LMFDB - Embedded newform 272.10.a.f.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [272,10,Mod(1,272)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(272, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("272.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 272 = 2^{4} \cdot 17 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	272.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:	$140.089747437$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 1596x^{3} + 5754x^{2} + 488987x - 2711704 $ x^5 - 2x^4 - 1596x^3 + 5754x^2 + 488987x - 2711704
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3 $
Twist minimal:	no (minimal twist has level 17)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$18.8209$ of defining polynomial
Character	$\chi$	$=$	272.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+67.6654 q^{3} +2390.67 q^{5} +11355.8 q^{7} -15104.4 q^{9} +O(q^{10})$	$q+67.6654 q^{3} +2390.67 q^{5} +11355.8 q^{7} -15104.4 q^{9} +17740.6 q^{11} -76108.0 q^{13} +161765. q^{15} -83521.0 q^{17} -661811. q^{19} +768396. q^{21} +1.64772e6 q^{23} +3.76216e6 q^{25} -2.35390e6 q^{27} +1.49876e6 q^{29} +4.40242e6 q^{31} +1.20042e6 q^{33} +2.71480e7 q^{35} -5.62188e6 q^{37} -5.14988e6 q^{39} +2.29725e7 q^{41} +7.92989e6 q^{43} -3.61096e7 q^{45} +5.69337e7 q^{47} +8.86009e7 q^{49} -5.65148e6 q^{51} +2.56355e6 q^{53} +4.24119e7 q^{55} -4.47817e7 q^{57} +6.87685e7 q^{59} -1.09661e8 q^{61} -1.71523e8 q^{63} -1.81949e8 q^{65} -1.44824e8 q^{67} +1.11494e8 q^{69} +6.07354e7 q^{71} -1.68554e8 q^{73} +2.54568e8 q^{75} +2.01459e8 q^{77} -1.00310e8 q^{79} +1.38022e8 q^{81} +5.82713e8 q^{83} -1.99671e8 q^{85} +1.01414e8 q^{87} -1.56066e7 q^{89} -8.64268e8 q^{91} +2.97891e8 q^{93} -1.58217e9 q^{95} +1.64798e9 q^{97} -2.67961e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 236 q^{3} + 1480 q^{5} + 13202 q^{7} + 10981 q^{9}+O(q^{10}) $ 5 * q + 236 * q^3 + 1480 * q^5 + 13202 * q^7 + 10981 * q^9	$ 5 q + 236 q^{3} + 1480 q^{5} + 13202 q^{7} + 10981 q^{9} + 68036 q^{11} - 158862 q^{13} + 687324 q^{15} - 417605 q^{17} + 370992 q^{19} + 1783880 q^{21} - 1645870 q^{23} + 3270239 q^{25} + 2998268 q^{27} + 3668616 q^{29} + 7262362 q^{31} - 11334900 q^{33} + 26503988 q^{35} - 31420708 q^{37} + 42449884 q^{39} - 7996938 q^{41} + 56908268 q^{43} + 12799536 q^{45} + 16903336 q^{47} - 11784059 q^{49} - 19710956 q^{51} - 83362982 q^{53} - 6363364 q^{55} + 136615904 q^{57} + 37946604 q^{59} - 77685452 q^{61} + 191945278 q^{63} - 40321288 q^{65} + 304503600 q^{67} - 333409272 q^{69} + 476602922 q^{71} - 289980486 q^{73} + 153685772 q^{75} - 143385648 q^{77} + 828240610 q^{79} + 891328609 q^{81} - 194681148 q^{83} - 123611080 q^{85} - 158149884 q^{87} + 376848106 q^{89} - 194543664 q^{91} + 3494835920 q^{93} - 1498679864 q^{95} + 692035246 q^{97} - 2027106408 q^{99}+O(q^{100}) $ 5 * q + 236 * q^3 + 1480 * q^5 + 13202 * q^7 + 10981 * q^9 + 68036 * q^11 - 158862 * q^13 + 687324 * q^15 - 417605 * q^17 + 370992 * q^19 + 1783880 * q^21 - 1645870 * q^23 + 3270239 * q^25 + 2998268 * q^27 + 3668616 * q^29 + 7262362 * q^31 - 11334900 * q^33 + 26503988 * q^35 - 31420708 * q^37 + 42449884 * q^39 - 7996938 * q^41 + 56908268 * q^43 + 12799536 * q^45 + 16903336 * q^47 - 11784059 * q^49 - 19710956 * q^51 - 83362982 * q^53 - 6363364 * q^55 + 136615904 * q^57 + 37946604 * q^59 - 77685452 * q^61 + 191945278 * q^63 - 40321288 * q^65 + 304503600 * q^67 - 333409272 * q^69 + 476602922 * q^71 - 289980486 * q^73 + 153685772 * q^75 - 143385648 * q^77 + 828240610 * q^79 + 891328609 * q^81 - 194681148 * q^83 - 123611080 * q^85 - 158149884 * q^87 + 376848106 * q^89 - 194543664 * q^91 + 3494835920 * q^93 - 1498679864 * q^95 + 692035246 * q^97 - 2027106408 * 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
$3$	67.6654	0.482304	0.241152	−	0.970487i	$-0.422475\pi$
$3$	67.6654	0.482304	0.241152	+	0.970487i	$0.422475\pi$
$4$	0	0
$5$	2390.67	1.71062	0.855311	−	0.518115i	$-0.173366\pi$
$5$	2390.67	1.71062	0.855311	+	0.518115i	$0.173366\pi$
$6$	0	0
$7$	11355.8	1.78763	0.893814	−	0.448438i	$-0.148020\pi$
$7$	11355.8	1.78763	0.893814	+	0.448438i	$0.148020\pi$
$8$	0	0
$9$	−15104.4	−0.767383
$10$	0	0
$11$	17740.6	0.365343	0.182672	−	0.983174i	$-0.441525\pi$
$11$	17740.6	0.365343	0.182672	+	0.983174i	$0.441525\pi$
$12$	0	0
$13$	−76108.0	−0.739069	−0.369535	−	0.929217i	$-0.620483\pi$
$13$	−76108.0	−0.739069	−0.369535	+	0.929217i	$0.620483\pi$
$14$	0	0
$15$	161765.	0.825040
$16$	0	0
$17$	−83521.0	−0.242536
$17$	−83521.0	−0.242536
$18$	0	0
$19$	−661811.	−1.16505	−0.582523	−	0.812814i	$-0.697934\pi$
$19$	−661811.	−1.16505	−0.582523	+	0.812814i	$0.697934\pi$
$20$	0	0
$21$	768396.	0.862180
$22$	0	0
$23$	1.64772e6	1.22774	0.613872	−	0.789405i	$-0.289611\pi$
$23$	1.64772e6	1.22774	0.613872	+	0.789405i	$0.289611\pi$
$24$	0	0
$25$	3.76216e6	1.92623
$26$	0	0
$27$	−2.35390e6	−0.852416
$28$	0	0
$29$	1.49876e6	0.393496	0.196748	−	0.980454i	$-0.436962\pi$
$29$	1.49876e6	0.393496	0.196748	+	0.980454i	$0.436962\pi$
$30$	0	0
$31$	4.40242e6	0.856177	0.428089	−	0.903737i	$-0.359187\pi$
$31$	4.40242e6	0.856177	0.428089	+	0.903737i	$0.359187\pi$
$32$	0	0
$33$	1.20042e6	0.176207
$34$	0	0
$35$	2.71480e7	3.05796
$36$	0	0
$37$	−5.62188e6	−0.493144	−0.246572	−	0.969125i	$-0.579304\pi$
$37$	−5.62188e6	−0.493144	−0.246572	+	0.969125i	$0.579304\pi$
$38$	0	0
$39$	−5.14988e6	−0.356456
$40$	0	0
$41$	2.29725e7	1.26964	0.634821	−	0.772659i	$-0.281074\pi$
$41$	2.29725e7	1.26964	0.634821	+	0.772659i	$0.281074\pi$
$42$	0	0
$43$	7.92989e6	0.353720	0.176860	−	0.984236i	$-0.443406\pi$
$43$	7.92989e6	0.353720	0.176860	+	0.984236i	$0.443406\pi$
$44$	0	0
$45$	−3.61096e7	−1.31270
$46$	0	0
$47$	5.69337e7	1.70188	0.850940	−	0.525262i	$-0.176033\pi$
$47$	5.69337e7	1.70188	0.850940	+	0.525262i	$0.176033\pi$
$48$	0	0
$49$	8.86009e7	2.19561
$50$	0	0
$51$	−5.65148e6	−0.116976
$52$	0	0
$53$	2.56355e6	0.0446272	0.0223136	−	0.999751i	$-0.492897\pi$
$53$	2.56355e6	0.0446272	0.0223136	+	0.999751i	$0.492897\pi$
$54$	0	0
$55$	4.24119e7	0.624964
$56$	0	0
$57$	−4.47817e7	−0.561906
$58$	0	0
$59$	6.87685e7	0.738849	0.369424	−	0.929261i	$-0.379555\pi$
$59$	6.87685e7	0.738849	0.369424	+	0.929261i	$0.379555\pi$
$60$	0	0
$61$	−1.09661e8	−1.01407	−0.507036	−	0.861925i	$-0.669259\pi$
$61$	−1.09661e8	−1.01407	−0.507036	+	0.861925i	$0.669259\pi$
$62$	0	0
$63$	−1.71523e8	−1.37179
$64$	0	0
$65$	−1.81949e8	−1.26427
$66$	0	0
$67$	−1.44824e8	−0.878017	−0.439008	−	0.898483i	$-0.644670\pi$
$67$	−1.44824e8	−0.878017	−0.439008	+	0.898483i	$0.644670\pi$
$68$	0	0
$69$	1.11494e8	0.592147
$70$	0	0
$71$	6.07354e7	0.283648	0.141824	−	0.989892i	$-0.454703\pi$
$71$	6.07354e7	0.283648	0.141824	+	0.989892i	$0.454703\pi$
$72$	0	0
$73$	−1.68554e8	−0.694684	−0.347342	−	0.937739i	$-0.612916\pi$
$73$	−1.68554e8	−0.694684	−0.347342	+	0.937739i	$0.612916\pi$
$74$	0	0
$75$	2.54568e8	0.929027
$76$	0	0
$77$	2.01459e8	0.653098
$78$	0	0
$79$	−1.00310e8	−0.289749	−0.144875	−	0.989450i	$-0.546278\pi$
$79$	−1.00310e8	−0.289749	−0.144875	+	0.989450i	$0.546278\pi$
$80$	0	0
$81$	1.38022e8	0.356259
$82$	0	0
$83$	5.82713e8	1.34773	0.673865	−	0.738854i	$-0.264633\pi$
$83$	5.82713e8	1.34773	0.673865	+	0.738854i	$0.264633\pi$
$84$	0	0
$85$	−1.99671e8	−0.414887
$86$	0	0
$87$	1.01414e8	0.189785
$88$	0	0
$89$	−1.56066e7	−0.0263665	−0.0131832	−	0.999913i	$-0.504196\pi$
$89$	−1.56066e7	−0.0263665	−0.0131832	+	0.999913i	$0.504196\pi$
$90$	0	0
$91$	−8.64268e8	−1.32118
$92$	0	0
$93$	2.97891e8	0.412938
$94$	0	0
$95$	−1.58217e9	−1.99295
$96$	0	0
$97$	1.64798e9	1.89008	0.945040	−	0.326956i	$-0.106023\pi$
$97$	1.64798e9	1.89008	0.945040	+	0.326956i	$0.106023\pi$
$98$	0	0
$99$	−2.67961e8	−0.280358
$100$	0	0
$101$	−9.32619e8	−0.891780	−0.445890	−	0.895088i	$-0.647113\pi$
$101$	−9.32619e8	−0.891780	−0.445890	+	0.895088i	$0.647113\pi$
$102$	0	0
$103$	1.88493e9	1.65017	0.825085	−	0.565009i	$-0.191127\pi$
$103$	1.88493e9	1.65017	0.825085	+	0.565009i	$0.191127\pi$
$104$	0	0
$105$	1.83698e9	1.47486
$106$	0	0
$107$	9.44012e7	0.0696226	0.0348113	−	0.999394i	$-0.488917\pi$
$107$	9.44012e7	0.0696226	0.0348113	+	0.999394i	$0.488917\pi$
$108$	0	0
$109$	−1.72481e9	−1.17037	−0.585183	−	0.810901i	$-0.698977\pi$
$109$	−1.72481e9	−1.17037	−0.585183	+	0.810901i	$0.698977\pi$
$110$	0	0
$111$	−3.80406e8	−0.237845
$112$	0	0
$113$	1.02688e9	0.592471	0.296235	−	0.955115i	$-0.404269\pi$
$113$	1.02688e9	0.592471	0.296235	+	0.955115i	$0.404269\pi$
$114$	0	0
$115$	3.93915e9	2.10021
$116$	0	0
$117$	1.14957e9	0.567149
$118$	0	0
$119$	−9.48449e8	−0.433563
$120$	0	0
$121$	−2.04322e9	−0.866524
$122$	0	0
$123$	1.55445e9	0.612354
$124$	0	0
$125$	4.32481e9	1.58442
$126$	0	0
$127$	−2.57593e9	−0.878653	−0.439327	−	0.898327i	$-0.644783\pi$
$127$	−2.57593e9	−0.878653	−0.439327	+	0.898327i	$0.644783\pi$
$128$	0	0
$129$	5.36579e8	0.170600
$130$	0	0
$131$	1.67438e9	0.496746	0.248373	−	0.968664i	$-0.420104\pi$
$131$	1.67438e9	0.496746	0.248373	+	0.968664i	$0.420104\pi$
$132$	0	0
$133$	−7.51540e9	−2.08267
$134$	0	0
$135$	−5.62740e9	−1.45816
$136$	0	0
$137$	9.77626e8	0.237099	0.118550	−	0.992948i	$-0.462176\pi$
$137$	9.77626e8	0.237099	0.118550	+	0.992948i	$0.462176\pi$
$138$	0	0
$139$	5.07318e8	0.115269	0.0576346	−	0.998338i	$-0.481644\pi$
$139$	5.07318e8	0.115269	0.0576346	+	0.998338i	$0.481644\pi$
$140$	0	0
$141$	3.85244e9	0.820824
$142$	0	0
$143$	−1.35020e9	−0.270014
$144$	0	0
$145$	3.58303e9	0.673123
$146$	0	0
$147$	5.99522e9	1.05895
$148$	0	0
$149$	−3.39357e9	−0.564052	−0.282026	−	0.959407i	$-0.591006\pi$
$149$	−3.39357e9	−0.564052	−0.282026	+	0.959407i	$0.591006\pi$
$150$	0	0
$151$	−2.54210e9	−0.397921	−0.198961	−	0.980007i	$-0.563757\pi$
$151$	−2.54210e9	−0.397921	−0.198961	+	0.980007i	$0.563757\pi$
$152$	0	0
$153$	1.26153e9	0.186118
$154$	0	0
$155$	1.05247e10	1.46460
$156$	0	0
$157$	−8.59628e9	−1.12918	−0.564588	−	0.825373i	$-0.690965\pi$
$157$	−8.59628e9	−1.12918	−0.564588	+	0.825373i	$0.690965\pi$
$158$	0	0
$159$	1.73464e8	0.0215239
$160$	0	0
$161$	1.87112e10	2.19475
$162$	0	0
$163$	−7.22903e8	−0.0802113	−0.0401057	−	0.999195i	$-0.512769\pi$
$163$	−7.22903e8	−0.0802113	−0.0401057	+	0.999195i	$0.512769\pi$
$164$	0	0
$165$	2.86982e9	0.301423
$166$	0	0
$167$	−5.77488e9	−0.574538	−0.287269	−	0.957850i	$-0.592747\pi$
$167$	−5.77488e9	−0.574538	−0.287269	+	0.957850i	$0.592747\pi$
$168$	0	0
$169$	−4.81207e9	−0.453776
$170$	0	0
$171$	9.99625e9	0.894036
$172$	0	0
$173$	−2.26224e9	−0.192014	−0.0960068	−	0.995381i	$-0.530607\pi$
$173$	−2.26224e9	−0.192014	−0.0960068	+	0.995381i	$0.530607\pi$
$174$	0	0
$175$	4.27224e10	3.44338
$176$	0	0
$177$	4.65325e9	0.356350
$178$	0	0
$179$	3.69322e9	0.268885	0.134443	−	0.990921i	$-0.457076\pi$
$179$	3.69322e9	0.268885	0.134443	+	0.990921i	$0.457076\pi$
$180$	0	0
$181$	1.91251e10	1.32449	0.662247	−	0.749285i	$-0.269603\pi$
$181$	1.91251e10	1.32449	0.662247	+	0.749285i	$0.269603\pi$
$182$	0	0
$183$	−7.42027e9	−0.489091
$184$	0	0
$185$	−1.34400e10	−0.843582
$186$	0	0
$187$	−1.48171e9	−0.0886088
$188$	0	0
$189$	−2.67305e10	−1.52380
$190$	0	0
$191$	4.41416e9	0.239993	0.119996	−	0.992774i	$-0.461712\pi$
$191$	4.41416e9	0.239993	0.119996	+	0.992774i	$0.461712\pi$
$192$	0	0
$193$	−5.98467e9	−0.310479	−0.155239	−	0.987877i	$-0.549615\pi$
$193$	−5.98467e9	−0.310479	−0.155239	+	0.987877i	$0.549615\pi$
$194$	0	0
$195$	−1.23116e10	−0.609762
$196$	0	0
$197$	2.74494e10	1.29848	0.649240	−	0.760584i	$-0.275087\pi$
$197$	2.74494e10	1.29848	0.649240	+	0.760584i	$0.275087\pi$
$198$	0	0
$199$	−3.34164e10	−1.51050	−0.755250	−	0.655436i	$-0.772485\pi$
$199$	−3.34164e10	−1.51050	−0.755250	+	0.655436i	$0.772485\pi$
$200$	0	0
$201$	−9.79955e9	−0.423471
$202$	0	0
$203$	1.70196e10	0.703425
$204$	0	0
$205$	5.49197e10	2.17188
$206$	0	0
$207$	−2.48878e10	−0.942150
$208$	0	0
$209$	−1.17409e10	−0.425641
$210$	0	0
$211$	−4.50084e9	−0.156323	−0.0781615	−	0.996941i	$-0.524905\pi$
$211$	−4.50084e9	−0.156323	−0.0781615	+	0.996941i	$0.524905\pi$
$212$	0	0
$213$	4.10969e9	0.136805
$214$	0	0
$215$	1.89577e10	0.605081
$216$	0	0
$217$	4.99930e10	1.53053
$218$	0	0
$219$	−1.14053e10	−0.335049
$220$	0	0
$221$	6.35662e9	0.179251
$222$	0	0
$223$	4.59996e10	1.24561	0.622805	−	0.782377i	$-0.285993\pi$
$223$	4.59996e10	1.24561	0.622805	+	0.782377i	$0.285993\pi$
$224$	0	0
$225$	−5.68252e10	−1.47815
$226$	0	0
$227$	−1.68405e10	−0.420958	−0.210479	−	0.977598i	$-0.567502\pi$
$227$	−1.68405e10	−0.420958	−0.210479	+	0.977598i	$0.567502\pi$
$228$	0	0
$229$	4.54814e10	1.09288	0.546442	−	0.837497i	$-0.315982\pi$
$229$	4.54814e10	1.09288	0.546442	+	0.837497i	$0.315982\pi$
$230$	0	0
$231$	1.36318e10	0.314992
$232$	0	0
$233$	−5.00961e10	−1.11353	−0.556766	−	0.830669i	$-0.687958\pi$
$233$	−5.00961e10	−1.11353	−0.556766	+	0.830669i	$0.687958\pi$
$234$	0	0
$235$	1.36110e11	2.91127
$236$	0	0
$237$	−6.78751e9	−0.139747
$238$	0	0
$239$	−4.75443e10	−0.942558	−0.471279	−	0.881984i	$-0.656208\pi$
$239$	−4.75443e10	−0.942558	−0.471279	+	0.881984i	$0.656208\pi$
$240$	0	0
$241$	8.24645e10	1.57467	0.787337	−	0.616523i	$-0.211459\pi$
$241$	8.24645e10	1.57467	0.787337	+	0.616523i	$0.211459\pi$
$242$	0	0
$243$	5.56712e10	1.02424
$244$	0	0
$245$	2.11815e11	3.75586
$246$	0	0
$247$	5.03691e10	0.861049
$248$	0	0
$249$	3.94295e10	0.650016
$250$	0	0
$251$	5.90523e10	0.939085	0.469543	−	0.882910i	$-0.344419\pi$
$251$	5.90523e10	0.939085	0.469543	+	0.882910i	$0.344419\pi$
$252$	0	0
$253$	2.92315e10	0.448548
$254$	0	0
$255$	−1.35108e10	−0.200102
$256$	0	0
$257$	3.35982e10	0.480415	0.240207	−	0.970722i	$-0.422785\pi$
$257$	3.35982e10	0.480415	0.240207	+	0.970722i	$0.422785\pi$
$258$	0	0
$259$	−6.38410e10	−0.881557
$260$	0	0
$261$	−2.26378e10	−0.301962
$262$	0	0
$263$	−3.91693e9	−0.0504830	−0.0252415	−	0.999681i	$-0.508035\pi$
$263$	−3.91693e9	−0.0504830	−0.0252415	+	0.999681i	$0.508035\pi$
$264$	0	0
$265$	6.12859e9	0.0763403
$266$	0	0
$267$	−1.05602e9	−0.0127167
$268$	0	0
$269$	6.33897e10	0.738131	0.369065	−	0.929403i	$-0.379678\pi$
$269$	6.33897e10	0.738131	0.369065	+	0.929403i	$0.379678\pi$
$270$	0	0
$271$	−4.27576e10	−0.481561	−0.240781	−	0.970580i	$-0.577403\pi$
$271$	−4.27576e10	−0.481561	−0.240781	+	0.970580i	$0.577403\pi$
$272$	0	0
$273$	−5.84811e10	−0.637211
$274$	0	0
$275$	6.67430e10	0.703734
$276$	0	0
$277$	−1.32848e11	−1.35580	−0.677899	−	0.735155i	$-0.737109\pi$
$277$	−1.32848e11	−1.35580	−0.677899	+	0.735155i	$0.737109\pi$
$278$	0	0
$279$	−6.64958e10	−0.657015
$280$	0	0
$281$	−1.43951e11	−1.37733	−0.688663	−	0.725082i	$-0.741802\pi$
$281$	−1.43951e11	−1.37733	−0.688663	+	0.725082i	$0.741802\pi$
$282$	0	0
$283$	−6.72871e10	−0.623581	−0.311790	−	0.950151i	$-0.600929\pi$
$283$	−6.72871e10	−0.623581	−0.311790	+	0.950151i	$0.600929\pi$
$284$	0	0
$285$	−1.07058e11	−0.961209
$286$	0	0
$287$	2.60872e11	2.26965
$288$	0	0
$289$	6.97576e9	0.0588235
$290$	0	0
$291$	1.11511e11	0.911593
$292$	0	0
$293$	8.76124e10	0.694482	0.347241	−	0.937776i	$-0.387119\pi$
$293$	8.76124e10	0.694482	0.347241	+	0.937776i	$0.387119\pi$
$294$	0	0
$295$	1.64403e11	1.26389
$296$	0	0
$297$	−4.17596e10	−0.311424
$298$	0	0
$299$	−1.25405e11	−0.907389
$300$	0	0
$301$	9.00504e10	0.632319
$302$	0	0
$303$	−6.31060e10	−0.430109
$304$	0	0
$305$	−2.62163e11	−1.73469
$306$	0	0
$307$	−2.30476e11	−1.48083	−0.740413	−	0.672153i	$-0.765370\pi$
$307$	−2.30476e11	−1.48083	−0.740413	+	0.672153i	$0.765370\pi$
$308$	0	0
$309$	1.27545e11	0.795884
$310$	0	0
$311$	−2.89033e10	−0.175197	−0.0875984	−	0.996156i	$-0.527919\pi$
$311$	−2.89033e10	−0.175197	−0.0875984	+	0.996156i	$0.527919\pi$
$312$	0	0
$313$	−2.95929e11	−1.74276	−0.871382	−	0.490605i	$-0.836776\pi$
$313$	−2.95929e11	−1.74276	−0.871382	+	0.490605i	$0.836776\pi$
$314$	0	0
$315$	−4.10054e11	−2.34662
$316$	0	0
$317$	−3.12812e11	−1.73987	−0.869934	−	0.493168i	$-0.835839\pi$
$317$	−3.12812e11	−1.73987	−0.869934	+	0.493168i	$0.835839\pi$
$318$	0	0
$319$	2.65888e10	0.143761
$320$	0	0
$321$	6.38769e9	0.0335793
$322$	0	0
$323$	5.52751e10	0.282565
$324$	0	0
$325$	−2.86331e11	−1.42362
$326$	0	0
$327$	−1.16710e11	−0.564473
$328$	0	0
$329$	6.46529e11	3.04233
$330$	0	0
$331$	−8.91016e9	−0.0408000	−0.0204000	−	0.999792i	$-0.506494\pi$
$331$	−8.91016e9	−0.0408000	−0.0204000	+	0.999792i	$0.506494\pi$
$332$	0	0
$333$	8.49150e10	0.378430
$334$	0	0
$335$	−3.46225e11	−1.50195
$336$	0	0
$337$	2.90899e10	0.122859	0.0614296	−	0.998111i	$-0.480434\pi$
$337$	2.90899e10	0.122859	0.0614296	+	0.998111i	$0.480434\pi$
$338$	0	0
$339$	6.94843e10	0.285751
$340$	0	0
$341$	7.81015e10	0.312799
$342$	0	0
$343$	5.47888e11	2.13731
$344$	0	0
$345$	2.66544e11	1.01294
$346$	0	0
$347$	−1.09628e11	−0.405918	−0.202959	−	0.979187i	$-0.565056\pi$
$347$	−1.09628e11	−0.405918	−0.202959	+	0.979187i	$0.565056\pi$
$348$	0	0
$349$	−6.35598e10	−0.229334	−0.114667	−	0.993404i	$-0.536580\pi$
$349$	−6.35598e10	−0.229334	−0.114667	+	0.993404i	$0.536580\pi$
$350$	0	0
$351$	1.79151e11	0.629995
$352$	0	0
$353$	7.79401e10	0.267162	0.133581	−	0.991038i	$-0.457352\pi$
$353$	7.79401e10	0.267162	0.133581	+	0.991038i	$0.457352\pi$
$354$	0	0
$355$	1.45198e11	0.485214
$356$	0	0
$357$	−6.41772e10	−0.209109
$358$	0	0
$359$	−3.06852e11	−0.974998	−0.487499	−	0.873124i	$-0.662091\pi$
$359$	−3.06852e11	−0.974998	−0.487499	+	0.873124i	$0.662091\pi$
$360$	0	0
$361$	1.15306e11	0.357331
$362$	0	0
$363$	−1.38255e11	−0.417928
$364$	0	0
$365$	−4.02957e11	−1.18834
$366$	0	0
$367$	2.13117e11	0.613225	0.306613	−	0.951834i	$-0.400804\pi$
$367$	2.13117e11	0.613225	0.306613	+	0.951834i	$0.400804\pi$
$368$	0	0
$369$	−3.46986e11	−0.974302
$370$	0	0
$371$	2.91112e10	0.0797769
$372$	0	0
$373$	2.27081e11	0.607423	0.303712	−	0.952764i	$-0.401774\pi$
$373$	2.27081e11	0.607423	0.303712	+	0.952764i	$0.401774\pi$
$374$	0	0
$375$	2.92640e11	0.764175
$376$	0	0
$377$	−1.14067e11	−0.290821
$378$	0	0
$379$	−3.53633e11	−0.880391	−0.440196	−	0.897902i	$-0.645091\pi$
$379$	−3.53633e11	−0.880391	−0.440196	+	0.897902i	$0.645091\pi$
$380$	0	0
$381$	−1.74301e11	−0.423778
$382$	0	0
$383$	6.69512e11	1.58988	0.794939	−	0.606689i	$-0.207503\pi$
$383$	6.69512e11	1.58988	0.794939	+	0.606689i	$0.207503\pi$
$384$	0	0
$385$	4.81621e11	1.11720
$386$	0	0
$387$	−1.19776e11	−0.271438
$388$	0	0
$389$	6.62054e11	1.46595	0.732976	−	0.680254i	$-0.238131\pi$
$389$	6.62054e11	1.46595	0.732976	+	0.680254i	$0.238131\pi$
$390$	0	0
$391$	−1.37619e11	−0.297772
$392$	0	0
$393$	1.13298e11	0.239583
$394$	0	0
$395$	−2.39808e11	−0.495651
$396$	0	0
$397$	−6.61102e11	−1.33571	−0.667854	−	0.744293i	$-0.732787\pi$
$397$	−6.61102e11	−1.33571	−0.667854	+	0.744293i	$0.732787\pi$
$398$	0	0
$399$	−5.08533e11	−1.00448
$400$	0	0
$401$	6.56727e11	1.26834	0.634169	−	0.773195i	$-0.281342\pi$
$401$	6.56727e11	1.26834	0.634169	+	0.773195i	$0.281342\pi$
$402$	0	0
$403$	−3.35059e11	−0.632774
$404$	0	0
$405$	3.29964e11	0.609424
$406$	0	0
$407$	−9.97354e10	−0.180167
$408$	0	0
$409$	−4.36145e11	−0.770684	−0.385342	−	0.922774i	$-0.625917\pi$
$409$	−4.36145e11	−0.770684	−0.385342	+	0.922774i	$0.625917\pi$
$410$	0	0
$411$	6.61515e10	0.114354
$412$	0	0
$413$	7.80923e11	1.32079
$414$	0	0
$415$	1.39307e12	2.30546
$416$	0	0
$417$	3.43279e10	0.0555948
$418$	0	0
$419$	−6.66344e11	−1.05617	−0.528087	−	0.849190i	$-0.677091\pi$
$419$	−6.66344e11	−1.05617	−0.528087	+	0.849190i	$0.677091\pi$
$420$	0	0
$421$	−4.68417e11	−0.726713	−0.363357	−	0.931650i	$-0.618369\pi$
$421$	−4.68417e11	−0.726713	−0.363357	+	0.931650i	$0.618369\pi$
$422$	0	0
$423$	−8.59949e11	−1.30599
$424$	0	0
$425$	−3.14220e11	−0.467179
$426$	0	0
$427$	−1.24529e12	−1.81278
$428$	0	0
$429$	−9.13619e10	−0.130229
$430$	0	0
$431$	3.43298e11	0.479207	0.239604	−	0.970871i	$-0.422982\pi$
$431$	3.43298e11	0.479207	0.239604	+	0.970871i	$0.422982\pi$
$432$	0	0
$433$	−1.30884e12	−1.78933	−0.894666	−	0.446735i	$-0.852587\pi$
$433$	−1.30884e12	−1.78933	−0.894666	+	0.446735i	$0.852587\pi$
$434$	0	0
$435$	2.42447e11	0.324650
$436$	0	0
$437$	−1.09048e12	−1.43038
$438$	0	0
$439$	−5.48226e11	−0.704481	−0.352241	−	0.935909i	$-0.614580\pi$
$439$	−5.48226e11	−0.704481	−0.352241	+	0.935909i	$0.614580\pi$
$440$	0	0
$441$	−1.33826e12	−1.68488
$442$	0	0
$443$	1.00051e10	0.0123426	0.00617130	−	0.999981i	$-0.498036\pi$
$443$	1.00051e10	0.0123426	0.00617130	+	0.999981i	$0.498036\pi$
$444$	0	0
$445$	−3.73101e10	−0.0451031
$446$	0	0
$447$	−2.29628e11	−0.272045
$448$	0	0
$449$	−8.97323e10	−0.104193	−0.0520967	−	0.998642i	$-0.516590\pi$
$449$	−8.97323e10	−0.104193	−0.0520967	+	0.998642i	$0.516590\pi$
$450$	0	0
$451$	4.07546e11	0.463855
$452$	0	0
$453$	−1.72012e11	−0.191919
$454$	0	0
$455$	−2.06618e12	−2.26004
$456$	0	0
$457$	−1.13166e12	−1.21365	−0.606823	−	0.794837i	$-0.707556\pi$
$457$	−1.13166e12	−1.21365	−0.606823	+	0.794837i	$0.707556\pi$
$458$	0	0
$459$	1.96600e11	0.206741
$460$	0	0
$461$	−1.76123e12	−1.81620	−0.908098	−	0.418757i	$-0.862466\pi$
$461$	−1.76123e12	−1.81620	−0.908098	+	0.418757i	$0.862466\pi$
$462$	0	0
$463$	6.00361e11	0.607152	0.303576	−	0.952807i	$-0.401819\pi$
$463$	6.00361e11	0.607152	0.303576	+	0.952807i	$0.401819\pi$
$464$	0	0
$465$	7.12159e11	0.706380
$466$	0	0
$467$	7.30949e11	0.711150	0.355575	−	0.934648i	$-0.384285\pi$
$467$	7.30949e11	0.711150	0.355575	+	0.934648i	$0.384285\pi$
$468$	0	0
$469$	−1.64459e12	−1.56957
$470$	0	0
$471$	−5.81671e11	−0.544607
$472$	0	0
$473$	1.40681e11	0.129229
$474$	0	0
$475$	−2.48984e12	−2.24414
$476$	0	0
$477$	−3.87208e10	−0.0342462
$478$	0	0
$479$	1.76035e12	1.52788	0.763938	−	0.645289i	$-0.223263\pi$
$479$	1.76035e12	1.52788	0.763938	+	0.645289i	$0.223263\pi$
$480$	0	0
$481$	4.27870e11	0.364467
$482$	0	0
$483$	1.26610e12	1.05854
$484$	0	0
$485$	3.93978e12	3.23321
$486$	0	0
$487$	−2.40112e12	−1.93434	−0.967171	−	0.254126i	$-0.918212\pi$
$487$	−2.40112e12	−1.93434	−0.967171	+	0.254126i	$0.918212\pi$
$488$	0	0
$489$	−4.89155e10	−0.0386862
$490$	0	0
$491$	3.93654e11	0.305666	0.152833	−	0.988252i	$-0.451160\pi$
$491$	3.93654e11	0.305666	0.152833	+	0.988252i	$0.451160\pi$
$492$	0	0
$493$	−1.25178e11	−0.0954368
$494$	0	0
$495$	−6.40605e11	−0.479587
$496$	0	0
$497$	6.89700e11	0.507057
$498$	0	0
$499$	1.16585e12	0.841763	0.420881	−	0.907116i	$-0.361721\pi$
$499$	1.16585e12	0.841763	0.420881	+	0.907116i	$0.361721\pi$
$500$	0	0
$501$	−3.90760e11	−0.277102
$502$	0	0
$503$	1.06497e12	0.741794	0.370897	−	0.928674i	$-0.379050\pi$
$503$	1.06497e12	0.741794	0.370897	+	0.928674i	$0.379050\pi$
$504$	0	0
$505$	−2.22958e12	−1.52550
$506$	0	0
$507$	−3.25611e11	−0.218858
$508$	0	0
$509$	−1.27920e12	−0.844711	−0.422355	−	0.906430i	$-0.638797\pi$
$509$	−1.27920e12	−0.844711	−0.422355	+	0.906430i	$0.638797\pi$
$510$	0	0
$511$	−1.91407e12	−1.24184
$512$	0	0
$513$	1.55784e12	0.993103
$514$	0	0
$515$	4.50625e12	2.82282
$516$	0	0
$517$	1.01004e12	0.621771
$518$	0	0
$519$	−1.53076e11	−0.0926089
$520$	0	0
$521$	1.01572e12	0.603955	0.301977	−	0.953315i	$-0.402353\pi$
$521$	1.01572e12	0.603955	0.301977	+	0.953315i	$0.402353\pi$
$522$	0	0
$523$	7.26782e10	0.0424763	0.0212381	−	0.999774i	$-0.493239\pi$
$523$	7.26782e10	0.0424763	0.0212381	+	0.999774i	$0.493239\pi$
$524$	0	0
$525$	2.89083e12	1.66076
$526$	0	0
$527$	−3.67694e11	−0.207653
$528$	0	0
$529$	9.13829e11	0.507358
$530$	0	0
$531$	−1.03871e12	−0.566980
$532$	0	0
$533$	−1.74839e12	−0.938354
$534$	0	0
$535$	2.25682e11	0.119098
$536$	0	0
$537$	2.49903e11	0.129684
$538$	0	0
$539$	1.57183e12	0.802153
$540$	0	0
$541$	−4.64227e11	−0.232993	−0.116496	−	0.993191i	$-0.537166\pi$
$541$	−4.64227e11	−0.232993	−0.116496	+	0.993191i	$0.537166\pi$
$542$	0	0
$543$	1.29411e12	0.638809
$544$	0	0
$545$	−4.12345e12	−2.00205
$546$	0	0
$547$	2.64589e12	1.26366	0.631828	−	0.775109i	$-0.282305\pi$
$547$	2.64589e12	1.26366	0.631828	+	0.775109i	$0.282305\pi$
$548$	0	0
$549$	1.65637e12	0.778181
$550$	0	0
$551$	−9.91894e11	−0.458441
$552$	0	0
$553$	−1.13910e12	−0.517964
$554$	0	0
$555$	−9.09425e11	−0.406863
$556$	0	0
$557$	−1.89427e11	−0.0833862	−0.0416931	−	0.999130i	$-0.513275\pi$
$557$	−1.89427e11	−0.0833862	−0.0416931	+	0.999130i	$0.513275\pi$
$558$	0	0
$559$	−6.03528e11	−0.261423
$560$	0	0
$561$	−1.00261e11	−0.0427364
$562$	0	0
$563$	−3.50385e12	−1.46980	−0.734900	−	0.678175i	$-0.762771\pi$
$563$	−3.50385e12	−1.46980	−0.734900	+	0.678175i	$0.762771\pi$
$564$	0	0
$565$	2.45493e12	1.01349
$566$	0	0
$567$	1.56735e12	0.636858
$568$	0	0
$569$	4.15736e12	1.66270	0.831348	−	0.555752i	$-0.187570\pi$
$569$	4.15736e12	1.66270	0.831348	+	0.555752i	$0.187570\pi$
$570$	0	0
$571$	−3.38832e12	−1.33390	−0.666948	−	0.745104i	$-0.732400\pi$
$571$	−3.38832e12	−1.33390	−0.666948	+	0.745104i	$0.732400\pi$
$572$	0	0
$573$	2.98686e11	0.115749
$574$	0	0
$575$	6.19899e12	2.36492
$576$	0	0
$577$	−7.93761e11	−0.298125	−0.149062	−	0.988828i	$-0.547626\pi$
$577$	−7.93761e11	−0.298125	−0.149062	+	0.988828i	$0.547626\pi$
$578$	0	0
$579$	−4.04955e11	−0.149745
$580$	0	0
$581$	6.61718e12	2.40924
$582$	0	0
$583$	4.54789e10	0.0163043
$584$	0	0
$585$	2.74823e12	0.970178
$586$	0	0
$587$	−3.50948e12	−1.22003	−0.610016	−	0.792389i	$-0.708837\pi$
$587$	−3.50948e12	−1.22003	−0.610016	+	0.792389i	$0.708837\pi$
$588$	0	0
$589$	−2.91357e12	−0.997485
$590$	0	0
$591$	1.85738e12	0.626262
$592$	0	0
$593$	1.01035e12	0.335527	0.167763	−	0.985827i	$-0.446346\pi$
$593$	1.01035e12	0.335527	0.167763	+	0.985827i	$0.446346\pi$
$594$	0	0
$595$	−2.26743e12	−0.741663
$596$	0	0
$597$	−2.26114e12	−0.728521
$598$	0	0
$599$	−1.26453e12	−0.401337	−0.200669	−	0.979659i	$-0.564311\pi$
$599$	−1.26453e12	−0.401337	−0.200669	+	0.979659i	$0.564311\pi$
$600$	0	0
$601$	4.11286e12	1.28591	0.642953	−	0.765906i	$-0.277709\pi$
$601$	4.11286e12	1.28591	0.642953	+	0.765906i	$0.277709\pi$
$602$	0	0
$603$	2.18747e12	0.673775
$604$	0	0
$605$	−4.88466e12	−1.48230
$606$	0	0
$607$	−2.94970e12	−0.881919	−0.440959	−	0.897527i	$-0.645362\pi$
$607$	−2.94970e12	−0.881919	−0.440959	+	0.897527i	$0.645362\pi$
$608$	0	0
$609$	1.15164e12	0.339265
$610$	0	0
$611$	−4.33311e12	−1.25781
$612$	0	0
$613$	−1.04819e12	−0.299825	−0.149912	−	0.988699i	$-0.547899\pi$
$613$	−1.04819e12	−0.299825	−0.149912	+	0.988699i	$0.547899\pi$
$614$	0	0
$615$	3.71616e12	1.04751
$616$	0	0
$617$	−2.24197e12	−0.622797	−0.311398	−	0.950279i	$-0.600797\pi$
$617$	−2.24197e12	−0.622797	−0.311398	+	0.950279i	$0.600797\pi$
$618$	0	0
$619$	4.31403e12	1.18107	0.590534	−	0.807013i	$-0.298917\pi$
$619$	4.31403e12	1.18107	0.590534	+	0.807013i	$0.298917\pi$
$620$	0	0
$621$	−3.87857e12	−1.04655
$622$	0	0
$623$	−1.77225e11	−0.0471335
$624$	0	0
$625$	2.99120e12	0.784124
$626$	0	0
$627$	−7.94454e11	−0.205289
$628$	0	0
$629$	4.69545e11	0.119605
$630$	0	0
$631$	5.04376e12	1.26655	0.633275	−	0.773927i	$-0.281710\pi$
$631$	5.04376e12	1.26655	0.633275	+	0.773927i	$0.281710\pi$
$632$	0	0
$633$	−3.04552e11	−0.0753953
$634$	0	0
$635$	−6.15819e12	−1.50304
$636$	0	0
$637$	−6.74324e12	−1.62271
$638$	0	0
$639$	−9.17371e11	−0.217666
$640$	0	0
$641$	4.31567e12	1.00969	0.504844	−	0.863210i	$-0.331550\pi$
$641$	4.31567e12	1.00969	0.504844	+	0.863210i	$0.331550\pi$
$642$	0	0
$643$	4.67632e12	1.07883	0.539417	−	0.842039i	$-0.318645\pi$
$643$	4.67632e12	1.07883	0.539417	+	0.842039i	$0.318645\pi$
$644$	0	0
$645$	1.28278e12	0.291833
$646$	0	0
$647$	2.41143e11	0.0541010	0.0270505	−	0.999634i	$-0.491389\pi$
$647$	2.41143e11	0.0541010	0.0270505	+	0.999634i	$0.491389\pi$
$648$	0	0
$649$	1.21999e12	0.269933
$650$	0	0
$651$	3.38280e12	0.738179
$652$	0	0
$653$	5.60932e12	1.20726	0.603631	−	0.797264i	$-0.293720\pi$
$653$	5.60932e12	1.20726	0.603631	+	0.797264i	$0.293720\pi$
$654$	0	0
$655$	4.00289e12	0.849744
$656$	0	0
$657$	2.54591e12	0.533088
$658$	0	0
$659$	5.89433e12	1.21745	0.608724	−	0.793382i	$-0.291682\pi$
$659$	5.89433e12	1.21745	0.608724	+	0.793382i	$0.291682\pi$
$660$	0	0
$661$	−6.88972e11	−0.140377	−0.0701883	−	0.997534i	$-0.522360\pi$
$661$	−6.88972e11	−0.140377	−0.0701883	+	0.997534i	$0.522360\pi$
$662$	0	0
$663$	4.30123e11	0.0864533
$664$	0	0
$665$	−1.79668e13	−3.56266
$666$	0	0
$667$	2.46953e12	0.483113
$668$	0	0
$669$	3.11258e12	0.600763
$670$	0	0
$671$	−1.94546e12	−0.370484
$672$	0	0
$673$	5.91814e12	1.11203	0.556016	−	0.831171i	$-0.312329\pi$
$673$	5.91814e12	1.11203	0.556016	+	0.831171i	$0.312329\pi$
$674$	0	0
$675$	−8.85577e12	−1.64195
$676$	0	0
$677$	9.75918e12	1.78552	0.892759	−	0.450534i	$-0.148766\pi$
$677$	9.75918e12	1.78552	0.892759	+	0.450534i	$0.148766\pi$
$678$	0	0
$679$	1.87142e13	3.37876
$680$	0	0
$681$	−1.13952e12	−0.203030
$682$	0	0
$683$	1.05702e12	0.185862	0.0929310	−	0.995673i	$-0.470376\pi$
$683$	1.05702e12	0.185862	0.0929310	+	0.995673i	$0.470376\pi$
$684$	0	0
$685$	2.33718e12	0.405587
$686$	0	0
$687$	3.07752e12	0.527103
$688$	0	0
$689$	−1.95107e11	−0.0329826
$690$	0	0
$691$	1.03154e12	0.172122	0.0860608	−	0.996290i	$-0.472572\pi$
$691$	1.03154e12	0.172122	0.0860608	+	0.996290i	$0.472572\pi$
$692$	0	0
$693$	−3.04291e12	−0.501176
$694$	0	0
$695$	1.21283e12	0.197182
$696$	0	0
$697$	−1.91869e12	−0.307934
$698$	0	0
$699$	−3.38978e12	−0.537061
$700$	0	0
$701$	5.14597e12	0.804889	0.402444	−	0.915444i	$-0.368161\pi$
$701$	5.14597e12	0.804889	0.402444	+	0.915444i	$0.368161\pi$
$702$	0	0
$703$	3.72062e12	0.574535
$704$	0	0
$705$	9.20991e12	1.40412
$706$	0	0
$707$	−1.05906e13	−1.59417
$708$	0	0
$709$	−1.28349e13	−1.90759	−0.953795	−	0.300458i	$-0.902861\pi$
$709$	−1.28349e13	−1.90759	−0.953795	+	0.300458i	$0.902861\pi$
$710$	0	0
$711$	1.51512e12	0.222348
$712$	0	0
$713$	7.25395e12	1.05117
$714$	0	0
$715$	−3.22788e12	−0.461892
$716$	0	0
$717$	−3.21711e12	−0.454600
$718$	0	0
$719$	6.51717e12	0.909451	0.454725	−	0.890632i	$-0.349737\pi$
$719$	6.51717e12	0.909451	0.454725	+	0.890632i	$0.349737\pi$
$720$	0	0
$721$	2.14050e13	2.94989
$722$	0	0
$723$	5.58000e12	0.759472
$724$	0	0
$725$	5.63857e12	0.757963
$726$	0	0
$727$	4.75481e12	0.631290	0.315645	−	0.948877i	$-0.397779\pi$
$727$	4.75481e12	0.631290	0.315645	+	0.948877i	$0.397779\pi$
$728$	0	0
$729$	1.05033e12	0.137737
$730$	0	0
$731$	−6.62313e11	−0.0857896
$732$	0	0
$733$	−4.70068e12	−0.601441	−0.300721	−	0.953712i	$-0.597227\pi$
$733$	−4.70068e12	−0.601441	−0.300721	+	0.953712i	$0.597227\pi$
$734$	0	0
$735$	1.43326e13	1.81147
$736$	0	0
$737$	−2.56926e12	−0.320778
$738$	0	0
$739$	−7.68600e12	−0.947983	−0.473992	−	0.880529i	$-0.657187\pi$
$739$	−7.68600e12	−0.947983	−0.473992	+	0.880529i	$0.657187\pi$
$740$	0	0
$741$	3.40825e12	0.415288
$742$	0	0
$743$	−1.54208e13	−1.85634	−0.928170	−	0.372156i	$-0.878618\pi$
$743$	−1.54208e13	−1.85634	−0.928170	+	0.372156i	$0.878618\pi$
$744$	0	0
$745$	−8.11290e12	−0.964879
$746$	0	0
$747$	−8.80152e12	−1.03423
$748$	0	0
$749$	1.07200e12	0.124459
$750$	0	0
$751$	6.56715e12	0.753351	0.376676	−	0.926345i	$-0.377067\pi$
$751$	6.56715e12	0.753351	0.376676	+	0.926345i	$0.377067\pi$
$752$	0	0
$753$	3.99580e12	0.452925
$754$	0	0
$755$	−6.07732e12	−0.680693
$756$	0	0
$757$	−1.40433e13	−1.55431	−0.777156	−	0.629308i	$-0.783338\pi$
$757$	−1.40433e13	−1.55431	−0.777156	+	0.629308i	$0.783338\pi$
$758$	0	0
$759$	1.97796e12	0.216337
$760$	0	0
$761$	−9.41842e12	−1.01800	−0.508999	−	0.860767i	$-0.669984\pi$
$761$	−9.41842e12	−1.01800	−0.508999	+	0.860767i	$0.669984\pi$
$762$	0	0
$763$	−1.95866e13	−2.09218
$764$	0	0
$765$	3.01591e12	0.318377
$766$	0	0
$767$	−5.23384e12	−0.546061
$768$	0	0
$769$	1.32235e13	1.36358	0.681788	−	0.731550i	$-0.261203\pi$
$769$	1.32235e13	1.36358	0.681788	+	0.731550i	$0.261203\pi$
$770$	0	0
$771$	2.27343e12	0.231706
$772$	0	0
$773$	−1.14561e13	−1.15406	−0.577031	−	0.816722i	$-0.695789\pi$
$773$	−1.14561e13	−1.15406	−0.577031	+	0.816722i	$0.695789\pi$
$774$	0	0
$775$	1.65626e13	1.64919
$776$	0	0
$777$	−4.31983e12	−0.425179
$778$	0	0
$779$	−1.52035e13	−1.47919
$780$	0	0
$781$	1.07748e12	0.103629
$782$	0	0
$783$	−3.52793e12	−0.335422
$784$	0	0
$785$	−2.05508e13	−1.93159
$786$	0	0
$787$	−4.46778e12	−0.415151	−0.207575	−	0.978219i	$-0.566557\pi$
$787$	−4.46778e12	−0.415151	−0.207575	+	0.978219i	$0.566557\pi$
$788$	0	0
$789$	−2.65041e11	−0.0243482
$790$	0	0
$791$	1.16611e13	1.05912
$792$	0	0
$793$	8.34610e12	0.749470
$794$	0	0
$795$	4.14694e11	0.0368193
$796$	0	0
$797$	−1.13816e13	−0.999177	−0.499589	−	0.866263i	$-0.666515\pi$
$797$	−1.13816e13	−0.999177	−0.499589	+	0.866263i	$0.666515\pi$
$798$	0	0
$799$	−4.75516e12	−0.412767
$800$	0	0
$801$	2.35728e11	0.0202332
$802$	0	0
$803$	−2.99026e12	−0.253798
$804$	0	0
$805$	4.47323e13	3.75439
$806$	0	0
$807$	4.28929e12	0.356003
$808$	0	0
$809$	−1.94386e13	−1.59550	−0.797748	−	0.602991i	$-0.793976\pi$
$809$	−1.94386e13	−1.59550	−0.797748	+	0.602991i	$0.793976\pi$
$810$	0	0
$811$	−1.02059e12	−0.0828431	−0.0414216	−	0.999142i	$-0.513189\pi$
$811$	−1.02059e12	−0.0828431	−0.0414216	+	0.999142i	$0.513189\pi$
$812$	0	0
$813$	−2.89321e12	−0.232259
$814$	0	0
$815$	−1.72822e12	−0.137211
$816$	0	0
$817$	−5.24809e12	−0.412099
$818$	0	0
$819$	1.30542e13	1.01385
$820$	0	0
$821$	2.88205e12	0.221390	0.110695	−	0.993854i	$-0.464692\pi$
$821$	2.88205e12	0.221390	0.110695	+	0.993854i	$0.464692\pi$
$822$	0	0
$823$	−1.16904e13	−0.888237	−0.444118	−	0.895968i	$-0.646483\pi$
$823$	−1.16904e13	−0.888237	−0.444118	+	0.895968i	$0.646483\pi$
$824$	0	0
$825$	4.51619e12	0.339414
$826$	0	0
$827$	−5.81279e12	−0.432126	−0.216063	−	0.976379i	$-0.569322\pi$
$827$	−5.81279e12	−0.432126	−0.216063	+	0.976379i	$0.569322\pi$
$828$	0	0
$829$	6.40775e12	0.471205	0.235603	−	0.971849i	$-0.424294\pi$
$829$	6.40775e12	0.471205	0.235603	+	0.971849i	$0.424294\pi$
$830$	0	0
$831$	−8.98920e12	−0.653907
$832$	0	0
$833$	−7.40004e12	−0.532514
$834$	0	0
$835$	−1.38058e13	−0.982817
$836$	0	0
$837$	−1.03629e13	−0.729819
$838$	0	0
$839$	3.65270e12	0.254498	0.127249	−	0.991871i	$-0.459385\pi$
$839$	3.65270e12	0.254498	0.127249	+	0.991871i	$0.459385\pi$
$840$	0	0
$841$	−1.22609e13	−0.845161
$842$	0	0
$843$	−9.74051e12	−0.664290
$844$	0	0
$845$	−1.15041e13	−0.776240
$846$	0	0
$847$	−2.32024e13	−1.54902
$848$	0	0
$849$	−4.55301e12	−0.300756
$850$	0	0
$851$	−9.26328e12	−0.605455
$852$	0	0
$853$	2.63808e13	1.70615	0.853076	−	0.521787i	$-0.174734\pi$
$853$	2.63808e13	1.70615	0.853076	+	0.521787i	$0.174734\pi$
$854$	0	0
$855$	2.38977e13	1.52936
$856$	0	0
$857$	−2.49098e13	−1.57745	−0.788726	−	0.614745i	$-0.789259\pi$
$857$	−2.49098e13	−1.57745	−0.788726	+	0.614745i	$0.789259\pi$
$858$	0	0
$859$	−2.76464e13	−1.73248	−0.866242	−	0.499625i	$-0.833471\pi$
$859$	−2.76464e13	−1.73248	−0.866242	+	0.499625i	$0.833471\pi$
$860$	0	0
$861$	1.76520e13	1.09466
$862$	0	0
$863$	−6.15764e12	−0.377890	−0.188945	−	0.981988i	$-0.560507\pi$
$863$	−6.15764e12	−0.377890	−0.188945	+	0.981988i	$0.560507\pi$
$864$	0	0
$865$	−5.40827e12	−0.328463
$866$	0	0
$867$	4.72017e11	0.0283708
$868$	0	0
$869$	−1.77956e12	−0.105858
$870$	0	0
$871$	1.10222e13	0.648915
$872$	0	0
$873$	−2.48918e13	−1.45041
$874$	0	0
$875$	4.91117e13	2.83236
$876$	0	0
$877$	1.26417e11	0.00721619	0.00360809	−	0.999993i	$-0.498852\pi$
$877$	1.26417e11	0.00721619	0.00360809	+	0.999993i	$0.498852\pi$
$878$	0	0
$879$	5.92833e12	0.334952
$880$	0	0
$881$	1.81563e13	1.01540	0.507699	−	0.861535i	$-0.330496\pi$
$881$	1.81563e13	1.01540	0.507699	+	0.861535i	$0.330496\pi$
$882$	0	0
$883$	3.40979e13	1.88758	0.943788	−	0.330553i	$-0.107235\pi$
$883$	3.40979e13	1.88758	0.943788	+	0.330553i	$0.107235\pi$
$884$	0	0
$885$	1.11244e13	0.609580
$886$	0	0
$887$	−1.64233e13	−0.890851	−0.445426	−	0.895319i	$-0.646948\pi$
$887$	−1.64233e13	−0.890851	−0.445426	+	0.895319i	$0.646948\pi$
$888$	0	0
$889$	−2.92518e13	−1.57071
$890$	0	0
$891$	2.44859e12	0.130157
$892$	0	0
$893$	−3.76794e13	−1.98277
$894$	0	0
$895$	8.82926e12	0.459961
$896$	0	0
$897$	−8.48556e12	−0.437637
$898$	0	0
$899$	6.59816e12	0.336902
$900$	0	0
$901$	−2.14110e11	−0.0108237
$902$	0	0
$903$	6.09330e12	0.304970
$904$	0	0
$905$	4.57217e13	2.26571
$906$	0	0
$907$	3.95675e12	0.194136	0.0970679	−	0.995278i	$-0.469054\pi$
$907$	3.95675e12	0.194136	0.0970679	+	0.995278i	$0.469054\pi$
$908$	0	0
$909$	1.40866e13	0.684337
$910$	0	0
$911$	−9.23006e12	−0.443989	−0.221994	−	0.975048i	$-0.571257\pi$
$911$	−9.23006e12	−0.443989	−0.221994	+	0.975048i	$0.571257\pi$
$912$	0	0
$913$	1.03377e13	0.492384
$914$	0	0
$915$	−1.77394e13	−0.836650
$916$	0	0
$917$	1.90140e13	0.887997
$918$	0	0
$919$	3.01409e13	1.39391	0.696957	−	0.717113i	$-0.254537\pi$
$919$	3.01409e13	1.39391	0.696957	+	0.717113i	$0.254537\pi$
$920$	0	0
$921$	−1.55953e13	−0.714208
$922$	0	0
$923$	−4.62245e12	−0.209635
$924$	0	0
$925$	−2.11504e13	−0.949907
$926$	0	0
$927$	−2.84708e13	−1.26631
$928$	0	0
$929$	3.04484e13	1.34120	0.670601	−	0.741818i	$-0.266036\pi$
$929$	3.04484e13	1.34120	0.670601	+	0.741818i	$0.266036\pi$
$930$	0	0
$931$	−5.86371e13	−2.55799
$932$	0	0
$933$	−1.95576e12	−0.0844981
$934$	0	0
$935$	−3.54228e12	−0.151576
$936$	0	0
$937$	2.04000e13	0.864572	0.432286	−	0.901736i	$-0.357707\pi$
$937$	2.04000e13	0.864572	0.432286	+	0.901736i	$0.357707\pi$
$938$	0	0
$939$	−2.00242e13	−0.840543
$940$	0	0
$941$	−2.55881e13	−1.06386	−0.531930	−	0.846789i	$-0.678533\pi$
$941$	−2.55881e13	−1.06386	−0.531930	+	0.846789i	$0.678533\pi$
$942$	0	0
$943$	3.78523e13	1.55880
$944$	0	0
$945$	−6.39037e13	−2.60665
$946$	0	0
$947$	−8.56722e12	−0.346151	−0.173075	−	0.984909i	$-0.555370\pi$
$947$	−8.56722e12	−0.346151	−0.173075	+	0.984909i	$0.555370\pi$
$948$	0	0
$949$	1.28283e13	0.513420
$950$	0	0
$951$	−2.11665e13	−0.839146
$952$	0	0
$953$	−3.84991e13	−1.51193	−0.755966	−	0.654611i	$-0.772832\pi$
$953$	−3.84991e13	−1.51193	−0.755966	+	0.654611i	$0.772832\pi$
$954$	0	0
$955$	1.05528e13	0.410537
$956$	0	0
$957$	1.79915e12	0.0693366
$958$	0	0
$959$	1.11017e13	0.423845
$960$	0	0
$961$	−7.05834e12	−0.266961
$962$	0	0
$963$	−1.42587e12	−0.0534272
$964$	0	0
$965$	−1.43073e13	−0.531112
$966$	0	0
$967$	−4.31991e13	−1.58875	−0.794375	−	0.607428i	$-0.792201\pi$
$967$	−4.31991e13	−1.58875	−0.794375	+	0.607428i	$0.792201\pi$
$968$	0	0
$969$	3.74021e12	0.136282
$970$	0	0
$971$	1.97835e13	0.714196	0.357098	−	0.934067i	$-0.383766\pi$
$971$	1.97835e13	0.714196	0.357098	+	0.934067i	$0.383766\pi$
$972$	0	0
$973$	5.76100e12	0.206058
$974$	0	0
$975$	−1.93747e13	−0.686616
$976$	0	0
$977$	−2.56325e13	−0.900047	−0.450024	−	0.893017i	$-0.648584\pi$
$977$	−2.56325e13	−0.900047	−0.450024	+	0.893017i	$0.648584\pi$
$978$	0	0
$979$	−2.76870e11	−0.00963281
$980$	0	0
$981$	2.60522e13	0.898119
$982$	0	0
$983$	3.03713e13	1.03746	0.518732	−	0.854937i	$-0.326404\pi$
$983$	3.03713e13	1.03746	0.518732	+	0.854937i	$0.326404\pi$
$984$	0	0
$985$	6.56224e13	2.22121
$986$	0	0
$987$	4.37476e13	1.46733
$988$	0	0
$989$	1.30662e13	0.434278
$990$	0	0
$991$	1.54521e13	0.508929	0.254465	−	0.967082i	$-0.418101\pi$
$991$	1.54521e13	0.508929	0.254465	+	0.967082i	$0.418101\pi$
$992$	0	0
$993$	−6.02910e11	−0.0196780
$994$	0	0
$995$	−7.98875e13	−2.58390
$996$	0	0
$997$	−7.77458e12	−0.249201	−0.124600	−	0.992207i	$-0.539765\pi$
$997$	−7.77458e12	−0.249201	−0.124600	+	0.992207i	$0.539765\pi$
$998$	0	0
$999$	1.32333e13	0.420364

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	272.10.a.f.1.3		5
4.3	odd	2		17.10.a.a.1.4	✓	5
12.11	even	2		153.10.a.c.1.2		5
68.67	odd	2		289.10.a.a.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.10.a.a.1.4	✓	5	4.3	odd	2
153.10.a.c.1.2		5	12.11	even	2
272.10.a.f.1.3		5	1.1	even	1	trivial
289.10.a.a.1.4		5	68.67	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:	\( N \)	\(=\)	\( 272 = 2^{4} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	272.a (trivial)

\(f(q)\)	\(=\)	\(q+67.6654 q^{3} +2390.67 q^{5} +11355.8 q^{7} -15104.4 q^{9} +O(q^{10})\)	\(q+67.6654 q^{3} +2390.67 q^{5} +11355.8 q^{7} -15104.4 q^{9} +17740.6 q^{11} -76108.0 q^{13} +161765. q^{15} -83521.0 q^{17} -661811. q^{19} +768396. q^{21} +1.64772e6 q^{23} +3.76216e6 q^{25} -2.35390e6 q^{27} +1.49876e6 q^{29} +4.40242e6 q^{31} +1.20042e6 q^{33} +2.71480e7 q^{35} -5.62188e6 q^{37} -5.14988e6 q^{39} +2.29725e7 q^{41} +7.92989e6 q^{43} -3.61096e7 q^{45} +5.69337e7 q^{47} +8.86009e7 q^{49} -5.65148e6 q^{51} +2.56355e6 q^{53} +4.24119e7 q^{55} -4.47817e7 q^{57} +6.87685e7 q^{59} -1.09661e8 q^{61} -1.71523e8 q^{63} -1.81949e8 q^{65} -1.44824e8 q^{67} +1.11494e8 q^{69} +6.07354e7 q^{71} -1.68554e8 q^{73} +2.54568e8 q^{75} +2.01459e8 q^{77} -1.00310e8 q^{79} +1.38022e8 q^{81} +5.82713e8 q^{83} -1.99671e8 q^{85} +1.01414e8 q^{87} -1.56066e7 q^{89} -8.64268e8 q^{91} +2.97891e8 q^{93} -1.58217e9 q^{95} +1.64798e9 q^{97} -2.67961e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 236 q^{3} + 1480 q^{5} + 13202 q^{7} + 10981 q^{9}+O(q^{10}) \) 5 * q + 236 * q^3 + 1480 * q^5 + 13202 * q^7 + 10981 * q^9	\( 5 q + 236 q^{3} + 1480 q^{5} + 13202 q^{7} + 10981 q^{9} + 68036 q^{11} - 158862 q^{13} + 687324 q^{15} - 417605 q^{17} + 370992 q^{19} + 1783880 q^{21} - 1645870 q^{23} + 3270239 q^{25} + 2998268 q^{27} + 3668616 q^{29} + 7262362 q^{31} - 11334900 q^{33} + 26503988 q^{35} - 31420708 q^{37} + 42449884 q^{39} - 7996938 q^{41} + 56908268 q^{43} + 12799536 q^{45} + 16903336 q^{47} - 11784059 q^{49} - 19710956 q^{51} - 83362982 q^{53} - 6363364 q^{55} + 136615904 q^{57} + 37946604 q^{59} - 77685452 q^{61} + 191945278 q^{63} - 40321288 q^{65} + 304503600 q^{67} - 333409272 q^{69} + 476602922 q^{71} - 289980486 q^{73} + 153685772 q^{75} - 143385648 q^{77} + 828240610 q^{79} + 891328609 q^{81} - 194681148 q^{83} - 123611080 q^{85} - 158149884 q^{87} + 376848106 q^{89} - 194543664 q^{91} + 3494835920 q^{93} - 1498679864 q^{95} + 692035246 q^{97} - 2027106408 q^{99}+O(q^{100}) \) 5 * q + 236 * q^3 + 1480 * q^5 + 13202 * q^7 + 10981 * q^9 + 68036 * q^11 - 158862 * q^13 + 687324 * q^15 - 417605 * q^17 + 370992 * q^19 + 1783880 * q^21 - 1645870 * q^23 + 3270239 * q^25 + 2998268 * q^27 + 3668616 * q^29 + 7262362 * q^31 - 11334900 * q^33 + 26503988 * q^35 - 31420708 * q^37 + 42449884 * q^39 - 7996938 * q^41 + 56908268 * q^43 + 12799536 * q^45 + 16903336 * q^47 - 11784059 * q^49 - 19710956 * q^51 - 83362982 * q^53 - 6363364 * q^55 + 136615904 * q^57 + 37946604 * q^59 - 77685452 * q^61 + 191945278 * q^63 - 40321288 * q^65 + 304503600 * q^67 - 333409272 * q^69 + 476602922 * q^71 - 289980486 * q^73 + 153685772 * q^75 - 143385648 * q^77 + 828240610 * q^79 + 891328609 * q^81 - 194681148 * q^83 - 123611080 * q^85 - 158149884 * q^87 + 376848106 * q^89 - 194543664 * q^91 + 3494835920 * q^93 - 1498679864 * q^95 + 692035246 * q^97 - 2027106408 * q^99

Introduction

L-functions

Modular forms

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Database

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