$433$	19.1311	0.919380	0.459690	+	0.888079i	$0.347960\pi$
$434$	0	0
$435$	0.914999	0.0438709
$436$	0	0
$437$	16.2491	0.777300
$438$	0	0
$439$	−5.96774	−0.284825	−0.142412	−	0.989807i	$-0.545486\pi$
$439$	−5.96774	−0.284825	−0.142412	+	0.989807i	$0.545486\pi$
$440$	0	0
$441$	−7.92535	−0.377398
$442$	0	0
$443$	13.6168	0.646953	0.323476	−	0.946236i	$-0.395148\pi$
$443$	13.6168	0.646953	0.323476	+	0.946236i	$0.395148\pi$
$444$	0	0
$445$	−10.3121	−0.488840
$446$	0	0
$447$	5.18123	0.245064
$448$	0	0
$449$	−21.1699	−0.999069	−0.499535	−	0.866294i	$-0.666496\pi$
$449$	−21.1699	−0.999069	−0.499535	+	0.866294i	$0.666496\pi$
$450$	0	0
$451$	−51.8832	−2.44308
$452$	0	0
$453$	−9.98702	−0.469231
$454$	0	0
$455$	−19.0424	−0.892722
$456$	0	0
$457$	−40.4939	−1.89422	−0.947112	−	0.320903i	$-0.896014\pi$
$457$	−40.4939	−1.89422	−0.947112	+	0.320903i	$0.896014\pi$
$458$	0	0
$459$	4.95040	0.231065
$460$	0	0
$461$	12.7827	0.595348	0.297674	−	0.954668i	$-0.403789\pi$
$461$	12.7827	0.595348	0.297674	+	0.954668i	$0.403789\pi$
$462$	0	0
$463$	14.1113	0.655807	0.327903	−	0.944711i	$-0.393658\pi$
$463$	14.1113	0.655807	0.327903	+	0.944711i	$0.393658\pi$
$464$	0	0
$465$	1.57836	0.0731946
$466$	0	0
$467$	8.84635	0.409360	0.204680	−	0.978829i	$-0.434385\pi$
$467$	8.84635	0.409360	0.204680	+	0.978829i	$0.434385\pi$
$468$	0	0
$469$	−27.5972	−1.27432
$470$	0	0
$471$	−8.82483	−0.406627
$472$	0	0
$473$	−72.9792	−3.35559
$474$	0	0
$475$	1.96671	0.0902388
$476$	0	0
$477$	−22.8815	−1.04767
$478$	0	0
$479$	−18.4501	−0.843005	−0.421502	−	0.906827i	$-0.638497\pi$
$479$	−18.4501	−0.843005	−0.421502	+	0.906827i	$0.638497\pi$
$480$	0	0
$481$	−5.87920	−0.268068
$482$	0	0
$483$	−22.8585	−1.04010
$484$	0	0
$485$	13.6946	0.621838
$486$	0	0
$487$	−6.81917	−0.309006	−0.154503	−	0.987992i	$-0.549378\pi$
$487$	−6.81917	−0.309006	−0.154503	+	0.987992i	$0.549378\pi$
$488$	0	0
$489$	−16.4670	−0.744664
$490$	0	0
$491$	39.0097	1.76048	0.880242	−	0.474526i	$-0.157380\pi$
$491$	39.0097	1.76048	0.880242	+	0.474526i	$0.157380\pi$
$492$	0	0
$493$	−1.17791	−0.0530504
$494$	0	0
$495$	14.4622	0.650026
$496$	0	0
$497$	−17.6237	−0.790531
$498$	0	0
$499$	22.1365	0.990965	0.495482	−	0.868618i	$-0.334991\pi$
$499$	22.1365	0.990965	0.495482	+	0.868618i	$0.334991\pi$
$500$	0	0
$501$	−15.8452	−0.707910
$502$	0	0
$503$	−12.1363	−0.541129	−0.270565	−	0.962702i	$-0.587210\pi$
$503$	−12.1363	−0.541129	−0.270565	+	0.962702i	$0.587210\pi$
$504$	0	0
$505$	−9.02875	−0.401774
$506$	0	0
$507$	18.4206	0.818086
$508$	0	0
$509$	−26.2508	−1.16355	−0.581774	−	0.813351i	$-0.697641\pi$
$509$	−26.2508	−1.16355	−0.581774	+	0.813351i	$0.697641\pi$
$510$	0	0
$511$	−47.9084	−2.11934
$512$	0	0
$513$	−8.85390	−0.390909
$514$	0	0
$515$	12.5149	0.551474
$516$	0	0
$517$	49.3029	2.16834
$518$	0	0
$519$	2.37897	0.104425
$520$	0	0
$521$	18.9798	0.831519	0.415759	−	0.909475i	$-0.363516\pi$
$521$	18.9798	0.831519	0.415759	+	0.909475i	$0.363516\pi$
$522$	0	0
$523$	−12.8159	−0.560402	−0.280201	−	0.959941i	$-0.590401\pi$
$523$	−12.8159	−0.560402	−0.280201	+	0.959941i	$0.590401\pi$
$524$	0	0
$525$	−2.76668	−0.120748
$526$	0	0
$527$	−2.03187	−0.0885098
$528$	0	0
$529$	45.2619	1.96791
$530$	0	0
$531$	11.0167	0.478083
$532$	0	0
$533$	47.8857	2.07416
$534$	0	0
$535$	7.47267	0.323072
$536$	0	0
$537$	16.3982	0.707635
$538$	0	0
$539$	−22.2363	−0.957785
$540$	0	0
$541$	13.8264	0.594441	0.297221	−	0.954809i	$-0.403940\pi$
$541$	13.8264	0.594441	0.297221	+	0.954809i	$0.403940\pi$
$542$	0	0
$543$	−12.3330	−0.529259
$544$	0	0
$545$	−15.7316	−0.673868
$546$	0	0
$547$	2.61296	0.111722	0.0558611	−	0.998439i	$-0.482210\pi$
$547$	2.61296	0.111722	0.0558611	+	0.998439i	$0.482210\pi$
$548$	0	0
$549$	−6.31062	−0.269331
$550$	0	0
$551$	2.10672	0.0897493
$552$	0	0
$553$	−22.3682	−0.951192
$554$	0	0
$555$	−0.854190	−0.0362583
$556$	0	0
$557$	−23.1268	−0.979914	−0.489957	−	0.871746i	$-0.662988\pi$
$557$	−23.1268	−0.979914	−0.489957	+	0.871746i	$0.662988\pi$
$558$	0	0
$559$	67.3563	2.84887
$560$	0	0
$561$	5.98326	0.252614
$562$	0	0
$563$	−28.8229	−1.21474	−0.607371	−	0.794419i	$-0.707776\pi$
$563$	−28.8229	−1.21474	−0.607371	+	0.794419i	$0.707776\pi$
$564$	0	0
$565$	13.0701	0.549862
$566$	0	0
$567$	−9.60547	−0.403392
$568$	0	0
$569$	19.3336	0.810507	0.405254	−	0.914204i	$-0.367183\pi$
$569$	19.3336	0.810507	0.405254	+	0.914204i	$0.367183\pi$
$570$	0	0
$571$	−22.3830	−0.936699	−0.468350	−	0.883543i	$-0.655151\pi$
$571$	−22.3830	−0.936699	−0.468350	+	0.883543i	$0.655151\pi$
$572$	0	0
$573$	−1.22967	−0.0513700
$574$	0	0
$575$	8.26208	0.344552
$576$	0	0
$577$	−4.61285	−0.192036	−0.0960178	−	0.995380i	$-0.530611\pi$
$577$	−4.61285	−0.192036	−0.0960178	+	0.995380i	$0.530611\pi$
$578$	0	0
$579$	−12.5201	−0.520319
$580$	0	0
$581$	48.8773	2.02777
$582$	0	0
$583$	−64.1991	−2.65885
$584$	0	0
$585$	−13.3479	−0.551867
$586$	0	0
$587$	−3.84800	−0.158824	−0.0794120	−	0.996842i	$-0.525304\pi$
$587$	−3.84800	−0.158824	−0.0794120	+	0.996842i	$0.525304\pi$
$588$	0	0
$589$	3.63406	0.149739
$590$	0	0
$591$	12.5689	0.517014
$592$	0	0
$593$	−14.7027	−0.603769	−0.301885	−	0.953344i	$-0.597616\pi$
$593$	−14.7027	−0.603769	−0.301885	+	0.953344i	$0.597616\pi$
$594$	0	0
$595$	3.56164	0.146013
$596$	0	0
$597$	18.4299	0.754285
$598$	0	0
$599$	−29.5129	−1.20586	−0.602932	−	0.797793i	$-0.706001\pi$
$599$	−29.5129	−1.20586	−0.602932	+	0.797793i	$0.706001\pi$
$600$	0	0
$601$	−30.5886	−1.24773	−0.623867	−	0.781530i	$-0.714439\pi$
$601$	−30.5886	−1.24773	−0.623867	+	0.781530i	$0.714439\pi$
$602$	0	0
$603$	−19.3444	−0.787764
$604$	0	0
$605$	29.5767	1.20247
$606$	0	0
$607$	−29.2968	−1.18912	−0.594560	−	0.804052i	$-0.702674\pi$
$607$	−29.2968	−1.18912	−0.594560	+	0.804052i	$0.702674\pi$
$608$	0	0
$609$	−2.96364	−0.120093
$610$	0	0
$611$	−45.5042	−1.84090
$612$	0	0
$613$	3.47701	0.140435	0.0702176	−	0.997532i	$-0.477631\pi$
$613$	3.47701	0.140435	0.0702176	+	0.997532i	$0.477631\pi$
$614$	0	0
$615$	6.95733	0.280547
$616$	0	0
$617$	−2.51630	−0.101302	−0.0506512	−	0.998716i	$-0.516130\pi$
$617$	−2.51630	−0.101302	−0.0506512	+	0.998716i	$0.516130\pi$
$618$	0	0
$619$	1.22180	0.0491082	0.0245541	−	0.999699i	$-0.492183\pi$
$619$	1.22180	0.0491082	0.0245541	+	0.999699i	$0.492183\pi$
$620$	0	0
$621$	−37.1949	−1.49258
$622$	0	0
$623$	33.4003	1.33816
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−10.7012	−0.427365
$628$	0	0
$629$	1.09963	0.0438450
$630$	0	0
$631$	8.85317	0.352439	0.176219	−	0.984351i	$-0.443613\pi$
$631$	8.85317	0.352439	0.176219	+	0.984351i	$0.443613\pi$
$632$	0	0
$633$	17.4426	0.693279
$634$	0	0
$635$	−13.3191	−0.528553
$636$	0	0
$637$	20.5230	0.813153
$638$	0	0
$639$	−12.3534	−0.488694
$640$	0	0
$641$	4.11909	0.162694	0.0813472	−	0.996686i	$-0.474078\pi$
$641$	4.11909	0.162694	0.0813472	+	0.996686i	$0.474078\pi$
$642$	0	0
$643$	31.9950	1.26176	0.630880	−	0.775880i	$-0.282694\pi$
$643$	31.9950	1.26176	0.630880	+	0.775880i	$0.282694\pi$
$644$	0	0
$645$	9.78621	0.385332
$646$	0	0
$647$	24.2341	0.952740	0.476370	−	0.879245i	$-0.341952\pi$
$647$	24.2341	0.952740	0.476370	+	0.879245i	$0.341952\pi$
$648$	0	0
$649$	30.9097	1.21331
$650$	0	0
$651$	−5.11222	−0.200364
$652$	0	0
$653$	−7.21767	−0.282449	−0.141225	−	0.989978i	$-0.545104\pi$
$653$	−7.21767	−0.282449	−0.141225	+	0.989978i	$0.545104\pi$
$654$	0	0
$655$	10.5612	0.412660
$656$	0	0
$657$	−33.5817	−1.31015
$658$	0	0
$659$	−2.15865	−0.0840890	−0.0420445	−	0.999116i	$-0.513387\pi$
$659$	−2.15865	−0.0840890	−0.0420445	+	0.999116i	$0.513387\pi$
$660$	0	0
$661$	−6.64726	−0.258549	−0.129274	−	0.991609i	$-0.541265\pi$
$661$	−6.64726	−0.258549	−0.129274	+	0.991609i	$0.541265\pi$
$662$	0	0
$663$	−5.52227	−0.214467
$664$	0	0
$665$	−6.37007	−0.247021
$666$	0	0
$667$	8.85025	0.342683
$668$	0	0
$669$	3.91861	0.151502
$670$	0	0
$671$	−17.7058	−0.683526
$672$	0	0
$673$	−10.2481	−0.395033	−0.197517	−	0.980300i	$-0.563288\pi$
$673$	−10.2481	−0.395033	−0.197517	+	0.980300i	$0.563288\pi$
$674$	0	0
$675$	−4.50189	−0.173278
$676$	0	0
$677$	40.6131	1.56089	0.780444	−	0.625225i	$-0.214993\pi$
$677$	40.6131	1.56089	0.780444	+	0.625225i	$0.214993\pi$
$678$	0	0
$679$	−44.3560	−1.70223
$680$	0	0
$681$	0.544280	0.0208569
$682$	0	0
$683$	4.93337	0.188770	0.0943850	−	0.995536i	$-0.469912\pi$
$683$	4.93337	0.188770	0.0943850	+	0.995536i	$0.469912\pi$
$684$	0	0
$685$	−3.01862	−0.115335
$686$	0	0
$687$	−2.10361	−0.0802577
$688$	0	0
$689$	59.2527	2.25735
$690$	0	0
$691$	47.9432	1.82384	0.911922	−	0.410363i	$-0.134598\pi$
$691$	47.9432	1.82384	0.911922	+	0.410363i	$0.134598\pi$
$692$	0	0
$693$	−46.8422	−1.77939
$694$	0	0
$695$	−5.39014	−0.204460
$696$	0	0
$697$	−8.95640	−0.339248
$698$	0	0
$699$	−18.2568	−0.690535
$700$	0	0
$701$	15.1679	0.572882	0.286441	−	0.958098i	$-0.407528\pi$
$701$	15.1679	0.572882	0.286441	+	0.958098i	$0.407528\pi$
$702$	0	0
$703$	−1.96671	−0.0741758
$704$	0	0
$705$	−6.61132	−0.248997
$706$	0	0
$707$	29.2437	1.09982
$708$	0	0
$709$	−13.1856	−0.495197	−0.247598	−	0.968863i	$-0.579641\pi$
$709$	−13.1856	−0.495197	−0.247598	+	0.968863i	$0.579641\pi$
$710$	0	0
$711$	−15.6791	−0.588012
$712$	0	0
$713$	15.2665	0.571736
$714$	0	0
$715$	−37.4504	−1.40057
$716$	0	0
$717$	−14.3220	−0.534864
$718$	0	0
$719$	−19.7531	−0.736665	−0.368333	−	0.929694i	$-0.620071\pi$
$719$	−19.7531	−0.736665	−0.368333	+	0.929694i	$0.620071\pi$
$720$	0	0
$721$	−40.5352	−1.50961
$722$	0	0
$723$	−9.41199	−0.350036
$724$	0	0
$725$	1.07119	0.0397830
$726$	0	0
$727$	23.3563	0.866238	0.433119	−	0.901337i	$-0.357413\pi$
$727$	23.3563	0.866238	0.433119	+	0.901337i	$0.357413\pi$
$728$	0	0
$729$	4.80338	0.177903
$730$	0	0
$731$	−12.5981	−0.465958
$732$	0	0
$733$	44.9384	1.65984	0.829919	−	0.557884i	$-0.188387\pi$
$733$	44.9384	1.65984	0.829919	+	0.557884i	$0.188387\pi$
$734$	0	0
$735$	2.98180	0.109985
$736$	0	0
$737$	−54.2749	−1.99924
$738$	0	0
$739$	−11.8588	−0.436232	−0.218116	−	0.975923i	$-0.569991\pi$
$739$	−11.8588	−0.436232	−0.218116	+	0.975923i	$0.569991\pi$
$740$	0	0
$741$	9.87671	0.362830
$742$	0	0
$743$	−11.7814	−0.432217	−0.216109	−	0.976369i	$-0.569337\pi$
$743$	−11.7814	−0.432217	−0.216109	+	0.976369i	$0.569337\pi$
$744$	0	0
$745$	6.06566	0.222229
$746$	0	0
$747$	34.2608	1.25354
$748$	0	0
$749$	−24.2036	−0.884380
$750$	0	0
$751$	20.9814	0.765622	0.382811	−	0.923827i	$-0.374956\pi$
$751$	20.9814	0.765622	0.382811	+	0.923827i	$0.374956\pi$
$752$	0	0
$753$	9.56791	0.348674
$754$	0	0
$755$	−11.6918	−0.425509
$756$	0	0
$757$	8.46096	0.307519	0.153759	−	0.988108i	$-0.450862\pi$
$757$	8.46096	0.307519	0.153759	+	0.988108i	$0.450862\pi$
$758$	0	0
$759$	−44.9554	−1.63178
$760$	0	0
$761$	0.203546	0.00737854	0.00368927	−	0.999993i	$-0.498826\pi$
$761$	0.203546	0.00737854	0.00368927	+	0.999993i	$0.498826\pi$
$762$	0	0
$763$	50.9539	1.84465
$764$	0	0
$765$	2.49655	0.0902629
$766$	0	0
$767$	−28.5282	−1.03009
$768$	0	0
$769$	40.3495	1.45504	0.727520	−	0.686087i	$-0.240673\pi$
$769$	40.3495	1.45504	0.727520	+	0.686087i	$0.240673\pi$
$770$	0	0
$771$	−1.12902	−0.0406605
$772$	0	0
$773$	−43.1915	−1.55349	−0.776745	−	0.629815i	$-0.783131\pi$
$773$	−43.1915	−1.55349	−0.776745	+	0.629815i	$0.783131\pi$
$774$	0	0
$775$	1.84779	0.0663744
$776$	0	0
$777$	2.76668	0.0992540
$778$	0	0
$779$	16.0187	0.573931
$780$	0	0
$781$	−34.6602	−1.24024
$782$	0	0
$783$	−4.82238	−0.172338
$784$	0	0
$785$	−10.3312	−0.368737
$786$	0	0
$787$	53.1183	1.89346	0.946731	−	0.322026i	$-0.104364\pi$
$787$	53.1183	1.89346	0.946731	+	0.322026i	$0.104364\pi$
$788$	0	0
$789$	1.81470	0.0646049
$790$	0	0
$791$	−42.3333	−1.50520
$792$	0	0
$793$	16.3416	0.580308
$794$	0	0
$795$	8.60884	0.305324
$796$	0	0
$797$	−10.3533	−0.366733	−0.183366	−	0.983045i	$-0.558699\pi$
$797$	−10.3533	−0.366733	−0.183366	+	0.983045i	$0.558699\pi$
$798$	0	0
$799$	8.51097	0.301097
$800$	0	0
$801$	23.4122	0.827228
$802$	0	0
$803$	−94.2207	−3.32498
$804$	0	0
$805$	−26.7604	−0.943182
$806$	0	0
$807$	−2.41201	−0.0849068
$808$	0	0
$809$	−33.6495	−1.18305	−0.591527	−	0.806286i	$-0.701474\pi$
$809$	−33.6495	−1.18305	−0.591527	+	0.806286i	$0.701474\pi$
$810$	0	0
$811$	19.0060	0.667391	0.333695	−	0.942681i	$-0.391704\pi$
$811$	19.0060	0.667391	0.333695	+	0.942681i	$0.391704\pi$
$812$	0	0
$813$	2.65868	0.0932440
$814$	0	0
$815$	−19.2779	−0.675277
$816$	0	0
$817$	22.5320	0.788296
$818$	0	0
$819$	43.2331	1.51069
$820$	0	0
$821$	34.5127	1.20450	0.602251	−	0.798307i	$-0.294271\pi$
$821$	34.5127	1.20450	0.602251	+	0.798307i	$0.294271\pi$
$822$	0	0
$823$	−33.9110	−1.18206	−0.591031	−	0.806649i	$-0.701279\pi$
$823$	−33.9110	−1.18206	−0.591031	+	0.806649i	$0.701279\pi$
$824$	0	0
$825$	−5.44118	−0.189437
$826$	0	0
$827$	5.39726	0.187681	0.0938406	−	0.995587i	$-0.470086\pi$
$827$	5.39726	0.187681	0.0938406	+	0.995587i	$0.470086\pi$
$828$	0	0
$829$	−21.2944	−0.739584	−0.369792	−	0.929114i	$-0.620571\pi$
$829$	−21.2944	−0.739584	−0.369792	+	0.929114i	$0.620571\pi$
$830$	0	0
$831$	19.3400	0.670898
$832$	0	0
$833$	−3.83857	−0.132998
$834$	0	0
$835$	−18.5499	−0.641947
$836$	0	0
$837$	−8.31852	−0.287530
$838$	0	0
$839$	31.6302	1.09200	0.545998	−	0.837787i	$-0.316151\pi$
$839$	31.6302	1.09200	0.545998	+	0.837787i	$0.316151\pi$
$840$	0	0
$841$	−27.8526	−0.960433
$842$	0	0
$843$	21.5188	0.741148
$844$	0	0
$845$	21.5649	0.741857
$846$	0	0
$847$	−95.7975	−3.29164
$848$	0	0
$849$	3.81215	0.130833
$850$	0	0
$851$	−8.26208	−0.283220
$852$	0	0
$853$	−34.4425	−1.17929	−0.589645	−	0.807662i	$-0.700732\pi$
$853$	−34.4425	−1.17929	−0.589645	+	0.807662i	$0.700732\pi$
$854$	0	0
$855$	−4.46514	−0.152704
$856$	0	0
$857$	46.0156	1.57186	0.785931	−	0.618315i	$-0.212184\pi$
$857$	46.0156	1.57186	0.785931	+	0.618315i	$0.212184\pi$
$858$	0	0
$859$	17.7505	0.605638	0.302819	−	0.953048i	$-0.402072\pi$
$859$	17.7505	0.605638	0.302819	+	0.953048i	$0.402072\pi$
$860$	0	0
$861$	−22.5344	−0.767972
$862$	0	0
$863$	−8.54323	−0.290815	−0.145408	−	0.989372i	$-0.546449\pi$
$863$	−8.54323	−0.290815	−0.145408	+	0.989372i	$0.546449\pi$
$864$	0	0
$865$	2.78506	0.0946950
$866$	0	0
$867$	−13.4884	−0.458089
$868$	0	0
$869$	−43.9911	−1.49230
$870$	0	0
$871$	50.0931	1.69734
$872$	0	0
$873$	−31.0916	−1.05229
$874$	0	0
$875$	−3.23895	−0.109496
$876$	0	0
$877$	−32.2455	−1.08885	−0.544427	−	0.838808i	$-0.683253\pi$
$877$	−32.2455	−1.08885	−0.544427	+	0.838808i	$0.683253\pi$
$878$	0	0
$879$	0.201544	0.00679792
$880$	0	0
$881$	−50.6144	−1.70524	−0.852621	−	0.522530i	$-0.824988\pi$
$881$	−50.6144	−1.70524	−0.852621	+	0.522530i	$0.824988\pi$
$882$	0	0
$883$	27.5424	0.926876	0.463438	−	0.886129i	$-0.346616\pi$
$883$	27.5424	0.926876	0.463438	+	0.886129i	$0.346616\pi$
$884$	0	0
$885$	−4.14486	−0.139328
$886$	0	0
$887$	−39.3815	−1.32230	−0.661151	−	0.750253i	$-0.729932\pi$
$887$	−39.3815	−1.32230	−0.661151	+	0.750253i	$0.729932\pi$
$888$	0	0
$889$	43.1399	1.44687
$890$	0	0
$891$	−18.8909	−0.632870
$892$	0	0
$893$	−15.2221	−0.509388
$894$	0	0
$895$	19.1974	0.641698
$896$	0	0
$897$	41.4917	1.38537
$898$	0	0
$899$	1.97933	0.0660143
$900$	0	0
$901$	−11.0824	−0.369210
$902$	0	0
$903$	−31.6970	−1.05481
$904$	0	0
$905$	−14.4382	−0.479943
$906$	0	0
$907$	−42.6218	−1.41523	−0.707617	−	0.706596i	$-0.750230\pi$
$907$	−42.6218	−1.41523	−0.707617	+	0.706596i	$0.750230\pi$
$908$	0	0
$909$	20.4985	0.679893
$910$	0	0
$911$	−37.7452	−1.25055	−0.625277	−	0.780403i	$-0.715014\pi$
$911$	−37.7452	−1.25055	−0.625277	+	0.780403i	$0.715014\pi$
$912$	0	0
$913$	96.1262	3.18131
$914$	0	0
$915$	2.37428	0.0784913
$916$	0	0
$917$	−34.2072	−1.12962
$918$	0	0
$919$	−21.1076	−0.696276	−0.348138	−	0.937443i	$-0.613186\pi$
$919$	−21.1076	−0.696276	−0.348138	+	0.937443i	$0.613186\pi$
$920$	0	0
$921$	28.6173	0.942971
$922$	0	0
$923$	31.9897	1.05296
$924$	0	0
$925$	−1.00000	−0.0328798
$926$	0	0
$927$	−28.4134	−0.933219
$928$	0	0
$929$	−34.3168	−1.12590	−0.562949	−	0.826491i	$-0.690333\pi$
$929$	−34.3168	−1.12590	−0.562949	+	0.826491i	$0.690333\pi$
$930$	0	0
$931$	6.86537	0.225003
$932$	0	0
$933$	21.0453	0.688993
$934$	0	0
$935$	7.00461	0.229075
$936$	0	0
$937$	8.12430	0.265409	0.132705	−	0.991156i	$-0.457634\pi$
$937$	8.12430	0.265409	0.132705	+	0.991156i	$0.457634\pi$
$938$	0	0
$939$	−13.5192	−0.441182
$940$	0	0
$941$	−47.4488	−1.54679	−0.773393	−	0.633927i	$-0.781442\pi$
$941$	−47.4488	−1.54679	−0.773393	+	0.633927i	$0.781442\pi$
$942$	0	0
$943$	67.2942	2.19140
$944$	0	0
$945$	14.5814	0.474332
$946$	0	0
$947$	−22.0372	−0.716113	−0.358057	−	0.933700i	$-0.616561\pi$
$947$	−22.0372	−0.716113	−0.358057	+	0.933700i	$0.616561\pi$
$948$	0	0
$949$	86.9613	2.82288
$950$	0	0
$951$	2.48280	0.0805104
$952$	0	0
$953$	−52.0188	−1.68505	−0.842526	−	0.538655i	$-0.818933\pi$
$953$	−52.0188	−1.68505	−0.842526	+	0.538655i	$0.818933\pi$
$954$	0	0
$955$	−1.43957	−0.0465834
$956$	0	0
$957$	−5.82853	−0.188410
$958$	0	0
$959$	9.77715	0.315721
$960$	0	0
$961$	−27.5857	−0.889861
$962$	0	0
$963$	−16.9657	−0.546711
$964$	0	0
$965$	−14.6573	−0.471836
$966$	0	0
$967$	5.32965	0.171390	0.0856950	−	0.996321i	$-0.472689\pi$
$967$	5.32965	0.171390	0.0856950	+	0.996321i	$0.472689\pi$
$968$	0	0
$969$	−1.84731	−0.0593441
$970$	0	0
$971$	−38.6826	−1.24138	−0.620691	−	0.784055i	$-0.713148\pi$
$971$	−38.6826	−1.24138	−0.620691	+	0.784055i	$0.713148\pi$
$972$	0	0
$973$	17.4584	0.559690
$974$	0	0
$975$	5.02195	0.160831
$976$	0	0
$977$	−5.63508	−0.180282	−0.0901411	−	0.995929i	$-0.528732\pi$
$977$	−5.63508	−0.180282	−0.0901411	+	0.995929i	$0.528732\pi$
$978$	0	0
$979$	65.6879	2.09939
$980$	0	0
$981$	35.7164	1.14034
$982$	0	0
$983$	−51.6119	−1.64616	−0.823082	−	0.567922i	$-0.807747\pi$
$983$	−51.6119	−1.64616	−0.823082	+	0.567922i	$0.807747\pi$
$984$	0	0
$985$	14.7144	0.468839
$986$	0	0
$987$	21.4137	0.681607
$988$	0	0
$989$	94.6563	3.00990
$990$	0	0
$991$	−18.7040	−0.594151	−0.297075	−	0.954854i	$-0.596011\pi$
$991$	−18.7040	−0.594151	−0.297075	+	0.954854i	$0.596011\pi$
$992$	0	0
$993$	9.86802	0.313152
$994$	0	0
$995$	21.5759	0.684001
$996$	0	0
$997$	32.3042	1.02309	0.511543	−	0.859258i	$-0.329074\pi$
$997$	32.3042	1.02309	0.511543	+	0.859258i	$0.329074\pi$
$998$	0	0
$999$	4.50189	0.142433

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2960.2.a.bc.1.4		6
4.3	odd	2		1480.2.a.k.1.3	✓	6
20.19	odd	2		7400.2.a.s.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1480.2.a.k.1.3	✓	6	4.3	odd	2
2960.2.a.bc.1.4		6	1.1	even	1	trivial
7400.2.a.s.1.4		6	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2960 = 2^{4} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2960.a (trivial)

\(f(q)\)	\(=\)	\(q+0.854190 q^{3} +1.00000 q^{5} -3.23895 q^{7} -2.27036 q^{9} +O(q^{10})\)	\(q+0.854190 q^{3} +1.00000 q^{5} -3.23895 q^{7} -2.27036 q^{9} -6.36999 q^{11} +5.87920 q^{13} +0.854190 q^{15} -1.09963 q^{17} +1.96671 q^{19} -2.76668 q^{21} +8.26208 q^{23} +1.00000 q^{25} -4.50189 q^{27} +1.07119 q^{29} +1.84779 q^{31} -5.44118 q^{33} -3.23895 q^{35} -1.00000 q^{37} +5.02195 q^{39} +8.14495 q^{41} +11.4567 q^{43} -2.27036 q^{45} -7.73987 q^{47} +3.49079 q^{49} -0.939290 q^{51} +10.0784 q^{53} -6.36999 q^{55} +1.67994 q^{57} -4.85239 q^{59} +2.77957 q^{61} +7.35358 q^{63} +5.87920 q^{65} +8.52041 q^{67} +7.05738 q^{69} +5.44118 q^{71} +14.7914 q^{73} +0.854190 q^{75} +20.6321 q^{77} +6.90600 q^{79} +2.96561 q^{81} -15.0905 q^{83} -1.09963 q^{85} +0.914999 q^{87} -10.3121 q^{89} -19.0424 q^{91} +1.57836 q^{93} +1.96671 q^{95} +13.6946 q^{97} +14.4622 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{3} + 6 q^{5} - 8 q^{7} + 15 q^{9}+O(q^{10}) \) 6 * q - q^3 + 6 * q^5 - 8 * q^7 + 15 * q^9	\( 6 q - q^{3} + 6 q^{5} - 8 q^{7} + 15 q^{9} + 10 q^{13} - q^{15} + 3 q^{17} + 6 q^{19} + 9 q^{21} - 4 q^{23} + 6 q^{25} - 7 q^{27} + 3 q^{29} + 3 q^{31} + 9 q^{33} - 8 q^{35} - 6 q^{37} + 16 q^{39} + 10 q^{41} + 11 q^{43} + 15 q^{45} - 23 q^{47} + 8 q^{49} + 16 q^{51} + 10 q^{53} + 4 q^{57} - 6 q^{59} + q^{61} - 23 q^{63} + 10 q^{65} + 4 q^{67} - 2 q^{69} - 9 q^{71} + 11 q^{73} - q^{75} + 32 q^{77} + 14 q^{79} + 46 q^{81} - 11 q^{83} + 3 q^{85} - 32 q^{87} + 30 q^{89} + 4 q^{91} + 2 q^{93} + 6 q^{95} + 29 q^{97} + 39 q^{99}+O(q^{100}) \) 6 * q - q^3 + 6 * q^5 - 8 * q^7 + 15 * q^9 + 10 * q^13 - q^15 + 3 * q^17 + 6 * q^19 + 9 * q^21 - 4 * q^23 + 6 * q^25 - 7 * q^27 + 3 * q^29 + 3 * q^31 + 9 * q^33 - 8 * q^35 - 6 * q^37 + 16 * q^39 + 10 * q^41 + 11 * q^43 + 15 * q^45 - 23 * q^47 + 8 * q^49 + 16 * q^51 + 10 * q^53 + 4 * q^57 - 6 * q^59 + q^61 - 23 * q^63 + 10 * q^65 + 4 * q^67 - 2 * q^69 - 9 * q^71 + 11 * q^73 - q^75 + 32 * q^77 + 14 * q^79 + 46 * q^81 - 11 * q^83 + 3 * q^85 - 32 * q^87 + 30 * q^89 + 4 * q^91 + 2 * q^93 + 6 * q^95 + 29 * q^97 + 39 * q^99

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