LMFDB - Embedded newform 2960.2.a.bb.1.3

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:	$5$
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:	no (minimal twist has level 1480)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.592644$ 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)

$f(q)$	$=$	$q+1.59264 q^{3} +1.00000 q^{5} -4.50235 q^{7} -0.463486 q^{9} +O(q^{10})$	$q+1.59264 q^{3} +1.00000 q^{5} -4.50235 q^{7} -0.463486 q^{9} +5.01600 q^{11} -0.0423816 q^{13} +1.59264 q^{15} -3.74503 q^{17} +5.42336 q^{19} -7.17064 q^{21} +4.97270 q^{23} +1.00000 q^{25} -5.51610 q^{27} +4.97270 q^{29} +4.62591 q^{31} +7.98870 q^{33} -4.50235 q^{35} -1.00000 q^{37} -0.0674988 q^{39} +5.01600 q^{41} -0.112259 q^{43} -0.463486 q^{45} +1.60991 q^{47} +13.2712 q^{49} -5.96450 q^{51} -10.8510 q^{53} +5.01600 q^{55} +8.63748 q^{57} +9.19351 q^{59} -8.80768 q^{61} +2.08678 q^{63} -0.0423816 q^{65} -0.203540 q^{67} +7.91974 q^{69} +13.3710 q^{71} +7.45311 q^{73} +1.59264 q^{75} -22.5838 q^{77} -6.13604 q^{79} -7.39472 q^{81} +17.6979 q^{83} -3.74503 q^{85} +7.91974 q^{87} +17.1170 q^{89} +0.190817 q^{91} +7.36742 q^{93} +5.42336 q^{95} -4.13233 q^{97} -2.32485 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.59264	0.919513	0.459757	−	0.888045i	$-0.347937\pi$
$3$	1.59264	0.919513	0.459757	+	0.888045i	$0.347937\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−4.50235	−1.70173	−0.850864	−	0.525385i	$-0.823921\pi$
$7$	−4.50235	−1.70173	−0.850864	+	0.525385i	$0.823921\pi$
$8$	0	0
$9$	−0.463486	−0.154495
$10$	0	0
$11$	5.01600	1.51238	0.756191	−	0.654352i	$-0.227058\pi$
$11$	5.01600	1.51238	0.756191	+	0.654352i	$0.227058\pi$
$12$	0	0
$13$	−0.0423816	−0.0117545	−0.00587727	−	0.999983i	$-0.501871\pi$
$13$	−0.0423816	−0.0117545	−0.00587727	+	0.999983i	$0.501871\pi$
$14$	0	0
$15$	1.59264	0.411219
$16$	0	0
$17$	−3.74503	−0.908304	−0.454152	−	0.890924i	$-0.650058\pi$
$17$	−3.74503	−0.908304	−0.454152	+	0.890924i	$0.650058\pi$
$18$	0	0
$19$	5.42336	1.24420	0.622102	−	0.782936i	$-0.286279\pi$
$19$	5.42336	1.24420	0.622102	+	0.782936i	$0.286279\pi$
$20$	0	0
$21$	−7.17064	−1.56476
$22$	0	0
$23$	4.97270	1.03688	0.518440	−	0.855114i	$-0.326513\pi$
$23$	4.97270	1.03688	0.518440	+	0.855114i	$0.326513\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.51610	−1.06157
$28$	0	0
$29$	4.97270	0.923407	0.461704	−	0.887034i	$-0.347238\pi$
$29$	4.97270	0.923407	0.461704	+	0.887034i	$0.347238\pi$
$30$	0	0
$31$	4.62591	0.830838	0.415419	−	0.909630i	$-0.363635\pi$
$31$	4.62591	0.830838	0.415419	+	0.909630i	$0.363635\pi$
$32$	0	0
$33$	7.98870	1.39065
$34$	0	0
$35$	−4.50235	−0.761036
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	0	0
$39$	−0.0674988	−0.0108085
$40$	0	0
$41$	5.01600	0.783368	0.391684	−	0.920100i	$-0.371893\pi$
$41$	5.01600	0.783368	0.391684	+	0.920100i	$0.371893\pi$
$42$	0	0
$43$	−0.112259	−0.0171194	−0.00855969	−	0.999963i	$-0.502725\pi$
$43$	−0.112259	−0.0171194	−0.00855969	+	0.999963i	$0.502725\pi$
$44$	0	0
$45$	−0.463486	−0.0690924
$46$	0	0
$47$	1.60991	0.234829	0.117415	−	0.993083i	$-0.462539\pi$
$47$	1.60991	0.234829	0.117415	+	0.993083i	$0.462539\pi$
$48$	0	0
$49$	13.2712	1.89588
$50$	0	0
$51$	−5.96450	−0.835197
$52$	0	0
$53$	−10.8510	−1.49050	−0.745248	−	0.666787i	$-0.767669\pi$
$53$	−10.8510	−1.49050	−0.745248	+	0.666787i	$0.767669\pi$
$54$	0	0
$55$	5.01600	0.676357
$56$	0	0
$57$	8.63748	1.14406
$58$	0	0
$59$	9.19351	1.19689	0.598446	−	0.801163i	$-0.295785\pi$
$59$	9.19351	1.19689	0.598446	+	0.801163i	$0.295785\pi$
$60$	0	0
$61$	−8.80768	−1.12771	−0.563854	−	0.825875i	$-0.690682\pi$
$61$	−8.80768	−1.12771	−0.563854	+	0.825875i	$0.690682\pi$
$62$	0	0
$63$	2.08678	0.262909
$64$	0	0
$65$	−0.0423816	−0.00525679
$66$	0	0
$67$	−0.203540	−0.0248664	−0.0124332	−	0.999923i	$-0.503958\pi$
$67$	−0.203540	−0.0248664	−0.0124332	+	0.999923i	$0.503958\pi$
$68$	0	0
$69$	7.91974	0.953425
$70$	0	0
$71$	13.3710	1.58685	0.793424	−	0.608670i	$-0.208297\pi$
$71$	13.3710	1.58685	0.793424	+	0.608670i	$0.208297\pi$
$72$	0	0
$73$	7.45311	0.872320	0.436160	−	0.899869i	$-0.356338\pi$
$73$	7.45311	0.872320	0.436160	+	0.899869i	$0.356338\pi$
$74$	0	0
$75$	1.59264	0.183903
$76$	0	0
$77$	−22.5838	−2.57366
$78$	0	0
$79$	−6.13604	−0.690359	−0.345179	−	0.938537i	$-0.612182\pi$
$79$	−6.13604	−0.690359	−0.345179	+	0.938537i	$0.612182\pi$
$80$	0	0
$81$	−7.39472	−0.821636
$82$	0	0
$83$	17.6979	1.94259	0.971297	−	0.237871i	$-0.0764495\pi$
$83$	17.6979	1.94259	0.971297	+	0.237871i	$0.0764495\pi$
$84$	0	0
$85$	−3.74503	−0.406206
$86$	0	0
$87$	7.91974	0.849085
$88$	0	0
$89$	17.1170	1.81439	0.907197	−	0.420706i	$-0.138218\pi$
$89$	17.1170	1.81439	0.907197	+	0.420706i	$0.138218\pi$
$90$	0	0
$91$	0.190817	0.0200030
$92$	0	0
$93$	7.36742	0.763966
$94$	0	0
$95$	5.42336	0.556425
$96$	0	0
$97$	−4.13233	−0.419574	−0.209787	−	0.977747i	$-0.567277\pi$
$97$	−4.13233	−0.419574	−0.209787	+	0.977747i	$0.567277\pi$
$98$	0	0
$99$	−2.32485	−0.233656
$100$	0	0
$101$	−5.09961	−0.507430	−0.253715	−	0.967279i	$-0.581653\pi$
$101$	−5.09961	−0.507430	−0.253715	+	0.967279i	$0.581653\pi$
$102$	0	0
$103$	−5.57483	−0.549304	−0.274652	−	0.961544i	$-0.588563\pi$
$103$	−5.57483	−0.549304	−0.274652	+	0.961544i	$0.588563\pi$
$104$	0	0
$105$	−7.17064	−0.699783
$106$	0	0
$107$	17.6507	1.70636	0.853178	−	0.521620i	$-0.174672\pi$
$107$	17.6507	1.70636	0.853178	+	0.521620i	$0.174672\pi$
$108$	0	0
$109$	4.92697	0.471918	0.235959	−	0.971763i	$-0.424177\pi$
$109$	4.92697	0.471918	0.235959	+	0.971763i	$0.424177\pi$
$110$	0	0
$111$	−1.59264	−0.151167
$112$	0	0
$113$	−3.61721	−0.340278	−0.170139	−	0.985420i	$-0.554422\pi$
$113$	−3.61721	−0.340278	−0.170139	+	0.985420i	$0.554422\pi$
$114$	0	0
$115$	4.97270	0.463707
$116$	0	0
$117$	0.0196433	0.00181602
$118$	0	0
$119$	16.8614	1.54569
$120$	0	0
$121$	14.1603	1.28730
$122$	0	0
$123$	7.98870	0.720317
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−6.16208	−0.546796	−0.273398	−	0.961901i	$-0.588148\pi$
$127$	−6.16208	−0.546796	−0.273398	+	0.961901i	$0.588148\pi$
$128$	0	0
$129$	−0.178789	−0.0157415
$130$	0	0
$131$	−5.85863	−0.511871	−0.255935	−	0.966694i	$-0.582383\pi$
$131$	−5.85863	−0.511871	−0.255935	+	0.966694i	$0.582383\pi$
$132$	0	0
$133$	−24.4179	−2.11730
$134$	0	0
$135$	−5.51610	−0.474750
$136$	0	0
$137$	5.93167	0.506777	0.253389	−	0.967365i	$-0.418455\pi$
$137$	5.93167	0.506777	0.253389	+	0.967365i	$0.418455\pi$
$138$	0	0
$139$	16.5338	1.40238	0.701189	−	0.712975i	$-0.252653\pi$
$139$	16.5338	1.40238	0.701189	+	0.712975i	$0.252653\pi$
$140$	0	0
$141$	2.56401	0.215929
$142$	0	0
$143$	−0.212586	−0.0177773
$144$	0	0
$145$	4.97270	0.412960
$146$	0	0
$147$	21.1362	1.74329
$148$	0	0
$149$	11.9854	0.981878	0.490939	−	0.871194i	$-0.336654\pi$
$149$	11.9854	0.981878	0.490939	+	0.871194i	$0.336654\pi$
$150$	0	0
$151$	12.7074	1.03411	0.517055	−	0.855952i	$-0.327028\pi$
$151$	12.7074	1.03411	0.517055	+	0.855952i	$0.327028\pi$
$152$	0	0
$153$	1.73577	0.140329
$154$	0	0
$155$	4.62591	0.371562
$156$	0	0
$157$	−19.2060	−1.53280	−0.766402	−	0.642361i	$-0.777955\pi$
$157$	−19.2060	−1.53280	−0.766402	+	0.642361i	$0.777955\pi$
$158$	0	0
$159$	−17.2817	−1.37053
$160$	0	0
$161$	−22.3888	−1.76449
$162$	0	0
$163$	−4.55387	−0.356687	−0.178343	−	0.983968i	$-0.557074\pi$
$163$	−4.55387	−0.356687	−0.178343	+	0.983968i	$0.557074\pi$
$164$	0	0
$165$	7.98870	0.621920
$166$	0	0
$167$	14.8724	1.15086	0.575429	−	0.817851i	$-0.304835\pi$
$167$	14.8724	1.15086	0.575429	+	0.817851i	$0.304835\pi$
$168$	0	0
$169$	−12.9982	−0.999862
$170$	0	0
$171$	−2.51365	−0.192224
$172$	0	0
$173$	10.1210	0.769488	0.384744	−	0.923023i	$-0.374290\pi$
$173$	10.1210	0.769488	0.384744	+	0.923023i	$0.374290\pi$
$174$	0	0
$175$	−4.50235	−0.340346
$176$	0	0
$177$	14.6420	1.10056
$178$	0	0
$179$	−19.6018	−1.46511	−0.732554	−	0.680709i	$-0.761672\pi$
$179$	−19.6018	−1.46511	−0.732554	+	0.680709i	$0.761672\pi$
$180$	0	0
$181$	10.9123	0.811107	0.405553	−	0.914071i	$-0.367079\pi$
$181$	10.9123	0.811107	0.405553	+	0.914071i	$0.367079\pi$
$182$	0	0
$183$	−14.0275	−1.03694
$184$	0	0
$185$	−1.00000	−0.0735215
$186$	0	0
$187$	−18.7851	−1.37370
$188$	0	0
$189$	24.8354	1.80651
$190$	0	0
$191$	−3.64884	−0.264021	−0.132011	−	0.991248i	$-0.542143\pi$
$191$	−3.64884	−0.264021	−0.132011	+	0.991248i	$0.542143\pi$
$192$	0	0
$193$	−13.6388	−0.981744	−0.490872	−	0.871232i	$-0.663322\pi$
$193$	−13.6388	−0.981744	−0.490872	+	0.871232i	$0.663322\pi$
$194$	0	0
$195$	−0.0674988	−0.00483369
$196$	0	0
$197$	−10.1556	−0.723554	−0.361777	−	0.932265i	$-0.617830\pi$
$197$	−10.1556	−0.723554	−0.361777	+	0.932265i	$0.617830\pi$
$198$	0	0
$199$	−19.1149	−1.35502	−0.677511	−	0.735512i	$-0.736942\pi$
$199$	−19.1149	−1.35502	−0.677511	+	0.735512i	$0.736942\pi$
$200$	0	0
$201$	−0.324167	−0.0228650
$202$	0	0
$203$	−22.3888	−1.57139
$204$	0	0
$205$	5.01600	0.350333
$206$	0	0
$207$	−2.30478	−0.160193
$208$	0	0
$209$	27.2036	1.88171
$210$	0	0
$211$	−23.7554	−1.63539	−0.817693	−	0.575655i	$-0.804747\pi$
$211$	−23.7554	−1.63539	−0.817693	+	0.575655i	$0.804747\pi$
$212$	0	0
$213$	21.2953	1.45913
$214$	0	0
$215$	−0.112259	−0.00765602
$216$	0	0
$217$	−20.8275	−1.41386
$218$	0	0
$219$	11.8701	0.802110
$220$	0	0
$221$	0.158720	0.0106767
$222$	0	0
$223$	4.44119	0.297404	0.148702	−	0.988882i	$-0.452490\pi$
$223$	4.44119	0.297404	0.148702	+	0.988882i	$0.452490\pi$
$224$	0	0
$225$	−0.463486	−0.0308991
$226$	0	0
$227$	8.06653	0.535394	0.267697	−	0.963503i	$-0.413737\pi$
$227$	8.06653	0.535394	0.267697	+	0.963503i	$0.413737\pi$
$228$	0	0
$229$	−22.7783	−1.50523	−0.752616	−	0.658460i	$-0.771208\pi$
$229$	−22.7783	−1.50523	−0.752616	+	0.658460i	$0.771208\pi$
$230$	0	0
$231$	−35.9679	−2.36652
$232$	0	0
$233$	−5.67263	−0.371626	−0.185813	−	0.982585i	$-0.559492\pi$
$233$	−5.67263	−0.371626	−0.185813	+	0.982585i	$0.559492\pi$
$234$	0	0
$235$	1.60991	0.105019
$236$	0	0
$237$	−9.77253	−0.634794
$238$	0	0
$239$	−17.5676	−1.13635	−0.568177	−	0.822906i	$-0.692351\pi$
$239$	−17.5676	−1.13635	−0.568177	+	0.822906i	$0.692351\pi$
$240$	0	0
$241$	13.9317	0.897418	0.448709	−	0.893678i	$-0.351884\pi$
$241$	13.9317	0.897418	0.448709	+	0.893678i	$0.351884\pi$
$242$	0	0
$243$	4.77114	0.306069
$244$	0	0
$245$	13.2712	0.847864
$246$	0	0
$247$	−0.229851	−0.0146250
$248$	0	0
$249$	28.1864	1.78624
$250$	0	0
$251$	−30.0917	−1.89937	−0.949685	−	0.313207i	$-0.898597\pi$
$251$	−30.0917	−1.89937	−0.949685	+	0.313207i	$0.898597\pi$
$252$	0	0
$253$	24.9431	1.56816
$254$	0	0
$255$	−5.96450	−0.373512
$256$	0	0
$257$	−15.3701	−0.958760	−0.479380	−	0.877607i	$-0.659138\pi$
$257$	−15.3701	−0.958760	−0.479380	+	0.877607i	$0.659138\pi$
$258$	0	0
$259$	4.50235	0.279762
$260$	0	0
$261$	−2.30478	−0.142662
$262$	0	0
$263$	1.94766	0.120098	0.0600488	−	0.998195i	$-0.480874\pi$
$263$	1.94766	0.120098	0.0600488	+	0.998195i	$0.480874\pi$
$264$	0	0
$265$	−10.8510	−0.666570
$266$	0	0
$267$	27.2612	1.66836
$268$	0	0
$269$	−12.3145	−0.750826	−0.375413	−	0.926858i	$-0.622499\pi$
$269$	−12.3145	−0.750826	−0.375413	+	0.926858i	$0.622499\pi$
$270$	0	0
$271$	−21.5055	−1.30637	−0.653184	−	0.757199i	$-0.726567\pi$
$271$	−21.5055	−1.30637	−0.653184	+	0.757199i	$0.726567\pi$
$272$	0	0
$273$	0.303903	0.0183931
$274$	0	0
$275$	5.01600	0.302476
$276$	0	0
$277$	3.60528	0.216620	0.108310	−	0.994117i	$-0.465456\pi$
$277$	3.60528	0.216620	0.108310	+	0.994117i	$0.465456\pi$
$278$	0	0
$279$	−2.14404	−0.128361
$280$	0	0
$281$	−21.7157	−1.29545	−0.647725	−	0.761875i	$-0.724279\pi$
$281$	−21.7157	−1.29545	−0.647725	+	0.761875i	$0.724279\pi$
$282$	0	0
$283$	33.2657	1.97744	0.988721	−	0.149768i	$-0.0478528\pi$
$283$	33.2657	1.97744	0.988721	+	0.149768i	$0.0478528\pi$
$284$	0	0
$285$	8.63748	0.511640
$286$	0	0
$287$	−22.5838	−1.33308
$288$	0	0
$289$	−2.97473	−0.174984
$290$	0	0
$291$	−6.58133	−0.385804
$292$	0	0
$293$	26.4145	1.54315	0.771576	−	0.636137i	$-0.219469\pi$
$293$	26.4145	1.54315	0.771576	+	0.636137i	$0.219469\pi$
$294$	0	0
$295$	9.19351	0.535267
$296$	0	0
$297$	−27.6688	−1.60550
$298$	0	0
$299$	−0.210751	−0.0121881
$300$	0	0
$301$	0.505431	0.0291325
$302$	0	0
$303$	−8.12186	−0.466588
$304$	0	0
$305$	−8.80768	−0.504326
$306$	0	0
$307$	−1.44489	−0.0824640	−0.0412320	−	0.999150i	$-0.513128\pi$
$307$	−1.44489	−0.0824640	−0.0412320	+	0.999150i	$0.513128\pi$
$308$	0	0
$309$	−8.87871	−0.505092
$310$	0	0
$311$	23.8734	1.35373	0.676867	−	0.736105i	$-0.263337\pi$
$311$	23.8734	1.35373	0.676867	+	0.736105i	$0.263337\pi$
$312$	0	0
$313$	13.0895	0.739860	0.369930	−	0.929060i	$-0.379382\pi$
$313$	13.0895	0.739860	0.369930	+	0.929060i	$0.379382\pi$
$314$	0	0
$315$	2.08678	0.117577
$316$	0	0
$317$	−13.9112	−0.781331	−0.390666	−	0.920533i	$-0.627755\pi$
$317$	−13.9112	−0.781331	−0.390666	+	0.920533i	$0.627755\pi$
$318$	0	0
$319$	24.9431	1.39654
$320$	0	0
$321$	28.1112	1.56902
$322$	0	0
$323$	−20.3106	−1.13011
$324$	0	0
$325$	−0.0423816	−0.00235091
$326$	0	0
$327$	7.84691	0.433935
$328$	0	0
$329$	−7.24837	−0.399616
$330$	0	0
$331$	−2.44882	−0.134599	−0.0672997	−	0.997733i	$-0.521438\pi$
$331$	−2.44882	−0.134599	−0.0672997	+	0.997733i	$0.521438\pi$
$332$	0	0
$333$	0.463486	0.0253989
$334$	0	0
$335$	−0.203540	−0.0111206
$336$	0	0
$337$	34.3987	1.87382	0.936908	−	0.349575i	$-0.113674\pi$
$337$	34.3987	1.87382	0.936908	+	0.349575i	$0.113674\pi$
$338$	0	0
$339$	−5.76093	−0.312891
$340$	0	0
$341$	23.2036	1.25654
$342$	0	0
$343$	−28.2350	−1.52455
$344$	0	0
$345$	7.91974	0.426385
$346$	0	0
$347$	4.92916	0.264611	0.132306	−	0.991209i	$-0.457762\pi$
$347$	4.92916	0.264611	0.132306	+	0.991209i	$0.457762\pi$
$348$	0	0
$349$	−15.3951	−0.824079	−0.412039	−	0.911166i	$-0.635183\pi$
$349$	−15.3951	−0.824079	−0.412039	+	0.911166i	$0.635183\pi$
$350$	0	0
$351$	0.233781	0.0124783
$352$	0	0
$353$	−1.82194	−0.0969722	−0.0484861	−	0.998824i	$-0.515440\pi$
$353$	−1.82194	−0.0969722	−0.0484861	+	0.998824i	$0.515440\pi$
$354$	0	0
$355$	13.3710	0.709660
$356$	0	0
$357$	26.8543	1.42128
$358$	0	0
$359$	22.3202	1.17802	0.589009	−	0.808127i	$-0.299518\pi$
$359$	22.3202	1.17802	0.589009	+	0.808127i	$0.299518\pi$
$360$	0	0
$361$	10.4128	0.548042
$362$	0	0
$363$	22.5523	1.18369
$364$	0	0
$365$	7.45311	0.390113
$366$	0	0
$367$	−35.4420	−1.85006	−0.925028	−	0.379898i	$-0.875959\pi$
$367$	−35.4420	−1.85006	−0.925028	+	0.379898i	$0.875959\pi$
$368$	0	0
$369$	−2.32485	−0.121027
$370$	0	0
$371$	48.8549	2.53642
$372$	0	0
$373$	17.1859	0.889853	0.444927	−	0.895567i	$-0.353230\pi$
$373$	17.1859	0.889853	0.444927	+	0.895567i	$0.353230\pi$
$374$	0	0
$375$	1.59264	0.0822438
$376$	0	0
$377$	−0.210751	−0.0108542
$378$	0	0
$379$	−18.8886	−0.970241	−0.485120	−	0.874447i	$-0.661224\pi$
$379$	−18.8886	−0.970241	−0.485120	+	0.874447i	$0.661224\pi$
$380$	0	0
$381$	−9.81399	−0.502786
$382$	0	0
$383$	−8.49437	−0.434042	−0.217021	−	0.976167i	$-0.569634\pi$
$383$	−8.49437	−0.434042	−0.217021	+	0.976167i	$0.569634\pi$
$384$	0	0
$385$	−22.5838	−1.15098
$386$	0	0
$387$	0.0520306	0.00264486
$388$	0	0
$389$	8.99097	0.455860	0.227930	−	0.973677i	$-0.426804\pi$
$389$	8.99097	0.455860	0.227930	+	0.973677i	$0.426804\pi$
$390$	0	0
$391$	−18.6229	−0.941802
$392$	0	0
$393$	−9.33070	−0.470672
$394$	0	0
$395$	−6.13604	−0.308738
$396$	0	0
$397$	−2.87698	−0.144391	−0.0721957	−	0.997390i	$-0.523001\pi$
$397$	−2.87698	−0.144391	−0.0721957	+	0.997390i	$0.523001\pi$
$398$	0	0
$399$	−38.8889	−1.94688
$400$	0	0
$401$	7.78489	0.388759	0.194379	−	0.980926i	$-0.437731\pi$
$401$	7.78489	0.388759	0.194379	+	0.980926i	$0.437731\pi$
$402$	0	0
$403$	−0.196053	−0.00976612
$404$	0	0
$405$	−7.39472	−0.367447
$406$	0	0
$407$	−5.01600	−0.248634
$408$	0	0
$409$	9.69631	0.479452	0.239726	−	0.970841i	$-0.422942\pi$
$409$	9.69631	0.479452	0.239726	+	0.970841i	$0.422942\pi$
$410$	0	0
$411$	9.44704	0.465988
$412$	0	0
$413$	−41.3924	−2.03679
$414$	0	0
$415$	17.6979	0.868754
$416$	0	0
$417$	26.3325	1.28951
$418$	0	0
$419$	−38.8423	−1.89757	−0.948785	−	0.315922i	$-0.897686\pi$
$419$	−38.8423	−1.89757	−0.948785	+	0.315922i	$0.897686\pi$
$420$	0	0
$421$	−9.73212	−0.474315	−0.237157	−	0.971471i	$-0.576216\pi$
$421$	−9.73212	−0.474315	−0.237157	+	0.971471i	$0.576216\pi$
$422$	0	0
$423$	−0.746170	−0.0362800
$424$	0	0
$425$	−3.74503	−0.181661
$426$	0	0
$427$	39.6553	1.91905
$428$	0	0
$429$	−0.338574	−0.0163465
$430$	0	0
$431$	30.4974	1.46901	0.734504	−	0.678605i	$-0.237415\pi$
$431$	30.4974	1.46901	0.734504	+	0.678605i	$0.237415\pi$
$432$	0	0
$433$	6.12103	0.294158	0.147079	−	0.989125i	$-0.453013\pi$
$433$	6.12103	0.294158	0.147079	+	0.989125i	$0.453013\pi$
$434$	0	0
$435$	7.91974	0.379723
$436$	0	0
$437$	26.9687	1.29009
$438$	0	0
$439$	−13.8560	−0.661309	−0.330654	−	0.943752i	$-0.607269\pi$
$439$	−13.8560	−0.661309	−0.330654	+	0.943752i	$0.607269\pi$
$440$	0	0
$441$	−6.15100	−0.292905
$442$	0	0
$443$	−10.5277	−0.500184	−0.250092	−	0.968222i	$-0.580461\pi$
$443$	−10.5277	−0.500184	−0.250092	+	0.968222i	$0.580461\pi$
$444$	0	0
$445$	17.1170	0.811422
$446$	0	0
$447$	19.0884	0.902850
$448$	0	0
$449$	18.5450	0.875193	0.437596	−	0.899172i	$-0.355830\pi$
$449$	18.5450	0.875193	0.437596	+	0.899172i	$0.355830\pi$
$450$	0	0
$451$	25.1603	1.18475
$452$	0	0
$453$	20.2383	0.950878
$454$	0	0
$455$	0.190817	0.00894563
$456$	0	0
$457$	24.1737	1.13080	0.565400	−	0.824817i	$-0.308722\pi$
$457$	24.1737	1.13080	0.565400	+	0.824817i	$0.308722\pi$
$458$	0	0
$459$	20.6580	0.964231
$460$	0	0
$461$	−2.32737	−0.108397	−0.0541983	−	0.998530i	$-0.517260\pi$
$461$	−2.32737	−0.108397	−0.0541983	+	0.998530i	$0.517260\pi$
$462$	0	0
$463$	33.3460	1.54972	0.774860	−	0.632133i	$-0.217820\pi$
$463$	33.3460	1.54972	0.774860	+	0.632133i	$0.217820\pi$
$464$	0	0
$465$	7.36742	0.341656
$466$	0	0
$467$	20.0405	0.927365	0.463683	−	0.886001i	$-0.346528\pi$
$467$	20.0405	0.927365	0.463683	+	0.886001i	$0.346528\pi$
$468$	0	0
$469$	0.916410	0.0423159
$470$	0	0
$471$	−30.5883	−1.40943
$472$	0	0
$473$	−0.563093	−0.0258910
$474$	0	0
$475$	5.42336	0.248841
$476$	0	0
$477$	5.02928	0.230275
$478$	0	0
$479$	−9.14244	−0.417729	−0.208864	−	0.977945i	$-0.566977\pi$
$479$	−9.14244	−0.417729	−0.208864	+	0.977945i	$0.566977\pi$
$480$	0	0
$481$	0.0423816	0.00193244
$482$	0	0
$483$	−35.6575	−1.62247
$484$	0	0
$485$	−4.13233	−0.187639
$486$	0	0
$487$	−6.04646	−0.273991	−0.136996	−	0.990572i	$-0.543745\pi$
$487$	−6.04646	−0.273991	−0.136996	+	0.990572i	$0.543745\pi$
$488$	0	0
$489$	−7.25269	−0.327978
$490$	0	0
$491$	−4.38395	−0.197845	−0.0989224	−	0.995095i	$-0.531540\pi$
$491$	−4.38395	−0.197845	−0.0989224	+	0.995095i	$0.531540\pi$
$492$	0	0
$493$	−18.6229	−0.838734
$494$	0	0
$495$	−2.32485	−0.104494
$496$	0	0
$497$	−60.2010	−2.70038
$498$	0	0
$499$	1.01021	0.0452233	0.0226117	−	0.999744i	$-0.492802\pi$
$499$	1.01021	0.0452233	0.0226117	+	0.999744i	$0.492802\pi$
$500$	0	0
$501$	23.6864	1.05823
$502$	0	0
$503$	−18.9199	−0.843598	−0.421799	−	0.906689i	$-0.638601\pi$
$503$	−18.9199	−0.843598	−0.421799	+	0.906689i	$0.638601\pi$
$504$	0	0
$505$	−5.09961	−0.226930
$506$	0	0
$507$	−20.7015	−0.919386
$508$	0	0
$509$	−20.3956	−0.904021	−0.452010	−	0.892013i	$-0.649293\pi$
$509$	−20.3956	−0.904021	−0.452010	+	0.892013i	$0.649293\pi$
$510$	0	0
$511$	−33.5565	−1.48445
$512$	0	0
$513$	−29.9158	−1.32081
$514$	0	0
$515$	−5.57483	−0.245656
$516$	0	0
$517$	8.07530	0.355151
$518$	0	0
$519$	16.1192	0.707554
$520$	0	0
$521$	−13.1431	−0.575812	−0.287906	−	0.957659i	$-0.592959\pi$
$521$	−13.1431	−0.575812	−0.287906	+	0.957659i	$0.592959\pi$
$522$	0	0
$523$	−32.7479	−1.43197	−0.715983	−	0.698118i	$-0.754021\pi$
$523$	−32.7479	−1.43197	−0.715983	+	0.698118i	$0.754021\pi$
$524$	0	0
$525$	−7.17064	−0.312952
$526$	0	0
$527$	−17.3242	−0.754653
$528$	0	0
$529$	1.72776	0.0751199
$530$	0	0
$531$	−4.26106	−0.184914
$532$	0	0
$533$	−0.212586	−0.00920813
$534$	0	0
$535$	17.6507	0.763105
$536$	0	0
$537$	−31.2187	−1.34719
$538$	0	0
$539$	66.5682	2.86729
$540$	0	0
$541$	−34.3959	−1.47880	−0.739398	−	0.673268i	$-0.764890\pi$
$541$	−34.3959	−1.47880	−0.739398	+	0.673268i	$0.764890\pi$
$542$	0	0
$543$	17.3794	0.745823
$544$	0	0
$545$	4.92697	0.211048
$546$	0	0
$547$	−13.0683	−0.558761	−0.279380	−	0.960181i	$-0.590129\pi$
$547$	−13.0683	−0.558761	−0.279380	+	0.960181i	$0.590129\pi$
$548$	0	0
$549$	4.08224	0.174226
$550$	0	0
$551$	26.9687	1.14891
$552$	0	0
$553$	27.6266	1.17480
$554$	0	0
$555$	−1.59264	−0.0676040
$556$	0	0
$557$	−2.85695	−0.121053	−0.0605265	−	0.998167i	$-0.519278\pi$
$557$	−2.85695	−0.121053	−0.0605265	+	0.998167i	$0.519278\pi$
$558$	0	0
$559$	0.00475773	0.000201231 0
$560$	0	0
$561$	−29.9179	−1.26314
$562$	0	0
$563$	−31.8292	−1.34144	−0.670719	−	0.741711i	$-0.734015\pi$
$563$	−31.8292	−1.34144	−0.670719	+	0.741711i	$0.734015\pi$
$564$	0	0
$565$	−3.61721	−0.152177
$566$	0	0
$567$	33.2936	1.39820
$568$	0	0
$569$	33.7191	1.41358	0.706789	−	0.707425i	$-0.250143\pi$
$569$	33.7191	1.41358	0.706789	+	0.707425i	$0.250143\pi$
$570$	0	0
$571$	16.9889	0.710963	0.355482	−	0.934683i	$-0.384317\pi$
$571$	16.9889	0.710963	0.355482	+	0.934683i	$0.384317\pi$
$572$	0	0
$573$	−5.81131	−0.242771
$574$	0	0
$575$	4.97270	0.207376
$576$	0	0
$577$	−7.17670	−0.298770	−0.149385	−	0.988779i	$-0.547729\pi$
$577$	−7.17670	−0.298770	−0.149385	+	0.988779i	$0.547729\pi$
$578$	0	0
$579$	−21.7218	−0.902727
$580$	0	0
$581$	−79.6820	−3.30577
$582$	0	0
$583$	−54.4285	−2.25420
$584$	0	0
$585$	0.0196433	0.000812150 0
$586$	0	0
$587$	−40.5510	−1.67372	−0.836860	−	0.547418i	$-0.815611\pi$
$587$	−40.5510	−1.67372	−0.836860	+	0.547418i	$0.815611\pi$
$588$	0	0
$589$	25.0880	1.03373
$590$	0	0
$591$	−16.1742	−0.665317
$592$	0	0
$593$	−42.9705	−1.76458	−0.882292	−	0.470702i	$-0.844001\pi$
$593$	−42.9705	−1.76458	−0.882292	+	0.470702i	$0.844001\pi$
$594$	0	0
$595$	16.8614	0.691252
$596$	0	0
$597$	−30.4433	−1.24596
$598$	0	0
$599$	10.2723	0.419716	0.209858	−	0.977732i	$-0.432700\pi$
$599$	10.2723	0.419716	0.209858	+	0.977732i	$0.432700\pi$
$600$	0	0
$601$	4.12080	0.168091	0.0840456	−	0.996462i	$-0.473216\pi$
$601$	4.12080	0.168091	0.0840456	+	0.996462i	$0.473216\pi$
$602$	0	0
$603$	0.0943381	0.00384174
$604$	0	0
$605$	14.1603	0.575696
$606$	0	0
$607$	32.5387	1.32071	0.660353	−	0.750956i	$-0.270407\pi$
$607$	32.5387	1.32071	0.660353	+	0.750956i	$0.270407\pi$
$608$	0	0
$609$	−35.6575	−1.44491
$610$	0	0
$611$	−0.0682305	−0.00276031
$612$	0	0
$613$	26.0635	1.05270	0.526348	−	0.850269i	$-0.323561\pi$
$613$	26.0635	1.05270	0.526348	+	0.850269i	$0.323561\pi$
$614$	0	0
$615$	7.98870	0.322136
$616$	0	0
$617$	2.64509	0.106487	0.0532436	−	0.998582i	$-0.483044\pi$
$617$	2.64509	0.106487	0.0532436	+	0.998582i	$0.483044\pi$
$618$	0	0
$619$	40.1959	1.61561	0.807804	−	0.589451i	$-0.200656\pi$
$619$	40.1959	1.61561	0.807804	+	0.589451i	$0.200656\pi$
$620$	0	0
$621$	−27.4299	−1.10072
$622$	0	0
$623$	−77.0666	−3.08761
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	43.3256	1.73026
$628$	0	0
$629$	3.74503	0.149324
$630$	0	0
$631$	−20.6336	−0.821411	−0.410706	−	0.911768i	$-0.634718\pi$
$631$	−20.6336	−0.821411	−0.410706	+	0.911768i	$0.634718\pi$
$632$	0	0
$633$	−37.8338	−1.50376
$634$	0	0
$635$	−6.16208	−0.244535
$636$	0	0
$637$	−0.562453	−0.0222852
$638$	0	0
$639$	−6.19728	−0.245160
$640$	0	0
$641$	−13.9879	−0.552488	−0.276244	−	0.961088i	$-0.589090\pi$
$641$	−13.9879	−0.552488	−0.276244	+	0.961088i	$0.589090\pi$
$642$	0	0
$643$	−19.2404	−0.758768	−0.379384	−	0.925239i	$-0.623864\pi$
$643$	−19.2404	−0.758768	−0.379384	+	0.925239i	$0.623864\pi$
$644$	0	0
$645$	−0.178789	−0.00703981
$646$	0	0
$647$	−28.0394	−1.10234	−0.551172	−	0.834392i	$-0.685819\pi$
$647$	−28.0394	−1.10234	−0.551172	+	0.834392i	$0.685819\pi$
$648$	0	0
$649$	46.1146	1.81016
$650$	0	0
$651$	−33.1707	−1.30006
$652$	0	0
$653$	−11.0352	−0.431841	−0.215920	−	0.976411i	$-0.569275\pi$
$653$	−11.0352	−0.431841	−0.215920	+	0.976411i	$0.569275\pi$
$654$	0	0
$655$	−5.85863	−0.228915
$656$	0	0
$657$	−3.45441	−0.134769
$658$	0	0
$659$	35.2250	1.37217	0.686085	−	0.727522i	$-0.259328\pi$
$659$	35.2250	1.37217	0.686085	+	0.727522i	$0.259328\pi$
$660$	0	0
$661$	3.62528	0.141007	0.0705034	−	0.997512i	$-0.477539\pi$
$661$	3.62528	0.141007	0.0705034	+	0.997512i	$0.477539\pi$
$662$	0	0
$663$	0.252785	0.00981736
$664$	0	0
$665$	−24.4179	−0.946884
$666$	0	0
$667$	24.7278	0.957463
$668$	0	0
$669$	7.07324	0.273467
$670$	0	0
$671$	−44.1793	−1.70552
$672$	0	0
$673$	−31.3762	−1.20947	−0.604733	−	0.796428i	$-0.706720\pi$
$673$	−31.3762	−1.20947	−0.604733	+	0.796428i	$0.706720\pi$
$674$	0	0
$675$	−5.51610	−0.212315
$676$	0	0
$677$	25.7609	0.990074	0.495037	−	0.868872i	$-0.335155\pi$
$677$	25.7609	0.990074	0.495037	+	0.868872i	$0.335155\pi$
$678$	0	0
$679$	18.6052	0.714002
$680$	0	0
$681$	12.8471	0.492302
$682$	0	0
$683$	33.8124	1.29379	0.646897	−	0.762577i	$-0.276066\pi$
$683$	33.8124	1.29379	0.646897	+	0.762577i	$0.276066\pi$
$684$	0	0
$685$	5.93167	0.226638
$686$	0	0
$687$	−36.2777	−1.38408
$688$	0	0
$689$	0.459882	0.0175201
$690$	0	0
$691$	−20.9425	−0.796692	−0.398346	−	0.917235i	$-0.630416\pi$
$691$	−20.9425	−0.796692	−0.398346	+	0.917235i	$0.630416\pi$
$692$	0	0
$693$	10.4673	0.397619
$694$	0	0
$695$	16.5338	0.627163
$696$	0	0
$697$	−18.7851	−0.711536
$698$	0	0
$699$	−9.03447	−0.341715
$700$	0	0
$701$	−6.38881	−0.241302	−0.120651	−	0.992695i	$-0.538498\pi$
$701$	−6.38881	−0.241302	−0.120651	+	0.992695i	$0.538498\pi$
$702$	0	0
$703$	−5.42336	−0.204546
$704$	0	0
$705$	2.56401	0.0965662
$706$	0	0
$707$	22.9602	0.863508
$708$	0	0
$709$	10.0528	0.377541	0.188770	−	0.982021i	$-0.439550\pi$
$709$	10.0528	0.377541	0.188770	+	0.982021i	$0.439550\pi$
$710$	0	0
$711$	2.84397	0.106657
$712$	0	0
$713$	23.0033	0.861479
$714$	0	0
$715$	−0.212586	−0.00795027
$716$	0	0
$717$	−27.9790	−1.04489
$718$	0	0
$719$	−46.4985	−1.73410	−0.867050	−	0.498220i	$-0.833987\pi$
$719$	−46.4985	−1.73410	−0.867050	+	0.498220i	$0.833987\pi$
$720$	0	0
$721$	25.0998	0.934767
$722$	0	0
$723$	22.1882	0.825188
$724$	0	0
$725$	4.97270	0.184681
$726$	0	0
$727$	25.2092	0.934956	0.467478	−	0.884005i	$-0.345163\pi$
$727$	25.2092	0.934956	0.467478	+	0.884005i	$0.345163\pi$
$728$	0	0
$729$	29.7829	1.10307
$730$	0	0
$731$	0.420415	0.0155496
$732$	0	0
$733$	6.03408	0.222874	0.111437	−	0.993772i	$-0.464455\pi$
$733$	6.03408	0.222874	0.111437	+	0.993772i	$0.464455\pi$
$734$	0	0
$735$	21.1362	0.779622
$736$	0	0
$737$	−1.02096	−0.0376075
$738$	0	0
$739$	−5.08559	−0.187077	−0.0935383	−	0.995616i	$-0.529818\pi$
$739$	−5.08559	−0.187077	−0.0935383	+	0.995616i	$0.529818\pi$
$740$	0	0
$741$	−0.366070	−0.0134479
$742$	0	0
$743$	−43.1636	−1.58352	−0.791760	−	0.610832i	$-0.790835\pi$
$743$	−43.1636	−1.58352	−0.791760	+	0.610832i	$0.790835\pi$
$744$	0	0
$745$	11.9854	0.439109
$746$	0	0
$747$	−8.20271	−0.300122
$748$	0	0
$749$	−79.4696	−2.90375
$750$	0	0
$751$	34.6969	1.26611	0.633054	−	0.774107i	$-0.281801\pi$
$751$	34.6969	1.26611	0.633054	+	0.774107i	$0.281801\pi$
$752$	0	0
$753$	−47.9253	−1.74650
$754$	0	0
$755$	12.7074	0.462468
$756$	0	0
$757$	39.1211	1.42188	0.710941	−	0.703252i	$-0.248269\pi$
$757$	39.1211	1.42188	0.710941	+	0.703252i	$0.248269\pi$
$758$	0	0
$759$	39.7254	1.44194
$760$	0	0
$761$	21.7692	0.789134	0.394567	−	0.918867i	$-0.370895\pi$
$761$	21.7692	0.789134	0.394567	+	0.918867i	$0.370895\pi$
$762$	0	0
$763$	−22.1830	−0.803077
$764$	0	0
$765$	1.73577	0.0627569
$766$	0	0
$767$	−0.389636	−0.0140689
$768$	0	0
$769$	27.7450	1.00051	0.500255	−	0.865878i	$-0.333239\pi$
$769$	27.7450	1.00051	0.500255	+	0.865878i	$0.333239\pi$
$770$	0	0
$771$	−24.4791	−0.881593
$772$	0	0
$773$	−33.1447	−1.19213	−0.596066	−	0.802935i	$-0.703270\pi$
$773$	−33.1447	−1.19213	−0.596066	+	0.802935i	$0.703270\pi$
$774$	0	0
$775$	4.62591	0.166168
$776$	0	0
$777$	7.17064	0.257245
$778$	0	0
$779$	27.2036	0.974669
$780$	0	0
$781$	67.0690	2.39992
$782$	0	0
$783$	−27.4299	−0.980265
$784$	0	0
$785$	−19.2060	−0.685491
$786$	0	0
$787$	37.2528	1.32792	0.663959	−	0.747769i	$-0.268875\pi$
$787$	37.2528	1.32792	0.663959	+	0.747769i	$0.268875\pi$
$788$	0	0
$789$	3.10192	0.110431
$790$	0	0
$791$	16.2859	0.579062
$792$	0	0
$793$	0.373284	0.0132557
$794$	0	0
$795$	−17.2817	−0.612920
$796$	0	0
$797$	0.704005	0.0249371	0.0124686	−	0.999922i	$-0.496031\pi$
$797$	0.704005	0.0249371	0.0124686	+	0.999922i	$0.496031\pi$
$798$	0	0
$799$	−6.02916	−0.213296
$800$	0	0
$801$	−7.93347	−0.280315
$802$	0	0
$803$	37.3848	1.31928
$804$	0	0
$805$	−22.3888	−0.789103
$806$	0	0
$807$	−19.6126	−0.690395
$808$	0	0
$809$	−46.4644	−1.63360	−0.816801	−	0.576919i	$-0.804255\pi$
$809$	−46.4644	−1.63360	−0.816801	+	0.576919i	$0.804255\pi$
$810$	0	0
$811$	8.07347	0.283498	0.141749	−	0.989903i	$-0.454727\pi$
$811$	8.07347	0.283498	0.141749	+	0.989903i	$0.454727\pi$
$812$	0	0
$813$	−34.2506	−1.20122
$814$	0	0
$815$	−4.55387	−0.159515
$816$	0	0
$817$	−0.608822	−0.0213000
$818$	0	0
$819$	−0.0884409	−0.00309038
$820$	0	0
$821$	−4.39564	−0.153409	−0.0767045	−	0.997054i	$-0.524440\pi$
$821$	−4.39564	−0.153409	−0.0767045	+	0.997054i	$0.524440\pi$
$822$	0	0
$823$	−4.13714	−0.144212	−0.0721059	−	0.997397i	$-0.522972\pi$
$823$	−4.13714	−0.144212	−0.0721059	+	0.997397i	$0.522972\pi$
$824$	0	0
$825$	7.98870	0.278131
$826$	0	0
$827$	−17.1322	−0.595745	−0.297872	−	0.954606i	$-0.596277\pi$
$827$	−17.1322	−0.595745	−0.297872	+	0.954606i	$0.596277\pi$
$828$	0	0
$829$	34.3574	1.19328	0.596641	−	0.802509i	$-0.296502\pi$
$829$	34.3574	1.19328	0.596641	+	0.802509i	$0.296502\pi$
$830$	0	0
$831$	5.74192	0.199185
$832$	0	0
$833$	−49.7009	−1.72204
$834$	0	0
$835$	14.8724	0.514680
$836$	0	0
$837$	−25.5170	−0.881996
$838$	0	0
$839$	5.08387	0.175515	0.0877574	−	0.996142i	$-0.472030\pi$
$839$	5.08387	0.175515	0.0877574	+	0.996142i	$0.472030\pi$
$840$	0	0
$841$	−4.27224	−0.147319
$842$	0	0
$843$	−34.5854	−1.19118
$844$	0	0
$845$	−12.9982	−0.447152
$846$	0	0
$847$	−63.7545	−2.19063
$848$	0	0
$849$	52.9805	1.81828
$850$	0	0
$851$	−4.97270	−0.170462
$852$	0	0
$853$	−14.1332	−0.483912	−0.241956	−	0.970287i	$-0.577789\pi$
$853$	−14.1332	−0.483912	−0.241956	+	0.970287i	$0.577789\pi$
$854$	0	0
$855$	−2.51365	−0.0859650
$856$	0	0
$857$	−22.9794	−0.784961	−0.392481	−	0.919760i	$-0.628383\pi$
$857$	−22.9794	−0.784961	−0.392481	+	0.919760i	$0.628383\pi$
$858$	0	0
$859$	−50.3663	−1.71848	−0.859239	−	0.511575i	$-0.829062\pi$
$859$	−50.3663	−1.71848	−0.859239	+	0.511575i	$0.829062\pi$
$860$	0	0
$861$	−35.9679	−1.22578
$862$	0	0
$863$	30.7879	1.04803	0.524016	−	0.851708i	$-0.324433\pi$
$863$	30.7879	1.04803	0.524016	+	0.851708i	$0.324433\pi$
$864$	0	0
$865$	10.1210	0.344125
$866$	0	0
$867$	−4.73769	−0.160900
$868$	0	0
$869$	−30.7784	−1.04409
$870$	0	0
$871$	0.00862637	0.000292293 0
$872$	0	0
$873$	1.91528	0.0648223
$874$	0	0
$875$	−4.50235	−0.152207
$876$	0	0
$877$	2.67081	0.0901868	0.0450934	−	0.998983i	$-0.485641\pi$
$877$	2.67081	0.0901868	0.0450934	+	0.998983i	$0.485641\pi$
$878$	0	0
$879$	42.0689	1.41895
$880$	0	0
$881$	−13.7272	−0.462480	−0.231240	−	0.972897i	$-0.574278\pi$
$881$	−13.7272	−0.462480	−0.231240	+	0.972897i	$0.574278\pi$
$882$	0	0
$883$	58.7206	1.97610	0.988052	−	0.154119i	$-0.0492540\pi$
$883$	58.7206	1.97610	0.988052	+	0.154119i	$0.0492540\pi$
$884$	0	0
$885$	14.6420	0.492185
$886$	0	0
$887$	32.0357	1.07565	0.537826	−	0.843056i	$-0.319246\pi$
$887$	32.0357	1.07565	0.537826	+	0.843056i	$0.319246\pi$
$888$	0	0
$889$	27.7438	0.930498
$890$	0	0
$891$	−37.0919	−1.24263
$892$	0	0
$893$	8.73111	0.292175
$894$	0	0
$895$	−19.6018	−0.655216
$896$	0	0
$897$	−0.335651	−0.0112071
$898$	0	0
$899$	23.0033	0.767202
$900$	0	0
$901$	40.6373	1.35382
$902$	0	0
$903$	0.804971	0.0267878
$904$	0	0
$905$	10.9123	0.362738
$906$	0	0
$907$	−29.5472	−0.981098	−0.490549	−	0.871414i	$-0.663204\pi$
$907$	−29.5472	−0.981098	−0.490549	+	0.871414i	$0.663204\pi$
$908$	0	0
$909$	2.36360	0.0783955
$910$	0	0
$911$	43.8920	1.45421	0.727103	−	0.686528i	$-0.240866\pi$
$911$	43.8920	1.45421	0.727103	+	0.686528i	$0.240866\pi$
$912$	0	0
$913$	88.7725	2.93794
$914$	0	0
$915$	−14.0275	−0.463735
$916$	0	0
$917$	26.3776	0.871065
$918$	0	0
$919$	4.92960	0.162612	0.0813062	−	0.996689i	$-0.474091\pi$
$919$	4.92960	0.162612	0.0813062	+	0.996689i	$0.474091\pi$
$920$	0	0
$921$	−2.30119	−0.0758268
$922$	0	0
$923$	−0.566685	−0.0186527
$924$	0	0
$925$	−1.00000	−0.0328798
$926$	0	0
$927$	2.58385	0.0848649
$928$	0	0
$929$	43.4664	1.42609	0.713044	−	0.701120i	$-0.247316\pi$
$929$	43.4664	1.42609	0.713044	+	0.701120i	$0.247316\pi$
$930$	0	0
$931$	71.9743	2.35886
$932$	0	0
$933$	38.0218	1.24478
$934$	0	0
$935$	−18.7851	−0.614338
$936$	0	0
$937$	−27.6880	−0.904527	−0.452264	−	0.891884i	$-0.649383\pi$
$937$	−27.6880	−0.904527	−0.452264	+	0.891884i	$0.649383\pi$
$938$	0	0
$939$	20.8469	0.680311
$940$	0	0
$941$	−41.0590	−1.33849	−0.669243	−	0.743044i	$-0.733381\pi$
$941$	−41.0590	−1.33849	−0.669243	+	0.743044i	$0.733381\pi$
$942$	0	0
$943$	24.9431	0.812258
$944$	0	0
$945$	24.8354	0.807896
$946$	0	0
$947$	52.5618	1.70803	0.854015	−	0.520248i	$-0.174161\pi$
$947$	52.5618	1.70803	0.854015	+	0.520248i	$0.174161\pi$
$948$	0	0
$949$	−0.315875	−0.0102537
$950$	0	0
$951$	−22.1556	−0.718445
$952$	0	0
$953$	−43.0856	−1.39568	−0.697840	−	0.716254i	$-0.745855\pi$
$953$	−43.0856	−1.39568	−0.697840	+	0.716254i	$0.745855\pi$
$954$	0	0
$955$	−3.64884	−0.118074
$956$	0	0
$957$	39.7254	1.28414
$958$	0	0
$959$	−26.7065	−0.862397
$960$	0	0
$961$	−9.60097	−0.309709
$962$	0	0
$963$	−8.18084	−0.263624
$964$	0	0
$965$	−13.6388	−0.439049
$966$	0	0
$967$	15.0297	0.483322	0.241661	−	0.970361i	$-0.422308\pi$
$967$	15.0297	0.483322	0.241661	+	0.970361i	$0.422308\pi$
$968$	0	0
$969$	−32.3476	−1.03916
$970$	0	0
$971$	−30.0841	−0.965444	−0.482722	−	0.875774i	$-0.660352\pi$
$971$	−30.0841	−0.965444	−0.482722	+	0.875774i	$0.660352\pi$
$972$	0	0
$973$	−74.4410	−2.38647
$974$	0	0
$975$	−0.0674988	−0.00216169
$976$	0	0
$977$	−43.7845	−1.40079	−0.700396	−	0.713755i	$-0.746993\pi$
$977$	−43.7845	−1.40079	−0.700396	+	0.713755i	$0.746993\pi$
$978$	0	0
$979$	85.8587	2.74406
$980$	0	0
$981$	−2.28358	−0.0729092
$982$	0	0
$983$	44.9580	1.43394	0.716969	−	0.697105i	$-0.245529\pi$
$983$	44.9580	1.43394	0.716969	+	0.697105i	$0.245529\pi$
$984$	0	0
$985$	−10.1556	−0.323583
$986$	0	0
$987$	−11.5441	−0.367452
$988$	0	0
$989$	−0.558232	−0.0177507
$990$	0	0
$991$	−18.7339	−0.595103	−0.297552	−	0.954706i	$-0.596170\pi$
$991$	−18.7339	−0.595103	−0.297552	+	0.954706i	$0.596170\pi$
$992$	0	0
$993$	−3.90010	−0.123766
$994$	0	0
$995$	−19.1149	−0.605985
$996$	0	0
$997$	−30.8930	−0.978392	−0.489196	−	0.872174i	$-0.662710\pi$
$997$	−30.8930	−0.978392	−0.489196	+	0.872174i	$0.662710\pi$
$998$	0	0
$999$	5.51610	0.174522

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2960.2.a.bb.1.3		5
4.3	odd	2		1480.2.a.g.1.3	✓	5
20.19	odd	2		7400.2.a.r.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1480.2.a.g.1.3	✓	5	4.3	odd	2
2960.2.a.bb.1.3		5	1.1	even	1	trivial
7400.2.a.r.1.3		5	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)

\(f(q)\)	\(=\)	\(q+1.59264 q^{3} +1.00000 q^{5} -4.50235 q^{7} -0.463486 q^{9} +O(q^{10})\)	\(q+1.59264 q^{3} +1.00000 q^{5} -4.50235 q^{7} -0.463486 q^{9} +5.01600 q^{11} -0.0423816 q^{13} +1.59264 q^{15} -3.74503 q^{17} +5.42336 q^{19} -7.17064 q^{21} +4.97270 q^{23} +1.00000 q^{25} -5.51610 q^{27} +4.97270 q^{29} +4.62591 q^{31} +7.98870 q^{33} -4.50235 q^{35} -1.00000 q^{37} -0.0674988 q^{39} +5.01600 q^{41} -0.112259 q^{43} -0.463486 q^{45} +1.60991 q^{47} +13.2712 q^{49} -5.96450 q^{51} -10.8510 q^{53} +5.01600 q^{55} +8.63748 q^{57} +9.19351 q^{59} -8.80768 q^{61} +2.08678 q^{63} -0.0423816 q^{65} -0.203540 q^{67} +7.91974 q^{69} +13.3710 q^{71} +7.45311 q^{73} +1.59264 q^{75} -22.5838 q^{77} -6.13604 q^{79} -7.39472 q^{81} +17.6979 q^{83} -3.74503 q^{85} +7.91974 q^{87} +17.1170 q^{89} +0.190817 q^{91} +7.36742 q^{93} +5.42336 q^{95} -4.13233 q^{97} -2.32485 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{3} + 5 q^{5} + 5 q^{7} + 6 q^{9}+O(q^{10}) \) 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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